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abstract: 'Self-supervised learning has proven to be invaluable in making best use of all of the available data in biomedical image segmentation. One particularly simple and effective mechanism to achieve self-supervision is inpainting, the task of predicting arbitrary missing areas based on the rest of an image. In this work, we focus on image inpainting as the self-supervised proxy task, and propose two novel structural changes to further enhance the performance of a deep neural network. We guide the process of generating images to inpaint by using supervoxel-based masking instead of random masking, and also by focusing on the area to be segmented in the primary task, which we term as the region-of-interest. We postulate that these additions force the network to learn semantics that are more attuned to the primary task, and test our hypotheses on two applications: brain tumour and white matter hyperintensities segmentation. We empirically show that our proposed approach consistently outperforms both supervised CNNs, without any self-supervision, and conventional inpainting-based self-supervision methods on both large and small training set sizes.'
author:
- 'Subhradeep Kayal, Shuai Chen'
- Marleen de Bruijne
bibliography:
- 'mybibliography.bib'
title: 'Region-of-interest guided Supervoxel Inpainting for Self-supervision'
---
Introduction and Motivation
===========================
*Self-supervised learning* points to methods in which neural networks are explicitly trained on large volumes of data, whose labels can be determined automatically and inexpensively, to reduce the need for manually labeled data. Many ways of performing self-supervision exist, amongst which a popular way is the *pre-train and fine-tune* paradigm where: (1) a convolutional neural network is pre-trained on a proxy task for which labels can be generated easily, and (2) it is then fine-tuned on the main task using labeled data. Utilizing a suitable and complex proxy task, self-supervision teaches the network robust and transferable visual features, which alleviates overfitting problems and aides its performance when fine-tuned on the main task [@DBLP:journals/corr/abs-1902-06162].
In the medical imaging domain a variety of proxy tasks have been proposed, such as sorting 2D slices derived from 3D volumetric scans [@7950587], predicting 3D distance between patches sampled from an organ [@DBLP:conf/miccai/SpitzerKAHD18], masking patches or volumes within the image and learning to predict them [@CHEN2019101539], and shuffling 3D blocks within an image and letting a network predict their original positions [@DBLP:conf/miccai/ZhuangLHMYZ19]. Recently, state-of-the-art results were achieved on several biomedical benchmark datasets by networks which were self-supervised using a sequence of individual proxy tasks [@10.1007/978-3-030-32251-9_42].
![**Proposed ROI-guided inpainting.** (a) Examples from BraTS 2018 dataset (left to right from top to bottom): original FLAIR image-slice, ground-truth segmentation map, FLAIR image-slice with superpixels overlaid, region-of-interest (ROI) influenced superpixels, examples of synthesized images to be inpainted. (b) Examples from White Matter Hyperintensities 2017 dataset. Notice that the ground-truth segmentations are much smaller in size. (c) first a U-net is pre-trained on the inpainting task with MSE loss, next it is fine-tuned on the main segmentation task with Dice loss.[]{data-label="figinpainting"}](ROI-inpainting.jpg){width="100.00000%"}
Prior works in self-supervision literature have designed the proxy task largely uninfluenced by the downstream task in focus. However, since the features that the network learns are dependent on where it is focusing on during the self-supervision task, it might be beneficial to bias or *guide* the proxy task towards areas that are of interest to the main task. Specifically for image segmentation, these would be the foreground areas to be segmented in the main task, which we term as the *region-of-interests or ROIs*.
We experiment with the proxy task of inpainting [@DBLP:conf/cvpr/PathakKDDE16], where the network must learn to fill-in artificially created gaps in images. In the context of biomedical imaging, a network that learns to inpaint healthy tissue will learn a different set of semantics than one which inpaints various kinds of tumours. Thus, if the main task is that of segmenting tumours, it can be hypothesized that having a network inpaint tumourous areas as a proxy task will likely teach it semantics attuned to segmenting tumours, and thereby be more beneficial for the main task than learning general semantics. In other words, by increasing the frequency of inpainting tumours, we can teach the network features which are more related to the tumour segmentation task.
Furthermore, in prior work the selection of regions to mask has largely been uninformed and random. We try to improve upon this situation by selecting regions which are homogeneous. Masking such regions could force the network to learn more about the anatomical meaning and relation to other structures of the masked tissue. For example, masking small regions in a lung CT scan would only require the model to correctly interpolate the structures (airways, vessels) around the masked region. In contrast, when a full airway or vessel branch is masked, inpainting requires understanding of the relation between branches in vessel or airway trees and/or the relation between airways and arteries, a piece of information that has been found to improve airway segmentation [@LO2010527].
The contributions of this work are stated next. Firstly, this paper demonstrates that guiding the inpainting process with the main class(es) of interest (,i.e., the segmentation foreground, interchangeably used with the term *ROI* in this paper) during the self-supervised pre-training of a network improves its performance over using random regions. Secondly, we show that instead of inpainting regions of regular shapes in an uninformed way, further performance gain is possible if the masked regions are chosen to be homogeneous. This is done by constructing supervoxels, in a preprocessing step, and using them as candidate regions to be inpainted. In order to show the efficiency of these proposed changes, we conduct empirical analyses on two popularly used public datasets for biomedical image segmentation.
Methods {#methodology}
=======
The proposed method (Figure \[figinpainting\]) utilizes supervoxelization to create candidate regions, followed by selecting only those supervoxels which have an overlap with (any of) the foreground class(es). The selected supervoxels are utilized in the inpainting process, where we use them as masks to suppress areas in an image to train a network to predict (or *inpaint*) them based on their surroundings. Since we control the parameters of this process, it can be used to create an arbitrarily large amount of synthetic training images for pre-training.
Region-of-interest guided inpainting
------------------------------------
Inpainting is an effective proxy task for self-supervision, which proceeds by training a network to reconstruct an image from a masked version of it. In this section, we explain our proposed masking approach, followed by the description of the network in Section \[trainingstrat\].
**Supervoxelization:** While previous works primarily use random grids and cubes as candidate regions to inpaint, the first step in our proposed approach is to select regions based on some notion of homogeneity. One way of achieving this is to construct supervoxels, which may be defined as homogeneous groups of voxels that share some common characteristics. A particularly efficient algorithm to construct such supervoxels is *SLIC* or *simple linear iterative clustering* [@6205760].
For 3D medical images, SLIC can cluster voxels based on their intensity values, corresponding to the various modalities, and spatial coordinates of the voxel within the image. SLIC has two main hyperparameters: one, *compactness*, controls the balance between emphasis on intensity values and spatial coordinates (larger values make square/cubic grids), and the other defines the maximum number of supervoxels. Examples in the second row of Figure \[figinpainting\], subfigure (a) and (b).
In this work, we use SLIC with intensity values corresponding to the two modalities we used in our experiments, FLAIR and T1 (or contrast enhanced T1), in order to construct supervoxels. The exact parameter settings for supervoxelization are described later in Section \[experiments:settings\].
**ROI-guided masking for inpainting image synthesis:** Once the supervoxel labels have been created, the next step is to retain only those supervoxels which have an overlap with the region-of-interest. To achieve this, we first convert the segmentation map to a binary one by considering all foreground areas to be a class with a label value as $1$ and the background as $0$, since there may be multiple regions-of-interest in a multi-class segmentation setting. Then an elementwise *and* operation is performed between the resulting binary segmentation map and the generated supervoxel. For all the supervoxels that remain, training images for the inpainting task can be synthesized by masking an area corresponding to such a *ROI-guided supervoxel*, with the original unmasked image being the target for the network. Some examples of this are in the second row of Figure \[figinpainting\], subfigure (a) and (b).
By constructing a training set for the inpainting task in this fashion, we are essentially increasing the frequency of inpainting regions which are important to the main task more than random chance. This is what, we posit, will bring about improvements in the performance of the network on the main task.
Formally, let $D_{train} = \{(I_i, S_i)\}_{i=1..n}$ be the training dataset containing $n$ images with $I_i$ being a 3D multi-modal training image and $S_i$ being the segmentation ground-truth label, containing zero values representing background. If $f$ is a supervoxelization algorithm (in our case, SLIC), then a ROI-guided supervoxelized image is given by ${R_i} = f(I_i) \odot S_i$, where $\odot$ signifies elementwise multiplication.
${R_i}$ contains supervoxel regions having non-zero labels corresponding to foregound supervoxels. Then, the synthetic dataset for inpainting, $D_{inp}$ is constructed as: $$D_{inp} = \Bigg\{ \Big\{ \big( I_i \odot r_{ij}^0, I_i \big) \Big\}_{r_{ij} \in R_i, j=1..{m_i}} \Bigg\}_{i=1..n}$$ where $r_{ij}$ is a single supervoxel region in the set ${R_i}$, which contains a total of $m_i$ supervoxels, and $r_{ij}^0$ is the corresponding inverted region-mask, containing 0 for voxels belonging to the region and 1 everywhere else. $I_i \odot r_{ij}$ is then the masked image input to the network and $I_i$ is the expected output to reconstruct, the target for the inpainting task. Thus, the maximum cardinality of $D_{inp}$ can be $n \times m_i$.
Examples $ D_{inp}$ are in the last row of Figure \[figinpainting\], subfigure (a) and (b).
Training Strategy {#trainingstrat}
-----------------
**Network:** For all the experiments, a shallow 3D U-net [@10.1007/978-3-319-24574-4_28; @10.1007/978-3-319-46723-8_49] containing 3 resolution levels has been used, with a batch-normalization layer after every convolution layer. In our experiments we find that 3 layers provide sufficient capacity for both the inpainting and the segmentation task. Since we use two modalities for our experiments, the U-net has two input channels.
If we were to use an image reconstruction proxy task, a U-net would learn to copy over the original image because of its skip connections, and would not be useful in learning features. In our task of inpainting the network never sees the masked regions and, therefore, cannot memorize it, making the use of a U-net reasonable.
**Pre-training:** In order to pre-train the network, it is fitted to the $D_{inp}$ dataset by minimizing the mean squared error (MSE) between the masked and the original images using the *Adam* [@DBLP:journals/corr/KingmaB14] optimizer. We call this model *inpainter U-net*.
**Fine-tuning:** The inpainter U-net is then fine-tuned on the (main) segmentation task using the original labeled training dataset, $D_{train}$, by optimizing the Dice segmentation overlap objective on the labeled images. If the data is multi-modal, the inpainter U-net will be trained to produce multi-channel outputs, in which case we would need to replace the last 3D convolutional layer to have a single-channel output for segmentation.
More details about the network parameters are provided in section \[experiments:settings\].
Experimental Settings {#experiments}
=====================
Data {#experiments:data}
----
For our experiments, we use two public datasets containing 3D MRI scans and corresponding manual segmentations.
**BraTS 2018** [@8669968]: 210 MRI scans from patients with high-grade glioma are randomly split three times into 150, 30 and 30 scans for training, validation and testing, respectively, using a 3-fold Monte-carlo cross-validation scheme. To be able to easily compare our method against baselines, we focus on segmenting the whole tumour and use two of the four modalities, *FLAIR* and *T1-gd*, which have been found to be the most effective at this task [@10.1007/978-3-319-24553-9_65].
**White Matter Hyperintensities (WMH) 2017** [@6975210]: The total size of the dataset is 60 FLAIR and T1 scans, coming from 3 different sites, with corresponding manual segmentations of white matter hyperintensities. We employ a 3-fold Monte-carlo cross-validation scheme again, splitting the dataset into 40, 10 and 10 for training, validation and testing, respectively, and use both of the available modalities for our experiments.
Baseline Methods {#experiments:baseline}
----------------
We term the technique proposed in this paper as *roi-supervoxel* to denote the use of the segmentation map and supervoxelization to guide the inpainting process used for pre-training. In order to validate its effectiveness, it is tested against the following baselines: *vanilla-unet*: a U-net without any pre-training; *restart-unet*: a U-net pre-trained on the main (segmentation) task and fine-tuned on the same task for an additional set of epochs; *noroi-grid*: the more traditional inpainting mechanism where random regular sized cuboids are masked; *roi-grid*: a similar process as *roi-supervoxel*, except for the use of regular cuboids overlapping with the segmentation map, instead of supervoxel regions, for masking; *noroi-supervoxel*: where random supervoxels are masked.
Settings {#experiments:settings}
--------
**Inpainting Parameters**: The inpainting process starts by creating the supervoxel regions using SLIC[^1]. We fix these the compactness value at 0.15 and choose the maximum number of supervoxels to be 400, by visual inspection of the nature of the supervoxels that contain the tumour and the white matter hyperintensities for the two datasets. For example, between a setting where one supervoxel is part tumour and part background, versus another where one supervoxel fully represents tumour, we choose the latter case.
We then use either the supervoxels or simple cuboids (for the baseline methods) as areas to be masked, and the question arises of how many and how large areas to choose as masks to construct synthetic images for $D_{inp}$. Too small a volume, and it might be trivial for a network to inpaint it; too large, and it might not be a feasible task. For our experiments, we choose masks whose volume is at least 1500 voxels. For constructing cuboids, we randomly generate cuboids which are at least 12 units in each dimension (as $12^3$ is more than 1500, but $11^3$ is not). Finally, we ensure that the size of $D_{inp}$ is roughly 10 times that of the $D_{train}$, by choosing masks which fit the volume criteria as they are generated, and producing at most 10 synthetic images on-the-fly for a single real input image.
**Network Parameters**: The input size to the 3D U-net is 160 $\times$ 216 $\times$ 32, such that each input image is centre-cropped to 160 $\times$ 216 ($X-Y$ axes) to tightly fit the brain region in the scan, while we use the overlapping tile strategy in the Z-axis as inspired by the original U-net. Each of the 3 resolution levels consists of two 3 $\times$ 3 $\times$ 3 convolution layers using zero-padding and *ReLU* activation, except for the last layer which is linear in the inpainter U-net and *sigmoid* in the fine-tuned U-net. The number of feature channels are 16, 32 and 64 at the varying resolution levels. The feature maps in the upsampling path are concatenated with earlier ones through skip-connections.
**Optimization Parameters**: The inpainter U-net is optimized on MSE while fine-tuning is performed using a Dice objective, both using *Adam*. The learning rate is 0.0001 and 0.001 for BraTS and WMH datasets, respectively. We used a batch-size of 4, as permitted by our GPU memory. For pre-training, we use 100 epochs while for fine-tuning we employ another 150, both without the possibility of early stopping, saving the best performing model based on the validation loss at every epoch.
To foster open-science, all of the code will be released[^2].
Results and Discussion {#experiments:results}
======================
To study the effect reduced dataset sizes on the proposed approach, experiments were performed on the full training dataset as well as smaller fractions of it. For BraTS, we perform experiments on 25%, 50% and 100% of the training data, while for WMH, which is much smaller in size, we only perform an extra set of experiments with 50% of the data. To keep the comparisons fair, we use the same subset of the training data in the pre-training procedure as well.
The segmentation results are shown in Table \[t1\]. It can be observed that the proposed method (*roi-supervoxel*) outperforms the basic U-net (*vanilla-unet*) by a large margin, and traditional inpainting based pre-training (*noroi-grid*) by a small, but significant, margin.
The deductions from the empirical results can be summarized as follows:
**Restarts improve U-net performance:** It can be observed that for both datasets, the performance of the *restart-unet* is better than that of the *vanilla-unet*. This is in line with observations in literature [@sgdr], where warm restarts have aided networks to find a more stable local minimum. Based on this observation, we argue that for any proposed method involving pre-training models, the results should always be compared to such a *restarted* model.
**Adding ROI information to the inpainting proxy task is helpful:** For both the datasets, the performance of the *roi-supervoxel* method exceeds that of all other baselines. Importantly, it exceeds the performance of the *restart-unet* and the *noroi-grid*, which is the traditional inpainting procedure, by 3.2% and 5% (relative) respectively for BraTS, and 4.9% and 2.9% for WMH, when all of the data is used. Also important to note is that the performance of methods which use the region-of-interest information to generate the masked areas is always better than those which do not.
[|c|clclc|ccc|]{}\
& &\
& **.25** & & **.50** & & **1.0** & **.50** & **** & **1.0**\
**vanilla-unet** & 0.257 (.05) & & 0.585 (.03) & & 0.784 (.02) & 0.576 (.05) & & 0.745 (0.02)\
**restart-unet** & 0.302 (.05) & & 0.607 (.03) & & 0.793 (.02) & 0.610 (.05) & & 0.776 (.03)\
**noroi-grid** & 0.311 (.06) & & 0.611 (.04) & & 0.780 (.03) & 0.632 (.04) & & 0.791 (.03)\
**roi-grid** & *0.354 (.06)* & & 0.620 (.04) & & *0.795 (.03)* & 0.653 (.04) & & **0.812 (.03)**\
**noroi-supervoxel** & 0.340 (.06) & & *0.621 (.04)* & & 0.791 (.02) & 0.650 (.04) & & 0.797 (.03)\
**roi-supervoxel** & **0.363 (.06)** & & **0.646 (.04)** & & **0.814 (.03)** & **0.671 (.04)** & & **0.814 (.03)**\
\[t1\]
**Inpainting is more beneficial when the size of the training set is smaller:** The difference in performance between the inpainting-assisted methods and *vanilla-unet* is larger when the size of the training dataset is smaller. For example, for BraTS, the difference between *vanilla-unet* and *roi-supervoxel* (our proposed approach) is as large as 41.2% (relative) when the size of the training dataset is 25% of the total. This trend is also observed between the methods with and without ROI information.
**Supervoxels help more when areas to be segmented are larger rather than finer:** ROI-guided inpainting can be postulated to have a better chance of affecting the downstream performance when the ROI itself is larger. Taking into account that tumours in BraTS are, on-average, larger than the hyperintensities to be segmented in the WMH dataset, it can be observed that the performance difference between the inpainting methods using supervoxels (*roi-supervoxel* and *noroi-supervoxel*) versus the ones which do not (*roi-grid* and *noroi-grid*) is smaller in the case of WMH than for BraTS. For example, when using all of the training data, the difference in performance between *roi-supervoxel* and *roi-grid* is 3% (relative) for BraTS but only 0.25% for WMH. This could likely be alleviated by problem specific selection of parameters for SLIC, which we did not explore. This would ensure that the supervoxels are not too large as compared to the ROI, in which case the effect of ROI would not be significant.
These results show that our approach is promising. An important point to note is that a similar approach may be valuable in other forms of local self-supervision techniques like jigsaw puzzle solving [@DBLP:conf/miccai/ZhuangLHMYZ19], where the shuffling could be guided by the ROI and the tiles could be picked by ensuring homogeneity constraints.
Although efficient, this method does have some limitations: firstly, its efficiency depends on the parameters of the supervoxelization process and a poor choice of parameters could lead to limited performance gain; secondly, although sizeable synthetic datasets can be created in this process, the reliance on ROI means that we still need segmentation annotations. Perhaps one way of solving the second problem would be using co-training [@10.1145/279943.279962] to label all of the data and then employ our method using the entire corpus.
Conclusion
==========
In summary, this work explores the use of supervoxels and foreground segmentation labels, termed the *region-of-interest (ROI)*, to guide the proxy task of inpainting for self-supervision. Together, these two simple changes have been found to add a significant boost in the performance of a convolutional neural network for segmentation (as much as a relative gain of 5% on the BraTS 2018 dataset), in comparison to traditional methods of inpainting-based self-supervision.
[^1]: We use the implementation in <https://scikit-image.org/docs/dev/api/skimage.segmentation.html?highlight=slic#skimage.segmentation.slic>
[^2]: url-masked-for-blind-review
|
---
abstract: 'Historically, light curve studies of supernovae (SNe) and other transient classes have focused on individual objects with copious and high signal-to-noise observations. In the nascent era of wide field transient searches, objects with detailed observations are decreasing as a fraction of the overall known SN population, and this strategy sacrifices the majority of the information contained in the data about the underlying population of transients. A population level modeling approach, simultaneously fitting all available observations of objects in a transient sub-class of interest, fully mines the data to infer the properties of the population and avoids certain systematic biases. We present a novel hierarchical Bayesian statistical model for population level modeling of transient light curves, and discuss its implementation using an efficient Hamiltonian Monte Carlo technique. As a test case, we apply this model to the Type IIP SN sample from the Medium Deep Survey, consisting of [18,837]{} photometric observations of [76]{} SNe, corresponding to a joint posterior distribution with [9,176]{} parameters under our model. Our hierarchical model fits provide improved constraints on light curve parameters relevant to the physical properties of their progenitor stars relative to modeling individual light curves alone. Moreover, we directly evaluate the probability for occurrence rates of unseen light curve characteristics from the model hyperparameters, addressing observational biases in survey methodology. We view this modeling framework as an unsupervised machine learning technique with the ability to maximize scientific returns from data to be collected by future wide field transient searches like LSST.'
author:
- 'N. E. Sanders, M. Betancourt, A. M. Soderberg'
title: Unsupervised Transient Light Curve Analysis Via Hierarchical Bayesian Inference
---
INTRODUCTION {#sec:intro}
============
The majority of luminous transients in the universe are core-collapse supernovae (CC-SNe), marking the explosive deaths of massive stars [@Heger03; @Smartt09]. Stellar evolution theory, as well as both detailed observations of the explosive transient and fortuitous pre-explosion observations of the progenitor star, point to progenitor initial mass as the primary factor determining stars’ eventual death state. Metallicity, rotation rate, binarity, and other properties play important secondary roles, and permutations of these parameters are likely responsible for the extreme diversity of core-collapse supernovae phenomenology observed in the universe [@Heger03; @Smartt09; @Smith11; @Ekstrom12; @Jerkstrand13]. The progenitor star mass distribution for each SN type, as well as the distribution of these secondary factors, have far reaching implications throughout astrophysics, influencing the theory of stellar evolution [@Groh13], galactic chemical evolution [@Nomoto13], hydrodynamic feedback in galaxy formation [@Stilp13], and astrobiology [@Lineweaver04].
Studies of individual transients typically focus on well observed cases within each object class, capitalizing on the availability of detailed and high signal-to-noise observations to facilitate comparison to finely tuned hydrodynamic explosion simulations and analytic light curve models (e.g. @Mazzali03 [@Utrobin08]). Syntheses of these observations, studies of large samples of SNe of a given class, are then typically composed of samples culled from these well observed cases (see e.g. @Nomoto06 [@Bersten09; @Jerkstrand13]). However, the properties of luminous and/or high signal-to-noise objects within a survey sample may be systematically different from their lower luminosity / signal-to-noise counterparts, and traditional targeted transient searches themselves are inherently biased towards particular SN progenitor properties like high metallicity [@SandersIbc; @nes2010ay]. To derive truly robust and unbiased inferences about SN progenitor populations, it is therefore necessary to study transient samples in a fashion as complete and observationally agnostic as possible.
Here we discuss a methodological framework for the simultaneous modeling of multi-band, multi-object photometric observations from wide field transient surveys, which addresses certain biasing factors inherent to transient searches. This method is rooted in “hierarchical” and “multi-level” Bayesian analysis, where information about similar events within a sample is partially pooled through a hierarchical structure applied to the joint prior distribution (see @BDA3 and references therein; see @Mandel09 for applications to SN Ia light curves). We adopt Hamiltonian Monte Carlo as a computational technique to efficiently explore the high-dimensional and strongly correlated posterior distribution of this hierarchical model [@Betancourt13]. The result of this modeling is simultaneous inference on physically-relevant light curve parameters describing individual objects in the sample, as well as the parameter distribution among the population, regularized by the application of minimal (“weakly informative”) prior information.
In Section \[sec:model\] we discuss the design and implementation of a hierarchical Bayesian model capable of simultaneously fitting large quantities of raw photometric data from wide field transient surveys to infer the population properties of the underlying SN sample. We test this model with a sample dataset of Type IIP SNe from the (PS1) survey (Section \[sec:data\]), previously published in [@Sanders14IIP]. We explore the results of this test in Section \[sec:results\], including comparison with inferences drawn from traditional modeling based on fits to individual light curves. We discuss the implications of this methodology for future wide field transient surveys in Section \[sec:disc\] and conclude in Section \[sec:conc\].
MODEL DESIGN {#sec:model}
============
We have designed a hierarchical Bayesian generalized linear model (GLM) to simultaneously describe the individual multi-band light curves of a set of optical transients, and the population distribution of their light curve parameters. Due to the nature of the sample dataset we discuss in Section \[sec:data\], the non-linear link function of our GLM is tailored for Type IIP SNe (see Section \[sec:model:lc\]), but the hierarchical structure of the model is generalizable to any transient class.
Type IIP SNe are particularly apt for a hierarchical modeling approach because their long lived light curves reduce the likelihood of any individual object to have fully identifiable light curve parameters. In particular, because the plateau phase of the SN IIP light curve has a duration ($\sim3$ months) similar to the length of observing seasons for typical pointings of ground based telescopes, individual light curves are typically incomplete. The detected SNe IIP have often exploded between observing seasons, when their field is behind the sun, or their field sets before the plateau phase has ended. As a result, individual objects in the data set do not have the temporal coverage needed to fully identify their light curve parameters. Partial pooling among objects in the sample can compensate, helping to identify unconstrained parameter values for individual objects, while applying information from well-constrained parameters of the individual light curves to all other objects in the sample.
Light curve model {#sec:model:lc}
-----------------
We have designed a physically motivated parameterized model for the SN IIP light curve, composed of 5 piecewise power law and exponential segments. The model is fully specified by a set of 12 independent parameters per optical filter and an explosion date ($t_0$). These parameters are 4 time durations ($t_1,t_p,t_2,t_d$) defining the knot locations of the segments, 5 rate parameters describing the slope of each light curve segment ($\alpha,\beta_1,\beta_2,\beta_{dN},\beta_{dC}$), a luminosity scale ($M_p$), a background level ($Y_b$) for the photometric data, and an intrinsic scatter ($V$) to encompass deviation from the model. The light curve model and its primary parameters are illustrated in Figure \[fig:modschem\]; a full mathematical description of the light curve model is given in [@Sanders14IIP].
Hierarchical structure {#sec:model:ml}
----------------------
We allow for partial pooling between the light curve parameters of this 5-component model using a linear hierarchical structure. This structure is illustrated in Figure \[fig:dagm7\]. For the time, rate, luminosity scale, and intrinsic scatter parameters of the model, this structure includes levels for individual photometric filters, individual SNe-filter combinations, and a top-level parameter; a 3-level structure. For the explosion date ($t_0$) parameter, which is not defined per-filter, we use a modified two-level structure. The structure includes separate hyperprior distributions for objects discovered within and between observing seasons, which will have significantly different delay times between explosion and the epoch of detection. We do not adopt a hierarchical structure for the background level $Y_b$ parameter, which should nominally be $0$ except in the presence of artifacts among the PS1 template images used in difference imaging.
In effect, this structure means that a single top level value is drawn for each rate parameter; separate filter-level rate parameters are drawn for each of the $grizy$ filters from the hyperprior distribution specified by the top level value; and bottom level rate parameters for each SN-filter combination are drawn from the hyperprior specified by the filter-level draw. In practice, this “centered” multi-level parameterization is non-optimal, because it introduces significant correlations between the hyperparameters in the model that decrease the efficiency of the MCMC sampler. Instead we use a modified, “non-centered parameterization,” where correlations between hyperparameters are exchanged for correlations between hyperparameters and data. This is a general technique applicable for any distribution in the location-scale family, and optimal when the data poorly identify the parameter values [@Papaspiliopoulos07; @Betancourt13]. We therefore adopt normal hyperprior distributions for all the location hyperparameters in our model, and half-cauchy distributions for all scale parameters (including hyperprior width parameters; @Gelman08).
The hierarchical modeling framework largely eschews the specification of prior information, instead allowing the model to set its own hyperprior distributions learned from the data during fitting. We view this process, as applied to transient optical light curve studies, as a form of unsupervised machine learning. In effect, the model is learning the shape and range of variation among light curves within the transient class, and applying that information to optimally interpret individual light curves.
However, it is necessary to set prior distributions for the top level hyperparameters, and we adopt weakly informative priors except where needed to enforce regularization of the light curve model. In particular, we assign mean values for the normal prior distribution on the filter-level parameter ($t_{hF,t_p}$) controlling the plateau phase rise time ($t_p$) to specify the within filter variation observed in [@Sanders14IIP]. We do the same for the filter-level priors controlling the plateau phase rise and decay rates ($\beta_1$ and $\beta_2$). We specify the prior on the explosion date hyperparameters with means of 1 and 100 days for the within- and between-season objects, respectively. We use a restrictive $\rm{cauchy}(0.001)$ hyperprior for the top-level intrinsic scatter parameter ($V_h$) to regularize its ability to dominate the likelihood evaluation. We note that narrow hyperprior distributions are needed here because the hierarchical model exponentially amplifies variances. Prior information is therefore needed to ensure a reasonable range of variation of the top level parameters and to avoid numerical overflow during sampling. The model then fits optimal values for each of these hyperparameters given the likelihood for the data, and these priors serve largely to regularize the results.
Stan implementation {#sec:model:stan}
-------------------
To sample from this model posterior, we employ the C++ library *Stan* [@STAN], which implements the adaptive Hamiltonian Monte Carlo (HMC) *No-U-Turn Sampler* (NUTS) of [@NUTS]. HMC is advantageous for inference on high dimensional multi-level models, because it capitalizes on the gradient of the posterior to efficiently traverse the joint posterior despite the presence of the highly correlated parameters inherent to hierarchical models [@Betancourt13]. In practice, HMC will achieve a significantly higher effective sample size ratio (i.e. lower autocorrelation in the trace) than traditional Gibbs samplers for models with highly correlated parameters [@Betancourt13; @stan-manual:2014].
NUTS operates in two phases; “adaptation” and “sampling.” During adaptation, the algorithm automatically tunes the temporal step size which controls the discretization of the Hamiltonian [@hgelman2013]. Additionally, the algorithm estimates a diagonal HMC mass matrix during adaptation, which effectively scales the global step size to the optimal value for each parameter (we do not configure *Stan* to estimate the full, “dense” mass matrix given the significant additional computational overhead). During the sampling phase, the step size and mass matrix are fixed.
We use *Stan* to construct [32]{} independent MCMC chains from the posterior distribution of the model.[^1] We have used the Harvard Faculty of Arts and Sciences “Odyssey” Research Computing cluster to run these chains in parallel, running for the cluster’s maximum job execution time of 3 days per chain, for a total utilization of $2,304$ cpu hrs. Given our total yield of [12651]{} samples, this represents an average chain length of [395]{} samples and an effective sampling rate of [[5.49]{}]{} samples per hour per chain. For the purposes of convergence testing (Section \[sec:res:converge\]), we consider the full chains including adaptation phase. For the purposes of light curve modeling, we exclude the adaptation phase as well as the first 20 iterations of the sampling phase, yielding [11265]{} total samples from the approximate posterior stationary distribution.
The high computational cost of sampling from the model posterior distribution is due to the small HMC step size emerging from the NUTS adaptation. Figure \[fig:treedepth\] illustrates this effect, comparing the Hamiltonian discretization step size to the number of leapfrog steps per iteration as the step size varies during NUTS adaptation. As the step size decreases, the number of leapfrog steps needed (the number of posterior calculations, and therefore the execution time) grows exponentially. The horizontal feature at the top of the this figure illustrates saturation of the leapfrog algorithm tree depth, suggesting that yet smaller step sizes may be needed to optimally sample from the posterior. However, given the onerous computation time required to iterate the NUTS algorithm (which is not immediately parallelizable) at the selected maximum tree depth ($\gtrsim1$ hour of CPU time at the maximum tree depth of 16), we have elected not to increase the maximum tree depth. As a result, the HMC sampler could potentially become stuck in local minima of the multi-dimensional posterior, biasing the resulting samples away from the tails of the true joint posterior distribution.
SAMPLE DATA {#sec:data}
===========
Optical Observations {#sec:data:PS1}
---------------------
Our Type IIP supernova light curve sample is selected from four years of systematic Medium Deep Field observations by the (PS1) telescope, as described in [@Sanders14IIP]. PS1 is a high-etendue wide-field imaging system, designed for dedicated survey observations and located on a mountaintop site in the Hawaiian island chain. Observations are conducted remotely, from the University of Hawaii–Institute for Astronomy Advanced Technology Research Center (ATRC) in Pukalani. A complete description of the PS1 system, both hardware and software, is provided by [@PS1]. The 1.8 m diameter primary mirror, $3.3^\circ$ field of view, and other PS1 optical design elements are described in [@PS1opt]; the array of 0.258pixel detectors, and other attributes of the PS1 imager is described in [@PS1cam]; and the survey design and execution strategy are described in [@PS_MDRM]. The PS1 Medium Deep Survey (MDS) consists of 10 pencil beam fields observed with a typical cadence of 3 d in each filter.
The PS1 observations are obtained through a set of five broadband filters, which we refer to interchangeably as as [$g_{\rm P1}$]{}, [$r_{\rm P1}$]{}, [$i_{\rm P1}$]{}, [$z_{\rm P1}$]{}, and [$y_{\rm P1}$]{}or simply $grizy$ [@PS1cal]. MDS achieves a $5\sigma$ depth of $\sim23.3$ mag in $griz$ filters, and $\sim21.7$ mag in the $y$-filter (with observations taken near full moon). Photometry presented here is in the “natural” PS1 system, $m = −2.5 \log(\rm flux) + m^\prime$, with a single zero-point adjustment $m^\prime$ made in each band to conform to the AB magnitude scale [@Schlafly12; @Tonry12; @Magnier13].[^2] We assume a systematic uncertainty of 1% for our PS1 observations due to the asymmetric PS1 point spread function and uncertainty in the photometric zero-point calibration [@Tonry12]. The standard reduction, astrometric solution, and stacking of the nightly images is done by the Pan-STARRS1 IPP system [@PS1_IPP; @PS1_astrometry], and the nightly MDS stacks are processed through a frame subtraction analysis using the *photpipe* image differencing pipeline [@Rest05; @Scolnic13].
\[sec:data:sample\]
We adopt the final spectroscopic SN IIP sample from [@Sanders14IIP], including all objects sub-classified using the Support Vector Machine machine learning classification method therein. This sample consists of [18,837]{} total photometric data points, including [5,056]{} robust detections, for [76]{} SNe IIP in the $grizy$ filters. We note that the photometric observations which are not robust detections still play a significant role in the likelihood of our model, serving to constrain the rise time and decay rate parameters of the model, as well as directly identifying the background parameter $Y_b$. The particular transients included in this sample and their properties are described in [@Sanders14IIP].
Posterior Probability Convergence {#sec:data:logprob}
---------------------------------
The HMC algorithm quickly and efficiently converges on a maximal value of the global posterior probability for the model by identifying optimal values for each bottom level light curve parameter for all the SNe and for the hyperparmeters. Given that the global model for the PS1 SNe IIP sample has a total of [9,176]{} individual parameters, this fast convergence is a significant testament to the efficiency of HMC as an optimization engine for high-dimensional functions.
Figure \[fig:trace:logprob\] shows the posterior probability evolution of the Markov chains as the NUTS sampler adapts and then reaches the sampling phase. Chains typically converge near the maximum achievable posterior probability during our warmup period of only 30 iterations.
RESULTS {#sec:results}
=======
Sampling Characteristics and Fit Convergence {#sec:res:converge}
--------------------------------------------
Figure \[fig:trace:beta2\] shows the MCMC trace for a well identified bottom level parameter; the values drawn from the HMC algorithm for the plateau phase decline rate ($\beta_2$) of an object ([PS1-12cey]{}) with sufficient $r$-band photometry to constrain this phase of the light curve. The sampler moves quickly in this dimension, with low autocorrelation between samples, and the parameter is acceptably convergent (with potential scale reduction factor $\hat{R}={1.14}$).
In contrast, Figure \[fig:trace:rhP\] shows the trace for a moderately well identified top level parameter ($r_{hP}$), controlling the global plateau phase decay rate across all filters. The trace indicates that the sampler is moving more slowly in this dimension, with significant autocorrelation between samples. This top level parameter has apparently not yet converged ($\hat{R}={1.77}$). This suggests that additional sampling is needed to achieve a desirable level of convergence among some hyperparameters, but the computational cost is prohibitive at this time; we discuss alternative methods for achieving convergence with HMC in Section \[sec:disc\].
The origin of the convergence challenges facing the HMC algorithm are illustrated in Figure \[fig:joint:rhSNF\]. The Figure, showing a slice from the joint posterior distribution, illustrates the high correlation between the hierarchically linked parameters in the model. In contrast, Figure \[fig:joint:thPthF\] shows a slice of the joint posterior along the dimensions of the top level and filter-level hyperparameters for the plateau phase time duration. Dependence between these hyperparameters was obviated via selection of the non-centered parameterization (Section \[sec:model:ml\]) and, indeed, their marginal posteriors have very low correlation.
Posterior Predictive Check Comparison {#sec:res:PPC}
-------------------------------------
We validate the success of our model in describing the light curve behavior of objects in our SN IIP sample using posterior predictive checks [@BDA3], comparing the distribution of luminosities predicted under our fitted light curve model to the observed photometric data. Figure \[fig:PPC:lq\] shows a posterior predictive check for [[PS1-10zu]{}]{}, whose poor temporal coverage illustrates the strengths of the hierarchical model. The figure compares the $r$-band light curve fit for this object to the fit under the individual-level model presented in [@Sanders14IIP], which uses an identical 5-component light curve model, but does not make use of no partial pooling between SNe. The hierarchical fit achieves significantly greater constraints on the parameters describing the rising phases of the SN, resulting in a much tighter distribution of explosion dates and plateau durations (a parameter critical for physical inference on the progenitor star). The improvement is due to the strongly identified plateau duration hyperparameters (Section \[sec:res:pop\]). In the individual-level model, a weakly informative prior distribution was established for this parameter based on the theoretically predicted range of plateau duration variation; in the multi-level model, the hyperparameters are inferred from the data themselves, resulting in much stronger prior information at the individual SN level. That the fit to the later phases of the light curve, where the data are strongly identifying, is indistinguishable from the fit obtained in the individual model is validation of the unbiased performance of the hierarchical model.
Figure \[fig:PPC:hq\] shows a comparison of fits for several additional objects, illustrating features of the hierarchical modeling framework under different data scenarios, which we describe here. [[PS1-11azd]{}]{} was observed only during the rise and initial stages of the plateau phase. The posterior predictive luminosity distribution of the individual and hierarchical models are similar, but the plateau decline phase duration ($t_2$) parameter is much more constrained in the hierarchical model (Figure \[fig:PPC:hq\] a).
[[PS1-12bku]{}]{} (Figure \[fig:PPC:hq\] b) was observed from explosion through the final, radioactive decay-dominated phase. Generally, this case confirms that, where the data is strongly identifying, the hierarchical model produces fits in agreement with the individual-level model. Interestingly, for this object there is a $r$-band photometric observation with relatively high uncertainty at $\sim+80$ days, which introduces a degeneracy in the posterior—whether this point should be assigned to the plateau or transition phase of the light curve. The fit for this SN under the hierarchical model looks similar to the individual-level model fit, exploring both forks of the degeneracy. However, the two fits favor opposite sides of the fork. The individual fit maximizes the likelihood of the $r$-band photometry for the object alone, placing the point on the transition phase, while the hierarchical fit prefers the solution where the point falls on the plateau. Because the fork favored by the hierarchical model is more consistent with the modeled distribution of plateau durations among SNe IIP (based on partial pooling from the other objects in the sample), it is the more well justified solution.
In Figure \[fig:PPC:hq\] c, the hierarchical $z$-band fit for the SN [[PS1-11ai]{}]{} is much more highly regularized to match the shape of other $z$-band light curves than the individual fit. This leads to significantly improved constraints on the peak magnitude and plateau duration for this object. For [[PS1-12wn]{}]{} ($y$-band; Figure \[fig:PPC:hq\] d), only 1 robust photometric detection is available, and all the photometry is highly uncertain ($\delta m\sim0.5$ mag at $1\sigma$). The individual light curve fit in this case is very poorly regularized, obeying the peak magnitude suggested by the detection and the limits suggested by the non-detections, but otherwise has very poorly constrained light curve properties like plateau duration and decline rate. The hierarchical fit in this case is far superior in regularization, showing a characteristic $y$-band SN IIP light curve shape matched to the available photometry.
Population Parameter Distribution Characteristics {#sec:res:pop}
-------------------------------------------------
To further the investigation of SN IIP plateau durations from [@Sanders14IIP], in Figure \[fig:tplatdist\] we compare the plateau duration inferences from that project to the duration distributions inferred from the hierarchical model. The plateau duration distribution hyperprior in the hierarchical model is a sum of the $t_p$ and $t_2$ hyperpriors. The hyperpriors include both location (e.g. $t_{hP}$) and width (e.g. $\sigma~t_{hP}$) hyperparameters, so we visualize the posterior distribution of hyperpriors by showing multiple lognormal hyperpriors corresponding to random draws of the hyperparameters. We focus on the $r$-band durations here, and so include both the $t_{hP}$ and $t_{hF}$ hyperparameters.
The distribution of bottom-level plateau duration parameters for our hierarchical model agree well with the individual light curve fits from [@Sanders14IIP]. Taking the median bottom level parameter from the MCMC chain, the distribution of values from the hierarchical model has a mean and standard deviation of ${90}\pm{6}$ days, compared to ${92}\pm{14}$ days for the individual model. Note that the variance in the hierarchical model distribution is significantly lower than from the individual fits, because partial pooling between objects constrains the bottom level posteriors.
By directly modeling the underlying population of transients, the hierarchical modeling framework allows us to overcome potential biases in transient search methodology. In particular, although long duration SNe IIP are less likely to be observed with full temporal coverage in ground based transient searches (Section \[sec:model\]), we can estimate the fraction of unseen, long-duration transients in the population from the hierarchical model posterior. This product of the posterior distribution of the hierarchical model is constrained by both the observed characteristics of objects in the sample, and characteristics allowed by pre-explosion and late time non-detections from the transient search at the same location.
Our hyperparameter posterior distributions suggest there is a [60]{}% probability that at least 10% of the underlying population of SNe IIP have $r$-band plateau durations longer than the bottom level parameter value for any individual object in the sample ($>{112}$ days). The probability that at least 20% of objects fall above this value is [30]{}%. Among the sampled hyperprior distributions, the median of the population standard deviation is [33]{} days. The standard deviation distribution has a strong tail at larger values, shown in Figure \[fig:tplatdist\]. These results emphasize and support the finding of [@Sanders14IIP], that the plateau duration distribution of SNe IIP has significant variance.
DISCUSSION {#sec:disc}
==========
The multi-level model developed and applied in this paper points to a methodological framework for the interpretation of SN light curves from the next generation of wide-field transient searches, such as the Large Synoptic Survey Telescope (LSST; @LSSTbook). In the coming era, the volume of available photometric data will increase dramatically, while the human and observational resources for follow-up of individual objects will not. In this regime, to capitalize on the larger SN sample sizes afforded by these next generation searches, it will be critical to apply population level light curve modeling. To do this, it is necessary to apply analytical methods that are robust to data sparseness and posterior non-identifiability for individual objects, and computational methods that are capable of generating inferences from large datasets. The combination of hierarchical Bayesian methodology and Hamiltonian Monte Carlo methods explored in this work are natural methods for addressing both these concerns.
This work also suggests future paths for improvement of the Bayesian light curve modeling framework presented here. First, to permit applications to purely photometric datasets (where the SN classification and redshift are not known or poorly constrained), the model must be generalized. The redshift can trivially be added to the multi-level model as a vector of free parameters, but it will introduce significant posterior correlations and interactions that will need to be tested and may require the assertion of significant prior information to aid identifiability. Additional model components are needed to allow for application to multiple SN classes simultaneously. This could be accomplished through categorical mixture modeling with several different, physically-motivated SN light curve prescriptions for each SN type or by using a more generic, non-parametric, or continuously expanding light curve model to allow fitting of diverse SN types. Both of these solutions would support classification inferences, by assessment of the categorical simplex parameter posteriors directly, or by a clustering analysis of the continuous model expansion parameters, respectively. Applications to transient searches delving to significantly higher redshifts (and also the non-parametric modeling approach) may require full three-dimensional modeling of the SN spectral energy distribution evolution, rather than two-dimensional light curve modeling in each filter, in order to permit $K$-corrections at all distances. Finally, incorporating host galaxy information will be critical to producing purely-photometric informative inferences across SN classes. This should include modeling of the distribution of host galaxy global properties per SN class to uncover and take advantage of differences in progenitor properties (see e.g. @Kelly12 [@SandersIbc; @galsnid; @Lunnan13b; @McCrum14]), as well as the line-of-sight host galaxy extinction and reddening law (e.g. @Mandel11).
Finally, we look to Riemannian manifold Hamiltonian Monte Carlo (RMHMC; @betancourt:2013) techniques for permitting posterior characterization in the future, in the face of these additional modeling complexities. Compared to traditional, Euclidian Hamiltonian Monte Carlo (EHMC; as applied for this work), RHMC samplers efficiently explore highly correlated and high-dimensional posterior functions by automatically adapting the Hamiltonian integration step size to a value optimal for local conditions [@Girolami11; @Betancourt13]. This capability would permit unbiased sampling even amidst models with joint posterior distributions with higher curvature than the multi-level model examined here, such as a model including interactions between filter-level parameters or interactions between light curve parameter groups (e.g. $t$-$r$ interactions). Our attempts to fit such a model with EHMC have not achieved convergence within reasonable integration times, with the high posterior curvature preventing the hyperparameters from moving at a sufficient rate to produce convergent chains. The addition of RHMC sampling capabilities to *Stan* in the near future [@betancourt:2013] will make these techniques accessible to astronomers in the LSST era.
CONCLUSIONS {#sec:conc}
===========
We have explored the use of Bayesian hierarchical modeling and Hamiltonian Monte Carlo (HMC) to enable population-level inference on multi-band transient light curves from comprehensive analysis of optical photometry from wide field transient searches. The primary conclusions of this work are:
- While computational limits still challenge the implementation of hierarchical models, due to the high curvature in their joint posterior distributions, sufficient convergence is achieved in the bottom level model parameters (Section \[sec:res:converge\]) to enable their immediate application for transient light curve studies.
- Comparisons between light curve posterior predictive distributions from our hierarchical model fit to the individual light curve fits of [@Sanders14IIP] show strong agreement for well identified parameters, and show an advantage for hierarchical models among poorly identified parameters (Section \[sec:res:PPC\]). In particular, partial pooling of parameter information between transients supports improved regularization of light curve shapes, and supports model selection between partially degenerate light curve parameter scenarios.
- By directly modeling the underlying transient population, hierarchical models permit inference on the occurrence of properties not observed within the dataset (Section \[sec:res:pop\]). This feature is of particular value in overcoming observational biases induced by ground based transient searches, such as the under-representation of long duration transients like some SNe IIP.
We have concluded with a discussion of future directions for this modeling (Section \[sec:disc\]), including applications to upcoming wide field transient searches, extensions to the hierarchical model structure developed here, and expanded capabilities to be enabled by the advent of Riemannian manifold Hamiltonian Monte Carlo.
\[sec:ackn\]
We thank K. Mandel for sage guidance and many helpful conversations; M. Brubaker, B. Carpenter, A. Gelman, and the *Stan* team for their excellent modeling language and HMC sampler and for thoughtful feedback on our model design; and the entire PS1 collaboration for their monumental efforts towards the collection of the SN IIP light curve dataset.
The Pan-STARRS1 Surveys (PS1) have been made possible through contributions of the Institute for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the Max-Planck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham University, the University of Edinburgh, Queen’s University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network Incorporated, the National Central University of Taiwan, the Space Telescope Science Institute, the National Aeronautics and Space Administration under Grant No. NNX08AR22G issued through the Planetary Science Division of the NASA Science Mission Directorate, the National Science Foundation under Grant No. AST-1238877, the University of Maryland, and Eotvos Lorand University (ELTE).
Support for this work was provided by the David and Lucile Packard Foundation Fellowship for Science and Engineering awarded to A.M.S. M.B. is supported under EPSRC grant EP/J016934/1. Computations presented in this paper were performed using the Odyssey supercomputing cluster supported by the FAS Science Division Research Computing Group at Harvard University.
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Appendix A: Hierarchical Light Curve Stan Model {#ap:stan}
===============================================
Below we reproduce the full hierarchical model for the 5 component piecewise SN IIP light curves in the *Stan* modeling language, as described in Section \[sec:model\]. The Stan model specification format is documented in the Stan Modeling Language Users Guide and Reference Manual [@stan-manual:2014].
The model takes the following data as input: `N_obs`, the total number of photometric data points; `N_filt`, the number of photometric filters; `t`, a vector of MJD dates of the photometric observations; `fL`, a vector of luminosities corresponding to the photometric observations (with units as described in @Sanders14IIP); `dfL` a corresponding vector of luminosity uncertainties; `z` the redshift; `t0_mean` an initial estimate of the explosion date (for initialization and for centering the explosion date prior distribution); `J` a vector of integers specifying the filter ID of each photometric observation; `Kcor_N`, a matrix of pre-computed $K$-corrections for each filter, in magnitudes with spacing of 1 day; `fluxscale` the zero-point of the luminosity unit system ($\texttt{fluxscale} = 10^7$ in the system we have employed); and `duringseason`, a boolean value specifying whether the object exploded within or between observing seasons, for selection of the explosion date prior distribution parameters. The calculation of the model light curve flux and application of the $K$-correction values is performed in the `transformed parameters` section, and the prior and likelihood calculations are performed in the `model` section. Certain vector-valued prior distribution parameters are specified in the `transformed data` section for convenience. We note that the higher level parameters for the five different light curve rates and four different phase durations are grouped together in vectors (e.g. $r_{hP}$ and $t_{hP}$ for the top level, and $r_{hF}$ and $t_{hF}$ for the filter level, respectively) for convenience.
The *Stan* model is then compiled and run [@stan-manual:2014] to yield MCMC samples from the posterior distribution of light curve parameters. We configured the *No-U-Turn Sampler* to use fixed $0$ initialization of the parameter values, an adaptation phase of 30 steps, a maximum treedepth of 16, and otherwise employed the default sampler parameters. We have used CmdStan version 2.2.0[^3].
data {
int<lower=0> N_obs;
int<lower=0> N_SN;
int<lower=0> N_filt;
vector[N_obs] t;
vector[N_obs] fL;
vector[N_obs] dfL;
vector[N_SN] z;
vector[N_SN] t0_mean;
int<lower=1,upper=N_filt> J[N_obs];
int<lower=1,upper=N_SN> SNid[N_obs];
int<lower=0> Kcor_N;
real Kcor[N_SN, N_filt,Kcor_N];
real<lower=0> fluxscale;
vector<lower=0,upper=1>[N_SN] duringseason;
}
transformed data {
vector[N_filt] prior_t_hF[4];
vector[N_filt] prior_t_hF_s[4];
vector[N_filt] prior_r_hF[5];
vector[N_filt] prior_r_hF_s[5];
for (i in 1:N_filt) {
prior_t_hF[1,i] <- 0;
prior_t_hF_s[1,i] <- 0.1;
}
prior_t_hF[2,1] <- -1;
prior_t_hF[2,2] <- -0.5;
prior_t_hF[2,3] <- 0;
prior_t_hF[2,4] <- 0.5;
prior_t_hF[2,5] <- 1;
for (i in 1:N_filt) {prior_t_hF_s[2,i] <- 0.1;}
for (i in 1:N_filt) {
prior_t_hF[3,i] <- 0;
prior_t_hF_s[3,i] <- 0.1;
}
for (i in 1:N_filt) {
prior_t_hF[4,i] <- 0;
prior_t_hF_s[4,i] <- 0.1;
}
for (i in 1:N_filt) {
prior_r_hF[1,i] <- 0;
prior_r_hF_s[1,i] <- 0.1;
}
prior_r_hF[2,1] <- 2;
prior_r_hF[2,2] <- 1;
prior_r_hF[2,3] <- 0;
prior_r_hF[2,4] <- -0.5;
prior_r_hF[2,5] <- -1;
for (i in 1:N_filt) {prior_r_hF_s[2,i] <- 0.1;}
prior_r_hF[3,1] <- 1;
prior_r_hF[3,2] <- 0.3;
prior_r_hF[3,3] <- 0;
prior_r_hF[3,4] <- -1;
prior_r_hF[3,5] <- -1;
for (i in 1:N_filt) {prior_r_hF_s[3,i] <- 0.1;}
for (i in 1:N_filt) {
prior_r_hF[4,i] <- 0;
prior_r_hF_s[4,i] <- 0.1;
}
for (i in 1:N_filt) {
prior_r_hF[5,i] <- 0;
prior_r_hF_s[5,i] <- 0.1;
}
}
parameters {
vector[4] t_hP;
vector<lower=0>[4] sig_t_hP;
vector[N_filt] t_hF[4];
vector<lower=0>[N_filt] sig_t_hF[4];
vector[N_SN * N_filt] t_hSNF[4];
vector<lower=0>[N_SN * N_filt] sig_t_hSNF[4];
vector[5] r_hP;
vector<lower=0>[5] sig_r_hP;
vector[N_filt] r_hF[5];
vector<lower=0>[5] sig_r_hF[5];
vector[N_SN * N_filt] r_hSNF[5];
vector<lower=0>[N_SN * N_filt] sig_r_hSNF[5];
real M_h;
real<lower=0> sig_M_h;
vector[N_filt] M_hF;
vector<lower=0>[N_filt] sig_M_hF;
vector[N_SN * N_filt] M_hSNF;
vector<lower=0>[N_SN * N_filt] sig_M_hSNF;
real Y_h;
real<lower=0> sig_Y_h;
vector[N_SN * N_filt] Y_hSNF;
vector<lower=0>[N_SN * N_filt] sig_Y_hSNF;
real t0s_h;
real<lower=0> sig_t0s_h;
vector[N_SN] t0s_hSN;
vector<lower=0>[N_SN] sig_t0s_hSN;
real t0l_h;
real<lower=0> sig_t0l_h;
vector[N_SN] t0l_hSN;
vector<lower=0>[N_SN] sig_t0l_hSN;
real<lower=0> V_h;
vector<lower=0>[N_filt] V_hF;
vector<lower=0>[N_SN * N_filt] V_hSNF;
}
transformed parameters {
vector[N_obs] mm;
vector[N_obs] dm;
vector<upper=0>[N_SN] pt0;
matrix<lower=0>[N_SN, N_filt] t1;
matrix<lower=0>[N_SN, N_filt] t2;
matrix<lower=0>[N_SN, N_filt] td;
matrix<lower=0>[N_SN, N_filt] tp;
matrix[N_SN, N_filt] lalpha;
matrix[N_SN, N_filt] lbeta1;
matrix[N_SN, N_filt] lbeta2;
matrix[N_SN, N_filt] lbetadN;
matrix[N_SN, N_filt] lbetadC;
matrix[N_SN, N_filt] Mp;
matrix[N_SN, N_filt] Yb;
matrix<lower=0>[N_SN, N_filt] V;
matrix<lower=0>[N_SN, N_filt] M1;
matrix<lower=0>[N_SN, N_filt] M2;
matrix<lower=0>[N_SN, N_filt] Md;
for (l in 1:N_SN) {
if (duringseason[l] == 1) {
pt0[l] <- -exp( t0s_h + sig_t0s_h * ( t0s_hSN[l] .* sig_t0s_hSN[l] ));
} else {
pt0[l] <- -exp( t0l_h + sig_t0l_h * ( t0l_hSN[l] .* sig_t0l_hSN[l] ));
}
}
for (i in 1:N_filt) {
for (j in 1:N_SN) {
t1[j,i] <- exp( log(1) + t_hP[1] + sig_t_hP[1] * (
t_hF[1,i] * sig_t_hF[1,i]
+ sig_t_hSNF[1,(i-1)*N_SN+j] * t_hSNF[1,(i-1)*N_SN+j]
));
tp[j,i] <- exp( log(10) + t_hP[2] + sig_t_hP[2] * (
t_hF[2,i] * sig_t_hF[2,i]
+ sig_t_hSNF[2,(i-1)*N_SN+j] * t_hSNF[2,(i-1)*N_SN+j]
));
t2[j,i] <- exp( log(100) + t_hP[3] + sig_t_hP[3] * (
t_hF[3,i] * sig_t_hF[3,i]
+ sig_t_hSNF[3,(i-1)*N_SN+j] * t_hSNF[3,(i-1)*N_SN+j]
));
td[j,i] <- exp( log(10) + t_hP[4] + sig_t_hP[4] * (
t_hF[4,i] * sig_t_hF[4,i]
+ sig_t_hSNF[4,(i-1)*N_SN+j] * t_hSNF[4,(i-1)*N_SN+j]
));
lalpha[j,i] <- -1 + ( r_hP[1] + sig_r_hP[1] * (
r_hF[1,i] * sig_r_hF[1,i]
+ sig_r_hSNF[1,(i-1)*N_SN+j] * r_hSNF[1,(i-1)*N_SN+j]
));
lbeta1[j,i] <- -4 + ( r_hP[2] + sig_r_hP[2] * (
r_hF[2,i] * sig_r_hF[2,i]
+ sig_r_hSNF[2,(i-1)*N_SN+j] * r_hSNF[2,(i-1)*N_SN+j]
));
lbeta2[j,i] <- -4 + ( r_hP[3] + sig_r_hP[3] * (
r_hF[3,i] * sig_r_hF[3,i]
+ sig_r_hSNF[3,(i-1)*N_SN+j] * r_hSNF[3,(i-1)*N_SN+j]
));
lbetadN[j,i] <- -3 + ( r_hP[4] + sig_r_hP[4] * (
r_hF[4,i] * sig_r_hF[4,i]
+ sig_r_hSNF[4,(i-1)*N_SN+j] * r_hSNF[4,(i-1)*N_SN+j]
));
lbetadC[j,i] <- -5 + ( r_hP[5] + sig_r_hP[5] * (
r_hF[5,i] * sig_r_hF[5,i]
+ sig_r_hSNF[5,(i-1)*N_SN+j] * r_hSNF[5,(i-1)*N_SN+j]
));
Mp[j,i] <- exp(M_h + sig_M_h * (
M_hF[i] * sig_M_hF[i]
+ sig_M_hSNF[(i-1)*N_SN+j] * M_hSNF[(i-1)*N_SN+j]
));
Yb[j,i] <- Y_h + sig_Y_h * (Y_hSNF[(i-1)*N_SN+j] .* sig_Y_hSNF[(i-1)*N_SN+j]);
V[j,i] <- V_h * V_hF[i] * V_hSNF[(i-1)*N_SN+j];
}
}
M1 <- Mp ./ exp( exp(lbeta1) .* tp );
M2 <- Mp .* exp( -exp(lbeta2) .* t2 );
Md <- M2 .* exp( -exp(lbetadN) .* td );
for (n in 1:N_obs) {
real N_SNc;
int Kc_up;
int Kc_down;
real t_exp;
int j;
int k;
real mm_1;
real mm_2;
real mm_3;
real mm_4;
real mm_5;
real mm_6;
j <- J[n];
k <- SNid[n];
t_exp <- ( t[n] - (t0_mean[k] + pt0[k]) ) / (1 + z[k]);
if (t_exp<0) {
mm_1 <- Yb[k,j];
} else {
mm_1 <- 0;
}
if ((t_exp>=0) && (t_exp < t1[k,j])) {
mm_2 <- Yb[k,j] + M1[k,j] * pow(t_exp / t1[k,j] , exp(lalpha[k,j]));
} else {
mm_2 <- 0;
}
if ((t_exp >= t1[k,j]) && (t_exp < t1[k,j] + tp[k,j])) {
mm_3 <- Yb[k,j] + M1[k,j] * exp(exp(lbeta1[k,j]) * (t_exp - t1[k,j]));
} else {
mm_3 <- 0;
}
if ((t_exp >= t1[k,j] + tp[k,j]) && (t_exp < t1[k,j] + tp[k,j] + t2[k,j])) {
mm_4 <- Yb[k,j] + Mp[k,j] * exp(-exp(lbeta2[k,j]) * (t_exp - t1[k,j] - tp[k,j]));
} else {
mm_4 <- 0;
}
if ((t_exp >= t1[k,j] + tp[k,j] + t2[k,j]) && (t_exp < t1[k,j] + tp[k,j] + t2[k,j] + td[k,j])) {
mm_5 <- Yb[k,j] + M2[k,j] * exp(-exp(lbetadN[k,j]) * (t_exp - t1[k,j] - tp[k,j] - t2[k,j]));
} else {
mm_5 <- 0;
}
if (t_exp >= t1[k,j] + tp[k,j] + t2[k,j] + td[k,j]) {
mm_6 <- Yb[k,j] + Md[k,j] * exp(-exp(lbetadC[k,j]) * (t_exp - t1[k,j] - tp[k,j] - t2[k,j] - td[k,j]));
} else {
mm_6 <- 0;
}
dm[n] <- sqrt(pow(dfL[n],2) + pow(V[k,j],2));
if (t_exp<0) {
N_SNc <- 0;
} else if (t_exp<Kcor_N-2){
Kc_down <- 0;
while ((Kc_down+1) < t_exp) {
Kc_down <- Kc_down + 1;
}
Kc_up <- Kc_down+1;
N_SNc <- Kcor[k,j,Kc_down+1] + (t_exp - floor(t_exp)) * (Kcor[k,j,Kc_up+1]-Kcor[k,j,Kc_down+1]);
} else {
N_SNc <- Kcor[k,j,Kcor_N];
}
mm[n] <- (mm_1+mm_2+mm_3+mm_4+mm_5+mm_6) / (pow(10, N_SNc/(-2.5)));
}
}
model {
t0s_h ~ normal(0, 0.5);
sig_t0s_h ~ cauchy(0, 0.1);
t0l_h ~ normal(log(100), 1);
sig_t0l_h ~ cauchy(0, 0.1);
V_h ~ cauchy(0, 0.001);
Y_h ~ normal(0, 0.1);
sig_Y_h ~ cauchy(0, 0.01);
M_h ~ normal(0, 1);
sig_M_h ~ cauchy(0, 0.1);
t_hP ~ normal(0,0.1);
sig_t_hP ~ cauchy(0, 0.1);
for (i in 1:4) {
t_hF[i] ~ normal(prior_t_hF[i], prior_t_hF_s[i]);
sig_t_hF[i] ~ cauchy(0, 0.1);
t_hSNF[i] ~ normal(0,1);
sig_t_hSNF[i] ~ cauchy(0, 0.1);
}
r_hP ~ normal(0,1);
sig_r_hP ~ cauchy(0, 0.1);
for (i in 1:5) {
r_hF[i] ~ normal(prior_r_hF[i], prior_r_hF_s[i]);
sig_r_hF[i] ~ cauchy(0, 0.1);
r_hSNF[i] ~ normal(0,1);
sig_r_hSNF[i] ~ cauchy(0, 0.1);
}
M_hF ~ normal(0,1);
sig_M_hF ~ cauchy(0, 0.1);
M_hSNF ~ normal(0,1);
sig_M_hSNF ~ cauchy(0, 0.1);
Y_hSNF ~ normal(0,1);
sig_Y_hSNF ~ cauchy(0, 0.1);
V_hF ~ cauchy(0, 0.1);
V_hSNF ~ cauchy(0, 0.1);
t0s_hSN ~ normal(0,1);
sig_t0s_hSN ~ cauchy(0, 0.1);
t0l_hSN ~ normal(0,1);
sig_t0l_hSN ~ cauchy(0, 0.1);
fL ~ normal(mm,dm);
}
[^1]: The full *Stan* code for our statistical model is discussed in Appendix \[ap:stan\].
[^2]: The magnitudes quoted throughout this paper are in the AB system, except where explicitly noted.
[^3]: <https://github.com/stan-dev/stan/releases/tag/v2.2.0>
|
---
abstract: 'We study a 2D cellular automaton (CA) model for the evolution of coronal loop plasmas. The model is based on the idea that coronal loops are made of elementary magnetic strands that are tangled and stressed by the displacement of their footpoints by photospheric motions. The magnetic stress accumulated between neighbor strands is released in sudden reconnection events or nanoflares that heat the plasma. We combine the CA model with the Enthalpy Based Thermal Evolution of Loops (EBTEL) model to compute the response of the plasma to the heating events. Using the known response of the XRT telescope on board Hinode we also obtain synthetic data. The model obeys easy to understand scaling laws relating the output (nanoflare energy, temperature, density, intensity) to the input parameters (field strength, strand length, critical misalignment angle). The nanoflares have a power-law distribution with a universal slope of -2.5, independent of the input parameters. The repetition frequency of nanoflares, expressed in terms of the plasma cooling time, increases with strand length. We discuss the implications of our results for the problem of heating and evolution of active region coronal plasmas.'
author:
- Marcelo López Fuentes
- 'James A. Klimchuk'
title: 2D cellular automaton model for the evolution of active region coronal plasmas
---
Introduction {#intro}
============
One of the most persistent conundrums of Solar Physics has been, and still is, the problem of coronal heating. The difficulties arise from both the theoretical and observational sides. Observationally, it is very difficult to explain with a single scenario the diverse set of observations obtained in different wavelengths. The first X-ray observations in the decade of 1970 suggested that coronal loops were in quasi-static equilibrium and that a steady or quasi-steady heating process was balanced by radiative and conductive losses (Rosner et al. 1978). However, the situation changed as soon as ultraviolet observations from space became available, especially when instruments like the Transition Region and Coronal Explorer (TRACE, Handy et al. 1999) began to produce higher resolution and cadence data. The evolution of many EUV loops was too dynamic and intermittent to be consistent with quasi-static evolutions. Furthermore, density determinations using EUV instruments showed that loops are too dense given their temperature and length and had scales heights that are too large to be quasi-static (Aschwanden et al. 2001, Winebarger et al. 2003). It was proposed then that EUV loops could be heated by impulsive mechanisms. Thermal evolution models based in this premise were successful in explaining many of the observed physical conditions and evolutions (see e.g., Klimchuk 2006, 2009; Reale 2010, and references therein). More recently it has been shown that impulsive heating plays an important role in the diffuse component of the corona as well (Viall & Klimchuk 2011, 2012, 2013; Bradshaw et al. 2012, Warren et al. 2012, Schmelz & Pathak 2012). The question is now if it is possible to understand X-ray loops, EUV loops, and the diffuse emission as part of the same phenomenon or if they are caused by completely different mechanisms. The problem has of course many different aspects: the emission evolution, geometry, location within active regions, physical conditions of the plasma, etcetera, and all of these have to be considered to provide an explanation. One proposed possibility is that all of the corona is heated by impulsive short duration events, but the rate of repetition is high at some locations (e.g., X-ray loops) and indiscernible from a continuous source (Warren et al. 2010).
From the theoretical side, and in particular regarding the heating mechanism itself, several models have been proposed. They can be grossly classified in two types: models based on the dissipation of waves and models based on the dissipation of magnetic stresses (see the reviews by Klimchuk 2006, Reale 2010, Parnell & De Moortel 2012). Both types can produce impulsive heating, but the best known idea is from the second category and came from Parker (1988). He proposed that loops are made of elementary magnetic strands that are shuffled and tangled by photospheric motions. Current sheets form at the boundaries between the strands, and energy is released by magnetic reconnection. Parker estimated that the energy content of a single impulsive event is roughly one-billionth of a large flare, so he coined the term “nanoflare." We now use this term generically to describe any small spatial scale impulsive event, irrespective of the physical cause. Parker’s mechanism, the basis of our investigation, could provide the impulsive events invoked in the previous paragraph to explain EUV loops, X-ray loops, and diffuse emission within the same phenomenological framework.
Several studies over the years analyzed different aspects of the nanoflare heating problem, such as the conditions for reconnection (Parker 1983a, 1983b; Priest et al. 2002, Darlburg et al. 2005), the thermal evolution of the plasma (Cargill 1994, Cargill & Klimchuk 2004) and the observed coronal emission (Warren et al. 2010, López Fuentes et al. 2007). Other relevant issues are the mechanism by which footpoint motions translate into magnetic stress, the geometrical and temporal characteristics of the interaction between strands and the energy distribution of the produced nanoflares. A series of models based on the concept of Self Organized Criticality (SOC, Bak et al. 1988) have been developed during the last 20 years to analyze this part of the problem (see e.g., Lu & Hamilton 1991, Lu 1995, Longcope & Noonan 2000, Morales & Charbonneau 2008). The idea of this kind of approach is to use simple sets of rules for the injection of energy simulating the effect of footpoint motions, the interaction between magnetic strands, and the existence of a magnetic stress threshold beyond which energy release occurs (pedagogical reviews on the subject can be found in Charbonneau et al. 2001, and the book by Aschwanden 2011).
In a recent paper we presented a simple pseudo-1D cellular automaton model to explain the observed evolution of soft X-ray loops (Lopez Fuentes & Klimchuk 2010). Here we develop a more sophisticated model based on a similar aproach. Simulating the motions of magnetic strand footpoints in a 2D array we establish a series of rules for the interaction between strands and critical conditions for the magnetic stress relaxation through reconnection. The consequent energy release takes the form of short duration events whose output heats the plasma in the strands. To simulate the plasma response to these events we use the EBTEL code (Enthalpy Based Thermal Evolution of Loops, Klimchuk et al. 2008, Cargill et al. 2012). We analyze the output of the model by studying its dependence on the model parameters, the presence of power-laws in the energy distribution and the temporal properties of the produced nanoflares. In a second paper we will compare the results of the model with observed X-ray and EUV loops.
The paper is organized as follows. In Section \[model\] we provide a detailed description of the model, the implementation of EBTEL to compute the plasma response and the obtainment of synthetic observations. In Section \[analysis\] we present and analyze the results and discuss some of their implications and conclude in Section \[conclusions\].
Description of the model {#model}
========================
The model studied here is a more sophisticated version of the simple 1D model presented in López Fuentes & Klimchuk (2010). In the present version magnetic strand footpoints are represented by moving elements on a 2D grid. The idea behind the model can be explained starting from the classic view (Parker 1988) shown in Figure \[cartoon\], that represents coronal magnetic flux tubes or “strands” as vertical structures connecting two remote sections of the photosphere (represented by horizontal planes). In the initial configuration of panel (a), all flux tubes and their corresponding axial magnetic fields are vertical. As the strand footpoints are dragged by photospheric motions, magnetic stress is induced in the configuration progressively increasing the free magnetic energy. In the geometrical representation of Figure \[cartoon\], the displacement of the footpoints (the distance $d$ shown in panel (b)) is associated with an inclination angle ($\theta$) and the appearence of a horizontal component ($B_h$) of the strand magnetic field. Naming $B_v$ the vertical field, it can be seen from Figure \[cartoon\], panel (b), that
$$B_h = B_v \tan\theta \approx B_v \frac{d}{L},$$
where $L$ is the strand length. The approximation is accurate as long as the strand is not strongly inclined, as we expect. When sections of neighboring strands are in contact, their mutual inclination ($\Delta\theta$, see panel (c)) implies a current sheet at the interface. Dahlburg et al. (2005, 2009) have shown that the current sheet is unstable to the secondary instability whenever the misalignment exceeds a critical angle. Explosive reconnection occurs releasing magnetic energy and heating the plasma impulsively.
In our model, the above scenario is reduced to 2 dimensions in the following way. As it is shown in Figure \[grid\], the strands are represented by moving elements in a plane grid. Initially, each grid node is occupied by a single strand element. To simulate the displacement of the strand footpoints by photospheric motions, at each time step each element is randomly displaced one grid step in any of 4 directions as shown in panel (a). The boundary conditions are periodic, meaning that each time an element crosses the mesh border it reenters the grid on the opposite side. The physical justification is that to keep the density of strands constant, each element that leaves the system will on average be replaced by a similar one. The distance between grid nodes corresponds to the typical size of a convective granule ($d\sim$1000 km) and we associate the time step duration with the convective turnover time scale ($\delta t \sim$1000 sec). After a certain number of time steps, strand element $i$ would have been displaced across the grid a distance $S_i$ (see panel (b)). Following the arguments of the previous paragraph, this displacement produces a horizontal component of the magnetic field that can be approximated as
$$B_h^i \approx \frac{B_v}{L} S_i.$$
As will become more clear in the following paragraphs, $S_i$ should not be regarded as a distance on the grid *per se*, but as a length variable associated with the inclination of the strand. Consider now the situation from panel (b), in which strand element $i$ encounters $j$, that has its own displacement variable $S_j$. In the model, we relate this encounter to the situation described in Figure \[cartoon\], panel (c). By similar arguments to those used for Equations 1 and 2, the misalignment angle can be approximated as
$$\label{delta_theta}
\Delta \theta \approx \tan^{-1} \left(\frac{S_i+S_j}{L}\right).$$
Note that the grid in Figure \[grid\] should [*not*]{} be thought of as the photosphere. It is simply a conceptual way of treating the chance (coronal) interaction of strands in a tangled magnetic field. We assume that the motions of the strands are uncorrelated. If this assumption were violated, the misalignment angle would be less than indicated in Equation 3.
Assuming the existence of a critical condition for the energy release, each time two strand elements occupy the same node we compare the inclination angle, $\Delta \theta$, with a given critical value $\theta_c$. If $\Delta \theta > \theta_c$, we simulate reconnection by relaxing the magnetic stress between the two strands through the transformation of variables $S_i$ and $S_j$, as follows:
$$\begin{aligned}
\label{relaxation1}
S_i' = \alpha (S_i-d) + (1-\alpha) (S_j-d),\\
\label{relaxation2}
S_j' = (1-\alpha) (S_i-d) + \alpha (S_j-d),\end{aligned}$$
where $\alpha$ is a random number between 0.2 and 0.8, that accounts for the fact that the reconnection between strands $i$ and $j$ is not symmetric in general. The strands essentially exchange legs, and these segments can be long or short depending on how high in the corona (at what location between the planes in Figure \[cartoon\]) the reconnection takes place. The transformation given by Equations \[relaxation1\] and \[relaxation2\] implies that each time a strand reconnects, its “displacement” variable $S_i$ is on average reduced by an amount $d$. The choice is not arbitrary, since after reconnection the expected entanglement with other strands will prevent the reconnecting pair from relaxing much more than the typical separation between strands ($d$). This average decrease coincides with the increase received at each time step due to the footpoint displacement process, which is therefore the typical approximate excess over the critical condition. The relaxation returns the strands to a state close to, and not far below, the critical level of stress.
With each iteration, we move each strand to an adjacent position on the grid and increase $S_i$ by an amount $d$. If there is an encounter with one or more other strands at that same position, we decrease $S_i$ as described above. However, since we do [*not*]{} move the strand back to its old position, after a few transformations, $S_i$ rapidly departs from the length of the path on the grid. (It is smaller than the path length.) As we stated above, $S_i$ should not be regarded as a distance, but as a variable associated with the strand inclination in the sense given by Equation \[delta\_theta\]. We emphasize that the grid does not represent the photosphere, or a physical surface at any height. Instead, it should be regarded as a convenient 2D version of the 3D scenario illustrated in Figure \[cartoon\]. Its purpose is to model the random aspect of strand pair interactions. At a given iteration, a strand may reconnect with one, several (rarely more than 3), or no other strands, depending on how many occupy the same node. This realistically captures the property that strands are in contact with multiple neighbors in a tangled 3D magnetic configuration.
Operationally, it is important to “order” the testing and reconnection of the strands in the system. If the strands are tested and reconnected in order, for instance, starting with strand 1, then 2, then 3, and so on, most of the nanoflares will occur only in the first strands. There can be strands that suffer less nanoflares because their interacting partners already reconnected with some of the first strands and relaxed. In the actual code, we solve this by identifying first all the critical pairs. We then pick one randomly, and once this first pair has been reconnected we retest for all the remaining critical pairs again, then we randomly pick and reconnect another one, and so on until no critical pairs remain. At that point the program advances one timestep. With this scheme if one strand of a pair remains critical after its first reconnection it will eventually be reconnected a second time in the same step. Of course, no more than one or two reconnections per strand are expected.
We remind the reader that the idea of the present approach is to reproduce in a physically reasonable, but still geometrically simple way, the scenario of footpoint motion producing magnetic stress between interacting strands that leads to energy release in the form of short duration reconnection events. The simple displacement of points in a 2D grid is meant to reproduce the random nature of the strand mixing in the corona. The main motivation for the model simplicity is to save computing time, so a very large number of strands and nanoflares can be easily handled within runs of durations from minutes to tens of minutes. The combination with a fast code like EBTEL makes this choice even more convenient, so diverse sets of parameter combinations can be tested in manageable times. Such a study is not possible with a full 3D MHD treatment.
Noting that $S_i'+S_j'=S_i+S_j-2d$, the transformation implies, through Equation 3, a relaxation of the mutual inclination of the strands. Considering the association between $B_h^i$ and $S_i$ (correspondingly $j$) given by Equation 2, it can be easily shown that the above transformation implies a difference of magnetic energy density,
$$\label{energy_eq}
\Delta e_{ij} = \frac{B_v^2}{8\pi L^2}\left[\left(S_i'^2+S_j'^2\right)
- \left(S_i^2+S_j^2\right)\right].$$
We consider that each of these energy releases heats the plasma in the strands in the form of nanoflares. Since we do not expect the heat to be equally distributed between the strands, we include a second random number $\beta$ (0.25 $<\beta<$ 0.75), so strand $i$ receives an energy $e_i = -\beta \Delta e_{ij}$ and strand $j$ receives $e_j = -(1-\beta) \Delta e_{ij}$. At each time step we record the total energy per unit volume received by each strand due to the interaction with all of its critical partners. As previously discussed, depending on the initial $S_i$ and $S_j$, the transformation from Equation \[relaxation1\] and \[relaxation2\] could be insufficient to fully remove the critical condition ($\Delta \theta > \theta_c$). Therefore, the procedure is repeated for all interacting strands until no critical pairs remain. Once this is accomplished, the system is allowed to evolve to the next time step. The process repeats for as many time steps as wished. The longer the evolution of the system, the larger the nanoflare statistics produced.
The main input parameters of the model are the strand length ($L$), the vertical magnetic field ($B_v$), the critical misalignment angle ($\theta_c$) and the number of strands ($N$). We set these parameters to reasonable solar values and vary them, each at a time, to investigate the effect that they have on the model’s output. As explained above, parameters such as the photospheric displacement of the footpoints ($d$) and the time step duration ($\delta t$) are set at 1000 km and 1000 sec respectively, for all runs. The output of the model is the recorded series of nanoflares suffered by each strand at each time step.
One of the main motivations of the present model is the comparison with actual coronal observations. To accomplish this we need to produce synthetic observations from the model’s output described above. To simulate the response of the plasma contained in each strand to the heating produced by the nanoflares, we use the 0D hydrodynamic code EBTEL (see Klimchuk et al. 2008, Cargill et al. 2012). Modeling each nanoflare as a triangular function of a given duration, the EBTEL code provides us the temperature and density evolution of the plasma, which we combine with the known response of particular coronal instruments to obtain simulated strand light-curves. Since an observation will include many unresolved strands, due to both the optically thin nature of the corona and the finite resolution of real instruments, we add the emission of all strands to obtain the evolution of the total intensity per pixel registered by the instrument. It is worth noting here that the cellular automaton model and the EBTEL model work on different temporal scales. While CA time steps last 1000 s, individual nanoflares are set to several hundred seconds, and EBTEL time steps last one second.
In this work we simulate observations obtained with the X-ray Telescope (XRT, see Golub et al. 2007) on board Hinode (Kosugi et al. 2007), to study the effect of different input parameter combinations on the modeled emission. In an accompanying observational paper we compare the model with coronal data in different wavelengths obtained with the Hinode/XRT and SDO/AIA (Pesnell et al. 2012, Lemen et al. 2012) instruments.
Analysis of the results {#analysis}
=======================
Scaling relations {#scalings}
-----------------
To study the effect of the main input parameters on the model’s output, we choose a set of values appropriate for the Sun, and we vary them one at a time leaving the rest fixed. In Table \[parameters\] we list the different parameters indicating in bold face the set of values that remain fixed during individual variations. As we show below, $B_v$, $L$ and $\tan \theta_c$, are the parameters that have the largest effect on the model’s output. We focus on how they affect the mean energy of the nanoflares produced by the model, the plasma density and temperature computed with EBTEL, and the synthetic XRT intensity obtained as described in Section \[model\]. For all the EBTEL simulations we use a time resolution of 1 s. Figures \[energy\], \[plasma\] and \[xint\], show plots of the corresponding mean values as a function of the different input parameters tested. In all cases the log-log plots indicate the clear presence of power scalings. The lines correspond to linear regressions of the data and their slopes are indicated in the panels. In what follows we analyze the physical meaning of the scaling relations.
Let us begin with the mean nanoflare energy density ($\Delta e$) that heats the strands (see Figure \[energy\]). If we consider that, on average, the “distance” variables of two interacting strands are similar ($S_i \approx S_j$), and that after reconnection each of them is roughly reduced by a distance $d$, we can approximate the energy expression from Equation \[energy\_eq\] as
$$\Delta e \approx \frac{B_v^2}{4\pi L^2}\left[\left(S_i-d\right)^2
- S_i^2\right] \approx \frac{B_v^2 d}{4\pi L^2}\left(-2S_i+d\right).$$
Since in the fully developed regime $2S_i>>d$,
$$|\Delta e| \approx \frac{B_v^2 d}{2\pi L^2}S_i.$$
It is easy to see from Equation \[delta\_theta\] that $S_i \approx (L/2) \tan \theta_c$. Therefore, we can write the following scaling relation for the mean nanoflare energy:
$$\label{e_scaling}
|\Delta e| \propto \frac{B_v^2}{L}\tan \theta_c.$$
The above expression explains the scalings obtained from the linear fittings shown in Figure \[energy\].
The explanation of the scalings shown in Figure \[plasma\] requires the analysis of approximate relations for the plasma response to the heating. As discussed by Vesecky et al. (1979), in quasi-static conditions, the average heating rate ($Q$), conductive loss rate, and radiative loss rate are all comparable:
$$\label{balance}
Q~\approx \frac{2}{7} \kappa_0 \frac{T^{7/2}}{L^2} \approx n^2 \Lambda(T).$$
The second term is an approximation for the divergence of the heat flux. Here, $\kappa_0$ is the coefficient of thermal conduction, $T$ and $n$ are the temperature and density of the plasma, and $\Lambda(T)$ is the radiative loss function. One might imagine that the above relation is only valid in the high-frequency nanoflare regime that resembles a quasi-steady state. However, it can also be reasonable if low-frequency nanoflares dominate. Cooling after a nanoflare occurs in three stages. Thermal conduction dominates early, radiation dominates late, and the two are comparable at intermediate times. This middle stage lasts the longest, so median values of temperature and density will be representative of this stage and therefore of quasi-static conditions. Also, depending on the temperature sensitivity range of the observing instrument and on the nanoflare energy, the strands may only be visible when they are in this middle stage of cooling. The radiation loss function is usually expressed as
$$\label{lambda}
\Lambda(T) \approx \Lambda_0 T^b,$$
where $\Lambda_0$ and $b$ are constants within given temperature intervals. From Rosner et al. (1978), for the temperature range of interest here, $b$ = -0.5. From Equation \[balance\] we can obtain the following relation between the plasma temperature and the heating rate:
$$\label{t_q_scaling}
T \propto Q^{2/7} L^{4/7}.$$
In the present model, the heating rate on each strand is the average nanoflare energy ($\Delta e$) divided by the mean waiting time between events. As discussed in Section \[model\], due to the transformations from Equations \[relaxation1\] and \[relaxation2\], a strand would need on average a single time step to recover the critical condition. It is expected then, that each strand experiences an average of one nanoflare per time step, regardless of the model parameters. We confirm this by directly counting, from the model output, the number of nanoflares that occur in each strand at each time step. Then, we conclude that $Q \propto \Delta e$. Replacing this in Equation \[t\_q\_scaling\] and using the scaling from Equation \[e\_scaling\], we finally have for the temperature:
$$\label{t_equation}
T \propto B_v^{0.57} L^{0.28} \left(\tan \theta_c \right)^{0.28}.$$
Here, we write the exponents in decimal form for an easier comparison with the line slopes provided in the corresponding panels of Figure \[plasma\]. Notice that the slope of the solid line in the $T$($L$) plot (second row, left panel), which includes all the $L$ values, slightly departs from the above expected exponent. The reason is that the quasi-static approximation assumed to obtain Equation \[t\_equation\] is less applicable for shorter loops, in which the evolution produced by the impulsive heating is much more dynamic. To confirm this, we make a second line fitting (dashed line) that does not include the $L$= 40 Mm case. We then obtain a slope (0.31) that is much closer to the expected value (0.28).
Using Equations \[balance\] and \[lambda\] it can be easily shown that $n \propto T^2/L$. Replacing $T$ from Equation \[t\_equation\] in the previous expression, we obtain for the density:
$$\label{n_equation}
n \propto B_v^{1.14} L^{-0.43} \left(\tan \theta_c \right)^{0.57}.$$
The line slopes provided in the right panels of Figure \[plasma\] are consistent with the expected exponents. Once again, small departures are due to the limitations of the quasi-static approximation used.
The previous relations can be used to explain the intensity scalings obtained from the line fittings shown in Figure \[xint\]. The simulated XRT intensities whose mean values are plotted in the figure are obtained using the equation
$$I_{XRT} = n^2 S(T),$$
where $n$ and $T$ are the density and temperature computed with EBTEL, and $S(T)$ is the known response of the XRT instrument in the Al-poly filter position. Fitting the instrument response with a function of the form $S(T) = S_0 T^a$, we find that for the temperature range of interest here, $a$=1.92. Using the $n$ and $T$ scalings from Equations \[t\_equation\] and \[n\_equation\] and the previous relations we finally obtain
$$I_{XRT} \propto B_v^{3.37} L^{-0.33} \left(\tan \theta_c \right)^{1.67}.$$
The comparison between the exponents of the previous relation and the computed slopes in Figure \[xint\], show that although the $B_v$ and $\tan \theta_c$ exponents are in good agreement, the one corresponding to $L$ is not. This is possibly due to the several approximations used to obtain the above expression.
We expect that fluctuations of the simulated XRT signal will depend on the number of strands, $N$, that are are included in the summation. The heating-cooling processes due to individual nanoflares acting in different strands produce much more variability in the intensity signal when $N$ is small than when it is large. In the latter case, the sum of many strand intensities tend to smear out the ups and downs of individual evolutions. To illustrate the effect of number of strands on the XRT synthetic intensity, in Figure \[nstrands\] we plot the relative amplitude of the intensity standard deviation versus $N$. As can be noticed from the figure, the relative size of the fluctuations reaches approximately 0.15 or less for $N \gtrsim 100$. As we show in the observational study, this is consistent with what is found in real observations. Note that the larger the number of strands, the smaller the strand cross section must be in order to fit into the same volume.
XRT signal fluctuations are also affected by the nanoflare duration, $\tau$, because shorter nanoflares tend to produce a more marked heating/cooling alternation than longer events. Thus, the amplitude of the intensity fluctuations is expected to decrease with an increase of the nanoflare duration. We analyzed this in detail in López Fuentes & Klimchuk (2010) for a broad range of $\tau$ values. In the present study though, we adopt the conservative approach to consider that the nanoflare durations are not expected to be longer than the CA time step duration ($\delta t \approx$ 1000 sec, see previous Section). We remind the reader that the plasma evolution is modeled with EBTEL using a time resolution of 1 s. For the range of $\tau$ values presented in Table \[parameters\] we do not find substantial variation of the intensity fluctuations.
The scaling relations found in this Section can be used as test predictions of the model to be compared with future observations.
Nanoflare energy distribution {#power_laws}
-----------------------------
Since the model presented here is based on cellular automata evolution, the question arises whether it shares some of the known properties of similar models that produce self organization. One of the most salient features of Self-Organized Critical models (SOC, see the review by Charbonneau et al. 2001) is the existence of power laws in the distributions of energies, durations and geometric properties of the events produced. These models have been invoked to explain the presence of power law distributions in several solar phenomena, from flare observations in different wavelengths and energy ranges to Solar Energetic Particle (SEP) events (for a full list of references see Aschwanden 2011). As part of our effort to explain the observed properties and evolution of the plasma in coronal loops, we here examine the presence of power law distributions in the nanoflare energies produced by the model.
For each of the different parameter combinations in Table \[parameters\], we produced a nanoflare energy distribution, i.e., log-log histogram of the number of events as a function of energy. We then identified (by eye) and fitted (by regression) the linear portion of the distribution to determine its slope. Figure \[power-law\] shows one of these distributions, with the linear part indicated with a thick line segment and the corresponding slope provided in the panel. We find that all of the models have approximately the same slope. There is no evidence for a dependence on the model parameters. The mean slope for all of the parameter combinations in Table \[parameters\] is -2.54 and the standard deviation is 0.17. The minimum and maximum values are -2.9 and -2.14. Different runs using the same set of parameters provide slopes with the same type of distribution as for the full parameter variation. This suggests that the mean slope of approximately -2.5 is a robust feature associated with the kind process described by the model, regardless of the particular choice of parameters. The slight differences in the slopes are likely due to the binning and selection procedure and the random nature of the modeled system.
We wish to know whether the power law is valid over a wider range of energies than are represented by the parameters in Table \[parameters\]. For those runs, $B_v$ and $L$ were varied independently. However, since magnetic fields generally weaken with distance from their source, we expect long strands to have statistically weaker fields than short strands. Mandrini, Démoulin & Klimchuk (2000) found that $B \propto L^{-0.88}$ in active regions. We also know from Equation \[e\_scaling\] that nanoflares tend to be more energetic in strands with shorter lengths and stronger fields. Combining these two results, it is clear that nanoflare energies will vary considerably over an active region. We performed five new simulations using Mandrini et al. to relate $B_v$ and $L$:
$$\label{b_lock}
B_v \approx 10^4 \times L^{-0.88},$$
where $B_v$ is expressed in Gauss and $L$ in Mm, and the constant of proportionality is courtesy of C. H. Mandrini (private communication). The other model parameters were fixed at $\tan \theta_c = 0.25$, $N = 49$ and $\tau = 200$ sec. Table \[parameters2\] gives the combinations of $B_v$ and $L$ and corresponding range of nanoflare energies $\Delta E$ over which the distribution exhibits a power law with the indicated index. Notice that $\Delta E$ corresponds to the total energy of the nanoflares, not the energy density, $\Delta e$, analyzed in previous sections. The energies have been integrated over the strand volume assuming a strand cross section that is consistent with the properties of observed coronal loops. We consider that the $N$ strands of the model fill a loop of a typical observed diameter. We see that the indices are very similar even though the energy ranges are much different. The mean is -2.5 and the standard deviation is 0.17, similar to the values obtained with the full set of parameters of Table \[parameters\]. This provides further support for the existence of a universal power law. The fluctuations of the index are once again likely due to the errors in the determination of the linear part of the distribution. To test this we repeat several times the run with $L = 80$ Mm ($B_v = 211$ G) obtaining a mean and standard deviation which are consistent with the previous results.
Each of the numerical experiments described in the above analysis corresponds to model runs of 200 time steps. Considering that each strand is heated at an average rate of one nanoflare per time step and that $N = 49$ for all the runs, each energy distribution includes approximately 9800 nanoflares. To test the statistical robustness of the power-law indices, we performed an experiment with 50000 time steps, using $L = 80$ Mm ($B_v = 211$ G) and the rest of the parameters as indicated in the above paragraphs. In this case, the number of nanoflares in the sample increases to approximately 2.45$\times 10^6$. The index is -2.57, in agreement with the previous results.
It is interesting to analyze the origin of the energy ranges shown in the third column of Table \[parameters2\]. For a given set of parameters, the reconnection rules imposed by Equations \[relaxation1\] and \[relaxation2\] imply that strands are never far from their critical state. The nanoflare energies are expected to be distributed around a value proportional to $\frac{B_v^2}{L}\tan \theta_c$ (see Equation \[e\_scaling\]), with variations due to the randomness of the strand mixing and the redistribution of strand lengths and released energies produced by the use of the random variables $\alpha$ and $\beta$. The combination of the 2D strand mixing and the random variables produce energy distributions with a prominent linear part. The constancy of the slope of the linear part for a wide range of model parameter combinations suggests the presence of a scale invariance associated with the procedure used for the strand interaction and reconnection in the model.
Nanoflare frequency {#frequency}
-------------------
As discussed in the Introduction, it has been argued that nanoflares can explain a variety of observations (X-ray loops, EUV loops, active region cores, etc.) depending on whether the delay between successive events is long or short compared to a cooling time. Long delays correspond to low-frequency nanoflares, and short delays correspond to high-frequency nanoflares. With low-frequency nanoflares, the plasma has time to cool fully before the next event occurs. This results in a broad distribution of temperatures. With high-frequency nanoflares, the plasma is reheated after only a small amount of cooling, so the temperature fluctuates about a mean value. There are actually three important timescales in determining the thermodynamic properties of the plasma: the nanoflare repetition time, or interval between the start of consecutive events, the cooling time, and the nanoflare duration. If the duration is long, there may be little time for the plasma to cool even if the repetition time would suggest low-frequency heating. In this case the nanoflares are effectively high frequency. We examine the influence of all three of these timescales.
We start by defining three categories of nanoflare frequency:
$$\begin{aligned}
\textrm{High frequency:} & t_{rep} < 0.5~t_{cool}, \\
\textrm{Intermediate frequency:} & 0.5~t_{cool} < t_{rep} < 2~t_{cool}, \\
\textrm{Low frequency:} & t_{rep} > 2~t_{cool},\end{aligned}$$
where $t_{rep}$ is the repetition time between consecutive events and $t_{cool}$ is the cooling time. The intermediate frequency interval has not been discussed much in the past, but there is a new appreciation that many nanoflares may be of this type (Cargill 2014). It is convenient to define the frequency categories in terms of the temperature change of the cooling plasma. The top panels of Figure \[temp\_evol\], explained below, show that the temperature evolution after the nanoflare can be approximated by an exponential: $T(t) = T_0~\exp(-t/t_{cool})$, where $T_0$ is the maximum temperature reached during the event (see also Terzo et al. 2011). The temperature ratios $T(t_{rep})/T_0$ for each of the above frequency categories are then
$$\begin{aligned}
\textrm{High frequency:} & T(t_{rep})/T_0 > 0.61, \\
\textrm{Intermediate frequency:} & 0.14 < T(t_{rep})/T_0 < 0.61, \\
\textrm{Low frequency:} & T(t_{rep})/T_0 < 0.14.\end{aligned}$$
We determine statistics on nanoflare frequency in our models using these temperature ratios.\
The repetition times in our models are usually close to the timestep of the driving phase, 1000 s, the time it takes to displace a footpoint and increase the stress variable $S$ by an amount $d$ = 1000 km. The reason for this is that most strands are close to the critical condition, so they are pushed over the limit during most timesteps. The strands are near critical because the destressing that occurs at the end of the timestep is also $d$. This causes $S$ to fluctuate above and below the critical value. It is relatively uncommon for a strand to experience many events at one time and to relax far below the critical condition, i.e., ”avalanches” are rare. The fact that strands fluctuate close to the critical condition also explains why the range of nanoflare energies is relatively narrow (Figure \[power-law\] and Table \[parameters2\]).
Table \[freq\_length\] indicates the relative occurrence of low, intermediate, and high frequency nanoflares as a function of loop length. We have assumed a nanoflare duration of 200 s and used Equation \[b\_lock\] to relate the magnetic field strength to loop length. We see that low-frequency nanoflares are more common in short loops, and high-frequency nanoflares are more common in long loops. This is a direct consequence of the dependence of the cooling time on temperature, since the repetition time is similar for all models, as discussed above. The initial cooling after a nanoflare is dominated by thermal conduction, for which the cooling timescale is proportional to $L^2 T^{-5/2}$ (López Fuentes, Klimchuk & Mandrini 2007). Short loops cool more quickly because of the length dependence, and also because they have a greater field strength and more energetic nanoflares, leading to higher temperatures. More rapid cooling at a fixed repetition time means lower frequency nanoflares as defined by $T(t_{rep})/T_0$. Note that by combining Equations \[e\_scaling\] and \[b\_lock\], the nanoflare energy scales as $L^{-2.76}$, so the dependence of frequency on loop length is strong, as seen in the table.
To illustrate this point, we examine two cases: a high energy nanoflare ($e =$ 900 erg cm$^{-3}$) in a short loop ($L =$ 40 Mm), and a low energy nanoflare ($e =$ 1 erg cm$^{-3}$) in a long loop ($L =$ 100 Mm). The nanoflare duration is 200 s. The $B$-$L$ relationship in these cases is stronger than given by Equations \[b\_lock\], so they should be considered rather extreme examples. The upper panels of Figure \[temp\_evol\] show the temperature evolution. For guidance, the dashed horizontal lines in the plots correspond to the 0.61 $T_0$ and 0.14 $T_0$ levels. It can be easily seen that in the high energy event (upper-left panel) the temperature falls off very rapidly, so that the cooling time is relatively short. If a second nanoflare were to follow 1000 s later, the temperature would have decreased below the 14% level relative to the maximum temperature. The situation is very different for the low energy case of the upper-right panel, in which the temperature reaches the “low frequency level” roughly 4000 s after the nanoflare started.
The difference between the cases is further illustrated in the lower panels of Figure \[temp\_evol\]. There, we show the same nanoflares as in the upper panels but repeating at a 1000 s rate, corresponding to the cellular automaton time step duration. This is the minimum time delay between consecutive nanoflares in our models, and comparable to the typical delay, as discussed above. Clearly, according to our frequency interval definitions, we would classify nanoflares in the lower-left panel as low frequency and those in the lower-right panel as high frequency. Note that the initial static equilibrium conditions in the loop at $t = 0$ s are different from the conditions at start of the subsequent nanoflares. It takes several cycles before knowledge of the initial conditions is lost, especially in the example of the lower-right.
Table \[freq\_tau\] indicates how the nanoflare frequency as defined by $T(t_{rep})/T_0$ depends on the nanoflare duration $\tau$. The runs used $L=80$ Mm and $B=211$ G. As expected, the percentage of high-frequency events increases with nanoflare duration. The longer the heating is switched on, the less time there is for the plasma to cool before the next event. The percentage of low-frequency nanoflares is small at all $\tau$ for this combination of $L$ and $B$. Smaller $L$ and/or larger $B$ would produce stronger nanoflares and therefore shorter cooling times and a higher percentage of low-frequency events.
Discussion and conclusions {#conclusions}
==========================
We study the problem of nanoflare heating of coronal loops using a 2D cellular automaton (CA) model based on Parker’s (1988) idea of footpoint shuffling and tangling of elementary magnetic strands. To determine the plasma response to the heating we use the Enthalpy Based Thermal Evolution of Loops (EBTEL) model. From the computed temperature and density evolutions and the known response of coronal observing instruments we simulate observed lightcurves. We study the dependence of the model’s output on the relevant physical parameters and we find and analyze a series of predicted scalings that can be compared with future observations.
Two primary results of our study concern the number distribution of nanoflares as a function of energy and the frequency with which nanoflares repeat on a given strand. We find that the number distribution obeys a power law with a slope of approximately -2.5. This is a robust result, with a standard deviation of 17% as we vary the model parameters. For many years researchers have extrapolated power laws measured for flares and other resolvable events to lower energies in order to determine whether the corona could be heated by nanoflares. As pointed out by Hudson (1991), the slope of the distribution must be steeper than -2 in order for nanoflares to be energetically important. The results have been mixed, ranging from less steep (Aschwanden & Parnell 2002 and references therein) to more steep (Krucker & Benz 1998; Parnell and Jupp 2000; Benz 2004; Pauluhn & Solanki 2007; Bazarghan et al. 2008), with shallower slopes tending to come from studies of flares and steeper slopes tending to come from studies of smaller impulsive events. The variation reflects the difficulty in measuring the slope, in part because of the assumptions that must be made in estimating the total energy that is released. We would also point out that there is no compelling reason to believe that the slope should be constant over the full range extending from large flares to nanoflares.
From an observational perspective, the nanoflare repetition frequency is important only insofar as the delay between successive events is longer than, shorter than, or comparable to the plasma cooling time. In our models, the nanoflare frequency increases with loop length. Since hot (X-ray) loops are best explained by high-frequency nanoflares, and warm (EUV) loops are best explained by low-frequency nanoflares, this would suggest that hot loops should be longer than warm loops. This is not observed to be the case, however. Hot loops are more prevalent in the cores of active regions, while warm loops are more prevalent outside the core. Most core emission is contained in a diffuse component, however, rather than in observationally distinct loops (Viall & Klimchuk 2011). Whether this diffuse emission is better explained by high or low frequency nanoflares is a matter of debate. See the studies summarized in Table 3 of Bradshaw et al. (2012) and the discussion of uncertainties in Guennou et al. (2013). Subramanian et al. (2014) have examined diffuse emission outside the core, and again the results are inconclusive regarding nanoflare frequency. Cargill (2014) recently suggested that all the results might be reconciled if nanoflares have random frequencies centered about a mean value that is comparable to a cooling time, i.e., intermediate frequency, and if there is a relationship between the event size and delay between events.
The predictions of our model of course depend upon the assumptions that go into it. We have three main assumptions. First, magnetic reconnection does not occur unless a critical condition corresponding the misalignment angle between adjacent magnetic strands is met. This is supported by theoretical studies of the secondary instability of current sheets (Dahlburg et al. 2005, 2009). Second, the test for criticality does not happen until after a full stressing step has been made. In other words, reconnection holds off at least until photospheric convection has transported the magnetic footpoint a characteristic distance $d = 1$ Mm. The critical condition can therefore be exceeded, though not by a large amount. Third, reconnection, once it occurs, releases an amount of stress equal to the stress added during a driving step. It does not cause the total stress to drop far below the critical value. A strand can reconnect with multiple strands during a single iteration, but in practice the number rarely exceeds three. The mean is close to one. As a result the stress tends to hover around the critical value. The energy of the nanoflare, which combines the energies of the separate reconnections, does not have a wide variation. The energy range is typically less than 1.0 in the logarithm, i.e., factor of 10 difference between the largest and smallest for a given set of model parameters.
Whether these assumptions are reasonable has yet to be determined. The detailed physical scenario of nanoflares is still unknown. We note that Ugarte-Urra & Warren (2014) used a combination of observations and hydro modeling to infer a nanoflare repetition rate of 2 to 3 per hour. This corresponds to a delay of $\sim$1400 s, which agrees well with our assumed delay of 1000 s. From another observational/modeling comparison, Cargill (2014) suggests a delay of a few hundred to somewhat over 2000 s. Finally, Dahlburg et al. (2005) suggest approximately 2000 s based on the MHD simulations of the secondary instability. It is clear that much more work is left to be done.
The authors acknowledge useful comments and suggestions from the anonymous referee. They also wish to thank Dr. Peter Cargill for useful discussions. JAK’s work was funded by the NASA Supporting Research and Technology and Guest Investigator programs. MLF acknowledges financial support from the Argentinean grants PICT 2012-0973 (ANPCyT), UBACyT 20020100100733 and PIP 2009-100766 (CONICET).
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\[parameters\] $\begin{array}{lcccccc}
\hline
B_{v} $(Gauss)$ & 40 & 100 & \bf{200} & 500 \\
L $(Mm)$ & 40 & 60 & \bf{80} & 100 & 120 \\
\tan \theta_{c} & 0.15 & 0.2 & \bf{0.25} & 0.3 & 0.35 \\
N & 9 & 25 & \bf{49} & 81 & 121 & 169 \\
\tau & 50 & 100 & \bf{200} & 500 & 1000 \\
\hline
\end{array}$
\[parameters2\] $\begin{array}{cccc}
\vspace{0.1cm}
L($Mm$) & B($G$) & \Delta $E range($10^{24}$erg)$ & $Power-law index$ \\
\hline
40 & 389 & 557-2465 & -2.43 \pm 0.14 \\
60 & 272 & 226-722 & -2.40 \pm 0.19 \\
80 & 211 & 139-394 & -2.73 \pm 0.14 \\
100 & 174 & 101-290 & -2.64 \pm 0.11 \\
120 & 148 & 70-174 & -2.32 \pm 0.14 \\
\hline
\end{array}$
\[freq\_length\] $\begin{array}{cccc}
\vspace{0.1cm}
L($Mm$) & $Low (%)$ & $Intermediate (%)$ & $High (%)$ \\
\hline
40 & 43 & 52 & 5 \\
60 & 33 & 51 & 16 \\
80 & 9 & 48 & 43 \\
100 & 7 & 50 & 43 \\
120 & 1 & 45 & 54 \\
\hline
\end{array}$
\[freq\_tau\] $\begin{array}{cccc}
\vspace{0.1cm}
\tau($sec$) & $Low (%)$ & $Intermediate (%)$ & $High (%)$ \\
\hline
200 & 7 & 61 & 32 \\
400 & 7 & 45 & 48 \\
600 & 5 & 42 & 53 \\
800 & 6 & 37 & 57 \\
1000 & 8 & 34 & 58 \\
\hline
\end{array}$
{width="16.cm"}
{width="15.cm"}
{width="9.cm"}
{width="16.cm"}
{width="9.cm"}
{width="13.cm"}
{width="13.cm"}
{width="17.cm"}
|
---
abstract: |
Let $V$ be a finite relational vocabulary in which no symbol has arity greater than 2. Let ${\mathcal{M}}$ be countable $V$-structure which is homogeneous, simple and 1-based. The first main result says that if ${\mathcal{M}}$ is, in addition, primitive, then it is strongly interpretable in a random structure. The second main result, which generalizes the first, implies (without the assumption on primitivity) that if ${\mathcal{M}}$ is “coordinatized” by a set with SU-rank 1 and there is no definable (without parameters) nontrivial equivalence relation on $M$ with only finite classes, then ${\mathcal{M}}$ is strongly interpretable in a random structure.\
[*Keywords*]{}: model theory, homogeneous structure, simple theory, 1-based theory, random structure.
address: 'Vera Koponen, Department of Mathematics, Uppsala University, Box 480, 75106 Uppsala, Sweden.'
author:
- Vera Koponen
title: 'Homogeneous 1-based structures and interpretability in random structures'
---
Introduction
============
A first-order structure ${\mathcal{M}}$ will be called [*homogeneous*]{} if it has a finite relational vocabulary and every isomorphism between finite substructures of ${\mathcal{M}}$ can be extended to an automorphism of ${\mathcal{M}}$. For surveys of homogeneous structures and their connections to other areas see [@BP; @Che98; @HN; @Mac10; @Nes]. Although there are $2^\omega$ countable nonisomorphic homogeneous structures for a vocabulary with only one binary relation symbol [@Hen72], it has been shown that in several cases, such as partial orders, undirected graphs, directed graphs and stable relational structures (with finite relational vocabulary), the countable homogeneous structures among them can be classified in a rather concrete way [@Che98; @Gar; @GK; @JTS; @Lach84; @Lach97; @LT; @LW; @Schm; @Shee]. The work on stable homogeneous structures together and on “geometric stability theory” (notably by Zilber [@Zil]) was useful to reach a good understanding of $\omega$-categorical $\omega$-stable structures [@CHL] and later of smoothly approximable structures [@CH; @KLM].
Stability theory was from the mid 90ies generalized to simplicity theory, where the class of simple theories is the largest class of complete theories $T$ such that in every ${\mathcal{M}}\models T$ there is a symmetric “independence relation” on subsets of $M$. The random graph (or Rado graph) is a standard example of a homogeneous simple structure (i.e. one which has simple theory) which is not stable, and the same is true of the “random structure” with respect to any finite relational vocabulary. Since the [*infinite*]{} countable stable homogeneous structures are classified [@Lach97], one may ask if it is possible to reach, if not a classification, at least some systematic understanding of (infinite countable) simple homogeneous structures. Besides the present work, [@AK14; @Kop15] and the dissertation of Aranda Lópes [@AL] the author is not aware of any results in this direction. The class of all simple homogeneous structures seems too wide to start with, so we focus on a subclass of it which seems easier to deal with. We consider homogeneous, simple and 1-based structures, where the property [*1-based*]{} implies (for structures in general) that the independence relation behaves “nicely” (like in a vector space or random structure, for example). In fact, it follows from work of Macpherson [@Mac91] and De Piro and Kim [@PK] that every homogeneous, simple and 1-based structure has trivial dependence. If a homogeneous structure has only unary and binary relation symbols, then it is 1-based if and only if it has trivial dependence (see Definition \[definition of trivial dependence\] and Fact \[homogeneity and 1-basedness implies finite rank and trivial dependence\] below).
Moreover, we focus on binary homogeneous structures, where a structure is called [*binary*]{} if its vocabulary is finite, relational and has no symbol with arity greater than 2. This simplifies the situation because the “independence theorem” of simple theories has strong consequences in the context of binary structures. For example it follows, rougly speaking, that every ${\emptyset}$-definable set of SU-rank 1 (in the extension by imaginaries) is – when viewed as a “canonically embedded structure” – a reduct of a binary random structure. The working hypothesis is that countable, binary, homogeneous, simple and 1-based structures are sufficiently uncomplicated that it should be possible to work out some sort of rather explicit understanding of them. Moreover, the author does not know of any example of a homogeneous and simple structure which is not 1-based. Such an example, particularly a binary one, would probably shed light on the understanding of simple homogeneous structures. A proof that it does not exist would also be striking.
We say that a structure is [*primitive*]{} if there is no nontrivial equivalence relation on its universe which is definable without parameters. (Nontrivial means that there are at least two classes and at least one has more than one element.) It is easy to see that if ${\mathcal{M}}$ is homogeneous (and simple and 1-based) and nonprimitive, then the substructure on any equivalence class of the equivalence relation that witnesses nonprimitivity is homogeneous (and simple and 1-based). So a good understanding of homogeneous simple structures requires an equally good understanding of primitive homogeneous simple structures. In particular we have the following problem, where the notion of a ‘binary random structure’ is explained in Section \[Random structures\]:
- [*Is every countable, binary, homogeneous, primitive, simple and 1-based structure a random structure?*]{}
The problem is open also if we remove the assumptions about binarity and 1-basedness. Our first main result can be seen as an “approximate solution”:
\[first main result\] Suppose that ${\mathcal{M}}$ is a structure which is countable, binary, homogeneous, primitive, simple and 1-based. Then ${\mathcal{M}}$ is strongly interpretable in a binary random structure.
The notion [*strongly interpretable*]{} (Definition \[definition of strong interpretability\]) implies the notion [*interpretable*]{} in its usual senses (e.g. [@Hod; @Mac91]). Hence Theorem \[first main result\] can be interpreted as saying that a countable, binary, homogeneous, primitive, simple and 1-based cannot be more complicated than a binary random structure. The theorem may also be of help to solve the problem stated above.
The second main result is a generalisation of the first one, but has a more complicated formulation, which is why we stated Theorem \[first main result\] first and not as a corollary of Theorem \[second main result\] below. Theorem \[second main result\] applies to some (but not all) [*non*]{}primitive binary homogeneous simple and 1-based structures. For example, let ${\mathcal{R}}$ be the random graph, let $M = R^2$ and let ${\mathcal{M}}$ be the structure with universe $M$ that has a binary relation symbol $R_p$ for every 4-type over ${\emptyset}$ of ${\mathcal{R}}$ and where $R_p$ is interpreted as $\{((a,b), (a',b')) : {\mathcal{R}}\models p(a,b,a',b')\}$. Then ${\mathcal{M}}$ satisfies the hypotheses of Theorem \[second main result\]. (The same holds if ${\mathcal{R}}$ is any binary random structure and we consider ordered $k$-tuples for any fixed $k > 1$.) On the other hand if, for example, ${\mathcal{M}}$ is the structure with infinite countable universe, two equivalence relations $R_1, R_2$, such that $R_2$ refines $R_1$ by splitting each $R_1$-class into infinitely many infinite parts, then ${\mathcal{M}}$ does not have height 1, so Theorem \[second main result\] does not apply.[^1] The concept of ‘height’, appearing below, is given by Definition \[definition of height\], and the notation ‘${\mathcal{M}}/ \approx$’ is explained in Definition \[definition of canonically embedded structure\]:
\[second main result\] Suppose that ${\mathcal{M}}$ is a structure which is countable, binary, homogeneous, simple and 1-based with height 1. Let $\approx$ denote the equivalence relation ‘${\mathrm{acl}}_{\mathcal{M}}(x) = {\mathrm{acl}}_{\mathcal{M}}(y)$’, where ‘${\mathrm{acl}}_{\mathcal{M}}$’ denotes algebraic closure in ${\mathcal{M}}$.\
(i) Then there is a binary random structure ${\mathcal{R}}$ such that ${\mathcal{M}}/ {{\negthickspace}\approx}$ is definable in ${\mathcal{R}}{^{\mathrm{eq}}}$.\
(ii) If, in addition, the equivalence relation ‘$\approx$’ is trivial, then ${\mathcal{M}}$ is strongly interpretable in a binary random structure.
The author hopes that this theorem can be generalised in a suitable way to ‘height $r$’ for any $0 < r < \omega$. One would then have a theorem which applies to all countable, binary, homogeneous, simple and 1-based structures (since ‘1-based’ implies ‘finite height’). But besides finding the “right” generalisation it seems like one has to overcome a number of technical difficulties.
This article is organised as follows. Section \[Preliminaries\] recalls the necessary background about homogeneous structures and $\omega$-categorical simple structures. The proofs of the above theorems are given in Section \[Binary homogeneous 1-based structures\], which ends with a recipe for making structures that satisfy the main results of this article. The proofs of the main results use the “coordinatization results” from [@Djo06], which do not need the assumption that the structure is binary. One also needs to know what a canonically embedded structure ${\mathcal{G}}$ in ${\mathcal{M}}{^{\mathrm{eq}}}$ may look like if its universe is a definable set with ${\mathrm{SU}}$-rank 1. It turns out that if ${\mathcal{M}}$ is countable, binary, homogeneous and simple with trivial dependence then ${\mathcal{G}}$ is a [*reduct*]{} (Definition \[definition of reduct\]) of a binary random structure. This is proved in [@AK14] and the result is refered to as Fact \[rank 1 sets are reducts of binary random structures\] below. The main technical result (besides Fact \[rank 1 sets are reducts of binary random structures\]) is Lemma \[crd(a) is rigid\] which is proved in Section \[Proof of the main technical result\]. The assumption that ${\mathcal{M}}$ in Theorems \[first main result\] and \[second main result\] is binary is only used in the application of Fact \[rank 1 sets are reducts of binary random structures\] and in the proof of Lemma \[crd(a) is rigid\].
Preliminaries {#Preliminaries}
=============
The prerequisites of this article are more or less the same as those of [@AK14]. We list, in this section, a number of definitions and facts, concerning homogeneous structures, $\omega$-categorical simple structures and imaginary elements, in order to make this article relatively self contained, but refer to [@AK14] for further explanations.
General notation and terminology
--------------------------------
We call a vocabulary (also called signature) [*relational*]{} if it only contains relation symbols. Given a finite relational vocabulary the [*maximal arity*]{} of it is the largest integer $k$ such that some relation symbol of it has arity $k$. If $V$ is a [*finite*]{} vocabulary and the maximal arity is 2 then we call $V$ [*binary*]{} (although it may contain unary relation symbols), and in this case a $V$-structure may be called a [*binary structure*]{}. We denote (first-order) structures by ${\mathcal{A}}, {\mathcal{B}}, \ldots, {\mathcal{M}}, {\mathcal{N}}, \ldots$ and their respective universes by $A, B, \ldots, M, N, \ldots$. Finite sequences (tuples) of elements of some structure (or set in general) will be denoted $\bar{a}, \bar{b}, \ldots$, while $a, b, \ldots$ ususally denote elements from the universe of some structure. The notation $\bar{a} \in A$ means that every element in the sequence $\bar{a}$ belongs to $A$. Sometimes we write $\bar{a} \in A^n$ to show that the length of $\bar{a}$, denoted $|\bar{a}|$, is $n$. The [*range of $\bar{a}$*]{}, denoted ${\mathrm{rng}}(\bar{a})$, is the set of elements that occur in $\bar{a}$. Notation regarding types, algebraic closure and definable closure is standard, where we may use a subscript or superscript to indicate which structure we have in mind. A structure ${\mathcal{M}}$ is called [*$\omega$-categorical*]{}, respectively [*simple*]{}, if its complete theory $Th({\mathcal{M}})$ has that property. (See [@Hod] and [@Cas; @Wag] for definitions.)
Homogeneous and $\omega$-categorical structures {#Homogeneous structures}
-----------------------------------------------
\[definition of homogeneous\][(i) Let $V$ be a finite relational vocabulary and ${\mathcal{M}}$ a $V$-structure. We call ${\mathcal{M}}$ [*homogeneous*]{} if for all finite substructures ${\mathcal{A}}$ and ${\mathcal{B}}$ of ${\mathcal{M}}$, every isomorphism from ${\mathcal{A}}$ to ${\mathcal{B}}$ can be extended to an automorphism of ${\mathcal{M}}$.\
(ii) We say that a structure ${\mathcal{M}}$, with any vocabulary, is [*$\omega$-homogeneous*]{} if whenever $n < \omega$, $a_1, \ldots, a_n, a_{n+1}, b_1, \ldots, b_n \in M$ and ${\mathrm{tp}}_{\mathcal{M}}(a_1, \ldots, a_n) = {\mathrm{tp}}_{\mathcal{M}}(b_1, \ldots, b_n)$, then there is $b_{n+1} \in M$ such that ${\mathrm{tp}}_{\mathcal{M}}(a_1, \ldots, a_{n+1}) = {\mathrm{tp}}_{\mathcal{M}}(b_1, \ldots, b_{n+1})$. ]{}
For standard results about homogeneous structures we refer to (for example) [@Hod Sections 7.1 and 7.4]. We mainly use the following implications of homogeneity (or $\omega$-categoricity), which follow from Corollary 7.4.2 in [@Hod] and from the well known characterization of $\omega$-categorical theories by Engeler, Ryll-Nardzewski and Svenonius:
\[fact about homogeneous structures\] (i) If ${\mathcal{M}}$ is $\omega$-categorical then it is $\omega$-homogeneous.\
(ii) Suppose that ${\mathcal{M}}$ is countable and $\omega$-homogeneous. Then for all $0 < n < \omega$ and all $a_1, \ldots, a_n, b_1, \ldots, b_n \in M$ such that ${\mathrm{tp}}_{\mathcal{M}}(a_1, \ldots, a_n) = {\mathrm{tp}}_{\mathcal{M}}(b_1, \ldots, b_n)$, there is an automorphism $f$ of ${\mathcal{M}}$ such that $f(a_i) = b_i$ for every $i$.\
(iii) Suppose that ${\mathcal{M}}$ is a countable $V$-structure where $V$ is a finite relational vocabulary. Then ${\mathcal{M}}$ is homogeneous if and only if ${\mathcal{M}}$ is $\omega$-categorical and has elimination of quantifiers.
Simple structures, imaginary elements, 1-basedness and triviality of dependence {#Simple omega-categorical structures}
-------------------------------------------------------------------------------
We assume familiarity with imaginary elements and ${\mathcal{M}}{^{\mathrm{eq}}}$, defined in [@Hod; @She] for example, and with basic simplicity theory, as found in [@Cas; @Wag] for example. Since the distinction between sorts (of imaginary elements) will be relevant here, and since some notions and results are simplified when considering $\omega$-categorical simple theories, compared with simple theories in general, we will nevertheless rehearse some notions and results that will be used.
Let $V$ be a vocabulary and ${\mathcal{M}}$ a $V$-structure. For every $0 < n < \omega$ and ${\emptyset}$-definable equivalence relation $E$ on $M^n$, $V{^{\mathrm{eq}}}$ (the vocabulary of ${\mathcal{M}}{^{\mathrm{eq}}}$) contains a unary relation symbol $P_E$ and a relation symbol $F_E$ of arity $n+1$ (both of which do not belong to $V$), where $P_E$ is interpreted in ${\mathcal{M}}{^{\mathrm{eq}}}$ as the set of $E$-equivalence classes and $F_E$ is interpreted as the graph of the function which sends every $\bar{a} \in M^n$ to its equivalence class. A [*sort*]{} of ${\mathcal{M}}{^{\mathrm{eq}}}$ is, by definition, a set of the form $S_E = \{a \in M{^{\mathrm{eq}}}: {\mathcal{M}}{^{\mathrm{eq}}}\models P_E(a) \}$ for some $E$ as above. If $A \subseteq M{^{\mathrm{eq}}}$ and there are only finitely many $E$ such that $A \cap S_E \neq {\emptyset}$, then we say that [*only finitely many sorts are represented in $A$*]{}. The identity relation, ‘=’, is clearly a ${\emptyset}$-definable equivalence relation on $M$ and every $=$-class is a singleton. Therefore $M$ can (and will) be identified with the sort $S_=$, which we call the [*real sort*]{}, so $M \subseteq M{^{\mathrm{eq}}}$. Below follow some facts and definitions. See for example [@AK14] for explanations or proofs of these facts.
\[facts about simple omega-categorical structures\] Suppose that ${\mathcal{M}}$ is $\omega$-categorical and countable, let $A \subseteq M{^{\mathrm{eq}}}$ and suppose that only finitely many sorts are represented in $A$.\
(i) For every $n < \omega$ and finite $B \subseteq {\mathcal{M}}{^{\mathrm{eq}}}$, only finitely many types from $S_n^{{\mathcal{M}}{^{\mathrm{eq}}}}({\mathrm{acl}}_{{\mathcal{M}}{^{\mathrm{eq}}}}(B))$ are realized by $n$-tuples in $A^n$.\
(ii) If $B \subseteq M{^{\mathrm{eq}}}$ is finite and $\bar{a} \in M{^{\mathrm{eq}}}$, then ${\mathrm{tp}}_{{\mathcal{M}}{^{\mathrm{eq}}}}(\bar{a} / {\mathrm{acl}}_{{\mathcal{M}}{^{\mathrm{eq}}}}(B))$ is isolated.\
(iii) If $B \subseteq {\mathcal{M}}{^{\mathrm{eq}}}$ is finite, $n < \omega$ and $p \in S_n^{{\mathcal{M}}{^{\mathrm{eq}}}}({\mathrm{acl}}_{{\mathcal{M}}{^{\mathrm{eq}}}}(B))$ is realized in some elementary extension of ${\mathcal{M}}{^{\mathrm{eq}}}$ by an $n$-tuple of imaginary elements (i.e. elements satisfying $P_E(x)$ for some, not necessarily the same, $E$), then $p$ is realized in ${\mathcal{M}}{^{\mathrm{eq}}}$.
\[definition of canonically embedded structure\][(i) We say that a structure ${\mathcal{N}}$ is [*canonically embedded*]{} in ${\mathcal{M}}{^{\mathrm{eq}}}$ if $N$ is a ${\emptyset}$-definable subset of $M{^{\mathrm{eq}}}$ and for every $0 < n < \omega$ and every relation $R \subseteq N^n$ which is ${\emptyset}$-definable in ${\mathcal{M}}{^{\mathrm{eq}}}$ there is a relation symbol in the vocabulary of ${\mathcal{N}}$ which is interpreted as $R$ and the vocabulary of ${\mathcal{N}}$ contains no other relation symbols (and no constant or function symbols).\
(ii) In particular, if $R$ is an equivalence relation on $M$ which is ${\emptyset}$-definable in ${\mathcal{M}}$, then ${\mathcal{M}}/R$ denotes the canonically embedded structure with universe $M/R$, where $M/R$ is the set of all equivalence classes of $R$ (which is a ${\emptyset}$-definable subset of $M{^{\mathrm{eq}}}$). ]{}
We immediately get the following:
\[first fact about canonically embedded structures\] If ${\mathcal{N}}$ is canonically embedded in ${\mathcal{M}}{^{\mathrm{eq}}}$, then for all $\bar{a}, \bar{b} \in N$ and all $C \subseteq N$, ${\mathrm{acl}}_{\mathcal{N}}(C) = {\mathrm{acl}}_{{\mathcal{M}}{^{\mathrm{eq}}}}(C) \cap N$ and ${\mathrm{tp}}_{\mathcal{N}}(\bar{a} / C) = {\mathrm{tp}}_{\mathcal{N}}(\bar{b} / C)$ if and only if ${\mathrm{tp}}_{{\mathcal{M}}{^{\mathrm{eq}}}}(\bar{a} / C) = {\mathrm{tp}}_{{\mathcal{M}}{^{\mathrm{eq}}}}(\bar{b} / C)$.
Suppose that $T$ is a complete simple theory. For every type $p$ (possibly over a set of parameters) with respect to $T$, there is a notion of SU-rank of $p$, denoted ${\mathrm{SU}}(p)$ (which is either an ordinal or undefined);
See for instance [@Cas; @Wag] for definitions and basic results about SU-rank. As usual, we abbreviate ${\mathrm{SU}}({\mathrm{tp}}_{\mathcal{M}}(\bar{a} / A))$ with ${\mathrm{SU}}(\bar{a} / A)$ and ${\mathrm{SU}}(\bar{a} / {\emptyset})$ with ${\mathrm{SU}}(\bar{a})$. (When using this notation there will be no ambiguity about which structure we work in.) If ${\mathrm{SU}}(\bar{a})$ is finite for every ${\mathcal{M}}\models T$ and every $\bar{a} \in M$, then we say that $T$ (and any ${\mathcal{M}}\models T$) has [*finite ${\mathrm{SU}}$-rank*]{}.
\[definition of trivial dependence\][Let $T$ be a complete simple theory.\
(i) We say that $T$ has [*trivial dependence*]{} if, whenever ${\mathcal{M}}\models T$, $A, B, C_1, C_2 \subseteq M{^{\mathrm{eq}}}$ and $A \underset{B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} (C_1 \cup C_2)$, then $A \underset{B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} C_i$ for $i = 1$ or $i = 2$.\
(ii) $T$ (as well as every model of it) is [*1-based*]{} if for every ${\mathcal{M}}\models T$ and all $A, B \subseteq M{^{\mathrm{eq}}}$, $A$ is independent from $B$ over ${\mathrm{acl}}_{{\mathcal{M}}{^{\mathrm{eq}}}}(A) \cap {\mathrm{acl}}_{{\mathcal{M}}{^{\mathrm{eq}}}}(B)$. ]{}
For homogeneous structures the notions of 1-basedness, triviality of dependence and finiteness of rank are fairly tightly connected, in particular in the binary case.
\[homogeneity and 1-basedness implies finite rank and trivial dependence\] (i) Suppose that ${\mathcal{M}}$ is homogeneous, simple and 1-based. Then $Th({\mathcal{M}})$ has trivial dependence and finite ${\mathrm{SU}}$-rank (so in particular it is supersimple).\
(ii) Suppose that ${\mathcal{M}}$ is binary, homogeneous and simple. Then ${\mathcal{M}}$ has finite SU-rank. Also, the following three conditions are equivalent: (a) ${\mathcal{M}}$ is 1-based, (b) ${\mathcal{M}}$ has trivial dependence, and (c) every type (over a finite set of parameters) of SU-rank 1 has trivial pregeometry (given by algebraic closure restricted to that type).
[**Proof.**]{} (i) Let ${\mathcal{M}}$ satisfy the premisses of part (i) of the lemma, so it follows that ${\mathcal{M}}$ is $\omega$-categorical. By Theorem 1.1 in [@Mac91], it is not possible to interpret an infinite group in ${\mathcal{M}}$, which, with the terminology of [@Mac91], means that it is not possible to define, with finitely many parameters, an infinite group in ${\mathcal{M}}{^{\mathrm{eq}}}$. Corollary 3.23 in [@PK] implies that if $Th({\mathcal{M}})$ does not have trivial dependence, then an infinite group is definable, with finitely many parameters, in ${\mathcal{M}}{^{\mathrm{eq}}}$. It follows that ${\mathcal{M}}$ must have trivial dependence. And finally, Corollary 4.7 in [@HKP] says that every simple, 1-based and $\omega$-categorical theory is supersimple with finite ${\mathrm{SU}}$-rank.
\(ii) The first claim is the main result of [@Kop15]. The second claim follows by combining results from [@PK; @HKP; @Mac91] and is explained in some more detail in the introduction to [@Kop15]. $\square$
\[definition of height\][Let ${\mathcal{M}}$ be a simple structure. We say that ${\mathcal{M}}$ has [*height 1*]{} if there is a ${\emptyset}$-definable $D \subseteq M{^{\mathrm{eq}}}$ in which only finitely many sorts are represented and $M \subseteq {\mathrm{acl}}_{{\mathcal{M}}{^{\mathrm{eq}}}}(D)$ and ${\mathrm{SU}}(d) = 1$ for every $d \in D$. ]{}
Random structures {#Random structures}
-----------------
\[definition of a binary random structure\][Let $V$ be a binary vocabulary and let ${\mathcal{M}}$ be an infinite homogeneous $V$-structure.\
(i) A finite $V$-structure ${\mathcal{A}}$ is called a [*forbidden structure with respect to ${\mathcal{M}}$*]{} if ${\mathcal{A}}$ cannot be embedded into ${\mathcal{M}}$.\
(ii) Suppose that ${\mathcal{A}}$ is a forbidden structure with respect to ${\mathcal{M}}$. We call ${\mathcal{A}}$ a [*minimal*]{} forbidden structure with respect to ${\mathcal{M}}$ if no proper substructure of ${\mathcal{A}}$ is a forbidden structure with respect to ${\mathcal{M}}$.\
(iii) We call [*${\mathcal{M}}$ a binary random structure*]{} if there does [*not*]{} exist a minimal forbidden structure ${\mathcal{A}}$ with respect to ${\mathcal{M}}$ such that $|A| \geq 3$. ]{}
\[remarks about binary random structures\][(i) It is straightforward to see that the above definition of a binary random structure is equivalent to the definition given in [@AK14 Section 2.3].\
(ii) The Rado graph, usually called random graph in model theory, is of course an example of a binary random structure.\
(iii) Suppose that ${\mathcal{M}}$ is a binary random structure. Let $x_1, \ldots, x_n$, where $n \geq 3$, be distinct variables and, for all $1 \leq i < j \leq n$, let $p_{i, j}(x_i, x_j) \in S^{\mathcal{M}}_2({\emptyset})$. Moreover assume that for all $i < j$ and all $i' < j'$, if $k \in \{i, j\} \cap \{i', j'\}$, then the restriction of $p_{i,j}$ to $x_k$ is identical to the restriction of $p_{i',j'}$ to $x_k$. It now follows straightforwardly from the definition of a binary random structure that $\bigcup_{1 \leq i < j \leq n} p_{i, j}(x_i, x_j)$ is consistent with $Th({\mathcal{M}})$ and realized in ${\mathcal{M}}$. Since ${\mathcal{M}}$ is homogeneous it also follows that, for any $1 \leq i < j \leq n$, if ${\mathcal{M}}\models p_{i,j}(a_i, a_j)$ then there are $a_k \in M$ for all $k \in \{1, \ldots, n\} \setminus \{i, j\}$ such that ${\mathcal{M}}\models \bigwedge_{1 \leq k < l \leq n} p_{k, l}(a_k, a_l)$. ]{}
\[properties of binary random structures\] Let ${\mathcal{M}}$ be a binary random structure. Then $Th({\mathcal{M}})$ is simple, has ${\mathrm{SU}}$-rank 1 and is 1-based with trivial dependence.
[**Proof sketch.**]{} Let ${\mathcal{M}}$ be a binary random structure, so it is homogeneous. That $Th({\mathcal{M}})$ is simple, has ${\mathrm{SU}}$-rank 1 and trivial dependence is proved in essentially the same way as the (folkore) result that the random graph has these properties. It now follows from Corollary 4.7 in [@HKP] (where the terminology ‘modular’ is used in stead of ’1-based’) that $Th({\mathcal{M}})$ is 1-based. $\square$\
The proofs of the main results, Theorems \[first main result\] and \[second main result\], use the notion of reduct and Fact \[rank 1 sets are reducts of binary random structures\], below, from [@AK14].
\[definition of reduct\][Let ${\mathcal{M}}$ and ${\mathcal{N}}$ be structures which need not have the same vocabulary. We say that ${\mathcal{M}}$ is a [*reduct*]{} of ${\mathcal{N}}$ if they have the same universe ($M = N$) and for every $0 < n < \omega$, if $R \subseteq M^n$ is ${\emptyset}$-definable in ${\mathcal{M}}$, then it is ${\emptyset}$-definable in ${\mathcal{N}}$. ]{}
\[rank 1 sets are reducts of binary random structures\] [[@AK14]]{} Let ${\mathcal{M}}$ be countable, binary, homogeneous and simple with trivial dependence. Suppose that $G \subseteq M{^{\mathrm{eq}}}$ is ${\emptyset}$-definable, only finitely many sorts are represented in $G$, and ${\mathrm{SU}}(a) = 1$ and ${\mathrm{acl}}_{{\mathcal{M}}{^{\mathrm{eq}}}}(a) \cap G = \{a\}$ for every $a \in G$. Let ${\mathcal{G}}$ denote the canonically embedded structure in ${\mathcal{M}}{^{\mathrm{eq}}}$ with universe $G$. Then ${\mathcal{G}}$ is a reduct of a binary random structure.
Interpretability {#Interpretability}
----------------
\[definition of strong interpretability\]
Let ${\mathcal{M}}$ and ${\mathcal{N}}$ be structures, possibly with different vocabularies.\
(i) We say that ${\mathcal{N}}$ is [*strongly interpretable in ${\mathcal{M}}$*]{} if there are
- $0 < n < \omega$,
- a formula $\chi(x_1, \ldots, x_n)$ without parameters in the language of ${\mathcal{M}}$,
- a bijective function $f : \chi({\mathcal{M}}) \to N$, and
- for every $0 < k < \omega$ and formula $\varphi(x_1, \ldots, x_k)$ without parameters in the language of ${\mathcal{N}}$, a formula $\psi_\varphi(\bar{y}_1, \ldots, \bar{y}_k)$ without parameters in the language of ${\mathcal{M}}$, such that, for all $\bar{a}_1, \ldots, \bar{a}_k \in \chi({\mathcal{M}})$, $${\mathcal{M}}\models \psi_\varphi(\bar{a}_1, \ldots, \bar{a}_k) \ \Longleftrightarrow \ {\mathcal{N}}\models \varphi(f(\bar{a}_1), \ldots, f(\bar{a}_k)).$$
\(ii) We say that ${\mathcal{N}}$ is [*definable in ${\mathcal{M}}$*]{} if ${\mathcal{N}}$ is strongly interpretable in ${\mathcal{M}}$ and it is possible to choose $n = 1$ in the definition of ‘strongly interpretable’.
It is immediate that if ${\mathcal{N}}$ is strongly interpretable in ${\mathcal{M}}$, then ${\mathcal{N}}$ is interpretable in ${\mathcal{M}}$ in the sense of Chapter 5.3 in [@Hod], and ${\mathcal{N}}$ is interpretable in ${\mathcal{M}}$ in the sense of [@Mac91], and it is definable in ${\mathcal{M}}{^{\mathrm{eq}}}$ in the sense of [@PK]. The following will be convenient to use.
\[equivalent condition of being strongly interpretable\] Suppose that there are positive integers $l, n_1, \ldots, n_l$, formulas $\chi_i(x_1, \ldots, x_{n_i})$ for $i = 1, \ldots, l$, a bijective function $f : \bigcup_{i=1}^l \chi_i({\mathcal{M}}) \to N$ and, for every $0 < k < \omega$, $1 \leq i \leq l$ and formula $\varphi(x_1, \ldots, x_k)$ without parameters in the language of ${\mathcal{N}}$, a formula $\psi_{\varphi,i}(\bar{y}_1, \ldots, \bar{y}_k)$ without parameters in the language of ${\mathcal{M}}$, such that, for all $\bar{a}_1, \ldots, \bar{a}_k \in \chi_i({\mathcal{M}})$, $${\mathcal{M}}\models \psi_{\varphi,i}(\bar{a}_1, \ldots, \bar{a}_k) \ \Longleftrightarrow \ {\mathcal{N}}\models \varphi(f(\bar{a}_1), \ldots, f(\bar{a}_k)).$$ Then ${\mathcal{N}}$ is strongly interpretable in ${\mathcal{M}}$.
[**Proof sketch.**]{} Let $n$ be the maximum of $n_1, \ldots, n_l$. Now the idea is that for every $i = 1, \ldots, l$ and $\bar{a} = (a_1, \ldots, a_i) \in \chi_i({\mathcal{M}})$, $\bar{a}$ can be “translated” into an $n$-tuple $\bar{a}' = (a'_1, \ldots, a'_n)$ where $a'_j = a_j$ for $j = 1, \ldots, n_i$ and $a'_j = a'_{n_i}$, for $j = n_i, \ldots, n$. In this way the set $\{ \bar{a}' : \bar{a} \in \chi_i({\mathcal{M}}) \}$ is ${\emptyset}$-definable in ${\mathcal{M}}$, and it follows that the union of these sets, for $i = 1, \ldots, l$, is also ${\emptyset}$-definable in ${\mathcal{M}}$. The rest is straightforward, via obvious modifications of $f$ and $\psi_{\varphi,i}$ for each $\varphi$. $\square$
Binary homogeneous 1-based structures {#Binary homogeneous 1-based structures}
=====================================
In this section we prove the main results, Theorems \[first main result\] and \[second main result\]. Throughout this section we make the following assumption, which is shared by both theorems:
- [**${\mathcal{M}}$ is countable, binary, homogeneous, simple and 1-based.**]{}
Now Fact \[homogeneity and 1-basedness implies finite rank and trivial dependence\] implies that
- [**$Th({\mathcal{M}})$ has trivial dependence and is supersimple with finite rank.**]{}
Throughout this section and Section \[Proof of the main technical result\] we use the following notational convention:
\[notation for acl etc\][The notations ${\mathrm{acl}}( \ )$, ${\mathrm{dcl}}( \ )$ and ${\mathrm{tp}}( \ )$ are abbreviations of ${\mathrm{acl}}_{{\mathcal{M}}{^{\mathrm{eq}}}}( \ )$, ${\mathrm{dcl}}_{{\mathcal{M}}{^{\mathrm{eq}}}}( \ )$ and ${\mathrm{tp}}_{{\mathcal{M}}{^{\mathrm{eq}}}}( \ )$, respectively. However, when speaking of algebraic closure, definable closure or types in some other structure (${\mathcal{M}}$ for example), then we will show this explicitly with a subscript. ]{}
The first step is is to use results from [@Djo06] to show that every $a \in M$ has finitely many “coordinates” of rank 1 in ${\mathcal{M}}{^{\mathrm{eq}}}$ which to a large extent determine the properties of $a$. By the results in Section 3 of [@Djo06], there is a [*self-coordinatized*]{} set $C \subseteq M{^{\mathrm{eq}}}$, in the sense of Definition 3.3 in [@Djo06], such that $$\begin{aligned}
\label{C_i}
&\text{ $M \subseteq C$, $C$ is ${\emptyset}$-definable;} \\
&\text{ there are $0 < r < \omega$ and ${\emptyset}$-definable sets
$C_0 \subseteq C_1 \subseteq \ldots \subseteq C_r \subseteq C$ such that} \nonumber \\
&\text{ $C_0 = {\emptyset}$, and for every $n < r$ and every $a \in C_{n+1}$, ${\mathrm{SU}}(a/C_n) = 1$;} \nonumber \\
&\text{ for all $n \leq r$, if $a \in C_n$, $b \in M{^{\mathrm{eq}}}$ and ${\mathrm{tp}}(a) = {\mathrm{tp}}(b)$, then $b \in C_n$;} \nonumber \\
&\text{ only finitely many sorts are represented in $C$; and} \nonumber \\
&\text{ $M \subseteq C \subseteq {\mathrm{acl}}(C_r)$.} \nonumber\end{aligned}$$
We assume that
- [**$C$ is chosen so that $r$ is minimal such that (\[C\_i\]) holds.**]{}
As explained in Remark 3.9 of [@Djo06], the number $r$ is an invariant of $Th({\mathcal{M}})$. The next lemma shows that the terminology “the height is 1” in the sense of [@Djo06] is equivalent with saying that “the height is 1” in the sense of Definition \[definition of height\].
\[height and existence of a coordinatizing rank one set\] The following are equivalent:
- $r = 1$.
- There is a ${\emptyset}$-definable $D \subseteq M{^{\mathrm{eq}}}$ in which only finitely many sorts are represented such that $M \subseteq {\mathrm{acl}}(D)$ and ${\mathrm{SU}}(d) = 1$ for every $d \in D$.
[**Proof.**]{} The implication from (1) to (2) is immediate, because if (1) holds, then $M \subseteq {\mathrm{acl}}(C_1)$ where ${\mathrm{SU}}(c) = 1$ for every $c \in C_1$ by (\[C\_i\]).
For the other direction, suppose that (2) holds. Then take $C_0 = {\emptyset}$, $C_1 = D$ and $C = C_1 \cup M$. Now it is straightforward to verify that (\[C\_i\]) holds for $r = 1$. $\square$
\[r=1\] If ${\mathcal{M}}$ is primitive, then $r = 1$, so $M \subseteq {\mathrm{acl}}(C_1)$.
[**Proof.**]{} Suppose that ${\mathcal{M}}$ is primitive. Towards a contradiction, suppose that $r > 1$. Consider the following equivalence relation on $M$: $$x \sim y \ \Longleftrightarrow \ {\mathrm{acl}}(x) \cap C_{r-1} = {\mathrm{acl}}(y) \cap C_{r-1}.$$ By Fact \[facts about simple omega-categorical structures\], this relation is ${\emptyset}$-definable in ${\mathcal{M}}$. By the assumption that $r > 1$ and that $r$ is minimal such that (\[C\_i\]) holds, there is $a \in M$ such that $a \notin {\mathrm{acl}}(C_{r-1})$. Hence $a \notin {\mathrm{acl}}({\mathrm{acl}}(a) \cap C_{r-1})$. Then (by Fact \[facts about simple omega-categorical structures\]) there is $a' \in M$ such that $a' \neq a$ and $${\mathrm{tp}}(a' / {\mathrm{acl}}(a) \cap C_{r-1}) = {\mathrm{tp}}(a / {\mathrm{acl}}(a) \cap C_{r-1}).$$ It follows that ${\mathrm{acl}}(a') \cap C_{r-1} = {\mathrm{acl}}(a) \cap C_{r-1}$, so $a \sim a'$. By the assumption that ${\mathcal{M}}$ is primitive, we must have $b \sim a$ for all $b \in M$. In other words, $$\label{all element have the same acl in C_r-1}
\text{ for all } b \in M, \ {\mathrm{acl}}(b) \cap C_{r-1} = {\mathrm{acl}}(a) \cap C_{r-1}.$$ Let $A = {\mathrm{acl}}(a) \cap C_{r-1}$, so $A$ is finite; let $\bar{a}$ enumerate $A$. Since, by (\[C\_i\]) (and properties of dividing), ${\mathrm{SU}}(b) \geq 1$ for every $b \in A$, there is $\bar{b} \in C_{r-1}$ such that ${\mathrm{rng}}(\bar{b}) \neq A$ and ${\mathrm{tp}}(\bar{b}) = {\mathrm{tp}}(\bar{a})$. Hence there is $a'' \in M$ such that $${\mathrm{acl}}(a'') \cap C_{r-1} = {\mathrm{rng}}(\bar{b}) \neq A = {\mathrm{acl}}(a) \cap C_{r-1},$$ which contradicts (\[all element have the same acl in C\_r-1\]). Hence $r = 1$ and now (\[C\_i\]) immediately gives that $M \subseteq {\mathrm{acl}}(C_1)$. $\square$\
Our aim is to prove Theorems \[first main result\] and \[second main result\] where it is assumed that ${\mathcal{M}}$ is primitive or has weight 1. It follows from the definition of having height 1 (Definition \[definition of height\]) and Lemmas \[height and existence of a coordinatizing rank one set\] and \[r=1\] that in either case we have $r=1$. Therefore
- [**we assume for the rest of this section that $r = 1$, so $M \subseteq {\mathrm{acl}}(C_1)$.**]{}
By (\[C\_i\]), for every $c \in C_1$, ${\mathrm{SU}}(c) = 1$. As dependence is trivial it follows that if $c \in C_1$, $A \subseteq C_1$ and $c \in {\mathrm{acl}}(A)$, then $c \in {\mathrm{acl}}(a)$ for some $a \in A$. Now consider the equivalence relation ‘${\mathrm{acl}}(x) \cap C_1 = {\mathrm{acl}}(y) \cap C_1$’ on $C_1$. By Fact \[facts about simple omega-categorical structures\] this relation is ${\emptyset}$-definable and there is $t < \omega$ such that every equivalence class has at most $t$ elements. Since ${\mathcal{M}}{^{\mathrm{eq}}}$ has elimination of imaginaries (see [@Hod; @She]) it follows that each equivalence class corresponds to an element in ${\mathcal{M}}{^{\mathrm{eq}}}$ in the following sense: There is a ${\emptyset}$-definable set $C'_1 \subseteq M{^{\mathrm{eq}}}$ in which only finitely many sorts are represented and a ${\emptyset}$-definable surjective function $f : C_1 \to C'_1$ (meaning that the graph of $f$ is a ${\emptyset}$-definable relation) such that if $c, c' \in C_1$ then $f(c) = f(c')$ if and only if ${\mathrm{acl}}(c) \cap C_1 = {\mathrm{acl}}(c') \cap C_1$. It follows that for every $c \in C_1$, $f(c) \in {\mathrm{dcl}}(c)$ and $c \in {\mathrm{acl}}(f(c))$. This implies that $M \subseteq {\mathrm{acl}}(C'_1)$ and if $c \in C'_1$, then ${\mathrm{SU}}(c) = 1$ and ${\mathrm{acl}}(c) \cap C'_1 = \{c\}$. Since dependence is trivial, it follows that ${\mathrm{acl}}(A) \cap C'_1 = A$ for every $A \subseteq C'_1$. In order not to switch from the notation ‘$C_1$’ to the notation ‘$C'_1$’, we may (by the above argument), without loss of generality, assume that $$\label{C_1 is a trivial geometry}
\text{ for every } A \subseteq C_1, \ {\mathrm{acl}}(A) \cap C_1 = A.$$
\[definition of crd\][For every $a \in M{^{\mathrm{eq}}}$, let ${\mathrm{crd}}(a) = {\mathrm{acl}}(a) \cap C_1$. We call ‘${\mathrm{crd}}(a)$’ the (set of) [*coordinates*]{} of $a$. (Our definition of ${\mathrm{crd}}$ corresponds to the notation ${\mathrm{crd}}_1$ in [@Djo06].) ]{}
From the definition we obviously have ${\mathrm{crd}}(a) \subseteq {\mathrm{acl}}(a)$ for every $a \in M$.
\[every element is in dcl of its coordinates\] (i) For every $a \in M$, $a \in {\mathrm{acl}}({\mathrm{crd}}(a))$ and hence ${\mathrm{acl}}(a) = {\mathrm{acl}}({\mathrm{crd}}(a))$.\
(ii) For all $a, a' \in M$, ${\mathrm{acl}}_{\mathcal{M}}(a) = {\mathrm{acl}}_{\mathcal{M}}(a')$ if and only if ${\mathrm{crd}}(a) = {\mathrm{crd}}(a')$.\
(iii) If the equivalence relation ${\mathrm{acl}}_{\mathcal{M}}(x) = {\mathrm{acl}}_{\mathcal{M}}(y)$ has only singleton classes, then $a \in {\mathrm{dcl}}({\mathrm{crd}}(a))$ for every $a \in M$.
[**Proof.**]{} Part (i) is a direct consequence of Lemma 5.1 in [@Djo06], but can easily be proved directly as follows. Let $a \in M$, so $a \in {\mathrm{acl}}(C_1)$. Let $B = {\mathrm{acl}}(a) \cap C_1$. Suppose that $a \notin {\mathrm{acl}}(B)$. Then $a \underset{B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} C_1$. As dependence is trivial there is $c \in C_1 \setminus B$ such that $a \underset{B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} c$ and hence $aB {\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }c$ Since $B \subseteq {\mathrm{acl}}(a)$ we get $a {\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }c$ which (as ${\mathrm{SU}}(c) = 1$) implies that $c \in {\mathrm{acl}}(a)$. Hence $c \in B$ which contradicts the choice of $c$.
Part (ii) follows immediately from part (i) and the definition of ${\mathrm{crd}}$. Hence it remains to prove (iii).
Suppose that $a, a' \in M$ and $a' \in {\mathrm{acl}}({\mathrm{crd}}(a))$. By the primitivity of ${\mathcal{M}}$ we have ${\mathrm{tp}}(a) = {\mathrm{tp}}(a')$, hence $|{\mathrm{crd}}(a)| = {\mathrm{crd}}(a')|$ and therefore ${\mathrm{crd}}(a) = {\mathrm{crd}}(a')$. Hence $a$ and $a'$ belong to the same equivalence class of ${\mathrm{acl}}_{\mathcal{M}}(x) = {\mathrm{acl}}_{\mathcal{M}}(y)$ and thus $a = a'$. $\square$
\[remark on lemma about properties of crd\][Let ‘$\approx$’ be the (${\emptyset}$-definable) equivalence relation ‘${\mathrm{acl}}_{\mathcal{M}}(x) = {\mathrm{acl}}_{\mathcal{M}}(y)$’ and ${\mathcal{M}}/ {{\negthickspace}\approx}$ the canonically embedded structure of ${\mathcal{M}}{^{\mathrm{eq}}}$ with universe $M / {{\negthickspace}\approx}$. For every $a \in M$ we have ${\mathrm{acl}}(a) = {\mathrm{acl}}([a]_\approx)$. Therefore ${\mathrm{crd}}(a) = {\mathrm{crd}}([a]_\approx)$ for every $a \in M$ and Lemma \[every element is in dcl of its coordinates\] holds if ‘$M$’ and ’${\mathcal{M}}$’ are replaced with ‘$M / {{\negthickspace}\approx}$’ and ‘${\mathcal{M}}/ {{\negthickspace}\approx}$’, respectively. Since the equivalence relation ‘${\mathrm{acl}}_{{\mathcal{M}}/ \approx}(x) = {\mathrm{acl}}_{{\mathcal{M}}/ \approx}(y)$ is trivial it follows that part (iii) simplifies to the statement: for every $a \in M / {{\negthickspace}\approx}$, $a \in {\mathrm{dcl}}({\mathrm{crd}}(a))$. ]{}
\[crd is unique up to isomorphism\] If $a, b \in M$ and ${\mathrm{tp}}(a) = {\mathrm{tp}}(b)$, then ${\mathrm{crd}}(a)$ and ${\mathrm{crd}}(b)$ can be ordered as $\bar{a}$ and $\bar{b}$, respectively, so that ${\mathrm{tp}}(\bar{a}) = {\mathrm{tp}}(\bar{b})$.
[**Proof.**]{} Suppose that $a, b \in M$ and ${\mathrm{tp}}(a) = {\mathrm{tp}}(b)$. Let $\bar{a}$ be an ordering of ${\mathrm{crd}}(a)$ and let $f$ be an automorphism of ${\mathcal{M}}{^{\mathrm{eq}}}$ such that $f(a) = b$. Then $f(\bar{a})$ is an ordering of ${\mathrm{crd}}(b)$ such that ${\mathrm{tp}}(\bar{a}) = {\mathrm{tp}}(f(\bar{a}))$. $\square$\
By considering a ${\emptyset}$-definable subset of $C_1$ if necessary, we may, in addition to previous assumptions, assume that $$\label{every member of C_1 is a coordinate of M}
\text{ for every $c \in C_1$, there is $a \in M$ such that $c \in {\mathrm{crd}}(a)$.}$$
Let ${\mathcal{C}}_1$ be the canonically embedded structure in ${\mathcal{M}}{^{\mathrm{eq}}}$ with universe $C_1$. By Fact \[rank 1 sets are reducts of binary random structures\], ${\mathcal{C}}_1$ is a reduct of a binary random structure ${\mathcal{R}}$, so in particular $R = C_1$, where $R$ is the universe of ${\mathcal{R}}$. Hence ${\mathrm{crd}}(a) \subseteq R$ for every $a \in M$.
\[M is definable in Req\] Let ‘$\approx$’ be the ${\emptyset}$-definable equivalence relation on $M$ defined by $x \approx y$ if and only if ${\mathrm{acl}}_{\mathcal{M}}(x) = {\mathrm{acl}}_{\mathcal{M}}(y)$. Then ${\mathcal{M}}/ {{\negthickspace}\approx}$ is definable in ${\mathcal{R}}{^{\mathrm{eq}}}$ and we can use a formula $\chi(x)$ and bijection $f : \chi({\mathcal{R}}{^{\mathrm{eq}}}) \to M / {{\negthickspace}\approx}$ as in Definition \[definition of strong interpretability\] with the properties that
1. for all $c \in \chi({\mathcal{R}}{^{\mathrm{eq}}})$, ${\mathrm{acl}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(c) \cap R = {\mathrm{crd}}(f(c))$ and ${\mathrm{acl}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(c) = {\mathrm{acl}}_{{\mathcal{R}}{^{\mathrm{eq}}}}({\mathrm{crd}}(f(c)))$,
2. for all $c, c' \in \chi({\mathcal{R}}{^{\mathrm{eq}}})$, $c = c'$ if and only if ${\mathrm{crd}}(f(c)) = {\mathrm{crd}}(f(c'))$, and
3. $c \in {\mathrm{dcl}}_{{\mathcal{R}}{^{\mathrm{eq}}}}({\mathrm{acl}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(c) \cap R)$ for all $c \in \chi({\mathcal{R}}{^{\mathrm{eq}}})$.
[**Proof.**]{} In order to simplify notation and make the argument more clear we prove the proposition under the extra assumption that $\approx$ is trivial, in other words, that it has only singleton classes. The general case is a straightforward modification of this special case, where we use Remark \[remark on lemma about properties of crd\] instead of Lemma \[every element is in dcl of its coordinates\]. The assumption that $\approx$ is trivial implies that ${\mathcal{M}}/ {{\negthickspace}\approx}$ is definable in ${\mathcal{M}}$ via the map taking every $a \in M$ to $[a]_\approx$, where $[a]_\approx$ is the (singleton) $\approx$-class to which $a$ belongs. Hence it suffices to show that ${\mathcal{M}}$ is definable in ${\mathcal{R}}{^{\mathrm{eq}}}$.
Observe that since ${\mathcal{C}}_1$ is a reduct of ${\mathcal{R}}$ we can (and will) view $({\mathcal{C}}_1){^{\mathrm{eq}}}$ as a reduct of ${\mathcal{R}}{^{\mathrm{eq}}}$. Also, since ${\mathcal{C}}_1$ is canonically embedded in ${\mathcal{M}}{^{\mathrm{eq}}}$ we can (and will) assume that $(C_1){^{\mathrm{eq}}}\subseteq (M{^{\mathrm{eq}}}){^{\mathrm{eq}}}$. Moreover, as ${\mathcal{C}}_1$ is canonically embedded in ${\mathcal{M}}{^{\mathrm{eq}}}$, for all $\bar{c}, \bar{c}' \in (C_1){^{\mathrm{eq}}}$, ${\mathrm{tp}}_{({\mathcal{C}}_1){^{\mathrm{eq}}}}(\bar{c}) = {\mathrm{tp}}_{({\mathcal{C}}_1){^{\mathrm{eq}}}}(\bar{c}')$ if and only if ${\mathrm{tp}}_{({\mathcal{M}}{^{\mathrm{eq}}}){^{\mathrm{eq}}}}(\bar{c}) = {\mathrm{tp}}_{({\mathcal{M}}{^{\mathrm{eq}}}){^{\mathrm{eq}}}}(\bar{c}')$.
Let $p_1, \ldots, p_s$ enumerate (without repetition) $S^{\mathcal{M}}_1({\emptyset})$. For each $p_i$ choose a realization $a_i \in M$ of $p_i$ and enumerate (without repetition) ${\mathrm{crd}}(a_i)$ as $\bar{b}_i$. Then let $q_i = {\mathrm{tp}}(\bar{b}_i)$. Since ${\mathcal{M}}$ is $\omega$-categorical and since ${\mathcal{C}}_1$ is canonically embedded in ${\mathcal{M}}{^{\mathrm{eq}}}$ and ${\mathcal{C}}_1$ is a reduct of ${\mathcal{R}}$ it follows that (for each $i$) the set $q_i({\mathcal{M}}{^{\mathrm{eq}}})$ is ${\emptyset}$-definable in ${\mathcal{M}}{^{\mathrm{eq}}}$, ${\mathcal{C}}_1$ and in ${\mathcal{R}}$.
For each $i$, define an equivalence relation on $q_i({\mathcal{M}}{^{\mathrm{eq}}})$ as follows: $\bar{x} \sim_i \bar{y}$ if and only if ${\mathrm{rng}}(\bar{x}) = {\mathrm{rng}}(\bar{y})$. (The relation $\sim_i$ can be extended to all $|\bar{b}_i|$-tuples of elements from $M{^{\mathrm{eq}}}$, $R$ or $C_1$, by saying that all $|\bar{b}_i|$-tuples outside of $q_i({\mathcal{M}}{^{\mathrm{eq}}})$ belong to the same class.) Note that $\sim_i$ is a ${\emptyset}$-definable relation in ${\mathcal{R}}$ as well as in ${\mathcal{C}}_1$ and in ${\mathcal{M}}{^{\mathrm{eq}}}$. Hence, for every $i$, the set of $\sim_i$-classes of tuples in $q_i({\mathcal{M}}{^{\mathrm{eq}}})$ is a ${\emptyset}$-definable subset of ${\mathcal{R}}{^{\mathrm{eq}}}$, of $({\mathcal{C}}_1){^{\mathrm{eq}}}$ and of $({\mathcal{M}}{^{\mathrm{eq}}}){^{\mathrm{eq}}}$.
For every $i$ and $\bar{b}$ realizing $q_i$ let $[\bar{b}]_i$ be its $\sim_i$-class. Then let $$X \ = \ \{ c : c = [\bar{b}]_i \text{ for some $i$ and $\bar{b}$} \}$$ so $X$ is a subset of $R{^{\mathrm{eq}}}$, $(C_1){^{\mathrm{eq}}}$ and of $(M{^{\mathrm{eq}}}){^{\mathrm{eq}}}$. Moreover, $X$ is ${\emptyset}$-definable in ${\mathcal{R}}{^{\mathrm{eq}}}$ by some formula $\chi(x)$.
Now we define a bijection $g : M \to X$ such that if $f = g^{-1}$ then $f$ has the required properties. For every $a \in M$ define $g(a)$ as follows: let $i$ be such that ${\mathcal{M}}\models p_i(a)$ and (using Lemma \[crd is unique up to isomorphism\]) let $\bar{b}$ enumerate ${\mathrm{crd}}(a)$ in such a way that $\bar{b}$ realizes $q_i$ and let $g(a) = [\bar{b}]_i$. The surjectivity of $g$ follows from the $\omega$-homogeneity of ${\mathcal{M}}{^{\mathrm{eq}}}$ and the definition of $X$. Observe that if $a \in M$ realizes $p_i$ and $\bar{b}$ enumerates ${\mathrm{crd}}(a)$ in such a way that it realizes $q_i$ then (by Lemma \[every element is in dcl of its coordinates\]) ${\mathrm{acl}}(\bar{b}) = {\mathrm{acl}}(a)$ and $[\bar{b}]_i \in {\mathrm{dcl}}_{({\mathcal{M}}{^{\mathrm{eq}}}){^{\mathrm{eq}}}}(a)$. Since each $\sim_i$-class is finite we get ${\mathrm{acl}}_{({\mathcal{M}}{^{\mathrm{eq}}}){^{\mathrm{eq}}}}([\bar{b}]_i) = {\mathrm{acl}}_{({\mathcal{M}}{^{\mathrm{eq}}}){^{\mathrm{eq}}}}(a)$. As we assume that the equivalence relation ${\mathrm{acl}}_{\mathcal{M}}(x) = {\mathrm{acl}}_{\mathcal{M}}(y)$ has only singleton classes (and hence the same holds for ${\mathrm{acl}}_{({\mathcal{M}}{^{\mathrm{eq}}}){^{\mathrm{eq}}}}(x) = {\mathrm{acl}}_{({\mathcal{M}}{^{\mathrm{eq}}}){^{\mathrm{eq}}}}(y)$ restricted to $M$) it follows that $g : M \to X$ is bijective and $a \in {\mathrm{dcl}}_{({\mathcal{M}}{^{\mathrm{eq}}}){^{\mathrm{eq}}}}(g(a))$ for all $a \in M$. (In the general case, note that the equivalence relation ${\mathrm{acl}}_{{\mathcal{M}}/ \approx}(x) = {\mathrm{acl}}_{{\mathcal{M}}/ \approx}(y)$ is trivial, by the definition of $\approx$.)
Let $0 < n < \omega$ and $a_1, \ldots, a_n, a'_1, \ldots, a'_n \in M$. By the observations already made (following from the fact that ${\mathcal{C}}_1$ is a reduct of ${\mathcal{R}}$ and ${\mathcal{C}}_1$ is canonically embedded in ${\mathcal{M}}{^{\mathrm{eq}}}$) and in particular since $a_i \in {\mathrm{dcl}}_{({\mathcal{M}}{^{\mathrm{eq}}}){^{\mathrm{eq}}}}(g(a_i))$ for all $i$ and similarly for $a'_i$ we get: $$\begin{aligned}
{\mathrm{tp}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(g(a_1), \ldots, g(a_n)) \ &= \ {\mathrm{tp}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(g(a'_1), \ldots, g(a'_n)) \ \ \Longrightarrow \\
{\mathrm{tp}}_{({\mathcal{C}}_1){^{\mathrm{eq}}}}(g(a_1), \ldots, g(a_n)) \ &= \ {\mathrm{tp}}_{({\mathcal{C}}_1){^{\mathrm{eq}}}}(g(a'_1), \ldots, g(a'_n)) \ \ \Longrightarrow \\
{\mathrm{tp}}_{({\mathcal{M}}{^{\mathrm{eq}}}){^{\mathrm{eq}}}}(g(a_1), \ldots, g(a_n)) \ &= \ {\mathrm{tp}}_{({\mathcal{M}}{^{\mathrm{eq}}}){^{\mathrm{eq}}}}(g(a'_1), \ldots, g(a'_n)) \ \ \Longrightarrow \\
{\mathrm{tp}}_{({\mathcal{M}}{^{\mathrm{eq}}}){^{\mathrm{eq}}}}(a_1, \ldots, a_n) \ &= \ {\mathrm{tp}}_{({\mathcal{M}}{^{\mathrm{eq}}}){^{\mathrm{eq}}}}(a'_1, \ldots, a'_n) \ \ \Longrightarrow \\
{\mathrm{tp}}_{\mathcal{M}}(a_1, \ldots, a_n) \ &= \ {\mathrm{tp}}_{\mathcal{M}}(a'_1, \ldots, a'_n).\end{aligned}$$ As ${\mathcal{R}}$ is homogeneous, every type over ${\emptyset}$ with respect to ${\mathcal{R}}{^{\mathrm{eq}}}$ which is realized by elements from $X$ is isolated. Moreover, for each $0 < n < \omega$, only finitely many types from $S^{{\mathcal{R}}{^{\mathrm{eq}}}}_n({\emptyset})$ are realized by $n$-tuples from $X^n$. It follows that for every $0 < n < \omega$ and $p \in S^{\mathcal{M}}_n({\emptyset})$ there is a formula $\varphi_p(x_1, \ldots, x_n)$ without parameters in the language of ${\mathcal{R}}{^{\mathrm{eq}}}$ such that for all $a_1, \ldots, a_n \in M$, $${\mathcal{M}}\models p(a_1, \ldots, a_n) \ \ \Longleftrightarrow \ \ {\mathcal{R}}{^{\mathrm{eq}}}\models \varphi_p(g(a_1), \ldots, g(a_n)).$$ This implies that ${\mathcal{M}}$ is definable in ${\mathcal{R}}{^{\mathrm{eq}}}$ via the map $f = g^{-1}$ as in Definition \[definition of strong interpretability\].
It remains to verify that $f$ has the other properties stated in the proposition. Let $c \in \chi({\mathcal{R}}{^{\mathrm{eq}}})$, so $c = [\bar{b}]_i$ for some $i$ and some $\bar{b} \in q_i({\mathcal{M}}{^{\mathrm{eq}}})$. From the definition of $f$ it follows that ${\mathrm{rng}}(\bar{b}) = {\mathrm{crd}}(f(c))$, so ${\mathrm{acl}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(\bar{b}) = {\mathrm{acl}}_{{\mathcal{R}}{^{\mathrm{eq}}}}({\mathrm{crd}}(f(c)))$. As $c = [\bar{b}]_i$ is a finite equivalence class we get ${\mathrm{acl}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(c) = {\mathrm{acl}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(\bar{b}) = {\mathrm{acl}}_{{\mathcal{R}}{^{\mathrm{eq}}}}({\mathrm{crd}}(f(c)))$. In particular, ${\mathrm{acl}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(c) \cap R = {\mathrm{acl}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(\bar{b}) \cap R$, and since $\bar{b} \in R$ where ${\mathcal{R}}$ is a binary random structure we get ${\mathrm{acl}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(\bar{b}) \cap R = {\mathrm{rng}}(\bar{b}) = {\mathrm{crd}}(f(c))$. Hence ${\mathrm{acl}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(c) \cap R = {\mathrm{crd}}(f(c))$. This proves (1). Since $f$ is bijective and the relation ${\mathrm{acl}}_{\mathcal{M}}(x) = {\mathrm{acl}}_{\mathcal{M}}(y)$ has only singleton classes (by assumption) it follows (using Lemma \[every element is in dcl of its coordinates\] (ii)) that for all $c, c' \in \chi({\mathcal{R}}{^{\mathrm{eq}}})$, $c = c'$ if and only if ${\mathrm{crd}}(f(c)) = {\mathrm{crd}}(f(c'))$, so (2) is proved. For (3), suppose that $c, c' \in \chi({\mathcal{R}}{^{\mathrm{eq}}})$ are distinct and that $c' \in {\mathrm{acl}}_{{\mathcal{R}}{^{\mathrm{eq}}}}({\mathrm{acl}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(c) \cap R) =
{\mathrm{acl}}_{{\mathcal{R}}{^{\mathrm{eq}}}}({\mathrm{crd}}(f(c)) = {\mathrm{acl}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(c)$. Then ${\mathrm{crd}}(f(c')) = {\mathrm{acl}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(c') \cap R \subseteq
{\mathrm{acl}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(c) \cap R = {\mathrm{crd}}(f(c))$, so by (2), $|{\mathrm{acl}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(c') \cap R| < |{\mathrm{acl}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(c) \cap R|$. This implies that ${\mathrm{tp}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(c') \neq {\mathrm{tp}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(c)$ whenever $c, c' \in \chi({\mathcal{R}}{^{\mathrm{eq}}})$ are distinct and $c' \in {\mathrm{acl}}_{{\mathcal{R}}{^{\mathrm{eq}}}}({\mathrm{acl}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(c) \cap R)$. It follows that $c \in {\mathrm{dcl}}_{{\mathcal{R}}{^{\mathrm{eq}}}}({\mathrm{acl}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(c) \cap R)$. $\square$\
[*Observe that Proposition \[M is definable in Req\] proves part (i) of Theorem \[second main result\].*]{} We continue by proving part (ii) of Theorem \[second main result\]. Therefore,
- [**we assume for the rest of the section that the equivalence relation ${\mathrm{acl}}_{\mathcal{M}}(x) = {\mathrm{acl}}_{\mathcal{M}}(y)$ is trivial.**]{}
Then there is an obvious bijection $h : M / {{\negthickspace}\approx}\ \to M$ such that for all $\bar{a}, \bar{b} \in M / {{\negthickspace}\approx}$, ${\mathrm{tp}}(\bar{a}) = {\mathrm{tp}}(\bar{b})$ if and only if ${\mathrm{tp}}(h(\bar{a})) = {\mathrm{tp}}(h(\bar{b}))$. Therefore Proposition \[M is definable in Req\] allows us to identify $M$ with $\chi({\mathcal{R}}{^{\mathrm{eq}}})$ via the bijection $h \circ f$ where $\chi$ and $f$ are as in that proposition. It follows that for every $0 < n < \omega$ and $D \subseteq M^n$, if $D$ is ${\emptyset}$-definable in ${\mathcal{M}}$, then it is ${\emptyset}$-definable in ${\mathcal{R}}{^{\mathrm{eq}}}$. So
- [**for the rest of this section we assume that ${\mathcal{M}}$ is a reduct of the canonically embedded structure of ${\mathcal{R}}{^{\mathrm{eq}}}$ with universe $M = \chi({\mathcal{R}}{^{\mathrm{eq}}})$.**]{}
Note that the identification $h(f(c)) = c$ for all $c \in \chi({\mathcal{R}}{^{\mathrm{eq}}})$ together with (1) of Proposition \[M is definable in Req\] implies that $$\text{ for all $a \in M$}, \ {\mathrm{crd}}(a) \ = \ {\mathrm{acl}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(a) \cap R.$$ Part (3) of Proposition \[M is definable in Req\] and the new assumptions imply that $$\label{a is in definable closure of acl(a) cut with R}
\text{for every $a \in M$, $a \in {\mathrm{dcl}}_{{\mathcal{R}}{^{\mathrm{eq}}}}({\mathrm{crd}}(a))$.}$$
\[crd(a) is rigid\] Let $a \in M$ and ${\mathrm{crd}}(a) = \{b_1, \ldots, b_m\}$, where the elements are enumerated without repetition. Then for every nontrivial permutation $\pi$ of $\{1, \ldots, m\}$, ${\mathrm{tp}}_{\mathcal{R}}(b_1, \ldots, b_m) \neq {\mathrm{tp}}_{\mathcal{R}}(b_{\pi(1)}, \ldots, b_{\pi(m)})$.
The proof of Lemma \[crd(a) is rigid\] is given in Section \[Proof of the main technical result\], but now we derive a corollary of it.
\[crd is in dcl of a\] For every $a \in M$, ${\mathrm{crd}}(a) \subseteq {\mathrm{dcl}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(a)$.
[**Proof.**]{} Let $a \in M$ and ${\mathrm{crd}}(a) = \{b_1, \ldots, b_m\}$. If $b_i \notin {\mathrm{dcl}}_{{\mathcal{M}}{^{\mathrm{eq}}}}(a)$ for some $i$, then ${\mathrm{tp}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(a, b_i) = {\mathrm{tp}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(a, b_j)$ for some $j \neq i$, from which it follows (using that ${\mathcal{R}}{^{\mathrm{eq}}}$ is $\omega$-homogeneous) that there is a nontrivial permutation $\pi$ of $\{1, \ldots, m\}$ such that $${\mathrm{tp}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(a, b_1, \ldots, b_m) \ = \ {\mathrm{tp}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(a, b_{\pi(1)}, \ldots, b_{\pi(m)}).$$ Then ${\mathrm{tp}}_{\mathcal{R}}(b_1, \ldots, b_m) = {\mathrm{tp}}_{\mathcal{R}}(b_{\pi(1)}, \ldots, b_{\pi(m)})$ which contradicts Lemma \[crd(a) is rigid\]. $\square$\
Now we are ready to prove the remaining parts of the main results (and when stating them we repeat the assumptions made in the beginning of this section).
[**Theorem \[second main result\]**]{}[* Suppose that ${\mathcal{M}}$ is a structure which is countable, binary, homogeneous, simple and 1-based with height 1. Let $\approx$ denote the equivalence relation ‘${\mathrm{acl}}_{\mathcal{M}}(x) = {\mathrm{acl}}_{\mathcal{M}}(y)$’.\
(i) Then there is a binary random structure ${\mathcal{R}}$ such that ${\mathcal{M}}/ {{\negthickspace}\approx}$ is definable in ${\mathcal{R}}{^{\mathrm{eq}}}$.\
(ii) If, in addition, the equivalence relation ‘$\approx$’ is trivial, then ${\mathcal{M}}$ is strongly interpretable in a binary random structure.* ]{}
[**Proof.**]{} As mentioned above, part (i) follows from Proposition \[M is definable in Req\]. So it remains to prove (ii) and for this we adopt the assumption that the equivalence relation $\approx$ is trivial, as well as all other assumptions that have been made earlier in this section. The proof is similar to the proof of Proposition \[M is definable in Req\]. The essential difference is that the assumption that $\approx$ is trivial together with Lemma \[crd(a) is rigid\] allows us to reach a stronger conclusion than in Proposition \[M is definable in Req\].
Let $p_1, \ldots, p_s$ enumerate $S^{\mathcal{M}}_1({\emptyset})$. For each $1 \leq i \leq s$ choose a realization $a_i \in M$ of $p_i$ and then choose an ordering $\bar{b}_i$ of ${\mathrm{crd}}(a_i)$. For each $i$, let $\theta_i$ isolate ${\mathrm{tp}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(a_i, \bar{b}_i)$ and let $\varphi_i$ isolate ${\mathrm{tp}}_{\mathcal{R}}(\bar{b}_i)$. Let $X = \varphi_1({\mathcal{R}}) \cup \ldots, \cup \varphi_s({\mathcal{R}})$.
By (\[a is in definable closure of acl(a) cut with R\]), for every $\bar{b} \in X$ there is a unique $a \in M$ such that ${\mathcal{R}}{^{\mathrm{eq}}}\models \theta_i(a, \bar{b})$ for some $i$. For every $\bar{b} \in X$ we let $f(\bar{b}) = a$ for the unique $a \in M$ such that ${\mathcal{R}}{^{\mathrm{eq}}}\models \theta_i(a, \bar{b})$ for some $i$. Since ${\mathrm{crd}}(a)$ exists as a subset of $C_1 = R$ for every $a \in M$ it follows that $f : X \to M$ is surjective. Now we claim that $f$ is injective. For if $\bar{b}, \bar{b}' \in X$ and $f(\bar{b}) = f(\bar{b}') = a \in M$, then ${\mathcal{R}}{^{\mathrm{eq}}}\models \theta_i(a, \bar{b}) \wedge \theta_j(a, \bar{b}')$ for some $i, j$. Then ${\mathrm{tp}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(a, \bar{b}) = {\mathrm{tp}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(a_i, \bar{b}_i)$ and ${\mathrm{tp}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(a, \bar{b}') = {\mathrm{tp}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(a_j, \bar{b}_j)$ from which it follows that ${\mathrm{tp}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(a_i) = {\mathrm{tp}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(a_j)$ and ${\mathrm{rng}}(\bar{b}) = {\mathrm{crd}}(a) = {\mathrm{rng}}(\bar{b}')$. From the construction it now follows that $i = j$ and hence ${\mathrm{tp}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(a, \bar{b}) = {\mathrm{tp}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(a, \bar{b}')$. By Corollary \[crd is in dcl of a\], ${\mathrm{rng}}(\bar{b}) \subseteq {\mathrm{dcl}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(a)$ and ${\mathrm{rng}}(\bar{b}') \subseteq {\mathrm{dcl}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(a)$ so we must have $\bar{b} = \bar{b}'$.
Let $0 < n < \omega$, $1 \leq j \leq s$ and $\bar{b}_1, \ldots, \bar{b}_n, \bar{b}'_1, \ldots, \bar{b}'_n \in \varphi_j({\mathcal{R}})$. From the definition of $f$, its graph is a ${\emptyset}$-definable relation, which implies (since $f$ is bijective between two ${\emptyset}$-definable subsets of $R{^{\mathrm{eq}}}$) that ${\mathrm{rng}}(\bar{b}_i) \subseteq {\mathrm{dcl}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(f(\bar{b}_i))$ and $f(\bar{b}_i) \in {\mathrm{dcl}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(\bar{b}_i)$ for each $i$, and similarly for each $\bar{b}'_i$. This implies that $$\begin{aligned}
{\mathrm{tp}}_{\mathcal{R}}(\bar{b}_1, \ldots, \bar{b}_n) \ &= \ {\mathrm{tp}}_{\mathcal{R}}(\bar{b}'_1, \ldots, \bar{b}'_n) \ \Longleftrightarrow \\
{\mathrm{tp}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(\bar{b}_1, \ldots, \bar{b}_n) \ &= \ {\mathrm{tp}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(\bar{b}'_1, \ldots, \bar{b}'_n) \ \Longleftrightarrow \\
{\mathrm{tp}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(f(\bar{b}_1), \ldots, f(\bar{b}_n)) \ &= \ {\mathrm{tp}}_{{\mathcal{R}}{^{\mathrm{eq}}}}(f(\bar{b}'_1), \ldots, f(\bar{b}'_n)) \ \Longrightarrow \\
{\mathrm{tp}}_{\mathcal{M}}(f(\bar{b}_1), \ldots, f(\bar{b}_n)) \ &= \ {\mathrm{tp}}_{\mathcal{M}}(f(\bar{b}'_1), \ldots, f(\bar{b}'_n)).\end{aligned}$$ Since ${\mathcal{R}}$ and ${\mathcal{M}}$ are homogeneous and hence $\omega$-categorical, there is for every $0 < n < \omega$ and formula $\xi(x_1, \ldots, x_n)$ in the language of ${\mathcal{M}}$ a formula $\xi'(\bar{x}_1, \ldots, \bar{x}_n)$ in the language of ${\mathcal{R}}$ such that for all $\bar{b}_1, \ldots, \bar{b}_n \in \varphi_i({\mathcal{R}})$, ${\mathcal{M}}\ \models \xi(f(\bar{b}_1), \ldots, f(\bar{b}_n))$ if and only if ${\mathcal{R}}\models \xi'(\bar{b}_1, \ldots, \bar{b}_n)$. By Lemma \[equivalent condition of being strongly interpretable\], ${\mathcal{M}}$ is strongly interpretable in ${\mathcal{R}}$. $\square$
[**Theorem \[first main result\]**]{}[* Suppose that ${\mathcal{M}}$ is a structure which is countable, binary, homogeneous, primitive, simple and 1-based. Then ${\mathcal{M}}$ is strongly interpretable in a binary random structure.* ]{}
[**Proof.**]{} Suppose that ${\mathcal{M}}$ satisfies the assumptions of the Theorem \[first main result\]. The primitivity of ${\mathcal{M}}$ implies that the equivalence relation ${\mathrm{acl}}_{\mathcal{M}}(x) = {\mathrm{acl}}_{\mathcal{M}}(y)$ is trivial, because it is ${\emptyset}$-definable in ${\mathcal{M}}$. Therefore Theorem \[first main result\] is a direct consequence of Lemma \[r=1\] and Theorem \[second main result\]. $\square$
\[remark on construction of line structures\][One can construct structures that satisfy the assumptions of Theorem \[second main result\] roughly as follows. Take a binary random structure ${\mathcal{R}}$. Choose some rigid nonisomorphic substructures ${\mathcal{L}}_1, \ldots, {\mathcal{L}}_k$ of ${\mathcal{R}}$ (where [*rigid*]{} means that there is no nontrivial automorphism) together with a fixed enumeration of $L_i$ for each $i$. Let $M$ be the set of substructures of ${\mathcal{R}}$ which are isomorphic to some ${\mathcal{L}}_i$. If ${\mathcal{L}}\cong {\mathcal{L}}_i$ then we think of $L$ as being enumerated so that this enumeration and the enumeration of $L_i$ induces an isomorphism from ${\mathcal{L}}$ to ${\mathcal{L}}_i$. For any ${\mathcal{A}}, {\mathcal{A}}', {\mathcal{B}}, {\mathcal{B}}' \in M$, let $({\mathcal{A}}, {\mathcal{A}}')$ and $({\mathcal{B}}, {\mathcal{B}}')$ have the same type in ${\mathcal{M}}$ if and only if $\bar{a}\bar{a}'$ and $\bar{b}\bar{b}'$ have the same type in ${\mathcal{R}}$ where $\bar{a}, \bar{a}', \bar{b}, \bar{b}'$ are the enumerations of the elements in the respective structure. By Theorem \[first main result\] and Lemma \[crd(a) is rigid\] it follows that if ${\mathcal{N}}$ is countable, binary, homogeneous, primitive, simple and 1-based, then ${\mathcal{N}}$ is a proper reduct of such a structure ${\mathcal{M}}$ (with $k = 1$). ]{}
Proof of Lemma \[crd(a) is rigid\] {#Proof of the main technical result}
==================================
In this section we prove Lemma \[crd(a) is rigid\], so all assumptions in Section \[Binary homogeneous 1-based structures\] up to the Lemma \[crd(a) is rigid\] apply in this section, including the conventions of Notation \[notation for acl etc\]. In particular we recall that ${\mathcal{R}}$ is a binary random structure and ${\mathcal{C}}_1$ is the canonically embedded structure of ${\mathcal{M}}{^{\mathrm{eq}}}$ with universe $C_1$.
The intuition behind the proof of Lemma \[crd(a) is rigid\] comes from the argument that the line graph over a complete graph with infinite countable vertex set is not homogeneous. Somewhat more precisely: Suppose that ${\mathcal{K}}$ is a complete graph with infinite countable vertex set $K$. The line graph over ${\mathcal{K}}$ has as its vertex set the set of all 2-subsets of $K$ and two 2-subsets are adjacent (in the line graph) if and only if they intersect in exactly one point. Then one can choose distinct $a, b, c \in K$ and distinct $d, e, e', e'' \in K$ and it is easy to see that the 3-tuples $(\{a, b\}, \{b, c\}, \{c, a\})$ and $(\{d, e\}, \{d, e'\}, \{d, e''\})$ satisfy the same quantifier free formulas in the line graph, but there is a formula which is satisfied by one of the tuples but not the other.
[**Lemma \[crd(a) is rigid\]**]{}[*Let $a \in M$ and ${\mathrm{crd}}(a) = \{b_1, \ldots, b_m\}$, where the elements are enumerated without repetition. Then for every nontrivial permutation $\pi$ of $\{1, \ldots, m\}$, ${\mathrm{tp}}_{\mathcal{R}}(b_1, \ldots, b_m) \neq {\mathrm{tp}}_{\mathcal{R}}(b_{\pi(1)}, \ldots, b_{\pi(m)})$.* ]{}
[**Proof.**]{} As usual we use the facts from Section \[Preliminaries\] without further reference. The lemma is trivial if $m = 1$, so we assume that $m \geq 2$. Let $a \in M$ and let $${\mathrm{crd}}(a) = \{b_1, \ldots, b_m\}.$$ Towards a contradiction, suppose that there is a nontrivial permutation $\pi$ of $\{1, \ldots, m\}$ such that $$\label{the permuted sequence has the same type}
{\mathrm{tp}}_{\mathcal{R}}(b_1, \ldots, b_m) = {\mathrm{tp}}_{\mathcal{R}}(b_{\pi(1)}, \ldots, b_{\pi(m)}).$$ To simplify notation and witout loss of generality, $$\label{pi(1) equals 2}
\text{ we assume that $\pi(1) = 2$. }$$ From (\[the permuted sequence has the same type\]) and (\[pi(1) equals 2\]) we get $${\mathrm{tp}}_{\mathcal{R}}(b_1) = {\mathrm{tp}}_{\mathcal{R}}(b_2).$$ As ${\mathcal{R}}$ is a binary random structure we can argue as in Remark \[remarks about binary random structures\] (with $n = 3$, $p_{1, 2}(x_1, x_2) = {\mathrm{tp}}_{\mathcal{R}}(b_1, b_2)$, $p_{1, 3}(x_1, x_3) = {\mathrm{tp}}_{\mathcal{R}}(b_1, b_2)$ and $p_{2, 3}(x_2, x_3) = {\mathrm{tp}}_{\mathcal{R}}(b_1, b_2)$) and find $b'_2 \in R$ such that $${\mathrm{tp}}_{\mathcal{R}}(b_1, b'_2) \ = \ {\mathrm{tp}}_{\mathcal{R}}(b_1, b_2) \ = \ {\mathrm{tp}}_{\mathcal{R}}(b_2, b'_2).$$ If $m > 2$, then, by using that ${\mathcal{R}}$ is a random structure again, we find [*distinct*]{} elements $b'_3, \ldots, b'_m, b''_3, \ldots, b''_m \in R$ such that if we let $$\label{identifications of elements}
b'_1 = b_1, \ b''_1 = b_2, \ b''_2 = b'_2,$$ then $$\begin{aligned}
\label{the other sequences have the same type}
{\mathrm{tp}}_{\mathcal{R}}(b_1, b_2, \ldots, b_m) \ = \
{\mathrm{tp}}_{\mathcal{R}}(b'_1, b'_2, \ldots, b'_m) \ = \
{\mathrm{tp}}_{\mathcal{R}}(b''_1, b''_2, \ldots, b''_m). \end{aligned}$$
From (\[the permuted sequence has the same type\]) and (\[the other sequences have the same type\]) we get $$\begin{aligned}
\label{the same type when the other sequences are permuted}
{\mathrm{tp}}_{\mathcal{R}}(b_1, b_2, \ldots, b_m) \ &= \ {\mathrm{tp}}_{\mathcal{R}}(b_{\pi(1)}, b_{\pi(2)}, \ldots, b_{\pi(m)}), \\
{\mathrm{tp}}_{\mathcal{R}}(b'_1, b'_2, \ldots, b'_m) \ &= \ {\mathrm{tp}}_{\mathcal{R}}(b'_{\pi(1)}, b'_{\pi(2)}, \ldots, b'_{\pi(m)}) \quad \text{ and} \nonumber \\
{\mathrm{tp}}_{\mathcal{R}}(b''_1, b''_2, \ldots, b''_m) \ &= \ {\mathrm{tp}}_{\mathcal{R}}(b''_{\pi(1)}, b''_{\pi(2)}, \ldots, b''_{\pi(m)}), \nonumber\end{aligned}$$ Since ${\mathrm{tp}}_{\mathcal{R}}(b_1) = {\mathrm{tp}}_{\mathcal{R}}(b_2) = {\mathrm{tp}}_{\mathcal{R}}(b'_2)$ and ${\mathcal{R}}$ is a binary random structure it follows from (\[the same type when the other sequences are permuted\]) that there are $c_2, \ldots, c_m \in R \setminus \{b_1, \ldots, b_m, b'_1, \ldots, b'_m\}$ such that if $c_1 = b_1 (= b'_1)$, then $$\begin{aligned}
\label{equal types of some long sequences}
{\mathrm{tp}}_{\mathcal{R}}(b_1, \ldots, b_m, c_1, \ldots, c_m) \ &= \ {\mathrm{tp}}_{\mathcal{R}}(b_{\pi(1)}, \ldots, b_{\pi(m)}, b''_1, \ldots, b''_m)
\quad \text{ and} \\
{\mathrm{tp}}_{\mathcal{R}}(b'_1, \ldots, b'_m, c_1, \ldots, c_m) \ &= \ {\mathrm{tp}}_{\mathcal{R}}(b'_{\pi(1)}, \ldots, b'_{\pi(m)}, b''_{\pi(1)}, \ldots, b''_{\pi(m)}).
\nonumber\end{aligned}$$
Since ${\mathcal{C}}_1$ is a reduct of ${\mathcal{R}}$ we can replace ‘${\mathrm{tp}}_{\mathcal{R}}$’ with ‘${\mathrm{tp}}_{{\mathcal{C}}_1}$’ everywhere in (\[the other sequences have the same type\]), (\[the same type when the other sequences are permuted\]) and (\[equal types of some long sequences\]). Moreover, as ${\mathcal{C}}_1$ is a canonically embedded structure of ${\mathcal{M}}{^{\mathrm{eq}}}$, we can replace ‘${\mathrm{tp}}_{{\mathcal{C}}_1}$’ with ‘${\mathrm{tp}}$’ (which abbreviates ‘${\mathrm{tp}}_{{\mathcal{M}}{^{\mathrm{eq}}}}$’), so altogether we get $$\begin{aligned}
\label{type identifications in Meq}
&{\mathrm{tp}}(b_1, b_2, \ldots, b_m) \ = \
{\mathrm{tp}}(b'_1, b'_2, \ldots, b'_m) \ = \
{\mathrm{tp}}(b''_1, b''_2, \ldots, b''_m), \\
&{\mathrm{tp}}(b_1, b_2, \ldots, b_m) \ = \ {\mathrm{tp}}(b_{\pi(1)}, b_{\pi(2)}, \ldots, b_{\pi(m)}), \nonumber \\
&{\mathrm{tp}}(b'_1, b'_2, \ldots, b'_m) \ = \ {\mathrm{tp}}(b'_{\pi(1)}, b'_{\pi(2)}, \ldots, b'_{\pi(m)}), \nonumber \\
&{\mathrm{tp}}(b''_1, b''_2, \ldots, b''_m) \ = \ {\mathrm{tp}}(b''_{\pi(1)}, b''_{\pi(2)}, \ldots, b''_{\pi(m)}), \nonumber \\
&{\mathrm{tp}}(b_1, \ldots, b_m, c_1, \ldots, c_m) \ = \ {\mathrm{tp}}(b_{\pi(1)}, \ldots, b_{\pi(m)}, b''_1, \ldots, b''_m)
\quad \text{ and} \nonumber \\
&{\mathrm{tp}}(b'_1, \ldots, b'_m, c_1, \ldots, c_m) \ = \ {\mathrm{tp}}(b'_{\pi(1)}, \ldots, b'_{\pi(m)}, b''_{\pi(1)}, \ldots, b''_{\pi(m)}). \nonumber\end{aligned}$$
In particular, (\[type identifications in Meq\]) implies that ${\mathrm{tp}}(c_1, \ldots, c_m) = {\mathrm{tp}}(b''_{\pi(1)}, \ldots, b''_{\pi(m)}) = {\mathrm{tp}}(b''_1, \ldots, b''_m) = {\mathrm{tp}}(b'_1, \ldots, b'_m) = {\mathrm{tp}}(b_1, \ldots, b_m)$. So, using the $\omega$-homogeneity of ${\mathcal{M}}{^{\mathrm{eq}}}$, there is $a^* \in M$ such that $${\mathrm{tp}}(a^*, c_1, \ldots, c_m) \ = \ {\mathrm{tp}}(a, b_1, \ldots, b_m)$$ and hence ${\mathrm{crd}}(a^*) = \{c_1, \ldots, c_m\}$. For the same reason there is $a' \in M$ such that $${\mathrm{tp}}(a', b'_1, \ldots, b'_m) \ = \ {\mathrm{tp}}(a, b_1, \ldots, b_m)$$ and hence ${\mathrm{crd}}(a') = \{b'_1, \ldots, b'_m\}$. By the $\omega$-homogeneity of ${\mathcal{M}}{^{\mathrm{eq}}}$ again and (\[type identifications in Meq\]), there are $a_0, a'', a'_0, a''_0 \in M$ such that $$\begin{aligned}
\label{the elements corresponding to a}
{\mathrm{tp}}(a, b_1, \ldots, b_m, a^*, c_1, \ldots, c_m) \ &= \ {\mathrm{tp}}(a_0, b_{\pi(1)}, \ldots, b_{\pi(m)}, a'', b''_1, \ldots, b''_m) \ \ \text{and} \\
{\mathrm{tp}}(a', b'_1, \ldots, b'_m, a^*, c_1, \ldots, c_m) \ &= \ {\mathrm{tp}}(a'_0, b'_{\pi(1)}, \ldots, b'_{\pi(m)}, a''_0, b''_{\pi(1)}, \ldots, b''_{\pi(m)}). \nonumber\end{aligned}$$ Then ${\mathrm{crd}}(a_0) = \{b_1, \ldots, b_m\}$, ${\mathrm{crd}}(a'_0) = \{b'_1, \ldots, b'_m\}$ and ${\mathrm{crd}}(a'') = {\mathrm{crd}}(a''_0) = \{b''_1, \ldots, b''_m\}$. As the equivalence relation ${\mathrm{acl}}_{\mathcal{M}}(x) = {\mathrm{acl}}_{\mathcal{M}}(y)$ is assumed to be trivial it follows from Lemma \[every element is in dcl of its coordinates\] that $a_0 = a$, $a'_0 = a'$ and $a''_0 = a''$. Therefore (\[the elements corresponding to a\]) implies that $$\begin{aligned}
&{\mathrm{tp}}(a, a'') = {\mathrm{tp}}(a, a^*) \text{ and } {\mathrm{tp}}(a', a'') = {\mathrm{tp}}(a', a^*), \text{ and hence} \\
&{\mathrm{tp}}_{\mathcal{M}}(a, a'') = {\mathrm{tp}}_{\mathcal{M}}(a, a^*) \text{ and } {\mathrm{tp}}_{\mathcal{M}}(a', a'') = {\mathrm{tp}}_{\mathcal{M}}(a', a^*).\end{aligned}$$ As ${\mathcal{M}}$ is a homogeneous and binary we get $${\mathrm{tp}}_{\mathcal{M}}(a, a', a'') = {\mathrm{tp}}_{\mathcal{M}}(a, a', a^*)$$ and hence ${\mathrm{tp}}(a, a', a'') = {\mathrm{tp}}(a, a', a^*)$. Since (by definition) ${\mathrm{crd}}(a) = {\mathrm{acl}}(a) \cap C_1$ and similarly for $a', a''$ and $a^*$, there are permutations $\sigma_1, \sigma_2, \sigma_3$ of $\{1, \ldots, m\}$ such that $$\begin{aligned}
\label{the long sequences have the same type}
&{\mathrm{tp}}(b_1, \ldots, b_m, b'_1, \ldots, b'_m, b''_1, \ldots, b''_m) \\
= \
&{\mathrm{tp}}(b_{\sigma_1(1)}, \ldots, b_{\sigma_1(m)}, b'_{\sigma_2(1)}, \ldots, b'_{\sigma_2(m)}, c_{\sigma_3(1)}, \ldots, c_{\sigma_3(m)}).
\nonumber\end{aligned}$$ We have $$\{b_1, \ldots, b_m\} \cap \{b'_1, \ldots, b'_m\} \cap \{c_1, \ldots, c_m\} \ \neq \ {\emptyset}$$ because $b_1 = b'_1 = c_1$ belongs to this intersection, while $$\{b_1, \ldots, b_m\} \cap \{b'_1, \ldots, b'_m\} \cap \{b''_1, \ldots, b''_m\} \ = \ {\emptyset}$$ by the choice of these elements. This contradicts (\[the long sequences have the same type\]). $\square$
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[^1]: This structure has height 2 in the sense of [@Djo06]. It is of course a structure which is easy to understand. But examples with height 2 and the other properties of ${\mathcal{M}}$ can be more complicated.
|
---
abstract: 'In this paper we use Riesz spectral Theory and Gershgorin Theory to obtain explicit information concerning the spectrum of pseudo-differential operators defined on the unit circle $\mathbb{T} := \mathbb{R}/ 2 \pi \mathbb{ Z}$. For symbols in the Hörmander class $S^m_{1 , 0} ({\mathbb{T}}\times {\mathbb{Z}})$, we provide a sufficient and necessary condition to ensure that the corresponding pseudo-differential operator is a Riesz operator in $L^p (\mathbb{T})$, $1< p < \infty$, extending in this way compact operators characterisation in [@shahla] and Ghoberg’s lemma in [@Molahajloo2010] to $L^p (\mathbb{T})$. We provide an example of a non-compact Riesz pseudo-differential operator in $L^p ({\mathbb{T}})$, $1<p<2$. Also, for pseudo-differential operators with symbol satisfying some integrability condition, it is defined its associated matrix in terms of the Fourier coefficients of the symbol, and this matrix is used to give necessary and sufficient conditions for $L^2$-boundedness without assuming any regularity on the symbol, and to locate the spectrum of some operators.'
address: |
Department of Mathematics\
Universidad del Valle\
Cali\
Colombia
author:
- 'Juan Pablo Velasquez-Rodriguez'
bibliography:
- 'main.bib'
date: 'August 10, 2018'
nocite: '[@*]'
title: 'ON SOME SPECTRAL PROPERTIES OF PSEUDO-DIFFERENTIAL OPERATORS ON $\mathbb{T}$'
---
**INTRODUCTION** {#introduction .unnumbered}
================
Pseudo-differential operators acting on functions defined on smooth manifolds are an important generalisation of differential operators, and their study is a very active branch of contemporary mathematics, among other reasons, because of its applications in various areas of physics, such as black-hole physics, quantum electrodynamics and quantum field theory [@QFT; @Geometry]. Pseudo-differential operators considered when the manifold is a compact Lie group have arisen as in interesting arena from the point of view of Ruzhansky-Turunen Theory, in which the representations of the group allow to define a global symbol instead of symbols depending on charts.
The simplest example in this direction is the case of the one dimensional torus, on which we have the concept of periodic pseudo-differential operators acting on periodic functions. These operators have been widely studied [@JulioLpbounds; @Duvan1; @Duvan2; @MolahajloS1; @Ghaemi2; @Ghaemi3; @Ghaemi2017; @Pirhayati2011; @delgado_ruzhansky_2017] and remarkable results have been obtained, being one of the most important, perhaps the most, the global definition of pseudo-differential operators on the unit circle in [@Agranovich1979], proposed by Agranovich in 1979 crediting L.R. Volevich. The definition is illuminating and readily generalisable for any torus ${\mathbb{T}^d}$, but it is a non-trivial fact that it is equivalent to the original one given by Hörmander. However, time after Agranovich published a definition, McLean in [@Mclean] proved the equivalence for operators with symbols in the Hörmander $(\rho , \delta)$- classes, and since then other proofs have been published.
In this paper we will be working within the framework of the global symbols introduced by Ruzhansky and Turunen with extension and periodisation techniques [@ruzhanskytor2008] since their approach is useful and elegant and that is why their book [@ruzhansky1] will be one of our main references. This work is organised as follows:
- In Section 1 we recall some of the basics on periodic pseudo-differential operators from the point of view of Ruzhansky and Turunen Theory.
- In Section 2, we treat Riez operators in the sense of Pietsch, which are those operators acting on a Banach space with spectrum behaviour identical to that of compact operators. Inspired by Molahajloo in [@shahla], we give the main theorem of the section, a necessary and sufficient condition for a pseudo-differential operator with symbol in the Hörmander class $S^0_{1,0} (\mathbb{T} \times \mathbb{Z} )$ to be a Riesz operator in $L^p (\mathbb{T})$, $1< p < \infty$ (Theorem 2.4), together with an example of a non-compact Riesz pseudo-differential operator.
- In Section 3 the interest is focused in the study of global pseudo-differential operators acting on $L^2 ({\mathbb{T}})$. It is defined the infinite matrix associated to the operator, and it is used to study operators in $L^2 ({\mathbb{T}})$ factorising in terms of operators in $L^2 ({\mathbb{Z}})$, which are generally easier to deal with. As a result we obtain necessary and sufficient conditions for boundedness (Theorems 3.1 and 3.2) without assuming any regularity on the symbol, and sufficient conditions for invertibility, and spectrum localisation (Theorems 3.4 and 3.5).
- In Section 4 we discuss a proof of Ghoberg’s lemma in $L^p ({\mathbb{T}})$ as an extension of [@Molahajloo2010 Theorem 3.3].
**Periodic Pseudo-differential Operators.**
=============================================
In this section we recall the definition of periodic global pseudo-differential operator, the periodic version of Hörmander classes and the basics for the corresponding pseudo-differential calculus in the sense of Ruzhansky-Turunen theory [@ruzhansky1]. We remark that it is possible to consider global pseudo-differential operators associated to full symbols which are $p$-integrable in the toroidal variable.
As usual for an integrable function $f: {\mathbb{T}}\to {\mathbb{C}}$ we denote its $k$-th Fourier coefficient by $\widehat{f} (k)$, which is defined by $$\widehat{f} (k) : = \int_{\mathbb{T}}f(x) e^{-i x \cdot k} dx,$$where $k \in {\mathbb{Z}}$, and $dx$ denotes the normalized Haar measure on ${\mathbb{T}}$. Let $\mathcal{P}({\mathbb{T}}):= Span\{e^{i x \cdot k} {\text{ }}: {\text{ }}k \in {\mathbb{Z}}\}$ denote the collection of trigonometric polynomials on the unit circle. Consider a given linear operator $A: \mathcal{P} ({\mathbb{T}}) \to L^p ({\mathbb{T}})$ for $1< p < \infty $. Following [@ruzhansky1 Theorem 10.4.6] , we associate to the linear operator $A$ the [*full symbol* ]{} $\sigma_A: {\mathbb{T}}\times {\mathbb{Z}}\to \mathbb{C}$, defined by $$\sigma_A (x , k) := e^{- i x \cdot k} (A e_k ) (x),$$ where $e_k (x):= e^{i x \cdot k}$. This full symbol satisfies that $$\begin{aligned}
(A_\sigma f)(x) = \sum_{k \in {\mathbb{Z}}} \sigma_A (x , k) \widehat{f} (k) e^{i x \cdot k},\end{aligned}$$for all $f \in \mathcal{P}({\mathbb{T}})$. On the other hand, given a measurable function $\sigma : {\mathbb{T}}\times {\mathbb{Z}}\to {\mathbb{C}}$ we can define using the equality above a linear operator acting, initially, on some subspace of $L^1({\mathbb{T}}).$ Our objective is to find suitable conditions on the function $\sigma$ in order to ensure that the domain of the corresponding pseudo-differential operator $A_\sigma$ can be extended to $L^p({\mathbb{T}})$. Let $m \in {\mathbb{R}}$, and $0 \leq \rho , \delta \leq 1$ given. The periodic Hörmander class $S^m_{\rho , \delta} ({\mathbb{T}}\times {\mathbb{Z}})$ of symbols $\sigma \in C^\infty ({\mathbb{T}}\times {\mathbb{Z}})$ is the set of functions satisfying the estimates $$|\Delta_{k}^{t} D_{x}^{r} \sigma (x, k)| \leq C_{t,r,\rho,\delta,m} \langle k \rangle^{m-\rho t + \delta r},$$for each $t,r \in {\mathbb{N}}$, $k \in {\mathbb{Z}}$, $x \in {\mathbb{T}}$, $C_{t,r,\rho,\delta,m} \in {\mathbb{R}}^+ $, and $\langle k \rangle := (1+k^2)^{1/2}$. Here the difference operator $\Delta_k^t \varphi$ acts on functions $\varphi: {\mathbb{Z}}\to {\mathbb{C}}$ as $$\Delta_k^t \varphi (k) := \sum_{h=0}^{t} (-1)^{t-h} { {t}\choose{h} }\varphi (k + h),$$and for functions $f\in C^r ({\mathbb{T}})$ the operator $D_x^r f$ is defined by $$D_x^r f (x) := (-i)^r \partial_x^r f (x).$$ Considering symbols in Hörmander classes allow us to treat with a graded $C^{*}$-algebra of pseudo-differential operators as a consequence of the pseudo-differential calculus summarised in [@ruzhansky1] .
We introduce the object of study in this paper, the periodic pseudo-differential operators with $p-$integrable symbols for $1<p< \infty$ fixed. This definition comes from the fact that, in order to obtain a well defined operator from $\mathcal{P}({\mathbb{T}})$ with the $L^p$-norm to $L^p ({\mathbb{T}})$, the linear operator defined by formula
$$(A_\sigma f)(x) = \sum_{k \in {\mathbb{Z}}} \sigma_A (x , k) \widehat{f} (k) e^{i x \cdot k},$$ should have symbol $\sigma_A (x , k)$ in $L^p ({\mathbb{T}})$ for every fixed $k$.
Let $1<p<\infty$ be given. Consider $\sigma : {\mathbb{T}}\times {\mathbb{Z}}\to {\mathbb{C}}$ be a measurable function such that $\sigma (\cdot , k) \in L^p ({\mathbb{T}})$ for each $k \in {\mathbb{Z}}$. The [*associated periodic pseudo-differential operator*]{} $T_\sigma: \mathcal{P} ({\mathbb{T}}) \to L^p ({\mathbb{T}})$ is defined by $$(T_\sigma f) (x) = \sum_{k \in {\mathbb{Z}}} \sigma (x , k) \widehat{f} (k) e^{i x \cdot k}.$$
In particular, if $\sigma (x,k):= \sigma(k)$ is a function depending only on $k$, the pseudo-differential operator $T_\sigma$ is called a Fourier multiplier.
Recall that for symbols in Hörmander classes, it is possible to provide a symbolic calculus to periodic pseudo-differential operators. This is stated formally in the following propositions [@ruzhansky1].
Let $T_\alpha$, $T_\beta$ be periodic pseudo-differential operators with symbols $\alpha \in S^m_{\rho , \delta} ({\mathbb{T}}\times {\mathbb{Z}}), \beta \in S^l_{\rho , \delta} ({\mathbb{T}}\times {\mathbb{Z}})$. Then $T_\alpha T_\beta$ is a pseudo-differential operator with symbol $ \sigma $ in the Hörmander class $ S^{m+l}_{\rho , \delta} ({\mathbb{T}}\times {\mathbb{Z}})$ and the toroidal symbol $\sigma$ has the following asymptotic expansion: $$\sigma (x , k ) \sim \sum_{h} \frac{1}{h !} \Delta_{k}^{h} \alpha (x, k) D_{x}^{h} \beta (x ,k) ,$$where the above means that for each $N \in {\mathbb{N}}$ we have
$$\sigma (x , k ) - \sum_{h< N} \frac{1}{h !} \Delta_{k}^{h} \alpha (x, k) D_{x}^{h} \beta (x ,k) \in S^{m-N}_{\rho,p \delta} ({\mathbb{T}}\times {\mathbb{Z}}).$$
Let $T_\sigma$ be a pseudo-differential operator with symbol $\sigma \in S^m_{\rho , \delta} ({\mathbb{T}}\times {\mathbb{Z}})$. Then the adjoint operator $(T_\sigma)^*$ is a pseudo-differential operator with symbol $\sigma^* \in S^m_{\rho , \delta} ({\mathbb{T}}\times {\mathbb{Z}})$ and it has the following asymptotic expansion
$$\sigma^* \sim \sum_h \frac{1}{h!} \Delta_k^h \partial_x^h \overline{\sigma (x,k)}.$$
**Riesz Pseudo-differential Operators**
========================================
In this section we extend the main result of [@shahla] on the compactness of periodic pseudo-differential operators. We will provide an alternative proof of main Theorem in that paper, which characterises the compact pseudo-differential operators on $L^2({\mathbb{T}})$, using different arguments not requiring the existence of an inner product, implying a result, the main result of this section, for the $L^p ({\mathbb{T}})$ case, where $1<p<\infty$. For this purpose we review the already known Gohberg’s Lemma, wich we extend to $L^p ({\mathbb{T}})$ in Section 4. In what follows, $\mathcal{L} (E,F)$ will denote the collection of all bounded linear operators from $E$ to $F$ normed spaces, and $\mathfrak{K} (E,F)$ denotes the collection of compact operators in $\mathcal{L} (E,F)$. For $E=F$ we write $\mathcal{L}(E)$ instead of $\mathcal{L} (E,E)$. For $T \in \mathcal{L}(E)$ the resolvent set of $T$ will be denoted by $$Res(T):\{\lambda\in \mathbb{C}:= (T-\lambda I)^{-1}\in \mathcal{L}(E) \},$$ and the spectrum by $Spec(T):=\mathbb{C}\setminus Res(T)$.
Let $T_\sigma$ be a pseudo-differential operator with symbol $\sigma \in S^0_{1,0} ({\mathbb{T}}\times {\mathbb{Z}})$, and take $1<p<\infty$ fixed. Then ${\left\lVertT_\sigma - K\right\rVert}_{\mathcal{L} (L^p ({\mathbb{T}}))} \geq d_\sigma $ for any compact operator $K \in \mathfrak{K}(L^p({\mathbb{T}}))$, where $$\begin{aligned}
d_\sigma:= \limsup_{|k| \to \infty} \{\sup_{x \in {\mathbb{T}}} |\sigma (x , k)|\}.\end{aligned}$$
A proof of the theorem above can be found in [@Molahajloo2010], where the authors study the case $p=2$. We assert that the proof also works if the $L^2$-norm is replaced by $L^p$-norm in each step. This statement will be proved in Section 4.
\[Molah\] Let $T_\sigma$ be a pseudo-differential operator with symbol $\sigma \in S^0_{1,0} ({\mathbb{T}}\times {\mathbb{Z}})$. Then $T_\sigma$ is a compact operator on $L^2 ({\mathbb{T}})$ if and only if
$$\begin{aligned}
d_\sigma:&= \limsup_{|k| \to \infty} \{ \sup_{x \in {\mathbb{T}}} |\sigma (x , k)|\} =\lim_{|k| \to \infty} \{ \sup_{x \in {\mathbb{T}}} |\sigma (x , k)| \} = 0.\end{aligned}$$
Assume that $d_\sigma =0$ and let $f \in C^2 ({\mathbb{T}})$. For all $x \in {\mathbb{T}}$
$$\begin{aligned}
(T_\sigma f) (x) &= \sum_{k \in {\mathbb{Z}}} \sigma(x , k) \widehat{f}(k) e^{i x \cdot k} \\ &= \sum_{k \in {\mathbb{Z}}} \Big( \sum_{m \in {\mathbb{Z}}} \widehat{\sigma} (m , k) e^{i x \cdot m}\Big) \widehat{f} (k) e^{i x \cdot k} \\
&= \sum_{m \in {\mathbb{Z}}} e^{i x \cdot m} \Big( \sum_{k \in {\mathbb{Z}}} \widehat{\sigma} (m , k) \widehat{f} (k) e^{i x \cdot k} \Big) \\
&= \sum_{m \in {\mathbb{Z}}} e^{i x \cdot m} (T_{\widehat{\sigma}_m} f) (x),\end{aligned}$$
where $\widehat{\sigma}_m (k) := \widehat{\sigma}(m,k)$ and the change in the order of summation is justified by Fubini–Tonelli’s theorem since
$$\begin{aligned}
\sum_{k \in {\mathbb{Z}}} \sum_{m \in {\mathbb{Z}}} |\widehat{\sigma} (m , k)| |\widehat{f}(k)| &= \sum_{k \in {\mathbb{Z}}} {\left\lVert\widehat{\sigma}(\cdot , k)\right\rVert}_{L^1 ({\mathbb{Z}})} \cdot |\widehat{f}(k) |\\
& \leq C_{0,2,0,0,0} \sum_{m \in {\mathbb{Z}}} \langle m \rangle^{-2} \cdot \sum_{k \in {\mathbb{Z}}} |\widehat{f}(k) | < \infty .\end{aligned}$$
By defining the operator $(A_m f) (x) := e^{i x \cdot m} f (x)$, a multiplication operator, we have
$$(T_\sigma f) (x) = \sum_{m \in {\mathbb{Z}}} (A_m T_{\widehat{\sigma}_m} f) (x),$$ and clearly $A_m \in \mathcal{L} (L^2({\mathbb{T}}))$. Now, for each $m \in {\mathbb{Z}}$, the operator $T_{\widehat{\sigma}_m}$ is a Fourier multiplier. As it is well known, a pseudo-differential operator with symbol depending only on the Fourier variable is a Fourier multiplier which is a compact operator in $L^2 ({\mathbb{T}})$ if and only if its symbol $\eta$ satisfy
$$\lim_{|k| \to \infty} |\eta (k)|=0.$$
Now, for each $m \in {\mathbb{Z}}$ we have that $$\lim_{|k| \to \infty} |\widehat{\sigma} (m,k)| \leq \lim_{|k| \to \infty} \sup_{x \in {\mathbb{T}}} |\sigma (x , k)| \leq \limsup_{|k| \to \infty} \sup_{x \in {\mathbb{T}}} |\sigma (x,k)|= 0.$$
This implies that each operator $T_{\widehat{\sigma}_m}$ is a compact operator. As a consequence each $A_m T_{\widehat{\sigma}_m}$ is compact and for all $N \in {\mathbb{N}}$ , the operator $$\displaystyle \sum_{|m| \leq N} A_m T_{\widehat{\sigma}_m},$$ is also compact since the set of compact operators $\mathfrak{K} (L^2 ({\mathbb{T}}))$ form a two sided ideal in $\mathcal{L} (L^2 ({\mathbb{T}}))$ (see [@vitali Proposition 4.3.4]) and this ideal of compact operators is a closed subset of $\mathcal{L} (L^2({\mathbb{T}}))$ in the operator norm topology. For this reason, if the series
$$\begin{aligned}
\sum_{m \in {\mathbb{Z}}} A_m T_{\widehat{\sigma}_m},\end{aligned}$$
converge in the operator norm topology then
$$\begin{aligned}
T_\sigma = \lim_{N \to \infty} \sum_{|k| \leq N} A_m T_{\widehat{\sigma}_m},\end{aligned}$$
is compact as it is the limit of a sequence of compact operators.
This argument is the most important part of the proof because it shows that it is not necessary to appeal to inner product or Calkin’s algebra to characterise compact operators. Moreover, it exposes, similar to Schatten classes [@schaten1DELGADO2014779; @schaten2], a relationship between operator ideals, approximation properties and pseudo-differential operators. In fact Theorem 2.1, seen from the perspective of operator ideals theory, is a characterisation of belonging to a certain closed ideal in terms of the symbol. This idea will be exploited later.
To conclude our proof, it is well known (See [@Molahajloo2010 Proposition 2.4]) that if $\sigma \in S^0_{1,0} ({\mathbb{T}}\times {\mathbb{Z}})$ then
$$\sum_{m \in {\mathbb{Z}}} {\left\lVertA_m T_{\widehat{\sigma}_m}\right\rVert}_{\mathcal{L}(L^2({\mathbb{T}}))} = \sum_{m \in {\mathbb{Z}}} {\left\lVertT_{\widehat{\sigma}_m}\right\rVert}_{\mathcal{L}(L^2({\mathbb{T}}))} < \infty.$$
In summary, $T_\sigma$ is a compact operator. Now, assume that $d_\sigma \neq 0$. We need only to show that $T_\sigma$ is not compact on $L^2 ({\mathbb{T}})$. Suppose that $T_\sigma$ is compact. If we set $T_\sigma = K$ in Theorem 2.1 then it contradicts our assumption that $d_\sigma \neq 0$.
Riesz Periodic Fourier Multipliers
----------------------------------
The previous proof of Theorem \[Molah\] essentially uses two facts: First, Gohber’s lemma which allow us to proof one implication. Second, the set of compact operators in $\mathcal{L} (L^2({\mathbb{T}}))$ is a closed operator ideal. In the case $L^p ({\mathbb{T}})$ $1<p<\infty$ we will use the same arguments as in Theorem \[Molah\] but, instead of the ideal of compact operators, we will use its radical.
We want to recall some basic definitions for the reader.
Let $\mathcal{L}$ be the collection of all bounded operators defined on Banach spaces. That is
$$\mathcal{L} := \bigcup_{E,F \text{ Banach Spaces}} \mathcal{L} (E,F).$$
An operator ideal is a sub-collection $\mathfrak{I}$ of $\mathcal{L}$ such that each one of its components
$$\mathfrak{I} (E,F) := \mathfrak{I} \cap \mathcal{L},$$ satisfy the following conditions:
1. For each one-dimensional Banach space $E$ the identity map $I_E$ belongs to $\mathfrak{I}(E)$.
2. If $T_1 , T_2 \in \mathfrak{I} (E,F)$ then $T_1 + T_2 \in \mathfrak{I}(E,F)$.
3. If $T \in \mathfrak{I}(E,F)$ and $X\in \mathcal{L}(F,F_0) , Y \in \mathcal{L} (E_0 , E)$ then their composition $X T Y \in \mathfrak{I} (E_0 , F_0)$, where $E_0 $ and $F_0$ are Banach spaces.
If additionally each component $\mathfrak{I} (E,F)$ is closed in the operator norm topology of $\mathcal{L} (E,F)$ then it is said that $\mathfrak{I}$ is a closed operator ideal.
Given a operator ideal $\mathfrak{I}$ it is said that $S \in \mathcal{L} (E,F)$ belongs to the radical of $\mathfrak{I}$ (see [@pietsch2 4.3.1]) if for all $L \in \mathcal{L} (F,E)$ there exist $U \in \mathcal{L} (E)$ and $X \in \mathfrak{I} (E)$ such that $$U(I_E - LS) = I_E -X.$$ The collection of operators that satisfy this condition is denoted by $\mathfrak{I}^{rad}$. It can be proved that $\mathfrak{I}^{rad}$ is a closed operator ideal so, to take advantage of this fact, we will work from now on with bounded operators in the radical of an special ideal. The radical of compact operators ideal is called the ideal of Gohberg or inessential operators, and is completely characterised by [@pietsch2 Theorem 26.7.2].
$\mathfrak{R} := \mathfrak{K}^{rad}$ is the largest operator ideal such that all components $\mathfrak{R} (E)$ consists of Riesz operators only.
Riesz operators are a generalisation of compact operators that has been widely studied ([@calkin; @pietsch; @West; @Ruston] and for that reason there are many equivalent definitions. One can use, for example, the definition given in [@pietsch].
Let $E$ be a Banach space. It is said that $T \in \mathcal{L} (E)$ is a [*Riesz operator*]{} if for all $\varepsilon > 0$ exist an exponent $s$ and points $v_1 ,..., v_h \in E$ which depend on $\varepsilon$ such that
$$T^s (B_E) \subseteq \bigcup_{j = 1}^{h} v_j + \varepsilon^s B_E,$$ where $B_E$ is the unit ball of $E$.
This definition does not provides too much information at first sight, but fortunately one can think of Riesz operators in a more intuitive way. In simple terms a bounded linear operator is a Riesz operator if its spectrum is countable, it contains only eigenvalues, each one of these has finite-dimensional associated eigen-space, and the number zero is the only possible accumulation point. In other words, Riesz operators are those bounded linear operators whose spectrum behaves like the spectrum of compact operators. They are generally easier to study than the compact operators because of their different characterisations. Several necessary and sufficient conditions to be a Riesz operator are listed in the following lemma, which is the collection of Propositions 3.2.13 and 3.2.24 in [@pietsch], 26.5.1 in [@pietsch2], and the corollary of Theorem 3.4.3 in [@calkin].
Let $E$ be a Banach space and suppose $T \in \mathcal{L} (E)$. The following statements are equivalent
1. $T$ is a Riesz operator.
2. $T^m$ is Riesz for some (for all) exponent $m$.
3. $\lambda T$ is iterative compact for all $\lambda \in {\mathbb{C}}$.
4. Non-zero points in $Spec(T)$ are isolated eigenvalues with finite-dimensional associated eigenspace and zero as the only cluster point. If $E$ is infinite dimensional then $0 \in Spec (T)$.
5. $T- \lambda I$ is a Fredholm operator for all $\lambda \in {\mathbb{C}}$.
In general, Riesz operators do not form an operator ideal. However, what we want to show now is that in certain subalgebras of $\mathcal{L} (L^p({\mathbb{T}}))$, $1<p<\infty$, there are ideals that only contain Riesz operators, and this will be a crucial fact in this work. The following two propositions are the first step in that direction.
Let $\sigma : {\mathbb{Z}}\to {\mathbb{C}}$ be a function such that $$\begin{aligned}
|\Delta_k^t \sigma (k)| \leq C \langle k \rangle^{-t} {\text{ }}, {\text{ }}k\in {\mathbb{Z}}\end{aligned}$$ for $0 \leq t \leq [1/2] + 1$. Then for $1<p< \infty$ the spectrum of its associated Fourier multiplier $T_\sigma$ as an operator in $\mathcal{L} (L^p ({\mathbb{T}}))$ is: $$Spec(T_\sigma)=\overline{ \{\sigma (k) {\text{ }}: {\text{ }}k \in {\mathbb{Z}}\}}$$
First, the condition $(1)$ guarantees that $T_\sigma$ satisfy the conditions of the toroidal version of Mikhlin multiplier Theorem (see [@wongdfa], Lemma 22.12) then $T_\sigma \in \mathcal{L} (L^p({\mathbb{T}}))$. Define $$\sigma_\lambda (k):= \sigma(k) - \lambda {\text{ }}{\text{ }}\text{and} {\text{ }}{\text{ }}\sigma_\lambda^{-1} (k) := \frac{1}{\sigma_\lambda (k)}$$for $\lambda \in {\mathbb{C}}\setminus \overline{ \{\sigma (k) {\text{ }}: {\text{ }}k \in {\mathbb{Z}}\}} $. Since ${\mathbb{C}}$ is a regular topological space, there exist a $d>0$ such that
$$\begin{aligned}
\inf_{k \in {\mathbb{Z}}} |\sigma(k) - \lambda| = d.\end{aligned}$$
Then $T_{\sigma_\lambda^{-1}}$ also satisfy the conditions of Mikhlin multiplier theorem because
$$\begin{aligned}
| \Delta_k \sigma_\lambda^{-1} (k) |&= \Big| \frac{1}{\sigma_\lambda (k+1)} - \frac{1}{\sigma_\lambda (k)}\Big| \\ &= \Big| \frac{\sigma (k+1) - \sigma (k)}{\sigma_\lambda (k+1) \sigma_\lambda (k)} \Big| \leq \frac{C}{d^2} \langle k \rangle^{-1}.
\end{aligned}$$
Finally, it is clear that $$(T_\sigma - \lambda I) (T_{\sigma_\lambda^{-1}}) = (T_{\sigma_\lambda}) (T_{\sigma_\lambda^{-1}}) =I,$$which proves that $Spec(T) \subseteq \overline{ \{\sigma (k) {\text{ }}: {\text{ }}k \in {\mathbb{Z}}\}}$. Now its easy to see that $\overline{ \{\sigma (k) {\text{ }}: {\text{ }}k \in {\mathbb{Z}}\}} \subseteq Spec(T)$ because the spectrum is closed and each number $\sigma (k)$ is an eigenvalue of $T_\sigma$ with associated eigenvector $e^{i x \cdot k}$, thus $$\{\sigma (k) {\text{ }}: {\text{ }}k \in {\mathbb{Z}}\} \subseteq Spec(T).$$
As an straightforward consequence we obtain the following.
Let $T_\sigma : L^p ({\mathbb{T}}) \to L^p ({\mathbb{T}})$ $1<p<\infty$ be a Fourier multiplier whose symbol satisfy the condition $(1)$. Then $T_\sigma$ is a Riesz operator if and only if
$$\lim_{|k| \to \infty} |\sigma (k)| = 0.$$
By the previous proposition and $(iv)$ in Lemma 2.1.
1. The collection of Fourier multipliers $$M_p := \{T_\sigma: L^p ({\mathbb{T}}) \to L^p ({\mathbb{T}}) {\text{ }}: {\text{ }}\sigma {\text{ }}\text{ satisfy (1) } {\text{ }}\},$$ is a sub-algebra of $\mathcal{L}(L^p({\mathbb{T}}))$, $1 < p<\infty$. This is a consequence of discrete Leibniz rule.
2. If $\mathcal{A}$ is an algebra and $\mathcal{B}$ is a sub-algebra of $\mathcal{A}$ then each ideal $\mathcal{I}$ of $\mathcal{A}$ defines an ideal of $\mathcal{B}$, this is $\mathcal{I} \cap \mathcal{B}$.
3. The component $\mathfrak{R} (L^p({\mathbb{T}}))$ is the largest ideal in $\mathcal{L} (L^p ({\mathbb{T}}))$ which consists only of Riesz operators. As an immediate consequence $\mathfrak{R} (L^p({\mathbb{T}})) \cap M_p$ is the largest ideal on $M_p$ that consists only of Riesz operators.
4. The set $$\mathfrak{R} (M_p):= \{ T_\sigma \in M_p {\text{ }}: {\text{ }}\lim_{|k| \to \infty} |\sigma (k)| = 0 \},$$ is an ideal of $M_p$, which, in virtue of Proposition 2.2, contains only Riesz operators, so $\mathfrak{R} (M_p) \subseteq \mathfrak{R} \cap M_p$.
5. Again, by Proposition 2.2, the symbol of every Fourier multiplier $T_\sigma \in \mathfrak{R} \cap M_p$ satisfy $\lim_{|k| \to \infty} |\sigma (k)| = 0$, so $\mathfrak{R} \cap M_p \subseteq \mathfrak{R} (M_p)$ and in conclusion $\mathfrak{R} \cap M_p = \mathfrak{R} (M_p)$, that is, all Riesz Fourier multipliers are inessential operators.
Riesz Periodic Pseudo-differential Operators
--------------------------------------------
We have constructed enough tools to prove the main theorem of this section. We will use the same techniques as in Theorem 2.2 but instead of compact operators we will use inessential operators.
Let $T_\sigma$ be a pseudo-differential operator with symbol $\sigma \in S^0_{1,0} ({\mathbb{T}}\times {\mathbb{Z}})$ and $1<p<\infty$. Then $T_\sigma$ is a Riesz operator on $L^p ({\mathbb{T}})$ if and only if $$d_\sigma' := \lim_{|k| \to \infty} \{ \sup_{x \in {\mathbb{T}}} |\sigma (x,k)| \} =0.$$
We assert that a pseudo-differential operator with symbol in $S^0_{1,0} ({\mathbb{T}}\times {\mathbb{Z}})$ is a Riesz operator if and only if $d_{\sigma^n}' = 0$ for some $n \in {\mathbb{N}}$, where $\sigma^n$ denotes the symbol of $(T_\sigma)^n$. Then to finish we will show that $d_{\sigma^n}' = 0$ if and only if $d_\sigma' = 0$.
First, suppose that $T_\sigma$ is a Riesz operator and $d_{\sigma^n}' > 0$ for all $n$. Using $(iii)$ in Lemma 2.1 we know that, for some $h \in {\mathbb{N}}$, $(T_\sigma)^h$ is a compact operator. Then, by Gohberg’s lemma, we have $$0 < d_{\sigma^h}' \leq d_{\sigma^h} \leq {\left\lVert(T_\sigma)^h - K\right\rVert}_{\mathcal{L} (L^p ({\mathbb{T}}))},$$ for all compact operators $ K $. If we set $K = (T_\sigma)^h$, the above chain of inequalities contradicts our assumption that $d_{\sigma^h}' \neq 0$.
Now suppose $d_{\sigma^n}' = 0$ for some $n \in {\mathbb{N}}$. Then for $f \in C^2 ({\mathbb{T}})$
$$\begin{aligned}
(T_{\sigma^n} f) (x) &= \sum_{k \in {\mathbb{Z}}} {\sigma^n (x , k)} \widehat{f} (k) e^{ix \cdot k} \\
&= \sum_{k \in {\mathbb{Z}}} \Big( \sum_{m \in {\mathbb{Z}}} {\widehat{\sigma}^n}(m ,k) e^{i x \cdot m} \Big) \widehat{f} (k) e^{ix \cdot k} \\
&=\sum_{m \in {\mathbb{Z}}} (A_m T_{\widehat{\sigma}_m^n} f) (x),\end{aligned}$$
and $$\lim_{|k| \to \infty} |\widehat{\sigma}^n (m,k)| \leq \lim_{|k| \to \infty} \sup_{x \in {\mathbb{T}}} |\sigma^n (x , k)|= 0,$$ so, for each $m \in {\mathbb{Z}}$, $T_{\sigma^n_m}$ is an operator in the ideal $\mathfrak{R} (M_p) $ which is a subset of the closed ideal $\mathfrak{R} (L^p ({\mathbb{T}}))$ of inessential operators. As a consequence each operator $A_m T_{\widehat{\sigma}_m^n}$ belongs to $\mathfrak{R} (L^p ({\mathbb{T}}))$ and for all $N \in {\mathbb{N}}$ the operator $$\sum_{|m| \leq N} A_m T_{\widehat{\sigma}_m^n},$$ belongs to $\mathfrak{R} (L^p ({\mathbb{T}}))$ as well. As already said, the series $$\sum_{m \in {\mathbb{Z}}} A_m T_{\widehat{\sigma}_m^n},$$ converge in the operator norm topology. In summary $T_{\sigma^n}$ belongs to $\mathfrak{R} (L^p ({\mathbb{T}}))$ as the limit of a sequence of inessential operators $$T_{\sigma^n} = \lim_{N \to \infty} \sum_{|m| \leq N} A_m T_{\widehat{\sigma}_m^n}.$$ Finally, let us see that $d_{\sigma^n}' = 0$ if and only if $d_\sigma'=0$. This is a simple consequence of toroidal composition formula. As we know, by toroidal composition formula, taking the first two terms in the asymptotic expansion of $T_{\sigma^n} =T_\sigma T_{\sigma^{n-1}}$ we have $$\sigma^{n} (x , k) = (\sigma(x, k))(\sigma^{n-1} (x , k)) + r_{n} (x, k),$$with $r_{n} (x, k) \in S^{-1}_{1,0} ({\mathbb{T}}\times {\mathbb{Z}})$. Repeating this process we obtain
$$\begin{aligned}
\sigma^{n} (x , k) &= (\sigma(x, k ))^2 (\sigma^{n-2} (x , k)) + (\sigma (x , k))(r_{n-1} (x , k)) + r_{n} (x , k) \\ &= ... \\ &= (\sigma(x , k))^n + \sum_{h=0 }^{n-1} (\sigma (x, k))^h (r_{n-h} (x, k)),\end{aligned}$$
with each $r_j (x, k) \in S^{-1}_{1,0} ({\mathbb{T}}\times {\mathbb{Z}})$. From this we have $$(\sigma(x , k))^n = \sigma^{n} (x , k) - \sum_{h=0 }^{n-1} (\sigma (x, k))^h (r_{n-h} (x, k)),$$which implies $$|\sigma(x , k)|^n \leq |\sigma^{n} (x , k)| + \sum_{h=0 }^{n-1} |\sigma (x, k)|^h |r_{n-h} (x, k)|,$$ and $$\sup_{x \in {\mathbb{T}}} |\sigma(x , k)|^n \leq \sup_{x \in {\mathbb{T}}} |\sigma^{n} (x , k)| + \sum_{k=0 }^{n-1} C_{\sigma}^h \sup_{x \in {\mathbb{T}}} |r_{n-k} (x, k)|,$$
where $C_\sigma$ is a constant that bounds uniformly $|\sigma (x , k)|$ and whose existence is clear because $\sigma \in S^{0}_{1,0} ({\mathbb{T}}\times {\mathbb{Z}})$. In conclusion $$\lim_{|k| \to \infty} \sup_{x \in {\mathbb{T}}} |\sigma(x , k)|^n \leq \lim_{|k| \to \infty} \sup_{x \in {\mathbb{T}}} |\sigma^{n} (x , k)| + \sum_{h=0 }^{n-1} \lim_{|k| \to \infty} C_{\sigma}^h \sup_{x \in {\mathbb{T}}} |r_{n-h} (x, k)|.$$
This finish the proof.
1. The collection of pseudo-differential operators $$\Psi_p^m := \{T_\sigma : L^p ({\mathbb{T}}) \to L^p ({\mathbb{T}}) {\text{ }}: {\text{ }}\sigma \in S^m_{1,0} ({\mathbb{T}}\times {\mathbb{Z}}) \},$$ is a sub-algebra of $\mathcal{L}(L^p ({\mathbb{T}}))$, this as a consequence of the toroidal composition formula.
2. The component $\mathfrak{R} (L^p({\mathbb{T}}))$ is the largest ideal in the algebra $\mathcal{L}(L^p ({\mathbb{T}}))$ consisting only of Riesz operators. As an immediate consequence $ \mathfrak{R} \cap \Psi_p^m$ is the largest ideal of $\Psi_p^m$ consisting only of Riesz operators
3. The set $$\text{ } \text{ } \text{ } \text{ } {\text{ }}{\text{ }}\text{ } \mathfrak{R} (\Psi_p^m) := \{T_\sigma : L^p({\mathbb{T}}) \to L^p({\mathbb{T}}) {\text{ }}: {\text{ }}\lim_{|k| \to \infty} \{ \sup_{x \in {\mathbb{T}}} |\sigma (x , k)| \} = 0 \},$$ is an ideal of $\Psi_p^m$ that, by the previous theorem , consist only of Riesz operators. Then $\mathfrak{R} (\Psi_p^m) \subseteq \mathfrak{R} \cap \Psi_p^m$.
4. Again, by the previous theorem, the symbol of any linear operator $T_\sigma \in \mathfrak{R} \cap \Psi_p^m$ satisfy $$\lim_{|k| \to \infty} \{ \sup_{x \in {\mathbb{T}}} |\sigma (x , k)| \} = 0.$$
In conclusion $\mathfrak{R} (\Psi_p^m) = \mathfrak{R} \cap \Psi_p^m$.
We finish this section with a remark about compact operators on $L^p ({\mathbb{T}})$. The statements of Proposition 2.2 and Theorem 2.4 could be more general. With sufficient conditions on the symbol for the boundedness of Fourier multipliers or pseudo-differential operators, let us call this conditions $(BF)$ and $(BP)$ respectively, and similar proofs as in Proposition 2.2 and Theorem 2.4, one can state:
Let $T_\sigma : L^p ({\mathbb{T}}) \to L^p ({\mathbb{T}})$ $1<p<\infty$ be a Fourier multiplier whose symbol $\sigma$ satisfy the condition $(BF)$. Then $T_\sigma$ is a Riesz operator if and only if $$\lim_{|k| \to \infty} |\sigma (k)| = 0.$$ Let $T_\sigma:L^p ({\mathbb{T}}) \to L^p ({\mathbb{T}})$ be a pseudo-differential operator whose symbol $\sigma$ satisfy the condition $(BP)$. Then $T_\sigma$ is a Riesz operator in $L^p ({\mathbb{T}})$ if and only if $$d_\sigma' := \lim_{|k| \to \infty} \{ \sup_{x \in {\mathbb{T}}} |\sigma (x,k)| \} =0.$$
Unfortunately conditions $(BF)$ and $(BP)$ are not easy to find, but we emphasise this more general form of Proposition 2.2 and Theorem 2.4 to give an example of a Riesz non-compact pseudo-differential operators, showing then that the problem of find necessary an sufficient conditions on the symbol for the compactness of the operator in $L^p ({\mathbb{T}})$ is still open. For this, we will exhibit a bounded pseudo-differential operator whose symbol tends to zero uniformly, and thus is a Riesz operator, but not a compact operator on $L^p ({\mathbb{T}})$.
As it is shown in [@ex], there exist strictly singular operators on $L^p [0,1]$, and then on $L^p ({\mathbb{T}})$, which are non-compact operators. Strictly singular operators are bounded linear operators between normed spaces which are not bounded below on any infinite-dimensional subspace, and the class of strictly singular operators is contained in the class of Riesz operators on any Banach space. For $L^p ({\mathbb{T}})$, $1 <p<2$, one can construct an example of a strictly singular non-compact operator through the following process: consider a complemented subspace $F_p$ of $L^p ({\mathbb{T}})$ isomorphic to $\ell^p({\mathbb{N}})$, generated by disjointly supported functions, and denote by $P_p$ a projection from $L^p ({\mathbb{T}})$ onto $F_p$. Take the inclusion $i_{ p,2}: \ell^p({\mathbb{N}}) \to \ell^2 ({\mathbb{N}})$, and define the operator $Q: \ell^2 ({\mathbb{N}}) \to L^p ({\mathbb{T}})$ by $$Q[(t_n)](x) = \sum_{n \in {\mathbb{N}}} t_n r_n (x),$$ where $r_n$ are the Rademacher functions $r_n (x) = sign(\sin (2^{k-1} x))$. By Khintchine’s inequality, the operator $Q$ is an isomorphic embedding of $\ell^2({\mathbb{N}})$ into $L^p({\mathbb{T}})$ for every $1 < p < \infty$, and the operator $A _p : L^p ({\mathbb{T}}) \to L^p ({\mathbb{T}})$ given by $$A = Q i_{p,2} P_{p},$$ is a non-compact strictly singular operator [@ex]. To construct our example let us define $$E_n := (\frac{2\pi}{2^{n+2}}, \frac{2 \pi}{2^{n+1}}), {\text{ }}{\text{ }}n \in {\mathbb{N}},$$ and let $\chi_{n} (x)$ be the characteristic function on $E_n$. Take the closed subspace $F_p$ as $$F_p := Span \{2^{(n+2)/2} \chi_n \}_{n \in {\mathbb{N}}},$$ which, as it is discussed in [@handbook1 Chapter 3, p.p. 126], is isomorphic to $\ell^p ({\mathbb{N}})$ and 1-complemented with the projection $P_p$ given by $$P_p [f](x) = \sum_{n \in {\mathbb{N}}} \Big( \int_{{\mathbb{T}}} 2^{(n+2)(p-1)/2} \chi_n (x) \overline{f(x)} dx\Big) 2^{(n+2)/2} \chi_{n} (x).$$ Thus the linear operator $A = Q i_{p,2} P_{p}$ is a pseudo-differential operator with symbol $$\begin{aligned}
\sigma_A (x,k) = e^{-i x \cdot k} A (e^{i x \cdot k}) &= e^{-i x \cdot k} \sum_{n \in {\mathbb{N}}} \Big( \int_{{\mathbb{T}}} 2^{(n+2)(p-1)/2} \chi_n (x) e^{-i x \cdot k} dx\Big) r_n (x) \\ &= e^{-i x \cdot k} \sum_{n \in {\mathbb{N}}} \Big( \int_{2 \pi/2^{n+2}}^{2 \pi/2^{n+1}} 2^{(n+2)(p-1)/2} e^{-i x \cdot k} dx\Big) r_n (x) \\ &= \frac{e^{- i x \cdot k}}{ik} \sum_{n \in {\mathbb{N}}} 2^{(n+2)(p-1)/2} e^{-i k \frac{2 \pi}{2^{n+2}}} (e^{-i k \frac{2 \pi}{2^{n+2}}} - 1)r_n (x). \end{aligned}$$
We note that each $k \in {\mathbb{Z}}$ fixed, and $n \in {\mathbb{N}}$ large enough, we get $$\begin{aligned}
|e^{-ik \frac{2 \pi}{2^{n+2}}} - 1|^2 &= 2|\cos{\frac{2 \pi k}{2^{n+2}}} - 1| = 2|\cos{\frac{2 \pi k + 0}{2^{n+2}}} - \cos{0}|\\ &= 4 |\sin{\frac{2 \pi k - 0}{2^{n+3}}} \sin{\frac{2 \pi k + 0}{2^{n+3}}}| \\ &\leq \frac{ \pi |k| }{2^{n+2}},\end{aligned}$$ which proves that there exist a constant $C>0$ such that $$S_k:=\sum_{n \in {\mathbb{N}}} 2^{(n+2)(p-1)/2} |e^{ik \frac{2 \pi}{2^{n+2}}} - 1| \leq C |k|^{1/2},$$ and then
$$\lim_{|k| \to \infty} \sup_{x \in {\mathbb{T}}} |\sigma_A (x , k)| \leq \lim_{|k| \to \infty} \frac{S_k}{|k|} \leq \lim_{|k| \to \infty} \frac{C}{|k|^{1/2}} = 0,$$ so $A$ is a non-compact Riesz operator.
**Gershgorin Theory for Periodic Pseudo-differential Operators**
================================================================
In this section we will leave aside the spaces $L^p ({\mathbb{T}})$, $p \neq 2$, and focus in the Hilbert space $L^2({\mathbb{T}})$. The main reason is that the Fourier transform is an isomorphism between $L^2({\mathbb{T}})$ and $L^2 ({\mathbb{Z}})$ which allows to factorise operators in $L^2 ({\mathbb{T}})$ through $L^2({\mathbb{Z}})$ a sequence space that, in some cases, it is easier to study. This factorisation allows one to obtain necessary and sufficient conditions for boundedness (Theorems 3.1 and 3.2) without assuming any regularity on the symbol, and sufficient conditions for invertibility, and spectrum localisation (Theorems 3.4 and 3.5). To start with , it is necessary to mention some details about the infinite matrices theory.
Infinite Matrices
-----------------
[An infinite matrix is a function $M: {\mathbb{Z}}\times {\mathbb{Z}}\to {\mathbb{C}}$ with matrix entries defined by $M_{jk} := M(j,k)$. If $M$ is an infinite matrix and $\varphi$ an infinite vector (or a function from ${\mathbb{Z}}$ to ${\mathbb{C}}$) then the product of the vector $\varphi$ an the matrix $M$ is defined as $$M\varphi (j) := \sum_{k \in {\mathbb{Z}}} M_{jk} \varphi (k).$$For infinite matrices $P$ and $Q$ their product is defined as the infinite matrix with entries $$PQ_{jk} = \sum_{h \in {\mathbb{Z}}} P_{jh} Q_{hk},$$and as usual, the adjoint of the infinite matrix $M$ is the infinite matrix $M^*$ with entries $$M^*_{jk} := \overline{(M_{kj})}.$$ ]{}
It is easy to see that with the above definition for any pair of infinite vectors (functions $\varphi_1, \varphi_2 : {\mathbb{Z}}\to {\mathbb{C}}$) and complex numbers $\lambda_1 , \lambda_2$ one has $$M(\lambda_1 \varphi_1 + \lambda_2 \varphi_2) = \lambda_1 M \varphi_1 + \lambda_2 M \varphi_2 ,$$So it is reasonable to think that an infinite matrix $ M $ can define a linear operator on some sequence space. However, not all infinite matrices define linear operators, and some conditions should be imposed on the matrix to be sufficiently well behaved to define a linear operator. In this case we are interested in linear operators on $L^2 ({\mathbb{Z}})$. Fortunately infinite matrices that define operators in $ L^2 ({\mathbb{Z}}) $ have already been studied and many results have been obtained. We state the most relevant for our work below. They can be found in [@Crone1971].
Let $M$ be an infinite matrix with rows and columns in $L^2({\mathbb{Z}})$ and define $P_n(x) := \sum_{|k| \leq n} \langle x , e_k \rangle e_k $, where $e_k$ is the $k$-th coordinate vector of $L^2 ({\mathbb{Z}})$. Then $M$ define a bounded operator in $L^2 ({\mathbb{Z}})$ if and only if $$\sup_{n \in {\mathbb{N}}} {\left\lVertP_n M^* M P_n\right\rVert}_{\mathcal{L}(L^2 ({\mathbb{Z}}))} < \infty
,$$when this happens $$\sup_{n \in {\mathbb{N}}} {\left\lVertP_n M^* M P_n\right\rVert}_{\mathcal{L}(L^2 ({\mathbb{Z}}))} = {\left\lVertM\right\rVert}_{\mathcal{L}(L^2 ({\mathbb{Z}}))}^2 .$$
Let $M$ be a infinite matrix. Then $M$ defines a bounded linear operator on $L^2({\mathbb{Z}})$ if and only if satisfy the following conditions
1. The rows of $M$ are functions in $L^2 ({\mathbb{Z}})$.
2. $(M^* M)^n$ is defined for all $n \in {\mathbb{N}}$.
3. $\sup_{n \in {\mathbb{N}}} \sup_{k \in {\mathbb{Z}}} |(M^* M)^n_{kk}|^{1/n} < \infty$
when this happens we get $$\sup_{n \in {\mathbb{N}}} \sup_{k \in {\mathbb{Z}}} |(M^* M)^n_{kk}|^{1/n} ={\left\lVertM\right\rVert}_{ \mathcal{L}(L^2({\mathbb{Z}}))}^2.$$
Next we will adapt these theorems to periodic pseudo-differential operators.
$L^2$-Boundedness
------------------
Let $\sigma : {\mathbb{T}}\times {\mathbb{Z}}\to {\mathbb{C}}$ be a measurable function such that $\sigma (\cdot , k) \in L^2 ({\mathbb{T}})$ for each $k \in {\mathbb{Z}}$ and $T_\sigma$ be its associated pseudo-differential operator. Then, for $f \in L^2 ({\mathbb{T}})$, we can write $$\begin{aligned}
(T_\sigma f) (x) &= \sum_{k \in {\mathbb{Z}}} \Big( \sum_{m \in {\mathbb{Z}}} \widehat{\sigma} (m , k) e^{i x \cdot m}\Big) \widehat{f} (k) e^{i x \cdot k} \\
&= \sum_{j \in {\mathbb{Z}}} \Big( \sum_{k \in {\mathbb{Z}}} \widehat{\sigma} (j-k , k) \widehat{f} (k) \Big) e^{i x \cdot j}.\end{aligned}$$ From this, it is straightforward that the $j$-th Fourier coefficient of $T_\sigma f$ is $$\sum_{k \in {\mathbb{Z}}} \widehat{\sigma} (j-k , k) \widehat{f}(k),$$ which is similar to the $j$-th entry of the matrix-vector product
$$\sum_{k \in {\mathbb{Z}}} M_{jk} \varphi (k).$$
This observation is the key fact of this section and is the motivation for the following definition.
[Let $\sigma : {\mathbb{T}}\times {\mathbb{Z}}\to {\mathbb{C}}$ be a measurable function such that $\sigma (\cdot , k) \in L^2 ({\mathbb{T}})$ for each $k \in {\mathbb{Z}}$ and $T_\sigma$ be its associated pseudo-differential operator. Then its]{} associated matrix
$M_\sigma$ is defined as the infinite matrix with entries
$$\begin{aligned}
(M_\sigma)_{jk} := \widehat{\sigma} (j-k , k) = \int_{\mathbb{T}}\sigma (x , k) e^{- i x \cdot (j-k)} dx .\end{aligned}$$
With this definition in mind, the operator $T_\sigma $ can be factored through $L^2 ({\mathbb{Z}})$ as the following diagram shows
L\^2 () & L\^2 ()\
L\^2() & L\^2()
where $\mathcal{F}_{\mathbb{T}}$ and $\mathcal{F}_{\mathbb{T}}^{-1}$ are the toroidal Fourier transform and toroidal inverse Fourier transform, defined by $$\mathcal{F}_{\mathbb{T}}(f) (k) := \int_{\mathbb{T}}f(x) e^{-ix \cdot k} dx, {\text{ }}{\text{ }}\text{ and } {\text{ }}{\text{ }}\mathcal{F}_{\mathbb{T}}^{-1} (\widehat{f}) (x) := \sum_{k \in {\mathbb{Z}}} \widehat{f} (k) e^{i x \cdot k}.$$
These linear operators extend to a unitary operator. For this reason the operator $T_\sigma$ is bounded in $L^2 ({\mathbb{T}})$ if and only if the infinite matrix $M_\sigma$ defines a bounded operator in $L^2 ({\mathbb{Z}})$ and ${\left\lVertT_\sigma\right\rVert}_{\mathcal{L} (L^2({\mathbb{T}}))} = {\left\lVertM_\sigma\right\rVert}_{\mathcal{L} (L^2({\mathbb{Z}}))}$ . This allow us to apply Lemmas 3.1 and 3.2 to give necessary and sufficient conditions for $L^2$-Boundedness of pseudo-differential operators.
Let $\sigma : {\mathbb{T}}\times {\mathbb{Z}}\to {\mathbb{C}}$ be a measurable function such that $\sigma (\cdot , k) \in L^2 ({\mathbb{T}})$ for each $k \in {\mathbb{Z}}$ and $T_\sigma$ be its associated pseudo-differential operator. Then $T_\sigma$ defines a bounded linear operator on $L^2({\mathbb{T}})$ if and only if rows of the associated matrix $M_\sigma$ are in $L^2 ({\mathbb{Z}})$, $(M_\sigma^* M_\sigma)^n$ is defined for all $n \in {\mathbb{N}}$, and
$$\sup_{n \in {\mathbb{N}}} \sup_{k \in {\mathbb{Z}}} \Big| \int_{{\mathbb{T}}} \tau^n (x , k) dx \Big|^{1/n} < \infty
,$$
where $\tau$ is the symbol of ${\text{ }}(T_\sigma)^* T_\sigma = T_{\sigma^*} T_\sigma$ and $\tau^n$ is the symbol of $(T_\tau)^n$. When this happens then
$$\sup_{n \in {\mathbb{N}}} \sup_{k \in {\mathbb{Z}}} \Big| \int_{{\mathbb{T}}} \tau^n (x , k) dx \Big|^{1/n} = {\left\lVertT_\sigma\right\rVert}_{\mathcal{L} (L^2 ({\mathbb{T}}))}^2.$$
Let $M_\sigma$ be the associated matrix of $T_\sigma$. Then, since rows of $M_\sigma$ are $L^2({\mathbb{Z}})$, the adjoint matrix is well defined as a densely defined operator in $L^2({\mathbb{Z}})$, and is a pseudo-differential operator with symbol $$\sigma^* (x, j)= \sum_{k \in {\mathbb{Z}}} \widehat{\sigma} (j - k , k) e^{i x \cdot k}.$$Now consider the operator $T_{\sigma^*} T_\sigma$. Since $(M_\sigma^* M_\sigma)^n$ is defined for all $n$, it is a pseudo-differential operator with symbol $\tau(\cdot , k) \in L^2({\mathbb{T}})$ for each $k$ and, again by the existence of $(M_\sigma^* M_\sigma)^n$ for all $n$, $T_\tau^n$ is a pseudo-differential operator with symbol $\tau^n (\cdot , k) \in L^2({\mathbb{T}})$ for each $k$ and for all $n \in {\mathbb{N}}$. We have already cover satisfy hypothesis $(i)$ and $(ii)$ of Lemma 3.2 and consequently $M_\sigma$ defines a bounded linear operator in $L^2({\mathbb{Z}})$ if and only if aditionally we have $$\sup_{n \in {\mathbb{N}}} \sup_{k \in {\mathbb{Z}}} |(M_{\sigma^*} M_\sigma )^n_{kk}|^{1/n} = \sup_{n \in {\mathbb{N}}} \sup_{k \in {\mathbb{Z}}} \Big| \int_{{\mathbb{T}}} \tau^n (x , k) dx \Big|^{1/n} < \infty.$$
Let $T_\sigma$ be a pseudo-differential operator with symbol $\sigma \in S^m_{\rho , \delta} ({\mathbb{T}}\times {\mathbb{Z}})$. Then $T_\sigma$ defines a bounded operator on $L^2({\mathbb{T}})$ if and only if $$\sup_{n \in {\mathbb{N}}} \sup_{k \in {\mathbb{Z}}} \Big| \int_{{\mathbb{T}}} \tau^n (x , k) dx \Big|^{1/n} < \infty
,$$
where $\tau$ is the symbol of ${\text{ }}(T_\sigma)^* T_\sigma = T_{\sigma^*} T_\sigma$ and $\tau^n$ is the symbol of $(T_\tau)^n$. When this happens then
$$\sup_{n \in {\mathbb{N}}} \sup_{k \in {\mathbb{Z}}} \Big| \int_{{\mathbb{T}}} \tau^n (x , k) dx \Big|^{1/n} = {\left\lVertT_\sigma\right\rVert}_{\mathcal{L} (L^2 ({\mathbb{T}}))}^2.$$
The existence of $(M_\sigma^* M_\sigma)^n$ for all $n \in {\mathbb{N}}$ follows from toroidal composition formula, an the fact that $(M_\sigma^* M_\sigma)^n$ is the matrix associated to the symbol $\tau^n$ of $ ((T_\sigma)^* T_\sigma)^n = (T_{\sigma^*} T_\sigma )^n$. Also, the $L^2({\mathbb{Z}})$-norm of the $j$-th row of $M_\sigma$ is
$$\begin{aligned}
\Big( \sum_{k \in {\mathbb{Z}}} |\widehat{\sigma} (j - k , k)|^2 \Big)^{1/2} &= \Big( \sum_{k \in {\mathbb{Z}}} |\overline{\widehat{\sigma}^* (k - j , j) }|^2 \Big)^{1/2} \\
&= \Big( \sum_{k \in {\mathbb{Z}}} |\widehat{\sigma}^* (k - j , j)|^2 \Big)^{1/2} \\
&= {\left\lVert\sigma^* (\cdot , j)\right\rVert}_{L^2({\mathbb{T}})} < \infty.\end{aligned}$$
It follows from Theorem 3.1 that $T_\sigma$ is bounded if and only if $$\sup_{n \in {\mathbb{N}}} \sup_{k \in {\mathbb{Z}}} \Big| \int_{{\mathbb{T}}} \tau^n (x , k) dx \Big|^{1/n} < \infty
.$$
Let $\sigma : {\mathbb{T}}\times {\mathbb{Z}}\to {\mathbb{C}}$ be a measurable function such that $\sigma (\cdot , k) \in L^2 ({\mathbb{T}})$ for each $k \in {\mathbb{Z}}$ and $T_\sigma$ be its associated pseudo-differential operator. Then $T_\sigma$ defines a bounded operator on $L^2 ({\mathbb{T}})$ if and only if rows of associated matrix $M_\sigma$ are in $L^2 ({\mathbb{Z}})$ and
$$\sup_{n \in {\mathbb{N}}} {\left\lVertM_{\sigma , n}\right\rVert}_{\mathcal{L}(\ell^2 ({\mathbb{C}}^{2n+1}))} < \infty,$$
where $M_{\sigma , n}$ is the $(2n+1)\times (2n+1)$ matrix with entries
$$\begin{aligned}
(M_{\sigma , n})_{jk} := \overline{\langle \sigma (x,j)e^{i x \cdot j} | \sigma (x,k)e^{i x \cdot k} \rangle}_{L^2 ({\mathbb{T}})} {\text{ }}, {\text{ }}|j|,|k| \leq 2n+1,\end{aligned}$$
when this happens
$$\sup_{n \in {\mathbb{N}}} {\left\lVertM_{\sigma , n}\right\rVert}_{\mathcal{L}(\ell^2 ({\mathbb{C}}^{2n+1}))} = {\left\lVertT_\sigma\right\rVert}_{\mathcal{L}(L^2 ({\mathbb{T}}))}^2 .$$
We just have to see that
$$\begin{aligned}
(M_{\sigma^*} M_\sigma)_{jk} :&= \sum_{h \in {\mathbb{Z}}} \widehat{\sigma^*} (j-h,h) \cdot \widehat{\sigma} (h-k, k) \\
&= \sum_{h \in {\mathbb{Z}}} \overline{\widehat{\sigma} (h-j,j) } \cdot \widehat{\sigma} (h-k, k) \\
&= \overline{ \langle \sigma(x , j) e^{i x \cdot j} | \sigma(x,k) e^{i x \cdot k} \rangle}_{L^2({\mathbb{T}})} ,\end{aligned}$$
and $${\left\lVertP_n M_{\sigma^*} M_\sigma P_n\right\rVert}_{\mathcal{L}(L^2 ({\mathbb{Z}}))} = {\left\lVertM_{\sigma , n}\right\rVert}_{\mathcal{L} (\ell^2 ({\mathbb{C}}^{2n+1}))}.$$
with this the result follows from Lemma 3.1.
Usually, sufficient conditions for boundedness of pseudo-differential operators are given appealing to the existence of a certain number of derivatives in the toroidal variable of the symbol [@ruzhansky1; @JulioLpbounds; @Duvan1]. However, Theorems 3.1 and 3.2 provide necessary and sufficient conditions for boundedness without any derivative. This allows one, for example, to recover the classical result about boundedness of a multiplication operators by a fixed function, which is the same as an operator with associated Toeplitz matrix, using Theorem 3.1.
Let us define $T_\phi f := \phi f$. Then $T_\phi$ is a pseudo-differential operator with symbol $\sigma(x , k)= \phi (x)$, and consequently $T_\phi^*$, $T_\phi^* T_\phi$ and $(T_\phi^* T_\phi)^n$ are pseudo-differential operators with symbols $\overline{\phi}$, $|\phi|^2$ and $|\phi|^{2n}$, respectively. This proves that rows of the associated matrix $M_\sigma$ are in $L^2 ({\mathbb{Z}})$, and that $(M_\sigma^* M_\sigma)^n$ is defined for all $n \in {\mathbb{N}}$ so, $T_\phi$ is bounded if and only if $$\sup_{n \in {\mathbb{N}}} \sup_{k \in {\mathbb{Z}}} \Big| \int_{{\mathbb{T}}} |\phi|^{2n} (x) dx \Big|^{1/n} < \infty
,$$ but this equivalent to $$|\phi|^2 \in L^p({\mathbb{T}}) {\text{ }}{\text{ }}\text{for all} {\text{ }}1\leq p < \infty, {\text{ }}{\text{ }}\text{and} {\text{ }}{\text{ }}\sup_{p \in [1,\infty)} ||(|\phi|^2)||_{L^p({\mathbb{T}})} < \infty,$$ which is equivalent to $\phi \in L^\infty({\mathbb{T}})$. Another interesting example is given by the pseudo-differential operator associated to the symbol $$\sigma(x,k) = \begin{cases}
0 & \text{if} {\text{ }}{\text{ }}\frac{1}{2^{|k|}} \leq x \leq 2\pi, \\ 1 & \text{if} {\text{ }}{\text{ }}0 \leq x < \frac{1}{2^{|k|}}.
\end{cases}$$ For this operator one has $$\overline{\langle \sigma (x,j)e^{i x \cdot j} | \sigma (x,k)e^{i x \cdot k} \rangle}_{L^2 ({\mathbb{T}})} = \int_{0}^{1/2^{\max(|j|, |k|)}}e^{-i x \cdot (j-k)}dx = \frac{-1}{i(j-k)} (e^{-i(j-k)/2^{\max(|j|,|k|)}} - 1),$$ and for $|j|,|k|$ large enough we have the inequality $$|\overline{\langle \sigma (x,j)e^{i x \cdot j} | \sigma (x,k)e^{i x \cdot k} \rangle}_{L^2 ({\mathbb{T}})}| = \frac{1}{|j-k|} |e^{-i(j-k)/2^{\max(|j|,|k|)}} - 1| \leq \frac{1}{2^{\max(|j|,|k|)}},$$ proving that the entries of the matrix $M_\sigma^* M_\sigma$ decay rapidly, and thus Theorem 3.2 ensures that $T_\sigma$ defines a bounded operator, even when $\sigma (x,k)$ has no derivatives in the toroidal variable..
Gershgorin Theory.
------------------
A beautiful and useful result about the spectrum of a matrix is the Gershgorin’s circle theorem which we enunciate below.
Let $M$ be a $n \times n$ matrix with entries $a_{jk}$ and define $r_j := \sum_{ k \neq j} |a_{jk}|$. Then each eigenvalue $\lambda$ of $M$ lies in one of the disks $\overline{B_{\mathbb{C}}(a_{jj} , r_j)}$.
This theorem can be extended to operators that act on a infinite dimensional space, particularly to infinite matrices. There are many papers on the subject and indeed the Gershgorin theorem gives rise to an entire theory, called the Gershgorin theory. We enunciate below a theorem pertaining to this theory which is a subtle variation Theorem 2 in [@SHIVAKUMAR198735]. See also [@ALEKSIC2014541].
Let $M$ be a infinite matrix. Define two new matrices $D$ and $F$ by $$D_{jk} := \delta_{jk} M_{jk} ,{\text{ }}{\text{ }}\text{and } {\text{ }}{\text{ }}F_{jk}:=(1-\delta_{jk}) M_{jk},$$ where $\delta_{jk}$ is the Kronecker delta. If the following conditions are met
1. $M_{kk} \neq 0 $ for all $k \in {\mathbb{Z}}$ and $\inf_{k \in {\mathbb{Z}}} |M_{kk}| > 0$.
2. $I + FD^{-1}$ defines a bounded operator in $L^2 ({\mathbb{Z}})$ with bounded inverse.
then $M$ is a invertible densely defined linear operator in $L^2 ({\mathbb{Z}})$ with bounded inverse. If in addition
$$\lim_{|k| \to \infty} |M_{kk}| = \infty,$$
then the inverse of $M$ is a compact operator.
We just have to write $M = D + F$. If $M$ satisfy condition $(i)$ then $D^{-1}$ exist and defines a bounded operator in $L^2 ({\mathbb{Z}})$. Clearly $$M = D + F = (I + FD^{-1}) D,$$ so the inverse of $ M $, if it exists, is $$M^{-1} = D^{-1} (I + FD^{-1})^{-1}.$$ Applying the hypothesis $(ii)$ we conclude that $ M $ is invertible with continuous inverse. In addition if $$\lim_{|k| \to \infty} |M_{kk}| = \infty,$$ then $$\lim_{|k| \to \infty} |M_{kk}|^{-1} = 0,$$ so $D^{-1}$ is a compact operator, and in consequence $M^{-1}$ is compact as well.
We will now apply this theorem to pseudo-differential operators to provide a way to locate the espectrum. For this the following proposition is needed.
Let $M$ be a infinite matrix such that $M$ and $M^*$ both define a bounded operator on $L^\infty ({\mathbb{Z}})$. Then $M$ defines a bounded operator in $L^2 ({\mathbb{Z}})$ and
$${\left\lVertM\right\rVert}_{\mathcal{L} (L^2 ({\mathbb{Z}}))} \leq \sqrt{{\left\lVertM\right\rVert}_{\mathcal{L} (L^\infty ({\mathbb{Z}}))} {\left\lVertM^*\right\rVert}_{\mathcal{L} (L^\infty ({\mathbb{Z}}))}}.$$
Let $\sigma : {\mathbb{T}}\times {\mathbb{Z}}\to {\mathbb{C}}$ be a measurable function such that $\sigma (\cdot , k) \in L^2 ({\mathbb{T}})$ for each $k \in {\mathbb{Z}}$ and $T_\sigma$ be its associated pseudo-differential operator. If $\sigma$ satisfy the following three properties
1. $$\int_{\mathbb{T}}\sigma (x, k) dx \neq 0{\text{ }}{\text{ }}\text{for all }{\text{ }}k \in {\mathbb{Z}}{\text{ }}, {\text{ }}\inf_{k \in {\mathbb{Z}}}\Big| \int_{\mathbb{T}}\sigma(x,j) dx \Big| > 0,$$
2. $$\sup_{k \in {\mathbb{Z}}} \Big| \int_{\mathbb{T}}\sigma(x,k) dx \Big|^{-1} \cdot \sum_{ j \neq k} |\widehat{\sigma}(j-k,k)| < 1,$$
3. $$\sup_{j \in {\mathbb{Z}}} \Big| \int_{\mathbb{T}}\sigma(x,j) dx \Big|^{-1} \cdot \sum_{k \neq j} |\widehat{\sigma}(j-k,k)| < 1,$$
or equivalently,
1. $$\int_{\mathbb{T}}\sigma (x, k) dx \neq 0 {\text{ }}{\text{ }}\text{for all }{\text{ }}k \in {\mathbb{Z}}{\text{ }}{\text{ }}, {\text{ }}\inf_{k \in {\mathbb{Z}}} \Big| \int_{\mathbb{T}}\sigma(x,j) dx \Big| > 0,$$
2. $$\sup_{k \in {\mathbb{Z}}} \Big| \int_{\mathbb{T}}\sigma(x,k) dx \Big|^{-1} \cdot {\left\lVert\widehat{\sigma}(\cdot , k)\right\rVert}_{L^1({\mathbb{Z}})} < 2,$$
3. $$\sup_{j \in {\mathbb{Z}}} \Big| \int_{\mathbb{T}}\sigma(x,j) dx \Big|^{-1} \cdot {\left\lVert\widehat{\sigma^*}(\cdot , j)\right\rVert}_{L^1({\mathbb{Z}})} < 2,$$
where $\sigma^* (x,k)$ is the symbol of adjoint operator, then $T_\sigma$ is an invertible linear operator with bounded inverse. In particular if $$\lim_{|k| \to \infty} \Big| \int_{\mathbb{T}}\sigma (x,k) dx \Big| = \infty$$ the inverse is a compact operator.
Let $M_\sigma$ be the associated matrix of $T_\sigma$. We will show that this infinite matrix satisfy the hypothesis of Lemma 3.3. First its easy to see that $(i)$ in previous statement is equivalent to $i$ in Lemma 3.3. For the remaining hypothesis define $D_\sigma$ and $F_\sigma$ as $$(D_\sigma)_{jk} := \delta_{jk} (M_\sigma)_{jk}, {\text{ }}{\text{ }}\text{and } {\text{ }}{\text{ }}(F_\sigma)_{jk}:=(1-\delta_{jk}) (M_\sigma)_{jk},$$then by Proposition 3.1 $$\begin{aligned}
{\left\lVertF_\sigma D_\sigma^{-1}\right\rVert}_{\mathcal{L}(L^2({\mathbb{Z}}))}^2 &= {\left\lVertI - (I + F_\sigma D_\sigma^{-1} )\right\rVert}_{\mathcal{L}(L^2({\mathbb{Z}}))}^2 \\
&\leq {\left\lVertF_\sigma D_\sigma^{-1}\right\rVert}_{\mathcal{L} (L^\infty ({\mathbb{T}}))} \cdot {\left\lVert(F_\sigma D_\sigma^{-1})^*\right\rVert}_{\mathcal{L} (L^\infty ({\mathbb{T}}))},\end{aligned}$$ but clearly $${\left\lVertF_\sigma D_\sigma^{-1}\right\rVert}_{\mathcal{L} (L^\infty ({\mathbb{T}}))} = \sup_{j \in {\mathbb{Z}}} \Big| \int_{\mathbb{T}}\sigma(x,j) dx \Big|^{-1} \cdot \sum_{k \neq j} |\widehat{\sigma}(j-k,k)| < 1,$$ and $${\left\lVert(F_\sigma D_\sigma^{-1})^*\right\rVert}_{\mathcal{L} (L^\infty ({\mathbb{T}}))} = \sup_{k \in {\mathbb{Z}}} \Big| \int_{\mathbb{T}}\sigma(x,k) dx \Big|^{-1} \cdot \sum_{ j \neq k} |\widehat{\sigma}(j-k,k)| < 1,$$
so ${\left\lVertF_\sigma D_\sigma^{-1}\right\rVert}_{\mathcal{L}(L^2({\mathbb{Z}}))} < 1$ and by Lemma 2.1 in [@conwayfa] the operator defined by $I + {F_\sigma}{D_\sigma}^{-1}$ is invertible with bounded inverse in $L^2 ({\mathbb{Z}})$.
Let $\alpha : {\mathbb{Z}}\to {\mathbb{C}}$ a measurable function that satisfy $\alpha(k) \neq 0$ for all $k$ and $V \in \mathcal{F}_{\mathbb{T}}^{-1} (L^1 ({\mathbb{Z}}))$ such that $\int_{\mathbb{T}}V(x) dx \neq 0$. Then for the symbol $\sigma (x,k) : = \alpha(k) V(x)$ one has
$$\widehat{\sigma} (j - k ,k) = \alpha(k) \widehat{V} (j-k)$$
and from this
$$\sup_{k \in {\mathbb{Z}}} \Big| \int_{\mathbb{T}}\sigma(x,k) dx \Big|^{-1} \cdot {\left\lVert\widehat{\sigma}(\cdot , k)\right\rVert}_{L^1({\mathbb{Z}})} = \Big| \int_{\mathbb{T}}V(x) dx \Big|^{-1} \cdot {\left\lVert\widehat{V}\right\rVert}_{L^1({\mathbb{Z}})},$$ $$\sup_{j \in {\mathbb{Z}}} \Big| \int_{\mathbb{T}}\sigma(x,j) dx \Big|^{-1} \cdot {\left\lVert\widehat{\sigma^*}(\cdot , j)\right\rVert}_{L^1({\mathbb{Z}})} = \sup_{j \in {\mathbb{Z}}} \Big| \alpha (j) \int_{\mathbb{T}}V(x) dx \Big|^{-1} \cdot \sum_{k \in {\mathbb{Z}}} \alpha(k) \widehat{V}(j-k).$$ so if $V$ satisfy the conditions $${\left\lVert\widehat{V}\right\rVert}_{L^1 ({\mathbb{Z}})} < 2 \Big| \int_{\mathbb{T}}V(x) dx \Big|,$$ and $$\Big| \alpha (j) \int_{\mathbb{T}}V(x) dx \Big|^{-1} \cdot \sum_{k \in {\mathbb{Z}}} \alpha(k) \widehat{V}(j-k) <1 {\text{ }}{\text{ }}\text{for all} {\text{ }}{\text{ }}j \in {\mathbb{Z}},$$ then $T_\sigma$ is a invertible linear operator with bounded inverse. If a additionally $$\lim_{|k| \to \infty} |\alpha(k) | = \infty,$$ then the inverse is a compact operator.
Let $\sigma : {\mathbb{T}}\times {\mathbb{Z}}\to {\mathbb{C}}$ be a measurable function such that $\sigma (\cdot , k) \in L^2 ({\mathbb{T}})$ for each $k \in {\mathbb{Z}}$ and $T_\sigma$ be its associated pseudo-differential operator. If the following conditions hold
1. $$\int_{\mathbb{T}}\sigma (x, k) dx - \lambda \neq 0 , {\text{ }}{\text{ }}\forall k \in {\mathbb{Z}}{\text{ }}{\text{ }}, \inf_{k \in {\mathbb{Z}}} \Big| \int_{\mathbb{T}}\sigma(x,j) dx - \lambda \Big| > 0,$$
2. $$\sup_{k \in {\mathbb{Z}}} \Big| \int_{\mathbb{T}}\sigma(x,k) dx - \lambda \Big|^{-1} \cdot \sum_{ j \neq k} |\widehat{\sigma}(j-k,k)| < 1,$$
3. $$\sup_{j \in {\mathbb{Z}}} \Big| \int_{\mathbb{T}}\sigma(x,j) dx - \lambda \Big|^{-1} \cdot \sum_{k \neq j} |\widehat{\sigma}(j-k,k)| < 1,$$
then $\lambda \in Res(T_\sigma)$.
We just have to see that for all $\lambda\in {\mathbb{C}}$ ${\text{ }}T_\sigma - \lambda I$ is a pseudo-differential operator with symbol $\sigma(x,k) - \lambda$.
Let $\alpha : {\mathbb{Z}}\to {\mathbb{C}}$ a measurable function and $V \in \mathcal{F}_{\mathbb{T}}^{-1} (L^1 ({\mathbb{Z}}))$. Then for the symbol $\sigma (x,k) := \alpha(k) + V(x)$ one has
$$\widehat{\sigma} (j-k,k) =
\begin{cases}
\widehat{V}(j-k) & j \neq k \\
\alpha(k) + \int_{\mathbb{T}}V(x) dx & j=k
\end{cases}$$and then if $$\sum_{k \neq j} |\widehat{V}(j-k)| < \inf_{k \in {\mathbb{Z}}} \Big| \alpha(k) + \int_{\mathbb{T}}V(x) dx - \lambda \Big|$$ $\lambda \in Res(T_\sigma)$ which proves that $$Spec(T_\sigma) \subseteq \bigcup_{k \in {\mathbb{Z}}} \overline{B_{\mathbb{C}}(c_k , r)}$$ where $$c_k = \alpha(k) + \int_{\mathbb{T}}V(x) dx {\text{ }}{\text{ }}\text{ and } {\text{ }}{\text{ }}r =\sum_{ j \neq k} |\widehat{V}(j-k)|$$
To finish this section we enunciate an improved version of Lemma 3.3, which can be found as the Theorem 3.1 in [@FARID19917]. We use it to prove an improved version of Theorem 3.4.
Let $M$ be an infinite matrix, considered as a linear operator on $L^p ({\mathbb{Z}})$ for $1\leq p <\infty$, with columns in $L^1 ({\mathbb{Z}})$. Define $r_k := \sum_{j \in {\mathbb{Z}}, j \neq k} |M_{jk}|$ and assume that
1. $M_{kk} \neq 0, {\text{ }}{\text{ }}\text{for all} {\text{ }}k \in {\mathbb{Z}}$ and $|M_{kk}| \to \infty$ as $|k| \to \infty$.
2. There exits $s \in [0,1)$ such that for all $k \in {\mathbb{Z}}$ $$r_k = s_k |M_{kk}|, {\text{ }}{\text{ }}s_k \in [0,s].$$
3. Either $FD^{-1}$ and $(I + \mu FD^{-1})^{-1}$ exists and are in $\mathcal{L}(L^p ({\mathbb{Z}}))$ for every $\mu \in (0,1]$, or $D^{-1}F$ and $(I + \mu D^{-1} F)^{-1}$ exists and are in $\mathcal{L}(L^p({\mathbb{T}}))$.
Then $M$ is a closed operator and the spectrum $Spec(M)$ is nonempty and consists of discrete nonzero eigenvalues, lying in the set
$$\bigcup_{k \in {\mathbb{Z}}} \overline{B_{\mathbb{C}}(M_{kk} , r_k)},$$
where the closed balls $\overline{B_{\mathbb{C}}(M_{kk} , r_k)}$ are called the Gershgorin discs. Furthermore, any set consisting of $n$ Gershgorin discs whose union is disjoint from all other Gersgorin discs intersects $Spec(M)$ in a finite set of eigenvalues of $M$ with total algebraic multiplicity $n$.
As an immediate consequence we have:
Let $\sigma : {\mathbb{T}}\times {\mathbb{Z}}\to {\mathbb{C}}$ be a measurable function such that $\sigma (\cdot , k) \in L^2({\mathbb{T}})$ for each $k \in {\mathbb{Z}}$ and $T_\sigma$ be its associated pseudo-differential operator. Let $M_\sigma$ be the associated matrix. Assume that
1. $\int_{\mathbb{T}}\sigma(x,k) dx \neq 0,$ for all $ k \in {\mathbb{Z}}$ and $\big| \int_{\mathbb{T}}\sigma(x,k) dx \big| \to \infty$ as $|k| \to \infty $.
2. Rows and columns of $M_\sigma$ are in $L^1 ({\mathbb{Z}})$.
3. $$\sup_{k \in {\mathbb{Z}}} \big| \int_{\mathbb{T}}\sigma(x,k) dx \big|^{-1} \cdot \sum_{ j \neq k} |\widehat{\sigma}(j-k,k)| < 1,$$ and$$\sup_{j \in {\mathbb{Z}}} \Big| \int_{\mathbb{T}}\sigma(x,j) dx \Big|^{-1} \cdot \sum_{k \neq j} |\widehat{\sigma}(j-k,k)| < 1.$$
Then $T_\sigma$ is a closed operator and the spectrum $Spec(T_\sigma)$ is nonempty and consists of discrete nonzero eigenvalues, lying in the set
$$\bigcup_{k \in {\mathbb{Z}}} \overline{B_{\mathbb{C}}(a_{kk} , r_k)} {\text{ }}{\text{ }}\text{where} {\text{ }}{\text{ }}a_{kk} = \int_{\mathbb{T}}\sigma(x,k) dx, {\text{ }}r_k = \sum_{ j \neq k} |\widehat{\sigma}(j-k,k)|.$$
Furthermore, any set of $n$ Gershgorin discs whose union is disjoint from all other Gersgorin discs intersects $Spec(T_\sigma)$ in a finite set of eigenvalues of $T_\sigma$ with total algebraic multiplicity $n$.
Condition $(i)$ in Theorem 3.5. is the condition $(i)$ of Lemma 3.3. Proposition 3.1. and conditions $(ii)$,$(iii)$ of Theorem 3.5 implies conditions $(ii)$ and $(iii)$ of Lemma 3.3.
**Gohberg’s Lemma in $L^p ({\mathbb{T}})$**
===========================================
In this section we discuss a proof of Gohberg’s Lemma in $L^p ({\mathbb{T}})$.
If $u$ is a nonzero function in $C^\infty ({\mathbb{T}})$ then
$$\begin{aligned}
T_\sigma u (x) &= \sum_{k \in {\mathbb{Z}}} \sigma (x,k) \widehat{u} (k) e^{i x \cdot k} \\
&= \sum_{k \in {\mathbb{Z}}} \int_{\mathbb{T}}\sigma(x,k) u(y) e^{i(x-y)k} dy \\
&= \int_{\mathbb{T}}(\mathcal{F}_{\mathbb{T}}^{-1} \sigma) (x, x-y) u(y) dy \\
&=\int_{\mathbb{T}}(\mathcal{F}_{\mathbb{T}}^{-1} \sigma) (x, y) u(x-y) dy,\end{aligned}$$
where $\mathcal{F}_{\mathbb{T}}^{-1} \sigma$ is the inverse Fourier transform of $\sigma$ with respect to the second variable in a distributional sense. Hence in a distributional sense $$T_\sigma u (x)=\int_{\mathbb{T}}(\mathcal{F}_{\mathbb{T}}^{-1} \sigma) (x, y) u(x-y) dy.$$ Since $\sigma$ is continuous it follows that for all $k \in {\mathbb{Z}}$ exists a $x_k \in [-\pi, \pi]$ such that $$|\sigma(x_k , k)| = \sup_{x \in {\mathbb{T}}} |\sigma(x,k)|.$$ By definition of $d_\sigma$, there exists a subsequence $\{(x_{k_l} , k_l)\}_{l \in {\mathbb{N}}}$ such that $$|k_l|\to \infty {\text{ }}{\text{ }}\text{and} {\text{ }}{\text{ }}|\sigma(x_{k_l} , k_l)|\to d_\sigma,$$ as $l \to \infty$. Then, since ${\mathbb{T}}$ is compact, the sequence $\{x_{k_l}\}_{l \in {\mathbb{N}}}$ must have an cluster point $x_0$ in ${\mathbb{T}}$ and a subsequence $\{x_{k_{l_m}}\}_{m \in {\mathbb{N}}}$ such that $x_{k_{l_m}} \to x_0$ as $m \to \infty$. For simplicity we will rename the sequence $\{(x_{k_{l_m}} , k_{l_m})\}_{m \in {\mathbb{N}}}$ as the original one $\{(x_{k_l} , k_l)\}_{l \in {\mathbb{N}}}$. Define the functions $u_{k_l}$ on ${\mathbb{T}}$ by $$u_{k_l} (x) = u(x-x_{k_l}) e^{ix \cdot k_l},$$ then wee see that ${\left\lVertu_{k_l}\right\rVert}_{L^p ({\mathbb{T}})} = {\left\lVertu\right\rVert}_{L^p ({\mathbb{T}})}$ for $l \in {\mathbb{N}}$. Let’s see that the sequence $\{u_{k_l}\}_{l \in {\mathbb{N}}}$ converges weakly to zero. First, for any $v \in L^q ({\mathbb{T}})$ $$\begin{aligned}
\int_{\mathbb{T}}u_{k_l} (x) v(x) dx =& \int_{\mathbb{T}}( u(x-x_{k_l}) - u(x - x_0) ) v(x) e^{ix \cdot k_l} dx \\
&+ \int_{\mathbb{T}}u(x-x_0) v(x) e^{ix \cdot k_l} dx.\end{aligned}$$
Since by Hölder inequality $u(x-x_0) v(x) \in L^1 ({\mathbb{T}})$ then we can apply the Riemann-Lebesgue lemma and consequently, for any $\varepsilon > 0$, there is a natural number $N_1$ such that $$l \geq N_1 \implies \Big| \int_{\mathbb{T}}u(x-x_0) v(x) e^{ix \cdot k_l} dx \Big|< \frac{\varepsilon}{2},$$and second, since $u$ is continuous, for any $\varepsilon>0$ there exist $N(\varepsilon) \in {\mathbb{N}}$, $\delta (\varepsilon) > 0$ such that $$\begin{aligned}
l \geq N(\varepsilon) &\implies |x_{k_l} - x_0| < \delta (\varepsilon) \\
&\implies |u(x-x_{k_l}) - u(x - x_0)| < \frac{\varepsilon}{2} \cdot {\left\lVertv\right\rVert}_{L^q ({\mathbb{T}})}^{-1},\end{aligned}$$ thus for any $\varepsilon>0$ $$l \geq \max(N_1 , N(\varepsilon)) \implies \Big| \int_{\mathbb{T}}u_{k_l} (x) v(x) dx \Big|< \varepsilon ,$$ as an immediate consequence $${\left\lVertK u_{k_l}\right\rVert}_{L^p ({\mathbb{T}})} \to 0 {\text{ }}{\text{ }}\text{as} {\text{ }}{\text{ }}l \to \infty,$$ for every $K \in \mathfrak{K} (L^p ({\mathbb{T}}))$. Then for an arbitrary positive number $\varepsilon$ and sufficiently large $l$
$$\begin{aligned}
{\left\lVertK u_{k_l}\right\rVert}_{L^p ({\mathbb{T}})} \leq \varepsilon {\left\lVertu_{k_l}\right\rVert}_{L^p ({\mathbb{T}})} \tag{G1}.\end{aligned}$$
On the other hand, we have the following lemma, whose proof can be found in [@Molahajloo2010].
${\left\lVert\sigma(\cdot,k_l) - T_\sigma u_{k_l}\right\rVert}_{L^p ({\mathbb{T}})} \to 0$ as $l \to \infty$.
Then for sufficiently large $l$,
$$\begin{aligned}
{\left\lVert\sigma(\cdot,k_l)\right\rVert}_{L^p ({\mathbb{T}})} - {\left\lVert T_\sigma u_{k_l}\right\rVert}_{L^p ({\mathbb{T}})} \leq \varepsilon {\left\lVertu\right\rVert}_{L^p ({\mathbb{T}})} .\tag{G2}\end{aligned}$$
By using again the continuity of $\sigma (\cdot , k_l)$ there exists a positive number $\delta$ such that for all $x$ in $[-\pi + x_{k_l}, \pi + x_{k_l}]$ with $|x - x_{k_l}|< \delta$, we have
$$\begin{aligned}
|\sigma(x, k_l) - \sigma (x_{k_l} , k_l)| < \varepsilon \tag{G3}.\end{aligned}$$
Let $u \in C^\infty ({\mathbb{T}})$ such that $u(x) =0$ for $|x| \geq \delta$. Then $u_{k_l} (x)=0$ for all $x$ in $[-\pi + x_{k_l}, \pi + x_{k_l}]$ with $|x - x_{k_l}| \geq \delta$. So
$$\begin{aligned}
{\left\lVert\sigma(x_{k_l},k_l) u_{k_l} \right\rVert}_{L^p ({\mathbb{T}})} &- {\left\lVert\sigma(\cdot, k_l) u_{k_l}\right\rVert}_{L^p ({\mathbb{T}})} \leq {\left\lVert\sigma(x_{k_l}) u_{k_l} -\sigma(\cdot, k_l) u_{k_l}\right\rVert}_{L^p ({\mathbb{T}})}\\
&= \Big( \int_{- \pi + x_{k_l}}^{\pi + x_{k_l}} |\sigma(x_{k_l},k_l) - \sigma (x,k_l)|^p |u_{k_l} (x)|^p dx \Big)^{1/p}\\
&= \Big( \int_{|x-x_{k_l}|<\delta} |\sigma(x_{k_l},k_l) - \sigma (x,k_l)|^p |u_{k_l} (x)|^p dx \Big)^{1/p},\end{aligned}$$
then by (G3)$$\begin{aligned}
| \sigma (x_{k_l} , k_l)| {\left\lVertu\right\rVert}_{L^p ({\mathbb{T}})} - {\left\lVert\sigma (\cdot , k_l) u_{k_l}\right\rVert}_{L^p ({\mathbb{T}})} < \varepsilon {\left\lVertu\right\rVert}_{L^p({\mathbb{T}})}. \tag{G4}\end{aligned}$$ Combining (G1), (G2) and (G4), we get for sufficiently large k,
$$\begin{aligned}
{\left\lVertu\right\rVert}_{L^p ({\mathbb{T}})} {\left\lVertT_\sigma - K\right\rVert}_{\mathcal{L} (L^p ({\mathbb{T}}))} &\geq {\left\lVert(T_\sigma - K)u\right\rVert}_{L^p ({\mathbb{T}})}\\
&\geq {\left\lVertT_\sigma u\right\rVert}_{L^p ({\mathbb{T}})} - {\left\lVertK u\right\rVert}_{L^p ({\mathbb{T}})}\\
&\geq {\left\lVertT_\sigma u\right\rVert}_{L^p ({\mathbb{T}})} - \varepsilon {\left\lVert u\right\rVert}_{L^p ({\mathbb{T}})}\\
&\geq {\left\lVert\sigma (\cdot, k_l) u_{k_l}\right\rVert}_{L^p ({\mathbb{T}})} - 2 \varepsilon {\left\lVertu\right\rVert}_{L^p ({\mathbb{T}})}\\
& \geq |\sigma (x_{k_l},k_l)| {\left\lVertu\right\rVert}_{L^p ({\mathbb{T}})} - 3 \varepsilon {\left\lVertu\right\rVert}_{L^p ({\mathbb{T}})}\\
&= (|\sigma (x_{k_l},k_l)| - 3 \varepsilon) {\left\lVertu\right\rVert}_{L^p ({\mathbb{T}})}.\end{aligned}$$
Letting $l \to \infty$, we get ${\left\lVertT_\sigma - K\right\rVert}_{\mathcal{L} (L^p ({\mathbb{T}}))} \geq d_\sigma - 3 \varepsilon,$ and finally, using the fact that $\varepsilon$ is an arbitrary positive number, we have $${\left\lVertT_\sigma - K\right\rVert}_{\mathcal{L} (L^p ({\mathbb{T}}))} \geq d_\sigma.$$
**Acknowledgments**. {#acknowledgments. .unnumbered}
====================
I sincerely thank the guidance of Carlos Andres Rodriguez Torijano who proposed me this research as the first step in my career as a mathematical researcher. I also want to thank professor Michael Ruzhansky for his comments.
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abstract: 'We study the continuous absorbing-state phase transition in the one-dimensional diffusive epidemic process via mean-field theory and Monte Carlo simulation. In this model, particles of two species (A and B) hop on a lattice and undergo reactions B $\to$ A and A + B $\to$ 2B; the total particle number is conserved. We formulate the model as a continuous-time Markov process described by a master equation. A phase transition between the (absorbing) B-free state and an active state is observed as the parameters (reaction and diffusion rates, and total particle density) are varied. Mean-field theory reveals a surprising, nonmonotonic dependence of the critical recovery rate on the diffusion rate of B particles. A computational realization of the process that is faithful to the transition rates defining the model is devised, allowing for direct comparison with theory. Using the quasi-stationary simulation method we determine the order parameter and the survival time in systems of up to 4000 sites. Due to strong finite-size effects, the results converge only for large system sizes. We find no evidence for a discontinuous transition. Our results are consistent with the existence of three distinct universality classes, depending on whether A particles diffusive more rapidly, less rapidly, or at the same rate as B particles.'
address: |
Departamento de Física, Instituto de Ciências Exatas,\
Universidade Federal de Minas Gerais\
C. P. 702, 30123-970, Belo Horizonte, Minas Gerais - Brazil
author:
- 'Daniel Souza Maia[^1] and Ronald Dickman[^2]'
title: 'Diffusive epidemic process: theory and simulation'
---
Introduction
============
This work is devoted to the one-dimensional diffusive epidemic process (DEP) [@kree], a model system in which two kinds of particles, A and B, diffuse on a lattice and undergo reactions B $\to$ A and A + B $\to$ 2B. There is no intrinsic limit on the number of particles that may be present at a given site; the total number of particles is conserved. In the epidemic interpretation A represents a healthy organism and B and infected one, with the reactions above corresponding, respectively, to spontaneous recovery and transmission of disease on contact. Other interpretations are possible, for example, A could represent a properly folded protein and B a misfolded one, etc.
The DEP is a nonequilibrium stochastic model exhibiting a phase transition to an absorbing state [@marro; @hinrichsen; @lubeck04; @odor04]. Such phase transitions arise in many models of epidemics, population dynamics and autocatalytic chemical reactions, and have attracted much interest in nonequilibrium statistical mechanics, in efforts to characterize the associated universality classes. The simplest example is the contact process (CP), or its discrete-time version, directed percolation (DP). In the CP, each site of a lattice is either vacant (0) or occupied by a particle ($X$). Particles die (i.e., the reaction $X \to 0$) at rate 1, independent of the configuration of the rest of the system, and reproduce at rate $\lambda$ (reaction $X + 0 \to
2X$). The offspring particle survives if and only if the site it is sent to (selected at random from the neighbors of the reproducing particle) is vacant. The CP is therefore a minimal model for birth-and-death processes with local competition for space. The particle-free configuration is absorbing. It is known that for reproduction rate $\lambda < \lambda_c$, the stationary density of particles $\rho$ is zero, and that (in the infinite system size limit), $\rho$ grows continuously from zero as $\lambda$ is increased beyond $\lambda_c$. Thus $\rho$ serves as the order parameter for this phase transition.
Critical scaling in the contact process and allied models has been studied extensively, both theoretically and numerically. A central conclusion deriving from these studies is that critical behavior of the DP type is generic for models exhibiting a continuous phase transition to an absorbing state, in the absence of any additional symmetries or conserved quantities. (Note that the absorbing state may in some cases encompass more that one configuration, as is the case in the pair contact process [@pcp]. If these configurations are not related by any symmetry the critical behavior is still expected to fall in the DP universality class.) Particularly relevant in the present context is the fact that the [*diffusive*]{} CP (in which particles hop at finite rate) also belongs to the DP class [@iwandcp].
The DEP offers a more complicated scenario of scaling. Let $D_A$ ($D_B$) denote the diffusion rate of A (B) particles. An especially interesting aspect of the DEP is that the absorbing-state phase transition appears to belong to three distinct classes, depending on whether $D_A < D_B$, $D_A = D_B$, or $D_A >
D_B$. The renormalization group (RG) analysis of Oerding et al. [@oerding] predicts a continuous phase transition in the first two cases. But for $D_A >
D_B$ there is no fixed point, leading to the conjecture that the transition is discontinuous in this case. These authors provide numerical evidence of such a transition in two dimensions; numerical studies of the one-dimensional DEP, however, show a continuous phase transition [@fulco].
The diffusive epidemic process was initially studied via RG methods [@kree; @oerding; @wijland]. Numerical simulations for equal diffusion rates were reported in Ref. [@freitas], yielding results in disagreement with the RG prediction $\nu_\perp = 2$ (see [@janssende; @freitasr]). Subsequently, the simulations reported in [@fulco] appeared to resolve this point, but suggested other departures from RG predictions in the one-dimensional case.
Given the disagreement between theory and simulation, it is of interest to perform further analysis of the DEP. In the present work we study a one-dimensional version of the process, formulated in continuous time, in a manner allowing for a simple formulation of the master equation. We study this process (defined in detail in the following section), using one- and two-site mean-field theory (Sec. III), and Monte Carlo simulation (Sec. IV) using the quasi-stationary simulation method. Conclusions and open questions are discussed in Sec. V.
Model
=====
The DEP is defined on a lattice of $L^d$ sites. A configuration is specified by the set of variables $a_j$ and $b_j$, denoting the number of A and B particles at each site $j$. There are no intrinsic restrictions on the number of particles at each site, making this process, in the language commonly employed in the literature, a “bosonic" model. (To avoid confusion we note that quantum statistics do not enter into the problem.) The model is a continuous-time Markov process characterized by four kinds of transitions:
- Hopping of A particles to a randomly chosen nearest neighbor (NN) site, at rate $D_A$.
- Hopping of B particles to a randomly chosen NN site, at rate $D_B$.
- Transformation of B particles to A particles, at rate $r$.
- Transformation of A particles to B particles, in the presence of a B particle at the same site, at a rate of $\lambda$ per A-B pair.
This means that a given site $j$ loses (via diffusion) an A particle at rate $D_A a_j$ (similarly for loss of a B particle), undergoes the process B $\to$ A at rate $r b_j$, and the process A + B $\to$ 2B at rate $\lambda a_j
b_j$. Note that all transitions conserve the total particle number $N = \sum_j
(a_j + b_j)$.
The process involves a rather large set of parameters: $D_A$, $D_B$, $r$, $\lambda$, and the particle density $\rho = N/L^d$. Since one of the rates may be eliminated through a suitable scaling of time, we set $\lambda = 1$ from here on. This still leaves four control parameters. In the studies described below, we fix the diffusion constants and study the behavior in the $r$-$\rho$ plane, or fix $D_A$ and $\rho$ and treat $D_B$ and $r$ as the control parameters.
Mean-field theory
=================
We begin by studying the model using dynamic mean-field theory (MFT) [@marro]. Here we present the results for one- and two-site approximations in one dimension, which follow from the master equation for the probability distribution $P(a_1,b_1;...;a_L,b_L,t)$. Consider the evolution of the one-site marginal distribution $P(a,b)$:
$$\begin{aligned}
\nonumber \dot{P}(a,b) &=& \!\! D_A [(a+1)P(a+1,b) - aP(a,b)] + D_A
\sum_{a',b'} a' [P(a-1,b;a',b') - P(a,b;a',b')]
\\
\nonumber &+& D_B [(b+1)P(a,b+1) - bP(a,b)] + D_B \sum_{a',b'} b'
[P(a,b-1;a',b') - P(a,b;a',b')]
\\
&+& r [(b\!+\!1) P(a\!-\!1,b\!+\!1) - b P(a,b)] +
(b\!-\!1)(a\!+\!1)P(a\!+\!1,b\!-\!1) - ab P (a,b) \label{mft1}\end{aligned}$$
Here $P(a,b;a',b')$ is the joint probability distribution for a pair of NN sites. Eq. (\[mft1\]) is the first in a hierarchy of equations for the probability distributions of 1, 2,...,$n$,... sites. At each level of the hierarchy, the diffusion terms couple the $n$-site distribution to that for $n+1$ sites. One-site MFT truncates this hierarchy at lowest order, via the factorization $P(a,b;a',b') = P(a,b)P(a',b')$, leading to,
$$\begin{aligned}
\nonumber \dot{P}(a,b) &=& \!\! D_A [(a+1)P(a+1,b) - aP(a,b)] + D_A \rho_A
[P(a-1,b) - P(a,b)]
\\
\nonumber &+& D_B [(b+1)P(a,b+1) - bP(a,b)] + D_B \rho_B [P(a,b-1) - P(a,b)]
\\
&+& r [(b\!+\!1) P(a\!-\!1,b\!+\!1) - b P(a,b)] +
(a\!+\!1)(b\!-\!1)P(a\!+\!1,b\!-\!1) - ab P(a,b) \label{mft2}\end{aligned}$$
where $\rho_A = \sum_{a,b} a P(a,b)$ is the density of A particles and similarly for $\rho_B$. An equation for $\rho_A$ is found by multiplying the above equation by $a$ and summing over all values of $a$ and $b$, giving, $$\dot{\rho}_A = r \rho_B - \langle ab \rangle \label{mft3}$$ where $\langle a^m b^n \rangle \equiv \sum_{a,b} a^m b^n P(a,b) $. (Thus $\langle a \rangle = \rho_A$ and similarly for $\rho_B$.) Note that, as discussed in detail below, the cross-moment $\langle a b \rangle$ is in general different from the simple product of A and B particle densities. If we nevertheless set $\langle a b \rangle = \rho_A \rho_B$, we have $$\dot{\rho}_A = r\rho_{B} - \rho_{A} \rho_{B} \label{mft4}$$ With the constraint $ \rho_A + \rho_B = \rho$, constant, we then find $$\dot{\rho}_B = (\rho-r)\rho_{B} - \rho_{B}^2 \label{mft5}$$ showing that at this (lowest) level of approximation the order parameter $\rho_B$ satisfies the Malthus-Verhulst equation with reproduction rate $\rho -
r$. At this level, which may be called a “rate equation", a continuous phase transition occurs at $\rho = r$, independent of the diffusion rates; the stationary density of $B$ particles follows $\overline{\rho}_B = \rho - r$. While the more detailed mean-field approximations described below yield a more reliable prediction for the phase boundary, we observe, in all cases, a continuous phase transition, and a linear relation between $\overline{\rho}_B$ and $\rho - r_c$ in the vicinity of the critical point (i.e., the usual mean-field critical exponent $\beta = 1$). We find no hint of a discontinuous transition.
A somewhat better result is obtained integrating (numerically) the full set of one-site MF equations, Eq. (\[mft2\]). In numerical analysis, it is necessary to truncate this set of equations at cutoff values $a_c$ and $b_c$. This is justified since the probability distribution falls off exponentially for large $a$ and/or $b$. The cutoff leads to certain technical restrictions in the numerical analysis. Naturally, transitions of the form $a_c \to a_c + 1$ must be excluded from consideration. Moreover, a transition of the form $a \to a-1$, due to an A particle hopping away from the site of interest, must have its rate multiplied by $1 - P_A(a_c)$, where $P_A(a) = \sum_b P(a,b)$ is the one-site marginal distribution for the number of A particles. (Similar restrictions apply to transitions involving B particles.)
Once we take the full one-site probability distribution into account, the results for the phase boundary $\rho_c(r)$ depend on the diffusion rates. Two factors enter into this dependence. First, in the vicinity of the phase transition, the reaction terms cause the marginal distribution for the number of B particles to deviate significantly from a Poisson distribution, while rapid diffusion tends to make the distribution more Poisson-like. Second, the reactions cause the variables $a$ and $b$ to be anti-correlated \[that is, cov($a,b$) = $\langle ab \rangle - \rho_A \rho_B < 0$\], whereas rapid diffusion tends to eliminate this correlation.
Figure \[mft1\] shows the critical line $\rho_c (r)$ as predicted by the one-site MFT for various combinations of $D_A$ and $D_B$. The higher the diffusion rates, the more closely $\rho_c$ approaches the simple rate equation result $\rho_c = r$. For finite diffusion rates $\rho_c$ is always greater than $\rho$, due again to the anti-correlation of $a$ and $b$. The critical value $\rho_c$ appears to be more sensitive to $D_B$ than $D_A$. In fact, for $D_B =
0.2$, the curves for $D_A = 0.2$ and $D_A = 1$ are virtually identical. (For larger values of $D_B$, increasing $D_A$ does reduce the critical density.)
A richer description is obtained when we extend the approximation to two sites. There are (in general) 16 transitions into (and out of) a given state . Using the symmetry $P(a,b;a',b') =P(a',b';a,b)$, and factoring three-site probabilities so that $P(a,b;a',b';a'',b'') =
P(a,b;a',b')P(a',b';a'',b'')/P(a',b')$, the equation governing the two-site joint probability can be written as $$\begin{aligned}
\frac{dP(a,b;a',b')}{dt}& = & \frac{D_A}{2}[(a+1)P(a+1,b;a',b') +(a'+1)P(a,b;a'+1,b') \nonumber \\
& & +(a'+1)P(a-1,b;a'+1,b') +(a+1)P(a+1,b;a'-1,b')] \nonumber \\
& & \frac{D_B}{2}[(b+1)P(a,b+1;a',b') +(b'+1)P(a,b;a',b'+1) \nonumber \\
& & +(b'+1)P(a,b-1;a',b'+1) +(b+1)P(a,b+1;a',b'-1)] \nonumber \\
& & + \frac{D_A}{2}\left[\Phi_A(a-1,b)P(a-1,b;a',b')
+ \Phi_A(a'-1,b')P(a,b;a'-1,b') \right] \nonumber \\
& & + \frac{D_B}{2}\left[\Phi_B(a,b-1)P(a,b-1;a',b')
+ \Phi_B(a',b'-1)P(a,b;a',b'-1) \right] \nonumber \\
& & + r[(b+1)P(a-1,b+1;a',b') + (b'+1)P(a,b;a'-1,b'+1)] \nonumber \\
& & + (a\!+\!1)(b\!-\!1)P(a\!+\!1,b\!-\!1;a',b') + (a'\!+\!1)(b'\!-\!1)P(a,b,a'\!+\!1,b'\!-\!1) \nonumber \\
& & - \left\{ D_A(a+a') + D_B (b+b') + \frac{D_A}{2} [\Phi_A(a,b) + \Phi_A(a',b')] \right. \nonumber \\
& & + \left. \frac{D_B}{2} [\Phi_B(a,b) + \Phi_B(a',b')] + r(b+b') + ab + a'b'
\right\} P(a,b;a',b')\end{aligned}$$
where $$\Phi_A(a,b) = \frac{\sum_{a'}\sum_{b'} a'\,P(a,b,a',b')}{P(a,b)} \label{defphi}$$
is the conditional A-particle density at a site, given that one of its nearest neighbors has occupancy $(a,b)$. ($\Phi_B$ is defined analogously.) The above equations are integrated numerically using the fourth-order Runge-Kutta method and a cutoff of 10 for the variables $a$, $b$, $a'$ and $b'$. For densities $\rho \leq 2$, the error incurred is negligible. Figure 1 shows that the two-site approximation predicts a larger value of $\rho_c$ than does the one-site approximation, other parameters being equal.
We compare the mean-field predictions for the critical recovery rate against simulation for three representative cases (all for density $\rho=1$) in Table I. The site approximation overestimates $r_c$ by a factor of up to 3.3; in each case the two-site approximation yields a substantial improvement, although it still overestimates the critical value by a factor of up to 2.3. In principle, further improvement could be furnished using higher order approximations, but in the present case the computational demands seem excessive. \[The number of equations to be integrated in the $n$-site approximation is $[(a_c+1)(
b_c+1)]^n$.\]
An important aspect of the model is the anti-correlation between variables $a$ and $b$ at a given site. To quantify this we study $$Q_A \equiv \frac{\langle a b \rangle}{\rho_B} - \rho_A =
\frac{\mbox{cov}(a,b)}{\rho_B} \label{defQA}$$ At the critical point ($\rho_B \to 0$), $Q_A$ represents the excess density of A particles at a site bearing a B particle. In the one-site approximation, we find (at the respective critical points), $Q_A$ = -0.583, -0.298, and -0.0895, for $D_A\!=\!D_B$ = 0.2, 1, and 5, respectively. That is, the species are anti-correlated, and the magnitude of the correlation decreases with increasing diffusion rate, as expected. This effect appears even more markedly in the two-site approximation, where, for example, $Q_A = -0.447$ at the critical point with $D_A = D_B = 1$. (All results quoted are for density $\rho=1$.) Similar values are found in simulations. The two-site approximation shows that the anti-correlation of the numbers of A and B particles extends to the nearest neighbor site: $\langle a_j b_{j+1}\rangle/\rho_B - \rho_A = -0.444$ for the parameters noted above. The variables $b_j$ and $b_{j+1}$, on the other hand, show a strong positive correlation.
Since diffusion of B particles is essential to the survival of the process, one might suppose that as $D_B$ is reduced, the critical density would increase (for fixed recovery rate $r$), or the critical recovery rate $r_c$ decrease (for fixed density). Mean-field theory (at both levels) predicts otherwise, as shown in Fig. \[epg\]: the critical recovery rate exhibits a minimum at a small value, $D_B^*$, of the B diffusion rate, but for even smaller values, it grows rapidly. For $D_B > D^*$ the critical recovery rate grows systematically, and appears to saturate at the rate-equation value, $r_c=\rho$. Fig. \[epg\] shows that $r_c$ follows the same qualitative trends in simulations as in the mean-field approximations, although its numerical value is (generally) considerably smaller.
The reason for the increase in $r_c$ at small values of $D_B$ appears to lie in the possibility of accumulating many B particles at a single site. Suppose that, by some fluctuation, a site acquires several B particles. Since $D_B \ll
D_A$, this accumulation is relatively long-lived, and any A particles at this site will rapidly change to B, since the effective rate for the reaction $A \to
B$, at this site, is $b$, which is large compared to $r$, the rate of the inverse reaction, and also large compared to $D_A$. (Note as well that A particles straying onto such a site will readily change to B, and tend to remain at this site for a long time.) Indeed, for $D_B < D^*$, mean-field theory reveals that the one-site marginal distribution $P(b)$, though quite small for $b \geq 1$ decays very slowly with increasing $b$. The distributions $P(a)$ and $P(b)$ are compared in Fig. \[dist002g\] (for the case $\rho=1$, $r \simeq r_c = 0.688$, $D_A = 0.5$ and $D_B = 0.02$). Although (globally) almost all particles belong to species A, $P(b)$ decays much more slowly than $P(a)$ for large occupancies. (In this regime we increase $a_c$ and $b_c$ to 40 in the one-site MF calculations, to ensure that the cutoff does not affect the result.) $P(a)$ is well approximated by a Poisson distribution, while $P(b)$ shows strong deviations from this from. As shown in Fig. \[dist002g\] (inset), similar distributions are observed in simulation, in the regime $D_B
\ll D_A$.
Simulations
===========
We perform Monte Carlo simulations using a simulation algorithm designed to reproduce faithfully the transition rates defining the process. Simpler, more efficient computational models involving the four types of transition \[A and B diffusion, recovery (R), and infection (I)\], are possible, but do not correspond to the same set of transition rates. Our simulation method permits quantitative comparison with theoretical predictions, including systematic expansions of the master equation [@inprog].
The simulation consists of a sequence of events, each of which involves choosing the type of transition and the site at which it occurs. The choice of event type depends the total transition rates for each of the four processes, given by:
- Hopping of A particles: total transition rate $W_A = N_A D_A$, where $N_A = \sum_j a_j$ is the total number of A particles.
- Hopping of B particles: total transition rate $W_B = N_B D_B$.
- Transformation of B particles to A particles: total transition rate $W_R = r N_B$.
- Transformation of A particles to B particles: total transition rate $W_I = \sum_j a_j b_j$.
If we let $W_T = W_A + W_B + W_R + W_I$ denote the total transition rate for all processes, then the probability that the next event is of type $m$ ($=$ A, B, R or I), is $P_m = W_m/W_T$, while the mean time to the next event is $\Delta t = 1/W_T$. (After each event the time is advanced by this amount.) The next event is chosen at random from this set of probabilities. Once the event type is determined, the site at which it occurs must be chosen. For this purpose a number of lists are maintained. For example, if the chosen event is hopping of an A particle, we select a site at random from a list of all sites (call it the A-list) having $a_j > 0$. Not all sites on the A-list, however, are equally likely to host the event: since each A particle has the same hopping rate, the probability of the event occurring at site $j$ is proportional to $a_j$. We therefore keep track of the number $a_{max}$ of A particles at the site with the largest number of such particles. When site $j$ is chosen from the A-list, it is accepted for the next event with probability $p_{acc} = a_j/a_{max}$. In case of rejection, we again select a site ($k$, say) from the A-list and compare $a_k$ with $a_{max}$. In his manner we ensure that each A particle has the same likelihood of hopping. In a diffusion event the target site is chosen at random from the nearest neighbors of the original site.
The same procedure is adopted for the other three processes, necessitating the maintenance of a B-list and an AB-list (the latter containing all sites such that $a_j b_j > 0$). The rejection procedure outlined above, while necessary to maintain fidelity to the original transition rates, is computationally expensive. For this reason we restrict the present study to a relatively low density, $\rho =1$, since in this case the maximum values, $a_{max}$, $b_{max}$, and $(ab)_{max}$ are generally not very large. (One could in fact use lists of the positions of all A and B [*particles*]{}, instead of A- and B-bearing sites. But since the A-B reaction step requires an AB site list, we adopt a uniform procedure for all processes. Using particle instead of site lists for all processes except the A-B reaction could improve efficiency, particularly for large diffusion rates.)
In the studies reported here we sample the [*quasi-stationary*]{} (QS) distribution of the process, (that is, conditioned on survival), an approach that has been found very useful in the study of systems with an absorbing state [@qss]. (In fact, conventional simulations of “stationary" properties of models with an absorbing state actually study the quasi-stationary regime, given that the only true stationary state for a finite system is the absorbing one). We employ a recently devised simulation method that yields quasi-stationary properties directly [@qssim]. This is done by maintaining, and gradually updating, a set of configurations visited during the evolution; when a transition to the absorbing state is imminent the system is instead placed in one of the saved configurations. Otherwise the evolution is identical to that of a conventional simulation. (The set of saved configurations is updated by replacing one of the saved configurations with the current one, with a small probability $p_{rep}$ at each time step.)
The above scheme was shown [@qssim] to yield precise results, in accord with the exact QS distribution for the contact process on a complete graph, and with conventional simulations of the same model on a ring [@qssim]. The scheme has also been shown to yield results that agree, to within uncertainty, with the corresponding results of conventional simulations for a sandpile model [@qssand]. The advantage of the method is that a realization of the process can be run to arbitrarily long times. Thus, whereas in conventional simulations a large number of realizations must be performed to have a decent sampling of the quasi-stationary state, here every realization provides useful information. This leads to an order of magnitude improvement in efficiency in the critical region. For further details on the method see [@qssim].
We performed extensive simulations of the DEP on rings of $L=200$, 500, 1000, 2000, and 4000 sites, using the QS method. The number of saved configurations $M$ ranges from 1000, for $L=200$, to $100$ for $L=4000$. Values of $p_{rep}$ range from $10^{-4}$ to $10^{-5}$ (smaller values for larger systems). Two time scales appear to be relevant in the choice of the replacement probability $p_{rep}$. One is the mean residence time of a configuration on the list, $\tau_L= M/p_{rep}$. The other is the QS lifetime $\tau$, i.e., the mean time between attempts to visit the absorbing state. We find that our results are independent of the choice of $p_{rep}$ provided $\tau/\tau_L < 1$. This appears to be associated with the need to preserve configurations visited prior to the last attempted transition to the absorbing state. Of course one could make $\tau_L$ arbitrarily large by reducing $p_{rep}$, but this would prolong the memory of the initial state. (For this reason, we use a $p_{rep}$ ten times larger during the relaxation phase. The latter represents about 10% of the total simulation time.)
Initially, half the particles are of type A and half B; the particles are distributed randomly and independently over the sites, so that the distribution of $a$ and $b$ at a given site is essentially Poisson with mean 1/2. Each realization of the process runs for a certain maximum time, $T_m$, of up to $10^8$ time units. The results reported here represent averages over 4 - 12 independent realizations for each set of parameter values. Averages are taken in the QS regime, after discarding an initial transient, with a duration that depends on the system size. In practice we accumulate histograms of the time during which the system has exactly 1, 2,...n,..., B particles. The histograms are used to evaluate the mean B particle density, $\rho_B$ and the moment ratio [@dic-jaff]
$$m = \frac{\langle b^2 \rangle}{\langle b \rangle^2} .\label{defm}$$
We determine the critical recovery rate $r_c(\rho,D_A,D_B)$ using the criteria of power-law dependence of $\rho_B$ and $\tau$ on system size $L$ (i.e., the usual finite-size scaling relations $\rho_B \sim L^{-\beta/\nu_\perp}$ and $\tau\sim L^z$). To probe the three characteristic regimes of the process, three cases are studied in detail: $D_A = D_B = 0.5$; $D_A = 0.5$, $D_B=0.25$; and $D_A = 0.25$, $D_B = 0.5$.
Consider first the case $D_A = 0.5$, $D_B = 0.25$. Since it has been suggested that the phase transition is discontinuous for $D_A > D_B$, it is of interest to study the behavior of the order parameter $\rho_B$ as a function of the recovery rate $r$. In Fig. \[epd525\] we plot estimates for the order parameter in the limit $L \to \infty$, based on our data for $L$ = 200 - 4000. (The extrapolation can only be performed reliably for $r \leq 0.22$ with the data at hand.) It appears that the order parameter decays [*continuously*]{} to zero as $r \to r_c \simeq 0.23$. (We cannot rule out a weakly discontinuous transition on the basis of these data, but a continuous transition seems the more natural interpretation.) The evidence for a continuous transition is greatly strengthened by our observation of critical scaling, as we now discuss.
Data for the QS order parameter $\rho_B$ versus system size $L$ are shown on log scales (for the case $D_A = 0.5$, $D_B=0.25$) in Fig. \[fssrho\]. The data for $r=0.231$ curve upward, while those for $r=0.234$ curve downward, leading to the estimate $r_c = 0.2325(10)$. For this range of values we see good evidence of power-law scaling, as expected at a critical point. Similarly, the data for the QS lifetime $\tau$ (see Fig. \[fsstau\]) show power-law scaling for $r \simeq r_c$, and significant curvature away from $r_c$.
As a check on our procedure for determining $r_c$, we also perform, for $D_A =
0.5$, $D_B=0.25$, initial decay studies [@kockelkoren; @parkpark06]. In these studies the order parameter $\rho_B$ is followed as a function of time, using the same initial condition as in the QS studies. (Due to the large system size, the process does not enter the absorbing state on the time scale of the simulation.) Here the expected critical behavior is $\rho_B \sim t^{-\theta}$. Using deviations from the power law to identify off-critical values, a study of systems of $10^6$ sites, to a maximum time of 10$^7$, enables us to restrict $r_c$ to the interval \[0.231, 0.239\]. This is consistent with, but less precise than, the QS results. Analysis of the data for $t \geq 5 \times 10^5$ furnishes $\theta \simeq 0.5$. The lack of precision in these studies and in our result for the moment ratio highlights the slow convergence of these simulations, presumably reflecting strong finite-size effects.
The studies described above lead to estimates for the critical exponent ratios $\beta/\nu_\perp$ and $z = \nu_{||}/\nu_\perp$. To obtain a direct estimate of $\nu_\perp$, we study the order parameter in the neighborhood of the critical point $r_c$ for various lattice sizes. This permits us to estimate the correlation length exponent $\nu_\perp$, using the finite-size scaling relation for the order parameter,
$$\rho_B(\Delta,L)\propto L^{-\beta/\nu_\perp}\mathcal{F}
(L^{1/\nu_\perp}\Delta),$$
where $\Delta = r-r_c$ and $\mathcal{F}$ is a scaling function. This implies $$\left| \frac{\partial \ln \rho_B}{\partial r}\right|_{r_c} \propto
L^{1/\nu_\perp}.$$
For the case $D_A = 0.5$, $D_B=0.25$, the data for $L=$ 500, 1000, 2000 and 4000 yield the estimate $\nu_\perp= 2.3(3)$. We performed similar finite-size scaling analyses for the other cases.
The moment ratio $m$ is also useful in characterizing critical behavior. Our results, shown in Fig. \[fssm\], lead to estimates for the limiting ($L \to
\infty$) value except in the case $D_A = 0.5$, $D_B =0.25$, for which the moment ratio decreases systematically with system size $L$. The trend to smaller values of $m$ with increasing $D_B/D_A$ may reflect the reduced tendency for B particles to accumulate at a given site. Our results for the critical parameters are summarized in Table II.
We close this section with the observation that the DEP is characterized by unusually large and long-lived fluctuations, which make it difficult to obtain precise results in simulations. In the case of the contact process, for example, a system size of $L=1000$ is sufficient for the determination of critical exponents to rather high precision. As in other models dominated by diffusion, such as conserved sandpiles [@qssand] and the diffusive pair contact process (PCPD) [@henkelhinrichsen; @pcpdqss], scaling properties of the DEP emerge clearly only at larger sizes. One signal of this is the slow convergence (with $L$) of the order parameter moment ratio $m$.
Discussion
==========
We study the one-dimensional diffusive epidemic process using mean-field approximations and Monte Carlo simulations. Mean-field theory provides a qualitative description of the phase diagram, and reveals a surprising, non-monotonic dependence of the critical recovery rate on the B particle diffusion rate, confirmed in simulation. This reflects the tendency (for $D_B
\ll D_A$) for many B particles to accumulate at a single site, as revealed in the one-site probability distribution $P(b)$. Mean-field theory also captures qualitatively the anti-correlation in the number of A and B particles at a single site.
Although our simulations do not extend to sufficiently large systems to furnish precise values of critical exponents, our results, especially for the ratio $\beta/\nu_\perp$, clearly support the scenario of distinct universality classes for the three cases, $D_A < D_B$, $D_A = D_B$, and $D_A > D_B$. The transition appears to be continuous in all cases. Renormalization group analysis [@kree; @wijland; @oerding; @janssende] predicts (independent of the relative magnitudes of $D_A$ and $D_B$) that $z=2$ and $\nu_\perp = 2/d$. The simulation results are consistent with these predictions [*except*]{} for the case $D_A = 0.25$, $D_B = 0.5$, for which our estimates are significantly smaller. This may be due to the slow convergence noted above. We hope to address this point in future studies of larger systems. It would also be of interest to verify that the critical exponents are in fact insensitive to changes in the diffusion rates, within the three universality regimes that have been established. We expect the present results to serve as a point of reference for studies based on systematic expansions of the master equation.
[**Acknowledgments**]{}
We are grateful to Hugues Chaté for helpful comments. This work was supported by CNPq and Fapemig, Brazil.
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[Table I. Critical recovery rate $r_c$ in one- and two-site\
mean-field approximations, compared with simulation.]{}
$D_A$ $D_B$ $\;\;\;\; r_c$ (1-site) $\;\;\;\; r_c$ (2-site) $\;\;\;\; r_c$ (sim)
------- ------- ------------------------- ------------------------- ----------------------
0.5 0.25 0.5420 0.411 0.2325(10)
0.5 0.5 0.5771 0.429 0.1921(5)
0.25 0.5 0.5144 0.368 0.1585(3)
[Table II. Critical parameters from simulation.]{}
$ D_A$ $D_B$ $\beta/\nu_\perp$ $z$ $\nu_\perp$ $m$
-------- ------- ------------------- --------- ------------- -----------
0.5 0.25 0.404(10) 2.01(4) 2.3(3) $<$ 1.15
0.5 0.5 0.192(4) 2.02(4) 2.0(2) 1.093(10)
0.25 0.5 0.113(8) 1.6(2) 1.77(3) 1.06(1)
[^1]: [email protected]
[^2]: [email protected]
|
---
abstract: 'Properties of solitary waves in pre-compressed Hertzian chains of particles are studied in the long wavelength limit using a well-known continuum model. Several main results are obtained by parameterizing the solitary waves in terms of their wave speed and their asymptotic amplitude. First, the asymptotic amplitude is shown to be directly related to the continuum sound speed, and the ratio of asymptotic amplitude to peak amplitude is shown to describe the degree of dynamical nonlinearity in the underlying discrete system. Second, an algebraic relation is derived that determines the dynamical nonlinearity ratio in terms of the ratio of the solitary wave speed to the sound speed. In particular, highly supersonic solitary waves correspond to highly nonlinear propagating pulses in weakly compressed systems, and slightly supersonic solitary waves correspond to weakly nonlinear propagating pulses in strongly compressed systems. Third, explicit formulas for the physical height, width, impulse and energy of the solitary waves are obtained in both the strongly nonlinear regime and the weakly nonlinear regime. Asymptotic expansions are used to show that in the strongly nonlinear regime, solitary waves are well-approximated by Nesterenko’s compacton (having the same wave speed), while in the weakly nonlinear regime, solitary waves coincide with solitons of the Korteweg-de Vries (KdV) equation, with the same wave speed. All of these results are illustrated by means of exact solitary wave solutions, including the physically important case that models a chain of spherical particles.'
author:
- |
Stephen C. Anco$^1$ Michelle Przedborski$^{2,3}$\
\
\
\
\
\
title: |
Long wavelength solitary waves in Hertzian chains\
and their properties in different nonlinearity regimes
---
email: [email protected], [email protected]
Introduction {#sec:intro}
============
There has been considerable interest in the study of one-dimensional chains of discrete macroscopic particles that interact by a power-law contact potential [@Nesterenko1983; @Nesterenko1985; @Nesterenko1994; @Sinkovits1995; @Nesterenko1995; @Sen1996; @Coste1997; @Sen1998; @Chatterjee1999; @Hinch1999; @Hong1999; @Ji1999; @Manciu1999a; @Manciu1999b; @Sen1999; @Hascoet2000; @Manciu2001; @Nesterenko2001; @Sen2001; @Rosas2003; @Rosas2004; @Daraio2005; @English2005; @Nesterenko2005; @Daraio2006; @Job2007; @Sokolow2007; @Zhen2007; @Porter2008; @Sen2008; @Herbold2009; @Porter2009; @Rosas2010; @Starosvetsky2010; @Santibanez2011; @James2012; @Khatri2012; @Stefanov2012; @Takato2012; @Vitelli2012; @Yasuda2017; @HerboldNesterenko2012; @HerboldNesterenko2013; @WangWensrichOoi]. These potentials have the form $V=a\delta^{k+1}H(\delta)$ where $H$ denotes the Heaviside step function, $\delta$ is the dynamical overlap distance between adjacent particles, $a$ is a constant which depends on their material properties, and $k>1$ is determined by the geometry of their contact surface [@Spence1968; @Johnson1985; @Johnson2005; @Persson2006]. In particular, spherical particles have $k=3/2$ [@Hertz1882], while $k=2$, and $k=3$ are Hertz exponents for more complicated contact geometries [@Goddard1990; @Sun2011].
One of the original motivations was the experimental discovery [@Nesterenko1983; @Nesterenko1985; @Nesterenko1994] that the dynamical strain in these discrete systems can exhibit solitary waves, which are propagating non-dispersive localized compressive pulses. The existence of such waves makes these systems useful for a variety of physical applications related to shock absorption [@Nakagawa2003; @Sokolow2005; @Hong2005; @Doney2006; @Melo2006; @Doney2009; @Breindel2011; @Przedborski2015b] and energy localization [@Vergara2006; @Daraio2006energy; @Job2009; @Theocharis2009; @Theocharis2010; @Boechler2010].
Experimental and numerical results [@Nesterenko2001; @Daraio2006Tunability; @Sun2013] indicate that the typical wavelength of solitary waves compared to the size of the particles in the discrete system is large enough to allow the use of a nonlinear continuum model for studying analytical properties of the solitary waves. The strongest nonlinearity arises when the discrete system is either uncompressed, with adjacent particles being just in contact, or weakly compressed, with the dynamical overlap $\delta$ being approximately at least the size of the initial overlap.
The nonlinear continuum model for arbitrary $k>1$ is given by a highly nonlinear, fourth-order wave equation [@Nesterenko1994; @Porter2008; @Porter2009; @Nesterenko2001], which we will refer to as the long-wavelength Hertzian continuum (LWHC) wave equation. In this model, the spatial gradient of solutions of the LWHC wave equation locally corresponds to the total strain exhibited by pulses propagating in the discrete system. The total strain includes a contribution from the pre-compression, which corresponds to the asymptotic value of the spatial gradient. In particular, solutions whose asymptotic spatial gradient is zero provide a model for long-wavelength pulses in a discrete system with no pre-compression, while solutions having a non-zero asymptotic spatial gradient represent a model for long-wavelength pulses in a discrete system with non-zero pre-compression.
The LWHC wave equation has a well-known explicit solution [@Nesterenko1983; @Nesterenko1985; @Nesterenko1994] whose gradient is a periodic travelling wave. A single arch of this periodic travelling wave can be cut off in a sufficiently smooth fashion when $1<k<5/3$ to yield an exact compact nonlinear wave solution, called a compacton [@Nesterenko1983; @Nesterenko1994; @James2012]. For $k>5/3$, the cutoff is singular, which is sometimes not emphasized in the literature. Since the spatial asymptotic amplitude of the compacton is zero, this compacton describes a compact solitary wave which models a propagating strictly localized pulse in a discrete system with no pre-compression. No explicit solitary wave solutions displaying a non-zero asymptotic spatial gradient were known until some recent work [@PrzAnc] in which we obtained explicit exact solution expressions when $k=2,3$. (We also obtained explicit periodic travelling wave solutions, some of which represent propagating non-dispersive localized rarefactive pulses.) These exact solitary waves model propagating solitary wave pulses in a discrete system that has an arbitrary non-zero pre-compression.
The main purpose of the present paper is to investigate the qualitative and quantitative properties of solitary wave solutions of the LWHC wave equation in different nonlinearity regimes, and to illustrate these properties by using the known exact solitary wave solutions in the cases $k=2,3$, as well as in the physically important case $k=3/2$ whose exact solution will be derived here. Several interesting results are obtained.
First, the background strain given by the asymptotic spatial gradient of solitary wave solutions is shown to be directly related to sound speed as defined by the dispersion relation for the linearized LWHC wave equation. This relationship is used to show that all solitary waves are supersonic.
Second, the ratio of the background strain to the peak strain in solitary wave solutions is shown to describe the degree of dynamical nonlinearity in the underlying discrete system. An algebraic relation is derived that determines this dynamical nonlinearity ratio in terms of the ratio of the solitary wave speed to the sound speed. Specifically, highly supersonic solitary waves correspond to highly nonlinear propagating localized pulses in weakly compressed discrete systems, while slightly supersonic solitary waves correspond to weakly nonlinear propagating localized pulses in strongly compressed discrete systems.
Third, expressions for the height, width, impulse and energy of solitary waves are derived and shown to depend principally on the ratio of the wave speed to the sound speed. The width expression, which comes from an asymptotic analysis of the tail of solitary waves, has not appeared previously in the literature. It yields a width approximately equal to 5 particle diameters when $k=3/2$ in the strongly nonlinear case, which has been experimentally verified and reported many times in the literature. Beyond this, the expressions for the height, width, impulse and energy yield scaling relations that agree with ones known to hold in discrete systems.
Fourth, the properties of highly supersonic solitary waves are shown to be close to the same properties of the compacton. In particular, the height and width are approximately equal to the compacton height and width. An approximate expression for highly supersonic solitary waves is obtained, which gives a good approximation of the peak and the tail of these waves.
Fifth, the profile of slightly supersonic solitary waves is shown to be approximately a sech-squared profile which is the same as solitons of the Korteweg-de Vries (KdV) equation. In addition, a two-scale asymptotic expansion of the LWHC equation around a fixed background strain is shown to yield the KdV equation with a scaled time variable and a scaled space variable in a reference frame moving with the sound speed.
Finally, solitary waves with the same speed are compared across different nonlinearity regimes. This type of comparison has not been carried out previously.
The paper is organized as follows.
In section \[model\], we give a full derivation of the LWHC wave equation from the equations of motion for a one-dimensional homogeneous chain of $N\gg 1$ discrete particles with arbitrary (non-zero) pre-compression. We discuss the total strain and background strain for solutions of the LWHC wave equation, and show how the regimes of weak and strong nonlinearity can be formulated directly in terms of these two strains. We also discuss linearized wave solutions and derive their dispersion relation from which the sound speed is obtained. Finally, we review the conservation laws for impulse momentum, energy, and momentum in the LWHC wave equation.
In section \[solns\], we first review an exact quadrature expression for all solitary wave solutions. In the two cases $k=2$ and $k=3$, we note that the solitary waves have explicit expressions in terms of elementary functions, which were derived in our previous work [@PrzAnc]. In the case $k=3/2$, we present an implicit expression for the solitary waves in terms of elliptic functions. This result has not appeared previously in the literature.
In section \[properties\], we give the expressions for the height, width, impulse and energy of the solitary waves. Using these expressions, we discuss physical properties of the solitary waves and their dependence on the wave speed. We examine the properties in more detail in the cases when the wave speed is slightly supersonic and highly supersonic.
In section \[compare\], we explain how slightly supersonic waves have a weak nonlinearity, and highly supersonic waves have a strong nonlinearity. In the strongly nonlinear case, we show that the solitary waves and the compacton have similar features, and we present a useful approximate expression for the profile of the solitary waves. In the weakly nonlinear case, we show that the solitary waves are scaled KdV solitons. Finally, we present and discuss some scaling relations that hold among the energy, impulse, height and speed of solitary waves in both nonlinearity regimes.
We make some concluding remarks in section \[remarks\].
Some analytical details of our results are given in three appendices.
Long-wavelength continuum wave equation and its conservation laws {#model}
=================================================================
The equations of motion for a one-dimensional homogeneous chain of $N\gg 1$ discrete particles interacting by a general power-law contact potential are given by $$\label{discrete-eom}
m \ddot U_{i} = (k+1) a \big( (\delta_0-(U_{i}-U_{i-1}))^k - (\delta_0 -(U_{i+1}-U_{i}))^k \big),
\quad
i=2,\ldots, N-1$$ in terms of the particle displacements $U_{j}(t)$, $j=1,\ldots,N$, relative to their initial (equilibrium) positions, where $\delta_0\geq 0$ is the initial overlap between adjacent particles due to pre-compression of the chain at $t=0$, and $k>1$ is determined by the geometry of their contact surface. Here $m$ is the particle mass, and $a$ is a constant which depends on the particles’ material properties. The particles at each end of the chain obey similar equations of motion with a different potential that takes into account the boundary conditions. Since we will be interested in the continuum limit, in which $N\to\infty$, we will only need to consider the equations of motion for the $N-2$ interior particles.
In this system, the initial particle displacements are $U_{j}(0)=0$, $j=1,\ldots,N$ (since the chain is initially in its equilibrium configuration) at $t=0$. The system is initially uncompressed if $\delta_0=0$, in which case the initial separation between the center of mass of adjacent particles is $2R$, where $R$ is the particle radius. Instead, if $\delta_0>0$, the system has a pre-compression, in which case the initial separation between the center of mass of adjacent particles is $2R-\delta_0$. For $t>0$, the dynamical overlap between adjacent particles is given by $$\label{overlap}
\delta(t;i) = \begin{cases}
\delta_0+U_{i}(t)-U_{i+1}(t) & \text{ if } U_{i+1}(t)-U_{i}(t)< \delta_0\\
0 & \text{ if } U_{i+1}(t)-U_{i}(t) \geq \delta_0
\end{cases}$$ where the property $$\label{contact}
\delta(t;i)\geq 0$$ enforces the contact nature of the potential.
In typical experiments and numerical simulations, the initial particle velocities are $\dot U_{j}(0)=0$ for the interior particles $j=2,\ldots,N-1$, while for the two end particles, the velocities $\dot U_{1}(0)$ and $\dot U_{N}(0)$ correspond to imparting an initial sharp (short duration) impulse at one end or both ends of the chain. Such an impulse gives the chain a fixed amount of energy $\tfrac{1}{2}m \big ( \dot U^2_{1}(0) + \dot U^2_{N}(0) \big )$ [@Nesterenko1985; @Sokolow2007; @Job2007]. In this situation, a propagating compressive pulse is produced at each end in the system, where the wave amplitude is described by the dynamical overlap variable , with $$\label{initialoverlap}
\delta(0;i)=\delta_0$$ being the initial (background) amplitude. Since the pulse is compressive, the moving particles in the pulse will satisfy $U_{i+1}-U_{i}<0$.
Rarefactive pulses [@YanLiuYanZhaDuaYan; @Yasuda2017; @PrzAnc] can be produced by imparting more complicated initial conditions to the discrete system. The moving particles in these pulses will satisfy $0\leq U_{i+1}-U_{i} <\delta_0$. Note that $U_{i+1}-U_{i} =\delta_0$ corresponds to a broken contact between the particles.
For any type of pulse, if the dynamical overlap is recorded for one (or a few) particle(s), it gives the wave amplitude $\delta(t;i)$ as a function of time $t$ at one (or a few) fixed value(s) of $i$. When travelling wave pulses are considered, the amplitude profile $\delta(t;i)$ in $t$ will have the same shape (up to scaling) as a snapshot of the amplitude $\delta(t_0;i)$ as a function of particle number $i$ at a fixed time $t=t_0$.
The dynamical state of a precompressed system is said to be *strongly compressed* if the relative displacement between adjacent moving particles is much smaller than the size of the initial overlap: $$\label{strong:U}
|U_{i+1}(t)-U_{i}(t)|=|\delta(t;i)-\delta_0| \ll \delta_0,
\quad
t>0.$$ In this case the dynamical overlap will be approximately the same magnitude as the initial overlap, $$\label{strong}
\delta(t;i)\approx \delta_0,$$ and the resulting motion of the particles in the system will be *weakly nonlinear*.
Instead if the dynamical overlap is large compared to the initial overlap, then the dynamical state of a precompressed system is said to be *weakly compressed*: $$\label{weak}
\delta(t;i)\gg \delta_0,
\quad
t>0.$$ In this case the relative displacement between adjacent moving particles will be at least the size of the initial overlap, $$U_{i}(t)-U_{i+1}(t) \gtrsim \delta_0.$$ and the resulting motion of the particles in the system will be *strongly nonlinear*.
To derive a continuum wave equation for long wavelength pulses in a discrete system with arbitrary pre-compression $\delta_0\geq 0$, it is useful to begin with a change of variables $$\label{displacement}
U_i = u_i +i\delta_0,
\quad
i=1,\dots,N .$$ Physically, $u_i$ is the displacement of the $i$th particle as measured with respect to a reference system that has zero pre-compression. In terms of this variable, the equations of motion of the precompressed system become $$\label{discrete-sys}
m \ddot u_{i} = (k+1) a \big( (u_{i-1} -u_{i})^k - (u_{i} -u_{i+1})^k \big),
\quad
i=2,\ldots, N-1.$$ Then the continuum limit for this system is obtained by putting $$u_{i}(t) \to u(t,x),
\quad
u_{i\pm1}(t) \to u(t,x\pm 2R) = e^{\pm2R\partial_x}u(t,x),
\quad
i=2,\dots,N-1,$$ with $u_{i-1} -u_{i} \to W_-$ and $u_{i} -u_{i+1} \to W_+$, where $$\pm W_\pm = u(t,x) - u(t,x\pm2R)
= u - e^{\pm2R\partial_x}u
= -\sum_{n=1}^{\infty} \tfrac{(\pm 2R)^{n}}{n!} \partial_x^{n} u.$$ This yields $$\label{CWE}
\tfrac{m}{a(k+1)} u_{tt} = W_-^k - W_+^k$$ which is a continuum wave equation for $u(t,x)$.
The continuum limit of the wave amplitude is given by $$\label{amplitude}
\delta(t;i) = u_{i}-u_{i+1} \to W_+$$ which is a nonlocal variable, while the continuum limit of the displacement variable is given by $$\label{U}
U_i(t) = u_i(t)+i\delta_0 \to U(t,x) = u(t,x) + \tfrac{\delta_0}{2R}x .$$ It will be natural to suppose that the linear mass density of the discrete system is preserved in the continuum limit. This yields $$\label{massdens}
\rho = m/(2R-\delta_0)$$ for the continuum mass density.
The main step now consists of making a long wavelength expansion of the continuum wave equation in terms of $u(t,x)$. Consider a pulse of wavelength $\ell$, where $u_x$ is taken to be $O(1)$ while the derivative of $u_x$ is $O(1/\ell)$. The condition for the wavelength to be long compared to the particle size is that $\ell\gg 2R$. Then the nonlinear terms in the wave equation can be expanded in terms of the parameter $\epsilon = 2R/\ell \ll 1$ by using the relation $(2R)^n\partial_x^n u_x = O(\epsilon^n)$. Thus, we have $$\begin{aligned}
W_\pm^k & =
(2R)^k\Big( -u_x -\sum_{n=1}^{\infty} \tfrac{(\pm 2R)^{n}}{(n+1)!} \partial_x^{n} u_x \Big)^k \\
& =
(2R)^k \Big(
(-u_x)^k \pm k R (-u_x)^{k-1} (-u_{xx})
+kR^2 (-u_x)^{k-2}\big( \tfrac{2}{3}(-u_x)(-u_{xxx})
\\&\qquad
+\tfrac{1}{2}(k-1)(-u_{xx})^2 \big)
\pm kR^3 (-u_x)^{k-3}\big( \tfrac{1}{3}(-u_x)^2(-u_{xxxx})
\\&\qquad
+\tfrac{2}{3}(k-1)(-u_x)(-u_{xx})(-u_{xxx})
+\tfrac{1}{6}(k-1)(k-2)(-u_{xx})^3 \big)
+O(\epsilon^4) \Big) .
\end{aligned}$$ This expansion yields $$\begin{aligned}
\tfrac{m}{a(k+1)} u_{tt} = W_-^k - W_+^k
& = -k(2R)^{k+1} \Big(
(-u_x)^{k-1} (-u_{xx})
+\tfrac{1}{6}R^2\big( (k-1)(k-2) (-u_x)^{k-3}(-u_{xx})^3
\\&\qquad
+4(k-1)(-u_x)^{k-2} (-u_{xx})(-u_{xxx}) +2(-u_x)^{k-1} (-u_{xxxx}) \big)
+O(\epsilon^4) \Big).
\end{aligned}$$ Truncating the expansion at this order produces the well-known highly nonlinear fourth-order wave equation [@Nesterenko1994; @Porter2008; @Porter2009; @Nesterenko2001] for $u(t,x)$: $$\label{LWCE-u}
c^{-2} u_{tt} = (-u_x)^{k-1} u_{xx}
+\alpha (-u_x)^{k-3}u_{xx}^3 -\beta (-u_x)^{k-2} u_{xx}u_{xxx} +\gamma (-u_x)^{k-1} u_{xxxx}$$ where $$\label{coeffs}
\alpha = \tfrac{1}{6}R^2(k-1)(k-2),
\quad
\beta = \tfrac{2}{3}R^2(k-1),
\quad
\gamma = \tfrac{1}{3}R^2$$ and $$\label{c}
c^2 = ak(k+1)(2R)^{k+1}/m.$$ Note, in these expressions –, the constant $c$ has units of speed, and the constants $\alpha$, $\beta$, $\gamma$ have units of length-squared, while $u$ has units of length.
We will call equation the *long-wavelength Hertzian continuum* (LWHC) wave equation. We emphasize that it is a valid long-wavelength continuum limit of the discrete system with arbitrary pre-compression $\delta_0\geq 0$.
In the same limit, the wave amplitude corresponds to the $O(1)$ term in the expansion $W_+ = (2R)\big( -u_x +O(\epsilon) \big)$. This term, given by $W_+ \approx -(2R)u_x$, is proportional to the dimensionless strain defined by $$\label{strain}
v = -u_x .$$ Note this expression physically represents the *total strain* (in the long-wavelength continuum limit) which includes a contribution from the precompression $\delta_0$. In particular, if we use the continuum limit of the particle displacement and define the corresponding strain variable $V=-U_x$, then we have the relation $$\label{dynstrain}
V = v-\tfrac{\delta_0}{2R}$$ which has the physical meaning of the *dynamical strain* (in the long-wavelength continuum limit).
In terms of the total strain , we see that the condition enforcing the contact nature of the underlying potential is simply given by $$\label{straincond}
v > 0 .$$ Note that we exclude the possibility $v=0$ because it would correspond to having no contact between adjacent particles in the underlying discrete system. The initial precompression corresponds to the initial condition $$\label{initialstrain}
v|_{t=0} =v_0= \tfrac{\delta_0}{2R}$$ which describes a background strain. On physical grounds, $$\label{initialstrain_conds}
0< v_0 < 1 .$$
Compressive pulses in the continuum limit are described by the property $V\geq 0$ for the dynamical strain, whereas rarefactive pulses have the property $0\geq V > -v_0$. The weakly nonlinear regime (corresponding to strong compression in the discrete system) is characterized by the dynamical condition $|V| \ll v_0$; the strongly nonlinear regime (corresponding to weak compression in the discrete system) is characterized by the dynamical condition $V \gtrsim v_0$. These dynamical conditions can be expressed in terms of the total strain by $$\label{weaknonlin:V}
|v- v_0| \ll v_0$$ for the weakly nonlinear regime, and $$\label{strongnonlin:V}
v \gg v_0$$ for the strongly nonlinear regime.
Finally, we note that the total strain satisfies a long-wavelength wave equation given by the $x$-derivative of the LWHC wave equation: $$\label{LWCE-v}
c^{-2} v_{tt} = (v^{k-1} v_{x}+\alpha v^{k-3}v_{x}^3 +\beta v^{k-2} v_{x}v_{xx} +\gamma v^{k-1} v_{xxx})_x$$ with $v$ satisfying the conditions and .
Linearized (sound) waves
------------------------
In the continuum limit, waves that have very small amplitude relative to the size of the pre-compression physically describe the linearized sound waves of the continuum system. Long wavelength sound waves satisfy the linearized approximation of the long-wavelength wave equation , which is given in terms of the dynamical strain by $$\label{LWCE-V}
c^{-2} V_{tt} = (v_0^{k-1} V_{x}+\gamma v_0^{k-1} V_{xxx})_x$$ where $\gamma=\tfrac{1}{3}R^2$. In this approximation, $V$ is small compared to the initial total strain $v_0$: $$0<V\ll v_0=\tfrac{\delta_0}{2R} .$$ To obtain the dispersion relation, we substitute a harmonic mode expression $V=e^{i(\kappa x -\omega t)}$ into this wave equation, yielding $c^{-2} \omega^2 =v_0^{k-1} \kappa^2 -\tfrac{1}{3}R^2 v_0^{k-1} \kappa^4$. Since the wavelength is $\ell = 2\pi/\kappa$, which obeys $2R/\ell \ll 1$, we see that $c^{-2} \omega^2 =v_0^{k-1} (2\pi/\ell)^2( 1 -\tfrac{\pi^2}{3}(2R/\ell)^2)$ can be approximated by $c^{-2} \omega^2 \simeq v_0^{k-1} (2\pi/\ell)^2$. This yields a linear dispersion relation $$\pm \omega \simeq v_0^{(k-1)/2} c\kappa.$$ Hence the sound speed is given by $$\label{soundspeed}
c_0 = \omega/\kappa = \big(\tfrac{\delta_0}{2R}\big)^{(k-1)/2} c.$$
Conservation laws
-----------------
A conservation law for the LWHC wave equation is a local continuity equation $$\label{conslaw}
D_t T + D_x X =0$$ holding for all solutions $u(t,x)$ of the wave equation, where $T$ is the conserved density and $X$ is the spatial flux, which are given by functions of $t$, $x$, $u$, $u_t$, and $x$-derivatives of $u$ and $u_t$. As shown in Ref. [@PrzAnc], all conservation laws with conserved densities of the first-order form $T(t,x,u,u_t,u_x)$ are given by a linear combination of the following four conservation laws: $$\begin{aligned}
& \begin{aligned}
& T_1= -u_t u_x,
\\
& X_1 = \tfrac{1}{2} u_t^2 -c^2 \big( \tfrac{1}{3}R^2 (-u_x)^{k} u_{xxx} -\tfrac{k-2}{6} R^2 (-u_x)^{k-1} u_{xx}^2 -\tfrac{1}{k+1} (-u_x)^{k+1} \big);
\end{aligned}
\label{TX1}\\
& \begin{aligned}
& T_2 = \tfrac{1}{2} u_t^2 - c^2 \big( \tfrac{1}{6}R^2 (-u_x)^{k-1} u_{xx}^2 - \tfrac{1}{k(k+1)} (-u_x)^{k+1} \big),
\\
& X_2 = -c^2\big( \tfrac{1}{3}R^2 (-u_x)^{k-1}(u_t u_{xxx} -u_{tx} u_{xx}) -\tfrac{k-1}{6}R^2 (-u_x)^{k-2}u_t u_{xx}^2 - \tfrac{1}{k} (-u_x)^k u_t \big) ;
\end{aligned}
\label{TX2}\\
&\begin{aligned}
& T_3 = u_t,
\\
& X_3 = c^2\big( -\tfrac{1}{3}R^2 (-u_x)^{k-1} u_{xxx} +\tfrac{k-1}{6}R^2 (-u_x)^{k-2} u_{xx}^2 +\tfrac{1}{k} (-u_x)^k \big);
\end{aligned}
\label{TX3}
\\
&\begin{aligned}
& T_4 = u -tu_t,
\\
& X_4 = c^2 t \big( \tfrac{1}{3}R^2 (-u_x)^{k-1} u_{xxx} - \tfrac{k-1}{6}R^2 (-u_x)^{k-2} u_{xx}^2 -\tfrac{1}{k} (-u_x)^k \big) .
\end{aligned}
\label{TX4}\end{aligned}$$ In each of these expressions, we have omitted an overall factor consisting of the continuum mass density . When this factor is restored, the conservation laws and respectively describe the total momentum and energy of solutions $u(t,x)$; the conservation law describes the total impulse, while the conservation law is connected with the mean value of $u(t,x)$.
Exact solitary waves {#solns}
====================
We are interested in travelling wave solutions $$\label{u-travelwave}
u= \f(\zeta),
\quad
\zeta = x-\V t$$ of the LWHC wave equation . Substitution of expression into equation yields a fourth-order differential equation $$\label{f-ode}
(\V/c)^2 \f'' = (-\f')^{k-1} \f'' + \alpha (-\f')^{k-3}(\f'')^3 -\beta (\f')^{k-2}\f''\f''' +\gamma (-\f')^{k-1} \f''''$$ where $\V$ is the (constant) wave velocity, and where $c$, $\alpha$, $\beta$, $\gamma$ are given by expressions –.
The physical variable that will support solitary waves is the dynamical strain . For a travelling wave , this variable has the form $$V = -\f'(\zeta) -v_0 .$$ Since a solitary wave has a localized profile in $\zeta$, we want solutions for which $$\lim_{\zeta\to\pm\infty} V =0 .$$ In terms of the total strain , solitary waves are described by $$\label{f-cond}
v = -\f'(\zeta) >0$$ and $$\label{f-bc}
\lim_{\zeta\to\pm\infty} v = v_0 >0 .$$ Here $v_0$ is related to the pre-compression through the initial condition and represents the asymptotic (background) value of $v$.
It will be useful, mathematically, to work with dimensionless scaled variables: $$\label{scal-g-xi}
g = -\f'/\lambda,
\quad
\xi =\zeta/l$$ with $$\label{scaling}
\lambda = \left(\tfrac{1}{2}k(k+1)(\V/c)^2 \right)^\frac{1}{k-1} ,
\quad
l = \sqrt{\tfrac{1}{6}k(k+1)}R .$$ The inverse transformation from $g(\xi)$ to $v(t,x)$ is given by $$\label{change_vars}
v(t,x)= \lambda g((x-\V t)/l).
%\big(\tfrac{1}{2} (\V/c)^2 k(k+1)\big)^{1/(k-1)} g\big((\sqrt{6}/(R\sqrt{k(k+1)}))(x-\V t)\big)$$
In terms of these scaled variables , the travelling wave equation becomes a third-order differential equation $$\label{g-ode}
0= 2 g^{k-1} g''' +4(k-1) g^{k-2}g'g'' +(k-1)(k-2)g^{k-3}g'{}^3 +k(k+1) g^{k-1} g' -2 g',$$ and the asymptotic condition on solutions $g(\xi)$ becomes $$\label{g-bc}
\lim_{\xi\to\pm\infty} g(\xi) = g_0 >0$$ where $$\label{g0}
g_0 = v_0/\lambda = \tfrac{\delta_0}{2R} \left(\tfrac{1}{2}k(k+1)(\V/c)^2 \right)^\frac{1}{1-k} .$$
The third-order differential equation can be directly reduced to a first-order separable differential equation by using first integrals that arise from the conservation laws – (which do not contain $t$ and $x$ explicitly) admitted by the LWHC wave equation , as explained in Ref. [@PrzAnc]. This yields the ODE $$\label{scal-ode}
(g')^2 = g^{1-k} (g^2 + C_1 g +C_2 - g^{1+k})$$ where $C_1$ and $C_2$ are dimensionless arbitrary constants which correspond to scaled first integrals. The physical meaning of the two first integrals are, respectively, the spatial flux of the impulse in the rest frame of the travelling wave, and the spatial flux of energy/momentum in the rest frame of the travelling wave. Thus, for any solution $g(\xi)$, $C_2=E$ represents a dimensionless energy and $C_1 =I$ represents a dimensionless impulse.
We now view the first-order differential equation as being analogous to the energy integral for motion in a potential well $$\label{ode_energy_form}
g^{k-1} g'{}^2 + \mathcal{V}(g) = E$$ where $g^{k-1} g'{}^2$ plays the role of the kinetic energy, and where the potential energy is given by $$\label{V}
\mathcal{V}(g) = g^{1+k} -g(g+I)$$ which depends on a free parameter $I$. A comprehensive qualitative analysis has been carried out in Ref. [@PrzAnc] to determine the values of $(E,I)$ that lead to solitary wave solutions for $g(\xi)$ with the asymptotic boundary condition , and also to single-arch travelling wave (compacton) solutions for $g(\xi)$. This analysis, which we will summarize next, depends crucially on the shape of the potential .
Quadrature formula for solitary waves {#sec:solitary-examples}
-------------------------------------
Solitary waves correspond to motion for $g$ in which there is one turning point $g=g_1$ and one asymptotic equilibrium point $g=g_0$. These two points are roots of the effective energy equation $\mathcal{V}(g)=E$ such that $\mathcal{V}'(g_0)=0$ and $\mathcal{V}'(g_1)>0$. The condition of positive pre-compression combined with the shape of the potential $\mathcal{V}(g)$ implies that $g_1$ and $g_0$ belong to the intervals $$\label{solitary_g0}
0<g_0<g^* < g_1< 1$$ where $$\label{gstar}
g^* = \big( \tfrac{2}{k(k+1)}\big)^\frac{1}{k-1}$$ is the inflection point of the potential, $V''(g^*)=0$. The relationship between the asymptotic equilibrium point $g_0$ and the parameters $(E,I)$ is given by $$\label{solitary_EI}
E = g_0^2(1- k g_0^{k-1}),
\quad
I = (k+1)g_0^k - 2g_0$$ which leads to the corresponding parameter ranges $$\label{solitary_EI_range}
E>0,
\quad
0>I>I^*$$ where $$\label{Istar}
I^* = (1 + k) g^*{}^{k} - 2g^* = -\tfrac{2(k-1)}{k}g^* <0 .$$ Note $g_1$ is a function of $g_0$ as given by the positive root of the algebraic equation $$\label{g1}
0=\mathcal{V}(g_1)-E= g_1^{k+1} -g_1^2 +g_0(2-(k+1)g_0^{k-1})g_1 + g_0^2(kg_0^{k-1}-1)$$ in the interval . As shown in , $g_1$ is a decreasing function of $g_0$, with $g_1\to 1$ when $g_0\to 0$, and $g_1\to g^*$ when $g_0\to g^*$.
It will be useful for mathematical purposes to note that $E-\mathcal{V}(g) = (g-g_0)^2 A(g,g_0)$ holds, where $$\label{A}
A(g,g_0)=1- \partial_{g_0}\Big(g_0 \frac{g^k-g_0^k}{g-g_0} \Big),$$ which has the properties $$\label{A_props}
A(g_0,g_0)= 1-(g_0/g^*)^{k-1},
\quad
A(g_1,g_0)=0 .$$
All solitary wave solutions $g(\xi)$ are then given by the quadrature of the ODE : $$\label{solitary_ode_integral}
\int_{g}^{g_1} \frac{\sqrt{g^{k-1}}}{(g-g_0)\sqrt{A(g,g_0)}} \,dg = |\xi| ,$$ where $g_1$ is determined by the algebraic equation in terms of $g_0$, and where $g_0$ obeys the inequality . From this equation , $g(\xi)$ has the following two main features.
First, $g(\xi)$ has a single peak $g=g_1$ at $\xi=0$ and an asymptotic tail with $g\to g_0$ as $|\xi|\to \infty$. To see why, note that the extrema of $g(\xi)$ are determined by the roots of $0=E-\mathcal{V}(g)= (g-g_0)^2 A(g,g_0)$ in the interval $g_0\leq g\leq g_1$. The roots consist of $g=g_0$ and $g=g_1$. From the ODE , note that $g''=-\tfrac{1}{2}\big( g\mathcal{V}'(g) +(k-1)(E-\mathcal{V}(g)) \big)g^{-k}$. The properties of the potential $\mathcal{V}(g)$ then show $g''<0$ when $g=g_1$ and $g''=0$ when $g=g_0$. Hence, $g_1$ represents the peak value of $g(\xi)$. When $g=g_0$, it is straightforward to see that $g''=0$, $g'''=0$, and so on. Moreover, the integral clearly diverges as $g\to g_0$, whereby $|\xi|\to \infty$. Hence, $g_0$ represents the asymptotic value of $g(\xi)$.
Second, the asymptotic tail of $g(\xi)$ exhibits exponential decay $g\sim g_0 + (g_1-g_0)e^{\xi_0/\chi} e^{-|\xi|/\chi}$ for $|\xi|\gg \xi_0 +\chi$, where the dimensionless scale for the decay is given by $$\label{decayscale_g0}
\chi = 1/\sqrt{g_0^{1-k} - {g^*}^{1-k}} ,$$ and the offset in $\xi$ is given by $$\label{decayoffset_g0}
\xi_0 = \int_{g_0}^{g_1} \frac{F(g,g_0)}{g-g_0} \,dg$$ with $$\label{F}
F(g,g_0) =
\frac{g^\frac{k-1}{2}}{\sqrt{A(g,g_0)}} - \frac{g_0{}^\frac{k-1}{2}}{\sqrt{A(g_0,g_0)}} .$$ To derive this asymptotic behaviour, we split up the integral by extracting the leading-order term that involves $g$, a subleading term that is constant, and a remainder term that vanishes when $g=g_0$: $$\int_{g}^{g_1} \frac{g^\frac{k-1}{2}}{(g-g_0)\sqrt{A(g,g_0)}} \,dg
= \frac{g_0{}^\frac{k-1}{2}}{\sqrt{A(g_0,g_0)}} \int_{g}^{g_1} \frac{dg}{g-g_0}
+\int_{g_0}^{g_1} \frac{F(g,g_0)}{g-g_0} \,dg
-\int_{g_0}^{g} \frac{F(g,g_0)}{g-g_0} \,dg .$$ The first and second integrals on the righthand side yield, respectively, $\chi\ln\big((g_1-g_0)/(g-g_0)\big)$ and $\xi_0$, while the remaining integral is $O(g-g_0)$ because $F(g,g_0)/(g-g_0)= O(1)$ due to $F(g_0,g_0)=0$. This gives $$\int_{g}^{g_1} \frac{g^\frac{k-1}{2}}{(g-g_0)\sqrt{A(g,g_0)}} \,dg
\simeq \chi\ln\Big(\frac{g_1-g_0}{g-g_0}\Big) + \xi_0$$ for $g\simeq g_0$. The quadrature then yields $\xi_0 -|\xi| \simeq \chi\ln\big((g-g_0)/(g_1-g_0)\big)$, which establishes the leading-order term in the asymptotic form for $g-g_0$.
Explicit solitary wave expressions
----------------------------------
We showed in Ref. [@PrzAnc] that the quadrature can be evaluated in terms of elementary functions when $k=2,3$ and in terms of elliptic functions when $k=\tfrac{3}{2},4,5$. All of these cases lead to implicit algebraic expressions for $g(\xi)$.
For the cases $k=2$ and $k=3$, the respective solutions $g(\xi)$ are given by $$\frac{g_0}{\sqrt{P(g_0)}} \arctanh\Big(Q(g)\Big)
+\arctan\bigg( \frac{\sqrt{P(g)}}{g + g_0-\tfrac{1}{2}} \bigg)
=|\xi|,
\label{solitarysol1}$$ with $$P(g) = (g_1-g)g,
\quad
g_1 = 1-2g_0,
\quad
g^*=1/3,
\label{solitarysol1-P}$$ and $$\frac{g_0}{\sqrt{P(g_0)}} \arctanh\Big(Q(g)\Big)
+\arctan\bigg( \frac{\sqrt{P(g)}}{g + g_0} \bigg)
=|\xi| ,
\label{solitarysol2}$$ with $$P(g) = (g_1-g)(g+g_1+2g_0),
\quad
g_1 = \sqrt{1-2g_0^2}-g_0,
\quad
g^*=1/\sqrt{6},
\label{solitarysol2-P}$$ where $$\label{Q}
Q(g)=\frac{2\sqrt{P(g)P(g_0)}}{P(g)+ P(g_0)+(g-g_0)^2} .$$ In both cases, $g_0$ is a free parameter in the range $0<g_0<g^*$, and $g_1$ is given explicitly in terms of $g_0$. (Note, here $\arctan$ is defined to be continuous in the given range for $g_0$.) See Fig. \[fig:solution\_plots\](b,c) for the solution profiles for different $g_0$.
For the case $k=\tfrac{3}{2}$, the solution $g(\xi)$ is given by $$\label{solitarysol3}
I(\sqrt{g}) + J(\sqrt{g}) = |\xi|,
\quad
g=h^2$$ where $$\label{solitarysol3-rat-terms}
\begin{aligned}
I(h) & =
\sigma \ln\bigg(\frac{2\sqrt{\Gamma}\big(Y(h)/Y_0+\sqrt{X(h)/X_0}\big)\big(\sqrt{X(h)}+\sqrt{X_0}\big)((h+h_2)^2+h_3^2)}{\Lambda(h-h_0)(1+Y(h)/Y_0)\sqrt{1+\varpi X(h)}}\bigg)
\\&\qquad
+ \arctan\bigg( \frac{2\sqrt{X(h)}}{1-X(h)} \bigg)
\end{aligned}$$ is a sum of elementary functions given in terms of the rational functions $$X(h)=\frac{h(h_1-h)}{(h+h_2)^2+h_3^2},
\quad
Y(h)=\frac{h-h_1/\Phi_+}{h +h_1/\Phi_-}$$ and where $$\label{solitarysol3-ellipt-terms}
\begin{aligned}
J(h) & = \eta \big( (\theta-\phi)Z(h) + \Pi\big({-\mu};Z(h)|\psi\big) -\theta \Pi\big(\nu;Z(h)|\psi\big) \big)
\end{aligned}$$ is a sum of Jacobi elliptic functions (of the third kind) [@Lawden1980] $$\begin{aligned}
&
\Pi(n;\theta|l)
%= \int_{0}^{\sn(\theta|l)} \frac{dz}{(1-nz^2)\sqrt{(1-z^2)(1-lz^2)}}
= \frac{1}{\sqrt{1+j}} \int_{\cn(\theta|l)}^{1} \frac{dz}{(mz^2-1)\sqrt{(1-z^2)(1+jz^2)}},
\\
& l=j/(j+1),
\quad
n =m/(m-1),
\quad
j>0,
\end{aligned}$$ with $$\label{solitary-sol3-cn_inverse}
Z(h)= \cn^{-1}\big((\Phi_+/\Phi_-)Y(h)|\psi\big)$$ being given by the inverse of the $\cn$ elliptic function. The constants in expressions – are given by $$\begin{gathered}
X_0= X(h_0),
\quad
Y_0= Y(h_0),
\quad
W_0 =(h_0+h_2)^2 +h_3^2,
\quad
W_1 =(h_1+h_2)^2 +h_3^2,
\label{solitarysol3-const1}
\\
\sigma =\frac{2h_0^2}{\sqrt{h_0(h_1-h_0)W_0}},
%=\frac{2h_0^2}{\sqrt{h_0(h_1-h_0)B_2(h_0)}}
\quad
\varpi
%= \frac{(\Phi_+h_0 -h_1)^2(1-B_2(0)\Phi_-^2/h_1^2)}{(\Phi_- h_0+h_1)^2}
%= \frac{Y(h_0)^2}{Y(h_1)^2}(1-B_2(0)\Phi_-^2/h_1^2)
= \frac{Y_0^2\Phi_+^2}{\Phi_-^2}(1-1/\Omega^2),
\label{solitarysol3-const2}
\\
\eta
%= 2\frac{\Phi_+h_1}{2\Phi_-\sqrt[4]{B_2(0)B_2(h_1)}}
= \frac{2\Omega\Phi_+}{\Phi_-\sqrt{2(\Phi_++\Phi_-)}},
\quad
\phi
%= \frac{(1-Y(h_0))\sqrt{B_2(0)}}{\sqrt{B_2(h_0)}}
= (1-Y_0)\Lambda,
\quad
\theta = \frac{h_0 Y_0}{h_1-h_0},
\label{solitarysol3-const3}
\\
\psi
%= \frac{h_1^2-B_2(0)\Phi_-^2}{4\sqrt{B_2(0)B_2(h_1)}},
= (\Omega^2-1)\frac{\Phi_-^2}{2(\Phi_++\Phi_-)},
\quad
\mu
%= \frac{\Phi_-^2\sqrt{B_2(0)}}{4\sqrt{B_2(h_1)}}
%= \frac{\Phi_-^2}{2(\Phi_++\Phi_-)}
= \frac{\psi}{\Omega^2-1},
\quad
\nu
%= \frac{B_1(h_0)(h_1^2/B_2(0) -\Phi_-^2)}{(\Phi_- h_0+h_1)^2}
%= (\Omega^2-1)\Phi_-^2\Gamma
=\frac{\psi\Gamma}{2(\Phi_+ +\Phi_-)},
\label{solitarysol3-const4}
\\
\Phi_\pm = \frac{\sqrt{W_1}}{\sqrt{h_2^2+h_3^2}} \pm 1,
%= \frac{\sqrt{B_2(h_1)}}{\sqrt{h_2^2+h_3^2}} \pm 1
%= \frac{\sqrt{B_2(h_1)}}{\sqrt{B_2(0)}} \pm 1
\quad
\Omega = \frac{h_1}{\Phi_-\sqrt{h_2^2+h_3^2}},
%= \frac{h_1}{\Phi_-\sqrt{B_2(0)}}
\quad
\Gamma = \frac{h_0(h_1-h_0)}{(\Phi_-h_0+h_1)^2},
\quad
\Lambda = \frac{\sqrt{h_2^2+h_3^2}}{\sqrt{W_0}},
\label{solitarysol3-const5}\end{gathered}$$ with $$\begin{aligned}
& h_1 = \tfrac{1}{3}-\tfrac{2}{3}h_0 + r_-^{1/3} - r_+^{1/3},
\label{h1-expr}\\
& h_2 = -\tfrac{1}{3}+\tfrac{2}{3}h_0 + \tfrac{1}{2}(r_-^{1/3} - r_+^{1/3}),
\label{h2-expr}\\
& h_2^2+h_3^2 = (h_0 -\tfrac{2}{3})h_0 + r_-^{2/3} +r_+^{2/3} + (\tfrac{2}{3}h_0-\tfrac{1}{3})( r_-^{1/3} - r_+^{1/3} ),
\label{h2h3-expr}\\
& r_{\pm} =
\tfrac{5}{12} \sqrt{h_0^3(h_0-\tfrac{2}{3})(h_0^2-\tfrac{2}{5}h_0-\tfrac{4}{25})} \pm \tfrac{5}{108}(h_0^3+\tfrac{24}{5}h_0^2-\tfrac{12}{5}h_0-\tfrac{4}{5}) .\end{aligned}$$ In this solution, all of these constants depend only on $h_0$ which is a free parameter in the range $0<h_0<h^*=\tfrac{8}{15}$, corresponding to $0<g_0<g^*=\tfrac{64}{225}$. Furthermore, the constants – are positive. (Some details of the derivation of the solution are shown in Appendix \[appendix:k=3/2\].) See Fig. \[fig:solution\_plots\](a) for an illustration of the solution profile for different values of $g_0$.
The solutions for the other two cases $k=4$ and $k=5$ are similar to the case $k=\tfrac{3}{2}$ and will be omitted.
![ Solitary wave solutions , and , for $k=\tfrac{3}{2}, 2$ and $3$, respectively. []{data-label="fig:solution_plots"}](hertz_solitarywave_g_soln_plots.eps){width="\textwidth"}
\[sec:compacton\]Single-arch travelling wave (compacton)
--------------------------------------------------------
The compacton corresponds to motion for $g$ in which there is one turning point $g=g_1=1$ and one equilibrium point $g=g_0=0$, with $\mathcal{V}'(0)=\mathcal{V}(0)=0$ and $\mathcal{V}'(1)>0$. The parameters $(E,I)$ are given by $$\label{compacton_EI}
E = I = 0$$ which are the limiting values of $E$ and $I$ in the case of solitary waves when $g_0$ approaches $0$. For these values , the solution $g(\xi)$ of the ODE is given by the quadrature $$\label{nodal_ode_integral}
\int_{g}^{1} \frac{\sqrt{g^{k-3}}}{\sqrt{1-g^{k-1}}} \,dg = |\xi| .$$ Here the integral can be evaluated explicitly for all $k>1$, yielding $$\label{periodicsol}
g(\xi) = \cos\big(\tfrac{1}{2}(k-1)|\xi-L\lfloor \tfrac{1}{2} + \xi/L\rfloor|\big)^\frac{2}{k-1}\geq 0,
\quad
L=\tfrac{2}{k-1}\pi$$ where $\lfloor x \rfloor$ denotes the floor function. The solution is a periodic function of $\xi$ having a peak $g=1$ at $\xi=0 \mod L$, and a node $g=0$ at $\xi=\pm L/2 \mod L$. The node is a minimum if $1<k<3$, or a corner if $k=3$, or a cusp if $k>3$. We remark that this solution has appeared in but with the phase shift omitted. Without the phase shift, the power $\tfrac{2}{k-1}$ must be interpreted as the $(k-1)^\th$ positive real root of the square of the cosine.
Since the solution has a node at $\xi=\pm L/2$, we can cut off the solution expression at these two points and take $g(\xi)$ to vanish outside the domain $-L/2\leq \xi \leq L/2$. This yields a piecewise expression $$\label{compactonsol}
g(\xi) =
\begin{cases}
\cos\big(\tfrac{1}{2}(k-1)\xi\big)^\frac{2}{k-1}, & |\xi|\leq \tfrac{1}{k-1}\pi
\\
0 , & |\xi|\geq \tfrac{1}{k-1}\pi
\end{cases}$$ which is continuous at the nodal points $\xi=\pm L/2$. However, this piecewise expression needs to be three-times differentiable for it to be an actual solution of the travelling wave ODE . In particular, we must have $0=g_\per(\pm L/2)=g_\per'(\pm L/2)=g_\per''(\pm L/2)=g_\per'''(\pm L/2)$, where $g_\per(\xi)$ denotes the periodic solution . These conditions are easily verified to hold if and only if $1<k<\tfrac{5}{3}$. More specifically, at the cutoff $\xi=\pm L/2$, we see $g_\per'(\xi)$ is discontinuous when $k\geq 3$; $g_\per''(\xi)$ is discontinuous when $k\geq 2$; and $g_\per'''(\xi)$ is discontinuous when $k\geq \tfrac{5}{3}$.
The resulting solution , which consists of a single arch of the periodic solution , is a compacton. It exists only for $k$ in the range $$1<k<\tfrac{5}{3} .$$ This condition on $k$ is often overlooked in the literature.
Physical properties of the solitary waves {#properties}
=========================================
We first express the quadrature formula for solitary waves in terms of the physical variables representing the total strain .
All solitary waves $v=v(x-\V t)$ are determined implicitly by $$\label{phys_solitarywave}
\frac{c}{\sqrt{3}|\V|}\int_{v}^{v_1} \frac{\sqrt{v^{k-1}}}{(v-v_0)\sqrt{\hat A(v,v_0;|\V|/c)}} \,dv = \frac{|x-\V t|}{R}$$ where $\hat A(v,v_0;|\V|/c)=A(v/\lambda,v_0/\lambda)$ is given by the function , and $v_1$ is the positive root of the algebraic equation $$\label{v1Aeqn}
\hat A(v_1,v_0;|\V|/c)=0 ,$$ such that $$\label{v0v1interval}
0<v_0 < v^* < v_1 < \lambda$$ with $$\label{vstar}
v^* = (|\V|/c)^\frac{2}{k-1} = \big(\tfrac{2}{k(k+1)}\big)^\frac{1}{k-1}\lambda .$$
The shape of the solitary waves consists of a single peak $v=v_1$ at $x=\V t$ and an asymptotic tail $v\to v_0$ as $|x|\to\infty$, with the value of $v_0$ determined by the initial (background) pre-compression . For $v_0\to 0$, we have $v_1\to \lambda$, while for $v_0\to v^*$, we have $v_1\to v^*$.
A useful observation is that the background strain $v_0$ is related to the sound speed $c_0$ through equations and : $$\label{v0c0rel}
c_0/c = v_0{}^\frac{k-1}{2} .$$ As a consequence, the inequality $v_0<v^*$ can be expressed as the corresponding physical relation $$\label{minspeed}
|\V| > c_0$$ between the solitary wave speed $\V$ and the sound speed $c_0$. This shows that all solitary waves are supersonic, which is a well-known statement in the literature in the case of weak compression. Here we see that it holds for arbitrary compression.
Height and width
----------------
Solitary waves describe physical compression waves measured by the total strain, where $v_1$ is the maximum (peak) strain of the wave and $v_0$ is the asymptotic background strain on which the wave is propagating. The amplitude $v$ of a solitary wave away from the peak exhibits a spatial decay to the background amplitude $v_0$. As shown by the results in Section \[solns\], this decay is exponential $$\label{expdecay}
v-v_0 \sim
%(v_1-v_0) e^{-(|x-\V t|-\zeta_0)/l_\s}
(v_1-v_0)e^{\zeta_0/l_\s} e^{-|x-\V t|/l_\s},
\quad
|x-\V t| \gg \zeta_0 + l_\s$$ with respect to the length scale $l_\s=l\chi$ and the offset $\zeta_0 = \xi_0 l$, given in terms of expressions , , . Explicit expressions for the length scale and the offset are given by $$\label{decayscale}
l_\s =\tfrac{1}{\sqrt{3}} R\big/\sqrt{(\V^2/c^2) v_0^{1-k} - 1}$$ and $$\label{decayoffset}
\zeta_0 = \sqrt{\tfrac{k(k+1)}{6}}R\int_{v_0}^{v_1} \frac{\hat F(v,v_0;|\V|/c)}{v-v_0} \,dv$$ where $\hat F(v,v_0;|\V|/c) = F(v/\lambda,v_0/\lambda)$ is given by the function .
In terms of the dynamical strain , solitary waves have a peak amplitude $$\label{height}
h_\s = v_1 - v_0$$ which defines the height of the solitary wave. For a fixed wave speed $\V$, this height has the properties $h_\s\to 0$ as $v_0\to v^*$, and $h_\s \to \lambda$ as $v_0\to 0$, where $\lambda = (\tfrac{k(k+1)}{2}\V^2/c^2)^\frac{1}{k-1}$.
The physical width of a solitary wave can be defined by $\ell_\s =2|\zeta|$ such that $(v(\zeta)-v_0)/h_\s = e^{-S} \ll 1$, for some choice of $S>0$. We will choose $S=\S$, which captures all of the hump-shaped part of the solitary waves. From the asymptotic tail of $v(\zeta)$, this yields $$\label{width}
\ell_\s = 2(\zeta_0 + \S l_\s)
= \tfrac{2}{\sqrt{3}}R\hat\ell(|\V|/c,v_0) ,$$ where $$\label{hatell}
\hat\ell(|\V|/c,v_0) = \sqrt{\tfrac{k(k+1)}{2}}\int_{v_0}^{v_1} \frac{\hat F(v,v_0;|\V|/c)}{v-v_0} \,dv + \S/\sqrt{(\V^2/c^2) v_0^{1-k} - 1}$$ is an explicit dimensionless scale depending on $v_0$ and $\V$. Note that for $|x-\V t| \gg \ell_\s/2$ the amplitude of the solitary wave decays exponentially with respect to the physical length scale .
For a fixed wave speed $\V$, the physical width has the properties $\ell_\s\to\infty$ as $v_0\to v^*$, and $\ell_\s\to \sqrt{\tfrac{2}{3}}\tfrac{\sqrt{k(k+1)}}{k-1} \pi R$ as $v_0\to 0$. The first property is an immediate consequence of the limit $\S/\sqrt{(v_0/v^*)^{1-k} - 1} \to \infty$ for the algebraic term in expression , while the integral term goes to $0$ due to $v_1\to v^*$. To derive the second property, we note first that the algebraic term in expression goes to $0$, and the integral term reduces to $\int_{0}^{\lambda} (\hat F(v,0;|\V|/c)/v) \,dv$, where $\hat F(v,0;|\V|/c)= (v/\lambda)^\frac{k-1}{2}/\sqrt{\hat A(v,0;|\V|/c)}$ with $\hat A(v,0;|\V|/c) = 1- (v/\lambda)^{k-1}$. We can directly evaluate this integral to get $\tfrac{\pi}{k-1}$, which gives $\hat\ell \to \sqrt{\tfrac{k(k+1)}{2}}\tfrac{\pi}{k-1}$.
Since the LWHC wave equation is valid only for long wavelength waves, we must have $\ell_\s\gg 2R$, which yields the condition $\tfrac{\sqrt{k(k+1)}}{k-1} \gg \sqrt{6}/\pi$. If we impose a lower bound $\ell_\s\gtrsim 4R$, then this implies $\tfrac{\sqrt{k(k+1)}}{k-1} \gtrsim 2\sqrt{6}/\pi$, which holds for all $1<k\lesssim 3.6$.
Impulse and energy
------------------
In experiments on pre-compressed chains, compressive waves are generated by striking an end particle in the chain. This corresponds to imparting a specified total impulse and total energy to the discrete system. For the continuum system, described by the LWHC wave equation , the total impulse and the total energy of travelling waves $v=v(\zeta)$ for the total strain in terms of $\zeta = x-\V t$ are given by the respective conserved integrals $$\label{unreg_impulse}
\I = \int_{-\infty}^{\infty} \rho\V v\,d\zeta$$ and $$\label{unreg_energy}
\E = \int_{-\infty}^{\infty} \rho\big( \tfrac{1}{2}\V^2 v^2 + c^2 ( \tfrac{1}{k(k+1)} v^{k+1} -\tfrac{1}{6}R^2 v^{k-1} {v'}^2 ) \big) \,d\zeta$$ which arise directly from the conservation laws and . Here $\rho$ is the continuum mass density . Both of these integrals are divergent for solitary waves, because $v(\zeta)$ has a non-zero asymptotic tail $v\to v_0>0$ as $|\zeta|\to \infty$. However, we can remove the divergent contribution by subtracting the respective impulse density and energy density obtained for $v=v_0$. This yields the regularized impulse integral $$\label{reg_impulse}
\I_\s = \int_{-\infty}^{\infty} \rho\V (v-v_0)\,d\zeta$$ and the regularized energy integral $$\label{reg_energy}
\E_\s = \int_{-\infty}^{\infty} \rho\big( \tfrac{1}{2}\V^2 (v^2-v_0^2) + c^2 ( \tfrac{1}{k(k+1)} (v^{k+1}-v_0^{k+1}) -\tfrac{1}{6}R^2 v^{k-1} {v'}^2 ) \big) \,d\zeta ,$$ both of which will be finite for all solitary waves. These regularized integrals can be simplified by using the method explained in . We obtain $$\label{impulse}
\I_\s = \tfrac{2}{\sqrt{3}} \rho Rc\, \sgn(\V)\hat\I(|\V|/c,v_0)$$ and $$\label{energy}
\E_\s = \tfrac{1}{\sqrt{3}} \rho Rc|\V|\hat\E(|\V|/c,v_0)$$ where $$\begin{aligned}
\hat\I(|\V|/c,v_0) & = \int_{v_0}^{v_1} \frac{\sqrt{v^{k-1}}}{\sqrt{\hat A(v,v_0;|\V|/c)}}\,dv
\label{impulse_integral}
\\
\hat\E(|\V|/c,v_0) & = \int_{v_0}^{v_1} \sqrt{v^{k-1}}\left( \frac{v^2-v_0^2 + \tfrac{2}{k(k+1)}(c/\V)^2( v^{k+1}-v_0^{k+1} )}{(v-v_0)\sqrt{\hat A(v,v_0;|\V|/c)}} - (v-v_0)\sqrt{\hat A(v,v_0;|\V|/c)} \right)\,dv
%\int_{v_0}^{v_1} v^{(k-1)/2}\left( \frac{v+v_0 + \tfrac{2}{k(k+1)}(c^2/\V^2)( \frac{v^{k+1}-v_0^{k+1}}{v-v_0} )}{\sqrt{\hat A(v,v_0;\V)}} - (v-v_0)\sqrt{\hat A(v,v_0;\V)} \right)\,dv
\label{energy_integral}\end{aligned}$$ are explicit dimensionless integrals depending on $v_0$ and $|\V|/c$.
Speed dependence of height, width, impulse, energy
--------------------------------------------------
When the background strain $v_0>0$ is fixed, all solitary waves comprise a one-parameter family given in terms of the wave speed $\V$ in the range . Note that fixing $v_0$ corresponds to fixing the sound speed (where $k$ is taken to be fixed). Then the impulse, energy, height and width of a solitary wave will depend solely on the ratio $|\V|/c_0$.
To understand the specific dependence of these physical properties on $|\V|/c_0$, we first need to determine the peak strain $v_1$ in terms of $c_0$ and $\V$. The algebraic equation for $v_1$ can be expressed in the explicit form $$\label{v1root}
0= \tfrac{k(k+1)}{2}(\V^2/c^2) (v_1-v_0)^2 -v_1^{k+1} +(k+1)v_0^{k}v_1 -kv_0^{k+1} .$$ For convenience, we will write $$\label{strainratio}
r=v_0/v_1 ,$$ which has the range $$\label{rrange}
0 <r < 1$$ given by the intervals . In terms of this strain ratio , the algebraic equation is given by $$\label{rroot}
0= k(r-1)r^{k-1}\big( \tfrac{k+1}{2}(\V^2/c_0^2) (r-1) -r \big) + r^{k} -1$$ by use of relation . Hence, for a given value of $k$, $r$ is a function only of the speed ratio $|\V|/c_0$. This function is shown in .
![ Strain ratio $r$, defined by , as a function of the speed ratio $|\V|/c_0$ and the exponent $k$ in the Hertz potential. []{data-label="fig:strainratio_plots"}](hertz_solitarywave_strainratio_plots.eps){width="45.00000%"}
It is useful to observe that scaling $c \to s^\frac{k-1}{2}c$ (with $s\neq 0$) implies $v_0\to s v_0$ and $v_1\to s v_1$, and thus $$\begin{aligned}
& \hat F(s v,s v_0;|\V|/c) =s^\frac{k-1}{2}\hat F(v,v_0;s^\frac{1-k}{2}|\V|/c),
\\
& \hat\I(|\V|/c,sv_0) = s^\frac{k+1}{2}\hat\I(s^\frac{1-k}{2}|\V|/c,v_0) ,
\\
& \hat\E(|\V|/c,sv_0) = s^\frac{k+3}{2}\hat\E(s^\frac{1-k}{2}|\V|/c,v_0) . \end{aligned}$$ Then, using these scaling properties combined with relation , we have the following properties of solitary waves expressed in terms of the strain ratio .
The height is given by $$\label{height:r}
h_\s = (c_0/c)^{\frac{2}{k-1}}(1/r-1) ,$$ while the width , impulse and energy are given in terms of the integrals $$\begin{aligned}
\hat\ell &= \sqrt{\tfrac{k(k+1)}{2}} \int_{1}^{1/r} \frac{\hat F(z,1;|\V|/c_0)}{z-1} \,dz +\S/\sqrt{\V^2/c_0^2 -1} ,
\label{width_integral:r}
\\
\hat\I & = (\tfrac{c_0}{c})^\frac{k+1}{k-1} \int_{1}^{1/r} \frac{\sqrt{z^{k-1}}}{\sqrt{\hat A(z,1;|\V|/c_0)}}\,dz ,
\label{impulse_integral:r}
\\
\hat\E & = (\tfrac{c_0}{c})^\frac{k+3}{k-1} \int_{1}^{1/r} \sqrt{z^{k-1}}\left( \frac{z^2-1 + \tfrac{2}{k(k+1)}(c_0/\V)^2( z^{k+1}-1 )}{(z-1)\sqrt{\hat A(z,1;|\V|/c_0)}} - (z-1)\sqrt{\hat A(z,1;|\V|/c_0)} \right)\,dz .
\label{energy_integral:r}\end{aligned}$$ Here $$\label{hatF}
\sqrt{\tfrac{k(k+1)}{2}} \hat F(z,1;|\V|/c_0) = (c_0/|\V|) \Bigg( \frac{\sqrt{z^{k-1}}}{\sqrt{\hat A(z,1;|\V|/c_0)}} - \frac{1}{\sqrt{\hat A(1,1;|\V|/c_0)}} \Bigg) ,$$ and $$\label{hatA}
\begin{aligned}
\hat A(z,1;|\V|/c_0) & = 1- \tfrac{2}{k(k+1)}(c_0/\V)^2 (k-z\partial_z)\Big(\frac{z^k-1}{z-1}\Big) ,
\\
\hat A(1,1;|\V|/c_0) & = 1- (c_0/\V)^2 .
\end{aligned}$$
The height, width, impulse, and energy are shown in for the explicit solitary wave solutions presented in Section \[sec:solitary-examples\], where the background strain $v_0$ has been fixed to be a typical value considered in experiments. It is evident from (a) that the physical height $h_\s$ of a solitary wave increases with increasing $|\V|/c_0$ (i.e., with decreasing $r$). Moreover, for a given speed ratio $|\V|/c_0$, $h_\s$ decreases with increasing $k$. In contrast, for a given value of $k$, the width $\ell_\s$ of a solitary wave increases with decreasing $|\V|/c_0$, implying that the width becomes larger when the initial pre-compression is increased.
From (b), we see that, when $|\V|/c_0 \gtrsim 1.5$, the width $\ell_\s$ decreases with increasing $k$ for a given $|\V|/c_0$. In contrast, when $|\V|/c_0 \lesssim 1.5$, this trend is inverted, with $\ell_\s$ increasing as $k$ increases. This means that solitary waves are wider for a larger values of $k$ when the wave speed $|\V|$ is less than $1.5$ times the sound speed $c_0$.
Similarly to the properties of the height and width, both the impulse and energy, (c) and (d), increase with increasing $|\V|/c_0$ for a given value of $k$. When $k$ and $|\V|/c_0$ are fixed, the impulse, and energy each increase if the background strain $v_0$ is increased, which can be inferred from expressions and . Physically, this means that a larger impulse and energy is needed to obtain waves with the same speed ratio when the initial pre-compression is increased.
![ Properties of the solitary wave solutions presented in Section \[sec:solitary-examples\], using a fixed background strain $v_0 = 0.02$. (a) Solitary wave height, ; (b) solitary wave width, ; (c) impulse carried by solitary waves, ; (d) energy carried by solitary waves, . []{data-label="fig:height_width_impulse_plots"}](hertz_solitarywave_height_width_impulse_energy.eps){width="95.00000%"}
We will now consider in more detail two interesting cases: wave speeds $|\V|$ close to the sound speed ; wave speeds $|\V|$ much larger than the sound speed . Note, since the background strain $v_0$ is fixed, specifying the speed ratio $|\V|/c_0$ corresponds to specifying $v^*$ through the relation $|\V|/c_0 = (v^*/v_0)^{(k-1)/2}$.
Slightly supersonic solitary waves
----------------------------------
When $|\V|$ is close to $c_0$, a solitary wave is slightly supersonic. To examine its properties, we first observe that $v^*$ will be close to $v_0$, whereby the strain ratio $r$ will be close to $1$ as seen from . By expanding equation for $r$ in a series in $|\V|/c_0 - 1$ around $r=1$, we obtain the asymptotic expression $$\label{r_near1}
r\simeq 1- \tfrac{6}{k-1}(|\V|/c_0 - 1)$$ valid for $|\V|/c_0 -1 \ll 1$.
A similar asymptotic expansion of the height expression and the width integral yields $h_\s \simeq \tfrac{6}{k-1} (c_0/c)^\frac{2}{k-1} (|\V|/c_0 - 1)$ and $\hat\ell \simeq \tfrac{\S}{\sqrt{2}}\big/\sqrt{|\V|/c_0 - 1}$. (see Appendix \[appendix:expansion\] for details). Thus, we see that the width of a slightly supersonic solitary wave becomes large in comparison to the particle size $2R$, $$\label{width:r_near_1}
\ell_\s/(2R) \simeq \tfrac{\S}{\sqrt{6}} \big/\sqrt{|\V|/c_0 - 1} \gg 1,$$ while the height becomes small compared to the background strain $v_0$, $$\label{height:r_near_1}
h_\s/v_0 \simeq \tfrac{6}{k-1} (|\V|/c_0 - 1) \ll \tfrac{6}{k-1} .$$ In particular, the height and width satisfy the proportionality relationship $$\label{heightwidthrel:r_near_1}
h_\s \propto 1/\ell_\s^2 .$$
The impulse integral and the energy integral have the respective asymptotic expansions $\hat\I \simeq \tfrac{3\sqrt{2}}{k-1}(\tfrac{c_0}{c})^\frac{k+1}{k-1}\sqrt{|\V|/c_0 - 1}$ and $\hat\E \simeq \tfrac{6\sqrt{2}(k+1)}{k(k-1)}(\tfrac{c_0}{c})^\frac{k+3}{k-1}\sqrt{|\V|/c_0 - 1}$ (see Appendix \[appendix:expansion\] for details). Thus the impulse and the energy of a slightly supersonic solitary wave are given by $$\label{impulse:r_near_1}
\I_\s \simeq \tfrac{2\sqrt{6}}{k-1} \rho Rc_0 (c_0/c)^\frac{2}{k-1} \sgn(\V)\sqrt{|\V|/c_0 - 1} ,$$ and $$\label{energy:r_near_1}
\E_\s \simeq \tfrac{2\sqrt{6}(k+1)}{k(k-1)} \rho Rc_0^2(c_0/c)^\frac{4}{k-1} \sqrt{|\V|/c_0 - 1} .$$ In particular, these two quantities are related by $\E_\s/\I_\s \simeq (1+\tfrac{1}{k})(c_0/c)^\frac{2}{k-1}c_0\,\sgn(\V)$, which depends only on the sound speed $c_0$ and the direction of the solitary wave (in addition to the constant $c$).
As a limiting case, for $|\V|\to c_0$, we have $\ell_\s/(2R)\to \infty$, $h_\s/v_0 \to 0$, $\I_\s \to 0$ and $\E_\s \to 0$.
Highly supersonic solitary waves
--------------------------------
When $|\V|$ is much larger than $c_0$, a solitary wave is highly supersonic. This implies $v^*$ will be much greater than $v_0$, whereby the strain ratio $r$ will be much less than $1$ as seen from . Consequently, by expanding equation for $r$ in a series in $(c_0/|\V|)^\frac{2}{k-1}$ around $r=0$, we obtain the asymptotic expression $$\label{r_near0}
r\simeq (\tfrac{2}{k(k+1)}c_0^2/\V^2)^\frac{1}{k-1}$$ valid for $|\V|/c_0 \gg 1$.
The properties of highly supersonic solitary waves can be determined using this asymptotic form of the strain ratio.
First, we asymptotically expand the height , which yields $$\label{height:strong}
h_\s \simeq (\tfrac{k(k+1)}{2})^\frac{1}{k-1} (|\V|/c)^\frac{2}{k-1} .$$ Hence, compared to the background strain $v_0$, the height becomes large, $$\label{height:r_gtr_1}
h_\s/v_0 \simeq (\tfrac{k(k+1)}{2})^\frac{1}{k-1} (|\V|/c_0)^\frac{2}{k-1} \gg (\tfrac{k(k+1)}{2})^\frac{1}{k-1} .$$
Next, an asymptotic expansion of the width integral yields $$\label{hatell:r_gtr_1}
\hat\ell \simeq \tfrac{\sqrt{k(k+1)}}{\sqrt{2}(k-1)}\pi$$ (as shown in Appendix \[appendix:expansion\]). Thus, the asymptotic expression for the width is given by $\ell_\s \simeq \sqrt{\tfrac{2}{3}}\tfrac{\sqrt{k(k+1)}}{k-1}\pi R$. In comparison to the particle size $2R$, the width is a finite multiple $$\label{width:r_gtr_1}
\ell_\s/(2R) \simeq \tfrac{1}{\sqrt{6}}\tfrac{\sqrt{k(k+1)}}{k-1}\pi ,$$ which depends only on $k$. This multiple is a decreasing function of $k$, such that $\ell_\s/(2R)\to \tfrac{1}{\sqrt{6}}\pi $ ($\simeq 1.28$) as $k\to\infty$ and $\ell_\s/(2R)\to \infty$ as $k\to 1$. The trend for large $k$ is in agreement with previous experimental findings [@Sen2008] for discrete systems, where it has been reported that $\ell_\s/(2R)$ tended to $1$.
In the physically important case $k=\tfrac{3}{2}$, the width is $\ell_\s/(2R)\simeq \tfrac{\sqrt{5}}{\sqrt{2}}\pi$ ($\simeq 4.97$). This agrees with the experimentally measured value [@Daraio2006Tunability] of approximately $10R$ for the physical width of solitary waves in weakly compressed granular chains when the contact geometry of the particles is elliptical, corresponding to $k=\tfrac{3}{2}$.
Similarly, we find that the impulse integral and the energy integral have the respective asymptotic expansions $\hat I \simeq \lambda^\frac{k+1}{2} \sqrt{\pi} \Gamma(\tfrac{1}{2} +\tfrac{1}{k-1})/\Gamma(\tfrac{1}{k-1})$ and $\hat E \simeq \lambda^\frac{k+3}{2}\tfrac{1}{4}\sqrt{\pi} \tfrac{k+3}{k+1}\Gamma(\tfrac{1}{2} + \tfrac{2}{k-1})/\Gamma(\tfrac{2}{k-1})$ (derived in Appendix \[appendix:expansion\]). Thus the impulse and the energy of a highly supersonic solitary wave are given by $$\label{impulse:r_gtr_1}
\I_\s \simeq \tfrac{2}{\sqrt{3}}\sqrt{\pi} K_1 \rho \V (|\V|/c)^\frac{2}{k-1} R ,
\quad
K_1 = \sqrt{\tfrac{k(k+1)}{2}}^\frac{k+1}{k-1} \tfrac{\Gamma(\frac{1}{2}+\frac{1}{k-1})}{\Gamma(\frac{1}{k-1})} ,$$ and $$\label{energy:r_gtr_1}
\E_\s \simeq \tfrac{1}{4\sqrt{3}}\sqrt{\pi} K_2 \rho \V^2 (|\V|/c)^\frac{4}{k-1} R ,
\quad
K_2 = \tfrac{k+3}{k+1}\sqrt{\tfrac{k(k+1)}{2}}^\frac{k+3}{k-1} \tfrac{\Gamma(\frac{1}{2} + \frac{2}{k-1})}{\Gamma(\frac{2}{k-1})} .$$ In particular, these two quantities are related by $\E_\s/\I_\s \simeq \tfrac{1}{8}(1+\tfrac{2}{k+1}) \hat K \V h_\s$ in terms of the height and the speed, with $\hat K = \Gamma(\tfrac{1}{k-1})\Gamma(\tfrac{1}{2}+\tfrac{2}{k-1})/\big(\Gamma(\tfrac{2}{k-1})\Gamma(\tfrac{1}{2}+\tfrac{1}{k-1})\big)$.
Comparison of solitary waves across different nonlinear regimes {#compare}
===============================================================
In discrete systems of pre-compressed particles, the features of compressive and rarefactive pulses are very dependent on the size of the dynamical overlap of adjacent particles compared to the size of the pre-compression. This dependence is measured in the continuum limit by the ratio of the dynamical strain to the background strain . Specifically, the dynamics of compressive long-wavelength pulses (in the continuum limit) is weakly nonlinear when the strain ratio satisfies the condition , or strongly nonlinear when the strain ratio satisfies the condition .
We will now explore how the main features of solitary waves described by the LWHC wave equation differs across these two different nonlinearity regimes.
To begin, we note that the peak total strain $v_1$ in a solitary wave is a function of the background strain $v_0$ and the wave speed $\V$, as given by equation , where $v_0>0$ and $\V >c_0$. Since the degree of dynamical nonlinearity is essentially determined by the ratio $1/r=v_1/v_0$, which has the range , we can regard solitary waves as being determined by $r$ and $\V$ as two independent parameters. Specifically, from the algebraic equation for $r$, combined with relation for $v_0$, we have $$\label{v0v1_r}
v_0 = \Big( \tfrac{k(k+1)}{2} \frac{(\V^2/c^2) (1-r)^2}{r^{1-k} -k(1-r)r-r} \Big)^\frac{1}{k-1},
\quad
v_1 = v_0/r ,$$ which explicitly expresses the background strain and the peak strain as functions of $r$ and $\V$. Since $v_0$ determines the sound speed $c_0$ through relation , note that expression yields $$\label{c0_r}
c_0 = \sqrt{\tfrac{k(k+1)}{2}} \frac{|\V| (1-r)}{\sqrt{r^{1-k} -k(1-r)r-r}}$$ as a function of $r$ and $|\V|$. Consequently, weak nonlinearity arises when solitary waves are slightly supersonic, whereas strong nonlinearity occurs when solitary waves are highly supersonic, which we will show explicitly from the sound speed equation later.
There are two ways in which we can compare different solitary waves: one way is to look at all solitary waves that have the same physical speed, namely a family of waves parameterized by $r$ with $\V$ being fixed; another way is to consider all solitary waves that have the same impulse $\I_\s$, since fixing the impulse will determine $\V$ in terms of $v_0$, which gives a family of waves parameterized by $r$ with $\I_\s$ being fixed.
For comparisons of solitary waves, a useful quantity to study is the scaled wave profile $(v-v_0)/h_\s$ as a function of the scaled travelling wave variable $\zeta/\ell_\s = (x-\V t)/\ell_\s$, where $h_\s$ is the peak height of the wave and $\ell_\s$ is the width of the wave. The scaled profile has a range $0$ to $1$, while the scaled travelling wave variable can be taken in the interval $(-\tfrac{1}{2},\tfrac{1}{2})$ to exclude the asymptotic tail of a solitary wave.
Strongly nonlinear regime
-------------------------
In the strongly nonlinear regime , the peak total strain $v_1$ in a solitary wave satisfies $v_1 \gg v_0$. In terms of the strain ratio , this regime is characterized by $1/r \gg 1$, namely $r=v_0/v_1$ is small. Using the sound speed equation , we then have $c_0/|\V| \simeq \sqrt{\tfrac{k(k+1)}{2}} r^\frac{k-1}{2}$. Hence, this regime coincides with the case of solitary waves that are highly supersonic $$\label{speedratio:strong}
|\V| \simeq \sqrt{\tfrac{2}{k(k+1)}} r^\frac{1-k}{2} c_0 \gg c_0$$ where $r\ll 1$. It will be useful to note $$\label{v1:strong}
v_1\simeq (\tfrac{k(k+1)}{2})^\frac{1}{k-1}(|\V|/c)^\frac{2}{k-1}$$ holds in this regime, as shown by combining the relations and .
The main properties of strongly nonlinear solitary waves in terms of $r$ and $\V$ are given by the expressions , , , for the height, width, impulse, and energy in the highly supersonic case. To leading order in $r$, these properties are asymptotically independent of $r$. In particular, the width is a constant (which depends on $k$), and thus strongly nonlinear solitary waves of different speeds have approximately the same width. In contrast, the height , impulse , and energy are functions of $\V$.
The expressions for the height and width of strongly nonlinear solitary waves are the same as the height and width of the compact solution $$\label{phys_compactonsol}
v_\c = (\tfrac{k(k+1)}{2})^\frac{1}{k-1} (|\V|/c)^\frac{2}{k-1} \cos\big( (x-\V t)\pi/l_\c \big)^\frac{2}{k-1} \Theta\big(\pi(\tfrac{1}{2}-|x-\V t|/l_\c)\big)$$ which exists in the limiting case $r=0$. Here $\Theta(z)$ denotes the Heaviside step function, and $$\label{coswidthscale}
l_\c = \tfrac{\sqrt{2k(k+1)}}{\sqrt{3}(k-1)} \pi R$$ is a constant (which depends on $k$ and $R$). The presence of the step function causes $v$ to vanish for $|x-\V t|\geq \tfrac{1}{2}l_\c$, whereby $v$ is compactly supported as a function of the travelling wave variable $x-\V t$.
At any fixed value of $r$, the width and height decrease with increasing $k$. Specifically, the width $\ell_\s\simeq \ell_\c$ approaches a limiting size $\tfrac{2}{\sqrt{6}}\pi R $ ($\simeq 2.56R$) which corresponds to a multiple $1.28$ of a particle diameter, while the height $h_\s$ approaches the limit $1$.
As discussed in Section \[sec:solitary-examples\], the compacton is an actual solution of the LWHC wave equation only for $1<k<\tfrac{5}{3}$, since the cutoff imposed by the step function lacks sufficient differentiability when $k\geq \tfrac{5}{3}$. Moreover, this solution holds only in the case when the background strain is zero, which describes a continuum system with *no* pre-compression, $v_0=0$.
In contrast, strongly nonlinear solitary waves have non-zero tails for all $0<r\ll 1$. We will now show that the shape of strongly nonlinear solitary waves near their peak is approximated by the upper part of the compacton profile: $$\label{archapprxsoln}
v/v_0 \simeq (\tfrac{k(k+1)}{2})^\frac{1}{k-1} (|\V|/c_0)^\frac{2}{k-1} \cos\big( (x-\V t)\pi/l_\c \big)^\frac{2}{k-1}$$ for $$|x-\V t| \ll \tfrac{1}{2}l_\c ,$$ where $l_\c\simeq l_\s$. In particular, the scaled wave profile as a function of the scaled travelling wave variable $|\zeta|/\ell_\s$ is approximately given by $(v-v_0)/h_\s \simeq \cos(|\zeta|\pi/\ell_\s)^\frac{2}{k-1}$ in the interval $0\leq |\zeta|/\ell_\s \ll \tfrac{1}{2}$, where the cosine function is not close to $0$.
This approximation holds only for the part of the solitary wave where the amplitude is $v/v_0 \gg 1$. When instead the amplitude is $v/v_0 \simeq 1$, the tail of the solitary wave is given by the decaying exponential with $\zeta_0\simeq \tfrac{1}{\sqrt{3}}R\hat\ell$. For the part of the wave profile where $v/v_0$ is neither large nor close to $1$, no explicit approximation appears to be possible because of the strong nonlinearity. A comparison of the exact solitary wave solution with $r\ll 1$ and the compacton solution for $r=0$ is shown in for $k=3/2, 2, 3$.
![ Comparison between the compacton solution and the solitary wave (SW) solutions given by the quadrature for various values of $r \ll 1$. (a) $k=3/2$; (b) $k=2$; (c) $k=3$. Here, $h_\c$ is the compacton height, given by (see ), which is equivalent to . []{data-label="fig:compacton_comparison_plots"}](hertz_solitarywave_compacton_solitary_comparison.eps){width="\textwidth"}
It is clear from that the compacton well-approximates a solitary wave when $r \ll 1$, and that the approximation becomes worse with increasing $r$. For a given value of $r$, the approximation is better for larger values of $k$. While the compacton is a good approximation for $k=2$ and $k=3$ solitary waves when $r \ll 1$, it should be noted that the compacton is not an actual solution to the LWHC when $k>5/3$, as explained in Section \[sec:compacton\]. Contact geometries with $k>5/3$ occur in several physically interesting discrete systems, particularly when the macroscopic particles in the system have rough contact surfaces [@Spence1968; @Goddard1990; @Persson2006].
Interestingly, for any $k>1$, we can use the exact quadrature formula for solitary waves to derive an approximate quadrature that describes the entire solitary wave profile and that can be evaluated explicitly when $v/v_0 \gg 1$ to obtain the approximation near the peak, as well as when $v/v_0\simeq 1$ to obtain the approximation for the tail.
We start by writing out the full form of the quadrature , specifically the square-root term in the integrand. This term is given by $\hat A(v,v_0;|\V|/c)= 1- \lambda^{1-k}\partial_{v_0}\big(v_0(v^k-v_0^k)/(v-v_0)\big)$, which has the approximation $\hat A(v,v_0;|\V|/c) \simeq 1- \lambda^{1-k}v^{k-1}$ to leading-order in $v_0/v$, with $\lambda$ given by the relation . Hence the quadrature can be approximated by $$\label{strong:apprxsol}
\int_{v}^{v_1} \frac{\sqrt{v^{k-1}}\,dv}{(v-v_0)\sqrt{1-\lambda v^{k-1}}}
\simeq \sqrt{3}(|\V|/c) |\zeta|/R ,$$ where we keep the exact pole term $v-v_0$ in the integrand because it yields the tail of the solitary wave to leading-order in $v_0/v$. This quadrature provides a useful approximation that encompasses the entire wave profile. It reduces to the exact compacton solution if we put $v_0=0$. Moreover, for $v_0>0$, if we restrict $v$ so that $v/v_0 \gg 1$, which holds near the peak of the wave profile, then we obtain the approximation given by the upper part of the arch of the compacton solution.
Unfortunately, the approximate quadrature cannot be evaluated explicitly for arbitrary $k>1$ when $v_0>0$. But we can evaluate it for the same special values of $k$ for which the exact quadrature was evaluated in Section \[sec:solitary-examples\].
Specifically, when $k=2$, we have $$\label{strong:apprxsol:k=2}
\arccos(2v/\lambda-1) + \sqrt{v_0/\lambda} \ln\Big(\frac{v_0 + v +2\sqrt{v_0}\sqrt{v(1-v/\lambda)}}{v-v_0}\Big)
\simeq \tfrac{2\pi}{k-1}|\zeta|/l_\c ,$$ where we have used relation for $v_1$, and where we have dropped all terms that are small due to $c_0/|\V| \ll 1$, which holds in the strongly nonlinear regime. When $k=3$, we obtain $$\label{strong:apprxsol:k=3}
\tfrac{1}{2}\arccos(2v^2/\lambda^2-1) + (v_0/\lambda) \ln\Big(\frac{1-v_0v/\lambda^2 +\sqrt{1-v^2/\lambda^2}}{v-v_0}\Big)
\simeq \tfrac{2\pi}{k-1}|\zeta|/l_\c .$$ Likewise, when $k=3/2$, we have $$\label{strong:apprxsol:k=3/2}
\begin{aligned}
2\arccos(2\sqrt{v/\lambda}-1) - \sqrt[4]{v_0/\lambda}\bigg(
& \ln\bigg(\frac{\sqrt{v_0} +\sqrt{v}+2\sqrt[4]{v_0v}\sqrt{1-\sqrt{v/\lambda}}}{v-v_0}\bigg)
\\&
-\arctan\bigg(\frac{2\sqrt[4]{v_0v}\sqrt{1-\sqrt{v/\lambda}}}{v-v_0}\bigg) \bigg)
\simeq \tfrac{2\pi}{k-1}|\zeta|/l_\c .
\end{aligned}$$
These explicit approximations – along with the corresponding exact solitary wave solutions given by for $r \ll 1$ are shown in Fig. \[fig:strongly\_nonlinear\_approx\_plots\]. The plots indicate a good agreement between the exact solitary wave solutions and the approximate solitary wave solutions when $r$ is very small, while the approximation gradually worsens with increasing $r$, particularly near the peak of the wave. Similar to the compacton solution , at fixed $r$ the approximate solutions – agree better with the exact solitary wave solutions for larger values of $k$. The compacton solution is shown for comparison in row (iv) of Fig. \[fig:strongly\_nonlinear\_approx\_plots\]. As illustrated by the plots, for all values of $r$, the shape of the peak of the approximate solution is nearly identical to the compacton; however, the tail of the approximate solution agrees much better with the exact solitary wave solution.
![ (Color online) Comparison of solutions of the approximate (approx.) quadrature with solitary wave (SW) solutions of for various values of $r\ll 1$. Column (a) $k=3/2$, ; column (b) $k=2$, ; column (c) $k=3$, . The solitary wave width is indicated by dashed vertical lines in the row (i). The compacton solution is shown for comparison in row (iv). Here, $h_{c}$ is the compacton height, given by . []{data-label="fig:strongly_nonlinear_approx_plots"}](hertz_solitarywave_approx_strongly_nonlinear.eps){width="108.00000%"}
Weakly nonlinear regime
-----------------------
In the weakly nonlinear regime , the peak total strain $v_1$ in a solitary wave satisfies $v_1-v_0\ll v_0$. This regime is characterized in terms of the strain ratio by $1/r - 1 \ll 1$, namely $r=v_0/v_1$ is close to $1$. From the sound speed equation , we then see $c_0/|\V| \simeq 1 - \tfrac{k-1}{6}(1-r)$, showing that this regime coincides with the case of solitary waves that are slightly supersonic $$\label{speedratio:weak}
|\V| \simeq (1 + \tfrac{k-1}{6}(1-r))c_0 \gtrsim c_0$$ where $1-r\ll 1$.
We can now obtain the main properties of weakly nonlinear solitary waves from the expressions , , , for the width, relative-height, impulse, and energy in the slightly supersonic case. In terms of $r$ and $\V$, this yields $$\begin{aligned}
\I_\s & \simeq \tfrac{2}{\sqrt{k-1}} \rho R \V (|\V|/c)^\frac{2}{k-1} \sqrt{1-r}
\label{impulse:weak}
\\
\E_\s & \simeq \tfrac{2(k+1)}{k\sqrt{k-1}} \rho R \V^2 (|\V|/c)^\frac{4}{k-1} \sqrt{1-r}
\label{energy:weak}\end{aligned}$$ for the impulse and the energy, respectively. Note both of these quantities scale like $\sqrt{1-r}$ when the wave speed is fixed.
The width and the height of weakly nonlinear solitary waves are given in terms of $r$ and $\V$ by $$\begin{aligned}
\ell_\s & \simeq \tfrac{\twoS}{\sqrt{k-1}} R\big/\sqrt{1-r}
\label{width:weak}
\\
h_\s & \simeq (|\V|/c)^\frac{2}{k-1} (1-r) .
\label{height:weak}\end{aligned}$$ At any fixed value of $r$, the width and height have an interesting dependence on the Hertz exponent $k$. Specifically, the width $\ell_\s$ decreases with increasing $k$ and goes to $0$, while the height $h_\s$ also decreases and approaches the limit $1-r \neq 0$.
More importantly, at any fixed wave speed $|\V|$, the width scales like $1/\sqrt{1-r}$ in comparison to the particle size $2R$, while the height scales like $1-r$ in comparison to the background strain $v_0$. The same scaling proportionality between width and height is well-known to occur for solitons of the KdV equation, which have a sech-squared profile. In fact, the KdV equation emerges in general for evolution systems that exhibit weak nonlinearity and dispersion [@Calagero].
We will now show that solitary waves in the weakly nonlinear regime approximately have the form of KdV solitons: $$\label{kdvsoln}
v/v_0 - 1 \simeq \tfrac{6}{k-1}(|\V|/c_0-1)\sech^2\Big( \sqrt{\tfrac{3}{2}}\sqrt{|\V|/c_0-1}\, (x-\V t)/R\Big)$$ where $$\label{kdvregime}
|\V|/c_0-1\simeq \tfrac{k-1}{6}(1-r) \ll 1 .$$ Thus, the scaled wave profile as function of the scaled travelling wave variable is simply $(v-v_0)/h_\s \simeq \sech^2(\S|\zeta|/\ell_\s)$ where $$\label{kdvheight}
h_\s\simeq h_\kdv = \tfrac{6}{k-1}(|\V|/c_0-1)$$ is the height of the KdV soliton. The approximation here holds to leading order in $1-r$ uniformly in $v$.
Fig. \[fig:weakly\_nonlinear\_approx\_plots\] shows the KdV soliton approximation, along with the corresponding exact solitary wave solutions from Section \[sec:solitary-examples\] for $1-r \ll 1$. It is evident that when $r$ is very close to $1$, the KdV soliton well-approximates the exact solitary wave solution for each value of $k$. The approximation worsens, particularly near the peak of the wave, as $r$ deviates further from $1$. In contrast to the quality of the approximations for strongly nonlinear solitary waves, which worsened with decreasing $k$ (cf. Figs. \[fig:compacton\_comparison\_plots\] and \[fig:strongly\_nonlinear\_approx\_plots\]), the quality of the normalized KdV approximation is roughly independent of $k$, for a given value of $r$.
![ (Color online) Comparison between KdV solitons and exact solitary wave (SW) solutions for $1-r \ll1$. Column (a) $k=3/2$; column (b) $k=2$; column (c) $k=3$. The solitary wave width is indicated by dashed vertical lines in row (i). for the height $h_\kdv$ is used to normalize both the KdV solitons and the exact solitary wave solutions. []{data-label="fig:weakly_nonlinear_approx_plots"}](hertz_solitarywave_approx_weakly_nonlinear_kdv_comparison.eps){width="108.00000%"}
This approximation result is a continuum counterpart of what occurs [@JamesPelinovsky; @YanLiuYanZhaDuaYan] in the weak nonlinearity limit for pre-compressed discrete chains . It is known that a two-scale expansion of the discrete equations of motion yields a discrete version of the KdV equation. We show in Appendix \[appendix:KdV\] that the same expansion can be applied to the LWHC wave equation to obtain the continuum KdV equation.
To derive expression , we asymptotically expand the quadrature for solitary waves $$\label{smallv}
v=v(\zeta)= v_0(1 + w(\zeta))$$ such that $0<w\ll 1$ in the weakly nonlinear regime $1-r \ll 1$. In terms of $w$, this quadrature is given by $$\label{solitarywave_smallv}
\int_{w}^{1/r-1} \frac{\sqrt{(1+w)^{k-1}}\, \,dw}{w\sqrt{\hat A(1+w,1;|\V|/c_0)}}
= \tfrac{\sqrt{6}}{\sqrt{k(k+1)}} |\zeta|/R$$ where $$\hat A(1+w,1;|\V|/c_0) = 1- \tfrac{2}{k(k+1)}(c_0/\V)^2 (k-(1+w)\partial_w)\Big(\frac{(1+w)^k-1}{w}\Big)$$ from expression . Expanding in powers of $w$ to leading non-trivial order, we get $(1+w)^{k-1}/\hat A(1+w,1;|\V|/c_0) \simeq 1/(B_0 - B_1w)$ where $B_0 = \tfrac{k(k+1)}{2}((|\V|/c_0)^2-1)$ and $B_1 = \tfrac{k(k^2-1)}{6}((|\V|/c_0)^2-\tfrac{2}{3})$. Then we use the asymptotic expansion of the speed ratio $|\V|/c_0$ in terms of $1-r$, which yields $B_0 - B_1w \simeq \tfrac{k(k^2-1)}{6}(1-r -w)$ and hence $$\frac{(1+w)^{k-1}}{\hat A(1+w,1;|\V|/c_0)} \simeq \tfrac{6}{k(k^2-1)} (1-r -w)^{-1} .$$ Thus, the asymptotic expansion of the quadrature to leading order is given by $$\int_{w}^{1-r} \frac{dw}{w\sqrt{1-r -w}}
\simeq \sqrt{k-1} |\zeta|/R .$$ This integral yields $\tfrac{2}{\sqrt{1-r}}\arctanh(\sqrt{1-r-w}/\sqrt{1-r})$, and thus we obtain $$w \simeq (1-r)\sech^2\big(\tfrac{1}{2}\sqrt{1-r}\sqrt{k-1} |\zeta|/R\big)$$ which is valid for $1-r \ll 1$. The resulting solitary wave $$v=v_0\left(1 + (1-r)\sech^2\big(\tfrac{1}{2}\sqrt{1-r}\sqrt{k-1} |\zeta|/R\big)\right)$$ yields expression when $1-r$ is expressed back in terms of the speed ratio .
Comparisons and scaling relations
---------------------------------
The dependence of the width, height, impulse, and energy of solitary waves on the wave speed $\V$ is the same in both nonlinearity regimes. Specifically, when $r$ is fixed in each regime, the width is independent of the speed, whereas the height scales like $h_\s\propto |\V|^\frac{2}{k-1}$ while the impulse and energy scale like $\I_\s\propto |\V|^\frac{k+1}{k-1}$ and $\E_\s\propto |\V|^\frac{2k+2}{k-1}$. Consequently, the latter three quantities satisfy the proportionality relationships: $$\label{solitary_rels1}
\I_\s \propto h_\s \V,
\quad
\E_\s \propto h_\s^2 \V^2$$ and thus $$\label{solitary_rels2}
\E_\s/\I_\s \propto h_\s \V \propto \sgn(\V) |\V|^\frac{k+1}{k-1},
\quad
\E_\s \propto \I_\s^2.$$
The same proportionality between energy and impulse holds for pre-compressed discrete chains when compressive waves are generated by sharply striking an end particle in the chain, as noted in Section \[model\]. Moreover, the two scaling relations $h_\s\propto |\V|^\frac{2}{k-1}$ and $\I_\s\propto |\V|^\frac{k+1}{k-1}$ agree with the ones reported in the literature [@Nesterenko2001; @Daraio2006; @Porter2008; @Sen2008] for weakly pre-compressed discrete chains.
Finally, we note that the proportionality $\I_\s\propto |\V|^\frac{k+1}{k-1}$ indicates that all of the results on the properties of solitary waves in the weakly nonlinear and strongly nonlinear regimes can be expressed by replacing $\V$ in terms of $\I_\s$, using the respective impulse expressions and . From a physical viewpoint, if we impart a specified sharp impulse to a pre-compressed continuum system, then in either the weakly or strongly nonlinear regimes this impulse determines a unique solitary wave whose speed and height are given by $|\V| \propto |\I_\s|^\frac{k-1}{k+1}$, $h_\s \propto |\I_\s|^\frac{2}{k+1}$.
It is more difficult to study solitary waves in the intermediate regime, where $r$ is neither close to $0$ nor close to $1$. By using expression for the sound speed in terms of $\V$ and $r$, combined with expressions , , and for the height $h_\s$, width $\ell_\s$, impulse $\I_\s$ and energy $\E_\s$, we obtain exact formulas that hold in any nonlinearity regime: $$\begin{aligned}
& \ell_\s = 2R\Big( \sqrt{\tfrac{k(k+1)}{6}} \int_{1}^{1/r} \frac{\hat F(z,1;\mu(r))}{z-1} \,dz +\frac{\sqrt{3}}{\sqrt{\mu(r)^2 -1}} \Big) ,
%%% uses \S
\\
& h_\s = \big(\tfrac{|\V|}{c}\mu(r)\big)^{\frac{2}{k-1}}(1/r-1) ,
\\
& \I_\s = \tfrac{2}{\sqrt{3}} \rho Rc\, \sgn(\V) \bigg( \big(\tfrac{|\V|}{c}\mu(r)\big)^\frac{k+1}{k-1} \int_{1}^{1/r} \frac{\sqrt{z^{k-1}}}{\sqrt{\hat A(z,1;\mu(r))}}\,dz \bigg) ,
\\
&\begin{aligned}
\E_\s & = \tfrac{1}{\sqrt{3}} \rho Rc|\V|\bigg( \big(\tfrac{|\V|}{c}\mu(r)\big)^\frac{k+3}{k-1} \int_{1}^{1/r} \sqrt{z^{k-1}}\bigg( \frac{z^2-1 + \tfrac{2}{k(k+1)}(1/\mu(r))^2( z^{k+1}-1 )}{(z-1)\sqrt{\hat A(z,1;\mu(r))}}
\\&\qquad\qquad
- (z-1)\sqrt{\hat A(z,1;\mu(r))} \bigg)\,dz \bigg) ,
\end{aligned}\end{aligned}$$ where $$\mu(r) = \sqrt{\tfrac{2}{k(k+1)}} \frac{\sqrt{r^{1-k} -k(1-r)r-r}}{1-r} .$$ These formulas do not exhibit any general scaling relations. Nevertheless, they can be evaluated straightforwardly for any $0<r<1$. Likewise, from the quadrature for solitary waves, we have the exact integral formula for $v(x-\V t)$: $$\label{quadrature-r}
\int_{v/v_0}^{1/r} \frac{\sqrt{z^{k-1}}}{(z-1)\sqrt{\hat A(z,1;\mu(r))}} \,dz = \frac{\sqrt{3}\mu(r)}{R}|\zeta|,
\quad
\zeta = x-\V t,$$ where $$v_0 = \Big( \frac{|\V|/c}{\mu(r)} \Big)^\frac{2}{k-1}$$ is the background strain. The transition between regimes of weak nonlinearity and strong nonlinearity can be seen in Fig. \[fig:solution\_plots\], using the explicit evaluation of this integral for $k=3/2,2,3$ presented in Section \[sec:solitary-examples\], where $r=g_0/g_1$ with $g_1$ given by the peak value of the scaled amplitude $g(\xi)=(g_0/v_0)v(\zeta)$.
We see how the (scaled) profile of the solitary waves makes a continuous transition from a compacton-like shape when $r=0.001$ ($g_0=0.001$, $g_1=0.999$) is small, to a KdV-soliton-like shape when $r=0.5$ ($g_0=0.25$, $g_1=0.5$ for $k=2$; $g_0=0.3$, $g_1=0.6$ for $k=3$) and $r=0.43$ ($g_0=0.2$, $g_1=0.47$ for $k=3/2$).
An analysis of the continuous transition from weak nonlinearity to strong nonlinearity for solitary waves has not previously appeared in the literature. We remark that the nonlinearity transition for shock waves in discrete systems have been studied in .
Concluding remarks {#remarks}
==================
In this paper we have presented a physical analysis of the properties of long wavelength solitary waves in the continuum model of Hertzian chains with arbitrary pre-compression.
We find that the ratio of the background strain to the peak strain in solitary wave solutions describes the degree of dynamical nonlinearity in the underlying discrete chain. This dynamical nonlinearity ratio is determined by a nonlinear algebraic relation given in terms of the ratio of the solitary wave speed to the sound speed. In particular, highly supersonic solitary waves correspond to highly nonlinear propagating localized pulses in weakly compressed discrete chains, while slightly supersonic solitary waves correspond to weakly nonlinear propagating localized pulses in strongly compressed discrete chains.
We have obtained explicit analytical expressions for the height, width, impulse and energy of solitary waves. The width expression is a new result, coming from an asymptotic analysis of the tail of solitary waves. Using this expression, we have shown that the width of all solitary waves decreases with $k$ at any fixed value of the nonlinearity ratio, and increases with the nonlinearity ratio, at any fixed value of $k>1$. In the physically interesting case when $k=3/2$ (corresponding to spherical particles), the minimum width is approximately $5$ times the particle size. This confirms the validity of the continuum model for studying solitary waves.
We have used the height, width, impulse and energy expressions to show that the main physical features of solitary waves depend principally on the ratio of the wave speed to the sound speed at any fixed $k>1$. Moreover, highly supersonic solitary waves are shown to be well-approximated by Nesterenko’s compacton, while slightly supersonic solitary waves are shown to coincide approximately with KdV solitons which have a well-known sech-squared profile. We further have shown that the KdV equation itself arises directly from the LWHC equation through a two-scale asymptotic expansion combined with a Galilean transformation to a reference frame moving with the sound speed.
Exact solitary wave solutions have been used to compare the features of solitary waves across different nonlinearity regimes. Specifically, we have derived an exact expression for the solitary waves that arise in the continuum model with a Hertz exponent $k=3/2$, corresponding to solitary wave pulses in a chain of spherical discrete particles. The shape of these solitary waves is highly sensitive to their speed.
Our results establish that a continuum system supports solitary waves having the same impulse momentum yet displaying marked different shapes and speeds, especially in comparison to a compacton.
The same conclusion will hold for physical discrete chains when the long wavelength regime is considered.
Our long wavelength continuum analysis can be extended to models of heterogeneous chains, in particular dimer chains consisting of alternating particles with different masses. There are several further interesting directions for future work.
One direction would be to study the properties of solitary waves in a finite-size continuum model with physical boundary conditions. This will yield analytical results concerning the reflection of solitary waves at the end point boundaries, which is important for understanding physical discrete chains.
Another direction would be to investigate the statistical properties of a dilute (rarified) ensemble of interacting solitary waves, known as a soliton gas. This is relevant for the dynamics of discrete chains after transients have decayed such that the chain contains propagating solitary waves with a wide spectrum of energies. Of notable interest here is the recent observation that rogue waves can form in such circumstances [@SluPel; @ShuPel; @PelShu].
Acknowledgments {#acknowledgments .unnumbered}
===============
S.C.A. is supported by an NSERC research grant. The work of M.P. was supported by a Vanier Canada Graduate Scholarship.
Derivation of $k=\tfrac{3}{2}$ solitary wave solution {#appendix:k=3/2}
=====================================================
Here we outline the main steps in the derivation of the solitary wave solution for $k=\tfrac{3}{2}$, starting from the quadrature , which is given by $$\label{solitary_ode_integral_2k=3}
\pm \int_{g}^{g_1} \frac{g^{1/4}}{\sqrt{(g-g_0)^2+\tfrac{3}{2}g_0^{3/2}(g-g_0)-g(g^{3/2}-g_0^{3/2})}} \,dg = |\xi| .$$
As shown in , the change of variable $g=h^2$ brings this integral to the form of an elliptic integral $$\label{solitary_elliptic_integral_2k=3}
\pm 2 \int_{h}^{h_1} \frac{h^2}{(h-h_0)\sqrt{B(h)}} \,dh = |\xi|$$ where $$B(h)=h((1-\tfrac{3}{2}h_0)h_0^2+(2-3h_0)h_0h+(1-2h_0)h^2 -h^3)$$ is a quartic polynomial. It is straightforward to show that this polynomial has the factorization $B(h)=B_1(h)B_2(h)$ with $B_1(h)=h(h_1-h)$ and $B_2(h)=(h+h_2)^2+h_3^2$, where $h_1,h_2,h_3$ are given by expressions –. Standard methods [@Abramowitz1964; @Lawden1980] can now be used to evaluate the elliptic integral explicitly.
The first step is to change variables by applying a linear fractional transformation $$\label{eq:y}
y= (h-y_+)/(h-y_-),$$ with $y_\pm$ being defined by the condition $B_1(h) -\lambda_\pm B_2(h)= -(1+\lambda_\pm)(h-y_\pm)^2$, where $\lambda_\pm$ are the roots of the discriminant of $B_1(h)-\lambda B_2(h)$. This leads to the expressions $$\label{eq:BYrel}
(y-1)^2R B_1(h) = \sqrt{PQ} Y_1(y),
\quad
(y-1)^2R B_2(h) = \sqrt{PQ} Y_2(y)$$ given in terms of the quadratic polynomials $$Y_1(y) = K_-^2 - K_+^2 y^2,
\quad
Y_2(y) = (K_+^2 -R)y^2 +R- K_-^2.$$ where $$K_{\pm} = \sqrt{Q}\pm\sqrt{P}$$ and $$P=h_2^2+h_3^2=B_2(0),
\quad
Q=(h_1+h_2)^2+h_3^2=B_2(h_1),
\quad
R=(h_1+2h_2)^2.$$ Carrying out the change of variable , and splitting up the integral into terms with even/odd parity under $y\rightarrow -y$, we get $$\label{eq:integralterms}
\int_{h}^{h_1} \frac{h^2\, dh}{(h-h_0)\sqrt{B(h)}}
= c_1 I_1+ c_2 I_2 +c_0 J_0 +c_1 J_1 +c_2 y_0 J_2,$$ where $$\begin{aligned}
&
I_1=\int_{y}^{y_1} \frac{y\,dy}{(y^2-1)\sqrt{Y_1(y)Y_2(y)}},
\
I_2 = \int_{y}^{y_1} \frac{y\,dy}{(y^2-y_0^2)\sqrt{Y_1(y)Y_2(y)}},
\label{eq:integral-I}
\\
&
J_0 = \int_{y}^{y_1} \frac{dy}{\sqrt{Y_1(y)Y_2(y)}},
\
J_1 = \int_{y}^{y_1} \frac{dy}{(y^2-1)\sqrt{Y_1(y)Y_2(y)}},
\
J_2 = \int_{y}^{y_1} \frac{dy}{(y^2-y_0^2)\sqrt{Y_1(y)Y_2(y)}}
\label{eq:integral-J}\end{aligned}$$ and $$c_0= \frac{y_-^2(y_0-1)R}{\sqrt{PQ}},
\quad
c_1= -\frac{(y_+-y_-)^2R}{\sqrt{PQ}},
\quad
c_2= \frac{h_0^2(y_0-1)^2R}{\sqrt{PQ}},
\label{eq:c1c2c3}$$ with $$\label{eq:ypm}
y_\pm = \sqrt{P}(\pm\sqrt{Q} -\sqrt{P})/\sqrt{R}$$ and $$\begin{aligned}
& y_1 = (h_1-y_+)/(h_1-y_-) = K_-/K_+,
\\
& y_0 = (h_0-y_+)/(h_0-y_-) = \frac{h_0\sqrt{R}-K_-\sqrt{P}}{h_0\sqrt{R}+K_+\sqrt{P}}.\end{aligned}$$ We note that the integral must diverge to $\infty$ when $h\to h_0$, since this limit corresponds to the tail of the solitary wave $|\xi|\to \infty$. In terms of the variable $y$, this divergence occurs for $y\to y_0$.
The five separate integrals – can be simplified by a further change of variables. We introduce $$\label{eq:z}
z=y/y_1,$$ and let $$\begin{gathered}
\tau= K_-\sqrt{R- K_-^2},
\\
k= y_1^2(K_+^2-R)/(R-K_-^2),
\quad
n=(y_1/y_0)^2,
\quad
m=y_1^2.\end{gathered}$$ Note $y\rightarrow y_0$ now corresponds to $z\rightarrow y_0/y_1 =1/\sqrt{n}$.
The first two integrals can be evaluated directly in terms of elementary functions: $$\label{eq:I1}
\begin{aligned}
\frac{\tau}{y_1^2} I_1 &
= \int_{z}^{1} \frac{z\, dz}{(y_1^2z^2-1)\sqrt{(1-z^2)(1+kz^2)}}
\\&
= \frac{-1}{2\sqrt{(1-m)(k+m)}}
\bigg( \frac{\pi}{2} + \arctan\bigg( \frac{(m-1)(1+kz^2) +(k+m)(1-z^2)}{2\sqrt{(k+m)(1-m)}\sqrt{(1-z^2)(1+kz^2)}} \bigg) \bigg),
\end{aligned}$$ and $$\label{eq:I2}
\begin{aligned}
\frac{y_0^2\tau}{y_1^2} I_2 &
= \int_{z}^{1} \frac{z\,dz}{((y_1/y_0)^2z^2-1)\sqrt{(1-z^2)(1+kz^2)}}
\\&
= \frac{1}{2\sqrt{(n-1)(k+n)}}
\ln\bigg( \frac{\big(\sqrt{n-1} \sqrt{1+kz^2} +\sqrt{k+n} \sqrt{1-z^2}\big)^2}{(k+1)(nz^2-1)} \bigg).
\end{aligned}$$ For $z\rightarrow y_0/y_1 =1/\sqrt{n}$, the integral $I_1$ is finite, while the integral $I_2$ has a logarithmic singularity $(y_0/y_1)^2\tau I_2 \to \tfrac{-1}{2\sqrt{(n-1)(k+n)}} \ln(nz^2-1) \to +\infty$.
The remaining three integrals can be evaluated in terms of Jacobi elliptic functions: $$\begin{aligned}
&
\tfrac{\tau}{y_1} J_0
= \int_{z}^{1} \frac{dz}{\sqrt{(1-z^2)(1+kz^2)}}
= \tfrac{1}{\sqrt{1+k}} \cn^{-1}(z|l),
\label{eq:J0}
\\
&
\tfrac{\tau}{y_1} J_1
=\int_{z}^{1} \frac{dz}{(y_1^2z^2-1)\sqrt{(1-z^2)(1+kz^2)}}
= \tfrac{1}{(m-1)\sqrt{1+k}} \Pi\big(\tfrac{m}{m-1};\cn^{-1}(z|l)|l\big),
\label{eq:J1}
\\
&
\tfrac{\tau y_0^2}{y_1} J_2
= \int_{z}^{1} \frac{dz}{((y_1/y_0)^2z^2-1)\sqrt{(1-z^2)(1+kz^2)}}
= \tfrac{1}{(n-1)\sqrt{k+1}} \Pi\big(\tfrac{n}{n-1};\cn^{-1}(z|l)|l\big),
\label{eq:J2}\end{aligned}$$ with $$l=k/(1+k).$$ For $z\rightarrow y_0/y_1 =1/\sqrt{n}$, the integrals $J_0$ and $J_1$ are finite, while the integral $J_2$ can be shown to have a logarithmic singularity.
The next step consists of extracting the singular part of $J_2$ by using the elliptic function identity [@Abramowitz1964] $$\Pi\big(N;\theta|l\big)
= -\Pi\big(l/N;\theta|l\big) + \theta + \tfrac{1}{2}\psi\ln\Big(\frac{\psi +\sn(\theta|l)/(\cn(\theta|l)\dn(\theta|l))}{\psi -\sn(\theta|l)/(\cn(\theta|l)\dn(\theta|l))}\Big),
\quad
N>1,$$ where $\psi = \tfrac{1}{\sqrt{(N-1)(1-l/N)}}$. This gives $$\label{eq:logterm}
\begin{aligned}
\Pi\big(n/(n-1);\cn^{-1}(z|l)|l\big) & =
\cn^{-1}(z|l) -\Pi\big(l(n-1)/n;\cn^{-1}(z|l)|l\big)
\\&\qquad
+ \tfrac{1}{2}\psi\ln\Big(\frac{\tilde\psi +\sqrt{1-z^2}/(z\sqrt{1+kz^2})}{\tilde\psi -\sqrt{1-z^2}/(z\sqrt{1+kz^2})}\Big)
\end{aligned}$$ with $\tilde\psi = \frac{\psi}{\sqrt{k+1}}= \tfrac{\sqrt{1-1/n}}{(1/\sqrt{n})\sqrt{1+k/n}}$. In the logarithm term in expression , the denominator vanishes at $z^2=1/n$, and so we can factorize $$\begin{aligned}
\frac{\tilde\psi +\sqrt{1-z^2}/(z\sqrt{1+kz^2})}{\tilde\psi -\sqrt{1-z^2}/(z\sqrt{1+kz^2})} & =
\frac{(n-1)( z\sqrt{1+kz^2} + (1/\tilde\psi)\sqrt{1-z^2} )^2}{(nz^2-1)( 1+kz^2 + (k/n)(1-z^2) )}.
\end{aligned}$$ Thus, we can write $$J_2 = J_{2,0}+J_{2,1}+I_3,$$ where $$\label{eq:J2-elliptic}
\frac{\tau y_0^2}{y_1} J_{2,0}
= \tfrac{1}{(n-1)\sqrt{k+1}} \cn^{-1}(z|l),
\quad
\frac{\tau y_0^2}{y_1} J_{2,1}
= \tfrac{-1}{(n-1)\sqrt{k+1}} \Pi\big(l(n-1)/n;\cn^{-1}(z|l)|l\big),$$ are finite for $z\rightarrow y_0/y_1 =1/\sqrt{n}$, and where $$\label{eq:J2-rat}
\frac{\tau y_0^2}{y_1} I_3
= \tfrac{\sqrt{n}}{2\sqrt{k+n}} \ln\Big(\frac{(n-1)( z\sqrt{1+kz^2} + (1/\tilde\psi)\sqrt{1-z^2} )^2}{(nz^2-1)( 1+kz^2 + (k/n)(1-z^2) )}\Big).$$ has the logarithmic singularity $(y_0^2/y_1)\tau I_3 \to \tfrac{-\sqrt{n}}{2\sqrt{k+n}} \ln(nz^2-1) \to +\infty$.
All of the integrals , , , , , and can be written explicitly in terms of $h$ through the relations $$z=y/y_1 = \frac{K_+ h-\sqrt{P} h_1}{K_- h +\sqrt{P} h_1},
\quad
Y_1(y)/Y_2(y)=B_1(h)/B_2(h)$$ obtained by inverting the change of variables and and by using equations , , , For the final step, the combined elementary integrals $c_1I_1 +c_2(I_2 +y_0 I_3) =I(h)$ can be expressed in the form after some simplifications using the following identities: First, from relation evaluated at $y=y_0$ and $y=y_1$, we have $$\begin{aligned}
& K_-^2 - K_+^2 y_0^2
= \frac{(y_0-1)^2 h_0(h_1-h_0)R}{\sqrt{PQ}},
\label{eq:B1x0}\\
& (K_+^2 -R)y_0^2 +R- K_-^2
= \frac{(y_0-1)^2 RS}{\sqrt{PQ}},
\label{eq:B2x0}\\
& (K_+^2 -R)y_1^2 +R- K_-^2
= \frac{(y_1-1)^2 R\sqrt{Q}}{\sqrt{P}}.
\label{eq:B2x1}\end{aligned}$$ Next, we find $$K_+K_- = h_1\sqrt{R},
\quad
K_++K_- = 2\sqrt{Q},
\quad
K_+-K_- = 2\sqrt{P},$$ and $$y_+-y_-=\frac{2\sqrt{PQ}}{\sqrt{R}},
\quad
1-y_1 = \frac{2\sqrt{P}}{K_+}.$$ Last, we can derive $$\begin{gathered}
1-m=\frac{4\sqrt{PQ}}{K_+^2},
\quad
k+m= \frac{4\sqrt{PQ}K_-^2}{R(K_+^2-h_1^2)},
\quad
k+1
%= \frac{R\sqrt{Q}(1-y_1)^2}{(R-K_-^2)\sqrt{P}}
= \frac{\sqrt{PQ}}{K_+^2-h_1^2},
\\
n-1= \frac{Rh_0(h_1-h_0)(1-1/y_0)^2}{K_+^2\sqrt{PQ}},
\quad
k+n
%= \frac{RS y_1^2(1-1/y_0)^2}{(R-K_-^2)\sqrt{PQ}}
= \frac{S K_-^2}{(K_+^2-h_1^2)\sqrt{PQ}} \end{gathered}$$ which yields $$\begin{gathered}
\frac{1-z^2}{1+kz^2} = \frac{K_+^2-h_1^2}{h_1^2} \frac{Y_1(y)}{Y_2(y)},
%\frac{R-K_-^2}{K_-^2}
\quad
\frac{k+m}{1-m} = \frac{h_1^2}{K_+^2-h_1^2},
%\frac{K_-^2}{R-K_-^2}
\quad
\frac{k+n}{n-1} = \frac{h_1^2S}{(K_+^2-h_1^2)h_0(h_1-h_0)}. \end{gathered}$$ Similarly, the combined elliptic integrals $c_0 J_0 +c_1 J_1 +c_2 y_0 (J_{2,0}+J_{2,1})=J(h)$ can be expressed in the form by using the relations $$\begin{gathered}
l = \frac{k}{k+1}
= \frac{h_1^2-K_-^2}{4\sqrt{PQ}},
\quad
\frac{m}{1-m} = \frac{K_-^2}{4\sqrt{PQ}},
\\
\frac{n}{n-1} = \frac{K_-^2\sqrt{PQ}}{Rh_0(h_1-h_0)(1-y_0)^2},
\quad
\frac{l(n-1)}{n} = \frac{(K_+^2-R)h_0(h_1-h_0)(1-y_0)^2}{4PQ}.\end{gathered}$$
Asymptotic expansions of width, impulse, and energy {#appendix:expansion}
===================================================
Here we explain the steps for asymptotically expanding the width integral , impulse integral and energy integral in the slightly supersonic/weakly nonlinear and highly supersonic/strongly nonlinear cases.
As a first step, it is very helpful to make a change of integration variable $z=g/g_0$, where $g_0=v_0/\lambda$ is expression . By combining this expression and the sound speed expression , we have the useful relations $$\label{speed_g:rel}
%g_0 = (c_0/|\V|)^\frac{2}{k-1} g^*
|\V|/c_0 = (g^*/g_0)^\frac{k-1}{2} = (g^*\lambda)^\frac{k-1}{2} c/c_0 ,$$ where $g^*$ is expression which involves only $k$. Then the integrals , , are respectively given by $$\begin{aligned}
\hat\ell & = \sqrt{\tfrac{k(k+1)}{2}}\int_{g_0}^{g_1} \frac{F(g,g_0)}{g-g_0} \,dg + \S/\sqrt{(g_0/g^*)^{1-k} - 1} ,
\label{width:g}
\\
\hat\I & = \lambda^\frac{k+1}{2} \int_{g_0}^{g_1} \frac{\sqrt{g^{k-1}}}{\sqrt{A(g,g_0)}}\,dg,
\label{impulse:g}
\\
\hat\E & = \lambda^\frac{k+3}{2} \int_{g_0}^{g_1} \sqrt{g^{k-1}}\left( \frac{g^2-g_0^2 + g^{k+1}-g_0^{k+1} }{(g-g_0)\sqrt{A(g,g_0)}} - (g-g_0)\sqrt{A(g,g_0)} \right)\,dg .
\label{energy:g}\end{aligned}$$
Slightly supersonic/weakly nonlinear case
-----------------------------------------
This case $|\V|/c_0-1 \ll 1$ corresponds to $(g^*/g_0)^\frac{k-1}{2} -1 = \epsilon \ll 1$ from the first relation . Hence we have $g_0 \simeq (1-\tfrac{2}{k-1}\epsilon)g^*$, and $g_1 = g_0/r \simeq (1+\tfrac{4}{k-1}\epsilon)g^*$ from the strain ratio . In particular, note $(g_1-g_0)/g^* \simeq \tfrac{6}{k-1}\epsilon$.
To asymptotically expand the integrals , , in terms of $\epsilon$, we first take the leading-order term in the integrand, which is given by substitution of $g=g^*$, and then we multiply this term by the endpoint difference $g_1-g_0 \simeq \tfrac{6}{k-1} g^*\epsilon$.
For the width integral , this expansion yields $$\begin{aligned}
\hat\ell - \S/\sqrt{(g_0/g^*)^{1-k} - 1}
& \simeq \sqrt{\tfrac{k(k+1)}{2}}(g_1-g_0) \frac{F(g^*,g_0)}{g^*-g_0} \\
& \simeq 3\sqrt{\tfrac{k(k+1)}{2}}F(g^*,g_0) \\
\end{aligned}$$ which is $O(\epsilon)$ since $F(g_0,g_0)=0$. But since $1/\sqrt{(g_0/g^*)^{1-k} - 1} \simeq 1/\sqrt{2\epsilon}$, the leading-order term in the width integral is given by $\hat\ell \simeq \S/\sqrt{2\epsilon}$.
Expansion of the impulse integral directly yields $$\hat\I \simeq \lambda^\frac{k+1}{2} (g_1-g_0) \sqrt{g^*{}^{k-1}/A(g^*,g_0)} \\
\simeq \tfrac{6}{k-1} \big( (g^*\lambda)^\frac{k+1}{2} /\sqrt{A(g_0,g_0)} \big)\epsilon$$ where, from expression , $A(g_0,g_0)\simeq 2\epsilon$. Moreover, we note $g^*\lambda \simeq (c_0/c)^\frac{2}{k-1}$ from the relations . Hence the leading-order term in the impulse integral is given by $\hat\I \simeq \tfrac{3\sqrt{2}}{k-1} (c_0/c)^\frac{k+1}{k-1} \sqrt{\epsilon}$.
Similarly, the energy integral has the expansion $$\begin{aligned}
\hat\E & \simeq \lambda^\frac{k+3}{2} (g_1-g_0) \sqrt{g^*{}^{k-1}}\left( \frac{g^*{}^2-g_0^2 + g^*{}^{k+1}-g_0^{k+1} }{(g^*-g_0)\sqrt{A(g^*,g_0)}} - (g^*-g_0)\sqrt{A(g^*,g_0)} \right) \\
& \simeq \tfrac{6}{k-1} \lambda^\frac{k+3}{2} \sqrt{g^*{}^{k+3}/A(g^*,g_0)} (2 + (k+1) g^*{}^{k-1}) \epsilon \\
& \simeq \tfrac{12}{k-1} (1 + \tfrac{1}{k}) \big( (g^*\lambda)^\frac{k+3}{2}\big/\sqrt{A(g_0,g_0)}\big) \epsilon .
\end{aligned}$$ Hence the leading-order term is given by $\hat\E \simeq \tfrac{6\sqrt{2}(k+1)}{k(k-1)} (c_0/c)^\frac{k+3}{k-1} \sqrt{\epsilon}$.
Highly supersonic/strongly nonlinear case
-----------------------------------------
This case $|\V|/c_0 \gg 1$ corresponds to $(g_0/g^*)^\frac{k-1}{2} =\epsilon \ll 1$ from the first relation . Hence we have $g_0 = \epsilon^\frac{2}{k-1} g^*$ and $g_1 = g_0/r \simeq 1$ from the strain ratio . For the subsequent steps, it will be helpful to note $$\label{g1g0rel}
1-g_1 \simeq \tfrac{2}{k-1}g_0 .$$ This is obtained by solving equation to second order in powers of $\epsilon^\frac{2}{k-1}$.
To asymptotically expand the impulse integral and energy integral , we use a Taylor series in $g_0$ and explicitly evaluate the first two terms $$\begin{aligned}
\hat\I & = \hat\I|_{g_0=0} + g_0\partial_{g_0}\hat\I|_{g_0=0} + O(g_0^2),
\\
\hat\E & = \hat\E|_{g_0=0} + g_0\partial_{g_0}\hat\E|_{g_0=0} + O(g_0^2) .\end{aligned}$$ Since $g_0 =O(\epsilon^\frac{2}{k-1})$, the first term in the Taylor series gives the leading order term for the asymptotic expansions, while the second term gives the subleading term.
The leading-order term in each expansion is given by evaluating $$\lambda^{-\frac{k+1}{2}} \hat\I|_{g_0=0}
%= \lambda^\frac{k+1}{2} \int_{g_0}^{g_1} \frac{\sqrt{g^{k-1}}}{\sqrt{A(g,g_0)}}\,dg
= \int_{0}^{1} \frac{\sqrt{g^{k-1}}}{\sqrt{A(g,0)}}\,dg
= \int_{0}^{1} \frac{dg}{\sqrt{g^{1-k}-1}}$$ and $$\begin{aligned}
\lambda^{-\frac{k+3}{2}} \hat\E|_{g_0=0}
% = \lambda^\frac{k+3}{2} \int_{g_0}^{g_1} \sqrt{g^{k-1}}\left( \frac{g^2-g_0^2 + g^{k+1}-g_0^{k+1} }{(g-g_0)\sqrt{A(g,g_0)}} - (g-g_0)\sqrt{A(g,g_0)} \right)\,dg
= \int_{0}^{1} \sqrt{g^{k+1}}\left( \frac{1 + g^{k-1}}{\sqrt{A(g,0)}} - \sqrt{A(g,0)} \right)\,dg
= 2 \int_{0}^{1} \frac{g^{k}\,dg}{\sqrt{g^{1-k}-1}} ,
\end{aligned}$$ where $A(g,0)=1-g^{k-1}$ from expression . This yields $$\begin{aligned}
\hat\I & =
\lambda^\frac{k+1}{2} \sqrt{\pi} \Gamma\big(\tfrac{k+1}{2k-2}\big)/\Gamma\big(\tfrac{1}{k-1}\big)
+O(\epsilon^\frac{2}{k-1}),
\label{impulse:leadingterm}
\\
\hat\E & =
\lambda^\frac{k+3}{2} \sqrt{\pi} \tfrac{k+3}{4k+4}\Gamma(\tfrac{k+3}{2k-2})/\Gamma(\tfrac{2}{k-1})
+O(\epsilon^\frac{2}{k-1}) .
\label{energy:leadingterm}\end{aligned}$$
For the width integral , we separately consider the algebraic term and the integral term. The algebraic term can be expanded in a Taylor series in $(g_0/g^*)^\frac{k-1}{2} =\epsilon$, which yields $$\S/\sqrt{(g_0/g^*)^{1-k} - 1} = \S\epsilon + O(\epsilon^2) .$$ In contrast, the integral term is more complicated to analyse, and compared to the impulse integral and the energy integral, it does not possess a Taylor series in $g_0$ beyond the leading order term. To obtain an asymptotic expansion, we will first split up the integrand into a term given by the limit $g_0\to 0$ and a remainder term given by subtracting off this limit term.
The limit $g_0\to 0$ of the integrand $F(g,g_0)/(g-g_0)$ yields $F(g,0)/g = \sqrt{g^{k-3}/(1-g^{k-1})}$ through expression . Hence we split up the integral $$\label{r_near0:scaledwidth_integral}
\int_{g_0}^{g_1} \frac{F(g,g_0)}{g-g_0} \,dg
= \int_{g_0}^{g_1} \frac{F_1(g)}{g} \,dg
+ \int_{g_0}^{g_1} \frac{g_0F_1(g)+gF_2(g,g_0)}{g(g-g_0)} \,dg$$ where $$\quad
F_1(g)=F(g,0),
\quad
F_2(g,g_0) = F(g,g_0)-F_1(g) .$$ The first integral term on the right-hand side of equation can be evaluated explicitly $$\int_{g_0}^{g_1} \frac{F(g,0)}{g} \,dg
= \int_{g_0}^{g_1} \frac{\sqrt{g^{k-3}}}{\sqrt{1-g^{k-1}}} \,dg
= \tfrac{2}{k-1} \big( \arcsin(g_1{}^\frac{k-1}{2}) - \arcsin(g_0{}^\frac{k-1}{2}) \big) ,$$ which gives $$\int_{g_0}^{g_1} \frac{F(g,0)}{g} \,dg \simeq \tfrac{\pi}{k-1} +O(\epsilon^\frac{1}{k-1},\epsilon) .$$ To analyse the remaining integral term on the right-hand side of equation , we need to take into account that the integrand has a square-root singularity because $$A(g,g_0)=(g_1-g)B(g,g_0),
\quad
B(g_1,g_0)= \frac{(k+1)(g_1^k-g_0^k)}{(g_1-g_0)^2} \neq 0,$$ and we also need to note that, through relation , $F_1(g_1)\simeq g_1{}^\frac{k-1}{2}/\sqrt{2g_0}$ is singular when $g_0\to 0$. Consequently, we first combine these singular terms by expressing $$\begin{aligned}
g_0F_1(g)+gF_2(g,g_0) =&
\frac{(g-g_0)C(g,g_0)}{\sqrt{g_1-g}\sqrt{B(g,g_0)}\big(g_0{}^\frac{k-1}{2}g\sqrt{A(g,g_0)} + g_0\sqrt{A(g_0,g_0)} g^\frac{k-1}{2}\big)}
\\&
+\frac{g^\frac{k-1}{2}(g-g_0)H(g,g_0)}{\sqrt{g_1-g}\sqrt{B(g,g_0)}\big(\sqrt{1-g^{k-1}} + \sqrt{A(g,g_0)}\big)}
\end{aligned}$$ where $$\label{r_near0:H}
H(g,g_0) = \sqrt{1-g^{k-1}} -\frac{A(g,g_0)}{\sqrt{1-g^{k-1}}}$$ and $$\label{r_near0:C}
C(g,g_0) = g_0^{k-1}\frac{(g+g_0)A(g,g_0)}{\sqrt{A(g_0,g_0)}} -
g_0^2\sqrt{A(g_0,g_0)}\Big( \frac{g^{k-1}-g_0^{k-1}}{g-g_0} + g_0^{k-1}\frac{1-A(g,g_0)/A(g_0,g_0)}{g-g_0} \Big) .$$ Hence we have $$\label{r_near0:scaledwidth_remainder}
\begin{aligned}
\frac{g_0F_1(g)+gF_2(g,g_0)}{g(g-g_0)} = &
\frac{1}{\sqrt{g_1-g}\sqrt{B(g,g_0)}}\bigg(
\frac{g^\frac{k-3}{2}H(g,g_0)}{\sqrt{1-g^{k-1}} + \sqrt{A(g,g_0)}}
\\&\qquad
+ \frac{C(g,g_0)}{g_0\big(g_0{}^\frac{k-3}{2}g\sqrt{A(g,g_0)} + \sqrt{A(g_0,g_0)} g^\frac{k-1}{2}\big)g}
\bigg) .
\end{aligned}$$ We will now use this expression to estimate the size of the corresponding integral term on the right-hand side of equation , which is given by the sum of the integrals $$\begin{aligned}
& \int_{g_0}^{g_1}
\frac{g^\frac{k-3}{2}H(g,g_0)}{\sqrt{g_1-g}\sqrt{B(g,g_0)}\big(\sqrt{1-g^{k-1}} + \sqrt{A(g,g_0)}\big)}\, dg ,
\label{r_near0:scaledwidth_remainder_integral:H}
\\
& \int_{g_0}^{g_1}
\frac{C(g,g_0)}{g_0\sqrt{g_1-g}\sqrt{B(g,g_0)}\big(g_0{}^\frac{k-3}{2}g\sqrt{A(g,g_0)} + \sqrt{A(g_0,g_0)} g^\frac{k-1}{2}\big)g} .
\label{r_near0:scaledwidth_remainder_integral:C}\end{aligned}$$ It is hard to obtain a good estimate that holds for arbitrary $k>1$, so we will examine two special cases $k=2,3$ for which both of the integrals can be evaluated explicitly in terms of elementary functions, with $g_0/g^* =\epsilon^\frac{2}{k-1} \ll 1$ and $g_1\simeq 1-\tfrac{2}{k-1}g^*\epsilon^\frac{2}{k-1}$. (The case $k=\tfrac{3}{2}$ can be be evaluated in terms of elliptic functions.)
For $k=2$ and $k=3$, we get $$\begin{aligned}
\int_{g_0}^{g_1} \frac{(g_0F_1(g)+gF_2(g,g_0))\,dg}{g(g-g_0)}
& \simeq \sqrt{g_0} \big( \ln(g_0) +2\ln 2 \big) +\arccos(1-4g_0)
\\
& \simeq \sqrt{g_0}\ln(g_0) + O(\sqrt{g_0}) ,
\end{aligned}$$ and $$\begin{aligned}
\int_{g_0}^{g_1} \frac{(g_0F_1(g)+gF_2(g,g_0))\,dg}{g(g-g_0)}
& \simeq g_0 \big( \ln(1-2g_0) +\ln 2 \big) +\arccos(1-g_0)
\\
& \simeq \sqrt{2g_0} + O(g_0) .
\end{aligned}$$ In these two cases, we see that the sum of the two remainder integrals and vanishes as $g_0\to 0$.
Finally, combining the asymptotic expansion of the integral part of the width integral and the algebraic part of the width integral , we have $$\hat\ell \simeq \sqrt{\tfrac{k(k+1)}{2}} \tfrac{\pi}{k-1} + o(\epsilon) .$$
Derivation of KdV equation in the weakly nonlinear regime {#appendix:KdV}
=========================================================
Starting from the LWHC wave equation and using the relation between the amplitude $u(t,x)$ and the continuum limit $U(t,x)$ of the particle displacement , we introduce a scaled amplitude by putting $$U(t,x) = \epsilon v_0 Z(\tau,\xi)$$ with $\epsilon\ll 1$ being an expansion parameter, where $v_0$ is the background strain , and where $$\label{2scale}
\tau= \epsilon_2 t,
\quad
\xi = \epsilon_1 (x-c_0t)$$ are a scaled time variable and a scaled space variable with respect to a reference frame moving with the sound speed . Here $\epsilon_1$ and $\epsilon_2$ are parameters that will be subsequently related to $\epsilon$. Note that the amplitude $u(t,x)$ is given by $$\label{w-kdv}
u(t,x)/v_0 = \epsilon Z(\tau,\xi) -x .$$
We will now show that an expansion of the LWHC wave equation leads to the KdV equation in potential form for $Z(\tau,\xi)$ with an appropriate choice of $\epsilon_1$ and $\epsilon_2$ in powers of $\epsilon$.
To proceed, we substitute expression into the LWHC wave equation and use the relations $$\begin{aligned}
& u_x/v_0 = \epsilon \epsilon_1 Z_\xi -1 ,
\quad
u_{xx}/v_0 = \epsilon \epsilon_1{}^2 Z_{\xi\xi},
\quad
u_{xxx}/v_0 = \epsilon \epsilon_1{}^3 Z_{\xi\xi\xi},
\quad
u_{xxxx}/v_0 = \epsilon \epsilon_1{}^3 Z_{\xi\xi\xi\xi},
\\
& u_t/v_0 = \epsilon (\epsilon_2 Z_\tau -\epsilon_1 c_0 Z_\xi),
\quad
u_{tt}/v_0 = \epsilon (\epsilon_2{}^2 Z_{\tau\tau} -2\epsilon_1\epsilon_2 c_0 Z_{\tau\xi} + \epsilon_1{}^2 c_0{}^2 Z_{\xi\xi}) .
\end{aligned}$$ Next we expand the resulting terms in a series in $\epsilon$, yielding $$\label{LHWC-expand}
\begin{aligned}
0=
& \epsilon \epsilon_1\epsilon_2 (2v_0/c^2) Z_{\tau\xi}
+ \epsilon^2 \epsilon_1^3 (1-k)v_0^k Z_\xi Z_{\xi\xi}
+ \epsilon \epsilon_1^4 \gamma v_0^k Z_{\xi\xi\xi\xi}
\\& -\epsilon \epsilon_2{}^2 (v_0/c^2) Z_{\tau\tau}
- \epsilon^2 \epsilon_1^5 \beta v_0^k Z_{\xi\xi}Z_{\xi\xi\xi}
+ \epsilon^3 \epsilon_1^6 \alpha v_0^k Z_{\xi\xi}^3
\\& +\text{ higher order terms }
\end{aligned}$$ after we use expression for the sound speed $c_0$. The three terms in the first line of this equation correspond to the terms in the KdV equation in potential form for $Z$. Hence, we balance these terms by putting $\epsilon \epsilon_1\epsilon_2= \epsilon^2 \epsilon_1^3 =\epsilon \epsilon_1^4$, which determines $$\epsilon_1=\epsilon,
\quad
\epsilon_2=\epsilon^3 .$$ Then we see that equation becomes $$0=
(2v_0/c^2) Z_{\tau\xi} + (1-k)v_0^k Z_\xi Z_{\xi\xi} + \gamma v_0^k Z_{\xi\xi\xi\xi}
+ O(\epsilon^2) .$$ Hence, in the limit $\epsilon\to 0$, we obtain $$(2v_0/c^2) Z_{\tau\xi} + (1-k)v_0^k Z_\xi Z_{\xi\xi} + \gamma v_0^k Z_{\xi\xi\xi\xi} \simeq 0$$ which is the KdV equation for $Z_\xi$.
The scaled amplitude $Z(\tau,\xi)$ can be directly expressed in terms of the strain variable by $v/v_0 = 1 - \epsilon^2 Z_\xi$, and hence $$\epsilon^2 Z_\xi = 1-v/v_0 =-w$$ where $w$ is the variable introduced in the expansion for solitary waves in the weakly nonlinear regime with $0<w\ll 1$.
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abstract: |
Obliquity variability could play an important role in the climate and habitability of a planet. Orbital modulations caused by planetary companions and the planet’s spin axis precession due to the torque from the host star may lead to resonant interactions and cause large-amplitude obliquity variability. Here we consider the spin axis dynamics of Kepler-62f and Kepler-186f, both of which reside in the habitable zone around their host stars. Using [*N*]{}-body simulations and secular numerical integrations, we describe their obliquity evolution for particular realizations of the planetary systems. We then use a generalized analytic framework to characterize regions in parameter space where the obliquity is variable with large amplitude. We find that the locations of variability are fine-tuned over the planetary properties and system architecture in the lower-obliquity regimes ($\lesssim 40^\circ$). As an example, assuming a rotation period of 24 hr, the obliquities of both Kepler-62f and Kepler-186f are stable below $\sim 40^\circ$, whereas the high-obliquity regions ($60^\circ - 90^\circ$) allow moderate variabilities. However, for some other rotation periods of Kepler-62f or Kepler-186f, the lower-obliquity regions could become more variable owing to resonant interactions. Even small deviations from coplanarity (e.g. mutual inclinations $\sim 3^\circ$) could stir peak-to-peak obliquity variations up to $\sim 20^\circ$. Undetected planetary companions and/or the existence of a satellite could also destabilize the low-obliquity regions. In all cases, the high-obliquity region allows for moderate variations, and all obliquities corresponding to retrograde motion (i.e. $> 90^\circ$) are stable.
[**[Keywords:]{}**]{} Exoplanets: dynamics – Exoplanets: habitability – Methods: numerical – Methods: analytical
author:
- Yutong Shan
- Gongjie Li
title: 'Obliquity Variations of Habitable Zone Planets Kepler-62f and Kepler-186f'
---
Introduction
============
The rapidly growing arsenal of exoplanet detections has greatly improved our understanding on the occurrence and orbital and structural properties of planetary systems [e.g., @Lissauer14; @Winn15]. In particular, the NASA [*Kepler*]{} mission has discovered thousands of planetary candidates and identified terrestrial planets in the habitable zone (HZ) of their host stars [e.g., @Barclay13; @Borucki13; @Quintana14; @Torres15]. The HZ is conventionally defined as the region where liquid water may exist on the surface of the planets with atmospheres similar to that of the Earth [@Kasting93; @Kopparapu13]. The [*Kepler*]{} Habitable Zone Working Group has provided a list of HZ planets, based on various HZ boundaries and planetary radii, which includes 104 planetary candidates in the optimistic HZ and 20 planets with radii less than 2 $R_{\oplus}$ in the conservative HZ [@Kane16]. Many of such systems contain multiple planets, and their dynamical interactions could play a critical role in determining the habitability of these systems. @Kane16 analyzed the dynamical stability of the potentially habitable multiplanetary systems, which also serves to validate the planetary systems.
In addition to orbital stability, planetary obliquity and its variations are also important considerations in the habitability of a planet. Obliquity (or axial tilt) measures the angle between a planet’s spin and orbital axes. These values are known for planets in the solar system. However, no reliable values for exoplanets have been claimed to date. Methods to infer exoplanetary spin axis direction from light curves via the effect of rotational flattening on transit depth and infrared phase curves [e.g. @Gaidos04; @Carter10], as well as from high-contrast direct imaging through observing seasonal variations [@Kane17], have been proposed. With very high quality photometric data and sophisticated modeling, obliquity measurements may be possible for the most favorable exoplanets in the future.
Obliquity determines the latitudinal distribution of solar radiation on a planet and affects the modulation of its climate [@Williams97; @Chandler00; @Jenkins00; @Spiegel09]. According to the Milankovitch theory, ice ages on the Earth are closely associated with variations in insolation at high latitudes, which depends on the orbital eccentricity and orientation of the spin axis [@Hays76; @Weertman76; @Imbrie82; @Berger92]. At present, the obliquity variation of the Earth is regular and only undergoes small oscillations between $22\fdg1$ and $24\fdg5$ with a 41,000 year period [@Vernekar72; @Laskar93b]. This is not to say that obliquity instability is wholly incompatible with life – based on a simple 1D energy-balance atmospheric model, @Armstrong14 suggest that large and frequent obliquity variations could help maintain higher surface temperatures and extend the outer boundary of the traditional HZ.
Evolution in a planet’s obliquity is governed by orbital perturbations from its companion planets, as well as torque from the host star and any moons acting on the planetary spin axis [e.g., @Ward74; @Laskar93a]. When the perturbing frequencies from the companion planets match with the precession frequency due to the torques, the obliquity variation amplitude will increase due to resonant interactions. For instance, without the Moon, the torquing frequency of the Sun would match that from the companion planets, and the obliquity variation of the hypothetical Earth would be large (though constrained between $0$ and $45^\circ$) over billion-year timescales [@Laskar93a; @Lissauer12; @Li14]. With the presence of the Moon, the obliquity variation of the Earth is significantly suppressed [@Laskar93b]. Other notable examples of planets whose spin axis dynamics have been thoroughly studied include Venus and Mars. For an early Venus with less atmospheric tides, obliquity variation in the low-obliquity range is small [@Barnes16]. On the other hand, the obliquity of Mars can vary with large amplitudes [@Ward73; @Laskar93b; @Touma93]. The obliquity changes of Mars likely resulted in runaway condensation of $\ce{CO_2}$ in the atmosphere, rendering Mars uninhabitable [@Toon80; @Fanale82; @Pollack82; @Francois90; @Nakamura03; @Soto12].
Prior to the advent of [*Kepler*]{}, @Atobe04 studied the obliquity variability of hypothetical HZ terrestrial planets co-inhabiting systems with giant planets. They characterized variation amplitudes as a function of rotation period and orbital distance for specific planet configurations. These include hypothetical systems with up to two giant planets with assorted masses and orbital elements, as well as real systems with known RV planets. Most known systems at the time were giant planets around sun-like stars. Though they did not explicitly consider compact systems with multiple small planets, @Atobe04 arrived at many important general realizations to which we will refer in this article.
Now, [*Kepler*]{} has provided us with concrete systems to study, the kind with real, potentially habitable terrestrial planets neighboring multiple other small planets. Here, we will consider the obliquity variation of Kepler-62f and Kepler-186f, which are likely terrestrial planets in the HZ orbiting around a K2V-type star and an M1-type star, respectively [@Borucki13; @Quintana14]. Both of these two planets stay far away from their host stars, where the tidal influences of their host stars are comparatively weak. In particular, @Bolmont14 [@Bolmont15] and @Shields16 demonstrated that, for reasonable assumptions of the planetary properties and system ages, it is possible that Kepler-62f and Kepler-186f have not yet evolved to be tidally locked. This would allow them to keep high obliquities and short rotation periods. Note that the tidal evolution of planetary spin axis for general configurations shows sensitive dependence on the planetary tidal $Q$-values, as well as other assumptions (for more discussion, see Section \[ss:tides\] and @Heller11). Using the Community Climate System Model, @Shields16 identified combinations of orbital and atmospheric properties that permit surface liquid water for Kepler-62f, exploring both low- and high-obliquity regimes. To evaluate the potential habitability of these planets, these pioneering works considered many relevant factors, including long-term spin evolution. However, the effect of planetary spin-orbit coupling has not been investigated in detail. In addition, the mutual inclination between the planets has been assumed to take the minimum values (i.e. taken directly from the line-of-sight inclination of these planets), which leads to small obliquity variations. In reality, a larger range of inclinations is permitted by observation. Existing studies are also confined by their reliance on expensive [*N*]{}-body simulations, which limits the set of parameter values subject to exploration.
In this article, we focus on the spin axis variability of the HZ planets on shorter timescales. We start by assuming that all planets in the system have been detected and study the evolution of the two five-planet systems. We relax the mutual inclination assumption and include a wide range of planetary system parameters consistent with observational constraints. We present a secular analytical framework applicable in the situation of small orbital eccentricities and inclinations. Such a framework is powerful because it allows us to visualize and predict the nature of obliquity variations in a large parameter space, as well as examine the sensitivity of conclusions to errors in the observed parameters. Specifically, we consider different planetary rotation rates and additional planets and satellites, and we characterize regions in this parameter space where the resonant interactions between the HZ planet and its companions may cause large obliquity variations. We briefly discuss the prospects of long-term obliquity evolution subjected to the gradual but inevitable tidal synchronization process.
This paper is organized as follows: planetary system properties used throughout the paper are explained in section \[s:sys-pars\]. In section \[s:nr\], we use [*N*]{}-body simulations coupled with secular integration to illustrate the evolution of obliquity. We consider variations in obliquity as a product of resonant interactions using an analytical approach and interpret our numerical results in section \[s:ar\]. Section \[s:dis\] explores the effects of undetected planets, satellites, and the possible path to tidal synchronization. A summary is presented in section \[s:conclusion\].
Planetary System Parameters {#s:sys-pars}
===========================
We anchor our analysis on the measured properties of the two [*Kepler*]{} systems with potentially habitable planets and infer the rest of the relevant parameters under assumptions discussed below. Parameters and their representative values used in this work are given in Table \[t:sys-pars\].
Most direct observables in exoplanet systems are combinations of properties of the planet and the host star. Therefore, any indeterminacy in the stellar properties is directly propagated into errors in the planet properties. For instance, the radius of a planet and its semi-major axis are derived from the transit depth and orbital period in concert with the stellar radius ($R_\star$) and mass ($M_\star$), respectively. We set the mass and radius of Kepler-62 to be $M_\star = 0.69\pm0.02 M_\sun$ and $R_\star = 0.63\pm0.02 R_\sun$ [@Borucki13], and that of Kepler-186 to be $M_\star = 0.544~\pm0.02 M_\sun$ and $R_\star = 0.523~\pm0.02 R_\sun$ [@Torres15]. Measurements of orbital periods ($P_{\rm orb}$) from the transiting technique tend to be highly accurate. The semi-major axes ($a$) can in turn be calculated from $M_\star$ and $P_{\rm orb}$ using Kepler’s third law. We adopt the semi-major axes for the Kepler-62 planets from @Borucki13, and we calculate the semi-major axes for Kepler-186 based on the updated stellar mass from @Torres15 and the orbital period measurements from @Quintana14. The age of the Kepler-62 system is determined to be $7 \pm 4$ Gyr [@Borucki13], though recently @Morton16 arrived at a much younger age of $2.34^{+2.15}_{-1.02}$ Gyr. Kepler-186 is estimated to have an age of $4 \pm 0.6$ Gyr [@Quintana14].
The line-of-sight orbital inclinations ($i_{\rm LoS}$) are computed from the measured impact parameters ($b$), as well as the stellar radius ($R_\star$) and semi-major axes: $$i_{\rm LoS} = \arccos\left[\frac{b R_\star}{a}\right].
\label{e:inclination}$$ We obtain the $i_{\rm LoS}$ for planets in Kepler-62 from @Borucki13 directly, and we calculate those for Kepler-186 based on the measurements of the $b$ values in @Quintana14 and the updated stellar parameters from @Torres15. Although the line-of-sight inclinations may not directly translate into the inclination of the planets measured from a reference plane, the fact that the planets transit means that they are most likely arranged in nearly coplanar configurations. We use $90^\circ - i_{\rm LoS}$ as the minimum initial orbital inclinations in our numerical study, consistent with previous works [e.g. @Bolmont14; @Shields16]. While the eccentricities are not well constrained, they are all reasonably close to 0. For simplicity, we assume that all orbits are initially circular in the numerical simulations. Since the planetary masses ($M_p$) have not been directly measured for these systems, we use the publicly available code of @ChenKip17 based on a probabilistic approach that predicts masses for a variety of celestial bodies from their radii. We directly use the published planetary radii ($R_p$) and their errors as given in Table 1 of @Borucki13 for Kepler-62, and we calculate $R_p$ using transit depth measurements from @Quintana14 and stellar parameters from @Torres15, propagating both errors for Kepler-186. For a given input radius, the output is a probability distribution of the possible masses, where has marginalized over a range of possible planetary compositions. The resultant uncertainty in the forecasted masses can be quite large and asymmetric (on order of $\sim 30 - 70\%$). While the measured radii for Kepler-62f ($1.41 \pm 0.07 R_\earth$) and Kepler-186f ($1.17 \pm 0.11 R_\earth$) are consistent with planets with rocky compositions ($< 1.62 R_\earth$, @Rogers15), in its standard implementation also considers Neptunian compositions. Since we study the habitability of Kepler-62f and Kepler-186f in this work, we presuppose that they are rocky. Thus, to enforce the condition that they be terrestrial planets, we inverted the Terran mass-radius power law given in @ChenKip17 to generate the default masses for Kepler-62f and Kepler-186f used in this work:
$$\left(\frac{M_p}{M_\earth}\right) = \left[\frac{1}{1.008} \left(\frac{R_p}{R_\earth}\right)\right]^{1/0.279}.$$
The resulting masses are 3.3 and 1.7 $M_\earth$ for Kepler-62f and Kepler-186f, respectively, similar to those considered in existing studies [@Bolmont14; @Bolmont15; @Shields16; @Quarlesinprep]. We feed the median radii into either (all planets except 62f and 186f) or the Terran power-law relation, where we marginalize over hyperparameter posteriors given in @ChenKip17 (62f and 186f), and tabulate the default and 68% symmetric confidence intervals for the planetary mass posteriors in Table \[t:sys-pars\].
The Evolution of Obliquity over Time: Numerical Results {#s:nr}
=======================================================
An illustration of the planetary spin-orbit misalignment (obliquity) is shown in Figure \[f:cartoon\]. $L_{\rm orb}$ and $L_{\rm rot}$ denote the angular momentum vector of the orbit and the planet, respectively, and the obliquity angle ($\epsilon$) represents the angle between $L_{\rm orb}$ and $L_{\rm rot}$. The inclination $i$ and the longitude of node $\Omega$ characterize the orientation of the orbital plane, or the direction of $L_{\rm orb}$. Similarly, the obliquity angle $\epsilon$ and the longitude of the spin axis $\psi$ determine the orientation of the planetary spin axis with respect to the orbital plane.
![Geometry of obliquity evolution. This schematic diagram depicts the orbital ($L_{\rm orb}$) and spin ($L_{\rm rot}$) axes and the angles involved. The combined evolution of these two angles forms the trajectory of precession and nutation. $L_{\rm orb}$ can be torqued by other bodies in the system, which could interact resonantly with $L_{\rm rot}$ and cause it to oscillate. See also Fig 1 in @Atobe04.[]{data-label="f:cartoon"}](Obliquity_cartoon_v4.pdf)
The phenomenon of planetary obliquity variations is induced by the torques of the the host star on the equatorial bulge of the planet, as well as by periodic forcing from other planets. Therefore, its dynamics depends on the orbital configuration of all the bodies in the system. In this section, we use a numerical approach to study short-term obliquity evolution. We present the framework for the numerical studies in section \[ss:ham\], and numerical results on the obliquity variation of a few representative manifestations for the two five-planet systems are shown in \[ss:eps-t\]. A qualitative comparison to existing spin-dynamical studies on these systems is given in \[ss:compare\].
Numerical Method Framework {#ss:ham}
--------------------------
To calculate the obliquity evolution of a planet, we first perform [*N*]{}-body simulations to obtain its orbital evolution. Then, we compute the spin axis dynamics, which is coupled to the planetary orbit, based on the [*N*]{}-body results and the Hamiltonian below (Eqn.(\[e:ham\])). This approach implicitly assumes that the planets’ orbital angular momenta are dominant over that of their spin, which allows us to ignore the feedback of the spin axis to the planetary orbit.
We use the hybrid integrator in the publicly available [*N*]{}-body code [@Chambers99] to simulate representative manifestations for each planet system for $10^7$ yr, stepping in increments of 0.5 days. The initial values of semi-major axes, eccentricities, and mutual inclinations are discussed in Section \[s:sys-pars\].
The spin-orbit coupling is described by the secular Hamiltonian of obliquity variation, which is well documented in the literature [e.g. @Goldreich66; @Wisdom84; @Laskar93a; @deSurgy97]:
$$\begin{aligned}
H(\chi,\psi,t) = & \frac{1}{2}\alpha_p\chi^2+\sqrt{1-\chi^2} \nonumber \\
& \times [A(t) \sin \psi + B(t) \cos \psi].
\label{e:ham}\end{aligned}$$
Here $\chi$ and $\psi$ are the Andoyer canonical variables, where $\chi=\cos{\epsilon}$ and $\psi$ is the longitude of the planet’s spin axis [@Andoyer23; @Kinoshita72]. $\epsilon$ is the obliquity, and $A(t)$ and $B(t)$ are functions of $p = \sin(i/2)\sin\Omega$ and $q=\sin(i/2)\cos\Omega$, where $i(t)$ represents orbital inclination and $\Omega(t)$ is the longitude of the ascending node of the planet:
$$A(t) = 2(\dot{q}+p(q\dot{p}-p\dot{q}))/\sqrt{1-p^2-q^2},
\label{e:at}$$
$$B(t) = 2(\dot{p}+q(p\dot{q}-q\dot{p}))/\sqrt{1-p^2-q^2}.
\label{e:bt}$$
$\alpha_p$ is the precession coefficient, defined for a given planet $p$ as
$$\label{e:alpha}
\alpha_p = \frac{3}{2\omega}\left[\frac{GM_\star}{(a_p\sqrt{1-e_p^2})^3}\right]E_d ,
$$
where $a_p$ and $e_p$ denote the semi-major axis and eccentricity of the planetary orbit around the star, respectively. $\omega$ is the angular velocity of the planet’s spin. $E_d = (C-1/2(A+B))/C$ is the dynamical ellipticity, where $A$, $B$ and $C$ are the moment of inertia along the three principal axes. $E_d$ is related to the oblateness (flattening) of the planet and generally scales with $\omega^2$ [@Lambeck80]. For a moonless Earth (i.e. only considering the torque from the Sun), $\alpha_\Earth = 17\farcs4 ~{\rm yr}^{-1}$ [@Laskar93a].
As the internal structure and the dynamical ellipticity of the exoplanets are unknown, we assume that the dynamical ellipticity is the same as that of Earth if the planet rotates with the same period as Earth’s.[^1] For planets on nearly circular orbits ($e_p \sim 0$), the relation for $\alpha_p$ is then given as
$$\label{e:alpha_earth}
\alpha_p = \left(\frac{P_{\rm rot}}{\rm day}\right)^{-1}\left(\frac{M_\star}{M_\sun}\right) \left(\frac{a_p}{\rm au}\right)^{-3} \alpha_\Earth, ~~ \alpha_\Earth = 17\farcs4 ~{\rm yr}^{-1}.$$
Note $\omega \propto P_{\rm rot}^{-1}$. Scaling the precession coefficient with the host star mass and the semi-major axis of each planet, Eqn.(\[e:alpha\_earth\]) gives $\alpha_{K62f} = 32\farcs2~{\rm yr}^{-1}$ and $\alpha_{K186f} = 137\farcs4~{\rm yr}^{-1}$ corresponding to planetary rotation periods of 24 hr. This approach yields similar results to that of @Lissauer12 [see also @Quarlesinprep], who assumed that the moment of inertia coefficient ($C/(M_pR_{eq}^2)$) of the planet is the same as Earth’s, where $M_p$ is the planetary mass and $R_{eq}$ is the equatorial radius of the planet. In particular, for the planetary masses and radii assumed in Table \[t:sys-pars\] and rotation periods of 24 hours, $\alpha_{K62f} = 31\farcs9~{\rm yr}^{-1}$ and $\alpha_{186f} = 134\farcs4~{\rm yr}^{-1}$ based on the approach by @Lissauer12. We set $\alpha_{K62f} = 32\farcs2~{\rm yr}^{-1}$ and $\alpha_{K186f} = 137\farcs4~{\rm yr}^{-1}$ as our default, Earth-like values. Of course, the rotation state of an arbitrary planet may not resemble that of the Earth. Planets formed from protoplanet accretion in [*N*]{}-body simulations tend to have rapid primordial spins [e.g. @Kokubo07]. Over time, $P_{\rm rot}$ is modified by tidal interaction with the host star and any satellites present. To explore rotation periods deviating from that of the Earth, we scale the $\alpha_{\rm{p}}$ values with the inverse of the planet’s rotation period ($P_{\rm rot}$), as in Eqn.(\[e:alpha\_earth\]).
Obliquity Evolution for Variable Rotation Periods, and Orbital Inclinations {#ss:eps-t}
---------------------------------------------------------------------------
Theoretically, a planet’s initial obliquity is often determined by stochastic impacts during formation and can be expected to take an isotropic distribution [e.g. @Kokubo07]. Therefore, we explore the dynamical evolution of the obliquity starting with a wide range of initial values, $\epsilon_0$, [^2], from prograde ($\epsilon < 90^\circ$) to retrograde ($\epsilon > 90^\circ$). Hereafter, references to ‘low’ and ‘high obliquity’ are relative to the prograde regime only.
For particular realizations of the Kepler-62 and Kepler-186 systems, the evolution of the obliquity angle is shown in Figure \[f:eps-t\]. The left and right columns correspond to the results for Kepler-62f and Kepler-186f, respectively. The top panels assume that the planetary rotation period is 24 hr, similar to our Earth. In this case, the low-obliquity regions have low variability for both Kepler-62f and Kepler-186f, while the high-obliquity regions allow small variabilities for Kepler-62f. The timescale of oscillation ranges from a fraction of to several megayears. In both cases, retrograde obliquities show little variation.
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For particular values of the planetary rotation period (i.e. proxy for $\alpha_p$), the lower region of the obliquity allows larger variabilities. Examples of such rotation periods are approximately $17$ hours ($\alpha \approx 45\farcs5~{\rm yr}^{-1}$) for Kepler-62f and $260$ hr for Kepler-186f ($\alpha = 12\farcs7~{\rm yr}^{-1}$). The resultant obliquity evolutions are shown in the middle panels of Figure \[f:eps-t\]. We will illustrate in section \[s:ar\] that the locations and natures of obliquity variation are determined by the properties of resonant interactions.
In the bottom panels, we consider slightly higher inclinations, requiring the mutual inclinations of the planets to be within $3^\circ$, in accordance with studies on the mutual inclination of multiplanetary systems [@Fang12; @Fabrycky14; @Ballard16; @Moriarty16]. We set the initial longitude of ascending nodes of the planets such that all of the planets transit over 4 yr timescales, i.e. over the tenure of the original [*Kepler*]{} mission. The maximum inclinations of Kepler-62f and Kepler-186f reach $\sim 1\fdg5$ over 10 Myr. We adopt the same precession coefficient (i.e. assume the same rotation periods and interior mass distribution) as those in the middle panels. With the slightly higher mutual inclinations, the perturbation to the planetary orbit is stronger, leading to enhanced obliquity variation (up to $\sim 20^\circ$ in peak-to-peak amplitude). We obtain an analytical expression for the amplitude of variabilities in section \[ss:res-wid\].
Obliquity is affected by both the orbital inclination and the orientation of the spin axis relative to a fixed reference plane. Naturally, the intrinsic variabilities of the two factors both contribute to the overall observed variability in the obliquity. In regions where the strongest variabilities occur (due to resonant spin-orbit coupling; see 4), the variation is dominated by that of the spin axis. Weaker variability can manifest in other regions (due to higher-order effects; 4), particularly visible in the bottom panels of Figure \[f:eps-t\]. In these cases, oscillation of the orbital inclination and spin axis orientation both contribute comparably to the obliquity variations.
Comparison to Previous Studies {#ss:compare}
------------------------------
The obliquity evolutions for these two exoplanets were previously explored, albeit usually for either a narrow range of initial conditions and/or for a much longer term under tidal influence. Directly comparable with our study are the 10 Myr results, in which @Bolmont14 find that Kepler-186f’s obliquity is stable if it starts with an Earth-like obliquity ($23^\circ$) and rotation period ($\sim$ 24 hr). @Bolmont15 reach the same conclusion for Kepler-62f. These are very consistent with the outcomes found in this work for the same set of assumptions, as shown in the top panels of Figure \[f:eps-t\]. However, more complex behavior is possible with different $\epsilon_0$, rotation periods, and mutual inclinations, as illustrated in the bottom panels in this figure.
During the final preparation of our manuscript, we noticed that @Quarlesinprep have been working on a similar problem related to the obliquity variation of Kepler-62f, using a different approach. Specifically, adopting direct [*N*]{}-body simulations including spin-orbit coupling, @Quarlesinprep find that the low-obliquity region of Kepler-62f is stable, assuming that it is an Earth analog. This is consistent with our results.
The Theory of Obliquity Instablity and Results from an Analytical Framework {#s:ar}
===========================================================================
The resonant interactions between the torque from the host star acting on the planetary spin axis and the orbital perturbation from the companion planets can be quantified in a straightforward, analytical framework. This can in turn be used to rapidly approximate the obliquity evolution for given properties of the orbital architecture and that of the planet of interest. In this section, we summarize the analytical approach to characterize the locations of the resonant regions (section \[ss:res-zones\]), and we derive an analytical expression for the size of the resonant region for Kepler-62f and Kepler-186f (section \[ss:res-wid\]). Then, we interpret the numerical results based on the analytical approach (section \[ss:num-freq\]). In the end, we characterize the parameter space of obliquity evolution considering the uncertainties in the observational orbital parameters and planetary masses based on the analytical approach (section \[ss:ll-th\]). The basis of this analysis lies in the simplified Hamiltonian in Eqn.(\[e:ham\]), which is analogous to that of a physical pendulum.
Location of Resonances {#ss:res-zones}
----------------------
The expected resonant locations can be calculated analytically by identifying the obliquity values that allow resonant coupling between the precession of the planetary spin axis and the oscillation of the planetary orbital plane, as discussed extensively in the literacture [e.g., @Laskar93a; @Touma93; @Lissauer12]. Specifically, the form of Eqn.(\[e:ham\]) is analogous to the Hamiltonian describing a physical pendulum with angular position $\theta$ and angular momentum $\mathcal{L}$:
$$H_{\rm pendulum}(\mathcal{L},\theta)=\frac{\beta}{2}\mathcal{L}^2 + c\cos \theta.
\label{e:pen}$$
The $\beta$ term here is akin to $\alpha_p$ in Eqn.(\[e:ham\]), while $c$ can be compared with $\sqrt{1-\chi^2}A(t) \sim \sqrt{1-\chi^2}B(t)$, as $A(t)$ and $B(t)$ are expected to be on the same order.
A natural frequency for the pendulum system is given by the characteristic rate of variation for $\theta$:
$$\dot{\theta} = \frac{\partial H_{\rm pen}}{\partial \mathcal{L}} = \beta \mathcal{L},$$
and resonance occurs when a perturbing angular frequency ($f$) coincides with this natural frequency. Comparing to Eqn.(\[e:ham\]), we see that $\alpha_p \chi$ is analogous to the natural frequency of this system (indeed, it is the frequency of the axial precession). Therefore, for obliquity $\epsilon$ to vary, this condition amounts to requiring $-f = \alpha_p \chi = \alpha_p \cos(\epsilon)$. In general, there can be multiple forcing frequencies. We use $f_k$ to denote the $k$th modal frequency of the forcing terms $A(t)$ and $B(t)$. Since $A(t)$ and $B(t)$ are functions of orbital inclination and the longitude of the ascending node, their characteristic frequencies should follow that of the inclination vector.
Therefore, the $k$th resonance occurs at obliquity angle $\epsilon_{\rm{res}, k}$, where
$$\cos(\epsilon_{{\rm res}, k}) = -f_k/\alpha_p.
\label{e:coseps}$$
It follows that negative frequencies correspond to obliquity resonances in the prograde regime ($\epsilon < 90^\circ$) and positive frequencies to retrograde ones ($\epsilon > 90^\circ$). Of course, only modes with frequency $f_k/\alpha_p \in [-1,1]$ lead to physical values of $\epsilon_{{\rm res}, k}$, and can result in resonant interactions. This equation can be inverted to read $\alpha_p = -f_k/\cos(\epsilon_{{\rm res}, k})$. Since, given $f_k$, there is a corresponding $\alpha_p$-value for every arbitrary $\cos(\epsilon_{\rm res})$, we conclude that, for every obliquity value, each mode can induce instability at one fine-tuned rotation rate. This will become visually apparent in section \[ss:ll-th\].
Resonant Widths {#ss:res-wid}
---------------
Resonances are not point-like – each has a finite extent generally centered around $\epsilon_{{\rm res}, k}$. The width of the resonance determines the variability amplitude of the obliquity. Although the actual variability amplitude may not span the full width of the associated resonance owing to a dependence on the initial longitude of the spin axis, $\psi_0$ (see also \[ss:eps-t\]), resonance widths can serve as order-of-magnitude proxies to guide expectations. In this section, we calculate the resonant widths analytically and use them to characterize the amplitude of variability.
To derive the width of the resonant zone, we again invoke the similarity between obliquity dynamics and that of the physical pendulum. For the physical pendulum (Eqn.\[e:pen\]), the half-width of a resonant zone in $\mathcal{L}$ is $\Delta \sim 2\sqrt{c/\beta} \sim 2 (\sqrt{1-\chi^2}A(t)/\alpha_p)^{1/2}$ [e.g. @Li14].
For the obliquity variation in a resonant region, $A(t)$ can be approximated as follows when the inclination is small:
$$A(t)\sim \alpha_p\chi_{{\rm res},k} i_k.
\label{e:atsim}$$
Thus, the half-width of the resonant region can be expressed as the following: $$\Delta_{\chi, k} \sim 2 ( \chi_{{\rm res},k}\sqrt{1-\chi_{{\rm res},k}^2} i_k)^{1/2},
\label{e:wid}$$ which is identical to Eqn (11) in @Atobe04, who derives this relation directly from the equation of motion for this problem. The expression for variability amplitudes at nonresonant obliquity values can be found in,e.g., @Ward73.
Note that in both Equations (\[e:atsim\]) and (\[e:wid\]) $i_k$ is measured in radians, and we assume $\chi$ to be near $\chi_{\rm res}$ in the resonant region. Thus, this approximation fails when the resonant zone is large ($\gtrsim 0.1 \chi_{\rm res}$). The width is independent of $\alpha_p$ because it is canceled out in the amplitude calculation. Note that Eqn.(\[e:wid\]) does not apply when $\alpha = 0$, which corresponds to a completely rigid sphere with zero dynamical ellipticity. In this case, no precession is expected (and hence no resonance), and the obliquity variation follows that of the orbital inclination.
![Approximate resonant width in resonant zones ($\Delta \epsilon$) vs. the resonant obliquity ($\epsilon_{\rm res}$), following Eqn.(\[e:at\]). Resonances tend to be wider at lower-obliquity values and for modes associated with larger inclination components. []{data-label="f:res-widths"}](res_widths.pdf)
Figure \[f:res-widths\] illustrates the half-width of the resonant zones as a function of the resonant obliquity values. The width is much larger in the low resonant obliquity region owing to the conversion from $\chi$ to $\epsilon$ being arc-cosine. The dashed, solid, and dotted lines correspond to mode amplitudes $i_k = 3^\circ$, $0\fdg3$, and $0\fdg03$, respectively. When the modal amplitude is small (e.g. $i_k \sim 0\fdg03$), the resonant width is also small. Therefore, the fact that a given obliquity value lies in the resonant region does not necessarily imply that its variation amplitude must be large – depending on the associated modal importance, the variability could very well be confined to within a few degrees. In general, planets whose orbits have a larger mutual inclination with a given planet will induce stronger modes on that planet’s inclination. This mechanism is responsible for the enhanced variability amplitudes found in the numerical results for the more inclined cases in Figure \[f:eps-t\].
Modal Properties Based on [*N*]{}-body Simulations and Predicted Obliquity Variations {#ss:num-freq}
-------------------------------------------------------------------------------------
As explained in section \[ss:res-zones\], obliquity resonance occurs where the precession rate of the spin axis coincides with a modal frequency of the orbital inclination. Therefore, the properties of the orbital inclination oscillation modes can be used to predict the behavior of obliquity variations. Empirically, the modes can be obtained from a Fourier transform (FT) on the time series of orbital inclination modulated by the ascending node, $i(t) e^{\sqrt{-1}\Omega(t)}$. Using the [*N*]{}-body results described in section \[s:nr\], we calculate the FTs for Kepler-62f and Kepler-186f in the near-coplanar cases and present them in Figure \[f:num-res\] in black. The locations of peaks correspond to the modal frequency values, $f_k$, which determine the centers of the obliquity resonances via Eqn.(\[e:coseps\]). The peak amplitude reflects the power associated with each mode, $i_k$, which in turn determines the width of the corresponding obliquity resonance through Eqn.(\[e:wid\]). Figure \[f:num-res\] focuses on the negative frequency regime because in neither scenario do we detect notable real structure (i.e. not attributable to aliasing) in the positive frequency regime. This explains our numerical finding that retrograde obliquities are stable for these configurations.
![Fourier transform spectrum of $i(t) e^{\sqrt{-1}\Omega(t)}$ from [*N*]{}-body simulations for the near-coplanar case of Kepler-62f (top) and Kepler-186f (bottom). The power peaks are highlighted with red circles. Dashed lines are drawn where frequencies are predicted using the Lagrange-Laplace (L-L) approach (section \[ss:ll-th\]) and follow the same color code as Figure \[f:5p-mad\]. The L-L predictions show high consistency with the [*N*]{}-body results overall. For Kepler-186f, the peak at $-223\arcsec yr^{-1}$ is an alias of the $-1520\arcsec yr^{-1}$ mode. We do not detect significant real peaks in the positive frequency regime, which agrees with the L-L analysis. []{data-label="f:num-res"}](K62_v21_resfreq_ys.pdf)
![Fourier transform spectrum of $i(t) e^{\sqrt{-1}\Omega(t)}$ from [*N*]{}-body simulations for the near-coplanar case of Kepler-62f (top) and Kepler-186f (bottom). The power peaks are highlighted with red circles. Dashed lines are drawn where frequencies are predicted using the Lagrange-Laplace (L-L) approach (section \[ss:ll-th\]) and follow the same color code as Figure \[f:5p-mad\]. The L-L predictions show high consistency with the [*N*]{}-body results overall. For Kepler-186f, the peak at $-223\arcsec yr^{-1}$ is an alias of the $-1520\arcsec yr^{-1}$ mode. We do not detect significant real peaks in the positive frequency regime, which agrees with the L-L analysis. []{data-label="f:num-res"}](K186_v21_resfreq_ys.pdf)
For Kepler-62f, we identify two dominant nonzero frequency modes (amplitude $> 0\farcs01$), with frequencies $f_k = -10\farcs0 ~{\rm yr}^{-1}$ and $-41\farcs8~{\rm yr}^{-1}$, and FT peak amplitudes $i_k = 0\fdg09$ and $i_k = 0\fdg014$ for the coplanar configuration. Following Eqn.(\[e:coseps\]), we can predict the locations of the resonant regions of $\cos(\epsilon)$ for given values of the precession coefficient $\alpha_p$. In particular, for an Earth-like $\alpha_p = 32\farcs2~{\rm yr}^{-1}$, we would expect Kepler-62f to exhibit a resonant region at $\epsilon = 71\fdg8$ induced by the $-10\farcs0~{\rm yr}^{-1}$ mode ($f_k = -41\farcs8~{\rm yr}^{-1}$ yields an unphysical $\epsilon_{\rm res}$ for this $\alpha_p$). Substituting the amplitude of the mode in Eqn.(\[e:wid\]), the half-width of the resonance is expected to be $\sim 2\fdg6$. The half-amplitude measured from the numerical simulation is $\sim2\fdg3$
With a larger precession coefficient of $\alpha_p = 45\farcs5~{\rm yr}^{-1}$ ($P_{\rm rot} = 17$ hr), corresponding to the faster rotator as presented in the middle and lower left panels in Figure \[f:eps-t\] for Kepler-62f, both modes yield physical resonant regions, at $\epsilon = 77\fdg3$ and $\epsilon = 23\fdg0$. In the coplanar case, the resonance half-widths are $2\fdg2$ and $2\fdg6$ respectively. The numerical results for the faster rotator show variability half-amplitudes at $\epsilon \sim 77^\circ$ and $\sim 23^\circ$ to be $1\fdg8$ and $2\fdg9$ respectively. Overall, these analytical results agree well with the numerical results shown in the top and middle left panels of Figure \[f:eps-t\]. In general, the resonance widths given in Equation (\[e:wid\]) are an upper limit for the variability amplitude because the extent to which the variability occupies the full width of the resonance depends on the exact initial obliquity, $\epsilon_0$, as well as the initial spin axis longitude, $\psi_0$. Underestimates by the analytical resonance-width equation using the sharp FT peak values as inputs could happen when the numerical FT does not resolve the modal frequencies, which underestimates the modal amplitudes.
Increasing the orbital mutual inclinations tends to boost the amplitude $i_k$ associated with each inclination mode, which leads to enlarged resonant widths (\[ss:res-wid\]). This phenomenon is exemplified in the [*N*]{}-body results corresponding to the bottom panels in Figure \[f:eps-t\], where the orbital inclination of Kepler-62f reaches $\sim 1\fdg4$. While the modal frequencies remain almost unchanged from that of the near-coplanar case ($-10\farcs0~{\rm yr}^{-1}$ and $-41\farcs8~{\rm yr}^{-1}$), the modal amplitudes increase to $0\fdg32$ and $0\fdg17$, respectively (not shown in Figure \[f:num-res\]). Thus, according to Eqn.(\[e:wid\]), the corresponding half-widths of the resonances for the faster rotator increase to $4\fdg0$ and $8\fdg2$. Again, the location of the resonant regions agrees very well with the numerical results shown in section \[s:nr\]. The numerical half-amplitudes are $2\fdg9$ and $10\fdg4$. Importantly, the expression for the half-width is no longer a good approximation when the variability is large (i.e. when $\epsilon$ deviate from $\epsilon_{\rm res}$ in the resonant region).
Since every dominant mode can in theory cause low-obliquity instability for an appropriate $\alpha_p$ value (see section \[ss:res-zones\] and \[ss:ll-th\]), we should expect Kepler-62f to have two rotation periods that lead to destabilized low-obliquity zones. One of the modes ($f_k=-41\farcs8 ~{\rm yr}^{-1}$) is responsible for the lower-obliquity instability at $P_{\rm rot} \sim 15-20$ hr shown in Figure \[f:eps-t\]. The other mode ($f_k=-10\farcs0~{\rm yr}^{-1}$) results in a similar instability region at $P_{\rm rot} \sim 70$ hr.
Via the same method, for Kepler-186 we can only identify one dominant mode (with amplitude $>0\fdg01$) at $-11\farcs7 $yr, where the amplitude is $0\fdg39$. Then, for an Earth-like $\alpha_p = 137\farcs4 ~{\rm yr}^{-1}$, the resonance is located at $\epsilon = 85\fdg1$ and the half-width of the resonance is $2\fdg76$, according to the analytical approximation in Eqn.(\[e:wid\]). From the numerical results, the half-amplitude is $1\fdg7$. For the slower rotator with $\alpha = 12\farcs7 ~{\rm yr}^{-1}$, the resonant region is at $\epsilon = 23\fdg0$ and its expected half-width is $11\fdg7$, while in the numerical tests the obliquity half-amplitude is $8\fdg8$. For the case with higher mutual inclinations, where the inclination of Kepler-186f reaches $\sim 1\fdg5$, the modal amplitude is $0\fdg72$, corresponding to a resonance half-width of $15\fdg2$, while the numerical half-amplitude is $9^\circ$. Similar to Kepler-62f, the analytical results of the resonant location are consistent with the numerical values (shown in Figure \[f:eps-t\]), though the analytical approximation somewhat overestimates the amplitude of the obliquity variations for reasons outlined earlier. Another small peak occurs at $\sim -210\arcsec~{\rm yr}^{-1}$ and is actually an alias of a more distant mode at $-1520\arcsec~{\rm yr}^{-1}$ (see \[ss:ll-th\] and Table \[t:table-ll\]).[^3]
Dependence on Planetary Rotation Rates and Observational Uncertainties {#ss:ll-th}
----------------------------------------------------------------------
In addition to Fourier transforming the [*N*]{}-body results, one can alternatively estimate the relevant fundamental forcing frequencies of the planetary systems using a completely analytical approach following the Laplace-Lagrange (L-L) secular theory [e.g. @Murray99], which allows us to estimate the characteristic oscillation frequencies $f_k$ and their approximate amplitudes $i_k$ in planet $p$’s orbital plane induced by the other planets in the system. This is done by solving for the eigenvalues and eigenvectors of a matrix constructed from the masses and orbital semi-major axes present in the planetary system. The L-L approximation is accurate for systems with no mean motion resonances and containing nearly circular and coplanar orbits, conditions that are satisfied here. Such an approach is powerful because it allows us to rapidly characterize vast swaths of parameter space and visualize the results.
We include the $f_k$ values computed using the L-L approach in Figure \[f:num-res\], shown by vertical dashed lines following a color code for direct comparison to Figure \[f:5p-mad\]. The modal properties are also summarized in Table \[t:table-ll\]. Notice that, consistent with Section \[ss:num-freq\], no positive frequencies exist, thereby explaining why retrograde obliquities have all been stable. According to the L-L approach, the most dominant modes for Kepler-62f are $f_k = -41\farcs8 ~{\rm yr}^{-1}$ and $f_k = -10\farcs0~{\rm yr}^{-1}$, with normalized eigenvector elements of $0.56$ and $0.29$ respectively.[^4] Solving for the amplitudes of the modes applying the same initial condition as those for the numerical simulations, we found the amplitudes of the dominant modes to be $i_k \sim 0\fdg015$ and $i_k \sim 0\fdg14$ respectively, which agree rather well with the results from the numerical Fourier transform. As discussed in section \[ss:num-freq\], the numerical peaks are typically underestimated due to the finite resolution of the FT. For Kepler-186f, $f_k = -11\farcs6~{\rm yr}^{-1}$ is the most dominant mode, with normalized eigenvector elements of $0.80$. Assuming the same initial condition as those for the numerical simulation, the modal amplitude is $i_k \sim 0\fdg39$, which is precisely what is observed in the Fourier transform. Therefore, there is good agreement on the behavior of the obliquity variations and the locations of the instability regions between the numerical results and the analytical ones. L-L uncovers additional nonzero modes that do not directly manifest in the FTs in section \[ss:num-freq\] because they are outside the Nyquist frequency of the [*N*]{}-body data sampling (except in the form of aliases). In any case, these missed modes are exceedingly minor and hence contribute negligibly to variability.
In the high-inclination examples depicted in Figure \[f:eps-t\], we observed small-scale variabilities between the resonant zones predicted by L-L. These small-scale variabilities do not correspond to any primary resonances. It is important to remember that L-L assumes that the system has low orbital inclination and eccentricity values. As configurations deviate from the small-inclination and small-eccentricity regime, higher-order effects can become more prominent, leading to higher-order resonant regions. For instance, Figure \[f:num-res\] illustrates hints of FT peaks missed by L-L even in the near-coplanar regime. These higher-order peaks grow as orbital mutual inclinations increase, and they are responsible for the small variabilities away from the L-L resonances.
![Resonant obliquities as a function of precession coefficient ($\alpha$) for Kepler-62f (top panel) and Kepler-186f (bottom panel). Each curve corresponds to a modal frequency on orbital inclination excited by the planetary bodies in the system (as labeled). Solid curves depict dominant modes (i.e. $i_k > 0\fdg01$) while minor modes are shown with dashed curves. The MAD ranges for each frequency are plotted as dotted curves around the thick solid curves, based on Monte Carlo simulations over 1000 system realizations varying the stellar and planetary masses within their measurement uncertainties. The green solid vertical line represents the $\alpha_p$ values applicable if the planets were Earth analogs with $P_{\rm rot} = 1$ day. The cyan dot-dotted line denotes the planets with hypothetical rotation rates illustrated in Figure \[f:eps-t\]. The pink dot-dotted line corresponds to breakup velocity. []{data-label="f:5p-mad"}](alpha_vs_eps_MAD_K62_5p.pdf){width="3.7in"}
![Resonant obliquities as a function of precession coefficient ($\alpha$) for Kepler-62f (top panel) and Kepler-186f (bottom panel). Each curve corresponds to a modal frequency on orbital inclination excited by the planetary bodies in the system (as labeled). Solid curves depict dominant modes (i.e. $i_k > 0\fdg01$) while minor modes are shown with dashed curves. The MAD ranges for each frequency are plotted as dotted curves around the thick solid curves, based on Monte Carlo simulations over 1000 system realizations varying the stellar and planetary masses within their measurement uncertainties. The green solid vertical line represents the $\alpha_p$ values applicable if the planets were Earth analogs with $P_{\rm rot} = 1$ day. The cyan dot-dotted line denotes the planets with hypothetical rotation rates illustrated in Figure \[f:eps-t\]. The pink dot-dotted line corresponds to breakup velocity. []{data-label="f:5p-mad"}](alpha_vs_eps_MAD_K186_5p.pdf){width="3.7in"}
[lcccc]{}\
$f_k$ ($\arcsec/yr$) & $\hat{I}_{k,5}$ & $I_{k,5; cop}$ ($^\circ$) & $I_{k,5; h}$ ($^\circ$) & $\epsilon_{\rm res}$ ($^\circ$)\
0 & 0.45 & 0.11 & 0.56 & 90\
-10.0 & -0.29 & -0.14 & -0.46 & 72\
-41.8 & 0.56 & 0.015 & 0.22 &\
-279 & -$3.5\times10^{-4}$ & $-1.9\times10^{-4}$ & $-8.0\times10^{-4}$ &\
-1857 & $2.2\times10^{-6}$ & $1.54\times10^{-7}$ & $3.8\times10^{-6}$ &\
\
$f_k$ ($\arcsec/yr$) & $\hat{I}_{k,5}$ & $I_{k,5; cop}$ ($^\circ$) & $I_{k,5; h}$ ($^\circ$) & $\epsilon_{\rm res}$ ($^\circ$)\
0 & 0.45 & 0.35 & 0.73 & 90\
-11.6 & 0.80 & 0.39 & 0.72 & 85.2\
-737 & 0.0027 & $1.1\times10^{-3}$ & $7.7\times10^{-3}$ &\
-1520 & $2.1\times10^{-4}$ & $1.1\times10^{-4}$ & $1.9\times10^{-4}$ &\
-2407 & $2.9\times10^{-6}$ & $1.0\times10^{-6}$ & $2.9\times10^{-6}$ &\
Note. Column headings: $\hat{I}_{k,5}$ – normalized eigenvector elements; $I_{k,5; {\rm cop}}$ – components of the inclination eigenvector, the absolute values of which characterize the size of $i_k$ values for the near-coplanar configuration; $I_{k,5; h}$ – same as $I_{k,5; {\rm cop}}$, but for the higher mutual inclination ($< 3^\circ$) configuration; $\epsilon_{\rm res}$ – center of obliquity resonance region corresponding to Earth-like $\alpha_p$.
As shown in Eqn.(\[e:alpha\]), the precession coefficient $\alpha_p$ is highly sensitive to the planet’s spin rate, as it determines the oblateness of the planet. To comprehensively investigate the dependence of obliquity variation as a function of the planetary spin rate, we calculate the resonant obliquity values as a function of the precession coefficient $\alpha_p$ using Eqn.(\[e:coseps\]). The result is shown in Figure \[f:5p-mad\] for Kepler-62 and Kepler-186. We label the corresponding planetary rotation period in the top x-axis of Figure \[f:5p-mad\], assuming $\alpha_p \propto \omega$ as discussed in Section \[ss:ham\]. In general, ignoring the oblateness of the host star, a system of $N$ planets will induce $N-1$ coupled nonzero frequency inclination modes. The colored solid curves represent the resonant obliquities corresponding to the five different inclination modes (including $f=0$). For a given value of $\alpha_p$, only a subset of the modes will correspond to physical values of $\epsilon_{\rm res}$. The valid resonant obliquities appear in Figure \[f:5p-mad\] as intersections between an $\alpha_p$ vertical and the curves. Chaotic zones, where the obliquity evolution becomes unpredictable, occur when two resonant zones overlap with each other, which becomes more likely where the resonant widths are large and curves are dense.
The values of $f_k$ depend on the stellar mass $M_\star$, as well as the masses and semi-major axes of each planet $j$ in the system ($m_j, a_j$). While the orbital period is measured with high precision, the masses are often underconstrained. To investigate the sensitivity of the resonance locations to measurement errors, we conduct Monte Carlo experiments over $1000$ system realizations to obtain a measure of the error in the determinations of the median resonant obliquity. Each realization is assigned a stellar mass and a radius for each planet. Both are drawn from Gaussian distributions defined by their respective medians and errors quoted in Section \[s:sys-pars\]. To generate the planet mass, each planet radius draw is fed into [@ChenKip17], which outputs a draw from the mass posterior, taken to be $m_p$ for that realization. The exceptions are Kepler-62f ($3.3^{+2.3}_{-0.6} M_\earth$) and Kepler-186f ($1.7^{+1.1}_{-0.3} M_\earth$), for which the mass posteriors are calculated from the Terran power law marginalized over the hyperparameter distributions. Essentially, such a procedure marginalizes over the full planetary mass distributions accounting for uncertainties in radius.
In Figure \[f:5p-mad\], the median absolute deviation (MAD) of the resonant obliquities are overplotted in dashed lines. The MADs of the modal frequencies only vary within $\sim 20-30\%$ of their median value, and the modes with frequencies close to $-41\farcs8~{\rm yr}^{-1}$ and $-10\farcs0~{\rm yr}^{-1}$ ($-11\farcs6~{\rm yr}^{-1}$) remain dominant for Kepler-62f (Kepler-186f) in the Monte Carlo simulations. It appears that the indeterminacy in these curves are relatively small in the log plots.
In Figure \[f:5p-mad\], we use green solid lines to indicate the precession frequency corresponding to Earth analogs (i.e. $P_{\rm{rot}} = 1$ day). Both Kepler-62f and Kepler-186f avoid instability zones at Earth-like obliquity values (rather narrowly for Kepler-62f). However, one can expect higher obliquities to undergo mild variations. Any moderate mutual orbital inclination could also enlarge the widths of each resonant zone. These predictions are consistent with the numerical results in Section \[s:nr\]. We also overplot with cyan dot-dotted lines $\alpha_p$ values corresponding to the bottom panels of Figure \[f:eps-t\], and show that they coincide with regions where lower obliquities undergo resonances. This picture indicates that the resonant regions in the low-obliquity region is highly fine-tuned in rotation period. One rotation rate of interest is that corresponding to the planet’s breakup velocity, which is 0.0586 days for Earth analogs (shown as pink dot-dotted lines in Figure \[f:5p-mad\]). Rotation periods shortward of this value are unphysical. If rotating close to breakup, both Kepler-62f and Kepler-186f intersect with multiple resonant curves at small obliquities. Nevertheless, since the resonant amplitudes are low for these modes (see Table \[t:table-ll\]), the associated variability would be quite limited. Of course, given the probable ages of the systems and tidal interactions over time, it is unlikely that the planets are such extreme rotators today anyway. Higher-precession coefficients than the one corresponding to the breakup velocity are allowed for planets with moons (see section \[ss:moon\]).
Another regime of relevance in the long term is that of synchronized rotation with the orbital periods of 267 days and 130 days for Kepler-62f and Kepler-186f, respectively (see also Section \[ss:tides\]). The synchronized rotation periods do not correspond to physical resonant obliquity values. As pointed out by @Bolmont14 [@Bolmont15] and @Shields16, planetary obliquities eventually settle to 0, though the rate at which tidal synchronization occurs is dependent on many factors, such as the planetary tidal $Q$-values [@Heller11]. Kepler-62f and Kepler-186f have likely not yet reached this state [@Bolmont14; @Bolmont15; @Shields16 see also Section \[ss:tides\]] but are inevitably marching toward it. From Figure \[f:5p-mad\] it is visually apparent that, en route to the synchronized states, the spin rate and therefore the precession frequency of the planet decline and allow the system to sweep across the resonant regions. This allows obliquity variations in a wide range of obliquity regions similar to the future evolution of the Earth as discussed by @deSurgy97.
Discussion {#s:dis}
==========
Accurate determination of many properties of the exoplanets remains out of reach with current techniques. As will be discussed below, a planet’s obliquity evolution can be affected by many factors that are not well constrained. This includes the presence of additional planets and satellites, as well as the rotation periods and the oblateness of the planets. Therefore, the parameter space governing a planet’s obliquity evolution is large, and to accurately pinpoint the location occupied by a given exoplanet is challenging.
What is possible is to map out representative regions of this parameter space and infer the general behaviors. Such characterization could provide guidance over the range of considerations toward assessing the planet’s obliquity evolution. In this section we investigate the effect of the existence of additional planets, including giant Jupiter/Saturn analogs as well as internal rocky planets. We also consider the influence of a satellite, which plays a critical role in stabilizing the Earth’s own obliquity [@Laskar93a; @deSurgy97]. Finally, we discuss the planets’ long-term obliquity dynamics as it gradually synchronizes its rotation with its orbital period as a result of tidal interactions.
Extra Planets {#ss:more-p}
-------------
Thus far, our analysis has been anchored on the assumption that all the relevant planets that exist in the systems have been detected. In reality, the sensitivity of the transiting technique falls with increasing planet distance or inclination. The presence of additional planets, especially at high orbital inclinations, can influence the spin dynamics in a given system dramatically through the introduction of additional, potentially strong modes. In this subsection we map the resonant zones for system configurations involving external giant planets (\[sss:ext-p\]), as well as an internal nontransiting planet (\[sss:int-p\]).
### External Giant Planets {#sss:ext-p}
The current estimate on the occurrence rate of Jupiter-massed planets orbiting at Jupiter-like distances around M dwarfs is $\sim6\%$ [@Clanton16; @Meyer17]. Therefore, while atypical, external giant planets are not wholly uncommon in these systems. Jointly motivated by the architecture of our own solar system, we characterize the influence of hypothetical external giant planets on the obliquity evolution of Kepler-62f and Kepler-186f using the analytic techniques outlined in previous sections, and we confirm them with numerical integration.
![Same as Figure \[f:5p-mad\], but now with distant giant planets in the form of Jupiter and Saturn analogs. The additional frequencies induced tend to be low and impact slow rotators more severely, but they also modify the locations of the other resonances. In this scenario, Kepler-62f with an Earth-like precession coefficient can expect to experience chaotic obliquity evolution even in the low-obliquity regime. []{data-label="f:7p-mad"}](alpha_vs_eps_MAD_K62_7p.pdf){width="3.7in"}
![Same as Figure \[f:5p-mad\], but now with distant giant planets in the form of Jupiter and Saturn analogs. The additional frequencies induced tend to be low and impact slow rotators more severely, but they also modify the locations of the other resonances. In this scenario, Kepler-62f with an Earth-like precession coefficient can expect to experience chaotic obliquity evolution even in the low-obliquity regime. []{data-label="f:7p-mad"}](alpha_vs_eps_MAD_K186_7p.pdf){width="3.7in"}
As an example, we place a Jupiter analog and a Saturn analog in each system and calculate the resonant obliquities induced by each using the L-L theory. In this context, ‘analog’ signifies a planet with identical mass, orbital inclination, and a similar semi-major axis, in the present day. For Jupiter and Saturn, the orbital distances are $\sim 5.2$ and $9.6$ au and their inclinations are small ($1\fdg304$ and $2\fdg485$). Including only a Jupiter-analogue, a new dominant inclination variation mode is introduced at $f_k \sim -2\farcs5~{\rm yr}^{-1}$ for Kepler-62f, and at $f_k \sim -0\arcsec8~{\rm yr}^{-1}$ for Kepler-186f. The corresponding normalized eigenvector elements for this mode are $\sim 0.3$ for Kepler-62f and $\sim 0.4$ for Kepler-186f. These results are qualitatively similar for a range of orbital distances of Jupiter between 3 and 7 au. Including both the Jupiter and Saturn analogs, there is another mode introduced at $f_k \sim 30 \arcsec ~{\rm yr}^{-1}$ for Kepler 62f and $f_k \sim 34 ~{\rm yr}^{-1}$ for Kepler 186f. The modes attributed to the Saturn-analogue are much weaker than those introduced by Jupiter, with normalized eigenvector elements of $\sim 0.04$ for Kepler-62f and $\sim 0.03$ for Kepler-186f.
The L-L resonant curves in the $\alpha$(or $P_{\rm rot}$)-$\epsilon$ plane are shown in Figure \[f:7p-mad\] for systems including both the Jupiter and the Saturn analogs, where the orbital distances are fixed to those of Solar System’s Jupiter and Saturn. The solid lines represent the default orbital parameters as presented in Table \[t:sys-pars\], and the dashed lines represent the MAD values based on the Monte Carlo simulations taking into account the observational uncertainties of the planetary masses. An Earth analog in such systems would encounter more obliquity instability zones occurring at lower obliquities. For Kepler-62f, the resonant regions caused by the dominant mode $f_k \sim -45\farcs0~{\rm yr}^{-1}$ and the less dominant mode introduced by Saturn $f_k \sim 30\farcs6~{\rm yr}^{-1}$ are only slightly separated in the low-obliquity regime. Thus, one might expect the obliquity evolution of such a planet to be chaotic owing to higher probability of overlap between the resonances. The other dominant modes for both Kepler-62f and Kepler-186f correspond to low frequencies, which implies a greater chance for instability and chaos at higher obliquity angles for Earth analogs, as well as more severe impact on lower-obliquity regions for slowly rotating planets.
![Obliquity variation of Kepler-62f (top panel) and Kepler-186f (bottom panel) with $P_{\rm rot} = 24$ hr in the presence of outer Jupiter- and Saturn-like planets. Variations can be large in the lower-obliquity range for Kepler-62f, consistent with the analytical estimates. For Kepler-186f with $\epsilon \sim 85^\circ$ in this hypothetical system, the obliquity evolution is chaotic owing to the overlap of the two resonant zones associated with the $-0\farcs8~{\rm yr}^{-1}$ and $-13\farcs6~{\rm yr}^{-1}$ frequencies. In all cases, retrograde obliquities are stable. []{data-label="f:obSJ"}](Eps_vs_t_Giants.pdf){width="3.6in"}
For comparison, we numerically compute the obliquity evolution for Earth analogs (i.e. $P_{\rm rot} = 24$ hr) in systems including both Jupiter and Saturn analogs, and we present them in Figure \[f:obSJ\]. The inner planets start nearly coplanar to each other and are stirred into more inclined configurations by the giant planets, which start with their present solar system orbital inclinations ($1\fdg304$ and $2\fdg485$). Consistent with the analytical expectations, the lower-obliquity regions of Kepler-62f allow larger variabilities owing to two closely spaced inclination oscillation modes. The higher-obliquity region also exhibits larger amplitude variabilities than the case without the outer giant planets (as shown in Figure \[f:eps-t\]). This is because the giant planets induce larger mutual inclinations between all the planets. For Kepler-186f, the variabilities at low obliquities are still low, due to the lack of additional resonant regions in that region. Similar to Kepler-62f, high-obliquity variation is enlarged by the presence of the giant planets. Interestingly, an example of chaotic obliquity evolution is found around $85^\circ$, which is likely caused by some overlap of the resonant zones associated with the $-0\farcs8~{\rm yr}^{-1}$ and $-13\farcs6~{\rm yr}^{-1}$ frequencies. In all cases, retrograde obliquities are stable. @Quarlesinprep also investigate the influence of outer giant planets on the spin axis variability of Kepler-62f. Our results are mostly consistent with their conclusions, except that @Quarlesinprep find obliquity variability in the retrograde regime for a specific realization of the system.
### Additional Planet Interior to Kepler-186f {#sss:int-p}
In addition to distant planets, it is also possible to have internal nontransiting planetary companions. In particular, the separation between Kepler-186e and Kepler-186f is large, and it is likely that an extra undetected planet exists between these two known planets, based on accretion disk simulations [@Bolmont14]. Using dynamical simulations, @Bolmont14 characterized the mass of the extra planet. Considering a planetary mass ranging from $0.1M_{\oplus}$ to $1M_{Jup}$, @Bolmont14 found that the planet cannot be more massive than the Earth, in order to keep the mutual inclinations between the rest of the planets low so as to allow them to transit. Here we adopt the orbital configuration of the extra planet assumed by @Bolmont14 to study the obliquity evolution of Kepler-186f in the presence of this additional, more inclined body interior to its orbit, where $a_{\rm ex} = 0.233$ au, $e_{\rm ex} = 0.01$ and $i_{\rm ex} = 2^\circ$ to avoid transiting. We assumed the extra planet to be Earth-massed ($1M_{\oplus}$).
![Obliquity evolution of Kepler-186f in the presence of an additional planet ($1 M_\earth$ between Kepler-186f and Kepler-186e. The low-obliquity region allows moderate variabilities, and the obliquity variation around $\sim 80^\circ$ is larger than the case without the additional planet.[]{data-label="f:eps186expl"}](K186_Eps_vs_t_intp.pdf){width="3.7in"}
To initialize the simulation, we adopt the nearly coplanar configuration of Kepler-186, as discussed in section \[s:nr\], and set the precession coefficient for planet f to be $\alpha = 137\farcs4~{\rm yr}^{-1}$, assuming that the planet is Earth-like with a rotation period of 24 hr. The obliquity evolution of Kepler-186f is shown in Figure \[f:eps186expl\]. The low-obliquity region allows some variabilities, since the extra planet introduces a modal frequency at $f = 122\farcs5~{\rm yr}^{-1}$, which leads to a resonant region at $\epsilon \sim 26\fdg9$.
Comparing with the obliquity variation without the extra planet (top right panel of Figure \[f:eps-t\]), the variability in the high-obliquity region is also larger. This is because the higher inclination of the extra planet also excites the mutual inclination between the planets, which leads to a stronger perturbation to the planetary spin axis. The increase in the obliquity variation due to the extra planet is consistent with the discussions by @Bolmont14.
Presence of a Satellite {#ss:moon}
-----------------------
The solar system is teeming with moons. However, little is known about moons elsewhere, in part due to the challenges associated with their low expected signal. Thorough searches in transiting exoplanet data have been conducted [e.g. @Kipping13] but thus far have revealed only one possible exomoon candidate [@Teachey17]. @Sasaki14 suggest that the tidal decay lifetimes of typical large moons around habitable planets of smaller stars tend to be shorter. However, for Kepler-62f and Kepler-186f, this timescale should exceed the best estimated system ages [@Shields16].
In any study of planetary obliquity evolution, it is important to consider the possible presence of moons, since moons cause additional torque to the planet’s spin axes, hence increasing the precession coefficient. The impact could be large and is described by the modified version of Eqn.(\[e:alpha\]) given below [e.g. @deSurgy97]:
$$\begin{aligned}
\label{e:alpha-moon}
\alpha_p = & \frac{3G}{2\omega}\biggl[\frac{M_\star}{(a_p\sqrt{1-e_p^2})^3} \\
+ & \frac{m_M}{(a_M\sqrt{1-e_M^2})^3}\left(1-\frac{3}{2}\sin^2 i_M \right) \biggr]E_d. \nonumber\end{aligned}$$
In the second term, $m_M$ is the mass of the moon. Similarly, the subscript $M$ on the orbital elements $a$, $e$, and $i$ indicates that these quantities pertain to the moon’s orbit around the planet. Each additional moon contributes one such term, and their collective effect is additive.
For the Earth, the precession due to the Moon is about twice that of the Sun. The same is true for Kepler-62f and Kepler-186f, if we assume that each harbors a moon analogous to that of Earth, that is, a satellite that preserves the mass ratio and orbits at the same fraction of the planetary Hill radius ($r_{\rm H}$) as that in the Earth-Moon system. For a given planet-star pair, a moon for which $q = m_M/m_p$ and $f = a_M/r_H$ boosts the moonless version of $\alpha_p$ by the following factor:
$$F_{\rm boost} (q,f) \equiv \frac{\alpha_M}{\alpha_\star} = 1 + 3 \frac{q}{f^3},
\label{e:moon-boost}$$
assuming $e_M$ and $i_M \ll 1$. Here $\alpha_{\rm Moon}$ and $\alpha_\star$ refer to $\alpha_p$ with and without the moon, respectively. Figure \[f:moon-alpha\] shows contours of $\log F_{\rm boost}$ in $\log f$ and $\log q$. The boost is enhanced at low $f$ and high $q$-values.
\[f:moon-alpha\]
The Earth-Moon arrangement turns out to be critical for the obliquity stability of the Earth. Without the Moon, the Earth sits in a large obliquity resonant zone spanning $0^\circ$ to $45^\circ$ [@Lissauer12; @Li14]. Our Moon pushes the Earth away from this hazardous region [@Laskar93a]. Figure \[f:moon-alpha\] shows that, in general, satellites with sufficiently large $q$ and/or small $a_M$ can push $\alpha_p$ beyond all the major frequencies present in the system, thereby stabilizing obliquity. This is helped by the fact that there exists upper bounds on the forcing frequencies from giant planets on HZ terrestrial planets due to orbital stability criteria [@Atobe04]. However, we point out that the same satellites could destabilize the obliquity for sufficiently slowly rotating planets, i.e. those associated with precession frequencies that are lower than some of the perturbing frequencies. Therefore, in general, changes in the lunar parameters could be fortuitous, catastrophic, or simply inconsequential, depending on the landscape of resonant obliquity regions (as illustrated in Figure \[f:5p-mad\] and \[f:7p-mad\]) and the $\alpha_p$ and $F_{\rm boost}$ involved. Moreover, a satellite could alter the rotation rate of the planet over time due to tidal interactions, as our Moon has done to Earth, which would further modify the oblateness of the planet, resulting in different $\alpha_p$ values (see \[ss:tides\] as well as a detailed investigation in @Atobe07).
Assuming that Kepler-62f is an Earth analog with rotation period of $24$ hr, the increase to its $\alpha_p$ due to the existence of moons could destabilize its obliquity, if the new precession rate leads to resonant interactions with the inclination oscillation mode of $f_k = -41\farcs8~{\rm yr}^{-1}$. On the other hand, the precession coefficient of Kepler-186f with $P_{\rm rot} = 24$ hr is already larger than that of the frequency of the dominant inclination oscillation mode. Thus, the existence of a moon would not cause obliquity variations for Kepler-186f.
Long-term Tidal Evolution {#ss:tides}
-------------------------
Tidal locking implies synchronization between a planet’s rotation and orbital periods, as well as the direction of its spin and orbit vector. Both Kepler-62f and Kepler-186f are relatively far from their host stars, thereby likely to have weak tidal interactions with their host stars. However, whether or not they have already become tidally locked depends sensitively on the detailed properties of the planets, as well as the amount of time available to synchronize. Such incertitude has been noted in the literature. According to @Bolmont14 and @Bolmont15, Kepler-62f and Kepler-186f may still be evolving toward a tidally locked state. @Shields16 showed that, depending on whether a constant phase lag or constant time lag model is used, the rotation of Kepler-62f may or may not be synchronized within few-gigayear timescales. They also illustrate a strong tidal timescale dependence on initial spin period and orbital eccentricity. Furthermore, the ages of the systems are highly uncertain: $7\pm4$ Gyr [@Borucki13] or $2.34^{+2.15}_{-1.02}$ Gyr [@Morton16] for Kepler-62 and $4\pm0.6$ Gyr [@Quintana14] for Kepler-186.
The tidal synchronization timescale assuming a constant time lag tidal model follows the equation below [e.g., @Hut82; @Heller11; @Ogilvie14]: $$\begin{aligned}
\frac{1}{\tau_{syn}} = \Big|\frac{\dot{\omega}}{\omega}\Big| = 3k_2 \Delta t |\Omega_{orb} - \omega| \frac{L_{orb}}{L_s}\frac{M_*}{M_p}\Big(\frac{R_p}{a}\Big)^5\Omega_{orb} ,\end{aligned}$$ where $\omega$ is the planetary rotational angular velocity, $\Omega_{\rm orb}$ is the orbital angular velocity, $L_{\rm orb}$ is the orbital angular momentum, and $L_s = C \omega$ is the spin angular momentum. If the planets have the same love number and tidal time lag as the Earth, i.e. $k_2 \Delta t = 213$s [@Lambeck80],[^5] then the characteristic tidal synchronization timescales for Kepler-62f and Kepler-186f with 24 hr rotation periods are $\sim 5.5$ and $\sim 0.3$ Gyr, respectively. Regardless of whether the planets are still en route to tidal locking or have completed the journey in the past, tidal interaction has influenced the history of, and may continue to affect the evolution of, the planets’ obliquities. As the rotation rate of the planet evolves, the oblateness of the planet changes, which leads to a varying precession coefficient ($\alpha_p$). Thus, it is possible for the planet to move across different resonant regions during the tidal synchronization, as illustrated in Figure \[f:5p-mad\] and \[f:7p-mad\]. The fact that the tidal timescale for the alignment of the planetary spin axis is much longer than the obliquity variation timescales in the resonant zones means that long-term tidal effects cannot suppress short-term obliquity fluctuations. Consequently, the obliquity can still vary owing to the resonant interactions. This would be similar to the future obliquity evolution of Earth as discussed in @deSurgy97. For Kepler-62f and Kepler-186f, their respective synchronized spin periods are too long to allow resonant interactions (see Figure \[f:5p-mad\] and \[f:7p-mad\]). Thus, the obliquities are stable for both Kepler-62f and Kepler-186f in the synchronized stage.[^6]
Conclusions {#s:conclusion}
===========
In this article, we have investigated the short-term obliquity variability of HZ planets in two multiplanet transiting systems, Kepler-62f and Kepler-186f, over a large parameter space of possible planet properties and orbital architectures allowed by observational constraints. Using [*N*]{}-body simulations coupled with secular spin-orbit coupling analysis, we have shown in section \[s:nr\] that low-obliquity regions of Kepler-62f and Kepler-186f are stable over $10$ Myr timescales while higher-obliquity regions allow small variabilities, assuming that the planets are Earth analogs (i.e. same rotation rate and interior structure, obeying Eqn.(\[e:alpha\_earth\])).
We have also presented an analytical framework to characterize the nature of obliquity instabilities from first principles (\[s:ar\]). The basic elements of the method are as follows:
1. Present the nature of obliquity instability as arising from resonant interactions between the planetary spin axis and the orbital axis [e.g., @Laskar93a]. Wherever there is a match between the spin axis precession and inclination oscillation frequency, obliquity variation could occur (\[ss:res-zones\]). Given a forcing frequency, the location at which a resonance occurs can be calculated from Eqn.(\[e:coseps\]).
2. Derive the expression for the width of the resonant zones from the modal amplitude of the forcing inclination vector (Eqn.(\[e:wid\]), \[ss:res-wid\]).
3. Deduce modal frequencies and amplitudes in the orbital inclination vector of the planet of interest (e.g., Kepler-62f and Kepler-186f). This could be done in two ways:
1. numerically through FT on the output of an [*N*]{}-body simulation of the system (\[ss:num-freq\]), or
2. analytically through the Lagrange-Laplace formalism (\[ss:ll-th\]). The only inputs required are the initial planetary system architecture (component masses and semi-major axes).
When applicable, the analytical technique has the decided advantage of being a rapid, straightforward, and transparent way to compute the regions harboring resonant obliquities over a large parameter space. It provides a visualization for the behavior of obliquity variablity as these parameters vary. For Kepler-62f and Kepler-186f, we have shown good agreement between the numerical and analytical approach. Different planetary spin rates and orbital configurations of the Kepler systems could affect the obliquity variations. For instance, for Kepler-62f, the lower-obliquity region ($\sim 20^\circ-40^\circ$) can be unstable when the rotation period is $\sim 15-20$ hr or $\sim 60-70$ hr. For Kepler-186f, the same lower-obliquity region can be unstable when the rotation period is $\sim 240-300$ hr. For both planets, instability in the higher-obliquity regions ($\gtrsim 60^\circ$) occurs for a wider range of rotation periods ($\sim 0.3 - 3$ days). The specific values of the rotation period also depend on the properties of the assumed planetary interior structure. In general, instability in the lower-obliquity region is fine-tuned, while the higher-obliquity region can be unstable for a wider parameter space. The amplitude of variability is dependent on the mutual orbital inclination of the planets in the system. Configurations deviating from coplanarity by $\sim 3^\circ$ can already generate appreciable ($\sim 20^\circ$) fluctuations in the low-obliquity ranges.
Orbital architectures and planet properties are often difficult to measure and/or subjected to update. Our analytical approach enables us to characterize the overall obliquity variations including observational uncertainties, different planetary oblateness (which leads to different precession coefficients), extra planets, and the existence of satellites. We find that the observational uncertainties in the stellar mass and in the estimates of the planetary mass do not change our conclusion qualitatively. In investigating the impact of extra planets, we find that Jupiter and Saturn analogs can induce larger obliquity variations in the lower-obliquity range for Kepler-62f, assuming an Earth analog. The obliquity variations of Kepler-186f are not strongly affected by this specific realization of external giant planets. However, an extra planet between Kepler-186e and Kepler-186f may induce stronger obliquity variations for Kepler-186f in the low-obliquity region. Assuming rotation rates similar to that of the Earth, the existence of moons for Kepler-62f could destabilize the spin axis of Kepler-62f, but they cannot destabilize the spin axis of Kepler-186f. Long-term tidal interactions between the planet and the host star will synchronize the planetary spin axis and reduce the oblateness of the planet and its precession coefficient, moving it across resonant regions. Thus, one would expect the obliquity of the planets to vary with large amplitude during tidal synchronization, before reaching the obliquity-stable regions at synchronization.
Based on a simplified energy-balance model, @Armstrong14 showed that rapid and large obliquity variability can be favorable to life by keeping a planet’s global average temperature higher than it would have been otherwise, thereby systematically extending the outer edge of a host star’s HZ by $\sim 20\%$. However, it is by no means clear whether large obliquity variation is necessarily beneficial to life under all circumstances. For instance, it is believed that large obliquity variation for Mars may have caused the collapse of its atmosphere and rendered Mars inhabitable [e.g., @Toon80; @Soto12]. At the very least, obliquity variability can substantially affect transitions between multiple climate states. Recently, @Kilic17 mapped out the various equilibrium climate states reached by an Earth-like planet as a function of stellar irradiance and obliquity. They find that, in this parameter space, the state boundaries (e.g. between cryo- and aqua-planets) are sharp and very sensitive to the climate history of the planet. This suggests that a variable obliquity can easily move the planet across state divisions, as well as alter the boundaries themselves, which would translate into a dramatic impact on instantaneous surface conditions and long-term climate evolution. Similarly, the dependence of surface incident flux on obliquity and eccentricity was studied by @Kane17. They found that nonzero obliquity values could potentially effect large variations in insolation flux across planetary latitudes and orbital phases. Incidentally, among their test cases was Kepler-186f. Coupling the evolution of flux maps with that of obliquity could yield further insight into climate development on exoplanets. The detailed effects on the climate due to obliquity variations still need more investigation. While atmospheric modeling is beyond the scope of this study, our work can help provide input parameters to existing global climate models (GCMs) as another factor influencing the habilitability in a multiplanet system.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to thank Konstantin Batygin, Guillermo Torres, Jack Lissauer and Billy Quarles for helpful discussions. We are also grateful to Jennifer Yee, Matthew Holman, and the anonymous referee for providing detailed feedback on our draft. This work and G.L. were supported in part by the Harvard William F. Milton Award. Y.S. is supported by a Doctoral Postgraduate Scholarship from the Natural Science and Engineering Research Council (NSERC) of Canada.
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[^1]: With very high precision light curves, it may become possible to measure exoplanetary oblateness, which is related to $E_d$, from transit depth variations [@Carter10; @Biersteker17] and ingress/egress anomalies [@Zhu14], though it would be very difficult in general.
[^2]: In all our integrations, we started with initial longitude of spin axis at $0^\circ$ (i.e. $\psi_0 = 0^\circ$). The amplitude of variability has some dependence on $\psi_0$ [e.g., @Atobe04; @Quarlesinprep]. Different $\psi_0$ can produce variation amplitudes that differ by a factor of up to $\sim 3$.
[^3]: The Nyquist frequency for our [*N*]{}-body sampling rate (1/1000 yr) is $-648\arcsec{\rm yr}^{-1}$. Therefore, frequencies outside of this window can manifest as aliases folded on this value.
[^4]: The magnitude of the eigenvector element corresponding to a given mode is an indicator for its relative dominance.
[^5]: The high dissipation rate of the Earth is largely due to its oceans, which may be a unique feature and unrepresentative of terrestrial planets in general.
[^6]: A synchronously rotating body could also sustain large obliquity variations if it lies in a chaotic zone [@Wisdom84].
|
---
abstract: 'We investigate the energetics of droplets sourced by the thermal fluctuations in a system undergoing a first-order transition. In particular, we confine our studies to two dimensions with explicit calulations in the plane and on the sphere. Using an isoperimetric inequality from the differential geometry literature and a theorem on the inequality’s saturation, we show how geometry informs the critical droplet size and shape. This inequality establishes a “mean field" result for nucleated droplets. We then study the effects of fluctuations on the interfaces of droplets in two dimensions, treating the droplet interface as a fluctuating line. We emphasize that care is needed in deriving the line curvature energy from the Landau-Ginzburg energy functional and in interpreting the scalings of the nucleation rate with the size of the droplet. We end with a comparison of nucleation in the plane and on a sphere.'
author:
- 'Eric M. Horsley'
- 'Maxim O. Lavrentovich'
- 'Randall D. Kamien[^1]'
bibliography:
- 'thebibs.bib'
title: Aspects of nucleation on curved and flat surfaces
---
We are familiar with nucleation in first-order phase transitions where a system in a particular phase becomes unstable to the spatially-localized nucleation and growth of a new, more favorable phase. Some common examples include the condensation of a vapor into liquid droplets or the freezing of a liquid upon cooling. We will be interested in the scenario where the preexisting phase is metastable, and the new, stable phase can only form if thermal fluctuations can overcome the energetic barrier associated with the formation of a nucleus of the stable phase, which may then grow. Although these familiar processes typically occur in three dimensions, there are many naturally-occurring and engineered systems which have ordered phases nucleate and grow on surfaces. Examples of such phase transitions include crystallization of glucose isomerase in the plane [@Sleutel2015], colloidal crystal assembly at a curved oil-water interface [@manoharan], phase separation in a lipid bilayer vesicle [@baumgart], and the ordering of a block copolymer film deposited on a curved substrate [@copolymerfilm]. These processes may also describe aspects of biological processes such as viral capsid [@Zandi2006; @capsid1; @capsid2] and pollen grain assembly [@pollen]. Luque [*et al.*]{} have studied the growth of shells from identical subunits on the sphere and how their behavior is controlled by an effective line tension[@Luque2012]. Finally, the processes discussed here are also relevant for the study of the fate of false vacua (metastable states in the language we adopt here) in cosmology [@Coleman1977; @Coleman1977erratum; @Callan1977].
In our analysis we will always consider systems in some “disordered” state driven ([*e.g.*]{}, by cooling) to a region of parameter space where the “ordered state” becomes more energetically favorable, but the disordered state remains metastable. In this context, fluctuations will drive the appearance of nuclei of the ordered state. One of the most important observables to calculate is the nucleation rate $\Gamma$, which we may argue on quite general grounds [@Langer1967] has the form $$\begin{aligned}
\Gamma=\Gamma_0e^{-\beta E^*}, \label{eq:nucrate}\end{aligned}$$ originally derived by Kramers in the context of diffusive escape over a potential barrier [@Kramers]. Here, $\beta\equiv (k_B T)^{-1}$ is the inverse temperature, $E^*$ is the difference between the energy of the metastable state and the critical nucleated cluster of the ordered state, and $\Gamma_0$ is a prefactor which, in a chemical system, is derived from the microscopic kinetics. Within a field theoretic view, it is possible to compute the rate $\Gamma$ directly from the imaginary part of the appropriate free energy. This is particularly surprising since nucleation is an essentially non-equilibrium process and the free energy used comes from doing an equilibrium calculation. The theoretical foundation for the form of these nucleation rates was developed by Langer in a series of papers [@Langer1967; @Langer1968; @Langer1969]. Furthermore, factors in $\Gamma_0$ scaling with the size of the droplet were found and identified as universal insofar as they did not depend on underlying model parameters [@Gunther1980; @Langer1967].
The focus of our study will be the nucleated cluster energy $E$. In many cases such as the pollen and copolymers, the ordered phases have interesting spatial structure which may have important consequences for the nucleation processes, such as the presence of anisotropic surface tensions in the nuclei [@anisotropictension]. In addition, it must be noted that for crystals nucleating on curved surfaces there are additional elastic effects arising from geometric frustration which at certain length scales in the nucleation process are non-negligible [@Meng2014; @Grason2016]. For simplicity, we ignore the fine structure of the ordered phase and associated energetic contributions. Then, within classical nucleation theory (CNT), the energy of a single nucleated cluster takes the phenomenological form $$\begin{aligned}
E=\gamma P - c A, \label{eq:dH}\end{aligned}$$ where $\gamma$ is the line tension, $P$ is the perimeter of the cluster, $c$ is the difference in the bulk energy density of the metastable state and ground state, and $A$ is the area of the cluster. The physical variables here are chosen to correspond to the two-dimensional problem, but the original development of CNT would refer to the three-dimensional analogs (see [@Kalikmanov2013] for a review of CNT and the relevant literature). There are two issues to address in arriving at an improved understanding and expression of : the possible dependence of the line tension $\gamma$ on the cluster size and the consequences of a fluctuating interface.
Tolman was one of the first to address the possibility of the size dependence of the surface tension [@Tolman1949], seeming to resolve the issue (see the early references in Tolman’s paper for the relevant discussions). Tolman’s result for the surface tension $\sigma$ of a nucleated cluster in three dimensions reads $$\begin{aligned}
\sigma \equiv \sigma(R)=\sigma_{\infty}\left(1-\frac{2\delta_T}{R}\right),\end{aligned}$$ where $R$ is the radius of the cluster, $\sigma_{\infty}$ is the surface tensions of the infinite, flat interface, and $\delta_T$ is the Tolman length. The Tolman length is generally small, nearing the scale of the molecules themselves, and therefore the correction only becomes significant for very small clusters. However, experimental work looking into the size-dependence of the surface tension has found conflicting results (see the work of Bruot and Caupin [@Bruot2016] and references therein). These more recent works have not agreed on the sign of the Tolman length, and it has been suggested by Bruot and Caupin that future work should consider higher order corrections in $1/R$. Despite over half a century of work, robust conclusions seem few and far between, and it may be valuable to reexamine Tolman’s arguments and conclusions [@Horsch2017]. We will explore the fate of the Tolman length for nucleation on two-dimensional surfaces.
The impact of fluctuations of an interface within a field theoretic context was touched upon by Langer [@Langer1967], but received a full treatment by Günther [*et al*]{} [@Gunther1980]. In this context, one shows that an effective membrane energy is achieved from a Landau-Ginzburg functional of the order parameter. Specifically, the surface tension and bending moduli appearing in the effective membrane energy are related to the derivatives of the soliton solutions of the Euler-Lagrange equations for the order parameter. It is also possible to perform a similar analysis in a curved background [@Garriga1994]. Our work considers a similar Landau-Ginzburg functional for a scalar order parameter in curved and flat geometries, which we use to derive an effective membrane energy reminiscent of the Canham-Helfrich Hamiltonian on both the plane and the sphere. We then use a more geometrical approach similar to Voloshin’s analysis of nucleation rates on the plane [@Voloshin1985] (rather than the functional analysis approaches of [@Gunther1980; @Garriga1994]) to study the nucleated droplet shape and the thermal fluctuations in curved and flat backgrounds.
The paper is organized as follows. In the first section, we consider the critical radius $R^*$ for a nucleating droplet, where the critical condition is that the energy $E(R)$ of the droplet satisfies $\left.dE/dR \right|_{R=R^*}=0$. We can ask if the critical droplet radius $R^*$ is smaller or larger as we change the background curvature. This question was addressed by Gómez et al. [@Gomez2015]. We show that these results follow from a reasonable phenomenological model and a general, yet simple, isoperimetric inequality. Furthermore, this result acts as a sort of mean field around which we include interface fluctuations.
In the next section we show schematically how one can relate the original CNT form of the energy with that of the Landau-Ginzburg functional evaluated at soliton solutions of the Euler-Lagrange equations. In addition, we provide the results of the Tolman calculation for the size dependence of the line tension in two dimensions. These results help in the interpretation of later results which include fluctuations.
In the final section, we derive an effective interface energy to look for corrections to due to thermal fluctuations. Our general approach is analogous to the three-dimensional analysis of Prestipino, Laio, and Tosatti [@Prestipino2012; @Prestipino2013; @Prestipino2014] and the two-dimensional analysis of Voloshin [@Voloshin1985], except that we extend the analysis to nucleation on curved surfaces and discuss different regularizations of the high energy fluctuating modes. We conclude with a discussion of possible future work and the implications of our results for CNT.
An inequality and a theorem
===========================
The nucleation droplet shape with a fixed area will naturally minimize the contribution from the perimeter. We’ll assume for now that this perimeter has an infinitesimal width and we will ignore any spatial variation of the order parameter. We can than think about the nucleation droplet phenomenologically, as an area $A$ with some perimeter of length $P$ living on some surface. Then, we may make use of isoperimetric inequalties relating the size of an object to the size of its boundary. The following inequality derived by Morgan, Hutchings, and Howards [@Morgan2000] fits the bill: $$P^2 \geq 4 \pi (\chi-f+1)A - 2 \int_0^A G(t)\,\mathrm{d}t, \label{eq:isoineq}$$ where $f$ is the number of components of the droplets, $\chi$ is the Euler characteristic, and $G(t)$ is the supremum of the total Gaussian curvature for a given region area $t$. It would be nice if we could saturate the inequality in Eq. to find regions which minimize the perimeter $P$ for a given area $A$. Thankfully, for a reasonably large set of surfaces, such a saturation is possible via the following theorem:
**Isoperimetric Theorem[@Morgan2000].** *Consider a plane, sphere, real projective plane, or closed disk $S$ with smooth, rotationally symmetric metric such that the Gauss curvature is a nonincreasing function of the distance from the origin. Then among disjoint unions of embedded disks of a given area, a round disk centered at the origin minimizes perimeter.\
It is unique, except of course that a circle in the interior of a ball about the origin of constant Gauss curvature may be replaced by a congruent circle in that ball.*
Therefore, for a surface with a constant Gaussian curvature, we know that perimeter-minimizing area is a disk and that its perimeter $P$ satisfies, via Eq. , $$P^2= 4 \pi A-K A^2, \label{eq:perim}$$ where $K$ is the constant Gaussian curvature of the surface. For a sphere, we would have $K=1/R_s^2$, with $R_s$ the sphere radius. We’ll now suppose that we have a line tension $\gamma$ penalizing the perimeter and a condensation energy area density $c$. Then, the phenomenological form in Eq. combined with Eq. yields $$E = \gamma \sqrt{4 \pi A-K A^2} - cA.$$ To find the critical droplet area, we have to specify how the droplet would evolve in time. Perhaps the simplest choice of the dynamics is that the droplet area will grow or shrink, driving the energy $E$ to a minimum. These relaxation dynamics are: $\partial_t A=- \omega (dE/dA)$, where $\omega$ is inversely proportional to a characteristic relaxation time. We would then set $\partial_t A=0$ to find the critical area $A^*$ at which the seed starts to grow. We find $$A^*= \frac{2 \pi}{K} \left[ 1 - \frac{c}{\sqrt{c^2+K \gamma^2}} \right].$$ This formula also works for nucleation on a flat plane: Taking the $K \rightarrow 0$ limit, we find $A^* \rightarrow \pi \gamma^2/c^2$. Also, the critical area $A^*$ may be related to the critical radius $R^*$ of the geodesic disc, since $A = 2 \pi[1-\cos (\sqrt{K}R)]/K$, where $R$ is the geodesic distance from the center of the droplet to the edge. We find that the critical droplet radius is given by $$R^*= \frac{1}{\sqrt{K}} \arctan \left( \frac{\sqrt{K} \gamma}{c} \right),$$ recapitulating the result in Eq. 11 in Ref. [@Gomez2015]. Let us take these mathematical results as our foundation and build up the theory of droplets in two dimensions.
Thermodynamic potentials and Landau-Ginzburg energies
=====================================================
We will now consider possible corrections to the phenomenological energy in Eq. . First, we consider the possibility that the line tension $\gamma$ depends on the shape of the droplet. To begin, we will follow Tolman’s classical thermodynamic analysis [@Tolman1949], but work in two dimensions. This analysis consists of writing down the change in free energy of the combined liquid-vapor system giving the usual expression for the Laplace pressure. Then, using the Gibbs adsorption equation and the Laplace pressure, one finds a relation between the line tension $\gamma$ and the radius $R$ of the nucleated drop: $$\begin{aligned}
\mathrm{d} \gamma= - \frac{\Upsilon}{\Delta \rho} \, \mathrm{d} \left( \frac{\gamma}{R}\right), \label{eq:Tolmanrelation}\end{aligned}$$ where $\Upsilon$ is the order parameter ([*e.g.*]{}, mass density) computed at the interface ([*i.e.*]{}, a per-unit-length density) and $\Delta \rho$ is the change in order parameter between the ordered and disordered state ([*i.e.*]{}, a per-unit-area density). Recapitulating Tolman’s arguments, one finds that the ratio $\Upsilon/\Delta \rho$ may be parameterized in terms of a length $\delta$ as follows: $\Upsilon/\Delta \rho = \delta+\delta^2/(2R) $. Substituting this parameterization into Eq. and integrating yields $$\begin{aligned}
\gamma(R)&=\gamma_{\infty}\frac{ e^{\frac{\pi}{4}-\tan^{-1}\left(1+\frac{\delta}{R}\right)}}{\sqrt{1+\frac{\delta}{R}+\frac{\delta^2}{2R^2}}}\\
&\approx \gamma_{\infty}\left(1-\frac{\delta}{R}+\mathcal{O}\left(R^{-2}\right)\right),\end{aligned}$$ where we may now identify the $\delta$ as a length associated with the first $1/R$ correction to the line tension. This is the so-called Tolman length, which has evidently the same character in two and three dimensions, at least within Tolman’s original thermodynamic analysis. We have therefore identified a potential correction to our theory.
We have until now considered droplets with infinitely sharp and static interfaces. However, realistic nuclei will have finite thickness and fluctuating interfaces. We may therefore consider a scalar order parameter $\psi(x)$ that can capture spatial variation in the interface between phases. The field $\psi(x)$ may represent the local density of material or perhaps the degree of crystallinity. We will suppose that $\psi(x)$ takes on constant ([*i.e.*]{}, spatially uniform) values $\psi_1>0$ in the ordered phase and $\psi_2<0$ in the disordered phase (see Fig. \[fig:asympotential\]). Since we will be interested in first-order transitions between these phases, we can set up a potential for $\psi(x)$ that has two potential wells at $\psi_{1,2}$, separated by an energy barrier. A simple form for that potential is $$\begin{aligned}
\mathcal{V}(\psi)=-\frac{m\psi^2}{2}+\frac{\lambda \psi^4}{4!}-h\psi, \label{eq:potential}
\end{aligned}$$ where $m,\lambda>0$ are phenomenological parameters and $h$ is a bias that we can tune to make the ordered or disordered phase more energetically favorable by setting $h>0$ or $h<0$, respectively. At coexistence when the two phases are equally favorable and $h=0$, we readily find that $\psi_{1,2}= \pm \sqrt{6 m/\lambda}$ for the potential parameterization in Eq. . Penalizing spatial variations in $\psi(x)$, the free energy functional $\mathcal{H}$ associated with the field $\psi(x)$ may be written as $$\begin{aligned}
\mathcal{H}&=\int {\mathrm{d}}^dx\left(\frac{\kappa}{2}(\nabla \psi)^2+\mathcal{V}(\psi) \right).\label{energyfunc}\end{aligned}$$ In the absence of thermal fluctuations, the equilibrium configuration of the field $\psi$ will extremize the functional $\mathcal{H}$. Such a saddle-point solution will necessarily satisfy the Euler-Lagrange equation$$\begin{aligned}
\frac{\delta \mathcal{H}}{\delta \psi}=0=-\kappa\nabla^2{\psi} +\frac{\partial\mathcal{V}}{\partial \psi}. \label{ELequ}\end{aligned}$$ Two obvious solutions to this equation are the spatially uniform states $\psi = \psi_{1,2}$. However, when $h=0$, we also have a different solution $\psi \equiv \psi_I(z)$ that varies along one direction $z$: $$\begin{aligned}
{\psi}_I(z) = \sqrt{\frac{6m}{\lambda}} \tanh\left(\sqrt{\frac{m}{2\kappa}}z\right) \label{eq:flatsoliton}\end{aligned}$$ Such a solution interpolates between the values of the order parameter at the two minima of the potential $\mathcal{V}(\psi)$ and has a $(d-1)$-dimensional, infinite interface centered at $z=0$ with a characteristic thickness $w \sim \sqrt{\kappa/m} $. What about a solution with a droplet of the $\psi_1$ phase inside a sea of $\psi_2$? Indeed, such solutions are possible if we solve Eq. using spherical coordinates. We would then find a spherically symmetric soliton solution.
![The dashed curve corresponds to the scenario where the minima are degenerate $h=0$. The solid curve depicts the development of a metastable and stable state, $\psi_2$ and $\psi_1$, respectively. For both curves $m>0$ and $\lambda > 0$.[]{data-label="fig:asympotential"}](asympotential.pdf){width="\linewidth"}
The spherically symmetric case is particularly simple when the characteristic thickness of $w$ is sufficiently small and there is a slight asymmetry in the energy of the two minima (a small non-zero $h$, for example). Then, the spherically symmetric soliton solution has the same form as Eq. , where we replace $z$ with $r-r_0$, the radial distance away from the droplet surface, taken to be a sphere with radius $r_0$. This solution, $\psi_I(r-r_0)$, interpolates between the stable phase, $\psi_1$, inside the droplet $(r<r_0)$ and the metastable state $\psi_2$ outside the droplet $(r>r_0)$. This scenario is depicted in Figure . Then, substituting this solution into Eq. , we find that the free energy associated with such a droplet is given by $$\begin{aligned}
E&=2\pi\kappa \int {\mathrm{d}}r\, r \left(\frac{d \psi_I}{d r}\right)^2+2\pi\int {\mathrm{d}}r\, r\left(\mathcal{V}(\psi_I)\right) \nonumber\\
&\approx 2\pi r_0\underbrace{\left(\frac{4m\kappa}{\lambda}\sqrt{\frac{m}{2\kappa}}\right)}_{\gamma}-\pi r_0^2\underbrace{(\mathcal{V}(\psi_2)-\mathcal{V}(\psi_1))}_c,\end{aligned}$$ where we have assumed a thin interface width $w \ll r_0$. We have now identified that the free energy has two terms: one proportional to the perimeter $2 \pi r_0$ of the droplet and one to the area $\pi r_0^2$. We may thus identify the associated line tension $\gamma$ and condensation energy $c$. We have now confirmed that the mean-field solution of this scalar field model is consistent with our phenomenological analysis with the isoperimetric inequality.
What about the effects of thermal fluctuations? This is a difficult question because the nucleation and growth process is a non-equilibrium phenomenon and using standard equilibrium techniques is problematic. One possibility is to look at this process near the critical droplet size, where we expect the droplet to remain roughly stationary. In this case, we may suppose that the energy of the droplet in Eq. establishes a Boltzmann distribution with which we may calculate a free energy. Specifically, we may introduce a small field perturbation $\delta \psi(x)$ away from the mean field solution $\psi_I^* \equiv \psi_I(r-r_0^*)$, where the droplet radius is set to the critical value $r_0=r_0^*$. The free energy, then, would schematically look like $$\mathcal{F} = - k_BT \ln \int [{\rm d}\delta \psi] \, e^{- \beta \mathcal{H}[\psi_I^*+\delta \psi]}, \label{eq:funcdetmethod}$$ where we would integrate over all possible fluctuations $\delta \psi$ away from the critical droplet solution $\psi_I^*$. The first correction amounts to evaluating a functional determinant, as discussed by Callan and Coleman [@Callan1977]. We will calculate such a correction using a more geometrical approach that treats the undulations of the critical nucleus interface, instead. The connection between these two approaches is discussed in some detail by Günther, Nicole, and Wallace [@Gunther1980] and Zia [@Zia1985].
Fluctuations near coexistence
=============================
In this section we derive expressions for the line tension of droplets in the plane and on the surface of a sphere including Gaussian fluctuations of the interface away from a midline. Our approach follows the general tack of Zia’s analysis [@Zia1985]. The derivation ends up being less heuristic in two dimensions, as compared to three dimensions, since the interfaces are curves and the arclength coordinate is easily defined on the entire droplet. Our procedure is:
1. Construct normal coordinates in the surface near the interface, and in these coordinates compute the metric and gradient operator.
2. Assume that the order parameter, $\psi$, depends only on the coordinate normal to the interface and expand the Euler-Lagrange equations up to $\mathcal{O}\!\left(k^2\right)$ ($k(s)$ being the relevant curvature appearing in the metric).
3. Use these Euler-Lagrange equations to eliminate the potential $\mathcal{V}(\psi(\xi))$ from the total energy $\mathcal{H}$.
4. With this energy compute the partition functions, and thereby the free energy and surface tension.
Curves on surfaces
------------------
Consider a two dimensional surface embedded in three dimensions. Within this surface lies a curve, the droplet interface, parameterized by its arclength, $\mathbf{R}(s)$. The Darboux frame is constructed from $\hat{\mathbf{t}}$, the unit tangent; $\hat{\pmb{\gamma}}$, the curve normal in the surface; and $\hat{\mathbf{N}}$, the surface normal. The unit vector $\hat{\pmb{\gamma}}$ is constructed from the other two by the cross-product, $\hat{\pmb{\gamma}}\equiv \hat{\mathbf{N}} \times \hat{\mathbf{t}}$. This frame is determined at every point of the curve by the following set of differential relations: $$\begin{aligned}
\frac{d}{ds} \! \! \begin{pmatrix} \hat{\mathbf{t}}(s) \\ \hat{\pmb{\gamma}}(s) \\ \hat{\mathbf{N}}(s) \end{pmatrix}=\begin{pmatrix}
0 & k_g(s) & k_n(s) \\ -k_g(s) & 0 & -\tau_g(s) \\ -k_n(s) & \tau_g(s) & 0
\end{pmatrix} \! \! \begin{pmatrix} \hat{\mathbf{t}}(s) \\ \hat{\pmb{\gamma}}(s) \\ \hat{\mathbf{N}}(s) \end{pmatrix},\label{dframe}\end{aligned}$$ where $k_g$ is the geodesic curvature, $k_n$ is the normal curvature, and $\tau_g$ is the geodesic torsion. We first construct normal coordinates in the vicinity of the curve and within the surface. A point on the surface away from the curve would have a position $\mathbf{r}$ given by $$\begin{aligned}
\mathbf{r}(s,\xi)=\mathbf{R}(s)+\xi\hat{\pmb{\gamma}}(s),\end{aligned}$$ with $\xi$ a distance away from the curve along its normal at an arclength coordinate $s$. Making use of as necessary, we compute the metric: $$\begin{aligned}
g_{ij}&\equiv \frac{d\mathbf{r}}{dq_i}\cdot \frac{d\mathbf{r}}{dq_j}=\begin{pmatrix}
\left(1-\xi k_g\right)^2+\xi^2\tau_g^2 & 0 \\ 0 & 1
\end{pmatrix},\end{aligned}$$ where $q=(s,\xi)$. The derivatives of the order parameter are $$\begin{aligned}
\nabla \psi=&=\frac{\partial \psi}{\partial q_\alpha}g^{\alpha \beta}\frac{\partial \mathbf{r}}{\partial q_\beta}\\
&=\frac{1}{\left(1-\xi k_g\right)^2+(\xi \tau_g)^2}\frac{\partial \psi}{\partial s}\frac{\partial \mathbf{r}}{\partial s}+\frac{\partial \psi}{\partial \xi}\hat{\pmb{\gamma}}\label{grad}.\end{aligned}$$ Now, assuming the order parameter, $\psi(\xi)$, depends on only the normal coordinate $\xi$, we can write the energy as $$\begin{aligned}
\mathcal{H}&=\int {\mathrm{d}}s\, {\mathrm{d}}\xi \sqrt{\left(1-\xi k_g\right)^2+(\xi \tau_g)^2}\left[\frac{\kappa}{2}(\psi')^2+\mathcal{V}(\psi) \right]. \label{eq:energyintermediate}\end{aligned}$$ The Euler-Lagrange equations can be used to eleminate the potential from the energy. However the first integral necessary to do so only exists for the flat soliton solution given in Eq. . A consistent expansion in the curvature would require knowledge of higher and higher curvature contributions to the Euler-Lagrange equations. For our purposes, the lowest order of the Euler-Lagrange equations suffices when considering just one power of the curvature in the energy. However, if one wishes to work at higher orders in the curvature, such as $k_g^2$, more work is required. We provide an example in the Appendix II.
Since we are going to consider nucleation in the plane and on the sphere, we set $\tau_g(s)=0$ (see the Appendix I for details). Moreover, we may now set $\psi(\xi,s)$ to the arc-length independent solution $\psi_0(\xi)$ to the Euler-Lagrange equations. The energy from Eq. then takes the form $$\begin{aligned}
\mathcal{H} \rightarrow \mathcal{H}_{k_g}=\int\!\!{\mathrm{d}}s \left( \gamma_0-\gamma_1 k_g(s)\right) \label{Helf}\end{aligned}$$ where $\gamma_0=\kappa\int \psi_0'^2 {\mathrm{d}}\xi$ and $\gamma_1=\kappa\int \xi \psi_0'^2 {\mathrm{d}}\xi$. Note that the curvature term proportional to $k_g$ will not contribute for *symmetric* interface profiles $\psi_0(\xi)$ which look the same both in the ordered and disordered phase. This is natural as the sign of the curvature $k_g$ must be defined with respect to either the disordered or ordered phases ([*e.g.*]{}, $k_g >0$ for an interface curving into the ordered phase). Indeed, if there is no difference between the ordered and disordered phases as measured by the distance $\xi$ away from the interface, then this term must be zero by symmetry.
To study the effects of fluctuations of interface we expand the geodesic curvature in small deviations about some reference. The geodesic curvature can be written as follows $$\begin{aligned}
k_g(s)=\ddot{\mathbf{R}}\cdot(\mathbf{N}\times\dot{\mathbf{R}}).\end{aligned}$$ We are going to expand the curvature in a geodesic polar parametrization, $$\begin{aligned}
r(\phi)=R(1+\epsilon(\phi)),\label{param}\end{aligned}$$ with $\epsilon(\phi)$ describing the small fluctuations around a droplet of radius $R$. We expand $\epsilon(\phi)$ in Fourier modes, $$\epsilon(\phi)=\sum\limits_{n \neq 0}a_n f_n(\phi)$$ where $n$ is an integer and the $f_n(\phi)$ are the set of real orthonormal basis functions (see Appendix I for details). Note that $f_0(\phi)$ is a constant so that we may absorb it into the radius $R$. So, our summations and products over $n$ in everything that follows are assumed to be over all integers except for zero.
Starting with interfaces in the plane, the normal curvature is zero and the geodesic curvature becomes the only curvature in the problem: $$\begin{aligned}
k_g=k(\phi)=\frac{r^2+2(r')^2-r r''}{\left(r^2+(r')^2\right)^{\frac{3}{2}}}.\end{aligned}$$ Substituting into the curvature, expanding to quadratic order, and then plugging the result into the curvature energy gives the energy on the plane: $$\begin{aligned}
\mathcal{H}_{\mathbb{R}^2}&=\int \gamma_0 R\left[ 1+\frac{1}{2}(\epsilon')^2\right] \, {\mathrm{d}}\theta -2\pi\gamma_1 \nonumber \\
& =2 \pi (\gamma_0R-\gamma_1)+\frac{\gamma_0 R}{2} \sum_n n^2 a_n^2. \label{eq:interfaceE}\end{aligned}$$ Note that the energy in Eq. is just for the interface of the droplet. In addition, we know that there is a condensation energy that sets the overall size of the droplet. Again, we will be working near critical droplet sizes so that we may treat them as stationary objects with a fluctuating perimeter.
With these developments of the interface description, we now turn to the thermal fluctuations. An ensemble of critical droplet shapes at some fixed area $A$ and temperature $T$ will have a partition function with a fixed area constraint as follows: $$\begin{aligned}
\mathcal{Z}_{\mathbb{R}^2}=\ell \int \left[{\mathrm{d}}\mathbf{R}\right]\delta\!\left(A-\mathcal{A}[\mathbf{R}]\right)e^{-\beta \mathcal{H}_{\mathbb{R}^2}}\end{aligned}$$ where $\ell$ is a microscopic length scale that will depend on the details of the transition. A similar consideration occurs for fluctuating lipid vesicles, where $\ell$ is of order of a few lipid molecules [@faragopincus]. Furthermore, if the transition occurring is one of crystallization, then the relevant length scale will be the lattice spacing. In any case, $\ell$ is at the scale of the (typically microscopic) basic constituents of the system. The measure $[{\mathrm{d}}\mathbf{R}]$ represents an integration over all droplet shapes. The area of a particular configuration, $\mathcal{A}[\mathbf{R}]$, in the parametrization is $$\begin{aligned}
\mathcal{A}[\mathbf{R}]=\pi R^2\left(1+\frac{1}{2\pi}\sum\limits_{n}a_n^2\right)\equiv \pi R^2 (1+\delta_a),\end{aligned}$$ where we have defined $\delta_a$, a convenient representation of the sum in the first equality. The measure, then, may be written as $$\begin{aligned}
\int \left[{\mathrm{d}}\mathbf{R}\right]\equiv \int \mathrm{d}R \int\prod\limits_{n}{\mathrm{d}}a_n . \label{eq:measure}\end{aligned}$$ Together with this definition and a change of variables in the delta function, we have $$\begin{aligned}
\mathcal{Z}_{\mathbb{R}^2} & =\ell \int \mathrm{d}R\int\prod\limits_{n \neq 0}^{}{\mathrm{d}}a_n \frac{e^{-\beta \mathcal{H}_{\mathbb{R}^2}}}{2 \pi R(1+\delta_a)} \delta\!\left(R-\sqrt{\frac{A/\pi}{1+\delta_a}}\right) \nonumber \\
& =\frac{\ell}{\sqrt{4 \pi A }}\int\prod\limits_{n\neq 0}{\mathrm{d}}a_n \frac{e^{-\beta \bar{\mathcal{H}}_{\mathbb{R}^2}}}{\sqrt{1+\delta_a}} ,\end{aligned}$$ where we resolved the delta function by performing the integral over $R$. The substitution of $R$ results in a new energy, $\bar{\mathcal{H}}_{\mathbb{R}^2}$, which expanded to quadratic order in $a_n$ is $$\begin{aligned}
\bar{\mathcal{H}}_{\mathbb{R}^2}= \gamma_0\sqrt{4\pi A}-2 \pi\gamma_1+\gamma_0 \sqrt{\frac{A}{4\pi}} \sum_n (n^2-1) a_n^2 \label{energyinftrans}.\end{aligned}$$ The occurrence of $(n^2-1)$ in is not just happenstance: the modes $a_1$ and $a_{-1}$ correspond to infinitesimal translations of the membrane, analogously to the translation modes of three-dimensional vesicles when expanded in spherical harmonics [@helfrichves; @seifertves]. Next, we write the prefactor of the Boltzmann weight as follows: $$\begin{aligned}
\frac{1}{\sqrt{1+\delta_a}}&\approx 1-\frac{1}{4\pi}\sum a_n^2 \approx e^{-\frac{1}{4\pi}\sum a_n^2}.\end{aligned}$$
With all of the considerations given thus far the partition function takes the form $$\begin{aligned}
\mathcal{Z}_{\mathbb{R}^2}=\frac{\ell}{\sqrt{4 \pi A }}\int\prod\limits_{n\neq 0}{\mathrm{d}}a_n e^{-\frac{1}{4\pi}\sum a_n^2-\beta \bar{\mathcal{H}}_{\mathbb{R}^2}}.\end{aligned}$$ As it stands now, this partition function is problematic for a number of reasons. First, it does not yield a factor of the area coming from integration over the infinitesimal translations. This occurs because the term in the exponential coming from the delta function ($e^{-\frac{1}{4\pi}\sum a_n^2}$), which we call the Jacobian factor, contains the translation modes. To deal with this we must also include Faddeev-Popov and Liouville corrections as discussed by Cai [*et al.*]{} [@Cai1994]. Ordering the calculation by powers of the temperature, the Faddeev-Popov and Liouville corrections come in at the same order as the Jacobian factor. This is an essential point: to calculate consistently to lowest order, we must neglect the Jacobian term along with these more subtle corrections. All of these additional corrections are higher order in temperature than the leading-order corrections we are interested in. However, if in the future we wished to study correlation functions, we would be forced to include the other corrections: they are necessary at the outset for a consistent calculation. With these observations in mind, we take our partition function to be $$\begin{aligned}
\mathcal{Z}_{\mathbb{R}^2}=\frac{\ell}{\sqrt{4 \pi A }}\int\prod\limits_{n\neq 0}{\mathrm{d}}a_n e^{-\beta \bar{\mathcal{H}}_{\mathbb{R}^2}}.\end{aligned}$$ Let us now go back to the translation modes $a_1$ and $a_{-1}$. As for analogous analyses of vesicle shape fluctuations [@helfrichves; @seifertves], we exclude these modes from our analysis as they contribute to an entropic factor associated with the center of the droplet. Similar zero-energy modes are found in the functional determinant method of calculating the free energy (see discussion around Eq. ), and such modes must be properly normalized [@Coleman1988chp7]. However, in our simple physical picture of the nucleus as a fluctuating membrane, we can fix the center and ignore these modes entirely, as done for theories of fluctuating vesicles [@helfrichves].
![A few examples of regularization functions are given. The red square curve corresponds to the hard cutoff defined by $f(x)=1-\Theta[x-1]$. The remaining blue curves are examples of the smooth regularization curves defined as follows: For $0\leq x <1$, $\rho(x)=\exp[ r(-(x-1)^{-2}+1)]$ and for $x\geq 1$, $\rho(x)=0$. The various blue curves have varying $r$ from 0.001 (more sharp) to 0.01 (more smooth).[]{data-label="fig:regcurves"}](regulator.pdf){width="3.5in"}
Now we are prepared to compute the free energy, $\beta\mathcal{F}_{\mathbb{R}^2}=-\ln \mathcal{Z}_{\mathbb{R}^2}$, excluding the translation modes. The rest of the integrations over $a_n$ may be performed as they are all Gaussian integrals. We find $$\begin{aligned}
\beta \mathcal{F}_{\mathbb{R}^2}
& = \ln \left( \frac{P}{\ell}\right)+\beta (\gamma_0 P-2 \pi \gamma_1)\nonumber \\ & \quad {} +\sum\limits_{n>1}\ln \left[\frac{\beta \gamma_0 P }{4\pi^2}(n^2-1) \right],\end{aligned}$$ where we have recognized that we can replace $A$ with the perimeter $P$ of a disk with the same area $A$, for which $4 \pi A = P^2$. This makes the various terms contributing to the free energy a little more transparent, such as the usual constant line tension term $\beta \gamma_0 P$. Consider that the first term in the free energy, when given a minus sign and exponentiated, yields the size dependent scaling of the nucleation rate prefactor (recall ) $\Gamma_0\propto 1/P$. This result matches the results of [@Gunther1980; @Voloshin1985], and a requirement for this to be the true scaling is that no term logarithmic in the perimeter can arise from the fluctuation mode sum. Indeed with a hard cutoff used in the sum, there is no additional logarithm of the perimeter. What we describe now is a different regularization procedure for which the fluctuation sum does contribute terms logarithmic in the perimeter.
To complete these sums we a introduce a cutoff function $\rho(n/N)$ where $\rho(0)=1$ and compactly supported. Consider $\rho(x)$ as a smoothed and compactly supported version of the function $f(x)=1-\Theta[x-1]$, where $\Theta[x]$ is the Heaviside step function. Note that $f(x)$ is the usual hard cutoff at $n=N$. Such an example is depicted in Figure . The regularity of the cutoff function is related to the asymptotic estimates for the sums: more regularity leads to better estimates in powers of $1/N$. For an accessible review and introduction to these methods, see Tao’s online notes [@Tao2010]. Within this regularization scheme, the free energy becomes (for $N \gg 1$): $$\begin{aligned}
\beta\mathcal{F}_{\mathbb{R}^2}&\approx \beta \gamma P \left(1-\frac{2 \pi \delta_T}{P}\right) + \ln\left( \frac{\pi P}{\ell}\right)\nonumber \\&\quad {} +\left(-\frac{3}{2}+ N \!\braket{\rho}\right) \ln\left(\frac{\beta \gamma P }{4 \pi^2} \right) \nonumber \\ & {} \quad +N \int_0^1 \ln x^2 \rho(x) \, \mathrm{d} x+ N \ln N^2 \langle \rho \rangle \label{eq:flatFE}\\
& \approx\beta \gamma P \left(1-\frac{2 \pi \delta_T}{P}\right) + \ln\left( \frac{\pi P}{\ell}\right)-\frac{3}{2}\ln\left(\frac{\beta \gamma P }{4 \pi^2} \right) \nonumber \\&\quad {} +N\left[\ln\left(\frac{\beta \gamma P }{4 \pi^2} \right) -2+ \ln N^2 \right]. \label{eq:flatFE2}\end{aligned}$$ where $\langle \rho \rangle \equiv \int_0^1\rho(x)\,\mathrm{d}x \approx 1$, and we redefined $\gamma_0=\gamma$, $\gamma_1=\gamma \delta_T$.
Equations , are our main result of the thermal fluctuation analysis. The first term proportional to $\beta$ is the usual mean-field result, which includes a Tolman length $\delta_T$. Note that the Tolman length $\delta_T$ is proportional to $\gamma_1$, which we argued vanishes for symmetric phases, which has also been argued on general grounds for the three-dimensional problem [@Fisher1984]. Assuming the highest accessible undulation mode is given by $N=P/\ell$, where, again, $\ell$ is the length scale of the microscopic constituents of the droplet, we see that the nucleation rate prefactor takes the form $\Gamma_0\propto P^{1/2}$. It is for this reason that we discuss the regularization dependence on the nucleation rate. Note that Günther [*et al*]{}. [@Gunther1980] identify the three-dimensional case as a special one because the equivalent of our fluctuating sum contributes non-trivially to the nucleation rate (and does not for two and four dimensions).
Note that the contribution to $\Gamma_0$ comes from the finite piece of the sum over modes $n$, which evidently depends on the choice of regularization procedure. It’s possible that our model would require modifications to the integral measure in Eq. to remove this dependence, as the issue of having an infinite number of modes $a_n$ already appears in this measure. These subtleties may be the source of discrepancy between our calculation and previous work such as Garriga’s calculation [@Garriga1994], done using zeta-function regularization, and Voloshin’s analysis [@Voloshin1985], which connects the integration over modes $a_n$ to the harmonic oscillator partition function via the path integral. These and other studies ended up yielding the $\Gamma_0 \propto 1/ P$ result we find with a hard cutoff. An additional subtlety is that these previous works do not fix the droplet area $A$ and deal with the resulting unstable mode (the $a_0$ mode which gets a negative eigenvalue) using analytic continuation. In our analysis, by fixing the droplet area, we do not have to worry about the unstable $a_0$ mode. Reconciling these approaches would be an interesting topic for further study.
Unlike the $\Gamma_0$ contributions, both the smooth and hard cutoffs yield the same divergent terms (terms proportional to $N$ in ), which we will study in more detail below. Indeed, aside from the consideration of the nucleation rate, we may treat the various fluctuation corrections in Eq. as renormalizations of the line tension $\gamma$. These terms will introduce weak, logarithmic dependence on the perimeter $P$. For reasonable parameter values, these corrections serve to *increase* the effective line tension compared to the bare value $\gamma$. This is analogous to the correction to the surface tension in two-dimensional membranes [@davidleibler].
We note that interest in these logarithmic corrections is not new. Schmitz [*et al.*]{} have studied the impact of interface fluctuations on logarithmic contributions to the surface tension [@Schmitz2014]. As previously noted, the interest in logarithmic corrections as it pertains to scaling in nucleation rates spans a wide range of fields: from cosmology to chemical physics.[@Coleman1977; @Callan1977; @Langer1967; @Gunther1980; @Binder2016] These same corrections appear throughout the membrane literature as thermal corrections to membrane moduli. [@Morse1995; @Peliti1985; @Cai1994; @davidleibler] Most of this work focuses on three dimensions, and in terms of the prefactor to the nucleation rate, a general consensus is that $\Gamma_0 \propto A^{7/6}$, where $A$ is the surface area of the nucleated droplet [@Garriga1994; @Gunther1980; @Prestipino2014].
![The red curve is a schematic of the fluctuation interface of a droplet on the sphere.[]{data-label="fig:spherewobble"}](spherewobble.pdf){width="2.1in"}
In the same fashion one may calculate the free energy for nucleation on the surface of a sphere of radius $R_s$. The curve on the sphere is parametrized in geodesic polar coordinates $r(\phi)=R(1+\epsilon(\phi))$, as shown in Fig. \[fig:spherewobble\]. The quantity $R$ is the geodesic distance from the center of the droplet to the midline of the droplet edge, analogous to the the radius parameter $R$ used in the flat case. The process for analyzing the fluctuations is the same as in the plane, including the integration over $R$. The resulting free energy reads, for large $N$,$$\begin{aligned}
\beta \mathcal{F}_{S^2}&\approx\beta \gamma P\left[1- \frac{2 \pi\delta_T}{P}(1-2\bar{a})\right]+\ln \left(\frac{\pi P}{\ell}\right)\nonumber\\
&\quad{}+\left(-\frac{3}{2}+N\langle \rho \rangle\right)\ln\left[\frac{\beta\gamma R_s ^2(\cos^{-1}(1-2\bar{a}))^2}{P}\right] \nonumber \\
& \quad {}+N \int \ln x^2 \rho(x) \, \mathrm{d} x+ N \ln N^2 \langle \rho \rangle \\
&\approx\beta \gamma P\left[1- \frac{2 \pi\delta_T}{P}(1-2\bar{a})\right]+\ln \left(\frac{\pi P}{\ell}\right)\nonumber\\
&\quad{}+\left(-\frac{3}{2}+N \right)\ln\left[\frac{\beta\gamma R_s ^2(\cos^{-1}(1-2\bar{a}))^2}{P}\right] \nonumber \\
& \quad {}-2N+ N \ln N^2, \end{aligned}$$ where $\bar{a}=A/4\pi R_s^2$ is a rescaled area of the nucleated region, and we still have $\gamma_0=\gamma$ and $ \delta_T=\gamma_1/\gamma$. We wrote our expression in terms of the perimeter $P$ of a geodesic disk with area $A$, which satisfies $P=4 \pi R_s \sqrt{\bar{a}(1-\bar{a})}$. Note an interesting property of the nuclei on the sphere: When $\bar{a}=1/2$, the Tolman length term vanishes, regardless of the value of the Tolman length! This makes intuitive sense because when $A = 2 \pi R_s^2$, the nucleus fills a hemisphere. Therefore, its boundary is on the equator of the sphere and is a perfectly straight interface! Indeed, unlike the plane, it is possible to have a finite-sized region on the sphere with a perfectly straight boundary.
![We compare the effective surface tension, renormalized by temperature fluctuations, between the flat and spherical case. The flat surface tension is always larger, as shown by the plotted ratio $\gamma_T^{\mathrm{flat}}/\gamma_T^{\mathrm{sphere}} > 1$. We compared the tensions at the critical droplet sizes for the two respective cases. We used the bare tension value $\gamma=2$, a condensation energy $c=1$, and $\ell=0.01$. As expected, the two tensions coincide as $R_s$ becomes large. Also, as $T \rightarrow 0$, both tensions will just reduce to their bare value of $\gamma$. []{data-label="fig:effectivetension"}](effectivetensions.pdf){width="\linewidth"}
Apart from the difference in the Tolman term, the sphere energy has a different fluctuation correction proportional to $N$. One can verify that for small areas $A \ll 4 \pi R_s^2$, that the two corrections coincide in the flat and sphere case. It’s interesting to compare the renormalized line tensions in the two cases. Setting $N = P/\ell$, we find that the renormalized tension $\gamma_T\equiv \gamma+\delta \gamma$ has $$\begin{aligned}
\delta \gamma= \begin{cases}
\dfrac{k_BT}{\ell}\ln\left[\dfrac{\beta\gamma P^3}{(2\pi e \ell )^2}\right] &\mbox{flat} \\[15pt]
\dfrac{k_BT}{\ell}\ln\left[\dfrac{\beta \gamma P R_s ^2(\cos^{-1}(1-2\bar{a}))^2 }{(e \ell)^2}\right] &\mbox{sphere}
\end{cases}.\end{aligned}$$ The tensions are similar. Note that we have to evaluate these tensions at the respective critical droplet sizes for the two cases. This is because, as discussed above, the free energy approach only makes sense near the critical droplet size where the system may be approximated as stationary, with a Boltzmann distribution over fluctuating interface modes.
We can look at the difference between the spherical and flat case by looking at the ratio $\gamma_T^{\mathrm{flat}}/\gamma_T^{\mathrm{sphere}}$, which is plotted in Fig. \[fig:effectivetension\]. The difference between the two for reasonable parameter values is of order 10% for the largest values. We see that the surface tension is always larger in the flat case. The behavior of the ratio is understandable. First, as $T \rightarrow 0$, both of the tensions go to their bare values $\gamma$, so that the ratio goes to one. Similarly, as $R$ increases, the spherical case must approach the flat one and we also get a ratio approach unity. The crossover lengthscale will be related to the critical droplet radius, which for the choice of parameters given in Fig. \[fig:effectivetension\] is given by $R^*= \gamma/c=2$. So, as the sphere radius $R$ becomes much larger than this critical droplet size, the flat and sphere case begin to coincide. This is evident from Fig. \[fig:effectivetension\].
Conclusions
===========
We established a mean-field result for the critical nucleus from an isoperimetric inequality. This inequality provides the intuition that nucleation on surfaces of positive Gaussian curvature is easier and harder on negative Gaussian curvature surfaces compared to the flat space case. We then considered a more proper model of nucleation which includes a spatially varying order parameter that describes the transition region between the two phases. We showed that the scaling of the nucleation rate prefactor may be sensitive to the precise way in which the nucleating droplet is described and may not be as universal as currently believed. Furthermore, we focused on comparing nucleation on a sphere versus the flat plane, taking into account thermal fluctuations. We found that the fluctuations serve to increase the effective line tension of the critical nucleus in both the sphere and flat cases. We also found that the thermal fluctuation renormalizations are less severe in the spherical case.
The number of examples of nucleation phenomena occuring in 2D is proliferating at a considerable rate. Our result on the nucleation rate prefactor is a testable prediction, and we would like to find an experiment to make that test. In the near future there is the possibility to numerically study the Ising model, which is known to obey the classical nucleation theory quite well [@IsingCNT]. An extension of a functioning simulation of the Ising model critical droplets in 2D would be to do the simulation on a sphere.
This work was supported in part through NSF DMR12-62047. This work was partially supported by a Simons Investigator award from the Simons Foundation to Randall D. Kamien. M.O.L. gratefully acknowledges partial funding from the Neutron Sciences Directorate (Oak Ridge National Laboratory), sponsored by the U.S. Department of Energy, Office of Basic Energy Sciences.
Appendix I
==========
When expanding the droplet interface around its midline by introducing a small deviation $\epsilon(\theta)$, it is convenient to use the following set of functions as the basis for expansions in Fourier series:
: && f\_n()&= &&\
: && f\_n()&= ,&&
where $\theta \in [0,2\pi)$ parameterizes the droplet edge. With these definitions the basis is orthonormal. Furthermore, such definitions will make the functional integral simpler. A general expansion for $\epsilon(\theta)$ now takes the form $$\begin{aligned}
\epsilon=\sum\limits_{|n|>0}a_n f_n\end{aligned}$$ so, $$\begin{aligned}
&\int {\mathrm{d}}\theta \,\epsilon^2=\sum\limits_{|n|>0}a_n^2\\
&\int {\mathrm{d}}\theta \,(\epsilon')^2=\sum\limits_{|n|>0}n^2a_n^2.\end{aligned}$$
We now consider the shape of the midline of the droplet on the surface of a sphere. The geodesic torsion is defined as follows $$\begin{aligned}
\tau_g(s)\equiv \frac{d{\hat{\mathbf{N}}}}{d s}\cdot\hat{\pmb{\gamma}}\end{aligned}$$ If we are on a sphere of radius $R_s$, $$\begin{aligned}
\frac{d{\hat{\mathbf{N}}}}{d s}\cdot\hat{\pmb{\gamma}}=\frac{1}{R_s}\frac{d{\mathbf{R}}}{d s}\cdot\hat{\pmb{\gamma}}=\frac{1}{R_s} \,\hat{\mathbf{t}}\cdot\hat{\pmb{\gamma}}=0.\end{aligned}$$ The geodesic torsion is also zero in the plane since $\hat{\mathbf{N}}$ is constant. The geodesic curvature takes the form $$\begin{aligned}
k_g(s)=\ddot{\mathbf{R}}\cdot(\mathbf{N}\times\dot{\mathbf{R}}).\end{aligned}$$ To make progress we need to convert the m[é]{}lange of vectors as functions of the arclength into a function of $\phi$ (the azimuthal angle on the sphere). Note that a dot indicates a derivative with respect to arclength while a prime indicates a derivative with respect to the angle $\phi$. $$\begin{aligned}
\frac{d\mathbf{R}}{ds}&=\left(\frac{ds}{d\phi}\right)^{-1}\frac{d\mathbf{R}}{d\phi}\\
\frac{d^2\mathbf{R}}{ds^2}&=\left(\frac{ds}{d\phi}\right)^{-1}\frac{d}{d\phi}\left(\left(\frac{ds}{d\phi}\right)^{-1}\frac{d\mathbf{R}}{d\phi}\right)\\
&=-\left(\frac{ds}{d\phi}\right)^{-3}\frac{d^2s}{d\phi^2}\frac{d\mathbf{R}}{d\phi}+\left(\frac{ds}{d\phi}\right)^{-2}\frac{d^2\mathbf{R}}{d\phi^2}\\
&=-\left(\frac{ds}{d\phi}\right)^{-2}\frac{d^2s}{d\phi^2}\mathbf{t}+\left(\frac{ds}{d\phi}\right)^{-2}\frac{d^2\mathbf{R}}{d\phi^2}\end{aligned}$$ We can parametrize the spherical curve by $$\begin{aligned}
\mathbf{R}=R_s\mathbf{N}=R_s\begin{pmatrix}
\cos\phi \sin\theta \\
\sin\phi \sin\theta \\
\cos\theta
\end{pmatrix}.\end{aligned}$$ The arclength obeys the differential equation $$\begin{aligned}
\frac{ds}{d\phi}=R_s\sqrt{\sin^2\theta+\left(\frac{d\theta}{d\phi}\right)^2}.\end{aligned}$$ Now we take the necessary derivatives of $\mathbf{R}$ and then substitute into the expression for the geodesic curvature. $$\begin{aligned}
k_g(\phi)=\ddot{\mathbf{R}}\cdot(\mathbf{N}\times\dot{\mathbf{R}})=\left(\frac{ds}{d\phi}\right)^{-3}\mathbf{R}''\cdot(\mathbf{N}\times\mathbf{R}')\end{aligned}$$ A short calculation gives $$\begin{aligned}
\mathbf{R}''\cdot(\mathbf{N}\times\mathbf{R}')=&R_s^2\left(2\cos\theta \left(\frac{d\theta}{d\phi}\right)^2\right.\\
+&\left.\sin^2\theta \cos\theta - \sin\theta \frac{d^2\theta}{d\phi^2}\right).\end{aligned}$$ The geodesic curvature then reads $$\begin{aligned}
k_g=\frac{2\cos\theta \left(\frac{d\theta}{d\phi}\right)^2+\sin^2\theta \cos\theta - \sin\theta \frac{d^2\theta}{d\phi^2}}{R_s\left(\sin^2\theta+\left(\frac{d\theta}{d\phi}\right)^2\right)^{3/2}}.\end{aligned}$$ As a check, we can take two limits to verify the expression for the geodesic curvature. In the first, we assume there is no variation in the interface, looking to find the constant geodesic curvature expression for circles on the sphere. $$\begin{aligned}
\text{circle: } k_g=\frac{1}{R_s}\frac{\sin^2\theta \cos\theta}{\sin^3\theta}=\frac{1}{R_s\tan\theta}\end{aligned}$$ If we let $\theta=r/R_s$, where $r$ is a geodesic polar radius, and let $R_s\rightarrow \infty$ $$\begin{aligned}
\lim\limits_{R_s\rightarrow \infty}k_g=\lim\limits_{R_s\rightarrow \infty}\frac{1}{r+\mathcal{O}(\frac{1}{R_s^2})}=\frac{1}{r}.\end{aligned}$$ The curvature of a circle of radius $r$ in the plane! Now let us put the derivatives back but with the substitution $\theta=r/R_s$. $$\begin{aligned}
k_g
&=\frac{2\cos \left(\frac{r}{R_s}\right)\left[ \left(\frac{dr}{d\phi}\right)^2+\frac{R_s^2 }{2}\sin^2\left(\frac{r}{R_s}\right) \right]- R_s \sin\left(\frac{r}{R_s}\right) \frac{d^2r}{d\phi^2}}{\left[ R_s^2\sin^2\left(\frac{r}{R_s}\right)+\left(\frac{dr}{d\phi}\right)^2\right]^{3/2}}\end{aligned}$$ When we take $R_s\rightarrow \infty$, $$\begin{aligned}
\lim\limits_{R\rightarrow \infty} k_g&=\lim\limits_{R\rightarrow \infty}\frac{2 \left(\frac{dr}{d\phi}\right)^2+r^2- r \frac{d^2r}{d\phi^2}+\mathcal{O}(\frac{1}{R_s^2})}{\left(r^2+\left(\frac{dr}{d\phi}\right)^2+\mathcal{O}(\frac{1}{R_s^2})\right)^{3/2}}\nonumber\\
&=\frac{2(r')^2+r^2-r r''}{\left(r^2+(r')^2\right)^{3/2}}\end{aligned}$$ This is exactly the expression for the curvature in polar coordinates in the plane.
Appendix II \[appx3\]
=====================
Here we establish that the infinite interface form for the order parameter is the correct substitution to order $k^2$. We work in the spirit of the calculation of Ref. [@Fisher1984].
Begin with the following functional: $$\begin{aligned}
\mathcal{H}=\int {\mathrm{d}}^2x \left(-\frac{\kappa}{2}\psi \nabla^2\psi+\frac{c}{2}\left(\nabla^2\psi\right)^2 +\mathcal{V}(\psi)\right)\end{aligned}$$ The extremizer of this functional is the solution to the following Euler-Lagrange equation: $$\begin{aligned}
-\kappa\nabla^2 \psi+c\nabla^4 \psi =-\frac{\partial \mathcal{V}}{\partial \psi} \label{eq:appxEL}\end{aligned}$$ In the normal coordinates used throughout these notes, the Laplacian takes the form $$\begin{aligned}
\nabla^2\psi=\frac{\partial^2 \psi}{\partial \xi^2}-\frac{k_g(s)}{\left(1-\xi k_g(s)\right)}\frac{\partial \psi}{\partial \xi}\end{aligned}$$ where the order parameter, $\psi(\xi)$, takes a particular form: $$\begin{aligned}
\psi(s,\xi)=\psi_0(\xi)+k(s)\psi_1(\xi)+k(s)^2\psi_2(\xi)\end{aligned}$$ If we apply the Laplacian again to get the 4th derivative terms, then we will generate derivatives of the geodesic curvature with respect to the arclength. To simplify the problem we assume that $k_g(s)$ is slowly varying and, therefore, the derivatives can be set to zero.
Given below are expansions in powers of the curvature for various quantities appearing in the energy functional.
$$\begin{aligned}
\nabla^2 \psi\approx\psi_0''+k\left(\psi_1''-\psi_0'\right)+k^2\left(\psi_2''-\psi_1'-\xi\psi_0'\right)\end{aligned}$$
$$\begin{aligned}
\nabla^4\psi\approx&\psi_0^{(4)}+k\left(\psi_1^{(4)}-2\psi_0^{(3)}\right)\nonumber\\
&+k^2\left(\psi_2^{(4)}-2\psi_1^{(3)}-2\xi\psi_0^{(3)}-\psi_0''\right)\end{aligned}$$
$$\begin{aligned}
\mathcal{V}(\psi)=&\mathcal{V}(\psi_0)+k\psi_1\left.\frac{\partial\mathcal{V}}{\partial \psi}\right|_{\psi_0}\nonumber\\
&+k^2\left(\psi_2\left.\frac{\partial\mathcal{V}}{\partial \psi}\right|_{\psi_0}+\frac{1}{2}\psi_1^2\left.\frac{\partial^2 \mathcal{V}}{\partial\psi^2}\right|_{\psi_0}\right) \end{aligned}$$
With these results in hand, we expand the Euler-Lagrange equation in powers of the curvature $k$: $$\begin{aligned}
\begin{cases}
-\kappa \psi_0''+c\psi_0^{(4)}+\left.\frac{\partial \mathcal{V}}{\partial \psi}\right|_{\psi_0}=0 & k^0 \\
-\kappa\left(\psi_1''-\psi_0'\right)+c\left(\psi_1^{(4)}-2\psi_0^{(3)}\right)+\psi_1\left.\frac{\partial^2\mathcal{V}}{\partial \psi^2}\right|_{\psi_0}=0 & k^1
\end{cases}\end{aligned}$$ There is a “first integral" for the $k^0$-order Euler-Lagrange equation: $$\begin{aligned}
\psi_0'\left(-\kappa \psi_0''+c\psi_0^{(4)}+\left.\frac{\partial \mathcal{V}}{\partial \psi}\right|_{\psi_0}\right)=0,\\
\Rightarrow \left(-\frac{\kappa}{2}(\psi_0')^2+c\psi_0'\psi_0'''-\frac{c}{2}(\psi_0'')^2+\mathcal{V}\right)'=0\end{aligned}$$ which is: $$\begin{aligned}
\left(-\frac{\kappa}{2}(\psi_0')^2+c\psi_0'\psi_0'''-\frac{c}{2}(\psi_0'')^2+\mathcal{V}\right)=A\end{aligned}$$ The constant $A$ is fixed by enforcing the boundary conditions. As $\xi\rightarrow -\infty$, $\psi'(\xi)\rightarrow 0$ (the reason for considering the metastable state as being at $\xi\rightarrow -\infty$ is from the construction of the normal to the curve in the surface, which results in $\xi>0$ inside the droplet) and we get $$\begin{aligned}
\mathcal{V}\left(\psi(-\infty)\right)=A\end{aligned}$$ The order parameter at infinity takes on the value corresponding to the metastable state. We can, without loss of generality, set the value of the potential at the metastable state to be zero. Now we are free to write the first integral as $$\begin{aligned}
-\frac{\kappa}{2}(\psi_0')^2+c(\psi_0'\psi_0'')'-c(\psi_0'')^2-\frac{c}{2}(\psi_0'')^2+\mathcal{V}(\psi_0)=0. \label{eq:appxFI}\end{aligned}$$
The energy functional as series in the curvature has a leading-order term given by $$\begin{aligned}
\mathcal{H}_{k^0}=\int{\mathrm{d}}s \, {\mathrm{d}}\xi \left(\frac{1}{2}\kappa(\psi_0')^2+\frac{1}{2}c(\psi_0'')^2+\mathcal{V}(\psi_0)\right)\end{aligned}$$ From the first integral of motion, we have $$\begin{aligned}
\mathcal{V}(\psi_0)=\frac{\kappa}{2}(\psi_0')^2-c(\psi_0'\psi_0'')'+\frac{3c}{2}(\psi_0'')^2.\end{aligned}$$ Now we can rewrite this leading order term as $$\begin{aligned}
\mathcal{H}_{2,k^0}=\int{\mathrm{d}}s \, {\mathrm{d}}\xi \left(\kappa(\psi_0')^2+2c(\psi_0'')^2\right)\end{aligned}$$ The next order in the curvature $k$ yields the contribution $$\begin{aligned}
\mathcal{H}_{k^1}=\int{\mathrm{d}}s \, {\mathrm{d}}\xi&\left[\psi_1\left.\frac{\partial\mathcal{V}}{\partial \psi}\right|_{\psi_0}+\kappa \psi_0'\psi_1'-c\psi_0'\psi_0''+c\psi_0''\psi_1''\right.\nonumber\\
&\left.-\xi\left(\mathcal{V}(\psi_0)+\frac{1}{2}\kappa(\psi_0')^2+\frac{1}{2}c(\psi_0'')^2\right)\right]k(s)\end{aligned}$$ By substituting in for the zeroth order solution $\psi_0$ to the Euler-Lagrange equation, and using the first integral in , we rewrite this contribution as $$\begin{aligned}
\mathcal{H}_{k^1}=\int{\mathrm{d}}s \, {\mathrm{d}}\xi \, \xi\left(\kappa(\psi_0')^2+2c(\psi_0'')^2\right)k(s).\end{aligned}$$ Finally, at order $k^2$, we find the contribution $$\begin{aligned}
\mathcal{H}_{k^2}=\frac{1}{2}\int\!{\mathrm{d}}s\, {\mathrm{d}}\xi \, f[\xi,{\psi_i(\xi)}]\,k^2(s),\end{aligned}$$ where $$\begin{aligned}
f[\xi,\{\psi_i\}]=&(-2\xi\psi_1+2\psi_2)\left.\frac{\partial\mathcal{V}}{\partial \psi}\right|_{\psi_0}+\psi_1^2\left.\frac{\partial^2\mathcal{V}}{\partial \psi^2}\right|_{\psi_0}\nonumber\\
&+c(\psi_0')^2-2\xi\kappa\psi_0'\psi_1'+\kappa(\psi_1')^2\nonumber\\
&-2c\psi_1'\psi_0''-2c\psi_0'\psi_1''-2c\xi\psi_0''\psi_1'' \nonumber\\
&+c(\psi_1'')^2+2c\psi_0''\psi_2''+2\kappa\psi_0'\psi_2'.\end{aligned}$$ This contribution may be simplified by using the Euler-Lagrange equations and integration by parts. The simplified expression reads $$\begin{aligned}
\mathcal{H}_{k^2}=\frac{1}{2}\int{\mathrm{d}}s \, {\mathrm{d}}\xi\left(\kappa\psi_1\psi_0'+2c\psi_1'\psi_0''+c(\psi_0')^2\right)k^2(s).\end{aligned}$$ The correction $\psi_1$ to the Euler-Lagrange equation starts to come in at order $k^2$ in the energy.
[^1]: Electronic address: [email protected]
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abstract: 'We developed a Statistical Automatic Post-Editing (SAPE) system that works on Machine Translation (MT) output. A hybrid word alignment model was employed into the SAPE system. The proposed hybrid approach combines different word alignment tables and provides the well estimated alignment links to the SAPE system. This also allows the proposed system to correct lexical errors, erroneous words using insertion and deletion, as well as word ordering. We carried out the experiments on parallel dataset consisting of English text, Spanish MT output and corresponding post-edited output. In this paper, we have also applied the Hierarchical Phrase Based SMT (HPBSMT) to the SAPE system. It has to be mentioned that the output of our SAPE system not only provides better translations than the standard MT output, but also reduces the post-editing efforts as per the evaluation done with respect to different MT evaluation metrics (BLEU, TER and METEOR).'
author:
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bibliography:
- 'amta2016.bib'
title: 'Normalyzing Numeronyms - A NLP approach'
---
Introduction {#intro}
============
In the context of Machine Translation (MT), the quality of the translated text is not always considered good as per human understanding. Post-editing is the process of manually converting the machine translated texts into human readable form. While human post editing process is useful for small sized documents, it is not at all a feasible solution for large sized documents since it involves a lot of human post editing effort and time. In general, the term “Post-Editing” (PE) is defined as the correction performed by humans over the translation produced by a MT system [@veale:1997], often with minimum amount of manual labor [@taus:2010] and as a process of modification rather than revision [@Loffler:1985].
In order to measure the human post editing effort accurately, we use TER (Translational Error Rate) value [@Snover:2006]. It calculates the number of editing operations, including phrasal shifts that are needed to change a hypothesis translation into an adequate and fluent sentence, normalized by the length of the final sentence. It is observed that the translations provided by current MT systems often fail to deliver desirable output. Thus, to improve the quality, translations are corrected or post-edited by human translators. One of the major goals of an automatic PE system is to reduce the effort of the human post-editors by automatically customizing the MT output to be suited for a particular translation domain.
In general, MT output suffers from a number of adequacy errors which include incorrect lexical choice, word ordering, insertion, deletion, etc. Thus, in the present attempt, we developed a SAPE system that covers the above-mentioned errors while dealing with the translated and post-edited MT outputs. In addition to SAPE system, we have also incorporated a Hierarchical Phrase-Based SMT (HPBSMT) [@Chiang:2007] which is capable to correct word ordering error. The performance of a SMT (as well as SAPE) system heavily relies on bilingual data and word alignment methods. Therefore, we present a hybrid word alignment method in order to provide well estimated word alignment links to be used by our SAPE system. On the other hand, during the phrase alignment step, the system automatically estimates the word insertion errors (by considering one-to-many alignment links between MT–PE aligned data), word deletion errors (by considering many-to-one alignment links between MT–PE aligned data). It also handles lexical errors (by estimating lexical weights during model estimation) and word ordering. It has to be mentioned that the HPBSMT facilitates word ordering, as it uses synchronous context free grammar (SCFG) [@Aho:1969] based on hierarchical phrases.
We evaluated our system by computing the scores using BLEU [@Papineni:2002], TER [@Snover:2006] and Meteor [@Denkowski:2011] metrics, which show that our SAPE system produces significant improvement over the raw MT output.
The remainder of the paper is organized as follows. Section \[rw\] gives an overview of the related work, Section \[sd\] describes the components of our system: preprocessing, hybrid word alignment method and HPBSMT system. In Section \[exp\], we present the experimental setup whereas Section \[eval\] provides the results with some analysis, followed by conclusions and further work described in Section \[conclude\].
Related Research {#rw}
================
In the recent trends, various works have been attempted in automated post editing process with the help of Statistical MT (SMT) or Phrase based SMT (PBSMT) systems, as suggested by [@Lagarda:2009:HLT]. This means that the translation from the source language to the target language can be done using Rule based Machine Translation (RBMT) or PBSMT systems and the output can be fed to another SMT or PBSMT system to get the post edited form. Therefore, one of the advantages of APE systems is that they can adapt any black-box MT engine output and provide automatic PE output without retraining or re-implementing the original MT engine.
APE approaches cover a wide range of methods. [@Simard:2007:NAACL] and [@Simard:2007:WMT] applied SMT for post-editing, handled the repetitive nature of errors typically made by rule-based MT systems. Similarly, [@Pal:2015] applied SMT method with enhanced word alignment strategies for automatic post-editing on MT output. [@Rosa:2012:SMT] and [@Marecek:2011:WMT] applied a rule-based approach to APE on the morphological level.
[@Knight:1994] argued in favor of using a separate APE module, which is then portable across multiple MT systems and language pairs, and suggested that the post-editing task could be performed using SMT techniques. In connection to that, [@Allen:2000] sketched the outline of such an automated post-editing (APE) system, which would automatically learn the post-editing rules from a tri-parallel corpora consisting of source, raw MT and post-edited output. [@Elming:2006] suggested the use of transformation-based learning in order to automatically acquire rules of error correction from such data; however, the present method is only applied to reduce the lexical choice errors.
The current post-editing work was done by keeping the concept of sentence structure in mind. By default, English sentences and Spanish sentences follow the SVO (Subject-Verb-Object) structure when composing sentences. Thus, such rule can be considered during post editing. Working with this concept might be easy as it doesn’t have the need for SMT or PBSMT systems to be fed with complex rules and makes the process of APE faster.
[@Denkowski:2015:PhD] developed a method for integrating real time post-edited MT output into a translation model, by extracting a grammar for each input sentence. In some cases, the studies have even shown that the quality of MT plus PE can exceed the quality of human translation [@Fiederer:2009; @Koehn:2009; @DePalma:2009] as well as the productivity [@Zampieri:2014]. Post-editing can become a MT evaluation method, implying some specific language skills, different from ranking, for which a homogeneous group seems to be enough to perform the task [@Vela:2015].
System Description {#sd}
==================
The SAPE system consists of three basic components: pre-processing, a hybrid word alignment model and a HPBSMT based PE system integrated with the hybrid word alignment model. The system has been trained on monolingual Spanish MT output as well as its corresponding manually post-edited data collected from the WMT-2015 APE task.
Pre-processing {#preprocess}
--------------
Initially, we pre-processed the training data in order to enhance the quality of the word alignment model. The Spanish MT output and corresponding manually post-edited versions are tokenized and part of speech (POS) tags were extracted using TreeTagger[^1] [@Schmid:1994]. We performed several pre-processing steps such as lower casing and POS matching of unigrams before using the word alignment model. We have also performed the POS based pattern alignment model at bigram level. Such aligned POS patterns of bigrams are replaced by their corresponding surface forms only when the alignment is established by METEOR. The following steps were followed to prepare the bigram level alignment based on POS tags of the machine translated text and the corresponding post edited version.
- Initially, the POS tags of each word from every sentence were extracted. This procedure was done for both the machine translated text and the post edited text. Example of the step is given below.
POS MT Text: CommanderX/NC :/COLON Toad/NC su/PPO gilipollas/NC ./FS
POS PE Text: CommanderX/NC :/COLON Sapo/NC eres/VSfin un/ART gilipollas/NC ./FS
- The POS tags of both the files were converted to bigrams in the next step. The example of the procedure is given below.
Bigram POS MT Text: NC-COLON COLON-NC NC-PPO PPO-NC NC-FS
Bigram POS PE Text: NC-COLON COLON-NC NC-VSfin VSfin-ART ART-NC NC-FS
- The bigrams form both the text files were supplied as input to METEOR alignment. The alignment links in terms of positional index for the above example are given below:
METEOR Alignment: 0-0 1-1 4-5
- We use the above alignment links to produce MT-PE alignment at bigram level. Finally, we replace the surface forms of the words to produce the surface level alignment of the bigrams.
$CommanderX : ||| CommanderX :$\
$: Toad ||| : Sapo$\
$gilipollas . ||| gilipollas$ .
It has been observed that the system is capable to correct wrong lexical choice in MT output(e.g., $: Toad ||| : Sapo$) using the bigram alignment method. Thus, we add these bigram aligned data as addition to the training data for training of the word alignment model.
Hybrid Word Alignment {#HWA}
---------------------
The monolingual (Spanish MT output–PE output) word alignment process has been carried out by using the word alignment method based on edit distance (METEOR word aligner [@Lavie:2007]). We also added an additional word alignment method provided by the Berkeley word aligner [@Liang:2006].
### METEOR Alignment {#MA}
METEOR is an automatic MT evaluation metric that provides alignment between the hypothesis and reference. Given a pair of strings such as $H$ and $R$ to be compared, METEOR initially establishes a word alignment relation between them. The alignment is a mapping method between $H$ and $R$, which is built incrementally by the following sequence of word-mapping modules:
- [**Exact:**]{} maps if they are exactly same.
- [**Porter stemming:**]{} maps if they are same on their stemmed output obtained using the Porter stemmer.
- [**WN synonymy:**]{} maps if they are appeared as synonyms in the WordNet.
If multiple alignments exist, METEOR selects the alignment for which the word order in two strings is similar (i.e. having the fewest crossing alignment links). The final alignment is produced with respect to H and R by considering the union of all the alignments of different stages (e.g., Exact, Porter Stemming and WN synonymy).
### Berkeley Aligner
The Berkeley Aligner is used to align words across sentence pairs in a parallel corpus (in these case, the parallel monolingual MT-PE corpus). We initially trained our model on the parallel corpus using the fully unsupervised method of producing Berkeley word alignments. The Berkeley aligner is an extension of the Cross Expectation Maximization word aligner. The aligner uses agreement between two simple sequence-based models by training and facilitating substantial error reductions over standard models. Moreover, it is jointly trained with HMM models, and as a result, the AER [@Vilar:2006] was reduced by 29%.
### Hybridization
The hybrid word alignment method consists of two steps of hybridization:
- It combines the statistical word alignment methods like Berkeley word alignment with Grow-Diag-Final-And (GDFA) heuristic [@Koehn:2010] as well as edit distance based aligner such as METEOR. We rely METEOR word alignment than Berkeley aligner because METEOR provides edit distance based monolingual alignment which is more informative and produce accurate alignment table compare to the Berkeley aligner. The additional alignment links which are failed by METEOR are collected from the alignment table provided by the Berkeley aligner.
- It combines different kind of alignments such as alignment of surface form, parts of speech form, stem form and bigram POS form (as described in \[preprocess\]) provided by the hybridization method 1. We applied union to combine different word alignment tables and hypothesize that all alignments are correct. All the alignment tables are joined together and duplicate entries are removed [@Pal:2013; @Pal:2015].
HPBSMT
------
The Hierarchical PB-SMT is based on Synchronous Context Free Grammar (SCFG) [@Aho:1969]. SCFG rewrites rules on the right-hand side by aligned pairs [@Chiang:2007]. $$\label{eq7}
X \rightarrow <\gamma, \alpha, \sim>$$ where X represents a non-terminal, $\gamma$, $\alpha$ represent sequences of both terminal and non-terminal strings and $\sim$ represents a one-to-one correspondence between the occurrences of non-terminals appearing in $\gamma$ and $\alpha$.
The weight of each rule is defined as : $$\label{eq8}
w( X \rightarrow <\gamma, \alpha, \sim>) = \prod_{i} \phi_i(X \rightarrow <\gamma, \alpha, \sim>)^{\lambda_{i}}$$
where $\phi_i$ are features defined on each of the rules and $\lambda_{i}$ is the corresponding weight of $\phi_i$. The features are associated with 4 probabilities: frequency probability $P(\gamma|\alpha)$, $P(\alpha|\gamma)$, lexical frequency probability $P_{w}(\gamma|\alpha)$, $P_{w}(\alpha|\gamma)$ and a Phrase penalty $exp(-1)$.
There exist two additional rules called “glue rule” or “glue grammar” : $$S \rightarrow < S X, S X >$$ $$S \rightarrow < X,X >$$
These rules are used when no rule could match or the span exceeds a certain length (search depth: set the same as the initial phrase length limit). These rules simply monotonically connect translations of two adjacent blocks together.
The weight of the above type of rule is defined as $$\label{eq8}
w( S \rightarrow < S X, S X >) = exp(-\lambda_{g})$$
where $\lambda_{g}$ controls the model’s preference for hierarchical phrases over serial combination of phrases.
The weight ($w(d_g)$) of the derivation grammar ($d_g$) for generated source ($f_d$) and target ($e_d$) string is the product of the weights of the rules used in translation $w(r)$, language model probability $P_{lm}$ and the word penalty $exp(-\lambda_{wp}|e|)$ with some control over the length of the target output ($e$). The representation of $d_g$ can be defined as a triplet $<r,i,j>$, where, $r$ stands for grammar rule to rewrite a non-terminal that extends span $f_{d_{i}}^j$ on the source side. $$\label{eq8}
w(d_g) =\prod_{<r,i,j>\in d_g} w(r) \times P_{lm}^{\lambda_{lm}} \times exp(-\lambda_{wp}|e|)$$
Experiments {#exp}
===========
We performed the experiments on the development set and test set provided by the organizers of the APE task in the WMT2015.
Data
----
The 3-way parallel development set consists of 1,000 sentence triplets containing 21,617 English words in the source, 23,213 words in machine translated Spanish output and 23,098 words in the post-edited translations, respectively.
The texts were provided separately, each English sentence being aligned to the corresponding Spanish MT output and the post-edited MT output. In case of the training data, the number of sentences was less, having an impact on the number of words. The number of parallel sentences in the training data was 11,272 having an impact on the number of 238, 335 words in English source text whereas the MT output had 257,644 words and the post-edited MT output had 238,335 words, respectively. Similarly, the test set provided by the WMT 2015 APE task consists of 1,817 parallel sentences containing a total of 38,244 words on English side and 40,925 words on the corresponding MT output. The data sets did not require any pre-processing in terms of encoding or alignment. The additional monolingual Spanish data obtained from the WMT-2013 archive containing 51,381,432 tokens after filtering. As the monolingual corpus contains some non-Spanish words and sentences, we apply the Language Identifier [@Shuyo:2010] on monolingual Spanish corpora. From the monolingual corpus, we discarded the sentences that belong to different languages or contain different language segments.
Experimental Settings
---------------------
The effectiveness of the present work is demonstrated by using HPBSMT model. For building our SAPE system, we experimented with respect to various phrase lengths for the translation model and $n$–gram settings for the language model. We found that, using a maximum phrase length of 7 for the translation model and a 5-gram language model, the SAPE model produces the best results in terms of BLEU [@Papineni:2002] scores.
The other experimental settings were concerned with the hybrid word alignment (as described in Section \[sd\]) and the phrase-extraction [@Koehn:2003] algorithms. The Phrase level pairs that occur only once in the training data are assigned an unduly high probability mass (i.e. 1). To alleviate this shortcoming, we performed smoothing of the phrase table using the Good-Turing smoothing technique [@Foster:2006]. System tuning was carried out using Minimum Error Rate Training (MERT) [@Och:2003] optimized with k-best MIRA [@Cherry:2012] on a held out development set. The 5-gram target language model was trained using KenLM [@Heafield:2011]. When the parameters were tuned, the decoding was carried out on the held out test set.
Evaluation {#eval}
==========
The evaluation process was carried out using three well known automatic MT evaluation metrics: BLEU, METEOR and TER. We assume that the MT output of the test set provided by the WMT-2015 APE task as our base system translation output and we consider the corresponding PE version as a reference set for the evaluation. We perform the following experiment specified in Table \[tab:1\] to systematically improve the baseline system. Table \[tab:1\] provides a systematic comparison between the baseline system and APE systems with three different evaluation metrics. In all cases, our proposed system performed better. Table \[tab:1\] also shows the relative improvement over the baseline system and obtains maximum value with respect to TER. In Table \[tab:1\], the Basic Moses system with vanilla setting (Giza++ word alignment), the performance has been degraded by 3.46 BLEU point while TER increases 2.0 that signifies poor translation. The similar results were also found with respect to PBSMT system with Berkeley word alignment (i.e., experiment 3). In experiment 4, we apply the PBSMT system with monolingual edit distance based word alignment such as METEOR that also fails to perform better than the baseline system. As PBSMT system fails to achieve our goal, we apply HPBSMT system with Hybrid Word alignment. The system reported in experiment 5, has successfully improved over the base system with respect to all automatic evaluation metrics.
**Exp** **Systems** **BLEU** **MET** **TER**
--------- ----------------- ----------- ----------- -----------
1 Baseline System 65.90 74.54 22.71
2 Basic Moses 62.44 72.80 24.71
3 BA\_PBSMT 62.52 72.74 24.53
4 MA\_PBSMT 65.13 74.13 23.11
5 Hyb\_HPBSMT **66.23** **74.73** **22.33**
: Automatic Evaluation of SAPE system, MET:METEOR.[]{data-label="tab:1"}
Conclusion {#conclude}
==========
Our English–-Spanish APE system has successfully improved the translation quality in terms of automatic evaluation metrics. The hybrid word alignment plays crucial role in this tasks. Edit-distance based monolingual aligner provides very well alignment link for our SAPE system. We achieve considerable improvement over the base system after incorporating the hybrid word alignment into the state-of-the-art HPBSMT pipeline.
[^1]: http://www.cis.uni-muenchen.de/ schmid/tools/TreeTagger/
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abstract: 'Let $C$ be a smooth irreducible projective curve and $E$ be a stable bundle of rank $2$ on $C$. Then one can associate a rank $4$ vector bundle $\mathcal{F}_2(E)$ on $S^2(C)$, the second symmetric power of $C$. Our goal in this article is to study semistability of this bundle.'
address:
- 'Institute of Mathematical Sciences, C.I.T Campus, Tharamani, Chennai-600113, India'
- 'Indian Statistical Institute, 8th Mile, Mysore Road, Bangalore- 560059, India'
author:
- Krishanu Dan
- Sarbeswar Pal
title: Semistability of certain bundles on second symmetric power of a curve
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Introduction
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It has been an interesting and important object to study vector bundles over smooth projective varieties. The moduli space of semistable vector bundles with fixed topological invariants is well understood for the case of curves. However the question of existence of such bundles is open for higher dimensional varieties. In this article we will study the semistability of certain vector bundles on second symmetric power of a smooth projective curve, which arises naturally.
Let $C$ be smooth irreducible projective curve over the fields $\mathbb{C}$ of complex numbers and $E$ be a rank $r$ vector bundle on $C$. There is a naturally associated vector bundle $\mathcal{F}_2(E)$ of rank $2r$ on the second symmetric power $S^2(C)$ which is defined in Section \[s1\]. The stability and semi-stability for case $r=1,$ i.e. when $E$ is a line bundle on $C,$ has been studied and well understood ([@BN2], [@LMN]). In this article we consider the case when rank $E$ is two.
Fixing a point $x \in C$, the image of $\{x\} \times C$ in $S^2(C)$ defines an ample divisor $H'$ on $S^2(C)$, which we denote by $x+C$. We prove the following:
Let $E$ be a rank two stable vector bundle of even degree $d \geq 2$ on $C$ such that $\mathcal{F}_2(E)$ is globally generated. Then the bundle $\mathcal{F}_2(E)$ on $S^2(C)$ is $\mu_{H'}$-semistable with respect to the ample class $H'=x+C$.
Assume the genus of $C$ greater than 2. Let $E$ be a rank two $(0, 1)$-stable bundle (defined in Section \[s3\]) of odd degree $d \geq 1$ on $C$ such that $\mathcal{F}_2(E)$ is globally generated. Then the bundle $\mathcal{F}_2(E)$ on $S^2(C)$ is $\mu_{H'}$-semistable with respect to the ample class $H'=x+C$.
Preliminaries {#s1}
=============
Let $C$ be a smooth irreducible projective curve over the field of complex numbers $\mathbb{C}$ of genus $g$. On the space $C \times C$, consider the following involution $C \times C \longrightarrow C \times C, (x, y) \mapsto (y, x)$. The resulting quotient space is denoted by $S^2(C)$, called the second symmetric power of $C$. It is a smooth irreducible projctive surface over $\mathbb{C}$. Note that, $S^2(C)$ is naturally identified with the set of all degree $2$ effective divisors of $C$. Set $$\Delta_2:= \{(D, p) \in S^2(C) \times C \rvert D= p + q, \text{for some } q \in C\}.$$ Then $\Delta_2$ is a divisor in $S^2(C) \times C,$ called the universal divisor of degree $2.$ Let $q_1$ and $q_2$ be the projections from $S^2(C)\times C$ onto the first and second factors respectively. Then the restriction of the first projection to $\Delta_2$ induces a morphism $$q: \Delta_2 \longrightarrow S^2(C),$$ which is a two sheeted ramified covering. For any vector bundle $E$ of rank $r$ on $C$ we constract a bundle $\mathcal{F}_2(E):= (q)_*(q_2^*(E)\mid_{\Delta_2})$ of rank $2r$ over $S^2(C).$ From the exact sequence $$0 \to \mathcal{O}_{S^2(C) \times C}(-\Delta_2) \to {\mathcal{O}}_{S^2(C) \times C} \to {\mathcal{O}}_{\Delta_2} \to 0$$ on $S^2(C) \times C$ we get the following exact sequence on $S^2(C)$ $$0 \to q_{1*}(q_2^*E \otimes {\mathcal{O}}_{S^2(C) \times C}(-\Delta_2)) \to q_{1*}q_2^*E \to \mathcal{F}_2(E).$$
Define $f: C\times C \to \Delta_2$ by $(x,y) \mapsto (x+y, x).$ Then $f$ is an identification. Let $p_i:C\times C \to C$ be the $i$-th coordinate projection and let $\pi: C\times C \to S^2(C)$ be the quotient map. Then it’s easy to check that $\pi=q\circ f$ and $\mathcal{F}_2(E) = \pi_*p_2^*E.$
Let $C$ be a smooth irreducible projective curve over $\mathbb{C}$ of genus $g$ and let $M$ be a line bundle on $C$ of degree $d$. Consider the rank two vector bundle $V(M) := \pi_* p_2^*M$ on $S^2(C)$. Using Grothendieck-Riemann-Roch, one can compute the Chern classes of $V(M)$: $$c_1(V(M)) = (d - g - 1)x + \theta$$ and $$c_2(V(M)) = {{d - g} \choose 2}x^2 + (d - g)x.\theta + \frac{\theta^2}{2}$$ where $x$ is the image of the cohomology class of $x + C$ in $S^2(C)$, $\theta$ is the cohomology class of the pull back of the theta divisor in Pic$^2(C)$ under the natural map of $S^2(C)$ to Pic$^2(C)$ [@ACGH Lemma 2.5, Chapter VIII ]. Note that the cohomology group $H^4(S^2(C), \mathbb{Z})$ is naturally isomorphic to $\mathbb{Z}$, and $x^2 = 1, x.\theta = g,
\theta^2 = g(g - 1)$.
To find the Chern character of $\mathcal{F}_2(E),$ for any rank $r$ vector bundle $E,$ first choose a filtration of $E$ such that the successive quotients are line bundles and use the fact that $\mathcal{F}_2(\oplus M_k) = \oplus \mathcal{F}_2(M_k)$ where $M_k$’s are line bundles over $C.$ Then the Chern character of $\mathcal{F}_2(E)$ has the follwing expression [@BL]: $$ch(\mathcal{F}_2(E)) = \text{degree}(E)(1- \text{exp}(-x)) -r(g-1) + r(1 +g + \theta)\text{exp}(-x).$$ From the above expression one can easily see that $ c_1(\mathcal{F}_2(E)) = (d -r(g+1))x + r \theta,$ where $d=$ degree $E.$
Semistability of $\mathcal{F}_2(E)$, for degree E even
======================================================
Let $C$ be a smooth irreducible projective curve over the field of complex numbers $\mathbb C$ of genus $g$ and let $E$ be a rank $r$ vector bundle on $C.$ In this section we will prove the semistability of the vector bundle $\mathcal{F}_2(E)$, when $r =2$ and degree $E$ is even. We start with the following definitions.
Let $C$ be a non-singular irreducible curve. For a vector bundle $F$ on $C$ we define $$\mu(F) := \frac{\text{degree}(F)}{\text{rank}(F)}.$$ A vector bundle $F$ on $C$ is said to be semistable (respectively, stable) if for every subbundle $F'$ of $F$ we have $$\mu(F') \leq \mu(F) (\text{respectively,} \mu(F') < \mu(F)).$$
Let $X$ be a smooth irreducible surface and let $H$ be an ample divisor on $X.$ For a coherent torsion free sheaf $F$ on $X,$ we set $$\mu_H(F):= \frac{\text{degree}_H(F)}{\text{rank}(F)}$$ where $\text{degree}_H(F) = c_1(F)\cdot H.$
A vector bundle $F$ on $X$ is said to be $\mu_H$-semistable (respectively, $\mu_H$-stable), if for every coherent torsion free subsheaf $F'$ of $F$ with $0 < \text{rank}(F) < \text{rank} (E),$ we have $$\mu_H(F') \leq \mu_H(F) (\text{respectively,} \mu_H(F') < \mu_H(F)).$$
\[T\] Let $E$ be a rank two stable vector bundle of even degree $d \geq 2$ on $C$ such that $\mathcal{F}_2(E)$ is globally generated. Then the bundle $\mathcal{F}_2(E)$ on $S^2(C)$ is $\mu_{H'}$-semistable with respect to the ample class $H' = x + C$.
If $E$ is an even degree vector bundle which is a quotient of direct sum of very ample line bundles, i.e. if there is a surjection $\oplus L_i \to E$ where each $L_i$ is a very ample line bundle on $C,$ then $E$ satisfies the property of Theorem $3.1$.
We recall some well known results.
Let $f : X \longrightarrow Y$ be a finite surjective morphism of non-singular surfaces, $F$ be a vector bundle on $Y$, and $H$ be an ample divisor on $Y$. Assume $f^*(F)$ is $\mu_{f^*(H)}$-semistable (repectively, $\mu_{f^*(H)}$-stable). Then $F$ is $\mu_H$-semistable (respectively, $\mu_H$-stable).
[**Proof**]{}: .
Let $C$ be a smooth irreducible curve of genus $g \geq 1$ and let $K_C$ be the canonical bundle of $C$. Let $J^{g - 1}(C)$ be the variety of line bundles of degree $g - 1$ of $C$, and let $\Theta$ be the divisor on $J^{g - 1}(C)$ consisting of line bundles with non-zero sections. Let $\xi$ be a line bundle on $C$ of degree $g - 3$ and $$\nu_{\xi} : C \times C \longrightarrow J^{g - 1}(C)$$ be the morphism $(x, y) \mapsto \mathcal{O}_{C \times C}(x + y) \otimes \xi$. Then $$\nu_{\xi}^*(\Theta) \cong p_1^*(K_C \otimes \xi^*) \otimes p_2^*(K_C \otimes \xi^*) \otimes \mathcal{O}_{C \times C}(- \Delta)$$ where $\Delta$ is the diagonal of $C \times C$ and $p_i : C\times C \longrightarrow C$ is the $i$-th coordinate projection.
[**Proof**]{}: [@LMN Lemma 4.5].
Using Lemma $3.5$, we see that, to prove the semistablity of $\mathcal{F}_2(E)$ on $S^2(C)$ with respect to the ample class $x + C$, it is sufficient to prove the semistability of $\pi^*(\mathcal{F}_2(E))$ on $C \times C$ with respect to the ample divisor $H:= \pi^*(H') = [x \times C + C \times x]$. By Lemma $3.6$, we have $\pi^*(\theta) =
(g + 1)[x \times C + C \times x] - \Delta$. Since $c_1(\mathcal{F}_2(E)) = (d - 2(g +1))x + 2\theta$, we have $$c_1(\pi^*(\mathcal{F}_2(E))) = d[x \times C + C \times x] - 2\Delta,$$ and $$\mu_{H}(\pi^*(\mathcal{F}_2(E))) = \frac{d - 2}{2}.$$
First note that the bundle $\pi^*(\mathcal{F}_2(E))$ fits in the following exact sequence on $C \times C$: $$0 \rightarrow \pi^*(\mathcal{F}_2(E)) \rightarrow p_1^*(E) \oplus p_2^*(E) \xrightarrow{q} E = p_1^*(E)|_{\Delta} = p_2^*(E)|_{\Delta} \rightarrow 0$$ where the map $q$ is given by $q : (u, v) \mapsto u|_{\Delta} - v|_{\Delta}$. Let $\phi_i : \pi^*(\mathcal{F}_2(E)) \rightarrow p_i^*(E)$ be the restriction of the projection $p_1^*(E) \oplus p_2^*(E) \longrightarrow p_i^*(E)$ to $ \pi^*(\mathcal{F}_2(E)) \subset p_1^*(E) \oplus p_2^*(E)$. Then from the exact sequence (1), we get the following two exact sequences: $$0 \rightarrow p_1^*(E) \otimes \mathcal{O}_{C \times C}(- \Delta) \rightarrow \pi^*(\mathcal{F}_2(E)) \xrightarrow{\phi_1} p_2^*(E) \rightarrow 0,$$ and $$0 \rightarrow p_2^*(E) \otimes \mathcal{O}_{C \times C}(- \Delta) \rightarrow \pi^*(\mathcal{F}_2(E)) \xrightarrow{\phi_2} p_1^*(E) \rightarrow 0$$ [@BN Section 3].
$p_i^*(E)$ is $\mu_{H}$-stable, $\forall i = 1, 2$.
Due to symmetry, we will do it only for ${p_2}^*E$. Since over a smooth irreducible projective surface double dual of a torsion free sheaf is free, by taking double dual if necessary, we see that to prove stability or semistability it is enough to consider subsheafs which are line bundles. Let $L$ be a line bundle on $C \times C$ which is a subsheaf of ${p_2}^*E$ such that the quotient, $M$ say, is torsion free. We have an exact sequence $$0 \longrightarrow L \longrightarrow {p_2}^*E \longrightarrow M \longrightarrow 0.$$ We restrict this exact sequence to $x \times C$ and $C \times x$, respectively, to obtain the following exact sequences $$0 \longrightarrow L|_{x \times C} \longrightarrow E \longrightarrow M|_{x \times C} \longrightarrow 0,$$ and $$0 \longrightarrow L|_{C \times x} \longrightarrow \mathcal{O}_C \oplus \mathcal{O}_C \longrightarrow M|_{C \times x} \longrightarrow 0.$$ From the first exact sequence we get, deg$(L|_{x\times C}) = c_1(L).[x \times C] < \mu(E) = \frac{d}{2}$, since $E$ is stable. And from the second exact sequence we get deg$(L|_{C \times x}) =
c_1(L).[C \times x] \leq 0$. Thus deg$(L) = c_1(L).[x \times C + C \times x] < \frac{d}{2}= \mu_H(p_2^*E)$, proving the Lemma.
[**Proof of Theorem \[T\]**]{}:\
Let $L$ be a line bundle which is a subsheaf of $\pi^*(\mathcal{F}_2(E)) $ such that the quotient is torsion free. Suppose there is a non-zero homomorphism from $L$ to $p_1^*(E)(- \Delta) :=
p_1^*(E) \otimes \mathcal{O}_{C \times C}(- \Delta)$. Then $\mu_{H}(L) < \mu_{H}(p_1^*(E)(- \Delta)) = \frac{d - 4}{2} < \frac{d - 2}{2}$. So assume that there is no non-zero map from $L$ to $p_1^*(E)(- \Delta)$. Thus there is an injection $L \rightarrow p_2^*(E)$ so that $\mu_{H}(L) < \mu_{H}(p_2^*(E)) =
\frac{d}{2}$. Since $d$ is even, $\mu_{H}(L) \leq \frac{d}{2} - 1 = \frac{d - 2}{2}$.
Now let $F$ be a rank two coherent subsheaf of $\pi^*(\mathcal{F}_2(E))$ such that quotient is torsion-free. Then we have the following commutative diagram:
(U1)[0]{}; (U2) \[right of = U1\][$p_1^*(E)(- \Delta)$]{}; (U3) \[right of = U2\][$\pi^*(\mathcal{F}_2(E))$]{}; (U4) \[right of = U3\][$p_2^*(E)$]{}; (U5) \[right of = U4\][$0$]{}; (L1) \[below=.6cm of U1\][$0$]{}; (L2) \[below=.5cm of U2\][$F'$]{}; (L3) \[below=.5cm of U3\][$F$]{}; (L4) \[below=.5cm of U4\][$F''$]{}; (L5) \[below=.6cm of U5\][$0$]{}; (L2) to node (U2); (L3) to node (U3); (L4) to node (U4); (U1) to node (U2); (U2) to node (U3); (U3) to node (U4); (U4) to node (U5); (L1) to node (L2); (L2) to node (L3); (L3) to node (L4); (L4) to node (L5);
where the vertical arrows are injections. Suppose that both $F'$ and $F''$ are non-zero. These two are rank $1$ coherent sheaf. So we have, deg$(F') = \mu_{H}(F') < \mu_{H}(p_1^*(E)(- \Delta))
= \frac{d - 4}{2}$ and deg$(F'') = \mu_{H}(F'') < \mu_{H}(p_2^*(E)) = \frac{d}{2}$. Thus $\mu_{H}(F) = \frac{1}{2}($deg$(F') +$ deg$(F'')) < \frac{d - 2}{2}$. Now assume at least one of $F'$ and $F''$ is zero. First let $F''$ be zero. Then we have an injection $F \rightarrow p_1^*(E)(- \Delta)$ and the cokernel is a torsion sheaf. If the cokernel is supported at only finitely many points, then $\mu_{H}(F) = \mu_{H}(p_1^*(E)(- \Delta)) < \frac{d - 2}{2}$. If the cokernel is supported at a co-dimension 1 subscheme, then $\mu_{H}(F) < \mu_{H}(p_1^*(E)(- \Delta)) < \frac{d - 2}{2}$. Now let $F'$ is zero. So we have an injection $F \rightarrow p_2^*(E)$ and the cokernel is a torsion sheaf. If the cokernel is supported at a subscheme of co-dimension 1, then $\mu_{H}(F) < \mu_{H}(p_2^*(E)) = \frac{d}{2}$ so that $\mu_{H}(F) \leq \frac{d - 1}{2}$. If $\mu_{H}(F) = \frac{d-1}{2}$, then the cokernel is supported on a divisor of degree one. Now an effective divisor of degree one on $C \times C$ is of the form $x \times C$ or $C \times x$, for some $x \in C$. Thus $c_1(F)$ is of the form $c_1(p_2^*(E)) + [-x \times C]$ or $c_1(p_2^*(E)) + [-C \times x]$. But $c_1(\pi^*(\mathcal{F}_2(E)) = d[C \times x + x \times C] - 2 \Delta$, therefore $c_1((\pi^*(\mathcal{F}_2(E)/F)) = (d+1)[ x \times C] -2\Delta \text{ or } d[x \times C] + [C \times x] - 2\Delta$. In both the cases the torsion free sheaf $\pi^*(\mathcal{F}_2(E)/F$ restricted to any curve of the form $x \times C$ has negative degree. This gives a contradiction to the fact that $\pi^*(\mathcal{F}_2(E)$ is generated by sections. Thus we have, $\mu_{H}(F) \le \frac{d-2}{2}$.
If the cokernel is supported only at finitely many points then $\mu_{H}(F) = \mu_{H}(p_2^*(E)) = \frac{d}{2}$. In this case, $F$ is a rank two stable sheaf and hence it is isomorphic to $p_2^*(E)$. So the exact sequence $(2)$ splits, i.e., $\pi^*(\mathcal{F}_2(E)) \cong p_1^*(E)(- \Delta) \oplus p_2^*(E)$. Since $p_1^*(E)|_{x \times C}$ is trivial, deg$(p_1^*(E)(- \Delta)|_{x \times C}) < 0$. This contradicts the fact that $\mathcal{F}_2(E)$ and hence $\pi^*(\mathcal{F}_2(E))$ is globally generated.
Let $F$ be a rank $3$ coherent subsheaf of $\pi^*(\mathcal{F}_2(E))$ such that the quotient is torsion free. Then we have the following commutative diagram:
(U1)[0]{}; (U2) \[right of = U1\][$p_1^*(E)(- \Delta)$]{}; (U3) \[right of = U2\][$\pi^*(\mathcal{F}_2(E))$]{}; (U4) \[right of = U3\][$p_2^*(E)$]{}; (U5) \[right of = U4\][0]{}; (L1) \[below=.6cm of U1\][0]{}; (L2) \[below=.5cm of U2\][$F'$]{}; (L3) \[below=.5cm of U3\][F]{}; (L4) \[below=.5cm of U4\][F”]{}; (L5) \[below=.6cm of U5\][0]{}; (L2) to node (U2); (L3) to node (U3); (L4) to node (U4); (U1) to node (U2); (U2) to node (U3); (U3) to node (U4); (U4) to node (U5); (L1) to node (L2); (L2) to node (L3); (L3) to node (L4); (L4) to node (L5);
where the vertical arrows are injections. We have two possibilities: (I) rank$F' = 2$ and rank$F'' = 1$; (II) rank$F' = 1$ and rank$F'' = 2$. Suppose that rank$F' = 2$ and rank$F'' = 1$. By the arguments above, we have, $\mu_{H}(F') \leq \frac{d - 4}{2}$ and $\mu_{H}(F'') < \frac{d}{2}$. So $$\mu_{H}(F) < \frac{3d - 8}{6} < \frac{d - 2}{2}.$$ Now assume that rank$F' = 1$ and rank$F'' = 2$. In this case, we have, $\mu_{H}(F') < \frac{d - 4}{2}$ and $\mu_{H}(F'') \leq \frac{d}{2}$. If $d$ is even, $\mu_{H}(F') \leq \frac{d - 4}{2} - 1$, hence $\mu_{H}(F) \leq \frac{3d - 6}{6}= \frac{d - 2}{2}$.
semistability of $\pi^*(\mathcal{F}_2(E))$ for degree E odd {#s3}
===========================================================
In this section we will prove that the semi-stability of $\pi^*(\mathcal{F}_2(E))$ when degree $E$ is odd. First let’s recall some definitions.
Let $E$ be a non-zero vector bundle on $C$ and $k \in \mathbb{Z}$, we denote by $\mu_k(E)$ the rational number
$$\mu_k(E) := \frac{\text{degree}(E) + k}{\text{rank}(E)}.$$ We say that the vector bundle $E$ is $(k, l)$-stable (resp. $(k, l)$-semistable) if, for every proper subbundle $F$ of $E$ we have $$\mu_k(F) < \mu_{-l}(E/F) (resp. \mu_k(F) \le \mu_{-l}(E/F)).$$
Note that usual Mumford stability is equivalent to $(0, 0)$-stability. If $g \ge 3$, then there always exists a $(0, 1)$-stable bundle and if $g \ge 4$, then the set of $(0, 1)$-stable bundles form a dense open subset of the moduli space of stable bundels over $C$ of rank $2$ and degree $d$. [@RN Section 5]
Assume the genus of $C$ greater than 2. Let $E$ be a rank two $(0, 1)$-stable bundle of odd degree $d \geq 1$ on $C$ such that $\mathcal{F}_2(E)$ is globally generated. Then the bundle $\mathcal{F}_2(E)$ on $S^2(C)$ is $\mu_{H'}$-semistable with respect to the ample class $H'=x+C$.
Let $L$ be a line bundle which is a subsheaf of $\pi^*(\mathcal{F}_2(E)) $ such that the quotient is torsion free. Suppose there is a non-zero homomorphism from $L$ to $p_1^*(E)(- \Delta) :=
p_1^*(E) \otimes \mathcal{O}_{C \times C}(- \Delta)$. Then $\mu_{H}(L) < \mu_{H}(p_1^*(E)(- \Delta)) = \frac{d - 4}{2} < \frac{d - 2}{2}$. So assume that there is no non-zero map from $L$ to $p_1^*(E)(- \Delta)$. Thus there is an injection $L \rightarrow p_2^*(E).$ Now consider the exact sequence, $$\label{eqA}
0 \longrightarrow L \longrightarrow p_2^*(E) \longrightarrow M \longrightarrow 0,$$ where $M$ is a sheaf of rank $1$. Restricting the above exact sequence to $C \times x$, we see that, $c_1(L).[C \times x] \le 0$. On the other hand, restricting the above exact sequence to $C \times x$ and using that $E$ is (0.1)-stable, we get that $c_1(L).[C\times x] < \frac{d-1}{2}.$ Since $L$ is a line bundle, $c_1(L).[C\times x] \leq \frac{d-3}{2}.$ So we have $\mu_H(L) \leq \frac{d-3}{2}<\frac{d-2}{2}.$
Let’s assume $F$ be a rank two coherent subsheaf of $\pi^*(\mathcal{F}_2(E))$ such that quotient is torsion-free. Then we have the following commutative diagram:
(U1)[0]{}; (U2) \[right of = U1\][$p_1^*(E)(- \Delta)$]{}; (U3) \[right of = U2\][$\pi^*(\mathcal{F}_2(E))$]{}; (U4) \[right of = U3\][$p_2^*(E)$]{}; (U5) \[right of = U4\][$0$]{}; (L1) \[below=.6cm of U1\][$0$]{}; (L2) \[below=.5cm of U2\][$F'$]{}; (L3) \[below=.5cm of U3\][$F$]{}; (L4) \[below=.5cm of U4\][$F''$]{}; (L5) \[below=.6cm of U5\][$0$]{}; (L2) to node (U2); (L3) to node (U3); (L4) to node (U4); (U1) to node (U2); (U2) to node (U3); (U3) to node (U4); (U4) to node (U5); (L1) to node (L2); (L2) to node (L3); (L3) to node (L4); (L4) to node (L5);
where the vertical arrows are injections. We need to consider three different cases: (I) rank $F' = 1=$ rank $F''$; (II) $F''=0$; (III) $F'=0$. In each of these cases, we can argue exaclty as in the case of even degree to conclude that $\mu_H(F) \leq \frac{d-2}{2} =\mu_H(\pi^*\mathcal{F}_2(E)).$
Now assume $F$ is subsheaf of $\pi^*\mathcal{F}_2(E)$ rank $3.$ Then again we have the following commutative diagram:
(U1)[0]{}; (U2) \[right of = U1\][$p_1^*(E)(- \Delta)$]{}; (U3) \[right of = U2\][$\pi^*(\mathcal{F}_2(E))$]{}; (U4) \[right of = U3\][$p_2^*(E)$]{}; (U5) \[right of = U4\][0]{}; (L1) \[below=.6cm of U1\][0]{}; (L2) \[below=.5cm of U2\][$F'$]{}; (L3) \[below=.5cm of U3\][F]{}; (L4) \[below=.5cm of U4\][F”]{}; (L5) \[below=.6cm of U5\][0]{}; (L2) to node (U2); (L3) to node (U3); (L4) to node (U4); (U1) to node (U2); (U2) to node (U3); (U3) to node (U4); (U4) to node (U5); (L1) to node (L2); (L2) to node (L3); (L3) to node (L4); (L4) to node (L5);
where the vertical arrows are injections. We have two possibilities: (I) rank$F' = 2$ and rank$F'' = 1$; (II) rank$F' = 1$ and rank$F'' = 2$. Using the same argument as in Theorem $3.3$, we can show that in the case of (I), $\mu_H(F) < \frac{d-2}{2}.$ Now consider the case (II). In this case, restircting the exact sequence $0 \to F' \to p_1^*(E)(- \Delta)$ to $x \times C$ and $C \times x,$ we get that $$0 \to F'|_{x\times C} \to \mathcal{O}_C(-x)$$ and $$0 \to F'|_{C \times x} \to E \otimes \mathcal{O}_C(-x).$$ From these two exact sequences and using the fact that $E$ is $(0,1)$-stable we see that $\mu_H(F') < \frac{d-4}{2}$ and hence $\mu_H(F') \leq \frac{d-6}{2}.$ Also using the same argument as above, we have, in any case, $\mu_H(F'') \leq \frac{d}{2}.$ Combining all these, we get that $\mu_H(F) < \frac{d-2}{2}.$
Restriction to curves of the form $x+C$
=======================================
In this section we will investigate the restriction of $\mathcal{F}_2(E)$ to the curves of the form $x+C$ where $x+C$ is the reduced divisor of $S^2(C)$ whose support equals to $\{x+c : c \in C\}.$ For this we have the follwoing theorem.
Let $C$ be a smooth irreducible projective curve over $\mathbb{C}$ of genus $g$ and let $E$ be a rank to vector bundle on $C$ of degree $d \geq 3.$ Then for any $x \in C, \mathcal{F}_2(E)|_{x+C}$ is not semistable.
First note that, since $E$ is locally free, $p_2^*E$ is flat over $S^2(C)$ and using the base change formula we get $$\mathcal{F}_2(E)|_{x+C}=\pi_*(p_2^*E|_{\pi^{-1}(x+C)}).$$ Also we have the following exact sequence $$0 \to p_2^*E|_{\pi^{-1}(x+C)} \to p_2^*E|_{x \times C} \oplus p_2^*E|_{C \times x} \to E|_{(x,x)} \to 0.$$ From this exact sequence and using the fact that $\pi|_{x \times C} : x\times C \to x+C$ and $\pi|_{C \times x} : C \times x
\to x+C$ are isomorphisms and $p_2^*E|_{x \times C} = E$ and $p_2^*E|_{C \times x} = E_x \otimes \mathcal{O}_C$, we get an injective map $$0 \to E \otimes \mathcal{O}_C(-x) \to \mathcal{F}_2(E)|_{x+C}.$$ Now the degree of $E \otimes \mathcal{O}_C(-x) = d-2$ and that of $\mathcal{F}_2(E)|_{x+C} =d-2.$ So the cokernel is rank $2$ coherent sheaf of degree zero. If it is torsion free then clearly $\mathcal{F}_2(E)|_{x+C}$ is not semistable. If the cokernel has torsion, then there is an effective divisor $D$ such that the above map factors through $E \otimes \mathcal{O}_C(-x) \otimes \mathcal{O}_C(D)$ and in this case the cokernel will be again torsion free. But in this case the degree of the cokernel will be of negetive degree. So in this case $\mathcal{F}_2(E)|_{x+C}$ has a torsion free quotient of negetive degree. Hence it is not semistable.
$Acknowledgements:$ We would like to thank Prof. D.S. Nagaraj for encouraging us to choose the problem and also letting us the helping hand whenever necessary. The first named author would like to thank ISI Bangalore for their hospitality during the stay where the work started.
[1111]{}
E. Arbarello, M. Cornalba, P. Griffiths and J. Harris: Geometry of Algebraic curves I. Grundl. der Math. W. 267, Berlin-Heidelberg-New York 1985.
El Mazouni, A.; Laytimi, F.; Nagaraj, D. S. $\textit{Secant bundles on second symmetric power of a curve}$. J. Ramanujan Math. Soc. 26 (2011), no. 2, 181-194.
Biswas, Indranil; Nagaraj, D. S. $\textit{Stability of secant bundles on second}$ $\textit{symmetric power of a curve}$. Commutative algebra and algebraic geometry (CAAG-2010), 13-18, Ramanujan Math. Soc. Lect. Notes Ser., 17, Ramanujan Math. Soc., Mysore, 2013.
Biswas, Indranil; Nagaraj, D. S. $\textit{Reconstructing vector bundles on}$ $\textit{curves from their direct image on symmetric powers}$. Arch. Math. (Basel) 99 (2012), no. 4, 327-331.
Biswas, Indranil; Laytimi, Fatima $\textit{Direct image and parabolic structure on}$ $\textit{symmetric product of curves}$. J. Geom. Phys. 61 (2011), no. 4, 773-780.
Narasimhan, M. S.; Ramanan, S. $\textit{Geometry of Hecke cycles. I}$. C. P. Ramanujam$-$a tribute, pp. 291-345, Tata Inst. Fund. Res. Studies in Math., 8, Springer, Berlin-New York, 1978.
|
---
abstract: 'Embedding microscopic sensors, computers and actuators into materials allows physical systems to actively monitor and respond to their environments. This leads to the possibility of creating smart matter, i.e., materials whose properties can be changed under program control to suit varying constraints. A key difficulty in realizing the potential of smart matter is developing the appropriate control programs. We present a market[–]{}based multiagent solution to the problem of maintaining a physical system near an unstable configuration, a particularly challenging application for smart matter. This market control leads to stability by focussing control forces in those parts of the system where they are most needed. Moreover, it does so even when some actuators fail to work and without requiring the agents to have a detailed model of the physical system.'
author:
- |
Oliver Guenther, Tad Hogg and Bernardo A. Huberman\
Xerox Palo Alto Research Center\
Palo Alto, CA 94304
title: Power Markets for Controlling Smart Matter
---
Introduction
=============
Embedding microscopic sensors, computers and actuators into materials allows physical systems to actively monitor and respond to their environments in precisely controlled ways. This is particularly so for microelectromechanical systems (MEMS) [@berlin95; @bryzek94; @web.mems96] where the devices are fabricated together in single silicon wafers. Applications include environmental monitors, drag reduction in fluid flow, compact data storage and improved material properties.
In many such applications the relevant mechanical processes are slow compared to sensor, computation and communication speeds. This gives a [*smart matter*]{} regime, where control programs execute many steps within the time available for responding to mechanical changes. A key difficulty in realizing smart matter[’]{}s potential is developing the control programs. This is due to the need to robustly coordinate a physically distributed real-time response with many elements in the face of failures, delays, an unpredictable environment and a limited ability to accurately model the system[’]{}s behavior. This is especially true in the mass production of smart materials where manufacturing tolerances and occasional defects will cause the physical system to differ somewhat from its nominal specification. These characteristics limit the effectiveness of conventional control algorithms, which rely on a single global processor with rapid access to the full state of the system and detailed knowledge of its behavior.
A more robust approach for such systems uses a collection of autonomous agents, that each deal with a limited part of the overall control problem. Individual agents can be associated with each sensor or actuator in the material, or with various aggregations of these devices, to provide a mapping between agents and physical location. This leads to a community of computational agents which, in their interactions, strategies, and competition for resources, resemble natural ecosystems [@Huberman88Eco]. Distributed controls allow the system as a whole to adapt to changes in the environment or disturbances to individual components [@hogg91a].
Multiagent systems have been extensively studied in the context of distributed problem solving [@durfee91; @gasser89; @lesser95]. They have also been applied to problems involved in acting in the physical world, such as distributed traffic control [@nagel94], flexible manufacturing [@upton92], the design of robotic systems [@sanderson83; @williams96a], and self-assembly of structures [@semela95]. However, the use of multiagent systems for controlling smart matter is a challenging new application due to the very tight coupling between the computational agents and their embedding in physical space. Specifically, in addition to computational interactions between agents from the exchange of information, there are mechanical interactions whose strength decreases with the physical distance between them.
In this paper we present a novel control strategy for unstable dynamical systems based on market mechanisms. This is a particularly challenging problem, for in the absence of controls, the physics of an unstable system will drive it rapidly away from the desired configuration. This is the case, for example, for a structural beam whose load is large enough to cause it to buckle and break. In such cases, weak control forces, if applied properly, can counter departures from the unstable configuration while they are still small. Successful control leads to a virtual strengthening and stiffening of the material. Intentionally removing this control also allows for very rapid changes of the system into other desired configurations. Thus an effective way of controlling unstable systems opens up novel possibilities for making structures extremely adaptive.
Dynamics of Unstable Smart Matter
==================================
The devices embedded in smart matter are associated with computational agents that use the sensor information to determine appropriate actuator forces. The overall system dynamics is a combination of the behavior at the location of these agents and the behavior of the material between the agent locations. In mechanical systems, displacements associated with short length scales involve relatively large restoring forces, high frequency oscillations and rapid damping. Hence, they are not important for the overall stability [@hogg96b]. Instead, stability is primarily determined by the lowest frequency modes. We assume that there are enough agents so that their typical spacing is much smaller than the wavelengths associated with these lowest modes. Hence, the lower frequency dynamics is sufficiently characterized by the displacements at the locations of the agents only. The high-frequency dynamics of the physical substrate between agents serves only to couple the agents[’]{} displacements.
The system we studied, illustrated in Fig. \[x:pendulum\]a, consists of [*n*]{} mass points connected to their neighbors by springs. In addition a destabilizing force proportional to the displacement acts on each mass point. This force models the behavior of unstable fixed points: the force is zero exactly at the fixed point, but acts to amplify any small deviations away from the fixed point. This system can be construed as a linear approximation to the behavior of a variety of dynamical systems near an unstable fixed point, such as the inverted pendulae shown in the Fig. \[x:pendulum\]b. In the absence of control, any small initial displacement away from the vertical position rapidly leads to all the masses falling over. In this case, the lowest mode consists of all the pendulae falling over in the same direction and is the most rapidly unstable mode of behavior for this system. By contrast, higher modes, operating at shorter length scales, consist of the masses falling in different directions so that springs between them act to reduce the rate of falling.
The system[’]{}s physical behavior is described by
1. the number of mass points [*n*]{}
2. the spring constant [*k*]{} of the springs
3. a destabilizing force coefficient [*f*]{}
4. a damping force coefficient [*g*]{}
We also suppose the mass of each point is equal to one. The resulting dynamics of the unstable chain is given by[^1] [@goldstein80]: $$\label{x:motion}\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr
$\displaystyle{{{dx_{i}}\over{dt}}}$\hfilneg&$\displaystyle{{}=
v_{i}\hskip 0.265em }$\cr
$\displaystyle{{{dv_{i}}\over{dt}}}$\hfilneg&$\displaystyle{{}=
k{\left( x_{i-1}\hskip -0.167em -\hskip -0.167em
x_{i}\right) }+k{\left( x_{i+1}
\hskip -0.167em -\hskip -0.167em x_{i}\right) }
+fx_{i}-gv_{i}+H_{i}}$\cr
}}$$ where $
x_{i}$ is the displacement of mass point [*i*]{}, $
v_{i}$ is the corresponding velocity, and $
x_{0}=x_{n+1}=0$ is the boundary condition. The $
H_{i}$ term in Eq. (\[x:motion\]) is the additional control force produced by the actuator attached to mass point [*i*]{}. We suppose the magnitude of this control force is proportional to the power $
P_{i}$ used by the actuator. For reasons of simplicity we use a proportionality factor of 1.
For these systems, the long time response to any initial condition is determined by the eigenvalues of the matrix corresponding to the right hand side of Eq. (\[x:motion\]). Specifically, if the control force makes all eigenvalues have negative real parts, the system is stable [@hogg96b]. The corresponding eigenvectors are the system[’]{}s modes. Thus to evaluate stability for [*all*]{} initial conditions, we can use any single initial condition that includes contributions from all modes. If there are any unstable modes, the displacements will then grow. We used this technique to evaluate stability in the experiments described below.
A Power Market for Control
===========================
The control problem is how hard to push on the various mass points to maintain them at the unstable fixed point. This problem can involve various goals, such as maintaining stability in spite of perturbations typically delivered by the system[’]{}s environment, using only weak control forces so the actuators are easy and cheap to fabricate, continuing to operate even with sensor noise and actuator failures, and being simple to program, e.g., by not requiring a detailed physical model of the system.
Computational markets are one approach to this control problem [@clearwater96; @Ferguson88; @huberman95b; @Kurose89; @Malone88; @sutherland68; @waldspurger92; @wellman93]. As in economics, the use of prices provides a flexible mechanism for allocating resources, with relatively low information requirements [@hayek78]: a single price summarizes the current demand for each resource.
In designing a market of computational agents, a key issue is to identify the consumers and producers of the goods to be traded. Various preferences and constraints are introduced through the definition of the agents[’]{} utilities. This ability to explicitly program utility functions is an important difference from the situation with human markets. Finally, the market mechanism for matching buyers and sellers must be specified.
In the market control of smart matter treated here, actuators, or the corresponding mass points to which they are attached, are treated as consumers. The external power sources are the producers and as such are separate from consumers. All consumers start with a specified amount of money. All the profit that the producers get from selling power to consumers is equally redistributed to the consumers. This funding policy implies that the total amount of money in the system will stay constant.
In the spirit of the smart matter regime, where control computations are fast compared to the relevant mechanical time scales, we assume a market mechanism that rapidly finds the equilibrium point where overall supply and demand are equal. Possible mechanisms include a centralized auction or decentralized bilateral trades or arbitrage. This equilibrium determines the price and the amount of power traded. Each actuator gets the amount of power that it offers to buy for the equilibrium price and uses this power to push the unstable chain.
The utility function for using power [*P*]{} reflects a trade-off between using power to act against a displacement and the loss of wealth involved. While a variety of utility functions are possible, a particularly simple one for agent [*i*]{}, expressed in terms of the price of the power, [*p*]{}, and the agent[’]{}s wealth, $
w_{i}$, is: $$\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr
$\displaystyle{U_{i}=-{{1}\over{2w_{i}}}
pP^{2}+bP{\left| X_{i}\right| }
}$\cr
}}$$ where $$\label{x:structure}\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr
$\displaystyle{X_{i}=\sum _{j=1}^{n}
a_{ij}x_{j}}$\cr
}}$$ is a linear combination of the displacements of all mass points that provides information about the chain[’]{}s state. The parameter [*b*]{} determines the relative importance to an agent of responding to displacements compared to conserving its wealth for future use.
Actuator [*i*]{} always pushes in the opposite direction of $
X_{i}$, i.e., it acts to reduce the value of $
X_{i}$. In this paper we focus on the simple case of purely local control where $
a_{ij}=1$ when $
i=j$ and is 0 otherwise. Thus, consumer [*i*]{} considers only its own displacement $
x_{i}$. For simplicity, we use an ideal competitive market in which each consumer and producer acts as though its individual choice has no affect on the overall price, and agents do not account for the redistribution of profits via the funding policy. Thus a consumer[’]{}s demand function is obtained by maximizing its utility function as a function of power: $$\label{x:demand}\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr
$\displaystyle{{{dU_{i}}\over{dP}}}$\hfilneg&$\displaystyle{{}=
-p{{P}\over{w_{i}}}+b{\left| X_{%
i}\right| }=0\hskip 0.212em {\ifmmode\Rightarrow\else$\Rightarrow$\fi}\hskip 0.212em
P_{i}{\left( p\right) }=b{\left|
X_{i}\right| }{{w_{i}}\over{%
p}}}$\cr
}}$$ This demand function causes the agent to demand more power when the displacement it tries to control is large. It also reflects the trade-off in maintaining wealth: demand decreases with increasing price and when agents have little wealth. The overall demand function for the system is just the sum of these individual demands, giving $$\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr
$\displaystyle{P^{{\rm demand}}{\left( p\right) }
={{b}\over{p}}\sum {\left|
X_{i}\right| }w_{i}}$\cr
}}$$
Similarly, each producer tries to maximize its profit $
\rho $ given by the difference between its revenue from selling power and its production cost $
C{\left( P\right) }$: $
\rho =pP-C{\left( P\right) }$. To provide a constraint on the system to minimize the power use, we select a cost function for which the cost per unit of power, $
C{\left( P\right) }/P$ increases with the amount of power. A simple example of such a cost function is $$\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr
$\displaystyle{C{\left( P\right) }={{1}\over{2
a}}P^{2}}$\cr
}}$$ The parameter [*a*]{} reflects the relative importance of conserving power and maintaining stability. We obtain the producer[’]{}s supply function by maximizing its profit: $$\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr
$\displaystyle{{{d\rho }\over{dP}}=p-{{dC}\over{%
dP}}=0\hskip 0.212em {\ifmmode\Rightarrow\else$\Rightarrow$\fi}\hskip 0.212em P{\left(
p\right) }=ap}$\cr
}}$$ This is the same for all producers, so the overall supply function is then just $$\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr
$\displaystyle{P^{{\rm supply}}{\left( p\right) }
=nap}$\cr
}}$$
From this the price and amount of traded power is determined by the point where the overall supply and demand curves intersect, i.e., $
P^{{\rm demand}}{\left( p\right) }
=P^{{\rm supply}}{\left( p\right) }
$. For our choices of the utility and cost functions, this condition can be solved analytically to give $$\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr
$\displaystyle{p_{trade}=\Rad{{{b}\over{na}}
\sum _{i=1}^{n}{\left| X_{%
i}\right| }w_{i}}\DoRad }$\cr
}}$$ Given this equilibrium price, agent [*i*]{} then gets an amount of power equal to $
P_{i}{\left( p_{trade}\right) }
$ according to Eq. (\[x:demand\]) and the resulting control force is directly proportional to received power.
We can also consider the case where the amount of power available to the system is limited to $
P^{{\rm global}}_{{\rm max}}$. This hard constraint has the effect of limiting the overall supply function when the price is high so it becomes $$\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr
$\displaystyle{P^{{\rm supply}}{\left( p\right) }
=
\left\{\matrix{nap&{\rm if}\hskip 0.212em p{\ifmmode<\else$<$\fi}P^{%
{\rm global}}_{{\rm max}}/na\cr
P^{{\rm global}}_{{\rm max}}&
{\rm otherwise}\cr }\right.}$\cr
}}$$
The final aspect of the market dynamics is how the wealth changes with time. This is given by $$\vcenter{\halign{\strut\hfil#\hfil&#\hfil\cr
$\displaystyle{{{dw_{i}}\over{dt}}}$\hfilneg&$\displaystyle{{}=
-pP_{i}{\left( p\right) }+{{1
}\over{n}}pP^{{\rm demand}}{\left(
p\right) }}$\cr
$\displaystyle{}$\hfilneg&$\displaystyle{{}=-b{\left| X_{i}\right| }
w_{i}+{{b}\over{n}}\sum _{%
j=1}^{n}{\left| X_{j}\right| }
w_{j}}$\cr
}}$$ because we use the funding policy that all expenditures are returned equally to the agents in the system.
Comparing with Local Controls
==============================
As a simple comparison for the market behavior, we also study a local control method. In this case, each actuator [*i*]{} pushes with a strength that is proportional to the displacement of its respective mass point, and ignores the displacements of all other mass points. Specifically, the local controls are simply given by $
H_{i}=-c\hskip 0.212em x_{i}$. Other control strategies [@hogg96b] can try to estimate the amplitude of the lowest modes and push only against these modes, since these are the ones most important for stability.
For comparison with the market, we restrict ourselves to the case where the amount of available power is limited. This is useful for evaluating the ability of different control methods to maintain stability using only weak forces. We distinguish two ways the power could be limited for the local control. In the first, each actuator is separately limited to use no more than $
P_{{\rm max}}$ power (local control 1), which corresponds to a situation where each actuator has a separate power source such as its own battery. Any actuator that requests more power than this maximum has its control force reduced to require only $
P_{{\rm max}}$, i.e., $
{\left| H_{i}\right| }=P_{{\rm max}
}$. The second local control allows available power to be moved among the different actuators and is limited only in that all actuators together cannot use more than $
P^{{\rm global}}_{{\rm max}}=nP_{{\rm
max}}$ power (local control 2). This overall limit is implemented by comparing the total power requested according to the local control, i.e., $
P_{{\rm request}}=c\sum _{i}
{\left| x_{i}\right| }$ to the maximum available. If the requested amount exceeds the maximum, each agent has its power reduced by the factor $
P^{{\rm global}}_{{\rm max}}/P_{%
{\rm request}}$ so that the overall used power equals the global limit. The corresponding market has a total available power of $
P^{{\rm global}}_{{\rm max}}=nP_{{\rm
max}}$.
Results
========
We studied a chain composed of 27 mass points, all of them having unit mass and connected by springs with a spring constant of value 1 and damping coefficient 0.1. The destabilizing force coefficient is 0.2, which is sufficient to make the system unstable when there is no control force. All agents start with an initial wealth of 50 money units and we are using the values $
a=0.05$ and $
b=0.001$ in the cost and utility functions. For definiteness, we chose an initial condition where the single element in the middle of the chain had a unit displacement and all other values were at zero. This configuration includes a contribution from all the modes of the system, which are just sinusoidal waves in this chain with uniform masses and spring constants [@goldstein80]. For the local control, we used $
c=0.2$, which is more than sufficient to ensure stable control when power is unlimited [@hogg96b].
With $
P_{{\rm max}}=0.012$, Fig. \[x:limit\] compares the performances of both local and market controls. We show both the total power use $
\sum _{i}P_{i}$ and the average displacement of the chain $
\sum _{i}{\left| x_{i}\right| }
/n$. As can be seen, for the chosen parameter values the market is able to control the unstable chain in spite of the fact that the power is limited to a global maximum. This limit is reached several times. The local controls (1 and 2), on the other hand, fail in both cases, as seen in the figure. These results were obtained in a simulation run that lasted 20 time units. A longer simulation shows that the overall power usage and average displacements decrease with time for the market control while displacement continues increasing for the local controls.
Since the power cost function $
C{\left( P\right) }$ does not change, the overall supply curve never changes, as shown in Fig. \[x:curves\] which displays the supply curve and some demand curves for different times. The demand curves depend on the displacements and wealth of the agents. Since these are dynamical variables, overall demand curve changes in time. In addition to the times I, II and III marked in Fig. \[x:limit\]a, we also plot the overall demand curves for later times IV, V and VI. This shows that the amount of traded power decreases with time while the unstable chain is controlled by the market.
To demonstrate how robust the market mechanism is, we show in Fig. \[x:failure\] the system[’]{}s response when an actuator breaks down. In this case we slightly increase the amount of power available compared to the simulation used for Fig. \[x:limit\] to $
P_{{\rm max}}=0.015$, so that the local controller 2 can also control the unbroken system. With the system initially functioning properly, we turned off the actuator in the middle of the chain after 10 time units and observed the consequent evolution. As can be seen, the market is still able to control the system whereas the local control fails to do so.
Discussion
===========
In this paper we presented a novel mechanism of controlling unstable dynamical systems by means of a multiagent system using a market mechanism. We described how we defined consumers[’]{} and producers[’]{} utility functions that lead to the overall supply and demand curves and evaluated the price and amount of traded power within the system.
We showed that the market approach is able to control an unstable dynamical system in the case of limited power whereas a traditional local control strategy fails under the same assumptions. We also demonstrated that a market control adapts better to cases when an actuator breaks during the controlling process. These results show that a market control can be more robust than a local control when operating with given power constraints by focussing the power in those parts of the system where it is most needed. This not only reduces total power use but, more importantly, also allows control with weaker, and thus easier to fabricate, actuators.
The power of market approaches to control lies in the fact that relatively little knowledge of the system to be controlled is needed. This is in stark contrast to traditional AI approaches, which use symbolic reasoning with extremely detailed models of the physical system. However, while providing a very robust and simple design methodology, a market approach suffers from the lack of a high level explanation for its global behavior. An interesting open issue is to combine this approach with the more traditional AI one.
Although we have chosen particular forms of utility, supply and demand functions, there are many other functional forms that can also control the system. These could include additional goals, such as faster recovery from sudden changes and minimizing the number of active actuators. Furthermore, different funding strategies are possible, where profits are shared unequally among agents or the funds are allocated by an external agent. A very promising approach is the possibility of improving the performance of the system by having different market organizations that change in time. In our system, this corresponds to the agents learning to use information on the displacements or velocities of their neighbors when making their control decisions. In this way the multiagent system would take advantage of the fact that markets are a simple and powerful discovery process: new methods for selecting trades can be tried by a few consumers or producers and, if successful relative to existing approaches, gradually spread to other agents. Such a learning mechanism could help the system discover those organizational structures that lead to improved performance and adaptability.
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[^1]: We used a standard ordinary-differential-equation solver [@shampine75] to determine the controlled system[’]{}s behaviors.
|
---
author:
- |
Helmut H. W. Hofer\
Institute for Advanced Study
title: Polyfolds and Fredholm Theory
---
Introduction {#SEC1}
============
In this paper we discuss the generalized Fredholm theory in polyfolds. The initial version of the paper, [@H2008], was written in 2008 on the occasion of a lecture given at the Clay Mathematical Institute (CMI), and described the theory as developed in [@HWZ1; @HWZ2; @HWZ3]. When Nick Woodhouse became the CMI president he found a folder of manuscripts from past CMI events and came up with the idea of having them published. Six years is a long time. Since then the theory considerably advanced and a comprehensive discussion of its more abstract part is contained in [@HWZ7; @HWZ8]. Currently nontrivial applications are being developed, most notably those to symplectic field theory (SFT), see [@FHWZ] and [@HWZ6]. In order to account for the developments, the paper was to a large extend rewritten. Rather than dealing with the general theory, which also allows for boundary with corners, we restrict ourselves to a special case and illustrate it with a discussion of stable maps, a topic closely related to Gromov-Witten theory. We also would like to mention the paper [@FFW], where the ideas of polyfold theory are explained as well. The abstract theory has been applied in [@HWZ6] as part of the general construction of SFT. In [@SZ] it was used to address the Weinstein conjecture in higher dimensions. An extensive study of the case with boundary and corners is contained in [@HWZ7; @HWZ8]. These extensions are crucial for an application to SFT. A basic paper in this direction is the upcoming [@FHWZ] in which the polyfolds relevant for SFT are being constructed. This is the place where the full power of the theory becomes apparent.
We shall start by introducing the category of stable maps. The construction of Gromov-Witten invariants can be understood as the geometric study of perturbations of the full subcategory of $J$-holomorphic stable maps. We take a more general viewpoint and study the whole category. Our initial discussion is entirely topological, and surprisingly the arising objects are so natural that one is forced to raise the question if these structures go beyond topology. Indeed they do, and they give rise to a generalized differential geometry as well as a generalized nonlinear Fredholm theory accompanied by its own blend of nonlinear analysis. The whole package is referred to as polyfold theory. The polyfold theory provides a language and a large body of results to address questions arising in the study of moduli problems in symplectic geometry. It is clear that its applicability goes far beyond the latter field. From a nonlinear analysis perspective the symplectic applications are concerned with the study of isomorphism classes of families of nonlinear first order elliptic equations with varying domains and varying targets. The domains even are allowed to change the topology, and bubbling-off phenomena will in general occur. This is the source for compactness and transversality issues which make an algebraic counting of solutions very difficult. In the case of the problems arising in symplectic geometry the polyfold theory overcomes these difficulties. There is no doubt that the theory should have applications in other parts of nonlinear analysis as well.
It is a basic observation in symplectic geometry/topology that geometric questions can be rephrased as questions about solution spaces of nonlinear first order elliptic partial differential equation (the solution spaces are by definition the moduli spaces). Very often just counting solutions suffices to answer a seemingly difficult geometric question. The construction of the moduli problems are difficult analytical problems involving analytical limiting behaviors, like bubbling-off, breaking of trajectories and stretching the neck. The naive solution sets are very often not compact and not being cut out by the differential equation in a generic way. In most applications there exist intriguing compactifications of the solution spaces, which, however, are usually not compatible with the standard versions of smooth nonlinear analysis based on the notion of Fréchet differentiability. Moreover, as a consequence of local symmetries, very often even a generic choice of geometric auxiliary data used to construct the partial differential equation, will never result in a generic solution set. The polyfold theory allows to view the partial differential equation within an abstract framework, which provides a Sard-Smale perturbation theory and deals with the transversality issues. The framework is so general that it also encompasses the geometric perturbation theory. Hence one can proceed geometrically as long as it is possible, and use the abstract perturbations only if the problems cannot be dealt with geometrically. In particular, whatever has been established to work classically, will also work in this extended framework.
The analytical limiting phenomena, which we mentioned above, even assuming a sufficient amount of genericity, do not look like smooth phenomena if smoothness refers to the usual concept. Indeed, the coordinate changes are from a classical perspective usually nowhere differentiable. However, it turns out that the notion of smoothness can be relaxed, and a generalization of differential geometry and nonlinear functional analysis can be developed, for which the limiting phenomena can be viewed as smooth phenomena, even if they are quite often obscured by transversality issues. In this generalized context the classical nonlinear Fredholm theory can be extended to a much larger class of spaces and operators, which can deal with the aforementioned problems.
A great example to explain the theory comes from a study of the category of stable maps associated to a symplectic manifold. The study of these objects goes back to the seminal work by Gromov on pseudoholomorphic curve theory, see [@G]. Gromov showed that the study of pseudoholomorphic maps is a powerful tool in symplectic geometry, by demonstrating its uses by many examples. Kontsevich, later pointed out the importance of the notion of stable pseudoholomorphic curves, [@Ko]. Our stable maps need not to be pseudoholomorphic. We find several natural constructions which at first glance are just of topological nature without exhibiting more regularity. However, as it turns out, these are the shadows of smooth constructions, once the notion of differentiability in finite dimensions, which usually is generalized as Fréchet differentiability to infinite-dimensional Banach spaces, is generalized in a quite different way. Such a generalization requires an additional piece of structure, called an sc-structure, which occurs in interpolation theory, [@Tr], albeit under a different name. In fact, we give a quite different interpretation of such a structure and make clear, that it can be viewed as a generalization of a smooth structure on a Banach space. We call this generalization sc-smoothness, and the generalization of differentiability of a map we refer to as sc-differentiability.
The very interesting aspect is then the following fact. There are many sc-differentiable maps $r:U\rightarrow U$ satisfying $r\circ r=r$, i.e. sc-smooth retractions. For Fréchet differentiability the image of such a retraction can be shown to be a submanifold of $U$. Incidentally this is H. Cartan’s last mathematical theorem, see [@H.Cartan]. However, the images of sc-smooth retractions can be much more general. Most strikingly, they can have locally varying dimensions. Of course, a good notion of differentiability comes with the chain rule so that from $r\circ r=r$ we deduce $Tr=(Tr)\circ (Tr)$. In other words, the tangent map of an sc-smooth retraction is again an sc-smooth retraction. If a subset $O$ of an sc-Banach space is the image of an sc-smooth retraction $r$, then $TO=Tr(TU)$ defines the tangent space, and it turns out, that the definition does not depend on the choice of $r$ as long as $r$ is sc-smooth and has $O$ as its image. So we obtain quite general subsets of Banach spaces which have tangent spaces. An sc-smooth map $f:O\rightarrow O'$, where $O\subset E$ and $O'\subset F$ are sc-smooth retracts, is a map such that $f\circ r:U\rightarrow F$ is sc-smooth, where $r$ is an sc-smooth retraction onto $O$. As it turns out the definition does not depend on the choice of $r$. Further, one verifies that $Tf:=T(f\circ r)|TO$ defines a map $TO\rightarrow TO'$ between tangent spaces and that the definition also does not depend on the choice of $r$.
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In summary, once we have a good notion of differentiability for maps between open sets, we also obtain a notion of differentiability for maps between sc-smooth retracts. However, for the usual notion of differentiability, smooth retracts are manifolds and one does not obtain anything beyond the usual differential geometry and its standard generalization to Banach manifolds. On the other hand sc-differentiability opens up new possibilities with serious applications. We generalize differential geometry by generalizing the notion of a manifold to that of an M-polyfold. These are metrizable spaces which are locally homeomorphic to retracts with sc-smooth transition maps. The theory as described in this paper even gives new objects in finite dimensions, see Figure \[porkbarrel\]. Most important for us is the fact, that the new local models for a vastly generalized differential geometry allow us to build spaces, which can be used to bring nonlinear partial differential equations which might show bubbling-off phenomena into an abstract geometric framework which allows for a very general nonlinear Fredholm theory with the usual expected properties.\
[**Acknowledgement:**]{} The author thanks the Clay Institute for the opportunity to present this material at the 2008 Clay Research Conference. Some of the initial work was done during a sabbatical at the Stanford mathematics department, and was supported in part by the American Institute of Mathematics. Thanks to P. Albers, Y. Eliashberg, J. Fish, E. Ionel, K. Wysocki and E. Zehnder for many stimulating discussions.
A Motivating Example and Natural Structures {#SEC22}
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The following discussion starts with a closer look at the category of stable Riemann surfaces and describes some of the interesting structures appearing as a result of the classical Deligne-Mumford theory. These structures and the associated viewpoint can be related to a topic arising in Gromov-Witten theory, namely the study of the category of stable maps which are not necessarily pseudoholomorphic. This category of stable maps has natural structures. Natural structures in mathematics are usually there for a good reason. In our case a deeper analysis reveals the existence of more general smooth models for an extended differential geometry as well as a vastly generalized Fredholm theory. The latter can be used to define the Gromov-Witten invariants. The same scheme also works suitably extended in the much more complicated framework of stable maps in SFT.
The Category of Stable Noded Riemann Surfaces
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A good starting point for explaining some of the later features in the polyfold theory is a study of stable noded Riemann surfaces with marked points.
A stable, possibly noded Riemann surface with marked points, is a tuple $(S,j,M,D)$, where $(S,j)$ is a closed smooth Riemann surface, $M\subset S$ is a finite collection of points, called marked points, and a finite collection $D$ of un-ordered pairs $\{x,y\}$ of points $x,y\in S\setminus M$ having the following properties.
- For $\{x,y\}\in D$ it holds $x\neq y$. If $\{x,y\}\cap \{x',y'\}\neq \emptyset$, then $\{x,y\}=\{x',y'\}$.
- The topological space $\bar{S}$ obtained by identifying $x\equiv y$ for $\{x,y\}\in D$ is connected.
- Define the subset $|D|$ by $|D|=\bigcup_{\{x,y\}\in D} \{x,y\}.$ For every connected component $C$ of $S$ having genus $g(C)$ with $n_C$ being the number of points in $C\cap (|D|\cup M)$ it holds $2g(C)+n_C\geq 3$.
Condition (iii) is called the stability condition. In our case the set $M$ is not ordered, but the following discussion could be carried out in the ordered case as well.
We can view $\alpha=(S,j,M,D)$ as an object in a category ${\mathcal R}$ where the morphisms $\Phi:\alpha\rightarrow \alpha'$ are given by $$\Phi=(\alpha,\phi,\alpha'):\alpha\rightarrow \alpha'$$ and $\phi:(S,j)\rightarrow (S',j')$ is a biholomorphic map satisfying $\phi(M)=M'$ and $\phi(D)=D'$, where $\phi(D):=\{\{\phi(x),\phi(y)\}\ |\ \{x,y\}\in D\}$.
We call ${\mathcal R}$ the category of stable, (possibly) noded Riemann surfaces with un-ordered marked points.
The following result is well known, and is part of the Deligne-Mumford theory. The theory has been described from a more differential geometric perspective in [@RS], and is being described in [@HWZ-DM] from a polyfold perspective.
\[THH1\] The category ${\mathcal R}$ has the following properties.
- Every morphism in the category ${\mathcal R}$ is an isomorphism.
- The stability condition implies that between any two objects there are at most finitely many morphisms.
- The orbit space $|{\mathcal R}|$ of the category ${\mathcal R}$, which is the set of isomorphism classes of objects in ${\mathcal R}$ carries a natural metrizable topology.
- The topological space $|{\mathcal R}|$ carries in a natural way the structure of a holomorphic orbifold for which each connected component is compact.
A basis for the topology is given in Proposition \[propp1\]. We shall need a variation of this result, and shall describe a particular approach to ${\mathcal R}$ which one might view as a toy case for the polyfold approach to stable maps in symplectic manifolds. We describe some of the results we shall need later on, and refer the reader for more details to [@HWZ6; @HWZ-DM], or the references mentioned therein.
Given an object $\alpha=(S,j,M,D)$ in ${\mathcal R}$ it is not just a point in the category, but it has geometry as well. In fact, the additional structures on the category come from the fact that every object has its own geometry, which is being exploited by constructing associated objects by a plumbing procedure. Recall that the automorphism group $G$ of $\alpha$ is the finite collection of all morphisms (which are all isomorphisms) $\Phi:\alpha\rightarrow \alpha$.
Let $\alpha=(S,j,M,D)$ be an object in ${\mathcal R}$ with automorphism group $G$. A small disk structure associated to $\alpha$ and denoted by ${\bf D}$ assigns to every $x\in |D|$ a compact disk-like neighborhood $D_x\subset S$ around the point $x$, so that the following holds.
- $D_x$ has a smooth boundary and the disks are mutually disjoint.
- $\bigcup_{x\in |D|} D_x$ is invariant under $G$.
- $D_x\cap M=\emptyset$ for all $x\in |D|$.
Given a small disk structure ${\bf D}$ we have for every $\{x,y\}\in D$ an associated noded disk pair $(D_x\cup D_y,\{x,y\})$. From this data we can construct for every $\{x,y\}$ a natural gluing parameter. This is done as follows. Denote for $x\in |D|$ by ${\widehat}{x}$ an oriented real line in $(T_xS,j)$. If ${\mathbb S}^1\subset {\mathbb C}$ denotes the unit circle, we observe that given $\theta\in {\mathbb S}^1$, we can define a new oriented real line $\theta{\widehat}{x}$, making use of the complex structure $j$. We introduce an equivalence class of unordered pairs of real oriented lines $\{{\widehat}{x},{\widehat}{y}\}$ associated to $\{x,y\}\in D$ as follows. We say $\{{\widehat}{x},{\widehat}{y}\}$ is equivalent to $\{{\widehat}{x}',{\widehat}{y}'\}$ (here the lines lie over $x$ and $y$) provided there exists $\theta\in {\mathbb S}^1$ satisfying $${\widehat}{x}=\theta{\widehat}{x}'\ \text{and}\ {\widehat}{y}=\theta^{-1}{\widehat}{y}'.$$ Equivalence classes will be written as $[{\widehat}{x},{\widehat}{y}]$. Clearly the collection of all equivalence classes associated to the pair $\{x,y\}$ can be parametrized by the unit circle via $${\mathbb S}^1\ni \theta\rightarrow [{\widehat}{x},\theta{\widehat}{y}].$$
An equivalence class $[{\widehat}{x},{\widehat}{y}]$ is called a decorated nodal pair with underlying nodal pair $\{x,y\}$.
Hence above every $\{x,y\}\in D$ there lies an ${\mathbb S}^1$-worth of decorated nodal pairs $[{\widehat}{x},{\widehat}{y}]$. We shall refer to $[{\widehat}{x},{\widehat}{y}]$ as a decoration for $\{x,y\}$.
Consider a formal expression $r[{\widehat}{x},{\widehat}{y}]$ with $r\in [0,1/2)$. Two such formal expressions are considered the same, i.e. $r[{\widehat}{x},{\widehat}{y}]=r'[{\widehat}{x}',{\widehat}{y}]$ provided one of the following holds:
- $r=r'=0$, or
- $r=r'\neq 0$ and $[{\widehat}{x},{\widehat}{y}]=[{\widehat}{x}',{\widehat}{y}']$.
A natural gluing parameter associated to the nodal pair $\{x,y\}$ is a formal expression $\mathfrak{a}^{\{x,y\}}=r[{\widehat}{x},{\widehat}{y}]$ where $r\in [0,1/2)$, with the notion of equality as defined above. In case $r=0$ we shall write $\mathfrak{a}^{\{x,y\}}=0$.
We define the set ${\mathbb B}^{\{x,y\}}(1/2)$ by $${\mathbb B}^{\{x,y\}}(1/2)=\{r[{\widehat}{x},{\widehat}{y}]\ |\ r\in [0,1/2),\ [{\widehat}{x},{\widehat}{y}]\ \text{a decoration of}\ \{x,y\}\}$$ and call it the set of natural gluing parameters associated to the nodal pair $\{x,y\}$. The modulus of $\mathfrak{a}^{\{x,y\}}=r[{\widehat}{x},{\widehat}{y}]$ is the number $r$ and we shall write $|\mathfrak{a}^{\{x,y\}}|=r$.
Given $\alpha$, a natural (total) gluing parameter is a map $$\mathfrak{a}: D\ni \{x,y\}\rightarrow \mathfrak{a}^{\{x,y\}}\in {\mathbb B}^{\{x,y\}}(1/2).$$ The collection of all gluing parameters associated to $\alpha$ will be denoted by ${\mathbb B}^\alpha$.
The dependency of ${\mathbb B}^\alpha$ on $\alpha$ is as follows. It only depends on the complex multiplication on the tangent spaces $T_xS$ for $x\in |D|$. We note the following result having an easy proof which is left to the reader.
Assume that $\alpha$ is given. For every $\{x,y\}\in D$ the set ${\mathbb B}^{\{x,y\}}(1/2)$ has in a natural way the structure of a holomorphic manifold. It is characterized by the following property. For every choice of oriented real lines ${\widehat}{x}$ and ${\widehat}{y}$ the map $${\mathbb B}^{\{x,y\}}(1/2)\rightarrow \{z\in {\mathbb C}\ |\ |z|<1/2\}:r[{\widehat}{x},\theta{\widehat}{y}]\rightarrow r\theta$$ is biholomorphic. In particular the set ${\mathbb B}^\alpha$, of all natural gluing parameters $\mathfrak{a}:\{x,y\}\rightarrow \mathfrak{a}^{\{x,y\}}$ has a natural holomorphic structure and is biholomorphic to a product of open disks.
We shall use the natural gluing parameters to modify $\alpha$ and produce new stable noded Riemann surfaces. In this context an important concept is that of a gluing profile.
A gluing profile is a smooth diffeomorphism $\varphi:(0,1]\rightarrow [0,\infty)$.
We shall refer to $$\varphi(r)=-\frac{1}{2\pi}\cdot \ln(r)$$ as the logarithmic gluing profile, and to $$\varphi(r)=e^{\frac{1}{r}}-e$$ as the exponential gluing profile. Restricting $r[{\widehat}{x},{\widehat}{y}]$ to the case with $r\in [0,1/2)$ has to do with the choice of these two gluing profiles. For an arbitrary gluing profile $\varphi$ we only have to require that $r\in [0,\varepsilon_\varphi)$, where $\varepsilon_\varphi >0$, depending on the profile, is small enough.
Assume we are given a small disk structure ${\bf D}$ and a gluing profile $\varphi$. We can define a gluing or plumbing construction as follows. For a gluing parameter $\mathfrak{a}^{\{x,y\}}=r[{\widehat}{x},{\widehat}{y}]$ associated to $\{x,y\}\in D$, we consider the disks $D_x$, $D_y$, and the nodal pair $\{x,y\}$. If $\mathfrak{a}^{\{x,y\}}=0$ we keep this data, i.e. with other words the gluing of $(D_x\cup D_y,\{x,y\})$ with $a^{\{x,y\}}=0$ produces the noded disk $(D_x\cup D_y,\{x,y\})$. If $\mathfrak{a}^{\{x,y\}}\neq 0$ we compute the value $R=\varphi(r)$ and take a representative $\{{\widehat}{x},{\widehat}{y}\}$ of the class $[{\widehat}{x},{\widehat}{y}]$. After this choice, there exist uniquely determined biholomorphic maps $$h_x:(D_x,x)\rightarrow ({\mathbb D},0),\ Th_x(x){\widehat}{x}={\mathbb R},$$ and $$h_y:(D_y,y)\rightarrow ({\mathbb D},0),\ Th_y(x){\widehat}{y}={\mathbb R}.$$ Here ${\mathbb D}$ is the closed unit disk in ${\mathbb C}$, and ${\mathbb R}\subset T_0{\mathbb D}$ is oriented by $1$. Recall that the modulus of the annulus $A_{r_1,r_2}:=\{z\in {\mathbb C}\ |\ r_1\leq |z|\leq r_2\}$ is defined by $$\text{modulus}(A_{r_1,r_2}) = \frac{1}{2\pi}\cdot \ln(r_2/r_1).$$ If we are given a Riemann surface $(A,j)$ where $A$ is compact, has smooth boundary components, and is diffeomorphic to an annulus, we know from the uniformization theorem that $(A,j)$ is biholomorphic to $A_{r,1}$ for a uniquely determined $0<r<1$. The modulus of $(A,j)$ by definition is given by $$\text{modulus}(A,j)=-\frac{1}{2\pi}\cdot \ln(r).$$ We note that the right-hand side is precisely $R=\varphi(r)$ for the logarithmic gluing profile. In the next step the choice of the gluing profile matters, and a remark, why we consider other gluing profiles than the logarithmic one, is in order.
In later constructions we shall see that the sc-smooth structure we are going to construct will depend on the choice of $\varphi$. The sc-smooth structure also influences if certain operators are sc-smooth Fredholm operators. The operators we are interested in will be sc-smooth for the exponential gluing profile, but not for the logarithmic one.
With some gluing profile $\varphi$ fixed, we pick concentric compact annuli $A_x\subset D_x$ and $A_y\subset D_y$ with one boundary component being the boundary of $D_x$ and $D_y$, respectively, so that $A_x$ and $A_y$ have modulus $R=\varphi(r)$. We discard the points $D_x\setminus A_x$ and $D_y\setminus A_y$ from $S$ and the nodal pair $\{x,y\}$. With $A_x\cup A_y\subset S\setminus ((D_x\setminus A_x)\cup (D_y\setminus A_y))$ we identify $A_x$ with $A_y$ by identifying $z\equiv z'$, where $(z,z')\in A_x\times A_y$ satisfying $$h_x(z)\cdot h_y(z')= e^{-2\pi\cdot R}.$$ The definition of the identification $z\equiv z'$ is independent of the choice of the representatives $\{{\widehat}{x},{\widehat}{y}\}$ in $[{\widehat}{x},{\widehat}{y}]$. Let us denote by ${\mathcal Z}_{\mathfrak{a}^{\{x,y\}}}$ the glued space obtained by identifying the two annuli. If $\mathfrak{a}^{\{x,y\}}=0$ this is the noded $(D_x\cup D_y,\{x,y\})$ and for $\mathfrak{a}^{\{x,y\}}\neq 0$ it is the space obtained by identifications just described. Clearly, in the latter case we have natural biholomorphic maps $$A_x\rightarrow {\mathcal Z}_{\mathfrak{a}^{\{x,y\}}}\ \text{and}\ A_y\rightarrow {\mathcal Z}_{\mathfrak{a}^{\{x,y\}}},$$ by associating to a point its equivalence class. Carrying out the above procedure for every noded disk pair $(D_x\cup D_y,\{x,y\})$ defines for a gluing parameter $\mathfrak{a}\in {\mathbb B}^\alpha$ a new noded stable Riemann surface $(S_\mathfrak{a},j_\mathfrak{a},M_\mathfrak{a},D_\mathfrak{a})$ with marked points. We just described how to obtain $S_\mathfrak{a}$. We denote by $j_\mathfrak{a}$ the natural smooth almost complex structure on the latter. Observe that $z\equiv z'$ is a holomorphic identification. By $M_\mathfrak{a}$ we just mean $M$ naturally identified with a subset of $S_\mathfrak{a}$. The collection of nodal points $D_\mathfrak{a}$ is obtained by first removing from $D$ all $\{x,y\}$ with $\mathfrak{a}^{\{x,y\}}\neq 0$ and then identifying this set with the obvious set of nodal pairs for $S_\mathfrak{a}$. For every $\{x,y\}\in D_\mathfrak{a}$ we still have $D_x\cup D_y\subset S_\mathfrak{a}$.
At this point we can exhibit a basis of open neighborhoods for a given point $|\alpha|\in |{\mathcal R}|$ for the natural metrizable topology. For a representative $\alpha$ and an associated gluing parameter $\mathfrak{a}$ we define $|\mathfrak{a}|$ by $$|\mathfrak{a}|=\text{max}_{\{x,y\}\in D} \left | \mathfrak{a}^{\{x,y\}}\right|.$$ Write $\alpha=(S,j,M,D)$ and fix a small disk structure ${\bf D}$. The set of all smooth almost complex structures $k$ on $S$, which define the same orientation as $j$, has a natural metrizable topology measuring the $C^\infty$-distance. Denote a choice of metric by $\rho$. For a given gluing profile $\varphi$ and $\varepsilon>0$ small enough define $${\mathcal U}(\alpha,{\bf D},\varphi,\varepsilon)=\{|(S_\mathfrak{a},k_\mathfrak{a},M_\mathfrak{a},D_\mathfrak{a})|\ |\ \rho(j,k)<\varepsilon,\ j=k\ \text{on}\ {\bf D},\ |\mathfrak{a}|<\varepsilon\}.$$
\[propp1\] Fix a gluing profile $\varphi$, pick for every class $c\in |{\mathcal R}|$ an object $\alpha_c$, and choose an associated small disk structure ${\bf D}_c$ for $\alpha_c$. Consider ${\mathcal U}(\alpha_c,{\bf D}_c,\varphi,\varepsilon)$ for $0<\varepsilon<\varepsilon_c$ and denote this collection by $\mathfrak{U}_c$. Then the union $\mathfrak{U}=\bigcup_{c\in |{\mathcal R}|} \mathfrak{U}_c$ is the basis for a metrizable topology ${\mathcal T}$ on $|{\mathcal R}|$. The topology ${\mathcal T}$ does not depend on the choices involved in constructing $\mathfrak{U}$.
The topology ${\mathcal T}$ is the one referred to in Theorem \[THH1\]. The automorphism group $G$ acts on the set of gluing parameters ${\mathbb B}^\alpha$ via biholomorphic maps $$G\times {\mathbb B}^\alpha\rightarrow {\mathbb B}^\alpha: (g,\mathfrak{a})\rightarrow g\ast\mathfrak{a},$$ in a natural way as follows. Namely $\mathfrak{b}=g\ast \mathfrak{a}$ is defined by $$\mathfrak{b}^{\{g(x),g(y)\}} = r\cdot [Tg(x){\widehat}{x},Tg(y){\widehat}{y}],$$ where $r[{\widehat}{x},{\widehat}{y}]=\mathfrak{a}^{\{x,y\}}$. With the above construction an element $g\in G$ induces a biholomorphic map $$g_\mathfrak{a}:(S_\mathfrak{a},j_\mathfrak{a},M_\mathfrak{a},D_\mathfrak{a})\rightarrow (S_\mathfrak{b},j_\mathfrak{b},M_\mathfrak{b},D_\mathfrak{b}),$$ where $\mathfrak{b}=g\ast \mathfrak{a}$. It also holds $$h_{g\ast\mathfrak{a}}\circ g_\mathfrak{a}= (h\circ g)_\mathfrak{a}.$$ Let us abbreviate $(S_\mathfrak{a},j_\mathfrak{a},M_\mathfrak{a},D_\mathfrak{a})$ by $\alpha_{\mathfrak{a}}$. Then $\alpha_\mathfrak{a}=\alpha_{{\mathfrak{a}'}}$ if and only if $\mathfrak{a}=\mathfrak{a}'$. Consider for these objects $\alpha_\mathfrak{a}$ with $|\mathfrak{a}|<1/2$ the associated full subcategory ${\mathcal A}={\mathcal A}_{\alpha,{\bf D},\varphi}$ of ${\mathcal R}$. This subcategory is small. Given the object $\alpha$ with automorphism group $G$, every choice of a small disk structure (a set of possible choices) produces such a subcategory. We also consider the translation groupoid $G\ltimes {\mathbb B}^{\alpha}$. The latter is a small category with object set ${\mathbb B}^\alpha$ and morphism set $G\times {\mathbb B}^{\alpha}$, where $(g,\mathfrak{a})$ is seen as a morphism $\mathfrak{a}\rightarrow g\ast\mathfrak{a}$. Observe that object set and morphism set have in our case both smooth manifold structures. There exists the obvious functor $$\gamma:G\ltimes {\mathbb B}^{\alpha}\rightarrow {\mathcal A}$$ which maps the object $\mathfrak{a}$ to $\alpha_\mathfrak{a}$ and the morphism $(g,\mathfrak{a}):\mathfrak{a}\rightarrow g\ast \mathfrak{a}$ to $$(\alpha_\mathfrak{a},g_\mathfrak{a},\alpha_{g\ast\mathfrak{a}}):\alpha_\mathfrak{a}\rightarrow \alpha_{g\ast\mathfrak{a}}.$$ We note that $\gamma$ is injective on objects. More is true, namely from the results of the Deligne-Mumford theory we obtain the following proposition which is true for any gluing profile. It has to be viewed as an intermediate result neglecting for the moment the possibility of deforming the (integrable) almost complex structure $j$ associated to $\alpha$.
For a suitable $G$-invariant open neighborhood $U$ of $0\in {\mathbb B}^\alpha$ depending on the choice of the gluing profile, the functor $$\gamma: G\ltimes U\rightarrow {\mathcal R}$$ is injective on objects and fully faithful.
The above proposition describes the family we can obtain just from gluing (or plumbing) without considering deformations of the (integrable almost) complex structure. We can also obtain from $\alpha$ new objects by changing the complex structure. Of course, a priori, we only would like to consider only those deformations which produce non isomorphic objects. There is, however, an inherent difficulty coming from the automorphisms. Given the object $\alpha$ in the category ${\mathcal R}$ consider the complex vector space $\Gamma_0(\alpha)$ consisting of smooth sections of $TS\rightarrow S$, which vanish at the points in $M\cup |D|$. By $\Omega^{0,1}(\alpha)\rightarrow S$ we denote the complex vector space of smooth sections of $\text{Hom}_{\mathbb R}(TS,TS)\rightarrow S$ which are complex anti-linear. The Cauchy-Riemann operator defines a linear operator $$\bar{\partial}:\Gamma_0(\alpha)\rightarrow \Omega^{0,1}(\alpha).$$ Define via the formula $$g_a:=1+\sharp D+\sum_C [g(C)-1],$$ where the sum is taken over all connected components $C$ of $S$, the arithmetic genus $g_a$. The basic fact is given by the following proposition.
The Cauchy-Riemann operator is a complex linear differential operator. The stability property of $\alpha$ implies that $\bar{\partial}$ is injective and as a consequence of Riemann-Roch the cokernel has complex dimension $3g_a+\sharp M -\sharp D-3$.
In the next step we consider the deformation space of $\alpha$ for fixed combinatorial type. This means we consider the deformation of the complex structure, the marked points, and the nodal pairs, but do not remove nodes. The natural deformation space introduced in the following is the infinitesimal version of this, if we divide out by isomorphisms, but keep track of the automorphisms.
The (infinitesimal) natural deformation space (of fixed combinatorial type) of $\alpha=(S,j,M,D)$ is by definition the complex vector space $H^1(\alpha)$ defined by $$H^1(\alpha)=\Omega^{0,1}/(\bar{\partial}\Gamma_0(\alpha)).$$
The construction $\alpha\rightarrow H^1(\alpha)$ is functorial. Namely we associate to an object $\alpha$ a finite-dimensional complex vector space, and to a morphism $\phi:\alpha\rightarrow \alpha'$ a complex vector space isomorphism $$H^1(\phi):H^1(\alpha)\rightarrow H^1(\alpha'):[\tau]\rightarrow [\phi\circ \tau \circ\phi^{-1}].$$ In particular the automorphism group $G$ of the object $\alpha$ has an action on $H^1(\alpha)$.
Assume that $V$ is an open subset of some vector space $E$ and $V\in v\rightarrow j(v)$ a smooth family of almost complex structures on $S$. Then it holds $j(v)^2=-Id$, and differentiating at $v$ into the direction $\delta v\in E$ gives the identity $$j(v)(Dj(v)\delta v)=-(Dj(v)\delta v)j(v).$$ This means that for every $z\in S$ the map $$(Dj(v)\delta v)_z:(T_zS,j(v))\rightarrow (T_zS,j(v))$$ is complex anti-linear and induces an element $[Dj(v)\delta v]\in H^1(S,j(v),M,D)$. Hence we obtain for every $v\in V$ a linear map $$[Dj(v)]:E\rightarrow H^1(S,j(v),M,D).$$ This map is called the Kodaira-Spencer differential associated to $v\rightarrow j(v)$. We define $\alpha_v$ by $$\alpha_v=(S,j(v),M,D).$$
Given a stable $\alpha$ and an associated small disk structure ${\bf D}$ we shall call a smooth family $v\rightarrow j(v)$ of almost complex structures on $S$ defined on a $G$-invariant open neighborhood $V$ of $0$ in $H^1(\alpha)$ a good deformation compatible with ${\bf D}$ provided it has the following properties.
- $j(0)=j$.
- $j(v)=j$ on all disks $D_x$ associated to ${\bf D}$.
- For every $v \in V$ the Kodaira-Spencer differential $$[Dj(v)]:H^1(\alpha)\rightarrow H^1(\alpha_v)$$ is a complex linear isomorphism.
- For every $g\in G$ and $v\in V$ the map $g:\alpha_v\rightarrow \alpha_{g\ast v}$ is biholomorphic.
A well-known result with a proof given in [@HWZ-DM] is the following.
Given a stable $\alpha$ in ${\mathcal R}$ with automorphism group $G$, and a small disk structure ${\bf D}$, there always exist good deformations compatible with ${\bf D}$.
Suppose we start with a stable $\alpha$ with automorphism group $G$ and have fixed a small disk structure ${\bf D}$. Then we can, as we have seen, construct the functor $$\gamma:G\ltimes {\mathbb B}^\alpha\rightarrow {\mathcal R}.$$ Now taking a good deformation as described in the previous definition, say $V\ni v\rightarrow j(v)$, we can construct the following functor, where we take either the logarithmic or exponential gluing profile. $$\Psi: G\ltimes (V\times {\mathbb B}^\alpha)\rightarrow {\mathcal R}.$$ To objects $(v,\mathfrak{a})$ we assign $$\alpha_{(v,\mathfrak{a})}=(S_\mathfrak{a},j(v)_\mathfrak{a},M_\mathfrak{a},D_\mathfrak{a}).$$ The morphism $(g,(v,\mathfrak{a})):(v,\mathfrak{a})\rightarrow (g\ast v,g\ast\mathfrak{a})$ is mapped to $$(\alpha_{(v,\mathfrak{a})},g_\mathfrak{a}, \alpha_{(g\ast v,g\ast\mathfrak{a})}):\alpha_{(v,\mathfrak{a})}\rightarrow \alpha_{(g\ast v,g\ast \mathfrak{a})}.$$ The basic result is the following. A proof is given in [@HWZ-DM].
\[thmm1\] Given the stable object $\alpha$ in ${\mathcal R}$ with automorphism group $G$, a small disk structure ${\bf D}$, and a good deformation $v\rightarrow j(v)$ compatible with ${\bf D}$, there exists an open $G$-invariant neighborhood $O$ of $(0,0)\in V\times {\mathbb B}^\alpha$, so that the following holds for a given gluing profile.
- $\Psi:G\times O\rightarrow {\mathcal R}$ is a fully faithfull functor.
- The map $|\Psi|:{_{G}\backslash} O \rightarrow |{\mathcal R}|$ induced on orbit spaces defines a homeomorphism onto an open neighborhood $U$ of $|\alpha|$.
- For every $(v,\mathfrak{a})$ the partial Kodaira-Spencer differentiable associated to $\alpha_{(v,\mathfrak{a})}$ is an isomorphism.
- For every point $q\in O$ there exists an open neighborhood $U(q)\subset O$ with the property that every sequence $(q_k)\subset U(q)$ for which $|\Psi(q_k)|$ converges in $|{\mathcal R}|$ there exists a subsequence of $(q_k)$ converging in ${\operatorname{cl}}_O(U(q))$.
For the definition of at the partial Kodaira-Spencer differentiable see [@HWZ6] or [@HWZ-DM]. The above construction is possible for every given gluing profile $\varphi$.
\[Basic\_def\] Assume a gluing profile $\varphi$ is fixed. A functor $\Psi:G\ltimes O\rightarrow {\mathcal R}$ constructed as described previously, associated to a stable $\alpha$ and a small disk structure ${\bf D}$, and satisfying the properties (i)–(iv) of Theorem \[thmm1\] is called a good uniformizer associated to $\alpha$.
For the following discussion we assume that a gluing profile $\varphi$ has been fixed. Let us observe that the construction of $\Psi:G\ltimes O\rightarrow {\mathcal R}$ is associated to a given object $\alpha$ in ${\mathcal R}$ and $$\Psi(0,0)=\alpha\ \text{and}\ \Psi((g,(0,0))=(\alpha,g,\alpha):\alpha\rightarrow \alpha.$$ The constructions requires the choice of a small disk structure ${\bf D}$, the good deformation $v\rightarrow j(v)$, and the open $G$-invariant subset $O$ of $V\times {\mathbb B}^\alpha$. Here ${\mathbb B}^\alpha$ is a well-defined set associated to $\alpha$ and $V$ is an open subset of the complex vector space $H^1(\alpha)$ associated to $\alpha$. All in all, our possible choices involved in constructing $\Psi$ for a fixed $\alpha$ constitute a set. Therefore we can make the following definition.
By $F(\alpha)$ we denote the set of good uniformizers associated to $\alpha$.
We shall elaborate next about the functorial properties of $F$.
By ${\mathcal R}^-$ we denote the category which has the same objects as ${\mathcal R}$, but as morphisms only the identities.
We can view the assignment $$\alpha\rightarrow F(\alpha).$$ as a functor defined on ${\mathcal R}^-$ with values in the category of sets $\text{SET}$. This is the ‘minimalistic version’ of our construction giving us the functor $$F:{\mathcal R}^-\rightarrow \text{SET}: \alpha\rightarrow F(\alpha)\ \text{(on objects)},$$ which on morphisms $${F}(Id_\alpha) = Id_{F(\alpha)}.$$ We said that this is the ‘minimalistic version’ since there is, in fact, much more structure. Let us explore this now.
Recall that for $\alpha=(S,j,M,D)$ an element $\Psi$ in the set $F(\alpha)$ is constructed by making the following choices.
- A small disk structure ${\bf D}$ for $(S,j,M,D)$.
- A deformation of $j$ with specific properties $v\rightarrow j(v)$ on $(S,M,D)$, i.e. which is constant on ${\bf D}$, has some symmetry properties, and is defined on a $G$-invariant open neighborhood $V$ of $0$ in $H^1(\alpha)$.
- From this data we obtain the natural gluing parameters ${\mathbb B}^\alpha$ and can construct the family $$\Psi: (v,\mathfrak{a})\rightarrow (S_\mathfrak{a},j(v)_\mathfrak{a},M_\mathfrak{a},D_\mathfrak{a}).$$
- A $G$-invariant open neighborhood $O$ of $(0,0)$ in $V\times B^{\alpha}$ so that the restriction of the above family satisfies a list of properties defining an element in $F(\alpha)$, see Definition \[Basic\_def\].
Hence the collection $F(\alpha)$ is obtained via a precise instruction which requires choices in its construction. When we consider isomorphic objects in ${\mathcal R}$, say $\alpha$ and $\alpha'$, there is a natural way of matching a specific choice for $\alpha$ with a specific choice for $\alpha'$. This matching depends on the choice of an isomorphism and has functorial properties, so that one can upgrade $F:{\mathcal R}^-\rightarrow \text{SET}$ to a functor $F:{\mathcal R}\rightarrow \text{SET}$. More precisely, assume that $$\Phi:=(\alpha,\phi,\alpha'):\alpha\rightarrow \alpha'$$ is a morphism. Then we define the following data for $\alpha'$.
- ${\bf D}':=\phi_\ast({\bf D}):=\{\phi(D_z)\ |\ z\in |D|\}$.
- The biholomorphic map $\phi$ defines a linear isomorphism $\phi_\ast :H^1(\alpha)\rightarrow H^1(\alpha')$ and we take $V'=\phi_\ast (V)$, and a biholomorphic map $\phi_\ast: {\mathbb B}^\alpha\rightarrow {\mathbb B}^{\alpha'}$.
- For $j'$ we take the map $V'\ni v'\rightarrow j'(v'):=T\phi\circ j(\phi_\ast^{-1} v')\circ T\phi^{-1}.$
- $O'=\phi_\ast(O)$
From the data ${\bf D}'$, $v'\rightarrow j'(v')$ and ${\mathbb B}^{\alpha'}$ we construct the element $\Psi'\in F(\alpha')$ corresponding to $\Psi$. This shows that a morphism $\Phi:=(\alpha,\phi,\alpha'):\alpha\rightarrow \alpha'$ defines a bijection $$F(\Phi):F(\alpha)\rightarrow F(\alpha').$$ Hence we obtain a functor $$F:{\mathcal R}\rightarrow \text{SET}.$$ Just saying that $F$ is functor from ${\mathcal R}$ into $\text{SET}$ is again quite minimalistic, since we suppress the various maps $\phi_\ast$ we constructed above. Hence, underlying this bijection there is in our case even some fine structure, which we shall describe now.
Assume that the automorphism group of $\alpha=(S,j,M,D)$ is $G$. Then the elements of $G$ have the form $(\alpha,g,\alpha)$, where $g:\alpha\rightarrow \alpha$ is a biholomorphic map. For simplicity of notation we identity $g=(\alpha,g,\alpha)$. Assume that we are given an isomorphism $$\Phi=(\alpha,\phi,\alpha'):\alpha\rightarrow \alpha'$$ Having the domains fixed (most importantly the almost complex structures) we identify $\phi=(\alpha,\phi,\alpha')$. Denote by $G'$ the automorphism group of $\alpha'$, and proceeding as in the case of $G$, the morphism $\Phi$ determines a group isomorphism $$\gamma_\Phi:G\rightarrow G':g\rightarrow \phi\circ g\circ \phi^{-1}.$$ Consider the, by the previous discussion, related elements $\Psi\in F(\alpha)$ and $\Psi'\in F(\alpha')$ given by $$\Psi:G\ltimes O\rightarrow {\mathcal R}\ \ \text{and}\ \ \Psi':G'\ltimes O\rightarrow {\mathcal R}.$$ Related means that $F(\Phi)(\Psi)=\Psi'$. The biholomorphic map $\phi$ defines an equivariant diffeomorphism $O\rightarrow O'$ which satisfies $$\phi_\ast (g\ast q)=\gamma_{\Phi} (g)\ast \phi_\ast(q),$$ where $q=(v,\mathfrak{a})$. This in particular means that $\phi_\ast$ defines a smooth equivalence between the translation groupoids, but also is a bijection on objects and morphisms $$f_\Phi :G\ltimes O\rightarrow G'\ltimes O' :(q,q)\rightarrow (\gamma_{\Phi}(g),\phi_\ast(q)).$$ From this we see that we have the two functor $$\Psi\ \text{and}\ \Psi'\circ f_\Phi$$ both defined on $G\ltimes O$. On objects, with $(v',\mathfrak{a}')=f_\Phi (v,\mathfrak{a})$, we have the biholomorphic map $\phi_\mathfrak{a}: \alpha_{(v,\mathfrak{a})}\rightarrow \alpha_{(v',\mathfrak{a}')}'$, and hence the isomorphism $$\Phi_{(v,\mathfrak{a})}:=(\alpha_{(v,\mathfrak{a})} ,\phi_\mathfrak{a},\alpha_{f_\Phi(v,\mathfrak{a})}'):\alpha_{(v,\mathfrak{a})}
\rightarrow \alpha_{f_\Phi(v,\mathfrak{a})}'.$$ Equivalently written this precisely means for an object $(v,\mathfrak{a})$ in $G\ltimes O$ $$\Phi_{(v,\mathfrak{a})}: \Psi(v,\mathfrak{a})\rightarrow \Psi'\circ f_\Phi(v,\mathfrak{a}).$$ We note that $\Phi_{(0,0)}=\Phi$ and $f_\Phi(0,0)=(0,0)$. Assume that $(g,q):q\rightarrow g\ast q$ is a morphism in $G\ltimes O$ with $(g',q')=f_\Phi(g,q)$ being the corresponding morphism in $G'\ltimes O'$. We obtain the following commutative diagram $$\begin{CD}
\Psi(q) @> \Phi_{q}>> \Psi'\circ f_\Phi(q)\\
@V \Psi(g,q) VV @V \Psi'\circ f_{\Phi}(g,q)VV\\
\Psi(g\ast q) @> \Phi_{g\ast q} >> \Psi'\circ f_\Phi(g\ast q).
\end{CD}$$ Hence the map $$\Gamma:O\rightarrow {\mathcal R}:q\rightarrow \Gamma(q)=\Phi_q$$ is a natural transformation (in fact an equivalence) $$\Psi\xrightarrow{\Gamma} \Psi'\circ f_\Phi.$$ This shows that there is more structure than justing having a functor $F:{\mathcal R}\rightarrow \text{SET}$. Hence we have proved the following.
With the functor $F:{\mathcal R}\rightarrow \text{SET}$ as just described, given an isomorphism $\Phi:\alpha\rightarrow \alpha'$, there exists for every element $\Psi\in F(\alpha)$ with corresponding element $\Psi'=F(\Phi)(\Psi)$ an uniquely determined smooth equivalence of categories $f_\Phi:G\ltimes O\rightarrow G'\ltimes O'$ and a natural equivalence $\Gamma:\Psi\rightarrow \Psi'\circ f_\Phi$.
At this point the good uniformizers in $F(\alpha)$ and $F(\alpha')$ are mostly unrelated with the exception when the objects are isomorphic. In the following, in particular for the abstract theory we shall always take the minimalistic viewpoint, even if in all applications there is additional structure along the lines as just described.
We need some additional structure to related $F(\alpha)$ and $F(\alpha')$ even if $\alpha$ and $\alpha'$ are not isomorphic. This needs some preparation.
Given the functor ${F}:{\mathcal R}^-\rightarrow \text{SET}$ and objects $\alpha$ and $\alpha'$ in ${\mathcal R}^-$, we define for $\Psi\in {F}(\alpha)$ and $\Psi'\in{F}(\alpha')$ the set ${\bf M}(\Psi,\Psi')$, called the associated transition set between $\Psi$ and $\Psi'$ by $${\bf M}(\Psi,\Psi')=\{(q,\Phi,q')\ |\ q\in O,\ q'\in O',\ \Phi:\Psi(q)\rightarrow \Psi'(q')\}.$$
The construction of the transition set comes with several structure maps.
The target map $t:{\bf M}(\Psi,\Psi')\rightarrow O'$ and the source map $s:{\bf M}(\Psi,\Psi')\rightarrow O$ are given by $$t(q,\Phi,q')=q'\ \text{and}\ \ s(q,\Phi,q')=q.$$ The inversion map $\iota:{\bf M}(\Psi,\Psi')\rightarrow {\bf M}(\Psi',\Psi)$ is defined by $\iota(q,\Phi,q')=(q',\Phi^{-1},q)$. The unit map $O\rightarrow {\bf M}(\Psi,\Psi)$ is given by $u(q)=(q,1_{\Psi(q)},q)$, and the multiplication map is defined as $$\begin{aligned}
& {\bf M}(\Psi',\Psi''){_{s}\times_t}{\bf M}(\Psi,\Psi')\rightarrow {\bf M}(\Psi,\Psi''):&\\
&m((q',\Phi',q''),(q,\Phi,q'))=(q,\Phi'\circ\Phi,q'').&
\end{aligned}$$
The following results hold for the logarithmic and the exponential gluing profile. There are similar results for other, but not all, gluing profiles. The logarithmic case is a reformulation of the classical Deligne-Mumford theory, the case of the exponential gluing profile can be reduced to this classical case. In [@HWZ-DM] we derive both cases using pde-methods, but also show that the case with exponential gluing profile can be reduced to the classical case.
Given the logarithmic or the exponential gluing profile, the transition sets ${\bf M}(\Psi,\Psi')$ carry a natural metrizable topology and in addition naturally oriented smooth manifold structures, so that all the structure maps are smooth, and $s$ and $t$ are orientation preserving local diffeomorphisms. In case of the logarithmic gluing profile the smooth manifold structure is underlying to a complex manifold structure and all structure maps are holomorphic. The object space of $G\ltimes O$ is oriented by the orientation coming from the fact that $O$ is an open subset of a complex vector space. The orientations of the manifolds ${\bf M}(\Psi,\Psi')$ are determined by the fact that the source and target map are orientation preserving.
In particular, at this point we have, after picking the exponential gluing profile $\varphi$, a natural construction $(F,{\bf M})$ which associates to every object a set ${F}(\alpha)$ of good uniformizers $\Psi$ at $\alpha$, and to a transition set ${\bf M}(\Psi,\Psi')$ an oriented manifold structure.
Let us just note that with $\Psi'$ corresponding to $\Psi$ if $\Phi:\alpha\rightarrow \alpha'$ is given, the natural transformation $\Gamma:\Psi\rightarrow \Psi'\circ f_\Phi$ defines a diffeomorphism $$G\ltimes O\rightarrow {\bf M}(\Psi,\Psi'):(g,q)\rightarrow (q,\Gamma(g\ast q)\circ \Psi(g,q),f_\Phi(g\ast q)).$$
Here is one thing we can do with this data. Take a family ${(\Psi_\lambda)}_{\lambda\in\Lambda}$ ($\Lambda$ a set), of good uniformizers so that $$|{\mathcal R}| =\bigcup_{\lambda\in\Lambda} |\Psi_\lambda(O_\lambda)|.$$ Then we define an oriented smooth manifold $X$ by $$X=\coprod_{\lambda\in\Lambda} O_\lambda,$$ and a smooth oriented manifold ${\bf X}$ by $${\bf X}=\coprod_{(\lambda,\lambda')\in\Lambda\times \Lambda} {\bf M}(\Psi_\lambda,\Psi_{\lambda'}).$$ We may view $X$ as the collection of objects in a small category and ${\bf X}$ as the set of morphisms. More precisely, if $q_\lambda\in O_\lambda$ and $q'_{\lambda'}\in O_{\lambda'}$, then the morphisms $q_\lambda\rightarrow q'_{\lambda'}$ are precisely all elements $\Phi\in {\bf M}(\Psi_\lambda,\Psi_{\lambda'})$ with $s(\Phi)=q_\lambda$ and $t(\Phi)=q_{\lambda'}'$.
Using property (iv) it follows immediately that ${\mathcal X}=(X,{\bf X})$ is a an étale proper Lie groupoid, see [@Mj] for a short description of the theory. More comprehensive treatments are given in [@AR], and see [@hae1; @hae2; @hae3] for the beginnings of this theory. Using the $\Psi_\lambda$ we can construct a functor $$\beta:{\mathcal X}\rightarrow {\mathcal R}$$ which is an equivalence of categories. Namely we map the object $q_\lambda\in O_\lambda\subset X$ to $\Psi_\lambda(q_\lambda)$, so that on objects $$\beta:X\rightarrow \text{obj}({\mathcal R}):\beta(q_\lambda)=\Psi_\lambda(q_\lambda),$$ and on morphisms $$\beta:{\bf X}\rightarrow \text{mor}({\mathcal R}): \beta(q_\lambda,\phi,q_{\lambda'}')= (\Psi_\lambda(q_\lambda),\phi,\Psi_{\lambda'}(q_{\lambda'}')).$$
Hence we might view ${\mathcal X} $ as a smooth (up to equivalence) model for ${\mathcal R}$. Of course, making different choices we obtain $\beta':X'\rightarrow {\mathcal R}$ having the same properties. Taking the union of the choices we obtain $\beta'':{\mathcal X}''\rightarrow {\mathcal R}$ and smooth equivalences of étale proper Lie groupoids via the inclusions ${\mathcal X}\rightarrow {\mathcal X}''$ and ${\mathcal X}'\rightarrow {\mathcal X}''$. Taking the weak fibered product associated to the diagram $${\mathcal X}\rightarrow {\mathcal X}''\leftarrow {\mathcal X}'$$ denoted by ${\mathcal X}\times_{{\mathcal X}''} {\mathcal X}'$ the projections onto the factors are equivalences of étale proper Lie groupoids. With other words ${\mathcal X}$ and ${\mathcal X}'$ are Morita equivalent. Moreover $\beta:{\mathcal X}\rightarrow {\mathcal R}$ and $\beta':{\mathcal X}'\rightarrow {\mathcal R}$ are naturally equivalent via the data from ${\mathcal X}''$. In summary we have constructed up to Morita equivalence smooth models for ${\mathcal R}$. Each pair $({\mathcal X},|\beta|)$ gives an orbifold structure on $|{\mathcal R}|$ and these orbifold structures are equivalent. The whole collection turns $|{\mathcal R}|$ into a smooth oriented orbifold, with the structure depending on the gluing profile $\varphi$. If the process is carried out with the logarithmic gluing profile we again obtain a natural orbifold $|{\mathcal R}|$, which, however this time is holomorphic. The underlying smooth structures for the two constructions mentioned are not the same, but they should be diffeomorphic. The latter is definitely true over the un-noded part. The reader is referred to [@Mj; @MM; @AR] for more reading on étale proper Lie groupoids.
The Category of Stable Maps {#SSS2}
---------------------------
The construction carried out in this subsection is formally very similar to the one we carried out for ${\mathcal R}$. Now the category ${\mathcal R}$ is replaced by a different category ${\mathcal S}$, the category of stable maps. Again $|{\mathcal S}|$ has a natural metrizable topology and we shall construct a pair $(F,{\bf M})$ similarly as before. However, the local models are infinite-dimensional and all ingredients have never any change to be classically smooth. In fact, the occurring maps are generally nowhere differentiable in any classical sense. Nevertheless, as it will turn out, there is a smoothness concept, called sc-smoothness, so that the constructions viewed within an associated differential geometry are smooth.
Let $(Q,\omega)$ be a compact symplectic manifold without boundary. We consider maps defined on Riemann surfaces with image in $(Q,\omega)$ having various regularity properties. We shall write $$u:{\mathcal O}(S,x)\rightarrow Q$$ for a mapping germ defined on a Riemann surface $S$ near $x$.
Let $m\geq 2$ be an integer and $\varepsilon>0$. We say a germ of continuous map $u:{\mathcal O}(S,x)\rightarrow Q$ is of class $(m,\varepsilon)$ at the point $x$ if for a smooth chart $\phi:U(u(0))\rightarrow {\mathbb R}^{2n}$ mapping $u(0)$ to $0$ and holomorphic polar coordinates $ \sigma:[0,\infty)\times
S^1\rightarrow S\setminus\{x\}$ around $x$, the map $$v(s,t)=\phi\circ u\circ \sigma(s,t),$$ defined for $s$ large, has partial derivatives up to order $m$, which weighted by $e^{\varepsilon s}$ belong to $L^2([s_0,\infty)\times S^1,{\mathbb R}^{2n})$ if $s_0$ is sufficiently large. We say the germ is of class $m$ around a point $z\in S$ provided $u$ is of class $H^m_{loc}$ near $z$.
We observe that the above definition does not depend on the choices involved, like charts and holomorphic polar coordinates. We consider now tuples $\alpha=(S,j,M,D,u)=(\alpha^\ast,u)$, where $\alpha^\ast=(S,j,M,D)$ is a noded Riemann surface with ordered marked points $M$ and nodal pairs $D$, and $u:S\rightarrow W$ is a continuous map having some additional regularity properties. We do not assume $\alpha^\ast$ to be stable, i.e. being an object in ${\mathcal R}$ after forgetting the order of $M$.
A [noded Riemann surface with ordered marked points]{} is a tuple $\alpha^\ast=(S,j,M,D)$, where $(S,j)$ is a closed Riemann surface, $M\subset S$ a finite collection of [ ordered marked points]{}, and $D$ is a finite collection of un-ordered pairs $\{x,y\}$ of points in $S$, called [nodal pairs]{}, so that $x\neq y$ and two pairs which intersect are identical. The union of all $\{x,y\}$, denoted by $|D|$ is disjoint from $M$. We call $D$ the set of nodal pairs and $|D|$ the set of nodal points.
As in the definition of the objects in ${\mathcal R}$ the Riemann surface $S$ might consist of different connected components $C$. We call $C$ a domain component of $S$. The special points on $C$ are as before the points in $C\cap (M\cup|D|)$. We say that $(S,j,M,D)$ is connected, provided the topological space $\bar{S}$ obtained by identifying $x\equiv y$ in the nodal pairs $\{x,y\}\in D$ is connected. With our terminology it is possible that $\alpha^\ast=(S,j,M,D)$ is connected but on the other hand $S$ may have several connected components, i.e. its domain components.
Next we describe the tuples $\alpha$ in more detail.
\[D1\]We say that $\alpha=(S,j,M,D,u)$ is a stable map of [class]{} $({m},{\delta})$ provided the following holds, here $m\geq 2$ and $\delta>0$.
- The underlying topological space obtained by identifying the two points in any nodal pair is connected.
- The map $u$ is of class $(m,\delta)$ around the points in $|D|$ and of class $m$ around all other points in $S$.
- $u(x)=u(y)$ for every nodal pair $\{x,y\}\in D$.
- If a domain component $C$ of $S$ has genus $g_C$, and $n_C$ special points so that $2\cdot g_C +n_C\leq 2$, then $\int_C u^\ast\omega >0$. Otherwise we assume that $\int_C u^\ast\omega\geq 0$. This is called the stability condition.
Next we introduce the category of stable maps of class $(3,\delta_0)$, where $\delta_0$ is a fixed umber in $(0,2\pi)$.
Fix a $\delta_0\in (0,2\pi)$. The category of stable maps (of class $(3,\delta_0)$), denoted by $${\mathcal S}^{3,\delta_0}(Q,\omega)$$ has as objects the tuples $\alpha$ of class $(3,\delta_0)$ and as morphisms $$\Phi:=(\alpha,\phi,\alpha'):\alpha\rightarrow \alpha',$$ where $\phi$ is a biholomorphic map $(S,j,M,D)\rightarrow (S',j',M',D')$ satisfying $u'\circ\phi=u$.
\(i) The category ${\mathcal S}^{3,\delta_0}(Q,\omega)$ has several interesting features. All the morphisms are isomorphisms, and between two objects the number of morphisms is finite. This is a consequence of the stability condition.
\(ii) If we identify isomorphic objects in ${\mathcal S}^{3,\delta_0}(Q,\omega)$ we obtain the orbit class $|{\mathcal S}^{3,\delta_0}(Q,\omega)|$ which is easily verified to be a set. An element in the orbit set is written as $|\alpha|$.
\(iii) The choice of $\delta_0\in (0,2\pi)$ is dictated by the fact that in any analytical treatment of stable maps (Fredholm theory), one has to derive elliptic estimates near the nodes which involve self-adjoint operators which have spectrum $2\pi {\mathbb Z}$. So $(0,2\pi)$ has to be understood as a spectral gap. This requirement is important for the Fredholm theory.
The first important fact is that the orbit set $|{\mathcal S}^{3,\delta_0}(Q,\omega)|$ carries a natural topology. More precisely:
\[th-top\] Given $\delta_0\in (0,2\pi)$ the orbit space $|{\mathcal S}^{3,\delta_0}(Q,\omega)|$ carries a natural Hausdorff, second countable, regular, and hence metrizable, topology.
We refer the reader to [@H2; @HWZ6] for the complete construction. However, we shall give some ideas later on. The key is a recipe to construct a basis for a topology. This recipe involves choices. Nevertheless the resulting topology is independent of them and this is what we mean by being natural. However, this is only the beginning and there is another natural construction. What we shall see next will remind the reader of the type of construction which already occurred in the treatment of ${\mathcal R}$. In the case of ${\mathcal R}$ we saw uniformizers defined on translation groupoids, where object and morphism sets carried natural smooth manifold structures. In the relevant constructions for ${\mathcal S}^{3,\delta_0}(Q,\omega)$ our translation groupoids will have much less structure (for the moment), namely they are just metrizable topological spaces. The reason is that all occurring maps have no chance of being classically smooth and the natural local models seem very often far away from open subsets of Banach spaces.
Recall, that given a metrizable space $O$ with the action of a group $G$ by continuous maps, we can construct the metrizable translation groupoid $G\ltimes O$. This is a small metrizable category with objects being the elements in $O$ and the morphisms being the tuples $(g,q)$, where $q\in O$ and $g\in G$. Here the source of $(g,q)$ is $q$ and the target is $g\ast q$, i.e. $$(g,q):q\rightarrow g\ast q.$$ A metrizable category is a small category, where object and morphism sets are metrizable spaces and all structure maps are continuous, namely associating to a morphism $\phi$ its source and target, the inversion map $\phi\rightarrow \phi^{-1}$, the unit map $q\rightarrow 1_q$, as well as the multiplications map $m(\phi,\psi)=\phi\circ \psi$ defined on the appropriate subspace of the product of two copies of the morphism space.
Let $\alpha$ be an object in ${\mathcal S}^{3,\delta_0}(Q,\omega)$ with automorphism group $G$. A good uniformizer around $\alpha$ is a functor $\Psi:G\ltimes O\rightarrow {\mathcal S}^{3,\delta_0}(Q,\omega)$ having the following properties.
- $\Psi$ is fully faithful and there exists an object $q_0\in O$ with $\Psi(q_0)=\alpha$.
- Passing to orbit spaces $|\Psi|:{_{G}\backslash }O\rightarrow |{\mathcal S}^{3,\delta_0}(Q,\omega)|$ is a homeomorphism onto an open neighborhood of $|\alpha|$ in $|{\mathcal S}^{3,\delta_0}(Q,\omega)|$.
- For every object $q\in O$ there exists an open neighborhood $U(q)\subset O$, so that every sequence $(q_k)\subset U(q)$, for which $|\Psi(q_k)|$ converges in $|{\mathcal S}^{3,\delta_0}(Q,\omega)|$, has a convergent subsequence in ${\operatorname{cl}}_O(U(q))$.
Given two good uniformizers $\Psi$ around $\alpha$ and $\Psi'$ around $\alpha'$, we can define the set ${\bf M}(\Psi,\Psi')$ consisting of all tuples $(q,\Phi,q')$, where $q\in O$, $q'\in O'$ and $\Phi:\Psi(\alpha)\rightarrow \Psi'(\alpha')$ is a morphism in ${\mathcal S}^{3,\delta_0}(Q,\omega)$. Associated to ${\bf M}(\Psi,\Psi')$ we have the source map $$s:{\bf M}(\Psi,\Psi')\rightarrow O:(q,\Phi,q')\rightarrow q$$ and the target map $$t:{\bf M}(\Psi,\Psi')\rightarrow O':(q,\Phi,q')\rightarrow q'.$$ In addition there is the inversion map $$\iota:{\bf M}(\Psi,\Psi')\rightarrow {\bf M}(\Psi',\Psi):(q,\Phi,q')\rightarrow (q',\Phi^{-1},q),$$ the unit map $$u:O\rightarrow {\bf M}(\Psi,\Psi):q\rightarrow (q,1_{\Psi(q)},q),$$ and, given a third uniformizer $\Psi''$, the multiplication map $$\begin{aligned}
&m:{\bf M}(\Psi',\Psi''){_{s}\times_t}{\bf M}(\Psi,\Psi')\rightarrow {\bf M}(\Psi,\Psi''):&\\
&m((q',\Phi',q''),(q,\Phi,q'))=(q,\Phi'\circ\Phi,q'').&\end{aligned}$$ The maps $s,t,\iota,u$ and $m$ are called the structure maps.
The next natural construction is summarized by the following theorem, where as before $\text{SET}$ is the category of sets.
\[thm1\] There exists a natural functor $F:({\mathcal S}^{3,\delta_0}(Q,\omega))^-\rightarrow \text{SET}$ which associates to an object $\alpha$ a set $F(\alpha)$ of good uniformizers around $\alpha$. Moreover, $F$ comes with a natural construction, which associates to every choice $\Psi\in F(\alpha) $ and $\Psi'\in F(\alpha')$ a metrizable topology on ${\bf M}(\Psi,\Psi')$ so that the source and target maps are local homeomorphisms and all structure maps are continuous.
It is again possible to lift the functor to ${\mathcal S}^{3,\delta_0}(Q,\omega)$ so that morphisms are mapped to bijections. One can be more explicit about the correspondence of constructions for $\alpha$ and the constructions for $\alpha'$, when we are given a morphism $\Phi:\alpha\rightarrow \alpha'$. This is in the spirit of the discussion for ${\mathcal R}$.
There are many consequences of this result. For example we can pick a family ${(\Psi_\lambda)}_{\lambda\in\Lambda}$, where $\Lambda$ is a set, of good uniformizers so that $$|{\mathcal S}^{3,\delta_0}(Q,\omega)|=\bigcup_{\lambda\in\Lambda} |\Psi(O_\lambda)|.$$ Then we can define the objects of a category $X$ as the disjoint union $$X=\coprod_{\lambda\in\Lambda} O_\lambda$$ and the morphism set ${\bf X}$ by $${\bf X} =\coprod_{(\lambda,\lambda')\in\Lambda\times\Lambda} {\bf M}(\Psi_\lambda,\Psi_{\lambda'}).$$ We note that $X$ and ${\bf X}$ are both metrizable, and the $\Psi_\lambda$ can be used to define a functor $\beta$ from ${\mathcal X}=(X,{\bf X})$ to ${\mathcal S}^{3,\delta_0}(Q,\omega)$ by $$X\rightarrow \text{obj}({\mathcal S}^{3,\delta_0}(Q,\omega)):O_\lambda\ni q\rightarrow \Psi_\lambda(q)$$ and $$\begin{aligned}
&{\bf X}\rightarrow \text{mor}({\mathcal S}^{3,\delta_0}(Q,\omega)):&\\
&{\bf M}(\Psi_\lambda,\Psi_{\lambda'})\ni(q,\Phi,q')\rightarrow \Phi.&\end{aligned}$$
The functor $\beta$ is an equivalence of categories.
With other words we can build a topological version of ${\mathcal S}^{3,\delta_0}(Q,\omega)$ up to equivalence. Of course, the construction of ${\mathcal X}$ involves a choice of a family $(\Psi_\lambda)$. As in the ${\mathcal R}$-case, if we make a different choice we obtain $$\beta':{\mathcal X}\rightarrow {\mathcal S}^{3,\delta_0}(Q,\omega)$$ and taking the disjoint union of the choices $\beta'':{\mathcal X}''\rightarrow {\mathcal S}^{3,\delta_0}(Q,\omega)$. The inclusions ${\mathcal X}\rightarrow {\mathcal X}''$ and ${\mathcal X}'\rightarrow {\mathcal X}''$ are local homeomorphisms on the object and morphism spaces, and equivalences of categories.
Starting from this situation and quite similar to the ${\mathcal R}$-case one can define a notion of equivalence of two pairs $({\mathcal X},\beta)$ and $({\mathcal X}',\beta')$, so that we might say that ${\mathcal S}^{3,\delta_0}(Q,\omega)$ up to Morita equivalence has a unique topological (metrizable) model. As it turns out the construction of $F$ in Theorem \[thm1\] has many more properties. Namely the domains $G\ltimes O$ of the functors $\Psi_\lambda$ are in some sense smooth spaces. Not in the usual way, but in a generalized differential geometry. This will also be explained later on. At the end of day there is a differential geometric version of Theorem \[thm1\], which one might view as the construction of a smooth structure on the category ${\mathcal S}^{3,\delta_0}(Q,\omega)$, and which allows to build small smooth versions of our category up to Morita equivalence.
A Bundle and the CR-Functor {#SSS23}
---------------------------
Assume next that a compatible almost complex structure $J$ has been fixed for $(Q,\omega)$. We consider tuples $${\widehat}{\alpha}=(\alpha,\xi)=(S,j,M,D,u,\xi),$$ where $\alpha$ is an object in ${\mathcal S}^{3,\delta_0}(Q,\omega)$, and $\xi(z):T_zS\rightarrow T_{u(z)}Q$ is complex anti-linear for the given structures $j$ and $J$. Further we assume that $$z\rightarrow \xi(z)$$ has Sobolev regularity $H^2$ away from the nodal points. At the nodal points we assume it to be of class $(2,\delta_0)$. To make this precise, pick a nodal point $x$ and take positive holomorphic polar coordinates around $x$, say $\sigma(s,t)$ with $x=\lim_{s\rightarrow \infty} \sigma(s,t)$. Then take a smooth chart $\phi$ around $u(x)$ with $\phi(u(x))=0$. Finally consider the principal part $$(s,t)\rightarrow pr_2\circ T\phi(u(\sigma(s,t)))\circ \xi(\sigma(s,t))\left(\frac{\partial\sigma(s,t)}{\partial s}\right),$$ which we assume for large $s_0$ to be in $H^{2,\delta_0}([s_0,\infty)\times S^1,{\mathbb R}^{2n})$. The definition of the decay property does not depend on the choice of $\sigma$ and $\phi$. Denote by $\text{BAN}_G$ the category where the objects are the Banach spaces and the morphisms are topological linear isomorphisms. The above discussion gives us a functor $$\mu:{\mathcal S}^{3,\delta_0}(Q,\omega)\rightarrow \text{BAN}_G.$$ We associate to an object $\alpha$ the Hilbert space of all $\xi$ of class $(2,\delta_0)$ as described above. To a morphism $\Phi=(\alpha,\phi,\alpha')$ we associate the topological linear isomorphism $$\mu(\alpha)\rightarrow \mu(\alpha'):\xi\rightarrow \xi\circ T\phi^{-1}.$$ We can now define a new category ${\mathcal E}^{2,\delta_0}(Q,\omega,J)$, whose objects are the tuples ${\widehat}{\alpha}$ of the form $${\widehat}{\alpha}=(\alpha,\xi),$$ where $\alpha$ is an object in ${\mathcal S}^{3,\delta_0}(Q,\omega)$ and $\xi\in \mu(\alpha)$ is a complex anti-linear $TQ$-valued $(0,1)$-form along $u$ of class $(2,\delta_0)$. A morphism $${\widehat}{\Phi}=({\widehat}{\alpha},\Phi,{\widehat}{\alpha}'):{\widehat}{\alpha}\rightarrow {\widehat}{\alpha}'$$ is a tuple with $\Phi=(\alpha,\phi,\alpha'):\alpha\rightarrow \alpha'$ being a morphisms in ${\mathcal S}^{3,\delta_0}(Q,\omega)$ so that in addition $$\mu(\Phi)(\xi)=\xi'.$$ There is the projection functor $$P:{\mathcal E}^{2,\delta_0}(Q,\omega,J)\rightarrow {\mathcal S}^{3,\delta_0}(Q,\omega),$$ which on objects maps $(\alpha,\xi)\rightarrow \alpha$ and on morphisms $((\alpha,\xi),\Phi,(\alpha',\xi'))$ to $\Phi$. On the object level the preimage under $P$ of an object $\alpha$ is a Hilbert space consisting of the elements $(\alpha,\xi)$. Given $\Phi:\alpha\rightarrow \alpha'$ the morphism $\Phi$ is lifted to a bounded linear isomorphism $$\boldsymbol{\Phi}:P^{-1}(\alpha)\rightarrow P^{-1}(\alpha'): (\alpha,\xi) \rightarrow (\alpha',\xi\circ T\phi^{-1}),$$ where the map $\xi\rightarrow \xi\circ T\phi^{-1}$ is linear. We may view the two categories fibering over each other via $P:{\mathcal E}^{2,\delta_0}(Q,\omega,J)\rightarrow {\mathcal S}^{2,\delta_0}(Q,\omega)$ as a ‘bundle’. Then we have the Cauchy-Riemann section functor $$\bar{\partial}_J(\alpha)=\left(\alpha,\frac{1}{2}\cdot\left[ Tu+J(u)\circ Tu\circ j\right]\right),$$ where $\alpha=(S,j,M,D,u)$. From the results in [@HWZ6] we can deduce the following theorem.
\[thm2\] The orbit space $|{\mathcal E}^{2,\delta_0}(Q,\omega,J)|$ carries a natural second countable metrizable topology. The induced maps on orbit spaces $|P|$ and $|\bar{\partial}_J|$ are continuous. The topology on the orbit space of the full subcategory associated to the objects $\alpha$ with $\bar{\partial}_J(\alpha)=0$ has compact connected components and coincides with the (union of all) Gromov compactified moduli space of $J$-pseudoholomorphic curves.
The previous discussion about the construction $F$ can be extended to cover $P$. Given an object $\alpha$ the notion of a good uniformizer generalizes by replacing $O$ by a continuous surjective map $p:W\rightarrow O$ between metrizable spaces, where the fibers are Hilbert spaces (or more generally Banach spaces), and where in addition we have a continuous $G$-action, linear between the fibers. Then the good bundle uniformizers are given by commutative diagrams $$\begin{CD}
G\ltimes W @> \bar{\Psi}>> {\mathcal E}^{2,\delta_0}(Q,\omega,J)\\
@VVV @VVV\\
G\ltimes O @>\Psi>> {\mathcal S}^{3,\delta_0}(Q,\omega)
\end{CD}$$ with the obvious properties. The bottom $\Psi$ is as described before and $\bar{\Psi}$ is a linear bounded isomorphism between the fibers. Again, as it will turn out, these constructions will fit into a scheme of generalized differential geometry, and the local representative of $\bar{\partial}_J$, a $G$-equivariant section of $W\rightarrow O$, will turn out to be a nonlinear Fredholm section in the extended framework. We refer the reader to [@FH; @HWZ6] for the constructions related to stable maps, and to [@HWZ7; @HWZ8] for the abstract theory. In the above extension we have a functorial construction $\bar{F}:({\mathcal S}^{3,\delta_0}(Q,\omega))^-\rightarrow \text{SET}$ covering $F$ in the following way. For every object $\alpha$ the set $\bar{F}(\alpha)$ consists of strong bundle uniformizers $\bar{\Psi}$, which together with the underlying $\Psi\in F(\alpha)$, fit into the above commutative diagram. Moreover, it comes with an associated construction of a metrizable topology on the transition set ${\bf M}(\bar{\Psi},\bar{\Psi}')$. There is a projection $${\bf p}:{\bf M}(\bar{\Psi},\bar{\Psi}')\rightarrow{\bf M}({\Psi},{\Psi}'):(k,{\widehat}{\Phi},k')\rightarrow (o,\Phi,o')$$ whose fibers are Hilbert spaces. Moreover, there exists the commutative diagram involving source and target maps $$\begin{CD}
K @<s<< {\bf M}(\bar{\Psi},\bar{\Psi}') @>t>> K\\
@V p VV @V {\bf p} VV @ V p' VV\\
O@< s<< {\bf M}(\Psi,\Psi') @> t >> O'.
\end{CD}$$ We leave the execution of the idea to the reader. The above follows from the results in [@HWZ6].
A Basic Construction
--------------------
The following discussion is carried out in detail in [@HWZ8.7], see also [@HWZ6]. Consider the Hilbert space $E$ consisting of pair of maps $(u^+,u^-)$ $$u^\pm:{\mathbb R}^\pm\times S^1\rightarrow {\mathbb R}^N,$$ where for each pair there exists a constant $c\in {\mathbb R}^N$, so that $u^\pm-c$ has partial derivatives up to order $3$ which weighted by $e^{\delta_0|s|}$ belong to $L^2({\mathbb R}^\pm\times S^1)$. The constant $c$ is called the common asymptotic limit of $(u^+,u^-)$.
Given a complex number $|a|<1/2$ we write $a=|a|\cdot e^{-2\pi i\theta}$. If $a\neq 0$ we define $R=\varphi(|a|)$, where $\varphi$ is the exponential gluing profile, which we recall, is defined by $$\varphi:(0,1]\rightarrow [0,\infty): \varphi(r)=e^{\frac{1}{r}}-e.$$ We construct for $0<|a|<1/2$ the finite cylinder $Z_a$ by identifying $(s,t)\in [0,R]\times S^1$ with $(s',t')\in [-R,0]\times S^1$ via $$s=s'+R\ \text{and}\ t=t'+\theta.$$ Denote by $[s,t]$ the equivalence class of the point $(s,t)\in [0,R]\times S^1$ and by $[s',t']'$ the equivalence class of $(s',t')\in [-R,0]\times S^1$. Then $$[s,t]=[s-R,t-\theta]'$$ and $Z_a\rightarrow [0,R]\times S^1$ and $Z_a\rightarrow [-R,0]\times S^1$ defined by $[s,t]\rightarrow (s,t)$ and $[s',t']'\rightarrow (s',t')$, respectively, are holomorphic coordinates, where the targets are equipped with the standard complex structures.
If $a=0$ we define $Z_0={\mathbb R}^+\times S^1\coprod {\mathbb R}^-\times S^1$. Pick a smooth map $\beta:{\mathbb R}\rightarrow [0,1]$ satisfying $\beta(s)=1$ for $s\leq -1$, $\beta'(s)<0$ for $s\in (-1,1)$ and $\beta(s)+\beta(-s) =1$ for all $s\in {\mathbb R}$.
Given $(u^+,u^-)\in E$ and $|a|<1/2$ the glued map $$\oplus_a(u^+,u^-):Z_a\rightarrow {\mathbb R}^N,$$ is defined for $a=0$ by $\oplus_0(u^+,u^-)=(u^+,u^-)$ and for $a\neq 0$ by $$\oplus_a(u^+,u^-)([s,t])=\beta\left(s-\frac{R}{2}\right) u^+(s,t) +\left(1-\beta\left(s-\frac{R}{2}\right)\right) u^-(s-R,t-\theta).$$
For fixed $a\neq 0$ many different $(u^+,u^-)$ will produce the same glued map. We can remove the ambiguity by introducing the anti-glued map $\ominus_a(u^+,u^-)$. It is defined on $C_a$ given by $C_0=\emptyset$ and for $0<|a|<1/2$ by gluing ${\mathbb R}^-\times S^1$ and ${\mathbb R}^+\times S^1$ along $Z_a$,. Note that $Z_a$ can be identified with an obvious subset of both half cylinders. Namely $C_a$ is given by $$C_a=(({\mathbb R}^-\times S^1)\coprod ({\mathbb R}\times S^1))/\sim,$$ where $(s',t')\equiv (s,t)$ provided $(s',t')\in [-R,0]\times S^1$, $(s,t)\in {\mathbb R}^+\times S^1$ and $s=s'+R$ and $t=t'+\theta$. We shall write $[s,t]$ or equivalently $[s',t']'$ for the points in the part being identified. Note that we have for $0<|a|<1/2$ a natural embedding $$Z_a\rightarrow C_a:[s,t]\rightarrow [s,t].$$ For $a\neq 0$ the cylinder $C_a$ has a natural complex manifold structure, for which the maps on $Z_a\subset C_a$ defined by $$[s,t]\rightarrow (s,t)\in {\mathbb R}\times S^1,$$ and $$[s',t']'\rightarrow (s',t')\in {\mathbb R}\times S^1$$ have natural extensions to biholomorphic maps $C_a\rightarrow {\mathbb R}\times S^1$. The extensions will be written as above. The transition map $${\mathbb R}\times S^1\rightarrow {\mathbb R}\times S^1:(s,t)\rightarrow (s',t')$$ is given by $(s,t)\rightarrow (s-R,t-\theta)$.
For $(u^+,u^-)\in E$ the anti-glued map $\ominus_a(u^+,u^-):C_a\rightarrow {\mathbb R}^N$ is defined for $a=0$ by $\ominus_0(u^+,u^-)=0$, i.e. the only map $\emptyset\rightarrow {\mathbb R}^N$, and otherwise by $$\begin{aligned}
\ominus_a(u^+,u^-)([s,t])
&=&\left(\beta\left(s-\frac{R}{2}\right)-1\right) \left( u^+(s,t)-\text{av}_a(u^+,u^-)\right)\\
&&+\beta\left(s-\frac{R}{2}\right)\left(u^-(s-R,t-\theta)-\text{av}_a(u^+,u^-)\right).\end{aligned}$$ Here $$\text{av}_a(u^+,u^-)=\frac{1}{2}\cdot \int_{S^1} (u^+(R/2,t)+u^-(-R/2,t)) dt.$$
If we start with $(u^+,u^-)\in E$ and pick any $0<|a|<1/2$ the total gluing map $$\boxdot_a(u^+,u^-)=(\oplus_a(u^+,u^-),\ominus_a(u^+,u^-)),$$ defines a bounded linear isomorphism from $E$ to $H^3(Z_a,{\mathbb R}^N)\oplus H^{3,\delta_0}_c(C_a,{\mathbb R}^N)$. Here $ H^{3,\delta_0}_c(C_a,{\mathbb R}^N)$ denotes the Hilbert space of maps $u:C_a\rightarrow {\mathbb R}^N$ which are of class $H^3_{loc}$ so that in addition there exists a constant $c\in {\mathbb R}^N$ depending on $u$ for which $u\pm c$ has partial derivatives up to order $3$ which weighted by $e^{\delta_0|s|}$ belong to $L^2([s_0,\infty)\times S^1)$ or $L^2(-\infty,-s_0]\times S^1)$, respectively, for a suitably large $s_0$. In particular we obtain for $a\neq 0$ a linear topological isomorphism $${$\text{ker}\ {(}$}\ominus_a)\rightarrow H^3(Z_a,{\mathbb R}^N):(u^-,u^+)\rightarrow \oplus_a(u^+,u^-).$$ Since $E$ decomposes as the topological direct sum $$E={$\text{ker}\ {(}$}\ominus_a)\oplus {$\text{ker}\ {(}$}\oplus_a)$$ there exists an associated family $a\rightarrow \pi_a$ of bounded projection operators $\pi_a:E\rightarrow E$ which project onto ${$\text{ker}\ {(}$}\ominus_a)$ along ${$\text{ker}\ {(}$}\oplus_a)$.
\[prop-cont\] For every $|a|<1/2$ the linear operator $\pi_a:E\rightarrow E$ is a bounded projection. However, the map $$\{a\in {\mathbb C}\ |\ |a|<1/2\}\rightarrow L(E):a\rightarrow \pi_a,$$ where the space of bounded operators $L(E)$ is equipped with the operator norm is nowhere continuous. The map $$\{a\in {\mathbb C}\ |\ |a|<1/2\}\times E\rightarrow E:(a,(u^+,u^-))\rightarrow \pi_a(u^+,u^-)$$ is continuous.
The map $r:\{|a|<1/2\}\times E\rightarrow \{|a|<1/2\}\times E$ defined by $$r(a,(u^+,u^-))=(a,\pi_a(u^+,u^-))$$ is continuous and satisfies $r\circ r=r$. It also preserves the fibers of $\{|a|<1/2\}\times E\rightarrow \{|a|<1/2\}$. The image of $r$ is then a fiber-wise retract. Define ${$\text{ker}\ {(}$}\ominus_.)$ as the subset of $\{a\in {\mathbb C}\ |\ |a|<1/2\}\times E$ consisting of all tuples $(a,(u^+,u^-))$ with $\ominus_a(u^+,u^-)=0$. Then we have continuous projection $${$\text{ker}\ {(}$}\ominus_.)\rightarrow \{a\in {\mathbb C}\ |\ |a|<1/2\},$$ where the fibers are sc-Hilbert spaces. Let us define the set $\bar{X}^{3,\delta_0}({\mathbb R}^N)$ by $$\bar{X}^{3,\delta_0}({\mathbb R}^N):=(\{0\}\times E)\bigcup\left(\bigcup_{0<|a|<1/2} \left(\{0\}\times H^3(Z_a,{\mathbb R}^N)\right)\right).$$ We obtain a fiber-preserving, fiber-wise linear, bijection $$\begin{CD}
{$\text{ker}\ {(}$}\ominus_.) @>\oplus_. >> \bar{X}^{3,\delta_0}({\mathbb R}^N)\\
@VVV @VVV\\
\{|a|<1/2\} @= \{|a|<1/2\}.
\end{CD}$$ We equip the right-hand side with the topology ${\mathcal T}$ which makes $\oplus_.$ a homeomorphism. We observe that ${$\text{ker}\ {(}$}\ominus_.)$ depends on the choice of the cut-off function $\beta$, whereas the right-hand side does not depend on any choice. So one might ask if ${\mathcal T}$ depends on the choice of $\beta$.
The definition of the topology ${\mathcal T}$ does not depend on the choice of the cut-off function $\beta$. Moreover ${\mathcal T}$ is a metrizable topology.
As a consequence of this proposition $\bar{X}^{3,\delta_0}({\mathbb R}^N)$ is in a natural way a metrizable topological space so that the projection $$p:\bar{X}^{3,\delta_0}({\mathbb R}^N)\rightarrow \{|a|<1/2\}:[u:Z_a\rightarrow {\mathbb R}^N]\rightarrow a,$$ is continuous. We can carry out this construction for every ${\mathbb R}^N$. Interestingly a smooth map $f:{\mathbb R}^N\rightarrow {\mathbb R}^M$ induces a continuous map $$\bar{X}^{3,\delta_0}({\mathbb R}^N)\rightarrow \bar{X}^{3,\delta_0}({\mathbb R}^M):u\rightarrow f\circ u.$$ Summarizing this means that the following assertion holds true.
The construction $\bar{X}^{3,\delta_0}$ defines a natural functor from the category of Euclidean spaces with objects ${\mathbb R}^N$ and smooth maps between them, into the category of metrizable topological spaces.
If $Q$ is a smooth connected manifold without boundary we can take for a sufficiently large $N$ a smooth proper embedding $j:Q\rightarrow {\mathbb R}^N$ and define $\bar{X}^{3,\delta_0}(Q)$ to consist of all continuous maps $u:Z_a\rightarrow Q$ such that $j\circ u\in X^{3,\delta_0}({\mathbb R}^N)$, and equip it with the topology making the map $u\rightarrow j\circ u$ a homeomorphism onto its image. The resulting topological space $\bar{X}^{3,\delta_0}(Q)$ does not depend on the choice of the embedding. Indeed, if $k:Q\rightarrow {\mathbb R}^M$ is another proper embedding, then $k\circ j^{-1}$ and $j\circ k^{-1}$ are restrictions of globally defined smooth maps, which implies the conclusion. If $Q$ does not have boundary but is not connected we can apply the above to every component. Consequently we have the following result.
The functor $\bar{X}^{3,\delta_0}$, defined for the spaces ${\mathbb R}^N$ and the smooth maps between them, has a natural extension to the category of smooth manifolds without boundaries and the smooth maps between them.
Given a finite-dimensional smooth manifold $\Lambda$ and a smooth map $$\Lambda:L\rightarrow \{a\in {\mathbb C}\ |\ |a|<1/2\}$$ we can consider the pull-back $\Lambda^\ast X^{3,\delta_0}({\mathbb R}^N)\subset \Lambda\times X^{3,\delta_0}({\mathbb R}^N)$ defined by the diagram $$\begin{CD}
@. X^{3,\delta_0}({\mathbb R}^N)\\
@. @VVV\\
L @>\Lambda>> \{|a|<1/2\}.
\end{CD}$$ The next result is more complicated to prove. Denote by $j$ the standard almost complex structure on $Z_a$. We assume that for $|a|<1/2$ we are given a family $$a\rightarrow j(a)$$ of smooth almost complex structures on $Z_a$. The $j(a)$ live on different domains and we require a certain form of smooth dependence on $a$. The first requirement is the following where $\varphi$ denotes the exponential gluing profile.
- There exist $\varepsilon\in (0,1/2)$ and $H>0$ so that $$j(a)=j\ \text{on}\ \{[s,t]\in Z_a\ |\ s\in [H,\varphi(|a|)-H]\}$$ provided $0<|a|<\varepsilon$, and if $a=0$ it holds that $j(0)|({\mathbb R}^\pm\times S^1)$ equals $j$ on $[H,\infty)\times S^1$ and $(-\infty,-H]\times S^1$. Here we assume that $\varphi(\varepsilon)> 2H$.
With other words, if the neck gets longer the structure will be $j$ in the necks, but can be different near the boundaries. In the next step we shall have to formulate the smoothness requirements near the boundaries. Consider for $|a|<\varepsilon$ the maps $$[0,\frac{3}{2}H]\times S^1\rightarrow Z_a:(s,t)\rightarrow [s,t]
\ \text{and}\ \ [-\frac{3}{2}H,0]\times S^1\rightarrow Z_a:(s',t')\rightarrow [s',t']',$$ which we can use to pull back $j(a)$ to obtain the families $j^+(a)$ and $j^-(a)$, respectively defined for $|a|<\varepsilon$.
- For $\varepsilon$ as in (i) the maps $a\rightarrow j^\pm(a)$, $|a|<\varepsilon$ are smooth.
Next we need to say something about the smoothness for $|a|<1/2$. If $\frac{1}{2}\varepsilon<|a|< 1/2$ we can identify $[0,1]\times S^1$ with $Z_a$ via the map $(s,t)\rightarrow [\varphi(|a|)\cdot s,t]$ and via pull-back obtain the family $j^+(a)$, $\frac{1}{2}\varepsilon<|a|<1/2$ on $[0,1]\times S^1$. Similarly we can identify $[-1,0]\times S^1$ with $Z_a$ via the map $(s',t')\rightarrow [-\varphi(|a_0|)s',t']'$ and obtain the pull-back family $a\rightarrow j^-(a)$ on $[-1,0]\times S^1$. We require the following.
- $a\rightarrow j^+(a)$ and $a\rightarrow j^-(a)$ for $\frac{1}{2}\varepsilon<|a|<1/2$ are smooth families.
A family $a\rightarrow j(a)$, which associates to $|a|<1/2$ a smooth almost complex structure on $Z_a$ and having the properties (i),(ii), and (iii) as described above will be called a good family (of smooth almost complex structures).
There are many equivalent formulations for defining a good family. The important result is concerned with the continuity of a certain class of maps.
\[prop1x\] Assume $j$ and $k$ are two good families as described above and $$\{|a|<1/2\}\rightarrow \{|a|<1/2\}: a\rightarrow b(a)$$ is a smooth map so that there exists a core-smooth family (defined below) $$a\rightarrow [ \phi_a:(Z_a,j(a))\rightarrow (Z_{b(a)},k(b(a))) ]$$ of biholomorphic maps. Then the map $$b^\ast\bar{X}^{3,\delta_0}(Q)\rightarrow \bar{X}^{3,\delta_0}(Q):(a,u)\rightarrow u\circ \phi_a$$ is continuous.
The family $a\rightarrow\phi_a$ being core-smooth just requires this map and its inverse to be smooth near the boundaries. This is well-defined, using the coordinates $[s,t]$ and $[s',t']'$. By unique continuation $\phi_a$ is entirely determined by knowing $a$, $b(a)$, and its values near the boundaries. So the fact that $\phi_a:Z_a\rightarrow Z_{b(a)}$ is biholomorphic also determines the behavior inside, which, of course, is crucial for the validity of the theorem.
The proof of this result requires some work. The above map will in fact be sc-smooth once we have introduced this concept. In [@HWZ8.7; @HWZ6] proofs of sc-smooth versions of Theorem \[prop1x\] can be found. We note that maps of the type occurring in Theorem \[prop1x\] usually will not be classically differentiable unless they are constant. In fact the expression $u\circ\phi_a$, when differentiating with respect to $a$ would loose a derivative of $u$.
The previous discussion can be extended. In a first step, rather than considering the half-cylinders ${\mathbb R}^\pm\times S^1$, we consider a noded disk pair $(D_x\cup D_y,\{x,y\})$. From all possible decorations $[{\widehat}{x},{\widehat}{y}]$ of $\{x,y\}$ we obtain the natural gluing parameters $r[{\widehat}{x},{\widehat}{y}]$ with $r\in [0,1/2)$ and define the set of gluing parameters associated to $\{x,y\}\in D$ as described before by $${\mathbb B}^{\{x,y\}}(1/2)=\{r[{\widehat}{x},{\widehat}{y}]\ |\ r\in [0,1/2),\ [{\widehat}{x},{\widehat}{y}]\ \text{decorated nodal pair}\}.$$ If we pick ${\widehat}{x}$ and ${\widehat}{y}$, so that $\{{\widehat}{x},{\widehat}{y}\}$ is a representative of $[{\widehat}{x},{\widehat}{y}]$, there are (unique) associated biholomorphic maps $h_x:(D_x,x)\rightarrow ({\bf D},0)$ and similarly $h_y$ so that $Th_x(x){\widehat}{x}={\mathbb R}$ and $Th_y(y){\widehat}{y}={\mathbb R}$. With this data we first obtain the map $$\phi:{\mathbb B}^{\{x,y\}}(1/2)\rightarrow \{z\in {\mathbb C}\ |\ |z|<1/2\}: r[{\widehat}{x},\theta{\widehat}{y}]\rightarrow r\theta,$$ which by the definition of the complex structure on ${\mathbb B}^{\{x,y\}}(1/2)$ is biholomorphic. If $\mathfrak{a}=[{\widehat}{x},\theta{\widehat}{y}]$ and $h_x$ and $h_y$ are as described above, then $h_x$ and $\theta^{-1}\cdot h_y$ is the uniquely determined choice of maps $(D_x,x)\rightarrow ({\mathbb D},0)$ and $(D_y,y)\rightarrow ({\mathbb D},0)$ with $Th_x(x){\widehat}{x}={\mathbb R}$ and $T(\theta^{-1}\cdot h_y)(y)(\theta\cdot {\widehat}{y})={\mathbb R}$. Hence the identification $z\equiv z'$ with $h_x(z)\cdot h_y(z') =\theta\cdot e^{-2\pi R}$ defines $Z_{r[{\widehat}{x},\theta{\widehat}{y}]}$, where $R=\varphi(r)$. With $h_x$ and $h_y$ fixed, let us define biholomorphic maps $$D_x\setminus\{x\}\rightarrow [0,\infty)\times S^1:z\rightarrow (s,t),\ \text{where}\ e^{-2\pi(s+it)}=h_x(z),$$ and $$D_y\setminus\{y\}\rightarrow (-\infty,0]\times S^1:z'\rightarrow (s',t'),\ \text{where}\ e^{2\pi(s'+it')}=h_y(z').$$ We would like to find the complex number $a=a(\mathfrak{a})$ so that we obtain a natural induced biholomorphic map from ${\mathcal Z}_\mathfrak{a}$ to $Z_a$. Clearly the identification $z\equiv z'$ associated to $h_x(z)\cdot (\theta^{-1}\cdot h_y(z'))=e^{-2\pi R}$, which defines ${\mathcal Z}_\mathfrak{a}$ if $\mathfrak{a}=r[{\widehat}{x},\theta\cdot {\widehat}{y}]$, is equivalent to the identification $(s,t)\equiv (s',t')$ given by $$e^{-2\pi(s+it)}\cdot e^{2\pi(s'+it')} =e^{-2\pi i\vartheta}\cdot e^{-2\pi R},$$ where we write $\theta$ as $e^{-2\pi i\vartheta}$. Hence $$s=s'+R\ \text{and}\ \ t=t'+\vartheta.$$ This is the identification defining $Z_a$. Hence with $a(r[{\widehat}{x},\theta{\widehat}{y}])= r\theta$ we obtain an induced biholomorphic map $${\mathcal Z}_\mathfrak{a}\rightarrow Z_{a(\mathfrak{a})}: [z=e^{-2\pi(s+it)}]\rightarrow [s,t].$$ This family of maps allows us to use the definition of $\bar{X}^{3,\delta_0}({\mathbb R}^N)$ to define a set of maps on the ${\mathcal Z}_\mathfrak{a}$ resulting in a set $\bar{X}^{3,\delta_0}_{\mathcal D}({\mathbb R}^N)$ with $${\mathcal D}=(D_x\cup D_y,\{x,y\}).$$ More precisely $$\bar{X}^{3,\delta_0}_{\mathcal D}({\mathbb R}^N)=H^{3,\delta_0}_c({\mathcal D},{\mathbb R}^N)\coprod\left[\coprod_{\mathfrak{a}\in {\mathbb B}^{\{x,y\}}(1/2)\ |\ |0<\mathfrak{a}|<1/2\}} H^3(Z_\mathfrak{a},{\mathbb R}^N)\right].$$ The elements in $ H^{3,\delta_0}_c({\mathcal D},{\mathbb R}^N)$ are pairs $(u^+,u^-)$ of continuous maps defined $D_x$ and $D_y$, respectively, so that $u^+(x)=u^-(y)$ and if $\sigma_x$ are positive polar coordinates on $D_x$ centered at $x$, and $\sigma_y$ negative ones on $D_y$ centered at $y$, the pair $(u^+\circ\sigma_x,u^-\circ \sigma_y)$ belongs to $E$. The previous construction might a priori depend on the choices involved. However making different initial choices of identifying $\bar{X}^{3,\delta_0}_{\mathcal D}({\mathbb R}^N)$ with $E$ the associated transition families induce continuous maps as a consequence of Theorem \[prop1x\]. This is a natural construction and the space itself is homeomorphic to the image of a continuous retraction. Moreover, it follows immediately from the definition that for a smooth map $f:{\mathbb R}^N\rightarrow {\mathbb R}^M$ the map $w\rightarrow f\circ w$ is continuous. From this we can deduce that the functor $\bar{X}^{3,\delta_0}_{\mathcal D}$ has a natural extension to all smooth manifolds $Q$ without boundary.
At this point we have a well-defined functorial construction which associates to a nodal disk pair ${\mathcal D}=(D_x\cup D_y,\{x,y\})$ and smooth manifold without boundary $Q$ a metrizable topological space $\bar{X}^{3,\delta_0}_{\mathcal D}(Q)$ and to a smooth map $f:Q\rightarrow Q'$ a continuous map $$\bar{X}^{3,\delta_0}_{\mathcal D}(Q)\rightarrow \bar{X}^{3,\delta_0}_{\mathcal D}(Q'):w\rightarrow f\circ w.$$ We can go one step further. This step is natural as well. Consider a noded Riemann surface $(S,j,D)$. We assume that $S$ has no boundary and is compact. We do not assume any stability assumption. Suppose that we are given a finite group $G$ acting by biholomorphic maps on $(S,j,D)$ and ${\bf D}$ is a small disk structure invariant under $G$. Then we can take the natural gluing parameters and consider the family $\mathfrak{a}\rightarrow (S_\mathfrak{a},j_\mathfrak{a},D_\mathfrak{a})$. We have dealt with problems arising near the nodes and the obtained glued necks. Away from these regions we are just dealing with ‘static’ domains and Sobolev class $H^3$ functions on them. We obtain the following result from the previous discussion.
Given a noded closed Riemann surface $(S,j,D)$ with an action by biholomorphic maps by a finite group and a $G$-invariant small disk structure, there exists a functorial construction of a metrizable topological space $\bar{X}^{3,\delta_0}_{(S,j,D),{\bf D}}(Q)$ for every smooth manifold $Q$ without boundary. Moreover $G$ acts continuously on these spaces. Associated to a smooth map $f:Q\rightarrow Q'$ there is a continuous equivariant map $$\bar{X}^{3,\delta_0}_{(S,j,D),{\bf D}}(Q)\rightarrow \bar{X}^{3,\delta_0}_{(S,j,D),{\bf D}}(Q'):w\rightarrow f\circ w.$$
There are some additional useful constructions which will be used later. Assume that $\Xi\subset S$ is a finite $G$-invariant subset in the complement of the disks coming from the small disk structure. Suppose that we have fixed for every $G$-orbit $[z]$ of a point $z\in \Xi$ a codimension two submanifold $H_{[z]}$ in $Q$. Denote the collection of all these constraints by ${\mathcal H}$. Define a subset $\bar{X}^{3,\delta_0}_{(S,j,D),{\bf D},{\mathcal H}}(Q)$ of $\bar{X}^{3,\delta_0}_{(S,j,D),{\bf D}}(Q)$ to consist of all $u$ intersecting $H_{[z]}$ at $z$ transversally for all $z\in \Xi$. If we have a map $u$ in $\bar{X}^{3,\delta_0}_{(S,j,D),{\bf D},{\mathcal H}}(Q)$ we find a small neighborhood around every $z\in\Xi$ and an open neighborhood $U$ around $u$ so that for every $z\in\Xi$ there is continuous map $U\ni w\rightarrow z_w$, uniquely determined by the requirement that $w$ intersects $H_{[z]}$ transversally at $z_w$. Denote by $\Xi'$ a deformation of $\Xi$. We can construct a continuous family of self-maps $\phi_{\Xi',\mathfrak{a}}:(S_\mathfrak{a},D_\mathfrak{a})\rightarrow (S_\mathfrak{a},D_\mathfrak{a})$, which in the necks are the identity and which move $\Xi'$ back to $\Xi$ and are supported in a small neighborhood around $\Xi$.. Then the map defined for $w\in U$ given by $$w\rightarrow w\circ \phi_{\Xi_w,\mathfrak{a}}^{-1},$$ where $\Xi_w$ is the deformation associated to $w$, i.e. the collection of all $z_w$, defines a continuous retraction on some open neighborhood $O=O(w)$ so that $r(O) =O\cap \bar{X}^{3,\delta_0}_{(S,j,D),{\bf D},{\mathcal H}}(Q)$. With other words the subset $ \bar{X}^{3,\delta_0}_{(S,j,D),{\bf D},{\mathcal H}}(Q)$ of $\bar{X}^{3,\delta_0}_{(S,j,D),{\bf D}}(Q)$ is locally a retract given by an retraction of a particular form. Since the ambient space was modeled locally on retracts the same is true for this subset.
All the constructions which we have carried out do not have a chance to be classically smooth. However, as we shall see later, they are in fact smooth construction in the sc-smooth world, i.e. they are differential geometric constructions in an extended differential geometry as described in [@HWZ7; @HWZ8].
We complete the discussion by adding some words about the construction of certain bundles over $\bar{X}^{3,\delta_0}_{(S,j,D),{\bf D},{\mathcal H}}(Q)$. First of all we would like to change the almost complex structure on the $(S_\mathfrak{a},D_\mathfrak{a})$ by deforming the $j_\mathfrak{a}$. Let us assume for the moment we are given a smooth family $$\mathfrak{j}:V\ni v\rightarrow j(v)$$ satisfying $j(0)=j$ and $j(v)=j$ on the disks of the small disk structure. Here $V$ is an open subset of some finite-dimensional vector space. In general we would have an action of $G$ on these spaces, but at the moment this is not relevant for the point we would like to make. Then $V\times \bar{X}^{3,\delta_0}_{(S,j,D),{\bf D},{\mathcal H}}(Q)$ is a metrizable topological space and we can view a point $(v,w)$ in this space as a map defined on $(S_\mathfrak{a},j(v)_\mathfrak{a},D_\mathfrak{a})$, i.e. defined on a deformation of $(S_\mathfrak{a},j_\mathfrak{a},D_\mathfrak{a})$, the space on which $w$ was originally defined. We consider next tuples $(v,u,\xi)$, where $z\rightarrow \xi(z)$ is of class $H^{2,\delta_0}$ and for fixed $z\in S_\mathfrak{a}$ it holds that $$\xi(z):(T_zS_{\mathfrak{a}},j(v)_{\mathfrak{a}})\rightarrow (T_{u(z)}Q,J)$$ is complex anti-linear. By $\bar{E}^{2,\delta_0}_{(S,j,D),{\bf D},{\mathcal H},\mathfrak{j}}(Q,\omega,J)$ we denote the collection of all these tuples. We have a natural projection $$p:\bar{E}^{2,\delta_0}_{(S,j,D),{\bf D},{\mathcal H},\mathfrak{j}}(Q,\omega,J)\rightarrow V\times \bar{X}^{3,\delta_0}_{(S,j,D),{\bf D},{\mathcal H}}(Q):(v,u,\xi)\rightarrow (v,u).$$ A similar construction involving retractions allows to equip the above with a natural metrizable topology and the local models for these spaces are $K\rightarrow O$, where $O$ is a retraction in an Hilbert space, and similarly for $K$ in a product of Hilbert spaces, where in this case the retraction is linear on fibers, with respect to a projection of the product onto the first factor. We refer the reader to [@HWZ6; @HWZ8.7] for details.
A Good Uniformizer
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In the following discussion we shall need the version of Deligne-Mumford theory of stable Riemann surfaces using the exponential gluing profile $\varphi$. The discussion is in spirit completely parallel to the one for the category ${\mathcal R}$. However, the difference is that all occurring maps in the discussion of ${\mathcal R}$ are either smooth or holomorphic (depending on the gluing profile), but in the stable map discussion they are only continuous.
We shall now describe the construction of the functors $\Psi$ which were introduced in Subsection \[SSS2\]. Fix an object $\alpha$ in ${\mathcal S}^{3,\delta_0}(Q,\omega)$ with automorphism group $G$ and write $\alpha=(S,j,M,D,u)$. As a consequence of the stability condition we obtain the following.
The group $G$ is a finite.
We need the notion of a domain stabilization.
A domain stabilization $\Xi$ for $\alpha$ consists of a finite subset $\Xi\subset S\setminus(M\cup|D|)$ so that the following holds.
- $(S,j,M^\ast,D)$ with $M^\ast$ being the unordered set $M\cup\Xi$ is a stable, perhaps noded, Riemann surface.
- $\Xi$ is an invariant set under the action of $G$.
- If $z,z'\in \Xi$ and $u(z)=u(z')$, then there exists a $g\in G$ with $g(z)=z'$.
- For every $z\in \Xi$ the map $Tu(z):T_zS\rightarrow T_{u(z)}Q$ is injective and pulls back $\omega$ to a non-degenerate two-form on $T_zS$ defining the same orientation as $j$.
One easily proves that domain stabilizations always exist as a consequence of the stability assumption.
Every object $\alpha$ in ${\mathcal S}^{3,\delta_0}(Q,\omega)$ has a domain stabilization.
After fixing a domain stabilization we have two stable objects, namely the stable map $\alpha$ in $\text{obj}({\mathcal S}^{3,\delta_0}(Q))$ and the stable noded Riemann surface with unordered marked points $\alpha^\ast=(S,j,M\cup\Xi,D)$, which is an object in ${\mathcal R}$. The automorphism group $G$ of $\alpha$ is contained in the automorphism group $G^\ast$ of $\alpha^\ast$, and both are finite groups.
We continue with $\alpha^\ast$ and pick a small disk structure ${\bf D}$ which is invariant under $G^\ast$. Then we fix a good deformation $v\rightarrow j(v)$, with $v\in V$, where $V\subset H^1(\alpha^\ast)$ is an open $G^\ast$-invariant subset. We can construct the functor associated to $$(v,\mathfrak{a})\rightarrow (S_\mathfrak{a},j(v)_\mathfrak{a},M^\ast_{\mathfrak{a}}, D_\mathfrak{a}),$$ where $M^\ast=M\cup\Xi$. Taking a sufficiently small $G^\ast$-invariant open neighborhood $O$ of $(0,0)$ in $ H^1(\alpha^\ast)\times{\mathbb B}^{\alpha^\ast}$ we obtain a good uniformizer $$\Psi^\ast:G^\ast\ltimes O\rightarrow {\mathcal R}.$$ Denote for $z\in\Xi$ by $[z]$ the $G$-orbit. It follows, that if $[z]\neq [z']$, then $u(z)\neq u(z')$. Of course, $u$ takes the same value on all the elements of $[z]$. For every $[z]$ we fix a smooth oriented submanifold $H_{[z]}$ in $Q$ of codimension two so that $u(z)\in H_{[z]}$ and the oriented $\text{range}(Tu(z))\oplus H_{[z]}$ equals the oriented $T_{u(z)}Q$. Then $u$ intersects $H_{[z]}$ transversally at $z$. We assume that $H_{[z]}$ does not have a boundary. It is even fine to assume that it is diffeomorphic to an open ball. Denote the whole collection ${(H_{[z]})}$ by ${\mathcal H}$.
Recall our topological space $\bar{X}^{3,\delta_0}_{(S,j,D),{\bf D}}(Q)$ which contains $u:(S,D)\rightarrow Q$. From the Sobolev embedding theorem we know that $H^3_{loc}\rightarrow C^1_{loc}$. As a consequence the subset $\bar{X}^{3,\delta_0}_{(S,j,D),{\bf D},{\mathcal H}}(Q)$, consisting of all $w:(S_\mathfrak{a},D_\mathfrak{a})\rightarrow Q$ which at $z\in\Xi$ intersects $H_{[z]}$ transversally, is well-defined. We also recall that if $w\in \bar{X}^{3,\delta_0}_{(S,j,D),{\bf D}}(Q)$ is close enough to $u$ there is a unique intersection point $z_w$ near $z$ so that $w$ intersects $H_{[z]}$ transversally. Moreover $w\rightarrow z_w$ is continuous. One can use this to prove the following before mentioned result.
For every map $w$ in $\bar{X}^{3,\delta_0}_{(S,j,D),{\bf D},{\mathcal H}}(Q)$ there exists an open neighborhood $U(w)$ in $\bar{X}^{3,\delta_0}_{(S,j,D),{\bf D}}(Q)$ and a continuous map $r:U(w)\rightarrow U(w)$ so that $r(U(w))=U(w)\cap \bar{X}^{3,\delta_0}_{(S,j,D),{\bf D},{\mathcal H}}(Q)$.
With other words the subspace associated to the constraints is a local continuous retract. This part is very important and ties in with the DM-theory. With our original object being the stable map $\alpha=(S,j,M,D,u)$ it holds that $(S,D,u)\in \bar{X}^{3,\delta_0}_{(S,j,D),{\bf D},{\mathcal H}}(Q)$. We also have the object $\alpha^\ast$ in ${\mathcal R}$ and with the small disk structure we obtained the good uniformizing family $$(\mathfrak{a},v)\rightarrow (S_\mathfrak{a},j(v)_\mathfrak{a},(M\cup\Xi)_\mathfrak{a},D_\mathfrak{a})$$ which was used to construct the good uniformizer $$\Psi: G^\ast\ltimes O^\ast\rightarrow {\mathcal R}.$$ Next we form the topological product space $V\times \bar{X}^{3,\delta_0}_{(S,j,D),{\bf D},{\mathcal H}}(Q)$ and bring the two separate discussions together. We define for its elements the map $$(v,(S_\mathfrak{a},D_\mathfrak{a},w))\rightarrow (S_\mathfrak{a},j(v)_\mathfrak{a},M_\mathfrak{a},D_\mathfrak{a},w).$$ The group $G$ acts on the topological space $V\times \bar{X}^{3,\delta_0}_{(S,j,D),{\bf D},{\mathcal H}}(Q)$ and the above map defines a functor $${\Psi}: G\ltimes (V\times \bar{X}^{3,\delta_0}_{(S,j,D),{\bf D},{\mathcal H}}(Q))\rightarrow {\mathcal S}^{3,\delta_0}(Q).$$ The fundamental observation is the following, which follows from the results in [@HWZ6].
\[IMP\] For a suitable $G$-invariant open neighborhood ${O}$ of $$(0,(S,D,u))\in V\times \bar{X}^{3,\delta_0}_{(S,j,D),{\bf D},{\mathcal H}}(Q)$$ the following properties hold.
- ${\Psi}:G\ltimes {O}\rightarrow {\mathcal S}^{3,\delta_0}(Q)$ is full and faithful.
- The map induced on orbit spaces is a homemomorphism onto an open neighborhood of $|\alpha|$.
- If $(v,(S_\mathfrak{a},D_\mathfrak{a},w))\in {O}$, then $(\mathfrak{a},v)\in O^\ast$, where we recall the good unifomizer for ${\mathcal R}$ denoted by $\Psi:G^\ast\ltimes O^\ast\rightarrow {\mathcal R}$.
- For every $q:=(v,(S_\mathfrak{a},D_\mathfrak{a},w))\in {O}$ there exists an open neighborhood $U({q})\subset {O}$ so that every sequence $({q}_k)$ in $U({q})$, for which $(|{\Psi}({q}_k)|)$ converges in $|{\mathcal S}^{3,\delta_0}(Q)|$, has a subsequence ${({q}_{k_\ell})}$ which converges in ${\operatorname{cl}}_{O}(U({q}))$.
We note that the construction of ${\Psi}$ requires several choices. However starting with an object $\alpha$ we obtain a set worth ${F}(\alpha)$ of functors having the properties stated in the theorem. Hence again we obtain a functorial construction $${F}:({\mathcal S}^{3,\delta_0}(Q,\omega))^-\rightarrow \text{SET}.$$
Moreover, if $\Phi:\alpha\rightarrow \alpha'$ is an isomorphism the possible choices made for $\alpha$ can be bijectively identified with choices for $\alpha'$. This means $F$ is, in fact, a functor ${\mathcal S}^{3,\delta_0}(Q,\omega)\rightarrow \text{SET}$. Further there is a precise geometric relationship between $\Psi$ and $\Psi'=F(\Phi)(\Psi)$, similar to the ${\mathcal R}$-case.
Assume that we are given two such functors ${\Psi}$ and ${\Psi}'$ associated to stable maps $\alpha$ and $\alpha'$ and suppose there exist ${q}_0$ and ${q}_0'$ for which there exists an isomorphism $$\Phi_0:{\Psi}({q}_0)\rightarrow {\Psi}'({q}_0').$$ Then $\Phi_0$ has an underlying biholomorphic map $\phi_0$ and, as we shall see, taking a variation ${q}'$ of ${q}'_0$ there is a uniquely determined core-continuous (see [@HWZ6]) germ ${q}'\rightarrow \phi_{q'}$, and a germ of homeomorphism ${q}\rightarrow {q}({q}')$ so that $\Phi_{{q}'}:=({\Psi}({q}({q'})),\phi_{q'},{\Psi}({q}'))$ satisfies $$\Phi_{{q}'}:{\Psi}({q}({q'}))\rightarrow {\Psi}({q'}).$$ One can use this fact to construct a topology on the transition sets ${\bf M}({\Psi},{\Psi'})$. For this topology the source and target maps will be local homeomorphisms and all structure maps will be continuous. One can find the details in [@HWZ6], where this program is carried out in the sc-smooth case, which we shall discuss later on. The current topological discussion is easier.
In summary, there is a natural construction $({F},{\bf M})$ for ${\mathcal S}^{3,\delta_0}(Q,\omega)$. Here $F$ associates to an object $\alpha$ a set ${F}(\alpha)$ of good uniformizers defined on metrizable translation groupoids, and ${\bf M}$ associates to the transition sets ${\bf M}(\Psi,\Psi')$ metrizable topologies, having the property that the structural maps become continuous and the source and target maps even local homeomorphisms.
The procedure we just outlined can be extended to deal with $$P:{\mathcal E}^{2,\delta_0}(Q,\omega,J)\rightarrow {\mathcal S}^{3,\delta_0}(Q,\omega).$$ In this case the construction $({F},{\bf M})$ can be extend to a construction $({\bar{F}},{\bf \bar{M}})$ covering the construction $({F},{\bf M})$, where ${\bar{F}}:({\mathcal S}^{3,\delta_0}(Q,\omega))^-\rightarrow \text{SET}$, but associates to an object $\alpha$ a good bundle uniformizer $$\begin{CD}
G\ltimes K @>\bar{\Psi}>> {\mathcal E}^{2,\delta_0}(Q,\omega,J)\\
@VpVV @V P VV\\
G\ltimes O @>\Psi>> {\mathcal E}^{3,\delta_0}(Q,\omega).
\end{CD}$$ It requires variations of the previous constructions and the necessary details can be found in [@HWZ6; @H2].
One might raise the question, given the naturality of the construction, if there is perhaps more structure to be found. In particular, since perturbations of the pseudoholomorphic part of ${\mathcal S}^{3,\delta_0}(Q,\omega)$ for a given compatible almost complex structure are used to define Gromov-Witten invariants. The answer is a resounding ‘yes’. In fact all the constructions which occured are smooth constructions in an extension of differential geometry which relies on a more flexible notion of differentiablity in Banach spaces. In this differential geometry there is a much larger library of smooth finite-dimensional or infinite-dimensional local models for smooth spaces. These local models can have locally varying dimensions but still have tangent spaces. Moreover, there is a nonlinear Fredholm theory with the usual expected properties. We shall describe this in the next section.
Smoothness {#SER2}
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It is an amazing fact that the construction $({F},{\bf M})$ is not just topological, but in fact a smooth construction within a suitable framework of smoothness, which is quite different from the classical one. It will be described next. We keep in mind the occurrence of continuous retractions when constructing the domains of the good uniformizers.
Sc-Structures and Sc-Smooth Maps
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Assume we are given two Banach spaces $E$ and $F$ for which we have as vector spaces a continuous inclusion $E\subset F$. In interpolation theory, [@Tr], general methods are developed to construct Banach spaces which interpolate between $E$ and $F$. We take the concept of a scale (with suitable properties) from interpolation theory, but give it a new interpretation as a generalization of a smooth structure. This study was initiated in [@HWZ1; @HWZ2; @HWZ3], and leads to a quite unexpectedly rich theory. In [@HWZ7; @HWZ8] we streamlined the presentation and added many further developments, which have not been published before.
\[d1\] Let $E$ be a Banach space. An sc-smooth structure (or [sc-structure]{} for short) for $E$ consists of a nested sequence of Banach spaces $E_0\supset E_1\supset E_2\supset \cdots$ with $E_0=E$ so that
- The inclusion $E_{i+1}\rightarrow E_i$ is a compact operator.
- $E_\infty=\bigcap_i E_i$ is dense in every $E_m$.
\[example32\] A typical example is $E=L^2({\mathbb R})$ with the sc-structure given by $E_m:=H^{m,\delta_m}({\mathbb R})$, where $H^{m,\delta_m}({\mathbb R})$ is the Sobolev space of functions in $L^2$ so that the derivatives up to order $m$ weighted by $e^{\delta_m |s|}$ belong to $L^2$. Here $\delta_m$ is a strictly increasing sequence starting with $\delta_0=0$.
If $E$ and $F$ are sc-Banach spaces, then $E\oplus F$ has a natural sc-structure given by $$(E\oplus F)_m=E_m\oplus F_m.$$ Every finite-dimensional vector space has a unique sc-structure, namely the constant one, where $E_i=E$. If $E$ is infinite-dimensional the constant sequence violates (1) of Definition \[d1\]. We continue with some considerations about linear sc-theory.
Let $E$ be an sc-Banach space and $F\subset E$ a linear subspace. We call $F$ an [ sc-subspace]{} provided the filtration $F_i=F\cap E_i$ turns $F$ into an sc-Banach space. If $F\subset E$ is an sc-Banach space, then we say that it has an sc-complement, provided there exists an sc-subspace $G$ such $F_i\oplus G_i=E_i$ as topological linear sum for all $i$.
Let us note that a finite-dimensional subspace $F$ of $E$ has an sc-complement if and only if $F\subset E_\infty$, see [@HWZ1].
The linear operators of interest are those linear operators $T:E\rightarrow F$, which map $E_m$ into $F_m$ for all $m$, such that $T:E_m\rightarrow F_m$ is a bounded linear operator. We call $T$ an [ sc-operator]{}. An [ sc-isomorphism]{} $T:E\rightarrow F$ is a bijective sc-operator so that its inverse is also an sc-operator. Of particular interest are the linear [sc-Fredholm operators]{}.
A sc-operator $T:E\rightarrow F$ is said to be [sc-Fredholm]{} provided there exist sc-splittings $E=K\oplus X$ and $F=Y\oplus C$ so that $C$ and $K$ are smooth and finite-dimensional, $Y=T(X)$, and $T:X\rightarrow Y$ defines a linear sc-isomorphism.
We note that the above implies that $E_m=X_m\oplus K$ and $F_m=T(X_m)\oplus C$ for all $m$. The Fredholm index is by definition $${\operatorname{Ind}}(T)=\dim(K)-\dim(C).$$ Let us also observe that for every $m$ we have a linear Fredholm operator (in the classical sense) $T:E_m\rightarrow F_m$, which in particular have the same index and identical kernels.
Next we begin with the preparations to introduce the notion of an sc-smooth map. If $U\subset E$ is an open subset we can define an sc-structure for $U$ by the nested sequence ${(U_i)}_{i=0}^\infty$ given by $U_i=E_i\cap U$. We note that $U_\infty=\bigcap U_i$ is dense in every $U_m$. Considering $U$ with its sc-structure we see that $U_{i_0}\subset E_{i_0}$ also admits an sc-structure defined by $${(U_{i_0})}_m:=U_{i_0+m}.$$ We write $U^{i_0}$ for $U_{i_0}$ equipped with this sc-structure. Given two such sc-spaces $U$ and $V$ we write $U\oplus V$ for $U\times V$ equipped with the obvious sc-structure. Now we can give the rigorous definition of the tangent $TU$ of an open subset $U$ in an sc-Banach space $E$.
The [ tangent]{} $TU$ of an open subset $U\subset E$ of the sc-Banach space $E$ is defined by $TU=U^1\oplus E$.
We note that $$(TU)_i=U_{1+i}\oplus E_i.$$ Continuing Example \[example32\] we have $$TL^2({\mathbb R}) = H^{1,\delta_1}({\mathbb R})\oplus L^2({\mathbb R})\ \ \hbox{and}\ \
(TL^2({\mathbb R}))_i = H^{i+1,\delta_{i+1}}({\mathbb R})\oplus H^{i,\delta_i}({\mathbb R}).$$
Given two open subsets $U$ and $V$ in sc-Banach spaces, a map $f:U\rightarrow V$ is said to be of class ${ sc}^0$ provided for every $m$ the map $f$ maps $U_m$ into $V_m$ and the map $f:U_m\rightarrow V_m$ is continuous.
The following example takes a little bit of work.
Take $L^2({\mathbb R})$ with the previously defined sc-structure and define $$\Phi:{\mathbb R}\oplus L^2({\mathbb R}) \rightarrow L^2({\mathbb R}):(t,u)\rightarrow \Phi(t,u),$$ where $\Phi(t,u)(s)=u(s+t)$. Then $\Phi$ is ${\text{sc}}^0$.
Next we define the notion of an ${\text{sc}}^1$-map.
Let $U\subset E$ and $V\subset F$ be open subsets in sc-Banach spaces. An ${\text{sc}}^0$-map $f:U\rightarrow V$ is said to be ${ sc}^1$ provided for every $x\in U_1$ there exists a continuous linear operator $Df(x):E_0\rightarrow F_0$ so that the following holds.
- For $h\in E_1$ with $x+h\in U$ we have $$\lim_{\parallel h\parallel_1\rightarrow 0} \frac{1}{\parallel h \parallel_1}\cdot \parallel f(x+h)-f(x)-Df(x)h\parallel_0 =0.$$
- The map $Tf$ defined by $Tf(x,h)=(f(x),Df(x)h)$ for $(x,h)\in TU$ defines an ${\text{sc}}^0$-map $Tf:TU\rightarrow TV$.
Inductively we can define the notion of a $sc^k$-map. A map is $sc^\infty$ provided it is $sc^k$ for all $k$. The following result shows that the chain-rule holds. This is quite unexpected since this fact looks not compatible with (1) of the previous definition. However, one is saved by the compactness of the inclusions stipulated by our definition of sc-structure.
Assume that $U,V$ and $W$ are open subsets in sc-Banach spaces, and $f:U\rightarrow V$ and $g:V\rightarrow W$ are ${\text{sc}}^1$-maps. Then $g\circ f$ is ${\text{sc}}^1$ and $T(g\circ f)=(Tg)\circ (Tf)$. The same holds for sc$^k$ and sc-smooth maps.
One can show, see [@HWZ8.7], that the map $\Phi$ from the previous example is sc-smooth. Classically it is nowhere differentiable.
Sc-Smooth Spaces and M-Polyfolds
--------------------------------
Now we are in the position to introduce new local models for smooth spaces. The interesting thing about sc-smoothness is the fact that there are many smooth retractions with complicated images, so that one obtains a large ‘library’ of smooth local models for spaces. This library is large enough to describe problems occurring when studying the nonlinear Cauchy-Riemann operators in symplectic geometry, which shows analytical limiting behavior allowing for bubbling-off and similar analytical phenomena.
Let $U\subset E$ be an open subset of the sc-Banach space $E$. A map $r:U\rightarrow U$ is called a [${\text{sc}}^\infty$-retraction]{} provided it sc-smooth and $r\circ r=r$.
The chain rule implies that for a ${\text{sc}}^\infty$-retraction $r$ its tangent map $Tr$ is again an ${\text{sc}}^\infty$-retraction. We call the image $O=r(U)$ of an ${\text{sc}}^\infty$-retraction $r:U\rightarrow U$ an [${\text{sc}}^\infty$]{}-[retract]{}. The crucial definition is the following.
A [local sc-model]{} is a pair $(O,E)$, where $E$ is an sc-Banach space and $O\subset E$ an ${\text{sc}}^\infty$-retract given as the image of an sc-smooth retraction $r:U\rightarrow U$ defined on an open subset $U$ of $E$.
The following lemma is easily established.
Assume that $(O,E)$ is a local sc-model and $r$ and $s$ are sc-smooth retractions defined on open subsets $U$ and $V$ of $E$, respectively, having $O$ as the image. Then $Tr(TU)=Ts(TV)$.
In view of this lemma we can define the tangent of a local sc-model which again is a local sc-model as follows.
The [ tangent]{} of the local sc-model $(O,E)$ is defined by $$T(O,E):= (TO,TE),$$ where $TO=Tr(TU)$ for any sc-smooth retraction $r:U\rightarrow U$ having $O$ as the image, where $U$ is open in $E$.
Let us observe that if $(O,E)$ is a local sc-model and $O'$ an open subset of $O$, then $(O',E)$ is again a local sc-model. Indeed, if $r:U\rightarrow U$ is an sc-smooth retraction with $O=r(U)$, then define $U'=r^{-1}(O')$, and $r'=r|U':U'\rightarrow U'$ is an sc-smooth retraction with image $O'$.
A map $f:O\rightarrow O'$ between two local sc-models is [ sc-smooth (or ${\text{sc}}^k$)]{} provided $f\circ r:U\rightarrow E'$ is sc-smooth (or ${\text{sc}}^k$). One easily verifies that the definition does not depend on the choice of $r$. We can define the tangent map $Tf:TO\rightarrow TO'$ by $$Tf:= T(f\circ r)|Tr(TU).$$ As it turns out this is well-defined and does not depend on the choice of $r$ as long as it is compatible with $(O,E)$.
Assume that $(O,E)$, $(O',E')$ and $(O'',E'')$ are local sc-models and $f:O\rightarrow O'$ and $g:O'\rightarrow O''$ are ${\text{sc}}^1$. Then $g\circ f:O\rightarrow O''$ is ${\text{sc}}^1$ and $$T(g\circ f) =(Tg)\circ (Tf).$$ Moreover if $f,g$ are sc$^k$ so is $g\circ f$. The same for $sc^\infty$.
The following Remark \[2.18\] explains how the current account is related to [@HWZ1; @HWZ2; @HWZ3].
\[2.18\] In the series of papers [@HWZ1; @HWZ2; @HWZ3] we developed a generalized Fredholm theory in a slightly more restricted situation, which however is more than enough for the applications. Namely rather than considering ${\text{sc}}$-smooth retractions and ${\text{sc}}$-smooth retracts, splicings and open subsets of splicing cores were considered, which one can view as a special case. A splicing consists of an open subset $V$ in some sc-Banach space $W$ and a family of bounded linear projections $\pi_v:E\rightarrow E$, $v\in V$, where $E$ is another sc-Banach space, so that the map $$V\oplus E\rightarrow E:(v,e)\rightarrow\pi_v(e)$$ is sc-smooth. Then the associated splicing core is $K$, defined by $$K=\{(v,e)\in V\oplus E \ |\ \pi_v(e)=e\}.$$ Clearly $V\oplus E$ is an open subset in $W\oplus E$ and $r(v,e):=(v,\pi_v(e))$ is an sc-smooth retraction. The associated retract is, of course, the splicing core $K$. If $O$ is an open subset of $K$ we know that it is again an sc-smooth retract. Let us note that in all our applications the retractions are obtained from splicings. The above modifications have been implemented in [@HWZ7; @HWZ8].
We demonstrate next how the definition of a manifold can be generalized. Let $Z$ be a metrizable topological space. A chart for $Z$ is a tuple $(\varphi,U,(O,E))$, where $\varphi:U\rightarrow O$ is a homeomorphism and $(O,E)$ is a local sc-model. We say that two such charts are sc-smoothly compatible provided $$\psi\circ\varphi^{-1}:\varphi(U\cap V)\rightarrow \psi(U\cap V)$$ is sc-smooth and similarly for $\varphi\circ \psi^{-1}$. Here $(\psi,V,(P,F))$ is the second chart. Note that the sets $\varphi(U\cap V)$ and $\psi(U\cap V)$ are sc-smooth retracts for sc-smooth retractions defined on open sets in $E$ and $F$, respectively. An sc-smooth atlas for $Z$ consists of a family of sc-smoothly compatible charts so that their domains cover $Z$. Two sc-smooth atlases are compatible provided their union is an sc-smooth atlas. This defines an equivalence relation.
Let $Z$ be a metrizable space. An [sc-smooth structure]{} on $Z$ is given by an sc-smooth atlas. Two sc-smooth structures are [equivalent]{} if the union of the two associated atlases defines again an sc-smooth structure. A M-polyfold is a metrizable space $Z$ together with an equivalence class of sc-smooth structures.
We note that these spaces have a natural filtration $Z_0\supset Z_1\supset Z_2\supset \cdots $. The points in $Z_i$ one should view as the points of some regularity $i$. The sc-smooth spaces are a very general type of space on which one can define sc-smooth functions.
It is possible to generalize many of the constructions from differential geometry to these spaces. If we have an sc-smooth partition of unity we can define Riemannian metrics and consequently a curvature tensor. Note, however, that curvature would only be defined at points of regularity at least $2$. The existence of an sc-smooth partition of unity depends on the sc-structure.
The [tangent space]{} at a point of level at least one is defined in the same way as one defines them for Banach manifolds, see [@LA]. Namely one considers tuples $(z,\varphi,U,(O,E),h)$, where $z\in Z_1$ and $(\varphi,U,(O,E))$ a chart, so that $z\in U$, and $h\in T_{\varphi(z)}O$. Two such tuples, say the second is $(z',\varphi',U',(O',E'),h')$, are said to be equivalent provided $z=z'$ and $T(\varphi'\circ \varphi^{-1})(\varphi(z))h=h'$. An equivalence class $[(z,\varphi,U,(O,E),h)]$ then by definition is a tangent vector at $z$. The tangent space at $z\in Z_1$ is denoted by $T_zZ$ and we define $TZ$ as $$TZ=\bigcup_{z\in Z_1} \{z\}\times T_zZ.$$ One can show that $TZ$ has a natural M-polyfold structure so that the natural map $TZ\rightarrow Z^1$ is sc-smooth, see [@HWZ7].
Consider the metrizable space $Z$ given as the subspace of ${\mathbb R}^2$ defined by $$Z=\{(s,t)\in {\mathbb R}^2\ |\ t=0\ \hbox{if}\ s\leq 0\}.$$ Then $Z$ admits the structure of an M-polyfold . In order to see this, one constructs a topological embedding into ${\mathbb R}\oplus L^2({\mathbb R})$, where $L^2({\mathbb R})$ has the previously introduced sc-structure, in such a way that the image is an sc-smooth retract. Here the idea of an sc-smooth splicing comes in handy! Take a smooth, compactly supported map $\beta:{\mathbb R}\rightarrow [0,\infty)$ with $\int \beta(t)^2 ds=1$. Denote by $f_s$, for $s\in (0,\infty)$ the unit length element in $L^2$ defined by $$f_s(t)=\beta(t+e^\frac{1}{s}).$$ For $s\in (-\infty,0]$ we define $f_s=0$, and denote by $\pi_s$ the $L^2$-orthogonal projection onto the subspace spanned by $f_s$. Then a somewhat lengthy computation shows that $$r:{\mathbb R}\oplus L^2\rightarrow {\mathbb R}\oplus L^2:r(s,u)=(s,\pi_s(u))$$ is an ${\text{sc}}$-smooth retraction, with obvious image $O$ being $$\{(s,t\cdot f_s)\ |\ (s,t)\in {\mathbb R}^2\}.$$ Hence $(O,{\mathbb R}\oplus L^2)$ is a local sc-model. We note that it has varying dimension. The map $$Z\rightarrow {\mathbb R}\oplus L^2:(s,t)\rightarrow (s, t\cdot f_s)$$ is a homeomorphic embedding onto $O$. The map is clearly continuous and injective and has image $O$. Define ${\mathbb R}\oplus L^2\rightarrow {\mathbb R}^2$ by $$(s,x)\rightarrow \left(s,\int_{\mathbb R} x(t)f_s(t)dt\right).$$ This map is continuous and its restriction to $O$ is the inverse of the previously defined map. Hence we obtain the structure of an M-polyfold on $Z$. This gives us the first example of a finite-dimensional space, with varying dimension, which has a generalized manifold structure. We also note that the induced filtration is constant, so that a tangent space is defined at all points. This is due to the fact that the local model $O$ lies entirely in the smooth part of ${\mathbb R}\oplus L^2$. It is instructive to study sc-smooth curves $\phi:(-\varepsilon,\varepsilon)\rightarrow O$ satisfying $\phi(0)=(0,0)$. Modifications of the above construction allow us to put smooth structures on the spaces shown in Figure \[porkbarrel\]. The M-polyfold does not allow an sc-smooth embedding into any ${\mathbb R}^N$, since then it would have be a smooth manifold by [@H.Cartan] . However, as seen in the construction, it can be sc-smoothly embedded into an infinite-dimensional space.
Strong Bundles
--------------
The notion of a strong bundle is designed to give additional structures in the Fredholm theory, which guarantee a compact perturbation and transversality theory. The crucial point is the fact that there will be a well-defined vector space of perturbations, which have certain compactness properties. On the other hand these perturbations are plentiful enough to allow for different versions of Sard-Smale type theorems, [@smale], in the Fredholm theory.
Let us start with a non-symmetric product $U\triangleleft F$, where $U$ is an open subset in some sc-Banach space $E$, and $F$ is also an sc-Banach space. By definition, as a set $U\triangleleft F$ is the product $U\times F$, but in addition it has a double filtration $$(U\triangleleft F)_{m,k}=U_m\oplus F_k$$ defined for all pairs $(m,k)$ satisfying $0\leq k\leq m+1$. We view $U\triangleleft F\rightarrow U$ as a bundle with base space $U$ and fiber $F$, where the double filtration has the interpretation that above a point $x\in U$ of regularity $m$ it makes sense to talk about fiber regularity of a point $(x,h)$ up to order $k$ provided $k\leq m+1$. At this point it is not clear why one introduces this non-symmetric product coming with a non-symmetric double filtration. We refer the reader to the later Example \[examp\] explaining why it is introduced.
Given $U\triangleleft F$, we might consider the associated sc-spaces $U\oplus F$ and $U\oplus F^1$. Of interest for us are the maps $$\Phi:U\triangleleft F\rightarrow V\triangleleft G$$ of the form $$\Phi(u,h)=(\varphi(u),\phi(u,h))$$ which are linear in $h$.
We say that the map $\Phi$ as described above is of class ${\text{sc}}^0_\triangleleft$, provided it induces ${\text{sc}}^0$-maps $U\oplus F^i\rightarrow V\oplus G^i$ for $i=0,1$.
We define the tangent $T(U\triangleleft F)$ by $$T(U\triangleleft F) = (TU)\triangleleft (TF).$$ Note that the order of the factors is different from the order in $T(U\oplus F)$. One has to keep this in mind. Indeed, $$T(U\triangleleft F)=U_1\oplus E\oplus F_1\oplus F\ \ \hbox{and}\ \ T(U\oplus F)=U_1\oplus F_1\oplus E\oplus F.$$
A map $\Phi:U\triangleleft F\rightarrow V\triangleleft G$ is of class ${\text{sc}}^1_\triangleleft$ provided the maps $\Phi:U\oplus F^i\rightarrow V\oplus G^i$ for $i=0,1$ are ${\text{sc}}^1$. Taking the tangents of the latter, gives after rearrangement, the ${\text{sc}}^0_\triangleleft$-map $$T\Phi:(TU)\triangleleft (TF)\rightarrow (TV)\triangleleft (TG).$$ Iteratively we can define what it means that a map is ${\text{sc}}^k_\triangleleft$ for $k=1,2,\ldots$ and we can also define ${\text{sc}}_\triangleleft$-smooth maps.
Given $U\triangleleft F\rightarrow U$, a sc-smooth section $f$ is map of the form $x\rightarrow (x,\bar{f}(x))$ such that the induced map $U\rightarrow U\oplus F$ is sc-smooth. In particular, $f$ is ‘horizontal’ with respect to the filtration, i.e. a point on level $m$ is mapped to a point of bi-level $(m,m)$. This can be considered as a convention, and it is precisely this convention which is responsible for the filtration constraint $k\leq m+1$. There is another class of sections called ${\text{sc}}^+$-sections. These are sc-smooth sections of $U\triangleleft F\rightarrow U$ which induce sc-smooth maps $U\rightarrow U\oplus F^1$. In particular, if $s$ is an ${\text{sc}}^+$-section of $U\triangleleft F\rightarrow U$ and $s(x)=(x,\bar{s}(x))$ for $x\in U_m$ then $\bar{s}(x)\in F_{m+1}$. This type of sections will be important for the perturbation theory. Indeed, it is a kind of compact perturbation theory since the inclusion $F_{m+1}\rightarrow F_m$ is compact. We give an example before we generalize an earlier discussion about retracts and retractions to bundles of the type $U\triangleleft F\rightarrow U$.
\[examp\] Let us denote by $E$ the Sobolev space $H^1(S^1,{\mathbb R}^n)$ of loops. We define an sc-structure by $E_m= H^{1+m}(S^1,{\mathbb R}^n)$. Further we define $F=L^2(S^1,{\mathbb R}^n)=H^0(S^1,{\mathbb R}^n)$ which we filter via $F_m=H^m(S^1,{\mathbb R}^n)$. Finally we introduce $E\triangleleft F\rightarrow E$. Then we can view the map $f:x\rightarrow\dot{x}$ as an sc-smooth section of $E\triangleleft F\rightarrow E$. In particular, $f$ maps $E_m$ into $E_m\oplus F_m$. We observe that the filtration of $F$ is picked in such a way that the first order differential operator $x\rightarrow \dot{x}$ is an sc-smooth section, in particular, it is horizontal, i.e. the choices are made in such a way that they comply with our convention that sc-smooth sections are index preserving. The map $x\rightarrow x$ can be viewed as an ${\text{sc}}^+$-section. Then $x\rightarrow \dot{x}+x$ is an sc-smooth section obtained from the sc-smooth section $x\rightarrow \dot{x}$ via the perturbation by an ${\text{sc}}^+$-section. Consider now a smooth vector bundle map $$\Phi:{\mathbb R}^n\oplus {\mathbb R}^n\rightarrow
{\mathbb R}^n\oplus {\mathbb R}^n$$ of the form $$\Phi(x,h)=(\varphi(x),\phi(x)h),$$ where $\varphi:{\mathbb R}^n\rightarrow {\mathbb R}^n$ is a diffeomorphism and for every $x\in {\mathbb R}^n$ the map $\phi(x):{\mathbb R}^n\rightarrow {\mathbb R}^n$ is a linear isomorphism. Then we define for $(x,h)\in E\oplus F$ the element $\Phi_\ast(x,h)(t)=(\varphi(x(t)),\phi(x(t))h(t))$. Note that if $x\in E_m$ and $h\in F_{k}$ for $k\leq m+1$, then $\Phi_\ast(x,h)=:(y,\ell)$ satisfies $y\in E_m$ and $\ell\in F_k$. However, if $x\in E_m$ and $y\in F_k$ for some $k>m+1$ we cannot conclude that $\ell\in F_k$. We can only say that $\ell\in F_{m+1}$. Now one easily verifies that $$\Phi_\ast:E\triangleleft F\rightarrow E\triangleleft F$$ is ${\text{sc}}_\triangleleft$-smooth. This justifies our constraint $k\leq m+1$ for the double filtration.
An [ ${\text{sc}}^\infty_\triangleleft$-retraction]{} is an ${\text{sc}}_\triangleleft$-smooth map $$R:U\triangleleft F\rightarrow U\triangleleft F$$ with the property $R\circ R=R$.
Of course, $R$ has the form $R(u,h)=(r(u),\phi(u,h))$ with $r$ being an ${\text{sc}}$-smooth retraction and $\phi(u,h)$ linear in the fiber. Given $R$, we can define its image $K=R(U\triangleleft F)$ and $O=r(U)$. Then we have a natural projection map $$p:K\rightarrow O.$$ We may view this as the local model for a strong bundle. Observe that $K$ has a double filtration and $p$ maps points of regularity $(m,k)$ to points of regularity $m$.
The tuple $(K,E\triangleleft F)$, where $K$ is a subset of $E\triangleleft F$, so that there exists an ${\text{sc}}^\infty_\triangleleft$-retraction $R$ defined on $U\triangleleft F$, where $U\subset E$ is open and $K=R(U\triangleleft F)$, is called a [local strong bundle model]{}.
Starting with $(K,E\triangleleft F)$ we have the projection $K\rightarrow E$ and denote its image by $O$ and the induced map by $p:K\rightarrow O$. One can define $T(K,E\triangleleft F)$ by $$T(K,E\triangleleft F)=(TK, TE\triangleleft TF),$$ where $TK$ is the image of $TR$. As before we can show that the definition does not depend on the choice of $R$.
Now we are in the position to define the notion of a strong bundle. Let $p:W\rightarrow X$ be a surjective continuous map between two metrizable spaces, so that for every $x\in X$ the space $W_x:=p^{-1}(x)$ comes with the structure of a Banach space. A [strong bundle chart]{} is a tuple $(\Phi,p^{-1}(U),E\triangleleft F))$, where $\Phi:p^{-1}(U)\rightarrow K$ is a homeomorphism, covering a homeomorphism $\varphi:U\rightarrow O$, which between each fiber is a bounded linear operator $$\begin{CD}
p^{-1}(U) @>\Phi >> K\\
@V p VV @ VVV\\
U @>\varphi >> O.
\end{CD}$$ We call two such charts [ ${\text{sc}}_\triangleleft$-smoothly equivalent]{} if the associated transition maps are ${\text{sc}}_\triangleleft$-smooth. We can define the notion of a [ strong bundle atlas]{} and can define the notion of equivalence of two such atlases.
Let $p:W\rightarrow X$ be as described before. A [ strong bundle structure]{} for $p$ is given by a strong bundle atlas. Two strong bundle structures are [equivalent]{} if the associated atlases are equivalent. Finally $p$ equipped with an equivalence class of strong bundle atlases is called a [strong bundle]{}.
Let us observe that a strong bundle $p:W\rightarrow X$ admits a double filtration $W_{m,k}$ with $0\leq k\leq m+1$. By forgetting part of this double filtration we observe that $W(0)$, which is $W$ filtered by $W(0)_m:=W_{m,m}$, has in a natural way the structure of an M-polyfold. The same is true for $W(1)$ which is the space $W_{0,1}$ equipped with the filtration $W(1)_m:=W_{m,m+1}$. Obviously the maps $p:W(i)\rightarrow X$ for $i=0,1$ are sc-smooth.
The previously introduced notions of ${\text{sc}}$-smooth sections and ${\text{sc}}^+$-sections for $U\triangleleft F\rightarrow U$ generalize as follows.
Let $p:W\rightarrow X$ be a strong bundle over the M-polyfold $X$ (without boundary).
- A [sc-smooth section]{} of the strong bundle $p$ is an ${\text{sc}}^0$-map $s:X\rightarrow W$ with $p\circ s=Id_X$ such that $s:X\rightarrow W(0)$ is sc-smooth. The vector space of all such sections is written as $\Gamma(p)$.
- A sc$^+$-section section of the strong bundle $p$ is a sc$^0$-map $s:X\rightarrow W(1)$ with $p\circ s=Id_X$, which in addition is sc-smooth. The vector space of ${\text{sc}}^+$-sections is denoted by $\Gamma^+(p)$.
In some sense ${\text{sc}}^+$-sections are compact perturbations, since the inclusion map $W(1)\rightarrow W(0)$ is fiber-wise compact. They are very important for the perturbation theory.
A Special Class of Sc-Smooth Germs
----------------------------------
The next goal is to define a suitable notion of Fredholm section of a strong bundle. The basic fact about the usual Fréchet differentiability is the following. If $f:U\rightarrow F$ is a smooth map (in the usual sense) between an open neighborhood $U$ of $0\in E$ with target the Banach space $F$, and satisfying $f(0)=0$, then we can describe the solution set of $f=0$ near $0$ by an implicit function theorem provided $df(0)$ is surjective and the kernel of $df(0)$ splits, i.e has a topological linear complement. So smoothness and some properties of the linearized operator at a solution give us always qualitative knowledge about the solution set near $0$. On the other hand $f:U\rightarrow F$ being only sc-smooth, $df(0)$ being surjective and its kernel having an sc-complement is not enough to conclude much about the solution space near $0$. However, as we shall see there is a large class of sc-smooth maps for which a form of the implicit function theorem holds. In applications the class is large enough to explain gluing constructions (à la Taubes and Floer) as smooth implicit function theorems in the sc-world.
One of the issues which has to be addressed at some point is the fact, that the spaces we are concerned with, have locally varying dimensions. Though it might sound like a major issue it will turn out that there is a simple way to deal with these type of problems. It is a crucial observation, that in applications base and fiber dimension change coherently. The sc-formalism incorporates this with a minimum amount of technicalities. One should remark that our presentation is slightly more general than the one given in [@HWZ2].
Let us begin with some notation. As usual $E$ is an sc-Banach space. We shall write $\mathcal{O}(E,0)$ for an unspecified nested sequence $U_0\supset U_1\supset U_2\supset \cdots$, where every $U_i$ is an open neighborhood of $0\in E_i$. Note that this differs from previous notation where $U_i=E_i\cap U$. When we are dealing with germs we always have the new definition in mind. A [sc-smooth germ]{} $$f:{\mathcal O}(E,0)\rightarrow F$$ is a map defined on $U_0$ so that for points $x\in U_1$ the tangent map $Tf:U_1\oplus E_0\rightarrow TF$ is defined, which again is a germ $$Tf:{\mathcal O}(TE,0)\rightarrow TF.$$ We introduce a [basic class]{} $\mathfrak{C}_{basic}$ of germs of maps as follows.
An element in $\mathfrak{C}_{basic}$ is an sc-smooth germ $$f:\mathcal{O}({\mathbb R}^{n}\oplus W,0)\rightarrow ({\mathbb R}^N\oplus W,0)$$ for suitable $n$ and $N$, and an sc-Banach space $W,$ so that the following holds. If $P:{\mathbb R}^N\oplus W\rightarrow W$ is the projection, then $P\circ f$ has the form $$P\circ f(r,w)=w-B(r,w)$$ for $(r,w)\in U_0\subset {\mathbb R}^{n}\oplus W$. Moreover, for every $\varepsilon>0$ and $m\in {\mathbb N}$ we have $${\left\VertB(r,w)-B(r,w')\right\Vert}_m\leq \varepsilon\cdot {\left\Vertw-w'\right\Vert}_m$$ for all $(r,w), (r,w')\in U_m$ close enough to $(0,0)$ on level $m$.
In [@HWZ2] the class of basic germs was slightly more general in the sense that it was not required that $f(0)=0$ in its definition. However, all important results were then proved under the additional assumption that $f(0)=0$. In the applications to SFT and the other mentioned theories one can bring the occurring nonlinear elliptic differential operators even at bubbling-off points (modulo a filling, which is a crucial concept in the polyfold theory and will be explained shortly) via sc-smooth coordinate changes into the above form, see [@HWZ6], [@H2] for Gromov-Witten theory, and [@FHWZ] for the operators in SFT. It is important to note that if $f$ is sc-smooth so that $f(0)=0$ and $Df(0)$ is sc-Fredholm, it is generally not true that after a change of coordinates $f$ can be pushed forward to an element which belongs to $\mathfrak{C}_{basic}$.
As shown in [@HWZ2; @HWZ7], basic germs admit something like an infinitesimal smooth implicit function theorem near $0$ (this is something intrinsic to sc-structures) which for certain maps can be ‘bound together’ to a local implicit function theorem.
To explain this, assume that $U\subset E$ is an open neighborhood of $0$ and $f:U\rightarrow F$ is an sc-smooth having the following properties, where $U_i=E_i\cap U$.
- $f(0)=0$ and $Df(0)$ is a surjective sc-Fredholm operator.
- $f$ is regularizing. This means if $x\in U_m$ and $f(x)\in F_{m+1}$, then $x\in U_{m+1}$.
- Viewing $f$ as a section of $U\triangleleft F\rightarrow U$, near every smooth point $x$, and for a suitable ${\text{sc}}^+$-section with $s(x)=f(x)$ the germ $$f-s:{\mathcal O}(E,0)\rightarrow F$$ is conjugated to a basic germ.
Under these conditions there is a local implicit function theorem near $0$, which guarantees a local solution set of dimension being the Fredholm index of $Df(0)$ at $0$, and in addition guarantees a natural manifold structure on this solution set.
The infinitesimal implicit function theorem refers to the following phenomena for basic germs. If $f\in \mathfrak{C}_{basic}$, then $Pf(a,w)=w-B(a,w)$, where $B$ is a family of contractions on every level $m$ near $(0,0)$. Hence, using Banach’s fixed point theorem we find a germ $\delta_m$ solving $\delta_m(a)=B(a,\delta_m(a))$ on level $m$ for $a$ near $0$. By uniqueness a solution on level $m$ also solves the problem on lower levels. This implies that we have a solution germ $a\rightarrow (a,\delta(a))$ of $Pf(a,w)=0$. The infinitesimal sc-smooth implicit function theorem gives the nontrivial fact that the germ $$\delta:{\mathcal O}({\mathbb R}^{n},0)\rightarrow (W,0)$$ is an sc-smooth germ, see [@HWZ2; @HWZ7].
In summary, as we shall discuss in more detail later, if we have a regularizing sc-smooth section which around every smooth point is conjugated mod a suitable ${\text{sc}}^+$-section to a basic germ, then the ‘infinitesimal’ implicit function theorems around points $y$ near $x$, combine together to give a ‘local’ implicit function theorem near a point $x$ where the linearization is surjective. We refer the reader to [@HWZ7] for a comprehensive discussion.
Sc-Fredholm Sections
--------------------
Assume next that $p:K\rightarrow O$ is a strong local bundle, i.e. $(K,E\triangleleft F)$ is a local strong bundle model. Suppose $f:{\mathcal O}(O,x)\rightarrow K$ is a germ which we shall write as $[f,x]$.
A [filling]{} for the germ $[f,x]$ consists of the following data.
- An sc-smooth germ $\bar{f}:{\mathcal O}(E,x)\rightarrow F$.
- A choice of strong bundle retraction $R:U\triangleleft F\rightarrow U\triangleleft F$ such that $K$ is the image of $R$.
Viewing $f$ as a map $O\rightarrow F$ such that $\phi(y)f(y)=f(y)$, where $R(y,h)=(r(y),\phi(y)h)$, we assume that the data satisfies the following properties:
- $\bar{f}(y)=f(y)$ for all $y\in O$ near $x$.
- $\bar{f}(y)=\phi(r(y))\bar{f}(y)$ for $y$ near $x$ in $U$ implies that $y\in O$.
- The linearisation of the map $$y\rightarrow (Id-\phi(r(y)))\bar{f}(y)$$ at $x$ restricted to the ${$\text{ker}\ {(}$}Dr(x))$ defines a linear topological isomorphism ${$\text{ker}\ {(}$}Dr(x))\rightarrow {$\text{ker}\ {(}$}\phi(x))$.
The germ $[f,x]$ is said to be [ fillable]{} provided there exists a germ of strong bundle map $\Phi$, covering a germ of (local) sc-diffeomorphism $\varphi$, so that the push-forward germ $[\Phi_\ast(f),\varphi(x)]$ has a filling. A [ filled version]{} of $[f,x]$ is an sc-smooth germ $[\bar{g},\bar{x}]$ obtained as a filling of a suitable push-forward.
In the definition the meaning that $\Phi$ is a germ of strong bundle map covering $\varphi$ is the following. For the given $(K,U\triangleleft F)$, $p:K\rightarrow O$, with $x\in O\subset U$, there exists $(K',U'\triangleleft F')$, $p':K'\rightarrow O'$, with $x'\in O'\subset E'$ and open neighborhoods $x\in V\subset O$, $x'\in V'\subset O'$, so that the following is a commutative diagram associated to a strong bundle isomorphism $$\begin{CD}
p^{-1}(V) @>\Phi >> {(p')}^{-1}(V')\\
@Vp VV @V p' VV\\
V @>\varphi >> V'.
\end{CD}$$ Moreover the size of $V$ and then of $V'$ is unspecified small, but fixed. Also in this case $V_i=O_i\cap V$. Only for the solution germs the neighborhoods in higher regularity shrink.
If $[f,x]$ has a filling $[\bar{f},x]$, the local study of $f(y)=0$ with $y\in O$ near $x$ is equivalent to the local study of $\bar{f}(y)=0$ where $y\in U$ close to $x$. Let us note that if $f(x)=0$ the linearisation $f'(x):T_xO\rightarrow K_x$ has the same kernel as $\bar{f}'(x):T_xU\rightarrow F_x$ and the cokernels are naturally isomorphic.
If $f$ is an sc-smooth section of a strong M-polyfold bundle $p:W\rightarrow X$ with $\partial X=\emptyset$, and $x$ is a smooth point, we say that the germ $[f,x]$ admits a filled version, provided in a suitable local coordinate representation $[f,x]$ admits a filled version as defined in the previous definition. We always may assume that the filled version has the form $g:{\mathcal O}(E,0)\rightarrow F$.
We recall the notion of a regularizing section, which we already mentioned before.
Let $p:W\rightarrow X$ be a strong bundle over the M-polyfold $X$ (without boundary) and $f$ an sc-smooth section. We say that $f$ is [regularizing]{} provided for a point $x\in X$ the assertion $f(x)\in W_{m,m+1}$ implies that $x\in X_{m+1}$.
Note that for a regularizing section $f$ a solution $x$ of $f(x)=0$ belongs necessarily to $X_\infty$. If $f$ is regularizing and $s\in\Gamma^+(p)$, then $f+s$ is regularizing. Now we come to the crucial definition.
We call the sc-smooth section $f$ of the strong bundle $p:W\rightarrow X$, over an M-polyfold with $\partial X=\emptyset$, an [sc-Fredholm section]{}, provided $f$ is regularizing, and around every smooth point $x$ the germ $[f,x]$ has a filled version $[g,0]$ so that for a suitable germ of ${\text{sc}}^+$-section $s$ with $s(0)=g(0)$ the germ $[g-s,0]$ is conjugated to an element in $\mathfrak{C}_{basic}$. We denote the collection of all sc-Fredholm sections of $p$ by ${\mathcal F}(p)$.
An sc-Fredholm section according to the above definition is slightly more general than the sc-Fredholm sections defined in [@HWZ2]. An additional advantage of the current definition is the stability result saying, that given an sc-Fredholm section for $p:W\rightarrow X$ and an ${\text{sc}}^+$-section $s$, then $f+s$ is an sc-Fredholm section for $p:W\rightarrow X$. With the version given in [@HWZ2] one can only conclude that $f+s$ is an sc-Fredholm section of $p^1:W^1\rightarrow X^1$. In applications the difference is only ‘academic’. However, as far as a presentation is concerned this new version is more pleasant. see [@HWZ7].
The following stability result is crucial for the perturbation theory and rather tautological. In the setup of [@HWZ2] it was a nontrivial theorem. However, some of the burden is now moved to the implicit function theorem, see [@HWZ7] for the proofs in this new setup.
Let $p:W\rightarrow X$ a strong bundle over the M-polyfold $X$. Then given $f\in {\mathcal F}(p)$ and $s\in \Gamma^+(p)$ we have that $f+s\in {\mathcal F}(p)$.
Fredholm sections allow for an implicit function theorem.
Assume that $p:W\rightarrow X$ is a strong bundle over the M-polyfold $X$ without boundary. Let $f$ be a sc-Fredholm section and $x$ a smooth point such that $f(x)=0$ and $f'(x):T_xX\rightarrow W_x$ is surjective. Then the solution set near $x$ carries in a natural way the structure of a smooth manifold with dimension being the Fredholm index of $f'(x)$. In addition there exists an open neighborhood $V$ of $x$, so that for every $y\in V$ with $f(y)=0$ the linearisation $f'(y)$ is surjective. Moreover, its kernel can be identified with the tangent spaces of the solution set at $y$.
Finally we introduce the notion of an auxiliary norm and give a useful compactness result.
\[AUXN\] Assume that $p:W\rightarrow X$ is a strong bundle over the M-polyfold $X$ without boundary. An auxiliary norm for $p$ is a map $N:W\rightarrow {\mathbb R}^+\cup\{+\infty\}$ having the following properties.
- The restriction $N|W_{0,1}$ is real valued and continuous, and on $W\setminus W_{0,1}$ the map $N$ takes the value $\infty$.
- $N$ restricted to any fiber of $p:W_{0,1}\rightarrow X$ is a complete norm.
- If $(h_k)\subset W_{0,1}$ is a sequence such that $p(h_k)\rightarrow x_0$ in $X$, and $N(h_k)\rightarrow 0$, then $h_k\rightarrow 0_{x_0}$ in $W_{0,1}$.
The existence of an auxiliary norm can be establishes using continuous partitions of unity.
Given a strong bundle $p:W\rightarrow X$ over the M-polyfold $X$ without boundary there exists an auxiliary norm $N$. For two given auxiliary norms $N_1$ and $N_2$ there exist continuous maps $f_1,f_2:X\rightarrow (0,\infty)$ such that $$f_1\cdot N_1 \leq N_2\leq f_2\cdot N_1.$$
Now we are in the position to state a useful compactness result for sc-Fredholm sections which is important for the perturbation theory, see [@HWZ7].
Let $p:W\rightarrow X$ be a strong bundle over the M-polyfold $X$ without boundary. Suppose $f$ is an sc-Fredholm section for which $f^{-1}(0)$ is compact.
- Given an auxiliary norm $N$ there exists an open neighborhood $U$ of $f^{-1}(0)$ so that for every sc$^+$-section $s\in\Gamma^+(p)$ with support in $U$, satisfying $N(s(x))\leq 1$ for all $x\in X$, the set $(f+s)^{-1}(0)$ is compact.
- If $X$ admits sc-smooth partitions of unity we find for every $\varepsilon\in (0,1]$ an sc$^+$-section $s$ with support in $U$ so that $N(s(x))<\varepsilon$ for all $x$, and the set $M=(f+s)^{-1}(0)$ has the structure of a compact smooth manifold without boundary, so that the linearization $f'(m):T_mM\rightarrow W_m$ for all $m\in M$ is surjective and the tangent space $T_mM$ can be canonically identified with ${$\text{ker}\ {(}$}f'(m))$.
Polyfold Structures and Consequences
====================================
At this point we have discussed a smooth theory for the category of stable Riemann surfaces based on the construction of good uniformizers, and a topological theory associated to the category of stable maps. In order transform the latter discussion into one taking place in a smooth world, we have generalized the finite- or infinite-dimensional classical differential geometry to a more general sc-smooth differential geometry and described some of the aspects of an associated nonlinear functional analysis. The differential geometric/nonlinear functionalanalytic theory is discussed in great detail in [@HWZ7; @HWZ8] and gives many more ideas about the framework outlined in Section \[SER2\]. In the following we show how to use this theory to equip certain categories with sc-smooth structures and illustrate the ideas with the stable map example.
Polyfold Structures on Certain Categories
-----------------------------------------
It is useful to start with a general definition taken from [@HWZ8].
A good category with metrizable orbit space is given by a pair $({\mathcal C},{\mathcal T})$, where ${\mathcal C}$ is a category with the following properties.
- Between any two objects there are only finitely many morphisms and every morphism is an isomorphism.
- The orbit space $|{\mathcal C}|$ is a set.
Moreover ${\mathcal T}$ is a metrizable topology on $|{\mathcal C}|$. We call $({\mathcal C},{\mathcal T})$ a GCT (G=good, C=category, T=topology).
By our discussion in the previous sections the categories ${\mathcal R}$ and ${\mathcal S}^{3,\delta_0}(Q,\omega)$ are GCT’s. Precisely for such categories a construction of type $(F,{\bf M})$ is important and very often naturally exists if ${\mathcal C}$ is a category coming from geometric considerations. Next we give a precise definition of the type of previously discussed constructions in the polyfold framework.
Let ${\mathcal C}$ be a GCT. A good uniformizer for ${\mathcal C}$ around an object $c$ with automorphism group $G$, written as $$\Psi: G\ltimes O\rightarrow {\mathcal C},$$ consists of a M-polyfold $O$ with an sc-smooth action of $G$ on $O$, where $G\ltimes O$ is the associated translation groupoid, and $\Psi$ is a functor with the following properties.
- $\Psi$ is full and faithful.
- There exists a $q_0\in O$ with $\Psi(q_0)=c$.
- Passing to orbit spaces, $|\Psi|:{_{G}\backslash}O\rightarrow |{\mathcal C}|$ is a homeomorphism onto an open neighborhood of $|\alpha|$.
- Given $q\in O$ there exists an open neighborhood $U(q)\subset O$ with the property, that every sequence $(q_k)\subset U(q)$ for which $(|\Psi(q_k)|)$ converges in $|{\mathcal C}|$, has a convergent subsequence with limit in ${\operatorname{cl}}_O(U(q))$.
As in the continuous case we can define for two good uniformizers the transition set ${\bf M}(\Psi,\Psi')$ and have the usual structural maps. Here comes the crucial definition, which is the minimalistic version. Denote for the category ${\mathcal C}$ by ${\mathcal C}^-$ the category having the same objects, but as morphisms only the identities.
Let ${\mathcal C}$ be a GCT. A polyfold structure for ${\mathcal C}$ is a pair $(F,{\bf M})$, where $F:{\mathcal C}^-\rightarrow \text{SET}$ is a functor associating to an object $c$ a set $F(c)$ of good uniformizers, and where ${\bf M}$ associates to two good uniformizers, say $\Psi\in F(c)$ and $\Psi'\in F(c')$, a M-polyfold structure to the transition set ${\bf M}(\Psi,\Psi')$, so that all structural maps are sc-smooth, and the source and target map are local sc-diffeomorphisms.
Instead of denoting the polyfold structure by $(F,{\bf M})$ we just write $F$, i.e. $F\equiv(F,{\bf M}).$ Let us emphasize that $F(c)$ is a set of good uniformizers, i.e. a specifically picked collection of good uniformizers constructed by a given procedure. Since classically smooth manifolds are in particular M-polyfolds we see that the constructions associated to ${\mathcal R}$ equip it with a polyfold structure.
There are several useful points one should make.\
(a) Note that from $F$ the category ${\mathcal C}$ can be recovered. First of all, since $F$ is defined on ${\mathcal C}^-$, we know all the objects of ${\mathcal C}$. For two objects $c$ and $c'$ pick $\Psi\in F(c)$ and $\Psi'\in F(c')$, and let $q\in O$, $q'\in O'$ with $F(q)=c$ and $F(q')=c'$. Then we can identify the elements $(q,\phi,q')$ in ${\bf M}(\Psi(q),\Psi'(q'))$ with the morphisms in the set $\text{mor}_{\mathcal C}(c,c')$ via $(q,\phi,q')\rightarrow \phi$.\
(b) We also note that having a polyfold structure on ${\mathcal C}$ there exists a functor $$\text{reg}:{\mathcal C}\rightarrow {\mathbb N}_0\cup\{\infty\}=:{\mathbb N}^\infty_0,$$ which associates to an object its regularity. Here ${\mathbb N}_0^\infty$ has as morphisms only has the identities. Given an object $c$ pick $\Psi\in F(c)$ and $q_0\in O$ so that $\Psi(q_0)=c$. Then define $$\text{reg}(c)=\sup \{k\in {\mathbb N}_0\ |\ q_0\in O_k\}.$$ This is well-defined. independent of the choices involved, and a morphism invariant. We can define the full subcategories ${\mathcal C}_r$ of ${\mathcal C}$ associated to the objects $c$ with $\text{reg}(c)\geq r$.\
(c) Given a polyfold structure $F$ for ${\mathcal C}$ we consider ${\mathcal C}_1$ and define $F^1:{\mathcal C}_1\rightarrow \text{SET}$ by $$F^1(c)=\{\Psi^1:G\ltimes O^1\rightarrow {\mathcal C}_1\ |\ \Psi\in F(c)\}.$$ One can equip $|{\mathcal C}_1|$ with a uniquely determined metrizable topology which makes every element in $F^1(c)$ a good uniformizer. Then one can show that $F^1$ defines a polyfold structure on ${\mathcal C}_1$. We shall denote by ${\mathcal C}^1$ the category ${\mathcal C}_1$ equipped with the polyfold structure.
We also need a bundle version. As in Subsection \[SSS23\] denote by $\text{BAN}_G$ the category whose objects are Banach spaces and the morphisms are invertible topological linear isomorphisms. Suppose we are given a category ${\mathcal C}$ and a functor $$\mu:{\mathcal C}\rightarrow \text{BAN}_G.$$ Then we can build a new category ${\mathcal E}={\mathcal E}_\mu$ whose objects are pairs $(c,h)$, where $c$ is an object in ${\mathcal C}$ and $h$ is a vector in $\mu(c)$. The morphisms $(c,h)\rightarrow (c',h')$ are the lifts of the morphisms $\phi:c\rightarrow c'$, namely $$(\phi,\mu(\phi),h):(c,h)\rightarrow (c',h')$$ provided $s(\phi)=c$, $t(\phi)=c'$ and $\mu(\phi)(h)=h'$. We shall abbreviate ${\widehat}{\phi}=(\phi,\mu(\phi))$, which is a linear topological isomorphism $${\widehat}{\phi}:P^{-1}(s(\phi))\rightarrow P^{-1}(t(\phi)).$$ The class of all morphisms for the category ${\mathcal E}_\mu$ we shall denote by $\boldsymbol{\mathcal E}_\mu$. Since $|{\mathcal C}|$ is a set the same is true for $|{\mathcal E}|$. We denote by $P=P_\mu:{\mathcal E}_\mu\rightarrow {\mathcal C }$ the functor which on objects $(c,h)\rightarrow c$ and on morphisms $(\phi,\mu(\phi),h)\rightarrow \phi$. We also have the source map $s:\boldsymbol{\mathcal E}_\mu\rightarrow {\mathcal E}_\mu$ defined by $s(\phi,\mu(\phi),h)=(s(\phi),h)$ and the target map $t(\phi,\mu(\phi),h)=(t(\phi),\mu(\phi)(h))$.
A bundle GCT is given by a tuple $({\mathcal C},\mu,{\mathcal T}_\mu)$, where ${\mathcal C}$ is a GCT, $\mu:{\mathcal C}\rightarrow \text{BAN}_G$ is a functor, and ${\mathcal T}_\mu$ a metrizable topology on $|{\mathcal E}_\mu|$, so that $|P_\mu|:|{\mathcal E}_\mu|\rightarrow |{\mathcal C}|$ is continuous and open.
We shall introduce the notion of strong bundle uniformizers. The strong bundle uniformizer are build on strong bundles $p:K\rightarrow O$ over M-polyfolds, equipped with an action of a finite group $G$ acting by sc-smooth strong bundle isomorphisms inducing an action of $G$ on $O$ by sc-diffeomorphisms, so that $p$ is equivariant.
A good strong bundle uniformizer for the bundle GCT $({\mathcal C},\mu,{\mathcal T}_\mu)$ around an object $c$ in ${\mathcal C}$ is a functor $\bar{\Psi} : G\ltimes K\rightarrow {\mathcal E}$ covering a functor $\Psi:G\ltimes O\rightarrow {\mathcal C}$ so that the following holds.
- The following diagram is commutative diagram $$\begin{CD}
G\ltimes K @>\bar{\Psi} >> {\mathcal E}_\mu\\
@V p VV @V P_\mu VV\\
G\ltimes O @> \Psi >> {\mathcal C}.
\end{CD}$$ and $\Psi(q_0)=c$ for some $q_0\in O$.
- $\bar{\Psi}$ is full and faithful.
- $|\bar{\Psi}|:|K|\rightarrow |{\mathcal E}_\mu|$ is a homeomorphism onto an open subset of $|{\mathcal E}_\mu|$ of the form $|P|^{-1}(U)$, where $U=|\Psi(O)|$.
- The map $\Psi$ is fiber-wise a topological linear Banach space isomorphism.
- Given $q\in O$ there exists an open neighborhood $U(q)\subset O$ so that every sequence $(h_k)\subset K$, with $q_k:=p(h_k)\in U(q)$, for which $(|\bar{\Psi}(h_k)|)$ converges in $|{\mathcal E}_\mu|$, has a convergent subsequence in $p^{-1}({\operatorname{cl}}_O(U(q)))$.
We observe that automatically $\Psi:G\ltimes O\rightarrow {\mathcal C}$ has to be a good uniformizer. Given two strong bundle uniformizers $\bar{\Psi}$ and $\bar{\Psi}'$, we can similarly as before define the transition set $${\bf M}(\bar{\Psi},\bar{\Psi}')=\{(h,\bar{\Phi},h')\ |\ h\in K,\ h'\in K',\ \bar{\Phi}\in \text{mor}(\bar{\Psi}(h),\bar{\Psi}'(h'))\}.$$ Observe that we have a natural map $${\bf M}(\bar{\Psi},\bar{\Psi}')\rightarrow {\bf M}(\Psi,\Psi'):(h,\bar{\Phi},h')\rightarrow (p(h),\Phi,p'(h')).$$ where a fiber has a a natural Banach space structure. We note that $s$ and $t$ in the following diagram are fiber-wise linear (the top row) $$\begin{aligned}
\label{diagg}
\begin{CD}
K @< s << {\bf M}(\bar{\Psi},\bar{\Psi}') @>t >> K'\\
@V p VV @VVV @V p' VV\\
O @< s<< {\bf M}(\Psi,\Psi') @> t >> O'
\end{CD}\end{aligned}$$
The main definition is now that of strong polyfold bundle structure for $({\mathcal C},\mu,{\mathcal T}_\mu)$.
A strong polyfold bundle structure for $({\mathcal C},\mu,{\mathcal T}_\mu)$ is given by $(\bar{F},{\bf M})$ where $\bar{F}:{\mathcal C}^-\rightarrow \text{SET}$ is a functor associating to every object $c$ in ${\mathcal C}$ a set of good strong bundle uniformizers and to every transition $${\bf M}(\bar{\Psi},\bar{\Psi}')\rightarrow {\bf M}(\Psi,\Psi')$$ a strong bundle structure so that source and target map define local strong bundle isomorphisms and all structural maps are strong bundle maps, see Diagram \[diagg\].
Tangent Category and Differential Forms
---------------------------------------
Assume ${\mathcal C}$ is a GCT equipped with a polyfold structure $(F,{\bf M})$. Then we have a filtration and inclusion functors, since as a consequence of this construction we can talk about the regularity of an object $${\mathcal C}_\infty .... \rightarrow {\mathcal C}_{i+1}\rightarrow {\mathcal C }_i...\rightarrow {\mathcal C}_0={\mathcal C}.$$ We can define the tangent category $T{\mathcal C}$ together with a projection functor $T{\mathcal C}\rightarrow {\mathcal C}^1$. Here ${\mathcal C}^1$ is the category ${\mathcal C}_1$ with filtration ${\mathcal C}^1_i:={\mathcal C}_{i+1}$ (and its polyfold structure). Consider tuples $(c,\Psi,(q,h))$, where $c$ is an object in ${\mathcal C}_1$, $\Psi\in F(c)$, say $\Psi:G\ltimes O\rightarrow {\mathcal C}$, $\Psi(q)=c$, and $h\in T_qO$. We shall introduce the notion of equivalence of two such tuples. For this consider a second one, say $(c',\Psi',(q',h'))$. Take suitable open neighborhoods $U(q,1_c,q')\subset {\bf M}(\Psi,\Psi')$, $U(q)\subset O$ and $U(q')\subset O'$, so that the source and target maps $$U(q)\xleftarrow{s} U(q,1_c,q')\xrightarrow{t} U(q')$$ are sc-diffeomorphisms and define $$L:U(q)\rightarrow U(q') : L(p)=(t\circ (s|U(q,1_c,q'))^{-1})(p).$$ We call the two tuples equivalent, written as $$(c,\Psi,(q,h))\simeq (c',\Psi',(q',h'))$$ provided the following holds $$c=c'\ \text{and}\ \ TL(q)(h)=(q',h').$$ We denote an equivalence class by $[(c,\Psi,(q,h))] $ and view them as objects in a category denoted by $T{\mathcal C}$. On the object level we have the projection functor $\tau:T{\mathcal C}\rightarrow {\mathcal C}^1$ given by $$\tau([c,\Psi,(q,h))]=c.$$ We observe that $\tau^{-1}(c)$ is a Banach space. Given a morphism $\phi:c\rightarrow c'$ we can define a topological linear isomorphism $$T\phi:\tau^{-1}(c)\rightarrow \tau^{-1}(c')$$ as follows $$\begin{aligned}
\label{hofer-x}
T\phi ([(c,\Psi,(q,h))]) =[(c',\Psi',TL(q,h)].\end{aligned}$$ The morphisms in $T{\mathcal C}$ are given by the tuples $$\Phi:= ([(c,\Psi,(q,h))],T\phi,[(c',\Psi',(q',h'))])$$ where $\phi:c\rightarrow c'$ and $T\phi([(c,\Psi,(q,h))]) =[(c',\Psi',(q',h'))]$. Here $$s(\Phi)=[(c,\Psi,(q,h))]\ \text{and}\ t(\Phi)=[(c',\Psi',(q',h'))].$$ The projection functor $\tau$ on the level of morphisms is defined by $$\tau( ([(c,\Psi,(q,h))],T\phi,[(c',\Psi',(q',h'))]) = \phi.$$ At this point we have shown the following.
Let ${\mathcal C}$ be a GCT equipped with a polyfold structure $(F,{\bf M})$. Then there is a well-defined tangent category $T{\mathcal C}$ with objects being the equivalence classes $[(c,\Psi,(q,h))]$, with $c$ being an object in ${\mathcal C}^1$, $(q,h)\in T_qO$, where $\Psi(q)=c$ and $\Psi\in F(c)$. The morphisms are given by the tuples $$\Phi=([(c,\Psi,(q,h))],T\phi,[(c',\Psi',(q',h'))]),$$ where $T\phi([(c,\Psi,(q,h))]=[(c',\Psi',(q',h'))]$. In addition the projection functor $\tau:T{\mathcal C}\rightarrow {\mathcal C}^1$ is defined by $\tau(\Phi)=\phi$ on morphisms and $\tau([(c,\Psi,(q,h))]=c$ on objects.
We shall show that we can equip $T{\mathcal C}$ with a polyfold structure as well. This needs some preparation. Fix an object $c_0$ in ${\mathcal C}^1$ and pick $\Psi_0\in F(c_0)$ which is given by $$\Psi_0: G\ltimes O\rightarrow {\mathcal C}.$$ There is an element $q_0\in O_1$ with $\Psi_0(q_0)=c_0$. For $q\in O_1$ let $c=\Psi_0(q)$ and note that $q$ is an object in ${\mathcal C}_1$. Pick $\Psi\in F(c)$, say $\Psi:G'\ltimes O'\rightarrow {\mathcal C}$ and $q'_0\in O'$ with $\Psi(q_0')=c=\Psi_0(q)$. Then $q_0'\in O_1'$. Now we us the transition M-polyfold ${\bf M}(\Psi_0,\Psi)$ and take open neighborhoods $U(q)$, $U(q_0')$ and $U(q,{1_c},q_0')$ so that we have a diagram of sc-diffeomorphisms $$U(q)\xleftarrow{s} U(q,1_c,q_0')\xrightarrow{t}U(q_0').$$ We define $$L:U(q)\rightarrow U(q_0'): p\rightarrow L(p):=L=t\circ {(s|U(q,1_c,q'))}^{-1}(p).$$ The we define for $(q,h)\in T_qO$, which is an object in $G\ltimes TO$ $$T\Psi_0(q,h) = [(c,\Psi, TL(q,h))]$$ which belongs to $\tau^{-1}(c)$. For a morphism $(g,(q,h)):(q,h)\rightarrow (g\ast q, g\ast h)$ in $G\ltimes TO$ we define the morphism $$T\Psi_0(g,(q,h)):T\Psi_0(q,h)\rightarrow T\Psi_0(g\ast q,g\ast h)$$ by $$\begin{aligned}
&&T\Psi_0(g,(q,h))\\
&=& ([(c,\Psi,TL(q,h))],T(\Psi_0(g,q)),T(\Psi_0(g,q))([(c,\Psi,TL(q,h))]))\\
&=&(T\Psi_0(q,h),T(\Psi_0(g,q)),T(\Psi_0(q,q))(T\Psi_0(q,h))).
\end{aligned}$$ Here $T(\Psi_0(g,q))$ is the tangent associated to the morphism $\Psi_0(g,q):\Psi_0(q)\rightarrow \Psi_0(g\ast q)$ defined as in (\[hofer-x\]).
We define for an object $c_0$ in ${\mathcal C}^1$. $$\bar{F}(c_0)=\{ T\Psi: G\ltimes TO\rightarrow T{\mathcal C}\ |\ \Psi\in F(c_0)\},$$ and view this definition as a functor $\bar{F}:{({\mathcal C}^1)}^-\rightarrow \text{SET}$. Then we define $$TF:(T{\mathcal C})^-\rightarrow \text{SET} : TF = \bar{F}\circ \tau,$$ i.e. $TF([(c_0,\Psi_0,(q_0,h_0))]):=\bar{F}(c_0)$. Given $T\Psi\in TF(([(c_0,\Psi_0,(q_0,h_0))])$ and $T\Psi'\in ([(c'_0,\Psi'_0,(q_0',h_0'))]$ we can build ${\bf M}(T\Psi,T\Psi')$. The basic result proved in [@HWZ8] is the following theorem.
Given a polyfold structure $F$ for the ${\mathcal C}$ there exists an associated natural polyfold structure $TF$ for $T{\mathcal C}\rightarrow {\mathcal C}^1$ covering the lifted one for ${\mathcal C}^1$ given by $F^1$, see the diagram below.
More precisely given $\Psi\in F(c)$, say $\Psi:G\ltimes O\rightarrow {\mathcal C}$ the associated $\Psi^1$ fits into the following commutative diagram $$\begin{CD}
G\ltimes TO @>T\Psi >> T{\mathcal C}\\
@V \tau_O VV @V \tau VV \\
O^1 @> \Psi^1 >> {\mathcal C}^1.
\end{CD}$$ Here we already have an example of an sc-smooth functor. Namely $\tau:T{\mathcal C}\rightarrow {\mathcal C}^1$ has for every $T\Psi$ a sc-smooth representative, namely $\tau_O$. One of the main points of having polyfold structures is to say that certain functors defining algebraic structures are sc-smooth, or that they are sc-Fredholm functors, which allows a perturbation theory. Also if ${\mathcal C}$ is equipped with a polyfold structure we can define sc-differential forms as certain kind of functors.
We can define the $k$-fold product category $T{\mathcal C}\times..\times T{\mathcal C}$ which projects to a $k$-fold product of ${\mathcal C}^1$ with itself and pull-back by the multi-diagonal which we denote by $$\oplus_{i=1}^k T{\mathcal C}\rightarrow {\mathcal C}^1.$$ The preimages of objects are $k$-fold products of Banach spaces. Viewing ${\mathbb R}$ as a category with only the identities as morphisms we are interested in certain functors $$\omega:\oplus_{i=1}^k T{\mathcal C}\rightarrow {\mathbb R},$$ which are multi-linear and skew-symmetric on the fibers. We assume $T{\mathcal C}$ equipped with its canonical polyfold structure $TF$. Given $T\Psi\in \bar{F}(c)$ we can pull back $\omega$ via $\oplus_{i=1}^k T\Psi$ and obtain $\omega_\Psi:TO\oplus..\oplus TO\rightarrow {\mathbb R}$. In [@HWZ7] we have introduced the notion of an sc-smooth differential form.
The functor $\omega$ is said to be sc-smooth provided there exist a family ${(\Psi_\lambda)}_{\lambda\in\Lambda}$ ($\Lambda$ a set) of good unifomizers associated to $F$ so that the collection of sets $|\Psi_\lambda(O_\lambda)|$ covers $|{\mathcal C}|$, and in addition all the $\omega_{\Psi_\lambda}$ are sc-smooth differential forms.
Several remarks are in order.
\(1) One can show that the definition does not depend on the choice of the family $(\Psi_\lambda)$, see [@HWZ7].\
(2) We view $\omega$ as associated to ${\mathcal C}$, despite the fact that it is defined on $T{\mathcal C}$ which lies over ${\mathcal C}^1$, and therefore seemingly only involves ${\mathcal C}^1$, However, by using the good uniformizers for ${\mathcal C}$ in the definition of $T{\mathcal C}$ incorporates the polyfold structure on ${\mathcal C}$ in a subtle way.\
(3) One is tempted to call $\omega$ as defined above a sc-smooth differential form on ${\mathcal C}$. If we call $\omega$ according to (2) as associated to ${\mathcal C}$ it will turn out that $d\omega$, the exterior differentiation can only be defined as a form associated to ${\mathcal C}^1$. Since we have the system of inclusion functors $...\rightarrow {\mathcal C}^{i+1}\rightarrow {\mathcal C}^i\rightarrow...{\mathcal C}$ one can take a direct limit for forms. The set (see below) of all $[\omega]$ defined by the direct limit for all $k\geq 0$ turns out to be invariant under $d$ defined by $d[\omega]=[d\omega]$ so that it is better to call $[\omega]$ an sc-smooth differential form. Using this we obtain a de Rham complex associated to ${\mathcal C}$ as we shall see below. For the moment we shall call $\omega$ an sc-differential form, and leave the name sc-smooth differential form for the result of a further construction.
The collection of all sc-smooth functors $\omega$ is a set, since it is completely determined by the set $(\omega_{\Psi_\lambda})$. Using the inclusion functors ${\mathcal C}^{i+1}\rightarrow {\mathcal C}^i$, we can pull-back a functor $\omega$ defined on $\oplus_{i=1}^k T({\mathcal C}^i)$ to $\oplus_{i=1}^k T({\mathcal C}^{i+1})$. This is nothing else but restricting $\omega$ to tangent vectors of specified higher regularity. Denote by $\Omega^k{(\mathcal C}^i)$ the set of differential $k$-forms on ${\mathcal C}^i$, i.e. defined on $\oplus_{j=1}^k T{\mathcal C}^i$. This set has the obvious structure as a real vector space. Then we have the direct system $$\rightarrow \Omega^k({\mathcal C}^i)\rightarrow \Omega^k({\mathcal C}^{i+1})\rightarrow..$$ and denote the direct limit by $\Omega^k_\infty({\mathcal C})$, and its elements by $[\omega]$. If $(X,\beta)$ is an ep-groupoid constructed from the polyfold structure $F$ on ${\mathcal C}$ we can use the equivalence $\beta:X\rightarrow {\mathcal C}$ to pull-back $[\omega]$ since $T\beta:TX\rightarrow T{\mathcal C}$ is well-defined. In fact this pull-back completely determines $[\omega]$ and the pull back is compatible with the exterior derivative defined on $X$, see [@HWZ5], [@HWZ7], and [@HWZ8]. As it turns out the exterior differential is well-defined, so that we obtain the de Rham complex $$(\Omega_\infty^\ast({\mathcal C},F),d).$$ So in particular there exists a de Rham cohomology. We refer the reader for details of this theory to [@HWZ7; @HWZ8] and [@HWZ5].
Finite-dimensional, Branched, Weighted Subpolyfolds
---------------------------------------------------
Suppose that ${\mathcal C}$ is a GCT equipped with a polyfold structure $F$. We are interested in certain full subcategories which arise when studying Fredholm functors later on. In a first step consider the non-negative rational numbers ${\mathbb Q}^+$ as objects in a category with the morphisms being the identities. Of interest for us are certain functors $$\Theta:{\mathcal C}\rightarrow {\mathbb Q}^+.$$ In order to define this class of functors we need the definition of a submanifold $M$ of a M-polyfold $X$.
\[DEX1\] Let $X$ be a M-polyfold and $M$ a subset. We say $M$ is a submanifold provided for every $m\in M$ there exists an open neighborhood $U=U(m)\subset X$ and an sc-smooth map $r:U\rightarrow U$ having the following properties.
- $r(U_i)\subset U_{i+1}$ for all $i$ and $r:U\rightarrow U^1$ is sc-smooth.
- $r\circ r=r$
- $r(U)=U\cap M$.
First of all we note that $r$ is an sc-smooth retraction, so that $M$ is a sub-polyfold. But the stronger requirement that $r:U\rightarrow U^1$ is sc-smooth, in fact, implies that the M-polyfold structure on $M$ induced from $ X$ is the sc-smoothly equivalent to the structure of a finite-dimensional smooth manifold, see [@HWZ7]. This, of course, justifies that we call $M$ a submanifold in the first place. More precisely we have the following result.
Let $M$ be a submanifold of the M-polyfold $X$ in the sense of Definition \[DEX1\]. Then $M$ is a sub-polyfold and its sc-smooth structure induced from $X$ is sc-smoothly equivalent to a classical manifold structure.
A hint that this is true is given by the following consideration. If $r(x)=x$ it follows that $x\in X_\infty$, so that for every $m\in M$ the tangent space $T_mM$ is defined and an sc-Banach space. Since $Tr(m):T_mX\rightarrow T_mX$ is an sc$^+$-operator its image is compact. Since $Tr(m)$ restricted to its image is the identity the image must be finite-dimensional.
If $M$, viewed as manifold, is equipped with an orientation we shall call it an oriented submanifold.
Suppose $\Theta:{\mathcal C}\rightarrow {\mathbb Q}^+$ is a functor and $c$ an object. Pick $\Psi\in F(c)$ so that we can take the functor $\Theta\circ \Psi: G\ltimes O\rightarrow {\mathbb Q}^+$. Let $\Psi(q_0)=c$ and assume there exists an open neighborhood $U(q_0)$ in $O$, finitely many submanifolds ${(M_i)}_{i\in I}$ and positive rational numbers ${(\sigma_i)}_{i\in I}$ such that $$\Theta\circ \Psi(q)=\sum_{\{i\in I\ |\ q\in M_i\}} \sigma_i$$ for all $q\in U(q_0)$. We say $\Theta$ has a smooth finite-dimensional representation with respect to $\Psi$ at $c$. We also say that the representation is $n$-dimensional provided every $M_i$ is $n$-dimensional.
If $\Psi'\in F(c)$ with $\Psi'(q_0')=c$ we consider ${\bf M}(\Psi,\Psi')$ and find open neighborhoods so that the source and target map are sc-diffeomorphisms $$V(q_0)\xleftarrow{s} V(q_0,1_c,q_0')\xrightarrow{t} V(q_0').$$ We may assume that $V(q_0)\subset U(q_0)$ and can map the $M_i\cap V(q_0)$ to $M_i'\subset V(q_0')$. Then $$\Theta\circ \Psi'(q')=\sum_{\{i\in I\ |\ q\in M_i'\}} \sigma_i$$ for $q'\in V(q_0')$. Hence if we have a smooth finite-dimensional representation at $c$ for some $\Psi\in F(c)$ it holds for all uniformizers in $F(c)$. The same argument goes through if $\phi:c\rightarrow c'$ is an isomorphism and shows if we have a smooth finite-dimensional representation at $c$ we also have this property at an isomorphic $c'$. Observe that if the representation at $c$ with respect to $\Psi$ is $n$-dimensional this will be true for every isomorphic $c$ as well. We can therefore say that $\Lambda$ is smooth at $|c|$ and has a $n$-dimensional representation at $|c|$. Since the collection of all $|c|$ is a set, the following makes sense.
Let ${\mathcal C}$ be a GCT equipped with a polyfold structure, and $\Theta:{\mathcal C}\rightarrow {\mathbb Q}^+$ a functor. We call $\Theta$ a smooth, weighted, branched subpolyfold of dimension $n$, provided $\Theta$ has a smooth $n$-dimensional representation at every $|c|\in |{\mathcal C}|$.
There is also a notion of orientation for $\Theta$, see [@HWZ5; @HWZ8].
Let ${\mathcal C}$ be a polyfold category and $\Theta:{\mathcal C}\rightarrow {\mathbb Q}^+$ a smooth, weighted, branched subpolyfold of dimension $n$. We say $\Theta$ is closed provided the orbit space associated to all objects $c$ with $\Theta(c)>0$ is a compact subset of $|{\mathcal C}|$.
An important result is that we can integrate sc-differential forms over closed, smooth, oriented, weighted branched subpolyfold of dimension $n$, see [@HWZ8; @HWZ5].
Let ${\mathcal C}$ be a GCT equipped with a polyfold structure $F$, and $\Theta:{\mathcal C}\rightarrow {\mathbb Q}^+$ a closed, smooth, oriented, weighted, branched subpolyfold of dimension $n$. Suppose further $[\omega]$ is a n-dimensional sc-smooth differential form on ${\mathcal C}$. Then there is a well-defined integral $$\oint_\Theta[\omega],$$ called the branched integral.
The integral is characterized uniquely by certain properties, and in the somewhat more general context with boundary with corners, even a version of Stokes Theorem holds. The basic observation is that $\Theta$ defines a compact subset of $|{\mathcal C}|$ equipped with, what is called a Lebesgue $\sigma$-algebra of measurable sets, and a measurable ${\mathbb Q}^+$-valued weight function $w$ . The differential form $[\omega]$ induces a signed measure $\mu_{[\omega]}$ and the integral is given by $\int wd\mu_{[\omega]}$. This construction is formidable and given in [@HWZ5] for ep-groupoids, but generalizes immediately to our context, since the ep-groupoid version is compatible with Morita equivalence, i.e. generalized isomorphisms associated to diagrams of sc-smooth equivalences between ep-groupoids, see [@HWZ8].
Fredholm Theory and Transversality
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Suppose we are given a bundle GCT $({\mathcal C},\mu,{\mathcal T})$ equipped with a strong polyfold bundle structure $\bar{F}:{\mathcal C}^-\rightarrow\text{SET}$, and a section functor $$f:{\mathcal C}\rightarrow {\mathcal E}_\mu.$$ If $c$ is an object in ${\mathcal C}$ and $\bar{\Psi}$ a strong bundle uniformizer around $c$ we obtain the commutative diagram. $$\begin{CD}
G\ltimes K @>\bar{\Psi} >> {\mathcal E}_\mu\\
@V p VV @V P_\mu VV\\
G\ltimes O @>\Psi >> {\mathcal C}
\end{CD}$$ Since $f$ maps an object to an element in the Banach space associated to $c$, i.e. $P^{-1}(c)$ it follows that an object $q\in O$ is mapped to $f\circ \Psi(q)\in P^{-1}(\Psi(q))$. However $\bar{\Psi}:p^{-1}(q)\rightarrow P^{-1}(\Psi(q))$ is a linear isomorphism and it follows that $f$ has a local representative $f_\Psi$ which is a section of $p$.
Let $f$ be a section functor of $P:{\mathcal E}_\mu \rightarrow {\mathcal C}$, where $({\mathcal C},\mu,{\mathcal T})$ is a bundle GCT equipped with a strong polyfold bundle structure $\bar{F}$. We say $f$ is an sc-Fredholm functor provided there exists a family $(\bar{\Psi}_\lambda)_{\lambda\in\Lambda}$, $\Lambda$ a set, of good strong bundle uniformizers, so that $(\Psi_\lambda)_{\lambda\in\Lambda}$ covers $|{\mathcal C}|$, and every $f_{\bar{\Psi}_\lambda}$ is an sc-Fredholm section of $K_\lambda\rightarrow O_\lambda$.
This definition does not depend on the choice of the strong bundle uniformizers taken from $\bar{F}$.
Of course, like in the classical situation, one is interested in a perturbation theory, which allows an sc-Fredholm section to be brought into a general position. Since we are dealing with functors, there is the added difficulty that symmetry, i.e. compatiblility with morphisms, and transversality are competing issues.
In order to achieve transversality we locally break the symmetry under a small perturbation by an sc$^+$-section, but keep track of the symmetry by introducing a finite family (correlated with the initial pertubation) of local sc-Fredholm problems invariant under the symmetry. In order to have still the right counts of the solutions the problems in the local family need to be weighted. Of course, these local modifications have to be done coherently, so that overlapping families can be patched together in a suitable way.
In this context it is very important to understand how big perturbations can be, in order to guarantee that the perturbed Fredholm section is again proper. In order to formulate some results we need some auxiliary structures. View ${\mathbb R}^+\cup \{+\infty\}=[0,+\infty]$ as a category only having the identities as morphisms.
Let $({\mathcal C},\mu,{\mathcal T})$ be a bundle GCT equipped with a strong polyfold bundle structure $\bar{F}:{\mathcal C}^-\rightarrow \text{SET}$. An [auxiliary norm]{} $N$ is a functor $N:{\mathcal E}_\mu \rightarrow {\mathbb R}^+\cup\{+\infty\}$ with the following properties. There exists a family of good strong bundle uniformizers ${(\bar{\Psi}_\lambda)}_{\lambda\in \Lambda}$, $\Lambda$ a set, so that the underlying ${(|\Psi_\lambda(O_\lambda)|)}_{\lambda\in\Lambda}$ cover $|{\mathcal C}|$ and for every $\lambda\in\Lambda$ $$N\circ \bar{\Psi}_\lambda:K_\lambda\rightarrow {\mathbb R}^+\cup\{+\infty\}$$ is an auxiliary norm according to Definition \[AUXN\].
The definition does not depend on the family of good strong bundle uniformizers.
Let $({\mathcal C},\mu,{\mathcal T})$ be a bundle GCT equipped with a strong polyfold bundle structure $\bar{F}:{\mathcal C}^-\rightarrow \text{SET}$. A functor $\Lambda:{\mathcal E}_\mu\rightarrow {\mathbb Q}^+$ is called an sc$^+$-smooth multisection functor provided there exists a set ${(\bar{\Psi}_\lambda)}_{\lambda\in A}$ of good strong bundle uniformizers so that the underlying $(\Psi_\lambda)$ have the property $$|{\mathcal C}|=\bigcup_{\lambda\in A} |\Psi_\lambda(O_\lambda)|.$$ with the following property:
- For every $\Lambda\circ \bar{\Psi}_\lambda: G_\lambda\ltimes K_\lambda\rightarrow {\mathbb Q}^+$ and given $q\in O_\lambda$ there exist an open neighborhood $U(q)\subset O_\lambda$, and sc$^+$-sections ${(s_i)}_{i\in I}$ defined for $K_\lambda|U(q)$, and rational weights $\sigma_i>0$, $i\in I$, with $\sum_{i\in I}\sigma_i=1$, so that for $e\in K_\lambda|U(q)$ $$\Lambda\circ \bar{\Psi}_\lambda(e)=\sum_{\{i\in I\ |\ e=s_i(p_\lambda(e))\}}\sigma_i.$$
The definition is independent of the choice of $(\bar{\Psi}_\lambda)$ as long as the associated open sets $(\Psi(O_\lambda))$ cover the orbit space of ${\mathcal S}^{3,\delta_0}(Q,\omega)$. The functor $\Lambda$ induces a map $|\Lambda|:|{\mathcal E}_\mu|\rightarrow {\mathbb Q}^+$.
The support of an sc$^+$-smooth multisection functor $\Lambda$ is the full subcategory $\text{supp}(\Lambda)$ associated to all objects $e$ in ${\mathcal E}_\mu$ with $\Lambda(e)>0$. The domain support $\text{dom-supp}(\Lambda)$ is the full subcategory associated to the closure of the open subset $U$ of $|{\mathcal C}|$ consisting of all $|c|$ so that there exists $|e|\in |P|^{-1}(|c|)$, $|e|\neq 0$, with $|\Lambda|(|e|)>0$.
We note that each fiber of $|P|$ has a distinguished $0$ element. Given $\Lambda:{\mathcal E}_\mu\rightarrow {\mathbb Q}^+$ and an auxiliary norm $N:{\mathcal E}_\mu\rightarrow [0,\infty]$ we can measure the size of $\Lambda$. Namely for an object $c$ there are finitely many points $e_i\in P_\mu^{-1}(c)$, for which $\Lambda(e_i)>0$. Moreover the bi-regularity is at least $(0,1)$, so that $N(e_i)$ is finite. Therefore we can define $\max\{N(e_i)\ |\ i\in I\}$ and obtain a functor ${\mathcal C}_\mu\rightarrow [0,\infty)$ by $$c\rightarrow \max\{N(e_i)\ |\ i\in I\}.$$ We can pass to orbit space and obtain a continuous map $N_\Lambda:|{\mathcal C}|\rightarrow [0,\infty)$.
Now we are in the position to describe a series of results in sc-Fredholm theory for our categorical setup. Without going into much detail there is the notion of orientation for an sc-smooth Fredholm section functor, see [@HWZ6; @HWZ7; @HWZ8]. The first result is concerned with a compactness assertion.
Let $({\mathcal C},\mu,{\mathcal T})$ be a bundle GCT equipped with a strong polyfold bundle structure $\bar{F}:{\mathcal C}^-\rightarrow \text{SET}$. Suppose $N:{\mathcal E}_\mu\rightarrow {\mathbb R}^+\cup \{+\infty\}$ is an auxiliary norm, and $f$ an sc-Fredholm section functor of $P_\mu$, having the property that the orbit space $|f^{-1}(0)|\subset |{\mathcal C}|$ is compact. Then there exists an open neighborhood $U$ of $|f^{-1}(0)|$ in $|{\mathcal C}|$ so that for every sc$^+$-multisection functor $\Lambda :{\mathcal E}_\mu\rightarrow {\mathbb Q}^+$ with domain support in ${\mathcal C}_U$ and satisfying $N_\Lambda(|c|)\leq 1$ for all $|c|\in |{\mathcal C}|$, the orbit space associated to $\text{supp}(\Lambda\circ f)$ is compact.
Note that $\Lambda\circ f :{\mathcal C}\rightarrow {\mathbb Q}^+$ is a functor and ${\operatorname{supp}}(\Lambda\circ f)$ consists of all objects $c$ with $\Lambda\circ f(c)>0$.
If $(U,N)$ is a pair consisting of an auxiliary norm $N$, and an open neighborhood $U$ of the orbit space associated to $f^{-1}(0)$, so that the conclusion of the previous theorem concerning compactness holds, we shall say that $(U,N)$ controls compactness.
In order to construct sc$^+$-multisection functors one needs that the underlying polyfold structure on ${\mathcal C}$ admits sc-smooth partitions of unity. For example if everything is build on Hilbert scales, or at least the zero-level, then these are available, see [@HWZ7; @HWZ8]. For the following denote by $H^\ast_{dR}({\mathcal C})$ the de Rham cohomology.
Let $({\mathcal C},\mu,{\mathcal T})$ be a bundle GCT equipped with a strong polyfold bundle structure $\bar{F}:{\mathcal C}^-\rightarrow \text{SET}$. Suppose $N:{\mathcal E}_\mu\rightarrow {\mathbb R}^+\cup \{+\infty\}$ is an auxiliary norm, and $f$ an sc-Fredholm section functor of $P_\mu$, having the property that the orbit space $|f^{-1}(0)|\subset |{\mathcal C}|$ is compact. We assume that the induced polyfold structure on ${\mathcal C}$ admits sc-smooth partitions of unity. Let $U$ be an open neighborhood around the orbit space associated to $f^{-1}(0)$, so that $(U,N)$ controls compactness. Then the following holds.
- Given any $\varepsilon\in (0,1)$ there exists a sc$^+$-multisection functor $\Lambda$ with domain support in ${\mathcal C}_U$ and $N_\Lambda(|c|)<\varepsilon$ for all $|c|$, so that $$\Theta:=\Lambda\circ f:{\mathcal C}\rightarrow {\mathbb Q}^+$$ is a smooth, closed, weighted, branched subpolyfold of dimension $n$.
- If $f$ is oriented, i.e. $(f,\mathfrak{o})$, then $\Theta$ is naturally oriented.
- Given $[\omega]\in H^n_{dR}({\mathcal C})$ the branched integral $\oint_\Theta[ \omega]$ does not depend on the choice of $\Lambda$ as long as it is generic and satisfies the above conditions.
- The value of the integral is independent of the choice $(U,N)$ as long as it is admissible.
As a consequence of this theorem the oriented sc-Fredholm section functor $(f,\mathfrak{o})$ defines a linear functional $$I_{(f,\mathfrak{o})}:H^\ast_{dR}({\mathcal C})\rightarrow {\mathbb R},$$ via $$I_{(f,\mathfrak{o})}(\boldsymbol{ [ }[\omega]\boldsymbol{ ]})=\oint_{\Lambda\circ f}[\omega],$$ for $\Lambda$ having support sufficiently small and being generic. This functional will stay the same under even large oriented deformations $$t\rightarrow (f_t,\mathfrak{o}_t)$$ as long as the orbit space of the solution set satisfies some compactness properties. During the deformation we can even change the $\mu_t$ as long as it is done sc-smoothly for an overall strong polyfold bundle structure. There are also appropriate versions where the underlying category changes (sc-smoothly) as well.
The Stable Map Example
----------------------
In some sense we just need to take a fresh look at what we did in Section \[SEC22\] and verify, implementing the discussion from Section \[SER2\], that the constructions are sc-smooth. We shall concentrate on the category ${\mathcal S}^{3,\delta_0}(Q,\omega)$ rather than the full problem $P:{\mathcal E}^{3,\delta_0}(Q,\omega,J)\rightarrow {\mathcal S}^{3,\delta_0}(Q,\omega)$. In the case of ${\mathcal S}^{3,\delta_0}(Q,\omega)$ our discussion so far gives a precise construction with established topological properties. From the discussion in [@HWZ6], based on results in [@HWZ8.7], see the forthcoming [@H2] for a comprehensive treatment, it follows that indeed all the constructions are sc-smooth.
It is very good exercise using [@HWZ8.7] and some of the results in [@HWZ6] to fill in the technical details.
We give a few useful comments. In the constructions mentioned in Theorem \[IMP\] we already used the exponential gluing profile. Given the strictly increasing sequence $\delta=(\delta_0,\delta_1,..)$ with all $0<\delta_i<2\pi$, it can be shown that $\bar{X}^{3,\delta_0}_{(S,j,D),{\bf D},{\mathcal H}}(Q)$ has a M-polyfold structure. It only depends on $\delta$ and we shall abbreviate this topological space equipped with this M-polyfold structure by $\bar{X}^{3,\delta}_{(S,j,D),{\bf D},{\mathcal H}}(Q)$. As a consequence the product $$V\times \bar{X}^{3,\delta}_{(S,j,D),{\bf D},{\mathcal H}}(Q)$$ also has a M-polyfold structure. For this structure the automorphism group $G$ acts by sc-diffeomorphisms. Taking a suitable open $G$-invariant neighborhood $O$ of $(0,(S,D,u))$ we obtain $$\Psi: G\ltimes O\rightarrow {\mathcal S}^{3,\delta_0}(Q,\omega)$$ which on objects is given by $$(v,(S_\mathfrak{a},D_\mathfrak{a},w))\rightarrow (S_\mathfrak{a},j(v)_\mathfrak{a},M_\mathfrak{a},D_\mathfrak{a},w).$$ Also recall that we had the underlying good uniformizers for ${\mathcal R}$ $$G\ltimes O^\ast\rightarrow {\mathcal R},$$ which on objects map $$(v,\mathfrak{a})\rightarrow (S_{\mathfrak{a}},j(v)_\mathfrak{a},M^\ast_\mathfrak{a},D_\mathfrak{a}).$$ Our aim is to define a polyfold structure on ${\mathcal S}^{3,\delta_0}(Q,\omega)$ utilizing the previous construction of $F$. Of course, we need to be able to equip ${\bf M}(\Psi,\Psi')$ with a M-polyfold structure. In order to achieve this we have to make an additional requirement which is achieved by possibly restricting $\Psi$ to a smaller $O$. This leads to the definition of a good (polyfold) uniformizer.
\[UNI\] Assume a sequence $\delta$ and the exponential gluing profile are given as previously described. Let $\alpha=(S,j,M,D,u)$ be an object in ${\mathcal S}^{3,\delta_0}(Q,\omega)$ with automorphism group $G$. A good polyfold uniformizer $\alpha$ for ${\mathcal S}^{3,\delta_0}(Q,\omega)$ is a functor $\Psi:G\ltimes O\rightarrow {\mathcal S}^{3,\delta_0}(Q,\omega)$ where $O$ is a $G$-invariant open neighborhood of $(0,(S,D,u))$ in $V\times \bar{X}^{3,\delta}_{(S,j,D),{\bf D},{\mathcal H}}(Q)$ so that the following holds.
- $\Psi$ is fully faithful and $\Psi(v,(S,D,u))=(S,j,M,D,u)$.
- Passing to orbit spaces, $|\Psi|:{_G\backslash}O\rightarrow |{\mathcal S}^{3,\delta_0}(Q,\omega)|$ is a homeomorphism onto an open neighborhoodof $|\alpha|$.
- The collection of all $(v,\mathfrak{a})$ occuring in elements $(v,(S_\mathfrak{a},D_\mathfrak{a},w))\in O$ are contained in $O^\ast$, and $O^\ast\ni (v,\mathfrak{a})\rightarrow (S_\mathfrak{a},j(v)_\mathfrak{a},M_\mathfrak{a}^\ast,D_\mathfrak{a})$ defines a good uniformizer for ${\mathcal R}$.
- For every object $q\in O$ there exists an open neighborhood $U(q)\subset O$, so that every sequence $(q_k)\subset U(q)$, for which $|\Psi(q_k)|$ converges in $|{\mathcal S}^{3,\delta_0}(Q,\omega)|$, has a convergent subsequence in ${\operatorname{cl}}_O(U(q))$.
\(a) The condition (iii) is very important and it summarizes four conditions from the definition of a good uniformizer for ${\mathcal R}$. Most important is that the partial Kodaira-Spencer differentials are isomorphisms. This is extensively used when putting a M-polyfold structure on ${\bf M}(\Psi,\Psi')$. It is crucial for making $s$ and $t$ local sc-diffeomorphisms, see [@HWZ6].\
(b) Let $F(\alpha)$ consists of the good uniformizers previously constructed and their domain $G\ltimes O$ equipped with the M-polyfold structures associated to a choice of $\delta$. In addition we assume that the set of all $(v,\mathfrak{a})$ coming from the $(v,(S_\mathfrak{a},D_\mathfrak{a},w))$ lies in $O^\ast$, so that we have the good uniformizer for ${\mathcal R}$ defined on $G\ltimes O^\ast$. With $F$ modified as just described it is possible to lift ${\bf M}$ to a functor which associates to $(\Psi,\Psi')$ not only a metrizable space, but in fact a M-polyfold structure. The details given in the ep-groupoid seen are presented in [@HWZ6].
The relevant construction $$F:{({\mathcal S}^{3,\delta_0}(Q,\omega))}^-\rightarrow \text{SET}$$ associates to an object $\alpha$ the set $F(\alpha)$ consisting of good polyfold uniformizers as described in Definition \[UNI\]. This leads to the following important result.
Given the exponential gluing profile $\varphi$ and an increasing sequence of weights $0<\delta_0<\delta_1<...<2\pi$, the construction $({F},{\bf M})$ given in Theorem \[IMP\], with the modification just mentioned, defines a polyfold structure, when the local models are equipped with the sc-structures associated to these weights.
We can also construct a bundle GCT $({\mathcal S}^{3,\delta_0}(Q,\omega),\mu,{\mathcal T})$ as follows. Given an object $\alpha=(S,j,M,D,u)$ we associate to it the Hilbert space $\mu(\alpha)$ consisting of all $(TQ,J)$-valued $(0,1)$-forms $\xi$ along $u$ of class $(2,\delta_0)$. In particular $\xi(z):(T_zS,j)\rightarrow (T_{u(z)}Q,J)$ is complex anti-linear. A morphism $\Phi:=(\alpha,\phi,\alpha'):\alpha\rightarrow \alpha'$ defines a linear topological isomorphism $$\mu(\Phi):\mu(\alpha)\rightarrow \mu(\alpha'): \xi\rightarrow \xi\circ T\phi^{-1}.$$ This allows us to define the category ${\mathcal E}_\mu$ with objects being the $(\alpha,\xi)$, where $\alpha$ is an object in ${\mathcal S}^{3,\delta_0}(Q,\omega)$, and $\xi$ is a vector in $\mu(\alpha)$. We shall denote ${\mathcal E}_\mu$ by ${\mathcal E}^{2,\delta_0}(Q,\omega,J)$. One can define a metrizable topology ${\mathcal T}$ for $|{\mathcal E}^{2,\delta_0}(Q,\omega,J)|$ and carry out a construction $(\bar{F},{\bf M})$, $$\bar{F}:({\mathcal S}^{3,\delta_0}(Q,\omega))^-\rightarrow \text{SET}$$ equipping $P$ with a strong polyfold bundle structure. Here $\bar{F}(\alpha)$ consists of good strong bundle uniformizers, where an element $\bar{\Psi}$ in the latter fits into a commutative diagram of functors with certain properties $$\begin{CD}
G\ltimes K @>\bar{\Psi}>> {\mathcal E}^{2,\delta_0}(Q,\omega,J)\\
@V p VV @V P VV\\
G\ltimes O @>\Psi >> {\mathcal S}^{3,\delta_0}(Q,\omega).
\end{CD}$$ On objects $\Psi$ has the form $$\Psi(v,(S_\mathfrak{a},D_\mathfrak{a},w))=(S_\mathfrak{a},j(v)_\mathfrak{a},M_\mathfrak{a},w)$$ and $\bar{\Psi}$ is given by $$\bar{\Psi}((v,(S_\mathfrak{a},D_\mathfrak{a},w),\xi)=(S_\mathfrak{a},j(v)_\mathfrak{a},M_\mathfrak{a},w,\xi).$$ The transition structure $${\bf M}(\bar{\Psi},\bar{\Psi}')\rightarrow {\bf M}(\Psi,\Psi')$$ has an sc-smooth strong bundle structure. These structures, as already explained, can be viewed as some kind of sc-smooth bundle structure for $P:{\mathcal E}^{2,\delta_0}(Q,\omega,J)\rightarrow {\mathcal S}^{3,\delta_0}(Q,\omega)$. The latter, equipped with this strong bundle structure, which depends on the weight sequence $\delta$, is written as $$P:{\mathcal E}^{2,\delta}(Q,\omega,J)\rightarrow {\mathcal S}^{3,\delta}(Q,\omega).$$
The local representative of the Cauchy-Riemann section takes the form $$\begin{aligned}
\label{display1}
&O\rightarrow K:&\\
&(v,(S_\mathfrak{a},D_\mathfrak{a},w))\rightarrow \left(v,\left(S_\mathfrak{a},D_\mathfrak{a},w\right),\frac{1}{2}\left[ Tw+J(w)\circ Tw\circ j(v)_\mathfrak{a}\right]\right).&\nonumber\end{aligned}$$ It has been proved in [@HWZ6] that the section just defined is sc-Fredholm.
The local representative of the section functor $\bar{\partial}_J$ of $P$ with respect to $\bar{\Psi}\in \bar{\bf F}(\alpha)$ for any object $\alpha$, as given in (\[display1\]) is sc-Fredholm. Hence $\bar{\partial}_J$ is an sc-Fredholm section of $P:{\mathcal E}^{2,\delta}(Q,\omega,J)\rightarrow {\mathcal S}^{3,\delta}(Q,\omega)$.
In the case of our Gromov-Witten example consider the full subcategory ${\mathcal S}_{g,m,A}^{3,\delta}(Q,\omega)$ of ${\mathcal S}^{3,\delta}(Q,\omega)$, where $g\geq 0$ and $m\geq 0$ are integers, and $A\in H_2(Q,{\mathbb Z})$, consisting of all stable maps of arithmetic genus $g$ with $m$ marked points in the homology class $A$.
The following holds.
- The orbit space $|{\mathcal S}_{g,m,A}^{3,\delta}(Q,\omega)|$ is an open and closed subset of $|{\mathcal S}^{3,\delta}(Q,\omega)|$ and consequently ${\mathcal S}_{g,m,A}^{3,\delta}(Q,\omega)$ has an induced polyfold structure. Moreover, ${\mathcal E}_{g,m,A}^{2,\delta}(Q,\omega,J): ={\mathcal E}^{2,\delta}(Q,\omega,J)|{\mathcal S}_{g,m,A}^{3,\delta}(Q,\omega)$ has an induced strong polyfold bundle structure.
- The orbit space of $(f|{\mathcal S}_{g,m,A}^{3,\delta}(Q,\omega))^{-1}(0)$ is compact.
- For every $1\leq i\leq m$ the evaluation map $$\begin{aligned}
\label{xst1}
ev_i:{\mathcal S}_{g,m,A}^{3,\delta}(Q,\omega)\rightarrow Q\end{aligned}$$ at the $i$-th marked point is an sc-smooth functor in the sense that $\text{ev}_i\circ \Psi$ is sc-smooth if $\Psi$ is a good uniformizer.
- If $2g+m\geq 3$ then the forgetful functor $$\begin{aligned}
\label{xst2}
\sigma:{\mathcal S}_{g,m,A}^{3,\delta}(Q,\omega)\rightarrow {\mathcal R}_{g,m}^{ord}\end{aligned}$$ into the stable Riemann surface category with ordered mark points is sc-smooth as well, where we can take good uniformizers for ${\mathcal S}^{3,\delta}_{g,m,A}(Q,\omega)$ and ${\mathcal R}_{g,m}^{ord}$ to see that the local representative is sc-smooth.
The consequence of (iii) is that the pull-back of a smooth differential form on $Q$ gives an sc-smooth differential form on ${\mathcal S}_{g,m,A}^{3,\delta}(Q,\omega)$. For the forgetful map the pull-back of a smooth differential form defines an sc-smooth differential form on ${\mathcal S}_{g,m,A}^{3,\delta}(Q,\omega)$ as a consequence of (iv).
Moreover $\bar{\partial}_J$ restricted to ${\mathcal S}_{g,m,A}^{3,\delta}(Q,\omega)$ is an sc-smooth Fredholm functor with a compact solution set, and it has a natural orientation, giving us $(\bar{\partial}_J,\mathfrak{o})$ and its restrictions $(\bar{\partial}_{J,(g,k,A)},\mathfrak{o})$. As a consequence we have the linear maps $I_{(g,m,A)}:=I_{(\bar{\partial}_{J,(g,k,A},\mathfrak{o})}$ $$I_{(g,m,A)}:H^\ast_{dR}({\mathcal S}_{g,k,A}^{3,\delta}(Q,\omega))\rightarrow {\mathbb R}.$$ These maps give precisely the data from which one can define the Gromov-Witten invariants, see [@HWZ6].
There is, of course, a large literature on Gromov-Witten invariants and further developments, see [@FO; @FOOO; @LiT; @LuT; @R1; @R2; @Tian; @McD; @MW12; @MW14; @MWss]. All the methods differ. In some sense, all the approaches had to come up with a fix for the fact that classical Fredholm theory doesn’t work.
The theory which we described here provides a very powerful language to deal with moduli problems in symplectic geometry. In [@FHWZ] we shall use this approach to define a Fredholm setup for SFT. The language for doing so is that developed in [@HWZ7; @HWZ8]. The necessary nonlinear analysis comes from [@HWZ8.7].
In [@H2] the ideas described here are carried out on the level of a graduate text describing the construction of Gromov-Witten theory and the verification of its axioms, see [@Man], in detail.
[99]{} A. Adem, J. Leida and Y. Ruan, Orbifolds and Stringy Topology, Cambridge Tracts in Mathematics, 171. Cambridge University Press, Cambridge, 2007. xii+149 pp. V. Borisovich, V. Zvyagin and V. Sapronov, Nonlinear Fredholm maps and Leray-Schauder degree, Russian Math. Survey’s 32:4 (1977), p 1-54. F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki and E. Zehnder, Compactness Results in Symplectic Field Theory, [*Geometry and Topology*]{}, Vol. 7, 2003, pp.799-888. H. Cartan, Sur les rétractions d’une variété, C. R. Acad.Sc. Paris, t. 303, Serie I, no 14, 1986, p. 715. K. Cieliebak, I. Mundet i Riera and D. A. Salamon, Equivariant moduli problems, branched manifolds, and the Euler class, Topology [**42**]{} (2003), no. 3, 641–700. S. Donaldson and P. Kronheimer, The geometry of four-manifolds, Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1990. Y. Eliashberg, A. Givental and H. Hofer, Introduction to Symplectic Field Theory, Geom. Funct. Anal. [**2000**]{}, Special Volume, Part II, 560–673. H. Eliasson, Geometry of manifolds of maps, J. Differential Geometry [**1**]{}(1967), 169–194. O. Fabert, J. W. Fish, R. Golovko, and K. Wehrheim, Polyfolds: A First and Second Look, arXiv:1210.6670. J. Fish and H. Hofer, Lectures on Polyfold Constructions in Symplectic Geometry [I]{}: The Polyfolds of Gromov-Witten Theory, in preparation. J. Fish, H. Hofer, K. Wysocki, and E. Zehnder, Applications of Polyfold Theory [II]{}: The Polyfolds of SFT, in preparation. A. Floer and H. Hofer, Coherent orientations for periodic orbit problems in symplectic geometry, Math. Z. [**212**]{} (1993), no. 1, 13–38. K. Fukaya and K. Ono, Arnold conjecture and Gromov-Witten invariants. Topology,Vol. 38 No 5, 1999.pp. 933-1048. K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Lagrangian intersection Floer theory-anomaly and obstruction, Part I. AMS/IP Studies in Advanced Mathematics, 46.1. American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2009. pp i-xii+396. K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Lagrangian intersection Floer theory-anomaly and obstruction, Part II. AMS/IP Studies in Advanced Mathematics, 46.2. American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2009. pp. i-xii + 397-805. P. Gabriel and M Zisman, Calculus of Fractions and Homotopy Theory, Ergebnisse Vol. 35, Springer (1967). M. Gromov, Pseudoholomorphic Curves in Symplectic Geometry, [*Inv. Math.*]{} Vol. 82 (1985), 307-347. A. Haefliger, Homotopy and integrability, in Manifolds (Amsterdam, 1970), 133–163, Springer Lecture Notes in Math., 197, 1971. A. Haefliger, Holonomie et classifiants, Asterisque 116 (1984), 70–97. A. Haefliger, Groupoids and foliations, Contemp. Math. 282 (2001), 83–100. H. Hofer, A General Fredholm Theory and Applications, Current Developments in Mathematics, edited by D. Jerison, B. Mazur, T. Mrowka, W. Schmid, R. Stanley, and S. T. Yau, International Press, 2006. H. Hofer, Polyfolds And A General Fredholm Theory, arxiv 0809.3753. H. Hofer, K. Wysocki and E. Zehnder, Deligne-Mumford-Type spaces with a View Towards Symplectic Field Theory, lecture note in preparation. H. Hofer, K. Wysocki and E. Zehnder, A General Fredholm Theory [I]{}: A Splicing-Based Differential Geometry, J. Eur. Math. Soc. (JEMS) 9 (2007), no. 4, 841–876. H. Hofer, K. Wysocki and E. Zehnder, A General Fredholm Theory [II]{}: Implicit Function Theorems, Geom. Funct. Anal. 19 (2009), no. 1, 206-293. H. Hofer, K. Wysocki and E. Zehnder, A General Fredholm Theory [III]{}: Fredholm Functors and Polyfolds, Geom. Topol. 13 (2009), no. 4, 2279-2387. H. Hofer, K. Wysocki and E. Zehnder, Integration Theory for zero sets of polyfold Fredholm sections, Math. Ann. 346 (2010), no. 1, 139-198. H. Hofer, K. Wysocki and E. Zehnder, Applications of Polyfold Theory [I]{}: Gromov-Witten Theory, arxiv 1107.2097, Memoirs of the AMS, to appear. H. Hofer, K. Wysocki and E. Zehnder, Polyfolds and Fredholm Theory [I]{}: Basic Theory in M-Polyfolds, arXiv:1407.3185 H. Hofer, K. Wysocki and E. Zehnder, Polyfolds and Fredholm Theory [II]{}: Basic Theory in Polyfolds, in preparation. H. Hofer, K. Wysocki and E. Zehnder, *Sc-Smoothness, Retractions and New Models for Smooth Spaces*, Discrete and Continuous Dynamical Sytems, Vl 28 (No 2), (2010), 665–788. D. Joyce, Kuranishi bordism and Kuranishi homology, preprint arxiv 0707.3572v1. M. Konsevich, Enumeration of rational curves via torus action, in “The Moduli Space of Curves” (R. Dijkgraaf, C. Faber and G. van der Geer, eds.) Birkhauser (1995), 335-568. S. Lang, [*Introduction to differentiable manifolds*]{}, Second edition, Springer, New York, 2002. J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds, in: Topics in symplectic 4-manifolds (Irvine, CA, 1996), 47-83, Int. Press (1998). G. Lu and G. Tian, Constructing virtual Euler cycles and classes, Int. Math. Res. Surv. IMRS 2007, 2008 in electronic version, Art. ID rym001, 220 pp. Y. Manin, [*Frobenius Manifolds, Quantum Cohomology, and Moduli Spaces*]{}, AMS Colloquium Publications, Volume 47. D. McDuff, Groupoids, Branched Manifolds and Multisection, J. Symplectic Geom. 4, 259-315 (2006). D. McDuff, Notes on Kuranishi Atlases, arxiv:1411.4306v1. D. McDuff and K. Wehrheim, Smooth Kuranishi atlases with trivial isotropy, arXiv:1208.1340; revision in progress. D. McDuff and K. Wehrheim, Smooth Kuranishi atlases with isotropy, work in progress. D. McDuff and K. Wehrheim, Stratied smooth Kuranishi atlases, work in progress. D. McDuff and D. Salamon, [*Introduction to symplectic topology*]{}, 2nd edition, Oxford University Press, 1998. D. McDuff and D. Salamon, [*J-holomorphic curves and symplectic topology*]{} , Colloquium Publications, vol. 52, Amer. Math. Soc., Providence, RI, 2004, xii+669 pp. I. Moerdijk, Orbifolds as Groupoids: An Introduction, Contemp. Math. 310, 205-222 (2002). I. Moerdijk and J. Mrčun, [*Introduction to Foliation and Lie Groupoids*]{}, Cambridge studies in advanced mathematics, Vol. 91, 2003. J. Robbin and D. Salamon, A construction of the Deligne-Mumford orbifold. J. Eur. Math. Soc. (JEMS) 8 (2006), no. 4, 611-699. Y. Ruan, Topological sigma model and Donaldson type invariants in Gromov theory, Duke Math. J. 83, 1996, 451-500. Y. Ruan, Symplectic Topology on Algebraic 3-folds, J. Diff. Geom. 39, 1994, 215-227. S. Smale, An infinite dimensional version of Sard’s theorem. Amer. J. Math. 87 (1965), 861–866. S. Suhr and K. Zehmisch, Polyfolds, Cobordisms, and the strong Weinstein conjecture, arXiv:1411.5016. G. Tian, Quantum cohomology and its associativity, Current Developments in Mathematics, 1995, p. 361-397, International Press H. Triebel, [*Interpolation theory, function spaces, differential operators*]{}, North-Holland, Amsterdam, 1978.
|
---
author:
- |
Aaron N. Siegel\
`[email protected]`
- |
David Wolfe\
`[email protected]`
title: The Package
---
The Symbol Font
===============
defines the following symbols.
---------------------------- ------------------------------ --------------------------- -- -- --
[$\cgup$ & [``]{}]{} [$\cgdown$ & [``]{}]{} [$\cgstar$ & [``]{}]{}
[$\cgdoubleup$ & [``]{}]{} [$\cgdoubledown$ & [``]{}]{} [$\cgneg$ & [``]{}]{}
[$\cgtripleup$ & [``]{}]{} [$\cgtripledown$ & [``]{}]{} [$\cgfarstar$ & [``]{}]{}
[$\cgquadup$ & [``]{}]{} [$\cgquaddown$ & [``]{}]{} [$\cgsunny$ & [``]{}]{}
[$\cgtiny$ & [``]{}]{} [$\cgminy$ & [``]{}]{} [$\cgloony$ & [``]{}]{}
[$\cgko$ & [``]{}]{} [$\cgKo$ & [``]{}]{}
[$\cgkobar$ & [``]{}]{} [$\cgKobar$ & [``]{}]{}
---------------------------- ------------------------------ --------------------------- -- -- --
: ordinary symbols.
------------------------- --------------------------- ----------------------------- -- -- --
[$\cglfuz$ & [``]{}]{} [$\cggfuz$ & [``]{}]{} [$\cgfuzzy$ & [``]{}]{}
[$\cgupsum$ & [``]{}]{} [$\cgdownsum$ & [``]{}]{} [$\cgnmultiply$ & [``]{}]{}
------------------------- --------------------------- ----------------------------- -- -- --
: binary relations.
A few notes.
- Usage of [``]{} for Norton multiplication has been largely supplanted by a simple [``]{}. Compare: $$G \cgnmultiply \cgup \qquad\qquad \textrm{versus} \qquad\qquad G \cdot \cgup$$
- This author dislikes the [``]{} symbol, precisely because there are already too many slashes in CGT. A reasonable alternative is [``]{}[``]{}, which is available in the package. Compare: $$G \cgfuzzy H \qquad\qquad \textrm{versus} \qquad\qquad G \not\gtrless H$$
- The package provides a [``]{} symbol that can be used as an alternative to [``]{}: $$\cgloony \qquad\qquad \textrm{versus} \qquad\qquad \textrm{\Moon}$$
Braces-and-Slashes Notation
===========================
also has a powerful facility for typesetting games using braces-and-slashes notation. The basic command is [``]{}. All braces and slashes used within the command will be properly spaced, and in displayed equations they’ll be sized to fit the surrounding material. Some examples:
[@l@c@]{}
& $\displaystyle \combgame{2||1|0}$\
& $\displaystyle \combgame{\{3,\{4||2|1\}|||0||||-8\}}$
\^T G =
$\displaystyle \int^T G = \combgame{\{T + \int^T G^L | -T + \int^T G^R\}}$
- There is also a starred form, [``]{}, that suppresses growth of the vertical bars:
[@l@c@]{}
& $\displaystyle \combgame{2|1||0|||-1||||-2}$\
& $\displaystyle \combgame*{2|1||0|||-1||||-2}$
- You should use [``]{} whenever appropriate, even for simple expressions, since it will typeset the result much more cleanly than otherwise. Compare the following two examples.
[l@l@l]{}
& $\combgame*{\{2||-1|-3\}}$ & Beautiful!\
{2||-1|-3}
& $\{2||-1|-3\}$ & Hideous!
- Within the `combgame` argument, the brace commands [``{]{} and [`}`]{} are redefined to mean [``]{}[``{]{} and [``]{}[`}`]{}. The old commands are still available as [``]{} and [``]{}.
- Due to the way TeX processes command inputs, the slash notation will not function correctly if used inside a macro. If you wish to define macros that refer to [``]{}, there is an alternative command `\cgslashes{n}` for an $n$-tuple slash. The commands [``]{} and [``]{} are shorthand for $n = 1$ and $2$, respectively. Here’s an example:
\newcommand\threeswitch[3]
{\combgame{\{#1 \cgsslash #2 \cgslash #3\}}}
Then `\threeswitch{2}{1}{0}` would typeset
$$\combgame{\{2||1|0\}}$$
- You can control the growth of the slashes by setting [``]{}. The default is `1.5pt`. For example:
[@l@c@]{}
& $\displaystyle \combgame{2|1||0|||-1||||-2}$\
\setlength\cgslashextension{3pt}
\combgame{2|1||0|||-1||||-2}
& $\displaystyle \setlength\cgslashextension{3pt} \combgame{2|1||0|||-1||||-2}$
Game Trees
==========
You can typeset game trees easily with the powerful [``]{} command. Trees should be placed inside a `pspicture` environment and can coexist with other pstricks objects. Here’s a simple example:
(5,4) (1,4)
The general syntax of a [``]{} argument is `\cgtree{node}` where `node` has the following specification:
node ::= label (subtree)?
| special
subtree ::= '(' (node)* '|' (node)* ')'
`label` may be either a single token, or a more complicated expression delineated by braces. The two nodelists in the subtree expression are typeset as left and right options of the parent node. Left options are laid out right-to-left; Right options left-to-right; always starting at the parent node. If the label is `.` then a node will be created with no label. `special` may be one of the following:
- `+` increases the space between options
- `-` decreases the space between options
- `:` creates a symbolic link (described below)
The `cgtree` command takes several options:
- `arrows` specifies the arrowheads used for tree edges, as in pstricks. Example: `arrows=->`
- `unit` specifies the scale for drawing the tree, e.g., `unit=.5cm` (default: 1cm)
- `nodesep` specifies the separation between edges and node boundaries. Default is `nodesep=.75ex`
In addition, each *node* may have several options. Node options should be written in brackets immediately following the node label (and before the associated subtree, if one exists).
- `arrow` specifies the arrowhead for the edge *to* this node
- `sep` specifies the separation for this node
- `name` gives a name for this node. *Names must consist only of letters and numbers*. They can be used in symbolic links (see below) or elsewhere in the `pspicture` environment.
- `ko` draws the edge *to* this node as a ko.
If a `:` is specified instead of a node, a symbolic link is created. The `:` must be followed by a list of options in brackets, which *must* include the `name` option. An edge will then be drawn to the previously named node. Be careful, as this will *not* take into account whether the edge points left or right!
(6,6) (1,6)
One last example: the following mess typesets a pretty figure from Bill Fraser’s thesis.
(19,6) (5,6)
Game Boards
===========
There is also an extensible facility for typesetting grid-based diagrams. The following examples illustrate usage for several of the built-in games.
=
You can declare new games easily with the [``]{} (equivalently, [``]{}) command. The syntax is very simple. The command argument consists of a list of allowed grid characters together with instructions for typesetting them. For example, the [``]{} command is declared as follows.
\newgridgame[unit=.3cm]{clobber}{
X {\ClobberX}
O {\ClobberO}
x {\ClobberX}
o {\ClobberO}
}
Here [``]{} and [``]{} are primitive commands for rendering clobber symbols. See the source file for more examples and a complete list of options. The file `board.tex` contains more examples.
Thermographs
============
provides a `thermoplot` environment that can be used to draw an arbitrary number of thermographs together on the same plot. For example:
(0,3)(2,1)(2,-1);(0,3)(-1,2)(-1,1)(-3,-1)
Coordinates must be specified as *rational numbers in the usual (CGT) coordinate space*. The [``]{} command is followed by an argument specifying the mast value. This is followed by any number of semicolon-separated trajectories, listing the $(\textit{v},\textit{t})$-coordinates of each critical point on the trajectory.
Multiple thermographs can be placed on the same plot:
(1/2,13/2)(2,5)(2,2)(8,-1); (1/2,13/2)(2,5)(2,2)(-1,-1) (-19/4,19/4)(1,-1); (-19/4,19/4)(-21,-1)
Note how, in these examples, each trajectory is carried down to $-1$ instead of $0$. This generates the “hooks” that extend below $t = 0$ on the thermographs.
You can also place individual trajectories instead of full thermographs. This can be useful if the mast has nonzero slope. For example:
(1,4)(-1,2)(-1,1)(-3,-1) (-3,2)(-2,1)(-2,-1);(-3,2)(-4,1)(-4,-1)
|
---
abstract: 'We analyze a simple Split Supersymmetry scenario where fermion masses come from anomaly mediation, yielding $m_s \sim 1000$ TeV, $m_{3/2} \sim 100$ TeV, and $m_f \sim 1 $ TeV. We consider non-thermal dark matter production in the presence of moduli, and we find that the decay chains of moduli $\to$ LSPs and moduli $\to$ gravitinos $\to$ LSPs generate dark matter more efficiently than perturbative freeze-out, allowing for a light, LHC visible spectrum. These decaying moduli can also weaken cosmological constraints on the axion decay constant. With squark masses of order $1000$ TeV, LHC gluinos will decay millimeters from their primary vertices, resulting in a striking experimental signature, and the suppression of Flavor Changing Neutral Currents is almost sufficient to allow arbitrary mixing in squark mass matrices.'
---
[**Split Supersymmetry and Dark Matter Generation**]{}
[**Jared Kaplan[^1]**]{}
[*Jefferson Laboratory of Physics, Harvard University,\
Cambridge, Massachusetts 02138, USA*]{}
Introduction
============
With the Large Hadron Collider only a few years away, a new division has arisen in the particle physics community – is the unnaturalness of the standard model a problem to be solved through clever model building, or is it a hint that physics at the TeV scale is different than we have imagined, so that the values of dimensionful parameters are determined by anthropic fine tuning? As physicists, we would prefer a ‘physical’ explanation of small numbers such as the cosmological constant and the higgs mass, but the success of Weinberg’s prediction of the cosmological constant [@wein] and the level of fine tuning necessary even in our best theories forces us to take the anthropic argument seriously. We will be taking it seriously for the present work.
Yet the mind of an anthropically-motivated model builder is a troubled one, for it seems that this profound shift in our worldview has had the unfortunate side effect of putting our whole enterprise out of business. If the weak scale is determined by anthropic selection, perhaps there is no new physics at the TeV scale, and the LHC and even the ILC will be colossal disappointments.
There are two simple reasons to remain hopeful about new TeV scale physics: gauge coupling unification and dark matter. The former requires new particles charged under the standard model gauge group, and the latter requires a new stable particle with the correct relic density. If we assume for maximal simplicity that these two problems have a common solution, then it is reasonable to assume that our dark matter candidate will have a weak interaction cross section. Furthermore, if we assume that the relic dark matter density is determined by perturbative freeze-out, then we can expect new physics at the TeV scale, tightly constrained by gauge coupling unification. Some of these assumptions may be wrong, but that is not necessarily a bad thing – the minimal model that follows from this reasoning will probably be invisible to the LHC [@mindmunif].
A very popular set of models that result from this methodology fall under the heading of Split Supersymmetry [@split], where the scalar superpartners necessitated by supersymmetry are very heavy, while the fermions, protected by chiral symmetry, lie near the TeV scale. The purpose of the present work is to show that in one particularly simple model of Split Supersymmetry, perturbative freeze-out is not the dominant mechanism for generating dark matter. However, the mechanism that will replace it is more efficient, allowing for an even lighter, more LHC-visible spectrum. This seems like good news, because in standard Split Supersymmetry scenarios [@aspsplit], [@wells], dark matter is expected to be too heavy to be seen at the LHC; however we will see that in our case there is a danger of overclosure. One of the most elegant scenarios is ruled out, but we explore several ideas that can rescue it, and the scenario of Moroi and Randall [@modtosm] remains a viable possibility.
To understand our mechanism, we first need to explain the spectrum of the model. As shown in [@anom], [@outworld], wherever supersymmetry is broken, there will be visible sector supersymmetry breaking effects from Anomaly Mediation. Furthermore, the methodology of effective field theory requires that we include in the Lagrangian all higher dimension operators allowed by symmetry, with appropriate suppression by inverse powers of the cutoff. When supersymmetry is broken in a hidden sector, F and D term VEVs in these operators will induce supersymmetry breaking in the visible sector. These two effects are completely generic, requiring no theoretical gymnastics, so a model where supersymmetry breaking arises only in this way would be particularly elegant. This is the model that we will analyze; it was studied for related reasons in [@wells].
With Anomaly Mediated Supersymmetry Breaking, gaugino masses are given by a loop suppression factor times $m_{3/2}$, so we expect that $m_{3/2} \sim 100$ TeV so that new fermions are near a TeV. Contributions from higher dimension operators suggest that the scalar mass scale $m_s \sim 1000$ TeV. In supersymmetric theories, there generically exist weakly interacting moduli that get masses of order $m_s$ after supersymmetry breaking. Now we see the complexity of the cosmology – not only do we have to account for the perturbative freeze out of the LSPs, but we also need to consider processes involving late-decaying moduli and gravitinos. These late decaying particles produce a great deal of entropy, potentially weakening cosmological constraints on the properties of axions.
We will see that with our mass spectrum, dark matter production is dominated by two potential decay chains: moduli $\to$ LSPs and moduli $\to$ gravitinos $\to$ LSPs. Only the first process is available for $m_\phi < 2m_{3/2}$,[^2] whereas both decay chains are open for $m_\phi > 2m_{3/2}$ – thus the physics is qualitatively different for different values of the modulus mass. The first process was studied in [@modtosm], and it naturally leads to the correct relic density in our scenario when $m_\phi \lesssim m_{3/2}$. The most naive version of the second decay chain is too efficient [@newdecay], [@newdecay2], [@dine], so we consider a variety of mechanisms to alleviate this problem, including symmetries in the modulus sector, scenarios with multiple moduli, KKLT type moduli, and supersymmetry breaking dominated by D-terms. Of course we do not need these mechanisms for a successful cosmology if $m_\phi \lesssim m_{3/2}$, but it is interesting to explore all reasonable options.
Our model also has an exciting LHC signature. The gluino must decay through a virtual squark, so with $1000$ TeV squarks, it is very likely that the LHC will see displaced gluino vertices if gluino production is kinematically allowed. In fact, we expect that gluinos will be copiously produced, since cosmology suggests a light spectrum.
We are also in an interesting region for Flavor Changing Neutral Currents. The strongest constraints from FCNC come from the $\epsilon$ parameter of the K-$\bar{\mathrm{K}}$ system, which depends on the imaginary parts of the mass insertion parameters. Setting all mass insertions equal to a common value $\delta$ (there are no significant cancellations), we find from [@FCNC] that $$\epsilon = 3 \times 10^{-3} \delta \left(\frac{1000 \ \mathrm{TeV}}{m_s}\right)^2$$ Experiments constrain $\epsilon < 2 \times 10^{-3}$, so for $m_s \approx 1000$ TeV, we find that as long as $\delta \lesssim 1/2$, our scalar mass scale does not conflict with observations of FCNC. We also expect that electric dipole moments induced by new interactions may be visible at next generation experiments, as shown in [@aspsplit], and our spectrum may help to explain neutrino masses, as examined in [@wells].
The outline of this paper is as follows. In section two we display the mass spectrum of our model. In section three we consider the cosmological implications of moduli, gravitinos, and LSP (Wino) dark matter. In section four we show that displaced gluino vertices will be a generic LHC signature for our model, and with section five we conclude.
The Mass Spectrum
=================
We obtain the fermion mass spectrum $$\begin{aligned}
m_{3/2} & = & \frac{\langle W \rangle}{M_p^2} = \langle F_\phi \rangle \sim 100 \ \mathrm{TeV} \\
m_{\tilde{g}} & = & \frac{\beta(g)}{2g} \langle F_\phi \rangle \sim 1 \ \mathrm{TeV}\end{aligned}$$ from anomaly mediated supersymmetry breaking [@anom], [@outworld]. The details of hidden sector supersymmetry breaking are unimportant because the hidden sector only communicates with standard model particles through the auxilary field of supergravity and through higher dimension operators in the Kahler potential. We assume that the scalars acquire masses from operators such as $$L \supset \int d^4 \theta \frac{1}{M_{GUT}^2} X^{\dag}X Q^{\dag} Q$$ where $X$ is a hidden sector field, because these terms cannot be forbidden by any symmetry. We expect these operators to arise generically when we integrate out GUT or string scale particles, making them a factor of $10$ - $100$ times larger than $m_{3/2}$. Thus we can estimate that $$m_s = m_{3/2} \frac{M_{pl}}{M_{GUT}} \sim 1000 \ \mathrm{TeV} ,$$ completing our rough picture of the mass spectrum.
It is also important to consider the generation and the effects of the $\mu$ and $B\mu$ terms. Will $\mu$ and $B \mu$ be near $m_s$, $m_{3/2}$, or at the TeV scale? First, there can be contributions from R-symmetry breaking spurionic operators $X = 1 + \theta^2 m_s$ such as $$\begin{aligned}
L & \supset & \int d^4 \theta \epsilon X^{\dag} X H_1 H_2 \\
L & \supset & \int d^4 \theta \epsilon X^{\dag} H_1 H_2\end{aligned}$$ where the factor of $\epsilon$ is included to parameterize a possible approximate PQ symmetry. This gives rise to $B \mu \sim \epsilon m_s^2$ and $\mu \sim \epsilon m_s$. In most split supersymmetry scenarios we use R-symmetry to prohibit such spurionic contributions because they produce a nearly degenerate spectrum. However, in our scenario $m_{3/2}/m_f \sim 100$, which can be conveniently explained as a loop factor from anomaly mediation. Thus such operators may be permitted. If supersymmetry breaking does not lead to R-symmetry breaking we only have spurions[^3] such as $$L \supset \int d^4 \theta \epsilon Y H_1 H_2$$ with $Y = 1 + \theta^4 m_s^2$, which contribute to $B \mu$ only. Finally, there are contributions from the conformal compensator of supergravity $$L \supset \int d^4 \theta \epsilon \phi^{\dag} \phi H_1 H_2$$ where $\phi = 1 + \theta^2 m_{3/2}$, so that $\mu \sim \epsilon m_{3/2}$. Thus we find that $B \mu \approx \epsilon m_s^2$ in all cases, and either $\mu \approx \epsilon m_s$ or $\mu \approx \epsilon m_{3/2}$ . We can take $\epsilon \approx 1/100$ to explain the top-bottom mass hierarchy, giving a $\mu$ term at the TeV scale.
The presence of the $\mu$ term modifies the gaugino masses [@anom], [@outworld] so that $$\begin{aligned}
\label{spectrum}
m_{\tilde{b}} & = & 8.9 \times 10^{-3} \left(1 - \frac{f(\mu^2/m_A^2)}{11} \right) m_{3/2} \\
m_{\tilde{w}} & = & 2.7 \times 10^{-3} \left(1 - f(\mu^2/m_A^2) \right) m_{3/2} \\
m_{\tilde{g}} & = & 2.6 \times 10^{-2} m_{3/2}\end{aligned}$$ where $m_A^2 \sin(2 \beta) = B \mu$ and the function $$f(x) = \frac{2 x \ln(x)}{x - 1}$$ For small values of $x$ (the expected case) this is a small effect. It increases the separation between the wino, bino, and gluino mass scales, but there is no real qualitative change in the spectrum. For large $x$ we can obtain a very light Wino LSP, although we would need to abandon the PQ symmetry; this will be breifly considered in section 3.3.3.
Cosmology
=========
Our variant of the Split Supersymmetry mass spectrum includes $100$-$10,000$ TeV moduli field(s) $\phi$ and a $100$ TeV gravitino, so we must check that these new ingredients do not disturb Big Bang Nucleosynthesis, and that the correct relic abundance of LSP dark matter candidates obtains. We begin with some general issues about our universe’s history, and then we consider the possibility that our moduli fields weaken cosmological bounds on the axion. Finally, we explain the details of our proposed LSP generation mechanism.
We can think of our modulus field as though it were an inflaton – it begins with a VEV $\phi_0$ that can be of order $M_{pl}$. Once the hubble constant decreases to $H \sim m_\phi$, the modulus begins to oscillate and its energy density red-shifts like that of matter. This occurs very early in the universe’s history, with $T_{roll} \sim \sqrt{M_{pl} m_{\phi}} \sim 10^9$ TeV. At a time $t_{eq}$ shortly after $\phi$ begins to roll (assuming $\phi$ begins with a Planckian VEV), $\rho_\phi = \rho_{Rad}$, and from this point until $t_{decay} = \Gamma_{tot}^{-1}$ the universe is modulus dominated. We take the modulus decay rate as[^4] $$\Gamma_{tot} = \frac{N}{16 \pi} \frac{m_\phi^3}{M_{pl}^2} =
\left(10^{-5} \ \mathrm{sec}\right)^{-1} \left( \frac{N}{10} \right) \left(\frac{m_\phi}{1000 \ \mathrm{TeV}}\right)^3$$ where $M_{pl} = 2.4 \times 10^{18}$ GeV is the reduced Planck mass. We can think of $N$ as the number of available light decay modes, although it is strongly dependent on unknown, $O(1)$ couplings. We see that the modulus decays well before BBN. The decaying modulus will dilute particles left over from the original inflaton decay, reheat the universe, and re-populate the universe with gravitinos.
Any thermal gravitinos produced during a prior inflationary reheating are diluted by the entropy from the modulus decay, which is given by $$\frac{s_{after}}{s_{before}} = \left(\frac{\rho_\phi(t_{decay})}{\rho_R(t_{decay})}\right)^{3/4}
\sim \frac{M_{pl}}{m_\phi} \left(\frac{\phi_0}{M_{pl}}\right)^2$$ where the ratio of the energy densities at the time of modulus decay simply comes from the two different equations of state. If we assume $\phi_0$ is in the neighborhood of $M_{pl}$, then the entropy released is very large. Modulus decay reheats the universe to a temperature [@modtogravmod] $$T_R \approx 1.1 g_*^{-1/4} \sqrt{\Gamma_{tot} M_{pl}}
= 180 \ \mathrm{MeV} \times \left( \frac{N}{10} \right)^{1/2} \left(\frac{g_*}{10.75}\right)^{-1/4} \left(\frac{m_\phi}{1000 \ \mathrm{TeV}}\right)^{3/2}$$ which is very low, so we certainly do not thermally regenerate heavy particles. To avoid disturbing BBN, we need $T_R \gg 1$ MeV, so $m_\phi \gg 30$ TeV. Similarly, the gravitino lifetime is [@moroithesis] $$\tau_{3/2} = 5 \times 10^{-2} \ \mathrm{sec} \left(\frac{100 \ \mathrm{TeV}}{m_{3/2}}\right)^3$$ so for $m_{3/2} > 60$ TeV gravitinos do not disturb BBN [@BBNgrav], [@newdecay] even if they are re-introduced by modulus decay. Note, however, that the lifetime is increased in split supersymmetry because the gravitino cannot decay to states involving the heavy scalars.
Dark matter is produced in two different ways: directly from decaying moduli, and indirectly from decaying gravitinos. Thermal relic gravitinos are diluted to negligible levels by modulus decay, so gravitinos can only arise from the decay of heavy moduli with $m_\phi > 2m_{3/2}$. Thus there are two qualitatively different types of moduli – those with $m_\phi > 2 m_{3/2}$ and those with $m_\phi < 2 m_{3/2}$. In the following subsection we consider direct modulus-LSP interactions, focusing on moduli with $m_\phi < 2 m_{3/2}$, and commenting on heavier moduli. Then in the next subsection we consider gravitino production from moduli with $m_\phi > 2 m_{3/2}$. It turns out that dark matter production from gravitinos is more efficient than direct production from modulus decay; this is essentially because the dark matter from gravitinos is produced later in the universe’s history, so it accounts for a larger fraction of the universe’s energy density.
Direct Modulus Decay to Dark Matter
-----------------------------------
We expect that the modulus will have generic string or Planck suppressed couplings to MSSM fields. For example, we may have interactions such as $$\int d^2\theta \frac{\Phi}{M_{pl}} W_\alpha W^\alpha \ \ \ \implies \ \ \
L \supset \frac{\phi}{M_{pl}} F_{\mu \nu}^2 + \frac{F_\phi}{M_{pl}} \lambda \lambda ,$$ where $\lambda$ may be any gaugino[^5]. If $\Phi$ has a supersymmetric mass, then $F_\phi \propto m_\phi \phi$, and both terms give large decay rates, but in other situations where $m_\phi$ only arises after supersymmetry breaking, we may have $\frac{d F_\phi}{d \phi} < m_\phi$ and the fermionic decay mode will be suppressed by the fermion mass. Due to R-Parity conservation, this implies that an order one fraction of the ‘modulus particles’ may eventually produce an LSP. The number density of ‘modulus particles’ relative to the entropy just after reheating is roughly $$Y_\phi = \left( \frac{m_{\phi}}{T_R} \right)^{-1} \sim \left( \frac{m_{\phi}}{M_{Pl}} \right)^{1/2}$$ If this were the whole story, we would overclose the universe. If the branching ratio to fermions is suppressed by $m_f^2/m_\phi^2$ an acceptable dark matter density might be possible.
However, the LSPs produced by modulus decay pair annihilate [@modtosm] until their annihilation rate is less than $H$, after which they freeze out. Although the Wino’s produced by the decaying modulus are ultra-relativistic, they acquire a thermal distribution with temperature $T_R$ from interactions with the radiation bath. For instance, the process $\tilde{W^0} \nu \to \tilde{W^0} \nu$ proceeds more quickly than $H$ for $T_R \gg 1$ MeV, rendering the Winos non-relativistic. The annihilation cross section for Wino dark matter [@modtosm] into $W^{\pm}$ pairs is $$\langle v_{rel} \sigma \rangle = \frac{g^2}{2 \pi} \frac{1}{m_{LSP}^2} \frac{(1-x_W)^{3/2}}{(2-x_W)^2}$$ in the non-relativistic limit, with $x_W = m_W^2/m_{LSP}^2$. Thus the maximal dark matter relic density is reduced to $$\label{annih}
n_{LSP} \sim \left. \frac{3 H}{2 \langle v \sigma \rangle} \right|_{T_R} \sim \frac{T_R^2}{M_{pl} \langle v \sigma \rangle} .$$ Note that this will apply to any late decay that produces dark matter, including gravitino decay. Thus we find [@modtosm] $$\frac{\rho}{s} \approx 4 \times 10^{-10} \ \mathrm{GeV} \times \left( \frac{10}{N} \right)
\left(\frac{g_*}{10.75} \right)^{-1/4} \frac{(2-x_W)^2}{(1-x_W)^{3/2}}
\left(\frac{m_{LSP}}{100 \ \mathrm{GeV}} \right)^3 \left(\frac{100 \ \mathrm{TeV}}{m_\phi} \right)^{3/2}$$ where we have taken $g_* = 10.75$. We need $\rho/s \approx 4 \times 10^{-10}$ GeV [@WMAP] to account for dark matter, so if there is a single light modulus with $m_\phi < 2m_{3/2}$, we can easily obtain the correct relic abundance [@modtosm]. Since the decay of this modulus will not produce gravitinos, we have a viable cosmology.
If $m_\phi \gg m_{3/2}$ then after modulus decay we are left with a negligable density of LSPs, so next we consider the consequences of gravitino production in scenarios with such a heavy modulus.
Moduli-Gravitino Interactions and Indirect Dark Matter Production
-----------------------------------------------------------------
There are two potential channels in which modulus decay produces gravitinos, $$\begin{aligned}
\phi & \to & \tilde{\phi} + \psi_{3/2} \\
\phi & \to & 2\psi_{3/2}\end{aligned}$$ where $\tilde{\phi}$ is the modulino, the superpartner of the modulus field. We do not expect the modulus to decay to a gravitino and another fermion because the other fermion would reside in the hidden sector, and we expect that hidden sector fields are very massive. For now we will assume that the modulino is very heavy, so that the first channel is also forbidden by kinematics – later we will see that relaxing this assumption does not help us to obtain a viable cosmology. In this section we consider the limit that $m_\phi \gg m_{3/2}$, and we begin with the two gravitino channel.
Gravitinos interact primarily through their longitudinal (helicity $\pm 1/2$) components, which correspond to the goldstino, the field that is ‘eaten’ in the superhiggs mechanism. Thus the goldstone boson equivalence theorem becomes the goldstino equivalence theorem, which simply tells us that amplitudes involving longitudinal gravitinos can be computed using the goldstino instead. In the appendix it is shown that the only two derivative operator involving a scalar and two non-linearly realized, derivatively coupled goldstinos (longitudinal gravitinos) is $$\frac{1}{F^\dag} \partial_\nu s_\chi^\dag \chi \sigma^\nu \bar{\sigma}^\mu \partial_\mu \chi
- \frac{1}{F} \partial_\nu s_\chi \chi^\dag \bar{\sigma}^\mu \sigma^\nu \partial_\mu chi^\dag .$$ where the ‘sgoldstino’ field is $$s_\chi = \sum_i \frac{F_i}{F} \phi_i ,$$ and $F^2 = \Sigma_i F_i^2 + \Sigma_a D_a^2$ gives the goldstino decay constant. The sgoldstino is generally not a mass eigenstate – it is a blend of many different scalar fields. Thus the rate for moduli decay to goldstino pairs is determined entirely by the overlap of the mass eigenstate modulus with the sgoldstino.
In supergravity one expects higher dimension operators to be present in the Kahler potential, with suppression of order the GUT, string, or Planck scale. These operators tend to shift the VEVs of scalar fields and auxiliary fields. We would expect the effect on a modulus to be proportional to $1/m_{\phi}$ because the mass tends to stabilize the field, so on dimensional grounds we can estimate that $$F_{\Phi} \sim \frac{\mu_{susy}^4}{m_{\phi} M_{pl}} \sim \frac{m_{3/2}}{m_\phi}F$$ for the auxiliary field of a heavy modulus. A more rigorous argument for this conclusion was given in [@newdecay2], and it seems to be generically true in specific models in the absence of special symmetries[^6]. Assuming that the $\phi$ field really is a mass eigenstate, this implies a rate for $\phi \to 2 \psi_{3/2}$ $$\Gamma_{3/2} = \frac{C}{16 \pi} \frac{m_\phi^3}{M_{pl}^2}$$ where $C$ is a dimensionless constant. The branching fraction for modulus decay to gravitino pairs is therefore $C/N$.
In the approximation of instantaneous modulus decay, the ratio of modulus particles before the decay to entropy immediately after the decay is simply $Y_\phi = T_R/m_\phi$. Thus we estimate that $$Y_{3/2} = \frac{C}{N} \frac{T_R}{m_{\phi}} = 6 \times 10^{-8} \frac{C}{\sqrt{N}}
\left( \frac{m_\phi}{1000 \ \mathrm{TeV}} \right)^{1/2}$$ Now conservation of R-parity implies that each gravitino will eventually decay into at least one LSP, so we can approximate $Y_{LSP} = Y_{3/2}$. If the LSPs do not pair annihilate, then we find $$\frac{\rho_{LSP}}{s} = 6 \times 10^{-6} \ \mathrm{GeV} \frac{C}{\sqrt{N}}
\left( \frac{m_{LSP}}{100 \ \mathrm{GeV}} \right) \left( \frac{m_\phi}{1000 \ \mathrm{TeV}} \right)^{1/2}$$ so we must take $C/\sqrt{N} \lesssim 10^{-4}$ to avoid overclosing the universe. This seems to be a very stringent constraint – it rules out the most elegant scenario with $C \sim 1$ – so in the next section we will consider whether a small $C$ can be achieved naturally, or if other processes can reduce the relic density of LSPs. Note that the relic density is proportional to $m_{LSP}$, so if this scenario can be made viable, then we expect a light spectrum.
One might hope that the LSPs produced by gravitino decay pair annihilate, decreasing their abundance to within acceptable levels. We can use equation (\[annih\]) together with the spectrum given in equation (\[spectrum\]) to write the resulting LSP density in terms of $m_{3/2}$ alone [@newdecay], [@newdecay2] $$\label{gravannih}
\frac{\rho_{ann}}{s} \approx 8 \times 10^{-9} \ \mathrm{GeV} \frac{(2-x_W)^2}{(1-x_W)^{3/2}}
\left(\frac{m_{3/2}}{100 \ \mathrm{TeV}} \right)^{3/2} ,$$ and this would require $m_{3/2} < 20$ TeV to obtain the correct relic density. Such a light gravitino would cause severe problems with BBN, and a Wino with $m_{LSP} < 50$ GeV would already have been detected. Thus if the gaugino masses are given by anomaly mediation (with $\mu \ll m_A$), then we cannot rely on Wino pair annihilation, and instead we need a mechanism for suppressing $C$ if this scenario is to be viable. However, because this result seems tantalizingly close to giving the correct relic density, we will consider below whether deviations from the basic anomaly mediated predictions are possible.
Possibilities for Suppressing Moduli-Induced Gravitinos
-------------------------------------------------------
### Approximate Symmetries
If the modulus is charged under an approximate symmetry, then its decay rate to gravitino pairs may be naturally suppressed. For this to make sense, the modulus cannot have a shift symmetry, so it cannot be a volume modulus as in KKLT, but it may be a shape modulus. It is also important that there exists a sector of light particles with the same symmetry properties, so that modulus decay to this sector is unsuppressed – otherwise the overall modulus decay rate would be small, but the branching fraction to gravitinos would be unchanged.
For instance, as a simple toy model we could consider $$\begin{aligned}
L & = & \int d^4\theta \left[ X^\dag X + \Phi^\dag \Phi + \Phi_c^\dag \Phi_c +
\epsilon \frac{\Phi^\dag X^2}{M_{pl}} + \mathrm{H.C.} - \frac{(X^\dag X)^2}{M^2} + ... \right] \nonumber \\
& & + \int d^2 \theta \left[ \Lambda + \mu^2 X + m_\phi \Phi \Phi_c \right]\end{aligned}$$ where $\Phi$ and $\Phi_c$ have opposite charge under an approximate $U(1)$ symmetry whose violation is parameterized by $\epsilon$. In the limit that $\epsilon \to 0$, the modulus cannot decay to gravitinos by charge conservation, so for small $\epsilon$ the decay rate is suppressed. We could easily include a sector of light ‘charged’ particles so that the overall modulus decay rate is not suppressed. Whether or not such a situation could arise depends on the details of very high-energy physics, but there is no reason to expect that such a setup is impossible.
### Light Moduli
If the modulus is light, with mass close to $2 m_{3/2}$, then the branching fraction to gravitinos will be phase-space suppressed. Thus parametrically we would have $$C \propto \frac{\sqrt{m_{\phi}^2 - 4 m_{3/2}^2}}{m_{\phi}} ,$$ but obtaining $C \sim 10^{-3}$ in this way would require a large, poorly motivated fine-tuning.
A more interesting possibility is that there are two or more moduli with different masses. If the lightest modulus is lighter than $2m_{3/2}$, then it will not produce gravitinos, but its decay will dilute the gravitinos from earlier processes. If the two moduli have masses $m_{\phi} < 2 m_{3/2} \ll m_{\Phi}$, then the lighter modulus begins rolling later in the universe’s history, so that $\rho_{\phi} \approx \rho_{\Phi}$ while they are both oscillating. After $\Phi$ decays, the lighter field dominates the energy density of the universe until it in turn decays, releasing more entropy. Between the time of $\Phi$ decay and the time of $\phi$ decay, radiation red-shifts as $R^{-4}$ but the energy in $\phi$ only red-shifts as $R^{-3}$, so $\phi$ decay dilutes heavy relics such as gravitinos by a factor $$\Delta \approx \left( \frac{R_{\phi}}{R_{\Phi}} \right)^{3/4} \approx \left( \frac{T_{RH-\Phi}}{T_{RH-\phi}} \right)^{3/4}
\approx \left( \frac{m_\Phi}{m_\phi} \right)^{9/8}$$ which could optimistically be as large as $1000$. For instance, with $m_\Phi = 5 \times 10^4$ TeV, $m_\phi = 100$ TeV, $m_{3/2} = 60$ TeV, and $N \sim 10$, one would need $C \sim 1/20$ to obtain the correct relic abundance of dark matter (with these values direct production of dark matter from $\phi$ decay is not a problem). We will see that the presence of such a light modulus is also useful for alleviating cosmological bounds on the axion.
Finally, we could simply return to the original scenario of Moroi and Randall [@modtosm], with a single modulus with $m_\phi < 2 m_{3/2}$. As they showed, the correct relic density can be obtained rather easily by relying on the pair annihilation studied in section 3.1 (in addition, they claimed that the modulus branching fraction to Winos was chirally suppressed, but this is false [@newdecay], [@dine]). It is certainly possible that moduli with masses of order $m_s$ simply do not exist, but this is a UV sensitive issue.
### Model Dependence
Although models with $C \sim 1$ are generic, there are models where $C$ is parametrically different. For instance, as shown in [@dine], in specific supergravity realizations of the KKLT scenario, one finds $$C \sim \left( \frac{m_{3/2}}{m_{\phi}} \right)^2 .$$ In this case $F_\phi \sim \frac{m_{3/2}}{m_\phi} F$ as usual, but ‘$\phi$’ is not a mass eigenstate. The mass eigenstate modulus mixes with the supersymmetry breaking sector, so that it effectively has $F_{eigen} \sim \frac{m_{3/2}^2}{m_\phi^2} F$ giving a suppressed $\phi \to 2 \psi_{3/2}$ decay rate. This result depends on specific properties of the supersymmetry breaking sector (the Polonyi field must be light, with mass of order $m_{3/2}$) that need not be true in more general KKLT-type scenarios, but these theories are technically natural, and in some cases the terms that would give $C \sim 1$ can be forbidden by global symmetries.
We have been assuming that the gauginos get there mass from anomaly mediation alone, but as mentioned above, it is possible that the same operators $$\int d^2\theta \frac{\Phi}{M_{pl}} W_\alpha W^\alpha \ \ \ \implies \ \ \
L \supset \frac{F_\phi}{M_{pl}} \lambda \lambda$$ that allow moduli to decay to gauginos also contribute to the mass of the gauginos. If $F_\phi/M_{pl} \sim m_{3/2}^2/m_\phi$ as expected generically, then the Wino mass can be altered, and for a light gravitino and heavy modulus the Wino could become lighter. In that case, Wino pair annihilation could be sufficient to achieve the correct relic density; for instance, with $m_{\tilde{W}} = 70$ GeV and $m_{3/2} = 100$ TeV, one obtains the correct abundance. Unfortunately this is a UV sensitive question, and it seems to require fine tuning to make the Wino light. Furthermore, there is the possibility of introducing CP violation through these types of interactions. Note that these operators are negligable when $F_\phi$ is small and the $\phi \to 2 \psi_{3/2}$ decay rate is small, which is the domain of the section 3.3.1.
A better solution is to use a large $\mu$ term in equation (\[spectrum\]) to make the Wino light. We would need to abandon the approximate PQ symmetry explaining the top-bottom hierarchy, but with $\mu \sim m_A$ one obtains a lighter Wino, making it possible to explain the dark matter relic abundance purely in terms of pair annihilation.
A final possibility is that supersymmetry breaking is dominated by $D$ terms. As shown in the appendix, in this case $C$ would be suppressed as $$C \propto \left( \frac{F}{D} \right)^2 .$$ This occurs because only the sgoldstino decays to gravitino pairs, and in the limit of $D$-breaking, there is no sgoldstino (instead the goldstino has a vector partner, which is not of interest here). As shown in [@Dterms] it is possible to obtain parametrically large $D$ terms, and $D$-breaking naturally gives a split supersymmetry type spectrum. This possibility may warrant further investigation, as it may help to alleviate the cosmological moduli/gravitino problem in more general circumstances.
The $\phi \to \tilde{\phi} + \psi_{3/2}$ Channel
------------------------------------------------
For completeness, we consider the decay channel $\phi \to \tilde{\phi} + \psi_{3/2}$ [@modtogravmod], although we will see that it is dangerous, so we will need to forbid it kinematically. It comes from operators of the form $$\begin{aligned}
L & \supset & \int d^4 \theta \frac{a}{M_{pl}} X \Phi^\dag \Phi \nonumber \\
& = & a \frac{m_{\tilde{\phi}}}{M_{pl}} \chi \tilde{\phi} \phi -
\sqrt{3} a m_{3/2} m_{\tilde{\phi}} \phi \phi + ...\end{aligned}$$ where $\chi$ is the goldstino, $X = x + \theta \chi + \theta^2 m_{3/2} M_{pl}$ is the field that dominantly breaks supersymmetry, and $a$ is an $O(1)$ coupling constant that must be relatively large so that the modulino will be lighter than the modulus. As a complex scalar field, $\phi$ actually has two real modes, with masses $$m_{\phi \pm} = \sqrt{m_{\tilde{\phi}}^2 \pm \sqrt{3} a m_{3/2} m_{\tilde{\phi}}}
\approx m_{\tilde{\phi}} \pm \frac{\sqrt{3}}{2} a m_{3/2}$$ Generically, both modes will be present, but $\phi_-$ cannot decay to a modulino and a gravitino. If we require that $m_{\phi +}$ is greater than the sum of the modulino and gravitino masses, then we must have $a > (2 + m_{3/2}/m_{\tilde{\phi}})/\sqrt{3}$; if this relation is violated, then the decay will be kinematically prohibited.
With these assumptions, one can calculate the modulus decay rate and branching fraction into modulinos and gravitinos [@modtogravmod], obtaining $$\begin{aligned}
\Gamma & = & \frac{a^2 m_\phi m_{\tilde{\phi}}^2}{8 \pi M_{pl}^2}
\left(1 -\frac{(m_{\tilde{\phi}} + m_{3/2})^2}{m_\phi^2}\right)^{3/2}
\left(1 -\frac{(m_{\tilde{\phi}} - m_{3/2})^2}{m_\phi^2} \right)^{1/2} \nonumber \\
& \approx & \frac{3 \sqrt{2} a^4 m_{\phi} m_{3/2}^2}{8 \pi M_{pl}^2}\end{aligned}$$ for the decay rate into a gravitino and a modulino, where in the second line we take $m_{3/2} \ll m_{\phi}$. Thus we find a relic abundance [@modtogravmod] $$Y_{3/2} \gtrsim 10^{-9} \left(\frac{m_{3/2}}{100 \ \mathrm{TeV}}\right)^2
\left(\frac{1000 \ \mathrm{TeV}}{m_{\phi}}\right)^{3/2}$$ We know that $Y_{LSP} \approx Y_{3/2}$, so we obtain $$m_{LSP} Y_{LSP} \gtrsim 10^{-7} \ \mathrm{GeV} \times \left(\frac{m_{LSP}}{100 \ \mathrm{GeV}}\right)
\left(\frac{m_{3/2}}{100 \ \mathrm{TeV}}\right)^2 \left(\frac{1000 \ \mathrm{TeV}}{m_{\phi}}\right)^{3/2}$$ for the dark matter energy density of the universe. We expect that $m_{LSP} Y_{LSP} = 4 \times 10^{-10}$ GeV [@WMAP] if the LSP accounts for all of the dark matter in the universe, so this decay mode will overproduce LSPs. Fortunately, we can prohibit this decay kinematically without introducing any fine-tuning.
Weakening Cosmological Bounds on the Axion
------------------------------------------
The axion [@origaxion] decay constant $F$ is bounded from below due to astrophysical constraints, and in generic cosmological scenarios it is bounded from above by the requirement that the axion does not overclose the universe. The moduli fields in our model decay when the universe is at a temperature near $\Lambda_{QCD}$, releasing a significant amount of entropy, and potentially relaxing constraints on $F$ by diluting the axions. Throughout we will be considering the relic axion energy density from misalignment production, which is by far the dominant method of production for large $F$.
As shown in [@axionblow], diluting relic axions is a bit delicate. This is because there are three processes that need to be considered: a particle or field is decaying, releasing entropy and making the universe cool more slowly; $H$ is decreasing, alleviating axion hubble friction; and the axion mass is increasing, since $m_a$ is strongly dependent on temperature because it arises from instanton effects. The axion only begins to roll as $H$ drops below $m_a$, so if our modulus decays before this point then the entropy released does not decrease the final axion density. In general, particles decaying after the universe has cooled below $1$ MeV are dangerous to BBN, leaving a narrow window of $1$ MeV $< T_{R} < T_h$ for reheating temperatures of decays that can dilute axions. Note that $T_R$ should be interpreted as the approximate temperature of the universe after modulus decay – the temperature of the universe never increases.
First we estimate $T_h$, the reheating temperature above which the modulus does not dilute the axion density at all. As the modulus decay completes, the universe will be radiation dominated with temperature $T_R$, so $H \sim T_R^2/M_{pl}$ at this time. The axion has mass $$m_a(0) = 13 \ \mathrm{MeV} \left( \frac{1 \ \mathrm{GeV}}{F} \right)$$ at temperatures less than about $\Lambda_{QCD}/\pi$ and $$m_a(T) = 0.1 m_a(0) \left( \frac{\Lambda_{QCD}}{T} \right)^{3.7}.$$ at temperatures above $\Lambda_{QCD}/\pi$. If $T_R = T_h$, then $3H(T_h) = m_a(T_h)$ just as the modulus decays, so we find that $$T_h = 1.5 \ \mathrm{GeV} \left(\frac{10^{12} \ \mathrm{GeV}}{F} \right)^{0.18}$$ where we have taken $\Lambda_{QCD} = 200$ MeV. We see that modulus decay will certainly dilute axions with $F$ near the current bound, but for $F = 10^{15}$ GeV, $T_h \sim T_R$ for the $1000$ TeV modulus field. However, we can do much better with a lighter modulus.
In order to obtain a relic axion energy density less than the current matter density in the universe, it was shown in [@axionblow] that we must have $$F < 2 \times 10^{14} \ \mathrm{GeV} \left(\frac{100 \ \mathrm{MeV}}{T_R} \right)^{1/2}
\left( \frac{m_a(T_a)}{m_a(0)} \right)^{1/2}$$ where $T_a$ is the temperature of the universe when $m_a(T) = 3H$. We can estimate $T_a$ as [@axionblow] $$T_a = 200 \ \mathrm{MeV} \left(\frac{T_R}{100 \ \mathrm{MeV}}\right)^{0.26} \left(\frac{10^{15} \ \mathrm{GeV}}{F}\right)^{0.13}$$ thus as anticipated above, $m_\phi = 1000$ TeV leads to no improvement. However, if we consider moduli with $m_\phi \sim 100$ TeV as considered in section 3.1 and 3.3.2 then we obtain $T_R \sim 6$ MeV, giving $F < 7 \times 10^{14}$ GeV.
Displaced Gluino Vertices at the LHC
====================================
We know that cosmology favors a light anomaly mediated spectrum, so we can be justifiably optimistic that our model will be tested at the LHC. In fact, we expect that gluinos will be copiously produced, and that gluino pairs will decay at secondary vertices separated by distances of order a millimeter or more. To begin to analyze this process, note that the resolution of the LHC’s ATLAS detector [@ATLAS] will be $$\begin{aligned}
\Delta d_0 & = & 11 + \frac{73}{(p_T / \mathrm{GeV}) \sqrt{\sin(\theta)}} \ \mu m \\
\Delta z_0 & = & 87 + \frac{115}{(p_T / \mathrm{GeV}) (\sin(\theta))^{\frac{3}{2}}} \ \mu m\end{aligned}$$ Since gluinos are produced by QCD reactions such as $gg \to \tilde{g} \tilde{g}$, we can expect a large $p_T$ and an order one $\theta$. Thus we will need gluino decay vertices displaced by $\gtrsim 10$ $\mu m$ to have a chance of distinguishing them from primary vertices. There are four factors relevant to the displacement distance: the gluino production rate, the gluino lifetime, the relativistic time dilation, and the possibility of detecting gluinos from the tail of their distribution, which decays exponentially with distance from the primary vertex.
We will not attempt to compute the the gluino production rate precisely [@gluinoprod], [@gluinoprod2], [@gluinoprod3], [@gluinoprod4], but we know that the answer is large, and our results will be very insensitive to the details. It was found in [@stopgluino] that for $m_{\tilde{g}} \sim 350$ GeV, the LHC will produce about one gluino per second, or about $3 \times 10^7$ per year of operation. For $m_{\tilde{g}} \sim 2$ TeV, the LHC will produce at least a thousand per year. Thus we make the conservative assumption that the LHC will produce $$N = 10^3 \left(\frac{2 \ \mathrm{TeV}}{m_{\tilde{g}}}\right)^4$$ gluinos per year.
Various groups [@gluinodecay], [@gluinochannels] have examined the decay of the gluino and computed its lifetime in Split Supersymmetry, including the relevant one-loop operator running from the SUSY breaking scale to the TeV scale. The result is $$\tau_{g} = \frac{4 \ \mathrm{sec}}{K} \left(\frac{m_s}{10^6 \ \mathrm{TeV}}\right)^4
\left(\frac{1 \ \mathrm{TeV}}{m_{\tilde{g}}}\right)^5$$ where $K \lesssim 1$ is a weakly varying function of $m_s$, $m_{\tilde{g}}$, and $\tan(\beta)$. This is extended by a factor of approximately $$\gamma \approx \frac{3 \ \mathrm{TeV}}{m_{\tilde{g}}}$$ due to relativistic time dilation. We translate the gluino lifetime into a displacement by multiplying by a factor of $c \sqrt{1-m^2/(3 \ \mathrm{TeV})^2} \approx c$.
{width="14cm" height="8cm"}
Putting these results together, we find that the number of gluinos with displacement in the interval $(R, R + dR)$ is $$N(R) = 1000 \left( \frac{2 \ \mathrm{TeV}}{m_{\tilde{g}}} \right)^4
\exp\left(-\frac{R}{D_0}\right) \frac{dR}{D_0}$$ where the canonical distance $D_0$ is given by $$D_0 = \left(7.2 \ \mathrm{mm}\right) \left(\frac{1 \ \mathrm{TeV}}{m_{\tilde{g}}}\right)^6
\left(\frac{m_s}{1000 \ \mathrm{TeV}}\right)^4$$ If we require the observation of at least a few displaced gluino vertices in one year’s worth of LHC data, we find the limit $$10^{-2} < \left(7 + 4 \log\left( \frac{2 \ \mathrm{TeV}}{m_{\tilde{g}}} \right) \right)
\left(\frac{1 \ \mathrm{TeV}}{m_{\tilde{g}}}\right)^6
\left(\frac{m_s}{1000 \ \mathrm{TeV}}\right)^4$$ on the mass parameters of the model. This result is only accurate to within a factor of a few, but this is unimportant because displaced gluino vertices will be visible for virtually the entirety of our anomaly-mediated parameter space, as seen in figure 1. For comparison, if we require the observation of $100$ displaced gluino vertices each year at the LHC, then the digit $7$ in the above equation changes into a $2$, but this is the only alteration. Note that backgrounds for this signal from muons and b-quarks will be small due to the large jet energies, large missing energy, and especially because of the large vertex displacements. It was shown in the detailed study of [@RHadron] that simply using cuts on $E_T$ and missing $E_T$, for $m_{\tilde{g}} < 1.4$ TeV, the signal from R-Hadronized gluinos could be seen in $30$ $fb^{-1}$ of LHC data.
Conclusions
===========
We have shown that a simple Split Supersymmetry spectrum based on anomaly mediation can satisfy cosmological constraints with a light spectrum. This is possible because the successive decays moduli $\to$ LSPs and moduli $\to$ gravitinos $\to$ LSPs generate dark matter very efficiently, in contrast with most anthropically motived models [@aspsplit], [@wells], [@mindmunif] based on perturbative freeze out, which often require $1$-$2$ TeV LSP masses ([@PTDM] is an exception). Unfortunately, in one of the most elegant scenarios dark matter is overproduced, but we found several mechanisms that can lead to the correct relic density, and the original mechanism of Randall and Moroi [@modtosm] remains a very viable possibility. We have also seen that our model will have a striking LHC signature – displaced gluino vertices.
As outlined in the introduction, there are only a few clues for would-be anthropic model builders. The two most tantalizing are probably gauge coupling unification and dark matter, and we must assume that these two issues are resolved in concert if we are to avoid an almost infinite set of possibilities for new physics. Here we have shown that retracting the usual assumptions about the generation of dark matter do not necessarily make models irrelevant for experimental collider physics, as we might have feared[^7]. Furthermore, we noticed a fortuitous accident – as a side effect, our mechanism weakens cosmological constraints on axions, expanding the rather narrow window on the axion decay constant. With so little information about new physics, we should be appreciative when a small piece of the strong CP problem falls into our lap.
The success of anthropic predictions of the cosmological constant, combined with the derth of electric dipole moment signatures and new flavor changing neutral current effects, the smallness of the S and T parameters, and the lack of new physics at LEP point toward an anthropic solution to the two naturalness problems of high-energy physics and cosmology. If the world is supersymmetric at high energy, then the spectrum of new particles that we have considered is an excitingly predictive model for LHC physics.
Appendix: Moduli-Gravitino Decay Rates
======================================
These results were obtained in collaboration with Aaron Pierce and Jesse Thaler. Similar methods can be found in [@lutygoldstino].
Consider an arbitrary, globally supersymmetric lagrangian $L(\phi, \psi, F, \lambda, A_\mu, D, \partial_\mu)$ that is a function of off-shell chiral multiplets and vector multiplets. Now assume that supersymmetry is spontaneously broken by some combination of $D$ component and $F$ component VEVs. The sgoldstino[^8] is the linear combination $$\phi_\chi = \sum_i \frac{F_i}{F} \phi_i ,$$ where $F^2 = \Sigma_i F_i^2 + \Sigma_a D_a^2$ is the goldstino decay constant. Note that there is no reason to expect that $\phi_\chi$ is a mass eigenstate. We will show that the only two derivative operator involving a scalar and two goldstinos is $$\frac{1}{F^\dag} \partial_\nu \phi_\chi^\dag \chi \sigma^\nu \bar{\sigma}^\mu \partial_\mu \chi
- \frac{1}{F} \partial_\nu \phi_\chi \chi^\dag \bar{\sigma}^\mu \sigma^\nu \chi^\dag .$$ Thus the rate of moduli decays to gravitino pairs is determined by the overlap of the mass-eigenstate moduli with $\phi_\chi$. As a simple corollary, we see that if supersymmetry were broken entirely by $D$ terms, then there would not be a sgoldstino, so the rate for modulus decay to gravitino pairs would be zero.
To prove this assertion, we need to do a field redefinition to introduce the non-linearly realized goldstino $\chi$. As in the conceptually simpler case of a goldstone boson, we can do this by choosing a ‘vacuum alignment’, and then parameterizing a supersymmetry transformation with the goldstino $\chi(x)$. For instance, if the theory consisted of a single chiral multiplet $\Phi = (\phi, \psi, F)$, we would represent $\Phi = e^{i \delta_\chi} (\phi, 0, F)$.
In the general case, when supersymmetry is linearly realized the goldstino is the combination $$\psi_g = \sum_i \frac{F_i}{F} \psi_i + \sum_a \frac{D_a}{F} \lambda_a .$$ To isolate it, we begin by rotating the fermions $\psi_i$, $\lambda_a$ into a mass eigenstate basis, so that the fermion lagrangian becomes $$L_f = i \psi_g^\dag \bar{\sigma}^\mu \partial_\mu \psi_g
+ \sum_\alpha \left[ i \psi_{\alpha}^\dag \bar{\sigma}^\mu \partial_\mu \psi_{\alpha}
+ \frac{m_\alpha}{2} ( \psi_{\alpha} \psi_{\alpha} + \psi_{\alpha}^\dag \psi_{\alpha}^\dag) \right] + ...$$ where the elipsis denotes interaction terms. Note that the superpartners of the fermions $\psi_\alpha$ have no auxiliary component VEVs.
Now we perform a $\chi$ parameterized supersymmetry transformation on $L$, and then choose the ‘vacuum alignment’ condition $\psi_g = 0$. We are only interested in two operators, the interaction term mentioned above and the kinetic term for $\chi$, which we require for canonical normalization. Both of these operators involve two goldstinos, so they can only come from $C \delta_\chi^2 A$ or $C \delta_\chi A \delta_\chi B$ where $A$, $B$, and $C$ are some combination of fields in the lagrangian. But the first type of term would be $\frac{\Delta L}{\Delta A} \delta_\chi^2 A$, and this vanishes on-shell because it is proportional to the $A$ equation of motion. Thus only terms of the second kind are relevant.
The supersymmetry transformation rules are the familiar $$\begin{aligned}
\delta_\chi \phi & = & \sqrt{2} \chi \psi \\
\delta_\chi \psi & = & i \sqrt{2} \sigma^\mu \chi^\dag D_\mu \phi + \sqrt{2} \chi F \\
\delta_\chi F & = & i \sqrt{2} \chi^\dag \bar{\sigma}^\mu D_\mu \psi \\
\delta_\chi \lambda & = & \sigma^{\mu \nu} \chi F_{\mu \nu} + i \chi D \\
\delta_\chi A_\mu & = & -i \lambda^\dag \bar{\sigma}_\mu \chi + i \chi^\dag \bar{\sigma}_\mu \lambda \\
\delta_\chi D & = & - \chi \sigma^\mu D_\mu \lambda^\dag - D_\mu \lambda \sigma^\mu \chi^\dag .\end{aligned}$$ where $D_\mu$ is the appropriate gauge covariant derivative. The $\chi$ kinetic term can only come from the transformation of the $\psi_g$ kinetic term, since the transformation of all other operators involve too many fields. Furthermore, it is remarkable that two derivative interactions between a scalar and two goldstinos also only come from the $\psi_g$ kinetic term, because the other terms either give too many fields or vanish on the equations of motion. Thus as claimed we find $$L \supset i \chi^\dag \bar{\sigma}^\mu \partial_\mu \chi
+ \frac{1}{F^\dag} \partial_\nu \phi_\chi^\dag \chi \sigma^\nu \bar{\sigma}^\mu \partial_\mu \chi
- \frac{1}{F} \partial_\nu \phi_\chi \chi^\dag \bar{\sigma}^\mu \sigma^\nu \chi^\dag + ...$$ after canonically normalizing $\chi$, where the elipsis denotes other interactions, including a two goldstino interaction with the vector superpartner of the goldstino. Although from the transformation rules it might seem that $\chi$ has non-derivative interactions, these all cancel – $\chi$ has a shift symmetry, just like a goldstone boson. Thus yukawa couplings with $\chi$ cannot be generated.
We should note that this is an effective field theory, and it is only valid for $E^2 \ll F = m_{3/2} M_{pl}$. In most examples of interest (KKLT moduli, or moduli that get masses from supersymmetry breaking), the modulus mass easily satisfies this criterion. In any case, above this scale supersymmetry breaking is a small effect, and in place of a single goldstino one would need to consider the detailed dynamics of the hidden sector.
Acknowledgements {#acknowledgements .unnumbered}
================
The results of Appendix A were obtained in collaboration with Aaron Pierce and Jesse Thaler. We would like to thank Lian-Tao Wang, Ben Lillie, Philip Schuster, Natalia Toro, and especially Nima Arkani-Hamed for discussions. This work was supported by the Fannie and John Hertz Foundation.
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[^1]: email: [email protected]
[^2]: Of course this depends on the available gravitino-producing modulus decay channels – the strict limit is $m_\phi < m_{3/2}$ – but under reasonable assumptions $\phi \to 2 \psi_{3/2}$ is the only available decay mode.
[^3]: As explained in [@aspsplit], this does not require that supersymmetry breaking comes from gauge superfields, but only that supersymmetry breaking is not accompanied by R-symmetry breaking.
[^4]: There is certainly a significant uncertainty because we do not know any of the $O(1)$ couplings involved. However, we do know that modulus decays to matter in the complex representation of a symmetry group [@modtosm], [@newdecay], [@dine] receive a suppression proportional to a power of $m_{mat}/m_\phi$, which means that moduli decay dominantly to gauge bosons, gauginos, and possibly the heavy scalars.
[^5]: Note that if $\Phi$ acquires a large $F$-component VEV, then this operator could give dangerously large gaugino masses, so we must assume that either $F_\phi$ is small or that the coefficient of this operator is suppressed.
[^6]: It continues to be true when the modulus has a shift symmetry, although as we will discuss later there are exceptions in some models, such as specific versions of KKLT [@dine].
[^7]: However, it is worth noting that when dark matter is not generated by perturbative freeze out, we lose the elegant parametric prediction that the weak scale is the geometric mean of the Planck and Cosmological Constant scales [@split]. The numerical prediction for the value of the cosmological constant is unchanged.
[^8]: If there are non-vanishing $D$ component VEVs then there will also be a vector partner of the goldstino, but since we are only interested in moduli-goldstino interactions, we are ignoring these fields.
|
---
abstract: |
In this note it is established that, for any finite set $A$ of real numbers, there exist two elements $a,b \in A$ such that
$$|(a+A)(b+A)| \gg \frac{|A|^2}{\log |A|}.$$
In particular, it follows that $|(A+A)(A+A)| \gg \frac{|A|^2}{\log |A|}$. The latter inequality had in fact already been established in an earlier work of the author and Rudnev [@rectangles], which built upon the recent developments of Guth and Katz [@GK] in their work on the Erdős distinct distance problem. Here, we do not use those relatively deep methods, and instead we need just a single application of the Szemerédi-Trotter Theorem. The result is also qualitatively stronger than the corresponding sum-product estimate from [@rectangles], since the set $(a+A)(b+A)$ is defined by only two variables, rather than four. One can view this as a solution for the pinned distance problem, under an alternative notion of distance, in the special case when the point set is a direct product $A \times A$. Another advantage of this more elementary approach is that these results can now be extended for the first time to the case when $A \subset \mathbb C$.
address: 'O. Roche-Newton: Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, 4040 Linz, Austria '
author:
- 'Oliver Roche-Newton'
title: 'A short proof of a near-optimal cardinality estimate for the product of a sum set'
---
Introduction
============
In this note, we consider a variation on the sum-product problem, in which the aim is to show that certain sets defined by a combination of additive and multiplicative operations will always be large. For example, given a finite set $A$ of real numbers, define $$(A-A)(A-A):=\{(a-b)(c-d):a,b,c,d \in A\}.$$ By the same heuristic arguments that support the Erdős-Szemerédi sum-product conjecture, one expects that $(A-A)(A-A)$ will always be large in comparison to the input set $A$. In [@rectangles], the following[^1] bound was established which showed that this is indeed the case: $$|(A-A)(A-A)| \gg \frac{|A|^2}{\log |A|}.
\label{sp1}$$ The same argument in [@rectangles] yields the same lower bound for $|(A+A)(A+A)|$. Some other interesting results in this direction can be found in [@ENR], [@TJthesis], [@TJ], [@MORNS], [@RSS] and [@ungar], amongst others.
In all of the aforementioned works, incidence geometry plays a central role. An extremely influential result in this area is the Szemerédi-Trotter Theorem, which says that, given finite sets $P$ and $L$ of points and lines respectively in $\mathbb R^2$, the number of incidences between $P$ and $L$ satisfies the upper bound $$|\{(p,l) \in P \times L : p \in l\}| \ll |P|^{2/3}|L|^{2/3}+|P|+|L|.
\label{ST}$$ The quantity on the left hand side of the above inequality is usually denoted by $I(P,L)$. Incidence geometry also played a central role in the recent landmark work of Guth and Katz [@GK] on the Erdős distinct distances problem. Guth and Katz established an incidence bound for points and lines in $\mathbb R^3$, which was then used to prove that for any finite set $P$ of points in $\mathbb R^2$, the set of distinct distances determined by $P$ has near-linear size. To be precise, they proved that $$|\{d(p,q):p,q \in P\}| \gg \frac{|P|}{\log |P|},
\label{GK}$$ where $d(p,q)$ denotes the Euclidean distance between $p$ and $q$. Note that the example $P=[N] \times [N]$, where $[N]=\{1,2,\dots,N\}$, illustrates that this bound is close to best possible.
One of the tools that Guth and Katz use in their analysis is the Szemerédi-Trotter Theorem. They also introduced polynomial partitioning, and utilise some non-trivial facts from algebraic geometry.
In [@rectangles] the authors considered the pseudo-distance $R(p,q)$ in place of $d(p,q)$, where $R(p,q)$ denotes the (signed) area of the axis-parallel rectangle with $p$ and $q$ at opposite corners. To be precise, for two points $p=(p_1,p_2)$ and $q=(q_1,q_2)$ in the plane, we define $$R(p,q):=(p_1-q_1)(p_2-q_2)$$ It was then possible to apply the incidence result of Guth and Katz to establish that $$|\{R(p,q):p,q \in P\}| \gg \frac{|P|}{\log |P|},
\label{recdis}$$ and followed as a corollary after taking $P= A \times A$. Once again, the example $P=[N] \times [N]$ shows that this bound is close to best possible.
In this note, we prove the following result which strengthens :
\[mainthm\] For any set $A \subset \mathbb R$, there exist elements $a,a' \in A$ such that $$|(A-a)(A-a')| \gg \frac{|A|^2}{\log |A|}.$$
Here, we obtain quadratic growth for a set which depends on only two variables. There are similarities here with the Erdős pinned distance problem, where the aim is to show that, for any finite set $P\subset \mathbb R^2$, there exists $p \in P$ such that $$|\{d(p,q):q \in P\}| \gg \frac{|P|}{\sqrt{\log |P|}}.$$ This harder version of the Erdős distinct distance problem remains open, with the current best-known result, due to Katz and Tardos [@KT], stating that there exists $p \in P$ such that $$|\{d(p,q):q \in P\}|\gg |P|^{\alpha},$$ where $\alpha \approx 0.864$. However, Theorem \[mainthm\] shows that, if we instead consider the pseudo-distance $R(p,q)$ then we have a near-optimal bound for the corresponding pinned distance problem, in the special case when $P=A \times A$ is a direct product. Such a result, even with the additional direct product restriction, is not currently known for Euclidean distance.
Another advantage of the approach in this paper is that the proof is relatively straightforward. In particular, we obtain a new proof of , and in fact a stronger result, without utilising the Guth-Katz machinery.
This paper is closely related to work contained in the PhD thesis of Jones [@TJthesis] on the growth of sets of real numbers. In fact, the main lemma here, the forthcoming Lemma \[mainlemma\], forms part of the proof of [@TJthesis Theorem 5.2], although it is expressed rather differently there in terms of the notion of the cross-ratio. Consequently, we are able to give a new proof of Theorem 5.2 from [@TJthesis]; that is we establish the following three-variable expander bound $$\left|\left\{\frac{a-b}{a-c}:a,b,c \in A\right\}\right| \gg \frac{|A|^2}{\log |A|}.$$
It appears that the proof here is more straightforward than the one originally given by Jones [@TJthesis].
The only major tool needed in this paper is the Szemerédi-Trotter Theorem. In particular, we use the following standard corollary of which bounds the number of rich lines in an incidence configuration:
\[STcor\]\[Szemerédi-Trotter Theorem\]
Let $P$ be a set of points in $\mathbb R^2$ and let $k \geq 2$ be a real number. Define $L_k$ to be the set of lines containing at least $k$ points from $P$. Then $$|L_k| \ll \frac{|P|^2}{k^3}+\frac{|P|}{k}.
\label{STcor1}$$ In particular, if $k \leq |P|^{1/2}$, then $$|L_k| \ll \frac{|P|^2}{k^3}.
\label{STcor2}$$
Energy bound
============
\[mainlemma\] Let $Q$ denote the number of solutions to the equation $$\label{energy}
(a-b)(a'-c')=(a-c)(a'-b')$$ such that $a,a',b,b',c,c' \in A$. Then $$Q \ll |A|^4\log |A|.$$
First of all, the number solutions to of the form $$(a-b)(a'-c')=(a-c)(a'-b')=0,$$ is at most $4|A|^4$. Also, there are at most $|A|^4$ trivial solutions whereby $b=c$. Now, let $Q^*$ denote the number of solutions to $$\label{energy*}
(a-b)(a'-c')=(a-c)(a'-b')\neq 0,\,\,\,\,\,\,\,\,\,\,\,\,b \neq c.$$ This is the same as the number of solutions to $$\label{energy**}
\frac{a-b}{a'-b'}=\frac{a-c}{a'-c'}\neq 0,\,\,\,\,\,\,\,\,\,\,\,\,\, b \neq c.$$ Let $P=A\times A$ and let $L(P)$ denote the set of lines determined by $P$. That is, $L(P)$ is the set of lines supporting $2$ or more points from the set. Note that $a,a',b,b',c$ and $c'$ satisfy only if the points $(a',a),(b',b)$ and $(c',c)$ from $P$ are collinear and distinct. Therefore, $$\begin{aligned}
Q^* &\leq \sum_{l\in L(P)}|l \cap P|^3
\\& \ll \sum_j \sum_{2^{j} \leq |l \cap P| < 2^{j+1}} |l \cap P|^3,\end{aligned}$$ where $j$ ranges over all positive integers such that $2^{j} \leq |A|$. Note that there are no lines in $L(P)$ which contain more than $|A|$ points from $P$, which is why this sum does not need to include any larger values of $j$.
For the aforementioned range of values for $j$, it follows from Corollary \[STcor\], and in particular bound , that $$|\{l: |l\cap P| \geq 2^{j}\}| \ll \frac{|P|^2}{(2^j)^3}.$$ Therefore, $$Q^* \ll \sum_j |P|^2 \ll |A|^4 \log|A|.$$ Finally, $Q \ll |A|^4 +Q^* \ll |A|^4 \log |A|$, as required.
\[TJ\] For any finite set $A \subset \mathbb R$, $$\left|\left\{\frac{a-b}{a-c}:a,b,c \in A\right\}\right| \gg \frac{|A|^2}{\log |A|}.$$
Let $$n(x):=\left| \left\{(a,b,c) \in A^3:\frac{a-b}{a-c}=x\right\}\right|$$ denote the number of representations of $x$ as an element of the set in question. We know that $$|A|^3 \ll |A|^3-|A|^2 = \sum_x n(x).$$ Also, the quantity $\sum_x n^2(x)$ is strictly[^2] less than the number of solutions to . Therefore, it follows from the Cauchy-Schwarz inequality and Lemma \[mainlemma\] that $$\begin{aligned}
|A|^6 \ll \left(\sum_x n(x)\right)^2 &\leq \left|\left\{\frac{a-b}{a-c}:a,b,c \in A\right\}\right| \sum_x n^2(x)
\\& \ll \left|\left\{\frac{a-b}{a-c}:a,b,c \in A\right\}\right| |A|^4 \log |A|,\end{aligned}$$ and the result follows after rearranging this inequality.
Remarks
-------
Let $E^*(A,B)$ be the *multiplicative energy* of $A$ and $B$; that is, the number of solutions to $$ab=a'b'$$ such that $a,a' \in A$ and $b,b' \in B$. Using this notation, Lemma \[mainlemma\] can be expressed in the form of the following bound: $$\sum_{a,a' \in A}E^*(a-A,a'-A) \ll |A|^4 \log |A|.
\label{energysum}$$ See [@MORNS Lemma 2.4] for a similar bound on the sum of multiplicative energies after different additive shifts.
The proof of Lemma \[mainlemma\] can undergo a number of small modifications in order to deduce slightly different results involving multiple sets $A,B,C, \dots \in \mathbb R$ of approximately the same size. For example, if we instead take $P=(A \cup B) \times (A \cup B)$, where $|B| \approx |A|$, then the number of solutions to such that $a,a' \in A $ and $b,b',c,c' \in B$ is less than the number of collinear triples in the point set $P$. After repeating the argument of Lemma \[mainlemma\], it follows that $$\sum_{a,a' \in A}E^*(a-B,a'-B) \ll |A|^4 \log |A|.
\label{energysum2}$$ In particular, if $B=-A$, this yields $$\sum_{a,a' \in A}E^*(a+A,a'+A) \ll |A|^4 \log |A|.
\label{energysum3}$$
Proof of Theorem \[mainthm\]
============================
It follows from that there exist $a,a' \in A$ such that $$E^*(a-A,a'-A) \ll |A|^2 \log |A|.
\label{exists}$$ We also have the following well-known bound for the multiplicative energy, which follows from an application of the Cauchy-Schwarz inequality: $$E^*(A,B) \geq \frac {|A|^2|B|^2}{|AB|}.
\label{CS}$$ After comparing and , it follows that $$|(A-a)(A-a')| \gg \frac{|A|^2}{\log |A|},$$ as required. $\square$
Remark
------
By the same argument, but utilising in place of , it also follows that there exist $a,a' \in A$ such that $$|(A+a)(A+a')| \gg \frac{|A|^2}{\log |A|}.$$
The complex setting
===================
As stated in the abstract, an advantage of this more straightforward approach is that it allows for results that were previously only known for sets of real numbers to be extended to the complex setting. The only tool used in the proofs of Theorem \[mainthm\] and Corollary \[TJ\] is the Szemerédi-Trotter Theorem. It is now known that this theorem holds for sets of points and lines in $\mathbb C^2$ (this was first proven by [@toth], with a more modern proof given by Zahl [@zahl]; see also Solymosi and Tao [@ST]).
One can therefore repeat the analysis of this paper verbatim in the complex setting, applying the complex Szemerédi-Trotter Theorem in place of the real version, and deduce exactly the same results for a set $A$ of complex numbers.
In particular, we deduce that for any finite set $A \subset \mathbb C$, there exist $a,a' \in A$ such that $$|(A-a)(A-a')| \gg \frac{|A|^2}{\log |A|}
\label{comp1}$$ and it follows that $$|(A-A)(A-A)| \gg \frac{|A|^2}{\log |A|}.
\label{comp2}$$ Since the earlier proof of for real $A$ in [@rectangles] was based on the three dimensional incidence bounds in [@GK], it was not previously known that this bound extended to the complex setting. Similarly, the approach in this paper can be used to show that for any finite set $A \subset \mathbb C$, we have $$\left|\left\{\frac{a-b}{a-c}:a,b,c \in A\right\}\right| \gg \frac{|A|^2}{\log |A|}.$$
Acknowledgements {#acknowledgements .unnumbered}
================
The author was supported by the Austrian Science Fund (FWF): Project F5511-N26, which is part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications". I am grateful to Brendan Murphy for his helpful feedback. I am also grateful to Ilya Shkredov for helpful conversations and especially for his interpretation of the work of Jones [@TJthesis].
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J. Zahl, ‘A Szemerédi-Trotter type theorem in $\mathbb R^4$’, *arXiv:1203.4600*.
[^1]: Here and throughout this paper, for positive values $X$ and $Y$ the notation $X \gg Y$ is used as a shorthand for $X\geq cY$, for some absolute constant $c>0$. If both $X \gg Y$ and $X \ll Y$ hold, we may write $X \approx Y$.
[^2]: The quantity $\sum_x n^2(x)$ is the number of solutions to , minus the number of solutions for which $a=c$ or $a'=c'$.
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abstract: 'We consider one single copy of a mixed state of two qubits and investigate how its entanglement changes under local quantum operations and classical communications (LQCC) of the type $\rho''\sim (A\otimes B)\rho(A\otimes B)^{\dagger}$. We consider a real matrix parameterization of the set of density matrices and show that these LQCC operations correspond to left and right multiplication by a Lorentz matrix, followed by normalization. A constructive way of bringing this matrix into a normal form is derived. This allows us to calculate explicitly the optimal local filterin operations for concentrating entanglement. Furthermore we give a complete characterization of the mixed states that can be purified arbitrary close to a Bell state. Finally we obtain a new way of calculating the entanglement of formation.'
address: |
Katholieke Universiteit Leuven, Department of Electrical Engineering, Research Group SISTA\
Kard. Mercierlaan 94, B-3001 Leuven, Belgium
author:
- 'Frank Verstraete, Jeroen Dehaene, Bart De Moor[@FJBmail]'
title: Local filtering operations on two qubits
---
[2]{}\[\] Entanglement of two separated quantum systems implies that there are non-local correlations between them. This feature of non-locality has found practical applications in quantum information theory (see for example [@Lo]). Most applications require that both parties share maximally entangled states. A realistic preparation and transmission of entangled states however yields mixed states. Therefore Bennett et al. [@Bennett] proposed a protocol which allows to obtain asymptotically a nonzero number of maximally entangled pure states by carrying out collective measurements on a large number of copies of entangled mixed states. Their scheme however required that the fidelity of the mixed states exceeded 1/2. The Horodecki’s subsequently showed how mixed states of arbitrary fidelity could be purified by applying first a filtering operation on each copy separately [@Horodecki2]. Linden et al. [@Linden] then asked the question if it is possible to obtain singlets out of mixed states by allowing only local operations on each copy separately. While this is possible for pure states, they proved that this is impossible in general for mixed states [@Linden; @Kent1], as the best state one can obtain is a Bell diagonal state [@Kent2]. The Horodecki’s however gave an example of a mixed state that could be purified arbitrary close to a singlet state by a process called quasi-distillation [@Horodecki3].
We shed new light on those results by observing that filtering operations on two qubits correspond to Lorentz transformations on a real parameterization of their density matrix. Using Lorentz transformations, this real parameterization can be brought into one of two types of normal forms, thus giving a characterization of all states that can be transformed into each other by local operations. Our scheme also yields a new way of calculating the entanglement of formation [@Wootters], with as a by-product a simple proof of the necessity and sufficiency of the partial transpose criterion of Peres [@Peres; @Horodecki]. The main result of this letter however is the fact that we provide a constructive way of finding the optimal POVM’s for concentrating the entanglement. We show that there exist two classes of states corresponding to the two normal forms : the ones that can be brought into Bell diagonal form leaving the rank of the density matrix constant, and the ones that can asymptotically be brought into Bell diagonal form with lower rank. This last class contains a subclass of mixed states that can be purified arbitrary close to the singlet state.
In this letter we will consider the filtering operations $$\rho'=\frac{(A\otimes B)\rho(A\otimes
B)^{\dagger}}{{\mathop{\rm Tr}\nolimits}((A\otimes B)\rho(A\otimes
B)^{\dagger})}\label{LQCC}$$ where $A^{\dagger}A\leq
I_2$, $B^{\dagger}B\leq I_2$. As a local projective measurement destroys all entanglement, we will only consider the cases ${\rm
det}(A)\neq 0$ and ${\rm det}(B)\neq 0$. Let us now calculate how the entanglement of formation (EoF) changes under these local operations. The EoF of a two qubit system can be calculated as a convex monotonously increasing function of the concurrence [@Wootters]. As shown in [@Verstraete1], the concurrence of $\rho$ is given by $\max(0,\tau_1-\tau_2-\tau_3-\tau_4)$ with $\{\tau_i\}$ the singular values of $X^T(\sigma_y\otimes\sigma_y)X$ with $\rho=XX^\dagger$. Under the filtering operations we have $X'=(A\otimes
B)X/\sqrt{{\mathop{\rm Tr}\nolimits}(A^\dagger A\otimes B^\dagger B\rho)}$. As $(A\otimes B)^T(\sigma_y\otimes\sigma_y)(A\otimes
B)=\det(A)\det(B)(\sigma_y\otimes\sigma_y)$, this proves the following theorem:
Under the filtering operations (\[LQCC\]), the concurrence changes as $$C'=C\frac{|\det(A)||\det(B)|}{{\mathop{\rm Tr}\nolimits}(A^\dagger
A\otimes B^\dagger B\rho)}.\label{C}$$ \[t3\]
It will turn out very usefull to introduce the real and linear parameterization of the density matrix [@Mahler] $$\rho=\frac{1}{4}\sum_{i,j}R_{ij}\sigma_i\otimes\sigma_j$$ where the summation extends from 0 to 3 and with $\sigma_0$ the 2x2 identity matrix and $\sigma_1,\sigma_2,\sigma_3$ the Pauli spin matrices. Below we will often leave out the normalization of $\rho$ and $R$. Note that normalization of $R$ is very simple since $R_{0,0}={\mathop{\rm Tr}\nolimits}(\rho)$.
We will now prove how $R$ transforms under the LQCC operations (\[LQCC\]).
The [4x4]{} matrix $R$ with elements $R_{ij}=\frac{1}{2}{\mathop{\rm Tr}\nolimits}(\rho(\sigma_i\otimes\sigma_j))$ transforms, up to normalization, under LQCC operations (\[LQCC\]) as $$R'=L_ARL_B^T$$ where $L_A$ and $L_B$ are proper orthochronous Lorentz transformations given by $$\begin{aligned}
&L_A&=T(A\otimes A^*)T^{\dagger}/|{\rm det}(A)|\\
&L_B&=T(B\otimes B^*)T^{\dagger}/|{\rm det}(B)|\\
T&=&\frac{1}{\sqrt{2}}\left(\begin{array}{cccc}
1&0&0&1\\0&1&1&0\\0&i&-i&0\\1&0&0&-1\end{array}\right).\end{aligned}$$ \[t1\]
This theorem can be proven by introducing the matrix $\tilde{\rho}_{kl,k'l'}=\rho_{kk',ll'}$ and noting that $R=4T\tilde{\rho}T^{T}$. It is easy to check that under the LQCC operations (\[LQCC\]) $\tilde{\rho}$ transforms as $\tilde{\rho}'=(A\otimes A^*)\tilde{\rho}(B\otimes B^*)^T$. Therefore $R$ transforms as $R'=L_A R L_B^T|{\rm det}(A)||{\rm
det}(B)|$ with $L_A=T(A\otimes A^*)T^\dagger/|{\rm det}(A)|$, $L_B=T(B\otimes B^*)T^\dagger/|{\rm det}(B)|$. Using the identities $A\sigma_y A^T={\rm det}(A)\sigma_y$ and $T^\dagger M
T^*=-\sigma_y\otimes\sigma_y$ with $M$ the matrix associated with the Lorentz metric $ M={\rm diag}[1,-1,-1,-1]$, it is easily checked that $L_A M L_A^T=M=L_B ML_B^T$. Furthermore the determinant of $L_A$ and $L_B$ is equal to +1, and the $(0,0)$ element of $L$ is positive, which completes the proof.[$\Box$]{}
As the complex 2$\times$2 matrices with determinant one indeed form the spinor representation of the Lorentz group, there is a 1 to 2 correspondence between each $L_A$ and $A/\sqrt{{\rm
det}(A)}$. It is interesting to note that when both $A$ and $B$ are unitary, the theorem reduces to the well-known fact [@Mahler] that the rows and columns of $R$ transform under $SO(3)$, which is indeed a subgroup of the Lorentz group.
With the above theorem in mind, a natural question is to find a decomposition of $R$ as $R=L_1\Sigma L_2^T$ with $\Sigma$ diagonal and $L_1,L_2$ proper orthochronous Lorentz transformations. This would be the analogue of a singular value decomposition but now in the Lorentz instead of the Euclidean metric.
The [4x4]{} matrix $R$ with elements $R_{ij}={\mathop{\rm Tr}\nolimits}(\rho\sigma_i\otimes\sigma_j)$ can be decomposed as $$R=L_1\Sigma L_2^T$$ with $L_1,L_2$ proper orthochronous Lorentz transformations, and $\Sigma$ either of diagonal form $\Sigma={\rm diag}[s_0,s_1,s_2,s_3]$ with $s_0\geq s_1\geq s_2\geq
|s_3|$, either of the form $$\Sigma=\left(\begin{array}{cccc}a&0&0&b\\0&d&0&0\\0&0&-d&0\\c&0&0&a+c-b\end{array}\right)\label{jb}$$ with $a,b,c,d$ real. \[t2\]
The proof of this theorem is quite technical. It heavily depends on results on matrix decompositions in spaces with indefinite metric [@Gohberg]. We first introduce the matrix $C=MRMR^T$ which is $M$-selfadjoint. Using theorem (5.3) in [@Gohberg], it follows that there exist matrices $X$ and $J$ with $C=X^{-1}JX$, $J$ consisting of a direct sum of real Jordan blocks and $XMX^T=N_J$ with $N_J$ a direct sum of symmetric nxn matrices of the form $[S_{ij}]=\pm[\delta_{i+j,n+1}]$ with $n$ the size of the corresponding Jordan block. Using Sylvester’s law of inertia, there exists orthogonal $O_J$ such that $N_J=O_J^TMO_J$. It is then easy to check that $O_JX=L_1^T$ is a Lorentz transformation. Therefore the relations $C=MRMR^T=ML_1MO_JJO_J^TL_1^T$ hold. Multiplying left by $M$, Sylvester’s law of inertia implies that there exist a matrix $\Sigma$ with the same rank as $J$ such that $MO_JJO_J^T=\Sigma M\Sigma^T$. Therefore we have the relation $RMR^T=L_1\Sigma M\Sigma^TL_1^T$. If $R$ has the same rank as $RMR^T$, this relation implies that there exists a Lorentz transformation $L_2$ such that $R=L_1\Sigma L_2^T$.
Let us now investigate the possible forms of $\Sigma$. As $N_J=O_J^TMO_J$ has signature $(+---)$, $J$ can only be a direct sum of the following form: 4 1x1 blocks; 1 orthogonal 2x2 block and 2 1x1 blocks; 1 2x2 Jordan block and 2 1x1 blocks; 1 3x3 Jordan block and 1 1x1 block. Noting the eigenvalues of $C$ as $\{\lambda_i\}$, it is easy to verify that a “square root” $\Sigma$ in the four cases is respectively given by
1. $\Sigma={\rm
diag}[\sqrt{|\lambda_0|},\sqrt{|\lambda_1|},\sqrt{|\lambda_2|},\sqrt{|\lambda_3|}]P$ with $P$ a permutation matrix permutating the first column with one other column;
2. $\Sigma={\rm
diag}\left[\sqrt{|\lambda_0|}\left(\begin{array}{cc}\cos(\phi)&\sin(\phi)\\
\sin(\phi)&-\cos(\phi)\end{array}\right),\sqrt{|\lambda_2|},\sqrt{|\lambda_3|}\right]$;
3. $\Sigma={\rm
diag}\left[\left(\begin{array}{cc}a&b\\c&a+c-b\end{array}\right),\sqrt{|\lambda_2|},\sqrt{|\lambda_3|}\right]$;
4. $\Sigma={\rm diag}\left[\left(\begin{array}{ccc}a
&0&0\\b&\sqrt{a^2+b^2}&0\\0&\frac{-ab}{\sqrt{a^2+b^2}}&\frac{a^2}{\sqrt{a^2+b^2}}\end{array}\right),\sqrt{|\lambda_3|}\right]$ with $a=\sqrt{|\lambda_0|}$ and $b=-1/\sqrt{2|\lambda_0|}$.
Now we go back to the relation $R=L_1^T\Sigma L_2$. $L_1$ and $L_2$ can be made proper and orthochronous by absorbing factors $-1$ into the rows and colums of $\Sigma$ yielding $\Sigma'$. Theorem (2) now implies that this $\Sigma'$ corresponds to an unnormalized physical state, which means that $\rho'$ corresponding to $\Sigma'$ has no negative eigenvalues. It is easy to show that this requirement excludes cases 2 and 4 of the possible forms of $\Sigma$. The third case corresponds to (\[jb\]). Furthermore in the first case the permutation matrix has to be the identity and $|\lambda_0|\geq\max(|\lambda_1|,|\lambda_2|,|\lambda_3|)$. Multiplying left and right by proper orthochronous Lorentz transformations, the elements $\{s_i\}$ of this diagonal $\Sigma$ can always be ordered as $s_0\geq s_1\geq s_2\geq |s_3|$.
The case where the rank of $C$ is lower then the rank of $R$ still has to be considered. This is only possible if the rowspace of $R$ has an isotropic subspace $Q$ for which $QMQ^T=0$. Some straightforward calculations reveal that the only physical states for which this hold have normal form (\[jb\]) with $a=b=c$ and $d=0$ or $a=b$ and $c=d=0$. This completes the proof.[$\Box$]{}
The two normal forms can be computed very efficiently by calculating the Jordan canonical decomposition of $C=MRMR^T$ and of $C'=MR^TMR$. It is easy indeed to show that for example in the case of diagonalizable $R$ the eigenvectors of $C$ form a Lorentz matrix, and $|s_i|=\sqrt{\lambda_i(C)}$. Note that we always order the diagonal elements such that $s_0\geq s_1\geq s_2\geq|s_3|$.
States that are diagonal in $R$ correspond to (unnormalized) Bell-diagonal states with ordered eigenvalues $$\begin{aligned}
\lambda_1&=&(s_0+s_1+s_2-s_3)/4\\
\lambda_2&=&(s_0+s_1-s_2+s_3)/4\\
\lambda_3&=&(s_0-s_1+s_2+s_3)/4\\
\lambda_4&=&(s_0-s_1-s_2-s_3)/4,\label{ebd}\end{aligned}$$ whereas states of type (\[jb\]) correspond to the rank deficient states $$\rho=\frac{1}{2}\left(\begin{array}{cccc}a+c&0&0&d\\0&0&0&0\\0&0&b-c&0\\d&0&0&a-b\end{array}\right)\label{n2}.$$ For both cases it is easy to calculate the entanglement of formation analytically, respectively given by [@Hill] $C=\max(0,(\lambda_1-\lambda_2-\lambda_3-\lambda_4)/(\lambda_1+\lambda_2+\lambda_3+\lambda_4))=\max(0,(-s_0+s_1+s_2-s_3)/(2s_1))$ and $C=\max(0,|d|/a)$.
Let us now consider an arbitrary state $\rho$ with corresponding $R$. Combining theorem (1),(2) and (3), it follows that the concurrence of $\rho$ is equal to the concurrence of the state corresponding to $\Sigma$ multiplied by $R_{00}$. We have therefore proven:
Given a state $\rho$ and associated with this state $R=L_1\Sigma L_2^T$, then the concurrence of $\rho$ is given by $C=\max(0,(-s_0+s_1+s_2-s_3)/2)$ or by $C=\max(0,|d|)$ depending on the normal form $\Sigma$.\[t10\]
We thus have obtained a new method of calculating the entanglement of formation of a system of two qubits. Interestingly, it turns out that this characterization relates the concepts of entanglement of formation and of partial transposition [@Peres]. Let us therefore define $R^{PT}_{ij}={\mathop{\rm Tr}\nolimits}(\rho^{PT}\sigma_i\otimes\sigma_j)$, which changes the sign of the third column of $R$. In the case of diagonal normal form of $R$, it is readily verified that the normal form of $R^{PT}$ equals that of $R$ except for the last element where $s_3^{PT}=-s_3$. Retransforming $\Sigma^{PT}$ to the $\rho^{PT}$-picture, we see that the corresponding Bell-diagonal partial transposed state has minimal eigenvalue $s_0-s_1-s_2+s_3$. We readily recognize the expression of the concurrence of theorem (\[t10\]) and therefore this eigenvalue is negative if and only if the concurrence exceeds 0. Moreover we know that $\rho^{PT}$ is related to this Bell-diagonal state by some similarity transformation $A\otimes B$ which cannot change the signature of a matrix due to the inertia law of Sylvester. In the case of normal form (\[jb\]), an analogue reasoning shows that $\rho^{PT}$ has a negative eigenvalue if and only if $|d|>0$, which again is necessary and sufficient to have entanglement. This completes the proof of:
Given a system of two qubits, this state is separable if and only if its partial transpose has a negative eigenvalue.\[t11\]
Although this result was already proven by Horodecki [@Horodecki], we believe the previous derivation is of interest as it connects the entanglement measures concurrence and negativity. Using this formalism, it indeed becomes possible to prove that the concurrence always exceeds the negativity, and it is furthermore possible to find a complete characterization of all states with maximal and minimal negativity for given concurrence [@Verstraete3]. This is important because in the two qubit case the negativity is a measure of the robustness of entanglement against noise.
Next we want to solve the problem of finding the POVM such as to have a non-zero chance to produce a new state with highest possible entanglement. From equation (\[C\]), the maximum EoF is obtained with $A,B$ minimizing the expression ${\mathop{\rm Tr}\nolimits}(A^\dagger A\otimes B^\dagger B\rho)/(|{\rm det(A)}{\rm
det}(B)|$. Absorbing the factors $|{\rm det}(A)|$ and $|{\rm
det}(B)|$ into $A$ and $B$, it is sufficient to consider $A$ and $B$ with determinant 1. In the $R$-picture, the optimization is then equal to minimizing the $(0,0)$ element of $R=L_1\Sigma
L_2^T$ by appropriate $L_A,L_B$. Absorbing $L_1$ and $L_2$ into $L_A'=L_AL_1^TM$ and $L_B'=L_BL_2^TM$, this is equivalent to finding the optimal vectors $l_A$ and $l_B$ such that $l_a^T\Sigma l_B$ is minimized under the constraints $l_A^TMl_A=1=l_B^TMl_B$.
Let us first consider the case of diagonal $\Sigma$ with elements $s_0\geq s_1\geq s_2\geq |s_3|$. Parameterizing $l_A$ as $(\sqrt{1+\|\vec{x}\|^2},\vec{x})$ and $l_B$ as $(\sqrt{1+\|\vec{y}\|^2},\vec{y})$, the following inequalities hold: $l_A^T\Sigma l_B\geq
s_0\sqrt{1+\|\vec{x}\|^2}\sqrt{1+\|\vec{y}\|^2}-s_1\|\vec{x}\|\|\vec{y}\|\geq
s_0$. Therefore the concurrence will be maximized for $\vec{x}=\vec{y}=0$, leaving $\Sigma$ into diagonal form. Collecting the previous results, it follows that if $R$ is diagonalizable, the state with maximal concurrence that can be obtained from it by single copy LQCC operations is the one corresponding to $\Sigma$ which is a Bell diagonal state. This is in complete accordance with the results of Kent et al. [@Kent2]. The optimal $A$ and $B$ are thus given by the 2x2 matrices corresponding to $L_1^TM$ and $L_2^TM$. The optimal POVM can then be obtained by dividing $A$ and $B$ by their largest singular value such that $A^{\dagger}A\leq 1$ and $B^{\dagger}B\leq 1$, followed by calculating the square roots $A_c=\sqrt{I_2-A^{\dagger}A}$ and $B_c=\sqrt{I_2-B^{\dagger}B}$ which are rank 1. The optimal POVM’s to be performed on the two qubits are then given by $\{A,A_c\}$ and $\{B,B_c\}$ respectively. Note that the probability of measuring $(A,B)$ is given by the inverse of the gain in concurrence divided by the product of the largest singular values of $A$ and $B$, and that the rank of the Bell diagonal state is equal to the rank of the original state.
If $\Sigma$ is of the form (\[jb\]) however, things get more complicated. An analogous reasoning as in the diagonal case leads to the conclusion that $l_A$ and $l_B$ are vectors associated with the Lorentz transformations bringing (\[jb\]) into diagonal form. This is however only possible in the limit where $l_A$ and $l_B$ contain factors $\lim_{t\rightarrow
\infty}[\sqrt{1+t^2},0,0,t]$ and $\lim_{t\rightarrow
\infty}[\sqrt{1+t^2},0,0,-t]$ respectively. This indeed allows to bring $R$ asymptotically into diagonal form with diagonal elements given by $[\sqrt{(a-b)(a+c)},d,-d,\sqrt{(a-b)(a+c)}]$ and off-diagonal elements of order $1/t^2$, yielding a state infinitesimally close to a Bell diagonal state. The probability to get this state during a measurement of the optimal POVM however scales as $\lim_{t\rightarrow\infty}1/t^2$. This is equivalent to the quasi-distillation protocol by Horodecki [@Horodecki]. In this limit of $t\rightarrow\infty$ the rank of the new state is less than the original one, and its concurrence is given by $|d|/\sqrt{(a-b)(a+c)}$.
In the case where $a-b=a+c=|d|$ we are therefore able to create a state arbitrary close to the singlet state. Therefore the only mixed states that can be quasi-purified to the singlet state by single copy LQCC operations are the rank two states having normal form (\[jb\]) with $a-b=a+c=|d|$.
In conclusion, we obtained new insight into the problem of local filtering on one copy of two qubits by introducing the notion of Lorentz transformations on a real matrix parameterization of their density matrix. This matrix can be brought into one of two types of normal forms. These normal forms contain all the information about the entanglement of formation and reveal an elementary connection between concurrence and the partial transpose criterion of Peres. Moreover, this new formalism enabled us to derive in a constructive way the optimal local filtering operations for concentrating entanglement on an arbitrary mixed state of two qubits. This could be of great interest in constructing optimal distillation protocols. We showed that states of the first type can be locally transformed into a Bell diagonal state of the same rank with finite probability, whereas states of the second kind can asymptotically be transformed into Bell diagonal states with lower rank. This last class is of special interest as is contains the mixed states that can be transformed arbitrary close to the singlet state.
Frank Verstraete is PhD student, Jeroen Dehaene is postdoctoral researcher and Bart De Moor is full Professor at the K.U.Leuven. [This work is supported by several institutions: 1. the Flemish Government: a. Research Council K.U.Leuven : Concerted Research Action Mefisto-666; b. the FWO projects G.0240.99, G.0256.97, and Research Communities: ICCoS and ANMMM; c. IWT projects: EUREKA 2063-IMPACT, STWW; 2. the Belgian State: a. IUAP P4-02 and IUAP P4-2; b. Sustainable Mobility Programme - Project MD/01/24; 3. the European Commission: a. TMR Networks: ALAPEDES and System Identification; b. Brite/Euram Thematic Network : NICONET;3. Industrial Contract Research : ISMC, Data4S, Electrabel, Laborelec, Verhaert, Europay; The scientific responsibility is assumed by the authors.]{}
[99]{} {frank.verstraete,jeroen.dehaene,bart.demoor}\
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abstract: 'The High-Z Supernova Search is an international collaboration to discover and monitor type Ia supernovae (SN Ia) at $z > 0.2$ with the aim of measuring cosmic deceleration and global curvature. Our collaboration has pursued a basic understanding of supernovae in the nearby Universe, discovering and observing a large sample of objects, and developing methods to measure accurate distances with SN Ia. This paper describes the extension of this program to $z \geq 0.2$, outlining our search techniques and follow-up program. We have devised high-throughput filters which provide accurate two-color restframe $B$ and $V$ light curves of SN Ia, enabling us to produce precise, extinction-corrected luminosity distances in the range $0.25 < z < 0.55$. Sources of systematic error from K-corrections, extinction, selection effects, and evolution are investigated, and their effects estimated. We present photometric and spectral observations of SN 1995K, our program’s first supernova, and use the data to obtain a precise measurement of the luminosity distance to the $z=0.479$ host galaxy. This object, when combined with a nearby sample of SN, yields an estimate for the matter density of the Universe of $\Omega_M = -0.2^{+1.0}_{-0.8}$ if $\Omega_\Lambda = 0$. For a spatially flat universe composed of normal matter and a cosmological constant, we find $\Omega_M = 0.4^{+0.5}_{-0.4}$, $\Omega_\Lambda = 0.6^{+0.4}_{-0.5}$. We demonstrate that with a sample of $\sim 30$ objects, we should be able to determine relative luminosity distances over the range $0 < z< 0.5$ with sufficient precision to measure $\Omega_M$ with an uncertainty of $\pm 0.2$.'
author:
- 'Brian P. Schmidt, Nicholas B. Suntzeff, M. M. Phillips, Robert A. Schommer, Alejandro Clocchiatti, Robert P. Kirshner, Peter Garnavich, Peter Challis, B. Leibundgut, J. Spyromilio, Adam G. Riess, Alexei V. Filippenko, Mario Hamuy, R. Chris Smith, Craig Hogan, Christopher Stubbs, Alan Diercks, David Reiss, Ron Gilliland, John Tonry, José Maza, A. Dressler, J. Walsh, and R. Ciardullo'
title: |
The High-Z Supernova Search: Measuring Cosmic Deceleration and Global Curvature of the Universe Using Type Ia Supernovae\
([*To appear in the Astrophysical Journal*]{})
---
Introduction
============
Measuring the cosmological parameters which describe the global properties of the Universe has been a fundamental quest in astronomy ever since Robertson (1936) and Walker (1936) formulated the metric for a homogeneous and isotropic Universe. By observing how a standard candle dims as a function of redshift, usually shown as a Hubble diagram, the effects of curvature and cosmic deceleration can be observed and quantified (Sandage 1961). Early luminosity distance investigations (Humason, Mayall, & Sandage 1956; Baum 1957; Minkowski 1960) used brightest cluster galaxies as standard candles and measured galaxy brightnesses in the range $0.01 < z < 0.5$. Attempts to trace luminosity distances versus redshifts with these galaxies at $z>0.1$ changed emphasis when it was realized both from theory (Tinsley 1972) and observation (e.g., Oke, Gunn, & Hoessel 1996) that the effects of galaxy evolution are much larger than the differences due to cosmology.
Although many other methods for measuring the global curvature and cosmic deceleration exist (e.g., Peebles 1993), none has yet delivered a definitive result. For example, measuring the number of galaxies as a function of magnitude maps out the volume of space as a function of redshift, and can be used to gauge the global geometry. Attempts to use this method (Shanks et al. 1984) have been hampered by galaxy evolution and merging, although some of the uncertainty may be eliminated by moving to the infrared (Yoshi & Peterson 1995). Another test examines the angular size of a standard rod as a function of redshift. Kellerman (1993) resolved a large sample of compact radio sources out to $z\approx3$ using Very Long Baseline Interferometry. Although the angular sizes increase at $z> 1$ as expected for a non-empty Universe, evolutionary effects are hard to quantify. Stepanas & Saha (1995) have also shown that the unknown intrinsic distribution of source sizes makes it difficult to obtain a statistically significant measurement of cosmological parameters. Guerra & Daly (1998) have also used extended radio galaxies as standard rods and show that the results are consistent with a low density universe.
Type Ia supernovae (SN Ia) have long been considered promising tools for measuring extragalactic luminosity distances, but only recent searches, the resulting sets of light curves and spectra, and new methods of analysis (Phillips 1993 \[P93\]; Hamuy et al. 1995, 1996a-d \[H95,H96a-d\]; Riess, Press, & Kirshner 1995, 1996a \[RPK95, RPK96\]) have quantified the nature, power, and limitations of SN Ia as distance indicators. SN Ia offer high intrinsic luminosity \[$M_B\approx-19.4$ mag (Saha et al. 1997)\] and as individual stars, may not be subject to the same evolutionary effects which plague galaxies (although this must be demonstrated). Even before this breakthrough in understanding SN Ia, two searches for distant SN Ia were initiated (Nørgaard-Nielsen et al. 1989; Perlmutter et al. 1995) to measure cosmological parameters. These searches demonstrate that it is possible to find SN at $z > 0.3$ with large-format CCD detectors, and give hope that a significant sample of SN Ia can be gathered in just a few years. Perlmutter et al. (1997, 1998) have already presented observations of 8 objects at $z > 0.35$. A sample of $\sim 30$ objects, if carefully measured and shrewdly analyzed, will provide a statistically interesting measurement of global cosmological parameters.
Recently, much effort has been put into examining how to use the power spectrum of fluctuations in the cosmic microwave background (CMB) to measure cosmological parameters (see Hu 1996 for a review). Future satellite missions, such as the Microwave Anisotropy Probe and Planck Missions, will measure temperature fluctuations across the sky on scales substantially less than a degree, determining the power spectrum of fluctuations out to a multipole expansion of $\ell >500$. Model fits to these observations promise to provide simultaneous measurement of ten cosmological parameters. However, because the fits are degenerate for certain combinations of interesting parameters such as $\Omega_M$ and $\Omega_\Lambda$, the CMB observations will need to be combined with other observational data, such as those from high-redshift supernovae (Zaldarriaga, Spergel, & Seljak 1997), to determine $\Omega_M$ and $\Omega_\Lambda$ individually.
This paper reports on the High-Z SN Search, a coordinated program to discover, spectroscopically classify, and measure photometrically in at least two filters a distant set of SN Ia. This enterprise aims to measure the deceleration parameter, $q_0$, with an uncertainty smaller than $0.1$, and will place strong limits on combinations of cosmological parameters such as $\Omega_M - \Omega_\Lambda$. The CMB observations provide a nearly orthogonal set of parameters, so we will be able to separate the effects of any exotic forms of matter-energy in the Universe from normal matter. We will be able to discern whether the Universe is open, closed, or has nearly zero global curvature. Preliminary results from our program were reported by Schmidt et al. (1996), Schmidt (1997), and Leibundgut & Spyromilio (1997). In addition, we have confirmed the predicted time-dilation of redshifted objects using SN Ia light curves (Leibundgut et al. 1996) and spectra (Riess et al. 1997 ), and presented observations of 3 objects observed with [*HST*]{} (Garnavich et al. 1998).
In §2 we describe how the expansion, deceleration, and curvature of the Universe are related to luminosity distances, and in §3 we discuss measuring distances with SN Ia at $z<0.1$. Our supernova search and observational follow-up program are outlined in §4. In §5 we present the techniques and limitations of using SN Ia to measure accurate luminosity distances at $z>0.2$. Observations of the first SN discovered by this program, SN 1995K at z=0.479, are presented in §6, with the techniques discussed in §3-5 applied to this object. We summarize the High-Z SN Search to date in §7 and use the results for SN 1995K to estimate the precision with which we will be able to constrain cosmological parameters.
Expansion, Deceleration, and Curvature
======================================
The precise large-scale isotropy of the microwave background confirms a picture in which the Universe is accurately described on large scales by the maximally symmetric, Robertson-Walker line element (e.g., Weinberg 1972). For events with time separation $dt$, radial coordinate separation $dr$, and angular separation $d\theta$, the line element $ds$ is given by $$ds^2=dt^2-a^2(t)\left[{dr^2\over{1-kr^2}}+r^2d\theta^2\right].\label{eq:RW}$$ The global spatial geometry has the character of a hypersphere of radius $k^{-1/2} a(t)$, where $a(t)$ is the cosmic scale factor which defines the physical scale of the hypersphere at each time. In these units the spatial curvature parameter $k$ can be 1, 0, or $-1$, corresponding to a closed, flat, or open Universe, respectively.
The complete spacetime metric, which depends on $a(t)$, is determined by the Friedmann equation $$H^2\equiv(\dot a/a)^2={8\pi G\rho\over 3} - {k\over a^2}, \label{eq:F}$$ where $\rho$ is the total density of all forms of matter-energy. Friedmann-Robertson-Walker (FRW) cosmologies are based on equations (\[eq:RW\]) and (\[eq:F\]), and provide a complete description of an isotropic and homogeneous universe.
We adopt a conventional model in which the matter content of the Universe is composed of a sum of components each having a fraction $\Omega_i$ of the current critical density $\rho_{crit}\equiv 3H_0^2/8\pi G$ and various equations of state with density $\rho_i\propto
({\rm volume})^{-(1+\alpha_i)}$ \[e.g., $\alpha=0$ for normal matter ($\Omega_M$), $\alpha=-1$ for a cosmological constant ($\Omega_\Lambda$), $\alpha=+1/3$ for radiation ($\Omega_{\rm rad}$), $\alpha=-1/3$ for non-commuting strings ($\Omega_{\rm S}$)\]. It is convenient to adopt a parameter $\kappa_0\equiv
k c^2/[a(t_0)^2H_0^2]$ representing the scalar curvature in units commensurate with the density parameters; the current physical radius of hypersphere curvature is $k^{-1/2} a(t_0)= \kappa_0^{-1/2} c H_0^{-1}$ and the definition of critical density gives $\kappa_0=\sum_i\Omega_i-1$. We can then write the Friedmann equation in terms of these model parameters, $$H^2= H_0^2[ \sum_i\Omega_i(1+z )^{3+3\alpha_i}-\kappa_0(1+z )^2].\label{eq:F1}$$ It is conventional to define a “deceleration parameter” $q_0\equiv -\ddot a(t_0) a(t_0)/\dot a^2(t_0)$, characterizing the low-redshift behavior, which can be expressed as $$q_0= {1\over 2}\sum_i\Omega_i(3+3\alpha_i)-\sum_i\Omega_i ={1\over 2}\sum_i\Omega_i(1+3\alpha_i).\label{eq:q}$$
Distance measurements based on SN Ia light curves are described as luminosity distances, $D_L$, and are defined by the ratio of the intrinsic luminosity $\cal L$ to the observed flux $\cal F$ as $$D_{L} = \left(\frac{{\cal L}}{4 \pi {\cal F}}\right)^{\frac{1}{2}}.\label{eq:DL}$$ In FRW cosmologies $D_L$ is derived by computing the area of the sphere over which the flux is distributed from a source at a radial coordinate fixed by the redshift. Including the effects of time dilation and redshift, the luminosity distance is $$D_L H_0= (1+z)\left | \kappa_0 \right |^{-1/2}S\lbrace\left | \kappa_0 \right |^{1/2}
\int_0^zdz'[\sum_i
\Omega_i(1+z')^{3+3\alpha_i}-\kappa_0(1+z')^2 ]^{-1/2}\rbrace, \label{eq:intDL}$$ where $S\{x\}\equiv \sin(x)$, $x$, or $\sinh(x)$ for $k= 1,0,-1$, respectively (Coles & Lucchin 1995).
Mattig (1958) showed that when normal matter is the only contributor to $\Omega$, $$D_L H_0 = {1 \over q_0^2}\left[q_0z +(q_0-1)(\sqrt{1+2q_0z}-1)\right].\label{eq:mattig}$$ Alternatively, equation (\[eq:intDL\]) can be expanded in $z$ to give $$D_L H_0= z+z^2\left({1-q_0\over 2}\right) +{\cal O}(z^{3}), \label{eq:expansion}$$ which is valid for all models. The linear term of the expansion is the Hubble law, and has been studied for many years. Its linear form has been verified to high precision in the nearby Universe using SN Ia with the same techniques we employ for this project (H96b; RPK96; Tammann & Leibundgut 1990), using brightest cluster galaxies (Lauer & Postman 1992), and again using SN Ia at larger redshifts (Kim et al. 1997). The current debate on the value of $H_0$ centers on obtaining an absolute calibration for these distance indicators in nearby galaxies through accurate absolute distances. The measurements of curvature and deceleration require only a relative distance indicator to obtain the shape of the ($D_L$,$z$) relation, and are not affected by current uncertainties in $H_0$ due to local calibration. Equation (\[eq:expansion\]) shows that departures in the luminosity distance from a pure Hubble law, to lowest order in $z$, are proportional to $q_0$ — they depend only on deceleration and not on curvature. With a distance modulus, $m-M = 5 \log_{10} \left(D_L/{\rm 10~parsec}\right)$, measured to precision $\Delta m$ mag for an object at redshift $z$, equation (\[eq:expansion\]) shows (using $\Delta m=5\Delta\log_{10}[H_0D_L]$) that we measure $q_0$ to a precision $\Delta q_0\approx 0.9 \Delta m/z$; thus a single well-observed SN Ia at $z=0.5$ with $\sigma = 0.15$ mag (H96b, RPK96) should yield a precision of about $\Delta q_0=0.27$, almost a $2\sigma$ discrimination between an empty ($q_0=0$) and a flat ($q_0=0.5$) Universe. However, we caution the reader that already at $z=0.5$, ${\cal O}(z^{3})$ terms cannot be neglected, especially in cosmologies with significant cosmological constants.
To illustrate the precise effects of cosmology on luminosity distance, we plot the differences in distance modulus, $m-M$, from an $\Omega_{\rm tot}=0$ Universe as a function of redshift for a set of universes composed of different amounts and types of matter-energy (Figure \[fig:dlvsz\]). Although first order deviations constrain only the linear combination of parameters corresponding to deceleration, data on objects over a range of redshifts up to $z\approx 1$ can separate out the effects of the various forms of mass-energy in the $(D_L,z)$ relation, and place limits on global curvature. In particular, it is possible to separate flat cosmological models with non-zero $\Omega_\Lambda$ from open universes containing only normal matter (Goobar and Perlmutter 1995).
Using Type Ia Supernovae to Measure Luminosity Distances
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SN Ia have been used as extragalactic distance indicators since Kowal (1968) published a Hubble diagram ($\sigma = 0.6$ mag) for SN I. We now recognize that the old SN I spectroscopic class conflated two distinct types of objects: SN Ib/c which are probably massive stars that undergo core collapse after losing their hydrogen atmospheres, and SN Ia which are most likely thermonuclear explosions of white dwarfs (see Filippenko 1997 for a review). Modern versions of the SN I (now SN Ia) Hubble diagram shows scatter about the inverse-square line of about $0.3$ to $0.5$ mag (Tammann & Leibundgut 1990; van den Bergh & Pazder 1992; Branch & Miller 1993; Sandage & Tammann 1993), which is remarkable given the heterogeneous sources and oftentimes poor observations upon which these diagrams are based.
The advent of precise observations of nearby SN Ia made with CCD detectors produced evidence for genuine differences in the luminosities, light curve shapes, and spectra among the Type Ia family. SN 1984A (Branch 1987; Barbon, Rosino, & Iijima 1989), SN 1986G (Phillips et al. 1987), SN 1991bg (Filippenko et al. 1992b; Leibundgut et al. 1993), and SN 1991T (Filippenko et al. 1992a; Phillips et al. 1992) provided proof that SN Ia were not all identical objects whose observed differences could be attributed to measurement errors, but that real differences among these explosions are undoubtedly present.
The problem of understanding SN Ia well enough to use them as cosmological probes despite their intrinsic variation was solved by assembling a sufficiently large, uniform, and well-observed data set. In 1990 a group of astronomers at CTIO and the University of Chile at Cerro Calán initiated a systematic photographic search for SN Ia using the Curtis Schmidt telescope at CTIO (Hamuy et al. 1993a). Their program, which discovered 30 SN Ia in $2.5$ years, also acquired high-quality spectral and photometric follow-up for these supernovae. The resulting data set (H96c) allows the precise determination of the properties of SN Ia as distance indicators. At maximum light, SN Ia have an intrinsic range of $>2$ mag in $B$ and $>1$ mag in $V$. Although this is an interesting result for supernova physics, it does not bode well for using SN Ia as high-precision distance indicators without additional information.
Although their brightness at maximum light has a moderately large scatter, SN Ia do exhibit a correlation ($\sigma \approx 0.15$ mag) between the rate at which their luminosity declines and absolute magnitude. P93 demonstrated this relationship by plotting the absolute magnitude of ten nearby SN Ia which had dense photoelectric or CCD coverage, versus the parameter $\Delta m_{15} (B)$, the amount by which the SN decreased in brightness in the $B$ band over the 15 days following maximum light. The sample showed a correlation, which when taken into account, dramatically improved the predictive power of SN Ia. The Calán/Tololo survey yielded an independent confirmation of the P93 absolute magnitude-decline rate relationship from the sample of 30 SN Ia by using a $\chi^2$ fitting technique to the $B,V$, and $I$ light curves (H95, H96a). When corrected for their rate of decline, H96c demonstrated that the scatter in the Hubble diagram could be lowered to $\sigma \sim 0.15$ mag in $V$ for a sample of nearly 30 SN Ia. Another technique, the Multicolor Light Curve Shape (MLCS) method, has been developed by RPK95 and RPK96. By “training” on a nearby set of objects (P93’s sample plus a few additions), they achieve $\sigma < 0.2$ mag on a sample of 20 objects (H95, augmented by 10 additional well observed SN Ia) in the Hubble flow. This result is encouraging because the Hubble diagram derived by RPK96 is independent of the objects on which their method was “trained,” and therefore provides an upper limit for the true dispersion of this distance measuring technique. Other methods to correct for intrinsic luminosity differences or limits on the input sample by various criteria have also been proposed to increase the precision of SN Ia as distance indicators (Tammann & Sandage 1995; Fisher et al. 1995; van den Bergh 1995; Branch et al. 1996; Perlmutter et al. 1997).
The analyses described above assume that all SNe Ia can accurately be described by a one-parameter family of light curves. We know this is not true because the scatter about the Hubble line in either H96c and RPK96 is larger than the observational errors would indicate (H96c,RPK96). The inferred scatter beyond the observational uncertainties is small, ($\sigma \approx 0.12$ mag), and the residuals (including observational uncertainties) are distributed about the mean with a distribution consistent with a Gaussian. To this date, no other observable has been shown to successfully account for the remaining small intrinsic scatter about the one parameter family of light curves. Unless supernovae are much different at high redshift, the imperfection of SN Ia as distance indicators will have a negligible impact on using SN Ia as cosmological probes.
Search and Follow-Up Program
============================
Many techniques have been successfully used to discover supernovae, including visual observations of nearby galaxies (Evans 1994), photographic surveys (Zwicky 1968; Mueller 1989, McNaught 1990, Hamuy et al. 1993a; Pollas 1992), and CCD surveys (Perlmutter et al. 1992, 1995; Treffers et al. 1993; Martin, Williams, & Woodings 1997; Reiss et al. 1998). Although it is possible to discover objects up to $z\approx1$ (Schmidt et al. 1997c) by using large format CCDs coupled with wide fields on telescopes with the best image quality, it is efficient to measure cosmological parameters by observing objects in the range $0.35 < z < 0.55$. When systematic effects are small, the leverage gained with high-redshift objects is offset by the difficulty in obtaining accurate measurements. It is challenging to obtain accurate restframe $B$ and $V$ photometry of objects observed at $ 0.55 < z < 0.9$ because they are outside the optimum K-correction window (Figure \[fig:Kcorrsig\]), and these SN are currently less powerful tools for measuring cosmological parameters than their lower redshift siblings. The [*Hubble Space Telescope*]{} ([*HST*]{}) + WFPC2/NICMOS could acquire accurate restframe $B$ and $V$ measurements for SN Ia at $z\approx 1$. These objects hold the promise of establishing powerful constraints on cosmology within this more distant observational window. From the ground, however, the band $0.35 < z < 0.55$ gives the best combination of measurements and systematics to investigate cosmology.
Observing Strategy
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To maximize the number of SN Ia discovered in our target redshift range, $0.35 < z < 0.55$, we observe a large area and aim to achieve a limiting magnitude of $m_R\approx23$ mag, which is $\sim 1$ magnitude fainter than the expected brightness of a $z=0.5$ SN Ia at maximum light (Figure \[fig:magvsz\]). Finding objects is not the only consideration; the objects must be found near or before maximum light, and we need to follow discoveries with spectra and multi-color photometry. To ensure our objects are discovered young, we use the technique described by Hamuy et al. (1993a) and Perlmutter et al. (1995), imaging fields near the end of a dark run, and then reimaging the fields at the beginning of the next dark run. These two runs, separated by approximately 21 days (close to the rise time of a time-dilated SN Ia at z=0.5), provide objects which are at, or before maximum light. Observations are generally made near the celestial equator, so that we can use telescopes in both hemispheres for follow-up spectroscopy and photometry. At least two observations are made at each search position, to detect motion of asteroids, eliminate cosmic rays, and remove chip defects.
As an outgrowth of this project, Riess et al. (1997) developed a method to measure the age of a SN Ia relative to maximum light from its spectrum alone. This technique is especially valuable because it provides another way to identify young objects, ideally while still observing at the telescope.
Since we need to schedule large blocks of telescope time months in advance to follow the supernovae, it is essential to have candidates after each supernova discovery run, and not be derailed by weather. During the summer months of December through March, the Chilean Atacama desert has nearly 100% clear weather. We have concentrated our search efforts at the CTIO 4 m telescope, the instrument which currently provides the widest field of view of any large telescope in Chile.
On the CTIO 4 m telescope we image approximately $3$ square degrees per night with a single 2048$^2$ detector, taking two consecutive 150 s $B45$ exposures of each field. In good conditions ($1^{\prime\prime}$ seeing), a combined frame has a limiting magnitude of $m_R\approx 23$, and provides a sufficiently long time baseline to remove Kuiper belt objects, which have a typical parallax motion of $3^{\prime\prime}~{\rm hr}^{-1}$. Since January 1997, the “Big Throughput Camera” (BTC) has been available at the CTIO 4 m telescope. This mosaic of 4 chips quadruples the imaging area, but has a somewhat longer readout time. We have recently used the BTC to obtain two consecutive 300s $R$ exposures at every pointing, enabling us to cover 7 square degrees per night to a depth of $m_R=23.5$ mag.
Search Software
---------------
Our supernova search is automated, with final cuts on potential candidates being made by eye. The automated processing program is written in PERL, and calls IRAF tasks, DoPHOT (Schechter, Mateo, & Saha 1993), and various programs written in C. In brief, the program aligns the second-epoch image with the first, initially finding the bright stars using DoPHOT, and then matching stars in the two frames using a triangle-matching algorithm similar to that described by Groth (1986). The images are then registered using the IRAF tasks “geomap” and “geotran.” After registration, we match the point spread functions (PSF) of the two epochs applying the method of Phillips & Davis (1995), which computes a convolution kernel in Fourier space, and fits the high frequencies with a Gaussian profile. The DoPHOT analytic PSF measurements show which image needs to be degraded, and indicates if the PSF matching cannot be made in a single convolution. This is the case when the images are elongated with respect to each other such that neither image has a FWHM which is smaller than the other at all position angles. In these cases, we convolve one of the images with an appropriate Gaussian, and then apply the Phillips & Davis method. After PSF matching, the images are scaled to the same intensity and sky brightness values by plotting the intensity of each pixel in one image against the intensity of the corresponding pixel in the other in a subraster centered on a star. We fit for an offset (difference in sky brightness) and a scale (differential atmospheric transparency), and then subtract the intensity-scaled images. This procedure is carried out on both second-epoch images, and these differenced images are averaged, rejecting any high pixels which are discordant by more than 3$\sigma$.
The resulting image is searched using a point-source detection algorithm. Our algorithm samples the combined difference image at many locations over the image, estimating the average noise within a PSF, and then scans the image for objects above this threshold by a certain number of $\sigma$ ($\sigma > 4$ being a typical choice). A list of candidates, eliminating those near known bright stars, is sorted by magnitude. This entire process takes about 6 minutes to run on a 170 MHz Sun UltraSparc for a pair of $2048^2$ images. For inspection of candidates, the examiner is presented with subrasters of the candidate’s region from the first epoch, both second epochs, and the subtracted image. These images can be viewed simultaneously as a mosaic, or stacked and blinked.
The number of candidates to examine depends on the detection threshold and the quality of the match between the two epochs. Typically, 5-50 objects are examined on each pair of search images, but most are easily eliminated by inspection. Our approach is to minimize false alarms from a night of observing. We usually have 5–20 possible SN candidates per night that are detected by our software filter and which are not discarded by visual inspection. When there is doubt about the reality of a candidate, we make a repeat observation of the field. At this point, the candidate list is sent to collaborators for spectroscopic observations that can show whether or not the object is a supernova, give some information on its type and age, and provide the redshift. Roughly 75% of these candidates are confirmed as SN Ia, the remainder consisting of other supernova types, AGNs, and occasional mystery objects. These mystery objects typically have no visible host galaxy, and fade by more than two magnitudes within 24 hours (and in one case, at least 2 magnitudes in 3 hours). It is conceivable that these are flare stars in the halo of the Milky Way, or the unbeamed optical counterparts to gamma-ray bursts (Rhoads 1997). There is no bias in our selection against SN in which there is no visible galaxy since the whole CCD field is searched.
Data Reduction Procedures
-------------------------
We extract the high-redshift photometry in the same way we have measured the $z< 0.1$ sample (H96c, RPK96). We first calibrate a local photometric sequence of stars which appear in the CCD field of each supernova. These stars are calibrated by observing standard stars on photometric nights, deriving color and atmospheric extinction transformations, and then applying these to the local sequence. The local photometric sequence, which typically spans a substantial range of color and brightness, is then combined, correcting for any color term of the system used in the observation, with relative photometry between the supernova and sequence stars to produce a standard magnitude for the supernova. In general, the color terms of our different systems are small, since most of the supernova observations are taken with identical filters as described in Appendix A.
To produce precise relative photometry for our high-redshift supernovae we follow the same procedure employed in the Calán/Tololo survey for galaxy subtraction (Hamuy et al. 1994). A template image, in which the supernova is absent, is required for every object/filter combination. Ideally, these images would have better seeing than any of the other observations, so that the observations are not degraded in the PSF matching process, and should have more than twice the signal-to-noise (S/N) ratio to minimize the addition of shot noise in the subtractions. In most cases images of our SN discovery regions which are appropriate for use as templates are not in hand before the time of explosion, so we must return to these fields after about a year to obtain templates. One of our most difficult tasks is to obtain good seeing images with long integration times to serve as acceptable templates. The analysis of our edata set will be continuously improved as we build up improved templates for the supernovae we follow.
We have developed a pipeline closely related to our search software for extracting photometry from our CCD frames. After normal processing in which the frames are bias subtracted and flat fielded, this pipeline uses the search software to align an image to the template, match the two images’ PSFs, scale the intensity of the observation to the template, and then subtract the template from the observation in a region around the galaxy. The software then runs DoPHOT on the galaxy-subtracted frame, using a fixed position of the supernova and a sequence of comparison stars, to extract the magnitude of the supernova. After measuring the SN’s brightness, the script builds a PSF from the comparison stars using DAOPHOT (Stetson 1987), and adds this PSF to user specified galaxies in the image so that they have the same brightness as the SN measurement. After subtracting the template from the areas containing the added PSFs, DoPHOT is once again run to extract the magnitudes of these artificial SN. These synthetic objects provide an effective method to estimate the relative photometric errors, and can give a strong indication if something has gone wrong in the reduction process.
Our spectra are reduced in a typical manner, except that we generally extract only a small region along the slit at the position of the supernova to minimize host galaxy contamination. We also bin the spectra as necessary in the dispersion direction, after final reduction, to increase the S/N ratio per pixel, to aid in classifying the objects.
Measuring Accurate Relative Distances Between Nearby and High-Z SN Ia
=====================================================================
The distances derived to SN Ia are well characterized and tested in the nearby Universe, but accurately comparing these objects to their more distant counterparts requires great care. Selection effects can introduce systematic errors as a function of redshift, as can uncertain K-corrections, or an evolution of the SN Ia progenitor population as a function of look-back time. These effects, if they are large, will limit our ability to measure the ($D_L$,$z$) relation accurately, and have the potential to sap the potency of high-redshift SN Ia for measuring cosmological parameters.
Extinction
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Most attempts at using SN Ia as distance indicators have not corrected for extinction in the supernova host galaxy. The majority of the SN Ia discovered at $z<0.1$ which comprise the current sample of well observed SN Ia have been detected using photography at Schmidt telescopes, and it is possible that many SN Ia embedded in dusty spiral arms or galaxy nuclei were missed by these searches. Distant SN Ia are discovered with CCD observations by subtracting a digital template from the data, and are more affected by galaxy surface brightness than galaxy morphology. Additional complications arise because magnitude-limited selection might prevent extinguished objects from being discovered with equal efficiency in both nearby and distant samples, or because the smaller angular size of distant objects makes it more difficult to detect SN deep inside spirals — those likely to be extinguished. In short, it would not be surprising to have systematic differences between the average extinctions of objects discovered in the nearby and distant supernova searches. Rather than attempt to untangle this hopelessly knotted set of selection effects, we believe it is more straightforward to correct for extinction of individual objects that are in the samples based on their colors.
Except for a small group of very rapidly fading supernovae, H95 show that SN Ia have a relatively small range in their $(B-V)$ color at maximum light which can be used to estimate their extinction. Even so, it is worthwhile to correct for extinction according to the prescription of P93/H95 or RPK95. Using the framework set out earlier in this section, but for two filters, a relative reddening estimate of any SN Ia can be extracted from its observations, by simultaneously fitting $B$ and $V$ (or any set of two or more photometric bands) and deriving the $E(B-V)$ offset. This procedure does not require the training set to be free of reddening, as the calculated color excess will be relative to the average extinction of the training set. In practice, using an iterative process, it is possible to define an essentially reddening-free set of synthetic light curves, weeding out the reddened objects on the basis of their color evolution. RPK96 have demonstrated that using $B,V,R,$ and $I$ light curves within their framework, dispersions of less than 0.15 mag are attainable. Furthermore, they show that their best estimate of the reddening law from a few heavily reddened objects in their sample of SN Ia is indistinguishable from the mean Galactic reddening law (Riess, Press, & Kirshner 1996b ). A recent study by Phillips et al. (1998) based on previous work by Lira (1996) on the intrinsic color evolution of SN Ia has also developed a consistent method to deredden type Ia light curves. Correcting SN Ia for extinction does not necessarily decrease their dispersion as distance indicators (extinction corrections only decrease the scatter when the error in the derived extinction is less than the dispersion caused by extinction to a sample of objects), but is essential to remove the systematic bias that absorption might introduce.
It is possible that the average properties of dust might evolve as a function of look-back time, e.g., the LMC and SMC have extinction laws which differ significantly from that of the Milky Way (Bouchet et al. 1985). Such an effect could significantly bias the comparison of nearby and distant objects if the average extinction to either the nearby or distant sample is large. Although the nearby sample of objects has few reddened objects (RPK96), and indications are that the high-z sample has low extinction as well (Riess et al. 1998), this possible bias warrants further investigation. For example, observations of reddened objects in rest-frame $BVR_CI_C$ at large redshifts could be used to investigate the extinction law at these look-back times, and limit uncertainties introduced by this effect.
K-corrections
-------------
When comparing objects at significant redshifts with nearby counterparts, it is necessary to account for the wavelength shift of light. The effect of redshift on luminosity distances, dubbed the K-correction, was tabulated for galaxies in modern photometric bands by Oke & Sandage (1968). They defined the K-correction $K_i$ such that an object’s magnitude $m_i$ in filter $i$ as a function of redshift $z$ is $$m_i(z) = m_i(z=0) + K_i(z).\label{eq: Kcorrdef}$$ An object with spectrum $F(\lambda)$ (in units of power per unit area per unit wavelength) observed in a filter with sensitivity function $S_i(\lambda)$ has $$m_i(z=0) = -2.5\log{{\int S_i(\lambda) F(\lambda)d\lambda} \over
{\int S_i(\lambda) d\lambda} } + {\cal Z}_i, \label{eq:magz=0}$$ where ${\cal Z}_i$ is the zero point of the filter. Oke & Sandage demonstrated that $$K_i = 2.5\log\left[ (1+z) {\int F(\lambda) S_i(\lambda)d\lambda \over
\int F(\lambda/(1+z))S_i(\lambda)d\lambda}\right],\label{eq:Kcorr}$$ where $K_i$ accounts for the $(1+z)$ shift of the photons in wavelength, and the $(1+z)$ increase in the unit $d\lambda$ which they occupy. In the case of SN Ia, K-corrections have an added complication caused by the changing spectral energy distribution of a supernova as it evolves. SN Ia change rapidly when they are near maximum, which translates to rapidly changing (and hence uncertain) K-corrections (Hamuy et al. 1993). This evolution is particularly pronounced blueward of 4000 Å due to line blanketing, and therefore using K-corrections in the traditional manner on the observed $B$ and $V$ light curves of SN Ia at $z=0.5$ is an extremely risky proposition. Following the precepts of Gunn (1978), Perlmutter et al. (1995) demonstrated that by observing distant SN Ia with the $R_C$ broadband filter, K-correction uncertainties could be substantially reduced, because at $z=0.5$ the $R_C$ filter approximates the restframe $B$ filter. Kim, Goobar, & Perlmutter (1996) show that this modified K-correction is
$$K_{ij} = 2.5\log\left[(1+z) {\int F(\lambda) S_i(\lambda)d\lambda \over
\int F(\lambda/(1+z))S_j(\lambda)d\lambda}\right] + {\cal Z}_j - {\cal Z}_i, \label{eq:newKcorr}$$
where $K_{ij}(z)$ is the correction for going from filter $i$ to filter $j$. Note that if filter $i$ is exactly the redshifted counterpart of filter $j$, then at this redshift $K_{ij}$ is not zero, but has a constant value corresponding to the $(1+z)$ stretching of the wavelength region $d\lambda$ photons occupy combined with the difference in the zero points of the filters.
Rather than constrain ourselves to existing broadband filter sets, we decided to define our own set of redshifted filters, created as broadband interference filters. We currently have defined four filters — pairs of $B$ and $V$ filters redshifted to $z=0.35$ and $z=0.45$ (henceforth referred to as the $B35$, $V35$, $B45$, and $V45$ filters). Although adopting this new photometric system adds the complication of defining standards for these filters, we believe the ability to translate SN Ia observed in the range $0.25 < z < 0.55$ to standard $B$ and $V$ filters with small systematic and statistical uncertainties makes it worthwhile. An overview of our system is given in Appendix A.
An important advantage in designing new filters is that modern transmission coatings have peak transmissions in excess of $85\%$, whereas traditional $R_C$ and $I_C$ filters are typically around $60\%$. In addition, our filters have a sharp cut-off on the long wavelength side, which results in significantly lower backgrounds from night sky lines than their Kron-Cousins counterparts. In Figure \[fig:Kcorrsig\] we have plotted the uncertainty in the K-corrections in translating to restframe $B$ and $V$ from both our specialized filters (Appendix A), the $R_C$ and $I_C$ filters (Bessell 1990), and selected [*HST*]{} filters. These corrections have been calculated using the series of SN Ia spectra in Hamuy et al. (1993b), augmented with additional spectra of SN 1994D by Filippenko (1997). The resulting K-corrections are fit as a function of SN age for each redshift with a low-order polynomial, and the residual scatter of this fit is used to gauge the uncertainty of the K-corrections. The estimated uncertainties are insensitive to the exact order of the polynomial used, and provide an estimate of the random component of K-correction uncertainties due to errors in our SN Ia spectrophotometry and intrinsic differences in supernovae. For proper redshift-filter matches, the resulting K-corrections in transforming to restframe $B$ and $V$ are accurate to better than 3% for SN Ia observed between 14 days before maximum and 50 days past maximum over the range of $0.25 < z < 0.55$ (and using [*HST*]{}, between $0.90<z<1.1$). For objects which are extinguished, it is necessary to modify the spectra used for determining the K-corrections iteratively with reddening curves, using fits to their light curves to estimate the amount of extinction. In addition, if the SN differs significantly from those used to construct the K-corrections, a substantial error can result, especially when the object is at a redshift where the filter is not a close match to its restframe counterpart. In these cases, the first order difference between the spectra is usually color, and the spectra used for determining the K-corrections can be adjusted with a power law to match the color evolution of the object. The systematic uncertainty in translating between filter systems limits the accuracy with which we are able to measure luminosity distances at high redshift. Note that K-correction zero point errors are magnified through extinction corrections: since $A_V=3.1 E(B-V)$, a $0.02$ mag uncertainty of $(B-V)$ translates to a 0.06 mag uncertainty in distance modulus. It is a very difficult task to ensure systematic errors in $(B-V) < 0.02$ mag in either the nearby or distant sample, and we believe that this uncertainty will be our largest source of systematic error.
Evolution
---------
An attractive feature of using SN Ia as distance indicators at significant redshifts is the possibility of minimizing the evolutionary effects which plague distance indicators based on the properties of galaxies. Initial models proposed that SN Ia arise from the explosions of carbon-oxygen white dwarfs as they reach the Chandrasekhar mass (Hoyle & Fowler 1960; Arnett 1969; Colgate & McKee 1969). This mechanism for SN Ia explosions leads to a burning front which propagates outwards from the white dwarf’s center, burning nearly the entire star to nuclear statistical equilibrium. A broader range of models for the presupernova star and for the behavior of the burning wave is needed to account for the intermediate-mass elements that are seen in the spectra of SN Ia (Wheeler & Harkness 1990), and to reproduce the observed range of SN Ia luminosities. Successful models have been produced by several groups employing a variety of mechanisms (Nomoto, Thielemann, & Yokoi 1984; Livne 1990; Khokhlov, Müller, & Höflich 1993; Woosley & Weaver 1994; Höflich, Khokhlov, & Wheeler 1995). Although the details of the explosion mechanism remain an active area for research, many plausible (but similar) models match the spectral features of SN Ia, and can produce the relation between light curve shape and luminosity detected by P93.
The local sample of SN Ia shows that the light curve shapes and luminosities are weakly correlated with the type of galaxy in which they occur (H96a). Spirals show a wide range of light curve shapes while ellipticals show a narrow range. If left untreated, the relation between stellar population and the luminosities of SN Ia could poison the inference of cosmological parameters by introducing a subtle drift in SN Ia properties with look-back time (von Hippel, Bothun, & Schommer 1997). We know that the rate of SN Ia per unit $B$ luminosity is almost twice as high in spirals as in ellipticals at the present epoch (Cappellaro et al. 1997). If we assume this increased rate of SN Ia production is related to the higher rate of star formation in spirals, then the SN Ia in spirals come from progenitors which are likely to be younger than the progenitors of SN Ia in ellipticals. Incidentally, this is the conclusion reached by Oemler and Tinsley (1979) nearly two decades ago.
However, our local sample calibrates the effect. Figure \[fig:evol\] shows that the calibration of RPK96, which does not use any information about galaxy type as an input, results in distances to early-type (8 objects) and late-type galaxies (19 objects) which are consistent to $0.006
\pm 0.07$ magnitudes (SN Ia with elliptical hosts give slightly closer distances). To obtain limits on the possible offsets between SN Ia with early-type and late-type progenitors, we perform a Monte Carlo simulation with the following assumptions. Late-type galaxies have SN Ia with young and old progenitors at a ratio of 1:1 (a conservative limit from their rates — sprials have twice the rate of supernovae as ellipticals, so 1/2 of the progenitors are old and 1/2 are young), and ellipticals contain only old progenitors. Simulated data sets demonstrate that with the above assumptions, and the observed offsets between distances to ellipticals and spirals, the allowed range in the distance offsets between objects with young and old progenitors is $0.02 \pm 0.15$ mag. If the ratio of young to old progenitors in spirals were to be 2:1, then the limits become $0.01 \pm 0.11$ mag. This unsophisticated analysis, while not providing hard limits on evolution, demonstrates there is no obvious dependance on SN Ia distances with respect to the age of the stellar population in which they reside. Since, at $z=0.5$, we still expect to see SN Ia originating from a mixture of young and old progenitors, the average evolution should be smaller than the maximum possible differences quoted above. A larger sample of objects and a better understanding of the local rates as a function of progenitor type, as will be provided by the Mount Stromlo Abell Cluster Supernova Search (Reiss et al. 1998), will increase the power of this type of test.
Theory provides another avenue by which to explore the possible effects of evolution. Höflich, Thielemann & Wheeler(1997) have calculate the differences in light curve shape, luminosity, and spectral characteristics of SN Ia as a function of the initial composition and metallicity of their white dwarf progenitors. Their calculations show that although changes in restframe $U$ can be considerable, the effects in the $B$ and $V$ bands are only $\sim 0.05$ mags, even when considerable changes are made to the metallicity and C/O ratio of the white dwarf progenitors. Changes to the light curve shapes of the objects may also occur, and in total their calculations suggest distances could drift by as much as 0.3 mag in $B$ and 0.15 mag in $V$. At what redshift such differences could appear is unclear, but these types of changes would be accompanied by significant spectral differences, and should not go undetected.
We assume that the relation between light curve shape and luminosity that holds for a wide range of stellar populations at low redshift also covers the range of stellar populations we encounter in our high-redshift sample. The range of ages for SN Ia progenitors in the nearby sample is likely to be larger than the look-back time to the galaxies in our high-redshift sample, so we expect that our local calibration will work well at eliminating any pernicious drift in the supernova distances between the local and distant samples. Although we expect this approach to be valid for joining our nearby and distant samples, until we know more about the stellar ancestors of SN Ia, we need to be vigilant for changes in the properties of the SN at significant look-back times. These might be detected as changes in the spectra or colors of SN Ia with redshift.
Selection Biases
----------------
Like every sample selected by a flux limit, our SN Ia sample will be affected by the shift of the mean to intrinsically brighter objects near our redshift limit due to the dispersion of their intrinsic luminosities; this is often referred to as Malmquist bias (e.g., Gonzalez & Faber 1997). Generally speaking, the bias is proportional to $\sigma^2$ of the distance indicator. Since we only suffer the bias left [*after*]{} the correction for light curve shape, the average bias is much smaller than for uncorrected SN Ia peak brightnesses. This is fortunate, because it is not a straightforward task to correct for luminosity biases in the limited sample of objects produced by a particular SN search.
In addition to classic Malmquist bias, the SN Ia we discover at high redshift are subject to a number of selection effects. These could affect the average properties of our sample as a function of redshift, and could possibly bias our measurement of the ($D_L$,$z$) relation. The supernovae discovered by the High-Z SN Search are limited by their detectability between two epochs (typically 25 days apart in the observer’s frame), and only those objects which increase in flux by an amount greater than our detection threshold are discovered. Therefore, our ability to discover a supernova depends not only on its brightness at maximum light, but also on its age, the light curve shape of the object, and redshift due to time dilation of the light curves.
We have performed Monte Carlo simulations in which we randomly explode SN Ia assuming their rate is constant with look-back time, identifying which objects would be discovered in a search by our supernova program. These simulations show that with SN Ia distances of precision $\sigma=0.15$ mag, our average measurement of $\Omega_M$ will be biased high by $\sim 0.05$ for $0 < \Omega_M < 1$ ( $\Omega_\Lambda=0$). Though it may seem surprising, selection biases are significantly less than other sources of error, such as the uncertainty in $(B-V)$, because the scatter in luminosity, after correction for light curve shape, is so small. Nevertheless, selection effects are a source of systemic error that should be removed.
Corrections could be computed through Monte Carlo simulations for equation (\[eq:intDL\]) provided that the selection function (completeness vs. magnitude) for finding objects were known. Unfortunately, the selection function for discovering SN Ia is not sharply defined; it depends critically on the PSF of both the pre-discovery and discovery images, and our procedure ultimately rests on a human decision of whether an object is real. Since false detections waste scarce observing time, observers are reluctant to follow the light curve of an object unless the probability the object is real is very high. Perlmutter et al. (1997) performed simulations to determine the magnitude limit for each of their discovery frames, adding stars to galaxies and then measuring their ability to recover them in software. Our experience with similar simulations is that our software selects fainter objects than human reviewers will accept as being real. By comparing Monte Carlo simulations of the expected redshift distribution of objects to the actual sample, it should be possible to estimate corrections to these detection thresholds.
Rather than apply corrections to individual objects, it would be better to employ a maximum likelihood approach in which one finds the most likely cosmological model, given the SN redshifts, estimated detection thresholds, SN distances and light curves, and detection simulations. This is an involved, computationally expensive procedure, which is probably not yet warranted given the small size of the corrections and the limited set of data.
Weak Gravitational Lensing
--------------------------
It has been pointed out by Kantowski, Vaughan, & Branch (1995) that large-scale structure could magnify (or demagnify) a SN’s light through weak gravitational lensing as it travels to an observer. Wambsganss et al. (1997) have computed the effect for $\Omega_\Lambda$-dominated and $\Omega_M$-dominated flat universes, and find that weak lensing could produce a modest increase in the dispersion of a distance indicator (approximately 5% at $z=1$). It also leads to a small systematic shift in observed SN brightnesses to fainter magnitudes — approximately 1% at $z=1$. The average line-of-sight to a SN is more likely to pass through voids than clusters and filaments, leading to an average demagnification. Holz and Wald (1997) have calculated a “worst case” scenario for the effect of gravitational lensing, where the universe’s mass is made up of randomly distributed objects with mass greater than $0.01 M_\odot$. At $z=0.5$ for $\Omega_M=1$, an average standard candle is made 0.15 mag fainter by the lensing. The effect is more pronounced at $z\approx1$, and is diminished as $\Omega_M$ approaches 0, as suggested by the SN observations presented here, by Garnavich et al. (1998), and by Perlmutter et al. (1998). Given the size of other systematic errors, uncertainties due to gravitational lensing are not likely to be of major concern up to $z\approx 1$. It is unlikely that we might find an event which has undergone significant lensing so that it can be readily separated from the intrinsic dispersion of SN Ia brightnesses, although Kolatt & Bartelmann (1998) suggest how to maximize these events by searching through galaxy clusters.
Summary of Uncertainties
------------------------
The data presented by Garnavich et al. (1998) and Riess et al. (1998) indicate that we can measure the distances to high-z supernovae with a statistical uncertainty of $\sigma = 0.2$ mag (10%) per object. With only 10 objects a comparison of $z\approx0$ and $z\approx0.5$ can be made to a precision of better than 5% — leaving systematic uncertainties as a major contributor to the total error budget. A summary of the contributions to high-redshift supernova distance uncertainties is given in Table \[tab:sys\]. This table shows that our program to measure cosmology will most likely be limited by the possibility of the evolution of SN Ia explosions with look-back time. Future work to address this possible problem will be as important as obtaining large numbers of objects at high redshift.
Observations and Analysis of SN 1995K
======================================
Photometry and Spectroscopy
----------------------------
In 1995 February and March our team imaged approximately 2.5 square degrees with the CTIO 4 m telescope + 2048$^2$ CCD. Our first attempts at searching for supernovae in these data were hampered by distortions caused by curvature in the CCD, and general inexperience. However, in an image taken on 30 March (UT dates are used throughout this paper), a new object was discovered in a galaxy at position (J2000.0) $\alpha=10^h50^m47^s.0$, $\delta=-09^\circ15^\prime07^{\prime\prime}.4$ (Figure \[fig:95Kbanda\]). This object was reported to the IAU and designated SN 1995K (Schmidt et al. 1995).
We obtained spectra of SN 1995K using the ESO NTT+EMMI on 1995 April 3. CCD images containing the SN spectra were bias subtracted, and the spectrum of the object extracted as a single pixel row, to minimize the contribution of the host galaxy. The spectra were then wavelength calibrated with comparison lamp spectra interspersed with the SN observations, and the resulting individual spectra were combined. The extracted host galaxy spectrum was scaled and subtracted. The host galaxy’s redshift, measured from its H$\alpha$ emission, is $z=0.479$. The spectrum was binned to increase the S/N ratio per resolution element, and the resulting spectrum is plotted in Figure \[fig:95Kspec\]. Also plotted is the spectrum of SN 1994D (Filippenko 1997) near maximum light, binned to the same resolution as SN 1995K. The spectrum of SN 1995K appears to be typical of SN Ia, exhibiting the characteristic Si II absorption at a rest wavelength of $\lambda6150$ Å and is incompatible with peculiar SN Ia such as SN 1991T (Filippenko et al. 1992a; Phillips et al. 1992; Ruiz-Lapuente et al. 1992; Ford et al. 1993) or SN 1991bg (Filippenko et al. 1992b; Leibundgut et al. 1993; Turatto et al. 1996). Using the spectral aging technique of Riess et al. (1997), we estimate the age of the SN on this date, 1995 April 3, to be $1\pm 2$ days past maximum light, [*independent*]{} of, and in agreement with its light curve. An additional spectrum of SN 1995K’s host galaxy was obtained on 1995 Apr 25 with the CTIO 4 m telescope (Figure \[fig:95Kspec\]). Comparison of the galaxy spectrum with the catalog of Kennicutt (1992) shows this galaxy to be consistent with a star-forming galaxy of type Sb/c.
CCD images of SN 1995K were obtained at several telescopes in $B45$ and $V45$ filters, and in some cases $R_C$ and $I_C$. The sequence of stars shown in Figure \[fig:95Kseq\] was calibrated on three photometric nights by observing spectrophotometric standards listed in Appendix A and standards of \[Landolt (1992a) for $R_C$ and $I_C$ measurements\], measuring the color and extinction transformation coefficients, and applying these to observations of the SN 1995K field. The resulting magnitudes were averaged from multiple nights and are given in Table \[tab:95Kseq\], with uncertainties calculated from the dispersion of the observations. Relative photometry between the supernova and the stellar sequence was carried out as prescribed in §4.3, using a template for galaxy subtraction obtained with the CTIO 4 m+2048$^2$ CCD on 1996 March 15. SN 1995K’s light curve was brought to the standard system by offsetting from the standard star sequence, and applying color corrections derived for each instrumental setup (these corrections ranged from $-0.01$ to $0.04$ mag per mag of $(B_{\rm 45}-V_{\rm 45})$). The resulting light curve is listed in Table \[tab:95Klc\].
The Luminosity Distance to SN 1995K
-----------------------------------
The K-corrections to translate $B45$ and $V45$ observations of SN Ia at a redshift of $z=0.479$ to restframe $B$ and $V$, respectively, are plotted in Figure \[fig:95KKcorr\]. The K-corrections are nearly constant, because of the close match of these redshifted filters to their restframe counterparts. Table \[tab:95Ktab\] lists each corrected photometric measurement, the time-dilated age of the SN with respect to maximum fit from the light curve (iteratively determined by applying K-corrections, fitting the light curve, and refitting the K-corrections, until convergence occurs), and the resulting restframe magnitudes for SN 1995K. The light curve of SN 1995K has already successfully been used to provide strong evidence for the predicted effects of time dilation (Leibundgut et al. 1996) in SN light curves (Leibundgut 1990).
The light curve of SN 1995K is plotted in Figure \[fig:95Klc\], and is of sufficient quality to provide an accurate luminosity distance to its $z=0.479$ host. Using the techniques described in H95 we derive $\Delta m_{15}(B) = 1.15\pm 0.1$ mag for SN 1995K, and restframe brightnesses at maximum light of $m_B=22.93\pm 0.08$ mag and $m_V=23.04 \pm 0.13$ mag. This decline rate and color at maximum are typical for the unreddened SN Ia we observe in the nearby Universe. The derived distance modulus, applying the $\Delta m_{15}(B)$ versus $M_V$ relations of H96a (their Table 3), is $(m-M)_V = 42.31 \pm 0.13$ mag, and the Hubble diagram which includes SN 1995K and the H96b distance measurements to 29 objects is shown in Figure \[fig:95KhubH96\]. Applying MLCS (RPK96) to SN 1995K indicates that this SN is slightly over-luminous relative to the average ($\Delta= 0.07$), is unreddened, and has a distance modulus of $42.40\pm0.25$ mag on the RPK96 scale. This object and the distance measurements of RPK96 are plotted in Figure \[fig:95KhubRPK96\]. Distances derived to the nearby and distant supernovae must be done in exactly the same manner, so the results of these two ways of using the light curve shape are not plotted on the same diagram. However, one can compare the resulting ($D_L$, $z$) relation, and the bottom halves of Figures \[fig:95KhubH96\] and \[fig:95KhubRPK96\] demonstrate that a consistent result is obtained.
With a single object it is difficult to make serious conclusions about cosmological parameters, regardless of the distance precision it offers, because there is no way to judge systematic errors in an empirical way. However, taken at face value, if $\Omega_\Lambda=0$, SN 1995K yields an estimate for the matter density of the Universe to be $\Omega_M = -0.2^{+1.0}_{-0.8}$, where we have taken the range to include both the H96 and RPK96 distance uncertainties, extinction uncertainties, and uncertainties due the sources of systematic error described in $\S$5. For a spatially flat universe composed of normal matter and a cosmological constant, we find $\Omega_M = 0.4^{+0.5}_{-0.4}$, $\Omega_\Lambda = 0.6^{+0.4}_{-0.5}$. An $\Omega=\Omega_M=1$ universe is excluded with greater than 80% confidence from this single distance estimate.
Perlmutter et al. (1997) present luminosity distances to 7 high-redshift supernovae. Direct comparison with our work is difficult, because not all of their objects were positively identified as SN Ia (two of their objects have decline rates more extreme than any other known SN Ia and do not have confirming spectra), and their single-color results cannot be corrected for extinction in the host galaxy. However, taken on equal terms, SN 1995K suggests that $\Omega_M$ may be lower than the central value of $\Omega_M= 0.88^{+0.69}_{-0.60}$ they find. Recent [*HST*]{} observations by the two groups (Garnavich et al. 1998; Perlmutter et al. 1998) seem to confirm this view, yielding values of $\Omega_M = -0.1 \pm 0.5$ and $\Omega_M = 0.2 \pm 0.4$, respectively. More data will elevate this discussion.
The Future
==========
SN 1995K is the first of more than 30 confirmed SN Ia discovered by the High-Z SN Search (Schmidt et al. 1995, 1997a, 1997b; Kirshner et al. 1995; Suntzeff et al. 1996; Garnavich et al. 1996a,b, 1997a,b). The observations presented here indicate that these objects, and those of Perlmutter et al. (1997), should provide an accurate method of tracing out luminosity distances to high redshifts. Figure \[fig:future\] shows the level of precision likely to be achieved by a sample of $N$ SN Ia with observations of comparable precision to those of SN 1995K; a measurement of $\Omega_M$ to $\pm 0.2$ should be achieved from objects at $z=0.5$ given our expected systematic uncertainties. The High-Z SN Search has been allocated sufficient orbits to follow about 8 SN Ia with the [*HST*]{} in two photometric bands. We have already obtained observations of three SN Ia, one of which lies at $z=0.97$ (Garnavich et al. 1998). As demonstrated in Figure \[fig:Kcorrsig\], this object lies in a window in which it would be possible to obtain accurate restframe $B$ and $V$ observations of SN Ia with [*HST*]{}. A sample of 10 such $z=1$ objects, coupled with the $z=0.5$ sample, would provide enough information to separate the individual effects of $\Omega_M$ and $\Omega_\Lambda$ (Goobar & Perlmutter 1995). Figure \[fig:future1\] shows the $1\sigma$ joint confidence region for $\Omega_M$ and $\Omega_\Lambda$ that is obtainable with a sample of 30 SN Ia observed at $z=0.5$ augmented with 10 SN Ia observed at $z=1$ with [*HST*]{}. As can be seen from the figure, although $\Omega_M$ and $\Omega_\Lambda$ are not individually measured accurately, the combination $\Omega_M - \Omega_\Lambda$ is tightly constrained. If we limit ourselves to flat cosmological models, $\Omega_M +\Omega_\Lambda=1$, the uncertainty in $\Omega_M$ and $\Omega_\Lambda$ would become small, with uncertainties in each quantity of approximately $0.1$.
Although measurements of the CMB anisotropy on small scales still have several years to reach maturity, the high-z SN Ia and CMB observations are complementary. The error ellipses of these two measurements in the $(\Omega_M$,$\Omega_\Lambda)$ plane will be nearly perpendicular to each other (Figure \[fig:future1\]). When combined, they could provide a definitive measurement of the global parameters of our Universe. We wish to thank Ed Carter at NOAO for tracing the redshifted filter set. AVF acknowledges support from NSF grant AST-9417213. RPK acknowledges support from NSF grants AST-9528899 and AST-9617058, and thanks the Institute for Theoretical Physics, University of California, Santa Barbara for their generous hospitality. AGR acknowledges support from the Miller Institute for Basic Research in Science, University of California, Berkeley. SN research at UW is supported by the NSF and NASA. CWS acknowledges the generous support of the Seaver Institute and the Packard Foundation. MH acknowledges support provided for this work by theNSF through grant number GF-1002-97 from the Association of Universities for Research in Astronomy, Inc., under NSF Cooperative Agreement No. AST-8947990 and from Fundación Andes under project C-12984; MH also acknowledges support by Cátedra Presidencial de Ciencias 1996-1997. Partial support for AC was provided by the NSF through grant GF-1001-95 from AURA, Inc., under NSF cooperative agreement AST-8947990, and from Fundación Antorchas Argentina under project A-13313. This research used IRAF, an astronomical reduction package distributed by the National Optical Astronomy Observatories, which is operated by the Association of Universities for Research in Astronomy, Inc. (AURA) under cooperative agreement with the NSF.
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Appendix: The High-Z Standard System
====================================
We have created a new set of broadband filters which represent the Johnson $B$ and $V$ filters redshifted to $z=0.35$ and $z=0.45$. We call these $B35$, $V35$, $B45$, and $V45$, respectively. Traces of each filter have been supplied by the manufacturer, [*Omega Optical*]{}, and these traces have been verified at KPNO using a spectrophotometer. The sensitivity functions $S_{B35}$, $S_{V35}$, $S_{B45}$, and $S_{V45}$ of our broadband filters have been derived by combining these filter traces with a quantum efficiency curve of a thinned SITE CCD and a normalized function which increases linearly with $\lambda$ (Table \[tab:sens\]). This latter function is included because CCDs are photon detectors and standard sensitivity functions (e.g., Bessell 1990) assume that a star with spectrum $F_{\lambda}$ in filter $i$ has magnitude $m_i$ given by $$m_i = -2.5\log{{\int S_i(\lambda) F_\lambda(\lambda)d\lambda} \over
{\int S_i(\lambda) d\lambda} } + {\cal Z}_i. \label{eq:mag}$$ For the purposes of the High-Z SN Search, the exact definition of our zero points ${\cal Z}_i$ is not important. As long as we use a consistent method in our calculations with equations (\[eq:newKcorr\]) and (\[eq:magz=0\]), our derived K-corrections are sensitive only to the difference in zero points between our redshifted filters and the Johnson filters, e.g. ${\cal Z}_{B} - {\cal Z}_{B45}$. Rather than establishing our standard system’s zero point using Vega, we use the Hamuy et al. (1994) spectrophotometric standards. These stars have accurate broadband photometry in the standard systems of Cousins (1980) and Landolt (1992b), the systems with which our nearby supernova observations have been calibrated. This consistency is important because our program’s goal is to make as accurate a differential measurement as possible of SN brightnesses as function of redshift.
We define ${\cal Z}_i$ of our broadband filters such that A-type stars with $(V-I)=0$ mag have $B35=V35=B45=V45 \equiv V$. Hamuy et al. (1994) have observed a set of A-type stars (referred to as secondary standards) which have $V-I_C \approx 0$. Since the Hamuy stars do not have exactly $V-I_C = 0$, we must apply small corrections to $m_V$ of the stars to obtain magnitudes in our filter system. We use equation (\[eq:mag\]) to derive the magnitudes of the tertiary stars of Hamuy et al. (1994), which have $-0.3 < (V-I_C) < 0.8$, in our bandpasses, and then obtain the linear transformation between our system and the Johnson/Kron-Cousins systems of the form $$m_i = a_{ij}(V-I_C) + m_j. \label{eq:trans}$$
The transformations used to convert the Johnson/Kron-Cousins system to our new system are listed in Table \[tab:stdsi\] and the fits shown in Figure (\[fig:HZphot\]). We use these values to transform the secondary standards of Hamuy to the high-redshift SN system, and these stars and their derived magnitudes are listed as the first 10 stars in Table \[tab:stds\]. Because of the near zero color of the Hamuy secondary stars, the corrections, even for the $B35$ and $V35$ filters which have large transformation coefficients, are very small ($ < 0.02$ mag). We compute the ${\cal Z}_i$ for our filters by comparing the magnitudes for the Hamuy secondary stars obtained from equation (\[eq:trans\]) with the magnitudes obtained from equation (\[eq:mag\]), and adjusting ${\cal Z}_i$ for each filter so that the average offset between the two sets of magnitudes is zero. The resulting ${\cal Z}_i$ are given in Table \[tab:stdsi\]. Table \[tab:stds\] lists the $B35, V35, B45, V45$ magnitudes of the Hamuy tertiary stars using these ${\cal Z}_i$ with equation (\[eq:mag\]). The Hamuy secondary and tertiary stars form the primary standards for the high-redshift SN photometric system. In a future paper we will present observations of these standard stars and use them to calibrate selected Landolt (1992a) fields.
[rc|rc]{} Photometric System Zero Point& 0.05 & Individual Zero Points & 0.02 Selection Effects & 0.02 & Shot noise & 0.15 Evolution & $<0.17$ & K-corrections & 0.03 Evolution of Extinction Law & 0.02 & Extinction & 0.10 Gravitational Lensing & 0.02 & $\sigma$ of SN Ia & 0.15
[lcccc]{} 1 & 21.72(03) & 20.91(04)& 21.66(03) & 20.85(04)2 & 20.06(01) & 19.60(01)& 20.03(02) & 19.57(03)3 & 17.05(01) & 16.40(01)& 17.01(02) & 16.36(02)4 & 20.51(02) & 18.88(04)& 20.40(03) & 18.77(04)5 & 17.28(02) & 16.51(03)& 17.22(03) & 16.45(04)6 & 17.18(02) & 16.83(03)& 17.15(03) & 16.80(04)7 & 20.72(03) & 20.31(03)& 20.69(04) & 20.28(05)8 & 19.75(02) & 19.40(03)& 19.73(03) & 19.37(03)9 & 18.16(02) & 17.72(03)& 18.13(03) & 17.69(03)10 &20.64(04) & 19.11(03)& 20.53(04) & 19.01(03)11 &18.57(02) & 17.60(02)& 18.50(02) & 17.54(03)12 &19.83(02) & 19.46(03)& 19.81(03) & 19.44(03)13 &19.63(02) & 17.99(03)& 19.52(03) & 17.88(03)
[llccccll]{} 2449774.6&1995 Feb 26& $>$24.5 & && & CTIO 4m & Hamuy et al. 2449783.6&1995 Mar 07& 24.09(38)& && & CTIO 4m & Hamuy et al. 2449801.7&1995 Mar 25& 22.26(15)& && & CTIO 4m & Hamuy et al. 2449806.6&1995 Mar 30& 22.19(06)& && & CTIO 4m & Hamuy et al. 2449810.6&1995 Apr 03& & 22.02(30)&22.37(12)& & ESO NTT & Leibundgut & Spyromilio 2449812.6&1995 Apr 05& & &22.28(15)& & ESO 3.6m & Walsh 2449812.6&1995 Apr 05& 22.23(09)& 22.18(10)&& & LCO 2.4m & Dressler 2449814.7&1995 Apr 07& 22.44(14)& 22.25(15)&& & LCO 2.4m & Dressler 2449815.6&1995 Apr 08& 22.39(12)& 22.48(22)&& & LCO 2.4m & Dressler 2449815.7&1995 Apr 08& & &22.64(14)& 22.36(14)& KPNO 4m & Ciardullo 2449816.6&1995 Apr 09& 22.62(12)& 22.48(12)&& & LCO 2.4m & Dressler 2449817.5&1995 Apr 10& 22.79(12)& && & LCO 2.4m & Dressler 2449830.6&1995 Apr 23& 23.62(30)& 22.95(30)&& & ESO NTT & Leibundgut 2449831.5&1995 Apr 24& & &23.70(25)& 22.95(30)& ESO NTT & Leibundgut 2449836.5&1995 Apr 29& 23.96(33)& 22.92(29)&& & ESO 1.5m & Leibundgut 2449866.4&1995 May 29& $>$24.30 & $>$23.50 && & ESO NTT & Spyromilio
[lrccccll]{} 2449774.6&-20.8&$>25.23$ &&-0.73&2449783.6&-14.7 &24.82(38) &&-0.73&2449801.7& -2.4 &23.01(15) &&-0.75&2449806.6& 0.9 &22.94(06) &&-0.75&2449810.6& 3.6 &23.11(12) &22.87(30) &-0.74&-0.85 2449812.6& 4.9 &23.02(15) &&-0.74&2449812.6& 4.9 &22.96(09) &23.02(10) &-0.73&-0.84 2449814.7& 6.4 &23.18(14) &23.09(15) &-0.74&-0.84 2449815.6& 7.0 &23.13(12) &23.32(22) &-0.74&-0.84 2449815.7& 7.0 &23.36(14) &23.22(14) &-0.72&-0.86 2449816.6& 7.6 &23.35(12) &23.32(12) &-0.73&-0.84 2449817.5& 8.2 &23.53(12) &&-0.74&2449830.6& 17.1 &24.34(28) &23.78(27) &-0.72&-0.83 2449831.5& 17.7 &24.43(25) &23.72(30) &-0.73&-0.77 2449836.5& 21.1 &24.69(33) &23.74(29) &-0.73&-0.82 2449866.4& 41.3 &$>25.02$ &$>24.31$ &-0.72&-0.81
[lllll]{} 5100.0& 0.002& 0.000& 0.000& 0.000 5200.0& 0.246& 0.000& 0.000& 0.000 5300.0& 0.708& 0.000& 0.000& 0.000 5400.0& 0.923& 0.000& 0.000& 0.000 5500.0& 0.996& 0.000& 0.000& 0.0005600.0& 1.000& 0.127& 0.000& 0.0005700.0& 0.987& 0.628& 0.000& 0.0005800.0& 0.953& 0.933& 0.000& 0.0005900.0& 0.904& 0.930& 0.000& 0.0006000.0& 0.938& 0.962& 0.000& 0.0006100.0& 0.964& 0.987& 0.000& 0.0006200.0& 0.884& 0.909& 0.000& 0.0006300.0& 0.878& 0.944& 0.000& 0.0006400.0& 0.924& 1.000& 0.000& 0.0006500.0& 0.864& 0.934& 0.000& 0.0006600.0& 0.702& 0.917& 0.128& 0.0006700.0& 0.440& 0.927& 0.606& 0.0006800.0& 0.237& 0.887& 0.868& 0.0006900.0& 0.121& 0.857& 0.950& 0.0007000.0& 0.064& 0.795& 0.965& 0.0007100.0& 0.041& 0.573& 1.000& 0.0807200.0& 0.027& 0.330& 0.962& 0.3297300.0& 0.017& 0.180& 0.892& 0.6497400.0& 0.013& 0.095& 0.861& 0.8607500.0& 0.009& 0.055& 0.837& 0.9787600.0& 0.007& 0.036& 0.817& 1.0007700.0& 0.004& 0.023& 0.832& 0.9527800.0& 0.002& 0.017& 0.849& 0.9087900.0& 0.000& 0.013& 0.814& 0.8828000.0&0.000 & 0.011 & 0.578& 0.8558100.0&0.000 & 0.000 & 0.323& 0.8498200.0&0.000 & 0.000 & 0.168& 0.829 8300.0&0.000 & 0.000 & 0.088& 0.719 8400.0&0.000 & 0.000 & 0.048& 0.500 8500.0&0.000 & 0.000 & 0.029& 0.312 8600.0&0.000 & 0.000 & 0.018& 0.171 8700.0&0.000 & 0.000 & 0.012& 0.097 8800.0&0.000 & 0.000 & 0.009& 0.054 8900.0&0.000 & 0.000 & 0.006& 0.034 9000.0&0.000 & 0.000 & 0.005& 0.021 9100.0&0.000 & 0.000 & 0.003& 0.014 9200.0&0.000 & 0.000 & 0.002& 0.009 9300.0&0.000 & 0.000 & 0.001& 0.006 9400.0&0.000 & 0.000 & 0.000& 0.003 9500.0&0.000 & 0.000 & 0.000& 0.001
[lrrrr]{} HR718 & 4.285& 4.299 & 4.329 & 4.340 HR1544& 4.352& 4.344 & 4.340 & 4.333 HR3454& 4.346& 4.378 & 4.448 & 4.480 HR4468& 4.711& 4.721 & 4.749 & 4.760 HR4963& 4.373& 4.369 & 4.373 & 4.370 HR5501& 5.699& 5.696 & 5.699 & 5.699 HR7596& 5.583& 5.548 & 5.499 & 5.473 HR7950& 3.765& 3.761 & 3.764 & 3.764 HR8634& 3.428& 3.439 & 3.474 & 3.489 HR9087& 5.147& 5.163 & 5.207 & 5.228 L377& 11.077& 10.947& 10.750& 10.656 L1020& 11.335& 11.180& 10.938& 10.820 EG21& 11.431& 11.468& 11.513& 11.536 L1788& 12.991& 12.854& 12.636& 12.527 L2415& 12.074& 11.955& 11.761& 11.662 H600& 10.380& 10.326& 10.245& 10.200 L3218& 11.818& 11.776& 11.698& 11.662 L3864& 12.009& 11.870& 11.649& 11.539 L4364& 11.435& 11.362& 11.266& 11.221 F56& 11.072& 11.085& 11.126& 11.141 L4816& 13.787& 13.791& 13.770& 13.758 CD32& 10.365& 10.290& 10.185& 10.135 L6248& 11.638& 11.493& 11.261& 11.147 EG274& 11.093& 11.140& 11.215& 11.258 L7379& 10.048& 9.892& 9.656& 9.543 L7987& 12.278& 12.311& 12.350& 12.372 L9239& 11.876& 11.711& 11.452& 11.326 F110& 11.893& 11.957& 12.081& 12.138 L9491& 14.093& 14.067& 14.043& 14.037
[ll]{} $B35 = -0.246(V-I_C)+{\rm V}$ & ZP$=-21.339$ $B45 = +0.027(V-I_C)+{\rm R_C}$& ZP$=-21.582$ $V35 =+0.198(V-I_C)+{\rm I_C}$ & ZP$=-22.045$ $V45 = +0.034(V-I_C)+{\rm I_C}$& ZP$=-22.292$
|
---
abstract: 'Following suggestions of T. H. Koornwinder [@b-tom], we give a new proof of Kummer’s theorem involving Zeilberger’s algorithm, the WZ method and asymptotic estimates. In the first section, we recall a classical proof given by L. J. Slater [@b-slater]. The second section discusses the new proof, in the third section sketches of similar proofs for Bailey’s and Dixon’s theorems are given.'
author:
- 'Bruno Gauthier [^1]'
title: 'A proof of Kummer’s theorem'
---
The author is grateful to Peter Paule for his helpful comments.
Slater’s proof
==============
In classical hypergeometric theory, formulae concerning summation, transformation and contiguous relations are provided in order to manipulate expressions that represent special functions. One of those is Kummer’s non-terminating summation theorem: $$\label{e:kummerth}
{}_2F_1\left[{a,b \atop 1+a-b};-1\right]
= \frac{ {\Gamma}(1+\frac{a}{2}) \: {\Gamma}(1+a-b) }
{ {\Gamma}(1+a) \: {\Gamma}(1+\frac{a}{2}-b) } \ \mbox{,}$$ in which $ Re(b) < 1 $ ensures the series to be convergent. The first step in Slater’s proof is to prove the following quadratic transformation, due to Kummer: $$\label{e:kummerqua}
{}_2F_1\left[{a,b \atop 1+a-b};z\right]
= (1-z)^{-a}
\: {}_2F_1\left[{\frac{a}{2},\frac{1}{2}+\frac{a}{2}-b
\atop 1+a-b};\frac{-4\:z}{(1-z)^2}\right] \ \mbox{.}$$ The proof of this transformation is as follows.\
The right-hand side of (\[e:kummerqua\]) can be written: $$\begin{aligned}
\displaystyle
& & (1-z)^{-a} \: \sum_{k=0}^\infty
\frac{(\frac{a}{2})_k \: (\frac{1}{2}+\frac{a}{2}-b)_k}
{(1+a-b)_k \: k!}
\: \frac{(-4)^k \: z^k}{(1-z)^{2\:k}}\\
&=& \sum_{k=0}^\infty
\frac{(\frac{a}{2})_k \: (\frac{1}{2}+\frac{a}{2}-b)_k \: (-4)^k}
{(1+a-b)_k \: k!}
\: z^k \: (1-z)^{-a-2\:k} \ \mbox{,}\\
&=& \sum_{k=0}^\infty
\frac{(\frac{a}{2})_k \: (\frac{1}{2}+\frac{a}{2}-b)_k \: (-4)^k}
{(1+a-b)_k \: k!}
\: z^k \: {}_1F_0\left[{a+2\:k \atop -};z\right]
\quad \mbox{by the binomial theorem,}\\
&=& \sum_{k=0}^\infty \sum_{n=0}^\infty
\frac{(\frac{a}{2})_k \: (\frac{1}{2}+\frac{a}{2}-b)_k \: (-4)^k}
{(1+a-b)_k \: k!}
\: \frac{(a+2\:k)_n}{n!} \: z^{k+n} \ \mbox{.}\end{aligned}$$ Extracting the coefficient of $z^N$ yields: $$\begin{aligned}
& & \sum_{k=0}^{N}
\frac{(\frac{a}{2})_k \: (\frac{1}{2}+\frac{a}{2}-b)_k
\: (-4)^k \: (a+2\:k)_{N-k}}
{(1+a-b)_k \: (N-k)! \: k!} \nonumber \\
\qquad & = & \frac{(a)_N}{N!}
\: \sum_{k=0}^{N}
\frac{(\frac{1}{2}+\frac{a}{2}-b)_k \: (a+N)_k \: (-N)_k}
{(1+a-b)_k \: (\frac{1}{2}+\frac{a}{2})_k \: k!}
\label{e:beforesaal}\end{aligned}$$ since: $$\frac{1}{(N-k)!} = \frac{(-1)^k \: (-N)_k}{N!}$$ and $$(a+2\:k)_{N-k} = \frac{(a)_{N+k}}{(a)_{2\:k}}
= \frac{(a)_N \: (a+N)_k}{4^k \: (\frac{a}{2})_k
\: (\frac{1}{2}+\frac{a}{2})_k} \ \mbox{.}$$ The right-hand side of (\[e:beforesaal\]) is summable by Saalsch[" u]{}tz theorem [@b-slater (III.2)]: $$\frac{(a)_N}{N!}
\: \frac{(\frac{1}{2}+\frac{a}{2})_N \: (1-b-N)_N}
{(1+a-b)_N \: (\frac{1}{2}-\frac{a}{2}-N)_N} \ \mbox{.}$$ As this is the coefficient of $z^N$ in the right-hand side of (\[e:kummerqua\]), we obtain: $$\begin{aligned}
\sum_{N=0}^{\infty} \frac{(a)_N}{N!}
\: \frac{(\frac{1}{2}+\frac{a}{2})_N \: (1-b-N)_N}
{(1+a-b)_N \: (\frac{1}{2}-\frac{a}{2}-N)_N} \: z^N
& = & \sum_{N=0}^{\infty} \frac{(a)_N \: (b)_N}{(1+a-b)_N \: N!} \: z^N \\
& = & {}_2F_1\left[{a,b \atop 1+a-b};z\right] \ \mbox{,}\end{aligned}$$ which proves Kummer’s quadratic transformation.\
The second step is to set $z$ to $-1$ in (\[e:kummerqua\]). Therefore $$\label{e:kummerquaspe}
{}_2F_1\left[{a,b \atop 1+a-b};-1\right]
= 2^{-a}
\: {}_2F_1\left[{\frac{a}{2},\frac{1}{2}+\frac{a}{2}-b
\atop 1+a-b};1\right] \ \mbox{.}$$ Note that these series are convergent if $ Re(b) < 1 $.\
We can sum the series on the right-hand side by Gauss’s theorem [@b-slater (III.3)]. This yields $${}_2F_1\left[{a,b \atop 1+a-b};-1\right]
= 2^{-a}
\: \frac{{\Gamma}(1+a-b) \: {\Gamma}(\frac{1}{2})}
{{\Gamma}(1+\frac{a}{2}-b) \: {\Gamma}(\frac{1}{2}+\frac{a}{2})} \ \mbox{.}$$ Since $ {\Gamma}(\frac{1}{2}+\frac{a}{2}) \: {\Gamma}(\frac{a}{2}+1)
= 2^{a} \: {\Gamma}(\frac{1}{2}) \: {\Gamma}(a+1) $, we finally obtain Kummer’s theorem: $${}_2F_1\left[{a,b \atop 1+a-b};-1\right]
= \frac{ {\Gamma}(1+a-b) \: {\Gamma}(1+\frac{a}{2})}
{ {\Gamma}(1+\frac{a}{2}-b) \: {\Gamma}(1+a)} \ \mbox{.}$$
A new proof
===========
In formula (\[e:kummerth\]), let us replace $a$ by $a+2\:n$ where $n$ is a new variable that stands for a nonnegative integer. The resulting formula $$\label{e:kummervar}
{}_2F_1\left[{a+2\:n,b \atop 1+a+2\:n-b};-1\right]
= \frac{ {\Gamma}(1+\frac{a}{2}+n) \: {\Gamma}(1+a+2\:n-b) }
{ {\Gamma}(1+a+2\:n) \: {\Gamma}(1+\frac{a}{2}+n-b) }$$ is clearly equivalent to (\[e:kummerth\]).\
The introduction of the free parameter $n$ is the key to this new computer-assisted proof.
Formula (\[e:kummervar\]) can be written as: $$\label{e:p1}
\frac{\sum_k f(n,k)}{S(n)} = 1 \ \mbox{,}$$ where $$\displaystyle f(n,k) = \frac{(a+2\:n)_k \: (b)_k}{(1+a+2\:n-b)_k}
\: \frac{(-1)^k}{k!}$$ and $$\displaystyle S(n) = \frac{{\Gamma}(1+\frac{a}{2}+n) \: {\Gamma}(1+a+2\:n-b)}
{{\Gamma}(1+a+2\:n) \: {\Gamma}(1+\frac{a}{2}+n-b)} \ \mbox{.}$$ (The coefficient $2$ in the substitution $a \rightarrow a+2\:n$ makes $S(n)$ hypergeometric in $n$.)\
Formula (\[e:p1\]) can be rewritten as: $$\label{e:p2}
\sum_k F(n,k) = 1 \qquad \mbox{where }
F(n,k) = \frac{f(n,k)}{S(n)} \mbox{.}$$ We shall actually prove this last formula. Using Gauthier’s Maple package `HYPERG` [@b-user], we apply Zeilberger’s algorithm to obtain: $$\label{e:kummerWZ}
F(n,k) - F(n+1,k) = G(n,k+1) - G(n,k) \ \mbox{,}$$ with $ G(n,k) = F(n,k) \: C(n,k) $ and $ \displaystyle C(n,k)
= -\frac{(b-1)\:k}{(1+a+2\:n-b+k) \: (a+2\:n)} $.\
This $C(n,k)$ is the so-called certificate of the WZ-pair $(F,G)$.\
After summing (\[e:kummerWZ\]) for $ k $ from $ 0 $ to $K$, the right-hand side telescopes: $$\begin{aligned}
\sum_{k=0}^{K} F(n,k) - \sum_{k=0}^{K} F(n+1,k)
& = & \sum_{k=0}^{K} G(n,k+1) - \sum_{k=0}^{K} G(n,k) \\
& = & G(n,K+1) - G(n,0) \\
\label{e:WZsum}
& = & F(n,K+1) \: C(n,K+1) \ \mbox{.}\end{aligned}$$ We now let $K$ tend to infinity. Clearly, $$\lim_{K \rightarrow +\infty} C(n,K+1) = -\frac{b-1}{a+2\:n} \ \mbox{.}$$ To determine $ \displaystyle \lim_{K\rightarrow+\infty} F(n,K+1) $, we use the well-known estimate (see e.g. [@b-tom]): $$\label{e:lim}
\frac{{\Gamma}(a+k)}{{\Gamma}(b+k)} \sim k^{a-b}
\quad \mbox{as } k \rightarrow \infty \ \mbox{.}$$ First, $$\begin{aligned}
F(n,k) & = & \frac{(a+2\:n)_k \: (b)_k}{(1+a+2\:n-b)_k}
\: \frac{(-1)^k}{k!} \: \frac{1}{S(n)} \ \mbox{,} \\
& = & \frac{{\Gamma}(a+2\:n+k) \: {\Gamma}(b+k) \: (-1)^k}
{{\Gamma}(1+a+2\:n-b+k) \: {\Gamma}(k+1)}
\: T(n) \ \mbox{,}\end{aligned}$$ where $\displaystyle T(n)=\frac{(a+2\:n)\:{\Gamma}(1+\frac{a}{2}+n-b)}
{{\Gamma}(b)\:{\Gamma}(1+\frac{a}{2}+n)}$ does not depend on $k$.\
Then, $$\begin{aligned}
|F(n,k)| & \sim & k^{a+2n-(1+a+2n-b)} \: k^{b-1} \: T(n) \ \mbox{,} \\
& \sim & k^{2b-2} \: T(n) \ \mbox{.}\end{aligned}$$ We know that $ Re(b)<1 $ (necessary condition for the convergence of the series in (\[e:kummerth\])).\
Therefore $$\lim_{K \rightarrow +\infty} F(n,K+1) = 0 \ \mbox{.}$$
Expressing as before $F(n,k)$ in terms of Gamma functions and using the same aymptotics, it is easy to find the limit of $F(n,k)$ when $n$ tends to infinity: $$\begin{aligned}
F(n,k) & = & \frac{(a+2\:n)_k \: (b)_k}{(1+a+2\:n-b)_k}
\: \frac{(-1)^k}{k!}
\: \frac{{\Gamma}(1+a+2\:n) \: {\Gamma}(1+\frac{a}{2}+n-b)}
{{\Gamma}(1+\frac{a}{2}+n) \: {\Gamma}(1+a+2\:n-b)} \\
& = & \frac{{\Gamma}(a+2\:n+k) \: {\Gamma}(1+a+2\:n-b)}
{{\Gamma}(a+2\:n)\:{\Gamma}(1+a+2\:n-b+k)}
\: \frac{(b)_k \: (-1)^k}{k!} \\
& & \times \frac{{\Gamma}(1+a+2\:n) \: {\Gamma}(1+\frac{a}{2}+n-b)}
{{\Gamma}(1+a+2\:n-b) \: {\Gamma}(1+\frac{a}{2}+n)} \\
|F(n,k)| &\sim& \left| (2\:n)^{k} \: (2\:n)^{-k} \: \frac{(b)_k \: (-1)^k}{k!}
\: (2\:n)^b \: n^{-b} \right| \ \mbox{.}\end{aligned}$$ So, $$\lim_{n \rightarrow +\infty} F(n,k)
= \frac{(b)_k \: (-1)^k}{k!} \: 2^b \ \mbox{.}$$
By (\[e:lim\]), we have $ 1/S(n) \sim 2^b $ when $n$ tends to infinity, so for all $n \geq n_0$, $1/S(n) < A$ where $A$ is a positive constant. Therefore, $$\label{e:convS}
|F(n,k)| = \left|\frac{f(n,k)}{S(n)}\right| \leq A \: |f(n,k)| \ \mbox{.}$$ Moreover, $$\label{e:majorF1}
|f(n,k)| = \frac{ |(a+2\:n)_k| \: |(b)_k| }{ |(1+a+2\:n-b)_k| \: k! }
\leq \frac{ |(b)_k| }{ k! } \ \mbox{,}$$ because $$\forall i \geq 0 \ \mbox{,} \qquad
\frac{|a+2\:n+i|}{|1+a+2\:n-b+i|} < 1
\quad \mbox{when } n \mbox{ is big enough,}$$ and $$\begin{aligned}
& & \frac{ |(a+2\:n)_k| }{ |(1+a+2\:n-b)_k| } \\[0.2cm]
& = & \frac{ | (a+2\:n) \: (a+2\:n+1) \ldots (a+2\:n+k-1) |}
{ | (1+a+2\:n-b) \: (1+a+2\:n-b+1) \ldots (1+a+2\:n+b+k-1) |}
\ \mbox{,} \\[0.2cm]
& \leq & \frac{ |a+2\:n| \: |a+2\:n+1| \ldots |a+2\:n+k-1| }
{ |1+a+2\:n-b| \: |1+a+2\:n-b+1| \ldots |1+a+2\:n+b+k-1| }
\ \mbox{,} \\[0.2cm]
& \leq & 1 \ \mbox{.}\end{aligned}$$ We can finally establish (use (\[e:convS\]), (\[e:majorF1\]) and (\[e:lim\])) that $$\label{e:majorF2}
|F(n,k)| \leq A \: \left| \frac{(b)_k}{k!} \right|
\sim A \: \left| \frac{k^{b-1}}{{\Gamma}(b)} \right| \mbox{.}$$ For $Re(b)<0$, this last expression is the general term of a convergent series, so $\sum_{k \geq 0} F(n,k) $ is dominated by a convergent series.
The limit of formula (\[e:WZsum\]) when $K$ tends to infinity gives $$\sum_{k=0}^{\infty} F(n,k) - \sum_{k=0}^{\infty} F(n+1,k) = 0 \ \mbox{.}$$ Hence, the sum $\sum_{k=0}^{\infty} F(n,k) $ is independent of $n$. Then $$\begin{aligned}
\sum_{k=0}^{\infty} F(n,k)
& = & \lim_{n \rightarrow +\infty} \sum_{k=0}^{\infty} F(n,k) \ \mbox{,} \\
& = & \sum_{k=0}^{\infty} \lim_{n \rightarrow +\infty} F(n,k) \ \mbox{,}\end{aligned}$$ where the last equality is justified by dominated convergence. Moreover, $$\begin{aligned}
\sum_{k=0}^{\infty} F(n,k)
& = & \sum_{k=0}^{\infty}
\lim_{n \rightarrow +\infty} (F(n,k) \ \mbox{,} \\
& = & \sum_{k=0}^{\infty} 2^b \: \frac{(b)_k \: (-1)^k}{(k)!} \\
& = & 2^b \: \sum_{k=0}^{\infty} \: \frac{(b)_{k} \: (-1)^k}{k!}
\ \mbox{,} \\
& = & 2^b \: (1-(-1))^b = 1 \ \mbox{,} \qquad \mbox{by binomial theorem.}\end{aligned}$$ This is precisely formula (\[e:p2\]). So, we proved — with the restriction $
Re(b)<0 $ — the validity of Kummer’s theorem (specializing $n$ by $0$ in (\[e:kummervar\])).
Kummer’s theorem (\[e:kummerth\]) can be rewritten as: $$\sum_k M(a,b,k) = \frac{ {\Gamma}(1+\frac{a}{2}) \: {\Gamma}(1+a-b) }
{ {\Gamma}(1+a) \: {\Gamma}(1+\frac{a}{2}-b) } \ \mbox{,}$$ with $\displaystyle M(a,b,k) = \frac{(a)_k \: (b)_k}{(1+a-b)_k \: k!} \: (-1)^k $.\
We derive by Zeilberger’s algorithm, a recurrence for $M$ with respect to the parameter $b$: $$(a-2\:b) \: M(a,b,k) + (-2\:a+2\:b) \: M(a,b+1,k)
= G'(a,b,k+1) - G'(a,b,k)$$ with $ G'(a,b,k) = M(a,b,k) \: \frac{(a-b+k)\:k}{b} $ and $ Re(b)<0 $. After summing both sides with respect to $k$ from $0$ to $K-1$, it gives $$(a-2\:b) \: \sum_{k=0}^{K-1} M(a,b,k)
+ (-2\:a+2\:b) \: \sum_{k=0}^{K-1} M(a,b+1,k)
= G'(a,b,K) - G'(a,b,0) \ \mbox{,}$$ and then we let $K$ tend to infinity to obtain $$(a-2\:b) \: \sum_{k=0}^{\infty} M(a,b,k)
+ (-2\:a+2\:b) \: \sum_{k=0}^{\infty} M(a,b+1,k)
= 0 \ \mbox{,}$$ because $ G'(a,b,0) = 0 $ and $ \displaystyle \lim_{K \rightarrow +\infty} G'(a,b,K) = 0 $.
We have now for any $b$ with $Re(b)<0$ $$\begin{aligned}
\sum_{k=0}^{\infty} M(a,b+1,k)
& = & \frac{\frac{a}{2}-b}{a-b} \: \sum_{k=0}^{\infty} M(a,b,k) \ \mbox{,} \\
& = & \frac{\frac{a}{2}-b}{a-b}
\: \frac{ {\Gamma}(1+\frac{a}{2}) \: {\Gamma}(1+a-b) }
{ {\Gamma}(1+a) \: {\Gamma}(1+\frac{a}{2}-b) } \ \mbox{,} \\
& = & \frac{ {\Gamma}(1+\frac{a}{2}) \: {\Gamma}(a-b) }
{ {\Gamma}(1+a) \: {\Gamma}(\frac{a}{2}-b) } \ \mbox{.}\end{aligned}$$ Let us replace $b+1$ by $B$, we finally conclude that $${}_2F_1\left[{a,B \atop 1+a-B};-1\right]
= \frac{ {\Gamma}(1+\frac{a}{2}) \: {\Gamma}(1-a-B) }
{ {\Gamma}(1+a) \: {\Gamma}(1+\frac{a}{2}-B) } \ \mbox{,}$$ for all values of $B$ such that $Re(B)<1$.
**Remark:** We can also derive this result directly from (\[e:majorF2\]) by a convergence-acceleration argument. For this, we define a new sequence $ H(n,k) := F(n,2\:k) + F(n,2\:k+1) $, and we find that $\sum_{k \geq 0} H(n,k)$ is dominated by a convergent series. Hence, we have $$\begin{aligned}
\sum_{k=0}^{\infty} F(n,k)
& = & \lim_{n \rightarrow +\infty} \sum_{k=0}^{\infty} F(n,k) \ \mbox{,} \\
& = & \lim_{n \rightarrow +\infty}
\sum_{k=0}^{\infty} (F(n,2\:k)+F(n,2\:k+1)) \ \mbox{,} \\
& = & \lim_{n \rightarrow +\infty} \sum_{k=0}^{\infty} H(n,k)
= \sum_{k=0}^{\infty} \lim_{n \rightarrow +\infty} H(n,k) \ \mbox{,} \\
& = & \sum_{k=0}^{\infty}
\lim_{n \rightarrow +\infty} (F(n,2\:k)+F(n,2\:k+1)) \ \mbox{,} \\
& = & \sum_{k=0}^{\infty} 2^b \:
\left( \frac{(b)_{2\:k} \: (-1)^{2\:k}}{(2\:k)!}
+ \frac{(b)_{2\:k+1} \: (-1)^{2\:k+1}}{(2\:k+1)!} \right) \\
& = & 2^b \: \sum_{k=0}^{\infty} \: \frac{(b)_{k} \: (-1)^k}{k!}
\ \mbox{,} \\
& = & 1 \ \mbox{.}\end{aligned}$$
Other summation theorems
========================
The combination of Zeilberger’s algorithm and asymptotic estimates suggests an approach to automatically prove identities involving hypergeometric series. It is well suited to summation theorems. As an example, we now give two apparently new proofs of classical hypergeometric identities then a list of summation theorems that the package `HYPERG` can prove automatically.
Bailey’s theorem
----------------
$$\label{e:baileyth}
{}_2F_1\left[{a,1-a \atop b};\frac{1}{2}\right]
= \frac{ {\Gamma}(\frac{b}{2}) \: {\Gamma}(\frac{1+b}{2}) }
{ {\Gamma}(\frac{a+b}{2}) \: {\Gamma}(\frac{1-a+b}{2}) } \ \mbox{,}$$
Following the same scheme, let $ b \rightarrow b+2\:n $. Formula (\[e:baileyth\]) can be written: $$\label{e:bai1}
\sum_k F(n,k) = 1 \ \mbox{,}$$ where $ \displaystyle F(n,k) = \frac{f(n,k)}{S(n)} $,\
with $ \displaystyle
f(n,k) = \sum_k \frac{(a)_k \: (1-a)_k}{(b+2\:n)_k}
\: \left(\frac{1}{2}\right)^k \quad \mbox{and} \quad
S(n) = \frac{{\Gamma}(\frac{b}{2}+n) \: {\Gamma}(\frac{1}{2}+\frac{b}{2}+n)}
{{\Gamma}(\frac{a+b}{2}+n) \: {\Gamma}(\frac{1-a+b}{2}+n)} \ \mbox{.} $\
Zeilberger’s algorithm applied to $F(n,k)$ succeeds to finding a WZ-pair: $$\label{e:baileyWZ}
F(n,k) - F(n+1,k) = G(n,k+1) - G(n,k) \ \mbox{,}$$ where $ \displaystyle G(n,k) = F(n,k) \: \left(-2\:\frac{k}{b+2\:n+k}\right) $.\
As with Kummer’s theorem, we have: $$\begin{aligned}
G(n,0) & = & 0 \ \mbox{,} \\
\lim_{k \rightarrow +\infty} G(n,k) & = & 0 \ \mbox{,}\end{aligned}$$ and so, $ \displaystyle \sum_k F(n,k) $ is independent of $n$.\
In this case, $ \displaystyle \lim_{n \rightarrow +\infty} F(n,k)
= \delta_{k,0} $, so that: $$\sum_{k=0}^{\infty} F(n,k)
= \lim_{n \rightarrow +\infty} \sum_{k=0}^{\infty} F(n,k)
= \sum_{k=0}^{\infty} \lim_{n \rightarrow +\infty} F(n,k)
= \sum_{k=0}^{\infty} \delta_{k,0} = 1 \ \mbox{,}$$ which proves Bailey’s theorem.
Dixon’s theorem
---------------
$$\begin{aligned}
\label{e:dixonth}
& & {}_3F_2\left[{a,b,c \atop 1+a-b,1+a-c};1\right] \nonumber \\
& = & \frac{ {\Gamma}(1+a-b) \: {\Gamma}(1+a-c) \: {\Gamma}(1+\frac{a}{2})
\: {\Gamma}(1+\frac{a}{2}-b-c) }
{ {\Gamma}(1+a) \: {\Gamma}(1+\frac{a}{2}-b)
\: {\Gamma}(1+\frac{a}{2}-c) \: {\Gamma}(1+a-b-c) } \ \mbox{,}\end{aligned}$$
where $ Re(2+a-2\:b-2\:c) > 0 $.\
Once again, we replace $ a $ by $a+2\:n$. With the same notation (\[e:bai1\]), we now have: $$\begin{aligned}
f(n,k) & = & \sum_k \frac{(a+2n)_k \: (b)_k \: (c)_k}
{(1+a+2n-b)_k \: (1+a+2n-c)_k \: k!} \ \mbox{,} \\
\mbox{and} \quad
S(n) & = & \frac{{\Gamma}(1+a+2n-b) {\Gamma}(1+a+2n-c)
{\Gamma}(1+\frac{a}{2}+n) {\Gamma}(1+\frac{a}{2}+n-b-c)}
{{\Gamma}(1+a+2n) {\Gamma}(1+\frac{a}{2}+n-b)
{\Gamma}(1+\frac{a}{2}+n-c) {\Gamma}(1+a+2n-b-c)} \ \mbox{.}\end{aligned}$$ The WZ-pair given by Zeilberger’s algorithm is $$\label{e:dixonWZ}
F(n,k) - F(n+1,k) = G(n,k+1) - G(n,k) \ \mbox{,}$$ with $ \displaystyle G(n,k) = F(n,k) \: C(n,k) $, where the computed certificat $C(n,k)$ is:
-(-2-8*a*n*c-a+4*b+4*c-2*n-2*n*b*c-2*a*b*k-2*a*k*c-a*b*c-8*a*n*b
-4*n*k*c-4*n*b*k+a*b+2*n*b-2*a^2*c-8*n^2*c+a*c+2*n*c-2*b^2+3*a^2
+12*a*n+12*n^2+12*a^2*n-2*a^2*b+24*a*n^2+a*k^2-8*n^2*b+2*n*k^2
+2*a^3+16*n^3-6*c*b+3*k*a^2+3*k*a+6*k*n+12*k*n^2+12*k*a*n+2*b^2*c
+2*b*c^2-2*c^2)*k/(2+a+2*n-2*b-2*c)/(1+a+2*n-c+k)/(1+a+2*n-b+k)
/(a+2*n)
It is easy to establish the following limits: $$\begin{aligned}
G(n,0) & = & 0 \ \mbox{,} \\
\lim_{k \rightarrow +\infty} G(n,k) & = &
\lim_{k \rightarrow +\infty} k^{2\:b+2\:c-2\:n-4-a} \: T(n) \\
& = & 0 \ \mbox{.}\end{aligned}$$ Again, $ \displaystyle \sum_k F(n,k) $ is independent of $n$.\
Since $ \displaystyle \lim_{n \rightarrow +\infty} F(n,k)
= \delta_{k,0} $, we conclude again that: $$\sum_{k=0}^{\infty} F(n,k)
= \lim_{n \rightarrow +\infty} \sum_{k=0}^{\infty} F(n,k)
= \sum_{k=0}^{\infty} \lim_{n \rightarrow +\infty} F(n,k)
= \sum_{k=0}^{\infty} \delta_{k,0} = 1 \ \mbox{.}$$
A list of theorems
------------------
These formulas can automatically be proved using the `HYPERG` package:
- Gauss’s theorem [@b-slater (III.3)]: $${}_2F_1\left[{a,b \atop c};1\right]
= \frac{ {\Gamma}(c) \: {\Gamma}(c-a-b) }
{ {\Gamma}(c-a) \: {\Gamma}(-b+c) } \ \mbox{.}$$
- another Dixon’s theorem [@b-slater (III.10)]: $${}_4F_3\left[{a,1+\frac{a}{2},b,c \atop \frac{a}{2},1+a-b,1+a-c};
-1\right]
=
\frac{{\Gamma}(1+a-b) \: {\Gamma}(1+a-c)}
{{\Gamma}(1+a) \: {\Gamma}(1+a-b-c)} \ \mbox{.}$$
- another Dixon’s theorem [@b-slater (III.12)]: $$\begin{aligned}
& & {}_5F_4\left[{a,\frac{a}{2}+1,b,c,d \atop \frac{a}{2},1+a-b,1+a-c,1+a-d};
1\right]
\\
& = &
\frac{{\Gamma}(1+a-b) \: {\Gamma}(1+a-c) \: {\Gamma}(1+a-d) \: {\Gamma}(1+a-b-c-d)}
{{\Gamma}(a+1) \: {\Gamma}(1+a-b-c) \: {\Gamma}(1+a-b-d) \: {\Gamma}(1+a-c-d)} \ \mbox{.}
\end{aligned}$$
[widest-label]{}
B. Gauthier, HYPERG, Maple package, User’s reference manual, Version 1.0, (182 pages), Electronic version available at `http://www-igm.univ-mlv.fr/’176 gauthier`, (1998). R. W. Gosper, Jr, Decision procedure for indefinite hypergeometric summation, [*Proceedings of the National Academy of Sciences of the United States of America*]{} [**75**]{} (1978) 40–42. T. H. Koornwinder, Identities of nonterminating series by Zeilberger’s algorithm, [*J. Comput. Appl. Math.*]{} [**99**]{} (1998) 449–461. C. Krattenthaler, HYP, Manual for a Mathematica package for handling hypergeometric series, 160 pages, (1994). P. Paule, M. Schorn, A Mathematica version of Zeilberger’s algorithm for proving binomial coefficient identities, RISC-Linz Report Series, [**93-36**]{} (1993), J. Kepler University of Linz. M. Petkovšek, H. S. Wilf, D. Zeilberger, [*A = B*]{} (A K Peters, Wellesley, Massachusetts, 1996). L. J. Slater, [*Generalized hypergeometric functions*]{}, (Cambridge University Press, Cambridge, 1966). D. Zeilberger, A fast algorithm for proving terminating hypergeometric identities, [*Discrete Math.*]{} [**80**]{} (1990) 207–211. D. Zeilberger, Three recitations on holonomic systems and hypergeometric series, [*Séminaire Lotharingien de Combinatoire*]{} [**B24a**]{} (1990). D. Zeilberger, The method of creative telescoping, [*J. Symbol. Comp.*]{} [**11**]{} (1991) 195–204.
[^1]: `[email protected]`, *Institut Gaspard Monge*, Université Marne-la-Vallée, Cité Descartes, 5 bd Descartes, Champs sur marne, F-77454 Marne-la-Vallée Cedex 2.
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abstract: 'Using the IRAM 30 m telescope, a mapping survey in optically thick and thin lines was performed towards 46 high mass star-forming regions. The sample includes UC H[ii]{} precursors and UC H[ii]{} regions. Seventeen sources are found to show “blue profiles”, the expected signature of collapsing cores. The excess of sources with blue over red profiles (\[$N_{\rm blue}$ – $N_{\rm red}$\]/$N_{\rm total}$) is 29% in the HCO$^+$ $J$=1–0 line, with a probability of 0.6% that this is caused by random fluctuations. UC H[ii]{} regions show a higher excess (58%) than UC H[ii]{} precursors (17%), indicating that material is still accreted after the onset of the UC H[ii]{} phase. Similar differences in the excess of blue profiles as a function of evolutionary state are not observed in low mass star-forming regions. Thus, if confirmed for high mass star-forming sites, this would point at a fundamental difference between low- and high-mass star formation. Possible explanations are inadequate thermalization, stronger influence of outflows in massive early cores, larger gas reserves around massive stellar objects or different trigger mechanisms between low- and high- mass star formation.'
author:
- 'Yuefang Wu, Christian Henkel, Rui Xue, Xin Guan, Martin Miller'
title: |
Signatures of inflow motion in cores of massive star formation:\
Potential collapse candidates
---
Introduction
============
Inflow motion is a fundamental phenomenon during stellar formation. Although the search for inflow is usually more difficult than that for outflow, studies of inflow have made great progress since the 1990s. In low-mass star forming regions, inflow motions were detected at different evolutionary stages, including Class –I, Class 0 and Class I cores [@zho93; @mmt97; @lee99; @gre00; @evans03]. Recently, a number of inflow candidates were found in high mass star formation regions. Among a sample of 28 massive cores, 12 were found to show line profiles that peak at blue-shifted velocities (hereafter “blue profiles”; see Sect.3.1), the expected signature of inflow [@we03]. @fws05 (hereafter FWS05) detected such asymmetric profiles in 22 cores within a sample of 77 high-mass proto-stellar objects (HMPOs). Most recently, @wyr06 detected 9 sources with a blue profile in a sample of 12 ultracompact (UC) H[ii]{} regions.
Variation of inflow motion with time is critical for high mass star formation. It has been indicated that when a protostar reaches $>$10 M$_{\odot}$ it can generate enough radiation pressure to halt spherical infall and inhibit its mass increasing[@wc87]. Observationally, however, it is not yet clear how inflow is related to the evolution of massive (proto)stars. To study this problem, we have carried out a survey for a sample including both cores of UC H[ii]{} regions and precursors of UC H[ii]{} regions.
While previous surveys using single point observations provided some statistical evidence for the occurrence of infall within massive cores, blue profiles can also be caused by rotation. Therefore maps of the molecular environment are indispensable. Mapping also allows us to locate the center of the inflow and to identify cores that are simultaneously showing evidence for in- and outflow.
Therefore, we conducted a mapping survey including 46 high mass star-forming regions which were selected applying three criteria: (1) The sources must have been mapped in the submillimeter or millimeter wavelengths with continuum or spectroscopy; (2) signal-to-noise ratios should be $>$5 at 350$\mu$m (Mueller et al. 2002) and higher at other wavelengths; (3) there should be no other core within one arcmin [@zhs97; @hnb98; @tie98; @hat00; @mbc00; @bsp02; @mse02]. With respect to their stellar content, we can divide the sample into two different groups of targets: (1) Thirty three sources lack 6 cm continuum emission and are precursors of UC H[ii]{} regions or HMPOs [@mbc00; @bsp02]. Among these, thirty are hosting a luminous IRAS source. The remaining three are associated with IRAC (the InfraRed Array Camera on the Spitzer Space Telescope) point sources (W3-W and W3-SE) or are not hosting an IRAC source (18454–3). All 33 cores comprise ‘group I’. (2) Thirteen UC H[ii]{} regions are assigned to ‘group II’. This letter presents a list of the identified collapse candidates and provides the statistics of blue excesses. Detailed properties of individual cores will be analyzed in a future paper.
Observations
============
The observations were performed with the IRAM 30m telescope at Pico Veleta, Spain, from July 28 to Aug. 1, 2005. Four receivers were used simultaneously, usually two at $\lambda$$\sim$3mm and two at $\lambda$$\sim$1.3mm (A/B configuration). For some sources, none of the four 3 and 1.3mm lines were optically thin. In these cases the tracer lines were changed employing two receivers at $\lambda$$\sim$2mm and the other two to cover the upper part of the 1.3mm window (C/D configuration). The lines and corresponding beam sizes, efficiencies, and channel widths are given in Table 1. The channel spacing and the bandwidth are 78.125 kHz and 105 MHz respectively. The weather was extremely good for summer conditions, allowing us to observe the HCO$^+$ $J$=3–2 transition at 268GHz and leading to 3 and 1.3mm (2 and 1.2mm) system temperatures of order 150 and 400K (200 and 550K) on a $T_{\rm A}^*$ scale. Pointing and calibration were checked by continuum measurements of the standard sources W3(OH), G34.24, and NGC 7027 and were found to be better than 4 and $\pm$20%, respectively. All observations were carried out in a position switching mode. For each source we observed a nine point map in a cross pattern with a spacing of 15. If inflow signature was detected, the map was enlarged in most cases to cover the entire region showing this signature. The on-source integration time per position was 1 minute, yielding a $T_{\rm A}$\* 1$\sigma$ noise level of 0.07K for the 3mm N$_2$H$^+$ (1–0) line. For the data analysis, the GILDAS software package (CLASS/GREG) was used [@gl00].
Results and discussion
======================
Blue profile identification
---------------------------
For self-absorbed optically thick lines, the classical signature of inflow is a double peaked profile with the blue-shifted peak being stronger, or a line asymmetry with the peak skewed to the blue side. While optically thin lines should show a single velocity component peaking at the line center.
Among the 46 cores observed, five (05490+2658, G31.41+0.31, 18454-3, 18454-4, 19266+1745) will be ignored because they show either too complex spectral profiles, inhibiting a detailed analysis, or a lack of optically thin lines. Estimates of optical depths were obtained from line ratios between different isotopomers of CO and CS and from the relative intensities of individual hyperfine components in the case of C$^{17}$O and N$_2$H$^+$. C$^{18}$O, C$^{17}$O, C$^{34}$S and N$_2$H$^+$ tend to be optically thin, while CS is optically thick. HCO$^+$ opacities could not be estimated. However, the similarity of HCO$^+$ and CS line shapes (see Sect.3.2) as well as the results of [@gre00] and FWS05 clearly indicate that HCO$^+$ is also optically thick.
The 41 remaining sources were detected in at least one optically thick and one optically thin line. A blue profile caused by inflow motion with velocity $v \propto r^{-1/2}$ in a region with higher excitation temperature ($T_{ex}$) inside requires $T_{\rm
A}$\*(B)/$T_{\rm A}$\*(R) $>$ 1. Here $r$ is the radius of the collapsing core [@zho93]. $T_{\rm A}$\*(B) and $T_{\rm A}$\*(R) are the blue and red peak intensities of the optically thick line. We also define a dimensionless asymmetry parameter following [@mmt97], $\delta V$ = ($V_{\rm thick}$-$V_{\rm
thin}$)/$\Delta$$V_{\rm thin}$. $V_{\rm thick}$ is the peak velocity of the opaque line, $V_{\rm thin}$ and $\Delta$$V_{\rm thin}$ denote the peak velocity and width of the optically thin line. Only for $\delta V < -0.25$ or $> 0.25$ the line profile is rated blue or red, respectively.
Our sources (Table 2) discriminate among five main types of line shapes: (1) cores with lines showing a “blue profile” (in the following denoted with B); (2) cores with lines showing a “red profile” (R); (3) cores exhibiting blue and red profiles at different spatial positions (BRS); (4) cores where some lines show a blue profile, while others display a red profile (BRL); (5) cores without obvious asymmetric lines (S). Only cores showing at least one line of type B, but no lines of type R are identified as targets potentially undergoing inflow motion.
Collapse candidates and their profile “excess” {#bozomath}
----------------------------------------------
With the criteria outlined in Sect.3.1, seventeen inflow candidates are identified (see Table 2). Ten belong to group I and seven are part of group II. To provide a typical example, Fig.1 shows the infall signature of the group I core W3-SE. Fig. 1a displays the HCO$^+$(1–0) spectra, showing the angular size of the core. Fig. 1b shows a number of profiles towards the central position. The HCO$^+$(1–0) and (3–2) lines as well as the CS(3–2) transition show the blue asymmetry. For the HCO$^+$(1–0) line this is also demonstrated in the position-velocity (P-V) diagram of Fig.1c. For comparison, Fig.1d shows a P-V diagram of the optically thin C$^{18}$O(1–0) emission.
The quantity “excess” as defined by [@mmt97] is $E$ = ($N_{\rm
B} - N_{\rm R}$)/$N_{\rm T}$, where $N_{\rm B}$ and $N_{\rm R}$ mark the numbers of sources with blue and red profiles. $N_{\rm T}$ is the total number of sources. For our survey the excess was calculated for the two HCO$^+$ transitions and the CS(3–2) line. Fig. 2 shows the log\[$T_{\rm A}$\*(B)/$T_{\rm A}$\*(R)\] and $\delta$V (see Sect.3.1) distributions of the three individual lines. Statistical results are given in Table 3. The observed excess derived from the HCO$^+$(1–0) and (3–2) lines is 0.29 and 0.11, respectively. Both are larger than those obtained by FWS05 for the same lines (0.15 and 0.04). For the CS transition we obtain 0.29. To evaluate the statistical significance of the determined values, we conducted the binomial test (see FWS05 and references therein). Probabilities that the excesses are a product of a random distribution are given in the last column of Table 3. These are 0.006 and 0.01 for HCO$^+$(1–0) and CS (3–2) respectively. Apparently, both lines are sensitive tracers of potential inflow motion in massive cores.
To evaluate differences between the two classes of cores (I and II; see Sect.1) with respect to the excess, we used the HCO$^+$(1–0) line, which was mapped in the largest number of sources. The results listed in the lower part of Table 3 include 16 sources with profiles of type B. The excesses observed for group I and II are 0.17 and 0.58, respectively.
Twenty of our 46 sources overlap with those of FWS05. Among them are 19 group I sources (out of 33), but only one source is from group II (out of 13). Our study includes various CO and CS lines. We also made maps. Thus we can view the common objects from a different perspective and can check, how far the choice of different molecular transitions and the presence of maps is leading to contradictions with previously published results. Differences are indeed significant. For eight of the 19 overlapping type I cores we obtain different line asymmetry classification, emphasizing the need for detailed maps. Nevertheless, the overall difference in the HCO$^+$(1–0) excess is negligible (0.17 versus 0.15).
To summarize, both data sets indicate that the HCO$^+$(1–0) excess is low for UC H[ii]{} precursors. For UC H[ii]{} regions, our results and those of [@wyr06] suggest that the excess is larger and more significant. From the binomial test for group I and II, the probability that the blue excesses (0.17 and 0.58 respectively) are arising by chance is 0.13 and 0.008, respectively.
A comparison with low mass star-forming surveys
-----------------------------------------------
While low mass star-forming regions show infall from the Class –I to the Class I stages of evolution, high mass star-forming regions also exhibit infall signatures from their earliest stages till a UC H[ii]{} region has formed [@wzx05; @bkl06; @qzm06]. In low mass cores the profile excess was found to be 0.30, 0.31 and 0.31 for Class –I, 0 and I core samples in the HCO$^+$(3–2) line ([@evans03] and references therein). There seem to be no significant differences among the cores in different evolutionary phases. However, our samples show the excess of UC H[ii]{} regions far surpassing that of the UC H[ii]{} precursors. This may point to fundamental differences between low and high mass star-forming conditions. Possible causes to the higher blue excess in Group II sources may be: (1) The molecular gas surrounding UC H[ii]{} regions may be more adequately thermalized to show the blue excess, i.e. the excitation temperature of specific lines may increase more monotonically towards the center. Thus all lines may produce blue profiles indicating infall motion, while in younger cores still some lines may show red profiles. (2) The amount of dense cool gas is larger towards younger objects. Outflows of dense molecular gas may be more active around Group I objects, shaping more red profiles. (3) Low mass cores are relatively isolated and their gas supply is limited. Simulations showed that this may halt the increase of inflow [@vor05]. However, high mass stars form in giant molecular clouds and their inflow motions are not easily halted by the exhaustion of molecular gas before most of it is dispelled. (4) In low mass cores, star formation may be spontaneous. In high mass cores, collapse may be trigged by extrinsic disturbances and the collapse may take more time to develop.
With respect to potential selection effects, we used the same criteria to identify the targets of the two separate groups of sources. Since this study is based on a limited number of sources, more data quantifying the blue excess as a function of evolutionary stage would be highly desirable.
Outlook
=======
We have carried out a mapping survey towards 46 molecular cores associated with massive star formation. Seventeen collapse candidates were identified. Among them are 10 UC H[ii]{} precursors and 7 UC H[ii]{} regions. Overall, statistical results indicate a predominance of blue over red profiles which is surprisingly similar to that obtained towards cores forming low mass stars. Among high mass star-forming sites, the probability to detect blue profiles seems to depend on evolutionary stage and increases from UC H[ii]{} precursors to UC H[ii]{} regions. Toward low mass star-forming sites, however, this effect is not observed, suggesting a more fundamental difference in the way stellar masses are assembled. Larger line surveys and more detailed maps in various molecular transitions are needed to improve statistical evidence in order to confirm or to reject this potentially important finding.
We are grateful to the IRAM staff for their assistance and F. Wyrowski for useful discussions. This research is supported by the Grant 10128306 and 10733030 of NSFC.
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[lllllll]{}
No. & Line & Frequency & HPBW & $\eta_{\rm mb}$ & $\Delta V_{\rm res}$\
& & (GHz) & (arcsec) & & (kms$^{-1}$)\
1 & $HCO^{+} (1-0)$ & 89.18852 & 27.6 & 0.77 & 0.263\
2 & $HCO^{+} (3-2)$ & 267.55763 & 9.2 & 0.45 & 0.112\
3 & $CS (3-2)$ & 146.96905 & 16.7 & 0.69 & 0.159\
4 & $CS (5-4)$ & 244.93561 & 10.0 & 0.49 & 0.122\
5 & $N_{2}H^{+} (1-0)$ & 93.17378 & 26.4 & 0.77 & 0.251\
6 & $C^{18}O (1-0)$ & 109.78218 & 22.4 & 0.75 & 0.213\
7 & $C^{18}O (2-1)$ & 219.56033 & 11.2 & 0.55 & 0.137\
8 & $C^{17}O (1-0)$ & 112.35928 & 21.9 & 0.74 & 0.209\
9 & $C^{17}O (2-1)$ & 224.71437 & 10.9 & 0.54 & 0.133\
10 & $C^{34}S (5-4)$ & 241.01618 & 10.2 & 0.50 & 0.122\
[lcccrc|lcccrc]{} Source & -0.2truecm $\alpha$$^b$ & -0.2truecm $\delta$$^b$ & -0.3truecm $D^{c}$ & -0.3truecm & -0.3truecm & Source & -0.2truecm $\alpha$$^b$ & -0.2truecm $\delta$$^b$ & -0.3truecm $D^{c}$ & -0.3truecm & -0.3truecm\
Name$^{a}$ & -0.2truecm J2000 & -0.2truecm J2000 & -0.3truecm (kpc) & -0.3truecm Profile$^{d}$ & -0.3truecm Ref. & Name$^{a}$ & -0.2truecm J2000 & -0.2truecm J2000 & -0.3truecm (kpc) & -0.3truecm Profile$^{d}$ & -0.3truecm Ref.\
W3-W$^I$ & -0.2truecm $02~25~32.4$ & -0.2truecm $+62~06~01$ & -0.2truecm $1.95$ & -0.3truecm $B$ & -0.2truecm 1 & 18488+0000SE$^I$ & -0.2truecm $18~51~25.6$ & -0.2truecm $+00~04~07$ & -0.2truecm $5.4$ & -0.3truecm $BRL$ & -0.2truecm 3,7\
W3-C$^{II}$ & -0.2truecm $02~25~39.5$ & -0.2truecm $+62~05~51$ & -0.2truecm $2.3$ & -0.3truecm $BRS$ & -0.2truecm 1 & G34.26+0.15$^{II}$ & -0.2truecm $18~53~18.4$ & -0.2truecm $+01~14~56$ & -0.2truecm $3.7$ & -0.3truecm $In,B$ & -0.2truecm 6\
W3-SE$^I$ & -0.2truecm $02~25~54.5$ & -0.2truecm $+62~04~11$ & -0.2truecm $2.3$ & -0.3truecm B & -0.2truecm 1 & 18521+0134$^I$ & -0.2truecm $18~54~40.8$ & -0.2truecm $+01~38~02$ & -0.2truecm $5.0$ & -0.3truecm $B$ & -0.2truecm 3,7\
05358+3543$^I$ & -0.2truecm $05~39~10.4$ & -0.2truecm $+35~45~19$ & -0.2truecm $1.8$ & -0.3truecm $BRS$ & -0.2truecm 3,7 & 18530+0215$^I$ & -0.2truecm $18~55~34.2$ & -0.2truecm $+02~19~08$ & -0.2truecm $5.1$ & -0.3truecm $S$ & -0.2truecm 3,7\
05490+2658$^I$ & -0.2truecm $05~52~12.9$ & -0.2truecm $+26~59~33$ & -0.2truecm $2.1$ & -0.3truecm $ ... $ & -0.2truecm 3,7 & S76E$^{II}$ & -0.2truecm $18~56~11.0$ & -0.2truecm $+07~53~28$ & -0.2truecm $2.1$ & -0.3truecm $S?$ & -0.2truecm 4,9\
G10.47+0.03$^{II}$ & -0.2truecm $18~08~38.2$ & -0.2truecm $-19~51~50$ & -0.2truecm $5.8$ & -0.3truecm $B$ & -0.2truecm 10 & 18553+0414NE$^I$ & -0.2truecm $18~57~53.4$ & -0.2truecm $+04~18~15$ & -0.2truecm $0.6$ & -0.3truecm $B$ & -0.2truecm 3,7\
G12.42+0.50$^{II}$ & -0.2truecm $18~10~51.8$ & -0.2truecm $-17~55~56$ & -0.2truecm $2.1$ & -0.3truecm $B$ & -0.2truecm 2 & 19012+0536$^I$ & -0.2truecm $19~03~45.1$ & -0.2truecm $+05~40~40$ & -0.2truecm $4.6$ & -0.3truecm $B$ & -0.2truecm 3,7\
G12.89+0.49$^{II}$ & -0.2truecm $18~11~51.3$ & -0.2truecm $-17~31~29$ & -0.2truecm $3.5$ & -0.3truecm $BRL$ & -0.2truecm 2,3,9 & 19092+0841SW$^I$ & -0.2truecm $19~11~36.7$ & -0.2truecm $+08~46~20$ & -0.2truecm $4.48$ & -0.3truecm $BRL$ & -0.2truecm 5\
G13.87+0.28$^{II}$ & -0.2truecm $18~14~35.4$ & -0.2truecm $-16~45~37$ & -0.2truecm $4.4$ & -0.3truecm $S?$ & -0.2truecm 10 & G43.89-0.79$^{II}$ & -0.2truecm $19~14~26.2$ & -0.2truecm $+09~22~34$ & -0.2truecm $4.2$ & -0.3truecm $B$ & -0.2truecm 10\
18144-1723NW$^I$ & -0.2truecm $18~17~23.8$ & -0.2truecm $-17~22~09$ & -0.2truecm $4.33$ & -0.3truecm $R$ & -0.2truecm 5 & 19217+1651N$^I$ & -0.2truecm $19~23~58.8$ & -0.2truecm $+16~57~45$ & -0.2truecm $10.5$ & -0.3truecm $B$ & -0.2truecm 3,7\
18182-1433$^I$ & -0.2truecm $18~21~07.9$ & -0.2truecm $-14~31~53$ & -0.2truecm $4.5$ & -0.3truecm $B$ & -0.2truecm 3,7 & 19266+1745$^I$ & -0.2truecm $19~28~54.0$ & -0.2truecm $+17~51~56$ & -0.2truecm $0.3$ & -0.3truecm $ ... $ & -0.2truecm 3,7\
G19.61$^{II}$ & -0.2truecm $18~27~37.9$ & -0.2truecm $-11~56~07$ & -0.2truecm $4.0$ & -0.3truecm $S?$ & -0.2truecm 2 & 19410+2336$^I$ & -0.2truecm $19~43~11.4$ & -0.2truecm $+23~44~06$ & -0.2truecm $2.1$ & -0.3truecm $B$ & -0.2truecm 3,7\
18264-1152$^I$ & -0.2truecm $18~29~14.3$ & -0.2truecm $-11~50~26$ & -0.2truecm $3.5$ & -0.3truecm $R$ & -0.2truecm 3,7 & S86SE$^I$ & -0.2truecm $19~43~49.7$ & -0.2truecm $+23~28~41$ & -0.2truecm $1.9$ & -0.3truecm $BRS$ & -0.2truecm 4\
18306-0835$^I$ & -0.2truecm $18~33~21.8$ & -0.2truecm $-08~33~38$ & -0.2truecm $4.9$ & -0.3truecm $R$ & -0.2truecm 3,7 & S87N$^I$ & -0.2truecm $19~46~20.6$ & -0.2truecm $+24~36~04$ & -0.2truecm $2.3$ & -0.3truecm $S?$ & -0.2truecm 4\
G24.49-0.04$^{II}$ & -0.2truecm $18~36~05.3$ & -0.2truecm $-07~31~23$ & -0.2truecm $3.5$ & -0.3truecm $B$ & -0.2truecm 2,11 & 20126+4104$^I$ & -0.2truecm $20~14~26.0$ & -0.2truecm $+41~13~32$ & -0.2truecm $1.7$ & -0.3truecm $S?$ & -0.2truecm 3,7\
18337-0743NE$^I$ & -0.2truecm $18~36~40.9$ & -0.2truecm $-07~39~20$ & -0.2truecm $4.0$ & -0.3truecm $BRL$ & -0.2truecm 3 & 20216+4107$^I$ & -0.2truecm $20~23~23.8$ & -0.2truecm $+41~17~40$ & -0.2truecm $1.7$ & -0.3truecm $S$ & -0.2truecm 3.7\
18355-0650$^{II,*}$ & -0.2truecm $18~38~14.2$ & -0.2truecm $-06~47~47$ & -0.2truecm $4.2$ & -0.3truecm $B$ & -0.2truecm 8 & 20319+3958$^I$ & -0.2truecm $20~33~49.3$ & -0.2truecm $+40~08~45$ & -0.2truecm $1.6$ & -0.3truecm $S$ & -0.2truecm 3,7\
18372-0541$^I$ & -0.2truecm $18~39~56.0$ & -0.2truecm $-05~38~49$ & -0.2truecm $1.8$ & -0.3truecm $B$ & -0.2truecm 3,7 & 22134+5834$^I$ & -0.2truecm $22~15~09.1$ & -0.2truecm $+58~49~09$ & -0.2truecm $2.6$ & -0.3truecm $BRS$ & -0.2truecm 3,7\
18385-0512E$^I$ & -0.2truecm $18~41~13.3$ & -0.2truecm $-05~09~06$ & -0.2truecm $2.0$ & -0.3truecm $R$ & -0.2truecm 3,7 & 23033+5951$^I$ & -0.2truecm $23~05~25.7$ & -0.2truecm $+60~08~08$ & -0.2truecm $3.5$ & -0.3truecm $S?$ & -0.2truecm 3,7\
G31.41+0.31$^{II}$ & -0.2truecm $18~47~34.7$ & -0.2truecm $-01~12~46$ & -0.2truecm $7.9$ & -0.3truecm $ ... $ & -0.2truecm 10 & NGC7538-11$^I$ & -0.2truecm $23~13~44.7$ & -0.2truecm $+61~26~54$ & -0.2truecm $2.8$ & -0.3truecm $B$ & -0.2truecm 2\
18454-3$^I$ & -0.2truecm $18~47~55.9$ & -0.2truecm $-01~53~35$ & -0.2truecm $5.6$ & -0.3truecm $ ... $ & -0.2truecm 3,7 & NGC7538-N$^{II}$ & -0.2truecm $23~13~45.4$ & -0.2truecm $+61~28~12$ & -0.2truecm $2.8$ & -0.3truecm $B$ & -0.2truecm 2\
18454-4$^I$ & -0.2truecm $18~48~01.4$ & -0.2truecm $-01~52~37$ & -0.2truecm $5.6$ & -0.3truecm $ ... $ & -0.2truecm 3,7 & 23139+5939$^I$ & -0.2truecm $23~16~09.3$ & -0.2truecm $+59~55~23$ & -0.2truecm $4.8$ & -0.3truecm $BRS$ & -0.2truecm 3,7\
18470-0044$^I$ & -0.2truecm $18~49~36.7$ & -0.2truecm $+00~41~05$ & -0.2truecm $8.2$ & -0.3truecm $R$ & -0.2truecm 3,7 & 23151+5912$^I$ & -0.2truecm $23~17~21.0$ & -0.2truecm$+59~28~49$ & -0.2truecm $5.7$ & -0.3truecm $BRS?$ & -0.2truecm 3,7\
[llllll]{}
Line/Source & $N_{\rm B}$ & $N_{\rm R}$ & $N_{\rm T}$ & $E$ & $p$\
\
HCO$^+$ (1–0) & 16 & 4 & 41 & 0.29 & 0.006\
HCO$^+$ (3–2) & 8 & 5 & 28 & 0.11 & 0.29\
CS (3–2) & 9 & 1 & 28 & $0.29$ & 0.01\
\
Group I & 9 & 4 & 29 & 0.17 & 0.13\
Group II & 7 & 0 & 12 & 0.58 & 0.008\
Total & 16 & 4 & 41 & 0.29 & 0.006\
|
---
abstract: 'Polymorphism in programming languages enables code reuse. Here, we show that polymorphism has broad applicability far beyond computations for technical computing: **parallelism** in distributed computing, **presentation** of visualizations of runtime data flow, and **proofs** for formal verification of correctness. The ability to reuse a single codebase for all these purposes provides new ways to understand and verify parallel programs.'
author:
- |
Jiahao Chen\
\
\
\
\
\
Alan Edelman\
\
\
\
\
bibliography:
- 'prefix.bib'
date: 15 October 2014
title: |
Parallel Prefix Polymorphism Permits\
Parallelization, Presentation & Proof
---
Introduction
============
Abstractions are the lifeblood of computer science [@Abelson1996]. Separating higher level concepts from lower level implementation details allows code to be more composable and reusable, and code using appropriate abstractions is simpler to understand and maintain. However, the desire for high level abstractions often seems inimical to writing high-performance code; oftentimes performance can only result from code that takes advantage of low-level implementation details and hence break abstractions. Thus areas like technical computing and parallel programming, where performance is a key consideration, exhibit a general dearth of abstraction and suffer the consequent complexities in large, inscrutable and nearly unmaintainable codebases. Such challenges are particularly acute for parallel programs, which are written primarily for performance reasons. However, parallel programs are notoriously prone to subtle, nondeterministic bugs arising from race conditions, and they are difficult to verify for correctness.
Programs for technical computing tend to sacrifice abstraction for performance, but pay the cost in expressiveness. In contrast, the mathematical structure of technical computations is inherently amenable to abstraction, particularly polymorphism. Just think of the myriad ways different mathematical objects can be multiplied together: one can multiply two numbers together, or two matrices together, or a number and a matrix. The notion of multiplication itself can also be extended to dot products, cross products, tensor products, wedge products, outer products, and smash products, to just name a few. All these operators share a common metaphor and it is natural to implement these products with polymorphic code that can express the mathematical commonality.
In this paper, we argue that high level languages like Julia provide a suitable framework for expressing high level abstractions that allow for extremely powerful code reuse, while affording a reasonable level of performance. In particular, we explore how Julia’s generic function system and type system are suitable for expressing polymorphism at the operator level, and show how polymorphism can be used not just to encompass different kinds of computations, but can be applied to parallelize code, enable visualizations of code implementations, and even verify correctness of code.
Multimethods in Julia
---------------------
In this paper, we use the Julia language[^1], a very high level dynamic language designed specifically for technical computing [@Bezanson2012]. Julia offers language constructs that support different mechanisms of polymorphism, which programming language theorists call universal polymorphism and *ad hoc* polymorphism, or overloading [@Strachey2000]. In this paper, we focus only on Julia’s overloading mechanism provided by multimethods.
*Ad hoc* polymorphism, or overloading, is a language construct that naturally expresses the polymorphism inherent in the mathematical structure of technical computing [@Bezanson2014]. Consider the [`*`]{} operator which represents multiplication: the product can be taken between two integers, two floating-point numbers, a scalar and a vector, or a matrix and a matrix, just to list a few typical possibilities. All these different semantics can be represented by with the same syntax, namely an expression of the form [`a*b`]{}. All programming languages resolve the ambiguity by considering the types of the arguments [`a`]{} and [`b`]{}, which is formally equivalent to specifying the domains of the operands [@Scott1976].
In practice, programming languages vary greatly in how they allow users to reason about types. Some languages, like Julia, offer the ability to define multimethods, where a single generic function like [`*`]{} can be defined with more than one method, each with a different type signature: in Julia notation, [`*(a::Number, b::Number)`]{} defines a method for scalar multiplication, whereas [`*(a::Matrix, b::Vector)`]{} defines a method for matrix-vector products, and so on. Closely related is the support for multiple dispatch, where the method chosen to match an expression like [`a*b`]{} can depend on the type of more than one argument. In this respect, multimethods differs greatly from more conventional languages that provide class-based objects; dispatch only occurs on the first argument, which is the type of the class.
In this paper, we demonstrate how multimethod polymorphism is far more general than just dispatching on computational kernels. We study one specific algorithm, namely scan, and show how the same exact code written in Julia for serial computation can be composed with appropriately overloaded operators to generate parallel code, visualizations, and also proofs of correctness.
The scan algorithm {#sec:prefix}
------------------
The basic problem of interest is to compute from some initial data [`y`]{} the partial sums [`z`]{} such that:
z[1] = y[1]
z[2] = y[1] + y[2]
z[3] = y[1] + y[2] + y[3]
...
One way to compute this sum efficiently is to note the prefix property, i.e.that the $k$th partial sum depends only on the $(k-1)$th partial sum and the $k$th element:
z[1] = y[1]
z[2] = z[1] + y[2]
z[3] = z[2] + y[3]
...
which leads to the simple algorithm:
function prefix_serial!(y, +)
for i=2:length(y)
y[i] = y[i-1] + y[i]
end
y
end
The cumulative sum problem generalizes to any associative operator; in this Julia function, the argument [`+`]{} specifies the operator of interest, allowing the same code to be reused for other operators like multiplication ([`*`]{}), maximization ([`max`]{}) [@Shah2013], or even string concatenation[^2]. The [`!`]{} suffix is a Julia convention denoting that the function mutates at least one of its arguments; in this case, the cumulative sums are computed in-place on [`y`]{}.
The general problem is called the prefix sum [@Blelloch1989; @Blelloch1993] or scan [@Iverson1962; @Iverson1979]. Nominally, it appears that the data has to be scanned in one sweep from first to last and is a naturally serial process. However, the insight behind parallel prefix algorithms [@Blelloch1989; @Brent1982; @Kogge1973; @Kruskal1985; @Ladner1980; @Sklansky1960] is that associativity allows the operations to regrouped in different ways which can expose potential for concurrent execution, which can be interpreted as generic divide-and-conquer strategies for recursive computation [@Smith1987].
In its general form, scan algorithms can be computed as a higher-order function which takes as input some associative operator. Table \[tab:prefixapps\] shows a representative list of applications of parallel prefix, showing the diversity of applications and associative operators associated with those applications [@Blelloch1990; @Blelloch1993]. Scan is therefore a prime example of an algorithm that can exploit polymorphism for genericity.
Application Operator
-------------------------------------------------- ---------------------- --
Addition
Poisson random variates [@Lu1996] sequence lengths
Minimal coverings [@Moitra1991] joining 2D regions
Stream reduction [@Horn2005] counting records
Maximization
Line of sight [@Blelloch1990] height
String alignment [@Hillis1986; @Chi1992] substring length
Multiplication
Binary addition [@Sklansky1960] Boolean matrices
Polynomial interpolation [@Egecioglu1990] scalars
Sorting [@Hillis1986; @Blelloch1989] permutations
Tridiagonal equations [@Mathias1995] matrices
Function composition
Finite state automata [@Ladner1980; @Hillis1986] transition functions
: Representative applications of the scan algorithm, employing four basic types of operations: addition, maximization, multiplication, and function composition.[]{data-label="tab:prefixapps"}
The Brent–Kung form of parallel prefix
--------------------------------------
In this paper, we focus on the Brent–Kung form [@Brent1982] of parallel prefix, where the computation is organized into two trees. For simplicity, we present first the special case of parallel prefix for $n=8$ data points.
function prefix8!(y, +)
length(y)==8 || error("length 8 only")
for i in [2,4,6,8] y[i] = y[i-1] + y[i] end
for i in [ 4, 8] y[i] = y[i-2] + y[i] end
for i in [ 8] y[i] = y[i-4] + y[i] end
for i in [ 6 ] y[i] = y[i-2] + y[i] end
for i in [ 3,5,7 ] y[i] = y[i-1] + y[i] end
y
end
Figure \[fig:gates\] illustrates the difference between the number and order of operations in [`prefix_serial!`]{} and [`prefix8!`]{}. Each vertical line represents a processor [`i`]{} operating on the data [`y[i]`]{}. Each operation of the form [`y[i] = y[j] + y[i]`]{} is represented by a gate with inputs on lines [`i`]{} and [`j`]{} and a single output on line [`i`]{}. The main idea is that even though it takes more operations to organize the computation in the double tree form of [`prefix8!`]{}, it is possible to execute each stage of the computation tree concurrently, and parallel speedup can be achieved if the depth of the resulting tree is shorter than the depth of the tree for the serial algorithm. Nevertheless, at this point we have not actually computed anything in parallel, merely organized the computation in a way that would *allow* for concurrent execution. Running the code as is on an [`Array`]{} object would run the operations sequentially, from left to right, then top to bottom of the computation tree.
render(prefix_serial!(AccessArray(8),+))
![Above: operation order generated from the left-associative algorithm [`prefix_serial!`]{}. Below: operation order generated from the tree algorithm [`prefix8!`]{}. The figures were generated directly from the same kernels used for computation in Section \[sec:parallel-prefix\]. The code listing for the [`render`]{} function is given in Section \[sec:render\]. This figure was rendered in Compose, a Julia package for declarative vector graphics [@Compose.jl].[]{data-label="fig:gates"}](serial)
render(prefix!(AccessArray(8),+))
![Above: operation order generated from the left-associative algorithm [`prefix_serial!`]{}. Below: operation order generated from the tree algorithm [`prefix8!`]{}. The figures were generated directly from the same kernels used for computation in Section \[sec:parallel-prefix\]. The code listing for the [`render`]{} function is given in Section \[sec:render\]. This figure was rendered in Compose, a Julia package for declarative vector graphics [@Compose.jl].[]{data-label="fig:gates"}](tree)
To conclude our exposition of the scan problem, we present the [`prefix!`]{} function that solves the general case of $n$ data points. While the indices are somewhat less clear than when explicitly written out in [`prefix8!`]{}, the [`prefix!`]{} function nonetheless preserves the double tree structure.
function prefix!(y, +)
l=length(y)
k=iceil(log2(l))
#The "reduce" tree
for j=1:k, i=2^j:2^j:min(l, 2^k)
y[i] = y[i-2^(j-1)] + y[i]
end
#The "broadcast" tree
for j=(k-1):-1:1, i=3*2^(j-1):2^j:min(l, 2^k)
y[i] = y[i-2^(j-1)] + y[i]
end
y
end
Again, at this point we have only written serial code that introduces more computations than the naive algorithm [`prefix_serial!`]{}. However, we will argue in Section \[sec:parallel-prefix\] that the exact same code in [`prefix!`]{} can be reused for parallel execution which can achieve speedup over [`prefix_serial!`]{}.
Operators for distributed computations
======================================
In this section we show how the prefix algorithm we wrote above can be run in a distributed setting without modification. The key is to make use of overloading using the multimethod dispatch feature of Julia.
Julia provides native support for multiprocess distributed computing based on one-sided message passing. The basic functionality is provided by the [`remotecall`]{} function, which initiates a nonblocking remote function call and returns an explicit future [@Friedman1976] (a remote pointer of type [`RemoteRef`]{}) whose value is retrieved by the [`fetch`]{} function, which is a blocking operation. Julia also provides more convenient syntax for [`remotecall`]{} with the [`@spawn`]{} and [`@spawnat`]{} macros, which automatically rewrite Julia expressions into [`remotecall`]{} function calls.
We can use Julia’s multiple dispatch feature to define associative operators which act on remote data rather than local data. Julia’s generic function system allows new methods which act on remote data to be defined for functions like [`+`]{} and [`*`]{}, which are simply functions for which the parser supports infix notation. In effect, we can overload addition and multiplication (or in general any binary associative function) transparently to work on remote data.
For example, we can run the following code:
#Start a Julia process on every available core
#addprocs(n) adds n processors
#Sys.CPU_CORES is the total number of available
#CPU cores
#nprocs() returns the total number of Julia
#processes attached to the current master
#(including itself)
addprocs(max(0, Sys.CPU_CORES-nprocs()))
import Base.* #Extend existing generic function
#Define elementary operations on remote data
*(r1::RemoteRef,r2::RemoteRef)=
@spawnat r2.where fetch(r1)*fetch(r2)
This one method defines multiplication on remote data by [`fetch`]{}ing the remote data from the process containing the data of [`r1`]{}, copying the data of [`fetch(r1)`]{} to the memory space of the process with id [`r2.where`]{}, which already stores the data of [`r2`]{}. The process [`r2.where`]{} now contains local copies of both operands. Assuming that the local data are of type [`T`]{}, the Julia code then invokes another round of method dispatch based on the method signature [`*(::T, ::T)`]{}. In this way, any data type [`T`]{} that supports multiplication will now also support remote multiplication, regardless of whether the data are scalar numbers, $N\times N$ matrices, or something else entirely.
The main point of this paper is that the very same function [`prefix!`]{} which was executed in serial in previous sections will now run in parallel, simply by passing to it an associative operator over remote data rather than local data. Julia’s multimethods and multiple dispatch semantics allow operations on remote data to share the same syntax as their corresponding operations on local data, thus removing any syntactic difference between remote and local operations. The new method for [`*`]{} defines new behavior specific to [`RemoteRef`]{}s, which are Julia’s explicit futures. With this new method defined in the current scope, running [`prefix!(y, *)`]{} will automatically compute cumulative products on remote data if [`y`]{} is an array of [`RemoteRef`]{}s. Julia will automatically dispatch on the [`*(r1::RemoteRef, r2::RemoteRef)`]{} method within the inner loops of [`prefix!`]{} by comparing the types of the data elements of [`y`]{} with method signatures defined for [`*`]{}.
Parallel prefix {#sec:parallel-prefix}
---------------
We now run the [`prefix!`]{} function in parallel. The remote operations [`*(r1::RemoteRef, r2::RemoteRef)`]{} contain blocking operations implied by [`fetch(r1)`]{}, and Julia dynamically schedules all remote operations simultaneously so long as they are not waiting on the result of a [`fetch`]{} operation. The scheduling and dependency structure of [`prefix!`]{} thus results in all operations in each stage of the tree being executed simultaneously. Neglecting overhead from communication latency and bandwidth, the total execution time of [`prefix!`]{} depends only on the depth of the trees defined by the inner loops of [`prefix!`]{} and visualized in Figure \[fig:gates\].
From the indices of each loop in [`prefix!`]{} for $l$ data points, the first tree has at least one operation at depth $k$ for $l \ge 2^k$, and therefore the depth of the entire tree is $k = {\left\lfloor \log_2 l \right\rfloor}$. Similarly, the second tree has at least one operation at depth $k$ for $l \ge 3\cdot2^{k-1}$, and hence has depth $k = 1 + {\left\lfloor log_2 \frac l 3 \right\rfloor}$. Adding these depths and assuming that we distribute one datum per processor, we therefore obtain the theoretical speedup ratio for $p$ processors running [`prefix!`]{} over [`prefix_serial!`]{} as:
$$r (p) = \frac {p-1} {{\left\lfloor \log_2 p \right\rfloor} + 1 + {\left\lfloor \log_2 \frac p 3 \right\rfloor}}.
\label{eq:scaling-theory}$$
Figure \[fig:scaling\] summarizes benchmark timings for a sample problem where we generated $p$ square random matrices with Gaussian entries of size $n
= 4096$ and timed how long it took to multiply these matrices together on an 80-core Xeon E7-8850 machine with 1TB of shared memory. We specifically left out the time needed to broadcast the data to the remote processes, so as to focus only on the execution times of the kernels of interest. We also took care to disable the garbage collector. Julia, like many high-level dynamic languages, provides a garbage collector to aid in memory management. Julia v0.3.1 uses a simple stop-the-world, non-moving, precise mark and sweep garbage collector, where deallocation and finalization of garbage objects may not happen immediately after objects become unused[^3] [@McCarthy1960]. Therefore, it becomes important to factor out the possible effects of stop-the-world garbage collection. We explicitly disabled garbage collection with [`gc_disable()`]{} before running each kernel, then re-enabled garbage collection with [`gc_enable()`]{} after running each kernel. As an additional precaution, we timed the kernels multiple times and took the minimum time for each kernel so as to reduce fluctuations due to general nondeterministic delays.
![Weak scaling of the prefix sum kernels. Speedup ratios are the timings for [`prefix!`]{} over [`prefix_serial!`]{}. Plotted as a solid line is the theoretical speedup ratio $r(p)$ of Equation \[eq:scaling-theory\]. This figure was rendered in Gadfly, a Julia package for native plotting and visualization [@Gadfly.jl].[]{data-label="fig:scaling"}](scaling){width="0.9\columnwidth"}
The empirical timings shown in Figure \[fig:scaling\] show excellent agreement with the theoretical prediction of Equation \[eq:scaling-theory\], with slight deterioration for $p>40$ cores reflecting the increased communication overhead. The steps in the graph are as predicted by theory, arising from the depth of the computation tree growing by one to accommodate the extra data.
Other variants of parallel prefix
---------------------------------
A wide variety of parallel prefix algorithms exist beyond the Brent–Kung form [@Blelloch1989; @Egecioglu1992; @Kogge1973; @Kruskal1985; @Ladner1980; @Ofman1963; @Sklansky1960; @Sanders2006; @Sengupta2007; @Wang1996]. All of these variants can also be expressed as generic higher-order functions analogously to [`prefix!`]{}; however, not all of them can be written as in-place computations [@Merrill2009]. Nevertheless, the general principle still holds that generic kernels can be written for an arbitrary parallel prefix computation tree, and that these generic kernels can be composed effortlessly in Julia to support exactly the same range of operations as the Brent–Kung form in [`prefix!`]{}.
The genericity of [`prefix!`]{} is also very useful for implementing so-called meta-strategies for parallel prefix such as scan-then-fan [@Merrill2009; @Wilt2013]. The scan-then-fan variant of parallel prefix is a divide-and-conquer algorithm, where parts of the scan are chunked and computed serially on each process, after which each chunk is offset by the correct result by the value of the last element in its preceding chunk. The offsets can be propagated simply by running [`prefix!`]{} on the distributed array of chunks, with the operator
+(a, b) = a[end] .+ b
This variant of parallel prefix avoids the extra work incurred by generating a computation tree that is wider than the total number of available processes, and thus is more efficient to compute.
Operator-level instrumentation
==============================
Earlier, we showed in Figure \[fig:gates\] visualizations demonstrating the double tree structure of the Brent–Kung parallel prefix algorithm and also the cascading or rippling structure of the serial scan. These figures were generated programmatically from the exact same kernels [`prefix!`]{} and [`prefix_serial!`]{} used to perform the computations.
Many visualizations of algorithms are bespoke; the representations are completely decoupled from executable implementations. Alternatively, one may envision generating visualizations of algorithms directly from code implementations. Visualizations of algorithms can be generated by static analysis: feed the compute kernel into another program as data to compute the internal data flow. The static approach, however, is tantamount to reimplementing the compiler to generate the correct execution trace, from which the data flow can be inferred. Instead, one can employ dynamic analysis, instrumenting the program much like a debugger. Conventional debuggers either work on modified code with explicit instrumentation hooks embedded into the original kernel, or run the program in a special virtual machine with instrumentation hooks built into the low-level machine architecture. In these dynamic analyses, the execution trace is reconstructed from the global machine state, and again the data flow is inferred from the execution flow.
In this section, we describe a simple way to generate visualizations programmatically by instrumenting the interface of specific data objects, namely arrays. Instrumentation at this level retains the advantage of composing with unmodified compute kernels, but does not require the sophisticated infrastructure of an instrumented virtual machine, and reuses the static analysis of the original compiler. Furthermore, the instrumentation occurs at the level of individual variables, enabling highly selective traces which are cheaper than conventional approaches which instrument the entire program state. Additionally, the instrumentation measures the data flow directly, rather than inferring it from global execution flow. The resulting visualization provides an individual variable’s point of view of what happens over the course of a computation.
Our implementation in Julia takes advantage of genericity in the object model. Unlike most traditional object-oriented paradigms, which focus on data encapsulation [@Cardelli1985], the object model in Julia focuses on the interface to objects provided by method calls [@Mitchell1988]. Making the behavior primary over the data contents lends more naturally to data abstraction [@Mitchell1988; @Abadi1996], and furthermore admits less conventional object models involving multimethods and multiple dispatch [@Castagna1997].
[`Array`]{}s in Julia are containers of a given size (possibly with multiple dimensions) and element type. The basic array interface for Julia provides size and indexing semantics [@Bezanson2014]. The basic interface is provided by three functions:
[`length(A)`]{}
: returns the number of elements in the array [`A`]{},
[`getindex(A, idx...)`]{}
: retrieves the element of the array [`A`]{} with index [`idx`]{},
[`setindex!(A, val, idx...)`]{}
: puts the value [`val`]{} in the array [`A`]{} at the index [`idx`]{}.
The Julia parser also provides syntax sugar for the latter two operations: code like
A[i] = A[j] + A[k]
is desugared into code of the form
x = getindex(A, j)
y = getindex(A, k)
z = x + y
setindex!(A, z, i)
All the operations in the prefix sum kernels presented have array access operations of this form: two [`getindex`]{} calls followed by one [`setindex!`]{}. Based on this observation, we can write a very simple data type that nominally provides exactly the same interface as an [`Array`]{}, but rather than actually storing data elements, merely records the indices accessed by indexing operations.
Here is the entire Julia code that implements [`AccessArray`]{}, an abstract array type which instruments its indexing operations:
import Base: getindex, setindex!, length
type AccessArray
length :: Int
read :: Vector
history :: Vector
AccessArray(length)=new(length, Any[], Any[])
end
length(A::AccessArray)=A.length
function getindex(A::AccessArray, i)
push!(A.read, i)
nothing
end
function setindex!(A::AccessArray, x, i)
push!(A.history, (A.read, Any[i]))
A.read = Any[]
end
#Dummy associative operator
+(a::Void, b::Void) = nothing
The [`AccessArray`]{} type contains three fields:
[`length`]{}
: the effective length of the array,
[`read`]{}
: the history of indices accessed by [`getindex`]{} that have yet to be followed by a [`setindex!`]{} call, and
[`history`]{}
: the history of indices accessed by ([`getindex`]{}, [`setindex!`]{}) calls.
The [`Any[]`]{} construct defines an empty array which is explicitly typed to allow elements of any type. [`getindex`]{} is defined to always return the value [`nothing`]{}[^4], while recording the index [`i`]{} into [`A.read`]{}. [`setindex!`]{} records the index [`i`]{}, pairs it with the current value of [`A.read`]{}, and stores the pair into [`A.history`]{}.
As implemented, the [`AccessArray`]{} type can only accurately trace code where a [`setindex!`]{} call uses all the data from previous [`getindex`]{} calls. Furthermore, it does not handle cases where execution flow depends on the values of the array elements. Nevertheless, the [`AccessArray`]{} type is sufficiently powerful to record transactions relevant for prefix sums, and can be extended to more general execution flow patterns if necessary by wrapping actual data elements. The Appendix further defines the [`render`]{} function used to construct vector graphical objects from the instrumentation data stored in an [`AccessArray`]{}, and Figure \[fig:gates\] shows graphical renderings of the access patterns produced by [`prefix_serial!`]{} and [`prefix!`]{}.
Operators for formal verification
=================================
In Section \[sec:prefix\] we introduced several different kernels to compute scans. But how do we know that these kernels compute the prefix sum correctly? Each of these kernels have exactly the same function signature [`(y, +)`]{} representing the data [`y`]{} and associative binary operator [`+`]{}. It turns out that the inputs [`(y, +)`]{} to the scan algorithm turn out to have exactly the algebraic structure of a monoid, if the domain of array elements [`y[i]`]{} contains an identity under the operation [`+`]{}. The monoidal structure has been used in at least two ways to prove correctness. First, [@Hinze2004] constructed a formal algebra that allows correctness of circuits to be proved by derivation: all circuits which are equivalent to a known correct circuit, up to certain algebraic transformations, will all be correct. However, the algebraic proof of correctness is not constructive and does not lend itself easily to programmatic verification. Second and more recently, [@Chong2014] proved that the correctness of a kernel can be demonstrated by proving correctness for the interval monoid (Definition \[def:intervalmonoid\]), which formalizes the notion of indexing the subarrays being accessed over the course of the prefix sum computation. The latter method of proof is easy to verify programmatically.
In this section, we show how polymorphism allows the same Julia code written in previous sections for practical computations to also be used in the formal setting of verifying correctness. For convenience, we quote the definition of the interval monoid:
[[@Chong2014 Definition 4.3]]{} \[def:intervalmonoid\]
The *interval monoid* $I$ has the elements
$$\mathbb S_I = \left\{ (i_1, i_2) \in \mathrm{Int} \times \mathrm{Int} \;\vert\; i_1 \le i_2 \right\} \cup \{\mathbf 1_I, \top \}$$
and a binary operator $\oplus_I$ defined by:
$$\begin{aligned}
\mathbf 1_I \oplus_I x = x \oplus_I \mathbf 1_I &= x \textrm{ for all } x \in \mathbb S_I \\
\top \oplus_I x = x \oplus_I \top &= \top \textrm{ for all } x \in \mathbb S_I \\
(i_1, i_2) \oplus_I (i_3, i_4) &= \begin{cases} (i_1, i_4) &\textrm{if } i_2 + 1 = i_3 \\
\top &\textrm{otherwise.}
\end{cases}
\label{eq:intervalplus}\end{aligned}$$
The elements $(i, j) \in \mathbb S_I$ are abstractions of array indexing operations which produce array slices; they are produced by Julia code like [`y[i:j]`]{} where [`i:j`]{} is of type [`UnitRange`]{} and is a range of unit stride representing the set $\{i, i+1, \dots, j\}$. The definition of $\oplus_I$ in formalizes the notion of combining the results from the subarrays [`y[i:j]`]{} and [`y[j+1:k]`]{} to get the result for the subarray [`y[i:k]`]{}. The identity element $\mathbf 1_I$ formalizes an empty interval, while the annihilator $\top$ encodes noncontiguous ranges, which correspond to partial sums which cannot be represented by slicing with a [`UnitRange`]{}.
The key insight of [@Chong2014] is that correct computations of prefix sums cannot generate noncontiguous elements $\top$, otherwise they would by definition violate the prefixing property [`prefix!(y[1:j+1], +) = prefix!(y[1:j], +) + y[j+1]`]{}.\
From this insight, the authors of [@Chong2014] derive two correctness results:
1. A function that computes the prefix sum in serial is correct for $n$ data points if and only if that function computes the correct answer for the input\
$\left(\left((1, 1), (2, 2), \dots, (n, n)\right),
\oplus_I\right)$[^5] [@Chong2014 Theorem 4.5]. Furthermore, the correct answer is $\left((1, 1), (1, 2), \dots, (1, n)
\right)$, as the $k$th partial sum involves summing the subarray [`y[1:k]`]{}.
2. A function that computes the prefix sum in parallel is correct if it is free of data races and its equivalent serialization is correct [@Chong2014 Theorem 5.3].
We can use these results directly to verify the correctness of the Julia code we have written in earlier sections. By construction, the [`fetch`]{}es on [`RemoteRefs`]{} insert implicit synchronization barriers and thus the parallel code is free of data races. Thus only the serial correctness result needs to be verified explicitly.
Julia allows us to encode the interval monoid directly from the definition, by making use of the rich type system which is exposed to the user. The type system is conventionally used in Julia for type inference and data abstraction; here, we exploit the Curry–Howard correspondence to use the type system as a computational resource that can be used to prove correctness [@Curry1958; @Tait1965; @Howard1980]. A convenient feature of Julia’s type system is the ability to use abstract data types as singleton values: Julia types are values, and types can be used as singleton values using the [`Type{T}`]{} construct. Thus, the domain $\mathbb S_I$ can be written as a Julia type [`S`]{}, which is the [`Union`]{} (type union) of:
- [`UnitRange`]{},
- [`Type{Id}`]{}, the identity singleton $\mathbf 1_I$, and
- [`Type{Tee}`]{}, the annihilator singleton $\top$.
With this mapping of the abstract interval monoid domain $\mathbb S_I$ onto Julia types, Definition \[def:intervalmonoid\] translates directly into the following code:
#S is the domain of the interval monoid, $\mathbb{S}_I$
abstract Tee #$\top$
abstract Id #$\mathbf{1}_I$
typealias S Union(UnitRange, Type{Tee}, Type{Id})
#+ is the operator of the interval monoid, $\oplus_I$
+(I::UnitRange, J::UnitRange) = #$+_1$
I.stop+1==J.start ? (I.start:J.stop) : Tee
+(::Type{Id}, ::Type{Id}) = Id #$+_2$
+(I::S, ::Type{Id}) = I #$+_3$
+(::Type{Id}, J::S) = J #$+_4$
+(I::S, J::S) = Tee #$+_5$
Figure \[fig:dispatch\] summarizes the method dispatch table for the interval monoid, which demonstrates the use of some interesting features of Julia’s method dispatcher [@Bezanson2012]. First, the Julia method dispatcher chooses the most specific method that matches the type signature of a given set of arguments. Thus even though [`+`]{} may appear ambiguous for inputs of type [`(::UnitRange, ::UnitRange)`]{}, which matches both $+_1$ and $+_5$ methods, Julia resolves the ambiguity in favor of $+_1$ which has the more specific type signature, since by definition [`UnitRange`]{} is a subtype of [`S`]{}. Second, Julia uses symmetric multiple dispatch: the positions of the arguments are not used to resolve ambiguities. Hence we need the special-case method $+_2$ with type signature [`(::Type{Id}, ::Type{Id})`]{}, which lies in the intersection of the type signatures of $+_3$ and $+_4$. Bearing these rules in mind, it is straightforward to verify that the definition of [`+`]{} in the code block above is equivalent to that of $\oplus_I$ in Definition \[def:intervalmonoid\]. Julia’s method dispatch rules allow [`+`]{} to be defined in a way that reveals the catch-all nature of $\top$: method $+_5$, which returns [`Tee`]{}, is dispatched only when none of the other methods matches the type signature of the given arguments.
![Operation table for the interval monoid $(\mathbb S_I, \oplus_I)$, showing the overlapping domains of the various methods $+_{i\in\{1,\dots,5\}}$. The dispatch rules in Julia choose the most specific method defined over the given cell [@Bezanson2012].[]{data-label="fig:dispatch"}](intervaldispatch){width=".9\columnwidth"}
Verifying some function [`kernel`]{} for the problem size [`n`]{} simply reduces to writing the assertion:
#Test that kernel is correct for problem size n
@assert kernel([k:k for k=1:n],+)==[1:k for k=1:n]
Attempting to verify an incorrect kernel results in at least one $\top$ being produced during the computation, thus poisoning the program state and precipitating type conversion errors of the form
{jlcon}
`convert` has no method matching
convert(::Type{UnitRange}, ::Type{Tee})
which arise from the inability of noncontiguous ranges to be expressed as [`UnitRange`]{}s.
The Curry–Howard correspondence allows us to verify correct programs as programs with no type errors; programs with type errors must necessarily be incorrect. Julia thus allows for the same kernels used for computation to be verified directly without any rewriting or translation, simply by exploiting the polymorphism arising from the generic nature of prefix sum kernels, and composing such generic functions with appropriate choices of input data types and associative operators over those types.
Related work
============
Julia does not provide parallel prefix in the base library; however, several languages like APL [@Iverson1962; @Iverson1979], Chapel [@Deitz2006], C\*\* [@Viswanathan1996] and ZPL [@Chamberlain2000; @Deitz2002] do provide scan primitives. Other languages can use commonly-used library routines for scans: MPI provides the [`MPI_scan`]{} primitive [@Snir1995; @MPI], and in MPI-2, also the [`MPI_Exscan`]{} primitive for exclusive scan [@MPI2]. Intel’s Threading Building Blocks (TBB) library provides similar functionality [@Reinders2007]. GPU-specific implementations also exist, such in Haskell’s Accelerate library [@Chakravarty2011] and the Thrust C++ library [@Bell2012]. Most of these implementations, however, either lack genericity or express genericity through cumbersome language semantics. APL does not provide generic parallel prefix, and parallelism is provided only by nonstandard implementations. C\*\* only supports user-definable parallel prefix for commutative operations [@Viswanathan1996]. ZPL allows only for a limited form of overloading in terms of existing operators [@Deitz2002; @Deitz2006]. MPI allows parallel prefix on any [`MPI_Datatype`]{} and [`MPI_op`]{}; user-defined operations and data types can be used, but must be explicitly wrapped to do so, and furthermore requires users to reason explicitly about low-level parallelism. The Haskell Accelerate library provides genericity by generating code from user-specified expressions into code that implements parallel prefix, but such code does not benefit from Haskell’s static type system since they are not statically analyzable. Thrust and TBB, being written in C++, requires user-specified functions to be specified as C++ functors, which are cumbersome to write as they must be expressed using C++ expression templates. Chapel [@Deitz2006] comes closest to our work in providing user-definable scan operators in convenient syntax; however, operators have to be explicitly defined as classes, which do not support the full polymorphic expressiveness of multimethods.
Our implementation of parallel prefix as a higher-order function in Julia is generic in that it makes use of duck typing: so long as the input operation is associative and the collection of data is indexable, then the prefix sum computation will simply work. The simplicity of this generic approach, however, is by design rather naive and does not account for the complexities in real world implementations, for example possible synchronicity issues produced by higher levels of the broadcast and reduce trees that could result in bus saturation. Nevertheless, we can handle resource constraints by using a more sophisticated dynamic scheduler.
Conclusions and outlook
=======================
We have demonstrated how polymorphism using multimethods allows us to write generic computation kernels for parallel prefix, which can then be composed seamlessly with appropriate types and associative operators not only for computation for a wide variety of problems, but also for various code introspection tasks such as visualization and formal verification using the interval monoid. Julia’s language features lends to very natural and minimalist code that takes full advantage of *ad hoc* polymorphism to provide a single set of computational kernels that can also be visualized directly and verified without the need for retranslation and reimplementation in a more specialized language.
Acknowledgments
===============
The ideas in this paper were developed from examples in 18.337, the MIT course on parallel computing, in the Fall 2013 semester. We gratefully acknowledge the Julia community, especially Jeff Bezanson and Jake Bolewski, for insightful discussions. Funding for this work was provided by the Intel Science and Technology Center for Big Data, the MIT Deshpande Center Innovation Grant, the DARPA XDATA Program, and the National Science Foundation Grant DMS-1312831.
The [`render`]{} function {#sec:render}
=========================
This appendix shows the [`render`]{} function used to generate the figures in Figure \[fig:gates\]. We use here the Compose package for declarative vector graphics in Julia [@Compose.jl].
[`render`]{} is defined with two methods. The first describes how to render each elementary operation is represented as a logic [`gate`]{} with inputs [`ins`]{} and outputs [`outs`]{}. The [`render(G::gate, ...)`]{} method draws the inputs as small circles at coordinates [`ipoints`]{} and links them to the outputs, which are drawn as large circles at coordinates [`opoints`]{}. The second method for [`render`]{} describes how to render the instrumentation data in an [`AccessArray`]{}: a first pass through the data computes the depth of the tree to draw, and the second pass actually places gates appropriately for each operation, and finally vertical guidelines for every processor are added.
The code as written uses a heuristic taking advantage of the sequential left-to-right access order of the serialized prefix sum kernels: if an operation accesses an index lower than the most recently accessed index, then the current operation defines a new layer of the computation tree. This simplifying assumption does not fundamentally change the main idea of rendering instrumentation data being acquired at the level of individual variables, and more sophisticated reasoning about tree depths can be used as necessary.
using Compose
type gate
ins :: Vector
outs:: Vector
end
function render(G::gate, x, y, y0; ri=0.1, ro=0.25)
ipoints = [(i, y0+ri) for i in G.ins]
opoints = [(i, y0+0.5) for i in G.outs]
igates = [circle(i..., ri) for i in ipoints]
ogates = [circle(i..., ro) for i in opoints]
lines = [line([i, j]) for i in ipoints,
j in opoints]
compose(context(units=UnitBox(0.5, 0, x, y+1)),
compose(context(), stroke("black"),
fill("white"), igates..., ogates...),
compose(context(), linewidth(0.3mm),
stroke("black"), lines...))
end
function render(A::AccessArray)
#Scan to find maximum depth
olast = depth = 0
for y in A.history
(any(y[1] .<= olast)) && (depth += 1)
olast = maximum(y[2])
end
maxdepth = depth
#Map each operation onto a gate
olast = depth = 0
C = Any[]
for y in A.history
(any(y[1] .<= olast)) && (depth += 1)
push!(C, render(gate(y...), A.length,
maxdepth, depth))
olast = maximum(y[2])
end
#Compose everything together with processor
#guidelines
push!(C, compose(context(
units=UnitBox(0.5, 0, A.length, 1)),
[line([(i,0), (i,1)]) for i=1:A.length]...,
linewidth(0.1mm), stroke("grey")))
compose(context(), C...)
end
[^1]: Julia is MIT-licensed open source software and can be downloaded freely from [julialang.org](julialang.org). We use v0.3.1 in this paper.
[^2]: As written, the [`prefix_serial!`]{} function assumes, but does not check, that the function passed to it is associative. If necessary, checks of the form
@assert (y\[1\]+y\[2\])+y\[3\] == y\[1\]+(y\[2\]+y\[3\]) can be included, but for simplicity of presentation, we omit such checks from the code presented in this paper. We also neglect concerns relating to *approximate* associativity, such as roundoff errors in floating-point addition or multiplication [@Mathias1995].
[^3]: The code for Julia’s garbage collector may be found at <https://github.com/JuliaLang/julia/blob/275afc8b74b9c6ea5d34aefb8085525ff5dfc239/src/gc.c>
[^4]: [`nothing`]{} is a value of the special singleton type [`Void`]{}, akin to Python’s [`none`]{} or Haskell’s [`Nothing`]{}.
[^5]: Our presentation differs from the original only in that Julia arrays are 1-based, in contrast to C/OpenCL arrays studied in the original [@Chong2014], which are 0-based.
|
---
abstract: 'Spectral properties of chromophores are used to probe complex biological processes in vitro and in vivo, yet how the environment tunes their optical properties is far from being fully understood. Here we present a method to calculate such properties on large scale systems, like biologically relevant molecules in aqueous solution. Our approach is based on many body perturbation theory combined with quantum-mechanics/molecular-mechanics (QM/MM) approach. We show here how to include quasi-particle and excitonic effects for the calculation of optical absorption spectra in a QM/MM scheme. We apply this scheme, together with the well established TDDFT approach, to indole in water solution. Our calculations show that the solvent induces a redshift in the main spectral peak of indole, in quantitative agreement with the experiments and point to the importance of performing averages over molecular dynamics configurations for calculating optical properties.'
author:
- 'A. Mosca Conte (1), E. Ippoliti (2), R. Del Sole (1), P. Carloni (2), O. Pulci (1)'
title: 'Many-Body meets QM/MM: Application to indole in water solution'
---
Optical properties of aromatic chromophores embody a key facet of cell biology, allowing for a precise interrogation of a variety of biochemical events, including signaling, metabolism and aberrant processes. These range from probing transient interactions between biomolecules (proteins and nucleic acids), to protein dynamics and fibrillation and plaque formation in neurodegenerative diseases. Understanding how the environment tunes such optical properties is therefore crucial in structural genomics, yet this information is so far mostly lacking. A powerful tool to address this issue is given by the so-called quantum-mechanics/molecular-mechanics (QM/MM) methods \[\]. In this approach, the aromatic moiety is treated at quantum mechanical level, whilst the environment is described with an effective potential: the influence of the MM (presumably very complex and very large) environment is basically included as an external potential, and, in case the molecule is covalently bound to MM region, by a mechanical coupling with the environment.
Most often the QM approach is solved within Density Functional Theory (DFT) \[\] to study ground state properties, and time-dependent DFT (TDDFT) \[\] when excited states are involved as in the case of the optical properties \[\]. TDDFT is computationally very efficient, yet its predictive power depends dramatically on the system and on the functional used to reproduce the exchange and correlation interactions.
Several approaches, including Post-Hartree-Fock ones \[\] (configuration interaction and similar methods), have been already used to predict optical properties of biomolecules. Quantum-many-body techniques (MBPT) \[\], are an attractive alternative, although of course they come with a much higher computational cost than TDDFT. Strikingly, however, biophysical applications of one of the most widely used scheme, the combination of the GW method \[\] with the Bethe-Salpeter Equation (BSE) \[\] are so far lacking. The GW method is used for the evaluation of the single quasiparticle energies, and the BSE to introduce excitonic effects. Keeping in mind future biological applications, it is imperative to assess the accuracy of a MBPT/MM approach versus the more conventional TDDFT/MM one.
The main assumption in interfacing a QM/MM method with TDDFT or MBPT approaches is that the optical properties of the chromophore do not involve the MM part’s electronic structure. Hence, special care has to be devoted to the choice of the two regions.
Here we present MBPT/MM calculations on the indole ring of the Tryptophan protein residue (Fig. \[chdens\]). This system appears ideal for such an approach in several respects. First, it is very relevant biologically, as the indole ring has been exploited as a spectroscopic tool to monitor changes in proteins \[\] and to yield information about local structure and dynamics. In fact, its spectral signatures allow it to be used as a structural probe in proteins. Second, it contains a relatively small number of atoms (16), which can still be treated at the GW-BSE level. Next, the optical gap of liquid water (7 eV \[\]) is larger than the gap of the indole molecule (4.3 eV \[\]). Under 7 eV the spectra of indole and water do not overlap, and it is justified to treat the solvent in a classical scheme. Finally, CASPT2 calculations \[\] and experimental data \[\] are available, and allow to compare the changes of the optical properties upon passing from the gas phase to aqueous solution.
We performed QM/MM Car-Parrinello \[\] simulations of indole in water by the fully Hamiltonian QM/MM scheme \[\]. Such scheme has been applied to a variety of biological systems \[\]. The biomolecule was treated, at this step, at the DFT level whilst the solvent with the Amber force field \[\]. The approach allows for an explicit treatment of solvation, in contrast to previous studies \[\].
Indole single quasiparticle energies have been then evaluated at the GW level for several snapshots. Finally, we solved the BSE to calculate the average absorption spectrum and compared the results with the ones obtained within TDDFT. We calculated the indole absorbance in water as well as in gas phase. The shift in the spectra gives the solvatochromism.
The GW approximation consists in setting the vertex in the Hedin equations \[\] equal to a delta function. Under this condition, the time-Fourier transform of the proper exchange-correlation self energy, $\Sigma(\mathbf{{r},{r^{\prime }},\omega )}$, is a convolution of the Green function $G(\mathbf{{r},{r^{\prime }},\omega )}$, with the screened Coulomb potential $W(\mathbf{{r},{r^{\prime }},\omega )}$. The electronic bands are obtained by solving the following eigenproblem:
$$\begin{aligned}
\left[ -\frac{\hbar^{2} \nabla^{2}}{2m} + U^{QM}(\mathbf{r})+ U^{QM/MM}(
\mathbf{r}) + V_{H}(\mathbf{r}) \right] \phi_{j}(\mathbf{r}) &+& \nonumber \\
\int d^{3}\mathbf{{r^{\prime}}} \Sigma(\mathbf{{r},{r^{\prime}}}
,\varepsilon^{QP}_{j}) \phi_{j}(\mathbf{r^{\prime}}) = \varepsilon^{QP}_{j}
\phi_{j}({\mathbf{r}}).&& \label{dyson}\end{aligned}$$
This expression is derived from the Dyson equation in Lehmann representation \[\]. $V_{H}$ is the Hartree potential of the QM part, $U^{QM}$ is the electron-ion potential of the QM part, while $U^{QM/MM}$ is the potential felt by the electrons due to the point charges of the MM part. Finally, $\varepsilon _{j}^{QP}$ are the quasi-particle eigenvalues. Eq. (\[dyson\]) has the same form as the KS equation \[\] in the presence of an external electric field, where the exchange-correlation potential $V_{xc}(\mathbf{r}$) is replaced by the self energy $\Sigma (\mathbf{{r},{r^{\prime
}}},\varepsilon _{j}^{QP})$ which acts as a non-local, energy-dependent potential. Therefore, the eigenvalue problem described above can be solved perturbatively considering the KS equation as an unperturbed Hamiltonian and $\Sigma -V_{xc}$ as a perturbative term. The quasi-particle eigenvalues are obtained in first order approximation: $$\varepsilon^{QP}_{j} = \varepsilon^{KS}_{j} + \frac{\langle
\phi_{j}^{KS}|\Sigma(\varepsilon_{j}^{KS}) - V_{xc}|\phi_{j}^{KS}\rangle} {%
1-\langle \phi_{j}^{KS}|\frac{d \Sigma(\varepsilon^{KS}_{j})}{d \omega}%
|\phi_{j}^{KS}\rangle}. \label{first_perturb}$$
All the Coulomb interactions, and hence also the one induced by the classical region, are included in the KS eigenvalues $\varepsilon^{KS}_{j}$, and eigenvectors $|\phi_{j}^{KS}\rangle$. In this GW/MM scheme we neglect the contribution of the classical atoms to the $\Sigma$ operator in the same way as it is neglected for $V_{xc}$ in the DFT/MM scheme.
As a bonus from GW calculations we obtain also the four-point independent quasi-particle polarizability required to solve the BSE and which reads, in transition space \[\]:
$$P^{0}_{(n1,n2)(n3,n4)}(\omega) = \frac{f_{n2}-f_{n1}}{\varepsilon^{QP}_{n2}-%
\varepsilon^{QP}_{n1}-\omega} \delta_{n1,n3} \delta_{n2,n4}
\label{polarizability}$$
where $f_{n}$ is the occupation numbers of level $n$: the quantum plus classical external potential is not explicitly present in the BSE, but indirectly determines all the ingredients.
![(color online) Indole in water solution. Colors correspond to the following atomic species: BLUE=N, CYAN=C, WHITE=H, RED=O.[]{data-label="chdens"}](Fig1.eps){width="7.0cm"}
We performed a 20 ps hybrid QM/MM Car-Parrinello simulation of an indole molecule (QM part) \[\] surrounded by 2000 water molecules treated classically (MM part), with the Amber force field \[\]. Such a large number of water molecules was necessary to correctly reproduce the physical properties of a disordered system such as liquid water at room temperature (300 K). We also verified that such a large number of water molecules used for the dynamics was also necessary to calculate the optical absorption spectrum. This became clear already at the DFT-independent particle approach (DFT-IPA), as shown in Fig.\[2w\_qmmm\]: we calculated the absorption spectrum for indole with no water (with indole in the same “distorted” geometrical configuration as if in water), for indole with 2 water molecules, and for indole with 2000 water molecules. The three spectra are all different. This is a strong indication that also for the absorption spectrum it is necessary to include many water molecules in the theoretical simulation. This is due to the long range electrostatic potential of water which acts on indole.
Next, we tested our assumption that the solvent can be treated classically in the calculation of absorption spectra by performing TDDFT calculations \[\] for a system where two water molecules were treated at quantum level, and the remaining 1998 classically. The position of absorption peak is the same (within 0.02 eV) as in the case where all waters were treated within MM. Our result supports the use of this approach for solutes in water \[\].
For ten snapshots of the QM/MM dynamics (one every two ps) we computed the optical spectra at the independent particle level (DFT-IPA) and within TDDFT. The final spectrum was obtained by an average over the snapshots. The convergence of the spectrum was reached already after 6 snapshots, hence the subsequent GW and BSE calculations have been performed on only 6 snapshots.
The calculated DFT and GW HOMO-LUMO gaps \[\], averaged over the QM/MM configurations, are 3.8 eV (with standard deviation $\sigma=\pm$ 0.1 eV) and 7.2($\sigma=\pm$ 0.2) eV, respectively. The GW corrections to the gap turned out to be practically constant in all the snapshots considered (3.4 $\pm$ 0.1 eV). This fact, already found for liquid water \[\], confirms that one can strongly reduce the computational effort, by performing a GW calculations for just one snapshot.
![(color online) DFT-IPA optical spectra of indole in 2000 water molecules (blue solid line), in 2 water molecules (red circles) and without water (black dashed line).[]{data-label="2w_qmmm"}](Fig2.eps){width="8.5cm"}
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![(color online) BSE (a) and TDDFT (b) spectra of indole in water. The tiny red dashed lines are the spectrum of each snapshot. The red solid line is obtained by an average over these spectra. The black dashed line is for indole in vapor phase.[]{data-label="solventshift"}](Fig3.eps "fig:"){width="8.5cm"}
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![(color online) BSE optical spectra. Solid blue line: Indole in vapor phase. Black dashed line: indole without water molecules, with distorted geometry taken from a snapshot corresponding to 13.08 ps of the dynamics. Circles: indole in water, spectrum calculated for the same snapshot.[]{data-label="fig-bse-geow"}](Fig4.eps){width="8.5cm"}
We finally calculate the low energy range of the optical spectrum of indole, by GW-BSE and TDDFT, always as result of an average over the QM/MM snapshots. In Fig.\[solventshift\] we report our results together with the calculated absorption spectrum in gas phase. We notice that, in both approaches the most intense peak ($^{1}L_{a}$ in the experiment) is red-shifted on passing from gas phase to the water solution. This agrees with experiments \[\]. Same conclusion was obtained by previous theoretical calculations of indole in water based on CASSCF method and CASPT2 \[\]. In these approaches, the solvent was simulated by a continuum model with a cavity containing the indole molecule. The value we calculate for such a redshift is $\sim$ 0.2 eV, in fair agreement with experiment (0.18 eV) \[\]. On the contrary the CASSCF and CASPT2 prediction for the solvent shift is about 0.06 eV only. Such an underestimation may depend on the geometrical distortion of indole molecule caused by temperature effects due to the solvent and by an explicit H-bonding between water molecules, which were not considered explicitly therein. To quantify the effect of the geometry distortion on such shift, we performed calculations of indole switching on and off the water field in order to separate the geometry effect from the electrostatic ones. The results are presented for a single snapshot in GW+BSE (Fig. \[fig-bse-geow\]) and are obtained by performing additional calculations for the same QM/MM configuration without the water field. The corresponding solvent-shift goes from -0.1 eV with water field to +0.2 eV (hence, a blue-shift) without water field. This emphasises the importance of taking into account explicitly the electrostatic interaction with the solvent, since the geometry distortion alone would give, at least for this snapshot, a wrong sign.
In addition, TDDFT underestimates the energy of the $^1L_a$ peak both in gas phase and in solution by $\sim $0.4 eV, and BSE-GW overestimates them by $\sim $0.3 eV. As expected \[\], CASSCF is much worse, it overestimates by $\sim $1 eV or more, whilst CASPT2 is more accurate ($\sim $0.13 eV or less).
In conclusion, we have included many-body perturbative techniques in a QM/MM scheme. We have applied it, together with a TDDFT/MM approach, to study the optical properties of indole in water solution. Both methods reproduce quantitatively the redshift induced by the solvent. Hence, the GW-BSE method could be applied to biomolecules in aqueous solution (i.e. in laboratory-realizable conditions) in cases where the TDLDA/GGA approach does not work \[\]. Our GW-BSE calculations further show that the solvent shift is a consequence of the combination of two effects: the geometrical distortion of indole molecule in the solvent and the electrostatic interaction with the water molecules electric dipoles. Both effects, and their sum, depend on the particular configuration of the system; this emphasizes the need of more than one snapshot (several, indeed) for carrying out accurate optical calculations.
This work opens the way to further applications in other bio-relevant molecules, such as proteins and cell membranes, for which the evaluation of the optical shift enables to understand the nature of their environment. This work was supported by the EU through the Nanoquanta NOE (NMP4-CT-2004-500198). Computer resources from INFM “Progetto Calcolo Parallelo” at CINECA are gratefully acknowledged. We also thank L. Guidoni for interesting discussions.
[99]{}
, **104**, 7764 (2007).
see for example *Quantum theory of many-particle systems*, A.L. Fetter, J.D. Walecka ,McGraw-Hill, San Francisco (1971).
.
.
The QM/MM code combines the CPMD3.11.1 \[\] and Gromos \[\] codes. The time step chosen for the simulations was about 0.1 fs. Since the indole’s N–H group forms H-bonds with water, we used an energy cutoff of 70 Ry with the BLYP recipe \[\] for the exchange-correlation functional, which have widely been used for biophysical applications. Norm-conserving pseudopotentials of Troullier-Martins type \[\] corrected for a better description of Van der Walls interactions \[\] have been used. A Nose-Hoover thermostat \[\] is applied throughout all simulations to keep the temperature constant.
J. Hutter, A. Alavi, T. Deutsch, P. Ballone, M. Bernasconi, P. Focher, S. Goedecker, M. Tuckerman, M Parrinello, CPMD, Copyright IBM Corp 1990-2006, Copyright MPI für Festkörperforschung, Stuttgart 1997-2001.
W. F. van Gunsteren, S. R. Billeter, A. A. Eising, P. H. Hünenberger, P. Krüger, A. E. Mark, W. R. P. Scott, I. G. Tironi, Biomolecular Simulation: The GROMOS96 Manual and User Guide; Vdf Hochschulverlag AG an der ETH Zürich: Zürich, 1996.
.
.
We used the Parm99 version of Amber force field with the TIP3P model \[\] for water molecules.
.
TDDFT calculations are obtained within the Tamm-Dancoff approximation \[\] as implemented in CPMD 3.11.1 package, using the PBE \[\] exchange-correlation functional. The same corrected Troullier-Martins pseudopotentials as for the dynamics have been used.
.
.
.
GW calculations have been done by using 12077 plane-waves and 500 electronic bands. Following \[\], we used a cutoff in the real space for the self energy to prevent that periodic images of the quantum part interact with each other. The screen function is calculated within a plasmon pole approximation.
.
.
|
---
abstract: 'Coronavirus Disease 2019 (COVID-19) is a rapidly emerging respiratory disease caused by the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). Due to the rapid human-to-human transmission of SARS-CoV-2, many healthcare systems are at risk of exceeding their healthcare capacities, in particular in terms of SARS-CoV-2 tests, hospital and intensive care unit (ICU) beds and mechanical ventilators. Predictive algorithms could potentially ease the strain on healthcare systems by identifying those who are most likely to receive a positive SARS-CoV-2 test, be hospitalised or admitted to the ICU. Here, we study clinical predictive models that estimate, using machine learning and based on routinely collected clinical data, which patients are likely to receive a positive SARS-CoV-2 test, require hospitalisation or intensive care. To evaluate the predictive performance of our models, we perform a retrospective evaluation on clinical and blood analysis data from a cohort of 5644 patients. Our experimental results indicate that our predictive models identify (i) patients that test positive for SARS-CoV-2 a priori at a sensitivity of 75% (95% CI: 67%, 81%) and a specificity of 49% (95% CI: 46%, 51%), (ii) SARS-CoV-2 positive patients that require hospitalisation with 0.92 AUC (95% CI: 0.81, 0.98), and (iii) SARS-CoV-2 positive patients that require critical care with 0.98 AUC (95% CI: 0.95, 1.00). In addition, we determine which clinical features are predictive to what degree for each of the aforementioned clinical tasks. Our results indicate that predictive models trained on routinely collected clinical data could be used to predict clinical pathways for COVID-19, and therefore help inform care and prioritise resources.'
author:
- 'Patrick Schwab, August DuMont Schütte, Benedikt Dietz, and Stefan Bauer [^1]'
bibliography:
- 'references.bib'
title: 'predCOVID-19: A Systematic Study of Clinical Predictive Models for Coronavirus Disease 2019'
---
SARS-CoV-2, COVID-19, Machine Learning, Artificial Intelligence, Interpretability
Introduction {#sec:introduction}
============
Disease 2019 (COVID-19) was first discovered in December 2019 in China, and has since rapidly spread to over 200 countries [@who2020covid]. The COVID-19 pandemic challenges healthcare systems worldwide as a high peak capacity for testing and hospitalisation is necessary to diagnose and treat affected patients, particularly if the spread of SARS-CoV-2 is not mitigated. To avoid exceeding the available healthcare capacities, many countries have adopted social distancing policies, imposed travel restrictions, and postponed non-essential care and surgeries in order to reduce peak demand on their healthcare systems [@chinazzi2020effect; @jernigan2020update; @lin2020conceptual].
![We study the use of predictive models (light purple) to estimate whether patients are likely (i) to be SARS-CoV-2 positive, and whether SARS-CoV-2 positive patients are likely (ii) to be admitted to the hospital and (iii) to require critical care based on clinical, demographic and blood analysis data. Accurate clinical predictive models stratify patients according to individual risk, and, in this manner, help prioritise healthcare resources, such as testing, hospital and critical care capacity.[]{data-label="fig:teaser"}](img/covid-teaser.pdf){width="0.67\linewidth"}
The adoption of clinical predictive models that accurately predict who is likely to require testing, hospitalisation and intensive care from routinely collected clinical data could potentially further reduce peak demand by ensuring resources are prioritised to those individuals with the highest risk (Figure \[fig:teaser\]). For example, a clinical predictive model that accurately identifies patients that are likely to test positive for SARS-CoV-2 a priori could help prioritise limited SARS-CoV-2 testing capacity. However, developing accurate clinical prediction models for SARS-CoV-2 is difficult as relationships between clinical data, hospitalisation, and intensive care unit (ICU) admission have not yet been established conclusively due to the recent emergence of SARS-CoV-2.
In this systematic study, we develop and evaluate clinical predictive models that use routinely collected clinical data to identify (i) patients that are likely to receive a positive SARS-CoV-2 test, (ii) SARS-CoV-2 positive patients that are likely to require hospitalisation, and (iii) SARS-CoV-2 positive patients that are likely to require intensive care. Using the developed predictive models, we additionally determine which clinical features are most predictive for each of the aforementioned clinical tasks. Our results indicate that predictive models could be used to predict clinical pathways for COVID-19 patients. Such predictive models may be of significant utility for healthcare systems as preserving healthcare capacity has been linked to successfully combating SARS-CoV-2 [@wang2020response; @lee2020interrupting].
This work contains the following contributions:
- We develop and systematically study predictive models for estimating the likelihoods of (i) a positive SARS-CoV-2 test in patients presenting at hospitals, (ii) hospital admission in SARS-CoV-2 positive patients, and (iii) critical care admission in SARS-CoV-2 positive patients.
- We validate the performance of the developed clinical predictive models in a retrospective evaluation using real-world data from a cohort of 5644 patients.
- We determine and quantify the predictive power of routinely-collected clinical, demographic, and blood analysis data for the aforementioned clinical prediction tasks.
Related Work
============
A substantial body of work is dedicated to the study, validation and implementation of predictive models for clinical tasks. Clinical predictive models have, for example, been used to predict risk of septic shock [@henry2015targeted; @horng2017creating], risk of heart failure [@wu2010prediction], readmission following heart failure [@frizzell2017prediction; @shameer2017predictive; @golas2018machine], false alarms in critical care [@schwab2018not], risk scores [@caruana2015intelligible], outcomes [@visweswaran2005patient] and mortality in pneumonia [@cooper1997evaluation; @wu2014using], and mortality risk in critical care [@clermont2001predicting; @ghassemi2014unfolding; @johnson2017reproducibility]. Predicting clinical outcomes for individual patients is difficult because a large number of confounding factors may influence patient outcomes, and collecting and accounting for these factors in an unbiased way remains an open challenge in clinical practice [@obermeyer2016predicting]. Systematic studies, such as the one presented in this work, enable medical practitioners to better understand, assess and potentially overcome these issues by systematically evaluating generalisation ability, expected predictive performance, and influential predictors of various clinical predictive models. Beyond the need for systematic evaluation, missingness [@faris2002multiple; @wells2013strategies; @lipton2016directly; @che2018recurrent], noise [@lasko2013computational; @schwab2017beat], multivariate input data [@schwab2018not; @schwab2019phonemd; @ghassemi2015multivariate; @schwab2020deepms], and the need for interpretability [@choi2016retain; @doshi2017towards; @ross2017right; @schwab2019cxplain] have been highlighted as particularly important considerations in healthcare settings. In this work, we build on recent methodological advances to develop and systematically study clinical predictive models that may aid in prioritising healthcare resources [@chen2017machine] for COVID-19, and thereby help prevent a potential overextension of healthcare system capacity.
Clinical Predictive Models for COVID-19
---------------------------------------
Several clinical predictive models have recently been proposed for COVID-19, for example, for predicting potential COVID-19 diagnoses using data from emergency care admission exams [@de2020covid] and chest imaging data [@wang2020covid; @narin2020automatic; @li2020artificial; @castiglioni2020artificial; @wang2020deep; @xu2020deep], for predicting COVID-19 related mortality from clinical risk factors [@sarkar2020machine; @yan2020prediction], and for predicting which patients will develop acute respiratory distress syndrome (ARDS) from patients’ clinical characteristics [@jiang2020towards]. [@siordia2020epidemiology] presented a review of epidemiology and clinical features associated with COVID-19, and [@wynants2020prediction] a critical review that assessed limitations and risk of bias in diagnostic and prognostic models for COVID-19. In addition, [@wang2020clinical] performed a cohort study for clinical and laboratory predictors of COVID-19 related in-hospital mortality that identified baseline neutrophil count, age and several other clinical features as top predictors of mortality. Beyond prediction, [@ienca2020responsible] have argued for the responsible use of data in tackling the challenges posed by SARS-CoV-2.
Owing to the recent emergence of SARS-CoV-2, there currently exists, to the best of our knowledge, no prior systematic study on clinical predictive models that predict likelihood of a positive SARS-CoV-2 test, hospital and intensive care unit admission from clinical, demographic and blood analysis data that accounts for the missingness that is characteristic for the clinical setting. We additionally assess the influence of various clinical, demographic, and blood analysis measurements on the predictions of the developed clinical predictive models.
![The presented multistage machine-learning pipeline consists of preprocessing (light purple) the input data $x$, developing multiple candidate models using the given dataset (orange), selecting the best candidate model for evaluation (blue), and evaluating the selected best model’s outputs $\hat{y}$.[]{data-label="fig:pipeline"}](img/pipeline.pdf){width="\linewidth"}
Methods and Materials
=====================
### Problem Setting
In the given setting, we are given 106 routine clinical, laboratory and demographic measurements, or features, $x_i \in x$ for presenting patients. Features may be discrete or continuous, and some features may be missing as not all tests are necessarily performed on all patients. The clinical predictive tasks consist of utilising the routine clinical features $x_i$ to predict, for a newly presenting patient, (i) the likelihood $\hat{y}_\text{SARS-CoV-2}$ of receiving a positive SARS-CoV-2 test result, (ii) the likelihood $\hat{y}_\text{admission}$ of requiring hospital admission, and the (iii) likelihood $\hat{y}_\text{ICU}$ of requiring intensive care. In addition, we are given a development dataset consisting of $N$ patients, their corresponding observed routine clinical features $x_i$, SARS-CoV-2 test results $y_\text{SARS-CoV-2} \in \{0,1\}$, hospital admissions $y_\text{admission} \in \{0,1\}$, and ICU admissions $y_\text{ICU} \in \{0,1\}$, where 1 indicates the presence of an outcome. Using this development dataset, our goal is to derive clinical predictive models $\hat{f}_\text{SARS-CoV-2}$, $\hat{f}_\text{admission}$ and $\hat{f}_\text{ICU}$ for the respective before-mentioned tasks in order to inform care and help prioritise scarce healthcare resources. $$\begin{aligned}
\hat{y}_\text{SARS-CoV-2} &= \hat{f}_\text{SARS-CoV-2}(x) \\
\hat{y}_\text{admission} &= \hat{f}_\text{admission}(x) \\
\hat{y}_\text{ICU} &= \hat{f}_\text{ICU}(x)\end{aligned}$$
### Methodology
To derive the clinical predictive models $\hat{f}_\text{SARS-CoV-2}$, $\hat{f}_\text{admission}$ and $\hat{f}_\text{ICU}$ from the given development dataset, we set up a systematic model development, validation, and evaluation pipeline (Fig. \[fig:pipeline\]). To evaluate the generalisation ability of the developed clinical predictive models and to rule out overfitting to patients in the evaluation cohort, the development data is initially split into independent and stratified training, validation, and test folds without any patient overlap. Concretely, the multistage pipeline consists of (i) preprocessing, (ii) model development, (iii) model selection, and (iv) model evaluation stages. For preprocessing and model development, only the training fold is used, and only the validation and test folds of the development data are used for model selection and model evaluation, respectively. We outline the pipeline stages in detail in the following paragraphs.
### Preprocessing
In the preprocessing stage, we first drop all input features that are missing for more than 99.8% of all training set patients to ensure we have a minimal amount of data for each feature. This removes a total of 9 features from the original 106 routine clinical, laboratory and demographic features. We then transform all discrete features for each patient into their one-hot encoded representation with one out of $p$ indicator variables set to 1 to indicate the discrete value for this patient, and all others set to 0 with $p$ being the number of unique values for the discrete feature. We defined those features as discrete that have fewer than 6 unique values across all patients in the training fold. For discrete features, missing features were counted as a separate category in the one hot representation. Next, we standardised all continuous features to have zero mean and unit standard deviation across the training fold data. Lastly, we performed multiple imputation by chained equations (MICE) to impute all missing values of every continuous feature from the respective other features in an iterative fashion [@white2011multiple]. We additionally added a missing indicator that indicates 1 if the feature was imputed by MICE and 0 if it was originally present in order to preserve missingness information in the data after imputation. After the preprocessing stage, continuous input features are standardised and fully imputed, and discrete input features are one-hot encoded. All preprocessing operations are derived only from the training fold, and naïvely applied without adjustment to validation and test folds in order to avoid information leakage.
### Model Development
In the model development stage, we train candidate clinical predictive models $\hat{f}_\text{SARS-CoV-2}$, $\hat{f}_\text{admission}$ and $\hat{f}_\text{ICU}$ using supervised learning on the training fold of the preprocessed data. To derive the models from the preprocessed training fold data, we optimise various types of predictive models, and perform a hyperparameter search with $m$ runs for each of them. The model development process yields $m$ candidate models with different hyperparameter choices and predictive performances for each model category.
### Model Selection
In order to select the best model amongst the set of candidate models, we evaluate their predictive performance against the held-out validation fold that had not been used for model development. We choose the top candidate model by ranking all models by their evaluated predictive performance. The model selection stage using the independent validation fold enables us to optimise hyperparameters without utilising test fold data.
### Model Evaluation
In the model evaluation stage, we evaluate the selected best clinical predictive model against the held-out test fold that had not been used neither for training nor model selection in order to estimate the expected generalisation error of the models on previously unseen data. Using this approach, every selected best model from the model selection stage is evaluated exactly once against the test fold.
Using the presented standardised model development, selection and evaluation pipeline, we compare various types of clinical predictive models in the same test setting, with exactly the same amount of hyperparameter optimisation and input features against the same test fold. This process enables us to systematically study the expected generalisation ability, predictive performance and influential features of clinical predictive models for predicting SARS-CoV-2 test results, hospital admission for SARS-CoV-2 positive patients, and ICU admission for SARS-CoV-2 positive patients.
Experiments
===========
We conducted retrospective experiments to evaluate the predictive performance of a number of clinical predictive models on each of the presented clinical prediction tasks using the standardised development, validation and evaluation pipeline. Concretely, our experiments aimed to answer the following questions:
1. What is the expected predictive performance of the various clinical predictive models in predicting (i) SARS-CoV-2 test results for presenting patients, (ii) hospital admission for SARS-CoV-2 positive patients, and (iii) ICU admission for SARS-CoV-2 positive patients?
2. Which clinical, demographic and blood analysis features were most important for the respective best encountered predictive models for each clinical prediction task?
The following subsections describe the conducted experimental evaluation in detail.
[l@[0.65ex]{}l@[0.65ex]{}l@[0.65ex]{}l@[0.65ex]{}r]{}\
Property & Training & Validation & Test\
Subjects (\#) & 2822 (50%) & 1129 (20%) & 1693 (30%)\
SARS-CoV-2 (%) & 9.85 & 9.92 & 9.92\
Admit (%) & 1.42 & 1.33 & 1.42\
ICU (%) & 1.59 & 1.68 & 1.59\
Age (quantile) & 9.0 (1.0, 17.0) & 9.0 (1.0, 18.0) & 9.0 (2.0, 17.0)\
\
\
Property & Training & Validation & Test\
Subjects (\#) & 279 (50%) & 112 (20%) & 167 (30%)\
SARS-CoV-2 (%) & 100. & 100. & 100.\
Admit (%) & 6.45 & 6.25 & 6.59\
ICU (%) & 2.87 & 2.68 & 2.99\
Age (quantile) & 10.0 (4.0, 17.0) & 11.5 (4.5, 18.5) & 10.0 (4.0, 17.5)\
Dataset and Study Cohort
------------------------
We used anonymised data from a cohort of 5644 patients seen at the Hospital Israelita Albert Einstein in São Paulo, Brazil in the early months of 2020[^2]. Over the data collection time frame, the rate of SARS-CoV-2 positive patients at the hospital was around 10% of which around 6.5% and 2.5% required hospitalisation and critical care, respectively (Table \[tb:dataset\]). Notably, younger patients were underrepresented in the SARS-CoV-2 positive group relative to the general patient population which may have been caused by the reportedly more severe disease progression in older patients [@remuzzi2020covid]. Information on patient sex was not included in our dataset. We randomly split the entire available patient cohort into training (50%), validation (20%) and test folds (30%) within strata of patient age, SARS-CoV-2 test result, hospital admission status, and ICU admission status. After stratification, the three folds were approximately balanced across the stratification dimensions.
Models
------
Using the presented systematic evaluation methodology, we trained five different model types: Logistic Regression (LR), Neural Network (NN), Random Forest (RF), Support Vector Machine (SVM), and Gradient Boosting (XGB) [@chen2016xgboost]. The NN was a multi-layer perceptron (MLP) consisting of $L$ hidden layers with $N$ hidden units each followed by a non-linear activation function (ReLU [@nair2010rectified], SELU [@klambauer2017self], or ELU [@clevert2015fast]) and batch normalisation [@ioffe2015batch], and was trained using the Adam optimiser [@kingma2014adam] for up to 300 epochs with an early stopping patience of 12 epochs on the validation set loss.
Hyperparameters
---------------
We followed an unbiased, systematic approach to hyperparameter selection and optimisation. For each type of clinical predictive model, we performed a maximum of 30 hyperparameter optimisation runs with hyperparameters chosen from predefined ranges (Table \[tb:hyperparameters\]). The performance of each hyperparameter optimisation run was evaluated against the validation cohort. After computing the validation set performance, we selected the best candidate predictive model across the 30 hyperparameter optimisation runs by area under the receiver operator curve for further evaluation against the test set.
Hyperparameter Range / Choices
-- --------------------------------- ------------------------------------------------------------------------------
Regularization strength $C$ 0.01, 0.1, 1.0, 10.0
Number of hidden units $N$ 16, 32, 64, 128
Number of layers $L$ 1, 2, 3
Activation $a$ ReLU [@nair2010rectified], SELU [@klambauer2017self], ELU [@clevert2015fast]
Batch size $B$ 16, 32, 64, 128
L$2$ regularisation $\lambda_2$ 0.0, 0.00001, 0.0001
Learning rate $\alpha$ 0.003, 0.03
Dropout percentage $p$ (0%, 25%)
Tree depth $D$ 3, 4, 5
Number of trees $T$ 32, 64, 128, 256
Regularization strength $C$ 0.01, 0.1, 1.0, 10.0
Kernel $k$ polynomial, RBF, sigmoid
Polynomial degree $d$ 3, 5, 7
Subsample ratio $r$ 0.25, 0.5, 0.75, 1.0
Max. tree depth $T$ 2, 3, 4, 5, 6, 7, 8
Min. partition loss $\gamma$ 0.0, 0.1, 1.0, 10.0
Learning rate $\alpha$ 0.003, 0.03, 0.3, 0.5
L1 regularisation $\lambda_1$ 1.0, 0.1, 0.001, 0.0
L2 regularisation $\lambda_2$ 1.0, 0.1, 0.001, 0.0
Num. boosting rounds $B$ 5, 10, 15, 20
: Hyperparameters. []{data-label="tb:hyperparameters"}
[l@[17.0ex]{}l@[5ex]{}l@[5ex]{}l@[5ex]{}l@[5ex]{}l]{}\
\
Model & AUC & AUPR & Sensitivity & Specificity & Spec.@95% Sens.\
XGB & **0.66** (0.63, 0.70) & (0.15, 0.28) & **0.75** (0.67, 0.81) & (0.46, 0.51) & **0.23** (0.07, 0.32)\
RF & [^$\dagger$^]{}[0.65]{} (0.62, 0.69) & [^$\dagger$^]{}[0.19]{} (0.14, 0.24) & [^$\dagger$^]{}[0.69]{} (0.61, 0.74) & [^$\dagger$^]{}[0.54]{} (0.51, 0.57) & [^$\dagger$^]{}[0.19]{} (0.10, 0.25)\
NN & [^$\dagger$^]{}[0.62]{} (0.57, 0.65) & [^$\dagger$^]{}**0.22** (0.15, 0.28) & [^$\dagger$^]{}[0.60]{} (0.52, 0.67) & [^$\dagger$^]{}[0.55]{} (0.53, 0.58) & [^$\dagger$^]{}[0.17]{} (0.14, 0.28)\
LR & [^$\dagger$^]{}[0.61]{} (0.57, 0.65) & [^$\dagger$^]{}[0.17]{} (0.13, 0.24) & [^$\dagger$^]{}[0.58]{} (0.51, 0.65) & [^$\dagger$^]{}[0.55]{} (0.52, 0.57) & [^$\dagger$^]{}[0.19]{} (0.16, 0.25)\
SVM & [^$\dagger$^]{}[0.61]{} (0.57, 0.65) & (0.15, 0.27) & [^$\dagger$^]{}[0.57]{} (0.51, 0.64) & [^$\dagger$^]{}**0.59** (0.56, 0.61) & [^$\dagger$^]{}[0.14]{} (0.06, 0.16)\
\
\
\
Model & AUC & AUPR & Sensitivity & Specificity & Spec.@95% Sens.\
RF & **0.92** (0.81, 0.98) & (0.19, 0.81) & (0.19, 0.85) & **0.96** (0.92, 0.98) & **0.34** (0.29, 0.97)\
XGB & (0.80, 0.98) & [^$\dagger$^]{}**0.52** (0.28, 0.84) & [^$\dagger$^]{}[0.64]{} (0.43, 0.95) & [^$\dagger$^]{}[0.94]{} (0.90, 0.97) & [^$\dagger$^]{}[0.00]{} (0.00, 0.94)\
LR & [^$\dagger$^]{}[0.88]{} (0.70, 0.98) & (0.18, 0.83) & [^$\dagger$^]{}**0.82** (0.52, 1.00) & [^$\dagger$^]{}[0.85]{} (0.79, 0.90) & [^$\dagger$^]{}[0.13]{} (0.08, 0.93)\
NN & [^$\dagger$^]{}[0.85]{} (0.68, 0.97) & [^$\dagger$^]{}[0.31]{} (0.13, 0.66) & [^$\dagger$^]{}[0.64]{} (0.33, 1.00) & [^$\dagger$^]{}[0.95]{} (0.91, 0.97) & [^$\dagger$^]{}[0.11]{} (0.06, 0.93)\
SVM & [^$\dagger$^]{}[0.85]{} (0.70, 0.98) & [^$\dagger$^]{}[0.35]{} (0.17, 0.77) & [^$\dagger$^]{}[0.64]{} (0.30, 1.00) & [^$\dagger$^]{}[0.95]{} (0.91, 0.97) & [^$\dagger$^]{}[0.21]{} (0.15, 0.96)\
\
\
\
Model & AUC & AUPR & Sensitivity & Specificity & Spec.@95% Sens.\
SVM & **0.98** (0.95, 1.00) & (0.14, 1.00) & **0.80** (0.36, 1.00) & (0.92, 0.98) & **0.95** (0.91, 1.00)\
LR & **0.98** (0.93, 1.00) & [^$\dagger$^]{}**0.67** (0.09, 1.00) & **0.80** (0.29, 1.00) & [^$\dagger$^]{}[0.93]{} (0.89, 0.96) & [^$\dagger$^]{}[0.91]{} (0.87, 1.00)\
NN & [^$\dagger$^]{}[0.97]{} (0.94, 0.99) & [^$\dagger$^]{}[0.35]{} (0.10, 0.88) & **0.80** (0.36, 1.00) & [^$\dagger$^]{}[0.95]{} (0.91, 0.99) & (0.90, 0.99)\
RF & (0.92, 1.00) & [^$\dagger$^]{}[0.56]{} (0.13, 1.00) & [^$\dagger$^]{}[0.60]{} (0.15, 1.00) & [^$\dagger$^]{}**0.98** (0.96, 1.00) & [^$\dagger$^]{}[0.90]{} (0.86, 1.00)\
XGB & [^$\dagger$^]{}[0.67]{} (0.53, 0.98) & [^$\dagger$^]{}[0.29]{} (0.01, 0.68) & [^$\dagger$^]{}[0.40]{} (0.00, 1.00) & [^$\dagger$^]{}[0.94]{} (0.91, 0.97) & [^$\dagger$^]{}[0.00]{} (0.00, 0.96)\
Metrics
-------
### Predictive Performance
To assess the predictive performance of each of the developed clinical predictive models, we evaluated their performance in terms of area under the receiver operator curve (AUC), area under the precision recall curve (AUPR), sensitivity, specificity, and specificity at greater than 95% sensitivity (Spec.@95%Sens.) on the held-out test set cohorts for each task (Table \[tb:dataset\]). After model development and hyperparameter optimisation, we evaluated each model type exactly once against the test set to calculate the final performance metrics. Operating thresholds for each model were the operating points on the receiver operator characteristic curve closest to the top left coordinate as calculated for the validation cohort. We chose a variety of complementary evaluation metrics in order to give a comprehensive picture of the expected performance of each clinical predictive model on the evaluated tasks. For each of the performance metrics, we additionally computed 95% confidence intervals (CIs) using bootstrap resampling with 100 bootstrap samples on the test set cohort in order to quantify the uncertainty of our analysis results. We also assessed whether differences between clinical predictive models were statistically significant at significance level $\alpha = 0.05$ using pairwise t-tests with the respective best models for each task as measured by AUC.
### Importance of Test Types
To quantify the importance of specific clinical, demographic and blood analysis features on each of the predicted outcomes, we utilised causal explanation (CXPlain) models [@schwab2019cxplain]. CXPlain provides standardised relative feature importance attributions for any predictive model by computing the marginal contribution of each input feature towards the predictive performance of a model [@schwab2018granger], and is therefore particularly well-suited for assessing feature importance in our diverse set of models. We used the test fold’s ground truth labels to compute the exact marginal contribution of each input feature without any estimation uncertainty.
Results
=======
Predictive Performance
----------------------
In terms of predictive performance (Table \[tb:results\_all\]), we found that the overall best identified models by AUC were XGB for predicting SARS-CoV-2 test results, RF for predicting hospital admissions for SARS-CoV-2 positive patients, and SVM for predicting ICU admission for SARS-CoV-2 positive patients with respective AUCs of 0.66 (95% CI: 0.63, 0.70), 0.92 (95% CI: 0.81, 0.98), and 0.98 (95% CI: 0.95, 1.00). Notably, we found that predicting positive SARS-CoV-2 results from routinely collected clinical measurements was a considerably more difficult task for clinical predictive models than predicting hospitalisation and ICU admission. Nonetheless, the best encountered clinical predictive model for predicting SARS-CoV-2 test results (XGB) achieved a respectable sensitivity of 75% (95% CI: 67%, 81%) and specificity of 49% (95% CI: 46%, 51%). After fixing the operating threshold of the model to meet a sensitivity level of at least 95% (Spec.@95% Sens.), the best XGB model for predicting SARS-CoV-2 test results would achieve a specificity of 23% (95% CI: 7%, 32%). We additionally found that the differences in predictive performance between the best XGB model for predicting SARS-CoV-2 test results and the other predictive models was significant at a pre-specified significance level of $\alpha = 0.05$ (t-test) for all but the AUPR metric, where NN achieved a significantly better AUPR of 0.22 and the difference to SVM was not significant at the pre-specified significance level. On the task of predicting hospital admissions for SARS-CoV-2 positive patients, the best encountered RF model achieved a sensitivity of 55% (95% CI: 19%, 85%), a high specificity of 96% (95% CI: 92%, 98%), and a specificity at a fixed sensitivity of at least 95% (Spec.@95% Sens.) of 34% (95% CI: 29%, 97%). Owing to the lower sample size due to the smaller cohort of SARS-CoV-2 positive patients, the performance results for predicting hospital admission generally had wider uncertainty bounds but were nonetheless significantly better for RF than the other predictive models at the pre-specified significance level of $\alpha = 0.05$ (t-test) for most performance metrics with the exception of AUC where XGB achieved an AUC of 0.91 and AUPR where LR achieved an AUPR of 0.44. On the task of predicting ICU admission for SARS-CoV-2 positive patients, SVM had a sensitivity of 80% (95% CI: 36%, 100%), a specificity of 96% (95% CI: 92%, 98%), and a specificity at a fixed sensitivity of at least 95% (Spec.@95% Sens.) of 95% (95% CI: 91%, 100%). Due to the small percentage of around 3% of SARS-CoV-2 positive patients that were admitted to the ICU (Table \[tb:dataset\]), uncertainty bounds were wider than for the models predicting hospital admissions, and the results of the best encountered SVM were found to be not significantly better than LR and RF in terms of AUC, LR and NN in terms of sensitivity, and NN in terms of Spec.@95% Sens. at the pre-specified significance level of $\alpha = 0.05$ (t-test).
Feature Importance
------------------
In terms of feature importance, we found that importance scores were distributed highly unequally, relatively uniform and highly uniform for the best models encountered for predicting SARS-CoV-2 test results, for predicting hospital admissions for SARS-CoV-2 positive patients, and for predicting ICU admission, respectively (Figure \[fig:importance\]). Most notably, we found that 71.7% of the importance for the best XGB model for predicting SARS-CoV-2 test results was assigned to the missing indicator corresponding to the Arterial Lactic Acid measurement, i.e. much of the marginal predictive performance gain of the XGB model was attributed to whether or not the Arterial Lactic Acid test had been ordered. Beyond Arterial Lactic Acid being missing, age, leukocyte count, platelet count, and creatinine were implied to be associated with a positive SARS-CoV-2 test result by the best encountered predictive model, which further substantiates recent independent reports of those factors being potentially associated with SARS-CoV-2 [@covid2020severe; @rothan2020epidemiology; @lippi2020thrombocytopenia; @cheng2020kidney; @wang2020clinical]. Similarly to the best encountered XGB model for predicting SARS-CoV-2 test results, the top encountered predictive models for hospital admission and ICU admission for SARS-CoV-2 positive patients assigned a considerable degree of importance to missingness patterns associated with a number of measurements. A possible explanation for missingness appearing as a top predictor across the different tasks is that decisions whether or not to order a certain test to be performed for a given patient were influenced by patient characteristics that were not captured in the set of clinical measurements that were available to the predictive models. A controlled setting with standardised testing guidelines would be required to determine which confounding factors are behind the predictive power of the missingness patterns that have been implied to be associated with COVID-19 by the predictive models. Beyond missingness patterns, top predictors for predicting hospital admission were lactate dehydrogenase [@zhou2020clinical], gamma-glutamyltransferase, which through abnormal liver function has been reported to be implicated in COVID-19 severity [@fan2020clinical], and HCO$_3$ [@dondorp2020respiratory]. For predicting ICU admission in SARS-CoV-2 positive patients, pCO$_2$ and pH [@wang2020clinical] were top predictors. Blood pH, and in particular respiratory alkalosis, has been reported to be associated with severe COVID-19 [@ashraf2020covid].
Discussion
==========
We presented a systematic study of predictive models that predict SARS-CoV-2 test results, hospital admission for SARS-CoV-2 positive patients, and ICU admission for SARS-CoV-2 positive patients using routinely collected clinical measurements. Models that predict SARS-CoV-2 test results could help prioritise scarce testing capacity by identifying those individuals that are more likely to receive a positive result. Similarly, predictive models that predict which SARS-CoV-2 positive patients would be most likely to require hospital and critical care beds could help better utilise existing hospital capacity by prioritising those patients that have the highest risk of deterioration. Facilitating the efficient utilisation of scarce healthcare resources is particularly important in dealing with SARS-CoV-2 as its rapid transmission significantly increases demand for healthcare services worldwide.
![A comparison of the top-10 features ranked by relative feature importance scores for the best encountered model for predicting SARS-CoV-2 test results (XGB, top), hospital admission for SARS-CoV-2 positive patients (RF, middle), and critical care admission for SARS-CoV-2 positive patients (SVM, bottom), respectively. The bar length corresponds to the relative marginal importance (in %) of the displayed features towards the predictive performance of the respective model. Features names that include an all-caps “MISSING” indicate that the given marginal contribution refers to the importance of the absence or presence of that feature, not the feature itself.[]{data-label="fig:importance"}](img/merged){width="0.91\linewidth"}
The main limitation of the presented study is that its experimental evaluation was based on data collected from a single study site, and its results may therefore not generalise to settings with significantly different patient populations, admission criteria, patterns of missingness, and testing guidelines. In addition, we did not have access to mortality data for the analysed cohort, and we were therefore not able to correlate our predicted individual risk scores with patient mortality, which is another related prediction task that may be of clinical importance. Future studies should include a broader set of clinical measurements and outcomes, cohorts from multiple distinct geographical sites and under varying patterns of missingness in order to determine the robustness of the clinical predictive models to these confounding factors. Finally, we believe that the inclusion of data from other modalities, such as genomic profiling and medical imaging, and data on co-morbidities, symptoms and treatment histories could potentially further improve predictive performance of clinical predictive models across the presented prediction tasks.
Conclusion
==========
We presented a systematic study in which we developed and evaluated clinical predictive models for COVID-19 that estimate (i) the likelihood of a positive SARS-CoV-2 test in patients presenting at hospitals, (ii) the likelihood of hospital admission and (iii) intensive care unit admission in SARS-CoV-2 positive patients. We evaluated our developed clinical predictive models in a retrospective evaluation using a cohort of 5644 hospital patients seen in São Paulo, Brazil. In addition, we determined the clinical, demographic and blood analysis measurements that were most important for accurately predicting SARS-CoV-2 status, hospital admissions, and ICU admissions. Our experimental results indicate that clinical predictive models may in the future potentially be used to inform care and help prioritise scarce healthcare resources by assigning personalised risk scores for individual patients using routinely collected clinical, demographic and blood analysis data. Furthermore, our findings on the importance of routine clinical measurements towards predicting clinical pathways for patients increase our understanding of the interrelations of individual risk profiles and outcomes in SARS-CoV-2. Based on our study’s results, we conclude that healthcare systems should explore the use of predictive models that assess individual COVID-19 risk in order to improve healthcare resource prioritisation and inform patient care.
![Receiver operator characteristic (ROC) curves for the best encountered model for predicting SARS-CoV-2 test results (XGB, left), hospital admissions for SARS-CoV-2 positive patients (RF, top right), and critical care admissions for SARS-CoV-2 positive patients (SVM, bottom right). Numbers in the bottom right of each subgraph show the respective model’s area under the curve (AUC). Solid dots on the curves indicate operating thresholds selected on the validation fold.[]{data-label="fig:importance"}](img/roc_merged){width="0.95\linewidth"}
Acknowledgments {#acknowledgments .unnumbered}
===============
The anonymised data used in this manuscript were generously contributed by patients at Hospital Israelita Albert Einstein in São Paulo, Brazil, and are freely available at https://www.kaggle.com/einsteindata4u/covid19.
[^1]: P. Schwab is with F. Hoffmann-La Roche Ltd, Basel, Switzerland, A. DuMont Schütte and B. Dietz are with ETH Zurich, Switzerland, and S. Bauer is with the Max Planck Institute of Intelligent Systems, Tübingen, Germany (correspondence to: [email protected]).
[^2]: Exact data collection dates are unknown. The dataset is available at https://www.kaggle.com/einsteindata4u/covid19
|
---
abstract: 'Einstein field equations with a cosmological constant are approximated to the second order in the perturbation to a flat background metric. The final result is a set of Einstein-Maxwell-Proca equations for gravity in the weak field regime. This approximation procedure implements the breaking of gauge symmetry in general relativity. A brief discussion of the physical consequences (Pioneer anomalous deceleration) is proposed in the framework of the gauge theory of gravity.'
author:
- 'C. J. de Matos[^1]'
title: Approximation to the Second Order Approximation of Einstein Field Equations with a Cosmological Constant in a Flat Background
---
Introduction
============
Einstein introduced by hand a Cosmological Constant (CC) $\Lambda$ in his field equations to compensate for the universe expansion, and predict a stationary universe. He made reference to the CC as being his biggest mistake, when Hubble discovered that our universe was in expansion. Modern observational cosmology reveals a flat universe in accelerated expansion, which might be explained through a positive CC different from zero. What Einstein considered to be his worst idea, might well become its smartest thought!
The only homogeneous and isotropic vacuum solution of the Einstein field equations (EFE) with CC is (anti) de Sitter space, not Minkowski space [@Higuchi]. However, since contemporary observational cosmology tend to measure a flat universe in accelerated expansion [@Spergel], although Minkowski metric is not a solution of the vacuum EFE with CC, in the present work we impose the *ansatz* of a flat metric for the background instead of the (anti) de Sitter background metric, and we apply an approximation procedure, closely related to the standard linearization method [@de; @Matos1] [@Peng] [@Campbell], leading to “massive” GravitoElectroMagnetism (GEM). The result is a set of Einstein-Maxwell-Proca type equations for GEM. The *ansatz* of a flat background metric to approximate EFE with a CC instead of the (anti) de Sitter background corresponds to the implementation of a spontaneous breaking of gauge invariance in general relativity [@de; @Matos2]. As we will see in the conclusion, the Gauge Theory of Gravity (GTG) is particularly well suited to discuss the physical consequences of this symmetry breaking [@Hestenes].
Second Order Approximation of Einstein Field Equations with a Cosmological Constant
===================================================================================
In the following we propose an approximation of the second order approximation of Einstein Field Equations with cosmological constant. The following assumptions underly the approximation procedure:
1. \[approx\_1\]the mass densities are normal (no dwarf stars), and correspond to *local* physical systems located in the Earth laboratory or in the solar system.
2. \[approx\_2\]All motions are much slower than the speed of light, so that special relativity can be neglected. (Often special relativistic effects will hide general relativistic effects), $v<<c$.
3. \[approx\_3\]The kinetic or potential energy of all the bodies being considered is much smaller than their mass energy, $T_{\mu\nu}<<\rho c^2$.
4. \[approx\_4\]The gravitational fields are always weak enough so that superposition is valid, $\phi<<c^2$.
5. \[approx\_5\]The distances between objects is not so large that we have to take retardation into account. (This can be ignored when we have a stationary problem where the fields have already been prescribed and are not changing with time.)
6. \[approx\_55\]We consider a running cosmological constant which depends on the local density of mass of the physical system being considered $$\Lambda=\frac{4\pi G}{c^2}\rho$$
7. \[approx 6\]We consider that the proposed approximation is only valid in the following range of distances: $$\sqrt{|\frac{h_{\alpha\beta}}{\Lambda}|}<<r<<\sqrt{|\frac{1}{\Lambda}|}$$ defining the characteristic length scale for the physical system being considered. This restriction allows to neglect second order terms in the perturbation to Minkowsky’s metric $|h_{\alpha\beta}|^2$, but we do not neglect terms involving simultaneously the perturbation to the metric and the cosmological constant, $|\Lambda h_{\alpha\beta}|$.
8. \[approx 7\] The approximated second order EFE are solved, by approximation, using the solutions for the perturbation, $|h_{\alpha\beta}|$, to Minkowsky’s metric obtained in the linear approximation, in which only first order terms in $|h_{\alpha\beta}|$ are considered.
We start with Einstein Field Equations (EFE)with a cosmological constant. $$R_{\alpha\beta}-\frac{1}{2}g_{\alpha\beta}R-\Lambda
g_{\alpha\beta}=\frac{8\pi G}{c^4} T_{\alpha\beta}\label{equ1}$$ The weak field approximation assumes small perturbations, $|h_{\alpha\beta}|<<1$, of Minkowsky’s metric $\eta_{\alpha\beta}(+---)$ (Landau-Lifschitz “timelike convention”). This approximation is deliberately kept also for the case of having a cosmological constant different from zero. $$g_{\alpha\beta}\approx\eta_{\alpha\beta}+h_{\alpha\beta}\label{equ2}$$ Doing Equ.(\[equ2\]) into Equ.(\[equ1\]) with the derivation indices obeying the same rule as the covariant indices, $f^{,\mu}=\eta^{\mu\nu}f_{,\nu}$, we obtain: $$-\frac{1}{2}\Big(\bar{h}^{,\mu}_{\alpha\beta,\mu}+\eta_{\alpha\beta}\bar{h}^{,\mu\nu}_{\mu\nu}-\bar{h}^{,\mu}_{\alpha\mu,\beta}-\bar{h}^{,\mu}_{\beta\mu,\alpha}\Big)
-\Lambda\Big(\eta_{\alpha\beta}+h_{\alpha\beta}\Big)=\frac{8\pi
G}{c^4} T_{\alpha\beta}\label{equ3}$$ As usual in order to simplify the linearization procedure we have introduced the intermediate tensor: $$\bar{h}_{\alpha\beta}=h_{\alpha\beta}-\frac{1}{2}\eta_{\alpha\beta}h\label{equ4}$$ where $h=h^{\mu}_{\mu}=\eta^{\mu\nu}h_{\mu\nu}=h_{00}-h_{11}-h_{22}-h_{33}$ is the trace of the perturbation tensor. Imposing the harmonic gauge condition $$\bar{h}^{\mu\nu}_{,\nu}=0\label{equ5}$$ Equ.(\[equ3\]) reduces to $$\bar{h}^{,\mu}_{\alpha\beta,\mu}+2\Lambda\Big(\eta_{\alpha\beta}+h_{\alpha\beta}\Big)=-\frac{16\pi
G}{c^4} T_{\alpha\beta}\label{equ6}$$ Equ.(\[equ6\]) can be written in function of the Dalembertian operator, $\triangle$. If $f$ is a given function, then $$\triangle{f}=f^{,\mu}_{,\mu}=\eta^{\mu\nu}f_{,\mu\nu}=\Bigg(\frac{\partial^2}{(\partial
x^0)^2}-\frac{\partial^2}{(\partial x^i)^2}\Bigg)f\label{equ7}$$ Where $x_0=ct$. Therefore Equ.(\[equ6\]) becomes $$\triangle\bar{h}_{\alpha\beta}+2\Lambda\Big(\eta_{\alpha\beta}+h_{\alpha\beta}\Big)=-\frac{16\pi
G}{c^4} T_{\alpha\beta}\label{equ8}$$ This is the approximated second order form of EFE with a cosmological constant, assuming a flat background for the metric. We will now solve these equations, by approximation, using the solutions of the perturbation to Minkowsky’s metric we obtain in the case of linear EFE without CC.
To split spacetime into gravitoelectric and gravitomagnetic parts we consider respectively the energy-momentum tensor components: $$T_{00}=\rho c^2,\label{equ9}$$ and $$T_{0i}=-\rho c v_i.\label{equ10}$$ The solution of EFE without cosmological constant (with a flat background), $\triangle\bar{h}_{00}=-\frac{16\pi G}{c^4} T_{00}$, for the energy momentum tensor component given by Equ.(\[equ9\])is: $$h_{00}=\frac{2\phi}{c^2}\label{equ11}$$ Where $\phi$ is the gravitational scalar potential. The solution of EFE without cosmological constant,$\triangle\bar{h}_{0i}=-\frac{16\pi G}{c^4} T_{0i}$, for the energy momentum tensor component of Equ.(\[equ10\]) is: $$h_{0i}=-\frac{4A_{gi}}{c}.\label{equ12}$$ Where $A_{gi}$ are the three components of the gravitomagnetic vector potential.
Writing the Einstein tensor in function of the intermediate tensor $\bar{h}_{\alpha\beta}$, and using the gauge condition of Equ.(\[equ5\]), Einstein tensor reduces to the tensor $G_{\alpha\beta\mu}$. $$G_{\alpha\beta\mu}=\frac{1}{4}\Bigg(\bar{h}_{\alpha\beta,\mu}-\bar{h}_{\alpha\mu,\beta}\Bigg)\label{equ13}$$ Using Equ.(\[equ13\]) one can re-write Equ(\[equ8\]) under the following form: $$\frac{\partial G_{\alpha\beta\mu}}{\partial
x^{\mu}}+\frac{1}{2}\Lambda\Big(\eta_{\alpha\beta}+h_{\alpha\beta}\Big)=-\frac{4\pi
G}{c^4} T_{\alpha\beta}\label{equ14}$$ We can also use the tensor Equ.(\[equ13\]) to express the gravitational field: $$g_i=-c^2G_{00i}.$$ Which can also be written in terms of the gravitational scalar potential $\phi$ and of the gravitomagnetic vector potential $\vec{A_g}$. $$\vec{g}=-\bigtriangledown \phi - \frac {\partial
\vec{A_g}}{\partial t}$$
Similarly we formulate the gravitomagnetic field as follows: $$cG_{0ij}=-(A_{gi,j}-A_{gj,i})$$ which obviously shows that the gravitomagnetic field $\vec{B_g}$ is generated by a vectorial potential vector $\vec{A_g}$. $$\vec{B_g}=\nabla\times \vec{A_g}\label{Bg}$$ We have now everything we need to derive Proca-type equations for gravity. For the energy momentum tensor component of $T_{00}$ of Equ.(\[equ9\]), Equ.(\[equ14\]) reduces to: $$\frac{\partial G_{00\mu}}{\partial
x^{\mu}}+\frac{1}{2}\Lambda\Big(\eta_{00}+h_{00}\Big)=-\frac{4\pi
G \rho}{c^2}\label{equ15}$$ Substituting the solution Equ(\[equ11\]) of EFE, into Equ.(\[equ15\]) we can approximate the divergent part of the gravitational field: $$\nabla\cdot\vec{g}=-4\pi G
\rho-\Lambda\phi-\frac{1}{2}c^2\Lambda\label{equ16}$$ For the energy momentum tensor component $T_{0i}$ of Equ.(\[equ10\]), Equ.(\[equ14\]) reduces to: $$\frac{\partial G_{0i\mu}}{\partial
x^{\mu}}+\frac{1}{2}\Lambda\Big(\eta_{0i}+h_{0i}\Big)= \frac{4\pi
G}{c^3} \rho v_i \label{equ17}$$ Substituting the solution Equ(\[equ12\]) of EFE, into Equ.(\[equ17\]) we can approximate the rotational part of the gravitomagnetic field: $$\nabla\times\vec{B_g}=-\frac{4\pi G}{c^2}
\vec{j_m}+\frac{1}{c^2}\frac{\partial\vec g}{\partial t}-2 \Lambda
\vec{A_g}\label{equ18}$$ Where $\vec{j_m}=\rho\vec{v}$ is the mass current. The tensor $G_{\alpha\beta\mu}$, Equ.(\[equ13\]), has the following property: $$G^{\alpha\beta\mu,\lambda}+G^{\alpha\lambda\beta,\mu}+G^{\alpha\mu\lambda,\beta}=0.\label{19}$$ which are equivalent to the two other set of Maxwell like equations for gravity, $$\nabla\cdot\vec{B_g}=0\label{equ20}$$ and $$\nabla\times\vec g=-\frac{\partial \vec{B_g}}{\partial
t}\label{equ21}$$ Note also that Equ.(\[equ20\]) is a direct and trivial corollary of the definition of the gravitomagnetic field Equ.(\[Bg\]). As we see, Equs. (\[equ20\]) and (\[equ21\]) are not affected by the cosmological constant.
In summary Equs (\[equ16\]) (\[equ18\]) (\[equ20\]) and (\[equ21\]) form a set of Einstein-Maxwell-Proca equations for gravity in the weak field regime: $$\begin{aligned}
\nabla\cdot\vec{g}&=& -4\pi G \rho-\Lambda\phi-\frac{1}{2}c^2\Lambda \label{equ22}\\
\nabla\cdot\vec{B_g}&=& 0 \label{equ23}\\
\nabla\times\vec g &=& -\frac{\partial \vec{B_g}}{\partial t} \label{equ24}\\
\nabla\times\vec{B_g} &=& -\frac{4\pi G}{c^2}
\vec{j_m}+\frac{1}{c^2}\frac{\partial\vec g}{\partial t}-2\Lambda
\vec{A_g}\label{equ25}\end{aligned}$$ These equations are closely analogous to the ones derived by Argyris to investigate the consequences of massive gravitons in general relativity [@Argyris].
Considering the case of an universe empty of material gravitational sources, $\rho=0$ and $\phi=0$, Equ.(\[equ22\]) reduces to: $$\nabla\vec g=-\frac{1}{2}c^2\Lambda \label{equ27}$$ Integrating this equation over a volume bounded by a sphere of radius $R$ we obtain a fundamental “Machian-type” accelerated contraction of that volume, which only depends on its radius and on the value of the CC, $\Lambda$. $$g=\frac{1}{6} c^2 \Lambda R \label{equ28}$$ This acceleration is directed inwards on the boundary of the sphere. Doing the radius of the observable universe expressed in function of the CC
$R=R_U=\sqrt{3/\Lambda}$ and using $\Lambda=1.29\times10^{-52}
[m^{-2}]$ derived from the value of $71 [Km.S^{-1}.Mpc^{-1}]$ for the Hubble constant $H=c\sqrt{\Lambda/3}$ assumed by Nottale [@Nottale], into Equ.(\[equ28\]) we obtain a fundamental cosmic deceleration. $$g=\frac{1}{2\sqrt 3} c^2 \Lambda^{1/2}=2.9\times10^{-10} m.s^{-2}
\label{equ29}$$ So the linearized acceleration is a deceleration,it is interesting to note that this value is in good agreement with the Pioneer anomalous deceleration, whose current measured value is $a_{Pio}=(8.5\pm1.3)\times10^{-10} m.s^{-2}$[@Pioneer]. If this is a correct contribution to the Pioneer anomaly, this would imply that this deceleration would be independent of the origin at which the physical observer, measuring the Pioneer deceleration, is located. The deceleration should always be directed towards the observer, because the universe has no preferred center, and the deceleration only depends on the universe radius. Similarly the acceleration of expansion of the universe is independant of the origin at which the astronomer measuring it is located. The Pioneer anomalous deceleration would be a kind of local “Machian-type back reaction”to the accelerated cosmological expansion.
It might seem odd that the linearized analysis predicts an inward acceleration $\vec{g}$ when $\Lambda >0$, as a positive value of $\Lambda $ is known to cause the universe to accelerate outwards. The acceleration in the second case is not $\vec{g}$ as expressed in Equ.(\[equ28\]), but the acceleration $\ddot{R}/R$ of the scale factor in the Friedmann Robertson Walker (FRW) metric: $$ds^{2}=c^{2}d\tau ^{2}-R^{2}(\tau )(dx^{2}+dy^{2}+dz^{2})$$ According to the Friedmann equation (the Einstein equation for $R_{00}$, with non-negligible isotropic pressure $p$), $$\ddot{R}/R=\frac{1}{3}c^{2}\Lambda -\frac{4}{3}\pi G(\rho
+3pc^{-2})$$ and so a positive $\Lambda $ increases the value of $\ddot{R}/R$. The two accelerations $g$ and $\ddot{R}/R$ are not directly comparable, as the FRW metric is not written in harmonic coordinates.
Spontaneous Breaking of Gauge Invariance in General Relativity
==============================================================
General Relativity is founded on the *principle of equivalence*, which rests on the equality between the inertial and the gravitational mass of any physical system, and formulates that *at every space-time point in an arbitrary gravitational field it is possible to choose a “locally inertial coordinate system” such that, within a sufficiently small region of the point in question, the laws of nature take the same form as in unaccelerated Cartesian coordinate systems in the absence of gravity*. In other words, The inertial frames, that is, the “freely falling coordinate systems”, are indeed determined by the local gravitational field, which arises from all the matter in the universe, far and near. However, once in an inertial frame, the laws of motion are completely unaffected by the presence of nearby masses, either gravitationally or in any other way.
Following Steven Weinberg, the *Principle of General Covariance* (PGC) is an alternative version of the principle of equivalence[@Weinberg], which is very appropriate to investigate the field equations for electromagnetism and gravitation. It states that *a physical equation holds in a general gravitational field, if two conditions are met*:
1. The equation holds in the absence of gravitation; that is, it agrees with the laws of special relativity when the metric tensor $g_{\alpha\beta}$ equals the Minkowsky tensor $\eta_{\alpha\beta}$ and when the affine connection $\Gamma_{\beta\gamma}^{\alpha}$ vanishes.
2. The equation is generally covariant; that is, it preserves its form under a general coordinate transformation $x \rightarrow
x'$.
It should be stressed that general covariance by itself is empty of physical content. The significance of the principle of general covariance lies in its statement about the effects of gravitation, that a physical equation by virtue of its general covariance will be true in a gravitational field if it is true in the absence of gravitation. The PGC is not an invariance principle, like the principle of Galilean or special relativity, but is instead a statement about the effects of gravitation, and about nothing else. In particular general covariance does not imply Lorentz invariance. Any physical principle such as the PGC, which takes the form of an invariance principle but whose content is actually limited to a restriction on the interaction of one particular field, is called a dynamic symmetry. Local gauge invariance, which governs the electromagnetic interaction is another important dynamical symmetry. We can actually say that the Principle of General Covariance in general relativity is the analogous of the Principle of Gauge Invariance in electrodynamics. Spontaneous breaking of gauge invariance in general relativity would therefore correspond to a breaking of the PGC.
In contrast to the Einstein-Maxwell type theory of linear gravitation [@Peng], in the Einstein-Maxwell-Proca type theory, Equs.(\[equ22\])-(\[equ25\]), the potentials $\phi$ and $\vec A_g$ are directly measurable quantities so that gauge invariance is not possible, and the Lorentz gauge condition $$\nabla.\vec A_g + \frac {1}{c} \frac{\partial \phi}{\partial
t}=0\label{equ26}$$ is required in order to conserve energy [@Argyris]. Since $\Lambda\neq0$ is not consistent with gauge invariance, Proca generalisation of gravitoelectromagnetism could be aesthetically defective in the eyes of many theoretical physicists. However, the only certain statements about the value of $\Lambda$ that can be made must be based on experiment, and cosmological observations.
It should be noted that Einstein-Maxwell-Proca equations, Equs.(\[equ22\])-(\[equ25\]), form a good phenomenological base to investigate the GEM properties of superconductors [@de; @Matos2]. They offer also an interesting perspective on Mach’s principle as formulated in the framework of relational mechanics[@Assis].
Conclusion
==========
The set of Einstein-Maxwell-Proca equations derived from EFE with CC assuming a flat background, can be understood as the result of spontaneous breaking of gauge invariance in general relativity, which is physically revealed through the violation of the principle of general covariance. This is elegantly expressed in the framework of gauge theory of gravity. The Gauge Theory of Gravity (GTG) is formulated in the language of Geometric Calculus (GC), initially discovered by Clifford and further developed by Hestenes [@Hestenes], Doran, Lasenby [@Doran]. GTG is fully compliant with all classical tests of the standard formulation of General Relativity (GR). However it is not based on the principle of equivalence. Gauge symmetry plays a more fundamental role in the theory than the spacetime metric. The following two gauge principles for gravitation form the base of the theory:
1. The Displacement Gauge Principle (DGP), which states that the equations of physics must be invariant under arbitrary smooth remappings of events onto spacetime.
2. The Rotation Gauge Principle (RGP), which formulates that the equations of physics must be covariant under local Lorentz rotations.
DGP is a vast generalization of “*translational invariance*” in special relativity, so it has a comparable physical interpretation. Accordingly, the DGP can be interpreted as asserting that “*spacetime is globally homogeneous*”. In other words, with respect to the equations of physics all spacetime points are equivalent. DGP throws new light on Einstein’s Principle of General Covariance (PGC). The problem with the PGC, as we saw above, is that it is not a true symmetry principle [@Weinberg]. For a transformation group to be a physical symmetry group, there must be a well defined “geometric object” that the group leaves invariant. Thus the “*displacement group*” of the DGP is a symmetry group, because it leaves the flat spacetime background invariant. Following Noether’s theorem, homogeneity of spacetime is associated with the conservation of 4-linear momentum.
In special relativity, Lorentz transformations are passive rotations expressing equivalence of physics with respect to different inertial reference frames. In RGP, however, covariance under active rotations expresses local physical equivalence of different directions in spacetime. In other words, *RGP asserts that spacetime is locally isotropic*. Thus “passive equivalence” is an equivalence of observers, while “active equivalence” is an equivalence of states. Noether’s theorem establishes the conservation of the 4-angular momentum from the local spacetime isotropy.
Therefore a violation of gauge symmetry in general relativity is associated with a violation of energy-momentum conservation, which naturally takes place in a non-homogeneous and anisotropic universe! This is indeed what tends to be confirmed by the latest cosmological observations [@carroll], which would also implicitly confirm the validity of Einstein-Maxwell-Proca equations.
[99]{}
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Spergel, D. N., et al., Astrophy. J. Suppl., **148**, 175, (2003)
de Matos, C. J., Becker, R.,gr-qc/9908001, pp 5-15
Peng, H., “On calculation of Magnetic-Type Gravitation and Experiments”, **15**(8), 725-735 (1983)
Campbell, W.B., Morgan, T.A., “Maxwell Form of the Linear Theory of Gravitation”, Am. J. Phys., **44**, 356-365 (1976)
de Matos, C. J., “Gravitoelectromagnetism and Dark Energy in Superconductors”, to appear in Int. J. Mod. Phys. D, (2006). (also available gr-qc:/0607004)
Nottale, L.,“The Pioneer anomalous acceleration: a measurement of the cosmological constant at the scale of the solar system”, 2003, gr-qc/0307042
Anderson,J.D., et al, “Study of the anomalous acceleration of Pioneer 10 and 11” 2202a, Phys.Rev.D 65, 082004/1-50, gr-qc/0104064
Weinberg, S., “Gravitation and Cosmology Principles and Applications of the General Theory of Relativity”, (John Wiley and Sons, New York, 1972) p.91, 111, 361.
Argyris, J., Ciubotariu, C., “Massive Gravitons in General Relativity”, Aust. J. Phys., 1997, **50**, 879-91
Assis, A.K.T., “Relational Mechanics”, Apeiron, 1999 pp 199-205
Doran, C., Lasenby A., “Geometric Algebra for Physicists”, Cambridge, 2003
Hestesnes, D., “Gauge Theory Gravity with Geometric Calculus”,Foundations of Physics, 35(6): 903-970 (2005).
Carroll, S., Living Rev. Rel., **4**, 1 (2001)
[^1]: ESA-HQ, European Space Agency, 8-10 rue Mario Nikis, 75015 Paris, France, e-mail: [email protected]
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---
abstract: 'Local and nonlocal contributions to the total strain rate tensor $S_{ij}$ at any point $\textbf{x}$ in a flow are formulated from an expansion of the vorticity field in a local spherical neighborhood of radius $R$ centered on $\textbf{x}$. The resulting exact expression allows the nonlocal (background) strain rate tensor $S_{ij}^{B}(\textbf{x})$ to be obtained from $S_{ij}(\textbf{x})$. In turbulent flows, where the vorticity naturally concentrates into relatively compact structures, this allows the local alignment of vorticity with the most extensional principal axis of the background strain rate tensor to be evaluated. In the vicinity of any vortical structure, the required radius $R$ and corresponding order $n$ to which the expansion must be carried are determined by the viscous lengthscale $\lambda_{\nu}$. We demonstrate the convergence to the background strain rate field with increasing $R$ and $n$ for an equilibrium Burgers vortex, and show that this resolves the anomalous alignment of vorticity with the intermediate eigenvector of the total strain rate tensor. We then evaluate the background strain field $S_{ij}^{B}(\textbf{x})$ in DNS of homogeneous isotropic turbulence where, even for the limited $R$ and $n$ corresponding to the truncated series expansion, the results show an increase in the expected equilibrium alignment of vorticity with the most extensional principal axis of the background strain rate tensor.'
author:
- 'Peter E. Hamlington$^1$[^1], Jörg Schumacher$^2$[^2], and Werner J. A. Dahm$^1$[^3]'
title: Local and Nonlocal Strain Rate Fields and Vorticity Alignment in Turbulent Flows
---
\[sec:level1\]Introduction
==========================
Vortex stretching is the basic mechanism by which kinetic energy is transfered from larger to smaller scales in three-dimensional turbulent flows [@Burgers1948; @Batchelor1964; @Lundgren1982; @Lundgren1993]. An understanding of how vortical structures are stretched by the strain rate field $S_{ij}({\bf x})$ is thus essential to any description of the energetics of such flows. Over the last two decades, direct numerical simulations (DNS) [@Ashurst1987; @She1991; @Nomura1998] and experimental studies [@Tsinober1992; @Buch1996; @Su1996; @Zeff2003; @Mullin2006] of the fine-scale structure of turbulence have revealed a preferred alignment of the vorticity with the intermediate eigenvector of the strain rate tensor. This result has been widely regarded as surprising. Indeed the individual components of the inviscid vorticity transport equation, in a Lagrangian frame that remains aligned with the eigenvectors of the strain rate tensor, are simply $$\label{bs0}
\frac{\mbox{D}\omega_1}{\mbox{D}t} = s_1 \omega_1\,,\;\;\;
\frac{\mbox{D}\omega_2}{\mbox{D}t} = s_2 \omega_2\,,\;\;\;
\frac{\mbox{D}\omega_3}{\mbox{D}t} = s_3 \omega_3\,,$$ where $s_1$, $s_2$ and $s_3$ are the eigenvalues of $S_{ij}$. For incompressible flow, $s_1+s_2+s_3\equiv 0$, and then denoting $s_1\ge s_2\ge s_3$ requires $s_1\geq0$ and $s_3\leq 0$. As a consequence, (\[bs0\]) would predict alignment of the vorticity with the eigenvector corresponding to the most extensional principal strain rate $s_1$. Yet DNS and experimental studies have clearly shown that the vorticity instead is aligned with the eigenvector corresponding to the [*intermediate*]{} principal strain rate $s_2$.
A key to understanding this result is that, owing to the competition between strain and diffusion, the vorticity in turbulent flows naturally forms into concentrated vortical structures. It has been noted, for example in Refs. [@Nomura1998; @Jimenez1992; @Brasseur2005], that the anomalous alignment of the vorticity with the strain rate tensor $S_{ij}({\bf x})$ might be explained by separating the local self-induced strain rate field created by the vortical structures themselves from the background strain field in which these structures reside. The total strain rate tensor is thus split into $$\label{bs1}
S_{ij}(\textbf{x}) = S^R_{ij}(\textbf{x})+S^B_{ij}(\textbf{x}) \:,$$ where $S_{ij}^R$ is the [*local*]{} strain rate induced by a vortical structure in its neighboring vicinity, and $S_{ij}^B$ is the [*nonlocal*]{} background strain rate induced in the vicinity of the structure by all the remaining vorticity. The vortical structure would then be expected to align with the principal axis corresponding to the most extensional eigenvalue of the background strain rate tensor $S_{ij}^B({\bf x})$.
![Decomposition of the vorticity field in the vicinity of any point $\textbf{x}$ into local and nonlocal parts; the Biot-Savart integral in (\[bs6\]) over each part gives the local and nonlocal (background) contributions to the total strain rate tensor $S_{ij}$ at $\textbf{x}$.[]{data-label="plane"}](Fig1){width="3.3in"}
In the following, we extend this idea and suggest a systematic expansion of the total strain rate field $S_{ij}({\bf x})$ that allows the background strain rate field $S_{ij}^B({\bf x})$ to be extracted. Our approach is based on an expansion of the vorticity over a local spherical region of radius $R$ centered at any point ${\bf x}$. This leads to an exact operator that provides direct access to the background strain rate field. The operator is tested for the case of a Burgers vortex, where it is shown that the local self-induced strain field produced by the vortex can be successfully removed, and the underlying background strain field can be increasingly recovered as higher order terms are retained in the expansion. The anomalous alignment of the vorticity with respect to the eigenvectors of the total strain field is shown in that case to follow from a local switching of the principal strain axes when the vortex becomes sufficiently strong relative to the background strain. Finally, the operator is applied to obtain initial insights into the background strain $S_{ij}^B({\bf x})$ in DNS of homogeneous isotropic turbulence, and used to compare the vorticity alignment with the eigenvectors of the total strain field and of this background strain field.
The Background Strain Field
===========================
The velocity $\textbf{u}$ at any point $\textbf{x}$ induced by the vorticity field $\bm{\omega}(\textbf{x})$ is given by the Biot-Savart integral $$\label{bs1a}
\textbf{u}(\textbf{x}) = \frac{1}{4\pi} \int_{\bm{\Lambda}}
\bm{\omega}(\textbf{x}^{\prime})\times\frac{\textbf{x}-\textbf{x}^{\prime}}
{|\textbf{x}-\textbf{x}^{\prime}|^3} d^3 \textbf{x}^{\prime}\:,$$ where the integration domain $\bm{\Lambda}$ is taken to be infinite or periodic. In index notation (\[bs1a\]) becomes $$\label{bs2}
u_i(\textbf{x})= \frac{1}{4\pi} \int_{\bm{\Lambda}}
\epsilon_{ilk} \omega_l(\textbf{x}^{\prime})\frac{\left(x_k-x_k^{\prime}\right)}
{|\textbf{x}-\textbf{x}^{\prime}|^3} d^3 \textbf{x}^{\prime}\:,$$ where $\epsilon_{ilk}$ is the cyclic permutation tensor. The derivative with respect to $x_j$ gives the velocity gradient tensor $$\label{bs3}
\frac{\partial }{\partial x_j}u_i(\textbf{x}) = \frac{1}{4\pi}
\int_{\bm{\Lambda}} \epsilon_{ilk} \omega_l
(\textbf{x}^{\prime})\left[
\frac{\delta_{kj}}{r^3} - 3\frac{r_k r_j}
{r^5}\right]d^3 \textbf{x}^{\prime}\:,$$ where $r\equiv|\textbf{x}-\textbf{x}^{\prime}|$ and $r_m \equiv
x_m-x_m^{\prime}$. The strain rate tensor $S_{ij}$ at $\textbf{x}$ is the symmetric part of the velocity gradient, namely $$\label{bs4}
S_{ij}(\textbf{x})\equiv \frac{1}{2}\left(
\frac{\partial u_i}{\partial x_j}+
\frac{\partial u_j}{\partial x_i}\right)\:.$$ From (\[bs3\]) and (\[bs4\]), $S_{ij}(\textbf{x})$ can be expressed [@Ohkitani1994] as an integral over the vorticity field as $$\label{bs5}
S_{ij}(\textbf{x})=\frac{3}{8\pi}
\int_{\bm{\Lambda}}
\left(\epsilon_{ikl}r_j+r_i\epsilon_{jkl}\right)\frac{r_k}{r^5}
\omega_l(\textbf{x}^{\prime})d^3\textbf{x}^{\prime}\:.$$ As shown in Fig. \[plane\], the total strain rate in (\[bs5\]) is separated into the local contribution induced by the vorticity within a spherical region of radius $R$ centered on the point $\textbf{x}$ and the remaining nonlocal (background) contribution induced by all the vorticity outside this spherical region. The strain rate tensor in (\[bs5\]) thus becomes $$\begin{aligned}
\label{bs6}
S_{ij}(\textbf{x}) = & \underbrace{\frac{3}{8\pi}\int_{r\leq R}\left[\cdots\right]
d^3\textbf{x}^{\prime}} +& \underbrace{\frac{3}{8\pi}\int_{r> R} \left[\cdots
\right] d^3\textbf{x}^{\prime}}\:, \\
& \equiv S_{ij}^R(\textbf{x}) & \qquad \equiv S_{ij}^B(\textbf{x}) \nonumber\end{aligned}$$ where $[\cdots]$ denotes the integrand in (\[bs5\]). The nonlocal background strain tensor at $\textbf{x}$ is then $$\label{bs7}
S^B_{ij}(\textbf{x}) = S_{ij}(\textbf{x}) - S_{ij}^R
(\textbf{x})\:.$$ The total strain tensor $S_{ij}(\textbf{x})$ in (\[bs7\]) is readily evaluated via (\[bs4\]) from derivatives of the velocity field at point $\textbf{x}$. Thus all that is required to obtain the background (nonlocal) strain rate tensor $S_{ij}^B(\textbf{x})$ via (\[bs7\]) is an evaluation of the local strain integral $S_{ij}^R(\textbf{x})$ in (\[bs6\]) produced by the vorticity field $\omega_l(\textbf{x}^{\prime})$ within $r \leq R$ in Fig. \[plane\].
Evaluating the Background Strain Rate Tensor
--------------------------------------------
The vorticity field within the sphere of radius $R$ can be represented by its Taylor expansion about the center point $\textbf{x}$ as $$\begin{aligned}
\label{bs8}
\left.\omega_l(\textbf{x}^{\prime})\right|_{r\leq R} &=&\omega_l(\textbf{x})+ \left(
x^{\prime}_m - x_m\right) \left.\frac{\partial \omega_l}{\partial x_m}\right|_\textbf{x}
\\ &+& \frac{1}{2}\left( x^{\prime}_m - x_m\right)\left( x^{\prime}_n - x_n\right)
\left.\frac{\partial^2 \omega_l}{\partial x_m \partial
x_n}\right|_\textbf{x}+\cdots\:.\nonumber
\end{aligned}$$ Recalling that $x_m-x_m^{\prime} \equiv r_m$ and using $a_l, b_{lm}, c_{lmn},\ldots$ to abbreviate the vorticity and its derivatives at $\textbf{x}$, we can write (\[bs8\]) as $$\label{bs9}
\left.\omega_l(\textbf{x}^{\prime})\right|_{r\leq R} \equiv a_l - r_m b_{lm} + \frac{1}{2} r_m
r_n c_{lmn} -\cdots\:.$$ Substituting (\[bs9\]) in the $S_{ij}^{R}$ integral in (\[bs6\]) and changing the integration variable to $\textbf{r} = \textbf{x} -
\textbf{x}^{\prime}$, the strain tensor at $\textbf{x}$ produced by the vorticity in $R$ is $$\begin{aligned}
\label{bs10}
S^R_{ij}(\textbf{x}) &=& \frac{3}{8\pi}
\int_{r\leq R}
\left(\epsilon_{ikl}r_j+r_i\epsilon_{jkl}\right)\frac{r_k}{r^5}\\
&\times&\left[ a_l - r_m b_{lm} + \frac{r_m
r_n}{2} c_{lmn} - \cdots\right]d^3\textbf{r}\:.\nonumber\end{aligned}$$ This integral can be solved in spherical coordinates centered on $\textbf{x}$, with $r_1 = r\sin{\theta} \cos{\phi}$, $r_2 =
r\sin{\theta} \sin{\phi}$, and $r_3 = r\cos{\theta}$ for $r\in
[0,R]$, $\theta \in[0,\pi]$, and $\phi \in[0,2\pi)$. To integrate (\[bs10\]) note that
\[bs11\] $$\begin{aligned}
&&\int_{r\leq R} \frac{r_k r_j}{r^5} d^3 \textbf{r}
= \frac{4\pi}{3} \delta_{jk}\int_0^R \frac{1}{r} dr\:,\\
&&\int_{r\leq R} \frac{r_k r_j r_m}{r^5} d^3 \textbf{r}=
0\:,\\
&&\int_{r\leq R} \frac{r_k r_j r_m r_n}{r^5} d^3
\textbf{r}=
\frac{2\pi}{15} R^2 \left(\delta_{mn}\delta_{jk} + \right. \\
&&\qquad\qquad\qquad\qquad\qquad\qquad\:\:\left.\delta_{mj}\delta_{kn}
+\delta_{mk}\delta_{jn}\right)\:.\nonumber\end{aligned}$$
The resulting local strain rate tensor at $\textbf{x}$ is then $$\begin{aligned}
\label{bs12}
&&S_{ij}^R(\textbf{x}) =\frac{R^2}{40}c_{lmn}\left(\epsilon_{ijl}\delta_{mn}+\epsilon_{jil}\delta_{mn}
+\epsilon_{inl}\delta_{mj}\right. \\
&&\qquad\qquad\qquad\left.+ \epsilon_{jnl} \delta_{mi}
+\epsilon_{iml}\delta_{nj} +
\epsilon_{jml}\delta_{ni}\right)+O(R^4)\:,\nonumber\end{aligned}$$ where the contribution from the $a_l$ term in (\[bs10\]) is zero since $\epsilon_{ijl}=-\epsilon_{jil}$. For the same reason the first two terms in (\[bs12\]) also cancel, giving $$\begin{aligned}
\label{bs13}
&&S_{ij}^R(\textbf{x}) =
\frac{R^2}{40}c_{lmn}\left(\epsilon_{inl}\delta_{mj} + \epsilon_{jnl}
\delta_{mi}\right. \\
&&\qquad\qquad\qquad\qquad\left.+\epsilon_{iml}\delta_{nj} +
\epsilon_{jml}\delta_{ni}\right)+O(R^4)\:.\nonumber\end{aligned}$$ Recalling that $c_{lmn} = c_{lnm} \equiv \partial^2 \omega_l /\partial x_m\partial x_n$, and contracting with the $\delta$ and $\epsilon$ in (\[bs13\]), gives $$\begin{aligned}
\label{bs14}
&&S_{ij}^R(\textbf{x})=\frac{R^2}{20}\left[
\frac{\partial}{\partial x_j}\left(\epsilon_{iml}\frac{\partial \omega_l}{\partial
x_m}\right)\right. \\
&&\qquad\qquad\qquad\qquad\left. + \frac{\partial}{\partial x_i}\left(\epsilon_{jml}\frac{\partial \omega_l}{\partial
x_m}\right)\right] +O(R^4)\:.\nonumber\end{aligned}$$ Note that $\epsilon_{iml}\,\partial \omega_l/\partial x_m \equiv
\left(\nabla\times \bm{\omega}\right)_i$ and $$\label{bs15}
\nabla\times\bm{\omega} =
\nabla\times\left(\nabla\times \textbf{u}\right) = \nabla\left(\nabla\cdot
\textbf{u}\right) - \nabla^2 \textbf{u}\:,$$ so for an incompressible flow ($\nabla\cdot \textbf{u} \equiv 0$) the local strain rate tensor at $\textbf{x}$ becomes $$\begin{aligned}
\label{bs16}
S_{ij}^R(\textbf{x}) =
-\frac{R^2}{20}\nabla^2\left(\frac{\partial u_i}{\partial x_j} +
\frac{\partial u_j}{\partial x_i}\right)
+O(R^4)\:.\end{aligned}$$ From (\[bs7\]), with $S_{ij}^{R}$ from (\[bs16\]) we obtain the background strain tensor as $$\label{bs17}
S_{ij}^B(\textbf{x}) =S_{ij}(\textbf{x})+\frac{R^2}{10}\nabla^2S_{ij}(\textbf{x})
+O(R^4)\:.$$ The remaining terms in (\[bs17\]) result from the higher-order terms in (\[bs9\]). The contributions from each of these can be evaluated in an analogous manner, giving $$\begin{aligned}
\label{bs18}
&&S_{ij}^B(\textbf{x})=\left[1+\frac{R^2}{10}\nabla^2
+\frac{R^4}{280}\nabla^2\nabla^2+\cdots\right.\\
&&\qquad\qquad+\left.
\frac{3R^{2n-2}}{\left(2n-2\right)!
(4n^2-1)}\left(\nabla^2\right)^{n-1}+\cdots\right]
S_{ij}(\textbf{x})\:,\nonumber\end{aligned}$$ where the terms shown in (\[bs18\]) correspond to $n=1,2,\dots$. The final result in (\[bs18\]) is an operator that extracts the nonlocal background strain rate tensor $S_{ij}^B$ at any point $\textbf{x}$ from the total strain rate tensor $S_{ij}$. For the Taylor expansion in (\[bs8\]), this operator involves Laplacians of the total strain rate field $S_{ij}(\textbf{x})$.
Practical Implementation
------------------------
When using (\[bs18\]) to examine the local alignment of any concentrated vortical structure with the principal axes of the background strain rate field $S_{ij}^B(\textbf{x})$ in which it resides, the radius $R$ must be taken sufficiently large that the spherical region $| {\bf x^{\prime}} - {\bf x} | \leq R$ encloses essentially all the vorticity associated with the structure, so that its local induced strain rate field is fully accounted for. Generally, as $R$ increases it is necessary in (\[bs18\]) to retain terms of increasingly higher order $n$ to maintain a sufficient representation of $\bm{\omega}(\textbf{x}^{\prime})$ over the spherical region. Thus for any vortical structure having a characteristic gradient lengthscale $\lambda_{\nu}$, it can be expected that $R$ must be of the order of $\lambda_{\nu}$, and $n$ will then need to be sufficiently large to adequately represent the vorticity field within this sphere. However, since the local gradient lengthscale in the vorticity field in a turbulent flow is determined by an equilibrium between strain and diffusion, the vorticity field over the lengthscale $\lambda_{\nu}$ will be relatively smooth, and thus relatively low values of $n$ may suffice to give a usable representation of $\bm{\omega}(\textbf{x}^{\prime})$. This is examined in the following Section.
![Equilibrium Burgers vortex with circulation $\Gamma$ and strain-limited viscous diffusion lengthscale $\lambda_{\nu}$ in a uniform, irrotational, axisymmetric background strain rate field $S_{ij}^{B}(\bf{x})$.[]{data-label="burgf1"}](Fig2){width="2.5in"}
Test Case: Burgers Vortex
=========================
The equilibrium Burgers vortex [@Burgers1948; @Lundgren1982; @Buch1996; @Gibbon1999] is formed from vorticity in a fluid with viscosity $\nu$ by a spatially uniform, irrotational, axisymmetric background strain rate field $S_{ij}^{B}$ that has a single extensional principal strain rate $S_{zz}$ directed along the $z$ axis, as shown in Fig. \[burgf1\]. This simple flow, often regarded as an idealized model of the most concentrated vortical structures in turbulent flows, provides a test case for the result in (\[bs18\]). The combined strain rate field $S_{ij}(\textbf{x})$ produced by the vortex and the background strain flow should, when applied in (\[bs18\]), produce the underlying background strain field $(S_{rr}^{B}, S_{\theta\theta}^{B}, S_{zz}^{B}) = (-\frac{1}{2},
-\frac{1}{2}, 1) S_{zz}$ at all $\textbf{x}$ when $R \to \infty$ and all orders $n$ are retained. For finite $R$ and $n$, the resulting $S_{ij}^B(\textbf{x})$ will reflect the convergence properties of (\[bs18\]).
Strain Rate Tensor
------------------
The equilibrium Burgers vortex aligned with the extensional principal axis of the background strain rate field has a vorticity field $$\label{burg1}
\bm\omega(\textbf{x}) = \omega_z(r)\hat{\textbf{z}} =
\frac{\alpha}{\pi} \frac{\Gamma}{\lambda_\nu^2}\exp\left(- \alpha \eta^2\right)\hat{\textbf{z}}\:,$$ where $\Gamma$ is the circulation, $\lambda_\nu$ is the viscous lengthscale that characterizes the diameter of the vortex, $\eta\equiv r/\lambda_\nu$ is the radial similarity coordinate, and the constant $\alpha$ reflects the chosen definition of $\lambda_{\nu}$. Following [@Buch1996], $\lambda_{\nu}$ is taken as the full width of the vortical structure at which $\omega_{z}$ has decreased to one-fifth of its peak value, for which $\alpha = 4 \ln 5$. When diffusion of the vorticity is in equilibrium [@Buch1996] with the background strain, then $$\label{equilib1}
\lambda_{\nu} = \sqrt{8\alpha}\left(\frac{\nu}{S_{zz}}\right)^{1/2}\:.$$ The combined velocity field $\textbf{u}(\textbf{x})$ produced by the vortex and the irrotational background strain is given by the cylindrical components
\[burg2\] $$\begin{aligned}
u_r(r,\theta,z) &=& -\frac{S_{zz}}{2} r \,,\\
u_\theta(r,\theta,z) &=& \frac{\Gamma}{2\pi\lambda_{\nu}} \frac{1}{\eta} \left[1-\exp\left(-\alpha\eta^2\right)\right] \,,\\
u_z(r,\theta,z) &=& S_{zz} z\,.\end{aligned}$$
The combined strain rate tensor for such a Burgers vortex is thus $$\label{burg3}
S_{ij}(\textbf{x}) = \left[\begin{array}{ccc}
-S_{zz}/2 & S_{r\theta}^{\smallskip v} & 0 \\
S_{r\theta}^{\smallskip v} & -S_{zz}/2 & 0 \\
0 & 0 & S_{zz}
\end{array}\right]\:,$$ where $S_{r\theta}^{\smallskip v}$ is the shear strain rate induced by the vortex, given by $$\begin{aligned}
S_{r\theta}^{\smallskip v}\left(\textbf{x}\right) =
\frac{\Gamma}{\pi\lambda_\nu^2}
\left[\left(\alpha+\frac{1}{\eta^2}\right)
\exp\left(-\alpha\eta^2\right) - \frac{1}{\eta^2}\right]\,.
\label{burg45}\end{aligned}$$ From (\[burg3\]), $S_{ij}(\textbf{x})$ has one extensional principal strain rate equal to $S_{zz}$ along the $\hat{\textbf{z}}$ axis, with the remaining two principal strain axes lying in the $r$-$\theta$ plane and corresponding to the principal strain rates $$\label{s23}
s = -\frac{1}{2}S_{zz} \pm |S_{r\theta}^{\smallskip v}| \:.$$ As long as the largest $s$ in (\[s23\]) is smaller than $S_{zz}$, the most extensional principal strain rate $s_{1}$ of $S_{ij}$ will be $S_{zz}$, and the corresponding principal strain axis will point in the $\hat{\textbf{z}}$ direction. The vorticity is then aligned with the most extensional eigenvector of $S_{ij}$. This remains the case until the vortex becomes sufficiently strong relative to the background strain rate that $s
> S_{zz}$, namely $$\label{switch1}
|S_{r\theta}^{\smallskip v}|\geq \frac{3}{2}S_{zz}\:,$$ which from (\[burg45\]) occurs wherever $$\label{axisswitch}
\left(\alpha+\frac{1}{\eta^2}\right)\exp(-\alpha\eta^2) -
\frac{1}{\eta^2}\geq
\frac{3\pi}{2}\left(\frac{\Gamma/\lambda_\nu^2}{S_{zz}}\right)^{-1}\:.$$ At any $\eta$ for which (\[axisswitch\]) is satisfied, the most extensional principal axis of the combined strain rate tensor $S_{ij}(\textbf{x})$ will switch from the $\hat{\textbf{z}}$ direction to instead lie in the $r$-$\theta$ plane. Since the vorticity vector everywhere points in the $\hat{\textbf{z}}$ direction, wherever (\[axisswitch\]) is satisfied the principal axis of $S_{ij}$ that is aligned with the vorticity will switch from the most extensional eigenvector to the intermediate eigenvector. This alignment switching is purely a result of the induced strain field $S_{ij}^{\smallskip v}(\textbf{x})$ locally dominating the background strain field $S_{ij}^{B}(\textbf{x})$.
The dimensionless vortex strength parameter $$\label{axisswitchparam}
\Omega \equiv \left[\frac{\Gamma/\lambda_{\nu}^{2}}{S_{zz}}\right]
= \frac{\pi}{\alpha} \frac{\omega_{max}}{S_{zz}}$$ on the right-hand side of (\[axisswitch\]) characterizes the relative strength of the background strain and the induced strain from the vortical structure, where $\omega_{max}$ is obtained from (\[burg1\]) at $\eta=0$. For $$\Omega < \Omega^{*} \approx 2.45\,,$$ the background strain rate $S_{zz}$ is everywhere larger than the largest $s$ in (\[s23\]), and thus no alignment switching occurs at any $\eta$. For $\Omega > \Omega^{*}$, alignment switching will occur over the limited range of $\eta$ values that satisfy (\[axisswitch\]). With increasing values of $\Omega$, more of the vorticity field will be aligned with the intermediate principal axis of the $\textit{combined}$ strain rate tensor, even though all of the vorticity field remains aligned with the most extensional principal axis of the $\textit{background}$ strain rate tensor.
Figure \[burgf2\] shows the vorticity $\omega_z$ and the induced shear strain component $-S_{r\theta}^{\smallskip v}$ as a function of $\eta$. The horizontal dashed lines correspond to three different values of $\Omega$, and indicate the range of $\eta$ values where the alignment switching in (\[axisswitch\]) occurs for each $\Omega$. Wherever $-S^{\smallskip v}_{r\theta}$ is above the dashed line for a given $\Omega$, the vorticity will be aligned with the local intermediate principal axis of the combined strain rate field.
In principle, regardless of the vortex strength parameter $\Omega$, at any $\eta$ the result in (\[bs18\]) can reveal the alignment of the vorticity with the most extensional principal axis of the background strain field $S_{ij}^B$. However, this requires $R$ to be sufficiently large that a sphere with diameter $2R$, centered at the largest $\eta$ for which $-S^{\smallskip
v}_{r\theta}$ in Fig. \[burgf2\] is still above the horizontal dashed line, will enclose essentially all of the vorticity associated with the vortical structure. As $\Omega$ increases, the required $R$ will increase accordingly as dictated by (\[axisswitch\]), and as $R$ is increased the required $n$ in (\[bs18\]) also increases.
Irrespective of the value of $\Omega$, when (\[bs18\]) is applied to the combined strain rate field $S_{ij}(\bf{x})$ in (\[burg3\]) and (\[burg45\]), if $\tilde{R}\equiv(R/\lambda_{\nu}) \rightarrow \infty$ and all orders $n$ are retained then the resulting $S^{B}_{ij}(\bf{x})$ should recover the background strain field, namely $$\label{convergence}
S_{r\theta}^B \rightarrow 0$$ for all $\bf{x}$, and the vorticity should show alignment with the most extensional principal axis of $S_{ij}^{B}$. For finite $R/\lambda_{\nu}$ and various orders $n$, the convergence of $S_{ij}^{B}$ from (\[bs18\]) to this background strain field is examined below.
![Similarity profiles of $\omega_{z}(\eta)$ and $S_{r\theta}(\eta)$ for any equilibrium Burgers vortex; wherever $-S_{r\theta}$ exceeds the horizontal line determined by the relative vortex strength parameter $\Omega$ in (\[axisswitchparam\]) the most extensional principal axis of the total strain rate $S_{ij}(\bf{x})$ switches from the $\hat{\textbf{z}}$-axis to lie in the $r$-$\theta$ plane.[]{data-label="burgf2"}](Fig3){width="3.5in"}
![Accuracy of the Taylor expansion for the local vorticity in (\[bs8\]) for a Burgers vortex, showing results for 6th order approximation. In each panel, solid black curve shows actual vorticity profile, and red dashed curve gives approximated vorticity from derivatives at location marked by square.[]{data-label="TaylorFit"}](Fig4){width="3.5in"}
![\[fig:wide\]Convergence of background strain rate field $S_{ij}^{B}(\bf{x})$ for a Burgers vortex, obtained from total strain rate field $S_{ij}(\bf{x})$ using (\[bs18\]) for various $(n,\tilde{R})$ combinations, where $\tilde{R}\equiv
R/\lambda_\nu$. Shown are effects of increasing $\tilde{R}$ for fixed $n = 6$ ($top$), increasing $n$ for fixed $\tilde{R} = 0.65$ ($middle$), and increasing $n$ and $\tilde{R}$ simultaneously ($bottom$). The dashed horizontal lines follow from (\[switch1\]) and (\[axisswitchparam\]).[]{data-label="burgf3"}](Fig5){width="3in"}
Convergence of the Background Strain
------------------------------------
The accuracy with which (\[bs18\]) can recover the background strain field $S_{ij}^B(\textbf{x})$ that acts on a concentrated vortical structure depends on how well the expansion in (\[bs8\]) represents the vorticity field within the local spherical neighborhood $R$. Figure \[TaylorFit\] shows the results of a local sixth-order Taylor series approximation for the vorticity in (\[burg1\]) at various radial locations across the Burgers vortex. In each panel, the blue square marks the location $\bf{x}$ at which the sphere is centered, and the red dashed curve shows the resulting Taylor series approximation for the vorticity. On the axis of the vortex, the approximated vorticity field correctly accounts for most of the circulation in the vortex, and thus the induced strain field from the vortex will be reasonably approximated. Off the axis, the approximation becomes increasingly poorer, but the $1/r^2$ decrease in the Biot-Savart kernel in (\[bs1a\]) nevertheless renders it adequate to account for most of the vortex-induced strain rate field. At the largest radial location, corresponding to the bottom right panel of Fig.\[TaylorFit\], the approximation becomes relatively poor, however at large $\eta$ values the vortex-induced strain is sufficiently small that it is unlikely to lead to alignment switching for typical $\Omega$ values.
Figure \[burgf3\] shows the shear component $S_{r\theta}^{B}(\eta)$ of the background strain rate tensor obtained via (\[bs18\]) for various $n$ and $\tilde{R}$ as a function of $\eta$. In each panel, the black curve shows the total strain rate $S_{ij}(\eta)$ and the colored curves show the background strain rate $S_{ij}^{B}(\eta)$ from (\[bs18\]) for the $(n,\tilde{R})$ combinations listed. The horizontal dashed line corresponding to $\Omega = (3/2)\Omega^{*}$ reflects the relative vortex strength, and shows the range of $\eta$ where the anomalous alignment switching occurs due to the vortex-induced strain field. Wherever the $-S^B_{r\theta}$ curves are above this line, the vorticity there will be aligned with the intermediate principal axis of the $combined$ strain rate tensor $S_{ij}$. Figure \[burgf3\]$(a)$ examines the effect of increasing the radius $\tilde{R}$ of the spherical region for fixed order $n =
6$. It is apparent that with increasing $\tilde{R}$ the resulting $-S_{r\theta}^{B}$ converges toward the correct background strain field in (\[convergence\]). For the value of $\Omega$ shown, it can be seen that for $R \gtrsim 0.5 \lambda_{\nu}$ the resulting $S_{r\theta}^{B}$ is everywhere below the horizontal dashed line, indicating that the vorticity everywhere is aligned with the most extensional principal axis of the resulting background strain rate tensor $S_{ij}^{B}(\textbf{x})$ from (\[bs18\]).
In Fig. \[burgf3\]$(b)$ similar results are shown for the effect of increasing the order $n$ of the expansion for the vorticity field for fixed $\tilde{R} = 0.65$. It is apparent that the effect of $n$ is somewhat smaller than for $\tilde{R}$ in Fig. \[burgf3\]$(a)$. Moreover, the results suggest that the series in (\[bs18\]) alternates with increasing order $n$. For this $\Omega$ and $\tilde{R}$, even $n = 3$ is seen to be sufficient to remove most of the vortex-induced shear strain, and thus reduce $S_{r\theta}^{B}(\textbf{x})$ below the horizontal dashed line. For these parameters, the $S_{r\theta}^{B}$ field from (\[bs18\]) would thus reveal alignment of the vorticity with the most extensional principal axis of the background strain tensor throughout the entire field.
Figure \[burgf3\]$(c)$ shows the combined effects of increasing both $\tilde{R}$ and $n$, in accordance with the expectation that larger $\tilde{R}$ should require a higher order $n$ to adequately represent the vorticity field within the spherical region. The shear strain rate field shows convergence to the correct background strain field in (\[convergence\]). The convergence of the shear strain rate $S_{r\theta}^B(\textbf{x})$ to zero in the vicinity of the vortex core is of particular importance. The systematic reduction in the peak remaining shear stress indicates that, even for increasingly stronger vortices or increasingly weaker background strain fields as measured by $\Omega$, the resulting $S_{r\theta}^B(\textbf{x})$ from (\[bs18\]) will reveal the alignment of all the vorticity in such a structure with the most extensional principal strain axis of the background strain field.
Vorticity Alignment in Turbulent Flows
======================================
Having seen in the previous Section how (\[bs18\]) is able to reveal the expected alignment of vortical structures with the most extensional eigenvector of the [*background*]{} strain rate in which they reside, in this Section we apply it to obtain insights into the vorticity alignment in turbulent flows. In particular, we examine the alignment at every point ${\bf x}$ of the vorticity $\bm{\omega}$ relative to the eigenvectors of the total strain rate tensor field $S_{ij}(\bf{x})$ and those of the background strain field data $S_{ij}^{B}(\bf{x})$. This analysis uses data from a highly-resolved, three-dimensional, direct numerical simulation (DNS) of statistically stationary, forced, homogeneous, isotropic turbulence [@Schumacher2007; @Schumacher2007a]. The simulations correspond to a periodic cube with sides of length of $2\pi$ resolved by $2048^3$ grid points. The Taylor microscale Reynolds number $R_{\lambda}$ is 107.
The DNS data were generated by a pseudospectral method with a spectral resolution that exceeds the standard value by a factor of eight. As a result, the highest wavenumber corresponds to $k_{max}\eta_K=10$, and the Kolmogorov lengthscale $\eta_{K}=\nu^{3/4}/\langle\epsilon\rangle^{1/4}$ is resolved with three grid spacings. This superfine resolution makes it possible to apply the result in (\[bs18\]) for relatively high orders $n$, which require accurate evaluation of high-order derivatives of the DNS data. In Schumacher [*et al.*]{} [@Schumacher2007] it was demonstrated that derivatives up to order six are statistically converged. More details on the numerical simulations are given in Refs. [@Schumacher2007; @Schumacher2007a].
Figure \[burgf4\] gives a representative sample of the DNS data, where the instantaneous shear component $S_{12}$ of the total strain rate tensor field $S_{ij}(\bf{x})$ is shown in a typical two-dimensional intersection through the $2048^3$ cube. The data can be seen to span nearly 700 $\eta_K$ in each direction. The $512^2$ box at the lower left of Fig. \[burgf4\] is used here to obtain initial results for alignment of the vorticity with the eigenvectors of the background strain rate tensor.
The background strain rate tensor field $S_{ij}^{B}(\bf{x})$ is first extracted via (\[bs18\]) from $S_{ij}({\bf x})$ for $n=3$ and various $\left(R/\eta_K\right)$. Higher-order evaluation of the background strain rate is not feasible, as the results in Ref. [@Schumacher2007] show that only spatial derivatives of the velocity field up to order six can be accurately obtained from these high-resolution DNS data. For $n=4$, the expansion in (\[bs18\]) involves seventh-order derivatives of the velocity field, and the background strain evaluation becomes limited due to the grid resolution. The results are shown and compared in Fig.\[burgf5\], where the shear component $S_{12}$ of the full strain rate tensor is shown at the top, and the corresponding nonlocal (background) component $S_{12}^{B}$ and local component $S_{12}^{R}$ are shown, respectively, in the left and right columns for $\left(R/\eta_K\right)=2.5$ ($top$ $row$), $3.5$ ($middle$ $row$), and $4.5$ ($bottom$ $row$). Consistent with the results from the Burgers vortex in Fig. \[burgf3\], as $\left(R/\eta_K\right)$ increases the magnitude of the extracted local strain rate in the right column increases. However, for the largest $\left(R/\eta_K\right)=4.5$ case, $n=3$ appears to be too small to adequately represent the local vorticity field. This leads to truncation errors which are manifested as strong ripples in the background and local strain fields (see panels (*f*) and (*g*)).
![Instantaneous snapshot of total strain rate component field $S_{12}(\bf{x})$ in a two-dimensional slice through a highly-resolved three-dimensional $(2048^3)$ DNS of homogeneous, isotropic turbulence [@Schumacher2007; @Schumacher2007a]. Axes are given both in grid coordinates ($i=1\ldots2048$) and normalized by the Kolmogorov length $\eta_K$. Box indicates region in which background strain rate field $S_{ij}^{B}(\bf{x})$ is computed in Fig. \[burgf5\].[]{data-label="burgf4"}](Fig6){width="3.5in"}
![Total strain rate component field $S_{12}(\bf{x})$ ($a$), with corresponding results from (\[bs18\]) for nonlocal (background) field $S_{12}^{B}(\bf{x})$ ($left$) and local field $S_{12}^{R}(\bf{x})$ ($right$) for $\left(R/\eta_K\right)=2.5$ ($b,c$), $\left(R/\eta_K\right)=3.5$ ($d,e$), and $\left(R/\eta_K\right)=4.5$ ($f,g$), all with $n=3$.[]{data-label="burgf5"}](Fig7){width="3.5in"}
The results in Fig. \[burgf5\] thus indicate that radii up to $\left(R/\eta_K\right)=3.5$ in combination with $n=3$ can be used to assess alignment of the vorticity vector with the eigenvectors of the background strain rate field. Figure \[alignments\] shows the probability densities of the alignment cosines for the vorticity vector with the total strain rate tensor and with the background strain rate tensors from (\[bs18\]). We compare $S_{ij}$ (Fig. \[alignments\]*a*) with $S_{ij}^B$ for $\left(R/\eta_K\right)=2.5, n=3$ (Fig.\[alignments\]*b*) and $S_{ij}^B$ for $\left(R/\eta_K\right)=3.5, n=3$ (Fig.\[alignments\]*c*). The results for alignment with the total strain rate tensor are essentially identical to the anomalous alignment seen in numerous other DNS studies [@Ashurst1987; @She1991; @Nomura1998] and experimental studies [@Tsinober1992; @Buch1996; @Su1996; @Zeff2003; @Mullin2006], which show the vorticity to be predominantly aligned with the eigenvector corresponding to the intermediate principal strain rate. However, the results for the Burgers vortex in the previous section show that such anomalous alignment with the eigenvectors of the total strain rate tensor is expected when the local vortex strength parameter $\Omega$ is sufficiently large to cause alignment switching.
{width="5in"}
By comparison, the results in Fig. \[alignments\] (*b*) and (*c*) obtained for the alignment cosines of the vorticity vector with the $background$ strain rate tensor $S_{ij}^{B}$ from (\[bs18\]) show a significant decrease in alignment with the intermediate eigenvector, and an increase in alignment with the most extensional eigenvector. While data in panel (*b*) show only a slight change compared to those in (*a*), the results in panel (*c*) demonstrate that our decomposition can indeed diminish the anomalous alignment significantly. This is consistent with the results for the Burgers vortex in the previous Section, and with the hypothesis that the alignment switching mechanism due to the local contribution $S_{ij}^{R}$ to the total strain rate tensor is the primary reason for the anomalous alignment seen in earlier studies. It is also consistent with the expected equilibrium alignment from (\[bs0\]). While a more detailed study is needed to examine possible nonequilibrium contributions to the alignment distributions associated with eigenvector rotations of the background strain field, as well as to definitively determine the $R$ and $n$ convergence of the background strain rate tensor in Fig. \[burgf5\], the present findings support both the validity of the result in (\[bs18\]) for extracting the background strain rate tensor field $S_{ij}^{B}(\bf{x})$ from the total strain rate tensor field $S_{ij}(\bf{x})$, and the hypothesis that at least much of the anomalous alignment of vorticity in turbulent flows is due to the differences between the total and background strain rate tensors and the resulting alignment switching noted herein.
Concluding Remarks
==================
We have developed a systematic and exact result in (\[bs18\]) that allows the local and nonlocal (background) contributions to the total strain rate tensor $S_{ij}$ at any point $\bf{x}$ in a flow to be disentangled. The approach is based on a series expansion of the vorticity field in a local spherical neighborhood of radius $R$ centered at the point $\bf{x}$. This allows the background strain rate tensor field $S_{ij}^{B}(\bf{x})$ to be determined via a series of increasingly higher-order Laplacians applied to the total strain rate tensor field $S_{ij}(\bf{x})$. For the Burgers vortex, with increasing radius $R$ relative to the local gradient lengthscale $\lambda_{\nu}$ and with increasing order $n$, we demonstrated convergence of the resulting background strain tensor field to its theoretical form. We also showed that even with limited $R$ and $n$ values, the local contribution to the total strain rate tensor field can be sufficiently removed to eliminate the anomalous alignment switching throughout the flow field. This conclusion is expected to also apply to the more realistic case of a non-uniformly stretched vortex where $S_{zz}=f(z)$ [@Gibbon1999; @Ohkitani2002; @Cuypers2003; @Rossi2004].
Consistent with the results for the Burgers vortex, when (\[bs18\]) was used to determine the background strain rate tensor field $S_{ij}^{B}(\bf{x})$ in highly-resolved DNS data for a turbulent flow, the anomalous alignment seen in previous DNS and experimental studies was substantially reduced. We conclude that (\[bs18\]) allows the local background strain rate tensor to be determined in any flow. Furthermore, we postulate that the vorticity vector field in turbulent flows will show a substantially preferred alignment with the most extensional principal axis of the background strain rate field, and that at least much of the anomalous alignment found in previous studies is simply a reflection of the alignment switching mechanism analyzed in Section III and conjectured by numerous previous investigators.
Lastly, the result in (\[bs18\]) is based on a Taylor series expansion of the vorticity within a spherical neighborhood of radius $R$ around any point $\bf{x}$. Such an expansion inherently involves derivatives of the total strain rate tensor field, which can lead to potential numerical limitations. If larger $R$ and correspondingly higher orders $n$ are needed to obtain accurate evaluations of background strain rate fields, then otherwise identical approaches based on alternative expansions may be numerically advantageous. For instance, an expansion in terms of orthonormal basis functions allows the coefficients to be expressed as integrals over the vorticity field within $r \le R$, rather than as derivatives evaluated at the center point $\bf{x}$. (For example, wavelets have been used to test alignment between the strain rate eigenvectors and the vorticity gradient in two-dimensional turbulence [@Protas2002].) This would allow a result analogous to (\[bs18\]) that can be carried to higher orders with less sensitivity to discretization error. The key conclusion, however, of the present study is that it is possible to evaluate the background strain tensor following the general procedure developed herein, and that when such methods are applied to assess the background strain rate fields in turbulent flows they reveal a substantial increase in the expected alignment of the vorticity vector with the most extensional principal axis of the background strain rate field.
PH and WD acknowledge support from the Air Force Research Laboratory (AFRL) under the Michigan AFRL Collaborative Center for Aeronautical Sciences (MACCAS), and by the National Aeronautics & Space Administration (NASA) Marshall and Glenn Research Centers and the Department of Defense (DoD) under the NASA Constellation University Institutes Project (CUIP) under Grant No. NCC3-989. JS acknowledges support by the German Academic Exchange Service (DAAD) and by the Deutsche Forschungsgemeinschaft (DFG) under grant SCHU 1410/2. The direct numerical simulations have been carried out within the Deep Computing Initiative of the DEISA consortium on 512 CPUs of the IBM-p690 cluster JUMP at the John von Neumann Institute for Computing at the Research Centre Jülich (Germany).
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[^1]: email: [email protected] (corresponding author)
[^2]: email: [email protected]
[^3]: email: [email protected]
|
---
abstract: 'We establish a hierarchy of quantum dilogarithm identities associated to a sequence of triangular shaped quivers. The tetrahedron equation plays a key role in our construction.'
---
[ **Tetrahedron equation\
and cyclic quantum dilogarithm identities** ]{}
Introduction. Main results.
===========================
Fix $q \in (0,1)$. The [*quantum exponential function*]{} is the following formal series $$\label{Sx}
{\left\langle x \right\rangle_{\!q}} = \sum_{n = 0}^{\infty}
\frac{(-x)^n }{(1-q)\ldots(1-q^{n})} \,.$$ It is well known that if $\SX$ and $\SY$ are two $q$–commuting indeterminates, that is they satisfy the commutation relation $$\label{qcomm}
\SY \SX = q \, \SX \SY \,,$$ then the following identities hold: $$\begin{aligned}
\label{qexp}
{}& {\left\langle \SX \right\rangle_{\!q}} \, {\left\langle \SY \right\rangle_{\!q}} =
{\left\langle \SX + \SY \right\rangle_{\!q}} \,, & \\[1mm]
{}& \label{pent}
{\left\langle \SX \right\rangle_{\!q}} \, {\left\langle \SX\SY \right\rangle_{\!q}} \, {\left\langle \SY \right\rangle_{\!q}} =
{\left\langle \SY \right\rangle_{\!q}} \, {\left\langle \SX \right\rangle_{\!q}} \,. &\end{aligned}$$ The first one is due to Schützenberger [@Sch1]. The second identity was found in [@FK1] and is now commonly called the [*pentagon identity*]{}. It is also often called a [*quantum dilogarithm identity*]{} because it is closely related to the five–term dilogarithm identity [@FK1; @Vo1].
Now take three pairwise $q$–commuting indeterminates $\SX$, $\SY$, and $\SZ$, $$\label{XYZ}
\SY \SX = q \, \SX \SY \,, \qquad
\SX \SZ = q \, \SZ \SX \,, \qquad
\SZ \SY = q \, \SY \SZ \,.$$ Following [@KV1; @Se1], we utilize the pentagon relation twice and find that $$\label{twice}
{\left\langle \SX \right\rangle_{\!q}} \, {\left\langle \SZ \right\rangle_{\!q}} \, {\left\langle \SX\SY \right\rangle_{\!q}} \, {\left\langle \SY \right\rangle_{\!q}} =
{\left\langle \SZ \right\rangle_{\!q}} \, {\left\langle \SZ\SX \right\rangle_{\!q}} \, {\left\langle \SX \right\rangle_{\!q}} \,
{\left\langle \SX\SY \right\rangle_{\!q}} \, {\left\langle \SY \right\rangle_{\!q}} =
{\left\langle \SZ \right\rangle_{\!q}} \, {\left\langle \SZ\SX \right\rangle_{\!q}} \, {\left\langle \SY \right\rangle_{\!q}} \, {\left\langle \SX \right\rangle_{\!q}} .$$ Note that $\SX\SY$ and $\SZ$ commute. Permuting ${\left\langle \SX\SY \right\rangle_{\!q}}$ and ${\left\langle \SZ \right\rangle_{\!q}}$ on the l.h.s, we bring identity (\[twice\]) to the form $\ST = \rho (\ST)$, where $\ST={\left\langle \SX \right\rangle_{\!q}} \, {\left\langle \SX\SY \right\rangle_{\!q}} \, {\left\langle \SZ \right\rangle_{\!q}} \, {\left\langle \SY \right\rangle_{\!q}}$ and $\rho: \SX \to \SZ, \SY \to \SX, \SZ \to \SY$ is a third order automorphism of the associative algebra defined by presentation (\[XYZ\]). Hence follows immediately the following triple identity $$\label{444}
\ST = \rho (\ST) = \rho \left( \rho (\ST) \right) \,,$$ or, explicitly, $$\label{44}
{\left\langle \SX \right\rangle_{\!q}} \, {\left\langle \SX\SY \right\rangle_{\!q}} \, {\left\langle \SZ \right\rangle_{\!q}} \, {\left\langle \SY \right\rangle_{\!q}} =
{\left\langle \SZ \right\rangle_{\!q}} \, {\left\langle \SZ\SX \right\rangle_{\!q}} \, {\left\langle \SY \right\rangle_{\!q}} \, {\left\langle \SX \right\rangle_{\!q}} =
{\left\langle \SY \right\rangle_{\!q}} \, {\left\langle \SY\SZ \right\rangle_{\!q}} \, {\left\langle \SX \right\rangle_{\!q}} \, {\left\langle \SZ \right\rangle_{\!q}} .$$ In view of (\[444\]), we will say that (\[44\]) is a [*cyclic*]{} quantum dilogarithm identity.
The goal of this paper is to obtain a hierarchy of cyclic quantum dilogarithm identities in which (\[44\]) would be the first nontrivial member. For this purpose we will introduce an algebra ${\mathcal T}_N$ with generators assigned to the vertices of a certain quiver $Q_N$. In what follows, $N$ stands for an integer number greater than one.
The quiver $Q_N$ is an oriented graph with vertices which are labelled by pairs of integer numbers $(i,j)$ such that $1\leq i <j \leq N$. The directed edges go from $(i,j)$ to $(i,j+1)$, from $(i,j)$ to $(i+1,j)$, and from $(i+1,j+1)$ to $(i,j)$.
Thus, $Q_N$ has $N \choose {2}$ vertices. For instance, the quivers $Q_3$ and $Q_4$ are
(300,90) (0,38)[$Q_3$:]{} (30,58)[(1,2)]{} (57,62)[(1,0)[16]{}]{} (77,58)[(1,3)]{} (54,35)[(-2,3)[12]{}]{} (52,24)[(2,3)]{} (86,53)[(-2,-3)[12]{}]{} (180,36)[$Q_4$:]{} (210,70)[(1,2)]{} (237,74)[(1,0)[16]{}]{} (257,70)[(1,3)]{} (237,47)[(-2,3)[12]{}]{} (260,13)[(-2,3)[12]{}]{} (284,47)[(-2,3)[12]{}]{} (280,36)[(2,4)]{} (283,74)[(1,0)[16]{}]{} (303,70)[(1,4)]{} (235,36)[(2,3)]{} (260,39)[(1,0)[16]{}]{} (257,1)[(3,4)]{} (288,31)[(-2,-3)[12]{}]{} (264,64)[(-2,-3)[12]{}]{} (310,64)[(-2,-3)[12]{}]{}
Given $Q_N$, one defines in the standard way its skew–symmetric incidence matrix $B$: $B_{(i,j),(i',j')}=-B_{(i',j'),(i,j)}=1$ if there is a directed edge going from $(i,j)$ to $(i',j')$, and $B_{(i,j),(i',j')}=0$ if the vertices $(i,j)$ and $(i',j')$ are not connected.
\[QA\] The algebra ${\mathcal T}_N$ associated with the quiver $Q_N$ is a unital associative algebra over ${\mathbb R}$ with ${N} \choose {2}$ generators $\SZ_{ij}$, $1\leq i <j \leq N$, and the following defining relations: $$\begin{aligned}
\label{ZZ}
\SZ_{ij} \, \SZ_{i'j'}=
q^{ B_{(i,j),(i',j')}} \, \SZ_{i'j'} \, \SZ_{ij} \,.\end{aligned}$$
In particular, ${\mathcal T}_2$ is generated by a single generator. The algebra ${\mathcal T}_3$ (with generators $\SZ_{12}$, $\SZ_{13}$, $\SZ_{23}$ renamed $\SX$, $\SZ$, $\SY$) coincides with the algebra defined by presentation (\[XYZ\]).
\[AAA\] Let $I$ be a subset of vertices of $Q_N$. Consider a linear homomorphism $\sigma_I$ such that $\sigma_I(\SZ_{ij})=0$ if $(i,j) \in I$ and $\sigma_I(\SZ_{ij})=\SZ_{ij}$ otherwise. Since relations (\[ZZ\]) are homogeneous, $\sigma_I$ is an algebra homomorphism from ${\mathcal T}_N$ to its subalgebra. In particular, sending $\SZ_{iN}$ to zero for all $i$, we reduce ${\mathcal T}_N$ to ${\mathcal T}_{N-1}$. Thus, we have a chain of subalgebra inclusions: ${\mathcal T}_2 \subset {\mathcal T}_3 \ldots
\subset {\mathcal T}_{N-1} \subset {\mathcal T}_N$.
\[remcent\] The center ${\mathcal Z}_N$ of the algebra ${\mathcal T}_N$ has dimension $N/2$ if $N$ is even and $(N-1)/2$ if $N$ is odd. (See Section 2.2 for more details.)
The quiver $Q_N$ is mapped to itself by a clockwise rotation about its geometric center by $2\pi/3$. Hence, the algebra ${\mathcal T}_N$ admits a third order automorphism $\rho$, $$\label{rhodef}
\rho \bigl(\SZ_{ij} \bigr)
= \SZ_{j-i,N+1-i} \,.$$
Furthermore, the quiver $Q_N$ is mapped into a dual quiver (were all the arrows are reverted) by a reflection with respect to either of its three axes of symmetry passing through the corner vertices. Hence, the algebra ${\mathcal T}_N$ admits three involutive anti–automorphisms: $$\label{mudef}
\mu_1 \bigl(\SZ_{ij} \bigr)
= \SZ_{j-i,j} \, \qquad
\mu_2 \bigl(\SZ_{ij} \bigr)
= \SZ_{N+1-j,N+1-i} \, \qquad
\mu_3 \bigl(\SZ_{ij} \bigr)
= \SZ_{i,N+1+i-j} \,.$$ Note that $$\label{murho}
\mu_1 \circ \mu_2 = \mu_2 \circ \mu_3
= \mu_3 \circ \mu_1 = \rho \,.$$
Let us adopt the following notations for products of non–commuting factors. Let $\prec$ be the lexicographic order relation of elements of a set $\Lambda \subset {\mathbb Z}^m$. Then $\prod\limits_{\lambda \in \Lambda}^{\rightarrow} f_\lambda$ and $\prod\limits_{\lambda \in \Lambda}^{\leftarrow} f_\lambda$ stand for ordered products, where $f_\lambda$ is put to the right (respectively, to the left) of all $f_{\lambda'}$ such that $\lambda' \prec \lambda$. In particular, if $\Lambda = [1,k] \subset {\mathbb Z}$, then $\prod\limits_{ \lambda \in \Lambda}^{\rightarrow} f_\lambda
= f_1 \ldots f_k$ and $\prod\limits_{\lambda \in \Lambda}^{\leftarrow} f_\lambda
= f_k \ldots f_1$.
Let $\Lambda_N \subset {\mathbb Z}^3$ be the following discrete tetrahedron containing ${{N+1} \choose {3}}$ points: $$\Lambda_N =
\bigl\{ \lambda=(a,b,c) \ | \
1 \leq a < b < c \leq N+1 \bigr\} \,.$$ To each point $\lambda=(a,b,c) \in \Lambda_N$ we associate the following element of ${\mathcal T}_{N}$: $$\label{Zkdef}
{{\mathbb R}_{ \lambda }} =
{\biggl\langle \,
\prod_{0 \leq k \leq c-b-1}^{\rightarrow} \SZ_{a+k,b+k} \biggr\rangle_{\!q}} \,.$$
Now we can define an analogue of the element $\ST$ used in (\[444\]).
\[defT\] $\ST_N \in {\mathcal T}_{N}$ is the following lexicographically ordered product: $$\label{TTlex}
\ST_N =
\prod_{\lambda \in \Lambda_N}^{\rightarrow}
{{\mathbb R}_{\lambda}} \,.$$
In particular, we have $\ST_2={{\mathbb R}_{ 123 }} = {\left\langle \SZ_{12} \right\rangle_{\!q}}$ and $\ST_3=
{{\mathbb R}_{ 123 }}{{\mathbb R}_{ 124 }}{{\mathbb R}_{ 134 }}{{\mathbb R}_{ 234 }} = \\
{\left\langle \SZ_{12} \right\rangle_{\!q}} {\left\langle \SZ_{12} \SZ_{23} \right\rangle_{\!q}}
{\left\langle \SZ_{13} \right\rangle_{\!q}}{\left\langle \SZ_{23} \right\rangle_{\!q}}$.
Some factors in the product (\[TTlex\]) mutually commute, as, e.g., ${{\mathbb R}_{ 124 }}$ and ${{\mathbb R}_{ 134 }}$ in $\ST_3$. Therefore, $\ST_N$ admits a number of equivalent expressions obtained by permutations of such factors. Some of these expressions are given in Lemma \[TW23\].
We will say that a quantum dilogarithm identity is of the type $n \sim m$ if it involves $n$ quantum exponentials with monomial arguments on one side and $m$ on the other. For instance, (\[pent\]) and (\[44\]) are of the type $2 \sim 3$ and $4 \sim 4$, respectively. The main result of the present work is a family of cyclic quantum dilogarithm identities of the type ${{N+1} \choose {3}} \sim {{N+1} \choose {3}}$.
\[main\] Identities $$\begin{aligned}
\label{Tmu}
{}&
\ST_N = \mu_1 \bigl( \ST_N \bigr) =
\mu_2 \bigl( \ST_N \bigr) =
\mu_3 \bigl( \ST_N \bigr) \,, & \\[2mm]
\label{Tr}
{}&
\ST_N = \rho \bigl( \ST_N \bigr)
= \rho \bigl( \rho \bigl( \ST_N \bigr) \bigr) \,, &\end{aligned}$$ hold for any integer $N \geq 2$.
Identities (\[Tmu\]) involve the anti–automorphism transformations $\mu_a$ corresponding to a reflection rather than a rotation symmetry of the quiver. Nevertheless, two of these identities are almost cyclic in the following sense. We will see below (cf. Lemma \[mu2T\]) that $\mu_2$ applied to $\ST_N$ acts almost as an identical transformation just permuting some commuting quantum exponentials. (As an example, apply to (\[44\]) an anti–automorphism which maps $\SZ$ to itself and exchanges $\SX$ and $\SY$.) Along with (\[murho\]) this implies that, again up to a permutation of commuting factors, $\mu_1$ applied to $\ST_N$ acts as $\rho$, and $\mu_3$ applied to $\ST_N$ acts as $\rho^{-1}=\rho \circ \rho$.
It is clear from the definition (\[Sx\]) of the quantum exponential that ${\left\langle 0 \right\rangle_{\!q}}=1$. This, along with Remark \[AAA\], implies the following.
\[QQ\] Let $I$ be a subset of the set of vertices of $Q_N$. In the identity (\[Tr\]), replace with unity every quantum exponential that contains at least one $\SZ_{ij}$ with $(i,j) \in I$. The result is a correct quantum dilogarithm identity.
Note that the resulting reduced identity is not necessarily cyclic. For instance, sending $\SZ$ to $0$ in the $4 \sim 4$ identity (\[44\]), we obtain the pentagon identity (\[pent\]).
Quantum dilogarithm identities associated with various quivers have close connections with (quantum) cluster algebras and Y–systems, see, e.g. [@BZ1; @FG1; @KN1; @Ke1]. Our motivation for studying a specific family of such identities is an observation that they are [*accessible*]{} from the $4 \sim 4$ identities (\[44\]). For instance, let us demonstrate how to derive the equality $\ST_4 = \rho \bigl( \ST_4 \bigr)$ without invoking the pentagon relation but using only the $4 \sim 4$ identities (\[44\]): $$\begin{aligned}
\ST_4
{}&= {\left\langle \SZ_{12} \right\rangle_{\!q}} {\left\langle \SZ_{12} \SZ_{23} \right\rangle_{\!q}}
{\left\langle \SZ_{12} \SZ_{23} \SZ_{34} \right\rangle_{\!q}}
{\left\langle \SZ_{13} \right\rangle_{\!q}} {\left\langle \SZ_{13} \SZ_{24} \right\rangle_{\!q}}
{\left\langle \SZ_{14} \right\rangle_{\!q}} {\left\langle \SZ_{23} \right\rangle_{\!q}}
{\left\langle \SZ_{23} \SZ_{34} \right\rangle_{\!q}}
{\left\langle \SZ_{24} \right\rangle_{\!q}} {\left\langle \SZ_{34} \right\rangle_{\!q}} \\
{}&= \underline{ {\left\langle \SZ_{12} \right\rangle_{\!q}} {\left\langle \SZ_{12} \SZ_{23} \right\rangle_{\!q}}
{\left\langle \SZ_{13} \right\rangle_{\!q}} {\left\langle \SZ_{23} \right\rangle_{\!q}} }
{\left\langle \SZ_{12} \SZ_{23} \SZ_{34} \right\rangle_{\!q}}
{\left\langle \SZ_{13} \SZ_{24} \right\rangle_{\!q}}
{\left\langle \SZ_{23} \SZ_{34} \right\rangle_{\!q}} {\left\langle \SZ_{14} \right\rangle_{\!q}}
{\left\langle \SZ_{24} \right\rangle_{\!q}} {\left\langle \SZ_{34} \right\rangle_{\!q}} \\
{}& = {\left\langle \SZ_{13} \right\rangle_{\!q}}
{\left\langle \SZ_{13}\SZ_{12} \right\rangle_{\!q}} {\left\langle \SZ_{23} \right\rangle_{\!q}}
\underline{ {\left\langle \SZ_{12} \right\rangle_{\!q}}
{\left\langle \SZ_{12} \SZ_{23} \SZ_{34} \right\rangle_{\!q}}
{\left\langle \SZ_{13} \SZ_{24} \right\rangle_{\!q}}
{\left\langle \SZ_{23} \SZ_{34} \right\rangle_{\!q}} }
{\left\langle \SZ_{14} \right\rangle_{\!q}}
{\left\langle \SZ_{24} \right\rangle_{\!q}} {\left\langle \SZ_{34} \right\rangle_{\!q}} \\
{}&= {\left\langle \SZ_{13} \right\rangle_{\!q}} {\left\langle \SZ_{23} \right\rangle_{\!q}}
{\left\langle \SZ_{13}\SZ_{12} \right\rangle_{\!q}}
{\left\langle \SZ_{13} \SZ_{24} \right\rangle_{\!q}} {\left\langle \SZ_{13} \SZ_{24} \SZ_{12} \right\rangle_{\!q}}
{\left\langle \SZ_{23} \SZ_{34} \right\rangle_{\!q}} {\left\langle \SZ_{12} \right\rangle_{\!q}}
{\left\langle \SZ_{14} \right\rangle_{\!q}}
{\left\langle \SZ_{24} \right\rangle_{\!q}} {\left\langle \SZ_{34} \right\rangle_{\!q}} \\
{}& ={\left\langle \SZ_{13} \right\rangle_{\!q}} {\left\langle \SZ_{23} \right\rangle_{\!q}}
{\left\langle \SZ_{13} \SZ_{24} \right\rangle_{\!q}}
\underline{ {\left\langle \SZ_{13}\SZ_{12} \right\rangle_{\!q}}
{\left\langle \SZ_{13} \SZ_{12} \SZ_{24} \right\rangle_{\!q}}
{\left\langle \SZ_{14} \right\rangle_{\!q}} {\left\langle \SZ_{24} \right\rangle_{\!q}} }
{\left\langle \SZ_{23} \SZ_{34} \right\rangle_{\!q}} {\left\langle \SZ_{34} \right\rangle_{\!q}} {\left\langle \SZ_{12} \right\rangle_{\!q}} \\
{}&= {\left\langle \SZ_{13} \right\rangle_{\!q}} {\left\langle \SZ_{23} \right\rangle_{\!q}}
{\left\langle \SZ_{13} \SZ_{24} \right\rangle_{\!q}}
{\left\langle \SZ_{14} \right\rangle_{\!q}}
{\left\langle \SZ_{14} \SZ_{13} \SZ_{12} \right\rangle_{\!q}}
{\left\langle \SZ_{24} \right\rangle_{\!q}} {\left\langle \SZ_{13}\SZ_{12} \right\rangle_{\!q}}
{\left\langle \SZ_{23} \SZ_{34} \right\rangle_{\!q}} {\left\langle \SZ_{34} \right\rangle_{\!q}} {\left\langle \SZ_{12} \right\rangle_{\!q}} \\
{}&= {\left\langle \SZ_{13} \right\rangle_{\!q}} {\left\langle \SZ_{13} \SZ_{24} \right\rangle_{\!q}}
{\left\langle \SZ_{14} \right\rangle_{\!q}}
{\left\langle \SZ_{14} \SZ_{13} \SZ_{12} \right\rangle_{\!q}}
\underline{ {\left\langle \SZ_{23} \right\rangle_{\!q}}
{\left\langle \SZ_{23} \SZ_{34} \right\rangle_{\!q}} {\left\langle \SZ_{24} \right\rangle_{\!q}}
{\left\langle \SZ_{34} \right\rangle_{\!q}} }
{\left\langle \SZ_{13}\SZ_{12} \right\rangle_{\!q}} {\left\langle \SZ_{12} \right\rangle_{\!q}}
\\
{}&=
\underline{ {\left\langle \SZ_{13} \right\rangle_{\!q}} {\left\langle \SZ_{13} \SZ_{24} \right\rangle_{\!q}}
{\left\langle \SZ_{14} \right\rangle_{\!q}} {\left\langle \SZ_{24} \right\rangle_{\!q}} }
{\left\langle \SZ_{14} \SZ_{13} \SZ_{12} \right\rangle_{\!q}}
{\left\langle \SZ_{24} \SZ_{23} \right\rangle_{\!q}}
{\left\langle \SZ_{34} \right\rangle_{\!q}} {\left\langle \SZ_{23} \right\rangle_{\!q}}
{\left\langle \SZ_{13}\SZ_{12} \right\rangle_{\!q}} {\left\langle \SZ_{12} \right\rangle_{\!q}}
\\
{}&=
{\left\langle \SZ_{14} \right\rangle_{\!q}}
{\left\langle \SZ_{14} \SZ_{13} \right\rangle_{\!q}} {\left\langle \SZ_{24} \right\rangle_{\!q}}
{\left\langle \SZ_{13} \right\rangle_{\!q}}
{\left\langle \SZ_{14} \SZ_{13} \SZ_{12} \right\rangle_{\!q}}
{\left\langle \SZ_{24} \SZ_{23} \right\rangle_{\!q}}
{\left\langle \SZ_{34} \right\rangle_{\!q}}
{\left\langle \SZ_{13}\SZ_{12} \right\rangle_{\!q}} {\left\langle \SZ_{23} \right\rangle_{\!q}} {\left\langle \SZ_{12} \right\rangle_{\!q}}
\\
{}& = {\left\langle \SZ_{14} \right\rangle_{\!q}}
{\left\langle \SZ_{14} \SZ_{13} \right\rangle_{\!q}}
{\left\langle \SZ_{14} \SZ_{13} \SZ_{12} \right\rangle_{\!q}}
{\left\langle \SZ_{24} \right\rangle_{\!q}} {\left\langle \SZ_{24} \SZ_{23} \right\rangle_{\!q}}
{\left\langle \SZ_{34} \right\rangle_{\!q}} {\left\langle \SZ_{13} \right\rangle_{\!q}}
{\left\langle \SZ_{13}\SZ_{12} \right\rangle_{\!q}} {\left\langle \SZ_{23} \right\rangle_{\!q}}
{\left\langle \SZ_{12} \right\rangle_{\!q}}
\\{}& = \rho \bigl(\ST_4 \bigr).\end{aligned}$$ The underlined terms were transformed by applying the $4\sim 4$ identities (\[44\]). The remaining transformations changed only the order of commuting factors or the order of commuting generators in the arguments of quantum exponentials.
We will give a proof of the identities listed in Theorem \[main\] which makes it evident that they are accessible from the $4 \sim 4$ identities (\[44\]) for all $N$. The origin of this accessibility is that these identities stem from identities for certain words in a group whose generators satisfy the [*tetrahedron equation*]{}, $$\label{RTE0}
R_{abc} \, R_{abd} \, R_{acd} \, R_{bcd} =
R_{bcd} \, R_{acd} \, R_{abd} \, R_{abc} \,.$$
The paper is organized as follows. Section 2 contains auxiliary statements which we need to combine together in order to prove Theorem \[main\]. Namely, in Section 2.1 we consider families of groups ${\mathcal B}(n,N)$ whose generators satisfy the Yang–Baxter equation (for $n=2$), the tetrahedron equation (for $n=3$), or their higher analogues (for ). The two key technical results here are an identity for certain words containing all the generators of the group ${\mathcal B}(n,N)$ and relation of these words to the element $\ST_N$. In Section 2.2 we describe the center of ${\mathcal T}_N$. In Section 2.3 we consider a local tensor space representation $\phi$ for ${\mathcal B}(3,N)$. In Section 2.4 we explain how evaluation of the above mentioned identity for words of ${\mathcal B}(3,N)$ in the representation $\phi$ yields ultimately the desired quantum dilogarithm identities for an arbitrary $N$. Appendix contains proofs of all statements given in Section 2.
Main technical ingredients
==========================
Below we assume that $N$ and $n$ are positive integers and $N \geq n$.
A group with $n$–simplex relations
----------------------------------
\[BRN\] ${\mathcal B}(n,N)$ is a group with ${N} \choose {n}$ generators $R_{a_1,\ldots,a_n}$, where $1 \leq a_1 < a_2 < \ldots < a_n \leq N$. The group is defined by the following presentation:
- The generators commute, $$\label{Rcomm}
R_{a_1,\ldots,a_n} \, R_{b_1,\ldots,b_n} =
R_{b_1,\ldots,b_n} \, R_{a_1,\ldots,a_n} \,,$$ unless the set $\{a_1,\ldots,a_n\} \bigcap \{b_1,\ldots,b_n\}$ contains exactly $(n-1)$ element.
- If $N > n$, the generators satisfy the following ${N} \choose {n+1}$ relations: $$\label{Rgen}
\prod_{1\leq j \leq n+1}^{\rightarrow}
R_{a_1,\ldots,\check{a}_j,\ldots,a_{n+1}}
= \prod_{1\leq j \leq n+1}^{\leftarrow}
R_{a_1,\ldots,\check{a}_j,\ldots,a_{n+1}} \,,$$ where $\check{a}_j$ is dropped.
For $n=1$, relations (\[Rgen\]) imply commutativity, $R_a R_b = R_b R_a$, so that ${\mathcal B}(1,N)$ is an abelian group with $N$ generators.
For $n=2$, relations (\[Rgen\]) have the form of the [*Yang–Baxter equation*]{}, $$\label{RYB}
R_{ab} \, R_{ac} \, R_{bc} = R_{bc} \, R_{ac} \, R_{ab} \,,
\qquad a< b < c \,,$$ and $R_{ab}$ commutes with $R_{a'b'}$ if they have no common index.
For $n=3$, relations (\[Rgen\]) have the form of the [*tetrahedron equation*]{}, $$\label{RTE}
R_{abc} \, R_{abd} \, R_{acd} \, R_{bcd} =
R_{bcd} \, R_{acd} \, R_{abd} \, R_{abc} \,,
\qquad a< b < c < d \,,$$ and $R_{abc}$ commutes with $R_{a'b'c'}$ unless they have exactly two common indices.
Recall that, given a set $\Lambda \subset {\mathbb Z}^m$, we use the symbol $\prod\limits^{\rightarrow}_{\lambda\in\Lambda} f_\lambda$ to denote the lexicographically ordered product of non–commuting factors.
\[defWW\] The word $W(n,N) \in {\mathcal B}(n,N)$ is the lexicographically ordered product of all generators of ${\mathcal B}(n,N)$, that is $$\label{lex}
W(n,N) = \prod_{1\leq a_1 <a_2 \ldots < a_n \leq N
}^{\rightarrow}
R_{a_1,\ldots,a_{n}} \,.$$
In particular, we have $W(n,n) = R_{1,\ldots,n}$ for all $n$ and $W(1,N) = R_1 \ldots R_N$ for all $N$.
Let $\prec'$ be the colexicographic order relation of elements of a set $\Lambda \subset {\mathbb Z}^m$. That is, components of elements of $\Lambda$ are compared starting from the right. For instance, $(a,b) \prec' (c,d)$ iff $b < d$ or $b=d$ and $a<c$. We will denote by $\mathop{{\prod}'}\limits^{\rightarrow}_{\lambda \in \Lambda} f_\lambda$ and $\mathop{{\prod}'}\limits^{\leftarrow}_{\lambda \in \Lambda} f_\lambda$ ordered products where $f_\lambda$ is put to the right (respectively, to the left) of all $f_{\lambda'}$ such that $\lambda' \prec' \lambda$.
\[defWalt\] Define $$\label{Wexp}
W'(n,N) =
\mathop{{\prod}'}\limits^{\rightarrow}_{1\leq a_1 <a_2 \ldots < a_n \leq N }
R_{a_1,\ldots,a_{n}} \,.$$ Then we have the equality $$\label{WWp}
W(n,N) = W'(n,N) \,.$$
For instance, (\[lex\]) yields $W(2,4) = R_{12} R_{13} R_{14} R_{23} R_{24} R_{34}$ while (\[Wexp\]) yields $W'(2,4) = R_{12} R_{13} R_{23} R_{14} R_{24} R_{34}$. These two words coincide since, by (\[Rcomm\]), $R_{14}$ and $R_{23}$ commute. It is also true in general that one needs to use only the commutativity relations (\[Rcomm\]) in order to change the order of factors in (\[Wexp\]) to match that in (\[lex\]).
Let $\eta$ be an involutive anti–automorphism of ${\mathcal B}(n,N)$ such that $$\label{eta}
\eta \bigl( R_{a_1,\ldots,a_{n}} \bigr)
= R_{a_1,\ldots,a_{n}}$$ for all the generators of ${\mathcal B}(n,N)$. Define $$\label{bomW}
\bar{W}(n,N) = \eta \bigl( W(n,N) \bigr) \,,$$ which is a word with the order of factors reverse to that of $W(n,N)$. Our first key technical statement is the following.
\[multitetra\] For all $N \geq n$, we have the equality $$\label{WWr}
W(n,N) = \bar{W}(n,N) \,.$$
A simple inspection of the proof given in the Appendix shows that the invertibility of $R$’s is not really needed. Thus, Theorem \[multitetra\] holds also if ${\mathcal B}(n,N)$ is a semigroup.
\[SYZ\] The presented proof provides a constructive recursive procedure that transforms $W(n,N)$ into $\bar{W}(n,N)$. The transformation involves trivial moves based on (\[Rcomm\]) and moves ${\mathcal R}_{a_1,\ldots,a_{n+1}}$ which transform the l.h.s. of (\[Rgen\]) into its r.h.s. For instance, ${\mathcal R}_{123}\bigl( W(2,4)\bigr)=
R_{23} R_{13} R_{12} R_{14} R_{24} R_{34}$. Let ${\mathcal W}(n,N)$ and $\bar{\mathcal W}(n,N)$ stand for the compositions of such moves in which $\mathcal R$’s are ordered in the same way as $R$’s are ordered in $W(n,N)$ and $\bar{W}(n,N)$. A simple inspection of the proof shows that the l.h.s. of (\[WWr\]) is transformed into its r.h.s. by ${\mathcal W}^*(n+1,N)$, where star means that trivial moves are included when necessary. Moreover, the same transformation is achieved by $\bar{\mathcal W}^*(n+1,N)$ if we start each reordering not from the left but from the right. Thus, the moves satisfy the identity ${\mathcal W}^*(n,N) = \bar{\mathcal W}^*(n,N)$ analogous to (\[WWr\]).
Eqs. (\[lex\]) and (\[Wexp\]) for $n=2$ and $n=3$ read $$\begin{aligned}
\label{W2lex}
& W(2,N) = \prod_{1\leq a < b \leq N}^{\rightarrow}
R_{ab} , \qquad\quad\
W'(2,N)=
\mathop{{\prod}'}\limits^{\rightarrow}_{1\leq a< b \leq N }
R_{ab} \,, \\
\label{W3lex}
& W(3,N) = \prod_{1\leq a < b < c \leq N }^{\rightarrow}
R_{abc} , \qquad
W'(3,N) =
\mathop{{\prod}'}\limits^{\rightarrow}_{1\leq a< b <c \leq N }
R_{abc} \,.\end{aligned}$$ Recall that ${{\mathbb R}_{abc}}$ was defined in (\[Zkdef\]). We introduce also the following elements of ${\mathcal T}_N$: $$\label{Rkdef}
{{\mathbb R}_{ ab }} \, = \!
\prod_{b+1 \leq c \leq N+1}^{\rightarrow} {{\mathbb R}_{ abc }} \,.$$
Let us introduce the following homomorphisms from ${\mathcal B}(2,N)$ and ${\mathcal B}(3,N+1)$ to ${\mathcal T}_N$ (it should be stressed that they are not algebra homomorphisms): $$\begin{aligned}
\label{RabZ}
{} \sharp : R_{ab} \to {{\mathbb R}_{ ab }} \,, \quad
{} \star : R_{abc} \to {{\mathbb R}_{ abc }} \,, \quad
\star\star : R_{abc} \to {{\mathbb R}_{ a,c+a-b,c }} \,.\end{aligned}$$
\[TW23\] The element $\ST_N$ defined in (\[TTlex\]) can be obtained as follows: $$\begin{aligned}
\label{TWRZa}
{}& \ST_N = \bigl( W(2,N) \bigr)^\sharp
= \bigl( W'(2,N) \bigr)^\sharp \,,\\
\label{TWRZb}
{}& \ST_N = \bigl( W(3,N+1) \bigr)^\star =
\bigl( W'(3,N+1) \bigr)^{\star} \,, \\
\label{TWRZc}
{}& \ST_N = \bigl( W(3,N+1) \bigr)^{\star\star} =
\bigl( W'(3,N+1) \bigr)^{\star\star} \,.
\end{aligned}$$
This observation suggests that identities (\[WWr\]) can be recast in some quantum dilogarithm identities. Furthermore, Remark \[SYZ\] indicates that one needs to use the $4 \sim 4$ relation (\[44\]) $N+1 \choose 4$ times in order to derive identity (\[Tr\]) for given $N$.
The center of ${\mathcal T}_N$
------------------------------
Let us assign to each vertex $(i,j)$ of the quiver $Q_N$ a non–negative integer weight $\alpha_{ij}$. Let $\alpha$ denote the vector comprised of those weights taken in the lexicographic order. For instance, $\alpha=(\alpha_{12},\alpha_{13},\alpha_{23})$ if $N=3$.
Monomials $$\label{mona}
\SM(\alpha) = \prod_{1\leq i < j \leq N}^{\rightarrow}
\SZ_{ij}^{\alpha_{ij}} \,, \qquad
\alpha \in {\mathbb Z}_{\geq 0}^{N(N-1)/2}$$ constitute a basis of ${\mathcal T}_N$. We will say that $\alpha_{ij}$ are the weights of $ \SM(\alpha)$.
The algebra ${\mathcal T}_N$ has a nontrivial center ${\mathcal Z}_N$. In particular, for any $N$, if $\alpha^0$ is a vector such that $\alpha^0_{ij}=1$, then $\SM(\alpha^0)$ belongs to ${\mathcal Z}_N$.
\[center\] Let $\chi(N)$ stand for $N/2$ if $N$ is even and for $(N-1)/2$ if $N$ is odd.
- If $ \SM(\alpha) \in {\mathcal Z}_N$, then its weights have the following symmetries $$\label{alsym}
\alpha_{ij} = \alpha_{j-i,N+1-i} =
\alpha_{j-i,j} =
\alpha_{N+1-j,N+1-i} = \alpha_{i,N+1+i-j} \,,$$ and $\SM(\alpha)$ is invariant under the action of $\rho$ and $\mu_k$ defined in (\[rhodef\]) and (\[mudef\]).
- If $ \SM(\alpha), \SM(\alpha') \in {\mathcal Z}_N$, and $\alpha_{1j}=\alpha'_{1j}$ for $j=2,\ldots,\chi(N) {+} 1$, then $\alpha=\alpha'$ and hence $ \SM(\alpha)=\SM(\alpha')$.
- Given $N$ and an arbitrary integer sequence $\beta_{1}, \ldots ,\beta_{\chi(N) }$, there exist $m \in {\mathbb Z}_{\geq 0}$ and a vector $\alpha \in {\mathbb Z}_{\geq 0}^{N(N-1)/2}$ such that $\SM(\alpha) \in {\mathcal Z}_N$ and $\alpha_{1j}=\beta_{j-1} +m$ for $j=2,\ldots,\chi(N) {+} 1$.
In other words, each monomial central element $ \SM(\alpha) \in {\mathcal Z}_N$ is uniquely determined by its weights assigned to the vertices comprising a half of a boundary side of the quiver $Q_N$. It follows also that the dimension of ${\mathcal Z}_N$ is $\chi(N)$.
For instance, for $N=3$ we have $\chi(3)=1$ and $ \SM(\alpha) \in {\mathcal Z}_3$ iff $ \alpha_{13}=\alpha_{23}=\alpha_{12}$. Thus, ${\mathcal Z}_3$ is generated by $\SZ_{12} \SZ_{13} \SZ_{23} $. For $N=4$ we have $\chi(4)=2$ and $ \SM(\alpha)\in {\mathcal Z}_4$ iff $ \alpha_{14}=\alpha_{34}=\alpha_{12}$ and $ \alpha_{23}=\alpha_{24}=\alpha_{13}$. Thus, ${\mathcal Z}_4$ is generated by $\SZ_{12} \SZ_{14} \SZ_{34} $ and $\SZ_{13} \SZ_{23} \SZ_{24} $.
For $N=5$ we have $\chi(5)=2$ and $ \SM(\alpha)\in {\mathcal Z}_5$ iff $ \alpha_{15}=\alpha_{45}=\alpha_{12}$, $ \alpha_{14}=\alpha_{25}=\alpha_{35}=
\alpha_{23}=\alpha_{34}=\alpha_{13}$, and $\alpha_{24}=2\alpha_{13} - \alpha_{12}$. The last relation requires that $2\alpha_{13} \geq \alpha_{12}$.
The last part of Theorem \[center\] shows that the $\chi(N)$ weights that define a central monomial can be taken almost arbitrary (possibly, up to a total shift by an integer $m$). A direct inspection up to $N=8$ suggests that in order to have $m=0$ it suffices to take a non–decreasing integer sequence, $0 \leq \beta_{1} \leq \ldots \leq \beta_{\chi(N) }$.
The generators $\SZ_{ij}$ of ${\mathcal T}_N$ can be constructed as exponential functions of canonically conjugate variables $p_k, x_k$ such that $[x_k, p_{k'}]=\sqrt{-1}\,\delta_{k k'}$. Theorem \[center\] implies that the number of [*degrees of freedom*]{} of ${\mathcal T}_N$, i.e., the minimal number of such pairs $(p_k, x_k)$ is $N(N-2)/4$ if $N$ is even and $(N-1)^2/4$ if $N$ is odd.
A local tensor space representation for tetrahedron equation
------------------------------------------------------------
Let $V$ be a vector space. For $N \geq 2$, we define $S_N = V^{\otimes N(N-1)/2}$. The tensor components of $S_N$ will be labelled in the lexicographic order by pairs of integers $(i,j)$, where $1 \leq i < j \leq N$. For instance, $S_3 = V_{12} \otimes V_{13} \otimes V_{23}$, where all $V_{ij}$ are isomorphic to $V$.
For a given $\SR \in \text{End}\,\bigl(S_3\bigr)$, we denote by $\SR_{abc}$ its canonical extension to a linear operator on $S_N$ which acts non–trivially only on the tensor components $V_{ab}$, $V_{ac}$, and $V_{bc}$. E.g., $\SR_{123}=\SR \otimes id \otimes id \otimes id$ if $N=4$.
Let $\SR \in \text{End}\,\bigl(S_3\bigr)$ be invertible and let its action canonically extended to $S_4$ satisfy the tetrahedron equation, $$\label{RRR}
\SR_{123} \, \SR_{124} \, \SR_{134} \, \SR_{234} =
\SR_{234} \, \SR_{134} \, \SR_{124} \, \SR_{123} \,.$$ A [*local tensor space representation*]{} of ${\mathcal B}(3,N)$ is a homomorphism ${\mathcal B}(3,N) \to \text{End}\,\bigl(S_{N}\bigr)$ sending $R_{abc}$ to $\SR_{abc}$.
Note that the commutativity relations (\[Rcomm\]) hold by construction. Indeed, if $\SR_{abc}$ and $\SR_{a'b'c'}$ have no common pair of indices then they act non–trivially on different tensor components of $S_N$ and hence they commute.
An equation formally identical to (\[RRR\]) was considered in [@FM1]. There, however, it was treated as an operator equation on $V^{\otimes 4}$ with the identification $\SR_{123} =\SR \otimes id$ etc. We treat (\[RRR\]) as an equation on $V^{\otimes 6}$ which is the standard Zamolodchikov’s tetrahedron equation [@Za1] but with a non–standard (double index) labelling of the tensor components.
Let $V$ be the vector space of formal series in $x,x^{-1}$. Then $S_N$ is the vector space of formal series in $x_{ij}, x_{ij}^{-1}$, where $1 \leq i < j \leq N$.
Define operators $\sx_{ij}, \sy_{ij} \in \text{End}\,\bigl(S_N\bigr)$, $1 \leq i < j \leq N$ such that $$\label{xyf}
\begin{aligned}
\bigl(\sx_{ij} \, f)(x_{12},\ldots,x_{ij},\ldots)
{} &= x_{ij} \, f(x_{12},\ldots,x_{ij},\ldots) \,, \\
\bigl(\sy_{ij} \, f)(x_{12},\ldots,x_{ij},\ldots)
{} &= f(x_{12},\ldots,q x_{ij},\ldots) \,,
\end{aligned}$$ for any $f \in S_N$. These operators comprise $N \choose 2$ $q$–commuting pairs, $$\label{qxy}
\sx_{ij} \, \sx_{i'j'} =
\sx_{i'j'} \, \sx_{ij} \,, \qquad
\sy_{ij} \, \sx_{i'j'} = q^{\delta_{ii'} \delta_{jj'}} \,
\sx_{i'j'} \, \sy_{ij} \,, \qquad
\sy_{ij} \, \sy_{i'j'} =
\sy_{i'j'} \, \sy_{ij} \,.$$
Consider $\SF\in \text{End}\,\bigl(S_3\bigr)$ whose action on monomials is given by $$\label{Fdef}
\SF : x_{12}^k \, x_{13}^l \, x_{23}^m \to
x_{12}^k \, x_{13}^{m+k} \, x_{23}^{l-k} \,,$$ or, equivalently, $(\SF f)(x_{12},x_{13},x_{23})
= f(\frac{x_{12}x_{13}}{x_{23}},x_{23},x_{13})$. It is easy to check that $$\begin{aligned}
\label{Fxx}
{}& \SF \, \sx_{12} = \sx_{12} \sx_{13} \sx_{23}^{-1} \SF \,, \qquad
\SF \, \sx_{13} = \sx_{23} \, \SF \,, \qquad
\SF \, \sx_{23} = \sx_{13} \, \SF \,, \\
\label{Fyy}
{}& \SF \, \sy_{12} = \sy_{12} \, \SF \,, \qquad
\SF \, \sy_{13} = \sy_{12} \sy_{23} \, \SF \,, \qquad
\SF \, \sy_{23} = \sy_{12}^{-1} \sy_{13} \, \SF \,.\end{aligned}$$ Clearly, $\SF$ is invertible and $\SF^2 = id$.
Let $\SR \in \text{End}\,\bigl(S_3\bigr)$ be the following operator $$\label{Rdef}
\SR = \SF \cdot
{\left\langle q^{1+\gamma} \sx_{12} \sx_{23}^{-1}
\sy_{12}^{\gamma} \sy_{13}^{-\gamma} \sy_{23}^{-1-\gamma} \right\rangle_{\!q}} \,,$$ where $\gamma\in \mathbb Z$. A reader familiar with the Yang–Baxter equation will see in (\[Rdef\]) an analogy with the standard ansatz for $R$–matrix, $\SR=\SP\cdot \check{\SR}$, where $\SP$ is the permutation, $\SP^2=id$.
\[RSR\] Let $\SF_{abc}$ and $\SR_{abc}$ stand for the canonical extensions of $\SF$ defined by (\[Fdef\]) and $\SR$ defined by (\[Rdef\]) to operators on $S_N$. The homomorphisms $\theta, \phi_\gamma :
{\mathcal B}(3,N) \to \text{End}\,\bigl(S_{N}\bigr)$ such that $\theta(R_{abc})=\SF_{abc}$ and $\phi_\gamma(R_{abc})=\SR_{abc}$ are local tensor space representations of ${\mathcal B}(3,N)$.
This Lemma generalizes two previously known solutions of the tetrahedron equation. Namely, the case corresponding to $\gamma=0$ was found in [@Se1], and the case corresponding to $\gamma=-1$ was considered in [@KV1]. Our proof follows closely that given in [@KV1].
\[Zrep\] Let $\sx_{ab}$, $\sy_{ab}$, $1 \,{\leq}\, a \,{<}\, b \,{\leq}\, N+1$ act on $S_{N+1}$ as defined in (\[xyf\]) and let $\sy_{ab} \equiv 1$ if $a=b$. Then, for all $N \geq 2$, the linear homomorphism $\tau : {\mathcal T}_{N} \to \text{End}\,\bigl(S_{N+1}\bigr)$ such that $$\label{Zxy}
\tau(\SZ_{ab} ) =
q\, \frac{\sx_{ab} }{ \sx_{a+1,b+1} }
\frac{ \sy_{a+1,b}}{\sy_{a+1,b+1}} \,
\qquad \text{for}\quad
1 \,{\leq}\, a \,{<}\, b \,{\leq}\, N \,,$$ is a faithful representation of ${\mathcal T}_{N}$.
Proof of Theorem \[main\]
-------------------------
One the three symmetries of $\ST_N$ presented in (\[Tmu\]) can be established by reordering commuting factors. Namely, applying $\mu_2$ to relations (\[TWRZb\]) and (\[TWRZc\]) and using Lemma \[mu2T\] (see Appendix A.4), we obtain the following statement.
\[Tmu2\] For all $N \geq 2$, we have $$\label{tmu2}
\mu_2 \bigl( \ST_N \bigr) = \ST_N \,.$$
Clearly, in order to establish an analogous equality involving $\mu_1$ or $\mu_3$ we will have to use the $4 \sim 4$ relation (\[44\]). To do it for $\mu_1$, we establish a connection between the element $\ST_N$ evaluated in the representation $\tau$ and words from ${\mathcal B}(3,N+1)$ evaluated in the representation $\phi_\gamma$ (we will take $\gamma=0$ for simplicity).
\[RRR’\] For all $N \geq 2$, we have the equalities $$\begin{aligned}
\label{RFFt}
{}& \phi_0 \bigl(W'(3,N+1)\bigr) =
\theta \bigl(W'(3,N+1)\bigr) \,
\tau \bigl( \ST_N \bigr) \,, \\[1mm]
\label{RFFtr}
{}& \phi_0 \bigl(\bar{W}'(3,N+1)\bigr) =
\theta \bigl(\bar{W}'(3,N+1)\bigr) \,
\tau \bigl( \mu_1 (\ST_N) \bigr) \,.\end{aligned}$$
Invoking Theorem \[multitetra\], we infer that $
\tau\bigl( \ST_N\bigr) =
\tau \bigl( \mu_1 \bigl(\ST_N \bigr) \bigr)
$. Which, by virtue of Lemma \[Zrep\], implies that $$\label{tmu1}
\mu_1 \bigl(\ST_N \bigr) = \ST_N \,.$$ Now, combining (\[tmu2\]) with (\[tmu1\]) and using (\[murho\]), we conclude that $\rho \bigl(\ST_N \bigr) =
(\mu_1 \circ \mu_2 )\bigl(\ST_N \bigr) = \ST_N$. And finally, using (\[murho\]) again, we obtain $\mu_3 \bigl(\ST_N \bigr) =
(\mu_2 \circ \rho )\bigl(\ST_N \bigr) = \ST_N$.
Thus, we have obtained all relations given in Theorem \[main\].
Appendix
========
Proofs of propositions of Section 2.1
-------------------------------------
[**Proof of Lemma \[defWalt\].**]{} Let $\bigl\lfloor W \bigr\rfloor_{N+1} \in {\mathcal B}(n,N+1)$ denote the word which is obtained from a word $W \in {\mathcal B}(n-1,N)$ by the replacement $R_{a_1,\ldots,a_{n-1}} \to R_{a_1,\ldots,a_{n-1},{N+1}}$ applied to all factors. A key step in proving Lemma \[defWalt\] is to observe that the following recursive relation holds $$\label{Wp}
W(n,N) \,
\bigl\lfloor W(n-1,N) \bigr\rfloor_{N+1}
= W(n,N+1) \,.$$ For instance, $$\begin{aligned}
\nonumber
{}&
W(2,3) \bigl\lfloor W(1,3) \bigr\rfloor_{4} =
R_{12} R_{13} R_{23} \bigl\lfloor R_{1} R_{2} R_3 \bigr\rfloor_4 =
R_{12} R_{13} R_{14} R_{23} R_{24} R_{34} = W(2,4) .\end{aligned}$$
For $n=2$, eq. (\[Wp\]) is almost trivial since $\lfloor W(1,N) \rfloor_{N+1} = R_{1,N+1} \ldots R_{N,N+1}$ and for each factor $R_{a,N+1}$, $a\neq N$ the rightmost factor in $W(2,N)$ with which it does not commute is $R_{a,N}$. Therefore, moving all factors from $\lfloor W(1,N) \rfloor_{N+1}$ to the left until they meet their non–commuting counterparts, we obtain the lexicographically ordered word $W(2,N+1)$. Note that only the last two factors in $\lfloor W(1,N) \rfloor_{N+1}$ have the same non–commuting counterpart, namely, $R_{N-1,N}$.
The proof for $n \geq 3$ is similar. Each factor $R_{a_1,\ldots,a_{n-1},N+1}$ in $\lfloor W(n-1,N) \rfloor_{N+1}$ belongs to a cluster, i.e. a product of consecutive lexicographically ordered factors. A cluster has length one if $a_{n-1} < N-1$. If $a_{n-1} \geq N-1$, then the length of a cluster is $k+1$, where $k=\max\{m : a_{n-m} \geq N-m \}$. If $R_{a_1,\ldots,a_{n-k-1},N-k,\ldots,N-1,N+1}$ is the leftmost factor of a cluster of length $k+1$, then the rightmost non–commuting counterpart in $W(n,N)$ for all factors of the cluster is $R_{a_1,\ldots,a_{n-k-1},N-k,\ldots,N-1,N}$ Therefore, moving each cluster from $\lfloor W(n-1,N) \rfloor_{N+1}$ to the left until it meets its counterpart, we achieve the lexicographic order of all factors, that is, we obtain the word .
The recursive relation (\[Wp\]) implies that $$\label{Wpp}
W(n,N) =
\prod_{n\leq k \leq N}^{\rightarrow}
\bigl\lfloor W(n-1,k-1)\bigr\rfloor_{k} \,,$$ which in turn leads to the expression (\[Wexp\]) for $W'(n,N)$.
To prove Theorem \[multitetra\] we will need also the following statement.
\[LEX\] Let $\omega$ be an involutive automorphism of ${\mathcal B}(n,N)$ such that $$\label{om}
\omega \bigl( R_{a_1,\ldots,a_{n}} \bigr) =
R_{N+1-a_{n},\ldots,N+1-a_{1}} \,,$$ for all generators of ${\mathcal B}(n,N)$. Then for all $N \geq n$ the equalities $$\label{omWW}
\omega \bigl( W(n,N) \bigr) =
\eta \bigl( W'(n,N) \bigr) \,, \qquad
\omega \bigl( W'(n,N) \bigr) =
\eta \bigl( W(n,N) \bigr)$$ hold in the strong sense, that is their r.h.s. coincide with their l.h.s. without a permutation of commuting factors.
[**Proof of Lemma \[LEX\].**]{}Denote $\bar{W}'(n,N)=\eta \bigl( W'(n,N)\bigr)$. Applying the anti–automorphism $\eta$ to (\[Wexp\]) and (\[Wp\]), we obtain $$\begin{aligned}
\label{barWexp}
{}& \bar{W}'(n,N) = \!
\mathop{{\prod}'}\limits^{\leftarrow}_{1\leq b_1< \ldots < b_n \leq N }
\! R_{b_1,\ldots,b_{n}} \,,\\
\label{Wm}
{}& \bar{W}'(n,N+1) =
\bigl\lfloor \bar{W}'(n-1,N)
\bigr\rfloor_{N+1} \, \bar{W}'(n,N) \,.\end{aligned}$$ Applying to (\[barWexp\]) the automorphism $\omega$, and relabelling the indices by $a_k = N+1 - b_{n+1-k}$, we recover the r.h.s of formula (\[lex\]). Hence we infer that $\omega \bigl( \eta \bigl( W'(n,N) \bigr) \bigr) = W(n,N)$, which proves the first equality in (\[omWW\]). The second equality follows then immediately since $\omega$ and $\eta$ mutually commute and both are involutive.
[**Proof of Theorem \[multitetra\].**]{}For $n=1$, the statement of the Theorem is obvious since all $R$’s commute. For $n=2$, the statement can be proven by induction on $N$. The base, for $N=3$, is simply the relation (\[Rgen\]). Assume that the equality $W(2,N)=\bar{W}(2,N)$ has been already established for some $N \geq 3$. Then, taking into account the recursive structure of (\[Wp\]) and (\[Wm\]), we have to prove that $$\label{W22}
\bigl\lfloor \prod_{1 \leq a \leq N }^{\rightarrow}
R_{a} \bigl\rfloor_{N+1} =
\bigl( W(2,N) \bigr)^{-1}
\bigl\lfloor \prod_{1 \leq a \leq N }^{\leftarrow}
R_{a} \bigl\rfloor_{N+1}
\, W(2,N) \,.$$ Observe that the Yang–Baxter equation (\[RYB\]) can be rewritten as the following “almost commutativity" relation for two $R$’s, $$\label{RYB2}
\bigl\lfloor R_a \, R_b \bigl\rfloor_{N+1} =
R^{-1}_{ab} \,\bigl\lfloor R_{b} \, R_{a} \bigl\rfloor_{N+1} R_{ab} \,.$$ Using this relation, we can move $R_{1,N+1}$ in the product $\prod\limits^{\rightarrow} R_{a, N+1} $ to the right, then move $R_{2,N+1}$, and so on until we obtain the reverse ordered product $\prod\limits^{\leftarrow} R_{a, N+1} $. Notice that all the extra “twisting" factors $R^{\pm 1}_{ab}$ arising in this process commute with any $R_{c,N+1}$, $c\neq a,b$. Therefore, at each step these twisting factors can be moved outside of the product $\bigl\lfloor \ldots \bigl\rfloor_{N+1}$. It is easy to see that these factors combine into $\bigl(W(2,N)\bigr)^{\pm 1}$. Thus, we have established the inductive step (\[W22\]) and hence the Theorem is proven for $n=2$.
For $n=3$, the Theorem can be proven along the same lines. The base, for $N=4$, is the relation (\[Rgen\]). Assuming that $W(3,N)=\bar{W}(3,N)$ has been already established for some $N \geq 4$ and taking into account (\[Wp\]) and (\[Wm\]), we have to prove the inductive step $$\label{W33}
\bigl\lfloor W(2,N) \bigl\rfloor_{N+1} =
\bigl( W(3,N) \bigr)^{-1}
\bigl\lfloor \bar{W}(2,N) \bigl\rfloor_{N+1} \,
W(3,N) \,.$$ Observe that the tetrahedron equation (\[RTE\]) can be rewritten as the following “almost Yang–Baxter" relation for three $R$’s, $$\label{RTE22}
\bigl\lfloor R_{ab} \, R_{ac} \, R_{bc} \bigl\rfloor_{N+1} =
R_{abc}^{-1} \,
\bigl\lfloor R_{bc} \, R_{ac} \, R_{ab} \bigl\rfloor_{N+1} \, R_{abc} \,,$$ where $a<b<c$. Notice that $R_{abc}$ commutes with any factor in $\bigl\lfloor W(2,N) \bigl\rfloor_{N+1}$ except those on the l.h.s of (\[RTE22\]). Therefore, the factors in $\bigl\lfloor W(2,N) \bigl\rfloor_{N+1}$ can be reordered exactly in the same way as the factors in $W(2,N)$ and the extra twisting factors $R_{abc}^{\pm 1}$ arising at each step can be moved outside of the product $\bigl\lfloor \ldots \bigl\rfloor_{N+1}$. It is easy to see that these factors combine into $\bigl(W(3,N)\bigr)^{\pm 1}$. Since we already know that $W(2,N)=\bar{W}(2,N)$, it follows that the inductive step (\[W33\]) holds and so the Theorem is proved for $n=3$.
It is now clear that the proof continues by the double induction on $n$ and $N$. For a given $n$, relation (\[Rgen\]) can be rewritten as “almost" the relation for $n-1$. Using it, we can reorder the factors in $\bigl\lfloor W(n-1,N) \bigl\rfloor_{N+1}$ exactly in the same way as the factors in $ W(n-1,N)$. At each step the extra twisting factors can be moved outside of the product $\bigl\lfloor \ldots \bigl\rfloor_{N+1}$ due to relation (\[Rcomm\]). Therefore, the same line of arguments as for $n=2,3$ proves the Theorem for an arbitrary $n$.
[**Proof of Lemma \[TW23\].**]{}\
$\bullet$The first equality in (\[TWRZa\]) is obvious from Definition \[defT\] and eq. (\[Rkdef\]). In order to prove the second equality in (\[TWRZa\]), we consider the following elements of ${\mathcal T}_{N}$ $$\label{Trm}
\ST_{N,b} =
\prod_{1\leq a < k \leq b }^{\rightarrow}
{{\mathbb R}_{ ak}} \,, \qquad 2 \leq b \leq N .$$ We observe that $$\label{Trb}
\ST_{N,b} = \ST_{N,b-1}
\prod_{1 \leq a < b}^{\rightarrow} {{\mathbb R}_{ ab }} \,.$$ Indeed, it follows from the definition (\[Rkdef\]) that the rightmost factor in the lexicographically ordered product $\ST_{N,b}$ which does not commute with ${{\mathbb R}_{ ab}}$ is ${{\mathbb R}_{ a,b-1}}$. Repeating the argument used in the proof of formula (\[Wp\]) for $n=2$, we conclude that (\[Trb\]) holds. And it is easy to see that (\[Trb\]) along with $\ST_N = \ST_{N,N}$ implies that $$\label{Trbb}
\ST_N = \ST_{N,N-1}
\prod_{1 \leq a < N}^{\rightarrow} {{\mathbb R}_{ aN }}
= \ldots =
\mathop{{\prod}'}\limits^{\rightarrow}_{1\leq a< b \leq N }
{{\mathbb R}_{ab}}
= \bigl( W'(2,N) \bigr)^\sharp \,.$$
$\bullet$The first equality in (\[TWRZb\]) is obvious from Definition \[defT\]. The second equality in (\[TWRZb\]) is equivalent to the statement that $$\label{TrR}
\ST_N = \ST_{N-1} \, \SW'(2,N) \,,$$ where $ \SW'(2,N)$ is given by (\[W2lex\]) with each $R_{ab}$ replaced with ${{\mathbb R}_{a,b,N+1}}$. Using relations (\[Wp\]) and (\[Trbb\]), we can rewrite the r.h.s. of (\[TrR\]) as follows $$\label{TrRR}
\tilde{\SW}'(2,N-1) \, \SW'(2,N-1) \, \Bigl(
\prod_{1 \leq e < N}^{\rightarrow} {{\mathbb R}_{ eN }} \Bigr) \,,$$ where $ \tilde{\SW}'(2,N)$ is given by (\[W2lex\]) with each $R_{ab}$ replaced with ${{\mathbb R}_{ ab }}'$ which is an element of ${\mathcal T}_{N-1}$ given by (\[Rkdef\]). Definition (\[Rkdef\]) implies that ${{\mathbb R}_{ ab }}'$ is the rightmost factor in $\tilde{\SW}'(2,N-1)$ which does not commute with the factor ${{\mathbb R}_{a,b,N+1}}$ in $\SW'(2,N-1)$. Therefore, moving all factors from $\SW'(2,N-1)$ to the left until they meet their non–commuting counterparts and noticing that ${{\mathbb R}_{ ab}}' {{\mathbb R}_{a,b,N+1}}=
{{\mathbb R}_{ ab }}$, we conclude that (\[TrRR\]) coincides with the second expression in (\[Trbb\]) and hence relation (\[TrR\]) holds.
$\bullet$To prove the last part of the Lemma we consider $ \SW'(2,N)$ introduced in (\[TrR\]). One can check that all its factors containing $\SZ_{N-1,N}$ can be moved to the right preserving their order. Making then the same procedure with factors containing $\SZ_{N-2,N}$, $\SZ_{N-3,N}$, etc., we conclude that $ \SW'(2,N)$ is given by the same expression in (\[W2lex\]) where each $R_{ab}$ is replaced with ${{\mathbb R}_{a,N+1+a-b,N+1}}$. With such a form of $ \SW'(2,N)$ formula (\[TrR\]) leads immediately to the second equality in (\[TWRZc\]).
Equivalence of the first and second expressions for $\ST_N$ in (\[TWRZc\]) is verified as follows $$\label{wwmm}
\bigl( W(3,N)\bigr)^{\star\star}
= \mu_2 \Bigl( \bigl( W'(3,N)\bigr)^{\star} \Bigr)
= \mu_2 \Bigl( \bigl( W(3,N)\bigr)^{\star} \Bigr)
= \bigl( W'(3,N)\bigr)^{\star\star} .$$ The middle equality is due to the second part of the Lemma. In the first and last equalities we used Lemma \[mu2T\], which is proven below in Section A.4.
Proofs of propositions of Section 2.2
-------------------------------------
[**Proof of Theorem \[center\].**]{}Recall that $\alpha \in {\mathbb Z}_{\geq 0}^{N(N-1)/2}$ is a vector comprised of the weights $\alpha_{ij}$ assigned to the vertices of $Q_N$ ordered lexicographically. By (\[mona\]), each $\alpha$ determines a monomial $\SM(\alpha) \in {\mathcal T}_N$. Relations (\[ZZ\]) imply that $\SM(\alpha)$ commutes with all generators of ${\mathcal T}_N$ iff $B \alpha =0$. Thus, we have to study the kernel of the incidence matrix $B$.
$\bullet$ Let $\mathcal U$ be the symmetry transformation of the vertices of $Q_N$ such that ${\mathcal U} (i,j)=(j-i,N+1-i)$, and $U$ be the matrix of the corresponding orthogonal transformation of the basis in ${\mathbb Z}_{\geq 0}^{N(N-1)/2}$. Applying $\rho$ to (\[ZZ\]), we infer that $B$ commutes with $U$. Taking into account that $U$ is orthogonal and $B$ is skew–symmetric, we conclude that if $\alpha \in \mathrm{Ker} B$, then $\alpha$ is an eigenvector of $U$. However, $U^3=1$ and so the only real eigenvalue of $U$ is one. Thus, $\SM(\alpha) \in {\mathcal Z}_N$ implies that $U \alpha = \alpha$. Hence $$\label{alrho}
\alpha_{{\mathcal U} (i,j)} = \alpha_{ij} \,,$$ which is the first symmetry in (\[alsym\]). Applying $\rho$ to a central monomial $\SM(\alpha)$ and taking this symmetry of its weights into account, we infer that $\rho \bigl( \SM(\alpha)\bigr) =
q^{\varepsilon(\alpha)} \SM(\alpha)$, where the multiplicative constant appears due to reordering of the generators. However, the property $\rho \circ \rho \circ \rho =id$ implies that $q^{\varepsilon(\alpha)}=1$ and thus $\rho \bigl( \SM(\alpha)\bigr) = \SM(\alpha)$.
Let $\mu$ be any of the anti–homomorphisms in (\[mudef\]), $\mathcal K$ be the corresponding symmetry transformation of the vertices of $Q_N$ (e.g. ${\mathcal K}(i,j)=(j-i,j)$ for $\mu_1$), and $K$ be the matrix of the corresponding orthogonal transformation of the basis in ${\mathbb Z}_{\geq 0}^{N(N-1)/2}$. Applying $\mu$ to (\[ZZ\]), we infer that $K B K = - B$. Hence $B^2$ commutes with $K$. Taking into account that both these matrices are real symmetric and $K^2=1$, we conclude that if $\alpha \in \mathrm{Ker} B$, then $K \alpha = \pm \alpha$. But $\alpha_{12}=\alpha_{1N}=\alpha_{N-1,N}$ thanks to (\[alrho\]). Therefore, $K \alpha = \alpha$. Hence $$\label{alrefl}
\alpha_{{\mathcal K}(i,j)} = \alpha_{ij} \,,$$ which yields the remaining symmetries in (\[alsym\]). Applying $\mu$ to a central monomial $\SM(\alpha)$ and taking these symmetries into account, we obtain that $\mu \bigl( \SM(\alpha)\bigr) =
q^{\varepsilon'(\alpha)} \SM(\alpha)$. The property $\mu \circ \mu =id$ implies that $q^{\varepsilon'(\alpha)}=1$ and thus $\mu \bigl( \SM(\alpha)\bigr) = \SM(\alpha)$.
$\bullet$ Consider monomials $\SM(\alpha)$ and $\SM(\alpha')$ such that $\alpha_{1j} = \alpha'_{1j}$ for $j=2,\ldots,N$. Then, in view of (\[alrho\]), all components of vector $\alpha''=\alpha-\alpha'$ corresponding to boundary vertices of $Q_N$ vanish. Suppose that $\SM(\alpha)$ and $\SM(\alpha')$ are central. Then $\SM(\alpha'')$ commutes with all generators of ${\mathcal T}_N$. In particular, the condition that $\SM(\alpha'')$ commutes with all $\SZ_{1j}$ is equivalent to a system of equations: $\alpha''_{2,j+1}=\alpha''_{2,j}$. However, we already have $\alpha''_{23}=0$. Therefore, we conclude that $\alpha''_{2j}=0$ for $j=3,\ldots,N$. This implies, in view of (\[alrho\]), that all components of $\alpha''$ corresponding to next to boundary vertices of $Q_N$ also vanish. Continuing this consideration similarly for $\SZ_{2j}$, $\SZ_{3j}$ etc., we conclude that $\alpha''=0$. Thus, $\SM(\alpha')=\SM(\alpha)$, i.e. two central monomials coincide iff they have coinciding weights at one boundary of $Q_N$. Thanks to the symmetry (\[alrefl\]), the latter condition is equivalent to a weaker condition: $\alpha'_{1j} = \alpha_{1j}$ for $j=2,\ldots,\chi(N)+1$.
$\bullet$ Let us show that, given arbitrary integers $\beta_1, \ldots, \beta_{\chi(N)}$, there exists a unique integer vector $\tilde{\alpha}$ such that $\tilde{\alpha}_{1j}=\beta_{j-1}$ for $j=2,\ldots,\chi(N)+1$ and $\SM(\tilde{\alpha})$ given by (\[mona\]) (where some $\tilde{\alpha}_{ij}$ can be negative) commutes with all generators $\SZ_{ij}$. Let $\partial Q_N$ stand for the set of all boundary vertices of $Q_N$. Forgetting about the edges, we have $Q_N =
\partial Q_N \cup \partial Q_{N-3} \cup \ldots$.
First, given the weights $\tilde{\alpha}_{1j}$ for $j=2,\ldots,\chi(N)+1$, we extend them to weights at other vertices of $\partial Q_N$ by the symmetries (\[alrho\]) and (\[alrefl\]). Now, the requirement that $\SM(\tilde{\alpha})$ commutes with all generators assigned to $\partial Q_N$ fixes uniquely all weights at $\partial Q_{N-3}$. Indeed, $\SM(\tilde{\alpha})$ commutes with $\SZ_{1j}$ if $\tilde{\alpha}_{2,j+1}-
\tilde{\alpha}_{2j}=\tilde{\alpha}_{1,j+1}-
\tilde{\alpha}_{1,j-1}$. Taken into account that $\tilde{\alpha}_{23}=\tilde{\alpha}_{13}$, this set of equations determines all $\tilde{\alpha}_{2j}$ uniquely. Moreover, the symmetry (\[alrefl\]) for $\tilde{\alpha}_{1j}$ induces the same symmetry for $\tilde{\alpha}_{2j}$. Finally, $\tilde{\alpha}_{2j}$ can be extended to weights at other vertices of $\partial Q_{N-3}$ by the symmetry (\[alrho\]).
Similarly, given weights at $\partial Q_{k+3}$ and $\partial Q_{k}$, the requirement that $\SM(\tilde{\alpha})$ commutes with all generators assigned to $\partial Q_k$ fixes uniquely all weights at $\partial Q_{k-3}$. Indeed, $\SM(\tilde{\alpha})$ commutes with all $\SZ_{ij}$ for a given $i$ if $\tilde{\alpha}_{i+1,j+1}-
\tilde{\alpha}_{i+1,j} =\tilde{\alpha}_{i,j+1}
-\tilde{\alpha}_{i,j-1}+\tilde{\alpha}_{i-1,j-1}-
\tilde{\alpha}_{i-1,j}$. These equations are resolved uniquely since we know the r.h.s. and several first values of $\tilde{\alpha}_{i+1,j}$ thanks to the symmetry (\[alrho\]).
Thus, the vector $\tilde{\alpha}$ is recovered uniquely from its first $\chi(N)$ components. However, it can happen that some $\tilde{\alpha}_{ij}$ are negative. In this case we take another vector, $\alpha=\tilde{\alpha} +m\, \alpha^0$, where $\alpha^0$ is a vector such that $\alpha^0_{ij}=1$, and $m$ is sufficiently large positive number to ensure positivity of all $\alpha_{ij}$. Then we have $\SM(\alpha) = q^{\varepsilon} \cdot
\SM(\tilde{\alpha}) \bigl( \SM(\alpha^0) \bigr)^m$. It remains to observe that $\SM(\alpha^0) \in {\mathcal Z}_N$ and hence $\SM(\alpha) \in {\mathcal Z}_N$.
Proofs of propositions of Section 2.3
-------------------------------------
[**Proof of Lemma \[RSR\].**]{}The statement of the Lemma for $\SF$ follows from that the action of $\SF_{123} \, \SF_{124} \, \SF_{134} \, \SF_{234}$ and $\SF_{234} \, \SF_{134} \, \SF_{124} \, \SF_{123}$ on the monomial $x_{12}^a x_{13}^b x_{23}^c x_{14}^d x_{24}^e x_{34}^f$, yields the same result.
Note that $\SR$ given by (\[Rdef\]) is invertible. If the argument of the quantum exponential in (\[Rdef\]) is denoted by $\SX$, then we have $\SR^{-1} = {\left\langle \SX \right\rangle_{\!q}}^{-1} \SF$, where $ {\left\langle x \right\rangle_{\!q}}^{-1}$ is the following formal series $$\label{Sxinv}
{\left\langle x \right\rangle_{\!q}}^{-1} = \sum_{n \geq 0}
\frac{ q^{n(n-1)/2}\, x^n}{(1-q)\ldots(1-q^{n})} \,.$$
In order to check that $\SR$ satisfies the tetrahedron equation, one has to substitute $\SR_{abc}$ into (\[RRR\]), move all $\SF$’s to the left, and cancel the products of $\SF$’s on the both sides of the equation by invoking the first part of the Lemma. Then one is left with the equality $$\label{4444}
{\left\langle \SX \right\rangle_{\!q}} \, {\left\langle \SX\SY \right\rangle_{\!q}} \, {\left\langle \SZ \right\rangle_{\!q}} \, {\left\langle \SY \right\rangle_{\!q}} =
{\left\langle \SZ \right\rangle_{\!q}} \, {\left\langle \SZ\SX \right\rangle_{\!q}} \, {\left\langle \SY \right\rangle_{\!q}} \, {\left\langle \SX \right\rangle_{\!q}} \,,$$ where $$\label{xyz44}
\begin{aligned}
{}& \SX = q^{1+\gamma} \sx_{12} \sx_{23}^{-1}
\sy_{12}^{\gamma} \sy_{13}^{-\gamma} \sy_{23}^{-1-\gamma} , \quad
\SY= q^{1+\gamma} \sx_{23} \sx_{34}^{-1}
\sy_{23}^{\gamma} \sy_{24}^{-\gamma} \sy_{34}^{-1-\gamma} , \\[1mm]
{}& \qquad\qquad\qquad
\SZ = q^{1+\gamma} \sx_{13} \sx_{24}^{-1}
\sy_{13}^{\gamma} \sy_{14}^{-\gamma}
\sy_{23}^{1+\gamma} \sy_{24}^{-1-\gamma} .
\end{aligned}$$ These operators satisfy relations (\[XYZ\]). Therefore, comparing the first equality in (\[44\]) with (\[4444\]), we conclude that equality (\[4444\]) holds.
[**Proof of Lemma \[Zrep\].**]{} Note that $\tau(\SZ_{ab})$ does not commute with $\tau(\SZ_{a'b'})$ only in the following six cases: 1) $a'=a$ and $b'=b \pm 1$; 2) $a'-a=\pm 1$ and $b'=b$ or $b'=b+a'-a$. It is easy to check that in these cases $\tau(\SZ_{ab})$ and $\tau(\SZ_{a'b'})$ satisfy the defining relations (\[ZZ\]).
Clearly, the representation (\[Zxy\]) is faithful for ${\mathcal T}_{2}$ since in this case the algebra has only one generator. The proof of faithfulness of this representation in other cases will use the inclusion ${\mathcal T}_{N-1} \subset {\mathcal T}_{N}$ (cf. Remark \[AAA\]). Let $N$ be the minimal positive number such that the representation $\tau$ is not faithful for ${\mathcal T}_{N}$. Then there exists a polynomial $H$ in ${N} \choose {2}$ variables such that $\SH \equiv H(\tau(\SZ_{12}),\ldots,\tau(\SZ_{N-1,N}))=0$.
Without a loss of generality we can assume that $\SH$ has the form $$\label{PZ}
\SH =
\sum_{k_1, \ldots, k_{N-1} \geq 0} B_{k_1 \ldots k_{N-1}} \,
\tau\bigl(\SZ_{1N}^{k_1} \ldots \SZ_{N-1,N}^{k_{N-1}} \bigr) \,
\SH_{k_1 \ldots k_{N-1}} \,,$$ where $B_{k_1 \ldots k_{N-1}} \neq 0$ at least for one set $\{k_1, \ldots, k_{N-1}\}$ such that $k_1 + \ldots + k_{N-1} >0$. $\SH_{k_1 \ldots k_{N-1}}$ stand for polynomials in the generators of ${\mathcal T}_{N-1}$ evaluated in the representation (\[Zxy\]).
Acting by $\SH$ on $f \in S_{N} \subset S_{N+1}$, we obtain $$\label{QZ}
\SH \cdot f =
\sum_{k_1 \ldots k_{N-1} \geq 0}
q^{\varepsilon(k_1,\ldots,k_{N-1})} B_{k_1 \ldots k_{N-1}} \,
\biggl(
\prod_{a=1}^{N-1} \Bigl( \frac{x_{aN}}{x_{a+1,N+1}} \Bigr)^{k_a}
\biggr) \,
\Bigl( \tilde{\SH}_{k_1 \ldots k_{N-1}} \cdot f \Bigr) \,,$$ where $\tilde{\SH}_{k_1 \ldots k_{N-1}}
= \SY_N \SH_{k_1 \ldots k_{N-1}}$ with $\SY_N=\sy_{2N} \ldots \sy_{N-1,N}$. Note that all terms in the sum in (\[QZ\]) are linearly independent monomials in $x^{-1}_{2,N+1},\ldots,x^{-1}_{N,N+1}$. Hence, $\SH=0$ implies that $\tilde{\SH}_{k_1 \ldots k_{N-1}} \cdot f =0$. Since $\SY_N$ is invertible, we conclude that $\SH_{k_1 \ldots k_{N-1}}$ annihilates an arbitrary $f$ from $S_N$ and, thus, $\SH_{k_1 \ldots k_{N-1}}=0$. But this contradicts the assumption that the representation $\tau$ is faithful for ${\mathcal T}_{N-1}$.
Proof of propositions of Section 2.4
-------------------------------------
\[mu2T\] For all $N \geq 3$, the following equalities $$\label{muW}
\mu_2 \Bigl( \bigl( W(3,N)\bigr)^{\star} \Bigr)
= \bigl( W'(3,N)\bigr)^{\star\star} , \qquad
\mu_2 \Bigl( \bigl( W'(3,N)\bigr)^{\star} \Bigr)
= \bigl( W(3,N)\bigr)^{\star\star} ,$$ hold in the strong sense that is their r.h.s. coincide with their l.h.s. without a permutation of commuting factors.
[**Proof of Lemma \[mu2T\].**]{}The first equality in (\[muW\]) is checked with the help of Lemma \[LEX\] as follows: $$\begin{aligned}
{}& \mu_2 \Bigl( \bigl( W(3,N) \bigr)^\star \Bigr) =
\mu_2 \Bigl(
\prod_{1\leq a < b < c \leq N}^{\rightarrow}
(R_{abc})^\star \Bigr)
=
\mathop{{\prod}'}\limits^{\rightarrow}_{1\leq a< b <c \leq N }
\mu_2 \Bigl( \bigl(\omega (R_{abc}) \bigr)^\star \Bigr) \\
{}& = \mathop{{\prod}'}\limits^{\rightarrow}_{1\leq a< b <c \leq N }
\mu_2 \Bigl( {{\mathbb R}_{ N+1-c,N+1-b,N+1-a }} \Bigr)
= \mathop{{\prod}'}\limits^{\rightarrow}_{1\leq a< b <c \leq N }
{{\mathbb R}_{ a,c+a-b,c }}
= \bigl( W'(3,N) \bigr)^{\star\star} .\end{aligned}$$ The second equality can be checked in a similar way.
Recall that the homomorphisms $\theta$ and $\phi_\gamma$ were introduced in Lemma \[RSR\]. In this subsection we will use the following notations: $$\label{Rabc}
\SR_{abc} = \phi_0(R_{abc}) = \SF_{abc} \cdot
\check{\SR}_{abc} , \qquad
\SF_{abc} = \theta \bigl( R_{abc} \bigr) , \qquad
\check{\SR}_{abc} =
{\left\langle q \sx_{ab} \sx_{bc}^{-1}
\sy_{bc}^{-1} \right\rangle_{\!q}} .$$ We will need also the following homomorphisms $\varphi, \varphi'$ from ${\mathcal B}(3,N)$ to $\text{End}\,\bigl(S_N\bigr)$: $$\begin{aligned}
\label{phi}
{}& \varphi ( R_{abc} ) =
{\left\langle \frac{q \sx_{ab} }{ \sx_{a+c-b,c} }
\prod_{k=1}^{c-b} \frac{ \sy_{a+k,b+k-1}}{\sy_{a+k,b+k}} \right\rangle_{\!q}}
, \\
\label{phihat}
{}& \varphi' ( R_{abc} ) =
{\left\langle q \Bigl(
\prod_{k=1}^{c-b} \frac{\sx_{b-a,b+k-1} }{ \sx_{b-a+1,b+k} }
\Bigr)
\frac{\sy_{b-a+1,b}}{\sy_{b-a+1,c}} \right\rangle_{\!q}} ,\end{aligned}$$ where $\sy_{ab} \equiv 1$ if $a=b$.
Recall that the word $W'(3,N)$ is given by (\[W3lex\]), and that $\bar{W}'(3,N)=\eta \bigl(W'(3,N)\bigr)$ is the word with reversely ordered factors. Let us introduce $$\label{Fr}
\SF_N = \theta \bigl(W'(3,N)\bigr) =
\theta \bigl(\bar{W}'(3,N)\bigr) ,$$ where the equality of the expressions is due to Theorem \[multitetra\].
\[RR’\] For all $N \geq 3$, we have the equalities $$\begin{aligned}
\label{RFF}
{}& \phi_0 \bigl(W'(3,N)\bigr) =
\SF_N \cdot
\varphi \bigl(W'(3,N)\bigr) \,, \\[1mm]
\label{RFFr}
{}& \phi_0 \bigl(\bar{W}'(3,N)\bigr) =
\SF_N \cdot
\varphi' \bigl(\bar{W}'(3,N)\bigr) \,.\end{aligned}$$
[**Proof of Lemma \[RR’\].**]{}Eq. (\[Wpp\]) implies that $$\label{w3t}
\phi_0 \bigl( W'(3,N) \bigr) =
\prod_{3\leq c \leq N}^{\rightarrow}
\phi_0 \Bigl( \bigl\lfloor W'(2,c-1)\bigr\rfloor_{c} \Bigr) \,.
$$ Consider $\check{\SR}_{abc}$ entering the $\phi_0\bigl(\lfloor\ldots\rfloor_c \bigr)$ part. In order to establish (\[RFF\]) it suffices to prove that:\
[**i**]{}) $\check{\SR}_{abc}$ transforms into $\varphi(\SR_{abc})$ when all $\SF$’s from $\phi_0\bigl(\lfloor\ldots\rfloor_c \bigr)$ are moved to the left of this part;\
[**ii**]{}) $\varphi(\SR_{abc})$ does not change when, for any $k>c$, all $\SF$’s from $\phi_0\bigl(\lfloor W'(2,k-1)\rfloor_k \bigr)$ are moved through it.
[**Proof of i).**]{}Consider $\check{\SR}_{abc}$ which is not the rightmost one (otherwise, it already has the form $\varphi(R)$). All $\SF$’s to the right of it have the form $\SF_{a'b'c}$, $\{a',b'\}\neq \{a,b\}$. By (\[Fxx\]), they do not change $\sx_{ab}$ but act on the first index of $\sx_{bc}$ as permutations $P_{a'b'}$. From the recursive structure of the word $W'(2,c-1)=\lfloor W(1,1) \rfloor_{2} \ldots
\lfloor W(1,c-2) \rfloor_{c-1} $ it follows that $\sx_{bc}$ has to be pulled through the following chain of permutations: $$P_{a+1,b}\ldots P_{b-1,b} \ P_{1,b+1} \ldots P_{b,b+1} \ldots
P_{1,c-1} \ldots P_{c-2,c-1} .$$ The part $P_{a+1,b}\ldots P_{b-1,b}$ here is present if $a<b-1$. It transforms $\sx_{bc}$ into $\sx_{a+1,c}$. The remaining permutations are grouped into $(c-1-b)$ shift operators $U_d =P_{d,d+1} \ldots P_{12}$, $b \leq d \leq c-2$. Each $U_d$ increases the first index of $\sx_{a+1,c}$ by one, thus, transforming it into $\sx_{a+c-b,c}$.
Consider now $\sy_{bc}$ entering the argument of ${\check\SR}_{abc}$. It is first transformed by $\SF_{a+1,b,c}$ into $\frac{\sy_{a+1,c}}{\sy_{a+1,b}}$. Note that, by (\[Fyy\]), all $\SF$’s entering $\phi_0\bigl(\lfloor W'(2,c-1) \rfloor_{c}\bigr)$ act non–trivially only on $\sy$’s whose second index is $c$. From the recursive structure of the word $W'(2,c-1)$ it follows that $\sy_{a+1,c}$ is transformed by consecutive action of pairs $\SF_{a+k,b+k,c} \SF_{a+1+k,b+k,c}$, $1 \leq k \leq c-b-1$. Each such pair transforms $\sy_{a+k,c}$ into $\frac{\sy_{a+k,b+k}}{\sy_{a+1+k,b+k}} \sy_{a+1+k,c}$, which yields the product of $\sy$’s in (\[phi\]).
[**Proof of ii).**]{} This part is trivial for $\sy$’s since, by (\[Fyy\]), $\sy_{ab}$ commutes with any $\SF_{a^{\prime} b^{\prime} k}$ if $k \neq a,b$. For $\sx$’s we have to consider the transformations of $\frac{\sx_{ab} }{ \sx_{a'c} }$, where $a' \equiv a+c-b$. Note that $a< a' <c$. Therefore, $\frac{\sx_{ab} }{ \sx_{a'c} }$ is first transformed by $\SF_{abk}$ into $\frac{\sx_{ab} \sx_{ak} }{ \sx_{a'c} \sx_{bk} }$. Then this expression is pulled through all $\SF$’s between $\SF_{abk}$ and $\SF_{a'ck}$. They act only on the first indices of $\sx_{ak}$ and $\sx_{bk}$ as the following chain of permutations: $$P_{a+1,b}\ldots P_{b-1,b} \ P_{1,b+1} \ldots P_{b,b+1}
\ldots P_{1,c} \ldots P_{a'-1,c} \,.$$ The part $P_{a+1,b}\ldots P_{b-1,b}$ transforms $\frac{ \sx_{ak} }{ \sx_{bk} }$ into $\frac{ \sx_{ak} }{ \sx_{a+1,k} }$. Then each of the shift operators $U_d$, $b \leq d \leq c-2$ increases the first indices of $\sx_{ak}$ and $\sx_{a+1,k}$ by one, which yields $\frac{ \sx_{a'-1,k} }{ \sx_{a'k} }$. The last part, $P_{1,c} \ldots P_{a'-1,c}$, transforms $\frac{ \sx_{a'-1,k} }{ \sx_{a'k} }$ into $\frac{ \sx_{ck} }{ \sx_{a'k} }$. Finally, $\frac{\sx_{ab} \sx_{ck} }{ \sx_{a'c} \sx_{a'k} }$ is transformed by $\SF_{a'ck}$ into $\frac{\sx_{ab} }{ \sx_{a'c} }$ which then is not affected by the remaining $\SF$’s.
Thus, we have verified [**i)**]{} and [**ii)**]{} and therefore proved relation (\[RFF\]).
In order to establish Eq. (\[RFFr\]) we will prove first that $$\label{RFFFr}
\phi_0 \bigl(\bar{W}'(3,N)\bigr) =
\varphi'' \bigl(\bar{W}'(3,N)\bigr) \cdot \SF_N \,,$$ where $\varphi''$ is a homomorphism from ${\mathcal B}(3,N)$ to $\text{End}\,\bigl(S_N\bigr)$, $$\label{phi3}
\varphi'' ( R_{abc} ) =
{\left\langle \frac{q \sx_{ab} }{ \sx_{a+c-b,c} }
\prod_{k=0}^{c-b-1} \frac{ \sy_{a+k,b+k}}{\sy_{a+k,b+k+1}} \right\rangle_{\!q}}.$$ Indeed, we have $$\label{Rhatabc}
\SR_{abc} = \phi_0(R_{abc}) =
\hat{\SR}_{abc} \cdot \SF_{abc} \,, \qquad
\hat{\SR}_{abc} =
{\left\langle q \sx_{ab} \sx_{bc}^{-1}
\sy_{ab} \sy_{ac}^{-1} \right\rangle_{\!q}} .$$ Note that the $\sx$’s arguments of $\hat{\SR}_{abc}$ and $\check{\SR}_{abc}$ are the same. Therefore, moving all $\SF$’s in $\phi_0 \bigl(\bar{W}'(3,N)\bigr) $ to the right results in the same transformation of these arguments as moving all $\SF$’s in $\phi_0 \bigl(W'(3,N)\bigr)$ to the left.
What the $\sy$’s arguments of $\hat{\SR}_{abc}$ are concerned, they are transformed only by those $\SF$’s that enter the $\phi_0\bigl(\lfloor \bar{W}'(2,c-1)\rfloor_c\bigr)$ part. Specifically, $\sy_{ab}$ does not change whereas $\sy_{ac}$ is transformed by consecutive action of pairs $\SF_{a+k,b+1+k,c}\SF_{a+1+k,b+1+k,c}$, $0 \leq k \leq c-b-1$, which yields the product of $\sy$’s in (\[phi3\]).
It remains to pull $\SF_N$ in (\[RFFFr\]) to the left. Note that $\SF_N^2 =1$. Using (\[Fxx\]) and (\[Fyy\]), one can verify that $$\begin{aligned}
\label{Frx}
{}& \SF_N \, \sx_{ab} \, \SF_N
= \sx_{b-a,b} \, \prod_{k=1}^{N-b}
\frac{\sx_{b-a,b+k}}{\sx_{b-a+1,b+k}} \,, \qquad
\SF_N \, \sy_{ab} \, \SF_N
= \prod_{k=1}^{b-a}
\frac{\sy_{k,a+k}}{\sy_{k,a+k-1}} \,.\end{aligned}$$ Whence $$\begin{aligned}
\label{Frxx}
{}& \frac{\sx_{ab}}{\sx_{a+1,b+1}} \, \SF_N
= \SF_N \, \frac{\sx_{b-a,b}}{\sx_{b-a+1,b+1}} \,,\qquad
\frac{\sy_{ab}}{\sy_{a,b+1}} \, \SF_N
= \SF_N \, \frac{\sy_{b-a+1,b}}{\sy_{b-a+1,b+1}} .\end{aligned}$$ Applying these relations to (\[phi3\]), we obtain (\[phihat\]).
[**Proof of Lemma \[RRR’\].**]{} Using (\[Zkdef\]), (\[RabZ\]), and (\[Zxy\]), we find $$\begin{aligned}
\label{tauphi1}
\tau \bigl( {{\mathbb R}_{ abc}} \bigr) =
\varphi \bigl( R_{abc} \bigr) \,, \qquad
\tau \bigl( \mu_1 \bigl( {{\mathbb R}_{abc}} \bigr) \bigr)
= \varphi' \bigl( R_{abc} \bigr) .\end{aligned}$$ Therefore, invoking Lemma \[TW23\], we conclude that $$\begin{aligned}
\label{tauphi2}
{}& \tau \bigl( \ST_N \bigr) =
\tau \Bigl( \bigl( W'(3,N+1)\bigr)^\star \Bigr)
= \varphi \bigl( W'(3,N+1)\bigr) \,, \\
{}& \tau \Bigl( \mu_1 \bigl( \ST_N \bigr) \Bigr) =
\tau \Bigl( \mu_1 \bigl( \bigl( W'(3,N+1)\bigr)^\star \bigr) \Bigr)
= \varphi' \bigl( \bar{W}'(3,N+1) \bigr) .\end{aligned}$$ These relations along with Lemma \[RR’\] yield the statement of Lemma \[RRR’\].
[**Acknowledgements.**]{} This work was supported in part by the Swiss National Science Foundation grants 200020-129609 (A.V.) and 200020-126817 (A.B.) and by the RFBR grant 11-01-00570. The authors thank Anton Alekseev, Ludwig Faddeev, Sergey Fomin, and Rinat Kashaev for helpful conversations.
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Keller B. “On cluster theory and quantum dilogarithm identities." In *Representations of algebras and related topics*, 85–116. EMS Series of Congress Reports, Eur. Math. Soc., Zürich, 2011. Schützenberger M.P. “Une interprétation de certaines solutions de l’équation fonctionnelle: $F(x+y)=F(x)F(y)$." C. R. Acad. Sci. Paris 236 (1953): 352–353.
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[Section of Mathematics, University of Geneva, C.P. 64, 1211 Genève 4, Switzerland\
Steklov Mathematical Institute, Fontanka 27, 191023, St. Petersburg, Russia]{}
|
---
abstract: 'The Krein–von Neumann extension is studied for Schrödinger operators on metric graphs. Among other things, its vertex conditions are expressed explicitly, and its relation to other self-adjoint vertex conditions (e.g. continuity-Kirchhoff) is explored. A variational characterisation for its positive eigenvalues is obtained. Based on this, the behaviour of its eigenvalues under perturbations of the metric graph is investigated, and so-called surgery principles are established. Moreover, isoperimetric eigenvalue inequalities are obtained.'
address: |
Matematiska institutionen\
Stockholms universitet\
106 91 Stockholm\
Sweden
author:
- Jacob Muller
- Jonathan Rohleder
title: 'The Krein–von Neumann extension for Schrödinger operators on metric graphs'
---
Introduction
============
It is an almost hundred-year-old story that many of the differential operators appearing in mathematical physics and their boundary conditions can be described conveniently in the framework of extension theory of symmetric operators. A complete description of all self-adjoint extensions of a symmetric operator was first given by von Neumann [@N30]. On the other hand, it turned out that a theory of self-adjoint extensions of symmetric operators that are semibounded from below can be done conveniently by means of semibounded sesquilinear forms; this originates from the work of Friedrichs [@F34]. However, it is due to Krein [@K47] (see also the works of Vishik [@V52] and Birman [@B56]) that among all non-negative extensions of a positive definite symmetric operator $S$, there are two extremal ones, the Friedrichs extension $S_{\rm F}$ and the (by now so-called) Krein–von Neumann extension $S_{\rm K}$, in the sense that each non-negative self-adjoint extension $A$ of $S$ satisfies $$\begin{aligned}
S_{\rm K} \leq A \leq S_{\rm F}.\end{aligned}$$ These inequalities may be understood in the sense of quadratic forms or via the involved operators’ resolvents. It is beyond the scope of this article to provide a complete historical review of the developments related to the Krein–von Neumann extension; for further reading we refer the reader to [@AN70] and the survey articles [@AS80; @AGMST13]. Among the abstract advancements on extremal extensions of positive definite symmetric operators (and, more generally, symmetric linear relations), we mention [@AHSS02; @AT02; @AT03; @CS78; @HMS04; @M92; @PS96; @SS03; @S96; @T81].
In the study of e.g. elliptic second order differential operators on Euclidean domains, the Friedrichs extension is a very natural object; for instance, for the minimal symmetric Laplacian on a bounded domain in $\R^n$ corresponding to both Dirichlet and Neumann boundary conditions, the Friedrichs extension is the self-adjoint Laplacian subject to Dirichlet boundary conditions. On the other hand, in the same setting, the Krein–von Neumann extension corresponds to certain non-local boundary conditions which can be described in terms of the associated Dirichlet-to-Neumann map; for properties of the Krein–von Neumann extension of elliptic differential operators and recent related developments, we refer the reader to [@AK11; @AGMT10; @BGMM16; @G83; @G12; @M94].
For differential operators on metric graphs, which we consider in the present paper, the situation is similar, yet different in some respects. If $\Gamma$ is a finite metric graph, then we take, as a starting point, the (negative) Laplacian (i.e. the negative second derivative operator on each edge) $S$ in $L^2 (\Gamma)$, which satisfies on each vertex both Dirichlet and Kirchhoff vertex conditions; that is, the functions in the domain of $S$ vanish and have derivatives which sum up to zero at each vertex. This symmetric operator is very natural to carry out extension theory, since its adjoint $S^*$ is the Laplacian on $\Gamma$ with continuity as its (only) vertex conditions. Therefore any self-adjoint extension of $S$ in $L^2 (\Gamma)$ (which, at the same time is a restriction of $S^*$) satisfies continuity conditions and thus reflects, at least to some extent, the connectivity of the graph. Nevertheless, the Friedrichs extension of $S$ in this setting is the Laplacian on functions which are zero at every single vertex, an operator which, despite continuity, is determined by the graph’s edges considered as separate intervals, instead of the actual graph structure. Other self-adjoint extensions of $S$ which are more suitable for the spectral analysis of network structures are the operator with continuity-Kirchhoff conditions (the so-called standard or natural Laplacian) or with $\delta$-type vertex conditions; the latter we will not discuss here further.
The Krein–von Neumann extension for a Laplacian on a metric graph has not been considered much in the literature so far; an attempt for a symmetric operator with vertex conditions different from the ones considered here was done in [@M15]. The Krein–von Neumann extension of our operator $S$ is, like for the minimal Laplacian on a Euclidean domain, an operator with non-local vertex conditions. Nevertheless, its domain is intimately connected to the structure of the underlying graph. In fact, we prove that the matrix that couples the values and the sums of derivatives at the vertices for functions in the domain of $S_{\rm K}$ is exactly the weighted discrete Laplacian on the underlying discrete graph, where the weights are the inverse edge lengths.
Our main focus in the present paper is on spectral properties of the operator $S_{\rm K}$, not only in the case of the Laplacian, but also for Schrödinger operators with nonnegative potentials $q_e$ on the edges. Namely, we consider the operator $S$ acting as $- \frac{\dd^2}{\dd x^2} + q_e$ on each edge $e$ of $\Gamma$, with Dirichlet and Kirchhoff vertex conditions as described above in the case of the Laplacian. Its Krein–von Neumann extension, the so-called [*perturbed Krein Laplacian*]{}, denoted by $- \Delta_{{\rm K}, \Gamma, q}$, is the main object of consideration in this article. We first describe the domain of $- \Delta_{{\rm K}, \Gamma, q}$ in terms of vertex conditions and establish Krein-type formulae for the resolvent differences with both the Friedrichs extension (the Schrödinger operator with Dirichlet vertex conditions) and the the Schrödinger operator $- \Delta_{{\rm st}, \Gamma, q}$ with standard vertex conditions. As a consequence, we obtain the formula $$\begin{aligned}
\dim \ran \Big[ \big(- \Delta_{{\rm K}, \Gamma, q} - \lambda \big)^{-1} & - \big(- \Delta_{{\rm st}, \Gamma, q} - \lambda \big)^{-1} \Big] = \begin{cases} V - 1 & \text{if}~q = 0~\text{identically}, \\ V, & \text{else}, \end{cases}\end{aligned}$$ in which $V$ denotes the number of vertices of $\Gamma$ and $\lambda$ takes appropriate complex values. This formula distinguishes the potential-free case clearly from the case influenced by a potential. It also sheds light on another interesting phenomenon: the Krein Laplacian may, in some rare occasions, coincide with the standard Laplacian, and this is the case if and only if $\Gamma$ has only one vertex (with possibly many loops attached to it) and thus is a so-called flower graph. Moreover, we use the Krein-type resolvent formulae to obtain some results on spectral asymptotics of the perturbed Krein Laplacian.
A further property of the perturbed Krein Laplacian on a metric graph $\Gamma$, which we establish, is the possibility to describe its positive eigenvalues variationally. In fact, the spectrum of $- \Delta_{{\rm K}, \Gamma, q}$ is purely discrete, and the lowest eigenvalue is always zero, with multiplicity equal to $V$, the number of vertices, as we show. All its positive eigenvalues $\lambda_j^+ (- \Delta_{{\rm K}, \Gamma, q})$, ordered nondecreasingly and counted with multiplicities, can be characterised by the variational principle $$\begin{aligned}
\label{eq:minMaxIntro}
\lambda_j^+ \big( - \Delta_{{\rm K}, \Gamma, q} \big) = \min_{\substack{F \subset \widetilde H_0^2 (\Gamma) \\ \dim F = j}} \max_{\substack{f \in F \\ f \neq 0}} \frac{\int_\Gamma \left|- f'' + q f\right|^2 \dd x}{\int_\Gamma |f'|^2 \dd x + \int_\Gamma q |f|^2 \dd x};\end{aligned}$$ here, $\widetilde H^2_0 (\Gamma)$ is the second-order Sobolev space on each edge, equipped with Dirichlet and Kirchhoff conditions on all edges. This formula is the exact counterpart of a variational description of the positive eigenvalues of the perturbed Krein Laplacian on a domain in $\R^n$, which was established in [@AGMT10 Proposition 7.5]. Before we derive , we first establish an abstract version of this principle; see Theorem \[thm:minMaxKrein\]. Its proof is along the lines of the result for the Laplacian in [@AGMT10]; however, we found it useful and of independent interest to have it at hand also abstractly for the Krein–von Neumann extension of any symmetric, positive definite operator $S$ for which $\dom S$ equipped with the graph norm of $S$ satisfies a compactness condition. As a consequence of the formulation for graphs , we easily obtain inequalities between the (positive) eigenvalues of the perturbed Krein Laplacian and other self-adjoint extensions of $S$.
An important field of application of the eigenvalue characterisation are so-called surgery principles. Such principles study the influence of geometric perturbations of a metric graph on the specta of associated Laplacians or more general differential operators. The reader may think of sugery operations such as joining two vertices into one or cutting through a vertex, or adding or removing edges (or even entire subgraphs). Such principles were studied in depth for the Laplacian or Schrödinger operators subject to standard (and some other local) vertex conditions; see [@BKKM19; @KKT16; @KKMM16; @KN14; @RS20]. As we point out, the eigenvalues of the perturbed Krein Laplacian behave in some respects in the same way as the eigenvalues of $- \Delta_{{\rm st}, \Gamma, q}$; for instance, when gluing vertices all eigenvalues increase (or stay the same), and adding pendant edges or graphs (a process which increases the “volume” of $\Gamma$) may only decrease the eigenvalues. On the other hand, in some respects the behaviour is different from what we are used to for standard vertex conditions. Let us only mention three examples: firstly, for the positive eigenvalues, gluing vertices has actually a non-increasing effect (but at the same time also the multiplicity of the eigenvalue 0 decreases), whilst for standard vertex conditions, the positive eigenvalues behave non-decreasingly and the dimension of the kernel remains the same. Secondly, removing a vertex of degree two (replacing the two incident edges by one) may change eigenvalues in a monotonous way, whilst it does not have any influence on the spectrum of an operator with standard vertex conditions. Thirdly, inserting an edge between two existing vertices makes all eigenvalues decrease (or stay the same); for standard vertex conditions, this is not necessarily the case; see e.g. [@KMN13].
A typical application of surgery principles for graph eigenvalues consists of deriving spectral inequalities in terms of geometric and topological parameters of the graph such as its total length, diameter, number of edges or vertices, or its first Betti number (or Euler characteristics, equivalently). For a few recent advances on spectral inequalities for quantum graphs, we refer to [@BL17; @BKKM17; @K20; @KN19; @MP20; @P20]. To demonstrate how surgery principles for the perturbed Krein Laplacian on a metric graph may be applied, we establish lower bounds for the positive eigenvalues, in terms of eigenvalues of a loop graph or edge lengths. For instance, for the first positive eigenvalue of the Krein Laplacian without potential the lower bound is explicit, $$\begin{aligned}
\lambda_1^+ (- \Delta_{\rm K, \Gamma}) \geq 4 \left( \frac{\pi}{\ell (\Gamma)} \right)^2,\end{aligned}$$ where $\ell (\Gamma)$ denotes the total length of $\Gamma$, and we specify the class of graphs for which this estimate is optimal.
Considering the Krein–von Neumann (and other) extensions of a Schrödinger operator with Dirichlet and Kirchhoff vertex conditions at all vertices is natural, as we pointed out above. However, it may also be useful to study extensions of a symmetric Schrödinger operator with different vertex conditions. We mention, as an example, the Laplacian with both Dirichlet and Neumann (Kirchhoff) vertex conditions at the “loose ends”, i.e. the vertices of degree one, but standard vertex conditions at all interior vertices. In this case, the vertex conditions of the Krein–von Neumann extension will still be standard at all interior vertices, but they will couple the vertices of degree one in a nonlocal way. We conclude our paper with a short section where we discuss such situations.
Let us briefly describe how this paper is organised. In Section \[sec:abstractKrein\], we review some background on the abstract Krein–von Neumann extension. Moreover, we provide a proof of the abstract counterpart of the variational principle and derive a few easy consequences. Additionally, we study some basic properties of boundary triples, which we use as a tool. The aim of Section \[sec:Krein\] is to introduce the perturbed Krein Laplacian on a metric graph and to study its properties, such as a description of its domain, Krein-type resolvent formulae and some consequences of the min-max principle. Section \[sec:surgery\] is devoted to a collection of surgery principles, whilst in Section \[sec:isoperimetric\], we apply some of them in order to obtain some isoperimetric inequalities. Finally, Section \[sec:moreGeneral\] deals with the more general setting where self-adjoint vertex conditions are fixed at some vertices, and extension theory is applied with respect to the remaining vertices.
The abstract Krein–von Neumann extension and its eigenvalues {#sec:abstractKrein}
============================================================
Preliminaries
-------------
Throughout this section we assume that $\cH$ is a separable complex Hilbert space with inner product $(\cdot, \cdot)$ and corresponding norm $\| \cdot \|$. For any closed linear operator $A$ in $\cH$, we denote by $\sigma (A)$ and $\rho (A)$ its spectrum and resolvent set respectively. If $A$ is self-adjoint and has a purely discrete spectrum, then we write $$\begin{aligned}
\lambda_1 (A) \leq \lambda_2 (A) \leq \dots\end{aligned}$$ for its eigenvalues, counted according to their multiplicities. If $\cG$ is a further Hilbert space, we denote by $\cB (\cG, \cH)$ the space of all bounded, everywhere-defined linear operators from $\cG$ to $\cH$ and abbreviate $\cB (\cG) := \cB (\cG, \cG)$.
We make the following assumption.
\[hyp:abs\] The operator $S : \cH \supset \dom S \to \cH$ is closed and symmetric with dense domain $\dom S$. Furthermore, $S$ has a positive lower bound, i.e. there exists $\mu > 0$ such that $$\begin{aligned}
\label{eq:mu}
(S f, f) \geq \mu \|f\|^2, \qquad f \in \dom S.\end{aligned}$$
Under Hypothesis \[hyp:abs\], the defect numbers $(n_-, n_+)$ of $S$ satisfy $n_- = n_+ = \dim \ker S^*$, where $S^*$ denotes the adjoint of $S$. Moreover, $\dom S \cap \ker S^* = \{0\}$ and the Krein–von Neumann extension of $S$ can be defined as follows.
The [*Krein–von Neumann extension*]{} of $S$ is the operator $S_{\rm K}$ in $\cH$ given by $$\begin{aligned}
\label{eq:AK}
S_{\rm K} f = S^* f, \qquad \dom S_{\rm K} = \dom S \dotplus \ker S^*.\end{aligned}$$
It is well-known that $S_{\rm K}$ is self-adjoint and is the smallest non-negative self-adjoint extension of $S$ in the sense of quadratic forms. Its counterpart, the Friedrichs extension of $S$, is the largest non-negative extension of $S$ and we denote it by $S_{\rm F}$. It can be defined via completion of the quadratic form induced by $S$; we do not go into the details but refer the reader to, e.g. the discussion in [@K95 Chapter VI]. For any self-adjoint, non-negative extension $A$ of $S$, the relation $$\begin{aligned}
\big( (S_{\rm F} - \lambda)^{-1} f, f \big) \leq \big( (A - \lambda)^{-1} f, f \big) \leq \big( (S_{\rm K} - \lambda)^{-1} f, f \big), \quad f \in \cH,\end{aligned}$$ holds for each $\lambda < 0$. The spectrum of the Friedrichs extension has a strictly positive lower bound; in fact, $\min \sigma (S_{\rm F})$ coincides with the supremum over all $\mu$ such that holds. Conversely, the Krein–von Neumann extension $S_{\rm K}$ has the point 0 as the bottom of its spectrum, and the corresponding eigenspace is given by $$\begin{aligned}
\ker S_{\rm K} = \ker S^*,\end{aligned}$$ which follows from the definition of $S_{\rm K}$ and the fact that 0 is not an eigenvalue of $S$. In particular, $\dim \ker S_{\rm K} = n_- = n_+$, the defect number of $S$. We refer the reader to, e.g. the survey [@AGMST13] for a more detailed discussion of the Krein–von Neumann extension.
A variational characterisation of the positive eigenvalues of the Krein–von Neumann extension
---------------------------------------------------------------------------------------------
The main goal of this subsection is to provide an abstract variational description of the eigenvalues different from 0 of the Krein–von Neumann extension. The credits for the arguments that lead to the min-max principle in Theorem \[thm:minMaxKrein\] below go to the articles [@AGMST10; @AGMST13; @AGMT10], where the abstract Krein–von Neumann extension and the perturbed Krein Laplacian on domains in $\R^n$ were studied. There, the min-max principle is stated in the context of the application, so for the convenience of the reader we state and prove this variational principle here abstractly.
Associated with the operator $S$ is the space $$\begin{aligned}
\cH_S := \dom S \qquad \text{with norm} \qquad \|f\|_S := \|S f\|, \quad f \in \cH_S.\end{aligned}$$ Due to , $\cH_S$ is a normed space, and as $S$ is closed, it follows that $\cH_S$ is a Banach space. The norm $\| \cdot \|_S$ corresponds to the inner product $(f, g)_S = (S f, S g)$; hence $\cH_S$ is a Hilbert space. Moreover, there exists a constant $\widetilde \mu > 0$ such that $$\begin{aligned}
\label{eq:alsoDoch}
\|f\|_S \geq \widetilde \mu \|f\|, \qquad f \in \cH_S.\end{aligned}$$ (Indeed, if not then for each $n \in \N$ there exists $f_n \in \cH_S$, w.l.o.g. $\|f_n\| = 1$, such that $\|S f_n\| < \frac{1}{n}$ and hence $\mu \leq (S f_n, f_n) \leq \|S f_n\| < \frac{1}{n}$ by , a contradiction to $\mu > 0$.) We further denote by $\cH_S^*$ the dual space of $\cH_S$ and write $(\cdot, \cdot)_{\cH_S^*, \cH_S}$ for the sesquilinear duality between $\cH_S^*$ and $\cH_S$, i.e. the continuous extension of $$\begin{aligned}
(h, f)_{\cH_S^*, \cH_S} := (h, f), \qquad h \in \cH, \quad f \in \cH_S,\end{aligned}$$ to all $h \in \cH_S^*$. (Note that $\cH$ is dense in $\cH_S^*$ as $\cH_S$ is dense in $\cH$.)
It will sometimes be useful to consider $S$ as an operator from $\cH_S$ to $\cH$ rather than as an operator in $\cH$. Therefore we define $$\begin{aligned}
\widetilde S : \cH_S \to \cH, \qquad \widetilde S f : = S f, \quad f \in \cH_S.\end{aligned}$$ Then $\widetilde S$ is bounded and its adjoint $\widetilde S^*$ is the unique bounded operator from $\cH$ to $\cH_S^*$ that satisfies $$\begin{aligned}
\big( \widetilde S f, g \big) = \big( f, \widetilde S^* g \big)_{\cH_S, \cH_S^*}, \qquad f \in \cH_S, g \in \cH.\end{aligned}$$ Note that on the left-hand side we might as well replace $\widetilde S$ by $S$. For later use, we remark also that $\widetilde S^* g \in \cH$ implies $g \in \dom S^*$ and $S^* g = \widetilde S^* g$. In particular, $$\begin{aligned}
\label{eq:Kernels!}
\ker \widetilde S^* = \ker S^*.\end{aligned}$$
The following lemma is a variant of [@AGMST10 Lemma 3.1]. For the convenience of the reader, we provide a complete proof.
\[lem:B\] Let Hypothesis \[hyp:abs\] be satisfied. Then the operator $\widetilde S^* S : \cH_S \to \cH_S^*$ is bijective, and $$\begin{aligned}
\label{eq:B}
B := (\widetilde S^* S)^{- 1} S : \cH_S \to \cH_S\end{aligned}$$ is a bounded, self-adjoint, non-negative operator with $\ker B = \{0\}$. Moreover, a number $\lambda > 0$ is an eigenvalue of $S_{\rm K}$ if and only if $\lambda^{- 1}$ is an eigenvalue of $B$.
The operator $\widetilde S^* S$ is injective as $\widetilde S^* S f = 0$ implies $$\begin{aligned}
0 = \big( \widetilde S^* S f, f \big)_{\cH_S^*, \cH_S} = (S f, S f) = \|S f\|^2,\end{aligned}$$ that is, $f \in \ker S$ which by implies $f = 0$. Furthermore, let $h \in \cH_S^*$. According to the Fréchet–Riesz theorem, there exists a unique $f \in \cH_S$ such that $$\begin{aligned}
(g, h)_{\cH_S, \cH_S^*} = (g, f)_S = (S g, S f) = \big( g, \widetilde S^* S f \big)_{\cH_S, \cH_S^*} \end{aligned}$$ holds for all $g \in \cH_S$, and hence $\widetilde S^* S f = h$. Thus $\widetilde S^* S$ is bijective and, by the open mapping theorem, has a bounded inverse. In particular, the operator $B$ in is well-defined and bounded as the product of two bounded operators.
Let us show next that $B$ is symmetric and thus self-adjoint. Indeed, for $f \in \cH_S$, we get $$\begin{aligned}
\label{eq:Bsym}
\begin{split}
(B f, f)_S & = \big( S ( \widetilde S^* S)^{- 1} S f, S f \big) = \big( \widetilde S^* S ( \widetilde S^* S)^{- 1} S f, f \big)_{\cH_S^*, \cH_S} \\
& = (S f, f) \geq \mu \|f\|^2
\end{split}\end{aligned}$$ by and, in particular, $(B f, f)_S \in \R$. Hence $B$ is self-adjoint and non-negative, and also implies that $\ker B = \{0\}$.
Now let $\lambda > 0$ be such that $S_{\rm K} g = \lambda g$ holds for some $g \in \dom S_{\rm K}$, $g \neq 0$. Define also $f := S_{\rm F}^{- 1} S_K g$, where $S_{\rm F}$ is the Friedrichs extension of $S$. As $0 \notin \sigma (S_{\rm F})$ by , $f$ is well-defined and belongs to $\dom S_{\rm F}$. Moreover, as $g \in \dom S_{\rm K}$, by we can write $g = g_S + g_*$ with $g_S \in \dom S$ and $g_* \in \ker S^*$ and get $$\begin{aligned}
\label{eq:klapptDoch}
f = S_{\rm F}^{- 1} S_{\rm K} g = S_{\rm F}^{- 1} S g_S + S_{\rm F}^{- 1} S^* g_* = S_{\rm F}^{- 1} S_{\rm F} g_S = g_S \in \dom S.\end{aligned}$$ Furthermore, $f \neq 0$ as otherwise $g \in \ker S_{\rm K}$, contradicting $S_{\rm K} g = \lambda g \neq 0$, and $S f = S_{\rm F} f = S_{\rm K} g = \lambda g$ together with yields $$\begin{aligned}
\widetilde S^* S f = \lambda \widetilde S^* g = \lambda S^* (g_S + g_*) = \lambda S g_S = \lambda S f.\end{aligned}$$ Thus $B f = \lambda^{- 1} f$, that is, $\lambda^{- 1}$ is an eigenvalue of $B$.
Conversely, let $B f = \lambda^{- 1} f$ for some $\lambda > 0$ and $f \in \cH_S$, $f \neq 0$. Then $\widetilde S^* S f = \lambda S f$, which can be rewritten as $\widetilde S^* (S - \lambda) f = 0$, that is, $(S - \lambda) f \in \ker \widetilde S^* = \ker S^*$; see . Define $g := \lambda^{- 1} S f$. Then $g$ is nonzero and $$\begin{aligned}
f + \lambda^{- 1} (S - \lambda) f = f + g - f = g,\end{aligned}$$ which, due to $f \in \dom S$ and $(S - \lambda) f \in \ker S^*$, implies $g \in \dom S_{\rm K}$. Finally, $$\begin{aligned}
S_{\rm K} g = \lambda^{- 1} \widetilde S^* S f = S f = \lambda g,\end{aligned}$$ that is, $\lambda$ is an eigenvalue of $S_{\rm K}$. This completes the proof.
We point out that Lemma \[lem:B\] describes, in an abstract setting, the coincidence between the positive eigenvalues of the Krein–von Neumann extension and the eigenvalues of an abstract buckling problem; the latter reads $\widetilde S^* S f = \lambda S f$ and is discussed in detail in [@AGMST10 Section 3].
Next we provide an abstract version of the min-max principle established for Krein Laplacians on domains in [@AGMT10 Proposition 7.5]. The Rayleigh quotient $$\begin{aligned}
R_{\rm K} [f] := \frac{\|S f\|^2}{(S f, f)}, \qquad f \in \dom S, f \neq 0,\end{aligned}$$ is well-defined due to .
\[thm:minMaxKrein\] Assume that Hypothesis \[hyp:abs\] is satisfied and that the embedding $\iota : \cH_S \to \cH$ is compact. Then $\sigma (S_{\rm K}) \setminus \{0\}$ is purely discrete, and the positive eigenvalues $\lambda_1^+ (S_{\rm K}) \leq \lambda_2^+ (S_{\rm K}) \leq \dots$ of $S_{\rm K}$, counted with multiplicities, satisfy $$\begin{aligned}
\lambda_j^+ (S_{\rm K}) = \min_{\substack{F \subset \dom S \\ \dim F = j}} \max_{\substack{f \in F \\ f \neq 0}} R_{\rm K} [f]\end{aligned}$$ for all $j \in \N$.
As the embedding $\iota : \cH_S \to \cH$ is compact, it follows that the Friedrichs extension $S_{\rm F}$ of $S$ has a compact resolvent, from which it can be deduced that $\sigma (S_{\rm K}) \setminus \{0\}$ is purely discrete; see, e.g., [@AGMST13 Theorem 2.10].
For the rest of this proof, we make the abbreviation $\lambda_j := \lambda_j^+ (S_{\rm K})$. Let $B : \cH_S \to \cH_S$ be the bounded, self-adjoint, nonnegative operator in Lemma \[lem:B\] whose eigenvalues coincide with $\{\lambda_j^{- 1} : j \in \N \}$. As $\iota$ is compact, the same holds for the embedding $\iota^* : \cH \to \cH_S^*$, and $B$ can be rewritten as $$\begin{aligned}
B = (\widetilde S^* S)^{- 1} \iota^* S,\end{aligned}$$ which is also then compact. In particular, we can choose an orthonormal basis $\{f_j : j \in \N \}$ of $\cH_S$ such that $\lambda_j B f_j = f_j$, or equivalently $S^* S f_j = \lambda_j S f_j$, holds for all $j \in \N$. (Here we are assuming $\dim \cH_S = \infty$; the finite-dimensional case is exactly the same with a finite orthonormal basis.) Then for each $j \in \N$, $$\begin{aligned}
R_{\rm K} [f_j] = \frac{\|S f_j\|^2}{(S f_j, f_j)} = \lambda_j \frac{\|S f_j\|^2}{(S^* S f_j, f_j)} = \lambda_j\end{aligned}$$ holds. Let us define $F_0 := \{0\}$ and $$\begin{aligned}
F_j := \operatorname{span}\left\{ f_k : k \leq j \right\}, \qquad j = 1, 2, \dots,\end{aligned}$$ and denote by $F_{j - 1}^\perp$ the orthogonal complement of $F_{j - 1}$ with respect to the inner product $(\cdot, \cdot)_S$ in $\cH_S$ for all $j \in \N$. Now fix $j \in \N$. Then any $f \in F_{j - 1}^\perp$ can be written as $f = \sum_{k = j}^\infty c_k f_k$ for appropriate $c_k \in \C$, where the sum converges in $\cH_S$ (and hence also in $\cH$ due to ). Then the continuity of $S$ with respect to the norm in $\cH_S$ implies $$\begin{aligned}
(S f, f) & = \sum_{k = j}^\infty c_k (S f_k, f) = \sum_{k = j}^\infty \lambda_k^{- 1} c_k (S^* S f_k, f) = \sum_{k = j}^\infty \lambda_k^{- 1} |c_k|^2 (f_k, f_k)_S \\
& \leq \lambda_j^{- 1} \|f\|_S^2,\end{aligned}$$ and thus $R_{\rm K} [f] \geq \lambda_j$ for all $f \in F_{j - 1}^\perp$, with equality for $f = f_j$. Consequently, $$\begin{aligned}
\label{eq:minV}
\min_{\substack{f \in F_{j - 1}^\perp \\ f \neq 0}} R_{\rm K} [f] = \lambda_j.\end{aligned}$$ By a similar calculation, one verifies $$\begin{aligned}
\label{eq:maxV}
\max_{\substack{f \in F_j \\ f \neq 0}} R_{\rm K} [f] = \lambda_j. \end{aligned}$$ Now let $G_j \subset \cH_S$ be a $j$-dimensional subspace with $G_j \neq F_j$. Then by a dimension argument, there exists $g_j \in (G_j \cap F_{j - 1}^\perp) \setminus \{0\}$, and gives $$\begin{aligned}
\lambda_j = \min_{\substack{f \in F_{j - 1}^\perp \\ f \neq 0}} R_{\rm K} [f] \leq R_{\rm K} [g_j] \leq \max_{\substack{f \in G_j \\ f \neq 0}} R_{\rm K} [f].\end{aligned}$$ Together with , this implies the assertion of the theorem.
As a direct consequence, one gets the following comparison principle for the positive eigenvalues of $S_{\rm K}$ and the eigenvalues of any self-adjoint extension of $S$. The inequality between eigenvalues of $S_{\rm F}$ and $S_{\rm K}$ is mentioned for completeness, but it has been known for a long time, see, e.g. [@AS80 Theorem 5.1]. However, it follows conveniently from the above min-max principle.
\[thm:inequalitiesAbstract\] Assume that Hypothesis \[hyp:abs\] is satisfied and that the embedding $\iota : \cH_S \to \cH$ is compact, and let $A$ be any self-adjoint extension of $S$ with a purely discrete spectrum. Moreover, let $d := \dim \ker A$. Then $$\begin{aligned}
\label{eq:indexShift}
\lambda_{j + d} (A) \leq \lambda_j^+ (S_{\rm K})\end{aligned}$$ holds for all $j \in \N$. In particular, $$\begin{aligned}
\label{eq:FriedrichsKreinIneq}
\lambda_j (S_{\rm F}) \leq \lambda_j^+ (S_{\rm K}) \end{aligned}$$ holds for all $j \in \N$. If $j \in \N$ is such that $\lambda_j (S_{\rm F})$ is not an eigenvalue of $S$, then the inequality is strict, that is, $\lambda_j (S_{\rm F}) < \lambda_j^+ (S_{\rm K})$.
Let us fix $j$ and choose a $j$-dimensional subspace $F$ of $\dom S$ such that $$\begin{aligned}
\|S f\|^2 \leq \lambda_j^+ (S_{\rm K}) (S f, f) \qquad \text{for all}~f \in F.\end{aligned}$$ Then for any $f \in F$ and $g \in \ker A$ we have $$\begin{aligned}
\big( A (f + g), f + g \big)^2 \leq \|A (f + g)\|^2 \|f + g\|^2,\end{aligned}$$ and hence $$\begin{aligned}
\label{eq:sum}
\frac{\big(A (f + g), f + g \big)}{\|f + g\|^2} \leq \frac{\|A (f + g)\|^2}{\big( A (f + g), f + g \big)} = \frac{\|A f\|^2}{(A f, f)} = \frac{\|S f\|^2}{(S f, f)} = \lambda_j^+ (S_{\rm K}). \end{aligned}$$ Due to , $\ker A \cap \dom S = \{0\}$ and, thus $\dim (F + \ker A) = j + d$. Therefore together with the usual min-max principle for $A$ implies the assertion . Note that by the compactness of the embedding $\iota$, the spectrum of $S_{\rm F}$ is purely discrete, and thus implies . Finally, assume that $\lambda_j (S_{\rm F})$ is not an eigenvalue of $S$, and let $g = 0$ in the estimate . Assuming $\lambda_j (S_{\rm F}) = \lambda_j^+ (S_{\rm K})$ for a contradiction, we get equality in , with $A = S_{\rm F}$ for some nontrivial $f \in \dom S$. Then $f \in \ker (S_{\rm F} - \lambda_j (S_{\rm F})) \cap \dom S = \ker (S - \lambda_j (S_{\rm F}))$ follows, a contradiction.
We wish to point out that compactness of the embedding of $\cH_S$ into $\cH$ does not imply that all self-adjoint extensions of $S$ have a purely discrete spectrum. An example is the Krein–von Neumann extension of the Laplacian with both Dirichlet and Neumann boundary conditions on a bounded, sufficiently smooth domain in $\R^m$, $m \geq 2$, where $\ker S_{\rm K} = \ker S^*$ consists of all harmonic functions, and thus is infinite-dimensional, see, e.g. [@AGMT10] for more details.
If the Krein–von Neumann extension of $S$ has purely discrete spectrum (in particular $d = \dim \ker S_{\rm K}$ is finite) we may choose $A = S_{\rm K}$ in Theorem \[thm:inequalitiesAbstract\]. As $\lambda_{j + d} (S_{\rm K}) = \lambda_j^+ (S_{\rm K})$, this shows that the inequality is not necessarily strict in general, not even if $S$ does not have any eigenvalues.
Given two symmetric operators $S, \widetilde S$ in $\cH$ such that $S \subset \widetilde S$, we get the following interlacing properties of the positive eigenvalues of their respective Krein–von Neumann extensions. We will apply it several times in subsequent sections.
\[thm:interlacingAbstract\] Let $S, \widetilde S$ be closed, densely defined, symmetric operators in $\cH$ with $S \subset \widetilde S$ such that holds for $S$ replaced by $\widetilde S$. Moreover, assume that the embedding $\widetilde \iota : \cH_{\widetilde S} \to \cH$ is compact, and denote by $S_{\rm K}$ and $\widetilde S_{\rm K}$ the Krein–von Neumann extensions of $S$ and $\widetilde S$ respectively. Then $\sigma (S_{\rm K}) \setminus \{0\}$ and $\sigma (\widetilde S_{\rm K}) \setminus \{0\}$ are purely discrete. If we assume, in addition, that $\dom S$ is a subspace of $\dom \widetilde S$ of co-dimension $k$, then the positive eigenvalues of $S_{\rm K}$ and $\widetilde S_{\rm K}$ satisfy the interlacing inequalities $$\label{eq:interlacingAbstract}
\lambda_j^+ (\widetilde S_{\rm K}) \leq \lambda_j^+ (S_{\rm K}) \leq \lambda_{j + k}^+ (\widetilde S_{\rm K}) \leq \lambda_{j + k}^+ (S_{\rm K})$$ for all $j \in \N$.
Firstly, the assumption $S \subset \widetilde S$ implies $\cH_S \subset \cH_{\widetilde S}$ algebraically, together with $$\begin{aligned}
\|f\|_S = \|f\|_{\widetilde S} \qquad \text{for all}~f \in \cH_S.\end{aligned}$$ Hence follows also for $S$, and compactness of the embedding $\widetilde \iota$ implies compactness of the embedding $\iota : \cH_S \to \cH$. With the help of the latter, the discreteness statement on the spectra of $S_{\rm K}$ and $\widetilde S_{\rm K}$ follows from Theorem \[thm:minMaxKrein\].
Secondly, the first and third inclusion in follow directly from the inclusion $S \subset \widetilde S$, and the min-max principle in Theorem \[thm:minMaxKreinV\]. It remains to prove the middle inequality in . Let $j \in \N$ and let $\widetilde F \subset \dom \widetilde S$ be any $(j + k)$-dimensional subspace of $\dom \widetilde S$ such that $$\begin{aligned}
\max_{0 \neq f \in \widetilde F}\frac{\|\widetilde S f\|^2}{(\widetilde S f, f)} = \lambda_{j + k}^+ (\widetilde S_{\rm K}).\end{aligned}$$ As $\dom S$ is a subspace of $\dom \widetilde S$ of co-dimension $k$, the subspace $F := \widetilde F \cap \dom S$ of $\dom S$ satisfies $\dom F \geq j$, and we have $$\lambda_j^+(S_{\rm K}) \leq \max_{0 \neq f \in F}\frac{\|S f\|^2}{(S f, f)} = \max_{0 \neq f \in F}\frac{\|\widetilde S f\|^2}{(\widetilde S f, f)} \leq \max_{0 \neq f \in \widetilde F}\frac{\|\widetilde S f\|^2}{(\widetilde S f, f)} = \lambda_{j + k}^+ (\widetilde S_{\rm K}),$$ which completes the proof.
We conclude this subsection with a comment on additive perturbations of the Krein–von Neumann extension.
\[rem:perturbation\] Assume that $Q = Q^*$ is a bounded, nonnegative, everywhere defined operator in $\cH$. If $S$ is closed, symmetric, densely defined, and satisfies then all these properties are also true for $S + Q$, and thus $S + Q$ has a Krein–von Neumann extension which we denote by $(S + Q)_{\rm K}$. It is remarkable that this operator does not coincide with $S_{\rm K} + Q$, the additively perturbed Krein–von Neumann extension of $S$. This is in contrast to the Friedrichs extension, for which $(S + Q)_{\rm F} = S_{\rm F} + Q$ holds. For instance, if $Q = I$ is the identity operator then $(S + I)_{\rm K}$ has a nontrivial kernel (coinciding with $\ker (S^* + I)$), whilst $S_{\rm K} + I$ is bounded from below by one. Nevertheless, $S_{\rm K} + Q$ is a self-adjoint, nonnegative extension of $S + Q$ and we know thus that $$\begin{aligned}
\lambda_j \big( (S + Q)_{\rm K} \big) \leq \lambda_j (S_{\rm K} + Q)\end{aligned}$$ holds for all $j \in \N$. On the other hand, by our Theorem \[thm:inequalitiesAbstract\] one has $$\begin{aligned}
\lambda_{j + d} (S_{\rm K} + Q) \leq \lambda_j^+ \big( (S + Q)_{\rm K} \big)\end{aligned}$$ for all $j \in \N$, where $d := \dim \ker (S_{\rm K} + Q) \leq \dim \ker S_{\rm K}$.
The Krein–von Neumann extension in the framework of boundary triples
--------------------------------------------------------------------
In this subsection, we review properties of the Krein–von Neumann extension in the framework of boundary triples. Our main focus is on a Krein–type formula that expresses the resolvent difference between the Krein–von Neumann extension and another self-adjoint extension of $S$ (as, e.g. the Friedrichs extension) in terms of abstract boundary operators. We assume Hypothesis \[hyp:abs\] throughout. First we recall the definition of a boundary triple.
Assume Hypothesis \[hyp:abs\]. A triple $\{\cG, \Gamma_0, \Gamma_1\}$ consisting of a Hilbert space $(\cG, (\cdot, \cdot)_\cG)$ and two linear mappings $\Gamma_1, \Gamma_2 : \dom S^* \to \cG$ is called [*boundary triple for $S^*$*]{} if the following conditions are satisfied:
1. the mapping $\{\Gamma_0, \Gamma_1\} : \dom S^* \to \cG \times \cG$ is surjective;
2. the [*abstract Green identity*]{} $$\begin{aligned}
(S^* f, g) - (f, S^* g) = (\Gamma_1 f, \Gamma_0 g)_\cG - (\Gamma_0 f, \Gamma_1 g)_\cG
\end{aligned}$$ holds for all $f, g \in \dom S^*$.
We remark that boundary triples exist for any symmetric, densely defined operator $S$ with equal defect numbers, even without the requirement . For a detailed review on boundary triples and literature references we refer the reader to, e.g. the recent monograph [@BHS20] or [@S12 Chapter 14].
For any given boundary triple, we have $S^* \upharpoonright (\ker \Gamma_0 \cap \ker \Gamma_1) = S$, and two self-adjoint extensions of $S$ are especially distinguished, namely $$\begin{aligned}
\label{eq:AB}
A := S^* \upharpoonright \ker \Gamma_0 \qquad \text{and} \qquad B := S^* \upharpoonright \ker \Gamma_1.\end{aligned}$$ A boundary triple comes with two operator-valued functions defined on the resolvent set $\rho (A)$ of $A$.
Let Hypothesis \[hyp:abs\] be satisfied, and let $\{\cG, \Gamma_0, \Gamma_1\}$ be a boundary triple for $S^*$. The mappings $$\begin{aligned}
\gamma : \rho (A) \to \cB (\cG, \cH) \qquad \text{and} \qquad M : \rho (A) \to \cB (\cG)\end{aligned}$$ defined as $$\begin{aligned}
\gamma (\lambda) \Gamma_0 f = f \qquad \text{and} \qquad M (\lambda) \Gamma_0 f = \Gamma_1 f\end{aligned}$$ for $f \in \ker (S^* - \lambda)$ are called [*$\gamma$-field*]{} and [*Weyl function*]{} respectively, associated with the boundary triple $\{\cG, \Gamma_0, \Gamma_1\}$.
The well-definedness of $\gamma (\lambda)$ and $M (\lambda)$ is due to the direct sum decomposition $$\begin{aligned}
\dom S^* = \dom A \dotplus \ker (S^* - \lambda), \qquad \lambda \in \rho (A).\end{aligned}$$ The operator $\gamma (\lambda)$ can be viewed as an abstract Poisson operator, and $M (\lambda)$ may be interpreted as an abstract Dirichlet-to-Neumann map. It is well-known that $\lambda \mapsto M (\lambda)$ is an operator-valued Herglotz–Nevanlinna–Pick function. In particular, $M (\lambda)$ is self-adjoint for $\lambda \in \rho (A) \cap \R$ (if such points exist, which is always the case if is assumed).
Boundary triples can be used to characterise e.g. self-adjoint extensions of $S$ in terms of abstract boundary conditions of the form $\Gamma_1 f = \Theta \Gamma_0 f$ with a self-adjoint parameter $\Theta$ acting in $\cG$. In order to actually describe all self-adjoint extensions of $S$, one needs to allow not only self-adjoint operators $\Theta$ but so-called self-adjoint linear relations (or multi-valued linear operators), and we do not go into these details here. For us it is sufficient to know the following; see e.g. [@BHS20 Theorems 2.1.3, 2.6.1, and 2.6.2].
\[prop:KreinFormula\] Let Hypothesis \[hyp:abs\] be satisfied, let $\{ \cG, \Gamma_0, \Gamma_1\}$ be a boundary triple for $S^*$, and let $\Theta$ be a self-adjoint operator in $\cG$. Then $$\begin{aligned}
A_\Theta := S^* \upharpoonright \big\{ f \in \dom S^* : \Gamma_1 f = \Theta \Gamma_0 f \big\}\end{aligned}$$ is a self-adjoint extension of $S$. Moreover, if we denote by $\lambda \mapsto \gamma (\lambda)$ and $\lambda \mapsto M (\lambda)$ the corresponding $\gamma$-field and Weyl function respectively, and $A$ is defined in , then the following assertions hold.
1. The point $\lambda \in \rho (A)$ is an eigenvalue of $A_\Theta$ if and only if 0 is an eigenvalue of $\Theta - M (\lambda)$.
2. The point $\lambda \in \rho (A)$ belongs to $\rho (A_\Theta)$ if and only if $0 \in \rho (\Theta - M (\lambda))$.
3. For all $\lambda \in \rho (A) \cap \rho (A_\Theta)$, $$\begin{aligned}
(A_\Theta - \lambda)^{-1} - (A - \lambda)^{-1} = \gamma (\lambda) \big( \Theta - M (\lambda) \big)^{-1} \gamma (\overline \lambda)^*
\end{aligned}$$ holds.
Characterisations analogous to item (i) in the previous theorem hold for other types of spectra too, such as the continuous or residual spectrum, but this is not of relevance for us in this work.
If the boundary triple is chosen such that $0 \in \rho (A)$, then the Krein–von Neumann extension of $S$ can be characterised in the following way; this is well-known, but for the convenience of the reader we repeat the short proof.
\[prop:KreinBT\] Let Hypothesis \[hyp:abs\] be satisfied, and let $\{ \cG, \Gamma_0, \Gamma_1\}$ be a boundary triple for $S^*$ such that $0 \in \rho (A)$. Moreover, let $\lambda \mapsto M (\lambda)$ denote the corresponding Weyl function. Then the Krein–von Neumann extension $S_{\rm K}$ of $S$ equals $$\begin{aligned}
S_{\rm K} = S^* \upharpoonright \big\{ f \in \dom S^* : \Gamma_1 f = M (0) \Gamma_0 f \big\}.\end{aligned}$$
Since $M (0)$ is self-adjoint, the restriction of $S^*$ to all $f$ which satisfy $\Gamma_1 f = M (0) \Gamma_0 f$ is a self-adjoint extension of $S$ by Proposition \[prop:KreinFormula\]. Moreover, by definition, each $f \in \dom S_{\rm K}$ can be written uniquely as $f = f_S + f_*$ with $f_S \in \dom S$ and $f_* \in \ker S^*$, and therefore $$\begin{aligned}
\Gamma_1 f = \Gamma_1 f_* = M (0) \Gamma_0 f_* = M (0) \Gamma_0 f,\end{aligned}$$ where we have used $\dom S = \ker \Gamma_0 \cap \ker \Gamma_1$. This completes the proof.
Now that we have this characterisation of the domain of $S_{\rm K}$ at hand, we may use the above Krein–type resolvent formula to express the difference to both the distinguished self-adjoint extensions $A$ and $B$ of $S$.
\[prop:Krein\] Assume that Hypothesis \[hyp:abs\] holds. Let $\{ \cG, \Gamma_0, \Gamma_1\}$ be a boundary triple for $S^*$, and let $\lambda \mapsto \gamma (\lambda)$ and $\lambda \mapsto M (\lambda)$ denote the corresponding $\gamma$-field and Weyl function respectively. Let $A$ and $B$ be given in , and assume that $0 \in \rho (A)$. Then the following identities hold.
1. For all $\lambda \in \rho (A) \cap \rho (S_{\rm K})$, $$\begin{aligned}
\label{eq:KreinA}
(S_{\rm K} - \lambda)^{-1} - (A - \lambda)^{-1} = \gamma (\lambda) \big( M (0) - M (\lambda) \big)^{-1} \gamma (\overline \lambda)^*
\end{aligned}$$ holds.
2. For all $\lambda \in \rho (B) \cap \rho (S_{\rm K}) \cap \rho (A)$, the operator $M (\lambda)$ is invertible with $M (\lambda)^{-1} \in \cB (\cG)$ and $$\begin{aligned}
\label{eq:KreinB}
(S_{\rm K} - \lambda)^{-1} - (B - \lambda)^{-1} = \gamma (\lambda) \big( M (0) - M (\lambda) \big)^{-1} M (0) M (\lambda)^{-1} \gamma (\overline \lambda)^*
\end{aligned}$$ holds.
Assertion (i) follows directly from plugging the result of Proposition \[prop:KreinBT\] into the resolvent formula of Proposition \[prop:KreinFormula\] (iii). On the other hand, the operator $B$ corresponds to the operator $A_\Theta$ with $\Theta = 0$, and hence $$\begin{aligned}
(A - \lambda)^{-1} - (B - \lambda)^{-1} = \gamma (\lambda) M (\lambda)^{-1} \gamma (\overline{\lambda})^*\end{aligned}$$ for all $\lambda \in \rho (A) \cap \rho (B)$. For those $\lambda$ which additionally belong to $\rho (S_{\rm K})$, we combine the latter formula with assertion (i) of the present proposition to get $$\begin{aligned}
(S_{\rm K} - \lambda)^{-1} - (B - \lambda)^{-1} = \gamma (\lambda) \Big[ \big( M (0) - M (\lambda) \big)^{-1} + M (\lambda)^{-1} \Big] \gamma (\overline \lambda)^*.\end{aligned}$$ From this, the assertion (ii) follows by an easy calculation left to the reader.
The resolvent formulae in the previous proposition may be used to determine the rank of the resolvent differences as follows.
\[cor:KreinB\] Assume that Hypothesis \[hyp:abs\] is satisfied. Let $\{ \cG, \Gamma_0, \Gamma_1\}$ be a boundary triple for $S^*$ with Weyl function $\lambda \mapsto M (\lambda)$, and let $A, B$ be as defined in . Moreover, let $0 \in \rho (A)$. Then the following hold.
1. For all $\lambda \in \rho (A) \cap \rho (S_{\rm K})$, $$\begin{aligned}
\dim \ran \left[ (S_{\rm K} - \lambda)^{-1} - (A - \lambda)^{-1} \right] = \dim \ker (S^*-\overline{\lambda}) = \dim \cG.\end{aligned}$$
2. For all $\lambda \in \rho (A) \cap \rho (B) \cap \rho (S_{\rm K})$, $$\begin{aligned}
\dim \ran \left[ (S_{\rm K} - \lambda)^{-1} - (B - \lambda)^{-1} \right] = \dim \ran M (0).\end{aligned}$$
This follows rather directly from formulas and in a way similar to the proof of [@BHS20 Theorem 2.8.3]. In fact, we use that $\gamma (\lambda) : \cG \to \ker (S^* - \lambda)$ is an isomorphism and that $$\begin{aligned}
\ker \gamma (\overline \lambda)^* = \left( \ran \gamma (\overline \lambda) \right)^\perp = \left( \ker (S^* - \overline{\lambda}) \right)^\perp.\end{aligned}$$ This implies that $$\begin{aligned}
\ran \left[ (S_{\rm K} - \lambda)^{-1} - (A - \lambda)^{-1} \right] = \ran \left[ (S_{\rm K} - \lambda)^{-1} - (A - \lambda)^{-1} \right] \upharpoonright \ker (S^* - \overline{\lambda}),\end{aligned}$$ for all $\lambda \in \rho (A) \cap \rho (S_{\rm K})$, with the same equation holding after replacing $A$ with $B$ for all $\lambda \in \rho (A) \cap \rho (B) \cap \rho (S_{\rm K})$. Finally, as $$\begin{aligned}
\gamma (\overline \lambda)^* \upharpoonright \ker (S^* - \overline{\lambda}) : \ker (S^* - \overline{\lambda}) \to \cG\end{aligned}$$ is an isomorphism and both $(M (0) - M (\lambda))^{-1}$ and $M (\lambda)^{-1}$ are isomorphisms of $\cG$, the desired result follows from and .
Perturbed Krein Laplacians on metric graphs {#sec:Krein}
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In this section and all sections which follow, we assume that $\Gamma$ is a metric graph consisting of a vertex set $\cV$, an edge set $\cE$, and a length function $\ell : \cE \to (0, \infty)$ which assigns a length to each edge. Every edge $e \in \cE$ is identified with the interval $[0, \ell (e)]$, and this parametrisation gives rise to a natural metric on $\Gamma$. We will always assume that $\Gamma$ is finite, i.e. $V := |\cV|$ and $E := |\cE|$ are finite numbers, and we consider only connected graphs.
We view a function $f : \Gamma \to \C$ as a collection of functions $f_e : (0, \ell (e)) \to \C$, $e \in \cE$, and say, accordingly, that $f$ belongs to $L^2 (\Gamma)$ if $f_e \in L^2 (0, \ell (e))$ for each $e \in \cE$. In order to define Schrödinger operators on metric graphs we make use of the Sobolev spaces $$\begin{aligned}
\widetilde H^k (\Gamma) := \left\{ f \in L^2 (\Gamma) : f_e \in H^k (0, \ell (e))~\text{for each}~e \in \cE \right\},\end{aligned}$$ $k \in \N$. For functions in $\widetilde H^1 (\Gamma)$, we may talk about continuity at a vertex $v$, meaning that for any two edges $e, \hat e$ incident with $v$, the limit values (or traces) of $f_e$ and $f_{\hat e}$ at the endpoints of the edges corresponding to $v$ coincide. In this sense, we make use of the function space $$\begin{aligned}
H^1 (\Gamma) := \left\{ f \in \widetilde H^1 (\Gamma) : f~\text{is continuous at each vertex} \right\}.\end{aligned}$$ Moreover, for $f \in \widetilde H^2 (\Gamma)$ and $v \in \cV$, we write $$\begin{aligned}
\partial_\nu f (v) := \sum \partial f_e (v),\end{aligned}$$ where the sum is taken over all edges $e$ incident with $v$, and $\partial f_e (v)$ is the derivative of $f_e$ at the endpoint corresponding to $v$, taken in the direction pointing towards $v$; if $e$ is a loop then both endpoints have to be taken into account.
We will consider Schrödinger operators on metric graphs with potentials that are, for simplicity, bounded. However, everything may be extended easily to form-bounded (i.e. $L^1$) potentials. We will always assume the following hypothesis.
\[hyp\] On the finite, connected metric graph $\Gamma$, the potential $q : \Gamma \to \R$ is measurable and bounded, and $q (x) \geq 0$ holds for almost all $x \in \Gamma$.
Under Hypothesis \[hyp\], we define the Schrödinger operator with potential $q$ subject to Dirichlet and Kirchhoff vertex conditions at all vertices, $$\begin{aligned}
\label{eq:SV}
\begin{split}
(S f)_e & = - f_e'' + q_e f_e \qquad \text{on each edge}~e \in \cE, \\
\dom S & = \widetilde H_0^2 (\Gamma) := \left\{ f \in \widetilde H^2 (\Gamma) \cap H^1 (\Gamma) : \partial_\nu f (v) = f (v) = 0~\text{for each}~v \in \cV \right\}.
\end{split}\end{aligned}$$ It is easy to see that $S$ is a symmetric, nonnegative, densely defined operator in the Hilbert space $L^2 (\Gamma)$. Since $\oplus_{e \in \cE} C_0^\infty (0, \ell (e)) \subset \dom S$, the Friedrichs extension of $S$ is the operator $- \Delta_{{\rm D}, \Gamma, q}$, called the *perturbed Dirichlet Laplacian*, given by $$\begin{aligned}
(- \Delta_{{\rm D},\Gamma, q} f)_e & = - f_e'' + q_e f_e \qquad \text{on each edge}~e \in \cE, \\
\dom (- \Delta_{{\rm D},\Gamma, q}) & = \left\{ f \in \widetilde H^2 (\Gamma) \cap H^1 (\Gamma) : f (v) = 0~\text{for each}~v \in \cV \right\};\end{aligned}$$ if $q = 0$ identically, we just write $- \Delta_{{\rm D}, \Gamma}$ and call it the *Dirichlet Laplacian*. The operator $- \Delta_{{\rm D}, \Gamma, q}$ has a purely discrete spectrum. In the case $q = 0$ identically, the latter is given by $$\begin{aligned}
\label{eq:VDirichleteigenvalues}
\sigma (- \Delta_{{\rm D},\Gamma}) = \left\{ \lambda = \frac{k^2 \pi^2}{\ell (e)^2} : e \in \cE, k = 1, 2, \dots \right\},\end{aligned}$$ where the multiplicity of an eigenvalue $\lambda$ coincides with the number of values $k$ and edges $e$ for which $\lambda = \frac{k^2 \pi^2}{\ell (e)^2}$. In particular, $$\begin{aligned}
\min \sigma (- \Delta_{{\rm D},\Gamma, q}) \geq \min \sigma (- \Delta_{{\rm D},\Gamma}) = \frac{\pi^2}{(\max_{e \in \cE} \ell (e))^2} =: \mu > 0,\end{aligned}$$ where we have used the assumption that $q$ is nonnegative, and the inclusion $S \subset - \Delta_{{\rm D},\Gamma, q}$ implies $$\begin{aligned}
(S f, f) \geq \mu \|f\|^2, \quad f \in \dom S,\end{aligned}$$ where $(\cdot, \cdot)$ and $\| \cdot \|$ denote the inner product and norm respectively in $L^2 (\Gamma)$. By an easy integration by parts, the adjoint of $S$ is given by $$\begin{aligned}
(S^* f)_e & = - f_e'' + q_e f_e \qquad \text{on each edge}~e \in \cE, \\
\dom S^* & = \widetilde H^2 (\Gamma) \cap H^1 (\Gamma).\end{aligned}$$ The two self-adjoint extensions of $S$ in focus here will be the Krein–von Neumann extension of $S$ and the Schrödinger operator with standard (also called continuity-Kirchhoff) vertex conditions.
We assume that Hypothesis \[hyp\] is satisfied.
1. The [*perturbed Krein Laplacian*]{} on $\Gamma$ is the Krein–von Neumann extension $$\begin{aligned}
- \Delta_{{\rm K}, \Gamma, q} := S_{\rm K}\end{aligned}$$ of $S$.
2. The *perturbed standard Laplacian* on $\Gamma$ is the operator given by $$\begin{aligned}
(- \Delta_{{\rm st},\Gamma,q} f)_e & = - f_e'' + q_e f_e \qquad \text{on each edge}~e \in \cE, \\
\dom (- \Delta_{{\rm st},\Gamma,q}) & = \left\{ f \in \widetilde H^2 (\Gamma) \cap H^1 (\Gamma) : \partial_\nu f (v) = 0~\text{for each}~v \in \cV \right\}.\end{aligned}$$ The corresponding vertex conditions are called *standard conditions*.
In the case that the potenial $q$ is identically zero, we write $ - \Delta_{{\rm K}, \Gamma} := - \Delta_{{\rm K}, \Gamma,0}$ and $- \Delta_{{\rm st},\Gamma} := - \Delta_{{\rm st}, \Gamma, 0}$ and call these operators [*Krein Laplacian*]{} and *standard Laplacian*, respectively.
We point out that, in general, $ - \Delta_{{\rm K}, \Gamma, q} \ne - \Delta_{{\rm K}, \Gamma} + q$ (where we interpret the latter as an additive perturbation of the Krein Laplacian); see the discussion in Remark \[rem:perturbation\]. On the other hand, it holds that $- \Delta_{{\rm st},\Gamma,q}=- \Delta_{{\rm st},\Gamma} +q$, by definition.
In what follows, it will be useful to embed the study of $- \Delta_{{\rm K}, \Gamma, q}$ in the framework of boundary triples. The following proposition can be found in [@EK14 Lemma 2.14 and Theorem 2.16]; see also [@BLLR18 Proposition 10.1]. For the statement on the weighted discrete Laplacian, see e.g. Step 2 in the proof of [@GR20 Proposition 3.1].
\[prop:graphBT\] Assume that Hypothesis \[hyp\] is satisfied, and let $S$ be defined in . For $f \in \dom S^* = \widetilde H^2 (\Gamma) \cap H^1 (\Gamma)$, define $$\begin{aligned}
\Gamma_0 f = \begin{pmatrix} f (v_1) \\ \vdots \\ f (v_V) \end{pmatrix} \quad \text{and} \quad \Gamma_1 f = \begin{pmatrix} - \partial_\nu f (v_1) \\ \vdots \\ - \partial_\nu f (v_V) \end{pmatrix},\end{aligned}$$ where $v_1, \dots, v_V$ is an enumeration of the vertices of $\Gamma$. Then $S$ is a closed operator and $\{ \C^V, \Gamma_0, \Gamma_1\}$ is a boundary triple for $S^*$; in particular, $S$ has defect numbers $$\begin{aligned}
\label{eq:defectV}
n_- = n_+ = V.\end{aligned}$$ The corresponding extensions $A$ and $B$ of $S$ defined in are given by $$\begin{aligned}
A = - \Delta_{{\rm D}, \Gamma, q} \quad \text{and} \quad B = - \Delta_{{\rm st}, \Gamma, q};\end{aligned}$$ in particular, $0 \in \rho (A)$. The value of the corresponding Weyl function at $\lambda = 0$ is $M (0) = - \Lambda_q$, where $\Lambda_q$ is the [*Dirichlet-to-Neumann matrix*]{} defined via the relation $$\begin{aligned}
\label{eq:DN}
\begin{pmatrix} \partial_\nu f_* (v_1) \\ \vdots \\ \partial_\nu f_* (v_V) \end{pmatrix} = \Lambda_q \begin{pmatrix} f_* (v_1) \\ \vdots \\ f_* (v_V) \end{pmatrix},\end{aligned}$$ where $f_* \in \ker S^*$ is arbitrary. In the potential-free case, $q = 0$ identically, the value of the corresponding Weyl function is $M (0) = - \Lambda_0 = - L$, where $L$ is the weighted discrete Laplacian $L$ defined as $$\begin{aligned}
\label{eq:L}
L_{i, j} = \begin{cases} - \sum_{e~\text{connects $v_i$ and $v_j$}} \frac{1}{L (e)} & \text{if}~v_i~\text{and}~v_j~\text{are adjacent}, i \neq j,\\ 0 & \text{if}~v_i, v_j~\text{are not adjacent}, \\ \sum_{e \in \cE (v_i),~e~\text{no loop}} \frac{1}{L (e)} & \text{if}~i = j. \end{cases}\end{aligned}$$
This proposition allows us to describe the domain of $- \Delta_{{\rm K}, \Gamma, q}$ in terms of its vertex conditions and to obtain some properties of the perturbed Krein Laplacian right away. The next proposition follows immediately from Proposition \[prop:graphBT\] together with Proposition \[prop:KreinBT\]. Furthermore, from we obtain the multiplicity of the zero eigenvalue of $- \Delta_{{\rm K}, \Gamma, q}$.
\[prop:KreinV\] Under Hypothesis \[hyp\] the perturbed Krein Laplacian acts as $$\begin{aligned}
\big( - \Delta_{{\rm K}, \Gamma, q} f \big)_e = - f_e'' + q_e f_e \qquad \text{on each edge}~e \in \cE,\end{aligned}$$ and its domain consists of all $f \in \widetilde H^2 (\Gamma) \cap H^1 (\Gamma)$ such that $$\begin{aligned}
\begin{pmatrix} \partial_\nu f (v_1) \\ \vdots \\ \partial_\nu f (v_V) \end{pmatrix} = \Lambda_q \begin{pmatrix} f (v_1) \\ \vdots \\ f (v_V) \end{pmatrix},\end{aligned}$$ where $\Lambda_q$ is the Dirichlet-to-Neumann matrix defined in . Moreover, $$\begin{aligned}
\label{eq:Vkernel}
\dim \ker \big( - \Delta_{{\rm K}, \Gamma, q} \big) = \dim\ker S^* = V.\end{aligned}$$ In the potential-free case $q = 0$ identically, the domain of $- \Delta_{{\rm K},\Gamma}$ consists of all $f \in \widetilde H^2 (\Gamma) \cap H^1 (\Gamma)$ which satisfy the vertex conditions $$\begin{aligned}
\label{eq:KreinConditionsVLaplace}
\begin{pmatrix} \partial_\nu f (v_1) \\ \vdots \\ \partial_\nu f (v_V) \end{pmatrix} = L \begin{pmatrix} f (v_1) \\ \vdots \\ f (v_V) \end{pmatrix},\end{aligned}$$ where $L$ is the weighted discrete Laplacian in .
The vertex conditions of $- \Delta_{{\rm K}, \Gamma, q}$ are nonlocal, i.e. they couple values of the function and its derivatives at different vertices. In the potential-free case it actually follows from that the vertex conditions of the Krein Laplacian couple each vertex with all of its neighbours.
We calculate the vertex conditions of the Krein Laplacian explicitly for two example graphs.
\[ex:interval\] Let $\Gamma= [0, \ell]$ be an interval, i.e. a graph consisting of two vertices and one edge between them. On this graph, the weighted discrete Laplacian $L$ defined in equals $$\begin{aligned}
L = \frac{1}{\ell} \begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix}\end{aligned}$$ and the vertex condition for the Krein Laplacian $-\Delta_{{\rm K},\Gamma}$ as described in Proposition \[prop:KreinV\] can be rewritten $$\begin{aligned}
f' (0) = f' (\ell), \qquad f (\ell) = f (0) + \ell f' (0).\end{aligned}$$
Our second example shows that the Krein Laplacian and the standard Laplacian may coincide in some cases; cf. Corollary \[cor:flower\] below.
\[ex:flower\] Let $\Gamma$ be a flower graph, i.e. a graph with one vertex and $E$ loops attached to it; special cases are loops ($E = 1$) and figure-8 graphs ($E = 2$); cf. Figure \[fig:flower\].
; (0,0) circle (2.5pt);
; (0,0) circle (2.5pt);
; (0,0) circle (2.5pt);
Then any function $f_*$ which is harmonic on every edge and belongs to $H^1 (\Gamma)$ is necessarily constant on all of $\Gamma$. Thus each $f \in \dom (- \Delta_{{\rm K},\Gamma})$ satisfies $f = f_S + c$ with $f_S \in \dom S$ and $c$ constant; in particular, $f \in \dom (- \Delta_{{\rm st},\Gamma})$, the domain of the standard Laplacian on $\Gamma$. As both $- \Delta_{{\rm K},\Gamma}$ and $- \Delta_{{\rm st},\Gamma}$ are self-adjoint operators, they coincide, $- \Delta_{{\rm K},\Gamma} = - \Delta_{{\rm st},\Gamma}$, on any flower graph $\Gamma$.
Next we compare the perturbed Krein Laplacian with the perturbed Dirichlet Laplacian and the perturbed standard Laplacian. We apply Proposition \[prop:KreinV\] and Corollary \[cor:KreinB\] to the boundary triple in Proposition \[prop:graphBT\] and get the following result.
\[thm:resDiffGraph\] Assume that Hypothesis \[hyp\] is satisfied. Let $\lambda \mapsto \gamma (\lambda)$ and $\lambda \mapsto M (\lambda)$ be the $\gamma$-field and Weyl function respectively corresponding to the boundary triple in Proposition \[prop:graphBT\].
1. For $\lambda \in \rho (- \Delta_{{\rm K}, \Gamma, q}) \cap \rho (- \Delta_{{\rm D}, \Gamma, q})$, the formula $$\begin{aligned}
\big(- \Delta_{{\rm K}, \Gamma, q} - \lambda \big)^{-1} - \big(- \Delta_{{\rm D}, \Gamma, q} - \lambda \big)^{-1} = -\gamma (\lambda) \big( \Lambda_q + M (\lambda) \big)^{-1} \gamma (\overline \lambda)^*
\end{aligned}$$ holds. In particular, $$\begin{aligned}
\dim \ran \Big[ \big(- \Delta_{{\rm K}, \Gamma, q} - \lambda \big)^{-1} & - \big(- \Delta_{{\rm D}, \Gamma, q} - \lambda \big)^{-1} \Big] = V.\end{aligned}$$
2. For $\lambda \in \rho (- \Delta_{{\rm K}, \Gamma, q}) \cap \rho(- \Delta_{{\rm st}, \Gamma, q}) \cap \rho (- \Delta_{{\rm D}, \Gamma, q})$, the formula $$\begin{aligned}
\big(- \Delta_{{\rm K}, \Gamma, q} - \lambda \big)^{-1} - \big(- \Delta_{{\rm st}, \Gamma, q} - \lambda \big)^{-1} = \gamma (\lambda) \big( \Lambda_q + M (\lambda) \big)^{-1} \Lambda_q M (\lambda)^{-1} \gamma (\overline \lambda)^*
\end{aligned}$$ holds. In particular, $$\begin{aligned}
\dim \ran \Big[ \big(- \Delta_{{\rm K}, \Gamma, q} - \lambda \big)^{-1} & - \big(- \Delta_{{\rm st}, \Gamma, q} - \lambda \big)^{-1} \Big] \\
& = \dim \ran \Lambda_q = \begin{cases} V - 1 & \text{if}~q = 0~\text{identically}, \\ V, & \text{else}. \end{cases}\end{aligned}$$
The only assertion to prove is that $\ran \Lambda_q$ has the dimension claimed in the theorem. For the potential-free case, where $\Lambda_q = L$, the weighted discrete Laplacian, it is well-known that the kernel is one-dimensional (consisting of the constant vectors), and hence its range has dimension $V - 1$. Now let $q \geq 0$ be a nontrivial function, and let $\phi \in \ker \Lambda_q$. Then by definition, there exists a unique $f \in \ker S^*$ such that $\Gamma_0 f = \phi$ and $\Gamma_1 f = 0$, i.e. $f \in \ker S^* \cap \dom (- \Delta_{{\rm st}, \Gamma, q})$. In other words, $f \in \ker (- \Delta_{{\rm st}, \Gamma, q})$. But then, by standard variational principles, $$\begin{aligned}
0 = \int_\Gamma |f'|^2 \dd x + \int_\Gamma q |f|^2 \dd x.\end{aligned}$$ Since both terms on the right-hand side are nonnegative, they are zero separately. From $\int_\Gamma |f'|^2 \dd x = 0$, it follows that $f$ is constant on each edge and, by continuity, constant on $\Gamma$. But then $\int_\Gamma q |f|^2 \dd x = 0$ yields $f = 0$ identically, as $q$ is nontrivial. Finally, $\phi = \Gamma_0 f = 0$, so that $\ker \Lambda_q = \{0\}$. Consequently, $\dim \ran \Lambda_q = V$, which yields the desired result.
Now the observation of Example \[ex:flower\] can be sharpened in the following way. Since flower graphs are the only graphs with $V = 1$, this is an immediate consequence of Theorem \[thm:resDiffGraph\].
\[cor:flower\] Under Hypothesis \[hyp\], the following statements are equivalent.
1. The perturbed Krein Laplacian and the perturbed standard Laplacian coincide, i.e. $- \Delta_{{\rm K}, \Gamma, q} = - \Delta_{{\rm st},\Gamma, q}$;
2. $\Gamma$ is a flower graph and $q = 0$ identically.
Theorem \[thm:resDiffGraph\] allows us to deduce eigenvalue asymptotics for the perturbed Krein Laplacian: for $\lambda \geq 0$ denote by $$\begin{aligned}
\mathcal{N} \big(\lambda; - \Delta_{\bullet, \Gamma, q} \big) := \big| \sigma \big( - \Delta_{\bullet, \Gamma, q} \big) \cap (-\infty,\lambda] \big|, \qquad \bullet = {\rm K, D, st},\end{aligned}$$ the number of eigenvalues of the respective operator up to $\lambda$. Under Hypothesis \[hyp\], it follows immediately from Theorem \[thm:resDiffGraph\] and the minimality property of the Krein–von Neumann extension that $$\begin{aligned}
\mathcal{N} \big(\lambda;- \Delta_{{\rm D}, \Gamma,q} \big) & \leq \mathcal{N}\big(\lambda;- \Delta_{{\rm K}, \Gamma,q}\big) \leq \mathcal{N} \big(\lambda;- \Delta_{{\rm D}, \Gamma,q} \big)+V, \label{eq:countDV}\\
\mathcal{N} \big(\lambda;- \Delta_{{\rm st}, \Gamma,q} \big) & \leq \mathcal{N}\big(\lambda;- \Delta_{{\rm K}, \Gamma,q} \big) \leq \mathcal{N} \big(\lambda;- \Delta_{{\rm st}, \Gamma,q} \big)+V.\nonumber\end{aligned}$$ In the case that $q$ is identically zero on $\Gamma$, the latter inequality may be strengthened, $$\begin{aligned}
\label{eq:countstVLaplace}
\mathcal{N} \big(\lambda;- \Delta_{{\rm st}, \Gamma} \big) & \leq \mathcal{N}\big(\lambda;- \Delta_{{\rm K}, \Gamma} \big) \leq \mathcal{N} \big(\lambda;- \Delta_{{\rm st}, \Gamma} \big) + V - 1.\end{aligned}$$ Morover, one can use the inequalities for $- \Delta_{{\rm D}, \Gamma}$ in this case to deduce the following.
\[cor:VWeyl\] In the case of zero potential $q\equiv0$, for any $\lambda\geq0$, $$\frac{\ell(\Gamma)}{\pi}\sqrt{\lambda}-E \leq \mathcal{N} \big(\lambda;- \Delta_{{\rm K}, \Gamma} \big) \leq \frac{\ell(\Gamma)}{\pi}\sqrt{\lambda}+V.$$
It is a straightforward exercise to show that $$\mathcal{N}(\lambda;- \Delta_{{\rm D}, \Gamma}) = \sum_{e\in \cE} \left\lfloor \frac{\ell(e)}{\pi}\sqrt{\lambda}\right\rfloor$$ follows from . In particular, this implies $$\frac{\ell(\Gamma)}{\pi}\sqrt{\lambda}-E \leq \mathcal{N}(\lambda;- \Delta_{{\rm D}, \Gamma}) \leq \frac{\ell(\Gamma)}{\pi}\sqrt{\lambda},$$ and then inserting this into yields the desired result.
One can immediately deduce from Corollary \[cor:VWeyl\] that the eigenvalues for $- \Delta_{{\rm K}, \Gamma}$ possess the Weyl asymptotics $$\lambda_j(- \Delta_{{\rm K}, \Gamma}) \sim \left(\frac{j \pi}{\ell(\Gamma)}\right)^2$$ as $j\to\infty$. However, we remark that in fact any self-adjoint extension of the operator $S$ given by possesses these same asymptotics.
In the following, we are going to state some eigenvalue inequalities for the perturbed Krein Laplacian. It follows directly from that $$\begin{aligned}
\lambda_{j + V} (- \Delta_{{\rm K}, \Gamma, q})=\lambda_j^+ (- \Delta_{{\rm K}, \Gamma, q})\end{aligned}$$ holds for all $j\in\mathbb{N}$. To investigate properties of the positive eigenvalues of $- \Delta_{{\rm K},\Gamma,q}$, we first formulate the abstract variational principle in Theorem \[thm:minMaxKrein\] in our specific situation.
\[thm:minMaxKreinV\] If Hypothesis \[hyp\] is satisfied, then the spectrum of $- \Delta_{{\rm K}, \Gamma, q}$ is purely discrete, and the positive eigenvalues $$\begin{aligned}
\lambda_1^+ \big( - \Delta_{{\rm K}, \Gamma, q} \big) \leq \lambda_2^+ \big( - \Delta_{{\rm K}, \Gamma, q} \big) \leq \dots\end{aligned}$$ of $- \Delta_{{\rm K}, \Gamma, q}$, counted with multiplicities, satisfy $$\begin{aligned}
\lambda_j^+ \big( - \Delta_{{\rm K}, \Gamma, q} \big) = \min_{\substack{F \subset \widetilde H_0^2 (\Gamma) \\ \dim F = j}} \max_{\substack{f \in F \\ f \neq 0}} \frac{\int_\Gamma \left|- f'' + q f\right|^2 \dd x}{\int_\Gamma |f'|^2 \dd x + \int_\Gamma q |f|^2 \dd x}\end{aligned}$$ for all $j \in \N$. In particular, in the potential-free case $q = 0$ identically, $$\begin{aligned}
\lambda_j^+ (- \Delta_{{\rm K},\Gamma}) = \min_{\substack{F \subset \widetilde H_0^2 (\Gamma) \\ \dim F = j}} \max_{\substack{f \in F \\ f \neq 0}} \frac{\int_\Gamma |f''|^2 \dd x}{\int_\Gamma |f'|^2 \dd x}\end{aligned}$$ holds for all $j \in \N$.
The following eigenvalue inequalities and equalities are direct consequences of Theorem \[thm:inequalitiesAbstract\] and .
\[thm:inequalitiesV\] Let Hypothesis \[hyp\] be satisfied, let $- \Delta_{\Gamma, q}$ be any self-adjoint extension of the operator $S$ in , and let $d := \dim \ker (- \Delta_{\Gamma, q})$. Then $$\begin{aligned}
\lambda_{j + d} (- \Delta_{\Gamma, q}) \leq \lambda_j^+ (- \Delta_{{\rm K}, \Gamma, q}) = \lambda_{j + V} (- \Delta_{{\rm K}, \Gamma, q})\end{aligned}$$ holds for all $j \in \N$. In particular, $$\begin{aligned}
\label{eq:DirichletKreinV}
\lambda_j (- \Delta_{{\rm D}, \Gamma, q}) \leq \lambda_j^+ (- \Delta_{{\rm K}, \Gamma, q}) = \lambda_{j + V} (- \Delta_{{\rm K}, \Gamma, q})\end{aligned}$$ holds for all $j \in \N$, and in the potential-free case we have $$\begin{aligned}
\label{eq:standardKreinV}
\lambda_{j + 1} (- \Delta_{{\rm st},\Gamma}) \leq \lambda_j^+ (- \Delta_{{\rm K},\Gamma}) = \lambda_{j + V} (- \Delta_{{\rm K},\Gamma})\end{aligned}$$ for all $j \in \N$.
The inequalities and can alternatively be deduced from Theorem \[thm:resDiffGraph\]; cf. and .
If the edge lengths in $\Gamma$ are rationally independent and $q \equiv 0$, then the inequality is strict for all $j \in \N$, as in this case it can be seen easily that $S$ does not possess any eigenvalues.
If $\Gamma$ is a tree graph, then it is known that for the Laplacian, $$\begin{aligned}
\label{eq:Friedlander}
\lambda_{j + 1} (- \Delta_{{\rm st},\Gamma}) \leq \lambda_j (- \Delta_{{\rm D},\Gamma})\end{aligned}$$ holds for all $j \in \N$; see e.g. [@R17 Theorem 4.1]. One may combine this with to obtain in an alternative way. However, it is worth pointing out that does not hold in general for graphs with cycles (see the discussion in [@KR20 Section 5]), but in this case is still true.
Spectral implications of graph surgery operations {#sec:surgery}
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Next, we investigate the effect of graph surgery operations on the eigenvalues of the perturbed Krein Laplacian $- \Delta_{{\rm K}, \Gamma, q}$. Graph surgery refers to the process of transforming the operator by making topological changes to the metric graph, such as gluing vertices together or adding edges, forming a new graph $\widetilde\Gamma$. One associates a potential $\widetilde q$ to the new graph $\widetilde \Gamma$ which will be determined by the type of surgery carried out. Given a surgery operation $- \Delta_{{\rm K}, \Gamma, q}\mapsto- \Delta_{{\rm K}, \widetilde\Gamma, \widetilde q}$, only the operators $- \Delta_{{\rm K}, \Gamma, q}$ and $- \Delta_{{\rm K}, \widetilde\Gamma, \widetilde q}$ will be of significance to us, and thus we use the following simplified notation for their eigenvalues throughout this section: $$\begin{aligned}
\lambda_j^+:= & \lambda_j^+(- \Delta_{{\rm K}, \Gamma, q}), & \widetilde \lambda_j^+:= & \lambda_j^+(- \Delta_{{\rm K}, \widetilde \Gamma, \widetilde q}), \\
\lambda_j:= & \lambda_j(- \Delta_{{\rm K}, \Gamma, q}), & \widetilde \lambda_j:= & \lambda_j(- \Delta_{{\rm K}, \widetilde \Gamma, \widetilde q}). \end{aligned}$$ In what follows, we always assume Hypothesis \[hyp\]; the new potential $\widetilde q$ will satisfy the analogue of Hypothesis \[hyp\] conditions for $\widetilde\Gamma$ by construction.
We begin with transformations which only affect the vertex conditions of the operator, or add new vertices. For such operations, the potential $\widetilde q\equiv q$ is unchanged (except possibly on a set of measure zero).
Let $\widetilde\Gamma$ be the graph formed from $\Gamma$ by identifying a number of its vertices, say $v_1,\dots,v_{k+1}$, to form a new vertex $v_0$. The total number of vertices is thereby reduced by $k$, and the potential $q$ associated with $\Gamma$ remains well-defined on $\widetilde\Gamma$. The transformation $- \Delta_{{\rm K}, \Gamma, q}\mapsto- \Delta_{{\rm K}, \widetilde \Gamma, q}$ is called *gluing vertices*, and the inverse operation is referred to as *cutting through vertices*; cf. Figure \[fig:gluingV\].
at (-4.5,0) [$\bullet$]{}; at (-3.5,1) [$\bullet$]{}; at (-3,0) [$\bullet$]{}; at (-2,-0.5) [$\bullet$]{}; at (-1.5,0.5) [$\bullet$]{};
at (-5.5,0.5) [$\bullet$]{}; at (-4.5,-1.5) [$\bullet$]{}; at (-2.5,1.5) [$\bullet$]{};
at (-3.5,1) \[yshift=5pt\] [$v_1$]{}; at (-3,0) \[yshift=-5pt\] [$v_2$]{}; at (-1.5,0.5) \[yshift=5pt\] [$v_3$]{};
(-4.5,0) – (-3.5,1) ; (-4.5,0) – (-3,0) ; (-3.5,1) – (-3,0) ; (-3,0) – (-2,-0.5) ; (-2,-0.5) – (-1.5,0.5) ;
(-4.5,0) – (-5.5,0.5) ; (-4.5,0) – (-4.5,-1.5) ; (-3.5,1) – (-2.5,1.5) ;
(-0.5,0) – (0.5,0) ;
at (2.5,0) [$\bullet$]{}; at (4,0) [$\bullet$]{}; at (5,-0.5) [$\bullet$]{};
at (1.5,0.5) [$\bullet$]{}; at (2.5,-1.5) [$\bullet$]{}; at (4.5,1) [$\bullet$]{};
at (4,0) \[xshift=5pt, yshift=-3pt\] [$v_0$]{};
(2.5,0) to \[bend left=90\] (4,0) ; (2.5,0) to \[bend right=90\] (4,0) ;
(4,0) to \[bend left=60\] (5,-0.5) ; (4,0) to \[bend right=60\] (5,-0.5) ;
(4,0) to \[bend left=90\] (3.5,0) ; (4,0) to \[bend right=90\] (3.5,0) ;
(2.5,0) – (1.5,0.5) ; (2.5,0) – (2.5,-1.5) ; (4,0) – (4.5,1) ;
\[thm:gluingV\] Let Hypothesis \[hyp\] be satisfied, and let $\widetilde{\Gamma}$ be the graph formed by gluing precisely $k + 1$ vertices of $\Gamma$. Then for the corresponding perturbed Krein Laplacians:
- the positive eigenvalues satisfy the interlacing inequalities $$\label{eq:gluingV+}
\widetilde \lambda_j^+ \leq \lambda_j^+\leq\widetilde\lambda_{j + k}^+ \leq \lambda_{j + k}^+ , \qquad j \in \N;$$
- the eigenvalues (counting ground states) satisfy the interlacing inequalities $$\label{eq:gluingV}
\lambda_j \leq \widetilde\lambda_j \leq\lambda_{j+k} \leq \widetilde\lambda_{j+k} , \qquad j \in \N.$$ In particular, $$\label{eq:gluingVV}
0=\lambda_V< \widetilde\lambda_{V-k+1}.$$
Denote by $S$ and $\widetilde S$ the symmetric operators in $L^2 (\Gamma)$ and $L^2 (\widetilde \Gamma)$, respectively, defined as in . Then $$\begin{aligned}
\dom S = \widetilde H_0^2 (\Gamma) \subset \widetilde H_0^2 (\widetilde \Gamma) = \dom \widetilde S,\end{aligned}$$ and the action of the two operators coincides on the smaller domain; we always identify functions on $\Gamma$ with functions on $\widetilde \Gamma$, and conversely, in the obvious way. Thus $S \subset \widetilde S$.
We show next that the co-dimension of $\dom S$ in $\dom \widetilde S$ is $k$, and we do this for the case $k = 1$ only; for higher $k$ this can be obtained by successively gluing vertices. For $k = 1$, denote by $v_1,v_2$ the vertices of $\Gamma$ that are glued to form the new vertex $v$. Let $f, g \in\dom \widetilde S$ and observe that the linear combination $$h:=(\partial_\nu g (v_1)) f - (\partial_\nu f(v_1)) g$$ satisfies both Dirichlet and Kirchhoff conditions at both $v_1$ and $v_2$, with the latter due to the fact that $f, g$ satisfy Kirchhoff conditions at $v$ (i.e. $\partial_\nu f(v_1) + \partial_\nu f (v_2) = 0$ and likewise for $g$). Then $h \in \dom S$, which proves the claim on the co-dimension. Thus we can apply Theorem \[thm:interlacingAbstract\] to obtain inequality .
For inequality , one applies , together with the fact that the number of vertices of $\widetilde \Gamma$ is $V - k$, to the chain of inequalities to obtain . Finally, inequality is a trivial consequence of .
Gluing vertices therefore increases the eigenvalues of the perturbed Krein Laplacian, with inequality providing bounds for this increase. Indeed, implies that eigenvalues $\lambda_{V-k+1},\dots,\lambda_V$ increase strictly. On the other hand, the increases are counteracted by the fact that the kernel of the operator shrinks after gluing, which explains why the positive eigenvalues actually decrease. By contrast, whilst the eigenvalues of the perturbed standard Laplacian increase by gluing, satisfying in particular the interlacing inequalities $$\lambda_j (- \Delta_{{\rm st},\Gamma,q}) \leq \lambda_j (- \Delta_{{\rm st},\widetilde \Gamma,q})\leq\lambda_{j+k} (- \Delta_{{\rm st},\Gamma,q})\leq \lambda_{j+k} (- \Delta_{{\rm st},\widetilde \Gamma,q}),$$ the kernel is unchanged, and thus the positive eigenvalues increase as well.
\[ex:glueInterval\] Let $\Gamma=[0,\ell]$ be the interval of length $\ell$. The vertex conditions for the Krein Laplacian $-\Delta_{{\rm K},\Gamma}$ were calculated in Example \[ex:interval\]. From this, one computes that the eigenvalues $\lambda=\kappa^2$ are given by the solutions of the equation $$\left[\cos\frac{\kappa\ell}{2}-\frac{2}{\kappa\ell}\sin\frac{\kappa\ell}{2}\right]\sin\frac{\kappa\ell}{2}=0.$$ The positive solutions to this are $$\kappa=\begin{cases}\frac{j\pi}{\ell} & \text{if $j=2,4,6,...$}\\\frac{j\pi}{\ell}-\eta_j & \text{if $j=3,5,7,...$}\end{cases}$$ where the numbers $\eta_j$ are such that $0<\eta_j\ll \frac{\pi}{\ell} $ and $\lim_{j\to\infty}\eta_j=0$.
Now let $\widetilde \Gamma$ be the loop of length $\ell$, formed by gluing together the two vertices of the interval $\Gamma$; see Figure \[fig:gluingExample\]. According to Corollary \[cor:flower\], the Krein Laplacian $- \Delta_{{\rm K}, \widetilde \Gamma}$ on the loop is identical to the standard Laplacian $- \Delta_{{\rm st}, \widetilde \Gamma}$, and thus they share the same eigenvalues.
at (-3.5,0) [$\bullet$]{}; at (-1.5,0) [$\bullet$]{};
at (-2.5,0) \[yshift=-7pt\] [ $\ell$]{};
(-3.5,0) – (-1.5,0) ;
(-0.5,0) – (0.5,0) ;
at (1.5,0) [$\bullet$]{};
at (2.774,0) \[xshift=5pt\] [ $\ell$]{};
(2.137,0) circle (0.637cm);
The following tables are demonstrative of Theorem \[thm:gluingV\] for these two graphs: the positive eigenvalues decrease by gluing, but when the ground states are included, they increase.
--------------- ------------------------------------------- ------------------------------------------------------
$j$ $\lambda_j^+(- \Delta_{{\rm K}, \Gamma})$ $\lambda_j^+(- \Delta_{{\rm K}, \widetilde \Gamma})$
\[0.5ex\]
$1$ $\left(\frac{2\pi}{\ell}\right)^2$ $\left(\frac{2\pi}{\ell}\right)^2$
\[0.5ex\] $2$ $\left(\frac{3\pi}{\ell}-\eta_3\right)^2$ $\left(\frac{2\pi}{\ell}\right)^2$
\[0.5ex\] $3$ $\left(\frac{4\pi}{\ell}\right)^2$ $\left(\frac{4\pi}{\ell}\right)^2$
\[0.5ex\] $4$ $\left(\frac{5\pi}{\ell}-\eta_5\right)^2$ $\left(\frac{4\pi}{\ell}\right)^2$
\[0.5ex\] $5$ $\left(\frac{6\pi}{\ell}\right)^2$ $\left(\frac{6\pi}{\ell}\right)^2$
\[0.5ex\] $6$ $\left(\frac{7\pi}{\ell}-\eta_7\right)^2$ $\left(\frac{6\pi}{\ell}\right)^2$
--------------- ------------------------------------------- ------------------------------------------------------
: All eigenvalues
--------------- ------------------------------------------- ----------------------------------------------------
$j$ $\lambda_j(- \Delta_{{\rm K}, \Gamma})$ $\lambda_j(- \Delta_{{\rm K}, \widetilde \Gamma})$
\[0.5ex\]
$1$ $0$ $0$
\[0.5ex\] $2$ $0$ $\left(\frac{2\pi}{\ell}\right)^2$
\[0.5ex\] $3$ $\left(\frac{2\pi}{\ell}\right)^2$ $\left(\frac{2\pi}{\ell}\right)^2$
\[0.5ex\] $4$ $\left(\frac{3\pi}{\ell}-\eta_3\right)^2$ $\left(\frac{4\pi}{\ell}\right)^2$
\[0.5ex\] $5$ $\left(\frac{4\pi}{\ell}\right)^2$ $\left(\frac{4\pi}{\ell}\right)^2$
\[0.5ex\] $6$ $\left(\frac{5\pi}{\ell}-\eta_5\right)^2$ $\left(\frac{6\pi}{\ell}\right)^2$
--------------- ------------------------------------------- ----------------------------------------------------
: All eigenvalues
Assume that Hypothesis \[hyp\] is satisfied, and let $e_0$ be an edge of $\Gamma$ with (possibly coincident) incident vertices $v_1,v_2$. Let $\widetilde\Gamma$ be the graph formed from $\Gamma$ by replacing $e_0$ with a path graph from $v_1$ to $v_2$, composed of two edges $e_1,e_2$, joined together by a degree-2 vertex $v_0$, and with total length $\ell(e_1)+\ell(e_2)=\ell(e_0)$. Parametrising $e_0$ by $[0,\ell(e_0)]$ and $e_1,e_2$ by $[0,\ell(e_1)],[\ell(e_1),\ell(e_0)]$ respectively, where the endpoint $\ell(e_1)$ in both of the latter is identified with $v_0$, the potential $\widetilde q$ associated with $\widetilde\Gamma$ is defined by $$\widetilde q_{e_1}:=\left.q_{e_0}\right|_{[0,\ell(e_1)]},\qquad\widetilde q_{e_2}:=\left.q_{e_0}\right|_{[\ell(e_1),\ell(e_2)]}$$ on $e_1,e_2$, and $\widetilde q_e \equiv q_e$ on all other edges $e$. The transformation $- \Delta_{{\rm K}, \Gamma, q}\mapsto- \Delta_{{\rm K}, \widetilde \Gamma, \widetilde q}$ is called *inserting a degree-2 vertex along an edge*, and the inverse operation is referred to as *removing a degree-2 vertex*; cf. Figure \[fig:deg2V\].
at (7-4.5,0) [$\bullet$]{}; at (7-3,0) [$\bullet$]{}; at (7-2,0.5) [$\bullet$]{};
at (7-3,0) \[yshift=5pt\] [$v_0$]{}; at (7-3.75,0) \[yshift=5pt\] [ $e_1$]{}; at (7-2.5,0.25) \[yshift=7pt\] [ $e_2$]{};
at (7-4.5,0) \[xshift=3pt, yshift=-5pt\] [$0$]{}; at (7-3,0) \[xshift=-8pt, yshift=-6pt\] [$\ell (e_1)$]{}; at (7-3,0) \[xshift=12pt, yshift=-3pt\] [$\ell (e_1)$]{}; at (7-2,0.5) \[xshift=-3pt, yshift=-10pt\] [$\ell (e_0)$]{};
(7-4.5,0) – (7-3,0) ; (7-3,0) – (7-2,0.5) ;
(7-4.5,0) – (7-5.5,0.5) ; (7-4.5,0) – (7-5,-1) ; (7-2,0.5) – (7-1.5,1.5) ; (7-2,0.5) – (7-1.5,0) ;
(-0.5,0) – (0.5,0) ;
at (-7+2.5,0) [$\bullet$]{}; at (-7+5,0.5) [$\bullet$]{};
at (-7+3.75,0.25) \[yshift=-7pt\] [ $e_0$]{};
at (-7+2.5,0) \[xshift=3pt, yshift=-5pt\] [$0$]{}; at (-7+5,0.5) \[xshift=-12pt, yshift=4pt\] [$\ell (e_0)$]{};
(-7+2.5,0) – (-7+5,0.5) ;
(-7+2.5,0) – (-7+1.5,0.5) ; (-7+2.5,0) – (-7+2,-1) ; (-7+5,0.5) – (-7+5.5,1.5) ; (-7+5,0.5) – (-7+5.5,0) ;
In the special case that $\Gamma$ is just one loop, it obviously does not make sense to remove the vertex of degree two, as the result would be a graph with one edge but no vertices. However, in this case the above procedure may just be understood as replacing the perturbed Krein Laplacian with the perturbed standard Laplacian. To replace “Krein vertex conditions” by standard conditions on arbitrary vertices, we refer to Theorem \[thm:addstandardB\].
\[thm:deg2V\] Let Hypothesis \[hyp\] be satisfied, and let $\widetilde{\Gamma}$ be the graph formed by inserting $k_0$ vertices of degree 2 along edges of $\Gamma$. Then for the corresponding Krein Laplacians:
- the positive eigenvalues satisfy $$\label{eq:deg2V+}
\lambda_j^+ \leq\widetilde\lambda_j^+ \leq \lambda_{j + k_0}^+ \leq \widetilde\lambda_{j + k_0}^+, \qquad j \in \N;$$
- the eigenvalues (counting ground states) satisfy $$\label{eq:deg2V}
\widetilde\lambda_{j}\leq \lambda_{j}\leq\widetilde \lambda_{j + k_0} \leq \lambda_{j + k_0} , \qquad j \in \N.$$
If we define $S$ and $\widetilde S$ corresponding to $\Gamma$ and $\widetilde \Gamma$ respectively, as in , then $\widetilde S \subset S$. Moreover, $\dom \widetilde S$ has co-dimension $k_0$ in $\dom S$; indeed, if $k_0 = 1$, then for any two linearly independent functions $f, g \in \dom S$, the function $f (v_1) g - g (v_1) f$ vanishes at $v_1$ and thus belongs to $\dom\widetilde S$. The case of arbitrary $k_0$ follows inductively. Then all estimates in follow directly from Theorem \[thm:interlacingAbstract\], noting that the roles of $S$ and $\widetilde S$ are reversed. After this, follows with the help of .
The following example shows that the positive eigenvalues of the Krein Laplacian may indeed increase strictly from adding a degree-2 vertex, in contrast with the standard Laplacian which does not feel degree-2 vertices at all.
Let $\Gamma$ be the interval of length two, and let $\widetilde\Gamma$ be the path graph formed by inserting a vertex of degree 2 at its midpoint, creating two intervals each of length one connected by a single vertex. A direct computation shows that the positive eigenvalues of the Krein Laplacian on $\widetilde\Gamma$ are the numbers $\kappa^2$ for which $\kappa$ is a root of $$\begin{aligned}
\kappa \big( (\kappa^2 - 2) \sin (2 \kappa) + \kappa + 4 \sin \kappa - 4 \kappa \cos \kappa + 3 \kappa \cos(2 \kappa) \big) = 0.\end{aligned}$$ The lowest two positive eigenvalues are then $\widetilde\lambda_1^+ \approx 4.5^2$ and $\widetilde\lambda_2^+ = (2 \pi)^2$. In contrast to this, the first two positive eigenvalues of the Krein Laplacian on $\Gamma$ are $\lambda_1^+=\pi^2$ and $\lambda_2^+<(3 \pi/2)^2$; cf. Example \[ex:glueInterval\].
We have seen in Theorem \[thm:gluingV\] how the eigenvalues change upon gluing vertices of $\Gamma$. It is also possible to glue arbitrary points of $\Gamma$ together. Again, as the (perturbed) Krein Laplacian distinguishes between vertices of degree two and non-vertex points on the graph, the following is more general than Theorem \[thm:gluingV\].
Assume that Hypothesis \[hyp\] is satisfied, and let $\mathcal{N}$ be a finite subset of points in $\Gamma$ (which may include both vertices and points along edges). Let $\widetilde\Gamma$ be the graph formed by first inserting a vertex at each of the points in $\mathcal{N}$ which are not already vertices, and then gluing all of these new vertices together with the remaining vertices in $\mathcal{N}$ to form a single point. The transformation $- \Delta_{{\rm K}, \Gamma, q}\mapsto- \Delta_{{\rm K}, \widetilde \Gamma, q}$ is called *gluing the points in $\cN$*.
This is evidently a two-step process, consisting of insterting degree-2 vertices along edges, and then gluing vertices. In general, one cannot determine the effect on individual eigenvalues since they increase during the first step but decrease during the second. Nevertheless, a direct application of Theorems \[thm:gluingV\] and \[thm:deg2V\] gives some insight into their behaviour.
\[cor:arbgluingV\] Assume that Hypothesis \[hyp\] is satisfied. Let $\mathcal{N}$ be a finite subset of $k+1$ points in $\Gamma$ of which $k_0\leq k+1$ are not vertices, and let $\widetilde{\Gamma}$ be the graph formed by gluing these points together. Then for the corresponding perturbed Krein Laplacians:
- the positive eigenvalues satisfy $$\begin{aligned}
\widetilde \lambda_{j}^+ \leq \lambda_{j + k_0}^+ \leq \widetilde\lambda_{j + k + k_0}^+ \leq \lambda_{j + k + 2 k_0}^+, \qquad j \in \N;\end{aligned}$$
- the eigenvalues (counting ground states) satisfy $$\begin{aligned}
\widetilde \lambda_{j} \leq \lambda_{j + k} \leq \widetilde\lambda_{j + k + k_0} \leq \lambda_{j + 2 k + k_0} , \qquad j \in \N.\end{aligned}$$
Next, we move on to transformations which change the volume of $\Gamma$. Here, the potential $q$ will not be well-defined on the new graph, for which the associated potential $\widetilde q$ is defined accordingly.
Assume that Hypothesis \[hyp\] is satisfied. Let $\widetilde\Gamma$ be the graph formed from $\Gamma$ by lengthening one of its edges, $e_0$, by a factor of $\alpha>1$, so that it has length $\widetilde \ell (e_0) = \alpha \ell (e_0)$ in $\widetilde \Gamma$. If there is a potential $q$ associated with $\Gamma$, then the potential $\widetilde q$ associated with $\widetilde\Gamma$ is defined via $$\label{lengthenedpotential}
\widetilde q_{e_0}(x):=\alpha^{-2}q_{e_0} (x/\alpha),$$ and $\widetilde q_e \equiv q_e$ on all other edges. The transformation $- \Delta_{{\rm K}, \Gamma, q}\mapsto- \Delta_{{\rm K}, \widetilde \Gamma, \widetilde q}$ is called *lengthening the edge $e_0$*, and the inverse operation is referred to as *shrinking the edge $e_0$*.
\[thm:lengthendgeV\] Let Hypothesis \[hyp\] be satisfied, and let $\widetilde{\Gamma}$ be the graph formed by lengthening one of the edges of $\Gamma$. Then for the corresponding perturbed Krein Laplacians:
- the positive eigenvalues satisfy $$\label{eq:lengthenedgeV+}
\widetilde \lambda_j^+\leq\lambda_j^+, \qquad j \in \N;$$
- the eigenvalues (counting ground states) satisfy $$\label{eq:lengthenedgeV}
\widetilde \lambda_j \leq \lambda_j , \qquad j \in \N.$$
Suppose that an edge $e_0$ of $\Gamma$ is lengthened by a factor of $\alpha>1$. Given $f\in\widetilde{H}_0^2(\Gamma)$, let $\widetilde{f}$ be the function such that $\widetilde{f}_{e_0} (x) = \alpha f_{e_0} (x/\alpha)$ and $\widetilde{f}_e (x) = f_e (x)$ for all other edges $e$. Now, $\widetilde{f}_{e_0}(0)=\widetilde{f}_{e_0}( \ell (e_0))=0$, preserving the Dirichlet conditions, and $\widetilde{f}_{e_0}'(0)=f_{e_0}'(0)$, $\widetilde{f}_{e_0}'( \ell (e_0)) = f_{e_0}' (\ell (e_0))$, preserving the Kirchhoff conditions, whence $\widetilde{f}\in\widetilde{H}_0^2(\widetilde{\Gamma})$. Notice that $\widetilde{H}_0^2({\Gamma})\to\widetilde{H}_0^2(\widetilde{\Gamma}):f\mapsto\widetilde{f}$ is a bijection. Then $$\begin{aligned}
\int_{0}^{\widetilde \ell (e_0)} |\widetilde{f}_{e_0}'|^2 \dd x + \int_{0}^{\widetilde \ell (e_0)}\widetilde{q}_{e_0}|\widetilde{f}_{e_0}|^2\dd x &= \alpha \int_0^{\ell (e_0)} |f_{e_0}'|^2\dd x + \alpha \int_0^{\ell (e_0)} q_{e_0} |f_{e_0}|^2 \dd x,\end{aligned}$$ and $$\begin{aligned}
\int_{0}^{\widetilde \ell (e_0)} \left|-\widetilde{f}_{e_0}'' + \widetilde{q}_{e_0}\widetilde{f}_{e_0} \right|^2 \dd x &= \frac{1}{\alpha} \int_0^{\ell (e_0)} \left|-f_{e_0}'' + q_{e_0} f_{e_0} \right|^2 \dd x,\end{aligned}$$ recalling that the potential is redefined by on the lengthened edge. Thus $$\frac{\int_{\widetilde{\Gamma}}|-\widetilde{f}''+\widetilde{q}\widetilde{f}|^2 \dd x}{\int_{\widetilde{\Gamma}}|\widetilde{f}'|^2\dd x + \int_{\widetilde{\Gamma}}\widetilde{q}|\widetilde{f}|^2\dd x}\leq\frac{\int_\Gamma |-f'+qf'|^2 \dd x}{\int_\Gamma |f'|^2 \dd x+\int_\Gamma q|f|^2 \dd x}.$$ Inequality follows from Theorem \[thm:minMaxKreinV\], and then from since the kernel of the operator is unchanged by the transformation.
The remaining surgery operation deals with expanding the graph by inserting a new finite, connected metric graph $\Gamma_0$ in some way to the original graph. If there is a potential $q_0$ associated with $\Gamma_0$, then we assume that it satisfies the following hypothesis in agreement with what is assumed for $q$ on $\Gamma$.
\[hyp0\] On the finite, connected metric graph $\Gamma_0$, the potential $q_0 : \Gamma_0 \to \R$ is measurable and bounded, and $q_0 (x) \geq 0$ holds for almost all $x \in \Gamma_0$.
As a rule, if no new potential is specified on the new edges, then it is reasonable to take the potential to be zero there. Nevertheless, the inequalities in Theorem \[thm:attachV\] hold for the potential chosen arbitrarily there under Hypothesis \[hyp0\].
Let Hypothesis \[hyp\] be satisfied, and let $\widetilde\Gamma$ be the graph formed from $\Gamma$ by gluing $m$ of the vertices of a finite, connected metric graph $\Gamma_0$ to distinct vertices of $\Gamma$. The new potential $\widetilde q$ associated with $\widetilde\Gamma$ is identical to $q$ on the edges inherited from $\Gamma$ and satisfies Hypothesis \[hyp0\] on the edges from $\Gamma_0$. The transformation $- \Delta_{{\rm K}, \Gamma, q}\mapsto- \Delta_{{\rm K}, \widetilde \Gamma, \widetilde q}$ is called *attaching a (connected) graph to $\Gamma$ (by $m$ vertices)*. The inverse operation may be referred to as *deleting a (connected) subgraph*; cf. Figure \[fig:attachV\].
at (-5.5,0) [$\bullet$]{}; at (-4.5,1) [$\bullet$]{}; at (-4,0) [$\bullet$]{}; at (-6.5,0.5) [$\bullet$]{}; at (-5.5,-1.5) [$\bullet$]{}; at (-3.5,1.5) [$\bullet$]{};
at (-3,-0.5) [$\bullet$]{}; at (-2,-1) [$\bullet$]{}; at (-1.5,0) [$\bullet$]{}; at (-3.5,0.5) [$\bullet$]{};
at (-4.65,0) \[yshift=10pt\] [$\Gamma$]{}; at (-2.25,-0.5) \[xshift=3pt\] [$\Gamma_0$]{};
(-5.5,0) – (-4.5,1) ; (-5.5,0) – (-4,0) ; (-4.5,1) – (-4,0) ; (-5.5,0) – (-6.5,0.5) ; (-5.5,0) – (-5.5,-1.5) ; (-4.5,1) – (-3.5,1.5) ;
(-3,-0.5) – (-2,-1) ; (-2,-1) – (-1.5,0) ; (-3,-0.5) – (-1.5,0) ; (-1.5,0) – (-3.5,0.5) ;
(-0.5,0) – (0.5,0) ;
at (2.5,0) [$\bullet$]{}; at (3.5,1) [$\bullet$]{}; at (4,0) [$\bullet$]{}; at (1.5,0.5) [$\bullet$]{}; at (2.5,-1.5) [$\bullet$]{}; at (4.5,1.5) [$\bullet$]{};
at (5,-0.5) [$\bullet$]{}; at (5.5,0.5) [$\bullet$]{};
at (3.35,0) \[yshift=10pt\] [$\widetilde\Gamma$]{};
(2.5,0) – (3.5,1) ; (2.5,0) – (4,0) ; (3.5,1) – (4,0) ; (2.5,0) – (1.5,0.5) ; (2.5,0) – (2.5,-1.5) ; (3.5,1) – (4.5,1.5) ;
(4,0) – (5,-0.5) ; (5,-0.5) – (5.5,0.5) ; (4,0) – (5.5,0.5) ; (5.5,0.5) – (3.5,1) ;
\[thm:attachV\] Assume that Hypotheses \[hyp\] and \[hyp0\] hold. Let $\widetilde{\Gamma}$ be the graph formed by attaching $\Gamma_0$ to $\Gamma$ by $m$ vertices. Then for the corresponding perturbed Krein Laplacians:
- the positive eigenvalues satisfy $$\label{eq:attachV+}
\widetilde\lambda_j^+\leq\lambda_j^+, \qquad j \in \N;$$
- the eigenvalues (counting ground states) satisfy $$\label{eq:attachV}
\widetilde\lambda_{j + V_0 - m} \leq \lambda_j, \qquad j \in \N;$$ here $V_0$ is the number of vertices of $\Gamma_0$.
Every function in $\widetilde H_0^2 (\Gamma)$ can be extended by zero to a function in $\widetilde H_0^2 (\widetilde{\Gamma})$, and this does not change the Rayleigh quotient. Thus inequality follows from Theorem \[thm:minMaxKreinV\]. Finally, is obtained from , since the dimension of the kernel of the operator increases by $V_0 - m$.
A special case of the previous theorem consists of inserting a single edge between two vertices of $\Gamma$, a process which does not change the dimension of the kernel of the perturbed Krein Laplacian.
\[cor:addedgeV\] Let Hypothesis \[hyp\] hold, and let $\widetilde{\Gamma}$ be the graph formed by inserting an edge $e_0$ between two (not necessarily distinct) vertices of $\Gamma$. Assume that the potential $q_0$ on $\Gamma_0=e_0$ satisfies Hypothesis \[hyp0\]. Then the eigenvalues (counting ground states) of the corresponding perturbed Krein Laplacians satisfy $$\begin{aligned}
\widetilde \lambda_j \leq \lambda_j , \qquad j \in \N.\end{aligned}$$
We emphasise that this behaviour differs substantially from the one for standard vertex conditions, where inserting an edge may either increase or decrease eigenvalues; cf. [@KMN13].
Isoperimetric inequalities {#sec:isoperimetric}
==========================
We now turn to estimates for the positive eigenvalues of the perturbed Krein Laplacian. We start with a lower estimate for the first positive eigenvalue, which we may call the [*spectral gap*]{}; cf. Remark \[rem:gap\] below.
\[thm:upperBound\] Assume Hypothesis \[hyp\], and denote by $\ell (\Gamma)$ the total length of $\Gamma$. Furthermore, let $\Lambda$ be the loop of length $\ell (\Gamma)$. Then $$\begin{aligned}
\label{eq:lowerBoundV}
\lambda_1^+ (- \Delta_{{\rm K}, \Gamma, q}) \geq \lambda_1^+ (- \Delta_{\delta, \Lambda, I})\end{aligned}$$ holds, where $- \Delta_{\Lambda, I}$ is the Laplacian on $\Lambda$ with a $\delta$-interaction of strength $I := \int_\Gamma q \,\dd x$ at one (arbitrary) point. In particular, $$\label{eq:1stlowerBound}
\lambda_1^+ (- \Delta_{{\rm K}, \Gamma}) \geq 4 \left(\frac{\pi}{\ell (\Gamma)}\right)^2.$$ Equality in holds if and only if $\Gamma$ is an interval, a loop, an equilateral 2-cycle, or an equilateral figure-8.
Let $\widetilde \Gamma$ be the flower graph formed from $\Gamma$ by gluing all vertices. Then by Theorem \[thm:gluingV\] and Theorem \[thm:inequalitiesV\], we have $$\begin{aligned}
\lambda_1^+ (- \Delta_{{\rm K}, \Gamma, q}) \geq \lambda_1^+ (- \Delta_{{\rm K}, \widetilde \Gamma, q}) \geq \lambda_1^+ (- \Delta_{{\rm st}, \widetilde \Gamma, q}).\end{aligned}$$ Moreover, as the only vertex of $\widetilde \Gamma$ has even degree (equal to twice the number of edges), we may cut through the vertex in such a way that we obtain an (Eulerian) cycle $\Lambda$ of length $\ell (\Gamma)$, and by surgery principles for the perturbed standard Laplacian $- \Delta_{{\rm st}, \widetilde \Gamma, q}$, see e.g. [@RS20 Theorem 4.1], we get $$\begin{aligned}
\lambda_1^+ (- \Delta_{{\rm st}, \widetilde \Gamma, q}) \geq \lambda_1^+ (- \Delta_{{\rm st}, \Lambda, q}).\end{aligned}$$ In the case $q = 0$ identically, we have hereby shown , and follows from a direct calculation. If $q$ is nontrivial, then $I > 0$, and for both operators $- \Delta_{{\rm st}, \Lambda, q}$ and $- \Delta_{\delta, \Lambda, I}$, the smallest eigenvalue is positive. Hence we may argue further as in the proof of [@KKT16 Theorem 1]: let $\psi$ be an eigenfunction of $- \Delta_{{\rm st}, \Gamma, q}$ corresponding to its lowest eigenvalue. Then $$\begin{aligned}
\lambda_1^+ (- \Delta_{{\rm st}, \Lambda, q}) = \frac{\int_\Gamma |\psi'|^2 \dd x + \int_\Gamma q |\psi|^2 \dd x}{\int_\Gamma |\psi|^2 \dd x} \geq \frac{\int_\Gamma |\psi'|^2 \dd x + \int_\Gamma q |\psi (x_{\min})|^2 \dd x}{\int_\Gamma |\psi|^2 \dd x},\end{aligned}$$ where $x_{\min}$ is any point on $\Gamma$ where $|\psi|$ takes its minimum. Since the last quotient is the Rayleigh quotient of the Laplacian with a $\delta$-vertex condition of strength $I = \int_\Gamma q \dd x$ at $x_{\min}$, the assertion follows also for nontrivial potentials.
In the case of equality in , all of the above inequalities must in fact be equalities. In particular, the standard Laplacian on the flower graph $\widetilde \Gamma$ in the above argument already has to have $4 \pi^2 / \ell (\widetilde \Gamma)^2$ as its first positive eigenvalue, which is only possible if on the loop $\Lambda$ resulting from splitting the central vertex of $\widetilde \Gamma$, there exists an eigenfunction for the first positive eigenvalue which has the same value at each point that was glued together previously (cf. [@KMN13 Theorem 1]). Since each eigenfunction of $- \Delta_{{\rm st}, \Lambda}$ corresponding to the first nonzero eigenvalue takes each of its values exactly twice on the loop – at two points with distance $\ell (\Gamma) / 2$ from each other – it follows that $\widetilde \Gamma$ can be recovered from $\Lambda$ by gluing at most two points. Hence $\widetilde \Gamma$ is either a loop itself or an equilateral figure-8. In other words, joining all vertices in the original graph $\Gamma$ leads to a loop or a figure-8, and this is only possible if $\Gamma$ is of one of the following six types: an interval, a path graph with two equal edges, a loop, an equilateral 2-cycle or an equilateral figure-8. Considering these graphs only, one finds by calculation that there exist eigenfunctions with corresponding eigenvalue $4 \pi^2 / \ell (\Gamma)^2$ if and only if $\Gamma$ is equilateral and has one of the four forms listed in the statement of the theorem.
\[rem:gap\] The interval $(0, 4 \pi^2 / \ell (\Gamma)^2)$ has empty intersection not only with the spectrum of the Krein Laplacian on one individual graph $\Gamma$. In fact, Theorem \[thm:upperBound\] asserts that, for fixed $\ell > 0$, the interval $(0, 4 \pi^2 / \ell^2)$ is free of spectrum for the Krein Laplacians on the whole class of metric graphs with total length $\ell$.
Alternatively, one may use in combination with known lower bounds on the eigenvalues of the standard Laplacian to obtain lower bounds for the positive Krein Laplacian eigenvalues. However, using the optimal lower bound from [@F05], one gets $$\begin{aligned}
\lambda_1^+ (- \Delta_{{\rm K}, \Gamma}) \geq \lambda_2 (- \Delta_{{\rm st}, \Gamma}) \geq \frac{\pi^2}{\ell (\Gamma)^2},\end{aligned}$$ which is weaker than the sharp bound .
The two crucial surgery operations used in the above proof are standard: gluing all vertices of a graph into one was used in [@KKMM16], and cutting through vertices to obtain an Eulerian cycle goes back at least to [@KN14]. Nevertheless, the above proof is slightly unusual: for the standard Laplacian, gluing vertices increases eigenvalues (the positive ones, as well as counting the ground state) whilst cutting vertices decreases them, so that both surgery operations used above – gluing all vertices into one and cutting vertices to obtain an Eulerian cycle – cannot be used within the same argument. However, in the present situation this works smoothly since gluing is performed on the positive eigenvalues of the perturbed Krein Laplacian and cutting is done only after transition to standard vertex conditions.
We point out that the exact same proof also yields an estimate for higher eigenvalues in the potential-free case:
Assume that Hypothesis \[hyp\] is satisfied with $q = 0$ identically, and that $\Lambda$ is a loop with the same length as for $\Gamma$. Then $$\begin{aligned}
\lambda_j^+ (- \Delta_{{\rm K}, \Gamma}) \geq \lambda_j^+ (- \Delta_{\rm st, \Lambda}) = \lambda_{j + 1} (- \Delta_{\rm st, \Lambda})\end{aligned}$$ holds for all $j \in \N$.
We conclude this section with a remark on how to apply the min-max principle to get upper spectral bounds. We do not go far into this and discuss only, very briefly, the special case of graphs which contain Eulerian cycles. We restrict ourselves here to the potential-free case, although natural generalisations for potentials exist (but their formulation may be less pleasant).
Suppose that $\Gamma$ contains an Eulerian cycle $\Sigma$ (obtained by cutting through vertices and removing edges not on the cycle), and let $\cE_\Sigma\subseteq\cE$ denote the set of edges belonging to $\Sigma$. Then the function $f$ which on each $e\in\cE_\Sigma$ takes the form $$f_{e}(x)=\pm\frac{\ell(e)}{n_e}\sin\left(\frac{n_e\pi x}{\ell(e)}\right),\qquad x\in [0,\ell(e)],$$ for some $n_e\in\mathbb{N}$, clearly satisfies Dirichlet conditions at all vertices of $\Sigma$, and, moreover, its derivatives have equal magnitude at all endpoints. Each $f_e$ contains $n_e/2$ periods of sine, and thus, by moving around the cycle, one can ensure that Kirchhoff conditions are satisfied at all vertices of $\Sigma$ by choosing appropriate signs for $f_e$ on adjacent pairs of edges; the only place where there could be a discrepancy is when one returns to the start of the cycle, as the function may end on a half-number of periods, but this problem is averted by imposing the further restriction that $\sum_{e\in\cE_\Sigma}n_e\in 2\mathbb{N}$. Now, $f$ satisfies Dirichlet-Kirchhoff conditions not only on $\Sigma$, but also on $\Gamma$, after extending it by zero on $\cE\backslash\cE_\Sigma$, so such functions provide upper estimates for the positive eigenvalues of $-\Delta_{{\rm K},\Gamma}$ via the min-max principle, Theorem \[thm:minMaxKreinV\]. The Rayleigh quotient for this $f$ is $$R_{\rm K}[f]=\frac{\pi^2}{\ell (\Sigma)}\sum_{e\in\cE_\Sigma}\frac{n_e^2}{\ell(e)},$$ which is an explicit upper bound for the first positive eigenvalue; the maximum value of $R_{\rm K}[f]$ among $j$ linearly independent functions of this type gives an upper estimate for $\lambda_j^+ (-\Delta_{{\rm K},\Gamma})$.
Of course, it is true in general, even with potentials, that for $\Gamma$ containing an Eulerian cycle $\Sigma$, one has $\lambda_j^+ (-\Delta_{{\rm K},\Gamma,q}) \leq \lambda_j^+ (-\Delta_{{\rm K},\Sigma,\widetilde q})$, where $\widetilde q:=\left.q\right|_\Sigma$, due to Theorems \[thm:gluingV\] and \[thm:attachV\].
More general perturbed Krein Laplacians {#sec:moreGeneral}
=======================================
Thus far, we have studied the Krein extension of the symmetric perturbed Laplacian with Dirichlet and Kirchhoff conditions at all vertices, but the abstract theory of Krein extensions of symmetric operators allows one to extend this work to cover symmetric perturbed Laplacians with more general vertex conditions. In this section we illustrate this by considering perturbed Laplacians with “Krein vertex conditions” on a selected subset of the vertex set, and standard (continuity-Kirchhoff) vertex conditions at all further vertices. We indicate in which form the results of the previous sections carry over to this setting. The proofs are analogous in the present case and are mostly left to the reader.
Let Hypothesis \[hyp\] be satisfied. For $\cB \subset \cV$, define the operator $S_\cB$ in $L^2 (\Gamma)$ by $$\begin{aligned}
\label{eq:SB}
\begin{split}
(S_\cB f)_e & = - f_e'' + q_e f_e \qquad \text{on each edge}~e \in \cE, \\
\dom S_\cB & = \Big\{ f \in \widetilde H^2 (\Gamma) \cap H^1 (\Gamma) : \partial_\nu f (v) = 0~\text{for each}~v \in \cV, \\
& \qquad \qquad \qquad \qquad \qquad \quad f (v) = 0~\text{for each}~v \in \cB \Big\}.
\end{split}\end{aligned}$$
A more general setting may be treated with the same methods, but we do not go into these details here: it is possible to replace the standard vertex conditions at the vertices in $\cV \setminus \cB$ by any self-adjoint, local vertex conditions. For the description of such conditions, we refer the reader to [@BK12].
The reader may think of the selected vertex set $\cB$ as a kind of boundary for $\Gamma$. One choice, which may be natural in some cases, is to let $\cB$ consist of all vertices of degree one. We are not restricted to this situation, but we may keep it in mind as a typical example.
The operator $S_\cB$ in is symmetric, closed, and densely defined. It has defect numbers $n_- = n_+ = |\cB|$, and is thus only self-adjoint if $\cB = \emptyset$. Furthermore, $S_\cB$ is clearly nonnegative, and its Friedrichs extension $S_{\cB, \rm F}$ is the perturbed Krein Laplacian subject to Dirichlet boundary conditions on $\cB$ and standard vertex conditions on $\cV \setminus \cB$. In particular, $$\begin{aligned}
\label{eq:muB}
(S_\cB f, f) \geq \mu \|f\|^2, \quad f \in \dom S_\cB\end{aligned}$$ holds, where $\mu > 0$ may be chosen as the lowest eigenvalue of $S_{\cB, \rm F}$. The adjoint of $S_\cB$ equals $$\begin{aligned}
(S_\cB^* f)_e & = - f_e'' + q_e f_e \qquad \text{on each edge}~e \in \cE, \\
\dom S^* & = \left\{ f \in \widetilde H^2 (\Gamma) \cap H^1 (\Gamma) : \partial_\nu f (v) = 0~\text{for all}~v \in \cV \setminus \cB \right\}.\end{aligned}$$
Due to , for nonempty $\cB \subset \cV$, we may consider the operator $$\begin{aligned}
- \Delta_{{\rm K}, \Gamma, q, \cB} := S_{\cB, \rm K},\end{aligned}$$ the Krein–von Neumann extension of $S_\cB$. If $q = 0$ identically, we write $- \Delta_{{\rm K}, \Gamma, \cB}$.
To derive some properties of the operator $- \Delta_{{\rm K}, \Gamma, q, \cB}$, constructing an appropriate boundary triple is useful.
\[prop:graphBTB\] Assume that Hypothesis \[hyp\] is satisfied, and let $\cB \subset \cV$ be nonempty. Let $S_\cB$ be defined in . For $f \in \dom S_\cB^*$, define $$\begin{aligned}
\Gamma_0 f = \begin{pmatrix} f (v_1) \\ \vdots \\ f (v_b) \end{pmatrix} \quad \text{and} \quad \Gamma_1 f = \begin{pmatrix} - \partial_\nu f (v_1) \\ \vdots \\ - \partial_\nu f (v_b) \end{pmatrix},\end{aligned}$$ where $\cB = \{v_1, \dots, v_b\}$ (and $b = |\cB|$). Then $\{ \C^b, \Gamma_0, \Gamma_1\}$ is a boundary triple for $S_\cB^*$; in particular, $S_\cB$ has defect numbers $$\begin{aligned}
n_- = n_+ = b.\end{aligned}$$ The corresponding extensions $A$ and $B$ of $S$ defined in are given by $$\begin{aligned}
A = S_{\cB, \rm F} \quad \text{and} \quad B = - \Delta_{{\rm st}, \Gamma, q};\end{aligned}$$ in particular, $0 \in \rho (A)$. The value of the corresponding Weyl function at $\lambda = 0$ is $M_\cB (0) = - \Lambda_{q, \cB}$, where $\Lambda_{q, \cB}$ is the [*Dirichlet-to-Neumann matrix for $\cB$*]{} defined via the relation $$\begin{aligned}
\begin{pmatrix} \partial_\nu f_* (v_1) \\ \vdots \\ \partial_\nu f_* (v_b) \end{pmatrix} = \Lambda_{q, \cB} \begin{pmatrix} f_* (v_1) \\ \vdots \\ f_* (v_b) \end{pmatrix},\end{aligned}$$ where $f_* \in \ker S_\cB^*$ is arbitrary.
The Weyl function $\lambda \mapsto M_\cB (\lambda)$ may be computed from the Weyl function $\lambda \mapsto M (\lambda)$ of the boundary triple in Proposition \[prop:graphBT\]. Indeed, if we write $\cV = \{v_1, \dots, v_b, v_{b + 1}, \dots, v_V\}$, where the vertices are ordered such that the first $b$ of them form $\cB$, and write $$\begin{aligned}
M (\lambda) = \begin{pmatrix} \widehat D (\lambda) & - B (\lambda)^\top \\ - B (\lambda) & \widehat L (\lambda) \end{pmatrix},\end{aligned}$$ where the block decomposition is taken according to the decomposition of the vertices into $\cB$ and $\cV \setminus \cB$, then we have $$\begin{aligned}
M_\cB (\lambda) = \widehat D (\lambda) - B (\lambda)^\top \widehat L (\lambda)^{-1} B (\lambda).\end{aligned}$$ The proof is straightforward; for a special case it may be found in [@GR20 Proposition 3.1]; see also [@KL20 Lemma 3.1]. In particular, in the potential-free case, $- M_\cB (0)$ is the Schur complement of the weighted discrete Laplacian $L$ in with respect to decomposition of the vertices into $\cB$ and $\cV \setminus \cB$.
From Proposition \[prop:graphBTB\], the following properties of $- \Delta_{{\rm K}, \Gamma, q, \cB}$ are immediate.
\[prop:KreinB\] Assume that Hypothesis \[hyp\] holds and that $\cB \subset \cV$ is nonempty. Then $- \Delta_{{\rm K}, \Gamma, q, \cB}$ acts as $$\begin{aligned}
\big( - \Delta_{{\rm K}, \Gamma, q, \cB} f \big)_e = - f_e'' + q_e f_e \qquad \text{on each edge}~e \in \cE,\end{aligned}$$ and its domain consists of all $f \in \widetilde H^2 (\Gamma) \cap H^1 (\Gamma)$ such that $$\begin{aligned}
\begin{pmatrix} \partial_\nu f (v_1) \\ \vdots \\ \partial_\nu f (v_d) \end{pmatrix} = \big( \widehat D - B^\top \widehat L^{-1} B \big) \begin{pmatrix} f (v_1) \\ \vdots \\ f (v_d) \end{pmatrix},\end{aligned}$$ where we have written $$\begin{aligned}
\Lambda_{q, \cB} = \begin{pmatrix} \widehat D & - B^\top \\ - B & \widehat L \end{pmatrix},\end{aligned}$$ in block matrix form with respect to the decomposition of $\cV$ into $\cB$ and $\cV \setminus \cB$. Moreover, $$\begin{aligned}
\dim \ker \big( - \Delta_{{\rm K}, \Gamma, q, \cB} \big) = \dim\ker S_\cB^* = b.\end{aligned}$$
Next, as an application of the abstract Theorem \[thm:minMaxKrein\], we obtain the following variational characterisation for the eigenvalues of $- \Delta_{{\rm K}, \Gamma, q, \cB}$.
\[thm:minMaxKreinB\] If Hypothesis \[hyp\] is satisfied and $\cB \subset \cV$ is nonempty, then the spectrum of $- \Delta_{{\rm K}, \Gamma, q, \cB}$ is purely discrete, and the positive eigenvalues $$\begin{aligned}
\lambda_1^+ \big( - \Delta_{{\rm K}, \Gamma, q, \cB} \big) \leq \lambda_2^+ \big( - \Delta_{{\rm K}, \Gamma, q, \cB} \big) \leq \dots\end{aligned}$$ of $- \Delta_{{\rm K}, \Gamma, q, \cB}$, counted with multiplicities, satisfy $$\begin{aligned}
\lambda_j^+ \big( - \Delta_{{\rm K}, \Gamma, q, \cB} \big) = \min_{\substack{F \subset \dom S_\cB \\ \dim F = j}} \max_{\substack{f \in F \\ f \neq 0}} \frac{\int_\Gamma \left|- f'' + q f\right|^2 \dd x}{\int_\Gamma |f'|^2 \dd x + \int_\Gamma q |f|^2 \dd x}\end{aligned}$$ for all $j \in \N$.
Analogously to Theorem \[thm:resDiffGraph\], one may express the resolvent differences of $- \Delta_{{\rm K}, \Gamma, q, \cB}$ with the Friedrichs extension of $S_\cB$ and the perturbed standard Laplacian. In particular, one gets the following.
\[thm:resDiffB\] Assume that Hypothesis \[hyp\] is satisfied and that $\cB \subset \cV$ is nonempty. Then $$\begin{aligned}
\dim \ran \Big[ \big(- \Delta_{{\rm K}, \Gamma, q, \cB} - \lambda \big)^{-1} & - \big(- \Delta_{{\rm st}, \Gamma, q} - \lambda \big)^{-1} \Big] \\
& = \dim \ran \Lambda_{q, \cB} = \begin{cases} b - 1 & \text{if}~q = 0~\text{identically}, \\ b, & \text{else}, \end{cases}\end{aligned}$$ where $b = |\cB|$.
In particular, in the potential-free case, if $b = |\cB| = 1$, then $- \Delta_{{\rm K}, \Gamma, \cB}$ equals the standard Laplacian. As a consequence of either Theorem \[thm:resDiffB\] or Theorem \[thm:minMaxKreinB\], we get, analogously to , $$\begin{aligned}
\lambda_{j + 1} \big(- \Delta_{{\rm st}, \Gamma} \big) \leq \lambda_j^+ \big(- \Delta_{{\rm K}, \Gamma, \cB} \big) = \lambda_{j + b} \big(- \Delta_{{\rm K}, \Gamma, \cB} \big), \quad j \in \N,\end{aligned}$$ in the case without potential.
The surgery principles of Section \[sec:surgery\] remain valid for the (positive) eigenvalues of the operator $- \Delta_{{\rm K}, \Gamma, q, \cB}$, provided that all vertices involved in the surgery operations belong to $\cB$; we leave it to the reader to formulate and prove the corresponding results. Instead we formulate a related result which deals with the transition between standard and “Krein vertex conditions”.
\[thm:addstandardB\] Let Hypothesis \[hyp\] be satisfied. Moreover, let $\widetilde \cB \subset \cB \subset \cV$ be sets of size $b = |\cB|$ and $\widetilde b = |\widetilde \cB|$, respectively, and let $k := b - \widetilde b$. Then for $$\begin{aligned}
& \lambda_j^+ := \lambda_j^+ \big( - \Delta_{{\rm K}, \Gamma, q, \cB} \big), \quad \widetilde \lambda_j^+ := \lambda_j^+ \big( - \Delta_{{\rm K}, \Gamma, q, \widetilde \cB} \big), \\
& \lambda_j := \lambda_j \big( - \Delta_{{\rm K}, \Gamma, q, \cB} \big), \quad \,\,\, \widetilde \lambda_j := \lambda_j \big( - \Delta_{{\rm K}, \Gamma, q, \widetilde \cB} \big),\end{aligned}$$ the following statements hold:
1. the positive eigenvalues satisfy $$\begin{aligned}
\label{eq:inclusion+}
\widetilde \lambda_j^+ \leq \lambda_j^+ \leq \widetilde \lambda_{j + k}^+ \leq \lambda_{j + k}^+, \quad j \in \N;
\end{aligned}$$
2. the eigenvalues (counting ground states) satisfy $$\begin{aligned}
\label{eq:inclusion}
\lambda_j \leq \widetilde \lambda_j \leq \lambda_{j + k} \leq \widetilde \lambda_{j + k}, \quad j \in \N.
\end{aligned}$$
If we denote by $S$ and $\widetilde S$ the symmetric operators defined in for the vertex subsets $\cB$ and $\widetilde B$ respectively, then $\widetilde \cB \subset \cB$ implies the operator inclusion $S \subset \widetilde S$. Moreover, it is easy to see that $\dom S$ has co-dimension $k = b - \widetilde b$ in $\dom \widetilde S$. Therefore follows directly from Theorem \[thm:interlacingAbstract\]. Using the fact that the perturbed Krein Laplacians for $\cB$ and $\widetilde \cB$ have respectively $b$ and $\widetilde b$ linearly independent functions in their kernels, follows from .
As a simple consequence, for any nonempty $\cB \subset \cV$, we have $$\begin{aligned}
\lambda_j^+ \big(- \Delta_{{\rm st}, \Gamma, q} \big) \leq \lambda_j^+ \big(- \Delta_{{\rm K}, \Gamma, q, \cB} \big) \leq \lambda_j^+ \big(- \Delta_{{\rm K}, \Gamma, q} \big)\end{aligned}$$ as well as $$\begin{aligned}
\lambda_j \big(- \Delta_{{\rm K}, \Gamma, q} \big) \leq \lambda_j \big(- \Delta_{{\rm K}, \Gamma, q, \cB} \big) \leq \lambda_j \big(- \Delta_{{\rm st}, \Gamma, q} \big),\end{aligned}$$ for all $j \in \N$.
The authors are grateful to Fritz Gesztesy for his comments on the literature. J.R. acknowledges financial support by grant no. 2018-04560 of the Swedish Research Council (VR).
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---
abstract: 'The compatibility “demonstrated” by Rovelli & al. in [@Rovelli] between various “gauge conditions” both in Classical Electromagnetism and General Relativity can be better understood if one distinguishes “gauge conditions” of the solution type and “gauge conditions” of the constraint type.'
author:
- Germain Rousseaux
title: |
Comment on the paper by Rovelli & al. about the compatibility\
of various “gauge conditions”
---
Recently, we showed that the well-known Coulomb and Lorenz “gauge conditions” (see [@Okun] for an introduction to the various “gauge conditions” encountered in the literature) were, in fact, not equivalent because they must be interpreted as physical constraints that is electromagnetic continuity equations [@AFLB03]. In addition, we were able to demonstrate that the Coulomb “gauge condition” is the galilean approximation of the Lorenz “gauge condition” within the magnetic limit of Lévy-Leblond & Le Bellac [@LBLL; @EPL05; @EJP06; @AJP07]. So, to “make a gauge choice” that is choosing a gauge condition is, as a consequence of our findings, not related to the fact of fixing a special couple of potentials. Gauge conditions are completely uncorrelated to the supposed indeterminacy of the potentials. Hence, we propose to rename the “gauge conditions” like the ones of Lorenz or Coulomb by physical “constraints”.
The purpose of our comment is to show that a “demonstration” of the compatibility between the Fock-Schwinger “gauge condition” and the Lorenz “gauge conditions” by Rovelli & al. in [@Rovelli] is blurred by a slip in terminology. As a matter of fact, the Lorenz equation $\partial _\mu A^\mu =0$ is according to us a constraint whereas the Fock-Schwinger equation $x_\mu A^\mu =0$ is what we called a “solution” submitted to the Lorenz constraint. If so, the “compatibility” is obvious...
First of all, let us illustrate what we mean by a “solution” under a physical constraint. One often finds in textbooks that we can describe a uniform magnetic field ${\bf B} =B {\bf e}_{z}$ by either the so-called symmetric “gauge” $\bf{A_1}=1/2\bf{B} \times \bf{r}$ or by the so-called Landau “gauge” [@Okun]. This two “gauges” are related by a gauge transformation :
$${\bf A}_{1}=\frac{1}{2} {\bf B} \times {\bf r} = \frac{1}{2} [-By,Bx,0]$$
becomes either : $${\bf A}_{2} =[0,Bx,0]
\quad
or
\quad
{\bf A}_{3}=[-By,0,0]$$ with the gauge functions $\pm f = \pm xy/2$.
However, there is no discussion in the litterature of the following issue. As a matter of fact, if we consider a solenoid with a current along $\bf{e_\theta}$, the magnetic field is uniform (along $\bf{e_z}$) and could be described by the symmetric “gauge” or the Landau “gauge”. Yet, the vector potential in the Landau “gauge” ${\bf A}_{2}$ is along $\bf{e_y}$ whereas the vector potential in the symmetric gauge is along $\bf{e_\theta}$. We advocate that only the symmetric “gauge” is valid in this case because it does respect the symmetry of the currents (${\bf J} = J {\bf e}_{\theta}$) whereas the Landau “gauge” does not. Moreover, the symmetric “gauge” (or the Landau “gauge”) is not, in fact, a gauge condition but a solution describing a uniform magnetic field under the Coulomb constraint ($\nabla .{\bf{A_1}}=\bf{0}$). In order to understand this last point, one can picture an analogy between Fluid Mechanics and Classical Electromagnetism. Indeed, the solenoid is analogous to a cylindrical vortex core with vorticity $\bf{w}$ and we know that the velocity inside the core is given by $\bf{u}=1/2\bf{w} \times \bf{r}$ which is analogous to the symmetric gauge for an incompressible flow ($\nabla .{\bf{u}}=\bf{0}$). Outside the vortex core, the velocity is given by [@GHP] : $${\bf u}=\frac {\Gamma \nabla \theta} {2
\pi}=\frac {\Gamma}{2 \pi r} {\bf e_\theta}$$ where $\Gamma$ is the flux of vorticity inside the vortex or the circulation of the velocity outside the vortex. One recovers the analogue formula for the vector potential outside a solenoid...
Of course, if the problem we are considering does not feature the cylindrical geometry (two horizontal plates with opposite surface currents for example, analogous to a plane Couette flow [@GHP]), one of the Landau “gauges” ${\bf A}_{2}$ or ${\bf A}_{3}$ submitted to the constraints $\nabla .{\bf{A_2}}=\bf{0}$ or $\nabla .{\bf{A_3}}=\bf{0}$ must be used instead of the symmetric “gauge” ${\bf A}_{1}$ according to the necessity of respecting the underlying distribution/symmetry of the currents which is at the origin of both the vector potential and the magnetic field. To give a magnetic vector field without specifying its current source is an ill-posed problem which was interpreted so far by attributing an indeterminacy to the vector potential which is wrong. Now, how can we test experimentally this argument based on symmetry ? If the current of the solenoid varies with time, it will create an electric field which is along $\bf{e_\theta}$ as the vector potential because the electric field is minus the time derivative of the vector potential. If the currents in the horizontal plates change with time, a horizontal electric field will appear for the same reason.
We come back now to the main claim of Rovelli & al. [@Rovelli]. We will argue that the Fock-Schwinger equation is a solution submitted to the Lorenz constraint. Let us write the Fock-Schwinger “gauge” $x_\mu A^\mu =0$ in projection for cartesian coordinates $x_0 A^0 +x_1 A^1+x_2 A^2 +x_3 A^3 =0$. The four-position and four-potential have components $x_\mu =\left (-ct,{\bf x} \right )$ and $A^\mu=\left (V/c,{\bf A} \right )$. For example, the Fock-Schwinger equation becomes $xA_x +yA_y +zA_z =tV$ in cartesian coordinates.
The Lorenz “gauge” $\nabla .{\bf{A}}+1/c^2 \partial V/\partial t =0$ becomes after a Fourier transformation $i{\bf k . A}-i\omega /c^2 V=0$. But a light wave is such that $|k|=\omega /c$.
A plane or spherical light wave has a null space-time interval ($ds^2 =dx_\mu .dx^\mu =0$): $x^2 +y^2 +z^2 =c^2 t^2$ or $r^2=c^2 t^2$. The Lorenz constraint for a plane or spherical light wave becomes: $V=cA_x$ or $V=cA_r$ where $A_x$ or $A_r$ is the component of the light wave which is parallel to the wave-vector ${\bf k}$. With $x=ct$ ($y=z=0$) or $r=ct$ for outgoing waves, one gets $V=cA_x = x/t A_x$ or $V=cA_r =r/t A_r$ that is the Fock-Schwinger solution: $xA_x =tV$ or $rA_r =tV$.
As a conclusion, the Fock-Schwinger solution describes the propagation of a light wave submitted to the Lorentz-covariant Lorenz constraint which is the continuity equation allowing the propagation. It is interesting to notice that V. A. Fock himself derived the Fock-Schwinger equation starting with the Lorenz equation and using a gauge transformation [@Fock]...
[10]{}
E. Magliaro, C. Perini & C. Rovelli, Compatibility of radial, Lorenz and harmonic gauges, preprint, May 2007.\
(Article available online at :\
$http://fr.arxiv.org/abs/0704.0992v2$)
J. D. Jackson & L. B. Okun, Historical roots of gauge invariance, Reviews of Modern Physics, Vol. 73, p. 663-680, 2001.
G. Rousseaux, On the physical meaning of the gauge conditions of Classical Electromagnetism : the hydrodynamics analogue viewpoint, Annales de la Fondation Louis de Broglie, Volume 28, Numéro 2, p. 261-270, 2003.\
(Article available online at :\
$http://www.ensmp.fr/aflb/AFLB-282/\\aflb282p261.pdf$)
M. Le Bellac & J.-M. Lévy-Leblond, Galilean Electromagnetism, Il Nuovo Cimento, Vol. 14B, N. 2, 11 Aprile, p. 217-233, 1973.
G. Rousseaux, Lorenz or Coulomb in Galilean Electromagnetism ?, EuroPhysics Letters, 71 (1), p. 15-20, July 2005.
M. de Montigny & G. Rousseaux, On the electrodynamics of moving bodies at low velocities, European Journal of Physics, Volume 27, Number 4, p. 755-768, July 2006.
M. de Montigny & G. Rousseaux, On some applications of the galilean electrodynamics of moving bodies, to be published, 2007.
E. Guyon, J.-P. Hulin, L. Petit L. & C. D. Mitescu, Physical Hydrodynamics, Oxford University Press, 2001.
V. A. Fock, Proper Time in Classical and Quantum Mechanics in Selected Works, Edited by L. D. Fadeev, L. A. Khalfin, I. V. Komarov, p. 421-439, Chapman & Hall/CRC, 2004.
|
---
address: 'Department of Mathematics, University of California, Irvine, Irvine, CA 92697'
author:
- Christopher Lin
- Zhiqin Lu
bibliography:
- 'new060923.bib'
- 'unp060923.bib'
- 'local.bib'
date: 'January 6, 2004'
title: |
Existence of Bound States for Layers Built Over Hypersurfaces in ${{{\mathchoice {\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\scriptstyle N\kern-0.3em R$}}
{\hbox{$\scriptscriptstyle R\kern-0.2em R$}}}}}^{n+1}$
---
[^1]
Introduction
============
In their study of the spectrum of quantum layers [@DEK], Duclos, Exner, and Krejčiř[í]{}k proved the existence of bound states for certain quantum layers [^2]. Part of their motivations to study the quantum layers is from mesoscopic physics. From the mathematical point of view, a quantum layer is a noncompact noncomplete manifold. For such a manifold, the spectrum of the Laplacian (with Dirichlet or Neumann boundary condition) is less understood. Nevertheless, from [@DEK] and this paper, we shall see that the spectrum of a quantum layer has very interesting properties not only from the point of view of physics but also from the point of view of mathematics.
Mathematicians are interested in the spectrum of two kinds of manifolds: compact manifolds (with or without boundary), and noncompact complete manifolds. For these two kinds of manifolds, one can prove [@Gaffney2; @Gaffney1] that the Laplacians can be uniquely extended as self-adjoint operators from operators on smooth functions with compact support. For a compact manifold, by the Hodge theorem, we can prove that the spectrum of the Laplacian is composed of only discrete spectrum. On the other hand, the spectrum of Laplacian of a noncompact complete manifold is rather complicated. In general it has both discrete and essential spectrum.
In general, it is rather difficult to prove the existence of discrete spectrum for a noncompact manifold, because the existence of an $L^2$ eigenfunction is a highly nontrivial fact. However, in the following special case, the discrete spectrum does exist.
We define the following two quantities:
Let $M$ be a manifold whose Laplacian $\Delta$ can be extended to a self-adjoint operator. Let $$\begin{aligned}
&\sigma_0=\inf_{f\in C_0^\infty(M)}\frac{-\int_M f\Delta
f}{\int_M f^2},\label{1-1}\\ &\sigma_{\rm ess}=
\sup_{K}\inf_{f\in C_0^\infty(M\backslash K)}
\frac{-\int_M f\Delta
f}{\int_M f^2},\label{1-2}\end{aligned}$$ where $K$ is running over all compact subsets of $M$.
$\sigma_0$ is the lower bound of the spectrum and $\sigma_{\rm ess}$ is the lower bound of the essential spectrum. In general, we have $\sigma_0\leq\sigma_{\rm
ess}$. If $\sigma_0<\sigma_{\rm ess}$, then the set of discrete spectrum must be nonempty. In particular, since the spectrum of a self-adjoint operator is a closed subset of the real line, there is an $L^2$ smooth function $f$ of $M$ such that $$\Delta f=-\sigma_0 f.$$ $(\sigma_0,f)$ is called the ground state of the Laplacian.
We don’t expect $\sigma_0<\sigma_{\rm ess}$ to be true in general. It seems that more often we would get the opposite result $\sigma_0=\sigma_{\rm ess}$. For example, by a theorem of Li and Wang [@Li-Wang Theorem 1.4], we know that if the volume growth of a complete manifold is sub-exponential and if the volume is infinite, then $\sigma_{\rm ess}=0$. Thus in that case, $\sigma_0=\sigma_{\rm ess}=0$ and we don’t know a general way to prove the existence of ground state.
Let ${\varSigma}$ be an oriented $n$-manifold and $\pi:{\varSigma}\rightarrow
{{{\mathchoice {\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\scriptstyle N\kern-0.3em R$}}
{\hbox{$\scriptscriptstyle R\kern-0.2em R$}}}}}^{n+1}$ be an immersion. Let $N$ be the unit normal vector field of ${\varSigma}$, we can define the following map $$p:{\varSigma}\times(-a,a)\rightarrow {{{\mathchoice {\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\scriptstyle N\kern-0.3em R$}}
{\hbox{$\scriptscriptstyle R\kern-0.2em R$}}}}}^{n+1},\quad
(x,u)\mapsto x+uN,$$ where $a$ is a small positive number such that the map $p$ is an immersion.
The quantum layer $\Omega$ built over ${\varSigma}$, as a differentiable manifold, is very simple: $\Omega={\varSigma}\times (-a,a)$. The Riemannian metric on $\Omega$ is defined by $p^*(ds_E^2)$, where $ds_E^2$ is the Euclidean metric of ${{{\mathchoice {\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\scriptstyle N\kern-0.3em R$}}
{\hbox{$\scriptscriptstyle R\kern-0.2em R$}}}}}^{n+1}$. The number $a$ is called the depth of the layer.
The setting above is a little bit more general than in the paper [@DEK], where the authors require that both ${\varSigma}$ and $\Omega$ are embedded. In particular, they assume that the quantum layers are not self-intersecting. There are some advantages of our treatment: first, all the theorems still remain true in immersed cases, and second, it is possible to estimate the range of the depth $a$ using the upper bound of the second fundamental form in the case of immersion, while in the embedded case, global conditions of $({\varSigma},\pi)$ must be imposed in order to keep the layers from self-intersecting.
The aim of this paper is to study the ground state of the noncompact noncomplete Riemannian manifold $(\Omega,p^*(ds_E^2))$, where we assume the Dirichlet boundary condition for the Laplacian. Our work is clearly motivated by the work of [@DEK].
The first main result of this paper is the existence of the ground state of layer over convex surface in ${{{\mathchoice {\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\scriptstyle N\kern-0.3em R$}}
{\hbox{$\scriptscriptstyle R\kern-0.2em R$}}}}}^3$. We are motivated by the following result in [@DEK]:
Let $\Omega$ be a layer of depth $a$ over a surface of revolution whose Gauss curvature is integrable. Suppose $\Omega$ is not self-intersecting, and suppose $a||A||<C_0<1$, where $||A||$ is the norm of the second fundamental form and $C_0$ is a constant. If the surface is not totally geodesic, then $\sigma_0
<\pi^2/(4a^2)$.
Overlapping with the above result, we proved the following
\[main3\] Let ${\varSigma}$ be a convex surface in ${{{\mathchoice {\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\scriptstyle N\kern-0.3em R$}}
{\hbox{$\scriptscriptstyle R\kern-0.2em R$}}}}}^3$ which can be represented by the graph of a convex function $z=f(x,y)$. Suppose $0$ is the minimum point of the function and suppose that at $0$, $f$ is strictly convex. Furthermore suppose that the second fundamental form goes to zero at infinity. Let $C$ be the supremum of the second fundamental form of ${\varSigma}$. Let $Ca<1$. Then the ground state of the quantum layer $\Omega$ exists.
We let ${\varSigma}$ to be the surface defined by the function $$f(x,y)=x^2+y^2.$$ A straightforward computation gives the mean curvature of ${\varSigma}$: $$H=4\cdot\frac{1-\frac{2(x^2+y^2)}{1+4(x^2+y^2)}}{\sqrt{1+4(x^2+y^2)}}.$$ Thus $H$ and then the second fundamental form goes to zero at infinity. By the above theorem, the quantum layer built from the above surface has a ground state.
The second main result of this paper is motivated by the following:
Let ${\varSigma}$ be a $C^2$-smooth complete simply connected noncompact surface with a pole embedded in ${{{\mathchoice {\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\scriptstyle N\kern-0.3em R$}}
{\hbox{$\scriptscriptstyle R\kern-0.2em R$}}}}}^3$. Let the layer $\Omega$ built over the surface be not self-intersecting. If the surface is not a plane but it is asymptotically planar, and if the Gauss curvature is integrable and the total Gauss curvature is nonpositive, then the ground state exists.
In a more recent paper [@CEK-1], Carron, Exner, and Krejčiř[í]{}k observed that the assumptions of simply-connectedness and the existence of a pole on ${\varSigma}$ can be removed. ${\varSigma}$ is allowed to have a rather complicated topology.
By a theorem of Huber [@Huber], ${\varSigma}$ is conformally equivalent to a compact Riemann surface with finitely many points removed. In particular, we have $$\int_{\varSigma}K\leq 2\pi e({\varSigma}),$$ where $e({\varSigma})$ is the Euler characteristic number of ${\varSigma}$. The deficit of the above inequality can be represented by isoperimetric constants. Let $E_1,\cdots,E_s$ be the ends of the surface ${\varSigma}$. For each $E_i$, we define $$\label{iso}
\lambda_i=\underset{r\rightarrow\infty}{\lim}\,\frac
{A_i(r)}{\pi r^2},$$ where $A_i(r)$ is the area of the ball $B(r)\cap
E_i$. We have the following
Using the above notations, we have $$\frac{1}{2\pi}\int_{\varSigma}K=e({\varSigma})-\sum\lambda_i.$$
We have the following [^3]
\[main2\] Suppose that ${\varSigma}$ is a complete immersed surface of ${{{\mathchoice {\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\scriptstyle N\kern-0.3em R$}}
{\hbox{$\scriptscriptstyle R\kern-0.2em R$}}}}}^3$ such that the second fundamental form $A\rightarrow
0$. Suppose that the Gauss curvature is integrable and suppose that $$\label{1-4}
e({\varSigma})-\sum\lambda_i\leq 0,$$ where $\lambda_i$ is the isoperimetric constant at each end defined in . Let $a$ be a positive number such that $a||A||<C_0<1$. If ${\varSigma}$ is not totally geodesic, then the ground state of the quantum layer $\Omega$ exists. In particular, if $e({\varSigma})\leq 0$, then the ground state exists.
More generally, one of the key observation of our paper is that in order to generalize the results in [@DEK] to high dimensions, we must assume the parabolicity of the hypersurface ${\varSigma}$. The parabolicity of complete manifold was introduced by Li and Tam [@Li-Tam] (see also the survey papers [@Li; @Li-1]). A surface with a pole and $L^1$ Gauss curvature is parabolic. Thus the following result is a high dimensional generalization of the above result of Duclos, Exner, and Krejčiř[í]{}k:
\[main1\] Let $n\geq 2$ be a natural number. Suppose ${\varSigma}\subset
{{{\mathchoice {\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\scriptstyle N\kern-0.3em R$}}
{\hbox{$\scriptscriptstyle R\kern-0.2em R$}}}}}^{n+1}$ is a complete immersed parabolic hypersurface such that the second fundamental form $A\rightarrow 0$ at infinity. Moreover, we assume that $$\label{133}
\sum_{k=1}^{[ n/2]}\mu_{2k}{\rm Tr}(\mathcal R^k)\,\,
\text{is integrable and }\quad
\int_{\varSigma}\sum_{k=1}^{[n/2
]}\mu_{2k}{\rm Tr}(\mathcal R^k)
d\varSigma \leq 0,$$ where $\mu_{2k}>0$ for $k\geq 1$ are coefficients defined in Lemma \[lem51\], $[ n/2 ] $ is the integer part of $n/2$, and ${\rm Tr}(\mathcal R^k)$ is defined in . Let $a$ be a positive real number such that $a||A||<C_0<1$ for a constant $C_0$. If ${\varSigma}$ is not totally geodesic, then the ground state of the quantum layer $\Omega$ exists.
\[main4\] Let $\rho$ be the scalar curvature of ${\varSigma}$. If $n=3$, then the main conditions in Theorem \[main1\] become
1. $\rho$ is integrable;
2. $\int_{\varSigma}\rho \,d{\varSigma}\leq 0$,
If $n=4$, and if the sectional curvature of ${\varSigma}$ is positive outside a compact set of ${\varSigma}$, then the conditions become
1. $\rho$ is integrable;
2. $\int_{\varSigma}\rho \,d{\varSigma}+16(\frac{\pi^2}{6}-1){a^3}
e({\varSigma})\leq 0$,
where $e({\varSigma})$ is the Euler characteristic number of ${\varSigma}$.
The organization of the paper is as follows: in §2, we define the quantum layers and give their basic properties; in §3, we give the lower bound of the essential spectrum of a quantum layer; in §4, the parabolicity of a submanifold of ${{{\mathchoice {\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\scriptstyle N\kern-0.3em R$}}
{\hbox{$\scriptscriptstyle R\kern-0.2em R$}}}}}^{n+1}$ is introduced; in §5, Theorem \[main2\], The main theorem (Theorem \[main1\]), and Corollary \[main4\] are proved; finally, in §6, Theorem \[main3\] is proved.
We end up this section by posing the following question:
Let ${\varSigma}$ be a noncompact complete Riemannian manifold of dimension $n$. Then what do we have to assume on ${\varSigma}$ so that when ${\varSigma}\rightarrow {{{\mathchoice {\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\scriptstyle N\kern-0.3em R$}}
{\hbox{$\scriptscriptstyle R\kern-0.2em R$}}}}}^{n+1}$ is an asymptotically flat but not totally geodesic immersion, the layer $\Omega$ built over ${\varSigma}$ has ground state?
In particular, if $n=2$, the works of [@DEK; @CEK-1] suggest that the quantum layer $\Omega$ should have ground state when the Gauss curvature is integrable[^4].
[**Acknowledgement.**]{} The authors thank P. Li for advising us to study the parabolicity of manifolds, which makes this paper possible. The first author thanks A. Klein for clarifying a lot of the subtleties of self-adjointness. Corollary \[main2\] is the outcome of a discussion with G. Tian. Finally, we give special thanks to P. Exner and D. Krejčiř[í]{}k for many useful suggestions to make the paper in its present form.
Geometry of quantum layers {#S2}
==========================
Let $n>1$ be an integer and let ${\varSigma}$ be an immersed (oriented) hypersurface of ${{{\mathchoice {\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\scriptstyle N\kern-0.3em R$}}
{\hbox{$\scriptscriptstyle R\kern-0.2em R$}}}}}^{n+1}$. Let $a>0$ be a real number. Heuristically speaking, the quantum layer $\Omega$ is obtained by fattening $\varSigma$ by a thickness of $a$ in the directions of $N$ and $-N$, respectively, where $N$ is the unit normal vector field. As a differentiable manifold, $\Omega$ is just ${\varSigma}\times (-a,a)$. We impose the following assumptions on ${\varSigma}$ and $\Omega$:
1. Let $A$ be the second fundamental form of ${\varSigma}$. We regard $A$ as a linear operators on $T_x{\varSigma}$ for every $x\in{\varSigma}$. We assume that there is a constant $C_0$ such that $a||A||(x)<C_0<1$.
2. $\lVert A\rVert(x)
\rightarrow 0$ as $d(x,x_0)\rightarrow \infty$, where $x_0\in {\varSigma}$ is a fixed point.
\[section\]
\[def21\] Let $x_1,\cdots,x_n$ be a local coordinate system of ${\varSigma}$. Then $(\frac{{\partial}}{{\partial}x_1},\cdots,\frac{{\partial}}{{\partial}x_n},\frac{{\partial}}{{\partial}u})$ is a local frame of $\Omega$, where $u\in(-a,a)$. Such a local coordinate system of $\Omega$ is referred as a standard coordinate system of $\Omega$ in this paper.
We consider the map $$p: {\varSigma}\times (-a,a)\rightarrow {{{\mathchoice {\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\scriptstyle N\kern-0.3em R$}}
{\hbox{$\scriptscriptstyle R\kern-0.2em R$}}}}}^{n+1},
\quad (x,u)\mapsto x+uN.$$
Let $y_1,\cdots,y_{n+1}$ be the Euclidean coordinates of ${{{\mathchoice {\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\scriptstyle N\kern-0.3em R$}}
{\hbox{$\scriptscriptstyle R\kern-0.2em R$}}}}}^{n+1}$. Let $$ds_E^2=dy_1^2+\cdots+ dy_{n+1}^2$$ be the Euclidean metric of ${{{\mathchoice {\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\scriptstyle N\kern-0.3em R$}}
{\hbox{$\scriptscriptstyle R\kern-0.2em R$}}}}}^{n+1}$. Let $G_{ij}$ ($i,j=1,\cdots,n+1$) be defined by $$\sum_{i,j=1}^n G_{ij} dx_idx_j+
\sum_{i=1}^nG_{i,n+1}dx_idu
+\sum_{i=1}^nG_{n+1,i}dudx_i
+G_{n+1,n+1}dudu=p^*(ds_E^2).$$ If $p$ is nonsingular at a point, then the matrix $G_{ij}$ is positive definite at that point. In order to express $G_{ij}$ in term of the geometry of ${\varSigma}$, we introduce the following notations:
Let $(h_{ij})$ ($i,j=1,\cdots n$) be the matrix representation of the second fundamental form $A$ with respect to the local frame $(\frac{{\partial}}{{\partial}x_1},
\cdots,\frac{{\partial}}{{\partial}x_n})$. Let $g_{ij}dx_idx_j=p^*(ds_E^2)$ be the Riemannian metric of ${\varSigma}$. Let $h^\sigma_j=g^{\sigma i}h_{ij}$. Then a straightforward computation gives (cf. [@DEK]):
$$\label{chris}
G_{ij} = \left\{
\begin{array}{ll}
(\delta_{i}^{\sigma} -
uh_{i}^{\sigma})(\delta_{\sigma}^{\rho} -
uh_{\sigma}^{\rho})g_{\rho j}& 1\leq i,j\leq n\\
0 & i {\text \,\, or \,\,} j=n+1\\
1& i=j=n+1
\end{array}
\right..$$
In particular, we have $$\label{2-1}
\det(G_{ij}) = (\det
(1-uA))^{2}\det(g_{ij}),$$ and $$\begin{aligned}
\begin{split}
&
\det(I-uA) = \prod_{i=1}^{n}(1-u\lambda_{i})\\
& = 1 -
u\sum_{i=1}^{n}\lambda_{i} + u^{2}\sum_{i<j}\lambda_{i}\lambda_{j} -
u^{3}\sum_{i<j<l}\lambda_{i}\lambda_{j}\lambda_{l} + \cdots +
(-1)^{n}u^{n}\prod_{i=1}^{n}\lambda_{i},
\end{split}\end{aligned}$$ where $\lambda_1,\cdots,\lambda_n$ are the eigenvalues, or the principal curvatures of the second fundamental form $A$. In a more intrinsic way, let $c_k(A)$ be the $k$-th elementary polynomial of $A$. Then we have $$\label{2-3}
\det(I-uA) = \sum_{i=0}^{n}(-1)^{k}u^{k}c_{k}(A),$$ where we define $c_{0}(A)= 1$.
The following lemma is elementary but important:
\[lem2-1\] Using the above notations and under Assumption A1), we have $$(1-|u|\cdot||A||)^n\leq |\det(I-uA)| \leq
(1+|u|\cdot||A||)^n.$$
The proof is elementary and is omitted.
\[cor21\] We adopt the above notations and Assumption A1). Then the map $p$ is an immersion. In that case, $p^*(ds_E^2)$ is a Riemannian metric on $\Omega$. Let $d\Omega$ be the measure defined by the metric and let $dud{\varSigma}$ be the product measure on $\Omega$. Then we have $$\label{26}
(1-|u|||A||)^ndud{\varSigma}\leq d\Omega\leq
(1+|u|||A||)^ndud{\varSigma}.$$
[**Proof.**]{} By Assumption A1), and Lemma \[lem2-1\], we know that $\det (G_{ij})>0$. Thus $p$ is nonsingular. follows from Lemma \[lem2-1\] directly.
We define the quantum layer to be the Riemannian manifold $(\Omega,
p^*(ds_E^2))$, where $ds_E^2$ is the standard Euclidean metric of ${{{\mathchoice {\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\scriptstyle N\kern-0.3em R$}}
{\hbox{$\scriptscriptstyle R\kern-0.2em R$}}}}}^{n+1}$. The real numbers $a$ and $d=2a$ are called the depth and the width of the quantum layer, respectively.
The Laplacian $\Delta=\Delta_\Omega$ can be written as $$\Delta=\frac{1}{\sqrt{\det(G_{kl})}}
\sum_{i,j=1}^n\frac{{\partial}}{{\partial}x_i}\left(
G^{ij}\sqrt{\det (G_{kl})}\frac{{\partial}}{{\partial}x_j}
\right)+\frac{1}{\sqrt{\det(G_{kl})}}\frac{{\partial}}{{\partial}u}
\left(\sqrt{\det (G_{kl})}\frac{{\partial}}{{\partial}u}\right),$$ where $(x_1,\cdots,x_n,u)$ is the local coordinates defined in Definition \[def21\]. We have $$(\Delta F,G)=(F,\Delta G) \quad \forall F, G \in
C_{0}^{\infty}(\Omega),$$ where $(\cdot,\cdot)$ is the $L^2$ inner product $$(F,G)=\int_\Omega FG d\Omega.$$ The norm $||F||$ is defined as $\sqrt{(F,F)}$. If $F,G$ are differentiable, we define $$(\nabla F,\nabla G)=\int_\Omega\left(\sum_{i,j=1}^{n}G^{ij}
\frac{{\partial}F}{{\partial}x_i}\frac{{\partial}G}{{\partial}x_j}+
\frac{{\partial}F}{{\partial}u}\frac{{\partial}G}{{\partial}u}\right) d\Omega.$$ Also, we define $||\nabla F||=\sqrt{(\nabla F,\nabla F)}$.
In the case of compact manifold or noncompact complete manifold, the self-adjointness of the Laplacians is classical [@Gaffney2; @Gaffney1]. A quantum layer is a noncompact noncomplete manifold. For such a manifold, we still have
$\Delta$ can be extended as a self-adjoint operator.
[**Proof.**]{} According to [@Reed-Simon-1], we define the Hilbert space $H_1$ to be the closure of the space $C^\infty_{0}(\Omega)$ under the norm $$||F||_{H_1}=\sqrt{||F||^2+||\nabla F||^2}.$$ We define the sesquilinear form $$Q(F,G)=(\nabla F, \nabla G),$$ for functions $F,G\in H_1$. By , $Q$ is the quadratic form of a unique self-adjoint operator. Such an operator is an extension of $\Delta$, which we still denote as $\Delta$. Furthermore, by the relation of $\Delta$ with the quadratic form, we can verify that $\sigma_0$ and $\sigma_{\rm ess}$ in , are the infimum of the spectrum and the essential spectrum of $\Delta$, respectively.
Lower bound of the essential spectrum.
======================================
The boundaries of $\Omega$ are ${\varSigma}\times \{{\pm}a\}$, which are smooth manifolds. It is not hard to see that can be written as $$\sigma_{\rm ess}
= \lim_{i\to \infty} \inf \{\frac{\int_\Omega \mid
\nabla f\mid^{2}}
{\int_\Omega f^{2}} \mid f\in C_{0}^{\infty}
(\Omega \setminus K_{i})\},$$ where $ \left\{x_{0}\right\} \subset K_{1} \subset K_{2}
\subset \cdots$ is any compact exhaustion of $\Omega$. For example, we can take $$K_{i} = \{ x + uN \mid x\in \overline{B_{x_o}(i)}
\subset {\varSigma},
u\in\left[\frac{-a(i-1)}{i}, \frac{a(i-1)}{i}\right] \}.$$
To perform our estimate we need to obtain a lower bound for the Rayleigh quotient $\frac{\int \mid
\nabla f\mid^{2}}{\int f^{2}}, \forall f\in
C_{0}^{\infty}(\Omega \setminus K_{i})$ for a large enough $i\in \mathbb{N}$.
We use the standard coordinate system $(x_1,\cdots,x_n,u)$ of Definition \[def21\]. Let $ f_j = \frac{\partial f}{\partial x_j}$, ($i=1,\cdots n)$ and $f_{n+1}=\frac{{\partial}f}{{\partial}u}$. Then
$$|\nabla f|^{2}
= |f_{n+1}|^{2} + \sum_{k,l \ne n+1}G^{kl}f_kf_l,$$ where $G^{ij}$ is the inverse of $G_{ij}$. In particular, we have $$\label{234}
|\nabla f|^{2} \geq \left|\frac{\partial f}{\partial
u}\right|^{2}.$$ Since $f=0$ on ${\partial}\Omega$, the Poincaré inequality gives $$\label{3-4}
\int_{-a}^{a}\left|\frac{\partial f}{\partial u}
\right|^{2}\, du
\geq\kappa_1^{2}\int_{-a}^{a}f(u)^{2}\, du,$$ where $\kappa_1=\pi/2a$.
\[thm31\] Under Assumptions A1) and A2), we have $\sigma_{\rm ess} \geq \kappa_{1}^{2}$.
[**Proof.**]{} We first observe that for arbitrary ${\varepsilon}>0$, there is an $i$ large enough such that $||A||<{\varepsilon}$ on ${\varSigma}\backslash
K_i$. By Corollary \[cor21\], we know that $$\label{3-3}
(1-a{\varepsilon})^n dud{\varSigma}\leq d\Omega\leq
(1+a{\varepsilon})^ndud{\varSigma}.$$ Thus we have $$\label{new}
\int_\Omega f^2d\Omega\leq(1+a{\varepsilon})^n
\int_{\varSigma}\int_{-a}^af^2 dud{\varSigma}.$$ On the other hand, by , and , we have $$\label{new1}
\int_\Omega|\nabla f|^2d\Omega\geq
(1-a{\varepsilon})^n\kappa_1^2\int_{\varSigma}\int^a_{-a}f^2dud{\varSigma}.$$ Comparing and , we have $$\sigma_{\rm ess}\geq\frac{(1-a{\varepsilon})^n}{(1+a{\varepsilon})^n}
\kappa_1^2.$$ Since ${\varepsilon}$ is arbitrary, we get the conclusion of the theorem.
Parabolicity of complete Riemannian manifolds
=============================================
Before giving the formal definition, we study the following example.
Suppose $n>1$ is an integer. Let $R>0$ be a big number. We are interested in the set of functions $$F(R)=\{f\in C_0^n({{{\mathchoice {\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\scriptstyle N\kern-0.3em R$}}
{\hbox{$\scriptscriptstyle R\kern-0.2em R$}}}}}^n)| f\equiv 1\,\, \text
{for }\,\,
|x|<R, \text{$f$ is rotationally symmetric.} \}.$$
We have the following
\[exp1\] If $n>2$, then for any $C>0$ there exists an $R_{0}$ such that for any $R>R_{0}$ we have $$\int_{\mathbb{R}^n}|\nabla f|^2 > C$$ for all $f\in F(R)$. If $n=2$, then for any ${\varepsilon}> 0$ there exists $R_{0}>0$ such that for any $R>R_{0}$, we can find an $f_{R} \in F(R)$ for which $$\int_{\mathbb{R}^2}|\nabla f|^2 < {\varepsilon}.$$
[**Proof.**]{} If $n>2$, then $$\int_{R}^\infty
1/r^{n-1}dr=\frac{1}{n-2}\cdot\frac{1}{R^{n-2}}.$$ Thus we have $$\int_{\mathbb R^n}|\nabla f|^2\geq
(n-2)cR^{n-2}\int_{ R}^\infty r^{n-1}\left|\frac{{\partial}f}{{\partial}r}
\right|^2dr
\int_{R}^\infty1/r^{n-1}dr\geq (n-2)cR^{n-2}
\rightarrow
+\infty$$ by Cauchy inequality, where $c$ is the volume of the unit $(n-1)$-sphere. However, for $n=2$, we let $f_R=\sigma_R(|x|)\in F(R)$, where $\sigma_R(t)$ is defined as $$\sigma_R(t)=
\left\{
\begin{array}{ll}
1& t\leq R\\
(1-\frac{\log R}{R})^{-1}(\frac{\log R}{\log t}-
\frac{\log R}{R}) & R<t\leq e^R\\
0& t\geq e^R
\end{array}
\right. .$$ A straightforward computation gives
$$\int_0^\infty t|\sigma_R'(t)|^2 dt\leq \frac 43\,\frac{1}{\log
R} \quad \text{for}\, R>3,$$ and thus $$\label{4-1}
\int_{\mathbb R^2}|\nabla f_R|^2\rightarrow 0,\quad
R\rightarrow
\infty.$$ This completes the proof.
The phenomenon in the above example can be explained by the result of Cheng-Yau [@CY2 Section 1]. In [@Li-Tam Definition 0.3], the authors defined the following
A complete manifold is said to be parabolic, if it does not admit a positive Green’s function. Otherwise it is said to be nonparabolic.
According to this definition, ${{{\mathchoice {\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\scriptstyle N\kern-0.3em R$}}
{\hbox{$\scriptscriptstyle R\kern-0.2em R$}}}}}^n$ is a parabolic manifold if and only if $n=2$. In particular, follows from Proposition \[prop41\] below, which is a result given in [@Li-Tam].
\[prop41\] Let ${\varSigma}$ be a parabolic manifold. Let $B(r)$ be the ball of radius $r$ in ${\varSigma}$ with respect to a reference point $x_0$. Let $R>r>1$. Consider the following Dirichlet problem $$\left\{
\begin{array}{ll}
\Delta f=0,&{\text on\,\,} B(R)\backslash B(r)\\
f=0, & {\text on\,\,} {\varSigma}\backslash B(R)\\
f=1, & {\text on\,\,} B(r).
\end{array}
\right.$$ Then we have $$\lim_{R\rightarrow \infty}\int_{\varSigma}|\nabla f|^2=0.$$
The functions $f$ serve as the high dimensional generalization of the MacDonald functions in the paper [@DEK page 21]. These functions will play an important role in the next section.
The following geometric criterion of parabolicity was proved by Grigor’yan [@Gro1; @Gro2] and Varopoulos [@varo] independently (cf. [@Li Equation (3.1)]):
\[thm4111\] Let $V(t)$ be the volume of the geodesic ball $B(t)$. If ${\varSigma}$ is nonparabolic, then $$\int_1^\infty\frac{tdt}{V(t)}<\infty.$$ In particular, if $V(t)$ is at most of quadratic growth, then ${\varSigma}$ is parabolic.
\[cor41\] Let $\varSigma$ be a smooth surface whose Gauss curvature $K \in
L^{1}(\varSigma)$. Then $\varSigma$ is a parabolic manifold of dimension $2$.
[**Proof.**]{} We wish to compare the volume growth rate of the geodesic ball $V(t)$ with $t$. To do so, first we assume that ${\varSigma}$ has a pole and we use the polar coordinate system given by the exponential map centered at the pole to write
$$V(t) = \int_{0}^{t} \int_{0}^{2\pi} f(r,\theta)\, dr
d\theta,$$ where under the polar coordinates, the expression of the metric becomes $ds_{\varSigma}^{2} = dr^{2} + f^{2}(r,\theta)d\theta^{2}$ on $\varSigma$.
It follows that $$V'(t) = \int_{0}^{2\pi}
f(t,\theta)\, d\theta.$$ The Jacobi equation for the exponential map gives $$f'' + Kf = 0 \hskip 0.1cm; \hskip 0.3cm f(0,\theta) = 0,
\hskip 0.1cm f'(0,\theta) = 1,$$ where the prime denotes derivative in the radial direction. Thus we have $$V'''(t)=-\int_0^{2\pi}Kf(t,\theta) d\theta.$$ Since $K$ is integrable, this implies that $$|V''(t)|\leq C$$ for some constant $C$. Consequently, $$\label{poiq}
V(t)\leq Ct^2$$ for $t$ large enough.
If the surface ${\varSigma}$ doesn’t have a pole, we get the similar estimate outside the cut locus with respect to a fixed reference point. Since the measure of the cut locus is zero, we get the same estimate . This is an observation of Gromov.
Thus the volume of ${\varSigma}$ is at most of quadratic growth and it must be parabolic by Theorem \[thm4111\].
The upper bound estimate of $\sigma_0$.
=======================================
The idea to estimate $\sigma_0$, the infimum of the spectrum of the Laplacian from above is to construct test functions which would provide the strict upper bound $\kappa_{1}^{2}$ (where $\kappa_1=\pi/2a$). We may construct test functions which are continuous everywhere on $\Omega$ and smooth everywhere on $\Omega$ except on a set of measure 0. Such functions must be in $H^{1,2}(\Omega)$, which serve our purpose.
We define the quadratic form $$\label{5-1-1}
Q(\xi,\xi)=\int_\Omega|\nabla \xi|^2d\Omega
-\kappa_1^2\int_\Omega \xi^2d\Omega,$$ for $\xi\in H^{1,2}(\Omega)$. By the nature of the metric on $\Omega$, we define $$\label{5-2-1}
Q_1(\xi,\xi)=\int_\Omega|\nabla'\xi|^2d\Omega,$$ where $$\label{5-2-2}
|\nabla'\xi|^2=\sum_{i,j=1}^n
G^{ij}\frac{{\partial}\xi}{{\partial}x_i}
\frac{{\partial}\xi}{{\partial}x_j},$$ and $$\label{5-3}
Q_2(\xi,\xi)=\int_\Omega\left(\frac{{\partial}\xi}{{\partial}u}\right)^2
d\Omega-\kappa_1^2\int_\Omega\xi^2d\Omega,$$ where $(x_1,\cdots,x_n,u)$ are the standard coordinates in Definition \[def21\]. It is clear that $$Q(\xi,\xi)=Q_1(\xi,\xi)+Q_2(\xi,\xi).$$
The test functions we shall construct will essentially be the product of a vertical function (depending only on $u$) and a horizontal one (depending only on $x\in\varSigma$). Let ${\varphi}=\psi\chi$ be a test function, where $\psi\in C_0^\infty ({\varSigma})$ and $\chi$ is a smooth function of $u$ such that $\chi(\pm a)=0$.
Note that $$\bigtriangledown (\chi\psi) = \chi\bigtriangledown \psi +
\psi\bigtriangledown \chi.$$ By , we have $<\nabla\psi,\nabla\chi>=0$. Thus we have $$\label{5-1}
\int_{\Omega}|\bigtriangledown(\chi\psi)|^{2}=
\int_{\Omega}\chi^{2}|\bigtriangledown \psi |^{2}
+ \int_{\Omega}\psi^{2}|\bigtriangledown \chi |^{2}.$$ We wish to prove, with the suitable choice of $\psi$ and $\chi$, that $$\label{5-2}
Q({\varphi},{\varphi}) =
\int_\Omega {\varphi}_{u}^{2}
-
\kappa_{1}^{2}\int_\Omega {\varphi}^{2}
+\int_\Omega\chi^2|\nabla\psi|^2
<0,$$
where ${\varphi}_{u}$ denotes $\frac{\partial {\varphi}}{\partial u}$ and ${\varphi}_i=\frac{{\partial}{\varphi}}{{\partial}x_i}$ for $i=1,\cdots,n$.
Like in the paper [@DEK], we choose $\chi=\cos
\kappa_1u$. We need the following elementary lemma:
\[lem51\] Let $a>0$ be a positive number and let $\kappa_1=\frac{\pi}{2a}$. Let $\chi(u) =
\cos{\kappa_1u}$, let $$\mu_k=
\int_{-a}^{a} u^{k}(\chi_{u}^{2} -
\kappa_{1}^{2}\chi^{2})\, du
,\, \forall k \geq 0.$$ Then $$\mu_k=\left\{
\begin{array}{ll}
0& \text{if $k$ is odd, or $k=0$};\\
\frac 12\frac{(k)!}{(2\kappa_1)^{k-1}}\sum_{l=1}^{k/2}
\frac{(-1)^{k/2-l}\pi^{2l-1}}{(2l-1)!},& \text{if $k\neq 0$
is even}.
\end{array}
\right.$$ Furthermore, $\mu_k>0$ if $k\neq 0$ is even.
\[thm51\] We assume that the hypersurface $\varSigma \subset
\mathbb{R}^{n+1}$ is parabolic satisfying Assumptions A1), A2). Moreover, we assume that $\sum_{k=1}^{[ n/2]}\mu_{2k}c_{2k}(A)$ is integrable and $$\int_{\varSigma}\sum_{k=1}^{[n/2
]}\mu_{2k}c_{2k}(A)
d\varSigma \leq 0,$$ where $A$ is the second fundamental form of ${\varSigma}$, $\mu_k$ for $k\geq 1$ is defined in Lemma \[lem51\], $[ n/2 ] $ is the integer part of $n/2$, and $c_k(A)$ is the $k$-th elementary symmetric polynomial of $A$. If ${\varSigma}$ is not totally geodesic, then $$\sigma_0 < \kappa_{1}^{2}
.$$
[**Proof.**]{} We first consider the test functions of the form ${\varphi}=
\psi\cdot\chi$. We define $\psi$ as follows: Let $x_0$ be a fixed point of ${\varSigma}$ and let $R>r>1$. Let $B(R)$ and $B(r)$ be two balls in ${\varSigma}$ of radius $R$ and $r$ centered at $x_0$ respectively. We define $\psi$ as $$\left\{
\begin{array}{ll}
\Delta\psi=0 &\text{on $B(R)-B(r)$};\\
\psi|_{B(r)}\equiv 1;\\
\psi|_{{\varSigma}-B(R)}\equiv 0,
\end{array}
\right.$$ and we define $\chi=\cos\kappa_1 u$.
By the definition of the functions $\chi$ and $\psi$, using Lemma \[lem2-1\], we know that there is a constant $C$ such that $$\label{5-8}
\int_{\Omega}\chi^{2}|\bigtriangledown \psi |^{2}
d\Omega
\leq C\int_{\varSigma}|\nabla_{\varSigma}\psi|^2d{\varSigma},$$ where $\nabla_{\varSigma}$ is the connection of ${\varSigma}$. We first assume that $$\label{5-9}
\int_{\varSigma}\sum_{k=1}^{[n/2
]}\mu_{2k}c_{2k}(A)
d\varSigma =-\delta<0.$$ By and Lemma \[lem51\], we have $$\label{5-10}
\int_\Omega \psi^{2}|\chi_u|^{2} -
\kappa_{1}^{2}\int_\Omega \psi^{2}\chi^{2}
= \int_{\varSigma}\psi^{2}\sum_{k=1}^{[ n/2
]}\mu_{2k}c_{2k}(A)\, d\varSigma.$$
By the maximum principle and the fact that $\psi|_{B(r)}
\equiv 1$, we have
$$\int_{\varSigma}\psi^{2}\sum_{k\geq 1} \mu_{2k}c_{2k}(A)
\leq \int_{B(r)}\sum_{k\geq 1}\mu_{2k}c_{2k}(A)
+\int_{\varSigma \setminus B(r)}|
\sum_{k\geq 1}
\mu_{2k}c_{2k}(A)|.$$
On the other side, since $\sum\mu_{2k}c_{2k}(A)$ is integrable, if $r$ is large enough, by the above inequality, we have $$\label{5-12}
\int_{\varSigma}\psi^2\sum
\mu_{2k}c_{2k}(A)d{\varSigma}<
-\frac{\delta}{2}.$$ By Proposition \[prop41\] and , if $R$ large enough, we have $$\label{5-13}
\int_\Omega\chi^2|\nabla\psi|^2\leq \frac{\delta}{4}.$$ Combining , and , we proved under the assumption .
Now we assume that
$$\label{5-14}
\int_{\varSigma}\sum_{k=1}^{[n/2
]}\mu_{2k}c_{2k}(A)
d\varSigma =0.$$
In this case, the test functions ${\varphi}=\psi\chi$ are not good enough to give the upper bound of $\sigma_0$. We shall use a trick in [@DEK] (see also [@DP; @RB]) to construct the test functions.
We let $${\varphi}_{\varepsilon}={\varphi}+{\varepsilon}j\chi_1,$$ where ${\varepsilon}$ is a small number, $j$ is a smooth function on ${\varSigma}$ whose support is contained is $B(r-1)$, and $\chi_1$ is a smooth function on $[-a,a]$ such that $\chi_1(\pm a)=0$. As a general fact, we have $$\label{5-15}
Q({\varphi}_{\varepsilon},{\varphi}_{\varepsilon})=Q({\varphi},{\varphi})+2{\varepsilon}Q({\varphi},j\chi_1)
+{\varepsilon}^2Q(j\chi_1,j\chi_1).$$ By , , and , we have $$\label{5-16}
Q({\varphi}_{\varepsilon},{\varphi}_{\varepsilon})\leq C\int_{\varSigma}|\nabla\psi|^2
d{\varSigma}+
\int_{\varSigma}\psi^{2}\sum_{k=1}^{[ n/2
]}\mu_{2k}c_{2k}(A)\, d\varSigma+2{\varepsilon}Q({\varphi},j\chi_1)
+{\varepsilon}^2Q(j\chi_1,j\chi_1).$$ Since ${\rm supp}\, j\subset B(r-1)$, we have $$Q({\varphi},j\chi_1)=\int_\Omega j(\chi_u(\chi_1)_u-\kappa_1^2
\chi\chi_1)d\Omega=\int_{\varSigma}j
\int_{-a}^a(\chi_u(\chi_1)_u-\kappa_1^2
\chi\chi_1)\det(1-uA)dud{\varSigma}.$$
Using integration by parts, we have $$Q({\varphi},j\chi_1)=-\int_{\varSigma}j\int_{-a}^a
\chi_u\frac{{\partial}}{{\partial}u}
\det (1-uA)\chi_1dud{\varSigma}.$$
Now we are able to choose suitable $j$ and $\chi_1$ for our purpose. By assumption, we know that ${\varSigma}$ is not totally geodesic. Thus at least there is a point $x\in
{\varSigma}$ such that ${\partial}_u\det(1-uA)\not\equiv 0$. We assume that $x\in B(r-1)$ without losing generality. We choose $\chi_1$ and $j$ such that the integral $Q({\varphi},j\chi_1)$ is not zero. Note that the choice of $j$ is independent of ${\varphi}$. We choose ${\varepsilon}$ (positive or negative) small enough so that $$2{\varepsilon}Q({\varphi},j\chi_1)
+{\varepsilon}^2Q(j\chi_1,j\chi_1)<0.$$ Finally, since $${\rm supp}\, j\subset B(r-1),$$ the above expression is independent to $r$ and $R$. By the parabolicity of ${\varSigma}$, if $r,R\rightarrow \infty$, then $$\int_{\varSigma}|\nabla\psi|^2d{\varSigma}\rightarrow 0,$$ and by the assumption , $$\int_{\varSigma}\psi^{2}\sum_{k=1}^{[ n/2
]}\mu_{2k}c_{2k}(A)\, d\varSigma\rightarrow 0.$$ Thus by , $Q({\varphi}_{\varepsilon},{\varphi}_{\varepsilon})$ is negative for $r, R$ large. This completes the proof of the theorem.
Let $\mathcal R=(R_{ijkl})$ be the curvature tensor of ${\varSigma}$. Define $$\label{trace-1}
{\rm tr}(\mathcal R^p)
=\sum_{i_s<j_s, k_s<l_s,
s=1,\cdots,p} (-1)^{sgn(\sigma)}
R_{i_1j_1k_1l_1}\cdots R_{i_pj_pk_pl_p},$$ where $\sigma$ is the permutation $(i_1,\cdots,j_p; k_1,\cdots,l_p)$. Then from Gray [@gray (4.15)], we have
\[lem63\] Using the above notations, we have $${\rm Tr}(\mathcal R^p)=c_{2p}(A)$$
If $n$ is even, then up to a constant, ${\rm Tr}(\mathcal
R^{n/2})=c_n(A)$ is the Gauss-Bonnet-Chern density.
[**Proof of Theorem \[main1\].**]{} By Theorem \[thm31\], Theorem \[thm51\], and Proposition \[lem63\], we have $$\sigma_0<\kappa_1^2\leq\sigma_{\rm ess}.$$ Thus the ground state exists.
[**Proof of Theorem \[main2\].**]{} By the Theorem of Hartman, we know that is equivalent to $$\int_{\varSigma}K\leq 0.$$ Thus the result follows from Theorem \[main1\] for $n=2$.
[**Proof of Corollary \[main4\].**]{} If $n=3$, then the conditions are $${\rm Tr}\,(\mathcal R^1) \text{ is integrable and}\quad
\int_{\varSigma}{\rm Tr}\,(\mathcal R^1)\leq 0.$$ But $\rho=2\,{\rm Tr}(\mathcal R^1)$.
If $n=4$, a tedious computation gives $${\rm Tr}\, (\mathcal R^2)=\frac{1}{24}(\rho^2-4|{\rm
Ric}|^2+|{\mathcal R}|^2),$$ where ${\rm Ric}$ is the Ricci curvature of ${\varSigma}$, and $|{\rm Ric}|, |\mathcal R|$ are the norms of the Ricci tensor and the curvature tensor, respectively. If the sectional curvature is positive outside a compact set, then by [@Greene-Wu Theorem 9], ${\rm Tr}\,(\mathcal R^2)$ is integrable and $$\int_{\varSigma}{\rm Tr}\,(\mathcal R^2)\leq \frac{4\pi^2}{3}
e({\varSigma}),$$ where $e({\varSigma})$ is the Euler characteristic number of ${\varSigma}$. The theorem follows from the above inequality, Proposition \[lem51\], and Theorem \[main1\].
Before finishing this section, we give the following example of the manifold ${\varSigma}$ of dimension $3$ satisfying the conditions in Theorem \[main1\]. Thus the theorem is not an empty statement for high dimensions.
There is a complete manifold ${\varSigma}$ of dimension $3$ immersed in ${{{\mathchoice {\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\scriptstyle N\kern-0.3em R$}}
{\hbox{$\scriptscriptstyle R\kern-0.2em R$}}}}}^4$ such that
1. It is parabolic;
2. $A\rightarrow 0$, where $A$ is the second fundamental form;
3. $\frac 12\int_{\varSigma}|\rho|=\int_{\varSigma}|c_2(A)|<+\infty$;
4. $\frac 12\int_{\varSigma}\rho=\int_{\varSigma}c_2(A)<0$.
[**Proof.**]{} Let ${\varSigma}=S^1\times {{{\mathchoice {\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\scriptstyle N\kern-0.3em R$}}
{\hbox{$\scriptscriptstyle R\kern-0.2em R$}}}}}^2$. We consider the immersion by $${\varSigma}\rightarrow{{{\mathchoice {\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\scriptstyle N\kern-0.3em R$}}
{\hbox{$\scriptscriptstyle R\kern-0.2em R$}}}}}^4, \quad
(\theta,t,{\varphi})\rightarrow
(\sigma(t)\cos\theta,\sigma(t)\sin\theta,
t\cos{\varphi},t\sin{\varphi}),$$ where $\sigma(t)$ is a smooth positive function defined in . Here we use $\theta$ as the local coordinate of $S^1$ and $(x,y)\in {{{\mathchoice {\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\scriptstyle N\kern-0.3em R$}}
{\hbox{$\scriptscriptstyle R\kern-0.2em R$}}}}}^2$ with $x=t\cos{\varphi}, y=t\sin{\varphi}$. The Riemannian metric of ${\varSigma}$ is $$ds^2=(1+\sigma'(t)^2)(dt)^2+\sigma^2(t)(d\theta)^2+t^2
(d{\varphi})^2.$$ We claim that ${\varSigma}$ is parabolic. In order to prove this, we let $x_0=(1,0,0)\in {\varSigma}$. Let $B(R)$ be the geodesic ball of radius $R$ centered at $x_0$. Then $B(R)\subset \{x\in \varSigma | t<R\}$. To see this, let $x\in B(R)$ such that ${\rm
dist}\,(x,x_0)=R'$, and let $\eta=(\eta_1(s),\eta_2(s),\eta_3(s))$ be the geodesic line of ${\varSigma}$ connecting $x_0$ and $x$, where $s$ is the arc length. Then we have $$R=R'\geq\int_0^{R'}(1+\sigma'(s)^2)^{\frac 12}
|\eta_1'(s)|ds\geq t.$$ From the above equation, we have $${\rm vol}\,B(R)\leq 4\pi^2\int_0^{R}t\sigma(t)
(1+\sigma'(t)^2)^{\frac 12}dt\leq CR^2\log R$$ for some constant $C$. Thus we have $$\int_0^\infty\frac{t}{{\rm vol}\,B(t)}dt
=+\infty,$$ and ${\varSigma}$ is parabolic by Theorem \[thm4111\].
The normal vector of ${\varSigma}$ in ${{{\mathchoice {\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\scriptstyle N\kern-0.3em R$}}
{\hbox{$\scriptscriptstyle R\kern-0.2em R$}}}}}^4$ is $$N=\frac{1}{\sqrt{1+\sigma'(t)^2}}
(\cos\theta,\sin\theta,-\sigma'\cos{\varphi},-\sigma'\sin{\varphi}).$$ The principal curvatures are $$\frac{\sigma''}{({1+\sigma'(t)^2})^{\frac
32}},\quad -\frac{1}{\sigma\sqrt{1+\sigma'(t)^2}},
\quad\frac{\sigma'}{t\sqrt{1+\sigma'(t)^2}}.$$ By the definition of $\sigma(t)$, all principal curvatures go to zero as $t\rightarrow \infty$. Thus $A\rightarrow 0$ at infinity. On the other hand $$\label{ex-n-2}
\int_{\varSigma}c_2(A)=4\pi^2
\int_0^\infty\left(\frac{\sigma\sigma'\sigma''}
{(1+\sigma'(t)^2)^{\frac 32}}
-\frac{t\sigma''}{(1+\sigma'(t)^2)^\frac 32}
-\frac{\sigma'}{\sqrt{1+\sigma'(t)^2}}
\right) dt$$
We let the function $\sigma(t)$ be a smooth increasing function such that $$\label{ex-n-1}
\left\{
\begin{array}{ll}
\sigma(t)=\log t& t>3+{\varepsilon}\\
\sigma(t)=\log 3 & t<3
\end{array}
\right.,$$ for ${\varepsilon}$ small. The last two terms of can be calculated easily: $$\label{ex-n-3}
\int_0^\infty\left(
-\frac{t\sigma''}{(1+\sigma'(t)^2)^\frac 32}
-\frac{\sigma'}{\sqrt{1+\sigma'(t)^2}}
\right) dt=\left.-\frac{t\sigma'}
{\sqrt{1+\sigma'(t)^2}}\right|_0^\infty=-1.$$ Let $R$ be a big number. We have $$\int_0^R\frac{\sigma\sigma'\sigma''}
{(1+\sigma'(t)^2)^{\frac 32}}
dt
=-\frac{\log R}{(1+\frac{1}{R^2})^\frac 12}
+\log 3 +\int_{3+{\varepsilon}}^R\frac{\sigma'(t)}
{(1+\sigma'(t)^2)^{\frac 12}} dt
+\int^{3+{\varepsilon}}_3\frac{\sigma'(t)}
{(1+\sigma'(t)^2)^{\frac 12}} dt.$$ The last term can be estimated by $$\int^{3+{\varepsilon}}_3\frac{\sigma'(t)}
{(1+\sigma'(t)^2)^{\frac 12}} dt
\leq \log (3+{\varepsilon})-\log 3.$$ Thus a straightforward computation gives $$\int_0^R\frac{\sigma\sigma'\sigma''}
{(1+\sigma'(t)^2)^{\frac 32}}
dt\leq
-\frac{\log R}{(1+\frac{1}{R^2})^\frac 12}
+\log( 3+{\varepsilon}) +\log (R+\sqrt{1+R^2})-\log (3+{\varepsilon}+\sqrt{1+(3+{\varepsilon})^2}).$$ We let $R\rightarrow \infty, {\varepsilon}\rightarrow 0$. Then we have $$\label{ex-n-5}
\int_0^\infty\frac{\sigma\sigma'\sigma''}
{(1+\sigma'(t)^2)^{\frac 32}}
dt\leq\log 6-\log (3+\sqrt{10})<0.$$ By and , we have $\int_M c_2(A)<0$. Finally, since $$\frac{\sigma\sigma'\sigma''}
{(1+\sigma'(t)^2)^{\frac 32}}\sim
O(\frac{\log t}{t^3}),$$ and $$-\frac{t\sigma''}{(1+\sigma'(t)^2)^\frac 32}
-\frac{\sigma'}{\sqrt{1+\sigma'(t)^2}}
\sim O(\frac{1}{t^3}),$$ we know that $c_2(A)$ is integrable.
Convex surfaces
===============
In this section, we consider the layer over a convex surface ${\varSigma}$ in ${{{\mathchoice {\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\scriptstyle N\kern-0.3em R$}}
{\hbox{$\scriptscriptstyle R\kern-0.2em R$}}}}}^3$. ${\varSigma}$ is defined by $$z=f(x,y),$$ where $f(x,y)$ is a smooth convex function, $f(0)=0$, $\nabla f(0)=0$, and $\nabla^2 f(0)>0$.
The main result of this section is the following:
\[thm61\] Let ${\varSigma}$ be defined as above. Suppose $\Omega={\varSigma}\times(-a,a)$ is the layer with depth $a>0$. Then the infimum of the spectrum $\sigma_0$ satisfies $$\sigma_0<\kappa_1^2,$$ where $\kappa_1=\pi/(2a)$.
We begin with the following
\[lem61\] With the assumptions on $f$, there is a number $\delta>0$ such that $$f_{r}=\frac{{\partial}f}{{\partial}r}>\delta$$ for $x^2+y^2\geq 1$, where $(r,\theta)$ is the polar coordinates defined by $x=r\cos\theta$ and $y=r\sin\theta$.
[**Proof.**]{} By the assumptions, there is a $\delta_0\in(0,1)$ such that $f_{r}|_{x^2+y^2=\delta_0}>0$. By convexity, we have $f_{rr}\geq 0$. Then since the circle $\{x^2+y^2=\delta_0\}$ is compact, we can conclude that $f_r\geq\delta$ for $x^2+y^2\geq 1$.
\[cor61\] Using the above notations, we have
1. $|\nabla f|\geq\delta$;
2. $f(x,y)\geq \delta\cdot (\sqrt{x^2+y^2}-1)$.
An interesting consequence of the above corollary is the following: Let $b$ be a large positive number. Let $C_b$ be the curve defined by the intersection of ${\varSigma}$ with respect to the plane $z=b$. Clearly $C_b$ is a convex curve. From the above corollary, $C_b$ is contained in a disk of radius $b/\delta+1$. In particular, we have the estimate of the length of the curve $$\label{6-2}
\int_{C_b}1\leq Cb$$ for a constant $C$.
For the manifold ${\varSigma}$, the mean curvature $H$ can be represented by $$H=\frac{(1+f_y^2)f_{xx}+(1+f_x^2)f_{yy}-2f_xf_yf_{xy}}
{(1+|\nabla f|^2)^{3/2}},$$ where $f_x=\frac{{\partial}f}{{\partial}x}$, $f_y=\frac{{\partial}f}{{\partial}y}$, $f_{xx}=
\frac{{\partial}^2 f}{{\partial}x^2}$, $f_{yy}=
\frac{{\partial}^2 f}{{\partial}y^2}$, $f_{xy}=
\frac{{\partial}^2 f}{{\partial}x{\partial}y}$, and $|\nabla f|^2=f_x^2
+f_y^2$. We compare the mean curvature to the curvature of the convex curve $f(x,y)=b$, which is given by $$k_b=\frac{f_{xx}f_y^2-2f_{xy}f_xf_y+f_{yy}f_x^2}{|\nabla
f|^{3}}.$$ By Corollary \[cor61\] and the convexity of $f$, we have $$\label{6-4}
H\geq \frac 12\delta^3 k_b,$$ if $\delta$ is small enough. Since $C_b:\{f(x,y)=b\}$ is a convex curve, we have $$\int_{f(x,y)=b}k_b=2\pi.$$ Thus by $$\label{6-7}
\int_{f(x,y)=b}H\geq\pi\delta^3.$$
By the co-area formula (cf. [@SY page 89]), we have
$$\int_{x^2+y^2\geq 1} Hd{\varSigma}\geq\int^\infty_{c}
(\int_{f=t}\frac{H}{|\tilde \nabla f|})
dt,$$
where $c$ is a positive real number, and $$\label{6-9}
|\tilde\nabla f|^2=\frac{|\nabla f|^2}
{1+|\nabla f|^2}$$ is the gradient of $f$ on the [*Riemannian manifold*]{} ${\varSigma}$. Thus by , and Corollary \[cor61\], $$\int_{x^2+y^2\geq 1} H=+\infty.$$
[**Proof of Theorem \[thm61\].**]{} We shall again use the trick introduced by [@DEK] (see also [@DP; @RB]) to perturb the “standard” test functions. However, our choices of perturbation functions are quite different from theirs in nature.
Let $K$ be the Gauss curvature of ${\varSigma}$. Then we have $K\geq 0$, and $$\label{6-11}
\int_{\varSigma}K\leq 2\pi$$ by the theorem of Huber [@Huber]. Since the Gauss curvature is nonnegative, the volume growth is at most quadratic. Thus ${\varSigma}$ is parabolic. For any $r_1>0$, we can find a function ${\varphi}$ such that
1. ${\varphi}\in C_0^\infty({\varSigma})$, $0\leq{\varphi}\leq 1$;
2. ${\varphi}\equiv 1$ on $B(r_1)$, where $B(r_1)$ is the geodesic ball of radius $r_1$ of ${\varSigma}$ centered at $0$;
3. $\int_{\varSigma}|\nabla{\varphi}|^2d{\varSigma}<1$.
The quadratic forms $Q$, $Q_1$, and $Q_2$ are defined in , , and . Let $\chi=\cos\kappa_1 u$. Then we have $$Q_1({\varphi}\chi,{\varphi}\chi)=\int_\Omega|\nabla{\varphi}|^2\chi^2d\Omega
\leq a{(1+C_0)^2},$$ where $C_0<1$ is defined in Assumption A1). We also have $$Q_2({\varphi}\chi,{\varphi}\chi)\leq
\mu_2\int_{{\varSigma}}K{\varphi}^2d{\varSigma}.$$ Combining the above two equations and using , we have $$\label{6-12}
Q({\varphi}\chi,{\varphi}\chi)=Q_1({\varphi}\chi,{\varphi}\chi)
+Q_2({\varphi}\chi,{\varphi}\chi)\leq C_1,$$ where $C_1$ is a constant depending only on ${\varSigma}$ and $a$.
Suppose $r_1$ is large enough such that $\{f(x,y)\leq 2R^2\}\subset B(r_1)$ for some large number $R>0$. We consider a function $\rho(t)$ on ${{{\mathchoice {\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\textstyle{\text{\strange R}}$}}
{\hbox{$\scriptstyle N\kern-0.3em R$}}
{\hbox{$\scriptscriptstyle R\kern-0.2em R$}}}}}$ such that
1. $\rho\equiv 1$, if $t\in[R,R^2]$;
2. $\rho \equiv 0$ if $t>R^2+1$ or $t<R-1$;
3. $0\leq\rho\leq 1$;
4. $|\rho'|\leq 4$.
We define $\psi(x,y)=\rho(f(x,y))$. $\psi$ is a smooth function of ${\varSigma}$. Let $\chi_1$ be an odd function of $u$ such that $\chi_1(\pm a)=0$, and $$\label{6-15}
\int_{-a}^a\chi_u(\chi_1)du=-\sigma<0,$$ where $\sigma>0$ is a positive number. We consider the function ${\varphi}\chi+{\varepsilon}\psi\chi_1/f$, where ${\varepsilon}$ is a small number to be determined. By the definition of $Q(\cdot,\cdot)$ and , we have $$\label{6-16}
Q({\varphi}\chi+{\varepsilon}\psi\chi_1/f, {\varphi}\chi+{\varepsilon}\psi\chi_1/f)
\leq
C_1+2{\varepsilon}Q({\varphi}\chi,\psi\chi_1/f)+{\varepsilon}^2
Q(\psi\chi_1/f,\psi\chi_1/f).$$ If $r_1$ and $R$ are big, then $${\rm supp}\,\psi\subset \{x\in{\varSigma}\mid {\varphi}(x)\equiv 1\}.$$ We thus have $$\int_\Omega\langle \nabla' ({\varphi}\chi),\nabla'(\psi\chi_1/f)\rangle d\Omega=0,$$ where $\nabla'$ is defined in . By , we have $$d\Omega=(1-Hu+Ku^2)d{\varSigma}.$$ Since $\chi_1$ is odd and $\chi$ is even, by the above equation, we have $$Q({\varphi}\chi,\psi\chi_1/f)=\int_\Omega(\chi_u(\chi_1)_u{\varphi}\psi/f-\kappa_1^2
\chi\chi_1{\varphi}\psi/f)d\Omega
=-\int^a_{-a}u(\chi_u(\chi_1)_u-\kappa_1^2\chi\chi_1) du\int_{\varSigma}\psi H/fd{\varSigma}.$$ Since $$\int^a_{-a}u(\chi_u(\chi_1)_u-\kappa_1^2\chi\chi_1) du=-\int_{-a}^a\chi_u(\chi_1)du,$$ we have $$\label{6-17}
Q({\varphi}\chi,\psi\chi_1/f)=-\sigma\int_{\varSigma}\frac{\psi H}{f}d{\varSigma},$$ where $\sigma$ is the number defined in . By the co-area formula and , , we have $$\label{6-18}
\int_{\varSigma}\frac{\psi H}{f}=\int_{\mathbb R}\frac
{\rho(t)}{t}
(\int_{f=t}\frac{H}{|\tilde\nabla
f|})dt\geq\pi\delta^3\int_{\mathbb R}\frac{\rho(t)} t
dt\geq\pi\delta^3\log R.$$
In order to estimate the last term of , we first note that $$|\tilde \nabla\frac\psi f|\leq C_2/f$$ for some constant $C_2$, where $\tilde\nabla$ is the covariant derivative of ${\varSigma}$. From this, we have $$\label{6-20}
Q_1(\frac{\psi\chi_1}{f},\frac{\psi\chi_1}{f})\leq
C_3\int_{R-1\leq f\leq R^2+1} 1/f^2d{\varSigma}$$ for some constant $C_3$. Using the same argument, we have $$\label{6-21}
Q_2(\frac{\psi\chi}{f},\frac{\psi\chi}{f})
\leq
C_4\int_{R-1\leq f\leq R^2+1} 1/f^2d{\varSigma}$$ for some constant $C_4$. We use the co-area formula again to estimate $$\int_{R-1\leq f\leq R^2+1} 1/f^2
=\int_{R-1}^{R^2+1}\frac{1}{t^2}(\int_{f=t}\frac{1}{|\tilde
\nabla f|})dt.$$ From Corollary \[cor61\], we know that $|\tilde\nabla f|$ has a lower bound. Thus by , there is a constant $C_5$ such that $$\label{6-23}
\int_{R-1\leq f\leq R^2+1} 1/f^2d{\varSigma}\leq
C_5\log R.$$ Thus by , , , , and , from , we have $$\label{6-25}
Q({\varphi}\chi+{\varepsilon}\psi\chi_1/f, {\varphi}\chi+{\varepsilon}\psi\chi_1/f)
\leq C_1 -2{\varepsilon}\pi\sigma\delta^3\log
R+{\varepsilon}^2C_5(C_3+C_4)\log R.$$ We choose ${\varepsilon}$ to be a small positive number such that $$-2{\varepsilon}\pi\sigma\delta^3+{\varepsilon}^2C_5(C_3+C_4)<0.$$ We then let $R$ large enough (which requires $r_1$ be large enough also). Then the left hand side of is negative. By the definition of $\sigma_0$, we know that $\sigma_0<\kappa_1^2$.
[**Proof of Theorem \[main3\].**]{} By Theorem \[thm31\] and Theorem \[thm61\], we have $$\sigma_0<\kappa_1^2\leq\sigma_{\rm ess}.$$ Thus the ground state exists.
[^1]: The first author is partially supported by NSF grant DMS-0202508. The second author is partially supported by NSF Career award DMS-0347033 and the Alfred P. Sloan Research Fellowship.
[^2]: Quantum layers were studied by many authors. An incomplete list of the works are [@BEGK-1; @CEK-1; @DEK-1; @EHL-1; @ESK-2; @ESK-1; @EK-1; @PS-1; @EV-1; @kr-1; @KA-1; @KK-1].
[^3]: There is an overlap of this result with the one in [@CEK-1]. The proofs are similar but not identical. In particular, we use the result of Hartman instead of the Kohn-Vossen inequality.
[^4]: This part of the question was implied in [@DEK]. By [@DEK] and this paper, we just need to show that for layers bulit over simply-connected surfaces with positive total Gauss curvature, the ground state exists.
|
---
abstract: 'We calculate the single transverse spin asymmetries(SSA) for forward inclusive particle production in pp and pA collisions using a hybrid approach. It is shown that the Sivers type contribution to the SSA drops out due to color entanglement effect, whereas the fragmentation contribution to the spin asymmetry is not affected by color entanglement effect. This finding offers a natural solution for the sign mismatch problem.'
author:
- Jian Zhou
title: 'Single spin asymmetries in forward p-p/A collisions revisited: the role of color entanglement'
---
[*I. Introduction.*]{}— During the past three decades, the studies of transverse single spin asymmetries in high energy scatterings have greatly deepened our understanding of some central aspects of Quantum Chromodynamics(QCD) factorization theorem, among which the universality issue attracted a lot of attentions. Within transverse momentum dependent(TMD) factorization framework [@Collins:1981uk], a TMD distribution, known as the Sivers function $f_{1T}^{\perp}$ [@Sivers:1989cc] was proposed to account for the observed large SSAs. It has been found that the Sivers function reverses sign between the semi-inclusive deeply inelastic scattering(SIDIS) and the Drell-Yan process [@Collins:1992kk; @Brodsky:2002cx; @Collins:2002kn]. The discovery of such novel and unique universality property has stimulated a lot of theoretical progress over the past decade. The preliminary results from the STAR collaboration and the COMPASS collaboration [@Adamczyk:2015gyk; @Aghasyan:2017jop] seem to confirm the sign change. This is undoubtedly one of the most remarkable achievements in high energy spin physics.
However, the situation with the SSA for the forward inclusive hadron production in $pp$ collisions(denoted as $A_N$) $p^\uparrow p\rightarrow h X$ is more complicated. Due to the lack of an additional hard scale, it is more appropriate to compute this observable using the collinear twist-3 approach [@Efremov:1981sh; @Qiu:1991pp; @Kouvaris:2006zy; @Yuan:2009dw; @Metz:2012ct] instead of TMD factorization. Phenomenologically, it was also studied in the generalized parton model [@Anselmino:1994tv; @Anselmino:2005sh]. The twist-3 effects leading to the SSA can be factorized into various three-parton correlation functions. One of these is the Qiu-Sterman function $T_F$ [@Qiu:1991pp] which can be related to the Sivers function [@Boer:2003cm], $$T_F(x,x)=-\!\int \! d^2 p_\perp \frac{p_\perp^2}{M} f_{1T}^{\perp}(x,p_\perp^2)|_\text{SIDIS}$$ where $M$ is the nucleon mass. Due to this relation, one can determine $T_F$ using the date on the SSA measured in SIDIS and compared with the Qiu-Sterman function extracted from inclusive hadron production in $pp$ collisions. Very surprisingly, $T_F$ extracted from these two observables actually differ in sign [@Kang:2011hk]. To resolve this sign mismatch problem, the authors of Ref. [@Kanazawa:2014dca] suggested that a genuine twist-3 function $ \text{Im} \hat E_{\text
F}$ [@Yuan:2009dw]($\hat H_{\text{FU}}^{\Im}$ in a different notation) instead of $T_F$ gives rise to the dominant contribution to $A_N$. It is worthy to mention that the data on the SSA in SIDIS [@Airapetian:2009ab; @Katich:2013atq] doesn’t disfavor this point of view because the Sivers function is not well constrained at large $x$ in SIDIS, allowing flexible parametrizations of $T_F$. Note that all other possible sources contributing to $A_N$ in the collinear twist-3 approach were shown to be small [@Kanazawa:2000hz; @Koike:2007rq; @Kanazawa:2010au].
The study of the SSA for inclusive hadron production in $pA$ collisions $p^\uparrow A\rightarrow h X$ could play an important role in pining down the true main cause of $A_N$ since the different sources contributing to $A_N$ is affected by saturation effect in the different ways. In fact, no strong nuclear suppression was observed in a recent measurement of $A_N$ in forward $pA$ collisions [@Heppelmann:2016siw]. This implies that the dominant pieces must be these which are not affected by saturation effect. It is thus of a great interest to take into account the saturation effect on the unpolarized target side. Some earlier work in this direction have been done in Refs. [@Boer:2006rj; @Kang:2011ni; @Kovchegov:2012ga; @Kang:2012vm].
In this paper, we compute $A_N$ using a hybrid approach [@Schafer:2014zea; @Zhou:2015ima] where target nucleus(or proton) is treated in the color glass condensate(CGC) framework [@McLerran:1993ni], while collinear twist-3 approach is applied to the transversely polarized projectile. It is a natural and powerful approach to take into account color entanglement effect that was first discovered in Ref. [@Rogers:2010dm](for the relevant work, see Refs. [@Buffing:2011mj; @Buffing:2012sz; @Rogers:2013zha]). Actually, it has been found that the SSAs for prompt photon production and photon-jet production in $pp$ or $pA$ collisions receive the contribution from color entanglement effect [@Schafer:2014zea; @Schafer:2014xpa]. In contrast, color entanglement effect is absent in the Drell-Yan process at low transverse momentum due to the trivial color flow [@Zhou:2015ima].
We notice that the hybrid approach has been used to compute $A_N$ in Refs. [@Hatta:2016wjz; @Hatta:2016khv] in the dilute limit. To include saturation effect, the authors of Ref. [@Hatta:2016wjz] derived the Wilson line structure using some heuristic argument, which, however differs from that we directly derived in the hybrid approach for the Sivers type contribution. To be more explicit, the Wilson line structure we obtained can be cast into the combination $G_{\text {DP}}- N_c^2 G_4$ where $G_{\text {DP}}$ is the normal dipole type gluon distribution, and $G_4$ is the gluon distribution that arises from color entanglement effect. Quit dramatically, the relation $G_{\text {DP}}= N_c^2 G_4$ holds in a quasi-classical model indicates that the Sivers type contribution completely drops out. As explained below, the heuristic argument used in Ref. [@Hatta:2016khv] to work out the Wilson line structure in the fragmentation case is well justified. It is shown [@Hatta:2016khv] that the contribution from the twist-3 fragmentation function related to the moment of the TMD Collins function [@Collins:1992kk; @Yuan:2009dw] is strongly suppressed by saturation effect. In view of the recent measurement at RHIC [@Heppelmann:2016siw], the genuine twist-3 fragmentation function turns out to be the only candidate for the main cause of $A_N$.
[*II. The computation of $A_N$ in the hybrid approach*]{}— We start the computation of $A_N$ in the hybrid approach by introducing the relevant kinematics. The dominant partonic channel for the spin independent forward particle production is, $$q_p(xP)+ g_A(x_g' \bar P+k_\perp) \to q(l_q)$$ which represents a quark $q_p$ from proton scattering off classical background gluon field $g_A$ inside the target. The light cone momenta are defined as $\bar P^\mu=\bar P^- n^\mu$ and $ P^\mu=
P^+ p^\mu$ with the usual light cone vectors $n^\mu$ and $p^\mu$, normalized according to $p \cdot
n=1$.
To generate an imaginary phase necessary for the nonvanishing spin asymmetry, one additional gluon attachment from the remanent of the polarized proton projectile must be taken into account. It is convenient to formulate such twist-3 calculation in the covariant gauge in which this extra gluon is longitudinally polarized. One then has to sum the multiple re-scattering of the incoming quark and the collinear gluon with small $x$ gluon field inside target to all orders simultaneously.
The incoming quark and gluon with physical polarization scattering off CGC state can be summed into a Wilson line in the fundamental and adjoint representation, respectively, $$\begin{aligned}
U(x_\perp)&=& {\cal P} {\rm exp} \left [ ig \int_{-\infty}^{+\infty} dz^+ A_A^-(z^+,x_\perp) \cdot t \right ]
\\
\tilde U(x_\perp)&=&{\cal P} {\rm exp} \left [ ig \int_{-\infty}^{+\infty} dz^+ A_A^-(z^+,x_\perp)
\cdot T \right ]\end{aligned}$$ with $T$ and $t$ being the generators in the adjoint and fundamental representation. However, the multiple scattering of a longitudinally polarized gluon with the background gluon field of target can not be simply described by a Wilson line in the CGC formalism. Instead, the expression for the gauge field created through the fusion of longitudinally polarized gluon from the proton and small $x$ gluons from the target takes a quite complicate form [@Blaizot:2004wu]. It contains both singular terms(proportional to $\delta(z^+)$) and the regular terms: $ A^\mu=A^\mu_{reg}+\delta^{\mu-} A^-_{sing} $, whose explicit expressions can be found in Refs. [@Blaizot:2004wu].
![ The contribution from the regular terms to the spin dependent amplitude. A black dot denotes a classical field $A_{reg}$ insertion.[]{data-label="1"}](fig1.eps){width="9"}
![ The contribution from the singular terms to the spin dependent amplitude. A black dot denotes a classical field $A_{sing}$ insertion. []{data-label="1"}](fig2.eps){width="9"}
When computing the spin dependent amplitude, all possible insertions of the fields $A^\mu_{reg}$ and $A^-_{sing} $ on the quark line must be taken into account as illustrated in Fig.1 and Fig.2 respectively. We calculate the contributions from Fig.1 and Fig.2 following the method outlined in Refs. [@Schafer:2014zea; @Zhou:2015ima]. Note that Fig.1(c) and Fig.1(d) don’t contribution to the amplitude because two poles are lying on the same half plane. The final expression for the spin dependent amplitude takes form, $$\begin{aligned}
{\cal M}\!\!&=&\!\!\!- g \!\! \int \!\! \frac{d k^-_1 d^2 k_{1\perp} d^2 x_\perp d^2 x_{1\perp}
}{(2\pi)^3}
e^{ix_\perp \! \cdot (k_\perp\! -k_{1\perp}\! -p_\perp)}e^{ix_{1\perp} \! \cdot k_{1\perp}}
\nonumber \\
&& \! \!\! \!\! \! \! \!\! \!\! \! \times \bar u(l_q)\!
\frac{C_U \!\!\!\!\!\!\!/ \ \ (q,p_\perp)}{q^2+i\epsilon} t^b S_{\!F}(l_q\!-\!q) n\!\!\!/U\!(x_{\perp})
u(xP) \!\! \left [ \tilde U\!(x_{1\perp})\!-\!1 \right ]_{ba}
\nonumber \\
&& \!\! \!\! \!\! \!\! \!\!\!\! \!\! \!+ g \!\! \int \!\! d^2 x_\perp e^{i(k_\perp\!-p_\perp)
\cdot x_\perp} \! \frac{\bar u(l_q) n \!\!\!/ t^b U\!( x_\perp) u(xP)} {x_gP^++i\epsilon}\! \!
\left [ \tilde U\!(x_\perp)\! -\!1 \right ]_{ba} \nonumber \\ && \!\! \!\! \!\! \!\! \!\!\!\!
\!\!\! + ig \!\! \int \!\! d^2 x_\perp e^{i(k_\perp\! -p_\perp) \cdot x_\perp} \! \bar u(l_q) t^a
p\!\!\!/ S_F(l_q\!\!-\!x_gP\!-\!p_\perp) n\!\!\!/ u(xP)
\nonumber \\ && \! \!\! \!\! \! \! \! \times
\left [ U(x_{\perp})-1 \right ]
\label{pamp}\end{aligned}$$ where the color index $a$ is associated with the collinear gluon from the polarized projectile which carries momentum $x_g P+p_\perp$. $S_F(l_q-q)$ and $S_F(l_q-x_gP-p_\perp)$ denote the standard quark propagators. The four vector $C_U^\mu(q,p_\perp)$ is defined as, $$\begin{aligned}
&& C^+_U(q,p_\perp)=-\frac{{ p}_\perp^2}{q^-+i\epsilon}, \ \
C_U^-(q,p_\perp)=\frac{{k}_{1\perp}^2-{q}_\perp^2}{q^++i\epsilon},
\\ \nonumber
&&
C^i_U(q,p_\perp)=-2{\bold p}_\perp^i \label{twist2}\end{aligned}$$ where $q^\mu=x_{g1}'\bar P^\mu+k_{1\perp}^\mu+x_g P^\mu+p_\perp^\mu$. The notation ${\bold
p}_{\perp}$ is used to denote four dimension vector with $p_{\perp}^2=-{\bold p}_\perp^2$.
It is worthy to point out that the second term in Eq. \[pamp\] which describes the interaction between the collinear gluon from the projectile and color source inside the target is missing in Ref. [@Hatta:2016wjz], and the Wilson line structure in the rest two terms are also organized in different ways as compared to that in Ref. [@Hatta:2016wjz]. Before computing the twist-3 piece, as a consistency check, let us first have a look at the twist-2 part of the derived amplitude by setting $p_\perp=0$. The first term vanishes due to $C_U^\mu(q,p_\perp=0)=0$. The leading twist contribution of the amplitude is simplified as, $$\begin{aligned}
{\cal M}_{\text{twist-2}} \!\! &=& \!\! \frac{g}{P^+} \!\! \int \!\! d^2 x_\perp e^{ik_\perp
\cdot x_\perp} \nonumber
\\ && \!\! \!\! \!\! \!\! \!\!\!\! \!\!\! \left \{ \left[ {\cal P}\frac{1}{x_g}+i \pi \delta(x_g) \right ]
\left [ U(x_{\perp})-1 \right ] \bar u(l_q) n \!\!\!/ t^a u(xP) \right .\
\nonumber
\\ && \!\! \!\! \!\! \!\! \!\!\!\! \left .\ -i \pi \delta(x_g)
\bar u(l_q) n \!\!\!/ t^a u(xP) 2 \left [ U(x_{\perp})-1 \right ] \right \}
\label{twist2}\end{aligned}$$ In arriving at the above expression, we used the algebraic identity, $U(x_{\perp})t^b
U^\dag(x_{\perp})\!=\! t^a \tilde U_{ba}(x_\perp) $. After integrating out the incoming quark transverse momentum, the contributions proportional to the delta function $\delta(x_g)$ are canceled out between the different cut diagrams. The additional gluon exchange from proton can be incorporated into the gauge link appears in the matrix element definition of quark PDF by carrying out the $x_g$ integration over the principal value part. As expected, the corresponding hard part is just the Born diagram contribution to a quark scattering off CGC state [@Dumitru:2002qt]. At this point, one can readily see that it is critical to keep the scattering amplitude gauge invariant by taking into account the initial interaction with the color source inside target. Note that the result derived in Ref. [@Hatta:2016wjz] fails to pass this consistency check.
If one applies TMD factorization on the polarized projectile side, the terms proportional to the delta function contributes to the gauge link in the Sivers TMD function. But unlike photon-jet production [@Schafer:2014xpa], such hybrid approach might not be well justified in the process under consideration because of the lack of an additional hard scale.
We now proceed to compute the spin dependent twist-3 contribution by first isolating imaginary part from different poles. We start with analyzing the pole structure in the first term in Eq. \[pamp\]. By carrying out $x_g$ and $k_1^-$ integration, two propagators are effectively put on shell, $$\begin{aligned}
q^2=0 \ , \ \ (l_q-q)^2=0\end{aligned}$$ Three particle lines connected by a quark-gluon vertex being simultaneously on shell implies that three momenta $q^\mu, l_q^\mu-q^\mu, l_q^\mu$ must be collinear to each other. This leads to, $$\begin{aligned}
\bar u(l_q) C_U \!\!\!\!\!\!\!/ \ \ (q,p_\perp) (l_q \!\!\!\!\!/-q\!\!\!/)=-\bar u(l_q) l_q
\!\!\!\!\!/ \ \ C_U \!\!\!\!\!\!\!/ \ \ (q,p_\perp) (1-\beta)=0\end{aligned}$$ where $q^\mu=\beta l_q^\mu$ for $0\leq\beta \leq1$. When commuting $C_U \!\!\!\!\!\!\!/$ with $l_q \!\!\!\!\!/-q\!\!\!/$ in the above formula, we used the property $C_U^\mu(q,p_\perp) \cdot
q_\mu=0$. One thus concludes that the hard gluon pole(or the soft fermion pole for $\beta=1$) contribution is completely washed out by saturation effect. This analysis is in agreement with that made in Ref. [@Hatta:2016wjz].
One should notice that the first term in Eq. \[pamp\] also contains the soft gluon pole(SGP) contribution which comes from the minus component of $C_U^\mu$. Combining it with the last two terms in Eq. \[pamp\], the SGP contribution is given by, $$\begin{aligned}
{\cal M}_{\text {SGP}}\!\!&=&\!\!\!-i\pi g \!\! \int \!\! \frac{ d^2 k_{1\perp} d^2 x_\perp d^2
x_{1\perp} }{(2\pi)^2}
e^{ix_\perp \! \cdot (k_\perp\! -k_{1\perp}\! -p_\perp)}e^{ix_{1\perp} \! \cdot k_{1\perp}}
\nonumber \\
&&\times \delta(x_g\!P^+) \frac{k_{1\perp}^2}{q_\perp^2} \ \bar u(l_q) n \!\!\!/ t^b U\!( x_\perp)
u(xP) \tilde U\!(x_{1\perp})_{ba}\!\!
\nonumber \\ && \!\! \!\! \!\!\!
+ i\pi g \!\! \int \!\! d^2 x_\perp e^{i(k_\perp\! -p_\perp) \cdot x_\perp}
\!\delta((l_q-x_gP-p_\perp)^2)
\nonumber \\ && \times
\bar u(l_q) t^a
p\!\!\!/ ( l_q\!\!\!\!\!/-\!x_g\!P\!\!\!\!/-\!p_\perp\!\!\!\!\!\!\!/ \ \ ) n\!\!\!/ u(xP) \left [
U(x_{\perp})-1 \right ]
\label{sfp}\end{aligned}$$ where the last term gives rise to the so-called derivative term contribution. At this point, we would like to mention that the spin dependent amplitude takes a slightly different form for the left cut diagrams due to the different $p_\perp$ flow. In the collinear twist-3 approach, the spin asymmetry arises from the interference between the imaginary part identified above and the conjugate Born scattering amplitude without an additional gluon attachment from the projectile. It is straightforward to compute the later in the CGC formalism [@Dumitru:2002qt]. Following the standard procedure, the next step is to make $p_\perp$ expansion, and factorize the soft part on the polarized proton side into the Qiu-Sterman function.
Finally, in order to express the spin dependent cross section in terms of the known gluon distributions, we simplify the relevant color structure, starting with the one associated with the delta function $\delta(x_g P^+)$, $$\begin{aligned}
&&\!\!\!\!\! \!\!\!\!\! {\rm Tr} \! \left [ t^a U^\dag(y_\perp) t^b U(x_\perp) \right ] \! \tilde
U(x_{1\perp})_{ba}\! = \! \frac{-1}{2N_c} {\rm Tr} \! \left [U^\dag(y_\perp) U(x_\perp) \right
]\nonumber \\ && \ \ \ \ \ \ \ \ \ \ \ \ +\frac{1}{2} {\rm Tr}\! \left[
U^\dag(y_\perp)U(x_{1\perp}) \right ] {\rm Tr} \left [ U^\dag(x_{1\perp})U(x_\perp) \right ]\end{aligned}$$ where $ U^\dag(y_\perp) $ is from the conjugate amplitude. Note that the forward scattering amplitude contribution has been neglected as we do so below. The contribution from $\left [
U^\dag(y_\perp) U(x_\perp) \right ]$ drops out because one can trivially carry out $x_{1\perp}$ integration, resulting in $k_{1\perp}=0$. In the large $N_c$ approximation, $\langle {\rm Tr}\!
\left[ U^\dag(y_\perp)U(x_{1\perp}) \right ] {\rm Tr} \left [ U^\dag(x_{1\perp})U(x_\perp) \right ]
\rangle$ can be related to the convolution of two dipole type gluon distributions. After summing the left and right cut diagrams contribution and making $p_\perp$ expansion, we encounter the following structure, $$\begin{aligned}
\int \!\! d^2 k_{1\perp} \!\! \left [
\frac{l_{q\perp}^\alpha-k_{1\perp}^\alpha}{(l_{q\perp}-k_{1\perp})^2}
F(l_{q\perp}^2)+ \frac{l_{q\perp}^\alpha}{2}
\frac{\partial F(l_{q\perp}^2)}{\partial l_{q\perp}^2} \right ] \!
F(k_{1\perp}^2)\end{aligned}$$ where $F(l_{q\perp}^2)$ is the Fourier transform of the dipole amplitude whose definition is given below. Using the method introduced in Ref. [@Hatta:2016khv], it is easy to verify that two terms are completely canceled out in the dilute limit, and are strongly suppressed in the saturation regime. One thus can safely neglect the SGP contribution induced by the initial state interaction.
We now turn to discuss the Wilson lines associated with the derivative term contribution, which reads, $$\begin{aligned}
{\rm Tr} \! \left [ t^a U^\dag(y_\perp) t^a U(x_\perp) \right ] \!\! &=& \!\!
\frac{1}{2} {\rm Tr}\! \left[ U^\dag(y_\perp) \right ] {\rm Tr}\! \left [U(x_{\perp}) \right ]
\nonumber \\ &-& \!\! \frac{1}{2N_c} {\rm Tr} \left [ U^\dag(y_{\perp})U(x_\perp) \right ]\end{aligned}$$ where the non-trivial color structure ${\rm Tr}\! \left[ U^\dag(y_\perp) \right ] {\rm Tr}\!
\left [U(x_{\perp}) \right ]$ arises from color entanglement effect as explained in Refs. [@Schafer:2014zea; @Schafer:2014xpa; @Zhou:2015ima]. The extra gluon attachment from the polarized proton plays a crucial role in yielding such unique structure. With all these calculation recipes, we derive the spin dependent partonic cross section, $$\begin{aligned}
&& \!\!\!\!\!\!\!\!\!\!\!\!
\frac{d \sigma}{dy d^2l_{q\perp}}=\frac{2\pi^2 \alpha_s x x_g'}{ N_c(N_c^2-1)}
\frac{\epsilon_{\alpha \beta} S_{\perp }^{\beta}l_{q\perp}^{ \alpha}}{l_{q\perp}^2}
\nonumber \\&& \!\!\!\! \times \! \left \{\! \frac{1}{l_{q\perp}^2} \left [ G_{\text{DP}}(x_g',l_{q\perp}^2)
\!-\!N_c^2 G_{4}(x_g',l_{q\perp}^2)\right ] x \frac{d T_F(x,x)}{d x} \right.\
\nonumber \\
&& \!\!\! \left .\ \ \ \ \ \ +\frac{\partial\left [ G_{\text {DP}}(x_g',l_{q\perp}^2)\!-\!N_c^2
G_{4}(x_g',l_{q\perp}^2)\right ]}{\partial l_{q\perp}^2 } T_F(x,x) \!\! \right \} \label{cs}\end{aligned}$$ where $S_\perp$ is the transverse spin vector of the proton. The momentum fractions $x$ and $x_g'$ are fixed according to $x=e^y|l_{q\perp}|/\sqrt{s} $ and $x_g'=e^{-y}|l_{q\perp}|/\sqrt{s} $ with $y$ being the outgoing quark rapidity. $G_{\text {DP}}$ is the normal dipole type gluon distribution, and related to the Fourier transform of the dipole amplitude $x_g'G_{\text
{DP}}(x_g',l_{q\perp}^2)=\frac{l_{q\perp}^2N_c}{2 \pi^2 \alpha_s} F(l_{q\perp}^2)$. $G_4$ introduced in Ref. [@Schafer:2014zea] is the gluon distribution that arises from color entanglement effect. Their operator definition are given by, $$\begin{aligned}
x_g' G_{\text {DP}}(x_g',l_{q\perp}^2)\!\! &= & \!\! \frac{l_{q\perp}^2N_c}{2 \pi^2 \alpha_s}
\int \! \frac{d^2 x_\perp d^2 y_\perp}{(2 \pi)^2 } e^{il_{q\perp} \! \cdot
(x_\perp-y_\perp)} \nonumber \\ && \times \frac{1}{N_c} \langle {\rm Tr} \left [
U^\dag(y_\perp)U(x_\perp) \right ]\rangle \nonumber
\\ x_g' G_{4}(x_g',l_{q\perp}^2)\!\!&= & \!\! \frac{l_{q\perp}^2N_c}{2 \pi^2 \alpha_s}
\int \! \frac{d^2 x_\perp d^2 y_\perp}{(2 \pi)^2 } e^{il_{q\perp} \! \cdot
(x_\perp-y_\perp)} \nonumber \\ && \times \frac{1}{N_c^2}\langle {\rm Tr} \left [
U^\dag(y_\perp)\right ] {\rm Tr} \left [ U(x_\perp) \right ]\rangle\end{aligned}$$ which can be evaluated and related to each other in the MV model [@Schafer:2014zea], $$\begin{aligned}
x_g' G_{4}(x_g',
l_{q\perp}^2)\! = \! \frac{1}{N_c^2}x_g'G_{\text {DP}}(x_g', l_{q\perp}^2) \label{mv}\end{aligned}$$ This simple relation leads to a complete cancelation between the contributions from $G_{\text
{DP}}$ and $G_4$ in Eq.\[cs\]. Therefore, the Sivers type contribution to $A_N$ drops out. Obviously, this conclusion remains true after promoting the partonic spin dependent cross section to the hardron production cross section.
We now comment on the twist-3 fragmentation function contribution to $A_N$. The derivative term contribution to $A_N$ in pp collisions was first computed in Ref. [@Yuan:2009dw] in the purely collinear twist-3 approach. The complete result was obtained in Ref. [@Metz:2012ct](see recent reviews Refs. [@Metz:2016swz; @Pitonyak:2016hqh]). In order to take into account multiple gluon rescattering effect on target side, the similar hybrid approach also can be applied in the fragmentation case [@Hatta:2016khv]. As well known, it is highly nontrivial to compute the SGP contribution in the light cone gauge [@Zhou:2009jm; @Zhou:2010ui]. Since the SGP contribution vanishes for the twist-3 fragmentation contribution [@Meissner:2008yf; @Gamberg:2010uw], it is more convenient to carry out the calculation in the light cone gauge where the additional gluon exchange from the twist-3 fragmentation function is physically polarized [@Hatta:2016khv]. A gluon with physical polarization scattering off the background gluon field can be summarized into a normal Wilson line in the adjoint representation. In this sense, the derivation of the Wilson line structure in Ref. [@Hatta:2016khv] is well justified. If one formulates such calculation in the covariant gauge, the fact that an imaginary phase from the scattering amplitude is not required in the twist-3 fragmentation case would make an essential difference in deriving the color structure. However, the detailed investigation is beyond the scope of the current work.
We close this section with few further remarks:\
(1) Following the standard procedure, one can derive the BK type evolution equation for the gluon distribution $G_4$, which will be presented in a separate publication. In the large $N_c$ limit, the relation Eq.\[mv\] holds under small $x$ evolution.\
(2) The relation Eq.\[mv\] is a model dependent result. In general case, an incomplete cancelation between two gluon distributions leaves some room for having tiny spin asymmetry for inclusive jet production in pp or pA collisions [@Nogach:2012sh].\
(3) If the $G_4$ contribution is neglected, Eq.\[cs\] is consistent with the collinear twist-3 result [@Kouvaris:2006zy] in the dilute limit.\
(4) Color entanglement effect is a leading power effect and should be taken into account in the genuine collinear twist-3 approach as well. We plan to redo calculation in the purely collinear framework by going beyond one gluon exchange approximation on target side.\
(5) T-even objects like the unpolarized twist-2 amplitude, are not affected by color entanglement effect. The observed color entanglement effect is the consequence of the non-trivial interplay among T-odd effect, multiple gluon re-scattering, and the non-Abelin feature of QCD [@Rogers:2010dm; @Schafer:2014zea; @Schafer:2014xpa; @Zhou:2015ima].
[*III. Summary.*]{}— Let us now summarize the recent progress on the topic addressed in this Letter. The sign mismatch problem was first observed in Ref. [@Kang:2011hk]. To find a way out, one naturally questions the dominance of the Sivers type contribution to $A_N$. It was indeed found that the genuine twist-3 fragmentation function could play an important role in generating the spin asymmetry [@Kanazawa:2014dca]. Later, the authors of Ref. [@Hatta:2016khv] have sorted out the piece of the contribution from the twist-3 fragmentation functions that is not suppressed by saturation effect using a hybrid approach first developed in Refs. [@Schafer:2014zea; @Zhou:2015ima]. The saturation suppressed fragmentation contribution being the major source of $A_N$ has been ruled out by the recent measurement [@Heppelmann:2016siw]. In this work, we demonstrate that the Sivers type contribution to the spin asymmetry drops out due to color entanglement effect. The nuclear independent part of the genuine twist-3 fragmentation contribution turns out to be the only candidate for the main cause of $A_N$. A recent work [@Gamberg:2017gle] shows that it is almost sufficient to account for $A_N$ by taking into account this fragmentation term alone with the input constrained by the Lorentz invariance relation [@Kanazawa:2015ajw]. We thus believe that the sign mismatch problem has been solved.
[***Acknowledgments:***]{} This work has been supported by the National Science Foundation of China under Grant No. 11675093, and by the Thousand Talents Plan for Young Professionals.
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author:
- |
Dmitrij V. Soroka and Vyacheslav A. Soroka\
Kharkov Institute of Physics and Technology, 61108 Kharkov, Ukraine\
E-mail: ,
title: Exterior Differentials in Superspace and Poisson Brackets
---
Introduction: two definitions of an exterior differential in superspace
=======================================================================
There exist two possibilities to define an exterior differential in superspace with coordinates $z^a$ having Grassmann parities $g(z^a)\equiv g_a$ and satisfying the permutation relations z\^az\^b=(-1)\^[g\_ag\_b]{}z\^bz\^a.The first one realized when we set the Grassmann parity of the exterior differential $d_{\sst0}=d_{\sst0}z^a\partial_{z^a}$ to be equal to zero g(d\_[0]{}z\^a)=g\_a,where $\partial_{z^a}\equiv\partial/\partial z^a$. In this case the symmetry property of an exterior product of two differentials is \[1\] d\_[0]{}z\^ad\_[0]{}z\^b= (-1)\^[g\_ag\_b+1]{}d\_[0]{}z\^bd\_[0]{}z\^a and a permutation relation of the exterior differential $d_{\sst0}z^a$ with the coordinate $z^b$ has the form d\_[0]{}z\^az\^b=(-1)\^[g\_ag\_b]{}z\^bd\_[0]{}z\^a. Note that relation (\[1\]) can be rewritten in the following form \[2\] (-1)\^[g\_a]{}d\_[0]{}z\^ad\_[0]{}z\^b= (-1)\^[(g\_a+1)(g\_b+1)]{}(-1)\^[g\_b]{}d\_[0]{}z\^bd\_[0]{}z\^a. By defining an exterior product of a differential $p$-form \^0=d\_[0]{}z\^[a\_p]{}d\_[0]{}z\^[a\_1]{} \^0\_[[a\_1]{}…[a\_p]{}]{},g(\^0\_[[a\_1]{}…[a\_p]{}]{})=g\_[a\_1]{}++g\_[a\_p]{}and a $q$-form \^0=d\_[0]{}z\^[b\_q]{}d\_[0]{}z\^[b\_1]{} \^0\_[[b\_1]{}…[b\_q]{}]{},g(\^0\_[[b\_1]{}…[b\_q]{}]{})=g\_[b\_1]{}++g\_[b\_q]{}as \^0\^0=(-1)\^[pq]{} d\_[0]{}z\^[b\_q]{}d\_[0]{}z\^[b\_1]{}d\_[0]{}z\^[a\_p]{}d\_[0]{}z\^[a\_1]{} \^0\_[[a\_1]{}…[a\_p]{}]{}\^0\_[[b\_1]{}…[b\_q]{}]{},we obtain the following symmetry property of this product \^0\^0=(-1)\^[pq]{}\^0\^0.By setting the exterior differential of the $p$-form $\Phi^0$ as follows \[3\] d\_[0]{}\^0= d\_[0]{}d\_[0]{}z\^[a\_p]{}d\_[0]{}z\^[a\_1]{} \^0\_[[a\_1]{}…[a\_p]{}]{}= (-1)\^pd\_[0]{}z\^[a\_p]{}d\_[0]{}z\^[a\_1]{}d\_[0]{}z\^b\_[z\^b]{}\^0\_[[a\_1]{}…[a\_p]{}]{}, we obtain the Leibnitz rule for the exterior differential of the exterior product of two forms d\_[0]{}(\^0\^0)=(d\_[0]{}\^0)\^0+ (-1)\^p\^0(d\_[0]{}\^0).Note that very often another definition for the exterior differential is adopted = d\_[0]{}z\^[a\_1]{}d\_[0]{}z\^[a\_p]{}d\_[0]{}z\^b\_[z\^b]{}\^0\_[[a\_p]{}…[a\_1]{}]{}which differs from (\[3\]) with the absence of the grading factor $(-1)^p$ and leads to the following form of the Leibnitz rule = (-1)\^q()\^0+ \^0().
Another definition for the exterior differential in superspace arises when the Grassmann parity of the exterior differential $d_{\sst1}=
d_{\sst1}z^a\partial_{z^a}$ is chosen to be equal to unit g(d\_[1]{}z\^a)=g\_a+1.Then the symmetry property of the exterior product for two differentials is defined as[^1] \[4\] d\_[1]{}z\^ad\_[1]{}z\^b= (-1)\^[(g\_a+1)(g\_b+1)]{}d\_[1]{}z\^bd\_[1]{}z\^a and a rule for the permutation of such a differential with the coordinate $z^b$ has to be d\_[1]{}z\^az\^b=(-1)\^[(g\_a+1)g\_b]{}z\^bd\_[1]{}z\^a. Relation (\[4\]) can be represented in the form \[5\] (-1)\^[g\_a]{}d\_[1]{}z\^ad\_[1]{}z\^b= (-1)\^[g\_ag\_b+1]{}(-1)\^[g\_b]{}d\_[1]{}z\^bd\_[1]{}z\^a. If the exterior product of the $p$-form \^1=d\_[1]{}z\^[a\_p]{}d\_[1]{}z\^[a\_1]{} \^1\_[[a\_1]{}…[a\_p]{}]{},g(\^1\_[[a\_1]{}…[a\_p]{}]{})=g\_[a\_1]{}++g\_[a\_p]{}and $q$-form \^1=d\_[1]{}z\^[b\_q]{}d\_[1]{}z\^[b\_1]{} \^1\_[[b\_1]{}…[b\_q]{}]{},g(\^1\_[[b\_1]{}…[b\_q]{}]{})=g\_[b\_1]{}++g\_[b\_q]{}is defined in the following way \^1\^1=(-1)\^[p(q+g\_[b\_1]{}++g\_[b\_q]{})]{} d\_[1]{}z\^[b\_q]{}d\_[1]{}z\^[b\_1]{}d\_[1]{}z\^[a\_p]{}d\_[1]{}z\^[a\_1]{} \^1\_[[a\_1]{}…[a\_p]{}]{}\^1\_[[b\_1]{}…[b\_q]{}]{},then the symmetry property of this product is \^1\^1=(-1)\^[pq]{}\^1\^1.Let us define in this case the exterior differential of the $p$-form $\Phi^1$ in the following form d\_[1]{}\^1= d\_[1]{}d\_[1]{}z\^[a\_p]{}d\_[1]{}z\^[a\_1]{}\^1\_[[a\_1]{}…[a\_p]{}]{}= (-1)\^[p+g\_[a\_1]{}++g\_[a\_p]{}]{} d\_[1]{}z\^[a\_p]{}d\_[1]{}z\^[a\_1]{}d\_[1]{}z\^b\_[z\^b]{}\^1\_[[a\_1]{}…[a\_p]{}]{}.Then the Leibnitz rule for the exterior differential of the exterior product of a $p$-form $\Phi^1$ and a $q$-form $\Psi^1$ will be d\_[1]{}(\^1\^1)=(d\_[1]{}\^1)\^1+ (-1)\^p\^1(d\_[1]{}\^1).Note that in this case the symmetry properties of the exterior product (\[4\]) coincide with the ones for the usual Grassmann product of two differentials for the coordinate $z^a$ \[6\] d\_[1]{}z\^ad\_[1]{}z\^b= (-1)\^[(g\_a+1)(g\_b+1)]{}d\_[1]{}z\^bd\_[1]{}z\^a.
The equivalence of the exterior calculi obtained with the use of the above mentioned different definitions for the exterior differential can be established as a result of the direct verification by taking into account relations (\[1\]) and (\[5\]) and by putting the following relations between coefficients of the corresponding $p$-forms $\Phi^0$ and $\Phi^1$ \^0\_[[a\_1]{}…[a\_p]{}]{}=(-1)\^[\_[k=1]{}\^[\[p/2\]]{} g\_[a\_[2k]{}]{}]{} \^1\_[[a\_1]{}…[a\_p]{}]{},where $[p/2]$ denotes a whole part of the quantity $p/2$. Thus, we proved that two definitions for the exterior differential, differed with the Grassmann parities, result in the same exterior calculus.
Poisson brackets related with the exterior differentials
========================================================
Now we show that application of these differentials leads, however, to the different results under construction from a given Poisson bracket with a Grassmann parity $\e=0,1$ of another one.
A Poisson bracket, having a Grassmann parity $\e$, written in arbitrary non-canonical variables $z^a$ \[in\] {A,B}\_=A\_[z\^a]{}ø\_\^[ab]{}(z)\_[z\^b]{}B has the following main properties: g({A,B}\_)g\_A+g\_B+2,{A,B}\_=-(-1)\^[(g\_A+)(g\_B+)]{} {B,A}\_,\_[(ABC)]{}(-1)\^[(g\_A+)(g\_C+)]{} {A,{B,C}\_}\_=0,which lead to the corresponding relations for the matrix $\o_\e^{ab}$ g(ø\_\^[ab]{})g\_a+g\_b+2,\[8\] ø\_\^[ab]{}=-(-1)\^[(g\_a+)(g\_b+)]{} ø\_\^[ba]{},\[9\] \_[(abc)]{}(-1)\^[(g\_a+)(g\_c+)]{} ø\_\^[ad]{}\_[z\^d]{} ø\_\^[bc]{}=0,\[10\] where $g_A\equiv g(A)$ and a sum with a symbol $(abc)$ under it designates a summation over cyclic permutations of $a, b$ and $c$.
The Hamilton equations for the phase variables $z^a$, which correspond to a Hamiltonian $H_\e$ ($g(H_\e)=\e$), ={z\^a,H\_}\_= ø\_\^[ab]{}\_[z\^b]{}H\_\[11\] can be represented in the form =ø\_\^[ab]{}\_[z\^b]{}H\_ø\_\^[ab]{}[(d\_H\_)(d\_z\^b)]{} (z\^a,d\_H\_)\_[+]{},where $d_{\sst\z}$ ($\z=0,1$) is one of the exterior differentials $d_{\sst0}$ or $d_{\sst1}$. By taking the exterior differential $d_{\sst\z}$ from the Hamilton equations (\[11\]), we obtain =(d\_ø\_\^[ab]{}) [(d\_H\_)(d\_z\^b)]{} +(-1)\^[(g\_a+)]{}ø\_\^[ab]{}\_[z\^b]{}(d\_H\_) (d\_z\^a,d\_H\_)\_[+]{}.As a result of two last equations we have by definition the following binary composition for functions $F$ and $G$ of the variables $z^a$ and their differentials $d_{\sst\z}z^a\equiv y_{\sst\z}^a$ \[fin\] (F,G)\_[+]{}= FG. In consequence of the grading properties (\[8\]) for the matrix $\o_\e^{ab}$ this composition has the Grassmann parity $\e+\z$ g\[(F,G)\_[+]{}\]g\_F+g\_G++2.By using the symmetry property (\[9\]) of $\o_\e^{ab}$ , we can establish the symmetry of the composition (\[fin\]) (F,G)\_[+]{}=-(-1)\^[(g\_F++)(g\_G++)]{} (G,F)\_[+]{}.At last, taking into account relations (\[9\]) and (\[10\]) for the matrix $\o_\e^{ab}$, we come to the Jacobi identities for this composition \_[(EFG)]{}(-1)\^[(g\_E++)(g\_G++)]{} (E,(F,G)\_[+ ]{})\_[+]{}=0.
We see that the composition (\[fin\]) satisfies all the main properties for the Poisson bracket with the Grassmann parity equal to $\e+\z$. Thus, the application of the exterior differentials of opposite Grassamann parities to the given Poisson bracket results in the brackets of the different Grassmann parities.
Note that by transition to the variables $y^{\sst\e+\sst\z}_a$, related with $y_{\sst\z}^a$ by means of the matrix $\o_\e^{ab}$ \[13\] y\_\^a=y\^[+]{}\_bø\_\^[ba]{}, the Poisson bracket (\[fin\]) takes a canonical form[^2] (F,G)\_[+]{}=FGthat can be proved with the use of the Jacobi identities (\[10\]).
In the case $\z=1$, due to relations (\[4\]), (\[6\]), the terms in the decomposition of a function $F(z^a,y_{\sst1}^a)$ into degrees $p$ of the variables $y_{\sst1}^a$ can be treated as $p$-forms and the bracket (\[fin\]) can be considered as a Poisson bracket on $p$-forms so that being taken between a $p$-form and a $q$-form results in a $(p+q-1)$-form[^3]. The bracket (\[fin\]) is a generalization of the bracket introduced in [@kar; @karmas] on the superspace case and on the case of the brackets (\[in\]) with arbitrary Grassmann parities.
Let us also note that if we take the bracket (\[fin\]) in the component form and rise low indexes with the use of the matrix $\o_\e^{ab}$ according to the rule (\[13\]) then we come to the generalizations of the Schouten-Nijenhuis brackets [@scho; @nij] (see also [@karmas; @nij1; @fr-nij; @kod-sp; @but; @bffls; @oz]) onto the cases of superspace and the brackets of diverse Grassmann parities. The details of this generalization will be given in a separate paper [@prep].
It follows from the structure of the bracket (\[fin\]) that if the initial bracket (\[in\]) is degenerate and possessed of a Casimir function $C(z)$ {…,C}\_=0,then the bracket (\[fin\]) has as Casimir functions this one (…,C)\_[+]{}=0and also a function of the form \[15\] C=y\_\^a\_[z\^a]{}C,(…,C)\_[+]{}=0.
Linear Poisson brackets related with semi-simple Lie groups
===========================================================
Here we apply the procedure described in the previous section to the linear even and odd brackets connected with a semi-simple Lie group having structure constants ${c_{\a\b}}^\g$ which obey the usual conditions \^=-[c\_]{}\^,\_[()]{}[c\_]{}\^ł[c\_[ł]{}]{}\^=0.
Let us take as an initial Poisson bracket (\[in\]) the linear even bracket given in terms of the commuting variables [^4] $x_\a$ (here $z^a=x_\a$) {A,B}\_0=A\_[x\_]{}[c\_]{}\^x\_\_[x\_]{}B.\[even\] In the case of a semi-simple Lie group, which hereafter will be considered, this bracket has a Casimir function C\_0=x\_x\_g\^,{…,C\_0}\_0=0,\[C0\] where $g^{\a\b}$ is an inverse tensor to the Cartan–Killing metric g\_=[c\_]{}\^ł[c\_[ł]{}]{}\^.By using the odd exterior differential $d_{\sst1}$, we obtain from the bracket (\[even\]) in conformity with the transition from the bracket (\[in\]) to the bracket (\[fin\]) the following linear odd bracket (F,G)\_1=F(\_[x\_]{}[c\_]{}\^x\_\_[\_]{}+ \_[\_]{}[c\_]{}\^x\_\_[x\_]{}+ \_[\_]{}[c\_]{}\^\_\_[\_]{})G,\[odd\] where $\t_\a=d_{\sst1}x_\a$ are Grassmann variables. In this case relation (\[13\]) has the form \_=\^\^x\_,where $\T^\b$ are also Grassmann variables in term of which the odd bracket (\[odd\]) takes a canonical form (F,G)\_1=F(\_[x\_]{}\_[\^]{}-\_[\^]{}\_[x\_]{})G.According to (\[15\]) the bracket (\[odd\]) has as Casimir functions apart from $C_0$ (\[C0\]) a nilpotent quantity C\_1=x\_\_g\^,(…,C\_1)\_1=0, (C\_1)\^2=0.
The odd bracket (\[odd\]) has two nilpotent Batalin-Vilkovisky type differential second order $\D$-operators [@bv1; @bv2] (see also [@schw; @schw1; @kn]) \_[-1]{}=-[12]{}\[\_[x\_]{}(x\_, )\_1+ \_[\_]{}(\_, )\_1\]=-[12]{}S\_\_[\_]{}, (\_[-1]{})\^2=0\[-1\] and =-[12]{}\[\_[x\_]{}(x\_, )\_1- \_[\_]{}(\_, )\_1\]= (T\_+[12]{}S\_)\_[\_]{},\^2=0,\[D\] where T\_=[c\_]{}\^x\_\_[x\_]{}\[T\] and S\_=[c\_]{}\^\_\_[\_]{}\[S\] are generators of the Lie group in the co-adjoint representation which obey the commutation relations =[c\_]{}\^T\_,=[c\_]{}\^S\_,=0.Note that (\_, )\_1=T\_+S\_Z\_and $Z_\a$ satisfy the relations =[c\_]{}\^Z\_.\[Z\]
Now let us take as an initial bracket (\[in\]) the linear odd bracket introduced in Refs. [@s; @ss] and given in terms of Grassmann variables $\t_\a$ (in this case $z^a=\t_\a$) {A,B}\_1=A\_[\_]{}[c\_]{}\^\_\_[\_]{}B,\[odd1\] which has in the case of the semi-simple Lie group a nilpotent Casimir function C\_1=\^\^\^c\_,{…,C\_1}\_1=0, (C\_1)\^2=0,\[C1\] where $\t^\a=g^{\a\b}\t_\b$ and $c_{\a\b\g}={c_{\a\b}}^\l g_{\l\g}$. With the help of the odd differential $d_{\sst1}$, according to the transition from (\[in\]) to (\[fin\]), we come from the bracket (\[odd1\]) to the even linear bracket of the form (F,G)\_0=F(\_[\_]{}[c\_]{}\^\_\_[x\_]{}+ \_[x\_]{}[c\_]{}\^\_\_[\_]{}+ \_[x\_]{}[c\_]{}\^x\_\_[x\_]{})G,\[even1\] where $x_\a=d_{\sst1}\t_\a$ are commuting variables. Relation (\[13\]) in this case has the form x\_=\^\^\_,where $\T^\b$ are Grassmann variables in term of which the bracket (\[even1\]) takes a canonical form for the Martin bracket [@mar] (F,G)\_0=F(\_[\_]{}\_[\^]{}+\_[\^]{}\_[\_]{})G.In accordance with (\[15\]) the even bracket (\[even1\]) has as Casimir functions besides $C_1$ (\[C1\]) the function C\_0=x\^\^\^c\_,(…,C\_0)\_0=0,where $x^\a=g^{\a\b}x_\b$.
The even bracket (\[even1\]), in contrast to the odd bracket, has no second order nilpotent differential $\D$-like operators. It is surprising enough that instead of this it has two [*nilpotent*]{} differential operators of the first order \_1=-[12]{}\[\^(x\_, )\_0+ x\^(\_, )\_0\]= -[12]{}\^S\_,(\_1)\^2=0. and Q=[12]{}\[\^(x\_, )\_0- x\^(\_, )\_0\]= \^(T\_+[12]{}S\_),Q\^2=0.\[Q\] In the papers [@s; @ss] the operators $\D_1$ and $\D_{-1}$, defined on the Grassmann algebra with generators $\t_\a$, have been introduced in connection with the linear odd Poisson bracket (\[odd1\]), which corresponds to a semi-simple Lie group, and the Lie superalgebra for them has been given. These operators are standard terms in the BRST and anti-BRST charges respectively.
Lie superalgebra for the BRST and anti-BRST charges
===================================================
Thus, in the superspace with coordinates $x_\a, \t_\a$ with the help of the linear even (\[even1\]) and odd (\[odd\]) Poisson brackets we constructed the operators $Q$ (\[Q\]) and $\D$ (\[D\]). These operators can be treated as the BRST and anti-BRST charges correspondingly (see, e.g., [@h]) if we consider $\t^\a$ and $\partial_{\t^\a}$ as representations for the ghosts and antighosts operators respectively. The operators $Q$ and $\D$ satisfy the following anticommutation relation: {Q,}=[12]{}(T\^T\_+Z\^Z\_),\[Q,D\] two terms in the right-hand side of which, because of the commutation relations =0,\[TT,Q\] =0,\[TT,D\] =0,\[Z,Q\] =0,\[Z,D\] =0,\[TT,Z\] are central elements of the Lie superalgebra formed by the quantities $Q$, $\D$, $T^\a T_\a$ and $Z^\a Z_\a$. The relations (\[Q,D\])–(\[TT,Z\]) remain valid if we take instead of the co-adjoint representation (\[T\]) an arbitrary representation for the generators $T_\a$. The quantity $Z^\a Z_\a$ contains the term $S^\a S_\a$ which can be written as S\^S\_=N-Kwhere N=\^\_[\^]{}, can be considered as a representation for the ghost number operator and the quantity $K$ has the form K = [12]{} \^\^\^łc\_[ł]{} \_[\_]{} \_[\_]{}.The operator $N$ has the following permutation relations with $Q$ and $\D$: =Q,\[N,Q\] =-\[N,D\] and commutes with the central elements $T^\a T_\a$ and $Z^\a Z_\a$ =0,\[N,T\] =0.\[N,Z\] We can add the commutation relations (\[Z\]) for the generators $Z_\a$ with the usual quadratic Casimir operator $Z^\a Z_\a$ for the semi-simple Lie group.
Note that the Lie superalgebra for the quantities $Q$, $\D$, $N$, $T^\a T_\a$ and $Z^\a Z_\a$ determined by the relations (\[Z\]), (\[Q,D\])–(\[N,Z\]) can be used for the calculation of the BRST operator cohomologies [@hv].
Conclusion
==========
Thus, we illustrated that in superspace the exterior differentials with opposite Grassmann parities give the same exterior calculus.
Then we introduced a prescription for the construction with the help of these differentials from a given Poisson bracket of the definite Grassmann parity of another one. We showed that the parity of the resulting bracket depends on the parities of the both initial bracket and exterior differential used.
It is also indicated that the resulting bracket is related with a generalization of the Schouten-Nijenhuis bracket on the superspace case and on the brackets of an arbitrary Grassmann parity.
By applying the prescription to the linear odd and even Poisson brackets, corresponding to a semi-simple Lie group with the structure constants ${c_{\a\b}}^\g$ and given respectively on the anticommuting $\t_\a$ and commuting $x_\a$ variables, we come with the help of the Grassmann-odd exterior differential to the correspondingly even and odd linear Poisson brackets which are both defined on the superspace with the coordinates $x_\a, \t_\a$ and related also with the same semi-simple Lie group.
We demonstrated that these resulting even and odd brackets are naturally connected with the BRST and anti-BRST charges respectively.
Acknowledgments {#acknowledgments .unnumbered}
===============
One of the authors (V.A.S.) is grateful to A.P. Isaev, J. Lukierski, M. Tonin and J. Wess for useful discussions and to S.J. Gates, Jr., P. Van Nieuwenhuizen, W. Siegel and B. Zumino for stimulating discussions and for hospitality respectively at the University of Maryland, SUNY (Stony Brook) and LBNL (Berkeley) where the parts of the work have been performed. V.A.S. thanks L. Bonora for fruitful discussions and for hospitality at SISSA (Trieste) where this work has been completed.
[99]{} S. Sternberg, *Lectures on differential geometry*, Prentice Hall, Inc. Englewood Cliffs, N.J. 1964. M.V. Karasev, *Proceeding of the Conference “Theory of group representations and its applications in physics”*, Tambov, 1989; Moscow, Nauka, 1990. M.V. Karasev and V.P. Maslov, *Non-linear Poisson brackets. Geometry and quantization*, Moscow, Nauka, 1991. J.A. Schouten, . A. Nijenhuis, . A. Nijenhuis, . A. Frohlicher and A. Nijenhuis, . K. Kodaira and D.C. Spencer, . C. Buttin, . F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, D. Sternheimer, *Deformation theory and quantization, 1. Deformations of symplectic structures*, . Z. Oziewicz, On Schouten-Nijenhuis and Frolicher-Nijenhuis Lie modules, *The lecture given at the XIX International Conference on Differential Geometric Methods in Theoretical Physics*, Rapallo (Genova) Italy, 1990. D.V. Soroka and V.A. Soroka, works in progress. I.A. Batalin and G.A. Vilkovisky, *Gauge algebra and quantization*, . I.A. Batalin, G.A. Vilkovisky, *Quantization of gauge theories with linearly dependent generators*, . A.S. Schwarz, *Geometry of Batalin-Vilkovisky quantization*, . A. Schwarz, *Semiclassical approximation in Batalin-Vilkovisky formalism*, . O.M. Khudaverdian, A.P. Nersessian, *On the geometry of Batalin-Vilkovisky formalism*, . V.A. Soroka, *Linear odd Poisson bracket on Grassmann variables*, . D.V. Soroka and V.A. Soroka, Lie-Poisson (Kirillov) odd bracket on Grassmann algebra, *Proceedings of the XXIII International Colloquium “Group Theoretical Methods in Physics”*, GROUP 23, July 31 – August 5, JINR, Dubna, Russia, 2000. J.L. Martin, . J.W. van Holten, *The BRST complex and the cohomology of compact Lie algebras*, . S.S. Horuzhy and A.V. Voronin, *A new approach to BRST operator cohomologies: exact results for the BRST-Fock theories*, .
[^1]: In this case we use another notation $\tilde\wedge$ for the exterior multiplication.
[^2]: There is no summation over $\e$ in relation (\[13\]).
[^3]: Concerning Poisson bracket between 1-forms and its relation with Lie bracket of vector fields see in the book [@stern].
[^4]: Lie-Poisson-Kirillov bracket.
|
---
abstract: 'Weak lensing provides a direct measure of the distribution of mass in the universe, and is therefore a uniquely powerful probe of dark matter. Weak lensing can also be used to measure the twin phenomenon of dark energy, via its effect upon the cosmological growth rate of structures. Essential for this technique are well-resolved images of background galaxies out to large distances. As a concrete example of the surveys that will become available by the end of the decade, we consider the planned [*Supernova/Acceleration Probe*]{} (SNAP) space telescope. Detailed simulations of space-based images, manufactured using the shapelets formalism, enable us to quantitatively predict the future sensitivity to weak lensing shear. The high number density of galaxies resolved from space will enable maps of dark matter to be produced in two and three dimensions, with a resolution superior to that from the ground. Such observations will also afford reduced systematics for high-precision measurements of weak lensing statistics. These will be used to set tight constraints on cosmological parameters. In particular, the parameter for equation of state of dark energy, $w$, will be measured using weak lensing with a precision comparable to and somewhat orthogonal to constraints from other methods.'
author:
- Richard Massey
- Alexandre Refregier
- Jason Rhodes
title: 'Probing Dark Matter and Dark Energy with Space-Based Weak Lensing'
---
\#1[[*\#1*]{}]{} \#1[[*\#1*]{}]{} =
\#1 1.25in .125in .25in
\[page:first\]
Introduction {#intro}
============
During its journey to our telescopes, the light from background galaxies is deflected by foreground mass concentrations, which act as gravitational lenses along the line-of-sight. The observed distortions in the shapes of distant galaxies are directly related to the gravitational potential of foreground structures, and therefore to their mass. This effect is independent of the nature or state of the foreground mass and therefore traces the distribution of otherwise invisible dark matter. In recent years, many groups have measured coherent distortions in the shapes of galaxies to measure the mass distribution of clusters ([*e.g.*]{} Joffre [*et al.*]{} 2000; Clowe, Trentham & Tonry 2001; Dahle [*et al.*]{} 2002). Other groups have measured the distribution of large-scale structure in random regions in the sky to set constraints on cosmological parameters ([*e.g.*]{} van Waerbeke [*et al.*]{} 2002; Bacon [*et al.*]{} 2002; Hoekstra [*et al.*]{} 2002; Jarvis [*et al.*]{} 2003). The second moment of the resulting “cosmic shear” field already provides a constraint on the amplitude of the mass power spectrum $\Omega_m \sigma_8^{0.5}$ which is comparable to the constraints set by more traditional methods such as the abundance of x-ray selected clusters ([*e.g.*]{} Viana & Liddle 1999; Pierpaoli, Scott & White 2001). The higher order moments of the shear field promise to break the $\Omega_m$-$\sigma_8$ degeneracy (Bernardeau et al. 1997) in the next generation of large cosmic shear surveys, [*e.g.*]{} CFHT Legacy Survey (CFHTLS, Mellier 2000), LSST (Tyson 2002a) and Pan-STARRS (Kaiser, Tonry & Luppino 2000). For reviews of the techniques and current status of weak lensing measurements, other than this volume, see Mellier et al. (2001), Hoekstra et al. (2002), Wittman (2002) and Refregier (2003).
The rapid growth of dedicated surveys and improvements in shear measurement methods has meant that errors on cosmological parameters from cosmic shear are now limited by systematic rather than statistical errors. Weak lensing has the advantage that its systematics arise from imperfect instruments and image analysis, rather than unknown physics (such as the mass-temperature relation which dominates $x$-ray selected cluster samples (Huterer & White, 2002)). Image analysis tools are currently making important advances, from the well-tested method of Kaiser, Squires & Broadhurst (1995; KSB), with work by Rhodes, Refregier & Groth (2000; RRG), Bernstein & Jarvis (2003) and “shapelets” by Refregier (2003a). Meeting the engineering challenge, however, will inevitably require rising above the atmosphere. Convolution with a large Point-Spread Function (PSF) that is larger than the majority of distant (and therefore small) objects irretrievably destroys any information their shapes had contained. This limits the number density and redshift of galaxies that can be used to measure cosmic shear with ground-based lensing surveys. Convolution with the time-varying atmospheric PSF further limits the recovery of the shape information of the remaining galaxies to the accuracy with which the PSF can be modelled from one exposure to the next.
In these proceedings, we discuss the potential advances that a wide-field imager from space offers for weak lensing. Adopting the planned [*Supernova/ Acceleration Probe*]{} (SNAP) mission as a concrete example, we present detailed shapelet-based simulations used to estimate the lensing sensitivity of such observations. Most interestingly for cosmology, SNAP’s enhanced spatial resolution will capture the shapes of background galaxies up to a much greater distance. Space-based surveys will therefore probe the evolution of the mass distribution, and the growth of structures, over a large fraction of time in the evolution of the universe. This evolution provides strong constraints on both $\Omega_m$ and the equation of state parameter of dark energy $w$. As discovered from observations of type Ia superovæ ([*e.g.*]{} Perlmutter 1999), this dark energy or “quintessence” is accelerating the expansion of the universe and hence retarding the growth of the mass power spectrum. Earlier studies of the constraints on dark energy from weak lensing surveys can be found in Hu (1999), Huterer (2001), Benabed & Bernardeau (2001), Hu (2001), Weinberg & Kamionkowski (2002), Munshi & Wang (2002), Benabed & van Waerbeke (2003). We also compare the quality of dark matter maps that will be possible from SNAP with those possible from current ground-based surveys. High resolution maps from space will enable the production of a fully mass-selected cluster catalogue, useful for a further constraint on cosmological parameters (Miyazaki 2002) but also for investigations into astrophysical effects during the non-linear infall and growth on smaller scales. Full details of our calculations are given in a series of papers by Rhodes (2004), Massey (2004b) and Refregier (2004b).
Principles of Weak Lensing
==========================
Gravitational lensing subjects the apparent images of background galaxies to a distortion that is characterized by the distortion matrix (see Figure 1 for an illustration, and Bartelmann & Schneider 1999 for a detailed review)
$$\label{eq:psi_def_theory}
\Psi_{ij} \equiv \frac{\partial (\delta\theta_{i})}{\partial \theta_{j}}
\equiv \left( \begin{array}{cc}
\kappa + \gamma_{1} & \gamma_{2}\\
\gamma_{2} & \kappa - \gamma_{1}\\
\end{array} \right) ~,$$
where $\delta \theta_{i}({\mathbf \theta})$ is the deflection vector produced by lensing on the sky. The convergence $\kappa$ describes overall dilations and contractions, and is proportional to the projected mass along the line of sight. The shear $\gamma_{1}$ ($\gamma_{2}$) describes stretches and compressions along (at $45^{\circ}$ from) the $x$-axis.
The distortion matrix is directly related to the matter density fluctuations along the line of sight by
$$\label{eq:lensing}
\Psi_{ij} = \int_{0}^{\chi_{h}} d\chi ~g(\chi)
\partial_{i}\partial_{j} \Phi~,$$
where $\Phi$ is the Newtonian potential, $\chi$ is the comoving distance, $\chi_{h}$ is the comoving distance to the horizon, and $\partial_{i}$ is the comoving derivative perpendicular to the line of sight. The radial weight function $g(\chi)$ reflects the fact that a lens is most effective when placed approximately half-way between the source and the observer. It is given by
$$g(\chi) = 2 \int_{\chi}^{\chi_{h}} d\chi'~n(\chi')
\frac{r(\chi)r(\chi'-\chi)}{r(\chi')} ~,$$
where $r$ is the comoving angular-diameter distance. The function $n(\chi)$ is the distribution of the galaxies as a function of the comoving distance $\chi$ from the observer and is assumed to be normalized as $\int
d\chi n(\chi)=1$.
Galaxies have an intrinsic distribution of shapes, but distant ones are randomly oriented in the absence of lensing. When a shear is applied to the galaxies in a particular patch of the sky, they become coherently distorted and their average shape changes from a circle to an ellipse. This observed ellipticity can be converted into the applied shear using a method like KSB, which properly takes into account their size distribution and radial profiles. Thus the cosmic shear field is an observable. By inverting the lensing equation (eqn. 2), the shear map can be converted into a map of the projected mass $\kappa$ and, therefore, of the dark matter distribution.
Cosmological models with Dark Energy
====================================
We consider a cosmological model with a matter component and a dark energy (or “quintessence”) component with present density parameters $\Omega_{m}$ and $\Omega_{q}$, respectively. The equation of state of the dark energy is parametrized by $w=p_{q}/\rho_{q}$, which we assume to be constant and is equal to $-1$ in the case of a cosmological constant. The evolution of the expansion parameter $a$ is given by the Hubble constant $H$ through the Friedmann equation
$$H=\frac{\dot{a}}{a}=H_{0}\left( \Omega_{m} a^{-3} + \Omega_{q}
a^{-3(1+w)} + \Omega_{\kappa} a^{-2} \right)^{\frac{1}{2}} ~,$$
where $\dot{a}=da/dt$ and the total and curvature density parameters are $\Omega$ and $\Omega_{\kappa}=1-\Omega$, respectively. The present value of the Hubble constant is parametrized as $H_{0}=100 h$ km s$^{-1}$ Mpc$^{-1}$.
As a reference model, we consider a fiducial $\Lambda$CDM cosmology with parameters $\Omega_{m}=0.30$, $\Omega_{b}=0.047$, $n=1$, $h=0.7$, $w=-1$, consistent with the recent CMB measurments from the WMAP experiment (Spergel 2003). In agreement with this experiment, we assume that the universe is flat, [*i.e.*]{} that $\Omega=\Omega_{m}+\Omega_{q}=1$.
Dark energy has several effects on weak lensing statistics (Ma 1999). First, it modifies the expansion history of the universe $a(t)$. As a result, both the angular-diameter distance and the growth rate of structures are modified. The latter effect is amplified by the non-linear evolution of structures. In some quintessence models, dark energy also modifies the linear power spectrum on large scales. We will ignore that effect since these scales are not easily probed by weak lensing surveys.
Cosmic shear statistics
=======================
The properties of the cosmic shear field can be quantified by several statistics. The most basic statistic is the weak lensing power spectrum. This is the equivalent in Fourier space of the shear-shear correlation functions, and is given by ([*e.g.*]{} Bartelmann & Schneider 1999; Hu & Tegmark 1999; see Bacon 2001 for conventions)
$$C_{\ell} = \frac{9}{16} \left( \frac{H_{0}}{c}
\right)^{4} \Omega_{m}^{2}
\int_{0}^{\chi_h} d\chi~\left[ \frac{g(\chi)}{a r(\chi)} \right]^{2}
P\left(\frac{\ell}{r}, \chi\right) ~,
\label{eq:cl}$$
where $r(\chi)$ is the comoving angular diameter distance, and $\chi_{h}$ corresponds to the comoving radius to the horizon. Considerable uncertainties remain for the non-linear power spectrum $P(k,z)$ in quintessence models (see discussion in Huterer 2001). Here, we will use the prescription of Peacock & Dodds (1996) to evaluate it from the linear power spectrum. The growth factor and COBE normalization for arbitrary values of $w$ can be computed using the fitting formulæ from Ma (1999).
The left-hand panel of figure 2 shows lensing power spectra for the fiducial $\Lambda$CDM model with background galaxies in two different shells, with median redshifts of 0.96 and 1.73. Deviations from the model are also shown, corresponding to variations in $\Omega_{m}$ and $w$. All models shown are COBE normalised. Note that non-linear corrections are dominant for $\ell \ga 100$.
Non-linear gravitational instability is known to produce non-Gaussian features in the cosmic shear field. The power spectrum therefore does not contain all the information available from weak lensing. We consider the most common measure of non-Gaussianity, namely the skewness $S_{3}$ which is defined as ([*e.g.*]{} Bernardeau 1997)
$$S_{3}(\theta) \equiv \frac{\langle \kappa^{3}
\rangle}{\langle \kappa^2 \rangle^2} ~,
\label{eq:s3}$$
where $\kappa$ is the convergence which can be derived from the shear field $\gamma_{i}$ and the angular brackets denote averages over circular top-hat cells of radius $\theta$. The denominator is the square of the convergence variance which is given by
$$\langle \kappa^{2} \rangle =
\langle \gamma^{2} \rangle \simeq
\frac{1}{2\pi} \int d\ell~\ell C_{\ell}|W_{\ell}|^{2} ~,$$
where $W_{\ell} \equiv 2J_{1}(\ell\theta)/(\ell\theta)$ is the window function for such cells and $C_{\ell}$ is the lensing power spectrum given by equation (5). To evaluate the numerator $\langle \kappa^{3}
\rangle$ of equation (6), we use the approximation of Hui (1999) who used the Hyperextended perturbation theory of Scoccimarro & Frieman (1999). While more accurate approximations for third order statistics now exist (see van Waerbeke 2001 and references therein), the present one suffices for our present purpose.
The right-hand panel of figure 2 shows the skewness as a function of scale for the same cosmological models considered in the left-hand panel. The skewness is only weakly dependent on the angular scale $\theta$, but depends more strongly on $\Omega_{m}$ and $w$.
SNAP: a wide field imager from space {#snap}
====================================
The [*Supernova/Acceleration Probe*]{} (SNAP) satellite has a planned launch date in 2010. The latest design of this wide-field 2m space telescope is shown in figure 3. Many of the stringent optical requirements for following the light curves of faint supernovæ are compatible with the desired instrumental properties for measurements of weak lensing. Indeed, most of SNAP’s limitations and trade-offs will be born by any similar wide-field imager from space. The detailed engineering models which are available for SNAP therefore act as a useful baseline for a generic space mission which will inevitably face similar engineering difficulties and reach similar solutions.
SNAP’s 0.7 square degree field of view will be covered by a mosaic of nine fixed filters spanning the optical and near IR from 350nm–1.7$\mu$m (Perlmutter 2003). These are arranged in such a pattern that scanning the telescope horizontally or vertically across the sky will build up an image of a contiguous survey region in all nine bands. At 800nm, the FWHM of the PSF will be approximately 0.15. The PSF, and any internal optical distortion (that could mimic cosmic shear), will remain stable because of the satellite’s high and therefore thermally stable orbit. Indeed, the telescope will rarely enter the shadow of the Earth and always maintain the same face pointing towards the sun (Rhodes 2004).
The SNAP survey strategy is divided into two primary missions. A deep survey to ABmag 30.2 in R (for a point source at 5$\sigma$), will cover 15 square degrees in all nine bands. This will be built up over a period of 32 months, by stacking observations taken once every four days in a search for type Ia supernovæ light curves. A second, wide survey to ABmag 27.7 will cover 300 square degrees. This will be gathered during a period of 5 months.
Two major weak lensing applications are enabled by the SNAP mission. The high number density of resolved galaxies (250 arcmin$^{-2}$) in the deep survey will be uniquely useful to construct high-resolution mass maps and search for galaxy clusters by mass. Complementing this, the wide survey is essential to reduce the impact of cosmic variance on cosmological parameter estimation. For this purpose, it is important that relatively high redshifts can still be reached in even a modest exposure time from space – and with sufficient colours, including near IR, to provide accurate photometric redshifts.
Shapelet-based image simulations {#shims}
================================
The image simulation method of Massey (2004a) was developed to mimic deep, high resolution space-based data. The simulations use the “shapelets” formalism of Refregier (2003a) to create realistic galaxies of all morphological types and with all the substructure and irregularity of faint HDF objects. The simulated objects possess a known size, magnitude and input shear, and may be used to calibrate the sensitivity of an instrument to weak lensing.
As a specific case, we have manufactured simulated images to the design specifications of the SNAP satellite. Since an object’s response to shear is a function of its overall shape, the presence of realistic and irregular galaxy morphologies is an important advance over earlier work (Bacon 2001, Erben 2001), which had used only azimuthally symmetric simulated galaxies with radial profiles parametrized as de Vaucouleurs $r^{1/4}$ laws or exponential discs. $30\arcmin\times30\arcmin$ sections of an example simulated image and a similarly scaled section of the HDF are shown in figure 4.
To generate the image, the galaxies in the HDF-North and HDF-South are first decomposed into shapelets. Shapelets are a complete basis set of two-dimensional, orthonormal basis functions, constructed from Laguerre functions multiplied by a Gaussian, as shown in figure 5. This basis is mathematically convenient for many aspects of image manipulation and analysis, including weak lensing shear and magnification. Linear combinations of shapelet basis states, weighted by “shapelet coefficients”, can be used to model any localised shape, rather like Fourier or wavelet synthesis. The shapelet coefficients are Gaussian-weighted multipole moments of the object, which can be used for quantitative shape measurement.
\[fig:flowchart\]
The shapelet coefficients of the real HDF galaxies populate an $n$-dimensional vector space, where $n$ is the maximum number of coefficients used. The vector space is illustrated in figure 6. It is analagous to the Hubble tuning fork: some regions of shapelet space represent spiral galaxies; other regions represent early-type elliptical galaxies. The underlying distribution of galaxy morphologies is only finitely sampled by the HDF, but may be recovered by smoothing the distribution in shapelet space. Massey (2004a) justify this process by comparing SExtractor and concentration/asymmetry morphology estimators of real data to the consistent statistics of shapelet-generated galaxies in the final simulated images.
An image of distant galaxies can then be manufactured by repeatedly resampling the recovered morphology distribution. Figure 7 illustrates the steps taken to produce a realistic simulated image. These steps mimic the processes acting on photons en route from a distant galaxy to a telescope. First, the galaxies are sheared by the input gravitational lensing signal. Then they are convolved with the SNAP PSF and slightly distorted through a model of the telescope’s internal optics. Typical observational noise is added at the end. These images can then be fed into a data reduction and shear measurement pipeline. As we show in the next section, comparing the known input shear with the output of this pipeline helps both to determine SNAP’s sensitivity to weak lensing and to calibrate and improve shear measurement algorithms.
Weak Lensing Measurement {#shims}
========================
We have used the RRG shear measurement method to attempt to recover the known level of shear input to the simulated images. Figure 8 shows the sensitivity to shear as a function of exposure time.
There are three main advantages of using space-based images for weak lensing measurements. Firstly, the smaller PSF means that there is a higher space density of galaxies sufficiently resolved for their shears to be measured, and shot noise is reduced on small scales. This is illustrated in the top panel of figure 8. Although the number of [*detected*]{} galaxies (dotted line) increases rapidly with increasing exposure time, the fraction of these which are resolved (solid line) begins to drop as the size distribution of the faint galaxies falls beneath the PSF size. Note that the simulations currently extend only to the depth of the HDF. The slope of the number counts beyond this, and the shape properties of these faint galaxies are unknown. We therefore await deeper observations with the ACS to probe this regime.
The second advantage of space is that the galaxies resolved from space are generally further away than those just resolved from the ground. The cosmic shear signal is predicted to increase with survey depth due to the cumulative lensing by more structures along the line of sight. The middle panel of figure 8 shows a prediction of the mean redshift of galaxies in the lensing sample as a function of exposure time. This is calculated via a simple model of median magnitude $vs$ median $z$, extrapolated from photometric observations of the HDF (Lilly 1996).
The third advantage is that the shape of each galaxy can be more reliably measured. Not only is the S/N higher on each galaxy, with more resolved morphological detail, but the correction for instrumental smearing and distortion can be much more accurate because of SNAP’s stable PSF. The PSF stability of SNAP will be superior even to that of HST, which suffers from telescope breathing due to the large changes in its temperature while passing in and out of direct sunlight during each orbit.
The bottom panel of figure 8 shows the resulting sensitivity to lensing of a space-based mission like SNAP. The error on the shear of galaxies binned into 1 arcmin$^2$ patches on the sky is shown as a function of exposure time. The dotted line shows the expected cosmic shear signal, which increases with exposure time due to the cumulative lensing by more and more objects as the survey depth increases. The S/N on scales of 1 arcmin$^2$ is about 1 and greater than about 1.8 for the wide and deep surveys respectively. As we discuss in the next section, this offers great prospects for mapping the dark matter.
In contrast to this precision from space, consider the limitations from a ground-based telescope. The ESI camera on Keck has proven itself to have minimal levels of internal optical distortion and telescope flexure, as required for high precision weak lensing measurements. The size of the primary mirror also lets it reach useful depths in exposures of only ten minutes (Bacon 2002). However, even in this short time, variations in the ground-based PSF seriously affect weak lensing observations with any ground-based telescope. Figure 9 shows examples of these changing atmospheric patterns, all taken with ESI on the same night. The line segments show the direction and ellipticity of the PSF, as measured from stars. Bacon (2001) and Erben (2001) demonstrated that the smearing of object shapes can be corrected at a level of $\sim90$% using the KSB method, and the corrected shapes of our stars are also shown in figure 9.
From space, the raw PSF ellipticities before correction are around the level on Keck after correction (Rhodes 2004). Coupled with more stable optics, and newer correction methods, this will result in a reduction in overall systematic contamination by at least an order of magnitude.
Cosmological constraints
========================
Two- and three-point statistics of the cosmic shear field in the wide SNAP survey will be used to constrain cosmological parameters. Figure 2 shows the effect of varying $\Omega_m$ and $w$ on the weak lensing power spectrum and skewness. The lensing power spectrum is shown for 2 bins of galaxy redshifts which can be derived from photometric redshifts. The error bars expected for the SNAP wide survey (using $A=200$ deg$^{2}$, $\sigma_{\gamma}=0.31$ and $n_{g}=100$ deg$^{-2}$) are displayed. The excellent precision afforded by SNAP will easily distinguish the models shown, and thus detect small changes in the properties of dark energy (Refregier 2004b).
We can compute the constraints which can be set on cosmological parameters using the Fisher matrix ([*e.g.*]{} Hu & Tegmark 1999)
$$F_{ij}= - \left\langle \frac{\partial \ln {\mathcal L}}
{\partial p_{i} \partial p_{j}} \right \rangle ~,$$
where ${\mathcal L}$ is the Likelihood function, and $p_{i}$ is a set of model parameters. The inverse ${\mathbf F}^{-1}$ provides a lower limit for the covariance matrix of the parameters.
Figure 10 shows the constraints on the $\Omega_m$–$w$ plane that will be possible from SNAP. The measurement of the weak lensing power spectrum at two different redshifts (or “redshift tomography”) provides an important lever arm upon the growth of structure and the evolution of the power spectrum, improving the constraints signicantly. The addition of the skewness (at a single angular scale) does not improve the constraints as much. Figure 11 compares the constraints from weak lensing (using the power spectrum in two redshift bins and the skewness) to that which can be derived from current and the future SNAP supernova surveys (Perlmutter 1999, 2003). Note however that the supernovæ constraints shown include systematic errors and a marginalisation over $w'$, the time derivative of $w$. The SNAP weak lensing survey will therefore provide constraints on the dark energy which are comparable with and complementary to those from the SNAP supernova survey.
\[fig:coscons2\]
\
\
\
Dark Matter Mapping
===================
From the observed shear field, one can reconstruct maps of the lensing convergence $\kappa$, which is proportional to the total mass projected along a given line of sight. The resolution of the maps depends on the size of the spatial element within which shear or convergence can be accurately measured to a S/N$\sim1$. The SNAP wide survey has been tailored to resolve the shapes of $\sim100$ background galaxies per square arcminute, over 300 square degrees. For the instrumental sensitivity to shear read from figure 8, this will achieve maps with a resolution of 1 arcmin$^2$ pixels ($\sim$250kpc at $z=0.3$). Extrapolating our simulations beyond the depth of the HDF, the SNAP deep survey will exceed even this density requirement by a factor of three or four (Massey 2004b). Thus, SNAP will open up a new regime of dark matter mapping, allowing direct comparison to be made between mass and light on fine scales over a very wide field of view.
In figure 12 we demonstrate the precision with which SNAP will be able to map the dark matter in an SCDM simulation from Jain, Seljak & White (2000). In this simulation, the source galaxies are assumed to lie on a single plane at $z=1$. Whilst only the most massive overdensities can be distinguished in ground-based shear data, the recovery is much improved from space. This is mainly due to the three- to four-fold increase in number density of resolved galaxies in even the SNAP wide survey. The statistical correlation of such maps will allow the bias between mass and light to be examined to high accuracy over a large range of scales.
Furthermore, simultaneously combining shear estimation with photometric redshifts will permit the use of a recently formulated direct 3D lensing inversion (Taylor 2001; Heavens in this volume). This method directly recovers the full 3D mass distribution without the need to slice projected maps into redshift bins. Applied to the SNAP deep survey, this technique will detect mass overdensities with a $1\sigma$ sensitivity lower than $10^{13}M_{\sun}$ at $z=0.25$. An unbiased, mass-selected cluster catalogue will trace the growth of mass structures in the universe (see Miyazaki 2002 and references therein).
Conclusions
===========
A space-based, wide-field imager is ideal for weak lensing measurements. Indeed, a systematic floor due to atmospheric seeing will be reached in the next generation of lensing surveys. This is due both to the isotropic PSF smearing that reduces the number of resolved galaxies for which shape measurements are possible; and anisotropic smearing which cannot be perfectly corrected when it varies from one exposure to the next. The proposed SNAP satellite will have both the wide field needed to survey large, representative cosmic volumes and the low level of instrumental systematics affecting galaxy shapes.
Through a dual wide and deep survey strategy, weak lensing with SNAP will be able to produce unique maps of the dark matter distribution, on small scales and in both 2D and 3D. Weak lensing with SNAP will be able to produce mass-selected cluster catalogues to a $1\sigma$ detection threshold of $10^{13}M_{\sun}$ at $z=0.25$.
Weak lensing primarily measures the distribution of the dark matter, but it is also a powerful probe of dark energy. A change in the equation of state $w$ of dark energy modifies the growth rate of structures and the angular-diameter distance. The resulting changes in the cosmic shear statistics will be easily detectable with future weak lensing surveys. In particular, weak lensing measurements with SNAP will be able to independently constrain cosmological parameters $\Omega_M$, $\sigma_8$ and $w$ at a level comparable with and somewhat orthogonal to those from future supernova searches and CMB experiments.
The authors would like to thank David Valls-Gabaud and Jean-Paul Kneib for organising a fine winter school. We thank the SNAP weak lensing working group, especially Richard Ellis, for our on-going collaboration and useful discussions.
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|
---
author:
- 'Elsayed Ahmed, Ahmed M. Eltawil, Zhouyuan Li, and Bedri A. Cetiner [^1]'
title: 'Full-Duplex Systems Using Multi-Reconfigurable Antennas'
---
[^1]: Elsayed Ahmed and Ahmed M. Eltawil are with the Department of Electrical Engineering and Computer Science at the University of California, Irvine, CA, USA (e-mail: {ahmede, aeltawil}@uci.edu). Zhouyuan Li and Bedri A. Cetiner are with the Department of Electrical and Computer Engineering at Utah State University, Logan, Utah, USA (e-mail: [email protected], [email protected]).
|
---
abstract: 'Diffusion of tracer particles in the cytoplasm of mammalian cells is often anomalous with a marked heterogeneity even within individual particle trajectories. Despite considerable efforts, the mechanisms behind these observations have remained largely elusive. We performed extensive single-particle tracking experiments on quantum dots in the cytoplasm of live mammalian cells (unperturbed or after disrupting the cytoskeleton). Analyses of the trajectories reveal a strong, microtubule-dependent subdiffusion with antipersistent increments and a substantial heterogeneity. Comparison to simulations highlight that the motion can be fully described as an intermittent fractional Brownian motion, alternating between two states of different mobility. Our data indicate transient associations with the endoplasmic reticulum (ER) network to be key for the low-mobility state. In addition, the ER couples the particle motion indirectly to active, cytoskeleton-based transport processes.'
author:
- 'Adal Sabri$^{1}$, Xinran Xu$^{2}$, Diego Krapf$\,^{2,3,*}$, and Matthias Weiss'
title: Elucidating the origin of heterogeneous anomalous diffusion in the cytoplasm of mammalian cells
---
The cytoplasm of mammalian cells is a complex aqueous environment, crowded with large amounts of macromolecules [@fulton1982; @Ellis] and a multitude of membrane-enveloped organelles of largely varying sizes. Diffusion of supposedly inert tracer particles in the cytoplasm of living cells has frequently been reported to be anomalous with a sublinear scaling of the mean square displacement (MSD), ${\langle r^2(\tau)\rangle}\sim t^\alpha$ ($\alpha<1$) on spatio-temporal scales below a few micrometers and several seconds [@franosch; @weiss2014; @norregaard2017]. The emergence of subdiffusive motion appears in many cases to be consistent with a stochastic process of the fractional Brownian motion (FBM) type [@kepten2011; @magdziarz2011; @metzler2014], i.e. a self-similar Gaussian process with stationary increments whose features are determined by the Hurst coefficient $H=\alpha/2$ [@mandelbrot1968]. FBM dynamics is subdiffusive for $0<H<1/2$ and trajectories are characterized by antipersistent, i.e. anticorrelated, increments. A plausible interpretation for such antipersistent memory effects is a viscoelastic environment [@Guigas2007; @jed; @weber2010; @sokolov2012; @ernst; @krapf2015] with a complex shear modulus that scales as $G(\omega)\sim\omega^\alpha$, where the elastic and the viscous parts are responsible for the FBM memory and for energy dissipation, respectively. Subdiffusion has long been recognized to emerge in solutions crowded with macromolecules, with an anomaly exponent $\alpha$ that decreases with crowder concentration [@WEK04; @banks2005]. However, the value of $\alpha$ is often observed to be considerably lower in the cytoplasm than in similarly crowded artificially fluids, e.g. $\alpha\approx0.6$ [@Guigas2007; @etoc] versus $\alpha\approx0.8$ [@jed; @ernst]. Therefore, it is currently understood that subdiffusion in the cytoplasm may not be caused solely by macromolecular crowding but also relies on additional mechanisms. As of yet, no general agreement exists for a physical model that can reliably describe cytoplasmic subdiffusion in detail. Further, subdiffusion is not universal but depends on tracer size, e.g. for particles in reconstituted entangled actin filament networks, where $\alpha$ can be continuously tuned between zero and unity as a function of particle radius and average mesh size [@wong2004]. Beyond such caging effects, it has also been proposed that non-inert crowders may strongly alter the dynamics of cytoplasmic particles [@nagle1992; @saxton1996]. Extensive Monte Carlo simulations have supported this hypothesis [@ghosh2015]. More recently, also experimental support has been obtained via single-particle tracking (SPT) on surface-modified tracer particles in the cytoplasm of HeLa cells: The emergence of subdiffusion and the value of $\alpha$ was shown to depend both on particle size and non-specific interactions to the cytoplasmic interior [@etoc]. Yet, the identity of the cytoplasmic binding partners that enforce the emergence of subdiffusive motion has remained elusive. Potential candidates include the cytoskeleton and organelles, e.g. the endoplasmic reticulum (ER) network that pervades the cytoplasm [@JLS]. Further, local variations in complex media are noticeable in the motion of particles therein: (Sub)diffusion in cellular fluids has been observed to be heterogeneous even within individual trajectories [@granick; @spako; @witzel], suggesting heterogeneous diffusion processes [@cherstvy2013] or spatiotemporal variations of transport coefficients [@slater; @chechkin2017; @cherstvy2016]. Despite the elegance of these theoretical models, it remains an open question how a distribution of apparent diffusivities emerges in the first place. A potential source might be the ambient active noise in the cytoplasm, i.e. the chemically induced rattling and shaking of the environment due to the non-equilibrium action of molecular motors and cytoskeletal filaments. In fact, breaking down cytoskeletal filaments alters the subdiffusive motion of organelle structures in mammalian cells [@SW2017; @SSW2018a] and also compromises the superdiffusive motion of beads in migrating amoebae [@witzel]. Taken together, it is currently neither clear (i) which mechanism regulates the value of the anomaly exponent $\alpha$ in the cytoplasm nor (ii) how one should picture the emergence of heterogeneous subdiffusion due to non-specific interactions in an actively driven environment. Here, we address these points by extensive SPT experiments on individual quantum dots loaded into the cytoplasm of living mammalian cells. In particular, we quantify the particles’ motion in the cytoplasm of untreated cells and in cells where the actin or microtubule cytoskeleton has been disrupted. Upon breaking down microtubules, a significant change towards more subdiffusive motion is observed. By comparing our experimental data to the dynamics of organelles, we arrive to the conclusion that non-specific binding of tracers to the vast ER network is responsible for the emergence of a marked cytoplasmic subdiffusion. The ER is further responsible for coupling the dynamics of cytoplasmic particles to active microtubule-based processes. Our experimental data are well described by an intermittent FBM model that switches stochastically between a higher or lower mobility while moving in the aqueous cytosol or on segments of the ER network, enforcing a heterogeneous FBM-like transport of tracers that is characteristic of the cytoplasmic subdiffusion in eukaryotic cells.
To explore the heterogeneous subdiffusion in the cytoplasm of mammalian cells, we performed extensive SPT on quantum dots that had been introduced into the cytoplasm of cultured HeLa cells by bead loading [@mcneil1987glass; @SM]. Measurements were performed with a sampling time of $\Delta t= 100$ ms, and quantum dot trajectories were first evaluated in terms of their time-averaged MSD (TA-MSD) using $N=100$ or $N=500$ positions, $$\label{tamsdt}
{{\langle r^2(\tau)\rangle}_t}= \frac{1}{N-k}\sum_{i=1}^{N-k}
\left[{\bf r}((i+k)\Delta t)-{\bf r}(i\Delta t)\right]^2 \,\,.$$ Following previous reports [@etoc; @manzo2015; @weron2017], individual TA-MSDs were fitted with a simple power law ${{\langle r^2(\tau)\rangle}_t}=K_\alpha\tau^\alpha$ in the range $\Delta t\le\tau\le 10\Delta t$ to extract the anomaly exponent $\alpha$ and the generalized diffusion coefficient $K_\alpha$. The resulting probability density function (PDF) of anomaly exponents, $p(\alpha)$, showed considerable trajectory-to-trajectory fluctuations around a mean ${\langle\alpha\rangle}\approx0.57$ ([Fig. \[fig01\]]{}a and Fig. S1a [@SM]) that slightly depends on the trajectory length $N$ (Table \[tab01\]).
![(a) The PDF of anomaly exponents $\alpha$, obtained from individual TA-MSDs ($N=100$), shows a broad variation around a mean ${\langle\alpha\rangle}=0.59$ in untreated cells (black histogram). Nocodazole-treated cells have a similarly broad PDF (red histogram) with a significantly lower mean (cf. Table \[tab01\]). Similar results are found for longer trajectories ($N=500$, Fig. S1a [@SM]). Using a bootstrapping approach with geometric averaging (diamonds with full lines being Gaussian fits) resulted in narrower PDFs with the same mean anomaly exponent. (b) PDFs of the generalized transport coefficient $K_\alpha$, obtained by the bootstrapping method, roughly have a lognormal shape (full lines) in untreated and nocodazole-treated cells. Short trajectories ($N=100$, histograms with color-coding as before) exhibit a small change upon disrupting microtubules whereas longer trajectories ($N=500$) appear more sensitive to treatment with nocodazole (squares and circles). Supposedly, a low $K_\alpha$ facilitates the acquisition of longer trajectories, biasing the data for $N=500$. []{data-label="fig01"}](adalfig01){width="8.5cm"}
To probe a potential perturbation of the power-law scaling due to static and dynamic localization errors [@moerner], and to validate the significance of the mean exponent ${\langle\alpha\rangle}$, we exploited a bootstrapping approach: From the whole set of calculated TA-MSDs we drew randomly a non-exhaustive ensemble of $100$ curves, averaged these geometrically, and used again a simple power-law fit to extract the scaling exponent $\alpha$ of the resulting ensemble-averaged TA-MSD (see [@SM] for details). Repeating this approach $M=200$ times, we noted that none of the ensemble-averaged TA-MSDs showed a significant offset in the extrapolated limit $t\to0$ (Fig. S2 [@SM]), i.e. static and dynamic localization errors appear to cancel each other in our data and therefore fitting with a simple power law gives meaningful results for $\alpha$. The PDF of $\alpha$ values obtained with the bootstrapping approach ([Fig. \[fig01\]]{}a) was very narrow with a mean ${\langle\alpha\rangle}$ that matched the respective value found before via individual TA-MSDs (Table \[tab01\]). Geometric averaging of TA-MSDs boils down to an arithmetic averaging of individual $\alpha$ values (but not of $K_\alpha$). Thus, the narrow width of $p(\alpha)$ after bootstrapping is determined by $\sigma/\sqrt{M}$, where $\sigma$ is the standard deviation of $\alpha$ derived from individual TA-MSDs. An arithmetic instead of a geometric averaging of TA-MSDs leads to an overestimation of the mean scaling exponent (Table \[tab01\] and Fig. S1b [@SM]).
---------------------- ----------------- ----------------- ----------------- -----------------
control noc cyto D lat A
\[1mm\]
\[-2mm\] TA-MSDs $0.59$ ($0.55$) $0.46$ ($0.36$) $0.58$ ($0.54$) $0.62$ ($0.58$)
\[1mm\] b.tr. geom. $0.58$ ($0.55$) $0.46$ ($0.36$) $0.58$ ($0.54$) $0.61$ ($0.57$)
\[1mm\] b.tr. arith. $0.79$ ($0.60$) $0.66$ ($0.43$) $0.82$ ($0.73$) $0.86$ ($0.76$)
\[1mm\]
---------------------- ----------------- ----------------- ----------------- -----------------
: Mean anomaly exponents ${\langle\alpha\rangle}$ for trajectories of length $N=100$ ($N=500$) without treatment and after application of nocodazole, cytochalasin D, or latrunculin A (named control and noc, cyto D, or lat A, respectively). Standard errors were in all cases smaller than $0.02$.[]{data-label="tab01"}
Being interested in how cytoplasmic diffusion is affected by the cytoskeleton, we applied either nocodazole to break down microtubules, or cytochalasin D or latrunculin A to disrupt actin filaments. Disrupting microtubules changed the diffusion anomaly substantially ([Fig. \[fig01\]]{}a and Table \[tab01\]) whereas disrupting actin networks had no significant effect (Table \[tab01\]). Transport coefficients $K_\alpha$ showed a higher sensitivity to microtubule disruption and also a stronger dependence on trajectory length ([Fig. \[fig01\]]{}b). Similar to previous observations on the dynamics of the ER [@SSW2018a], the effect of nocodazole on $K_\alpha$ was not particularly strong for short trajectories. For longer trajectories ($N=500$), however, a marked shift to smaller transport coefficients was visible upon microtubule disruption. This puts up a caveat that longer trajectories may represent a distinct subset of the acquired data, e.g. a lower mobility facilitating the tracking, but it also indicates that microtubule-associated processes significantly contribute to the diffusion anomaly in untreated cells beyond a change in the scaling of MSDs. Going beyond the MSD, we analyzed the ensemble average of the velocity autocorrelation function (VACF), $$\label{vacf}
C_v(\tau)=\langle {\bf v}(t){\bf v}(t+\tau)\rangle_{t,E}$$ that is highly sensitive to the nature of unconfined anomalous diffusion processes [@burov2011; @weber2012]. Here, ${\bf v}(t)=[{\bf r}(t+\delta t)-{\bf r}(t)]/\delta t$ is the velocity at time $t$, given via the increments in a period $\delta t$. Varying $\delta t=k\Delta t$ in multiples of the sampling time $\Delta t$, the VACFs showed in all cases a pronounced negative peak for $\tau=\delta t$ as expected for antipersistent random walks. By rescaling the times as $\xi=\tau/\delta t$, all VACF traces collapse to a single master curve that agrees with the analytical predictions for FBM ([Fig. \[fig02\]]{} and Fig. S3a [@SM]), namely $$\label{vacfFBM}
C_v(\xi)=\frac{(\xi+1)^\alpha+|\xi-1|^\alpha-2\xi^\alpha}{2}\,\,,$$ with $\alpha$ being set to the value ${\langle\alpha\rangle}$ found with the bootstrapping protocol (Table \[tab01\]). We emphasize the exceptional agreement of the experimental data with [Eq. (\[vacfFBM\])]{} without any fitting parameters since other antipersistent random walk data, e.g. from membrane proteins, can deviate significantly from the FBM prediction (see Fig. S3b [@SM] for an example).
![The rescaled normalized VACF \[[Eq. (\[vacf\])]{}\] of all experimental trajectories with $N=100$ at different $\delta t$ agrees with the predicted analytical form for FBM (full lines, [Eq. (\[vacfFBM\])]{}), both without treatment (grey symbols) and after nocodazole-treatment (red symbols). For better visibility, untreated cell data have been shifted upwards. No significant differences are seen for longer trajectories ($N=500$, Fig. S3a [@SM]). Inset: VACFs of simulated intermittent FBM realizations with $N=100$ and indicated anomaly values $\alpha_0$ also agree with [Eq. (\[vacfFBM\])]{} full lines).[]{data-label="fig02"}](adalfig02){width="8.5cm"}
Next we inspected the PDF of the normalized increments $\chi_{\delta t}$ within a time lag $\delta t$ [@spako]. The time series $\Delta x_i=x_{i+k}-x_i$ and $\Delta y_i=y_{i+k}-y_i$ were calculated and normalized by their mean step lengths $\langle |\Delta x_i|\rangle$ and $\langle |\Delta y_i|\rangle$ for each trajectory. Since no systematic differences were observed between $x$- and $y$-directions, all normalized increments were combined into a single set of $\chi_{\delta t}$. For a homogeneous FBM, a Gaussian PDF $p(\chi_{\delta t})$ is expected for all $\delta t$. Yet, for small $\delta t$ our data showed significant deviations from a Gaussian in the tails of the distribution ([Fig. \[fig03\]]{}). This suggests that individual trajectories are heterogeneous, i.e. the particle mobility changes within the trajectory. For $\delta t=10\Delta t$, this heterogeneity subsides, collapsing the increment statistics to the anticipated Gaussian (Fig. S4 [@SM]).
![The PDF of normalized increments $\chi_{\delta t}$ from trajectories with $N=100$ for a time lag $\delta t=\Delta t$ follows Gaussian (green dashed line) for small $\chi$ but shows significant deviations at large $\chi$, suggesting a heterogeneous random walk process (black circles and red crosses: untreated and nocodazole-treated, respectively). The data are in good agreement with simulations of an intermittent FBM model ($\alpha_0=0.5$ and $\alpha_0=0.3$: coinciding light and dark blue lines). For $\delta t=10\Delta t$, experimental data and simulations follow a Gaussian for all $\chi$ (cf. Fig. S4 [@SM]). []{data-label="fig03"}](adalfig03){width="8.5cm"}
Based on the observation of a heterogeneous and cytoskeleton-dependent subdiffusion of quantum dots in the cytoplasm, we hypothesized that an intermittent FBM model can correctly describe the data. Moreover, the similarities of ${\langle\alpha\rangle}$ in Table \[tab01\] with previously reported data for ER structures [@SSW2018a] triggered the idea that quantum dots transiently associate with the ER network between periods of exploring the surrounding aqueous cytosol. We therefore modeled the dynamics of individual particles as FBM with fixed anomaly $\alpha_0$ and a transport coefficient that randomly switches within each trajectory (see [@SM] for technical details). Particles were assumed to exist in ’on’ and ’off’ states with coefficients $K^{\text{on}}_\alpha <K^{\text{off}}_\alpha$, representing ER-associated and free motion. Dichotomous switching between these states was modeled as a Markov process with transition rates $k_{\text{on}}$ and $k_{\text{off}}$. In our simulations we kept these rates and the ratio $s=K^{\text{on}}_\alpha /K^{\text{off}}_\alpha$ fixed, and chose $\alpha_0=0.5$ ($\alpha_0=0.3$) for untreated (nocodazole-treated) cells, in accordance with the previously reported anomaly values for ER junctions [@SSW2018a]. Despite the simplicity of this model, we observed a surprisingly good overlap with our experimental data for $s=3.5$, $k_{\text{on}}\approx0.27$ s$^{-1}$, and $k_{\text{off}}\approx0.01$ s$^{-1}$: First, the mean anomaly of simulated realizations, extracted from TA-MSDs, was ${\langle\alpha\rangle}\approx0.55$ and ${\langle\alpha\rangle}\approx0.37$, respectively, in agreement with the experimental observations (Table \[tab01\]). The slightly larger value as compared to the imposed value $\alpha_0$ is a consequence of the dichotomous switching that perturbs the pure FBM behavior. Second, when using the respective value ${\langle\alpha\rangle}$, the VACF showed the same agreement with [Eq. (\[vacfFBM\])]{} as the experimental data (insets of [Fig. \[fig02\]]{} and Fig. S3a). Third, the non-Gaussian shape of the increment statistics $\chi_{\delta t}$ for $\delta t=\Delta t$ and the more Gaussian shape for $\delta t=10\Delta t$ are almost perfectly matched ([Fig. \[fig03\]]{} and Fig. S4 [@SM]). Hence, a switching between just two states is sufficient for reproducing key features of our data.
![The autocorrelation function of fluctuations in the squared increments, $G(\tau)$, of short trajectories ($N=100$) shows a non-trivial decay for untreated and nocodazole-treated cells (black and red symbols, respectively), indicating a temporal change of transport parameters. These data are in very good agreement with simulation results of an intermittent FBM model (light and dark blue lines). Inset: Also for longer trajectories ($N=500$), a non-trivial decay of $G(\tau)$ is observed that is well matched by simulation results of an intermittent FBM model. []{data-label="fig04"}](adalfig04){width="8.5cm"}
The dichotomous process predicts that the ensemble-averaged autocorrelation function of fluctuations in the squared increments $\Delta r^2(t)=|{\bf r}(t+\Delta t)-{\bf r}(t)|^2$, defined as $$\label{gtau}
G(\tau) = \left\langle\frac{\langle\Delta r^2(t)\cdot \Delta r^2(t+\tau)\rangle_t -
\langle \Delta r^2(t)\rangle_t^2}{\langle \Delta r^2(t)\rangle_t^2 }\right\rangle_E\,\,,$$ should show a characteristic long-lasting decay that depends only on the dwell times in the two states. Essentially, $G(\tau)$ describes how long a random walk is fueled by a homogenous PDF of steps with a given mean length before switching to a different mean step length. For pure FBM trajectories, $G(\tau)=0$ for $\tau>\Delta t$ (Fig. S5 [@SM]) whereas the intermittent model yields $G(\tau)\sim\exp[ -(k_{\text{on}}+k_{\text{off}})\tau ]$ for sufficiently long trajectories [@cao]. For small $N$, however, the fairly slow switching is not sampled well within each trajectory, leading to strongly fluctuating asymptotic zero lines that are eventually ensemble-averaged. As a result, a long-lasting decay emerges that appears to be unbounded. This prediction of the intermittent FBM model also agrees surprisingly well with the behavior of the experimental data ([Fig. \[fig04\]]{}). To probe more directly the apparent mobility switching within individual trajectories, we employed a recently developed analysis based on a trajectory’s local convex hull [@lch] (see [@SM] for details). This analysis confirmed the existence of at least two mobility states (Fig. S6a) and indicated a markedly larger mean residence time in the low-mobility state (Fig. S6b) for untreated and nocodazole-treated cells with an apparently Markovian switching between these two states. Moreover, the obtained PDFs for the residence times of experimental trajectories were in favorable agreement with results obtained from trajectories of the intermittent FBM model, giving further support to the validity of the model. What do our results imply and how can they be interpreted? Our experimental data are well described by an intermittent FBM model in which we have set $\alpha_0=0.5$ for untreated cells and $\alpha_0=0.3$ in the nocodazole-treated case. The same anomaly values have been observed experimentally for the motion of ER nodes in (un)treated cells [@SSW2018a], suggesting that interactions with the ER are the dominant cause of a marked subdiffusion in cytoplasm. Transient association with the ER hampers free diffusion but also couples the particle motion to active microtubule-based processes. As a result, microtuble-mediated active noise leads to enhanced fluctuations in the motion of tracer particles transiently bound to the ER network, without binding of the particles to microtubules. The resulting diffusion heterogeneity does not require a full and elaborate model but rather a switching between two modes of motion (ER-associated and free) is sufficient to reproduce the experimental results. Overall, subdiffusion in the cytoplasm is indeed a considerably more complex phenomenon than anomalous diffusion in artificial fluids crowded with passive macromolecules.
We thank Mike Tamkun for discussions on experimental design, O’Neil Wiggan for providing HeLa cells, and Ashok Prasad and Wenlong Xu for providing the bead loader assembly and helping with the bead loading protocol. This work was financially supported by the German Academic Exchange Service (PPP USA grant No. 57315749). AS and MW also gratefully acknowledge financial support by the VolkswagenStiftung (Az. 92738).
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abstract: 'A fundamental axiom of quantum mechanics requires the Hamiltonians to be Hermitian which guarantees real eigen-energies and probability conservation. However, a class of non-Hermitian Hamiltonians with Parity-Time ($\mathcal{PT}$) symmetry can still display entirely real spectra [@PRL_Bender_1998]. The Hermiticity requirement may be replaced by $\mathcal{PT}$ symmetry to develop an alternative formulation of quantum mechanics[@PRL_Bender_2002; @RPP_Bender]. A series of experiments have been carried out with *classical* systems including optics[@NP_Ruter], electronics[@PRL_N_Bender; @Nature_Assawaworrarit; @NC_Choi], microwaves[@PRL_Bittner], mechanics[@AJP_Bender] and acoustics[@PRX_Zhu; @NC_Popa; @NC_Fleury]. However, there are few experiments to investigate $\mathcal{PT}$ symmetric physics in quantum systems. Here we report the first observation of the $\mathcal{PT}$ symmetry breaking in a single spin system. We have developed a novel method to dilate a *general* $\mathcal{PT}$ symmetric Hamiltonian into a Hermitian one, which can be realized in a practical quantum system. Then the state evolutions under $\mathcal{PT}$ symmetric Hamiltonians, which range from $\mathcal{PT}$ symmetric unbroken to broken regions, have been experimentally observed with a single nitrogen-vacancy (NV) center in diamond. Due to the universality of the dilation method, our result opens a door for further exploiting and understanding the physical properties of $\mathcal{PT}$ symmetric Hamiltonian in quantum systems.'
author:
- Yang Wu
- Wenquan Liu
- Jianpei Geng
- Xingrui Song
- Xiangyu Ye
- 'Chang-Kui Duan'
- Xing Rong
- Jiangfeng Du
title: 'Observation of parity-time symmetry breaking in a single spin system'
---
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In quantum mechanics, the real energies of a system are guaranteed by a fundamental axiom associated with the Hermiticity of physical observables. However, a class of non-Hermitian Hamiltonians satisfying $\mathcal{PT}$ symmetry can still exhibit real eigenenergies[@PRL_Bender_1998]. In principle, wider range of systems can be described by $\mathcal{PT}$ symmetric Hamiltonians comparing to Hermitian ones. A Hamiltonian $H$ is considered to be $\mathcal{PT}$ symmetric if $[\mathcal{PT}, H] = 0$, where $\mathcal{P}$ and $\mathcal{T}$ denote the parity and time-reversal operators, respectively. A sufficient condition for a $\mathcal{PT}$ symmetric Hamiltonian $H$ to exhibit entire real eigenvalues is that $H$ corresponds to the region of unbroken $\mathcal{PT}$ symmetry, where any eigenfunction of $H$ is simultaneously an eigenfunction of the $\mathcal{PT}$ operator. Otherwise, if the $\mathcal{PT}$ symmetric Hamiltonian $H$ and the $\mathcal{PT}$ operator possess different eigenfunctions, $H$ corresponds to the region of broken $\mathcal{PT}$ symmetry. A Hamiltonian $H$ with a broken $\mathcal{PT}$ symmetry is typically associated with the presence of complex eigenenergies. An alternative formulation of quantum mechanics can be established in which the axiom of Hermiticity is replaced by the condition of $\mathcal{PT}$ symmetry[@PRL_Bender_2002; @RPP_Bender].
The rich physics associated with $\mathcal{PT}$ symmetric Hamiltonian have aroused considerable experimental interest[@NP_El_Ganainy]. A series of experiments have been performed with *classical* approaches. The optical analog of $\mathcal{PT}$ symmetric quantum mechanics was firstly proposed[@NP_Ruter], then the concept was quickly extended to other systems, such as electronics[@PRL_N_Bender; @Nature_Assawaworrarit; @NC_Choi], microwaves[@PRL_Bittner], mechanics[@AJP_Bender], acoustics[@PRX_Zhu; @NC_Popa; @NC_Fleury], and optical systems with atomic media[@PRL_Hang; @NP_Peng_2016; @PRL_Zhang]. In particular, the experimental research on $\mathcal{PT}$ symmetric classical optical systems has been honored as a most important accomplishment in the past decade[@NP_Cham] and has stimulated many applications such as unidirectional light transport[@Science_Feng_2011; @NP_Peng_2014] and single-mode lasers[@Science_Feng_2014; @Science_Hodaei]. As a contrast, it is still of challenge to experimentally investigate $\mathcal{PT}$ symmetric Hamiltonian related physics in quantum systems. This is because that experimental quantum systems are governed by Hermitian Hamiltonians. A possible approach is to realize a $\mathcal{PT}$ symmetric Hamiltonian in an open quantum system, but it is generally difficult to realize a controllable $\mathcal{PT}$ symmetric Hamiltonian by controlling the environment[@PRA_Gardas]. Some progress has been made with this approach in the system of light-matter quasiparticles[@Nature_Gao; @NC_Zhang]. A lossy Hamiltonian has been constructed to simulate the quantum dynamics under a corresponding $\mathcal{PT}$ symmetric Hamiltonian[@arxiv_Li]. In other experiments, non-unitary operators are designed to circumvent engineering $\mathcal{PT}$ symmetric Hamiltonians[@NPhoton_Tang; @NP_Xiao]. Recently, an approach has been developed to dilate a single-qubit $\mathcal{PT}$ symmetric Hamiltonian into a Hermitian Hamiltonian in a higher dimensional Hilbert space[@PRL_Gunther]. This method was further developed for dilation of arbitrary-dimensional Hamiltonians[@PRL_Kawabata]. However, these methods can not be utilized to dilate the $\mathcal{PT}$ broken Hamiltonian. Thus to observe the broken of $\mathcal{PT}$ symmetry in a single quantum system, such as a single spin, remains elusive.
In this paper, we report the first observation of the broken of $\mathcal{PT}$ symmetry in a single spin system. We develop a universal method to dilate a *general* $\mathcal{PT}$-symmetric Hamiltonian into a Hermitian one with an ancilla. This method is capable of Hermitian dilation of general $\mathcal{PT}$ symmetric Hamiltonian with arbitrary dimension, while only one ancilla qubit is required. A single nitrogen-vacancy center in diamond has been utilized as a platform to demonstrate our method. Both the state evolutions under $\mathcal{PT}$ symmetric broken and unbroken Hamiltonians have been successfully observed.
We consider a quantum system, $s$, which is driven by a $\mathcal{PT}$ symmetric Hamiltonian $H_s$. The quantum state of $s$ is denoted by $|\psi(t)\rangle$, which satisfies the Schr$\ddot{\text{o}}$dinger type equation, $i\frac{d}{dt}|\psi(t)\rangle=H_s|\psi(t)\rangle$. To realize $H_s$ in a quantum system, an ancilla qubit $a$ is introduced to dilate $H_s$ into a Hermitian Hamiltonian $H_{s,a}(t)$. The state of the combined system, $|\Psi(t)\rangle$, is a dilation of $|\psi(t)\rangle$ with the form $$|\Psi(t)\rangle=|\psi(t)\rangle|-\rangle+\eta(t)|\psi(t)\rangle|+\rangle,$$ where $|-\rangle=(|0\rangle-i|1\rangle)/\sqrt{2}$ and $|+\rangle=-i(|0\rangle+i|1\rangle)/\sqrt{2}$ are the eigenstates of $\sigma_y$ forming an orthonormal basis of the ancilla qubit and $\eta(t)$ is a linear operator. When a measurement is applied on the ancilla qubit and $|-\rangle$ is postselected, the evolution of quantum state, $|\psi(t)\rangle$, driven by $\mathcal{PT}$ symmetric Hamiltonian $H_s$ is produced. Now the key is to derive the expression of $H_{s,a}(t)$. The evolution governed by the Hermitian Hamiltonian $H_{s,a}(t)$ can be described by the Schr$\ddot{\text{o}}$dinger equation, $$i\frac{d}{dt}|\Psi(t)\rangle=H_{s,a}(t)|\Psi(t)\rangle.$$ The Hamiltonian, $H_{s,a}(t)$, can be designed flexibly according to practical physical systems for the reason that $H_{s,a}(t)$ is not uniquely determined (see Supplementary Material for details). For example, $H_{s,a}(t)$ can be designed to be $$H_{s,a}(t) = \Lambda(t) \otimes I + \Gamma(t) \otimes \sigma_z,$$ where $\Lambda(t)=\{H_s(t)+[i\frac{d}{dt}\eta(t)+\eta(t)H_s(t)]\eta(t)\}M^{-1}(t)$, and $\Gamma(t)=i[H_s(t)\eta(t)-\eta(t)H_s(t)-i\frac{d}{dt}\eta(t)]M^{-1}(t)$. The time-dependent operator $M(t)$ have the form $M(t)=\eta^{\dagger}(t)\eta(t)+I$, where $\sigma_x$, $\sigma_y$ and $\sigma_z$ are Pauli operators and $I$ is the identity matrix. This derivation of $H_{s,a}(t)$ holds for *arbitrary* Hamiltonians $H_s$ (see Supplementary Material for the proof). Our method can be utilized to Hermitianly dilate a *general* $\mathcal{PT}$ symmetric Hamiltonian. Thus it paves a way to a direct experimental investigate *general* $\mathcal{PT}$ symmetric related physics in quantum systems.
For clarity and without loss of generality, the $\mathcal{PT}$ symmetric Hamiltonian with the form, $$H_s = \left[ \begin{array}{cc}
ir & 1\\
1 & -ir
\end{array}
\right ],$$ is taken as an example, where $r$ is a real number. The eigenvalues of $H_s$ are $E=\pm\sqrt{1-r^2}$. In the region $|r|<1$, the eigenvalues $E$ are real and the system is in a unbroken-symmetry region. Especially, when $r=0$, the Hamiltonian $H_s$ is Hermitian. When $|r|>1$, the imaginary part of $E$ appears and the system is in a broken-symmetry region. The point $|r|=1$ is known as the exceptional point. The dilated Hermitian Hamiltonian of the $H_s$ can be derived from equation (3) by taking $\eta(t) = \eta_0(t) I$, where $\eta_0(t)$ is a real parameter. By expanding $\Lambda(t)$ and $\Gamma(t)$ in terms of Pauli operators, the dilated Hamiltonian has the form (see Supplementary Material for details) $$\begin{aligned}
H_{s,a}(t) &= A_1(t)\sigma_x \otimes I + A_2(t)I \otimes \sigma_z\\
&+A_3(t)\sigma_y \otimes \sigma_z + A_4(t)\sigma_z \otimes \sigma_z,
\end{aligned}$$ where $A_1(t)$, $A_2(t)$, $A_3(t)$ and $A_4(t)$ are real parameters corresponding to the $\mathcal{PT}$ symmetric Hamiltonian $H_s$ shown in equation (4).
A solid-state spin system based on NV center in diamond is utilized to demonstrate our proposal. As depicted in Fig. \[Fig1\]a, the NV center consists of a substitutional nitrogen atom with an adjacent vacancy site in the diamond crystal lattice. The Hamiltonian of the electron spin and the $^{14}$N nuclear spin system is $$H_{\mathrm{NV}} = 2\pi(DS_z^2 + \omega_eS_z + QI_z^2 + \omega_nI_z + AS_zI_z),$$ where $S_z$ and $I_z$ are the spin operators of the electron spin (spin-1) and the nuclear spin (spin-1). The electronic zero-field splitting is $D=2.87$ GHz and the nuclear quadrupolar interaction is $Q=-4.95$ MHz. The two spins are coupled by an hyperfine interaction $A=-2.16$ MHz. A magnetic field is applied along the NV symmetry axis (\[1 1 1\] crystal axis) to remove the degeneracy of the $|m_S=\pm1\rangle$ states, yielding the electron and nuclear Zeeman frequencies $\omega_e$ and $\omega_n$, respectively. A subspace of the total system is utilized to consist a two-qubit system, which is spanned by the four energy levels $|m_S=0, m_I=+1\rangle$, $|m_S=0, m_I=0\rangle$, $|m_S=-1, m_I=+1\rangle$, and $|m_S=-1, m_I=0\rangle$ labeled by $|0\rangle_e |1\rangle_n$, $|0\rangle_e |0\rangle_n$, $|-1\rangle_e |1\rangle_n$, and $|-1\rangle_e |0\rangle_n$ as shown in Fig. \[Fig1\]b. The electron spin qubit is treated as the system qubit while the nuclear spin qubit is served as the ancilla qubit. The dilated Hamiltonian $H_{s,a}(t)$ is achieved by two selective microwave (MW) pulses which are simultaneously applied on the electron spin. The control Hamiltonian can be written as $$\begin{aligned}
H_c &= 2\pi\sqrt{2}\Omega_1(t)\cos[\int_0^t\omega_1(\tau)d\tau+\phi_1(t)] S_x \otimes |1\rangle_{nn}\langle1| \\
&+2\pi\sqrt{2}\Omega_2(t)\cos[\int_0^t\omega_2(\tau)d\tau+\phi_2(t)] S_x \otimes |0\rangle_{nn}\langle0|,
\end{aligned}$$ where $\Omega_1(t)$, $\omega_1(t)$ and $\phi_1(t)$ ($\Omega_2(t)$, $\omega_2(t)$ and $\phi_2(t)$) correspond to the Rabi frequency, angular frequency, and phase of the selective control pulses on the electron spin which drive the spin transition if the state of the nuclear spin is $|1\rangle_n$ ($|0\rangle_n$). The frequencies of the two MW pulses are set to be $\omega_1(t)=\omega_{\mathrm{MW1}}+2A_4(t)$ and $\omega_2(t)=\omega_{\mathrm{MW2}}-2A_4(t)$, respectively. In the interaction picture, the two-qubit subspace of the total Hamiltonian, $H_{\mathrm{NV}}+H_c$, can be written as (see Supplementary Material for details) $$\begin{aligned}
H_{\mathrm{rot}} &= \pi\Omega(t)\cos[\phi(t)]\sigma_x\otimes I + A_2(t)I\otimes\sigma_z\\ &+ \pi\Omega(t)\sin[\phi(t)]\sigma_y\otimes\sigma_z + A_4(t)\sigma_z\otimes\sigma_z
\end{aligned}$$ by choosing $\Omega_1(t)=\Omega_2(t)=\Omega(t)$ and $-\phi_1(t)=\phi_2(t)=\phi(t)$. The dilated Hamiltonian $H_{s,a}(t)$ can realized when $\Omega(t)=\sqrt{A_1^2(t)+A_3^2(t)}/\pi$ and $\phi(t)=\arctan(A_3(t)/A_1(t))$.
Our experiment was implemented on a NV center in $[100]$ face bulk diamond which was isotopically purified (\[$^{12}$C\]=99.9%). The dephasing time $T_2^*$ of the electron spin is 19 (2) $\mathrm{\mu s}$. The 532 nm green laser pulses were modulated by an acousto-optic modulator (ISOMET). The laser beam traveled twice through the acousto-optic modulator before going through an oil objective (Olympus, PLAPON 60\*O, NA 1.42). The phonon sideband fluorescence (wavelength, 650-800nm) went through the same oil objective and was collected by an avalanche photodiode (Perkin Elmer, SPCM-AQRH-14) with a counter card. The magnetic field of 506 G was provided by a permanent magnet along the NV symmetry axis and the state of the two-qubit system can be effectively polarized to $|0\rangle_e|1\rangle_n$ by laser pumping. An arbitrary waveform generator (Keysight M8190A) generated microwave and radio-frequency pulses to manipulate the states of the two-qubit system. The microwave pulses were amplified by power amplifiers (Mini Circuits ZHL-30W-252-S+) and fed by a broadband coplanar waveguide with $15~$GHz bandwidth. The radio-frequency pulses were carried by a home-built coil with dual resonance frequencies after a power amplifier (Mini Circuits LZY-22+).
The experiment was preformed on a home-built optical detected magnetic resonance setup. When the strength of the static magnetic field was set to 506 Gauss, optical pumping laser pulses polarized the electron spin and nuclear spin simultaneously into the state $|0\rangle_e |1\rangle_n$ owing to resonant polarization exchange with the electronic spin in the excited state[@PRL_Wrachtrup]. The initial state of the two-qubit is $|\Psi\rangle = |0\rangle_e |-\rangle_n + \eta_0(0)|0\rangle_e |+\rangle_n$, which was obtained by the single-qubit rotation $Y(\theta)$ followed by the rotation $X(\pi/2)$ on nuclear spin as show in Fig. \[Fig1\]c. The operator, $Y(\theta)$, stands for the rotation around *y* axis with the rotation angle, $\theta = 2\arctan[\eta_0(0)]$. The rotation angle, $\theta$, varies with different $\mathcal{PT}$ symmetric Hamiltonian $H_s$. The rotation $X(\pi/2)$ is the single-qubit rotations around *x* axis to realize transformation between the basis spanned by $\{|0\rangle_n, |1\rangle_n\}$ and the basis spanned by $\{|+\rangle_n, |-\rangle_n\}$ of the nuclear spin qubit. Two selective MW pulses were applied on the electron spin to realize the dilation Hamiltonian $H_{s,a}(t)$. The parameters $\Omega_1(t)$, $\Omega_2(t)$, $\omega_1(t)$, $\omega_2(t)$, $\phi_1(t)$ and $\phi_2(t)$ are chosen corresponding to $A_1(t)$, $A_2(t)$, $A_3(t)$ and $A_4(t)$ as mentioned above. The MW pulses were generated by an arbitrary waveform generator and fed by a coplanar waveguide. The nuclear spin rotation, $X(-\pi/2)$, transform the state $|\Psi(t)\rangle = |\psi(t)\rangle_e |-\rangle_n + \eta_0(t)|\psi(t)\rangle_e |+\rangle_n$ into $|\Phi(t)\rangle = |\psi(t)\rangle_e |1\rangle_n + \eta_0(t)|\psi(t)\rangle_e |0\rangle_n$. Then the populations of each energy level of the two-qubit system are detected (see Supplementary Material for details). All the single nuclear spin rotations were realized with two channel radio frequency (RF) pulses applied simultaneously on the nuclear spin as illustrated in Fig. \[Fig1\]b. The frequency of the RF pulses are 2.9 MHz and 5.1 MHz corresponded to the nuclear spin transitions. The RF pulses were fed on the nuclear spin by a home-built coil with dual resonance frequencies. The Rabi frequency of the RF pulses were calibrated to 25 kHz.
![Constructing of $\mathcal{PT}$ symmetric Hamiltonian in NV center. **a**, Schematic atomic structure and energy levels of the NV center. **b**, Hyperfine structure of the coupling system with NV electron spin and $^{14}$N nuclear spin. The experiments are implemented on the two-qubit system composed of four energy levels $|m_S=0, m_I=+1\rangle$, $|m_S=-1, m_I=+1\rangle$, $|m_S=0, m_I=0\rangle$, and $|m_S=-1, m_I=0\rangle$ labeled by $|0\rangle_e |1\rangle_n$, $|0\rangle_e |0\rangle_n$, $|-1\rangle_e |1\rangle_n$, and $|-1\rangle_e |0\rangle_n$. The electronic zero-filed splitting is $D=2.87$ GHz and the Zeeman splitting of the electron spin is $\omega_e$. The two-qubit system is controlled by two microwave (MW) pulses (blue arrows) and two radio-frequency (RF) pulses (orange arrows), which selectively drive the two electron-spin transitions and the two nuclear-spin transitions, respectively. **c**, Quantum circuit of the experiment. The electron spin qubit is taken as the system qubit while the nuclear spin qubit is served as the ancilla qubit. X and Y denote the single nuclear spin qubit rotation around the *x* and *y* axes. The two-qubit system is prepared to $|\Psi(0)\rangle = |0\rangle_e |-\rangle_n + \eta_0(0)|0\rangle_e |+\rangle_n$ by rotations $\mathrm{Y}(\theta)$ and $\mathrm{X}(\pi/2)$. Then the two-qubit system evolve under the dilation Hamiltonian $H_{s,a}(t)$. The populations of the four energy levels are measured after the rotation $\mathrm{X}(-\pi/2)$. []{data-label="Fig1"}](fig1.pdf){width="0.9\columnwidth"}
The state evolution under the $\mathcal{PT}$ symmetric Hamiltonian $H_s$ is explored by monitoring $P_0$, i.e., the renormalized population of the state $|m_S=0\rangle$ of the electron spin when the nuclear spin state is in the selected state $|m_I=+1\rangle$ (see supplementary materials for details). The time of the state evolution is varied from 0 to 8 $\mathrm{\mu s}$. Figure \[Fig2\]a-d show the state evolution under $\mathcal{PT}$ symmetric Hamiltonian with the Hermitian case $r=0$ ( Fig. \[Fig2\]a ), $\mathcal{PT}$ unbroken case $r=0.6$ ( Fig. \[Fig2\]b ), exceptional point case $r=1.0$ ( Fig. \[Fig2\]c ) and $\mathcal{PT}$ broken case $r=1.4$ ( Fig. \[Fig2\]d ). All errors are one standard deviation with repeating the experiments for 0.5 million times. In the $\mathcal{PT}$ unbroken region, the time evolution of the state is periodic oscillation, while the oscillation breaks down in the $\mathcal{PT}$ broken region. The state evolutions under $\mathcal{PT}$ symmetric Hamiltonian with various values of the parameter, $r$, are shown in Fig. \[Fig2\]e-f. The $\mathcal{PT}$ phase transition threshold is evidently revealed at the exceptional point $r=1.0$. The experimental results (Fig. \[Fig2\]f) show good agreement with the corresponding theoretical predictions (Fig. \[Fig2\]e).
![State evolution under $\mathcal{PT}$ symmetric Hamiltonian. **a**-**d**, Experimental state evolution under $\mathcal{PT}$ symmetric Hamiltonians with different parameter a of the Hamiltonian, where $r=0$ (**a**), $r=0.6$ (**b**), $r=1.0$ (**c**) and $r=1.4$ (**d**) correspond to the Hermitian, $\mathcal{PT}$ unbroken, exceptional point and $\mathcal{PT}$ broken case, respectively. $P_0$ is the renormalized population of the state $|m_S=0\rangle$ of the electron spin when the nuclear spin state is in the selected state $|m_I=+1\rangle$. Blue dots are experimental results, and red lines are the theoretical predictions. **e** (theoretical results ) and **f** (experimental results) plot results for various $r$ values. The color bar stands for the population $P_0$. []{data-label="Fig2"}](fig2.pdf){width="0.9\columnwidth"}
The $\mathcal{PT}$ phase transition is also characterized by the eigenvalues of the $\mathcal{PT}$ symmetric Hamiltonians as shown in Fig. \[Fig3\]. The eigenvalues of the $H_s$ can be achieved by $E_\pm=\pm\sqrt{1-r_{\mathrm{exp}}^2}$. The parameter $r_{\mathrm{exp}}$ is obtained by curve fitting the experimental time evolution of the population $P_0$ to theoretical predictions under $\mathcal{PT}$ symmetric Hamiltonian $H_s$ (see supplementary materials for the details). When $r<1$, the system is in a $\mathcal{PT}$ symmetry unbroken region. The eigenvalues $E$ are still real as $r$ approaches 1 from 0. At the exceptional point $r=1$, the eigenvalues $E$ coalesce to 0. The system is in the $\mathcal{PT}$ symmetry broken region when $r>1$. The real parts of the eigenvalues $E$ coalesce and the imaginary parts appears. The experimental results show excellent agreement with the corresponding theoretical predictions.
![Experimental observation the breaking of the $\mathcal{PT}$ symmetry. **a** Real part and **b** imaginary part of the eigenvalues of the $\mathcal{PT}$ symmetric Hamiltonian. The blue dots are experimental data, and the red lines are the theoretical predictions of the eigenvalues. The $0<r<1$ regime represents the $\mathcal{PT}$ unbroken case and the $r>1$ regime represents the $\mathcal{PT}$ broken case. The exceptional point occurs at $r=1$. []{data-label="Fig3"}](fig3.pdf){width="0.9\columnwidth"}
In summary, we have experimentally demonstrate the state evolution under $\mathcal{PT}$ symmetric Hamiltonian by the Hermitian dilation method. The breaking of $\mathcal{PT}$ symmetry has been observed in a single electron spin. The universal dilation method present here is compatible for Hermitian dilation of an arbitrary non-Hermitian Hamiltonian, thus our work opens a door for future experimental study of the intriguing non-Hermitian and $\mathcal{PT}$ symmetric physics with quantum systems.
This work was supported by the National Key R$\&$D Program of China (Grants No. 2018YFA0306600 and No. 2016YFB0501603), the CAS (Grants No. GJJSTD20170001, No.QYZDY-SSW-SLH004 and No.QYZDB-SSW-SLH005), and Anhui Initiative in Quantum Information Technologies (Grant No. AHY050000). X.R. thanks the Youth Innovation Promotion Association of Chinese Academy of Sciences for the support.
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**APPENDIX**
. Universal Hermitian dilation of non-Hermitian Hamiltonians
=============================================================
This section mainly focus on universally dilating a non-Hermitian Hamiltonian into a Hermitian one. Subsection A demonstrates the dilation method, followed by some derivation used in the dilation in subsection B and subsection C. Finally, in subsection D the universality of this method is proved.
A. Universal dilation method
-----------------------------
Our target is to realize the dynamics of a quantum system $s$, which is described by the evolution $\varepsilon_1(t)$ governed by Hamiltonian $H_s(t)$. $H_s(t)$ is non-Hermitian, arbitrary dimensional and time-dependent in general. The state evolution of system $s$ is described by $|\psi(t)\rangle$, so the corresponding Schrödinger equation can be written as (natural unit are chosen so that $\hbar=1$ in this supplementary material) $$\label{Eq1}
i\frac{d}{dt}|\psi(t)\rangle=H_s(t)|\psi(t)\rangle.$$
To realize $\varepsilon_1(t)$ in a quantum system, an ancilla qubit is introduced to dilate $H_s(t)$ into a Hermitian Hamiltonian $H_{s,a}(t)$. The evolution of the combined system under $H_{s,a}(t)$ is $\varepsilon_2(t)$. The state evolution of the combined system is described by $|\Psi(t)\rangle$, which satisfies the following Schrödinger equation $$\label{Eq2}
i\frac{d}{dt}|\Psi(t)\rangle=H_{s,a}(t)|\Psi(t)\rangle.$$ $|\Psi(t)\rangle$ is a dilation of $|\psi(t)\rangle$ and can be written as $$\label{Eq3}
\left\{
\begin{aligned}
& |\Psi(t)\rangle=|\psi(t)\rangle|-\rangle+|\chi(t)\rangle|+\rangle, \\
& |\chi(t)\rangle=\eta(t)|\psi(t)\rangle, \\
\end{aligned}
\right.$$ where $|-\rangle=\frac{|0\rangle-i|1\rangle}{\sqrt{2}}$ and $|+\rangle=-i\frac{|0\rangle+i|1\rangle}{\sqrt{2}}$ are the eigenstates of Pauli operator $\sigma_y$, which forms an orthonormal basis of the ancilla qubit, and $\eta(t)$ is a linear operator. When a measurement is applied on the ancilla qubit and $|-\rangle$ is postselected, the evolution $\varepsilon_1(t)$ governed by the non-Hermitian Hamiltonian $H_s(t)$ can be produced.
Now the key is to derive the expression of $H_{s,a}(t)$. The mathematical form of $H_{s,a}(t)$ can be written as $$\label{Eq4}
H_{s,a}(t)=H_{s,a}^{(++)}(t)\otimes|+\rangle\langle+|+H_{s,a}^{(+-)}(t)\otimes|+\rangle\langle-|+H_{s,a}^{(-+)}(t)\otimes|-\rangle\langle+|+H_{s,a}^{(--)}(t)\otimes|-\rangle\langle-|,$$ where $H_{s,a}^{(++)}(t),H_{s,a}^{(+-)}(t),H_{s,a}^{(-+)}(t)$ and $H_{s,a}^{(--)}(t)$ have the same dimension as $H_s(t)$. Due to the Hermiticity of $H_{s,a}(t)$, we have $$\label{Eq5}
\left\{
\begin{aligned}
&H_{s,a}^{(++)\dagger}(t)=H_{s,a}^{(++)}(t), \\
&H_{s,a}^{(+-)\dagger}(t)=H_{s,a}^{(-+)}(t), \\
&H_{s,a}^{(--)\dagger}(t)=H_{s,a}^{(--)}(t). \\
\end{aligned}
\right.$$ By substituting equations \[Eq1\], \[Eq3\] and \[Eq4\] into equation \[Eq2\] and rearranging terms, we obtain $$\label{Eq6}
\left\{
\begin{aligned}
&H_{s,a}^{(--)}(t)+H_{s,a}^{(-+)}(t)\eta(t)-H_s(t)=0, \\
&H_{s,a}^{(+-)}(t)+H_{s,a}^{(++)}(t)\eta(t)-i\frac{d}{dt}\eta(t)-\eta(t)H_s(t)=0. \\
\end{aligned}
\right.$$ After taking Hermitian transpose of operators in equation \[Eq6\] we can get $$\label{Eq7}
\left\{
\begin{aligned}
& H_{s,a}^{(--)}(t)+\eta^\dag(t)H_{s,a}^{(+-)}(t)-H_s^\dag(t)=0, \\
& H_{s,a}^{(-+)}(t)+\eta^\dag(t)H_{s,a}^{(++)}(t)+i\frac{d}{dt}\eta^\dag(t)-H_s^\dag(t)\eta^\dag(t)=0, \\
\end{aligned}
\right.$$ where equation \[Eq5\] has been taken into consideration. Finally, equations \[Eq6\] and \[Eq7\] reduce to $$\label{Eq8}
\left\{
\begin{aligned}
&H_{s,a}^{(--)}(t)=H_s(t)-[H_s^\dag(t)\eta^\dag(t)-i\frac{d}{dt}\eta^\dag(t)-\eta^\dag(t)H_{s,a}^{(++)}(t)]\eta(t), \\
&H_{s,a}^{(-+)}(t)=H_s^\dag(t)\eta^\dag(t)-i\frac{d}{dt}\eta^\dag(t)-\eta^\dag(t)H_{s,a}^{(++)}(t), \\
&H_{s,a}^{(+-)}(t)=\eta(t)H_s(t)+i\frac{d}{dt}\eta(t)-H_{s,a}^{(++)}(t)\eta(t), \\
\end{aligned}
\right.$$ together with an equation that $\eta(t)$ should satisfy, $$\label{Eq9}
i\frac{d}{dt}[\eta^\dag(t)\eta(t)]=H_s^\dag(t)[\eta^\dag(t)\eta(t)+I]-[\eta^\dag(t)\eta(t)+I]H_s(t).$$ To solve equation \[Eq9\]£©, we can define a Hermitian operator $M(t)$ as follows $$\label{Eq10}
M(t)\equiv\eta^\dag(t)\eta(t)+I,$$ where $I$ is the identity operator. Then equation \[Eq9\] can be rewritten as $$\label{Eq11}
i\frac{d}{dt}M(t)=H_s^\dag(t)M(t)-M(t)H_s(t).$$ The solution of equation \[Eq11\] takes the form $$\label{Eq12}
M(t)=\mathcal{T}e^{-i\int_{0}^{t}H_s^\dag(t)dt}M(0)\overline{\mathcal{T}}e^{i\int_{0}^{t}H_s(t)dt},$$ where $\mathcal{T}$ and $\overline{\mathcal{T}}$ are time-ordering and anti-time-ordering operators, respectively. $M(0)$ is an initial operator of operator $M(t)$, which is chosen to ensure that $M(t)-I$ keeps positive for all $t$. According to equation \[Eq10\], an expression of $\eta(t)$ can be written as $$\label{Eq13}
\eta(t)=U(t)[M(t)-I]^\frac{1}{2},$$ where $U(t)$ is an arbitrary differentiable unitary operator. Equation \[Eq4\], \[Eq8\], \[Eq12\] together with equation \[Eq13\] give an explicit expression of the dilated Hamiltonian $H_{s,a}(t)$, in which $H_{s,a}^{(++)}(t)$ is an arbitrary Hamiltonian operator.
Note that $H_{s,a}(t)$ is not uniquely determined since $H_{s,a}^{(++)}(t)$ can be an arbitrary Hermitian operator and $U(t)$ can be an arbitrary differentiable unitary operator. This arbitrariness makes it possible to flexible design $H_{s,a}(t)$ according to different experimental systems. For example, to make $H_{s,a}(t)$ easy to be constructed in NV center, we can choose $$\label{Eq14}
\left\{
\begin{aligned}
& U(t)=I, \\
& H_{s,a}^{(++)}(t)=\{H_s(t)+[i\frac{d}{dt}\eta(t)+\eta(t)H_s(t)]\eta(t)\}M^{-1}(t).
\end{aligned}
\right.$$ The Hermiticity of $H_{a,s}^{(++)}(t)$ will be proved in subsection B of this section. According to equation \[Eq14\], equation \[Eq13\] reduces to $$\label{Eq15}
\eta(t)=[M(t)-I]^\frac{1}{2},$$ which means $\eta(t)$ is also a Hermitian operator. In this case the Hermitian operator, $$\label{Eq16}
M(t)=\eta^\dag(t)\eta(t)+I=\eta^2(t)+I,$$ commutes with $\eta(t)$, and the inverse operator of $M(t)$, $M^{-1}(t)$, is Hermitian and commutes with $\eta(t)$ as well. By substituting equation \[Eq14\] into equation \[Eq8\] and considering the Hermiticity of $\eta(t)$, we have $$\label{Eq17}
\left\{
\begin{aligned}
&H_{s,a}^{(--)}(t)=H_s(t)-H_s^\dag(t)\eta^2(t)+i[\frac{d}{dt}\eta(t)]\eta(t)+\eta(t)\{H_s(t)+[i\frac{d}{dt}\eta(t)+\eta(t)H_s(t)]\eta(t)\}M^{-1}(t)\eta(t), \\
&H_{s,a}^{(-+)}(t)=H_s^\dag(t)\eta(t)-i\frac{d}{dt}\eta(t)-\eta(t)\{H_s(t)+[i\frac{d}{dt}\eta(t)+\eta(t)H_s(t)]\eta(t)\}M^{-1}(t), \\
&H_{s,a}^{(+-)}(t)=\eta(t)H_s(t)+i\frac{d}{dt}\eta(t)-\{H_s(t)+[i\frac{d}{dt}\eta(t)+\eta(t)H_s(t)]\eta(t)\}M^{-1}(t)\eta(t). \\
\end{aligned}
\right.$$ Equation \[Eq17\] can be simplified to $$\label{Eq18}
\left\{
\begin{aligned}
&H_{s,a}^{(--)}(t)=\{H_s(t)+[i\frac{d}{dt}\eta(t)+\eta(t)H_s(t)]\eta(t)\}M^{-1}(t), \\
&H_{s,a}^{(-+)}(t)=[H_s(t)\eta(t)-\eta(t)H_s(t)-i\frac{d}{dt}\eta(t)]M^{-1}(t), \\
&H_{s,a}^{(+-)}(t)=-[H_s(t)\eta(t)-\eta(t)H_s(t)-i\frac{d}{dt}\eta(t)]M^{-1}(t). \\
\end{aligned}
\right.$$ The detail of the derivation from equation \[Eq17\] to equation \[Eq18\] is given in subsection C of this section. Substituting equation \[Eq18\] and equation \[Eq14\] into equation \[Eq4\], we obtain $$\label{Eq19}
\begin{aligned}
H_{s,a}(t)=
&\{\{H_s(t)+[i\frac{d}{dt}\eta(t)+\eta(t)H_s(t)]\eta(t)\}\otimes|+\rangle\langle+|-[H_s(t)\eta(t)-\eta(t)H_s(t)-i\frac{d}{dt}\eta(t)]\otimes|+\rangle\langle-| \\
&+[H_s(t)\eta(t)-\eta(t)H_s(t)-i\frac{d}{dt}\eta(t)]\otimes|-\rangle\langle+|+\{H_s(t)+[i\frac{d}{dt}\eta(t)+\eta(t)H_s(t)]\eta(t)\}\otimes|-\rangle\langle-|\}M^{-1}(t),
\end{aligned}$$ or $$\label{Eq20}
H_{s,a}(t)=\Lambda(t)\otimes I+\Gamma(t)\otimes\sigma_z,$$ with $$\label{Eq21}
\left\{
\begin{aligned}
&\Lambda(t)=\{H_s(t)+[i\frac{d}{dt}\eta(t)+\eta(t)H_s(t)]\eta(t)\}M^{-1}(t), \\
&\Gamma(t)=i[H_s(t)\eta(t)-\eta(t)H_s(t)-i\frac{d}{dt}\eta(t)]M^{-1}(t). \\
\end{aligned}
\right.$$
B. Proof of the Hermiticity of $H_{s,a}^{(++)}(t)$
--------------------------------------------------
Utilizing the commutation relation between $M(t)$ and $\eta(t)$, the second formula of equation \[Eq14\] can be rewritten as $$\label{Eq22}
H_{s,a}^{(++)}(t)=M^{-1}(t)\{M(t)H_s(t)+\eta(t)M(t)H_s(t)\eta(t)+iM(t)[\frac{d}{dt}\eta(t)]\eta(t)\}M^{-1}(t).$$ By substituting equation \[Eq16\] into the third term of the right-hand side of equation \[Eq22\], we obtain $$\label{Eq23}
H_{s,a}^{++}(t)=M^{-1}(t)\{M(t)H_s(t)+\eta(t)M(t)H_s(t)\eta(t)+i\eta^2(t)[\frac{d}{dt}\eta(t)]\eta(t)+i[\frac{d}{dt}\eta(t)]\eta(t)\}M^{-1}(t).$$ Considering the Hermiticity of $\eta(t)$, $M(t)$ and $M^{-1}(t)$, the Hermitian conjunction operator of $H_{a,s}^{(++)}(t)$ can be written as $$\label{Eq24}
H_{s,a}^{(++)\dag}(t)=M^{-1}(t)\{H_s^\dag(t)M(t)+\eta(t)H_s^\dag(t)M(t)\eta(t)-i\eta(t)[\frac{d}{dt}\eta(t)]\eta^2(t)-i\eta(t)[\frac{d}{dt}\eta(t)]\}M^{-1}(t).$$ Combining equation \[Eq23\] and equation \[Eq24\], we have $$\label{Eq25}
\begin{aligned}
&H_{s,a}^{(++)\dag}(t)-H_{s,a}^{++}(t) \\
=&M^{-1}(t)[H_s^\dag(t)M(t)-M(t)H_s(t)]M^{-1}(t)+M^{-1}(t)\eta(t)[H_s^\dag(t)M(t)-M(t)H_s(t)]\eta(t)M^{-1}(t) \\
&-iM^{-1}(t)\eta(t)\{[\frac{d}{dt}\eta(t)]\eta(t)+\eta(t)[\frac{d}{dt}\eta(t)]\}\eta(t)M^{-1}(t)-iM^{-1}(t)\{\eta(t)[\frac{d}{dt}\eta(t)]+\frac{d}{dt}\eta(t)]\eta(t)\}M^{-1}(t).
\end{aligned}$$ According to equation \[Eq11\] and equation \[Eq16\], we have $$\label{Eq26}
i\frac{d}{dt}M(t)=H_s^\dag(t)M(t)-M(t)H_s(t)=i\{[\frac{d}{dt}\eta(t)]\eta(t)+\eta(t)[\frac{d}{dt}\eta(t)]\}.$$ Then, equation \[Eq25\] vanishes, so the Hermiticity of $H_{s,a}^{(++)}(t)$ is proved.
C. Derivation of the form of $H_{s,a}^{(--)}(t), H_{s,a}^{(-+)}(t)$ and $H_{s,a}^{(+-)}(t)$
-------------------------------------------------------------------------------------------
The first formula in equation \[Eq17\] reads $$\label{Eq27}
H_{s,a}^{(--)}(t)=H_s(t)-H_s^\dag(t)\eta^2(t)+i[\frac{d}{dt}\eta(t)]\eta(t)+\eta(t)\{H_s(t)+[i\frac{d}{dt}\eta(t)+\eta(t)H_s(t)]\eta(t)\}M^{-1}(t)\eta(t).$$ Substitute equation \[Eq16\] into equation \[Eq27\] and consider the commutation of $M^{-1}(t)$ and $\eta(t)$ we have $$\label{Eq28}
\begin{aligned}
H_{s,a}^{(--)}(t)=&-H_s^\dag(t)M(t)+H_s^\dag(t)+i[\frac{d}{dt}\eta(t)]\eta(t)+\eta(t)H_s(t)\eta(t)M^{-1}(t)\\ &+i\eta(t)[\frac{d}{dt}\eta(t)][M(t)-I]M^{-1}(t)+[M(t)-I]H_s(t)[M(t)-I]M^{-1}(t)+H_s(t).
\end{aligned}$$ Simplify and rearrange terms in equation \[Eq28\], we obtain $$\label{Eq29}
\begin{aligned}
H_{s,a}^{(--)}(t)=&-[H_s^\dag(t)M(t)-M(t)H_s(t)]+i\{[\frac{d}{dt}\eta(t)]\eta(t)+\eta(t)[\frac{d}{dt}\eta(t)]\} \\ & +\{H_s(t)+\eta(t)H_s(t)\eta(t)-i\eta(t)[\frac{d}{dt}\eta(t)]+[H_s^\dag(t)M(t)-M(t)H_s(t)]\}M^{-1}(t),
\end{aligned}$$ then, by substituting equations \[Eq26\] into equation \[Eq29\] and rearranging terms, equation \[Eq29\] reduces to $$\label{Eq30}
H_{s,a}^{(--)}(t)=\{H_s(t)+[i\frac{d}{dt}\eta(t)+\eta(t)H_s(t)]\eta(t)\}M^{-1}(t),$$ which is the first formula in equation \[Eq18\].
The third formula in equation \[Eq17\] is given by $$\label{Eq31}
H_{s,a}^{(+-)}(t)=\eta(t)H_s(t)+i\frac{d}{dt}\eta(t)-\{H_s(t)+[i\frac{d}{dt}\eta(t)+\eta(t)H_s(t)]\eta(t)\}M^{-1}(t)\eta(t).$$ Considering the commutation of $M^{-1}(t)$ and $\eta(t)$, and substituting equation \[Eq16\], we have $$\label{Eq32}
H_{s,a}^{(+-)}(t)=\eta(t)H_s(t)+i\frac{d}{dt}\eta(t)-H_s(t)\eta(t)M^{-1}(t)-[i\frac{d}{dt}\eta(t)+\eta(t)H_s(t)][M(t)-I]M^{-1}(t).$$ Equation \[Eq32\] reduces to $$\label{Eq33}
H_{s,a}^{(+-)}(t)=-[H_s(t)\eta(t)-\eta(t)H_s(t)-i\frac{d}{dt}\eta(t)]M^{-1}(t),$$ so the third formula in equation \[Eq18\] is proved.
To prove the second formula in equation \[Eq18\], we rewrite equation \[Eq33\] as $$\label{Eq34}
H_{s,a}^{(+-)}(t)=-M^{-1}(t)[M(t)H_s(t)\eta(t)-\eta(t)M(t)H_s(t)-i\eta^2(t)\frac{d}{dt}\eta(t)-i\frac{d}{dt}\eta(t)]M^{-1}(t),$$ where equation \[Eq16\] and commutation of $M(t)$ and $\eta(t)$ have been utilized. According to equation \[Eq5\], $H_{s,a}^{(-+)}(t)=H_{s,a}^{(+-)\dag}(t)$, we have $$\label{Eq35}
H_{s,a}^{(-+)}(t)=-M^{-1}(t)\{\eta(t)H_s^\dag(t)M(t)-H_s^\dag(t)M(t)\eta(t)+i[\frac{d}{dt}\eta(t)]\eta^2(t)+i\frac{d}{dt}\eta(t)\}M^{-1}(t),$$ where the Hermiticity of $\eta(t)$, $M(t)$ and $M^{-1}(t)$ have been considered. Combining equation \[Eq34\] and equation \[Eq35\], we obtain $$\label{Eq36}
\begin{aligned}
H_{s,a}^{(-+)}(t)+H_{s,a}^{(+-)}(t)=& -M^{-1}(t)\{\eta(t)[H_s^\dag(t)M(t)-M(t)H_s(t)]- \\ &[H_s^\dag(t)M(t)-M(t)H_s(t)]\eta(t)+i[\frac{d}{dt}\eta(t)]\eta^2(t)-i\eta^2(t)\frac{d}{dt}\eta(t)\}M^{-1}(t).
\end{aligned}$$ Substituting equation \[Eq26\] into equation \[Eq36\], we have $$\label{Eq37}
\begin{aligned}
H_{s,a}^{(-+)}(t)+H_{s,a}^{(+-)}(t)=
&-M^{-1}(t)\{i\eta(t)[\frac{d}{dt}\eta(t)]\eta(t)+i\eta^2(t)[\frac{d}{dt}\eta(t)]-i[\frac{d}{dt}\eta(t)]\eta^2(t)-i\eta(t)[\frac{d}{dt}\eta(t)]\eta(t) \\
&+i[\frac{d}{dt}\eta(t)]\eta^2(t)-i\eta^2(t)\frac{d}{dt}\eta(t)\}M^{-1}(t).
\end{aligned}$$ By canceling terms, equation \[Eq37\] vanishes, that is $$\label{Eq38}
H_{s,a}^{(-+)}(t)=-H_{s,a}^{(+-)}(t)=[H_s(t)\eta(t)-\eta(t)H_s(t)-i\frac{d}{dt}\eta(t)]M^{-1}(t),$$ which is the second formula in equation \[Eq18\].
D. Proof of the universality of this dilation method
----------------------------------------------------
Here we show that our method can be utilized for Hermitian dilation of an arbitraty Hamiltonian $H_s(t)$. Without loss of generality, the evolution time is denoted by $T$, so $t\in[0,T]$ during the evolution. Since $H_s(t)$ is a Hamiltonian governing the evolution of a quantum system, given the state $|\psi(t_1)\rangle=|\psi_1\rangle$ at any time point $t_1\in[0,T]$, the subsequent state evolution is uniquely determined by $H_s(t)$. That is, the Schrödinger equation with an initial value, $$\label{Eq39}
\left\{
\begin{aligned}
& i\frac{d}{dt}|\psi(t)\rangle=H_s(t)|\psi(t)\rangle, \\
&|\psi(t_1)\rangle=|\psi_1\rangle,
\end{aligned}
\right.$$ has a unique solution on $[t_1,T]$. Stated another way, the evolution operator $$\label{Eq40}
\varepsilon_1(t;t_1)=\mathcal{T}e^{-i\int_{t_1}^{t}H_s(t)dt},$$ exist on $[t_1,T]$. Obviously, $\varepsilon_1(t_1;t_1)=I$ for any $t_1\in[0,T]$.
It can be proved that $\varepsilon_1(t;t_1)$ is an invertible operator for $t\in[0,T]$ as follows. Suppose that $\varepsilon_1(t;t_1)$ is not invertible, then we can divide the evolution time into several segments, for example, $[t_1,t]=[t_1,t_2]\cup[t_2,t_3]\cup...\cup[t_{k-1},t_k]\cup[t_k,t]$, where $t_1<t_2<t_3<...<t_k<t$, in this case, we have $$\label{Eq41}
\varepsilon_1(t;t_1)=\varepsilon_1(t;t_k)\varepsilon_1(t_k;t_{k-1})...\varepsilon_1(t_3;t_2)\varepsilon_1(t_2;t_1).$$ If $\varepsilon_1(t;t_1)$ is not invertible, there must be at least a segment corresponding to which the evolution operator is not invertible. Without loss of generality, suppose $\varepsilon_1(t_{n+1};t_n)$, the evolution operator on $[t_n,t_{n+1}]$ where $t_1<t_n<t_{n+1}<t$, is not invertible. The segment $[t_n,t_{n+1}]$ can be further divided into several subsegments. Similarity, there is at least a subsegment corresponding to which the evolution operator is not invertible. The division can be implemented for arbitrary times. As a result, there is a segment with infinitesimal length and the corresponding evolution operator is not invertible. However, an evolution operator with infinitesimal time duration tends to $I$, which is invertible. The contradiction shows that $\varepsilon_1(t;t_1)$ must be an invertible operator,then the inverse operator of $\varepsilon_1(t;t_1)$ can be written as $$\label{Eq42}
\varepsilon_1^{-1}(t;t_1)=\overline{\mathcal{T}}e^{i\int_{t_1}^{t}H_s(t)dt}.$$
According to the derivation of $H_{s,a}(t)$ in part one, the dilated Hamiltonian $H_{s,a}(t)$ can be obtained once a differentiable $\eta(t)$, which is a solution of equation \[Eq9\], is derived. Equations \[Eq12\] and \[Eq13\] provide the solution to equation \[Eq9\], given that $M(t)$ is a positive operator with all eigenvalues larger than $1$ during the evolution. It will be shown in the following that, by appropriate selection of the initial operator $M(0)$, the aforementioned requirement can be always satisfied for an arbitrary Hamiltonian $H_s(t)$. To start with, select a positive operator $M_0^\prime$, of which all the eigenvalues are larger than $1$. Then $M(0)^\prime$ is invertible and can be expressed as $$\label{Eq43}
M_0^\prime=\zeta^\dag\zeta.$$ Define $$\label{Eq44}
M^\prime(t)\equiv\mathcal{T}e^{-i\int_{0}^{t}H_s^\dag(t)dt}M(0)^\prime\overline{\mathcal{T}}e^{i\int_{0}^{t}H_s(t)dt}$$ then $$\label{Eq45}
M^\prime(t)=(\zeta\overline{\mathcal{T}}e^{i\int_{0}^{t}H_s(t)dt})^\dag(\zeta\overline{\mathcal{T}}e^{i\int_{0}^{t}H_s(t)dt}),$$ is a positive operator. Because $\overline{\mathcal{T}}e^{i\int_{0}^{t}H_s(t)dt}=\varepsilon_1^{-1}(t;0)$, $M_0^\prime$, and $\mathcal{T}e^{-i\int_{0}^{t}H_s^\dag(t)dt}=[\varepsilon_1^{-1}(t;0)]^\dag$ are all invertible operators, then according to equation \[Eq45\], $M^\prime(t)$ is an invertible operator. Therefore, all the eigenvalues of $M^\prime(t)$ are larger than $0$. Suppose the minimum of the eigenvalues of $M^\prime(t)$ is $\mu^\prime$, then $\mu^\prime>0$. Take $0<\mu<\mu^\prime$ and $M(0)=M_0^\prime/\mu$, then $$\label{Eq46}
M(t)=\frac{1}{\mu}M^\prime(t),$$ is a positive operator with all eigenvalues larger than $1$ during the evolution. Therefore, the dilated Hamiltonian $H_{s,a}(t)$ can be obtained with our method for an arbitrary Hamiltonian $H_s(t)$.
. Construct Hamiltonian $H_{s,a}(t)$ in NV center
=================================================
The dilated Hamiltonian shown in equation \[Eq20\] takes the form $$\label{Eq47}
H_{s,a}(t)=\Lambda(t)\otimes I+\Gamma(t)\otimes\sigma_z,$$ where operators $\Lambda(t)$ and $\Gamma(t)$ are given in equation \[Eq21\]. By expanding $\Lambda(t)$ and $\Gamma(t)$ in terms of Pauli operators, we can rewrite $H_{s,a}(t)$ as $$\label{Eq48}
\begin{aligned}
H_{s,a}(t)=
&B_1(t)I\otimes I+A_1(t)\sigma_x\otimes I + B_2(t)\sigma_y\otimes I+B_3(t)\sigma_z\otimes I+ \\
&A_2(t)I\otimes\sigma_z+B_4(t)\sigma_x\otimes\sigma_z +A_3(t)\sigma_y\otimes\sigma_z+A_4(t)\sigma_z\otimes\sigma_z,
\end{aligned}$$ where $A_i(t)$ and $B_i(t)$, $i\in[1,4]$, are the corresponding decomposition coefficients (real parameters). According to our numerical calculation, coefficients $B_i(t), i\in[1,4],$ vanish because of the form of the $H_s$ we choose. In this case, $H_{s,a}(t)$ reduces to $$\label{Eq49}
H_{s,a}(t)=A_1(t)\sigma_x\otimes I+A_2(t)I\otimes\sigma_z+A_3(t)\sigma_y\otimes\sigma_z +A_4(t)\sigma_z\otimes\sigma_z.$$
Fig. \[FigS1\] shows the parameters $A_i(t)$ ($i=1,2,3,4$) of the corresponding dilated Hamiltonian $H_{s,a}(t)$ when $H_s$ is in the region of unbroken $PT$ symmetry (Fig. \[FigS1\].(a)), at the exceptional point (Fig. \[FigS1\].(b)), and in the region of broken $PT$ symmetry (Fig. \[FigS1\].(c)), respectively.
![**Parameters $A_i(t)$ in the dilated Hamiltonian.** Parameters $A_i(t)$ ($i=1,2,3,4$) in the dilated Hamiltonian $H_{s,a}(t)$ as a function of t for (a) $r=0.6$, (b) $r=1.0$ and (c) $r=1.4$.[]{data-label="FigS1"}](Supplementary_Fig1.pdf){width="15cm"}
NV center is a kind of point defect in diamond consisting of a substitutional nitrogen atom and an adjoint vacancy. The electron spin and nuclear spin in NV center consist a highly controllable two-qubit solid-state system. The electron spin with a spin triple ground state ($S=1$) is coupled with the nearby $^{14}\mathrm{N}$ nuclear spin. By applying an external magnetic field $B_0$ along the NV axis, the Hamiltonian of NV center can be written as $$\label{Eq50}
H_{\mathrm{NV}}=2\pi(DS_z^2 + \omega_eS_z + QI_z^2 + \omega_nI_z + AS_zI_z),$$ where $\omega_e=-\gamma_eB_0/2\pi$ ($\omega_n=-\gamma_nB_0/2\pi$) is the Zeeman splitting of the electron ($^{14}\mathrm{N}$ nuclear) spin, with $\gamma_e$ ($\gamma_n$) being the electron ($^{14}\mathrm{N}$ nuclear) gyromagnetic ratio. $S_z$ and $I_z$ are the spin operators of the electron spin (spin-1) and the $^{14}\mathrm{N}$ nuclear spin (spin-1), respectively. $D=2.87$ GHz is the axial zero-field splitting parameter for the electron spin. $Q=-4.95$ MHz is the quadrupole splitting parameter of the $^{14}\mathrm{N}$ nuclear spin. $A=-2.16$ MHz is the hyperfine coupling parameter.
The experiment is performed in the two qubit subspace spanned by $|m_s=0,m_I=1\rangle, |m_s=0,m_I=0\rangle, |m_s=-1,m_I=1\rangle,$ and $|m_s=-1,m_I=0\rangle$, labeled by $|0\rangle_e|1\rangle_n, |0\rangle_e|0\rangle_n, |-1\rangle_e|1\rangle_n$ and $|-1\rangle_e|0\rangle_n$, in which the Hamiltonian can be simplified as $$\label{Eq51}
H_0=\pi [-(D-\omega_e-\frac{A}{2})\sigma_z\otimes I+(Q+\omega_n-\frac{A}{2})I\otimes\sigma_z+\frac{A}{2}\sigma_z\otimes\sigma_z].$$
To construct $H_{s,a}(t)$ in NV center, we can apply two slightly detuned microwave (MW) pulses to selectively drive the two electron spin transitions, as depicted in Fig. 1b in the main text. The total Hamiltonian in the two qubit subspace when applying MW pulses can be written as $$\label{Eq52}
\begin{aligned}
H_{\mathrm{tot}}=\ &\pi [-(D-\omega_e-\frac{A}{2})\sigma_z\otimes I+(Q+\omega_n-\frac{A}{2})I\otimes\sigma_z+\frac{A}{2}\sigma_z\otimes\sigma_z] \\
&+2\pi\Omega_1(t)\cos[\int_0^t\omega_1(\tau)d\tau+\phi_1(t)]\sigma_x\otimes|1\rangle_n~_n\langle1| +2\pi\Omega_2(t)\cos[\int_0^t\omega_2(\tau)d\tau+\phi_2(t)]\sigma_x\otimes|0\rangle_n~_n\langle0|,
\end{aligned}$$ where $\Omega_1(t)$, $\omega_1(t)$ and $\phi_1(t)$ ($\Omega_2(t)$, $\omega_2(t)$ and $\phi_2(t)$) are the Rabi frequency, angular frequency and phase of the MW pulses which drive the electron spin transition if the nuclear spin is $|1\rangle_n$ ($|0\rangle_n$). By choosing interaction picture $$\label{Eq53}
U_{rot}=e^{i\int_0^t[H_0-A_2(\tau)I\otimes\sigma_z-A_4(\tau)\sigma_z\otimes\sigma_z]d\tau},$$ the total Hamiltonian transforms to $$\label{Eq54}
\begin{aligned}
H_{rot}=\ &U_{rot}H_{tot}U_{rot}^\dag-iU_{rot}\frac{dU_{rot}^\dag}{dt} \\ =\ &A_2(t)I\otimes\sigma_z+A_4(t)\sigma_z\otimes\sigma_z \\
&+2\pi\Omega_1(t)\cos[\int_0^t\omega_1(\tau)d\tau+\phi_1(t)]\{\cos[\omega_{MW1}t+\int_0^t2A_4(\tau)d\tau]\sigma_x+\sin[\omega_{MW1}t+\int_0^t2A_4(\tau)d\tau]\sigma_y\}\otimes|1\rangle_n~_n\langle1| \\
&+2\pi\Omega_2(t)\cos[\int_0^t\omega_2(\tau)d\tau+\phi_2(t)]\{\cos[\omega_{MW2}t-\int_0^t2A_4(\tau)d\tau]\sigma_x+\sin[\omega_{MW2}t-\int_0^t2A_4(\tau)d\tau]\sigma_y\}\otimes|0\rangle_n~_n\langle0|,
\end{aligned}$$ with $\omega_{MW1}$($\omega_{MW2}$) being the energy difference between $|0\rangle_e|1\rangle_n$ and $|-1\rangle_e|1\rangle_n$ ($|0\rangle_e|0\rangle_n$ and $|-1\rangle_e|0\rangle_n$). By choosing $$\label{Eq55}
\left\{
\begin{aligned}
&\omega_1(t)=\omega_{MW1}+2A_4(t) \\
&\omega_2(t)=\omega_{MW2}-2A_4(t) \\
&\Omega_1(t)=\Omega_2(t)=\Omega(t) \\
&-\phi_1(t)=\phi_2(t)=\phi(t)
\end{aligned},
\right.$$ then in the condition of rotating wave approximation, $H_{rot}$ can be reduced to $$\label{Eq56}
H_{rot}= A_2(t)I\otimes\sigma_z+A_4(t)\sigma_z\otimes\sigma_z
+\pi\Omega(t)\cos[\phi(t)]\sigma_x\otimes I+\pi\Omega(t)\sin[\phi(t)]\sigma_y\otimes I.$$ Comparing equation \[Eq56\] with equation \[Eq51\], we can choose $$\label{Eq57}
\left\{
\begin{aligned}
&\Omega(t)=\frac{\sqrt{A_1^2(t)+A_3^2(t)}}{2\pi},\\
&\phi(t)=\arctan\frac{A_3(t)}{A_1(t)},\\
\end{aligned}
\right.$$ to realize Hamiltonian $H_{s,a}(t)$.
. Characterization of $\psi(t)$
===============================
The state of the system after the evolution governed by the dilated Hamiltonian $H_{s,a}(t)$ is $$\label{Eq58}
|\Psi(t)\rangle=|\psi(t)\rangle_e|-\rangle_n+\eta_0(t)|\psi(t)\rangle_e|+\rangle_n.$$ By applying a rotation of $\pi/2$ along $\mathrm{-X}$ axis on the nuclear spin qubit, as depected in Fig. 1c in the main text, the state evolutes to $$\label{Eq59}
|\Phi(t)\rangle=|\psi(t)\rangle_e|1\rangle_n+\eta_0(t)|\psi(t)\rangle_e|0\rangle_n.$$ Thus the evolution $|\psi(t)\rangle_e$ governed by $\mathcal{PT}$ symmetric Hamiltonian $H_s$ can be characterized by $P_0$, the normalized population of state $|0\rangle_e|1\rangle_n$ after selecting $|1\rangle_n$. $P_0$ can be calculated by $P_0=P_{|0\rangle_e|1\rangle_n}/(P_{|0\rangle_e|1\rangle_n}+P_{|-1\rangle_e|1\rangle_n})$, with $P_{|0\rangle_e|1\rangle_n}$ ($P_{|-1\rangle_e|1\rangle_n}$) being the population of state $|0\rangle_e|1\rangle_n$ ($|-1\rangle_e|1\rangle_n$) of state $|\Phi(t)\rangle$. This can be acquired by measuring the population distributon of the final state.
The four levels we choose to perform our experiment give rise to different photoluminescence(PL) rates\[\], providing information of the population distribution of the final state. However, different distributions can present the same PL rate, thus a set of pulse sequences are needed to determine the population distribution.
![ **Schematic normalization and measurement sequences.** Here $\rho_\textrm{i}$ denotes the initialized state after a optical excitation, $\rho_\textrm{f}$ denotes the final state after the nuclear $\frac{\pi}{2}$ pulse in readout (main text figure1). The length of MW pulse ($\pi_{MW1}$) is $896.5~$ns while the length of RF pulse ($\pi_{RF1},\pi_{RF2}$) is $20~\mathrm{\mu s}$. $N^x$ indicates the detected photoluminescence intensity after applying the pulse sequence $x$.[]{data-label="FigS2"}](Supplementary_Fig2.pdf){width="15cm"}
The PL rates of the four levels are measured by the pulse sequences shown in Fig. \[FigS2\](a). Optical excitation at a static magnetic field of about 506 G can efficiently polarize the system to $|0\rangle_e|1\rangle_n$ via electron spin$-$nuclear spin flip-flop-processes in the electronic excited state of the NV center\[\], in combination with an electron-spin dependent relaxation mechanics. This metnod is used to initialize the system to get the state $\rho_i$ in Fig. \[FigS2\](a). The number of photons we detect upon PL readout the initialized state is $$\label{Eq60}
N^0=P_eP_nN_{|0\rangle_e|1\rangle_n}+P_e(1-P_n)N_{|0\rangle_e|0\rangle_n}+(1-P_e)P_nN_{|-1\rangle_e|1\rangle_n}+(1-P_e)(1-P_n)N_{|-1\rangle_e|0\rangle_n},$$ where $N_i$ is the number of detected photons if all of the population occupies level $i$, and $P_e$($P_n$) is the population of $|0\rangle_e$ ($|1\rangle_n$) after the optical excitation. Because of the high polarization of the nuclear spin, $P_n$ is regared as 1 during the calculation. Considering all the pulse sequences in Fig. \[FigS2\](a), we can obtain the following equations $$\label{Eq61}
\begin{bmatrix}
P_e & 0 & 1-P_e & 0 \\
1-P_e & 0 & P_e & 0 \\
0 & P_e & 1-P_e & 0 \\
0 & 1-P_e & P_e & 0 \\
1-P_e & 0 & 0 & P_e \\
\end{bmatrix}
\begin{bmatrix}
N_{|0\rangle_e|1\rangle_n} \\ N_{|0\rangle_e|0\rangle_n} \\ N_{|-1\rangle_e|1\rangle_n} \\ N_{|-1\rangle_e|0\rangle_n} \\
\end{bmatrix}
= \begin{bmatrix}
N^0 \\ N^{\pi_{MW1}} \\ N^{\pi_{RF1}} \\ N^{\pi_{MW1}\pi_{RF1}} \\ N^{\pi_{MW1}\pi_{RF2}} \\ \end{bmatrix},$$ where $N^x$ indicates the detected PL after applying the pulse sequence $x$. $\pi_{MW1}$ ($\pi_{RFR}$,$\pi_{RF2}$) denotes a selective $\pi$ pulse between $|0\rangle_e|1\rangle_n$ and $|-1\rangle_e|1\rangle_n$ ($|0\rangle_e|1\rangle_n$ and $|0\rangle_e|0\rangle_n$, $|-1\rangle_e|1\rangle_n$ and $|-1\rangle_e|0\rangle_n$), which flips the poulations in these levels. By solving equation \[Eq61\], the PL rate of each level can be obtained.
Knowing $N_i$ we can determine the occupation probabilits $P_i$ of the final state $\rho_f$ by pulse sequences shown in Fig. \[FigS2\](b). The PL of an final state with level occupation probabilities $P_i$ is $$\label{Eq62}
N^0=P_{|0\rangle_e|1\rangle_n}N_{|0\rangle_e|1\rangle_n}+P_{|0\rangle_e|0\rangle_n}N_{|0\rangle_e|0\rangle_n}+P_{|-1\rangle_e|1\rangle_n}N_{|-1\rangle_e|1\rangle_n}+P_{|-1\rangle_e|0\rangle_n}N_{|-1\rangle_e|0\rangle_n}.$$ By flipping populations within the two-qubit subspace using the pulse sequences in Fig. \[FigS2\](b) and measuring the resulting PL we can calculate the $P_i$ from $$\label{Eq63}
\begin{bmatrix}
N_{|0\rangle_e|1\rangle_n} & N_{|0\rangle_e|0\rangle_n} & N_{|-1\rangle_e|1\rangle_n} & N_{|-1\rangle_e|0\rangle_n} \\
N_{|-1\rangle_e|1\rangle_n} & N_{|0\rangle_e|0\rangle_n} & N_{|0\rangle_e|1\rangle_n} & N_{|-1\rangle_e|0\rangle_n} \\
N_{|0\rangle_e|0\rangle_n} & N_{|0\rangle_e|1\rangle_n} & N_{|-1\rangle_e|1\rangle_n} & N_{|-1\rangle_e|0\rangle_n} \\
1 & 1 & 1 & 1 \\
\end{bmatrix}\begin{bmatrix} P_{|0\rangle_e|1\rangle_n} \\ P_{|0\rangle_e|0\rangle_n} \\ P_{|-1\rangle_e|1\rangle_n} \\ P_{|-1\rangle_e|0\rangle_n} \\ \end{bmatrix} = \begin{bmatrix} N^0 \\ N^{\pi_{MW1}} \\ N^{\pi_{RF1}} \\ 1 \\ \end{bmatrix}.$$
. Experimental acquisition of the eigenvalues of $\mathcal{PT}$ symmetric Hamiltonians
======================================================================================
By fitting the experimental evolution curve to the theoretical evolution curve we can get the parameter $r_{exp}$, then the eigenvalues of $\mathcal{PT}$ symmetric Hamiltonian $H_s$ can be calculated by $E_\pm=\pm\sqrt{1-r_{exp}^2}$. The Hamiltonian we realized in our experimental system is $$\label{Eq64}
H_s=
\begin{pmatrix}
& ir &1 \\
&1 &-ir \\
\end{pmatrix}.$$ The time evolution operator corresponding to $H_s$ takes the form $$\label{Eq65}
\begin{aligned}
U(t)&=e^{-iH_st} \\
&=
\begin{pmatrix}
&\frac{e^{t\sqrt{r^2-1}}(r+\sqrt{r^2-1}) -e^{-t\sqrt{r^2-1}}(r-\sqrt{r^2-1})}{2\sqrt{r^2-1}}
&\frac{ie^{-t\sqrt{r^2-1}}-ie^{t\sqrt{r^2-1}}}{2\sqrt{r^2-1}} \\
&\frac{ie^{-t\sqrt{r^2-1}}-ie^{t\sqrt{r^2-1}}}{2\sqrt{r^2-1}}
&\frac{e^{t\sqrt{r^2-1}}(-r+\sqrt{r^2-1}) -e^{-t\sqrt{r^2-1}}(-r-\sqrt{r^2-1})}{2\sqrt{r^2-1}} \\
\end{pmatrix}.
\end{aligned}$$ The initial state we choose is $|\psi(0)\rangle=|0\rangle=(1,0)^T$. Then the final state at time $t$, after the evolution governed by Hamiltonian $H_s$, is $$\label{Eq66}
|\psi(t)\rangle=\frac{1}{2\sqrt{r^2-1}}
\begin{pmatrix}
&e^{t\sqrt{r^2-1}}(r+\sqrt{r^2-1})-e^{-t\sqrt{r^2-1}}(r-\sqrt{r^2-1}) \\
&ie^{-t\sqrt{r^2-1}}-ie^{t\sqrt{r^2-1}} \\
\end{pmatrix}.$$ So the population of state $|0\rangle$ at moment $t$ is $$\label{Eq67}
P_0=\frac{|e^{t\sqrt{r^2-1}}(r+\sqrt{r^2-1}) -e^{-t\sqrt{r^2-1}}(r-\sqrt{r^2-1})|^2}{|e^{t\sqrt{r^2-1}}(r+\sqrt{r^2-1}) -e^{-t\sqrt{r^2-1}}(r-\sqrt{r^2-1})|^2 +|ie^{-t\sqrt{r^2-1}}-ie^{t\sqrt{r^2-1}}|^2}.$$ We use formula \[Eq67\] to fit our experimental data to get the parameter $r_{exp}$. The fitting result is shown on TABLE \[table1\].
$r$ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5
--------------------------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- -------
$r_{\mathrm{exp}}$ 0.006 0.099 0.191 0.328 0.416 0.472 0.616 0.713 0.800 0.906 1.002 1.079 1.170 1.321 1.418 1.509
$\delta r_{\mathrm{exp}}$ 0.018 0.024 0.014 0.016 0.009 0.015 0.006 0.006 0.003 0.006 0.010 0.015 0.021 0.019 0.038 0.001
: **Parameter $r_{\mathrm{exp}}$ obtained from the time evolution under $\mathcal{PT}$ symmetric Hamiltonian.** $r$ is the parameter at which the experiment implemented. $r_{\mathrm{exp}}$ is obtained by fitting the evolution curve. $\delta r_{\mathrm{exp}}$ is the fitting error.
\[table1\]
. $T_2^\star$ of NV center
==========================
The sample used in our experiment is isotopically purified (\[$^{12}$C\]=99.9%). So the coherence time of the electron spin, which is the system qubit in this experiment, is prolonged. Ramsey sequence is utilized to measure the $T_2^\star$ of the electron spin, and the result is demonstrated in Fig. \[FigS3\]. The measurement shows that the $T_2^\star$ of our sample is $19(2)~\mathrm{\mu s}$.
![**Coherence time of NV center.** Result of the Ramsey experiment (insert, pulse sequence) for the electron spin. The solid red line is fit to the experiment data (black square), the red dashed line is the fit to the envelope curve. The decay time of FID is measured to be $T_2^\star=19(2)~\mathrm{\mu s}$.[]{data-label="FigS3"}](Supplementary_Fig3.pdf){width="15cm"}
. Nuclear spin qubit operation
==============================
The single qubit operations of the nuclear spin qubit in the state prepartion and read out are realized by applying two channel radio frequency (RF) pulses simultaneously. The frequencies of the RF pulses are $2.9~\mathrm{MHz}$ and $5.1~\mathrm{MHz}$, corresponding to the nuclear spin transition frequencies. The RF pulses are generated by AWG and then carried by a homebuild RF coil with dual resonance frequencies. The coil is designed according to ref.\[\]. There are two input ports at the coil to input the RF pulses, and the return loss of the two ports is shown in Fig. \[FigS4\]. The Rabi frequencies of the nuclear spin transitions at different electron subspace are both calibrated to be 25 $\mathrm{kHz}$.
![**Return loss of the RF Coil.** Return loss of the RF coil, red line stands for the low frequency port, at $2.9~\mathrm{MHz}$, the return loss is $-9.1~\mathrm{dB}$, bule line stands for the high frequency port, at $5.1~\mathrm{MHz}$, the retirn loss is $-6.5~\mathrm{dB}$. []{data-label="FigS4"}](Supplementary_Fig4.pdf){width="15cm"}
[99]{}
M.Steiner, P.Neumann, *et al.* Universal Enhancement of the optical readout fidelity of single electron spins at nitrogen-vacancy centers in diamond. *Phys. Rev. B* **81**, 035205 (2010).
V.Jacques, P.Neumann *et al.* Dynamic Polarization of Single Nuclear Spin by Optical Pumping of Nitrogen-Vacancy Color Centers in Diamonds at room temperature. *Phys. Rev. Lett.* **102**, 057403 (2009).
A.Dabirzadeh, RF coil design for multi-frequency magnitic resonance imaging and spectroscopy. Texas, A$\&$M university, December 2008.
[^1]: These authors contributed equally to this work.
[^2]: These authors contributed equally to this work.
|
---
abstract: 'We study the properties of time evolution of the $K^{0}-\bar{K}^{0} $ system in spectral formulation. Within the one–pole model we find the exact form of the diagonal matrix elements of the effective Hamiltonian for this system. It appears that, contrary to the Lee–Oehme–Yang (LOY) result, these exact diagonal matrix elements are different if the total system is CPT–invariant but CP–noninvariant.'
author:
- |
Justyna Jankiewicz\
Instytute of Physics\
University of Zielona Gora\
Prof. Z. Szafrana 4a, Zielona Gora, Poland\
[email protected]
date: '09-05-2004 - 12-04-2004'
title: |
Properties of the Effective Hamiltonian\
for the System of Neutral Kaons[^1]
---
Introduction
============
Following the LOY approach, a nonhermitian Hamiltonian $H_{\parallel} $ is usually used to study the properties of the particle-antiparticle unstable system [@p:5] - [@p:11] $$H_{\parallel}\equiv M-\frac{i}{2}\Gamma, \label{j1-5}$$ where $$M=M^{+} \ , \ \Gamma = \Gamma^{+} \label{j1-6}$$ are $(2\times2) $ matrices acting in a two-dimensional subspace $\mathcal{H}_{\parallel} $ of the total state space $\mathcal{H}$. The $M $-matrix is called the mass matrix and $\Gamma$ is the decay matrix. Lee, Oehme and Yang derived their approximate effective Hamiltonian $H_{\parallel}\equiv H_{LOY} $ by adapting the one-dimensional Weisskopf-Wigner (WW) method to the two-dimensional case corresponding to the neutral kaon system.\
Almost all properties of this system can be described by solving the Schrödinger-like equation [@p:5] - [@p:8] $$i\frac{\partial }{\partial t}|\psi
;t\rangle_{\parallel}=H_{\parallel }|\psi ;t\rangle_{\parallel}, \
\ \ (t\geq t_{0}> -\infty ) \label{j1-7}$$ (where we have used $\hbar =c=1 $) with the initial conditions $$\parallel |\psi ;t=t_{0}\rangle_{\parallel} \parallel =1,
\ \ |\psi ;t_{0}=0\rangle_{\parallel} =0, \label{j1-8}$$ for $|\psi ;t=t_{0}\rangle_{\parallel} $ belonging to the subspace of states $\mathcal{H}_{\parallel} $ ($\mathcal{H}_{\parallel}\subset \mathcal{H} $), spanned by, e.g., orthonormal neutral kaons states $K^{0} $ and $\bar{K}^{0} $. The solutions of Eq. (\[j1-7\]) may be written in a matrix form, which may be used to define the time evolution operator $U_{\parallel}(t) $ acting in subspace $\mathcal{H}_{\parallel} $ $$|\psi ;t\rangle_{\parallel}=U_{\parallel}(t)|\psi
;t_{0}=0\rangle_{\parallel}\equiv U_{\parallel}(t)|\psi
\rangle_{\parallel}, \label{j1-9}$$ where $$|\psi \rangle_{\parallel}\equiv a_{1}|\textbf{1}\rangle +
a_{2}|\textbf{2}\rangle \label{j1-10}$$ and $|\textbf{1}\rangle $ denotes particle “1” – in the present case $|K^{0}\rangle $ whereas $|\textbf{2}\rangle $ corresponds to the antiparticle state for particle ”1”: $|\bar{K}^{0}\rangle,
$ $\langle\textbf{j}|\textbf{k}\rangle =\mathcal{\delta}_{jk}, $ $j,k=1,2 $. It is usually assumed that the real parts of the diagonal matrix elements of $H_{\parallel}$, namely $\Re (\cdot),
$ $$\Re (h_{jj})\equiv M_{jj} \ \ (j=1,2), \label{j1-11}$$ where $$h_{jk}=\langle \textbf{j}|H_{\parallel}|\textbf{k} \rangle \
\ \ \ (j,k=1,2) \label{j1-12}$$ correspond to the masses of the particle “1” and its antiparticle “2” [@p:5] - [@p:11]. $\Im (\cdot) $ is the imaginary part of $h_{jj}$ $$\Im (h_{jj})\equiv \Gamma_{jj} \ \ \ (j=1,2) \label{j1-13}$$ and $\Gamma_{jj}$ are interpreted as the decay widths of the particles. According to the standard result of the LOY approach, in a CPT invariant system, i.e. when $$\Theta H \Theta^{-1}=H, \label{j1-3}$$ (where $\Theta = CPT$, $H=H^{+}$ is the Hamiltonian of the total system under consideration)\
we have $$h_{11}^{LOY}=h_{22}^{LOY}. \label{j1-14}$$
The universal properties of the unstable particle-antiparticle subsystem described by the $H$ fulfilling the condition (\[j1-3\]), may be investigated by using the matrix elements of the exact $U_{\parallel}, $ instead of the approximate one used in the LOY theory. The exact $U_{\parallel} $ can be written as follows $$U_{\parallel}(t)=PU(t)P, \label{j1-17}$$ where $$P\equiv |\textbf{1}\rangle \langle\textbf{1}|+ |\textbf{2}\rangle \langle\textbf{2}|,
\label{j1-18}$$ and $U(t) $ is the exact evolution operator acting in the whole state space. This operator is the solution of the Schrödinger equation $$i\frac{\partial }{\partial t}U(t)|\phi \rangle = HU(t)|\phi
\rangle , \ \ U(0)=I. \label{j1-19}$$ $I $ is the unit operator in the $\mathcal{H}$ space and $|\phi \rangle \equiv | \phi
; t_{0}=0\rangle \in \mathcal{H} $ is the initial state of the system.\
In the remaining part of the poster we will be using the following matrix representation of the evolution operator $$U_{\parallel}(t)\equiv \left( \begin{array}{cc} \textbf{A(t)} &
\textbf{0}
\\ \textbf{0} & \textbf{0}
\end{array} \right), \label{j1-21}$$ where $\textbf{0} $ denotes the zero submatrices of the suitable dimension, and the $\textbf{A(t)} $ is a $(2\times 2) $ matrix acting in $\mathcal{H}_{\parallel} $ $$\textbf{A(t)}= \left( \begin{array}[c]{ll} A_{11}(t) & A_{12}(t) \\
A_{21}(t)& A_{22}(t)
\end{array} \right), \label{j1-22}$$ where $$A_{jk}(t)=\langle
\textbf{j}|U_{\parallel}(t)|\textbf{k}\rangle \equiv \langle
\textbf{j}|U(t)|\textbf{k}\rangle \ \ (j,k=1,2). \label{j1-23}$$ Assuming that the property (\[j1-3\]) holds and using the following definitions $$\Theta |\textbf{1}\rangle \equiv e^{-i\theta } |\textbf{2}\rangle,
\ \ \ \Theta |\textbf{2}\rangle \equiv e^{-i\theta }
|\textbf{1}\rangle, \label{j1-24}$$ it can be shown that $$A_{11}(t)=A_{22}(t). \label{j1-25}$$
A very important relation between the amplitudes $A_{12}(t) $ and $A_{21}(t) $ follows from the famous Khalfin Theorem [@p:16] - [@p:18] $$r(t)\equiv \frac{A_{12}(t)}{A_{21}(t)}=const\equiv r \ \
\Rightarrow \ \ |r|=1. \label{j1-26}$$ General conclusions concerning the properties of the matrix elements of $H_{\parallel}$ can be drawn by analyzing the following identity [@p:7; @p:1] $$H_{\parallel}(t)\equiv i\frac{\partial \textbf{A}(t)}{\partial
t}[\textbf{A}(t)]^{-1}. \label{j1-37}$$
Using Eq. (\[j1-37\]) we can easily find the general formulae for the diagonal matrix elements $h_{jj}$, of $H_{\parallel}(t)$ and next assuming (\[j1-3\]) and using relation (\[j1-25\]) which follows from our earlier assumptions, we get $$h_{11}(t)-h_{22}(t)=\frac{i}{det\textbf{A}(t)}\Biggl(\frac{\partial
A_{21}(t) }{\partial t}A_{12}(t)-\frac{\partial A_{12}(t)
}{\partial t}A_{21}(t)\Biggl). \label{j1-40}$$ In [@p:1] it was shown, by using relation (\[j1-40\]), that this result means that in the considered case (with CPT conserved) for $t>0$ we get the following theorem $$h_{11}(t)-h_{22}(t)=0 \ \ \ \Leftrightarrow \ \ \
\frac{A_{12}(t)}{A_{21}(t)}=const \ \ \ (t>0) . \label{j1-42}$$ Thus, for $t>0 $ the problem under study is reduced to the Khalfin Theorem (see relation (\[j1-26\])) [@p:1].
Having noticed this, let us now turn our attention to the conclusions following from Khalfin’s Theorem. $CP $ noninvariance requires that $|r|\neq 1$ [@p:5; @p:6; @p:7; @p:8; @p:16; @p:18; @p:14; @p:2]. This means that in this case the following condition must be fulfilled: $r=r(t)\neq const. $ Consequently, if in the considered system property (\[j1-3\]) holds, but at the same time $$[\mathcal{CP},H]\neq 0 \label{j1-45}$$ and the unstable states “1” i “2” are connected by (\[j1-24\]), then in this system for $t>0 $ [@p:1] $$h_{11}(t)-h_{22}(t)\neq 0. \label{j1-46}$$ So, in the exact quantum theory the difference $(h_{11}(t)-h_{22}(t))$ cannot be equal to zero with CPT conserved and CP violated.
A model: one pole approximation
===============================
While describing the two and three pion decay we are mostly interested in the $|K_{S}\rangle $ and $|K_{L}\rangle $ superposition of $|K^{0}\rangle$ and $|\bar{K}^{0}\rangle.$ These states correspond to the physical $|K_{S}\rangle $ and $|K_{L}\rangle $ neutral kaon states [@p:2; @p:26] $$\begin{aligned}
|K_{S}\rangle =p|K^{0}\rangle +q|\bar{K}^{0}\rangle,
\ \ |K_{L}\rangle =p|K^{0}\rangle
-q|\bar{K}^{0}\rangle. \label{j1-47}\end{aligned}$$ Using the spectral formalism we can write an unstable state $|\lambda (t)\rangle $ as $$\begin{aligned}
|\lambda (t)\rangle =\sum_{q}|q (t)\rangle \omega_{\lambda}(q),
\label{j1-62}\end{aligned}$$ where $|q (t)\rangle =e^{-itH}|q \rangle $, vectors $|q \rangle $ form a complete set of eigenvectors of the hermitian, quantum-mechanical Hamiltonian $H$ and $ \omega_{\lambda}(q) =
\langle q|\lambda \rangle.$ If the continuous eigenvalue is denoted by $m$, we can define the survival amplitude $A(t)$ (or the transition amplitude in the case of $K^{0}\leftrightarrow
\bar{K}^{0} $ ) in the following way: $$\begin{aligned}
A(t)=\int\limits_{Spec(H)}dm \; e^{-imt}\rho (m), \label{j1-62a}\end{aligned}$$ where the integral extends over the whole spectrum of the Hamiltonian and density $\rho (m)$ is defined as follows $$\begin{aligned}
\rho (m)=|\omega_{\lambda}(m)|^{2}, \label{j1-62b}\end{aligned}$$ where $\omega_{\lambda}(m)=\langle m |\lambda \rangle . $
In accordance with formula (\[j1-62\]) the unstable states $K_{S} $ and $K_{L} $ may now be written as a superposition of the eigenkets $$\begin{aligned}
|K_{S}\rangle =\int_{0}^{\infty }dm \;
\sum_{\alpha}\omega_{S,\alpha}(m)|\phi_{\alpha}(m)\rangle ;\label{j1-65}\\
|K_{L}\rangle =\int_{0}^{\infty }dm \;
\sum_{\beta}\omega_{L,\beta}(m)|\phi_{\beta}(m)\rangle
.\label{j1-66}\end{aligned}$$
The Breit-Wigner ansatz [@p:27] $$\begin{aligned}
\rho_{WB}(m)=\frac{\Gamma}{2\pi
}\frac{1}{(m-m_{0})^{2}+\frac{\Gamma^{2}}{4}} \equiv |\omega
(m)|^{2} \label{j1-63}\end{aligned}$$ leads to the well known exponential decay law which follows from the survival amplitude $$\begin{aligned}
A_{BW}(t)=\int_{-\infty}^{\infty}dm \;
e^{-imt}\rho_{WB}(m)=e^{-im_{0}t}e^{-\frac{1}{2}\Gamma|t|}.
\label{j1-64}\end{aligned}$$ (Note that the existence of the ground state induces non-exponential corrections to the decay law and to the survival amplitude (\[j1-64\]) — see [@p:2] ). It is therefore reasonable to assume a suitable form for $\omega_{S,\beta } $ and $\omega_{L,\beta }$. More specifically, we write [@p:2] $$\omega_{S,\beta}(m)= \sqrt{\frac{\Gamma_{S}}{2\pi}}
\frac{A_{S,\beta}(K_{S}\rightarrow\beta)}{m-m_{S}+ i\frac{\Gamma_{S}}{2}}, \ \
\omega_{L,\beta }(m)=\sqrt{\frac{\Gamma_{L}}{2\pi}}
\frac{A_{L,\beta}(K_{L}\rightarrow\beta)}{m-m_{L}+ i\frac{\Gamma_{L}}{2}}
\label{j1-73}$$ where $A_{S,\beta} $ and $A_{L,\beta} $ are decay (transition) amplitudes, end thus $$\rho_{x,\beta}(m)= \frac{\Gamma_{x}}{2\pi}
\frac{(A_{x,\beta}(K_{x}\rightarrow\beta))^{2}}{(m-m_{x})^{2}+
\frac{(\Gamma_{x})^{2}}{4}}, \label{j1-73a}$$where $x=L,S$.
In the one-pole approximation (\[j1-73\]) $A_{K^{0}K^{0}}(t) $ can be conveniently written as $$\begin{aligned}
A_{K^{0}K^{0}}(t)\nonumber&=&A_{\bar{K}^{0}\bar{K}^{0}}(t)=\nonumber
\\
&=&-\frac{1}{2\pi}\Biggl\{e^{-im_{S}t}
\left(-\int_{0}^{-\frac{m_{S}}{\gamma_{S}}}dy
\frac{e^{-i\gamma_{S}ty}}{y^{2}+1} +\int_{0}^{\infty}dy
\frac{e^{-i\gamma_{S}ty}}{y^{2}+1}\right)+\nonumber\\
&&+e^{-im_{L}t} \left(-\int_{0}^{-\frac{m_{L}}{\gamma_{L}}}dy
\frac{e^{-i\gamma_{L}ty}}{y^{2}+1}+ \int_{0}^{\infty}dy
\frac{e^{-i\gamma_{L}ty}}{y^{2}+1}\right)\Biggl\}. \label{j1-77}\end{aligned}$$ Collecting only exponential terms in (\[j1-77\]) one obtains an expression analogous to the WW approximation $$A_{K^{0}K^{0}}(t)=A_{\bar{K}^{0}\bar{K}^{0}}(t)=
\frac{1}{2}\left(e^{-im_{S}t}e^{-\gamma_{S}t}+
e^{-im_{L}t}e^{-\gamma_{L}t}\right)+N_{K^{0}K^{0}}(t).
\label{j1-78}$$ Here $N_{K^{0}K^{0}}(t)$ denotes all non-oscillatory terms present in the integral (\[j1-77\]).
Diagonal matrix elements of the effective\
Hamiltonian
==========================================
Using the decomposition of type (\[j1-78\]) and the one-pole ansatz (\[j1-73\]), we find the difference (\[j1-46\]), which is now formulated for the $K^{0} - \bar{K}^{0} $ system. Here it has the following form: $$\begin{aligned}
h_{11}(t)-h_{22}(t)=\frac{X(t)}{Y(t)}, \label{j1-79}\end{aligned}$$ where $$\begin{aligned}
X(t)= i\left(\frac{\partial A_{\bar{K}^{0}K^{0}}(t)}{\partial
t}A_{K^{0}\bar{K}^{0}}(t)-\frac{\partial
A_{K^{0}\bar{K}^{0}}(t)}{\partial t}A_{\bar{K}^{0}K^{0}}(t)\right)
\label{j1-79a}\end{aligned}$$ and $$\begin{aligned}
Y(t)= A_{K^{0}K^{0}}(t)A_{\bar{K}^{0}\bar{K}^{0}}(t)
-A_{K^{0}\bar{K}^{0}}(t)A_{\bar{K}^{0}K^{0}}(t). \label{j1-79b}\end{aligned}$$
Using the above mentioned spectral formulae in the one - pole approximation (\[j1-73\]) we get $A_{K^{0}\bar{K}^{0}}(t) $ and $A_{\bar{K}^{0}K^{0}}(t) $ $$\begin{aligned}
\nonumber&&A_{K^{0}\bar{K}^{0}}(t)= \frac{1+\pi }{8\pi
p^{\ast}q}\Biggl\{e^{-im_{S}t}e^{-\gamma_{S}t} \Biggl[
1+\nonumber\\
&&+\frac{\sqrt{\gamma_{S}\gamma_{L}}}{ \gamma_{S}} \Biggl(-2 \, i
\, \gamma_{S}C_{I} +D'_{I}-F'_{I}\Biggl)
\Biggl]+\nonumber\\
&&+e^{-im_{L}t}e^{-\gamma_{L}t}
\Biggl[-1+\nonumber\\
&&+\frac{\sqrt{\gamma_{S}\gamma_{L}}}{ \gamma_{L}} \Biggl(2 \, i
\, \gamma_{L}C_{I}-D'_{I}+F'_{I}\Biggl)
\Biggl]\Biggl\}+\nonumber\\
&&+N_{K^{0}\bar{K}^{0}}(t) \label{j1-87}\end{aligned}$$ and $$\begin{aligned}
\nonumber&&A_{\bar{K}^{0}K^{0}}(t)= \frac{1+\pi }{8\pi
pq^{\ast}}\Biggl\{e^{-im_{S}t}e^{-\gamma_{S}t} \Biggl[
1+\nonumber\\
&&+\frac{\sqrt{\gamma_{S}\gamma_{L}}}{ \gamma_{S}} \Biggl(2 \, i
\, \gamma_{S}C_{I}-D'_{I}+F'_{I}\Biggl)
\Biggl]+\nonumber\\
&&+e^{-im_{L}t}e^{-\gamma_{L}t}
\Biggl[-1+\nonumber\\
&&+\frac{\sqrt{\gamma_{S}\gamma_{L}}}{ \gamma_{L}} \Biggl(-2 \, i
\, \gamma_{L}C_{I}+D'_{I}-F'_{I}\Biggl)
\Biggl]\Biggl\}+\nonumber\\
&&+N_{\bar{K}^{0}K^{0}}(t), \label{j1-89}\end{aligned}$$ where $N_{K^{0}\bar{K}^{0}}(t)$, $N_{\bar{K}^{0}K^{0}}(t)$ denotes all non-oscillatory terms and $C_{I}, D'_{I}, F'_{I}$ are defined in [@p:2].
Using the expression for the derivative of $E_{i}$ we can find the derivatives which will be necessary for the following calculations $\frac{\partial
A_{K^{0}\bar{K}^{0}}(t)}{\partial t} $ and $\frac{\partial
A_{\bar{K}^{0}K^{0}}(t)}{\partial t} :$ $$\begin{aligned}
\frac{\partial A_{K^{0}\bar{K}^{0}}(t)}{\partial
t}\nonumber&=&\frac{1+\pi } {8\pi
p^{\ast}q}\Biggl\{e^{-im_{S}t}e^{-\gamma_{S}t}
\Biggl[-im_{S}-\gamma_{S}+\nonumber\\
&&+\sqrt{\gamma_{S}\gamma_{L}} \Biggl(2 i \gamma_{S}
C_{I}-D'_{I}+F'_{I}\Biggl)\Biggl]+\nonumber\\
&&+e^{-im_{L}t} e^{-\gamma_{L}t}\Biggl[im_{L}-\gamma_{L}+\nonumber\\
&&+\sqrt{\gamma_{S}\gamma_{L}} \Biggl(-2 i \gamma_{L}
C_{I}+D'_{I}-F'_{I}\Biggl)\Biggl]\Biggl\}+\nonumber\\
&&+\Delta N_{K^{0}\bar{K}^{0}}(t) \label{j1-91}\end{aligned}$$ and $$\begin{aligned}
\frac{\partial A_{\bar{K}^{0}K^{0}}(t)}{\partial
t}\nonumber&=&\frac{1+\pi } {8\pi
pq^{\ast}}\Biggl\{e^{-im_{S}t}e^{-\gamma_{S}t}
\Biggl[-im_{S}-\gamma_{S}+\nonumber\\
&&+\sqrt{\gamma_{S}\gamma_{L}} \Biggl(-2 i \gamma_{S}
C_{I}+D'_{I}-F'_{I}\Biggl)\Biggl]+\nonumber\\
&&+e^{-im_{L}t} e^{-\gamma_{L}t}\Biggl[im_{L}-\gamma_{L}+\nonumber\\
&&+\sqrt{\gamma_{S}\gamma_{L}} \Biggl(2 i \gamma_{L}
C_{I}-D'_{I}+F'_{I}\Biggl)\Biggl]\Biggl\}+\nonumber\\
&&+\Delta N_{\bar{K}^{0}K^{0}}(t), \label{j1-93}\end{aligned}$$ where $\Delta N_{K^{0}\bar{K}^{0}}(t)$, $\Delta
N_{\bar{K}^{0}K^{0}}(t)$ denotes all non-oscillatory terms.
The states $|K_{L}\rangle$ and $|K_{S}\rangle$ are superpositions of $|K^{0}\rangle$ and $|\bar{K}^{0}\rangle$. The lifetimes of particles $|K_{L}\rangle$ and $|K_{S}\rangle$ may be denoted by $\tau_{L}$ and $\tau_{S}$, respectively, $\tau_{L}=\frac{1}{\gamma_{L}}=5,183\cdot 10^{-8}s$ being much longer than $\tau_{S}=\frac{1}{\gamma_{S}}=0,8923\cdot 10^{-10}s.$
Below we calculate the difference (\[j1-79\]) for $t\sim
\tau_{L}$ $$\begin{aligned}
h_{11}(t\sim \tau_{L})-h_{22}(t\sim
\tau_{L})=\frac{X(t\sim\tau_{L})}{Y(t\sim\tau_{L})}. \label{j1-95}\end{aligned}$$ If we only consider the long living states $|K_{L}\rangle$ we may drop all the terms containing $e^{-\gamma_{S}t}|_{t\sim \tau_{L}}$ as they are negligible in comparison with elements involving the factor $e^{-\gamma_{L}t}|_{t\sim \tau_{L}}.$ We also drop all the non-oscillatory terms $N_{K^{0}K^{0}}(t),$ $N_{\bar{K}^{0}K^{0}}(t)$, $N_{K^{0}\bar{K}^{0}}(t)$ present in $A_{K^{0}K^{0}}(t)$, $A_{\bar{K}^{0}K^{0}}(t)$ and $A_{K^{0}\bar{K}^{0}}(t),$ that is in integrals (\[j1-77\]), (\[j1-87\]) and (\[j1-89\]), because they are extremally small in the region of time $ t\sim
\tau_{L}$ [@p:2; @p:28; @p:29]. Similarly, because of the properties of the exponential integral function $E_{i},$ we can drop terms like $\Delta
N_{\bar{K}^{0}K^{0}}$ and $\Delta N_{K^{0}\bar{K}^{0}}$ present in $\frac{\partial
A_{\bar{K}^{0}K^{0}}}{\partial t}$ (\[j1-91\]) and $\frac{\partial
A_{K^{0}\bar{K}^{0}}}{\partial t}$ (\[j1-93\]).This conclusion follows from the asymptotic properties of the exponential integral function $E_{i}$ and the fact that $\Delta N_{\bar{K}^{0}K^{0}},$ $\Delta N_{K^{0}\bar{K}^{0}}$ only contain expressions proportional to $E_{i}$.
We may now calculate the products $A_{K^{0}K^{0}}(t)A_{\bar{K}^{0}\bar{K}^{0}}(t), $ $A_{K^{0}\bar{K}^{0}}(t)A_{\bar{K}^{0}K^{0}}(t), $\
$\frac{\partial A_{\bar{K}^{0}K^{0}}}{\partial
t}(t)A_{K^{0}\bar{K}^{0}}(t), $ $\frac{\partial
A_{K^{0}\bar{K}^{0}}}{\partial t}(t)A_{\bar{K}^{0}K^{0}}(t)$, which, after using the above mentioned properties of $N_{K^{0}K^{0}}(t)$, $\Delta N_{K^{0}K^{0}}(t)$ and performing some algebraic transformations, leads to the following form of the difference (\[j1-95\]): $$\begin{aligned}
h_{11}(t\sim \tau_{L})-h_{22}(t\sim\tau_{L}))=
\Biggl(\frac{2\pi^{2}\sqrt{\gamma_{S}\gamma_{L}}}{
\pi^{2}+2\pi+1}\Biggl)\cdot \frac{Z}{W}\neq 0, \label{j1-97a}\end{aligned}$$ where $$\begin{aligned}
Z=\nonumber&& 4|p|^{2}|q|^{2}-\frac{\pi^{2}+2\pi
+1 }{4\pi^{2}}\Biggl[1+\nonumber\\
&&+\gamma_{S}\Biggl(4\gamma_{L}C_{I}^{2}+
\frac{1}{\gamma_{L}}(-D_{I}^{'2} -F_{I}^{'2}
+4D_{I}^{'}F_{I}^{'})+\nonumber\\
&&+4 i C_{I}(D_{I}^{'}-F_{I}^{'})\Biggl)\Biggl]\neq
0\label{j1-97b}\end{aligned}$$ $$\begin{aligned}
W=\nonumber&&2\Biggl(-C_{I}m_{L}+D_{I}^{'}-
F_{I}^{'}\Biggl)+\nonumber\\
&&+i\Biggl[-4C_{I}\gamma_{L}+\frac{m_{L}}{\gamma_{L}}
\Biggl(-D_{I}^{'}+F_{I}^{'}\Biggl)\Biggl]\neq 0. \label{j1-97c}\end{aligned}$$
Final remarks
=============
- Our results presented in the present poster have shown that in a CPT invariant and CP noninvariant system in the case of the exactly solvable one-pole model, the diagonal matrix elements do not have to be equal. In the general case the diagonal elements depend on time and their difference, for example at $t\sim \tau_{L}$, is different from zero. Z and W in (\[j1-97a\]) are different from zero, so the difference $(h_{11}(t)-h_{22}(t))|_{t\sim\tau_{L}}\neq 0.$ From this observation a conclusion of major importance can be drawn, namely that the measurement of the mass difference $(m_{K^{0}}-m_{\bar{K}^{0}}) $ should not be used while designing CPT invariance tests. This runs counter to the general conclusions following from the Lee, Oehme and Yang theory.
- A detailed analysis of $h_{jk}(t)$, $(j,k = 1,2)$ shows that the non-oscillatory elements $N_{\alpha ,\beta}(t), \Delta N_{\alpha ,\beta}(t)$ (where $\alpha , \beta = K^{0}, {\overline{K}}^{0}$) is the source of the non-zero difference $(h_{11}(t) - h_{22}(t))$ in the model considered. It is not difficult to verify that dropping all the terms of $N_{\alpha ,\beta}(t), \Delta N_{\alpha ,\beta}(t)$ type in the formula for $(h_{11}(t) - h_{22}(t))$ gives $(h_{11}^{osc}(t) - h_{22}^{osc}(t)) =
0$, where $h_{jj}^{osc}(t)$, $(j = 1,2)$, stands for $h_{jj}(t)$ without the non-oscillatory terms.
- The result ($h_{11}(t)-h_{22}(t))\neq 0$ seems to be very important as it has been obtained within the exactly solvable one-pole model based on the Breit-Wigner ansatz, i.e. the same model as used by Lee, Oehme and Yang.\
Acknowledgements {#acknowledgements .unnumbered}
================
The author wishes to thank Professor Krzysztof Urbanowski for many helpful discussions.
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M.K. Gaillard, M. Nicolic (Eds.), Weak Interaction, INPN et de Physique des Particules, Paris, 1977, Ch. 5, Appendix A;\
S.M. Bilenkij, in: Particles and Nucleus, Vol. 1. (1), 1970, p. 227, \[in Russian\]. L.P. Horwitz, J.P. Marchand, Helv. Phys. Acta, [**42**]{} (1969) 801. J.W. Cronin, Acta Phys.Polon., [**B 15**]{} (1984) 419;\
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L.A. Khalfin, A new CP-violation effect and new possibility for investigation of $K_{S}^{0}, K_{L}^{0} ( K^{0}, \bar{K}^{0}) $ decay modes (Report DOE-ER40200-247, Frebruary 1991).\
P.K. Kabir, A. Pilaftsis, Phys. Rev., [**A 53**]{} (1996) 66. L.A. Khalfin, Found. Phys., [**27**]{} (1997) 1549, and references one can find therein. O. Nachtmann, Elementary Particle Physics, Springer Verlag, Berlin 1990. K. Urbanowski, Phys. Lett., [**B 540**]{} (2002) 89. K. Hagiwara [*et al*]{}, Review of Particle Physics, Physical Review [**D 66**]{}, Part 1, No 1–I, (2002), 010001. M.Nowakowski, Time Evolution of $K^{0}-\bar{K}^{0} $ System in Spectral Formulation, SIS-Pubblicazioni, LNF-96/004(P); M.Nowakowski, Int. J. Mod. Phys., [**A 14**]{} (1999) 589 . P.K. Kabir, The CP Puzzle, Academic Press, London 1968. A. Bohm, Quantum Mechanics: Foundations and Applications, Springer Verlag, Berlin 1986. I.S. Gradshteyn, I.M. Ryzhik, Tables of Integrals, Series and Products, 4th edition, Academic Press, London 1965. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series - 55, Issued June 1964 Tenth Printing, December 1972, with corrections.
[^1]: Paper presented at **The XXXVI Symposium on Mathematical Physics**, *Poster session*, Toruń, Poland, June 9-12, 2004 . This is shortened version of [@p:44]
|
The properties of the vortex liquid in high-temperature superconductors, HTS, in the presence of strong disorder is one of the most challenging problems in the physics of the vortex state. It is well established now that in pristine ${\rm Bi_2Sr_2CaCu_2O_{8+\delta }}$, Bi-2212, crystals vortices form a weakly $c$-axis correlated pancake liquid. Thermodynamics and intralayer dynamics of this liquid are defined mostly by pancake concentration, i.e., by the $c$-axis magnetic field $B_{\bot}$ [@kes-vl]. This behavior is however, strongly affected by the presence of correlated disorder. The most effective pinning centers are produced by heavy ion irradiation. Irradiation of the HTS by energetic ions produces amorphous tracks, where the superconductivity is suppressed. Such columnar defects, CDs, with radii comparable with the superconducting coherence length, are ideal for pinning vortex lines, but their effect on pancake vortices is less obvious. The main questions are: a) are pancake vortices positioned mainly inside columnar defects in the liquid phase?, and b) are $c$-axis correlations enhanced in the presence of CDs or do they remain similar to that in the liquid phase of pristine crystals?
Important information about the effect of CDs on the pancake liquid was obtained from reversible magnetization measurements [@kees-Mrev]. In pristine crystals reversible magnetization $M$ monotonically increases with the magnetic field $B_{\bot}$. In irradiated crystals, due to gain in pinning energy, penetration of vortices into crystals becomes more favorable than in pristine crystals, and the diamagnetic moment drops in the presence of CDs at low fields $B_{\bot}\ll B_{\Phi}$ when all vortices may occupy CDs. However, as $B_{\bot}$ increases towards the matching field $B_{\Phi}$, interstitial vortices start to appear, losing the advantage in pinning energy associated with the CDs. This results in a drop of magnetization (i.e., in an increase in the diamagnetic moment) in the field interval between $\approx B_{\Phi}/4$ and $B_{\Phi}$. At larger fields the difference between pristine and irradiated crystals almost vanishes. Such an anomaly in $M(B_{\bot})$ dependence was observed in the vortex liquid up to rather high temperature, indicating that mobile pancakes, inherent to the vortex liquid, are localized largely onto CDs, even at temperatures close to $T_{c}$. Recent study of the vortex-lattice melting in weakly irradiated Bi-2212 also confirmed this picture [@borya-melt]. However, the thermodynamics of the vortex state depends weakly on $c$-axis correlation of pancakes; it is determined mainly by intralayer interactions, which are much stronger that those associated with the $c$-axis vortex structure [@lev-magn; @kosugi-bul]. For this reason reversible magnetization is determined by pancake concentration and scales with $B_{\bot}$. Thus magnetization measurements show that pancakes are positioned mainly inside CDs at $B_{\bot}<B_{\Phi}/3$ but they do not provide information on $c$-axis correlations of pancakes inside CDs.
In contrast, Josephson interlayer properties of highly anisotropic Bi-2212 are extremely sensitive to the $c$-axis correlations of the pancakes, because Josephson current depends on the interlayer phase difference. Pancakes, aligned along the $c$-axis do not contribute to the phase difference, but those, shifted due to thermal fluctuations or pinning do. This leads to larger phase difference in the uncorrelated liquid and thus to the suppression of the $c$-axis superconducting current. Measurements of Josephson plasma resonance, JPR, reveal for the first time that $c$-axis correlation in the vortex liquid in the presence of CDs depends on $B_{\bot}$ nonmonotonically in the temperature interval 60 - 69 K, showing enhancement at $B_{\bot} \sim
B_{\Phi}/3$ [@matsuda-JPR], while in the liquid phase in pristine crystals correlation drops with $B_{\bot}$ at all temperatures. Nonmonotonic behavior with $B_{\bot}$ was found also in the dependence of the $c$-axis resistivity, $\rho_c$, which is also determined by interlayer Josephson current [@nick-rhoc]. Namely, $\rho_{c}$ in the same temperature interval exhibits an increase with $B_{\bot}$ at $B_{\bot}\ll B_{\Phi}$, a flattening or even a decrease in the field interval between 0.2 $ B_{\Phi}$ and 0.4 $ B_{\Phi}$ followed by further increase of $\rho_{c}$ at higher $B_{\bot}$. Unfortunately, neither JPR nor $\rho_{c}(B_{\bot}$) measurements provide sufficient information to estimate the enhancement of the $c$-axis correlation quantitatively.
It was shown in [@koshelev-JPR; @nick-bx] that dependence of JPR frequency and $\rho_{c}$ on the parallel component of the magnetic field provides complete information about $c$-axis phase correlations. Based on this idea, in this paper we study the angular dependence of $\rho_{c}$ at different orientation of the magnetic field $B$ in the correlated pancake liquid phase in the presence of CDs. We show that scaling with $B_{\bot}$ is absent in this phase. Moreover, we found, that at $B=B_{cr}\approx B_{\Phi}/2$, this angular dependence is accurately compensated, providing a crossing point where $\rho_{c}$ is independent of the orientation $\theta$ of the magnetic field with respect to the crystal $c$-axis over a wide range of $\theta$. Such an anomaly gives an opportunity to estimate the correlation length of pancake positions along the $c$ axis at the crossing field.
For our experiments high quality Bi-2212 crystals $(T_{c}\simeq {\rm
85\:K})$ of about $1\times 1.5\times 0.02\:\ mm^{3}$ were used. The irradiation by 1.2 GeV ${\rm U^{238}}$-ions was performed on the ATLAS accelerator (Argonne National Lab.). According to TRIM calculations these high energy ions produce in Bi-2212 crystals continuous amorphous tracks with diameter 4-8 nm and length 25-30 $\mu$m. Below we present the results for samples irradiated with an effective density of columnar defects corresponding to the matching field $B_{\Phi } = 1$ T. Another pristine sample was used as a reference. We checked also several crystals from a different synthesis for universality of the obtained results.
=7.7cm
Our measurements were performed using a warm bore insert in a liquid helium cryostat with a 9 T superconducting magnet, and also in a liquid nitrogen, LN, dewar installed into an electromagnet that provides a magnetic field up to 1T. The sample was attached to a sample holder with goniometric stage, allowing rotation of the sample with respect to the applied field with an accuracy of about $\pm
0.1^{\circ }$. The sample was immersed in a LN bath in order to provide accurate temperature stabilization and to avoid heating effects. Magnetic fields applied along the [*c*]{}-axis of the sample were monitored by a Hall sensor attached to the sample stage. Two pair of silver contacts pads were deposited on both sides of the samples for standard 4-probe transport measurements and fine gold leads were attached using silver epoxy. The resistivity for the contact pair at room temperature was about $1-3~\Omega $. We measured $R_c$ as a function of applied magnetic field $B$ at different angles $\theta $ with respect to the [*c*]{}-axis of the crystal (see insert in Fig. 1).
In Fig. 1 we present the normalized magnetoresistance $\tilde{\rho_{c}}\equiv\rho_{c}/\rho_{n}$ as a function of applied field along the [*c*]{}-axis measured at $T=68$ K. Here $\rho_n$ is the normal state resistivity at $T=120$ K. After the sharp onset at the irreversibility line, $\tilde{\rho_c}$ for irradiated crystals displays a pronounced dip due to enhanced vortex correlation associated with filling of the CDs reported in Ref. . At fields $B\gg B_{\Phi}$ the magnetoresistance approaches the normal state value. In the pristine crystal the resistance increases with $B_{\bot}$ monotonically.
= 9.7cm
In the left panel of Fig. 2 the family $\tilde{\rho_c}$ vs. $B$ at different angles is presented. The magnetoresistance decreases with the angle at $B>B_{\Phi}/2$ (as in the pristine crystals), reflecting decrease of $B_{\bot}$, but [*increases*]{} with the angle at low fields $(0.1B_{\Phi}<B<B_{\Phi}/2)$, in the region where reentrant enhancement of $c$-axis correlation was observed in $\rho_{c}(B_{\bot})$ dependence.
In our experiment the inclination of the sample in the applied field results in two effects: decrease of the [*c*]{}-axis field $B_{\bot}=B \,{\rm \cos} \theta $ and increase of the in-plane field $B_{\parallel}=B \, {\rm \sin} \theta $. In order to separate these two effects we present $\tilde{\rho}_c$ as a function of the [*c*]{}-axis component $B_{\bot}$ in the right panel of Fig. 2. The magnetoresistance of the sample scales with $B_{\bot}$ at low and at high fields. However, in the range $B_{\Phi}/4 < B < B_{\Phi}$ this [*scaling is broken*]{}. This anomalous behavior occurs in the range of field and temperature where enhanced vortex correlations in the pancake liquid were observed in JPR and in $\rho_{c}(B_{\bot})$.
In Fig. 3 we blow up the region where the scaling is broken. It is clearly seen on the right panel that $\tilde{\rho_c}$ here depends on the in-plane field, increasing gradually with $B_{\parallel}$. The most outstanding feature can be seen on the left panel of Fig. 3. Here $\tilde{\rho_{c}}$ curves at different angles are presented as a function of $B$. They all cross at a single point, which we denote as $B_{cr}$. This crossing point is observed for all the crystals which display the dip of $\rho_{c}$. It means, that at $B=B_{cr}$, [*angular dependence of magnetoresistance practically vanishes*]{}, i.e., decrease of $\rho_{c}$ due to decrease of $B_{\bot}$ is accurately compensated by increase of the magnetoresistance with increasing $B_{\parallel}$. This effect can be explained in the framework of the approach developed in [@koshelev-JPR; @nick-bx] and will be used below to estimate quantitatively the characteristic length $R_{1}$ of the phase difference in-plane correlation function.
= 9.7cm
Let us first discuss how the in-plane field component affects the [*c*]{}-axis magnetoresistance. An approach developed initially for the field behavior of JPR frequency [@koshelev-JPR] was extended recently for $c$-axis transport properties [@nick-bx]. In Josephson coupled superconductors in the presence of a $c$-axis current, the voltage $V_{n,n+1}$ is induced by slips of the phase difference $\varphi_{n,n+1}({\bf r},t)$ between the layers $n$ and $n+1$, as described by the Josephson relation $V_{n,n+1}=(\hbar/2e)\dot{\varphi}_{n,n+1}$. Here ${\bf r}=x,y$ are coordinates in the $ab$ plane, and $t$ denotes the time. The $c$-axis conductivity in the vortex liquid state $\sigma_{c}=1/\rho_{c}$ is determined by the Kubo formula [@koshelev-kubo] $$\begin{aligned}
\sigma_c(B_{\bot},B_{\parallel})=(s{\cal J}_0^2/2T)\int_0^{\infty}dt
\int d{\bf r}S({\bf r},t),
\label{sig} \\
S({\bf r},t)=2\langle\sin\varphi_{n,n+1}(0,0)
\sin\varphi_{n,n+1}({\bf r},t)\rangle
\nonumber \\
\approx\langle\cos[\varphi_{n,n+1}({\bf r},t)-\varphi_{n,n+1}(0,0)]\rangle,
\label{1}\end{aligned}$$ where ${\cal J}_0$ is the Josephson critical current, $s$ is the interlayer distance, and $<\ldots>$ means thermal average and average over disorder. Time variations of the phase difference are caused mainly by mobile pancakes [@koshelev-SC] induced by $B_{\bot}$, while the parallel field component induces a stationary phase difference in the lowest order in Josephson coupling. We split $[\varphi_{n,n+1}({\bf r},t)-\varphi_{n,n+1}(0,0)]$ into the contribution induced by pancakes and that caused by the unscreened parallel component $B_{\parallel}$. Assuming that $B_{\parallel}$ is along the $x$ axis, we obtain: $$\begin{aligned}
&&\varphi_{n,n+1}(0,0)-\varphi_{n,n+1}({\bf r},t)= \label{me} \\
&&[\varphi_{n,n+1}(0,0)-\varphi_{n,n+1}({\bf r},t)]_{B_{\parallel}=0}-
2\pi sB_{\parallel}y/\Phi_0.
\nonumber\end{aligned}$$ In a single point-like junction the phase difference induced by $B_{\parallel}$ results in the Fraunhofer pattern of Josephson critical current as a function of the magnetic field parallel to the junction. In our case of a multilayer superconductor the phase difference induced by $B_{\parallel}$ interferes with that induced by pancakes. Thus we obtain $$\begin{aligned}
&&\sigma_c(B_{\bot},B_{\parallel})=(\pi s{\cal J}_0^2/T)\int
drr\tilde{G}(r,B_{\bot})
J_0(\alpha B_{\parallel}r),
\label{me1} \\
&&\tilde{G}(r,B_{\bot})=\int_0^{\infty}dtS(r,t,B_{\bot}), \nonumber\end{aligned}$$ where $J_0(x)$ is the Bessel function, $\alpha=2\pi s/\Phi_0$ ($\Phi_{0}$ is a flux quantum), and the function $\tilde{G}({\bf r},B_{\bot})$ depends on correlations of the phase difference induced by pancake vortices. For small $B_{\parallel}$ (small angles) we expand the Bessel function in $B_{\parallel}$: $$\begin{aligned}
&&\sigma_{c}(B_{\bot},B_{\parallel}) \approx
\sigma_{c}(B_{\bot},0)\left[ 1-\frac{1}{4}\alpha^{2}{R_{1}^{2}(B_{\bot})}B_{\parallel}^2
\right], \label{rel}
\\
&&R_1^2(B_{\bot})=\int drr^3\tilde{G}(r,B_{\bot})/\int drr\tilde{G}(r,B_{\bot}).
\nonumber\end{aligned}$$ Here the in-plane correlation length $R_{1}(B_{\bot})$ describes the decay of the phase difference correlation function. When pancakes are positioned mainly inside CDs, the characteristic length of this decay, $R_1$, gives direct information on the $c$-axis correlation of pancake positions because drop of the phase difference correlations in the $ab$ plane is caused by interruptions in the pancake arrangement along CDs [@koshelev-JPR]. The characteristic length, $L$, of the pancake density correlation function is related to $R_1$ as $L/s\approx R_1^2/10a^2$, where $a=(\Phi_0/B_{\bot})^{1/2}$ is the intervortex distance [@nick-bx]. This expression and Eq. (\[rel\]) are a key points for further discussion.
It is clear from Eq. (\[rel\]) that generally $\sigma_{c}$ depends upon both components of the field. However, in an uncorrelated liquid, when $R_{1} \approx a$, the effect of the in-plane field is small and can be observed only in high fields $B_{\parallel}\gtrsim\Phi_{0}/sa$. As a result, in the uncorrelated pancake liquid the [*c*]{}-axis conductivity scales with $B_{\bot}$ in fields $B\ll\Phi_{0}/sa$. For an irradiated sample, as filling of CDs progresses, vortices start to form stacks and the correlation length $R_{1}$ significantly exceeds $a$, reaching a maximum value near $B_{\Phi}/3$. Here the effect of the in-plane field becomes significant and scaling of magnetoresistance with $B_{\bot}$ breaks. As the field $B_{\bot}$ further increases, the fraction of the interstitial vortices increases. These vortices introduce additional disorder to the system which results in decay of correlations, in reduction of $R_{1}$ down to $\sim a$ and, consequently, the effect of $B_{\parallel}$ drops. Then, at elevated fields above $B_{\Phi}$ scaling of $\rho_{c}$ with $B_{\bot}$ is restored. This scenario describes qualitatively well our experimental results.
Now we show that the crossing point in Fig 3 allows us to estimate the correlation radius $R_{1}(B_{cr})$. Using Eq. (\[rel\]) for the variation of $\sigma_{c}$ at small angles we expand as: $$\delta \sigma _{c}(B_{\parallel},B_{\bot}) \approx
\frac{\partial^2\sigma_{c}}{\partial B^{2}_{\parallel}}\:\frac{(\delta
B_{\parallel})^2}{2}+
\frac{\partial\sigma_c}{\partial B_{\bot}} \delta B_{\bot}.$$ Substituting $B_{\parallel} \approx B\theta$ and $B_{\bot} \approx B(1-\theta^2/2)$ we obtain $$\delta \sigma _{c}(B_{\parallel},B_{\bot}) \approx \frac{1}{2}
B\left(\frac{\partial^2\sigma_{c}}{\partial B^{2}_{\parallel}}\:B-
\frac{\partial\sigma_c}{\partial B_{\bot}}\right) (\delta\theta)^2.
\label{cross}$$
Independence of $\sigma_{c}$ on the angle $\theta$ at $B_{cr}$ occurs when the expression in the brackets becomes zero. With the help of Eq. (\[rel\]) we obtain the correlation radius $R_{1}$: $$R_1^2(B_{cr})=-\frac{1}{2B_{cr}}\: \frac{\Phi_0^2}{\pi^2s^2 } \:
\left[\frac{\partial \ln \sigma_c (B_{\bot},0)}{\partial B_{\bot}}\right]_{B_{\bot}
=B_{cr}}.
\label{radius}$$ From the data presented in Fig. 3, we obtain $(\ln \sigma_c (B_{\bot},0)
)^\prime \approx -23 \: {\rm T^{-1}}$ at $B=B_{cr} \approx 0.47$ T. Using Eq. (\[radius\]) we calculate $R_{1}/a \approx 32$. This value is in reasonable agreement with the result obtained in [@nick-bx], where $R_{1}\sim 10a$ was found. Thus the correlations length $L$ of pancakes positions along the $c$ axis is $\approx 100s$.
We would like to emphasize the difference between our observation and results obtained in flux-transformer geometry [@richard]. In those latter experiments current was applied to the top surface of the sample and voltage was measured in the both top and bottom layers. In some range of the magnetic fields and temperatures top and bottom voltages coincide, indicating a similar motion of vortices in all layers. This was considered as evidence of pancake coupling along the $c$-axis. Unfortunately, the situation here is rather complicated because of mixing of the in-plane and $c$-axis resistivities. Namely, the current applied along the top layer penetrates deep into the sample along the $c$ axis and flows along the bottom layer as well as along the top layer. This results in a similar motion of the pancakes through the full sample thickness, even without $c$-axis correlation provided $\rho_{c}$ is small enough. As we have shown, the $c$-axis transport measurements as a function of in-plane field component are free of this additional effect and are very sensitive to pancake correlation between adjacent layers.
To conclude, we have presented evidence for the presence of a partially aligned vortex liquid in irradiated Bi-2212 from the angular dependence of [*c*]{}-axis magnetoresistance. In the range of temperatures and magnetic fields where $c$-axis correlations develop, interlayer transport becomes much more sensitive to the in-plane component of the magnetic field in comparison with the uncorrelated liquid in pristine crystals. Then scaling of $\rho_c$ with $B_{\bot}$ breaks, and, at the field $B_{cr}$, angular dependence becomes very weak. From this crossing point we estimate the $c$-axis correlation length of pancake positions $L/s\approx 100$ at $B_{\bot}\approx B_{\Phi}/2$. The origin of this alignment can be due to magnetic intralayer interaction of pancakes which favors similar filling of CDs due to their geometry. However an additional effect of interlayer magnetic interaction of pancakes cannot be excluded so far.
Fruitful discussions with H. Safar, and Y. Matsuda are greatly acknowledged. We thank J. H. Cho for providing the Bi-2212 single crystals. We also thank K.E. Gray, J.U. Lee, D.H. Kim, and D.J. Hofman of Argonne National Lab. for the help in irradiation of the samples. This work is supported by U.S. DOE.
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\[pagini\]
New solutions for the modified generalized
Degasperis$-$Procesi equation
Alvaro H. Salas
Department of Mathematics
Universidad de Caldas, Manizales, Colombia.
Universidad Nacional de Colombia, Manizales.
*email* : [email protected]
José Gonzalo Escobar Lugo
Universidad Libre, Bogotá, Colombia.
Universidad Cooperativa de Colombia, Bogotá.
*email* : [email protected]
Introduction
============
Degasperis and Procesi [@degas] showed, using the method of asymptotic integrability, that the PDE $$\label{eq00}
u_t - u_{xxt} + (b+1)uu_x = bu_xu_{xx} + uu_{xxx}$$ cannot be completely integrable unless $b = 2$ or $b = 3$. The case $b = 2$ is the Camassa– Holm (CH) shallow water equation [@cama], which is well known to be integrable and to possess multi-soliton (weak) solutions with peaks, so called multi-peakons. Degasperis, Holm and Hone [@degas1; @degas2] proved that the case $b = 3$, which they called the Degasperis– Procesi (DP) equation, is also integrable and admits multi-peakon solutions. They found the two-peakon solution explicitly by direct computation.
In this paper we show some exact solutions for the generalized mDP equation [@liu; @wazwaz] $$\label{equ00}
u_t - u_{xxt} + (b+1)u^2u_x = bu_xu_{xx} + uu_{xxx}$$ for any $b\neq -1$ and $b\neq -2$ by using three distinct methods: The Cole-Hopf transformation, the rational hyperbolic ansatz and the improved tanh-coth method.
The Cole-Hopf transformation
============================
We seek solutions to (\[equ00\]) in the form of a generalized Cole-Hopf transformation [@cole][@hopf] $$\label{equ01}
u=A\dfrac{\partial^2}{\partial x^2}\log(1+\varphi(x,t))+B,$$ for the special choice $$\label{equ02}
\varphi(x,t)=\exp(\mu x+\lambda t+\delta),\quad (A\neq 0,
\quad\lambda\neq0\quad\text{and}\quad\mu\neq 0.)$$ From (\[equ01\]) and (\[equ02\]) it follows that solutions have the form $$\label{equ03}
u=\frac{A \mu ^2}{2 (1+\cosh (\mu\,x+\lambda\,t))}+B.$$ Applying this transformation and defining $\zeta=\exp(k x+\omega
t+\delta)$, we obtain a polynomial equation in the variable $\zeta$. Equating the coefficients of the different powers of $\zeta$ to zero, we obtain the following algebraic system :
- $B\mu ^{3}+\lambda \mu ^{2}-\left( bB^{2}+B^{2}\right) \mu -\lambda =0.
$
- $B \mu ^3+\lambda \mu ^2+\left(b B^2+B^2\right) \mu +\lambda =0.$
- $(bA+A)\mu ^{5}-(2AB+2AbB+9B)\mu ^{3}-9\lambda \mu ^{2}-\left(
3bB^{2}+3B^{2}\right) \mu -3\lambda =0.$
- $(b A+A) \mu ^5+(2 A B+2 A b B+9 B) \mu ^3+9 \lambda \mu ^2+\left(3
b B^2+3 B^2\right) \mu +3 \lambda =0.$
- $\left(b A^2+A^2+5 b A+11 A\right) \mu ^5+(2 A B+2 A b B+10 B) \mu
^3+10 \lambda \mu ^2+\left(2 b B^2+2 B^2\right) \mu +2 \lambda
=0.$
- $\left(b A^2+A^2+5 b A+11 A\right) \mu ^5+(2 A B+2 A b B+10 B) \mu
^3+10 \lambda \mu ^2+\left(2 b B^2+2 B^2\right) \mu +2 \lambda
=0$.
Solving this system gives following solutions :
- $A=-\frac{6(b+2)}{b+1},~B=\frac{2\mu ^{2}-1+b\left( \mu ^{2}-1\right) +
\sqrt{1-b(b+2)\left( \mu ^{4}-1\right) }}{2(b+1)},~\lambda
=-\frac{1}{2}\mu \left( b+1-\sqrt{1-b(b+2)\left( \mu ^{4}-1\right)
}\right) :$ $$\begin{aligned}
u_1(x,t) &=&\frac{2\mu ^{2}-1+b\left( \mu ^{2}-1\right)
+\sqrt{1-b(b+2)\left( \mu
^{4}-1\right) }}{2(b+1)}- \\
&&\frac{6(b+2)\mu ^{2}}{2(b+1)\left( 1+\cosh \left( \mu\,
x-\frac{1}{2}\left( b+1-\sqrt{1-b(b+2)\left( \mu ^{4}-1\right)
}\right) \mu\, t\right) \right) }.\end{aligned}$$
- $A=-\frac{6(b+2)}{b+1},~B=\frac{b\mu ^{2}+2\mu ^{2}-1-b-\sqrt{
1-b(b+2)\left( \mu ^{4}-1\right) }}{2(b+1)},~\lambda
=-\frac{1}{2}\mu \left( b+1+\sqrt{1-b(b+2)\left( \mu ^{4}-1\right)
}\right) :$ $$\begin{aligned}
&&u_2(x,t)=\frac{b\mu ^{2}+2\mu ^{2}-1-b-\sqrt{1-b(b+2)\left( \mu
^{4}-1\right) }}{
2(b+1)}- \\
&&\frac{6(b+2)\mu ^{2}}{2(b+1)\left( 1+\cosh \left( \mu\,
x-\frac{1}{2}\left( b+1+\sqrt{1-b(b+2)\left( \mu ^{4}-1\right)
}\right) \mu\, t\right) \right) }.\end{aligned}$$
The rational hyperbolic ansatz
==============================
According this method, we seek solutions in the form $$\label{equ04}
u(x,t)=u(\xi)=\dfrac{a_0+a_1\sinh(\xi)+a_2\cosh(\xi)}{1+c_1\sinh(\xi)+c_2\cosh(\xi)},$$ where $\xi=x+\lambda t$ and $\lambda$, $a_0$, $a_1$, $a_2$, $c_1$ and $c_2$ are constants to be determined. Changing the hyperbolic functions $\sinh(\xi)$ and $\cosh(\xi)$ to their $\exp$ form and substituting the resulting expression into (\[equ00\]), we obtain a polynomial equation in the variable $\zeta=\exp(\xi)$. Equating the coefficients of the powers of $\zeta$ to zero, we obtain an algebraic system in the variables $\lambda$, $a_0$, $a_1$, $a_2$, $c_1$ and $c_2$. After solving this system, we get the following solutions.
- $\lambda =-\frac{b}{2},\,\,a_{0}=-\frac{3b+5}{b+1},\,\,a_{1}=0,\,
\,a_{2}=\frac{1}{b+1},\,\,c_{1}=0,\,\,c_{2}=1$ : $$u_{3}(x,t)=-\frac{3b+5-\cosh \left( x-\frac{bt}{2}\right)
}{(b+1)\left( 1+\cosh \left( x-\frac{bt}{2}\right) \right) }.$$
- $\lambda =-\frac{b}{2},\,\,a_{0}=-\frac{3b+5}{b+1},\,\,a_{1}=0,\,
\,a_{2}=-\frac{1}{b+1},\,\,c_{1}=0,\,\,c_{2}=-1$ : $$u_{4}(x,t)=-\frac{3b+5+\cosh \left( x-\frac{bt}{2}\right)
}{(b+1)\left( 1-\cosh \left( x-\frac{bt}{2}\right) \right) }.$$
- $\lambda =-\frac{b}{2}-1,\,\,a_{0}=-\frac{3(b+2)}{b+1}
,\,\,a_{1}=0,\,\,a_{2}=0,\,\,c_{1}=0,\,\,c_{2}=-1$ : $$u_{5}(x,t)=-\frac{3(b+2)}{(b+1)\left( 1-\cosh \left(
x-(1+\frac{b}{2} )t\right) \right) }.$$
- $\lambda =-\frac{b}{2}-1,\,\,a_{0}=-\frac{3(b+2)}{b+1}
,\,\,a_{1}=0,\,\,a_{2}=0,\,\,c_{1}=0,\,\,c_{2}=1$ : $$u_{6}(x,t)=-\frac{3(b+2)}{(b+1)\left( 1+\cosh \left(
x-(1+\frac{b}{2} )t\right) \right) }.$$
- $\lambda =-\frac{b}{2},\,\,a_{0}=-\frac{3b+5}{b+1},\,\,a_{1}=-\frac{
\sqrt{(b+1)^{2}a_{2}^{2}-1}}{b+1},\,\,c_{2}=a_{2}(b+1),\,\,c_{1}=-\sqrt{
(b+1)^{2}a_{2}^{2}-1}$ : $$u_{7}(x,t)=\frac{-3b-5+(b+1)\cosh \left( x-\frac{bt}{2}\right)
a_{2}-\sqrt{ (b+1)^{2}a_{2}^{2}-1}\sinh \left( x-\frac{bt}{2}\right)
}{(b+1)\left( 1- \sqrt{(b+1)^{2}a_{2}^{2}-1}\sinh \left(
x-\frac{bt}{2}\right) +(b+1)a_{2}\cosh \left( x-\frac{bt}{2}\right)
\right) }.$$
- $\lambda =-\frac{b}{2},\,\,a_{0}=-\frac{3b+5}{b+1},\,\,a_{1}=\frac{
\sqrt{(b+1)^{2}a_{2}^{2}-1}}{b+1},\,\,\,c_{2}=a_{2}(b+1),\,\,c_{1}=\sqrt{
(b+1)^{2}a_{2}^{2}-1}$ : $$u_{8}(x,t)=\frac{-3b-5+(b+1)\cosh \left( x-\frac{bt}{2}\right)
a_{2}+\sqrt{ (b+1)^{2}a_{2}^{2}-1}\sinh \left( x-\frac{bt}{2}\right)
}{(b+1)\left( 1+ \sqrt{(b+1)^{2}a_{2}^{2}-1}\sinh \left(
x-\frac{bt}{2}\right) +(b+1)a_{2}\cosh \left( x-\frac{bt}{2}\right)
\right) }.$$
- $\lambda =-\frac{b}{2}-1,\,\,a_{0}=-\frac{3(b+2)}{b+1}
,\,\,a_{1}=0,\,\,a_{2}=0,\,\,c_{1}=\sqrt{c_{2}^{2}-1}$ : $$u_{9}(x,t)=-\frac{3(b+2)}{(b+1)\left( 1+\sqrt{c_{2}^{2}-1}\sinh
\left( x-(1+ \frac{b}{2})t\right) +c_{2}\cosh \left(
x-(1+\frac{b}{2})t\right) \right) }.$$
- $\lambda =-\frac{b}{2}-1,\,\,a_{0}=-\frac{3(b+2)}{b+1}
,\,\,a_{1}=0,\,\,a_{2}=0,\,\,c_{1}=-\sqrt{c_{2}^{2}-1}$ : $$u_{10}(x,t)=\frac{3(b+2)}{(b+1)\left( 1+\sqrt{c_{2}^{2}-1}\sinh
\left( x-(1+ \frac{b}{2})t\right) -c_{2}\cosh \left(
x-(1+\frac{b}{2})t\right) \right) }.$$
The improved tanh-coth method
=============================
The traveling wave transformation $$\label{9}
\xi=x+\lambda t$$ reduces (\[eq00\]) to the nonlinear ode $$\label{equ05}
(b+1) u'(\xi ) u(\xi
)^2-u'''(\xi ) u(\xi )-\lambda
u'''(\xi)+\lambda u'(\xi )-b
u'(\xi ) u''(\xi )=0.$$ We seek solutions to (\[equ05\]) in the form $$\label{equ06}
u(\xi)=a_0+\sum _{i=1}^m \left(a_i \varphi
^i+c_i
\varphi ^{-i}\right),\\\\$$ where $m$ is a positive integer that will be determined and $\varphi=\varphi(\xi)$ satisfies the Riccati equation $$\label{equ07}
\varphi'(\xi)=\alpha+\beta \varphi(\xi)+\gamma \varphi(\xi)^2,$$ where $\alpha$, $\beta$, $\gamma$ are constants. The solutions of (\[equ07\]) are given by:
1. If $\alpha=0$ and $\beta\neq 0$, $$\label{13}
\varphi(\xi)=\dfrac{\beta}{-\gamma+\beta e^{-\beta \xi}}.$$
2. If $\beta=\alpha=0$ and $\gamma\neq 0$, $$\label{16}
\varphi(\xi)=-\dfrac{1}{\gamma \xi}.$$
3. If $\gamma=0$ and $\beta\neq 0$, $$\label{15}
\varphi(\xi)=\dfrac{-\alpha+\beta e^{\beta \xi}}{\beta}.$$
4. If $\beta=0$ and $\alpha\gamma\neq 0$, $$\label{14}
\begin{cases}
\varphi(\xi)=\dfrac{\sqrt{\alpha \gamma}}{\gamma}\tan(\sqrt{\alpha
\gamma}\xi), & \alpha>0,\gamma>0\\
\varphi(\xi)=\dfrac{\sqrt{\alpha \gamma}}{\gamma}\tanh(\sqrt{\alpha
\gamma}\xi), & \alpha>0,\gamma<0\\
\varphi(\xi)=\dfrac{\sqrt{-\alpha
\gamma}}{\gamma}\tanh(-\sqrt{-\alpha \gamma}\xi), &
\alpha<0,\gamma>0\\
\varphi(\xi)=\dfrac{\sqrt{\alpha \gamma}}{\gamma}\tan(-\sqrt{\alpha
\gamma}\xi), & \alpha<0,\gamma<0.
\end{cases}$$
5. If $\beta\neq 0$ and $\beta^2=4\alpha\gamma$, $$\label{18}
\varphi(\xi)=-\frac{2 \alpha (\beta \xi
+2)}{\beta ^2 \xi }$$
6. If $\beta^2<4 \alpha\gamma$, $$\label{16a}
\varphi(\xi)=\frac{\sqrt{4 \alpha \gamma
-\beta ^2} \tan
\left(\frac{1}{2} \sqrt{4
\alpha \gamma -\beta ^2}
\xi \right)-\beta }{2
\gamma }.$$
7. If $\beta^2>4 \alpha\gamma$ and $\gamma\neq 0$, $$\label{16b}
\varphi(\xi)=\frac{\sqrt{\beta ^2-4 \alpha
\gamma } \tanh
\left(\frac{1}{2}
\sqrt{\beta ^2-4 \alpha
\gamma } \xi \right)-\beta
}{2 \gamma }.$$
Substituting (\[equ06\]), along with (\[equ07\]) into the left hand of (\[equ05\]) and collecting all terms with the same power in $\varphi(\xi)$, we get a polynomial in the variable $\varphi=\varphi(\xi)$. This polynomial has the form $$\label{eq05}
p\varphi^{3m+1}(\xi)+q\varphi^{2m+1}(\xi)+r\varphi^{m+3}(\xi)+\text{lower
degree terms}$$ With $m$ determined, we equate each coefficient of the polynomial to zero. This will give an overdetermined system of algebraic equations involving the parameters $\lambda$, $a_0$, $a_i$ and $c_i$ ($i = 0,\ldots,m$). Having determined these parameters, and using (\[equ06\]) and (\[13\])-(\[16b\]), we obtain analytic solutions $u(x, t)$ in a closed form.\
To determine the parameter $m$, we usually balance the terms of highest order in the resulting polynomial. In our case, we determine $m$ from equations $3m+1=2m+1$, $3m+1=m+1$, $2m+1=m+3$. This gives $m=0$ or $m=2$. We define $m=2$ to avoid trivial solutions, so solutions have the form $$\label{eq06}
u(\xi)=a_0+a_1\varphi(\xi)+\dfrac{c_1}{\varphi(\xi)}+a_2\varphi^2(\xi)+\dfrac{c_2}{\varphi^2(\xi)}.$$ After solving the algebraic system , following solutions are obtained : **First Case** : $\beta \neq
0$ and $\beta ^{2}=4\alpha \gamma .$ $\bullet $ $a_{0}=\frac{3(b+2)\beta ^{2}}{2(b+1)}-1$, $\,\,a_{1}=0$, $ \,\,a_{2}=0$, $\,\,c_{1}=\frac{6(b+2)\alpha \beta }{b+1}$, $\,\,c_{2}=\frac{ 6(b+2)\alpha ^{2}}{b+1}$, $\,\,\lambda =-b-1:$ $$u_{11}(x,t)=\frac{6(b+2)\beta ^{2}}{(b+1)(\beta \xi +2)^{2}}-1.$$ **Second Case** : $\alpha =0$ and $\beta \neq 0.$$\bullet $ $\,\,a_{0}=-\frac{-b\beta ^{2}-2\beta
^{2}+b+\sqrt{1-b(b+2)\left( \beta ^{4}-1\right) }+1}{2(b+1)}$, $\,a_{1}=\frac{6(b+2)\beta \gamma }{b+1}$, $\,\,a_{2}=\frac{6(b+2)\gamma ^{2}}{b+1}$, $\,\,c_{1}=0$,$\,\,c_{2}=0$, $\lambda =\frac{1}{2}\left( -b-\sqrt{1-b(b+2)\left( \beta
^{4}-1\right) } -1\right) :$ $$\begin{aligned}
\,u_{12}(x,\,\,t) &=&\frac{1}{2(b+1)}\left( (b+2)\beta
^{2}-b-1-\sqrt{
1-b(b+2)\left( \beta ^{4}-1\right) }\right) + \\
&&\frac{6(b+2)\gamma \beta ^{2}}{b+1}\left( \frac{1}{e^{-\beta \xi
}\beta -\gamma }+\frac{\gamma }{\left( e^{-\beta \xi }\beta -\gamma
\right) ^{2}} \right) .\end{aligned}$$ $\bullet $ $\,a_{0}=\frac{2\beta ^{2}+b\left( \beta ^{2}-1\right)
+\sqrt{ 1-b(b+2)\left( \beta ^{4}-1\right)
}-1}{2(b+1)}$,$\,a_{1}=\frac{6(b+2)\beta \gamma
}{b+1}$,$\,\,a_{2}=\frac{6(b+2)\gamma ^{2}}{b+1}$, $\,c_{1}=0$,$
\,\,c_{2}=0$,$\,$$\,\ \lambda =\frac{1}{2}\left(
-b+\sqrt{1-b(b+2)\left( \beta ^{4}-1\right) } -1\right) :$ $$\begin{aligned}
\,u_{13}(x,\,\,t) &=&\frac{1}{2(b+1)}\left( (b+2)\beta
^{2}-b-1+\sqrt{
1-b(b+2)\left( \beta ^{4}-1\right) }\right) + \\
&&\frac{6(b+2)\gamma \beta ^{2}}{b+1}\left( \frac{1}{e^{-\beta \xi
}\beta -\gamma }+\frac{\gamma }{\left( e^{-\beta \xi }\beta -\gamma
\right) ^{2}} \right) .\end{aligned}$$ **Third Case** : $\beta =0$ and $\alpha \gamma \neq 0:$ $\bullet $ $a_{0}=-\frac{-8\alpha \gamma b+b-16\alpha \gamma +\sqrt{
b(b+2)\left( 1-256\alpha ^{2}\gamma ^{2}\right)
+1}+1}{2(b+1)}$,$\,\,a_{1}=0$ ,$\,\,a_{2}=\frac{6(b+2)\gamma
^{2}}{b+1}$,$\,\,c_{1}=0$,$\,\,c_{2}=\frac{ 6(b+2)\alpha
^{2}}{b+1}$,$\lambda =\frac{1}{2}\left(
-b-\sqrt{b(b+2)\left( 1-256\alpha ^{2}\gamma ^{2}\right)
+1}-1\right) :$ $$u_{14}(x,t)=\frac{-(b+1)-16\alpha \gamma (b+2)-\sqrt{b(b+2)\left(
1-256\alpha ^{2}\gamma ^{2}\right) +1}+48\alpha \gamma (b+2)\csc
^{2}\left( 2 \sqrt{\alpha \gamma }\xi \right) }{2(b+1)}.$$ $\bullet $ $a_{0}=\frac{8\alpha \gamma b-b+16\alpha \gamma +\sqrt{
b(b+2)\left( 1-256\alpha ^{2}\gamma ^{2}\right)
+1}-1}{2(b+1)}$,$\,\,a_{1}=0$ ,$\,\,a_{2}=\frac{6(b+2)\gamma
^{2}}{b+1}$,$\,\,c_{1}=0$,$\,\,c_{2}=\frac{ 6(b+2)\alpha
^{2}}{b+1}$,$\lambda =\frac{1}{2}\left(
-b+\sqrt{b(b+2)\left( 1-256\alpha ^{2}\gamma ^{2}\right)
+1}-1\right) :$ $$u_{15}(x,t)=\frac{-(b+1)-16\alpha \gamma (b+2)+\sqrt{b(b+2)\left(
1-256\alpha ^{2}\gamma ^{2}\right) +1}+48\alpha \gamma (b+2)\csc
^{2}\left( 2 \sqrt{\alpha \gamma }\xi \right) }{2(b+1)}.$$ $\bullet $ $a_{0}=-\frac{-8\alpha \gamma b+b-16\alpha \gamma +\sqrt{
b(b+2)\left( 1-16\alpha ^{2}\gamma ^{2}\right)
+1}+1}{2(b+1)}$,$\,\,a_{1}=0$, $\,\,a_{2}=0$,$\,\,c_{1}=0$,$\,\,c_{2}=\frac{6(b+2)\alpha
^{2}}{b+1}$, $\lambda =\frac{1}{2}\left( -b-\sqrt{b(b+2)\left( 1-16\alpha
^{2}\gamma ^{2}\right) +1}-1\right) :$ $$u_{16}(x,t)=\frac{-(b+1)+8\alpha \gamma (b+2)-\sqrt{b(b+2)\left(
1-16\alpha ^{2}\gamma ^{2}\right) +1}+12\alpha \gamma (b+2)\cot
^{2}\left( \sqrt{\alpha \gamma }\xi \right) }{2(b+1)}.$$ $\bullet $ $a_{0}=-\frac{-8\alpha \gamma b+b-16\alpha \gamma +\sqrt{
b(b+2)\left( 1-16\alpha ^{2}\gamma ^{2}\right)
+1}+1}{2(b+1)}$,$\,\,a_{1}=0$, $\,\,a_{2}=\frac{6(b+2)\gamma
^{2}}{b+1}$,$\,\,c_{1}=0$,$\,\,c_{2}=0$, $\lambda =\frac{1}{2}\left( -b-\sqrt{b(b+2)\left( 1-16\alpha
^{2}\gamma ^{2}\right) +1}-1\right) :$ $$u_{17}(x,t)=\frac{-(b+1)+8\alpha \gamma (b+2)-\sqrt{b(b+2)\left(
1-16\alpha ^{2}\gamma ^{2}\right) +1}+12\alpha \gamma (b+2)\tan
^{2}\left( \sqrt{\alpha \gamma }\xi \right) }{2(b+1)}.$$ $\bullet $ $a_{0}=\frac{8\alpha \gamma b-b+16\alpha \gamma +\sqrt{
b(b+2)\left( 1-16\alpha ^{2}\gamma ^{2}\right)
+1}-1}{2(b+1)}$,$\,\,a_{1}=0$, $\,\,a_{2}=0$,$\,\,c_{1}=0$,$\,\,c_{2}=\frac{6(b+2)\alpha
^{2}}{b+1}$, $\lambda =\frac{1}{2}\left( -b+\sqrt{b(b+2)\left( 1-16\alpha
^{2}\gamma ^{2}\right) +1}-1\right) :$ $$u_{18}(x,t)=\frac{-(b+1)+8\alpha \gamma (b+2)+\sqrt{b(b+2)\left(
1-16\alpha ^{2}\gamma ^{2}\right) +1}+12\alpha \gamma (b+2)\cot
^{2}\left( \sqrt{\alpha \gamma }\xi \right) }{2(b+1)}.$$ $\bullet $ $a_{0}=\frac{8\alpha \gamma b-b+16\alpha \gamma +\sqrt{
b(b+2)\left( 1-16\alpha ^{2}\gamma ^{2}\right)
+1}-1}{2(b+1)}$,$\,\,a_{1}=0$, $\,\,a_{2}=\frac{6(b+2)\gamma
^{2}}{b+1}$,$\,\,c_{1}=0$,$\,\,c_{2}=0$, $\lambda =\frac{1}{2}\left( -b+\sqrt{b(b+2)\left( 1-16\alpha
^{2}\gamma ^{2}\right) +1}-1\right) :$ $$u_{19}(x,t)=\frac{-(b+1)+8\alpha \gamma (b+2)+\sqrt{b(b+2)\left(
1-16\alpha ^{2}\gamma ^{2}\right) +1}+12\alpha \gamma (b+2)\tan
^{2}\left( \sqrt{\alpha \gamma }\xi \right) }{2(b+1)}.$$ **Fourth Case** : $\Delta \neq 0$, where $\Delta =\beta
^{2}-4\alpha \gamma :$$\bullet $ $a_{0}=\frac{24\alpha \gamma +2\Delta +b(12\alpha \gamma
+\Delta -1)-\sqrt{1-b(b+2)\left( \Delta ^{2}-1\right)
}-1}{2(b+1)}$,$\,\,a_{1}=\frac{ 6(b+2)\beta \gamma
}{b+1}$,$\,\,a_{2}=\frac{6(b+2)\gamma ^{2}}{b+1}$,$
\,\,c_{1}=0$,$\,\,c_{2}=0$,$\,\,\lambda =\frac{1}{2}\left(
-b-\sqrt{1-b(b+2)\left( \Delta ^{2}-1\right) }-1\right) :$ $$u_{20}(x,t)=-\frac{2\Delta (b+2)+b+1+\sqrt{1-b(b+2)\left( \Delta
^{2}-1\right) }}{2(b+1)}+\frac{3(b+2)\Delta }{2(b+1)}\tanh
^{2}\left( \frac{1 }{2}\sqrt{\Delta }\xi \right) .$$ $\bullet $ $a_{0}=\frac{24\alpha \gamma +2\Delta +b(12\alpha \gamma
+\Delta -1)+\sqrt{1-b(b+2)\left( \Delta ^{2}-1\right)
}-1}{2(b+1)}$,$\,\,a_{1}=\frac{ 6(b+2)\beta \gamma
}{b+1}$,$\,\,a_{2}=\frac{6(b+2)\gamma ^{2}}{b+1}$,$
\,\,c_{1}=0$,$\,\,c_{2}=0$,$\,\,\lambda =\frac{1}{2}\left(
-b+\sqrt{1-b(b+2)\left( \Delta ^{2}-1\right) }-1\right) :$ $$u_{21}(x,t)=-\frac{2\Delta (b+2)+b+1-\sqrt{1-b(b+2)\left( \Delta
^{2}-1\right) }}{2(b+1)}+\frac{3(b+2)\Delta }{2(b+1)}\tanh
^{2}\left( \frac{1 }{2}\sqrt{\Delta }\xi \right) .$$ $\bullet $ $a_{0}=\frac{24\alpha \gamma +2\Delta +b(12\alpha \gamma
+\Delta -1)+\sqrt{1-b(b+2)\left( \Delta ^{2}-1\right)
}-1}{2(b+1)}$,$\,\,a_{1}=0$,$
\,\,a_{2}=0$,$\,\,c_{1}=\frac{6(b+2)\alpha \beta
}{b+1}$,$\,\,c_{2}=\frac{ 6(b+2)\alpha ^{2}}{b+1}$,$\,\,\lambda =\frac{1}{2}\left( -b+\sqrt{1-b(b+2)\left( \Delta
^{2}-1\right) }-1\right) :$ $$\begin{aligned}
u_{22}(x,t) &=&\frac{1}{2(b+1)}((\Delta +12\alpha \gamma
)(b+2)-(b+1)+\sqrt{
1-b(b+2)\left( \Delta ^{2}-1\right) })- \\
&&\frac{12(b+2)\alpha \gamma \left( \beta ^{2}+\sqrt{\Delta }\tanh
\left( \frac{1}{2}\sqrt{\Delta }\xi \right) \beta -2\alpha \gamma
\right) }{ (b+1)\left( \beta +\sqrt{\Delta }\tanh \left(
\frac{1}{2}\sqrt{\Delta }\xi \right) \right) ^{2}}.\end{aligned}$$ $\bullet $ $a_{0}=\frac{24\alpha \gamma +2\Delta +b(12\alpha \gamma
+\Delta -1)-\sqrt{1-b(b+2)\left( \Delta ^{2}-1\right)
}-1}{2(b+1)}$,$\,\,a_{1}=0$,$
\,\,a_{2}=0$,$\,\,c_{1}=\frac{6(b+2)\alpha \beta
}{b+1}$,$\,\,c_{2}=\frac{ 6(b+2)\alpha ^{2}}{b+1}$,$\,\,\lambda =\frac{1}{2}\left( -b-\sqrt{1-b(b+2)\left( \Delta
^{2}-1\right) }-1\right) :$ $$\begin{aligned}
u_{23}(x,t) &=&\frac{1}{2(b+1)}\left( (\Delta +12\alpha \gamma
)(b+2)-(b+1)-
\sqrt{1-b(b+2)(\Delta ^{2}-1)}\right) - \\
&&\frac{12(b+2)\alpha \gamma \left( \beta ^{2}+\sqrt{\Delta }\tanh
\left( \frac{1}{2}\sqrt{\Delta }\xi \right) \beta -2\alpha \gamma
\right) }{ (b+1)\left( \beta +\sqrt{\Delta }\tanh \left(
\frac{1}{2}\sqrt{\Delta }\xi \right) \right) ^{2}}.\end{aligned}$$
Conclusions
===========
In this paper, by using three distinct methods and the help of a symbolic computation engine, we obtain some exact solutions for the equation (\[eq00\]). The methods we employ certainly work well for a large class of very interesting nonlinear equations. The main advantage of these methods is that their capability of greatly reducing the size of computational work compared to existing techniques such as the pseudo spectral method, the inverse scattering method, Hirota’s bilinear method, and the truncated Painlevé expansion. We think that some of the solutions we obtained are new in the literature.
[99]{} , *Asymptotic integrability Symmetry and Perturbation Theory*, (Rome, 1998) ed A Degasperis and G Gaeta (River Edge, NJ: World Scientific Publishing), pp. 23 - 37.
, *An integrable shallow water equation with peaked solitons*, Phys. Rev. Lett. **71**(1993) 1661–4. , A quasi-linear parabolic equation occuring in aerodynamics,*Quart. Appl. Math.*, 1951, 9, pages 225-236.
, *A new integrable equation with peakon solutions Theoretical and Mathematical Physics*, **133**(2002) 1463–1474 Preprint nlin.SI/0205023.
, *Integrable and non-integrable equations with peakons Nonlinear Physics*, Theory and Experiment (Gallipoli, 2002), vol II eds M.J. Ablowitz, M. Boiti, F. Pempinelli and B. Prinari (Singapore: World Scientific Publishing) pp 37–43. Preprint nlin.SI/0209008. , The partial differential equation $u_t+u
u_x=u_{xx}$, *Comm. Pure Appl. Math.*, **3**, 1950, 201-230. , *A note on solitary waves for modified forms of Camassa–Holm and Degasperis$–$Procesi equations*, Physics Letters A, **366**(2007) 377 - 381. , *Solitary wave solutions for modified forms of Degasperis$–$Procesi and Camassa–Holm equations*, Physics Letters A, **352**(2006) 500 - 504. , Modified extended tanh-function method and its application on nonlinear physical equations, Physics Letters A 353 (2006) 487 - 492.
|
---
abstract: 'Polarization transfer from the longitudinally polarized positron to the final-state quark is considered for the underlying QCD subprocesses giving rise to the reaction $e^+_{\uparrow} + p \to e^+ + \Lambda_{\uparrow} + X $ at the HERMES energy.'
address: |
Institute for High Energy Physics\
142284 Protvino, Moscow Region, Russia
author:
- Yuri Arestov
title: |
Polarization transfer in SIDIS\
for $\Lambda$ production$^{\dag}$
---
\#1[[1 2]{} ]{} \#1[ Tr]{} \#1\#2\#3 [\#1 [\^3 \#2 \^3 \#3]{}]{} \#1\#2\#3\#4\#5 [\#1 \#2 [\^6 \#3 \^3 \#4 \^3 \#5 ]{}]{}
Measuring the longitudinal $\Lambda$ polarization in the semi-inclusive deeply inelastic scattering (SIDIS) $$\begin{aligned}
e^+_{\uparrow} + p \to e^+ + \Lambda_{\uparrow} + X \, ,
\label{sidis}\end{aligned}$$ that is under study in the experiment HERMES at DESY, may give important information for the determination of the twist-2 polarized fragmentation function $G_1 (z,\mu )$. The latter can be found through the relation $$\begin{aligned}
D_{LL} \sim F_q (x)\cdot\hat\sigma
\cdot\hat d_{LL} \cdot G_1 (z,\mu )\, , \label{G1}\end{aligned}$$ where $F_q(x)$ is the unpolarized parton (quark/gluon) distribution function in the proton and the polarization transfer $D_{LL} $ is defined as $$D_{LL}=\frac{\sigma_{++}+\sigma_{--}-\sigma_{+-}-\sigma_{-+}}
{\sigma_{++}+\sigma_{--}+\sigma_{+-}+\sigma_{-+}} \,\equiv \,
\frac{{\rm d}\Delta\sigma}{{\rm d}\sigma} \, , \label{Dll}$$ and it can be measured in the experiment. The hatted entities in (\[G1\]) relate to the underlying QCD subprocess. The subscripts in $\sigma$’s in (\[Dll\]) denote the positron and $\Lambda$ helicities.
The published HERMES measurement in the reaction (\[sidis\]) is $D_{LL}=0.11\pm 0.17\pm 0.03 $[@exper].
As is seen from (\[G1\]), it is very instructive to study for the purpose of the data handling and analysis. The parameter can also be expressed as = ${P^z_{out}/ P^z_{in}}$ ——————————————————-\
[**$\dag$ Talk given at IX Workshop on High Energy Spin Physics, Dubna, Aug. 2001**]{}\
where $P$’s are the longitudinal components of the initial and final polarization vectors.
The elementary subprocesses under study are the elastic quark probing (Fig. \[sub1\]) $$\begin{aligned}
e^+_{\uparrow} ~+~ q \to e^+ ~+~ q_{\uparrow}' ~~~~({\rm LO})
\label{LO}\end{aligned}$$ and the diagrams with the gluon correction (Fig. \[sub2\]) $$\begin{aligned}
e^+_{\uparrow} ~+~ q \to e^+ ~+~ q_{\uparrow}' ~+~G ~~~~({\rm NLO}).
\label{NLO}\end{aligned}$$ Since we are interested mainly in the central quark production region, here we\
neglect the contribution of the process $e^+_{\uparrow} ~+~ G \to e^+ ~+~ q_{\uparrow} ~+~\bar q $ with the quark pair production by $\gamma^{\ast}G$ coupling. The spin transfer properties in the production of heavy $Q\bar Q$ quarks are discussed in ref.[@YuA].
The purpose of this study is to look for kinematic regions where the polarization transfer gets enhanced in the subprocesses of fig. 1 and fig. 2. Being translated onto the hadronic level, these kinematic regions, if any, can serve as guidelines for the experiment HERMES. The interval of DIS variables ($Q^2 ,\,\, y$ and other) was chosen to be typical for HERMES.
The calculations are based on the following expression for the scattered quark vector polarization $$P_{out}^z = \frac{\left( 1+P_z^{in}\right)
\left( \vert M_{++} \vert^2 - \vert M_{-+} \vert^2 \right) +
\left( 1-P_z^{in}\right)
\left( \vert M_{+-} \vert^2 - \vert M_{--} \vert^2 \right) }
{\left( 1+P_z^{in}\right)
\left( \vert M_{++} \vert^2 + \vert M_{-+} \vert^2 \right) +
\left( 1-P_z^{in}\right)
\left( \vert M_{+-} \vert^2 + \vert M_{--} \vert^2 \right) }\, ,$$ which can be derived from the final-state spin-density matrix.
The properties of the elastic subprocess (\[LO\]) are plotted in fig. \[eqeq\]. Both the differential cross section ${\rm d}\sigma$ and polarized differential cross section ${\rm d}\Delta\sigma$ decrease with increasing $Q^2$ so that the polarization transfer increases rapidly. It achieves the value $\sim$0.8 at the extreme value of $Q^2 \sim 4$GeV$^2$. For modest momentum transfers, say $Q^2 \sim 2$GeV$^2$, has the ’normal’ values of $\sim$0.2.
Fig.\[eqeqg\] represents the parameter calculated for the reaction (\[NLO\]) at four different values of the angle $\phi$ between two characteristic planes of the reaction – leptonic plane and the plane formed by the virtual photon and the outgoing quark. As is known, the polarization properties of the reactions 2$\to$3 depend essentially, in general, on the rotational angle $\phi$.
The distributions of in fig.\[eqeqg\] are shown for four rotations of the planes with an increment equal to 90$^o$. The lowest values of are obtained at $\phi$=180$^o$ when the normals to the said plains are anti-collinear. The more sizeable values of appear when the normals make right angles (90$^o$ or 270$^o$) – up to $\sim$0.4. And the most favorite kinematic region is predicted for the collinar normals ($\phi$=0) where the polarization transfer increases up to 0.6.
So coming down to the experiment (\[sidis\]), one can expect the sizeable longitudinal component of the $\Lambda$ polarization induced by the polarized positron under the kinematic configurations in which the normals to the leptonic plane and the plane ($\gamma^{\ast}$,$\Lambda$) are close to be collinear. The numeric estimates made at the HERMES energy can certainly be affected by the higher-order corrections.
A.Airapetian et al, [*Phys. Rev.*]{} D (in press); e-print hep-ex/9911017. Yu.Arestov, Twist-2 polarized fragmentation function in the open charm production in DIS, [*Proc. Spin’2000*]{}, Oct. 2000, Osaka, Japan.
|
---
abstract: 'The goal of the present paper is to investigate the algebraic structure of global conformal invariants of submanifolds. These are defined to be conformally invariant integrals of geometric scalars of the tangent and normal bundle. A famous example of a global conformal invariant is the Willmore energy of a surface. In codimension one we classify such invariants, showing that under a structural hypothesis the integrand can only consist of an intrinsic scalar conformal invariant, an extrinsic scalar conformal invariant and the Chern-Gauss-Bonnet integrand. In particular, for codimension one surfaces, we show that the Willmore energy is the unique global conformal invariant, up to the addition of a topological term (the Gauss curvature, giving the Euler Characteristic by the Gauss Bonnet Theorem). A similar statement holds also for codimension two surfaces, once taking into account an additional topological term given by the Chern-Gauss-Bonnet integrand of the normal bundle. We also discuss existence and properties of natural higher dimensional generalizations of the Willmore energy.'
address:
- ' ETH, Rämistrasse 101, Zurich, Switzerland'
- |
School of Mathematics and Physics\
The University of Queensland\
St Lucia, Brisbane, Australia, 4072
author:
- Andrea Mondino
- Huy The Nguyen
bibliography:
- 'conformalgeometry.bib'
title: Global Conformal Invariants of Submanifolds
---
Introduction
============
Let us consider an $m$-dimensional Riemannian manifold $ ( {{\mathcal}{M}}^m, g ^m )$ isometrically immersed in a Riemannian manifold $ ( \bar {{\mathcal}{M}}^n , \bar g ^ n )$. The fundamental objects describing the *intrinsic* geometry of $( {{\mathcal}{M}}^m, g^m )$ are the metric $ g^m$, the curvature tensor $ R _{ ijkl} $, and the Levi-Civita connection. On the other hand, the fundamental quantities describing the *extrinsic* geometry of $ ( {{\mathcal}{M}}^m, g^m) $ as submanifold of $ ( \bar {{\mathcal}{M}}^n , \bar g ^ n ) $ are the second fundamental form $ h _{ij } ^ \alpha $, the normal connection $ \nabla ^ \perp$, and the normal curvature $\bar R ^ \perp_{ ij\alpha\beta} $, where Roman indices indicate tangential directions and Greek indices indicate normal ambient directions. It is well known (see Section \[sec:BM\] for more details) that these geometric quantities are not mutually independent but must satisfy some compatibility conditions, the so called Gauss-Codazzi-Mainardi-Ricci equations. A natural way to define *geometric scalars* out of this list of tensors is by taking tensor products and then contracting using the metric $\bar{g}$. More precisely, we first take a finite number of tensor products, say $$\nonumber
R_{i_1 j_1 k_1 l_1} \otimes \ldots \otimes R^\perp_{i_r, j_r, \alpha_r, \beta_r} \otimes \ldots \otimes h_{i_s j_s} \quad,$$ thus obtaining a tensor of rank $4+\ldots+4+\ldots+2+\ldots+2$. Then, we repeatedly pick out pairs of indices in the above expression and contract them against each other using the metric $\bar{g}^{\alpha \beta}$ (of course, in case of contractions not including the normal curvature $R^\perp$ it is enough to contract using $g^{ij}$). This can be viewed in the more abstract perspective of Definition \[def:ComplContr\] by saying that we consider a *geometric complete contraction* $$\nonumber \label{eq:ContrGen0}
C(\bar{g},R,R^\perp,h) = \operatorname{contr}(\bar{g}^{\alpha_1 \beta_1}\otimes \ldots \otimes R_{i_1 j_1 k_1 l_1} \otimes \ldots \otimes R^\perp_{i_r, j_r, \alpha_r, \beta_r} \otimes \ldots \otimes h_{i_s j_s}) \quad.$$ Let us stress that a complete contraction is determined by the *pattern* according to which different indices contract agains each other; for example, the complete contraction $R_{ijkl}\otimes R^{ijkl}$ is different from $R^i_{ikl} \otimes R_{s}^{ksl}$. By taking linear combinations of geometric complete contractions (for the rigorous meaning see Definition \[def:LinCombCC\]), we construct *geometric scalar quantities* $$\nonumber \label{eq:PgRh0}
P(\bar{g},R, R^\perp, h):= \sum_{ l \in L } a _ l C ^l (\bar{g},R, R^\perp, h) \quad.$$ The goal of the present paper is to classify those geometric scalar quantities which, once integrated over arbitrary submanifolds $({{\mathcal}{M}}^m, g^m)$ of arbitrary manifolds $(\bar{{{\mathcal}{M}}}^n,\bar{g}^n)$, give rise to *global conformal invariants*. More precisely, we say that the geometric scalar quantity $P(\bar{g},R, R^\perp, h)$ is a *global conformal invariant for $m$-submanifolds in $n$-manifolds* if the following holds: for any ambient Riemannian manifold $\bar{{{\mathcal}{M}}}^n$, any compact orientable $m$-dimensional immersed submanifold ${{\mathcal}{M}}^m$ of $\bar{{{\mathcal}{M}}}^n$ and any $\phi \in C^\infty(\bar{{{\mathcal}{M}}})$, if one considers the conformal deformation $\hat{\bar{g}}:= e ^{2 \phi(x)} \bar{g}$ and calls $\hat{R}, \hat{R}^\perp, \hat{h}$ the tensors computed with respect to the conformal metric $\hat{\bar{g}}$, then $$\nonumber \label{eq:ConInv0}
\int_{{{\mathcal}{M}}^m} P(\hat{\bar{g}},\hat{R}, \hat{R}^\perp, \hat{h}) \, d \mu_{\hat{g}}= \int_{{{\mathcal}{M}}^m} P(\bar{g},R, R^\perp, h)\, d \mu_g \quad.$$
Let us mention that the corresponding classification for *intrinsic* global conformal invariants of Riemannian manifolds was a classical problem in conformal geometry motivated also by theoretical physics (the goal being to understand the so called conformal anomalies): indeed it is the celebrated Deser-Schwimmer conjecture [@DeserSchwimmer] which has recently been solved in a series of works by Alexakis [@AlexI; @AlexII; @AlexPf1; @AlexPf2; @AlexIV; @AlexBook]. Inspired by the aforementioned papers, we address the problem of an analogous classification for global conformal invariants, but this time, *for submanifolds*. Of course, as explained above, if one considers global conformal invariants for *submanifolds* many other *extrinsic* terms appears, namely the second fundamental form, the curvature of the normal bundle, and the normal connection; therefore the zoology of global conformal invariants is more rich and the classification more complicated.
A well-known example of a global conformal invariant for two-dimensional submanifolds (called from now on surfaces) is the *Willmore energy*. For an immersed surface $ f : {{\mathcal}{M}}^ 2 \rightarrow ( \bar {{\mathcal}{M}}^ n , \bar g^n ) $ this is defined by $$\label{eq:defW}
{\mathcal}{W} ( f ) = \int _{ {{\mathcal}{M}}} | H | ^ 2 d \mu _ g + \int _{ {{\mathcal}{M}}} \bar {K}_{ \bar {{\mathcal}{M}}} (Tf({{\mathcal}{M}})) d \mu _g \quad,$$ where $H= \frac{1}{2} g^{ij} h_{ij}$ is the mean curvature vector and $\bar {K}_{ \bar {{\mathcal}{M}}} (Tf({{\mathcal}{M}}))$ is the sectional curvature of the ambient manifold computed on the tangent space of $f({{\mathcal}{M}})$. Clearly, in case $\bar{{{\mathcal}{M}}}^n={\mathbb{R}}^n$, reduces to the familiar Euclidean Willmore energy as $ \bar K _{ {\mathbb{R}}^ n } = 0$. It is well known that the Willmore energy in Euclidean space is invariant under conformal transformations of the ambient manifolds, that is Möbius transformations where the inversion is centered off the submanifold. In fact, more generally the conformal Willmore energy is invariant under conformal deformations of the ambient background metric. This can be seen by the following decomposition, $$\begin{aligned}
\|h^\circ \| ^ 2 = \| h \| ^ 2 - 2 |H| ^ 2, \quad K_{{{\mathcal}{M}}} = \frac{1}{2} (4 |H| ^ 2 - \| h \| ^ 2 ) + \bar K _{ \bar {{\mathcal}{M}}}, \end{aligned}$$ where $h^\circ_{ij}:=h_{ij}-H g_{ij}$ is the traceless second fundamental form, $K_{{{\mathcal}{M}}}$ is the Gauss curvature of $({{\mathcal}{M}},g)$, and in the second identity we just recalled the classical Gauss equation. It follows that the conformal Willmore energy can be written as $$\begin{aligned}
|H| ^ 2 + \bar K_{\bar {{\mathcal}{M}}} = \frac 12 \| h^\circ \| ^ 2 + K_{{\mathcal}{M}}.\end{aligned}$$ Since $ \| h^\circ \| ^ 2 d \mu_g $ is a pointwise conformal invariant and $ \int _{{{\mathcal}{M}}} K_{{{\mathcal}{M}}} d \mu_ g= 2 \pi \chi({{\mathcal}{M}})$ is a topological (hence, a fortiori, global conformal) invariant by the Gauss-Bonnet theorem, clearly any linear combination of the two is a global conformal invariant. A natural question is whether the Willmore functional is the unique energy having such an invariance property, up to topological terms. Let us briefly mention that the Willmore energy has recently received much attention [@Blaschke; @CaMo; @BeRi; @KMS; @KS; @LMS; @LY; @Mon2; @MR2; @MontielUrbano; @Riv; @Sim; @Will], and in particular the Willmore conjecture in codimension one has been solved [@MN].
We will show that, for codimension one surfaces, any global conformal invariant of a surface must be a linear combination of the norm squared of the traceless second fundamental form and the intrinsic Gauss curvature, that is the Willmore energy is the unique global conformal invariant up to the Gauss-Bonnet integrand which is a topological quantity (see Theorem \[thm:CD1surf\]).
A similar statement holds also for codimension two surfaces (see Theorem \[thm:CD2surf\]), once taking into account an additional topological term given by the Chern-Gauss-Bonnet integrand of the normal bundle.
For general submanifolds of codimension one, we show that if the global conformal invariant is a polynomial in the second fundamental form only, then it must be a contraction of the *traceless* second fundamental form, that is it must be the integral of a *pointwise* conformal invariant (see Theorem \[thm:P(g,h)\]). Combining this with a theorem of Alexakis [@AlexI Theorem 1], we show that if $m$ is even and the global conformal invariant has no mixed contractions between the intrinsic and the extrinsic curvatures then the integrand must be a linear combination of contractions of the (intrinsic) Weyl curvature, contractions of the traceless second fundamental form and the integrand of the Chern-Gauss-Bonnet formula, see Theorem \[thm:P=P1+P2\].
As an application of these ideas, in the last Section \[sec:genWill\], we introduce a higher dimensional analogue of the Willmore energy for hypersurfaces in Euclidean spaces. Such new energies are conformally invariant and attain the strictly positive lower bound only at round spheres, with rigidity, regardless of the topology of the hypersurface (see Theorem \[thm:3\] and Theorem \[thm:4\]).
[**Acknowledgments**]{}\
This work was written while the first author was visiting the [*Forschungsinstitut für Mathematik*]{} at the ETH Zürich. He would like to thank the Institut for the hospitality and the excellent working conditions.\
The second author is supported by the ETH fellowship.
Background Material {#sec:BM}
===================
Complete contractions: abstract definition
------------------------------------------
Following [@AlexI], in this short section we define the notion of complete contractions.
\[def:ComplContr\] Any complete contraction $$\begin{aligned}
C = \operatorname{contr}( ( A ^ 1 ) _{ i_ 1\dots i_s} \otimes ( A ^ t ) _{ j_1 \dots j _ q }) \end{aligned}$$ will be seen as a formal expression. Each factor $ ( A ^ l ) _{ i _ 1\dots i_s } $ is an ordered set of slots. Given the factors $ A ^ 1 _{ i _1\dots i _ s }, A ^ l _{ j_1\dots j _ q } $ a complete contraction is then a set of pairs of slots $ ( a _1, b_1 ), \dots, (a _ w, b _ w ) $ with the following properties: if $ k \neq l , \{a _ l, b _ l \}\cap \{ a _k ,b _k \} = \emptyset $, $ a _k \neq b _k $ and $ \bigcup_{ i = 1 } ^ { w } \{ a _i, b _i \} = \{ i _ 1 , \dots, j_q \}.$ Each pair corresponds to a particular contraction.
Two complete contractions $$\begin{aligned}
\operatorname{contr}(( A^1 ) _{ i_1\dots i _ s } \otimes \dots \otimes ( A ^ t ) _ { j _1\dots j _w} ) \end{aligned}$$ and $$\begin{aligned}
\operatorname{contr}( (B^1)_{f _1\dots f _ q} \otimes \dots \otimes ( B ^ {t'} )_{v _ 1\dots v _ z } ) \end{aligned}$$ will be identical if $ t = t ', A ^ l = B ^ l$ and if the $ \mu$-th index in $ A ^ l$ contracts against the $ \nu$-th index in $ A^r$ then the $\mu$-th index in $ B^ l $ contracts against the $\nu$-th in $ B ^ r $.
For a complete contraction, the *length* refers to the number of factors.
\[def:LinCombCC\] Linear combinations of complete contractions are defined as expressions of the form $$\begin{aligned}
\sum_{ l \in L } a _ l C ^l_1, \quad \sum_{ r \in R } b _r C_2 ^ r \quad,\end{aligned}$$ where each $ C ^ l _i$ is a complete contraction. Two linear combinations are identical if $ R = L$, $ a _ l = b _l $ and $ C_1 ^ l = C_2 ^ l$. A linear combination of complete contractions is identically zero if for all $ l \in L $ we have $ a _l =0 $. For any complete contraction, we will say that a factor $ ( A)_{ r _1\dots r _{s_l} } $ has an internal contraction if two indices in $$\begin{aligned}
A _{ r_1\dots r _ {s_l} } \end{aligned}$$ are contracting amongst themselves.
Riemannian and Submanifold Geometry {#SSS:RSG}
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Consider an $n$-dimensional Riemannian manifold $(\bar{{{\mathcal}{M}}} ^ n ,\bar{g} ^ n )$. Given $ x _ 0 \in \bar{{{\mathcal}{M}}}^ n$, let $ ( x ^1, \dots , x ^ n)$ be a local coordinate system with associated coordinate vector fields denoted by $ X^ \alpha $, that is $ X ^ \alpha = \frac { \partial }{ \partial x ^ \alpha } $. Called $\bar{\nabla}$ the Levi-Civita connection associated to $(\bar{{{\mathcal}{M}}} ^ n ,\bar{g} ^ n )$, the covariant derivative $ \bar{\nabla} _{ \frac { \partial }{\partial x ^ \alpha} }$ will be shortly denoted by $\bar{\nabla}_\alpha$.
The curvature tensor $ \bar{R} _{ \alpha \beta \gamma \eta } $ of $ \bar{g} ^ {n} _{ \alpha \beta } $ is given by the commutator of the covariant derivatives, that is $$\begin{aligned}
\label{eq:defR}
[ \bar{\nabla} _\alpha \bar{\nabla} _\beta - \bar{\nabla} _\beta \bar{\nabla} _\alpha ] X_\gamma = \bar{R} _{ \alpha\beta \gamma \eta} X ^ \eta \quad,\end{aligned}$$ which in terms of coordinate systems may be expressed by Christoffel symbols, $$\begin{aligned}
\bar{R} _{ \alpha \beta \gamma} ^ \eta = \partial_{ \beta } \Gamma _{ \alpha \gamma } ^ \eta - \partial _\gamma \Gamma ^ \eta _{ \alpha \beta } + \sum_{ \mu } (\Gamma_{ \alpha \gamma } ^ \mu \Gamma _{ \mu \beta } ^ \eta -\Gamma _{ \alpha \beta } ^ \mu \Gamma _{ \mu \gamma } ^ \eta ). \end{aligned}$$ The two Bianchi identities are then $$\begin{aligned}
&\bar{R}_{ \alpha \beta \gamma\eta} + \bar{R} _{ \gamma\alpha \beta \eta } + \bar{R} _{\beta \gamma\alpha \eta } = 0\\
&\bar{\nabla} _ \alpha \bar{R} _{ \beta \gamma\eta \mu } + \bar{\nabla} _\gamma \bar{R}_{\alpha \beta \eta \mu } + \bar{\nabla} _{ \beta } \bar{R} _{ \gamma\alpha \eta \mu } = 0. \end{aligned}$$ Recall that, under a conformal change of metric $\hat{\bar{g}}^n= e ^{2 \phi (x)} \bar{g}^n(x)$, the curvature transforms as follows (see for instance [@Eastwood]): $$\begin{aligned}
\bar{R}^{\hat{g}^n}_{\alpha \beta \gamma\eta }&=& e ^{2\phi (x)} \big[\bar{R}^{\bar{g}^n}_{\alpha \beta \gamma\eta }+\bar{\nabla}_{\alpha \eta }\phi \bar{g}^n_{\beta \gamma}+ \bar{\nabla}_{\beta \gamma} \phi \bar{g}^n_{\alpha \eta }-\bar{\nabla}_{\alpha \gamma}\phi \bar{g}^n_{\beta \eta }- \bar{\nabla}_{\beta \eta } \phi \bar{g}^n_{\alpha \gamma} \nonumber \\
&& \quad \quad \quad+ \bar{\nabla}_\alpha \phi \bar{\nabla}_\gamma \phi \bar{g}^n_{\beta \eta } + \bar{\nabla}_\beta \phi \bar{\nabla}_\eta \phi \bar{g}^n_{\alpha \gamma} - \bar{\nabla}_\alpha \phi \bar{\nabla}_\eta \phi \bar{g}^n_{\beta \gamma} - \bar{\nabla}_\beta \phi \bar{\nabla}_\gamma \phi \bar{g}^n_{\alpha \eta } \nonumber \\
&&\quad \quad \quad + |\bar{\nabla} \phi|^2 \bar{g}^n_{\alpha \eta } \bar{g}^n_{\beta \gamma} - |\bar{\nabla} \phi |^2 \bar{g}^n_{\alpha \gamma} g_{\eta \beta } \big] \quad .\label{eq:Rhatg} $$ Now let briefly introduce some basic notions of submanifold geometry. Given an $m$-dimensional manifold ${{\mathcal}{M}}^m$, $2\leq m <n$, we consider $f:{{\mathcal}{M}}^m \hookrightarrow \bar{{{\mathcal}{M}}}^n$, a smooth immersion. Recall that for every fixed $\bar{x} \in {{\mathcal}{M}}^m$ one can find local coordinates $(x^1, \ldots , x^n)$ of $\bar{{{\mathcal}{M}}}^n$ on a neighborhood $V_{f(\bar{x})}^{\bar{{{\mathcal}{M}}}}$ of $f(\bar{x})$ such that $\big((x^1\circ f), \ldots, (x^m\circ f) \big)$ are local coordinates on a neighborhood $U_{\bar{x}}^{{{\mathcal}{M}}}$ of $\bar{x}$ in ${{\mathcal}{M}}^m$ and such that $$f\big(U_{\bar{x}}^{{{\mathcal}{M}}}\big)=\Big\{(x^1, \ldots, x^n)\in V_{f(\bar{x})}^{\bar{{{\mathcal}{M}}}} \; : \; x^{m+1}=\ldots =x^n=0\Big \} \quad.$$ Such local coordinates on $\bar{{{\mathcal}{M}}}$ are said to be *adapted* to $f({{\mathcal}{M}})$. We use the convention that latin index letters vary from $1$ to $m$ and refer to geometric quantities on ${{\mathcal}{M}}^m$, while greek index letters vary from $1$ to $n$ (or sometimes from $m+1$ to $n$ if otherwise specified) and denote quantities in the ambient manifold $\bar{{{\mathcal}{M}}}^n$ (or in the orthogonal space to $f({{\mathcal}{M}})$ respectively). In adapted coordinates, it is clear that $X_1, \ldots, X_m$ are a bases for the tangent space of $f({{\mathcal}{M}}^m)$ and that the restriction of the ambient metric $\bar{g}^n$ defines an induced metric on ${{\mathcal}{M}}^m$, given locally by $$g^m_{i j}:= \bar{g}^n(X_i, X_j)\quad .$$ Using standard notation, $(g^m)^{ij}$ denotes the inverse of the matrix $(g_m)_{ij}$, that is $(g^m)^{ik} (g^m)_{kj}=\delta_{ij}$. For every $\bar{x} \in {{\mathcal}{M}}$, the ambient metric $\bar{g}^n$ induces the orthogonal splitting $$T_{f(\bar{x})} \bar{{{\mathcal}{M}}}= T_{f(\bar{x})} f({{\mathcal}{M}}) \oplus [T_{f(\bar{x})} f({{\mathcal}{M}})]^\perp \quad,$$ where, of course, $ [T_{f(\bar{x})} f({{\mathcal}{M}})]^\perp$ is the orthogonal complement of the $m$-dimensional subspace $T_{f(\bar{x})} f({{\mathcal}{M}}) \subset T_{f(\bar{x})} \bar{{{\mathcal}{M}}}$. We call $\pi_T: T_{f(\bar{x})} \bar{{{\mathcal}{M}}} \to T_{f(\bar{x})} f({{\mathcal}{M}})$ and $\pi_N=Id-\pi_T: T_{f(\bar{x})} \bar{{{\mathcal}{M}}} \to [T_{f(\bar{x})} f({{\mathcal}{M}})]^\perp$ the tangential and the normal projections respectively, one can define the *tangential* and the *normal connections* (which correspond to the Levi-Civita connections on $({{\mathcal}{M}}, g)$ and on the normal bundle respectively) by $$\begin{aligned}
\label{eq:tgNConn}
\nabla_{X_i} X_j&:=&\pi_T(\bar{\nabla}_{X_i} X_j), \; i,j=1,\ldots,m, \label{eq:tgConn} \\
\nabla^\perp _{X_i} X_{\alpha} &:=&\pi_N(\bar{\nabla}_{X_i} X_\alpha), \; i=1,\ldots,m, \alpha=m+1, \ldots, n. \label{eq:NConn}\end{aligned}$$ Associated to the tangential and normal connections we have the tangential and normal Riemann curvature tensors (which correspond to the curvature of $({{\mathcal}{M}}, g)$ and of the normal bundle respectively) defined analogously to : $$\begin{aligned}
[ \nabla _i \nabla _j - \nabla _j \nabla _i ] X_k &= &R _{ i j k l} X ^ l \;, \quad i,j,k,l=1,\ldots, m \label{eq:defRt} \\
\left[{\nabla^\perp_i \nabla^\perp_j - \nabla^\perp_j \nabla^\perp_i} \right] X_{\alpha} &= &R^\perp _{ i j \alpha \beta} X ^ \beta \;, \quad i,j=1,\ldots, m, \; \alpha, \beta=m+1, \ldots, n. \label{eq:defRn} \end{aligned}$$ The transformation of $R_{ijkl}$ and $R^\perp_{ij\alpha \beta}$ under a conformal change of metric is analogous to , just replacing $\bar{\nabla}$ with $\nabla$ or with $\nabla^{\perp}$ respectively. The second fundamental form $h$ of $f$ is defined by $$\label{eq:defh}
h(X_i, X_j):= \pi_{N} (\bar{\nabla}_{ X_i} X_j)= \bar{\nabla}_{X_i} X_j- \nabla_{X_i} X_j \quad .$$ It can be decomposed orthogonally into its trace part, the *mean curvature* $$\label{eq:defH}
H:=\frac{1}{m} (g^m)^{ij} h_{ij} \quad,$$ and its trace free part, the *traceless second fundamental form* $$\label{eq:hH}
h^\circ_{ij}:=h_{ij}-H g^m_{ij} \quad,$$ indeed it is clear from the definitions that $$\label{eq:Splith}
h_{ij}= h_{ij}^\circ + H g^m_{ij} \quad \text{ and } \quad \langle h^\circ, Hg^m \rangle = (g^m)^{ik} (g^m)^{jl} \; h^\circ_{ij} \; H g_{kl}=H \; \operatorname{Tr}_{g^m} (h^\circ) =0 \quad.$$ Under a conformal change of the ambient metric $\hat{\bar{g}}^n= e ^{2 \phi (x)} \bar{g}^n(x)$, the above quantities change as follow: $$\label{eq:CChH}
h^{\hat{g}}_{ij}= e^{\phi} \big[h_{ij}-g^{m}_{ij}\; \pi_{N} (\bar{\nabla} \phi) \big], \quad H^{\hat{g}}=e^{- \phi} \left[ H - \pi_{N} (\bar{\nabla} \phi) \right] \quad\text{and}\quad (h^{\hat{g}})^{\circ}_{ij}= e^{\phi}\; h^{\circ}_{ij} \quad.$$ Observe that, in particular, the endomorphism of $T f({{\mathcal}{M}})$ associated to $h^{\circ}$ is invariant under conformal deformation of the ambient metric, that is $$[(h^{\hat{g}})^{\circ}]^i_j= [h^{\circ}]^i_j \quad.$$ Finally let us recall the fundamental equations of Gauss and Ricci which link the ambient curvature $\bar{R}$ computed on $T f ({{\mathcal}{M}})$ (respectively on $T f ({{\mathcal}{M}})^{\perp}$) with the second fundamental form and the intrinsic curvature $R$ (respectively with the second fundamental form and the normal curvature $R^\perp$): $$\begin{aligned}
\bar{R}_{ijkl}&=&R_{ijkl}+(h_{il})_{\alpha} \; (h_{jk})^{\alpha}- (h_{ik})_{\alpha} \; (h_{jl})^{\alpha} \quad, \label{eq:Gauss} \\
\bar{R}_{ij \alpha \beta}&=&R_{ij\alpha \beta}^\perp + (h_{ik})_\alpha \; (h_j^k)_{\beta}-(h_{ik})_{\beta} \; (h_j^k)_{\alpha} \quad, \label{eq:Ricci}
\end{aligned}$$ where, of course, $(h_{il})^{\alpha}$ denotes the $\alpha$-component of the vector $h_{il}\in T f ({{\mathcal}{M}})^{\perp}\subset T \bar{{{\mathcal}{M}}}$ and where we adopted Einstein’s convention on summation of repeated indices.
Geometric complete contractions and global conformal Invariants of Submanifolds
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Given an immersed submanifold $f:{{\mathcal}{M}}^m\hookrightarrow \bar{{{\mathcal}{M}}}^n$, of course the above defined Riemannian curvature $R$, the curvature of the normal bundle $R^\perp$, and the extrinsic curvatures $h, h^\circ, H$ are geometric objects, that is they are invariant under change of coordinates and under isometries of the ambient manifold. So they give a list of *geometric tensors*. A natural way to define *geometric scalars* out of this list of tensors is by taking tensor products and then contracting using the metric $\bar{g}$. More precisely, we first take a finite number of tensor products, say, $$R_{i_1 j_1 k_1 l_1} \otimes \ldots \otimes R^\perp_{i_r, j_r, \alpha_r, \beta_r} \otimes \ldots \otimes h^{\circ}_{i_s j_s} \otimes \ldots \otimes H g_{i_t j_t} \quad,$$ thus obtaining a tensor of rank $4+\ldots+4+\ldots+2+\ldots+2$. Then, we repeatedly pick out pairs of indices in the above expression and contract them against each other using the metric $\bar{g}^{\alpha \beta}$ (of course, in case of contractions not including the normal curvature $R^\perp$ it is enough to contract using $g^{ij}$). This can be viewed in the more abstract perspective of Definition \[def:ComplContr\] by saying that we consider a *geometric complete contraction* $$\label{eq:ContrGen}
C(\bar{g},R,R^\perp,h) = \operatorname{contr}(\bar{g}^{\alpha_1 \beta_1}\otimes \ldots \otimes R_{i_1 j_1 k_1 l_1} \otimes \ldots \otimes R^\perp_{i_r, j_r, \alpha_r, \beta_r} \otimes \ldots \otimes h^{\circ}_{i_s j_s} \otimes \ldots \otimes H g_{i_t j_t} ) \quad.$$ Let us stress that a complete contraction is determined by the *pattern* according to which different indices contract agains each other; for example, the complete contraction $R_{ijkl}\otimes R^{ijkl}$ is different from $R^i_{ikl} \otimes R_{s}^{ksl}$. By taking linear combinations of geometric complete contractions (for the rigorous meaning see Definition \[def:LinCombCC\]), we construct *geometric scalar quantities* $$\label{eq:PgRh}
P(\bar{g},R, R^\perp, h):= \sum_{ l \in L } a _ l C ^l (\bar{g},R, R^\perp, h) \quad.$$
Notice that thanks to the Gauss and Ricci equations, one can express the ambient curvature $\bar{R}$ restricted on the tangent space of ${{\mathcal}{M}}$ or restricted to the normal bundle (that is $\bar{R}_{ijkl}$ and $\bar{R}_{ij\alpha \beta}$) as quadratic combination of $R, R^\perp$ and $h$. This is the reason why we can assume it is not present in the complete contractions without losing generality. Analogously, thanks to , we can assume that $h$ is not present in the complete contractions but just $h^\circ$ and $H g$.
Let $P(\bar{g},R, R^\perp, h)$ be a geometric scalar quantity as in and consider the homothetic rescaling $\bar{g} \mapsto t^2 \bar{g}$ of the ambient metric $\bar{g}$, for $t \in {\mathbb{R}}_{+}$. By denoting $R_{t^2\bar{g}}, R^\perp_{t^2\bar{g}}, h_{t^2\bar{g}}$ the tensors computed with respect to the rescaled metric $t^2 \bar{g}$, if $$P(t^2\bar{g},R_{t^2\bar{g}}, R^\perp_{t^2\bar{g}}, h_{t^2\bar{g}})= t^K P(\bar{g},R_{\bar{g}}, R^\perp_{\bar{g}}, h_{\bar{g}}), \quad \text{for some } K \in {\mathbb Z}\quad,$$ we then say that $P(\bar{g},R, R^\perp, h)$ is a geometric scalar quantity of *weight $K$*.
Recall that, under a general conformal deformation $\hat{\bar{g}}= e ^{2 \phi(x)} \bar{g}$ of the ambient metric $\bar{g}$ on $\bar{{{\mathcal}{M}}}^n$, the volume form of the immersed $m$-dimensional submanifold $f({{\mathcal}{M}})$ rescales by the formula $d\mu_{\hat{g}}=e ^{m \phi(x)} d\mu_{g}$, where of course $\hat{g}= e ^{2 \phi(x)} g$ is the induced conformal deformation on $f({{\mathcal}{M}})$. In particular, for every constant $t\in {\mathbb{R}}_+$, we have $d\mu_{t^2 g}= t^m d \mu_{g}$. Thus, for any scalar geometric quantity $P(\bar{g},R, R^\perp, h)$ of weight $-m$, the integral $\int_{{{\mathcal}{M}}^m} P(\bar{g},R, R^\perp, h) \, d\mu_g$ is scale invariant for all compact orientable $m$-dimensional immersed submanifolds in any $n$-dimensional ambient Riemannian manifold. We are actually interested in those scalar geometric quantity $P(\bar{g},R, R^\perp, h)$ of weight $-m$ which give rise to integrals which are invariant not only under constant rescalings, but under general conformal rescalings. Let us give a precise definition.
\[def:GCI\] Let $P(\bar{g},R, R^\perp, h)$ be a geometric scalar quantity as in and consider the conformal rescaling $\hat{\bar{g}}:= e ^{2 \phi(x)} \bar{g}$ of the ambient metric $\bar{g}$, for $\phi \in C^\infty(\bar{{{\mathcal}{M}}})$. By denoting $\hat{R}, \hat{R}^\perp, \hat{h}$ the tensors computed with respect to the conformal metric $\hat{\bar{g}}$, if $$\label{eq:ConInv}
\int_{{{\mathcal}{M}}^m} P(\hat{\bar{g}},\hat{R}, \hat{R}^\perp, \hat{h}) \, d \mu_{\hat{g}}= \int_{{{\mathcal}{M}}^m} P(\bar{g},R, R^\perp, h)\, d \mu_g$$ for any ambient Riemannian manifold $\bar{{{\mathcal}{M}}}^n$, any compact orientable $m$-dimensional immersed submanifold $f({{\mathcal}{M}}^m)\subset \bar{{{\mathcal}{M}}}^n$ and any $\phi \in C^\infty(\bar{{{\mathcal}{M}}})$, we then say that $\int_{{{\mathcal}{M}}^m} P(\bar{g},R, R^\perp, h)\, d \mu_g$ is a *global conformal invariant for $m$-submanifolds in $n$-manifolds*.
In this paper we address the question of classifying (at least in some cases) such global conformal invariants of submanifolds.
The Operator $ I _{\bar{g},R, R^\perp, h} (\phi) $ and its Polarisations {#SS:Ig}
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Inspired by the work of Alexakis [@AlexI; @AlexII; @AlexPf1; @AlexPf2; @AlexIV; @AlexBook] on the classification of global conformal invariants of Riemannian manifolds, we introduce the useful operator $I_{\bar{g},R, R^\perp, h}(\phi)$, which “measures how far the scalar geometric invariant $P(\bar{g},R, R^\perp, h)$ is to give rise to a global conformal invariant”.
\[def:Iphi\] Let $P(\bar{g},R, R^\perp, h)$ be a linear combination $$P(\bar{g},R, R^\perp, h):= \sum_{ l \in L } a _ l C ^l (\bar{g},R, R^\perp, h) \quad,$$ where each $C ^l (\bar{g},R, R^\perp, h)$ is in the form and has weight $-m$, and assume that $P(\bar{g},R, R^\perp, h)$ gives rise to a global conformal invariant for $m$-submanifolds, that is it satisfies . We define the differential operator $I _{\bar{g},R, R^\perp, h} (\phi)$, which depends both on the geometric tensors $\bar{g},R, R^\perp, h$ and on the auxiliary function $\phi\in C^{\infty}(\bar{{{\mathcal}{M}}}^n)$ as $$\label{eq.defI}
I _{\bar{g},R, R^\perp, h} (\phi)(x):= e^{m \phi(x)} P(\hat{\bar{g}},\hat{R}, \hat{R}^\perp, \hat{h})(x)-P(\bar{g},R, R^\perp, h)(x)\quad,$$ where we use the notation of Definition \[def:GCI\].
Notice that, thanks to , it holds $$\label{eq:IntIInv}
\int_{{{\mathcal}{M}}^m} I _{\bar{g},R, R^\perp, h} (\phi) \, d \mu_g =0 \quad,$$ for every Riemannian $n$-manifold $\bar{{{\mathcal}{M}}}^n$, every compact orientable $m$-submanifold $f({{\mathcal}{M}}^m)\subset \bar{{{\mathcal}{M}}}^n$, and every function $\phi \in C^\infty(\bar{{{\mathcal}{M}}})$.\
By using the transformation laws for $R, R^\perp, h$ under conformal rescalings recalled in Section \[SSS:RSG\], it is clear that $I _{\bar{g},R, R^\perp, h} (\phi)$ is a differential operator acting on the function $\phi$. In particular we can polarize, that is we can pick any $A>0$ functions $\psi_1(\cdot), \ldots, \psi_{A}(\cdot)$, and choose $$\phi(x):=\sum_{l=1}^A \psi_l(x)\quad.$$ Thus, we have a differential operator $I _{\bar{g},R, R^\perp, h} (\psi_1, \ldots, \psi_A)(\cdot)$ so that, by , it holds $$\int_{{{\mathcal}{M}}^m} I _{\bar{g},R, R^\perp, h} (\psi_1, \ldots, \psi_A) \, d \mu_g =0 \quad,$$ for every Riemannian $n$-manifold $\bar{{{\mathcal}{M}}}^n$, every compact orientable $m$-submanifold $f({{\mathcal}{M}}^m)\subset \bar{{{\mathcal}{M}}}^n$, and any functions $\psi_1, \ldots, \psi_A \in C^\infty(\bar{{{\mathcal}{M}}})$.\
Now, for any given functions $\psi_1(\cdot),\ldots, \psi_{A}(\cdot)$, we can consider the rescalings $$\lambda_1 \psi_1(\cdot), \ldots, \lambda_{A} \psi_{A}(\cdot),$$ and, as above, we have the equation $$\label{eq:IntLambda}
\int_{{{\mathcal}{M}}^m} I _{\bar{g},R, R^\perp, h} (\lambda_1 \psi_1, \ldots, \lambda_A \psi_A) \, d \mu_g =0 \quad.$$ The trick here is to see $\int_{{{\mathcal}{M}}^m} I _{\bar{g},R, R^\perp, h} (\lambda_1 \psi_1, \ldots, \lambda_A \psi_A) \, d \mu_g$ as a polynomial $\Pi(\lambda_1, \ldots, \lambda_A)$ in the independent variables $\lambda_1, \ldots, \lambda_A$. But then, equation implies that such polynomial $\Pi(\lambda_1, \ldots, \lambda_A)$ is identically zero. Hence, each coefficient of each monomial in the variables $\lambda_1, \ldots, \lambda_A$ must vanish.
We will see later in the proofs of the results how to exploit this crucial trick.
Global Conformal Invariants of Surfaces
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It is well known that in the Euclidean space $ {\mathbb{R}}^ n $, the Willmore energy $ {\mathcal}{W} ( f):=\int |H|^2 d \mu_g$ of a surface is invariant under conformal maps, that is under Möbius transformations with inversion centred off the surface. It can be shown that this is a consequence of the fact that the conformal Willmore energy of a surface immersed in a general Riemannian manifold $ f :{{\mathcal}{M}}^2 \hookrightarrow \bar{{{\mathcal}{M}}} ^ n $ $$\begin{aligned}
{\mathcal}W_{conf}(f) = \int_{ {{\mathcal}{M}}} | H| ^ 2 d \mu _g + \int _{ {{\mathcal}{M}}} \bar K d \mu_g \end{aligned}$$ where $ {\overline}K $ is the sectional curvature of the ambient space restricted to the surface, is invariant under conformal deformations of the ambient metric. Recall that by the Gauss equation one can write the intrinsic Gauss curvature $K$ of $({{\mathcal}{M}}^2,g)$ as $K = ( |H| ^ 2 - \frac{1}{2} |h^\circ|^2 ) + {\overline}K $, where $ | h^\circ | ^ 2 = g^{ik}g^{jl} h^\circ_{ij} h^\circ_{kl}$ is the squared norm of the traceless second fundamental form, we can rewrite ${\mathcal}W_{conf}$ as $${\mathcal}W_{conf}(f)=\frac{1}{2} \int _{ {{\mathcal}{M}}} | h^\circ | ^ 2 d \mu _g + \int _{ {{\mathcal}{M}}} K d \mu _g \quad .$$ Notice that both $\int _{ {{\mathcal}{M}}} | h^\circ | ^ 2 d \mu _g$ and $\int _{ {{\mathcal}{M}}} K d \mu _g$ are natural conformal invariants: the first integral is conformally invariant as $ |h^\circ |^ 2 d \mu _g $ is a pointwise conformally invariant thanks to formula , and the second integral is conformally invariant by the Gauss-Bonnet theorem (more generally it is a *topological* invariant). It trivially follows that any linear combination of the two integrands gives rise to a global conformal invariant. Our next result is that actually in codimension one there are no other global conformal invariants and in codimension two the situation is analogous once also the normal curvature is taken into account.
Global Conformal Invariants of Codimension One Surfaces
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As announced before, as first result we show that, for codimension one surfaces, any global conformal invariant must be a linear combination of the squared norm of the traceless second fundamental form and the intrinsic Gauss curvature.
\[thm:CD1surf\] Let $ P ( \bar{g} ,R, R^\perp, h )=\sum_{l\in L} a_l C^l(\bar{g},R,R^\perp,h) $ be a geometric scalar quantity for two-dimensional submanifolds of codimension one made by linear combinations of complete contractions in the general form $$\label{eq:ContrGen1}
C^l(\bar{g},R,R^\perp,h) = \operatorname{contr}(\bar{g}^{\alpha_1 \beta_1}\otimes \ldots \otimes R_{i_1 j_1 k_1 l_1} \otimes \ldots \otimes R^\perp_{i_r, j_r, \alpha_r, \beta_r} \otimes \ldots \otimes h^{\circ}_{i_s j_s} \otimes \ldots \otimes H g_{i_t j_t} ) \quad,$$ and assume that $ \int _{{{\mathcal}{M}}}P ( \bar{g} ,R, R^\perp, h ) d \mu_{g}$ is a global conformal invariant, in the sense of Definition \[def:GCI\].
Then there exist $a,b \in {\mathbb{R}}$ such that $ P$ has the following decomposition $$\begin{aligned}
P ( \bar{g} ,R, R^\perp, h )= a K + b | h^\circ |^ 2 \quad.\end{aligned}$$
First of all notice that since by assumption here we are working in codimension one, the normal curvature $R^\perp$ vanishes identically so it can be suppressed without losing generality. Recall that by taking a constant rescaling function $\phi\equiv \log t$, for some $t>0$, the volume form rescales as $t^2$: $d\mu_{t^2 g}= t^2 d\mu_{g}$. Therefore, if $P ( \bar{g} ,R ,h ) $ gives rise to a global conformal invariant it must necessarily be a linear combination of complete contractions $C^ l (\bar{g},R,h)$ each of weight $-2$. Observing that any contraction of $g^{-1}\otimes g^{-1} \otimes R$ is already of weight $-2$ and that any contraction of $g^{-1} \otimes h$ is of weight $-1$, the only possibility for $P ( \bar{g} ,R ,h )$ to be of weight $-2$ is that it decomposes as $$\begin{aligned}
P(\bar{g},R,h)&=& \quad a \operatorname{contr}(g^{i_1j_1}\otimes g^{i_2 j_2} \otimes R_{i_3 j_3 i_4 j_4}) \nonumber \\
&&+ b \operatorname{contr}(g^{i_1j_1}\otimes g^{i_2 j_2}\otimes H g_{i_{3} j_{3}}\otimes \ldots \otimes H g_{i_{r+2} j_{r+2}}\otimes h^\circ_{i_{r+3} j_{r+3}}\otimes \ldots \otimes h^\circ_{i_{r+4} j_{r+4}} ), \label{eq:Psplits}\end{aligned}$$ where in the above formula we intend that $r=0,1,2$ is the number of $H$ factors, and $2-r$ is the number of $h^\circ$ factors. Clearly, since by assumption ${{\mathcal}{M}}$ is a 2-d manifold, the term $\operatorname{contr}(g^{i_1j_1}\otimes g^{i_2 j_2} R_{i_3 j_3 i_4 j_4})$ is a (possibly null) multiple of the Gauss curvature. To get the thesis it is therefore sufficient to prove that $r=0$, that is the second summand in is completely expressed in terms of the traceless second fundamental form. Indeed any complete contraction of $g^{-1}\otimes g^{-1} \otimes h^\circ$ is a (possibly null) multiple of $|h^\circ|^2$.
To this aim observe that, since by the Gauss Bonnet Theorem $\int_{{{\mathcal}{M}}} K d \mu_g$ is a global conformal invariant, called $$\begin{aligned}
P_1(g, h)&:=& P(\bar{g},R,h)- a K \nonumber \\
&=& b \operatorname{contr}(g^{i_1j_1}\otimes g^{i_2 j_2}\otimes H g_{i_{3} j_{3}}\otimes \ldots \otimes H g_{i_{r+2} j_{r+2}}\otimes h^\circ_{i_{r+3} j_{r+3}}\otimes \ldots \otimes h^\circ_{i_{r+4} j_{r+4}} ), \label{eq:defP1gh}\end{aligned}$$ we have that $\int_{{{\mathcal}{M}}} P_1(g,h) d \mu_g$ is a global conformal invariant for compact surfaces immersed in 3-manifolds, as difference of such objects.\
Consider then an arbitrary compact surface $f({{\mathcal}{M}}^2)$ immersed into an arbitrary Riemannian $3$-manifold $(\bar{{{\mathcal}{M}}}^3, \bar{g}^3)$, and an arbitrary conformal rescaling $\hat{\bar{g}}^3=e^{2 \phi(x)} \bar{g}^3$ of the ambient metric by a smooth function $\phi\in C^\infty(\bar{{{\mathcal}{M}}})$. All the hatted geometric tensors $\hat{g}, \hat{h}, \hat{h}^\circ, \hat H$ denote the corresponding tensors computed with respect to the rescaled metric $\hat{\bar{g}}$. In Section \[SS:Ig\] we defined the operator $$I^1_{(g,h)}(\phi):= e^{2 \phi} P_1(\hat{g}, \hat{h})- P_1(g,h)\quad ,$$ and we observed that the conformal invariance of the integrated quantities implies (see ) $$\label{eq:IntI=00}
\int_{{{\mathcal}{M}}} I^1_{(g,h)}(\phi) \, d \mu_{g}=0 \quad.$$ From the formulas of the change of $h^\circ$ and $H$ under a conformal deformation of metric, it is clear that $I^1_{(g,h)}(\phi)$ does not depend directly on $\phi$ but just on $\pi_N(\bar{\nabla} \phi)= (\partial_N \phi) N $, the normal derivative of $\phi$. More precisely $I^1_{(g,h)}(\phi)$ is a polynomial in $\partial_N \phi$ exactly of the same degree $0\leq r\leq 2$ as $P_1(g, h^\circ, H)$ seen as a polynomial in $H$.
By considering $t\phi$ for $t \in {\mathbb{R}}$, we therefore get that $$I^1_{g, h}(t \phi)= \sum_{k=1}^r a_k C^k(g, h^\circ, H) \; t^k \left(\partial_N \phi \right)^k \quad.$$ Recalling now , we obtain that $ \int_{{{\mathcal}{M}}} I^1_{(g,h)}(t\phi) \, d \mu_{g} $ vanishes identically as a polynomial in $t$, so $$\label{eq:IntIk0}
0=\frac{d^k}{dt^k}\big|_{{t=0}} \int_{{{\mathcal}{M}}} I^1_{(g,h)}(t\phi) \, d \mu_{g} = k! \int_{{{\mathcal}{M}}} a_k C^k(g, h^\circ, H) \left( \partial_N \phi \right)^k \, d \mu_{g},\quad \forall k=0,\ldots,r.$$ Pick an arbitrary point $x \in {{\mathcal}{M}}$; by choosing local coordinates in $\bar{{{\mathcal}{M}}}^3$ adapted to $f({{\mathcal}{M}})$ at $f(x)$, it is easy to see that for any given function $\psi \in C^{\infty}_c({{\mathcal}{M}})$ supported in such coordinate neighborhood of $x$, there exists $\phi \in C^\infty(\bar{{{\mathcal}{M}}})$ such that $$\psi = \frac{\partial \phi}{\partial x^3} \circ f= \partial_N \phi \circ f \quad.$$ By plugging such arbitrary $C^\infty_c({{\mathcal}{M}})$ function $\psi$ in place of $\partial_N \phi$ in , we obtain that not only the integrals but the integrands themselves must vanish, that is $a_k C^k(g, h^\circ, H) \equiv 0$ on ${{\mathcal}{M}}$. It follows that $I^1_{g,h}(\phi)\equiv 0$ or, in other words, the degree of $I^1_{g, h}(t \phi)$ as a polynomial in $t$ is $0$. By the above discussion we have then that $r=0$, which was our thesis.
Global Conformal Invariants of Codimension Two Surfaces
-------------------------------------------------------
Let $ (\bar{{{\mathcal}{M}}} ^ 4, \bar{g} _{ \alpha \beta } ) $ be a four dimensional Riemannian manifold and $ f : {{\mathcal}{M}}\hookrightarrow \bar{{{\mathcal}{M}}}$ an immersion of an oriented compact surface ${{\mathcal}{M}}^2$. In local coordinates, if $ \{ e _1, e _ 2, e _ 3 , e _ 4 \} $ is an adapted orthonormal frame, so that $ \{ e _1, e _2 \} $ is a frame on $ T f({{\mathcal}{M}})$ and $ \{ e _3, e _ 4\} $ is a frame on $ [T f ({{\mathcal}{M}})]^\perp$ we define $$\begin{aligned}
K ^ \perp = R ^ \perp( e _1, e _2, e _3, e _ 4 )\quad,\end{aligned}$$ where $ R ^ \perp$ is the curvature tensor of the normal connection defined in . Note that $K^\perp$ is well defined up to a sign depending on orientation, indeed by the symmetries of the normal curvature tensor, for a codimension two surface $R^\perp$ has only one non-zero component, namely $\pm K ^ \perp$. We also denote by $$\bar K^ \perp = \bar R ( e _1, e _2 , e _ 3 ,e _4 )$$ the ambient curvature evaluated on the normal bundle. For a codimension two surface, we note that we have two topological invariants: the integral of the Gauss curvature $K$ giving the Euler Characteristic of ${{\mathcal}{M}}$ via the Gauss-Bonnet Theorem, and the integral of the normal curvature $K^\perp$ which gives the Euler characteristic of the normal bundle, $$\label{eq:TopInv}
\int_{{{\mathcal}{M}}} K d \mu _g = 2 \pi \chi({{\mathcal}{M}}) , \quad \int _{{{\mathcal}{M}}} K ^ \perp d \mu _g = 2 \pi \chi^ \perp (f({{\mathcal}{M}})).$$ As already observed, $|h^ \circ|^2 d \mu_g$ is a *pointwise* conformal invariant. Hence any linear combination of these three integrands is a global conformal invariant and in the next theorem we show that there are no others.
\[thm:CD2surf\] Let $ P ( \bar{g} ,R, R^\perp, h )=\sum_{l\in L} a_l C^l(\bar{g},R,R^\perp,h) $ be a geometric scalar quantity for two-dimensional submanifolds of codimension two made by linear combinations of complete contractions in the general form $$\label{eq:ContrGen1}
C^l(\bar{g},R,R^\perp,h) = \operatorname{contr}(\bar{g}^{\alpha_1 \beta_1}\otimes \ldots \otimes R_{i_1 j_1 k_1 l_1} \otimes \ldots \otimes R^\perp_{i_r, j_r, \alpha_r, \beta_r} \otimes \ldots \otimes h^{\circ}_{i_s j_s} \otimes \ldots \otimes H g_{i_t j_t} ) \quad,$$ and assume that $ \int _{{{\mathcal}{M}}}P ( \bar{g} ,R, R^\perp, h ) d \mu_{g}$ is a global conformal invariant, in the sense of Definition \[def:GCI\].
Then there exist $a,b,c \in {\mathbb{R}}$ such that $ P$ has the following decomposition $$\begin{aligned}
P ( \bar{g} ,R, R^\perp, h )= a K +b K^\perp + c | h^\circ |^ 2 \quad.\end{aligned}$$
Exactly as in the proof of Theorem \[thm:CD1surf\], if $P ( \bar{g} ,R, R^\perp ,h ) $ gives rise to a global conformal invariant it must necessarily be a linear combination of complete contractions $C^ l (\bar{g},R, R^\perp, h)$ each of weight $-2$. Observing that any contraction of $g^{-1}\otimes g^{-1} \otimes R$ and of $\bar{g}^{-1}\otimes \bar{g}^{-1} \otimes R^\perp$ is already of weight $-2$, and that any contraction of $g^{-1} \otimes h$ is of weight $-1$, the only possibility for $P ( \bar{g} ,R, R^\perp, h )$ to be of weight $-2$ is that it decomposes as $$\begin{aligned}
P(\bar{g},R, R^\perp, h)&=& \quad a \operatorname{contr}(g^{i_1j_1}\otimes g^{i_2 j_2} \otimes R_{i_3 j_3 i_4 j_4}) + b \operatorname{contr}(\bar{g}^{\alpha_1 \beta_1}\otimes g^{\alpha_2 \beta_2} \otimes R^\perp_{\alpha_3 \beta_3 \alpha_4 \beta_4}) \nonumber \\
&&+ c \operatorname{contr}(g^{i_1j_1}\otimes g^{i_2 j_2}\otimes H g_{i_{3} j_{3}}\otimes \ldots \otimes H g_{i_{r+2} j_{r+2}} \nonumber \\
&& \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \; \otimes h^\circ_{i_{r+3} j_{r+3}}\otimes \ldots \otimes h^\circ_{i_{r+4} j_{r+4}} ), \nonumber \end{aligned}$$ where in the above formula we intend that $r=0,1,2$ is the number of $H$ factors, and $2-r$ is the number of $h^\circ$ factors. Clearly, since by assumption ${{\mathcal}{M}}$ is a 2-d manifold, the term $\operatorname{contr}(g^{i_1j_1}\otimes g^{i_2 j_2} R_{i_3 j_3 i_4 j_4})$ is a (possibly null) multiple of the Gauss curvature. Analogously, since $f({{\mathcal}{M}})\subset \bar{{{\mathcal}{M}}}$ is a codimension two submanifold, the term $\operatorname{contr}(\bar{g}^{\alpha_1 \beta_1}\otimes g^{\alpha_2 \beta_2} \otimes R^\perp_{\alpha_3 \beta_3 \alpha_4 \beta_4}) $ is a (possibly null) multiple of the normal curvature $K^\perp$. But, by , we already know that $\int_{{{\mathcal}{M}}} K d \mu_g$ and $\int_{{{\mathcal}{M}}} K^\perp d \mu_g$ are global conformal invariants so, called $$\begin{aligned}
P_1(g, h)&:=& P(\bar{g},R,R^\perp, h)- a K- bK^\perp \nonumber \\
&=& b \operatorname{contr}(g^{i_1j_1}\otimes g^{i_2 j_2}\otimes H g_{i_{3} j_{3}}\otimes \ldots \otimes H g_{i_{r+2} j_{r+2}}\otimes h^\circ_{i_{r+3} j_{r+3}}\otimes \ldots \otimes h^\circ_{i_{r+4} j_{r+4}} ), \nonumber\end{aligned}$$ also $\int_{{{\mathcal}{M}}} P_1(g,h) d \mu_g$ is a global conformal invariant for compact surfaces immersed in 3-manifolds, as difference of such objects. The thesis can be now achieved by repeating verbatim the second part of the proof of Theorem \[thm:CD1surf\].
### Two examples: the complex projective plane and the complex hyperbolic plane
Two particular cases of interest (apart from the spaces forms) are $ {\mathbb}{CP } ^ 2 $ and $ {\mathbb}{CH}^ 2$ the complex projective plane and the complex hyperbolic plane respectively. These are Kähler manifolds with their standard Kähler form $ \Omega$ of constant holomorphic sectional curvature and unlike their real counterparts, ${\mathbb}{S}^4$ and $ {\mathbb}H ^ 4$, they are not locally conformal to $ {\mathbb}{C} ^ 2 $.
Let us consider an immersion $ \phi : {{\mathcal}{M}}\hookrightarrow \bar{{{\mathcal}{M}}}$ of an oriented surface, where $ \bar{{{\mathcal}{M}}} = {\mathbb}{CP}^ 2, {\mathbb}{CH}^2$. The Kähler function $C$ on ${{\mathcal}{M}}$ is defined by $ \phi ^ * \Omega = C d \mu_g$. Only the sign of $ C$ depends on the orientation, hence $ C ^2 $ and $ |C|$ are well defined for non-orientable surfaces. The Kähler function satisfies $$\begin{aligned}
-1 \leq C \leq 1 \quad.\end{aligned}$$ By direct computation, we find that the Willmore functional is equal to $$\begin{aligned}
{\mathcal}{W}_{{\mathbb}{CP}^ 2 }( \phi) = \int _{{{\mathcal}{M}}} \left( | H | ^ 2 + {\overline}K \right) \, d \mu_g = \int _{ {{\mathcal}{M}}} \left( | H| ^ 2 + 1 + 3 C ^ 2 \right) d \mu_g \quad ,\end{aligned}$$ and $$\begin{aligned}
{\mathcal}{W}_{{\mathbb}{CH} ^ 2 } ( \phi )= \int _{{{\mathcal}{M}}} \left( | H | ^ 2 + {\overline}K \right) d \mu_g = \int _{ {{\mathcal}{M}}} \left( | H| ^ 2 - 1 -3 C ^ 2 \right) d \mu_g \quad.\end{aligned}$$ By the Ricci equation , we also have $$\begin{aligned}
{\overline}K ^ \perp = {\overline}R_{1234} = K^\perp - \left(( {\accentset{\circ}}h_{1p})_{3} ({\accentset{\circ}}h _{ 2p})_{4} - ( {\accentset{\circ}}h _{ 2 p})_{ 3 }( {\accentset{\circ}}h _{ 1 p})_{4 } \right)\end{aligned}$$ which, applying the symmetries of the curvature tensor, can be written as a complete contraction. Thanks to the last formula, it is clear that $\int {\overline}K ^ \perp d \mu_g$ is a global conformal invariant for codimension two surfaces (since linear combination of such objects); therefore, the following energies are all global conformal invariant:
$$\begin{aligned}
{\mathcal}W ^ +_{{\mathbb}{CP}^2} ( \phi) &= \int _{ {{\mathcal}{M}}} | H| ^ 2 + {\overline}K - {\overline}K ^ \perp = \int_{{{\mathcal}{M}}} (|H|^ 2 + 6 C ^ 2 ) d \mu_g \label{eq:WCP21} \\
{\mathcal}W ^ -_{{\mathbb}{CP}^2} ( \phi) &= \int _{ {{\mathcal}{M}}} | H| ^ 2 + {\overline}K + {\overline}K ^ \perp = \int_{{{\mathcal}{M}}} (|H|^ 2 + 2) d \mu_g \quad, \label{eq:WCP22}\end{aligned}$$
and $$\begin{aligned}
{\mathcal}W ^ +_{{\mathbb}{CH}^2} ( \phi) &= \int _{ {{\mathcal}{M}}} | H| ^ 2 + {\overline}K - {\overline}K ^ \perp = \int_{{{\mathcal}{M}}} (|H|^ 2 - 6 C ^ 2 ) d \mu_g \label{eq:WCH21} \\
{\mathcal}W ^ -_{{\mathbb}{CH}^2} ( \phi) &= \int _{ {{\mathcal}{M}}} | H| ^ 2 + {\overline}K + {\overline}K ^ \perp = \int_{{{\mathcal}{M}}} (|H|^ 2 - 2) d \mu_g \quad. \label{eq:WCH22}\end{aligned}$$ Let us remark that the energies , have already been object of investigation in [@MontielUrbano], where it was shown that $ {\mathcal}{W}^-_{{\mathbb}{CP} ^ 2} ( \phi) \geq 4 \pi \mu - 2 \int_{\Sigma}|C| d \mu_g $ where $\mu$ is the maximum multiplicity. Moreover in the same paper it was shown that equality holds if and only if $ \mu =1 $ and ${{\mathcal}{M}}$ is either a complex projective line or a totally geodesic real projective plane, or $ \mu = 2 $ and $ {{\mathcal}{M}}$ is a Lagrangian Whitney sphere.
Since the goal of the present paper is to investigate the structure of global conformal invariants, we recalled the definition of ${\mathcal}W ^{\pm}_{{\mathbb}{CP}^2}$ given in [@MontielUrbano] and we defined the new functionals ${\mathcal}W ^{\pm}_{{\mathbb}{CH}^2}$ in order to give interesting examples of Willmore-type energies in codimension 2. In a forthcoming work we will address the question wether ${\mathcal}W ^{\pm}_{{\mathbb}{CH}^2}$ satisfy analogous properties as ${\mathcal}W ^{\pm}_{{\mathbb}{CP}^2}$.
Global Conformal Invariants of Submanifolds
===========================================
Let us consider a geometric scalar quantity $ P (g^m, h^m)$ of the form $$\begin{aligned}
\label{eqn_star}
\sum_{ l \in L } a _l C^ l ( g^m , h^m) \quad,\end{aligned}$$ where each $C ^ l$ is a complete contraction $$\begin{aligned}
\label{eqn_contract}
\operatorname{contr}(g^{i_1 j_1}\otimes\ldots \otimes g^{i_s j_s} \otimes h_{ i _{s+1} j_{s+1}}\otimes \dots \otimes h _{ i _{2s} j_{2s}}) \quad ; \end{aligned}$$ that is we consider complete contractions as defined in but depending just on the second fundamental form $h^m$ and the induced metric $g^m$ for immersed $m$-submanifolds $f({{\mathcal}{M}}^m)$ in Riemannian $n$-manifolds $(\bar{{{\mathcal}{M}}}^n,\bar{g}^n)$.
Our first goal in this section is to understand the structure of geometric scalar quantities giving rise to global conformal invariants for submanifolds, in the sense of Definition \[def:GCI\]. This is exactly the content of the next result.
\[thm:P(g,h)\] Let $ P ( g^m ,h^m ) $ be as in with each $ C^l$ of the form , and assume that $ \int _{{{\mathcal}{M}}}P( g ^ m ,h ^ m) d \mu_{g^m}$ is a global conformal invariant, in the sense of Definition \[def:GCI\]. Then there exists a *pointwise* conformal invariant $W ( g ^ m, {\accentset{\circ}}h^m )$ of weight $ -m$ depending only on the traceless second fundamental form ${\accentset{\circ}}h$ contracted with the induced metric $g^m$ so that $$\begin{aligned}
P( g ^ m , h ^ m ) = W ( g ^ m, {\accentset{\circ}}h ^m )\quad .\end{aligned}$$ In other words, for every $l \in L$ one has that $ C ^l$ in is a complete contraction of weight $-m$ of the form $$\begin{aligned}
\label{eqn_contract}
\operatorname{contr}(g^{i_1 j_1}\otimes\ldots \otimes g^{i_m j_m} \otimes {\accentset{\circ}}h_{ i _{m+1} j_{m+1}}\otimes \dots \otimes {\accentset{\circ}}h _{ i _{2m} j_{2m}}) \quad . \end{aligned}$$
First of all notice that by taking a constant rescaling function $\phi\equiv \log t$, for some $t>0$, the volume form rescales with a power $m$, that is $d\mu_{{t^2 g}^m}= t^m d\mu_{{g}^m}$. Therefore, if $P ( g^m ,h^m ) $ gives rise to a global conformal invariant it must necessarily be a linear combination of complete contractions $C^ l ( g^m , h^m)$ each of weight $-m$. But observing that any contraction of $g^{-1} \otimes h$ is of weight $-1$, this implies that in we have $s=m$, that is $C ^ l(g^m, h^m)$ is a complete contraction of the form $$\begin{aligned}
\label{eqn_contract1}
\operatorname{contr}(g^{i_1 j_1}\otimes\ldots \otimes g^{i_m j_m} \otimes h_{ i _{m+1} j_{m+1}}\otimes \dots \otimes h _{ i _{2m} j_{2m}}) \quad .\end{aligned}$$ Recalling , that is the orthogonal splitting $h_{ij}=H g_{ij}+h _{ij}^\circ$ , we can rewrite such a complete contraction as $$\begin{aligned}
\label{eqn_contract1}
\operatorname{contr}(g^{i_1 j_1}\otimes\ldots \otimes g^{i_m j_m} \otimes H g_{ i _{m+1} j_{m+1}}\otimes \dots \otimes H g_{ i_{m+r} j_{m+r}} \otimes h_{ i _{m+r+1} j_{m+r+1}}^\circ \otimes \dots \otimes h _{ i _{2m} j_{2m}}^\circ ) \quad .\end{aligned}$$ Our goal is to prove that $r=0$, that is there are no $H$ factors, so $P(g^m,h^m)=P(g^m,{\accentset{\circ}}h^m)$ is expressed purely as complete contractions of traceless fundamental forms, which are *pointwise* conformal invariants once multiplied by $d\mu_g$ thanks to .\
To that aim consider an arbitrary compact $m$-dimensional immersed submanifold $f({{\mathcal}{M}}^m)$ of an arbitrary Riemannian $n$-manifold $(\bar{{{\mathcal}{M}}}^n, \bar{g}^n)$, and an arbitrary conformal rescaling $\hat{\bar{g}}^n=e^{2 \phi(x)} \bar{g}^n$ of the ambient metric by a smooth function $\phi\in C^\infty(\bar{{{\mathcal}{M}}})$. All the hatted geometric tensors $\hat{g}, \hat{h}, \hat{h}^\circ, \hat H$ denote the corresponding tensors computed with respect to the deformed metric $\hat{\bar{g}}$. In Section \[SS:Ig\] we defined the operator $$I_{(g,h)}(\phi):= e^{m \phi} P(\hat{g}, \hat{h})- P(g,h)\quad ,$$ and we observed that the conformal invariance of the integrated quantities implies (see ) $$\label{eq:IntI=0}
\int_{{{\mathcal}{M}}} I_{(g,h)}(\phi) \, d \mu_{g}=0 \quad.$$ From the formulas of the change of $h^\circ$ and $H$ under a conformal deformation of metric, it is clear that $I_{(g,h)}(\phi)$ does not depend directly on $\phi$ but just on $\pi_N(\bar{\nabla} \phi)$, the projection of $\bar{\nabla} \phi$ onto the normal space of $f({{\mathcal}{M}})$. More precisely $I_{(g,h)}(\phi)$ is polynomial in the components of $\pi_N(\bar{\nabla} \phi)$ exactly of the same degree $0\leq r\leq m$ as $P(g, h^\circ, H)$ seen as a polynomial in $H$.
By considering $t\phi$ for $t \in {\mathbb{R}}$, we get that $$I_{g, h}(t \phi)= \sum_{k=1}^r a_k C^k(g, h^\circ, H, \pi_N(\bar{\nabla} \phi)) \; t^k\quad ,$$ where $C^k(g, h^\circ, H, \pi_N(\bar{\nabla} \phi))$ is an homogeneous polynomial of degree $k$ in the components of $\pi_N(\bar{\nabla} \phi)$. Recalling now , we obtain that $ \int_{{{\mathcal}{M}}} I_{(g,h)}(t\phi) \, d \mu_{g} $ vanishes identically as a polynomial in $t$, so $$\label{eq:IntIk}
0=\frac{d^k}{dt^k}\mid_{{t=0}} \int_{{{\mathcal}{M}}} I_{(g,h)}(t\phi) \, d \mu_{g} = k! \int_{{{\mathcal}{M}}} a_k C^k(g, h^\circ, H, \pi_N(\bar{\nabla} \phi)) \, d \mu_{g},\quad \forall k=0,\ldots,r.$$ Pick an arbitrary point $x \in {{\mathcal}{M}}$; by choosing local coordinates in $\bar{{{\mathcal}{M}}}^n$ adapted to $f({{\mathcal}{M}})$ at $f(x)$, it is easy to see that for any given functions $\psi^i,\ldots, \psi^{n-m} \in C^{\infty}_c({{\mathcal}{M}})$ supported in such a coordinate neighborhood of $x$, there exists $\phi \in C^\infty(\bar{{{\mathcal}{M}}})$ such that $$\psi^i= (\bar{\nabla} \phi)^{m+i} \circ f \quad,$$ where of course thanks to this choice of coordinates we have $$\pi_N(\bar{\nabla} \phi)=\Big((\bar{\nabla} \phi)^{m+1}, \ldots, (\bar{\nabla} \phi)^{n}\Big)\quad.$$ By plugging such arbitrary $C^\infty_c({{\mathcal}{M}})$ functions $\psi^i,\ldots, \psi^{n-m}$ in place of $\pi_N(\bar{\nabla} \phi)$ in , we obtain that not only the integrals but the integrands themselves must vanish, that is $a_k C^k(g, h^\circ, H, \pi_N(\bar{\nabla} \phi))\equiv 0$ on ${{\mathcal}{M}}$. It follows that $I_{g,h}(\phi)\equiv 0$ or, in other words, the degree of $I_{g, h}(t \phi)$ as a polynomial in $t$ is $0$. By the above discussion we have that $r=0$, which was our thesis.
We pass now to consider the more general geometric scalar quantity $ P (g^m, R^m, h^m)$, for $m\in {\mathbb N}$ *even*, of the form $$\label{eq:PP1P2}
P (g^m, R^m, h^m)=P_1(g^m, h^m)+ P_2(g^m,R^m)\quad,$$ where $$\begin{aligned}
\label{eq:P1P2}
P_1(g^m,h^m)=\sum_{ l \in L } a _l C^ l ( g^m , h^m) \quad \text{and} \quad P_2(g^m,R^m)=\sum_{ s \in S } b _s C^ s ( g^m , R^m) \quad,\end{aligned}$$ where each $C ^ l( g^m , h^m)$ is a complete contraction $$\begin{aligned}
\label{eq:Cl}
\operatorname{contr}(g^{i_1 j_1}\otimes\ldots \otimes g^{i_s j_s} \otimes h_{ i _{s+1} j_{s+1}}\otimes \dots \otimes h _{ i _{2s} j_{2s}}) \quad \end{aligned}$$ and each $C^ s ( g^m , R^m)$ is a complete contraction $$\begin{aligned}
\label{eq:Cs}
\operatorname{contr}(g^{i_1 j_1}\otimes\ldots \otimes g^{i_{2r} j_{2r}} \otimes R_{ i _{2r+1} j_{2r+1} k_{2r+1} l_{2r+1} }\otimes \dots \otimes R_{ i _{3r} j_{3r} k_{3r} l_{3r} }) \quad. \end{aligned}$$ In other words we consider complete contractions as defined in which split in two parts: one depending just on the second fundamental form $h^m$ and the other one just on the intrinsic curvature $R^m$, for immersed $m$-submanifolds $(f({{\mathcal}{M}}^m), g^m)$ in Riemannian $n$-manifolds $(\bar{{{\mathcal}{M}}}^n,\bar{g}^n)$. As usual, the goal is to understand the structure of geometric scalar quantities giving rise to global conformal invariants for submanifolds, in the sense of Definition \[def:GCI\]. This is exactly the content of the next result.
\[thm:P=P1+P2\] Let $m \in {\mathbb N}$ be *even* and let $P( g ^ m , R^m, h ^m )=P_1(g^m, h^m)+ P_2(g^m,R^m)$ be a geometric scalar quantity as above. Assume that $ \int _{{{\mathcal}{M}}}P( g ^ m, R^m, h ^ m) d \mu_{g^m}$ is a global conformal invariant, in the sense of Definition \[def:GCI\].
Then both $ \int _{{{\mathcal}{M}}}P_1( g ^ m, h ^ m) d \mu_{g^m}$ and $ \int _{{{\mathcal}{M}}}P_2( g ^ m, R ^ m) d \mu_{g^m}$ are global conformal invariants. It follows that
i\) There exists a *pointwise* conformal invariant $W_1 ( g ^ m, {\accentset{\circ}}h^m )$ of weight $ -m$ depending only on the traceless second fundamental form ${\accentset{\circ}}h$ contracted with the induced metric $g^m$ so that $$\begin{aligned}
P_1( g ^ m , h ^ m ) = W_1 ( g ^ m, {\accentset{\circ}}h ^m )\quad ;\end{aligned}$$ or, in other words, for every $l \in L$ one has that $ C ^l$ in is a complete contraction of weight $-m$ of the form $$\begin{aligned}
\label{eqn_contract2}
\operatorname{contr}(g^{i_1 j_1}\otimes\ldots \otimes g^{i_m j_m} \otimes {\accentset{\circ}}h_{ i _{m+1} j_{m+1}}\otimes \dots \otimes {\accentset{\circ}}h _{ i _{2m} j_{2m}}) \quad . \end{aligned}$$
ii\) Called $ \operatorname{Pfaff}(R^m)$ the Pfaffian of the intrinsic Riemann tensor $R^m$ and $W^m$ the Weyl tensor of $g^m$, $P_2(g^m, R^m)$ is of the form $$\begin{aligned}
P_2(g^m, R^m)= \tilde{P}_2(g^m, W^m)+ c \operatorname{Pfaff}(R^m) \;, \quad \text{for some } c\in {\mathbb{R}}\quad, \end{aligned}$$ where $\tilde{P}_2(g^m, W^m)$ is a *pointwise* conformal invariant of weight $-m$ expressed as a linear combination of complete contractions of the form $$\operatorname{contr}(g^{i_1 j_1}\otimes\ldots \otimes g^{i_{m} j_{m}} \otimes W_{ i _{m+1} j_{m+1} k_{m+1} l_{m+1} }\otimes \dots \otimes W_{ i _{\frac{3m}{2}} j_{\frac{3m}{2}} k_{\frac{3m}{2}} l_{\frac{3m}{2}} }) \quad.$$
It is well known that
- The Weyl tensor $W_{ijkl}(g^m)$ is a pointwise scalar conformal invariant of weight $2$, that is it satisfies $W_{ijkl}(e^{2\phi(x)} g^m(x))= e^{2 \phi(x) } W(g^m)(x)$ for every $\phi\in C^\infty({{\mathcal}{M}})$ and every $x\in {{\mathcal}{M}}^m$. It follows that any complete contraction of the tensor $g^{-1}\otimes g^{-1} \otimes W$ is a pointwise scalar conformal invariant of weight $-2$.
- The Pfaffian $\operatorname{Pfaff}(R_{ijkl})$ of the intrinsic curvature $R_{ijkl}=R^m$ integrated over the manifold gives rise to a *topological* invariant: $$\int_{{{\mathcal}{M}}^m} \operatorname{Pfaff}(R_{ijkl}) \, d\mu_g= \frac{2^m \pi^{m/2} (\frac{m}{2}-1)!}{2(m-1)!} \chi({{\mathcal}{M}}^m)\quad,$$ where $\chi({{\mathcal}{M}}^m)$ is the Euler Characteristic of ${{\mathcal}{M}}^m$.
Therefore, recalling the conformal invariance of the traceless second fundamental form , any linear combination of complete contractions $$P(g^m,h^m,R^m)=W_1(g^m, {\accentset{\circ}}h^m)+ \tilde{P}_2(g^m, W^m)+ c \operatorname{Pfaff}(R^m)$$ as in the thesis of Theorem \[thm:P=P1+P2\] gives rise to a “trivial” global conformal invariant. Thereom \[thm:P=P1+P2\] states that, under the assumption that $P$ splits into the sum of an intrinsic part and of an extrinsic part depending just on the second fundamental form, this is actually the only possibility.
Since by assumption $P( g ^ m , R^m, h ^m )=P_1(g^m, h^m)+ P_2(g^m,R^m)$ gives rise to a global conformal invariant, if we show that $\int_{{{\mathcal}{M}}} P_1(g^m, h^m) \, d \mu_g$ is a global conformal invariant, the same will be true for $\int_{{{\mathcal}{M}}} P_2(g^m, R^m) \, d \mu_g$. In order to prove that, consider an arbitrary compact $m$-dimensional immersed submanifold $f({{\mathcal}{M}}^m)$ of an arbitrary Riemannian $n$-manifold $(\bar{{{\mathcal}{M}}}^n, \bar{g}^n)$, and an arbitrary conformal rescaling $\hat{\bar{g}}^n=e^{2 \phi(x)} \bar{g}^n$ of the ambient metric by a smooth function $\phi\in C^\infty(\bar{{{\mathcal}{M}}})$. All the hatted geometric tensors $\hat{g}, \hat{h}, \hat{h}^\circ, \hat H, \hat{R}$ denote the corresponding tensors computed with respect to the rescaled metric $\hat{\bar{g}}$. Analogously to Section \[SS:Ig\], define the operators $$\begin{aligned}
I^1_{g,h}(\phi)&:=& e^{m \phi} P_1(\hat{g}, \hat{h})- P_1(g,h)\quad , \label{eq:I1} \\
I^2_{g,R}(\phi)&:=& e^{m \phi} P_2(\hat{g}, \hat{R})- P_2(g,R)\quad , \label{eq:I2} \\
I_{g,R,h}(\phi)&:=& I^1_{g,h}(\phi)+I^2_{g,R}(\phi) \quad . \label{eq:I=I1+I2}
\end{aligned}$$ As already observed in the proof of Theorem \[thm:P(g,h)\], $I^1_{g,h}(\phi)$ does not depend directly on $\phi$ but only on $\pi_N(\bar{\nabla} \phi)$, the projection of $\bar{\nabla} \phi$ onto the normal space of $f({{\mathcal}{M}})$. More precisely $I^1_{g,h}(\phi)$ is polynomial in the components of $\pi_N(\bar{\nabla} \phi)$ exactly of the same degree $0\leq r\leq m$ as $P_1(g, h^\circ, H)$ seen as a polynomial in $H$.\
On the other hand, recalling the formulas of the change of the intrinsic curvature $R_{ijkl}$ of $({{\mathcal}{M}}^m, g^m)$ under conformal deformation of the metric, it is clear that $I^2_{g,R}$ does not depend directly on $\phi$ but only on $\nabla_{{\mathcal}{M}}\phi$, the projection of $\bar{\nabla} \phi$ onto the tangent space of $f({{\mathcal}{M}})$, and on $\nabla^2_{{\mathcal}{M}}\phi$, the covariant Hessian of $\phi|_{{{\mathcal}{M}}}$. More precisely $I^2_{g,R}(\phi)$ is polynomial in the components of $\nabla_{{\mathcal}{M}}\phi, \nabla^2 _{{\mathcal}{M}}\phi$.
By considering $t\phi$ for $t \in {\mathbb{R}}$, we get that $$I^1_{g, h}(t \phi)= \sum_{k=1}^r a_k C^k(g, h^\circ, H, \pi_N(\bar{\nabla} \phi)) \; t^k\quad ,\quad I^2_{g,R}(t \phi)= \sum_{l=1}^s b_l C^l(g, R, \nabla_{{\mathcal}{M}}\phi, \nabla^2 _{{{\mathcal}{M}}} \phi) \; t^k\quad$$ where $C^k(g, h^\circ, H, \pi_N(\bar{\nabla} \phi))$ (respectively $C^l(g, R, \nabla_{{\mathcal}{M}}\phi, \nabla^2 _{{{\mathcal}{M}}} \phi)$) is an homogeneous polynomial of degree $k$ in the components of $\pi_N(\bar{\nabla} \phi)$ (respectively of $\nabla_{{\mathcal}{M}}\phi, \nabla^2 _{{\mathcal}{M}}\phi$). Recalling now , we obtain that $ \int_{{{\mathcal}{M}}} I_{(g,h)}(t\phi) \, d \mu_{g} $ vanishes identically as a polynomial in $t$, so $$\begin{aligned}
0&=&\frac{d^k}{dt^k}\bigg|_{{t=0}} \int_{{{\mathcal}{M}}} I_{(g,h)}(t\phi) \, d \mu_{g} \nonumber \\
&=& k! \int_{{{\mathcal}{M}}} a_k C^k(g, h^\circ, H, \pi_N(\bar{\nabla} \phi)) + b_k C^k(g, R, \nabla_{{\mathcal}{M}}\phi, \nabla^2 _{{{\mathcal}{M}}} \phi) \, d \mu_{g},\quad \forall k\in {\mathbb N}. \label{eq:akbk}\end{aligned}$$ Pick an arbitrary point $x \in {{\mathcal}{M}}$; by choosing local coordinates in $\bar{{{\mathcal}{M}}}^n$ adapted to $f({{\mathcal}{M}})$ at $f(x)$, it is easy to see that for any given functions $\psi^i,\ldots, \psi^{n-m} \in C^{\infty}_c({{\mathcal}{M}})$ supported in such coordinate neighborhood of $x$, there exists $\phi \in C^\infty(\bar{{{\mathcal}{M}}})$ such that $$\psi^i= (\bar{\nabla} \phi)^{m+i} \circ f \quad \text{and} \quad \nabla_{{{\mathcal}{M}}} (\phi\circ f)=0 \quad,$$ where of course thanks to the this choice of coordinates we have $$\pi_N(\bar{\nabla} \phi)=\Big((\bar{\nabla} \phi)^{m+1}, \ldots, (\bar{\nabla} \phi)^{n}\Big)\quad.$$ With this choice of $\phi$, the second summand in the integral of disappears, and thanks to the arbitrariness of the $C^\infty_c({{\mathcal}{M}})$ functions $\psi^i,\ldots, \psi^{n-m}$ in place of $\pi_N(\bar{\nabla} \phi)$ in , we obtain that not only the integrals but the integrands themselves must vanish, that is $a_k C^k(g, h^\circ, H, \pi_N(\bar{\nabla} \phi))\equiv 0$ on ${{\mathcal}{M}}$, so $I^1_{g,h}(\phi)\equiv 0$. In particular $\int_{{{\mathcal}{M}}} P_1(g^m,h^m) d \mu_g$ is a global conformal invariant for $m$-dimensional submanifolds which implies that also $\int_{{{\mathcal}{M}}} P_2(g^m,R^m) d \mu_g$ is a global conformal invariant, since by assumption $\int_{{{\mathcal}{M}}} [P_1(g^m,h^m) + P_2(g^m,R^m) ] d \mu_g $ is that as well.
Claim i) follows then directly from Theorem \[thm:P(g,h)\]. To get claim ii) observe that by construction $P_2(g^m, R^m)$ depends just on the *intrinsic* Riemannian structure $({{\mathcal}{M}},g)$ and not on the immersion $f$ into an ambient manifold $\bar{{{\mathcal}{M}}}$. Therefore we proved that $\int_{{{\mathcal}{M}}} P_2(g^m, R^m) d \mu_g$ is an intrinsic Riemannian conformally invariant quantity, which enters into the framework of the papers of Alexakis [@AlexI], [@AlexII]. More precisely, by applying [@AlexI Theorem 1], we obtain claim ii) and the proof is complete.
Generalized Willmore Energies in Higher Dimensions {#sec:genWill}
==================================================
In this final section, we will introduce a higher dimensional analogue of the Willmore energy (actually we will construct a two-parameters family of such functionals). This new energy is conformally invariant and only attains its strictly positive lower bound at a round sphere. Let us start with some preliminaries about the Willmore functional.
Given an immersion $f:{{\mathcal}{M}}^2 \hookrightarrow {\mathbb{R}}^3$ of a closed surface ${{\mathcal}{M}}^2$ into the Euclidean space ${\mathbb{R}}^3$, the Willmore functional ${\mathcal}W(f)$ is defined by $${\mathcal}W (f):=\int_{{{\mathcal}{M}}} |H|^2 d \mu_g\quad.$$ A natural way to introduce such functional is via conformal invariance: by Gauss-Bonnet Theorem, $\int_{{{\mathcal}{M}}} K d \mu_g$ is a topological hence a fortiori conformal invariant quantity; moreover, by the formula , $|h^\circ|^2 d \mu_g$ is a *pointwise* conformal invariant. It follows that *any* functional $$W_{\alpha}(f):=\int_{{{\mathcal}{M}}} \left( K + \alpha |h^\circ|^2 \right) d \mu_g \text{ is a global conformally invariant quantity}$$ in the sense of Definition \[def:GCI\]. For immersions into ${\mathbb{R}}^3$ we have that $K+\frac{1}{2} |h^\circ|^2= |H|^2 $, so $${\mathcal}W(f) = W_{1/2}(f) \text{ is invariant under conformal trasformations of ${\mathbb{R}}^3$,}$$ that is under Moebius transformations centred out of $f({{\mathcal}{M}})$. Observing that $K=\operatorname{det_g}(h)$ and that $\frac{1}{2}|h^\circ|^2=-\operatorname{det_g}(h^\circ)$, we can also write $$\label{eq:Wdeth}
{\mathcal}W(f)=\int_{{{\mathcal}{M}}} \left( \operatorname{det_g}(h) - \operatorname{det_g}(h^\circ) \right) \, d \mu_g \quad.$$ Of course the functional ${\mathcal}W$ is non-negative and vanishes exactly on minimal surfaces, which are therefore points of strict global minimum; the critical points of ${\mathcal}W$ can therefore be seen in a natural way as “generalized conformal minimal surfaces”; this was indeed the starting point in the ’20ies of the theory of Willmore surfaces by Blaschke [@Blaschke], who was looking for a natural conformally invariant class of immersions which included minimal surfaces. Let us mention that such functional was later rediscovered in the 1960’s by Willmore [@Will] who proved that round 2-spheres are the points of strict global minimum for ${\mathcal}W$.
Motivated by this celebrated two dimensional theory, our goal is to investigate the case of 4-d hypersurfaces in ${\mathbb{R}}^5$, that is $f:({{\mathcal}{M}}^4,g)\hookrightarrow ({\mathbb{R}}^5, \delta_{\mu\nu})$ isometric immersion.\
\
We address the following natural questions:
1\) Is it possible to “perturb” the Pfaffian of the Riemann tensor of the induced metric $g$ on ${{\mathcal}{M}}^4$ in order to get a conformally invariant functional vanishing on minimal surfaces?
2\) Is that functional positive definite? If not, how can we preserve the conformal invariance and make it positive definite?
3\) Are round spheres of ${\mathbb{R}}^5$ strict global minimum of this conformally invariant functional?
As in the 2-d case, the starting point is the Gauss-Bonnet Theorem. For $4$-d smooth closed (that is compact without boundary) immersed hypersurfaces $f:({{\mathcal}{M}}^4,g) \hookrightarrow ({\mathbb{R}}^5, \delta_{\mu \nu})$, a well known generalization of the Gauss-Bonnet Theorem states that $$\label{eq:GenGB}
\int_{{{\mathcal}{M}}^4} \operatorname{det_g}(h) \, d \mu_g=\frac{4 \pi^2 }{3} \; \chi({{\mathcal}{M}}) = \frac{8 \pi^2 }{3} \deg(\gamma) \quad,$$ where $\operatorname{det_g}(h):=\det(g^{-1} h)$, $\chi({{\mathcal}{M}})$ is the Euler Characteristic of ${{\mathcal}{M}}$ and $\gamma: {{\mathcal}{M}}^4 \to S^4\subset {\mathbb{R}}^5$ is the Gauss map associated to the immersion $f$. By applying the classical Newton’s identities for symmetric polynomials to the symmetric polynomials of the principal curvatures of the immersion $f$, it is an easy exercise to write $\det(g^{-1} h)$ as $$\label{eq:detgh}
\operatorname{det_g}(h)=\frac{32}{3} H^4- 4 H^2 |h|^2+\frac{4}{3} H \operatorname{Tr}_g(h^3)+\frac{1}{8}|h|^4-\frac{1}{4} \operatorname{Tr}_g(h^4) \quad,$$ where $\operatorname{Tr}_g(h^p):=\operatorname{Tr}[(g^{-1} h)^p]$. By using the orthogonal decomposition $h={\accentset{\circ}}h+ H g$, see , iteratively we get that $$\begin{aligned}
|h|^2&=& \operatorname{Tr}_g(h^2) = |h^\circ|^2 + 4 H^2 \nonumber \\
\operatorname{Tr}_{g}(h^3)&=& \operatorname{Tr}_{g}({\accentset{\circ}}h^3)+ 3 H |{\accentset{\circ}}h|^2 + 4 H^3 \nonumber\\
\operatorname{Tr}_{g}(h^4)&=& \operatorname{Tr}_{g}({\accentset{\circ}}h^4)+ 4 H \operatorname{Tr}_{g}({\accentset{\circ}}h^3) + 6 H^2 |{\accentset{\circ}}h|^2 + 4 H^4 \quad, \nonumber
\end{aligned}$$ which, plugged into , give $$\label{eq:detg-1h}
\operatorname{det_g}(h)= H^4- \frac{1}{2} H^2 |{\accentset{\circ}}h|^2+\frac{1}{3} H \operatorname{Tr}_g({\accentset{\circ}}h ^3)+\frac{1}{8}|{\accentset{\circ}}h|^4-\frac{1}{4} \operatorname{Tr}_g({\accentset{\circ}}h^4) \quad.$$ Inspired by the 2-d case, in particular by formula , we wish to make appear a term $\operatorname{det_g}({\accentset{\circ}}h)$, this is possible thanks to the following lemma.
\[lem:Trdet\] For any $4$-d hypersurface immersed into ${\mathbb{R}}^5$, we have that $$\label{eq:Trdet}
\operatorname{Tr}_g({\accentset{\circ}}h^4)-\frac{1}{2} |{\accentset{\circ}}h|^4 = -4 \operatorname{det_g}({\accentset{\circ}}h) \quad.$$
Since ${\accentset{\circ}}h$ is bilinear symmetric with respect to $g$ it can be diagonalized and its eigenvalues $\{\lambda_i\}_{i=1,\ldots,4} \subset {\mathbb{R}}$ satisfy $\lambda_4= -\lambda_1-\lambda_2-\lambda_3$, since $\operatorname{Tr}_g({\accentset{\circ}}h)=0$. We then have that $$\begin{aligned}
\operatorname{Tr}_g({\accentset{\circ}}h^4)& = &\lambda_1^4+ \lambda_2^4 + \lambda_3^4 + (\lambda_1+\lambda_2+\lambda_3)^4 \nonumber\\
&=& 2 \sum_{i=1}^3 \lambda_i^4 + 4 \sum_{1\leq i\neq j \leq 3} \lambda_i^3 \lambda_j + 6 \sum_{1\leq i\neq j \leq 3} \lambda_i^2 \lambda_j^2 + 12 \sum_{1\leq i\neq j \neq k \leq 3} \lambda_i^2 \lambda_j \lambda_k \quad.\label{eq:Trh4} \end{aligned}$$ On the other hand, $$\begin{aligned}
|{\accentset{\circ}}h|^4 & = & \left[\lambda_1^2+ \lambda_2^2 + \lambda_3^2 + (\lambda_1+\lambda_2+\lambda_3)^2\right]^2 \nonumber\\
&=& 4 \sum_{i=1}^3 \lambda_i^4 + 8 \sum_{1\leq i\neq j \leq 3} \lambda_i^3 \lambda_j + 12 \sum_{1\leq i\neq j \leq 3} \lambda_i^2 \lambda_j^2 + 16 \sum_{1\leq i\neq j \neq k \leq 3} \lambda_i^2 \lambda_j \lambda_k \quad.\label{eq:h4} \end{aligned}$$ Combining and and recalling that $\lambda_4= -\lambda_1-\lambda_2-\lambda_3$ we can conclude: $$\operatorname{Tr}_g({\accentset{\circ}}h^4)- \frac{1}{2}|{\accentset{\circ}}h|^4 = 4 \sum_{1\leq i\neq j \neq k \leq 3} \lambda_i^2 \lambda_j \lambda_k = 4 \lambda_1 \lambda_2 \lambda_3 (\lambda_1+ \lambda_2+\lambda_3)= -4 \operatorname{det_g}({\accentset{\circ}}h) \quad.$$
Thanks to the identity , we can rewrite as $$\label{eq:dethH}
\operatorname{det_g}(h)- \operatorname{det_g}({\accentset{\circ}}h)= H^4- \frac{1}{2} H^2 |{\accentset{\circ}}h|^2+\frac{1}{3} H \operatorname{Tr}_g({\accentset{\circ}}h ^3)\quad.$$
Since for even dimensional hypersurfaces in the Euclidean space it is well know that the Pfaffian of the intrinsic Riemann tensor is a multiple of $\operatorname{det_g}(h)$, we have just answered to question 1):
\[prop:1\] Given an isometric immersion $f:({{\mathcal}{M}}^4,g)\hookrightarrow ({\mathbb{R}}^5, \delta_{\mu\nu})$ of a closed $4$-manifold $({{\mathcal}{M}}^4,g)$, define $${\mathcal}P (f):= \int_{{{\mathcal}{M}}} \left[ \operatorname{det_g}(h)- \operatorname{det_g}({\accentset{\circ}}h) \right] d \mu_g \quad.$$ Then, the functional ${\mathcal}P$ is invariant under conformal transformations of ${\mathbb{R}}^5$ centered off of $f({{\mathcal}{M}}^4)$.\
Moreover ${\mathcal}P $ vanishes identically on minimal hypersurfaces, and the minimal hypersurfaces of ${\mathbb{R}}^5$ satisfying $\operatorname{Tr}_g({\accentset{\circ}}h^3)\equiv 0$ are critical points for ${\mathcal}P$.
By using , we have that $\operatorname{det_g}(h)- \operatorname{det_g}({\accentset{\circ}}h) $ clearly vanishes if $H\equiv 0$. Observe that the right hand side of has a linear term in $H$, so a priori a minimal hypersurface is not a critical point for ${\mathcal}P$, but if we assume also that $\operatorname{Tr}_g({\accentset{\circ}}h ^3)$ vanishes, of course we obtain criticality. Since $\left[ \frac{1}{4} \operatorname{Tr}_g({\accentset{\circ}}h^4)- \frac{1}{8}|{\accentset{\circ}}h|^4 \right] d \mu_g$ is a pointwise conformal invariant and $\operatorname{det_g}(h)$ is a topological invariant by , the functional ${\mathcal}P$ is then invariant under conformal transformation of ${\mathbb{R}}^5$ preserving the topology of $f({{\mathcal}{M}})$, that is under conformal transformations of ${\mathbb{R}}^5$ centered off of $f({{\mathcal}{M}}^4)$.
Notice that, formally, the integrand of the 4-d functional ${\mathcal}P$ is exactly the same as the 2-d Willmore functional ${\mathcal}W$, written as in .
Since in 2-d the quantity $\operatorname{det_g}(h)-\operatorname{det_g}(h)=H^2$ is non-negative, it is natural to ask if the same is true in the 4-d case, that is if $\operatorname{det_g}(g)-\operatorname{det_g}({\accentset{\circ}}h)$ is non negative. Surprisingly this is not the case, for instance it is not difficult to compute that if the principal curvatures are $1,1,6,6,$ one has $\operatorname{det_g}(h)-\operatorname{det_g}({\accentset{\circ}}h)=-\frac{49}{16}<0$. A natural question is then if we can manipulate ${\mathcal}P$ it in order to obtain a new functional which is still conformally invariant but this time is nonnegative definite; this is exactly question 2) above. To this aim observe that, by Young’s inequality, we have that $$\begin{aligned}
-\frac{1}{2} H^2 |{\accentset{\circ}}h|^2 &\geq& -\frac{\alpha}{4} H^4 - \frac{1}{4 \alpha} |{\accentset{\circ}}h|^4 \quad \text{for every } \alpha>0 \quad\text{and }\nonumber \\
\frac{1}{3} H \operatorname{Tr}_g({\accentset{\circ}}h ^3) &\geq& -\frac{\beta}{12} H^4 - \frac{1}{4 \beta^{1/3}} [\operatorname{Tr}_g({\accentset{\circ}}h ^3)]^{4/3} \quad \text{for every } \beta>0 \quad. \nonumber\end{aligned}$$ The last two lower bounds combined with give $$\label{eq:Pab}
P_{\alpha \beta} (g,h):= \operatorname{det_g}(h)+ \frac{1}{4} \operatorname{Tr}_g({\accentset{\circ}}h^4)+\left(\frac{1}{4 \alpha} -\frac{1}{8} \right) |{\accentset{\circ}}h|^4 + \frac{1}{4 \beta^{1/3}} [\operatorname{Tr}_g({\accentset{\circ}}h ^3)]^{4/3} \geq \left(1- \frac{3\alpha+\beta}{12} \right) H^4.$$ We have therefore answered to question 2) above:
\[prop:2\] Given an isometric immersion $f:({{\mathcal}{M}}^4,g)\hookrightarrow ({\mathbb{R}}^5, \delta_{\mu\nu})$ of a closed $4$-manifold $({{\mathcal}{M}}^4,g)$, and let $ P_{\alpha \beta}$ be the following expression $$\label{eq:defPab}
P_{\alpha \beta} (g,h):= \operatorname{det_g}(h)+ \frac{1}{4} \operatorname{Tr}_g({\accentset{\circ}}h^4)+\left(\frac{1}{4 \alpha} -\frac{1}{8} \right) |{\accentset{\circ}}h|^4 + \frac{1}{4 \beta^{1/3}} [\operatorname{Tr}_g({\accentset{\circ}}h ^3)]^{4/3} \quad.$$ Then, the functional $${\mathcal}P_{\alpha \beta} (f):= \int_{{{\mathcal}{M}}} P_{\alpha \beta} (g,h) d \mu_g$$ is invariant under conformal transformations of ${\mathbb{R}}^5$ centered off of $f({{\mathcal}{M}}^4)$.\
Moreover $ P_{\alpha \beta} (g,h) \geq 0$ for every immersed hypersurface $f({{\mathcal}{M}}^4)\subset {\mathbb{R}}^5$, provided $\alpha, \beta>0$ satisfy $3\alpha+\beta \leq 12$.
We now answer the last question 3) by proving that round spheres are the points of strict global minimum for the functional ${\mathcal}P_{\alpha, \beta}$, for $\alpha, \beta>0$ with $\alpha \leq 2$. This result may be seen as the 4-d analogue of the celebrated 2-d theorem of Willmore [@Will] asserting that the Willmore functional is strictly minimized by embedded round 2-d spheres of ${\mathbb{R}}^3$.
\[thm:3\] For any isometric immersion $f:({{\mathcal}{M}}^4,g)\hookrightarrow ({\mathbb{R}}^5, \delta_{\mu\nu})$ of a closed $4$-manifold $({{\mathcal}{M}}^4,g)$, let $$F(g,h):=\operatorname{det_g}(h)+Z({\accentset{\circ}}h),$$ where $Z({\accentset{\circ}}h)$ is a non negative real function of the eigenvalues of ${\accentset{\circ}}h$, homogeneous of degree $4$ (that is $Z(t {\accentset{\circ}}h)=t^4 Z({\accentset{\circ}}h)$, for every $t \in {\mathbb{R}}$), and such that $Z({\accentset{\circ}}h)=0$ if and only if ${\accentset{\circ}}h=0$. Then the functional $${\mathcal}F(f):= \int_{{{\mathcal}{M}}} \left[ \operatorname{det_g}(h)+Z({\accentset{\circ}}h) \right] \, d\mu_g$$ is invariant under conformal transformations of ${\mathbb{R}}^5$ centered off of $f({{\mathcal}{M}}^4)$ and $$\begin{aligned}
{\mathcal}F (f) \geq \frac{ 8 \pi ^ 2 } { 3} \quad,\end{aligned}$$ with equality if and only if $f({{\mathcal}{M}})$ is embedded in $ {\mathbb{R}}^ 5 $ as a round sphere.\
In particular, this holds for the functional ${\mathcal}P_{\alpha \beta}$ above provided $\alpha, \beta>0$ and $\alpha \leq 2$.
Since by assumption the function $Z({\accentset{\circ}}h)$ is non-negative, denoting $${{\mathcal}{M}}^+:=\{x \in {{\mathcal}{M}}: \operatorname{det_g}(h)\geq 0\} \subset {{\mathcal}{M}}\quad,$$ we have $$\label{eq:intPab}
{\mathcal}F (f)=\int_{{{\mathcal}{M}}} \operatorname{det_g}(h)+Z({\accentset{\circ}}h) d \mu_g \geq \int_{{{\mathcal}{M}}} \operatorname{det_g}(h) \, d \mu_g \geq \int_{{{\mathcal}{M}}^+} \operatorname{det_g}(h) \, d \mu_g \quad,$$ with equality in the first estimate if and only if ${\accentset{\circ}}h\equiv 0$, but this happens if and only if $f({{\mathcal}{M}}^4)$ is embedded as a round sphere (since by assumption ${{\mathcal}{M}}^4$ is closed so $f({{\mathcal}{M}})$ cannot be an affine hyperplane).
Observing that $|\operatorname{det_g}(h)|=|\operatorname{det_g}(D\gamma)|= |J(\gamma)|$, where $\gamma: {{\mathcal}{M}}^4 \to S^4$ is the Gauss map of the immersion $f$ and $J(\gamma)$ is its Jacobian, we get $$\label{eq:gammaJ}
\int_{{{\mathcal}{M}}^+} \operatorname{det_g}(h) \, d \mu_g = \int_{{{\mathcal}{M}}^+} |J(\gamma)| d \mu_g \geq Vol_{S^4}(\gamma({{\mathcal}{M}}^+)) \quad,$$ where $Vol_{S^4}(\gamma({{\mathcal}{M}}^+))$ is the volume of $\gamma({{\mathcal}{M}}^+)\subset S^4$ with respect to the volume form of $S^4$. But now it is well known that $\gamma({{\mathcal}{M}}^+)=S^4$, since $f({{\mathcal}{M}}^4)$ is a *closed* hypersurface (the standard argument is to consider, for every $\nu \in S^4$, an affine hyperplane of ${\mathbb{R}}^5$ orthogonal to $\nu$ and very far from $f({{\mathcal}{M}}^4)$; then one translates such affine hyperplane towards $f({{\mathcal}{M}})$, keeping the orthogonality with $\nu$, up to the first tangency point $f(x)$. There, one has $\gamma(x)=\nu$ and moreover, by construction, $f({{\mathcal}{M}}^4)$ must lie on just one side of its affine tangent space at $f(x)$, namely such translated hyperplane, therefore $\operatorname{det_g}(h)|_{x}\geq 0$. Hence $\nu \in \gamma({{\mathcal}{M}}^+)$).
Combining this last fact with and , we get that $${\mathcal}F (f) \geq |S^4| =\frac{8 \pi^2}{3}\quad.$$ Equality in the last formula of course implies equality in and but, as already observed above, equality in implies that $f({{\mathcal}{M}}^4)$ is embedded as a round sphere in ${\mathbb{R}}^5$.
To see that ${\mathcal}P_{\alpha, \beta}$ satisfies the assumptions, observe that if $\alpha, \beta>0$ and $\alpha \leq 2$, then $$P_{\alpha \beta} (g,h):= \operatorname{det_g}(h)+ \frac{1}{4} \operatorname{Tr}_g({\accentset{\circ}}h^4)+\left(\frac{1}{4 \alpha} -\frac{1}{8} \right) |{\accentset{\circ}}h|^4 + \frac{1}{4 \beta^{1/3}} [\operatorname{Tr}_g({\accentset{\circ}}h ^3)]^{4/3} \geq \operatorname{det_g}(h),$$ with equality if and only if $\operatorname{Tr}_g({\accentset{\circ}}h^4)=|{\accentset{\circ}}h^2|^2 \equiv 0$; but that happens if and only if ${\accentset{\circ}}h\equiv 0$.
By a similar proof one can show the following higher dimensional generalization, however obtaining a comparably explicit formula for $F(g,h)$ is less clear.
\[thm:4\] For any isometric immersion $f:({{\mathcal}{M}}^{2n},g)\hookrightarrow ({\mathbb{R}}^{2n+1}, \delta_{\mu\nu})$ of a closed $2n$-manifold $({{\mathcal}{M}}^{2n},g)$, let $$F(g,h):=\operatorname{det_g}(h)+Z({\accentset{\circ}}h),$$ where $Z({\accentset{\circ}}h)$ is a non negative real function of the eigenvalues of ${\accentset{\circ}}h$, homogenous of degree $2n$ (that is $Z(t {\accentset{\circ}}h)=t^{2n} Z({\accentset{\circ}}h)$, for every $t \in {\mathbb{R}}$), and such that $Z({\accentset{\circ}}h)=0$ if and only if ${\accentset{\circ}}h=0$. Then the functional $${\mathcal}F(f):= \int_{{{\mathcal}{M}}} \left[ \operatorname{det_g}(h)+Z({\accentset{\circ}}h) \right] \, d\mu_g$$ is invariant under conformal transformations of ${\mathbb{R}}^{2n+1}$ centered off of $f({{\mathcal}{M}}^{2n})$ and $$\begin{aligned}
{\mathcal}F (f) \geq \omega_{2n} \quad,\end{aligned}$$ where $\omega_{2n}$ is the surface area of a unit sphere $ {\mathbb}{S}^{2n}\subset {\mathbb{R}}^{2 n+1}$ with equality if and only if $f({{\mathcal}{M}})$ is embedded in $ {\mathbb{R}}^ {2n+1} $ as a round sphere.\
Moreover there exists $C=C(n)>0$ such that, setting $Z({\accentset{\circ}}h)=C \| {\accentset{\circ}}h \|^{2n}$, it holds $$F(g,h)= \operatorname{det_g}(h)+ C \| {\accentset{\circ}}h \|^{2n} \geq 0\quad .$$
The argument for the first part of the theorem is analogous as above, so we will just show the last statement.\
Using the orthogonal decomposition for the second fundamental form $ h = {\accentset{\circ}}h + \frac { H}{2n} g $, clearly one has $$\begin{aligned}
\operatorname{det_g}(h) = \operatorname{det_g}\left( {\accentset{\circ}}h + \frac{ H}{ 2n} g \right).\end{aligned}$$ Now, expanding the determinant, we have that $$\begin{aligned}
\operatorname{det_g}\left( {\accentset{\circ}}h + \frac{ H}{ 2n} g \right) = \operatorname{det_g}({\accentset{\circ}}h) + \sum_{k = 1 } ^ { 2n -2 } H ^ k P_{k}({\accentset{\circ}}h ) + \frac { H ^ {2n} }{ (2n) ^ { 2n-1}}\quad,\end{aligned}$$ where $ P_{k}( {\accentset{\circ}}h ) $ is a complete contraction of order $ 2n-k$. Note that the term corresponding to $k =2n-1$ does not appear as $\operatorname{tr}_g ({\accentset{\circ}}h) = 0 $. We can now apply Young’s inequality to conclude $$\begin{aligned}
\sum_{ k =1 } ^{2n-2} H ^ k P_{ k } ({\accentset{\circ}}h ) \geq - {\epsilon}H ^ {2n} - C_{\epsilon}\|{\accentset{\circ}}h \| ^ {2n}. \end{aligned}$$ Furthermore, as $ \operatorname{det_g}( {\accentset{\circ}}h) \geq - C ' \| {\accentset{\circ}}h \| ^ {2n}$ we find $$\begin{aligned}
\operatorname{det_g}(h) \geq \left( \frac { 1 }{( 2n) ^{2n-1} } - {\epsilon}\right) H ^ {2n} - C_{\epsilon}' \|{\accentset{\circ}}h \| ^ {2n } .\end{aligned}$$ Choosing ${\epsilon}=(2n)^{-2n}$ we then get that there exists $C=C(n)$ such that $\operatorname{det_g}(h) + C \|{\accentset{\circ}}h \| ^ {2n } \geq 0$, as desired.
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author:
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A. Abdel-Rehim\
CaSToRC, The Cyprus Institute, Nicosia, Cyprus\
E-mail:
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F. Burger, B. Kostrzewa\
Humboldt-Universit[ä]{}t zu Berlin, Institut f[ü]{}r Physik, Berlin, Germany\
E-mail:
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A. Deuzeman\
Albert Einstein Center for Fundamental Physics - University of Bern, Switzerland\
E-mail:
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K. Jansen\
NIC, DESY, Zeuthen, Germany\
E-mail:
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L. Scorzato\
Trento Institute for Fundamental Physics and Application (TIFPA), Trento, Italy\
E-mail:
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\
HISKP (Theory), Rheinische Friedrich-Wilhelms Universit[ä]{}t Bonn, Germany\
E-mail:
bibliography:
- 'bibliography.bib'
title: Recent developments in the tmLQCD software suite
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Introduction
============
The Lattice QCD community relies to a large extent on the efficient usage of available computer resources. It is, therefore, mandatory to optimise existing codes for new supercomputer architectures as well as commodity systems newly appearing on the market.
Due to the increasing complexity and diversity of modern computer architectures there is need for flexible software which allows one to quickly implement and test new developments. Furthermore, lattice QCD actions nowadays simulated have also reached a high level of diversity, which needs to be mirrored by the software. Writing, debugging and implementing such software requires, hence, a non-negligible amount of manpower.
From this perspective it might appear sensible to develop only one, community wide code basis. However, for scientific hygiene there should be clearly at least two or three implementations available such that cross-checks are possible. Ideally, all the different implementations are publicly available, which increases the chance to find mistakes in the codes and allows any lattice QCD practitioner to re-use them (c.f. Ref. [@albert:lat2013]).
By now there are several lattice QCD software suites available as open source, among others the MILC code [@milc], Chroma [@Edwards:2004sx], openQCD [@openqcd] and bQCD [@Nakamura:2010qh]; another one is tmLQCD [@Jansen:2009xp] obtained from github [@tmlqcdgithub]. tmLQCD started as a code for simulations using the Wilson twisted mass formulation of lattice QCD, but includes by now a much wider range of actions and lattice Dirac operators. Moreover, tmLQCD is fully parallelised and includes optimisations for most modern supercomputer architectures.
tmLQCD: General Overview
========================
The tmLQCD software is written in the C programming language following the C99 standard. It ships with an [autoconf]{} configuration script, which makes it relatively easy to compile the code on most modern computer platforms. It also comes with documentation as a LaTeXdocument.
Once compiled, tmLQCD offers two executables: firstly an inverter offering a range of iterative solvers needed for computing propagators. The second executable implements a Hybrid Monte Carlo (HMC) algorithm [@Duane:1987de] for generating gauge configurations using Wilson twisted mass actions, also including the clover term.
The various physical and algorithmic parameters of both programmes can by chosen by the user using an input file, which has a simple and human readable syntax. An example of the general section of such an input file for the HMC might look as follows:
``` {frame="single"}
L=4 # spacial lattice extend
T=8 # time extend
Measurements = 1000 # no. of trajectories
StartCondition = hot
ReversibilityCheck = yes # perform reversibility check
ReversibilityCheckIntervall=2 # every second traj.
```
The code comes with a selection of sample input files.
The executables can be compiled as scalar or parallel programmes to be decided at configure time. The parallelisation is implemented using a hybrid approach with the Message Passing Interface (MPI) and openMP. The MPI topology and the number of (openMP) threads per MPI task can be specified in the input file.
tmLQCD directly reads and writes the ILDG gauge configuration format [@DeTar:2007au] and the SCIDAC propagator format using the LIME library [@lime]. There is, therefore, full compatibility to the Chroma software. Moreover, tmLQCD can be configured to use the Lemon library [@Deuzeman:2011wz], which is a parallelised replacement of LIME using the MPI parallel I/O capabilities. Lemon significantly increases the I/O performance on massively parallel machines. In particular, the propagator computation, usually heavily I/O bound, benefits from the usage of Lemon.
Iterative Solvers
=================
One of the main tasks in lattice QCD is solving $$\label{eq:dirac}
D\ \cdot\ \psi\ =\ \eta$$ for $\psi$, where $D$ is some discretisation of the gauge covariant Dirac operator. Note that we have suppressed all indices for simplicity. The lattice Dirac operator can be viewed as a large sparse matrix, which makes iterative solvers and in particular Krylov space solvers like the conjugate gradient (CG) most suited for solving equation \[eq:dirac\], see Ref. [@saad:2003a] for a general discussion.
Several discretisations of the Dirac operator are implemented in tmLQCD: the Wilson and Wilson twisted mass Dirac operators both with and without clover term, the non-degenerate Wilson twisted mass Dirac operator [@Frezzotti:2000nk] with and without clover term and the overlap operator [@Neuberger:1997fp; @Neuberger:1998wv]. For Wilson type operators also even/odd preconditioning [@DeGrand:1990dk] is implemented. The clover operators are currently only available with even/odd preconditioning. The operator can be specified in the input file like in the following example for a (mass degenerate) even/odd preconditioned Wilson twisted mass Dirac operator:
``` {frame="single"}
BeginOperator TMWILSON
2kappaMu = 0.05
kappa = 0.177
UseEvenOdd = yes
Solver = CG
SolverPrecision = 1e-14
MaxSolverIterations = 1000
EndOperator
```
The physical parameters are the $\kappa=0.177$ hopping parameter and the twisted mass parameter $\mu$, the latter specified as $2\kappa\mu=0.05$. In the above listing the iterative solver to be used is CG, the solver precision is set to $\|D\psi - \eta\|^2\leq
10^{-14}$ and not more than $1000$ iterations should be used.
The list of available solvers includes CG, BiCG, BiCGstab, FGMRES, CGS, EigCG [@Stathopoulos:2007zi] and GCR. A FGMRES solver applying inexact deflation as discussed in Ref. [@Luscher:2007se] is also available, as well as a multiple mass CG solver for the twisted mass Dirac operator [@Chiarappa:2006hz]. Note that the optimal (Krylov) solver depends on the particular discretisation of the lattice Dirac operator.
Lattice Actions
===============
Lattice QCD actions are usually split into a sum of a gauge and a fermionic part. Concerning the gauge part, tmLQCD implements the Wilson plaquette gauge action and the family of gauge actions including an additional planar $2\times1$ rectangular Wilson loop. These include the tree level Symanzik improved, the Iwasaki and the DBW2 gauge actions.
For the fermionic part we have implemented several so-called pseudofermion actions. They are based on the stochastic representation of a determinant of a matrix $Q^2$ $$\det(Q^2) = \int\mathcal{D}\phi^\dagger\, \mathcal{D}\phi\ e^{-\phi^\dagger\frac{1}{Q^2}\phi}\,,$$ where the pseudofermion fields $\phi^\dagger,\phi$ follow bosonic statistics. Following the notation of Chroma, we call one such term a monomial and an example for an input file is as follows:
``` {frame="single"}
BeginMonomial DET
Timescale = 1 # time scale to integrate on
2KappaMu = 0.177
kappa = 0.177
Solver = CG
AcceptancePrecision =1e-20 # accept/reject precision
ForcePrecision = 1e-12 # MD evolution precision
EndMonomial
```
It corresponds to a two flavour, mass degenerate Wilson twisted mass Dirac operator pseudofermion monomial. You may specify the solver used in the HMC update, as well as the precisions for the molecular dynamics (MD) evolution and the accept/reject step.
There is a list of other monomials supported: for two mass degenerate Wilson and Wilson clover twisted mass fermions there are besides the “$\det$” monomials also ratios of determinants needed for Hasenbusch mass preconditioning with multiple timescales [@Hasenbusch:2001ne; @Hasenbusch:2002ai; @Urbach:2005ji]. For the mass non-degenerate Wilson and Wilson clover twisted mass doublet there is a polynomial [@Frezzotti:1998eu] and a rational monomial [@Clark:2006fx] implemented. Finally, for a single Wilson clover fermion a rational monomial can be used.
tmLQCD offers different schemes for integrating the MD equations of motion: the simple leap-frog integration scheme, the second order minimal norm scheme and a fourth order Omelyan scheme [@Sexton:1992nu; @Omelyan:2003:SAI]. They can be combined on different time scales, as can be seen from the following input file listing:
``` {frame="single"}
BeginIntegrator
Type0 = LEAPFROG # integrator on timescale 0
Type1 = 2MN # integrator on timescale 1
IntegrationSteps0 = 1
IntegrationSteps1 = 2
Tau = 1 # trajectory length
NumberOfTimescales = 2
EndIntegrator
```
The zeroth time scale of the two is the finest, the total trajectory length is specified using $\tau=1$ and the number of steps for time scale [N]{} by [IntegrationStepsN]{}. The step numbers are defined recursively. Therefore, the step length on timescale zero is given by $\Delta\tau_0=\tau/N_1/N_0=\Delta\tau_1/N_0$, where $N_i$ is the number of steps on timescale $i$. Note that there are additional factors of $1/2$ for the higher order schemes.
Optimisation: Example BG/Q
==========================
As mentioned before, tmLQCD includes optimisations for several modern supercomputer architectures, IBMs Blue Gene/Q, Intels SSE instruction set and the Aurora architecture. We also have an inverter and parts of the HMC implemented for NVIDIA GPUs.
Here we will discuss as an example the BG/Q architecture. The BG/Q compute nodes consist of one CPU with 16 cores with four hardware threads each. Hence, in total there are 64 hardware threads per node, which can be divided into MPI tasks and/or (openMP) threads. The nodes are connected via a five dimensional torus network. For a first discussion on how to port lattice QCD codes for BG/Q see Ref. [@Boyle:2012iy]. The floating point unit (FPU) includes a four double wide SIMD vector unit (QPX). For maximal performance it is mandatory to utilise it appropriately. In total one node of the BG/Q has a peak performance of $204.8$ Gflop/s.
For the following discussion we used a hybrid MPI/openMP implementation with always $64$ openMP threads and one MPI task per BG/Q node. For a more detailed discussion on how to optimally use openMP see the contribution [@bartekcode:lat2013] at this conference. In figure \[fig:bgq\] we show the performance of the tmLQCD Wilson Dirac hopping matrix in Gflop/s per BG/Q node as a function of the node local lattice extent $L_\mathrm{local}$. Each node worked on a local lattice volume of $L_\mathrm{local}^4$.
First we investigated the code performance [*without internode communication*]{}. The plain C99 implementation is shown as black diamonds leading to less than 5% of peak performance, almost independently of the local volume. This result points towards a badly saturated FPU. The red squares represent the code including the QPX instruction set utilising the intrinsic functions provided by the IBM C compiler. A strong improvement is visible, with up to 25% of peak at $L_\mathrm{local}=12$. For $L_\mathrm{local}\geq14$ the local problem does no longer fit into the cache leading to a plateau in the performance around $20$ Gflop/s per node.
The purple circles in figure \[fig:bgq\] represent the hopping matrix with QPX instructions, but now [*with internode communication switched on*]{}. The MPI overhead turns out to be significant, more than halving the performance where the local volume fits into cache.
Circumventing this problem is possible by overlapping communication and computation. For this purpose we use an implementation of the hopping matrix where in a first step we project to half size spinors $\phi^+$, $\phi^-$ for all $x$ and $\mu$ $$\phi^+(x-\hat\mu, \mu) =
U_\mu(x-\hat\mu)P_\mu^{(\textrm{full}\rightarrow\textrm{half})}(1-\gamma_\mu)\psi(x)\,,\
\phi^-(x+\hat\mu,\mu) =
P_\mu^{(\textrm{full}\rightarrow\textrm{half})}(1+\gamma_\mu)\psi(x)\,.$$ In a second step the communication is performed and finally the result is generated by reconstructing the full spinor for all $x$ $$\eta(x) = \sum_\mu\left[ P_\mu^{(\textrm{half}\rightarrow\textrm{full})}\phi^+(x,\mu) +
P_\mu^{(\textrm{half}\rightarrow\textrm{full})}U_\mu^\dagger(x-\hat\mu)\phi^-(x,\mu)\right]\,.$$ The field $\phi^{+,-}$ requires only half the amount of data to be communicated compared to the full spinor field $\psi$. Overlapping communication and computation can now be achieved by dividing all $x$ into one set $x_\mathrm{surface}$ residing on the local surface and a second set $x_\mathrm{bulk}$. Then the projection to $\phi^{+,-}$ is first done for all $x_\mathrm{surface}$, then the communication is started non-blocking in parallel to the the computation for $x_\mathrm{bulk}$. When the latter computation is finished it is checked whether the communication is finished also (c.f. also [@Brambilla:2012jc]).
It turns out that this does not work as expected using the MPI library provided on the BG/Q. All the communication is effectively done in the [MPI\_Wait]{} and, therefore, there is no gain. However, IBM provides a lower level communication library called SPI. Replacing the MPI calls with SPI calls leads to the light blue triangles in figure \[fig:bgq\] and, thus, to a significant improvement. The blue inverted triangles finally include also the correct mapping of the lattice QCD geometry to the five dimensional BG/Q network. So, eventually we obtain almost the same performance as in the case with communication switched off.
![Single BG/Q node double precision performance of the hopping matrix in Gflop/s as a function of $L_\mathrm{local}$. “Plain C” and “QPX” correspond to measurements with communication switched off using a plain C implementation and one including QPX instructions, respectively. The other points include communication, see text for details.[]{data-label="fig:bgq"}](qpxspimap.eps){width=".65\linewidth"}
Acknowledgements {#acknowledgements .unnumbered}
================
We thank all members of ETMC for the most enjoyable collaboration. B.K. is supported by the National Research Fund, Luxembourg. This work is supported in part by DFG and NSFC (CRC 110) and by DFG SFB/TR9. A. A.-R. acknowledges support from the PRACE-2IP project under grant number EC-RI-283493. LS thanks the SUMA project for partial support.
|
---
abstract: 'In this paper, we design and implement time exchange (TE) based cooperative forwarding where nodes use transmission time slots as incentives for relaying. We focus on distributed joint time slot exchange and relay selection in the sum goodput maximization of the overall network. We formulate the design objective as a mixed integer nonlinear programming (MINLP) problem and provide a polynomial time distributed solution of the MINLP. We implement the designed algorithm in the software defined radio enabled USRP nodes of the ORBIT indoor wireless testbed. The ORBIT grid is used as a global control plane for exchange of control information between the USRP nodes. Experimental results suggest that TE can significantly increase the sum goodput of the network. We also demonstrate the performance of a goodput optimization algorithm that is proportionally fair.'
author:
-
bibliography:
- 'Globecomm\_bibtex.bib'
title: Implementation of Distributed Time Exchange Based Cooperative Forwarding
---
Resource Delegation, Cooperative Forwarding, Global Control Plane, Testbed Implementation, GNUradio.
Introduction
============
Cooperative forwarding improves the connectivity and throughput in wireless networks [@Tse]. However, forwarding always incurs some costs, e.g., delay/power, at the forwarder node. Hence, there have been recent studies on resource delegation based incentivized forwarding [@Zhang; @Baochun:a; @Ray; @Nazmul2; @Zhang2; @Lindstorm]. Resource delegation based forwarding allows the sender node to delegate a portion of its allotted resource to the forwarder node as an immediate incentive for relaying. Previous works on incentivized forwarding have focused on resource exchange from a theoretical perspective and used numerical simulations to justify the effectiveness of the approach. The main contribution of our work is to demonstrate the advantages of incentivized forwarding using the ORBIT indoor wireless testbed [@ORBIT].
We specifically focus on the uplink of an $N$ node time division multiple access (TDMA) network where each node receives an initial number of time slots and transmits data to the base station (BS) through the direct path. In this context, we focus on a two hop time exchange (TE) based forwarding scheme where the sender node transfers a portion of its allotted time slots to the forwarder node as an incentive for relaying. The basic idea of TE is illustrated in Fig. \[fig:TimeExchange\]. In this example, node $1$ and $2$ initially receive $4$ time slots and transmit through the direct path. In TE, node $1$ performs a half duplex decode and forward (DF) relaying of node $2$’s data and node $2$ delegates one time slot to node $1$ as an incentive for relaying. Node $2$ may attain higher data rate through this cooperation since its data goes to the BS through two different paths. Node $1$ may also get higher data rate since it has more transmission time slots. The optimal time slot delegation is an important question in this context.
A sender node can delegate time slots to multiple forwarder nodes and transmit data to the BS through multiple paths. However, the authors of [@Ravi] have shown that for a source-destination pair, in the presence of multiple relay nodes, it is sufficient to select the “best relay node" to achieve full diversity order. On the other hand, the assumption of one sender node for one forwarder node reduces the relay selection complexity. Therefore, we assume one forwarder node for one sender node and vice versa in this work. The optimal distributed sender-forwarder pair selection problem becomes another important question in this context.
\[fig:TimeExchange\]
In this work, we address the joint time slot delegation and sender-forwarder pair selection question in the context of sum goodput maximization of the overall network. We formulate the joint optimization problem as a mixed integer nonlinear programming (MINLP) problem. Using our relay selection work of [@Nazmul2], we show that the distributed solution of the MINLP requires $O(k^{tot})$ computational complexity and at most $N^2$ message passing where $k^{tot}$ and $N$ denote the total time slots and nodes in the network. The designed algorithm maximizes the sum goodput of the network while preserving the local goodput of the individual nodes. We implement the designed algorithm in the ORBIT indoor wireless testbed. We use software defined radio enabled USRP [@USRP] nodes of the ORBIT [@ORBIT] testbed. The ORBIT grid serves as the global coordination plane [@Narayan] to exchange the protocol information between the nodes. The data transmission through the air is processed using the GNUradio codes [@GNUradio].
Previous works in resource exchange [@Zhang; @Baochun:a; @Ray; @Nazmul2; @Zhang2; @Lindstorm] have focused on developing information theoretical algorithms. We design the theoretical framework in accordance with the testbed constraints and then implement the framework in the ORBIT testbed. Our theoretical analysis has similarities to the classical maximum weighted scheduling study of [@Ephremides]. However, unlike [@Ephremides], we consider a network where each node starts with a fixed number of time slots and then tries to find the optimal number of time slot transfers. Our proposed framework can be applied to TDMA based commerical (GSM & Edge [@Goldsmith], 802.16 Wireless MAN [@WirelessMAN]) and tactical (Joint Tactical Radio System [@JTRS] and Link16 [@Link16]) networks.
This paper is organized as follows. Sec. \[sec:Model\] and \[sec:Theory\] illustrate the system model and design objective respectively. We solve the optimization problem in Sec. \[sec:Solution\]. After describing the experimental setup in Sec. \[sec:Setup\], we demonstrate the testbed results in Sec. \[sec:Results\]. We conclude the work in \[sec:Conclusion\].
\[fig:packet\_transmission\]
System Model {#sec:Model}
============
We consider the uplink of an $N$ node single cell TDMA network. Let $\mathcal{V} = \{1, \, 2, \, \cdots, \, N\}$ denote the set of $N$ nodes that transmit data to the BS (node $0$). Each node uses the same bandwidth. Node $i \in \mathcal{V}$ is initially allotted $k_i^{in}$ time slots per second. Without loss of generality, assume one packet is transmitted per time slot. Each node is assumed to employ a fixed modulation scheme. This assumption is based on the testbed implementation constraint and will be further explained in the experimental setup section. Due to the fixed modulation and total bandwidth usage, the data transmission rate of each node depends on the number of allotted time slots.
We use packet loss probability as the channel strength indicator. Let $Pe_{ij}$ denote the packet loss probability in the $ij$ path. Define goodput of node $i$ by the number of packets of node $i$ that successfully reach the BS. Node $i$ initially transmits through the direct link $i0$. Hence, the initial goodput of node $i$, $R_i^{in}$, can be found as: $$R_i^{in} = R_{i0} = k_i^{in} * (1 - Pe_{i0}) \label{eq:Packet_Direct}$$
Goodput Analysis in TE
----------------------
In TE, nodes perform two hop half duplex decode and forward (DF) relaying. Let, $\mathcal{SF} = \{\mathcal{SF}_1, \cdots, \mathcal{SF}_k\}
=\{(s_1,f_1),(s_2,f_2),\cdots,(s_K,f_K)\}$ denote the sender-forwarder pairing sets, i.e., the forwarder node $f_i$ relays the sender node $s_i$’s data, along with transmitting $f_i$’s own data. Let $\mathcal{D} = {d_1, d_2, \cdots, d_L}$ denotes the direct set, i.e., the set of remaining nodes that transmit data without cooperation.
The left and right hand side of Fig. \[fig:packet\_transmission\] shows the direct transmission and TE model of an arbitrary sender-forwarder pair $(s,\,f)$. Node $s$ and $f$ initially receive $k_s^{in}$ and $k_f^{in}$ time slots and obtain $R_s^{in}$ and $R_f^{in}$ goodput respectively.
Notation Meaning
------------------ ------------------------------------------------
$Pe_{ij}$ Packer error probability of the $ij$ link
$k_i^{in}$ Initial transmission time slot of node $i$
$k_i^{te}$ Node $i$’s btransmission time slot in BE
$R_i^{in}$ Initial goodput of node $i$
$R_i^{te}$ Node $i$’s goodput in TE
$R_{ij}$ Goodput in the $ij$ link
$\mathcal{V}$ Set of $N$ nodes
$\mathcal{D}$ Set of nodes that transmit without cooperation
$\mathcal{SF}$ Set of sender-forwarder pairs
$\mathcal{SF}_i$ $i^{th}$ sender-forwarder pair $(s_i,f_i)$
$s$ Sender node
$f$ Forwarder Node
$k^{tot}$ Total time slots in $1$ second
$n_i$ Number of neighbouring nodes of node $i$
: Summary of used notations[]{data-label="Tab:one"}
In TE, node $s$ transmits for $k_s^{te}$ time slots. During $k_s^{te}$ time slots, node $s$ transmits $k_s^{te}$ packets. Among these packets, the BS and node $f$ receive $R_{s0}$ and $R_{sf}$ ‘error-free’ packets respectively. Following the analysis of , $$R_{sf} = k_s^{te} * (1 - Pe_{sf}) \, , \, R_{s0} = k_s^{te} * (1 - Pe_{s0}) \label{eq:Packet1}$$ Assume $Pe_{sf} \leq Pe_{s0}$. Hence, $R_{sf} \geq R_{f0}$. Node $f$ acts as a forwarder of node $s$ and transmits for $k_f^{te}$ time slots. During these time slots, $k_f^{te} * (1 - Pe_{f0})$ packets ‘successfully’ reach the BS. Assume, $$R_f^{te} + R_c = k_f^{te} * (1 - Pe_{f0}) \label{eq:Packet3}$$ Here, $R_f^{te}$ denotes the number of ’error-free’ packets that contain node $f$’s own data. $R_c$ represents the number of ‘error-free’ packets that contain node $s$’s data and are forwarded by node $f$.
Let $R_s^{te}$ denote the goodput of node $s$ in TE. Also assume that when node $s$ transmits, the packets that get ‘lost’ at node $f$ (the closer node), also get ‘lost’ at node $0$ (the far node). Based on this assumption and using the max-flow-min-cut theorem [@Cover2], $$R_s^{te} \leq \min \bigl(R_{sf}, R_{s0} + R_c \bigr) \label{eq:Packet_flow}$$ Based on this goodput analysis, we focus on the distributed joint optimal time slot exchange and relay selection in the sum goodput maximization of a TE network.
Design Objective {#sec:Theory}
================
*Problem I*
$$\mathbf{\max.} \, \sum_{d \in \mathcal{D}} R_d^{te} + \sum_{(s, \, f) \in \mathcal{SF}}
\bigl(R_f^{te} + R_s^{te})
\label{eq:MainObjective}$$
$$\mathbf{s.t.} \, \, (R_f^{te},R_s^{te}) \, \in \, conv(k_f^{te},k_s^{te})
\, \, \forall \, (s,f) \, \in \, \mathcal{SF} \label{eq:RateConstraint1}$$
$$R_f^{te} \geq R_f^{in} \, , \, R_s^{te} \geq R_s^{in}
\, \, \forall \, (s,f) \, \in \, \mathcal{SF} \label{eq:RateConstraint2}$$
$$k_f^{te} + k_s^{te} \leq k_f^{in} + k_s^{in}, \, (k_f^{te}, k_s^{te}) \in \mathcal{Z}^+,
\, \forall \, (s,f) \, \in \, \mathcal{SF} \label{eq:BandwidthConstraint1}$$
$$R_d^{te} = k_d^{te} * (1 - Pe_{d0}), \, k_d^{te} = k_d^{in}, \, \forall d \, \in \, \mathcal{D}
\label{eq:BandwidthConstraint2}$$
$$\mathcal{D} \subseteq \mathcal{V} \, , \, \mathcal{SF} \in \mathcal{V} \times \mathcal{V} \, \, , \, \,
\mathcal{SF}_i \cap \mathcal{SF}_j = \emptyset \, \, \forall i \neq j \label{eq:RelaySelection1}$$
$$\mathcal{SF}_i \cap \mathcal{D} = \emptyset \, \forall \, i \, \in [1,K]
\label{eq:RelaySelection2}$$
$$\mathcal{SF}_1 \cup \mathcal{SF}_2 \cdots \cup \mathcal{SF}_K \cup \mathcal{D} = \mathcal{V}
\label{eq:RelaySelection3}$$
$$Variables \, \, \mathcal{D}, \mathcal{SF}, R_s^{te}, R_f^{te}, k_s^{te}, k_f^{te}
\nonumber$$
Equation and suggest that $R_d^{te} = R_d^{in}$. Therefore, the goodputs of the nodes in the direct set $\mathcal{D}$ are not optimization variables. Equation denotes that the goodputs of the forwarder and sender remain in the convex hull of the allotted time slots, $k_s^{te}$ and $k_f^{te}$. This convex hull is governed by and . Equation ensures that the goodputs of the sender and the forwarder through TE don’t drop below their initial goodputs. Equation shows that the total time slots used by the sender and forwarder are constrained by the summation of the initial time slots allotted to those nodes. Equation - denote that the direct node set and the sender-forwarder pairs cannot have any common node and together, they form the overall set $\mathcal{V}$.
Problem I is similar to the design objective of our earlier work on bandwidth exchange [@Nazmul2]. However, we focused on information theoretic capacity [@Cover] based resource allocation and relay selection in [@Nazmul2]. In this work, we focus on packet error probability based goodput maximization, which is a tangible objective in an indoor wireless testbed.
The convex constraint of , the discrete time slot allocation and the sender-forwarder pair selection objectives make problem I a mixed integer nonlinear programming (MINLP) problem. The solution of this MINLP involves an exponential number of variables and constraints. In the next section, we focus on designing a polynomial time distributed solution of problem I.
Optimization Problem Solution {#sec:Solution}
=============================
Let $R_{tot} = \sum_{i \in \mathcal{V}} R_i^{in}$ denote the summation of the initial goodputs of the nodes. For a fixed $\mathcal{SF}$, $R_{tot}$ can be expressed in the following form: $$\begin{aligned}
R_{tot} & = & \sum_{i \in \mathcal{V}} R_i^{in} \nonumber \\
& = & \sum_{d \in \mathcal{D}} R_d^{in} + \sum_{(s, \, f) \in \mathcal{SF}}
\bigl(R_f^{in} + R_s^{in} \bigr) \label{eq:Modified1} \\
& = & \sum_{d \in \mathcal{D}} R_d^{te} + \sum_{(s, \, f) \in \mathcal{SF}}
\bigl(R_f^{in} + R_s^{in} \bigr) \label{eq:Modified2} \end{aligned}$$ Equation uses the fact that $R_d^{te} = R_d^{in} \, \forall \, d \in D$. Subtracting $R_{tot}$ from the objective function of I, we find the following optimization problem:
*Problem II*
$$\mathbf{\max.} \, \sum_{(s, \, f) \in \mathcal{SF}}
\bigl(R_f^{te} + R_s^{te} - R_f^{in} - R_s^{in})
\label{eq:ModifiedObjective}$$
$$\mathbf{s.t.} \, \, (R_f^{te},R_s^{te}) \, \in \, conv(k_f^{te},k_s^{te})
\, \, \forall \, (s,f) \, \in \, \mathcal{SF}
\label{eq:constraint1}$$
$$R_f^{te} \geq R_f^{in} \, , \, R_s^{te} \geq R_s^{in}
\, \, \forall \, (s,f) \, \in \, \mathcal{SF}
\label{eq:constraint2}$$
$$k_f^{te} + k_s^{te} \leq k_f^{in} + k_s^{in}, \, (k_f^{te}, k_s^{te}) \in \mathcal{Z}^+,
\, \forall \, (s,f) \, \in \, \mathcal{SF}
\label{eq:constraint3}$$
$$\mathcal{D} \subseteq \mathcal{V} \, , \, \mathcal{SF} \in \mathcal{V} \times \mathcal{V} \, \, , \, \,
\mathcal{SF}_i \cap \mathcal{SF}_j = \emptyset \, \, \forall i \neq j$$
$$\mathcal{SF}_i \cap \mathcal{D} = \emptyset \, \forall \, i \, \in [1,K]$$
$$\mathcal{SF}_1 \cup \mathcal{SF}_2 \cdots \cup \mathcal{SF}_K \cup \mathcal{D} = \mathcal{V}$$
$$Variables \, \, \mathcal{D}, \mathcal{SF}, R_s^{te}, R_f^{te}, k_s^{te}, k_f^{te}
\nonumber$$
The inclusion of constant terms does not change the optimal variables of an optimization problem [@Boyd]. Therefore, the optimal variables of both problem I and II are same. We focus on solving problem II in the subsequent analysis and use the optimal variables to find the solution of problem I.
Problem II involves both sender-forwarder pair selection and discrete time slot allocation features. For a fixed set of sender-forwarder pairs, the constraints in - ensure that the discrete time slot exchange in one pair does not affect the other pairs. Hence, we now focus on an arbitrary sender-forwarder pair $(s,f)$ and find the discrete time slot exchange in this pair.
Time Slot Allocation for a Fixed Sender Forwarder Pair
------------------------------------------------------
*Problem III*
$$\max \, \, \bigl(R_s^{te} - R_s^{in} \bigr) + \bigl(R_f^{te} - R_f^{in} \bigr)
\label{eq:PacketSubObjective}$$
$$R_{sf} = k_s^{te} * (1 - Pe_{sf}) \, , \, R_{s0} = k_s^{te} * (1 - Pe_{s0})
\label{eq:PacketSubConstraint1}$$
$$R_f^{te} + R_c = k_f^{te} * (1 - Pe_{f0}) \, , \,
R_s^{te} \leq \min \bigl(R_{sf}, R_{s0} + R_c \bigr)
\label{eq:PacketSubConstraint2}$$
$$R_f^{te} \geq R_f^{in} \, \, , \, \, R_s^{te} \geq R_s^{in}
\label{eq:PacketSubConstraint4}$$
$$k_f^{te} + k_s^{te} \, \leq \, k_f^{in} + k_s^{in} \,
, \, (k_f^{te}, \, k_s^{te}) \, \in \mathcal{Z}^+ \, ,
\label{eq:PacketSubTimeConstraint}$$
$$variables \, \,
R_c, \, R_{sf}, \, R_{s0}, \, R_s^{te}, \, R_f^{te}, \, k_s^{te},
\, k_f^{te}$$
Problem III is concave if $(k_f^{te},k_s^{te}) \in \mathcal{R}^+$.
*Proof:* The objective function in and the constraints in , and are linear. Minimum of linear (concave) functions is concave [@Boyd]. Hence, the constraint in is convex. Thus, problem III is a concave maximization problem if $(k_f^{te},k_s^{te}) \in \mathcal{R}^+$. $\blacksquare$
The internal concave structure of problem III allows us to generate an upper and lower bound of problem III.
### Upper Bound
Let us modify problem III by relaxing the integer time slot constraint, i.e., let us assume $(k_f^{te},k_s^{te}) \in \mathcal{R}^+$. Let’s call it problem IV. Now, the feasible region of the modified problem is a superset of that of problem III. Hence, the optimal solution of the modified problem is an upper bound of problem III. Denote this upper bound by $u^0$.
### Lower Bound
Let $k_s^{te,*}$ and $k_f^{te,*}$ denote the optimal time slot solutions of problem IV. Now, if $k_s^{te,*}$ and $k_f^{te,*}$ are integers, they are also the optimal solutions of problem III. Otherwise, convert $k_s^{te,*}$ and $k_f^{te,*}$ to the nearest integers and find the corresponding goodputs. If the corresponding solution is feasible for problem III, it can serve as a lower bound to the optimal solution of problem III. Denote this lower bound by $l^0$.
### Computational Complexity of Problem III
Probelm III can be optimally solved by searching over $k_s^{in}$ time slot transfers. However, one can further reduce the complexity by solving the convex programming based upper and lower bounds of problem III. Let $\epsilon$ denote the tolerance of the optimal solution. If $(u_0 - l_0) \leq \epsilon$, we can use the lower bound as the solution of problem III.
Optimal Sender Forwarder Pair Selection
---------------------------------------
The optimal solution of problem III denotes the sender-forwarder pair’s goodput gain through cooperation, over non-cooperation. The optimal relay selection part of problem II is to find the set of sender-forwarder pairs that maximizes the summation of the goodput gain. Now, consider a graph $ \mathcal{G} = (\mathcal{V},\mathcal{E})$ where the vertices $\mathcal{V}$ represent the set of $N$ nodes under consideration and $\mathcal{E}$ denotes the edges between these nodes. Define the edge weight of any $(i,j)$ pair by $R_i^{te} + R_j^{te} - R_i^{in} - R_j^{in}$, i.e., the difference, in terms of goodput, between the cooperation and non-cooperation scenario. Using the interference-free scheduling algorithm of [@Ephremides], the optimal sender-forwarder set selection problem can be shown to be equivalent to solving the maximum weighted matching (MWM) algorithm [@Edmonds] in the above graph. A detailed proof of the equivalence between the relay selection problem and the MWM algorithm can be found in [@Nazmul2]. The next section will illustrate the use of MWM in the optimal sender-forwarder pair selection among $3$ testbed nodes.
The MWM algorithm can be distributively solved using the distributed local greedy MWM [@Preis]. Distributed MWM finds the pairs by selecting the locally heaviest edges and guarantees at least $50\%$ performance of the optimal solution of problem I [@Preis]. Our distributed TE implementation protocol is based on the distributed MWM and will be described in the next section.
Distributed TE Protocol
-----------------------
\[fig:Orbit\]
1. Node $i \, \in \, \mathcal{V}$ has $k_i^{in}$ transmission time slots and knows the packet error probability of its direct path, $Pe_{i0}$,
2. Let $\mathcal{J}$ be the set of neighbours of $i$. In a wireless environment, neighbouring nodes can hear each other. Node $j \, \in \, \mathcal{J}$ receives node $i$’s packets and can calculate the inter-node packet error probability, $Pe_{ij}$.
3. Node $i$ sends an omnidirectional message containing $Pe_{i0}$ and $k_i^{in}$ to its neighbours.
4. Node $i$ solves problem III for all $j \, \in \, \mathcal{J}$ and finds the goodput gain for each of its neighbours. Thus, node $i$ knows its adjacent link weights.
5. Node $i$ picks the “candidate" node $j$, based on the heaviest adjacent link weight and sends “add" request.
6. If node $i$ receives an “add" request from node $j$, $i$ and $j$ form a cooperative pair. Node $i$ sends “drops" request to its other neighbours.
7. If node $i$ receives a “drop" request from $j$, node $i$ removes the $(i,j)$ link from its adjacent edge set. Node $i$ returns to step $5$.
8. The pair selection process converges after at most $N^2$ message passings. The ‘matched’ nodes form the set of cooperative pairs $\mathcal{SF}$. The ‘unmatched’ nodes form $\mathcal{D}$ and transmit data without cooperation.
9. The sender node $s$ of a cooperative pair transmits its own data during $k_s^{te}$ time slots.
10. The BS sends ACK of the ‘correctly’ received packets to the sender node. The forwarder node also hears these ACK messages. Based on this information, node $f$ finds the packets of $s$ that got ‘lost’ at the BS.
11. Node $f$ transmits for $k_f^{te}$ time slots. During these slots, node $f$ forwards the ‘lost’ packets of $s$ and then transmits its own data packets to the BS.
Computational Complexity of the Distributed Algorithm
-----------------------------------------------------
Each node $i \, \in \, \mathcal{V}$ solves problem III for each of its neighbours. Let $n_i$ be the number of neighbours of node $i$. Since problem III can be optimally solved using $O(k_i^{in})$ searches, each node performs $O(n_i k_i^{in})$ computations. The total number of computations in the $N$ node network is $O(n_i k_i^{in} N)$.
The number of time slots allotted to node $i$, $k_i^{in}$, can be approximated as, $k_i^{in} \approx \frac{k^{tot}}{N}$. Since $n_i << k^{tot}$, the overall complexity is, $O(k^{tot})$.
Thus, the proposed TE protocol solves an approximated distributed version of problem I with $O(k^{tot})$ complexity and at most $N^2$ message passings.
Experimental Setup {#sec:Setup}
==================
ORBIT Testbed & USRP Nodes
--------------------------
\[fig:USRP\]
We implement the proposed TE based incentivized algorithm among the USRP nodes of ORBIT, an indoor wireless testbed of Wireless Information Network Laboratory (WINLAB), Rutgers University. ORBIT has $400$ nodes, overall, in a $20m \times 20m$ square grid. Fig. \[fig:Orbit\] shows a snapshot of the ORBIT testbed.
ORBIT has $15$ USRP nodes that can be used in software defined radio based experiments. Fig. \[fig:USRP\] shows the snapshots of two USRP daughter boards. We use the GNU radio software toolkil [@GNUradio] to run experiments in these USRP nodes. Specifically, we use the benchmark-tx.py and benchmark-rx.py codes to transmit and receive packets between two USRP nodes [@GNUradio]. The flexibility of GNUradio allows us to change the transmission power level and packet sizes through software. This variable power capability of GNUradio, along with the spatial separation among the nodes, allow us to create links with different strengths between different node pairs.
As shown in Fig. \[fig:node\_distance\], we use four USRP nodes of the ORBIT testbed to conduct the TE based cooperative forwarding experiments. Fig. \[fig:node\_distance\] also shows the spatial separation of the selected nodes. Here, node $1$, $2$ and $3$ constitute the user set $\mathcal{V}$ and node $0$ serves as the BS. The ORBIT grid is used as a global control plane to exchange the control information between the nodes.
\[fig:node\_distance\]
Selection of Parameters
-----------------------
The benchmark-tx.py and benchmark-rx.py codes of GNUradio allow the following four modulation schemes: a) GMSK, b) differential binary phase shift keying (DBPSK), c) differential quadrature phase shift keying (DQPSK) and d) differential 8 phase shift keying (D8PSK). DBPSK, DQPSK and D8PSK are found to be very sensitive to peak power clippings due to their variable envelope waveform. Therefore, we use a fixed modulation scheme, GMSK, in our experiments.
Each node transmits at $1$ Mbps and each packet contains $1500$ bytes. As a result, it takes $(1500 * 8)/(1 * 10^6)$, i.e., 0.012 second to transmit one packet. We assume each time slot to be $0.012$ second long, i.e., one packet is transmitted in each slot.
The total transmission time is assumed to be $3$ second. Each node is initially allotted $1$ second transmission time, i.e., $1/0.012$ or $83$ time slots. We approximate the number of time slots since fractional packet transmission is not considered.
We also add $32$ bit CRC sequence in each packet and make it similar to the Ethernet packet structure [@Ethernet]. Note that, we do not use error control coding in these experiments. Therefore, the presence of a single bit error leads to the ‘loss’ of the whole packet due to CRC.
Experimental Evaluation {#sec:Results}
=======================
Illustration of MWM in Relay Selection
--------------------------------------
\[fig:MWM\]
Fig. \[fig:MWM\] shows the use of MWM in the optimal sender-forwarder pair selection among $3$ testbed nodes. The left figure of the top row shows the packet loss probability between the inter-node pairs. These packet error probabilities were based on $1500$ byte packet length, GMSK modulation, some fixed power level and CRC checking. The middle figure of the top row focuses on the direct transmission scenario and shows the goodput (in packet/3 second) of each node. Each node initially receives $83$ time slots and transmits one packet at each slot through the direct path. The packet error probability in link $30$ is $0\%$. Therefore, all transmitted packets of node $3$ reach the BS. Node $1$ and $2$’s goodputs are considerably lower due to the high packet error probability in link $10$ and $20$ respectively.
The top right, the bottom left and the bottom middle figure show the goodput (in packet/3 second) of different sender-forwarder cooperation scenarios. The top right figure focuses on the TE based cooperation between node $1$ and $3$. Here, node $1$ and $3$ solve the two node time slot allocation optimization of problem III. The cooperation allows node $3$ to achieve a goodput of $132$ packets and ensures that node $1$’s goodput does not drop below $18$ packets, its initial value. Therefore, the overall goodput gain obtained through the cooperation of node $1$ and $3$ is $49$ packets. As a result, the $13$ link of the MWM graph, shown in the bottom right figure, is assigned a weight of $49$.
The bottom left and bottom middle figures demonstrate the cooperation scenario in node 1–2 and 2–3 respectively. The bottom right figure shows the link weight of the corresponding cooperation pairs. The distributed local greedy MWM selects link $13$. Therefore, node $1$ and $3$ cooperate using TE, whereas, node $2$ transmits without cooperation.
\[fig:Sumrate\]
Sum Goodput Maximization
------------------------
Fig. \[fig:Sumrate\] compares the sum goodput (in kilo bit per second (kbps)) of TE and direct path transmission. Fig. \[fig:MWM\] shows that node $1$ and $3$ get selected as the cooperative pair due to the MWM algorithm. Therefore, node $1$ and $3$ solve problem III to find the optimal time slot transfers. Due to the sum goodput maximization objective, the benefits of cooperation go to node 3, i.e., the node with the better channel. Node $3$’s goodput increases by $70\%$. The constraint of ensures that node $1$ gets its initial goodput, at least. On the other hand, node $2$ transmits without cooperation and its goodput does not change from the initial value.
Proportional Fair Maximization of Goodput
-----------------------------------------
Fig. \[fig:Fair\] compares the proportional fair maximization performance of direct transmission and TE. Here, the selected cooperative nodes, $s$ and $f$, solve a modified version of problem III. In this modified problem, $s$ and $f$ maximize $(R_s^{te} - R_s^{in}) * (R_f^{te} - R_f^{in})$ instead of maximizing $(R_s^{te} + R_f^{te})$. Hence, the goodput of both nodes increase due to cooperation. Fig. \[fig:Fair\] shows that the goodputs of node $1$ and $3$ increase by $70\%$ and $30\%$ respectively.
Conclusion {#sec:Conclusion}
==========
We designed and implemented TE based cooperative forwarding among the USRP nodes in the ORBIT indoor wireless testbed. We solved the joint time slot allocation and sender-forwarder pair selection problem in this setup. Our proposed algorithm maximizes the global goodput of the network while ensuring that no node’s goodput drops below its initial value. The ORBIT grid is used as a global control plane to exchange the control information between the USRP nodes. Experimental results suggest that resource delegation based cooperative forwarding can significantly improve the sum goodput and proportional fair goodput performance of the network.
The use of adaptive modulation and signal to noise ratio based resource allocation in testbed implementation remains an area of future research.
Acknowledgements
================
This work is supported by the Office of Naval Research under grant N00014-11-1-0132. We thank Kush Patel, Sid Paradkar and Hakim Ergaibi for their assistance in GNUradio coding and testbed implementation.
\[fig:Fair\]
|
---
abstract: |
[ We consider a discrete model of population with interaction where the birth and death rates are non linear functions of the population size. After proceeding to renormalization of the model parameters, we obtain in the limit of large population that the population size evolves as a diffusion solution of the SDE $$\begin{aligned}
Z^x_t =x+\int_0^t f(Z^x_s)ds+2\int_0^t\int_0^{Z^x_s}W(ds,du),\end{aligned}$$ where $W(ds,du)$ is a time space white noise on $([0,\infty))^2$.\
We give a Ray-Knight representation of this diffusion in terms of the local times of a reflected Brownian motion $H$ with a drift that depends upon the local time accumulated by $H$ at its current level, through the function $f'/2$. ]{}
author:
- Mamadou Ba
- Etienne Pardoux
---
[LATP-UMR 6632, CMI, Université de Provence, 39 rue F. Joliot Curie,\
Marseille cedex 13, FRANCE.\
[email : [email protected] ; [email protected]]{} ]{} \
\
\
=
[^1] =
.
=
[^2] =
Introduction {#introduction .unnumbered}
============
Consider a population evolving in continuous time with $m$ ancestors at time $t=0$, in which each individual, independently of the others, gives birth to children at a constant rate $\mu$, and dies after an exponential time with parameter $\lambda$. For each individual we superimpose additional birth and death rates due to interactions with others at a certain rate which depends upon the size of the total population. For instance, we might decide that each individual dies because of competition at a rate equal to $\gamma$ times the number of presently alive individuals in the population, which amounts to add a global death rate equal to $\gamma (X^m_t)^2$, if $X^m_t$ denotes the total number of alive individuals at time $t$. If we consider this population with $m=[Nx]$ ancestors at time $t=0$, weight each individual with the factor $1/N$, and choose $\mu_N=2N+\theta$, $\lambda_N=2N$ and $\gamma_N=\gamma/N$, then it is shown in Le, Pardoux and Wakolbinger [@4LPW] in the above particular case of a quadratic competition term that the “total population mass process” converges weakly to the solution of the Feller SDE with logistic drift $$\begin{aligned}
\label{4eqZ}
dZ^x_t=\left[ \theta Z^x_t- \gamma (Z^x_t)^2\right] dt+2\sqrt{Z^x_t}dW_t,~ Z^x_0=x.\end{aligned}$$ The diffusion $Z^x$ is called Feller diffusion with logistic growth and models the evolution of the size of a large population with competition. In this model $\theta$ represent the supercritical branching parameter while $\gamma$ is the rate at which each individual is killed by any one of his contemporaneans. This model has been studied in Lambert [@4LA], who shows in particular that its extinction time is finite almost surely.
We generalize the logistic model by replacing the quadratic function $\theta z-\gamma z^2$ by a more general nonlinear function $f$ of the population size. We then obtain in the continuous setting a diffusion which is the solution of the SDE $$\begin{aligned}
\label{4fA}
Z^x_t &=x+\int_0^t f(Z^x_s)ds+2\int_0^t\int_0^{Z^x_s}W(ds,du),\end{aligned}$$ where the function $f$ satisfies the following hypothesis.\
**Hypothesis A:** $f\in C({\mathbb{R}}_+;{\mathbb{R}})$, $f(0)=0$ and $\exists \beta\ge 0$ such that $$\begin{aligned}
f(x+y)-f(x)\le \beta y\quad \forall x,y\ge 0.\end{aligned}$$ The equation has a unique strong solution (see [@4DL]). Note that the hypothesis **A** implies that $$\begin{aligned}
\forall x\ge 0, f(x)\le \beta x.\end{aligned}$$ An equivalent way to write is the following. $$\begin{aligned}
\label{4fA1}
Z^x_t =x+\int_0^t f(Z^x_s)ds+2\int_0^t \sqrt{Z^x_s}dW^x_s,\end{aligned}$$ where $W^x$ is a standard Brownian motion. However, the joint evolution of the various population sizes $\left\{ Z^x_t,t\ge 0\right\}$ corresponding to different initial population sizes $x$ would necessitate a complicated description of the joint law of the $\left\{ W^x_., x\ge 0\right\}$. Whereas the formulation due to Dawson, Li [@4DL] with one unique space-time white noise $W$, describes exactly the joint evolution of $\left\lbrace Z^x_t,t\ge 0,x\ge 0\right\rbrace$ which we have in mind. We call this diffusion the generalized Feller diffusion. In order to derive this continuous model, we first define a discrete model. For defining jointly the discrete model for all initial population sizes, we need as in [@4PW] to impose a non symmetric competition rule between the individuals, which we will describe in section 1 below. We do a suitable renormalization of the parameters of the discrete model in order to obtain in section 2 a large population limit of our model which is a generalized Feller diffusion. Section 3 is devoted to give a Ray Knight representation for such a generalized Feller diffusion. The proof of this representation uses tools from stochastic analysis, in particular the “excursion filtration", following an analogous proof of another generalized Ray Knight theorem in [@4NRW].
Discrete model with a general interaction {#4discrete}
=========================================
In this section we set up a discrete mass continuous time approximation of the generalized Feller diffusion. We consider a discrete model of population with interaction in which each individual, independently of the others, gives birth naturally at rate $\lambda$, dies naturally at rate $\mu$. Moreover, we suppose that each individual gives birth and dies because of interaction with others at rates which depend upon the current population size. Moreover, we exclude multiple births at any given time and we define the interaction rule through a function $f$ which satisfies hypothesis **A**.
In order to define our model jointly for all initial sizes, we need to introduce a non symmetric description of the effect of the interaction as in [@4BP] and [@4LPW], but here we allow the interaction to be favorable to some individuals.
The model
---------
We consider a continuous time ${\mathbb{Z}}_+$–valued population process $\{X^m_t,\ t\ge0\}$, which starts at time zero from $m$ ancestors who are arranged from left to right, and evolves in continuous time. The left/right order is passed on to their offsprings: the daughters are placed on the right of their mothers and if at a time $t$ the individual $i$ is located at the left of individual $j$, then all the offsprings of $i$ after time $t$ will be placed on the left of all the offsprings of $j$. Since we have excluded multiple births at any given time, this means that the forest of genealogical trees of the population is a planar forest of trees, where the ancestor of the population $X^1_t$ is placed on the far left, the ancestor of $X^2_t-X^1_t$ immediately on his right, etc... Moreover, we draw the genealogical trees in such a way that distinct branches never cross. This defines in a non–ambiguous way an order from left to right within the population alive at each time $t$. See Figure 1.
![Planar forest with five ancestors[]{data-label="4fig1"}](dessord){width="85.00000%"}
We decree that each individual feels the interaction with the others placed on his left but not with those on his right. Precisely, at any time $t$, the individual $i$ has an interaction death rate equal to $\left( f({\mathcal{L}}_i(t)+1)-f({\mathcal{L}}_i(t))\right)^-$ or an interaction birth rate equal to $\left( f({\mathcal{L}}_i(t)+1)-f({\mathcal{L}}_i(t))\right)^+$, where ${\mathcal{L}}_i(t)$ denotes the number of individuals alive at time $t$ who are located on the left of $i$ in the above planar picture. This means that the individual $i$ is under attack by the others located at his left if $f({\mathcal{L}}_i(t)+1)-f({\mathcal{L}}_i(t)) <0$ while the interaction improve his fertility if $f({\mathcal{L}}_i(t)+1)-f({\mathcal{L}}_i(t))>0$. Of course, conditionally upon ${\mathcal{L}}_i(\cdot)$, the occurence of a “competition death event” or an “interaction birth event" for individual $i$ is independent of the other birth/death events and of what happens to the other individuals. In order to simplify our formulas, we suppose moreover that the first individual in the left/right order has a birth rate equal to $\lambda+f^+(1)$ and a death rate equal to $\mu+f^-(1)$.
The resulting total interaction death and birth rates endured by the population $X^m_t$ at time $t$ is then $$\sum_{k=1}^{X^m_t}[(f(k)-f(k-1))^+ -(f(k)-f(k-1)^-]=\sum_{k=1}^{X^m_t}(f(k)-f(k-1))= f(X^m_t).$$
As a result, $\{X^m_t,\ t\ge0\}$ is a continuous time ${\mathbb{Z}}_+$–valued Markov process, which evolves as follows. $X^m_0=m$. If $X^m_t=0$, then $X^m_s=0$ for all $s\ge t$. While at state $k\ge1$, the process $$X^m_t\text{ jumps to }\begin{cases}k+1,&\text{ at rate $\lambda k+\sum_{\ell=1}^{k}(f(\ell)-f(\ell-1))^+$};\\
k-1,&\text{ at rate $\mu k+\sum_{k=1}^{k}(f(\ell)-f(\ell-1))^-$}.
\end{cases}$$
Coupling over ancestral population size
---------------------------------------
The above description specifies the joint evolution of all $\{X^m_t,\ t\ge0\}_{m\ge 1}$, or in other words of the two–parameter process $\{X^m_t,\ t\ge0, m\ge1\}$. In the case of a linear function $f$, for each fixed $t>0$, $\{X^m_t,\ m\ge1\}$ is an independent increments process. In the case of a nonlinear function $f$, we believe that for $t$ fixed $\{X^m_t,\ m\ge1\}$ is not a Markov chain. That is to say, the conditional law of $X^{n+1}_t$ given $X^n_t$ differs from its conditional law given $(X^1_t,X^2_t,\ldots,X^n_t)$. The intuitive reason for that is that the additional information carried by $(X^1_t,X^2_t,\ldots,X^{n-1}_t)$ gives us a clue as to the fertility or the level of competition that the progeny of the $n+1$st ancestor had to beneficit or to suffer from, between time 0 and time $t$.
However, $\{X^m_\cdot,\ m\ge1\}$ is a Markov chain with values in the space $D([0,\infty);{\mathbb{Z}}_+)$ of càdlàg functions from $[0,\infty)$ into ${\mathbb{Z}}_+$, which starts from 0 at $m=0$. Consequently, in order to describe the law of the whole process, that is of the two–parameter process $\{X^m_t,\ t\ge0, m\ge1\}$, it suffices to describe the conditional law of $X^n_\cdot$, given $\{X^{n-1}_\cdot\}$. We now describe that conditional law for arbitrary $1\le m<n$. Let $V^{m,n}_t:=X^n_t-X^m_t$, $t\ge0$. Conditionally upon $\{X^\ell_\cdot,\ \ell\le m\}$, and given that $X^m_t=x(t)$, $t\ge0$, $\{V^{m,n}_t,\ t\ge0\}$ is a ${\mathbb{Z}}_+$–valued time inhomogeneous Markov process starting from $V^{m,n}_0=n-m$, whose time–dependent infinitesimal generator $\{Q_{k,\ell}(t),\ k,\ell\in{\mathbb{Z}}_+\}$ is such that its off–diagonal terms are given by $$\begin{aligned}
Q_{0,\ell}(t)&=0,\quad \forall \ell\ge1,\quad \text{ and for any } k\ge1,\\
Q_{k,k+1}(t)&=\mu k+\sum_{\ell=1}^{k}(f(x(t)+\ell)-f(x(t)+\ell-1))^+, \\
Q_{k,k-1}(t)&=\lambda k+\sum_{\ell=1}^{k}(f(x(t)+\ell)-f(x(t)+\ell-1))^-,\\
Q_{k,\ell}(t)&=0,\quad\forall\ell\not\in\{k-1,k,k+1\}.
\end{aligned}$$
The reader can easily convince himself that this description of the conditional law of $\{X^n_t-X^m_t,\ t\ge0\}$, given $X^m_\cdot$ is prescribed by what we have said above, and that $\{X^m_\cdot,\ m\ge 1\}$ is indeed a Markov chain.
Note that if the function $f$ is increasing on \[0, $a$\], $a>0$ and decreasing on $[a,\infty)$, the interaction improves the rate of fertility in a population whose size is smaller than $a$ but for large size the interaction amounts to competition within the population. This is reasonable because when the population is large, the limitation of resources implies competition within the population. For a positive interaction (for moderate population sizes) one can realize that an increase in the population size allows a more efficient organization of the society, with specalisation among its members, thes resulting in better food production, health care, etc... We are mainly interested in the model with interaction defined with functions $f$ such that $\lim_{x\to\infty}f(x)=-\infty.$ Note also that we could have generalized our model to the case $f(0)\ge 0$. $f(0)>0$ would mean an immigration flux. The reader can easily check that results in section 2 would still be valid in this case. However in Proposition \[subcritic\] and in section 3.3 below, assumption $f(0)=0$ is crucial, since we need the population to get extinct in finite time $a.s.$.
The associated exploration process in the discrete model
--------------------------------------------------------
The just described reproduction dynamics gives rise to a forest $\mathcal{F}^m$ of $m$ trees of descent, drawn into the plane as sketched in Figure 2. Note also that, with the above described construction, the $\left( \mathcal{F}^m, m\ge 1\right)$, are coupled: the forest $\mathcal{F}^{m+1}$ has the same law as the forest $\mathcal{F}^m$ to which we add a new tree generated by an ancestor placed at the $(m+1)$st position. If the function $f$ tends to $-\infty$ and $m$ is large enough, the trees further to the right of the forest $\mathcal{F}^m$ have a tendency to stay smaller because of the competition : they are “under attack" from the trees to their left. From $\mathcal{F}^m$ we read off a continuous and piecewise linear ${\mathbb{R}}_+ $-valued path $H^{m} = \left( H^{m}_s \right)$ (called the exploration process of $\mathcal{F}^m$) which is described as follows.
Starting from the initial time $s=0$ the process $H^m$ rises at speed $p$ until it hits the top of the first ancestor branch (this is the leaf marked with $D$ in Figure 2). There it turns and goes downwards, now at speed $-p$, until arriving at the next branch point (which is $B$ in Figure 2). From there it goes upwards into the (yet unexplored) next branch, and proceeds in a similar fashion until being back at height 0, which means that the exploration of the leftmost tree is completed. Then explore the next tree, and so on. See Figure 2.
{width="85.00000%"}
\[4fig2\]
We define the local time $L^m_s(t)$ accumulated by the process $H^m$ at level $t$ up to time $s$ by: $$L^m_s(t)=\lim_{\epsilon\rightarrow 0}\frac{1}{\epsilon}\int_{0}^s 1_{t \leq H^m_r<t+\epsilon} dr.$$ The process $H^m$ is piecewise linear, continuous with derivative $\pm p$ : at any time $s\ge 0$, the rate of appearance of minima (giving rise to new branches) is equal $$p\mu+\left[f(\lfloor \frac{p}{2}L^m_s(H^m_s)\rfloor+1)-f(\lfloor \frac{p}{2}L^m_s(H^m_s)\rfloor)\right]^+ ,$$ and the rate of appearance of maxima (describing deaths of branches) is equal to $$p\lambda +\left[f(\lfloor \frac{p}{2}L^m_s(H^m_s)\rfloor+1)-f(\lfloor \frac{p}{2}L^m_s(H^m_s)\rfloor )\right]^-.$$ Let $S^m$ be the time needed in order to explore the forest $\mathcal{F}^m$. We have $$S^m=\inf\lbrace s>0; \frac{p}{2}L^m_s(0)\ge m \rbrace .$$ Under the assumption that $S^m<\infty$ $a.s.$ for all $m\ge 1$, we have the following discrete Ray Knight representation (see Figure \[4fig4\]). $$\begin{aligned}
\left( X^m_t, t\ge 0, m\ge 1\right)\equiv\left( \frac{p}{2}L^m_{S^m}(t),t\ge 0, m\ge 1\right).\end{aligned}$$
{width="85.00000%"}
\[4fig4\]
Renormalized discrete model {#4secRenorm}
---------------------------
Now we proceed to a renormalization of this model. For $x\in {\mathbb{R}}_+$ and $N\in{\mathbb{N}}$, we choose $m=\lfloor Nx\rfloor$, $\mu=2N$, $\lambda=2N$, we multiply $f$ by N and divide by $N$ the argument of the function $f$. We affect to each individual in the population a mass equal to $1/N$. Then the total mass process $Z^{N,x}$, which starts from $\frac{\lfloor Nx\rfloor}{N}$ at time $t=0$, is a Markov process whose evolution can be described as follows.\
$Z^{N,x}$ jumps from $\frac{k}{N}$ to $\begin{cases} \frac{k+1}{N} \text{ at rate } 2Nk+N\sum_{i=1}^k\left(f(\frac{i}{N})-f(\frac{i-1}{N})\right)^+\\
\frac{k+1}{N} \text{at rate } 2Nk+N\sum_{i=1}^k\left((f(\frac{i}{N})-f(\frac{i-1}{N})\right)^-,
\end{cases}$\
Clearly there exist two mutually independent standard Poisson processes $P_1$ and $P_2$ such that $$\begin{aligned}
Z^{N,x}_t&=\frac{\lfloor Nx\rfloor}{N}+\frac{1}{N}P_1\left(\int_0^t \left(2N^2 Z^{N,x}_r+N\sum_{i=1}^{NZ^{N,x}_r}(f(\frac{i}{N})-f(\frac{i-1}{N}))^+\right)dr\right)\\
&-\frac{1}{N}P_2\left(\int_0^t\left(2N^2 Z^{N,x}_r+N\sum_{i=1}^{NZ^{N,x}_r}(f(\frac{i}{N})-f(\frac{i-1}{N}))^-\right)dr\right).\end{aligned}$$ Consequently there exists a local martingale $M^{N,x}$ such that $$\label{4fAN}
Z^{N,x}_t=\frac{\lfloor Nx\rfloor}{N}+\int_0^t f(Z^{N,x}_r) dr+M^{N,x}_t.$$ Since $M^{N,x}$ is a purely discontinous local martingale, its quadratic variation $[M^{N,x}]$ is given by the sum of the squares of its jumps, i.e. $$\begin{aligned}
\label{4crodroit}[M^{N,x}]_t&=\frac{1}{N^2}\left[P_1\left(\int_0^t \left(2N^2 Z^{N,x}_r+N\sum_{i=1}^{NZ^{N,x}_r}(f(\frac{i}{N})-f(\frac{i-1}{N}))^+\right)dr\right)\right. \nonumber\\
&\left.+ P_2\left(\int_0^t\left(2N^2 Z^{N,x}_r+N\sum_{i=1}^{NZ^{N,x}_r}(f(\frac{i}{N})-f(\frac{i-1}{N}))^-\right)dr\right)\right]. \end{aligned}$$
We deduce from that the conditional quadratic variation $\langle M^{N,x}\rangle$ of $M^{N,x}$ is given by $$\label{4croMNx}
\langle M^{N,x}\rangle_t=\int_0^t \left\lbrace 4Z^{N,x}_r+\frac{1}{N}\vert\vert f\vert\vert_{N,0,Z^{N,x}_r}\right\rbrace dr,$$ where for any $z=\frac{k}{N}$, $z'=\frac{k'}{N}$, $k\in{\mathbb{Z}}_+$ such that $k\le k'$, $$\vert\vert f\vert\vert_{N,z,z'}=\sum_{i=k+1}^{k'}\vert f(\frac{i}{N})-f(\frac{i-1}{N})\vert .$$ Now we precise the law of the pair $\left(Z^{N,x},Z^{N,y}\right) $, for any $x,y\in\mathbb{R}_+$ such that $x\le y$. Let $V^{N,x,y}:=Z^{N,y}-Z^{N,x}$, and consider the pair of process $\left(Z^{N,x},V^{N,x,y}\right) $, which starts from $\left(\frac{\lfloor Nx\rfloor}{N} ,\frac{\lfloor Ny\rfloor-\lfloor Nx \rfloor}{N} \right)$ at time $t=0$, and whose dynamic is described by: $\left( Z^{N,x},V^{N,x,y}\right) $ jumps\
from $(\frac{i}{N},\frac{j}{N})$ to $\begin{cases} (\frac{i+1}{N},\frac{j}{N})\text{ at rate } 2Ni +\sum_{k=1}^{i}(f(\frac{k}{N})-f(\frac{k-1}{N})^+\\
(\frac{i-1}{N},\frac{j}{N}) \text{ at rate } 2Ni+\sum_{k=1}^{i}(f(\frac{k}{N})-f(\frac{k-1}{N}))^-\\
(\frac{i}{N},\frac{j+1}{N}) \text{ at rate } 2Nj +\sum_{k=1}^{j}(f(\frac{i+k}{N})-f(\frac{i+k-1}{N})^+\\
(\frac{i}{N},\frac{j-1}{N}) \text{ at rate } 2Nj+\sum_{k=1}^{j}(f(\frac{i+k}{N})-f(\frac{i+k-1}{N})^-
\end{cases}$.\
The process $V^{N,x,y}$ can be expressed as follows. $$\begin{aligned}
\label{4fVN}
V^{N,x,y}_t=\frac{\lfloor Ny\rfloor -\lfloor Nx\rfloor}{N}+\int_0^t \left[f(Z^{N,x}_r+V^{N,x,y}_r)-f(Z^{N,x}_r)\right] dr+M^{N,x,y}_t,\end{aligned}$$ where $M^{N,x,y}$ is a local martingale whose conditional quadratic variation $\langle M^{N,x,y}\rangle$ is given by $$\begin{aligned}
\label{4croMNxy}
\langle M^{N,x,y}\rangle_t =\int_0^t \left\lbrace 4V^{N,x,y}_r+\frac{1}{N}\vert\vert f\vert\vert_{N,Z^{N,x}_r,V^{N,x,y}+Z^{N,x}_r}\right\rbrace dr.\end{aligned}$$ Since $Z^{N,x}$ and $V^{N,x,y}$ never jump at the same time, $$\begin{aligned}
[M^{N,x}, M^{N,x,y}]=0, \text{ hence } \langle M^{N,x},M^{N,x,y}\rangle =0,\label{4croch}\end{aligned}$$ which implies that the martingales $M^{N,x}$ and $M^{N,x,y}$ are orthogonal.
Consequently, $Z^{N,x}+V^{N,x,y}$ solves the SDE $$\begin{aligned}
Z^{N,x}_t+V^{N,x,y}_t=&\frac{\lfloor Ny\rfloor}{N}+\int_0^t f(Z^{N,x}_r+V^{N,x,y}_r)dr+\tilde{M}^{N,x,y}_t.\end{aligned}$$ where $\tilde{M}^{N,x,y}$ is a local martingale with $\langle \tilde{M}^{N,x,y}\rangle$ given by $$\begin{aligned}
\langle \tilde{M}^{N,x,y}\rangle_t &=\langle M^{N,x}\rangle_t+\langle M^{N,x,y}\rangle_t = \langle M^{N,x+y}\rangle_t,\quad \forall t\ge 0.\end{aligned}$$ We then deduce that for any $x,y\in{\mathbb{R}}_+$ such $x\le y, $ $$Z^{N,x}+V^{N,x,y}\stackrel{(d)}{=}Z^{N,y}.$$ In fact, we have that $$\begin{aligned}
V^{N,x,y}_t&=\frac{\lfloor Ny\rfloor -\lfloor Nx\rfloor}{N}+\frac{1}{N}P^1\left(N\int_0^t\sum_{k=1}^{NV^{N,x,y}_s}\left(f(Z^{N,x}_s+\frac{k}{N})-f(Z^{N,x}_s+\frac{k}{N})\right)^+ds\right)\\
&-\frac{1}{N}P^2\left(N\int_0^t\sum_{k=1}^{NV^{N,x,y}_s}\left(f(Z^{N,x}_s+\frac{k}{N})-f(Z^{N,x}_s+\frac{k}{N})\right)^-ds\right)\\
&+\frac{1}{N}P^3\left(2N^2\int_0^tV^{N,x,y}_sds\right)-\frac{1}{N}P^4\left(2N^2\int_0^tV^{N,x,y}_sds\right),\end{aligned}$$ where $P^1$, $P^2$, $P^3$ and $P^4$ are mutually independent standard Poisson processes which are all independent of $Z^{N,x}_.$. It follows that conditionally upon $\left\lbrace Z^{N,x'},x'\le x\right\rbrace$, $M^{N,x,y}$ is a local martingale.
We can also proceed to the renormalisation of the exploration process to provide a discrete Ray Knight representation of the process $Z^{N,x}$. We choose the slope $p=2N$ and we denote by $H^N$ the exploration processes associated to the forest $\mathcal{F}^{N,x}$ of $\lfloor Nx\rfloor$ trees. Let $L^N_s(t)$ be the local time of the process $H^N$ at level $t$ up to time $s$. At any time $s$, the rate of minima of $H^N$ is equal to $$4N^2+N\left[f(\frac{\lfloor NL^N_s(H^N_s)\rfloor}{N}+1/N)-f(\frac{\lfloor NL^N_s(H^N_s)\rfloor}{N})\right]^+,$$ and the rate of maxima is equal to $$4N^2 +N\left[f(\frac{\lfloor NL^N_s(H^N_s)\rfloor}{N}+1/N)-f(\frac{ \lfloor NL^N_s(H^N_s)\rfloor }{N})\right]^-.$$ Let $S^{N,x}$ be the time to explore the forest $\mathcal{F}^{N,x}$. We have that $$S^{N,x}=\inf\lbrace s>0; L^N_s(0)\ge \frac{\lfloor Nx\rfloor}{N}\rbrace .$$ Under the assumption that $S^{N,x}<\infty$ $a.s.$ for all $x>0$, the discrete Ray Knight representation with the renormalization becomes: $$\begin{aligned}
\left( Z^{N,x}_t, t\ge 0, x\ge 0\right)\equiv\left( L^N_{S^{N,x}}(t),t\ge 0, x\ge 0\right).\end{aligned}$$ One could probably deduce from this discrete approximation the Ray Knight representation of the general Feller diffusion by a limiting argument, as it is done in [@4BPS] in the linear case and in [@4LPW] in the quadratic case. But in this work we use stochastic analysis tools for proving our extended Ray Knight theorem.
Continous model with a general competition {#4contm}
------------------------------------------
Given a space-time white noise $W(ds,du)$, we now define an ${\mathbb{R}}_+$–valued two–parameter stochastic process $\lbrace Z^{x}_t,\ t\ge 0,$ $x\ge 0\rbrace$ which is such that for each fixed $x>0$, $\left\lbrace Z^{x}_t,t\ge 0\right\rbrace $ is a continuous process, solution of the SDE . We have that for any $0<x<y$, $\{V^{x,y}_t:=Z^{y}_t-Z^x_t,\ t\ge0\}$ solves the SDE $$\begin{aligned}
\label{4fAV}
V^{x,y}_t= y-x+\int_0^t \left[ f(Z^x_s+V^{x,y}_s) -f(Z^x_s)\right] ds+2\int_0^t\int_{Z^x_s}^{Z^x_s+V^{x,y}_s}W(ds,du)\end{aligned}$$ The process $V^{x,y}$ is nonnegative almost surely. We have that $\int_0^t\int_{0}^{Z^x_s}W(ds,du)$ and $\int_0^t\int_{Z^x_s}^{Z^x_s+V^{x,y}_s}W(ds,du)$ are orthogonal since $$[0,Z^x_s]\cap(Z^x_s, Z^x_s+V^{x,y}_s]=\emptyset,$$ and $$\int_0^t\int_{0}^{Z^x_s}W(ds,du)+\int_0^t\int_{Z^x_s}^{Z^x_s+V^{x,y}_s}W(ds,du)=\int_0^t\int_{0}^{Z^x_s+V^{x,y}_s}W(ds,du)\text{ a.s.}$$ This implies that $Z^{y}=Z^x+V^{x,y}$ a.s. It follows that, for each $t\ge 0$, the process $\{Z^x_t, x\ge0\}$ is almost surely non decreasing and for $0\le x<y$, the conditional law of $Z^y_\cdot$, given $\{Z^{x'}_t, x'\le x, t\ge 0\}$ and $Z^x_t=z(t)$, $t\ge0$, is the law of the sum of $z$ plus the solution of with $Z^x_t$ replaced by $z(t)$. Note that when $Z^x_.$ is replaced by a deterministic trajectory $z$, the solution of is independent of $\{Z^{x'}_., x'< x \}$ . Hence the process $\{Z^x_\cdot,\ x\ge0\}$ is a Markov process with values in $C([0,\infty),{\mathbb{R}}_+)$, the space of continuous functions from $[0,\infty)$ into ${\mathbb{R}}_+$, starting from $0$ at $x=0$. In the case $f$ linear, the increments of the mapping $x\to Z^x_t$ are independent, for each $t>0$.
For $x\ge 0$, define $T^x_0$ the extinction time of the process $Z^x$ by: $$T^x_0=\inf \left\lbrace t>0; Z^x_t=0\right\rbrace.$$ For any $x\ge 0$, we call the process $Z^x$ subcritical if it goes extinct almost surely in finite time i.e if $T^x_0$ is finite almost surely. The assumption **A** implies that $\frac{f(x)}{x}$ is bounded. Let us introduce the notation $$\label{lambda}
\Lambda(f):=\int_1^{\infty}\exp\left(-\frac{1}{2}\int_{1}^u \frac{f(r)}{r}dr\right)du.$$ We have the following Proposition.
\[subcritic\] Suppose that $f$ satisfies hypothesis **A**. For any $x\ge 0$, $Z^x$ is subcritical if and only if $\Lambda(f)=\infty$. In particular we have:
- A sufficient condition for ${\mathbb{P}}\left(T^x_0<\infty\right)=1$ is: there exists $z_0\ge 1$ such that $f(z)\le 2$, $\forall z\ge z_0$,
- A sufficient condition for ${\mathbb{P}}\left(T^x_0=\infty\right)>0$ is: there exists $z_0>1$ and $\delta>0$ such that $f(z)\ge 2+\delta$, $\forall$ $z\ge z_0$.
Let $S\in C^2({\mathbb{R}}_+)$ and $0\le a< x< b$. By Itô’s formula applied to the process $Z^x$ and the function $S$, we have that for any $t\ge0$, $$\begin{aligned}
\label{itoz}
S(Z^x_t)=S(x)+\int_0^t\left(S'(Z^x_s)f(Z^x_s)+2S''(Z^x_s)Z^x_s\right)ds+2\int_0^tS'(Z^x_s)\sqrt{Z^x_s}dW_s.\end{aligned}$$ Let us denote by $\mathcal{A}$ the generator of $Z^x$. If we can find a strictly increasing function $S$ on the interval $[a,b]$ such that $\mathcal{A} S\equiv 0$, then the drift term in vanishes and so $Z^x$ will be just a time changed Brownian motion in \[$S(a)$, $S(b)$\]. Such a function $S$ is called a scale function of the diffusion $Z^x$. We choose as scale function: for any $z\ge 0$, $$S(z)=\int_1^z\exp\left(-\frac{1}{2}\int_1^u \frac{f(r)}{r}dr\right)du.$$ Let us denote by $T^x_y$ the random time at which $Z^x$ hits $y$ for the first time. We have for any $0\le a< x< b$$$\begin{aligned}
{\mathbb{P}}(T^x_a<T^x_b)=\frac{S(b)-S(x)}{S(b)-S(a)}, \text{ and } {\mathbb{P}}(T^x_a<\infty)=\lim_{b\to \infty}{\mathbb{P}}(T^x_a<T^x_b).\end{aligned}$$ If the function $S(z)$ tends to infinity as $z$ goes to infinity, then ${\mathbb{P}}(T^x_a<\infty)=1$. Otherwise $0<{\mathbb{P}}(T^x_a<\infty)<1$. From this we deduce that $Z^x$ goes extinct almost surely in finite time if and only if $\lim_{z\to \infty}S(z)=\infty$ i.e. if and only if $\Lambda(f) =\infty$. The rest of the Proposition is immediate.
Convergence as $N\to\infty$
===========================
The aim of this section is to prove the convergence in law as $N\to\infty$ of the two–parameter process $\{Z^{N,x}_t,\ t\ge0, x\ge0\}$ defined in section \[4secRenorm\] towards the process $\{Z^{x}_t,\ t\ge0, x\ge0\}$ defined in section \[4contm\]. We need to make precise the topology for which this convergence will hold. We note that the process $Z^{N,x}_t$ (resp. $Z^x_t$) is a Markov processes indexed by $x$, with values in the space of càdlàg (resp. continuous) functions of $t$ $D(([0,\infty);{\mathbb{R}}_+)$ (resp. $C(([0,\infty);{\mathbb{R}}_+)$). So it will be natural to consider a topology of functions of $x$, with values in functions of $t$.
For each fixed $x$, the process $t\to Z^{N,x}_t$ is càdlàg, constant between its jumps, with jumps of size $\pm N^{-1}$, while the limit process $t\to Z^{x}_t$ is continuous. On the other hand, both $Z^{N,x}_t$ and $Z^{x}_t$ are discontinuous as functions of $x$. $x\to Z^x_\cdot$ has countably many jumps on any compact interval, but the mapping $x\to\{Z^x_t,\ t\ge\epsilon\}$, where $\epsilon>0$ is arbitrary, has finitely many jumps on any compact interval, and it is constant between its jumps. Recall that $D([0,\infty);{\mathbb{R}}_+)$ equipped with the distance $d^0_{\infty}$ defined by $(16.4)$ in [@4BI] is separable and complete, see Theorem $16.3$ in [@4BI]. We have the following statement
\[4thConv\] Suppose that the Hypothesis $\bf{A}$ is satisfied. Then as $N\to\infty$, $$\{Z^{N,x}_t,\ t\ge0, x\ge0\}\Rightarrow\{Z^{x}_t,\ t\ge0, x\ge0\}$$ in $D([0,\infty);D([0,\infty);{\mathbb{R}}_+))$, equipped with the Skohorod topology of the space of càdlàg functions of $x$, with values in the Polish space $D([0,\infty);{\mathbb{R}}_+)$ equipped with the metric $d^0_{\infty}$.
Proof of the theorem {#proof-of-the-theorem .unnumbered}
--------------------
To prove the theorem, we first show that for fixed $x\ge 0$ the sequence $\left\lbrace Z^{N,x},~N\geq 0\right\rbrace$ is tight in $D([0,\infty);{\mathbb{R}}_+)$.
Tightness of $Z^{N,x}$
----------------------
For this end, we first establish a few lemmas.
\[4lm1\] For all $T>0$, $x\ge 0$, there exist a constant $C_0>0$ such that for all $N\ge 1$, $$\begin{aligned}
\sup_{0\leq t\leq T}{\mathbb{E}}\left( Z^{N,x}_t\right) \le C_0.\end{aligned}$$ Moreover, for all $t\ge 0$, $N\ge1$, $$\begin{aligned}
{\mathbb{E}}\left(-\int_0^t f(Z^{N,x}_r)dr\right) \le x.\end{aligned}$$
Let $\left(\tau_n,n\ge 0\right)$ be a sequence of stopping times such that $\tau_n$ tends to infinity as $n$ goes to infinity and for any $n$, $\left(M^{N,x}_{t\wedge \tau_n},t\ge 0\right)$ is a martingale and $Z^{N,x}_{t\wedge\tau_n}\le n$. Taking the expectation on both sides of equation at time $t\wedge\tau_n$, we obtain $$\begin{aligned}
\label{4equ1}
{\mathbb{E}}\left(Z^{N,x}_{t\wedge \tau_n}\right)=\frac{\lfloor Nx\rfloor}{N}+{\mathbb{E}}\left(\int_0^{t\wedge \tau_n} f(Z^{N,x}_r)dr\right).\end{aligned}$$ It follows from the hypothesis **A** on $f$ that $$\begin{aligned}
{\mathbb{E}}\left(Z^{N,x}_{t\wedge \tau_n}\right) \le \frac{\lfloor Nx\rfloor}{N}+\beta \int_0^t{\mathbb{E}}(Z^{N,x}_{r\wedge \tau_n})dr\end{aligned}$$ From Gronwall and Fatou Lemmas, we deduce that there exists a constant $C_0>0$ which depends only upon $x$ and $T$ such that $$\sup_{N\ge 1}\sup_{0\le t\le T}{\mathbb{E}}\left(Z^{N,x}_t\right)\le C_0.$$ From , we deduce that $$-{\mathbb{E}}\left(\int_0^{t\wedge \tau_n} f(Z^{N,x}_r)dr\right)\le \frac{\lfloor Nx\rfloor}{N} .$$ Since $-f(Z^{N,x}_r )\ge -\beta Z^{N,x}_r$ , the second statement follows using Fatou’s Lemma and the first statement.
We now have the following lemma.
\[4lemcro\] For all $T>0$, $x\ge 0$, there exists a constant $C_1>0$ such that : $$\sup_{N\geq 1}{\mathbb{E}}\left(\langle M^{N,x}\rangle_T\right)\le C_1.$$
For any $N\ge 1$ and $k,k' \in{\mathbb{Z}}_+$ such that $k\le k'$, we set $z=\frac{k}{N}$ and $z'=\frac{k'}{N}$. We deduce from hypothesis **A** on $f$ that $$\begin{aligned}
\vert \vert f\vert\vert_{N,z,z'}&=\sum_{i=k+1}^{k'}\left\lbrace\left( f(\frac{i}{N})-f(\frac{i-1}{N})\right)^++\left( f(\frac{i}{N})-f(\frac{i-1}{N})\right)^-\right\rbrace\\
&=\sum_{i=k+1}^{k'}\left\lbrace 2\left( f(\frac{i}{N})-f(\frac{i-1}{N})\right)^+-\left( f(\frac{i}{N})-f(\frac{i-1}{N})\right)\right\rbrace.\end{aligned}$$ Consequentely, $$\begin{aligned}
\label{4bornvaria}
\vert \vert f\vert\vert_{N,z,z'} \le 2\beta(z'-z)+f(z)-f(z').\end{aligned}$$ We deduce from , and Lemma \[4lm1\] that $$\begin{aligned}
{\mathbb{E}}\left(\langle M^{N,x}\rangle_T\right)& \le \int_0^T \left\lbrace (4+\frac{2\beta}{N}){\mathbb{E}}(Z^{N,x}_r)- \frac{1}{N} {\mathbb{E}}\left(f(Z^{N,x}_r\right)\right\rbrace dr\\
&\le (4+\frac{2\beta}{N})C_0T+\frac{x}{N}.\end{aligned}$$ Hence the lemma. It follows from this that $M^{N,x}$ is in fact a square integrable martingale. We also have
\[4lm2\] For all $T>0$, $x\ge 0$, there exist two constants $C_2,C_3>0$ such that : $$\begin{aligned}
\sup_{N\geq 1}\sup_{0\leq t\leq T}{\mathbb{E}}\left( Z^{N,x}_t\right)^2 &\le C_2,\\
\sup_{N\geq 1}\sup_{0\leq t\leq T}{\mathbb{E}}\left( -\int_0^t Z^{N,x}_r f(Z^{N,x}_r)dr \right)& \leq C_3.\end{aligned}$$
We deduce from and Itô’s formula that $$\begin{aligned}
\label{4carre}
\left(Z^{N,x}_t\right)^2 = \left(\frac{ \lfloor Nx\rfloor }{N}\right)^2+2\int_0^t Z^{N,x}_r f(Z^{N,x}_r)dr+\langle M^{N,x}\rangle_t +M^{N,x,(2)}_t,
$$ where $M^{N,x,(2)}$ is a local martingale. Let $\left( \sigma_n,n\ge 1\right)$ be a sequence of stopping times such that $\lim_{n\to\infty}\sigma_n=+\infty$ and for each $n\ge 1$, $\left( M^{N,x,(2)}_{t\wedge\sigma_n}, t\ge 0\right)$ is a martingale. Taking the expectation on the both sides of at time $t\wedge \sigma_n$ and using hypothesis **A**, Lemma \[4lemcro\], Gronwall and Fatou lemmas we obtain that for all $T>0$, there exists a constant $C_2>0$ such that : $$\sup_{N\geq 1}\sup_{0\leq t\leq T}{\mathbb{E}}\left( Z^{N,x}_t\right)^2 dr \leq C_2.$$ We also have that $$\begin{aligned}
2{\mathbb{E}}\left(-\int_0^{t\wedge\sigma_n} Z^{N,x}_r f(Z^{N,x}_r)dr \right) \le \left(\frac{ \lfloor Nx\rfloor }{N}\right)^2+ C_1\end{aligned}$$ From Hypothesis $\textbf{A}$, we have $- Z^{N,x}_r f(Z^{N,x}_r)\ge - \beta(Z^{N,x}_r)^2$. The result now follows from Fatou’s Lemma.
We want to check tightness of the sequence $\left\lbrace Z^{N,x},~N\geq 0\right\rbrace $ using Aldous’ criterion. Let $\left\lbrace \tau_N,~N\geq 1\right\rbrace$ be a sequence of stopping time in $[0,T]$. We deduce from Lemma \[4lm2\]
\[4prop2\] For any $T>0$ and $\eta$, $\epsilon >0$, there exists $\delta>0$ such that $$\sup_{N\ge 1}\sup_{0\le \theta\le \delta}{\mathbb{P}}\left(\left\vert \int_{\tau_N}^{(\tau_N+\theta)\wedge T} f(Z^{N,x}_r)dr\right\vert \ge \eta \right)\le \epsilon .$$
Let $c$ be a non negative constant. We have $$\begin{aligned}
\left\vert \int_{\tau_N}^{(\tau_N+\theta)\wedge T}f(Z^{N,x}_r)dr\right\vert &\le \sup_{0\le r \le c }\vert f(r)\vert\delta +\int_{\tau_N}^{\tau_N+\theta}{\bf1}_{ \lbrace Z^{N,x}_r>c\rbrace }\vert f(Z^{N,x}_r)\vert dr\end{aligned}$$ But $$\begin{aligned}
\int_{\tau_N}^{\tau_N+\theta}{\bf1}_{ \lbrace Z^{N,x}_r>c\rbrace }\vert f(Z^{N,x}_r)\vert dr & \le c^{-1}\int_0^T Z^{N,x}_r \left(f^+(Z^{N,x}_r)+f^-(Z^{N,x}_r)\right)dr\\
&\le c^{-1}\int_0^T\left(2Z^{N,x}_r f^+(Z^{N,x}_r)-Z^{N,x}_rf(Z^{N,x}_r) \right)dr \\
&\le c^{-1}\int_0^T\left(2\beta (Z^{N,x}_r)^2-Z^{N,x}_rf(Z^{N,x}_r) \right)dr.\end{aligned}$$ From this and Lemma \[4lm2\], we deduce that $\forall$ $N\ge 1$ $$\begin{aligned}
\sup_{0\le \theta\le \delta}{\mathbb{P}}\left(\Big\vert \int_{\tau_N}^{(\tau_N+\theta)\wedge T} f(Z^{N,x}_r)dr\Big\vert\ge \eta \right) &\le \eta^{-1}{\mathbb{E}}\left(\left\vert \int_{\tau_N}^{(\tau_N+\theta)\wedge T}
f(Z^{N,x}_r)dr\right\vert \right)\\
& \le \sup_{0\le r \le c}\frac{\vert f(z)\vert\delta}{ \eta }+\frac{A}{c\eta},\end{aligned}$$ with $A=2\beta C_2T+C_3$. The result follows by choosing $c=2A/\epsilon\eta$, and then $\delta=\epsilon\eta/2\sup_{0\le r\le c}\vert f(z)\vert $. From Proposition \[4prop2\], the Lebesgue integral term in the right hand side of satisfies Aldou’s condition $[A]$, see [@AD]. The same Proposition, Lemma \[4lm1\], and imply that $<M^{N,x}>$ satisfies the same condition, hence so does $M^{N,x}$, according to Rebolledo’s theorem, see [@4JM]. Since all jumps are of size $\frac{1}{N}$, tightness follows. We have proved
\[4tightxfixed\] For any fixed $x\ge 0$, the sequence of processes $\left\lbrace Z^{N,x},~N\ge 1\right\rbrace $ is tight in $D\left([0,\infty);{\mathbb{R}}_+ \right) $.
We deduce from Proposition \[4tightxfixed\] the following Corollary.
\[4cortight\] For any $0\le x<y$ the sequence of processes $\left\lbrace V^{N,x,y},~N\ge 1\right\rbrace$ is tight in $D\left([0,\infty);{\mathbb{R}}_+ \right) $
For any $x$ fixed the process $Z^{N,x}$ has jumps equal to $\pm\frac{1}{N}$ which tends to zero as $N\to\infty$. It follows from that and equation that any weak limit of a converging subsequence of $Z^{N,x}$ is continuous and is the unique weak solution of equation . We deduce that for any $x,y\ge 0$, the sequence $\left\lbrace Z^{N,y}-Z^{N,x},N\ge 1\right\rbrace$ is tight since $\left\lbrace Z^{N,x},N\ge 1\right\rbrace$ and$\left\lbrace Z^{N,y},N\ge 1\right\rbrace$ are tight and both have a continuous limit as $N\to\infty$.
Proof of Theorem \[4thConv\]
----------------------------
From Theorem 13.5 in [@4BI], Theorem \[4thConv\] follows from the two next Propositions
\[4findimconv\] For any $n\in{\mathbb{N}}$, $0\le x_1<x_2<\cdots <x_n$, $$\left( Z^{N,x_1},Z^{N,x_2},\cdots,Z^{N,x_n}\right)\Rightarrow \left( Z^{x_1},Z^{x_2},\cdots,Z^{x_n}\right)$$ as $N\to\infty$, for the topology of locally uniform convergence in $t$.
We prove the statement in the case $n=2$ only. The general statement can be proved in a very similar way. For $0\le x_1<x_2$, we consider the process $\left(Z^{N,x_1},V^{N,x_1,x_2}\right)$, using the notations from section \[4discrete\]. The argument preceding the statement of Proposition \[4tightxfixed\] implies that the sequences of martingales $M^{N,x_1}$ and $M^{N,x_1,x_2}$ are tight. Hence\
$\left( Z^{N,x_1},V^{N,x_1,x_2}, M^{N,x_1},M^{N,x_1,x_2}\right)$ is tight. Thanks to , , , and , any converging subsequence of\
$\left\lbrace Z^{N,x_1},V^{N,x_1,x_2}, M^{N,x_1},M^{N,x_1,x_2}, N\ge 1\right\rbrace $ has a weak limit\
$\left(Z^{x_1},V^{x_1,x_2}, M^{x_1},M^{x_1,x_2}\right)$ which satisfies $$\begin{aligned}
Z^{x_1}_t&=x_1+\int_0^tf(Z^{x_1}_s)ds+M^{x_1}_t\\
V^{x_1,x_2}_t & =x_2-x_1+\int_0^tf\left[f(Z^{x_1}_s+V^{x_1,x_2}_s)-f(Z^{x_1}_s)\right]ds+M^{x_1,x_2}_t,
\end{aligned}$$ where the continuous martingales $M^{x_1}$ and $M^{x_1,x_2}$ satisfy $$\begin{aligned}
\langle M^x\rangle_t=4\int_0^t Z^{x_1}_sds, ~\langle M^{x_1,x_2}\rangle_t=4\int_0^tV^{x_1,x_2}_sds, ~
\langle M^{x_1},M^{x_1,x_2}\rangle_t =0.\end{aligned}$$ This implies that the pair $\left(Z^{x_1},V^{x_1,x_2}\right)$ is a weak solution of the system of SDEs and , driven by the same space-time white noise. The result follows from the uniqueness of the system, see [@4DL].
\[4tightD\] There exists a constant $C$, which depends only upon $\theta$ and $T$, such that for any $0\le x<y<z$, which are such that $y-x\le1$, $z-y\le 1$, $${\mathbb{E}}\left[\sup_{0\le t\le T}\vert Z^{N,y}_t-Z^{N,x}_t\vert^2\times
\sup_{0\le t\le T}\vert Z^{N,z}_t-Z^{N,y}_t\vert^2\right]\le C|z-x|^2.$$
We first prove the
\[4lmVN\] For any $0\le x<y$, we have $$\sup_{0\le t\le T}{\mathbb{E}}\left(Z^{N,y}_t-Z^{N,x}_t\right)=\sup_{0\le t\le T}{\mathbb{E}}(V^{N,x,y}_t)\le \left(\frac{\lfloor Ny \rfloor}{N}-\frac{\lfloor Nx \rfloor }{N}\right)e^{\beta T},$$
Let $\left( \sigma_n,n\ge 0\right)$ be a sequence of stopping times such that $\lim_{n\to\infty}\sigma_n=+\infty$ and $\left( M^{N,x,y}_{t\wedge\sigma_n}\right)$ is a martingale. Taking the expectation on the both sides of at time $t\wedge \sigma_n$ we obtain that $$\begin{aligned}
\label{4vb}
{\mathbb{E}}(V^{N,x,y}_{t\wedge \sigma_n})\le\left(\frac{\lfloor Ny \rfloor}{N}-\frac{\lfloor Nx \rfloor }{N}\right)+\beta\int_0^t {\mathbb{E}}(V^{N,x,y}_{r\wedge \sigma_n})dr \end{aligned}$$ Using Gronwall and Fatou lemmas, we obtain that $$\sup_{0\le t\le T}{\mathbb{E}}(V^{N,x,y}_t)\le\left(\frac{\lfloor Ny \rfloor}{N}-\frac{\lfloor Nx \rfloor }{N}\right)e^{\beta T}.$$ [Proof of Proposition \[4tightD\]]{} Using equation , a stopping time argument as above, Lemma \[4lmVN\] and Fatou’s lemma, where we take advantage of the inequality $f(Z^{N,x}_r)-f(Z^{N,x}_r+V^{N,x,y}_r)\ge -\beta V^{N,x,y}_r$, we deduce that $$\begin{aligned}
\label{4ine2}
{\mathbb{E}}\left(\int_0^t \left[f(Z^{N,x}_r)-f(Z^{N,x}_r+V^{N,x,y}_r) \right]dr\right) \le \frac{\lfloor Ny \rfloor}{N}-\frac{\lfloor Ny \rfloor }{N}.\end{aligned}$$ We now deduce from , Lemma \[4lmVN\], inequalities and that for each $t>0$, there exists a constant $C(t)>0$ such that $$\begin{aligned}
\label{4corbor}
{\mathbb{E}}\left( \langle M^{N,x,y}\rangle_t \right) \le C(t)\left(\frac{\lfloor Ny \rfloor}{N}-\frac{\lfloor Ny \rfloor }{N}\right). \end{aligned}$$ This implies that $M^{N,x,y}$ is in fact a square integrable martingale. For any $0\le x<y<z$, we have $Z^{N,z}_t-Z^{N,y}_t=V^{N,y,z}_t$ and $Z^{N,y}_t-Z^{N,x}_t=V^{N,x,y}_t$ for any $t\ge0$. On the other hand we deduce from and the hypothesis $\bf{A}$ $$\begin{aligned}
\sup_{0\le t\le T}(V^{N,x,y}_t )^2 & \le 3\left(\frac{\lfloor Ny \rfloor}{N}-\frac{\lfloor Nx \rfloor }{N}\right)^2+3\beta^2T\int_0^T \sup_{0\le s\le r}(V^{N,x,y}_s)^2dr \\
&+3\sup_{0\le t\le T}\left(M^{N,x,y}_t\right)^2,\end{aligned}$$ $$\begin{aligned}
\sup_{0\le t\le T}(V^{N,y,z}_t )^2 & \le 3\left(\frac{\lfloor Nz \rfloor}{N}-\frac{\lfloor Ny \rfloor }{N}\right)^2+3\beta^2T\int_0^t \sup_{0\le s\le r}(V^{N,y,z}_s)^2dr \\
&+3\sup_{0\le t\le T}\left(M^{N,y,z}_t\right)^2.\end{aligned}$$ Now let $\mathcal{G}^{x,y}:=\sigma\left( Z^{N,x}_t, Z^{N,y}_t,t\ge 0\right)$ be the filtration generated by $Z^{N,x}$ and $Z^{N,y}$. It is clear that for any $t$, $V^{N,x,y}_t$ is measurable with respect to $\mathcal{G}^{x,y}$. We then have $$\begin{aligned}
{\mathbb{E}}\left[\sup_{0\le t\le T}\vert V^{N,x,y}_t\vert^2\times \sup_{0\le t\le T}\vert V^{N,y,z}_t\vert^2\right]={\mathbb{E}}\left[\sup_{0\le t\le T}\vert V^{N,x,y}_t\vert^2 {\mathbb{E}}\left(\sup_{0\le t\le T}\vert V^{N,y,z}_t\vert^2 \vert \mathcal{G}^{x,y}\right) \right].\end{aligned}$$ Conditionally upon $Z^{N,x}$ and $Z^{N,y}=u(.)$, $V^{N,y,z}$ solves the following SDE $$\begin{aligned}
V^{N,y,z}_t=\frac{\lfloor Nz\rfloor -\lfloor Ny\rfloor}{N}+\int_0^t \left[f(V^{N,y,z}_r+u(r))-f(u(r))\right]dr+M^{N,y,z}_t,\end{aligned}$$ where $M^{N,y,z}$ is a martingale conditionally upon $\mathcal{G}^{x,y}$, hence the arguments used in Lemma \[4lmVN\] lead to $$\sup_{0\le t\le T}{\mathbb{E}}\left(V^{N,y,z}_t\vert \mathcal{G}^{x,y}\right) \le\left(\frac{\lfloor Nz \rfloor}{N}-\frac{\lfloor Ny \rfloor }{N}\right)e^{\beta T},$$ and those used to prove yield $${\mathbb{E}}\left(\int_0^t f(Z^{N,y}_r)-f(Z^{N,y}_r+V^{N,y,z}_r) dr\vert \mathcal{G}^{x,y}\right) \le \frac{\lfloor Nz \rfloor}{N}-\frac{\lfloor Ny \rfloor }{N}.$$ From this we deduce (see the proof of ) that $${\mathbb{E}}\left( \langle M^{N,y,z}\rangle_t \vert \mathcal{G}^{x,y}\right) \le C(t)\left(\frac{\lfloor Nz \rfloor}{N}-\frac{\lfloor Ny \rfloor }{N}\right).$$ From Doobs’s inequality we have $$\begin{aligned}
{\mathbb{E}}\left(\sup_{0\le t\le T}\vert M^{N,y,z}_t\vert^2 \vert \mathcal{G}^{x,y}\right)&={\mathbb{E}}\left( \langle M^{N,y,z}\rangle_T\vert \mathcal{G}^{x,y}\right) \\
& \le C(T)\left(\frac{\lfloor Nz \rfloor}{N}-\frac{\lfloor Ny \rfloor }{N}\right).\end{aligned}$$ Since $0<z-y<1$, we deduce that $$\begin{aligned}
{\mathbb{E}}\left(\sup_{0\le t\le T}\vert V^{N,y,z}_t\vert^2 \vert \mathcal{G}^{x,y}\right)&\le 3(1+C(T))\left(\frac{\lfloor Nz \rfloor}{N}-\frac{\lfloor Ny \rfloor }{N}\right)\\
&+3\beta^2T\int_0^T {\mathbb{E}}\left( \sup_{0\le s \le r}(V^{N,y,z}_s)^2\vert \mathcal{G}^{x,y}\right)dr,\end{aligned}$$ From this and Gronwall’s lemma we deduce that there exists a constant $K_1>0$ such that $$\begin{aligned}
{\mathbb{E}}\left(\sup_{0\le t\le T}\vert V^{N,y,z}_t\vert^2 \vert \mathcal{G}^{x,y}\right)\le K_1\left(\frac{\lfloor Nz \rfloor}{N}-\frac{\lfloor Ny \rfloor }{N}\right).\end{aligned}$$ Similary we have $${\mathbb{E}}\left[\sup_{0\le t\le T}\left(V^{N,x,y}_s\right)^2\right]\le K_1\left(\frac{\lfloor Ny \rfloor}{N}-\frac{\lfloor Nx \rfloor }{N}\right),$$ Since $0\le y-x<z-x$ and $0\le z-y<z-x$, we deduce that $$\begin{aligned}
{\mathbb{E}}\left[\sup_{0\le t\le T}\vert V^{N,x,y}_t\vert^2\times \sup_{0\le t\le T}\vert V^{N,y,z}_t\vert^2\right]\le K_1^2\left(\frac{\lfloor Nz \rfloor}{N}-\frac{\lfloor Nx \rfloor }{N}\right)^2 ,\end{aligned}$$ hence the result.\
[Proof of Theorem \[4thConv\]]{} We now show that for any $T>0$, $$\{Z^{N,x}_t,\ 0\le t\le T,\ x\ge0\}\Rightarrow\{Z^x_t,\ 0\le t\le T,\ x\ge0\}$$ in $D([0,\infty);D([0,T],{\mathbb{R}}_+))$. From Theorems 13.1 and 16.8 in [@4BI], since from Proposition \[4findimconv\], for all $n\ge1$, $0<x_1<\cdots< x_n$, $$(Z^{N,x_1}_\cdot,\ldots,Z^{N,x_n}_\cdot)\Rightarrow
(Z^{x_1}_\cdot,\ldots,Z^{x_n}_\cdot)$$ in $D([0,T];{\mathbb{R}}^n)$, it suffices to show that for all $\bar{x}>0$, $\epsilon$, $\eta>0$, there exists $N_0\ge1$ and $\delta>0$ such that for all $N\ge N_0$, $$\label{4w''}
{\mathbb{P}}(w_{\bar{x},\delta}(Z^N)\ge\epsilon)\le\eta,$$ where for a function $(x,t)\to z(x,t)$ $$w_{\bar{x},\delta}(z)=\sup_{0\le x_1\le x\le x_2\le\bar{x},x_2-x_1\le\delta}
\inf\left\{\|z(x,\cdot)-z(x_1,\cdot)\|,\|z(x_2,\cdot)-z(x,\cdot)\|\right\},$$ with the notation $\|z(x,\cdot)\|=\sup_{0\le t\le T}|z(x,t)|$. But from the proof of Theorem 13.5 in [@4BI], for $Z^N$ follows from Proposition \[4tightD\]
Ray Knight representation of a general Feller diffusion
=======================================================
In this section we establish a Ray-Knight representation of Feller’s branching diffusion solution of , in terms of the local time of a reflected Brownian motion $H$ with a drift that depends upon the local time accumulated by $H$ at its current level, through the function $f'$ where $f$ is a function satisfying the following hypothesis.\
**Hypothesis B:** $f\in C^1(\mathbb{R}_+;\mathbb{R_+})$, $f(0)=0$ and there exist a constant $\beta >0$ such that $$\begin{aligned}
f'(x)\le \beta \quad \forall x\ge0.\end{aligned}$$ Note that hypothesis **B** follows from hypothesis **A** if we assume that $f$ is differentiable.
The proof we give here is purely in terms of stochastic analysis, and is inspired by previous work of Norris, Rogers and Williams [@4NRW] and Pardoux, Wakolbinger [@4PW]. We specify an SDE for a process $\left(H_s\right)$, from which the generalized Feller’s diffusion solution of can be read off from reflected Brownian motion with a drift that is a function of the local time accumulated by $H$ at its current level. One way to understand the form of the drift is to see $\left(H_s\right)$ as the limit of the exploration process $H^N$ of the forest of random trees associated to $Z^{N,x}$. Precisely, fix $z\in C(\mathbb{R}_+;\mathbb{R_+})$, the set of continuous functions from ${\mathbb{R}}_+$ into ${\mathbb{R}}_+$ and consider the stochastic differential equation $$\label{4eqHz}
H^z_s=B_s+\frac{1}{2}\int_0^s f'(z(H^z_r)+L^z_r(H^z_r))dr+\frac{1}{2}L^z_s(0),$$ where $B$ is a standard Brownian motion, and for $s, t \ge 0$ $L^z_s(t)$ is the local time accumulated by $H^z$ at level t up to time $s$. For $x>0$ define $$S_x=\inf \left\lbrace r>0: L^z_s(0)>x\right\rbrace \text{ and } S=\sup_{x>0}S_x.$$ We first suppose that $f$ satisfies hypothesis **B** and the following.\
**Hypothesis C:** $$\begin{aligned}
\exists ~a,~ b\in {\mathbb{R}}: \forall z\ge 0, \quad \vert f'(z)\vert \le a z +b. \end{aligned}$$
Case where $f'$ satisfies hypothesis **C**.
-------------------------------------------
In this subsection we suppose that $f$ verifies hypothesis **C**. We have
\[4probor\] For any $z\in C(\mathbb{R}_+;\mathbb{R_+})$, equation has a unique weak solution.
Let $H$ denote Brownian motion reflected above 0 defined on a probability space $\left(\Omega,\mathcal{F}, {\mathbb{P}}\right)$. $H$ solves the following equation $$H_s=B_s+\frac{1}{2}L_s(0),$$ where $B$ is a $\mathcal{F}_s$ standard Brownian motion, and $L$ is the local time of $H$. Let $$M_s:=\frac{1}{2}\int_0^s f'(z(H_r)+L_r(H_r))dB_r \text{ and } G_s=\exp\left( M_s-\frac{1}{2}\langle M\rangle_s \right).$$ We will show below that ${\mathbb{E}}(G_s)=1$, for all $s\ge 0$, which is a sufficient condition for $G$ to be a martingale. By application of the Girsanov theorem, there exists a new probability $\tilde{{\mathbb{P}}^z}$ on $\left(\Omega,\mathcal{F}\right)$ such that $$\dfrac{d\tilde{{\mathbb{P}}^z}}{d{\mathbb{P}}}\vert_{\mathcal{F}_s}=G_s, s\ge 0,$$ where $\left( \mathcal{F}_s,s\ge 0\right)$ is the natural filtration of $H$. Moreover under $\tilde{{\mathbb{P}}^z}$, $$\tilde{B}^z_s:=B_s-\frac{1}{2}\int_0^sf'(z(H_r)+L_r(H_r))dr, ~~s\ge 0$$ is a standard Brownian motion. The fact that ${\mathbb{E}}(G_s)=1$ for any $s\ge 0$ follows thanks to assumption $\bf{C}$ from the existence of a constant $c$ such that $$\begin{aligned}
\label{4condg}\sup_{0\le r\le s}{\mathbb{E}}(\exp(c (L_r(H_r))^2)<\infty .\end{aligned}$$ The inequality is estabilished in [@4PW], see Lemma 2 and Lemma 3. The uniqueness is also proved in [@4PW] and that argument does not make use of hypothesis **C**.
For $K>0$, we now consider Brownian motion reflected in the interval $[0,K]$ $$H^K=B_s+\frac{1}{2}L^K_s(0)-\frac{1}{2}L^K_s(K^-),$$ defined on $\left(\Omega,\mathcal{F},{\mathbb{P}}\right)$, where $L^K$ denotes the local time of $H^K$. Define for $x>0$ $$S^K_x=\inf\left\lbrace s>0;L^K_s(0)>x\right\rbrace .$$ We again define $$M^K_s:=\frac{1}{2}\int_0^sf'(z(H^K_r)+L^K_r(H^K_r))dB_r \text{ and } G^K_s=\exp\left( M^K_s-\frac{1}{2}\langle M^K\rangle_t\right).$$ The same argument as above shows that ${\mathbb{E}}(G^K_s)=1$ for all $s\ge 0$. This implies that there exists a probability $\tilde{{\mathbb{P}}}^{K,z}$ defined on the measurable space $\left(\Omega,\mathcal{F}\right) $ under which $$\begin{aligned}
\tilde{B}^z_s=H^K-\frac{1}{2}L^K_s(0)+\frac{1}{2}L^K_s(K^-)-\frac{1}{2}\int_0^sf'(z(H^K_r)+L^K_r(H^K_r))dr,\end{aligned}$$ is a $\tilde{{\mathbb{P}}}^{K,z}$-Brownian motion. That is the equation $$\label{4eqHK}
H^K=\tilde{B}_s+\frac{1}{2}L^K_s(0)-\frac{1}{2}L^K_s(K^-)+\frac{1}{2}\int_0^sf'(z(H^K_r)+L^K_r(H^K_r))dr$$ admits a weak solution. Uniqueness of the weak solution of is obtained in a similar way as concerning . Moreover we have that (see again [@4PW]) $$\tilde{{\mathbb{P}}}^{K,z}\left(S_x^K<\infty\right)=1.$$ We now have the following Ray Knight representation.
\[4RNb\] For any $K>0$ and $z\in C(\mathbb{R}_+;\mathbb{R_+})$, the law of $\Big\lbrace L^K_{S^K_x}(t),$\
$0\le t<K \Big\rbrace$ under $\tilde{{\mathbb{P}}}^{K,z}$ is the same as the law of $ \left\lbrace Z^{x,z}_t,0\le t<K\right\rbrace$, where $Z^{x,z}$ solves the SDE $$\begin{aligned}
\label{4Zz}
dZ^{x,z}_t= \left[ f(Z^{x,z}_t+z(t)) -f(z(t))\right] dt+2\sqrt{Z^{x,z}_t}dW_t,~Z^{x,z}_0=x, \end{aligned}$$ and $W$ is a standard Brownian motion.
The proof of this Proposition is similar to that done in [@4PW] in the quadratic case. We will give below some details of the proof in the more general case without hypothesis **C**.
Existence and uniqueness of weak solution of without hypothesis **C**
---------------------------------------------------------------------
Now we do not suppose anymore that $f$ satisfies hypothesis **C**. We have
\[4sol\] For any $K>0$, $z\in C(\mathbb{R}_+;\mathbb{R_+})$, there exists a probability $\tilde{{\mathbb{P}}}^{K,z}$ under which equation has a unique weak solution on the random interval $[0,S^K)$, where $S^K=\sup_{x\ge 0}S^K_x$.
Consider again, for $K>0$, the Brownian motion $H^K$ reflected in the interval $[0,K]$ defined on $\left(\Omega, \mathcal{F},{\mathbb{P}}\right)$. $$H^K=B_s+\frac{1}{2}L^K_s(0)-\frac{1}{2}L^K_s(K^-).$$ For $n\ge 1$, we define the function $g_n(r)=f'(n\wedge r)$. It is clear that there exist two constants $a,b\ge 0$ such that $\vert g_n(r)\vert \le ar+b$. Thanks to Proposition \[4probor\], there exits for each $n\ge1$ a probability $\tilde{{\mathbb{P}}}^{K,z,n}$ such, $$\dfrac{ d\tilde{{\mathbb{P}}}^{K,z,n} }{d{\mathbb{P}}}\vert_{\mathcal{F}_s}=\exp\left\lbrace M^{K,n}_t-\frac{1}{2}\langle M^{K,n} \rangle_t\right\rbrace , s\ge 0,$$ where $M^{K,n}_s:=\frac{1}{2}\int_0^s g_n(z(H^K_r)+L^K_r(H^K_r))dB_r$. Under $\tilde{{\mathbb{P}}}^{K,z,n}$, $$\tilde{B}^{z,n}_s=H^K_s-\frac{1}{2}L^K_s(0)-\frac{1}{2}\int_0^s g_n(z(H^K_r)+L^K_r(H^K_r))dr+\frac{1}{2}L^K_s(K^-),\forall s\ge 0,$$ is a standard Brownian motian. For $n\ge 1$, we define the stopping time $$T_n=\inf \left\lbrace s>0; \sup_{0\le t<K}\left[z(t)+L^K_s(t)\right]>n \right\rbrace .$$ We need the following result which is a variant of Theorem 1.3.5 from Stroock-Varadhan [@4SV], whose proof is very similar to that in [@4SV].
\[4thr\] Let $\Omega=C({\mathbb{R}}_+,{\mathbb{R}}_+)$ be the canonical path space with its canonical filtration $\left\lbrace \mathcal{F}_t\right\rbrace$, and let $\left( T_n\right) $ be an increasing sequence of stopping times satisfying $T_n\le S^K$ $a.s.$ $\forall n\ge1$. Suppose there is a sequence $\left( {\mathbb{P}}_n\right)$ of probabilities on $(\Omega,\mathcal{F})$ such that
- ${\mathbb{P}}_{n+1}$ agrees with ${\mathbb{P}}_n$ on $\mathcal{F}_{T_n}$;\
- for each $x>0$, $${\mathbb{P}}_n \left( T_n< S^K_x\right)\rightarrow 0 \text{ as } n\rightarrow\infty.$$
Then there exists a probability ${\mathbb{P}}$ on $(\Omega,\mathcal{F}_{S^K})$ such that for each $n$, $${\mathbb{P}}={\mathbb{P}}_{n} \text{ on } \mathcal{F}_{T_n}.$$
This proves the existence of a probability $\tilde{{\mathbb{P}}}^{K,z}$ on $(\Omega,\mathcal{F}_{S^K})$, provided we show that for all $x>0$, $$\tilde{{\mathbb{P}}}^{K,z,n}\left(T_n<S^K_x\right)\rightarrow 0 \text{ as } n\rightarrow\infty.$$ We have $$\tilde{{\mathbb{P}}}^{K,z,n}\left( T_n<S^K_x\right)=\tilde{{\mathbb{P}}}^{K,z,n}\left( \sup_{0\le t<K}L^K_{S^K_x}(t)>n\right).$$ From Propostion \[4RNb\], under $\tilde{{\mathbb{P}}}^{K,z,n}$, $(L^K_{S^K_x}(t),0\le t< K )$ has the same law as $(Z^{x,z,n}_t,0\le t< K)$ solution of the SDE $$\begin{aligned}
dZ^{x,z,n}_t=\left(\int_{z(t)}^{z(t)+Z^{x,z,n}_t}g_n(u)du\right)dt+2\sqrt{Z^{x,z,n}_t}dW_t,~~Z^{x,n}_0=x.\end{aligned}$$ For any $x\ge 0$, consider the process $\tilde{Z}^x$, which is solution of the SDE $$\begin{aligned}
\tilde{Z}^x_t=x+\beta\int_0^t\tilde{Z}^x_rdr+2\int_0^t\sqrt{\tilde{Z}^x_r}dWr.\end{aligned}$$ By a well known comparison theorem for one dimensional SDEs, see [@4RY] theorem $X.3.7$, we have that for any $x\ge 0$ and $z\in C(\mathbb{R}_+;\mathbb{R_+})$, $Z^{x,z,n}\le \tilde{Z}^x$ $a.s.$ . We then have that $$\begin{aligned}
\tilde{{\mathbb{P}}}^{K,z,n}\left( T_n<S^K_x\right)&=\tilde{{\mathbb{P}}}^{K,z,n}\left( \sup_{0\le t<K}L^K_{S^K_x}(t)>n\right) \\
&={\mathbb{P}}\left( \sup_{0\le t<K}Z^{x,z,n}_t>n\right) \\
&\le {\mathbb{P}}\left( \sup_{0\le t<K}\tilde{Z}^x_t>n\right)\rightarrow 0 \text{ as } n\to\infty.\end{aligned}$$ We thus have proved for all $K>0$ and $z\in C(\mathbb{R}_+;\mathbb{R_+})$, the existence of a probability $\tilde{{\mathbb{P}}}^{K,z}$ under which on $[0,S^K)$, $$\begin{aligned}
\tilde{B}^z_s=H^K_s-\frac{1}{2}L^K_s(0)+\frac{1}{2}\int_0^sf'(z(H^K_r)+L^K_r(H^K_r))dr+\frac{1}{2}L^K_s(K^-)\end{aligned}$$ is a standard Brownian motion. Uniqueness is obtained in a similar way as in [@4PW]. Hence the Proposition. For any $z\in C({\mathbb{R}}_+;{\mathbb{R}}_+)$, we have the following Ray Knight representation.
\[p3.5\] For any $K>0$, $z\in C({\mathbb{R}}_+;{\mathbb{R}}_+)$ and $x\ge 0$, the law of $\left( L^K_{S^K_x}(t),0\le t<K\right)$ under $\tilde{{\mathbb{P}}}^{K,z}$ is the same as the law of $\left( Z^{x,z}_t,0\le t<K\right)$
For $K>0$ and $z\in C({\mathbb{R}}_+)$, we work under the probability measure $\tilde{{\mathbb{P}}}^{K,z}$. Using Tanaka’s formula, we have for any $r\ge0$ and $0\le t<K$, the following identity $$\begin{aligned}
\label{4tanfor} (H^K_r-t)^-=(-t)^-+\int_0^r{\bf1}_{\lbrace H^K_s\le t\rbrace}dH^K_s+\frac{1}{2}L^K_r(t) \end{aligned}$$ Recall that for any $x\ge 0$, ${\mathbb{P}}^{K,z}(S^K_x<\infty)=1$. Hence from , $$L^K_{S^K_x}(t)=2\int_0^{S^K_x}{\bf1}_{\lbrace H^K_s\le t\rbrace}dH^K_s .$$ Combining with equation , we get $$\begin{aligned}
L^K_{S^K_x}(t)=x+2\int_0^{S^K_x}{\bf1}_{ \lbrace H^K_s\le t\rbrace}d\tilde{B}_s+\int_0^{S^K_x}{\bf1}_{\lbrace H^K_s\le t\rbrace }f'(z(H^K_s)+L^K_s(H^K_s) )ds.\end{aligned}$$ From the generalized occupation time formula (see Exercise $VI.1.15$ in [@4RY]), we obtain $$\begin{aligned}
\int_0^{S^K_x}{\bf1}_{\lbrace H^K_s\le t\rbrace }f'(z(H^K_s)+L^K_s(H^K_s))ds &=\int_0^t\int_0^{S^K_x}f'(z(r)+L^K_s(r))dL^K_s(r)dr\\
&= \int_0^t \left(f(z(r)+L^K_{S^K_x}(r))-f(z(r))\right)dr .\end{aligned}$$ The key idea of the proof is now to introduce the “excursion filtration", as in [@4NRW] and [@4PW]. For any $0\le t<K$ and $s\ge0$, let define $$\begin{aligned}
A_s(t)&:=\int_0^s {\bf1}_{\lbrace H^K_r\le t\rbrace}dr, ~~\tau(r,t)=\inf\left\lbrace s>0; A_s(t)>r\right\rbrace ,\\
J(s,t)&:=\int_0^s {\bf1}_{\lbrace H^K_r\le t\rbrace}d\tilde{B}_r~~~~\xi(r,t):=J(\tau(r,t),t)\\
\mathcal{F}_{(r,t)}&:=\sigma(\xi(r,t):0\le u\le r),~~ \varsigma_t=\mathcal{F}_{(\infty,t)},\\
N_t &:=\int_0^{S^K_x}{\bf1}_{\lbrace H^K_s\le t\rbrace}d\tilde{B}_s.\end{aligned}$$ For fixed $t$, the process $\left( J(s,t),s\ge 0\right) $ is a martingale with respect to $\mathcal{F}_s$, while $\left( \xi(r,t),r\ge 0\right)$ is a $\mathcal{F}_{(r,t)}$-martingale and its quadratric variation equals $r$. Consequently $\left( \xi(r,t),r\ge 0\right)$ is a $\mathcal{F}(r,t)$-Brownian motion. We then have the
The process $\left( N_t,0\le t<K\right)$ is a continuous $\varsigma_t$-martingale with its quadratic variation given by $$\langle N \rangle_t=4\int_0^t L^K_{S^K_x}(r)dr.$$
This result is an easy consequence of Theorem 1 in [@4NRW]. By the martingale representation theorem, we deduce that there exists a Brownian motion $W$ such that $$N_t=2\int_0^t\sqrt{L^K_{S^K_x}(r)}dW_r.$$ Consequently for all $0\le t< K$, $$L^K_{S^K_x}(t)= x+\int_0^t\left( f(z(r)+L^K_{S^K_x}(r))-f(z(r))\right)dr+2\int_0^t\sqrt{L^K_{S^K_x}(r)}dW_r.$$
Ray Knight theorem in the subcritical case
------------------------------------------
We first prove the following proposition (recall the definition of $\Lambda(f)$)
Suppose that $f$ satisfies hypothesis **B** and $\Lambda(f) =\infty$. Then equation admits a unique weak solution on $[0,S)$.
For $x>0$ and $K>0$, let define $$\Omega ^{K,x}= \left\lbrace \sup_{[0,S^K_x]} H^{K'}<K,~~\forall~ K'>K \right\rbrace .$$ For any $x\ge 0$ and $z\in C$, since we are in the subcritical case, there exists $T_{x,z}<\infty$ $a.s.$ such that , $Z^{x,z}_t=0$, $\forall$ $t\ge T_{x,z}$. We deduce from this and Proposition \[p3.5\] that for any fixed $x\ge 0$, $$\Omega=\cup_{K>0}\Omega^{K,x} \text{ a.s. } .$$ Note that the family of events $\left\lbrace \Omega^{K,x},~K>0\right\rbrace $ is increasing, and on $\Omega^{K,x}$, $H^{K'}=H$ a.s., for any $K' >K$. We can define a probability $\tilde{{\mathbb{P}}}^{z,x}$ on $\left(\Omega, \mathcal{F}_{S_x}\right)$ such that $\tilde{{\mathbb{P}}}^{z,x}$=$\tilde{{\mathbb{P}}}^{K,z}$ on $\Omega^{K,x}$. Under $\tilde{{\mathbb{P}}}^{z,x}$, on $[0,S_x]$ $$\tilde{B}^z_s=H_s-\frac{1}{2}\int_0^s f'(z(r)+L_r(H_r))dr-\frac{1}{2}L_s(0)$$ is a standard Brownian motion. This proves that has a weak solution on $[0,S_x]$ whose uniqueness can be proved as in [@4PW]. ON A A NOUVEAU BESOIN DU TH DE SV ! We can deduce that there exists a probabilty $\tilde{{\mathbb{P}}}^z$ under which, on $[0,S)$ $\tilde{B}^z$ is a standard Brownian motion, where $$S=\sup_{x\ge 0}S_x .$$ For $z\equiv 0$, we write $\tilde{{\mathbb{P}}}=\tilde{{\mathbb{P}}}^{z}$. The following statement is a generalized Ray Knight theorem in the subcritical case.
\[4RN\] Suppose that $f$ satisfies Hypothesis **B** and $\Lambda(f) =\infty$. Then the law of the random fields $\left\lbrace L_{S_x}(t), t\ge 0, x\ge 0\right\rbrace$ under the probability $\tilde{{\mathbb{P}}}$ is the same as the law of $\left\lbrace Z^x_t,t\ge 0, x\ge 0\right\rbrace $.
We first establish the following Proposition.
\[4RNx\] Assume that the two assumptions of Theorem \[4RN\] holds. Then for any $x$ and $z\in C({\mathbb{R}}_+;{\mathbb{R}}_+)$ fixed, the law of $\left \lbrace L_{S_x}(t), t\ge 0\right\rbrace$ under $\tilde{{\mathbb{P}}}^z$ coincides with is the law of $\left\lbrace Z^{x,z}_t,t\ge 0\right\rbrace$.
We have that for any $K>0$, $z\in C({\mathbb{R}}_+;{\mathbb{R}}_+)$, under $\tilde{{\mathbb{P}}}^{K,z}$, $\Big( L^K_{S^K_x}(t),$\
$0\le t<K\Big)$ has the same law as $\left( Z^{x,z}_t,0\le t<K\right)$. A consequence of this is that for any $0<K<K',$ $$\begin{aligned}
\label{4idenlaw}
\left\lbrace L^K_{S^K_x}(t),0\le t<K \right\rbrace \stackrel{(d)}=\left\lbrace L^{K'}_{S^{K'}_x}(t),0\le t<K \right\rbrace. \end{aligned}$$ It now follows that for any $K$, under $\tilde{{\mathbb{P}}}^z$, $\left( L_{S_x}(t), 0\le t<K \right)$ has the same law as $\left( L^K_{S^K_x}(t), 0\le t<K \right)$ under $\tilde{{\mathbb{P}}}^{K,z,x}$. We then obtain that for any $K>0$ $$\left( L_{S_x}(t),0\le t<K \right) \stackrel{(d)}=\left( Z^{x,z}_t,0\le t<K \right).$$ Hence the proposition, letting $K$ go to $\infty$. In particular, for $x$ fixed, the law of $\left \lbrace L_{S_x}(t), t\ge 0\right\rbrace$ under $\tilde{{\mathbb{P}}}$ is the same as the law of $\left\lbrace Z^x_t,t\ge 0\right\rbrace$.
The identity could also obtained from a generalization of Lemma 2.1 in Delmas [@4DJ]. For $0<a<b$, we define the application $\pi^{a,b}$ with maps continuous trajectories with value in $[0,b]$ into trajectories with values in $[0,a]$ as follows. If $u\in C({\mathbb{R}}_+,[0,b])$, $$\rho_u(s)=\int_0^s{\bf1}_{u(r)<a}dr,~~~~~~\pi^{a,b}(u)(s)=u(\rho^{-1}_u(s)).$$ The following equality in law holds. $$\pi^{a,b}(H^b)\stackrel{(d)}=H^a.$$ This identity together with the strong Markov property of the Brownian motion implies .
[Proof of Theorem \[4RN\]]{} Recall that $\left( Z^x_., x\ge 0\right) $ is a Markov process with value in the space of continuous paths from ${\mathbb{R}}_+$ into ${\mathbb{R}}_+$ with compact support. From Proposition \[4RNx\] with $z\equiv 0$, its marginal laws coincide with those of $L_{S_x}(.)$. We now check that $\left(L_{S_x}(.),x\ge 0\right)$ is a Markov process. This follows readily from the fact that for any $0\le x<y $, conditionnaly upon $\left(L_{S_{x'}}(.),x' \le x\right)$ and given $L_{S_x}(.)=z(.)$, on $[0,S_y]$ the process $H^x_s:=H_{S_x+s}$ solves the SDE $$\begin{aligned}
H^x_s=\bar{B}_s+\frac{1}{2}\int_0^s\left(f'(z(H^x_r)+L^z_r(H^x_r))\right)dr+\frac{1}{2}L^z_s(0),\end{aligned}$$ where $\bar{B}$ is a Brownian motion independent of $\left( L_{S_{x'}}(t),x'\le x,0\le t\le S_x\right)$ and $L^z$ denotes the local time of $H^x$, which is also the additional local time accumulated by $H$ after time $S_x$. To complete the proof of the theorem it now suffices to prove that for any $x,y\ge 0$ the conditional law of $\left( L_{S_{x+y}}(t),t\ge 0\right)$ given $\left( L_{S_x}(t),t\ge0\right)$ is the same as the conditional law of $\left( Z^{x+y}_t,t\ge\right)$ given $\left( Z^x_t,t\ge 0\right)$. Conditioned upon $L_{S_x}(.)=z(.)$, $L_{S_{x+y}}(.)-L_{S_x}()$ is the collection of local times accumulated by $H^x$ up to time $S_{y}$, and it has the same law as $L^z_{S_y}(.)$ while conditionally upon $Z^x=z(.)$, $Z^{x+y}-Z^x$ has the same law as $Z^{y,z}$. The identity of those two laws has been established in Proposition \[4RNx\]. We can deduce from the Proposition \[4RNx\] and the occupation time formula that
Suppose that $f$ satisfies Hypothesis **B** and $\Lambda(f) =\infty$. We have $$\forall x\ge0, \quad \tilde{{\mathbb{P}}}\left( S_x<\infty\right)=1.$$
Let $g(h)=1$, for any $h>0$. By the occupation times’s formula, we have $$\begin{aligned}
S_x& =\int_0^{S_x}g(H_r)dr\\
&=\int_0^{\infty}L_{S_x}(t)dt=\int_0^{T_0^x}Z^x_rdr<\infty \text{ a.s.} \end{aligned}$$ Note that $S_x$ is the total mass of the process $\left(Z^x_t,t\ge0\right)$.
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[^1]: [Keywords]{} : Galton–Watson processes with interaction, generalized Feller diffusion
[^2]: *AMS 2000 subject classification.* (Primary) [60J80, 60F17]{} (Secondary) [92D25]{}.
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---
abstract: 'In models of modified gravity, extra degrees of freedom usually appear. They must be removed from the spectrum because they may indicate the presence of instabilities and because otherwise the model might not agree with observation. In the present letter, we will discuss a model that modifies General Relativity through the addition of a Lorentz-violating potential-like term for the metric. No additional propagating modes and no classical instabilities are present. The model departs from GR only in the extreme infrared region, and the cosmological evolution contains a bounce when the size of the Universe is of the same order of the added deformation.'
bibliography:
- 'LVdS.bib'
nocite:
- '[@Ganor:2007qh; @Dubovsky:2006vk; @Dubovsky:2004sg; @Bluhm:2007bd; @Bluhm:2007gs]'
- '[@Ganor:2007qh; @Dubovsky:2006vk; @Dubovsky:2004sg; @Bluhm:2007bd; @Bluhm:2007gs]'
---
[NYU-TH-07/11/27]{}
0.9cm
**Lorentz-Violating Massive Gravity**
0.2cm
**in Curved Space**
0.7cm
Luca Grisa
0.3cm
*Center for Cosmology and Particle Physics*
*Department of Physics, New York University, New York, NY 10003, USA*
1.9cm
Introduction
============
The gravitational interaction is well described by General Relativity within a vast range of distance scales: from solar system to galaxy cluster sizes. Still questions arise on its applicability at microscopic, where quantum effects are no longer negligible, and super-horizon distances. At the far end of the length scale, in the deep infrared region, it is not obvious whether gravity should behave the same as it does at much shorter distances. Rather, the discovery of (recent) cosmic acceleration from supernovae data opens up the possibility that gravity could be very different at these scales from what we experience every day. It could be that the gravitational field is very slightly massive, and the fact was simply overlooked, because the effects are naïvely negligible at scales shorter than the characteristic Compton wavelength. But, it was shown [@Boulware:1973my] that a simple mass term, quadratic in the fluctuation of the metric, is not acceptable. The classical Hamiltonian – constructed in the ADM formalism – is not bounded from below. The model is not classically stable.
Modifying gravity in the IR regime seems to be rather non-trivially constrained. In the past, several attempts to build classically stable models of IR modified gravity have been made. For instance, extra-dimensions may manifest themselves only at very large distances, like in the DGP model [@Dvali:2000hr]. Or, without invoking extra-dimensions, the presence of non-local interactions could change the gravitational field in the IR regime, as in [@ArkaniHamed:2002fu]. Or again, a similar effect can be induced by Lorentz-violating terms, like in [@ArkaniHamed:2003uy; @Rubakov:2004eb; @Blas:2007zz; @Rubakov:2008nh].
In [@Gabadadze:2004iv], we considered a class of models of the latter group. We were able to construct models that are four-dimensional and local, and have the property of being both stable in the IR and of departing from standard gravitational interaction only at large distances. But this was only possible at the price of explicitly breaking the Lorentz symmetry of the action, which had the effect of introducing new phenomena, like instantaneous interactions. These effects can be within the experimental bound by tuning the parameters, and more importantly they do not violate causality.
In the present letter, we will extend the previous discussion by looking at effective models over a constantly curved background. GR will be deformed with a small Lorentz-violating term. The model is classically stable and behaves like a de Sitter space-time at short scales. At larger and larger distances, the departure from de Sitter becomes more and more pronounced.
A rather generic feature of the model is the tendency to stop the cosmological evolution and to lead towards a contracting phase. The effect of the deformation is opposite to that of the cosmological constant, for, as a cosmological constant increases the expansion speed, the deformation decreases it. After the expansion is stopped, [*i.e.*]{}, when the Hubble parameter goes to zero, the Universe will go through a contracting phase, leading to a late-time (contracting) de Sitter phase.
This picture will be modified by quantum corrections. In particular, out of the gravitational field, particles are produced when the Hubble parameter changes. The effect of quantum produced particles is to create at late times a space-time singularity of the same kind as the one of a supercritical Universe. In the latter case, if the matter energy density is greater than a critical value, the scale factor grows up to a maximum size and then contracts towards a “Big Crunch” singularity: at some finite moment in the future the scalar curvature diverges. In the present case, instead, no criticality condition is present. The cosmological evolution is not stopped by the matter density, but by the deformation we introduced, therefore, no matter how few particles are produced by quantum effects, a singularity will always be reached.
The model, which is classically stable, is unstable under quantum correction. This instability is milder, in fact its time scale can be made parametrically much greater than the present age of our Universe. The present letter is organized as follows: we will firstly discuss the general properties of the gravitational field in models, where a Lorentz-violating interaction for the metric is added, explicitly showing the absence of additional degrees of freedom and the presence of instantaneous interaction.
We will then study the cosmological solution of the (deformed) Einstein equation for two specific choices of the Lorentz-violating interaction. One will give rise to a bouncing Universe, while the other to a cyclic Universe of parametrically long period. In both cases, we will consider the quantum production that occurs at the bounce, and its effect on the cosmological evolution.
The cyclic model will then be discussed in the context of inflation. We will find that the parameters of the model are constrained by the usual cosmological bounds. In particular, they have to be exponentially smaller than the Hubble scale during inflation for matching density perturbation with the values observed in the CMB.
Finally, we will see the effects of the present class of deformations when in conjunction with the massive term studied in our previous work [@Gabadadze:2004iv].
General Overview {#gen.overview}
================
From a particle physicist’s point of view, gravity is the interaction that emerges upon gauging the Lorentz symmetry of Special Relativity. As for any gauge theory, it should be possible to describe the low energy effective theory, which arises from the breaking of a part (or the whole) of the gauge symmetry as a result of some high energy dynamics.
In gravity though, unexpected constraints arise for the low energy theory. It was noted by Boulware and Deser [@Boulware:1973my] that, have the classical non-linearities of the gravitational self-interaction taken into account, the Hamiltonian would generally not be bounded in models of massive gravity. Thus the Boulware-Deser (BD) instability appears.
The origin of the instability – and therefore how to render stable a model of modified gravity as originally described in [@Gabadadze:2004iv] – can be easily understood in the ADM formalism [@Arnowitt:1962hi], that is the Hamiltonian formalism for gravity.
Let us construct the GR Hamiltonian. By foliating the space-time with hypersurfaces $\Sigma_t$ for a time variable $t$, we can replace the four-dimensional metric with the following three-dimensional variables $$\gamma_{ij}\equiv g_{ij}\,,\quad N\equiv(-^{(4)}g^{00})^{-1/2}\quad\mbox{and}\quad
N_i\equiv\phantom{.}^{(4)}g_{0i}\,.
\label{3D.vars}$$ $N$ is known as the lapse function, and $N_i$ as the shift function; $\gamma_{ij}$ is the induced metric on $\Sigma_t$. In term of these variables, we can write the four-dimensional ones as $$\begin{aligned}
\sqrt{-^{(4)}g}&\equiv& N\sqrt{\gamma}\,,\\
\phantom{.}^{(4)}{\mathcal{R}}&\equiv&\phantom{.}^{(3)}{\mathcal{R}}+{\mathcal{K}}_{ij}{\mathcal{K}}^{ij}-{\mathcal{K}}^2\,,
\label{3D.defs}\end{aligned}$$ where ${\mathcal{K}}_{ij}$ is the extrinsic curvature on $\Sigma_t$, defined as $${\mathcal{K}}_{ij}\equiv\frac{1}{2N}\left[ \dot\gamma_{ij}-\nabla_iN_j-\nabla_jN_i \right]\,.
\label{K.extrcurv}$$ The canonical momentum $\pi^{ij}\equiv\delta{\mathcal{L}}/\delta\dot\gamma_{ij}$ is related to ${\mathcal{K}}_{ij}$ by the relation $$\pi^{ij}=\sqrt{\gamma}\left[ {\mathcal{K}}^{ij}-{\mathcal{K}}\gamma^{ij} \right]\,.
\label{pi.canmom}$$ We now have all the ingredients to write the Hamiltonian for the Einstein-Hilbert Lagrangian $${\mathcal{L}}=\sqrt{g}{\mathcal{R}}\quad\rightarrow\quad{\mathcal{H}}\equiv\pi^{ij}\dot\gamma_{ij}-
{\mathcal{L}}|_{\dot\gamma_{ij}\mapsto\pi_{ij}}
=\sqrt{\gamma}\left[ NR^0+N_iR^i \right]\,,
\label{GR.H}$$ where $$\begin{aligned}
R^0&\equiv&-^{(3)}{\mathcal{R}}+\gamma^{-1}\left( \pi_{ij}\pi^{ij}-\frac{1}{2}\pi^2 \right)\,,\nonumber\\
R^i&\equiv&-2D_j(\gamma^{-1/2}\pi^{ij})\,,
\label{R0.Ri}\end{aligned}$$ and $D_j$ is the covariant derivative defined with respect to $\gamma_{ij}$.\
Both $N$ and $N_i$ appear linearly in the Hamiltonian, thus they are Lagrange multipliers. The variation with respect to them leads to the constraints – $R^0=0$ and $R^i=0$ – on the propagating degrees of freedom. The Hamiltonian is exactly zero on the surface of the constraints, hence the theory is trivially stable: the energy density of the system is bounded from below.
We will show the emergence of the BD instability for deformed Einstein-Hilbert actions. For sake of definiteness, we shall consider the Pauli-Fierz (PF) model [@Fierz:1939ix].
The PF term is the most generic deformation that is quadratic in the fluctuation of the metric over a particular background, and it is Lorentz-symmetric. It describes a mass for the gravitational field $$ -\frac{1}{2}m_{\mathrm{PF}}^2[h_{\mu\nu}^2-({h^\mu_{\phantom.\mu}})^2]=
-\frac{1}{2}m_{\mathrm{PF}}^2[h_{ij}^2-h^2-2N_i^2+2h(1-N^2-N_i^2)]\,.
\label{PF.mass}$$ In the equality, the field $h_{\mu\nu}$ is expressed in terms of the three-dimensional variables . The tensor field is defined as $h_{\mu\nu}\equiv g_{\mu\nu}-\hat{g}_{\mu\nu}$ over a particular background metric $\hat{g}_{\mu\nu}$. The indices are contracted using $\hat{g}^{\mu\nu}$, [*i.e.*]{}, $h^\mu_{\phantom.\mu}\equiv\hat{g}^{\mu\nu}h_{\mu\nu}$, $h\equiv \hat{g}^{ij}h_{ij}$, and so on.
It is evident that the lapse and the shift functions cease to be Lagrange multipliers, and the variations with respect to them lead to algebraic equations for them, rather than constraints on the propagating degrees of freedom, as in GR. This is hardly unexpected. A massive field is known, from the Lorentz group representation, to propagate a number of degrees of freedom different from that of a massless field.
But, if we now study closely the equations $$\begin{aligned}
N&=&\frac{\sqrt{\gamma}R^0}{2m_{\mathrm{PF}}^2h}\,,\\
N^i&=&\frac{1}{2m_{\mathrm{PF}}^2}(\hat{g}^{ij}-h\gamma^{ij})^{-1}R^j\,,
\label{PF.NNi}\end{aligned}$$ we notice that the Hamiltonian, after substituting the above values for $N$ and $N_i$, $${\mathcal{H}}=\frac{1}{4m_{\mathrm{PF}}^2}\left[ \frac{(\sqrt{\gamma}R^0)^2}{h}
+\gamma R^i(\hat{g}^{ij}-h\gamma^{ij})^{-1}R^j \right]
+\frac{1}{2}m_{\mathrm{PF}}^2(h_{ij}^2-h^2+2h)\,,
\label{PF.H}$$ is unbounded, as it is readily seen by considering the limit $h\rightarrow0^-$, while keeping $\sqrt{\gamma}R^0$ and $R^i=0$ fixed.
It appears that models with $N^2$-terms – such as PF – are generally unstable. An easy way-out is obviously to consider more general classes of deformations, in particular the ones linear in $N$. It should be noted though, that the $N^2$-term in the PF Hamiltonian comes from the time component of the tensor $h_{\mu\nu}$. From the definition of the lapse function , it follows that $h_{00}=g_{00}-\hat{g}_{00}\sim N^2$. Thus, removing such a term would lead to an explicit breaking of the Lorentz symmetry in the action. For a detailed discussion of a PF-like model with such a property we remind to our previous letter [@Gabadadze:2004iv].
The class of models we would like to discuss in the present work has the following Hamiltonian $${\mathcal{H}}=\sqrt{\gamma}\left[ NR^0+N_iR^i+2\Lambda N-2m^2Nf(\sqrt{\gamma}) \right]\,,
\label{H.deform}$$ where $f(\sqrt\gamma)$ is some function of the determinant of the spatial metric $\gamma_{ij}$ as in , and we have assumed the presence of a cosmological constant $\Lambda$. The deformation we added is modelled to be linear in $N$, thus the lapse function is still a Lagrange multiplier. As in GR, the Hamiltonian is exactly zero on the solution of the constraints. The model is therefore (classically) stable.
This model, as well as the ones presented in [@Gabadadze:2004iv], should be thought of as an effective low energy theory. Like in the Higgs mechanism, the gauge symmetry is broken at low energy, hence we are assuming the presence of some UV-physics that spontaneously breaks the Lorentz symmetry of the action, like in the recent models [@ArkaniHamed:2003uy],[@Ganor:2006ub]-[@Bluhm:2008rf].
The introduction of a deformation could, in principle, lead to the propagation of more degrees of freedom, some of which may develop into instabilities for the theory. We will show that only a tranverse-traceless tensor mode is propagating. The explicit breaking of the Lorentz symmetry will instead show up as instanteneous interactions, as first noticed in [@Gabadadze:2004iv]. To explicitly study the degrees of freedom, we turn now to the Lagrangian formalism.
The Lagrangian can be found by performing a Legendre transformation on $${\mathcal{L}}=\sqrt{-g}\left[ {\mathcal{R}}-2\Lambda+2m^2f(\sqrt{\gamma}) \right]\,,
\label{L.deform}$$ and it should be noted that we are forced to keep a somewhat mixed formalism. The deformation is written in terms of the determinant of $\gamma_{ij}$, hence retaining in part the notion of the three-dimensional variables used in the ADM formalism.
The equations of motion are $$G_{\mu\nu}+\left[ \Lambda-m^2f(\sqrt{\gamma})
\left( 1+\sqrt{\gamma}\frac{f'(\sqrt{\gamma})}{f(\sqrt\gamma)} \right) \right]g_{\mu\nu}
-m^2\frac{\sqrt{\gamma}f'(\sqrt\gamma)}{|g^{00}|}\delta^0_\mu\delta^0_\nu=0\,,
\label{eom.deform}$$ where $G_{\mu\nu}$ is the Einstein tensor defined as $G_{\mu\nu}\equiv{\mathcal{R}}_{\mu\nu}-1/2\,{\mathcal{R}}\,g_{\mu\nu}$, and the last term is zero for $\mu,\nu\ne0$.
The deformed action is not invariant under Lorentz transformations anymore. The determinant of $\gamma_{ij}$ will transform under $x_\mu\rightarrow\Lambda_\mu^\alpha\,x_\alpha$ as it can be explicitly checked. The introduced deformation breaks the Lorentz invariance of the action down to the rotational group. The measure on the hypersurface $\Sigma_t$ is invariant under diffeomorphisms acting on its own world-volume.
The breaking of the Lorentz symmetry stems out from the presence of a preferred frame in the model. In the construction of the GR Hamiltonian, a frame is chosen when picking up a particular foliation $\Sigma_t$ of the space-time. The choice is formal in GR, where the action is invariant under the entire group of the diffeomorphisms, but it is not in the theory at hand.
The deformation we consider is a function of the spatial part $\gamma_{ij}$ of the metric tensor only. To define $\gamma_{ij}$, we need to choose a direction $u_\mu$ and $\gamma_{ij}$ be the induced three-dimensional metric on the hypersurface orthogonal to $u_\mu$. The preferred frame is defined therefore by $u_\mu$. Any transformation orthogonal to $u_\mu$ is a symmetry of the model, as briefly stated before.
Because of this remaining invariance of the action, some components of the perturbation $h_{\mu\nu}$ over a background metric $\hat{g}_{\mu\nu}$ are not independent to the others. For sake of simplicity, we shall consider $\hat{g}_{\mu\nu}$ to be the Minkowski metric $\eta_{\mu\nu}$. It can be shown that the results we are presenting do not depend on this choice.
The redundancy in the components of $h_{\mu\nu}$ can be fixed by a gauge choice, for instance $\partial^ih_{ij}=0$. It fixes the longitudinal modes of $h_{ij}$, but not its trace. The action is indeed not invariant under the transformation $h^i_{\phantom.i}\rightarrow h^i_{\phantom.i}+2\partial^i\xi_i$, hence the trace $h^i_{\phantom.i}$ can not be fixed by a gauge choice.[^1]
The gauge fixing is not the only condition we have to impose on $h_{\mu\nu}$. The requirement that the equations of motion are covariantly conserved $$(2f'+\sqrt{\gamma}f'')\partial_\mu\sqrt{\gamma}+
\delta^0_\mu g^{0\alpha}\partial_\alpha\frac{\sqrt{\gamma}f'}{|g^{00}|}-
g^{\alpha\beta}(\Gamma^0_{\alpha\beta}\delta^0_\mu+\Gamma^0_{\alpha\mu}\delta^0_\beta)
\frac{\sqrt{\gamma}f'}{|g^{00}|}=0\,,
\label{bianchi.deform}$$ which is known as Proca condition for massive vector fields, constraints even further the independent components of $h_{\mu\nu}$.
For perturbations over a Minkowski background, becomes $$\begin{aligned}
f_0'\partial^ih_{0i}&=0\,,
\label{proca.deform.1}\\
(f_0'+\frac{1}{2}f_0'')\partial_kh^i_{\phantom.i}+\frac{1}{2}f_0'\partial_kh^0_{\phantom.0}&=0\,,
\label{proca.deform.2}\end{aligned}$$ where $f_0'\equiv f'(\sqrt{\hat{\gamma}})$ and $f_0''\equiv f''(\sqrt{\hat{\gamma}})$.
The study can be done easily using the following decomposition, common in the study of cosmological perturbations [@Brandenberger:1993zc], $$h_{\mu\nu}^{(s)}=
\left( \begin{array}{cc}
\phi & \partial_iB \\
\partial_jB & \eta_{ij}\chi+\partial_i\partial_jE
\end{array}\right)\,,\quad
h_{\mu\nu}^{(v)}=
\left( \begin{array}{cc}
0 & \psi_i \\
\psi_j & \partial_{(i}F_{j)}
\end{array} \right)\,,\quad
h_{\mu\nu}^{(t)}=
\left( \begin{array}{cc}
0 & 0 \\
0 & h_{ij}^{\mathrm{TT}}
\end{array} \right)\,,$$ where $h_{ij}^{\mathrm{TT}}$ is a transverse and traceless tensor, $\psi_i$, $F_j$ transverse vectors, and the rest scalars.
The scalar $E$ and the vector $F_j$ are fixed by our choice of gauge $\partial^ih_{ij}=0$. The analogous , of the Proca conditions fix instead the longitudinal component of $h_{0i}$, [*i.e.*]{}, the scalar $B$, and a combination of $h^0_{\phantom.0}$ and $h^i_{\phantom.i}$, that is the solution of .
The independent components of $h_{\mu\nu}$ are therefore: a transverse-traceless tensor $h_{ij}^{\mathrm{TT}}$, a transverse vector $\psi_i$, and a scalar, combination of $\phi$ and $\chi$.
Not all of these five independent components are propagating degrees of freedom. It was noticed in [@Gabadadze:2004iv] that a common feature of Lorentz-violating theories is the presence of an instantaneous interaction. An explicit study of the equations of motion for each and all components of the perturbation $h_{\mu\nu}$ would show that both the vector and the scalar appear without time derivatives in their equations of motion. Hence, they can not be identified as propagating degrees of freedom, rather as an instantaneous background.
The propagating degrees of freedom of the model are equivalent to the ones of standard GR, [*i.e.*]{}, a transverse-traceless tensor field. The difference is the presence of instantaneous interactions. It should not surprise, in fact breaking the gauge invariance of the action does not allow to remove from scattering cross sections components of the metric, like the Newtonian potential $h^0_{\phantom.0}$, which is instantaneous in nature. Thus, they would appear not only in the exchange of virtual particles, but they would also manifest as physical phenomena.
Stopping the Cosmological Expansion {#sect:stopping.exp}
===================================
Having established the properties of the model under discussion, we shall now consider more specific examples. We choose $f(\sqrt{\gamma})\equiv\gamma^{\alpha/2}$.
We are interested in particular to IR-modifications of gravity. We would like the deformation to be dominant only at late times during the cosmological evolution, [*i.e.*]{}, when $a(t)\gg1$. This can be achieved by assuming $\alpha>0$.
The equations of motion for this particular choice of $f(\sqrt{\gamma})$ are $$G_{\mu\nu}+\left[ \Lambda-(\alpha+1)\,m^2\,\gamma^{\alpha/2} \right]g_{\mu\nu}-
\alpha\,m^2\,\frac{\gamma^{\alpha/2}}{|g^{00}|}\,\delta_\mu^{\,0}\,\delta_\nu^{\,0}=0\,.
\label{eom.N}$$
On the FRW ansatz with zero spatial curvature, ${\mathrm{d}}s^2=-{\mathrm{d}}t^2+a(t)^2{\mathrm{d}}\vec x^2$, they become $$\frac{\dot a^2}{a^2}-\frac{\Lambda}{3}+\frac{m^2}{3}a^{3\alpha}=0\,.
\label{H}$$ the space-space component of is proportional to the time derivative of as it can be checked explicitly.
The additional term $a^{3\alpha}$ in the Friedman equation can be mimicked by a field with equation of state $w=-1-\alpha$. For $\alpha>0$, this fictitious field has $w<-1$.
When $m^2>0$, the energy density of the “field” has a wrong sign, while if $m^2<0$ its energy density is well-behaved. The former case will be discussed in detail in the following of the present section. The latter instead, being of a matter field with $w<-1$, will drive the cosmological expansion at an ever increasing acceleration rate, as opposed to de Sitter space-time of constant acceleration rate. Such a matter field (or deformation) will drive the Universe towards a final state sometimes called “Big Rip”: the scale factor will diverge in a finite time.
Our intuition on the cosmological evolution for the $m^2>0$ case is scarcer, and this case should not be treated as for a matter field. Let us therefore investigate the solution step by step[^2].
It might be proven useful to discuss the solution of the (modified) Friedmann equation , as of a classical point-particle moving in a potential $V(a)=(-\Lambda/3+m^2/3\,a^{3\alpha})a^2$ with zero total energy. The potential is pictured in Fig. \[fig:potential\], and it should be noticed the presence of a turning point at $a^\star=(\Lambda/m^2)^{1/3\alpha}$.
A classical point-particle moves down the potential hill from an initial position at $a=0$ until stops at $a^\star$, because of the “attractive force" generated by the $m^2$-term. After that position is reached, it rolls down in opposite direction towards $a=0$, which is reached in an infinite time.
This classical analog is easily translated into the cosmological evolution of the Universe. For small scale factor $a\sim0$, the dynamics is dominated by the cosmological constant and the Universe is in an approximate de Sitter phase. During this time of exponential expansion, the “attractive force” of the $m^2$-term will grow in intensity, until it will become dominant driving the cosmological evolution to a bounce at $a(t^\star)=a^\star=(\Lambda/m^2)^{1/3\alpha}$. Then, it will contract approaching at late times a (contracting) de Sitter phase.
We can solve analytically the equation of motion $$a(t)=\left(\frac{\Lambda}{m^2}\right)^{\frac{1}{3\alpha}}
\left(\cosh\frac{\sqrt{3\Lambda}\,\alpha}{2}(t-t_0)\right)^{-\frac{2}{3\alpha}}\,,
\label{sola.N}$$ where $t_0$ is an integration constant to be fixed by imposing the initial condition $a|_{t=0}=1$: $t_0=\frac{2}{\sqrt{3\Lambda}\alpha}\cosh^{-1}\sqrt{\frac{\Lambda}{m^2}}$.
This exact solution is plotted for some positive value of $\alpha$ alongside with the Ricci curvature $\mathcal{R}(t)$ in Fig. \[fig:aR\]. The features previously described are easily recognizable: at early and late time, the expansion is dominated by the cosmological constant and the Universe exponentially expands and contracts respectively.
The dynamics presented here is classical, and would be modified by quantum corrections. In particular it is known that, whenever the scalar curvature changes in time, particles are created via a phenomenon similar to the Hawking radiation for a Black Hole.
To understand it, let us consider a basis of particle creation/annihilation operators $\{A_k^\dagger\,,A_k\}$. The Fock space is defined by determining the vacuum state $|\Omega\rangle$, which is destroyed by all the annihilation operators, $A_k|\Omega\rangle=0$, and then populating it by acting with the creation operators onto $|\Omega\rangle$.
The operators are constructed by canonically quantizing the fields of the particular theory at hand, but in doing so we should assume a particular background metric. Hence, if the metric is time-dependent, so are the operators. The vacuum $|\Omega\rangle$ is annihilated by all $A_k$ at a given time, but not, in general, at every time. Thus as time goes by, the vacuum state will be in a superposition of particles, for $A_k(t)|\Omega\rangle\ne0$ at a generic late time.
Since the background we found is time-dependent, we would like to study the effects quantum particle production has on the cosmological evolution.
For sake of clarity we specialize to $\alpha=1$. Following [@Ford:1986sy], the energy density $\rho_\mathrm{q}$ created up to a time $\bar t$ is given by the following expression $$\rho_\mathrm{q}=-\frac{1}{32\pi^2a(\bar t)^4}\int_{-\infty}^{\bar t}{\mathrm{d}}t_1\int_{-\infty}^{\bar t}{\mathrm{d}}t_2
\log\left|\frac{\eta_1-\eta_2}{\eta_0}\right|\,{\mathcal{V}}'(t_1){\mathcal{V}}'(t_2)\,,
\label{partprod}$$ where ${\mathcal{V}}'(t)=(1-6\xi)(\dot a^2+a\,\ddot a)$ for the FRW ansatz and $\xi=1/6$ for conformally coupled fields; $\eta$ is the conformal time defined as $d\eta=dt/a(t)$ $$\eta=\eta_0\,I\left(\frac{1}{\cosh^2\frac{\sqrt{3\Lambda}}{2}(t-t_0)};-\frac{1}{3},\frac{1}{2}\right)
\quad\mbox{with }
\eta_0\equiv\left(\frac{m^2}{\Lambda}\right)^{1/3}
\frac{B(-\frac{1}{3},\frac{1}{2})}{\sqrt{3\Lambda}}\,,
\label{conftime}$$ and $I(z;a,b)$ is the regularized beta function $I(z;a,b)=B(z;a,b)/B(a,b)$, with $B(z;a,b)$ and $B(a,b)$ being the incomplete and complete beta function respectively.
For sake of simplicity we approximate the quantum corrected dynamics as if particles were generated all at once when $\mathcal{R}(\bar t)=0$. This rough approximation will not change the qualitative picture we will describe.
By evaluating the integral of up to $\bar t$, we find the energy density to be $$\rho_\mathrm{q}=\frac{\mathcal{I}}{288\pi^2}\Lambda^2\sim\Lambda^2\,,
\label{partprod.value}$$ where $\mathcal{I}$ is the numerical result of the (adimensional) integral of .
As it was noticed in [@Ford:1986sy], $\rho_\mathrm{q}$ does not depend on the time when the de Sitter expansion stops, but solely on the change in the scalar curvature. In our model, this means $\rho_\mathrm{q}$ is proportional to $\Lambda$, but not to $m^2$.
After $\bar t$, the cosmological evolution will change to include the presence of matter. The analytical solution of the Friedmann equation is $$a\!=\!\left(\frac{\Lambda}{2m^2}+\sqrt{\frac{\Lambda^2}{4\mu^4}+\frac{\rho_\mathrm{q}}{{M_\mathrm{Pl}}^2m^2}}\right)^{1/3}\!\!\!
{\operatorname{sn}}^{2/3}\!\left(\frac{\sqrt\frac{3m^2\rho_\mathrm{q}}{2{M_\mathrm{Pl}}^2\Lambda^2}(t-\bar t)}
{\sqrt{1+\!\sqrt{1\!+\!\frac{4m^2\rho_\mathrm{q}}{{M_\mathrm{Pl}}^2\Lambda^2}}}},
\frac{1+\!\sqrt{1\!+\!\frac{4m^2\rho_\mathrm{q}}{{M_\mathrm{Pl}}^2\Lambda^2}}}{1\!-\!\sqrt{1+\!\frac{4m^2\rho_\mathrm{q}}{{M_\mathrm{Pl}}^2\Lambda^2}}}\right),
\label{solarho.N}$$ where ${\operatorname{sn}}(u,\mu)$ is one of the Jacobi elliptic functions.
The plot of the scale factor $a(t)$ obtained by matching with at $t=\bar t$ is shown in Fig. \[fig:reheating\].
At $t\gtrsim\bar t$, the $m^2$-term is driving the cosmological evolution, in fact to have it to stop the exponential expansion driven by $\Lambda$ – that is to approach ${\mathcal{R}}(\bar t)=0$ – its “strength" has to be of the same order of the cosmological constant. Because $m^2$ is dominant, the cosmological evolution does not change significantly from the one we described without matter. The scale factor reaches a maximum, and then starts to contract. The dynamics starts to depart from at this point. Instead of approaching a contracting de Sitter phase with constant curvature $\Lambda$, at a distant but finite time in the future the Ricci curvature diverges: a “Big Crunch" occurs. At late times, when – during the contracting phase – $a(t)\ll1$, the matter density is driving the evolution, and therefore the singularity is unavoidable.
This picture is similar to a supercritical Universe, with the difference that no criticality condition for $\rho_\mathrm{q}$ is present. The expansion is not stopped by the (supercritical) matter density, but by the IR modification we introduced. Thus no matter how small $\rho_\mathrm{q}$ is, the dynamics ends with a singularity at a finite time.
The model, despite being classically stable, is unstable under quantum corrections.
Moreover, even considering that the “Big Crunch" will occur in a (parametrically) very distant future, the exit of the de Sitter expansion, which could be associated with a period of inflation, coincides with an era dominated by the $m^2$-term. Thus, the model would not follow the known cosmological evolution of our Universe.
Cyclic Universe {#sect:cyclic}
===============
As we have discussed in the previous section, a “Big Crunch" is unavoidable once quantum corrections are taken into account. One way to avoid the singularity could be to modify once more the gravitational Lagrangian.
By studying the effect the $m^2$-term has on the cosmological evolution, we understood that the modification we introduced slows down the expansion and ultimately stops it, after which the Universe goes through a contracting phase. If we could have a term that acts as the $m^2$-term, namely that slows down the cosmological evolution, but is dominant only when the size of the Universe is small, we might stop the collapse before the scale factor reaches zero, and the curvature diverges. This can be attained by having chosen the exponent $\alpha$ to be negative.
The most general (and minimal) Lagrangian is $${\mathcal{L}}=\sqrt{-g}\left[\mathcal{R}-2\Lambda+2m^2\gamma^{\alpha/2}+2k^2\gamma^{\beta/2}\right]\,,
\label{lagr.cyclic}$$ where $\alpha$ is positive and $\beta$ negative definite. On the FRW ansatz, the equation of motion is $$\frac{\dot a^2}{a^2}+\frac{m^2}{3}a^{3\alpha}+\frac{k^2}{3}a^{3\beta}-\frac{\Lambda}{3}=0\,.
\label{eom.cyclic}$$
We can study the dynamics of $a(t)$ by analogy with a classical particle moving in the potential $V=(m^2a^{3\alpha}-\Lambda+k^2a^{3\beta})a^2/3$. The potential is plotted in Fig. \[fig:pot-muk\], for $\beta<-2/3$, $\beta=-2/3$ and $\beta>-2/3$. The common feature of all the plotted potentials is the presence of two turning points $a_{(\pm)}$, between which the point-particle would oscillate back and forth.
The model is of a cyclic Universe that “eternally" oscillates between a minimum and a maximum size. Between the two extrema, the Universe is in a de Sitter phase, either contracting or expanding.
We can study analytically the dynamics for the choice of $\alpha=2/3$ and $\beta=-2/3$; the solution, shown in Fig. \[fig:cyclic\], is $$a(t)=a_{(-)}\left[1-{\operatorname{sn}}^2\left({\mathrm{i}}\sqrt{\frac{a_{(+)}^2-a_{(-)}^2}{3}}m(t-t_0),
\frac{a_{(-)}^2}{a_{(-)}^2-a_{(+)}^2}\right)\right]^{1/2}\,,
\label{sol.cyclic}$$ where $a_{(\pm)}\equiv(\Lambda\pm\sqrt{\Lambda^2-4k^2m^2})/2m^2$, and ${\operatorname{sn}}(u,\mu)$ is the Jacobi ${\operatorname{sn}}$ elliptic function with periodicity[^3] $$T=\frac{4\sqrt{3}}{(\Lambda^2-4k^2m^2)^{1/4}}
\left[{\mathrm{i}}K\left(\frac{a_{(-)}^2}{a_{(-)}^2-a_{(+)}^2}\right)
+K\left(\frac{a_{(+)}^2}{a_{(-)}^2-a_{(+)}^2}\right)\right]\,,
\label{period.cyclic}$$ where $K(\mu)$ is the complete elliptic integral of first kind.
The presence of the $k^2$-term effectively “screens" regions of small size. It slows and stops the contraction up to a non-zero scale factor $a_{(-)}$, in the same way that the $m^2$-term stops the expansion at large scale.
Naïvely we would expect this model to be stable under quantum correction, for $k^2$-term could stop the collapse in presence of matter too. It is like the new term creates a potential barrier at small scales, preventing the scale factor to reach zero size. The presence of matter would lower the barrier, but by tuning $k^2$ we can avoid its disappearance.
The loop-hole in this argument is the periodicity of the model. At each cycle new matter is generated by quantum corrections as seen in the previous section. Eventually $\rho_\mathrm{q}$ will “overcome" the barrier leading towards a “Big Crunch", unless the $k^2$-deformation is always dominant at small scale factors, no matter how big $\rho_\mathrm{q}$ is. This is obtained by taking $\beta<-1-w$, where $w$ is the equation of state for the particles generated through quantum effects. For radiation $w$ is equal to $1/3$, thus if $\beta<-4/3$ a minimal size $a_{(-)}$ is always present.
In the approximation the $m^2$-deformation and cosmological constant are negligible, which is always the case for $a(t)\ll1$, the minimal size is $$a_{(-)}\sim\left( \frac{k^2}{\rho_\mathrm{q}} \right)^{1/3|\beta+1+w|}\,.
\label{cyclic.minsize}$$
Obviously, because more and more matter is generated at each cycle, a time will come when $a_{(-)}$ will be of Planck size, and therefore our semi-classical description will break down.
Embedding Inflation
===================
In the following, we would like to depart slightly from the previous discussions, that have considered the effects of gravity modifications in a de Sitter space-time.
We will consider a generic inflationary model, and describe how the terms we introduced would effect inflation, and what kind of bounds we could have. We will, therefore, consider modified gravity coupled to a scalar field, the inflaton. The plot of a typical inflaton potential $V(\varphi)$ is sketched in Fig. \[fig:inflaton\].
When the inflaton is atop the plateau, its energy density is dominated by the potential, namely $\dot\varphi^2\ll V(\varphi)$, and approximately constant. This plays the role of the cosmological constant $\Lambda=V(\varphi)$. The space-time is exponentially expanding as long as the inflaton sits on the plateau.
This configuration is not stable, in fact it is energetically favorable for the scalar field to condense at the minimum of the potential, $\langle\varphi\rangle=\varphi_0$. When the inflaton condenses on its true vacuum, inflation ends, because the energy density is no longer a non-zero constant.
Once inflation is over, the inflaton decays more or less efficiently, depending on its coupling, into the Standard Model fields, reheating the Universe. From this point on, the cosmological evolution follows the Standard Model of Cosmology.
We can embed this general picture into our model and describe what constraints arise from cosmology and general requirements.
The Universe described by – solution of the Friedmann equation – goes through periodic exponential expansions and contractions. The requirement to have the usual description from the inflation framework within this model demands the scalar field $\varphi$ to condense into its true vacuum during an exponential expansion. Conversely, if it condensed during a contraction, the density perturbation originated during the inflationary period would be washed out at the unavoidable bounce.
Thus the inflaton can condense any time between $a_{(-)}$ and $a_{(+)}$, and this in turn provides a bound on the number of e-foldings $N_e$ as a function of the parameters $m^2$ and $k^2$ $$N_e\geqslant\log\frac{a_{(+)}}{a_{(-)}}=\log\frac{\Lambda+\sqrt{\Lambda^2-4k^2m^2}}{\Lambda-\sqrt{\Lambda^2-4k^2m^2}}\,.$$
This relation can also be read as a bound on the parameter $m^2$. Assuming $m^2$ to be much smaller than $\Lambda$ and $k^2$, we find the following bound $$m^2\lesssim\frac{\Lambda^2}{k^2}\exp[-2\,N_e]\,.
\label{mu2bound.cyclic}$$
The modification we introduced must be exponentially smaller than the de Sitter curvature in order to satisfy the cosmological bound on $N_e$ as it arises from matching the inflaton density perturbations with the anisotropies observed in theCMB ($N_e\gtrsim60$).
Few comments should follow. Firstly, is strictly an upper bound, for, if inflation end when the $m^2$-term is dominant, that is when $a(t)\sim a_{(+)}$, the cosmological evolution could not start in a radiation-dominated epoch, as in the Standard Model of Cosmology.
Secondly, whether in our model inflation could be eternal – see, for instance [@Linde:2004kg; @Guth:2007ng]. As we have throughly discussed in the previous sections, the additional potential-like term acts to stop the cosmological evolution bringing to an end any inflationary periods. It is not obvious how inflation could be eternal in a model that does not allow the space time to expand indefinitely.
In the standard picture of eternal inflation, false vacuum bubbles nucleate due to quantum fluctuations of the inflaton field. If the size of these bubbles is larger than their Hubble radius, they are causally disconnected from the “ambient” space. They will independently evolve and eventually pinch off.
As long as the deformation is negligible, the previous picture applies to our model as well. From the point of view of an observer living in one of these bubbles, they have no way to know anything about what is happening outside the horizon. It should be natural to expect that, once the “ambient” space bounces back, it keeps collapsing towards a de Sitter contracting phase, while the bubbles, unaware of anything happening outside their Hubble radius, expand until they pinch off. Hence new “baby” Universes are generated, and they will follow the same evolution of their “parent” Universe: the Universe will keep self-replicating and at any time at least one patch will be in an inflationary regime.
Even though every bubble can expand only till reaching a finite size before bouncing back towards a contracting phase, the total number of nucleated bubbles will be infinite, hence eternal inflation is a possible scenario in our model.
A Curiosity
-----------
Something curious happens for a particular, non-small value of $km/\Lambda$.
If $\Lambda=2km$, the two turning points $a_{(\pm)}$ are equal as it follows from their definitions. At this particular value, the solution of the Friedmann equation is of a flat space-time.
For the tuned value of $m^2$, an otherwise de Sitter space-time turns out to be effectively flat. This solution can be nicely understood from the analogous classical point-particle description. As we have already stressed, the cosmological constant acts as a repulsive force, whilst the $m^2$-term as an attractive one. When $\Lambda=2km$, those two forces balance exactly leading to the allowed range of $a(t)$ to shrink down to a point. From the cosmological view point, the scale factor is time-independent and therefore the space-time is effectively flat.
Some fine-tuning is required for this solution, therefore it would be interesting to understand how (if) the solution is reached in a generic inflationary model.
Let us consider a linear potential $V(\varphi)=\Lambda_0-\eta^2\varphi$ with $\eta^2\ll\Lambda_0$ to satisfy the slow-roll condition. In a usual inflationary model, inflation would be eternal, for no minimum of the potential of the scalar field is present. But an exponential expansion always stops when the $m^2$-term becomes dominant, so it is not clear what dynamics will follow in our model.
The scalar field follows its own equations of motion coupled to gravity $$\begin{aligned}
\ddot\varphi+3\frac{\dot a}{a}\dot\varphi+V'(\varphi)&=&0\,,\nonumber\\
\frac{\dot a^2}{a^2}+m^2a^2+k^2a^{-2}-\left[ \frac{1}{2}\dot\varphi^2+V(\varphi) \right]&=&0\,,
\label{eom.slowroll}\end{aligned}$$ where $H=\dot a/a$, the Hubble constant, is a friction force for $\varphi$. In the slow-roll approximation, the inflaton reaches critical but small velocity $|\dot\varphi|\sim|V'(\varphi)/3H|=\eta^2/3H\ll V(\varphi)$.
The effective cosmological constant is $\Lambda\sim V(\varphi)=\Lambda_0-\eta^2\varphi$. Since $\varphi$ is changing in time, so is $\Lambda$, but we can assume the slow-roll approximation to hold at any time, $\dot\varphi^2\ll V(\varphi)$.
The Hubble constant $H^2=(\dot a/a)^2$ is monotonically decreasing from its initial value $\Lambda_0$. The lower limit is at $\Lambda\sim2km$, where $\dot\varphi$ diverges. Thus, at this point, the inflaton is in a fast-roll regime with the kinetic energy dominating over the potential: $\dot\varphi^2\gg V(\varphi)$.
We can drop the potential from the equations of motion $$\begin{aligned}
\ddot\varphi+3\frac{\dot a}{a}\dot\varphi&=&0\,,\label{phi.fastroll}\\
\frac{\dot a^2}{a^2}+m^2a^2+k^2a^{-2}-\frac{1}{2}\dot\varphi^2&=&0\,.\label{a.fastroll}\end{aligned}$$
The dominant term is the kinetic energy – $\dot\varphi^2=\dot\varphi^2_0a^{-6}$ from – that drives the expansion until, at very late times, the $m^2$-term becomes dominant and leads to a contracting phase.
Because $\dot\varphi_0^2\gg m^2,k^2$, the dynamics is that of a supercritical system, and the Universe would eventually collapse in a “Big Crunch" singularity.
Massive Modification
====================
In the present section, we will describe the effects a massive modification – of the kind first described in [@Gabadadze:2004iv] – has when it is considered in conjunction with the deformation described in section \[sect:stopping.exp\].
Before starting on describing the cosmological evolution that arises when both deformations are taken into account, we should emphasize that the present modification is very different from the one so far described. We showed that the number of propagating degrees of freedom is unaltered when $\sqrt{-g}f(\sqrt\gamma)$ is introduced. Instead, in the case we shall present the number of degrees of freedom will be different.
The Lagrangian of the model we would like to discuss is $${\mathcal{L}}= \sqrt{-g}[{\mathcal{R}}-2\Lambda+2\mu_1^2\frac{(1-N)^2}{N}+2\mu_2^2\gamma^{\alpha/2}]\,,
\label{lagr.tuttifrutti}$$ with the following equations of motion $$\begin{aligned}
G_{\mu\nu}
&+&\left[ \Lambda-\mu_1^2\frac{(1-N)^2}{N}-\mu_2^2(\alpha+1)\gamma^{\alpha/2}
\right]g_{\mu\nu}+\nonumber\\
&+&\left[ \mu_1^2N(N^2-1)-\mu_2^2\alpha N^2\gamma^{\alpha/2}
\right]\delta_\mu^0\delta_\nu^0=0\,.
\label{eom.tuttifrutti}\end{aligned}$$ Notation is consistent with the one used before. $N$ is the lapse function and $\gamma_{ij}$ the induced spatial metric defined in . As we discussed in our previous letter [@Gabadadze:2004iv], the modification $\sqrt{-g}(1-N)^2/N$ is the only one quadratic in the lapse function, over which fluctuations do not present instabilities like tadpoles and the Hamiltonian of is bounded from below.
It is evident from the Hamiltonian $${\mathcal{H}}=\sqrt{\gamma}
\left[ NR^0+N_jR^j+2\Lambda N-2\mu_1^2(1-N)^2-2\mu_2^2N\gamma^{\alpha/2} \right]\,,
\label{ham.tuttifrutti}$$ that the shift function $N_j$ remains a Lagrange multiplier, while the lapse $N$ ceases to be it. The algebraic equation for $N$, that is $2\mu_1^2\,N=R^0/2+\Lambda+\mu_1^2-\mu_2^2\,\gamma^{\alpha/2}$, can be seen as a constraint for the degrees of freedom, thus on this background the gravitational field propagates three degrees of freedom, instead of the two as in the previous cases.
For this, the newly added modification is different from the ones we discussed previously: despite our analysis will follow closely the one of the previous sections, we should bear in mind that the models describe two very different fields. On the one hand, in section \[sect:stopping.exp\] the field is a massless tensor field propagating on a deformed (with respect to standard GR) background; on the other, in the present section we will discuss a model for a massive[^4] tensor field. The presence of the deformations will modify the cosmological evolution away from GR, as we will readily see in a moment.
On the ansatz ${\mathrm{d}}s^2=-N(t)^2{\mathrm{d}}t^2+a(t){\mathrm{d}}\vec x^2$ and after a bit of algebraic manipulation of the equations of motion , we find the following equation for the scale factor $a$ $$\dot{a}^2-\frac{\Lambda}{3}a^2
\left[ 1+2\frac{\mu_1^2}{\Lambda}(1-\sqrt{1+a^{-3}})-\frac{\mu_2^2}{\Lambda}a^{3\alpha} \right]=0\,,
\label{H.tuttifrutti}$$ where the derivative $\dot{a}$ is in respect to the proper time $d\tau=N\,dt$ and $N=\sqrt{1+a^{-3}}$, relation that can be derived directly from the equations of motion.
It is useful to discuss the solution of as for an analogous one-dimensional point particle moving in the potential $V(a)=-\frac{\Lambda}{3}a^2
\left[ 1+2\frac{\mu_1^2}{\Lambda}(1-\sqrt{1+a^{-3}})-\frac{\mu_2^2}{\Lambda}a^{3\alpha} \right]$.
The potential is depicted in Fig. \[fig:tuttifrutti\] for various values of $\mu_1$ and $\mu_2$. In particular, the continuous line is for $\mu_1=\mu_2=0$ and the dynamics is that of a de Sitter space-time with cosmological constant $\Lambda$, as expected. The other two limiting cases are when one of the two deformations is absent; we find that for $\mu_1=0$ – short dash curve – the cosmological expansion proceeds as for a de Sitter universe until a maximum size $a_{(+)}$, after which it goes through an exponentially contracting phase. That is, the solution we discussed previously in so much detail is recovered.
For $\mu_2=0$ – long dash curve – a minimum size $a_{(-)}$ for the scale factor emerges. The dynamics is very much similar to the one we described for the cyclic universe model of section \[sect:cyclic\] when only the $k^2$-deformation of is present: the cosmological evolution is that of a contracting de Sitter universe until the Universe reaches size $a_{(-)}$, after which it bounces back toward an expanding de Sitter phase.
When, instead, both deformations are present, the analogous potential $V(a)$ displays two turning points $a_{(\pm)}$. The dynamics can be read easily from it, and it is like the cyclic Universe model of section \[sect:cyclic\]: the cosmological evolution goes through subsequent periods of expansion and contraction, between the minimum and maximum sizes $a_{(-)}$ and $a_{(+)}$. Those values are the zeros of $V(a)$, and given that, when the size of the Universe is of order of $a_{(-)}$ the $\mu_2^2$-term is sub-dominant in respect to both the cosmological constant $\Lambda$ and the $\mu_1^2$-term (and vice-versa for $a_{(+)}$), they are approximately: $$\begin{aligned}
a_{(-)}&\sim&\left( \frac{4\mu_1^4}{\Lambda(\Lambda+4\mu_1^2)} \right)^\frac{1}{3}\,,\nonumber\\
a_{(+)}&\sim&\left( \frac{\Lambda}{\mu_2^2} \right)^\frac{1}{3\alpha}\,.
\label{minmax.tuttifrutti}\end{aligned}$$ The turning point $a_{(+)}$ is the same as the one we found in the model of section \[sect:stopping.exp\]. On the other hand, $a_{(-)}\sim(\mu_1/\Lambda)^{2/3}$, when $\mu_1^2\ll\Lambda$, is what would be the turning point for a deformation of the kind $\gamma^{\alpha/2}$ with $\alpha=-1/2$. This is not surprising. In the regime when $\mu_2$ is sub-dominant, the cosmic evolution is driven by the $\mu_1^2$-deformation. From the equations of motion , we find cosmology being driven by a term – $\sqrt{1+a^{-3}}\sim a^{-3/2}$ – that acts like $a^{3\alpha}$ for $\alpha=-1/2$, [*i.e.*]{}, equivalent to the deformation of section \[sect:stopping.exp\] for a particular choice of $\alpha$.
Again we should stress that, even though the backgrounds are alike, the fields propagating over them act very differently, for one carries two degrees of freedom, while the other three. Therefore, they describe two very different gravitational models.
Conclusions
===========
Models, that describe gravity beyond Einstein GR, have been focussed mainly on higher derivative deformations, like $f({\mathcal{R}})$ gravity for instance. In the present letter, we discussed a class of models, that modify gravity via potential-like terms.
The $f(\sqrt{\gamma})$ deformation we introduced does not add any derivatives of the metric to the action. Because of this, we argued that the propagating degrees of freedom are of a transverse-traceless tensor field, as in GR. We arrived to this conclusion by studying the perturbations of the metric in the Lagrangian formalism. Therein, it was also evidenced the presence of instantaneous interactions. This a-casual effect is characteristic of the models at hand, where Lorentz symmetry is broken explicitly. Their presence was firstly noted in [@Gabadadze:2004iv], and we remind to it for a more detailed discussion.
We then studied the exact cosmological solution for some particular choices of $f(\sqrt{\gamma})$. The main feature is that the introduced term acts to generically stop the cosmological evolution. Depending on the details of the model, a bounce is generated during either a period of contraction or one of expansion. Independently from these details though, particles are produced via quantum effects at the bounce. Quantum corrections destabilize an otherwise stable classical solution. The result is to create a future “Big Crunch” singularity: after a finite time the scalar curvature will diverge and the Universe will shrink to zero size.
In more specific sections, we discussed the effects and bounds on the parameters that arise when the model is embedded into a generic model of inflation. In particular, we noticed that the dimension-full parameter $m^2$ has to be exponentially smaller than the Hubble radius during inflation, so to satisfy the bounds imposed by the CMB.
Left to future investigations is the study of Schwarzschild solutions for the present model, like in [@Gabadadze:2005qy; @Gabadadze:2007as; @Dubovsky:2007zi]. It would be interesting to know what kind of effects the $f(\sqrt{\gamma})$-term has on a Schwarzschild-like solution, and to see how (if) it screens the gravitational field of a massive point-particle.
Also left out is the understanding of the UV completion of the model. We have always stressed that the introduced deformation $f(\sqrt{\gamma})$ should be thought as an effective term rising from some UV phenomena. The question is to exactly determine what kind of phenomena. Studies in this direction have flourished in the past few years, [@ArkaniHamed:2003uy] and [@Ganor:2006ub]-[@Bluhm:2008rf], and it would be interesting to have a fully consistent mechanism that breaks spontaneously the Lorentz symmetry and generates $f(\sqrt{\gamma})$-term at low energies.
Acknowledgements {#acknowledgements .unnumbered}
================
I would like to thank Gia Dvali, Gregory Gabadadze and Oriol Pujolàs for useful discussions. This work is supported by Physics Department Graduate Student Fellowship at NYU.
[^1]: Álvarez et al. in [@Alvarez:2006uu] studied a similar model, where the action is symmetric under transverse diffeomorphism (TDiff), that is $h_{\mu\nu}\rightarrow h_{\mu\nu}+\partial_{(\mu}\xi_{\nu)}$ with $\partial_\mu\xi^\mu=0$. They found that TDiff invariant theories contain an additional scalar field; in our case the action is invariant under spatial TDiff and no additional propagating degrees of freedom are present.
[^2]: The study of the cosmology dynamics in the presence of a matter field with negative energy density can be found in [@Babichev:2004qp].
[^3]: more exactly, the Jacobi ${\operatorname{sn}}(u,\mu)$ is doubly periodic in the complex plane, that is ${\operatorname{sn}}(u+4(K(\mu)+{\mathrm{i}}\,n\,K(1-\mu)),\mu)={\operatorname{sn}}(u,\mu)$ where $n\in\mathbb{Q}$. In the present case, $n$ is fixed by requiring the period to be real.
[^4]: in this context, we call “massive” a field that propagates a number of degrees of freedom different than two; it should be noted also that the action is not symmetric under the Lorentz symmetry and therefore the intuition of a massive tensor field propagating five degrees of freedom is not necessarily respected.
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abstract: 'Let $\mathbb{G}=\left(\mathbb{V},\mathbb{E}\right)$ be the graph obtained by taking the cartesian product of an infinite and connected graph $G=(V,E)$ and the set of integers $\mathbb{Z}$. We choose a collection $\mathcal{C}$ of finite connected subgraphs of $G$ and consider a model of Bernoulli bond percolation on $\mathbb{G}$ which assigns probability $q$ of being open to each edge whose projection onto $G$ lies in some subgraph of $\mathcal{C}$ and probability $p$ to every other edge. We show that the critical percolation threshold $p_{c}\left(q\right)$ is a continuous function in $\left(0,1\right)$, provided that the graphs in $\mathcal{C}$ are “well-spaced” in $G$ and their vertex sets have uniformly bounded cardinality. This generalizes a recent result due to Szabó and Valesin.'
author:
- 'Bernardo N. B. de Lima[^1]'
- 'Humberto C. Sanna$^*$'
title: A note on inhomogeneous percolation on ladder graphs
---
\#1\#2\#3[\_[[\#1]{}]{}([\#2]{},[\#3]{})]{}
[MSC numbers: 60K35, 82B43]{}
Introduction
============
In this note we address a particular case of the following problem: let $\mathbb{G}=\left(\mathbb{V},\mathbb{E}\right)$ be an infinite, connected graph, and $\mathbb{E}',\mathbb{E}''$ a decomposition of the edge set $\mathbb{E}$. Consider the Bernoulli percolation model in which the edges of $\mathbb{E}'$ are open with probability $p$ and the edges of $\mathbb{E}''$, regarded as the set of inhomogeneities, are open with probability $q$. If we define the quantity $p_{c}(q)$ as the supremum of the values of $p$ for which percolation with parameters $p,q$ does not occur, what can we say about the behavior of the function $q\mapsto p_{c}(q)$?
Perhaps one of the earliest works concerning this type of problem is due to Kesten, presented in [@Ke]. Considering the square lattice $\mathbb{L}^{2}=\left(\mathbb{Z}^{2},\mathbb{E}\right)$ and choosing $\mathbb{E}''$ and $\mathbb{E}'$ to be respectively the sets of vertical and horizontal edges, he proves that $p_{c}(q)=1-q$. Later on, in [@Z], Zhang also considers the square lattice, but with the edge set $\mathbb{E}''$ being only the vertical edges within the $y$-axis and $\mathbb{E}'=\mathbb{E}\setminus\mathbb{E}''$. He proves that for any $q<1$ there is no percolation at $p=1/2$, which implies that $p_{c}(q)$ is constant in the interval $[0,1)$. In the context of long-range percolation, the authors in [@LRV] consider an oriented, $d$-regular, rooted tree $\mathbb{T}_{d,k}$, where besides the usual set of “short bonds” $\mathbb{E}'$, there is a set $\mathbb{E}''$ of “long edges” of length $k\in\mathbb{N}$, pointing from each vertex $x$ to its $d^{k}$ descendants at distance $k$. They show that $q\mapsto p_{c}(q)$ is continuous and strictly decreasing in the region where it is positive. This conclusion is also achieved in [@CLS], where the authors consider the slab of thickness $k$ induced by the vertex set $\mathbb{Z}^{2}\times\{0,\ldots,k\}$, with $\mathbb{E}'$ and $\mathbb{E}''$ being respectively the sets of edges parallel and perpendicular to the $xy$-plane.
Another work that we mention is that of Iliev, Janse van Rensburg and Madras, [@IRM]. In the context of bond percolation in $\mathbb{Z}^{d}$, they define $\mathbb{E}''$ to be the set of edges within the subspace $\mathbb{Z}^{s}\times\{0\}^{d-s}$, $2{\leqslant}s<d$, and study the behavior of the quantity $q_{c}(p)$, defined analogously to $p_{c}(q)$. Among other standard results, the authors prove that $q_{c}(p)$ is strictly decreasing in the interval $[0,p_{c}]$, where $p_{c}$ is the percolation threshold in the homogeneous case.
More recently, in [@SV], Szabó and Valesin consider the same framework for $\mathbb{G}$, $\mathbb{E}'$ and $\mathbb{E}''$ and prove that, under this setting, $p_{c}(q)$ is continuous in the interval $(0,1)$. In their model, the graph $\mathbb{G}$ is obtained by taking the cartesian product of an infinite and connected graph $G=(V,E)$ and the set of integers $\mathbb{Z}$. The set of inhomogeneities $\mathbb{E}''$ is constructed by selecting a finite number of infinite “columns” and “ladders” and considering all the edges within it, and $\mathbb{E}'=\mathbb{E}\setminus\mathbb{E}''$.
It is in the spirit of [@SV] that we approach the aforementioned problem. More specifically, we extend their result in the sense that the continuity of $p_{c}(q)$ also holds when we set parameter $q$ on infinitely many “ladders” and “columns”, as long as they are “well spaced”.
\[subsec:ladder-graphs-result\]Inhomogeneous percolation on ladder graphs: definitions and result
-------------------------------------------------------------------------------------------------
Let $G=(V,E)$ be an infinite, connected and bounded degree graph with vertex set $V$ and edge set $E$. Starting from $G$, we define the graph $\mathbb{G}=(\mathbb{V},\mathbb{E})$, where $\mathbb{V}:= V\times\mathbb{Z}$ and $$\mathbb{E}:=\bigl\{\left\langle \left(u,n\right),\left(v,n\right)\right\rangle ;\left\langle u,v\right\rangle \in E,n\in\mathbb{Z}\bigr\}\cup\bigl\{\left\langle \left(w,n\right),\left(w,n+1\right)\right\rangle ; w\in V,n\in\mathbb{Z}\bigr\}.$$
Consider the Bernoulli percolation process on $\mathbb{G}$ described as follows. Every edge of $\mathbb{E}$ can be *open* or *closed*, states which shall be represented by $1$ and $0$, respectively. Hence, a typical percolation configuration is an element of $\Omega=\{0,1\}^{\mathbb{E}}$. As usual, the underlying $\sigma$-algebra is the one generated by the finite-dimensional cylinder sets of $\Omega$. For the probability measure of the process, we shall define it based on the rule specified below:
Fix a family of subgraphs $\left\{ G^{(r)}=\left(U^{(r)},E^{(r)}\right)\right\} _{r\in\mathbb{N}}$ of $G$, such that:
- $G^{(r)}$ is finite and connected for every $r\in\mathbb{N}$;
- $\dist G{U^{(i)}}{U^{(j)}}{\geqslant}3,\forall i\neq j$ (where $\dist G{.}{.}$ denotes the graph distance).
For each $r\in\mathbb{N}$, let $$\begin{split}\mathbb{E}^{\text{in},(r)} & :=\left\{ \left\langle \left(u,n\right),\left(v,n\right)\right\rangle ;\langle u,v\rangle\in E^{(r)},n\in\mathbb{Z}\right\} \\
& \quad\cup\left\{ \left\langle \left(w,n\right),\left(w,n+1\right)\right\rangle ; w\in U^{(r)},n\in\mathbb{Z}\right\}
\end{split}
\label{eq:internal-cylinder-infinite}$$
Given $p\in\left[0,1\right]$ and $q\in(0,1)$, declare each edge of $\mathbb{E}^{\text{in},(r)}$ open with probability $q$, independently of all other edges, for every $r\in\mathbb{N}$. Likewise, declare each edge of $\mathbb{E}\setminus\left(\cup_{r\in\mathbb{N}}\mathbb{E}^{\text{in},(r)}\right)$ open with probability $p$, also independently of any other edge. Let $\mathbb{P}_{q,p}$ be the law of the open edges for the process just described.
Having established our model, we turn our attention to state the main result of this section. First, a few definitions are required.
An *open path* in $\mathbb{G}$ is a set of distinct vertices $\left(v_{0},n_{0}\right),\left(v_{1},n_{1}\right),\ldots,\left(v_{m},n_{m}\right)$ such that for every $i=0,\ldots,m-1$, $\left\{ \left(v_{i},n_{i}\right),\left(v_{i+1},n_{i+1}\right)\right\} \in\mathbb{E}$ and is open. Given $\omega\in\Omega$ and $\left(v_{0},n_{0}\right),(v,n)\in\mathbb{E}$, we say that $(v,n)$ can be reached from $\left(v_{0},n_{0}\right)$ in the configuration $\omega$ either if the two vertices are equal or if there is an open path from $\left(v_{0},n_{0}\right)$ to $(v,n)$. Denote this event by $\left(v_{0},n_{0}\right)\leftrightarrow(v,n)$; we also use the notation $(v_0,n_0)\stackrel{S}{\leftrightarrow}(v,n)$ to denote the event where there exists an open path connecting $(v_{0},n_{0})$ and $(v,n)$ with all vertices belonging to the set $S$. The *cluster* $C_{(v,n)}$ of $(v,n)$ in the configuration $\omega$ is the set of vertices that can be reached from $(v,n)$. That is, $$C_{(v,n)}:=\left\{ \left(u,m\right)\in\mathbb{V}; (v,n)\leftrightarrow(u,m)\right\} .$$ In particular, we denote $C_{v}=C_{(v,0)}$. If $\left|C_{(v,n)}\right|=\infty$, we say that the vertex $(v,n)$ *percolates* and write $\left\{ (v,n)\leftrightarrow\infty\right\} $ for the set of such realizations.
Now, fix $v\in V$ and note that whether or not $\mathbb{P}_{q,p}\left((v,0)\leftrightarrow\infty\right)>0$ depends on the values of the parameters $p$ and $q$. With this in mind, we define the *critical curve* of our model as a function of $q$, namely $$p_{c}(q):=\sup\left\{ p\in[0,1];\mathbb{P}_{q,p}\left((v,0)\leftrightarrow\infty\right)=0\right\} .$$ One should observe that although the probability $\mathbb{P}_{q,p}\left((v,0)\leftrightarrow\infty\right)$ may vary from vertex to vertex, the value of $p_{c}(q)$ does not depend on the choice of $v\in V$, since $\mathbb{G}$ is connected.
What we shall prove in the next section is, in some sense, a generalization of Theorem 1 in [@SV]. It states that the continuity of $p_{c}(q)$ still holds, provided that the cardinality of the sets $U^{(r)}$ are uniformly bounded.
\[thm:continuity-pcrit\] If $\sup_{r\in\mathbb{N}}\left|U^{(r)}\right|<\infty$ and $\dist G{U^{(i)}}{U^{(j)}}{\geqslant}3,\forall i\neq j$, then $q\mapsto p_{c}(q)$ is continuous in $(0,1)$.
Just as we have based our non-oriented percolation model upon the one of Szabó and Valesin, we can generalize the oriented model also present in [@SV] in an analogous manner. By the same reasoning we shall present in the sequel, the continuity of the critical parameter for this new model also holds.
Proof of Theorem \[thm:continuity-pcrit\]
=========================================
Theorem \[thm:continuity-pcrit\] is a consequence of the following proposition:
\[prop:worse-q-better-p\]Fix $p,q\in(0,1)$ and $\lambda=\min(p,1-p)$. If $\sup_{r\in\mathbb{N}}\left|U^{(r)}\right|<\infty$ and $\dist G{U^{(i)}}{U^{(j)}}{\geqslant}3,\forall i\neq j$, for all $\varepsilon\in(0,\lambda)$, there exists $\eta=\eta(q,p,\varepsilon)>0$ such that if $\delta\in(0,\eta)$ then $$\mathbb{P}_{q+\delta,p-\varepsilon}\left((v,0)\leftrightarrow\infty\right){\leqslant}\mathbb{P}_{q-\delta,p+\varepsilon}\left((v,0)\leftrightarrow\infty\right)$$ for every $v\in V\setminus\left(\cup_{r\in\mathbb{N}}U^{(r)}\right)$.
Since $q\mapsto p_{c}(q)$ is non-increasing, any discontinuity, if exists, must be a jump. Suppose $p_{c}$ is discontinuous at some point $q_{0}\in(0,1)$, let $a=\lim_{q\downarrow q_{0}}p_{c}(q)$ and $b=\lim_{q\uparrow q_{0}}p_{c}(q)$. Then, for any $p\in(a,b)$, we can find an $\varepsilon>0$ such that for every $\delta>0$ we have $$\mathbb{P}_{q_{0}-\delta,p+\varepsilon}\left((v,0)\leftrightarrow\infty\right)=0<\mathbb{P}_{q_{0}+\delta,p-\varepsilon}\left((v,0)\leftrightarrow\infty\right)$$ for every $v\in V$, a contradiction according to Proposition \[prop:worse-q-better-p\].
The proof of Proposition \[prop:worse-q-better-p\] is based on the construction of a coupling which allows us to understand how a small change in the parameters of the model affects the percolation behavior. This construction is done in several steps. First, we split our edge set $\mathbb{E}$ in an appropriate disjoint family of subsets. Second, we define coupling measures on each of these sets in such a way that the increase of one parameter compensates an eventual decrease of the other in the sense of preserving the connections between boundary vertices of some “well chosen sets”, which will play an important role when we consider percolation on the graph $\mathbb{G}$ as a whole. Third, we verify that we can set the same parameters for each coupling provided that we can limit the size of the sets in which the inhomogeneities are introduced. Finally, we merge these couplings altogether by considering the product measure of each one. Most of these ideas are the same as in [@LRV] and [@SV]. To put it rigorously, we begin with some definitions.
For $r\in\mathbb{N}$, $n\in\mathbb{Z}$, let $L_{r}:=\left|U^{(r)}\right|$ and $$\begin{aligned}
\mathbb{V}_{n}^{(r)} & :=\left\{ (v,m)\in\mathbb{V}; \dist Gv{U^{(r)}}{\leqslant}1,(2L_{r}+2)n{\leqslant}m{\leqslant}(2L_{r}+2)(n+1)\right\} ;\\
\mathbb{E}_{n}^{(r)} & :=\left\{ e\in\mathbb{E}; \text{\ensuremath{e} has both endvertices in \ensuremath{\mathbb{V}_{n}^{(r)}}}\right\} \\
& \qquad\setminus\left\{ e\in\mathbb{E}; e=\langle(u,(2L_{r}+2)(n+1)),(v,(2L_{r}+2)(n+1))\rangle,\langle u,v\rangle\in\mathbb{E}\right\} ;\\
\mathbb{E}^{(r)} & :=\cup_{n\in\mathbb{Z}}\mathbb{E}_{n}^{(r)}.\end{aligned}$$ Note that
- $G$ has bounded degree and $\left|U^{(r)}\right|<\infty$ implies $\left(\mathbb{V}_{n}^{(r)},\mathbb{E}_{n}^{(r)}\right)$ is finite;
- $\mathbb{E}_{n}^{(r)}\cap\mathbb{E}_{n'}^{(r)}=\emptyset, \forall n\neq n'$;
- For any $n,n'\in\mathbb{Z},\ \mathbb{E}_{n}^{(r)}\cap\mathbb{E}_{n'}^{(r')}=\emptyset,\forall r\neq r'$. This is true since we are assuming $\dist G{U^{(r)}}{U^{(r')}}{\geqslant}3$, which implies $\dist{\mathbb{G}}{\mathbb{V}_{n}^{(r)}}{\mathbb{V}_{n'}^{(r')}}{\geqslant}1$.
Next, recall the definition of $\mathbb{E}^{\text{in},(r)}$ in (\[eq:internal-cylinder-infinite\]) and define $$\begin{aligned}
\mathbb{E}_{n}^{\partial,(r)} & :=\mathbb{E}_{n}^{(r)}\setminus\mathbb{E}^{\text{in},(r)}, & \mathbb{E}_{n}^{\text{in},(r)} & :=\mathbb{E}_{n}^{(r)}\cap\mathbb{E}^{\text{in},(r)}, & \mathbb{E}_{\mathcal{O}} & :=\mathbb{E}\setminus\left(\cup_{r\in\mathbb{N}}\mathbb{E}^{(r)}\right).\end{aligned}$$ One should also observe that $\mathbb{E}$ is a disjoint union of the sets defined above: $$\begin{aligned}
\mathbb{E} & =\mathbb{E}_{\mathcal{O}}\cup\bigcup_{r\in\mathbb{N}}\mathbb{E}^{(r)}\\
& =\mathbb{E}_{\mathcal{O}}\cup\bigcup_{r\in\mathbb{N}}\bigcup_{n\in\mathbb{Z}}\mathbb{E}_{n}^{(r)}\\
& =\mathbb{E}_{\mathcal{O}}\cup\bigcup_{r\in\mathbb{N}}\bigcup_{n\in\mathbb{Z}}\left(\mathbb{E}_{n}^{\partial,(r)}\cup\mathbb{E}_{n}^{\text{in},(r)}\right).\end{aligned}$$
Thus, letting $$\begin{aligned}
\Omega_{\mathcal{O}} & =\{0,1\}^{\mathbb{E}_{\mathcal{O}}}, & \Omega_{n}^{(r)} & =\{0,1\}^{\mathbb{E}_{n}^{(r)}}, & \Omega_{n}^{\partial,(r)} & =\{0,1\}^{\mathbb{E}_{n}^{\partial,(r)}}, & \Omega_{n}^{\text{in},(r)} & =\{0,1\}^{\mathbb{E}_{n}^{\text{in},(r)}},\end{aligned}$$ we can write $$\begin{aligned}
\Omega & =\Omega_{\mathcal{O}}\times\prod_{r\in\mathbb{N}}\prod_{n\in\mathbb{Z}}\Omega_{n}^{(r)}\\
& =\Omega_{\mathcal{O}}\times\prod_{r\in\mathbb{N}}\prod_{n\in\mathbb{Z}}\left(\Omega_{n}^{\partial,(r)}\times\Omega_{n}^{\text{in},(r)}\right).\end{aligned}$$
Denote $\partial\mathbb{V}_{n}^{(r)}$ to indicate the vertex boundary of $\mathbb{V}_{n}^{(r)},$ that is, $$\begin{aligned}
\partial\mathbb{V}_{n}^{(r)} & :=\left\{ (v,m)\in\mathbb{V}_{n}^{(r)}; \dist Gv{U^{(r)}}=1\right\} \\
& \;\cup\left( U^{(r)}\times\{(2L_{r}+2)n\}\right) \cup\left(U^{(r)}\times\{(2L_{r}+2)(n+1)\}\right) .\end{aligned}$$
Finally, for $A\subset\partial\mathbb{V}_{n}^{(r)}$ and $\omega_{n}^{(r)}\in\Omega_{n}^{(r)}$, define $$\begin{aligned}
C_{n}^{(r)}\left(A,\omega_{n}^{(r)}\right) :=\left\{ (v,m)\in\partial\mathbb{V}_{n}^{(r)}; \exists (v_{0},n_{0})\in A,(v,m)\stackrel{\mathbb{V}_{n}^{(r)}}{\leftrightarrow}(v_{0},n_{0})\right\} .\end{aligned}$$
Given any $A\subset\mathbb{E}$, let $\mathbb{P}_{.,.}\restriction_{A}$ the measure $\mathbb{P}_{.,.}$ restricted to the sample space $\{0,1\}^A$. With these definitions in hand, we are ready to establish the facts necessary for the proof of Proposition \[prop:worse-q-better-p\].
\[claim:coupling-world-outside\]Let $p,q\in(0,1)$, $\lambda=\min(p,1-p)$. For any $\varepsilon\in(0,\lambda)$ and $\delta\in(0,1)$ such that $(q-\delta,q+\delta)\subset [0,1]$, there exists a coupling $\mu_{\mathcal{O}}=(\omega_{\mathcal{O}},\omega'_{\mathcal{O}})$ on $\Omega_{\mathcal{O}}^{2}$ such that
- $\omega_{\mathcal{O}} \overset{(d)}{=}\mathbb{P}_{q+\delta,p-\varepsilon}\restriction_{\mathbb{E}_{\mathcal{O}}}$;
- $\omega'_{\mathcal{O}} \overset{(d)}{=}\mathbb{P}_{q-\delta,p+\varepsilon}\restriction_{\mathbb{E}_{\mathcal{O}}}$;
- $\omega_{\mathcal{O}} {\leqslant}\omega'_{\mathcal{O}}\quad\text{a.s.}$.
This construction is standard. Let $Z=(Z_{1},Z_{2})\in\Omega_{\mathcal{O}}^{2}$ be a pair of random elements defined in some probability space, such that the marginals $Z_{1}$ and $Z_{2}$ are independent on every edge of $\mathbb{E}_{\mathcal{O}}$ and assign each edge to be open with probabilities $p-\varepsilon$ and $\frac{2\varepsilon}{1-p+\varepsilon}$, respectively. Taking $\omega_{\mathcal{O}}=Z_{1}$ and $\omega'_{\mathcal{O}}=Z_{1}\lor Z_{2}$, define $\mu_{\mathcal{O}}$ to be the distribution of $(\omega_{\mathcal{O}},\omega'_{\mathcal{O}})$ and the claim readily follows.
The next lemma is one of the fundamental facts established in [@SV], so we refer the reader to the paper for a proof of the statement.
\[claim:coupling-inside-cylinders\]Let $p,q\in(0,1)$, $\lambda=\min(p,1-p)$, $r\in\mathbb{N}$. For any $\varepsilon\in(0,\lambda)$, there exists an $\eta^{(r)}>0$ such that if $\delta\in\left(0,\eta^{(r)}\right)$, there is a coupling $\mu_{n}^{(r)}=(\omega_{n}^{(r)},{\omega'}_{n}^{(r)})$ on $\Omega_{n}^{(r)}\times\Omega_{n}^{(r)}$ with the following properties:
- $\omega_{n}^{(r)} \overset{(d)}{=}\mathbb{P}_{q+\delta,p-\varepsilon}\restriction_{\mathbb{E}_{n}^{(r)}}$;
- ${\omega'}_{n}^{(r)} \overset{(d)}{=}\mathbb{P}_{q-\delta,p+\varepsilon}\restriction_{\mathbb{E}_{n}^{(r)}}$;
- $C_{n}^{(r)}\left(A,\omega_{n}^{(r)}\right)\subset C_{n}^{(r)}\left(A,{\omega'}_{n}^{(r)}\right)$ for every $A\in\partial\mathbb{V}_{n}^{(r)}$ almost surely.
Moreover, the value $\eta^{(r)}>0$ depends only on the choice of $q$, $p$, $\varepsilon$ and the graph $\left(\mathbb{V}_{0}^{(r)},\mathbb{E}_{0}^{(r)}\right)$.
The last ingredient used in the proof Proposition \[prop:worse-q-better-p\] is the following fact:
\[claim:uniform-bound\]If $\sup_{r\in\mathbb{N}}\left|U^{(r)}\right|<\infty$ then for any $\epsilon>0$ fixed, the sequence $\left\{ \eta^{(r)}\right\} _{r\in\mathbb{N}}$ in Lemma \[claim:coupling-inside-cylinders\] may be chosen bounded away from $0$.
From Lemma \[claim:coupling-inside-cylinders\], it follows that, for every $r\in\mathbb{N}$, the value $\eta^{(r)}>0$ depends on the choice of $q$, $p$, $\varepsilon$ and the graph $\left(\mathbb{V}_{0}^{(r)},\mathbb{E}_{0}^{(r)}\right)$. Note that while the values of $q$, $p$ and $\varepsilon$ are the same for different values of $r\in\mathbb{N}$, the graphs $\left(\mathbb{V}_{0}^{(r)},\mathbb{E}_{0}^{(r)}\right)$ may differ. However, there are only a finite number of possible graphs for $\left(\mathbb{V}_{0}^{(r)},\mathbb{E}_{0}^{(r)}\right)$ to assume. As a matter of fact, the graph $\left(\mathbb{V}_{0}^{(r)},\mathbb{E}_{0}^{(r)}\right)$ is obtained from the vertex set $U^{(r)}\cup\partial U^{(r)}$ and from the edges with both endpoints in $U^{(r)}\cup\partial U^{(r)}$. Since $\sup_{r\in\mathbb{N}}\left|U^{(r)}\right|<\infty$ and $G$ is of limited degree, we have that $M:=\sup_{r\in\mathbb{N}}\left|U^{(r)}\cup\partial U^{(r)}\right|<\infty$. Since there are only a finite number of graphs of limited degree with at most $M$ vertices, the claim regarding $\left(\mathbb{V}_{0}^{(r)},\mathbb{E}_{0}^{(r)}\right)$ follows, that is, $\eta:=\inf_{r\in\mathbb{N}}\eta^{(r)}>0$.
From Lemmas \[claim:coupling-inside-cylinders\] and \[claim:uniform-bound\] we have the following result: Let $p,q\in(0,1)$, $\lambda=\min(p,1-p)$. For any $\varepsilon\in(0,\lambda)$, there exists an $\eta>0$ such that if $\delta\in(0,\eta)$, there is a family of couplings $\left\{ \mu_{n}^{(r)}\right\} _{\substack{r\in\mathbb{N}\\
n\in\mathbb{Z}
}
}$, with each $\mu_{n}^{(r)}=(\omega_{n}^{(r)},{\omega'}_{n}^{(r)})$ defined on $\Omega_{n}^{(r)}\times\Omega_{n}^{(r)}$ and having the following property:
- $\omega_{n}^{(r)} \overset{(d)}{=}\mathbb{P}_{q+\delta,p-\varepsilon}\restriction_{\mathbb{E}_{n}^{(r)}}$;
- ${\omega'}_{n}^{(r)} \overset{(d)}{=}\mathbb{P}_{q-\delta,p+\varepsilon}\restriction_{\mathbb{E}_{n}^{(r)}}$;
- $C_{n}^{(r)}\left(A,\omega_{n}^{(r)}\right)\subset C_{n}^{(r)}\left(A,{\omega'}_{n}^{(r)}\right)$ for every $A\in\partial\mathbb{V}_{n}^{(r)}$ almost surely.
Let $\mu_{\mathcal{O}}$ be the coupling of Lemma \[claim:coupling-world-outside\]. Defining the coupling measure $\mu$ on $\Omega^{2}$ by $$\mu=\mu_{\mathcal{O}}\times\prod_{r\in\mathbb{N}}\prod_{n\in\mathbb{Z}}\mu_{n}^{(r)},$$ it is clear that if $(\omega,\omega')\sim\mu$, then $\omega\overset{(d)}{=}\mathbb{P}_{q+\delta,p-\varepsilon}$, $\omega'\overset{(d)}{=}\mathbb{P}_{q-\delta,p+\varepsilon}$, and almost surely $(v,0)\leftrightarrow\infty$ in $\omega$ implies $(v,0)\leftrightarrow\infty$ in ${\omega'}$, for every $v\in V\setminus\left(\cup_{r\in\mathbb{N}}U^{(r)}\right)$.
Acknowledgements {#acknowledgements .unnumbered}
================
BNBL is partially suported by CNPq. Both authors would like to thank CAPES for the financial support.
[999]{}
R.G. do Couto, B.N.B. de Lima and R. Sanchis, *Anisotropic Percolation on slabs*, Markov Processes and Related Fields [**20**]{}, 145-154, (2014). B.N.B. de Lima, L.T. Rolla and D. Valesin, *Monotonicity and phase transition diagram for multirange percolation on oriented trees*, To appear in Random Structures & Algorithms, (2019). G.K. Iliev, E.J. Janse van Rensburg and N. Madras, *Phase diagram of inhomogeneous percolation of a defect plane*, Journal of Statistical Physics [**158**]{}, 255-299, (2015).
Kesten H., *Percolation Theory for Mathematicians*, Birkhäuser, Boston, (1982). R. Szabó and D. Valesin, *Inhomogeneous percolation on ladder graphs*, ArXiv:1805.03419, (2018). Y. Zhang, *A note on inhomogeneous percolation*, Annals of Probability [**22**]{}, 803-819, (1994).
[^1]: Departamento de Matem[á]{}tica, Universidade Federal de Minas Gerais, Av. Antônio Carlos 6627 C.P. 702 CEP 30123-970 Belo Horizonte-MG, Brazil
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---
abstract: 'It is classically known that the proportion of lattice points visible from the origin via functions of the form $f(x)=nx$ with $n\in \mathbb{Q}$ is $\frac{1}{\zeta(2)}$ where $\zeta(s)$ is the classical Reimann zeta function. Goins, Harris, Kubik and Mbirika, generalized this and determined that the proportion of lattice points visible from the origin via functions of the form $f(x)=nx^b$ with $n\in \mathbb{Q}$ and $b\in\mathbb{N}$ is $\frac{1}{\zeta(b+1)}$. In this article, we complete the analysis of determining the proportion of lattice points that are visible via power functions with rational exponents, and simultaneously generalize these previous results.'
address:
- 'Department of Mathematics and Statistics, Williams College, United States'
- 'Department of Mathematics and Statistics, Harvey Mudd College, United States'
author:
- 'Pamela E. Harris'
- Mohamed Omar
title: Lattice point visibility on power functions
---
Introduction
============
In classical lattice point visibility, a point $(r,s)\in\mathbb{Z}\times\mathbb{Z}$ is said to be visible (from the origin) if there are no other integer lattice points on the line segment joining $(0,0)$ and $(r,s)$. One early result in this field showed that determining the proportion of lattice points visible from the origin is equivalent to determining the probability that two integers are relatively prime, which is classically known to be $1/\zeta(2) = 6/\pi^2$, where $$\zeta(s) = \sum_{n=1}^\infty 1/n^s = \prod_{p\;\text{prime}}\left(1-1/p^s\right)^{-1}$$ is the classical Riemann zeta function, as was first established (independently) by Cesàro and Sylvester in 1883 [@Ces1883; @Syl1883].
Since the introduction of lattice point visibility by Herzog and Stewart in 1971 [@HerzogStewart], the field and its generalizations continues to intrigue present day mathematicians [@Adhikari; @Adhikari.vip; @Apostol2000; @Chen; @Goins2017; @Laishram; @Laison; @Nicholson; @Schumer]. One recent generalization, by Goins, Harris, Kubik, and Mbirika, fixed a positive integer $b$ and defined a lattice point $(r,s)$ to be $b$-visible (from the origin) if the point lies on the graph of a power function $f(x)=nx^b$ with $n\in\mathbb{Q}$ and no other integer lattice point lies on this curve between $(0,0)$ and $(r,s)$ [@Goins2017]. Note that when $b=1$ this is the classical lattice point visibility setting. One of their main results ([@Goins2017 Theorem 1]) established that the proportion of $b$-visible integer lattice points is given by $1/\zeta(b+1)$. In this short note, we complete the analysis by determining the proportion of visible lattice points when the lines of sight are power functions with rational exponents. Our main result is as follows.
\[thm:main\] Fix a rational $b/a>0$ with $\gcd(a,b)=1$. Let ${\mathbb{N}}=\{1,2,3,\ldots\}$ and $\mathbb{N}_a$ be the set of integers of the form $\ell^a$ with $\ell \in \mathbb{N}$. Then the proportion of points in $\mathbb{N}_a\times\mathbb{N}$ that are $(b/a)$-visible is $\frac{1}{\zeta(b+1)}$, and the proportion of points in $\mathbb{N}_a\times\mathbb{N}$ that are $(-b/a)$-visible is $\frac{1}{\zeta(b)}$.
We note that we consider the density of visible points with respect to the set $\mathbb{N}_a\times\mathbb{N}$ because a point $(r,s)$ lies on the graph of the function $f(x)=nx^{b/a}$ with $n,b/a\in\mathbb{Q}$ only when $r=\ell^a$ for some integer $\ell\in\mathbb{N}$. If we instead considered the density of visible points with respect to ${\mathbb{N}}\times {\mathbb{N}}$, the density would be $0$. Indeed, since the points $(r,s)$ that are visible must have that $r$ is an $a$-th power, the visible points are a subset of the set $\{(\ell^a,s) \ : \ \ell,s \in {\mathbb{N}}\}$. This set has density $\frac{\sqrt[a]{N}}{N}$ if we restrict to points in the grid $[N] \times [N]$ (here $[N]:=\{1,2,\ldots,N\}$) and this tends to $0$ as $N \to \infty$ for values $a \geq 2$. However, when $a=1$, ${\mathbb{N}}_a = {\mathbb{N}}$ so there is no difference.
Main Result {#sec:2}
===========
It is important to note that the graph of $f(x)=nx^{b/a}$ with $n,b/a\in\mathbb{Q}$ passes through the origin only when $b/a>0$. In this case, we continue to consider lattice point visibility from the origin. In the case where $b/a<0$, we define visibility from a point at “infinity.” That is, since $f(x)=nx^{b/a}$ monotonically decreases to $0$ as $x$ goes to positive infinity we think of visibility from the point at infinity on the positive $x$-axis. We make these definitions precise shortly, but illustrate the concepts in Figure \[fig:two\_lines\_of\_sights\] where we provide lines of sight $f_1(x)=3x^{1/2}$ in blue and $f_2(x)=10x^{-1}$ in red. Note that the point $(1,3)$ is the only visible point on $f_1(x)$ (from the origin) and $(10,1)$ is the only visible point on $f_2(x)$ (from infinity).
; ;
coordinates [ (1,10) \[c\] (2,5) \[c\] (5,2) \[c\] (10,1) \[d\] (1,3) \[d\] (4,6) \[c\] (9,9) \[c\] ]{};
We begin by recalling the definition of a lattice point being $b$-visible when $b\in\mathbb{N}$.
\[def:bvisible\] Fix $b\in \mathbb{N}$. A point $(r,s)\in \mathbb{N}\times\mathbb{N}$ is said to be *$b$-visible* if it lies on the graph of $f(x)=nx^b$ for some $n\in \mathbb{Q}$ and there does not exist another point in $\mathbb{N} \times \mathbb{N}$ on the graph of $f(x)$ lying between $(0,0)$ and $(r,s)$.
Observe that for a fixed value of $b$, the point $(r,s)\in{\mathbb{N}}\times{\mathbb{N}}$ lies on exactly one power function $f(x)=nx^b$, the one in which $n=s/r^b$. This makes the previous definition well-defined because for a fixed $b$ it is impossible for the point $(r,s)$ to be $b$-visible with respect to one such power function and not $b$-visible with respect to another (since there is only one such function $f$). A similar observation holds for power functions with more general exponents (see Definition \[def:2\] and Definition \[def:3\]).
Moreover, suppose $(r,s)$ on the graph of $f(x)=nx^b$ is $b$-visible. Then any other point in $\mathbb{N} \times \mathbb{N}$ on the graph of $f$ has larger $y$-coordinate (when $f(x)$ is graphed in the $xy$-plane) because $f$ is monotonically increasing. This observation holds for Definition \[def:2\] as well, but is slightly different for Definition \[def:3\], as we address below. Such a perspective will be useful in our subsequent proofs.
We will also need the following proposition that gives a number-theoretic characterization of a point being $b$-visible.
\[prop:1\] Fix $b\in{\mathbb{N}}$. Then the lattice point $(r,s) \in \mathbb{N} \times \mathbb{N}$ is $b$-visible if and only if $s=nr^b$ for some $n\in\mathbb{Q}$ and there does not exists a prime $p$ such that $p|r$ and $p^b|s$.
Proposition \[prop:1\] is useful in the computations leading up to the proof of Theorem \[thm:main\]. The statement in this proposition is equivalent to the definition of $b$-visibility in [@Goins2017], but the link between the intuitive definition provided in Definition \[def:bvisible\] and the mathematical implications of Proposition \[prop:1\] was not established. We provide a proof of this result for sake of completion.
Suppose $(r,s) \in {\mathbb{N}}\times {\mathbb{N}}$ lies on the curve $f(x)=nx^{b}$.
For the forward direction, we prove the contrapositive and suppose that there is a prime $p$ for which $p|r$ and $p^b|s$. Then the point $(r/p,s/p^b)$ lies on the graph of $f(x)=nx^b$ and is between $(0,0)$ and $(r,s)$. Thus $(r,s)$ is not $b$-visible.
For the backward direction, we prove the contrapositive and suppose that $(r,s)$ is not $b$-visible. So there exists $(r',s')\in{\mathbb{N}}\times{\mathbb{N}}$ on the curve $f(x)=nx^b$ and lying between (0,0) and $(r,s)$. Observe that $$s'=f(r')=n(r')^b=\frac{s(r')^b}{r^b}.$$ Since $s'<s$, it must be the case that $\frac{r'}{r}<1$ so we can write $\frac{r'}{r}$ as a fraction $\frac{\alpha}{\beta}$ where $\alpha,\beta \in {\mathbb{N}}$, $\gcd(\alpha,\beta)=1$ and $\beta \geq 2$. Furthermore, we can assume $\alpha|r'$ and $\beta|r$. From this, $s' = \frac{s \alpha^b}{\beta^b}.$ Since $\beta \geq 2$ it has some prime factor $p$. Since $s'$ is an integer and $\gcd(\alpha,\beta)=1$, it must be the case that $p^b$ divides $s$. Furthermore, since $p|\beta$, $p|r$.
For the remainder of this article we assume that $b/a$ is rational with $\gcd(a,b)=1$. A natural definition for $(b/a)$-visibility when $b/a>0$ is as follows.
\[def:2\] Fix $b/a>0$. Suppose the point $(r,s)$ lies on the curve $f(x)=nx^{b/a}$. The point $(r,s)$ is said to be $(b/a)$-visible if there does not exist another point in $\mathbb{N} \times \mathbb{N}$ on the graph of $f(x)$ lying between $(0,0)$ and $(r,s)$.
Similar to how Proposition \[prop:1\] gave a number-theoretic characterization of Definition \[def:bvisible\], a similar paradigm occurs for Definition \[def:2\] and Proposition \[prop:2\].
\[prop:2\] Fix $b/a>0$. Then the lattice point $(r,s) \in \mathbb{N} \times \mathbb{N}$ is $(b/a)$-visible if and only if $s=nr^{b/a}$ for some $n \in \mathbb{Q}$, $r=\ell^a$ for some $\ell\in\mathbb{N}$, and $(\ell,s)$ is $b$-visible.
Notice first that if $(r,s)$ is $(b/a)$-visible then $r=\ell^a$ for some positive integer $\ell$ because $ r^{b/a}=s/n$ is rational. Now, if $(r',s')$ is another point with integer coordinates on the graph of $f(x)=nx^{b/a}$, then similarly $r'=(\ell')^a$ for some positive integer $\ell'$. We observe that $(\ell',s')$ lies on $g(x)=nx^b$. Now notice that $(\ell',s')$ lies on the graph of $g(x)=nx^b$ between $(0,0)$ and $(\ell,s)$ if and only if $(r',s')$ lies on the graph of $f(x)=nx^{b/a}$ between $(0,0)$ and $(r,s)$. So $(r,s)$ is $(b/a)$-visible if and only if $(\ell,s)$ is $b$-visible.
We now determine the proportion of $(b/a)$-visible points in ${\mathbb{N}}_a \times {\mathbb{N}}$ for $b/a>0$.
\[lem:2\] Fix $b/a>0$. Then the proportion of points in $\mathbb{N}_a\times\mathbb{N}$ that are $(b/a)$-visible is $\frac{1}{\zeta(b+1)}$.
Define $[N]:=\{1, 2, \ldots, N\} $ and $ [N]_a := \{1^a,2^a,\ldots,{\lfloor \sqrt[a]{N} \rfloor}^a\}.$ Let $r,s$ be two numbers picked independently with uniform probability in $[N]_a$ and $[N]$ respectively, and fix a prime $p$ in $[N]$. Let $P_{p,N}$ denote the probability that $r=\ell^a$ for some $\ell\in{\mathbb{N}}$, $p$ divides $\ell$ and $p^b$ divides $s$. By Propositions \[prop:1\] and \[prop:2\], and the independence of divisibility by different primes, it suffices to compute $\displaystyle \lim_{N \rightarrow \infty} \prod_{\substack{p\;\text{prime}\\p\leq N}} \left( 1 - P_{p,N} \right)$.
The integers $r \in [N]_a$ for which $p$ divides $r$ are precisely the integers in $\{1, 2, \ldots, {\lfloor \sqrt[a]{N} \rfloor}\}$ that are divisible by $p$, because $\ell^a$ is divisible by $p$ if and only if $\ell$ is. The number of such integers is $ \left\lfloor\frac{\lfloor \sqrt[a]{N} \rfloor}{p}\right\rfloor$. Thus the probability that $r \in [N]_a$ and $p$ divides $\ell$ is $\frac{1}{\lfloor \sqrt[a]{N} \rfloor}\left\lfloor\frac{\lfloor \sqrt[a]{N} \rfloor}{p}\right\rfloor$. There are $\left\lfloor{\frac{N}{p^b}}\right\rfloor$ integers in $[N]$ that are divisible by $p^b$; namely $p^b, 2p^b, \ldots, \left\lfloor{\frac{N}{p^b}}\right\rfloor p^b$. Thus the probability that $p^b$ divides $s$ is $\frac{1}{N}\left\lfloor{\frac{N}{p^b}}\right\rfloor$.
By mutual independence, the probability that $(r,s) \in [N]_a \times [N]$ has $p$ dividing $r$ and $p^b$ dividing $s$ is $P_{p,N} = \frac{1}{N \lfloor \sqrt[a]{N} \rfloor} \left\lfloor{\frac{\lfloor\sqrt[a]{N}\rfloor}{p}}\right\rfloor \left\lfloor{\frac{N}{p^b}}\right\rfloor$. Therefore, the probability that the pair $(r,s) \in [N]_a \times [N]$ has $p$ not dividing $r$, or $p^b$ not dividing $s$ is $1 - P_{p,N}$. Since $P_{p,N} \rightarrow \frac{1}{p^{b+1}}$ as $N \rightarrow \infty$, by multiplying over all of the primes we have that the probability that all primes $p$ satisfy that $p$ does not divide $r$ or that $p^b$ does not divide $s$ is $$\label{eqprime_new}
\displaystyle \lim_{N \rightarrow \infty} \prod_{\substack{p\;\text{prime}\\p\leq N}} \left( 1 - P_{p,N} \right) =
\prod_{p\;\text{prime}} \left(1-\frac{1}{p^{b+1}}\right)=\frac{1}{\zeta(b+1)}.\qedhere$$
In determining the density of visible points, we computed the limit as $N \to \infty$ of densities in the rectangles $[N]_a \times [N]$. One might suspect that determining the density by approximating ${\mathbb{N}}_a \times {\mathbb{N}}$ by other regions might give a different limit. Though this might be the case, our approach is consistent with similar density computations throughout the literature. For such examples, see [@Goins2017; @Pinsky].
We now consider rational exponents $-b/a<0$. Note that the corresponding power functions $f(x)=nx^{-b/a}$ with $n\in\mathbb{Q}$ do not go through the origin. To stay consistent with the pictorial interpretation that a visible lattice point should obstruct the visibility of all lattice points behind it, we think of viewing lattice points from $(\infty,0)$ instead of $(0,0)$. In this case, a visible point will consequently have a $y$-coordinate that is minimal among all lattice points lying on the graph of $f(x)=nx^{-b/a}$. This is because $f(x)$ is monotonically decreasing. Note that one could consider augmenting the perspective by viewing visibility from $(0,\infty)$ instead. This will recover an integer point whose $x$-coordinate is minimal. However, by replacing $-b/a$ with $-a/b$, this becomes equivalent to our perspective of viewing lattice points from $(\infty,0)$.
\[def:3\] Fix $-b/a<0$. Suppose the point $(r,s)$ lies on the curve $f(x)=nx^{-b/a}$. The point $(r,s)$ is said to be $(-b/a)$-visible if there does not exist another point in $\mathbb{N} \times \mathbb{N}$ on the graph of $f(x)$ lying between $(r,s)$ and $(\infty,0)$.
We begin analyzing $(-b/a)$-visibility when $a=1$ with the following number-theoretic characterization of $(-b)$-visibility.
\[prop:3\] Let $b$ be a positive integer. Then the lattice point $(r,s) \in \mathbb{N} \times \mathbb{N}$ is $(-b)$-visible from $(\infty,0)$ if and only if there does not exist a prime $p$ such that $p^b|s$.
The proof is very similar to that of Proposition \[prop:1\] so we omit it.
We now determine the proportion of $(-b)$-visible points in ${\mathbb{N}}\times {\mathbb{N}}$.
\[lem:3\] Fix an integer $-b<0$. Then the proportion of points in $\mathbb{N}\times\mathbb{N}$ that are $(-b)$-visible is $\frac{1}{\zeta(b)}$.
Fix a prime $p$ in $[N]$ and let $s$ be a number picked independently with uniform probability in $[N]$. Let $P_{p,N}$ denote the probability that $p^b$ divides $s$. By Proposition \[prop:3\], and the independence of divisibility by different primes, it suffices to compute $\displaystyle \lim_{N \rightarrow \infty} \prod_{\substack{p\;\text{prime}\\p\leq N}} \left( 1 - P_{p,N} \right)$.
There are $\left\lfloor{\frac{N}{p^b}}\right\rfloor$ integers in $[N]$ that are divisible by $p^b$; namely $p^b, 2p^b, \ldots, \left\lfloor{\frac{N}{p^b}}\right\rfloor p^b$. Thus the probability that $p^b$ divides $s$ is $\frac{1}{N}\left\lfloor{\frac{N}{p^b}}\right\rfloor$. Therefore, the probability that the pair $(r,s) \in [N] \times [N]$ has $p^b$ not dividing $s$ is $1 - P_{p,N}$. Since $P_{p,N} \rightarrow \frac{1}{p^{b}}$ as $N \rightarrow \infty$, by multiplying over all of the primes we have that the probability that all primes $p$ satisfy that $p^b$ does not divide $s$ is $$\displaystyle \lim_{N \rightarrow \infty} \prod_{\substack{p\;\text{prime}\\p\leq N}} \left( 1 - P_{p,N} \right) =
\prod_{p\;\text{prime}} \left(1-\frac{1}{p^{b}}\right)=\frac{1}{\zeta(b)}.\qedhere$$
We now prove our main theorem.
By Lemmas \[lem:2\] and \[lem:3\] the only remaining case to consider is negative non-integer exponents $-b/a$ where $a>1$. Suppose we have a point $(r,s)$ that is $(-b/a)$-visible. Then it would need to be the case that there is some rational $n$ such that $s=nr^{-b/a}$, which is equivalent to $sr^{b/a}=n$. Consequently $r=\ell^a$ for some positive integer $\ell$. From this, $s\ell^b=n$. By an argument similar to that in Proposition \[prop:2\], $(r,s)$ will then be $(-b/a)$-visible if and only if $(\ell,s)$ is $(-b)$-visible. A similar argument as in Lemma \[lem:2\] extends Lemma \[lem:3\] to give us a density of $\frac{1}{\zeta(b)}$.
[99]{}
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abstract: 'We investigate a cognitive radio system where secondary user (SU) relays primary user (PU) packets using two-phase relaying. SU transmits its own packets with some access probability in relaying phase using time sharing. PU and SU have queues of finite capacity which results in packet loss when the queues are full. Utilizing knowledge of relay queue state, SU aims to maximize its packet throughput while keeping packet loss probability of PU below a threshold. By exploiting structure of the problem, we formulate it as a linear program and find optimal access policy of SU. We also propose low complexity sub-optimal access policies, namely constant probability transmission and step transmission. Numerical results are presented to compare performance of proposed methods and study effect of queue sizes on packet throughput.'
author:
- 'Kedar Kulkarni, and Adrish Banerjee, [^1]'
bibliography:
- 'database\_full.bib'
title: '[On Optimal Spectrum Access of Cognitive Relay With Finite Packet Buffer[^2] ]{}'
---
Blocking probability, cognitive radio, finite capacity queue, optimal access, relaying
Introduction\[sec:Introduction\]
================================
In cognitive radio (CR) networks, secondary users (SUs) access spectrum allocated to primary users (PUs) in such a way that given quality-of-service (QoS) requirement of PUs is satisfied. Users store packets arriving from higher layers in queues before transmission over wireless link. Various works have studied SU packet throughput for non-cooperation scenarios where SU’s access probability is optimized under queue stability constraint of PU [@Simeone_stable_throughput; @Jeon; @fanous]. Cooperation between SU and PU improves throughput of both users as shown in [@Kompella] and [@Ashour]. These works consider queues of infinite storing capacity. In practice, queues are of finite size. If a queue is full, new packets cannot be admitted to the queue and are lost. Queueing performance of finite sized SU queue was studied in [@Chu]. In [@Krikidis_buffer; @Schober_buffer2; @Schober_buffer; @Teh_buffer; @Krikidis_buffer_survey], authors studied relay selection problem for finite buffer-aided relaying systems. These works considered dedicated relay nodes that do not have their own data to transmit. In [@Shafie] and [@Elmahdy], authors considered cooperative CR networks with finite sized relay queue and proposed packet admission control assuming that relay queue information is available at SU. However, underlying assumption in these works is that PU queue has infinite buffer length. Also, whole slot is used by the relay either for transmission or for reception. A relaying protocol where packet reception and transmission takes place in the same slot using time sharing, may enable the relay to improve its throughput by transmitting own packets more frequently.
In this paper, we investigate SU throughput in a cooperative CR system where SU relays failed packets of PU using two-phase relaying. SU transmits its own packets in the relaying phase using time sharing, with some access probability. Furthermore, we consider that both PU and SU have finite capacity queues. SU’s finite queue size affects cooperation offered to PU. Thus, queue sizes at both PU and SU impact PU’s packet loss. Our aim is to find optimal access policy of SU that maximizes SU packet throughput while satisfying PU’s packet loss constraint. Specifically, our contribution is as follows.
- We model PU and relay queues as discrete time Markov chains (DTMC). Using DTMC analysis, we characterize packet loss probability of PU and SU packet throughput.
- We formulate the problem of maximizing SU throughput under PU packet loss constraint, which is non-convex. By exploiting structure of the problem, we transform it into a linear programming (LP) problem over the feasible range of PU packet throughput. We also propose two low-complexity suboptimal access methods that transform original multi-dimensional problem into one dimensional problem.
- Finally, we present numerical results to study effect of queue sizes, path loss and time sharing on SU packet throughput. We also compare the performance with infinite capacity queue system under queue stability constraint.
System Model\[sec:System-Model\]
================================
As shown in Fig. \[fig:SysMod\], a PU source $\mathcal{P}$ transmits packets to PU destination $\mathcal{D}$ with assistance of a SU node $\mathcal{S}$ using two-phase relaying as done in [@Schober_buffer2]. Nodes $\mathcal{P}$ and $\mathcal{S}$ are equipped with packet queue $Q_{\mathcal{P}}$ of capacity $N_{P}$ and relay queue $Q_{\mathcal{S}}$ of capacity $N_{S}$ respectively. In a slot of duration $T$, $\mathcal{P}$ transmits its packet with power $P_{\mathcal{P}}$ for time $\beta T,\,\beta\in\left[0,\,1\right]$. If $\mathcal{D}$ fails to receive the packet, it is admitted to the relay queue at $\mathcal{S}$, provided that the packet is correctly received at $\mathcal{S}$ and the relay queue is not full. In relaying phase of duration $\left(1-\beta\right)T$, $\mathcal{S}$ relays the PU packet to $\mathcal{D}$ with power $P_{\mathcal{S}}$. With some access probability, SU also transmits its own packets to SU destination $\mathcal{R}$ using time sharing, that is, SU relays PU packet for duration $\alpha\left(1-\beta\right)T$ and transmits its own packet for duration $\left(1-\alpha\right)\left(1-\beta\right)T$, $\alpha\in\left[0,\,1\right]$. The access probability is $p_{n}$ when there are $n,\,0\leq n\leq N_{S}$ packets in $Q_{\mathcal{S}}$. If $Q_{\mathcal{S}}$ is empty, whole relaying duration $\left(1-\beta\right)T$ is used to transmit SU packet, with probability $p_{0}=1$.
We assume that all channels are independent block-fading in nature, that is, channel gains remain constant during a slot and vary independently from slot to slot. Channel power gain between source $s$ and destination $d$ is denoted as $g_{sd}$ and is exponentially distributed with mean $\sigma_{sd}^{2},\,s,d\in\left\{ \mathcal{P},\,\mathcal{S},\,\mathcal{D},\,\mathcal{R}\right\} $. The distance between $s$ and $d$ is denoted by $r_{sd}$ and path-loss exponent is denoted by $\kappa$. Additive white Gaussian noise (AWGN) at receivers has power $\sigma_{\mathcal{N}}^{2}$. PU and SU packets have fixed length of $\mathcal{B}$ bits. A packet is assumed to be delivered successfully to intended receiver if instantaneous channel capacity is greater than required transmission rate. Then probability of successful packet transmission is given by [@fanous]
$$\theta_{sd}\!=\!\Pr\!\left[\!\log_{2}\left(1+\frac{g_{sd}P_{s}r_{sd}^{-\kappa}}{\sigma_{\mathcal{N}}^{2}}\right)\!\geq\!\frac{\mathcal{B}}{WT_{s}}\!\right]\!\!=\!\exp\!\left(\!\!\frac{\!-\sigma_{\mathcal{N}}^{2}\!\left(\!2^{\frac{\mathcal{B}}{WT_{s}}}\!-\!1\!\right)}{P_{s}r_{sd}^{-\kappa}\sigma_{sd}^{2}}\!\!\right)\!\!,$$
where $P_{s}$ is transmit power, $T_{s}$ is transmission duration and $W$ is channel bandwidth. We denote probability of successful packet transmission on $s-d$ link without time sharing by $\theta_{sd},\,s,d\in\left\{ \mathcal{P},\,\mathcal{S},\,\mathcal{D},\,\mathcal{R}\right\} $. In case of time sharing, time available for relaying/transmission of PU and SU packets is less than $\left(1-\beta\right)T$. We use $\overline{\theta_{\mathcal{SD}}}$ and $\overline{\theta_{\mathcal{SR}}}$ to denote successful transmission probabilities in case of time sharing[^3]. As required transmission rate is higher, probabilities of successful packet transmission decrease. Thus, we have $\overline{\theta_{\mathcal{SD}}}<\theta_{\mathcal{SD}}$ and $\overline{\theta_{\mathcal{SR}}}<\theta_{\mathcal{SR}}$. Successful transmission probabilities on all links are known to the SU [@Jeon; @fanous; @Kompella].
![System model of SU relaying PU packets and transmitting own packets using two-phase relaying and time sharing\[fig:SysMod\]](SysMod2)
Queue blocking and packet loss
------------------------------
Packet arrival process at PU queue $Q_{\mathcal{P}}$ is Bernoulli with average rate $\lambda_{P}\in\left[0,\,1\right]$ packets/slot. A packet is removed from $Q_{\mathcal{P}}$ only when it is received at $\mathcal{D}$ or $\mathcal{S}$. A PU packet is admitted to the relay queue $Q_{\mathcal{S}}$ when all of the following events are true– 1) Packet transmission on $\mathcal{P}-\mathcal{D}$ link fails, 2) PU packet is successfully received at $\mathcal{S}$, and 3) $Q_{\mathcal{S}}$ is not full. Thus, packet departure rate at $Q_{\mathcal{P}}$, denoted as $\mu_{P}$, depends on channel between $\mathcal{P}-\mathcal{S}$ and state of $Q_{\mathcal{S}}$. When $Q_{\mathcal{P}}$ is full, new packets cannot be admitted to the queue and are dropped.
PU queue can be modeled as a discrete time Markov chain (DTMC) as shown in Fig \[fig:DTMC\](a) where states denote number of packets in PU queue. Let $w_{n},\,n=0,\,1,\dots,\,N_{P}$ be steady state probability of PU queue being in state $n$. Also let $\gamma=\frac{\lambda_{P}\left(1-\mu_{P}\right)}{\left(1-\lambda_{P}\right)\mu_{P}}$. Then we can write local balance equations for DTMC of $Q_{\mathcal{P}}$ as $$\begin{aligned}
w_{1} & =\frac{\gamma}{\left(1-\mu_{P}\right)}w_{0},\\
w_{n+1} & =\gamma\,w_{n},\,n=1,\,2,\dots,\,N_{P}-1.\end{aligned}$$ Noting that $w_{n}=\gamma^{n-1}w_{1},\,n>1$ and $\sum_{n=0}^{N_{P}}w_{n}=1$, we get probability of $Q_{\mathcal{P}}$ being empty as $$w_{0}=\begin{cases}
\frac{\left(1-\mu_{P}\right)\left(1-\gamma\right)}{1-\mu_{P}\left(1-\gamma\right)-\gamma^{N_{P}+1}} & \,\,\mbox{for}\,\,\gamma\neq1\\
\frac{1-\mu_{P}}{N_{P}+1-\mu_{P}} & \,\,\mbox{for}\,\,\gamma=1.
\end{cases}$$
Then probabilities of $Q_{\mathcal{P}}$ being non-empty and $Q_{\mathcal{P}}$ being full are given by $\nu_{1}=1-w_{0}$ and $\nu_{N_{P}}=w_{N_{P}}=\frac{1}{\left(1-\mu_{P}\right)}\gamma^{N_{P}}w_{0}$ respectively. To keep PU packet loss below a limit, probability of $Q_{\mathcal{P}}$ being full should remain below a threshold $\epsilon$, i.e. $\nu_{N_{P}}\leq\epsilon$. From Fig. \[fig:DTMC\](a), we observe that, any increase in $\mu_{P}$ would decrease probability of $Q_{\mathcal{P}}$ being full, that is, $\nu_{N_{P}}$ monotonically decreases in $\mu_{P}$. Thus, for given value of $\lambda_{P}$, we can find $\overline{\mu_{P}}\in\left[0,\,1\right]$ such that $\nu_{N_{P}}\left(\overline{\mu_{P}}\right)=\epsilon$, using Bisection method. Packet loss constraint $\nu_{N_{P}}\leq\epsilon$ can now be written as $\mu_{P}\geq\overline{\mu_{P}}$. SU should choose its access probability in such a way that packet loss constraint of PU is satisfied.
![Discrete time Markov chain (DTMC) model of (a) PU queue $Q_{\mathcal{P}}$ and (b) relay queue $Q_{\mathcal{S}}$\[fig:DTMC\]](Relay_markov3_2){width="8.2cm" height="5.4cm"}
Optimal spectrum access
=======================
DTMC of the relay queue $Q_{\mathcal{S}}$ is as shown in Fig. \[fig:DTMC\](b) where state $n$ denotes number of packets in relay queue at the end of receiving phase. Probability of a PU packet arriving at $Q_{\mathcal{S}}$ is $q$. When $Q_{\mathcal{S}}$ is in state $n$, SU transmits its own packets with probability $p_{n}$ using time sharing. Thus, with probability $\left(1-p_{n}\right)$, PU packet is relayed for duration $\left(1-\beta\right)T$ and with probability $p_{n}$, PU packet is relayed for duration $\alpha\left(1-\beta\right)T$. Then probability of a PU packet departing $Q_{\mathcal{S}}$ in state $n>0$ is $$r_{n}=\left(1-p_{n}\right)\theta_{\mathcal{SD}}+p_{n}\overline{\theta_{\mathcal{SD}}}=\theta_{\mathcal{SD}}-p_{n}\left(\theta_{\mathcal{SD}}-\overline{\theta_{\mathcal{SD}}}\right).\label{eq:mu_relay}$$
When PU is present, a packet is received at $\mathcal{S}$ with probability $\left(1-\theta_{\mathcal{PD}}\right)\theta_{\mathcal{PS}}$. Thus, we have $q=\nu_{1}\theta_{\mathcal{PS}}\left(1-\theta_{\mathcal{PD}}\right)$. For $1\leq n<N_{S}$, state transition from $n$ to $\left(n+1\right)$ occurs when packet transmission of a packet in $Q_{\mathcal{S}}$ fails and a new PU packet is received, which happens with probability $q\left(1-r_{n}\right)$. State transition from $n$ to $\left(n-1\right)$ occurs when a packet is successfully relayed and no new packet arrives, which happens with probability $\left(1-q\right)r_{n}$. Let $\pi_{n},\,n=0,\,1,\dots,\,N_{S}$ be steady state probability of $Q_{\mathcal{S}}$ being in state $n$. Then we write local balance equations as $$\begin{aligned}
\pi_{1} & =\frac{q}{\left(1-q\right)r_{1}}\pi_{0},\label{eq:balance1}\\
\pi_{n+1} & =\frac{q\left(1-r_{n}\right)}{\left(1-q\right)r_{n+1}}\pi_{n},\,\,n=1,\,2,\dots,\,N_{S}-1.\label{eq:balance2}\end{aligned}$$ For given values of $\lambda_{P}$ and $\mu_{P}$, steady state probabilities of relay queue can be calculated from (\[eq:balance1\]), (\[eq:balance2\]) using $$\sum_{n=0}^{N_{S}}\pi_{n}=1.\label{eq:normalize}$$
At the start of receiving phase, $Q_{\mathcal{S}}$ is full with probability $\pi_{N_{S}}\left(1-r_{N_{S}}\right)$. As a PU packet is admitted to $Q_{\mathcal{S}}$ only when $Q_{\mathcal{S}}$ is not full, we obtain packet departure rate of PU queue as $$\mu_{P}=\theta_{\mathcal{PD}}+\theta_{\mathcal{PS}}\left(1-\theta_{\mathcal{PD}}\right)\left[1-\pi_{N_{S}}\left(1-r_{N_{S}}\right)\right].\label{eq:mup_relay}$$
SU throughput maximization
--------------------------
When there are $n>0$ packets in $Q_{\mathcal{S}}$, SU transmits its own packet for duration $\left(1-\alpha\right)\left(1-\beta\right)T$ with probability $p_{n}$. If $Q_{\mathcal{S}}$ is empty, whole duration $\left(1-\beta\right)T$ is used to transmit SU packet with probability $p_{0}=1$. Given PU packet arrival rate $\lambda_{P}$, our objective is to maximize SU packet throughput while ensuring that packet loss probability of PU is kept below specified threshold. Thus, the optimization problem is written as $$\begin{aligned}
\max_{\boldsymbol{p},\,\boldsymbol{\pi},\,\mu_{P}} & \quad\mu_{S}=\theta_{\mathcal{SR}}\pi_{0}+\overline{\theta_{\mathcal{SR}}}\sum_{n=1}^{N_{S}}\pi_{n}p_{n}\label{eq:Opt_problem}\\
\mbox{s. t.} & \quad\mu_{P}\geq\overline{\mu_{P}},\label{eq:blocking_constraint}\\
& \quad0\leq p_{n},\pi_{n}\leq1,\,\,n=0,\,1,\dots,\,N_{S},\nonumber \\
& \quad p_{0}=1\nonumber \\
& \quad\mu_{P}=\theta_{\mathcal{PD}}+\theta_{\mathcal{PS}}\left(1-\theta_{\mathcal{PD}}\right)\left[1-\pi_{N_{S}}\left(1-r_{N_{S}}\right)\right],\nonumber \\
& \quad(\ref{eq:balance1}),\,(\ref{eq:balance2}),\,(\ref{eq:normalize}),\nonumber \end{aligned}$$ where $\boldsymbol{p}=\left[p_{0},\dots,\,p_{N_{S}}\right]^{T}$ and $\boldsymbol{\pi}=\left[\pi_{0},\dots,\,\pi_{N_{S}}\right]^{T}$. Optimization problem in (\[eq:Opt\_problem\]) is non-convex due to product terms of optimization variables $\pi_{n}$ and $p_{n}$. We transform it into a linear programming (LP) problem by exploiting structure of the problem.
Let $a_{n}=\pi_{n}p_{n}$. Then we have $a_{0}=\pi_{0}$ and $0\leq a_{n}\leq\pi_{n},\,\,n=1,\,2,\dots,\,N_{S}$. From (\[eq:normalize\]), we have $$0\leq\sum_{n=0}^{N_{S}}a_{n}\leq1.\label{eq:normalize_new}$$ Using (\[eq:mu\_relay\]), we can transform balance equations (\[eq:balance1\]) and (\[eq:balance2\]) as given in (\[eq:balance\_new\_1\]) and (\[eq:balance\_new\_2\]) on next page.
$$\theta_{\mathcal{SD}}\left(1-q\right)\pi_{1}-q\pi_{0}=\left(\theta_{\mathcal{SD}}-\overline{\theta_{\mathcal{SD}}}\right)\left(1-q\right)a_{1},\label{eq:balance_new_1}$$
$$\theta_{\mathcal{SD}}\left(1-q\right)\pi_{n+1}-q\left(1-\theta_{\mathcal{SD}}\right)\pi_{n}=\left(\theta_{\mathcal{SD}}-\overline{\theta_{\mathcal{SD}}}\right)\left(1-q\right)a_{n+1}+q\left(\theta_{\mathcal{SD}}-\overline{\theta_{\mathcal{SD}}}\right)a_{n},\,\,n=1,\dots,\,N_{S}-1.\label{eq:balance_new_2}$$
------------------------------------------------------------------------
Similarly, constraint in (\[eq:mup\_relay\]) can be written as $$\left(1-\theta_{\mathcal{SD}}\right)\pi_{N_{S}}+\left(\theta_{\mathcal{SD}}-\overline{\theta_{\mathcal{SD}}}\right)a_{N_{S}}=1-\frac{\mu_{P}-\theta_{\mathcal{PD}}}{\theta_{\mathcal{PS}}\left(1-\theta_{\mathcal{PD}}\right)}.\label{eq:mup_relay_new}$$ Thus, constraints (\[eq:balance1\]), (\[eq:balance2\]), (\[eq:mup\_relay\]) are transformed into constraints (\[eq:normalize\_new\]), (\[eq:balance\_new\_1\]), (\[eq:balance\_new\_2\]) and (\[eq:mup\_relay\_new\]) which are affine in $\pi_{n}$ and $a_{n}$. The optimization problem in (\[eq:Opt\_problem\]) is still non-convex in $\mu_{P}$. However, for a given value of $\mu_{P}$, the problem becomes a LP problem in variables $\boldsymbol{\pi}$ and $\boldsymbol{a}=\left[a_{0},\,a_{1},\dots,\,a_{N_{S}}\right]^{T}$ and is written as $$\begin{aligned}
\max_{\boldsymbol{\pi},\,\boldsymbol{a}} & \quad\theta_{\mathcal{SR}}a_{0}+\overline{\theta_{\mathcal{SR}}}\sum_{n=1}^{N_{S}}a_{n}\label{eq:opt_prob_new}\\
\mbox{s. t.} & \quad0\leq\pi_{n}\leq1,\,\,n=0,\thinspace1,\dots,\,N_{S},\nonumber \\
& \quad a_{0}=\pi_{0},\,\,0\leq a_{n}\leq\pi_{n},\,\,n=1,\,2,\dots,\,N_{S},\nonumber \\
& \quad(\ref{eq:normalize}),\,(\ref{eq:normalize_new}),\,(\ref{eq:balance_new_1}),\,(\ref{eq:balance_new_2}),\,(\ref{eq:mup_relay_new}).\nonumber \end{aligned}$$ From (\[eq:mup\_relay\]) and (\[eq:blocking\_constraint\]), we see that the feasible values of $\mu_{P}$ are $$\begin{aligned}
\max\left\{ \overline{\mu_{P}},\,\theta_{\mathcal{PD}}+\theta_{\mathcal{PS}}\overline{\theta_{\mathcal{SD}}}\left(1-\theta_{\mathcal{PD}}\right)\right\} \nonumber \\
& \hspace{-3cm}\leq\mu_{P}\leq\theta_{\mathcal{PD}}+\theta_{\mathcal{PS}}\left(1-\theta_{\mathcal{PD}}\right).\label{eq:feasible}\end{aligned}$$ The linear program in (\[eq:opt\_prob\_new\]) is solved over feasible values of $\mu_{P}$. Value of $\mu_{P}$ that corresponds to the maximum SU packet throughput is chosen. From optimal $a_{n}$ and $\pi_{n}$, optimal SU access probabilities are found as $p_{n}=\frac{a_{n}}{\pi_{n}},\,\,n=0,\,1,\dots,\,N_{S}$. We have used CVX package for MATLAB [@cvx] to solve (\[eq:opt\_prob\_new\]) in polynomial complexity.
Suboptimal methods
------------------
From (\[eq:mu\_relay\]), (\[eq:balance1\]), (\[eq:balance2\]) and (\[eq:normalize\]), we get steady state probabilities for $Q_{\mathcal{S}}$ as $$\begin{aligned}
\pi_{0} & =\left[1+\frac{1}{r_{1}}\sum_{n=1}^{N_{S}}\left(\frac{q}{1-q}\right)^{n}\prod_{m=1}^{n-1}\left(\frac{1-r_{m}}{r_{m+1}}\right)\right]^{-1},\label{eq:relay_empty}\\
\pi_{n} & =\left[\left(\frac{q}{1-q}\right)^{n}\frac{1}{r_{1}}\prod_{m=1}^{n-1}\left(\frac{1-r_{m}}{r_{m+1}}\right)\right]\pi_{0},\,\,n>0.\end{aligned}$$ It can be proven that $\pi_{0}$ is monotonically decreasing function of access probability $p_{n},\,n=1,\,2,\dots,\,N_{S}$. Also we can prove that $\pi_{n},\,0<n\leq N_{S}$ is a monotonically increasing function of $p_{m},\,m\leq n$ and a monotonically decreasing function of $p_{m},\,m>n$. Intuitively, this can be explained from Fig. \[fig:DTMC\](b) as follows. As access probability $p_{m}$ increases, packet departure rate of $Q_{\mathcal{S}}$ decreases. Thus, more packets get queued up in $Q_{\mathcal{S}}$. Hence, the probability of $Q_{\mathcal{S}}$ having more than $m$ packets increases, while probability of $Q_{\mathcal{S}}$ having packets less than or equal to $m$ decreases. Using this nature, we propose low complexity suboptimal methods that simplify $\left(N_{S}+1\right)$ dimensional problem in (\[eq:Opt\_problem\]) to a one-dimensional problem.
### Constant probability transmission (CPT)
In this method, SU transmits its own packets with a fixed probability $p$ when there are $n>0$ packets in relay queue. Thus, we have $$p_{n}=\begin{cases}
1 & \,\mbox{for}\,n=0\\
p & \,\mbox{otherwise}.
\end{cases}\label{eq:prob_const_trans}$$ In this case, SU packet throughput is $\mu_{S}=\theta_{\mathcal{SR}}\pi_{0}+\overline{\theta_{\mathcal{SR}}}p\sum_{n=1}^{N_{S}}\pi_{n}$. Using $\sum_{n=1}^{N_{S}}\pi_{n}=1-\pi_{0}$, we can write the problem of maximizing $\mu_{S}$ for fixed $\mu_{P}$ as $$\begin{aligned}
\max_{p\in\left[0,\,1\right]} & \quad\overline{\theta_{\mathcal{SR}}}p+\pi_{0}\left(\theta_{\mathcal{SR}}-\overline{\theta_{\mathcal{SR}}}p\right)\\
\mbox{s. t.} & \quad(\ref{eq:mup_relay}),\,(\ref{eq:balance1}),\,(\ref{eq:balance2}),\,(\ref{eq:normalize}),\,(\ref{eq:prob_const_trans}).\end{aligned}$$ The term $\pi_{0}\left(\theta_{\mathcal{SR}}-\overline{\theta_{\mathcal{SR}}}p\right)$ is monotonically decreasing in $p$ while term $\overline{\theta_{\mathcal{SR}}}p$ is increasing in $p$. Thus, there exists a unique $p$ that maximizes $\mu_{S}$. Optimal solution can be found using Interval halving method with complexity $\mathcal{O}\left(1\right)$.
### Step transmission (ST)
In this method, SU transmits its own packets using time sharing with probability $1$ until length of $Q_{\mathcal{S}}$ reaches a threshold $N_{th}$. Once it crosses $N_{th}$, the relaying phase duration of $\left(1-\beta\right)T$ is used only to relay PU packets. Thus, we have $$p_{n}=\begin{cases}
1 & \,\mbox{for}\,n\leq N_{th}\\
0 & \,\mbox{otherwise}.
\end{cases}\label{eq:prob_step_trans}$$ In this case, the objective is $$\begin{aligned}
\max_{N_{th}\in\left\{ 0,\,1,\dots,\,N_{S}\right\} } & \quad\theta_{\mathcal{SR}}\pi_{0}+\overline{\theta_{\mathcal{SR}}}\sum_{n=1}^{N_{th}}\pi_{n}\\
\mbox{s. t.} & \quad(\ref{eq:balance1}),\,(\ref{eq:balance2}),\,(\ref{eq:normalize}),\,(\ref{eq:mup_relay}),\,(\ref{eq:prob_step_trans}).\end{aligned}$$ With increasing $N_{th}$, $\pi_{0}$ decreases while number of terms in summation increase. If value of $\overline{\theta_{\mathcal{SR}}}$ is very low compared to $\theta_{\mathcal{SR}}$, decrease in $\pi_{0}$ is significant and $\mu_{S}$ initially decreases. But as $\pi_{0}$ approaches zero, $\mu_{S}$ increases due to increasing value of $\overline{\theta_{\mathcal{SR}}}\sum_{n=1}^{N_{th}}\pi_{n}$. For high value of $\overline{\theta_{\mathcal{SR}}}$, $\mu_{S}$ increases with increasing $N_{th}$. Throughput drops to zero when $\pi_{N_{S}}$ increases to such a value that constraint (\[eq:mu\_relay\]) cannot be satisfied. Value of $N_{th}$ that maximizes $\mu_{S}$ can be found by linear search with complexity $\mathcal{O}\left(N_{S}\right)$.
Suboptimal methods run over all feasible values of $\mu_{P}$ given in (\[eq:feasible\]) and the value that corresponds to maximum SU packet throughput is chosen.
Numerical results and discussion
================================
Parameter values used to plot results are as follows. Transmit power is $P_{\mathcal{P}}=P_{\mathcal{S}}=0.1\,\mbox{W}.$ Frame duration is $T=100\,\mbox{ms}$. Time sharing factors are $\beta=\alpha=0.5$ unless stated otherwise. All channels have average channel gain $\sigma_{sd}^{2}=-10\,\mbox{dB},\,s,d\in\left\{ \mathcal{P},\,\mathcal{S},\,\mathcal{D},\,\mathcal{R}\right\} $. Noise power is $\sigma_{\mathcal{N}}^{2}=10^{-5}\,\mbox{W}$. We take $\mathcal{B}/W=3\times10^{-3}\,\mbox{bits/Hz}$. We consider $r_{\mathcal{PS}}=r_{\mathcal{SD}}=r_{\mathcal{SR}}=100\,\mbox{m}$. Path loss exponent is $\kappa=2$. Packet loss probability threshold is $\epsilon=0.01$.
### Throughput region
![SU packet throughput $\mu_{S}$ versus PU packet arrival rate $\lambda_{P}$ for $N_{S}=10$ and $N_{P}\rightarrow\infty$. \[fig:SUvsLem\_eps\]](MuVsLemp)
Fig. \[fig:SUvsLem\_eps\] plots throughput region of proposed cooperation model. As $\lambda_{P}$ increases, higher $\mu_{P}$ is required to satisfy PU packet loss constraint. To support high $\mu_{P}$, SU lowers its access probability. Thus, $\mu_{S}$ decreases with increase in $\lambda_{P}$. As $\lambda_{P}$ increases further, constraint (\[eq:blocking\_constraint\]) becomes infeasible, at which point $\mu_{S}$ drops to zero. We also see that performance of constant probability transmission (CPT) method and step transmission (ST) method is close to that of optimal method. Hence, the suboptimal methods are good low-complexity alternatives to the optimal method.
As a baseline for comparison, we also plot throughput region of cooperative relaying method (CRM) in [@Elmahdy]. In CRM, PU utilizes whole undivided frame duration $T$ for transmission/reception and optimizes SU access probability under PU queue stability constraint. In contrast, two-phase relaying model dedicates $\beta T$ duration for reception in each slot. Thus, in CRM, probabilities of successful transmission on $\mathcal{P}-\mathcal{D}$ and $\mathcal{S}-\mathcal{R}$ links are higher, resulting in better performance of CRM at low and high values of $\lambda_{P}$. But in mid-range of $\lambda_{P}$, two-phase relaying benefits by gaining time to transmit own packets as SU relays PU packets in the same slot.
### Effect of queue sizes
![Effect of queue sizes on SU packet throughput $\mu_{S}$ for (a) $N_{S}=10$ (b) $N_{P}=100$ and $\lambda_{P}=0.5$ \[fig:SUvsLem\_NP\]](MuVsN)
Fig. \[fig:SUvsLem\_NP\](a) plots SU packet throughput $\mu_{S}$ against PU queue capacity $N_{P}$ for different values of $\mathcal{P}-\mathcal{D}$ channel gains. Low values of $N_{P}$ cannot satisfy packet loss constraint in (\[eq:blocking\_constraint\]). Packet throughput achieved in such infeasible cases in zero. Increasing $N_{P}$ decreases $\overline{\mu_{P}}$ which is the minimum PU departure rate required to satisfy packet loss constraint. This allows SU to transmit its own packets with higher access probabilities. Thus, $\mu_{S}$ increases with increase in $N_{P}$. As $N_{P}$ increases further, decrease in $\overline{\mu_{P}}$ is insignificant. Access probabilities of SU become constant and in turn $\mu_{S}$ becomes constant. For high value of $\sigma_{\mathcal{PD}}^{2}$, PU packet arrival rate at $Q_{\mathcal{S}}$ is less which allows higher SU access probabilities. Thus, $\mu_{S}$ increases as $\sigma_{\mathcal{PD}}^{2}$ increases.
Fig. \[fig:SUvsLem\_NP\](b) shows an interesting tradeoff involving relay queue capacity $N_{S}$. Increase in $N_{S}$ allows SU to transmit its own packets with higher probability. Also, from (\[eq:relay\_empty\]), we observe that increase in $N_{S}$ decreases probability of relay queue being empty $\pi_{0}$. For high values of $\sigma_{\mathcal{PD}}^{2}$, PU packet arrival rate at $Q_{\mathcal{S}}$ is less. In this case, decrease in $\pi_{0}$ (and subsequent decrease in $\theta_{\mathcal{SR}}\pi_{0}$) is significant compared to increase in $\mu_{S}$ due to higher access probability. Thus, $\mu_{S}$ decreases with increase in $N_{S}$. For low values of $\sigma_{\mathcal{PD}}^{2}$, PU packet arrival rate at $Q_{\mathcal{S}}$ is more. In this case, increase in SU throughput due to higher access probability is significant. But as $N_{S}$ increases further, $\pi_{0}$ approaches zero and $\sum_{n=1}^{N_{S}}\pi_{n}p_{n}$ decreases. Thus, with increasing $N_{S}$, $\mu_{S}$ initially increases and then gradually decreases.
### Effect of distance
![Effect of distance between PU source and relay $r_{\mathcal{PS}}$ on SU packet throughput $\mu_{S}$ for $N_{P}=100$, $N_{S}=10$, $\lambda_{P}=0.5$ and $r_{\mathcal{PD}}=200\,\mbox{m}$.\[fig:SUvsDist\]](MuVsDistance)
Fig. \[fig:SUvsDist\] plots SU throughput against distance between PU source and SU source $r_{\mathcal{PS}}$. Here, we assume that $\mathcal{D}$ and $\mathcal{R}$ are in close vicinity and lie on the line connecting $\mathcal{P}$ and $\mathcal{S}$. Then for given $r_{\mathcal{PD}}$, we have $r_{\mathcal{SD}}=r_{\mathcal{SR}}=r_{\mathcal{PD}}-r_{\mathcal{PS}}$. When $\mathcal{P}-\mathcal{D}$ channel is weak, PU packet arrival rate at SU is high. Thus, probability of $Q_{\mathcal{S}}$ being full is high. As SU moves away from PU source, $\theta_{\mathcal{PS}}$ decreases, while $\theta_{\mathcal{SD}}$, $\theta_{\mathcal{SR}}$, $\overline{\theta_{\mathcal{SD}}}$ and $\overline{\theta_{\mathcal{SR}}}$ increase. This increases SU throughput. As $r_{\mathcal{PS}}$ increases further, $\mu_{P}$ decreases to such a point that queue blocking constraint cannot be satisfied for given $\lambda_{P}$. In this infeasible region, $\mu_{S}$ is zero. When $\mathcal{P}-\mathcal{D}$ channel is strong, decrease in $\mu_{P}$ due to increasing $r_{\mathcal{PS}}$ is insignificant. Thus, SU throughput $\mu_{S}$ keeps increasing as $\mathcal{S}$ moves closer to $\mathcal{R}$.
### Effect of time sharing
![Effect of time sharing factors $\beta$ and $\alpha$ on SU packet throughput $\mu_{S}$ for $\lambda_{P}=0.5$, $N_{S}=10$, $N_{P}=50$ and $\sigma_{\mathcal{PD}}^{2}=-20\,\mbox{dB}$.\[fig:SUvsLem\_alph\]](MuVsAlpBet)
Fig. \[fig:SUvsLem\_alph\] plots $\mu_{S}$ against time sharing factors $\beta$ and $\alpha$. When $\beta$ is low, values of $\theta_{\mathcal{PD}}$ and $\theta_{\mathcal{PS}}$ are low. This results in lower value of $\mu_{P}$ that cannot support given $\lambda_{P}$. As $\beta$ increases, PU packet departure rate increases. This allows SU to transmit its own packets with non-zero probability. Thus, $\mu_{S}$ increases. As $\beta$ increases further, less time is available for SU to transmit its own packets which decreases $\theta_{\mathcal{SR}}$. Thus, $\mu_{S}$ decreases at high value of $\beta$. As $\alpha$ increases, $\overline{\theta_{\mathcal{SD}}}$ increases while $\overline{\theta_{\mathcal{SR}}}$ decreases. Increase in departure rate of PU packets from $Q_{\mathcal{S}}$ allows SU to transmit its own packet with higher access probability. Thus, $\mu_{S}$ increases with increasing $\alpha$. But as $\alpha$ increases further, decrease in $\overline{\theta_{\mathcal{SR}}}$ becomes dominant, in turn decreasing $\mu_{S}$. This indicates that there is scope to improve $\mu_{S}$ by optimizing $\beta$ and $\alpha$. However, the problem is difficult to solve as objective in (\[eq:Opt\_problem\]) is non-convex in $\beta$ and $\alpha$.
Conclusion
==========
We studied a CR system where SU employs two-phase relaying to relay failed PU packets. Both users have packet queues of finite capacity which results in packet loss. We proposed optimal as well as suboptimal access methods for SU to maximize its packet throughput under packet loss constraint of PU. We observed that two-phase relaying model performs better than cooperation model without time sharing for mid-range values of PU packet arrival rate. Suboptimal methods have negligible loss in the performance and are good low complexity alternatives to the optimal method. Furthermore, results revealed that as relay queue size increases, SU throughput improves initially but then decreases. PU queue size limits maximum supported PU packet arrival rate.
[^1]: The authors are with Department of Electrical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, INDIA (e-mail: [email protected]; [email protected]).
[^2]: Copyright (c) 2015 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected].
[^3]: Note that notation $\overline{\theta_{sd}}$ only signifies success probability with time sharing and $\overline{\theta_{sd}}\neq1-\theta_{sd}$.
|
---
author:
- Biswajoy Brahmachari
- Probir Roy
title: 'Testable constraint on near-tribimaximal neutrino mixing'
---
Introduction
============
The phenomenon of mixing between different generations of quarks/leptons has now been experimentally studied fairly well [@nf-1]. The three quark mixing angles are at present quite well-measured. Though the leptonic mixing angles are not known as well, one has credible nonzero $3 \sigma$ upper and lower bounds on them. CP violation has been investigated quite thoroughly in the quark sector, but as yet there is no reliable observation of CP-violation involving only leptons. Quark mixing angles are known to become progressively smaller in order of magnitude as one moves from $1$-$2$ to $2$-$3$ and $1$-$3$ generation mixing. This fact can be understood qualitatively in terms of a hierarchical quark mass matrix. The mixing angles, that emerge from such a mass matrix, are small and turn out to be given roughly by the mass ratios of relevant generations of quarks. Since the masses of both up- and down-type quarks are strongly hierarchical with respect to generations, this ties in with observation. In complete contrast, the leptonic mixing angles have been found to be much larger and show a different pattern. The qualitative difference between quark and lepton mixing patterns is made starkly evident by a quantitative comparison of the approximate magnitudes [@nf-2; @nf-3a; @nf-3b; @nf-3c] of the elements of the respective unitary matrices $V_{CKM}$ and $U_{PMNS}$: $$|V_{CKM}| \sim \begin{pmatrix}0.9 & 0.2 & 0.004 \cr
0.2 & 0.9 & 0.01 \cr
0.008 & 0.04 & 0.9 \end{pmatrix},
|U_{PMNS}| \sim \begin{pmatrix}0.8 & 0.5 & 0.2 \cr
0.4 & 0.6 & 0.7 \cr
0.4 & 0.6 & 0.7 \end{pmatrix}. \label{eqn-1}$$
Though the masses of the charged leptons $l~(=e,\mu,\tau)$ show a pronounced hierarchical pattern with respect to generations, one suspects that such may not be the case with neutrinos. What operates for the mixing the latter, possibly related to their presumed Majorana nature[^1] originating, say from some kind of a seesaw mechanism [@nf-2], is perhaps some underlying family symmetry. Though one need not make any specific assumption on the neutrino mass hierarchy, such considerations are most natural for quasi-degenerate neutrinos. Even if there is any mass hierarchy among neutrinos, it can be presumed to be quite mild. Thus we separate the issue of the mixing of neutrinos from that of their mass hierarchy. More definitely, the family symmetry controlling their mixing can be taken to be independent of the neutrino mass hierarchy.
For fermions of type $t~(=u,d,l,\nu)$, we can define the mass basis as one in which the corresponding mass matrix $M_t$ is diagonal. We can also consider the flavor basis in which the fermions $|\chi^t \rangle$ are flavor eigenstates but the mass matrix $M_{t f}$ is not necessarily diagonal. The hermitian squared mass matrix $M^\dagger M$ in each basis is related by a unitary transformation $U_t$: $$U^\dagger_t M^\dagger_{t f} M_{t f} U_t=M^\dagger_t M_t. \label{eqn-2}$$ We subscribe to the following viewpoint. While each of $U_u,U_d,U_\ell$ shows a hierarchical structure, this is not true of $U_\nu$ which is governed by a different principle. The way to gain new insights into this principle is through more precise measurements of the leptonic mixing angles and of the associated CP-violating Dirac phase $\delta_{CP}$ as well as of the concerned neutrino masses. These can test mixing constraints from specific theoretical ideas. Our aim in this paper is to derive some such constraint which is experimentally testable. This we do by considering lowest order perturbation theory in the additive breaking of tribimaximal (TBM) neutrino mixing for neutrino and charge lepton mass matrices in the flavor basis. The additively broken TBM paradigm is explained in detail below. From our consideration, we obtain two alternative experimentally testable possibilities, at least one of which is obligatory. Though our result is derived by use of general arguments, we check it in specific flavor models.
The rest of the paper is organized as follows. Section 2 is devoted to a discussion of tribimaximal mixing and its breaking. In section 3 we set up our basic lowest order perturbative formalism which is meant to compute the deviations away from tribimaximality. Section 4 contains the derivation of the theoretical consequences of the said formalism. In Section 5 we discuss the experimentally testable constraint arising therefrom. Section 6 includes a comparative study of our result with those of various flavor models incorporating deviations from TBM. The final section 7 summarizes our conclusions.
Broken tribimaximal mixing and its effects
==========================================
There is a vast literature [@nf-2; @nf-3a; @nf-3b; @nf-3c; @nf-4] covering theoretical ideas on the principle governing $M_{\nu f}$ and $U_\nu$. Our focus, however, is on tribimaximal (TBM) mixing [@nf-4; @nf-5a; @nf-5b; @nf-5c] which is elegant, predictive and can be given a solid theoretical foundation [@nf-6; @nf-7] from specific realizations of discrete family symmetries such as $A_4,S_3$ and $\Delta_{27}$. Some of the latter have also suggested a few neutrino mixing sum-rules [@nf-8na; @nf-8nb; @nf-8a; @nf-8b; @nf-8c; @nf-8nc]. We henceforth use the superscript zero to denote the TBM limit. In this limit we choose to work in the weak flavor basis in which the charged leptons have a diagonal Dirac mass matrix $$M^0_\ell={\rm diag.}~(m^0_e, m^0_\mu, m^0_\tau). \label{eqn-3}$$ The TBM limit of the neutrino mass matrix $M^0_{\nu f}$ in the flavor basis is characterized by certain linear relations among elements of $M^0_{\nu f}$: $$\begin{aligned}
(M^0_{\nu f})_{12} &=& -(M^0_{\nu f})_{13}, \label{eqn-4}\\
(M^0_{\nu f})_{22} &=& (M^0_{\nu f})_{33}, \label{eqn-5}\\
(M^0_{\nu f})_{11}-(M^0_{\nu f})_{13} &=&
(M^0_{\nu f})_{22}-(M^0_{\nu f})_{23}. \label{eqn-6}\end{aligned}$$ Given (\[eqn-4\]), (\[eqn-5\]) and (\[eqn-6\]), the most general form of $M^0_{\nu f}$ is $$M^0_{\nu f} = \begin{pmatrix}X & Y & -Y \cr
Y & X+Z & -Y+Z \cr
-Y & -Y+Z & X+Z \end{pmatrix}, \label{eqn-7}$$ where $X,Y,Z$ are unknown complex mass dimensional parameters. Now the TBM pattern obtains with the three emergent pairwise mixing angles, that appear [@nf-1] in $U_{PMNS}$, being fixed at $
\theta^0_{12}=\sin^{-1} \sqrt{1 \over 3} \simeq 35.3^\circ,
\theta^0_{23}=\sin^{-1} \sqrt{1 \over 2} = 45^\circ,
\theta^0_{13}=0
$ independent of whether the neutrino mass ordering is normal or inverted.
We can compare the TBM-predicted values of the three mixing angles with their current $3 \sigma$ allowed ranges. Recent global fits yield [@nf-9; @nf-10; @nf-11] $31^\circ \le \theta_{12} \le 36^\circ,
36^\circ \le \theta_{23} \le 55^\circ, 7.2^\circ \le \theta_{13} \le 10^\circ$. Thus while $\theta_{12}$ and $\theta_{23}$ are certainly compatible with TBM values within their measured ranges, $\theta_{13}$ a fortiori is not. Indeed, the measurement of a significantly nonzero value of $\theta_{13}$ has been a major experimental advance recently [@nf-12a; @nf-12b; @nf-12c; @nf-12d] with a tremendous theoretical impact. This is due to two reasons. First, CP-violation, that is observable in neutrino oscillations, enters through the terms $s_{13}~e^{\pm i \delta_{CP}}$; thus $s_{13} \sim 0.12$-$0.17$ is very encouraging to that end. Second, it means that any symmetry, leading to TBM, must be a broken symmetry. The next natural question is: how quantitative is this breaking and is TBM still relevant in an approximate sense ?
We make an attempt to answer this last question. Our approach is to add small general perturbations to the TBM limits of hermitian squared mass matrices $M^\dagger_{\ell f} M_{\ell f}$ and $M^\dagger_{\nu f} M_{\nu f}$. We take both sets of perturbations to be of the same order of magnitude and treat them to the lowest order. Much effort [@nf-13new; @nf-13a; @nf-13b; @nf-13c] has already been expended in this direction. However, we do have something new and interesting to say. We bring out a novel feature of the near-TBM mixing of neutrinos in terms of an analytically derived constraint which merits being highlighted. The constraint implies that at least one of two conditions, that are testable in forthcoming neutrino oscillation experiments, must hold. Either the deviation $|s_{23}-\sqrt{1 \over 2}|$ from the maximal value of $\theta_{23}$ or the measure of CP violation $|s_{13} \sin \delta_{CP}|$ has to be quite small ( $< 3 \%$ as opposed to[^2] $12$-$17 \%$ for the value of $s_{13}$ as compared with unity), the latter meaning that CP will be conserved at the lowest order. This conclusion is a consequence of the fact that the perturbed eigenstates $|\chi^{\ell,\nu}_i \rangle$ for $i=1,2,3$ make up the columns of the matrices $U_{\ell,\nu}$ to the lowest order. Hence any observation in the near future of both a sizable deviation from maximal atmospheric neutrino mixing and a large amount of CP-violation in neutrino oscillations would go against the idea of lowest order additive perturbation to TBM-invariant neutrino and charged lepton mass matrices.
In deriving the above conclusion, we do not assume any additional model either at a high or at a low scale, or any specific discrete family symmetry. In fact, we perform a lowest order model independent analysis with the most general TBM violating perturbation matrices whose nonzero elements are expected to be of the same order of magnitude. Moreover, our results on neutrino mixing do not need to assume anything about the neutrino mass ordering. This is since the perturbations are expected to be some kind of symmetry breaking terms, which characterize their contributions to $|\chi^{\ell,\nu} \rangle$ by a set of small dimensionless coefficients $\{\epsilon^{\ell,\nu}\}$. All members of the subset $\{\epsilon^\nu\}$ in the neutrino sector are taken to be typically of magnitude $\sim s_{13} \equiv \sin \theta_{13} \sim 0.12$-$0.17$, i.e. of the order of $12$ -$17 \%$ or thereabouts of the unperturbed quantities. On the other hand, in the charged lepton sector, arguements are given why $\{\epsilon^\ell\}$ are much less in magnitude than $\{\epsilon^\nu\}$ on account of the strongly mass hierarchical nature of the charged leptons. This will be shown to follow from all nonzero perturbation matrix elements being taken to be of the same order of magnitude. Of course, the neglected $O(\epsilon^2)$ terms are estimated to be only at a $2$-$3 \%$ level which is below [@nf-new-1; @nf-new-2] the accuracy of the measurement of TBM deviants in ongoing and forthcoming neutrino oscillation experiments[^3].
Lowest order perturbation away from tribimaximality
===================================================
For charged leptons $\ell$ the normalized eigenvectors in the mass basis and the flavor basis are identical in the TBM limit. Thus we can take $$|\chi^{\ell 0}_1 \rangle=|\chi^{\ell 0}_1 \rangle_f=\begin{pmatrix}
1 \cr 0 \cr 0 \end{pmatrix},
|\chi^{\ell 0}_2 \rangle=|\chi^{\ell 0}_2 \rangle_f=
\begin{pmatrix}0 \cr 1 \cr 0 \end{pmatrix},
|\chi^{\ell 0}_3 \rangle=|\chi^{\ell 0}_3 \rangle_f=
\begin{pmatrix}0 \cr 0 \cr 1 \end{pmatrix}. \label{eqn-8}$$ Moreover, the charged lepton mass matrix is identical in each basis in the same limit, namely $$M^0_{\ell f}=M^0_{\ell}. \label{eqn-9}$$ Adding a perturbation $M^\prime_{\ell f}(\equiv \lambda_{ij})$ to $M^0_{\ell f}$ so that $M_{\ell f}=M^0_{\ell f}+ M^\prime_{\ell f}$, we can construct the corresponding matrix $M^\prime_\ell$ in the mass basis as $$M_\ell=M^0_\ell+M^\prime_\ell \label{eqn-10}$$ via, $$M^\dagger_\ell M_\ell=U^\dagger_\ell M^\dagger_{\ell f} M_{\ell f} U_\ell. \label{eqn-11}$$
Turning to neutrinos in the TBM limit, we can write $${U^0_\nu}^\dagger {M^0_{\nu f}}^\dagger M^0_{\nu f} U^0_\nu
={\rm diag}.~(|m^0_{\nu 1}|^2,|m^0_{\nu 2}|^2,|m^0_{\nu 3}|^2) \label{eqn-12}$$ with $$U^0_\nu= \begin{pmatrix} \sqrt{2/3} & \sqrt{1/3} & 0 \cr
-\sqrt{1/6} & \sqrt{1/3} & \sqrt{1/2} \cr
\sqrt{1/6} & -\sqrt{1/3} & \sqrt{1/2} \end{pmatrix}. \label{eqn-13}$$ The normalized flavor eigenvectors $|\chi^{\nu 0}_i \rangle$ of ${M^0}^\dagger_{\nu f}M^0_{\nu f}$ for $i=1,2,3$ are the columns of $U^0_\nu$ while those in the mass basis are identical to the charged lepton ones. Thus $$|\chi^{\nu 0}_1 \rangle=\begin{pmatrix}1 \cr 0 \cr 0 \end{pmatrix},
|\chi^{\nu 0}_2 \rangle=\begin{pmatrix} 0 \cr 1 \cr 0 \end{pmatrix},
|\chi^{\nu 0}_3 \rangle=\begin{pmatrix}0 \cr 0 \cr 1 \end{pmatrix}, \label{eqn-14}$$ whereas $$|\chi^{\nu 0}_1 \rangle_f=\begin{pmatrix}\sqrt{2 \over 3} \cr -\sqrt{1 \over 6} \cr \sqrt{1 \over 6} \end{pmatrix},
|\chi^{\nu 0}_2 \rangle_f=\begin{pmatrix}\sqrt{1 \over 3} \cr \sqrt{1 \over 3} \cr -\sqrt{1 \over 3} \end{pmatrix},
|\chi^{\nu 0}_3 \rangle_f=\begin{pmatrix}0 \cr \sqrt{1 \over 2} \cr \sqrt{1 \over 2} \end{pmatrix}. \label{eqn-15}$$ Once the perturbation is introduced, we have $M_{\nu f}=M^0_{\nu f} + M^\prime_{\nu f}$, where $M^0_{\nu f}$ obey the TBM conditions (\[eqn-4\])–(\[eqn-6\]) while $(M^\prime_{\nu f})_{ij} \equiv \mu_{ij}=\mu_{ji}$ violate them. The violation in TBM conditions is given by, $$\begin{aligned}
(M_{\nu f})_{12}+(M_{\nu f})_{13} &=& \mu_{12}+\mu_{13}, \label{new1}\\
(M_{\nu f})_{22}-(M_{\nu f})_{33} &=& \mu_{22}-\mu_{33}, \label{new2}\\
(M_{\nu f})_{11}-(M_{\nu f})_{13}-(M_{\nu f})_{22}+(M_{\nu f})_{23} &=& \mu_{11}-\mu_{13}-\mu_{22}+\mu_{23}. \label{new3}\end{aligned}$$ Note that, unlike the real diagonal $M^0_\ell$ and the general $M^\prime_\ell$, both $M^0_{\nu f}$ and $M^\prime_{\nu f}$ have to be complex symmetric matrices in order to make the corresponding neutrinos Majorana particles.
We now expand the perturbed eigenstates for both charged leptons and neutrinos at the lowest order. We choose to use a compact notation covering both cases by introducing perturbation parameters $\epsilon^{\nu,\ell}_{ik} $ (for $i,k=1,2,3$). Thus we can write the ith first order perturbed eigenvectors of $M^\dagger_{\nu f}M_{\nu f}$ on one hand and of $M^\dagger_{\ell f}M_{\ell f}$ on the other as $$|\chi^{\nu,\ell}_{i}\rangle_{f} = |\chi^{0 \nu,\ell}_{i}\rangle_{f}
+\sum_{k \ne i}\epsilon^{\nu,\ell}_{ik} | \chi^{0 \nu,\ell}_{k} \rangle_{f}
+ O(\epsilon^2).\label{eqn-16}$$ Two new quantities have been introduced in (\[eqn-16\]). They are defined by $$\begin{aligned}
\epsilon^{\nu,\ell}_{ik}
&=& -{\epsilon^{\nu,\ell}}^*_{ki}=
(|m^0_{\nu,\ell i}|^2-|m^0_{\nu,\ell k}|^2)^{-1}p^{\nu,\ell}_{ki},\label{eqn-17}\\
p^{\nu,\ell}_{ik} &=& \langle \chi^{0 \nu,\ell}_i|{M^0_{\nu,\ell}}^\dagger M^\prime_{\nu,\ell}
+ {M^\prime_{\nu,\ell}}^\dagger M^0_{\nu,\ell}|\chi^{0 \nu,\ell}_k \rangle. \label{eqn-18}\end{aligned}$$ Note that (\[eqn-17\]) and (\[eqn-18\]) have been written in the mass basis utilizing the fact that $\epsilon^{\nu,\ell}_{ik}$, as well as $p^{\nu,\ell}_{ik}$, do not change from one basis to the other. We can also comment on the lack of dependence of the epsilon parameters on the yet unknown overall neutrino mass scale. If $M^0_\nu$ and $M^\prime_\nu$ are both scaled by a factor $\alpha$, the unperturbed eigenvalues $\{ m^0_{\nu i} \}$ will also be scaled similarly. As a result, $\epsilon^\nu_{ik}$ will remain invariant under an overall mass scaling. On the other hand, suppose two of the mass eigenvalues are large but close to one another, as is the case with $\nu_1$ and $\nu_2$ with an inverted mass hierarchy, and this is not much affected by the perturbations. In such a case the corresponding $\epsilon^\nu_{12}$ will get enhanced.
Turning to (\[eqn-16\]), we see that its LHS for $i=1,2,3$ can be identified with three corresponding columns of $U_{\nu,\ell}$. Thus $$U_{\nu,l}=(|\chi^{\nu,\ell}_1 \rangle_f ~~|\chi^{\nu,\ell}_2 \rangle_f ~~|\chi^{\nu,\ell}_3 \rangle_f).
\label{eqn-19}$$ Neglecting $O(\epsilon^2)$ terms, it follows from (\[eqn-19\]) that $$U_\ell=\begin{pmatrix}1 & -\epsilon^{\ell *}_{12} & -\epsilon^{\ell *}_{13} \cr
\epsilon^{\ell }_{12} & 1 & -\epsilon^{\ell *}_{23} \cr
\epsilon^{\ell }_{13} & \epsilon^{\ell }_{23} & 1 \end{pmatrix}
\label{eqn-20}$$ and $$U_\nu=
\begin{pmatrix}
\sqrt{\frac{2}{3}}+ \sqrt{\frac{1}{3}} \epsilon^\nu_{12}
& \sqrt{\frac{1}{3}}-\sqrt{\frac{2}{3}} \epsilon^{\nu *}_{12}
& -\sqrt{\frac{2}{3}}\epsilon^{\nu *}_{13} -\sqrt{\frac{1}{3}} \epsilon^{\nu *}_{23} \cr
-\sqrt{\frac{1}{6}}+ \sqrt{\frac{1}{3}} \epsilon^\nu_{12}+\sqrt{\frac{1}{2}} \epsilon^\nu_{13}
& \sqrt{\frac{1}{3}}+\sqrt{\frac{1}{6}} \epsilon^{\nu *}_{12}+\sqrt{\frac{1}{2}} \epsilon^\nu_{23}
& \sqrt{\frac{1}{2}} + \sqrt{\frac{1}{6}}\epsilon^{\nu *}_{13} -\sqrt{\frac{1}{3}} \epsilon^{\nu *}_{23} \cr
\sqrt{\frac{1}{6}}- \sqrt{\frac{1}{3}} \epsilon^\nu_{12}+\sqrt{\frac{1}{2}} \epsilon^\nu_{13}
& -\sqrt{\frac{1}{3}}-\sqrt{\frac{1}{6}} \epsilon^{\nu *}_{12}+\sqrt{\frac{1}{2}} \epsilon^\nu_{23}
& \sqrt{\frac{1}{2}}-\sqrt{\frac{1}{6}} \epsilon^{\nu *}_{13} + \sqrt{\frac{1}{3}} \epsilon^{\nu *}_{23}
\end{pmatrix}.
\label{eqn-21}$$
Let us define the Majorana phase matrix $$K \equiv {\rm diag}.~(1,e^{i \alpha_{21} \over 2},e^{i \alpha_{31} \over 2}).
\label{eqn-22}$$ Then $U_{PMNS}K^{-1}$ can be written in the PDG convention [@nf-1] as $$U_{PMNS}K^{-1} =
\begin{pmatrix}
c_{12}c_{13} & s_{12}c_{13} & s_{13}e^{-i \delta_{CP}} \cr
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i \delta_{CP}} & c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i \delta_{CP}} & s_{23} c_{13} \cr
s_{12} s_{23} -c_{12} c_{23} s_{13} e^{i \delta_{CP}} & -c_{12}s_{23}-s_{12}c_{23}s_{13} e^{i \delta_{CP}} & c_{23}c_{13}
\end{pmatrix}. \label{eqn-23}$$ We can now make the identification $$U_{PMNS}K^{-1}=U^\dagger_\ell U_\nu \label{eqn-24}$$ and work out the consequences from (\[eqn-20\]), (\[eqn-21\]) and (\[eqn-23\]).
Consequences of lowest order perturbation
=========================================
Let us define $L \equiv U^\dagger_\ell U_\nu$ and $N \equiv U_{PMNS} K^{-1}$. The identification $L_{ij}=N_{ij}$ as per (\[eqn-24\]) leads to nine equations which are detailed in convenient combinations in the Appendix. Not all of these are independent, but they lead to four independent constraint conditions and three equations for the three TBM-deviants $c_{12}-\sqrt{2 \over 3}, c_{23}-s_{23}$ and $s_{13} e^{i \delta_{CP}}$. The constraint conditions follow from the fact that four of the elements of $N$ are real. They are given by $$\begin{aligned}
&& {\rm Im}~\epsilon^\nu_{12} = O(\epsilon^2), \label{eqn-25} \\
&& {\rm Im}~(\epsilon^\nu_{13}-\sqrt{2} \epsilon^\nu_{23}) = O(\epsilon^2), \label{eqn-26}\\
&& {\rm Im}~\epsilon^l_{23} = O(\epsilon^2) \label{eqn-27}, \\
&& {\rm Im}~(\epsilon^l_{12}-\epsilon^l_{13}) = O(\epsilon^2). \label{eqn-28}\end{aligned}$$ Neglecting $O(\epsilon^2)$ terms, the three measurable TBM-deviants are linear in the $\epsilon$ coefficients and may be given as $$\begin{aligned}
&& c_{12}-\sqrt{2 / 3}
=\sqrt{1 / 2}\left(\sqrt{1 / 3} -s_{12} \right)
=\sqrt{1 / 3} ~\epsilon^\nu_{12}
- \sqrt{1 / 6}\left( \epsilon^l_{12}-\epsilon^l_{13} \right),\label{eqn-29} \\
&& c_{23}-s_{23}
=-\sqrt{2 / 3}\left(\epsilon^\nu_{13}-\sqrt{2}
~\epsilon^\nu_{23}\right)-\sqrt{2}~ \epsilon^l_{23}, \label{eqn-30} \\
&& s_{13}~e^{i \delta_{CP}}
=-\sqrt{1 / 3}~\left(\sqrt{2}~ \epsilon^\nu_{13}
+ \epsilon^\nu_{23}\right)
+\sqrt{1 / 2}\left(\epsilon^l_{12}+\epsilon^l_{13}\right). \label{eqn-31}\end{aligned}$$ The derivation of Eqs. (\[eqn-25\])–(\[eqn-31\]) appears in the Appendix.
Because of (\[eqn-26\]), the real and imaginary parts of (\[eqn-31\]) enable us to write, modulo $O(\epsilon^2)$ terms, that $$\tan \delta_{CP}
= \frac{3 ~{\rm Im}~\epsilon^\nu_{23} -\sqrt{3 /2} ~{\rm Im}~(\epsilon^\ell_{12}+\epsilon^\ell_{13})
}{
{\rm Re}~(\sqrt{2} ~\epsilon^\nu_{13} + \epsilon^\nu_{23})
-\sqrt{3 / 2} ~{\rm Re}~(\epsilon^l_{12} + \epsilon^l_{13})}. \label{eqn-32}$$ The above equation may be recast in terms of the basis independent Jarlskog invariant $J$ which equals $${\rm Im}~[(U_\ell^\dagger U_\nu)_{e 1}(U_\ell^\dagger U_\nu)_{\mu 2}
(U_\ell^\dagger U_\nu)^*_{e 2}(U_\ell^\dagger U_\nu)^*_{\mu 1}].$$ We then have $$J=-{\frac{1}{\sqrt{6}}}~{\rm Im}~[\epsilon^\nu_{23}-{\frac{1}{\sqrt{6}}}
(\epsilon^l_{12} + \epsilon^l_{13})]+O(\epsilon^2). \label{eqn-33}$$
Let us now explore, to the lowest order in $\epsilon$, the consequences of (\[eqn-17\]) and (\[eqn-18\]) by explicitly taking elements of the respective perturbing mass matrices for neutrinos and charged leptons. We take $$(M^\prime_{\nu f})_{ij}= \mu_{ij}=\mu_{ji} \label{eqn-34}$$ and $$(M^\prime_{\ell f})_{ij}=(M^\prime_\ell)_{ij}+O(\epsilon^2) \label{eqn-35}
=\lambda_{ij}$$ with $\lambda_{ij}$ and $\mu_{ij}=\mu_{ji}$ as complex mass dimensional parameters naturally expected to be of the same order of magnitude. The identity of the charged lepton mass basis and flavor basis in the TBM limit makes the calculations in this case quite straightforward. From (\[eqn-17\]) and (\[eqn-18\]), we can easily derive $$\begin{aligned}
\epsilon^l_{12} &=& ({m^0_e}^2-{m^0_\mu}^2)^{-1}~(m^0_\mu \lambda_{21} + m^0_e \lambda^*_{12}), \label{eqn-36}\\
\epsilon^l_{23} &=& ({m^0_\mu}^2-{m^0_\tau}^2)^{-1}~(m^0_\tau \lambda_{32} + m^0_\mu \lambda^*_{23}), \label{eqn-37}\\
\epsilon^l_{13} &=& ({m^0_e}^2-{m^0_\tau}^2)^{-1}~(m^0_\tau ~\lambda_{31}+m^0_e{\lambda^*_{13}}). \label{eqn-38}\end{aligned}$$ We want to comment on the magnitudes of $\epsilon^\ell_{23}$ and $\epsilon^\ell_{13}$. In order for them to be large, the relevant $\lambda$ parameters would need to be of order $m_\tau$. That is not in conformity with our premise that nonzero charged lepton perturbation mass matrix elements ( i.e. $\lambda_{ij}$) cannot be very different in order of magnitude from those for neutrinos (i.e. $\mu_{ij}$). Thus we expect that $|\epsilon^\ell_{23}|$ and $|\epsilon^\ell_{13}|$ to be quite small. In any event, because of the strongly hierarchical nature of charged lepton masses, (\[eqn-27\]) and (\[eqn-28\]) can be satisfied without unnatural cancellations only by $\lambda_{12},\lambda_{21},\lambda_{13},\lambda_{31},\lambda_{23},\lambda_{32}$ all being real to order $\epsilon$. One then automatically obtains that $${\rm Im} ~\epsilon^l_{12}=O(\epsilon^2)={\rm Im}~ \epsilon^l_{13}.\label{eqn-39}$$ Feeding this information, we can simplify (\[eqn-32\]) and (\[eqn-33\]) to $$\tan \delta_{CP}
=\frac{3 ~{\rm Im}~\epsilon^\nu_{23}
}{
{\rm Re}~(\sqrt{2} ~\epsilon^\nu_{13} + \epsilon^\nu_{23})
-\sqrt{3 / 2} ~ {\rm Re}~(\epsilon^l_{12} + \epsilon^l_{13})},
\label{eqn-40}$$ $$J=-\frac{1}{\sqrt{6}}~{\rm Im}~\epsilon^\nu_{23} +O(\epsilon^2)\label{eqn-41}$$ respectively.
Turning to neutrinos next, the relevant off-diagonal elements of $M^\prime_\nu={U^0_\nu}^T M^\prime_{\nu f} U^0_\nu$ are $$\begin{aligned}
(M^\prime_\nu)_{12} &=&
\frac{1}{3 \sqrt{2}}(2 \mu_{11}+\mu_{12}-\mu_{13}-\mu_{22}+2 \mu_{23}-\mu_{33}), \label{eqn-42}\\
(M^\prime_\nu)_{23}
&=& \frac{1}{\sqrt{6}}(\mu_{12}+ \mu_{13}+ \mu_{22} - \mu_{33}), \label{eqn-43}\\
(M^\prime_\nu)_{13}
&=& \frac{1}{\sqrt{3}}(\mu_{12}+ \mu_{13}-\frac{1}{2} \mu_{22} + \frac{1}{2}
\mu_{33}). \label{eqn-44}\end{aligned}$$ It is now convenient to define $$\begin{aligned}
&& \Delta^0_{ij} \equiv |m^0_{\nu i}|^2-|m^0_{\nu j}|^2, \label{eqn-45} \\
&& a^\mp_{ij} \equiv m^0_{\nu i} \mp m^0_{\nu j}. \label{eqn-46}\end{aligned}$$ Then we take (\[eqn-17\]) and (\[eqn-18\]) and successively consider the index combinations $i=1,~k=2$ and $i=2,~k=3$ as well as $i=1,~k=3$. Separating the real and imaginary parts and using (\[eqn-45\]) and (\[eqn-46\]), we obtain the following six equations $$\begin{aligned}
2~\Delta^0_{12}~
\begin{pmatrix} {\rm i ~Im}~\epsilon^\nu_{12} \cr
{\rm Re}~\epsilon^\nu_{12} \end{pmatrix}
&&=
{a^\mp}^*_{21}~{(M^\prime_\nu)}_{12}
\mp c.c.,\label{eqn-47}\\
&& \nonumber\\
2~\Delta^0_{23}~
\begin{pmatrix} {\rm i~Im}~\epsilon^\nu_{23} \cr
{\rm Re}~ \epsilon^\nu_{23} \end{pmatrix}
&&= {a^\mp}^*_{32}{(M^\prime_\nu)}_{23}
\mp c.c., \label{eqn-48} \\
&& \nonumber\\
2 ~ \Delta^0_{13}~
\begin{pmatrix} {\rm i~Im}~\epsilon^\nu_{13} \cr
{\rm Re}~\epsilon^\nu_{13} \end{pmatrix}
&&= {a^\mp}^*_{31}(M^\prime_\nu)_{13} \mp c.c. \label{eqn-49}\end{aligned}$$ Needless to add, order $\epsilon^2$ terms have been neglected in deriving the above results.
Results and discussion
======================
Eq. (\[eqn-47\]) has a simple consequence if we exclude unnatural cancellations. In conjunction with (\[eqn-25\]), it forces the combination of $\mu_{ij}$, occurring in $(M^\prime_\nu)_{12}$, i.e. $2 \mu_{11}+ \mu_{12}-\mu_{13}-\mu_{22} + 2 \mu_{23}-\mu_{33}$, to be real. It also implies that $m^0_{\nu_2}-m^0_{\nu_1}$ is real, the latter forcing $\alpha^0_{21}$ to be $0$ or $\pi$. However, our key observation follows from combining (\[eqn-48\]) and (\[eqn-49\]) with (\[eqn-26\]). That procedure yields the equality $$\begin{aligned}
&& {\rm Im}~[ ({m^{0 *}_{\nu_3}}-{m^{0 *}_{\nu_2}})(\mu_{12}+\mu_{13}+\mu_{22}-\mu_{33})] \nonumber\\
&& =
{\rm Im}~[ ({m^{0 *}_{\nu_3}}-{m^{0 *}_{\nu_1}})(\mu_{12}+\mu_{13}+{1 \over 2}\mu_{22}-{1 \over 2}\mu_{33})].
\label{eqn-50}\end{aligned}$$ There are two ways to satisfy (\[eqn-50\]) without any unnatural cancellation, at least one of which is obligatory. Either we must have option [**(1)**]{}, namely that $m^0_{\nu_2}=m^0_{\nu_1}$ and $\mu_{22}=\mu_{33}$ or there must be option [**(2)**]{}, namely that $m^0_{\nu_3},m^0_{\nu_2},m^0_{\nu_1},\mu_{12}+\mu_{13}$ and $\mu_{22}-\mu_{33}$ are all real so that each side of (\[eqn-50\]) vanishes. Take [**(1)**]{} first. Since $m^0_{\nu 1}=|m^0_{\nu 1}|$ by choice, we now have $|m^0_{\nu_1}|=|m^0_{\nu_2}|$ and $\alpha^0_{21}=0$, i.e. $\Delta_{21} \equiv |m_{\nu 2}^2|-|m_{\nu 1}|^2$ arises solely from TBM breaking. Further, with $\mu_{22}=\mu_{33}$, the implication from from (\[eqn-48\]) and (\[eqn-49\]) is that $\sqrt{2}~{\rm Re}~\epsilon^\nu_{23}={\rm Re}~\epsilon^\nu_{13}
+ O(\epsilon^2)$. Consequently, it follows from (\[eqn-27\]) and (\[eqn-30\]) that $c_{23}-s_{23}=-\sqrt{2} \epsilon^{\ell}_{23}+O(\epsilon^2)$ which leads to the result $|s_{23}-{1 \over \sqrt{2}}|={1 \over \sqrt{2}} |\epsilon^\ell_{23}|
+|O(\epsilon^2)| << |O(\epsilon^\nu)|$. The strong inequality in the last step has been based on the discussion which followed (\[eqn-38\]). Thus option [**(1)**]{} says that the magnitude of any deviation from maximal atmospheric mixing, being of order $|\epsilon^\ell_{23}|$ and small, will not be easily observed in forthcoming experiments. Let us turn to alternative [**(2)**]{}. Now we have $\alpha^0_{21}$ and $\alpha^0_{31}$ equalling $0$ or $\pi$. Further, by use of (\[eqn-43\]) and (\[eqn-48\]), we derive that ${\rm Im}~\epsilon^\nu_{23}=O(\epsilon^2)$. As a result, by virtue of (\[eqn-40\]) as well as (\[eqn-41\]), one concludes that $s_{13}\sin \delta_{CP}=O(\epsilon^2)$ and $J=O(\epsilon^2)$, so that both would be small and hard to detect in experiments planned for the near future. The implication of option [**(2)**]{} is that CP violation in neutrino oscillations may not be seen in those experiments. It may be noted that the assumption $|\epsilon^\ell| << |\epsilon^\nu|$ is unnecessary for this option.
It is also noteworthy that in option ([**1**]{}) one needs to use degenerate perturbation theory [@nf-14a; @nf-14b; @nf-14c] with respect to the TBM limit for the 1-2 sector of neutrinos. In the latter case, the perturbation splits the 1-2 mass degeneracy and generates the solar neutrino mass difference with $m^0_{\nu 1}=m^0_{\nu 2}=m^0_\nu$. One then obtains $$\Delta_{21}=\sqrt{(p^\nu_{11}-p^\nu_{22})^2+{p^\nu_{12}}^2},$$ as calculated using (\[eqn-18\]). Additionally, to order $\epsilon^\nu$ and $\epsilon^\ell$, $s_{13} e^{i \delta_{CP}}$ can be obtained in terms of $m^0_{\nu 3},m^0_\nu,m^0_e,m^0_\mu,m^0_\tau$ as well as the $\mu$ and $\lambda$ parameters by using (\[eqn-31\]) and employing the expressions for the $\epsilon$ parameters. We choose not write that full expression here.
Some comments on the issue of unnatural cancellations are in order. The TBM breaking terms in the mass matrix of charged leptons do not leave any residual symmetry except possibly some rephasing invariances. As stated earlier, given that $m_e << m_\mu << m_\tau$, the cancellations required to avoid the reality condition on all $\lambda_{ij}$ (for $i \ne j$) cannot be effected by any such invariance. In the neutrino case, there generally is a residual $Z_2$ symmetry [@nf-5a; @nf-5b; @nf-5c; @nf-6; @nf-7; @nf-8a; @nf-8b] after TBM is broken. Even such a discrete symmetry does not generally enable one to obtain the concerned complicated equality between specific combinations of TBM violating perturbation parameters, TBM invariant neutrino masses as well as Majorana phases. We feel, therefore, that our argument ruling out such cancellations is sound and our conclusions are reliable.
Let us finally remark on the relevance of our result to planned experiments at the proton beam intensity frontier. The determination of the sign of the neutrino mass ordering is one of their aims. It is noteworthy that the constraint on neutrino mixing parameters, derived by us, is independent of this issue [*just as the consequences of exact TBM are*]{}. Those experiments will also investigate neutrino mixing parameters. A combination [@nf-15a; @nf-15b; @nf-15c] of data from the ongoing and upcoming runs of T2K and NO$\nu$A experiments would probe $|s_{23}-{1 \over \sqrt{2}}|$ from the conversion probability $P(\nu_\mu \rightarrow \nu_e)$. Now, in case a sizable nonzero value of that quantity is measured, being of magnitude comparable in percentage terms to (100 $s_{13}$)$\%$ of the maximal value of $s_{23}$, our condition ([**2**]{}) would hold and predict a small amount of CP violation in neutrino oscillations from the above data. Contrariwise, the failure to measure any deviation from maximal atmospheric neutrino mixing outside error bars would mean that our condition ([**1**]{}) would operate with $s_{13}\sin \delta_{CP} = O(\epsilon^\nu)$, $J=O(\epsilon^\nu)$ permitted; that would bolster the hope of detecting CP nonconservation for oscillating neutrinos from the difference in conversion probabilities $P(\nu_\mu \rightarrow \nu_e)
-P(\overline{\nu_\mu} \rightarrow \overline{\nu_e})$. The latter would be good news not only for a combined analysis of data from forthcoming runs of [@nf-16] of T2K and NO$\nu$A but also for future experiments with superbeams, such as LBNF [@nf-17], LBNO [@nf-18a; @nf-18b] or a neutrino factory at 10 GeV [@nf-19]. Current hints, either for a non-maximal $\theta_{23}$ or a nonzero $\sin \delta_{CP}/J$, by no means constitute any robust evidence and an experimental resolution of these two issues is urgently called for.
Comparative studies with specific flavor models of broken TBM
=============================================================
In the present analysis we have used first order perturbation theory to analytically establish relations between basis independent sets of small coefficients $\{\epsilon^\ell, \epsilon^\nu\}$ and TBM deviant measurables. In doing so, we have been able to establish the relations given in Eq.(\[eqn-25\]) to Eq.(\[eqn-28\]). Physical observables partaining to CP violation have also been related analytically to these basis independent $\epsilon$ coefficients. For these relations to remain valid, TBM symmetry should be broken weakly so that one could jusify first order perturbation theory. If that symmetry is broken strongly, in other words, if the Lagrangian contains large terms violating TBM symmetry, then these relations would fail to be true [^4] . In that case direct numerical diagonalization would need to be made. On the other hand, numerical diagonalization cannot be done in a model independent way; consequently, a case by case study, depending on the model of TBM symmetry breaking, would be required.
Given our two assumptions, namely (1) $|\epsilon^\ell| << |\epsilon^\nu|$ and (2) the absence of unnatural cancellations, it is desirable to cross check our result with specific flavor symmetry models which break tribimaximality by some amount. We consider below several such proposed models in a representative but not comprehensive survey. Most (though not all) of these are variations of a basic family symmetry model [@nf-39] utilizing the discrete group $A_4$ along with gauge singlet Higgs fields called flavons which transform as specified $A_4$ representations. Not every such model can be cast within the framework of additive perturbations to $M^0_{\nu f}$ and $M^0_{\ell f}$. Nonetheless, we deem it useful to make this comparison. In these models, if some flavons develop VEVs aligned in appropriately chosen directions in the corresponding $A_4$ representation space, TBM obtains in the neutrino sector with mass diagonal charged leptons. Certain higher mass dimensional terms are entered into the Lagrangian containing ratios of flavon VEVs divided by a much larger cut-off scale. If some slight misalignment is then introduced in these VEV directions, deviations result from exact TBM.
The first example of this type on our list is that of Ref.[@nf-40] which utilizes two $A_4$ triplet and three $A_4$ singlet flavons. An analytical study of this model was made here while the TBM deviants were investigated numerically. The magnitude of the dominant TBM breaking parameters was restricted to small values by taking $|U_{13}| < 0.2$. This analysis took care to ensure unitary implementation of broken TBM symmetry, i.e. that perturbed eigenstates of type $t$ do make up the columns of $U_t$. A revealing facet of this model is that the misalignment induced coefficients in the perturbed charged lepton eigenstates turn out to be significantly less in magnitude than the corresponding ones for neutrinos. This is since the latter get enhanced by mass ratio factors such as $(m^0_{\nu 1} + m^0_{\nu 3})(m^0_{\nu 2}-m^0_{\nu 1})^{-1},
(m^0_{\nu 1} + m^0_{\nu 3})|m^0_{\nu 2}-m^0_{\nu 3}|^{-1}$ and $(m^0_{\nu 3} + m^0_{\nu 1})|m^0_{\nu 1}-m^0_{\nu 3}|^{-1}$ in our notation, from the imposed unitary implementation. [*The corresponding factors in the charged lepton case are non-enhancing because of the hierarchical nature of the charged lepton masses.*]{} Thus the model manifestly satisfies our condition $|\epsilon^\ell| < |\epsilon^\nu|$. The computed numerical values of $J$ are found to go all the way up to $0.046$ when the full parameter space is scanned, cf. Table II of Ref.[@nf-40]. This means that $(s_{13} \sin \delta_{CP})_{\rm max} \sim 0.195$, allowing substantial possible CP violation in neutrino oscillations. However, throughout the parameter space, one always has $\sin^2 2 \theta_{23} < 0.994$ i.e. $|s_{23} -\sqrt{1 \over 2}| < 0.03$, which permits only a tiny deviation from maximality in atmospheric atmospheric neutrino mixig. Therefore this model satisfies our option (1). The second model [@nf-41] that we consider is very similar to that of Ref.[@nf-40] except that the perturbations can be arbitrarily large and real flavon VEVs were chosen; consequently, there is no CP violation to be observed in neutrino oscillations. The deviation from maximality in $s_{23}$ can be made large only by chosing the TBM breaking perturbation parameter $|U_{e3}| \sim 0.4$. If $|U_{e3}|$ is restricted to $<0.2$, as dictated by later experiments, once again the numerical constraint $|s_{23} - \sqrt{1 \over 2}| < 0.03$ is seen to operate in agreement with Ref.[@nf-40], i.e. the deviation from maximal neutrino mixing is small by our critarion. Hence our option (1) is maintained here with the additional proviso of a nonexistent $J$. Significant deviations $|s_{23}-\sqrt{2 \over 3}|$ can occur for very large perturbations which are beyond the scope of our work.
We then consider the study of $A_4$ and $S_4$ based flavor symmetry models with perturbed lepton mass matrices reported in Ref.[@nf-42]. In particular, for the $S_4$ based model investigated, $s^2_{23}$ gets fixed at $1/2$ and there is no deviation from maximality in atmospheric neutrino mixing; moreover, $\delta_{CP}$ is preferred to be near $\pi$, i.e. no significant CP-violation in neutrino oscillations is predicted. So this model is compatible with both our options $1$ and $2$. In the $A_4$ based model considered (with just two $A_4$ singlet flavons), the authors derive the sum rule $s^2_{13} \sin^2 \delta_{CP}=s^2_{13} -2 (s^2_{23}-1/2)^2$. It is noteworthy that both our options $1$ and $2$ are compatible with this result. This is since, according to the sum rule, CP violation in neutrino oscillations is largest when $s^2_{23}=1/2$ while the deviation from maximal atmospheric neutrino mixing is greatest when the Dirac phase $\delta_{CP}=0$ or $\pi$ i.e. there is no CP violation.
The next analysis in our menu is that of Ref.[@nf-43]. Here again $A_4$-based models are considered with the number of $A_4$ singlet flavons varying from one to three and with the possibility of including the see-saw mechanism for neutrino mass generation. Additive perturbations are considered $vis$-$a$-$vis$ TBM invariant charged lepton and neutrino mass matrices and numerical diagonalization is carried out. The parameter spaces of the models considered here allow both a substantial $J \sim 0.02$ (i.e. $\delta_{CP} \sim 30^\circ$) and a sizable $|s^2_{23} -1/2| \ge 0$. However, unlike in Ref.[@nf-40], very large perturbation parameters have been alowed here. For instance, the charged lepton perturbations $\epsilon^{ch}$ have been taken upto $0.3$ while the corresponding neutrino ones have been kept completely free in the numerical scan with large allowed values. Thus lowest order perturbation theory does not apply to a considerable region of their parameter space. We expect that their results should agree with those of Ref.[@nf-40] once the smallness criterion is imposed on the perturbations.
The final analysis within the ambit of our comparative study is that of Ref.[@nf-44]. This work is somewhat different from the previously considered models in that no specific flavor symmetry such as $A_4$ for the Lagrangian is assumed. Instead, three separate mechanisms of TBM breaking are considered $per~se:$ $(1)$ corrections to $U_\ell$ in the charged lepton sector while keeping $U^0_\nu$ unchanged, $(2)$ renormalization group corrections (with supersymmetry) starting from exact TBM and nearly mass degenerate neutrinos at a very high scale and $(3)$ explicit TBM breaking terms added to $M^0_{\nu f}$ in the neutrino sector only. For $(1)$, the authors find that $J$ approaches a near maximum with $\delta_{CP} \sim \pi/2$ but the deviation from maximal atmospheric neutrino mixing is small with $s^2_{23}=1/2+ O(|U_{e3}|^2)$. This respects our option $(1)$. For cases $(2)$ and $(3)$ of Ref.[@nf-44], sizable such deviations in the latter are possible with $|s_{23}^2-1/2| \sim 0.1$-$0.2$; however, $J$ was not investigated. For case $(2)$, in particular, exact TBM at a high scale makes the starting boundary value of $\delta_{CP}$ indeterminate and an unambiguous answer is not possible.
Concluding summary
==================
In this paper we have considered general perturbations at the lowest order to hermitian squared mass matrices $M^\dagger_{\ell f} M_{\ell f}$ and $M^\dagger_{\nu f} M_{\nu f }$ respectively for charged leptons and neutrinos in the flavor basis of each and away from their TBM limits by carefully taking into account the unitary relation between the mass basis and the flavor basis. We have utilized the fact that columns of the said unitary matrix are the perturbed eigenstates. We have derived linear expressions for the three measurable TBM deviants in terms of the dimensionless coefficients that appear in the perturbed charged lepton and neutrino eigenstates. We have further derived four independent constraints on the imaginary parts of the latter from the requirement that four of the elements of $U^\dagger_\ell U_\nu$ have to be real. With the plausible arguements of the mixing caused by the strongly mass hierarchical charged leptons being significantly smaller than that due to neutrinos and no unnatural cancellations, we have derived a result, forcing one of two possibilities, which should be testable in the foreseeable future. This main result of ours can be stated succintly in the language of mathematical logic. Proposition A: an accurate description of neutrino mixing is given by the lowest order of additively perturbed tribimaximality without unnatural cancellations and with the mixing from the strongly mass hierarchical charged leptons being significantly smaller than that from neutrinos. Proposition B: $|s_{23}-\sqrt{1 \over 2}|=O(\epsilon^\nu)$. Proposition C: $s_{13} \sin \delta_{CP}/J=O(\epsilon^\nu)$. Then $A \cap ( B \cup C) = \emptyset$.
Appendix: Derivation of mixing constraints {#appendix-derivation-of-mixing-constraints .unnumbered}
==========================================
Neglecting $O(\epsilon^2)$ terms, we may write, $$\begin{aligned}
|\psi^{\nu,l}_1\rangle_f &=& |\psi^{0 \nu,l}_1\rangle_f +
\epsilon^{\nu,l}_{12}|\psi^{0 \nu,l}_2\rangle_f+
\epsilon^{\nu,l}_{13}|\psi^{0 \nu,l}_3\rangle_f, \\
|\psi^{\nu,l}_2\rangle_f &=& -{\epsilon^{\nu,l}_{12}}^*|\psi^{0 \nu,l}_1\rangle_f +
|\psi^{0 \nu,l}_2\rangle_f+
\epsilon^{\nu,l}_{23}|\psi^{0 \nu,l}_3\rangle_f, \\
|\psi^{\nu,l}_3\rangle_f &=& -{\epsilon^{\nu,l}_{12}}^* |\psi^{0 \nu,l}_1\rangle_f -
{\epsilon^{\nu,l}_{23}}^* |\psi^{0 \nu,l}_2\rangle_f+
|\psi^{0 \nu,l}_3\rangle_f ,\end{aligned}$$ where $$|\psi^{0 \nu}_1 \rangle_f=\begin{pmatrix}
\sqrt{2 \over 3} \cr -\sqrt{1 \over 6} \cr \sqrt{1 \over 6} \end{pmatrix},
|\psi^{0 \nu}_2 \rangle_f=\begin{pmatrix}\sqrt{1 \over 3} \cr \sqrt{1 \over 3} \cr -\sqrt{1 \over 3} \end{pmatrix},
|\psi^{0 \nu}_3 \rangle_f=\begin{pmatrix} 0 \cr \sqrt{1 \over 2} \cr \sqrt{1 \over 2} \end{pmatrix}$$ and $$|\psi^{0 l}_1 \rangle_f=\begin{pmatrix}1 \cr 0 \cr 0 \end{pmatrix},
|\psi^{0 l}_2 \rangle_f=\begin{pmatrix}0 \cr 1 \cr 0 \end{pmatrix},
|\psi^{0 l}_3 \rangle_f=\begin{pmatrix}0 \cr 0 \cr 1 \end{pmatrix}.$$
By using $U_\ell=(|\psi^\ell_1\rangle_f~|\psi^\ell_2\rangle_f~|\psi^\ell_2\rangle_f)$ and $U_\nu=(|\psi^\nu_1\rangle_f~|\psi^\nu_2\rangle_f~|\psi^\nu_3\rangle_f)$, one is led to the respective expressions for $U_\ell$ and $U_\nu$, as given in the text. If we define $L \equiv U^\dagger_\ell U_\nu$, then neglecting $O(\epsilon^2)$ terms, the nine elements of the $L$ matrix are $$\begin{aligned}
L_{11} &=&
\sqrt{2 \over 3}
+ \sqrt{1 \over 3} \epsilon^\nu_{12}
-\sqrt{1 \over 6} \epsilon^{\ell *}_{12}
+\sqrt{1 \over 6} \epsilon^{\ell *}_{13}, \\
L_{12} &=&
\sqrt{1 \over 3}
- \sqrt{2 \over 3} \epsilon^{\nu *}_{12}
+\sqrt{1 \over 3} \epsilon^{\ell *}_{12}
-\sqrt{1 \over 3} \epsilon^{\ell *}_{13},\\
L_{13} &=&
-\sqrt{2 \over 3} \epsilon^{\nu *}_{13}
-\sqrt{1 \over 3} {\epsilon^{\nu *}_{23}}
+\sqrt{1 \over 2} {\epsilon^{\ell *}_{12}}
+\sqrt{1 \over 2} {\epsilon^{\ell *}_{13}} ,\\
L_{21} &=&
-\sqrt{1 \over 6}
+ \sqrt{1 \over 3} \epsilon^\nu_{12}
+\sqrt{1 \over 2} \epsilon^{\nu}_{13}
-\sqrt{2 \over 3} \epsilon^{\ell}_{12}
+\sqrt{1 \over 6} {\epsilon^{\ell *}_{23}},\\
L_{22} &=&
\sqrt{1 \over 3}
+ \sqrt{1 \over 6} \epsilon^{\nu *}_{12}
+\sqrt{1 \over 2} \epsilon^{\nu}_{23}
-\sqrt{1 \over 3} \epsilon^{\ell}_{12}
-\sqrt{1 \over 3} \epsilon^{\ell *}_{12}, \\
L_{23} &=&
\sqrt{1 \over 2}
+ \sqrt{1 \over 6} \epsilon^{\nu *}_{13}
-\sqrt{1 \over 3} \epsilon^{\nu *}_{23}
+\sqrt{1 \over 2} \epsilon^{\ell *}_{23}, \\
L_{31} &=&
\sqrt{1 \over 6}
- \sqrt{1 \over 3} \epsilon^\nu_{12}
+\sqrt{1 \over 2} {\epsilon^{\nu}_{13}}
-\sqrt{2 \over 3} {\epsilon^{\ell}_{13}}
+\sqrt{1 \over 6} \epsilon^\ell_{23}, \\
L_{32} &=&
-\sqrt{1 \over 3}
- \sqrt{1 \over 6} {\epsilon^{\nu *}_{12}}
+\sqrt{1 \over 2} {\epsilon^{\nu}_{23}}
-\sqrt{1 \over 3} {\epsilon^{\ell}_{13}}
-\sqrt{1 \over 3} \epsilon^\ell_{23}, \\
L_{33} &=&
\sqrt{1 \over 2}
- \sqrt{1 \over 6} {\epsilon^{\nu *}_{13}}
+ \sqrt{1 \over 3} {\epsilon^{\nu *}_{23}}
-\sqrt{1 \over 2} {\epsilon^{\ell }_{23}}. \end{aligned}$$ Similarly, defining $N \equiv U_{PMNS}K^{-1}$ and again neglecting $O(\epsilon^2)$ terms $$N=\begin{pmatrix}c_{12} & s_{12} & s_{13}~e^{-i\delta_{CP}} \cr
-s_{12}c_{23}-\sqrt{1 \over 3}s_{13}e^{i \delta_{CP}}
& c_{12}c_{23} -\sqrt{1 \over 6} s_{13}e^{i \delta_{CP}} & s_{23} \cr
s_{12}s_{23}-\sqrt{1 \over 3} s_{13} e^{i \delta_{CP}} & -c_{12}s_{23}-\sqrt{1 \over 6} s_{13}e^{i \delta_{CP}}
& c_{23} \end{pmatrix}.$$
Expanding in $\epsilon$, the relations $\sqrt{2}c_{12}+s_{12}=\sqrt{3}+O(\epsilon^2)$ and $c_{23}+s_{23}=\sqrt{2} + O(\epsilon^2)$ are automatic. The equality $L=N$ leads to the mixing constraint relations. Specifically, the identification of elements or their combinations $$\begin{aligned}
&& L_{11}=N_{11},~L_{21}-L_{31}=N_{21}-N_{31},L_{21}+L_{31}=N_{21}+N_{31},\nonumber\\
&& L_{12}=N_{12}, L_{22}+L_{32}=N_{22}+N_{32},~L_{22}-L_{32}=N_{22}-N_{32}, \nonumber\\
&& L^*_{13}=N^*_{13},~L_{33}+L_{23}=N_{33}+N_{23},~L_{33}-L_{23}=N_{33}-N_{23}, \nonumber\end{aligned}$$ neglecting $O(\epsilon^2)$ terms, lead respectively to the equations $$\begin{aligned}
&& c_{12}-\sqrt{2 \over 3} = {1 \over \sqrt{2}}~(\sqrt{1 \over 3}-s_{12})
=\sqrt{1 \over 3} \epsilon^\nu_{12}-\sqrt{1 \over 6}({\epsilon^{\ell *}_{12}} -{\epsilon^{\ell *}_{13}} ),\label{a16}\\
&& -\sqrt{1 \over 3}(c_{23}-s_{23})-{2 \over \sqrt{3}}s_{13}e^{i \delta_{CP}} = \sqrt{2}
\epsilon^\nu_{13}-\sqrt{2 \over 3}(\epsilon^\ell_{12}+\epsilon^\ell_{13})
+\sqrt{1 \over 6}(\epsilon^\ell_{23}+ {\epsilon^{\ell *}_{23}}),\label{a17}\\
&&-\sqrt{2}s_{12} = -2 \sqrt{1 \over 6}+2\sqrt{1 \over 3}\epsilon^\nu_{12}
-\sqrt{2 \over 3}(\epsilon^\ell_{12}-\epsilon^\ell_{13})
+\sqrt{1 \over 6}(\epsilon^\ell_{23}-{\epsilon^{\ell *}_{23}}),\label{a18}\\
&& s_{12}= \sqrt{1 \over 3}-\sqrt{2 \over 3}{\epsilon^{\nu *}_{12}}
+\sqrt{1 \over 3}({\epsilon^{\ell *}_{12}}-{\epsilon^{\ell *}_{13}}),\label{a19}\\
&&\sqrt{2 \over 3}\left( c_{23} -s_{23} -s_{13} e^{i \delta_{CP}}\right)
= \sqrt{2}\epsilon^\nu_{23}-\sqrt{1 \over 3}
(\epsilon^\ell_{12}+ \epsilon^\ell_{13}+{\epsilon^{\ell *}_{23}} + \epsilon^\ell_{23}),\label{a20}\\
&& \sqrt{2}c_{12}=2 \sqrt{1 \over 3} + 2 \sqrt{1 \over 6}\epsilon^{\nu *}_{12}
-\sqrt{1 \over 3}({\epsilon^{\ell *}_{23}}-\epsilon^\ell_{23})
-\sqrt{1 \over 3}(\epsilon^\ell_{12}-\epsilon^\ell_{13}),\label{a21}\\
&& s_{13}e^{i \delta_{CP}}=-\sqrt{1 \over 3}(\sqrt{2} \epsilon^\nu_{13} + \epsilon^\nu_{23})
+\sqrt{1 \over 2}(\epsilon^\ell_{12}+ \epsilon^\ell_{13})
\label{a22}\\
&&c_{23}+s_{23}=\sqrt{2}+{1 \over \sqrt{2}}({\epsilon^{\ell *}_{23}}-\epsilon^\ell_{23}),
\label{a23}\\
&&c_{23}-s_{23}=-\sqrt{2 \over 3}{\epsilon^{\nu *}_{13}}
-\sqrt{1 \over 2}(\epsilon^\ell_{23}+\epsilon^{\ell *}_{23})
+ {2 \over \sqrt{3}} \epsilon^{\nu *}_{23}. \label{a24}\end{aligned}$$
Eq. (\[eqn-27\]) is a direct conseqence of (\[a23\]). Eq. (\[eqn-26\]) is easily derived from (\[a20\]) and (\[a22\]), while Eq. (\[eqn-25\]) follows from (\[a19\]) and (\[a21\]). Now Eq. (\[eqn-28\]) obtains from (\[a21\]), whereas Eq. (\[eqn-29\]) is just a rewritten form of (\[a16\]) with the input of Eq. (\[eqn-28\]). Eq. (\[eqn-30\]) follows from (\[a17\]) and (\[a20\]). Finally, Eq. (\[eqn-31\]) is the same as (\[a22\]).
Acknowledgement {#acknowledgement .unnumbered}
================
A part of this work was done at the WHEPP13 workshop, Puri, India. We thank A. Dighe, P. Byakti, S. Choubey, A. Raychaudhuri and S. Uma Sankar for their comments. PR acknowledges a Senior Scientistship of Indian National Science Academy.
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[^1]: We follow the procedure of Ref. [@nf-2] and take neutrinos to be light Majorana particles occurring in three generations. Consequently, we take a complex symmetric mass matrix for them. In the mass basis, that is $M_\nu={\rm diag}.~(m_{\nu 1}, m_{\nu 2}, m_{\nu 3})$ with $m_{\nu 1}=|m_{\nu 1}|,~
m_{\nu 2}= |m_{\nu 2}| e^{- i \alpha_{21}},~ m_{\nu 3}=|m_{\nu 3}| e^{-i \alpha_{31}}$ and $\alpha_{21},~\alpha_{31}$ as Majorana phases. We also use $c_{ij} \equiv \cos \theta_{ij}$ and $s_{ij} \equiv \sin \theta_{ij}$ for the angle of mixing $\theta_{ij}$ between neutrino flavors $i$ and $j$.
[^2]: We shall throughout refer to a TBM deviating effect as (1) “very large” if it is $>>$ $100 s_{13} \% \sim $12-17$\%$ so that higher order perturbations cannot be ignored, (2) “large” if it is in the ballpark of $100 s_{13}\%$ so that it should be soon measurable as well as computable with only lowest order perturbations and (3) negligibly “small” if it is $< 0.03$ which is $O(s^2_{13})$.
[^3]: Experiments in the far future with neutrino factories may probe such a level and, for such measurements, the neglected $O(\epsilon^2)$ effects as well as those due to renornalization group evolution from an assumed high scale symmetry would be relevant.
[^4]: TBM braking in general is naturally expected to be under control for lowest order perturbation theory since $s_{13}$ has been observed to be $<0.18$.
|
---
abstract: 'The main objective of this paper is to introduce a new method for qualitative analysis of various designs of robot arms. To this end we define the complexity of a map, examine its main properties and develop some methods of computation. In particular, when applied to a forward kinematic map associated to some robot arm structure, the complexity measures the inherent discontinuities that arise when one attempts to solve the motion planning problem for any set of input data. In the second part of the paper, we consider instabilities of motion planning in the proximity of singular points, and present explicit computations for several common robot arm configurations.'
address: 'Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia'
author:
- Petar Pavešić
title: COMPLEXITY OF THE FORWARD KINEMATIC MAP
---
Introduction
============
In this paper we introduce and discuss a new qualitative measure of the complexity of a forward kinematic map from the configuration space of the robot arm joints to the working space of the end-effector. Let us illustrate the problem on a familiar example of a robot arm with $n$ revolute joints. The position of the $i$-th joint is uniquely determined by some angle of rotation $\theta_i$, so we may identify the position of each joint with a point on the unit circle $T$, and the combined position of all $n$ joints with an $n$-tuple of values $(\theta_1,\ldots,\theta_n)\in T^n=T\times\ldots\times T$ ($n$ factors). The position of the end-effector is determined by the spatial location and the orientation, so it corresponds to a point $(\vec{r},R)$ in the cartesian product ${\mathord{\mathbb{R}}}^3\times SO(3)$ (here we identify the space of all possible orientation of a rigid body with the set $SO(3)$ of all orthogonal $3\times 3$ matrices of determinant 1). The exact form of the resulting forward kinematic map $F\colon T^n\to{\mathord{\mathbb{R}}}^3\times SO(3)$ depends on the lengths and the respective inclinations of the axes, and is usually given in terms of Denavit-Hartenberg matrices but the explicit formulae will not be relevant for the main part of our discussion.
The motion planning problem in this setting may be stated as follows: given an initial state of joint parameters $(\theta_1,\ldots,\theta_n)\in T^n$ and a required end-effector position $(\vec{r},R)\in{\mathord{\mathbb{R}}}^3\times SO(3)$, find motions of the joints starting at $(\theta_1,\ldots,\theta_n)$ and ending in a position of joints such that the corresponding end-effector position is $(\vec{r},R)$. The problem may be modelled as follows: let $P(T^n)$ denote the space of all possible paths in $T^n$ (i.e. continuous maps $\alpha\colon [0,1]\to T^n$ from the interval $[0,1]$ to the joint parameter space), and let $\pi\colon P(T^n)\to T^n\times ({\mathord{\mathbb{R}}}^3\times SO(3))$ be the map that to each path assigns its starting and ending position, $\pi(\alpha):=\big(\alpha(0),F(\alpha(1))\big)$. Then the solution of the motion planning problem can be viewed as an inverse map $\rho \colon T^n\times ({\mathord{\mathbb{R}}}^3\times SO(3))\to P(T^n)$, with the property $\pi(\rho((\theta_1,\ldots,\theta_n),(\vec{r},R)))=((\theta_1,\ldots,\theta_n),(\vec{r},R))$. We are mainly interested in robust motion plans, such that a small perturbation of the initial data results in a comparatively small perturbation of the corresponding motion plan. In other words, we normally require that the map $\rho$ is continuous with respect to the input data.
The starting point of our investigation is the following fundamental observation (see Theorem \[thm: cx 1 implies section\]):
*If there exists a robust global solution of the motion planning problem for the map $F$, then there also exists a continuous global solution of the inverse kinematic problem for the map $F$.*
Since in practice inverse kinematic solutions can be found only for a very restricted class of simple manipulators, it follows that a solution of the motion planning problem will almost always require a partition of the input data space into smaller domains, over which a robust motion plan can be constructed. The minimal number of domains that is needed to cover all possible input data measures the complexity of the motion planning for a given robot arm configuration. We are going to describe a mathematical model that will allow a clear definition of the complexity and develop several methods for its computation.
Prior work
----------
Motion planning is one of the basic problems in robotics, it has been extensively studied under all possible aspects, and there exists a vast literature on the topic. We are going to rely on [@Hollerbach; @89] and [@Kavraki-LaValle] as basic references. In the standard formulation of the motion planning problem one specifies the *configuration space* ${\mathcal{C}}$ of the robot device (which normally correspond to the set of joint parameters but may also take into account various limitations), the *working space* ${\mathcal{W}}$ (spatial position and orientation that can be reached by the robot), and *obstacle regions* in ${\mathcal{W}}$. Then one considers *queries* that consist of an *initial configuration* $q_I\in{\mathcal{C}}$ and a *goal configuration* $q_G\in{\mathcal{C}}$ (corresponding to the desired position and orientation of the end-effector), and asks for a path $\rho\colon [0,1]\to{\mathcal{C}}$ that avoids obstacles, and for which $\rho(0)=q_I$ and $\rho(1)=q_G$.
The complexity of motion planning was mostly considered within the context of computational complexity: indeed, the search for explicit algorithms aimed to the solution of a given motion planning problem was often accompanied by more general considerations regarding the algorithmic complexity of various solutions - see [@Reif; @79] and [@Canny; @88]. For a more recent study that extensively uses topological methods and is similar in spirit to our approach see [@Erdmann; @10].
A more geometrical measure for the complexity of motion planning was introduced by M. Farber [@Farber:TCMP] who observed that algorithmic solutions normally yield robust mapping plans. He then defined the concept of the *topological complexity* of motion planning in the working space of a mechanical device as the minimal number of continuous partial solutions to the motion planning problem. In many cases the computation of the topological complexity of a space may be reduced to the computation of a very classical numerical invariant, called the Lusternik-Schnirelmann category. Indeed, in certain sense the study of the complexity of the geometric motion planning can be traced back to the 1930’s, to the work in variational calculus by L. Lusternik and L. Schnirelmann. They introduced what is today called the *Lusternik-Shnirelmann category* of a space, denoted ${\mathord{\mathrm{cat}}}(X)$, as a tool to estimate the number of critical points of a smooth map. Their work was widely extended both in analysis, most notably by J. Schwartz [@Schwartz] and R. Palais [@Palais], and in topology, by R. Fox [@Fox], T. Ganea [@Ganea], I. James [@James] and many others. Today Lusternik-Schnirelmann category is a well-developed theory with many ramifications and methods of computation techniques that allow to systematically determine the category for most of the spaces that will appear in this paper - see [@CLOT]. It is interesting to see how this very classical and independently developed theory found its application in the study of problems in robotics. For an overview of principal results on topological complexity see [@Farber:ITR], for a less technical and very readable account, see also [@Ghrist; @14 Chapter 8].
The study of the complexity of a map is a natural continuation of Farber’s work and was suggested as a problem by A. Dranishnikov during the conference on Applied Algebraic Topology in Castro Urdiales (Spain, 2014). In spite of strong formal similarities, the flavour of this new theory is quite different from the topological complexity. A partial explanation can be found in some earlier work by J. Hollerbach [@Hollerbach] and D. Gottlieb [@Gottlieb:IEEE], who studied the possibility to avoid singularities of the forward kinematic map by introducing additional joints. They proved that under standard assumptions a forward kinematic map always has singularities and (with rare exceptions) does not admit global inverse kinematics. As a consequence, the study of the complexity of a map seems to be less amenable to purely homotopy-theoretical methods.
Our contribution
----------------
We introduce a general framework for the study of the complexity of a continuous map. Roughly speaking, the complexity of a map $F\colon{\mathcal{C}}\to{\mathcal{W}}$ is the minimal number of robust rules that take as input pairs of points $(c,w)\in{\mathcal{C}}\times{\mathcal{W}}$, and yield paths $\rho=\rho(c,w)$ starting at $c$ and ending at some $c'$ that is mapped by $F$ to $w$. We first show that under the assumption that $F$ is regular and admits a right inverse (section), the computation of the complexity of $F$ can be reduced to the topological complexities of the spaces ${\mathcal{C}}$ and ${\mathcal{W}}$ in the sense of Farber. However, a forward kinematic map usually satisfy these assumptions only locally, which causes considerable difficulties and require the development of entirely new methods. As a main computational tool we introduce the concept of relative complexity of $F$ with respect to suitably chosen subspaces of ${\mathcal{C}}\times{\mathcal{W}}$. In specific situations we normally proceed in two stages, by first finding a suitable decomposition of ${\mathcal{C}}\times{\mathcal{W}}$ and then using different estimates to determine the complexity of $F$ over each piece. In this way we are able to obtain good estimates for some important practical cases, especially various combinations of revolute joints.
Outline
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In the next section we define the complexity of a forward kinematic map and compare it with some related concepts. The central part of the paper are Sections 3 and 4. Section 3 contains the theoretical background and is divided into four subsections, where we consider kinematic maps that admit inverses, regular kinematic maps, lower bounds related to some cohomological obstructions, and finally the general situation, where the kinematic map is not assumed to be regular or possess a global inverse. In Section 4 we apply the theory to estimate the complexity of several important examples, including the system of two parallel joints, the universal joint, the triple-roll wrist and the 6-DOF joint. We suggest to the reader to read Section 4 in parallel with Section 3 in order to appreciate the significance of various theoretical results.
Throughout the paper we use the following standard notation: $T$ is the unit circle in the plane, $T^n$ is the cartesian product of $n$ circles, $S^2$ is the two-dimensional sphere, and $SO(3)$ is the space of rotations of the three-dimensional space, or equivalently, of orthogonal $3\times 3$-matrices with determinant 1. Moreover ${\rm pr}_{\mathcal{C}}$ denotes projection of a product to the factor ${\mathcal{C}}$ and ${\rm gr}(F)$ denotes the graph of the map $F$.
Complexity of a map
===================
As mentioned in the Introduction, the basic example that we have in mind is the forward kinematic map of a robot arm with $n$ revolute joints. However, in order to allow other types of joints and arm configurations, we are going to work in a more general setting. Let us therefore consider an arbitrary forward kinematic map $F\colon {\mathcal{C}}\to {\mathcal{W}}$ from some *configuration space* ${\mathcal{C}}$ to a *working space* ${\mathcal{W}}$. Normally the space ${\mathcal{C}}$ is the cartesian product of the parameter spaces for the individual joints (but the space may also be restricted to reflect various obstacles and other conditions), while ${\mathcal{W}}$ is a subspace of ${\mathord{\mathbb{R}}}^3\times SO(3)$ (but may be enlarged to include the velocity and angular momentum of the end-effector). A motion of the arm is then simply a path in ${\mathcal{C}}$, which we model by a continuous map $\rho\colon [0,1]\to {\mathcal{C}}$. All theoretically possible motions of the arm are described by the set of all paths in the joint parameter space, which we denote by ${\mathrm{P}(\mathcal{C})}$. The setting of time-scale as an interval between 0 and 1 allows a simple description of the initial and final stage of the motion, so we have a map $\pi\colon {\mathrm{P}(\mathcal{C})}\to {\mathcal{C}}\times {\mathcal{W}}$, given by $\pi(\rho)=(\rho(0),F(\rho(1))$, i.e. to each movement of the arm we assign its initial position of the joints and the goal position of the end-effector. The pair $(c,w)\in{\mathcal{C}}\times {\mathcal{W}}$ representing an initial configuration $c$ and a goal position $w$ is often called a *query*, and ${\mathcal{C}}\times {\mathcal{W}}$ is the *query space*. A map $\rho\colon{\mathcal{C}}\times {\mathcal{W}}\to {\mathrm{P}(\mathcal{C})}$ for which $\pi(\rho(c,w))=(c,w)$ is called a *roadmap*, because it may be interpreted as a rule that to each initial configuration $c$ and goal position $w$ of the end-effector assigns a movement of joints $\rho(c,w)$ that starts in the configuration $c$ and ends in the position $w$. More formally, a roadmap is a right inverse $\rho$ to the projection $\pi$. We require that the map $\rho$ is robust, which means that a small perturbation of the initial data results in a small change of the path performed by the robot arm. In more mathematical terms, this amounts to the requirement that the map $\rho$ must be continuous.
We are now ready to state our first conclusion that may be viewed as the starting point of this study.
\[thm: cx 1 implies section\] If a given forward kinematic map $F\colon {\mathcal{C}}\to {\mathcal{W}}$ admits a robust roadmap, then it also admits an continuous inverse kinematic map.
Let $\rho\colon {\mathcal{C}}\times {\mathcal{W}}\to {\mathrm{P}(\mathcal{C})}$ be a robust roadmap for $F$, and let $c_0$ be some initial configuration of the robot arm. Then the formula $I(w):=\rho(c_0,w)(1)$ satisfies the relation $F(I(w))=w$ for every $w\in{\mathcal{W}}$, which means that $I\colon {\mathcal{W}}\to {\mathcal{C}}$ is the inverse kinematic map for $F$.
We are, of course, mostly interested in the negative aspect of the above result: since most forward kinematic maps that appear in practice do not admit a continuous inverse kinematic map, it follows that most motion planning problems cannot have a global robust solution. We are therefore forced to look for robust solutions on subspaces of the query space ${\mathcal{C}}\times {\mathcal{W}}$. The minimal number of domains with robust roadmaps that cover all possible queries in ${\mathcal{C}}\times{\mathcal{W}}$ will be called the *complexity of the forward kinematic map* $F$ and will be denoted ${\mathord{\mathrm{cx}}}(F)$.
Observe that if we assume that ${\mathcal{C}}$ and ${\mathcal{W}}$ are the same space and that $F$ is the identity map on ${\mathcal{C}}$, then ${\mathord{\mathrm{cx}}}({\mathord{\mathrm{Id}}}_{\mathcal{C}})$ coincides with ${\mathord{\mathrm{TC}}}({\mathcal{C}})$, the topological complexity of ${\mathcal{C}}$ that we mentioned in the Introduction. Indeed, that case corresponds to motion planning within the configuration space of the robot arm, without taking into account the relation to the working space, given by the forward kinematic map. The topological complexity of most of the spaces that are of interest for us has already been computed and can be found in the literature so we will systematically reduce the computation of ${\mathord{\mathrm{cx}}}(F)$ to the computation of the topological complexity of ${\mathcal{C}}$ and ${\mathcal{W}}$. Moreover, we will simplify the notation and write ${\mathord{\mathrm{cx}}}({\mathcal{C}})$ and ${\mathord{\mathrm{cx}}}({\mathcal{W}})$ instead of ${\mathord{\mathrm{cx}}}({\mathord{\mathrm{Id}}}_{\mathcal{C}})$ and ${\mathord{\mathrm{cx}}}({\mathord{\mathrm{Id}}}_{\mathcal{W}})$.
Similarly as in the case of the topological complexity, we will not try to compute the complexity of $F$ directly but rather by finding suitable upper and lower estimates. Upper estimates are mostly based on explicit partitioning of the query space and description of corresponding robust roadmaps. Lower estimates are more subtle, as they require to theoretically demonstrate the impossibility to find a smaller number of robust roadmaps.
Estimates of ${\mathord{\mathrm{cx}}}(F)$
=========================================
Let us introduce a more formal definition of the complexity that will ensure mathematical correctness of our conclusions. A space is said to be an *Euclidean Neighbourhood Retract* (short: ENR) if it can be obtained as a retract of some open subspace of ${\mathord{\mathbb{R}}}^n$. This class includes most interesting geometric objects like manifolds, polyhedra, algebraic sets and other spaces that arise as configuration and working spaces of mechanical systems. We will assume that ${\mathcal{C}}$ and ${\mathcal{W}}$ are ENR spaces and will consider only zzs of ${\mathcal{C}}$ and ${\mathcal{W}}$ that are also ENR. A *partial roadmap* for the forward kinematic map $F\colon{\mathcal{C}}\to{\mathcal{W}}$ is a continuous map $\rho\colon {\mathcal{Q}}\to {\mathrm{P}(\mathcal{C})}$ whose domain ${\mathcal{Q}}$ is an ENR subspace of the query space ${\mathcal{C}}\times {\mathcal{W}}$, and $\rho$ is a right inverse for $\pi$, i.e. $\pi(\rho(c,w))=(c,w)$ for every query $(c,w)\in{\mathcal{Q}}$. The *complexity* of $F$ is the minimal integer $n$ for which ${\mathcal{C}}\times{\mathcal{W}}$ can be covered by $n$ partial roadmaps. We will usually take the domains of roadmaps to be disjoint but that requirement is not part of the definition, and sometimes it may even be more natural to allow overlapping (for example, when we want to take into account various inaccuracies and noise that arise in real-world situations).
Invertible kinematic maps
-------------------------
In this subsection we will work under the assumption that $F\colon {\mathcal{C}}\to {\mathcal{W}}$ admits a continuous inverse kinematic map $I\colon {\mathcal{W}}\to{\mathcal{C}}$, such that $F(I(w))=w$ for all $w\in {\mathcal{W}}$. This assumption is rarely satisfied in practice, but we will be still able to apply the results when the inverse kinematic map can be computed over parts of ${\mathcal{W}}$ (i.e. avoiding singular points and gimbal lock positions). We have the following basic result that gives good upper and lower estimates for the complexity of $F$.
\[thm: sectioned F\] If the map $F\colon {\mathcal{C}}\to {\mathcal{W}}$ admits a right inverse $I\colon {\mathcal{W}}\to {\mathcal{C}}$ then $${\mathord{\mathrm{cx}}}({\mathcal{W}})\le {\mathord{\mathrm{cx}}}(F) \le {\mathord{\mathrm{cx}}}({\mathcal{C}}).$$
To prove that ${\mathord{\mathrm{cx}}}({\mathcal{W}})\le {\mathord{\mathrm{cx}}}(F)$ we will show that a partition of ${\mathcal{C}}\times{\mathcal{W}}$ into $n$ partial roadmaps for $F$ allows to generate a partition of ${\mathcal{W}}\times{\mathcal{W}}$ into $n$ partial roadmaps for ${\mathord{\mathrm{Id}}}_{\mathcal{W}}$. For every partial roadmap $\rho\colon {\mathcal{Q}}\to {\mathrm{P}(\mathcal{C})}$ the following formula $$\bar \rho(w,w'):=F\circ \rho(I(w),w')$$ clearly determines a partial roadmap on $\overline {\mathcal{Q}}=\{(w,w')\mid (I(w),w')\in {\mathcal{Q}}\}$. Moreover, if the domains ${\mathcal{Q}}_1,\ldots,{\mathcal{Q}}_n$ cover ${\mathcal{C}}\times {\mathcal{W}}$ then the corresponding domains $\overline {\mathcal{Q}}_1,\ldots,\overline {\mathcal{Q}}_n$ cover ${\mathcal{W}}\times{\mathcal{W}}$.
Similarly, given a subspace ${\mathcal{Q}}\subseteq {\mathcal{C}}\times{\mathcal{C}}$ and a partial roadmap $\rho\colon{\mathcal{Q}}\to{\mathrm{P}(\mathcal{C})}$ for ${\mathord{\mathrm{Id}}}_{\mathcal{C}}$, the formula $$\bar \rho(c,w):=\rho(c,I(w))$$ determines a partial roadmap on $$\overline {\mathcal{Q}}=\{(c,w)\mid (c,I(w))\in {\mathcal{Q}}\}.$$ If the domains ${\mathcal{Q}}_1,\ldots,{\mathcal{Q}}_n$ cover ${\mathcal{C}}\times {\mathcal{C}}$ then the corresponding domains $\overline {\mathcal{Q}}_1,\ldots,\overline {\mathcal{Q}}_n$ cover ${\mathcal{C}}\times{\mathcal{W}}$, therefore ${\mathord{\mathrm{cx}}}(F)\le{\mathord{\mathrm{cx}}}({\mathcal{C}})$.
The configuration space of a $n$-joint robot arm is the cartesian product of $n$-circles, ${\mathcal{C}}=T^n$, whose topological complexity is exactly $n+1$ (see [@Farber:TCMP Theorem 13]), so if there exists a global robust inverse kinematic map for $F$, then ${\mathord{\mathrm{cx}}}(F)\le n+1$. On the other side, the complexity of the standard working space ${\mathcal{W}}={\mathord{\mathbb{R}}}^3\times SO(3)$ is 4 (see [@Farber:ITR Theorem 4.61]), so in the presence of a global inverse kinematic map the motion planning requires at least 4 robust partial roadmaps. This surprising result explains many of the practical difficulties that arise when we try to construct explicit motion planning algorithms. We will see later, that for systems that do not admit a global inverse kinematics the minimal number of robust roadmaps may be even bigger.
We have seen in Theorem \[thm: cx 1 implies section\] that if the complexity of a forward kinematic map $F$ is 1 then $F$ admits a global inverse kinematic map, and so by Theorem \[thm: sectioned F\] the complexity of the working space is also 1. By [@Farber:TCMP Theorem 1] spaces with complexity one are contractible (i.e continuously deformable to a point). This is possible only if the robot working space is a spatial domain without obstacles, and its movement does not involve planar or spatial rotations (cf. [@Handbook Section 5.6.1]).
Regular kinematic maps
----------------------
In this subsection we take into account the analytic properties of the forward kinematic map $F\colon{\mathcal{C}}\to{\mathcal{W}}$. In fact, $F$ is normally a smooth mapping so we may consider its Jacobian $J(F)$ which can at every point be represented by an $m\times n$ matrix, where $m$ and $n$ are respectively the dimensions of the spaces ${\mathcal{W}}$ and ${\mathcal{C}}$. In particular the dimension of the configuration space corresponds to the *degrees of freedom* of the robot system. If at some point $c\in{\mathcal{C}}$ the rank of the Jacobian matrix of $F$ is not maximal, then from that point the device cannot move in all possible directions in the working space. This is a common problem in robotic systems, known as a *gimbal lock*. The forward kinematic map $F$ is *regular* at $c\in{\mathcal{C}}$ if the Jacobian matrix of $F$ at that point has maximal rank, otherwise $F$ is *singular* at $c$. A forward kinematic map $F$ is *regular* if it is regular at all points. Hollerbach [@Hollerbach] proved that a forward kinematic map whose working space allows arbitrary rotations of the end-effector always has singular points, even if the system is redundant (cf. also Gottlieb [@Gottlieb]). Nevertheless, in this section we are going to consider the complexity of regular forward kinematic maps as an intermediate step toward the general case.
For our purposes, the main property of regular kinematic maps is that they allow to lift paths from the working space to the configuration space in the following sense. Let $c$ be a configuration in ${\mathcal{C}}$, and let $\alpha$ be a path in ${\mathcal{W}}$ (corresponding to a sequence of movements of the end-effector) that starts at $\alpha(0)=F(c)$. We may interpret the pair $(c,\alpha)$ as an input datum for the following task: find a sequence of motions of the joints that starts from the joint configuration $c$, and such that the end-effector performs exactly the movements prescribed by the path $\alpha$. It is a standard fact of differential topology due to Ehresmann [@Ehresmann] that if $F$ is regular, then this task has a robust solution. More formally, let us denote by ${\mathcal{C}}\sqcap P({\mathcal{W}})=\{(c,\alpha)\in{\mathcal{C}}\times P({\mathcal{W}})\mid F(c)=\alpha(0)\}$ the space of input tasks for a robot system. Then Ehresmann’s theorem may be stated as follows.
If the forward kinematic map $F\colon{\mathcal{C}}\to{\mathcal{W}}$ is regular, then there exists a continuous map $\Gamma\colon{\mathcal{C}}\sqcap P({\mathcal{W}})\to{\mathrm{P}(\mathcal{C})}$ such that for all $(c,\alpha)\in {\mathcal{C}}\sqcap P({\mathcal{W}})$ we have $\Gamma(c,\alpha)(0)=c$ and $F\circ\Gamma(c,\alpha)=\alpha$.
The property stated in the proposition essentially means that we may solve every motion task in the configuration space, provided we are able to solve the corresponding task in the working space. Thus the following result does not come as a surprise.
\[thm: F fibration\] If the forward kinematic map $F\colon{\mathcal{C}}\to{\mathcal{W}}$ is regular then $${\mathord{\mathrm{cx}}}(F)\le{\mathord{\mathrm{cx}}}({\mathcal{W}}).$$
Let ${\mathcal{Q}}\subseteq {\mathcal{W}}\times{\mathcal{W}}$ and let $\rho\colon{\mathcal{Q}}\to P({\mathcal{W}})$ be a partial roadmap for ${\mathord{\mathrm{Id}}}_{\mathcal{W}}$. Then the formula $$\bar \rho(c,w):=\Gamma(c,\rho(F(c),w))$$ determines a partial roadmap on $\overline {\mathcal{Q}}=\{(c,w)\mid (F(c),w)\in {\mathcal{Q}}\}$. If the domains ${\mathcal{Q}}_1,\ldots,{\mathcal{Q}}_n$ cover ${\mathcal{W}}\times {\mathcal{W}}$ then the corresponding domains $\overline {\mathcal{Q}}_1,\ldots,\overline {\mathcal{Q}}_n$ cover ${\mathcal{C}}\times{\mathcal{W}}$, therefore ${\mathord{\mathrm{cx}}}(F)\le{\mathord{\mathrm{cx}}}({\mathcal{W}})$.
By combining Theorems \[thm: sectioned F\] and \[thm: F fibration\] we obtain the following corollary:
\[cor: sectioned fibration\] If the forward kinematic map is regular and admits an inverse kinematic map, then ${\mathord{\mathrm{cx}}}(F)={\mathord{\mathrm{cx}}}({\mathcal{W}})$.
In the next result we give a precise characterization of a query set that can admit a robust roadmap.
\[prop: cx=1\] Let $F\colon {\mathcal{C}}\to{\mathcal{W}}$ be a regular forward kinematic map. Then a set of queries ${\mathcal{Q}}\subset{\mathcal{C}}\times{\mathcal{W}}$ admits a robust partial roadmap $\rho \colon{\mathcal{Q}}\to{\mathrm{P}(\mathcal{C})}$ if, and only if ${\mathcal{Q}}$ can be deformed (within ${\mathcal{C}}\times{\mathcal{W}}$) to the graph of $F$.
Given a robust roadmap $\rho\colon{\mathcal{Q}}\to{\mathrm{P}(\mathcal{C})}$ we can define a deformation $D\colon{\mathcal{Q}}\times [0,1]\to {\mathcal{C}}\times{\mathcal{W}}$ by the formula $$D(c,w,t):=(\rho(c,w)(t),w).$$ Clearly, the initial stage of deformation is $D(c,w,0)=(\rho(c,w)(0),w)=(c,w)$, and the final stage is $D(c,w,1)=(\rho(c,w)(1),w)=(c',w)$, where $F(c')=w$, therefore $D({\mathcal{Q}}\times\{1\})$ is contained in the graph of $F$.
Conversely, if $D\colon{\mathcal{Q}}\times[0,1]\to{\mathcal{C}}\times{\mathcal{W}}$ is a deformation of ${\mathcal{Q}}$ to the graph of $F$, then the projections of $D(c,w,t)$ to ${\mathcal{C}}$ and ${\mathcal{W}}$ yield paths $\alpha$ from $c$ to $c'$ in ${\mathcal{C}}$ and $\alpha'$ from $w$ to $w'$ in ${\mathcal{W}}$, such that $F(c')=w'$. Therefore, we may join the path $\alpha$ with the reverse of the lifting of $\alpha'$ to obtain a motion plan from $c$ to $w$. The corresponding formula for the roadmap $\rho\colon{\mathcal{Q}}\to{\mathrm{P}(\mathcal{C})}$ is thus $$\rho(c,w)(t):=\left\{\begin{array}{ll}
{\rm pr}_{\mathcal{C}}(D(c,w,2t)) &;\ \ 0\le t\le\frac{1}{2}\\
\Gamma \big(({\rm pr}_{\mathcal{C}}(D(c,w,1)),{\rm pr}_{\mathcal{W}}(D(c,w,-))\big)(2-2t)&;\ \ \frac{1}{2}\le t\le 1\end{array}\right.$$
Observe that the regularity of $F$ was used only in the second half of the proof. In fact, a roadmap on ${\mathcal{Q}}$ always determines a deformation of ${\mathcal{Q}}$ to the graph of $F$, and moreover, during the deformation the ${\mathcal{W}}$-component is preserved. We will say that the roadmap defines a *horizontal* deformation of ${\mathcal{Q}}$ to the graph.
Cohomological lower bound {#subsec: cohomology}
-------------------------
The upper bounds for the complexity of $F$ that we obtained in the last two subsections are actually constructive, being derived from the complexities of ${\mathcal{C}}$ and ${\mathcal{W}}$ for which suitable roadmaps can be explicitly described. On the other side, the lower bound in Theorem \[thm: sectioned F\] depends indirectly on the lower bound for ${\mathord{\mathrm{cx}}}({\mathcal{W}})$, which is in turn based on some cohomological estimates as in [@Farber:ITR Section 4.5]. In this subsection we will obtain better estimates by considering the homomorphism in cohomology induced by the forward kinematic map.
Let $H^*$ be any cohomology theory (e.g. de Rham, singular, Čech...) and assume that a set of queries ${\mathcal{Q}}$ admits a roadmap $\rho\colon{\mathcal{Q}}\to{\mathrm{P}(\mathcal{C})}$. Then by the above discussion ${\mathcal{Q}}$ may be continuously deformed to the graph of $F$ which may be expressed by the following diagram $$\xymatrix{
{{\mathcal{C}}} \ar[rr]^{{\mathord{\mathrm{Id}}}_{\mathcal{C}}\times F} & & {{\mathcal{C}}\times{\mathcal{W}}}\\
& {{\mathcal{Q}}} \ar[lu]^{d} \ar@{^(->}[ru]_i}$$ where $d(c,w)=\rho(c,w)(1)$, the image of ${\mathord{\mathrm{Id}}}_{\mathcal{C}}\times F$ is exactly the graph of $F$, and $({\mathord{\mathrm{Id}}}_{\mathcal{C}}\times F)\circ d$ is homotopic to the inclusion $i$. By applying the contravariant cohomology functor $H^*$ we obtain a commutative diagram of respective cohomology groups $$\xymatrix{
H^*({{\mathcal{C}}}) \ar[rd]_{d^*} & & {H^*({\mathcal{C}}\times{\mathcal{W}})}\ar[ll]_{{\mathord{\mathrm{Id}}}\times F^*}\ar[ld]^{i^*}\\
& {H^*({\mathcal{Q}})} }$$ If a cohomology class $u$ is in the kernel of ${\mathord{\mathrm{Id}}}\times F^*$ then clearly $i^*(u)=0$. The kernel of $i^*$ coincides with the image of the homomorphism $$j^* \colon H^*({\mathcal{C}}\times{\mathcal{W}},{\mathcal{Q}})\to H^*({\mathcal{C}}\times{\mathcal{W}}),$$ therefore one can find a relative cohomology class $\bar u\in H^*({\mathcal{C}}\times{\mathcal{W}},{\mathcal{Q}})$ such that $u=j^*(\overline u)$. In particular, if ${\mathord{\mathrm{cx}}}(F)=1$ then every element of ${\mathord{\mathrm{Ker}}}({\mathord{\mathrm{Id}}}\times F^*)$ is the image of some class in $H^*({\mathcal{C}}\times{\mathcal{W}},{\mathcal{C}}\times{\mathcal{W}})=0$, and hence must be trivial. In other words, non-triviality of ${\mathord{\mathrm{Ker}}}({\mathord{\mathrm{Id}}}\times F^*)$ implies that motion planning in ${\mathcal{C}}\times{\mathcal{W}}$ requires more than one robust roadmap. We are going to estimate the minimal number of necessary roadmaps by considering products of cohomology classes.
Let $u_1,\ldots,u_n$ be elements of the kernel of ${\mathord{\mathrm{Id}}}\times F^*$, and let $Q_1,\ldots,Q_n$ be query sets that admit robust roadmaps and cover the entire query set ${\mathcal{C}}\times{\mathcal{W}}$. By the above argument one can choose representatives $\bar u_k\in H^*({\mathcal{C}}\times{\mathcal{W}},{\mathcal{Q}}_k)$ such that $u_k=j^*(\bar u_k)$. Then the cohomology product $$u_1\cdot\ldots\cdot u_n=j^*(\bar u_1)\cdot\ldots\cdot j^*(\bar u_n)=j^*(\bar u_1\cdot\ldots\cdot \bar u_n)=0,$$ because $\bar u_1\cdot\ldots\cdot \bar u_n\in H^*({\mathcal{C}}\times{\mathcal{W}},{\mathcal{Q}}_1\cup\ldots{\mathcal{Q}}_n)=H^*({\mathcal{C}}\times{\mathcal{W}},{\mathcal{C}}\times{\mathcal{W}})=0$. Therefore, if ${\mathcal{C}}\times{\mathcal{W}}$ can be covered by $n$ robust roadmaps, then every product of $n$ elements of the kernel of ${\mathord{\mathrm{Id}}}\times F^*$ must be 0. The minimal $n$ for which the product of any $n$ elements in the ideal ${\mathord{\mathrm{Ker}}}({\mathord{\mathrm{Id}}}\times F^*)$ is zero is called the *nilpotency* of the ideal, and is denoted ${\mathord{\mathrm{nil}}}({\mathord{\mathrm{Ker}}}({\mathord{\mathrm{Id}}}\times F^*))$. We may now state the main result of this section.
\[thm: nilpotency\] The complexity of the map $F\colon{\mathcal{C}}\to{\mathcal{W}}$ is bounded below by $${\mathord{\mathrm{cx}}}(F)\ge{\mathord{\mathrm{nil}}}({\mathord{\mathrm{Ker}}}({\mathord{\mathrm{Id}}}\times F^*)).$$
See Section \[subsec: 6DOF\] for an application of the cohomological estimate.
General case
------------
Let us now consider the general case that arises in practice, i.e. a forward kinematic map $F\colon{\mathcal{C}}\to{\mathcal{W}}$ that may have some singularities and that admit only partially defined inverse kinematic maps. The results and methods obtained in the previous sections will produce estimates of the complexity of $F$ over subspaces of the query space. In order to combine these into global estimates for the complexity of $F$, we will need a concept of relative complexity of $F$ over subspaces of the query space.
Let ${\mathcal{A}}$ be any ENR subspace of the query space ${\mathcal{C}}\times{\mathcal{W}}$. The *relative complexity* ${\mathord{\mathrm{cx}}}(F|{\mathcal{A}})$ of $F$ over ${\mathcal{A}}$ is the minimal number of robust roadmaps that are needed to cover all points of ${\mathcal{A}}$.
The relative complexity subsume as special instance several previously studied concepts. If we take the identity map on ${\mathcal{C}}$ and a subset ${\mathcal{A}}\subseteq{\mathcal{C}}\times{\mathcal{C}}$ then ${\mathord{\mathrm{cx}}}({\mathord{\mathrm{Id}}}_{\mathcal{C}}|{\mathcal{A}})={\mathord{\mathrm{cx}}}({\mathcal{C}}|{\mathcal{A}})$ coincides with the relative topological complexity (cf. [@Farber:ITR Section 4.3]). In many applications ${\mathcal{A}}$ will be a product of the form ${\mathcal{C}}'\times{\mathcal{W}}'$ where ${\mathcal{C}}'\subseteq{\mathcal{C}}$ and ${\mathcal{W}}'\subseteq{\mathcal{W}}$, corresponding to the complexity of the task to navigate from configurations in ${\mathcal{C}}'$ to end-effector positions in ${\mathcal{W}}'$. Another important special case is ${\mathord{\mathrm{cx}}}({\mathcal{C}}|{c_0}\times {\mathcal{C}})$ which coincides with the *Lusternik-Schnirelmann category* of ${\mathcal{C}}$ (cf. [@Farber:ITR Lemma 4.29]).
In the next proposition we collect the main properties of the relative complexity.
\[prop: rel cx\] Let $F\colon{\mathcal{C}}\to{\mathcal{W}}$ be a map, whose graph we denote by ${\rm gr}(F)$, and let ${\mathcal{A}},{\mathcal{A}}'$ be ENR subspaces of ${\mathcal{C}}\times {\mathcal{W}}$. Then the relative complexity of $F$ satisfies the following relations.
1. ${\mathord{\mathrm{cx}}}(F|{\rm gr}(F))=1$;
2. ${\mathcal{A}}\subseteq{\mathcal{A}}' \ \ \implies \ \ {\mathord{\mathrm{cx}}}(F|{\mathcal{A}})\le{\mathord{\mathrm{cx}}}(F|{\mathcal{A}}')$;
3. ${\mathord{\mathrm{cx}}}(F|{\mathcal{A}}\cup{\mathcal{A}}')\le{\mathord{\mathrm{cx}}}(F|{\mathcal{A}})+{\mathord{\mathrm{cx}}}(F|{\mathcal{A}}')$;
4. If ${\mathcal{A}}'$ can be horizontally deformed into ${\mathcal{A}}$ then ${\mathord{\mathrm{cx}}}(F|{\mathcal{A}})\ge{\mathord{\mathrm{cx}}}(F|{\mathcal{A}}')$.
The first three statements are self-evident, and only the last requires some proof. A horizontal deformation of ${\mathcal{A}}'$ into ${\mathcal{A}}$ is a continuous map $D\colon {\mathcal{A}}'\times [0,1]\to{\mathcal{C}}\times {\mathcal{W}}$, such that $D(c,w,0)=(c,w)$, $D(c,w,1)\in{\mathcal{A}}$ and ${\rm pr}_{\mathcal{W}}(D(c,w,t))=w$ for all $(c,w)\in{\mathcal{A}}'$ and $t\in[0,1]$. Let $\rho\colon {\mathcal{Q}}\to{\mathrm{P}(\mathcal{C})}$ be a robust roadmap for some ${\mathcal{Q}}\subseteq{\mathcal{A}}$. Then we may define a robust roadmap $\rho'\colon{\mathcal{Q}}'\to{\mathrm{P}(\mathcal{C})}$, where ${\mathcal{Q}}'=\{(c,w)\in{\mathcal{A}}'\mid D(c,w,1)\in{\mathcal{Q}}\}$ and $$\rho'(c,w)(t)=\left\{\begin{array}{ll}
{\rm pr}_{\mathcal{C}}(D(c,w,2t)) &;\ \ 0\le t\le \frac{1}{2}\\
\rho(D(c,w,1))(2t-1) &; \ \ \frac{1}{2}\le t\le 1 \end{array}\right.$$ If the domains ${\mathcal{Q}}_1,\ldots,{\mathcal{Q}}_n$ cover ${\mathcal{A}}$, then the corresponding domains ${\mathcal{Q}}'_1,\ldots,{\mathcal{Q}}'_n$ cover ${\mathcal{A}}'$, therefore ${\mathord{\mathrm{cx}}}(F|{\mathcal{A}}')\le{\mathord{\mathrm{cx}}}(F|{\mathcal{A}})$.
Our next objective is to extend the results of the previous subsections to forward kinematic maps that have singular points. Let ${\mathcal{W}}^r$ denote the set of *regular values* of the forward kinematic map $F\colon{\mathcal{C}}\to{\mathcal{W}}$, i.e. the set of $w\in{\mathcal{W}}$ such that all configurations in the pre-image $F^{-1}(w)\subset {\mathcal{C}}$ are regular for $F$. In practice this means that whenever the robot device position is in ${\mathcal{W}}^r$ it can been moved in all directions in the working space, regardless of the position of joints. Moreover, let ${\mathcal{C}}^r:=F^{-1}({\mathcal{W}}^r)$ the subspace of regular configurations that map to positions in ${\mathcal{W}}^r$. Then the restriction $F\colon{\mathcal{C}}^r\to{\mathcal{W}}^r$ is regular and we may extend our previous results on the complexity of regular maps.
\[thm: rel F fibration\] Let $F\colon{\mathcal{C}}^r\to{\mathcal{W}}^r$ be the restriction of the forward kinematic map to the subspace of ${\mathcal{C}}$ where $F$ has regular values. Then
1. ${\mathord{\mathrm{cx}}}(F|{\mathcal{C}}^r\times{\mathcal{W}}^r)\le {\mathord{\mathrm{cx}}}({\mathcal{W}}^r)$;
2. ${\mathord{\mathrm{cx}}}(F|{\mathcal{C}}\times{\mathcal{W}}^r)\le {\mathord{\mathrm{cat}}}({\mathcal{C}}\times{\mathcal{W}}^r)$.
Statement (1) is a direct application of Theorem \[thm: F fibration\]. As for the second claim, recall that an ENR subset $A\subseteq X$ is *categorical* if it can be deformed to a point within $X$, and ${\mathord{\mathrm{cat}}}(X)$ is the minimal number of categorical subsets needed to cover $X$. Therefore, in order to prove (2) it is sufficient to show that every categorical subset ${\mathcal{A}}\subseteq{\mathcal{C}}\times{\mathcal{W}}^r$ admits a roadmap $\rho\colon{\mathcal{A}}\to{\mathrm{P}(\mathcal{C})}$. Let $D\colon{\mathcal{A}}\times [0,1] \to{\mathcal{C}}\times{\mathcal{W}}^r$ be a deformation of ${\mathcal{A}}$ to a point $(c_0,w_0)$ (and we may assume without loss of generality that $(c_0,w_0)\in{\rm gr}(F)$). For every $(c,w)\in{\mathcal{A}}$ we define $$\rho(c,w)(t):=\left\{\begin{array}{ll}
{\rm pr}_{\mathcal{C}}(D(c,w,2t)) &; \ \ 0\le t\le\frac{1}{2}\\
\Gamma(c_0,{\rm pr}_{\mathcal{W}}(D^-(c,w,-))(2t-1) &; \ \ \frac{1}{2} \le t \le 1
\end{array}\right.$$ where $\Gamma$ is the path-lifting function for the regular map $F\colon{\mathcal{C}}^r\to{\mathcal{W}}^r$ and $D^-(c,w,-)$ is the reverse of the path $D(c,w,-)$. Note that $\Gamma$ may be applied because the path ${\rm pr}_{\mathcal{W}}(D^-(c,w,-))$ is entirely contained in ${\mathcal{W}}^r$. It is easy to verify that $\rho\colon {\mathcal{A}}\to{\mathrm{P}(\mathcal{C})}$ is a robust roadmap. As every categorical subset of ${\mathcal{C}}\times{\mathcal{W}}^r$ admits a roadmap, we conclude that the complexity of $F$ over ${\mathcal{C}}\times {\mathcal{W}}^r$ does not exceed the category of ${\mathcal{C}}\times{\mathcal{W}}^r$.
Note that if $F$ is regular then part (2) gives the estimate ${\mathord{\mathrm{cx}}}(F)\le{\mathord{\mathrm{cat}}}({\mathcal{C}}\times{\mathcal{W}})$, that we haven’t mention previously because it is usually weaker than the estimate that we proved in Theorem \[thm: F fibration\]. For relative complexity the situation is different as the two estimates refer to different sets of queries.
Inverse kinematics for kinematic maps with singularities can be quite complicated. In fact, even if the map $F\colon{\mathcal{C}}\to{\mathcal{W}}$ is regular one cannot expect to find in general a global inverse kinematic map - cf. [@Gottlieb:IEEE] for a discussion of obstructions to the existence of inverse kinematic maps. Thus we will usually first partition the set of regular values of $F$ into subspaces that admit robust inverse kinematic maps, and then study separately the possibility to extend motion plans near the singular values of $F$.
Let us assume that the forward kinematic map $F\colon{\mathcal{C}}\to{\mathcal{W}}$ admits a continuous inverse kinematic map $I\colon{\mathcal{W}}'\to{\mathcal{C}}$ over some subspace ${\mathcal{W}}'$ of the working space of the robot. The next theorem gives a general estimate of the relative complexity and two important cases when equality holds.
\[thm: relative sectioned fibration\] Assume that the map $F\colon{\mathcal{C}}\to{\mathcal{W}}$ admits a continuous partial right inverse $I\colon{\mathcal{W}}'\to{\mathcal{C}}$ over a subset ${\mathcal{W}}'\subseteq{\mathcal{W}}$, and let ${\mathcal{C}}':=F^{-1}({\mathcal{W}}')$. Then
1. ${\mathord{\mathrm{cx}}}({\mathcal{W}}|{\mathcal{W}}'\times{\mathcal{W}}')\le {\mathord{\mathrm{cx}}}(F|{\mathcal{C}}\times{\mathcal{W}}')\le {\mathord{\mathrm{cx}}}({\mathcal{C}}|{\mathcal{C}}\times I({\mathcal{W}}'));$
2. If ${\mathcal{C}}'$ can be deformed to $I({\mathcal{W}}')$, then ${\mathord{\mathrm{cx}}}(F|{\mathcal{C}}\times{\mathcal{W}}')={\mathord{\mathrm{cx}}}({\mathcal{C}}|{\mathcal{C}}\times I({\mathcal{W}}'));$
3. If ${\mathcal{W}}'\subseteq{\mathcal{W}}^r$, then ${\mathord{\mathrm{cx}}}(F|{\mathcal{C}}'\times{\mathcal{W}}')={\mathord{\mathrm{cx}}}({\mathcal{W}}|{\mathcal{W}}'\times{\mathcal{W}}').$
To verify statement (1) consider $${\mathord{\mathrm{cx}}}({\mathcal{W}}|{\mathcal{W}}'\times{\mathcal{W}}')\le {\mathord{\mathrm{cx}}}(F|I({\mathcal{W}}')\times{\mathcal{W}}')\le{\mathord{\mathrm{cx}}}(F|{\mathcal{C}}\times{\mathcal{W}}')\le {\mathord{\mathrm{cx}}}({\mathcal{C}}|{\mathcal{C}}\times I({\mathcal{W}}')),$$ where the first and third inequalities are proved by the same argument as in Theorem \[thm: sectioned F\] while the second inequality follows from Proposition \[prop: rel cx\](2). Similarly, the statement (3) is analogous to Corollary \[cor: sectioned fibration\] and is proved by combining relative versions of Theorems \[thm: sectioned F\] and \[thm: F fibration\].
As for (2) observe that in general the requirement that a roadmap for a query $(c,w)$ ends in $I(w)$ is quite restrictive and we will give an example in Section \[subsec universal joint\] where ${\mathord{\mathrm{cx}}}({\mathcal{C}}|{\mathcal{C}}\times I({\mathcal{W}}'))$ is strictly bigger than ${\mathord{\mathrm{cx}}}(F|{\mathcal{C}}\times{\mathcal{W}}')$. However, if ${\mathcal{C}}'$ can be deformed to $I({\mathcal{W}}')$, then we will show that ${\mathord{\mathrm{cx}}}(F|{\mathcal{C}}\times{\mathcal{W}}')\ge {\mathord{\mathrm{cx}}}({\mathcal{C}}|{\mathcal{C}}\times I({\mathcal{W}}'))$, and so the two complexities coincide. Let $D\colon {\mathcal{C}}'\times [0,1]\to {\mathcal{C}}$ be a deformation such that $D(c,0)=c$ and $D(c,1)=I(F(D(c,1)))$ for every $c\in{\mathcal{C}}'$, and assume that ${\mathcal{A}}\subseteq {\mathcal{C}}\times {\mathcal{W}}'$ admits a robust roadmap $\rho\colon{\mathcal{A}}\to{\mathrm{P}(\mathcal{C})}$. Then the formula $$\bar\rho(c,c')(t):=\left\{\begin{array}{ll}
\rho(c,F(c'))(2t) &; \ \ 0\le t \le\frac{1}{2}\\
D\big(\rho(c,F(c'))(1),2t-1\big) &; \ \ \frac{1}{2}\le t \le 1
\end{array}\right.$$ defines a roadmap $\bar\rho\colon\bar{\mathcal{A}}\to{\mathrm{P}(\mathcal{C})}$ where $\bar{\mathcal{A}}=\{(c,c')\in {\mathcal{C}}\times I({\mathcal{W}}')\mid (c,F(c'))\in{\mathcal{A}}\}$. If the domains ${\mathcal{A}}_1,\ldots,{\mathcal{A}}_n$ cover ${\mathcal{C}}\times{\mathcal{W}}'$, then the corresponding domains $\bar{\mathcal{A}}_1,\ldots,\bar{\mathcal{A}}_n$ cover ${\mathcal{C}}\times I({\mathcal{W}}')$, therefore ${\mathord{\mathrm{cx}}}(F|{\mathcal{C}}\times{\mathcal{W}}')\ge {\mathord{\mathrm{cx}}}({\mathcal{C}}|{\mathcal{C}}\times I({\mathcal{W}}'))$.
In many applications the configurations space ${\mathcal{C}}$ is given as the cartesian product of circles (namely, parameter spaces of individual joints), therefore it possesses the structure of a topological group. The following result then allows a precise computation of the upper bound in the above theorem.
\[thm: top group\] If ${\mathcal{C}}$ is a topological group, then $${\mathord{\mathrm{cx}}}({\mathcal{C}}|{\mathcal{C}}\times{\mathcal{C}}')={\mathord{\mathrm{cx}}}({\mathcal{C}})={\mathord{\mathrm{cat}}}({\mathcal{C}})$$ for every nonempty subspace ${\mathcal{C}}'\subseteq{\mathcal{C}}$.
If ${\mathcal{C}}$ is a topological group, then the complexity of ${\mathcal{C}}$ is equal to its Lusternik-Schnirelmann category - see [@Farber:IRM Lemma 8.2 ]. Let $c_0$ be any configuration in ${\mathcal{C}}'$. Then ${\mathord{\mathrm{cat}}}({\mathcal{C}})={\mathord{\mathrm{cx}}}({\mathcal{C}}|{\mathcal{C}}\times\{c_0\})$ by [@Farber:ITR Lemma 4.29] so we obtain the following chain of (in)equalities $${\mathord{\mathrm{cat}}}({\mathcal{C}})={\mathord{\mathrm{cx}}}({\mathcal{C}}|{\mathcal{C}}\times\{c_0\})\le{\mathord{\mathrm{cx}}}({\mathcal{C}}|{\mathcal{C}}\times{\mathcal{C}}')\le{\mathord{\mathrm{cx}}}({\mathcal{C}})={\mathord{\mathrm{cat}}}({\mathcal{C}}).$$
In our final result we will relate the complexity of $F$ to the number of partial right inverses (also called partial *sections*) that are needed to cover all points in its codomain. Let $\rho\colon{\mathcal{Q}}\to {\mathrm{P}(\mathcal{C})}$ be a partial robust roadmap over some ${\mathcal{Q}}\subseteq {\mathcal{C}}\times{\mathcal{W}}$, and let ${\mathcal{Q}}':=\{w\in{\mathcal{W}}\mid (c_0,w)\in{\mathcal{Q}}\}$ for some fixed element $c_0\in{\mathcal{C}}$. Then the formula $I_\rho(w):=\rho(c_0,w)(1)$ defines a robust map $I_\rho\colon {\mathcal{Q}}'\to{\mathcal{C}}$ such that $F(I_\rho(w))=w$ for all $w\in{\mathcal{Q}}'$, therefore $I_\rho$ is a partial section of $F$. Moreover, one can define a deformation $D_\rho\colon I_\rho({\mathcal{Q}}')\times [0,1]\to{\mathcal{C}}$ as $D_\rho(c,t):=\rho(c_0,F(c))(t)$, so that $D_\rho(c,0)=c_0$ and $D_\rho(c,1)=c$ for all $c\in I_\rho({\mathcal{Q}}')$. In other words, $I_\rho$ has the additional property that its image can be deformed within ${\mathcal{C}}$ to a point. We will say that $I_\rho$ is a *categorical partial section* for $F$. Let ${\mathord{\mathrm{sec}}}(F)$ denote the minimal number of partial sections of $F$ that are needed to cover all points of ${\mathcal{W}}$, and let ${\mathord{\mathrm{csec}}}(F)$ be the minimal number of categorical partial sections of $F$ that are needed to cover all points of ${\mathcal{W}}$. Clearly ${\mathord{\mathrm{sec}}}(F)\le{\mathord{\mathrm{csec}}}(F)$.
The complexity of a map $F\colon{\mathcal{C}}\to{\mathcal{W}}$ satisfies the following inequality $${\mathord{\mathrm{csec}}}(F)\le{\mathord{\mathrm{cx}}}(F)\le\sum_{k=1}^{{\mathord{\mathrm{sec}}}(F)} {\mathord{\mathrm{cx}}}({\mathcal{C}}|{\mathcal{C}}\times I_k({\mathcal{Q}}_k)),$$ where $I_k\colon{\mathcal{Q}}_k\to{\mathcal{C}}$ are partial sections of $F$ and ${\mathcal{W}}={\mathcal{Q}}_1\cup\ldots\cup{\mathcal{Q}}_{{\mathord{\mathrm{sec}}}(F)}$. In particular, if ${\mathord{\mathrm{cx}}}({\mathcal{C}}|{\mathcal{C}}\times I_k({\mathcal{Q}}_k))=1$ for all $k$, then ${\mathord{\mathrm{cx}}}(F)={\mathord{\mathrm{sec}}}(F)={\mathord{\mathrm{csec}}}(F)$.
Let $\rho_1,\ldots,\rho_{{\mathord{\mathrm{cx}}}(F)}$ be some minimal set of roadmaps that cover ${\mathcal{C}}\times{\mathcal{W}}$. By the above discussion, there exist categorical partial sections $I_{\rho_1},\ldots,I_{\rho_{{\mathord{\mathrm{csec}}}(F)}}$ whose domains cover ${\mathcal{W}}$, therefore ${\mathord{\mathrm{csec}}}(F)\le{\mathord{\mathrm{cx}}}(F)$.
The second inequality follows from Proposition \[prop: rel cx\](3) and Theorem \[thm: relative sectioned fibration\](1) as we have $${\mathord{\mathrm{cx}}}(F)\le\sum_{k=1}^{{\mathord{\mathrm{sec}}}(F)} {\mathord{\mathrm{cx}}}(F|{\mathcal{C}}\times {\mathcal{Q}}_k)\le \sum_{k=1}^{{\mathord{\mathrm{sec}}}(F)} {\mathord{\mathrm{cx}}}({\mathcal{C}}|{\mathcal{C}}\times I_k({\mathcal{Q}}_k)).$$
Note that in the last theorem we did not assume that $F$ is regular at any point, so the inequality may be applied to estimate the complexity of arbitrary continuous maps.
Examples and computations
=========================
In this section we consider several examples of robot mechanisms that arise in practice and apply our results to estimate the motion planning complexity of their forward kinematic maps.
One revolute joint
------------------
Let us begin with a system consisting of one revolute joint as in Fig. \[fig: one joint\]. Its configuration space ${\mathcal{C}}$ is the set of all possible angles of rotation of the joint, and so it may be identified with the unit circle $T$. If the end-effector is the tip of the arm, then the working space ${\mathcal{W}}$ is also the circle and the forward kinematic map is the identity. The complexity ${\mathord{\mathrm{cx}}}({\mathord{\mathrm{Id}}}_{\mathcal{C}})={\mathord{\mathrm{cx}}}(T)$ which is known to be equal 2 (see [@Farber:TCMP p.213] for explicit description of roadmaps).
![Position of the arm is completely described by the displacement angle $\theta$. Configuration space is $T$.[]{data-label="fig: one joint"}](fig1.png)
If instead of the identity map we imagine the end-effector to be connected to the joint by some transmission mechanism so that one full rotation of the joint correspond to several rotations of the end-effector then the configuration space and working space are still equal to $T$ but the kinematic map may be described as $F(\theta)=k\theta$ where $k$ is the transmission ratio. It is clear that the kinematic map is regular but, somewhat surprisingly, it does not admit an inverse kinematic map (unless $k=1$). In fact, to define the inverse kinematic map $I\colon T\to T$ one must first define the value of $I(0)=\theta_0$ among angles $\theta$ for which $k\theta$ is a multiple of $2\pi$. This choice then uniquely defines $I(\theta)=\theta_0+\theta/k$ for $\theta\in [0,2\pi)$, but if $k\ne 1$ the resulting map is clearly not continuous when $\theta$ approaches $2\pi$. By Theorems \[thm: cx 1 implies section\] and \[thm: F fibration\] we have $$1<{\mathord{\mathrm{cx}}}(F)\le{\mathord{\mathrm{cx}}}(T)=2,$$ and so ${\mathord{\mathrm{cx}}}(F)=2$. Observe that the proof of Theorem \[thm: F fibration\] also provides explicit roadmaps for $F$.
It often happens in practice that the configuration space or the working space are constrained for various reasons. Imagine for example that the robot arm can rotate only some finite amount of full circles (e.g. because of the wiring). Then the configuration space is just an interval, say of the form ${\mathcal{C}}=[-\theta_{max},\theta_{max}]$. We may interpret this situation as an instance of relative complexity ${\mathord{\mathrm{cx}}}(F|[-\theta_{max},\theta_{max}]\times{\mathcal{W}})$ which is as before bounded above by ${\mathord{\mathrm{cx}}}(T)=2$, but it cannot be equal to 1 because $F$ does not admit a section.
Finally, assume that both the configuration and working space are restricted to intervals, and the forward kinematic map is a bicontinuous bijection. Then the complexity of $F$ equals the complexity of the interval, which is 1 by [@Farber:TCMP Theorem 1]. However, if the end-effector is connected to a transmission mechanism so that it takes several full rotations of the joint to move the end-effector between two position in the working space, then there does not exist a continuous inverse kinematic map and one still needs two robust roadmaps to navigate this simple mechanism.
Two planar revolute joints
--------------------------
Given two revolute joints, they may be pinned together so that they rotate in the same plane or in different (usually perpendicular) planes. We begin with the planar case, which is simpler - see Fig. \[fig: two planar joints\]. If the rotation around both joints is unobstructed then the configuration space may be identified with the cartesian product of two circles, ${\mathcal{C}}=T^2$. If, as is normally the case, the first arm is longer than the second, then the working space of the end-effector is easily seen to be an annulus ${\mathcal{W}}=\{(x,y)\in{\mathord{\mathbb{R}}}^2\mid R_1-R_2\le \sqrt{x^2+y^2}\le R_1+R_2\}$. The forward kinematic map $F\colon{\mathcal{C}}\to{\mathcal{W}}$ can be described using polar coordinates as $$F(\theta_1,\theta_2)=\big(R_1\cos\theta_1+R_2\cos(\theta_1+\theta_2),R_1\sin\theta_1+R_2\sin(\theta_1+\theta_2)\big).$$ To each position in the working space correspond two joint configurations, except when both arms are parallel, so it is easy to explicitly determine an inverse kinematic function $I\colon{\mathcal{W}}\to{\mathcal{C}}$, e.g. by always choosing the ‘elbow down’ joint position.
![Position of the arm is completely described by the angles $\theta_1$ and $\theta_2$. Configuration space is $T^2$, while the working space is the annulus $T\times[R_1-R_2,R_1+R_2]$.[]{data-label="fig: two planar joints"}](fig2.png)
Thus, by Theorem \[thm: sectioned F\], together with the computations of complexity for $T$ and $T^2$ (see [@Farber:TCMP Theorem 12]) we have $$2={\mathord{\mathrm{cx}}}(T\times [R_1-R_2,R_1+R_2])\le {\mathord{\mathrm{cx}}}(F) \le {\mathord{\mathrm{cx}}}(T^2)=3.$$ To obtain the precise value, observe that the restriction of $F$ on ${\mathcal{C}}'=\{(\theta_1,0)\mid\theta_1\in T\}\subset{\mathcal{C}}$ is injective, so a roadmap from ${\mathcal{C}}$ to $F({\mathcal{C}}')$ is essentially the same as a roadmap in ${\mathcal{C}}$ from ${\mathcal{C}}$ to ${\mathcal{C}}'$. Then we may apply Proposition \[prop: rel cx\] and Theorem \[thm: top group\] to get $${\mathord{\mathrm{cx}}}(F)\ge{\mathord{\mathrm{cx}}}(F|{\mathcal{C}}\times F({\mathcal{C}}'))={\mathord{\mathrm{cx}}}({\mathcal{C}}|{\mathcal{C}}\times{\mathcal{C}}')={\mathord{\mathrm{cx}}}({\mathcal{C}})=3,$$ therefore the complexity of $F$ is 3. Explicit roadmaps for $F$ can be derived from the proof of Theorem \[thm: sectioned F\].
We may extend the above reasoning to a system of $n$ planar joints. In fact, the configuration space is the cartesian product of $n$ circles, ${\mathcal{C}}=T^n$, while the working space ${\mathcal{W}}$ is either a disk or an annulus, depending on the relative lengths of the robot arms. The forward kinematic map $F\colon{\mathcal{C}}\to{\mathcal{W}}$ is given by $$\begin{aligned}
F(\theta_1,\ldots,\theta_n)=&\big(R_1\cos\theta_1+R_2\cos(\theta_1+\theta_2)+\ldots+R_n\cos(\theta_1+\ldots+\theta_n),\\
& \ \, R_1\sin\theta_1+R_2\sin(\theta_1+\theta_2)+\ldots+R_n\sin(\theta_1+\ldots+\theta_n)\big),\end{aligned}$$ and it is easy to see that it admits an inverse kinematic map $I\colon{\mathcal{W}}\to{\mathcal{C}}$. For example, if the length of the first arm exceeds the sum of the lengths of the remaining arms (so that ${\mathcal{W}}$ is an annulus), then one may define inverse kinematics by letting $\theta_3=\ldots=\theta_n=0$ and choosing $\theta_1$ and $\theta_2$ as in the two-arm case. Furthermore, the restriction of $F$ on ${\mathcal{C}}'=\{(\theta_1,0,\ldots,0)\mid \theta_1\in T\}\subset{\mathcal{C}}$ is injective, so we have $${\mathord{\mathrm{cx}}}({\mathcal{C}})={\mathord{\mathrm{cx}}}({\mathcal{C}}|{\mathcal{C}}\times{\mathcal{C}}')= {\mathord{\mathrm{cx}}}(F|{\mathcal{C}}\times F({\mathcal{C}}'))\le{\mathord{\mathrm{cx}}}(F)\le{\mathord{\mathrm{cx}}}({\mathcal{C}}).$$ Therefore, by [@Farber:TCMP Theorem 12], ${\mathord{\mathrm{cx}}}(F)={\mathord{\mathrm{cx}}}(T^n)=n+1$.
Universal joint {#subsec universal joint}
---------------
Universal joint (also called Cardan joint) consists of two revolute joints whose axes intersect orthogonally, as in Fig. \[fig: universal joint\]. The joint has two degrees of freedom, its configurations space is the cartesian product of two circles, ${\mathcal{C}}=T^2$, and the space of positions that can be reached by its end-effector can be parametrized by the points on the two-dimensional sphere, so ${\mathcal{W}}=S^2$.
![Position of the arm is completely described by the angles $\theta_1$ and $\theta_2$. The end-effector can reach every point on the sphere, centered at the second joint. Configuration space is $T^2$ and the working space is the sphere ${\mathcal{W}}=S^2$.[]{data-label="fig: universal joint"}](fig3.png)
The forward kinematic map can be described using spherical coordinates $$F(\theta_1,\theta_2)=(R\cos\theta_1\cos\theta_2,R\sin\theta_1\cos\theta_2,R\sin\theta_2).$$ The computation of the Jacobian matrix detects the two well-known gimbal lock positions, namely when $\theta_2=\pm\frac{\pi}{2}$ and the end-effector points at the north ($N$) or south ($S$) pole of the sphere. Therefore ${\mathcal{W}}^r=S^2-\{N,S\}$, and ${\mathcal{C}}^r=T\times \big(T-\{-\frac{\pi}{2},\frac{\pi}{2}\}\big)$. The map $I\colon{\mathcal{W}}^r\to{\mathcal{C}}^r$ that to each point assigns its unique longitude $\theta_1$ and latitude $\theta_2$ satisfying the requirement that $\theta_2\in (-\frac{\pi}{2},\frac{\pi}{2})$, is clearly a robust inverse kinematic map for $F$. In order to compute the complexity of $F$ we begin with the following estimate based on properties (2) and (3) of Proposition \[prop: rel cx\]: $${\mathord{\mathrm{cx}}}(F|{\mathcal{C}}\times {\mathcal{W}}^r)\le{\mathord{\mathrm{cx}}}(F)\le {\mathord{\mathrm{cx}}}(F|{\mathcal{C}}\times {\mathcal{W}}^r)+{\mathord{\mathrm{cx}}}(F|{\mathcal{C}}\times \{N,S\}).$$ Note that we can define another inverse kinematic map $I'\colon{\mathcal{W}}^r\to{\mathcal{C}}^r$ by choosing latitude in the interval $(\frac{\pi}{2},\frac{3\pi}{2})$, and that ${\mathcal{C}}^r=I({\mathcal{W}}^r)\cup I'({\mathcal{W}}^r)$. It is easy to see that $I'({\mathcal{W}}^r)$ can be horizontally deformed within ${\mathcal{C}}$ to $I({\mathcal{W}}^r)$. Therefore ${\mathcal{C}}^r$ can be deformed into $I({\mathcal{W}}^r)$, so we may apply Theorem \[thm: relative sectioned fibration\](2), and then Theorem \[thm: top group\] and [@Farber:TCMP Theorem 12] to compute $${\mathord{\mathrm{cx}}}(F|{\mathcal{C}}\times {\mathcal{W}}^r)={\mathord{\mathrm{cx}}}({\mathcal{C}}|{\mathcal{C}}\times I({\mathcal{W}}^r))={\mathord{\mathrm{cx}}}({\mathcal{C}})=3.$$ To determine ${\mathord{\mathrm{cx}}}(F|{\mathcal{C}}\times \{N,S\})$ we first assume that ${\mathord{\mathrm{cx}}}(F|{\mathcal{C}}\times \{N\})=1$. Then, by the proof of Proposition \[prop: cx=1\] and the comments after it, there exists a horizontal deformation of ${\mathcal{C}}\times\{N\}$ to the graph of $F$. But a horizontal deformation would contract ${\mathcal{C}}$ to a point within ${\mathcal{C}}\times{\mathcal{W}}$, which is clearly impossible, therefore ${\mathord{\mathrm{cx}}}(F|{\mathcal{C}}\times \{N\})>1$. On the other side, we may define two explicit roadmaps over ${\mathcal{C}}\times\{N\}$: let $${\mathcal{Q}}_1:=\left\{(\theta_1,\theta_2,N)\in {\mathcal{C}}\times\{N\}\mid \theta_2\neq -\frac{\pi}{2}\right\},$$ $$\rho_1(\theta_1,\theta_2,N)(t):=\left(\theta_1,(1-t)\theta_2+\frac{t\pi}{2}\right),$$ and $${\mathcal{Q}}_2:=\left\{(\theta_1,\theta_2,N)\in {\mathcal{C}}\times\{N\}\mid \theta_2= -\frac{\pi}{2}\right\},$$ $$\rho_2(\theta_1,-\frac{\pi}{2},N)(t):=\left(\theta_1,-\frac{\pi}{2}+t\pi\right).$$ Analogous formulas define roadmaps $\rho'_1\colon{\mathcal{Q}}'_1\to{\mathrm{P}(\mathcal{C})}$ and $\rho'_2\colon{\mathcal{Q}}'_2\to{\mathrm{P}(\mathcal{C})}$ for ${\mathcal{C}}\times\{S\}$. Since ${\mathcal{C}}\times\{N\}$ and ${\mathcal{C}}\times\{S\}$ are disjoint, we may combine $\rho_1$ and $\rho'_1$ into a robust roadmap on ${\mathcal{Q}}_1\cup{\mathcal{Q}}'_1$, and similarly $\rho_2$ and $\rho'_2$ into a robust roadmap on ${\mathcal{Q}}_2\cup{\mathcal{Q}}'_2$, which implies that ${\mathord{\mathrm{cx}}}(F|{\mathcal{C}}\times\{N,S\})=2$. Note that the map $F$ admits an obvious inverse kinematics over the one point space $\{N\}$ (namely, choose any point with $\theta_2=\frac{\pi}{2}$), but ${\mathord{\mathrm{cx}}}({\mathcal{C}}|{\mathcal{C}}\times\{I(N)\})={\mathord{\mathrm{cx}}}({\mathcal{C}})=3$, so this gives an example where ${\mathord{\mathrm{cx}}}({\mathcal{C}}|{\mathcal{C}}\times I({\mathcal{W}}'))$ is strictly bigger than ${\mathord{\mathrm{cx}}}(F|{\mathcal{C}}\times{\mathcal{W}}')$.
At this point we know that the complexity of $F$ is between 3 and 5. We are going to examine the instability of the roadmaps around the singular points of $F$ and show that ${\mathord{\mathrm{cx}}}(F)$ is in fact at least 4. Assume that there exists a motion plan for $F$ that consists of robust roadmaps $\rho_i\colon{\mathcal{Q}}_i\to{\mathrm{P}(\mathcal{C})}$ for $i=1,2,3$. By restriction we obtain 3 roadmaps on ${\mathcal{C}}\times{\mathcal{W}}^r$, which is by above computation also the minimal number of roadmaps necessary to cover ${\mathcal{C}}\times{\mathcal{W}}^r$. Therefore, we may find robust roadmaps $\bar\rho_i\colon\overline{\mathcal{Q}}_i\to{\mathrm{P}(\mathcal{C})}$ that cover ${\mathcal{C}}\times I({\mathcal{W}}^r)$ and for which $\rho_i(c,w)=\bar\rho_i(c,I(w))$ for every $i$ and every $(c,w)\in{\mathcal{Q}}_i\cap({\mathcal{C}}\times{\mathcal{W}}^r)$. Moreover, if we choose a small ball $B$ around $N\in{\mathcal{W}}$, the complexity ${\mathord{\mathrm{cx}}}({\mathcal{C}}|{\mathcal{C}}\times I(B-\{N\}))$ is still equal to 3, so $\overline{\mathcal{Q}}_i\cap ({\mathcal{C}}\times I(B-\{N\}))\ne\emptyset$ for $i=1,2,3$. It follows that for each $i=1,2,3$ we may find sequences $(c_j,w_j)$ and $(c'_j,w'_j)$ in ${\mathcal{Q}}_i$ converging to $(c,N)$, but such that $(c_j,I(w_j))$ and $(c'_j,I(w'_j))$ converge to different points in ${\mathcal{C}}\times F^{-1}(N)$ (i.e. $(w_j)$ and $(w'_j)$ converge to $N$ from different directions). But then we would have $$\lim_j \rho_i(c_j,w_j)=\lim_j\bar\rho_i(c_j,I(w_j))\ne\lim_j\bar\rho_i(c'_j,I(w'_j))=\lim_j \rho_i(c'_j,w'_j),$$ which is a contradiction. We conclude that the motion planning for the universal joint requires at least four robust roadmaps.
Triple-roll wrist
-----------------
Triple-roll wrist is a compound joint consisting of three revolute joints whose axes pass through a common point - see Fig. \[fig: triple joint\]. By rotating the individual joints the end-effector can assume any orientation in the three-dimensional space. The device has three degrees of freedom, its configuration space is the cartesian product of three circles, ${\mathcal{C}}=T^3$, and the working space consists of all possible orientations of the end-effector, ${\mathcal{W}}=SO(3)$. There exist several ways to relate the positions of the joints to the resulting orientation, the most common being through the Euler angles, see [@Handbook Chapter 1.2] for explicit formulas for the forward kinematic map $F\colon{\mathcal{C}}\to{\mathcal{W}}$.
![Position of the arm is completely described by three angles of rotation around respective joints. The end-effector can assume arbitrary orientation in the space. Configuration space is $T^3$ and the working space is $SO(3)$.[]{data-label="fig: triple joint"}](fig4.png)
The computation of ${\mathord{\mathrm{cx}}}(F)$ is similar to the one for the universal joint, just more complicated. We are going to sketch the main steps and leave the detailed verification to the reader. Let ${\mathcal{C}}=T^3$, the cartesian product of three circles, ${\mathcal{W}}=SO(3)$ the set of orthogonal matrices with determinant 1, and the forward kinematic map $F\colon{\mathcal{C}}\to{\mathcal{W}}$ given by Euler angles, following the $X-Z-X$ convention. Then
1. The set of singular values ${\mathcal{W}}^s$ of $F$ consists of all matrices of the form $$R=\left[\begin{array}{ccc}
\cos\theta & -\sin\theta & 0 \\
\sin\theta & \cos\theta & 0\\
0 & 0 & 1 \end{array}\right]$$ They correspond to rotations whose $Z$-axis coincide with the $Z$-axis of the reference frame. Clearly, the set of singular values ${\mathcal{W}}^s$ may be viewed as a circle embedded in $SO(3)$, and ${\mathcal{W}}^r$ is its complement in $SO(3)$.
2. The singular points of $F$ are Euler triples of the form $(\theta_1,0,\theta_3)$ or $(\theta_1,\pi,\theta_3)$, because under $X-Z-X$ convention the second angle corresponds to the rotation that moves the $Z$-axis of the reference frame to that of the represented rotation. We may therefore identify the set of singular points with the cartesian product $T\times\{0,\pi\}\times T\subset T^3={\mathcal{C}}$. Geometrically speaking, that is a disjoint union ot two two-dimensional tori in ${\mathcal{C}}$.
3. To every regular value of $F$ corresponds a unique Euler triple $(\theta_1,\theta_2,\theta_3)$ where $\theta_1,\theta_3\in T$ and $\theta_2\in (0,\pi)$. This correspondence determines an inverse kinematic map $I\colon{\mathcal{W}}^r\to{\mathcal{C}}$. Alternatively, if we choose $\theta_2\in(\pi,2\pi)$ then we get another inverse kinematic map $I'\colon{\mathcal{W}}^r\to{\mathcal{C}}$, and ${\mathcal{C}}^r=I({\mathcal{W}}^r)\cup I'({\mathcal{W}}^r)$. As in the universal joint case, ${\mathcal{C}}^r$ may be deformed within ${\mathcal{C}}$ to $I({\mathcal{W}}^r)$.
We may now proceed to the computation of ${\mathord{\mathrm{cx}}}(F)$: by Proposition \[prop: rel cx\] $${\mathord{\mathrm{cx}}}(F|{\mathcal{C}}\times{\mathcal{W}}^r) \le {\mathord{\mathrm{cx}}}(F) \le {\mathord{\mathrm{cx}}}(F|{\mathcal{C}}\times{\mathcal{W}}^r)+{\mathord{\mathrm{cx}}}(F|{\mathcal{C}}\times{\mathcal{W}}^s).$$ Furthermore, by Theorems \[thm: relative sectioned fibration\](2), \[thm: top group\] and [@Farber:TCMP Theorem 12] $${\mathord{\mathrm{cx}}}(F|{\mathcal{C}}\times{\mathcal{W}}^r)={\mathord{\mathrm{cx}}}({\mathcal{C}}|{\mathcal{C}}\times I({\mathcal{W}}^r))={\mathcal{C}}(T^3)=4.$$ Finally, one may use a similar approach as in the previous subsection to construct two roadmaps from $T^3$ to $T^2\times \{0,\pi\}$ and show that ${\mathord{\mathrm{cx}}}(F|{\mathcal{C}}\times{\mathcal{W}}^s)=2$. Thus we may conclude that the complexity of the triple-roll wrist is between 4 and 6 (and we believe that by a closer analysis of the singular points of $F$ one may actually prove that the complexity of $F$ is at least 5).
6-DOF joint {#subsec: 6DOF}
-----------
Six-degree-of-freedom serial manipulators are among the most common robot arm structure used in various applications. The precise analysis of the corresponding forward kinematic map $F\colon T^6\to{\mathord{\mathbb{R}}}^3\times SO(3)$ is difficult and depends on the exact configuration of the joints. Nevertheless, the underlying topology of the configuration and working spaces allows an estimate of the complexity of $F$, regardless of the specific joint configuration. To this end we use the cohomological lower bound that was described in \[subsec: cohomology\].
The following results can be found in [@Hatcher; @02]. The cohomology with ${\mathord{\mathbb{Z}}}_2$ coefficients of the six-dimensional torus $T^6$ and of the space of rotations $SO(3)$ are given as follows: the cohomology ring of $T^6$ is $H^*(T^6)=\wedge(x_1,\ldots,x_6)$, the exterior ${\mathord{\mathbb{Z}}}_2$-algebra on 6 generators in dimension 1, while the cohomology ring of $SO(3)$ is $H^*(SO(3))={\mathord{\mathbb{Z}}}_2[u]/(u^4)$, the truncated polynomial algebra with a 1-dimensional generator and the relation $u^4=0$. A full rotation around some fixed axis represents a homotopically non-trivial loop in $SO(3)$ (in fact the non-trivial element of its fundamental group). Since a 6-DOF mechanism allows full rotations we may conclude that the induced homomorphism $F^*\colon H^*(SO(3))\to H^*(T^6)$ is non-trivial, so the image of the generator $F^*(u)=s\in H^1(T^6)$ is a non-trivial sum of generators $x_1,\ldots,x_6$.
Based on these facts, it is easy to check that the cohomology class $s\times 1+ 1\times u\in H^*(T^6\times SO(3))$ is contained in the kernel of the homomorphism $(1\times F)^*\colon H^*(T^6\times)\to H^*(T^6)$. Moreover, by taking into account that $s^2=0$ and that the addition is modulo 2, we obtain $$(s\times 1+ 1\times u)^2=1\times u^2\ \ \text{and}\ \ (s\times 1+ 1\times u)^3=s\times u^2+1\times u^3\ne 0,$$ which by Theorem \[thm: nilpotency\] implies that the complexity of $F$ is at least 4.
Conclusion
==========
The applicability of topological methods in robotics is not as surprising as it may appear at first sight, especially when one considers qualitative questions regarding the possibility to find suitable inverse kinematic maps or to avoid singular points of certain configurations. In this paper we introduced a new topological measure for the complexity of motion planning in robotic systems. Unlike the previous approaches to the complexity as developed by [@Farber:TCMP] and other authors, where one studies the motion within a single space (either the configuration or the working space of a system), we constructed a more realistic model that takes into account the forward kinematic map. We considered queries consisting of two sets of data - the initial configuration of the joints, and the requested final position of the end-effector. Then we studied obstructions to the existence of robust algorithms that take those queries as input, and return movements of the joints that start from a given joint configuration and end with the required position of the end-effector.
It turned out that in most cases one needs several robust algorithms in order to cover all possible queries. In particular, we proved that the complexities (i.e. the minimal number of distinct robust algorithms that are needed to cover all queries) of some basic joint configurations like the universal joint, the triple-roll wrist and the 6-DOF joint configurations are at least 4.
We believe that this new invariant reflects and explains many of the difficulties in the computation of the inverse kinematic maps and of the construction of explicit motion plans, and is thus one of the factors that should be taken into account in the design of specific robot-arm configurations.
[99]{} M. Brady, J.M. Hollerbach, T.L. Johnson, T. Lozano-Pérez, M.T. Mason (eds.), *Robot Motion: Planning and Control* (MIT Press, 1982). M. Brady (ed.), *Robotics Science* (MIT Press, 1989). J.F. Canny: *The Complexity of Robot Motion Planning*, (MIT Press, Cambridge 1988) O. Cornea, G. Lupton, J. Oprea, D. Tanré, *Lusternik-Schnirelmann category*, Mathematical Surveys and Monographs [**103**]{} (American Mathematical Society, Providence, RI, 2003). C. Ehresmann, Les connexions infinitésimales dans un espace fibré différentiable, in *Colloque de Topologie*, Bruxelles (1950), 29-55. M. Erdmann, On the Topology of Discrete Strategies, *The Int’l Journal of Robotics Research*, [**29**]{} (2010), 855-896, M. Farber, Topological Complexity of Motion Planning, *Discrete Comput Geom*, [**29**]{} (2003), 211-221. M. Farber, Instabilities of robot motion, *Top. Appl.* [**140**]{} (2004), 245-266 M. Farber, *Invitation to topological robotics*, (EMS Publishing House, Zurich, 2008). R.H. Fox, On the Lusternik-Schnirelmann category, *Ann. of Math.* [**42**]{} (1941), 333-370. T. Ganea, Lusternik-Schnirelmann category and cocategory, *Proc. Lond. math. Soc.* [**10**]{} (1960), 623-639 D. Gottlieb, Robots and topology, *Proc. 1986 IEEE International Conference on Robotics and Automation*, Vol 3., 1689-1691. D. Gottlieb, Topology and the Robot Arm, *Acta Applicandae Math.* [**11**]{} (1988), 117-121. R. Ghrist, *Elementary Applied Topology*, ISBN 978-1502880857, 2014. A. Hatcher: *Algebraic Topology*, (Cambridge Univ. Press, Cambridge 2002). J.M. Hollerbach, *Optimal kinematic design for a seven degree of freedom manipulator*, 2nd International Symposium on Robotics Research, Kyoto, (Japan, 1984). J.M. Hollerbach, *Kinematics and Dynamics for Control*, in M. Brady (ed.), *Robotics Science*, Chapter 10 (MIT Press, 1989). I.M. James, On category, in the sense of Lusternik-Schnirelmann, *Topology* [**17**]{} (1978) 331-348. L.E. Kavraki, S.M. LaValle, *Motion Planning*, in B. Siciliano, O. Khatib (eds.), *Springer Handbook of Robotics*, Chapter 5, (Springer, Berlin, 2008). L. Lusternik, L. Schnirelmann, *Methodes Topologiques dans les Problemes Variationnels* (Hermann, Paris, 1934). R.S. Palais, Lusternik-Schnirelmann theory on Banach manifolds, *Topology* [**5**]{} (1966), 115-132. J.H. Reif: *Complexity of the mover’s problem and generalizations*, IEEE Symp. Found. Comput. Sci. (1979), 421-427. J.T. Schwartz, Generalizing the Lusternik-Schnirelmann theory of critical points, *Comm. Pure Appl. Math.* [**17**]{} (1964), 307-315. B. Siciliano, O. Khatib (eds.), *Springer Handbook of Robotics*, (Springer, Berlin, 2008). K. Waldron, J. Schmiedeler, *Kinematics*, in B. Siciliano, O. Khatib (eds.), *Springer Handbook of Robotics*, Chapter 1, (Springer, Berlin, 2008).
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abstract: |
The yields of the [$\mathrm{K^{*}{\ensuremath{\kern-0.09em}}(892)^{0}}$]{}and [${\ensuremath{\upphi}\xspace}(1020)$]{}resonances are measured in collisions at through their hadronic decays using the ALICE detector. The measurements are performed in multiple centrality intervals at mid-rapidity $(|y|<0.5)$ in the transverse-momentum ranges $0.3<{\ensuremath{p_{\mathrm{T}}}\xspace}<5$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}for the [$\mathrm{K^{*}{\ensuremath{\kern-0.09em}}(892)^{0}}$]{}and $0.5<{\ensuremath{p_{\mathrm{T}}}\xspace}<5$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}for the [${\ensuremath{\upphi}\xspace}(1020)$]{}. The yields of [$\mathrm{K^{*}{\ensuremath{\kern-0.09em}}(892)^{0}}$]{}are suppressed in central collisions with respect to pp and peripheral collisions (perhaps due to re-scattering of its decay products in the hadronic medium), while the longer lived [${\ensuremath{\upphi}\xspace}(1020)$]{}meson is not suppressed. These particles are also used as probes to study the mechanisms of particle production. The shape of the [$p_{\mathrm{T}}$]{}distribution of the [${\ensuremath{\upphi}\xspace}(1020)$]{}meson, but not its yield, is reproduced fairly well by hydrodynamic models for central collisions. In central collisions at low and intermediate [$p_{\mathrm{T}}$]{}, the [$\mathrm{p}\kern-0.1em/{\ensuremath{\kern-0.18em}}{\ensuremath{{\ensuremath{\upphi}\xspace}(1020)}\xspace}$]{}ratio is flat in [$p_{\mathrm{T}}$]{}, while the [$\mathrm{p}\kern-0.05em/{\ensuremath{\kern-0.09em}}{\ensuremath{\uppi}\xspace}$]{}and [${\ensuremath{{\ensuremath{\upphi}\xspace}(1020)}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}{\ensuremath{\uppi}\xspace}$]{}ratios show a pronounced increase and have similar shapes to each other. These results indicate that the shapes of the [$p_{\mathrm{T}}$]{}distributions of these particles in central collisions are determined predominantly by the particle masses and radial flow. Finally, [${\ensuremath{\upphi}\xspace}(1020)$]{}production in collisions is enhanced, with respect to the yield in pp collisions and the yield of charged pions, by an amount similar to the $\Lambda$ and $\Xi$.
*PACS numbers*: 25.75.Dw, 13.85.Ni, 14.40.Df, 14.40Be
bibliography:
- 'refs.bib'
title: 'K$\boldsymbol{^{*}}$(892)$\boldsymbol{^{0}}$ and $\boldsymbol{\upphi}$(1020) production in collisions at $\boldsymbol{\sqrt{s_{\mathrm{NN}}}}$ = 2.76 TeV'
---
Introduction\[sec:intro\]
=========================
Ultrarelativistic heavy-ion collisions are expected to produce a hot and dense state of matter, the quark-gluon plasma [@Petreczky_ConfinementX; @Borsanyi_2010b; @Borsanyi_TC]. At a critical temperature of $T_{\mathrm{c}}\approx 160$ MeV [@Borsanyi_TC; @Aoki_TC1; @Aoki_TC2] a cross-over transition between the partonic (*i.e.*, a system with deconfined quarks) and hadronic phases is expected to take place. Statistical models [@Cleymans_2006; @Rafelski_2005; @Petran_Rafelski_2013a; @Petran_Rafelski_2013b; @Andronic2009; @Andronic2009_Erratum; @AndronicQM2011; @Becattini_2004; @Becattini_2010] have been successfully applied to particle yields in order to estimate the values of the chemical freeze-out temperature and the baryochemical potential. However, resonance yields may deviate from the values expected from thermal models due to hadronic processes (re-scattering and regeneration) that might change the reconstructible resonance yields even after chemical freeze-out. Resonance yields may be regenerated through pseudo-elastic scattering, in which particles scatter through a resonance state \[*e.g.*, ${\ensuremath{{\ensuremath{\uppi}\xspace}^{-}}\xspace}{\ensuremath{\mathrm{K}^{+}}\xspace}\rightarrow{\ensuremath{\mathrm{K^{*}{\ensuremath{\kern-0.09em}}(892)^{0}}}\xspace}\rightarrow{\ensuremath{{\ensuremath{\uppi}\xspace}^{-}}\xspace}{\ensuremath{\mathrm{K}^{+}}\xspace}$ and ${\ensuremath{\mathrm{K}^{-}}\xspace}{\ensuremath{\mathrm{K}^{+}}\xspace}\rightarrow{\ensuremath{{\ensuremath{\upphi}\xspace}(1020)}\xspace}\rightarrow{\ensuremath{\mathrm{K}^{-}}\xspace}{\ensuremath{\mathrm{K}^{+}}\xspace}$\] [@Bleicher_Stoecker; @Markert_thermal; @Vogel_Bleicher]. Pseudo-elastic scattering does not change the abundances of the scattered particles, but may increase the measured yield of the resonance state through which they scattered. If a resonance has a short enough lifetime, it may decay during the hadronic phase and its decay products may undergo elastic or pseudo-elastic scatterings. Information about the resonance may be lost if at least one of its decay products elastically scatters in the hadronic medium or undergoes pseudo-elastic scattering via a different resonance state \[*e.g.*, a pion from a [$\mathrm{K^{*}{\ensuremath{\kern-0.09em}}(892)^{0}}$]{}decay scatters with another pion, ${\ensuremath{{\ensuremath{\uppi}\xspace}^{-}}\xspace}{\ensuremath{{\ensuremath{\uppi}\xspace}^{+}}\xspace}\rightarrow\uprho(770)^{0}\rightarrow{\ensuremath{{\ensuremath{\uppi}\xspace}^{-}}\xspace}{\ensuremath{{\ensuremath{\uppi}\xspace}^{+}}\xspace}$\] [@Bliecher_Aichelin]. The net effect of pseudo-elastic scattering on the yield of a resonance will depend on whether regeneration of that resonance is outweighed by re-scattering of its decay products through other resonances. In the case of the [$\mathrm{K^{*}{\ensuremath{\kern-0.09em}}(892)^{0}}$]{}, the ${\ensuremath{\uppi}\xspace}$K interaction cross section [@Matison_Kpi] is smaller than the ${\ensuremath{\uppi}\xspace}{\ensuremath{\uppi}\xspace}$ cross section [@Protopopescu_pipi], so re-scattering may dominate and the measured [$\mathrm{K^{*}{\ensuremath{\kern-0.09em}}(892)^{0}}$]{}yield may be smaller than the yield at chemical freeze-out. Calculations using Ultrarelativistic Quantum Molecular Dynamics (UrQMD) [@UrQMD_Bass; @UrQMD] predict that both regeneration and re-scattering affect the resonance yields predominantly for transverse momenta ${\ensuremath{p_{\mathrm{T}}}\xspace}\lesssim2$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{} [@Bleicher_Stoecker; @Bliecher_Aichelin]. The final reconstructible resonance yields depend on the chemical freeze-out temperature, the scattering cross sections of its decay products, and the timescale during which re-scattering and regeneration are active in the hadronic phase, *i.e.*, the time between chemical and kinetic freeze-out. The model described in [@Markert_thermal; @Torrieri_thermal; @Torrieri_thermal_2001b; @Torrieri_thermal_2001b_erratum] combines thermal-model calculations with re-scattering effects in the hadronic phase. It predicts the ratios of ([$p_{\mathrm{T}}$]{}-integrated) resonance yields to the yields of stable particles as a function of both the chemical freeze-out temperature and the lifetime of the hadronic phase. While this model was derived for a Relativistic Heavy Ion Collider (RHIC) collision energy (), its predictions span a wide range of freeze-out temperatures and hadronic lifetimes and remain valid at Large Hadron Collider (LHC) energies.
Chiral symmetry is expected to be restored [@Petreczky] above the chiral transition temperature; resonances that decay when chiral symmetry was at least partially restored are expected to exhibit mass shifts and/or width broadening [@Brown_Rho; @Rapp2009; @Brodsky_chiral; @Eletsky]. Regeneration of resonances in the late hadronic phase increases the fraction of resonances with vacuum masses and widths and may inhibit the observation of the signatures of chiral symmetry restoration. Since model calculations indicate that re-scattering and regeneration modify the resonance signal more strongly for ${\ensuremath{p_{\mathrm{T}}}\xspace}\lesssim 2$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}, signatures of chiral symmetry may be difficult to observe in the case of low-[$p_{\mathrm{T}}$]{}resonances which are reconstructed via hadronic decays.
This article presents measurements of the [$\mathrm{K^{*}{\ensuremath{\kern-0.09em}}(892)^{0}}$]{}, [$\mathrm{\overline{K}^{*}(892)^{0}}$]{}, and [${\ensuremath{\upphi}\xspace}(1020)$]{}mesons performed in multiple centrality intervals for collisions at using the ALICE detector. The focus here is on low and intermediate [$p_{\mathrm{T}}$]{}\[$0.3<{\ensuremath{p_{\mathrm{T}}}\xspace}<5$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}for the [$\mathrm{K^{*}{\ensuremath{\kern-0.09em}}(892)^{0}}$]{}and $0.5<{\ensuremath{p_{\mathrm{T}}}\xspace}<5$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}for the [${\ensuremath{\upphi}\xspace}(1020)$]{}\] and the integrated yields; results for high [$p_{\mathrm{T}}$]{}will be presented in a future article. All measurements of the [$\mathrm{K^{*}{\ensuremath{\kern-0.09em}}(892)^{0}}$]{}and [$\mathrm{\overline{K}^{*}(892)^{0}}$]{}are averaged and these mesons are collectively referred to as [$\mathrm{K^{*0}}$]{}. The [${\ensuremath{\upphi}\xspace}(1020)$]{}meson is referred to as [$\upphi$]{}. The ALICE detector is described in Sec. \[sec:alice\], with the emphasis on the sub-detectors used in this analysis. The data-analysis procedure is described in Secs. \[sec:pid\]-\[sec:corr\]. Results, including resonance yields, masses, widths, mean transverse momenta, ratios to non-resonances, comparisons to predicted [$p_{\mathrm{T}}$]{}distributions, and the [$\upphi$]{}enhancement ratio are presented in Sec. \[sec:results\].
ALICE Experiment\[sec:alice\]
=============================
A comprehensive description of the ALICE detector can be found in [@ALICE_detector]. The main detector components used in this analysis are the V0 detector, the Inner Tracking System (ITS), and the Time Projection Chamber (TPC), which are located inside a 0.5 T solenoidal magnetic field. The V0 detector [@ALICE_VZERO] consists of two scintillator hodoscopes placed on either side of the interaction point covering the pseudorapidity ranges $-3.7<\eta<-1.7$ and $2.8<\eta<5.1$. A combination of hits in the V0 detector and the two innermost layers of the ITS is used is used as a minimum-bias trigger for collisions [@ALICE_multiplicity]. Collision centrality is determined by using the multiplicity measured in the V0 detector along with Glauber-model simulations to describe the multiplicity distribution as a function of the impact parameter [@ALICE_multiplicity; @ALICE_centrality]. These simulations give [$\langle N_{\mathrm{part}}\rangle$]{}, the mean number of nucleons which participated in collisions in a given centrality interval. The ITS is made up of six cylindrical layers of silicon detectors with radii between 3.9 and 43 cm from the beam axis, covering the full azimuth. The pseudorapidity range $|\eta|<0.9$ is covered by all six layers, with some of the individual layers covering larger ranges in pseudorapidity. The TPC [@ALICE_TPC], which is the main tracking detector, is a large cylindrical drift detector that covers the pseudorapidity range $|\eta|<0.9$ with full azimuthal acceptance. Multi-wire proportional chambers with cathode pad readout are arranged in 159 pad rows located at the ends of the TPC. Hits in the ITS and TPC are used to reconstruct charged particle tracks, which are used in the final determination of the primary collision vertex. The position resolution for the primary vertex in both the longitudinal direction and the transverse plane is $\sim10\;\upmu\mathrm{m}$ for heavy-ion collisions. The TPC is also used to identify particles through their [${\ensuremath{\mathrm{d}}}E{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}{\ensuremath{\mathrm{d}}}x$]{}(specific energy loss) in the TPC gas. The value of [${\ensuremath{\mathrm{d}}}E{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}{\ensuremath{\mathrm{d}}}x$]{}is calculated using a truncated-mean procedure in which the average is evaluated using only the 60% of points with the lowest [${\ensuremath{\mathrm{d}}}E{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}{\ensuremath{\mathrm{d}}}x$]{}values measured along a given track. The measured [${\ensuremath{\mathrm{d}}}E{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}{\ensuremath{\mathrm{d}}}x$]{}is then compared to the expected [${\ensuremath{\mathrm{d}}}E{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}{\ensuremath{\mathrm{d}}}x$]{}for a given particle species using a Bethe-Bloch parametrization. The deviation from the expected [${\ensuremath{\mathrm{d}}}E{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}{\ensuremath{\mathrm{d}}}x$]{}value is expressed in units of the energy-loss resolution [$\sigma_{\mathrm{TPC}}$]{}, which is 5% for isolated tracks and 6.5% for central collisions [@ALICE_piKp_PbPb]. The TPC allows kaons to be distinguished from pions for momenta $p<0.7$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}and (anti)protons to be distinguished from pions and kaons for $p<1$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}(with a separation power of 2$\sigma$ in both cases).
Event and Track Selection\[sec:pid\]
====================================
The yields of [$\mathrm{K^{*0}}$]{}and [$\upphi$]{}mesons are measured in about 13 million collisions recorded in 2010 in the 0-90% centrality interval. The position of the primary vertex along the beam axis is required to be within 10 cm of the center of the ALICE detector. The [$\mathrm{K^{*0}}$]{}and [$\upphi$]{}mesons are identified by reconstruction of their respective hadronic decays: ${\ensuremath{\mathrm{K^{*0}}}\xspace}\kern-0.15em\rightarrow\kern-0.15em{\ensuremath{{\ensuremath{\uppi}\xspace}^{\pm}}\xspace}\mathrm{K}^{\mp}$ (branching ratio 0.666) and ${\ensuremath{\upphi}\xspace}\kern-0.15em\rightarrow\kern-0.15em{\ensuremath{\mathrm{K}^{-}}\xspace}{\ensuremath{\mathrm{K}^{+}}\xspace}$ (branching ratio 0.489) [@PDG]. The lifetimes in the vacuum of the [$\mathrm{K^{*0}}$]{}and [$\upphi$]{}are $4.16\pm0.05$ fm/$c$ and $46.3\pm0.4$ fm/$c$, respectively [@PDG]. High-quality tracks are selected by requiring at least 70 reconstructed TPC clusters out of a possible 159 and requiring that the $\chi^{2}$ per cluster of the reconstructed tracks be less than 4. Track momenta and pseudorapidity are restricted to the ranges ${\ensuremath{p_{\mathrm{T}}}\xspace}>150$ [$\mathrm{MeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}and $|\eta|<0.8$, respectively. To reduce the number of secondary particles from weak decays, each track is required to have at least one hit in the innermost layer of the ITS and a small distance of closest approach (DCA) to the primary vertex in the $xy$ plane: $\mathrm{DCA}_{xy}<(0.0182+0.035{\ensuremath{p_{\mathrm{T}}}\xspace}^{-1.01})$ cm. The distance of closest approach in the $z$ direction is also restricted: $\mathrm{DCA}_{z}<2$ cm. The $\mathrm{DCA}_{z}$ cut is wider not because of the vertex resolution (which is similar for the longitudinal and transverse directions), but because of the tracking resolution, which is less precise for the $z$ direction than the transverse plane. This is because the positions of points in the Silicon Pixel Detector (the innermost part of the ITS) are determined more precisely in the $xy$ plane. The wide $\mathrm{DCA}_{z}$ cut is intended to remove particles that are highly displaced from the vertex. Finally, in the [$\mathrm{K^{*0}}$]{}and [$\upphi$]{}analyses, pion and kaon tracks are required to be within $2{\ensuremath{\sigma_{\mathrm{TPC}}}\xspace}$ of the expected [${\ensuremath{\mathrm{d}}}E{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}{\ensuremath{\mathrm{d}}}x$]{}values for each particle species.
Signal Extraction\[sec:signal\]
===============================
{width="38pc"}
{width="38pc"}
The [$\mathrm{K^{*0}}$]{}and [$\upphi$]{}resonances are reconstructed through their invariant mass via identified decay-product candidates. For each centrality and [$p_{\mathrm{T}}$]{}interval, the invariant-mass distribution of pairs of unlike-charge resonance decay products from the same event is constructed \[as an example, see Figs. \[fig:signal:examples\_ks\] and \[fig:signal:examples\_phi\], panels (a) and (c)\]. It is required that the rapidity of the pair lies within the range $|y_{\mathrm{pair}}|<0.5$. In the construction of the [$\mathrm{K^{*0}}$]{}invariant-mass distributions, it is possible that a track will be designated as both a pion candidate and a kaon candidate because it passes both identification cuts (especially at high [$p_{\mathrm{T}}$]{}). In this event, such a track is assigned the kaon mass for some pairs and the pion mass for other pairs. First, the track will be assigned the kaon mass and pairs will be formed with each of the pion candidate tracks. Then the track will be assigned the pion mass and paired with each of the kaon candidates. (The track will never be paired with itself.) The [$\mathrm{K^{*0}}$]{}([$\upphi$]{}) peak has a signal-to-background ratio[^1] that ranges from $1.1{\ensuremath{\kern-0.1em\times\kern-0.2em}}10^{-4}$ $(1.4{\ensuremath{\kern-0.1em\times\kern-0.2em}}10^{-3})$ to 0.049 (1.7), depending on the [$p_{\mathrm{T}}$]{}interval analyzed. For the full [$p_{\mathrm{T}}$]{}range and centrality 0-80%, the [$\mathrm{K^{*0}}$]{}([$\upphi$]{}) peak has a signal-to-background ratio of $2.7{\ensuremath{\kern-0.1em\times\kern-0.2em}}10^{-4}$ ($4.4{\ensuremath{\kern-0.1em\times\kern-0.2em}}10^{-3}$).
The combinatorial background is estimated with an event-mixing technique by forming pairs using particles from different events. Each decay-product candidate track is combined with tracks from five other events to build uncorrelated pairs. Events for mixing are grouped based on the following similarity criteria: the difference in the vertex $z$ position is less than 2 (5) cm for the [$\mathrm{K^{*0}}$]{}([$\upphi$]{}) and the difference in the centrality percentile is required to be less than 10%. For the [$\mathrm{K^{*0}}$]{}analysis, the difference in the event plane azimuthal angles between the two events is required to be less than $30^{\circ}$. The signal-to-background ratio is lower for the [$\mathrm{K^{*0}}$]{}than the [$\upphi$]{}and the residual background for the [$\mathrm{K^{*0}}$]{}also tends to have a larger slope or greater curvature than for the [$\upphi$]{}. For these reasons, and in order to provide a mixed-event combinatorial background which is a good representation of the true combinatorial background, the event mixing similarity criteria are somewhat stricter for the [$\mathrm{K^{*0}}$]{}. The [$\mathrm{K^{*0}}$]{}mixed-event combinatorial background is normalized such that its integral in the region of $1.1<{\ensuremath{m_{{\ensuremath{\uppi}\xspace}\mathrm{K}}}\xspace}<1.3$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c^{2}$]{}is the same as the integral of the unlike-charge distribution over the same interval. The [$\upphi$]{}mixed-event combinatorial background is normalized to a region that surrounds, but excludes, the [$\upphi$]{}peak ($1<{\ensuremath{m_{\mathrm{KK}}}\xspace}<1.01~{\ensuremath{\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c^{2}}\xspace}$ and $1.03<{\ensuremath{m_{\mathrm{KK}}}\xspace}<1.06~{\ensuremath{\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c^{2}}\xspace}$). The boundaries of the normalization regions are changed and the resulting variations in the experimental results (*e.g.*, average values of 2.2% for the [$\mathrm{K^{*0}}$]{}yield and 0.4% for the [$\upphi$]{}yield) are incorporated into the systematic uncertainties (see “Combinatorial background" in Table \[table:sys\]). The combinatorial background is also estimated from the invariant-mass distribution of like-charge pairs from the same event. However, the resulting yields have larger statistical uncertainties and larger bin-to-bin fluctuations than the mixed-event background; the latter is therefore used for this analysis. Due to its lower signal-to-background ratio, the analysis of the [$\mathrm{K^{*0}}$]{}is performed in four centrality intervals from 0-80%, while the [$\upphi$]{}analysis is performed in narrower centrality intervals.
After the normalized combinatorial background has been subtracted from the unlike-charge distribution, [$\mathrm{K^{*0}}$]{}and [$\upphi$]{}peaks can be observed on top of a residual background \[as an example, see Figs. \[fig:signal:examples\_ks\] and \[fig:signal:examples\_phi\], panels (b) and (d)\]. The residual background may be due to correlated ${\ensuremath{\uppi}\xspace}\mathrm{K}$ or KK pairs emitted within a jet, correlated pairs from particle decays (with three or more stable particles at the end of the decay chain), or misidentified correlated pairs (*e.g.*, a $\uprho\rightarrow{\ensuremath{\uppi}\xspace}{\ensuremath{\uppi}\xspace}$ decay being misidentified as a ${\ensuremath{\mathrm{K^{*0}}}\xspace}\rightarrow{\ensuremath{\uppi}\xspace}\mathrm{K}$ decay). Differences in the structure of the two mixed events, including differences in the event planes, elliptic flow, primary vertices, and multiplicities, can also lead to an imperfect combinatorial background (if necessary, such differences can be reduced through the use of similarity criteria for the mixed events as described above). Figures \[fig:signal:examples\_ks\] and \[fig:signal:examples\_phi\] show invariant-mass distributions for the [$\mathrm{K^{*0}}$]{}and [$\upphi$]{}mesons, respectively (two [$p_{\mathrm{T}}$]{}intervals each). Integrated over the full transverse-momentum range and using the same centrality interval of 0-80% for both particles, the uncorrected [$\mathrm{K^{*0}}$]{}([$\upphi$]{}) yield is 27.4 (5.9) million, or 2.2 (0.47) per event, with a significance of 86 (146). For each [$p_{\mathrm{T}}$]{}and centrality interval, the background-subtracted invariant-mass distributions are fitted by using a combined function to describe the residual background and the signal peak (the peak fitting functions are described below). The fitting regions are $0.77<{\ensuremath{m_{{\ensuremath{\uppi}\xspace}\mathrm{K}}}\xspace}<1.02$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c^{2}$]{}for the [$\mathrm{K^{*0}}$]{}and $1<{\ensuremath{m_{\mathrm{KK}}}\xspace}<1.07~{\ensuremath{\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c^{2}}\xspace}$ for the [$\upphi$]{}. The boundaries of the fitting region are varied by 10-50 [$\mathrm{MeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c^{2}$]{}for [$\mathrm{K^{*0}}$]{}and 5-30 [$\mathrm{MeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c^{2}$]{}for the [$\upphi$]{}. The variation in the yields does not increase if the fitting region boundaries are varied by larger amounts. Varying the boundaries of the fitting region produces average variations in the [$\mathrm{K^{*0}}$]{}([$\upphi$]{}) yield of 9.9% (3.5%), which are added to the systematic uncertainties (“Fitting region" in Table \[table:sys\]). The systematic uncertainties also include variations due to the order of the residual background polynomial (first-, second-, or third-order). Varying the residual background polynomial changes the [$\mathrm{K^{*0}}$]{}([$\upphi$]{}) yield by 5.8% (2.7%) on average (“Residual background shape" in Table \[table:sys\]).
For each [$p_{\mathrm{T}}$]{}and centrality interval, the [$\mathrm{K^{*0}}$]{}mass and width are extracted from a relativistic $p$-wave Breit-Wigner function with a Boltzmann factor.
$$\label{eq:signal:pwavebw}
\frac{{\ensuremath{\mathrm{d}}}N}{{\ensuremath{\mathrm{d}}}m_{{\ensuremath{\uppi}\xspace}\mathrm{K}}}=\frac{Cm_{{\ensuremath{\uppi}\xspace}\mathrm{K}}\Gamma M_{0}}{(m_{{\ensuremath{\uppi}\xspace}\mathrm{K}}^{2}-M_{0}^{2})^{2}+M_{0}^{2}\Gamma^{2}}\left[\frac{m_{{\ensuremath{\uppi}\xspace}\mathrm{K}}}{\sqrt{m_{{\ensuremath{\uppi}\xspace}\mathrm{K}}^{2}+{\ensuremath{p_{\mathrm{T}}}\xspace}^{2}}}\exp\left(-\frac{\sqrt{m_{{\ensuremath{\uppi}\xspace}\mathrm{K}}^{2}+{\ensuremath{p_{\mathrm{T}}}\xspace}^{2}}}{T}\right)\right].$$
Here, $C$ is an overall scale factor and $M_{0}$ is the pole mass. The Boltzmann factor \[in square brackets in Eq. (\[eq:signal:pwavebw\])\] is based on the assumption that in collisions the [$\mathrm{K^{*0}}$]{}resonance is predominantly produced through scattering (*e.g.*, ${\ensuremath{\uppi}\xspace}\mathrm{K}\rightarrow{\ensuremath{\mathrm{K^{*0}}}\xspace}$) in a thermalized medium rather than directly from string fragmentation. The factor accounts for the phase-space population of the parent pions and kaons [@STAR_Kstar_200GeV_2005; @ZXu_resonances_RHIC; @Kolb; @Shuryak]. The temperature $T$ is fixed to 160 MeV; this is approximately equal to the chemical freeze-out temperature and varying this temperature by $\pm30$ MeV does not produce a significant change in the [$\mathrm{K^{*0}}$]{}mass position. The parameter $\Gamma$ in Eq. (\[eq:signal:pwavebw\]) is not constant, but depends on $m_{{\ensuremath{\uppi}\xspace}\mathrm{K}}$, the pole mass $M_{0}$, the resonance width $\Gamma_{0}$, and the vacuum masses of the charged pion and charged kaon ($M_{{\ensuremath{\uppi}\xspace}}$ and $M_{\mathrm{K}}$, respectively)
$$\label{eq:signal:Gamma}
\Gamma=\Gamma_{0}\frac{M_{0}^{4}}{m_{{\ensuremath{\uppi}\xspace}\mathrm{K}}^{4}}\left[\frac{\left(m_{{\ensuremath{\uppi}\xspace}\mathrm{K}}^{2}-M_{{\ensuremath{\uppi}\xspace}}^{2}-M_{\mathrm{K}}^{2}\right)^{2}-4M_{{\ensuremath{\uppi}\xspace}}^{2}M_{\mathrm{K}}^{2}}{\left(M_{0}^{2}-M_{{\ensuremath{\uppi}\xspace}}^{2}-M_{\mathrm{K}}^{2}\right)^{2}-4M_{{\ensuremath{\uppi}\xspace}}^{2}M_{\mathrm{K}}^{2}}\right]^{3/2}.$$
The [$\mathrm{K^{*0}}$]{}yield is determined by integrating the background-subtracted invariant-mass distribution over the range $0.77<{\ensuremath{m_{{\ensuremath{\uppi}\xspace}\mathrm{K}}}\xspace}<1.02$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c^{2}$]{}, removing the integral of the residual background fit over the same range, and correcting the result to account for the yield outside that range. For this purpose, the [$\mathrm{K^{*0}}$]{}peak is fitted with a non-relativistic Breit-Wigner function with the width fixed to the vacuum value, allowing the yield in the tails outside the range of integration to be calculated. This corresponds to $\sim 9\%$ of the total [$\mathrm{K^{*0}}$]{}yield. As an alternative, the [$\mathrm{K^{*0}}$]{}yield is also found by integrating the peak fitting functions. The systematic uncertainties of the [$p_{\mathrm{T}}$]{}-differential [$\mathrm{K^{*0}}$]{}yield, the [$p_{\mathrm{T}}$]{}-integrated yield [${\ensuremath{\mathrm{d}}}N{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}{\ensuremath{\mathrm{d}}}y$]{}, and the mean transverse momentum [$\langle{\ensuremath{p_{\mathrm{T}}}\xspace}\rangle$]{}account for variations due to the two methods applied in extracting the yield. This variation is 2.5% for the [$p_{\mathrm{T}}$]{}-differential [$\mathrm{K^{*0}}$]{}yield (“Yield extraction" in Table \[table:sys\]). The [$\mathrm{K^{*0}}$]{}yield is also extracted from a relativistic Breit-Wigner function and a non-relativistic Breit-Wigner function with a free width. Changes in the experimental results due to these different peak fitting functions are incorporated into the systematic uncertainties. The [$\mathrm{K^{*0}}$]{}yield varies by 5.2% on average when different peak fitting functions are used (“Peak shape" in Table \[table:sys\]).
To find the [$\upphi$]{}mass and width for each [$p_{\mathrm{T}}$]{}and centrality interval, the peak is fitted by using a Voigtian function.[^2] This is the convolution of a non-relativistic Breit-Wigner peak and a Gaussian, which accounts for the detector resolution
$$\label{eq:signal:voigt}
\frac{{\ensuremath{\mathrm{d}}}N}{{\ensuremath{\mathrm{d}}}m_{\mathrm{KK}}}=\frac{C\Gamma_{0}}{(2\pi)^{3/2}\sigma}\!\int\limits_{-\infty}\limits^{\infty}\exp\left[-\frac{\left(m_{\mathrm{KK}}-m^{\prime}\right)^{2}}{2\sigma^{2}}\right]\frac{1}{\left(m^{\prime}-M_{0}\right)^{2}+\Gamma_{0}^{2}/4}{\ensuremath{\mathrm{d}}}m^{\prime}.$$
The mass resolution parameter $\sigma$, which has been shown to be independent of collision centrality, has been constrained to the value extracted from fits of simulated [$\upphi$]{}signal peaks. This value is about 1.2 [$\mathrm{MeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c^{2}$]{}for ${\ensuremath{p_{\mathrm{T}}}\xspace}\approx 0.6$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}. It reaches a minimum of about 1 [$\mathrm{MeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c^{2}$]{}for ${\ensuremath{p_{\mathrm{T}}}\xspace}\approx 1.2$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}and increases to about 1.5 [$\mathrm{MeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c^{2}$]{}for ${\ensuremath{p_{\mathrm{T}}}\xspace}=4-5$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}. To estimate $\sigma$, the production and decay of [$\upphi$]{}mesons are simulated using HIJING [@HIJING], while the propagation of the decay products through the ALICE detector material is described using GEANT 3 [@GEANT3]. The [$\upphi$]{}yield is determined through the same procedure used for the [$\mathrm{K^{*0}}$]{}. The range of integration is $1.01<{\ensuremath{m_{\mathrm{KK}}}\xspace}<1.03$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c^{2}$]{}. The yield in the tails is about $\sim 13\%$ of the total [$\upphi$]{}yield, which is computed using the same Voigtian fits that are used to find the mass and width. Average variations in the [$\upphi$]{}yield of 1.2% are observed for the two different yield extraction methods. Different peak shapes are used in order to obtain alternate measurements of the yield, mass, and width. The resolution $\sigma$ is varied within the range of values observed in the simulation. Fits are also performed with the width fixed to the vacuum value while the resolution is kept as a free parameter. On average, the [$\upphi$]{}yield varies by 3.3% when different peak fitting functions are used.
----------------------------------- ------ ---------------------------------- ------ ---------------------------------- ------ ---------------------------------- ----- ---------------------------------------- ------ ----------------------------------
Type ${\ensuremath{\upphi}\xspace}\;$ ${\ensuremath{\upphi}\xspace}\;$ ${\ensuremath{\upphi}\xspace}\;$ ${\ensuremath{\upphi}\xspace}\;\;\;\;$ ${\ensuremath{\upphi}\xspace}\;$
Combinatorial background 2.2 0.4 1.0 0.4 0.01 0.3 0.1 0.0001 2.0 0.4
Fitting region 9.9 3.5 6.2 2.7 5.7 0.9 0.4 0.0023 18.2 4.4
Residual background shape 5.8 2.7 2.1 1.2 2.2 0.7 0.3 0.0025 15.7 3.9
Yield extraction 2.5 1.2 1.4 0.6 1.2 0.3 – – – –
Peak shape 5.2 3.3 2.7 2.7 1.8 0.5 0.3 0.0007 10.0 7.8
Particle identification 2.7 6.2 1.2 2.3 1.1 2.1 0.3 0.0130 4.4 10.3
Tracking/track selection 10.0 10.0 10.0 10.0 – – 0.4 0.0038 3.0 5.5
Material budget 1.0 1.0 1.0 1.0 – – 0.2 0.0100 – –
[$p_{\mathrm{T}}$]{}extrapolation – – 1.2 2.1 2.1 1.2 – – – –
Normalization 2.7 3.3 2.7 3.3 – – – – – –
Total 17.3 14.0 12.2 11.9 7.2 3.1 0.7 0.0192 26.4 16.7
----------------------------------- ------ ---------------------------------- ------ ---------------------------------- ------ ---------------------------------- ----- ---------------------------------------- ------ ----------------------------------
Yield Corrections\[sec:corr\]
=============================
{width="38pc"}
To obtain the corrected resonance yields, the raw yields are divided by the decay branching ratios [@PDG], the acceptance $A$, the resonance reconstruction efficiency [$\varepsilon_{\mathrm{rec}}$]{}, and the particle identification (PID) efficiency [$\varepsilon_{\mathrm{PID}}$]{}. The acceptance accounts for the geometrical acceptance of the ALICE detector, the $|y|<0.5$ resonance rapidity cut, and in-flight decays of the pions and kaons used to reconstruct the resonances. The PID efficiency accounts for the particle identification cuts used to identify the species of the decay-product candidates, *i.e.*, the [${\ensuremath{\mathrm{d}}}E{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}{\ensuremath{\mathrm{d}}}x$]{}cuts in the TPC. The factor [$\varepsilon_{\mathrm{rec}}$]{}accounts for the remainder of the efficiency, including the tracking efficiency and the cuts used to select good-quality tracks coming from the primary vertex. The product $A\times{\ensuremath{\varepsilon_{\mathrm{rec}}}\xspace}$ is extracted from the same HIJING simulations that are used to estimate the mass resolution (with $9{\ensuremath{\kern-0.1em\times\kern-0.2em}}10^{5}$ generated [$\mathrm{K^{*0}}$]{}and $4{\ensuremath{\kern-0.1em\times\kern-0.2em}}10^{5}$ generated [$\upphi$]{}mesons). The factor $A\times{\ensuremath{\varepsilon_{\mathrm{rec}}}\xspace}$ is the fraction of simulated resonances for which both decay products are reconstructed in the ALICE detector and pass the track selection cuts (PID cuts excluded). Figure \[fig:corr:efficiency\] shows $A\times{\ensuremath{\varepsilon_{\mathrm{rec}}}\xspace}$ for [$\mathrm{K^{*0}}$]{}and [$\upphi$]{}mesons as a function of [$p_{\mathrm{T}}$]{}in different centrality intervals. The efficiency [$\varepsilon_{\mathrm{PID}}$]{}is the product of the independent [${\ensuremath{\mathrm{d}}}E{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}{\ensuremath{\mathrm{d}}}x$]{}-cut efficiencies for each decay product. The [${\ensuremath{\mathrm{d}}}E{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}{\ensuremath{\mathrm{d}}}x$]{}distributions of the decay-product candidates are Gaussians with resolution [$\sigma_{\mathrm{TPC}}$]{}. When PID cuts of $2{\ensuremath{\sigma_{\mathrm{TPC}}}\xspace}$ are applied to the [${\ensuremath{\mathrm{d}}}E{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}{\ensuremath{\mathrm{d}}}x$]{}values of the pion and kaon candidates (*i.e.*, for both resonance decay products) ${\ensuremath{\varepsilon_{\mathrm{PID}}}\xspace}=91.1\%$. The use of different [${\ensuremath{\mathrm{d}}}E{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}{\ensuremath{\mathrm{d}}}x$]{}cuts (1.5[$\sigma_{\mathrm{TPC}}$]{}and 2.5[$\sigma_{\mathrm{TPC}}$]{}) can result in large changes in the shape of the residual background, which affects the extracted resonance signal. The [$\mathrm{K^{*0}}$]{}([$\upphi$]{}) yield varies by 2.7% (6.2%) on average and these variations are incorporated into the systematic uncertainties (see “Particle identification" in Table \[table:sys\]). A systematic uncertainty of 10% (for all [$p_{\mathrm{T}}$]{}and centrality intervals), adapted from the analysis described in [@ALICE_charged_highpT_2760GeV], accounts for variations in the yields due to the tracking efficiency and different choices of track quality cuts (“Tracking/track selection" in Table \[table:sys\]). A systematic uncertainty of 1% (for all [$p_{\mathrm{T}}$]{}and centrality intervals), which accounts for the uncertainty in the yield due to the uncertainty in the material budget of the ALICE detector (“Material budget" in Table \[table:sys\]), is estimated based on [@ALICE_strange_900GeV]. The uncertainties in the branching ratios [@PDG] are negligible in comparison to the total systematic uncertainties. The yields extracted with different cuts on the primary vertex $z$ position are found to be consistent with each other.
Results and Discussion\[sec:results\]
=====================================
Transverse-Momentum Distributions\[sec:results:spectra\]
--------------------------------------------------------
{width="35pc"}
The [$p_{\mathrm{T}}$]{}distributions of the [$\mathrm{K^{*0}}$]{}and [$\upphi$]{}mesons for $|y|<0.5$, normalized to the number of events and corrected for the efficiency, acceptance, and branching ratio of the decay channel, are shown in Fig. \[fig:spectra\]. For central (peripheral) collisions the statistical uncertainty is approximately 3% (4%) near the maximum of the [$p_{\mathrm{T}}$]{}distribution and increases to approximately 7% (10%) in the highest [$p_{\mathrm{T}}$]{}bin; the systematic uncertainties are summarized in Table \[table:sys\]. In order to extract the values of the mean transverse momentum [$\langle{\ensuremath{p_{\mathrm{T}}}\xspace}\rangle$]{}and the [$p_{\mathrm{T}}$]{}-integrated particle yield [${\ensuremath{\mathrm{d}}}N{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}{\ensuremath{\mathrm{d}}}y$]{}, these [$p_{\mathrm{T}}$]{}distributions are fitted with a Boltzmann-Gibbs blast-wave function [@BoltzmannGibbsBlastWave], which assumes that the emitted particles are locally thermalized in a uniform-density source at a kinetic freeze-out temperature $T_{\mathrm{kin}}$ and move with a common collective transverse radial flow velocity field. In this parametrization,
$$\label{eq:results:spectra:blastwave}
\frac{1}{{\ensuremath{p_{\mathrm{T}}}\xspace}}\frac{{\ensuremath{\mathrm{d}}}N}{{\ensuremath{\mathrm{d}}}{\ensuremath{p_{\mathrm{T}}}\xspace}}\propto \int_{0}^{R}r\;{\ensuremath{\mathrm{d}}}r\;{\ensuremath{m_{\mathrm{T}}}\xspace}\; I_{0}\negthickspace\left(\frac{{\ensuremath{p_{\mathrm{T}}}\xspace}\;\mathrm{sinh}\rho}{T_{\mathrm{kin}}}\right)K_{1}\negthickspace\left(\frac{{\ensuremath{m_{\mathrm{T}}}\xspace}\;\mathrm{cosh}\rho}{T_{\mathrm{kin}}}\right).$$
Here, the transverse mass ${\ensuremath{m_{\mathrm{T}}}\xspace}=\sqrt{m^{2}+{\ensuremath{p_{\mathrm{T}}}\xspace}^{2}}$, $I_{0}$ and $K_{1}$ are modified Bessel functions, $R$ is the fireball radius, and $r$ is the radial distance in the transverse plane. The velocity profile $\rho$ is
$$\label{eq:results:spectra:rho}
\rho=\mathrm{tanh}^{-1}\beta_{\mathrm{T}}=\mathrm{tanh}^{-1}\left[\left(\frac{r}{R}\right)^{n}\beta_{\mathrm{s}}\right],$$
where $\beta_{\mathrm{T}}$ is the average transverse expansion velocity and $\beta_{\mathrm{s}}$ is the transverse expansion velocity at the surface. The free parameters in the fits are $T_{\mathrm{kin}}$, $\beta_{\mathrm{s}}$, and the velocity profile exponent $n$. These fits have ${\ensuremath{\chi^{2}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}n_{\mathrm{dof}}}\xspace}<1.3$ for all centrality intervals. Between central and peripheral collisions, it is observed that the temperature and the velocity profile exponent $n$ increase, while the expansion velocity decreases, trends which are also observed in blast-wave fits of [${\ensuremath{\uppi}\xspace}^{\pm}$]{}, [$\mathrm{K}^{\pm}$]{}, and (anti)proton [$p_{\mathrm{T}}$]{}distributions in the same collision system [@ALICE_piKp_PbPb]. The behavior of $T_{\mathrm{kin}}$ and $\beta_{\mathrm{s}}$ as a function of centrality is also observed at RHIC [@STAR_overview_2005; @PHENIX_phi_AuAu_2005]. These trends are consistent with a scenario in which the fireballs created in peripheral collisions have shorter lifetimes than in central collisions, with higher freeze-out temperatures and less time to build up radial flow [@Heinz_Lecture2003].
In order to find [${\ensuremath{\mathrm{d}}}N{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}{\ensuremath{\mathrm{d}}}y$]{}the measured resonance [$p_{\mathrm{T}}$]{}distributions are integrated, while the fits are used to estimate the resonance yields at low and high [$p_{\mathrm{T}}$]{}, where no signal could be measured. The low-[$p_{\mathrm{T}}$]{}extrapolation region \[${\ensuremath{p_{\mathrm{T}}}\xspace}({\ensuremath{\mathrm{K^{*0}}}\xspace})<0.3$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}and ${\ensuremath{p_{\mathrm{T}}}\xspace}({\ensuremath{\upphi}\xspace})<0.5$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}\] accounts for 5% (14%) of the total [$\mathrm{K^{*0}}$]{}([$\upphi$]{}) yield, while the high-[$p_{\mathrm{T}}$]{}extrapolation region (${\ensuremath{p_{\mathrm{T}}}\xspace}>5$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}) accounts for $\sim 0.1\%$ $(<0.5\%)$ of the total yield. Alternate functions are also used to fit the resonance [$p_{\mathrm{T}}$]{}distributions: Lévy-Tsallis functions [@Tsallis; @STAR_strange_pp_2007] for both resonances as well as exponential functions in transverse mass for the calculation of [${\ensuremath{\mathrm{d}}}N{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}{\ensuremath{\mathrm{d}}}y$]{}for [$\upphi$]{}. Variations in [${\ensuremath{\mathrm{d}}}N{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}{\ensuremath{\mathrm{d}}}y$]{}and [$\langle{\ensuremath{p_{\mathrm{T}}}\xspace}\rangle$]{}due to the choice of the fitting function are incorporated into the systematic uncertainties (“[$p_{\mathrm{T}}$]{}extrapolation" in Table \[table:sys\]). The values of [${\ensuremath{\mathrm{d}}}N{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}{\ensuremath{\mathrm{d}}}y$]{}for [$\mathrm{K^{*0}}$]{}([$\upphi$]{}) vary by 1.2% (2.1%) on average when the alternate fitting functions are used. Uncertainties in the boundaries of the centrality percentiles result in a normalization uncertainty for the particle yields. The values of the normalization uncertainty reported in [@ALICE_piKp_PbPb] (ranging from 0.5% for central collisions to $_{-8.5}^{+12}\%$ for peripheral collisions) are also used for [$\mathrm{K^{*0}}$]{}and [$\upphi$]{}(“Normalization" in Table \[table:sys\]).
Mass and Width\[sec:results:mass\_width\]
-----------------------------------------
{width="38pc"}
The masses and widths of the [$\mathrm{K^{*0}}$]{}and [$\upphi$]{}resonances \[*i.e.*, the fit parameters $M_{0}$ and $\Gamma_{0}$ from Eqs. (\[eq:signal:pwavebw\])-(\[eq:signal:voigt\])\] are shown in Fig. \[fig:results:mass\_width\] as a function of [$p_{\mathrm{T}}$]{}for multiple centrality intervals. The systematic uncertainties in the masses and widths are evaluated as described in Secs. \[sec:signal\] and \[sec:corr\]. In addition, variations in the masses and widths of the resonances due to changes in the track selection cuts (on $\mathrm{DCA}_{xy}$ and the number of TPC clusters) are incorporated into the systematic uncertainties (average values of 0.4% for the [$\mathrm{K^{*0}}$]{}mass and 0.0038% for the [$\upphi$]{}mass). Uncertainties in the material budget of the ALICE detector introduce a further systematic uncertainty of approximately 0.2% (0.01%) in the [$\mathrm{K^{*0}}$]{}([$\upphi$]{}) mass. The measured [$\mathrm{K^{*0}}$]{}mass has uncertainties of 5-10 [$\mathrm{MeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c^{2}$]{}(an uncertainty of approximately 4 [$\mathrm{MeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c^{2}$]{}is correlated between [$p_{\mathrm{T}}$]{}bins) and is consistent with the mass values found in the HIJING simulation. The measured [$\mathrm{K^{*0}}$]{}width has uncertainties of 10-20 [$\mathrm{MeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c^{2}$]{}(2 [$\mathrm{MeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c^{2}$]{}correlated between [$p_{\mathrm{T}}$]{}bins) and is also consistent with the values found in the simulation. The width of the [$\upphi$]{}meson is an order of magnitude smaller than the width of the [$\mathrm{K^{*0}}$]{}. The [$\upphi$]{}mass is therefore measured with better precision than the [$\mathrm{K^{*0}}$]{}, with systematic uncertainties of $\sim0.2$ [$\mathrm{MeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c^{2}$]{}. A mass shift, due to detector effects, is observed in the HIJING simulation. This shift ranges from $-0.35$ [$\mathrm{MeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c^{2}$]{}at low [$p_{\mathrm{T}}$]{}to +0.05 [$\mathrm{MeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c^{2}$]{}at high [$p_{\mathrm{T}}$]{}. The measured [$\upphi$]{}mass is corrected to account for this shift. The corrected [$\upphi$]{}mass, shown in Fig. \[fig:results:mass\_width\](c), has uncertainties of 0.15-0.5 [$\mathrm{MeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c^{2}$]{}(0.1 [$\mathrm{MeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c^{2}$]{}correlated between [$p_{\mathrm{T}}$]{}bins). The [$\upphi$]{}mass is observed to be consistent with the vacuum value. The [$\upphi$]{}width has uncertainties of 0.7-2 [$\mathrm{MeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c^{2}$]{}(0.3 [$\mathrm{MeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c^{2}$]{}correlated between [$p_{\mathrm{T}}$]{}bins) and is consistent with the width observed in the HIJING simulation.[^3] Neither the mass nor the width of either resonance varies with centrality and no evidence is seen for a modification of the mass or width in collisions at . The masses and widths of these resonances have also been studied at lower collision energies. No significant change in the mass or width of the [$\mathrm{K^{*0}}$]{}meson is observed by the STAR Collaboration in and collisions at and [@STAR_Kstar_2011]. The STAR Collaboration observes that the measured mass and width of the [$\upphi$]{}meson deviate from the values extracted from simulations at low [$p_{\mathrm{T}}$]{}($\lesssim1.5$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}) in pp, , and collisions at and collisions at [@STAR_phi_2009]. However, the deviations do not appear to depend on the size of the collision system and are likely due to detector effects that are not properly reproduced in the simulations. No clear evidence is observed for changes in the [$\upphi$]{}mass or width by the PHENIX Collaboration in collisions at [@PHENIX_phi_AuAu_2005], nor by the NA49 Collaboration in collisions at [@NA49_phi_2008].
[ r c c c ]{}\
Centrality & [${\ensuremath{\mathrm{d}}}N{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}{\ensuremath{\mathrm{d}}}y$]{}& [${\ensuremath{\mathrm{K^{*0}}}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}& [$\langle{\ensuremath{p_{\mathrm{T}}}\xspace}\rangle$]{}([$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{})\
0-20% & $16.6 \pm 0.6 \pm 2.5 \pm 0.1$ & $0.20 \pm 0.01 \pm 0.03$ & $1.31 \pm 0.04 \pm 0.11$\
20-40% & $9.0 \pm 0.8 \pm 1.1 \pm 0.1$ & $0.24 \pm 0.02 \pm 0.03$ & $1.29 \pm 0.04 \pm 0.11$\
40-60% & $3.9 \pm 0.3 \pm 0.4 \pm 0.1$ & $0.28 \pm 0.02 \pm 0.03$ & $1.16 \pm 0.04 \pm 0.08$\
60-80% & $1.13 \pm 0.09 \pm 0.11 \pm0.07$ & $0.31 \pm 0.02 \pm 0.03$ & $1.08 \pm 0.03 \pm 0.07$\
\
\
Centrality & [${\ensuremath{\mathrm{d}}}N{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}{\ensuremath{\mathrm{d}}}y$]{}& [${\ensuremath{\upphi}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}& [$\langle{\ensuremath{p_{\mathrm{T}}}\xspace}\rangle$]{}([$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{})\
0-5% & $13.8 \pm 0.5 \pm 1.7 \pm 0.1$ & $0.127 \pm 0.004 \pm 0.014$ & $1.31 \pm 0.04 \pm 0.06$\
5-10% & $11.7 \pm 0.4 \pm 1.4 \pm 0.1$ & $0.130 \pm 0.004 \pm 0.014$ & $1.34 \pm 0.04 \pm 0.06$\
10-20% & $9.0 \pm 0.2 \pm 1.0 \pm 0.1$ & $0.134 \pm 0.003 \pm 0.013$ & $1.34 \pm 0.03 \pm 0.04$\
20-30% & $7.0 \pm 0.1 \pm 0.8 \pm 0.1$ & $0.152 \pm 0.003 \pm 0.015$ & $1.29 \pm 0.02 \pm 0.03$\
30-40% & $4.28 \pm 0.09 \pm 0.48 \pm 0.09$ & $0.144 \pm 0.003 \pm 0.014$ & $1.25 \pm 0.03 \pm 0.03$\
40-50% & $2.67 \pm 0.05 \pm 0.30 \pm 0.06$ & $0.148 \pm 0.003 \pm 0.014$ & $1.22 \pm 0.02 \pm 0.05$\
50-60% & $1.49 \pm 0.03 \pm 0.16 \pm 0.05$ & $0.145 \pm 0.003 \pm 0.014$ & $1.20 \pm 0.02 \pm 0.04$\
60-70% & $0.72 \pm 0.02 \pm 0.08 \pm 0.04$ & $0.140 \pm 0.004 \pm 0.013$ & $1.17 \pm 0.03 \pm 0.05$\
70-80% & $0.30 \pm 0.01 \pm 0.04 \pm 0.02$ & $0.133 \pm 0.005 \pm 0.015$ & $1.12 \pm 0.03 \pm 0.03$\
80-90% & $0.097 \pm 0.004 \pm 0.012 _{-0.008}^{+0.012}$ & $0.113 \pm 0.005 \pm 0.014$ & $1.14 \pm 0.05 \pm 0.06$\
 Ratios of [$p_{\mathrm{T}}$]{}-integrated particle yields [${\ensuremath{\mathrm{K^{*0}}}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}and [${\ensuremath{\upphi}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}as a function of [$({\ensuremath{{\ensuremath{\mathrm{d}}}N_{\mathrm{ch}}\kern-0.06em /\kern-0.13em{\ensuremath{\mathrm{d}}}\eta}\xspace})^{1/3}$]{} [@ALICE_multiplicity; @ALICE_multiplicity_pp7TeV] for collisions at and pp collisions at [$\sqrt{s}=$ 7 TeV]{} [@ALICE_kstar_phi_7TeV; @ALICE_piKp_7TeV]. The values of [${\ensuremath{\mathrm{d}}}N_{\mathrm{ch}}\kern-0.06em /\kern-0.13em{\ensuremath{\mathrm{d}}}\eta$]{}were measured at mid-rapidity. The statistical uncertainties are shown as bars. The shaded boxes show systematic uncertainties that are not correlated between centrality intervals, while the open boxes show the total systematic uncertainties including both correlated and uncorrelated sources. The values given by a grand-canonical thermal model with a chemical freeze-out temperature of 156 MeV are also shown [@Stachel_SQM2013].](Figure06){width="19.1pc"}
Particle Ratios and Interactions in the Hadronic Phase\[sec:results:rescattering\]
----------------------------------------------------------------------------------
Table \[table:results\] gives the values of [${\ensuremath{\mathrm{d}}}N{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}{\ensuremath{\mathrm{d}}}y$]{}, the [$p_{\mathrm{T}}$]{}-integrated particle yields for $|y|<0.5$, for the [$\mathrm{K^{*0}}$]{}and [$\upphi$]{}resonances in different centrality intervals. This table also includes the ratios of [$p_{\mathrm{T}}$]{}-integrated particle yields [${\ensuremath{\mathrm{K^{*0}}}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}and [${\ensuremath{\upphi}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}, which are calculated using the [${\ensuremath{\mathrm{d}}}N{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}{\ensuremath{\mathrm{d}}}y$]{}values for [$\mathrm{K}^{-}$]{}from [@ALICE_piKp_PbPb]. These ratios are shown in Fig. \[fig:results:ratios\_cent\] for collisions at and pp collisions at [$\sqrt{s}=$ 7 TeV]{} [@ALICE_kstar_phi_7TeV; @ALICE_piKp_7TeV]. These ratios are presented as a function of [$({\ensuremath{{\ensuremath{\mathrm{d}}}N_{\mathrm{ch}}\kern-0.06em /\kern-0.13em{\ensuremath{\mathrm{d}}}\eta}\xspace})^{1/3}$]{}(the cube root of the charged-particle multiplicity density measured at mid-rapidity) [@ALICE_multiplicity; @ALICE_multiplicity_pp7TeV] for reasons discussed below. The [${\ensuremath{\mathrm{K^{*0}}}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}ratio is observed to be lower in central collisions \[larger values of [$({\ensuremath{{\ensuremath{\mathrm{d}}}N_{\mathrm{ch}}\kern-0.06em /\kern-0.13em{\ensuremath{\mathrm{d}}}\eta}\xspace})^{1/3}$]{}\] than in pp and peripheral collisions. When the [${\ensuremath{\mathrm{K^{*0}}}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}ratio in central collisions is divided by the [${\ensuremath{\mathrm{K^{*0}}}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}ratio in peripheral collisions the result[^4] is $0.65\pm0.11$, which is different from unity at the $3.2\sigma$ level. On the other hand, the [${\ensuremath{\upphi}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}ratio does not depend strongly on collision centrality and may be enhanced in mid-central collisions with respect to peripheral and pp collisions. The value of the [${\ensuremath{\upphi}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}ratio in central collisions is consistent with the value measured in pp collisions.
As discussed in Sec. \[sec:intro\], it is possible that resonance yields are modified during the hadronic phase by re-scattering (which would reduce the measured yields) and regeneration (which would increase the yields). The observed suppression of the [${\ensuremath{\mathrm{K^{*0}}}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}ratio may be the result of these effects, with re-scattering dominating over regeneration. The fact that the [${\ensuremath{\upphi}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}ratio does not exhibit suppression for central collisions suggests that the [$\upphi$]{}(which has a lifetime an order of magnitude larger than the [$\mathrm{K^{*0}}$]{}) might decay predominantly outside the hadronic medium. Of [$\mathrm{K^{*0}}$]{}mesons with momentum $p=1$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}, 55% will decay within 5 fm/$c$ of production (a typical estimate for the time between chemical and kinetic freeze-out in heavy-ion collisions [@Bliecher_Aichelin; @Bass_Freezeout_1999]), while only 7% of [$\upphi$]{}mesons with $p=1$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}will decay within that time. It should be noted that elastic scattering of the resonance decay products might be expected to broaden the measured [$\mathrm{K^{*0}}$]{}invariant-mass distribution, which is not observed. The simultaneous observation of [${\ensuremath{\mathrm{K^{*0}}}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}suppression but no [$\mathrm{K^{*0}}$]{}width modification could be explained by decay-product re-scattering if that process were to take place predominantly through elastic scattering with large momentum transfers (which would make the modified signal indistinguishable from the background) or through pseudo-elastic scattering via other resonances.
In Fig. \[fig:results:ratios\_cent\] the [${\ensuremath{\mathrm{K^{*0}}}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}and [${\ensuremath{\upphi}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}ratios have been plotted as a function of [$({\ensuremath{{\ensuremath{\mathrm{d}}}N_{\mathrm{ch}}\kern-0.06em /\kern-0.13em{\ensuremath{\mathrm{d}}}\eta}\xspace})^{1/3}$]{}in order to study whether the strength of the suppression might be related to the system radius. It is an established practice in femtoscopy studies to plot the HBT radii as a function of [$({\ensuremath{{\ensuremath{\mathrm{d}}}N_{\mathrm{ch}}\kern-0.06em /\kern-0.13em{\ensuremath{\mathrm{d}}}\eta}\xspace})^{1/3}$]{} [@ALICE_HBT_2011]. In some cases these radii have been observed to increase approximately linearly with [$({\ensuremath{{\ensuremath{\mathrm{d}}}N_{\mathrm{ch}}\kern-0.06em /\kern-0.13em{\ensuremath{\mathrm{d}}}\eta}\xspace})^{1/3}$]{} [@ALICE_HBT_2011; @Lisa_FemtoscopyReview], suggesting that [$({\ensuremath{{\ensuremath{\mathrm{d}}}N_{\mathrm{ch}}\kern-0.06em /\kern-0.13em{\ensuremath{\mathrm{d}}}\eta}\xspace})^{1/3}$]{}might be used as a proxy for the system radius. If it is assumed that the suppression of the [$\mathrm{K^{*0}}$]{}yield is due to re-scattering and that the strength of re-scattering effects is proportional to the distance which the decay products travel through the hadronic medium, the [${\ensuremath{\mathrm{K^{*0}}}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}ratio would be expected to decrease as a decaying exponential in [$({\ensuremath{{\ensuremath{\mathrm{d}}}N_{\mathrm{ch}}\kern-0.06em /\kern-0.13em{\ensuremath{\mathrm{d}}}\eta}\xspace})^{1/3}$]{}. The observed dependence of the [${\ensuremath{\mathrm{K^{*0}}}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}ratio on the multiplicity is consistent with the behavior that would be expected if re-scattering were the cause of the suppression.
Figure \[fig:results:ratios\_cent\] also includes the values given by a thermal model [@Stachel_SQM2013] for the [${\ensuremath{\mathrm{K^{*0}}}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}and [${\ensuremath{\upphi}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}ratios in central collisions at , with a chemical freeze-out temperature of 156 MeV and a baryochemical potential of 0 MeV. This thermal model does not include re-scattering effects. These results were obtained by fitting a variety of particle yields measured in this collision system. The [$\upphi$]{}yield was included in the fit, but the [$\mathrm{K^{*0}}$]{}was excluded due to the possibility that its yield could be modified as discussed above. The [$\upphi$]{}yield from the fit agrees with the measured yield within 0.5 times the uncertainties and the fit results are not expected to change significantly if the [$\upphi$]{}is excluded. The [${\ensuremath{\mathrm{K^{*0}}}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}ratio given by the thermal model is about 50% larger than the measured ratio. The thermal-model [${\ensuremath{\upphi}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}ratio for central collisions at is consistent with the measured value.
The measured [${\ensuremath{\mathrm{K^{*0}}}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}and [${\ensuremath{\upphi}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}ratios are compared in Fig. \[fig:results:ratios\_full\] to results for different collision systems and energies, plotted as a function of [$({\ensuremath{{\ensuremath{\mathrm{d}}}N_{\mathrm{ch}}\kern-0.06em /\kern-0.13em{\ensuremath{\mathrm{d}}}\eta}\xspace})^{1/3}$]{}and [$\sqrt{s_{\mathrm{NN}}}$]{}. This figure also includes the same thermal-model ratios for central collisions shown in Fig. \[fig:results:ratios\_cent\]. The [${\ensuremath{\mathrm{K^{*0}}}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}ratio is compared in Fig. \[fig:results:ratios\_full\](a) and \[fig:results:ratios\_full\](b) to results for different collision systems at RHIC[^5] [@STAR_Kstar_200GeV_2005; @STAR_Kstar_2011; @STAR_resonances_dAu_2008; @STAR_piKp_CuCu] and LHC [@ALICE_piKp_PbPb; @ALICE_kstar_phi_7TeV; @ALICE_piKp_7TeV] energies. The [${\ensuremath{\mathrm{K^{*0}}}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}ratio is plotted as function of [$({\ensuremath{{\ensuremath{\mathrm{d}}}N_{\mathrm{ch}}\kern-0.06em /\kern-0.13em{\ensuremath{\mathrm{d}}}\eta}\xspace})^{1/3}$]{} [@ALICE_multiplicity; @ALICE_multiplicity_pp7TeV; @STAR_piKp_CuCu; @STAR_centrality_2009] in panel (a). In general, these values appear to follow a single trend independent of collision energy, tending to exhibit suppression in central collisions with respect to pp, , and peripheral collisions. The decrease in the [${\ensuremath{\mathrm{K^{*0}}}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}ratio between pp and central collisions is similar at both RHIC and LHC energies. Refs. [@STAR_Kstar_200GeV_2005] and [@STAR_Kstar_2011] also suggest that the decrease in this ratio for collisions at may be due to re-scattering of the [$\mathrm{K^{*0}}$]{}decay products in the hadronic medium. The same ratio is shown in panel (b) as a function of [$\sqrt{s_{\mathrm{NN}}}$]{}for pp collisions, as well as central and collisions. The [${\ensuremath{\mathrm{K^{*0}}}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}ratio is higher in pp collisions than in central and collisions. The value of the [${\ensuremath{\mathrm{K^{*0}}}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}ratio is larger in central than in central collisions, which is expected due to the smaller size of the collision system.
{width="38pc"}
The [${\ensuremath{\upphi}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}ratio is compared in Fig. \[fig:results:ratios\_full\](c) and \[fig:results:ratios\_full\](d) to results for different collision systems at SPS [@NA49_phi_2008; @NA49_phi_2000; @NA49_piK_2002], RHIC [@PHENIX_phi_AuAu_2005; @STAR_phi_2009; @STAR_phi_200GeV_2005; @STAR_phi_130GeV; @PHENIX_mesons_pp_2011; @PHENIX_piKp_pp_2011], and LHC [@ALICE_strange_900GeV; @ALICE_kstar_phi_7TeV; @ALICE_piKp_7TeV; @ALICE_piKp_900GeV] energies. The [${\ensuremath{\upphi}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}ratio is plotted as a function of [$({\ensuremath{{\ensuremath{\mathrm{d}}}N_{\mathrm{ch}}\kern-0.06em /\kern-0.13em{\ensuremath{\mathrm{d}}}\eta}\xspace})^{1/3}$]{} [@ALICE_multiplicity; @ALICE_multiplicity_pp7TeV; @STAR_piKp_CuCu; @STAR_centrality_2009] in panel (c) for collisions at and LHC energies. The measured [${\ensuremath{\upphi}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}ratio for collisions tends to be larger at than at for similar values of [$({\ensuremath{{\ensuremath{\mathrm{d}}}N_{\mathrm{ch}}\kern-0.06em /\kern-0.13em{\ensuremath{\mathrm{d}}}\eta}\xspace})^{1/3}$]{}; however, the values are consistent within their uncertainties. As observed at LHC energies, the [${\ensuremath{\upphi}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}ratio at does not exhibit a strong centrality dependence, though there are indications of a small enhancement (not beyond the uncertainties) for mid-central and central collisions. The [${\ensuremath{\upphi}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}ratio is shown in panel (d) as a function of [$\sqrt{s_{\mathrm{NN}}}$]{}for pp collisions and for central and collisions. The [${\ensuremath{\upphi}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}ratio is independent of collision energy and system from RHIC to LHC energies,[^6] while at SPS energies the ratio measured in collisions is a factor of two larger than the ratio in pp collisions.
The measured [$p_{\mathrm{T}}$]{}distributions and yields may reflect elastic and pseudo-elastic interactions in the hadronic phase, with the magnitude of the change depending on the resonance lifetime. Thermal models, which give particle yields at chemical freeze-out, do not include these effects. Therefore, including the yields of short-lived resonances like [$\mathrm{K^{*0}}$]{}in thermal-model fits might give misleading results. The model described in [@Markert_thermal; @Torrieri_thermal; @Torrieri_thermal_2001b; @Torrieri_thermal_2001b_erratum] is based on a thermal-model framework, but includes, in addition, re-scattering effects which modify the resonance yields after chemical freeze-out. This model predicts particle ratios, including [${\ensuremath{\mathrm{K^{*0}}}\xspace}{\ensuremath{\kern-0.18em}}/\mathrm{K}$]{}, as a function of the chemical freeze-out temperature and the lifetime of the hadronic phase. If an assumption is made about the value of the chemical freeze-out temperature, a measured [${\ensuremath{\mathrm{K^{*0}}}\xspace}{\ensuremath{\kern-0.18em}}/\mathrm{K}$]{}ratio can be used to extract an estimate of the lifetime. Assuming a chemical freeze-out temperature of 156 MeV (based on thermal-model fits of ALICE data [@Stachel_SQM2013]) and using the measured [${\ensuremath{\mathrm{K^{*0}}}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}ratio for the 0-20% centrality interval, it is possible to estimate a lower limit of 2 fm/$c$ for the time between chemical and kinetic freeze-out. Only a lower limit can be extracted because the model does not include regeneration of resonances in the hadronic medium. This limit on the hadronic lifetime is the same order of magnitude as the [$\mathrm{K^{*0}}$]{}lifetime, but 23 times shorter than the [$\upphi$]{}lifetime. This value can be compared to the hadronic lifetime of $>4$ fm/$c$ extracted using the same model and the $\Lambda(1520){\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}\Lambda$ ratio measured in collisions at [@STAR_baryon_resonances_2006]. If a constant chemical freeze-out temperature is assumed, the increase in [${\ensuremath{\mathrm{K^{*0}}}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}from central to peripheral collisions (see Fig. \[fig:results:ratios\_cent\]) corresponds to a decreasing hadronic-phase lifetime and, equivalently, a larger kinetic freeze-out temperature. This is in qualitative agreement with results from blast-wave fits of particle [$p_{\mathrm{T}}$]{}distributions [@ALICE_piKp_PbPb], which also exhibit an increase in the kinetic freeze-out temperature for more peripheral collisions. Alternatively, if no hadronic lifetime or no re-scattering is assumed, the model predicts a freeze-out temperature for the [$\mathrm{K^{*0}}$]{}of about $120\pm7$ MeV.
It should be noted that these estimates of the temperature or the lifetime of the hadronic phase are model-dependent. The estimate of 2 fm/$c$ for the lower limit of the lifetime of the hadronic phase is only valid insofar as the model described in [@Markert_thermal; @Torrieri_thermal; @Torrieri_thermal_2001b; @Torrieri_thermal_2001b_erratum] is valid. Later work by one of the same authors [@Petran_Rafelski_2013a; @Petran_Rafelski_2013b] uses a non-equilibrium thermal model to extract an estimate of 138 MeV for the freeze-out temperature with no time difference between chemical and kinetic freeze-out. However, this non-equilibrium model predicts a [${\ensuremath{\mathrm{K^{*0}}}\xspace}{\ensuremath{\kern-0.18em}}/\mathrm{K}$]{}ratio that is essentially independent of centrality, which appears to disagree with the results reported above.
{width="38pc"}
According to UrQMD [@UrQMD_Bass; @UrQMD] calculations for RHIC energies, the hadronic re-scattering effect is expected to be momentum-dependent, with greater strength for low-[$p_{\mathrm{T}}$]{}resonances (${\ensuremath{p_{\mathrm{T}}}\xspace}\lesssim2$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}) [@Bleicher_Stoecker; @Bliecher_Aichelin]. To investigate the [$p_{\mathrm{T}}$]{}dependence of the observed suppression, the blast-wave model is used to generate expected transverse-momentum distributions without re-scattering effects for the [$\mathrm{K^{*0}}$]{}and [$\upphi$]{}resonances at kinetic freeze-out. The kinetic freeze-out temperature $T_{\mathrm{kin}}$, velocity profile exponent $n$, and surface expansion velocity $\beta_{\mathrm{s}}$ (radial flow) are taken from simultaneous blast-wave fits of [${\ensuremath{\uppi}\xspace}^{\pm}$]{}, [$\mathrm{K}^{\pm}$]{}, and (anti)proton [$p_{\mathrm{T}}$]{}distributions in collisions at [@ALICE_piKp_PbPb]. For each centrality interval, these fits were performed over the ranges $0.5<{\ensuremath{p_{\mathrm{T}}}\xspace}<1$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}for [${\ensuremath{\uppi}\xspace}^{\pm}$]{}, $0.2<{\ensuremath{p_{\mathrm{T}}}\xspace}<1.5$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}for [$\mathrm{K}^{\pm}$]{}, and $0.3<{\ensuremath{p_{\mathrm{T}}}\xspace}<3$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}for (anti)protons. The simultaneous fits provide good descriptions of the particle [$p_{\mathrm{T}}$]{}distributions within these fit ranges. The parameters used in the present study are the averages (weighted by the number of events multiplied by [${\ensuremath{\mathrm{d}}}N_{\mathrm{ch}}\kern-0.06em /\kern-0.13em{\ensuremath{\mathrm{d}}}\eta$]{} [@ALICE_multiplicity]) of the values reported for the narrower centrality intervals in [@ALICE_piKp_PbPb]. For the 0-20% (60-80%) centrality interval, $T_{\mathrm{kin}}$ is 0.097 GeV (0.13 GeV), $n$ is 0.73 (1.38), and $\beta_{\mathrm{s}}$ is 0.88 (0.80). For ${\ensuremath{p_{\mathrm{T}}}\xspace}<3$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}, these parameters are used with the blast-wave model to generate the shapes, but not the total yields, of expected [$p_{\mathrm{T}}$]{}distributions for the [$\mathrm{K^{*0}}$]{}and [$\upphi$]{}mesons. The expected [$\mathrm{K^{*0}}$]{}([$\upphi$]{}) distribution is normalized so that its integral is the [$\mathrm{K}^{\pm}$]{}yield [${\ensuremath{\mathrm{d}}}N{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}{\ensuremath{\mathrm{d}}}y$]{}in collisions [@ALICE_piKp_PbPb] multiplied by the [${\ensuremath{\mathrm{K^{*0}}}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}([${\ensuremath{\upphi}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}) ratio given by a thermal-model fit to ALICE data [@Stachel_SQM2013] (with a chemical freeze-out temperature of 156 MeV). These are taken to be the expectations if hydrodynamics, as parameterized by the blast-wave model, describes the [$p_{\mathrm{T}}$]{}distributions of the stable particles and the resonances simultaneously and if the [$\mathrm{K^{*0}}$]{}and [$\upphi$]{}meson [$p_{\mathrm{T}}$]{}distributions are not modified by any additional effects (*e.g.*, re-scattering or regeneration). The normalization depends on the parameters of the thermal model: if a temperature of 164 MeV [@AndronicQM2011] is used instead, the expected [$\mathrm{K^{*0}}$]{}([$\upphi$]{}) yield is 5% (6%) greater.
Figure \[fig:results:blast\] shows these expected [$\mathrm{K^{*0}}$]{}and [$\upphi$]{}distributions (as solid lines), the measured resonance [$p_{\mathrm{T}}$]{}distributions, and the ratios of the measurement to the model for central (0-20%) and peripheral (60-80%) collisions. The ratio of the measured [$\upphi$]{}meson [$p_{\mathrm{T}}$]{}distribution to the expected distribution is around unity and no significant difference is observed in central collisions, nor in peripheral collisions for ${\ensuremath{p_{\mathrm{T}}}\xspace}<2$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}. On the other hand, the average measured/expected ratio for the [$\mathrm{K^{*0}}$]{}is $0.6\pm0.1$ for ${\ensuremath{p_{\mathrm{T}}}\xspace}<3$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}in central collisions, a deviation from unity of about four times larger than the uncertainties. In peripheral collisions, the measured/expected ratio for the [$\mathrm{K^{*0}}$]{}does not appear to deviate significantly from unity for ${\ensuremath{p_{\mathrm{T}}}\xspace}<2$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}. For central collisions, the shape of the [$p_{\mathrm{T}}$]{}distribution of [$\mathrm{K^{*0}}$]{}for ${\ensuremath{p_{\mathrm{T}}}\xspace}<3$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}is consistent with the blast-wave parameterization of radial flow within uncertainties. Figure \[fig:results:blast\](a) shows a [$\mathrm{K^{*0}}$]{}suppression of $\sim 40\%$ in the measured low-[$p_{\mathrm{T}}$]{}range and does not indicate the strongly momentum-dependent modification which is predicted by UrQMD for ${\ensuremath{p_{\mathrm{T}}}\xspace}<2$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{} [@Bleicher_Stoecker; @Bliecher_Aichelin]. However, this UrQMD calculation counts the suppression relative to the sum of both primary as well regenerated [$\mathrm{K^{*0}}$]{}resonances and therefore cannot be compared directly to the data.
The suggestion that [$\mathrm{K^{*0}}$]{}suppression does not have a strong [$p_{\mathrm{T}}$]{}dependence for ${\ensuremath{p_{\mathrm{T}}}\xspace}<3$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}might be interpreted as evidence that the reduction observed in the [$\mathrm{K^{*0}}$]{}yield is not due to re-scattering. However, it should be noted that regeneration is also expected to be more important at low [$p_{\mathrm{T}}$]{}, which could counteract some of the low-[$p_{\mathrm{T}}$]{}suppression that would be expected from re-scattering alone. Furthermore, there is evidence for some increase in the measured/expected ratio for [$\mathrm{K^{*0}}$]{}from ${\ensuremath{p_{\mathrm{T}}}\xspace}=1.2$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}to ${\ensuremath{p_{\mathrm{T}}}\xspace}=3$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}for central collisions. Further theoretical studies of the [$p_{\mathrm{T}}$]{}dependence of [$\mathrm{K^{*0}}$]{}suppression, with a full treatment of both re-scattering and regeneration, would be helpful in determining the likelihood that re-scattering is responsible for the observed decrease in the [$\mathrm{K^{*0}}$]{}yield.
Mean Transverse Momentum\[sec:results:mpt\]
-------------------------------------------
{width="38pc"}
The mean transverse momentum [$\langle{\ensuremath{p_{\mathrm{T}}}\xspace}\rangle$]{}values for the [$\mathrm{K^{*0}}$]{}and [$\upphi$]{}resonances are presented in Table \[table:results\] for different centrality intervals. Figure \[fig:results:mpt\] shows [$\langle{\ensuremath{p_{\mathrm{T}}}\xspace}\rangle$]{}for the [$\mathrm{K^{*0}}$]{}and [$\upphi$]{}resonances in collisions at as a function of [$\langle N_{\mathrm{part}}\rangle$]{} [@ALICE_centrality]. Also shown are measurements of [$\langle{\ensuremath{p_{\mathrm{T}}}\xspace}\rangle$]{}in pp collisions at [$\sqrt{s}=$ 7 TeV]{} [@ALICE_kstar_phi_7TeV] and in pp and collisions at [@STAR_Kstar_200GeV_2005; @STAR_Kstar_2011; @STAR_phi_2009; @STAR_phi_200GeV_2005; @STAR_centrality_2009]. The values of [$\langle{\ensuremath{p_{\mathrm{T}}}\xspace}\rangle$]{}for the [$\mathrm{K^{*0}}$]{}([$\upphi$]{}) meson increase by about 20% (15%) in central collisions relative to peripheral collisions. The values of [$\langle{\ensuremath{p_{\mathrm{T}}}\xspace}\rangle$]{}in pp collisions at [$\sqrt{s}=$ 7 TeV]{}are consistent with the values observed in peripheral collisions at . For central collisions, [$\langle{\ensuremath{p_{\mathrm{T}}}\xspace}\rangle$]{}of the [$\mathrm{K^{*0}}$]{}([$\upphi$]{}) resonance measured in collisions at is about 20% (30%) higher than the values measured in collisions at . This is consistent with the observation [@ALICE_piKp_PbPb] of increased radial flow in collisions at the LHC relative to RHIC.
The values of [$\langle{\ensuremath{p_{\mathrm{T}}}\xspace}\rangle$]{}for [${\ensuremath{\uppi}\xspace}^{+}$]{}, [$\mathrm{K}^{+}$]{}, [$\mathrm{K^{*0}}$]{}, p, and [$\upphi$]{}in collisions at are shown in Fig. \[fig:results:mpt\_X\] for different centrality intervals [@ALICE_centrality; @ALICE_piKp_PbPb]. The values of [$\langle{\ensuremath{p_{\mathrm{T}}}\xspace}\rangle$]{}for the resonances in pp collisions at [$\sqrt{s}=$ 7 TeV]{}are also shown [@ALICE_kstar_phi_7TeV]. All particles exhibit an increase in [$\langle{\ensuremath{p_{\mathrm{T}}}\xspace}\rangle$]{}from peripheral to central collisions, but this increase is greatest for the protons. While the increase in [$\langle{\ensuremath{p_{\mathrm{T}}}\xspace}\rangle$]{}from the most peripheral to most central measured interval is about 20% for [${\ensuremath{\uppi}\xspace}^{+}$]{}, [$\mathrm{K}^{+}$]{}, [$\mathrm{K^{*0}}$]{}, and [$\upphi$]{}, the value of [$\langle{\ensuremath{p_{\mathrm{T}}}\xspace}\rangle$]{}for protons increases by 50%. For the 0-40% most central collisions $({\ensuremath{\langle N_{\mathrm{part}}\rangle}\xspace}\gtrsim100)$ the [$\langle{\ensuremath{p_{\mathrm{T}}}\xspace}\rangle$]{}values of the [$\mathrm{K^{*0}}$]{}, proton, and [$\upphi$]{}all appear to follow the same trend. Within a given centrality interval, the [$\langle{\ensuremath{p_{\mathrm{T}}}\xspace}\rangle$]{}values of these three particles are consistent with each other within uncertainties. It should be noted that the masses of these three particles are similar: 896 [$\mathrm{MeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c^{2}$]{}for the [$\mathrm{K^{*0}}$]{}, 938 [$\mathrm{MeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c^{2}$]{}for the p, and 1019 [$\mathrm{MeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c^{2}$]{}for the [$\upphi$]{} [@PDG]. The similarity in [$\langle{\ensuremath{p_{\mathrm{T}}}\xspace}\rangle$]{}values is consistent with expectations from a hydrodynamic framework, in which [$p_{\mathrm{T}}$]{}distributions would be determined predominantly by the particle masses. This is discussed further in the context of the [$p_{\mathrm{T}}$]{}-dependent [$\mathrm{p}\kern-0.1em/{\ensuremath{\kern-0.18em}}{\ensuremath{\upphi}\xspace}$]{}ratio in Sec. \[sec:discussion:prod:phip\].
Particle Production\[sec:discussion:prod\]
------------------------------------------
In this section, the [$p_{\mathrm{T}}$]{}distributions and total yields of [$\upphi$]{}mesons are compared to theoretical models and other particle species (with different baryon number, mass, or strange quark content) to study particle production mechanisms. The [$\upphi$]{}meson is used for these studies because it lives long enough that its yields and [$p_{\mathrm{T}}$]{}distributions do not appear to be affected by re-scattering or regeneration in the hadronic phase. The possibility that such effects might change the [$\mathrm{K^{*0}}$]{}[$p_{\mathrm{T}}$]{}distributions and yields complicates any attempt to use that particle to study particle production. The predictions of hydrodynamic models, which have described the yields and [$p_{\mathrm{T}}$]{}distributions of other hadrons with fair accuracy [@ALICE_piKp_PbPb; @ALICE_multistrange_PbPb; @ALICE_multistrange_PbPb_erratum], are compared in Sec. \[sec:discussion:prod:theory\] to the [$p_{\mathrm{T}}$]{}distribution of [$\upphi$]{}mesons in central collisions. Differences in the production mechanisms of baryons and mesons can be studied through baryon-to-meson ratios. The [$p_{\mathrm{T}}$]{}-dependent [$\Omega\kern-0.05em/{\ensuremath{\kern-0.09em}}{\ensuremath{\upphi}\xspace}$]{}ratio, which compares baryons and mesons containing only strange (anti)quarks, is compared in Sec. \[sec:discussion:prod:theory\] to theoretical predictions and measurements in other collision systems. If hadron production can be explained in a hydrodynamic framework, the particle mass plays an important role in determining the shape of the [$p_{\mathrm{T}}$]{}distribution. To study this aspect of particle production, in Sec. \[sec:discussion:prod:phip\] [$p_{\mathrm{T}}$]{}distributions of [$\upphi$]{}mesons are compared to protons, which are baryons with a different quark content but a very similar mass to the [$\upphi$]{}. The dependence of particle production on the strange quark content is explored in Sec. \[sec:discussion:prod:enhancement\]. Here the enhancement of the [$\upphi$]{}(which consists entirely of strange quarks but has no net strangeness) is compared to particles with 1, 2, and 3 units of open strangeness: the $\Lambda$, $\Xi$, and $\Omega$ baryons, respectively.
 Mean transverse momentum of [${\ensuremath{\uppi}\xspace}^{+}$]{}, [$\mathrm{K}^{+}$]{}, [$\mathrm{K^{*0}}$]{}, p, and [$\upphi$]{}in collisions at (filled symbols) [@ALICE_piKp_PbPb] as a function of [$\langle N_{\mathrm{part}}\rangle$]{} [@ALICE_centrality]. Also shown are [$\langle{\ensuremath{p_{\mathrm{T}}}\xspace}\rangle$]{}values for the resonances in pp collisions at [$\sqrt{s}=$ 7 TeV]{} [@ALICE_kstar_phi_7TeV] (open symbols). The measurements for central and mid-central [$\upphi$]{}and p have been shifted horizontally for visibility.](Figure10){width="19.1pc"}
### Comparisons to Theoretical Models\[sec:discussion:prod:theory\]
The measured [$\upphi$]{}meson [$p_{\mathrm{T}}$]{}distribution for collisions at (centrality 0-10%) is compared in Fig. \[fig:results:theory\] to five predicted distributions from hydrodynamic models. The measured and predicted distributions are shown in panel (a), while the ratio of the predicted distributions to the measured distribution is shown in panel (b). VISH2+1 is a (2+1)-dimensional viscous hydrodynamic model [@VISH2p1_MCGlb; @VISH2p1_MCKLN]. It has been observed to reproduce the total yields and the shapes of the [$p_{\mathrm{T}}$]{}distributions of [$\uppi$]{}and K within about 25% for ${\ensuremath{p_{\mathrm{T}}}\xspace}<2$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}in central collisions [@ALICE_piKp_PbPb]. VISH2+1 overestimates the total yields of the $\Xi$ and $\Omega$ baryons, though it provides a fair description of the shape of the $\Xi$ [$p_{\mathrm{T}}$]{}distribution [@ALICE_multistrange_PbPb; @ALICE_multistrange_PbPb_erratum]. The VISH2+1 predictions shown in Fig. \[fig:results:theory\] are for two different sets of initial conditions: Monte-Carlo Kharzeev-Levin-Nardi initial conditions (MC-KLN) with $\eta/s=0.2$ [@VISH2p1_MCKLN] and Monte-Carlo Glauber (MC-Glb) initial conditions with $\eta/s=0.08$ [@VISH2p1_MCGlb]. These predictions are larger than the measured [$\upphi$]{}yield at low [$p_{\mathrm{T}}$]{}. If the VISH2+1 MC-KLN prediction is fitted to the measured [$\upphi$]{}data through multiplication by a [$p_{\mathrm{T}}$]{}-independent factor (0.74), it reproduces the shape of the measured [$p_{\mathrm{T}}$]{}distribution for ${\ensuremath{p_{\mathrm{T}}}\xspace}<3$ with a $\chi^{2}$ per degrees of freedom ([$\chi^{2}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}n_{\mathrm{dof}}$]{}) value of 0.52 and no deviations beyond the experimental uncertainties. Similarly, the VISH2+1 MC-Glb prediction can be fitted to the measured [$\upphi$]{}[$p_{\mathrm{T}}$]{}distribution, with a multiplicative constant of 0.74, ${\ensuremath{\chi^{2}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}n_{\mathrm{dof}}}\xspace}=1.1$, and no deviations beyond the experimental uncertainties.
The VISHNU model [@VISHNU_2013; @VISHNU] is a hybrid model which connects the VISH2+1 hydrodynamic description of the QGP to a microscopic hadronic transport model (UrQMD) [@UrQMD_Bass; @UrQMD] to describe the hadronic phase. The VISNHU prediction for the [$\upphi$]{}yield is larger than the measured data, and does not appear to reproduce the shape of the [$p_{\mathrm{T}}$]{}distribution (for ${\ensuremath{p_{\mathrm{T}}}\xspace}<3$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}, ${\ensuremath{\chi^{2}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}n_{\mathrm{dof}}}\xspace}=2.6$ when the prediction is fitted to the data with a multiplicative constant of 0.53). It is also larger than either of the pure VISH2+1 predictions, which is attributed to the production of additional [$\upphi$]{}mesons in the hadronic phase through ${\ensuremath{\mathrm{K}^{-}}\xspace}{\ensuremath{\mathrm{K}^{+}}\xspace}$ scattering while [$\upphi$]{}decays are turned off [@VISHNU_2013].
 (a): Comparison of the measured [$\upphi$]{}meson [$p_{\mathrm{T}}$]{}distribution in collisions at (centrality 0-10%) to the distributions predicted by the Kraków model [@KRAKOW], two versions of the VISH2+1 model [@VISH2p1_MCGlb; @VISH2p1_MCKLN], the VISHNU [@VISHNU_2013; @VISHNU] model, and the HKM [@HKM_2011; @HKM_2013]. The curves show the original predictions, while the horizontal lines show the predicted distributions re-binned so that they have compatible [$p_{\mathrm{T}}$]{}bins with the measured distribution. (b): The ratio of the re-binned predictions to the measured distribution for [$\upphi$]{}mesons. The shaded band shows the fractional uncertainty of the measured data points.](Figure11){width="19.1pc"}
The hydrokinetic model (HKM) [@HKM_2011; @HKM_2013] combines an ideal hydrodynamic phase with a hadronic cascade (UrQMD) after the hydrodynamic description of the partonic phase. Additional radial flow is built up during the hadronic phase and particle yields are affected by hadronic interactions (including baryon-antibaryon annihilation). HKM has been observed to reproduce the measured [$\uppi$]{}, K, proton, $\Xi$ and $\Omega$ data [@ALICE_piKp_PbPb; @ALICE_multistrange_PbPb; @ALICE_multistrange_PbPb_erratum] better than VISH2+1, though it overestimates the yields of the multi-strange baryons. The [$\upphi$]{}yield is overestimated by HKM, though by a smaller amount than the VISH2+1 predictions. The HKM prediction can be fitted to the measured [$\upphi$]{}data through multiplication by a constant (0.80) for ${\ensuremath{p_{\mathrm{T}}}\xspace}<$ 3 [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}(its full range) with ${\ensuremath{\chi^{2}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}n_{\mathrm{dof}}}\xspace}=0.53$ and no deviations beyond the experimental uncertainties.
The Kraków model [@KRAKOW] is a hydrodynamic model which introduces a bulk viscosity in the transition from the partonic to the hadronic phase, producing deviations from local equilibrium within the fluid elements, thereby affecting the hadron [$p_{\mathrm{T}}$]{}distributions and yields. This model reproduces the [$\uppi$]{}, K, and (anti)proton [$p_{\mathrm{T}}$]{}distributions within 20% for ${\ensuremath{p_{\mathrm{T}}}\xspace}<3$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}in central collisions [@ALICE_piKp_PbPb] and reproduces the $\Xi$ [$p_{\mathrm{T}}$]{}distributions within 30% in the same [$p_{\mathrm{T}}$]{}range for the centrality range 0-60% [@ALICE_multistrange_PbPb; @ALICE_multistrange_PbPb_erratum]. It does not, however, describe the shape of the $\Omega$ [$p_{\mathrm{T}}$]{}distribution. The Kraków model over-predicts the [$\upphi$]{}yield for $1<{\ensuremath{p_{\mathrm{T}}}\xspace}<4$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}; however, it does not deviate from the measured yield by more than twice the uncertainty. The Kraków model prediction can be fitted to the measured [$\upphi$]{}meson [$p_{\mathrm{T}}$]{}distribution through multiplication by a constant (0.85) for ${\ensuremath{p_{\mathrm{T}}}\xspace}<4$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}with ${\ensuremath{\chi^{2}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}n_{\mathrm{dof}}}\xspace}=1.1$ and no deviations beyond the uncertainties.
The hydrodynamic models considered above describe the measured [$\upphi$]{}meson [$p_{\mathrm{T}}$]{}distribution with varying degrees of success. All of these models over-predict the [$\upphi$]{}yield, while all except the Kraków model predict softer [$p_{\mathrm{T}}$]{}distributions for the [$\upphi$]{}meson than was measured. The best descriptions of the shape of the [$\upphi$]{}meson [$p_{\mathrm{T}}$]{}distribution are given by the HKM and the Kraków model. Coupling hydrodynamics to a hadronic cascade, as is done in the KHM and VISHNU, has produced widely different results. For the [$\upphi$]{}, the two implementations of the VISH2+1 model produce similar results for ${\ensuremath{p_{\mathrm{T}}}\xspace}<2$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}, despite having different initial conditions and viscosities.
 Ratio [$(\Omega^{-}+\overline{\Omega}^{+})\kern-0.05em/{\ensuremath{\kern-0.09em}}{\ensuremath{\upphi}\xspace}$]{}as a function of [$p_{\mathrm{T}}$]{}for collisions at (centrality 0-10%) [@ALICE_multistrange_PbPb; @ALICE_multistrange_PbPb_erratum], pp collisions at [$\sqrt{s}=$ 7 TeV]{} [@ALICE_kstar_phi_7TeV; @ALICE_multistrange_7TeV] and collisions at (centrality 0-12%) [@STAR_phi_2009]. The statistical uncertainties are shown as bars, systematic uncertainties (including [$p_{\mathrm{T}}$]{}-uncorrelated and [$p_{\mathrm{T}}$]{}-correlated components) are shown as shaded boxes, and the sum in quadrature of the statistical and systematic uncertainties (for the pp data) is shown as open boxes. Also shown are predictions of this ratio made by various models for central collisions at (centrality 0-20% for HIJING/B$\overline{\mathrm{B}}$, centrality 0-10% for the other models) [@VISH2p1_MCGlb; @VISH2p1_MCKLN; @KRAKOW; @HKM_2011; @HKM_2013; @HIJINGBBbar_2011a; @HIJINGBBbar_2011b].](Figure12){width="19.1pc"}
The [$\upphi$]{}and $\Omega$ are, respectively, a meson and a baryon made up entirely of strange (anti)quarks. In some particle production models, such as the HIJING/B$\overline{\mathrm{B}}$ model [@HIJINGBBbar_2011a; @HIJINGBBbar_2011b], soft particles are produced through string fragmentation. The string tension is predicted [@HIJINGBBbar_2011a] to influence the yields of strange particles, with multi-strange baryons and the [$\Omega\kern-0.05em/{\ensuremath{\kern-0.09em}}{\ensuremath{\upphi}\xspace}$]{}ratio being particularly sensitive to the tension [@LHC_predictions_2008]. Figure \[fig:results:Omega2phi\_pt\] shows the ${\ensuremath{\Omega\kern-0.05em/{\ensuremath{\kern-0.09em}}{\ensuremath{\upphi}\xspace}}\xspace}\equiv{\ensuremath{(\Omega^{-}+\overline{\Omega}^{+})\kern-0.05em/{\ensuremath{\kern-0.09em}}{\ensuremath{\upphi}\xspace}}\xspace}$ ratio as a function of [$p_{\mathrm{T}}$]{}in collisions at (centrality 0-10%) [@ALICE_multistrange_PbPb; @ALICE_multistrange_PbPb_erratum], pp collisions at [$\sqrt{s}=$ 7 TeV]{} [@ALICE_kstar_phi_7TeV; @ALICE_multistrange_7TeV], and collisions at (centrality 0-12%) [@STAR_phi_2009]. The ratio measured in collisions at is consistent with the ratio measured in collisions at for ${\ensuremath{p_{\mathrm{T}}}\xspace}\lesssim 3$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}, but is larger than the measurement at high [$p_{\mathrm{T}}$]{}. Predictions from the HIJING/B$\overline{\mathrm{B}}$ and hydrodynamic models are also shown. None of these models is able to predict the measured [$\Omega\kern-0.05em/{\ensuremath{\kern-0.09em}}{\ensuremath{\upphi}\xspace}$]{}ratio. HKM provides a better description of the $\Omega$ [$p_{\mathrm{T}}$]{}distributions than VISH2+1; however, it overestimates the total yield [@ALICE_multistrange_PbPb; @ALICE_multistrange_PbPb_erratum]. The VISH2+1 and HKM predictions are consistent with the measured [$\Omega\kern-0.05em/{\ensuremath{\kern-0.09em}}{\ensuremath{\upphi}\xspace}$]{}ratio for ${\ensuremath{p_{\mathrm{T}}}\xspace}<2$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}, but increase faster with [$p_{\mathrm{T}}$]{}than the data for ${\ensuremath{p_{\mathrm{T}}}\xspace}>2$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}. The HKM does appear to provide a better description of the slope of the measured [$\Omega\kern-0.05em/{\ensuremath{\kern-0.09em}}{\ensuremath{\upphi}\xspace}$]{}ratio. The Kraków model under-predicts the measured data at low [$p_{\mathrm{T}}$]{}, but is consistent with the data for $2<{\ensuremath{p_{\mathrm{T}}}\xspace}<3.5$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}. This model is able to reproduce the measured $\Omega$ yield within about 30% [@ALICE_multistrange_PbPb; @ALICE_multistrange_PbPb_erratum], but does not reproduce the shape of the [$p_{\mathrm{T}}$]{}distribution. The [$\Omega\kern-0.05em/{\ensuremath{\kern-0.09em}}{\ensuremath{\upphi}\xspace}$]{}ratio predicted by the HIJING/B$\overline{\mathrm{B}}$ v2.0 model [@HIJINGBBbar_2011a; @HIJINGBBbar_2011b; @HIJINGBBbar_2012; @HIJINGBBbar_2014], with a strong color field and a string tension of $\kappa=1.8$ GeV/fm, reproduces neither the shape nor the values of the measured data. A larger string tension of $\kappa=5.1$ GeV/fm gives a predicted [$\Omega\kern-0.05em/{\ensuremath{\kern-0.09em}}{\ensuremath{\upphi}\xspace}$]{}ratio (not shown) that is at least a factor of three larger than the measured ratio. The same model can reproduce the [$\Omega\kern-0.05em/{\ensuremath{\kern-0.09em}}{\ensuremath{\upphi}\xspace}$]{}ratio observed in pp collisions at [$\sqrt{s}=$ 7 TeV]{} [@ALICE_kstar_phi_7TeV; @ALICE_multistrange_7TeV][^7] with a string tension of $\kappa=2$ GeV/fm, and describes the [$\Omega\kern-0.05em/{\ensuremath{\kern-0.09em}}{\ensuremath{\upphi}\xspace}$]{}ratio observed in collisions at [@LHC_predictions_2008; @STAR_phi_200GeV_2006] with a string tension of $\kappa=3$ GeV/fm.
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### Particles with Similar Masses\[sec:discussion:prod:phip\]
The proton and [$\upphi$]{}have similar masses, but different baryon numbers and quark content. If production of these particles is described within a hydrodynamic framework, the [$p_{\mathrm{T}}$]{}distributions of these species are expected to have similar shapes, despite their different quantum numbers. Figure \[fig:results:ratios\_pt\](a) shows the ratio ${\ensuremath{\mathrm{p}\kern-0.1em/{\ensuremath{\kern-0.18em}}{\ensuremath{\upphi}\xspace}}\xspace}\equiv{\ensuremath{(\mathrm{p}+\bar{\mathrm{p}}){\ensuremath{\kern-0.09em}}/{\ensuremath{\kern-0.09em}}{\ensuremath{\upphi}\xspace}}\xspace}$ as a function of [$p_{\mathrm{T}}$]{}for collisions at [@ALICE_piKp_PbPb; @ALICE_piKp_PbPb_combined] for different centralities.[^8] For central collisions, the [$\mathrm{p}\kern-0.1em/{\ensuremath{\kern-0.18em}}{\ensuremath{\upphi}\xspace}$]{}ratio is flat over the entire measured range. However, for non-central collisions, this ratio is observed to decrease with [$p_{\mathrm{T}}$]{}. This behavior can also be seen in the [$\langle{\ensuremath{p_{\mathrm{T}}}\xspace}\rangle$]{}values of p and [$\upphi$]{}in Fig. \[fig:results:mpt\_X\]: these values are consistent with each other in central collisions, but [$\langle{\ensuremath{p_{\mathrm{T}}}\xspace}\rangle$]{}is lower for p than for [$\upphi$]{}in peripheral collisions. The flat [$\mathrm{p}\kern-0.1em/{\ensuremath{\kern-0.18em}}{\ensuremath{\upphi}\xspace}$]{}ratio in central collisions indicates that, at LHC energies, the shapes of the [$p_{\mathrm{T}}$]{}distributions of the p and [$\upphi$]{}at low and intermediate [$p_{\mathrm{T}}$]{}are determined by the particle masses. One possible explanation for the non-constant [$\mathrm{p}\kern-0.1em/{\ensuremath{\kern-0.18em}}{\ensuremath{\upphi}\xspace}$]{}ratio in peripheral collisions would be that the particles have a production mechanism in which the quark content is an important factor in determining the shapes of the [$p_{\mathrm{T}}$]{}distributions. At RHIC energies, a splitting in the nuclear modification factor [$R_{\mathrm{CP}}$]{}(the ratio of central to peripheral particle yields scaled by the number of binary collisions in the two centrality intervals), with baryons being less suppressed than mesons at intermediate [$p_{\mathrm{T}}$]{} [@STAR_Kstar_200GeV_2005; @STAR_pip_2006; @STAR_K0sLambda_2004; @STAR_phi_2007; @STAR_XiOmega_2007], has been taken as evidence in favor of recombination models [@Coalescence_Review_2008]. However, at LHC energies, the flat [$\mathrm{p}\kern-0.1em/{\ensuremath{\kern-0.18em}}{\ensuremath{\upphi}\xspace}$]{}ratio suggests that recombination might not be suited to explain the shapes of the observed particle [$p_{\mathrm{T}}$]{}distributions in central collisions at low and intermediate [$p_{\mathrm{T}}$]{}.
The ${\ensuremath{\mathrm{p}\kern-0.05em/{\ensuremath{\kern-0.09em}}{\ensuremath{\uppi}\xspace}}\xspace}\equiv{\ensuremath{(\mathrm{p}+\bar{\mathrm{p}}){\ensuremath{\kern-0.09em}}/{\ensuremath{\kern-0.09em}}({\ensuremath{{\ensuremath{\uppi}\xspace}^{-}}\xspace}+{\ensuremath{{\ensuremath{\uppi}\xspace}^{+}}\xspace})}\xspace}$ ratio [@ALICE_piKp_PbPb; @ALICE_piKp_PbPb_combined] is shown in Fig. \[fig:results:ratios\_pt\](b). When this ratio was first reported [@ALICE_piKp_PbPb], it was not clear if the observed increase in [$\mathrm{p}\kern-0.05em/{\ensuremath{\kern-0.09em}}{\ensuremath{\uppi}\xspace}$]{}with transverse momentum is due to hydrodynamic effects or quark recombination. As shown in Fig. \[fig:results:ratios\_pt\], the baryon-to-meson ratio [$\mathrm{p}\kern-0.05em/{\ensuremath{\kern-0.09em}}{\ensuremath{\uppi}\xspace}$]{}has a very similar shape to the meson-to-meson ratio ${\ensuremath{{\ensuremath{\upphi}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}{\ensuremath{\uppi}\xspace}}\xspace}\equiv{\ensuremath{{\ensuremath{\upphi}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}({\ensuremath{{\ensuremath{\uppi}\xspace}^{-}}\xspace}+{\ensuremath{{\ensuremath{\uppi}\xspace}^{+}}\xspace})}\xspace}$ for ${\ensuremath{p_{\mathrm{T}}}\xspace}<3$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}. This indicates that the number of quarks is not the main factor that determines the shapes of particle [$p_{\mathrm{T}}$]{}distributions at low and intermediate [$p_{\mathrm{T}}$]{}in central collisions. This is contrary to the expectations from recombination, but consistent with hydrodynamic models.
### Strangeness Content\[sec:discussion:prod:enhancement\]
The enhancement ratio is defined as the yield ([${\ensuremath{\mathrm{d}}}N{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}{\ensuremath{\mathrm{d}}}y$]{}) of a particle in collisions normalized to [$\langle N_{\mathrm{part}}\rangle$]{}and divided by the same quantity in pp collisions[^9] at the same energy. This ratio has been the traditional way of presenting strangeness production in heavy-ion collisions [@ALICE_multistrange_PbPb; @ALICE_multistrange_PbPb_erratum; @STAR_XiOmega_2007; @NA57_SE_40AGeV; @NA57_SE_158AGeV; @WA97_enhancement; @NA49_Omega_2005; @NA49_Lambda_Xi_2008; @NA49_Lambda_Xi_2009; @STAR_multistrange_2004; @STAR_SE_200GeV]. However, given the fact that charged-particle production increases in a non-linear way with the number of participants [@ALICE_multiplicity], part of the enhancement observed using this ratio cannot be attributed to strangeness. A way to avoid this bias is to normalize to the pion yield. In order to allow for an easy comparison to previous measurements both approaches are discussed in this section.
The [$\upphi$]{}yield in pp collisions at [$\sqrt{s}=$ 2.76 TeV]{} has been estimated by interpolating between the measured yields at [$\sqrt{s}=$ 900 GeV]{} [@ALICE_strange_900GeV] and [$\sqrt{s}=$ 7 TeV]{} [@ALICE_kstar_phi_7TeV], assuming that the yield varies as $s^{n}$. Given the measured [$\upphi$]{}yields at [$\sqrt{s}=$ 900 GeV]{} and [$\sqrt{s}=$ 7 TeV]{}, the value of the power $n$ was found to be 0.10. For comparison, the calculation of the enhancement values of multi-strange baryons at uses an energy dependence of $s^{0.13}$ to find the interpolated pp reference values [@ALICE_multistrange_PbPb; @ALICE_multistrange_PbPb_erratum]. The charged-particle pseudorapidity density is observed to vary as $s^{0.11}$ [@ALICE_multiplicity_PbPb]. The systematic uncertainty in the interpolated [$\upphi$]{}yield is estimated by successively increasing, then decreasing each of the two measured points by its own uncertainty and repeating the interpolation procedure. The resulting variations in the interpolated yield are incorporated into the systematic uncertainty. The systematic uncertainty in the interpolated [$\upphi$]{}reference yield is 13%. Including the [$\upphi$]{}meson yields measured in pp collisions at [$\sqrt{s}=$ 200 GeV]{} [@STAR_phi_200GeV_2005; @PHENIX_mesons_pp_2011] in the interpolation does not significantly alter the result. The $\Lambda$ enhancement in collisions at , calculated using the yields reported in Ref. [@ALICE_k0s_Lambda_PbPb], is also reported below for the purpose of comparison with the [$\upphi$]{}. The reference $\Lambda$ yield in pp collisions at [$\sqrt{s}=$ 2.76 TeV]{} is estimated by extrapolating from the measured yield in (inelastic) pp collisions at [$\sqrt{s}=$ 900 GeV]{} [@ALICE_strange_900GeV], assuming the same energy dependence as [${\ensuremath{\mathrm{d}}}N_{\mathrm{ch}}\kern-0.06em /\kern-0.13em{\ensuremath{\mathrm{d}}}\eta$]{}. The systematic uncertainty in this extrapolation is estimated by using the energy dependence of the $\Lambda+\overline{\Lambda}$ yield in non-single-diffractive pp collisions at [$\sqrt{s}=$ 200 GeV]{}, 900 GeV, and 7 TeV [@STAR_strange_pp_2007; @CMS_strange_900GeV_7TeV]. The uncertainty in the extrapolated $\Lambda$ reference yield is 19%.
{width="38pc"}
The enhancement values for [$\upphi$]{}for different centrality intervals are shown in Fig. \[fig:results:se\](a) along with the enhancement values for $\Lambda$, $\Xi$, and $\Omega$ [@ALICE_multistrange_PbPb; @ALICE_multistrange_PbPb_erratum; @ALICE_k0s_Lambda_PbPb]. Enhancement values for [$\upphi$]{}in collisions at [@STAR_phi_2009b] are also shown. The [$\upphi$]{}enhancement ratio decreases from to , a trend that has been observed for the other particles as well [@ALICE_multistrange_PbPb; @ALICE_multistrange_PbPb_erratum; @STAR_SE_200GeV]. The values of the [$\upphi$]{}and $\Lambda$ enhancement for the 80-90% centrality interval (${\ensuremath{\langle N_{\mathrm{part}}\rangle}\xspace}=7.5$) are consistent with unity, *i.e.*, the yields per participant nucleon of these particles in peripheral collisions are consistent with the estimated yields in pp collisions. The yields of [$\upphi$]{}, $\Lambda$, $\Xi$, and $\Omega$ at LHC energies increase faster than linearly with [$\langle N_{\mathrm{part}}\rangle$]{}until ${\ensuremath{\langle N_{\mathrm{part}}\rangle}\xspace}\approx 100$, while the enhancement values seem to be saturated for higher values of [$\langle N_{\mathrm{part}}\rangle$]{}. The enhancement values increase with the number of strange valence (anti)quarks, a trend which is also observed at lower energies. For collisions at , the [$\upphi$]{}enhancement is consistent with the enhancement values of $\Lambda$ (one strange valence quark), as well as $\Xi^{-}$ and $\overline{\Xi}^{+}$ (two strange valence quarks or antiquarks). The central values of the [$\upphi$]{}enhancement tend to be between the $\Lambda$ and $\Xi$ enhancement values. A similar behavior is observed when the [$\upphi$]{}is compared to $\Lambda$, $\overline{\Lambda}$, and $\overline{\Xi}^{+}$ in collisions at [@STAR_SE_200GeV; @STAR_phi_2009b].
As an alternative to the standard [$\langle N_{\mathrm{part}}\rangle$]{}-based enhancement ratio discussed above, the yields of particles containing strange quarks are compared to pion yields. This is shown in Fig. \[fig:results:se\](b) for and pp collisions at RHIC and LHC energies [@ALICE_piKp_PbPb; @ALICE_strange_900GeV; @STAR_strange_pp_2007; @STAR_phi_2009; @ALICE_kstar_phi_7TeV; @ALICE_piKp_7TeV; @STAR_centrality_2009; @STAR_phi_200GeV_2005; @ALICE_piKp_900GeV; @ALICE_multistrange_PbPb; @ALICE_multistrange_PbPb_erratum; @ALICE_multistrange_7TeV; @STAR_XiOmega_2007; @ALICE_k0s_Lambda_PbPb]. The ratios shown are ${\ensuremath{{\ensuremath{\upphi}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}{\ensuremath{\uppi}\xspace}}\xspace}\equiv{\ensuremath{\upphi}\xspace}/({\ensuremath{{\ensuremath{\uppi}\xspace}^{-}}\xspace}+{\ensuremath{{\ensuremath{\uppi}\xspace}^{+}}\xspace})$, ${\ensuremath{\Xi\kern-0.1em/{\ensuremath{\kern-0.09em}}{\ensuremath{\uppi}\xspace}}\xspace}\equiv(\Xi^{-}+\overline{\Xi}^{+})/({\ensuremath{{\ensuremath{\uppi}\xspace}^{-}}\xspace}+{\ensuremath{{\ensuremath{\uppi}\xspace}^{+}}\xspace})$, and ${\ensuremath{\Omega\kern-0.05em/{\ensuremath{\kern-0.09em}}{\ensuremath{\uppi}\xspace}}\xspace}\equiv(\Omega^{-}+\overline{\Omega}^{+})/({\ensuremath{{\ensuremath{\uppi}\xspace}^{-}}\xspace}+{\ensuremath{{\ensuremath{\uppi}\xspace}^{+}}\xspace})$. For collisions at , ${\ensuremath{\Lambda/{\ensuremath{\kern-0.09em}}{\ensuremath{\uppi}\xspace}}\xspace}\equiv 2\Lambda/({\ensuremath{{\ensuremath{\uppi}\xspace}^{-}}\xspace}+{\ensuremath{{\ensuremath{\uppi}\xspace}^{+}}\xspace})$, but otherwise ${\ensuremath{\Lambda/{\ensuremath{\kern-0.09em}}{\ensuremath{\uppi}\xspace}}\xspace}\equiv(\Lambda+\overline{\Lambda})/({\ensuremath{{\ensuremath{\uppi}\xspace}^{-}}\xspace}+{\ensuremath{{\ensuremath{\uppi}\xspace}^{+}}\xspace})$. While the [$\Lambda/{\ensuremath{\kern-0.09em}}{\ensuremath{\uppi}\xspace}$]{}, [$\Xi\kern-0.1em/{\ensuremath{\kern-0.09em}}{\ensuremath{\uppi}\xspace}$]{}, and [$\Omega\kern-0.05em/{\ensuremath{\kern-0.09em}}{\ensuremath{\uppi}\xspace}$]{}ratios in pp collisions are higher at LHC energies than at RHIC energies, the [${\ensuremath{\upphi}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}{\ensuremath{\uppi}\xspace}$]{}ratio in pp collisions does not exhibit a significant change from [$\sqrt{s}=$ 200 GeV]{} to 7 TeV. Relative to pp collisions, strangeness production in collisions at at first increases with centrality and appears to saturate for ${\ensuremath{\langle N_{\mathrm{part}}\rangle}\xspace}\gtrsim 100$. A decrease in the [${\ensuremath{\upphi}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}{\ensuremath{\uppi}\xspace}$]{}, [$\Lambda/{\ensuremath{\kern-0.09em}}{\ensuremath{\uppi}\xspace}$]{}, and [$\Xi\kern-0.1em/{\ensuremath{\kern-0.09em}}{\ensuremath{\uppi}\xspace}$]{}ratios for the 0-20% most central collisions (${\ensuremath{\langle N_{\mathrm{part}}\rangle}\xspace}\gtrsim 230$) may be present; however, the trend is flat within systematic uncertainties. The increase in these ratios from pp to central collisions at LHC energies is $\sim 3.3$ for [$\Omega\kern-0.05em/{\ensuremath{\kern-0.09em}}{\ensuremath{\uppi}\xspace}$]{}, $\sim 1.6$ for [$\Xi\kern-0.1em/{\ensuremath{\kern-0.09em}}{\ensuremath{\uppi}\xspace}$]{}, $\sim 1.2$ for [$\Lambda/{\ensuremath{\kern-0.09em}}{\ensuremath{\uppi}\xspace}$]{}, and $\sim 1.4$ for [${\ensuremath{\upphi}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}{\ensuremath{\uppi}\xspace}$]{}. These values are about one half of the enhancement ratios discussed above. The fractional increase in the [${\ensuremath{\upphi}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}{\ensuremath{\uppi}\xspace}$]{}ratio is similar to the increases observed in both the [$\Lambda/{\ensuremath{\kern-0.09em}}{\ensuremath{\uppi}\xspace}$]{}and [$\Xi\kern-0.1em/{\ensuremath{\kern-0.09em}}{\ensuremath{\uppi}\xspace}$]{}ratios, a trend which is also observed in the standard enhancement ratios presented in the previous paragraph. At SPS energies, a study of the [${\ensuremath{\upphi}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}{\ensuremath{\uppi}\xspace}$]{}, $\mathrm{K}\kern-0.05em/{\ensuremath{\kern-0.09em}}{\ensuremath{\uppi}\xspace}$, and $(\mathrm{K}\kern-0.05em/{\ensuremath{\kern-0.09em}}{\ensuremath{\uppi}\xspace})^{2}$ ratios suggests that the [$\upphi$]{}behaves as a particle with an effective strangeness quantum number between 1 and 2 [@Kraus_SPS_2007]. The values of the [${\ensuremath{\upphi}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}{\ensuremath{\uppi}\xspace}$]{}, [$\Lambda/{\ensuremath{\kern-0.09em}}{\ensuremath{\uppi}\xspace}$]{}, [$\Xi\kern-0.1em/{\ensuremath{\kern-0.09em}}{\ensuremath{\uppi}\xspace}$]{}, and [$\Omega\kern-0.05em/{\ensuremath{\kern-0.09em}}{\ensuremath{\uppi}\xspace}$]{}ratios obtained from grand-canonical thermal models with temperatures of 170 MeV [@Cleymans_2006] (upper dashed lines), 164 MeV [@Andronic2009; @Andronic2009_Erratum] (solid lines), and 156 MeV [@Stachel_SQM2013] (lower dashed lines) are also shown. It should be noted that the model using a temperature of 164 MeV gives a [$\mathrm{p}\kern-0.05em/{\ensuremath{\kern-0.09em}}{\ensuremath{\uppi}\xspace}$]{}ratio that is about 50% greater than the measured value.
Conclusions\[sec:conclusions\]
==============================
The [$p_{\mathrm{T}}$]{}distributions, masses, and widths of [$\mathrm{K^{*0}}$]{}and [$\upphi$]{}mesons have been measured at mid-rapidity ($|y|<0.5$) in collisions at using the ALICE detector. The masses and widths of these resonances, reconstructed via their hadronic decays, are consistent with the vacuum values. The measured [$\langle{\ensuremath{p_{\mathrm{T}}}\xspace}\rangle$]{}is 15-20% higher in central collisions than in peripheral collisions and it is found to be higher in collisions at LHC energies than at RHIC energies. This suggests stronger radial flow at the LHC, which has also been concluded based on previous measurements of pion, kaon, and proton [$p_{\mathrm{T}}$]{}distributions. Relative to the yields of charged kaons, the total yield ([${\ensuremath{\mathrm{d}}}N{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}{\ensuremath{\mathrm{d}}}y$]{}) of [$\mathrm{K^{*0}}$]{}is observed to be suppressed in central collisions. When plotted as a function of [$({\ensuremath{{\ensuremath{\mathrm{d}}}N_{\mathrm{ch}}\kern-0.06em /\kern-0.13em{\ensuremath{\mathrm{d}}}\eta}\xspace})^{1/3}$]{}, the [${\ensuremath{\mathrm{K^{*0}}}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}ratio appears to follow a single trend for both RHIC and LHC energies and for different collision systems. In contrast, no suppression is observed for the [$\upphi$]{}. When the [$p_{\mathrm{T}}$]{}distributions of the [$\mathrm{K^{*0}}$]{}and [$\upphi$]{}mesons are compared to expected distributions based on the blast-wave model (using parameters taken from fits to other hadrons), [$\mathrm{K^{*0}}$]{}suppression is observed in central collisions for transverse momenta ${\ensuremath{p_{\mathrm{T}}}\xspace}<3$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}. The suppression of the integrated [$\mathrm{K^{*0}}$]{}yield might be taken to suggest that re-scattering of resonance decay products in the hadronic phase reduces the measurable yield of [$\mathrm{K^{*0}}$]{}mesons. However, it is unclear if such a scenario can fully explain the observed [$p_{\mathrm{T}}$]{}dependence of the [$\mathrm{K^{*0}}$]{}suppression or the absence of broadening in its invariant-mass distribution. The lack of suppression for the [$\upphi$]{}meson could indicate that this particle decays outside the fireball due to its longer lifetime. The measured [${\ensuremath{\mathrm{K^{*0}}}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}ratio is compared to an extended thermal-model prediction [@Markert_thermal; @Torrieri_thermal; @Torrieri_thermal_2001b; @Torrieri_thermal_2001b_erratum] that includes re-scattering effects. By assuming a chemical freeze-out temperature of 156 MeV, a model-dependent estimate of 2 fm/$c$ as the lower limit of the time between the chemical and kinetic freeze-out is extracted. The measurement of at least one more resonance-to-stable ratio \[such as $\Lambda(1520){\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}\Lambda$\] will allow both the lifetime of the hadronic phase and the chemical freeze-out temperature to be estimated simultaneously within the framework of this model. At LHC energies the [$\upphi$]{}, which has hidden strangeness, is enhanced by an amount similar to particles with one or two units of open strangeness. While a hydrodynamic framework can roughly describe the measured particle [$p_{\mathrm{T}}$]{}distributions in collisions at , inconsistencies nevertheless remain. For central collisions the [$\mathrm{p}\kern-0.1em/{\ensuremath{\kern-0.18em}}{\ensuremath{\upphi}\xspace}$]{}ratio is flat as a function of transverse momentum for ${\ensuremath{p_{\mathrm{T}}}\xspace}\lesssim 3$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c$]{}. This is consistent with hydrodynamic models, thereby suggesting that mass and hence radial flow plays a dominant role in the determination of the shapes of [$p_{\mathrm{T}}$]{}distributions at low and intermediate [$p_{\mathrm{T}}$]{}. Models based on hydrodynamics (Kraków, HKM, and VISH2+1) are able to reproduce the shape of the [$\upphi$]{}meson [$p_{\mathrm{T}}$]{}distribution fairly well, but overestimate the [$\upphi$]{}yield. These models describe the [$p_{\mathrm{T}}$]{}distributions of other particles, such as [$\uppi$]{}, K, and protons, reasonably well, but they encounter difficulties in describing the [$p_{\mathrm{T}}$]{}distribution of the $\Omega$ and the [$\Omega\kern-0.05em/{\ensuremath{\kern-0.09em}}{\ensuremath{\upphi}\xspace}$]{}ratio.
Acknowledgements {#acknowledgements .unnumbered}
================
The ALICE Collaboration {#app:collab}
=======================
[^1]: The signal-to-background ratio is evaluated by comparing the integrals of the signal and background over the ranges $0.77<{\ensuremath{m_{{\ensuremath{\uppi}\xspace}\mathrm{K}}}\xspace}<1.02$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c^{2}$]{}for the [$\mathrm{K^{*0}}$]{}and $1.01<{\ensuremath{m_{\mathrm{KK}}}\xspace}<1.03$ [$\mathrm{GeV}{\ensuremath{\kern-0.18em}}/{\ensuremath{\kern-0.09em}}c^{2}$]{}for the [$\upphi$]{}.
[^2]: The choice of fitting functions for the two resonances is driven by the different widths. The [$\mathrm{K^{*0}}$]{}has a width much larger than the resolution; therefore, a Voigtian fit is not necessary. However, since the [$\mathrm{K^{*0}}$]{}is broad enough, its shape may be influenced by phase-space effects. The [$\upphi$]{}has a width of the same order of magnitude as the resolution and phase-space effects can be neglected.
[^3]: No centrality dependence is observed for the [$\upphi$]{}width in the simulation, so the average width for centrality 0-80% is plotted in Fig. \[fig:results:mass\_width\](d).
[^4]: This calculation excludes the tracking/track selection and material budget systematic uncertainties, which are assumed to be correlated between centrality intervals.
[^5]: For collisions [@STAR_resonances_dAu_2008] at the ratio $\mathrm{K}^{*}{\ensuremath{\kern-0.18em}}/\mathrm{K}^{-}$ is plotted instead, where the yield in the numerator is calculated from a combination of all four $\mathrm{K}^{*}{\ensuremath{\kern-0.09em}}(892)$ states.
[^6]: For collisions at , the [${\ensuremath{\upphi}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}ratio measured by the PHENIX Collaboration [@PHENIX_phi_AuAu_2005] is $\sim40\%$ less than (and not consistent with) the [${\ensuremath{\upphi}\xspace}{\ensuremath{\kern-0.18em}}/{\ensuremath{\mathrm{K}^{-}}\xspace}$]{}ratio measured by the STAR Collaboration [@STAR_phi_2009]. A possible explanation for this discrepancy is discussed in [@Rafelski_2005].
[^7]: The prediction was calculated for [$\sqrt{s}=$ 5 TeV]{}.
[^8]: The values of the [$\mathrm{p}\kern-0.1em/{\ensuremath{\kern-0.18em}}{\ensuremath{\upphi}\xspace}$]{}ratio for the 10-20% and 40-60% centrality intervals, which are not shown here, are available in the Durham Reaction Database.
[^9]: Reference yields measured in p–Be collisions have also been used [@NA57_SE_40AGeV; @NA57_SE_158AGeV].
|
---
abstract: |
Current-voltage characteristics of a system with a variable number of phase-slip centers resulting from phase separation in a tin whisker under external microwave field with a frequency $\Omega /2
\pi \simeq 35-45$ GHz have been studied experimentally. Emergence and disappearance of steps with zero slope in a whisker’s current-voltage characteristic at $U_{m/n}=(m/n)U_{\Omega}$, where $m$ and $n$ are integers and $U_{\Omega}$ is determined by Josephson’s formula $\hbar \Omega=2eU_{\Omega}$, have been investigated. Microwave field generated by phase-slip centers is nonharmonic, and the system of phase-slip centers permits synchronization of internal oscillations at a microwave frequency by an external field with a frequency which is the $n$-th harmonic of internal oscillations. The estimated microwave power generated by a whisker is $10^{-8}$ W. Stimulation of superconductivity in a current-carrying whisker has been detected.
author:
- 'V. I. Kuznetsov'
- 'V. A. Tulin'
title: 'Synchronization of high-frequency oscillations of phase-slip centers in a tin whisker under microwave radiation'
---
\[sec:level1\]INTRODUCTION
==========================
Microwave generation in a Josephson junction (a weak-coupling element in a superconducting circuit) under a dc voltage has attracted researchers’ attention since the time when the ac Josephson effect was discovered. The simple structure of the experimental device and easy control of the generated frequency are the most attractive features of the effect. The frequency generated by the junction is determined by the formula $$\omega=2Ue/ \hbar \;, \nonumber$$ where $U$ is the voltage drop across the junction, $e$ is the electron charge, and $\hbar$ is the Planck constant. The disadvantages of these devices are their low output and difficulties in matching the superconducting circuits containing Josephson junctions to the microwave circuits. Attempts have been made to overcome these difficulties using circuits of short junctions [@c1; @c2; @c3; @c4].
Josephson junctions have a typical linear size in the direction perpendicular to the supercurrent density vector, namely the Josephson penetration depth $\lambda_{j}$. If the junction dimension in the direction perpendicular to the supercurrent satisfies the condition $d<\lambda_{j}$ (a short junction), the phase variation is uniform over the junction volume, and one has a single source of microwave radiation. In the case of a network of short synchronized junctions, it seems possible to derive a high microwave output close to the sum of powers generated by each element.
A long uniform superconducting channel with phase-slip centers can be classified with such systems. Phase-slip centers occur in resistive states of a long narrow channel carrying a constant current at a temperature close to the superconducting transition ($I>I_{c}$, $T<T_{c}$) [@c5; @c6]. Real structures in which phase-slip centers have been detected are thin films with width $w$ and single-crystal wires (whiskers) with diameter $d$ smaller than the superconductor coherence length $\xi$. From the viewpoint of experimenters dealing with superconducting channels, whiskers (thin crystalline wires) are preferable because their uniformity over the length is higher. But thin films have some advantages when applications are concerned, since their dimensions are directly controlled during their manufacture. On the other hand, microscopic inhomogeneities due to fabrication technologies can lead to considerable degradation of parameters of phase-slip centers, and a thin film may behave like a system of weak superconducting bounds localized along the narrow film.
An isolated phase-slip center is an nonstationary, inhomogeneous entity “localized in the space” and containing an internal region with a size of about $\xi$ where the superconducting order parameter oscillates at the Josephson frequency $$\omega=2U_{\Omega} e/ \hbar \;. \nonumber$$ At temperatures near the transition point, the voltage averaged over the oscillation period, $U_{\omega}$, in the phase-slip center is due to penetration of a non-uniform longitudinal electric field into the outer region of the center through a distance of about $l_{E}$ (the electric field penetration range), and the electric resistance of each phase-slip center is $$R_{0}=2\rho_{N}l_{E}/S \;, \nonumber$$ where $\rho_{N}$ is the material resistivity in the normal state and $S$ is the channel cross section [@c5; @c6; @c7; @c8]. At the moment when the absolute value of the order parameter vanishes, the phase difference over the center jumps by $2 \pi$. Current-voltage characteristics (CVC) of such superconducting channels contain a set of sloped linear sections corresponding to resistances $$R_{n}=nR_{0} \;, \nonumber$$ (where $n$ is an integer) connected by sections of curves with current jumps. Extrapolations of these linear sections cross the current axis at approximately the same point $I_{0}$ (an excess current) [@c5; @c6; @c7; @c9].
Although the number of publications dedicated to phase-slip centers is fairly large [@c5; @c6], the dynamics of systems with phase-slip centers has been studied insufficiently [@c7; @c8; @c10; @c11; @c12]. The reversed ac Josephson effect under external electromagnetic radiation was detected in thin tin films at a frequency of 10 GHz [@c7] and in single-crystal wires (whiskers) at frequencies of up to 900 MHz [@c13; @c14; @c15]. In both these cases, a CVC contains, in addition to sloping steps, a fundamental step with a zero slope at voltage $U_{\Omega}$ in the region of parameters corresponding to one phase-slip center and associated with high-frequency oscillations of the order parameter in the center, and “weak” steps at $$U_{m/n}=(m/n)U_{\Omega} \;,$$ where $U_{\Omega}$ is the voltage corresponding to the external field frequency and $m$ and $n$ are integers. Ivlev and Kopnin [@c16] analyzed the ac Josephson effect in terms of the microscopic theory. The pattern of various zero-slope steps at different direct currents and microwave frequencies in systems with variable numbers of phase-slip centers has not been investigated in full. A current-carrying whisker under an electromagnetic field with a frequency higher than 900 MHz has never been studied.
SAMPLES AND EXPERIMENTAL DETAILS
================================
In the reported work, we have studied the effect of microwave fields with frequencies ranging between 35 and 45 GHz on CVCs of tin whiskers in the regime when several phase-slip centers exist in a sample at voltages of order of $U_{\Omega}$. In previous experiments [@c7; @c13; @c14; @c15] the parameters $T$ and $\Omega/2
\pi$ were selected so that the mean voltage $U_{\omega}$ across one center, which determined the frequency of proper high-frequency oscillations, $$\omega=2Ue/ \hbar \;, \nonumber$$ could be tuned to $U_{\Omega}$, i.e., the frequency $\omega$ of internal oscillations should be equal to that of applied microwave field. We have used higher microwave frequencies $\Omega /2 \pi$ and temperatures at a greater distance from $T_{c}$ than Tidecks et al. [@c13; @c14; @c15] so that to satisfy the condition $$U_{\omega}=U_{\Omega}/n \;, \nonumber$$ i.e., $n \omega=\Omega$ ($n>1$) when a sample contained several phase-slip centers at voltages about $U_{\Omega}$. It follows from the microscopic theory [@c16] that this is the condition under which induced steps on a CVC are generated at voltages $U_{\Omega}/n$. The presence of such steps means that the radiation generated by the system of phase-slip centers is nonharmonic. Given the higher uniformity of whiskers over their lengths and smaller number of structural defects than in films, they are preferable for such experiments. Moreover, zero-slope steps on a CVC of an irradiated whisker [@c13; @c14; @c15] are considerably wider than in narrow films [@c7]. In many experiments (see for example Ref. [@c17]), low-frequency oscillations instead of high-frequency oscillations were detected in narrow films. Whiskers grown from thin tin films deposited on silicon substrates had diameters $d=0.2-0.8$ $\mu$m, lengths of about 1 mm, resistance ratio $R_{300} /R_{4.2}<100$, and $T_{c}
\approx 3.1$ K. A whisker was set across a 300-$\mu$m gap in a thin tin film about 1000 [Å]{} thick. A whisker was attached to electrodes by electrostatic forces at the initial moment, then, apparently, by the Van der Waals forces. It is not easy to remove a whisker from the substrate surface. The heat-sinking conditions, probably, were fairly good because the greater part of the sample was in contact with the polished substrate surface, therefore measured CVCs did not exhibit a notable hysteresis in the studied temperature range, unlike CVCs reported in Refs. [@c13; @c14; @c15]. CVCs were measured using either the two-terminal configuration (this was possible because $T_{c}$ of films was higher than that of whiskers) or the four-terminal configuration. The substrate supporting the whisker was placed in a copper waveguide and insulated from environment by a superconducting lead shield. The curves of the critical current and resistance versus temperature for the case of a single phase-slip center at $T_{C}-T<10$ mK had shapes typical of whiskers [@c13]: $I_{c} \sim (1-T/T_{c})^{3/2}$, $R_{0} \sim
(1-T/T_{c})^{-1/4}$ .
EXPERIMENTAL RESULTS
====================
Current-voltage characteristics of all samples are piece-wise linear, i.e., they are composed of linear sections connected by nonlinear sections with larger slopes.
![\[fig:1\] CVC of the Sn3 whisker ($R_{0} \approx
0.19$ $\Omega$, $T_{c} \approx 3.71$ K, $R_{300}/R_{4.2} \approx
73$) without irradiation by an external microwave field at $T
\approx 3.56$ K. The insert shows the CVC of the Sn2 sample ($R_{0} \approx 0.21$ $\Omega$, $T_{c} \approx 3.72$ K, $d \approx
0.8$ $\mu$m, $R_{300}/R_{4.2} \approx 50$) without irradiation at $T \approx 3.63$ K.](jtpfig1.eps){width="1.0\linewidth"}
![\[fig:2\] Set of CVCs of the Sn1 sample ($R_{0}=0.79-0.63$ $\Omega$, $T_{c} \approx 3.69$ K, $d \approx
0.3$ $\mu$m, $R_{300}/R_{4.2} \approx 20$) at different powers of external microwave irradiation at frequency $\Omega /2 \pi=40.62$ GHz and $T \approx 3.62$ K: (a) 70 dB (dashed line); (b) 36 dB; (c) 31 dB, (d) 30 dB, (e) 28.6 dB. The insert shows low-current sections of CVCs of the Sn1 sample at approximately equal parameters in another cycle of measurements: (a) 32.6 dB; (b) 30.6 dB; (c) 29 dB; (d) 28 dB; (e) 25.2 dB.](jtpfig2.eps){width="1.0\linewidth"}
Figure 1 shows the examples of CVCs of superconducting whiskers with microwave radiation off. The initial parts of the whisker CVCs without microwaves are also shown in Figs. 2-4 by dashed lines. The numbers near the linear sections of the whisker CVCs indicate the ratios between their resistances and that of a single phase-slip center, $R_{0}$. The latter parameter was determined as the largest common divisor of differential resistance values of all linear CVC sections and compared to an estimate derived from the size and resistivity of the whisker. The CVC linear sections are connected by nonlinear sections, which are reproducible and reversible in the range of studied frequencies. Note that in most experiments, the initial CVC sections at $I>I_{c}$ (curves (a) in Figs. 2 and 3) without radiation are nonlinear, and the first reproducible linear sections correspond to states with several phase-slip centers (the linear section $3R_{0}$ on curve (a) in Fig. 2 and $5R_{0}$ on curve (a) of Fig. 3). In earlier experiments [@c13; @c14; @c15], the states with one phase-slip center could be regularly produced. In contrast to those experiments, where the temperature difference $T_{c}-T$ was less than 10 mK, we measured CVCs mostly at temperatures 70-160 mK below $T_{c}$. In this case, states with several phase-slip centers were stable at notably larger temperature differences $T_{c}-T$. This can be seen by comparing the CVC shown in Fig. 1 with the CVC in the insert to this graph. Moreover, the linear sections with the same resistance (such as $3R_{0}$ sections on curve (a) in Fig. 2 for Sn1 and $2R_{0}$ in curve (d) in Fig. 4 for Sn3) separated by voltage jumps were recorded many times. On the basis of these observations, we have come to the conclusion that, in spite of some complications in interpreting CVCs of our whiskers, they are superconducting channels with phase-slip centers at appropriate temperatures and transport currents.
When samples are exposed to microwave radiation, their CVCs contain, in addition to linear sloping sections due to the presence of certain numbers of phase-slip centers, steps with a zero slope at voltages $$U_{m/n}=(m/n)U_{\Omega} \;. \nonumber$$
![\[fig:3\] Set of CVCs of the Sn2 sample ($R_{0}=0.23-0.18$ $\Omega$, $T_{c} \approx 3.72$ K, $d \approx
0.8$ $\mu$m, $R_{300}/R_{4.2} \approx 50$) at different powers of microwaves at frequency $\Omega /2 \pi = 40.62$ GHz at $T \approx
3.63$ K: (a) 70 dB (dashed line); (b) 30.2 dB; (c) 30 dB; (d) 28.2 dB; (e) 26.1 dB.](jtpfig3.eps){width="1.0\linewidth"}
At low microwave powers, the channel critical current was higher, i.e., stimulation of superconductivity theoretically described by Eliashberg [@c18] took place. Instead of the emergence of the zero-slope step first at $U_{\Omega}$ [@c13; @c14; @c15], we observed the sequential appearance of steps at $2U_{\Omega}$, $U_{\Omega}$, $3U_{\Omega}$, and $U_{\Omega}/2$ for a sample Sn1 (Fig. 2), and in Sn2 (Fig. 3) we first observed a step at $U_{\Omega}/2$ and then at $U_{\Omega}$ (not shown in the graph). At lower temperatures the unusual shapes of the CVCs at zero radiation intensity with linear sections of the same slope (curve (d) in Fig. 4) or the lowest linear sections corresponding to several phase-slip centers were replaced by more common CVC shapes. The sequence of microwave-induced steps in whiskers’ CVCs emerging with increasing microwave power also became more like the usual sequence at lower temperatures, namely, the step at $U_{\Omega}$ was detected first, then the step at $2U_{\Omega}$, and at still higher microwave power at $U_{\Omega}/2$ (Fig. 4). The curve became similar to those given in Refs. [@c13; @c14; @c15]. As the microwave power increased, the sloping linear sections due to the phase-slip centers became more pronounced on CVCs (Figs. 2 and 3).
Steps with zero slope emerge on linear sections of CVCs, which either exist in the samples not exposed to microwaves or appear in the samples irradiated by the microwave field. For example, the step of zero slope on the CVC of the Sn1 whisker at $U_{\Omega}$ (insert to Fig. 2) appears after the emergence of a linear section on the curve, its growth, and the shift of its lower edge to the required voltage (curves (a), (b), and (c) in the insert to Fig. 2). As soon as the edge of the linear section achieves $U_{\Omega}$, a zero-slope step is produced (curve (d), and its width increases with the microwave power curve (e)). The steps at $U_{\Omega}/2$ (curves (c), (d), and (e) in Fig. 2) emerge in a similar manner. The step at $3U_{\Omega}$ (curve (c) in Fig. 2) appears when the sloping linear section with differential resistance $3R_{0}$ extends to this region. A zero-slope step can disappear at a higher microwave power (for example, the steps at $3U_{\Omega}$, $U_{\Omega}$, and $U_{\Omega}/2$) when the upper edge of the linear section shifts below the respective voltage, and a vertical CVC section moves to this region. In this case, the differential resistance of the linear section can have a jump (curves (d) and (e) in Fig. 2), namely, the linear section at about $3U_{\Omega}$ changed its factor from 3 to 2.
![\[fig:5\] Normalized critical current $I_{c}/I_{m}$ versus the relative amplitude of external microwave field at frequency $\Omega /2 \pi=40.62$ GHz at $T \approx 3.62$ K for the Sn1 whisker in two different cycles of measurements (full squares are the data of the first cycle and empty squares correspond to the second cycle), $I_{m} \approx 107$ $\mu$A. Normalized widths of zero-slope steps on a CVC as functions of relative microwave field amplitude at voltage $U_{\Omega}$ (crosses plot data of the second cycle) and at voltage $2U_{\Omega}$ (full circles are the data of the first cycle and empty circles correspond to the second cycle). The dashed, dash-dotted, and solid lines show absolute values of Bessel functions $J_{0}(x)$, $J_{1}(x)$, and $J_{2}(x)$, respectively; $x=100(P/P_{m})^{1/2}$, $P$ is the power, and $P_{m}$ is the maximal output of the microwave generator.](jtpfig5.eps){width="1.0\linewidth"}
Thus, a linear section on a CVC of a sample with or without microwave pumping at $U_{m/n}$ is a necessary condition for formation of a zero-slope step, i.e., for the existence of the required number of phase-slip centers in the sample.
By tuning the incident microwave frequency $\Omega$, we could detect zero-slope steps not observed previously when voltage $U_{m/n}$ coincided with a linear section of a CVC recorded without irradiation.
Sloping linear sections in a CVC of a whisker containing a certain number of phase-slip centers and exposed to microwaves of a fixed power could decrease their resistance factor with respect to the resistance of an isolated phase-slip center if the direct transport current increased (see curve (b) in Fig. 2). The resistance factor could also remain unchanged (curve (e) in Fig. 2, section 2). An increase in the incident microwave power could cause, in addition to suppression of both the critical and excess current at a fixed voltage, a switch-over to a linear section with a lower differential resistance. On the curves in Fig. 2, the resistance factor dropped from four to two, and in Fig. 3 from five to two. The CVCs of the Sn2 whisker (Fig. 3) initially contained a linear section with resistance $5R_{0}$ at voltages above $U_{\Omega}$, and under microwave irradiation this parameter dropped to $3R_{0}$ and then $2R_{0}$. At higher microwave powers the length of the $2R_{0}$ section increased at a constant resistance factor. Note that $R_{0}$ could vary under microwave radiation within 20% . Thus, microwaves not only produce horizontal steps on CVCs, but also strongly affect CVCs of tin whiskers.
We have also measured the widths of microwave induced steps as functions of the incident power over the interval of their existence. The experimental dependencies of current-normalized widths of zero-slope steps at voltages $U_{\Omega}$ and $2U_{\Omega}$, and of the critical current for a sample Sn1 obtained in different measurement cycles at approximately equal parameters as functions of the relative microwave amplitude are given in Fig. 5. The graph also shows as an illustration the absolute values of Bessel functions of order 0, 1, and 2 ($J_{0}(x)$, $J_{1}(x)$, and $J_{2}(x)$) although we believe that the experimental curves are not directly related to these functions. Note the main features of the curves in Fig. 5. (1) The microwave stimulation of superconductivity led to an increase in the critical current of about 20% . (2) The zero-slope step at $2U_{\Omega}$ emerged at a lower microwave power and had the maximum width of about $0.5I_{c}$. (3) The step at $U_{\Omega}$ observed in the second cycle of measurements (it was too small in the first cycle and its width is not shown in Fig. 5) appeared at a higher microwave power, and in its presence the width of the $2U_{\Omega}$ step and the critical current as functions of the microwave field amplitude changed considerably. In this case the critical current and width of the $2U_{\Omega}$ step vanished at a notably higher microwave field amplitude than in the first cycle. (4) Induced $U_{\Omega}$ and $2U_{\Omega}$ steps appeared at a finite microwave power, i.e., there is a certain threshold microwave power needed for formation of these steps. This threshold is related to the extension of the linear CVC sections to voltages $U_{\Omega}$ and $2U_{\Omega}$. (5) There is only one interval of the microwave field amplitude on which the critical current and CVC steps exist. No oscillations have been detected on the curves of critical current and step width.
In studying step widths as functions of the microwave power, we recorded (in several cases) nonmonotonic curves with relatively narrow down-peaks against the background of wide bell-shaped curves.
DISCUSSION OF RESULTS
=====================
The current-voltage characteristic of a uniform superconducting channel, which is our model for a whisker, depends on its length. In the case of a short whisker section through which current is fed, $l \approx l_{E}$, the presence of one phase-slip center allows the sample to conduct a current higher than the critical value. If $l \gg l_{E}$, the exponentially decaying parameters of phase-slip centers have little effect on the channel properties, therefore it should contain several phase-slip centers, whose number is determined by the channel length. In our samples, the condition $l \gg l_{E}$ was satisfied ($l \approx (10-20)l_{E}$), therefore we assume that several phase-slip centers were necessary to conduct a current slightly higher than the critical value. In the process of generation of the required number of centers, the instantaneous number of centers can be unstable and variable in both time and space.
The CVCs of our samples have piecewise linear shapes with sections characterized by differential resistance $R=nR_{0}$, where $n$ is an integer. These sections correspond to definite numbers of phase-slip centers, which can be derived from the sample sizes. In addition, there are the nonlinear sections on which the number of centers is probably unstable and varies with time. A dedicated investigation is needed to verify this hypothesis. The linear sections of CVCs of the superconducting channel in the simplified model [@c7] are described by the formula $$U=nR_{0}(I-I_{0}) \;. \nonumber$$ The excess current $I_{0}$ is usually related to the average superconducting component of the total current. This formula is not universal for all linear sections, because $I_{0} \neq const$ for all groups of linear sections [@c9]. The CVCs of our samples contain neighboring linear sections with equal $n$ but different $I_{0}$.
Microwave irradiation of our samples has a dual effect on their CVCs. The first effect is the generation of constant voltage steps, which was the main subject of the reported study. The second effect is the change in the number of phase-slip centers under microwave radiation and stabilization of CVC sections with definite numbers of these centers. This shows up in the extension of linear sections and transformation of some nonlinear CVC portion to linear.
The existence of constant-voltage steps under microwave radiation indicates that there are currents of microwave frequencies with spectral components $$\omega=2enU_{\omega}/\hbar \;,
n=1,2,3,...$$ in the regions of phase-slip centers. When the external frequency equals that of one of these harmonics, several centers are synchronized, which shows up in the form of constant-voltage steps at $$U_{m}= mU_{\omega} \;,$$ where $m$ is the number of phase-slip centers, $$U_{\omega} = \hbar \Omega /2en \;,$$ and $\Omega$ is the external radiation frequency.
![ \[fig:6\] Low-current sections of CVCs of the Sn2 sample at different microwave field powers at frequency $\Omega /2
\pi=40.62$ GHz in the second cycle of measurements at $T \approx
3.60$ K: (a) 70 dB; (b) 24.49 dB; (c) 22 dB.](jtpfig6.eps){width="1.0\linewidth"}
As a result, steps can occur at $$U_{m/n}=(m/n) \hbar \Omega /2e$$ if this voltage coincides with an inherent or microwave-induced linear section of CVC with a definite (integral) number of phase-slip centers.
Unfortunately, it is difficult to determine $m$ and $n$ with certainty using CVCs. Linear sections from which the number of centers could be exactly determined could be seen near constant-voltage steps only at certain values of parameters. We believe that the step at $U_{\Omega}/2$ in Fig. 2 is due to the synchronization of two phase-slip centers by the fourth harmonic of proper oscillations, i.e., $U_{\Omega}/2 \to 2U_{\Omega}/4$, similarly $U_{\Omega} \to 4U_{\Omega}/4$, $2U_{\Omega} \to
4U_{\Omega}/2$, 3$U_{\Omega} \to 6U_{\Omega}/2$, the step at $4U_{\Omega} \to 8U_{\Omega}/2$ is not shown; in Fig. 4 $U_{\Omega}/2 \to 2U_{\Omega}/4$, $U_{\Omega} \to 2U_{\Omega}/2$, and $2U_{\Omega} \to 4U_{\Omega}/2$; in Fig. 3, $U_{\Omega}/2 \to
2U_{\Omega}/4$; in Fig. 6, $U_{\Omega}/2 \to 3U_{\Omega}/6$, $2U_{\Omega}/3 \to 4U_{\Omega}/6$, and $U_{\Omega} \to 6U_{\Omega}
/6$. At other values of parameters this sample demonstrated steps at $5U_{\Omega}/6 \to 5U_{\Omega}/6$ and $U_{\Omega}/3 \to
2U_{\Omega}/6$ (not shown in the graphs of this paper).
Thus, at certain positions of these voltages in the whisker CVCs, microwave field synchronizes oscillations of the order parameter in all phase-slip centers present in a sample, which results in constant voltage drops across isolated centers and across the entire sample. States with synchronized phase-slip centers under microwave radiation emerge predominantly at corresponding locations in the CVCs. Other CVC sections may correspond to states in which some phase-slip centers are synchronized by external field and the rest are not. This conjecture allows us to interpret the drop in the differential resistance of linear sections (and the behavior of the differential resistance in general) when the current increases under microwave radiation. The existence of neighboring sloping steps with equal resistance but different excess current can also be interpreted in these terms. A similar effect without microwave radiation can be attributed to a different but, in a sense, similar phenomenon. So-called Fiske steps [@c19] were detected in experiments with tunneling Josephson junctions when the frequency generated by the junction was locked to the resonant frequency of the structural cavity in the experimental device. In this case, constant-voltage steps determined by the Josephson formula with the resonant cavity frequency could be seen on CVCs. The gap in the tin film on which the whisker was mounted could act as a structural resonator. The length of this gap was about 5 mm, and, given the silicon substrate dielectric constant ($\varepsilon \approx 12$), we have a resonant frequency in the studied microwave band. In this case, a section with a constant voltage due to synchronization of a group of phase-slip centers (Fiske step) can occur. The centers whose oscillations are not locked to the resonant frequency should demonstrate a linear behavior. As a result, the CVC of the sample should have a linear section with the resistance corresponding to the number of unlocked centers, which is smaller than the total number. The question why horizontal steps have not been observed remains unanswered. Doubts in this interpretation could be eliminated by directly measuring microwaves generated in the sample.
Figure 5 shows the widths of constant-voltage steps as functions of the microwave field amplitude in relative units. The maximal width of these steps allows us to estimate the microwave power generated by the whisker: $$P \approx (\Delta I)^{2}mR_{0} \;, \nonumber$$ where $\Delta I$ is the step width in terms of current. Hence, $P\approx 10^{-8}$ W.
The microwave generation in the phase-slip centers can be interpreted in terms of the order parameter versus time, which vanishes at some moment and then increases to some value. At the moment when the order parameter is zero, the difference between the phases on different sides of the phase-slip center drops by $2\pi$. It would be interesting to estimate the times of these processes and compare their reciprocal values with the frequencies of the order parameter oscillations and external radiation. The most important parameter is the time $\tau_{\Delta}$ in which the order parameter recovers. When $\tau_{\Delta}$ is much longer than the order parameter oscillation period determined by the Josephson formula, both the mean and instantaneous absolute values of the order parameter within the center are much smaller than the equilibrium value in other regions of the superconducting channel. If $\tau_{\Delta}$ is comparable to or smaller than the period of the order parameter oscillations, the instantaneous value of the gap in the phase-slip center can be large and comparable to the gap in the surrounding regions. The spectra of normal excitations in phase-slip centers should be notably different in these two cases, which can lead to differences in some electrical properties of phase-slip centers. Since the energy relaxation time of current carriers in tin is $3 \times 10^{-10}$ s, the first case is realized in the microwave frequency band.
The behavior of the step width is determined by two factors. The first is the width of the step against the background of an infinite linear CVC section with a definite number of phase-slip centers as a function of the microwave field amplitude. The second is the limitation of the constant-voltage step by the length of the CVC linear sloping section, whose positions, as follows from experimental data, are also functions of the microwave power. A change in the number of phase-slip centers breaks the initial synchronization condition, and the system can switch to either a totally un-synchronized state, or a partially synchronized state, or fully synchronized state at a different harmonic and with a different number of phase-slip centers (for example, the zero-slope step at $2U_{\Omega}$ in Fig. 2 can be due to synchronization of four centers by the second harmonic or six centers by the third harmonic). Given these two effects, we could not determine the constant-voltage step width as a function of the microwave amplitude unambiguously and compare it to the theoretical model. The existence of the microwave power threshold at which induced steps appear and the absence of oscillations in both the zero-slope step width at $U_{m/n}$ and critical current as functions of the microwave field amplitude are due to a definite number of phase-slip centers required at these voltages.
CONCLUSION
==========
In the reported work, we have studied the effect of microwave radiation on current-voltage characteristics of whiskers with submicron diameters. Such whiskers can serve as microwave oscillators at frequencies of up to 40 GHz with an output of about $10^{-8}$ W. The spectrum of generated waves contains many harmonics, and the generation occurs on CVC sections with stable numbers of phase-slip centers. Features of CVCs of our samples under microwave radiation are determined by changes in the number of phase-slip centers and the synchronization degree of generation in these centers.
The work was supported by the *Superconductivity* subprogram of the *Physics of Condensed State* program sponsored by the Russian government (Project No. 95021), and by the *Physics of Solid-State Nanostructures* program (Project No. 1-084/4).
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abstract: 'This paper proposes a distributed algorithm for a set of tiny unit disc shaped robot to form a straight line. The robots are homogeneous, autonomous, anonymous. They observe their surrounding up to a certain distance, compute destinations to move to and move there. They do not have any explicit message sending or receiving capability. They forget their past observed or computed data. The robots do not have any global coordinate system or origin. Each robot considers its position as its origin. However, they agree on the X and Y axis. The robots are not aware of the total number of robots in the system. The algorithm presented in this paper assures collision free movements of the robots. To the best of our knowledge this paper is the first reported result on line formation by fat robots under limited visibility.'
author:
- Arijit Sil$^1$
- Sruti Gan Chaudhuri$^2$
title: Line Formation by Fat Robots under Limited Visibility
---
Introduction
============
One of the current trends of research in the field of Robotics is to replace a big robot by a group of small autonomous robots who work in coordination between themselves. The group of robots may perform many real time jobs like moving a big object, cleaning a big surface, guarding a geographical area etc. In theoretical point of view, one of the fundamental tasks for executing this kind of jobs is to form the geometric patterns on the plane by the robots. In this paper we address the problem of line formation by unit disc shaped robots or [*fat robots*]{}. The robots can sense their surrounding up to a certain range. They compute their destination locations by our proposed algorithm and move there. After reaching their destinations they forget all their past sensed and computed data.
All reported line formation algorithms [@FPS2012book] for mobile robots considers that the robots as points and they are able to sense all other robots. A point robot neither creates any visual obstruction nor acts as an obstacle in the path of other robots. Czyzowicz et. al,[@Reference14] extended the traditional weak model of robots by replacing the point robots with unit disc robots (fat robots). Only some solutions on gathering problem has been reported for fat robots [@AGM2013PODC; @GM2015JDA; @GM2016book]. Under limited visibility gathering is solved for point robots [@FPSW2005TCS] and fat robots [@BKF2012SE]. Dutta et. al [@Reference15] proposed a circle formation algorithm for fat robots assuming common origin and axes for the robots. Here the robots are assumed to be transparent in order to avoid visibility block. However, a robot acts as an physical obstacle if it falls in the path of other robots. The visibility range/radius of the robots is assumed to be limited. Datta et. al[@Reference16], proposed another distributed algorithm for circle formation by a system of mobile asynchronous transparent fat robots with unlimited visibility.
In this paper we consider fat robots and propose a collision free movement strategy to form a line where the robots can only sense other robots up to a finite distance.
Underlying Model
----------------
The robot model used in this paper is describes as follows.
- The robots are autonomous.
- Robots are anonymous and homogeneous i.e., they are not uniquely identifiable.
- A robot is represented as a transparent disc with unit radius. The robots are transparent or see-through in order to ensure full visibility, but they act as physical obstructions for other robots.
- The robots do not have any global coordinate system. Each robot considers its position as its origin. They agree on the direction of $XY$ axes. The robots also can agree on unit distance (the radius of the robots can be considered as unit).
- Every robot executes a cycle of three phases:
- [**Look**]{} - the robot takes a snapshot around itself up to a finite range and identifies the other robots’ positions w.r.t its own coordinate system;
- [**Compute**]{} - based on other robot positions, the observer robot computes its destination;
- [**Move**]{} - the robot moves to the destination point calculated in the previous phase.
- The robots execute this cycle in semi-synchronous scheduling where an arbitrary set of robots look, compute and move simultaneously. This scheduling assures that when a robot is moving no other robot is observing it.
- The robots do not stop before reaching its destination ([*rigid*]{} motion).
- The robots are oblivious in the sense that they cannot remember any data from previous cycle.
- A robot can see up to a fixed distance around itself comprised of a circular area centered at the center of the robot having radius length $rad_{v}$ on the 2D plane.
- The robots do not know about the total number of robots in the system.
- Robots cannot communicate using explicit messages.
- The robots form a graph $G \big(V, E\big)$. Every robot is a vertex $v$ in $G$, where $v \in V$ of graph $G$. There exists an edge $e$, $e \in E$ between the robots $r_{i}$ and $r_{j}$ if and only if they can see each other. Initially, the graph $G$ is assumed to be connected which implies that every robot can see at least one other robot.
- Initially the robots are stationary and the mutual distance between two robots is atleast $\delta > 2 units$.
Overview of the problem
=======================
Let $R$ be a set of $n$ robots under the model described in previous section. The robots are assumed to be transparent in order to ensure full visibility, but they act as physical obstacles for other robots. A robot is represented by its center, i.e., by $r$ we mean a robot whose center is $r$. The robots in $R$ have to move in such a way that after a finite number of execution cycles, the robots in $R$ will form a straight line.
When a robot $r\in R$ becomes active it first enters into look state. In this state $r$ takes a snapshot of the robots that are present within its visibility circle and plots those robots in its local coordinate system $Z_{r}$. This set of robots visible to $r$ is called the neighbours of $r$. With respect to $Z_{r}$, the set of robots visible to $r$ can be divided into eight distinct and non-overlapping sets. Refer to Fig. \[fig1\],
- Set $A$ consists of the robots partially or fully present in the area of unit distance around the positive $Y$ axis but it does not contain robot $r$ itself.
- Set $C$ consists of the robots partially or fully present in the area of unit distance around the positive $X$ axis.
- The set $F$ represents the robots partially or fully present in the area of unit distance around the negative $X$ axis and set $I$ contains the robots partially or fully present in the area of unit distance around negative $Y$ axis.
- Set $B$ and $D$ account for the robots that fall within the first and second quadrant of the local coordinate system respectively (leaving the robots in $A$, $C$ and $I$).
- The robots which are in third and fourth quadrant of the coordinate system $Z_{r}$ make the sets $E$ and $G$ respectively (leaving the robots on $A$, $F$ and $I$).
In the first step of the computation phase $r$ calls a routine to check whether it should make a move in this cycle or not. The routine considers all possible scenarios and determines a destination point for $r$.
In the second step of the computation phase $r$ calls a second routine that returns the amount of horizontal and /or vertical shift that is allowed for $r$ in accordance with within $r$’s visibility circle.
In the final phase of the current cycle $r$ moves to the destination point computed in the previous phase. The movement ensures that it preserves connectivity and avoids collision with the neighbouring robots.
#### **Vacant Point:**
A point is vacant if there exist no parts of another robot around a circular region of radius 1 around this point.
#### **Free path:**
A path of a robot is called free path, if from source to destination point (Refer to Fig. \[freepath\]) the rectangular area having length as the source to destination distance and width as two units, is not contained any part of another robot.
![An example of free path of robot $r$ and Vacant point $p$[]{data-label="freepath"}](freepath)
Description of the Algorithm
============================
The algorithm we have developed is executed by the robots in semi- synchronous manner in computation phase and determines the distance and direction of their movement. The algorithm $LineForm()$ is divided into two subroutines. One subroutine, $NoMovement()$ identifies the cases where the robot will not move. whereas the other subroutine $getDestination()$ computes the destination points of the robots. Finally the robots move to this computed destinations
#### **NoMovement(r):**
$r$ will not move for the following configurations.
- If, $r$ does not find any other robot in its visibility circle [^1](Fig. \[nomove\](a)).
- If $F$ is not empty (Fig \[nomove\](b)).
- If sets $E \cup F \cup G$ are not empty (Fig \[nomove\](c)).
- If $B \cup C \cup D \cup E \cup F \cup G$ is empty and $A \cup I$ is not empty. (Fig \[nomove\](d)).
#### **getDestination(r):**
This subroutine considers all the robots that are visible to $r$ and ahead of it in the direction of positive $X$ axis (hereinafter referred as RIGHT). If $E \cup F \cup G$ is not empty the robot $r$ does not move. Hence, this routine takes into account only the robots in the sets $B$, $C$ and $D$. If $B \cup C \cup D$ is empty then the subroutine returns the current location of $r$ as the destination point. But if $B \cup C \cup D$ is not empty then the algorithm finds the nearest axis vertical to $X$ axis that contains one or more robots from $B \cup C \cup D$. Let, $\psi_{right}$ be that axis. It then considers four different scenarios.
Suppose, the robots partially of fully present in the area of unit distance around the axis $\psi_{right}$ that are within the visibility range of $r$ forms the set $R_{\psi_{right}}$. The coordinate of $r$ is taken as $(0,0)$ w.r.t. its local coordinate system.
If $R_{\psi_{right}} \cap C$ is empty, (Fig. \[sc1\]), which means none of the robots in $R_{\psi_{right}}$ resides on $X$ axis, then the coordinates of the destination point $p$ is given by:
$x_{p} = 0 + \Delta x$ \[where $\Delta x$ = offset along $X$ axis from current position of $r$ to the intersection point between $X$ axis and $\psi_{right}$\].
$y_{p} = 0$.
[**Scenario 2:**]{} If $R_{\psi_{right}} \cap C$ is not empty and $R_{\psi_{right}} \cap B$ is not empty and$R_{\psi_{right}} \cap D$ is empty then $R_{\psi_{right}}$ has a robot on $X$ axis, it has got one or more robot that belongs to set $B$, but does not have any robot that belongs to set $D$. So, if $r$ only moves horizontally it will collide with the robot already present on the intersection point of $\psi_{right}$ axis and $X$ axis. So, its horizontal movement must be followed by a vertical movement towards the negative $Y$ direction. As the set $D$ does not contain any robot on $\psi_{right}$ axis, $r$ does not have to face collision with any other robot as it moves vertically. So the coordinates of the destination point $p$ is given by:
$x_{p} = 0 + \Delta x$
$y_{p} = 0 - \Delta y$ \[where $\Delta y$ = the radius of robots\]
If $p$ is not a vacant point, the subroutine re-computes $y_p$ as,
$y_{p} = 0 - i\Delta y$ \[where $\Delta y$ = the radius of robots, i=2,3..\].
This process continues till a vacant point is found.
If no vacant point is found in $R_{\psi_{Right}}\cap B$ using this procedure, then the vacant point can be found in the same manner in $R_{\psi_{Right}}\cap D$.
[**Scenario 3:**]{} If $R_{\psi_{right}} \cap C$ is not empty and $R_{\psi_{right}} \cap B$ is empty and $R_{\psi_{right}} \cap D$ is not empty then $R_{\psi_{right}}$ has a robot on $X$ axis, it has no robot that belongs to set $B$, but does have one or more robot that belongs to set $D$. So, if $r$ only moves horizontally, just as in the previous scenario, it will collide with the robot already present on the intersection point of $R_{\psi_{right}}$ axis and $X$ axis. So, its horizontal movement must be followed by a vertical movement towards the positive $Y$ direction.
So the coordinates of the destination point $p$ is given by:
$x_{p} = 0 + \Delta x$
$y_{p} = 0 + \Delta y$ \[where $\Delta y$ = the radius of robots\]
If $p$ is not a vacant point, the subroutine re-computes $y_p$ as,
$y_{p} = 0 + i\Delta y$ \[where $\Delta y$ = the radius of robots, i=2,3..\].
This process continues till a vacant point is found.
If no vacant point is found in $R_{\psi_{Right}}\cap D$ using this procedure, then the vacant point can be found in the same manner in $R_{\psi_{Right}}\cap B$.
[**Scenario 4:**]{} If there is no vacant point in $R_{\psi_{right}} \cap B$ and $R_{\psi_{right}} \cap C$ and $R_{\psi_{right}} \cap B$, then the robot $r$ computes its destination point as follows.
- If $I$ is completely empty, then $r$ moves towards either $-Y$ axis such that,
$x_p = 0$.
$y_p = 0 - rad_v +1$.
- If $I$ is not empty, then $r$ moves along $-Y$ axis such that,
$x_p = 0$.
$y_p = 0 - d+1$ where $d=$ the vertical distance with $r$ and the robot with maximum $y$ value in $I$.
Correctness
===========
The robots successfully form a straight line in finite time using our proposed algorithm. The algorithm is correct as it gives assurance of the following facts.
- The visibility graph $G$ does not becomes disconnected.
- The robots do not collide due to their movement strategies.
- The robots do not fall into deadlock and form the line in finite time.
Following lemmas are presented to prove these facts.
\[Lemma 1\] The connectivity graph $G$ remains connected.
Consider scenario 1. The robots in $B$ or $C$ or $D$ will not move due to the presence of $r$ according to our $NoMovement()$ subroutine. Now we will show that when $r$ is moving to its destination it gets closer to the robots in $B$ or $C$ or $D$. Without loss of generality let us prove this by taking the existence of any robot in $B$. The same arguments hold for the presence of robots in $B$ or $C$ or $D$.
Let $s$ be the starting location of $r$ (Fig. \[cor1\]). Let there exists a robot $r_b$ in $B$ at $b$. Let $t$ be the destination of $r$. $r$ moves along the edge $st$ of the triangle $stb$. In triangle $sbt$ $|tb|<|sb|$ (since $sb$ is the diagonal.) Hence, when $r$ moves to $t$, it becomes closer to the robot present in $B$.
Using the similar argument we can prove that under scenario 2 and 3, due to the movement of $r$, it becomes closer to the other robots present in $B$, $C$ and $D$, Hence, there is no chance to get disconnected with any robot.
Now consider Scenario 4. No robots in $B$ or $C$ or $D$ moves following $No$ $Movement()$ subroutine. First consider the case when $I$ are empty. Then $r$ moves along $-Y$, $rad_v$ distance. Note that $r$ in its new position is connected with the robots in $D$. Note that the robots in $D$ in connected with $B$, $C$, $D$, $A$. Hence $G$ remains connected. Now suppose $I$ is not empty. $r$ moves towards $-Y$ till it touches the robots in $I$. Due to this movement $G$ does not become disconnected.
\[Lemma 4\] The robots never collide.
In scenario 1, 2 and 3 the robot $r$ moves to its next visible vertical line $\psi_{right}$ only when there is a [*vacant point*]{} on it.
Consider scenario 1, the $r$ moves to the intersection point of $+X$ axis and $\psi_{right}$. This point is vacant according to the algorithm. The path towards this point from $r$ is also a it [*free path*]{} as there is no vertical line between $y$ axis and $\psi_{right}$.
Consider scenario 2, the $r$ moves to the vacant point on $\psi_{right}$ at $B$. The path towards this point from $r$ is also a free path as there is no vertical line between $y$ axis and $\psi_{right}$.
Consider scenario 3, the $r$ moves to the vacant point on $\psi_{right}$ at $D$. The path towards this point from $r$ is also a free path as there is no vertical line between $y$ axis and $\psi_{right}$.
In scenario 1,2 and 3 no robot in the visibility circle moves other than $r$. $r$ moves in such a way that it does not collide with any other robot.
Consider scenario 4, the $r$ moves down along $-Y$ axis if it has free path. Otherwise it does not move. Hence, there is no chance for collision.
Note that the destination point for $r$ is chosen in such a way that no robot which is not visible to $r$, can come in $r$’s path.
Hence for all movements of $r$, it does not collide with any other robot.
\[Lemma 2\] There exists always a robot which will move unless the robots in $R$ forms a straight line.
If a robot see any robot at its right side it will move. If it does not see any robot at $B \cup C \cup D$, it does not move. This is possible for following two cases.
- There is a single robot.
- The robots have formed a straight line.
- The robots do not form a straight line but there exist another robot in $A$ or $I$ which has the connectivity with the right side or left side of the $Y$ axis. For both the cases there exists robot other than $r$, which will move.
Hence, the lemma is true.
If the robots do not form a line yet it will leave its $Y$ axis after a finite time and move in the $+X$ direction.
Follows from lemma \[Lemma 2\].
Given a set of robots $R$ on the 2D plane in its initial configuration, we may assume the existence of a line passing through the global right most robots. Let this line be the Right Most Axis (RMA) of initial configuration. In fact through our algorithm the robots finally are placed on the RMA and form the required straight line.
\[movetosir\] Each robot will move closer to RMA in finite time interval.
If a robot finds any robot at its right side it moves in $+X$ direction following scenarios 1, 2, 3 and moves to its $\psi_{right}$. As the robots do not stay idle for infinite time, it is guaranteed that the robots will reach to its $\psi_{right}$ in finite time, i.e., closer to RMA.
In scenario 4, $r$ moves down along $-Y$ axis if there is a free path. When $r$ moves down its visibility circle also moves down and it covers a new set of robots. Eventually $r$ moves towards right and placed on the next vertical line nearer to RMA. If $r$ does not move down, there exists another robot to move and eventually $r$ gets its chance to move unless the straight line is already formed.
\[Lemma 3\] None of the robots ever crosses the RMA of the set of robot.
Suppose there is a robot $r$ which has crossed RMA. $r$ can do that in two different ways. If $r$ was initially on RMA then it has left that axis to move to RIGHT or $r$ was initially LEFT of RMA and has crossed it while going towards RIGHT. In the first case to leave RMA $r$ has to observe an axis containing robots towards RIGHT. But as $r$ was sitting on RMA no such axis can exist and therefore it contradicts our assumption and therefore once on RMA, $r$ cannot leave it anymore. In the second case $r$ was LEFT of RMA and in order to go past RMA it has to find RMA to be the nearest axis containing robots towards RIGHT as there is no other axis with robots RIGHT of RMA. But if RMA is the nearest axis then the maximum horizontal shift would take $r$ up to RMA and not beyond that and once it reaches RMA, $r$ cannot leave it anymore. Therefore, the scenario contradicts our assumption. So, by contradiction we can say that none of the robots ever crosses RMA.
\[finite\] All the robots in $R$ will be on the RMA in finite time.
According to lemma \[movetosir\], each robot reach its $\psi_{right}$ in finite time. This implies that after a finite time there will be configuration when RMA will be the $\psi_{right}$ for each robot. After this configuration in finite time all robots will move to RMA.
Conclusion
==========
Finally we can summarize the result in the following theorem
A set of asynchronous, oblivious fat robots can form a straight line under limited visibility and one axis agreement without collision.
The future scope of this work would be to find the possibility of other pattern formation by fat robots or study the same problem removing the axis agreement or placing the robots uniformly distributed on the line.
[6]{}
C. Agathangelou, C. Georgiou, M. Mavronicolas: A distributed algorithm for gathering many fat mobile robots in the plane, Proceedings of the 2013 ACM symposium on Principles of distributed computing, 250-259
K. Bolla, T. Kovacs, G. Fazekas: Gathering of Fat Robots with Limited Visibility and without Global Navigation, Swarm and Evolutionary Computation, LNCS vol. 7269, 30-38.
J. Czyzowicz, L. Gasieniec, A. Pelc:Gathering few fat mobile robots in the plane. Theoretical Computer Science, vol. 410, no. 6–7, pages 481 499, 2009. Principles of Distributed Systems.
A. Dutta, S. Gan Chaudhuri, S. Datta, K. Mukhopadhyaya: Circle formation by asynchronous fat robots with limited visibility; International Conference on Distributed Computing and Internet Technology; Springer Berlin Heidelberg; 83-9.
S. Datta, A. Dutta, S. Gan Chaudhuri, K. Mukhopadhyaya: Circle Formation by Asynchronous Transparent Fat Robots; International Conference on Distributed Computing and Internet Technology Springer-Verlag Berlin Heidelberg 2013; 195-207.
P. Flocchinia, G. Prencipeb, N. Santoro, P. Widmayerd: Gathering of Asynchronous Robots with Limited Visibility, Theoretical Computer Science, vol. 337 (2005) 147 – 168
P. Flocchini, G. Prencipe, and N. Santoro: Distributed Computing by Oblivious Mobile Robots, Synthesis Lectures on Distributed Computing Theory2012 3:2, 1-185
S. Gan Chaudhuri, K. Mukhopadhyaya: Leader election and gathering for asynchronous fat robots without common chirality, Journal of Discrete Algorithms, vol. 33, July 2015, 171-192.
S. Gan Chaudhuri, K. Mukhopadhyaya: Distributed Algorithms for Swarm Robots, Handbook of Research on Design, Control, and Modeling of Swarm Robotic, IGI Global, Copyright@2016, page 26.
[^1]: Which happens if and only if there is only one robot.
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---
abstract: 'This paper proposes a charge-conserving, variational, spatio-temporal discretization for the drift-kinetic Vlasov-Maxwell system, utilizing finite-elements for the electromagnetic fields and the particle-in-cell approach for the Vlasov distribution. The proposed scheme is fully electromagnetic, dealing with fields instead of potentials, and includes the effects of polarization and magnetization in the Gauss and Ampère-Maxwell laws, a consequence of reducing the full particle dynamics to drift-center dynamics. There is, however, no need to invert the Gauss law: it is satisfied automatically at every time-step as a result of a discrete Noether symmetry, and the electric field is updated directly from the Ampère-Maxwell equation. The method provides an update for the magnetic field that is fully explicit, involving only local operations. The update for particles is implicit for each particle individually, also leading to local operations only. The update for the electric field is linearly implicit due to the presence of a finite-element mass matrix and polarization and magnetization effects in the Ampère-Maxwell equation, hence involving a sparse matrix inversion once at every time step. Because the scheme deals with the electromagnetic fields and not the potentials, it also provides the first serious attempt at constructing a structure-preserving numerical scheme for the mixed kinetic-ion–drift-kinetic-electron Vlasov-Maxwell model. Consequently, the proposed method could be used to simulate electromagnetic turbulence in fusion experiments or space plasmas that exhibit a strong background magnetic field while retaining all of the ion physics, most of the necessary electron physics, yet eliminating perhaps the biggest obstacle in reaching macroscopic transport time scales in kinetic simulations, namely the electron cyclotron time scale.'
author:
- Eero Hirvijoki
bibliography:
- 'references.bib'
title: 'Charge-conserving, variational particle-in-cell method for the drift-kinetic Vlasov-Maxwell system'
---
Summary and discussion
======================
This paper was devoted to investigating the possibility of a variational algorithm for the drift-kinetic Vlasov-Maxwell system. As it was demonstrated, such a discrete scheme was indeed found and, furthermore, guarantees a discrete charge-conservation law as a consequence of the discrete electromagnetic gauge invariance of the action. The key to the presented results was the recent discovery of a gauge-free electromagnetic gyrokinetic theory which allows one to express the drift-kinetic perturbation terms in the action in terms of the perturbed electric and magnetic field instead of the perturbed electromagnetic potentials. Consequently, also the discrete equations involve only the electromagnetic fields. Perhaps the most important effect of this fact is that it opens up the possibility to construct a structure-preserving variational scheme that couples fully kinetic ions and drift-kinetic electrons to study electromagnetic turbulence and the associated transport in steep background gradients where the gyrokinetic assumptions for ions might not necessarily be valid.
Finally, we stress that our choice for the discretization is not unique. The only guiding principle was to retain the fully discrete action gauge invariant. For example, the choice for discretizing the guiding-center part of the action was based purely on the aesthetically appealing looks of the resulting discrete equations for advancing the individual drift-center coordinates. Future studies into the topic should focus on discretizations that contain only polynomial dependencies on the drift-center degrees of freedom for efficient numerical integration of the line integrals and, especially, on the possible stability issues in degenerate variational phase-space discretizations. In the end, this paper presents only the first attempt at constructing a structure-preserving integrator for the drift-kinetic plasma model. Hopefully more will come, and the superior long-time stability properties of the new algorithms find their way to production-level codes within the fusion-research community. At least one such code is to be launched with the next years, to reboot [ELMFIRE](http://elmfire.eu) full-$f$ gyrokinetic programme currently developed and maintained at Aalto University.
The author is grateful to Joshua W. Burby and Alain J. Brizard for the numerous discussions over the years regarding action principles and the gyrokinetic theory, and for the encouragement from the ELMFIRE group. Financial support for the research was provided by the Academy of Finland grant no. 315278. Any subjective views or opinions expressed herein do not necessarily represent the views of the Academy of Finland or Aalto University.
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---
abstract: 'We obtain a system for the spatial metric and extrinsic curvature of a spacelike slice that is hyperbolic non-strict in the sense of Leray and Ohya and is equivalent to the Einstein equations. Its characteristics are the light cone and the normal to the slice for any choice of lapse and shift functions, and it admits a well-posed causal Cauchy problem in a Gevrey class of index $\alpha=2$. The system becomes quasidiagonal hyperbolic if we posit a certain wave equation for the lapse function, and we can then relate the results to our previously obtained first order symmetric hyperbolic system for general relativity.'
author:
- |
Andrew Abrahams, Arlen Anderson, Yvonne Choquet-Bruhat[@YCBadd]\
and James W. York, Jr.\
[*Department of Physics and Astronomy*]{}\
[*University of North Carolina, Chapel Hill 27599-3255 USA*]{}
date: 'June 26, 1996'
title: 'A non-strictly hyperbolic system for the Einstein equations with arbitrary lapse and shift'
---
=0.3in =-0.0in =8.5in =6.25in
IFP-UNC-518
TAR-UNC-054
gr-qc/9607006
Introduction
============
We consider, as in previous works[@CBY95; @aacby], the dynamics of General Relativity as the evolution of the geometry of spacelike slices of a spacetime manifold $M\times R$. We take as coframe $$\theta^0=dt,\quad \theta^i=dx^i +\beta^i dt,$$ and write the spacetime metric $$ds^2=-N^2 (\theta^0)^2 + g_{ij} \theta^i \theta^j.$$ The geometry of a slice $M_t$, $x^0=t$, is determined by its metric ${\bf \bar g}=g_{ij} dx^i dx^j$ and extrinsic curvature $K_{ij}$. The operator $\dzeroh$, acting on time-dependent space tensors, is defined by $$\dzeroh = {\partial\over \partial t}-{\cal L}_\beta,$$ where ${\cal L}_\beta$ is the Lie derivative with respect to the spatial shift vector $\beta^k$. We have $$\label{d0g}
\dzeroh g_{ij}=-2N K_{ij}.$$
By using a combination $\Omega_{ij}=\dzeroh R_{ij}- 2\bnabla_{(i}R_{j)0}$ of the $\dzeroh$ derivative of $R_{ij}$ with space derivatives of the momentum constraints, we have obtained a second-order system for $K_{ij}$. This system reduces to a quasi-diagonal system for any choice of shift when the lapse $N$ is chosen so that either the time slicing is harmonic or the slices have [*a priori*]{} a given mean extrinsic curvature, $H\equiv K^k\mathstrut_k=h(x,t)$. The system obtained for $({\bf \bar g, K})$ is hyperbolic and causal in the first case. It is a mixed hyperbolic (causal) and elliptic system in the second case, where one uses the $R_{00}$ equation to determine $N$. We have shown moreover that the system with the harmonic time slicing (and also generalizations of this slicing) can be written as a first order symmetric hyperbolic system if one uses also the $R_{00}$ equation.
In this paper we use a combination of $\dzeroh\dzeroh R_{ij}$ and derivatives of the constraints $R_{i0}$ and $R_{00}$ to obtain a system for $({\bf \bar g, K})$ whose characteristics are the light cone and the normal to the spatial slice, for any choice of lapse and shift (as functions of spacetime). This system is hyperbolic non-strict in the sense of Leray-Ohya[@LeO]. It admits a well-posed causal Cauchy problem in a Gevrey class of index $\alpha=2$.
We also show that if the lapse $N$ satisfies a wave equation whose source is an arbitrary (smooth) function of spacetime, ${\bf \bar g,\ K},\ N$ and first derivatives of $N$, the system becomes hyperbolic in the usual sense, with solutions in local Sobolev spaces. This system can be put in first order form, but it is not symmetric because the symmetrizing matrix is semi-definite. However, for an appropriate choice of the wave equation for the lapse, one can write our new system as a symmetric hyperbolic first order system that is the $\dzeroh$ derivative of the one previously obtained[@CBY95; @aacby]. (We can also show that if the lapse is required to satisfy an appropriate elliptic equation, there is a well-posed hyperbolic-elliptic formulation[@CBY96b].)
Non-strictly hyperbolic system
==============================
We consider the combination $$\Lambda_{ij}\equiv \dzeroh\dzeroh R_{ij} -2\dzeroh \bnabla_{(i}R_{j)0}
+\bnabla_i \bnabla_j R_{00}\equiv \dzeroh \Omega_{ij}
+\bnabla_i \bnabla_j R_{00},$$ where $(ij)={1\over 2}(ij +ji)$. We use the expression for $\Omega_{ij}$, in the form given in [@aacby] from the 3+1 decomposition of the Ricci tensor, together with the expression for $R_{00}$, to obtain the following identity $$\begin{aligned}
\label{Lam}
\Lambda_{ij}&\equiv& \dzeroh (N\Boxh K_{ij}) +\dzeroh J_{ij}+
\bnabla_i \bnabla_j(N\Boxh N-N^2 K^{mk}K_{mk})+
{\cal C}_{ij}, \nonumber\end{aligned}$$ where $$\Boxh =-N^{-1}\dzeroh N^{-1} \dzeroh
+ \bnabla^k \bnabla_k ,$$ $$\begin{aligned}
\label{Jij}
J_{ij}&=& \hat\partial_0 (H K_{ij} - 2 K_{i}\vphantom{|}^{k} K_{jk})
+(N^{-2}\hat\partial_0 N+ H)\bar\nabla_i \bar\nabla_j N \nonumber \\
&&\hspace{-0.75cm}
-2N^{-1}(\bar\nabla_k N) \bar\nabla_{(i}(N K^{k}\vphantom{|}_{j)})
+3 (\bar\nabla^k N) \bar\nabla_k K_{ij} \\
&&\hspace{-0.75cm} +N^{-1}K_{ij} \bar\nabla^k (N\bar\nabla_k N)
-2 \bar\nabla_{(i}(K_{j)}\vphantom{|}^{k}\bar\nabla_k N)
+N^{-1} H \bar\nabla_i\bar\nabla_j N^2
\nonumber \\
&&\hspace{-0.75cm}
+2 N^{-1}(\bar\nabla_{(i} H)(\bar\nabla_{j)}N^2 )
-2N K^{k}\vphantom{|}_{(i}\bar R_{j)k}
-2N \bar R_{kijm}K^{km}, \nonumber\end{aligned}$$ and $${\cal C}_{ij}\equiv \bnabla_i\bnabla_j(\dzeroh (N^{-1}\dzeroh N) +
N \dzeroh H) -
\dzeroh (N^{-1} \bnabla_i\bnabla_j (\dzeroh N+N^2 H)).$$ We see that ${\cal C}_{ij}$ contains terms of at most second order in ${\bf K}$, and first order in ${\bf \bar g}$, after replacing $\dzeroh g_{ij}$ by $-2N K_{ij}$.
The identity given above shows that for a solution of the Einstein equations $$R_{\alpha\beta}= \rho_{\alpha\beta},$$ the extrinsic curvature ${\bf K}$ satisfies, for [*any choice of lapse $N$ and shift $\beta$*]{}, a third order differential system which is quasidiagonal with principal part the hyperbolic operator $\dzeroh \Boxh$. The other unknown ${\bf \bar g}$ appears at second order except for third derivatives occuring in $ \bnabla_j\bnabla_i \Boxh N$. To avoid an explicit discussion of properties of matter evolution equations, we consider below the vacuum case.
Any solution of the vacuum Einstein equations $R_{\alpha\beta}=0$ satisfies the following system of partial differential equations $$\label{sys}
\Lambda_{ij}=0, \quad \dzeroh g_{ij}=-2NK_{ij}.$$ For any choice of shift $\beta$ and lapse $N>0$, (\[sys\]) is a non-strictly hyperbolic system with domain of dependence determined by the light cone when the metric $\bar g$ is properly Riemannian.
If the Cauchy data ${\bf \bar g}(.,0)={\bf \gamma},\ {\bf K}(.,0)={\bf k}$ belong to the Gevrey class $G^{2,{\rm loc}}_2(M_0)$, with ${\bf \gamma}$ properly Riemannian, there exists a solution of the system (\[sys\]) in a neighborhood $U$ of $M_0$, for any $N,\beta\in G_2^{2,{\rm loc}}(U)$, whose initial data $\dzeroh {\bf K}(.,0)$ and $\dzeroh\dzeroh{\bf K}(.,0)$ are determined by the restrictions $R_{ij}(,.0)=0$ and $(\dzeroh R_{ij})(.,0)=0$.
[*Proof.*]{} The principal matrix is triangular with diagonal elements either $\dzeroh$ or $\dzeroh \Boxh$. Both of these operators are (strictly) hyperbolic, but the system is not (quasi-diagonal) hyperbolic, due to the presence of non-diagonal terms in the principal matrix. It is, however, hyperbolic non-strict in the sense of Leray-Ohya [@LeO]. The Gevrey class in which the system is well-posed is obtained by quasidiagonalization [@CB1] and study of the multiplicity of the characteristics of the resulting diagonal elements. Here we obtain a quasidiagonal fourth order system for ${\bf \bar g}$, equivalent to the original system for ${\bf \bar g,\ K}$, by substituting $K_{ij}= -(2N)^{-1}\dzeroh g_{ij}$ into $\Lambda_{ij}$. The principal operators on the diagonal are then $\dzeroh\dzeroh\Boxh$. They are hyperbolic non-strict with multiplicity two for $\dzeroh$. The existence and uniqueness theorems of Leray-Ohya [@LeO] give the result.
Equivalence to the Einstein equations
=====================================
Let $N>0$ and $\beta$ be arbitrary in the Gevrey class $G_2^{2,{\rm loc}}$. Let $({\bf \bar g,\ K})$ be a solution of the system $$\label{mat}
\Lambda_{ij} = \Theta_{ij} \equiv \dzeroh\dzeroh \rho_{ij}
-2\dzeroh \bnabla_{(i} \rho_{j)0} + \bnabla_j\bnabla_i \rho_{00},$$ where $\rho_{\alpha\beta}$ is some symmetric 2-tensor belonging to $G_2^{2,{\rm loc}}$ that satisfies the conservation laws $$\nabla_\alpha (\rho^{\alpha \beta} -{1\over 2} g^{\alpha\beta}
g^{\lambda \mu} \rho_{\lambda \mu})=0.$$ Suppose the Cauchy data $({\bf \bar g,\ K},\ \dzeroh {\bf \bar K},
\ \dzeroh\dzeroh {\bf \bar K})_{M_0}$ satisfy on $M_0$ the Einstein equations $(R_{\alpha\beta}-\rho_{\alpha\beta})_{M_0}=0$ together with $(\dzeroh(R_{ij}-\rho_{ij}))_{M_0}=0$. Then the metric $({\bf \bar g}, N, \beta)$ satisfies the Einstein equations $R_{\alpha\beta}-\rho_{\alpha\beta}=0$ in the domain of dependence of $M_0$.
[*Note.*]{} The equations $(R_{ij}-\rho_{ij})_{M_0}=0$ and $(\dzeroh(R_{ij}-\rho_{ij}))_{M_0}=0$ determine $(\dzeroh K_{ij})_{M_0}$ and $(\dzeroh\dzeroh K_{ij})_{M_0}$ when the data $({\bf \bar g, K})_{M_0}$ are given. The equations $(R_{0\alpha}-\rho_{0\alpha})_{M_0}=0$ are then equivalent to the usual constraints on $({\bf \bar g, K})_{M_0}$, while the conditions $(\dzeroh (R_{0\alpha}-\rho_{0\alpha}))_{M_0}=0$ are a consequence of the restriction to $M_0$ of the (twice-contracted) Bianchi identities and conservation laws.
[*Proof.*]{} We set $X_{\alpha\beta}=R_{\alpha\beta}-\rho_{\alpha\beta}$. The Bianchi identities, together with the conservation laws, imply $$\label{bi2}
\nabla_i X^{i}\mathstrut_0 -{1\over 2}( N^{-2} \nabla_0 X_{00}
+g^{ij} \nabla_0 X_{ij})=0.$$ We rewrite (\[mat\]) $$\label{mat2}
\nabla_0\nabla_0 X_{ij} -\nabla_0 \nabla_{(i}X_{j)0} +
\nabla_j \nabla_i X_{00}= {\rm lhe_1},$$ where ${\rm lhe}_n$ means linear and homogeneous in ${\bf X}$ and derivatives of order less than or equal to $n$.
The sum of twice the $\nabla_0$ derivative of (\[bi2\]) and the contraction with $g^{ij}$ of (\[mat2\]) gives the equation $$\label{boxX00}
-N^{-2} \nabla_0\nabla_0 X_{00} + g^{ij} \nabla_j\nabla_i X_{00}=
{\rm lhe_1}.$$ The $\nabla_0^2$ derivative of the Bianchi identity-conservation law with free index $j$ reads $$\label{bi3}
\nabla_0^2 \nabla_0 X^{0j} + \nabla_0^2 \nabla_i X^{ij}
-{1\over 2} g^{ij} \nabla_0^2 \nabla_i(-N^{-2} X_{00} +
g^{mk} X_{mk})=0.$$ We use (\[mat2\]) and the Ricci identity to obtain $$\begin{aligned}
\label{boxXj0}
\nabla_0^2 \nabla_0 X^{0j} + 2\nabla_0 \nabla_k \nabla^{(k}
X^{j)}\mathstrut_0 - \nabla^j \nabla^k \nabla_k X_{00} +
{1\over 2} N^{-2} \nabla^j \nabla_0^2 X_{00}\hspace{1cm}&& \\
-{1\over 2} \nabla^j[2\nabla_0\nabla_k X^{k}\mathstrut_0-
\nabla^k \nabla_k X_{00}] &&\nonumber \\
\equiv\nabla_0(-N^{-2} \nabla_0^2 X^j\mathstrut_0 + \nabla^k \nabla_k
X^j\mathstrut_0) -{1\over 2} \nabla^j(-N^{-2} \nabla_0^2 X_{00}
+\nabla^k \nabla_k X_{00})&=&{\rm lhe_2}. \nonumber\end{aligned}$$ The equations (\[mat2\]), (\[boxX00\]), (\[boxXj0\]), with unknowns $X_{\alpha\beta}$ have a triangular principal matrix with diagonal principal operators $\Boxh$ and $\dzeroh \Boxh$. It is a non-strictly hyperbolic system. It is equivalent to the quasidiagonal system consisting of (\[boxX00\]), (\[bi3\]) and the following equation obtained by combination of the previous ones $$\label{boxXij}
\nabla_0^2(-N^{-2} \nabla_0^2 + \nabla^k \nabla_k) X_{ij}=
{\rm lhe_3}.$$ The system (\[boxX00\]), (\[boxXj0\]), (\[boxXij\]) is hyperbolic non-strict in the Gevrey class $G_2^{2,{\rm loc}}$. The result follows from the Leray-Ohya uniqueness theorem.
Hyperbolic system for $({\bf \bar g, K}, N)$
============================================
We add to the system (\[sys\]) a wave equation for $N$ with arbitrary right hand side. Namely, we set $$\label{boxN}
\Boxh N\equiv -N^{-1}\dzeroh (N^{-1} \dzeroh N) + \bnabla^k \bnabla_k N=F,$$ with $F$ a given function of spacetime, $N,{\bf \bar g,K}$ and first derivatives of $N$. Therefore, the equation $\Lambda_{ij}=0$, reduced by the substitution of (\[boxN\]) and the replacement of $\dzeroh g_{ij}$ by $-2N K_{ij}$ wherever it occurs, takes the form $$\label{d0boxK}
\dzeroh \Boxh K_{ij}=f_{ij}({\rm 2\ in\ }{\bf \bar g,K};{\rm 3\ in\ }N).$$ The left hand side is third order in ${\bf K}$ and of order less than 3 in ${\bf\bar g}$ and $N$. The numbers in $f_{ij}$ denote the order of the highest derivatives occuring there. Because $\Boxh N$ involves first derivatives of ${\bf \bar g}$, to obtain a hyperbolic system we differentiate (\[boxN\]) by $\dzeroh$ and use $\dzeroh g_{ij}=-2N K_{ij}$. This gives an equation which is third order in $N$ and first order in ${\bf \bar g}$ and ${\bf K}$, $$\label{d0boxN}
\dzeroh \Boxh N= \dzeroh F.$$
The system (\[d0g\]), (\[d0boxK\]), (\[d0boxN\]), is a quasi-diagonal hyperbolic system.
We attribute to the unknowns and equations the following Leray-Volevic indices: $$\begin{aligned}
m({\bf K})=3,\quad m({\bf \bar g})=3, \quad m(N)=4, \\
n(\ref{d0boxK})=0,\quad n(\ref{d0g})=2,\quad n(\ref{d0boxN})=1. \nonumber\end{aligned}$$ The system assumes then a quasidiagonal form[@prin] with principal operators $\dzeroh$ and $\dzeroh \Boxh$. It is a hyperbolic system.
We remark that a mixed hyperbolic-elliptic system for $({\bf \bar g,
K},N)$, with $N$ determined by an elliptic equation is also possible [@CBY96b]. It is a straightforward extension of the result in [@CBY96a].
First order system
==================
The system (\[d0g\]), (\[boxN\]), (\[d0boxK\]) can be put in first order form by a straightforward procedure. The system can be symmetrized, but the symmetrizing matrix is found to be semi-definite, not positive definite, so the usual existence and uniqueness theorems do not apply. It is an open question whether the system in first-order form is well-posed.
To achieve a symmetric hyperbolic system, the source $F$ of the wave equation (\[boxN\]) for $N$ can be chosen so that no second spatial derivatives of ${\bf K}$ appear in the first order system: $$\label{boxNsymm}
\Boxh N -NK_{mk}K^{mk} +N H^2=\tilde F({\bf \bar g},N,b_0,a_i;x,t).$$ This equation is closely related to a form of harmonic time slicing that is allowed in our second order system for ${\bf K}$ [@CBY95; @aacby], namely, $$\label{d0N}
\dzeroh N + N^2 H=Nf(\sqrt{{\bf \bar g}}\alpha^{-1},N;x,t),$$ where $\alpha(x,t)$ is an arbitrary positive scalar density of weight one and $f$ is a scalar. Using (\[d0N\]) and the Einstein equation for $R_{00}$ gives (\[boxNsymm\]) with $\tilde F=\tilde F(
\sqrt{{\bf \bar g}}\alpha^{-1},N,b_0;x,t)$, where $b_0=N^{-1}\dzeroh N$ is related to $H$ by the harmonic condition (\[d0N\]). In this case, the symmetric system obtained using (\[boxNsymm\]) is the time derivative of the first order symmetric hyperbolic system previously obtained in [@CBY95; @aacby].
Acknowledgments. A.A., A.A., and J.W.Y. were supported by National Science Foundation grants PHY-9413207 and PHY-9318152/ASC-9318152 (ARPA supplemented).
[99]{}
Permanent address: Gravitation et Cosmologie Relativiste, t.22-12, Un. Paris VI, Paris 75252 France.
Y. Choquet-Bruhat and J. W. York, C. R. Acad. Sci. Paris, [**t. 321**]{}, Série I, 1089 (1995).
A. Abrahams, A. Anderson, Y. Choquet-Bruhat, and J.W. York, Phys. Rev. Lett. [**75**]{}, 3377 (1995).
J. Leray and Y. Ohya, Math. Ann. [**170**]{}, 167 (1967).
Y. Choquet-Bruhat and J. W. York, submitted to Banach Center Publications, gr-qc/9606001.
Y. Bruhat, J. Maths Pures et Appl. [**45**]{}, 371-387 (1966).
The principal operator for the $p^{\rm th}$ unknown in the $q^{\rm th}$ equation is of order $m(p)-n(q)$.
Y. Choquet-Bruhat and J. W. York, to appear in [*Gravitation, Electromagnetism and Geometric Structures*]{}, edited by G. Ferrarese (Pythagoras Editrice, Bologna, Italy, 1996), gr-qc/9601030 (1996).
|
---
abstract: 'Considering the dipole-dipole coupling intensity between two atoms and the field in the Fock state, the entanglement dynamics between two atoms that are initially entangled in the system of two two-level atoms coupled to a single mode cavity in the presence of phase decoherence has been investigated. The two-atom entanglement appears with periodicity without considering phase decoherence, however, the phase decoherence causes the decay of entanglement between two atoms, with the increasing of the phase decoherence coefficient, the entanglement will quickly become a constant value, which is affected by the two-atom initial state, Meanwhile the two-atom quantum state will forever stay in the maximal entangled state when the initial state is proper even in the presence of phase decoherence. On the other hand, the Bell violation and the entanglement does not satisfy the monotonous relation, a large Bell violation implies the presence of a large amount of entanglement under certain conditions, while a large Bell violation corresponding to a little amount of entanglement in certain situations. However, the violation of Bell-CHSH inequality can reach the maximal value if two atoms are in the maximal entangled state, or vice versa.'
author:
- 'C. J. Shan[^1]'
- 'W. W. Cheng'
- 'T. K. Liu[^2]'
- 'D. J. Guo'
- 'Y. J. Xia'
title: 'Entanglement dynamics and Bell Violations of two atoms in Tavis-Cummings model with phase decoherence [^3]'
---
**Introduction**
================
Quantum entanglement is one of the most striking features of quantum mechanics, and plays an important role in quantum information processing, such as quantum teleportation\[1\], quantum dense coding\[2\], quantum cryptography\[3\] and quantum computation\[4\]. Therefore quantum entanglement has been viewed as an essential resource for quantum information process, and a great deal of effort has been devoted to study and characterize the entanglement. Cavity quantum electrodynamics (QED) techniques has been recognized as a promising candidate for the physical realization of quantum information processing. Quantum entanglement based cavity QED was generated by sending two atoms being present simultaneously in the cavity \[5\] or the two atoms interacting consecutively with the cavity \[6\]. However, the above preparation processes are considered in closed system and the influences of environment are neglected. Time evolution of isolated quantum systems is followed by the Schrodinger equation. But a quantum system unavoidably interacts with the environment. The decoherence effect of this interaction will lead to the degradation of quantum coherence and entanglement. The entangled state will loss purity and become mixed. Entanglement dynamics behavior of a quantum system coupled to its environment can reflect the details of the decoherence effect\[7,8\]. On the other hand, entanglement can exhibit the nature of a nonlocal correlation between quantum systems. Bell’s theorem\[9\] provides a effective way to test quantum nonlocality\[10\], quantum nonlocality will be exhibited if Bell-type inequality is violated for a given quantum state. Namely, a violation of any Bell-type inequality gives a quantitative confirmation that a state behaves quantum nonlocality.\
In the original papers, researchers investigated the entanglement in the JCM\[11\], a damped JCM\[12\] and two-atom Tavis-Cummings model\[13\]. Recently Hein etal.\[14\] investigate entanglement properties of multipartite states under the influence of decoherence. Reference \[7\] shows that quantum mechanical entanglement can prevail in noisy open quantum systems at high temperature and far from thermodynamical equilibrium, despite the deteriorating effect of decoherence. Reference \[8\]considers the interaction of a single two-level atom with one of two coupled microwave cavities and shows analytically that the atom-cavity entanglement increases with cavity leakage.We investigate the entanglement time evolution of two entangled two-level atoms that interact resonantly with a single-mode field in the Fock state\[15\]. In Ref.\[16\], the author investigated two two-level atoms coupled to a single mode optical cavity with the phase decoherence and showed the rich dynamical features of entanglement arising between atoms and cavity or between two atoms, however the two-atom dipole-dipole coupling intensity is neglected, the two atoms are initially in a separate state and the cavity field is initially prepared in the vacuum state. In order to study explicitly the entanglement dynamics of the two-atom system, therefore, in this paper we investigate the entanglement dynamics between two atoms that are initially in entangled state in Tavis-Cummings model introducing dipole-dipole coupling intensity and the field in the Fock state with phase decoherence, to our knowledge, which has not been reported so far. In addition quantum nonlocality has been widely studied for the two-atom entanglement system using Bell-CHSH inequality. Our studies show that the entanglement between two atoms and Bell-CHSH inequality decay with phase decoherence and disappear in a constant, which is affected by two-atom initial state and dipole-dipole coupling intensity. Meanwhile many new interesting phenomena are exhibited, e.g., the two-atom quantum state will forever stay in the maximal entangled state when the initial state is proper even in the presence of phase decoherence. These interesting phenomena result from two-atom initial state and dipole-dipole coupling intensity. The phase decoherence can be used to play a constructive role and generate the controllable stable entanglement by adjusting two-atom initial state and dipole-dipole coupling intensity.\
This paper is organized as follows. We introduce the model and calculate the reduced density matrices of two two-level atoms in the next section. In Sec. 3, Entanglement dynamics of two atoms with phase decoherence have been studied. Sec. 4 gives the relations between entanglement and Bell violations, and Sec. 5 is the conclusions.
The model and reduced density matrices of two two-level atoms
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Consider two two-level atoms interacting resonantly with a single-mode cavity field initially prepared in the Fock state. In the rotating-wave approximation the Hamiltonian of the atom-field system reads $$H=\omega_{0}\sum_{j=1}^{2}S_{j}^{z}+\omega_{a}a^{\dag}a+\sum_{j=1}
^{2}g(a^{\dag}S_{j}^{-}+aS_{j}^{+})+\sum_{i,j=1;{i}\neq{j}}^{2}\Omega
S_{i}^{+} S_{j}^{-}$$ where $a$ ($a^{\dagger}$) denotes the annihilation (creation) operator of the resonant single-mode field, $\omega_{0}$, $\omega_{a}$ are atomic transition frequency, cavity frequency, respectively, $g$ is the coupling constant between atoms and cavity, $S_{j}^{+}=|e\rangle_{j}\langle
g|$, $S_{j}^{-}=|g\rangle_{j}\langle e|$, $
S_{j}^{z}=\frac{1}{2}(|e\rangle_{j}\langle e|-|g\rangle_{j}\langle
g|)$ are atomic operators, and $\Omega$ is atomic dipole-dipole coupling constant. In this paper, we investigate the entanglement between two atoms by only considering the phase decoherence. In this situation, the master equation governing the time evolution of the system under the Markovian approximation is given by\[17\] $$\frac{d\rho}{dt}=-i[H,\rho]-\frac{\gamma}{2}[H,[H,\rho]]$$ where$\gamma$ is the phase decoherence coefficient. The equation with the similar form has been proposed to describe the intrinsic decoherence \[18\]. The formal solution of the master equation (2) can be expressed as follows \[19\]: $$\rho(t)=\sum_{k=0}^{\infty}\frac{(\gamma
t)^{k}}{k!}M^{k}(t)\rho(0)M^{\dagger k}(t)$$ where $\rho(0)$ is the density operator of the initial atom-field system and $M^{k}(t)$ is defined by $$M^{k}(t)=H^{k}exp(-iHt)exp(-\frac{\gamma t}{2}H^{2})$$ We assume $\omega_{0}$ = $\omega_{a}$, the cavity field is prepared initially in the Fock state $|n\rangle$, atom A and atom B are prepared in the entangled state $\cos\theta|eg\rangle+\sin\theta|ge\rangle$, then the initial density operation for the whole atom-field system is $$\rho(0)=(\cos\theta|eg\rangle+\sin\theta|ge\rangle)(\cos\theta\langle
eg|+\sin\theta\langle ge|)\otimes|n\rangle\langle n|$$ In the subspace of $K=a^{\dag}a+\frac{1}{2}(S_{1}^{z}+S_{2}^{z})\equiv n$, the eigenvectors and eigenvalues of Hamiltonian (1) can be written as\[20\] $$\begin{aligned}
&|E_{0}\rangle=-\sqrt{\frac{1+n}{1+2n}}|n-1\rangle|ee\rangle+\sqrt{\frac{n}{1+2n}}|n+1\rangle|gg\rangle,
E_{0}=n\omega&\nonumber\\
&|E_{1}\rangle=\frac{1}{\sqrt{2}}(|n\rangle|ge\rangle-|n\rangle|eg\rangle),
E_{1}=n\omega-\Omega&\nonumber\\
&|E_{2}\rangle=\frac{1}{2}\sqrt{\frac{\Delta-\Omega}{\Delta}}(\frac{4\sqrt{n}g}{\Delta-\Omega}|n-1\rangle|ee\rangle-|n\rangle|ge\rangle-|n\rangle|eg\rangle+
\frac{4\sqrt{n+1}g}{\Delta-\Omega}|n+1\rangle|gg\rangle),\nonumber\\
&E_{2}=\frac{1}{2}(2n\omega+\Omega-\Delta)&\\
&|E_{3}\rangle=\frac{1}{2}\sqrt{\frac{\Delta+\Omega}{\Delta}}(\frac{4\sqrt{n}g}{\Delta+\Omega}|n-1\rangle|ee\rangle+|n\rangle|ge\rangle+|n\rangle|eg\rangle+
\frac{4\sqrt{n+1}g}{\Delta+\Omega}|n+1\rangle|gg\rangle),\nonumber\\
& E_{3}=\frac{1}{2}(2n\omega+\Omega+\Delta)&\nonumber\end{aligned}$$ Where $\Delta=\sqrt{8(1+2n)g^{2}+\Omega^{2}}$\
Substituting $\rho(0)$ into the Eq.(3), the exact time-dependent density operation can be expressed as $$\begin{aligned}
\rho(t)=&C_{1}|E_{1}\rangle\langle E_{1}|+C_{2}|E_{2}\rangle\langle
E_{2}|+C_{3}|E_{3}\rangle\langle E_{3}|+C_{4}|E_{1}\rangle\langle
E_{2}|+\nonumber\\&C_{5}|E_{2}\rangle\langle
E_{1}|+C_{6}|E_{1}\rangle\langle E_{3}|+C_{7}|E_{3}\rangle\langle
E_{1}|+C_{8}|E_{2}\rangle\langle E_{3}|+C_{9}|E_{3}\rangle\langle
E_{2}|\end{aligned}$$ where $$\begin{aligned}
&C_{1}=\frac{1}{2}(1-\sin2\theta),C_{2}=\frac{1}{4}(1+\sin2\theta)
\frac{\Delta-\Omega}{\Delta},C_{3}=\frac{1}{4}(1+\sin2\theta)
\frac{\Delta+\Omega}{\Delta}&\nonumber\\
&C_{4}=\frac{1}{2\sqrt{2}}\cos2\theta\sqrt{\frac{\Delta-\Omega}{\Delta}}
exp(-\frac{(E_{2}-E_{1})^{2}}{2}\gamma t)exp(i(E_{2}-E_{1})t)&\nonumber\\
&C_{5}=\frac{1}{2\sqrt{2}}\cos2\theta\sqrt{\frac{\Delta-\Omega}{\Delta}}
exp(-\frac{(E_{2}-E_{1})^{2}}{2}\gamma t)exp(-i(E_{2}-E_{1})t)&\nonumber\\
&C_{6}=-\frac{1}{2\sqrt{2}}\cos2\theta\sqrt{\frac{\Delta+\Omega}{\Delta}}
exp(-\frac{(E_{3}-E_{1})^{2}}{2}\gamma
t)exp(i(E_{3}-E_{1})t)&\nonumber\\
&C_{7}=-\frac{1}{2\sqrt{2}}\cos2\theta\sqrt{\frac{\Delta+\Omega}{\Delta}}
exp(-\frac{(E_{3}-E_{1})^{2}}{2}\gamma t)exp(-i(E_{3}-E_{1})t)&\nonumber\\
&C_{8}=\frac{1}{\sqrt{2}}(1+\sin2\theta)\frac{g\sqrt{1+2n}}{\Delta}
exp(-\frac{(E_{3}-E_{2})^{2}}{2}\gamma t)exp(i(E_{3}-E_{2})t)&\nonumber\\
&C_{9}=\frac{1}{\sqrt{2}}(1+\sin2\theta)\frac{g\sqrt{1+2n}}{\Delta}
exp(-\frac{(E_{3}-E_{2})^{2}}{2}\gamma
t)exp(-i(E_{3}-E_{2})t)&\nonumber\end{aligned}$$ The reduced density matrices of the subsystem composed of two two-level atoms is $$\rho_{AB}(t)=a_{1}|gg\rangle\langle gg|+a_{2}|ge\rangle\langle ge|+
a_{3}|ge\rangle\langle eg|+a_{4}|eg\rangle\langle
ge|+a_{5}|eg\rangle\langle eg|+a_{6}|ee\rangle\langle ee|$$ Where $$\begin{aligned}
a_{1}=(1+\sin2\theta)\frac{2(n+1)g^{2}}{\Delta^{2}}
(1-exp(-\frac{(E_{3}-E_{2})^{2}}{2}\gamma t)\cos(E_{3}-E_{2})t)\end{aligned}$$ $$\begin{aligned}
a_{2}=&\frac{1}{2}+(1+\sin2\theta)
\frac{(1+2n)g^{2}}{\Delta^{2}}(-1+
exp(-\frac{(E_{3}-E_{2})^{2}}{2}\gamma
t)\cos(E_{3}-E_{2})t)\nonumber\\& -\frac{1}{4}\cos2\theta
\frac{\Delta-\Omega}{\Delta} exp(-\frac{(E_{2}-E_{1})^{2}}{2}\gamma
t)\cos(E_{2}-E_{1})t\nonumber\\&-\frac{1}{4}\cos2\theta
\frac{\Delta+\Omega}{\Delta} exp(-\frac{(E_{3}-E_{1})^{2}}{2}\gamma
t)\cos(E_{3}-E_{1})t\end{aligned}$$ $$\begin{aligned}
a_{3}=a_{4}^{*}=&\frac{\sin2\theta}{2}+(1+\sin2\theta)
\frac{(1+2n)g^{2}}{\Delta^{2}}(-1+exp(-\frac{(E_{3}-E_{2})^{2}}{2}\gamma
t)\cos(E_{3}-E_{2})t )\nonumber\\& -\frac{i}{4}\cos2\theta
\frac{\Delta-\Omega}{\Delta} exp(-\frac{(E_{2}-E_{1})^{2}}{2}\gamma
t)\sin(E_{2}-E_{1})t\nonumber\\&-\frac{i}{4}\cos2\theta
\frac{\Delta+\Omega}{\Delta} exp(-\frac{(E_{3}-E_{1})^{2}}{2}\gamma
t)\sin(E_{3}-E_{1})t\end{aligned}$$ $$\begin{aligned}
a_{5}=&\frac{1}{2}+(1+\sin2\theta)
\frac{(1+2n)g^{2}}{\Delta^{2}}(-1+
exp(-\frac{(E_{3}-E_{2})^{2}}{2}\gamma
t)\cos(E_{3}-E_{2})t)\nonumber\\& +\frac{1}{4}\cos2\theta
\frac{\Delta-\Omega}{\Delta} exp(-\frac{(E_{2}-E_{1})^{2}}{2}\gamma
t)\cos(E_{2}-E_{1})t\nonumber\\&+\frac{1}{4}\cos2\theta
\frac{\Delta+\Omega}{\Delta} exp(-\frac{(E_{3}-E_{1})^{2}}{2}\gamma
t)\cos(E_{3}-E_{1})t\end{aligned}$$ $$\begin{aligned}
&a_{6}=(1+\sin2\theta)\frac{2n g^{2}}{\Delta^{2}}
(1-exp(-\frac{(E_{3}-E_{2})^{2}}{2}\gamma t)\cos(E_{3}-E_{2})t)&\end{aligned}$$
Entanglement dynamics of two atoms with phase decoherence
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In order to discuss the entanglement dynamics in the above system, we adopt the negative eigenvalues of the partial transposition to quantify the degree of entanglement. The idea of this measure of the entanglement is the Peres-Horodecki criterion for the separability of bipartite systems \[21\]. The state is separable if the partial transposition is a positive operator, however, if one of the eigenvalues of the partial transposition is negative then the state is entangled. For a two-qubit system described by the density operator, the negativity can be defined by:\[22\] $$E_{AB}=-2\sum_{i}\mu_{i}$$ where $\mu_{i}$ are the negative eigenvalues of the partial transposition of $\rho_{AB}^{\Gamma}$. When $E_{AB}=0$, the two qubits are separable and $E_{AB}=1$ indicates maximal entanglement between them.\
We can make a partial transposition for atom B and work out the eigenvalues of the partial transposition $\rho_{AB}^{\Gamma}$. The four eigenvalues are $a_{2}$, $a_{5}$, $\frac{1}{2}(a_{1}+a_{6}+\sqrt{a_{1}^{2}-2a_{1}a_{6}+a_{6}^{2}+4a_{3}a_{4}})$, $\frac{1}{2}(a_{1}+a_{6}-\sqrt{a_{1}^{2}-2a_{1}a_{6}+a_{6}^{2}+4a_{3}a_{4}})$. Substitute them into Eq.(14), the explicit expression of $E_{AB}$ characterizing the entanglement of two atoms can be found to be $$\begin{aligned}
&E_{AB}=|a_{2}|+|a_{5}|+|\frac{1}{2}(a_{1}+a_{6}+\sqrt{a_{1}^{2}-2a_{1}a_{6}+a_{6}^{2}+4a_{3}a_{4}})|+
\nonumber\\&|\frac{1}{2}(a_{1}+a_{6}-\sqrt{a_{1}^{2}-2a_{1}a_{6}+a_{6}^{2}+4a_{3}a_{4}})|
-(a_{2}+a_{5}+\frac{1}{2}(a_{1}+a_{6}+\sqrt{a_{1}^{2}-2a_{1}a_{6}+a_{6}^{2}+4a_{3}a_{4}})\nonumber\\&
+\frac{1}{2}(a_{1}+a_{6}-\sqrt{a_{1}^{2}-2a_{1}a_{6}+a_{6}^{2}+4a_{3}a_{4}}))\end{aligned}$$ In the following, we analyze the numerical results for the time evolution of the two-atomic entanglement.\
We firstly consider the case of $\gamma=0$, i.e., the absence of phase decoherence. The time evolution behaviors of the entanglement are showed in Fig.1-Fig.3(assuming g=1 in all the figures in this paper) with different initial state and dipole-dipole coupling intensity for $g=1,n=0$.\
Figure 1 depicts the time evolution of the entanglement when the pair of atoms are initially prepared in the different states. It is observed that the entanglement evolves periodically in the absence of phase decoherence. We consider three cases of the initial state, i.e., the disentangled of the two atoms ((a)and(c), solid line ), not maximal entangled state ((a), (b), (c) and(d), dashed line) and maximal entangled state ((b)and(d), solid line). In the first case, we can observe that the two atoms that are initially separate can generate entanglement by the atom-field interaction and atom-atom interaction. At certain time the entanglement evolves to its zero and the two two-level atoms are disentangled, while at the large time scale the two atoms are entangled. In a period, the degree of the entanglement increases gradually to a larger value(about 0.5), then decreases to a smaller value(about 0.2), then again increases and finally decreases to zero. In the second case, the two atoms own the same entanglement at $t=0$, but have different phase angles. It is the phase angle that leads to considerable different time evolution of the entanglement. One case is that the degree of the two-atom entanglement is no more than that of the initial entanglement, as is shown in Fig.1((a)and (b), dashed line), the other case is that the degree of the two-atom entanglement is more than that of the initial entanglement all the time during the interaction, the peak of the entanglement increases, as is shown in Fig.1((c)and (d), dashed line), which means the larger entangled state can be prepared by choosing the initial phase angle. The third case is that the two atoms are initially in the maximal entangled state. In Fig.1((c), solid line), the time evolution is similar to the above case, however, from Fig.1((d), solid line), we can find the two-atom quantum state will forever stay in the maximum entangled state when the initial state is proper, this corresponding to the fact that the two atoms do not show any dynamic evolution and remain the initial state.\
Figure 2 displays the time evolution of the entanglement for two values of no and weak dipole-dipole interaction. Fig.2((a), solid line) corresponding to the case of being no dipole-dipole interaction, the peak of the maximum entanglement becomes small comparing with the case of that in Fig.1((a),solid line, $\Omega=1$), Since there is no dipole-dipole interaction between the two atoms, it is very clear that this entanglement is induced purely by atom-field interaction. This is consistent with Ref. \[11\]. The dipole-dipole interaction plays a constructive role in the entanglement formation between two atoms. From these figures, we can see that the degree of the entanglement is not necessarily increases with the increase of dipole-dipole interaction. In Fig.2(c), the degree of the entanglement can reach the maximum value 1 and the range of the oscillation becomes larger comparing with the situation in Fig.1(a), while the value of the dipole-dipole interaction in Fig.2(c) is less than that in Fig.1(a). It is interesting to find that the two atoms can generate maximal entangled state even they are separate initially by adjusting the dipole-dipole interaction.\
In Fig.3, we consider the situation of strong dipole-dipole interaction. With the increase of dipole-dipole interaction, the period of the oscillation becomes short. The time evolution character is similar to the case of the weak dipole-dipole interaction for the separate initial state. However, for the entangled initial state, that is not the case. An interesting result is the entanglement between the two atoms increases to a larger value than the initial entanglement in Fig.3((a)and (c), dashed line), while the entanglement decreases in in Fig.1((a), dashed line). In the strong coupling case, i.e., $\Omega \gg g $, from Eq.(9-13), we can see that dipole-dipole interaction $\Omega$ plays a key role in the quantum entanglement between the atom. Atom-atom interaction reduces the atom-field interaction. That is to say, strong dipole-dipole interaction is helpful for the entanglement production.\
Let us now turn to discuss the condition of existing phase decoherence($\gamma\neq0$). The phase decoherence causes the decay of the entanglement between two atoms, which is shown in Figs.4(a)and 4(b). With the increase of phase decoherence coefficient, the initial entanglement oscillates with time and will gradually become a constant value, which depends on the initial state of the two atoms. That is to say, the phase decoherence in the atom-field interaction suppresses the entanglement, but the phase decoherence can not fully destroy the entanglement between two atoms. From Figs.4(a)and 4(b), we can also see that the pairwise entanglement between two atoms can achieve a very large value even in the presence of phase decoherence, which is similar to the case without phase decoherence. For the proper initial state, their entanglement can be preserved during the time evolution as its initial value with phase decoherence. The above time evolution character arises due to in the time evolution the additional term in Eq.(2) leads to the appearance of the decay factor, which are responsible for the destruction of the entanglement. In order to discuss how the entanglement changes with the dipole-dipole interaction, in Figs.4(c)and 4(d) we give the plot of the entanglement for $\Omega=0.5$ and $\Omega=5$ in the present of $\gamma=0.1$. The result is that more stronger the dipole-dipole interaction is, more faster the entanglement does oscillate. As for the situation of strong dipole-dipole interaction, the entanglement decreases rapidly, then approaches to a stable value, which is different from the case in the absent of phase decoherence. what affects the stable value? From Eq.(15), it is easy to verify that $E_{AB}$ in the case of $\gamma\neq 0$ for given long time, $$E_{AB}=\frac{-2(1+2n)g^{2}(\sin\theta+\cos\theta)^{2}+\sqrt{4g^{4}(\sin\theta+\cos\theta)^{4}+
(1+2n)^{2}(-2g^{2}+6(g^{2}+\Omega^{2})\sin2\theta)^{2}}}
{8(1+2n)g^{2}+\Omega^{2}}$$ which means that the entanglement of stationary state depends on the initial state, the dipole-dipole interaction and the field in the Fock state. One may question whether there exists a situation in which two atoms can forever achieve maximal entanglement in the present of phase decoherence. Fig.4((a)and(b), dash dot line) give the answer. What is the reason why the two atoms can stay the maximal entanglement in the present of phase decoherence ? From Eq.(9-13), we can see $a_{1}=a_{6}=0$, $a_{2}=a_{5}=\frac{1}{2}$, $a_{3}=a_{4}=-\frac{1}{2}$ if the angles satisfy the following relation $\theta=\frac{(4k-1)\pi}{4}, k=1, 2, \cdots$. The two atoms are in the maximally entangled state $\frac{1}{\sqrt{2}}(eg\rangle-|ge\rangle)$, so the entanglement has nothing with the phase decoherence coefficient, the two-atom initial state, the dipole-dipole coupling intensity between two atoms and the field in the Fock state.\
At the end of this section, we discuss to achieve entanglement between the two atoms if the initial atoms are prepared in different states and the cavity field is in the Fock state. In Fig.5, we plot the entanglement as the function of time t for different values of phase decoherence rate $\gamma$ and dipole-dipole coupling intensity $\Omega$ if the field in the $|1\rangle$. Two cases are shown in Fig.5(a) for different dipole-dipole coupling intensity if $\gamma=0$, i.e the entanglement between two atoms of being no dipole-dipole interaction falls off while $E_{AB}$ increases having dipole-dipole interaction as n increases for the initial separate two-atom state. The influence is completely different compared to that for the $n=0$ case. For the initial entangled two-atom state, the notable difference here is that the peak of the entanglement becomes larger than that in Fig.1, while at some time, the two atoms stay in the separate state. The photon number n helps to increase the peak value of entanglement. Figs.5(c) and 5(d) corresponding to the case of phase decoherence $\gamma=0.1$. An interesting comparison can be made with the case of the field in the vacuum state. The entanglement decays sharply as n increases and the stationary state entanglement is affected by the Fock state, so we can get two-atom entanglement mediated by the Fock state cavity field.\
From the above analysis, it is clear to note that the phase decoherence coefficient, the two-atom initial state, the dipole-dipole coupling intensity between two atoms and the field in the Fock state have notable influence on the entanglement of two atoms.
Bell violations and the relations between entanglement and Bell violations
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The quantum nonlocal property can be characterized by the maximal violation of Bell’s inequality. Jeong etal.\[23\] have defined the maximal violation of Bell’s inequality as mearurement of the degree of quantum nonlocality. Here we discuss the CHSH inequality. The CHSH operator is defined by\[24\] $$\vec{B}=({\vec{a}}\cdot{\vec{\sigma}})\otimes(\vec{b}\cdot{\vec{\sigma}})
+({\vec{a}}\cdot{\vec{\sigma}})\otimes(\vec{b'}\cdot{\vec{\sigma}})
+ ({\vec{a'}}\cdot{\vec{\sigma}})\otimes(\vec{b}\cdot{\vec{\sigma}})
+({\vec{a'}}\cdot{\vec{\sigma}})\otimes(\vec{b'}\cdot{\vec{\sigma}})$$ where $\vec{a}$, $\vec{a'}$, $\vec{b}$, $\vec{b'}$ are unit vectors. The hidden variable theories impose the Bell-CHSH inequality $|<\vec{B}>|\leq2$ where $<\vec{B}>$ is the mean value of the bell operation for a given quantum state. However, in the quantum theory it is found that $|<\vec{B}>|\leq2\sqrt{2}$, which implies the Bell-CHSH inequality is violated. The maximal amount of Bell’s violation of a state $\rho$ is given by \[25\] $$<B>=2\sqrt{\lambda+\lambda'}$$ Where $\lambda$, $\lambda'$ are the two largest eigenvalues of $T_{\rho}^{\dag}T_{\rho}$, the elements of matrix $T_{\rho}$ are $(T_{\rho})_{nm}=Tr(\rho\sigma_{n}\otimes\sigma_{m})$, here $\sigma_{1}=\sigma_{x}$, $\sigma_{2}=\sigma_{y}$, and $\sigma_{3}=\sigma_{z}$ denote the usual Pauli matrices. For the density operator in Eq. characterizing the time evolution of two atoms, $\lambda+\lambda'$ can be written as follows: $$\lambda+\lambda'=4a_{3}a_{4}+\max[4a_{3}a_{4},(a_{1}+a_{6}-a_{2}-a_{5})^{2}]$$ it is easy to draw the violation of Bell¡¯s inequality for two atoms. $$<B>=2\sqrt{4a_{3}a_{4}+\max[4a_{3}a_{4},(a_{1}+a_{6}-a_{2}-a_{5})^{2}]}$$ Similarly, Figs.6-8 display the numerical results of the analytical expression of maximal violation of Bell’s inequality for the field in the vacuum state. In Fig.6, we plot the time evolution of the maximal violation of Bell’s inequality for $\Omega=1$ and $\Omega=0.5$ when the two atoms are prepared in different states. For the sepatate initial state, our calculations show that two atoms cannot violate the CHSH inequality in this case, which is seen in Fig.6((a)dashed line). If we appropriately choose the value of the dipole-dipole interaction $\Omega$, From Fig.6(c)(dashed line), an interesting result is that two atoms can violate the CHSH inequality in certain time. Even the two atoms have the same entanglement and the phase angle, it is the dipole-dipole interaction that makes the CHSH inequality of the two atoms evolve in different ways. The violation of the CHSH inequality increases firstly in Fig.6(b)(solid line), while the violation the CHSH inequality decreases firstly in Fig.6(d)(dashed line). In addition the violation of Bell-CHSH inequality can stay in the maximal value when the entanglement angle satisfies $\theta=3\pi/4$. Fig.7 corresponding to the time evolution of Bell-CHSH inequality in the present of phase decoherence. Fig.8 depicts the time evolution of Bell-CHSH inequality against the strong dipole-dipole interaction with the phase decoherence and without the phase decoherence. The result is expected as it is shown in Figs.8(a) and 8(b) that the strong dipole-dipole interaction maximize the violation of the CHSH inequality, in this case the larger violation of Bell-CHSH inequality can be achieved. Similar to the influence of phase decoherence on the entanglement, the violation of Bell-CHSH inequality is very fragile against the phase decoherence and finally disappears in the different stationary state with different initial state and dipole-dipole coupling intensity.\
In the following, we are devoted to settling the relationship between entanglement, measured in terms of the negativity, and the Bell violations in the system \[1\]. And although the quantitative relations have never been investigated in detail, it is quite often suggested that a large Bell violation implies the presence of a large amount of entanglement and vice versa. Recently, Verstraete et al. investigated the relations between the violation of the CHSH inequality and the concurrence for systems of two qubits\[26\]. For the pure states and some Belldiagonal states, the maximal value of B for given concurrence C is $2\sqrt{1+C^{2}}$. If the concurrence $C\geq\sqrt{2}/2$, the minimal value of B is $2\sqrt{2}C$, furthermore, the entangled two-qubits state may not violate any CHSH inequality with the concurrence C$\leq\sqrt{2}/2$, except their Belldiagonal normal form does violate the CHSH inequalities. Comparing Fig.1((a) solid line) with Fig.5((a) solid line), we can find that though two atoms get entangled in the time evolution, two atoms cannot violate the CHSH inequality in this case. Fig.5 shows two atoms can violate the CHSH inequality in the case that the entanglement is larger than a certain value. Under certain condition, the more Bell violation, the larger amount of entanglement. However, the violation of Bell¡¯s inequality is not a sufficient condition for the entanglement, that is to say, a large Bell violation is not necessarily with a large amount of entanglement, which can be seen in Fig.1((c) dashed line), Fig.2((d) solid line), Fig.6((b)solid line, (d)dashed line). In Fig.3((a) solid line) and Fig.8((a) solid line). The dipole-dipole interaction decreases the degree of violation while increases the amount of entanglement. One interesting point is that the entanglement degree is initially very little, while the violation of Bell¡¯s inequality can be generated, according to Re.\[23\], we can know the Bell diagonal normal form in system (1) does violate the CHSH inequalities. Our calculations also show that the condition of the maximal violation is that the entanglement degree is maximal. In a word, the Bell violation and entanglement does not satisfy the monotonous relation. This is consist with Re.\[13\]. So this phenomenon is still valid for the form of Bell’s inequality and the entanglement measurement in this paper.\
Conclusion
==========
In summary, we have studied quantum entanglement and quantum nonlocality of two atoms in Tavis-Cummings model with phase decoherence. It is shown that the phase decoherence causes the decay of entanglement between two atoms. With the increasing of the phase decoherence coefficient, the entanglement will quickly become a constant value, which is affected by the two-atom initial state, the dipole-dipole coupling intensity and the field in the Fock state. Therefore, the amount of the entanglement can be increased by adjusting the two-atom initial state, the dipole-dipole coupling intensity and the field in the Fock state. The violation of Bell-CHSH inequality is very fragile against the phase decoherence and finally disappears in the different stationary state in the absence of phase decoherence. In addition, the relationship between the entanglement and the nonlocality of two atoms is investigated, under certain conditions either a larger violation or a less violation can be generated with the increasing of entanglement. We hope that the results obtained in this paper would find their applications in quantum information processing and the test of quantum nonlocality.\
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![The entanglement between the two atoms ($E_{AB}$) is plotted as a function of time t with $g=1,\Omega=1,\gamma=0,n=0$ when the two-atomic state is initially prepared in the different state[]{data-label="Fig.1.EPS"}](Fig.1.EPS "fig:"){width="100.00000%"}\
![The entanglement between the two atoms ($E_{AB}$) is plotted as a function of time t with $g=1,\gamma=0,n=0$.[]{data-label="Fig.2.EPS"}](Fig.2.EPS "fig:"){width="100.00000%"}\
![The entanglement between the two atoms ($E_{AB}$) is plotted as a function of time t with $g=1,\gamma=0,n=0$.[]{data-label="Fig.3.EPS"}](Fig.3.EPS "fig:"){width="100.00000%"}\
![The entanglement between the two atoms ($E_{AB}$) is plotted as a function of time t with $g=1,n=0$ in the present of phase decoherence.[]{data-label="Fig.4.EPS"}](Fig.4.EPS "fig:"){width="100.00000%"}\
![TThe entanglement between the two atoms ($E_{AB}$) is plotted as a function of time t with $g=1,n=1$, (a) and (b)$\gamma=0$, while (c) and (d)$\gamma=0.1$\
Fig.6 The time evolution of maximal violation of Bell-CHSH inequality for $g=1,\gamma=0,n=0$.[]{data-label="Fig.5.EPS"}](Fig.5.EPS "fig:"){width="100.00000%"}\
![The time evolution of maximal violation of Bell-CHSH inequality for $g=1,\gamma=0,n=0$.[]{data-label="Fig.6.EPS"}](Fig.6.EPS "fig:"){width="100.00000%"}\
![The time evolution of maximal violation of Bell-CHSH inequality for $g=1,\gamma=0.1,n=0$.[]{data-label="Fig.7.EPS"}](Fig.7.EPS "fig:"){width="100.00000%"}\
![The time evolution of maximal violation of Bell-CHSH inequality for $g=1,\Omega=5,n=0$.[]{data-label="Fig.8.EPS"}](Fig.8.EPS "fig:"){width="100.00000%"}\
[^1]: E-mail: [email protected]
[^2]: Corresponding author. E-mail: [email protected]
[^3]: Supported by the Key Higher Education Programme of Hubei Province under Grant No Z20052201, the Natural Science Foundation of Hubei Province, China under Grant No 2006ABA055, and the Postgraduate Programme of Hubei Normal University under Grant No 2007D20.
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abstract: 'There is a countable metrizable group acting continuously on the space of rationals in such a way that the only equivariant compactification of the space is a singleton. This is obtained by a recursive application of a construction due to Megrelishvili, which is a metric fan equipped with a certain group of homeomorphisms. The question of existence of a topological transformation group with the property in the title was asked by Yu.M. Smirnov in the 1980s.'
address:
- 'Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Ave., Ottawa, Ontario, Canada K1N 6N5'
- 'Departamento de Matemática, Universidade Federal de Santa Catarina, Trindade, Florianópolis, SC, 88.040-900, Brazil'
author:
- 'Vladimir G. Pestov'
title: 'A topological transformation group without non-trivial equivariant compactifications '
---
[^1]
Introduction
============
Let a (Hausdorff) topological group $G$ act continuously on a (Tychonoff) topological space $X$. An [*equivariant compactification*]{} of $X$ is a compact space $K$, equipped with a continuous action by $G$, together with a continuous equivariant map $i\colon X\to K$ having a dense image in $K$. (See [@dV-2; @dV-1; @dV; @stoyanov; @megrelishvili; @megrelishvili07; @vM].) The question of interest is whether, given $G$ and $X$, there exists a compactification into which $X$ embeds topologically, that is, $i$ is a homeomorphism onto its image.
For instance, this is so if a topological group $G$ acts on itself by left translations, in which case the compactification is the greatest ambit of $G$ ([@brook]; [@P06], p. 42). If the acting group $G$ is discrete, then its action on $\beta X$, the Stone-Čech compactification of $X$, is clearly continuous, so $i$ is the canonical homeomorphic embedding $X\hookrightarrow\beta X$. For non-discrete $G$, the action on the Stone-Čech compactification $\beta X$ is continuous just in some exceptional cases. Nevertheless, if the acting group $G$ is locally compact, then every $G$-space $X$ admits an equivariant compactification into which $X$ embeds homeomorphically, this is an important result by de Vries [@dV-1; @dV]. (For compact Lie groups, the result was known to Palais much earlier, [@palais].)
For a long time it was unknown whether the same conclusion holds in the case of every acting topological group (the question was advertised by de Vries in 1975, see [@dV-2]). However, Megrelishvili [@megrelishvili; @megrelishvili89] has shown it is not so, by constructing an example where both $G$ and $X$ are Polish, yet the embedding $i$ is never topological for any equivariant compactification of $X$. More examples can be found in the work of Megrelishvili and Scarr [@MegS] and Sokolovskaya [@sokolovskaya]. (The earliest claim of an example [@AS] was withdrawn by its authors.) In some examples of Megrelishvili, the mapping $i$ is not even injecive.
Some time in the 1980s, Yuri M. Smirnov asked about the existence of a topological group $G$ acting on a (non-trivial) space $X$ in such a way that the only equivariant compactification of $X$ is the singleton. The question was apparently never mentioned in Smirnov’s papers; however, it was well known among the Moscow general and geometric topologists, and later documented in Megrelishvili’s papers [@megrelishvili89; @megrelishvili07]. (See also a discussion in [@P06], Rem. 3.1.6.) Here we notice that a topological transformation group conjectured by Smirnov indeed exists.
The construction is based on the example of Megrelishvili [@megrelishvili; @megrelishvili89], the first ever in which the mapping $i\colon X\to K$ is not a topological embedding. In this construction, the space $X$ is the metric fan, joining the base point $\ast$ with countably many endpoints $x_n$ with the help of intervals of unit length, and equipped with the graph metric. The group $G$ is a certain subgroup of the group of homeomorphisms of $X$ with the compact-open topology, chosen in a certain subtle way. Megrelishvili has shown that in every equivariant compactification of $X$, the images of the points $x_n$ converge to the image of the base point $\ast$, so the compactification mapping $i$ is not homeomorphic. A further observation by Megrelishvili is that if one joins two copies of such a fan by identifying their respective endpoints and extends the group action over the resulting space, then the image of the sequence of $(x_n)$ will converge to the image of each the two base points, thus these two points have a common image, and consequently $i$ is not even injective.
The idea of our example is, starting with a topological transformation group, to attach to it a copy of the double Megrelishvili fan (in its rational version) at every pair of distinct points $x,y$. As a result, in any equivariant compactification of the resulting space, $X$ collapses to a point. If we repeat the same construction countably many times, the union of the increasing chain of resulting spaces has no nontrivial compactifications. The topological space that we obtain is countably infinite, metrizable, and has no isolated points, and so is homeomorphic to the space of rational numbers.
The article is concluded with a discussion of some open questions.
Universal equivariant compactifications
=======================================
In this and the next section we will discuss known results, with a view to make the presentation self-sufficient.
All our actions will be on the left. If a topological group $G$ acts continuously on a compact space $K$, then the associated representation of $G$ by linear isometries on the space $C(K)$, $${^gf}(x) = f(g^{-1}x),~~g\in G,~x\in K,~f\in C(K),$$ is continuous. Now let $G$ also act continuously on a Tychonoff space $X$, and let $i\colon X\to K$ be an equivariant continuous map with a dense image; that is to say, the $G$-space $K$ is an [*equivariant compactification*]{} of the $G$-space $X$. For every $f\in C(K)$, the composition $f\circ i$ is a bounded continuous function on $X$, and as a consequence of the continuity of the action of $G$ on $C(K)$, the orbit map $$G\ni g\mapsto {^gf\circ i}={^g(f\circ g)}\in \CB(X)$$ is continuous as a mapping from $G$ to the Banach space (moreover, a $C^\ast$-algebra) $\CB(X)$ of all continuous bounded complex functions on $X$ with the supremum norm.
This allows one to retrieve the [*maximal*]{} (or: [*universal*]{}) equivariant compactification of $X$, as follows. Denote $\UCB_G(X)$ the family of all functions $f\in \CB(X)$ for which the orbit map $g\mapsto {^gf}$ is continuous. In other words, one has: $$\forall \e>0,~~\exists V\ni e,~\forall x\in X~\forall v\in V~~{\lvertf(x)-f(vx)\rvert}<\e.$$ (Such functions are sometimes called either [*$\pi$-uniform*]{} or [*$\alpha$-uniform*]{} on $X$, where $\pi$ or $\alpha$ is the symbol for the action of $G$ on $X$. Another possibility is to simply call those functions [*right uniformly continuous*]{} on $X$.) This $\UCB_G(X)$ is a $C^\ast$-subalgebra of $\CB(X)$, and so $G$ acts continuously on the corresponding maximal ideal space, $\alpha_G(X)$. Equipped with the canonical continuous map $X\mapsto \alpha_G(X)$ having a dense image (evaluations at points $x\in X$), this is the universal equivariant compactification of $X$: indeed, for any other compactification $K$, the functions $f\circ i$, $f\in C(X)$, form a $C^\ast$-subalgebra of $\UCB_G(X)$, meaning that $X$ is an equivariant factor-space of $\alpha_G(X)$.
Thus, in order to prove that every equivariant compactification of a $G$-space $X$ is trivial, it is enough to show that $\alpha_G(X)$ is a singleton, or, the same, $\UCB_G(X)$ consists of constant functions only.
All of the above facts are quite easy observations, and the construction appears not only in the context of continuous actions, but also Borel and measurable ones. (Cf. e.g. [@GTW; @Gl-W3]; we will allude to this parallel again in the Conclusion.)
For a more detailed treatment with numerous related results, references, and open problems, see Megrelishvili’s survey paper [@megrelishvili07].
Equivariant metric fan
======================
Here we revise Megrelishvili’s important construction from [@megrelishvili; @megrelishvili89].
Denote $M$ the usual metric fan, that is, the union of countably many copies of the unit interval, $[0,1]\times\N$, with the base $\{0\}\times\N$ glued to a single point, $\ast$, and the topology given by the graph metric. In particular, $M$ is homeomorphic to the union of straight line intervals $[0,e_n]$ in the Hilbert space $\ell^2$ joining zero with the basic vectors $e_n$, $n\in\N_+$, with the norm topology. This is the cone over the discrete space $\N$ of the positive integers, except that the topology is weaker than that in the cone construction as used in algebraic topology.
The main invention of Megrelishvili is the clever choice of an acting group, $G$. It consists of all homeomorphisms of the fan keeping fixed each point located at a distance $1/k$, $k=1,2,3,\ldots$ from the base point $\ast$. (We will refer to such points in the future as [*marked points.*]{}) Thus, each element of $G$, when restricted to an interval of the form $[1/(k+1),1/k]\times\{n\}$, $k,n\in\N_+$, is an orientation-preserving self-homeomorhism of the interval. This means that $G$, as a group, is the product of a countably infinite family of copies of the group of homeomorphisms of the interval. Equip the group of homeomorphisms of every interval with the usual compact-open topology (that is, the topology of uniform convergence on the interval), and $G$, with the corresponding product topology. Now one can easily verify that this topology is the compact-open topology, or the topology of compact convergence on the fan.
The action of $G$ on $M$ is continuous. At every point $x\in M\setminus\{\ast\}$ (including the marked points) this is so because $x$ has a $G$-invariant closed neighbourhood homeomorphic to the unit interval, and locally the action of $G$ is just the action of some group of homeomorphisms with the compact-open topology which is continuous. At the base point $\ast$, the action is continuous namely due to all those fixed points converging to it in all directions. If $\e>0$, one can select $k$ with $1/k<\e$, then the union $[0,1/k]\times\N$ is a $G$-invariant neighbourhood of $\ast$, contained in the $\e$-ball around $\ast$.
Now let $f$ be a real-valued $\pi$-uniform function on $M$. One can assume that $f$ takes values in the interval $[0,1]$. Fix an $\e>0$. Since $f$ is continuous at the base point, there is $k$ with ${\lvertf(x)-f(\ast)\rvert}<\e/2$ whenever $d(\ast,x)\leq 1/k$, that is, if $x$ belongs to one of the intervals $[0,1/k]\times\{n\}$.
For each one of the fixed points $(1/k,n),(1/(k-1),n),\ldots,(1,n)$, $n\in\N$, because of the continuity of $f$, there is $\e_{i,n}>0$ such that the difference of value of $f$ at the points $\left(\frac 1i\pm \e_{i,n},n\right)$ is less than $\e/2k$. We can assume $\e_{i,n}$ to be so small that the open interval of radius $\e_{i,n}$ around $(1/k,n)$ contains no other marked points. Since $f$ is $\pi$-uniform, for some neighbourhood $V$ of the identity in $G$ one has ${\lvertf(x)-f(vx)\rvert}<\e/2k$ for each $x\in M$ and $v\in V$. In view of the definition of the topology of $G$, for all $n$ sufficiently large, $n\geq N$, the restriction of $V$ to every interval between two consequent fixed points along $[0,1]\times \{n\}$ contains all homeomorphisms of this interval. In particular, suitable elements of $V$ take $\left(\frac 1k+\e_{k,n},n\right)$ to $\left(\frac 1{k-1}-\e_{k,n},n\right)$, $\left(\frac 1{k-1}+\e_{k,n},n\right)$ to $\left(\frac 1{k-2}-\e_{k,n},n\right)$, and so on. The triangle inequality implies that ${\lvertf(1,n)-f(\ast)\rvert}<\e$ for all $n\geq N$.
By the definition of the topology of the universal compactification $\alpha_G(M)$, the images $i(1,n)$, $n\to\infty$ converge to $i(\ast)$. The continuous equivariant mapping $i$ is no not a homeomorphic embedding.
The above construction was further modified by Megrelishvili as follows. Let $M(x)$ and $M(y)$ be two copies of the metric fan, having base points $x$ and $y$ respectively, where $x\neq y$. The same group $G$ acts continuously on both spaces. Now glue $M(x)$ and $M(y)$ together along the respective endpoints $(1,n)$, $n\in\N$, with the quotient topology (the same: the topology given by the graph metric). Thus, we obtain a “metric suspension” over $\N$. Since the endpoints are all fixed, the action of $G$ on the resulting space is well-defined and continuous. In the universal $G$-compactification of the “double fan” $M(x,y)$, one has $i(1,n)\to i(x)$ and $i(1,n)\to i(y)$, and so $i(x)=i(y)$. The mapping $i$ is not even injective.
In this paper we find it more convenient to replace the acting group, $G$, with a product of two copies of the same topological group, $G_x\times G_y$, where $G_x$ acts on $M(x)$ and $G_y$ acts on $M(y)$. The action by each of the respective groups, $G_x$ and $G_y$, extend over the disjoint union $M(x)\sqcup M(y)$ by acting on the “complementary part” in a trivial way, that is, leaving each point fixed. This way, we obtain a continuous action of the product group $G_x\times G_y$. After we glue $M(x)$ and $M(y)$ together along the endpoints $(1,n)$, $n\in\N$, the action of the product group $G_x\times G_y$ on $M(x,y)$ is still well-defined and it is easy to see that it is continuous. Again, in the universal $G_x\times G_y$ equivariant compactification of $M(x,y)$, the images of the points $x$ and $y$ are identical.
Megrelishvili’s double metric fan, or metric suspension, will be the key element in the construction below. However, we need to introduce slight modifications.
First of all, notice that the topology on the group $G$ can be strengthened to the topology of pointwise convergence on $M$ viewed as discrete. In other words, the basic system of neighbourhoods of identity is formed by finite intersections of stabilizers $\St_{x_1}\cap\ldots\cap \St_{x_n}$, for all finite collections $x_1,x_2,\ldots,x_n\in M$. This is a group topology. On the group of homeomorphisms of the interval, this topology is easily seen to be finer than the compact-open topology: indeed, given $\e>0$, find $k>\e^{-1}$, and if a homeomorphism stabilizers every point $1/k, 2/k,\ldots, (k-1)/k$, then it moves any point of the interval by less than $\e$. And on the group $G$, the pointwise topology is that of the product of groups of homeomorphisms of the individual intervals between marked points, if each one of them is equipped with the pointwise topology. Consequently, our topology is strictly finer than the topology used by Megrelishvili, and so the action is still continuous. This fact is easily seen directly as well: for points $x$ other than the base point, one can have $x$ “boxed in” on both sides by means of two points $y,z$ stabilized by an open neighbourhood, $V\ni e$, and as a result, every element of $V$ will take the interval $(y,z)$ to itself. At the base point, the continuity of the action is, as before, a consequence of the existence of the uniformly convergent sequences of fixed points in all directions.
When this topology is used, the interval $[0,1]$ can be replaced just with a linearly ordered set, $(X,\leq)$. The only restrictions are: $X$ contains the smallest element to which there converges a strictly decreasing countable sequence, and the order-preserving bijections stabilizing elements of this sequence (of marked points) act transitively on each interval between two consecutive marked points. In this paper, we will make the most economical selection, namely, the interval $[0,1]_{\Q}$ of the rational numbers. Also, technically it would be more convenient for us to fix as the convergent sequence $(2^{-k})_{k=0}^{\infty}$ rather than $(1/k)_{k=1}^{\infty}$.
To sum up, we will denote $M(x,y)$ the [*rational Megrelishvili double fan*]{}. As a set, it is obtained from $[0,1]_\Q\times \N\times \{x,y\}\times \{(x,y)\}$ by identifying
- all the points $(0,n,x,(x,y))$, $n\in\N_+$, with the point $x$,
- all the points $(0,n,y,(x,y))$, $n\in\N_+$, with the point $y$, and
- every point $(1,n,x,(x,y))$ with the point $(1,n,y,(x,y))$, $n\in\N_+$.
Notice that the order of $x$ and $y$ matters, which is why we have included the indexing pair $(x,y)$ in the definition. However, if it is clear what pair we are talking about, we will still simply talk of the points $(t,n,x)$ and $(t,n,y)$, suppressing the index $(x,y)$.
The topology on the space $M(x,y)$ is given by the graph distance, that is, the length of the shortest path joining two points. A path between two points $a,b$ is a sequence of points $x_0=a, x_1,\ldots,x_n=b$, where any two adjacent points belong to the same edge. Thus, $M(x,y)$ is a countable metric space.
A topological realization of $M(x,y)$ can be obtained inside the space $\ell^2(\N)$ by identifying $x$ with $e_0$, $y$ with $-e_0$, and by joining $e_0$ and $-e_0$ with each vector $e_n$, $n\in\N_+$, by means of a straight line segment in the space $\ell^2$ viewed as a vector space [*over the rationals,*]{} $\Q$. Then induce the norm topology from $\ell^2$.
The group $G=G_{(x,y)}$ of transformations of $M(x,y)$ consists of all homeomorphisms which keep each point of the form $$\left(2^{-k},n,x\right), ~\left(2^{-k},n,y\right),~k\in\N,~n\in\N_+,$$ fixed. The restriction of the action on each interval with the endpoints $(2^{-k-1},n,a)$ and $(2^{-k},n,a)$, $a\in \{x,y\}$, consists of all order-preserving bijections of the interval. The topology on $G$ is that of a simple convergence on $M(x,y)$ viewed as discrete, that is, induced by the embedding of $G$ into the space $M(x,y)$ with the discrete topology raised to the power $M(x,y)$.
Since the space is countable, the group $G$ is separable metrizable, and in fact Polish. Indeed, one can easily see that $G$ is the product of countably many copies of the well-known Polish group $\Aut(\Q,\leq)$ of all order-preserving bijections of the rationals, with its standard topology of simple convergence on $\Q$ as discrete. The group $G$ acts continuously on $M(x,y)$, and for every equivariant compactification of this space, one has $i(x)=i(y)$, with the same exact argument as in the original Megrelishvili’s example.
The rational version of Megrelishvili’s metric fan is not new. It was first described by J. van Mill ([@vM], Remark 3.4), see also [@kozlov], example 4 in §5.
Attaching a double fan to every pair of points of a $G$-space
=============================================================
Given a topological $G$-space $X$, the space that we will denote $M(X)$ is obtained from $X$ by attaching to every ordered pair of distinct points $x,y\in X$ a double rational fan $M(x,y)$, and subsequently enlarging the transformation group so as to bring in all the copies of $G_{(x,y)}$ acting on each $M(x,y)$. Notice that we attach two different fans to each pair $\{x,y\}$, one corresponding to the ordered pair $(x,y)$, and the other, to the pair $(y,x)$, this will be necessary later on for the symmetry reasons. Under some restrictions on $(X,G)$, the resulting action is continuous. As a consequence, within every equivariant compactification of the resulting space $M(X)$, the original space $X$ will collapse to a point. Here is a more precise description.
One of the oldest results of uniform topology (see e.g. a preliminary draft [@bourbaki] of the corresponding chapter of Bourbaki’s [*Topologie Générale*]{}) states that a topological space is Tychonoff if and only if the topology is generated by a separated uniform structure, or, equivalently, by continuous pseudometrics separating points.
Let $X$ be a Tychonoff topological space. As usual, $X^2\setminus\Delta$ denotes the collection of all ordered pairs of different points of $X$.
Consider the disjoint sum $\sqcup_{(x,y)\in X^2\setminus\Delta} M(x,y) $ of copies of the double rational fan, and glue this sum to $X$ along the canonical map sending each fan endpoint to the corresponding point of $X$. Denote the resulting set $M(X)$.
The first task is to topologize $M(X)$. Let $d$ is a pseudometric on $X$. Denote $\bar d$ the corresponding path pseudometric on $M(X)$, that is, $\bar d(a,b)$ is the smallest among the numbers $\sum_{i=0}^n d(x_i,x_{i+1})$, where $x_0=a$, $x_n=b$, and the distance between every two consecutive points $x_i,x_{i+1}$ is calculated either to $X$, or in some $M(x,y)$.
In general, the path length can be arbitrarily long, but if $d\leq 2$ (which is not a restrictive condition from the view of topology that $d$ generates), then $n\leq 3$ suffices. For instance, in this case $$\bar d(x,y) = d(x,y)\mbox{ for }x,y\in X,$$ $$\bar d((t,n,x,(x,y)),z) = \min\{t + d(x,z),2-t+d(y,z)\}\mbox{ if }x,y,z\in X,$$ and so forth. Also, one has $\bar d\leq 4$.
We will topologize $M(X)$ with the set of all pseudometrics of the form $\bar d$, as $d$ runs over all continuous pseudometrics on $X$ with $d\leq 2$. The following should be now obvious.
\[l:4.1\] The space $M(X)$ is a Tychonoff space containing $X$ and each fan $M(x,y)$ as closed subspaces. Every set $M(x,y)\setminus\{x,y\}$ is open in $M(X)$.
The resulting topology does not depend on the choice of a family of pseudometrics $d$ once it generates the topology of $X$.
Let $D$ be some family of pseudometrics generating the topology of $X$ and satisfying $d\leq 2$ for all $d\in D$. The pseudometrics $\bar d$, $d\in D$ generate the same topology on $M(X)$ as above. \[l:D\]
We need to verify that the same topology as defined by us previously is generated at every point of $M(x,y)$. For the interior points of each fan, this is clear. Let therefore $x\in X$ and let $V$ be a neighbourhood of $x$ in $M(X)$. There is a continuous pseudometric $\rho$ on $X$ and $\e$, $0<\e<1$, so that the corresponding open ball around $x$, formed in $M(X)$, is contained in $V$, $$V\supseteq B(x,\e,{\bar \rho}).$$ By our assumption on $D$, there is $d\in D$ and a $\delta>0$ with the property that $B(x,\delta,d)\subseteq B(x,\e/2,\rho)$ (the balls in $X$). Also, without loss in generality, one can assume that $\delta<\e/2$. If now $z\in B(x,\delta,\bar d)$ (the ball formed in $M(X)$), then for some unique $y,w\in X$ one has either $z=(t,n,y,(y,w))$ or $z=(t,n,y,(w,y))$ and $$\bar d(x,z) \leq t + d(x,y) < \delta<\frac{\e}2,$$ and $$\begin{aligned}
\bar\rho(x,z) &\leq & t + \rho(x,y) \\
&<& \frac{\e}2 + \frac{\e}2 = \e.
\end{aligned}$$
Next we will extend the action of the group $G$ from $X$ over $M(X)$.
Let first $g$ be an arbitrary self-homeomorphism of $X$. Then $g$ extends to a homeomorphism of $X^2\setminus\Delta$, as $(x,y)\mapsto (gx,gy)$. Given any $(x,y)$ and a point $(t,n,a)\in M(x,y)$, $a\in\{x,y\}$, define $$g(t,n,a,(x,y)) = (t,n,ga,(gx,gy))\in M(gx,gy).$$ It is clear that this defines a homeomorphism of $M(x,y)$ onto $M(gx,gy)$. In particular, if $g$ swaps $x$ and $y$, then $M(x,y)$ is being homeomorphically mapped onto $M(y,x)$. In particular, the extension of $g$ over $M(X)$ is bijective.
Every extension of a homeomorphism $g$ of $X$ as above is a homeomorphism of $M(X)$. Also, the extended homeomorphism preserves the family of the marked points of all the double fans, $(2^{-k},n,a)$. \[l:extensionofg\]
Again, it is quite obvious that $g$ is a local homeomorphism (in fact, a local isometry) at the interior point of every double fan. It is also clear that the extension of $g^{-1}$ is the inverse to the extension of $g$. It remains to verify that $g$ is continuous at an arbitrary point $x\in X$. Given a neighbourhood $V$ of $gx$, find a continuous pseudometric $d$ on $X$ and an $\e>0$ with $B(gx,\e,\bar d)\subseteq V$. Define a continuous pseudometric $\rho$ on $X$ by $\rho(a,b) = d(ga,gb)$. Now it is easy to see that the image of the ball $B(x,\e,\bar\rho)$ under $g$ is contained in (in fact, coincides with) $B(gx,\e,\bar d)$. The second statement is clear.
In this way, the group $G$ acts on $M(X)$ by homeomorphisms.
Assume the topology of the group $G$ is that of simple convergence on $X$ as discrete. Then it is also the topology of simple convergence on $M(X)$ as discrete. If in addition $G$ acts on $X$ continuously, then $G$ acts on $M(X)$ continuously as well.
Let $\tau$ denote the original topology on $G$ (that is, the topology of simple convergence on $X$ viewed as a discrete space), and $\varsigma$, the topology of simple convergence on $M(X)$ equipped with the discrete topology. Then $\tau\subseteq\varsigma$. For every point of the form $(t,n,a)\in M(x,y)$, with $a\in\{x,y\}$, its stabilizer in $G$ consists exactly of all $g$ that leave both $x$ and $y$ fixed. This implies $\varsigma\subseteq\tau$.
The continuity of the action of $G$ on $M(X)$ needs to be verified separately at the interior points of the fans and at the points of $X$. Let $x,y\in X$, $(t,n,a)\in M(x,y)$ with $0<t\leq 1$ and $a\in\{x,y\}$. Take an arbitrary neighbourhood of this point, $W$, and set $U=W\cap M(x,y)\setminus\{x,y\}$. The set $\St_x\cap\St_y$ is an open neighbourhood of the identity element in $G$ and it sends the open neighbourhood $U$ of $(t,n,a)$ to itself.
Now let $x\in X$. It is enough to look at a neighbourhood of $x$ of the form $B(x,\e,\bar d)$, where $d$ is a continuous pseudometric on $X$ and $\e<1$. Since the action of $G$ on $X$ is continuous, there is a continuous pseudometric $\rho$ and $\delta>0$, as well as a neighbourhood $V$ of the identity in $G$, satisfying $vy\in B(x,\e/2,d)$ once $v\in V$, $y\in X$, and $\rho(x,y)<\delta$. We can further assume that $\delta\leq \e/2$ and $\delta<1$. Now let $z\in B(x,\delta,\bar\rho)$, where the ball is formed in $M(X)$. Then $z$ is of the form $(t,n,y,(a,b))$, where $y\in\{a,b\}$, $t\in [0,1]$, and $\rho(x,y)+t<\delta$. In particular, $t<\delta\leq \e/2$ and $\rho(x,y)<\delta$. Let $v\in V$. Then $vz = (t,n,vy,(va,vb))$, and we conclude: $$\bar d(vz,x)\leq t + d(vy,x) <\e.$$
Suppose a topological group $G$ act continuously on a topological space $X$ in such a way that the topology of $G$ is the topology of pointwise convergence on $X$ viewed as discrete. We have seen that the action of $G$ extends over $M(X)$, and the topology of $G$ is that of simple convergence on $M(X)$ with the discrete topology. Next we will add to the acting group $G$ all the topological groups $G_{(x,y)}$ acting on each double fan and keeping the rest of the space $M(X)$ fixed.
Of course if two different topological groups continuously act on the same topological space, it is not in general true that they can be jointly embedded in a larger topological group which still acts continuously. Such examples can be found even if one of the two acting groups is discrete. Therefore, in our case, we need to construct this larger group explicitely. It is going to be a generalized wreath product of $G$ with $G_{(\ast,\star)}$, where $\ast,\star$ are two distinct abstract points.
Consider the product group $G_{(\ast,\star)}^{X^2\setminus\Delta}$ with the product topology. We will canonically identify it with the product group $\prod_{(x,y)\in X^2\setminus\Delta}G_{(x,y)}$. For every $h\in G_{(\ast,\star)}^{X^2\setminus\Delta}$, $h=(h_{(x,y)})_{(x,y)\in X^2\setminus\Delta}$, define a self-bijection of $M(X)$ as follows: if $z\in M(x,y)$, $z=(t,n,a)$, $a\in\{x,y\}$, then $$h(z) = \left(h_{(x,y)}(t,n,a) \right)\in M(x,y).$$
In this way, we obtain an action of the group $G_{(\ast,\star)}^{X^2\setminus\Delta}$ on $M(X)$. Our aim is to show that it is a continuous action by homeomorphisms, and the topology of the group is that of pointwise convergence on $M(X)$ viewed as discrete. We start with a technical observation.
Let $z\in X$ and $0<\e<1$, and let $d$ be a continuous pseudometric on $X$. Then the image of the ball $B(z,\e/2,\bar d)$ formed in $M(X)$ under every transformation $h\in G_{(\ast,\star)}^{X^2\setminus\Delta}$ is contained in $B(z,\e,\bar d)$. \[l:sepres\]
Every point $x\in X$ is fixed by our group, as are the points of each convergent sequence $(2^{-k},n,x)$, $k\in\N$, inside of $M(a,b)$, where $x\in\{a,b\}$. The intersections of $V=B(z,\e,\bar d)$ and of $U=B(z,\e/2,\bar d)$ with the interval $[0,1]\times \{(n,x)\}$, provided they are non-empty, are the semi-open intervals $[0, \e-d(z,x)) \times \{(n,x)\}$ and $[0, \e/2-d(z,x)) \times \{(n,x)\}$ respectively, and there is at least one marked point of the form $(2^{-k},n,x)$ between the endpoints of the two intervals. Since the marked point is fixed by every $h\in G_{(\ast,\star)}^{X^2\setminus\Delta}$, the conclusion follows.
The above action of the group $G_{(\ast,\star)}^{X^2\setminus\Delta}$ on $M(X)$ is a continuous action on $M(X)$ by homeomorphisms. In addition, the product topology on this group is the topology of simple convergence on $M(X)$ viewed as discrete.
To verify that every $h$ gives a homeomorphism of $M(X)$, it is enough to check that $h$ is continuous at every point. For the interior points of the fans, it is clear from the definition, and for $x\in X$, it follows from Lemma \[l:sepres\].
Let us verify the continuity of the action at an arbitrary point $z\in M(X)$. For $z\in X$, this is again a consequence of Lemma \[l:sepres\]. Now let $z$ be an interior point of some double fan, $M(x,y)$. Let $V$ be a neighbourhood of $z$. There is a neighbourhood $O$ of the identity in the group $G_{(x,y)}$ and an open neighbourhood $U$ of $z$ in $M(x,y)$ with $O\cdot U\subseteq V$. Denote $\tilde O$ the standard basic neighbourhood of the identity in the product group $G_{(\ast,\star)}^{X^2\setminus\Delta}$ which is a cylinder set over $O$, that is, $h\in\tilde O$ if and only if $h_{(x,y)}\in O$. Then $\tilde O\cdot U\subseteq V$.
To verify the last statement of Lemma, if $F$ is a finite subset of $M(X)$, then for every pair $(x,y)\in X^2\setminus\Delta$ the stabilizer of $F\cap M(x,y)$ in $G_{(x,y)}$ is an open subgroup, and all but finitely many such stabilizers coincide with the entire group $G_{(x,y)}$. It follows that the pointwise stabilizer of $F$ is a standard basic neighbourhood of the identity element in the product. Since the topology of each $G_{(x,y)}$ is that of simple convergence on $M(x,y)$ with the discrete topology, we conclude that the pointwise stabilizers of finite subsets of $M(X)$ form a basic neighbourhood system for the product topology.
The group $G$ acts on $X$, and hence on $X^2\setminus\Delta$, and under our assumptions, the topology of $G$ is that of simple convergence on $X^2\setminus\Delta$ viewed as discrete. Further, $G$ acts on $G_{(\ast,\star)}^{X^2\setminus\Delta}$ by coordinate permutations: if $g\in G$, $h\in G_{(\ast,\star)}^{X^2\setminus\Delta}$, and $(x,y)\in X^2\setminus\Delta$, then $$({^g h})_{(x,y)} = h_{(g^{-1}x,g^{-1}y)}.$$ This is an action of $G$ by automorphisms of the topological group $G_{(\ast,\star)}^{X^2\setminus\Delta}$.
\[ Recall that if a topological group $G$ acts by automorphisms on a topological group $H$, and the action is continuous, then the semidirect product $G\ltimes H$ is the cartesian product $G\times H$ with the product topology and the group operation $$(a,b)(c,d)=(ac,b\cdot{^ad})$$ is a topological group. It contains both $G$ and $H$ as closed topological subgroups in a canonical way, and $H$ is normal, with $$ghg^{-1} = {^gh}.~~]$$
Under our assumptions on $G$, the above action is continuous, and so the corresponding semidirect product $G\ltimes G_{(\ast,\star)}^{X^2\setminus\Delta}$ is a topological group.
Let $h\in G_{(\ast,\star)}^{X^2\setminus\Delta}$ and let $V$ be a neighbourhood of $h$. It is enough to consider in place of $V$ a basic product neighbourhood: $v\in V$ if and only if $v_x\in V_x$ for all $x$ belonging to a finite set $F\subseteq X$, where each $V_x$ is a neighbourhood of $h_x$ in $G_{(\ast,\star)}$. The open subgroup $O=\cap_{x\in F}\St_x$ of $G$ has the property $O\cdot V=V$.
The topological group $\tilde G = G\ltimes G_{(\ast,\star)}^{X^2\setminus\Delta}$ acts continuously on $M(X)$ extending the actions of $G$ and of $G_{(\ast,\star)}^{X^2\setminus\Delta}$. The topology of $\tilde G$ is that of pointwise convergence on $M(X)$ viewed as discrete.
To verify that the actions of $G$ and of $G_{(\ast,\star)}^{X^2\setminus\Delta}$ combine together to give an action of the semidirect product, it is enough to verify that for each $g\in G$ and $h\in G_{(\ast,\star)}^{X^2\setminus\Delta}$, the action by the element ${^g h}$ on $M(X)$ equals the composition of actions of three elements, $g\circ h\circ g^{-1}$. For every pair $(x,y)$ and each point $(t,n,a,(x,y))$, $a\in\{x,y\}$, $t\in [0,1]$, one has $$\begin{aligned}
g\circ h\circ g^{-1}(t,n,a,(x,y)) &=& g\circ h (t,n,g^{-1}(a),(g^{-1}x,g^{-1}y)) \\
&=& g\left(h_{(g^{-1}x,g^{-1}y)}(t,n, g^{-1}(a),(g^{-1}x,g^{-1}y))\right) \\
&=& h_{(g^{-1}x,g^{-1}y)}(t,n,a,(x,y))) \\
&=& {^g h}(t,n,a,(x,y)).
\end{aligned}$$ Every element $(g,h)$ of $\tilde G$ can be uniquely written as the product $(e,h)(g,e)$, where $g\in G$, $h\in G_{(\ast,\star)}^{X^2\setminus\Delta}$, and the rule $$(g,h)(x) = h(g(x))$$ consistently defines an action. Indeed, $$\begin{aligned}
(a,b)(c,d)(x)&=& badc(x) \\
&=& bada^{-1}ac(x) \\
&=& (e,b)(e,{^ad})(ac,e)(x) \\
&=& (e,b{^ad})(ac,e)(x) \\
&=& (ac,b\,{^ad})(x).
\end{aligned}$$
Since the actions of both $G$ and $G_{(\ast,\star)}^{X^2\setminus\Delta}$ are continuous, it follows immediately that so is the action of the semidirect product.
The topologies of both groups are those of simple convergence on $M(X)$ viewed as discrete. Let $F$ be a finite subset of $M(X)$. Denote $\Phi$ the union of all sets $\{x,y\}$ having some $z\in F$ belonging to $M(x,y)$. Then the stabilizer of $F$ in $\tilde G$ is the cartesian product of the stabilizer of $\Phi$ in $G$ and a standard open neighbourhood of the identity in the product group, $U= \prod_{x\in X} U_x$, where each $U_x$ is a neighbourhood of the identity in the group $G_{(a_x,b_x)}$, stabilizing a finite set, and $U_x$ equals the entire group whenever $x\notin \Phi$. These observations implies the last statement of Lemma.
Iterating the procedure
=======================
Let $X$ be a $G$-space such that the topology of $G$ is that of simple convergence on $X$ as discrete. Repeat iteratively the extension ($M(X)$, $\tilde G$) countably infinitely many times. Denote the corresponding $n$-th iterations by $M^n(X)$ and $\tilde G^{(n)}$.
Each space $M^n(X)$ is contained inside $M^{n+1}(X)$, so we can form the union $M^{\infty}(X)$. Let us equip it with a topology. If $d$ is a continuous pseudometric on $X$ with the property $d\leq 2$, the graph metric extension $\bar d$ over $M(X)$ satisfies $\bar d\leq 4$. The pseudometric $\tilde d = \min\{\bar d, 2\}$ is continuous and induces the same topology as $\bar d$. Getting back to our construction, denote the $n$-th iteration of $d$ corresponding to the extension $\tilde d$ by $\tilde d^{(n)}$. This is a pseudometric on $M^{\infty}(X)$, and we equip the space with the topology generated by all such pseudometrics. Notice that the restriction of $\tilde d^{(\infty)}$ to each $M^{(n)}(X)$ equals $\tilde d^{(n)}$.
The action of each group $\tilde G^{(n)}$ on $M^n(X)$ lifts recursively to an action of $\tilde G^{(n)}$ on $M^{\infty}(X)$. Notice that each subspace $M^{(m)}(X)$ is invariant under the action of every group $\tilde G^{(n)}$ for all $m,n$.
From now on, we will be only working with a particular case of interest to us. Namely, we will start with the two-point space $X=\{\ast,\star\}$ such that $d(\ast,\star)=2$, equipped with an action of a trivial group $G=\{e\}$. The resulting space $M^{\infty}(X)$ is a countable metric space with the metric $\tilde d^{\infty}$, a graph with rational edges, infinitely branching at every point, and equipped with the graph distance. Because of Lemma \[l:D\], the topology induced by the metric $\tilde d^{\infty}$ on every subspace $M^{\infty}(X)$ is the same as the topology we have defined elsewhere.
Let $n\in\N$ and $g\in \tilde G^{(n)}$. The recursive extension of $g$ over $M^{\infty}(X)$ is a homeomorphism.
We have noted before that the extension of $g^{-1}$ from $X$ over $M(X)$ is the inverse of the extension of $g$ for every topological space $X$, therefore the same holds for the extension over $M^{\infty}(X)$. It is enough to verify that $g$ is continuous at an arbitrary point $z\in M^{\infty}(X)$.
Suppose first $z\notin M^n(X)$. Then $\e=\tilde d^{(\infty)}(z, M^n(X))>0$, and the restriction of $g$ to the open $\e$-ball around $z$ in $M^{\infty}(X)$ is an isometry, as seen from the recursive way in which $g$ is being extended.
Now suppose $z\in M^n(X)$. Given a neighbourhood $V$ of $gz$, find an $\e$ with $0<\e<1$ and $B(gz,\e,\tilde d^{(\infty)})\subseteq V$, where the open ball is formed in the metric space $M^{\infty}(X)$. There is $\delta$ satisfying $0<\delta<\e/2$ and $gB(z,\delta,\tilde d^{(n)})\subseteq B(gz,\e/2,\tilde d^{(n)})$ (the balls in $M^n(X)$). If $x\in B(z,\delta,\tilde d^{(\infty)})$, then there are $a,b\in M^n(X)$ and $w\in\{a,b\}$ with $\tilde d^{(\infty)}(x,z) = \tilde d^{(\infty)}(x,w) + \tilde d^{(n)}(w,z)$, and we have $$\tilde d^{(\infty)}(gx,gz) \leq \tilde d^{(\infty)}(gx,gw) + \tilde d^{(n)}(gw,gz) < \tilde d^{(\infty)}(x,w) +\e/2 < \e.$$
Thus, the group $\tilde G^{(\infty)}=\cup_{n\in\N} \tilde G^{(n)}$ acts on the countable metric space $M^{\infty}(X)$ by homeomorphisms. We equip it with the topology of simple convergence on $M^{\infty}(X)$ viewed as discrete. This is a separable, metric group topology, inducing the usual topology on every subgroup $\tilde G^{(n)}$.
The action of the topological group $\tilde G^{(\infty)}$ on the space $M^{\infty}(X)$ is continuous.
We will verify the continuity of the action at an arbitrary point $z\in M^{\infty}(X)$. Let $0<\e<1$. Denote $n=\min\{m\colon z\in M^{(m)}(X)\}$. Find a finite set $F\subseteq M^{(n)}(X)$ and a $\delta$ with $0<\delta<\e/4$, so that one has $gB(z,\delta,\tilde d^{(n)})\subseteq B(z,\e/2, \tilde d^{(n)})$ as soon as $g\in \tilde G^{(n)}$ and $g$ stabilizes each point of $F$.
We claim that $g^\prime B(z,\delta,\tilde d^{(\infty)})\subseteq B(z,\e/2, \tilde d^{(\infty)})$ as soon as $g^\prime\in \tilde G^{(\infty)}$ and $g^\prime$ stabilizes each point of $F$. Indeed, such a $g^{\prime}$ can be written $g^{\prime}=gh$, where $g\in \tilde G^{(n)}$, $g$ stabilizes each point of $F$, and $h$ stabilizes each point of $M^{(n)}(X)$. Let $x\in B(z,\delta,\tilde d^{(\infty)})$. Then for some $m\geq n$, $$\tilde d^{(\infty)}(x,z) = \sum_{i=n}^m {\lvertx_i-x_{i+1}\rvert} + \tilde d^{(n)}(x_n,z),$$ where $x_i\in M^{(i)}(X)$, and the notation is slightly abused to stress the fact that the distance between each two consecutive points is taken within a suitable rational interval. In particular, $\sum_{i=n}^m {\lvertx_i-x_{i+1}\rvert}<\delta$. For every $i$, there is a marked point along the edge starting at $x_i$ and going towards $x_{i+1}$ and beyond, at some distance $2^{-k_i}$ from $x_i$ so that ${\lvertx_i-x_{i+1}\rvert}\leq 2^{-k_i}< 2{\lvertx_i-x_{i+1}\rvert}$. This implies that the distance between $x_i$ and $x_{i+1}$ cannot be increased by more than twice under any transformation, and we have $$\begin{aligned}
\tilde d^{(\infty)}(g^{\prime}x,g^{\prime}z) + \tilde d^{(n)}(gx_n,gz)&\leq &
\sum_{i=n}^m {\lvertg^{\prime}x_i-g^{\prime}x_{i+1}\rvert}\leq \sum_{i=n}^m 2^{-k_i} + d^{(n)}(gx_n,gz) \\ &<& 2\delta + \frac{\e}2 < \e.
\end{aligned}$$
Consider the universal equivariant compactification of $M^{\infty}(X)$ under the action of $\tilde G^{(\infty)}$. For any $x,y\in M^{\infty}(X)$, there is $n$ with $x,y\in M^{(n)}(X)$. Therefore, $x$ and $y$ are joined by an equivariant Megrelishvili double fan $M(x,y)$ inside $M^{(n+1)}(X)$. The topological group $G_{x,y}$ is a topological subgroup of $\tilde G^{(n+1)}$, therefore of $\tilde G^{(\infty)}$, with the standard action on $M(x,y)$. Under the mapping $i$, one has $i(x)=i(y)$. We conclude: the entire universal compactification of $M^{\infty}(X)$ is a singleton.
Finally, notice that, according to a well-known result of Sierpinski [@sierpinski], a countable metrizable space without isolated points is homeomorphic to the space of rational numbers, $\Q$, with the usual topology. Thus, topologically, $M^{\infty}(X)$ is just $\Q$.
Concluding remarks
==================
The referee suggested that the recent preprint [@GJ] constructing universal compact metrizable $\R$-spaces can be used to give an explicit proof of de Vries’ theorem [@dV-1; @dV] in the particular case $G=\R$.
Both $X$ and $G$ in our example are separable metrizable. The author does not know if the example can be modified so as to have both $X$ and $G$ Polish (separable completely metrizable).
The example constructed here is of course artificial. However, if the story of extreme amenability is any indication (see e.g. [@P06]), the phenomenon may one day be naturally found. One needs also compare the phenomenon discovered and explored by Glasner, Tsirelson and Weiss [@GTW] and Glasner and Weiss [@Gl-W3] whereby a weakly continuous action of a Polish group on a standard probability measure space cannot be realized spatially, that is, as a set-theoretic action on a compact space equipped with an invariant measure. Here, the phenomenon occurs very naturally: every Lévy group [@GrM] of measure-preserving transformations behaves in such a way.
This reinforces a feeling that there might exist numerous [*natural*]{} examples of topological transformation groups whose universal equivariant compactification is a singleton.
For instance, can such an example be realized as the group of automorphisms of a suitable (discrete or continuous) ultrahomogeneous Fraïssé structure?
The [*Gromov compactification*]{} of a bounded metric space $X$ corresponds to the $C^\ast$-algebra generated by all distance functions $x\mapsto d(x,-)$; under the action of the isometry group with the pointwise topology, such functions are always $\pi$-uniform, and so the Gromov compactification is equivariant, and it is a homeomorphic embedding (see e.g. [@megrelishvili07], sect. 2). This means that the topology on the conjectural Fraíssé structure needs to be modified, but again, in some “natural” way (given by a suitable partial order for instance, as it is esentially the case in our example).
An elegant “natural” example where the universal equivariant compactification has been calculated explicitely is that of the unit sphere $\s^{\infty}$ in the Hilbert space $\ell^2$ under the action of the unitary group $U(\ell^2)$ with the strong operator topology. This compactification is the unit ball ${\mathbb{B}}^{\infty}$ with the weak topology. This is a result by Stojanov [@stoyanov].
This motivated Megrelishvili to ask whether the same conclusion holds for the unit sphere in every separable reflexive Banach space $E$ under the action of the group of isometries of $E$ with the strong operator topology, see Question 2.5 (2005) in [@megrelishvili07]. However, it appears to us the question needs to be adjusted.
According to Jarosz [@jarosz], every Banach space can be renormed in such a way that the group of isometries consists of constant multiples of the identity. For such a renormed space, the Stone-Čech compactification of the projective space is an equivariant compactification of the sphere, because the action of the group of isometries is trivial. Since for $E$ infinite dimensional the Stone-Čech compactification is non-metrizable, the answer to the question as stated is in the negative.
However, it makes sense to reformulate the question for separable reflexive Banach spaces whose group of isometries has a dense orbit in the unit sphere. For instance, it appears the answer is already unknown for $L^p(0,1)$, $1<p<\infty$, $p\neq 2$.
Even for non-reflexive spaces the question makes sense. In particular, the group of isometries of the Gurarij space has attacted plenty of interest recently [@BY]. Is it true that the universal equivariant compactification of the unit sphere in the Gurarij space under the action of the group of isometries with the pointwise topology is the Gromov compactification of the sphere?
The same question, for the Holmes space [@holmes] (see also [@P06], pp. 112–113).
Notice in this connection that the closed ball of $\ell^2$ in the weak topology is the Gromov compactification of the unit sphere.
The analogous question, for the Urysohn sphere $\s_{\mathbb U}$ [@NVTS]. More precisely: is the universal equivariant compactification of the Urysohn sphere under the action of the group of isometries with the pointwise topology equal to the Gromov compactification of $\s_{\mathbb U}$? Probably the answer is in the positive and should not be very difficult to obtain, as the group of isometries of the Urysohn sphere is by now pretty well understood.
In conclusion, here is a question suggested by Furstenberg and Scarr (see [@megrelishvili07], question 2.6 (2006)): does there exist a topological transformation group $(G,X)$ whose universal equivariant compactification is a singleton, yet the action of $G$ on $X$ is transitive?
Such an example cannot be Polish because of Effros’ Microtransivitity Theorem combined with the observation that for a closed subgroup $H$ of a topological group $G$, the equivariant compactification $\alpha_G(G/H)$ is always a homeomorphic embedding of $G/H$ [@dV-2].
As noted by Jan van Mill ([@vM], question 3.5) the Furstenberg–Scarr question remains open already for groups acting on the space of rational numbers, $\Q$. Of course in our example the action on $\Q$ is highly non-transitive in view of all those marked points.
Can there be a suitable group of homeomorphisms of the Hilbert space $\ell^2$ whose equivariant compactification is trivial?
Acknowledgements
================
I am very indebted to Michael Megrelishvili from whom over the years I have learned many things, including Smirnov’s question. Dana Bartošová and Micheal Pawliuk were considering my invitation to join as collaborators at the early stages of this project, and the ensuing discussions with them helped to have some of my initial ideas discarded. I am grateful to Konstantin Kozlov for providing some references, and for the anonymous referee for a number of comments.
Much of this work has been done while the author was a 2012–2015 Special Visiting Researcher of the program Science Without Borders of CAPES (Brazil), processo 085/2012.
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[^1]: [*2000 Mathematics Subject Classification:*]{} 37B05; 54H15
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abstract: |
Using the techniques connected with the measure of noncompactness we investigate the neutral difference equation of the following form $$\Delta \left( r_{n}\left( \Delta \left( x_{n}+p_{n}x_{n-k}\right) \right)
^{\gamma }\right) +q_{n}x_{n}^{\alpha }+a_{n}f(x_{n+1})=0,$$where $x:{\mathbb{N}}_{k}\rightarrow {\mathbb{R}}$, $a,p,q:{\mathbb{N}}_{0}\rightarrow {\mathbb{R}}$, $r:{\mathbb{N}}_{0}\rightarrow {\mathbb{R}}\setminus \{0\}$, $f\colon {\mathbb{R}}\rightarrow {\mathbb{R}}$ is continuous and $k$ is a given positive integer, $\alpha \geq 1$ is a ratio of positive integers with odd denominator, and $\gamma \leq 1$ is ratio of odd positive integers; ${\mathbb{N}}_{k}:=\left\{ k,k+1,\dots \right\} $. Sufficient conditions for the existence of a bounded solution are obtained. Also a special type of stability and asymptotic stability are studied. Some earlier results are generalized. We note that the solution which we obtain does not directly correspond to a fixed point of a certain continuous operator since it is partially iterated. The method which we develop allows for considering through techniques connected with the measure of noncompactness also difference equations with memory.
author:
- 'Marek Galewski[^1]'
- 'Robert Jankowski[^2]'
- 'Magdalena Nockowska–Rosiak[^3]'
- 'Ewa Schmeidel[^4]'
title: On the existence of bounded solutions for nonlinear second order neutral difference equations
---
Introduction
============
As it is well known difference equations serve as mathematical models in diverse areas, such as economy, biology, physics, mechanics, computer science, finance, see for example [@bib1B], [@bib2B]. One of such models is the Emden–Fowler equation which originated in the gaseous dynamics in astrophysics and further was used in the study of fluid mechanics, relativistic mechanics, nuclear physics and in the study of chemically reacting systems, see [@wong]. For the reader’s convenience, we note that the background for difference equations theory can be found in numerous well–known monographs: Agarwal [@bib1B], Agarwal, Bohner, Grace and O’Regan [@ABGO], Agarwal and Wong [@AW] Elaydi [@bib2B], Kelley and Peterson [@bib4B], and Kocić and Ladas [@bib3B].
In the present paper we study using techniques connected with the measure of noncompactness the existence of a bounded solution and some type of its asymptotic behavior to a nonlinear second order difference equation, which can be viewed as a generalization of a discrete Emeden–Fowler equation or else it can be viewed as a second order difference equation with memory. This makes a problem which we consider different from those already investigated via techniques of measure of noncompactness, see for example [@SchmeildelZbaszyniakCAMW] since we do not expect a direct connection between a fixed point of a suitably defined operator investigated on a non-reflexive space $l^{\infty }$ a solution to the problem under consideration. Indeed, this is the case. What we obtain is that starting from some index which we define the solution is taken from the fixed point while the previous terms have to be iterated. This also makes the definition of the operator different from this which would be used had the problem been without dependence on previous terms. It seems that the method which we sketch here would prove applicable for several other problems. We also note that due to the type of space which we use, namely $l^{\infty }$ we cannot apply standard fixed point techniques such us Banach Theorem or Schauder Theorem and related results. We expect that our method would apply for systems of difference equations. However, what we cannot obtain here is the asymptotic stability of the solution. This no surprise since solution which we get is not is a fixed point of some mapping. We will use axiomatically defined measures of noncompactness as presented in paper [@bib1] by Banaś and Rzepka.
The problem we consider is as follows $$\Delta \left( r_{n}\left( \Delta \left( x_{n}+p_{n}x_{n-k}\right) \right)
^{\gamma }\right) +q_{n}x_{n}^{\alpha }+a_{n}f(x_{n+1})=0. \label{e0}$$where $\alpha \geq 1$ is a ratio of positive integers with odd denominator, $\gamma \leq 1$ is ratio of odd positive integers, $x:{\mathbb{N}}_{k}\rightarrow {\mathbb{R}}$ $a,p,q:{\mathbb{N}}_{0}\rightarrow {\mathbb{R}}
$, $r:{\mathbb{N}}_{0}\rightarrow {\mathbb{R}}\setminus \{0\}$, and $f:{\mathbb{R}}\rightarrow {\mathbb{R}}$ is a locally Lipschitz function with no further growth assumptions. Here ${\mathbb{N}}_{0}:=\left\{ 0,1,2,\dots
\right\} $, ${\mathbb{N}}_{k}:=\left\{ k,k+1,\dots \right\} $ where $k$ is a given positive integer, and ${\mathbb{R}}$ is a set of all real numbers. By a solution of equation we mean a sequence $x:{\mathbb{N}}_{k}\rightarrow {\mathbb{R}}$ which satisfies for every $n\in {\mathbb{N}}_{k}$.
There has been an interest of many authors to study properties of solutions of the second–order neutral difference equations attract attention; see the papers [@DGJ2002], [@bibGL]–[@JS2], [@bibLG]–[@bibmm2], [@S1]–[@S3], [@TKP1]–[@TKP2] and the references therein. The interesting oscillatory results for first order and even order neutral difference equations can be found in [@LadasQian], [@MM2004] and [@MM2009].
Preliminaries
=============
Let $(E,\left\Vert \cdot \right\Vert )$ be an infinite–dimensional Banach space. If $X$ is a subset of $E$, then $\bar{X}$, $ConvX$ denote the closure and the convex closure of $X$, respectively. Moreover, we denote by ${\mathcal{M}}_{E}$ the family of all nonempty and bounded subsets of $E$ and by ${\mathcal{N}}_{E}$ the subfamily consisting of all relatively compact sets.
A mapping $\mu\colon{\mathcal{M}}_E \to [0,\infty)$ is called a measure of noncompactness in $E$ if it satisfies the following conditions:
$1^0$
: $\ker \mu = \left\{X \in {\mathcal{M}}_E \colon \mu(X)=0
\right\}\neq \emptyset \mbox{ and } \ker \mu \subset {\mathcal{N}}_E, $
$2^0$
: $X \subset Y \Rightarrow \mu(X) \leq \mu(Y), $
$3^0$
: $\mu(\bar{X})=\mu(X)=\mu(Conv \,\, X), $
$4^0$
: $\mu( c X +(1- c)Y ) \leq c \mu(X)+(1-c) \mu(Y) \mbox{ for }
0 \leq c \leq 1, $
$5^0$
: If $X_n \in {\mathcal{M}}_E, \,\,\, X_{n+1}\subset X_n, \,\,\,
X_n=\bar{X_n} \mbox{ for }n=1,2,3,\dots $and $\lim\limits_{n\to\infty} \mu(X_n)=0 \mbox{ then } \bigcap
\limits_{n=1}^{\infty} X_n\neq \emptyset$.
The following Darbo’s fixed point theorem given in [@bib1] is used in the proof of the main result.
\[D\] Let $M$ be a nonempty, bounded, convex and closed subset of the space $E$ and let $T:M\rightarrow M$ be a continuous operator such that $\mu
(T(X))\leq k\mu (X)$ for all nonempty subset $X$ of $M$, where $k\in \lbrack
0,1)$ is a constant. Then $T$ has a fixed point in the subset $M$.
We consider the Banach space $l^{\infty }$ of all real bounded sequences $x\colon {\mathbb{N}}_{k}\rightarrow {\mathbb{R}}$ equipped with the standard supremum norm, i.e. $$\Vert x\Vert =\sup_{n\in {\mathbb{N}}_{k}}|x_{n}|\text{ for }x\in \
l^{\infty }.$$Let $X$ be a nonempty, bounded subset of $l^{\infty }$, $X_{n}=\left\{
x_{n}:x\in X\right\} $ (it means $X_{n}$ is a set of $n$-th terms of any sequence belonging to $X$), and $$diam\,\,X_{n}=\sup \left\{ \left\vert x_{n}-y_{n}\right\vert \colon x,y\in
X\right\} .$$We use a following measure of noncompactness in the space $l^{\infty }$ (see [@bibM1B]) $$\mu (X)=\limsup_{n\rightarrow \infty }diam\,\,X_{n}.$$
Main Result
===========
In this section, sufficient conditions for the existence of a bounded solution of equation are derived.
\[L2\] Assume that $a,p,q:{\mathbb{N}}_{0}\rightarrow {\mathbb{R}}$, $r:{\mathbb{N}}_{0}\rightarrow {\mathbb{R}}\setminus \{0\}$, and $f:{\mathbb{R}}\rightarrow {\mathbb{R}}$. Let $$\label{alfa}
\alpha \geq 1 \,\,
\mbox{is a ratio of positive integers with odd
denominator,}$$ $$\label{gama}
\gamma \in \left( 0,1 \right] \,\,
\mbox{ is a ratio of odd positive
integers,}$$ and let $k$ be a fixed positive integer. Assume that $$f:{\mathbb{R}}\rightarrow {\mathbb{R}}\text{ is a locally Lipschitz function,} \label{z1}$$ and that the sequences $r:{\mathbb{N}}_{0}\rightarrow {\mathbb{R}}\setminus
\{0\}$, $a,q:{\mathbb{N}}_{0}\rightarrow {\mathbb{R}}$ satisfy $$\sum\limits_{n=0}^{\infty }\left\vert \frac{1}{r_{n}}\right\vert ^{\frac{1}{\gamma }}\sum\limits_{i=n}^{\infty }\left\vert a_{i}\right\vert <+\infty
\,\, \text{ and }\,\, \sum\limits_{n=0}^{\infty } \left\vert \frac{1}{r_{n}}\right\vert ^{\frac{1}{\gamma }}\sum\limits_{i=n}^{\infty }\left\vert
q_{i}\right\vert <+\infty . \label{z2}$$ Let the sequence $p:{\mathbb{N}}_{0}\rightarrow {\mathbb{R}}$ satisfies the following condition $$-1<\liminf\limits_{n\rightarrow \infty }p_{n}\leq
\limsup\limits_{n\rightarrow \infty }p_{n}<1. \label{z3}$$ Assume additionally that $$\sum\limits_{i=0}^{\infty }\left\vert a_{i}\right\vert <+\infty, \,\,\,
\sum\limits_{i=0}^{\infty }\left\vert q_{i}\right\vert <+\infty.
\label{add_series}$$ Then, there exists a bounded solution $x:{\mathbb{N}}_{k}\rightarrow {\
\mathbb{R}}$ of equation .
Condition implies that there exist $n_{0}\in {\mathbb{N}}_{0}$ and a constant $P\in \lbrack 0,1)$ such that $$\left\vert p_{n}\right\vert \leq P<1,\text{ for }n\geq n_{0}. \label{z5}$$By condition there exists $n_{1}\in {\mathbb{N}}_{0}$ such that $\sum\limits_{i=n_{1}}^{\infty }\left\vert a_{i}\right\vert <1.$ Hence, by , $$\sum\limits_{n=0}^{\infty }\left( \left\vert \frac{1}{r_{n}}\right\vert
\sum\limits_{i=n}^{\infty }\left\vert a_{i}\right\vert \right) ^{\frac{1}{\gamma }}\leq \sum\limits_{n=0}^{\infty }\left\vert \frac{1}{r_{n}}\right\vert ^{\frac{1}{\gamma }}\sum\limits_{i=n}^{\infty }\left\vert
a_{i}\right\vert .$$The above and condition , imply that $$\sum\limits_{n=0}^{\infty }\left( \left\vert \frac{1}{r_{n}}\right\vert
\sum\limits_{i=n}^{\infty }\left\vert a_{i}\right\vert \right) ^{\frac{1}{\gamma }}<+\infty . \label{z31}$$Analogously, we get $$\sum\limits_{n=0}^{\infty }\left( \left\vert \frac{1}{r_{n}}\right\vert
\sum\limits_{i=n}^{\infty }\left\vert q_{i}\right\vert \right) ^{\frac{1}{\gamma }}\leq \sum\limits_{n=0}^{\infty }\left\vert \frac{1}{r_{n}}\right\vert ^{\frac{1}{\gamma }}\sum\limits_{i=n}^{\infty }\left\vert
q_{i}\right\vert . \label{z211}$$Recalling that remainder of a series is the difference between the $n$–th partial sum and the sum of a series, we denote by $\alpha _{n}$ and by $\beta _{n}$ the following remainders $$\alpha _{n}=\sum\limits_{j=n}^{\infty }\left( \left\vert \frac{1}{r_{j}}\right\vert \sum\limits_{i=j}^{\infty }\left\vert a_{i}\right\vert \right) ^{\frac{1}{\gamma }}\text{ and }\,\,\beta _{n}=\sum\limits_{j=n}^{\infty
}\left( \left\vert \frac{1}{r_{j}}\right\vert \sum\limits_{i=j}^{\infty
}\left\vert q_{i}\right\vert \right) ^{\frac{1}{\gamma }} \label{ab}$$We see, by and that $$\lim\limits_{n\rightarrow \infty }\alpha _{n}=0\,\,\text{ and }\lim\limits_{n\rightarrow \infty }\beta _{n}=0. \label{z7}$$Fix any number $d>0$. From , there exists a constant $M_{d}>0$ such that $$\left\vert f\left( x\right) \right\vert \leq M_{d}\mbox{ for all }x\in \left[ -d,d\right] . \label{Md}$$Chose a constant $C$ such that $$0<C\leq \frac{d-Pd}{\left( 2^{\frac{1}{\gamma }-1}\left( M_{d}\right) ^{\frac{1}{\gamma }}+2^{\frac{1}{\gamma }-1}\left( d^{\alpha }\right) ^{\frac{1}{\gamma }}\right) }. \label{C}$$By condition there exists a positive integer $n_{2}$ such that $$\alpha _{n}\leq C\text{ and }\beta _{n}\leq C\mbox{ for }n\in {\mathbb{N}}_{n_{2}}. \label{z8}$$
Define set $B$ as follows $$B\colon =\left\{ (x_{n})_{n=0}^{\infty }:\left\vert x_{n}\right\vert \leq d,\text{ for }n\in {\mathbb{N}}_{n_{0}}\right\} ,$$Define a mapping $T\colon B\rightarrow l^{\infty }$ as follows $$(Tx)_{n}=\begin{cases}
-p_{n}x_{n-k}-\sum\limits_{j=n}^{\infty }\left( \frac{1}{r_{j}}\sum\limits_{i=j}^{\infty }\left( a_{i}f(x_{i+1})+q_{i}x_{i}^{\alpha
}\right) \right) ^{\frac{1}{\gamma }}, & \text{for any}\ n\geq n_{3}, \\
x_{n}, & \text{for any}\ 0\leq n<n_{3}\end{cases}
\label{z10}$$where $n_{3}=\max \left\{ n_{1},n_{2}\right\} +k$. Observe that $B$ is a nonempty, bounded, convex and closed subset of $l^{\infty }$. We will prove that the mapping $T$ has a fixed point in $B$. This proof will follow in several subsequent steps.
**Step 1. Firstly, we show that** $T(B)\subset B$.
We will use classical inequality $$\label{ci}
\left( a+b\right) ^{s}\leq 2^{s-1}\left( a^{s}+b^{s}\right), \,\, a,b>0,
\,\, s\geq 1$$ and the fact $t\rightarrow t^{1/\gamma }$ is nondecreasing. If $x\in B$, then for $n< n_3$ $\left\vert (Tx)_{n}\right\vert=\left\vert
x_{n}\right\vert\leq d$ and by , we get for any $n\geq n_3$ $$\left\vert (Tx)_{n}\right\vert \,\,\leq \left\vert p_{n}\right\vert
\left\vert x_{n-k}\right\vert +\left\vert \sum\limits_{j=n}^{\infty }\left(
\frac{1}{r_{j}}\sum\limits_{i=j}^{\infty }\left(
a_{i}f(x_{i+1})+q_{i}x_{i}^{\alpha }\right) \right) ^{\frac{1}{\gamma }
}\right\vert$$ $$\leq \left\vert p_{n}\right\vert \left\vert x_{n-k}\right\vert
+\sum\limits_{j=n}^{\infty }\left( \left\vert \frac{1}{r_{j}}
\sum\limits_{i=j}^{\infty }\left( a_{i}f(x_{i+1})+q_{i}x_{i}^{\alpha
}\right) \right\vert \right) ^{\frac{1}{\gamma }}$$ $$\leq \left\vert p_{n}\right\vert \left\vert x_{n-k}\right\vert
+\sum\limits_{j=n}^{\infty }\left( \left\vert \frac{1}{r_{j}}\right\vert
\sum\limits_{i=j}^{\infty }\left( \left\vert a_{i}\right\vert \left\vert
f(x_{i+1})\right\vert +\left\vert q_{i}\right\vert \left\vert
x_{i}\right\vert ^{\alpha }\right) \right) ^{\frac{1}{\gamma }}.$$ From , taking into account that $x_{n-k} \in B$, and because of $x_i \in B$ we have $\left\vert x_{i}\right\vert^\alpha \leq d^\alpha$. Thus $$\left\vert (Tx)_{n}\right\vert \,\,\leq \left\vert p_{n}\right\vert d
+\sum\limits_{j=n}^{\infty }\left( \left\vert \frac{1}{r_{j}}\right\vert
\sum\limits_{i=j}^{\infty }\left( \left\vert a_{i}\right\vert
M_{d}+\left\vert q_{i}\right\vert d^{\alpha }\right) \right) ^{\frac{1}{\gamma }}$$ $$\leq \left\vert p_{n}\right\vert d +\sum\limits_{j=n}^{\infty }\left(
\left\vert \frac{1}{r_{j}}\right\vert \left( \sum\limits_{i=j}^{\infty
}\left\vert a_{i}\right\vert M_{d}+\sum\limits_{i=j}^{\infty }\left\vert
q_{i}\right\vert d^{\alpha } \right) \right) ^{\frac{1}{\gamma }}.$$ By inequality , we have $$\left\vert (Tx)_{n}\right\vert \,\,\leq \left\vert p_{n}\right\vert d +2^{\frac{1}{\gamma }-1}\sum\limits_{j=n}^{\infty }\left( \left( \left\vert
\frac{1}{r_{j}}\right\vert \sum\limits_{i=j}^{\infty }\left\vert
a_{i}\right\vert M_{d}\right) ^{\frac{1}{\gamma }}+\left( \left\vert \frac{1}{r_{j}}\right\vert \sum\limits_{i=j}^{\infty }\left\vert q_{i}\right\vert
d^{\alpha }\right) ^{\frac{1}{\gamma }}\right)$$ $$\leq\left\vert p_{n}\right\vert d+2^{\frac{1}{\gamma}-1}\left(M_{d}\right)^{\frac{1}{\gamma}}\sum\limits_{j=n}^{\infty}\left(\left\vert \frac{1}{r_{j}}\right\vert \sum\limits_{i=j}^{\infty}\left\vert a_{i}\right\vert \right)^{\frac{1}{\gamma}} + 2^{\frac{1}{\gamma}-1}\left( d^{\alpha}\right)^{\frac{1}{
\gamma}}\sum\limits_{j=n}^{\infty }\left( \left\vert \frac{1}{r_{j}}\right\vert \sum\limits_{i=j}^{\infty}\left\vert q_{i}\right\vert \right)^{\frac{1}{\gamma}}.$$ By using , and , we estimate $$\left\vert (Tx)_{n}\right\vert \,\,\leq Pd+2^{\frac{1}{\gamma }-1}\left(
M_{d}^{{}}\right) ^{\frac{1}{\gamma }}C+2^{\frac{1}{\gamma }-1}\left(
d^{\alpha }\right) ^{\frac{1}{\gamma }}C$$ $$\leq Pd+\left( 2^{\frac{1}{\gamma }-1}\left( M_{d}^{{}}\right) ^{\frac{1}{\gamma }}+2^{\frac{1}{\gamma }-1}\left( d^{\alpha }\right) ^{\frac{1}{\gamma
}}\right) \frac{d-Pd}{\left( 2^{\frac{1}{\gamma }-1}\left( M_{d}^{{}}\right)
^{\frac{1}{\gamma }}+2^{\frac{1}{\gamma }-1}\left( d^{\alpha }\right) ^{\frac{1}{\gamma }}\right) }=d. \label{calc}$$ From above, we have estimation $$\label{d}
\left\vert (Tx)_{n}\right\vert \leq d, \mbox{ for }n\in {\mathbb{N}}_{n_{3}}.$$
**Step 2.** $T$** is continuous**
By assumption , , and by definition of $B$, there exists a constant $c^*>0$ such that $$\sum\limits_{i=j}^{\infty }\left\vert a_{i}f(x_{i+1})+q_{i}x_{i}^{\alpha
}\right\vert \leq c^*$$ for all $x\in B$. From , $t\rightarrow t^{1/\gamma }$ is locally Lipschitz then it is Lipschitz on closed and bounded intervals. Hence, there exists a constant $L_{\gamma }$ such that $$\label{Lgama}
\left\vert t^{1/\gamma }-s^{1/\gamma }\right\vert \leq L_{\gamma }\left\vert
t-s\right\vert \text{ for all }t,s\in \left[ -c^*,c^*\right].$$ From , function $f$ is Lipschitz on $\left[ -d,d\right] $. So, there is a constant $L_{d}>0$ such that $$\left\vert f\left( x\right) -f\left( y\right) \right\vert \leq
L_{d}\left\vert x-y\right\vert \label{lipschitz}$$ for all $x, y\in \left[ -d,d\right] $. From , $x\rightarrow
x^{\alpha }$ is also Lipschitz on $\left[ -d,d\right] $. Then there is a constant $L_{\alpha }$ such that $$\label{Lalfa}
\left\vert x^{\alpha }-y^{\alpha }\right\vert \leq L_{\alpha }\left\vert
x-y\right\vert \text{ for all }x,y\in \left[ -d,d\right].$$
Let $(y^{(p)})$ be a sequence in $B$ such that $\left\Vert
y^{(p)}-x\right\Vert \rightarrow 0$ as $p\rightarrow \infty $. Since $B$ is closed, $x\in B$. By and , we get $$\begin{array}{l}
\forall\, n\geq n_3\quad \left\vert (Tx)_{n}-(Ty^{(p)})_{n}\right\vert
\,\,\leq \left\vert p_{n}\right\vert \left\vert
x_{n-k}-y_{n-k}^{(p)}\right\vert \\
\\
+\sum\limits_{j=n}^{\infty }\left\vert \frac{1}{r_{j}}\right\vert ^{\frac{1}{
\gamma }}\left\vert \left(\sum\limits_{i=j}^{\infty }\left(
a_{i}f(x_{i+1})+q_{i}\left( x_{i}\right) ^{\alpha }\right) \right) ^{\frac{1}{\gamma }}-\left( \sum\limits_{i=j}^{\infty }\left(
a_{i}f(y_{i+1}^{(p)})+q_{i}\left( y_{i}^{\left( p\right) }\right) ^{\alpha
}\right) \right) ^{\frac{1}{\gamma }}\right\vert.\end{array}$$ From , we have $$\begin{array}{l}
\forall\, n\geq n_3\quad \left\vert (Tx)_{n}-(Ty^{(p)})_{n}\right\vert
\,\,\leq \bigskip \left\vert p_{n}\right\vert \left\vert
x_{n-k}-y_{n-k}^{(p)}\right\vert \\
\\
+\sum\limits_{j=n}^{\infty }\left\vert \frac{1}{r_{j}}\right\vert ^{\frac{1}{\gamma }}L_{\gamma }\left\vert \sum\limits_{i=j}^{\infty
}a_{i}f(x_{i+1})+\sum\limits_{i=j}^{\infty }q_{i}\left( x_{i}\right)
^{\alpha }-\sum\limits_{i=j}^{\infty }a_{i}f(y_{i+1}^{\left( p\right)
})-\sum\limits_{i=j}^{\infty }q_{i}\left( y_{i}^{\left( p\right) }\right)
^{\alpha }\right\vert \\
\\
\leq \bigskip \left\vert p_{n}\right\vert \left\vert
x_{n-k}-y_{n-k}^{(p)}\right\vert +L_{\gamma }\sum\limits_{j=n}^{\infty
}\left\vert \frac{1}{r_{j}}\right\vert ^{\frac{1}{\gamma }}\sum\limits_{i=j}^{\infty }\left\vert a_{i}\right\vert \left\vert
f(x_{i+1})-f(y_{i+1}^{\left( p\right) })\right\vert \\
\\
+L_{\gamma }\sum\limits_{j=n}^{\infty }\left\vert \frac{1}{r_{j}}\right\vert
^{\frac{1}{\gamma }}\sum\limits_{i=j}^{\infty }\left\vert q_{i}\right\vert
\left\vert \left( x_{i}\right) ^{\alpha }-\left( y_{i}^{\left( p\right)
}\right) ^{\alpha }\right\vert.\end{array}$$ Hence, by and , we obtain $$\begin{array}{l}
\forall\, n\geq n_3\quad \left\vert (Tx)_{n}-(Ty^{(p)})_{n}\right\vert
\,\,\leq \left\vert p_{n}\right\vert \left\vert
x_{n-k}-y_{n-k}^{(p)}\right\vert \bigskip \\
+L_{\gamma }L_{d}\sum\limits_{j=n}^{\infty }\left\vert \frac{1}{r_{j}}\right\vert ^{\frac{1}{\gamma }}\sum\limits_{i=j}^{\infty }\left\vert
a_{i}\right\vert \left\vert x_{i+1}-y_{i+1}^{\left( p\right) }\right\vert
+L_{\gamma }L_{\alpha }\sum\limits_{j=n}^{\infty }\left\vert \frac{1}{r_{j}}\right\vert ^{\frac{1}{\gamma }}\sum\limits_{i=j}^{\infty }\left\vert
q_{i}\right\vert \left\vert x_{i}-y_{i}^{\left( p\right) }\right\vert
\bigskip \\
\leq \sup\limits_{i \in \mathbb{N}_0}\left\vert y^{(p)}_i-x_i\right\vert
\left( \left\vert p_{n}\right\vert +L_{\gamma
}L_{d}\sum\limits_{j=n}^{\infty }\left\vert \frac{1}{r_{j}}\right\vert ^{\frac{1}{\gamma }}\sum\limits_{i=j}^{\infty }\left\vert a_{i}\right\vert
+L_{\gamma }L_{\alpha }\sum\limits_{j=n}^{\infty }\left\vert \frac{1}{r_{j}}\right\vert ^{\frac{1}{\gamma }}\sum\limits_{i=j}^{\infty }\left\vert
q_{i}\right\vert \right).\end{array}$$ Moreover, $$\forall\, 0\leq n< n_3\quad \left\vert (Tx)_{n}-(Ty^{(p)})_{n}\right\vert
\,\,\leq \left\Vert y^{(p)}-x\right\Vert$$ Thus, by and , we have $$\lim\limits_{p\rightarrow \infty }\left\Vert Ty^{(p)}-Tx\right\Vert =0\text{
as }\lim\limits_{p\rightarrow \infty }\left\Vert y^{(p)}-x\right\Vert =0.$$ This means that $T$ is continuous.
**Step 3. Comparison of the measure of noncompactness**
Now, we need to compare a measure of noncompactness of any subset $X$ of $B$ and $T(X)$. Let us fix any nonempty set $X\subset B$. Take any sequences $x,y\in X$. Following the same calculations which led to the continuity of the operator $T$ we see that $$\forall \,n\geq n_{3}\quad \left\vert (Tx)_{n}-(Ty)_{n}\right\vert \leq
\left\vert p_{n}\right\vert \left\vert x_{n-k}-y_{n-k}\right\vert +L_{\gamma
}L_{\alpha }\beta _{n}\left\vert x_{n}-y_{n}\right\vert +L_{\gamma
}L_{d}\alpha _{n}\left\vert x_{n+1}-y_{n+1}\right\vert .$$Taking sufficiently large $n$, by and , we get $$L_{\gamma }L_{d}\alpha _{n}\leq c_{1}<\frac{1-P}{4}\,\,\,\mbox{ and }\,\,\,L_{\gamma }L_{\alpha }\beta _{n}\leq c_{2}<\frac{1-P}{4}$$Here $c_{1},\,\,c_{2}$ are some real constants. From , we have $$P+c_{1}+c_{2}<\frac{1+P}{2}.$$We see that exists $n_{5}$ such that $$\forall \,n\geq n_{5}\quad diam\,\,(T(X))_{n}\leq
Pdiam\,\,X_{n-k}+c_{1}diam\,\,X_{n}+c_{2}diam\,\,X_{n+1}.$$This yields by the properties of the upper limit that $$\limsup_{n\rightarrow \infty }diam\,\,(T(X))_{n}\leq
\,P\,\limsup_{n\rightarrow \infty
}diam\,\,X_{n-k}+c_{1}\,\limsup_{n\rightarrow \infty
}diam\,\,X_{n}+c_{2}\limsup_{n\rightarrow \infty }diam\,\,X_{n+1}.$$From above, for any $X\subset B$, we have $\mu (T(X))\leq \left(
c_{1}\,+c_{2}+P\right) \mu (X)$.
**Step 4. Relation between fixed points and solutions**
By Theorem \[D\] we conclude that $T$ has a fixed point in the set $B$. It means that there exists $x\in B$ such that $$x_{n}=(Tx)_{n}.$$Thus $$x_{n}=-p_{n}x_{n-k}-\sum\limits_{j=n}^{\infty }\left( \frac{1}{r_{j}}\sum\limits_{i=j}^{\infty }\left( a_{i}f(x_{i+1})+q_{i}x_{i}^{\alpha
}\right) \right) ^{\frac{1}{\gamma }},\,\text{ for }\,\,n\in {\mathbb{N}}_{n_{3}} \label{z12}$$
To show that there exists a correspondence between fixed points of $T$ and solutions to we apply operator $\Delta $ to both sides of the following equation $$x_{n}+p_{n}x_{n-k}=-\sum\limits_{j=n}^{\infty }\left( \frac{1}{r_{j}}\sum\limits_{i=j}^{\infty }\left( a_{i}f(x_{i+1})+q_{i}x_{i}^{\alpha
}\right) \right) ^{\frac{1}{\gamma }},$$which is obtained from . We find that $$\Delta (x_{n}+p_{n}x_{n-k})=\left( \frac{1}{r_{n}}\sum\limits_{i=n}^{\infty
}\left( a_{i}f(x_{i+1})+q_{i}x_{i}^{\alpha }\right) \right) ^{\frac{1}{\gamma }},\,\,\,n\in {\mathbb{N}}_{n_{3}}.$$and next $$\left( \Delta (x_{n}+p_{n}x_{n-k})\right) ^{\gamma }=\frac{1}{r_{n}}\sum\limits_{i=n}^{\infty }\left( a_{i}f(x_{i+1})+q_{i}x_{i}^{\alpha
}\right) ,\,\,\,n\in {\mathbb{N}}_{n_{3}}.$$Taking operator $\Delta $ again to both sides of the above equation we obtain $$\Delta \left( r_{n}\left( \Delta (x_{n}+p_{n}x_{n-k})\right) ^{\gamma
}\right) =-a_{n}f(x_{n+1})-q_{n}x_{n}^{\alpha },\,\,\,n\in {\mathbb{N}}_{n_{3}}.$$So, we get equation for $n\in {\mathbb{N}}_{n_{3}}$. Sequence $x$, which is a fixed point of mapping $T$, is a bounded sequence which fulfills equation for large $n$. If $n_{3}\geq k$ the proof is ended. We find previous $n_{3}-k+1$ terms of sequence $x$ by formula $$x_{n-k+l}=\frac{1}{p_{n+l}}\left( -x_{n+l}+\sum\limits_{j={n+l}}^{\infty
}\left( \frac{1}{r_{j}}\sum\limits_{i=j}^{\infty }\left(
a_{i}f(x_{i+1})+q_{i}x_{i}^{\alpha }\right) \right) ^{\frac{1}{\gamma }}\right) ,$$where $l\in \left\{ 0,1,2,\dots ,k-1\right\} $, which results leads directly from . It means that equation has at least one bounded solution $x:{\mathbb{N}}_{k}\rightarrow {\mathbb{R}}$.
This completes the proof.
**Remark.** *We note the previous terms of the solution sequence are not obtained through a fixed point method, but through backward iteration. It is common that one has a* $1-1$* correspondence between fixed points to a suitably chosen operator and solutions to the problem under consideration. Here we get as a fixed point solution some sequence which starting from some index is a solution to the given problem and in which the first terms must be iterated. This procedure must be applied since we see that in equation we have to know also earlier terms in order to start iteration; this is the so called iteration with memory. We recall that in recent works concerning application of the measure of noncompactness to discrete equations, only problems without memory have been considered. That is why we had to alter to established procedure to overcome the difficulty arising in this problem. We believe our method would be applicable for several other problems*
A special type stability
========================
The type of stability investigated in this paper is contained in the following theorem.
\[T1\] Assume that $$q_{n}\equiv 0, \label{e}$$and conditions and – are held. Assume further that there exists a positive constant $D$ such that $$\left\vert f(u)-f(v)\right\vert \leq D\left\vert u-v\right\vert$$for any $u,v\in {\mathbb{R}}$. Then equation has at least one solution $x:{\mathbb{N}}_{k}\rightarrow {\mathbb{R}}$ with the following stability property: given any other solution $y:{\mathbb{N}}_{k}\rightarrow {\mathbb{R}}$ and $\varepsilon >0$ there exists $T>$ $n_{3}$ such that for every $t\geq T$ the following inequality holds $$\left\vert x(t)-y(t)\right\vert \leq \varepsilon .$$
From Theorem \[L2\], equation has at least one bounded solution $x:{\mathbb{N}}_{0}\rightarrow {\mathbb{R}}$ which can be rewritten in the form $$x_{n}=(Tx)_{n},$$where mapping $T$ is defined by for $n\geq n_{3}$. By Definition \[def2\], sequence $x$ is an asymptotically stable solution of equation $x_{n}=(Tx)_{n}$ From the above and condition , analogously as the steps in the proof of Theorem \[L2\], we see that $$\begin{array}{l}
\left\vert x_{n}-y_{n}\right\vert =\left\vert (Tx)_{n}-(Ty)_{n}\right\vert
\,\leq \bigskip \\
\left\vert p_{n}\right\vert \left\vert x_{n-k}-y_{n-k}\right\vert +L_{\gamma
}D\sum\limits_{j=n}^{\infty }\left\vert \frac{1}{r_{j}}\right\vert ^{\frac{1}{\gamma }}\sum\limits_{i=j}^{\infty }\left\vert a_{i}\right\vert \left\vert
x_{i+1}-y_{i+1}\right\vert .\end{array}$$Note that for $n$ large enough, say $n\geq n_{4}\geq n_{3}$, we have $$\vartheta :=\left\vert p_{n}\right\vert +L_{\gamma
}D\sum\limits_{j=n}^{\infty }\left\vert \frac{1}{r_{j}}\right\vert ^{\frac{1}{\gamma }}\sum\limits_{i=j}^{\infty }\left\vert a_{i}\right\vert <1$$Let us denote $$\limsup_{n\rightarrow \infty }\left\vert x_{n}-y_{n}\right\vert =l,$$and observe that $$\limsup_{n\rightarrow \infty }\left\vert x_{n}-y_{n}\right\vert
=\limsup_{n\rightarrow \infty }\left\vert x_{n-k}-y_{n-k}\right\vert
=\limsup_{n\rightarrow \infty }\left\vert x_{n+1}-y_{n+1}\right\vert .$$Thus, from the above, we have $$l\leq \vartheta \cdot l.$$This means that $\limsup\limits_{n\rightarrow \infty }\left\vert
x_{n}-y_{n}\right\vert =0$. This completes the proof since for $\varepsilon
>0$ there exists $n_{4}\in {\mathbb{N}}_{0}$ such that for every $n\geq
n_{4}\geq n_{3}$ the following inequality holds $$\left\vert x_{n}-y_{n}\right\vert \leq \varepsilon .$$
Comments and an example
=======================
In [@SchmeildelZbaszyniakCAMW], the authors consider a special type of problem , namely they investigate the existence of a solution and Lyapunov type stability to the following equation $$\Delta \left( r_{n}\Delta x_{n}\right) =a_{n}f(x_{n+1}). \label{ESZZ_RPW}$$ Their main assumption is the linear growth assumption on nonlinear term $f$. More precisely, they assume that there exists a positive constant $M$ such that $\left\vert f(x_{n})\right\vert \leq M$ $\left\vert x_{n}\right\vert $ for all $x\in N_{0}$. Using ideas developed in this paper we get the following result.
Assume that $f:{\mathbb{R}}\rightarrow {\mathbb{R}}$ satisfies the condition and the sequences $r:{\mathbb{N}}_{0}\rightarrow {\mathbb{R}}\setminus \{0\}$, $a:{\mathbb{N}}_{0}\rightarrow {\mathbb{R}}$ are such that $$\sum\limits_{n=0}^{\infty }\left\vert \frac{1}{r_{j}}\right\vert
\sum\limits_{i=n}^{\infty }\left\vert a_{i}\right\vert <+\infty.$$ Then, there exists a bounded solution $x:{\mathbb{N}}_{0}\rightarrow {\mathbb{R}}$ of equation .
Finally, we give an example of equation which can be considered by our method.
Take $k=3$, an arbitrary $C^{1}$ function $f:{\mathbb{R\rightarrow R}}$ and consider the following problem $$\Delta \left( \left( -1\right) ^{n}\Delta \left( x_{n}+\frac{1}{2}x_{n-3}\right) ^{1/3}\right) +\frac{1}{2^{n}} \left( \left( x_{n}\right)
^{5}+f\left( x_{n+1}\right)\right) =0. \label{prob}$$ Taking $\gamma =\frac{1}{3}$, $\alpha =5$, $r_{n}=\left( -1\right) ^{n}$, $p_{n}=\frac{1}{2}$, $a_{n}=q_{n}=\frac{1}{2^{n}}$ with $f\left( x\right)
=x^5 $ we see that $x_{n}=\left( -1\right) ^{n}$ is a bounded solution to . By Theorem \[T1\], this solution is asymptotically stable.
[99]{}
[^1]: Lodz University of Technology, Poland, email: [email protected]
[^2]: University of Bialystok and Lodz University of Technology, Poland, email: [email protected]
[^3]: Lodz University of Technology, Poland, email: [email protected]
[^4]: University of Bialystok, Poland, email: [email protected]
|
---
author:
- |
Keith Briggs[^1], Linlin Song (BT Research, Martlesham)\
& Thomas Prellberg (Mathematics, QMUL)
date:
title: A note on the distribution of the maximum of a set of Poisson random variables
---
Given a set of independent Poisson random variableswith common mean, we study the distribution of their maximum and obtain an accurate asymptotic formula to locate the most probable value of the maximum. We verify our analytic results with very precise numerical computations.
We deal with a set of independent Poisson random variables $\{X_1,X_2,\dots, X_n\}$ with common mean $\lambda$, so that $\pr[X_{i}=k]={e^{-\lambda}\lambda^{k}}/{k!}$. We let $M_{n}=\max(X_i)$ and wish to describe the distribution of $M_n$. Our motivation is a problem in random graph theory, where we were interested in the distribution of maximum degree in graphs with Poisson degree distribution.
We have $$\begin{aligned}
{\ensuremath{\text{Pr}[X_i<k]}} = Q(k,\lambda)\equiv\Gamma(k,\lambda)/\Gamma(k)\nonumber
$$ where $Q$ and $\Gamma(\cdot,\cdot)$ are incomplete Gamma functions; that is, $$\Gamma(a,x)=\int_{x}^{\infty}t^{a-1}e^{-t}{\text{d}}t.\nonumber$$ From the independence of the Poisson variables, $$\label{ccdf}
{\ensuremath{\text{Pr}[M_n\leqslant k]}}={\ensuremath{\text{Pr}[X_1 \leqslant k]}}^n=Q(k+1,\lambda)^n={\Gamma(k+1,\lambda)^n/\Gamma(k+1)}^n.\nonumber$$ Our aim is to approximate the distribution of $M_n$. We have $$\begin{aligned}
\label{PrMn=k}
{\ensuremath{\text{Pr}[M_n=k]}}&=&{\ensuremath{\text{Pr}[M_n\leqslant k]}}-{\ensuremath{\text{Pr}[M_n\leqslant
k-1]}}\nonumber\\
&=&Q(k+1,\lambda)^n-Q(k,\lambda)^n\nonumber\end{aligned}$$ Examples of these distributions are shown in Figure \[distribution\]. These numerical results demonstrate the so-called *focussing* effect; the maxima $M_n$ are concentrated on at most two adjacent integers for large $n$; we call them *modal values*. It is this focussing that allows us to characterize the distributions very precisely by a single asymptotic estimate.
![From left to right: the distribution of the maximum of Poisson variables for $\lambda=1/2, 1, 2, 5$ (left to right) and $n=10^0, 10^2, 10^4, \dots,
10^{24}.$ Note that there is an error in Fig 1 in @Anderson1997, where the curves labelled $k=6$ and $k=8$ are incorrect. []{data-label="distribution"}](figs/poisson_maxima_pdf_lambda=half.pdf "fig:"){width="0.243\hsize"} ![From left to right: the distribution of the maximum of Poisson variables for $\lambda=1/2, 1, 2, 5$ (left to right) and $n=10^0, 10^2, 10^4, \dots,
10^{24}.$ Note that there is an error in Fig 1 in @Anderson1997, where the curves labelled $k=6$ and $k=8$ are incorrect. []{data-label="distribution"}](figs/poisson_maxima_pdf_lambda=1.pdf "fig:"){width="0.23\hsize"} ![From left to right: the distribution of the maximum of Poisson variables for $\lambda=1/2, 1, 2, 5$ (left to right) and $n=10^0, 10^2, 10^4, \dots,
10^{24}.$ Note that there is an error in Fig 1 in @Anderson1997, where the curves labelled $k=6$ and $k=8$ are incorrect. []{data-label="distribution"}](figs/poisson_maxima_pdf_lambda=2.pdf "fig:"){width="0.23\hsize"} ![From left to right: the distribution of the maximum of Poisson variables for $\lambda=1/2, 1, 2, 5$ (left to right) and $n=10^0, 10^2, 10^4, \dots,
10^{24}.$ Note that there is an error in Fig 1 in @Anderson1997, where the curves labelled $k=6$ and $k=8$ are incorrect. []{data-label="distribution"}](figs/poisson_maxima_pdf_lambda=5.pdf "fig:"){width="0.23\hsize"}
![The maximal probability (with respect to $I_n$) that $M_n \in \{I_n,I_n+1\}$ for $\lambda=1/2, 1, 2, 5$ (left to right) and $10^0\leqslant n\leqslant 10^{40}$. The curves show the probability that $M_n$ takes either of its two most frequently occurring values.[]{data-label="prob"}](figs/poisson_maxima_approach_lambda=half.pdf "fig:"){width="0.243\hsize"} ![The maximal probability (with respect to $I_n$) that $M_n \in \{I_n,I_n+1\}$ for $\lambda=1/2, 1, 2, 5$ (left to right) and $10^0\leqslant n\leqslant 10^{40}$. The curves show the probability that $M_n$ takes either of its two most frequently occurring values.[]{data-label="prob"}](figs/poisson_maxima_approach_lambda=1.pdf "fig:"){width="0.23\hsize"} ![The maximal probability (with respect to $I_n$) that $M_n \in \{I_n,I_n+1\}$ for $\lambda=1/2, 1, 2, 5$ (left to right) and $10^0\leqslant n\leqslant 10^{40}$. The curves show the probability that $M_n$ takes either of its two most frequently occurring values.[]{data-label="prob"}](figs/poisson_maxima_approach_lambda=2.pdf "fig:"){width="0.23\hsize"} ![The maximal probability (with respect to $I_n$) that $M_n \in \{I_n,I_n+1\}$ for $\lambda=1/2, 1, 2, 5$ (left to right) and $10^0\leqslant n\leqslant 10^{40}$. The curves show the probability that $M_n$ takes either of its two most frequently occurring values.[]{data-label="prob"}](figs/poisson_maxima_approach_lambda=5.pdf "fig:"){width="0.23\hsize"}
In previous work on this problem, @Anderson1970 proved the existence of integers $I_n$ such that ${\ensuremath{\text{Pr}[M_n \in (I_n,I_n+1)]}} \to 1 $ as $n\rightarrow\infty$ for fixed $\lambda>0$; and that $I_n\sim\beta_n$, where $\beta_n$ is defined as the unique solution of $Q(\beta_n,\lambda)=1/n$.
Following this work, @Kimber1983 computed an asymptotic result; he showed $I_n\sim\log n/{\log\log n}$ and $P_n \sim (k/{I_n})^{1+B_n}$ with $B_n$ dense in $[-1/2, 1/2]$, and concluded that to the first order, the rate of growth of $I_n$ is independent of the Poisson parameter $\lambda$. He concluded that $P_n$, defined as $P_n={\ensuremath{\text{Pr}[M_n \in (I_n, I_n+1)]}}$, oscillates and the oscillation persists for arbitrarily large $n$. We illustrate in Figure \[prob\] exactly how this probability oscillates as $n
\to \infty$. Our numerical experiments show that $\log n/\log\log n$ estimates $I_n$ very poorly. We aim to improve this asymptotic formula.
Our method is a refinement of that of Kimber; that is, we consider a continuous distribution $g$ which interpolates the Poisson maximum distribution, and we solve $g(x)=1/n$. Consider $g_\lambda(x)\equiv
1-\Gamma(x+1,\lambda)/\Gamma(x+1)$ for fixed $\lambda \in \mathbb{R}^{+}$, which is a strictly decreasing function on $(0,\infty)$. If $\epsilon=1/n$ is a small positive real, then $g_\lambda(x)$ has a unique root $x(\epsilon)$ which increases as $\epsilon \to 0^{+}$. We will develop an asymptotic expansion (as $\epsilon \to 0$) of this root $x(\epsilon)$.
We have $$g_\lambda(x)=\exp(-\lambda)\;\lambda^x\;\sum_{i=1}^{\infty}\,\frac{\lambda^i}{\Gamma(x+i+1)}\nonumber$$ and we will work with $$\begin{gathered}
\log(g_\lambda(x))=-x\log(x)+(1+\log\lambda)x -\frac{3}{2}\log(x)\\+\left(\log\lambda-\lambda-\frac{\log(2\pi)}{2}\right)+\frac{\lambda-13/12}{x}+{\cal O}(x^{-2}).\label{gen}\end{gathered}$$ A first approximation to the solution of $\log(g_\lambda(x))=-\log n$ large and negative is given by keeping only the dominant first two terms in Equation (\[gen\]): $$M_n\sim x_0\equiv\frac{\log n}{W\!\left(\frac{\log n}{\exp(1)\lambda}\right)},\nonumber$$ where $W(\cdot)$ is the principal branch of Lambert’s $W$ function [@W]. That this is already quite accurate can be seen form the dark blue curves in Figure \[fig3\]. However, we would like to do better; ideally the error should be less than unity so that the mode of the distribution is correctly identified. A refinement $x_1$ may be generated by making a single Newton correction step; that is, $x_1=x_0-(h(x_0)+\log n)/h'(x_0)$, where $h$ is some approximation to $\log(g_\lambda)$. For example, by keeping all terms in $\log(g_\lambda(x))$ and $\log(g_\lambda(x))'$ which do not vanish as $n\rightarrow\infty$, we obtain $$M_n\sim x_1=x_0+\frac{\log\lambda-\lambda-{\log(2\pi)}/{2}-3\log(x_0)/2}{\log(x_0)-\log\lambda}.\nonumber$$ This appears to have error less than unity for all values of $n$ and $\lambda$ considered in Figure \[fig3\], and so is probably sufficiently precise for all practical purposes. If further accuracy is needed, it may be obtained by additional Newton steps. In any case, both $x_0$ and $x_1$ are considerably more precise than Kimber’s approximation.
![Exact values and asymptotics of $I_n$ for $\lambda=1/2, 1, 2, 5$ (left to right) and $n=10^0, \dots, 10^{40}$. The staircase red line (almost hidden by the cyan line) represents the exact mode $I_n$; the other lines represent asymptotic approximations: green for the result of @Kimber1983 (which is independent of $\lambda$), dark blue and cyan for our new results $x_0$ and $x_1$ respectively. The cyan curve always sits between the steps of $I_n$, meaning that $x_1$ has error less than unity.[]{data-label="fig3"}](figs/poisson_maxima_asymptotics_lambda=half.pdf "fig:"){width="0.243\linewidth"} ![Exact values and asymptotics of $I_n$ for $\lambda=1/2, 1, 2, 5$ (left to right) and $n=10^0, \dots, 10^{40}$. The staircase red line (almost hidden by the cyan line) represents the exact mode $I_n$; the other lines represent asymptotic approximations: green for the result of @Kimber1983 (which is independent of $\lambda$), dark blue and cyan for our new results $x_0$ and $x_1$ respectively. The cyan curve always sits between the steps of $I_n$, meaning that $x_1$ has error less than unity.[]{data-label="fig3"}](figs/poisson_maxima_asymptotics_lambda=1.pdf "fig:"){width="0.23\linewidth"} ![Exact values and asymptotics of $I_n$ for $\lambda=1/2, 1, 2, 5$ (left to right) and $n=10^0, \dots, 10^{40}$. The staircase red line (almost hidden by the cyan line) represents the exact mode $I_n$; the other lines represent asymptotic approximations: green for the result of @Kimber1983 (which is independent of $\lambda$), dark blue and cyan for our new results $x_0$ and $x_1$ respectively. The cyan curve always sits between the steps of $I_n$, meaning that $x_1$ has error less than unity.[]{data-label="fig3"}](figs/poisson_maxima_asymptotics_lambda=2.pdf "fig:"){width="0.23\linewidth"} ![Exact values and asymptotics of $I_n$ for $\lambda=1/2, 1, 2, 5$ (left to right) and $n=10^0, \dots, 10^{40}$. The staircase red line (almost hidden by the cyan line) represents the exact mode $I_n$; the other lines represent asymptotic approximations: green for the result of @Kimber1983 (which is independent of $\lambda$), dark blue and cyan for our new results $x_0$ and $x_1$ respectively. The cyan curve always sits between the steps of $I_n$, meaning that $x_1$ has error less than unity.[]{data-label="fig3"}](figs/poisson_maxima_asymptotics_lambda=5.pdf "fig:"){width="0.23\linewidth"}
[xx]{}
Anderson, C. W. 1970, ‘[Extreme value theory for a class of discrete distributions with applications to some stochastic processes]{}’, [*Journal of Applied Probability*]{} [**Vol. 7**]{}, 99–113. Anderson, C. W., Coles, S. G. Hüsler, J. 1997, ‘[Maxima of Poisson-like variables and related triangular arrays]{}’, [*The Annals of Applied Probability*]{} [ **7**]{}, 953–971. Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J. Knuth, D. E. 1996, ‘[On the Lambert $W$ Function]{}’, [*Advances in Computational Mathematics*]{} [**5**]{}, 329–359.
Kimber, A. C. 1983, ‘[A note on Poisson maxima]{}’, [*Probability Theory and Related Fields*]{} [**63**]{}(4), 551–552.
[^1]: corresponding author; `[email protected]`
|
---
abstract: |
We investigate the scaling of the escape rate from piecewise-linear dynamical systems displaying intermittency due to the presence of an indifferent fixed-point. Strong intermittent behaviour in the dynamics can result in the system preserving an infinite measure. We define a neighbourhood of the indifferent fixed point to be a hole through which points escape and investigate the scaling of the rate of this escape as the length of the hole decreases, both in the finite measure preserving case and infinite measure preserving case. In the infinite measure preserving systems we observe logarithmic corrections to and polynomial scaling of the escape rate with hole length. Finally we conjecture a relationship between the wandering rate and the observed scaling of the escape rate.\
\
PACS numbers: 05.45.Ac, 05.60.Cd
author:
- 'Sara Munday[^1] and Georgie Knight[^2]'
bibliography:
- 'Farey\_Bib.bib'
title: Escape rate scaling in infinite measure preserving systems
---
Introduction {#Sec:Intro}
============
We investigate discrete time dynamical systems that have a hole $H$ in their phase space $X$. In other words, the iterates of points $x \in X$ are considered only up until they enter $H$, after which they leave the system. In this setting an object of particular interest is the survival probability $Pr(n)$ which is the measure $\mu(S_n)$ of the set of surviving points $S_n$ up to time $n$, and $\mu$ is the measure of initial conditions in the system. When the dynamics are strongly chaotic, that is, when they exhibit exponential decay of correlations, we expect the survival probability to decay exponentially. The rate of this decay is defined to be $$\gamma:= \lim_{n \rightarrow \infty} -\frac{\ln(Pr(n))}{n},
\label{Eq:EscapeRateDef}$$ where $\gamma$, which is the object of investigation here, is called the [*escape rate*]{}. While Eq.(\[Eq:EscapeRateDef\]) tells us that the escape rate is dependent on the initial distribution $\mu$, note that the escape rate $\gamma$ can also be calculated as the largest eigenvalue $\lambda_H$ of the Perron-Frobenius transfer operator of the open system $$\gamma= -\ln(\lambda_H).
\label{Eq:EscapeRateDefEig}$$ Therefore, being a spectral quantity, the escape rate is equal for a large class of initial distributions. See [@DeYo06] for a discussion.
Systems where iterates can escape through holes were first studied in the context of Markov chains with absorbing states (see [@FeEtal95] and references therein) before being discussed in the theory of dynamical systems in [@PiYo79]. In this setting initial work was on the study of the conditionally invariant measures of the open system [@ColletEtal97]. These are the eigenvectors of the transfer operator of the open system. In particular, one question asked is which conditionally invariant measures are physical, or natural in that they somehow capture the statistics of the open system, see [@LivMau03] for more recent work and [@DeYo06] for a review. In addition to the mathematical interest, there is much attention given to the escape rate and its direct applications by the physics community. See [@AltRev13; @DetChapter] for recent reviews.
A topic which has recently received a lot of interest has been the study of the escape rate as a function of certain system parameters, for instance studying the behaviour of the escape rate as a function of hole position, or size. Initial numerical work in [@PaPa97] uncovered a nontrivial relationship between the periodic orbit structure and the lifetime of transients in chaotic maps. Related rigorous results from [@KeLi09] show that the asymptotic behaviour of the escape rate as the hole shrinks to a point is dependent upon the stability of the limiting point. This is independently shown in [@BuYu11] for the doubling map; they also show that finite size holes of equal measure can have different escape rates, among other results. Further work on the asymptotic behaviour of the escape rate can be found in [@Cr13] and [@Det13] where higher order logarithmic corrections are found. In [@DeWr12] it is shown that for a large class of systems the escape rate as a function of the continuous varying of the hole through phase space is a devil’s staircase.
Much work has been done on the uniformly hyperbolic setting. However, intermittency in open dynamical systems can lead to some interesting properties. Systems which display intermittency but still preserve finite measures are studied in [@Dahl99] [@FrMuSt11] [@DeFe14], where it is shown that the intermittency can lead to polynomial rather than exponential rates of escape and the non-existence of a natural conditionally invariant measure, which means that the escape rate depends heavily on the initial set-up of the system. Here we intend to study deviations from the standard linear scaling (see for example [@FrMuSt11]) of the escape rate with hole length. It is shown in [@KeLi09] that for a particular example, the escape rate does not scale linearly with the length of the hole when the measure of the closed system vanishes at the limiting point of the hole. Here our main goal is to examine and understand the scaling when the underlying closed map preserves an infinite invariant measure and when the measure diverges at the limit point. Rather than approach this problem with the elegant perturbation theory used in [@KeLi09], where the small hole is considered to be a perturbation of the dynamics of the closed system, we approach the problem using the [*top down*]{} techniques from [@Cr13] where one starts with a large hole and shrinks to the closed system.
The paper is organised as follows. In Section \[Sec:Prelim\] we define the setting of a family of piecewise-linear, $\alpha$-Farey maps, where $\alpha$ is an infinite partition of the unit interval. Depending on the type of partition, the $\alpha$-Farey map preserves either an infinite or finite Lebesgue-absolutely continuous measure. In Section \[Sec:Derive-zeta\] we then study the escape rate in these systems with a hole by deriving the dynamical zeta function, a polynomial which encodes important properties of the dynamics. Via the zero of this polynomial, in Section \[Sec:Asymp\] we study the small hole asymptotic behaviour of the escape rate and derive how the escape rate scales with the length of the hole for some particular examples of $\alpha$-Farey maps and for a parameter-dependent version which incorporates both finite and infinite measure preserving dynamics as a function of the control parameter. Finally in Section \[Sec:Conclusion\] we make some concluding remarks and observe that the scaling found in this particular setting is intimately related to a quantity known as the wandering rate, we furthermore conjecture that this relationship will hold in more general systems.
Preliminaries {#Sec:Prelim}
=============
Let $\alpha:=\{A_n:n\in\N\}$ denote a countably infinite partition of the interval $(0, 1]$, consisting of non-empty, right-closed and left-open intervals, and let $a_n:=\varepsilon(A_n)$, where $\varepsilon$ denotes the Lebesgue measure, and $$t_n:=\sum_{k=n}^\infty a_k.
\label{Eq:tn}$$ It is assumed throughout that the elements of $\alpha$ are ordered from right to left, starting from $A_1$, and that these elements accumulate only at the origin. Let us denote the set of all such partitions by $\mathcal{A}$. Then, for a given partition $\alpha\in \mathcal{A}$, the map $F_{\alpha}:[0,1] \to [0,1]$ is given by $$F_{\alpha}(x):=\left\{
\begin{array}{ll}
(1-x)/a_1 & \hbox{if $x\in A_1$,} \\
{a_{n-1}}(x-t_{n+1})/a_{n}+t_n & \hbox{if $x\in A_n$, for $n\geq2$.}
\\
0 & \hbox{if $x=0$. }
\end{array}
\right.
\label{Eq:A-Farey-map}$$ For example, let us consider the dyadic partition $\alpha_D:=\left\{\left(1/2^{n},1/2^{n-1}\right]:n\in\N\right\}$. One can immediately verify that the map $F_{\alpha_D}$ coincides with the tent map $T:[0,1]\to[0,1]$, which is given by $$T(x):=\left\{
\begin{array}{ll}
2x, & \hbox{for $x\in[0,1/2)$;} \\
2-2x, & \hbox{for $x\in [1/2, 1]$.}
\end{array}
\right.
\label{Eq:Tentmap}$$ To see this, it is enough to note that for each $n\in\N$ we have that $a_n=2^{-n}$ and $t_n=2^{-(n-1)}$.
Alternatively consider the partition $\alpha_H:=\left\{A_n:=\left(1/(n+1),1/n\right]:n\in\N\right\}$, which we will call the [*harmonic partition*]{}. Here, we obtain the map $F_{\alpha_H}$ which is given explicitly by $$F_{\alpha_H}(x):=\left\{
\begin{array}{ll}
2-2x & \hbox{for $x\in (1/2, 1]$;} \\
\frac{n+1}{n-1}x-\frac1{n(n-1)} & \hbox{for $x\in (1/(n+1),1/n]$ .}
\end{array}
\right.
\label{Eq:Farey-map}$$ The graphs of the maps $F_{\alpha_D}$ and $F_{\alpha_H}$ are shown in figure \[Fig:alphafarey\]. For more details about the $\alpha$-Farey systems, we refer the reader to [@KMS].
![The $\alpha_H$-Farey map, where $t_{n}=1/n$, $n\in \N$, and the $\alpha_D$-Farey map, where $t_n=2^{-(n-1)}$.[]{data-label="Fig:alphafarey"}](fig1a "fig:"){width="33.00000%"} ![The $\alpha_H$-Farey map, where $t_{n}=1/n$, $n\in \N$, and the $\alpha_D$-Farey map, where $t_n=2^{-(n-1)}$.[]{data-label="Fig:alphafarey"}](fig1b "fig:"){width="33.00000%"}
It was shown in [@KMS] that the Lebesgue-absolutely continuous measure $\nu_\alpha$ defined by the density $\phi_{\alpha}$ which is given by $$\phi_{\alpha}:=\frac{d\nu_\alpha}{d\varepsilon}=\sum_{n=1}^{\infty} \frac{t_n}{a_n} \cdot \mathbbm{1}_{A_{n}},$$ is the unique, up to multiplication by a constant, Lebesgue-absolutely continuous measure (ACIM) preserved by $F_\alpha$. Furthermore, this measure is infinite (non-normalisable) if and only if $$\lim_{n \rightarrow \infty}\sum_{k=1}^n t_k = \infty.
\label{Eq:Infinite-measure-property}$$ For the two examples mentioned above, we have that for $F_{\alpha_D}$, the density is the function constantly equal to 2 (in keeping with the fact that we know already in this case that the invariant measure is simply the Lebesgue measure), and for $F_{\alpha_H}$, the density is the simple function given by $\phi_{\alpha_H}= \sum_{n=1}^{\infty} (n+1)\cdot \mathbbm{1}_{A_n}$. We open the system $([0, 1], F_\alpha)$, for an arbitrary $\alpha\in \mathcal{A}$ by defining the interval $[0, t_n]$ to be the hole. The length of the hole will take on a discrete set of value s, $\varepsilon([0,t_n])$, depending on $n$, which tend to zero as $n$ goes to infinity (we will write $\varepsilon$ in the following to refer to the length of the hole). Here, the Perron-Frobenius transfer operator of the closed system $P$ acts on functions that are piecewise constant on the intervals $A_n$. As mentioned already above, there exists a unique ACIM $\nu_\alpha$ of the closed system, with corresponding eigenvalue $1$, that is $P \nu_\alpha= \nu_\alpha$.
The corresponding transfer operator of the open system $P_{[0,1]\backslash H}$ acts on functions that are piecewise constant on the intervals $A_1,..,A_k$ and the interval $[0,t_n]$. In this setting of sub-shift of finite type with a cylindrical hole, we have that the transfer operator $P_{[0,1]\backslash H}$ has an eigenfunction correponding to an absolutely continuous with respect to Lebesgue, conditionally invariant measure (ACCIM) $\varphi_H$ and corresponding eigenvalue $\lambda_H$. That is $P_{[0,1]\backslash H} \varphi_H=\lambda_H \varphi_H$ [@Collet00]. The behaviour of $\lambda_H$, and hence the escape rate, as a function of hole size is the subject of this paper.
Deriving the dynamical zeta function of the open system {#Sec:Derive-zeta}
=======================================================
The Perron-Frobenius transfer operator $P_{[0,1]\backslash H}$ in the setting we consider, for the hole $[0, t_n]$, takes the form of a transfer matrix with the the following structure $$\mathbf{P}_n=\left( \begin{array}{cccccc}
0 &0&0&0&0 &t_n \\
0 &0&0&0&0 & a_{n-1} \\
0 &1&0&0&0 & a_{n-2} \\
0 &0&1&0&0 & a_{n-3} \\
\vdots& & &\ddots&\vdots & \vdots \\
0 &0&0&0&1 &a_1 \\
\end{array} \right)
\label{Eq:P}$$ Note that the eigenvalue $\lambda_H$ is equal to the inverse of the zero $z_0$ of the polynomial $\det(1-z\mathbf{P}_n)$. This polynomial is also known as the dynamical zeta function, $\zeta_{op}(z):=\zeta_{op, n}(z)$, where the subscript [*op*]{} refers to the open system and, to lighten the notation, we drop the dependence on $n$. Let us now derive an expression for $\zeta_{op}(z)$. We have that $$\zeta_{op}(z)=\left| \begin{array}{cccccc}
1 &0&0&0&0 &-zt_n \\
0 &1&0&0&0 & -za_{n-1} \\
0 &-z&1&0&0 & -za_{n-2} \\
0 &0&-z&1&0 & -za_{n-3} \\
\vdots& & &\ddots&\vdots & \vdots \\
0 &0&0&0&-z &1-za_1 \\
\end{array} \right|
\label{Eq:1-zp}$$ Multiplying the second row by z and adding to the third, and continuing this process to the $n^{th}$ row we obtain $$\zeta_{op}(z)=\left| \begin{array}{cccccc}
1 &0&0&0&0 &-zt_n \\
0 &1&0&0&0 & -za_{n-1} \\
0 &0&1&0&0 & -za_{n-2}-z^2 a_{n-1} \\
0 &0&0&1&0 & -za_{n-3} -z^2a_{n-2}-z^3a_{n-1} \\
\vdots& & &\ddots&\vdots & \vdots \\
0 &0&0&0&0 &1-\sum_{k=1}^{n-1}z^ka_k \\
\end{array} \right|
\label{Eq:1-zp2}$$ Expanding the determinant along the first column and then the last row we obtain the dynamical zeta function $$\zeta_{op}(z)=\det(1-z\mathbf{P}_n)= 1-\sum_{k=1}^{n-1}z^ka_k.
\label{Eq:zeta}$$
Asymptotic behaviour and scaling {#Sec:Asymp}
================================
We remind the reader that we will use $\varepsilon$ to refer to the length of the hole $\varepsilon:=\varepsilon([0,t_n])$. In [@KeLi09] it was shown that for a large class of hyperbolic maps $$\gamma=\varepsilon(1-C)+o(\varepsilon)
\label{Eq:KLResult}$$ where $C$ is a constant that depends upon the stability eigenvalue of the limiting point of the hole. We recall that the notation $g(\varepsilon)=o(f(\varepsilon))$ means that $\lim_{\varepsilon \rightarrow 0} g(\varepsilon)/f(\varepsilon)=0$. For example in the doubling map, Eq.(\[Eq:KLResult\]) translates as $$\gamma=\varepsilon(1-2^{-p})+o(\varepsilon)
\label{Eq:BSResult}$$ where $p$ is the period of the limiting point. This first order result states that for small holes the escape rate behaves linearly in $\varepsilon$. For results on higher order terms see [@Det13], [@Cr13]. We will investigate deviations from this linear asymptotic behaviour in the family of $\alpha$-Farey maps defined above in Eq.(\[Eq:A-Farey-map\]) using the dynamical zeta function derived in Eq.(\[Eq:zeta\]). To this end, we aim to find a function $f:[0,1]\rightarrow [0,\infty)$ with the properties that $\lim_{\varepsilon \rightarrow 0}f(\varepsilon)=0$ and $$\gamma=f(\varepsilon)+o(\varepsilon).
\label{Eq:Escape_f}$$ That is, the function $f$ gives the small-hole behaviour of the escape rate. Given the results described by Eq.(\[Eq:KLResult\]) and Eq.(\[Eq:BSResult\]), in the case of hyperbolic maps $f$ will be a linear function of $\varepsilon$. Using the relationship between the zero $z_0$ of the polynomial in Eq.(\[Eq:zeta\]) and the escape rate, namely, $\gamma=\ln(z_0)$, we can write the zero $z_0$ in terms of the function $f$ $$z_0=1+f(\varepsilon) + o(f(\varepsilon)).
\label{Eq:Zeta-zero}$$ The explicit form of the function $f(\varepsilon)$ remains to be derived and will depend on the map $F_{\alpha}$. We will use our expression for the dynamical zeta function Eq.(\[Eq:zeta\]) and Eq.(\[Eq:Zeta-zero\]) to do this. Using Eq.(\[Eq:Zeta-zero\]) in Eq.(\[Eq:zeta\]) gives, $$\zeta_{op}(z_0)=1-\sum_{k=1}^{n-1}(1+f(\varepsilon)+o(f(\varepsilon))^ka_k=0.
\label{Eq:Zeta_with_zero}$$ Solving Eq. (\[Eq:Zeta\_with\_zero\]) and letting $\varepsilon$ tend to 0 (or, in other words, letting $n\to\infty$), we will be able to determine the function $f(\varepsilon)$, where we remind the reader that this function depends upon the partition $\alpha$. Using, $$\left(1+f(\varepsilon)+o(f(\varepsilon)\right)^k=\sum_{j=0}^k {k \choose j} (f(\varepsilon))+o(f(\varepsilon))^j,
\label{Eq:expansion_polynomial}$$ rearranging the order of the summations and extracting the $j=0$ term we obtain, $$\zeta_{op}(z_0) = 1-\sum_{k=1}^{n-1}a_k-\sum_{j=1}^{n-1}\sum_{k=j}^{n-1}{k \choose j}(f(\varepsilon))+ o(f(\varepsilon))^ja_k=0. \label{Eq:Zeta-approx}$$ Note that from the definition of the partition $\alpha$ we have that $$\sum_{k=1}^{\infty}a_k=1.
\label{Eq:ak_full_sum}$$ From the definition of $t_n$ given in Eq.(\[Eq:tn\]) and of the length of the hole $\varepsilon$ it immediately follows that $$\sum_{k=1}^{n-1}a_k=1-\varepsilon.
\label{Eq:aksum}$$ Using Eq.(\[Eq:aksum\]) in Eq.(\[Eq:Zeta-approx\]) we therefore obtain $$\varepsilon-\sum_{j=1}^{n-1}\sum_{k=j}^{n-1}{k \choose j}(f(\varepsilon))+ o(f(\varepsilon))^ja_k=0.
\label{Eq:s1Sum}$$ Also, we note here one further relation which will be useful in what follows: $$\sum_{k=j}^{n-1}{k \choose j} a_k= \sum_{k=j}^{n-1}{k \choose j}(t_k-t_{k+1})= -{n \choose j}t_n+ \sum_{k=j}^{n}{k-1 \choose j-1}t_k.
\label{Eq:kakSum}$$ Let us now consider some particular examples, including cases where the invariant measure of the closed system is finite and where it is infinite, in order to illustrate the usefulness of Eqs. (\[Eq:zeta\]) and (\[Eq:s1Sum\]).
Tent map {#SubSec:tentmap}
--------
Consider the dyadic partition, that is, the partition defined by $t_k=2^{-(k-1)}$ as described above. For each $n$ we have the system open on the interval $[0,2^{-(n-1)}]$ giving $\varepsilon=2^{-(n-1)}$, implying that $n=\ln(1/\varepsilon)/\ln(2)+1$. Using Eq.(\[Eq:s1Sum\]), Eq.(\[Eq:kakSum\]) and writing in terms of $n$ for simplicity (as $\varepsilon$ is a function of $n$) we derive, $$2^{-(n-1)}-\sum_{j=1}^{n-1}(f(n)+of(n))^j\left(-{n \choose j}2^{-(n-1)} +\sum_{k=j}^n{k-1 \choose j-1}2^{-(k-1)}\right)=0
\label{Eq:TentmapkakSum}$$ [Multiplying Eq.(\[Eq:TentmapkakSum\]) by $2^{n-1}$ and solving the $j=1$ term in the sum we obtain that]{} $$\begin{aligned}
\nonumber
1- 2^{n-1}(f(n)+of(n))(-n2^{-(n-1)} +2-2^{1-n}) &-&\\
2^{(n-1)}\sum_{j=2}^{n-1}(f(n)+of(n))^j\left(-{n \choose j}2^{-(n-1)} +\sum_{k=j}^n{k-1 \choose j-1}2^{-(k-1)}\right)&=&0
\label{Eq:TentmapkakSum2}\end{aligned}$$ Letting $f(n)=C2^{-(n-1)}$ in Eq.(\[Eq:TentmapkakSum2\]) with $C$ a constant and letting $n$ tend to infinity shows us that all other terms go to zero whilst $C=0.5$. Converting back into terms of $\varepsilon$ we obtain that $f(\varepsilon)=\varepsilon/2$, as expected [@KeLi09]. In other words, the escape rate scales linearly with the length of the hole. We further note that Eq.(\[Eq:TentmapkakSum\]) implies that there are logarithmic corrections in the higher order terms, a result which is well known (see [@Cr13], [@Det13]). See figure \[Fig:asymptoticbehaviour\] for an illustration of this result.
Piecewise Farey map {#SubSec:PFM}
-------------------
Let us now consider the piecewise linearisation $F_{\alpha_H}$ of the Farey map given by the harmonic partition. As noted above, this map preserves an infinite measure. Here $t_k=1/k$ and $\varepsilon=1/n$. Again from Eq.(\[Eq:s1Sum\]) and Eq.(\[Eq:kakSum\]) we have in this case $$\frac{1}{n}-\sum_{j=1}^{n-1}(f(n)+of(n))^j\left(-{n \choose j}\frac{1}{n} +\sum_{k=j}^n{k-1 \choose j-1}\frac{1}{k}\right)=0 \label{Eq:HarmonicSum}$$ Multiplying Eq.(\[Eq:HarmonicSum\]) by $n$ and extracting the $j=1$ term we obtain, $$1-n(f(n)+of(n))(H_n-1)-\sum_{j=2}^{n-1}(f(n)+of(n))^j\left(-{n \choose j} +n\sum_{k=j}^n{k-1 \choose j-1}\frac{1}{k}\right)=0
\label{Eq:HarmonicmapkakSum2}$$ Here, $H_n$ is the $n^{th}$ harmonic number which grows logarithmically with $n$, that is, logarithmically inversely with $\varepsilon$. Where $C$ is a constant to be determined, we now let $f(n)=C/n(H_n-1)$ and let $n$ go to infinity in Eq.(\[Eq:HarmonicmapkakSum2\]). Expanding the binomial coefficient as a polynomial and applying the Euler-Maclaurin formula we see that the higher order summation terms decay to zero as $n$ goes to infinity leaving $C=1$. In terms of $\varepsilon$ we therefore have $$f(\varepsilon)=\frac{\varepsilon}{H_n-1}.
\label{Eq:s1Harmonic}$$ However, given that $H_n$ grows logarithmically as $\varepsilon$ goes to zero, we see that the escape rate does not scale linearly with $\varepsilon$, rather there are logarithmic corrections present. See figure \[Fig:asymptoticbehaviour\].
![Escape rate as a function of hole length. In this figure the escape rate $\gamma$ is plotted as a function of hole size $\varepsilon$ , computed from the smallest zero of the dynamical zeta function (red crosses) along with the approximation $f(\varepsilon)$ (black line) which gives the small hole scaling and asymptotic behaviour. In $\mathbf{(a)}$ these are illustrated for the tent map which admits a finite invariant measure whilst in $\mathbf{(b)}$ the $\alpha_{H}$-Farey map which has an infinite invariant measure.[]{data-label="Fig:asymptoticbehaviour"}](fig2a.eps "fig:"){width="7cm"} ![Escape rate as a function of hole length. In this figure the escape rate $\gamma$ is plotted as a function of hole size $\varepsilon$ , computed from the smallest zero of the dynamical zeta function (red crosses) along with the approximation $f(\varepsilon)$ (black line) which gives the small hole scaling and asymptotic behaviour. In $\mathbf{(a)}$ these are illustrated for the tent map which admits a finite invariant measure whilst in $\mathbf{(b)}$ the $\alpha_{H}$-Farey map which has an infinite invariant measure.[]{data-label="Fig:asymptoticbehaviour"}](fig2b.eps "fig:"){width="7cm"}
Parameter dependence {#subsec:pd}
--------------------
The two examples given above, namely the $\alpha$-Farey maps arising from the dyadic and harmonic partitions, shows the existence of a fundamental difference in the scaling of the escape rate depending on whether the closed system preserves an infinite invariant measure or a finite one. We now explore this further. Let us consider the case that $$t_k=\frac{1}{k^\theta},\ \ k>1,
\label{Eq:t_kParameter}$$ where $\theta \in {{\mathbb R}}$ is a control parameter of the system. We have $\varepsilon=1/n ^{\theta}$ and from the considerations of Eq.(\[Eq:Infinite-measure-property\]) for $\theta \leq 1$ the ACIM is infinite whilst for $\theta>1$ it is finite. Note that the sum $\sum_{k=1}^{\infty} t_k$ is the value of the Riemann zeta function $\zeta(\theta)$ at $\theta$ and that for $\theta=1$ we recover the $\alpha_{H}$-Farey map. For connections between the Riemann hypothesis and the escape rate see [@BuDett05], [@DeRa14]. From Eq.(\[Eq:s1Sum\]) and Eq.(\[Eq:kakSum\]) we have, $$1-\sum_{j=1}^{n-1}(f(n)+of(n))^j\left(-{n \choose j}+n^{\theta}\sum_{k=j}^n{k-1 \choose j-1}\frac{1}{k^{\theta}}\right)=0
\label{Eq:PArSum}$$
Finite ACIM
-----------
Using Stieltjes integration, one can write the Riemann zeta function in the following way: $$\zeta(\theta):=\sum_{k=1}^\infty\frac{1}{k^\theta} = \frac{\theta}{1-\theta} - \theta\int_1^\infty \frac{\{x\}}{x^{\theta+1}}\ dx,
\label{Eq:Zeta-expression}$$ where $\{x\}$ denotes the fractional part of $x$. This relation is valid for all $\theta>1$. It therefore follows, after a straightforward calculation, that for $\theta>1$ we have $$\sum_{k=1}^n \frac{1}{k^\theta}= \zeta(\theta) +\frac{1}{n^{\theta-1}}- \theta\int_{n}^\infty\frac{\lfloor x\rfloor}{x^{\theta+1}}\ dx.
\label{Eq:Sum-finite-case}$$ For the integral in Eq.(\[Eq:Sum-finite-case\]), we have an upper bound given by $$\theta\int_{n}^\infty\frac{\lfloor x\rfloor}{x^{\theta+1}}\ dx <\frac{\theta}{\theta-1}n^{1-\theta}.
\label{Eq:Finite-integral-upperbound}$$ Considering the $j=1$ term in Eq.(\[Eq:PArSum\]), and again with $C$ a constant to be determined, we let $f(n)=C/n^{\theta}$. We let $n$ go to infinty and as before we expand the binomial coefficient as a polynomial and use the Euler-Maclaurin summation formula to check that the remaining terms is Eq.(\[Eq:PArSum\]) go to zero. This leaves us with $C=\zeta(\theta)$
Hence, as in the example of the Tent map above, the escape rate scales linearly with $\varepsilon$ with constant $\zeta(\theta)$. That is, $f(\varepsilon)$ is given by $$f(\varepsilon)=\varepsilon/\zeta(\theta).
\label{Eq:s1-finite}$$ This is illustrated in figure \[Fig:asymptotic-behaviour-parameter\].
![Escape rate as a function of hole length. In this figure the escape rate $\gamma$ is plotted as a function of hole size $\varepsilon$ for the $\alpha$ Farey map for parameter value $2$ (red diagonal crosses) and $3$ (blue crosses) along with the small $\varepsilon$ scaling and asymptotic behaviour as given by Eq.(\[Eq:s1-finite\]) for $\theta=2$ (dot-dash grey line) and $\theta=3$ (dashed grey line). []{data-label="Fig:asymptotic-behaviour-parameter"}](fig3a.eps "fig:"){width="7cm"} ![Escape rate as a function of hole length. In this figure the escape rate $\gamma$ is plotted as a function of hole size $\varepsilon$ for the $\alpha$ Farey map for parameter value $2$ (red diagonal crosses) and $3$ (blue crosses) along with the small $\varepsilon$ scaling and asymptotic behaviour as given by Eq.(\[Eq:s1-finite\]) for $\theta=2$ (dot-dash grey line) and $\theta=3$ (dashed grey line). []{data-label="Fig:asymptotic-behaviour-parameter"}](fig3b.eps "fig:"){width="7cm"}
Infinite ACIM {#theta<1}
-------------
When considering the case $\theta <1$ we can use the Euler-Maclaurin formula to rewrite the sum $ \sum_{k=1}^n \frac{1}{k^{\theta}}$ as $$\sum_{k=1}^n\frac{1}{k^\theta}= \frac{n^{1-\theta}}{1-\theta}+\frac{n^{-\theta}}{2} -\frac{\theta }{12}n^{-\theta-1}+\frac{(\theta+2)(\theta+3)}{12(\theta-1)}+\frac{\theta(\theta+1)}{2} \int_1^n x^{-\theta-2}(\{x\}^2 -\{x\}+\frac{1}{6})dx.
\label{Eq:SumGrowth}$$ The integral in Eq.(\[Eq:SumGrowth\]) has the following bound $$\frac{\theta(\theta+1)}{2} \int_1^n x^{-\theta-2}(\{x\}^2 -\{x\}+\frac{1}{6})dx \leq \frac{\theta(\theta+1)}{12} \left( \frac{-n^{-\theta -1}}{\theta+1} +\frac{1}{\theta+1}\right),
\label{Eq:Bound}$$ which (using $\varepsilon=1/n ^{\theta}$) implies that for $\theta <1$ we have $$\sum_{k=1}^n\frac{1}{k^\theta}= \frac{n^{1-\theta}}{1-\theta}+ o\left(n^{1-\theta}\right).
\label{Eq:SumGrowth2}$$ Considering then the $j=1$ term in Eq.(\[Eq:PArSum\]) we let $f(n)=C/n$, with $C$ a constant. We then use $$\sum_{k=j}^n{k-1 \choose j-1}\frac{1}{k^{\theta}}= \sum_{k=j}^n\frac{k^{j-1-\theta}}{(j-1)!}+o(k^{j-1-\theta})=\frac{n^{j-\theta}}{(j-1)!(j-\theta)}+o(n^{j-\theta}),
\label{Eq:Insumident}$$ and $${n \choose j}=\frac{n^j}{j!}+o(n^j),
\label{Eq:binomIdent}$$ in Eq.(\[Eq:PArSum\]), whilst letting $n$ go to infinity. which leaves us with $$1-\theta\sum_{j=1}^{\infty}\frac{C^j}{j!(j-\theta)}=0 \label{Eq:ConstantFunc}$$ In terms of $\varepsilon$ we have $$f(\varepsilon)=C\varepsilon^{\frac{1}{\theta}},
\label{Eq:f-infinite}$$ with $C$ a solution of Eq.(\[Eq:ConstantFunc\]). From which we see that the escape rate scales polynomially with the hole length $\varepsilon$. See figure \[Fig:asymptotic-infinite\] for an illustration of this.
![Escape rate scaling with hole length. In this figure the numerically computed escape rate $\gamma$ is plotted as a function of hole size $\varepsilon$ for four values of the map parameter $\theta$; $0.4$ (red crosses), $0.5$ (green asterixs), $0.6$ (blue diagonal crosses) and $0.8$ (yellow diamonds). Also illustrated is the scaling and asymptotic behaviour with $\varepsilon$ as given by Eq.(\[Eq:f-infinite\]) (black solid lines). []{data-label="Fig:asymptotic-infinite"}](fig4.eps){width="7cm"}
Conclusion {#Sec:Conclusion}
==========
Here we investigated the small hole scaling of the escape rate, in a class of piecewise linear dynamical systems where the hole is placed in the neighbourhood of an indifferent fixed point, so as the hole shrinks the system converges to a closed system with intermittent behaviour. In some cases the intermittency can lead to the closed system preserving an infinite absolutely continuous invariant measure. By the introduction of a parameter dependence of the intermittency, we were able to control this particular property. In the finite measure preserving case we saw that the escape rate scaled linearly with the hole size and derived the asymptotic behaviour explicitly. In the infinite measure preserving case we saw that the escape rate scaled polynomially with the hole length whilst on the border between the two (where the measure is sometimes called “barely infinite”), we saw logarithmic corrections to the scaling. We saw that there is a non-trivial relationship between the properties of the absolutely continuous invariant measure of the closed system and the scaling of the escape rate given a suitably positioned hole in phase space. We would like to extend these results to more general systems, potentially allowing the study of the properties of infinite invariant measures via tools based upon escape rate theory and vice-versa.
A possible direction for further study in this direction is the following sequence of observations. One way to answer the question of “how infinite” is a given system $(X, T, \mu)$, is to consider the wandering rate for some “good” measurable set $E$, which is defined to be the sequence $(w_n(E))_{n\geq1}$, where $w_n(E):=\mu\left( \bigcup_{k=0}^{n-1}T^{-k}(E)\right)$. In our case, we let the set $E$ be $A_1$, the first partition interval. Then, if $T=F_\alpha$, it is an easy calculation to see that the wandering rate is given by $w_n(A_1) = \sum_{k=1}^nt_k$. So, we have that for the $\alpha_H$-Farey map, the wandering rate is asymptotic to $\log n$, and for the examples from Section \[theta<1\], we have (either by comparing the sum to the integral or by considering the more careful calculation done above) that the wandering rate is asymptotic to $n^{1-\theta}$. One of the foundational results in infinite ergodic theory is Aaronson’s ergodic theorem [@Aar97], which states that there is no equivalent to Birkhoff’s ergodic theorem, that is, that for infinite systems the pointwise behaviour of the ergodic averages is so complicated that it is impossible to find a normalising sequence to estimate their actual size. On the other hand, the same author showed that in the situation of a system with a wandering rate for a “very good” set that is regularly varying, there exists a sequence $(v_n)_{n\geq1}$ with the property that the ergodic averages rescaled using $v_n$ do have an asymptotic distribution. Slightly more precisely, if $\theta\in[0,1]$ and if $w_n(E)$ is regularly varying with exponent ${1-\theta}$, then the scaling sequence $v_n(E)$ is regularly varying with exponent $\theta$. Notice that for our examples in Section \[theta<1\] we have that the scaling of the escape rate is the asymptotic inverse of this, i.e., it is equal to $1/\theta$. We tentatively conjecture that this relation might hold more generally.
Acknowledgments {#Sec:Acknowledgements .unnumbered}
===============
The authors would like to thank Giampaolo Cristadoro at the University of Bologna, Carl Dettmann at the University of Bristol and Alan Haynes at the University of York for enlightening conversations. GK would like to thank the University of Bristol for their hospitality during his visit there, similarly SM extends her thanks to the University of Bologna. The authors would also like to thank the referees for their careful reading and helpful comments which greatly improved the presentation.
[^1]: [email protected]
[^2]: [email protected]
|
---
abstract: 'We prove that an integral Jacobson radical ring is always nil, which extends a well known result from algebras over fields to rings. As a consequence we show that if every element $x$ of a ring $R$ is a zero of some polynomial $p_x$ with integer coefficients, such that $p_x(1)=1$, then $R$ is a nil ring. With these results we are able to give new characterizations of the upper nilradical of a ring and a new class of rings that satisfy the Köthe conjecture, namely the integral rings.'
address: 'Faculty of Electrical Engineering, University of Ljubljana, Tržaška cesta 25, 1000 Ljubljana, Slovenia'
author:
- Nik Stopar
title: 'Nilpotent, algebraic and quasi-regular elements in rings and algebras'
---
[*Key Words:* $\pi$-algebraic element, nil ring, integral ring, quasi-regular element, Jacobson radical, upper nilradical]{}
Introduction
============
Let $R$ be an associative ring or algebra. Every nilpotent element of $R$ is quasi-regular and algebraic. In addition the quasi-inverse of a nilpotent element is a polynomial in this element. In the first part of this paper we will be interested in the connections between these three notions; nilpotency, algebraicity, and quasi-regularity. In particular we will investigate how close are algebraic elements to being nilpotent and how close are quasi-regular elements to being nilpotent. We are motivated by the following two questions:
**Q1.** Algebraic rings and algebras are usually thought of as nice and well behaved. For example an algebraic algebra over a field, which has no zero divisors, is a division algebra. On the other hand nil rings and algebras, which are of course algebraic, are bad and hard to deal with. It is thus natural to ask what makes the nil rings and algebras bad among all the algebraic ones.
The answer for algebras over fields is well known, namely they are Jacobson radical. We generalize this to rings (and more generally to algebras over Jacobson rings) in two different ways; firstly we show that nil rings are precisely those that are integral and Jacobson radical (see Theorem \[ringI\]), and secondly, we show that the only condition needed for an algebraic ring to be nil, is that its elements are zeros of polynomials $p$ with $p(1)=1$ (see Theorem \[ring\]).
**Q2.** Can nilpotent elements be characterized by the property “quasi-inverse of $a$ is a polynomial in $a$”?
It is somewhat obvious that element-by-element this will not be possible, however we are able to characterize the upper nilradical in this way (see Corollary \[nilradical\]).
One of the most important problems concerning nil rings is the Köthe conjecture. In 1930 Köthe conjectured that if a ring has no nonzero nil ideals, then it has no nonzero nil one-sided ideals. The question whether this is true is still open. There are many statements that are equivalent to the Köthe conjecture and many classes of rings and algebras that are known to satisfy the Köthe conjecture (see [@Kre], [@Smo2], and [@Smo3] for an overview). We give yet another class of such rings, namely the integral rings (see Corollary \[kothe\]).
In the second part of this paper we investigate the structure of certain sets of elements of rings and algebras. In particular we show that a subgroup of the group of quasi-regular elements (equipped with quasi-multiplication) is closed for ring addition if and only if it is closed for ring multiplication. This gives us some information on the structure of the set of all elements of a ring which are zeros of polynomials $p$ with $p(1)=1$.
Preliminaries
=============
Throughout this paper we are dealing with associative rings and algebras, possibly nonunital and noncommutative. Given a ring or algebra $(R,+,\cdot)$, we define an operation $\circ$ on $R$, called *quasi-multiplication*, by $$a \circ b=a+b-ab.$$ It is easy to see that $(R,\circ)$ is a monoid with identity element $0$. An element $a \in R$ is called *quasi-regular* if it is invertible in $(R,\circ)$, i.e. if there exists $a' \in R$ such that $a \circ a'=a' \circ a=0$. In this case we say that $a'$ is the *quasi-inverse* of $a$. If $R$ is unital then this is equivalent to $1-a$ being invertible in $(R,\cdot)$ with inverse $1-a'$. In fact the map $f: (R,\circ) \to (R,\cdot)$ given by $x \mapsto 1-x$ is a monoid homomorphism, since $1-a \circ b=(1-a)(1-b)$. The set of all quasi-regular elements of $R$ will be denoted by $Q(R)$. Clearly $(Q(R),\circ)$ is a group, since this is just the group of invertible elements of the monoid $(R,\circ)$. For every $a \in Q(R)$ and every $n \in {\mathbb{Z}}$ the $n$-th power of $a$ in $(Q(R),\circ)$ will be denoted by $a^{(n)}$ to distinguish it from $a^n$, the $n$-th power of $a$ in $(R,\cdot)$. In particular $a^{(0)}=0$ and $a^{(-1)}$ is the quasi-inverse of $a$. If $R$ is unital, then $1-a^{(-1)}=(1-a)^{-1}$. A subset $S \subseteq R$ is called *quasi-regular* if $S \subseteq Q(R)$. The *Jacobson radical* of $R$ is the largest quasi-regular ideal of $R$ and will be denoted by $J(R)$.
The set of all nilpotent elements in $R$ will be denoted by $N(R)$. Every nilpotent element is quasi-regular, so $N(R) \subseteq Q(R)$. In fact if $x^n=0$ then $-x-x^2-\ldots-x^{n-1}$ is the quasi-inverse of $x$. A subset $S \subseteq R$ is called *nil* if $S \subseteq N(R)$. The *upper nilradical* of $R$ is the largest nil ideal of $R$ and will be denoted by $Nil^\ast(R)$. If $R$ is commutative then $Nil^\ast(R)=N(R)$.
The *lower nilradical* of $R$ (also called the *prime radical*) is the intersection of all prime ideals of $R$ and will be denoted by $Nil_\ast(R)$. It can also be characterized as the lower radical determined by the class of all nilpotent rings (see [@Gar] for details). For any ring $R$ we have $Nil_\ast(R) \subseteq Nil^\ast(R) \subseteq J(R)$.
Let $K$ be a commutative unital ring and $R$ a $K$-algebra, possibly noncommutative and nonunital. An element $a \in R$ is *algebraic* over $K$ if there exists a nonzero polynomial $p \in K[x]$ such that $p(0)=0$ and $p(a)=0$. If in addition $p$ can be chosen monic (i.e. the leading coefficient of $p$ is equal to $1$), then $a$ is called *integral* over $K$. The condition $p(0)=0$ is necessary only because $R$ may be nonunital, in which case only polynomials with zero constant term can be evaluated at elements of $R$. The set of all algebraic elements of $R$ will be denoted by $A_K(R)$, the set of all integral elements of $R$ will be denoted by $I_K(R)$. A $K$-algebra $R$ is *algebraic* (resp. *integral*) over $K$ if every element in $R$ is algebraic (resp. integral) over $K$. Note the special case of the above definitions when $R$ is just a ring, in which case we considder it as an algebra over $K={\mathbb{Z}}$. In this case we will also write $A(R)=A_{\mathbb{Z}}(R)$ and $I(R)=I_{\mathbb{Z}}(R)$. Clearly, every nilpotent element of $R$ is integral, so $N(R) \subseteq I_K(R) \subseteq A_K(R)$. If $K=F$ is a field then $I_F(R)=A_F(R)$.
$\pi$-algebraic rings and algebras
==================================
Throughout this section $K$ will always denote a commutative unital ring, $F$ a field, and $R$ an algebra over $K$ or $F$, unless specified otherwise. The two questions from the introduction motivate the following definition, which will play a crucial role in our considerations.
An element $a$ of a $K$-algebra $R$ is *$\pi$-algebraic* (over $K$) if there exists a polynomial $p \in K[x]$ such that $p(0)=0$, $p(1)=1$ and $p(a)=0$. In this case we will also say that $a$ is *$\pi$-algebraic with polynomial $p$*. A subset $S \subseteq R$ is *$\pi$-algebraic* if every element in $S$ is $\pi$-algebraic. The set of all $\pi$-algebraic elements of a $K$-algebra $R$ will be denoted by $\pi_K(R)$.
When $R$ is just a ring, we considder it as an algebra over $K={\mathbb{Z}}$, and write $\pi(R)=\pi_{\mathbb{Z}}(R)$. The crucial condition in this definition is the condition $p(1)=1$. The condition $p(0)=0$ is there simply because $R$ may not be unital, in which case only polynomials with zero constant term can be evaluated at an element of $R$.
We first present some basic properties of $\pi$-algebraic elements along with some examples.
\[inclusion\] If $R$ is a $K$-algebra then $N(R) \subseteq \pi_K(R) \subseteq A_K(R) \cap Q(R)$. If $R$ is an $F$-algebra then $N(R) \subseteq \pi_F(R)=A_F(R) \cap Q(R)$. The quasi-inverse of a $\pi$-algebraic element is a polynomial in this element.
Clearly every nilpotent element is $\pi$-algebraic and every $\pi$-algebraic element is algebraic. Suppose $a \in R$ is $\pi$-algebraic with polynomial $p$. Then $P(x)=1-(1-p(x))/(1-x)$ is again a polynomial with $P(0)=0$ (and proper coefficients). Hence we may define $a'=P(a)$. Since $x \circ P(x)=x+P(x)-xP(x)=p(x)$, we have $a \circ a'=0$. Similarly we get $a' \circ a=0$. Hence $a'$ in the quasi-inverse of $a$ and it is a polynomial in $a$. Now suppose $R$ is an $F$-algebra and $a$ is an element of $A_F(R) \cap Q(R)$. Let $r \in F[x]$ be the minimal polynomial of $a$ (if $R$ is not unital then $r(0)$ must be zero) and let $a'$ be the quasi-inverse of $a$. Suppose $r(1)=0$. Then $r(x)=(1-x)q(x)=q(x)-xq(x)$ for some polynomial $q \in F[x]$ of degree less then that of $r$. If $R$ is not unital then $q(0)=0$, so we may evaluate $q$ at $a$ in any case. Hence $0=r(a)-a'r(a)=q(a)-aq(a)-a'q(a)+a'aq(a)=q(a)-(a' \circ a)q(a)=q(a)$, which is a contradiction since $r$ was the minimal polynomial for $a$. Thus $r(1)$ is an invertible element of $F$ and hence the element $a$ is $\pi$-algebraic with polynomial $p(x)=r(1)^{-1}r(x)x$.
We shall see in the examples that the inclusion $\pi_K(R) \subseteq A_K(R) \cap Q(R)$ may be strict.
\[2minus\] If $R$ is a unital $K$-algebra then $2-\pi_K(R) \subseteq \pi_K(R)$. In particular $0,2 \in \pi_K(R)$ and $1 \notin \pi_K(R)$. If $R$ is a unital $F$-algebra then $2-\pi_F(R) \subseteq \pi_F(R)$. In addition $F\backslash \{1\} \subseteq \pi_F(R)$ and $1 \notin \pi_F(R)$.
If $a$ is $\pi$-algebraic with polynomial $p$ then $2-a$ is $\pi$-algebraic with polynomial $q(x)=p(2-x)x$. We always have $0 \in \pi_K(R)$, hence $2 \in \pi_K(R)$. The identity element is never $\pi$-algebraic since it is not quasi-regular. If $R$ is a unital $F$-algebra and $\lambda \neq 1$ is a scalar then $\lambda$ is $\pi$-algebraic with polynomial $p(x)=(1-\lambda)^{-1}(x-\lambda)x$.
Next we give a few examples.
For a finite ring $R$, $\pi(R)=Q(R)$ and $J(R)=Nil^\ast(R)$. To verify the first part observe that $(Q(R),\circ)$ is a finite group, say of order $n$. So for every $a \in Q(R)$ we have $a^{(n)}=0$, hence every $a \in Q(R)$ is $\pi$-algebraic with polynomial $p(x)=x^{(n)}=1-(1-x)^n$. The second part is well known and it also follows from the first part and Theorem \[ring\].
\[rac\] For any field $F$, $\pi_F(F)=F\backslash \{1\}=Q(F)$ by Lemma \[2minus\]. In particular $\pi_{{\mathbb{Q}}}({\mathbb{Q}})={\mathbb{Q}}\backslash \{1\}=Q({\mathbb{Q}})$. On the other hand we have $\pi({\mathbb{Q}})={\{1+\frac{1}{n} \ ; \ n \in {\mathbb{Z}}\backslash \{0\}\}}$. Indeed, if $n$ is a nonzero integer then $1+\frac{1}{n}$ is $\pi$-algebraic over ${\mathbb{Z}}$ with polynomial $s(x)=(1-n(x-1))x$. Conversely, suppose $\frac{a}{b} \in {\mathbb{Q}}$ with $a$ and $b$ coprime, is $\pi$-algebraic with polynomial $p \in {\mathbb{Z}}[x]$ of degree $d$. Then $q(x)=b^dp(\frac{x}{b})$ is a polynomial with integer coefficients. Hence $a-b$ divides $q(a)-q(b)=b^dp(\frac{a}{b})-b^dp(1)=-b^d$. Since $a$ and $b$ are coprime, this is only possible if $a-b=\pm 1$ (any prime that would divide $a-b$ would divide $b$ and hence $a$). Thus $\frac{a}{b}=1\pm \frac{1}{b}$ as needed. Obviously $A({\mathbb{Q}})={\mathbb{Q}}$, so the inclusion $\pi({\mathbb{Q}}) \subseteq A({\mathbb{Q}}) \cap Q({\mathbb{Q}})$ from Lemma \[inclusion\] is strict here.
\[mat\] Let $F \subseteq E$ be fields and $M_n(E)$ the ring of $n \times n$ matrices over $E$. Then $$\begin{aligned}
N(M_n(E)) &=& \textrm{ matrices with eigenvalues $0$}, \\
\pi_F(M_n(E)) &=& \textrm{ matrices with eigenvalues in $\overline{F}\backslash \{1\}$}, \\
Q(M_n(E)) &=& \textrm{ matrices with eigenvalues in $\overline{E}\backslash \{1\}$},\end{aligned}$$ where $\overline{F}\subseteq\overline{E}$ are algebraic closures of $F$ and $E$. A matrix is quasi-regular iff it has no eigenvalue equal to $1$. So in view of Lemma \[inclusion\], to verify the above, we only need to prove that $$A_F(M_n(E))= \textrm{ matrices with eigenvalues in $\overline{F}$}.$$ If $A \in M_n(E)$ is algebraic over $F$, it clearly has eigenvalues in $\overline{F}$. So suppose $A \in M_n(E)$ has eigenvalues $\lambda_1,\lambda_2,\ldots,\lambda_n \in \overline{F}$. For every $i=1,2,\ldots,n$, let $p_i$ be the minimal polynomial of $\lambda_i$ over $F$. Then the minimal polynomial $m_A$ of $A$ over $E$ divides $P(x)=\prod_{i=1}^{n}p_i(x)$, hence $P(A)=0$. Since $P$ has coefficients in $F$, $A$ is algebraic over $F$.
The following proposition gives a connection between $\pi$-algebraic and integral elements.
\[integral\] Let $R$ be a $K$-algebra and $a$ an element of $R$. The following are equivalent:
1. \[int1\] $a$ is $\pi$-algebraic,
2. \[int2\] $a$ is quasi-regular and $a^{(-1)}$ is integral,
3. \[int3\] $a$ is quasi-regular and $a^{(-1)}$ is a polynomial in $a$.
By Lemma \[inclusion\] implies . On the other hand, if $a^{(-1)}=P(a)$ where $P$ is a polynomial in $K[x]$, then $a+P(a)-aP(a)=0$, so $a$ is $\pi$-algebraic with polynomial $(x+P(x)-xP(x))x$. It remains to prove the equivalence of and .
For a polynomial $p \in K[x]$ define $\widehat{p}(x)=(x-1)^{\deg p}p(\frac{x}{x-1})$, which is again a polynomial in $K[x]$. Notice that $\widehat{p}(1)$ equals the leading coefficient of $p$ and the leading coefficient of $\widehat{p}$ equals $p(1)$ (the sum of all coefficients of $p$) if $p(1)\neq 0$. In addition $\widehat{p}(0)=0$ iff $p(0)=0$.
We may assume that $R$ is unital, otherwise we just adjoin a unit to $R$. Let $a$ be a quasi-regular element. Then the inverse of $1-a$ is $1-a^{(-1)}$, so the term $\frac{x}{x-1}$ evaluated at $a$ equals $-a(1-a^{(-1)})=-a+aa^{(-1)}=a^{(-1)}$. Thus $\widehat{p}(a)=(a-1)^{\deg p}p(a^{(-1)})$. This shows that $\widehat{p}(a)=0$ iff $p(a^{(-1)})=0$, since $1-a$ is invertible. Similarly $\widehat{p}(a^{(-1)})=0$ iff $p(a)=0$.
If $p$ is a monic polynomial such that $p(0)=0$ and $p(a^{(-1)})=0$ then $a$ is $\pi$-algebraic with polynomial $\widehat{p}$. If $a$ is $\pi$-algebraic with polynomial $p$ then $\widehat{p}$ is a monic polynomial such that $\widehat{p}(0)=0$ and $\widehat{p}(a^{(-1)})=0$, so $a^{(-1)}$ is integral.
In particular Proposition \[integral\] states that $\pi_K(R)=(Q(R) \cap I_K(R))^{(-1)}$ (compare with Lemma \[inclusion\]).
By Lemma \[inclusion\] an algebra over a field $F$ is $\pi$-algebraic if and only if it is algebraic and Jacobson radical. So the following proposition is just a restatement of a well known fact that any algebraic Jacobson radical $F$-algebra is nil (see for example [@Sza p. 144]). In fact, every algebraic element in the Jacobson radical of an $F$-algebra is nilpotent and its nilindex is equal to its degree.
\[alg\] Every $\pi$-algebraic $F$-algebra is nil.
We now extend this result to algebras over Jacobson rings. Recall that a commutative unital ring $K$ is a *Jacobson ring* (or a *Hilbert ring*) if every prime ideal of $K$ is an intersection of maximal ideals of $K$. Examples of Jacobson rings are fields and polynomial rings over fields in finitely many commutative variables. In addition, any principal ideal domain with infinitely many irreducible elements is also a Jacobson ring. In particular, the ring of integers ${\mathbb{Z}}$ is a Jacobson ring.
\[ringI\] If $K$ is a Jacobson ring then every integral Jacobson radical $K$-algebra is nil.
Let $R$ be an integral Jacobson radical $K$ algebra and $a \in R$. Consider $R$ as a subalgebra of some unital $K$-algebra $R^1$. Let $K[a]$ be a unital subalgebra of $R^1$ generated by $a$. Since $K$ is a Jacobson ring and $K[a]$ is a finitely generated (commutative unital) $K$-algebra, $K[a]$ is a Jacobson ring by a version of Hilbert’s Nullstellensatz [@Eis Theorem 4.19]. Hence $J(K[a])=Nil_\ast(K[a])$ is a nil ideal. It suffices to prove that $a \in J(K[a])$. Take any $r \in K[a]$. Since $ar$ is an element of $R$, it is quasi-regular in $R$ and its quasi-inverse $(ar)^{(-1)} \in R$ is integral. By Proposition \[integral\], $(ar)^{(-1)}$ is a polynomial in $ar$. But $ar$ is a polynomial in $a$, hence $(ar)^{(-1)} \in K[a]$, i.e. $ar$ is quasi-regular in $K[a]$. Since $r$ was arbitrary, we conclude that $a \in J(K[a])$.
Without the assumption that the ring $K$ is Jacobson, Theorem \[ringI\] fails.
\[ringI-converse\] If $K$ is not a Jacobson ring then there exists an integral Jacobson radical $K$-algebra which is not nil.
Let $P$ be a prime ideal of $K$ which is not an intersection of maximal ideals. Then $J(K/P)$ is a nonzero $K$-algebra. In addition, it is Jacobson radical and integral because an element $k+P \in J(K/P)$ is integral over $K$ with polynomial $x^2-kx$. Since $K$ is commutative and $P$ is a prime ideal, the algebra $K/P$ has no nonzero nilpotent elements, hence $J(K/P)$ is not nil.
The assumption that the algebra is integral in Theorem \[ringI\] is also crucial. A merely algebraic Jacobson radical algebra over a Jacobson ring need not be nil.
Consider $R={\{\frac{2m}{2n-1} \ ; \ m,n \in {\mathbb{Z}}\}}$ as a subring of rational numbers. The quasi-inverse of $\frac{2m}{2n-1}$ is $\frac{2m}{2m-2n+1}$, which is again an element of $R$. So $R$ is a Jacobson radical ring algebraic over ${\mathbb{Z}}$, but it is not nil.
As a direct consequence of Theorem \[ringI\] and Proposition \[integral\] we get
\[ring\] If $K$ is a Jacobson ring then every $\pi$-algebraic $K$-algebra is nil.
This answers question Q1 in two ways: the fact that distinguishes nil rings and algebras from all other algebraic ones is firstly that they are integral and Jacobson radical, and secondly that the polynomials ensuring algebraicity in the nil case have the sum of their coefficients equal to $1$. It is perhaps interesting that this rather large family of polynomials with the sum of coefficients equal to $1$ produces the same effect as the rather restrictive family $\{x,x^2,x^3,x^4,\ldots\}$.
Observe that in an algebraic division $F$-algebra only the identity is not $\pi$-algebraic (since all other elements are quasi-regular). So if only one element in an algebra is not $\pi$-algebraic then the algebra may be very nice instead of nil.
Next corollary addresses question Q2, giving new characterizations of the upper nilradical of a ring in the process. We formulate it only for rings, though it is valid for all algebras over Jacobson rings.
\[nilradical\] For a ring $R$ the following hold:
1. $Nil^\ast(R)$ is the largest $\pi$-algebraic ideal of $R$,
2. $Nil^\ast(R)$ is the largest integral quasi-regular ideal of $R$,
3. $Nil^\ast(R)$ is the largest quasi-regular ideal of $R$ such that the quasi-inverse of each element is a polynomial in this element.
If $I$ is an ideal of $R$ satisfying any of the above conditions then $I$ is $\pi$-algebraic by Proposition \[integral\] and thus nil by Theorem \[ring\]. Hence $Nil^\ast(R)$ is the largest such ideal.
\[JN\] If $R$ is an integral ring then $J(R)=Nil^\ast(R)$.
A ring $R$ is said to satisfy the Köthe conjecture if every nil one-sided ideal of $R$ is contained in a nil two-sided ideal of $R$ (cf. [@Yon]). If $J(R)=Nil^\ast(R)$ for a ring $R$, then $R$ satisfies the Köthe conjecture since $J(R)$ contains every nil one-sided ideal. Corollary \[JN\] thus implies
\[kothe\] Every integral ring satisfies the Köthe conjecture.
In what follows we will exhibit an even stronger connection between $\pi$-algebraic and nilpotent elements than that given by Theorem \[ring\], in case the ring $K$ satisfies certain properties given by the following definition.
We shall say that a principal ideal domain $K$ is *exceptional* if there is no nonconstant polynomial $p \in K[x]$ such that $p(k)$ would be invertible in $K$ for all $k \in K$.
Exceptional PIDs are quite common, here are some examples.
\[exceptional\]
1. \[ex1\] A field is an exceptional PID if and only if it is algebraically closed.
2. \[ex2\] The ring of integers ${\mathbb{Z}}$ and the ring of Gaussian integers ${\mathbb{Z}}[i]$ are exceptional PIDs.
3. \[ex3\] For any field $F$ the polynomial ring $F[x]$ is an exceptional PID.
4. \[ex4\] If $K$ is an exceptional PID and $S \subseteq K$ is a multiplicatively closed subset multiplicatively generated by a finite number of elements, then the localization $S^{-1}K$ is an exceptional PID.
Claim is clear.
: Let $p$ be a polynomial in ${\mathbb{Z}}[x]$ such that $p(k)$ is invertible for all $k \in {\mathbb{Z}}$. Since there are only finitely many invertible elements in ${\mathbb{Z}}$, there exist an invertible element $u \in {\mathbb{Z}}$, such that $p(k)=u$ for infinitely many $k \in {\mathbb{Z}}$. But then the polynomial $p(x)-u$ has infinitely many zeros, so it must be zero. Hence $p$ is a constant polynomial. The same proof works for ${\mathbb{Z}}[i]$.
: Let $F$ be a field and $P(y)$ a nonconstant polynomial in $(F[x])[y]$. Write $P(y)=p_0(x)+p_1(x)y+\ldots+p_n(x)y^n$ where $p_n(x) \neq 0$ and $n \geq 1$. Denote $d_i=\deg p_i$ and $d=\max{\{d_i \ ; \ i=0,1,2,\ldots,n\}}$, where the degree of the zero polynomial is equal to $-\infty$. Let $p(x)=x^{d+1}$. The degree of $p_i(x)(p(x))^i$ is equal to $d_i+i(d+1)$. Since $d_n,d \neq -\infty$, we have $d_n+n(d+1) \geq n(d+1)>d+(n-1)(d+1) \geq d_i+i(d+1)$ for all $i<n$. This implies that the degree of $P(p(x))$ is equal to $d_n+n(d+1) \geq 1$, hence $P(p(x))$ is not invertible in $F[x]$.
: A localization of a PID is again a PID. Factor each generator of $S$ into irreducible factors and let $S' \subseteq K$ be a multiplicatively closed subset multiplicatively generated by all the irreducible elements appearing in these factorizations. Since the localizations of $K$ at $S$ and $S'$ are isomorphic, we may assume that $S=S'$, i.e. $S$ is generated by a finite number of irreducible elements. Now suppose $p(x) \in S^{-1}K[x]$ is a polynomial, such that $p(\widehat{k})$ is invertible in $S^{-1}K$ for all $\widehat{k} \in S^{-1}K$. Take any $s \in S$, such that the coefficients of $sp(x)$ are elements of $K$. Let $t$ be the product of all irreducible elements in $S$. Observe that the coefficients of the polynomial $sp(sp(0)tx)$ are elements of $K$ and are all divisible by $sp(0) \in K$. Hence, $P(x)=\frac{s}{sp(0)}p(sp(0)tx)=\frac{1}{p(0)}p(sp(0)tx)$ is a polynomial with coefficients in $K$. Now take any $k \in K$. By assumption, $p(0)$ and $p(sp(0)tk)=p(0)P(k)$ are invertible in $S^{-1}K$, hence so is $P(k)$. But $P(k) \in K$, so the only irreducible elements that may divide $P(k)$ are those that lie in $S$. However, any such irreducible element divides $t$, hence it divides all coefficients of $P$ except $P(0)=1$, so it cannot divide $P(k)$. This shows that $P(k)$ is invertible in $K$. Since $K$ is an exceptional PID, $P(x)$, and consequently $p(x)$, must be a constant polynomial.
In a PID every nonzero prime ideal is maximal, so a PID is a Jacobson ring if and only if $0$ is an intersection of maximal ideals, i.e. the Jacobson radical is $0$.
\[J-semisimple\] If $K$ is an exceptional PID then $J(K)=0$, i.e. $K$ is a Jacobson ring. In particular, if $K$ is not a field then $K$ has infinitely many nonassociated irreducible elements.
Let $K$ be an exceptional PID. Suppose $J(K) \neq 0$ and take $0 \neq a \in J(K)$. Since $K$ is commutative and unital, this implies that $1-ak$ is invertible in $K$ for every $k \in K$. But then the polynomial $p(x)=1-ax$ contradicts the definition of an exceptional PID. Hence $J(K)=0$. Since $K$ is commutative and unital, $J(K)$ is just the intersection of all maximal ideals of $K$. If $K$ is not a field then the maximal ideals of $K$ are the principal ideals generated by the irreducible elements. If there are only finitely many such ideals then their intersection is nonzero.
The converse of Proposition \[J-semisimple\] does not hold. There exist PIDs which are Jacobson rings but are not exceptional. The simplest example is given by any field that is not algebraically closed, however fields are rather extremal among all PID, since they have no irreducible elements. Hence, we give an example which is not a field.
Let $S \subseteq {\mathbb{Z}}$ be a multiplicatively closed subset multiplicatively generated by all primes $p$ with $p=2$ or $p \equiv 1 \pmod 4$ and let $K=S^{-1}{\mathbb{Z}}$ be the localization of ${\mathbb{Z}}$ at $S$. Then $K$ has infinitely many nonassociated irreducible elements, represented by the primes $p$ with $p \equiv 3 \pmod 4$, hence $J(K)=0$ and $K$ is a Jacobson ring. Now let $p(x)=x^2+1$. To see that $K$ is not exceptional, we will show that $p(k)$ is invertible in $K$ for all $k \in K$. For $k=\frac{m}{n} \in K$ we have $p(k)=\frac{m^2+n^2}{n^2}$. To see that this is invertible in $K$ we need to show that any prime dividing $m^2+n^2$ is contained in $S$. Suppose $p$ is a prime with $p \equiv 3 \pmod 4$ that divides $m^2+n^2$. Then $m^2 \equiv -n^2 \pmod p$. Since $n \in S$, this implies that both $m$ and $n$ are coprime to $p$. Hence we have $1 \equiv m^{p-1} \equiv (m^2)^{\frac{p-1}{2}} \equiv (-n^2)^{\frac{p-1}{2}} \equiv (-1)^{\frac{p-1}{2}}n^{p-1} \equiv (-1)^{\frac{p-1}{2}} \equiv -1 \pmod p$. This is a contradiction since $p \neq 2$, which finishes the proof.
Theorem \[ring\] implies that if the subalgebra generated by an element $a$ is $\pi$-algebrac, then $a$ is a nilpotent element. The next proposition, which was our main motivation for the introduction of exceptional PIDs, considers the situation when only the submodule generated by $a$ is assumed to be $\pi$-algebraic. It thus gives a stronger connection between $\pi$-algebraic and nilpotent elements for algebras over exceptional PIDs.
\[na\] Let $K$ be an exceptional principal ideal domain and $R$ a $K$-algebra. If $a$ is an element of $R$ such that $Ka \subseteq \pi_K(R)$, then there exists $0 \neq k \in K$ such that $ka$ is nilpotent. In particular, if $R$ has no $K$-torsion, then $a$ is nilpotent.
For a nonzero polynomial $f \in K[x]$, let $\delta(f)$ denote the greatest common divisor of all coefficients of $f$. First we show that for any $\pi$-algebraic element $r$ there exists a nonzero polynomial $f \in K[x]$ and a nonzero element $c \in K$ such that $f(1)=1$, $cf(r)=0$, and $f$ divides (within $K[x]$) any polynomial that annihilates $r$. So let $r$ be $\pi$-algebraic with polynomial $h \in K[x]$. Choose a nonzero polynomial $p \in K[x]$ of minimal degree such that $p(r)=0$ and let $c=\delta(p)$ and $f(x)=\frac{p(x)}{c} \in K[x]$. So $cf(r)=0$ and $\delta(f)=1$. Suppose $P \in K[x]$ is a polynomial with $P(r)=0$. By the division algorithm there exists $0 \neq m \in K$ and polynomials $s,t \in K[x]$ with $\deg t<\deg f=\deg p$ such that $mP(x)=s(x)f(x)+t(x)$ (divide in $F[x]$, where $F$ is the field of fractions of $K$, and multiply by a common denominator of all fractions). Multiplying by $c$ we get $cmP(x)=cs(x)f(x)+ct(x)=s(x)p(x)+ct(x)$. The minimality of $p$ now implies $ct(x)=0$, hence $t(x)=0$ and $mP(x)=s(x)f(x)$. By Gauss’s lemma this implies $\delta(s)=m\delta(P)$ up to association, so $m$ divides $\delta(s)$. Thus the polynomial $\frac{s(x)}{m}$ has coefficients in $K$ and $P(x)=\frac{s(x)}{m}f(x)$, i.e. $f$ divides $P$. In particular $f$ divides $h$, so there is a polynomial $S$ such that $h(x)=S(x)f(x)$. Evaluating at $1$ we get $1=S(1)f(1)$, so $f(1)$ is invertible in $K$. We may assume that $f(1)=1$, otherwise we just multiply $f$ by $f(1)^{-1}=S(1)$.
Now let $R$ be a $K$-algebra and $a$ an element of $R$ with $Ka \subseteq \pi_K(R)$. By the above, for any $k \in K$ there exists $0 \neq c_k \in K$ and $0 \neq f_k \in K[x]$ such that $f_k(1)=1$, $c_kf_k(ka)=0$, and $f_k$ divides any polynomial that annihilates $ka$. Let $k \neq 0$. Then $f_1$ divides $c_kf_k(kx)$, since $c_kf_k(kx)$ annihilates $a$. Similarly $c_1k^{\deg f_1}f_1(\frac{x}{k})$ is a polynomial in $K[x]$ that annihilates $ka$, so $f_k$ divides $c_1k^{\deg f_1}f_1(\frac{x}{k})$. This in particular implies that all these polynomials have the same degree, so there exists $d_k \in K$ such that $c_1k^{\deg f_1}f_1(\frac{x}{k})=d_kf_k(x)$. We have $f_k(1)=1$, hence $\delta(f_k)=1$. Consequently, $c_1\delta(k^{\deg f_1}f_1(\frac{x}{k}))=d_k$ up to association. If $k$ is coprime to the leading coefficient of $f_1$ then $\delta(k^{\deg f_1}f_1(\frac{x}{k}))=1$ since $\delta(f_1)=1$. For such $k$ we have $c_1=d_k$ up to association, hence $c_1$ divides $d_k$ and $u_k=\frac{d_k}{c_1}$ is invertible. In addition $k^{\deg f_1}f_1(\frac{x}{k})=u_kf_k(x)$. Evaluating at $1$, we get $k^{\deg f_1}f_1(\frac{1}{k})=u_k$. Now $p(x)=x^{\deg f_1}f_1(\frac{1}{x})$ is a polynomial in $K[x]$ with $p(0)$ equal to the leading coefficient of $f_1$. Hence, we have proved above that $p(k)$ is invertible for every $k \neq 0$ coprime to $p(0)$. If we define $t(x)=p(p(0)x-1) \in K[x]$, then $t(k)$ is invertible for all $k \in K$ ($p(0)k-1=0$ means that $p(0)$ is invertible). Since $K$ is exceptional, it follows that $t$ is a constant polynomial and so is $p$. Hence, there exists $u \in K$ such that $f_1(\frac{1}{x})=\frac{u}{x^{\deg f_1}}$, i.e. $f_1(x)=ux^{\deg f_1}$. Consequently, $c_1ua^{\deg f_1}=0$ and $c_1ua$ is nilpotent. Clearly $c_1u \neq 0$.
\[rem-na\] We shall later need a slightly modified version of Proposition \[na\] with $K={\mathbb{Z}}$. Observe that the conclusion still holds if we assume just ${\mathbb{N}}a \subseteq \pi(R)$ instead of ${\mathbb{Z}}a \subseteq \pi(R)$. Indeed, one just has to replace polynomial $t(x)=p(p(0)x-1)$ in the proof with the polynomial $\widehat{t}(x)=p((p(0)x-1)^2)$.
Without the assumption that $K$ is exceptional, Proposition \[na\] fails.
Let $K$ be a principal ideal domain which is not exceptional. Then there exists a $K$-algebra $R$ and an element $a \in R$ such that $Ka \subseteq \pi_K(R)$, but $ka$ is not nilpotent for any $0 \neq k \in K$.
Choose a nonconstant polynomial $p \in K[x]$, such that $p(k)$ is invertible in $K$ for all $k \in K$. Let $F$ be the algebraic closure of the field of fractions of $K$. Clearly $F$ is a $K$-algebra. Since polynomial $p$ is nonconstant, the polynomial $P(x)=x^{\deg p}p\left(\frac{1}{x}\right) \in K[x]$ has a nonzero root $a \in F$. Clearly, $ka$ is not nilpotent for any $0 \neq k \in K$. To finish the proof we show, that $Ka \subseteq \pi_K(F)$. The zero element is always $\pi$-algebraic, so take any $0 \neq k \in K$. Observe that $\deg P=\deg p$, because $p(0)$ is invertible. Hence $Q(x)=k^{\deg p}P\left(\frac{x}{k}\right) \in K[x]$. Since $Q(1)=p(k)$ is invertible in $K$, the element $ka$ is $\pi$-algebraic over $K$ with polynomial $Q(1)^{-1}Q(x)x \in K[x]$.
Recall that an algebra $R$ is called *nil of bounded index $\leq n$* if $a^n=0$ for all $a \in R$. $R$ is called *nil of bounded index* if there exists an integer $n$ such that $R$ is nil of bounded index $\leq n$. Similarly we will say that a $K$-algebra $R$ is *$\pi$-algebraic of bounded degree $\leq n$* (resp. *integral of bounded degree $\leq n$*) if every element of $R$ is $\pi$-algebraic (resp. integral) over $K$ with some polynomial of degree $\leq n$. $R$ is *$\pi$-algebraic of bounded degree* (resp. *integral of bounded degree*) if there exists an integer $n$ such that $R$ is $\pi$-algebraic of bounded degree $\leq n$ (resp. integral of bounded degree $\leq n$).
It follows from the proof of Proposition \[integral\] that an algebra is $\pi$-algebraic of bounded degree $\leq n$ if and only if it is Jacobson radical and integral of bounded degree $\leq n$. Theorem \[ring\] raises the following natural question. If an algebra $R$ over a Jacobson ring $K$ is $\pi$-algebraic of bounded degree, is it nil of bounded index? The answer is positive for algebras with no $K$-torsion.
\[bounded\] Let $K$ be a Jacobson ring. If $R$ is a $\pi$-algebraic $K$-algebra of bound degree $\leq n$ with no $K$-torsion, then $R$ is nil of bounded index $\leq n$.
Let $R$ be a $\pi$-algebraic $K$-algebra of bounded degree $\leq n$ with no $K$-torsion. By the remark above, $R$ is integral of bounded degree $\leq n$, and by Theorem \[ring\], $R$ is nil. Take any $a \in R$. Let $p \in K[x]$ be a monic polynomial of degree $\leq n$, such that $p(a)=0$, and let $m$ be the smallest integer such that $a^m=0$. Suppose $m>n$. Write $p$ in the form $p(x)=t(x)x^k$, where $t(0) \neq 0$ and $k\leq n<m$. Multiplying the equality $0=t(a)a^k$ by $a^{m-k-1}$, we get $0=t(a)a^{m-1}=t(0)a^{m-1}$, because $a^m=0$. Since $R$ has no $K$-torsion, this implies $a^{m-1}=0$, which is in contradiction with the choice of $m$. Thus $m \leq n$ as needed.
Perhaps surprisingly, the answer for general algebras over Jacobson rings is negative as the following example shows.
Let $K$ be a Jacobson PID, which is not a field. Then $K$ has infinitely many nonassociated irreducible elements. Choose a countable set of nonassociated irreducible elements $\{p_1,p_2,p_3,\ldots\}$ and let $R=\bigoplus_{i=1}^{\infty} p_iK/p_i^iK$. Clearly, $R$ is nil, but not of bounded index. Let $a=(a_i)_i$ be an element of $R$. By the Chinese remainder theorem there is an element $k \in K$ such that $k \equiv a_i \pmod{p_i^i}$ for all $i$ with $a_i \neq 0$. Thus $a$ is a zero of the monic polynomial $x^2-kx$. This shows that $R$ is integral of bounded degree $\leq 2$, hence it is also $\pi$-algebraic of bounded degree $\leq 2$.
Nevertheless the following holds for arbitrary algebras over Jacobson rings.
Let $K$ be a Jacobson ring. If $R$ is a $\pi$-algebraic $K$-algebra of bounded degree then $Nil_\ast(R)=R$. In particular, $R$ is locally nilpotent.
Suppose $P$ is a prime ideal of $R$. We want to apply Corollary \[bounded\] to $R/P$. $K$-algebra $R/P$ is again $\pi$-algebraic of bounded degree. Let $I={\{k \in K \ ; \ k(R/P)=0\}}$. Clearly, $I$ is an ideal of $K$ and $R/P$ becomes a $K/I$-algebra if we define $(k+I)(r+P)=k(r+P)=kr+P$. Observe that $R/P$ is $\pi$-algebraic of bounded degree also over $K/I$. In addition, $R/P$ has no $K/I$-torsion. Indeed, if $(k+I)(r+P)=0$ for some $k \in K$ and $r \in R$ with $r+P \neq 0$, then $J={\{x+P \in R/P \ ; \ k(x+P)=0\}}$ is a nonzero ideal of $R/P$. But $k(R/P) \cdot J=0$ and $R/P$ is a prime $K$-algebra, so $k(R/P)=0$, i.e. $k+I=0$ in $K/I$ as needed. $K/I$ is again a Jacobson ring, hence Corollary \[bounded\] implies that $R/P$ is nil of bounded index. Thus, by a result of Levitzki [@Lev Theorem 4], we have $Nil_\ast(R/P)=R/P$, but on the other hand, $Nil_\ast(R/P)=0$ since $P$ is a prime ideal. So $P=R$, which shows that $Nil_\ast(R)=R$.
The structure of $\pi(R)$
=========================
In this section we investigate the structure of the set of all $\pi$-algebraic elements of an algebra. We restrict ourselves to algebras over fields and to rings. Throughout the section, $F$ will always denote a field and $R$ an $F$-algebra or a ring.
Recall that $(Q(R),\circ)$ is a group and by Lemma \[inclusion\] we have $N(R) \subseteq \pi(R) \subseteq Q(R)$. It is thus natural to ask under what conditions $N(R)$ and $\pi(R)$ are subgroups of $Q(R)$ and more generally what can be said about the structure of $\pi(R)$. In general $\pi(R)$ will not be closed under $\circ$. We give a concrete example later (see Example \[notcl\]), but the reason for this is that the integral elements of $R$ do not have any structure in general (they do not form a subring). However, if $R$ is commutative, then $\pi(R)$ will be closed under $\circ$. From here on $Q(R)$ will always be considered as a group with operation $\circ$.
\[aut\] For a quasi-regular element $r \in R$ the map $x \mapsto r\circ x\circ r^{(-1)}$ is an automorphism of $R$.
The proof of this lemma is an easy calculation. In fact, if $R$ is unital then $r\circ x\circ r^{(-1)}=(1-r)x(1-r^{(-1)})$, so the map is just the usual conjugation by $1-r$.
\[subgroup\]
1. If $R$ is a ring or an $F$-algebra then $N(R)$ is closed under conjugation and inversion. If $R$ is commutative then $N(R)$ is a subgroup of $Q(R)$.
2. If $R$ is an $F$-algebra, then $\pi_F(R)$ is closed under conjugation and inversion. If $R$ is commutative then $\pi_F(R)$ is a subgroup of $Q(R)$.
3. If $R$ is a ring, then $\pi(R)$ is closed under conjugation. If $R$ is commutative then $\pi(R)$ is a submonoid of $Q(R)$.
Let $a \in R$, $r \in Q(R)$, and let $p$ be a polynomial. Then by Lemma \[aut\] $r\circ p(a)\circ r^{(-1)}=p(r\circ a\circ r^{(-1)})$, so $r\circ a\circ r^{(-1)}$ is annihilated by the same polynomials as $a$. This shows that $N(R)$ and $\pi(R)$ (resp. $\pi_F(R)$) are closed under conjugation.
The group inverse (quasi-inverse) of a nilpotent element is a polynomial in this element, so it is again nilpotent. Thus $N(R)$ is closed under inversion. If $R$ is commutative then $N(R)$ is a subring of $R$, so it is closed under $\circ$ as well.
Let $R$ be an $F$-algebra. If $R$ is commutative then $A_F(R)$ is a subalgebra of $R$. Thus $A_F(R)$ is closed under $\circ$ and by Lemma \[inclusion\] so is $\pi_F(R)$. If $a \in \pi_F(R)$ then the quasi-inverse of $a$ is a polynomial in $a$, so it is algebraic and hence contained in $A_F(R) \cap Q(R)=\pi_F(R)$.
If $R$ is a ring then by Proposition \[integral\] $\pi(R)^{(-1)}=I(R)\cap Q(R)$. If $R$ is commutative then $I(R)$ is a subring of $R$ and hence closed under $\circ$. So $\pi(R)^{(-1)}$ and consequently $\pi(R)$ is closed under $\circ$.
For a ring $R$ the set $\pi(R)$ need not be closed under inversion. For example the quasi-inverse of $1+\frac{1}{2} \in \pi({\mathbb{Q}})$ is $1+2$ and is not contained in $\pi({\mathbb{Q}})$. In fact, we know that $\pi(R)^{(-1)}=I(R)\cap Q(R)$.
\[notcl\] Let $F$ be an algebraically closed field and $E=F(x)$ the field of rational functions over $F$. By Example \[mat\], $\pi_F(M_2(E))$ consists of matrices with eigenvalues in $F \backslash\{1\}$. Take matrices $$A=\left[\begin{array}{cc}0&x\\0&0 \end{array}\right] \qquad\textrm{and}\qquad B=\left[\begin{array}{cc}0&0\\1&0 \end{array}\right],$$ which both lie in $\pi_F(M_2(E))$, since they are nilpotent. Then $$A \circ B=\left[\begin{array}{cc}-x&x\\1&0 \end{array}\right]$$ does not have eigenvalues in $F$, since its trace is $-x \notin F$. So $\pi_F(M_2(E))$ is not closed under $\circ$.
For a subset $S$ of $Q(R)$ let ${\langle S \rangle}$ denote the normal subgroup of $Q(R)$ generated by $S$. By Proposition \[subgroup\] we have: $$\begin{aligned}
{\langle N(R) \rangle} &=& \textup{ finite products of elements of } N(R),\\
{\langle \pi_F(R) \rangle} &=& \textup{ finite products of elements of } \pi_F(R),\\
{\langle \pi(R) \rangle} &=& \textup{ finite products of elements of } \pi(R) \cup \pi(R)^{(-1)},\\
{\langle \pi(R)\cap\pi(R)^{(-1)} \rangle} &=& \textup{ finite products of elements of } \pi(R)\cap I(R),\end{aligned}$$ where products means products in operation $\circ$.
From Example \[rac\] it is easy to calculate that we have ${\langle \pi({\mathbb{Q}}) \rangle}=Q({\mathbb{Q}})={\mathbb{Q}}\backslash\{1\}$ and ${\langle \pi({\mathbb{Q}})\cap\pi({\mathbb{Q}})^{(-1)} \rangle}=\{0,2\}$.
Recall that a complex matrix $A$ is called *unipotent* if $I-A$ is nilpotent, where $I$ denotes the identity matrix. In [@Wan] it was shown that a complex matrix is a finite product of unipotent matrices iff it has determinant $1$. This shows that $${\langle N(M_n({\mathbb{C}})) \rangle}={\{A \in M_n({\mathbb{C}}) \ ; \ \det(I-A)=1\}},$$ which is a proper subgroup of $$\pi_{\mathbb{C}}(M_n({\mathbb{C}}))=Q(M_n({\mathbb{C}}))={\{A \in M_n({\mathbb{C}}) \ ; \ \det(I-A) \neq 0\}}.$$
Next we investigate what can be said about addition. We will need the following proposition which may be of independent interest. Recall that an integral domain $K$ is called a *factorization domain* (also an *atomic domain*) if every nonzero nonunit of $K$ can be written as a finite product of irreducible elements.
\[div\] Let $R$ be a unital ring and $K$ a commutative subring of $R$ with $1 \in K$ such that $R \backslash K \subseteq R^{-1}$. If $K$ is a factorization domain then one of the following holds:
1. $R=K$,
2. $R$ is a local ring with maximal ideal $m \subseteq K$ and $K$ is a local ring with maximal ideal $m$,
3. $R$ is a division ring.
Suppose that $R \neq K$ and $R$ is not a division ring. Then there exist $r \in R\backslash K \subseteq R^{-1}$ and $0 \neq a \in K \backslash R^{-1}$. Since $K$ is a factorization domain, we may assume that the element $a$ is irreducible. We will prove that $K^{-1}=K\cap R^{-1}$. Let $x$ be arbitrary element of $K$ that is invertible in $R$ and set $y=x^{-1}a$. Then $y$ is not invertible in $R$, since $a$ is not. But $R\backslash K \subseteq R^{-1}$, so $y \in K$. Thus $a=xy$ is a factorization of $a$ in $K$. Since $a$ was irreducible and $y$ is not invertible, $x$ must be invertible in $K$, as needed. Now let $m$ be the set of all elements of $K$ that are not invertible in $K$. Since $R \backslash K \subseteq R^{-1}$, $m$ is also the set of all non-invertible elements of $R$. If $x \in m$ and $k \in K$ then $xk$ is not invertible in $K$, otherwise $x$ would be invertible due to the commutativity of $K$. So $mK \subseteq m$. If $x,y \in m$ then by the above $x$ and $y$ are not invertible in $R$. By the choice of $r$ this implies that $xr$ and $yr$ are not invertible in $R$, so $xr,yr \in K$. Thus $(x-y)r \in K$. But $x-y \in K$ and $r \notin K$, hence $x-y$ cannot be invertible in $K$, so $x-y \in m$. This proves that $m$ in an ideal in $K$, so $K$ is local with maximal ideal $m$. Now let $x \in m$ and $s \in R$, so by the above $x$ is not invertible in $R$. If $s \in K$ then $sx,xs \in m$ by what we have just proved. If $s \notin K$ then $s$ is invertible in $R$. So $sx$ and $xs$ are not invertible in $R$, hence $sx,xs \in m$. This shows that $m$ is also an ideal of $R$ and $R$ is local with maximal ideal $m$.
There exist examples where case (ii) of Proposition \[div\] occurs in a nontrivial way. Take for example $R=E[[x]]$ and $K=F+E[[x]]x \subseteq R$ where $F \varsubsetneq E$ are fields. Every nonzero nonunit in $K$ is contained in $E[[x]]x$ and factors as $x^ng(x)$ for some nonnegative integer $n$ and some $g(x)$ of the form $\alpha_1 x+\alpha_2x^2+\alpha_3x^3+\ldots$ with $\alpha_1 \neq 0$.
\[add\] Let $R$ be a ring. For any subgroup $S$ of $Q(R)$ the following are equivalent:
1. $S$ is closed under addition,
2. $S$ is closed under multiplication,
3. $S$ is a subring of $R$.
We can verify by a short calculation that for any $x,y \in Q(R)$ we have $$xy=x\circ (x^{(-1)}+y^{(-1)})\circ y \qquad\textup{and}\qquad x+y=x\circ (x^{(-1)}y^{(-1)})\circ y.$$ This shows that (i) and (ii) are equivalent. For $x \in Q(R)$ we also have $$-x=(2x^{(-1)})\circ x,$$ so (i) implies that $S$ is closed under negation as well, which implies (iii). Clearly (iii) implies (i).
As a corollary to Theorem \[add\] we have the following.
\[cor1\] Let $F$ be a field of characteristic $0$ and $R$ a commutative $F$-algebra. If $\pi_F(R)$ is closed under addition then $\pi_F(R)=N(R)$.
Since $R$ is commutative, $\pi_F(R)$ is a subgroup of $Q(R)$ by Proposition \[subgroup\]. If $\pi_F(R)$ is closed under addition then it is a subring of $R$ by Theorem \[add\]. Let $a \in R$ be $\pi$-algebraic with polynomial $p$ and let $\lambda$ be a nonzero scalar. Since $F$ is of characteristic $0$ there exists a positive integer $n$ such that $n\lambda^{-1}$ is not a zero of $p$. Hence $n^{-1}\lambda a$ is $\pi$-algebraic with polynomial $p(n\lambda^{-1})^{-1}p(n\lambda^{-1}x)$. Since $\pi_F(R)$ is closed under addition and $\lambda a$ is a multiple of $n^{-1}\lambda a$, $\lambda a$ is $\pi$-algebraic as well. So $\pi_F(R)$ is in fact a subalgebra of $R$. Thus $\pi_F(R)$ is nil by Proposition \[alg\] and $\pi_F(R)=N(R)$ follows.
The conclusion of Corollary \[cor1\] also holds for rings.
\[pro1\] Let $R$ be a commutative ring. If $\pi(R)$ is closed under addition then $\pi(R)=N(R)$.
Suppose $\pi(R)$ is closed under addition. First we show that $\pi(R)$ is closed also under negation. If $a$ is $\pi$-algebraic, then ${\mathbb{N}}a \subseteq \pi(R)$ since $\pi(R)$ is closed under addition. By Proposition \[na\] and Remark \[rem-na\] there exists a nonzero integer $n$ such that $na$ is nilpotent. Thus $-|n|a$ is nilpotent and hence $\pi$-algebraic. So $-a=-|n|a+(|n|-1)a$ is $\pi$-algebraic as well, since $(|n|-1)a$ is a nonnegative multiple of $a$. The commutativity of $R$ implies that $\pi(R)$ is closed under $\circ$. Since $xy=x+y-x\circ y$, $\pi(R)$ is closed under multiplication as well. So $\pi(R)$ is a $\pi$-algebraic subring of $R$, hence it is nil by Theorem \[ring\].
We are now left with the case of algebras over fields of prime characteristic. We were not able to obtain an analogue of Corollary \[cor1\] for arbitrary fields of prime characteristic, but only for algebraic extensions of prime fields.
\[cor2\] Let $p$ be a prime number, $F$ an algebraic field extension of the prime field ${\mathbb{Z}}/p{\mathbb{Z}}$, and $R$ a commutative $F$-algebra. If $\pi_F(R)$ is closed under addition then $\pi_F(R)=N(R)$.
Since $F$ is algebraic over ${\mathbb{Z}}/p{\mathbb{Z}}$, we have $A_F(R)=A_{{\mathbb{Z}}/p{\mathbb{Z}}}(R)$, so $\pi_F(R)=\pi_{{\mathbb{Z}}/p{\mathbb{Z}}}(R)$ by Lemma \[inclusion\]. Now let $a \in R$ be $\pi$-algebraic over ${\mathbb{Z}}/p{\mathbb{Z}}$ with polynomial $\widehat{f}$ and let $f$ be a polynomial with integer coefficients that represents $\widehat{f}$. Since $\widehat{f}(1)=1$, there exists an integer $k$ such that $f(1)=kp+1$. If we set $F(x)=f(x)-kpx$, then $F(0)=0$, $F(1)=1$ and $F(a)=0$, since $pa=0$. So $a$ is $\pi$ algebraic over ${\mathbb{Z}}$. Hence $\pi_{{\mathbb{Z}}/p{\mathbb{Z}}}(R) \subseteq \pi(R)$ and clearly $\pi(R) \subseteq \pi_{{\mathbb{Z}}/p{\mathbb{Z}}}(R)$. This implies $\pi_F(R)=\pi(R)$ and so $\pi_F(R)=N(R)$ by Proposition \[pro1\].
This was one extremal situation, when every $\pi$-algebraic element is in fact nilpotent. The other extremal situation would be when there are no nilpotent elements, but many $\pi$-algebraic ones. As we have mentioned before, in an algebraic division algebra there are no nonzero nilpotent elements although all elements except the unit are $\pi$-algebraic. Next we investigate when something similar happens in general algebras. The question is whether $\pi_F(R) \cup \{1\}$ will form a division subring of a unital $F$-algebra $R$. When $R$ is just a ring, we can ask a similar question, however it seems more natural to consider the set ${\langle \pi(R) \rangle} \cup ({\mathbb{Z}}\cdot 1)$ in this case, since the elements in $({\mathbb{Z}}\cdot 1) \backslash \{1\}$ need not be automatically contained in ${\langle \pi(R) \rangle}$. In certain situations though, they are.
\[addunitzero\] Let $R$ be a unital ring of characteristic $0$. For any subgroup $S$ of $Q(R)$ with $\{0,2\} \varsubsetneq S$ the following are equivalent:
1. \[auz1\] $S \cup {\mathbb{Z}}$ is closed under addition,
2. \[auz2\] $S \cup \{1\}$ is a division subring of $R$.
Clearly implies , since in this case $S \cup \{1\}=S \cup {\mathbb{Z}}$. So assume holds. First we show that $S \cup {\mathbb{Z}}$ is a subring. If $x \in S \cup {\mathbb{Z}}$ then $2 \circ x=2-x \in S \cup {\mathbb{Z}}$, since $2 \in S \cap {\mathbb{Z}}$. So if $x \in S \cup {\mathbb{Z}}$ then $-x=2-(2+x) \in S \cup {\mathbb{Z}}$ by . Thus $S \cup {\mathbb{Z}}$ is closed under negation. $S$ and ${\mathbb{Z}}$ are both closed under $\circ$. If $x \in S$ and $n \in {\mathbb{Z}}$ then $x \circ n=n \circ x=n+x-nx \in S \cup {\mathbb{Z}}$, since $nx$ is a multiple of $x$ or $-x$ and $S \cup {\mathbb{Z}}$ is closed under addition. So $S \cup {\mathbb{Z}}$ is closed under $\circ$ and also under multiplication since $xy=x+y-x\circ y$. This shows that $S \cup {\mathbb{Z}}$ is a subring of $R$. Now every element in $S$ is quasi-regular with quasi-inverse in $S$, thus every element in $1-S$ is invertible in $S \cup {\mathbb{Z}}$. Since $S \cup {\mathbb{Z}}$ is a subring, we have $1-S \backslash {\mathbb{Z}}=S \backslash {\mathbb{Z}}$. So every element in $S \backslash {\mathbb{Z}}$ is invertible in $S \cup {\mathbb{Z}}$. By Proposition \[div\] either $S \subseteq {\mathbb{Z}}$ or $S \cup {\mathbb{Z}}$ is a division ring. Suppose $S \subseteq {\mathbb{Z}}$. Then the quasi-inverse of every element in $S \subseteq {\mathbb{Z}}$ lies again in $S \subseteq {\mathbb{Z}}$, so $S\subseteq Q({\mathbb{Z}})=\{0,2\}$, which contradicts our assumption. Therefore $S \cup {\mathbb{Z}}$ is a division ring. It remains to prove that ${\mathbb{Z}}\backslash \{1\} \subseteq S$. Let $n \in {\mathbb{Z}}\backslash \{1\}$. If $n=0$ or $n=2$ then $n \in S$ by assumption. So suppose $n \neq 0,2$. Since $S \cup {\mathbb{Z}}$ is a division ring, $1-n$ is invertible in $S \cup {\mathbb{Z}}$. Since $1-n \neq \pm 1$, the fact that the characteristic of $R$ is $0$ implies $(1-n)^{-1} \notin {\mathbb{Z}}$, i.e. $1-(1-n)^{-1} \in S$. Consequently $n=(1-(1-n)^{-1})^{(-1)} \in S$, since $S$ is a subgroup of $Q(R)$.
\[addunitprime\] Let $R$ be a unital ring of prime characteristic $p$. For any subgroup $S$ of $Q(R)$ the following are equivalent:
1. $S \cup {\mathbb{Z}}/p{\mathbb{Z}}$ is closed under addition,
2. $S \cup {\mathbb{Z}}/p{\mathbb{Z}}$ is a division subring of $R$.
In this case $S \cup {\mathbb{Z}}/p{\mathbb{Z}}$ is automatically closed under negation, since $-x=(p-1)x$ is a multiple of $x$. The proof is now the same as that of Theorem \[addunitzero\] except for the case $S \subseteq {\mathbb{Z}}/p{\mathbb{Z}}$, but in this case $S \cup {\mathbb{Z}}/p{\mathbb{Z}}={\mathbb{Z}}/p{\mathbb{Z}}$ is automatically a division ring.
Let $F$ be a field and $R$ a unital commutative $F$-algebra. If $\pi_F(R) \cup \{1\}$ is closed under addition then it is a subfield of $R$.
This follows directly from Proposition \[subgroup\] and Theorems \[addunitzero\] and \[addunitprime\], since $({\mathbb{Z}}\cdot 1) \backslash \{1\} \subseteq \pi_F(R)$ by Lemma \[2minus\].
Let $R$ be a unital commutative ring of prime or $0$ characteristic with $\pi(R) \neq \{0,2\}$. If $\pi(R)\circ \pi(R)^{(-1)} \cup ({\mathbb{Z}}\cdot 1)$ is closed under addition then it is a subfield of $R$.
The commutativity of $R$ implies $\langle \pi(R)\rangle=\pi(R)\circ \pi(R)^{(-1)}$. Since $2 \in \pi(R)$, the result follows from Theorems \[addunitzero\] and \[addunitprime\].
[99]{} D. Eisenbud: *Commutative Algebra with a View Toward Algebraic Geometry*, Springer-Verlag, New York, 1995 B.J. Gardner, R. Wiegandt: *Radical Theory of Rings*, Marcel Dekker, New York, 2004 J. Krempa: Logical connections between some open problems concerning nil rings, *Fund. Math.* 76 (1972), 121-130 J. Levitzki: A theorem on polynomial identities, *Proc. Amer. Math. Soc.* 1 (1950), 334-341 A. Smoktunowicz: Some open results related to Köthe’s conjecture, *Serdica Math. J.* 27 (2001), 159-170 A. Smoktunowicz: Some results in noncommutative ring theory, *Proceedings of the International Congress of Mathematicians*, Madrid, Spain, 2006 F.A. Szász: *Radicals of rings*, John Wiley & Sons, Chichester, 1981 J.H. Weng, P.Y. Wu: Products of unipotent matrices of index 2, *Linear Algebra Appl.* 149 (1991), 111-123 X. Yonghua: On the Koethe problem and the nilpotent problem, *Sci. Sin., Ser. A* 26 (1983), 901-908
|
---
abstract: 'With the development of Semantic Web, entity summarization has become an emerging task to generate concrete summaries for real world entities. To solve this problem, we propose an approach named MPSUM that extends a probabilistic topic model by integrating the idea of predicate-uniqueness and object-importance for ranking triples. The approach aims atwebview-panel:webview-panel/webview-0fb291c9-310d-42e2-a420-ed1a22b0a5ac generating brief but representative summaries for entities. We compare our approach with the state-of-the-art methods using DBpedia and LinkedMDB datasets. The experimental results show that our work improves the quality of entity summarization. The source code and outputs are available at <https://github.com/WeiDongjunGabriel/MPSUM>[^1].'
author:
- Dongjun Wei
- Shiyuan Gao
- Yaxin Liu
- Zhibing Liu
- 'Longtao Hang[^2]'
bibliography:
- 'ref.bib'
title: '**MPSUM: Entity Summarization with Predicate-based Matching**'
---
Introduction
============
Linked Open Data (LOD) can describe entities on the Semantic Web using Uniform Resource Identifiers (URIs) or Resource Description Framework (RDF). Therefore, LOD is regarded as a collection of entity descriptions and has formed many public datasets, such as DBpedia[@bizer2009dbpedia] and LinkedMDB[@consens2008managing]. An RDF triple is in the form of $<subject, predicate,
object>$. However, lengthy descriptions will take much time for users to comprehend and identify the underlying entities. To solve this problem, entity summarization has been proposed to generate a set of descriptions that are brief but effective to acquire enough information for quick comprehension. In this paper, we propose a method called MPSUM based on LDA model to identify the $top-k$ representative triples as summaries for entities. Apart from ranking triples based on their probability distributions, we propose a novel method for triples ranking with consideration of the importance of objects and uniqueness of predicates in RDF data.
Related Work
============
RELIN[@cheng2011relin] is a variant of the random surfer model for ranking features mainly based on relatedness and informativeness for quick identification of entities. DIVERSUM[@sydow2010diversum] solves the problem of diversified entity summarization in RDF-like knowledge graphs by incorporating the notion of diversification into the summarizing algorithm. FACES-E[@gunaratna2016gleaning] is able to match a suitable class from existing ontology classes set, which extends FACES to generate entity summaries in the way of gleaning and ranking object and datatype properties. CD[@xu2016cd] formulates entity summarization as a binary quadratic knapsack problem to solve. FACES[@gunaratna2015faces] improves the quality of entity summaries by taking the diversification of the relation types into consideration, introduces the concept of Cobweb clustering algorithm to partition features and rank them. LinkSUM[@thalhammer2016linksum] partitions the semantic links of each entity to rank features and is interfaced via the SUMMA entity summarization API. ES-LDA[@pouriyeh2017lda] is a probabilistic topic model based on LDA to generate representative summaries for entities and outperforms LinkSUM and FACES.
Preliminaries
=============
Resource Description Framework (RDF)
------------------------------------
RDF is a data modeling language of Semantic Web and is widely used to describe entities or resources. An RDF data graph is a set of entities (nodes) and relations (edges) between them, which is referred to a collection of triples where each triple $t$ consists of a subject $s$, predicate $p$, object $o$, in the form of $<s, p, o>$.
[**Document**]{} \[df:doc\] A document $d$ is defined as a collection of triples, $d = \{ t_1, t_2, ..., t_n \}$, that describes a single entity $e$. Therefore, all triples of a document $d$ have the same subject.
[**Entity summarization**]{} \[df:es\] Given an entity $e$ and a positive integer $k$, a summary of the entity $e$ is $ES(e, k)$, is the $top-k$ subset of all predicates and corresponding objects that are most relevant to that entity.
Latent Dirichlet Allocation (LDA)
---------------------------------
The Latent Dirichlet Allocation (LDA)[@wei2006lda] is an unsupervised machine learning technique that can be used to identify latent topics from a collection of documents. It uses a “bag of words” approach that regards each document as a word frequency vector, transforming the text information into digital information for modeling. LDA generates the words in a two-stage process: words are generated from topics and topics are generated from documents.
Problem Statement
=================
Problem Definition
------------------
An RDF document is in the form of a set of triples consisting of a subject with all predicates and corresponding objects related to a specific entity. In this paper, given an RDF dataset, entities are described by sets of their properties and corresponding objects. Our objective is to select $top-k$ representative triples to best describe each entity.
Supplementing RDF Data
----------------------
Based on two methods proposed to supplement the RDF data in ES-LDA[@pouriyeh2017lda], we enrich the information of each document by adding categories of the objects in DBpedia directly to the document and expanding each document by increasing the frequency of each object with the number of its categories.
Proposed Model
--------------
MPSUM is an extended model of LDA. The key idea behind our model is two-fold: (1) we apply the concept of LDA topic modeling into entity summarization; and (2) a novel method is proposed to rank triples and improve the performance of LDA in entity summarization.
The number of topics in MPSUM $K$ is set to be the number of unique predicates in the corpus. Let $D=\{d_1, d_2, ..., d_{\left|D\right|}\}$ be a corpus of documents. In our model, each document $d$ is a multinomial distribution over the predicate $r$ and each predicate is a multinomial distribution over the object $o$ from the Dirichlet prior $\alpha$ and $\beta$ respectively, in addition, $\theta$ and $\phi$ are latent variables. The plate notation is shown in Figure \[fig:ESM\].
![Entity Summarization Using LDA[]{data-label="fig:ESM"}](./ESM.png){width="5cm"}
ES-LDA[@pouriyeh2017lda] only uses probability distributions when ranks RDF triples. However, Bian et al.[@bian2014research] combined topic-importance and topic-distribution for sentence-ranking problem and got a better performance. Since each element of an RDF triple has different role, the influence of various elements in RDF triples cannot be regarded as the same. Each document can be deemed as “a bag of objects” because it owns the same subjects as mentioned above in [*D*efinition \[df:doc\]]{}. Therefore, a method called MP (match up objects and RDF triples based on predicates) has been proposed to rank the RDF triples via taking the predicate-uniqueness and object-importance into consideration.
Retrieve topic words $words$ and RDF triples $triples$ in document $d$ from trained model Initialize predicates’ list $predicates$ in $d$ predicate $p \leftarrow extractor(rp)$ Add $p$ into $predicates$ Output $rp$ and remove it from $triples$ predicate $p \leftarrow extractor(rp$) Add $p$ to $predicates$ Output $rp$ and remove it from $triples$ Output $rp$ and remove it from $triples$
Our model repeats the following process of MP for each document.
[**Step 1:**]{} Initialize the predicate set and enumerate all the RDF triples of an entity to identify the triples of which the objects are ranked based on probabilities. Then extract the corresponding predicate and add it into the aforementioned predicate set and output the triple when the predicate is not in the collection.
[**Step 2:**]{} Select the next triple from the remaining ones until all the objects complete the matching work, extract their predicates to compare with the predicate set generated from Step 1. If the predicate is not in the collection, add it into the current predicate set and output the related triple.
[**Step 3:**]{} Output the rest triples in order.
For example, after applying LDA method for training, top-10 predicates are selected according to the probability of each object within the subject which has the highest probability and corresponding triples of a given subject, including *broadcastArea, broadcastArea, callsignMeaning, programmeFormat, type, type, label, name, type* and *homepage*. However, apart from the probabilities of objects, MP method takes the information of subjects which is based on predicates into consideration, and generates better training effects, since the top-10 predicates are *broadcastArea, callsignMeaning, programmeFormat, label, name, type, subject, homepage, slogan* and *type*. The generative process of MP is shown in [**Algorithm 1**]{}.
Estimating Posterior Inference
------------------------------
Since it’s difficult to acquire the posterior inference of the LDA, it needs to find an algorithm for estimating the posterior inference. There are many existing methods including variational EM[@blei2003latent] and Gibbs sampling[@robert2013monte]. EM[@blei2003latent] is a maximum likelihood estimation method including probabilistic model parameters of latent variables. Gibbs sampling[@robert2013monte] is a Markov Chain Monte Carlo algorithm, which constructs a Markov chain over the latent variables in the model and converges to the posterior distribution, after a number of iterations. However, TF-IDF[@huang2011text] is a statistical method to assess the importance of a word for a file set or one of the files in a corpus. In our case, we evaluated our model using EM, Gibbs sampling and Gibbs sampling with TF-IDF to estimate posterior inference and the results demonstrate that the Gibbs sampling shows the best performance.
Experiments and Results
=======================
[**Data Preprocessing:**]{} Excessive work on RDF triples would introduce more corresponding triples to the topics that have higher probabilities and reduce the precision of MP method. Then we apply a concise extraction algorithm for objects to adapt to the algorithm for estimating the posterior inference. The extraction algorithm first acquires RDF triples which conclude the selected objects, then intercepts the part from the last ’\#’ or ’/’ of RDF to the end. For example, we extract *“broadcaster”* after lowering the capital letters from ’$http://dbpedia.org/ontology/Broadcaster$’.
[**Corpus Enlarging:** ]{}As mentioned in section 4.2, we supplement the RDF data via adding categories of objects and repeating the topics in each document to deal with common RDF data problems including *sparseness, lack of context*, etc. Compared with the methods of supplementing RDF data in ES-LDA model, the results of our model outperforms better.
[**Inference Algorithm:**]{} In section 4.4, we have introduced three algorithms to estimate the posterior inference including EM, Gibbs sampling and Gibbs sampling with TF-IDF. All of the above methods have been used in our experiments respectively, and the Gibbs sampling shows the best results.
[**Model Training:**]{} In the training process, it’s essential to set proper hyperparameters, $\alpha$ and $\beta$. We conduct experiments and finally find that when $\alpha=(E/20)/R$, , we can get a satisfying result. $E$ is the total number of unique entities and$R$ is the total number of unique predicates. We test 3 configurations as table \[tab:Mdd\]. As the results show, taking the total number of entities into account can optimize training effects. When the value of $\beta$ is set to $0.01$, DBpedia has the best result since it contains enrich corpus. However, for LinkedMDB, its corpus is insufficient, when $\beta = 50 / R$, the result is better.
[p[2cm]{}p[1.5cm]{}p[1.5cm]{}]{} dbpedia: & top5 & top10\
config\_1: & 0.389 & 0.463\
config\_2: & [**0.396**]{} & [**0.568**]{}\
config\_3: & 0.379 & 0.554\
\
lmdb: & top5 & top10\
config\_1: & 0.370 & 0.474\
config\_2: & 0.370 & 0.476\
config\_3: & [**0.371**]{} & [**0.576**]{}\
The DBpedia and LinkedMDB datasets are chosen for our experiments. We evaluate the F-measures (the harmonic average of the precision and recall) and MAP (Mean Average Precision) to compare our MPSUM model with other state-of-the-art approaches including RELIN, DIVERSUM, FACES-E, CD, FACES and LinkSUM ,and the results are in Table \[tab:Fm\] and Table \[tab:MAP\] respectively. From the results, we can observe that MPSUM performs best on all cases except the top-10 in DBpedia.As the results show,RELIN and LinkSUM could not meet the diversity requirement in the summarization process. FACES discount literals in entity summarization while FACES-E and RELIN take literals into account. Our approach maintains both diversity and relevancy, while representing each entity through $top-k$ predicates. Above all, MPSUM outperforms the selected approaches on overall datasets.
------------ ------------------------ ------------------------------- ------------------------ ------------------------ ------------------------ ------------------------
k=5 k=10 k=5 k=10 k=5 k=10
RELIN 0.250$_{\lambda=1.00}$ 0.468$_{\lambda=1.00}$ 0.210$_{\lambda=1.00}$ 0.260$_{\lambda=1.00}$ 0.239$_{\lambda=1.00}$ 0.409$_{\lambda=1.00}$
DIVERSUM 0.260 0.522 0.222 0.365 0.239 0.477
CD 0.299$_{\gamma=0.47}$ [**0.531**]{}$_{\gamma=0.23}$ 0.215$_{\gamma=1.00}$ 0.326$_{\gamma=1.00}$ 0.267$_{\gamma=0.52}$ 0.467$_{\gamma=0.16}$
FACES-E 0.285 0.527 0.252 0.348 0.276 0.476
FACES 0.272 0.439 0.160 0.259 0.240 0.388
LinkSUM 0.290$_{\alpha=0.01}$ 0.498$_{\alpha=0.04}$ 0.117$_{\alpha=1.00}$ 0.255$_{\alpha=1.00}$ 0.240$_{\alpha=0.01}$ 0.428$_{\alpha=0.04}$
MPSUM [**0.313**]{} 0.522 [**0.270**]{} [**0.440**]{} [**0.300**]{} [**0.499**]{}
Better[^3] 0.014 - 0.018 0.075 0.024 0.022
------------ ------------------------ ------------------------------- ------------------------ ------------------------ ------------------------ ------------------------
------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------
k=5 k=10 k=5 k=10 k=5 k=10
RELIN 0.348$_{\lambda=1.00}$ 0.532$_{\lambda=1.00}$ 0.243$_{\lambda=1.00}$ 0.337$_{\lambda=1.00}$ 0.318$_{\lambda=1.00}$ 0.476$_{\lambda=1.00}$
DIVERSUM 0.316 0.511 0.269 0.388 0.302 0.476
CD - - - - - -
FACES-E 0.354 0.529 0.258 0.361 0.326 0.481
FACES 0.247 0.386 0.140 0.261 0.261 0.351
LinkSUM 0.246$_{\alpha=0.25}$ 0.386$_{\alpha=0.03}$ 0.120$_{\alpha=1.00}$ 0.254$_{\alpha=1.00}$ 0.210$_{\alpha=0.25}$ 0.348$_{\alpha=0.03}$
MPSUM [**0.396**]{} [**0.568**]{} [**0.371**]{} [**0.476**]{} [**0.389**]{} [**0.542**]{}
Better[^4] 0.042 0.036 0.102 0.088 0.063 0.061
------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------
Conclusion
==========
In this paper, we put forward an LDA-based model MPSUM for entity summarization. In our method, we propose to increase the frequency of words by adding categories of the objects to supplement RDF data, which is an improvement of ES-LDA[@pouriyeh2017lda]. Besides, a novel method called MP has been proposed to rank triples with consideration of the importance of objects and uniqueness of predicates in RDF data. We utilize three algorithms (EM, Gibbs sampling, Gibbs sampling with TF-IDF) to estimate posterior inference and finally take advantage of Gibbs sampling for experiments and comparisons. The experimental results of our approach for entity summarization are quite promising. It performs better than 5 state-of-the-art techniques to generate summaries.
Future Work
===========
In this paper, we select the topic with the maximum probability. For further work, the selection of topics can be implemented by proportion to apply more topics for ranking RDF triples. Moreover, other proper methods to enlarge RDF data remain to be explored to improve the quality of representative triples for entity summarization.
Acknowledgements {#acknowledgements .unnumbered}
================
This research is supported in part by the National Natural Science Foundation of China under Grant No. 61702500.
[^1]: This paper was accepted in EYRE@CIKM’2018.
[^2]: Correspondence to Longtao Huang.
[^3]: By how much we are better than the best result of all other methods.
[^4]: By how much we are better than the best result of all other methods.
|
---
nocite:
- '[@Whittaker:53:1]'
- '[@Bozhevolnyi:95:4]'
- '[@Wolf:82:1]'
---
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[**Authors note to the electronic version**]{} The author hereby grant you to redistribute and print this file for your own personal use, provided that no modifications are made. If you would like to cite any of the results published herein you are free to do so, but if it is possible please make your citation to one of the papers below:
[Andersen, T. and O. Keller (1998a).]{}
: Local-field theory for optical phase conjugation by degenerate four wave mixing in mesoscopic interaction volumes of condensed media. [*Phys. Scr.*]{} [*58*]{}, 132–144.
[Andersen, T. and O. Keller (1998b).]{}
: Optical phase conjugation in a single-level metallic quantum well. [*Phys. Rev. B*]{} [*57*]{}, 14793–14808.
[Andersen, T. and O. Keller (1998c).]{}
: Two-dimensional confinement of light in front of a single level quantum well phase conjugator. [*Opt. Commun.*]{} [*155*]{}, 317–322.
[Andersen, T. and O. Keller (1999).]{}
: Local-field study of phase conjugation in nonmagnetic multi-level metallic quantum wells with probe fields of both propagating and evanescent character. [ *Phys. Rev. B*]{}, (to be published).
The first of these articles contains most of Chapter 3 and Parts II and III. The second contains most of Chapters 10 and 11 and part of Appendix B. The third article consists of the results presented in Chapter 12. The last of the four articles is based on the analysis presented in Part V and Appendices B and C. It is, however, greatly expanded with numerical results.
If you plan to include some of this work (figures, tables, equations) into your own work, please contact the author. Current (as of October 1998) contact information is:\
\
Dr. Torsten Andersen\
Max-Planck-Institut f[ü]{}r Mikrostrukturphysik\
Weinberg 2\
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Germany\
\
e-mail: [email protected]\
www: http://www.geocities.com/CapeCanaveral/Lab/9700/index.html\
\
A limited number of printed copies of the thesis are available. Please contact the author or the publisher if you are interested.
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[Theoretical Study of Phase Conjugation\
in Mesoscopic Interaction Volumes.\
Teoretisk unders[ø]{}gelse af fasekonjugation\
i mesoskopiske vekselvirkningsrumfang.\
]{}
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and the Institute of Physics, Aalborg University.\
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Preface {#preface .unnumbered}
=======
The present monograph describes results of my work carried out at the Institute of Physics at Aalborg University, Aalborg, Denmark in the period from February 1995 to May 1998 and at the Department of Physics and Astronomy at Mississippi State University, Starkville, Mississippi, United States of America in the period from August 1996 to January 1997. This monograph is submitted as a Ph.D. thesis to the Faculty of Engineering and Science at Aalborg University.
With this dissertation, I intend to describe in a unified fashion the work done in the time frame of the programme. It is my intention that the contents of this monograph should go deeper and broader into the material that has been processed for publishing in articles, and thus include results and comments on material not suitable for publication as parts of an article. The work is divided into six main parts, each consisting of some separate chapters. The outline of this monograph is presented in the following.
Outline of the dissertation {#outline-of-the-dissertation .unnumbered}
---------------------------
The motivation for carrying out the present study is given in Part I, divided into three separate chapters. In Chapter 1 a brief summary of the contributions to scientific progress within the last century, which I believe are the most important for my work, is presented. The historical summary includes remarks on nonlinear optics in general, optical phase conjugation in particular, near-field optics, and mesoscopic optics. Chapter 2 presents the theoretical model usually adopted in optical phase conjugation (standard theory). Chapter 3 consists of a discussion of the limitations of the standard theory as well as the requirements to a theoretical model that can be used when interaction takes place on small length scales and/or in the optical near-field region.
In Part II the task of developing a nonlocal theoretical description of phase conjugation of optical near-fields by degenerate four wave mixing is undertaken. It consists of three chapters. In Chapter 4 the basic working frame for the present treatment is established, starting from Maxwell’s equations. Chapter 5 sets up the first and third order responses of an electron using the density matrix formalism starting from the Liouville equation of motion. In Chapter 6 the general conductivity response tensors for degenerate four-wave mixing excluding spin-effects are established, and their symmetries are discussed. Part II is concluded with a small discussion.
Part III discusses degenerate four-wave mixing in quantum-well structures on a somewhat general level. For this purpose, two chapters are written. In Chapter 8 the conductivity response tensors in the case where a system has broken translational invariance in one spatial direction are established and discussed, and Chapter 9 is devoted to a discussion of the consequences of scattering in a plane using polarized light.
In Part IV the optical phase conjugation response of a single-level quantum well is studied in four chapters. In Chapter 10 the theoretical considerations necessary to describe the phase conjugation response from a single-level quantum well are discussed, and in Chapter 11 the numerical results for the phase conjugated response from a copper quantum well are discussed. Chapter 12 consists of a discussion of two-dimensional confinement of light in front of the single-level quantum-well phase conjugator considered in Chapters 10 and 11. Chapter 13 concludes this part with a short discussion.
Part V takes a similar point of view as Part IV, but for the two-level quantum well. Theoretical considerations are presented in Chapter 14, while the numerical results are discussed in Chapter 15. A short discussion concludes this part in Chapter 16.
Part VI contains a concluding discussion on the developed theory and the numerical work followed by an outlook.
There are five Appendices included, consisting of calculations not suitable for the main text. Appendix A is a calculation of the linear and nonlinear conductivity tensors relevant for studying degenerate four-wave mixing in quantum well structures. Appendix B contains the principal analytic solutions to the integrals over the states parallel to the plane of translational invariance in the quantum well structures. In Appendix C, the absolute solution to the integrals over the states parallel to the plane of translational invariance in the quantum well structures are presented in terms of the principal solutions given in Appendix B. Appendix D contains a small calculation of the Fermi energy for a quantum well in the low-temperature limit and a calculation of the minimal and maximal values of the thickness of a quantum well given the desired number of occupied eigenstates across the quantum well. Appendix E contains some intermediate results in the calculation of the integrals over the source region in Chapter 14.
References used in this work are listed in the bibliography at the end of the dissertation according to the recommendations by the thirteenth edition of [*The Chicago Manual of Style*]{} with author(s), title, and publication data in alphabetic order after the first authors surname.
Notation {#notation .unnumbered}
--------
Footnotes are marked using a superscript number in the text, and the footnote itself is found at the bottom of the page. Citations to other people’s work are made with reference to the authors surname(s) followed by the year of publication. The international system of units (SI) has been adopted throughout the work, except that the unit [Å]{}ngstr[ö]{}m ([Å]{}) is used to denote certain distances ($1$[Å]{}$=10^{-10}$m).
Vector quantities are denoted with a unidirectional arrow above them, i.e., $\vec{\kappa}$. Likewise, tensor quantities are denoted using a bidirectional arrow, i.e., $\stensor{\sigma}$. Integrations over vector quantities are denoted “$\int{f(\vec{\kappa})}d^n\kappa$”, where $n$ is the number of elements in the vector $\vec{\kappa}$, and $f(\vec{\kappa})$ is an arbitrary function of the integration variable $\vec{\kappa}$. Unit vectors are denoted $\vec{e}$ with an index indicating which direction is taken. The unit tensor is denoted by $\tensor{\openone}$, and is usually a $3\times3$ tensor.
Latin indices $\{i,j,k,h\}$ generally refers to the three spatial coordinate labels $\{x,y,z\}$, and the latin indices $\{n,m,v,l\}$ generally refers to quantum states. Exceptions from this are (i) when the letter “i” appears in formulae and is not an index, it is the complex number ${\rm{i}}^2=-1$, (ii) when the letter “$k$” appears in formulae and is not an index, it is the wavenumber $k=|\vec{k}|$, (iii) the letter “$m$” with an index “$e$” is the electron mass. Summations over repeated indices are stated explicitly whenever it should be performed.
To avoid confusion regarding the placement of $2\pi$’s in the Fourier integral representation, the Fourier transform pair $${\cal{F}}(t)={1\over2\pi}\int{}{\cal{F}}(\omega)e^{-{\rm{i}}\omega{}t}d\omega
\qquad\Longleftrightarrow\qquad
{\cal{F}}(\omega)=\int{}{\cal{F}}(t)e^{{\rm{i}}\omega{}t}dt$$ is adopted between the time- and frequency domains, and thus to be consistent the transform pair $${\cal{F}}(\vec{r})
={1\over(2\pi)^3}\int{}{\cal{F}}(\vec{k})e^{{\rm{i}}\vec{k}\cdot\vec{r}}d^3k
\qquad\Longleftrightarrow\qquad
{\cal{F}}(\vec{k})=
\int{}{\cal{F}}(\vec{r})e^{-{\rm{i}}\vec{k}\cdot\vec{r}}d^3r$$ is adopted between real space and $k$-space, here shown in three dimensions.
Furthermore, the complex conjugate and the Hermitian adjoint of a quantity $A$ are denoted $A^{*}$ and $A^{\dag}$, respectively. The phrases “+” and “$+\mbox{H.a.}$” at the end of an equation indicates the addition of the complex conjugate or the Hermitian adjoint of the foregoing terms. The phrase “$+\mbox{i.t.}$” at the end of an equation denotes the addition of a term in which the wave-vector $\vec{k}$ is replaced by $-\vec{k}$. The Laplacian is denoted $\nabla^2$. The Heaviside unit step function $\Theta(x)$ has the value $+1$ for $x>0$ and $0$ for $x<0$, and the Kronecker delta $\delta_{ij}$ has the value $+1$ for $i=j$ and $0$ for $i\neq{}j$. The Ludolphine number $3.14159265\dots$ is denoted by the greek letter $\pi$.
Scientific papers and presentations based on this work {#scientific-papers-and-presentations-based-on-this-work .unnumbered}
------------------------------------------------------
Parts of the work presented in this dissertation has been or will be published separately in the form of proceedings papers, articles, and letters. They are as follows:
[Andersen, T.]{}
: and O. Keller (1995a). Optical near-field phase conjugation: A nonlocal DFWM response tensor. In E. G. Bortchagovsky (Ed.), [*Proceedings of the International Autumn School-Conference for Young Scientists “Solid State Physics: Fundamentals & Applications” (SSPFA’95)*]{}, Kiev, pp. R5–R6. Institute of Semiconductor Physics of NASU. ISBN 5-7702-1199-7.
[Andersen, T. and O. Keller (1996a).]{}
: Phase Conjugation of Optical Near Fields: A new Nonlocal Microscopic Response Tensor. In O. Keller (Ed.), [*Notions and Perspectives of Nonlinear Optics*]{}, pp. 566–573. Singapore: World Scientific. ISBN 981-02-2627-6.
[Andersen, T. and O. Keller (1998a).]{}
: Local-field theory for optical phase conjugation by degenerate four wave mixing in mesoscopic interaction volumes of condensed media. [*Phys. Scr.*]{} [*58*]{}. In press.
[Andersen, T. and O. Keller (1998b).]{}
: Optical phase conjugation in a single-level metallic quantum well. [*Phys. Rev. B*]{} [*57*]{}, 14793–14808.
[Andersen, T. and O. Keller (1998c).]{}
: Two-dimensional confinement of light in front of a single level quantum well phase conjugator. [*Opt. Commun.*]{} Submitted.
Furthermore, an article on optical phase conjugation in a two-level (resonant) metallic quantum well is in preparation. In addition to the above-mentioned publications, presentations of abstracts has been given at conferences. They are:
[Keller, O., M. Xiao and T. Andersen (1994).]{}
: Phase conjugation and Near-Field Microscopy. Poster presented at the annual meeting of the Danish Optical Society, Lyngby, Denmark, November 24.
[Andersen, T.]{}
: and O. Keller (1995b) Random-phase-approximation study of the response function describing phase conjugation by degenerate four wave mixing. Poster presented at the annual meeting of the Danish Physical Society, Odense, Denmark, May 31–June 2.
[Andersen, T.]{}
: and O. Keller (1995c) Phase conjugation of optical near-fields: A new nonlocal response tensor allowing degenerate four wave mixing studies with probe beams strongly decaying in space. Talk given at the Third International Aalborg Summer School on Nonlinear Optics, Aalborg, Denmark, August 7–12.
[Andersen, T.]{}
: and O. Keller (1995d) Optical near-field phase conjugation: A nonlocal DFWM response tensor. Talk given at the International Autumn School-Conference for Young Scientists “Solid State Physics: Fundamentals & Applications” (SSPFA’95) in Uzhgorod, Ukraine, September 19–26.
[Andersen, T.]{}
: and O. Keller (1996b) Optical Phase Conjugation by Degenerate Four Wave Mixing in a Single Level Quantum Well. Poster presented at the annual meeting of the Danish Physical Society, Nyborg, Denmark, May 23–24.
[Andersen, T.]{}
: and O. Keller (1996c) Microscopic Description of Optical Near-Field Phase Conjugation. Talk given at the South Eastern Section Meeting of The American Physical Society, Atlanta-Decatur, Georgia (USA), November 14–16. [*Bulletin of the American Physical Society*]{} [*41*]{}, p. 1660.
[Andersen, T.]{}
: and O. Keller (1997a) Optical Near-Field Phase Conjugation: A Microscopic Description. Poster presented at the Fourth International Conference on Near-Field Optics (NFO-4) Jerusalem, Israel, February 9–13.
[Andersen, T.]{}
: and O. Keller (1997b) Focusing of classical light beyond the diffraction limit. Poster presented at the annual meeting of the Danish Optical Society, Lyngby, Denmark, November 18–19.
The abstracts are printed in the relevant meeting programmes.
Software used in this project {#software-used-in-this-project .unnumbered}
-----------------------------
Creation of the numerical results presented in this work has been done through development of computer programs, mainly in Fortran 90 [@Metcalf:96:1]. The final set of programs consists of approximately 6000 lines of code developed by the present author. Because of the size I have chosen not to include a reprint of the code in this monograph. The presentation of the calculated data is done using gnuplot pre-3.6 with some 400 lines of code to generate the plots as encapsulated PostScript files. This dissertation has been typed entirely in [@Goossens:94:1; @Goossens:97:1], an enhanced version of the typesetting program TeX, originally developed by .
Acknowledgements {#acknowledgements .unnumbered}
----------------
The topic of this work has been very fascinating to explore, and I would therefore like to thank professor, Dr. Scient. Ole Keller for introducing me to the world of mesoscopic physics, his huge support and the many inspiring discussions we have had throughout the period of this work. I would also like to thank professor, Dr. H. F. Arnoldus and the Department of Physics and Astronomy at Mississippi State University for kindly providing facilities and inspiration during my stay there. Additionally, I would like to thank the scientific staff at the Institute of Physics, Aalborg University for providing an inspiring atmosphere in general, and Dr. Brian Vohnsen in particular for many discussions on near-field optics and for reviewing parts of the text appearing in this dissertation.
My deepest thanks goes to my parents Maren and Agner Andersen for their moral and financial support, and to my brother Jens and my sisters Else, Karen, and Birgit. In addition, I would like to thank my friends around the globe who—together with my family—have enriched my life outside the academic world.
Finally, I wish to thank (i) the Faculty of Engineering and Science and the Institute of Physics at Aalborg University for providing the basic frame for my research work, (ii) the Mississippi Center for Supercomputing Research at the University of Mississippi, Jackson, Mississippi, for providing computer resources during my stay at Mississippi State University, and (iii) the people at Aalborg University Library for their excellent assistance in obtaining articles from far away in time and space.
1.5cm to
List of acronyms {#list-of-acronyms .unnumbered}
----------------
-------- -----------------------------------------------------
DC Direct current.
DFWM Degenerate four-wave mixing.
ED Electric dipole.
IB Infinite barrier.
PCDFWM Phase conjugation (by) degenerate four-wave mixing.
SVE Slowly varying envelope.
SVEA Slowly varying envelope approximation.
YAG Y$_3$Al$_5$O$_{12}$ [@Yariv:84:1].
-------- -----------------------------------------------------
(-60,-50)(60,65)
\[Part:I\]
Natura inest\
in mentibus nostrum\
insatiabilis quaedam\
cupiditas veri videndi\
([*Marcus Tullius Cicero*]{})
Historical perspective {#Ch:1}
======================
Indeed, as the great orator expressed it more than two millenia ago, nature has planted in our minds an insatiable longing to see the truth. This natural curiosity, I believe, has been the driving force behind scientific investigations in the history of mankind, and thus also behind the evolution of electromagnetic theory. But since the electrodynamic theory as we know it was initiated by [-@Maxwell:64:1; -@Maxwell:91:1], I will in the following historical remarks concentrate on the physics of the past century. Readers who want an overview of the evolution of electromagnetic theories before this century are referred to , and the comprehensive survey of .
The work described in the present dissertation is mainly concerned with a theoretical description of a nonlinear type of electromagnetic interactions called degenerate four-wave mixing (DFWM), particularly in the case where phase conjugation is obtained. The main interest behind this study is to model the behaviour of the DFWM interaction in mesoscopic volumes and in the optical near-field zone. Thus, in relation to established branches of modern optics, this work belongs to the fields of nonlinear optics (especially four-wave mixing), near-field optics, and mesoscopic systems. In the remaining of this chapter I therefore intend to describe briefly the contributions to scientific progress within the last century, which I believe are the most significant for the present study.
Although Einstein already in 1916 predicted the existence of stimulated emission ( ), the main objective of optics remained linear observations until after the development of the maser in the early 1950’s, where Townes and co-workers at Columbia University used stimulated emission for amplification of an electromagnetic field in combination with a resonator (, ). An application of the principles of the maser in the optical region of the electromagnetic spectrum was proposed in 1958 by , and in 1960 constructed the first laser—a pulsed ruby laser. The first laser delivering a continous-wave output was constructed in 1961 using a mixture of Helium and Neon gasses [@Javan:61:1]. The laser rapidly became of significant importance in optical physics, where the field of nonlinear optics was ignited by the successful observation by of radiation of light at the second harmonic frequency (with a wavelength $\lambda$ of $3472$[Å]{}) generated by a quartz crystal illuminated with light from a ruby laser ($\lambda=6943$[Å]{}). Since then nonlinear optics has been of interest to many researchers around the globe exploring a large number of different nonlinear phenomena, such as second-, third-, and higher order harmonic generation, optical rectification, sum and difference frequency generation, three-, four-, six-, and higher-number wave-mixing, laser cooling, laser induced atomic fusion, stimulated Raman- and Brillouin scattering, to mention a few \[see, e.g., , , , , , , and \].
The optical effect of interest in this work, optical phase conjugation, is nowadays usually produced by means of nonlinear optics, although the problem of reconstructing electromagnetic wavefronts started in the linear optical regime. The pioneering work on optical wavefront reconstruction (holography) was carried out several years before the invention of the laser by Gabor () with the purpose of improving the resolving power of the electron microscope \[see also \]. But only with the high intensities and with the degree of temporal and spatial coherence provided by the laser, holographic imaging became of practical importance. Such experiments were first reported by (). Soon thereafter used a hologram to correct static phase distortions introduced onto an optical wavefront. In this experiment a photosensitive film was used for holographic recording of an image, and the film had to be developed prior to its application for phase correction. This experiment of appears to be the first account on optical phase conjugation. However, since a new film has to be developed every time the phase distortion changes, this technique becomes rather cumbersome if the phase distortions changes frequently. A key discovery of made it possible to store holograms dynamically in crystals with an intensity-dependent refractive index, thereby extending the applicability domain of optical phase conjugation to cover descriptions where phase distortions are varying in time. Experimentally, the first real-time optical phase conjugation are credited to Zel’dovich and co-workers [@Zeldovich:72:1; @Nosach:72:1], in an experiment based on stimulated Brillouin scattering. In the late 1970’s, suggested the use of a degenerate four-wave mixing process to produce the phase conjugated field. Immediately thereafter , and independently, , further analyzed the optical phase conjugation via DFWM, resulting in predictions of amplified reflection, coherent image amplification and oscillation. Over the past twentyfive years, thousands of scientific papers, several books and review articles describing different aspects and applications of optical phase conjugation have been published, and phase conjugation in the form of DFWM is now an established discipline in modern experimental optics. The theoretical treatments of optical phase conjugation are usually based upon the work of , using the phase conjugating system as a device in studies of other processes. A comprehensive and coherent introduction to the field of optical phase conjugation can be found in the books by and , while a more specialized introduction can be achieved through collections of review papers appearing in books by and , or separately, by (), , and . Other collections of papers can be found in, e.g., , and . Within the last few years, DFWM has been used for creation of optical phase conjugation in configurations where the probe field and the detector are within subwavelength distances from the phase-conjugating medium [@Bozhevolnyi:94:1; @Vohnsen:97:1; @Bozhevolnyi:97:1].
The first account of attention to subwavelength (optical near-field) interaction of light with matter seems to be (), who proposed an apparatus, in which a sample is illuminated through a small aperture in an opaque screen, the area of the aperture being substantially smaller than the diffraction limit of the light used for the illumination (of subwavelength size). The aperture should be moved in small increments (scanned) over the sample by use of a piezo-electric crystal. At every step of the scanning procedure the light transmitted through the sample should be collected and the intensity measured. The resolving power of such an instrument should be limited by the size of the aperture and the distance from the aperture to the sample rather than by the wavelength of the illuminating light. For whatever reason, the proposal of was forgotten, and even though and () discussed the problem of diffraction by small holes, the idea of an optical near-field microscope remained forgotten until made the proposal, apparently without any knowledge of the instrument proposed by . The first demonstration of an image obtained with scanning in the electromagnetic near-field zone was given by , who used microwaves of wavelength of 3cm to resolve metallic gratings with linewidths down to 0.5mm, corresponding to $1/60$-th of a wavelength. Another twelve years should pass before near-field electrodynamics was adressed again. Near-field optics evolved in the mid-eighties in the wake of the experimental works by the groups of Pohl, Lewis, and Fischer [@Pohl:84:1; @Lewis:84:1; @Fischer:85:1; @Fischer:89:1]. The main efforts of this new branch of modern optics is concentrated on the original idea of subwavelength imaging \[see, for example, the recently published book by , the proceedings of the first conference on near-field optics [@Pohl:93:1], or proceedings from later conferences in near-field optics [@Isaacson:95:1; @Paesler:95:1; @Nieto-Vesperinas:96:1; @Hulst:98:1]\].
The appearance of microscopes with subwavelength resolution inevitably poses the questions of the resolution limit and the degree of spatial confinement of light—two inseparable questions in near-field optics. Fundamentally, the spatial confinement problem is linked to the field-matter interaction in the vicinity of the source emitting the field and in the near-field region of the detector. In classical optics, near-field effects traditionally have played a minor role, and the possibilities for studying material properties on a small length scale usually are judged in relation to the diffraction limit criterion attributed to Ernst and the third baron Rayleigh [-@Rayleigh:96:1]. The Rayleigh criterion, though mainly invoked in the context of spatial resolution, also sets the limit for the possibilities of light compression in far-field studies. As already emphasized by and later discussed, for instance, by , the resolution problem is not a simple one, even in classical optics. A recent survey of the resolution problem within the framework of classical optics has been given by .
When the interaction length of an electromagnetic field across the individual structures in a condensed matter system is on the order of an optical wavelength (typically a few atomic distances), the theoretical description of the field-matter interaction belongs to the field of mesoscopic electrodynamics. Within the last two decades studies of the optical properties of mesoscopic systems, such as quantum wells (single and multiple), -wires and -dots, surfaces, interfaces, and more exotic geometries have drawn the attention of many researchers. Because of the immediate potential for industrial application many of these studies have been concentrated on the properties of semiconductors (see, e.g., , and references herein). In recent years in particular investigations of the nonlinear electrodynamics have been in focus. Among the many nonlinear phenomena studies of second harmonic generation [@Sipe:82:1; @Richmond:88:1; @Heinz:91:1; @Reider:95:1; @Liebsch:95:1; @Pedersen:95:1], sum- and difference frequency generation [@Reider:95:1; @Bavli:91:1], photon drag [@Keller:93:1; @Vasko:96:1; @Chen:97:1; @Keller:97:2], DC-electric-field induced second harmonic generation [@Aktsipetrov:95:1; @Aktsipetrov:96:1], the Kerr effect [@Pustogowa:94:1; @Liu:95:1; @Rasing:95:1; @Rasing:96:1], electronic and vibrational surface Raman scattering [@Nkoma:89:1; @Mischenko:95:1; @Garcia-Vidal:96:1], two-photon photoemission [@Haight:95:1; @Fauster:95:1; @Georges:95:1; @Shalaev:96:1; @Tergiman:97:1], and generation of higher harmonics [@vonderLinde:96:1; @Garvila:92:1] have played a prominent role. Among the more exotic phenomena, studies of the Aharonov-Bohm effect in mesoscopic rings [@Wang:97:1] and whispering-gallery modes in microspheres [@Knight:95:1] have also been carried out lately.
From a theoretical point of view the refractive index concept becomes meaningless for structures of mesoscopic size. Therefore, macroscopic approaches to describe the field-matter interaction have to be abandoned from the outset, and the theoretical analyses have to be based on the microscopic Maxwell equations combined with the Schr[ö]{}dinger equation. The Schr[ö]{}dinger equation describes the quantum state of the condensed matter system, and is a fundamental part of the quantum mechanics initiated in the beginning of the 20th century by such scientists as Planck, Einstein, Bohr, Heisenberg, Born, Jordan, de Broglie, Schr[ö]{}dinger and Dirac. Even an attempt to give a satisfactory historical survey of the development of quantum mechanics at this point will fail because of the almost universal status quantum mechanics has reached in the description of modern physics. Instead, for the history of quantum mechanics including a description of the mathematical foundation, please consult for example or . A modern and comprehensive description of quantum mechanics is given by , where also a comprehensive list of references to key papers can be found. An example of interesting papers is the series of articles by [-@Schrodinger:26:0; -@Schrodinger:26:5].
Standard theory of\
optical phase conjugation by degenerate four-wave mixing {#Ch:2}
========================================================
Optical phase conjugation is a nonlinear optical phenomenon, in which an incoming optical field is reflected in such a manner that the wavefronts of the reflected field coincide with the incoming field, hence also the name “wavefront inversion”, frequently used in the literature. The principle of optical phase conjugation has gained widespread attention because of its ability to correct for distortions introduced in a path traversed by an optical signal. In principle, it works like this: An optical source is placed on one side of a distorting medium (crystal, waveguide, atmosphere, etc.). A system in which phase conjugation takes place (called the phase conjugator) is placed on the other side of the distorting medium. A field emitted from the source in the direction of the phase conjugator then travel through the distorting medium, and is reflected by the phase conjugator. The phase conjugator reverses the wavefront of the incoming (probe) field, and when the reflected light comes back through the distorting medium, the wavefront is (ideally) exactly reversed, compared to that originally emitted by the source. Since it is possible to see how the light was originally emitted by the source by looking at the phase conjugated replica, it is sometimes also given the somewhat misleading term “time reversal” [@Yariv:78:1]. Several schemes exist to achieve phase conjugation, the most widely used called “degenerate four-wave mixing” (DFWM). Optical phase conjugation in the form of degenerate four-wave mixing (DFWM) is a nonlinear third order effect, where mixing of two counterpropagating “pump” fields and a “probe” (or “signal”) field—all with the same frequency $\omega$—results in, among other signals, a generated field (the “conjugate”) with frequency $\omega=\omega+\omega-\omega$, which is counterpropagating to the probe field.
(0,0)(120,40) (40,10)(80,30) (40,5)(40,9) (40,31)(40,35) (80,5)(80,9) (40,33)(50,33) (40,0)[(0,4)\[c\][$z=0$]{}]{} (80,0)[(0,4)\[c\][$z=L$]{}]{} (51,31)[(0,4)\[l\][$z$]{}]{} (60,22.5)[(0,5)\[c\][Nonlinear]{}]{} (60,17.5)[(0,5)\[c\][optical]{}]{} (60,12.5)[(0,5)\[c\][medium]{}]{} (29,6)(39,11) (81,29)(91,34) (29,17)(39,17) (29,23)(39,23) (28,15)[(0,4)\[r\][$\vec{E}_{3}$ (probe)]{}]{} (28,21)[(0,4)\[r\][$\vec{E}_{4}$ (conjugate)]{}]{} (28,3)[(0,4)\[r\][$\vec{E}_{1}$ (pump 1)]{}]{} (92,33)[(0,4)\[l\][$\vec{E}_{2}$ (pump 2)]{}]{}
In the following, I present the theoretical model usually adopted in studies of optical phase conjugation by degenerate four-wave mixing, and consequently this chapter will consist mainly of textbook material. The treatment roughly follows that of and of . The DFWM geometry suggested by is shown in Fig. \[fig:2.1\]. In this configuration, a lossless nonlinear optical medium is illuminated by two strong counterpropagating pump fields $\vec{E}_{1}$ and $\vec{E}_{2}$ and by a weak signal (probe) wave $\vec{E}_{3}$. The pump fields are usually taken to be plane waves, although they in principle are allowed to have any kind of wavefront as long as their amplitudes are complex conjugates of each other. The probe field can have a more complex wavefront. Resulting from the mixing process in the medium a conjugate field appears, propagating in the direction oppositely to the probe.
In order to describe the electromagnetic field we first establish the wave equation for the interacting fields from the macroscopic Maxwell equations. It is thereafter reduced to its slowly varying envelope approximation (SVEA) form. The four macroscopic Maxwell equations are $$\begin{aligned}
\vec{\nabla}\times\vec{E}(\vec{r},t)&=&
-{\partial\vec{B}(\vec{r},t)\over\partial{}t},
\label{eq:macromax1}
\\
\vec{\nabla}\times\vec{H}(\vec{r},t)&=&\vec{J}(\vec{r},t)
+{\partial\vec{D}(\vec{r},t)\over\partial{}t},
\label{eq:macromax2}
\\
\vec{\nabla}\cdot\vec{D}(\vec{r},t)&=&\rho(\vec{r},t),
\label{eq:macromax3}
\\
\vec{\nabla}\cdot\vec{B}(\vec{r},t)&=&0.
\label{eq:macromax4}\end{aligned}$$ We now assume that the material is homogeneous, nonmagnetic ($\vec{B}=\mu_0\vec{H}$), and nonconducting ($\vec{J}=\vec{0}$) and that there are no free charges ($\rho=0$). We write the displacement vector $\vec{D}(\vec{r},t)$ as $$\vec{D}(\vec{r},t)=\epsilon_0\vec{E}(\vec{r},t)
+\epsilon_0\vec{P}(\vec{r},t),$$ where $\vec{P}(\vec{r},t)$ is the polarization, which we split into its linear, $\vec{P}_{\rm{L}}$, and nonlinear, $\vec{P}_{\rm{NL}}$, components $$\vec{P}(\vec{r},t)=\vec{P}_{\rm{L}}(\vec{r},t)+\vec{P}_{\rm{NL}}(\vec{r},t).$$ Above, the linear polarization describes the material response due to interaction with the field of first order. We thus define the linear susceptibility $\btensor{\chi}^{(1)}$ from the linear polarization in the manner $\vec{P}_{\rm{L}}=\btensor{\chi}^{(1)}\cdot\vec{E}$. The linear permittivity $\stensor{\epsilon}_{\!r}$ is then found from the linear part of the displacement, giving $\stensor{\epsilon}_{\!r}=\tensor{\openone}+\btensor{\chi}^{(1)}$. Taking the curl of Eq. (\[eq:macromax1\]) and inserting Eq. (\[eq:macromax2\]) into the resulting equation, we obtain (by use of the operator identity $\vec{\nabla}\times\vec{\nabla}\times=-\nabla^2+\vec{\nabla}\vec{\nabla}\cdot$) the following wave equation $$\left[\tensor{\openone}\nabla^2-{\stensor{\epsilon}_{\!r}\over{}c^2}
{\partial^2\over\partial{}t^2}\right]\cdot\vec{E}(\vec{r},t)
={1\over{}c^2}{\partial^2\vec{P}_{\rm{NL}}(\vec{r},t)\over\partial{}t^2},$$ where we have assumed that the electric fields are perpendicular to their corresponding wavevector (transversality). The nonlinear polarization is usually described as a power series in the electric field, $$\vec{P}=\btensor{\chi}^{(1)}\cdot\vec{E}+\btensor{\chi}^{(2)}:\vec{E}\vec{E}
+\btensor{\chi}^{(3)}\vdots\vec{E}\vec{E}\vec{E}+\dots,$$ where $\btensor{\chi}^{(1)}$ is the linear susceptibility, $\btensor{\chi}^{(2)}$ is the second-order nonlinear susceptibility tensor, $\btensor{\chi}^{(3)}$ is the third-order nonlinear susceptibility tensor, etc. This expansion is of course only of interest if we can assume that $\vec{P}^{(1)}\gg\vec{P}^{(2)}\gg\vec{P}^{(3)}\gg\cdots$ (the parametric approximation). Since we have assumed the medium to be lossless, the susceptibility tensors are real time-independent quantities, and hence also $\stensor{\epsilon}_{\!r}$ is a real quantity. The linear susceptibility tensor is included in the linear polarization ($\vec{P}_{\rm{L}}$) above, and the lowest order nonlinear polarization of interest to DFWM is the third-order one, i.e., $$\vec{P}_{\rm{NL}}(\vec{r},t)=\btensor{\chi}^{(3)}\vdots
\vec{E}(\vec{r},t)\vec{E}(\vec{r},t)\vec{E}(\vec{r},t).
\label{eq:2.9}$$ Above, the sum-product operator “$\vdots$” is defined such that element $i$ of the nonlinear polarization is $$P_{\rm{NL},i}(\vec{r},t)=\sum_{jkh}\chi_{ijkh}^{(3)}
E_{h}(\vec{r},t)E_{k}(\vec{r},t)E_{j}(\vec{r},t).
\label{eq:2.10}$$ The total electric field is a sum of the four individual fields in the DFWM process, $$\vec{E}(\vec{r},t)=\sum_{\alpha=1}^{4}\vec{E}_{\alpha}(\vec{r},t)=
{1\over2}\sum_{\alpha=1}^{4}\vec{E}_{\alpha}(\vec{r})
e^{{\rm{i}}(\vec{k}_{\alpha}\cdot\vec{r}-\omega{}t)}+\mbox{c.c.},
\label{eq:2.11}$$ where $\vec{E}_{\alpha}(\vec{r})$ are slowly varying quantities, and the wavevector $\vec{k}_{\alpha}$ is real. Since we assumed that the pump fields $\vec{E}_{1}(\vec{r},t)$ and $\vec{E}_{2}(\vec{r},t)$ are counterpropagating, the sum of their wavevectors is zero, i.e., $\vec{k}_{1}+\vec{k}_{2}=\vec{0}$. Inserting Eq. (\[eq:2.11\]) into Eq. (\[eq:2.9\]), a large number of terms are generated. In the phase conjugation configuration we are particularly interested in the terms related to the first harmonic in the cyclic frequency $\omega$. Among these terms are terms that can act as phase-matched source terms for the conjugate wave $\vec{E}_{4}(\vec{r},t)$ when the probe and conjugate fields are counterpropagating, i.e., when $\vec{k}_{3}+\vec{k}_{4}=\vec{0}$. Using these two properties of the wavevectors, terms with a spatial dependence of the form $e^{{\rm{i}}\vec{k}_{\alpha}\cdot\vec{r}}$ are particularly important because they produce the phase-matched terms for the four interacting electric fields. The polarizations associated with these phase-matched contributions (at $\omega$) become $$\begin{aligned}
\vec{P}_{\rm{NL}}^{(1)}(\vec{r},t)&=&{3\over8}\btensor{\chi}^{(3)}\vdots
\left[\vec{E}_{1}\vec{E}_{1}\vec{E}_{1}^{*}
+2\sum_{\alpha\in\{2,3,4\}}\vec{E}_{\alpha}\vec{E}_{1}\vec{E}_{\alpha}^{*}
+2\vec{E}_{2}^{*}\vec{E}_{3}\vec{E}_{4}\right]
e^{{\rm{i}}(\vec{k}_{1}\cdot\vec{r}-\omega{}t)}+\mbox{c.c.},
\label{eq:2.12}
\\
\vec{P}_{\rm{NL}}^{(2)}(\vec{r},t)&=&{3\over8}\btensor{\chi}^{(3)}\vdots
\left[\vec{E}_{2}\vec{E}_{2}\vec{E}_{2}^{*}
+2\sum_{\alpha\in\{1,3,4\}}\vec{E}_{\alpha}\vec{E}_{2}\vec{E}_{\alpha}^{*}
+2\vec{E}_{1}^{*}\vec{E}_{3}\vec{E}_{4}\right]
e^{{\rm{i}}(\vec{k}_{2}\cdot\vec{r}-\omega{}t)}+\mbox{c.c.},
\label{eq:2.13}
\\
\vec{P}_{\rm{NL}}^{(3)}(\vec{r},t)&=&{3\over8}\btensor{\chi}^{(3)}\vdots
\left[\vec{E}_{3}\vec{E}_{3}\vec{E}_{3}^{*}
+2\sum_{\alpha\in\{1,2,4\}}\vec{E}_{\alpha}\vec{E}_{3}\vec{E}_{\alpha}^{*}
+2\vec{E}_{1}\vec{E}_{2}\vec{E}_{4}^{*}\right]
e^{{\rm{i}}(\vec{k}_{3}\cdot\vec{r}-\omega{}t)}+\mbox{c.c.},
\label{eq:2.14}
\\
\vec{P}_{\rm{NL}}^{(4)}(\vec{r},t)&=&{3\over8}\btensor{\chi}^{(3)}\vdots
\left[\vec{E}_{4}\vec{E}_{4}\vec{E}_{4}^{*}
+2\sum_{\alpha\in\{1,2,3\}}\vec{E}_{\alpha}\vec{E}_{4}\vec{E}_{\alpha}^{*}
+2\vec{E}_{1}\vec{E}_{2}\vec{E}_{3}^{*}\right]
e^{{\rm{i}}(\vec{k}_{4}\cdot\vec{r}-\omega{}t)}+\mbox{c.c.},
\label{eq:2.15}\end{aligned}$$ in a short notation where $\vec{E}\equiv\vec{E}(\vec{r})$. With this splitting of the nonlinear polarization, the wave equation is satisfied, when each of the four fields and their related polarizations satisfy the wave equation separately. Next, assuming that the pump fields are much stronger than the probe and the conjugate fields, we can drop the terms in Eqs. (\[eq:2.12\])–(\[eq:2.15\]) containing more than one weak-field component, thus obtaining $$\begin{aligned}
\vec{P}_{\rm{NL}}^{(1)}(\vec{r},t)&=&{3\over8}\btensor{\chi}^{(3)}\vdots
\left[\vec{E}_{1}\vec{E}_{1}\vec{E}_{1}^{*}
+2\vec{E}_{2}\vec{E}_{1}\vec{E}_{2}^{*}\right]
e^{{\rm{i}}(\vec{k}_{1}\cdot\vec{r}-\omega{}t)}+\mbox{c.c.},
\label{eq:2.16}
\\
\vec{P}_{\rm{NL}}^{(2)}(\vec{r},t)&=&{3\over8}\btensor{\chi}^{(3)}\vdots
\left[\vec{E}_{2}\vec{E}_{2}\vec{E}_{2}^{*}
+2\vec{E}_{1}\vec{E}_{2}\vec{E}_{1}^{*}\right]
e^{{\rm{i}}(\vec{k}_{2}\cdot\vec{r}-\omega{}t)}+\mbox{c.c.},
\label{eq:2.17}
\\
\vec{P}_{\rm{NL}}^{(3)}(\vec{r},t)&=&{3\over4}\btensor{\chi}^{(3)}\vdots
\left[\vec{E}_{1}\vec{E}_{3}\vec{E}_{1}^{*}
+\vec{E}_{2}\vec{E}_{3}\vec{E}_{2}^{*}
+\vec{E}_{1}\vec{E}_{2}\vec{E}_{4}^{*}\right]
e^{{\rm{i}}(\vec{k}_{3}\cdot\vec{r}-\omega{}t)}+\mbox{c.c.},
\label{eq:2.18}
\\
\vec{P}_{\rm{NL}}^{(4)}(\vec{r},t)&=&{3\over4}\btensor{\chi}^{(3)}\vdots
\left[\vec{E}_{1}\vec{E}_{4}\vec{E}_{1}^{*}
+\vec{E}_{2}\vec{E}_{4}\vec{E}_{2}^{*}
+\vec{E}_{1}\vec{E}_{2}\vec{E}_{3}^{*}\right]
e^{{\rm{i}}(\vec{k}_{4}\cdot\vec{r}-\omega{}t)}+\mbox{c.c.},
\label{eq:2.19}\end{aligned}$$ again in short notation. Note that by this approximation the polarizations associated with the pump fields have been decoupled from the probe and conjugate fields. Then we may first solve the wave equations for the pump fields, and thereafter insert the result into the wave equations for the probe and conjugate fields. Following this insertion, the probe and conjugate fields can be found. Since $\nabla^2\{\vec{E}(\vec{r})e^{{\rm{i}}(\vec{k}\cdot\vec{r}-\omega{}t)}\}=\{(\nabla^2+2{\rm{i}}[\vec{k}\cdot\vec{\nabla}]-k^2)\vec{E}(\vec{r})\}e^{{\rm{i}}(\vec{k}\cdot\vec{r}-\omega{}t)}$, the wave equation for pump field 1 can be written $$\left[\tensor{\openone}\left(\nabla^2+2{\rm{i}}[\vec{k}_{1}\cdot\vec{\nabla}]
-k^2\right)+{\omega^2\over{}c^2}\stensor{\epsilon}_{\!r}\right]
\cdot\vec{E}_{1}=-{3\omega^2\over8c^2}\btensor{\chi}^{(3)}\vdots
\left[\vec{E}_{1}\vec{E}_{1}\vec{E}_{1}^{*}
+2\vec{E}_{2}\vec{E}_{1}\vec{E}_{2}^{*}\right],
\label{eq:2.20}$$ still in the short notation from above. Assuming now that we have an isotropic medium, the susceptibility tensors must be invariant to inversion and rotation around any axis in the chosen Cartesian coordinate system. The demand of inversion symmetry leaves all tensor elements with an odd number of $x$’s, $y$’s, or $z$’s zero, and thus only diagonal elements survive in the linear susceptibility tensor, and only the $21$ elements in the nonlinear susceptibility tensor of the form $\chi_{iijj}^{(3)}$, $\chi_{ijji}^{(3)}$, and $\chi_{ijij}^{(3)}$ are nonzero, $i$ and $j$ being any $x$, $y$, or $z$. The demand of invariance to rotational transformations results in the demand that the three remaining nonzero elements of the linear susceptibility tensor are equal, and thus we find that $\stensor{\epsilon}_{\!r}=\tensor{\openone}\epsilon_r$. In terms of the refractive index $n$ of the medium, that is $\epsilon_r=n^2$. For the nonlinear susceptibility tensor this demand implies that the nonzero elements can be written $$\chi_{ijkh}^{(3)}=
\chi_{xxyy}^{(3)}\delta_{ij}\delta_{kh}
+\chi_{xyxy}^{(3)}\delta_{ik}\delta_{jh}
+\chi_{xyyx}^{(3)}\delta_{ih}\delta_{jk}.
\label{eq:2.21}$$ In the DFWM case, permutation symmetry between the two fields without the complex conjugation makes $k$ and $h$ interchangeable, and thus $\chi_{xyxy}^{(3)}$ is equal to $\chi_{xyyx}^{(3)}$, leaving the nonzero elements $$\chi_{ijkh}^{(3)}=\chi_{xxyy}^{(3)}\delta_{ij}\delta_{kh}
+\chi_{xyyx}^{(3)}(\delta_{ih}\delta_{jk}+\delta_{ik}\delta_{jh})
\label{eq:2.22}$$ in the nonlinear susceptibility tensor. Furthermore, inside the medium the modulus of the wavevectors are the same, $k_1=k_2=k_3=k_4=k=n\omega/c$. Under the above assumptions, Eq. (\[eq:2.20\]) takes the form $$\begin{aligned}
\lefteqn{
\left(\nabla^2+2{\rm{i}}[\vec{k}_{1}\cdot\vec{\nabla}]\right)
\vec{E}_{1}(\vec{r})=-{3\omega^2\over8c^2}\left\{
\chi_{xxyy}^{(3)}
\left[[\vec{E}_{1}(\vec{r})\cdot\vec{E}_{1}(\vec{r})]\vec{E}_{1}^{*}(\vec{r})
+2[\vec{E}_{2}(\vec{r})\cdot\vec{E}_{1}(\vec{r})]\vec{E}_{2}^{*}(\vec{r})
\right]
\right.}\nonumber\\ &\quad&\left.\!
+2\chi_{xyyx}^{(3)}
\left[\vec{E}_{1}(\vec{r})[\vec{E}_{1}(\vec{r})\cdot\vec{E}_{1}^{*}(\vec{r})]
+2\vec{E}_{2}(\vec{r})[\vec{E}_{1}(\vec{r})\cdot\vec{E}_{2}^{*}(\vec{r})]
\right]
\right\}.
\label{eq:2.23}\end{aligned}$$ Now the simplest assumption is that the four fields travel in a direction almost parallel to the $z$-axis (the paraxial approximation), that they have the same state of polarization, and that the pump waves have plane wavefronts (independent of $x$ and $y$). Then instead of Eq. (\[eq:2.23\]), Eq. (\[eq:2.20\]) is rewritten into the form $$\left({d^2\over{}dz^2}+2{\rm{i}}k{d\over{}dz}\right)
E_{1}(z)=-{3\omega^2\over8c^2}\chi_{xxxx}^{(3)}
\left[|E_{1}(z)|^2+2|E_{2}(z)|^2\right]E_{1}(z),
\label{eq:2.24}$$ still for an isotropic medium, and now having $k_1=k_z$. Introducing into Eq. (\[eq:2.24\]) the slowly varying envelope approximation (SVEA), in which it is assumed that $|k(dE/dz)|$ $\gg|d^2E/dz^2|$, we obtain $${dE_1(z)\over{}dz}={3{\rm{i}}\omega\over16nc}\chi_{xxxx}^{(3)}
\left[|E_1(z)|^2+2|E_2(z)|^2\right]E_1(z)\equiv{\rm{i}}\kappa_1E_1(z).
\label{eq:2.25}$$ In a similar fashion we find that the pump field going in the negative $z$-direction is descibed by the equation $${dE_2(z)\over{}dz}=-{3{\rm{i}}\omega\over16nc}\chi_{xxxx}^{(3)}
\left[|E_2(z)|^2+2|E_1(z)|^2\right]E_2(z)\equiv-{\rm{i}}\kappa_2E_2(z),
\label{eq:2.26}$$ since $k_2=-k_z$. Since $\chi_{xxxx}^{(3)}$ and $n$ are real quantities (from the assumption of a lossless medium), $\kappa_1$ and $\kappa_2$ are also real quantities. Eqs. (\[eq:2.25\]) and (\[eq:2.26\]) have solutions on the form $E_1(z)=E_1(0)e^{{\rm{i}}\kappa_1z}$ and $E_2(z)=E_2(0)e^{-{\rm{i}}\kappa_2z}$, respectively.
Next, we consider the probe and conjugate fields. If we assume that the incident probe wave can be decomposed into plane waves we can for simplicity consider only one of these at a time. Under this assumption, and keeping the approximations mentioned before, the wave equations for the probe and conjugate fields are $$\begin{aligned}
\lefteqn{
{dE_3(z)\over{}dz}={3{\rm{i}}\omega\over8nc}\chi_{xxxx}^{(3)}
\left\{\left[|E_1(0)|^2+2|E_2(0)|^2\right]E_3(z)+E_1(0)E_2(0)E_4^*(z)
e^{{\rm{i}}(\kappa_1-\kappa_2)z}\right\}
}\nonumber\\ &\quad&\equiv
{\rm{i}}\kappa_3E_3(z)+{\rm{i}}\kappa{}E_4^*(z),
\label{eq:2.27}
\\
\lefteqn{
{dE_4(z)\over{}dz}=-{3{\rm{i}}\omega\over8nc}\chi_{xxxx}^{(3)}
\left\{\left[|E_1(0)|^2+2|E_2(0)|^2\right]E_4(z)+E_1(0)E_2(0)E_3^*(z)
e^{{\rm{i}}(\kappa_1-\kappa_2)z}\right\}
}\nonumber\\ &&\equiv
-{\rm{i}}\kappa_3E_4(z)-{\rm{i}}\kappa{}E_3^*(z).
\label{eq:2.28}\end{aligned}$$ To achieve perfect phase matching between the probe and the conjugate field, $\kappa$ has to be constant along $z$, requiring that $\kappa_1=\kappa_2$, which means that the intensity of the two pump fields must be the same ($|E_1(z)|^2=|E_2(z)|^2$). If we additionally introduce a change of variables by letting $E_3(z)=E_3'(z)e^{{\rm{i}}\kappa_3z}$ and $E_4(z)=E_4'(z)e^{-{\rm{i}}\kappa_3z}$, Eqs. (\[eq:2.27\]) and (\[eq:2.28\]) become $$\begin{aligned}
{dE_3'(z)\over{}dz}&=&{\rm{i}}\kappa{}E_4^{\prime{}*}(z),
\label{eq:2.29}\\
{dE_4'(z)\over{}dz}&=&-{\rm{i}}\kappa{}E_3^{\prime{}*}(z),
\label{eq:2.30}\end{aligned}$$ and we notice in passing that the primed and the unprimed variables coincide in the input plane of the interaction region, i.e., at $z=0$. Eqs. (\[eq:2.29\]) and (\[eq:2.30\]) shows why degenerate four-wave mixing leads to phase conjugation, since the generated field $E_4'(z)$ is driven only by the complex conjugate of the probe field amplitude. Differentiation of Eq. (\[eq:2.29\]) and insertion of Eq. (\[eq:2.30\]), an vice versa, we get $$\begin{aligned}
{d^2E_3'(z)\over{}dz^2}+\kappa^2E_3'(z)&=&0,
\label{eq:2.31}\\
{d^2E_4'(z)\over{}dz^2}+\kappa^2E_4'(z)&=&0.
\label{eq:2.32}\end{aligned}$$ The characteristic equation is $\lambda^2+\kappa^2=0$, which has solutions $\lambda=\pm{\rm{i}}\kappa$. The general solution to Eqs. (\[eq:2.31\]) and (\[eq:2.32\]) is then $$\begin{aligned}
E_3'(z)&=&C_1e^{{\rm{i}}\kappa{}z}+C_2e^{-{\rm{i}}\kappa{}z},
\label{eq:2.33}\\
E_4'(z)&=&C_3e^{{\rm{i}}\kappa{}z}+C_4e^{-{\rm{i}}\kappa{}z}.
\label{eq:2.34}\end{aligned}$$ Assuming that we know the values $E_3'(0)$ and $E_4'(L)$, we can then find $E_3'(z)$ and $E_4'(z)$ as a function of these two boundary values. Then the solutions to the coupled differential equations, Eqs. (\[eq:2.29\]) and (\[eq:2.30\]), describing the electric field inside the phase conjugating medium, become $$\begin{aligned}
E_3'(z)&=&E_3'(0){\cos[\kappa(L-z)]\over\cos(\kappa{}L)}
-{\rm{i}}E_4^{\prime{}*}(L){\sin(\kappa{}z)\over\cos(\kappa{}L)},
\label{eq:2.35}\\
E_4'(z)&=&E_4'(L){\cos(\kappa{}z)\over\cos(\kappa{}L)}
-{\rm{i}}E_3^{\prime{}*}(0){\sin[\kappa(L-z)]\over\cos(\kappa{}L)}.
\label{eq:2.36}\end{aligned}$$ In the practical case, $E_3'(0)$ (the probe field coming into the medium) is finite and $E_4'(L)$ (the phase conjugated field at the other end of the medium) is zero. The phase conjugated field coming out of the medium at $z=0$ is then $$E_4'(0)=-{\rm{i}}E_3^{\prime{}*}(0)\tan(\kappa{}L).
\label{eq:2.37}$$ Thus the phase conjugated field depends on (i) the intensity of the pump fields, (ii) the length of the active medium, and (iii) the incoming probe field, and we notice that the magnitude of the phase conjugated field can be larger than the magnitude of the incoming probe field.
Discussion {#Ch:3}
==========
The theoretical description given in the preceding chapter is not the only existing description of phase conjugation by DFWM in the macroscopic sense, but it illustrates quite well the usual line of thought when considering optical phase conjugation. As examples on theoretical papers going beyond the description in Chapter \[Ch:2\], let us mention that (i) polarization properties have been studied by , (ii) descriptions taking into account the vectorial properties \[see Eq. (\[eq:2.20\])\] have been given, e.g., by , (iii) improvements to the standard theory in the form of abandoning the slowly varying envelope approximation (SVEA) have also been discussed \[see, e.g., and \]. A feature of the standard theory \[see Eq. (\[eq:2.37\])\] is that the phase conjugated response depends on the length of the nonlinear crystal used (infinite at $\kappa{}L=(2p+1)\pi/2$ for any integer value of $p$). The standard theory has proven to be a satisfactory description for spatially nondecaying fields containing no evanescent components.
Though the overwhelming majority of optical phase conjugation experiments can be described without inclusion of evanescent components of the electromagnetic field, the possible phase conjugation of these components has been discussed from time to time. With the experimental observation of [-@Bozhevolnyi:94:1; -@Bozhevolnyi:95:2], the need for inclusion of near-field components and thus evanescent modes in the description of optical phase conjugation has drawn renewed attention.
In an important paper by the treatment was focused on an analysis of the phase conjugated replica produced by a so-called ideal phase conjugator, characterized phenomenologically by a polarization- and angle of incidence independent nonlinear amplitude reflection coefficient, and in recent articles by [-@Keller:96:3; -@Keller:96:2] attention was devoted to an investigation of the spatial confinement problem of the phase conjugated field. Macroscopic theories including near-field components in the optical phase conjugation process have also appeared recently [@Bozhevolnyi:95:3; @Arnoldus:95:2].
In their work used degenerate four-wave mixing (DFWM) produced by a 10mW HeNe laser with a wavelength of 633nm in an iron-doped lithium-niobate (Fe:LiNbO$_3$) crystal and an external-reflection near-field optical microscope to achieve phase conjugated light foci, which with a diameter of $\sim180$nm were well below the classical diffraction limit. The main conclusion of their experiments was that to achieve a spot size as small as 180nm phase conjugation of at least parts of the optical near-field emitted from the source must have taken place.
In the present work we go one step further in the theoretical study of the phase conjugation of optical signals which include near-field components by abandoning the ideal phase conjugator assumption. For simplicity our description is limited to cover only the degenerate four-wave mixing configuration for which the interacting optical fields all have the same cyclic frequency $\omega$.
Because of the small range of the optical evanescent fields from the (mesoscopic) source a substantial part of the near-field phase conjugation process is bound to take place in the surface region of the phase conjugating medium. It is thus from the very outset necessary to focus the attention on the surface region of the nonlinear mirror and investigate the phase conjugation process on a length scale (much) smaller than the optical wavelength. This fact in itself makes use of the ideal phase conjugator assumption doubtful. For a bulk phase conjugator it may furthermore be difficult to assure an effective nonlinear mixing in a surface layer as thin as the field penetration depth. Thus, experimentally it might be advantageous to use a thin film or even a quantum well as the nonlinear medium (see Fig. \[fig:PS1\]). From a different perspective the use of a thin film as the nonlinear medium has already drawn attention [@Montemezzani:96:1].
The present theory has been constructed in such a manner that it offers a framework for microscopic studies of degenerate four-wave mixing at surfaces of bulk media, in thin films and quantum wells, and in small particles. To carry out in detail a rigorous microscopic numerical analysis of the DFWM process it is, however, necessary to consider mesoscopic media with a particularly simple electronic structure, and we shall demonstrate later how the present theory can be applied to a simple quantum well structure.
(0,0)(4,2.5) (0,-0.5)[ (0.25,2.55)[0.05]{} (0.25,2.55)[0.15]{}[10]{}[35]{} (0.25,2.55)[0.15]{}[55]{}[80]{} (0.25,2.55)[0.15]{}[100]{}[125]{} (0.25,2.55)[0.15]{}[145]{}[170]{} (0.25,2.55)[0.15]{}[190]{}[215]{} (0.25,2.55)[0.15]{}[235]{}[260]{} (0.25,2.55)[0.15]{}[280]{}[305]{} (0.25,2.55)[0.15]{}[325]{}[350]{} (0.25,2.55)[0.25]{}[10]{}[35]{} (0.25,2.55)[0.25]{}[55]{}[80]{} (0.25,2.55)[0.25]{}[100]{}[125]{} (0.25,2.55)[0.25]{}[145]{}[170]{} (0.25,2.55)[0.25]{}[190]{}[215]{} (0.25,2.55)[0.25]{}[235]{}[260]{} (0.25,2.55)[0.25]{}[280]{}[305]{} (0.25,2.55)[0.25]{}[325]{}[350]{} (0.7,2.25)[(0,0.4)\[l\][$=$$\displaystyle\sum_{q_{\|}\leq{\omega\over{}c_0}}$]{}]{} (1.7,2.75)(1.7,2.25) (1.6875,2.75)(2.2,2.75) (1.7,2.75)(2.2,2.25) (1.7,2.25)(2.2,2.25) (2.2,2.25)(2.2,2.75) (1.75,2.60)(1.85,2.70) (1.80,2.55)(1.90,2.65) (1.85,2.50)(1.95,2.60) (1.90,2.45)(2.00,2.55) (1.95,2.40)(2.05,2.50) (2.00,2.35)(2.10,2.45) (2.5,2.25)[(0,0.4)\[l\][$+$$\displaystyle\sum_{q_{\|}>{\omega\over{}c_0}}$]{}]{} (3.5,2.75)(3.5,2.25) (3.4875,2.75)(4.0,2.75) (3.60,2.65)(3.60,2.85) (3.65,2.65)(3.65,2.85) (3.70,2.65)(3.70,2.85) (3.75,2.65)(3.75,2.85) (3.80,2.65)(3.80,2.85) (3.4,2.65)(3.6,2.65) (3.4,2.60)(3.6,2.60) (3.4,2.55)(3.6,2.55) (3.4,2.50)(3.6,2.50) (3.4,2.45)(3.6,2.45) ]{} (2,1)(1.61,0.9)(1.37,0.8)(1.22,0.7)(1.14,0.6)(1.082,0.5)(1.050,0.4)(1.030,0.3)(1.018,0.2)(1.011,0.1)(1.007,0.0) (1,1.1)(1,0) (0.8,1)(2.2,1) (0,0.75)(4,0.75) (0,0.35)(4,0.35) (1,1)[0.05]{} (1.3,1.3)(1.1,1.1) (1.325,1.20)[(0,0.4)\[l\][Mesoscopic source]{}]{} (3.9,0.8)[(0,0.4)\[r\][Vacuum]{}]{} (3.9,0.35)[(0,0.4)\[r\][Quantum well/Thin film]{}]{} (3.9,0)[(0,0.4)\[r\][Substrate]{}]{}
In conventional descriptions of optical phase conjugation by DFWM it is assumed that the interaction length is long compared to the wavelength of the probe fields, thus building up pictorially speaking from one of the pump beams and the probe beam a grating, from which the other pump beam is scattered into a phase conjugated replica (the ‘real-time holography’ picture). Furthermore it is assumed that the amplitudes of the fields are slowly varying on the optical wavelength scale \[Slowly varying envelope (SVE) approximation\] and thus also constant across the individual scattering units (atoms, molecules, …) \[Electric dipole (ED) approximation\] of the phase conjugating medium. Considering optical near fields, which contain components varying rapidly in space, the aforementioned approximations do not hold and we thus exclude them in the present formalism. We also avoid other approximations often made in the literature, namely (i) the assumption of a lossless medium, (ii) the [*ab initio*]{} requirement of phase matching between the interacting optical signals, and (iii) the assumption that the probe field is weak compared to the pump fields.
To illustrate the need for a theory going beyond the SVE and ED approximations, we have in Fig. \[fig:PS2\] shown the component of the probe wavevector perpendicular to the surface, inside as well as outside the phase conjugator, as a function of its parallel component. When the parallel component of the probe wavevector becomes larger than $\omega/c_0$, the perpendicular component of the wavevector becomes purely imaginary in the vacuum, but it is still real inside the phase conjugating mirror. A purely imaginary wavevector component means that the electromagnetic field is evanescent, whereas a real component indicates that the field is propagating and nondecaying (in the absense of absorption). When the parallel component becomes larger than $n\omega/c_0$ (where $n$ is the refractive index of the substrate) the perpendicular component of the probe wavevector becomes evanescent also inside the phase conjugating mirror, and the larger the parallel component, the more wrong the SVE and ED approximations become. Thus to study, for instance, the phase conjugation of all field components possibly emitted from a mesoscopic source in the vicinity of the phase conjugator it is necessary to abandon these approximations.
(-0.5,-1.0)(4,4) (0,0)(4,4) (0,0)[1.51]{}[0]{}[90]{} (1.51,0.00)(1.60,0.53)(2.00,1.31)(2.50,1.99)(3.00,2.59)(3.50,3.16)(4.00,3.70) (0,0)[1.0]{}[0]{}[90]{} (1.00,0.00)(1.10,0.46)(1.30,0.83)(1.50,1.12)(2.00,1.73)(2.50,2.29)(3.00,2.83)(3.50,3.35)(4.00,3.87) (-0.05,0)(4,0) (0,-0.05)(0,4) (-0.1,3.75)[(0.0,0.0)\[r\][$|q_{\perp}|$]{}]{} (-0.05,1.51)(0.05,1.51) (-0.1,1.51)[(0.0,0.0)\[r\][${n\omega\over{}c_0}$]{}]{} (-0.05,1.0)(0.05,1.0) (-0.1,1.0)[(0.0,0.0)\[r\][${\omega\over{}c_0}$]{}]{} (-0.1,0.0)[(0.0,0.0)\[r\][$0$]{}]{} (0.0,-0.1)[(0.0,0.0)\[t\][$0$]{}]{} (1.0,-0.05)(1.0,0.05) (1.0,-0.1)[(0.0,0.0)\[t\][${\omega\over{}c_0}$]{}]{} (1.51,-0.05)(1.51,0.05) (1.51,-0.1)[(0.0,0.0)\[t\][${n\omega\over{}c_0}$]{}]{} (3.75,-0.1)[(0.0,0.0)\[t\][$q_{\|}$]{}]{} (0.25,1.0)[(0.0,0.0)\[b\][$q_{\perp}^{0}$]{}]{} (0.5,1.51)[(0.0,0.0)\[b\][$q_{\perp}^{n}$]{}]{} (1.67,1.0)[(0.0,0.0)\[b\][$\alpha_{\perp}^{0}$]{}]{} (2.35,1.5)[(0.0,0.0)\[b\][$\alpha_{\perp}^{n}$]{}]{} (0,-0.65)(1,-0.65) (1,-0.65)(4,-0.65) (0,-0.7)(0,-0.6) (1,-0.7)(1,-0.6) (0.5,-0.625)[(0,0)\[b\][$q_{\perp}^{0}$]{}]{} (1.5,-0.625)[(0,0)\[bl\][$q_{\perp}^{0}={\rm{i}}\alpha_{\perp}^{0}$]{}]{} (0,-1)(1.51,-1) (1.51,-1)(4,-1) (0,-1.05)(0,-0.95) (1.51,-1.05)(1.51,-0.95) (0.75,-0.975)[(0,0)\[b\][$q_{\perp}$]{}]{} (2.0,-0.975)[(0,0)\[bl\][$q_{\perp}^{n}={\rm{i}}\alpha_{\perp}^{n}$]{}]{} (0.4875,3.50)(1.5,3.50) (0.5,3.50)(1.5,3.00) (0.5,3.50)(0.5,3.00) (1.55,3.50)[(0.0,0.0)\[l\][$q_{\|}$]{}]{} (1.55,3.00)[(0.0,0.0)\[l\][$q$]{}]{} (0.45,3.00)[(0.0,0.25)\[r\][$q_{\perp}^{n}$]{}]{} (0.45,3.25)[(0.0,0.25)\[r\][$q_{\perp}^{0}$]{}]{} (0.5,3.00)(1.5,3.00) (1.5,3.00)(1.5,3.50) (2.4875,1.0)(3.5,1.0) (2.5,1.0)(2.5,0.5) (3.55,1.0)[(0.0,0.0)\[l\][$q_{\|}$]{}]{} (2.45,0.5)[(0.0,0.25)\[r\][$\alpha_{\perp}^{n}$]{}]{} (2.45,0.75)[(0.0,0.25)\[r\][$\alpha_{\perp}^{0}$]{}]{}
As already mentioned, the present theory not only allows one to investigate the optical phase conjugation of evanescent waves with small penetration depths, it also enables one to investigate the possibility of achieving DFWM in mesoscopic films (quantum wells), a subject of interest in its own right. The main reason that the present formulation may be used in near-field optics as well as in mesoscopic-film electrodynamics originates in the fact that in both cases the microscopic local-field calculation is the crucial quantity. A further advantage of the present theory is that it allows us to study phase conjugation when one or more of the interacting fields are surface-wave fields.
The construction of such a theoretical model begins with the microscopic Maxwell–Lorentz equations, which combined with the nonlocal linear and third-order nonlinear constitutive equations are used to set up the basic wave equation for the phase conjugated field. The linear and nonlinear conductivity responses of the electrons will be calculated within the framework of the random-phase-approximation theory, the starting point being the Liouville equation of motion for the density matrix operator. In the description we include in the interaction Hamiltonian not only the standard $\vec{p}\cdot\vec{A}$ term ($\vec{p}$ being the momentum operator and $\vec{A}$ being the vector potential) but also the term proportional to the square of the vector potential, i.e., $\vec{A}\cdot\vec{A}$. For a monochromatic driving field (of cyclic frequency $\omega$), this term contains $2\omega$- and DC-parts, and both of these are in general important for the description of the microscopic phase conjugation process. In the current density operator we include the term containing the vector potential. This term, needed in order to ensure the gauge invariance of quantum electrodynamics, also turns out to be of importance in some cases. Starting from a dipolar interaction Hamiltonian the first explicit microscopic derivation of the third-order conductivity (susceptibility) response appears to be due to . The result of is based on a $\vec{r}\cdot\vec{E}$ calculation and only the vector-potential independent part of the current density operator is kept. Apart from a single study dealing with the electromagnetic self-action in a BCS-paired superconductor [@Keller:95:7], it seems that in all theoretical investigations of the DFWM-process in which microscopic considerations have appeared, the expression has been used. We cannot use this expression here, however, since we need to address a local-field problem when dealing with mesoscopic interaction volumes, and such a problem necessitates that we take into account values of $q_{\|}$ much larger than $\omega/c_0$, and thus that the calculation goes beyond the ED approximation. To account for local-field effects it is necessary to perform a spatially nonlocal calculation of the third-order conductivity, and this is most adequately done beginning with the minimal coupling interaction Hamiltonian which contains both the $\vec{p}\cdot\vec{A}$ and $\vec{A}\cdot\vec{A}$ terms. In the local limit where the vector potential only depends on time our expression for the nonlinear conductivity and the $\vec{r}\cdot\vec{E}$ based one of are physically equivalent, [*provided*]{} the terms stemming from the gauge conserving vector potential dependent part of the current density operator are neglected. Though physically equivalent, the explicit forms of the relation between the nonlinear current density and the electric field only coincides after having performed a relevant unitary transformation on the minimal coupling Hamiltonian and the related electronic wave functions. In stead of using the minimal coupling Hamiltonian to describe the nonlocal dynamics one could in principle have used the multipolar Hamiltonian. In practice this is less convenient for the present purpose due to the fact that the pronounced nonlocality we sometimes are facing in mesoscopic media would require that many multipole terms were kept in the Hamiltonian. The essentially nonlocal terms in the nonlinear conductivity are included in our treatment because they in certain cases—especially for very small interaction volumes—are the only contributing ones, and in other cases they dominate the phase conjugated response. Since we deal with a spatially nonlocal description it is important to characterize the spatial structure involved in the physical processes behind the phase conjugation, and the various physical processes hidden in the nonlinear and nonlocal constitutive equation are therefore identified. Following the identification of the physical processes, an expression for the so-called conductivity response tensor describing the nonlinear material response in the DFWM process is established, and the eigensymmetries of the conductivity tensors belonging to each of the processes occuring in the DFWM process are discussed. Rather than solving the full spatial problem (which would be cumbersome, if not impossible), we consider a simplified system possessing infinitesimal translational invariance in two directions. For such a system the potential of the related Schr[ö]{}dinger equation only varies in the direction perpendicular to the plane of translational invariance. The fundamental solutions to the time-independent Schr[ö]{}dinger equation are inserted into the linear and nonlinear conductivity tensors thus giving us the framework for a theoretical description of the DFWM process in mesoscopic films (quantum wells) as well as for evanescent waves. Compared to conventional descriptions of optical phase conjugation in bulk media the concept of phase matching (momentum conservation) now appears only in [*two*]{} dimensions. The lack of translational invariance in the third dimension implies that no phase matching occurs in this dimension. Phase matching (in one, two or three dimensions) is not a precondition set on our theory, it follows to the extent that the phase conjugating medium exhibits infinitesimal translational invariance. Despite the fact that the phase matching is lost in the third dimension, phase conjugation may still take place in quantum wells and thin films, and with evanescent fields, just as second harmonic generation can occur in quantum-well systems, at metallic (and semiconducting) surfaces and from nonlinear (sub)monolayer films deposited on linear substrates [@Richmond:88:1]. To complete our local-field calculation of the optical phase conjugation by degenerate four-wave mixing in mesoscopic interaction volumes, we use a Green’s function formalism to establish new integral equations for the phase conjugated field in the general case, and in the case where the nonlinear medium exhibits translational invariance in two dimensions. The microscopic local-field theory thus established is then used to describe the DFWM in one- and two-level quantum-well phase conjugators.
\[part:II\]
The electromagnetic field {#ch:4}
=========================
In this chapter a description of the electromagnetic field from a phase conjugating medium is established, starting from the microscopic Maxwell equations. First, we derive the relevant wave equation for the phase conjugated field. The field–matter interaction is then described through the use of constitutive relations. In the final step of this basic framework, a self-consistent description of the phase conjugated field is established.
Wave equation for the phase conjugated field
--------------------------------------------
As a starting point we take the microscopic Maxwell-Lorentz equations, in which the material response at the space-time point $(\vec{r},t)$ is completely described via the microscopic current density $\vec{J}(\vec{r},t)$, and the related charge density, $\rho(\vec{r},t)$. They are $$\begin{aligned}
\vec{\nabla}\times\vec{E}(\vec{r},t)&=&
-{\partial\vec{B}(\vec{r},t)\over\partial{}t},
\label{eq:Max1}\\
\vec{\nabla}\times\vec{B}(\vec{r},t)&=&\mu_{0}\vec{J}(\vec{r},t)
+{1\over{}c_0^2}{\partial\vec{E}(\vec{r},t)\over\partial{}t},
\label{eq:Max2}\\
\vec{\nabla}\cdot\vec{E}(\vec{r},t)&=&{1\over\epsilon_{0}}\rho(\vec{r},t),
\label{eq:Max3}\\
\vec{\nabla}\cdot\vec{B}(\vec{r},t)&=&0,
\label{eq:Max4}\end{aligned}$$ $\vec{E}(\vec{r},t)$ and $\vec{B}(\vec{r},t)$ being the electric and magnetic fields prevailing at the space point $\vec{r}$ at the time $t$.
Taking the curl of Eq. (\[eq:Max1\]) and inserting the result into Eq. (\[eq:Max2\]) we obtain the following wave equation for the prevailing local electric field $\vec{E}(\vec{r},t)$: $$\left({\tensor{\openone}}{\Box}
+\vec{\nabla}\otimes\vec{\nabla}\right)\cdot\vec{E}(\vec{r},t)
=-\mu_0{\partial\vec{J}(\vec{r},t)\over\partial{}t},\label{eq:Wave}$$ where ${\Box}={1\over{}c^{2}}{\partial^2\over\partial{}t^2}-\nabla^2$ is the d’Alembertian operator, $\tensor{\openone}$ is the ($3\times3$) unit tensor, and $\otimes$ is the outer (dyadic) product operator.
Introducing the electric field as a Fourier series in the cyclic frequency $\omega$, viz. $$\vec{E}(\vec{r},t)={1\over2}\sum_{\alpha=0}^{\infty}
\vec{E}_{-\alpha\omega}(\vec{r}\,)e^{-{\rm{i}}\alpha\omega{}t}+\mbox{c.c.},
\label{eq:Ert}$$ where $\alpha$ is an integer and “c.c.” denotes the complex conjugate of the first terms, we subsequently can limit ourselves to a harmonic analysis. Since $\vec{E}(\vec{r},t)$ is a real quantity, $\vec{E}^{*}_{-\alpha\omega}(\vec{r}\,)=\vec{E}_{\alpha\omega}(\vec{r}\,)$.
Likewise, we write the current density as a Fourier series in $\omega$, in which each component implicitly is expressed as a power series in the electric field. Thus $$\vec{J}(\vec{r},t)=
{1\over2}\sum_{\alpha=0}^{\infty}\sum_{\beta=0}^{\infty}
\left(\vec{J}_{-\alpha\omega}^{\,(\alpha+2\beta)}(\vec{r}\,)
e^{-{\rm{i}}\alpha\omega{}t}+\mbox{c.c.}\right),
\label{eq:J-power}$$ where $\alpha$ and $\beta$ are integers. Looking for solutions at the cyclic frequency $\omega$, only fields and current densities with $\alpha=1$ in Eqs. (\[eq:Ert\]) and (\[eq:J-power\]) contributes. Accordingly we in the following write the phase conjugated (PC) electric field without the reference to the cyclic frequency, i.e., $\vec{E}_{-\omega}(\vec{r}\,)\equiv\vec{E}_{\rm{PC}}(\vec{r}\,)$. In the case of DFWM we will assume that the lowest order nonlinear interaction dominates over higher order mixing processes. Thus, in order to describe the DFWM response of our medium we retain only the two currents of lowest order in $\beta$, namely the linear contribution $\vec{J}_{-\omega}^{\,(1)}(\vec{r}\,)$ and the lowest order nonlinear contribution $\vec{J}_{-\omega}^{\,(3)}(\vec{r}\,)$. The wave equation for the negative frequency part of the phase conjugated response hence takes the form $$\left[\tensor{\openone}\left({\omega^2\over{}c^2}+{\nabla}^2\right)
-\vec{\nabla}\otimes\vec{\nabla}\right]
\cdot\vec{E}_{\rm{PC}}(\vec{r}\,)
=-i\mu_0\omega\left(\vec{J}_{-\omega}^{\,(1)}(\vec{r}\,)
+\vec{J}_{-\omega}^{\,(3)}(\vec{r}\,)\right).
\label{eq:Wave2}$$
Constitutive relations for the current densities
------------------------------------------------
To close the loop for the calculation of the phase conjugated field, the microscopic current densities $\vec{J}_{-\omega}^{\,(1)}(\vec{r}\,)$ and $\vec{J}_{-\omega}^{\,(3)}(\vec{r}\,)$ are given in terms of the local electric field through constitutive relations describing the field–matter interaction in a perturbative manner. Choosing a gauge where the time-dependent part of the scalar potential is zero, the electric field is related to the vector potential via $\vec{E}(\vec{r}\,)={\rm{i}}\omega\vec{A}(\vec{r}\,)$. Thus the microscopic current densities can be related via the constitutive relations to the vector potentials of the phase conjugated field ($\vec{A}_{\rm{PC}}$) and the fields driving the process ($\vec{A}\,$). The linear constitutive relation we therefore write in the form $$\vec{J}_{-\omega}^{\,(1)}(\vec{r}\,)={i\omega}\int
\stensor{\sigma}(\vec{r},\vec{r}\,')
\cdot\vec{A}_{\rm{PC}}(\vec{r}\,')d^3r',
\label{eq:Jr}$$ where $\stensor{\sigma}(\vec{r},\vec{r}\,')\equiv
\stensor{\sigma}(\vec{r},\vec{r}\,';\omega)$ is the linear conductivity tensor. The $i$’th element of the first order current density is proportional to the integral of $[\stensor{\sigma}\cdot\vec{A}_{\rm{PC}}]_{i}=
\sum_{j}{\sigma}_{ij}{A}_{{\rm{PC}},j}$. The nonlinear DFWM constitutive relation is written in a similar fashion, i.e., $$\vec{J}_{-\omega}^{\,(3)}(\vec{r}\,)=(i\omega)^3\int\int\int
\tensor{\Xi}(\vec{r},\vec{r}\,',\vec{r}\,'',\vec{r}\,''')
\vdots\,\vec{A}(\vec{r}\,''')\vec{A}(\vec{r}\,'')\vec{A}^{*}(\vec{r}\,')
d^3r'''d^3r''d^3r',
\label{eq:J3r}$$ where $\tensor{\Xi}(\vec{r},\vec{r}\,',\vec{r}\,'',\vec{r}\,''')\equiv
\tensor{\Xi}(\vec{r},\vec{r}\,',\vec{r}\,'',\vec{r}\,''';\omega)$ is the nonlocal third order conductivity tensor. The three-dimensional sum-product operator “$\vdots$” is here meant to be interpreted for the $i$’th element of the third order current density in the following way: $$\begin{aligned}
\left[\tensor{\Xi}(\vec{r},\vec{r}\,',\vec{r}\,'',\vec{r}\,''')\vdots
\,\vec{A}(\vec{r}\,''')\vec{A}(\vec{r}\,'')\vec{A}^{*}(\vec{r}\,')\right]_{i}
=
\sum_{jkh}{\Xi}_{ijkh}(\vec{r},\vec{r}\,',\vec{r}\,'',\vec{r}\,''')
{A}_{h}(\vec{r}\,'''){A}_{k}(\vec{r}\,''){A}^{*}_{j}(\vec{r}\,').
\nonumber\\
\label{eq:sumprod3}\end{aligned}$$ By inserting Eqs. (\[eq:Jr\]) and (\[eq:J3r\]), with $\vec{A}_{\rm{PC}}=\vec{E}_{\rm{PC}}/({\rm{i}}\omega)$, into Eq. (\[eq:Wave2\]) the loop for the phase conjugated field is closed.
The phase conjugated field {#sec:field}
--------------------------
From the outset we assume that the parametric approximation can be adopted, i.e., we assume that the generated phase conjugated field does not affect the dynamics of the pump and signal fields. In the present case, where the phase conjugated field originates mainly in evanescent modes or from a quantum well, the interaction volume is small and the magnitude of the phase conjugated field thus very limited so that one may expect the parametric approximation to be quite good. The inherent spatial nonlocality of the processes which underlies the microscopic calculation of the local fields and currents is crucial and must be kept throughout the following analysis.
Above we used the microscopic Maxwell-Lorentz equations to establish a wave equation \[Eq. (\[eq:Wave2\])\] for the phase conjugated electric field. Since this equation holds not only inside the phase conjugator but also in the medium possibly in contact with the phase conjugator, it is adequate to divide the linear part of the induced current density into two, i.e., $$\vec{J}(\vec{r}\,)=\vec{J}_{\rm{cont}}(\vec{r};\omega)
+\vec{J}_{\rm{PC}}^{\,(1)}(\vec{r};\omega),$$ where $\vec{J}_{\rm{cont}}(\vec{r};\omega)$ is the linear current density of the medium in contact (cont) with the phase conjugator, and $\vec{J}_{\rm{PC}}^{\,(1)}(\vec{r};\omega)$ is the linear current density of the phase conjugator. In setting up the above-mentioned equation we have implicitly assumed that there is no (significant) electronic overlap between the phase conjugator and the contact medium. The two electron distributions can still be electromagnetically coupled, of course. In the quantum well case, $\vec{J}_{\rm{cont}}(\vec{r};\omega)$ is to be identified as the current density induced in the (assumed linear) response of the substrate. To deal with the evanescent response of a (semiinfinite) phase conjugator one just puts $\vec{J}_{\rm{cont}}(\vec{r};\omega)=\vec{0}$.
Instead of proceeding directly with the differential equation \[Eq. (\[eq:Wave2\])\] for the phase conjugated local field we convert it into an integral relation between the phase conjugated electric field and the prevailing current density, namely $$\vec{E}_{\rm{PC}}(\vec{r};\omega)=\vec{E}^{\rm{ext}}_{\rm{PC}}(\vec{r};\omega)
-{\rm{i}}\mu_{0}\omega
\int\tensor{G}_{0}(\vec{r},\vec{r}\,';\omega)
\cdot\left[\vec{J}_{\rm{cont}}(\vec{r}\,';\omega)
+\vec{J}_{\rm{PC}}^{\,(1)}(\vec{r}\,';\omega)\right]d^3r',
\label{eq:EPCrw}$$ where $\vec{E}^{\rm{ext}}_{\rm{PC}}(\vec{r};\omega)$ is the so-called external (ext) field driving the phase conjugation process, and $\tensor{G}_{0}(\vec{r},\vec{r}\,';\omega)$ is the electromagnetic vacuum propagator. Instead of proceeding with Eq. (\[eq:EPCrw\]) as it stands, if possible, it is often advantageous to eliminate the current density of the contact medium in favour of a so-called pseudo-vacuum (or contact-medium) propagator, $\tensor{G}(\vec{r},\vec{r}\,';\omega)$. Doing this, one obtains $$\vec{E}_{\rm{PC}}(\vec{r};\omega)=\vec{E}^{\rm{B}}_{\rm{PC}}(\vec{r};\omega)
-{\rm{i}}\mu_{0}\omega
\int\tensor{G}(\vec{r},\vec{r}\,';\omega)
\cdot\vec{J}_{\rm{PC}}^{\,(1)}(\vec{r}\,';\omega)d^3r',
\label{eq:EPCrwB}$$ where $\vec{E}^{\rm{B}}_{\rm{PC}}(\vec{r};\omega)$ is the so-called background (B) response of the phase conjugator. The background field is effectively the field driving the phase conjugated response. From a knowledge of the nonlinear part, $\vec{J}_{-\omega}^{\,(3)}(\vec{r}\,)$, of the current density of the phase conjugator, the background field can be calculated from the integral relation $$\vec{E}_{\rm{PC}}^{\rm{B}}(\vec{r};\omega)=-{\rm{i}}\mu_0\omega\int
\tensor{G}(\vec{r},\vec{r}\,';\omega)\cdot
\vec{J}_{-\omega}^{\,(3)}(\vec{r}\,')d^3r'.
\label{eq:EPCB}$$ In the parametric approximation adopted here the background field can be considered as a prescribed quantity. By inserting the linear constitutive equation $$\vec{J}_{\rm{PC}}^{\,(1)}(\vec{r};\omega)=
\int\stensor{\sigma}(\vec{r},\vec{r}\,';\omega)\cdot
\vec{E}_{\rm{PC}}(\vec{r}\,';\omega)d^3r'$$ into Eq. (\[eq:EPCrwB\]) one obtains the following integral equation for the phase conjugated field: $$\vec{E}_{\rm{PC}}(\vec{r};\omega)=\vec{E}_{\rm{PC}}^{\rm{B}}(\vec{r};\omega)
-{\rm{i}}\mu_{0}\omega\int\int\tensor{G}(\vec{r},\vec{r}^{\,\prime\prime};\omega)
\cdot\stensor{\sigma}(\vec{r}\,'',\vec{r}\,';\omega)
\cdot\vec{E}_{\rm{PC}}(\vec{r}\,';\omega)d^3r''d^3r'.
\label{eq:Inte}$$ The formal solution of this equation is given by $$\vec{E}_{\rm{PC}}(\vec{r};\omega)=
\int\tensor{\Gamma}(\vec{r},\vec{r}\,';\omega)
\cdot\vec{E}_{\rm{PC}}^{\rm{B}}(\vec{r}\,';\omega)d^3r',
\label{eq:Sol}$$ where the nonlocal field-field response tensor $\tensor{\Gamma}(\vec{r},\vec{r}\,';\omega)$ is to be derived from the dyadic integral equation $$\tensor{\Gamma}(\vec{r},\vec{r}\,';\omega)=\tensor{\openone}\delta(\vec{r}-\vec{r}\,')
+\int\tensor{K}(\vec{r},\vec{r}\,'';\omega)
\cdot\tensor{\Gamma}(\vec{r}\,'',\vec{r}\,';\omega)d^3r''.
\label{eq:Gamma}$$ In Eq. (\[eq:Gamma\]) the tensor $$\tensor{K}(\vec{r},\vec{r}\,'';\omega)=-{\rm{i}}\mu_{0}\omega
\int\tensor{G}(\vec{r},\vec{r}\,';\omega)
\cdot\stensor{\sigma}(\vec{r}\,',\vec{r}\,'';\omega)d^3r'
\label{eq:Kernel}$$ is the kernel of the integral equation in Eq. (\[eq:Inte\]). This kernel formally is identical to the one playing a prominent role in the electrodynamics of mesoscopic media and small particles \[see , section 4\].
By inserting Eq. (\[eq:EPCB\]) into Eq. (\[eq:Sol\]) and thereafter making use of Eq. (\[eq:J3r\]), the phase conjugated field may in principle be calculated from known quantities. In practice it is not so easy, since the integral equation in Eq. (\[eq:Inte\]) for the phase conjugated field in general is too difficult to handle numerically even if rather simple linear conductivity response tensors are used, the reason being the inherent three-dimensional ($\vec{r}\,$) nature of the problem. One therefore has to resort to one sort of approximation or another. Just as in other linear and nonlinear studies of mesoscopic media, or media with a small interaction volume, a tractable problem is obtained if the medium in question possesses translational invariance in two directions as discussed in Part \[part:III\].
Single-electron current density response {#ch:5}
========================================
In this chapter the Liouville equation of motion for the single-body density matrix operator is used together with the single-particle Hamiltonian to establish a more general quantum mechanical expression for the third-order current density than those hitherto found in the literature. The generalisation is of significant importance for the theory of near-field phase conjugation and for DFWM in mesoscopic films. Following the derivation of the linear and the DFWM responses, we end this chapter by a discussion of the underlying physical processes.
Density matrix operator approach
--------------------------------
The starting point for this calculation is the Liouville equation of motion for the single-body density matrix operator $\rho$, i.e., $${\rm{i}}\hbar{\partial\rho\over\partial{}t}=\left[{\cal{H}},\rho\right].$$ In the equation above, the single-particle Hamiltonian ${\cal{H}}$ appearing in the commutator $\left[{\cal{H}},\rho\right]$ in the present description is given by $${\cal{H}}={\cal{H}}_{\,0}+{\cal{H}}_{\,\rm{R}}+{\cal{H}}_{\,0}^{(2)}
+{1\over2}\sum_{\alpha=1}^{2}\left(
{\cal{H}}_{-\alpha\omega}^{(\alpha)}e^{-{\rm{i}}\alpha\omega{}t}
+\mbox{H.a.}\right),$$ where ${\cal{H}}_{0}$ is the Hamiltonian operator for the electron in the material when the perturbing optical field is absent, ${\cal{H}}^{(1)}$ is the interaction Hamiltonian of first order in the vector potential $\vec{A}(\vec{r}\,)$, ${\cal{H}}^{(2)}$ is the interaction Hamiltonian of second order in $\vec{A}(\vec{r}\,)$, ${\cal{H}}_{\,\rm{R}}$ represents the irreversible coupling to the “surroundings”, and “H.a.” denotes the Hermitian adjoint. Although the spin and spin-orbit dynamics may be included in the formalism in a reasonably simple fashion we have omitted to do so because spin effects are judged to be significant only for nonlinear phenomena of even order. Hence $$\begin{aligned}
{\cal{H}}_{\,0}&=&{1\over{2m_{e}}}\vec{p}\cdot\vec{p}+V(\vec{r}\,),
\label{eq:H0}\\
{\cal{H}}_{-\omega}^{(1)}=\left({\cal{H}}_{\,\omega}^{(1)}\right)^{\dag}&=&
{e\over2m_{e}}\left(\vec{p}\cdot\vec{A}(\vec{r}\,)+
\vec{A}(\vec{r}\,)\cdot\vec{p}\,\right),
\label{eq:H-+1}\\
{\cal{H}}_{-2\omega}^{(2)}=\left({\cal{H}}_{\,2\omega}^{(2)}\right)^{\dag}&=&
{e^2\over4m_{e}}\vec{A}(\vec{r}\,)\cdot\vec{A}(\vec{r}\,),
\label{eq:H-+2}\\
{\cal{H}}_{\,0}^{(2)}&=&
{e^2\over4m_{e}}\vec{A}(\vec{r}\,)\cdot\vec{A}^{*}(\vec{r}\,),
\label{eq:H02}\end{aligned}$$ where $\dag$ stands for Hermitian adjugation, $V(\vec{r}\,)$ is the scalar potential of the field-unperturbed Schr[ö]{}dinger equation, $\vec{p}=-{\rm{i}}\hbar\vec{\nabla}$ denotes the momentum operator, $m_{e}$ is the mass of the electron, and $-e$ is its electric charge.
As often is the practice in optics we assume that the irreversible coupling to the surrounding reservoir can be described using a phenomenological relaxation-time ansatz in the Liouville equation, so that $${1\over{}{\rm{i}}\hbar}\left[{\cal{H}}_{\,\rm{R}},\rho_{nm}\right]=
{\rho_{nm}^{(0)}-\rho_{nm}\over\tau_{nm}},\qquad n\neq{}m,$$ $\rho_{nm}^{(0)}$ being the $nm$’th element of the thermal equilibrium density matrix operator, and $\tau_{nm}$ the associated relaxation time.
In the present harmonic analysis we also use a combined Fourier and power series expansion of the density matrix operator, namely $${\rho}={1\over2}\sum_{\alpha=0}^{\infty}\sum_{\beta=0}^{\infty}\left(
\rho_{-\alpha\omega}^{(\alpha+2\beta)}e^{-{\rm{i}}\alpha\omega{}t}
+\mbox{H.a.}\right),
\label{eq:rho}$$ where $\alpha$ and $\beta$ are integers, as before. The density matrix operator is Hermitian, i.e., $(\rho_{-\alpha\omega}^{(\alpha+2\beta)})^{\dag}=
\rho_{\alpha\omega}^{(\alpha+2\beta)}$, and we solve the Liouville equation of motion in the usual iterative manner.
To determine the conductivity response tensors, $\stensor{\sigma}(\vec{r},\vec{r}\,')$ and $\tensor{\Xi}(\vec{r},\vec{r}\,',\vec{r}\,'',\vec{r}\,''')$, appropriate for describing the phase conjugation process, we consider the ensemble average $\vec{J}(\vec{r},t)$ of the microscopic single-body current-density operator $\vec{j}(\vec{r},t)$. This ensemble average is obtained as the trace of ${\rho}\vec{j}$, carried out in the usual manner as a quantum mechanical double sum over states, i.e., $$\vec{J}(\vec{r},t)=\mbox{Tr}\left\{{\rho}\vec{j}\,\right\}
\equiv\sum_{nm}\rho_{nm}\vec{j}_{mn}.
\label{eq:Trrhoj}$$ In Eq. (\[eq:Trrhoj\]) and hereafter the $ab$’th matrix element of a single-body operator ${\cal{O}}$ as usual is denoted by ${\cal{O}}_{ab}=\langle{a}|{\cal{O}}|{b}\rangle$. In the absence of spin effects the microscopic current-density operator is given by [@Bloembergen:65:1] $$\vec{j}(\vec{r},t)=\vec{j}^{\,(0)}(\vec{r}\,)
+{1\over2}\left(\vec{j}_{-\omega}^{\,(1)}e^{-{\rm{i}}\omega{t}}
+\mbox{H.a.}\right),
\label{eq:jrt}$$ where $$\begin{aligned}
\vec{j}^{\,(0)}(\vec{r}\,)&=&
-{e\over2m_{e}}\Bigl(\vec{p}(\vec{r}_e)\delta(\vec{r}-\vec{r}_e)
+\delta(\vec{r}-\vec{r}_e)\vec{p}(\vec{r}_e)\Bigr)
\label{eq:j(0)r}\\
\vec{j}_{-\omega}^{\,(1)}&=&
-{e^2\over{}m_{e}}\vec{A}(\vec{r}_e)\delta(\vec{r}-\vec{r}_e).
\label{eq:j(1)r}\end{aligned}$$
Linear response
---------------
Because of its usefulness for a subsequent comparison to the forced DFWM current density we first present the well known result for the linear response (, ). Thus, by using the expressions for the current density \[Eq. (\[eq:jrt\])\] and density matrix \[Eq. (\[eq:rho\])\] operators it is realised that the linear current density is to be obtained from $$\vec{J}_{-\omega}^{\,(1)}(\vec{r}\,)=
\mbox{Tr}\left\{\rho^{(0)}\vec{j}^{\,(1)}_{-\omega}\right\}
+\mbox{Tr}\left\{\rho^{(1)}_{-\omega}\vec{j}^{\,(0)}\right\}.
\label{eq:J1Tr}$$ In explicit form the two traces are $$\begin{aligned}
\mbox{Tr}\left\{\rho^{(0)}\vec{j}^{\,(1)}_{-\omega}\right\}&=&
\sum_{n}f_{n}\vec{j}^{\,(1)}_{-\omega,nn}.
\label{eq:J1-1}
\\
\mbox{Tr}\left\{\rho^{(1)}_{-\omega}\vec{j}^{\,(0)}\right\}&=&
\sum_{nm}{f_{n}-f_{m}\over\hbar}{{\cal{H}}^{(1)}_{-\omega,nm}
\over\tilde{\omega}_{nm}-\omega}\vec{j}^{\,(0)}_{mn}.
\label{eq:J1-2}\end{aligned}$$ In the equations above, we have introduced the complex cyclic transition frequency $\tilde{\omega}_{nm}=\omega_{nm}-{\rm{i}}\tau_{nm}^{-1}$ between states $n$ and $m$. The respective energies ${\cal{E}}_{\,n}$ and ${\cal{E}}_{\,m}$ of these states appear in the usual transition frequency $\omega_{nm}=\left({\cal{E}}_{\,n}-{\cal{E}}_{\,m}\right)/\hbar$. The quantity $$f_{a}=\left[1+\exp\left({{\cal{E}}_{\,a}-\mu\over{}k_{B}T}\right)\right]^{-1}
\label{eq:fn}$$ denotes the Fermi–Dirac distribution function for state $a$ ($a\in\{m,n\}$ above), $k_{B}$ being the Boltzmann constant, $\mu$ the chemical potential of the electron system, and $T$ the absolute temperature.
DFWM response
-------------
The nonlinear current density at $-\omega$, which originates in third order effects in the electric field, and which is the driving source for the DFWM process is given by $$\vec{J}_{-\omega}^{\,(3)}(\vec{r}\,)={1\over2}
\mbox{Tr}\left\{{\rho}_{-2\omega}^{(2)}\vec{j}_{\omega}^{\,(1)}\right\}
+\mbox{Tr}\left\{{\rho}_{0}^{(2)}\vec{j}_{-\omega}^{\,(1)}\right\}
+\mbox{Tr}\left\{{\rho}_{-\omega}^{(3)}\vec{j}^{\,(0)}\right\},
\label{eq:J3Tr}$$ as one readily realises from Eqs. (\[eq:J-power\]), (\[eq:rho\]), and (\[eq:jrt\]). The tedious calculation of the three traces can be carried out in a fashion similar to that used for the linear case, finally leading to $$\begin{aligned}
\lefteqn{
{1\over2}\mbox{Tr}\left\{\rho_{-2\omega}^{(2)}\vec{j}_{\omega}^{\,(1)}\right\}
=\sum_{nm}{f_{n}-f_{m}\over2\hbar}
{{\cal{H}}_{-2\omega,nm}^{(2)}\vec{j}_{\omega,mn}^{\,(1)}
\over\tilde{\omega}_{nm}-2\omega}
}\nonumber\\ &\quad&
+\sum_{nmv}\left({f_{m}-f_{v}\over\tilde{\omega}_{v{}m}-\omega}
+{f_{n}-f_{v}\over\tilde{\omega}_{nv}-\omega}\right)
{{\cal{H}}_{-\omega,nv}^{(1)}{\cal{H}}_{-\omega,v{}m}^{(1)}
\vec{j}_{\omega,mn}^{(1)}
\over4\hbar^2(\tilde{\omega}_{nm}-2\omega)},
\label{eq:J3-1} \\
\lefteqn{
\mbox{Tr}\left\{\rho_{0}^{(2)}\vec{j}_{-\omega}^{\,(1)}\right\}
=\sum_{nm}{f_{n}-f_{m}\over\hbar}
{{\cal{H}}_{\,0,nm}^{(2)}\vec{j}_{-\omega,mn}^{\,(1)}
\over\tilde{\omega}_{nm}}
+\sum_{nmv}\left\{
\left({f_{m}-f_{v}\over\tilde{\omega}_{v{}m}-\omega}
+{f_{n}-f_{v}\over\tilde{\omega}_{nv}+\omega}\right)
\right.}\nonumber\\ &&\times\left.\!
{{\cal{H}}_{\,\omega,nv}^{(1)}{\cal{H}}_{-\omega,v{}m}^{(1)}
\vec{j}_{-\omega,mn}^{\,(1)}\over4\hbar^2\tilde{\omega}_{nm}}
+\left({f_{m}-f_{v}\over\tilde{\omega}_{v{}m}+\omega}
+{f_{n}-f_{v}\over\tilde{\omega}_{nv}-\omega}\right)
{{\cal{H}}_{-\omega,nv}^{(1)}{\cal{H}}_{\,\omega,v{}m}^{(1)}
\vec{j}_{-\omega,mn}^{\,(1)}\over4\hbar^2\tilde{\omega}_{nm}}\right\},
\label{eq:J3-2} \\
\lefteqn{
\mbox{Tr}\left\{\rho_{-\omega}^{(3)}\vec{j}^{\,(0)}\right\}
=\sum_{nmv}{1\over2\hbar^2(\tilde\omega_{nm}-\omega)}
\left\{\left({f_{m}-f_{v}\over2(\tilde{\omega}_{v{}m}-2\omega)}
+{f_{n}-f_{v}\over2(\tilde{\omega}_{nv}+\omega)}\right)
{\cal{H}}_{\,\omega,nv}^{(1)}{\cal{H}}_{-2\omega,v{}m}^{(2)}
\right.}\nonumber\\ &\quad&
+\left({f_{n}-f_{v}\over2(\tilde{\omega}_{nv}-2\omega)}
+{f_{m}-f_{v}\over2(\tilde{\omega}_{v{}m}+\omega)}\right)
{\cal{H}}_{-2\omega,nv}^{(2)}{\cal{H}}_{\,\omega,v{}m}^{(1)}
+\left({f_{m}-f_{v}\over\tilde{\omega}_{v{}m}}
+{f_{n}-f_{v}\over\tilde{\omega}_{nv}-\omega}\right)
\nonumber\\ &&\times\left.\!
{\cal{H}}_{-\omega,nv}^{(1)}{\cal{H}}_{\,0,v{}m}^{(2)}
+\left({f_{n}-f_{v}\over\tilde{\omega}_{nv}}
+{f_{m}-f_{v}\over\tilde{\omega}_{v{}m}-\omega}\right)
{\cal{H}}_{\,0,nv}^{(2)}{\cal{H}}_{-\omega,v{}m}^{(1)}
\right\}\vec{j}_{mn}^{\,(0)}
\nonumber\\ &&
+\sum_{nmvl}{1\over2\hbar(\tilde\omega_{nm}-\omega)}
\left\{
\left[
\left({f_{l}-f_{m}\over\tilde{\omega}_{l{}m}-\omega}
+{f_{l}-f_{v}\over\tilde{\omega}_{v{}l}-\omega}\right)
{1\over4\hbar^2(\tilde{\omega}_{v{}m}-2\omega)}
\right.\right.\nonumber\\ &&\left.
+\left({f_{l}-f_{v}\over\tilde{\omega}_{vl}-\omega}
+{f_{n}-f_{v}\over\tilde{\omega}_{nv}+\omega}\right)
{1\over4\hbar^2\tilde{\omega}_{nl}}
\right]
{\cal{H}}_{\,\omega,nv}^{(1)}{\cal{H}}_{-\omega,v{}l}^{(1)}
{\cal{H}}_{-\omega,l{}m}^{(1)}
\nonumber\\ &&
+\left[
\left({f_{l}-f_{m}\over\tilde{\omega}_{l{}m}-\omega}
+{f_{l}-f_{v}\over\tilde{\omega}_{v{}l}+\omega}\right)
{1\over4\hbar^2\tilde{\omega}_{v{}m}}
+\left({f_{l}-f_{v}\over\tilde{\omega}_{vl}+\omega}
+{f_{n}-f_{v}\over\tilde{\omega}_{nv}-\omega}\right)
{1\over4\hbar^2\tilde{\omega}_{nl}}
\right]
\nonumber\\ &&\times
{\cal{H}}_{-\omega,nv}^{(1)}{\cal{H}}_{\,\omega,v{}l}^{(1)}
{\cal{H}}_{-\omega,l{}m}^{(1)}
+\left[
\left({f_{l}-f_{m}\over\tilde{\omega}_{l{}m}+\omega}
+{f_{l}-f_{v}\over\tilde{\omega}_{v{}l}-\omega}\right)
{1\over4\hbar^2\tilde{\omega}_{v{}m}}
\right.\nonumber\\ &&\left.\!\left.\!
+\left({f_{l}-f_{v}\over\tilde{\omega}_{vl}-\omega}
+{f_{n}-f_{v}\over\tilde{\omega}_{nv}-\omega}\right)
{1\over4\hbar^2(\tilde{\omega}_{nl}-2\omega)}
\right]
{\cal{H}}_{-\omega,nv}^{(1)}{\cal{H}}_{-\omega,v{}l}^{(1)}
{\cal{H}}_{\,\omega,l{}m}^{(1)}
\right\}\vec{j}_{mn}^{\,(0)}.
\label{eq:J3-3}\end{aligned}$$ Though quite complicated in its appearence the expression for the driving current density of the DFWM process is needed in order to understand the near-field phase conjugation process from a general point of view.[^1] Special scattering configurations of course can lead to analytical simplifications of the general result. The above result also enables us to establish a microscopic theory for DFWM in quantum wells and thin films as described in Parts \[part:III\], \[part:IV\], and \[part:V\].
Physical processes underlying the current densities
---------------------------------------------------
To gain insight into the physics underlying the nonlinear constitutive equation, given implicitly in Eqs. (\[eq:J3-1\])–(\[eq:J3-3\]), we next discuss the processes connecting in a nonlocal fashion the current density at a given point in space to the field points of the surroundings. To facilitate the understanding of the nonlinear response we start by a brief summary of the linear response.
### Linear part
The electrodynamic coupling connecting a source point for the field to an observation point for the current density associated to each of the two linear processes underlying Eqs. (\[eq:J1-1\]) and (\[eq:J1-2\]) is adequately illustrated in diagrammatic form as shown in Fig. \[fig:2Tr\].
(0,0)(80,30) (0,0)[ (15,25)[(0,5)\[br\][(A)]{}]{} (15,15)[7.07]{}[-30]{}[330]{} (10,10)[1.5]{} (7,7)[(0,0)\[c\][$\vec{r}$]{}]{} ]{} (40,0)[ (15,25)[(0,5)\[br\][(B)]{}]{} (25,25)[1.5]{} (5,5)[1.5]{} (5,5)(13,17)(25,25)(25,25) (5,5)(5,5)(13,17)(25,25) (2,2)[(0,0)\[c\][$\vec{r}\,'$]{}]{} (28,28)[(0,0)\[c\][$\vec{r}$]{}]{} ]{}
Hence, Fig. \[fig:2Tr\].A represents a picture of the well known diamagnetic process originating in the quantity $\vec{j}^{\,(1)}_{-\omega,nn}$ in Eq. (\[eq:J1-1\]). In this process, a photon is absorbed at the observation point $\vec{r}$ for the current density. Fig. \[fig:2Tr\].B is a picture of the paramagnetic process stemming from the term ${\cal{H}}^{(1)}_{-\omega,nm}$ appearing in Eq. (\[eq:J1-2\]). In this case, a photon is absorbed at space point $\vec{r}\,'$, and observation takes place at $\vec{r}$.
### Nonlinear part
In the DFWM process, the coupling between the three source points for the field and the observation point for the current density, described in Eqs. (\[eq:J3-1\])–(\[eq:J3-3\]), can be pictured in diagrammatic form as shown in Fig. \[fig:3Tr\].
(-4,0)(170,80) (0,40)[ (15,25)[(0,5)\[br\][(A)]{}]{} (30,30)[7.07]{}[-30]{}[330]{} (25,25)[1.5]{} (5,5)[1.5]{} (6,5)(6,5)(17,13)(26,25) (6,5)(6,5)(17,13)(26,25) (5,6)(5,6)(13,17)(25,26) (5,6)(5,6)(13,17)(25,26) (1.5,2)[(0,0)\[c\][$\vec{r}\,''$]{}]{} (28,28)[(0,0)\[c\][$\vec{r}$]{}]{} ]{} (39,40)[ (15,25)[(0,5)\[br\][(B)]{}]{} (30,30)[7.07]{}[-30]{}[330]{} (25,25)[1.5]{} (5,5)[1.5]{} (38,5)[1.5]{} (5,5)(5,5)(14,16)(25,25) (5,5)(5,5)(14,16)(25,25) (38,5)(38,5)(29,15)(25,25) (38,5)(38,5)(29,15)(25,25) (1.5,2)[(0,0)\[c\][$\vec{r}\,''$]{}]{} (28,28)[(0,0)\[c\][$\vec{r}$]{}]{} (34,2)[(0,0)\[c\][$\vec{r}\,'''$]{}]{} ]{} (88,40)[ (15,25)[(0,5)\[br\][(C)]{}]{} (30,30)[7.07]{}[-30]{}[330]{} (25,25)[1.5]{} (5,5)[1.5]{} (6,5)(17,13)(26,25)(26,25) (6,5)(6,5)(17,13)(26,25) (5,6)(5,6)(13,17)(25,26) (5,6)(5,6)(13,17)(25,26) (2,2)[(0,0)\[c\][$\vec{r}\,'$]{}]{} (28,28)[(0,0)\[c\][$\vec{r}$]{}]{} ]{} (127,40)[ (15,25)[(0,5)\[br\][(D)]{}]{} (30,30)[7.07]{}[-30]{}[330]{} (25,25)[1.5]{} (5,5)[1.5]{} (38,5)[1.5]{} (5,5)(5,5)(14,16)(25,25) (5,5)(5,5)(14,16)(25,25) (38,5)(29,15)(25,25)(25,25) (38,5)(38,5)(29,15)(25,25) (2,2)[(0,0)\[c\][$\vec{r}\,'$]{}]{} (28,28)[(0,0)\[c\][$\vec{r}$]{}]{} (34.5,2)[(0,0)\[c\][$\vec{r}\,''$]{}]{} ]{} (0,0)[ (15,25)[(0,5)\[br\][(E)]{}]{} (25,25)[1.5]{} (5,5)[1.5]{} (45,5)[1.5]{} (6,5)(6,5)(17,13)(26,25) (6,5)(6,5)(17,13)(26,25) (5,6)(5,6)(13,17)(25,26) (5,6)(5,6)(13,17)(25,26) (45,5)(34,14)(25,25)(25,25) (45,5)(45,5)(34,14)(25,25) (1.5,2)[(0,0)\[c\][$\vec{r}\,''$]{}]{} (28,28)[(0,0)\[c\][$\vec{r}$]{}]{} (42,2)[(0,0)\[c\][$\vec{r}\,'$]{}]{} ]{} (60,0)[ (15,25)[(0,5)\[br\][(F)]{}]{} (25,25)[1.5]{} (5,5)[1.5]{} (45,5)[1.5]{} (6,5)(17,13)(26,25)(26,25) (6,5)(6,5)(17,13)(26,25) (5,6)(5,6)(13,17)(25,26) (5,6)(5,6)(13,17)(25,26) (45,5)(45,5)(34,14)(25,25) (45,5)(45,5)(34,14)(25,25) (2,2)[(0,0)\[c\][$\vec{r}\,'$]{}]{} (28,28)[(0,0)\[c\][$\vec{r}$]{}]{} (41.5,2)[(0,0)\[c\][$\vec{r}\,''$]{}]{} ]{} (120,0)[ (15,25)[(0,5)\[br\][(G)]{}]{} (25,30)[1.5]{} (5,10)[1.5]{} (45,10)[1.5]{} (25,5)[1.5]{} (25,5)(25,5)(26,18)(25,30) (25,5)(25,5)(26,18)(25,30) (5,10)(5,10)(13,22)(25,30) (5,10)(5,10)(13,22)(25,30) (45,10)(37,22)(25,30)(25,30) (45,10)(45,10)(37,22)(25,30) (2,7)[(0,0)\[c\][$\vec{r}\,'$]{}]{} (21.5,2)[(0,0)\[c\][$\vec{r}\,''$]{}]{} (41,7)[(0,0)\[c\][$\vec{r}\,'''$]{}]{} (28,32)[(0,0)\[c\][$\vec{r}$]{}]{} ]{}
Hence, in Fig. \[fig:3Tr\].A the mixing process contained in the product ${\cal{H}}_{-2\omega,nm}^{(2)} \vec{j}_{\omega,mn}^{\,(1)}$ in Eq. (\[eq:J3-1\]) is illustrated. Here, two photons are [ *simultaneously*]{} absorbed at space point $\vec{r}\,''$, and one photon is emitted at the point of observation $\vec{r}$ for the current density. Fig. \[fig:3Tr\].B pictures the other mixing process in Eq. (\[eq:J3-1\]), namely that associated with the product ${\cal{H}}_{-\omega,nv}^{(1)}
{\cal{H}}_{-\omega,v{}m}^{(1)}\vec{j}_{\omega,mn}^{\,(1)}$. In this process, one photon is absorbed at $\vec{r}\,''$, another at $\vec{r}\,'''$, and the last one is emitted at $\vec{r}$. Fig. \[fig:3Tr\].C gives a view of the mixing process from the terms in Eq. (\[eq:J3-2\]) containing the product ${\cal{H}}_{\,0,nm}^{(2)} \vec{j}_{-\omega,mn}^{\,(1)}$. In this case a photon is absorbed and another is emitted [*simultaneously*]{} at space point $\vec{r}\,'$, and the third photon is absorbed at $\vec{r}$. Fig. \[fig:3Tr\].D shows the other type of mixing process occuring in Eq. (\[eq:J3-2\]). This process is described by the products ${\cal{H}}_{\,\omega,nv}^{(1)}
{\cal{H}}_{-\omega,v{}m}^{(1)}\vec{j}_{-\omega,mn}^{\,(1)}$ and ${\cal{H}}_{-\omega,nv}^{(1)}{\cal{H}}_{\,\omega,v{}m}^{(1)}
\vec{j}_{-\omega,mn}^{\,(1)}$. Here, photons are absorbed at $\vec{r}\,'$ and at the point of observation $\vec{r}$, while a photon is emitted at $\vec{r}\,''$. Fig. \[fig:3Tr\].E represents the diagram for the mixing process appearing in the terms containing the product ${\cal{H}}_{\,\omega,nv}^{(1)}
{\cal{H}}_{-2\omega,v{}m}^{(2)}$ (and the equivalent product ${\cal{H}}_{-2\omega,v{}m}^{(2)}{\cal{H}}_{\,\omega,nv}^{(1)}$) in Eq. (\[eq:J3-3\]). Here two photons are [*simultanously*]{} absorbed at $\vec{r}\,''$, and one is emitted at $\vec{r}\,'$. Fig. \[fig:3Tr\].F is the diagrammatic representation of the terms containing the product ${\cal{H}}_{\,0,nv}^{(2)}
{\cal{H}}_{-\omega,v{}m}^{(1)}$ (and the equivalent one ${\cal{H}}_{-\omega,nv}^{(1)}{\cal{H}}_{\,0,v{}m}^{(2)}$) in Eq. (\[eq:J3-3\]). In these terms a photon is absorbed at $\vec{r}\,'$ and at the [*same time*]{} one is emitted from there. The last photon is absorbed at $\vec{r}\,''$. Finally, Fig. \[fig:3Tr\].G gives a picture of one of six equivalent products of the last type appearing in Eq. (\[eq:J3-3\]). These are of the form ${\cal{H}}_{\,\omega,nv}^{(1)}
{\cal{H}}_{-\omega,vl}^{(1)}{\cal{H}}_{-\omega,l{}m}^{(1)}$ or equivalent forms (all six possible permutations of one “$\omega$”-term and two “$-\omega$”-terms). Here, a photon is absorbed at $\vec{r}\,'$, another at $\vec{r}\,''$, and the last photon is emitted at $\vec{r}\,'''$.
At this stage it is fruitful to compare the nonlocal result for the DFWM current density, shown in diagrammatic form in Fig. \[fig:3Tr\], with the commonly used standard (textbook) result. In the standard description all diamagnetic effects are neglected from the outset. The diamagnetic process is hidden in the diagrams containing a closed loop, cf. Fig. \[fig:2Tr\]. This means that all the processes depicted in Figs. \[fig:3Tr\].A–\[fig:3Tr\].D are absent in the standard description. Omission of diamagnetic effects in the nonlinear optics of quantum wells and in mesoscopic near-field optics is known to be dangerous [@Keller:96:1], and thus we [ *cannot*]{} omit these terms here. We shall substantiate on this point later. Also the interaction channels given by the diagrams in Figs. \[fig:3Tr\].E and \[fig:3Tr\].F are absent in textbook formulations. This is so because simultaneous two-photon processes originating in the $\vec{A}\cdot\vec{A}$ part of the interaction Hamiltonian are left out from the beginning. These processes however are known to be important in mesoscopic electrodynamics and can not be omitted [*a priori*]{}.\[Local\] In the local limit the result given by the diagram in Fig. \[fig:3Tr\].G is identical to the $\vec{r}\cdot\vec{E}$ dipolar interaction Hamiltonian, since a unitary transformation of the form $S=\exp(-{\rm{i}}e\vec{A}(t)\cdot\vec{r}/\hbar)$ performed on the wave functions and the minimal coupling Hamiltonian would display the equivalence of the two formalisms [@Ackerhalt:84:1; @Milonni:89:1].
Conductivity tensors for DFWM response {#ch:6}
======================================
In the preceding two chapters, we have found expressions for the phase conjugated field and the single-electron current density response. In the present chapter the connection between the single-electron current densities \[Eqs. (\[eq:J1Tr\]) and (\[eq:J3Tr\])\] and their related conductivity tensors \[Eqs. (\[eq:Jr\]) and (\[eq:J3r\])\] is established. First, the matrix elements of the Hamiltonian and the current density operator are written in terms of the vector potential. Then the symmetries of the various contributions to the conductivity tensors are studied. Finally, the expressions for the nonzero and independent elements of the conductivity tensors are written on explicit form.
General considerations
----------------------
In order to determine (i) the linear conductivity response tensor $\stensor{\sigma}(\vec{r},\vec{r}\,')$ introduced in Eq. (\[eq:Jr\]) from the expression for the linear current density in Eq. (\[eq:J1Tr\]) \[with insertion of Eqs. (\[eq:J1-1\]) and (\[eq:J1-2\])\], and (ii) the nonlinear conductivity response function $\tensor{\Xi}(\vec{r},\vec{r}\,',\vec{r}\,'',\vec{r}\,''')$ introduced in Eq. (\[eq:J3r\]) from the expression for the DFWM current density given in Eq. (\[eq:J3Tr\]) \[with Eqs. (\[eq:J3-1\])–(\[eq:J3-3\]) inserted\] we by now essentially just need to relate the various matrix elements appearing in Eqs. (\[eq:J1-1\]), (\[eq:J1-2\]), and (\[eq:J3-1\])–(\[eq:J3-3\]) to the vector potential.
Taking the $nm$ matrix element of the “$-\omega$” part of the part of the Hamiltonian which is linear in the vector potential one finds on integral form $${\cal{H}}^{(1)}_{-\omega,nm}=\left({\cal{H}}^{(1)}_{\omega,mn}\right)^{*}=
-\int\vec{J}_{mn}(\vec{r}\,)\cdot\vec{A}(\vec{r}\,)d^3r,
\label{eq:H-w1nm}\label{eq:Hw1nm}$$ where we have introduced the transition current density [*from*]{} state $m$ [*to*]{} state $n$, i.e., $\vec{j}^{\,(0)}_{nm}\equiv\vec{J}_{mn}$, in its explicit form, viz. $$\vec{J}_{mn}(\vec{r}\,)={e\hbar\over2{\rm{i}}m_{e}}
\left(\psi_{m}(\vec{r}\,)\vec{\nabla}\psi_{n}^{*}(\vec{r}\,)
-\psi_{n}^{*}(\vec{r}\,)\vec{\nabla}\psi_{m}(\vec{r}\,)\right),
\label{eq:Jm->n}$$ $\psi_{a}$ ($a\in\{m,n\}$) being the electronic eigenstate satisfying the unperturbed Schr[ö]{}dinger equation ${\cal{H}}_{\,0}\psi_{a}=
{\cal{E}}_{\,a}\psi_{a}$. From Eq. (\[eq:Jm->n\]) we note that $\vec{J}_{nm}(\vec{r}\,)=\vec{J}_{mn}^{*}(\vec{r}\,)$. Similarly, the $nm$ matrix elements of the “$-2\omega$” part of the Hamiltonian becomes $${\cal{H}}^{(2)}_{-2\omega,nm}
={e^2\over4m_{e}}\int\psi_{n}^{*}(\vec{r}\,)\psi_{m}(\vec{r}\,)
\vec{A}(\vec{r}\,)\cdot\vec{A}(\vec{r}\,)d^3r
\label{eq:H-2w2nm}$$ on integral form. Next, the matrix elements of the part of the Hamiltonian which is proportional to $\vec{A}\cdot\vec{A}^{*}$ are given by $${\cal{H}}^{(2)}_{\,0,nm}={e^2\over4m_{e}}
\int\psi_{n}^{*}(\vec{r}\,)\psi_{m}(\vec{r}\,)\vec{A}(\vec{r}\,)
\cdot\vec{A}^{*}(\vec{r}\,)d^3r.
\label{eq:H02nm}$$ Finally, the matrix elements of the current density operator $\vec{j}^{\,(1)}$ are found to be $$\vec{j}_{-\omega,nm}^{\,(1)}=\vec{j}_{\omega,mn}^{\,(1)*}=
-{e^2\over{}m_{e}}\psi_{n}^{*}(\vec{r}\,)\psi_{m}(\vec{r}\,)
\vec{A}(\vec{r}\,).
\label{eq:j-w1}\label{eq:jw1}$$
The calculation of the DFWM conductivity tensor is finalized in two steps. Thus we start by inserting Eqs. (\[eq:H-w1nm\])–(\[eq:j-w1\]) into the three traces in Eqs. (\[eq:J3-1\])–(\[eq:J3-3\]), and thereafter we extract the vector potential in such a manner that the result takes the general form given in Eq. (\[eq:J3r\]). For convenience, we in the following divide the nonlinear conductivity tensor into a sum of subparts A–G referring to the processes (A)–(G) shown in Fig. \[fig:3Tr\].
Since we are using the linear response function in the description of the phase conjugated field it is adequate for consistency again to describe the linear process, although it is already well known. The calculation is done in a similar manner as for the DFWM response, by inserting Eqs. (\[eq:H-w1nm\]), (\[eq:Jm->n\]) and (\[eq:j-w1\]) into the two traces in Eqs. (\[eq:J1-1\]) and (\[eq:J1-2\]), and thereafter isolating the vector potential so that the result takes the form of Eq. (\[eq:Jr\]). In the following the linear conductivity tensor is divided into a sum of subparts A–B referring to the two processes shown in Fig. \[fig:2Tr\].
Symmetry properties of the conductivity tensors
-----------------------------------------------
In order to study the symmetries of the various contributions to the conductivity tensors $\sigma(\vec{r},\vec{r}\,')$ and $\tensor{\Xi}(\vec{r},\vec{r}\,',\vec{r}\,'',\vec{r}\,''')$ one notices that the vector potential only appears via ${\cal{H}}^{(1)}_{-\omega}$, ${\cal{H}}^{(1)}_{\,\omega}$, ${\cal{H}}^{(2)}_{-2\omega}$, ${\cal{H}}^{(2)}_{\,0}$, ${\vec{j}}^{\,(1)}_{-\omega}$, and ${\vec{j}}^{\,(1)}_{\omega}$ of Eqs. (\[eq:H-w1nm\]) and (\[eq:H-2w2nm\])–(\[eq:j-w1\]). One further observes from Eq. (\[eq:H-w1nm\]) that the matrix elements of ${\cal{H}}^{(1)}_{-\omega}$ and ${\cal{H}}^{(1)}_{\omega}$ contain inner products between a transition current density and a vector potential and that those of ${\cal{H}}^{(2)}_{-2\omega}$ and ${\cal{H}}^{(2)}_{\,0}$ involve inner products between two vector potentials, see Eqs. (\[eq:H-2w2nm\]) and (\[eq:H02nm\]). These last inner products may conveniently be written in the form $\tensor{\openone}:\vec{A}\vec{A}$ and $\tensor{\openone}:\vec{A}\vec{A}^{*}$, respectively. The matrix elements of the current densities $\vec{j}_{-\omega}^{\,(1)}$ and $\vec{j}_{\omega}^{\,(1)}$ are directly proportional to the vector potential and may thus for the present purpose adequately be written in the forms $\tensor{\openone}\cdot\vec{A}$ and $\tensor{\openone}\cdot\vec{A}^{*}$, respectively.
### Linear conductivity tensor
In the view of the aforementioned remarks it is concluded that part A of the linear conductivity tensor, given by Eq. (\[eq:J1-1\]) has the symmetry of the unit tensor $\tensor{\openone}$. Part A thus have $3$ nonzero elements. Furthermore, of these $3$ nonzero elements only $1$ is independent, since the Cartesian index of the linear current density follows that of the vector potential appearing in $\vec{j}_{\omega}^{\,(1)}$, and thus $i=j$.
(0,4)(70,29) (2,0)(40,0)[2]{}[ (6,6)(4,6)(4,22)(6,22) (22,6)(24,6)(24,22)(22,22) [ (0,22.5)[(0,5)\[l\][$[ij]$]{}]{} ]{} ]{} (10,8)(40,0)[2]{}[ (0,0)(6,0)[3]{}[(0,0)(0,6)[3]{}[ ]{}]{}]{} (27,6)[(0,5)\[bl\][(A)]{}]{} (45,6)[(0,5)\[br\][(B)]{}]{} (10,20) (16,14) (22,8) (50,8)(6,0)[3]{}[(0,0)(0,6)[3]{}]{} (10,20)(22,8)
The other part of the linear conductivity tensor (part B) is extracted from Eq. (\[eq:J1-2\]), and it shows a symmetry to the outer product $\vec{J}_1\otimes\vec{J}_2$, where $\vec{J}_1$ and $\vec{J}_2$ in general are different, and part B of the linear conductivity tensor thus has $9$ independent nonzero elements. The symmetry schemes of the linear conductivity tensor are shown in Fig. \[fig:Sigma\].
(0,4)(170,240) (0,160)[ (75,6)[(10,5)\[bl\][(A)]{}]{} (2,0)(24,0)[3]{}[(0,0)(0,24)[3]{}[ (6,6)(4,6)(4,22)(6,22) (22,6)(24,6)(24,22)(22,22) ]{}]{} (10,8)(24,0)[3]{}[(0,0)(0,24)[3]{}[ (0,0)(6,0)[3]{}[(0,0)(0,6)[3]{}[ ]{}]{}]{}]{} [ (2,70)[(0,5)\[l\][$[xxkh]$]{}]{} (2,46)[(0,5)\[l\][$[yxkh]$]{}]{} (2,22)[(0,5)\[l\][$[zxkh]$]{}]{} (40,70)[(0,5)\[c\][$[xykh]$]{}]{} (40,46)[(0,5)\[c\][$[yykh]$]{}]{} (40,22)[(0,5)\[c\][$[zykh]$]{}]{} (78,70)[(0,5)\[r\][$[xzkh]$]{}]{} (78,46)[(0,5)\[r\][$[yzkh]$]{}]{} (78,22)[(0,5)\[r\][$[zzkh]$]{}]{} ]{} (10,68) (16,62) (22,56) (34,44) (40,38) (46,32) (58,20) (64,14) (70,8) (10,68)(70,8) ]{} (90,160)[ (5,6)[(0,5)\[br\][(B)]{}]{} (2,0)(24,0)[3]{}[(0,0)(0,24)[3]{}[ (6,6)(4,6)(4,22)(6,22) (22,6)(24,6)(24,22)(22,22) ]{}]{} (10,8)(24,0)[3]{}[(0,0)(0,24)[3]{}[ (0,0)(6,0)[3]{}[(0,0)(0,6)[3]{}[ ]{}]{}]{}]{} [ (2,70)[(0,5)\[l\][$[xxkh]$]{}]{} (2,46)[(0,5)\[l\][$[yxkh]$]{}]{} (2,22)[(0,5)\[l\][$[zxkh]$]{}]{} (40,70)[(0,5)\[c\][$[xykh]$]{}]{} (52,46)[(0,5)\[r\][$[yykh]$]{}]{} (28,22)[(0,5)\[l\][$[zykh]$]{}]{} (78,70)[(0,5)\[r\][$[xzkh]$]{}]{} (78,46)[(0,5)\[r\][$[yzkh]$]{}]{} (78,22)[(0,5)\[r\][$[zzkh]$]{}]{} ]{} (10,56)(0,6)[3]{}[(0,0)(6,0)[3]{}]{} (34,32)(0,6)[3]{}[(0,0)(6,0)[3]{}]{} (58,8)(0,6)[3]{}[(0,0)(6,0)[3]{}]{} (10,56)(20,42)(34,32) (34,32)(48,22)(58,8) (10,62)(20,48)(34,38) (34,38)(48,28)(58,14) (10,68)(20,54)(34,44) (34,44)(48,34)(58,20) (16,56)(26,42)(40,32) (40,32)(54,22)(64,8) (16,62)(26,48)(40,38) (40,38)(54,28)(64,14) (16,68)(26,54)(40,44) (40,44)(54,34)(64,20) (22,56)(32,42)(46,32) (46,32)(60,22)(70,8) (22,62)(32,48)(46,38) (46,38)(60,28)(70,14) (22,68)(32,54)(46,44) (46,44)(60,34)(70,20) ]{} (0,80)[ (75,6)[(10,5)\[bl\][(C)]{}]{} (2,0)(24,0)[3]{}[(0,0)(0,24)[3]{}[ (6,6)(4,6)(4,22)(6,22) (22,6)(24,6)(24,22)(22,22) ]{}]{} (10,8)(24,0)[3]{}[(0,0)(0,24)[3]{}[ (0,0)(6,0)[3]{}[(0,0)(0,6)[3]{}[ ]{}]{}]{}]{} [ (2,70)[(0,5)\[l\][$[xxkh]$]{}]{} (2,46)[(0,5)\[l\][$[yxkh]$]{}]{} (2,22)[(0,5)\[l\][$[zxkh]$]{}]{} (40,70)[(0,5)\[c\][$[xykh]$]{}]{} (40,46)[(0,5)\[c\][$[yykh]$]{}]{} (40,22)[(0,5)\[c\][$[zykh]$]{}]{} (78,70)[(0,5)\[r\][$[xzkh]$]{}]{} (78,46)[(0,5)\[r\][$[yzkh]$]{}]{} (78,22)[(0,5)\[r\][$[zzkh]$]{}]{} ]{} (10,68) (34,62) (58,56) (16,44) (40,38) (64,32) (22,20) (46,14) (70,8) (10,68)(58,56) (16,44)(64,32) (22,20)(70,8) (10,68)(22,20) ]{} (90,80)[ (5,6)[(0,5)\[br\][(D)]{}]{} (2,0)(24,0)[3]{}[(0,0)(0,24)[3]{}[ (6,6)(4,6)(4,22)(6,22) (22,6)(24,6)(24,22)(22,22) ]{}]{} (10,8)(24,0)[3]{}[(0,0)(0,24)[3]{}[ (0,0)(6,0)[3]{}[(0,0)(0,6)[3]{}[ ]{}]{}]{}]{} [ (2,70)[(0,5)\[l\][$[xxkh]$]{}]{} (1,46)[(0,5)\[l\][$[yxkh]$]{}]{} (2,22)[(0,5)\[l\][$[zxkh]$]{}]{} (40,70)[(0,5)\[c\][$[xykh]$]{}]{} (50,46)[(0,5)\[r\][$[yykh]$]{}]{} (30,22)[(0,5)\[l\][$[zykh]$]{}]{} (78,70)[(0,5)\[r\][$[xzkh]$]{}]{} (78,46)[(0,5)\[r\][$[yzkh]$]{}]{} (79,22)[(0,5)\[r\][$[zzkh]$]{}]{} ]{} (10,68) (10,62) (10,56) (34,68) (34,62) (34,56) (58,68) (58,62) (58,56) (16,44) (16,38) (16,32) (40,44) (40,38) (40,32) (64,44) (64,38) (64,32) (22,20) (22,14) (22,8) (46,20) (46,14) (46,8) (70,20) (70,14) (70,8) (10,68)(24,0)[3]{}[ (0,0)(0,-6)[3]{}[ (0,0)(12,-48) ]{}]{} ]{} (0,0)[ (75,6)[(10,5)\[bl\][(E)]{}]{} (2,0)(24,0)[3]{}[(0,0)(0,24)[3]{}[ (6,6)(4,6)(4,22)(6,22) (22,6)(24,6)(24,22)(22,22) ]{}]{} (10,8)(24,0)[3]{}[(0,0)(0,24)[3]{}[ (0,0)(6,0)[3]{}[(0,0)(0,6)[3]{}[ ]{}]{}]{}]{} [ (2,70)[(0,5)\[l\][$[xxkh]$]{}]{} (2,46)[(0,5)\[l\][$[yxkh]$]{}]{} (2,22)[(0,5)\[l\][$[zxkh]$]{}]{} (40,70)[(0,5)\[c\][$[xykh]$]{}]{} (40,46)[(0,5)\[c\][$[yykh]$]{}]{} (40,22)[(0,5)\[c\][$[zykh]$]{}]{} (78,70)[(0,5)\[r\][$[xzkh]$]{}]{} (78,46)[(0,5)\[r\][$[yzkh]$]{}]{} (78,22)[(0,5)\[r\][$[zzkh]$]{}]{} ]{} (10,68) (16,62) (22,56) (34,68) (40,62) (46,56) (58,68) (64,62) (70,56) (10,44) (16,38) (22,32) (34,44) (40,38) (46,32) (58,44) (64,38) (70,32) (10,20) (16,14) (22,8) (34,20) (40,14) (46,8) (58,20) (64,14) (70,8) (10,68)(24,0)[3]{}[ (0,0)(0,-24)[3]{}[ (0,0)(12,-12) ]{}]{} ]{} (90,0)[ (5,6)[(0,5)\[br\][(F)]{}]{} (2,0)(24,0)[3]{}[(0,0)(0,24)[3]{}[ (6,6)(4,6)(4,22)(6,22) (22,6)(24,6)(24,22)(22,22) ]{}]{} (10,8)(24,0)[3]{}[ (0,0)(0,24)[3]{}[ (0,0)(6,0)[3]{}[ (0,0)(0,6)[3]{}[ ]{}]{}]{}]{} [ (2,70)[(0,5)\[l\][$[xxkh]$]{}]{} (2,46)[(0,5)\[l\][$[yxkh]$]{}]{} (2,22)[(0,5)\[l\][$[zxkh]$]{}]{} (40,70)[(0,5)\[c\][$[xykh]$]{}]{} (40,46)[(0,5)\[c\][$[yykh]$]{}]{} (40,22)[(0,5)\[c\][$[zykh]$]{}]{} (78,70)[(0,5)\[r\][$[xzkh]$]{}]{} (78,46)[(0,5)\[r\][$[yzkh]$]{}]{} (78,22)[(0,5)\[r\][$[zzkh]$]{}]{} ]{} (10,20)(0,24)[3]{}[ (0,0)(24,-6)[3]{}[ (0,0)(6,0)[3]{}[ ]{}]{}]{} (10,68)(0,-24)[3]{}[ (0,0)(6,0)[3]{}[ (0,0)(48,-12) ]{}]{} ]{}
### DFWM conductivity tensor
Taking our symmetry analysis to the DFWM conductivity tensor we conclude that part A, given by the first sum on the right hand side of Eq. (\[eq:J3-1\]), has a symmetry given by the outer product $\tensor{\openone}\otimes\tensor{\openone}$. Part A thus have $9$ nonzero elements. Furthermore, since the Cartesian index of the DFWM current density follows that of the vector potential appearing in $\vec{j}_{\omega}^{\,(1)}$, $i=j$ in the index notation of Eq. (\[eq:sumprod3\]). From the form of the ${\cal{H}}^{(2)}_{-2\omega}$ term we next conclude that $k=h$. Altogether it is realised that the $9$ nonzero elements are identical. To get an overview of the conclusion, we show in Fig. \[fig:6.2\].A the result in terms of a symmetry scheme.
Utilising the same type of arguments it is concluded that each term in the second sum in Eq. (\[eq:J3-1\]), which gives rise to part B of the conductivity tensor, when written in the form of Eq. (\[eq:sumprod3\]) has a symmetry identical to the outer product $\tensor{\openone}\otimes\vec{J}_{1}\otimes\vec{J}_{2}$, where $\vec{J}_{1}$ and $\vec{J}_{2}$ are two generally different transition current densities. The form of this outer product leaves us with $27$ nonzero elements. Also here the coordinate convention of Eq. (\[eq:sumprod3\]) implies that $i=j$. Furthermore we observe that elements with $i=x$, $i=y$, and $i=z$ are identical, since the two ${\cal{H}}^{(1)}_{-\omega}$ terms essentially produces numbers. Finally, we see that the independent nature of the two ${\cal{H}}^{(1)}_{-\omega}$ terms makes them interchangeable, and thus gives us two different ways of constructing the sum in Eq. (\[eq:sumprod3\]). Of the $27$ nonzero elements only $9$ are independent, since as we have realised, $\Xi_{xxkh}^{\rm{B}}=\Xi_{yykh}^{\rm{B}}=\Xi_{zzkh}^{\rm{B}}$ for all permutations of $k$ and $h$ in the three Cartesian coordinates $\{x,y,z\}$. Expressed in terms of a symmetry scheme, the deductions above lead to the symmetry scheme shown in Fig. \[fig:6.2\].B.
The first sum in Eq. (\[eq:J3-2\]) gives rise to part C of the DFWM conductivity tensor, and the second sum in this equation leads to part D. Looking at the first sum it appears that this is proportional to $(\tensor{\openone}:\vec{A}\vec{A}^{*})\vec{A}$, a fact which in relation to the form given in Eq. (\[eq:sumprod3\]) implies that the symmetry of the conductivity tensor is given by the outer product $\vec{e}_{A}\otimes\tensor{\openone}\otimes\vec{e}_{A}$, with $\vec{e}_{A}=\vec{A}/A$. This product form leaves us with $9$ nonzero elements. As far as the Cartesian indices are concerned the above symmetry implies that $j=h$ and $i=k$. Finally we observe that the same constant appears in front of the vector potential indexed $k$. This leads to the conclusion that the cases $i=x$, $i=y$, and $i=z$ are equal, leaving at the end only one independent nonzero element of part C of the conductivity tensor. The symmetry scheme for part C is shown in Fig. \[fig:6.2\].C.
In the second sum of Eq. (\[eq:J3-2\]) the symmetry is proportional to the outer product $\vec{e}_{A}\otimes\vec{J}_{1}\otimes\vec{J}_{2}\otimes\vec{e}_{A}$ and thus we are left with $27$ nonzero elements. Then, in the form of Eq. (\[eq:sumprod3\]), $i=h$, and the permutations over $i$ are seen to be equal, so that we end up with only $9$ independent elements. Using the fact that $\Xi_{xjkx}^{\rm{D}}=\Xi_{yjky}^{\rm{D}}=\Xi_{zjkz}^{\rm{D}}$ one obtains the symmetry scheme shown in Fig. \[fig:6.2\].D.
Conductivity Tensor symmetry
-------------- -----------------
: The tensor symmetries of the various parts A–B of the linear, and A–G of the DFWM conductivity. As explained in the text, $\vec{J}_{1}$–$\vec{J}_{4}$ are four in general different vectors obtained by a weighted superposition of single-particle transition current densities, and $\vec{e}_{A}=\vec{A}/A$.\[fig:TableSym\]\[tab:1\]
0.25mm
------------------------------------------------------- ---------------------------------
$\hfil \stensor{\sigma}^{\rm{A}}(\vec{r},\vec{r}\,')$
$\stensor{\sigma}^{\rm{B}}(\vec{r},\vec{r}\,')$ $\vec{J}_{1}\otimes\vec{J}_{2}$
------------------------------------------------------- ---------------------------------
: The tensor symmetries of the various parts A–B of the linear, and A–G of the DFWM conductivity. As explained in the text, $\vec{J}_{1}$–$\vec{J}_{4}$ are four in general different vectors obtained by a weighted superposition of single-particle transition current densities, and $\vec{e}_{A}=\vec{A}/A$.\[fig:TableSym\]\[tab:1\]
0.25mm
---------------------------------------------------------------------- ---------------------------------------------------------------------
$\tensor{\Xi}^{\rm{A}}(\vec{r},\vec{r}\,',\vec{r}\,'',\vec{r}\,''')$ $\tensor{\openone}\otimes\tensor{\openone}$
$\tensor{\Xi}^{\rm{B}}(\vec{r},\vec{r}\,',\vec{r}\,'',\vec{r}\,''')$ $\tensor{\openone}\otimes\vec{J}_{1}\otimes\vec{J}_{2}$
$\tensor{\Xi}^{\rm{C}}(\vec{r},\vec{r}\,',\vec{r}\,'',\vec{r}\,''')$ $\vec{e}_{A}\otimes\tensor{\openone}\otimes\vec{e}_{A}$
$\tensor{\Xi}^{\rm{D}}(\vec{r},\vec{r}\,',\vec{r}\,'',\vec{r}\,''')$ $\vec{e}_{A}\otimes\vec{J}_{1}\otimes\vec{J}_{2}\otimes\vec{e}_{A}$
$\tensor{\Xi}^{\rm{E}}(\vec{r},\vec{r}\,',\vec{r}\,'',\vec{r}\,''')$ $\vec{J}_{1}\otimes\tensor{\openone}\otimes\vec{J}_{2}$
$\tensor{\Xi}^{\rm{F}}(\vec{r},\vec{r}\,',\vec{r}\,'',\vec{r}\,''')$ $\vec{J}_{1}\otimes\vec{J}_{2}\otimes\tensor{\openone}$
$\tensor{\Xi}^{\rm{G}}(\vec{r},\vec{r}\,',\vec{r}\,'',\vec{r}\,''')$ $\vec{J}_{1}\otimes\vec{J}_{2}\otimes\vec{J}_{3}\otimes\vec{J}_{4}$
---------------------------------------------------------------------- ---------------------------------------------------------------------
: The tensor symmetries of the various parts A–B of the linear, and A–G of the DFWM conductivity. As explained in the text, $\vec{J}_{1}$–$\vec{J}_{4}$ are four in general different vectors obtained by a weighted superposition of single-particle transition current densities, and $\vec{e}_{A}=\vec{A}/A$.\[fig:TableSym\]\[tab:1\]
Let us now take a closer look at the third trace in Eq. (\[eq:J3-3\]). It is convenient to split the first sum in this equation into two parts related to the two different processes that occur. The first part of the sum, which refers to the ${\cal{H}}^{(2)}_{-2\omega,nv}{\cal{H}}^{(1)}_{\,\omega,v{}m}$-type of terms, gives rise to part E of the third order conductivity tensor corresponding to process (E) of Fig. \[fig:3Tr\]. The second part of the first sum is related to part F of the third order conductivity tensor \[process (F) of Fig. \[fig:3Tr\]\]. Finally, the second sum on the right side of Eq. (\[eq:J3-3\]) produces part G of the third order conductivity tensor, corresponding to process (G) of Fig. \[fig:3Tr\].
The first part of the first sum, in relation to the representation in Eq. (\[eq:sumprod3\]), has a symmetry which can be represented by the outer product $\vec{J}_{1}\otimes\tensor{\openone}\otimes\vec{J}_{2}$, leaving $27$ nonzero elements. From the term ${\cal{H}}_{-2\omega}^{(2)}$, we see that $k=h$ in the chosen representation of coordinate sets, and furthermore we realise that elements with $k=x$, $k=y$, and $k=z$ are equal. These deductions reduce the number of independent nonzero elements to $9$, which fulfills $\Xi_{ijxx}^{\rm{E}}=\Xi_{ijyy}^{\rm{E}}=\Xi_{ijzz}^{\rm{E}}$ for all permutations of $i$ and $j$ in the three Cartesian coordinates $\{x,y,z\}$. The result is shown on schematic form in terms of the symmetry scheme in Fig. \[fig:6.2\].E.
In the second part of the first sum in Eq. (\[eq:J3-3\]) we observe that the symmetry of part F of the conductivity tensor, in relation to the form of Eq. (\[eq:sumprod3\]), is proportional to the outer product $\vec{J}_{1}\otimes\vec{J}_{2}\otimes\tensor{\openone}$, again leaving $27$ nonzero elements. Due to the chosen convention of the coordinate sets, we realise from the ${\cal{H}}_{\,0}^{(2)}$ term that the condition $k=j$ applies. We furthermore notice from this that terms with $j=x$, $j=y$, and $j=z$ are equal, leaving $9$ independent nonzero elements related by $\Xi_{ixxh}^{\rm{F}}=\Xi_{iyyh}^{\rm{F}}=\Xi_{izzh}^{\rm{F}}$ for all permutations of $i$ and $h$ in the three Cartesian coordinates $\{x,y,z\}$. This means that the symmetry scheme is as shown in Fig. \[fig:6.2\].F.
Part G of the third order conductivity tensor, which originates in the second sum in Eq. (\[eq:J3-3\]), obviously has the tensor form $\vec{J}_{1}\otimes\vec{J}_{2}\otimes\vec{J}_{3}\otimes\vec{J}_{4}$, and there will hence in general be $81$ independent nonzero elements in the associated symmetry scheme.
The considerations laying the foundations for the symmetry schemes of the various parts of the linear and nonlinear conductivity tensors are displayed in Tab. \[fig:TableSym\], where the relevant combinations of $\vec{J}$’s and $\tensor{\openone}$’s are given.
Expressions for the conductivity tensors
----------------------------------------
We end this chapter by giving the explicit expressions for the independent tensor elements of $\stensor{\sigma}(\vec{r},\vec{r}\,')$ and $\tensor{\Xi}(\vec{r},\vec{r}\,',\vec{r}\,'',\vec{r}\,''')$. Thus, the only independent tensor element in part A of the linear conductivity tensor is $${\sigma}^{\rm{A}}_{xx}(\vec{r},\vec{r}\,')=
{2{\rm{i}}\over{}\omega}{e^2\over{}m_{e}}\sum_{n}f_{n}
|\psi_{n}|^2\delta(\vec{r}-\vec{r}\,'),
\label{eq:SigmaAArz}$$ and the nine independent elements of part B are $${\sigma}^{\rm{B}}_{ij}(\vec{r},\vec{r}\,')=
{2{\rm{i}}\over\omega}{1\over\hbar}\sum_{nm}
{f_{n}-f_{m}\over\tilde{\omega}_{nm}-\omega}J_{j,mn}'J_{i,nm}.
\label{eq:SigmaABrz}$$ The only independent tensor element in part A of the third order conductivity tensor thus is $$\begin{aligned}
\lefteqn{
\Xi_{xxxx}^{\rm{A}}(\vec{r},\vec{r}\,',\vec{r}\,'',\vec{r}\,''')=
{2{\rm{i}}\over\omega^3}{e^4\over8m_{e}^2\hbar}\sum_{nm}
{f_{n}-f_{m}\over\tilde{\omega}_{nm}-2\omega}
\psi_{n}^{*}(\vec{r}\,'')\psi_{m}(\vec{r}\,'')
\psi_{m}^{*}(\vec{r}\,)\psi_{n}(\vec{r}\,)
}\nonumber\\ &\quad&\times
\delta(\vec{r}-\vec{r}\,')\delta(\vec{r}\,''-\vec{r}\,'''),
\label{eq:XiAArz}\end{aligned}$$ and the nine independent elements of part B are $$\begin{aligned}
\lefteqn{
\Xi_{xxkh}^{\rm{B}}(\vec{r},\vec{r}\,',\vec{r}\,'',\vec{r}\,''')=
{2{\rm{i}}\over\omega^3}{e^2\over4m_{e}\hbar^2}\sum_{nmv}
{1\over\tilde{\omega}_{nm}-2\omega}
\left({f_{m}-f_{v}\over\tilde{\omega}_{v{}m}-\omega}
+{f_{n}-f_{v}\over\tilde{\omega}_{nv}-\omega}\right)
}\nonumber\\ &\quad&\times
J_{h,v{}n}(\vec{r}\,''')J_{k,mv}(\vec{r}\,'')
\psi_{m}^{*}(\vec{r}\,)\psi_{n}(\vec{r}\,)
\delta(\vec{r}-\vec{r}\,').
\label{eq:XiABrz}\end{aligned}$$ The only independent nonzero element of part C of the third order conductivity tensor is given by $$\begin{aligned}
\lefteqn{
\Xi_{xxxx}^{\rm{C}}(\vec{r},\vec{r}\,',\vec{r}\,'',\vec{r}\,''')=
{2{\rm{i}}\over\omega^3}{e^4\over4m_{e}^{2}\hbar}\sum_{nm}
{f_{n}-f_{m}\over\tilde{\omega}_{nm}}\psi_{n}^{*}(\vec{r}\,')\psi_{m}(\vec{r}\,')
\psi_{m}^{*}(\vec{r}\,)\psi_{n}(\vec{r}\,)
}\nonumber\\ &\quad&\times
\delta(\vec{r}\,'-\vec{r}\,''')\delta(\vec{r}-\vec{r}\,''),
\label{eq:XiACrz}\end{aligned}$$ and the nine independent nonzero elements of part D of the third order conductivity tensor are $$\begin{aligned}
\lefteqn{
\Xi_{xjkx}^{\rm{D}}(\vec{r},\vec{r}\,',\vec{r}\,'',\vec{r}\,''')=
{2{\rm{i}}\over\omega^3}{e^2\over4m_{e}\hbar^2}\sum_{nmv}
{1\over\tilde{\omega}_{nm}}\left\{
\left({f_{m}-f_{v}\over\tilde{\omega}_{v{}m}-\omega}
+{f_{n}-f_{v}\over\tilde{\omega}_{nv}+\omega}\right)
J_{j,v{}n}(\vec{r}\,')J_{k,mv}(\vec{r}\,'')
\right.}\nonumber\\ &\quad&\left.\!
+\left({f_{m}-f_{v}\over\tilde{\omega}_{v{}m}+\omega}
+{f_{n}-f_{v}\over\tilde{\omega}_{nv}-\omega}\right)
J_{k,v{}n}(\vec{r}\,'')J_{j,mv}(\vec{r}\,')
\right\}
\psi_{m}^{*}(\vec{r}\,)\psi_{n}(\vec{r}\,)
\delta(\vec{r}-\vec{r}\,''').
\label{eq:XiADrz}\end{aligned}$$ The nine independent elements of part E have the explicit form $$\begin{aligned}
\lefteqn{
\Xi_{ijxx}^{\rm{E}}(\vec{r},\vec{r}\,',\vec{r}\,'',\vec{r}\,''')=
{2{\rm{i}}\over\omega^3}{e^2\over16m_{e}\hbar^2}
}\nonumber\\ &\quad&\times
\sum_{nmv}
{1\over\tilde{\omega}_{nm}-\omega}\left\{
\left({f_{m}-f_{v}\over\tilde{\omega}_{v{}m}-2\omega}
+{f_{n}-f_{v}\over\tilde{\omega}_{nv}+\omega}\right)
J_{j,v{}n}(\vec{r}\,')\psi_{v}^{*}(\vec{r}\,'')\psi_{m}(\vec{r}\,'')
\right.\nonumber\\ &&\left.\!
+\left({f_{n}-f_{v}\over\tilde{\omega}_{nv}-2\omega}
+{f_{m}-f_{v}\over\tilde{\omega}_{v{}m}+\omega}\right)
J_{j,mv}(\vec{r}\,')\psi_{n}^{*}(\vec{r}\,'')\psi_{v}(\vec{r}\,'')
\right\}J_{i,nm}(\vec{r}\,)\delta(\vec{r}\,''-\vec{r}\,'''),
\label{eq:XiAErz}\end{aligned}$$ and the nine independent elements of part F of the third order conductivity tensor are $$\begin{aligned}
\lefteqn{
\Xi_{ixxh}^{\rm{F}}(\vec{r},\vec{r}\,',\vec{r}\,'',\vec{r}\,''')=
{2{\rm{i}}\over\omega^3}{e^2\over8m_{e}\hbar^2}
}\nonumber\\ &\quad&\times
\sum_{nmv}
{1\over\tilde{\omega}_{nm}-\omega}\left\{
\left({f_{m}-f_{v}\over\tilde{\omega}_{v{}m}}
+{f_{n}-f_{v}\over\tilde{\omega}_{nv}-\omega}\right)
J_{h,v{}n}(\vec{r}\,'')\psi_{v}^{*}(\vec{r}\,')\psi_{m}(\vec{r}\,')
\right.\nonumber\\ &&\left.
+\left({f_{n}-f_{v}\over\tilde{\omega}_{nv}}
+{f_{m}-f_{v}\over\tilde{\omega}_{v{}m}-\omega}\right)
J_{h,mv}(\vec{r}\,'')\psi_{n}^{*}(\vec{r}\,')\psi_{v}(\vec{r}\,')
\right\}J_{i,nm}(\vec{r}\,)\delta(\vec{r}\,'-\vec{r}\,''').
\label{eq:XiAFrz}\end{aligned}$$ Finally, the eighty-one independent elements of part G of the third order conductivity tensor are given by $$\begin{aligned}
\lefteqn{
\Xi_{ijkh}^{\rm{G}}(\vec{r},\vec{r}\,',\vec{r}\,'',\vec{r}\,''')=
{2{\rm{i}}\over\omega^3}{1\over8\hbar^3}\sum_{nmvl}
{1\over\tilde{\omega}_{nm}-\omega}
\left\{
\left[
\left({f_{l}-f_{m}\over\tilde{\omega}_{l{}m}-\omega}
+{f_{l}-f_{v}\over\tilde{\omega}_{vl}-\omega}\right)
{1\over\tilde{\omega}_{v{}m}-2\omega}
\right.\right.}\nonumber\\ &\quad&\left.\!
+\left({f_{l}-f_{v}\over\tilde{\omega}_{vl}-\omega}
+{f_{n}-f_{v}\over\tilde{\omega}_{nv}+\omega}\right)
{1\over\tilde{\omega}_{nl}}
\right]
J_{h,ml}(\vec{r}\,''')J_{k,lv}(\vec{r}\,'')J_{j,v{}n}(\vec{r}\,')
\nonumber\\ &&
+\left[
\left({f_{l}-f_{m}\over\tilde{\omega}_{l{}m}-\omega}
+{f_{l}-f_{v}\over\tilde{\omega}_{vl}+\omega}\right)
{1\over\tilde{\omega}_{v{}m}}
+\left({f_{l}-f_{v}\over\tilde{\omega}_{vl}+\omega}
+{f_{n}-f_{v}\over\tilde{\omega}_{nv}-\omega}\right)
{1\over\tilde{\omega}_{nl}}
\right]
\nonumber\\ &&\times
J_{h,ml}(\vec{r}\,''')J_{k,v{}n}(\vec{r}\,'')J_{j,lv}(\vec{r}\,')
+\left[
\left({f_{l}-f_{m}\over\tilde{\omega}_{l{}m}+\omega}
+{f_{l}-f_{v}\over\tilde{\omega}_{vl}-\omega}\right)
{1\over\tilde{\omega}_{v{}m}}
\right.\nonumber\\ &&\left.\!\left.\!
+\left({f_{l}-f_{v}\over\tilde{\omega}_{vl}-\omega}
+{f_{n}-f_{v}\over\tilde{\omega}_{nv}-\omega}\right)
{1\over\tilde{\omega}_{nl}-2\omega}
\right]
J_{h,lv}(\vec{r}\,''')J_{k,v{}n}(\vec{r}\,'')J_{j,ml}(\vec{r}\,')
\right\}J_{i,nm}(\vec{r}\,).
\label{eq:XiAGrz}\end{aligned}$$ The number $2$ appearing in the first fraction of each part of the conductivity tensors represents the degeneracy of the spin energies, thus giving two electrons in each energy eigenstate.
Discussion {#ch:7}
==========
In Chapters \[ch:4\]–\[ch:6\] we have established a spatially nonlocal theoretical model for optical phase conjugation in mesoscopic media. The comparison to the existing (local) descriptions of the degenerate four-wave mixing response can be made by taking the local limit of our nonlocal response tensor and abandoning the contributions stemming from the microscopic current density of first order in $\vec{A}$.
In the local limit, the amplitudes of the interacting vector potentials are assumed not to vary with the spatial coordinates, and thus the expressions for the nonlocal DFWM conductivity tensor can be integrated over the $\vec{r}\,'''$, $\vec{r}\,''$, and $\vec{r}\,'$ spaces to obtain the local DFWM conductivity tensor, i.e., $$\tensor{\Xi}(\vec{r})=
\iiint\tensor{\Xi}(\vec{r},\vec{r}\,',\vec{r}\,'',\vec{r}\,''')
d^3r'''d^3r''d^3r'.$$ Using an orthogonal set of wave equations, parity teaches $$\int\psi_n^*(\vec{r}\,)\psi_m(\vec{r}\,)d^3r=\delta_{nm},$$ where $\delta_{nm}$ is the Kronecker delta. The integrals over the current densities gives zero if the two quantum numbers are identical, otherwise they depend on the individual wave functions. The consequences are the following: (i) Integration over the spatial coordinates $\vec{r}\,'''$ and $\vec{r}\,''$ in Eq. (\[eq:XiAArz\]) gives $n=m$, and thus $f_n=f_m$, such that part A of the DFWM conductivity tensor vanish. (ii) In part C, given by Eq. (\[eq:XiACrz\]), the effect is similar, but is here obtained after integration over $\vec{r}\,'''$, $\vec{r}\,''$, and $\vec{r}\,'$. (iii) In part E, integration over $\vec{r}\,'''$ and $\vec{r}\,''$ in Eq. (\[eq:XiAErz\]) makes two terms disappear immediately, and an inspection of the remaining two terms shows that they are of equal magnitude, but with opposite sign, ultimately cancelling the rest of part E. (iv) In part F, integration over $\vec{r}\,'''$ and $\vec{r}\,'$ in Eq. (\[eq:XiAFrz\]) gives a result similar in consequences as for part E. Thus, parts A, C, E, and F of the DFWM conductivity tensor are inherently nonlocal, while parts B, D, and G also contributes to the response in the local limit.
Abandoning parts B and D of the DFWM conductivity tensor because they are based on the response of the microscopic current density of first order in $\vec{A}$, we conclude that only the local contribution from part G is included in the previous descriptions of the DFWM response [@Bloembergen:78:1], as postulated on page .
The single-electron model for degenerate four-wave mixing established in the previous chapters can be used to study the four-wave mixing response from a number of different materials. For example, one could study (i) systems built from molecules or atoms with no electronic overlap (dielectrics), in which case the response from each molecule (atom) can be found separately. The coupling between the individual molecules would then be described using electromagnetic propagators. Another approach (ii) can be taken for studies of the response from metals, where a number of electrons from each ion in the metallic structure is shared with the other ions in a free-electron-like cloud, or (iii) one could study semiconductors, in which the behaviour of the electrons are strongly coupled.
In the present work, we will concentrate on the metallic case, and we proceed to give a simplified description of potential interest for DFWM in mesoscopic films and in near-field optics.
\[part:III\]
DFWM in two-dimensionally translational invariant media {#sec:Film}
=======================================================
\[sec:pot\] After having established and analyzed the DFWM conductivity response in its most general form we now turn the attention towards the specific case in which the medium under consideration [ *effectively*]{} exhibits translational invariance in two directions, say $x$ and $y$ in a Cartesian $(x,y,z)$ coordinate system. We study such a case because it appears to be of particular importance for optical phase conjugation (i) in mesoscopic films (quantum wells), and (ii) related to evanescent waves in near-field optics. In neither of these cases a microscopic theory exists today to our knowledge. For mesoscopic films the dynamics perpendicular to the film plane (here, the $x$-$y$-plane) has to be treated from a microscopic nonlocal point of view, whereas the dynamics in the plane of the film often is well modelled by a local conductivity (dielectric) function. In the following we assume for simplicity that the electron motion in the plane of the film is free-electron-like. It is possible to replace the free-electron-like behaviour with extended Bloch-function (or tight-binding) dynamics if necessary but we shall not do this here, since after all, in the local limit only matrix elements are changed in the oscillator model when the free-electron dynamics is replaced by a more complicated one. In the optical near-field case where evanescent waves with extremely small penetration depths in say the $z$-direction appear, it is crucial to keep the microscopic dynamics perpendicular to the surface of the phase conjugating mirror when calculating the DFWM response. So far, four-wave mixing in media with two-dimensional translational invariance has only been studied in the context of phase conjugation of electromagnetic surface waves (, ), and of a bulk wave by surface waves [@Zeldovich:80:1; @Ujihara:83:1; @Stegeman:83:1; @Ninzi:84:1; @Mamaev:84:1; @Mukhin:85:1; @Arutyunyan:87:1; @Pilipetskii:87:1]. In these investigations macroscopic approaches was used.
General DFWM response
---------------------
The assumed two-dimensional translational invariance against displacements parallel to the $x$-$y$-plane makes it natural to express the various vector and tensor quantities in a mixed Fourier representation. Thus, by a Fourier analysis in the $x$- and $y$-coordinates, the vector potential is $$\vec{A}(z,\vec{r}_{\|})={1\over(2\pi)^2}\int
\vec{A}(z;\vec{q}_{\|})e^{{\rm{i}}\vec{q}_{\|}\cdot\vec{r}_{\|}}d^2q_{\|},
\label{eq:Azr}$$ where $\vec{q}_{\|}=(q_x,q_y,0)$ and $\vec{r}_{\|}=(x,y,0)$. Likewise, the inverse relation reads for the current density of order $\alpha$ and linear in the cyclic frequency $\omega$ $$\vec{J}^{\,(\alpha)}_{-\omega}(z;\vec{q}_{\|})=
\int\vec{J}^{\,(\alpha)}_{-\omega}(z,\vec{r}_{\|})
e^{-{\rm{i}}\vec{q}_{\|}\cdot\vec{r}_{\|}}d^2r_{\|}.
\label{eq:Jzq}$$ In the mixed Fourier representation the relevant constitutive relations takes the form $$\begin{aligned}
\lefteqn{
\vec{J}_{-\omega}^{\,(1)}(z;\vec{q}_{\|})={{\rm{i}}\omega}
\int\stensor{\sigma}(z,z';\vec{q}_{\|})\cdot\vec{A}(z';\vec{q}_{\|})dz',
}\label{eq:J1zq}
\\
\lefteqn{
\vec{J}_{-\omega}^{\,(3)}(z;\vec{q}_{\|})={({\rm{i}}\omega)^3\over(2\pi)^4}
\int\cdots\int\tensor{\Xi}(z,z',z'',z''';\vec{q}_{\|},\vec{q}_{\|}^{\,\prime},
\vec{q}_{\|}^{\,\prime\prime},\vec{q}_{\|}^{\,\prime\prime\prime})
}\nonumber\\ &\quad&
\vdots\,
\vec{A}(z''';\vec{q}_{\|}^{\,\prime\prime\prime})
\vec{A}(z'';\vec{q}_{\|}^{\,\prime\prime})
\vec{A}^{*}(z';\vec{q}_{\|}^{\,\prime})
d^2q_{\|}'''d^2q_{\|}''d^2q_{\|}'
dz'''dz''dz'.
\label{eq:J3zq}\end{aligned}$$ Due to the manner in which the nonlinear conductivity response tensor was constructed in Chapter \[ch:5\], the various components parallel to the $x$-$y$-plane are not completely independent but satisfy the momentum conservation criterion $$\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}
-\vec{q}_{\|}^{\,\prime}-\vec{q}_{\|}=\vec{0}.
\label{eq:momentum}$$ In passing we stress again that Eq. (\[eq:momentum\]) is [*not*]{} an extra condition put on the dynamics, the equation is [ *derived*]{} from the general theory \[see Appendix \[ch:ConCalc\]\]. To study the phase conjugated response originating in the mixing of three incoming waves one must choose for the fields of the two pump waves, denoted by $(1)$ and $(2)$, the vector potentials with the double and triple primes in Eq. (\[eq:J3zq\]). The incoming probe field \[indexed $(p)$\] is represented via the vector potential with the single prime.
Phase conjugation DFWM response
-------------------------------
So far, we have not utilized the translational invariance condition on the properties of the medium. We do this first indirectly by assuming that each of the three incoming electromagnetic fields contains only one plane-wave component parallel to the $x$-$y$-plane. Further limiting our study to the case where the DFWM response becomes the phase conjugated response, i.e., the wavevector of the response must be counterpropagating to the probe field, conservation of pseudomomentum requires that the two pump fields are counterpropagating. Thus we take for the pump fields $$\begin{aligned}
\vec{A}(z''';\vec{q}_{\|}^{\,\prime\prime\prime})
&\equiv&\vec{A}(z''';-\vec{k}_{\|})
\delta(\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{k}_{\|}),
\label{eq:Apump-1a}\\
\vec{A}(z'';\vec{q}_{\|}^{\,\prime\prime})
&\equiv&\vec{A}(z'';\vec{k}_{\|})
\delta(\vec{q}_{\|}^{\,\prime\prime}-\vec{k}_{\|}),
\label{eq:Apump-2a}\end{aligned}$$ where $\vec{k}_{\|}$ is the common wavevector for the two pump fields. With these substitutions we can perform the integrals over $q_{\|}'''$ and $q_{\|}''$ in Eq. (\[eq:J3zq\]), and the conservation of pseudo-momentum is reduced from its general degenerate four-wave mixing form, $\vec{q}_{\|}^{\,\prime\prime\prime}
+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime}-\vec{q}_{\|}=\vec{0}$, to $\vec{q}_{\|}^{\,\prime}+\vec{q}_{\|}=\vec{0}$. This allows us also to solve the integral over $q_{\|}'$ in Eq. (\[eq:J3zq\]).
In relation to the conventional theory of three-dimensional (bulk) phase conjugation, the relation $\vec{q}_{\|}^{\,\prime}+\vec{q}_{\|}=\vec{0}$ expresses the fact that the two-dimensional wavevector $\vec{q}_{\|}$ of the phase conjugated field is equal in magnitude to the two-dimensional probe wavevector ($\vec{q}_{\|}^{\,\prime}$) but points in the opposite direction. Using the aforementioned criteria, the nonlinear constitutive equation is reduced to the form $$\begin{aligned}
\lefteqn{
\vec{J}_{-\omega}^{\,(3)}(z;\vec{q}_{\|})=
{({\rm{i}}\omega)^3\over(2\pi)^4}\int\int\int
\tensor{\Xi}(z,z',z'',z''';\vec{q}_{\|},\vec{k}_{\|})
}\nonumber\\ &\quad&
\vdots\,
\vec{A}(z''';-\vec{k}_{\|})\vec{A}(z'';\vec{k}_{\|})
\vec{A}^{*}(z';-\vec{q}_{\|})
dz'''dz''dz'+\mbox{i.t.},
\label{eq:J3-wzq}\end{aligned}$$ where appropriate integration over $\vec{q}_{\|}^{\,\prime\prime\prime}$, $\vec{q}_{\|}^{\,\prime\prime}$, and $\vec{q}_{\|}^{\,\prime}$ has been performed. The term “i.t.” denotes the so-called “interchanged term”. This term is obtained from the first one by interchanging the two pump fields. The reason that such a term has to be added arises from the fact that each of the vector potentials basically consists of a sum of all three incoming fields, and that the phase conjugated term from the product of the three vector potentials thus must include both permutations of the pump fields. The new phase conjugation DFWM (PCDFWM) conductivity tensor appearing after integration over $\vec{q}_{\|}^{\,\prime\prime\prime}$, $\vec{q}_{\|}^{\,\prime\prime}$, and $\vec{q}_{\|}^{\,\prime}$ is denoted $\tensor{\Xi}(z,z',z'',z''';\vec{q}_{\|},\vec{k}_{\|})$.
In order to calculate the nonlinear conductivity tensor $\tensor{\Xi}(z,z',z'',z''';\vec{q}_{\|},\vec{k}_{\|})$ in the mixed Fourier representation \[as well as the linear one, $\stensor{\sigma}(z,z';\vec{q}_{\|})$\], we begin by looking at the energy eigenstates for the light-unperturbed Schr[ö]{}dinger equation. Hence, since the potential energy of the individual electrons is independent of $x$ and $y$, i.e., $V(\vec{r}\,)=V(z)$ under our translational invariance assumption, the basis set may be taken in the generic form $$\psi_{n}(z,\vec{r}_{\|})
\equiv\psi_{n,\vec{\kappa}_{\|}}(z,\vec{r}_{\|})
={1\over2\pi}\psi_{n}(z)e^{{\rm{i}}\vec{\kappa}_{\|}\cdot\vec{r}_{\|}}
\label{eq:eigenstate}$$ where $\vec{\kappa}_{\|}=(\kappa_x,\kappa_y,0)$ is the wavevector describing the free-particle motion perpendicular to the $z$-direction. For a medium of macroscopic extension in the $x$- and $y$-directions, the set of wavevectors commonly denoted by $\vec{\kappa}_{\|}$ forms a two-dimensional quasi-continuum. Albeit the index $n$ in the wave function $\psi_{n}(z,\vec{r}_{\|})$ stands for a triple set of quantum numbers we also use this index to classify the various wave function parts, $\psi_{n}(z)$, belonging to the single indexed $z$-dynamics. In a readily understandable notation the energy eigenstates, ${\cal{E}}_{\,n}$, associated with the generic solution in Eq. (\[eq:eigenstate\]) is $${\cal{E}}_{\,n}=\varepsilon_{n}
+{\hbar^2\over2m_{e}}|\vec{\kappa}_{\|}|^2,$$ where we have introduced $\varepsilon_{n}$ as the energy of state $n$ in the solution dependent on the $z$-coordinate only. In the view of the abovementioned considerations the cyclic transition frequency becomes $$\omega_{nm}={1\over\hbar}\left[\varepsilon_{n}-\varepsilon_{m}+
{\hbar^2\over2m_{e}}\left(|\vec{\kappa}_{\|,\bar{n}}|^2
-|\vec{\kappa}_{\|,\bar{m}}|^2\right)\right],
\label{eq:transition-w}$$ in a notation where adequate subscripts $\bar{n}$ and $\bar{m}$ have been put on the wavevectors. In abbreviated form the complex transition frequency, which includes the relaxation time, is for the sake of the following analysis written in the form $$\tilde{\omega}_{nm}=\tilde{\omega}_{nm}(\vec{\kappa}_{\|,\bar{n}},
\vec{\kappa}_{\|,\bar{m}}),
\label{eq:transitionw}$$ omitting the reference to $\varepsilon_{n}$ and $\varepsilon_{m}$, since this is already implicitly given by the $nm$ subscript. The Fermi-Dirac distribution function we also present in an abbreviated form, viz. $$f_{n}({\cal{E}}_{\,n})=f_{n}\left(\varepsilon_{n}
+{\hbar^2\kappa_{\|,\bar{n}}^2\over2m_{e}}\right)
\equiv f_{n}(\vec{\kappa}_{\|,\bar{n}}).
\label{eq:f_n}$$ By inserting the generic solution in Eq. (\[eq:eigenstate\]) into the expression for the transition current density in Eq. (\[eq:Jm->n\]), we obtain $$\begin{aligned}
\lefteqn{
\vec{J}_{mn}(\vec{r}\,)=-{e\hbar\over2{\rm{i}}m_{e}}{1\over(2\pi)^2}
\Biggl[{\rm{i}}(\vec{\kappa}_{\|,\bar{m}}
+\vec{\kappa}_{\|,\bar{n}})\psi_{n}^{*}(z)\psi_{m}(z)
}\nonumber\\ &\quad&\left.\!
+\vec{e}_{z}\left(
\psi_{n}^{*}(z){\partial\psi_{m}(z)\over\partial{}z}-
\psi_{m}(z){\partial\psi_{n}^{*}(z)\over\partial{}z}\right)\right]
e^{{\rm{i}}(\vec{\kappa}_{\|,\bar{m}}-\vec{\kappa}_{\|,\bar{n}})\cdot\vec{r}_{\|}}
\nonumber\\ &&
\equiv{1\over(2\pi)^2}
\vec{j}_{mn}(z;\vec{\kappa}_{\|,\bar{m}}+\vec{\kappa}_{\|,\bar{n}})
e^{{\rm{i}}(\vec{\kappa}_{\|,\bar{m}}-\vec{\kappa}_{\|,\bar{n}})\cdot\vec{r}_{\|}},
\label{eq:Jm->n||}
\label{eq:j_hnm}\end{aligned}$$ where for convenience we have defined a new transition current density $\vec{j}_{mn}(z;\vec{\kappa}_{\|,\bar{m}}+\vec{\kappa}_{\|,\bar{n}})$ to separate out the dependence on the Cartesian coordinates $\vec{r}_{\|}$. For the various Cartesian components of this current density, we use the notation $j_{i,nm}(z;\vec{\kappa}_{\|,\bar{m}}+\vec{\kappa}_{\|,\bar{n}})$, $i\in\{x,y,z\}$.
Conductivity tensors
--------------------
The explicit expression for the phase conjugation degenerate four-wave mixing (PCDFWM) conductivity tensor $\tensor{\Xi}(z,z',z'',z''';\vec{q}_{\|},\vec{k}_{\|})$ is calculated by insertion of (i) the solutions to the time-independent Schr[ö]{}dinger equation given in Eq. (\[eq:eigenstate\]), (ii) the Fourier representation of the vector potential given by Eq. (\[eq:Azr\]), and (iii) the new form of the transition current given in Eq. (\[eq:Jm->n||\]) into the nonlinear DFWM constitutive relation in Eq. (\[eq:J3r\]) with the phase conjugation conductivity tensor in real space given by Eqs. (\[eq:XiAArz\])–(\[eq:XiAGrz\]), and thereafter inserting the outcome of these steps into the expression for the nonlinear current density in the mixed Fourier representation in Eq. (\[eq:Jzq\]). Finally, we perform the integrals over the two dimensions ($x$ and $y$) in real space and over relevant sets of $\vec{\kappa}_{\|}$-states. Altogether we are left with an expression on the form of Eq. (\[eq:J3-wzq\]). For the processes in Figs. \[fig:2Tr\] and \[fig:3Tr\], the abovementioned calculations are supplied in Appendix \[ch:ConCalc\], where also the general DFWM conductivity tensors are given. The nonzero element of the linear conductivity tensor part A become $${\sigma}_{xx}^{\,\rm{A}}(z,z';\vec{q}_{\|})=
{2{\rm{i}}\over\omega}{e^2\over{}m_{e}}{1\over(2\pi)^2}
\sum_{n}\int{}f_{n}(\vec{\kappa}_{\|})
d^2\kappa_{\|}
|\psi_{n}(z)|^2\delta(z-z'),
\label{eq:SigmaAA}$$ and the nine nonzero elements of part B are $$\begin{aligned}
\lefteqn{
{\sigma}_{ij}^{\,\rm{B}}(z,z';\vec{q}_{\|})=
{2{\rm{i}}\over\omega}{1\over\hbar}{1\over(2\pi)^2}\sum_{nm}\int
{f_{n}(\vec{\kappa}_{\|}+\vec{q}_{\|})-f_{m}(\vec{\kappa}_{\|})
\over\tilde{\omega}_{nm}(\vec{\kappa}_{\|}+\vec{q}_{\|},
\vec{\kappa}_{\|})-\omega}
}\nonumber\\ &\quad&\times
j_{j,mn}(z';2\vec{\kappa}_{\|}+\vec{q}_{\|})
j_{i,nm}(z;2\vec{\kappa}_{\|}+\vec{q}_{\|})
d^2\kappa_{\|}.
\label{eq:SigmaAB}\end{aligned}$$ The nonzero element of the PCDFWM conductivity tensor part A are $$\begin{aligned}
\lefteqn{
{\Xi}_{xxxx}^{\rm{A}}(z,z',z'',z''';\vec{q}_{\|},\vec{k}_{\|})=
{e^4\over8m_{e}^2\hbar}{1\over(2\pi)^2}
{2{\rm{i}}\over\omega^3}
\sum_{nm}
\psi_{n}^{*}(z'')\psi_{m}(z'')\psi_{m}^{*}(z)\psi_{n}(z)
}\nonumber\\ &\quad&\times
\delta(z-z')\delta(z''-z''')\int
{f_{n}(\vec{\kappa}_{\|})-f_{m}(\vec{\kappa}_{\|})
\over\tilde{\omega}_{nm}(\vec{\kappa}_{\|},\vec{\kappa}_{\|})-2\omega}
d^2\kappa_{\|},
\label{eq:XiAA}\end{aligned}$$ and the nine nonzero elements of part B become $$\begin{aligned}
\lefteqn{
{\Xi}_{xxkh}^{\rm{B}}(z,z',z'',z''';\vec{q}_{\|},\vec{k}_{\|})=
{e^2\over4m_{e}\hbar^2}{1\over(2\pi)^2}
{2{\rm{i}}\over\omega^3}\sum_{nmv}\psi_{m}^{*}(z)\psi_{n}(z)\delta(z-z')
}\nonumber\\ &&\times
\int
{1\over\tilde{\omega}_{nm}(\vec{\kappa}_{\|},\vec{\kappa}_{\|})-2\omega}
\left({f_{m}(\vec{\kappa}_{\|})-f_{v}(\vec{\kappa}_{\|}+\vec{k}_{\|})
\over\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|}+\vec{k}_{\|},
\vec{\kappa}_{\|})-\omega}
+{f_{n}(\vec{\kappa}_{\|})-f_{v}(\vec{\kappa}_{\|}+\vec{k}_{\|})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|},
\vec{\kappa}_{\|}+\vec{k}_{\|})-\omega}\right)
\nonumber\\ &&\times
j_{h,v{}n}(z''';2\vec{\kappa}_{\|}+\vec{k}_{\|})
j_{k,mv}(z'';2\vec{\kappa}_{\|}+\vec{k}_{\|})
d^2\kappa_{\|}.
\label{eq:XiAB}\end{aligned}$$ In part C of the nonlinear conductivity tensor the nonzero element is $$\begin{aligned}
\lefteqn{
{\Xi}_{xxxx}^{\rm{C}}(z,z',z'',z''';\vec{q}_{\|},\vec{k}_{\|})=
{e^4\over4m_{e}^2\hbar}{1\over(2\pi)^2}
{2{\rm{i}}\over\omega^3}\sum_{nm}
\psi_{n}^{*}(z')\psi_{m}(z')\psi_{m}^{*}(z)\psi_{n}(z)
}\nonumber\\ &&\times
\delta(z'-z''')\delta(z-z'')\int
{f_{n}(\vec{\kappa}_{\|}-\vec{k}_{\|}+\vec{q}_{\|})
-f_{m}(\vec{\kappa}_{\|})\over\tilde{\omega}_{nm}(\vec{\kappa}_{\|}
-\vec{k}_{\|}+\vec{q}_{\|},\vec{\kappa}_{\|})}
d^2\kappa_{\|},
\label{eq:XiAC}\end{aligned}$$ and the nine nonzero tensor elements in part D become $$\begin{aligned}
\lefteqn{
{\Xi}_{xjkx}^{\rm{D}}(z,z',z'',z''';\vec{q}_{\|},\vec{k}_{\|})=
{e^2\over4m_{e}\hbar^2}{1\over(2\pi)^2}
{2{\rm{i}}\over\omega^3}\sum_{nmv} \psi_{m}^{*}(z)\psi_{n}(z)
\delta(z-z''')
}\nonumber\\ &&\times
\int
{1\over\tilde{\omega}_{nm}(\vec{\kappa}_{\|}+\vec{k}_{\|}+\vec{q}_{\|}
,\vec{\kappa}_{\|})}
\left\{
\left({f_{m}(\vec{\kappa}_{\|})-f_{v}(\vec{\kappa}_{\|}+\vec{k}_{\|})
\over\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|}+\vec{k}_{\|},
\vec{\kappa}_{\|})-\omega}
\right.\right.\nonumber\\ &&\left.\!
+{f_{n}(\vec{\kappa}_{\|}+\vec{k}_{\|}+\vec{q}_{\|})
-f_{v}(\vec{\kappa}_{\|}+\vec{k}_{\|})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|}+\vec{k}_{\|}+\vec{q}_{\|},
\vec{\kappa}_{\|}+\vec{k}_{\|})+\omega}\right)
j_{j,v{}n}(z';2\vec{\kappa}_{\|}+2\vec{k}_{\|}+\vec{q}_{\|})
j_{k,mv}(z'';2\vec{\kappa}_{\|}+\vec{k}_{\|})
\nonumber\\ &&
+
\left({f_{m}(\vec{\kappa}_{\|})-f_{v}(\vec{\kappa}_{\|}+\vec{q}_{\|})
\over\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|}+\vec{q}_{\|},
\vec{\kappa}_{\|})+\omega}
+{f_{n}(\vec{\kappa}_{\|}+\vec{k}_{\|}+\vec{q}_{\|})
-f_{v}(\vec{\kappa}_{\|}+\vec{q}_{\|})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|}+\vec{k}_{\|}+\vec{q}_{\|},
\vec{\kappa}_{\|}+\vec{q}_{\|})-\omega}\right)
\nonumber\\ &&\times
j_{k,v{}n}(z'';2\vec{\kappa}_{\|}+\vec{k}_{\|}+2\vec{q}_{\|})
j_{j,mv}(z';2\vec{\kappa}_{\|}+\vec{q}_{\|})
\Bigr\}
d^2\kappa_{\|}.
\label{eq:XiAD}\end{aligned}$$ The nonlinear conductivity tensor part E has the nine nonzero elements $$\begin{aligned}
\lefteqn{
{\Xi}_{ijxx}^{\rm{E}}(z,z',z'',z''';\vec{q}_{\|},\vec{k}_{\|})=
{e^2\over16m_{e}\hbar^2}{1\over(2\pi)^2}
{2{\rm{i}}\over\omega^3}\sum_{nmv}\delta(z''-z''')\int
{1\over\tilde{\omega}_{nm}(\vec{\kappa}_{\|}+\vec{q}_{\|},\vec{\kappa}_{\|})
-\omega}
}\nonumber\\ &&\times
\!\left\{\!
\left(\!{f_{m}(\vec{\kappa}_{\|})-f_{v}(\vec{\kappa}_{\|})
\over\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|},\vec{\kappa}_{\|})-2\omega}
+{f_{n}(\vec{\kappa}_{\|}+\vec{q}_{\|})-f_{v}(\vec{\kappa}_{\|})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|}+\vec{q}_{\|},\vec{\kappa}_{\|})
+\omega}\right)
j_{j,v{}n}(z';2\vec{\kappa}_{\|}+\vec{q}_{\|})\psi_{v}^{*}(z'')\psi_{m}(z'')
\right.\nonumber\\ &&
+
\left({f_{m}(\vec{\kappa}_{\|})-f_{v}(\vec{\kappa}_{\|}+\vec{q}_{\|})\over
\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|}+\vec{q}_{\|},\vec{\kappa}_{\|})
+\omega}+{f_{n}(\vec{\kappa}_{\|}+\vec{q}_{\|})
-f_{v}(\vec{\kappa}_{\|}+\vec{q}_{\|})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|}+\vec{q}_{\|},\vec{\kappa}_{\|}
+\vec{q}_{\|})-2\omega}\right)
\nonumber\\ &&\times
j_{j,mv}(z';2\vec{\kappa}_{\|}+\vec{q}_{\|})
\psi_{n}^{*}(z'')\psi_{v}(z'')
\Bigr\}j_{i,nm}(z;2\vec{\kappa}_{\|}+\vec{q}_{\|})d^2\kappa_{\|}.
\label{eq:XiAE}\end{aligned}$$ Part F also has nine nonzero elements, which are $$\begin{aligned}
\lefteqn{
{\Xi}_{ixxh}^{\rm{F}}(z,z',z'',z''';\vec{q}_{\|},\vec{k}_{\|})=
{e^2\over8m_{e}\hbar^2}{1\over(2\pi)^2}
{2{\rm{i}}\over\omega^3}\sum_{nmv}\delta(z'-z''')\int
{1\over\tilde{\omega}_{nm}(\vec{\kappa}_{\|}+\vec{q}_{\|},\vec{\kappa}_{\|})
-\omega}
}\nonumber\\ &&\times
\left\{
\left({f_{m}(\vec{\kappa}_{\|})-f_{v}(\vec{\kappa}_{\|}-\vec{k}_{\|}
+\vec{q}_{\|})\over\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|}
-\vec{k}_{\|}+\vec{q}_{\|},\vec{\kappa}_{\|})}
+{f_{n}(\vec{\kappa}_{\|}+\vec{q}_{\|})
-f_{v}(\vec{\kappa}_{\|}-\vec{k}_{\|}+\vec{q}_{\|})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|}+\vec{q}_{\|},\vec{\kappa}_{\|}
-\vec{k}_{\|}+\vec{q}_{\|})-\omega}\right)
\right.\nonumber\\ &&\times
j_{h,v{}n}(z'';2\vec{\kappa}_{\|}-\vec{k}_{\|}+\vec{q}_{\|})
\psi_{v}^{*}(z')\psi_{m}(z')
\nonumber\\ &&
+
\left(
{f_{m}(\vec{\kappa}_{\|})-f_{v}(\vec{\kappa}_{\|}+\vec{k}_{\|})\over
\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|}+\vec{k}_{\|},
\vec{\kappa}_{\|})-\omega}+{f_{n}(\vec{\kappa}_{\|}+\vec{q}_{\|})
-f_{v}(\vec{\kappa}_{\|}+\vec{k}_{\|})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|}+\vec{q}_{\|},\vec{\kappa}_{\|}
+\vec{k}_{\|})}\right)
\nonumber\\ &&\times
j_{h,mv}(z'';2\vec{\kappa}_{\|}+\vec{k}_{\|})
\psi_{n}^{*}(z')\psi_{v}(z')
\Bigr\}j_{i,nm}(z;2\vec{\kappa}_{\|}+\vec{q}_{\|})
d^2\kappa_{\|}.
\label{eq:XiAF}\end{aligned}$$ Finally, the PCDFWM conductivity tensor part G has the eightyone nonzero elements $$\begin{aligned}
\lefteqn{
{\Xi}_{ijkh}^{\rm{G}}(z,z',z'',z''';\vec{q}_{\|},\vec{k}_{\|})=
{1\over8\hbar^3}{1\over(2\pi)^2}
{2{\rm{i}}\over\omega^3}\sum_{nmvl}\int
{1\over\tilde{\omega}_{nm}(\vec{\kappa}_{\|}+\vec{q}_{\|},\vec{\kappa}_{\|})
-\omega}
}\nonumber\\ &&\times
\left\{
\left[
\left({f_{l}(\vec{\kappa}_{\|}-\vec{k}_{\|})
-f_{m}(\vec{\kappa}_{\|})\over\tilde{\omega}_{l{}m}(\vec{\kappa}_{\|}
-\vec{k}_{\|},\vec{\kappa}_{\|})-\omega}
+{f_{l}(\vec{\kappa}_{\|}-\vec{k}_{\|})-f_{v}(\vec{\kappa}_{\|})
\over\tilde{\omega}_{vl}(\vec{\kappa}_{\|},
\vec{\kappa}_{\|}-\vec{k}_{\|})-\omega}\right)
{1\over\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|},\vec{\kappa}_{\|})-2\omega}
\right.\right.\nonumber\\ &&\left.
+\left({f_{l}(\vec{\kappa}_{\|}-\vec{k}_{\|})
-f_{v}(\vec{\kappa}_{\|})\over
\tilde{\omega}_{vl}(\vec{\kappa}_{\|},
\vec{\kappa}_{\|}-\vec{k}_{\|})-\omega}
+{f_{n}(\vec{\kappa}_{\|}+\vec{q}_{\|})-f_{v}(\vec{\kappa}_{\|})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|}+\vec{q}_{\|},\vec{\kappa}_{\|})
+\omega}\right)
{1\over\tilde{\omega}_{nl}(\vec{\kappa}_{\|}+\vec{q}_{\|},\vec{\kappa}_{\|}-\vec{k}_{\|})}
\right]
\nonumber\\ &&\times
j_{h,ml}(z''';2\vec{\kappa}_{\|}-\vec{k}_{\|})
j_{k,lv}(z'';2\vec{\kappa}_{\|}-\vec{k}_{\|})
j_{j,v{}n}(z';2\vec{\kappa}_{\|}+\vec{q}_{\|})
\nonumber\\ &&
+
\left[
\left({f_{l}(\vec{\kappa}_{\|}-\vec{k}_{\|})
-f_{m}(\vec{\kappa}_{\|})\over\tilde{\omega}_{l{}m}(\vec{\kappa}_{\|}
-\vec{k}_{\|},\vec{\kappa}_{\|})-\omega}
+{f_{l}(\vec{\kappa}_{\|}-\vec{k}_{\|})-f_{v}(\vec{\kappa}_{\|}
-\vec{k}_{\|}+\vec{q}_{\|})\over
\tilde{\omega}_{vl}(\vec{\kappa}_{\|}-\vec{k}_{\|}+\vec{q}_{\|},
\vec{\kappa}_{\|}-\vec{k}_{\|})+\omega}\right)
\right.\nonumber\\ &&\times
{1\over\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|}-\vec{k}_{\|}
+\vec{q}_{\|},\vec{\kappa}_{\|})}
+\left({f_{l}(\vec{\kappa}_{\|}-\vec{k}_{\|})
-f_{v}(\vec{\kappa}_{\|}-\vec{k}_{\|}+\vec{q}_{\|})\over
\tilde{\omega}_{vl}(\vec{\kappa}_{\|}-\vec{k}_{\|}+\vec{q}_{\|},
\vec{\kappa}_{\|}-\vec{k}_{\|})+\omega}
\right.\nonumber\\ &&\left.\!\left.\!
+{f_{n}(\vec{\kappa}_{\|}+\vec{q}_{\|})
-f_{v}(\vec{\kappa}_{\|}-\vec{k}_{\|}+\vec{q}_{\|})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|}+\vec{q}_{\|},\vec{\kappa}_{\|}
-\vec{k}_{\|}+\vec{q}_{\|})-\omega}
\right)
{1\over\tilde{\omega}_{nl}(\vec{\kappa}_{\|}
+\vec{q}_{\|},\vec{\kappa}_{\|}-\vec{k}_{\|})}
\right]
\nonumber\\ &&\times
j_{h,ml}(z''';2\vec{\kappa}_{\|}-\vec{k}_{\|})
j_{k,v{}n}(z'';2\vec{\kappa}_{\|}-\vec{k}_{\|}+2\vec{q}_{\|})
j_{j,lv}(z';2\vec{\kappa}_{\|}-2\vec{k}_{\|}+\vec{q}_{\|})
\nonumber\\ &&
+
\left[
\left({f_{l}(\vec{\kappa}_{\|}+\vec{q}_{\|})-f_{m}(\vec{\kappa}_{\|})
\over\tilde{\omega}_{l{}m}(\vec{\kappa}_{\|}+\vec{q}_{\|},
\vec{\kappa}_{\|})+\omega}
+{f_{l}(\vec{\kappa}_{\|}+\vec{q}_{\|})-f_{v}(\vec{\kappa}_{\|}
-\vec{k}_{\|}+\vec{q}_{\|})\over
\tilde{\omega}_{vl}(\vec{\kappa}_{\|}-\vec{k}_{\|}+\vec{q}_{\|},
\vec{\kappa}_{\|}+\vec{q}_{\|})-\omega}\right)
\right.\nonumber\\ &&\times
{1\over\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|}-\vec{k}_{\|}
+\vec{q}_{\|},\vec{\kappa}_{\|})}
+\left({f_{l}(\vec{\kappa}_{\|}+\vec{q}_{\|})
-f_{v}(\vec{\kappa}_{\|}-\vec{k}_{\|}+\vec{q}_{\|})\over
\tilde{\omega}_{vl}(\vec{\kappa}_{\|}-\vec{k}_{\|}+\vec{q}_{\|},
\vec{\kappa}_{\|}+\vec{q}_{\|})-\omega}
\right.\nonumber\\ &&\left.\!\left.\!
+{f_{n}(\vec{\kappa}_{\|}+\vec{q}_{\|})
-f_{v}(\vec{\kappa}_{\|}-\vec{k}_{\|}+\vec{q}_{\|})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|}+\vec{q}_{\|},\vec{\kappa}_{\|}
-\vec{k}_{\|}+\vec{q}_{\|})-\omega}\right)
{1\over\tilde{\omega}_{nl}(\vec{\kappa}_{\|}
+\vec{q}_{\|},\vec{\kappa}_{\|}+\vec{q}_{\|})-2\omega}
\right]
\nonumber\\ &&\times
j_{h,lv}(z''';2\vec{\kappa}_{\|}-\vec{k}_{\|}+2\vec{q}_{\|})
j_{k,v{}n}(z'';2\vec{\kappa}_{\|}-\vec{k}_{\|}+2\vec{q}_{\|})
j_{j,ml}(z';2\vec{\kappa}_{\|}+\vec{q}_{\|})
\Bigr\}
\nonumber\\ &&\times
j_{i,nm}(z;2\vec{\kappa}_{\|}+\vec{q}_{\|})
d^2\kappa_{\|}.
\label{eq:XiAG}\end{aligned}$$ In Eqs. (\[eq:SigmaAA\])–(\[eq:XiAG\]) above we have dropped the now superfluous index on $\vec{\kappa}_{\|}$.
Phase conjugated field
----------------------
After having sketched the calculation of the nonlinear DFWM response we turn our attention to the phase conjugated electric field. In the present case where the main parts of the interaction takes place in very small interaction volumes, we can expect that the generated phase conjugated field does not affect the dynamics of the pump and probe fields much, and thus take the parametric approximation.
Then the loop equation in Eq. (\[eq:Inte\]) is reduced to the single-coordinate form in the two-dimensional phase matching case [@Keller:96:1] $$\begin{aligned}
\lefteqn{
\vec{E}_{\rm{PC}}(z;\vec{q}_{\|},\omega)=
\vec{E}_{\rm{PC}}^{\rm{B}}(z;\vec{q}_{\|},\omega)
}\nonumber\\ &\quad&
-{\rm{i}}\mu_{0}\omega\int\int
\tensor{G}(z,z'';\vec{q}_{\|},\omega)\cdot
\stensor{\sigma}(z'',z';\vec{q}_{\|},\omega)\cdot
\vec{E}_{\rm{PC}}(z';\vec{q}_{\|},\omega)dz''dz',
\label{eq:Loophole}
\label{eq:loop}\end{aligned}$$ possibly with $\tensor{G}(z,z'';\vec{q}_{\|},\omega)$ replaced by $\tensor{G}_{0}(z,z'';\vec{q}_{\|},\omega)$. In the quantum-well case the explicit form of $\tensor{G}(z,z'';\vec{q}_{\|},\omega)$ is known [@Bagchi:79:1], and also $\tensor{G}_{0}(z,z'';\vec{q}_{\|},\omega)$, adequate in near-field optics, is of course known. For few (one, two, three, …)-level quantum wells several schemes exist for the handling of the integral equation problem in Eq. (\[eq:Loophole\]), cf., e.g., . The only factor which in the parametric approximation makes the DFWM loop problem different from those hitherto investigated is the background field. In the present case this is given by $$\vec{E}_{\rm{PC}}^{\rm{B}}(z;\vec{q}_{\|},\omega)=-{\rm{i}}\mu_{0}\omega\int
\tensor{G}(z,z';\vec{q}_{\|},\omega)\cdot
\vec{J}_{-\omega}^{\,(3)}(z';\vec{q}_{\|},\omega)dz',$$ with $\vec{J}_{-\omega}^{\,(3)}(z';\vec{q}_{\|},\omega)$ taken from Eq. (\[eq:J3-wzq\]) in the case of simple two-dimensional plane-wave mixing, or in general from Eq. (\[eq:J3zq\]).
In the quantum well case, the pseudo-vacuum propagator $\tensor{G}(z,z'';\vec{q}_{\|},\omega)$ can be written as a sum of three terms $$\tensor{G}(z,z';\vec{q}_{\|},\omega)=
\tensor{D}(z-z';\vec{q}_{\|},\omega)+
\tensor{I}(z+z';\vec{q}_{\|},\omega)+
\btensor{g}(z-z';\omega),
\label{eq:Green1}$$ where the first two are named after the processes they describe. Thus the term $\tensor{D}(z-z';\vec{q}_{\|},\omega)$ describes the direct propagation of the electromagnetic field from a source point at $z'$ to the observation point at $z$. It is given by $$\tensor{D}(z-z';\vec{q}_{\|},\omega)=
{e^{{\rm{i}}q_{\perp}|z-z'|}\over2{\rm{i}}q_{\perp}}\left[
\vec{e}_{y}\otimes\vec{e}_{y}+\Theta(z-z')\vec{e}_{i}\otimes\vec{e}_{i}
+\Theta(z'-z)\vec{e}_{r}\otimes\vec{e}_{r}\right].
\label{eq:Green2}$$ The indirect term, $\tensor{I}(z+z';\vec{q}_{\|},\omega)$, describes the propagation from the source point of the part of the electromagnetic field that is going to the point of observation via the surface of the bulk medium. The expression for the indirect term reads $$\tensor{I}(z+z';\vec{q}_{\|},\omega)=
{e^{-{\rm{i}}q_{\perp}(z+z')}\over2{\rm{i}}q_{\perp}}\left[
r^s\vec{e}_{y}\otimes\vec{e}_{y}+r^p\vec{e}_{r}\otimes\vec{e}_{i}\right].
\label{eq:Green3}$$ Finally, the self-field term characterizes the field generated at the observation point by the current density at the same point. The self-field part of the propagator is given by $$\btensor{g}(z-z';\omega)=q^{-2}\delta(z-z')\vec{e}_{z}\otimes\vec{e}_{z},
\label{eq:Green4}$$ where $q=\omega/c_0$ is the vacuum wavenumber. In the above equations, $q_{\perp}=[q^2-q_{\|}^2]^{1/2}$, $\vec{e}_{i}=q^{-1}(q_{\perp},0,-q_{\|})$, and $\vec{e}_{r}=q^{-1}(-q_{\perp},0,-q_{\|})$, taking $\vec{q}_{\|}=q_{\|}\vec{e}_{x}$. The quantities $r^s$ and $r^p$ are the amplitude reflection coefficients of the vacuum/substrate interface in the absence of the quantum well. In general these are functions of $\vec{q}_{\|}$. The appropriate propagators for a single quantum well system are shown in Fig. \[fig:1\].
(0,0)(80,25) (1,15)(29,15) (1,0)(29,5) (5,7)(8,5)(25,19) (5,7)(25,19) (5,7)[0.5]{} (25,19)[0.5]{} (5,7.5)[(0,0)\[b\][$\vec{r}\,'$]{}]{} (25,20)[(0,0)\[b\][$\vec{r}$]{}]{} (11,13.5)[(0,0)\[tr\][$D$]{}]{} (14,7.5)[(0,0)\[bl\][$I$]{}]{} (1,25)[(0,0)\[tl\][(a)]{}]{} (51,15)(79,15) (51,0)(79,5) (55,7)(60,5)(75,13) (55,7)(75,13) (75,13)(74,8)(76,8)(75,13) (55,7)[0.5]{} (75,13)[0.5]{} (55,7.5)[(0,0)\[b\][$\vec{r}\,'$]{}]{} (77,13.5)[(0,0)[$\vec{r}$]{}]{} (63,12.5)[(0,0)\[tr\][$D$]{}]{} (67,6.5)[(0,0)\[bl\][$I$]{}]{} (77,9)[(0,0)\[l\][$g$]{}]{} (79,25)[(0,0)\[tr\][(b)]{}]{} (31,5)(49,5) (34,25)(34,0) (34,5)[1.0]{} (34,5)[0.25]{} (47,6)[(0,0)\[rb\][$x$]{}]{} (33,6)[(0,0)\[rb\][$y$]{}]{} (33,0)[(0,0)\[rb\][$z$]{}]{} (33,15)(35,15) (35.5,13)[(0,4)\[l\][$-d$]{}]{} (33,23)(35,23) (35.5,21)[(0,4)\[l\][$-z_0$]{}]{} (15,23)[0.5]{} (65,23)[0.5]{}
Some limits of the PCDFWM conductivity tensor
---------------------------------------------
### Local limit in the $z$-coordinates {#sec:local-z}
In the local limit the three interacting fields are independent of the $z$-coordinate, and thus we may calculate the local PCDFWM response tensor as $$\tensor{\Xi}(z;\vec{q}_{\|},\vec{k}_{\|})=
\iiint\tensor{\Xi}(z,z',z'',z''';\vec{q}_{\|},\vec{k}_{\|})dz'''dz''dz'.$$ Since the dependence on the three coordinates $z'''$, $z''$, and $z'$ are fairly simple we may draw some conclusions directly from looking at Eqs. (\[eq:XiAA\])–(\[eq:XiAG\]). Using an orthogonal set of wave functions, parity teaches $$\int\psi_n(z)\psi_m(z)dz=\delta_{nm},$$ where $\delta_{nm}$ is the Kronecker delta. By inspection of Eq. (\[eq:j\_hnm\]), this is the type of integral appearing when considering the $x$ and $y$ coordinates of this current density.
Then we may conclude that (i) the only independent element of part A of the PCDFWM conductivity tensor is zero in the local limit, since the two Fermi-Dirac distribution functions in Eq. (\[eq:XiAA\]) becomes identical for $n=m$. This occurs when taking the local limit in the coordinate $z''$. In addition, (ii) the five independent elements of part E of the PCDFWM conductivity tensor also becomes zero, since the two pure interband terms are zero by themselves, and the two other terms are of the same magnitude but with opposite sign. This occurs when taking the local limit in the coordinates $z'''$ and $z''$. Furthermore, (iii) for part G of the PCDFWM conductivity tensor, elements with the Cartesian index $i=z$ and the other indices different from $z$ becomes zero, since the other indices implies that all quantum numbers in the summation become identical. Finally, (iv) the only independent element of part C of the PCDFWM conductivity tensor is reduced to a pure intraband contribution. The same reduction appears in tensor elements of parts D, F, and G with no Cartesian coordinate index $z$ in indices $jk$, $ih$, and $ijkh$, respectively. In part B of the PCDFWM conductivity tensor, the elements with no Cartesian index $z$ in $kh$ apparantly gives the same result, but the following integration over $\vec{\kappa}_{\|}$ makes them vanish. These conclusions are shown in the form of symmetry schemes in Fig. \[fig:8.2\].
(0,4)(170,160) (0,80)[ (75,6)[(0,5)\[bl\][(B)]{}]{} (2,0)(24,0)[3]{}[(0,0)(0,24)[3]{}[ (6,6)(4,6)(4,22)(6,22) (22,6)(24,6)(24,22)(22,22) ]{}]{} (10,8)(24,0)[3]{}[(0,0)(0,24)[3]{}[ (0,0)(6,0)[3]{}[(0,0)(0,6)[3]{}[ ]{}]{}]{}]{} [ (2,70)[(0,5)\[l\][$[xxkh]$]{}]{} (2,46)[(0,5)\[l\][$[yxkh]$]{}]{} (2,22)[(0,5)\[l\][$[zxkh]$]{}]{} (40,70)[(0,5)\[c\][$[xykh]$]{}]{} (52,46)[(0,5)\[r\][$[yykh]$]{}]{} (28,22)[(0,5)\[l\][$[zykh]$]{}]{} (78,70)[(0,5)\[r\][$[xzkh]$]{}]{} (78,46)[(0,5)\[r\][$[yzkh]$]{}]{} (78,22)[(0,5)\[r\][$[zzkh]$]{}]{} ]{} (10,56)(0,6)[3]{}[(0,0)(6,0)[3]{}]{} (34,32)(0,6)[3]{}[(0,0)(6,0)[3]{}]{} (58,8)(0,6)[3]{}[(0,0)(6,0)[3]{}]{} (10,56)(20,42)(34,32) (34,32)(48,22)(58,8) (16,56)(26,42)(40,32) (40,32)(54,22)(64,8) (22,56)(32,42)(46,32) (46,32)(60,22)(70,8) (22,62)(32,48)(46,38) (46,38)(60,28)(70,14) (22,68)(32,54)(46,44) (46,44)(60,34)(70,20) (10,62)(24,-24)[3]{}[(0,0)(6,0)[2]{}[(0,0)(0,6)[2]{}[ (0,0)[1.5]{} ]{}]{}]{} ]{} (90,80)[ (5,6)[(0,5)\[br\][(D)]{}]{} (2,0)(24,0)[3]{}[(0,0)(0,24)[3]{}[ (6,6)(4,6)(4,22)(6,22) (22,6)(24,6)(24,22)(22,22) ]{}]{} (10,8)(24,0)[3]{}[(0,0)(0,24)[3]{}[ (0,0)(6,0)[3]{}[(0,0)(0,6)[3]{}[ ]{}]{}]{}]{} [ (2,70)[(0,5)\[l\][$[xxkh]$]{}]{} (1,46)[(0,5)\[l\][$[yxkh]$]{}]{} (2,22)[(0,5)\[l\][$[zxkh]$]{}]{} (40,70)[(0,5)\[c\][$[xykh]$]{}]{} (50,46)[(0,5)\[r\][$[yykh]$]{}]{} (30,22)[(0,5)\[l\][$[zykh]$]{}]{} (78,70)[(0,5)\[r\][$[xzkh]$]{}]{} (78,46)[(0,5)\[r\][$[yzkh]$]{}]{} (79,22)[(0,5)\[r\][$[zzkh]$]{}]{} ]{} (22,8)(-6,24)[3]{}[(0,0)(24,0)[3]{}[ (0,0)(0,6)[3]{}[ ]{}]{}]{} (10,68)(24,0)[3]{}[ (0,0)(0,-6)[3]{}[ (0,0)(12,-48) ]{}]{} (22,14)(24,0)[2]{}[(0,0)(-6,24)[3]{}[ (0,0)(0,6)[2]{}[ (0,0)[1]{} (0,0)[1]{} ]{}]{}]{} ]{} (0,0)[ (75,6)[(0,5)\[bl\][(F)]{}]{} (2,0)(24,0)[3]{}[(0,0)(0,24)[3]{}[ (6,6)(4,6)(4,22)(6,22) (22,6)(24,6)(24,22)(22,22) ]{}]{} (10,8)(24,0)[3]{}[(0,0)(0,24)[3]{}[ (0,0)(6,0)[3]{}[(0,0)(0,6)[3]{}[ ]{}]{}]{}]{} [ (2,70)[(0,5)\[l\][$[xxkh]$]{}]{} (2,46)[(0,5)\[l\][$[yxkh]$]{}]{} (2,22)[(0,5)\[l\][$[zxkh]$]{}]{} (40,70)[(0,5)\[c\][$[xykh]$]{}]{} (40,46)[(0,5)\[c\][$[yykh]$]{}]{} (40,22)[(0,5)\[c\][$[zykh]$]{}]{} (78,70)[(0,5)\[r\][$[xzkh]$]{}]{} (78,46)[(0,5)\[r\][$[yzkh]$]{}]{} (78,22)[(0,5)\[r\][$[zzkh]$]{}]{} ]{} (10,20)(0,24)[3]{}[(0,0)(24,-6)[3]{}[ (0,0)(6,0)[3]{}[ ]{}]{}]{} (10,68)(0,-24)[3]{}[ (0,0)(6,0)[3]{}[ (0,0)(48,-12) ]{}]{} (10,44)(0,24)[2]{}[(0,0)(24,-6)[3]{}[ (0,0)(6,0)[2]{}[ (0,0)[1]{} (0,0)[1]{} ]{}]{}]{} ]{} (90,0)[ (5,6)[(0,5)\[br\][(G)]{}]{} (2,0)(24,0)[3]{}[(0,0)(0,24)[3]{}[ (6,6)(4,6)(4,22)(6,22) (22,6)(24,6)(24,22)(22,22) ]{}]{} (10,8)(24,0)[3]{}[(0,0)(0,24)[3]{}[ (0,0)(6,0)[3]{}[ ]{}]{}]{} (10,8)(24,0)[3]{}[(12,6)(0,24)[3]{}[ (0,0)(0,6)[2]{}[ ]{}]{}]{} (58,8)(0,24)[3]{}[(0,6)(0,6)[2]{}[ (0,0)(6,0)[2]{}[ ]{}]{}]{} (10,32)(24,0)[2]{}[(0,6)(0,24)[2]{}[(0,0)(6,0)[2]{}[ (0,0)(0,6)[2]{}[ (0,0)[1]{} (0,0)[1]{} ]{}]{}]{}]{} (10,8)(24,0)[2]{}[(0,6)(0,6)[2]{}[ (0,0)(6,0)[3]{}[ ]{}]{}]{} [ (2,70)[(0,5)\[l\][$[xxkh]$]{}]{} (2,46)[(0,5)\[l\][$[yxkh]$]{}]{} (2,22)[(0,5)\[l\][$[zxkh]$]{}]{} (40,70)[(0,5)\[c\][$[xykh]$]{}]{} (40,46)[(0,5)\[c\][$[yykh]$]{}]{} (40,22)[(0,5)\[c\][$[zykh]$]{}]{} (78,70)[(0,5)\[r\][$[xzkh]$]{}]{} (78,46)[(0,5)\[r\][$[yzkh]$]{}]{} (78,22)[(0,5)\[r\][$[zzkh]$]{}]{} ]{} ]{}
### Local limit along the surface {#sec:local-xy}
Taking the local limit along the surface coordinates, the wavenumbers are considered to be much less than the Fermi wavenumber, i.e., we take the limit where $\vec{q}_{\|}\rightarrow\vec{0}$ and $\vec{k}_{\|}\rightarrow\vec{0}$ (the dipole limit).
Then from Eq. (\[eq:transition-w\]) we observe that the transition frequencies become independent of $\vec{\kappa}_{\|}$, and thus we conclude that this approximation makes the integration over $\vec{\kappa}_{\|}$ particularly simple in the low temperature limit, since no integration variables appears in any of the denominators in Eqs. (\[eq:XiAA\])–(\[eq:XiAG\]). We further observe that only interband contributions are left compared to the full description.
### Local limit in three coordinates {#sec:local-xyz}
Locality in all three coordinates is achieved by a combination of the two limits mentioned in Secs. \[sec:local-z\] and \[sec:local-xy\] above. Thus, in this limit (i) parts A and E of the DFWM conductivity tensor does not contribute for the same reasons as before. Furthermore (ii) the integration over $z'$ in part C makes this part vanish, (iii) the integration over $z'$ in part F of the DFWM conductivity tensor cancels two terms, and the other two are of the same magnitude, but with opposite signs, resulting in the fact that part F does not contribute to the response in this limit. Finally (iv), all purely intraband contributing tensor elements found in the local limit vanish. All in all we are left with five independent nonzero elements of parts B and D, and fiftyseven independent nonzero elements of G, all labeled with a “$\bullet$” in Fig. \[fig:8.2\].
### Single level quantum well {#sec:OneLevel}
A substantial simplification of Eqs. (\[eq:XiAA\])–(\[eq:XiAG\]) occur in [*one*]{} special case, namely in the case where the thin film is a single level quantum well in the $z$-direction. In the single level quantum well, the summation indices are all equal to $1$.
Special attention is in this case devoted to the current density defined in Eq. (\[eq:j\_hnm\]), which in a single level quantum well is reduced to $$j_{h,11}(z;{\kappa}_{h})=-{e\hbar\over2m_{e}}
(\delta_{hx}+\delta_{hy})
{\kappa}_{h}|\psi_{1}(z)|^2,$$ for $h\in\{x,y\}$, since the $z$-dependent part vanish for any $n=m$. This observation leads to a drastic reduction of the number of contributing elements in most of the symmetry schemes associated with the occuring processes.
The only nonzero element in part A of the nonlinear conductivity tensor vanish for $n=m$. In part B of the conductivity tensor all elements with $k$ or $h$ equal to $z$ vanish for $n=m=v$, and the rest of the elements vanish by integration over $\vec{\kappa}_{\|}$. The only nonzero element of part C of the nonlinear conductivity tensor is conserved, but simplified. Part D of the nonlinear conductivity tensor is reduced somewhat, since either combination of $j=z$ or $k=z$ gives zero. Then we are left with four nonzero independent elements, as shown in Fig. \[fig:8.2\].D. Part E does not give any contributions to the intraband transitions, since two terms in the sum gives no intraband contributions in general, and the other two terms cancel each other. Part F is reduced in a manner similar to parts B and D, since any combination of $i=z$ or $h=z$ gives zero. The resulting four nonzero independent elements are shown in Fig. \[fig:8.2\].F. In the last part (G) of the nonlinear conductivity tensor any combination of $i=z$, $j=z$, $k=z$, or $h=z$ gives zero. As a consequence of this rather drastic reduction we are left with sixteen nonzero elements, as shown in Fig. \[fig:8.2\].G.
Polarized light in the $x$-$z$-plane {#ch:9}
====================================
\[ch:system\] Restricting ourselves to consider light propagating in the $x$-$z$-plane, which furthermore is polarized either in the $x$-$z$-plane ($p$-polarized) or perpendicular to the $x$-$z$-plane ($s$-polarized), the treatment can be split into eight separate parts related to the possible combinations of polarization of the three different incident fields. In this scattering geometry $\vec{q}_{\|}$ (and $\vec{k}_{\|}$) lie along the $x$-axis, giving a mirror plane at $y=0$. Consequently, only tensor elements in the ($3\times3\times3\times3$) PCDFWM response tensor with a Cartesian index even numbered in $y$ contributes, and the $81$ tensor elements generally appearing are reduced to $41$.
(80,76)(0,4) (2,0)(24,0)[3]{}[(0,0)(0,24)[3]{}[ (6,6)(4,6)(4,22)(6,22) (22,6)(24,6)(24,22)(22,22) ]{}]{} [ (2,70)[(0,5)\[l\][$[xxkh]$]{}]{} (2,46)[(0,5)\[l\][$[yxkh]$]{}]{} (2,22)[(0,5)\[l\][$[zxkh]$]{}]{} (40,70)[(0,5)\[c\][$[xykh]$]{}]{} (40,46)[(0,5)\[c\][$[yykh]$]{}]{} (40,22)[(0,5)\[c\][$[zykh]$]{}]{} (78,70)[(0,5)\[r\][$[xzkh]$]{}]{} (78,46)[(0,5)\[r\][$[yzkh]$]{}]{} (78,22)[(0,5)\[r\][$[zzkh]$]{}]{} ]{} (40,38) (10,68) (10,56) (10,20) (10,8) (22,68) (22,56) (22,20) (22,8) (58,68) (58,56) (58,20) (58,8) (70,68) (70,56) (70,20) (70,8) (16,62)[(0,0)\[c\][$\blacksquare$]{}]{} (16,14)[(0,0)\[c\][$\blacksquare$]{}]{} (64,62)[(0,0)\[c\][$\blacksquare$]{}]{} (64,14)[(0,0)\[c\][$\blacksquare$]{}]{} (34,32)[(0,0)\[c\][$\square$]{}]{} (34,44)[(0,0)\[c\][$\square$]{}]{} (46,32)[(0,0)\[c\][$\square$]{}]{} (46,44)[(0,0)\[c\][$\square$]{}]{} (40,8)[(0,0)\[c\][$\clubsuit$]{}]{} (40,20)[(0,0)\[c\][$\clubsuit$]{}]{} (40,56)[(0,0)\[c\][$\clubsuit$]{}]{} (40,68)[(0,0)\[c\][$\clubsuit$]{}]{} (16,32)[(0,0)\[c\][$\diamondsuit$]{}]{} (16,44)[(0,0)\[c\][$\diamondsuit$]{}]{} (64,32)[(0,0)\[c\][$\diamondsuit$]{}]{} (64,44)[(0,0)\[c\][$\diamondsuit$]{}]{} (34,14)[(0,0)\[c\][$\heartsuit$]{}]{} (46,14)[(0,0)\[c\][$\heartsuit$]{}]{} (34,62)[(0,0)\[c\][$\heartsuit$]{}]{} (46,62)[(0,0)\[c\][$\heartsuit$]{}]{} (10,38)[(0,0)\[c\][$\spadesuit$]{}]{} (22,38)[(0,0)\[c\][$\spadesuit$]{}]{} (58,38)[(0,0)\[c\][$\spadesuit$]{}]{} (70,38)[(0,0)\[c\][$\spadesuit$]{}]{} (10,14) (10,32) (10,44) (10,62) (16,8) (16,20) (16,38) (16,56) (16,68) (22,14) (22,32) (22,44) (22,62) (34,8) (34,20) (34,38) (34,56) (34,68) (40,14) (40,32) (40,44) (40,62) (46,8) (46,20) (46,38) (46,56) (46,68) (58,14) (58,32) (58,44) (58,62) (64,8) (64,20) (64,38) (64,56) (64,68) (70,14) (70,32) (70,44) (70,62)
Applying the two polarization states $s$ and $p$ chosen above to the three interacting fields, the resulting eight different combinations uses different matrix elements in the nonlinear conductivity tensor, and (as would be expected) these eight combinations together make use of [*all*]{} elements of the nonlinear conductivity tensor. This division is shown in Fig. \[fig:polarize\] for the $41$ contributing tensor elements as described in the following. The noncontributing elements of the nonlinear conductivity tensor is denoted using the symbol “$\cdot$” in Fig. \[fig:polarize\].
(0,0)(115,20) (0,0)(30,0)[4]{}[ (0,10)(25,10) (0,5)(11.5,5) (25,5)(13.5,5) (25,0)[(0,4)\[r\][(1)]{}]{} (0,0)[(0,4)\[l\][(2)]{}]{} (1,13)[(0,4)\[l\][(p)]{}]{} (3.4,19.1)(11.5,11) ]{} (0,0)[ (6.5,5)[1.0]{} (6.5,5)[0.25]{} (18.5,5)[1.0]{} (18.5,5)[0.25]{} (8,14.5)[ (0,0)[1.0]{} (0,0)[0.25]{} ]{} (25,20)[(0,0)\[tr\][(a)]{}]{} ]{} (30,0)[ (6.5,2)(6.5,8) (18.5,2)(18.5,8) (8,14.5)[ (0,0)[1.0]{} (0,0)[0.25]{} ]{} (25,20)[(0,0)\[tr\][(b)]{}]{} ]{} (60,0)[ (6.5,5)[1.0]{} (6.5,5)[0.25]{} (18.5,5)[1.0]{} (18.5,5)[0.25]{} (5.9,12.4)(10.1,16.6) (25,20)[(0,0)\[tr\][(c)]{}]{} ]{} (90,0)[ (6.5,2)(6.5,8) (18.5,2)(18.5,8) (5.9,12.4)(10.1,16.6) (25,20)[(0,0)\[tr\][(d)]{}]{} ]{}
(0,0)(115,20) (0,0)[ (0,10)(25,10) (0,5)(11.5,5) (25,5)(13.5,5) (6.5,2)(6.5,8) (18.5,5)[1.0]{} (18.5,5)[0.25]{} (25,0)[(0,4)\[r\][(1)]{}]{} (0,0)[(0,4)\[l\][(2)]{}]{} (1,13)[(0,4)\[l\][(p)]{}]{} (3.4,19.1)(11.5,11) (8,14.5)[ (0,0)[1.0]{} (0,0)[0.25]{} ]{} (25,20)[(0,0)\[tr\][(a)]{}]{} ]{} (30,0)[ (0,10)(25,10) (0,5)(11.5,5) (25,5)(13.5,5) (6.5,5)[1.0]{} (6.5,5)[0.25]{} (18.5,2)(18.5,8) (25,0)[(0,4)\[r\][(1)]{}]{} (0,0)[(0,4)\[l\][(2)]{}]{} (1,13)[(0,4)\[l\][(p)]{}]{} (3.4,19.1)(11.5,11) (8,14.5)[ (0,0)[1.0]{} (0,0)[0.25]{} ]{} (25,20)[(0,0)\[tr\][(b)]{}]{} ]{} (60,0)[ (0,10)(25,10) (0,5)(11.5,5) (25,5)(13.5,5) (6.5,2)(6.5,8) (18.5,5)[1.0]{} (18.5,5)[0.25]{} (20,0)[(0,4)\[r\][(1)]{}]{} (0,0)[(0,4)\[l\][(2)]{}]{} (1,13)[(0,4)\[l\][(p)]{}]{} (3.4,19.1)(11.5,11) (5.9,12.4)(10.1,16.6) (25,20)[(0,0)\[tr\][(c)]{}]{} ]{} (90,0)[ (0,10)(25,10) (0,5)(11.5,5) (25,5)(13.5,5) (6.5,5)[1.0]{} (6.5,5)[0.25]{} (18.5,2)(18.5,8) (25,0)[(0,4)\[r\][(1)]{}]{} (0,0)[(0,4)\[l\][(2)]{}]{} (1,13)[(0,4)\[l\][(p)]{}]{} (3.4,19.1)(11.5,11) (5.9,12.4)(10.1,16.6) (25,20)[(0,0)\[tr\][(d)]{}]{} ]{}
Eight sets of contributing matrix elements
------------------------------------------
From the point of view of the probe, the eight different combinations of polarized light can be divided into two groups of four, namely four giving a PCDFWM response with the same polarization as the probe and four giving the other polarization as the PCDFWM response. In the four combinations giving response of the same polarization as the probe, the two pump fields have the same polarization states. These configurations are sketched in Fig. \[fig:equalpumps\]. The other four combinations, where the two pump fields are differently polarized are sketched in Fig. \[fig:diff.pumps\].
Two out of the first four combinations describe $s$ to $s$ transitions, seen from the point of view of the probe. (i) The simplest combination arises when both pump fields and the probe field are $s$-polarized, as shown in Fig. \[fig:equalpumps\].a. In this case, only the $yyyy$ element of the nonlinear conductivity tensor is present. In Fig. \[fig:polarize\] it is marked with a “$\bullet$”. It should be noted that this is the [*only*]{} case of the eight, in which a single matrix element can be determined independently in an actual experiment. (ii) When both pump fields are $p$-polarized and the probe field is $s$-polarized (see Fig. \[fig:equalpumps\].b), the four contributing matrix elements in the nonlinear conductivity tensor have indices $i$ and $j$ equal to $y$ and indices $k$ and $h$ different from $y$. Each of these four elements is marked with a “$\square$” in Fig. \[fig:polarize\].
From the same point of view the other two of the first four combinations describe $p$ to $p$ transitions. (iii) If both pump fields are $s$-polarized and the probe field is $p$-polarized the configuration is sketched in Fig. \[fig:equalpumps\].c, and four matrix elements in the nonlinear conductivity tensor contribute to the solution. They have indices $k$ and $h$ equal to $y$ and indices $i$ and $j$ different from $y$. In Fig. \[fig:polarize\], each of these elements is marked with a “$\blacksquare$”. (iv) The other extreme case \[the simple extreme has been described in item (i)\] is the combination where both pump fields and the probe field are $p$-polarized, as shown in Fig. \[fig:equalpumps\].d. In order to obtain the solution for this combination as many as sixteen elements of the nonlinear conductivity tensor are required, since every element with an index without $y$’s in it contributes. Each of these elements is marked in Fig. \[fig:polarize\] with a “$\circ$”.
Still taking the “probe to response” point of view, two of the remaining four cases represent a probe to response transition from $s$ to $p$. (v) If pump field $1$ is $s$-polarized, pump field $2$ is $p$-polarized, and the probe field is $s$-polarized (the corresponding diagram is showed in Fig. \[fig:diff.pumps\].a), the four contributing matrix elements have indices $j$ and $k$ equal to $y$ and indices $i$ and $h$ different from $y$. In Fig. \[fig:polarize\] each of these elements is marked with the symbol “$\clubsuit$”. In the other of these cases, (vi), pump field $1$ is $p$-polarized and pump field $2$ is $s$-polarized, and we take the probe field to be $s$-polarized. This combination is sketched in Fig. \[fig:diff.pumps\].b, and the four contributing elements in the nonlinear conductivity tensor then have indices $j$ and $k$ equal to $y$ and indices $i$ and $h$ different from $y$. In Fig. \[fig:polarize\] the symbol “$\heartsuit$” is used to show these elements. As a direct consequence of the conservation of momentum criterion these two combinations are equivalent, since by replacing $\vec{q}_{\|}$ with $-\vec{q}_{\|}$ and $\vec{k}_{\|}$ with $-\vec{k}_{\|}$ the situation in (vi) changes to the situation in (v).
The last two combinations represent a transition from $p$ to $s$ in the picture from probe to response. (vii) If pump field $1$ is $s$-polarized and pump field $2$ is $p$-polarized, but the probe field is $p$-polarized, the situation is as sketched in Fig. \[fig:diff.pumps\].c. Then again four elements of the nonlinear conductivity tensor contribute to the solution. These four elements have indices $i$ and $h$ equal to $y$ and indices $j$ and $k$ different from $y$, and each element is marked using the symbol “$\diamondsuit$” in Fig. \[fig:polarize\]. Finally, (viii), when pump field $1$ is $p$-polarized and pump field $2$ is $s$-polarized, but the probe field is $p$-polarized, the configuration appears as shown in Fig. \[fig:diff.pumps\].d, which again gives four elements of the nonlinear conductivity tensor contributing to the solution, the indices $i$ and $k$ being equal to $y$ and the indices $j$ and $h$ being different from $y$. We mark each of these elements with the symbol “$\spadesuit$” in Fig. \[fig:polarize\]. For the same reason as before, cases (vii) and (viii) are equivalent.
Simplified description by choice of pump fields
-----------------------------------------------
Although the two pump fields has been drawn parallel to the interface in Figs. \[fig:equalpumps\] and \[fig:diff.pumps\], this is not a requirement, as long as they are counterpropagating to each other. If we are looking for a simplification in the treatment of optical phase conjugation from a quantum well structure, a reduction in the number of tensor elements to be calculated could be one alternative. Two immediate possibilities comes to mind. In the first case, the pump fields are taken to be parallel to the $x$-axis. The second case has the pump fields parallel to the $z$-axis. The consequences of these two cases are described in the following.
If the pump fields propagate in a direction parallel to the $x$-axis, the number of contributing tensor elements in the nonlinear conductivity tensor $\tensor{\Xi}(z,z',z'',z''';\vec{q}_{\|},\vec{k}_{\|})$ is reduced from $41$ to $18$ when considering $s$- and $p$-polarized light only. The surviving elements can be divided into four cases following the four possible combinations of polarization of the pump fields. When (i) both pump fields are $p$-polarized, their respective electric fields have only a $z$-component, and hence $h=k=z$. Similarly (ii), when both pump fields are $s$-polarized, their electric fields only have a $y$-component, that is, $h=k=y$. In case (iii) pump field 1 is $p$-polarized while pump field 2 is $s$-polarized, giving $h=z$ and $k=y$. In the final case (iv) the pump fields are polarized oppositely to those in case (iii), i.e., $h=y$ and $k=z$.
If we choose the pump fields to propagate in a direction parallel to the $z$ axis instead, the number of contributing tensor elements in the nonlinear conductivity tensor $\tensor{\Xi}(z,z',z'',z''';\vec{q}_{\|},\vec{k}_{\|})$ is again reduced from $41$ to $18$ when considering $s$- and $p$-polarized light, and again the surviving elements are divided into four groups following the four possible combinations of polarization the pump fields can have. Thus, (i) when both pump fields are $p$-polarized, the electric fields representing them have only $x$-components, i.e., $h=k=x$, (ii) for the pump fields both being $s$-polarized, the same elements as when the pump fields are parallel to the $x$-axis contributes to the solution, giving again $h=k=y$. In the cases of differently polarized pump fields, (iii) $h=x$ and $k=y$ when pump field 1 is $p$-polarized and pump field 2 is $s$-polarized, and (iv) $h=y$ and $k=x$ when the opposite polarizations occur.
In conclusion, these two possibilities of choice have five common contributing elements, namely the ones where $k=h=y$. At the same time, ten elements of the nonlinear conductivity tensor does not contribute to either simplification. They have $kh\in\{xz,zx\}$.
\[part:IV\]
Theoretical considerations {#ch:10}
==========================
Having discussed the properties of optical phase conjugation in quantum well structures in general, let us consider here the simplest configuration of a mesoscopic metallic optical quantum-well phase conjugator. In this case only a single bound state exists below the Fermi level and it is assumed that no levels above the Fermi level can be reached with the applied optical field. Such a quantum well is called a single-level quantum well.
Phase conjugated field
----------------------
In a mesoscopic film the electric field generated via the direct and indirect processes at a given point is roughly speaking of the order $(\mu_0\omega/q_{\perp})\int\vec{J}_{-\omega}^{\,(3)}dz'$, whereas the self-field has the magnitude $(\mu_0\omega/q^{2})\vec{J}_{-\omega}^{\,(3)}$. Since $qd\ll1$, where $d$ is the thickness of the film, we judge the self-field term to dominate the phase conjugated field inside the quantum well, at least for single-level metallic quantum wells which have thicknesses on the atomic length scale. In the following we therefore use the so-called self-field (electrostatic) approximation to calculate the phase conjugated field inside the quantum well. With the propagator $\tensor{G}(z,z'';\vec{q}_{\|},\omega)$ replaced by $\btensor{g}(z-z';\omega)$, the phase conjugated field fulfills the integral equation $$\vec{E}_{\rm{PC}}(z;\vec{q}_{\|},\omega)=
\vec{E}_{\rm{PC}}^{\rm{B}}(z;\vec{q}_{\|},\omega)
+{\vec{e}_{z}\otimes\vec{e}_{z}\over{}{\rm{i}}\varepsilon_0\omega}
\cdot\int\stensor{\sigma}(z,z';\vec{q}_{\|},\omega)\cdot
\vec{E}_{\rm{PC}}(z';\vec{q}_{\|},\omega)dz'
\label{eq:ping}$$ inside the well, and the background field is now $$\vec{E}_{\rm{PC}}^{\rm{B}}(z;\vec{q}_{\|},\omega)=
{\vec{e}_{z}\otimes\vec{e}_{z}\over{}{\rm{i}}\varepsilon_0\omega}
\vec{J}_{-\omega}^{\,(3)}(z;\vec{q}_{\|},\omega).
\label{eq:pong}$$ In the self-field approach the phase conjugated field has only a component perpendicular to the surface (the $z$-component) inside the well and only the $z$-component of the nonlinear current density $\vec{J}_{-\omega}^{\,(3)}$ drives the process.
Once the phase conjugated field inside the quantum well has been determined in a self-consistent manner from Eq. (\[eq:ping\]), it can be determined outside using Eq. (\[eq:loop\]). The self-field does of course not contribute to the exterior field, and no loop problem is involved. All that need to be done is to integrate known quantities in the $z$-direction over the well.
Nonlinear conductivity tensor
-----------------------------
As we may recall, the nonlinear conductivity tensor appearing in Eq. (\[eq:J3-wzq\]) may in general be written as a sum of seven parts (A–G) after the physical processes they describe. These have the tensor symmetries shown in Tab. \[tab:1\]. In this chapter we use this conductivity tensor in the form it takes for media with two-dimensional translational invariance as it was developed in Part \[part:II\], but for quantum wells so thin that only a single bound level exists. The quantum well may be free standing, or it may be deposited on a substrate that can be described by a refractive index $n$ relative to the vacuum on the other side of the film. The surface of the film is parallel to the $x$-$y$-plane in a Cartesian coordinate system, and the interface between the film and the substrate is placed at $z=0$ as shown in Fig. \[fig:1\]. We further limit our study to the case where (i) all scattering takes place in the $x$-$z$-plane, (ii) the interacting fields are linearly polarized in ($p$) or perpendicular to ($s$) the scattering plane, (iii) the pump fields in the phase conjugating system are counterpropagating monochromatic plane waves with a uniform amplitude along the $z$-axis and propagating in a direction parallel to the $x$-axis, and (iv) the field is calculated within the self-field approximation.
From (i) above we get a mirror plane at $y=0$, leaving only tensor elements of the conductivity tensors with an even number $(0,2,4)$ of $y$’s in the Cartesian index nonzero. Condition (iii) implies as a consequence of condition (ii) that no tensor elements of the nonlinear conductivity tensor with one or both of the last two Cartesian indices as $x$ contributes to the phase conjugated response. Requirement (iv) above implies that the first Cartesian index of a tensor element should be $z$ in order to contribute to the phase conjugated response. The choice of a single level quantum well in itself restricts the transition current density to contain $x$- and $y$- components only. Together with the fact that part B gives zero after integration over $\vec{\kappa}_{\|}$ and E gives pure interband contributions, these choices leave two nonzero elements of the nonlinear conductivity tensor, namely $$\begin{aligned}
\lefteqn{
{\Xi}_{zyyz}^{\rm{C}}(z,z',z'',z''';q_{\|}-k_{\|})=
{\Xi}_{zzzz}^{\rm{C}}(z,z',z'',z''';q_{\|}-k_{\|})=
}\nonumber\\ &\quad&
{e^4\over2^4\pi^2{\rm{i}}\hbar\omega^3m_{e}^2}
{\cal{C}}(q_{\|}-k_{\|})
\delta(z'-z''')\delta(z-z'')
|\psi(z')|^2|\psi(z)|^2,
\label{eq:XiEC}
\\
\lefteqn{
{\Xi}_{zyyz}^{\rm{D}}(z,z',z'',z''';q_{\|},k_{\|})=
{e^4\over2^6\pi^2i\omega^3m_e^3}
{\cal{D}}(q_{\|},k_{\|})
\delta(z-z''')
|\psi(z'')|^2|\psi(z')|^2|\psi(z)|^2,
}\nonumber\\
\label{eq:XiED}\end{aligned}$$ where $$\begin{aligned}
\lefteqn{
{\cal{C}}(q_{\|}-k_{\|})=2
\int{f(\vec{\kappa}_{\|}+[{q}_{\|}-{k}_{\|}]\vec{e}_{x})-f(\vec{\kappa}_{\|})
\over\hbar(q_{\|}-k_{\|})[2\kappa_x+q_{\|}-k_{\|}]/(2m_e)-{\rm{i}}/\tau}
d^2\kappa_{\|},
}\label{eq:C}
\\
\lefteqn{
{\cal{D}}(q_{\|},k_{\|})=2\int
{\kappa_y^2\over\hbar(q_{\|}+k_{\|})[2\kappa_x+q_{\|}+k_{\|}]/(2m_e)-{\rm{i}}/\tau}
}\nonumber\\ &\quad&\times
\left({f(\vec{\kappa}_{\|})-f(\vec{\kappa}_{\|}+{k}_{\|}\vec{e}_{x})
\over\hbar{}k_{\|}[2\kappa_x+k_{\|}]/(2m_e)-{\rm{i}}/\tau-\omega}
+{f(\vec{\kappa}_{\|}+[{k}_{\|}+{q}_{\|}]\vec{e}_{x})
-f(\vec{\kappa}_{\|}+{k}_{\|}\vec{e}_{x})
\over\hbar{}q_{\|}[2\kappa_x+q_{\|}+2k_{\|}]/(2m_e)-{\rm{i}}/\tau+\omega}
\right.\nonumber\\ &&\left.
+{f(\vec{\kappa}_{\|})-f(\vec{\kappa}_{\|}+{q}_{\|}\vec{e}_{x})
\over\hbar{}q_{\|}[2\kappa_x+q_{\|}]/(2m_e)-{\rm{i}}/\tau+\omega}
+{f(\vec{\kappa}_{\|}+[{k}_{\|}+{q}_{\|}]\vec{e}_{x})
-f(\vec{\kappa}_{\|}+{q}_{\|}\vec{e}_{x})
\over\hbar{}k_{\|}[2\kappa_x+k_{\|}+2q_{\|}]/(2m_e)-{\rm{i}}/\tau-\omega}
\right)
d^2\kappa_{\|}.
\nonumber\\
\label{eq:D}\end{aligned}$$ The number $2$ appearing in front of the integrals above represents the summation over the degenerate spin energies.
The free-particle character of the electron motion in the plane of the quantum well enables us to write the solutions to the light-unperturbed Schr[ö]{}dinger equation in the form $\Psi(\vec{r}\,)=(2\pi)^{-1}\psi(z)\exp({\rm{i}}\vec{\kappa}_{\|}\cdot\vec{r}\,)$, where $\vec{\kappa}_{\|}=(\kappa_x,\kappa_y,0)$ is the wavevector of the electron in consideration and $\psi(z)$, appearing in Eqs. (\[eq:XiEC\]) and (\[eq:XiED\]), is the $z$-dependent part of the wave function, common to all electrons. The $x$-$y$-dependent parts of the wave functions, $(2\pi)^{-1}\exp({\rm{i}}\vec{\kappa}_{\|}\cdot\vec{r}\,)$, are orthonormalized in the Dirac sense, i.e., they obey the equation $(2\pi)^{-2}\int\exp[{\rm{i}}(\vec{\kappa}_{\|}-\vec{\kappa}_{\|}')\cdot\vec{r}\,]d^2r=\delta(\vec{\kappa}_{\|}-\vec{\kappa}_{\|}')$, and the $z$-dependent part fulfills the separate normalization condition $\int|\psi(z)|^2dz=1$. In Eqs. (\[eq:C\]) and (\[eq:D\]) the response of all electrons is taken into account by integrating over all possible $\vec{\kappa}_{\|}$ wavevectors. The eigenenergy ${\cal{E}}(\vec{\kappa}_{\|})$ belonging to the state $\Psi(\vec{r}\,)$ is obtained by adding to the common bound-state energy $\varepsilon$, the kinetic energy in the parallel motion. Thus $${\cal{E}}(\vec{\kappa}_{\|})=\varepsilon+{\hbar^2\over2m_{e}}\kappa_{\|}^2.$$ The quantity $f(\vec{\kappa}_{\|})=[1+\exp\{({\cal{E}}(\vec{\kappa}_{\|})-\mu)/(k_{\rm{B}}T)\}]^{-1}$ denotes the Fermi-Dirac distribution function for this eigenstate, $\mu$ being the chemical potential of the electron system, $k_{\rm{B}}$ the Boltzmann constant, and $T$ the absolute temperature.
Probe with single Fourier component
-----------------------------------
In the following we calculate the phase conjugated field generated by a probe field which consists of only one plane-wave component of wavevector $\vec{q}=(q_{\|},0,q_{\perp})$. A probe field of the form $\vec{E}(z;\vec{q}_{\|})=\vec{E}e^{{\rm{i}}q_{\perp}z}$ is hence inserted in Eq. (\[eq:J3-wzq\]).
(0,0)(85,22) (0,0)[ (0,10)(25,10) (0,5)(11.5,5) (25,5)(13.5,5) (6.5,2)(6.5,8) (18.5,2)(18.5,8) (25,0)[(0,4)\[r\][(1)]{}]{} (0,0)[(0,4)\[l\][(2)]{}]{} (1,13)[(0,4)\[l\][(p)]{}]{} (3.4,19.1)(11.5,11) (5.9,12.4)(10.1,16.6) (25,20)[(0,0)\[tr\][(a)]{}]{} ]{} (30,0)[ (0,10)(25,10) (0,5)(11.5,5) (25,5)(13.5,5) (6.5,2)(6.5,8) (18.5,5)[1.0]{} (18.5,5)[0.25]{} (25,0)[(0,4)\[r\][(1)]{}]{} (0,0)[(0,4)\[l\][(2)]{}]{} (1,13)[(0,4)\[l\][(p)]{}]{} (3.4,19.1)(11.5,11) (8.0,14.5)[ (0,0)[1.0]{} (0,0)[0.25]{} ]{} (25,20)[(0,0)\[tr\][(b)]{}]{} ]{} (60,0)[ (0,10)(25,10) (0,5)(11.5,5) (25,5)(13.5,5) (6.5,5)[1.0]{} (6.5,5)[0.25]{} (18.5,2)(18.5,8) (25,0)[(0,4)\[r\][(1)]{}]{} (0,0)[(0,4)\[l\][(2)]{}]{} (1,13)[(0,4)\[l\][(p)]{}]{} (3.4,19.1)(11.5,11) (8.0,14.5)[ (0,0)[1.0]{} (0,0)[0.25]{} ]{} (25,20)[(0,0)\[tr\][(c)]{}]{} ]{}
Then, when using linearly polarized light, three different combinations of polarization gives a nonlinear current density, namely (i) the one in which all participating fields are $p$-polarized ($ppp$), and (ii) the two combinations where the pump fields are differently polarized and the probe field is $s$-polarized ($sps$ and $pss$). In all cases, the phase conjugated response is $p$-polarized, and thus characterized in terms of the polarization states of the probe and phase conjugated fields, case (i) may be classified as a $p$ to $p$ transition, and cases (ii) as $s$ to $p$ transitions. A schematic illustration of these interaction configurations is shown in Fig. \[fig:2\]. Defining the $z$-independent quantity $${\cal{J}}_{-\omega,z}^{(3)}(\vec{q}_{\|})
\equiv{J_{-\omega,z}^{(3)}(z;\vec{q}_{\|})\over|\psi(z)|^2},$$ the above conditions yields for the $p$ to $p$ transition $$\begin{aligned}
\lefteqn{
{\cal{J}}_{-\omega,z}^{(3)}(\vec{q}_{\|})=
{e^4\over2^8\pi^6{\rm{i}}\hbar\omega^3m_{e}^2}
\left[{\cal{C}}(q_{\|}-k_{\|})
+{\cal{C}}(q_{\|}+k_{\|})\right]
E_z^{(1)}E_z^{(2)}E_z^{*}
}\nonumber\\ &\quad&\times
\int|\psi(z')|^2e^{-{\rm{i}}q_{\perp}^{*}z'}dz'
\label{eq:J3-ppp-0}\end{aligned}$$ and for the $s$ to $p$ transitions $$\begin{aligned}
\lefteqn{
{\cal{J}}_{-\omega,z}^{(3)}(\vec{q}_{\|})=
{e^4\over2^8\pi^6{\rm{i}}\hbar\omega^3m_e^2}
\left[{\cal{C}}(q_{\|}+k_{\|})
+{\hbar\over4m_e}{\cal{D}}(\vec{q}_{\|},-\vec{k}_{\|})\right]
E_y^{(1)}E_z^{(2)}E_y^{*}
}\nonumber\\ &\quad&\times
\int|\psi(z')|^2 e^{-{\rm{i}}q_{\perp}^{*}z'}dz',
\label{eq:J3-sps-0}
\\
\lefteqn{
{\cal{J}}_{-\omega,z}^{(3)}(\vec{q}_{\|})=
{e^4\over2^8\pi^6{\rm{i}}\hbar\omega^3m_e^2}
\left[{\cal{C}}(q_{\|}-k_{\|})
+{\hbar\over4m_e}{\cal{D}}(q_{\|},k_{\|})\right]
E_z^{(1)}E_y^{(2)}E_y^{*}
}\nonumber\\ &&\times
\int|\psi(z')|^2 e^{-{\rm{i}}q_{\perp}^{*}z'}dz'.
\label{eq:J3-pss-0}\end{aligned}$$ In the above three equations, the superscript $(1)$ refers to the pump field propagating along the $x$-axis in the positive direction ($\vec{k}_{\|}=k_{\|}\vec{e}_{x}$), and the superscript $(2)$ refers to the other pump field. The $s$ to $p$ transitions are symmetric in the sense that if the probe wavevector $\vec{q}_{\|}$ is replaced by $-\vec{q}_{\|}$ in Eq. (\[eq:J3-sps-0\]), then the result of Eq. (\[eq:J3-pss-0\]) is obtained, and vice versa. The $p$ to $p$ transition is symmetric to itself in this sense.
For a single-level quantum well, the $zz$-component of the linear conductivity tensor is given by [@Feibelman:82:1] $$\sigma_{zz}(z,z';\vec{q}_{\|})={{\rm{i}}e^2{\cal{N}}\over{}m_e(\omega+{\rm{i}}/\tau)}
|\psi(z)|^2\delta(z-z'),
\label{eq:sigma-zz}$$ where $${\cal{N}}={2\over(2\pi)^{2}}\int{}f(\vec{\kappa}_{\|})d^2\kappa_{\|}.
\label{eq:N}$$ In order to take into account the coupling to surroundings we have introduced a phenomenological relaxation time $\tau$ in the diamagnetic expression for $\sigma_{zz}$ \[Eq. (\[eq:sigma-zz\])\] [@Feibelman:82:1]. A factor of two in this equation again stems from the spin summation, and the quantity ${\cal{N}}|\psi(z)|^2$ is the conduction electron density. The phase conjugated field inside the quantum well has a $z$-component, $E_{{\rm{PC}},z}(z;\vec{q}_{\|})$, only, and by combining Eqs. (\[eq:ping\]), (\[eq:pong\]), and (\[eq:sigma-zz\]) it appears that this is given by $$E_{{\rm{PC}},z}(z;\vec{q}_{\|})=
{im_e(\omega+{\rm{i}}/\tau)\over{}e^2{\cal{N}}|\psi(z)|^2
-\epsilon_0m_e\omega(\omega+{\rm{i}}/\tau)}
J_{-\omega,z}^{(3)}(z;\vec{q}_{\|}).$$ Using now Eq. (\[eq:loop\]), the $z$-components of the phase conjugated field outside the quantum well can be calculated, and the result is $$\begin{aligned}
E_{{\rm{PC}},z}(z;\vec{q}_{\|})=
{\cal{J}}_{-\omega,z}^{(3)}(\vec{q}_{\|})
e^{-{\rm{i}}q_{\perp}z}
{q_{\|}^2m_e(\omega+{\rm{i}}/\tau)\over2q_{\perp}}
\int{(e^{{\rm{i}}q_{\perp}z'}+r^pe^{-{\rm{i}}q_{\perp}z'})|\psi(z')|^2\over
e^2{\cal{N}}|\psi(z')|^2-\epsilon_0m_e\omega(\omega+{\rm{i}}/\tau)}dz',
\nonumber\\
\label{eq:EPC-IInt-na}\end{aligned}$$ where the relevant expression for ${\cal{J}}_{-\omega,z}^{(3)}(\vec{q}_{\|})$ is taken from Eq. (\[eq:J3-ppp-0\]), (\[eq:J3-sps-0\]), or (\[eq:J3-pss-0\]). Given the $z$-component of the phase conjugated field, the $x$-component may be found from $${E}_{{\rm{PC}},x}(z;\vec{q}_{\|})
={q_{\perp}\over{}q_{\|}}{E}_{{\rm{PC}},z}(z;\vec{q}_{\|}),
\label{eq:EPCx-zq}$$ which follows from the expression for the electromagnetic propagator, or equivalently from the demand that the phase conjugated field must be transverse in vacuum.
The integral in Eq. (\[eq:EPC-IInt-na\]) is different from zero only in the region of the quantum well \[from around $z'=-d$ to around $z'=0$ in the chosen coordinate system, the exact domain depending on the extent of the electronic wave function $\psi(z')$\]. Since the width ($\sim{}d$) of a single-level metallic quantum well is in the [Å]{}ngstr[ö]{}m range, and $q_{\perp}$ is typically in the micrometer range for optical signals such that $q_{\perp}d\ll1$, it is a good approximation to put $\exp(\pm{}{\rm{i}}q_{\perp}z')=1$ in Eq. (\[eq:EPC-IInt-na\]). For electromagnetic frequencies so high that $q_{\perp}\sim{}d^{-1}$, the present theory would anyway be too simple to rely on \[the Bloch function character of the wave functions along the surface and excitation to the continuum (photoemission) should be incorporated at least\]. With the above-mentioned approximation, Eq. (\[eq:EPC-IInt-na\]) is reduced to $$E_{{\rm{PC}},z}(z;\vec{q}_{\|})=
{\cal{J}}_{-\omega,z}^{(3)}(\vec{q}_{\|})
e^{-{\rm{i}}q_{\perp}z}
{(1+r^p)q_{\|}^2\over2\epsilon_0\omega{}q_{\perp}}
\int{|\psi(z')|^2\over{}\gamma|\psi(z')|^2-1}dz',
\label{eq:EPC-IInt}$$ where $\gamma=e^2{\cal{N}}/[\epsilon_0m_e\omega(\omega+{\rm{i}}/\tau)]$. Using the approximation $\exp({\rm{i}}q_{\perp}z')=1$ and the normalization condition on $\psi(z')$, Eqs. (\[eq:J3-ppp-0\])–(\[eq:J3-pss-0\]) are reduced to $$\begin{aligned}
{\cal{J}}_{-\omega,z}^{(3)}(\vec{q}_{\|})&=&
{e^4\over2^8\pi^6{\rm{i}}\hbar\omega^3m_{e}^2}
\left[{\cal{C}}(q_{\|}-k_{\|})
+{\cal{C}}(q_{\|}+k_{\|})\right]
E_z^{(1)}E_z^{(2)}E_z^{*},
\label{eq:J3-ppp-0-a}
\\
{\cal{J}}_{-\omega,z}^{(3)}(\vec{q}_{\|})&=&
{e^4\over2^8\pi^6{\rm{i}}\hbar\omega^3m_e^2}
\left[{\cal{C}}(q_{\|}+k_{\|})
+{\hbar\over4m_e}{\cal{D}}(\vec{q}_{\|},-\vec{k}_{\|})\right]
E_y^{(1)}E_z^{(2)}E_y^{*},
\label{eq:J3-sps-0-a}\end{aligned}$$ and $$\begin{aligned}
{\cal{J}}_{-\omega,z}^{(3)}(\vec{q}_{\|})&=&
{e^4\over2^8\pi^6{\rm{i}}\hbar\omega^3m_e^2}
\left[{\cal{C}}(q_{\|}-k_{\|})
+{\hbar\over4m_e}{\cal{D}}(q_{\|},k_{\|})\right]
E_z^{(1)}E_y^{(2)}E_y^{*},
\label{eq:J3-pss-0-a}\end{aligned}$$ respectively.
Thus the phase conjugated field from a single-level quantum well is described in the mixed Fourier space by Eq. (\[eq:EPC-IInt\]) with insertion of Eq. (\[eq:J3-ppp-0-a\]), (\[eq:J3-sps-0-a\]), or (\[eq:J3-pss-0-a\]), the expressions for ${\cal{C}}$ \[Eq. (\[eq:C\])\] and ${\cal{D}}$ \[Eq. (\[eq:D\])\] carrying the information on the two-dimensional electron dynamics.
So far, the description of the phase conjugated response has been independent of the actual wave functions in the active medium, and thus independent of the form of the quantum well potential. In order to prepare our theory for a numerical study we now introduce a model potential in our quantum well system, namely the infinite barrier potential.
Infinite barrier model
----------------------
To achieve a qualitative impression of the phase conjugation from a single-level metallic quantum well it is sufficient to carry out numerical calculations on the basis of the simple infinite barrier (IB) model. In this model the one-dimensional potential $V(z)$ is taken to be zero in the interval $-d\leq{}z\leq{}0$ (inside the quantum well) and infinite elsewhere. The stationary state wave function now is given by $\psi(z)=\sqrt{2/d}\sin(\pi{}z/d)$ inside the well and $\psi(z)=0$ outside, and the associated energy is $\varepsilon=(\pi\hbar)^2/(2m_ed^2)$. In the IB model the number of bound states is of course infinite, and to use this model in the context of a single level calculation, one must be sure that only one of the bound states (the ground state) has an energy below the Fermi energy, and that the optical frequency is so low that interlevel excitations are negligible.
For a metallic quantum well one may even at room temperature approximate the Fermi-Dirac distribution function appearing in the expressions for ${\cal{C}}$, ${\cal{D}}$, and ${\cal{N}}$ in Eqs. (\[eq:C\]), (\[eq:D\]), and (\[eq:N\]) by its value at zero temperature, i.e., $$\lim_{T\rightarrow0}f(\vec{\kappa}_{\|})=
\Theta\left\{{\cal{E}}_{F}-{\hbar^2\over2m_e}
\left[\left(\pi\over{}d\right)^2+\kappa_{\|}^2\right]\right\},
\label{eq:fermi-T=0}$$ where $\Theta$ is the Heaviside step function and ${\cal{E}}_{F}$ is the Fermi energy of the system. In the low temperature limit it is possible to find analytical solutions to the integrals over $\vec{\kappa}_{\|}$ appearing in Eqs. (\[eq:C\]) and (\[eq:D\]). This is adequately achieved by performing a coordinate transformation into cylindrical coordinates, since each Heaviside step function gives nonzero values in the $\kappa_x$-$\kappa_y$-space only inside a circle with radius, say, $\alpha$. The explicit calculations are tedious but trivial to carry out, and since the final expressions for ${\cal{C}}$ and ${\cal{D}}$ are rather long we do not present them here. For the interested reader these calculations are reproduced in Appendices \[ch:Solve-Q\] and \[app:C\] \[specifically, Section \[sec:CDN\]\].
The Fermi energy is calculated from the global charge neutrality condition \[see and the calculation performed in Appendix \[app:D\]\], which for a single level quantum well takes the form $${\cal{N}}=ZN_+d,$$ where $N_+$ is the number of positive ions per unit volume and $Z$ is the valence of these ions. Since ${\cal{N}}=m_e({\cal{E}}_{F}-\varepsilon)/(\pi\hbar^2)$, cf. the calculation in the Section \[sec:CDN\], one gets $${\cal{E}}_{F}={\pi\hbar^2\over{}m_e}\left[ZN_+d+{\pi\over2d^2}\right].
\label{eq:Fermi}$$ In order that just the ground state (energy $\varepsilon$) has an energy less than the Fermi energy, the film thickness must be less than a certain maximum value $d_{\rm{max}}$. When the thickness of the well becomes so large that the Fermi energy equals the energy $\varepsilon_2=(2\pi\hbar)^2/(2m_ed^2)$ of the first excited state a second bound state of energy less than ${\cal{E}}_{F}$ will appear. From the condition ${\cal{E}}_{F}(d_{\rm{max}})=\varepsilon_2(d_{\rm{max}})$, $d_{\rm{max}}$ can be calculated, and one gets by means of Eq. (\[eq:Fermi\]) $$d_{\rm{max}}=\sqrt[3]{3\pi/(2ZN_+)}\,,
\label{eq:dmax}$$ i.e., a result which depends on the number of conduction electrons in the film. The minimum thickness is in the IB model zero, but in reality the smallest thickness is a single monolayer.
Inserting the IB model into the integral over the source region appearing in Eq. (\[eq:EPC-IInt\]) we get $$\begin{aligned}
\int{|\psi(z')|^2\over{}\gamma|\psi(z')|^2-1}dz'
=\int_{-d}^{0}{2\sin^2({\pi{}z'/d})
\over2\gamma\sin^2({\pi{}z'/d})-d}dz',\end{aligned}$$ which by substitution of $\theta=\pi{}z'/d$, addition and subtraction of $d$ in the nominator of the integral, and use of $2\gamma\sin^2\theta-d=2\gamma[\sqrt{1-d/(2\gamma)}-\cos\theta][\sqrt{1-d/(2\gamma)}+\cos\theta]$ gives $$\begin{aligned}
{d\over\pi\gamma}\left[\pi-{d\over4\gamma}{1\over\sqrt{1-d/(2\gamma)}}
\int_{0}^{2\pi}{d\theta\over\sqrt{1-d/(2\gamma)}+\cos\theta}\right]
={d\over\gamma}\left[1-{1\over\sqrt{2\gamma/d-1}}\right]\approx{d\over\gamma}.
\nonumber\\
\label{eq:intdg}\end{aligned}$$ The solution to the integral in Eq. (\[eq:intdg\]) is obtained by use of Eq. (\[eq:Ang-nocos\]), and since $2|\gamma|/d\gg1$ \[for metals, $|\gamma|$ lies typically between $1$ and $100$ in the optical region (e.g., for copper $|\gamma|\approx85$ in the present study) and $d$ is in the [Å]{}ngstr[ö]{}m range\]. Using this result and the expression for the Fermi energy given in Eq. (\[eq:Fermi\]), we obtain by insertion into Eq. (\[eq:EPC-IInt\]) the result $$E_{{\rm{PC}},z}(z;\vec{q}_{\|})=
{q_{\|}^2m_e(\omega+{\rm{i}}/\tau)(1+r^p)\over2q_{\perp}e^2ZN_+}
{\cal{J}}_{-\omega,z}^{(3)}(\vec{q}_{\|})
e^{-{\rm{i}}q_{\perp}z}.$$ By insertion of the relevant expressions for ${\cal{J}}_{-\omega,z}^{(3)}(\vec{q}_{\|})$ we finally obtain the following results for the $z$-component of the phase conjugated field outside the quantum well: $$\begin{aligned}
\lefteqn{
E_{{\rm{PC}},z}(z;\vec{q}_{\|})=
{e^2(\omega+{\rm{i}}/\tau)(1+r^p)\over2^9\pi^6\hbar\omega^3ZN_+m_e}
{q_{\|}^2\over{}{\rm{i}}q_{\perp}}
\left[{\cal{C}}(q_{\|}-k_{\|})
+{\cal{C}}(q_{\|}+k_{\|})\right]
}\nonumber\\ &\quad&\times
E_z^{(1)}E_z^{(2)}E_z^{*}e^{-{\rm{i}}q_{\perp}z},
\label{eq:PC-ppp}\end{aligned}$$ for the purely $p$-polarized configuration, and $$\begin{aligned}
\lefteqn{
E_{{\rm{PC}},z}(z;\vec{q}_{\|})=
{e^2(\omega+{\rm{i}}/\tau)(1+r^p)\over2^9\pi^6\hbar\omega^3ZN_+m_e}
{q_{\|}^2\over{}{\rm{i}}q_{\perp}}
\left[{\cal{C}}(q_{\|}+k_{\|})
+{\hbar\over4m_e}{\cal{D}}(q_{\|},-k_{\|})\right]
}\nonumber\\ &\quad&\times
E_y^{(1)}E_z^{(2)}E_y^{*}
e^{-{\rm{i}}q_{\perp}z},
\label{eq:PC-sps}
\\
\lefteqn{
E_{{\rm{PC}},z}(z;\vec{q}_{\|})=
{e^2(\omega+{\rm{i}}/\tau)(1+r^p)\over2^9\pi^6\hbar\omega^3ZN_+m_e}
{q_{\|}^2\over{}{\rm{i}}q_{\perp}}
\left[{\cal{C}}(q_{\|}-k_{\|})
+{\hbar\over4m_e}{\cal{D}}(q_{\|},k_{\|})\right]
}\nonumber\\ &&\times
E_z^{(1)}E_y^{(2)}E_y^{*}
e^{-{\rm{i}}q_{\perp}z}
\label{eq:PC-pss}\end{aligned}$$ for the configurations with mixed polarization of the pump fields. The $x$-component of the phase conjugated field is obtained using Eq. (\[eq:EPCx-zq\]).
Numerical results {#sec:num}
=================
\[Ch:11\] The theoretical description presented in the previous chapter resulted in expressions for the phase conjugated field from a single level quantum well. Thus for the numerical work, the phase conjugated field is given completely by Eqs. (\[eq:PC-ppp\])–(\[eq:PC-pss\]) and (\[eq:EPCx-zq\]) with the insertion of the expressions for the electron dynamics parallel to the surface plane, given by Eqs. (\[eq:C-solved\])–(\[eq:D-solved\]) in Appendix \[app:C\]. In the following we will present the phase conjugation reflection coefficient, succeeded by a discussion of a possible excitation scheme which might be adequate for studies of phase conjugation of optical near fields [@Bozhevolnyi:94:1].
Phase conjugation reflection coefficient {#sec:11.1}
----------------------------------------
To estimate the amount of light we get back through the phase conjugated channel, we define the phase conjugation (energy) reflection coefficient as $$R_{\rm{PC}}(z;\vec{q}_{\|})={I_{\rm{PC}}(z;\vec{q}_{\|})
\over{}I^{(1)}I^{(2)}I_{\rm{Probe}}(-d;\vec{q}_{\|})},
\label{eq:RPC}$$ in which $I^{(1)}$, $I^{(2)}$, $I_{\rm{Probe}}$, and $I_{\rm{PC}}$ are the intensities of the two pump beams, the probe and the phase conjugated field, respectively. Each of the intensities are given by $$I={\epsilon_0c_0\over2}{\vec{E}\cdot\vec{E}^{*}\over(2\pi)^4},
\label{eq:RPC-I}$$ where the factor of $(2\pi)^{-4}$ originates from the manner in which we have introduced the Fourier amplitudes of the fields. If the probe field is evanescent the intensity of the phase conjugated field, $I_{\rm{PC}}(z;\vec{q}_{\|})$, will depend on the distance from the surface, and consequently the reflection coefficient is $z$-dependent in such a case.
For the remaining part of this work we choose a copper quantum well with $N_+=8.47\times10^{28}$m$^{-3}$ and $Z=1$ \[data taken from \]. Then from Eq. (\[eq:dmax\]), the maximal thickness becomes $d_{\rm{max}}=3.82$[Å]{}, which is more than two monolayers and less than three. Thus we have two obvious choices for the thickness of the quantum well, namely a single monolayer or two monolayers. We thus take a look at both possibilities in the following, corresponding to a thickness of $d=1.8$[Å]{} for one monolayer and $d=3.6$[Å]{} for two monolayers. The Cu quantum well can adequately be deposited on a glass substrate for which we use a refractive index $n$ of 1.51. With this substrate, a reasonable description of the linear vaccum/substrate amplitude reflection coefficient $r^p$ is obtained by use of the classical Fresnel formula $$r^p={n^2q_{\perp}-({n^2q^2-q_{\|}^2})^{1\over2}\over
n^2q_{\perp}+({n^2q^2-q_{\|}^2})^{1\over2}},
\label{eq:rp}$$ $q=\omega/c_0$ being the vacuum wavenumber, as before. Then, having the pump fields parallel to the $x$-axis gives a pump wavenumber $k_{\|}=1.51q$. The wavelength $\lambda$ of the light is chosen to be $\lambda=1061$nm.
(0,0)(127,96) (-8,2) (0,53)[$R_{\rm{PC}}(-d;\vec{q}_{\|})$ \[m$^4$/W$^2$\]]{} (71.5,0)[(0,0)\[b\][$q_{\|}/q$]{}]{} (43.75,57)(43.75,53) (43.75,17.5)(43.75,21.5) (43.75,58)[(0,0)\[b\][$n$]{}]{} (115,4)(115,96) (115,0)[(0,0)\[b\][$k_F/q$]{}]{}
(0,0)(127,96) (-8,2) (0,53)[$R_{\rm{PC}}(-d;\vec{q}_{\|})$ \[m$^4$/W$^2$\]]{} (71.5,0)[(0,0)\[b\][$q_{\|}/q$]{}]{} (43.75,57)(43.75,53) (43.75,20.5)(43.75,24.5) (43.75,58)[(0,0)\[b\][$n$]{}]{} (113,4)(113,96) (113,0)[(0,0)\[b\][$k_F/q$]{}]{}
(0,0)(127,143) (-8,2) (0,76.5)[$R_{\rm{PC}}(-d;\vec{q}_{\|})$ \[m$^4$/W$^2$\]]{} (71.5,0)[(0,0)\[b\][$q_{\|}/q$]{}]{} (10,85) (43.75,38.5)(43.75,34.5) (43.75,23)(43.75,27) (53.75,100)(53.75,104) (53.75,127)(53.75,123) (43.75,39.5)[(0,0)\[b\][$n$]{}]{} (53.75,128)[(0,0)\[b\][$n$]{}]{} (99.75,134)(99.75,130) (99.75,87)(99.75,91) (103,129)(103,125) (103,82)(103,86) (123,14)[(0,0)\[br\][$sps$]{}]{} (82,96)[(0,0)\[br\][$ppp$]{}]{}
(0,0)(127,143) (-8,2) (0,76.5)[$R_{\rm{PC}}(-d;\vec{q}_{\|})$ \[m$^4$/W$^2$\]]{} (71.5,0)[(0,0)\[b\][$q_{\|}/q$]{}]{} (10,85) (43.75,39)(43.75,35) (43.75,23)(43.75,27) (53.75,101.5)(53.75,105.5) (53.75,127)(53.75,123) (43.75,40)[(0,0)\[b\][$n$]{}]{} (53.75,128)[(0,0)\[b\][$n$]{}]{} (99.75,139)(99.75,135) (99.75,92)(99.75,96) (103,132)(103,128) (103,84)(103,88) (124,14)[(0,0)\[br\][$sps$]{}]{} (82,96)[(0,0)\[br\][$ppp$]{}]{}
The phase conjugation reflection coefficient at the vacuum/film interface, $R_{\rm{PC}}(-d;\vec{q}_{\|})$ is plotted in Figs. \[fig:11.1\] and \[fig:3\] as a function of the parallel component ($q_{\|}$) of the wavevector for both the $p$ to $p$ transition and the two $s$ to $p$ transitions. The reason that the two curves for the $s$ to $p$ transitions appear the same in the high end of the $q_{\|}/q$ spectrum is due to the fact that for $k_{\|}\ll{}q_{\|}$ we have ${\cal{C}}(q_{\|}-k_{\|})\simeq{\cal{C}}(q_{\|}+k_{\|})$ and ${\cal{D}}(q_{\|},k_{\|})\simeq{\cal{D}}(q_{\|},-k_{\|})$. The “bubble” appearing on the $sps$ and $pss$ curves from around $q_{\|}/q\sim100$ to $q_{\|}/q\sim{}k_F/q$ is due to the two-dimensional electron dynamics hidden in ${\cal{D}}(q_{\|},k_{\|})$. To be a little more specific, the left of the two peaks stems from the second term, while the peak to the right in the bubble stems from the third term.
To illustrate the similarity between the two possible $s$ to $p$ transitions, we can take Eq. (\[eq:PC-sps\]) to describe the phase conjugated field, which for positive values of $q_{\|}/q$ gives the result in Fig. \[fig:3\] ($sps$). Using the other $s$ to $p$ transition, given by Eq. (\[eq:PC-pss\]), instead we get the result in Fig. \[fig:3\] ($pss$) for positive values of $q_{\|}/q$. The symmetry between the two configurations is obtained by looking at the negative values of $q_{\|}/q$, since Eq. (\[eq:PC-sps\]) plotted for negative values of $q_{\|}/q$ gives the ($pss$) curve in Fig. \[fig:3\]. Similarly, by starting with Eq. (\[eq:PC-pss\]), the resulting curve for negative values of $q_{\|}/q$ gives the ($sps$) result in Fig. \[fig:3\].
The choice of an adequate relaxation time $\tau$ is a difficult problem and it appears from Figs. \[fig:11.3\] and \[fig:4\] that the value of the relaxation time has a great impact on the phase conjugation reflection coefficient. We have plotted the reflection coefficent for three values of the relaxation time, namely (i) 30fs and (ii) 200fs, which are typical values one would find for bulk copper [@Ashcroft:76:1] at (i) room temperature and (ii) at 77K, and (iii) 3fs. The value in case (iii) is obtained by a conjecture based on the difference between measured data for a lead quantum well [@Jalochowski:97:1] and the bulk value for lead at room temperature. The difference between the relaxation time measured by is for two monolayers approximately one order of magnitude. Based on the results of we have for the data presented in this work chosen the value of the relaxation time to be 3fs. As it can be seen from Fig. \[fig:4\], the bubble in the curve corresponding to the $sps$ configuration appears earlier in the $q_{\|}/q$-spectrum for higher values of $\tau$. For the $ppp$ configuration the lower end of the spectrum is damped when $\tau$ becomes smaller.
So what is the difference between using a single monolayer or two monolayers in the quantum well? In the single monolayer quantum well, the distance between the occupied energy level and the first free energy level in the infinite barrier model, and between the occupied energy level and the continuum states in a finite barrier model is larger than for a two monolayer well. Thus the single-monolayer well should behave more ideally like a single-level quantum well at higher frequencies than the two-monolayer well. If we take a look at Figs. \[fig:11.1\] and \[fig:11.2\] we observe that the bubble in the $sps$ and $pss$ curves has the highest maximal magnitude for the two-monolayer well, and the earliest falloff in the high end of the $q_{\|}/q$-spectrum. The value of each of the two peaks in the bubble is reached at the same $q_{\|}/q$-value in the two cases, as is also evident from Figs. \[fig:11.3\] and \[fig:11.4\] (shown using a set of arrows for each peak). From these two figures we also observe that the relaxation-time dependent low-$q_{\|}/q$ beginning of the bubble occurs a little earlier and is increasing faster in the two-monolayer well compared to the other. In the low end of the $q_{\|}/q$-spectrum the $sps$ and $pss$ curves are of equal magnitude. Looking at the $ppp$ curve, we observe that it is damped roughly by a factor of two in the low end of the $q_{\|}/q$-spectrum using a single-monolayer film in stead of two monolayers. In the high end it takes its maximal value for the single-monolayer well at rougly twice the value of $q_{\|}/q$ than for the two-monolayer film. In conclusion, the differences between the phase conjugated response for a single-monolayer film and a two-monolayer film will probably be very difficult, if not impossible, to observe in an experiment with single mode excitation. In the rest of this chapter we thus present results for the two-monolayer film only.
(0,0)(127,117) (0,60)[ (-8,2) (0,34)[$R_{\rm{PC}}(z;\vec{q}_{\|})$ \[m$^4$/W$^2$\]]{} (71.5,0)[(0,0)\[b\][$q_{\|}/q$]{}]{} (20,54)[(0,0)\[tl\][$ppp$]{}]{} ]{} (0,0)[ (-8,2) (0,34)[$R_{\rm{PC}}(z;\vec{q}_{\|})$ \[m$^4$/W$^2$\]]{} (71.5,0)[(0,0)\[b\][$q_{\|}/q$]{}]{} (20,54)[(0,0)\[tl\][$sps$]{}]{} ]{}
We have in Fig. \[fig:5\] plotted the phase conjugation reflection coefficient for the $p$ to $p$ transition and one of the $s$ to $p$ transitions, respectively, for different distances from the surface of the phase conjugator. Due to our particular interest in the phase conjugation of the evanescent modes in the Fourier spectrum the chosen distances are fractions of the vacuum wavelength. In Fig. \[fig:6\] we have plotted the part of the Fourier spectrum for all three configurations which is judged to be the most easily accessible to single-mode excitation in experimental investigations.
It appears from Fig. \[fig:5\] ($ppp$) that the phase conjugation reflection coefficient is independent of the distance from the metal film in the region where $q_{\|}/q\leq1$. This is so because the probe field, and hence also the phase conjugated field, are of propagating character ($q_{\perp}=[q^2-q_{\|}^2]^{1/2}$ is real). In the region where $q_{\|}/q>1$, both the probe field and the phase conjugated field are evanescent ($q_{\perp}={\rm{i}}[q_{\|}^2-q^2]^{1/2}$ is a purely imaginary quantity), and in consequence the reflection coefficient decreases rapidly with the distance from the phase conjugator. Already a single wavelength away from the surface of the phase conjugator the evanescent modes of the phase conjugated field have essentially vanished and only propagating modes are detectable. Although the evanescent Fourier components of the phase conjugated field are present only less than an optical wavelength from the surface, this [*does not*]{} imply that the nonlinear mixing of the electromagnetic waves is less effective in the regime of the evanescent modes. It is in fact opposite, as may be seen for instance from Fig. \[fig:3\]. The maximum coupling for the $p$ to $p$ transition is obtained for $q_{\|}/q\simeq500$, and in comparison with $R_{\rm{PC}}$ at $q_{\|}/q\simeq1$, the maximum in $R_{\rm{PC}}$ is nine orders of magnitude larger, and seven, respectively eight orders of magnitude larger for the two $s$ to $p$ transitions, which have their maxima at around $q_{\|}/q\simeq700$. As we observe from Fig. \[fig:5\], as the distance from the film increases the maximum value decreases and is shifted downwards in the $q_{\|}/q$ spectrum. But only when the distance from the phase conjugator becomes larger than $\sim\lambda/10$ ($ppp$) respectively $\sim\lambda/60$ ($sps$), the phase conjugated signal is largest at $q_{\|}/q\approx1$.
(0,1)(127,177) (0,120)[ (-8,2) (0,34)[$R_{\rm{PC}}(z;\vec{q}_{\|})$ \[m$^4$/W$^2$\]]{} (71.5,0)[(0,0)\[b\][$q_{\|}/q$]{}]{} (20,54)[(0,0)\[tl\][$ppp$]{}]{} ]{} (0,60)[ (-8,2) (0,34)[$R_{\rm{PC}}(z;\vec{q}_{\|})$ \[m$^4$/W$^2$\]]{} (71.5,0)[(0,0)\[b\][$q_{\|}/q$]{}]{} (20,54)[(0,0)\[tl\][$sps$]{}]{} ]{} (0,0)[ (-8,2) (0,34)[$R_{\rm{PC}}(z;\vec{q}_{\|})$ \[m$^4$/W$^2$\]]{} (71.5,0)[(0,0)\[b\][$q_{\|}/q$]{}]{} (20,54)[(0,0)\[tl\][$pss$]{}]{} ]{}
The absolute value of the reflection coefficients may seem very small, but utilizing a high-power Nd:YAG laser with, say an energy of 100mJ per pulse available for each of the three incoming fields, a pulse (assumed square for simplicity) duration of 4ns and an interaction area of 25mm$^2$, the intensity of each of these fields will be in the order of $1$TW/m$^2$, and the phase conjugated intensity lies between 100pW/m$^2$ and 1W/m$^2$ in the full range of $q_{\|}/q$ for which the reflection coefficient has been plotted in Fig. \[fig:5\] ($ppp$), and between 1$\mu$W/m$^2$ and 1kW/m$^2$ in relation to the data in Fig. \[fig:5\] ($sps$).
In many theoretical studies of the properties of phase conjugated fields it is assumed that the phase conjugator is ideal [@Hendriks:89:1; @Agarwal:95:1; @Keller:96:2]. By this is meant that the phase conjugation reflection coefficient is independent of the angle of incidense of the (propagating) probe field (and maybe also of the state of polarization). In the present case, the ideal phase conjugator assumption is certainly not good. Prior to the observation that evanescent fields could be phase conjugated [@Bozhevolnyi:94:1] it was often assumed in theory [@Yariv:82:1] that $R_{\rm{PC}}=0$ in the region $q_{\|}/q>1$, and in later studies [@Agarwal:95:1; @Keller:96:2] it has been assumed that also the phase conjugation of evanescent waves is ideal, i.e., independent of $q_{\|}/q$ ($\gtrsim1$). When it comes to the phase conjugation from quantum well systems our analysis indicates that use of an energy reflection coefficient independent of $q_{\|}/q$ in general is bad. Only at specific distances the ideal phase conjugator assumption might be justified, see, e.g., the results representing $R_{\rm{PC}}$ at $|z+d|=\lambda/8$ in Fig. \[fig:6\]. The kink in the reflection coefficient (which is most pronounced close to the metal/vacuum interface) found at $q_{\|}/q=n$ ($=1.51$) appears when the probe field changes from being propagating to being evanescent inside the substrate.
Above we have discussed the nonlinear reflection coefficient for the $p$ to $p$ configuration. It appears from Figs. \[fig:5\] and \[fig:6\] that the quantitative picture is the same for the $s$ to $p$ cases, though the reflection coefficient for the $s$ to $p$ transitions roughly speaking are five orders of magnitude larger in the experimentally most adequate evanescent region of the Fourier spectrum ($1\leq{}q_{\|}/q\lesssim2.5$) for single mode excitation.
The IB model only offers a crude description of the electronic properties of a quantum well.\[kF\] Among other things, the electron density profile at the ion/vacuum edge is poorly accounted for in this model, which gives too sharp a profile and underestimates the spill-out of the wave function. Altogether one should be careful to put too much reality into the IB model when treating local-field variations (related to, say, $q_{\|}$ or $q_{\perp}$) on the atomic length scale. Also the neclect of the Bloch character of the wave functions accounting for the dynamics in the plane of the well is doubtful in investigations of the local field among the atoms of the quantum well. The crucial quantity in the above-mentioned context is the Fermi wavenumber $k_F=(2m_e{\cal{E}}_{F})^{1/2}/\hbar$, and in relation to Fig. \[fig:5\], only results for $q_{\|}/q$ ratios less than approximately $${k_F\over{}q}=\lambda\sqrt{{ZN_+d\over2\pi}+{1\over4d^2}},$$ appears reliable. Insertion of the appropriate values for copper: $ZN_+=8.47\times10^{28}$, $d=1.8$[Å]{} (single monolayer film) or $d=3.6$[Å]{} (two-monolayer film), and the wavelength $\lambda=1061$nm gives $k_F=3.38\times10^3q$ for a single monolayer of copper, and $k_F=2.76\times10^3q$ for two monolayers of copper, respectively. The data presented in Fig. \[fig:5\] should therefore be well within this limit of our model.
Returning to the curve in Fig. \[fig:5\] ($ppp$) which represents the reflection coefficient closest to the surface of the phase conjugator ($|z+d|=\lambda/256$) one finds approximately a relation of the form $R_{\rm{PC}}=b(q_{\|}/q)^a$ with $a\simeq5$ in the lower wavenumber end of the evanescent region. The falloff of $R_{\rm{PC}}$ with $q_{\|}/q$ after the maximum (located at $q_{\|}/q\sim50$) is much stronger than the increase towards the maximum. As the distance from the phase conjugator is increased the value of $a$ gradually decreases. In Fig. \[fig:5\] (b) we observe a similar behaviour, but this time the value of $a$ in the approximate relation in the low end of the evanescent part of the Fourier spectrum is smaller, namely $a\simeq1.5$.
The energy reflection coefficient calculated at the vacuum/quantum well interface, $R_{\rm{PC}}(-d;\vec{q}_{\|})$, characterizes the effectiveness with which a given ($q_{\|}$) plane-wave probe field (propagating or evanescent) may be phase conjugated, and the results presented in Fig. \[fig:3\] indicate that this effectiveness (nonlinear coupling) is particularly large for (part of the) evanescent modes. The maximum in the effectivity is reached for a value of $q_{\|}/q$ as large as $\sim500$–$700$. The strong coupling in part of the evanescent region does not necessarily reflect itself in any easy manner experimentally. First of all, one must realize that the strong coupling effect only may be observed close to the quantum well, i.e., at distances $z\lesssim\lambda$. Secondly, one must be able to produce evanescent probe fields with relatively large values of $q_{\|}/q$. This is in itself by no means simple outside the range when the standard Otto [-@Otto:68:1; -@Otto:76:1] \[or possibly Kretschmann [@Kretschmann:68:1; @Raether:88:1]\] techniques can be adopted. Roughly speaking, this range coincides with the ones shown in Fig. \[fig:6\]. To create probe fields with larger $q_{\|}/q$ values other kinds of experimental techniques must be used, and in the following we shall consider a particular example and in a qualitative manner discuss the resulting Fourier spectrum of the phase conjugated field.
Phase conjugated response using a wire source
---------------------------------------------
In near-field optics evanescent fields with relatively large values of $q_{\|}/q$ are produced by various methods, all aiming at compressing the source field to subwavelength spatial extension \[see, e.g., and \]. From a theoretical point of view the radiation from a subwavelength source may in some cases be modelled by the radiation from an (electric) point-dipole source, or an assembly of such sources. It is a straightforward matter to decompose an electric point-dipole field into its relevant evanescent and propagating modes, and thereby estimate the intensity of the phase conjugated field in each of the $q_{\|}$-components. However, in order to determine the characteristics of the phase conjugated light focus generated by the quantum well one would have to calculate the four-wave mixing also for probe fields with wavevectors not confined to the $x$-$z$-plane, and to do this our theory must first be generalized to non-planar phase conjugation.
Within the framework of the present theory, it is possible, however, to study the spatial confinement (focusing) of the phase conjugated field generated by a quantum wire adequately placed above the surface of the quantum well [@Keller:98:1], and let us therefore as an example consider the case where the source of the probe field is a (quantum) wire. We imagine that the axis of the wire is placed parallel to the $y$-axis and cuts the $x$-$z$-plane in the point $(0,-z_0)$, cf. Fig. \[fig:1\]. Under the assumption that the spatial electron confinement in the wire is perfect (complete) and the wire current density is the same all along the wire at a given time, the harmonic source current density is given by $$\vec{J}(\vec{r};\omega)=\vec{J}_{0}(\omega)\delta(x)\delta(z+z_0),
\label{eq:Jrw}$$ where $\vec{J}_{0}(\omega)$ is its possibly frequency dependent vectorial amplitude. The spatial distribution of the field from this source is $$\begin{aligned}
\lefteqn{
\vec{E}(x,z;\omega)={1\over(2\pi)^2}\int_{-\infty}^{\infty}
\vec{E}(z;\vec{q}_{\|},\omega)e^{{\rm{i}}\vec{q}_{\|}\cdot\vec{r}}
\delta(q_{\|,y})d^2q_{\|},
}\nonumber\\ &\quad&=
{1\over(2\pi)^2}\int_{-\infty}^{\infty}
\vec{E}(z;\vec{q}_{\|},\omega)e^{{\rm{i}}q_{\|}x}dq_{\|},\end{aligned}$$ where $$\vec{E}(z;\vec{q}_{\|},\omega)=
-{e^{{\rm{i}}q_{\perp}(z+z_0)}\over2\epsilon_0\omega{}q_{\perp}}
\left[\begin{array}{ccc}
q_{\perp}^2 & 0 & -q_{\|}q_{\perp} \\
0 & q^2 & 0 \\
-q_{\|}q_{\perp} & 0 & q_{\|}^2
\end{array}\right]
\cdot\vec{J}_{0}(\omega),
\label{eq:Wire-z}$$ where as hitherto $q_{\|}^2+q_{\perp}^2=q^2$. At the phase conjugating mirror, the Fourier components of the wire probe are $\vec{E}(-d;\vec{q}_{\|},\omega)$.
(0,0)(127,95) (-3,2) (0,51)[$\displaystyle{\Delta{}I_{\rm{probe}}(-d;\vec{q}_{\|})\over|J_{0}(\omega)|^2}$ \[V/A\]]{} (73,0)[(0,0)\[b\][$q_{\|}/q$]{}]{}
To illustrate the angular spectral distribution of the field from this kind of wire source at the phase conjugator, we look closer at the cases, where (i) the current density is polarized along the $x$-axis, and (ii) along the $y$-axis. Thus, in case (i) we use $\vec{J}_{0}(\omega)=J_{0}(\omega)\vec{e}_{x}$, and by normalizing the electric fields to the amplitude of the current density, the corresponding normalized differential intensity \[$\Delta{}I_{\rm{Probe}}\equiv{1\over2}\epsilon_0c_0\vec{E}(-d;\vec{q}_{\|},\omega)\cdot\vec{E}^{*}(-d;\vec{q}_{\|},\omega)/(2\pi)^4$\] becomes $$\begin{aligned}
\lefteqn{
{\Delta{}I_{\rm{Probe}}(-d;\vec{q}_{\|})\over|{J}_{0}(\omega)|^2}=
{1\over2^7\pi^4\epsilon_0c_0}\left\{\Theta\left({1-(q_{\|}/q)}\right)
+\Theta\left((q_{\|}/q)-1\right)
\left[2({q_{\|}/q})^2-1\right]
\right.}\nonumber\\ &\quad&\times\left.
\exp\left({-2(z_0-d)q\sqrt{(q_{\|}/q)^2-1}}\right)\right\}
\label{eq:Ip-i}\end{aligned}$$ is shown in Fig. \[fig:7\] for different values of the distance $z_0-d$ from the wire to the vacuum/film interface. In case (ii), $\vec{J}_{0}(\omega)=J_{0}(\omega)\vec{e}_{y}$, and the associated normalized intensity which is given by $$\begin{aligned}
\lefteqn{
{\Delta{}I_{\rm{Probe}}(-d;\vec{q}_{\|})\over|{J}_{0}(\omega)|^2}=
{1\over2^7\pi^4\epsilon_0c_0}\left\{
{\Theta\left({1-(q_{\|}/q)}\right)\over1-(q_{\|}/q)^2}
+{\Theta\left((q_{\|}/q)-1\right)\over({q_{\|}/q})^2-1}
\right.}\nonumber\\ &\quad&\left.\!\times
\exp\left({-2(z_0-d)q\sqrt{(q_{\|}/q)^2-1}}\right)\right\},
\label{eq:Ip-ii}\end{aligned}$$ is also presented in Fig. \[fig:7\], for the same distances as in case (i). The third curve in Fig. \[fig:7\] represents the case where $\vec{J}_{0}(\omega)=J_{0}(\omega)\vec{e}_{z}$, and is shown for reference.
Looking at the curve in Fig. \[fig:7\] corresponding to $\vec{J}_{0}(\omega)=J_{0}(\omega)\vec{e}_{y}$ (and the curve corresponding to $\vec{J}_{0}(\omega)=J_{0}(\omega)\vec{e}_{z}$), we notice that a singularity occurs when $q_{\|}/q=1$, or equivalently where $q_{\perp}=0$. The presence of this singularity is an artifact originating in the (model) assumption that the electron confinement is complete in the $x$- and $z$-directions (see Eq. (\[eq:Jrw\])). If we had started from a quantum wire current density of finite (but small) extension in $x$ and $z$ the singularity would have been replaced by a (narrow) peak of finite height. Not only in quantum wire optics, but also in optical studies of quantum dots and wells singularities would appear if complete electron confinement was assumed (in 3D and 1D, respectively). In the present context the assumption of perfect electron confinement works well because we only consider the generated field outside the self-field region of the wire \[see, e.g., \]. In an experiment one would always end up integrating over some finite interval of $q_{\|}$ around the singularity, and this integral can in all cases be proven finite. At each distance of the wire from the phase conjugator the two curves $J_{0}\vec{e}_{x}$ and $J_{0}\vec{e}_{z}$ in Fig. \[fig:7\] becomes identical when $(q_{\|}/q)^2\gg{}1$, since from Eq. (\[eq:Wire-z\]) we may draw the relation $E_z=-(q_{\|}/q_{\perp})E_x$, and since $q_{\|}/q_{\perp}\simeq1$ when $(q_{\|}/q)^2\gg{}1$.
When the current oscillates in the direction of the wire, it appears that the field intensity in the evanescent probe modes is very small. An appreciable amount of the radiated energy is stored in components in the region $q_{\|}/q\sim1$ (and in the propagating modes). To study the phase conjugation of evanescent modes it is therefore better to start from $\vec{J}_{0}(\omega)=J_{0}(\omega)\vec{e}_{x}$ or from $\vec{J}_{0}(\omega)=J_{0}(\omega)\vec{e}_{z}$ because these two probe current densities give rise to significant probe intensities in the evanescent regime. If we look at the curve in Fig. \[fig:7\] representing the field at the surface of the phase conjugator when the probe is placed at $z_0-d=\lambda/256$, $I_{\rm{Probe}}$ peaks in both these cases at $q_{\|}/q\sim50$ in the evanescent regime. When the current density oscillates along the surface (in the $x$-direction) there is no singularity (and no peak) at $q_{\|}/q\sim1$, and the maximum value of $I_{\rm{Probe}}$, occuring at $q_{\|}/q\sim50$, is three orders of magnitude larger than the probe intensities of every one of the propagating modes. Above $q_{\|}/q\approx50$ the amplitude of the $q_{\|}$ components descends rapidly and has lost six orders of magnitude within the next order of magnitude of $q_{\|}/q$. At larger probe to surface distances the maximum in the $q_{\|}/q$ spectrum of the probe field at the vacuum/film interface is shifted downwards, and the magnitude becomes smaller, too. That is, compared to the raw $p$ to $p$ reflection coefficient, the intensity of each of the Fourier components available from the probe field begin their own falloff about one to two orders of magnitude before the reflection coefficient descends, depending on the distance from the probe to the surface of the phase conjugator. The $s$-polarized probe field starts the descending tendency already where the character of the Fourier components shifts from being propagating to evanescent ($q_{\perp}$ becoming imaginary), cf. the remarks above.
(0,0)(127,117) (0,60)[ (-3,2) (0,34)[$\displaystyle{I_{\rm{PC}}(-d;\vec{q}_{\|})\over{}I^{(1)}I^{(2)}|J_{0}(\omega)|^2}$ \[m$^2$/WA$^2$\]]{} (71.5,0)[(0,0)\[b\][$q_{\|}/q$]{}]{} (24,52)[(0,0)\[tl\][$J_{0}\vec{e}_{x}$]{}]{} ]{} (0,0)[ (-3,2) (0,34)[$\displaystyle{I_{\rm{PC}}(-d;\vec{q}_{\|})\over{}I^{(1)}I^{(2)}|J_{0}(\omega)|^2}$ \[m$^2$/WA$^2$\]]{} (71.5,0)[(0,0)\[b\][$q_{\|}/q$]{}]{} (24,52)[(0,0)\[tl\][$J_{0}\vec{e}_{y}$]{}]{} ]{}
Using a quantum wire as the source for the probe field, the angular spectrum of the phase conjugated response, normalized to the pump fields and the absolute square of the amplitude of the wire current density, is given by $$R_{\rm{PC}}(z;\vec{q}_{\|})
{I_{\rm{Probe}}(-d;\vec{q}_{\|})\over|J_{0}(\omega)|^2}=
{I_{\rm{PC}}(z;\vec{q}_{\|})\over{}I^{(1)}I^{(2)}|J_{0}(\omega)|^2},
\label{eq:RI/J}$$ and is obtained numerically by multiplying the energy reflection coefficient, $R_{\rm{PC}}(z;\vec{q}_{\|})$, with the normalized probe intensity, $I_{\rm{Probe}}(-d;\vec{q}_{\|})/|J_{0}(\omega)|^2$. In Fig. \[fig:8\], the angular spectrum at the vacuum/quantum-well interface ($z=-d$) given by Eq. (\[eq:RI/J\]) is shown for the cases where $\vec{J}_{0}(\omega)=J_{0}(\omega)\vec{e}_{x}$ and $\vec{J}_{0}(\omega)=J_{0}(\omega)\vec{e}_{y}$. It is plotted for the two-monolayer film, but since the main contribution is in the low end of the $q_{\|}/q$-spectrum, the similar curves for the single-monolayer film would be indistinguishable from the ones plotted (apart from a factor of two in the $ppp$ case). In both cases data are presented for the wire placed at different distances from the vacuum/film interface. By comparison with the raw reflection data in Fig. \[fig:3\] it appears that the high end of the reflected $q_{\|}$-spectrum is strongly damped. For the $s$ to $p$ transition we see that the energy of the phase conjugated signal is concentrated around $q_{\|}/q=1$, which is mainly due to the fact that the concentration of the radiated energy spectrum from the wire lies around that same point. In the $p$ to $p$ transition the evanescent components are still by far dominating the response at the place of the wire compared to the propagating components.
Two-dimensional confinement of light\
in front of a single-level quantum-well phase conjugator {#ch:12}
========================================================
The possibility of compressing light in space to a degree (much) better than predictable by classical diffraction theory has gained widespread attention only with the birth of near-field optics [@Pohl:93:1; @Nieto-Vesperinas:96:1].
As stated in Chapter \[Ch:1\], sub-wavelength electrodynamics was discussed only sporadically until near-field optics evolved in the mid-eighties in the wake of the experimental works by the groups of Pohl, Lewis, and Fischer [@Pohl:84:1; @Lewis:84:1; @Fischer:85:1]. The first investigations are usually attributed to , who presented a proposal for sub-wavelength microscopy as early as in 1928. The subject was studied again in 1944 by , by in 1950, and a proposal much similar to that of Synge was made by in 1956. Using microwaves, resolved a grating with a linewidth of $\lambda/60$ in 1972.
In the wake of theoretical studies of the possibility for phase conjugating the field emitted from a mesoscopic object carried out by , creation of light foci with a diameter below the classical diffraction limit was demonstrated experimentally by [-@Bozhevolnyi:94:1; -@Bozhevolnyi:95:2], who used the fibre tip of a near field optical microscope to create source spots of red light (633nm) in front of a photorefractive Fe:LiNbO$_3$ crystal, which acted as a phase conjugator. After creation, the phase conjugated replica of these light spots could be detected using the near field microscope, since they were maintained for approximately ten minutes because of the long memory of the phase conjugation process in the crystal. The resulting phase conjugated light foci had diameters of around 180nm, and the conclusion of their work was therefore that at least some of the evanescent field components of the source also must have been phase conjugated in order to achieve the observed size of the phase conjugated image.
The above-mentioned observation drew renewed attention to the description of focusing of electromagnetic fields in front of phase conjugating mirrors, and required inclusion of evanescent modes in the description of the optical phase conjugation process. In an important paper, , extending an original idea of , undertook an analysis of the phase conjugated replica of the field from a point particle as it is produced by a so-called ideal phase conjugator, and in recent articles by [-@Keller:96:3; -@Keller:96:2] attention was devoted to an investigation of microscopic aspects of the spatial confinement problem of the phase conjugated field.
In the previous chapters, we developed a microscopic theory for optical phase conjugation by degenerate four-wave mixing in mesoscopic interaction volumes, with the aim of establishing a theoretical framework for inclusion of near field components in the analysis. In order for near field components to give a significant contribution to the phase conjugated response, the phase conjugation process must be effective in a surface layer of thickness (much) less than the optical wavelength. This makes quantum well systems particularly adequate candidates for the nonlinear mixing process.
In this chapter we employ the developed microscopic theory to a study of the spatial confinement of an electromagnetic field emitted from an ideal line source (quantum wire with complete electron confinement), paying particular attention to the evanescent part of the angular spectrum. As phase conjugator a single-level metallic quantum well, particularly effective in phase conjugating evanescent field components, albeit with an overall small conversion efficiency, is used. The relevant expression for the phase conjugated field is given and the result of a numerical calculation of the nonlinear energy reflection coefficient for a copper well presented. Finally, the intensity distribution of the phase conjugated field in the region between the line source and the phase conjugator is calculated and the two-dimensional spatial focusing investigated.
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Quantum wire as a two-dimensional point source
----------------------------------------------
As starting point, we consider a line source (quantum wire) placed parallel to the $y$-axis of a Cartesian coordinate system and cutting the $x$-$z$-plane in the point $(0,-z_0)$. In the description of the source we assume (i) perfect spatial electron confinement in the wire and (ii) constant current density along the wire at a given time. Choosing the wire current to oscillate along the $x$-direction, the above-mentioned assumptions lead to a harmonic source current density given by $$\vec{J}(\vec{r};\omega)={J}_{0}(\omega)\delta(x)\delta(z+z_0)\vec{e}_{x},$$ where $\vec{J}_{0}(\omega)={J}_{0}(\omega)\vec{e}_{x}$ is its possibly frequency dependent vectorial amplitude, $\vec{e}_{x}$ being a unit vector in the $x$-direction. In order to calculate the phase conjugated response using the developed microscopic model we perform a Fourier analysis of the source field along the $x$-axis. The electric field of the quantum wire at the surface ($z=-d$) of the phase conjugating mirror thus becomes $$\vec{E}(x,-d;\omega)={1\over(2\pi)^2}\int_{-\infty}^{\infty}
\vec{E}(-d;\vec{q}_{\|},\omega)e^{iq_{\|}x}dq_{\|},
\label{eq:Wire-xz}$$ where the parallel component of the probe wavevector lies along the $x$-axis, i.e., $\vec{q}_{\|}=q_{\|}\vec{e}_{x}$, and [@Keller:98:1] $$\vec{E}(-d;\vec{q}_{\|},\omega)=
-{e^{iq_{\perp}(z_0-d)}\over2\epsilon_0\omega}
\left[\begin{array}{c} q_{\perp} \\ 0 \\ -q_{\|} \end{array}\right]
{J}_{0}(\omega).
\label{eq:Wire-z2}$$ In Eq. (\[eq:Wire-z2\]), $q_{\perp}$ is determined from the vacuum dispersion relation for the field, i.e., $(q_{\perp}^2+q_{\|}^2)^{1/2}=q$, where $q=\omega/c_0$ is the vacuum wavenumber. For propagating modes, satisfying the inequality $|q_{\|}|<q$, $q_{\perp}=(q^2-q_{\|}^2)^{1/2}$ is real (and positive), whereas for evanescent modes having $|q_{\|}|>q$, $q_{\perp}=i(q_{\|}^2-q^2)^{1/2}$ is purely imaginary. To illustrate the angular spectral distribution of the field from the wire at the phase conjugator, we calculate the magnitude of the differential source (probe) field intensity, i.e., $\Delta
I_{\rm{probe}}(-d;\vec{q}_{\|},\omega)= {1\over2}\epsilon_0c_0
\vec{E}(-d;\vec{q}_{\|},\omega)\cdot
\vec{E}^{*}(-d;\vec{q}_{\|},\omega)/(2\pi)^4$. From Eq. (\[eq:Wire-z2\]) one obtains \[Eq. (\[eq:Ip-i\])\] $$\begin{aligned}
\lefteqn{
{\Delta I_{\rm{probe}}(-d;\vec{q}_{\|})\over|{J}_{0}(\omega)|^2}=
{1\over2^7\pi^4\epsilon_0c_0}
\Bigl\{\Theta\left({1-(q_{\|}/q)}\right)
+\Theta\left((q_{\|}/q)-1\right)
\left[2({q_{\|}/q})^2-1\right]
}\nonumber\\ &\quad&\times
\exp\left({-2(z_0-d)q\sqrt{(q_{\|}/q)^2-1}}\right)\Bigr\},
\label{eq:Ip-i2}\end{aligned}$$ where $\Theta$ is the Heaviside unit step function. It appears from Eq. (\[eq:Ip-i2\]), that $\Delta{}I_{\rm{probe}}$ is independent of $q_{\|}$ for the propagating modes. In Fig. \[fig:wire\], the normalized differential probe distribution $\Delta{}I_{\rm{probe}}/|J_0(\omega)|^2$ is shown as a function of $q_{\|}/q$ for various distances $z_0-d$ between the wire and the vaccum/phase-conjugator interface. It is seen that the evanescent components tend to dominate the angular spectrum when $z_0-d\lesssim\lambda/4$.
Single-level metallic quantum-well phase conjugator
---------------------------------------------------
We take as the active medium a metallic quantum well, and we describe the conduction electron dynamics using the infinite barrier model potential. While such a model potential from a quantitative point of view of course is too na[ï]{}ve, in particular in cases where the conduction electrons of the well are allowed to mix with those of a semiconducting or metallic substrate, it suffices in the present context. We further assume that only the lowest lying band is below the Fermi energy and that the photon energy is so small that interband transitions do not contribute to the electrodynamics. The quantum well is deposited on a substrate that can be described alone by its refractive index $n$. Because of the chosen polarization of the wire current density we have limited the description to cover only the case where all interacting electric fields are polarized in the scattering plane ($p$-polarization). Then, within the limits of a self-field approximation, the $z$-component of the phase conjugated field becomes \[Eq. (\[eq:PC-ppp\])\] $$\begin{aligned}
\lefteqn{
E_{{\rm{PC}},z}(z;\vec{q}_{\|},\omega)=
{e^2(\omega+i/\tau)(1+r^p)\over2^9\pi^6\hbar\omega^3ZN_+m_e}
{q_{\|}^2\over{}iq_{\perp}}
\left[{\cal{C}}(q_{\|}-k_{\|})
+{\cal{C}}(q_{\|}+k_{\|})\right]
}\nonumber\\ &\quad&\times
E_z^{(1)}E_z^{(2)}E_z^{*}e^{-iq_{\perp}z},
\label{eq:PC-ppp2}\end{aligned}$$ where $${\cal{C}}(q_{\|}\pm{}k_{\|})=
{4\pi\over{}a^2}\left\{\sqrt{b^2-a^2\alpha^2}-a(q_{\|}\pm{}k_{\|})
-\sqrt{\left[b-a(q_{\|}\pm{}k_{\|})\right]^2-a^2\alpha^2}\right\},$$
(0,0)(127,117) (0,60)[ (-1,2) (0,32)[$R_{\rm{PC}}({q}_{\|})$ \[$10^{-36}$m$^4$/W$^2$\]]{} (71.5,0)[(0,0)\[b\][$q_{\|}/q$]{}]{} (115,4)(115,56) (115,0)[(0,0)\[b\][$k_F/q$]{}]{} (16,51)[(0,0)\[tl\][1ML Cu]{}]{} ]{} (0,0)[ (-1,2) (0,32)[$R_{\rm{PC}}({q}_{\|})$ \[$10^{-37}$m$^4$/W$^2$\]]{} (71.5,0)[(0,0)\[b\][$q_{\|}/q$]{}]{} (113,4)(113,56) (113,0)[(0,0)\[b\][$k_F/q$]{}]{} (16,51)[(0,0)\[tl\][2ML Cu]{}]{} ]{}
with $a=\hbar(q_{\|}\pm{}k_{\|})/m_e$ and $b=\hbar(q_{\|}\pm{}k_{\|})^2/(2m_e)-i/\tau$. The quantity $\alpha$ is the radius of the (two-dimensional) Fermi circle, given by $\alpha=[k_F^2-(\pi/d)^2]^{1/2}$. Given the $z$-component of the phase conjugated field, the $x$-component is also known, since the electric field must be transverse in vacuum. Above, $-e$ and $m_e$ are the electron charge and mass, respectively, $\omega$ and $\tau$ are the cyclic frequency of the optical field and the electron relaxation time, and $k_{\|}>0$ is the parallel component of the wavevector of pump field number $1$ having a $z$-component $E_z^{(1)}$. The corresponding quantities for pump field number $2$ are $-k_{\|}$ and $E_z^{(2)}$. When the two pump fields have numerically equal wavevector components in the plane of the phase conjugator, conservation of momentum parallel to the surface implies that given angular components of the probe and phase conjugated fields are counterpropagating along the surface. Moreover, $Z$ is the number of conduction electrons each atom in the quantum well contributes to the assumed free-electron gas, $N_+$ is the number of atoms per unit volume in the quantum well, and $k_F$ is the Fermi wavenumber. Since the two pump fields are counterpropagating, the wavevectors of the probe field ($\vec{q}_{\|}^{\,\prime}$) and the phase conjugated field ($\vec{q}_{\|}$) are related through the conservation of momentum, $\vec{q}_{\|}^{\,\prime}+\vec{q}_{\|}=\vec{0}$. This property was used in the derivation of Eq. (\[eq:PC-ppp2\]). Thus, the $z$-component of the probe field in Eq. (\[eq:Wire-z2\]), $E_z$, is given by $E_z=\int{}E_z(-d;\vec{q}_{\|}^{\,\prime},\omega)
\delta(\vec{q}_{\|}^{\,\prime}+\vec{q}_{\|})d^2q_{\|}'$, where $E_z(-d;\vec{q}_{\|}^{\,\prime},\omega)$ is taken from Eq. (\[eq:Wire-z2\]).
Numerical results and discussion
--------------------------------
For the numerical calculation we consider a copper quantum well \[$d=1.8$[Å]{} (for single-monolayer film), respectively $3.6$[Å]{} (two monolayers), $N_+=8.47\times10^{28}$m$^{-3}$ and $Z=1$ [@Ashcroft:76:1]\] deposited on a glass substrate with $n=1.51$, giving $k_{\|}=1.51q$ when the wavevectors of the pump fields are parallel to the $x$-axis. The wavelength of the light is chosen to be $\lambda=1061$nm. For a glass substrate it is adequate to calculate the linear vaccum/substrate amplitude reflection coefficient $r^p$ by means of the classical Fresnel formula $r^p=[{n^2q_{\perp}-({n^2q^2-q_{\|}^2})^{1\over2}}]/
[{n^2q_{\perp}+({n^2q^2-q_{\|}^2})^{1\over2}}]$. In the view of the recent experimental data discussed in Section \[sec:11.1\] [@Jalochowski:97:1], we have chosen an intraband relaxation time of $\tau=3$fs for the electrons in the ultrathin quantum-well film.
To give an impression of the efficiency of the phase conjugation process for the various evanescent modes, the nonlinear energy reflection coefficient of the phase conjugator at the vacuum/film interface, $R_{\rm{PC}}({q}_{\|})$, is shown in Fig. \[fig:RPC\] as a function of the parallel component ($q_{\|}$) of the probe wavevector for the two possible single-level quantum wells. It appears from this figure that in particular high spatial frequency components ($10^2\lesssim{}q_{\|}/q\lesssim{}10^3$) are phase conjugated in an effective manner. This is associated with the fact that in a single-level quantum well the two-dimensional intraband electron dynamics along the plane of the well is responsible for the main part of the phase conjugation process.
Using a square-potential barrier model to describe the quantum well the integration limits should not extend beyond the (two-dimensional) Fermi wavenumber $k_F=[2\pi{}ZN_+d+(\pi/d)^2]^{1/2}$ for the single level quantum well. Looking at the phase conjugation reflection coefficient shown in Fig. \[fig:RPC\], we not only notice that the main contribution to the phase conjugated signal is well above the point where the probe field becomes evanescent in vaccum, but also does not extend beyond the Fermi wavenumber. The phase conjugated image of our quantum-wire source field is then given by $$\vec{E}_{\rm{PC}}(x,z;\omega)={1\over2\pi}\int_{-k_F}^{k_F}
\vec{E}_{\rm{PC}}(z;\vec{q}_{\|},\omega)e^{iq_{\|}x}dq_{\|},$$ in the $x$-$z$-plane.
It appears from Eq. (\[eq:PC-ppp2\]), that the individual angular components of the phase conjugated field decay exponentially with the distance from the phase conjugator in the evanescent part of the Fourier spectrum. The evanescent components of the source likewise decay exponentially with the distance from the quantum wire. Therefore, the contribution from the evanescent components to the total phase conjugated field is expected to increase significantly when the distance between the source and the phase conjugator becomes smaller. Experimentally, it is feasible presently to carry out measurements at distances from the surface down to $\sim40$[Å]{} (using near-field microscopes). For the chosen system this leads to an intensity distribution in the $x$-$z$-plane between the probe and the phase conjugator as shown in Figs. \[fig:12.3\] and \[fig:12.4\] for different distances between the probe and the film, $z_0-d\in\{\lambda/4$, $3\lambda/16$, $\lambda/8$, $3\lambda/32$, $\lambda/16\}$, Fig. \[fig:12.3\] corresponding to the single-monolayer film and Fig. \[fig:12.4\] to the two-monolayer film. For $z_0-d\gtrsim\lambda/2$, the effect of the near-field components are negligible. In both Figs. \[fig:12.3\] and \[fig:12.4\], the figures to the left show by equal-intensity contours the intensity distribution of the phase conjugated field in the area of the $x$-$z$-plane between the quantum wire and the surface of the phase conjugator. The width (along the $x$-axis) of the shown area is in all cases twice the height (along the $z$-axis) on both sides of $x=0$. The contours are drawn in an exponential sequence, so that if the first contour corresponds to the intensity $I_{\rm{PC}}^{(1)}$ the $n$-th contour is associated with the intensity $I_{\rm{PC}}^{(n)}=I_{\rm{PC}}^{(1)}\exp[(1-n)\alpha]$, $\alpha$ varying from figure to figure. To further illustrate the capabilities of light focusing the chosen system possess, we have to the right on a linear scale shown the phase conjugated intensity at (i) the surface of the phase conjugator (solid lines) and (ii) along an axis parallel to the $x$-axis placed at the same distance from the phase conjugator as the wire (dashed lines). The two curves in each of the plots to the right are adjusted by multiplication of the curve in case (ii) by a factor of (a) $2590$, (b) $545$, (c) $271$, (d) $353$, and (e) $586$ in Fig. \[fig:12.3\], and by a factor of (a) $555$, (b) $212$, (c) $204$, (d) $322$, and (e) $528$ in Fig. \[fig:12.4\], respectively, so that the maximum values coincide in the plots.
It is seen from Figs. \[fig:12.3\] and \[fig:12.4\] that the width of the focus created by the phase conjugated field in all cases is smallest at the surface of the phase conjugator. Furthermore one observe that the focus becomes narrower when the distance between the source and the film becomes smaller, as one should expect when evanescent field components give a significant contribution to the process. At the surface the width of the main peak decreases roughly by a factor of two every time the source-film distance decreases by the same factor. This tendency continues closer to the surface of the phase conjugator, at least down to around $\lambda/256\approx4$nm, where the structure of the intensity distribution looks more or less like Fig. \[fig:12.3\].e \[and Fig. \[fig:12.4\].e\], scaled appropriately with the distance.
The difference between the ability of the single-monolayer film and the two-monolayer film to focus the field emitted from the source is small, as we would expect from the analysis in the previous chapter. We observe from Fig. \[fig:12.3\] that when the source is far away from the film (a–b), the intensity of the phase conjugated field decays faster in the case of a single-monolayer film than in the other case. Furthermore, we observe that the width of the centre peak is somewhat smaller for the focus in front of the single-monolayer phase conjugator, and that the height of the first sidelobe is slightly higher compared to the centre peak. Taking the source closer to the surface (c–e) we observe that the distance between minima in Fig. \[fig:12.3\] is still smaller than in Fig. \[fig:12.4\], but that the width of the centre peak becomes more and more equal in the two cases.
To give an impression of the size of the phase conjugated focus, let us take a look at Fig. \[fig:intensity-ptop\].e, where the distance from the probe to the surface is $\lambda/16\approx66$nm. The distance between the two minima at the surface of the phase conjugator is in this case around $40$nm (approximately $\lambda/25$). In the plane of the probe the distance between the two minima is around $100$nm ($\sim\lambda/10$).
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(60,-1.5)[(0,0)\[t\]]{} (19,-1.5)[(0,0)\[t\]]{} (42,20.5)[(0,0)\[bl\]]{}
(0,120)(0,-60)[3]{}
(50,-1.5)[(0,0)\[t\]]{} (29,-1.5)[(0,0)\[t\]]{} (70,-1.5)[(0,0)\[t\]]{} (9,-1.5)[(0,0)\[t\]]{}
(0,90)(0,-60)[2]{}
(50,-1.5)[(0,0)\[t\]]{} (29,-1.5)[(0,0)\[t\]]{} (70,-1.5)[(0,0)\[t\]]{} (9,-1.5)[(0,0)\[t\]]{}
(0,120)(0,-60)[3]{}
(80,-1.5)[(0,0)\[t\]]{} (-1,-1.5)[(0,0)\[t\]]{}
(0,90)(0,-60)[2]{}
(80,-1.5)[(0,0)\[t\]]{} (-1,-1.5)[(0,0)\[t\]]{}
(80,120)(0,-30)[5]{}
(1,22)[(0,0)\[tl\][()]{}]{}
(88,0)
(0,120) (0,90) (0,60) (0,30) (0,0) (0,120) (0,90) (0,60) (0,30) (0,0) (0,120)(0,-30)[5]{}[ (18,-2)(18,22) (-2,0)(40,0) (0,0)(9,0)[5]{}[(0,-1.25)(0,0)]{} (0,0)(2.25,0)[17]{}[(0,-0.75)(0,0)]{} (40,1.5)[(0,0)\[br\][$x$]{}]{} (16.5,22)[(0,0)\[tr\][$I_{\rm{PC}}$]{}]{} (18.5,-2)[(0,0)\[tl\]]{} ]{} (0,120)(0,-60)[3]{}
(27,-1.5)[(0,0)\[t\]]{} (8.5,-1.5)[(0,0)\[t\]]{}
(0,90)(0,-60)[2]{}
(27,-1.5)[(0,0)\[t\]]{} (8.5,-1.5)[(0,0)\[t\]]{}
(0,120)(0,-60)[3]{}
(36,-1.5)[(0,0)\[t\]]{} (-0.5,-1.5)[(0,0)\[t\]]{}
(0,90)(0,-60)[2]{}
(36,-1.5)[(0,0)\[t\]]{} (-0.5,-1.5)[(0,0)\[t\]]{}
Discussion {#discussion}
==========
We have in the previous three chapters discussed the phase conjugation response of a single-level quantum well, where only intraband transitions are possible. It is evident from our analysis that in this case the phase conjugated response depends strongly on the component of the probe wavevector that is parallel to the surface of the phase conjugator ($q_{\|}$). Consequently, the assumption of an ideal phase conjugator with constant reflection coefficient throughout the full $q_{\|}$-spectrum must be abandoned, at least when a single-level quantum-well phase conjugator is considered. The nonlinear coupling is strongest in the evanescent part of the $q_{\|}$-spectrum above the point up to which the probe field is propagating in the substrate ($q_{\|}/q=n$). As a consequence, if one wants to observe the phase conjugation of a broad Fourier spectrum of evanescent modes, both the observation and the excitation are required to take place near the surface of the phase conjugator.
As a possible method to excite the Fourier components in the high end of the $q_{\|}$-spectrum we have analyzed the consequences of using a quantum wire. When the quantum wire is placed close to the phase conjugator the phase conjugated response contains a broad range of evanescent components. This property made it a good candidate for investigations of the problem of focusing light to a spatial extent less than the Rayleigh limit. The spatial focusing of the phase conjugated response from a quantum wire was studied, and the problem of the resolution limit has been addressed. The conclusion of this study is in agreement with previous studies and mainly shows that the focus gets narrower when the distance from the quantum wire to the phase conjugator gets shorter. Judging from this we may conclude that in order to establish a better estimate of the limit of resolution the present model has to be improved, since continuing to get closer becomes meaningless at some point.
In examining the phase conjugation response of the single-level quantum well we have chosen a specific frequency of the interacting electromagnetic fields, which should be feasible for an experiment. It would be interesting to give a more detailed account of how (i) the phase conjugated response and (ii) the focusing of the phase conjugated field varies with the frequency. These properties will not be discussed in detail in this dissertation, but let us at this point just mention that (i) it seems like the evanescent components in the high end of the Fourier spectrum becomes better phase conjugated when the interacting fields move to longer wavelengths, and (ii) the spatial distance between minima in the intensity of the phase conjugated response when using a quantum wire becomes smaller when measured in fractions of the wavelength. Going to shorter wavelengths, the tendency goes in the opposite direction.
\[part:V\]
Theoretical properties {#ch:14}
======================
From the simple description of a quantum well where only intraband transitions contributed to the phase conjugated response we now turn our attention to the case where transitions between energy levels in the quantum well (interband transitions) can take place. In this chapter we therefore give the theoretical description that is necessary to describe the phase conjugated response from a quantum well where both interband and intraband transitions contribute to the response. In the following we adopt the same scattering geometry as in the previous treatment, i.e., scattering takes place in the $x$-$z$-plane and we use light that is polarized either in ($p$-polarized) or perpendicular to ($s$-polarized) the scattering plane.
Phase conjugated field
----------------------
Unlike in the case of a single-level quantum well, we cannot rely on the self-field approximation when considering multi-level quantum wells (with resonances). We therefore begin this treatment with the loop equation for the phase conjugated field in the two-dimensional Fourier space \[Eq. (\[eq:loop\])\]. It is repeated here for convenience: $$\begin{aligned}
\lefteqn{
\vec{E}_{\rm{PC}}(z;\vec{q}_{\|},\omega)=
\vec{E}_{\rm{PC}}^{\rm{B}}(z;\vec{q}_{\|},\omega)
}\nonumber\\ &\quad&
-{\rm{i}}\mu_0\omega
\iint
\tensor{G}(z,z'';\vec{q}_{\|},\omega)
\cdot\stensor{\sigma}(z'',z';\vec{q}_{\|},\omega)
\cdot \vec{E}_{\rm{PC}}(z';\vec{q}_{\|},\omega)dz''dz'.
\label{eq:PC-loop}\end{aligned}$$ The background field in Eq. (\[eq:PC-loop\]) is given by $$\vec{E}_{\rm{PC}}^{\rm{B}}(z;\vec{q}_{\|},\omega)=
-{\rm{i}}\mu_0\omega\int
\tensor{G}(z,z';\vec{q}_{\|},\omega)
\cdot\vec{J}_{-\omega}^{\,(3)}(z';\vec{q}_{\|},\omega)dz',$$ and can be determined from the previous analysis. The linear conductivity tensor consists in general of a diamagnetic and a paramagnetic part (see the discussion in Chapters \[ch:5\] and \[ch:6\]). It is, however, possible to combine the two parts in such a way that the diamagnetic conductivity tensor can be written as a correction to the paramagnetic one (, ). Then the expression for the total linear conductivity tensor becomes $$\begin{aligned}
\stensor{\sigma}(\vec{r},\vec{r}\,';\omega)=
-{2\over{}i\omega}{1\over\hbar}\sum_{nm}{\omega\over\tilde{\omega}_{nm}}
{f_n-f_m\over\tilde{\omega}_{nm}-\omega}
\vec{J}_{nm}(\vec{r})\otimes\vec{J}_{mn}(\vec{r}\,'),\end{aligned}$$ the correction from the diamagnetic term to the paramagnetic response being the factor $\omega/\tilde{\omega}_{nm}$, which close to resonance becomes $1$. In the two-dimensional mixed Fourier space this is $$\begin{aligned}
\lefteqn{
\stensor{\sigma}(z,z';\vec{q}_{\|},\omega)=
-{2\over{}i\omega}{1\over\hbar}{1\over(2\pi)^2}\sum_{nm}\int{\omega\over
\tilde{\omega}_{nm}(\vec{\kappa}_{\|}+\vec{q}_{\|},\vec{\kappa}_{\|})}
{f_{n}(\vec{\kappa}_{\|}+\vec{q}_{\|})-f_{m}(\vec{\kappa}_{\|})\over
\tilde{\omega}_{nm}(\vec{\kappa}_{\|}+\vec{q}_{\|},\vec{\kappa}_{\|})-\omega}
}\nonumber\\ &\quad&\times
\vec{j}_{nm}(z;2\vec{\kappa}_{\|}+\vec{q}_{\|})\otimes
\vec{j}_{mn}(z';2\vec{\kappa}_{\|}+\vec{q}_{\|})
d^2\kappa_{\|}.
\label{eq:sigma-zz'qw}\end{aligned}$$ The transition frequency appearing in Eq. (\[eq:sigma-zz’qw\]) is $$\tilde{\omega}_{nm}(\vec{\kappa}_{\|}+\vec{q}_{\|},\vec{\kappa}_{\|})=
{1\over\hbar}\left[\varepsilon_n-\varepsilon_m+{\hbar^2\over2m_{e}}
\left(2\kappa_xq_x+q_x^2\right)\right]-{i\over\tau_{nm}},$$ which by insertion into Eq. (\[eq:sigma-zz’qw\]) gives the five nonzero elements of the linear conductivity tensor $$\begin{aligned}
\sigma_{xx}(z,z';\vec{q}_{\|},\omega)&=&
\sum_{nm}{\cal{Q}}_{\,\,\,nm}^{xx}(\vec{q}_{\|},\omega)
{\cal{Z}}^{x}_{nm}(z){\cal{Z}}^{x}_{mn}(z'),
\label{eq:s-xx}\\
\sigma_{xz}(z,z';\vec{q}_{\|},\omega)&=&-i
\sum_{nm}{\cal{Q}}_{\,\,\,nm}^{xz}(\vec{q}_{\|},\omega)
{\cal{Z}}^{x}_{nm}(z){\cal{Z}}^{z}_{mn}(z'),
\label{eq:s-xz}\\
\sigma_{yy}(z,z';\vec{q}_{\|},\omega)&=&
\sum_{nm}{\cal{Q}}_{\,\,\,nm}^{yy}(\vec{q}_{\|},\omega)
{\cal{Z}}^{x}_{nm}(z){\cal{Z}}^{x}_{mn}(z'),
\label{eq:s-yy}\\
\sigma_{zx}(z,z';\vec{q}_{\|},\omega)&=&-i
\sum_{nm}{\cal{Q}}_{\,\,\,nm}^{xz}(\vec{q}_{\|},\omega)
{\cal{Z}}^{z}_{nm}(z){\cal{Z}}^{x}_{mn}(z'),
\label{eq:s-zx}\\
\sigma_{zz}(z,z';\vec{q}_{\|},\omega)&=&-
\sum_{nm}{\cal{Q}}_{\,\,\,nm}^{zz}(\vec{q}_{\|},\omega)
{\cal{Z}}^{z}_{nm}(z){\cal{Z}}^{z}_{mn}(z'),
\label{eq:s-zz}\end{aligned}$$ where we for the sake of notational simplicity have divided the total expression for each element into a $z$-dependent part and a $z$-independent part, the $z$-independent quantities being $$\begin{aligned}
\lefteqn{ {\cal{Q}}_{\,\,\,nm}^{xx}(\vec{q}_{\|},\omega)={2i\hbar\over(2\pi)^2}
\left({e\hbar\over2m_{e}}\right)^2
\int
{4\kappa_x^2+4\kappa_xq_x+q_x^2
\over\varepsilon_n-\varepsilon_m+\hbar(2\kappa_xq_x+q_x^2)/(2m_{e})
-i\hbar/\tau_{nm}}
}\nonumber\\ &\quad&\times
{f_{n}(\vec{\kappa}_{\|}+\vec{q}_{\|})-f_{m}(\vec{\kappa}_{\|})
\over\varepsilon_n-\varepsilon_m+\hbar(2\kappa_xq_x+q_x^2)/(2m_{e})
-i\hbar/\tau_{nm}-\hbar\omega}d^2\kappa_{\|},
\label{eq:A-xx}\\
\lefteqn{ {\cal{Q}}_{\,\,\,nm}^{xz}(\vec{q}_{\|},\omega)={2i\hbar\over(2\pi)^2}
\left({e\hbar\over2m_{e}}\right)^2
\int
{2\kappa_x+q_x
\over\varepsilon_n-\varepsilon_m+\hbar(2\kappa_xq_x+q_x^2)/(2m_{e})
-i\hbar/\tau_{nm}}
}\nonumber\\ &&\times
{f_{n}(\vec{\kappa}_{\|}+\vec{q}_{\|})-f_{m}(\vec{\kappa}_{\|})
\over\varepsilon_n-\varepsilon_m+\hbar(2\kappa_xq_x+q_x^2)/(2m_{e})
-i\hbar/\tau_{nm}-\hbar\omega}d^2\kappa_{\|},
\label{eq:A-xz}\\
\lefteqn{ {\cal{Q}}_{\,\,\,nm}^{yy}(\vec{q}_{\|},\omega)={2i\hbar\over(2\pi)^2}
\left({e\hbar\over2m_{e}}\right)^2
\int
{4\kappa_y^2
\over\varepsilon_n-\varepsilon_m+\hbar(2\kappa_xq_x+q_x^2)/(2m_{e})
-i\hbar/\tau_{nm}}
}\nonumber\\ &&\times
{f_{n}(\vec{\kappa}_{\|}+\vec{q}_{\|})-f_{m}(\vec{\kappa}_{\|})
\over\varepsilon_n-\varepsilon_m+\hbar(2\kappa_xq_x+q_x^2)/(2m_{e})
-i\hbar/\tau_{nm}-\hbar\omega}d^2\kappa_{\|},
\label{eq:A-yy}\\
\lefteqn{ {\cal{Q}}_{\,\,\,nm}^{zz}(\vec{q}_{\|},\omega)={2i\hbar\over(2\pi)^2}
\left({e\hbar\over2m_{e}}\right)^2
\int
{1\over\varepsilon_n-\varepsilon_m+\hbar(2\kappa_xq_x+q_x^2)/(2m_{e})
-i\hbar/\tau_{nm}}
}\nonumber\\ &&\times
{f_{n}(\vec{\kappa}_{\|}+\vec{q}_{\|})-f_{m}(\vec{\kappa}_{\|})
\over\varepsilon_n-\varepsilon_m+\hbar(2\kappa_xq_x+q_x^2)/(2m_{e})
-i\hbar/\tau_{nm}-\hbar\omega}d^2\kappa_{\|},
\label{eq:A-zz}\end{aligned}$$ since ${\cal{Q}}_{\,\,\,nm}^{xz}(\vec{\kappa}_{\|},\omega)=
{\cal{Q}}_{\,\,\,nm}^{zx}(\vec{\kappa}_{\|},\omega)$. The $z$-dependent quantities in Eqs. (\[eq:s-xx\])–(\[eq:s-zz\]) above are $$\begin{aligned}
{\cal{Z}}^{x}_{nm}(z)&=&{\cal{Z}}^{\,y}_{nm}(z)=\psi_m^*(z)\psi_n(z), \\
{\cal{Z}}^{z}_{nm}(z)&=&\psi_m^*(z){\partial\psi_n(z)\over\partial{}z}
-\psi_n(z){\partial\psi_m^*(z)\over\partial{}z}.\end{aligned}$$ Eqs. (\[eq:A-xx\])–(\[eq:A-zz\]) has the solutions given in Appendix \[app:C\], section \[sec:Q\] in terms of the analytic solution to the integrals given in Appendix \[ch:Solve-Q\]. Inserting this solution into Eq. (\[eq:PC-loop\]), we get $$\vec{E}_{\rm{PC}}(z;\vec{q}_{\|},\omega)=
\vec{E}_{\rm{PC}}^{\rm{B}}(z;\vec{q}_{\|},\omega)+\sum_{nm}
\tensor{F}_{nm}(z;\vec{q}_{\|},\omega)
\cdot \vec{\Gamma}_{mn}(\vec{q}_{\|},\omega),
\label{eq:loop-FG}$$ in which we have introduced the $3\times3$ tensor $\tensor{F}_{nm}(z;\vec{q}_{\|},\omega)$ with the nonzero elements $$\begin{aligned}
\lefteqn{
F_{nm}^{xx}(z;\vec{q}_{\|},\omega)=
-{\rm{i}}\mu_0\omega
\left\{
{\cal{Q}}_{\,\,\,nm}^{xx}(\vec{q}_{\|},\omega)
\int G_{xx}(z,z'';\vec{q}_{\|},\omega){\cal{Z}}^{x}_{nm}(z'')dz''
\right.}\nonumber\\ &\quad&\left.\!
-i{\cal{Q}}_{\,\,\,nm}^{xz}(\vec{q}_{\|},\omega)
\int G_{xz}(z,z'';\vec{q}_{\|},\omega){\cal{Z}}^{z}_{nm}(z'')dz''
\right\},
\label{eq:F-xx}\\
\lefteqn{
F_{nm}^{xz}(z;\vec{q}_{\|},\omega)=
{\rm{i}}\mu_0\omega
\left\{
i{\cal{Q}}_{\,\,\,nm}^{xz}(\vec{q}_{\|},\omega)
\int G_{xx}(z,z'';\vec{q}_{\|},\omega){\cal{Z}}^{x}_{nm}(z'')dz''
\right.}\nonumber\\ &&\left.\!
+{\cal{Q}}_{\,\,\,nm}^{zz}(\vec{q}_{\|},\omega)
\int G_{xz}(z,z'';\vec{q}_{\|},\omega){\cal{Z}}^{z}_{nm}(z'')dz''
\right\},
\label{eq:F-xz}\\
\lefteqn{
F_{nm}^{yy}(z;\vec{q}_{\|},\omega)=
-{\rm{i}}\mu_0\omega {\cal{Q}}_{\,\,\,nm}^{yy}(\vec{q}_{\|},\omega)
\int G_{yy}(z,z'';\vec{q}_{\|},\omega){\cal{Z}}^{x}_{nm}(z'')dz'',
}
\label{eq:F-yy}\\
\lefteqn{
F_{nm}^{zx}(z;\vec{q}_{\|},\omega)=
{q_{\|}\over{}q_{\perp}}F_{nm}^{xx}(z;\vec{q}_{\|},\omega),
}\label{eq:F-zx}\\
\lefteqn{
F_{nm}^{zz}(z;\vec{q}_{\|},\omega)=
{q_{\|}\over{}q_{\perp}}F_{nm}^{xz}(z;\vec{q}_{\|},\omega),
}\label{eq:F-zz}\end{aligned}$$ and the vector $$\vec{\Gamma}_{mn}(\vec{q}_{\|},\omega)=\left(
\begin{array}{c}
\displaystyle
\int{\cal{Z}}^{x}_{mn}(z')E_{{\rm{PC}},x}(z';\vec{q}_{\|},\omega)dz' \\[1mm]
\displaystyle
\int{\cal{Z}}^{x}_{mn}(z')E_{{\rm{PC}},y}(z';\vec{q}_{\|},\omega)dz' \\[1mm]
\displaystyle
\int{\cal{Z}}^{z}_{mn}(z')E_{{\rm{PC}},z}(z';\vec{q}_{\|},\omega)dz'
\end{array}\right)$$ can be determined from the following set of algebraic equations: $$\vec{\Gamma}_{mn}(\vec{q}_{\|},\omega)
-\sum_{vl}\tensor{K}_{mn}^{vl}(\vec{q}_{\|},\omega)
\cdot\vec{\Gamma}_{vl}(\vec{q}_{\|},\omega)
=\vec{\Omega}_{mn}(\vec{q}_{\|},\omega).
\label{eq:Gammafind}$$ Since we may now determine the different $\Gamma$ values independently of their dependence on the phase conjugated field, $\vec{E}_{\rm{PC}}(z;\vec{q}_{\|},\omega)$, we have by this operation kept the self-consistency in Eq. (\[eq:loop-FG\]), but the problem of solution has been reduced to a problem of solving a linear algebraic set of equations with just as many unknowns. This problem can be treated as a matrix problem and is thus in principle fairly easy to solve numerically. In Eq. (\[eq:Gammafind\]) above, the vectorial quantity $\vec{\Omega}_{mn}$ is given by $$\vec{\Omega}_{mn}(\vec{q}_{\|},\omega)=
\left(\begin{array}{c}
\displaystyle
\int{\cal{Z}}^{x}_{mn}(z)E_{{\rm{PC}},x}^{\rm{B}}(z;\vec{q}_{\|},\omega)dz \\[1mm]
\displaystyle
\int{\cal{Z}}^{x}_{mn}(z)E_{{\rm{PC}},y}^{\rm{B}}(z;\vec{q}_{\|},\omega)dz \\[1mm]
\displaystyle
\int{\cal{Z}}^{z}_{mn}(z)E_{{\rm{PC}},z}^{\rm{B}}(z;\vec{q}_{\|},\omega)dz
\end{array}\right)$$ and the $3\times3$ tensorial quantity $\tensor{K}_{mn}^{vl}(\vec{q}_{\|},\omega)$ has the five nonzero elements $$\begin{aligned}
K_{xx,mn}^{vl}(\vec{q}_{\|},\omega)&=&
\int{\cal{Z}}^{x}_{mn}(z)F_{lv}^{xx}(z;\vec{q}_{\|},\omega)dz,
\label{eq:Kxx}
\\
K_{xz,mn}^{vl}(\vec{q}_{\|},\omega)&=&
\int{\cal{Z}}^{x}_{mn}(z)F_{lv}^{xz}(z;\vec{q}_{\|},\omega)dz,
\\
K_{yy,mn}^{vl}(\vec{q}_{\|},\omega)&=&
\int{\cal{Z}}^{x}_{mn}(z)F_{lv}^{yy}(z;\vec{q}_{\|},\omega)dz,
\\
K_{zx,mn}^{vl}(\vec{q}_{\|},\omega)&=&{q_{\|}\over{}q_{\perp}}
\int{\cal{Z}}^{z}_{mn}(z)F_{lv}^{xx}(z;\vec{q}_{\|},\omega)dz,
\\
K_{zz,mn}^{vl}(\vec{q}_{\|},\omega)&=&{q_{\|}\over{}q_{\perp}}
\int{\cal{Z}}^{z}_{mn}(z)F_{lv}^{xz}(z;\vec{q}_{\|},\omega)dz.
\label{eq:Kzz}\end{aligned}$$ If we limit our treatment to polarized light perpendicular to the scattering plane ($s$) and in the scattering plane ($p$), we get for $s$-polarized light the set of equations $$\Gamma_{y,mn}-\sum_{vl}K_{yy,mn}^{vl}\Gamma_{y,vl}
=\Omega_{y,mn},
\label{eq:Gamma-y}$$ which is $m\times n$ equations with just as many unknowns, and for $p$-polarized light the set of equations $$\begin{aligned}
\Gamma_{x,mn}-\sum_{vl}\left(K_{xx,mn}^{vl}\Gamma_{x,vl}
+K_{xz,mn}^{vl}\Gamma_{z,vl}\right)&=&\Omega_{x,mn},
\label{eq:Gamma-x}
\\
\Gamma_{z,mn}-\sum_{vl}\left(K_{zx,mn}^{vl}\Gamma_{x,vl}
+K_{zz,mn}^{vl}\Gamma_{z,vl}\right)&=&\Omega_{z,mn},
\label{eq:Gamma-z}\end{aligned}$$ which is $2m\times n$ equations with just as many unknowns.
Infinite barrier quantum well
-----------------------------
Applying the infinite barrier quantum well to the above formalism, we are able to determine the integrals over the Cartesian coordinates in explicit form. The wave function constructs ${\cal{Z}}(z)$ becomes in the infinite barrier model $$\begin{aligned}
{\cal{Z}}^{x,\rm{IB}}_{nm}(z)&=&
{1\over{}d}\left[\cos\left({(n-m)\pi{}z\over{}d}\right)
-\cos\left({(n+m)\pi{}z\over{}d}\right)\right],
\label{eq:ZxIB}
\\
{\cal{Z}}^{z,\rm{IB}}_{nm}(z)&=&
{\pi\over{}d^2}\left[(n-m)\sin\left({(n+m)\pi{}z\over{}d}\right)
-(n+m)\sin\left({(n-m)\pi{}z\over{}d}\right)\right].
\label{eq:ZzIB}\end{aligned}$$ With this result the integrals over the source region apperaing in Eqs. (\[eq:F-xx\])– (\[eq:F-zz\]) and (\[eq:Kxx\])–(\[eq:Kzz\]) can be solved (see Appendix \[app:E\]), and the $K$ quantities thus become $$\begin{aligned}
\lefteqn{K_{xx,mn}^{vl}(\vec{q}_{\|},\omega)=
-{8\pi^4nmlv{}q_{\perp}d\left[1-e^{iq_{\perp}d}(-1)^{n+m}\right]
\over\epsilon_0\omega
[(iq_{\perp}d)^2+\pi^2(n-m)^2][(iq_{\perp}d)^2+\pi^2(n+m)^2]}
}\nonumber\\ &\quad&\times
{1+r^p-\left(e^{-iq_{\perp}d}+r^pe^{iq_{\perp}d}\right)(-1)^{l+v}
\over[(iq_{\perp}d)^2+\pi^2(l-v)^2][(iq_{\perp}d)^2+\pi^2(l+v)^2]}
\nonumber\\ &&\times
\left\{
{\cal{Q}}_{\,\,\,lv}^{xx}(\vec{q}_{\|},\omega)
q_{\perp}^2d
+{\cal{Q}}_{\,\,\,lv}^{xz}(\vec{q}_{\|},\omega)
{\pi^2(l^2-v^2)q_{\|}\over{}d}
\right\},
\\
\lefteqn{K_{xz,mn}^{vl}(\vec{q}_{\|},\omega)=
{8\pi^4inmlv{}q_{\perp}d\left[1-e^{iq_{\perp}d}(-1)^{n+m}\right]
\over\epsilon_0\omega
[(iq_{\perp}d)^2+\pi^2(n-m)^2][(iq_{\perp}d)^2+\pi^2(n+m)^2]}
}\nonumber\\ &&\times
{1+r^p-\left(e^{-iq_{\perp}d}+r^pe^{iq_{\perp}d}\right)(-1)^{l+v}
\over[(iq_{\perp}d)^2+\pi^2(l-v)^2][(iq_{\perp}d)^2+\pi^2(l+v)^2]}
\nonumber\\ &&\times
\left\{
{\cal{Q}}_{\,\,\,lv}^{xz}(\vec{q}_{\|},\omega)
q_{\perp}^2d
+{\cal{Q}}_{\,\,\,lv}^{zz}(\vec{q}_{\|},\omega)
{\pi^2(l^2-v^2)q_{\|}\over{}d}
\right\},
\\
\lefteqn{K_{yy,mn}^{vl}(\vec{q}_{\|},\omega)=
{\cal{Q}}_{\,\,\,lv}^{yy}(\vec{q}_{\|},\omega)
{8\pi^4\mu_0nmlv\omega{}q_{\perp}d^2[e^{iq_{\perp}d}(-1)^{n+m}-1]
\over[(iq_{\perp}d)^2+\pi^2(n-m)^2][(iq_{\perp}d)^2+\pi^2(n+m)^2]}
}\nonumber\\ &&\times
{1-r^s-\left(e^{-iq_{\perp}d}-r^se^{iq_{\perp}d}\right)(-1)^{l+v}
\over[(iq_{\perp}d)^2+\pi^2(l-v)^2][(iq_{\perp}d)^2+\pi^2(l+v)^2]}
\\
\lefteqn{K_{zx,mn}^{vl}(\vec{q}_{\|},\omega)=
-{2\pi^2ilv\over\epsilon_0\omega}
{4\pi^4nm(n^2-m^2)[e^{iq_{\perp}d}(-1)^{n+m}-1]
\over{}d[(iq_{\perp}d)^2+\pi^2(n-m)^2][(iq_{\perp}d)^2+\pi^2(n+m)^2]}
}\nonumber\\ &\quad&\times
{1+r^p-\left(e^{-iq_{\perp}d}+r^pe^{iq_{\perp}d}\right)(-1)^{l+v}
\over[(iq_{\perp}d)^2+\pi^2(l-v)^2][(iq_{\perp}d)^2+\pi^2(l+v)^2]}
\nonumber\\ &&\times
\left\{
{\cal{Q}}_{\,\,\,lv}^{xx}(\vec{q}_{\|},\omega)
{q_{\perp}^2d}
+{\cal{Q}}_{\,\,\,lv}^{xz}(\vec{q}_{\|},\omega)
{\pi^2(l^2-v^2)q_{\|} \over{}d}
\right\}
\\
\lefteqn{K_{zz,mn}^{vl}(\vec{q}_{\|},\omega)=
{2\pi^2ilv\over\epsilon_0\omega}
{4\pi^4nm(n^2-m^2)[e^{iq_{\perp}d}(-1)^{n+m}-1]
\over{}d[(iq_{\perp}d)^2+\pi^2(n-m)^2][(iq_{\perp}d)^2+\pi^2(n+m)^2]}
}\nonumber\\ &&\times
{1+r^p-\left(e^{-iq_{\perp}d}+r^pe^{iq_{\perp}d}\right)(-1)^{l+v}
\over[(iq_{\perp}d)^2+\pi^2(l-v)^2][(iq_{\perp}d)^2+\pi^2(l+v)^2]}
\nonumber\\ &&\times
\left\{
i{\cal{Q}}_{\,\,\,lv}^{xz}(\vec{q}_{\|},\omega)
q_{\perp}^2d
-{\cal{Q}}_{\,\,\,lv}^{zz}(\vec{q}_{\|},\omega)
{\pi^2(l^2-v^2)q_{\|}\over{}id}
\right\}.\end{aligned}$$ To find $\vec{\Omega}_{mn}(\vec{q}_{\|},\omega)$ is in general a much more difficult task, but insertion of the expression for $\vec{E}_{\rm{PC}}^{\rm{B}}(z;\vec{q}_{\|},\omega)$ gives $$\begin{aligned}
\lefteqn{
{\Omega}_{x,mn}(\vec{q}_{\|},\omega)=-{\rm{i}}\mu_0\omega
\int_{-d}^{0}{\cal{Z}}^{x}_{mn}(z)\int_{-d}^{0}
\left[G_{xx}(z,z';\vec{q}_{\|})J_{-\omega,x}^{(3)}(z';\vec{q}_{\|})
\right.}\nonumber\\ &\quad&\left.\!
+G_{xz}(z,z';\vec{q}_{\|})J_{-\omega,z}^{(3)}(z';\vec{q}_{\|})\right]dz'dz,
\\
\lefteqn{
{\Omega}_{y,mn}(\vec{q}_{\|},\omega)=-{\rm{i}}\mu_0\omega
\int_{-d}^{0}{\cal{Z}}^{x}_{mn}(z)\int_{-d}^{0}
G_{yy}(z,z';\vec{q}_{\|})J_{-\omega,y}^{(3)}(z';\vec{q}_{\|})dz'dz,
}
\\
\lefteqn{
{\Omega}_{z,mn}(\vec{q}_{\|},\omega)=-{\rm{i}}\mu_0\omega
\int_{-d}^{0}{\cal{Z}}^{z}_{mn}(z)\int_{-d}^{0}
{q_{\|}\over{}q_{\perp}}
\left[G_{xx}(z,z';\vec{q}_{\|})J_{-\omega,x}^{(3)}(z';\vec{q}_{\|})
\right.}\nonumber\\ &\quad&\left.\!
+G_{xz}(z,z';\vec{q}_{\|})J_{-\omega,z}^{(3)}(z';\vec{q}_{\|})\right]dz'dz.\end{aligned}$$ These integrals can by insertion of the propagators and the wave functions be solved for the integral over $z$, and thus we find $$\begin{aligned}
\lefteqn{ {\Omega}_{x,mn}(\vec{q}_{\|},\omega)=
-{2\pi^2inmq_{\perp}d\left[e^{iq_{\perp}d}(-1)^{n+m}-1\right]
\over\epsilon_0\omega
[(iq_{\perp}d)^2+\pi^2(n+m)^2][(iq_{\perp}d)^2+\pi^2(n-m)^2]}
}\nonumber\\ &\quad&\times
\!\int_{-d}^{0}\left[q_{\perp}
\left(e^{iq_{\perp}z}-r^pe^{-iq_{\perp}z}\right)
J_{-\omega,x}^{(3)}(z;\vec{q}_{\|})
+q_{\|}
\left(e^{iq_{\perp}z}+r^pe^{-iq_{\perp}z}\right)
J_{-\omega,z}^{(3)}(z;\vec{q}_{\|})\right]dz,
\nonumber\\
\label{eq:14.44}
\\
\lefteqn{ {\Omega}_{y,mn}(\vec{q}_{\|},\omega)=
-{2\pi^2{\rm{i}}\mu_0\omega{}nmd\left[e^{iq_{\perp}d}(-1)^{n+m}-1\right]
\over[(iq_{\perp}d)^2+\pi^2(n+m)^2][(iq_{\perp}d)^2+\pi^2(n-m)^2]}
}\nonumber\\ &&\times
\int_{-d}^{0}
\left(e^{iq_{\perp}z}+r^se^{-iq_{\perp}z}\right)
J_{-\omega,y}^{(3)}(z;\vec{q}_{\|})dz,
\\
\lefteqn{ {\Omega}_{z,mn}(\vec{q}_{\|},\omega)=
-{2\pi^4q_{\|}nm(n^2-m^2)\left[e^{iq_{\perp}d}(-1)^{m+n}-1\right]
\over\epsilon_0\omega{}d
[(iq_{\perp}d)^2+\pi^2(n+m)^2][(iq_{\perp}d)^2+\pi^2(n-m)^2]}
}\nonumber\\ &&\times
\int_{-d}^{0}\left[
\left(e^{iq_{\perp}z}-r^pe^{-iq_{\perp}z}\right)
J_{-\omega,x}^{(3)}(z;\vec{q}_{\|})
+{q_{\|}\over{}q_{\perp}}
\left(e^{iq_{\perp}z}+r^pe^{-iq_{\perp}z}\right)
J_{-\omega,z}^{(3)}(z;\vec{q}_{\|})\right]dz,
\nonumber\\
\label{eq:14.46}\end{aligned}$$ where we have dropped the now superfluous marking $z'$ in favor of a new $z$. Since the $z$-dependence of $\vec{J}_{-\omega}^{\,(3)}(z;\vec{q}_{\|})$ is expressed via the interacting fields and the wave functions, and we are limiting ourselves to studies where (i) the pump fields are parallel to either the $x$-axis or the $z$-axis and with uniform amplitude profile along that axis, and (ii) the probe field has only one plane-wave component on the form $\vec{E}(z;\vec{q}_{\|})=\vec{E}e^{iq_{\perp}z}$, the last integral above can be solved. This solution is discussed in Appendix \[app:C\], sections \[sec:Z\] and \[sec:Ps\]. Thus, in Eq. (\[eq:Gammafind\]), all $K$’s and $\Omega$’s are numbers with no inline integrals to solve numerically.
Numerical results for a two-level quantum well {#ch:15}
==============================================
Besides calculation of the nonlinear current densities, the main numerical work consists of finding the solution to the appropriate sets of equations, given by Eq. (\[eq:Gamma-y\]) for processes with $s$-polarized response, and Eqs. (\[eq:Gamma-x\]) and (\[eq:Gamma-z\]) for processes with $p$-polarized response. Computational procedures to solve this kind of problems are well known (see, e.g., , for description and Fortran routines).
Phase conjugation reflection coefficient {#phase-conjugation-reflection-coefficient}
----------------------------------------
To estimate the amount of light we get back through the phase conjugated channel, we use the phase conjugation reflection coefficient $R_{\rm{PC}}(z;\vec{q}_{\|})$ defined in Eq. (\[eq:RPC\]) together with the expression for the intensities given by Eq. (\[eq:RPC-I\]). As before, the reflection coefficient at the surface of the quantum well is thus $R_{\rm{PC}}(-d;\vec{q}_{\|})$.
In order to give an impression of the difference between the calculation where only intraband contributions were taken into account (chapter \[Ch:11\]) the present calculation is also based on the data for a two-monolayer thick copper quantum well \[$N_+=8.47\times10^{28}$m$^{-3}$, $Z=1$, and $d=3.8$[Å]{} [@Ashcroft:76:1]\]. As was the case for the single-level Cu quantum well, the two-level Cu quantum well can adequately be deposited on a glass substrate for which we use a refractive index $n$ of 1.51. With this substrate, a reasonable description of the linear vaccum/substrate amplitude reflection coefficients ($r^p$ for the $p$-polarized light and $r^s$ for the $s$-polarized light) can be obtained by use of the classical Fresnel formulae, given by Eq. (\[eq:rp\]) and $$\begin{aligned}
r^s&=&{q_{\perp}-({n^2q^2-q_{\|}^2})^{1\over2}\over
q_{\perp}+({n^2q^2-q_{\|}^2})^{1\over2}}.
\label{eq:rs}\end{aligned}$$ Keeping the pump fields parallel to the $x$-axis, we get a pump wavenumber $k_{\|}=1.51q$.
(0,0)(127,96) (-8,2) (0,53)[$R_{\rm{PC}}(-d;\vec{q}_{\|})$ \[m$^4$/W$^2$\]]{} (71.5,0)[(0,0)\[b\][$q_{\|}/q$]{}]{} (43.75,84)(43.75,80) (43.75,41)(43.75,45) (43.75,85)[(0,0)\[b\][$n$]{}]{} (113,4)(113,96) (113,0)[(0,0)\[b\][$k_F/q$]{}]{}
(0,6.55)(127,187.9) (0,94)[ (-8,2) (0,53)[$R_{\rm{PC}}(-d;\vec{q}_{\|})$ \[m$^4$/W$^2$\]]{} (71.5,2)[(0,0)\[b\][$q_{\|}/q$]{}]{} (43.75,70)(43.75,66) (43.75,38)(43.75,42) (43.75,71)[(0,0)\[b\][$n$]{}]{} (113,6)(113,95) (113,2)[(0,0)\[b\][$k_F/q$]{}]{} ]{} (0,0)[ (-8,2) (0,53)[$R_{\rm{PC}}(-d;\vec{q}_{\|})$ \[m$^4$/W$^2$\]]{} (71.5,2)[(0,0)\[b\][$q_{\|}/q$]{}]{} (43.75,66)(43.75,62) (43.75,30)(43.75,34) (43.75,67)[(0,0)\[b\][$n$]{}]{} (113,6)(113,95) (113,2)[(0,0)\[b\][$k_F/q$]{}]{} ]{}
In Figs. \[fig:15.1\] and \[fig:15.2\] we have plotted the phase conjugation reflection coefficient at the interface between the vacuum and the quantum well as a function of the parallel component of the wavevector normalized to the vacuum wavenumber, $q_{\|}/q$. The wavelength has in these plots been fixed to $\lambda=1061$nm (the same as in the single-level case). The plots have been divided into two sets, together covering all eight different combinations of polarization of the interacting fields. In Fig. \[fig:15.1\] is plotted the four combinations leading to a response with the same state of polarization as the probe, i.e., (i) the purely $p$-polarized case where all interacting fields are polarized in the scattering plane (denoted $ppp$), (ii) the purely $s$-polarized ($sss$) case where all three interacting fields are polarized perpendicular to the scattering plane, (iii) the case where the two pump fields both are $p$-polarized and the probe field is $s$-polarized ($pps$), and (iv) the case where the probe field is $p$-polarized and the two pump fields are $s$-polarized ($ssp$). The results for the other four combinations of polarization has been plotted in pairs in Fig. \[fig:15.2\]. The upper figure in Fig. \[fig:15.2\] shows the two cases where the pump fields are differently polarized and the probe field is $s$-polarized, while in the lower figure, the probe field is $p$-polarized, still with differently polarized pump fields. The vertical line inserted into Figs. \[fig:15.1\] and \[fig:15.2\] indicates the normalized Fermi wavenumber, which for the two-monolayer Cu quantum well is $2.78\times10^{3}$. The discussion of this quantity has been given in chapter \[Ch:11\] (in the paragraph starting at the end of page ).
(0,3)(127,96) (-8,2) (0,53)[$R_{\rm{PC}}(-d;\vec{q}_{\|})$ \[m$^4$/W$^2$\]]{} (98,3)[(0,0)\[b\][$\omega/\omega_{12}$]{}]{} (27,6)(27,96) (23,3)[(0,0)\[bl\][$\lambda=1061$nm]{}]{}
In addition to the plots in Figs. \[fig:15.1\] and \[fig:15.2\], where the phase conjugation reflection coefficient was plotted as a function of $q_{\|}/q$, we have in Figs. \[fig:15.3\] and \[fig:15.4\] plotted the phase conjugation reflection coefficient as a function of the optical frequency normalized to the interband transition frequency, $\omega/\omega_{12}$. The parallel component of the wavevector has in this case been fixed at $q_{\|}=0.8q$ (in the propagating regime). Again, the four cases of pump fields having the same polarization are plotted in the first of the two figures (Fig. \[fig:15.3\]), and the remaining four in the other figure (Fig. \[fig:15.4\]).
(0,6.55)(127,187.9) (0,94)[ (-8,2) (0,53)[$R_{\rm{PC}}(-d;\vec{q}_{\|})$ \[m$^4$/W$^2$\]]{} (98,3)[(0,0)\[b\][$\omega/\omega_{12}$]{}]{} (27,6)(27,95) (23,3)[(0,0)\[bl\][$\lambda=1061$nm]{}]{} ]{} (0,0)[ (-8,2) (0,53)[$R_{\rm{PC}}(-d;\vec{q}_{\|})$ \[m$^4$/W$^2$\]]{} (98,3)[(0,0)\[b\][$\omega/\omega_{12}$]{}]{} (27,6)(27,95) (23,3)[(0,0)\[bl\][$\lambda=1061$nm]{}]{} ]{}
All plots in Figs. \[fig:15.1\]–\[fig:15.4\] have been plotted using a relaxation time of $200$fs in the interband transition from the occupied state to the unoccupied state and a relaxation time of $3$fs within the intraband transitions of the occupied state. Unlike in the case of pure intraband response, the choice of adequate relaxation times seem less important in the two-level quantum-well case. Changing either of the relaxation times (or both) an order of magnitude up or down doesn’t change the results shown in Figs. \[fig:15.1\]–\[fig:15.4\] so much that the two curves for the respective choices of relaxation times would differ from each other, as it was the case in the single-level quantum well (see Fig. \[fig:11.4\]).
Returning our attention to Fig. \[fig:15.1\], we observe that the purely $p$-polarized combination of polarization gives the strongest phase conjugated response while the purely $s$-polarized combination gives the weakest response of the four. Looking at the $ppp$ curve, we see that the maximum value is reached in the propagating regime of the $q_{\|}/q$-spectrum, where two peaks occur approximately at the values of $q_{\|}=0.18q$ and $q_{\|}=0.22q$. These peaks must be due to the pump waves being $p$-polarized, since they also occur in the $pps$ configuration, but in neither of the $sss$ and $ssp$ configurations. In the evanescent regime the $ppp$ curve is increasing from the point $q_{\|}=nq$ until it reaches its maximal value at around $q_{\|}=10^3q$. Above $q_{\|}=10^3q$, the $ppp$ response starts decaying again. The $pps$ curve in Fig. \[fig:15.1\] has, apart from the two peaks discussed above, two additional peaks occuring symmetrically around the point $nq$ in the $q_{\|}$-spectrum, at approximately $q_{\|}=1.1q$ and $q_{\|}=1.9q$, respectively. After the second of these peaks, the amplitude of the response fades away with growing values of $q_{\|}$. The $ssp$ curve has a maximum when the probe field is perpendicularly incident on the phase conjugator ($q_{\|}/q=0$), and another one where the probe field becomes evanescent in the substrate, i.e., at $q_{\|}=nq$. In the evanescent regime of the $q_{\|}/q$-spectrum the response is increasing, with two small narrow peaks occuring at $q_{\|}\approx10q$ and $q_{\|}\approx13q$, and it reaches a maximum at $q_{\|}\approx2\times10^3$, and after going down to a minimum right after $k_F/q$ it increases again. This indicates that if we are able to produce probe fields with a significant amount of evanescent modes above $k_F/q$, the present model is probably not sufficient to describe the $ssp$ response (and maybe not sufficient to describe the $ppp$ response either). The last of the curves in Fig. \[fig:15.1\] represents the purely $s$-polarized case ($sss$). It has maxima at $q_{\|}/q=0$, and again at $q_{\|}/q=n$. Above $q_{\|}=nq$ it falls off rapidly.
Looking at the $q_{\|}/q$-spectrum of the phase conjugated response in the other four combinations of polarization, depicted in Fig. \[fig:15.2\], we see that both pairs have peaks in the propagating regime of the $q_{\|}/q$-spectrum at the same places as the $ppp$ and $pps$ curves of Fig. \[fig:15.1\] had. Another peak appears when $q_{\|}=nq$, and in the purely evanescent part of the $q_{\|}/q$-spectrum peaks appear at $q_{\|}\approx10q$ and $q_{\|}\approx13q$, the same places as in the $ssp$ response shown in Fig. \[fig:15.1\]. After these two peaks the responses of the $pss/sps$-pair increases until they reach their maximum at around $q_{\|}=5\times10^{3}$, after which they decrease again. This maximum is comparable in magnitude to the peaks in the propagating end of the $q_{\|}/q$-spectrum. We observe that the two curves in each of the pairs shown in Fig. \[fig:15.2\] becomes identical for high values of $q_{\|}$, as they should from the previous analysis. The $spp/psp$-pair of curves also starts increasing in magnitude in the high end of the $q_{\|}/q$-spectrum showed. They reach their maximum at around $q_{\|}=2.5\times10^3q$, after which the magnitude decreases again. In this case, however, the magnitude of this maximum is some fifteen orders of magnitude less than the magnitude of the peaks in the propagating regime. The problem in the $sps/pss$-pair of curves is that the maximal value is reached after the point $q_{\|}=k_F/q$, and the conclusion must therefore be the same as in the $ssp$ case, namely that if the probe has components of significance in the high end of the $q_{\|}/q$-spectrum, then the model should probably be extended in one way or another.
Continuing to the frequency plots, we observe from Fig. \[fig:15.3\] that the two cases where both pump fields are $s$-polarized has no resonances at all. Their decrease in magnitude as the frequency increases is mainly due to the factor of $\omega^{-3}$ occuring in the nonlinear conductivity tensor. The $ppp$ curve has a peak of high magnitude at $\omega\approx\omega_{12}/3$, a small one at $\omega=\omega_{12}$, and a large one again at $\omega\approx1.1\omega_{12}$. The $pps$ curve has a peak of small magnitude at $\omega=0.3\omega_{12}$, two large ones at $\omega\approx\omega_{12}/3$ and at $\omega\approx1.1\omega_{12}$, and finally a small one at $\omega\approx3.2\omega_{12}$. The peaks around $\omega\approx\omega_{12}$ probably arise from the combination of the denominators in the nonlinear conductivity tensors, but none of the peaks have been clearly identified from the formulas yet. Going to any of the two sides in the frequency spectrum away from this group of resonances, the curves behave like the $sss$ and $ssp$ curves, with the magnitude proportional to $\omega^{-3}$. The frequency plots for the two pairs of polarization combinations where the pump fields are differently polarized (Fig. \[fig:15.4\]) have resonances with the approximate values of $\omega/\omega_{12}$ of $1/3$, $1/2$, $2/3$, $1$, $1.1$, and $2$. Like in the previous case, these peaks have not been clearly identified from the formulas yet. Again one might assume that the peaks at $1$ and $1.1$ arise directly from (some of) the denominators in the nonlinear conductivity tensors. As before, outside of the shown frequency range, the behaviour of the response is proportional to $\omega^{-3}$. The results presented in this chapter will be treated more thoroughly in a forthcoming paper.
Discussion {#discussion-1}
==========
In the past two chapter we have briefly shown how to calculate the phase conjugated response from a multilevel quantum well and given numerical results for a two-level quantum well. As we concluded in the single-level quantum-well case, the assumption of an ideal phase conjugator also does not hold for a two-level quantum-well phase conjugator. In this case, however, it is not so much because of the efficiency in the high end of the $q_{\|}$-spectrum, since in most of the cases shown in chapter \[ch:15\] the efficiency in the evanescent regime is not so much larger than in the propagating regime. It is more because the two-level phase conjugator is much more efficient for certain values of $q_{\|}$ than for the rest of the spectrum.
However, before we can give a full description of the phase conjugated response from multi-level quantum wells, some aspects has to be addressed. Among the important ones are the fact that we need to identify (i) which terms of the nonlinear conductivity tensor that are dominating the phase conjugation response, and if it is possible by a careful choice of the system to make different terms dominate. Furthermore (ii) it is desirable to find out more precisely which individual terms in the nonlinear conductivity tensor gives rise to each of the peaks occuring in the curves in Figs. \[fig:15.1\]–\[fig:15.4\]. Many other things has to be investigated, for example (i) the behaviour of the response close to the peaks in Figs. \[fig:15.1\]–\[fig:15.4\], both in the frequency spectrum and in the $q_{\|}$-spectrum, (ii) the frequency dependence in general, and (iii) the response to sources with a large number of Fourier components in the $q_{\|}$-spectrum, such as the quantum wire discussed in Chapter \[ch:12\].
Summary of conclusions {#summary-of-conclusions .unnumbered}
----------------------
We have developed a spatially nonlocal theoretical model of degenerate four-wave mixing of electromagnetic fields on the mesoscopic length scale. We have analyzed the physical processes involved in creating the DFWM response and identified the independent nonzero elements of the related conductivity tensor for each type of process. Following the more general treatment in real space we have specialized the treatment to take into account only cases where translational invariance against displacements in two of the three spatial dimensions occur, thereby favouring a description in which the optical processes occur in surfaces and thin films of condensed matter.
As a consequence of this choice we have transformed the response function into Fourier space in two spatial coordinates, keeping the real-space coordinate in the third dimension. From there, the emphasis has been laid on phase conjugation, although the more general DFWM response tensor has been carried out in this mixed Fourier space as well. The emphasis on phase conjugation was realized by the choice of letting two of the interacting fields be spatially counterpropagating. Letting the two counterpropagating fields act as pump fields in the phase conjugation process, the third of the interacting fields became what we have referred to as the probe field.
The choice of a scattering geometry in which the pump fields were taken to be undamped plane waves traveling parallel to the translationally invariant plane resulted in a description where the main effort could be concentrated on studying the response due to the probe field, thus letting the pump fields effectively being a part of the phase conjugator. We concluded that using different combinations of light polarized in the scattering plane or perpendicular to this plane lead to different properties of the phase conjugated field compared to the incoming probe field, including changes in polarization in some cases.
Using the developed model on a single-level metallic quantum-well phase conjugator we have shown that the phase-conjugation reflection coefficient behaves quite differently from the uniform reflection coefficient that has often been assumed in previous studies where evanescent components have been included [@Agarwal:95:1; @Keller:92:1]. The response in the high end of the $q_{\|}$-spectrum turned out to be as much as ten orders of magnitude larger than in the propagating regime. Subsequently, it was shown that by use of the single-level phase conjugator it was possible to phase conjugate light emitted from a subwavelength source in the vicinity of the phase conjugator, and that the phase conjugated light at the plane parallel to the phase conjugator, where the source has been placed, has a subwavelength distance between the minima in the intensity. Consistent with another recent prediction [@Bozhevolnyi:97:1; @Bozhevolnyi:97:2] we have observed that the smallest distance between the two minima surrounding the main lobe in the phase conjugated field occur at the surface of the phase conjugator.
The theoretical model was concluded with a description of a quantum well with an arbitrary number of bound states, followed by a numerical calculation of the response from a two-level quantum well. We have shown that also the two-level quantum well does not come close to an ideal phase conjugator with a uniform reflection coefficient in the $q_{\|}$-spectrum. Furthermore, it does not behave the same way as the single-level quantum well, even though the combinations of polarization for the interacting fields that gives a $p$-polarized response lead to similar results in the high end of the $q_{\|}$-spectrum. In the low end of the $q_{\|}$-spectrum the phase conjugated response from a two-level quantum well is several orders of magnitude stronger in a small number of very narrow ranges in $q_{\|}$ than in the rest.
Finally, we concluded that if one is able to excite the two-level quantum well in the $q_{\|}$-range around the point of the Fermi wave number the present model could prove insufficient, because the maximum value of the phase conjugation reflection coefficient in the high end of the $q_{\|}$-spectrum in several cases is above the Fermi wave number.
Discussion and outlook {#discussion-and-outlook .unnumbered}
----------------------
With respect to the single-level quantum well several properties would be interesting to examine from a fundamental point of view, including (i) the response in the far-infrared and ultraviolet parts of the frequency spectrum, (ii) how the response can be divided into an electrostatic and an electrodynamic part, (iii) how the width of the phase conjugated focus from a quantum wire scales with the wavelength of the electromagnetic field used in the interaction, and (iv) the problem of a three-dimensional source.
For the two-level quantum well plenty of work remains to be done before it would be wise to take up some of the above-mentioned properties. First of all, we have to determine how much each of the terms in the nonlinear conductivity tensor contributes to the phase conjugated response. Also, the problem of phase conjugating a broad angular band should be addressed in order to study, for instance, focusing of light in front of a two-level phase conjugator.
The problem of focusing has to be addressed more carefully, since the present study has revealed only that when the mesoscopic source is moved closer to the phase conjugator the focus is narrowed. I imagine that this problem could be addressed properly using a pure engineering approach to make an adjustment of the present model by (i) abandoning the infinite barrier model by insertion of a more sophisticated potential across the barrier, and (ii) abandoning the point-source description of the probe field. Thereby one would also be able to discuss the problem of resolution in a near-field optical microscope.
Furthermore, it could be interesting to establish a model which provides a temporal resolution. It could be used to study, for example, the time delay and distortion of an electromagnetic pulse (wave packet) being phase conjugated, initially using, for example, plane waves as pumps, and ultimately to give an understanding of four-wave mixing using pulsed interacting fields. Such a model would provide a framework for a description of time-resolved optical phase conjugation in, for example, communications systems.
On the more sophisticated fundamental side it could be interesting to investigate the phase conjugated response when the phase conjugator is, for example, a mesoscopic ring, cylinder, sphere, or a quantum wire. Another possibility is to take into account spin effects in order to treat the phase conjugation response from magnetic materials. We also believe that there is a connection between the model for electromagnetic phase conjugation presented in this work and phase conjugation of electrons and atoms \[for an introduction to atomic phase conjugation and nonlinear atom optics, see [-@Lenz:93:1; -@Lenz:94:1] and \].
A problem that has to be taken into account when using the present formulation to describe nonlinear optical processes is the apparantly divergent behaviour in the long-wavelength (low-frequency) limit stemming (in third-order problems) from the $\omega^{-3}$-term in the beginning of the nonlinear conductivity tensor. This problem is a general one in the theoretical model, and although the linear problem has been solved [@Keller:96:1], the problem has still not been solved for any nonlinear case, including DFWM.
If time permits, I shall substantiate on some of these points in future work. Otherwise, it will be left for others to do.
to to
Calculation of linear and nonlinear conductivity tensors in two-dimensionally translational invariant systems \[$(z;\vec{q}_{\|})$-space\] {#ch:ConCalc}
==========================================================================================================================================
In this appendix I present a calculation of linear and nonlinear conductivity tensors suitable for calculation of the linear and nonlinear current densities in a physical system with translational invariance only in the $x$ and $y$ directions of the Cartesian $x$-$y$-$z$-coordinate system. The basic ingredients in this calculation consists of (i) the Fourier integral representation of the vector potential in the $x$- and $y$-coordinates given by Eq. (\[eq:Azr\]), (ii) the inverse relation for the current densities linear in the cyclic frequency $\omega$ (appeared as Eq. (\[eq:Jzq\])), (iii) the basis set of the wave functions taken on the form of Eq. (\[eq:eigenstate\]), and (iv) the corresponding transition current density in Eq. (\[eq:Jm->n||\]). Using these ingredients we start from the three-dimensional expressions in real space, which in the linear case are given by Eq. (\[eq:Jr\]) with insertion of Eqs. (\[eq:SigmaAArz\])–(\[eq:SigmaABrz\]) and in the nonlinear case are given by Eq. (\[eq:J3r\]) with insertion of Eqs. (\[eq:XiAArz\])–(\[eq:XiAGrz\]). The results of these calculations are presented as expressions for the individual matrix elements according to the definitions given by Eq. (\[eq:J1zq\]) for the linear conductivity tensor and Eq. (\[eq:J3-wzq\]) for the nonlinear conductivity tensor, the cyclic transition frequencies being expressed in the form of Eq. (\[eq:transitionw\]).
For convenience, we in the following treatment divide the linear current density into two parts following the two processes shown in Fig. \[fig:2Tr\]. If we define the linear current density as $J_{i}(z;\vec{q}_{\|})=\sum_jJ_{ij}(z;\vec{q}_{\|})$, the result of calculating the linear conductivity tensor is presented as the individual nonzero matrix elements corresponding to the symmetry analysis presented in Chapter \[ch:6\]. Like in the linear case, it is convenient in the nonlinear case to define the nonlinear current density as $J_{i}(\vec{r}\,)\equiv\sum_{jkh}J_{ijkh}(\vec{r}\,)$, with $J_{ijkh}(\vec{r}\,)=\Xi_{ijkh}(\vec{r},\vec{r}\,',\vec{r}\,'',\vec{r}\,''')
A_h(\vec{r}\,''')A_k(\vec{r}\,'')A_j(\vec{r}\,')$, and then split the treatment of the nonlinear current density in such a way that each of the processes mentioned in Fig. \[fig:3Tr\] is treated separately.
Linear process A
----------------
From Eq. (\[eq:SigmaAArz\]) we have the $xx$ element of part A of the linear current, in which we insert the expressions for the wave function and the vector potential in the two-dimensional Fourier representation, giving the result $$\begin{aligned}
\lefteqn{
J_{xx}^{\rm{A}}(z;\vec{q}_{\|})=
-{e^2\over{}m_{e}}
{2\over(2\pi)^4}
\iiint
\sum_{n}f_{n}|\psi_{n}(z)|^2\delta(\vec{r}-\vec{r}\,')
A_{x}(z';\vec{q}_{\|}^{\,\prime})e^{{\rm{i}}\vec{q}_{\|}^{\,\prime}
\cdot\vec{r}_{\|}^{\,\prime}}d^2q_{\|}'d^3r'
}\nonumber\\ &\quad&\times
e^{-{\rm{i}}\vec{q}_{\|}\cdot\vec{r}_{\|}}d^2r_{\|}.\end{aligned}$$ Solving the integrals over the real-space coordinates in the $x$-$y$-directions, $\int{}d^2r_{\|}'$ and $\int{}d^2r_{\|}$, we obtain $$\begin{aligned}
\lefteqn{ J_{xx}^{\rm{A}}(z;\vec{q}_{\|})=
-{e^2\over{}m_{e}}{2\over(2\pi)^2}
\iiint
\sum_{n}f_{n}(\vec{\kappa}_{\|,\bar{n}})
|\psi_{n}(z)|^2\delta(z-z')
\delta(\vec{q}_{\|}^{\,\prime}-\vec{q}_{\|})
}\nonumber\\ &\quad&\times
A_{x}(z';\vec{q}_{\|}^{\,\prime})
d^2\kappa_{\|,\bar{n}}
d^2q_{\|}'dz',
\label{eq:J1xx-A}\end{aligned}$$ where the (infinite) sum over the $\kappa_{\|}$ coordinates has been replaced by an integral. From this expression we extract part A of the linear conductivity tensor as $${\sigma}_{xx}^{\,\rm{A}}(z,z';\vec{q}_{\|},\vec{q}_{\|}^{\,\prime})=
-{2\over{}{\rm{i}}\omega}{e^2\over{}m_{e}}{1\over(2\pi)^2}
\sum_{n}\int{}f_{n}(\vec{\kappa}_{\|})
d^2\kappa_{\|}
|\psi_{n}(z)|^2\delta(z-z')
\delta(\vec{q}_{\|}^{\,\prime}-\vec{q}_{\|}),$$ where we have omitted the now superfluous reference to $n$ from $\vec{\kappa}_{\|}$. Taking into account the conservation of momentum given by the Dirac delta function $\delta(\vec{q}_{\|}^{\,\prime}-\vec{q}_{\|})$ we may integrate over $q_{\|}'$ in Eq. (\[eq:J1xx-A\]), and thereafter extract part A of the linear conductivity tensor as presented in Eq. (\[eq:SigmaAA\]).
Linear process B
----------------
Taking from Eq. (\[eq:SigmaABrz\]) element $ij$ of part B of the linear current density and inserting the expressions for the wave function \[Eq. (\[eq:eigenstate\])\] and the transition current density \[Eq. (\[eq:j\_hnm\])\] in the two-dimensional Fourier representation, we get $$\begin{aligned}
\lefteqn{
J_{ij}^{\rm{B}}(z;\vec{q}_{\|})=
-{1\over\hbar}{2\over(2\pi)^6}\iiint
\sum_{nm}
{f_{n}-f_{m}\over\tilde{\omega}_{nm}-\omega}
j_{j,mn}(z';\vec{\kappa}_{\|,\bar{m}}+\vec{\kappa}_{\|,\bar{n}})
j_{i,nm}(z;\vec{\kappa}_{\|,\bar{n}}+\vec{\kappa}_{\|,\bar{m}})
}\nonumber\\ &\quad&\times
e^{{\rm{i}}(\vec{\kappa}_{\|,\bar{m}}-\vec{\kappa}_{\|,\bar{n}})
\cdot\vec{r}_{\|}^{\,\prime}}
e^{{\rm{i}}(\vec{\kappa}_{\|,\bar{n}}-\vec{\kappa}_{\|,\bar{m}})
\cdot\vec{r}_{\|}}
A_{j}(z';\vec{q}_{\|}^{\,\prime})
e^{{\rm{i}}\vec{q}_{\|}^{\,\prime}\cdot\vec{r}_{\|}^{\,\prime}}
d^2q_{\|}'d^3r'e^{-{\rm{i}}\vec{q}_{\|}\cdot\vec{r}_{\|}}d^2r_{\|}.\end{aligned}$$ Solving the Cartesian integrals $\int{}d^2r_{\|}'$ and $\int{}d^2r_{\|}$ along the surface and replacing the infinite sum over $\vec{\kappa}_{\|}$ with an integral, this is $$\begin{aligned}
\lefteqn{
J_{ij}^{\rm{B}}(z;\vec{q}_{\|})=
-{1\over\hbar}
{2\over(2\pi)^2}
\idotsint
\sum_{nm}
{f_{n}(\vec{\kappa}_{\|,\bar{n}})-f_{m}(\vec{\kappa}_{\|,\bar{m}})
\over\tilde{\omega}_{nm}(\vec{\kappa}_{\|,\bar{n}},\vec{\kappa}_{\|,\bar{m}})-\omega}
j_{j,mn}(z';\vec{\kappa}_{\|,\bar{m}}+\vec{\kappa}_{\|,\bar{n}})
}\nonumber\\ &\quad&\times
j_{i,nm}(z;\vec{\kappa}_{\|,\bar{n}}+\vec{\kappa}_{\|,\bar{m}})
\delta(\vec{\kappa}_{\|,\bar{m}}-\vec{\kappa}_{\|,\bar{n}}+\vec{q}_{\|}^{\,\prime})
\delta(\vec{\kappa}_{\|,\bar{n}}-\vec{\kappa}_{\|,\bar{m}}-\vec{q}_{\|})
\nonumber\\ &&\times
A_{j}(z';\vec{q}_{\|}^{\,\prime})
d^2\kappa_{\|,\bar{n}}d^2\kappa_{\|,\bar{m}}
d^2q_{\|}'dz'.\end{aligned}$$ Of these two integrals over the surface states, we can solve one because of the coupling between the surface states and the wavevectors introduced by the Dirac delta functions appearing. Keeping the $\vec{\kappa}_{\|,\bar{m}}$ set, and thus we solve for the $n$ set. Solving for this set, we find that $\vec{\kappa}_{\|,\bar{n}}$ is replaced by $\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime}$, thus giving $$\begin{aligned}
\lefteqn{
J_{ij}^{\rm{B}}(z;\vec{q}_{\|})=
-{1\over\hbar}
{2\over(2\pi)^2}
\iiint
\sum_{nm}
{f_{n}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime})-f_{m}(\vec{\kappa}_{\|,\bar{m}})
\over\tilde{\omega}_{nm}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime},
\vec{\kappa}_{\|,\bar{m}})-\omega}
j_{j,mn}(z';2\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime})
}\nonumber\\ &\quad&\times
j_{i,nm}(z;2\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime})
\delta(\vec{q}_{\|}^{\,\prime}-\vec{q}_{\|})
A_{j}(z';\vec{q}_{\|}^{\,\prime})
d^2\kappa_{\|,\bar{m}}
d^2q_{\|}'dz'.
\label{eq:J1ij-B}\end{aligned}$$ From the above expression we extract part B of the linear conductivity tensor as $$\begin{aligned}
\lefteqn{
{\sigma}_{ij}^{\,\rm{B}}(z,z';\vec{q}_{\|},\vec{q}_{\|}^{\,\prime})=
-{2\over{}{\rm{i}}\omega}{1\over\hbar}{1\over(2\pi)^2}\sum_{nm}\int
{f_{n}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime})-f_{m}(\vec{\kappa}_{\|})
\over\tilde{\omega}_{nm}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime},
\vec{\kappa}_{\|})-\omega}
j_{j,mn}(z';2\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime})
}\nonumber\\ &\quad&\times
j_{i,nm}(z;2\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime})
\delta(\vec{q}_{\|}^{\,\prime}-\vec{q}_{\|})
d^2\kappa_{\|},\end{aligned}$$ where we have omitted the now superfluous index $m$ from the surface states $\vec{\kappa}_{\|}$. Again we take into account the conservation of pseudo-momentum, $\delta(\vec{q}_{\|}^{\,\prime}-\vec{q}_{\|})$, letting us perform the integration over $q_{\|}'$ in Eq. (\[eq:J1ij-B\]). From this result the linear conductivity tensor part B is extracted on the form shown in Eq. (\[eq:SigmaAB\]).
Nonlinear process A
-------------------
Inserting Eq. (\[eq:XiAArz\]) into Eq. (\[eq:J3r\]), we take element $xxxx$ of part A of the nonlinear current density. In the result we insert the expressions for the wave function and the vector potential in the two-dimensional Fourier representation \[Eqs. (\[eq:eigenstate\]) and (\[eq:Azr\]), respectively\]. Then by use of Eq. (\[eq:Jzq\]) we find $$\begin{aligned}
\lefteqn{
J_{xxxx}^{\rm{A}}(z;\vec{q}_{\|})=
-{e^4\over8m_{e}^2\hbar}
{2\over(2\pi)^{10}}\idotsint
\sum_{nm}
{f_{n}-f_{m}\over\tilde{\omega}_{nm}-2\omega}
\psi_{n}^{*}(z'')\psi_{m}(z'')\psi_{m}^{*}(z)\psi_{n}(z)
}\nonumber\\ &\quad&\times
A_{x}(z''';\vec{q}_{\|}^{\,\prime\prime\prime})
A_{x}(z'';\vec{q}_{\|}^{\,\prime\prime})
A_{x}^{*}(z';\vec{q}_{\|}^{\,\prime})
e^{{\rm{i}}(\vec{\kappa}_{\|,\bar{m}}-\vec{\kappa}_{\|,\bar{n}})\cdot\vec{r}_{\|}^{\,\prime\prime}}
e^{{\rm{i}}(\vec{\kappa}_{\|,\bar{n}}-\vec{\kappa}_{\|,\bar{m}})\cdot\vec{r}_{\|}}
e^{{\rm{i}}\vec{q}_{\|}^{\,\prime\prime\prime}\cdot\vec{r}_{\|}^{\,\prime\prime\prime}}
e^{{\rm{i}}\vec{q}_{\|}^{\,\prime\prime}\cdot\vec{r}_{\|}^{\,\prime\prime}}
e^{-{\rm{i}}\vec{q}_{\|}^{\,\prime}\cdot\vec{r}_{\|}^{\,\prime}}
\nonumber\\ &&\times
\delta(\vec{r}-\vec{r}\,')
\delta(\vec{r}\,''-\vec{r}\,''')
d^2q_{\|}'''d^2q_{\|}''d^2q_{\|}'
d^3r'''d^3r''d^3r'e^{-{\rm{i}}\vec{q}_{\|}\cdot\vec{r}_{\|}}d^2r_{\|}.\end{aligned}$$ In this equation, we first solve the integrals $\int{}d^2r_{\|}'$ and $\int{}d^2r_{\|}'''$, thereafter the remaining Cartesian integrals $\int{}d^2r_{\|}''$ and $\int{}d^2r_{\|}$, and finally replace the infinite sums over the different $\vec{\kappa}_{\|}$ coordinates with integrals, thereby obtaining $$\begin{aligned}
\lefteqn{
J_{xxxx}^{\rm{A}}(z;\vec{q}_{\|})=
-{e^4\over8m_{e}^2\hbar}
{2\over(2\pi)^6}
\idotsint
\sum_{nm}
{f_{n}(\vec{\kappa}_{\|,\bar{n}})-f_{m}(\vec{\kappa}_{\|,\bar{m}})
\over\tilde{\omega}_{nm}(\vec{\kappa}_{\|,\bar{n}},\vec{\kappa}_{\|,\bar{m}})-2\omega}
\psi_{n}^{*}(z'')\psi_{m}(z'')\psi_{m}^{*}(z)\psi_{n}(z)
}\nonumber\\ &\quad&\times
A_{x}(z''';\vec{q}_{\|}^{\,\prime\prime\prime})
A_{x}(z'';\vec{q}_{\|}^{\,\prime\prime})
A_{x}^{*}(z';\vec{q}_{\|}^{\,\prime})
\delta(\vec{\kappa}_{\|,\bar{m}}-\vec{\kappa}_{\|,\bar{n}}+\vec{q}_{\|}^{\,\prime\prime\prime}
+\vec{q}_{\|}^{\,\prime\prime})
\delta(\vec{\kappa}_{\|,\bar{n}}-\vec{\kappa}_{\|,\bar{m}}-\vec{q}_{\|}^{\,\prime}-\vec{q}_{\|})
\nonumber\\ &&\times
\delta(z-z')
\delta(z''-z''')
d^2\kappa_{\|,\bar{n}}d^2\kappa_{\|,\bar{m}}
d^2q_{\|}'''d^2q_{\|}''d^2q_{\|}'
dz'''dz''dz'.\end{aligned}$$ Of the two integrals over the $\kappa_{\|}$ quantities, we can solve one because of the coupling of these to the wavevectors introduced by the Dirac delta functions appearing. Keeping the $\vec{\kappa}_{\|,\bar{m}}$ set we thus solve the integrals for the $n$ set. Solving for this set, we find that $\vec{\kappa}_{\|,\bar{n}}$ is replaced by $\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime}+\vec{q}_{\|}$, which gives $$\begin{aligned}
\lefteqn{
J_{xxxx}^{\rm{A}}(z;\vec{q}_{\|})=
-{e^4\over8m_{e}^2\hbar}
{2\over(2\pi)^6}
\idotsint
\sum_{nm}
{f_{n}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime}+\vec{q}_{\|})
-f_{m}(\vec{\kappa}_{\|,\bar{m}})
\over\tilde{\omega}_{nm}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime}+\vec{q}_{\|},
\vec{\kappa}_{\|,\bar{m}})-2\omega}
\psi_{n}^{*}(z'')\psi_{m}(z'')
}\nonumber\\ &\quad&\times
\psi_{m}^{*}(z)\psi_{n}(z)
A_{x}(z''';\vec{q}_{\|}^{\,\prime\prime\prime})
A_{x}(z'';\vec{q}_{\|}^{\,\prime\prime})
A_{x}^{*}(z';\vec{q}_{\|}^{\,\prime})
\delta(\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime}-\vec{q}_{\|})
\delta(z-z')
\nonumber\\ &&\times
\delta(z''-z''')
d^2\kappa_{\|,\bar{m}}
d^2q_{\|}'''d^2q_{\|}''d^2q_{\|}'
dz'''dz''dz'.
\label{eq:J3A}\end{aligned}$$ From this we may extract part A of the nonlinear conductivity tensor as defined in Eq. (\[eq:J3zq\]) as $$\begin{aligned}
\lefteqn{
{\Xi}_{xxxx}^{\rm{A}}(z,z',z'',z''';\vec{q}_{\|},\vec{q}_{\|}^{\,\prime},
\vec{q}_{\|}^{\,\prime\prime},\vec{q}_{\|}^{\,\prime\prime\prime})=
{2{\rm{i}}\over\omega^3}{e^4\over8m_{e}^2\hbar}{1\over(2\pi)^2}
\sum_{nm}\int
{f_{n}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime}+\vec{q}_{\|})
-f_{m}(\vec{\kappa}_{\|})
\over\tilde{\omega}_{nm}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime}+\vec{q}_{\|},
\vec{\kappa}_{\|})-2\omega}
}\nonumber\\ &\quad&\times
\psi_{n}^{*}(z'')\psi_{m}(z'')\psi_{m}^{*}(z)\psi_{n}(z)
\delta(\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime}-\vec{q}_{\|})
\delta(z-z')\delta(z''-z''')
d^2\kappa_{\|},\end{aligned}$$ where we have omitted the now superfluous index $m$ from the surface states $\vec{\kappa}_{\|}$. Taking into account the fact that we look for the phase conjugation response we restrict ourselves to the case where the pump fields are counterpropagating, thus taking $$\begin{aligned}
\vec{A}(z''';\vec{q}_{\|}^{\,\prime\prime\prime})&\equiv&\vec{A}(z''';-\vec{k}_{\|})
\delta(\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{k}_{\|}),
\label{eq:Apump-1}\\
\vec{A}(z'';\vec{q}_{\|}^{\,\prime\prime})&\equiv&\vec{A}(z'';\vec{k}_{\|})
\delta(\vec{q}_{\|}^{\,\prime\prime}-\vec{k}_{\|}),
\label{eq:Apump-2}\end{aligned}$$ where $\vec{k}_{\|}$ is the common wavevector for the two pump fields. With these substitutions we can perform the integrals over $q_{\|}'''$ and $q_{\|}''$ in Eq. (\[eq:J3A\]), and the conservation of pseudo-momentum is reduced from its general degenerate four-wave mixing form, $\delta(\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime}-\vec{q}_{\|})$, to $\delta(\vec{q}_{\|}^{\,\prime}+\vec{q}_{\|})$. This allows us also to solve the integral over $q_{\|}'$ in Eq. (\[eq:J3A\]), and on the form of Eq. (\[eq:J3-wzq\]) we can extract the PCDFWM conductivity tensor part A, appearing as Eq. (\[eq:XiAA\]).
Nonlinear process B
-------------------
Inserting Eq. (\[eq:XiABrz\]) into Eq. (\[eq:J3r\]), we take element $xxkh$ of part B of the nonlinear current density. In the result we insert the expressions for the wave function, the vector potential and the transition current density in the two-dimensional Fourier representation \[Eqs. (\[eq:eigenstate\]), (\[eq:Azr\]) and (\[eq:Jm->n||\]), respectively\]. Then by use of Eq. (\[eq:Jzq\]) we find $$\begin{aligned}
\lefteqn{
J_{xxkh}^{\rm{B}}(z;\vec{q}_{\|})=
-{e^2\over4m_{e}\hbar^2}
{2\over(2\pi)^{12}}
\idotsint
\sum_{nmv}
{1\over\tilde{\omega}_{nm}-2\omega}
\left({f_{m}-f_{v}\over\tilde{\omega}_{v{}m}-\omega}
+{f_{n}-f_{v}\over\tilde{\omega}_{nv}-\omega}\right)
}\nonumber\\ &\quad&\times
j_{h,v{}n}(z''';\vec{\kappa}_{\|,\bar{v}}+\vec{\kappa}_{\|,\bar{n}})
j_{k,mv}(z'';\vec{\kappa}_{\|,\bar{m}}+\vec{\kappa}_{\|,\bar{v}})
\psi_{m}^{*}(z)
\psi_{n}(z)
\nonumber\\ &&\times
A_{h}(z''';\vec{q}_{\|}^{\,\prime\prime\prime})
A_{k}(z'';\vec{q}_{\|}^{\,\prime\prime})
A_{x}^{*}(z';\vec{q}_{\|}^{\,\prime})
e^{{\rm{i}}(\vec{\kappa}_{\|,\bar{n}}-\vec{\kappa}_{\|,\bar{m}})\cdot\vec{r}_{\|}}
e^{{\rm{i}}(\vec{\kappa}_{\|,\bar{v}}-\vec{\kappa}_{\|,\bar{n}})\cdot\vec{r}_{\|}^{\,\prime\prime\prime}}
e^{{\rm{i}}(\vec{\kappa}_{\|,\bar{m}}-\vec{\kappa}_{\|,\bar{v}})\cdot\vec{r}_{\|}^{\,\prime\prime}}
\nonumber\\ &&\times
e^{{\rm{i}}\vec{q}_{\|}^{\,\prime\prime\prime}\cdot\vec{r}_{\|}^{\,\prime\prime\prime}}
e^{{\rm{i}}\vec{q}_{\|}^{\,\prime\prime}\cdot\vec{r}_{\|}^{\,\prime\prime}}
e^{-{\rm{i}}\vec{q}_{\|}^{\,\prime}\cdot\vec{r}_{\|}^{\,\prime}}
d^2q_{\|}'''d^2q_{\|}''d^2q_{\|}'
\delta(\vec{r}-\vec{r}\,')
d^3r'''d^3r''d^3r'e^{-{\rm{i}}\vec{q}_{\|}\cdot\vec{r}_{\|}}d^2r_{\|}.\end{aligned}$$ Solving first the integral $\int{}d^2r_{\|}'$, then the integrals $\int{}d^2r_{\|}'''$, $\int{}d^2r_{\|}''$, and $\int{}d^2r_{\|}$, and finally replacing the sums over the $\vec{\kappa}_{\|}$ quantities with integrals, we obtain $$\begin{aligned}
\lefteqn{
J_{xxkh}^{\rm{B}}(z;\vec{q}_{\|})=
-{e^2\over4m_{e}\hbar^2}
{2\over(2\pi)^6}
\idotsint
\sum_{nmv}
{1\over\tilde{\omega}_{nm}(\vec{\kappa}_{\|,\bar{n}},\vec{\kappa}_{\|,\bar{m}})-2\omega}
}\nonumber\\ &\quad&\times
\left({f_{m}(\vec{\kappa}_{\|,\bar{m}})-f_{v}(\vec{\kappa}_{\|,\bar{v}})\over
\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|,\bar{v}},\vec{\kappa}_{\|,\bar{m}})-\omega}
+{f_{n}(\vec{\kappa}_{\|,\bar{n}})-f_{v}(\vec{\kappa}_{\|,\bar{v}})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|,\bar{n}},\vec{\kappa}_{\|,\bar{v}})-\omega}
\right)
j_{h,v{}n}(z''';\vec{\kappa}_{\|,\bar{v}}+\vec{\kappa}_{\|,\bar{n}})
\nonumber\\ &&\times
j_{k,mv}(z'';\vec{\kappa}_{\|,\bar{m}}+\vec{\kappa}_{\|,\bar{v}})
\psi_{m}^{*}(z)
\psi_{n}(z)
A_{h}(z''';\vec{q}_{\|}^{\,\prime\prime\prime})
A_{k}(z'';\vec{q}_{\|}^{\,\prime\prime})
A_{x}^{*}(z';\vec{q}_{\|}^{\,\prime})
\nonumber\\ &&\times
\delta(\vec{\kappa}_{\|,\bar{v}}-\vec{\kappa}_{\|,\bar{n}}+\vec{q}_{\|}^{\,\prime\prime\prime})
\delta(\vec{\kappa}_{\|,\bar{m}}-\vec{\kappa}_{\|,\bar{v}}+\vec{q}_{\|}^{\,\prime\prime})
\delta(\vec{\kappa}_{\|,\bar{n}}-\vec{\kappa}_{\|,\bar{m}}-\vec{q}_{\|}^{\,\prime}-\vec{q}_{\|})
\nonumber\\ &&\times
\delta(z-z')
d^2\kappa_{\|,\bar{n}}d^2\kappa_{\|,\bar{m}}d^2\kappa_{\|,\bar{v}}
d^2q_{\|}'''d^2q_{\|}''d^2q_{\|}'
dz'''dz''dz'.\end{aligned}$$ Of these three integrals over $\vec{\kappa}_{\|}$, we can solve two because of the coupling to the wavevectors introduced by the Dirac delta functions appearing. Keeping the $\vec{\kappa}_{\|,\bar{m}}$ set of surface states, we thus solve the integrals for the $v$ and $n$ sets (in that order). Solving for the $v$ set, we find that $\vec{\kappa}_{\|,\bar{v}}$ is replaced by $\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime}$, which then allows us to solve the $n$ set by replacing $\vec{\kappa}_{\|,\bar{n}}$ by $\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}$. Then we get $$\begin{aligned}
\lefteqn{
J_{xxkh}^{\rm{B}}(z;\vec{q}_{\|})=
-{e^2\over4m_{e}\hbar^2}
{2\over(2\pi)^6}
\idotsint
\sum_{nmv}
{1\over\tilde{\omega}_{nm}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime},
\vec{\kappa}_{\|,\bar{m}})-2\omega}
}\nonumber\\ &\quad&\times
\left({f_{m}(\vec{\kappa}_{\|,\bar{m}})-f_{v}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime})
\over\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime},
\vec{\kappa}_{\|,\bar{m}})-\omega}
+{f_{n}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime})
-f_{v}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime},
\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime})-\omega}\right)
\nonumber\\ &&\times
j_{h,v{}n}(z''';2\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+2\vec{q}_{\|}^{\,\prime\prime})
j_{k,mv}(z'';2\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime})
\psi_{m}^{*}(z)\psi_{n}(z)
\nonumber\\ &&\times
A_{h}(z''';\vec{q}_{\|}^{\,\prime\prime\prime})
A_{k}(z'';\vec{q}_{\|}^{\,\prime\prime})
A_{x}^{*}(z';\vec{q}_{\|}^{\,\prime})
\delta(\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime}-\vec{q}_{\|})
\nonumber\\ &&\times
\delta(z-z')d^2\kappa_{\|,\bar{m}}
d^2q_{\|}'''d^2q_{\|}''d^2q_{\|}'dz'''dz''dz'.
\label{eq:J3B}\end{aligned}$$ On the form of Eq. (\[eq:J3zq\]) we thus get part B of the conductivity tensor as $$\begin{aligned}
\lefteqn{
{\Xi}_{xxkh}^{\rm{B}}(z,z',z'',z''';\vec{q}_{\|},\vec{q}_{\|}^{\,\prime},
\vec{q}_{\|}^{\,\prime\prime},\vec{q}_{\|}^{\,\prime\prime\prime})=
}\nonumber\\ &\quad&
{2{\rm{i}}\over{}\omega^3}
{1\over(2\pi)^2}{e^2\over4m_{e}\hbar^2}
\sum_{nmv}\int
{1\over\tilde{\omega}_{nm}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime},
\vec{\kappa}_{\|})-2\omega}
\left({f_{m}(\vec{\kappa}_{\|})-f_{v}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime})\over
\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime},\vec{\kappa}_{\|})
-\omega}
\right.\nonumber\\ &&\left.\!
+{f_{n}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime})
-f_{v}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime},
\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime})-\omega}\right)
j_{h,v{}n}(z''';2\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}+2\vec{q}_{\|}^{\,\prime\prime})
\nonumber\\ &&\times
j_{k,mv}(z'';2\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime})
\psi_{m}^{*}(z)\psi_{n}(z)
\delta(\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime}-\vec{q}_{\|})
\delta(z-z')d^2\kappa_{\|},\end{aligned}$$ where we have omitted the now superfluous index $m$ from the surface states $\vec{\kappa}_{\|}$. Looking for the phase conjugation response the pump fields take the form of Eqs. (\[eq:Apump-1\])–(\[eq:Apump-2\]), and integration over $q_{\|}'''$ and $q_{\|}''$ in Eq. (\[eq:J3B\]) can be performed. Thereby the Dirac delta function accounting for conservation of pseudo-momentum is reduced from its general DFWM form, $\delta(\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime}-\vec{q}_{\|})$, to $\delta(\vec{q}_{\|}^{\,\prime}+\vec{q}_{\|})$. Thus performing also the integral over $q_{\|}'$, the PCDFWM conductivity tensor on the form of Eq. (\[eq:J3-wzq\]) can be extracted, and Eq. (\[eq:XiAB\]) appear.
Nonlinear process C
-------------------
Inserting Eq. (\[eq:XiACrz\]) into Eq. (\[eq:J3r\]), we take element $xxxx$ of part C of the nonlinear current density. In the result we insert the expressions for the wave function and the vector potential in the two-dimensional Fourier representation \[Eqs. (\[eq:eigenstate\]) and (\[eq:Azr\]), respectively\]. Then by use of Eq. (\[eq:Jzq\]) we find $$\begin{aligned}
\lefteqn{
J_{xxxx}^{\rm{C}}(z;\vec{q}_{\|})=
-{e^4\over4m_{e}^2\hbar}
{2\over(2\pi)^{10}}
\idotsint
\sum_{nm}
{f_{n}-f_{m}\over\tilde{\omega}_{nm}}
\psi_{n}^{*}(z')\psi_{m}(z')
\psi_{m}^{*}(z)\psi_{n}(z)
A_{x}(z''';\vec{q}_{\|}^{\,\prime\prime\prime})
}\nonumber\\ &\quad&\times
A_{x}(z'';\vec{q}_{\|}^{\,\prime\prime})
A_{x}^{*}(z';\vec{q}_{\|}^{\,\prime})
e^{{\rm{i}}(\vec{\kappa}_{\|,\bar{m}}-\vec{\kappa}_{\|,\bar{n}})\cdot\vec{r}_{\|}^{\,\prime}}
e^{{\rm{i}}(\vec{\kappa}_{\|,\bar{n}}-\vec{\kappa}_{\|,\bar{m}})\cdot\vec{r}_{\|}}
e^{{\rm{i}}\vec{q}_{\|}^{\,\prime\prime\prime}\cdot\vec{r}_{\|}^{\,\prime\prime\prime}}
e^{{\rm{i}}\vec{q}_{\|}^{\,\prime\prime}\cdot\vec{r}_{\|}^{\,\prime\prime}}
e^{-{\rm{i}}\vec{q}_{\|}^{\,\prime}\cdot\vec{r}_{\|}^{\,\prime}}
\delta(\vec{r}\,'-\vec{r}\,''')
\nonumber\\ &&\times
\delta(\vec{r}-\vec{r}\,'')
d^2q_{\|}'''d^2q_{\|}''d^2q_{\|}'
d^3r'''d^3r''d^3r'e^{-{\rm{i}}\vec{q}_{\|}\cdot\vec{r}_{\|}}d^2r_{\|}.\end{aligned}$$ Solving first the integrals $\int{}d^2r_{\|}'''$ and $\int{}d^2r_{\|}''$, then the integrals $\int{}d^2r_{\|}'$ and $\int{}d^2r_{\|}$, and finally replacing the sum over the $\vec{\kappa}_{\|}$ quantities with integrals, we get $$\begin{aligned}
\lefteqn{
J_{xxxx}^{\rm{C}}(z;\vec{q}_{\|})=
-{e^4\over4m_{e}^2\hbar}
{2\over(2\pi)^6}
\idotsint
\sum_{nm}
{f_{n}(\vec{\kappa}_{\|,\bar{n}})-f_{m}(\vec{\kappa}_{\|,\bar{m}})\over
\tilde{\omega}_{nm}(\vec{\kappa}_{\|,\bar{n}},\vec{\kappa}_{\|,\bar{m}})}
\psi_{n}^{*}(z')\psi_{m}(z')
\psi_{m}^{*}(z)\psi_{n}(z)
}\nonumber\\ &\quad&\times
A_{x}(z''';\vec{q}_{\|}^{\,\prime\prime\prime})
A_{x}(z'';\vec{q}_{\|}^{\,\prime\prime})
A_{x}^{*}(z';\vec{q}_{\|}^{\,\prime})
\delta(\vec{\kappa}_{\|,\bar{m}}-\vec{\kappa}_{\|,\bar{n}}+\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|}^{\,\prime})
\delta(\vec{\kappa}_{\|,\bar{n}}-\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|})
\nonumber\\ &&\times
\delta(z'-z''')
\delta(z-z'')
d^2\kappa_{\|,\bar{n}}d^2\kappa_{\|,\bar{m}}
d^2q_{\|}'''d^2q_{\|}''d^2q_{\|}'
dz'''dz''dz'.\end{aligned}$$ Of the two integrals over the $\vec{\kappa}_{\|}$ quantities, we can solve one because of the coupling to the wavevectors introduced by the Dirac delta functions appearing. We aim at keeping the $\vec{\kappa}_{\|,\bar{m}}$ set, and thus we solve the integrals for the $n$ set. Solving for this set, we find that $\vec{\kappa}_{\|,\bar{n}}$ is replaced by $\vec{\kappa}_{\|,\bar{m}}-\vec{q}_{\|}^{\,\prime\prime}+\vec{q}_{\|}$, thus leading to the result $$\begin{aligned}
\lefteqn{
J_{xxxx}^{\rm{C}}(z;\vec{q}_{\|})=
-{e^4\over4m_{e}^2\hbar}
{2\over(2\pi)^6}
\idotsint
\sum_{nm}
{f_{n}(\vec{\kappa}_{\|,\bar{m}}-\vec{q}_{\|}^{\,\prime\prime}+\vec{q}_{\|})
-f_{m}(\vec{\kappa}_{\|,\bar{m}})\over\tilde{\omega}_{nm}(\vec{\kappa}_{\|,\bar{m}}
-\vec{q}_{\|}^{\,\prime\prime}+\vec{q}_{\|},\vec{\kappa}_{\|,\bar{m}})}
\psi_{n}^{*}(z')\psi_{m}(z')
}\nonumber\\ &\quad&\times
\psi_{m}^{*}(z)\psi_{n}(z)
A_{x}(z''';\vec{q}_{\|}^{\,\prime\prime\prime})A_{x}(z'';\vec{q}_{\|}^{\,\prime\prime})
A_{x}^{*}(z';\vec{q}_{\|}^{\,\prime})
\delta(\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime}-\vec{q}_{\|})
\delta(z'-z''')
\nonumber\\ &&\times
\delta(z-z'')
d^2\kappa_{\|,\bar{m}}d^2q_{\|}'''d^2q_{\|}''d^2q_{\|}'dz'''dz''dz'.
\label{eq:J3C}\end{aligned}$$ On the form of Eq. (\[eq:J3zq\]) we thus get part C of the conductivity tensor as $$\begin{aligned}
\lefteqn{
{\Xi}_{xxxx}^{\rm{C}}(z,z',z'',z''';\vec{q}_{\|},
\vec{q}_{\|}^{\,\prime},\vec{q}_{\|}^{\,\prime\prime},
\vec{q}_{\|}^{\,\prime\prime\prime})=
{2{\rm{i}}\over\omega^3}{e^4\over4m_{e}^2\hbar}{1\over(2\pi)^2}
\sum_{nm}\int
{f_{n}(\vec{\kappa}_{\|}-\vec{q}_{\|}^{\,\prime\prime}+\vec{q}_{\|})
-f_{m}(\vec{\kappa}_{\|})\over\tilde{\omega}_{nm}(\vec{\kappa}_{\|}
-\vec{q}_{\|}^{\,\prime\prime}+\vec{q}_{\|},\vec{\kappa}_{\|})}
}\nonumber\\ &\quad&\times
\psi_{n}^{*}(z')\psi_{m}(z')
\psi_{m}^{*}(z)\psi_{n}(z)
\delta(\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime}-\vec{q}_{\|})
\delta(z'-z''')\delta(z-z'')
d^2\kappa_{\|}.\end{aligned}$$ where we have omitted the now superfluous index $m$ from the surface states $\vec{\kappa}_{\|}$. The phase conjugation response is found using the same procedure as before, since using the pump fields defined in Eqs. (\[eq:Apump-1\])–(\[eq:Apump-2\]) the integrals over $q_{\|}'''$ and $q_{\|}''$ in Eq. (\[eq:J3C\]) can be performed. Then (again) the conservation of pseudo-momentum, $\delta(\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime}-\vec{q}_{\|})$, is reduced to $\delta(\vec{q}_{\|}^{\,\prime}+\vec{q}_{\|})$, and after integration over $q_{\|}'$ we obtain on the form of Eq. (\[eq:J3-wzq\]) the PCDFWM conductivity tensor part C, appearing as Eq. (\[eq:XiAC\]).
Nonlinear process D
-------------------
Inserting Eq. (\[eq:XiADrz\]) into Eq. (\[eq:J3r\]), we take element $xjkx$ of part D of the nonlinear current density. In the result we insert the expressions for the wave function, the vector potential and the transition current density in the two-dimensional Fourier representation \[Eqs. (\[eq:eigenstate\]), (\[eq:Azr\]) and (\[eq:Jm->n||\]), respectively\]. Then by use of Eq. (\[eq:Jzq\]) we find $$\begin{aligned}
\lefteqn{
J_{xjkx}^{\rm{D}}(z;\vec{q}_{\|})=
-{e^2\over4m_{e}\hbar^2}
{2\over(2\pi)^{12}}
\idotsint
\sum_{nmv}
{1\over\tilde{\omega}_{nm}}\left\{
\left({f_{m}-f_{v}\over\tilde{\omega}_{v{}m}-\omega}
+{f_{n}-f_{v}\over\tilde{\omega}_{nv}+\omega}\right)
\right.}\nonumber\\ &\quad&\times
j_{j,v{}n}(z';\vec{\kappa}_{\|,\bar{v}}+\vec{\kappa}_{\|,\bar{n}})
j_{k,mv}(z'';\vec{\kappa}_{\|,\bar{m}}+\vec{\kappa}_{\|,\bar{v}})
e^{{\rm{i}}(\vec{\kappa}_{\|,\bar{v}}-\vec{\kappa}_{\|,\bar{n}})\cdot\vec{r}_{\|}^{\,\prime}}
e^{{\rm{i}}(\vec{\kappa}_{\|,\bar{m}}-\vec{\kappa}_{\|,\bar{v}})\cdot\vec{r}_{\|}^{\,\prime\prime}}
\nonumber\\ &&
+\left({f_{m}-f_{v}\over\tilde{\omega}_{v{}m}+\omega}
+{f_{n}-f_{v}\over\tilde{\omega}_{nv}-\omega}\right)
j_{k,v{}n}(z'';\vec{\kappa}_{\|,\bar{v}}+\vec{\kappa}_{\|,\bar{n}})
j_{j,mv}(z';\vec{\kappa}_{\|,\bar{m}}+\vec{\kappa}_{\|,\bar{v}})
e^{{\rm{i}}(\vec{\kappa}_{\|,\bar{v}}-\vec{\kappa}_{\|,\bar{n}})\cdot\vec{r}_{\|}^{\,\prime\prime}}
\nonumber\\ &&\times\left.\!
e^{{\rm{i}}(\vec{\kappa}_{\|,\bar{m}}-\vec{\kappa}_{\|,\bar{v}})\cdot\vec{r}_{\|}^{\,\prime}}
\right\}
\psi_{m}^{*}(z)
\psi_{n}(z)
\delta(\vec{r}-\vec{r}\,''')
A_{x}(z''';\vec{q}_{\|}^{\,\prime\prime\prime})
A_{k}(z'';\vec{q}_{\|}^{\,\prime\prime})
A_{j}^{*}(z';\vec{q}_{\|}^{\,\prime})
e^{{\rm{i}}(\vec{\kappa}_{\|,\bar{n}}-\vec{\kappa}_{\|,\bar{m}})\cdot\vec{r}_{\|}}
\nonumber\\ &&\times
e^{{\rm{i}}\vec{q}_{\|}^{\,\prime\prime\prime}\cdot\vec{r}_{\|}^{\,\prime\prime\prime}}
e^{{\rm{i}}\vec{q}_{\|}^{\,\prime\prime}\cdot\vec{r}_{\|}^{\,\prime\prime}}
e^{-{\rm{i}}\vec{q}_{\|}^{\,\prime}\cdot\vec{r}_{\|}^{\,\prime}}
d^2q_{\|}'''d^2q_{\|}''d^2q_{\|}'
d^3r'''d^3r''d^3r'e^{-{\rm{i}}\vec{q}_{\|}\cdot\vec{r}_{\|}}d^2r_{\|}.\end{aligned}$$ Solving in this equation the integral $\int{}d^2r_{\|}'''$, and then the integrals $\int{}d^2r_{\|}''$, $\int{}d^2r_{\|}'$, and $\int{}d^2r_{\|}$ we get, after having replaced the sums over the various $\vec{\kappa}_{\|}$ quantities with integrals as before, $$\begin{aligned}
\lefteqn{
J_{xjkx}^{\rm{D}}(z;\vec{q}_{\|})=
-{e^2\over4m_{e}\hbar^2}
{2\over(2\pi)^6}
\idotsint
\sum_{nmv}
{1\over\tilde{\omega}_{nm}(\vec{\kappa}_{\|,\bar{n}},\vec{\kappa}_{\|,\bar{m}})}\left\{
\left({f_{m}(\vec{\kappa}_{\|,\bar{m}})-f_{v}(\vec{\kappa}_{\|,\bar{v}})\over
\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|,\bar{v}},\vec{\kappa}_{\|,\bar{m}})-\omega}
\right.\right.}\nonumber\\ &\quad&\left.\!
+{f_{n}(\vec{\kappa}_{\|,\bar{n}})-f_{v}(\vec{\kappa}_{\|,\bar{v}})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|,\bar{n}},\vec{\kappa}_{\|,\bar{v}})+\omega}
\right)
j_{j,v{}n}(z';\vec{\kappa}_{\|,\bar{v}}+\vec{\kappa}_{\|,\bar{n}})
j_{k,mv}(z'';\vec{\kappa}_{\|,\bar{m}}+\vec{\kappa}_{\|,\bar{v}})
\delta(\vec{\kappa}_{\|,\bar{m}}-\vec{\kappa}_{\|,\bar{v}}+\vec{q}_{\|}^{\,\prime\prime})
\nonumber\\ &&\times
\delta(\vec{\kappa}_{\|,\bar{v}}-\vec{\kappa}_{\|,\bar{n}}-\vec{q}_{\|}^{\,\prime})
+\left({f_{m}(\vec{\kappa}_{\|,\bar{m}})-f_{v}(\vec{\kappa}_{\|,\bar{v}})\over
\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|,\bar{v}},\vec{\kappa}_{\|,\bar{m}})+\omega}
+{f_{n}(\vec{\kappa}_{\|,\bar{n}})-f_{v}(\vec{\kappa}_{\|,\bar{v}})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|,\bar{n}},\vec{\kappa}_{\|,\bar{v}})-\omega}
\right)
\nonumber\\ &&\times\left.
j_{k,v{}n}(z'';\vec{\kappa}_{\|,\bar{v}}+\vec{\kappa}_{\|,\bar{n}})
j_{j,mv}(z';\vec{\kappa}_{\|,\bar{m}}+\vec{\kappa}_{\|,\bar{v}})
\delta(\vec{\kappa}_{\|,\bar{v}}-\vec{\kappa}_{\|,\bar{n}}+\vec{q}_{\|}^{\,\prime\prime})
\delta(\vec{\kappa}_{\|,\bar{m}}-\vec{\kappa}_{\|,\bar{v}}-\vec{q}_{\|}^{\,\prime})
\right\}
\nonumber\\ &&\times
\psi_{m}^{*}(z)\psi_{n}(z)
\delta(z-z''')
\delta(\vec{\kappa}_{\|,\bar{n}}-\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|})
A_{x}(z''';\vec{q}_{\|}^{\,\prime\prime\prime})
A_{k}(z'';\vec{q}_{\|}^{\,\prime\prime})
\nonumber\\ &&\times
A_{j}^{*}(z';\vec{q}_{\|}^{\,\prime})
d^2\kappa_{\|,\bar{n}}d^2\kappa_{\|,\bar{m}}d^2\kappa_{\|,\bar{v}}
d^2q_{\|}'''d^2q_{\|}''d^2q_{\|}'
dz'''dz''dz'.\end{aligned}$$ Of the three integrals over $\vec{\kappa}_{\|}$ quantities, we can solve two because of the coupling to the wavevectors introduced by the Dirac delta functions appearing. We aim at keeping the $\vec{\kappa}_{\|,\bar{m}}$ set, and thus we solve the integrals for the $v$ and $n$ sets (in that order). Solving for the $v$ set, we find that $\vec{\kappa}_{\|,\bar{v}}$ is replaced by $\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime}$ in the first part of the sum and by $\vec{\kappa}_{\|,\bar{m}}-\vec{q}_{\|}^{\,\prime}$ in the second part of the sum, which then allows us to solve the $n$ set by replacing $\vec{\kappa}_{\|,\bar{n}}$ with $\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime}$ in general, giving the result $$\begin{aligned}
\lefteqn{
J_{xjkx}^{\rm{D}}(z;\vec{q}_{\|})=
-{e^2\over4m_{e}\hbar^2}
{2\over(2\pi)^6}
\idotsint
\sum_{nmv}
{1\over\tilde{\omega}_{nm}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime}
,\vec{\kappa}_{\|,\bar{m}})}
}\nonumber\\ &\quad&\times
\left\{
\left({f_{m}(\vec{\kappa}_{\|,\bar{m}})-f_{v}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime})
\over\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime},
\vec{\kappa}_{\|,\bar{m}})-\omega}
+{f_{n}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime})
-f_{v}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime},
\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime})+\omega}\right)
\right.\nonumber\\ &&\times
j_{j,v{}n}(z';2\vec{\kappa}_{\|,\bar{m}}+2\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime})
j_{k,mv}(z'';2\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime})
\nonumber\\ &&
+\left({f_{m}(\vec{\kappa}_{\|,\bar{m}})-f_{v}(\vec{\kappa}_{\|,\bar{m}}-\vec{q}_{\|}^{\,\prime})
\over\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|,\bar{m}}-\vec{q}_{\|}^{\,\prime},
\vec{\kappa}_{\|,\bar{m}})+\omega}
+{f_{n}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime})
-f_{v}(\vec{\kappa}_{\|,\bar{m}}-\vec{q}_{\|}^{\,\prime})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime},
\vec{\kappa}_{\|,\bar{m}}-\vec{q}_{\|}^{\,\prime})-\omega}\right)
\nonumber\\ &&\times\left.\!
j_{k,v{}n}(z'';2\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime}-2\vec{q}_{\|}^{\,\prime})
j_{j,mv}(z';2\vec{\kappa}_{\|,\bar{m}}-\vec{q}_{\|}^{\,\prime})
\right\}
\psi_{m}^{*}(z)\psi_{n}(z)
\delta(z-z''')
\nonumber\\ &&\times
\delta(\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime}-\vec{q}_{\|})
A_{x}(z''';\vec{q}_{\|}^{\,\prime\prime\prime})
A_{k}(z'';\vec{q}_{\|}^{\,\prime\prime})
A_{j}^{*}(z';\vec{q}_{\|}^{\,\prime})
d^2\kappa_{\|,\bar{m}}
d^2q_{\|}'''d^2q_{\|}''d^2q_{\|}'
\nonumber\\ &&\times
dz'''dz''dz'.
\label{eq:J3D}\end{aligned}$$ On the form of Eq. (\[eq:J3zq\]) we thus get part D of the conductivity tensor as $$\begin{aligned}
\lefteqn{
{\Xi}_{xjkx}^{\rm{D}}(z,z',z'',z''';\vec{q}_{\|},\vec{q}_{\|}^{\,\prime},
\vec{q}_{\|}^{\,\prime\prime},\vec{q}_{\|}^{\,\prime\prime\prime})=
{2{\rm{i}}\over\omega^3}{e^2\over4m_{e}\hbar^2}{1\over(2\pi)^2}\sum_{nmv}\int
{1\over\tilde{\omega}_{nm}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime}
,\vec{\kappa}_{\|})}
}\nonumber\\ &\quad&\times
\left\{
\left({f_{m}(\vec{\kappa}_{\|})-f_{v}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime})
\over\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime},
\vec{\kappa}_{\|})-\omega}
+{f_{n}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime})
-f_{v}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime},
\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime})+\omega}\right)
\right.\nonumber\\ &&\times
j_{j,v{}n}(z';2\vec{\kappa}_{\|}+2\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime})
j_{k,mv}(z'';2\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime})
\nonumber\\ &&
+\left({f_{m}(\vec{\kappa}_{\|})-f_{v}(\vec{\kappa}_{\|}-\vec{q}_{\|}^{\,\prime})
\over\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|}-\vec{q}_{\|}^{\,\prime},
\vec{\kappa}_{\|})+\omega}
+{f_{n}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime})
-f_{v}(\vec{\kappa}_{\|}-\vec{q}_{\|}^{\,\prime})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime},
\vec{\kappa}_{\|}-\vec{q}_{\|}^{\,\prime})-\omega}\right)
\nonumber\\ &&\times\left.\!
j_{k,v{}n}(z'';2\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime}-2\vec{q}_{\|}^{\,\prime})
j_{j,mv}(z';2\vec{\kappa}_{\|}-\vec{q}_{\|}^{\,\prime})
\right\}
\psi_{m}^{*}(z)\psi_{n}(z)
\delta(z-z''')
\nonumber\\ &&\times
\delta(\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime}-\vec{q}_{\|})
d^2\kappa_{\|},\end{aligned}$$ where we have omitted the now superfluous index $m$ from the surface states $\vec{\kappa}_{\|}$. Again, when looking for the DFWM response tensor we insert the pump fields defined in Eqs. (\[eq:Apump-1\])–(\[eq:Apump-2\]) and integrate over $q_{\|}'''$ and $q_{\|}''$ in Eq. (\[eq:J3D\]), again reducing the Dirac delta function accounting for conservation of pseudo-momentum to $\delta(\vec{q}_{\|}^{\,\prime}+\vec{q}_{\|})$. After integration over $q_{\|}'$ and separation according to Eq. (\[eq:J3-wzq\]), Eq. (\[eq:XiAD\]) appear as the PCDFWM conductivity tensor part D.
Nonlinear Process E
-------------------
Inserting Eq. (\[eq:XiAErz\]) into Eq. (\[eq:J3r\]), we take element $ijxx$ of part E of the nonlinear current density. In the result we insert the expressions for the wave function, the vector potential and the transition current density in the two-dimensional Fourier representation \[Eqs. (\[eq:eigenstate\]), (\[eq:Azr\]) and (\[eq:Jm->n||\]), respectively\]. Then by use of Eq. (\[eq:Jzq\]) we find $$\begin{aligned}
\lefteqn{
J_{ijxx}^{\rm{E}}(z;\vec{q}_{\|})=
-{e^2\over16m_{e}\hbar^2}
{2\over(2\pi)^{12}}
\idotsint
\sum_{nmv}
{1\over\tilde{\omega}_{nm}-\omega}\left\{
\left({f_{m}-f_{v}\over\tilde{\omega}_{v{}m}-2\omega}
+{f_{n}-f_{v}\over\tilde{\omega}_{nv}+\omega}\right)
\right.}\nonumber\\ &&\times
j_{j,v{}n}(z';\vec{\kappa}_{\|,\bar{v}}+\vec{\kappa}_{\|,\bar{n}})
\psi_{v}^{*}(z'')
\psi_{m}(z'')
e^{{\rm{i}}(\vec{\kappa}_{\|,\bar{v}}-\vec{\kappa}_{\|,\bar{n}})\cdot\vec{r}_{\|}^{\,\prime}}
e^{{\rm{i}}(\vec{\kappa}_{\|,\bar{m}}-\vec{\kappa}_{\|,\bar{v}})\cdot\vec{r}_{\|}^{\,\prime\prime}}
\nonumber\\ &&
+\left({f_{n}-f_{v}\over\tilde{\omega}_{nv}-2\omega}
+{f_{m}-f_{v}\over\tilde{\omega}_{v{}m}+\omega}\right)
j_{j,mv}(z';\vec{\kappa}_{\|,\bar{m}}+\vec{\kappa}_{\|,\bar{v}})
\psi_{n}^{*}(z'')
\psi_{v}(z'')
e^{{\rm{i}}(\vec{\kappa}_{\|,\bar{m}}-\vec{\kappa}_{\|,\bar{v}})\cdot\vec{r}_{\|}^{\,\prime}}
\nonumber\\ &&\times\left.\!
e^{{\rm{i}}(\vec{\kappa}_{\|,\bar{v}}-\vec{\kappa}_{\|,\bar{n}})\cdot\vec{r}_{\|}^{\,\prime\prime}}
\right\}
j_{i,nm}(z;\vec{\kappa}_{\|,\bar{n}}+\vec{\kappa}_{\|,\bar{m}})
\delta(\vec{r}\,''-\vec{r}\,''')
A_{x}(z''';\vec{q}_{\|}^{\,\prime\prime\prime})
A_{x}(z'';\vec{q}_{\|}^{\,\prime\prime})
A_{j}^{*}(z';\vec{q}_{\|}^{\,\prime})
\nonumber\\ &&\times
e^{{\rm{i}}(\vec{\kappa}_{\|,\bar{n}}-\vec{\kappa}_{\|,\bar{m}})\cdot\vec{r}_{\|}}
e^{{\rm{i}}\vec{q}_{\|}^{\,\prime\prime\prime}\cdot\vec{r}_{\|}^{\,\prime\prime\prime}}
e^{{\rm{i}}\vec{q}_{\|}^{\,\prime\prime}\cdot\vec{r}_{\|}^{\,\prime\prime}}
e^{-{\rm{i}}\vec{q}_{\|}^{\,\prime}\cdot\vec{r}_{\|}^{\,\prime}}
d^2q_{\|}'''d^2q_{\|}''d^2q_{\|}'
d^3r'''d^3r''d^3r'e^{-{\rm{i}}\vec{q}_{\|}\cdot\vec{r}_{\|}}d^2r_{\|}.\end{aligned}$$ In this equation, we first solve the integral $\int{}d^2r_{\|}'''$, and then the integrals $\int{}d^2r_{\|}''$, $\int{}d^2r_{\|}'$, and $\int{}d^2r_{\|}$, which together with replacement of the sums over the different $\vec{\kappa}_{\|}$ quantities with integrals yields the result $$\begin{aligned}
\lefteqn{
J_{ijxx}^{\rm{E}}(z;\vec{q}_{\|})=
-{e^2\over16m_{e}\hbar^2}
{2\over(2\pi)^6}
\idotsint
\sum_{nmv}
{1\over\tilde{\omega}_{nm}(\vec{\kappa}_{\|,\bar{n}},\vec{\kappa}_{\|,\bar{m}})-\omega}
\left\{\!\!\!
\left({f_{m}(\vec{\kappa}_{\|,\bar{m}})-f_{v}(\vec{\kappa}_{\|,\bar{v}})\over
\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|,\bar{v}},\vec{\kappa}_{\|,\bar{m}})-2\omega}
\right.\right.}\nonumber\\ &\quad&\left.\!
+{f_{n}(\vec{\kappa}_{\|,\bar{n}})-f_{v}(\vec{\kappa}_{\|,\bar{v}})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|,\bar{n}},\vec{\kappa}_{\|,\bar{v}})+\omega}
\right)
j_{j,v{}n}(z';\vec{\kappa}_{\|,\bar{v}}+\vec{\kappa}_{\|,\bar{n}})
\psi_{v}^{*}(z'')
\psi_{m}(z'')
\delta(\vec{\kappa}_{\|,\bar{v}}-\vec{\kappa}_{\|,\bar{n}}-\vec{q}_{\|}^{\,\prime})
\nonumber\\ &&\times
\delta(\vec{\kappa}_{\|,\bar{m}}-\vec{\kappa}_{\|,\bar{v}}
+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime})
+\left({f_{n}(\vec{\kappa}_{\|,\bar{n}})-f_{v}(\vec{\kappa}_{\|,\bar{v}})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|,\bar{n}},\vec{\kappa}_{\|,\bar{v}})-2\omega}
+{f_{m}(\vec{\kappa}_{\|,\bar{m}})-f_{v}(\vec{\kappa}_{\|,\bar{v}})\over
\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|,\bar{v}},\vec{\kappa}_{\|,\bar{m}})+\omega}
\right)
\nonumber\\ &&\times\left.\!
j_{j,mv}(z';\vec{\kappa}_{\|,\bar{m}}+\vec{\kappa}_{\|,\bar{v}})
\psi_{n}^{*}(z'')
\psi_{v}(z'')
\delta(\vec{\kappa}_{\|,\bar{m}}-\vec{\kappa}_{\|,\bar{v}}-\vec{q}_{\|}^{\,\prime})
\delta(\vec{\kappa}_{\|,\bar{v}}-\vec{\kappa}_{\|,\bar{n}}+\vec{q}_{\|}^{\,\prime\prime\prime}
+\vec{q}_{\|}^{\,\prime\prime})
\right\}
\nonumber\\ &&\times
j_{i,nm}(z;\vec{\kappa}_{\|,\bar{n}}+\vec{\kappa}_{\|,\bar{m}})
\delta(z''-z''')
A_{x}(z''';\vec{q}_{\|}^{\,\prime\prime\prime})
A_{x}(z'';\vec{q}_{\|}^{\,\prime\prime})
A_{j}^{*}(z';\vec{q}_{\|}^{\,\prime})
\nonumber\\ &&\times
\delta(\vec{\kappa}_{\|,\bar{n}}-\vec{\kappa}_{\|,\bar{m}}-\vec{q}_{\|})
d^2\kappa_{\|,\bar{n}}d^2\kappa_{\|,\bar{m}}d^2\kappa_{\|,\bar{v}}
d^2q_{\|}'''d^2q_{\|}''d^2q_{\|}'
dz'''dz''dz'.\end{aligned}$$ Of the three integrals over $\vec{\kappa}_{\|}$ quantities, we can solve two because of the coupling to the wavevectors introduced by the Dirac delta functions appearing. We aim at keeping the $\vec{\kappa}_{\|,\bar{m}}$ set, and thus we solve the integrals for the $v$ and $n$ sets (in that order). Solving for the $v$ set, we find that $\vec{\kappa}_{\|,\bar{v}}$ is replaced by $\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}$ in the first part of the sum and by $\vec{\kappa}_{\|,\bar{m}}-\vec{q}_{\|}^{\,\prime}$ in the second part of the sum, which then allows us to solve the $n$ set by replacing $\vec{\kappa}_{\|,\bar{n}}$ with $\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime}$ in general, giving $$\begin{aligned}
\lefteqn{
J_{ijxx}^{\rm{E}}(z;\vec{q}_{\|})=
-{e^2\over16m_{e}\hbar^2}
{2\over(2\pi)^6}
\idotsint
\sum_{nmv}
{1\over\tilde{\omega}_{nm}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}
-\vec{q}_{\|}^{\,\prime},\vec{\kappa}_{\|,\bar{m}})-\omega}
}\nonumber\\ &\quad&\times
\left\{
\left({f_{m}(\vec{\kappa}_{\|,\bar{m}})-f_{v}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}
+\vec{q}_{\|}^{\,\prime\prime})\over\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|,\bar{m}}
+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime},\vec{\kappa}_{\|,\bar{m}})-2\omega}
\right.\right.\nonumber\\ &&\left.\!
+{f_{n}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime})
-f_{v}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}
-\vec{q}_{\|}^{\,\prime},\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime})+\omega}
\right)
\nonumber\\ &&\times
j_{j,v{}n}(z';2\vec{\kappa}_{\|,\bar{m}}+2\vec{q}_{\|}^{\,\prime\prime\prime}+2\vec{q}_{\|}^{\,\prime\prime}
-\vec{q}_{\|}^{\,\prime})
\psi_{v}^{*}(z'')
\psi_{m}(z'')
\nonumber\\ &&
+\left({f_{n}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}
-\vec{q}_{\|}^{\,\prime})-f_{v}(\vec{\kappa}_{\|,\bar{m}}-\vec{q}_{\|}^{\,\prime})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}
-\vec{q}_{\|}^{\,\prime},\vec{\kappa}_{\|,\bar{m}}-\vec{q}_{\|}^{\,\prime})-2\omega}
+{f_{m}(\vec{\kappa}_{\|,\bar{m}})-f_{v}(\vec{\kappa}_{\|,\bar{m}}-\vec{q}_{\|}^{\,\prime})\over
\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|,\bar{m}}-\vec{q}_{\|}^{\,\prime},\vec{\kappa}_{\|,\bar{m}})
+\omega}\right)
\nonumber\\ &&\times\left.
j_{j,mv}(z';2\vec{\kappa}_{\|,\bar{m}}-\vec{q}_{\|}^{\,\prime})
\psi_{n}^{*}(z'')
\psi_{v}(z'')
\right\}
j_{i,nm}(z;2\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime})
\delta(z''-z''')
\nonumber\\ &&\times
A_{x}(z''';\vec{q}_{\|}^{\,\prime\prime\prime})
A_{x}(z'';\vec{q}_{\|}^{\,\prime\prime})
A_{j}^{*}(z';\vec{q}_{\|}^{\,\prime})
\delta(\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime}-\vec{q}_{\|})
d^2\kappa_{\|,\bar{m}}
d^2q_{\|}'''d^2q_{\|}''d^2q_{\|}'
\nonumber\\ &&\times
dz'''dz''dz'.
\label{eq:J3E}\end{aligned}$$ On the form of Eq. (\[eq:J3zq\]) we thus get part E of the conductivity tensor as $$\begin{aligned}
\lefteqn{
{\Xi}_{ijxx}^{\rm{E}}(z,z',z'',z''';\vec{q}_{\|},\vec{q}_{\|}^{\,\prime},
\vec{q}_{\|}^{\,\prime\prime},\vec{q}_{\|}^{\,\prime\prime\prime})=
}\nonumber\\ &\quad&
{2{\rm{i}}\over\omega^3}{e^2\over16m_{e}\hbar^2}{1\over(2\pi)^2}
\sum_{nmv}\int
{1\over\tilde{\omega}_{nm}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}
-\vec{q}_{\|}^{\,\prime},\vec{\kappa}_{\|})-\omega}
\nonumber\\ &&\times
\!\left\{\!\!
\left(\!{f_{m}(\vec{\kappa}_{\|})-f_{v}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}
+\vec{q}_{\|}^{\,\prime\prime})\over\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|}
+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime},\vec{\kappa}_{\|})-2\omega}
+{f_{n}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime})
-f_{v}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}
-\vec{q}_{\|}^{\,\prime},\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime})+\omega}
\!\right)
\right.\nonumber\\ &&\times
j_{j,v{}n}(z';2\vec{\kappa}_{\|}+2\vec{q}_{\|}^{\,\prime\prime\prime}+2\vec{q}_{\|}^{\,\prime\prime}
-\vec{q}_{\|}^{\,\prime})
\psi_{v}^{*}(z'')\psi_{m}(z'')
\nonumber\\ &&
+\left({f_{n}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}
-\vec{q}_{\|}^{\,\prime})-f_{v}(\vec{\kappa}_{\|}-\vec{q}_{\|}^{\,\prime})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}
-\vec{q}_{\|}^{\,\prime},\vec{\kappa}_{\|}-\vec{q}_{\|}^{\,\prime})-2\omega}
+{f_{m}(\vec{\kappa}_{\|})-f_{v}(\vec{\kappa}_{\|}-\vec{q}_{\|}^{\,\prime})\over
\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|}-\vec{q}_{\|}^{\,\prime},\vec{\kappa}_{\|})
+\omega}\right)
\nonumber\\ &&\times\left.\!
j_{j,mv}(z';2\vec{\kappa}_{\|}-\vec{q}_{\|}^{\,\prime})
\psi_{n}^{*}(z'')\psi_{v}(z'')
\right\}
j_{i,nm}(z;2\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime})
\delta(z''-z''')
\nonumber\\ &&\times
\delta(\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime}-\vec{q}_{\|})
d^2\kappa_{\|},\end{aligned}$$ where we have omitted the now superfluous index $m$ from the surface states $\vec{\kappa}_{\|}$. The PCDFWM response tensor part E we find by insertion of the pump fields defined by Eqs. (\[eq:Apump-1\]) and (\[eq:Apump-2\]) into Eq. (\[eq:J3E\]) and perform the integrals over $q_{\|}'''$ and $q_{\|}''$, followed by integration over $q_{\|}'$ because of the reduction in the Dirac delta function accounting for conservation of pseudo-momentum. After these operations, the resulting expression is separated in the form of Eq. (\[eq:J3-wzq\]), and Eq. (\[eq:XiAE\]) is obtained.
Nonlinear process F
-------------------
Inserting Eq. (\[eq:XiAFrz\]) into Eq. (\[eq:J3r\]), we take element $ixxh$ of part F of the nonlinear current density. In the result we insert the expressions for the wave function, the vector potential and the transition current density in the two-dimensional Fourier representation \[Eqs. (\[eq:eigenstate\]), (\[eq:Azr\]) and (\[eq:Jm->n||\]), respectively\]. Then by use of Eq. (\[eq:Jzq\]) we find $$\begin{aligned}
\lefteqn{
J_{ixxh}^{\rm{F}}(z;\vec{q}_{\|})=
-{e^2\over8m_{e}\hbar^2}
{2\over(2\pi)^{12}}
\idotsint
\sum_{nmv}
{1\over\tilde{\omega}_{nm}-\omega}\left\{
\left({f_{m}-f_{v}\over\tilde{\omega}_{v{}m}}
+{f_{n}-f_{v}\over\tilde{\omega}_{nv}-\omega}\right)
\right.}\nonumber\\ &&\times
j_{h,v{}n}(z'';\vec{\kappa}_{\|,\bar{v}}+\vec{\kappa}_{\|,\bar{n}})
\psi_{v}^{*}(z')
\psi_{m}(z')
e^{{\rm{i}}(\vec{\kappa}_{\|,\bar{v}}-\vec{\kappa}_{\|,\bar{n}})\cdot\vec{r}_{\|}^{\,\prime\prime}}
e^{{\rm{i}}(\vec{\kappa}_{\|,\bar{m}}-\vec{\kappa}_{\|,\bar{v}})\cdot\vec{r}_{\|}^{\,\prime}}
\nonumber\\ &&\left.\!
+\!\left(\!{f_{n}-f_{v}\over\tilde{\omega}_{nv}}
+{f_{m}-f_{v}\over\tilde{\omega}_{v{}m}-\omega}\right)
j_{h,mv}(z'';\vec{\kappa}_{\|,\bar{m}}+\vec{\kappa}_{\|,\bar{v}})
\psi_{n}^{*}(z')
\psi_{v}(z')
e^{{\rm{i}}(\vec{\kappa}_{\|,\bar{m}}-\vec{\kappa}_{\|,\bar{v}})\cdot\vec{r}_{\|}^{\,\prime\prime}}
e^{{\rm{i}}(\vec{\kappa}_{\|,\bar{v}}-\vec{\kappa}_{\|,\bar{n}})\cdot\vec{r}_{\|}^{\,\prime}}
\!\right\}\!
\nonumber\\ &&\times
j_{i,nm}(z;\vec{\kappa}_{\|,\bar{n}}+\vec{\kappa}_{\|,\bar{m}})
\delta(\vec{r}\,'-\vec{r}\,''')
A_{h}(z''';\vec{q}_{\|}^{\,\prime\prime\prime})
A_{x}(z'';\vec{q}_{\|}^{\,\prime\prime})
A_{x}^{*}(z';\vec{q}_{\|}^{\,\prime})
e^{{\rm{i}}(\vec{\kappa}_{\|,\bar{n}}-\vec{\kappa}_{\|,\bar{m}})\cdot\vec{r}_{\|}}
\nonumber\\ &&\times
e^{{\rm{i}}\vec{q}_{\|}^{\,\prime\prime\prime}\cdot\vec{r}_{\|}^{\,\prime\prime\prime}}
e^{{\rm{i}}\vec{q}_{\|}^{\,\prime\prime}\cdot\vec{r}_{\|}^{\,\prime\prime}}
e^{-{\rm{i}}\vec{q}_{\|}^{\,\prime}\cdot\vec{r}_{\|}^{\,\prime}}
d^2q_{\|}'''d^2q_{\|}''d^2q_{\|}'
d^3r'''d^3r''d^3r'e^{-{\rm{i}}\vec{q}_{\|}\cdot\vec{r}_{\|}}d^2r_{\|}.\end{aligned}$$ Solving in this equation first the integral $\int{}d^2r_{\|}'''$, and then the integrals $\int{}d^2r_{\|}''$, $\int{}d^2r_{\|}'$, and $\int{}d^2r_{\|}$, followed by a replacement of the sums over the various $\vec{\kappa}_{\|}$ quantities with integrals, we get $$\begin{aligned}
\lefteqn{
J_{ixxh}^{\rm{F}}(z;\vec{q}_{\|})=
-{e^2\over8m_{e}\hbar^2}
{2\over(2\pi)^6}
\idotsint
\sum_{nmv}
{1\over\tilde{\omega}_{nm}(\vec{\kappa}_{\|,\bar{n}},\vec{\kappa}_{\|,\bar{m}})-\omega}
\left\{
\left({f_{m}(\vec{\kappa}_{\|,\bar{m}})-f_{v}(\vec{\kappa}_{\|,\bar{v}})\over
\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|,\bar{v}},\vec{\kappa}_{\|,\bar{m}})}
\right.\right.}\nonumber\\ &\quad&\left.\!
+{f_{n}(\vec{\kappa}_{\|,\bar{n}})-f_{v}(\vec{\kappa}_{\|,\bar{v}})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|,\bar{n}},\vec{\kappa}_{\|,\bar{v}})-\omega}
\right)
j_{h,v{}n}(z'';\vec{\kappa}_{\|,\bar{v}}+\vec{\kappa}_{\|,\bar{n}})
\psi_{v}^{*}(z')
\psi_{m}(z')
\delta(\vec{\kappa}_{\|,\bar{m}}-\vec{\kappa}_{\|,\bar{v}}+\vec{q}_{\|}^{\,\prime\prime\prime}
-\vec{q}_{\|}^{\,\prime})
\nonumber\\ &&\times
\delta(\vec{\kappa}_{\|,\bar{v}}-\vec{\kappa}_{\|,\bar{n}}+\vec{q}_{\|}^{\,\prime\prime})
+\left({f_{n}(\vec{\kappa}_{\|,\bar{n}})-f_{v}(\vec{\kappa}_{\|,\bar{v}})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|,\bar{n}},\vec{\kappa}_{\|,\bar{v}})}
+{f_{m}(\vec{\kappa}_{\|,\bar{m}})-f_{v}(\vec{\kappa}_{\|,\bar{v}})\over
\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|,\bar{v}},\vec{\kappa}_{\|,\bar{m}})-\omega}
\right)
\nonumber\\ &&\times\left.
j_{h,mv}(z'';\vec{\kappa}_{\|,\bar{m}}+\vec{\kappa}_{\|,\bar{v}})
\psi_{n}^{*}(z')
\psi_{v}(z')
\delta(\vec{\kappa}_{\|,\bar{v}}-\vec{\kappa}_{\|,\bar{n}}+\vec{q}_{\|}^{\,\prime\prime\prime}
-\vec{q}_{\|}^{\,\prime})
\delta(\vec{\kappa}_{\|,\bar{m}}-\vec{\kappa}_{\|,\bar{v}}+\vec{q}_{\|}^{\,\prime\prime})
\right\}
\nonumber\\ &&\times
j_{i,nm}(z;\vec{\kappa}_{\|,\bar{n}}+\vec{\kappa}_{\|,\bar{m}})
\delta(z'-z''')
A_{h}(z''';\vec{q}_{\|}^{\,\prime\prime\prime})
A_{x}(z'';\vec{q}_{\|}^{\,\prime\prime})
A_{x}^{*}(z';\vec{q}_{\|}^{\,\prime})
\nonumber\\ &&\times
\delta(\vec{\kappa}_{\|,\bar{n}}-\vec{\kappa}_{\|,\bar{m}}-\vec{q}_{\|})
d^2\kappa_{\|,\bar{n}}d^2\kappa_{\|,\bar{m}}d^2\kappa_{\|,\bar{v}}
d^2q_{\|}'''d^2q_{\|}''d^2q_{\|}'
dz'''dz''dz'.\end{aligned}$$ Of the three integrals over $\vec{\kappa}_{\|}$ quantities, we can solve two because of the coupling to the wavevectors introduced by the Dirac delta functions appearing. We aim at keeping the $\vec{\kappa}_{\|,\bar{m}}$ set, and thus we solve the integrals for the $v$ and $n$ sets (in that order). Solving for the $v$ set, we find that $\vec{\kappa}_{\|,\bar{v}}$ is replaced by $\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|}^{\,\prime}$ in the first part of the sum and by $\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime}$ in the second part of the sum, which afterwards allows us to solve the $n$ set by replacing $\vec{\kappa}_{\|,\bar{n}}$ with $\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime}$ in general, giving $$\begin{aligned}
\lefteqn{
J_{ixxh}^{\rm{F}}(z;\vec{q}_{\|})=
-{e^2\over8m_{e}\hbar^2}
{2\over(2\pi)^6}
\idotsint
\sum_{nmv}
{1\over\tilde{\omega}_{nm}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}
-\vec{q}_{\|}^{\,\prime},\vec{\kappa}_{\|,\bar{m}})-\omega}
}\nonumber\\ &\quad&\times
\left\{
\left({f_{m}(\vec{\kappa}_{\|,\bar{m}})-f_{v}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}
-\vec{q}_{\|}^{\,\prime})\over\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|,\bar{m}}
+\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|}^{\,\prime},\vec{\kappa}_{\|,\bar{m}})}
\right.\right.\nonumber\\ &&\left.\!
+{f_{n}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime})
-f_{v}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|}^{\,\prime})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}
-\vec{q}_{\|}^{\,\prime},\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|}^{\,\prime})-\omega}
\right)
\nonumber\\ &&\times
j_{h,v{}n}(z'';2\vec{\kappa}_{\|,\bar{m}}+2\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}
-\vec{q}_{\|}^{\,\prime})
\psi_{v}^{*}(z')
\psi_{m}(z')
\nonumber\\ &&
+\left({f_{n}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}
-\vec{q}_{\|}^{\,\prime})-f_{v}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}
-\vec{q}_{\|}^{\,\prime},\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime})}
+{f_{m}(\vec{\kappa}_{\|,\bar{m}})-f_{v}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime})\over
\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime},
\vec{\kappa}_{\|,\bar{m}})-\omega}
\right)
\nonumber\\ &&\times\left.\!
j_{h,mv}(z'';2\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime})
\psi_{n}^{*}(z')
\psi_{v}(z')
\right\}
j_{i,nm}(z;2\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime})
\delta(z'-z''')
\nonumber\\ &&\times
A_{h}(z''';\vec{q}_{\|}^{\,\prime\prime\prime})
A_{x}(z'';\vec{q}_{\|}^{\,\prime\prime})
A_{x}^{*}(z';\vec{q}_{\|}^{\,\prime})
\delta(\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime}-\vec{q}_{\|})
d^2\kappa_{\|,\bar{m}}
d^2q_{\|}'''d^2q_{\|}''d^2q_{\|}'
\nonumber\\ &&\times
dz'''dz''dz'.
\label{eq:J3F}\end{aligned}$$ On the form of Eq. (\[eq:J3zq\]) we thus get part F of the conductivity tensor as $$\begin{aligned}
\lefteqn{
{\Xi}_{ixxh}^{\rm{F}}(z,z',z'',z''';\vec{q}_{\|},\vec{q}_{\|}^{\,\prime},
\vec{q}_{\|}^{\,\prime\prime},\vec{q}_{\|}^{\,\prime\prime\prime})=
}\nonumber\\ &\quad&
{2{\rm{i}}\over\omega^3}{e^2\over8m_{e}\hbar^2}{1\over(2\pi)^2}\sum_{nmv}\int
{1\over\tilde{\omega}_{nm}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}
-\vec{q}_{\|}^{\,\prime},\vec{\kappa}_{\|})-\omega}
\nonumber\\ &&\times
\left\{
\left({f_{m}(\vec{\kappa}_{\|})-f_{v}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}
-\vec{q}_{\|}^{\,\prime})\over\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|}
+\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|}^{\,\prime},\vec{\kappa}_{\|})}
+{f_{n}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime})
-f_{v}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|}^{\,\prime})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}
-\vec{q}_{\|}^{\,\prime},\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|}^{\,\prime})-\omega}
\right)
\right.\nonumber\\ &&\times
j_{h,v{}n}(z'';2\vec{\kappa}_{\|}+2\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}
-\vec{q}_{\|}^{\,\prime})
\psi_{v}^{*}(z')\psi_{m}(z')
\nonumber\\ &&
+\left({f_{n}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}
-\vec{q}_{\|}^{\,\prime})-f_{v}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}
-\vec{q}_{\|}^{\,\prime},\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime})}
+{f_{m}(\vec{\kappa}_{\|})-f_{v}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime})\over
\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime},
\vec{\kappa}_{\|})-\omega}
\right)
\nonumber\\ &&\times\left.\!
j_{h,mv}(z'';2\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime})
\psi_{n}^{*}(z')\psi_{v}(z')
\right\}
j_{i,nm}(z;2\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime})
\delta(z'-z''')
\nonumber\\ &&\times
\delta(\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime}-\vec{q}_{\|})
d^2\kappa_{\|},\end{aligned}$$ where we have omitted the now superfluous index $m$ from the surface states $\vec{\kappa}_{\|}$. Inserting the DFWM pump fields defined by Eqs. (\[eq:Apump-1\]) and (\[eq:Apump-2\]) into Eq. (\[eq:J3F\]), the integrals over $q_{\|}'''$ and $q_{\|}''$ can be solved. The resulting expression can then be solved for $q_{\|}'$ for the same reason as before, and on the form of Eq. (\[eq:J3-wzq\]), the PCDFWM conductivity tensor part F appears as Eq. (\[eq:XiAF\]).
Nonlinear process G
-------------------
Inserting Eq. (\[eq:XiAGrz\]) into Eq. (\[eq:J3r\]), we take element $ijkh$ of part G of the nonlinear current density. In the result we insert the expressions for the wave function, the vector potential and the transition current density in the two-dimensional Fourier representation \[Eqs. (\[eq:eigenstate\]), (\[eq:Azr\]) and (\[eq:Jm->n||\]), respectively\]. Then by use of Eq. (\[eq:Jzq\]) we find $$\begin{aligned}
\lefteqn{
J_{ijkh}^{\rm{G}}(z;\vec{q}_{\|})=
-{1\over8\hbar^3}
{2\over(2\pi)^{14}}
\idotsint
\sum_{nmvl}
{1\over\tilde{\omega}_{nm}-\omega}
\left\{
\left[
\left({f_{l}-f_{m}\over\tilde{\omega}_{l{}m}-\omega}
+{f_{l}-f_{v}\over\tilde{\omega}_{vl}-\omega}\right)
{1\over\tilde{\omega}_{v{}m}-2\omega}
\right.\right.}\nonumber\\ &\quad&\left.\!
+\left({f_{l}-f_{v}\over\tilde{\omega}_{vl}-\omega}
+{f_{n}-f_{v}\over\tilde{\omega}_{nv}+\omega}\right)
{1\over\tilde{\omega}_{nl}}
\right]
j_{h,ml}(z''';\vec{\kappa}_{\|,\bar{m}}+\vec{\kappa}_{\|,\bar{l}})
j_{k,lv}(z'';\vec{\kappa}_{\|,\bar{l}}+\vec{\kappa}_{\|,\bar{v}})
\nonumber\\ &&\times
j_{j,v{}n}(z';\vec{\kappa}_{\|,\bar{v}}+\vec{\kappa}_{\|,\bar{n}})
e^{{\rm{i}}(\vec{\kappa}_{\|,\bar{m}}-\vec{\kappa}_{\|,\bar{l}})\cdot\vec{r}_{\|}^{\,\prime\prime\prime}}
e^{{\rm{i}}(\vec{\kappa}_{\|,\bar{l}}-\vec{\kappa}_{\|,\bar{v}})\cdot\vec{r}_{\|}^{\,\prime\prime}}
e^{{\rm{i}}(\vec{\kappa}_{\|,\bar{v}}-\vec{\kappa}_{\|,\bar{n}})\cdot\vec{r}_{\|}^{\,\prime}}
\nonumber\\ &&
+
\left[
\left({f_{l}-f_{m}\over\tilde{\omega}_{l{}m}-\omega}
+{f_{l}-f_{v}\over\tilde{\omega}_{vl}+\omega}\right)
{1\over\tilde{\omega}_{v{}m}}
+\left({f_{l}-f_{v}\over\tilde{\omega}_{vl}+\omega}
+{f_{n}-f_{v}\over\tilde{\omega}_{nv}-\omega}\right)
{1\over\tilde{\omega}_{nl}}
\right]
\nonumber\\ &&\times
j_{h,ml}(z''';\vec{\kappa}_{\|,\bar{m}}+\vec{\kappa}_{\|,\bar{l}})
j_{k,v{}n}(z'';\vec{\kappa}_{\|,\bar{v}}+\vec{\kappa}_{\|,\bar{n}})
j_{j,lv}(z';\vec{\kappa}_{\|,\bar{l}}+\vec{\kappa}_{\|,\bar{v}})
e^{{\rm{i}}(\vec{\kappa}_{\|,\bar{m}}-\vec{\kappa}_{\|,\bar{l}})\cdot\vec{r}_{\|}^{\,\prime\prime\prime}}
\nonumber\\ &&\times
e^{{\rm{i}}(\vec{\kappa}_{\|,\bar{v}}-\vec{\kappa}_{\|,\bar{n}})\cdot\vec{r}_{\|}^{\,\prime\prime}}
e^{{\rm{i}}(\vec{\kappa}_{\|,\bar{l}}-\vec{\kappa}_{\|,\bar{v}})\cdot\vec{r}_{\|}^{\,\prime}}
+
\left[
\left({f_{l}-f_{m}\over\tilde{\omega}_{l{}m}+\omega}
+{f_{l}-f_{v}\over\tilde{\omega}_{vl}-\omega}\right)
{1\over\tilde{\omega}_{v{}m}}
\right.\nonumber\\ &&\left.\!
+\left({f_{l}-f_{v}\over\tilde{\omega}_{vl}-\omega}
+{f_{n}-f_{v}\over\tilde{\omega}_{nv}-\omega}\right)
{1\over\tilde{\omega}_{nl}-2\omega}
\right]
j_{h,lv}(z''';\vec{\kappa}_{\|,\bar{l}}+\vec{\kappa}_{\|,\bar{v}})
j_{k,v{}n}(z'';\vec{\kappa}_{\|,\bar{v}}+\vec{\kappa}_{\|,\bar{n}})
\nonumber\\ &&\times\left.\!
j_{j,ml}(z';\vec{\kappa}_{\|,\bar{m}}+\vec{\kappa}_{\|,\bar{l}})
e^{{\rm{i}}(\vec{\kappa}_{\|,\bar{l}}-\vec{\kappa}_{\|,\bar{v}})\cdot\vec{r}_{\|}^{\,\prime\prime\prime}}
e^{{\rm{i}}(\vec{\kappa}_{\|,\bar{v}}-\vec{\kappa}_{\|,\bar{n}})\cdot\vec{r}_{\|}^{\,\prime\prime}}
e^{{\rm{i}}(\vec{\kappa}_{\|,\bar{m}}-\vec{\kappa}_{\|,\bar{l}})\cdot\vec{r}_{\|}^{\,\prime}}
\right\}
\nonumber\\ &&\times
j_{i,nm}(z;\vec{\kappa}_{\|,\bar{n}}+\vec{\kappa}_{\|,\bar{m}})
A_{h}(z''';\vec{q}_{\|}^{\,\prime\prime\prime})
A_{k}(z'';\vec{q}_{\|}^{\,\prime\prime})
A_{j}^{*}(z';\vec{q}_{\|}^{\,\prime})
e^{{\rm{i}}(\vec{\kappa}_{\|,\bar{n}}-\vec{\kappa}_{\|,\bar{m}})\cdot\vec{r}_{\|}}
\nonumber\\ &&\times
e^{{\rm{i}}\vec{q}_{\|}^{\,\prime\prime\prime}\cdot\vec{r}_{\|}^{\,\prime\prime\prime}}
e^{{\rm{i}}\vec{q}_{\|}^{\,\prime\prime}\cdot\vec{r}_{\|}^{\,\prime\prime}}
e^{-{\rm{i}}\vec{q}_{\|}^{\,\prime}\cdot\vec{r}_{\|}^{\,\prime}}
d^2q_{\|}'''d^2q_{\|}''d^2q_{\|}'
d^3r'''d^3r''d^3r'e^{-{\rm{i}}\vec{q}_{\|}\cdot\vec{r}_{\|}}d^2r_{\|}.\end{aligned}$$ In the above equation, we may immediately solve the integrals $\int{}d^2r_{\|}'''$, $\int{}d^2r_{\|}''$, $\int{}d^2r_{\|}'$, and $\int{}d^2r_{\|}$, and by replacing the sums over the various $\vec{\kappa}_{\|}$ quantities with integrals, as before, we get $$\begin{aligned}
\lefteqn{
J_{ijkh}^{G}(z;\vec{q}_{\|})=
-{1\over8\hbar^3}{2\over(2\pi)^6}
\idotsint\sum_{nmvl}
{1\over\tilde{\omega}_{nm}(\vec{\kappa}_{\|,\bar{n}},
\vec{\kappa}_{\|,\bar{m}})-\omega}
}\nonumber\\ &&\times
\left\{
\left[
\left({f_{l}(\vec{\kappa}_{\|,\bar{l}})-f_{m}(\vec{\kappa}_{\|,\bar{m}})
\over\tilde{\omega}_{l{}m}(\vec{\kappa}_{\|,\bar{l}},
\vec{\kappa}_{\|,\bar{m}})-\omega}
+{f_{l}(\vec{\kappa}_{\|,\bar{l}})-f_{v}(\vec{\kappa}_{\|,\bar{v}})
\over\tilde{\omega}_{vl}(\vec{\kappa}_{\|,\bar{v}},
\vec{\kappa}_{\|,\bar{l}})-\omega}
\right)
{1\over\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|,\bar{v}},
\vec{\kappa}_{\|,\bar{m}})-2\omega}
\right.\right.\nonumber\\ &\quad&\left.\!
+\left({f_{l}(\vec{\kappa}_{\|,\bar{l}})
-f_{v}(\vec{\kappa}_{\|,\bar{v}})\over
\tilde{\omega}_{vl}(\vec{\kappa}_{\|,\bar{v}},
\vec{\kappa}_{\|,\bar{l}})-\omega}
+{f_{n}(\vec{\kappa}_{\|,\bar{n}})-f_{v}(\vec{\kappa}_{\|,\bar{v}})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|,\bar{n}},\vec{\kappa}_{\|,\bar{v}})
+\omega}
\right)
{1\over\tilde{\omega}_{nl}(\vec{\kappa}_{\|,\bar{n}},
\vec{\kappa}_{\|,\bar{l}})}
\right]
\nonumber\\ &&\times
j_{h,ml}(z''';\vec{\kappa}_{\|,\bar{m}}+\vec{\kappa}_{\|,\bar{l}})
j_{k,lv}(z'';\vec{\kappa}_{\|,\bar{l}}+\vec{\kappa}_{\|,\bar{v}})
j_{j,v{}n}(z';\vec{\kappa}_{\|,\bar{v}}+\vec{\kappa}_{\|,\bar{n}})
\nonumber\\ &&\times
\delta(\vec{\kappa}_{\|,\bar{m}}-\vec{\kappa}_{\|,\bar{l}}+\vec{q}_{\|}^{\,\prime\prime\prime})
\delta(\vec{\kappa}_{\|,\bar{l}}-\vec{\kappa}_{\|,\bar{v}}
+\vec{q}_{\|}^{\,\prime\prime})
\delta(\vec{\kappa}_{\|,\bar{v}}-\vec{\kappa}_{\|,\bar{n}}-\vec{q}_{\|}^{\,\prime})
\nonumber\\ &&
+
\left[
\left({f_{l}(\vec{\kappa}_{\|,\bar{l}})-f_{m}(\vec{\kappa}_{\|,\bar{m}})
\over\tilde{\omega}_{l{}m}(\vec{\kappa}_{\|,\bar{l}},
\vec{\kappa}_{\|,\bar{m}})-\omega}
+{f_{l}(\vec{\kappa}_{\|,\bar{l}})-f_{v}(\vec{\kappa}_{\|,\bar{v}})
\over\tilde{\omega}_{vl}(\vec{\kappa}_{\|,\bar{v}},
\vec{\kappa}_{\|,\bar{l}})+\omega}
\right)
{1\over\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|,\bar{v}},
\vec{\kappa}_{\|,\bar{m}})}
\right.\nonumber\\ &&\left.
+\left({f_{l}(\vec{\kappa}_{\|,\bar{l}})
-f_{v}(\vec{\kappa}_{\|,\bar{v}})\over
\tilde{\omega}_{vl}(\vec{\kappa}_{\|,\bar{v}},
\vec{\kappa}_{\|,\bar{l}})+\omega}
+{f_{n}(\vec{\kappa}_{\|,\bar{n}})-f_{v}(\vec{\kappa}_{\|,\bar{v}})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|,\bar{n}},\vec{\kappa}_{\|,\bar{v}})
-\omega}
\right)
{1\over\tilde{\omega}_{nl}(\vec{\kappa}_{\|,\bar{n}},
\vec{\kappa}_{\|,\bar{l}})}
\right]
\nonumber\\ &&\times
j_{h,ml}(z''';\vec{\kappa}_{\|,\bar{m}}+\vec{\kappa}_{\|,\bar{l}})
j_{k,v{}n}(z'';\vec{\kappa}_{\|,\bar{v}}+\vec{\kappa}_{\|,\bar{n}})
j_{j,lv}(z';\vec{\kappa}_{\|,\bar{l}}+\vec{\kappa}_{\|,\bar{v}})
\nonumber\\ &&\times
\delta(\vec{\kappa}_{\|,\bar{m}}-\vec{\kappa}_{\|,\bar{l}}+\vec{q}_{\|}^{\,\prime\prime\prime})
\delta(\vec{\kappa}_{\|,\bar{v}}-\vec{\kappa}_{\|,\bar{n}}+\vec{q}_{\|}^{\,\prime\prime})
\delta(\vec{\kappa}_{\|,\bar{l}}-\vec{\kappa}_{\|,\bar{v}}-\vec{q}_{\|}^{\,\prime})
\nonumber\\ &&
+
\left[
\left({f_{l}(\vec{\kappa}_{\|,\bar{l}})-f_{m}(\vec{\kappa}_{\|,\bar{m}})
\over\tilde{\omega}_{l{}m}(\vec{\kappa}_{\|,\bar{l}},
\vec{\kappa}_{\|,\bar{m}})+\omega}
+{f_{l}(\vec{\kappa}_{\|,\bar{l}})-f_{v}(\vec{\kappa}_{\|,\bar{v}})
\over\tilde{\omega}_{vl}(\vec{\kappa}_{\|,\bar{v}},
\vec{\kappa}_{\|,\bar{l}})-\omega}
\right)
{1\over\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|,\bar{v}},
\vec{\kappa}_{\|,\bar{m}})}
\right.\nonumber\\ &&\left.
+\left({f_{l}(\vec{\kappa}_{\|,\bar{l}})
-f_{v}(\vec{\kappa}_{\|,\bar{v}})\over
\tilde{\omega}_{vl}(\vec{\kappa}_{\|,\bar{v}},
\vec{\kappa}_{\|,\bar{l}})-\omega}
+{f_{n}(\vec{\kappa}_{\|,\bar{n}})-f_{v}(\vec{\kappa}_{\|,\bar{v}})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|,\bar{n}},\vec{\kappa}_{\|,\bar{v}})
-\omega}
\right)
{1\over\tilde{\omega}_{nl}(\vec{\kappa}_{\|,\bar{n}},
\vec{\kappa}_{\|,\bar{l}})-2\omega}
\right]
\nonumber\\ &&\times
j_{h,lv}(z''';\vec{\kappa}_{\|,\bar{l}}+\vec{\kappa}_{\|,\bar{v}})
j_{k,v{}n}(z'';\vec{\kappa}_{\|,\bar{v}}+\vec{\kappa}_{\|,\bar{n}})
j_{j,ml}(z';\vec{\kappa}_{\|,\bar{m}}+\vec{\kappa}_{\|,\bar{l}})
\nonumber\\ &&\times\left.
\delta(\vec{\kappa}_{\|,\bar{l}}-\vec{\kappa}_{\|,\bar{v}}+\vec{q}_{\|}^{\,\prime\prime\prime})
\delta(\vec{\kappa}_{\|,\bar{v}}-\vec{\kappa}_{\|,\bar{n}}+\vec{q}_{\|}^{\,\prime\prime})
\delta(\vec{\kappa}_{\|,\bar{m}}-\vec{\kappa}_{\|,\bar{l}}-\vec{q}_{\|}^{\,\prime})
\right\}
j_{i,nm}(z;\vec{\kappa}_{\|,\bar{n}}+\vec{\kappa}_{\|,\bar{m}})
\nonumber\\ &&\times
A_{h}(z''';\vec{q}_{\|}^{\,\prime\prime\prime})
A_{k}(z'';\vec{q}_{\|}^{\,\prime\prime})
A_{j}^{*}(z';\vec{q}_{\|}^{\,\prime})
\delta(\vec{\kappa}_{\|,\bar{n}}-\vec{\kappa}_{\|,\bar{m}}-\vec{q}_{\|})
d^2\kappa_{\|,\bar{n}}d^2\kappa_{\|,\bar{m}}d^2\kappa_{\|,\bar{v}}d^2\kappa_{\|,\bar{l}}
\nonumber\\ &&\times
d^2q_{\|}'''d^2q_{\|}''d^2q_{\|}'
dz'''dz''dz'.\end{aligned}$$ Of the four integrals over $\vec{\kappa}_{\|}$ quantities, we can solve three because of the coupling to the wavevectors introduced by the Dirac delta functions appearing. We aim at keeping the $\vec{\kappa}_{\|,\bar{m}}$ set, and thus we solve the integrals for the $l$, $v$, and $n$ sets. Thus (i), in the first part of the sum, we find that $\vec{\kappa}_{\|,\bar{l}}$ can be replaced by $\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}$, then letting us replace $\vec{\kappa}_{\|,\bar{v}}$ by $\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}$, which again let us replace $\vec{\kappa}_{\|,\bar{n}}$ by $\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime}$. (ii) In the second part of the sum, we find that $\vec{\kappa}_{\|,\bar{l}}$ can be replaced by $\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}$, then letting us replace $\vec{\kappa}_{\|,\bar{v}}$ by $\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|}^{\,\prime}$, which again let us replace $\vec{\kappa}_{\|,\bar{n}}$ by $\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime}$. (iii) In the third part of the sum, we find that $\vec{\kappa}_{\|,\bar{l}}$ can be replaced by $\vec{\kappa}_{\|,\bar{m}}-\vec{q}_{\|}^{\,\prime}$, then letting us replace $\vec{\kappa}_{\|,\bar{v}}$ by $\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|}^{\,\prime}$, which again let us replace $\vec{\kappa}_{\|,\bar{n}}$ by $\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime}$. Finally (iv), we observe that the substitution of $\vec{\kappa}_{\|,\bar{n}}$ by $\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime}$ is global, and we thus get the resulting current density element $$\begin{aligned}
\lefteqn{
J_{ijkh}^{\rm{G}}(z;\vec{q}_{\|})=
-{1\over8\hbar^3}{2\over(2\pi)^6}
\idotsint\sum_{nmvl}
{1\over\tilde{\omega}_{nm}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}
-\vec{q}_{\|}^{\,\prime},\vec{\kappa}_{\|,\bar{m}})-\omega}
}\nonumber\\ &\quad&\times
\left\{
\left[
\left({f_{l}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime})
-f_{m}(\vec{\kappa}_{\|,\bar{m}})\over\tilde{\omega}_{l{}m}(\vec{\kappa}_{\|,\bar{m}}
+\vec{q}_{\|}^{\,\prime\prime\prime},\vec{\kappa}_{\|,\bar{m}})-\omega}
+{f_{l}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime})-f_{v}(\vec{\kappa}_{\|,\bar{m}}
+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime})\over
\tilde{\omega}_{vl}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime},
\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime})-\omega}\right)
\right.\right.\nonumber\\ &&\times
{1\over\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}
+\vec{q}_{\|}^{\,\prime\prime},\vec{\kappa}_{\|,\bar{m}})-2\omega}
+\left({f_{l}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime})
-f_{v}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime})\over
\tilde{\omega}_{vl}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime},
\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime})-\omega}
\right.\nonumber\\ &&\left.\!
+{f_{n}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime})
-f_{v}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}
-\vec{q}_{\|}^{\,\prime},\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime})+\omega}
\right)
\nonumber\\ &&\times\left.\!
{1\over\tilde{\omega}_{nl}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}
+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime},\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime})}
\right]
j_{h,ml}(z''';2\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime})
\nonumber\\ &&\times
j_{k,lv}(z'';2\vec{\kappa}_{\|,\bar{m}}+2\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime})
j_{j,v{}n}(z';2\vec{\kappa}_{\|,\bar{m}}+2\vec{q}_{\|}^{\,\prime\prime\prime}+2\vec{q}_{\|}^{\,\prime\prime}
-\vec{q}_{\|}^{\,\prime})
\nonumber\\ &&
+
\left[
\left({f_{l}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime})
-f_{m}(\vec{\kappa}_{\|,\bar{m}})\over\tilde{\omega}_{l{}m}(\vec{\kappa}_{\|,\bar{m}}
+\vec{q}_{\|}^{\,\prime\prime\prime},\vec{\kappa}_{\|,\bar{m}})-\omega}
+{f_{l}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime})-f_{v}(\vec{\kappa}_{\|,\bar{m}}
+\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|}^{\,\prime})\over
\tilde{\omega}_{vl}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|}^{\,\prime},
\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime})+\omega}\right)
\right.\nonumber\\ &&\times
{1\over\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}
-\vec{q}_{\|}^{\,\prime},\vec{\kappa}_{\|,\bar{m}})}
+\left({f_{l}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime})
-f_{v}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|}^{\,\prime})\over
\tilde{\omega}_{vl}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|}^{\,\prime},
\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime})+\omega}
\right.\nonumber\\ &&\left.\!
+{f_{n}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime})
-f_{v}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|}^{\,\prime})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}
-\vec{q}_{\|}^{\,\prime},\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|}^{\,\prime})-\omega}
\right)
\nonumber\\ &&\times\left.
{1\over\tilde{\omega}_{nl}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}
+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime},\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime})}
\right]
j_{h,ml}(z''';2\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime})
\nonumber\\ &&\times
j_{k,v{}n}(z'';2\vec{\kappa}_{\|,\bar{m}}+2\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}
-2\vec{q}_{\|}^{\,\prime})
j_{j,lv}(z';2\vec{\kappa}_{\|,\bar{m}}+2\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|}^{\,\prime})
\nonumber\\ &&
+
\left[
\left({f_{l}(\vec{\kappa}_{\|,\bar{m}}-\vec{q}_{\|}^{\,\prime})-f_{m}(\vec{\kappa}_{\|,\bar{m}})
\over\tilde{\omega}_{l{}m}(\vec{\kappa}_{\|,\bar{m}}-\vec{q}_{\|}^{\,\prime},
\vec{\kappa}_{\|,\bar{m}})+\omega}
+{f_{l}(\vec{\kappa}_{\|,\bar{m}}-\vec{q}_{\|}^{\,\prime})-f_{v}(\vec{\kappa}_{\|,\bar{m}}
+\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|}^{\,\prime})\over
\tilde{\omega}_{vl}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|}^{\,\prime},
\vec{\kappa}_{\|,\bar{m}}-\vec{q}_{\|}^{\,\prime})-\omega}\right)
\right.\nonumber\\ &&\times
{1\over\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}
-\vec{q}_{\|}^{\,\prime},\vec{\kappa}_{\|,\bar{m}})}
+\left({f_{l}(\vec{\kappa}_{\|,\bar{m}}-\vec{q}_{\|}^{\,\prime})
-f_{v}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|}^{\,\prime})\over
\tilde{\omega}_{vl}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|}^{\,\prime},
\vec{\kappa}_{\|,\bar{m}}-\vec{q}_{\|}^{\,\prime})-\omega}
\right.\nonumber\\ &&\left.\!
+{f_{n}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime})
-f_{v}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|}^{\,\prime})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}
-\vec{q}_{\|}^{\,\prime},\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|}^{\,\prime})-\omega}
\right)
\nonumber\\ &&\times\left.\!
{1\over\tilde{\omega}_{nl}(\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}
+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime},\vec{\kappa}_{\|,\bar{m}}-\vec{q}_{\|}^{\,\prime})-2\omega}
\right]
j_{h,lv}(z''';2\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}-2\vec{q}_{\|}^{\,\prime})
\nonumber\\ &&\times\left.\!
j_{k,v{}n}(z'';2\vec{\kappa}_{\|,\bar{m}}+2\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}
-2\vec{q}_{\|}^{\,\prime})
j_{j,ml}(z';2\vec{\kappa}_{\|,\bar{m}}-\vec{q}_{\|}^{\,\prime})
\right\}
\nonumber\\ &&\times
j_{i,nm}(z;2\vec{\kappa}_{\|,\bar{m}}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime})
\delta(\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime}-\vec{q}_{\|})
A_{h}(z''';\vec{q}_{\|}^{\,\prime\prime\prime})
A_{k}(z'';\vec{q}_{\|}^{\,\prime\prime})
\nonumber\\ &&\times
A_{j}^{*}(z';\vec{q}_{\|}^{\,\prime})
d^2\kappa_{\|,\bar{m}}
d^2q_{\|}'''d^2q_{\|}''d^2q_{\|}'
dz'''dz''dz'.
\label{eq:J3G}\end{aligned}$$ On the form of Eq. (\[eq:J3zq\]) we thus get part G of the conductivity tensor as $$\begin{aligned}
\lefteqn{
{\Xi}_{ijkh}^{\rm{G}}(z,z',z'',z''';\vec{q}_{\|},\vec{q}_{\|}^{\,\prime},
\vec{q}_{\|}^{\,\prime\prime},\vec{q}_{\|}^{\,\prime\prime\prime})=
{2{\rm{i}}\over\omega^3}{1\over(2\pi)^2}{1\over8\hbar^3}
}\nonumber\\ &\quad&\times
\sum_{nmvl}\int
{1\over\tilde{\omega}_{nm}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}
-\vec{q}_{\|}^{\,\prime},\vec{\kappa}_{\|})-\omega}
\left\{
\left[
{1\over\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}
+\vec{q}_{\|}^{\,\prime\prime},\vec{\kappa}_{\|})-2\omega}
\right.\right.\nonumber\\ &&\times
\left({f_{l}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime})
-f_{m}(\vec{\kappa}_{\|})\over\tilde{\omega}_{l{}m}(\vec{\kappa}_{\|}
+\vec{q}_{\|}^{\,\prime\prime\prime},\vec{\kappa}_{\|})-\omega}
+{f_{l}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime})-f_{v}(\vec{\kappa}_{\|}
+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime})\over
\tilde{\omega}_{vl}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime},
\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime})-\omega}\right)
\nonumber\\ &&
+
{1\over\tilde{\omega}_{nl}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}
+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime},\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime})}
\left({f_{l}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime})
-f_{v}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime})\over
\tilde{\omega}_{vl}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime},
\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime})-\omega}
\right.\nonumber\\ &&\left.\!\left.\!
+{f_{n}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime})
-f_{v}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}
-\vec{q}_{\|}^{\,\prime},\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime})+\omega}
\right)
\right]
j_{h,ml}(z''';2\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime})
\nonumber\\ &&\times
j_{k,lv}(z'';2\vec{\kappa}_{\|}+2\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime})
j_{j,v{}n}(z';2\vec{\kappa}_{\|}+2\vec{q}_{\|}^{\,\prime\prime\prime}+2\vec{q}_{\|}^{\,\prime\prime}
-\vec{q}_{\|}^{\,\prime})
\nonumber\\ &&
+
\left[
\left({f_{l}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime})
-f_{m}(\vec{\kappa}_{\|})\over\tilde{\omega}_{l{}m}(\vec{\kappa}_{\|}
+\vec{q}_{\|}^{\,\prime\prime\prime},\vec{\kappa}_{\|})-\omega}
+{f_{l}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime})-f_{v}(\vec{\kappa}_{\|}
+\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|}^{\,\prime})\over
\tilde{\omega}_{vl}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|}^{\,\prime},
\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime})+\omega}\right)
\right.\nonumber\\ &&\times
{1\over\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}
-\vec{q}_{\|}^{\,\prime},\vec{\kappa}_{\|})}
+\left({f_{l}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime})
-f_{v}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|}^{\,\prime})\over
\tilde{\omega}_{vl}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|}^{\,\prime},
\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime})+\omega}
\right.\nonumber\\ &&\left.\!\left.\!
+{f_{n}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime})
-f_{v}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|}^{\,\prime})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}
-\vec{q}_{\|}^{\,\prime},\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|}^{\,\prime})-\omega}
\right)
{1\over\tilde{\omega}_{nl}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}
+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime},\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime})}
\right]
\nonumber\\ &&\times
j_{h,ml}(z''';2\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime})
j_{k,v{}n}(z'';2\vec{\kappa}_{\|}+2\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}
-2\vec{q}_{\|}^{\,\prime})
j_{j,lv}(z';2\vec{\kappa}_{\|}+2\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|}^{\,\prime})
\nonumber\\ &&
+
\left[
\left({f_{l}(\vec{\kappa}_{\|}-\vec{q}_{\|}^{\,\prime})-f_{m}(\vec{\kappa}_{\|})
\over\tilde{\omega}_{l{}m}(\vec{\kappa}_{\|}-\vec{q}_{\|}^{\,\prime},
\vec{\kappa}_{\|})+\omega}
+{f_{l}(\vec{\kappa}_{\|}-\vec{q}_{\|}^{\,\prime})-f_{v}(\vec{\kappa}_{\|}
+\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|}^{\,\prime})\over
\tilde{\omega}_{vl}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|}^{\,\prime},
\vec{\kappa}_{\|}-\vec{q}_{\|}^{\,\prime})-\omega}\right)
\right.\nonumber\\ &&\times
{1\over\tilde{\omega}_{v{}m}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}
-\vec{q}_{\|}^{\,\prime},\vec{\kappa}_{\|})}
+\left({f_{l}(\vec{\kappa}_{\|}-\vec{q}_{\|}^{\,\prime})
-f_{v}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|}^{\,\prime})\over
\tilde{\omega}_{vl}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|}^{\,\prime},
\vec{\kappa}_{\|}-\vec{q}_{\|}^{\,\prime})-\omega}
\right.\nonumber\\ &&\left.\!
+{f_{n}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime})
-f_{v}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|}^{\,\prime})\over
\tilde{\omega}_{nv}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}
-\vec{q}_{\|}^{\,\prime},\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}-\vec{q}_{\|}^{\,\prime})-\omega}
\right)
\nonumber\\ &&\times\left.
{1\over\tilde{\omega}_{nl}(\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}
+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime},\vec{\kappa}_{\|}-\vec{q}_{\|}^{\,\prime})-2\omega}
\right]
j_{h,lv}(z''';2\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}-2\vec{q}_{\|}^{\,\prime})
\nonumber\\ &&\times\left.
j_{k,v{}n}(z'';2\vec{\kappa}_{\|}+2\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}
-2\vec{q}_{\|}^{\,\prime})
j_{j,ml}(z';2\vec{\kappa}_{\|}-\vec{q}_{\|}^{\,\prime})
\right\}
\nonumber\\ &&\times
j_{i,nm}(z;2\vec{\kappa}_{\|}+\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime})
\delta(\vec{q}_{\|}^{\,\prime\prime\prime}+\vec{q}_{\|}^{\,\prime\prime}-\vec{q}_{\|}^{\,\prime}-\vec{q}_{\|})
d^2\kappa_{\|},\end{aligned}$$ where we have omitted the now superfluous index $m$ from the surface states $\vec{\kappa}_{\|}$. As was the case with parts A–F, we are particularly interested in finding the PCDFWM response tensor, and thus we insert the DFWM pump fields given by Eqs. (\[eq:Apump-1\]) and (\[eq:Apump-2\]) into Eq. (\[eq:J3G\]). This allows us to carry out the integrals over $q_{\|}'''$ and $q_{\|}''$, consequentally followed by solution to the integral over $q_{\|}'$. The resulting expression is then split according to Eq. (\[eq:J3-wzq\]), and the PCDFWM conductivity tensor part G appear as Eq. (\[eq:XiAG\]).
Principal analytic solution\
to the integrals over $\vec{\kappa}_{\|}$ in the low temperature limit {#ch:SolveQ}
======================================================================
\[ch:Solve-Q\]\[app:B\] In this appendix we discuss the analytic solution to the integrals over $\vec{\kappa}_{\|}$ appearing in the linear and nonlinear conductivity tensor. The discussion is limited to cover the low temperature limit, and it is presented as a principal solution to all integrals over $\vec{\kappa}_{\|}$ appearing in Eqs. (\[eq:SigmaAA\])–(\[eq:XiAG\]).
General type of integrals
-------------------------
Every integral over $\vec{\kappa}_{\|}$ in both the linear conductivity tensor, Eqs. (\[eq:SigmaAA\]) and (\[eq:SigmaAB\]), and the nonlinear conductivity tensor, Eqs. (\[eq:XiAA\])–(\[eq:XiAG\]), can when scattering takes place in the $x$-$z$-plane be expressed as a sum over terms of the general type $${\cal{F}}_{pq}^{\beta}(n,\{a\},\{b\},s)
=\int\int
{{\kappa}_{x}^{p}\kappa_{y}^{q}f_{n}(\vec{\kappa}_{\|}+s\vec{e}_{x})
\over\prod_{k=1}^{\beta}[a_k\kappa_{x}+b_k]}
d\kappa_{x}d\kappa_{y},
\label{eq:FpqB-indef}$$ where $p,k,\beta$ are nonnegative integers, and $q$ is an even nonnegative integer. The functions in general depends on (i) the quantum number $n$, which is a positive nonzero integer, (ii) a set of real quantities, $\{a\}\equiv\{a_1,\dots,a_\beta\}$ appearing in front of the integration variable $\kappa_x$ in the denominator, (iii) a set of complex nonzero quantities, $\{b\}\equiv\{b_1,\dots,b_\beta\}$ apearing as the other quantity in each term of the denominator, and (iv) the real quantity $s$ representing the displacement of the center of the Fermi-Dirac distribution function from $(\kappa_x,\kappa_y)=(0,0)$. The quantity $s$ together with each element in the set $\{a\}$ is in general a function of the parallel components of the probe and pump wavevectors, $\vec{q}_{\|}$ and $\vec{k}_{\|}$. Each element in the set $\{b\}$ is furthermore a function of $\tau$, the relaxation time.
In the low temperature limit the Fermi-Dirac distribution function is zero outside the Fermi sphere and equal to one inside, and it is therefore advantageous to shift $\kappa_x$ by $-s$, followed by a one-to-one mapping of the $x$-$y$-plane into polar coordinates ($r$-$\theta$-plane). Using in this way $\kappa_x=r\cos\theta$, $\kappa_y=r\sin\theta$, and thus $d\kappa_xd\kappa_y=rd\theta{}dr$, the indefinite integral in Eq. (\[eq:FpqB-indef\]) is turned into the definite integral $${\cal{F}}_{pq}^{\beta}(n,\{a\},\{b\},s)
=\int_{0}^{{\alpha}(n)}\int_{0}^{2\pi}{r(r\cos\theta-s)^{p}(r\sin\theta)^{q}
\over\prod_{k=1}^{\beta}[a_k(r\cos\theta-s)+b_k]}d\theta dr.
\label{eq:FpqB}$$ The quantity $\alpha(n)=\sqrt{k_F^2-(\pi{}n/d)^2}$ is the radius of the (two-dimensional) Fermi circle for state $n$, given by Eq. (\[eq:alpha(n)\]). The Fermi wavenumber $k_F$ obeys the relation $k_F>\pi{}n/d$, since the Fermi-Dirac distribution function is zero for $k_F<\pi{}n/d$, and thus in that case the integral would vanish.
(80,76)(0,4) (2,0)(24,0)[3]{}[(0,0)(0,24)[3]{}[ (6,6)(4,6)(4,22)(6,22) (22,6)(24,6)(24,22)(22,22) ]{}]{} [ (2,70)[(0,5)\[l\][$[xxkh]$]{}]{} (2,46)[(0,5)\[l\][$[yxkh]$]{}]{} (2,22)[(0,5)\[l\][$[zxkh]$]{}]{} (40,70)[(0,5)\[c\][$[xykh]$]{}]{} (40,46)[(0,5)\[c\][$[yykh]$]{}]{} (40,22)[(0,5)\[c\][$[zykh]$]{}]{} (78,70)[(0,5)\[r\][$[xzkh]$]{}]{} (78,46)[(0,5)\[r\][$[yzkh]$]{}]{} (78,22)[(0,5)\[r\][$[zzkh]$]{}]{} ]{} (70,8)[(0,0)\[c\][$\bigtriangleup$]{}]{} (46,32)[(0,0)\[c\][$\otimes$]{}]{} (70,38)[(0,0)\[c\][$\otimes$]{}]{} (64,32)[(0,0)\[c\][$\otimes$]{}]{} (46,14)[(0,0)\[c\][$\otimes$]{}]{} (40,8)[(0,0)\[c\][$\otimes$]{}]{} (64,14)[(0,0)\[c\][$\otimes$]{}]{} (40,38)[(0,0)\[c\][$\star$]{}]{} (70,56)[(0,0)\[c\][$+$]{}]{} (22,8)[(0,0)\[c\][$+$]{}]{} (70,20)[(0,0)\[c\][$+$]{}]{} (58,8)[(0,0)\[c\][$+$]{}]{} (46,62) (40,56) (64,62) (16,14) (40,20) (34,14) (22,38) (46,44) (34,32) (58,38) (16,32) (64,44) (22,56) (70,68) (58,56) (22,20) (10,8) (58,20) (16,62)[(0,0)\[c\][$\times$]{}]{} (40,68)[(0,0)\[c\][$\times$]{}]{} (34,62)[(0,0)\[c\][$\times$]{}]{} (16,44)[(0,0)\[c\][$\times$]{}]{} (10,38)[(0,0)\[c\][$\times$]{}]{} (34,44)[(0,0)\[c\][$\times$]{}]{} (22,68)[(0,0)\[c\][$\oplus$]{}]{} (10,56)[(0,0)\[c\][$\oplus$]{}]{} (58,68)[(0,0)\[c\][$\oplus$]{}]{} (10,20)[(0,0)\[c\][$\oplus$]{}]{} (10,68)[(0,0)\[c\][$\bigtriangledown$]{}]{} (10,14) (10,32) (10,44) (10,62) (16,8) (16,20) (16,38) (16,56) (16,68) (22,14) (22,32) (22,44) (22,62) (34,8) (34,20) (34,38) (34,56) (34,68) (40,14) (40,32) (40,44) (40,62) (46,8) (46,20) (46,38) (46,56) (46,68) (58,14) (58,32) (58,44) (58,62) (64,8) (64,20) (64,38) (64,56) (64,68) (70,14) (70,32) (70,44) (70,62)
Specific integrals to be solved
-------------------------------
The necessary combinations of $p$ and $q$ in Eq. (\[eq:FpqB\]) to be calculated in order to solve the integrals over $\vec{\kappa}_{\|}$ in the nonlinear conductivity tensor are summarized in Fig. \[fig:pq-distribution\]. From Fig. \[fig:pq-distribution\] we observe that a total of nine different combinations of $p$ and $q$ need to be calculated, namely when $(p,q)$ takes the values $(0,0)$, $(0,2)$, $(0,4)$, $(1,0)$, $(1,2)$, $(2,0)$, $(2,2)$, $(3,0)$, or $(4,0)$, and it is seen from Eqs. (\[eq:SigmaAA\])–(\[eq:XiAG\]) that $\beta$ can take the values $\beta\in\{1,2,3\}$. However, the complexity of the total solution can be reduced, since functions with $\beta=2$ can be expressed in terms of functions with $\beta=1$ in the following way $${\cal{F}}_{pq}^{2}(n,a_1,a_2,b_1,b_2,s)=
{a_1{\cal{F}}_{pq}^{1}(n,a_1,b_1,s)-a_2{\cal{F}}_{pq}^{1}(n,a_2,b_2,s)
\over{}a_1b_2-a_2b_1},
\label{eq:Fpq2}$$ $a_k\neq0, k\in\{1,2\}$. In a similar fashion, the functions with $\beta=3$ can be written in terms of functions with $\beta=1$, namely $$\begin{aligned}
\lefteqn{
{\cal{F}}_{pq}^{3}(n,a_1,a_2,a_3,b_1,b_2,b_3,s)=
{a_1^2{\cal{F}}_{pq}^{1}(n,a_1,b_1,s)
\over(a_2b_1-b_2a_1)(a_3b_1-b_3a_1)}
}\nonumber\\ &\quad&
+{a_2^2{\cal{F}}_{pq}^{1}(n,a_2,b_2,s)
\over(a_2b_1-b_2a_1)(a_3b_2-b_3a_2)}
+{a_3^2{\cal{F}}_{pq}^{1}(n,a_3,b_3,s)
\over(a_3b_1-b_3a_1)(a_3b_2-b_3a_2)},
\label{eq:Fpq3}\end{aligned}$$ provided $a_k\neq0$, $k\in\{1,2,3\}$. If any $a_k$, for instance $a_1$, becomes zero, we observe from Eq. (\[eq:FpqB\]) that the order of the denominator becomes smaller by one. This implies in Eq. (\[eq:Fpq2\]) that ${\cal{F}}_{pq}^{2}(n,0,a_2,b_1,b_2,s)={\cal{F}}_{pq}^{1}(n,a_2,b_2,s)/b_1$. The similar conclusion with respect to Eq. (\[eq:Fpq3\]) is ${\cal{F}}_{pq}^{3}(n,0,a_2,a_3,b_1,b_2,b_3,s)={\cal{F}}_{pq}^{2}(n,a_2,a_3,b_2,b_3,s)/b_1$. A corresponding reduction applies for any other $a_k=0$ in Eqs. (\[eq:Fpq2\]) and (\[eq:Fpq3\]).
As a consequence of Eqs. (\[eq:Fpq2\]) and (\[eq:Fpq3\]), the integrals appearing in Eqs. (\[eq:SigmaAA\])–([\[eq:XiAG\]]{}) can now be written in terms of functions of the type $${\cal{F}}_{pq}^{1}(n,a,b,s)
=\int_{0}^{{\alpha}(n)}\int_{0}^{2\pi}{r(r\cos\theta-s)^{p}(r\sin\theta)^{q}
\over{}b-as+ar\cos\theta}d\theta dr,
\label{eq:Fpq1}$$ dropping the now superfluous index on $a$ and $b$. Since the following treatment is a formal solution of Eq. (\[eq:Fpq1\]), we will also drop the reference to $n$ for brevity, letting $\alpha\equiv\alpha(n)$.
Solution when $a=0$
-------------------
Before carrying on with the solution to Eq. (\[eq:Fpq1\]) in the appropriate cases, we take a look at it in the case where $a=0$ (as would be the case in the local limit, for example). Then the only term left in the denominator is $b$, which is constant with respect to the integration variables. The solution to the remaining thus becomes trivial, with the results $$\begin{aligned}
\lefteqn{
{\cal{F}}_{00}^{1}(n,0,b,s)={2\over{}b}\int_{0}^{\alpha}\int_{0}^{\pi}
r d\theta dr
={\pi\alpha^2\over{}b},
}\label{eq:F-00-1-a=0}\\
\lefteqn{
{\cal{F}}_{02}^{1}(n,0,b,s)={2\over{}b}\int_{0}^{\alpha}\int_{0}^{\pi}
r^3\sin^2\theta d\theta dr
={\pi\alpha^4\over4b},
}\label{eq:F-02-1-a=0}\\
\lefteqn{
{\cal{F}}_{04}^{1}(n,0,b,s)={2\over{}b}\int_{0}^{\alpha}\int_{0}^{\pi}
r^5\sin^4\theta d\theta dr
={\pi\alpha^6\over8b},
}\label{eq:F-04-1-a=0}\\
\lefteqn{
{\cal{F}}_{10}^{1}(n,0,b,s)={2\over{}b}\int_{0}^{\alpha}\int_{0}^{\pi}
\left[r^2\cos\theta-rs\right]d\theta dr
=-{\pi{}s\alpha^2\over{}b},
}\label{eq:F-10-1-a=0}\\
\lefteqn{
{\cal{F}}_{12}^{1}(n,0,b,s)={2\over{}b}\int_{0}^{\alpha}\int_{0}^{\pi}
\left[r^4\cos\theta\sin^2\theta-r^3s\sin^2\theta\right]d\theta dr
=-{\pi{}s\alpha^4\over4b},
}\label{eq:F-12-1-a=0}\\
\lefteqn{
{\cal{F}}_{20}^{1}(n,0,b,s)={2\over{}b}\int_{0}^{\alpha}\int_{0}^{\pi}
\left[r^3\cos^2\theta+rs^2-2sr^2\cos\theta\right]d\theta dr
={\pi\alpha^4\over4b}+{\pi{}s^2\alpha^2\over{}b},
}\label{eq:F-20-1-a=0}\\
\lefteqn{
{\cal{F}}_{22}^{1}(n,0,b,s)={2\over{}b}\int_{0}^{\alpha}\int_{0}^{\pi}
\left[r^5\cos^2\theta\sin^2\theta+r^3s^2\sin^2\theta
-2r^4s\cos\theta\sin^2\theta\right]d\theta dr
}\nonumber\\ &\quad&
={\pi\alpha^6\over24b}+{\pi{}s^2\alpha^4\over4b},
\label{eq:F-22-1-a=0}\\
\lefteqn{
{\cal{F}}_{30}^{1}(n,0,b,s)={2\over{}b}\int_{0}^{\alpha}\int_{0}^{\pi}
\left[r^4\cos^3\theta-3r^3s\cos^2\theta
+3r^2s^2\cos\theta-rs^3\right]d\theta dr
}\nonumber\\ &&
=-{3\pi{}s\alpha^4\over4b}-{\pi{}s^3\alpha^2\over{}b},
\label{eq:F-30-1-a=0}\\
\lefteqn{
{\cal{F}}_{40}^{1}(n,0,b,s)={2\over{}b}\int_{0}^{\alpha}\int_{0}^{\pi}
\left[r^5\cos^4\theta-4r^4s\cos^3\theta
+6r^3s^2\cos^2\theta-4r^2s^3\cos\theta
\right.}\nonumber\\ &&\left.
+rs^4\right]d\theta dr
={\pi\alpha^6\over8b}+{3\pi{}s^2\alpha^4\over2b}
+{\pi{}s^4\alpha^2\over{}b},
\label{eq:F-40-1-a=0}\end{aligned}$$ where we have made use of the facts that (i) $q$ is an even integer, and (ii) $\cos\theta$ and $\sin^2\theta$ are symmetric around $\theta=\pi$ in solving the angular integrals.
General solution
----------------
When $a\neq0$, we have to consider the full solution to Eq. (\[eq:Fpq1\]) for the nine different combinations of $p$ and $q$ we need. To solve Eq. (\[eq:Fpq1\]), let us make the substitutions $$\eta\equiv{b-as\over{}a\alpha},\quad
r\equiv\alpha{}u,
\label{eq:Substitute}$$ giving $dr=\alpha{}du$. Thereby Eq. (\[eq:Fpq1\]) is turned into the nine functions $$\begin{aligned}
\lefteqn{
{\cal{F}}_{00}^{1}(n,a,b,s)={\alpha\over{}a}\int_{0}^{1}\int_{0}^{2\pi}
{u\over\eta+u\cos\theta}d\theta du,
}\label{eq:F00}
\\
\lefteqn{
{\cal{F}}_{02}^{1}(n,a,b,s)=
{\alpha^3\over{}a}\int_{0}^{1}\int_{0}^{2\pi}
\left[{u^3\over\eta+u\cos\theta}-{u^3\cos^2\theta\over\eta+u\cos\theta}\right]
d\theta du,
}\label{eq:F02}
\\
\lefteqn{
{\cal{F}}_{04}^{1}(n,a,b,s)={\alpha^5\over{}a}\int_{0}^{1}\int_{0}^{2\pi}
\left[{u^5\over\eta+u\cos\theta}-{2u^5\cos^2\theta\over\eta+u\cos\theta}
+{u^5\cos^4\theta\over\eta+u\cos\theta}\right]
d\theta du,
}\label{eq:F04}
\\
\lefteqn{
{\cal{F}}_{10}^{1}(n,a,b,s)={\alpha^2\over{}a}\int_{0}^{1}\int_{0}^{2\pi}
\left[{u^2\cos\theta\over\eta+u\cos\theta}
-{s\over\alpha}{u\over\eta+u\cos\theta}\right]
d\theta du,
}\label{eq:F10}
\\
\lefteqn{
{\cal{F}}_{12}^{1}(n,a,b,s)={\alpha^4\over{}a}\int_{0}^{1}\int_{0}^{2\pi}
\left[{u^4\cos\theta\over\eta+u\cos\theta}
-{u^4\cos^3\theta\over\eta+u\cos\theta}
-{s\over\alpha}{u^3\over\eta+u\cos\theta}
\right.}\nonumber\\ &\quad&\left.\!
+{s\over\alpha}{u^3\cos^2\theta\over\eta+u\cos\theta}
\right]d\theta du,
\label{eq:F12}
\\
\lefteqn{
{\cal{F}}_{20}^{1}(n,a,b,s)={\alpha^3\over{}a}\int_{0}^{1}\int_{0}^{2\pi}
\left[{u^3\cos^2\theta\over\eta+u\cos\theta}
-{2s\over\alpha}{u^2\cos\theta\over\eta+u\cos\theta}
+{s^2\over\alpha^2}{u\over\eta+u\cos\theta}
\right]d\theta du,
}\nonumber\\
\label{eq:F20}
\\
\lefteqn{
{\cal{F}}_{22}^{1}(n,a,b,s)={\alpha^5\over{}a}\int_{0}^{1}\int_{0}^{2\pi}
\left[{u^5\cos^2\theta\over\eta+u\cos\theta}
-{u^5\cos^4\theta\over\eta+u\cos\theta}
-{2s\over\alpha}{u^4\cos\theta\over\eta+u\cos\theta}
\right.}\nonumber\\ &&\left.\!
+{2s\over\alpha}{u^4\cos^3\theta\over\eta+u\cos\theta}
+{s^2\over\alpha^2}{u^3\over\eta+u\cos\theta}
-{s^2\over\alpha^2}{u^3\cos^2\theta\over\eta+u\cos\theta}
\right]d\theta du,
\label{eq:F22}
\\
\lefteqn{
{\cal{F}}_{30}^{1}(n,a,b,s)={\alpha^4\over{}a}\int_{0}^{1}\int_{0}^{2\pi}
\left[{u^4\cos^3\theta\over\eta+u\cos\theta}
-{3s\over\alpha}{u^3\cos^2\theta\over\eta+u\cos\theta}
+{3s^2\over\alpha^2}{u^2\cos\theta\over\eta+u\cos\theta}
\right.}\nonumber\\ &&\left.\!
-{s^3\over\alpha^3}{u\over\eta+u\cos\theta}
\right]d\theta du,
\label{eq:F30}
\\
\lefteqn{
{\cal{F}}_{40}^{1}(n,a,b,s)={\alpha^5\over{}a}\int_{0}^{1}\int_{0}^{2\pi}
\left[{u^5\cos^4\theta\over\eta+u\cos\theta}
-{4s\over\alpha}{u^4\cos^3\theta\over\eta+u\cos\theta}
+{6s^2\over\alpha^2}{u^3\cos^2\theta\over\eta+u\cos\theta}
\right.}\nonumber\\ &&\left.\!
-{4s^3\over\alpha^3}{u^2\cos\theta\over\eta+u\cos\theta}
+{s^4\over\alpha^4}{u\over\eta+u\cos\theta}
\right]d\theta du,
\label{eq:F40}\end{aligned}$$ where we have made use of the relations $$\begin{aligned}
\sin^2\theta&=&1-\cos^2\theta,
\label{eq:angle1}
\\
\cos\theta\sin^2\theta&=&\cos\theta-\cos^3\theta,
\\
\cos^2\theta\sin^2\theta&=&\cos^2\theta-\cos^4\theta,
\\
\sin^4\theta&=&1-2\cos^2\theta+\cos^4\theta.
\label{eq:angle4}\end{aligned}$$
### Solution to the angular integrals
Next, to carry out the angular integrals, we put $$t=e^{{\rm{i}}\theta}$$ so that these integrals become $$\int_0^{2\pi}{\cos^h\theta\over\eta+u\cos\theta}d\theta
={1\over{}2^h{\rm{i}}u}\oint{(1+t^2)^h\over{}t^h(t-t_+)(t-t_-)}dt,\!\!
\label{eq:t+t-h}$$ where $h\in\{0,1,2,3,4\}$. In Eq. (\[eq:t+t-h\]), the poles at $t_{\pm}$ in the $t$-plane are located at $$t_{\pm}=-{\eta\over{}u}\pm\sqrt{\left({\eta\over{}u}\right)^2-1},$$
(0,-5)(80,30) (40,0)[ (6,10)(34,10) (20,-4)(20,24) (20,10)[10]{}[0]{}[60]{} (20,10)[10]{}[50]{}[120]{} (20,10)[10]{}[110]{}[0]{} (29,19) (23,13) (30,19)[(0,0)\[tl\][$t_-$]{}]{} (24,13)[(0,0)\[bl\][$t_+$]{}]{} ]{} (0,0)[ (6,10)(34,10) (20,-4)(20,24) (20,10)[10]{}[0]{}[60]{} (20,10)[10]{}[50]{}[120]{} (20,10)[10]{}[110]{}[0]{} (29,19) (23,13) (30,19)[(0,0)\[tl\][$t_-$]{}]{} (24,13)[(0,0)\[bl\][$t_+$]{}]{} (20,10) (20,10) (19,9)[(0,0)\[tr\][$0$]{}]{} ]{}
and the integration runs along the unit circle. Since we have $t_+t_-=1$, one of these poles is inside the unit circle while the other is outside. When $h>0$ there are an additional pole of order $h$ at $t=0$. Using the unit circles shown in Fig. \[fig:21\] as the integration paths, we find by a residue calculation $$\begin{aligned}
\int_{0}^{2\pi}{1\over\eta+u\cos\theta}d\theta
&=&{2\pi\over\sqrt{\eta^2-u^2}},
\label{eq:Ang-nocos}
\\
\int_{0}^{2\pi}{\cos\theta\over\eta+u\cos\theta}d\theta
&=&{2\pi\over{}u}\left[1-{\eta\over\sqrt{\eta^2-u^2}}\right],
\\
\int_{0}^{2\pi}{\cos^2\theta\over\eta+u\cos\theta}d\theta
&=&{2\pi\eta\over{}u^2}\left[{\eta\over\sqrt{\eta^2-u^2}}-1\right],
\\
\int_{0}^{2\pi}{\cos^3\theta\over\eta+u\cos\theta}d\theta
&=&{\pi\over{}u}+{2\pi\eta^2\over{}u^3}
\left[1-{\eta\over\sqrt{\eta^2-u^2}}\right],
\\
\int_{0}^{2\pi}{\cos^4\theta\over\eta+u\cos\theta}d\theta
&=&{2\pi\eta^3\over{}u^4}\left[{\eta\over\sqrt{\eta^2-u^2}}-1\right]
-{\pi\eta\over{}u^2}.\end{aligned}$$ Inserting these results into Eqs. (\[eq:F00\])–(\[eq:F02\]), we get $$\begin{aligned}
\lefteqn{
{\cal{F}}_{00}^{1}(n,a,b,s)=
{2\pi\alpha\over{}a}\int_{0}^{1}
{u\over\sqrt{\eta^2-u^2}}du,
}
\label{eq:F00R}
\\
\lefteqn{
{\cal{F}}_{02}^{1}(n,a,b,s)=
{2\pi\alpha^3\over{}a}\int_{0}^{1}
\left[{u^3\over\sqrt{\eta^2-u^2}}
-{\eta^2u\over\sqrt{\eta^2-u^2}}+\eta{}u\right]
du,
}\label{eq:F02R}
\\
\lefteqn{
{\cal{F}}_{04}^{1}(n,a,b,s)=
{\pi\alpha^5\over{}a}\int_{0}^{1}
\left[{2u^5\over\sqrt{\eta^2-u^2}}
-{4\eta^2u^3\over\sqrt{\eta^2-u^2}}
+{2\eta^4u\over\sqrt{\eta^2-u^2}}
+3\eta{}u^3
-2\eta^3u
\right] du,
}\nonumber\\
\label{eq:F04R}
\\
\lefteqn{
{\cal{F}}_{10}^{1}(n,a,b,s)=
{2\pi\alpha^2\over{}a}\int_{0}^{1}
\left[u-\left(\eta+{s\over\alpha}\right){u\over\sqrt{\eta^2-u^2}}\right]
du,
}\label{eq:F10R}
\\
\lefteqn{
{\cal{F}}_{12}^{1}(n,a,b,s)=
{\pi\alpha^4\over{}a}\int_{0}^{1}
\left[{u^3}
-\left({2\eta^2}+{2s\eta{}\over\alpha}\right)u
-\left({2\eta}+{2s\over\alpha}\right){u^3\over\sqrt{\eta^2-u^2}}
\right.}\nonumber\\ &\quad&\left.\!
+\left({2\eta^3}+{2s\eta^2\over\alpha}\right){u\over\sqrt{\eta^2-u^2}}
\right] du,
\label{eq:F12R}
\\
\lefteqn{
{\cal{F}}_{20}^{1}(n,a,b,s)=
{2\pi\alpha^3\over{}a}\int_{0}^{1}
\left[\left(\eta^2+{2s\eta\over\alpha}+{s^2\over\alpha^2}\right)
{u\over\sqrt{\eta^2-u^2}}-\left(\eta+{2s\over\alpha}\right)u
\right] du,
}\label{eq:F20R}
\\
\lefteqn{
{\cal{F}}_{22}^{1}(n,a,b,s)=
{\pi\alpha^5\over{}a}\int_{0}^{1}
\left[\left({2\eta^2}+{4\eta{}s\over\alpha}
+{2s^2\over\alpha^2}\right){u^3\over\sqrt{\eta^2-u^2}}
-\left(\eta+{2s\over\alpha}\right)u^3
\right.}\nonumber\\ &&\left.\!
-\left({2\eta^4}+{4\eta^3s\over\alpha}
+{2\eta^2s^2\over\alpha^2}\right){u\over\sqrt{\eta^2-u^2}}
+\left({2\eta^3}+{4s\eta^2\over\alpha}+{2s^2\eta{}\over\alpha^2}\right)u
\right] du,
\label{eq:F22R}
\\
\lefteqn{
{\cal{F}}_{30}^{1}(n,a,b,s)=
{\pi\alpha^4\over{}a}\int_{0}^{1}
\left[{u^3}
+\left({2\eta^2}+{6s\eta\over\alpha}+{6s^2\over\alpha^2}\right)u
\right.}\nonumber\\ &&\left.\!
-\left({2\eta^3}+{6s\eta^2\over\alpha}+{6s^2\eta\over\alpha^2}
+{2s^3\over\alpha^3}\right){u\over\sqrt{\eta^2-u^2}}
\right] du,
\label{eq:F30R}
\\
\lefteqn{
{\cal{F}}_{40}^{1}(n,a,b,s)=
{\pi\alpha^5\over{}a}\int_{0}^{1}
\left[
\left({2\eta^4}+{8s\eta^3\over\alpha}+{12s^2\eta^2\over\alpha^2}
+{8s^3\eta\over\alpha^3}+{2s^4\over\alpha^4}\right){u\over\sqrt{\eta^2-u^2}}
\right.}\nonumber\\ &&\left.\!
-\left(\eta+{4s\over\alpha}\right)u^3
-\left(2\eta^3+{8s\eta^2\over\alpha}+{12s^2\eta\over\alpha^2}
+{8s^3\over\alpha^3}\right)u
\right] du.
\label{eq:F40R}\end{aligned}$$
### Solution to the radial integrals
This result leaves us with radial integrals of the type $$\int_0^1{u^h\over\sqrt{\eta^2-u^2}}du,\qquad h\in\{1,3,5\},
\label{eq:IntSq}$$ apart from the trivial $u^n$-type of integrals. The three integrals in Eq. (\[eq:IntSq\]) can be found for example in as Eqs. (2.271.7), (2.272.7), and (2.273.8), which by insertion of $u$ and $\eta$ becomes $$\begin{aligned}
\int{u\over\sqrt{\eta^2-u^2}}du&=&-\sqrt{\eta^2-u^2},
\\
\int{u^3\over\sqrt{\eta^2-u^2}}du&=&-{1\over3}\Bigl[u^2+2\eta^2\Bigr]
\sqrt{\eta^2-u^2},
\\
\int{u^5\over\sqrt{\eta^2-u^2}}du&=&
-{1\over15}\Bigl[3u^4+8\eta^4+4u^2\eta^2\Bigr]\sqrt{\eta^2-u^2},\end{aligned}$$ in that order, verified by differentiation, and after correction of the misprint appearing in Eq. (2.273.8). By insertion of the limits we get $$\begin{aligned}
\int_0^1{u\over\sqrt{\eta^2-u^2}}du&=&\eta-\sqrt{\eta^2-1},
\label{eq:2.271.7}
\\
\int_0^1{u^3\over\sqrt{\eta^2-u^2}}du&=&
{2\eta^3\over3}-{1+2\eta^2\over3}\sqrt{\eta^2-1},
\label{eq:2.272.7}
\\
\int_{0}^{1}{u^5\over\sqrt{\eta^2-u^2}}du&=&
{8\over15}\eta^5-{1\over15}\Bigl[3+8\eta^4+4\eta^2\Bigr]\sqrt{\eta^2-1}.
\label{eq:2.273.8}\end{aligned}$$ By insertion of Eqs. (\[eq:2.271.7\])–(\[eq:2.273.8\]) into Eqs. (\[eq:F00R\])–(\[eq:F40R\]) and solving the trivial $u^n$-type of integrals, the resulting expressions for the nine different cases of Eq. (\[eq:Fpq1\]) thus become $$\begin{aligned}
\lefteqn{
{\cal{F}}_{00}^{1}(n,a,b,s)=
{2\pi\alpha\over{}a}
\left[\eta-\sqrt{\eta^2-1}\right]
,}
\label{eq:F00Rs}
\\
\lefteqn{
{\cal{F}}_{02}^{1}(n,a,b,s)=
{2\pi\alpha^3\over{}a}
\left[
{\eta^2-1\over3}\sqrt{\eta^2-1}-{\eta^3\over3}+{\eta\over2}
\right],
}\label{eq:F02Rs}
\\
\lefteqn{
{\cal{F}}_{04}^{1}(n,a,b,s)=
{\pi\alpha^5\over{}a}
\left[
\left({4\over5}\eta^2-{2\over5}\eta^4-{2\over5}\right)\sqrt{\eta^2-1}
+{2\over5}\eta^5-\eta^3+{3\over4}\eta
\right]
}\label{eq:F04Rs}
\\
\lefteqn{
{\cal{F}}_{10}^{1}(n,a,b,s)=
{2\pi\alpha^2\over{}a}
\left[{1\over2}-\left(\eta+{s\over\alpha}\right)
\left(\eta-\sqrt{\eta^2-1}\right)
\right],
}\label{eq:F10Rs}
\\
\lefteqn{
{\cal{F}}_{12}^{1}(n,a,b,s)=
{\pi\alpha^4\over{}a}
\left[
\left({2s\over3\alpha}+{2\over3}\eta-{2s\over3\alpha}\eta^2-{2\over3}\eta^3
\right)\sqrt{\eta^2-1}
+{2\over3}\eta^4+{2s\over3\alpha}\eta^3
\right.}\nonumber\\ &\quad&\left.\!
-\eta^2-{s\over\alpha}\eta+{1\over4}
\right],
\label{eq:F12Rs}
\\
\lefteqn{
{\cal{F}}_{20}^{1}(n,a,b,s)=
{2\pi\alpha^3\over{}a}
\left[\left(\eta^2+{2s\eta\over\alpha}+{s^2\over\alpha^2}\right)
\left(\eta-\sqrt{\eta^2-1}\right)
-{1\over2}\left(\eta+{2s\over\alpha}\right)
\right],
}\label{eq:F20Rs}
\\
\lefteqn{
{\cal{F}}_{22}^{1}(n,a,b,s)=
{\pi\alpha^5\over{}a}
\left[
\left({2\over3}\eta^4+{4s\over3\alpha}\eta^3
+\left({2s^2\over3\alpha^2}-{2\over3}\right)\eta^2-{4s\over3\alpha}\eta
-{2s^2\over3\alpha^2}\right)\sqrt{\eta^2-1}
\right.}\nonumber\\ &&\left.
-{2\over3}\eta^5-{4s\over3\alpha}\eta^4
+\left(1-{2s^2\over3\alpha^2}\right)\eta^3
+{2s\over\alpha}\eta^2+\left({s^2\over\alpha^2}-{1\over4}\right)\eta
-{s\over2\alpha}
\right],
\label{eq:F22Rs}
\\
\lefteqn{
{\cal{F}}_{30}^{1}(n,a,b,s)=
{\pi\alpha^4\over{}a}
\left[{1\over4}
+{1\over2}\left({2\eta^2}+{6s\eta\over\alpha}+{6s^2\over\alpha^2}\right)
\right.}\nonumber\\ &&\left.\!
-\left({2\eta^3}+{6s\eta^2\over\alpha}+{6s^2\eta\over\alpha^2}
+{2s^3\over\alpha^3}\right)
\left(\eta-\sqrt{\eta^2-1}\right)
\right],
\label{eq:F30Rs}
\\
\lefteqn{
{\cal{F}}_{40}^{1}(n,a,b,s)=
{\pi\alpha^5\over{}a}
\left[
\left({2\eta^4}+{8s\eta^3\over\alpha}+{12s^2\eta^2\over\alpha^2}
+{8s^3\eta\over\alpha^3}+{2s^4\over\alpha^4}\right)
\left(\eta-\sqrt{\eta^2-1}\right)
\right.}\nonumber\\ &&\left.\!
-{1\over4}\left(\eta+{4s\over\alpha}\right)
-{1\over2}\left(2\eta^3+{8s\eta^2\over\alpha}+{12s^2\eta\over\alpha^2}
+{8s^3\over\alpha^3}\right)
\right].
\label{eq:F40Rs}\end{aligned}$$
Verification
------------
In order to verify the result, we take the limit where $a\rightarrow0$, we use the binomial series expansion $$\begin{aligned}
\lefteqn{
\sqrt{\eta^2-1}=
\eta-{1\over2\eta}-\sum_{n=2}^{\infty}{(2n-3)!!\over(2n)!!\eta^{2n-1}}
}\nonumber\\ &\quad&=
\eta
-{1\over2\eta}
-{1\over8\eta^3}
-{1\over16\eta^5}
-{5\over128\eta^7}
-{7\over256\eta^9}
-{21\over1024\eta^{11}}
-{33\over2048\eta^{13}}
\nonumber\\ &&
-{429\over32768\eta^{15}}
-{715\over65536\eta^{17}}-\dots,
\label{eq:binomial}\end{aligned}$$ where ‘$x!!$’ is the ‘double factorial’ operator taken for the integer $x$, given by $x!!\equiv{}x(x-2)(x-4)\cdots(x-k)$ for $x>k$. Thus by insertion of Eq. (\[eq:binomial\]) into Eqs. (\[eq:F00Rs\])–(\[eq:F40Rs\]) we get for small values of $a$ $$\begin{aligned}
\lefteqn{
{\cal{F}}_{00}^{1}(n,a,b,s)=
{2\pi\alpha\over{}a}
\left[{1\over2\eta}+{1\over8\eta^3}+{1\over16\eta^5}+{5\over128\eta^7}+\cdots
\right]
,}
\label{eq:F00Ra}
\\
\lefteqn{
{\cal{F}}_{02}^{1}(n,a,b,s)=
{2\pi\alpha^3\over{}a}
\left[{1\over8\eta}+{1\over48\eta^3}+{3\over384\eta^5}+{3\over768\eta^7}
+\cdots
\right],
}\label{eq:F02Ra}
\\
\lefteqn{
{\cal{F}}_{04}^{1}(n,a,b,s)=
{\pi\alpha^5\over{}a}
\left[
{1\over8\eta}+{1\over320\eta^3}+{19\over640\eta^5}+{21\over2560\eta^7}
+\cdots
\right]
}\label{eq:F04Ra}
\\
\lefteqn{
{\cal{F}}_{10}^{1}(n,a,b,s)=
-{2\pi\alpha^2\over{}a}
\left[{s\over2\alpha\eta}+{1\over8\eta^2}+{s\over8\alpha\eta^3}
+{1\over16\eta^4}+\cdots
\right],
}\label{eq:F10Ra}
\\
\lefteqn{
{\cal{F}}_{12}^{1}(n,a,b,s)=
-{\pi\alpha^4\over{}a}
\left[{s\over4\alpha\eta}+{1\over24\eta^2}+{s\over24\alpha\eta^3}
+{3\over192\eta^4}+\cdots
\right],
}\label{eq:F12Ra}
\\
\lefteqn{
{\cal{F}}_{20}^{1}(n,a,b,s)=
{2\pi\alpha^3\over{}a}
\left[\left({s^2\over2\alpha^2}+{1\over8}\right){1\over\eta}
+{s\over4\alpha\eta^2}
+\left({s^2\over8\alpha^2}+{1\over16}\right){1\over\eta^3}
+{s\over8\alpha\eta^4}+\cdots
\right],
}\nonumber\\
\label{eq:F20Ra}
\\
\lefteqn{
{\cal{F}}_{22}^{1}(n,a,b,s)=
{\pi\alpha^5\over{}a}\!
\left[\!\left(\!{s^2\over4\alpha^2}+{1\over24}\!\right)\!{1\over\eta}
+{s\over12\alpha\eta^2}
+\!\left(\!{s^2\over24\alpha^2}+{7\over192}\!\right)\!{1\over\eta^3}
+{7s\over96\alpha\eta^4}+\cdots
\right]\!,
}\nonumber\\
\label{eq:F22Ra}
\\
\lefteqn{
{\cal{F}}_{30}^{1}(n,a,b,s)=
-{\pi\alpha^4\over{}a}
\left[\left({s^3\over\alpha^3}+{3s\over4\alpha}\right){1\over\eta}
+\left({3s^2\over4\alpha^2}+{1\over8}\right){1\over\eta^2}
+\left({s^3\over4\alpha^3}+{3s\over8\alpha}\right){1\over\eta^3}
\right.}\nonumber\\ &\quad&\left.\!
+\left({3s^2\over8\alpha^2}+{5\over64}\right){1\over\eta^4}+\cdots
\right],
\label{eq:F30Ra}
\\
\lefteqn{
{\cal{F}}_{40}^{1}(n,a,b,s)=
{\pi\alpha^5\over{}a}
\left[
\left({s^4\over\alpha^4}+{3s^2\over2\alpha^2}+{1\over8}\right){1\over\eta}
+\left({s^3\over\alpha^3}+{s\over2\alpha}\right){1\over\eta^2}
\right.}\nonumber\\ &\quad&\left.\!
+\left({s^4\over4\alpha^4}+{3s^2\over4\alpha^2}+{5\over64}\right)
{1\over\eta^3}
+\left({s^3\over2\alpha^3}+{5s\over16\alpha}\right){1\over\eta^4}
+\cdots
\right],
\label{eq:F40Ra}\end{aligned}$$ which by insertion of $\eta=(b-as)/(a\alpha)$ and subsequently letting $a=0$ reduces to the results of Eqs. (\[eq:F-00-1-a=0\])–(\[eq:F-40-1-a=0\]).
Result
------
In terms of the original $a$ and $b$ quantities, the resulting expressions for the specific integrals are: $$\begin{aligned}
\lefteqn{
{\cal{F}}_{00}^{1}(n,a,b,s)={2\pi\over{}a^2}
\left[b-as-\sqrt{(b-as)^2-a^2\alpha^2}\right],
}\label{eq:F00s}
\\
\lefteqn{
{\cal{F}}_{02}^{1}(n,a,b,s)=
{2\pi\over3a^4}\left[
\left((b-as)^2-{\alpha^2a^2}\right)^{3\over2}
-(b-as)^3
+{3\over2}{\alpha^2a^2(b-as)}\right].
}\\
\lefteqn{
{\cal{F}}_{04}^{1}(n,a,b,s)=
{2\pi\over5a^6}
\left[
(b-as)^5-{5\over2}a^2\alpha^2(b-as)^3+{15\over8}a^4\alpha^4(b-as)
\right.}\nonumber\\ &\quad&\left.\!
-\left((b-as)^2-a^2\alpha^2\right)^{5\over2}
\right]
\\
\lefteqn{
{\cal{F}}_{10}^{1}(n,a,b,s)=
{2\pi\alpha^2\over{}a}
\left[{1\over2}-{b\over{}a^2\alpha^2}
\left({b-as}-\sqrt{(b-as)^2-a^2\alpha^2}\right)
\right],
}\\
\lefteqn{
{\cal{F}}_{12}^{1}(n,a,b,s)=
{\pi\over{}a^5}\left[{1\over4}a^4\alpha^4
-b\left(a^2\alpha^2(b-as)-2(b-as)^2\right)
\right.}\nonumber\\ &&\left.\!
-{2b\over3}\left((b-as)^2-a^2\alpha^2\right)^{3\over2}\right]
\\
\lefteqn{
{\cal{F}}_{20}^{1}(n,a,b,s)=
{2\pi\over{}a^4}
\left[b^2\left(b-as-\sqrt{(b-as)^2-a^2\alpha^2}\right)
-{1\over2}a^2\alpha^2(b+as)\right],
}\\
\lefteqn{
{\cal{F}}_{22}^{1}(n,a,b,s)=
{2\pi\over3a^6}\left[b^2\left(6a^2\alpha^2(b-as)-4a\alpha(b-as)^3
+\left((b-as)^2-a^2\alpha^2\right)^{3\over2}\right)
\right.}\nonumber\\ &&\left.\!
-{3\over8}a^4\alpha^4(b+as)\right]
\\
\lefteqn{
{\cal{F}}_{30}^{1}(n,a,b,s)=
{\pi\alpha^4\over{}a}
\left[{1\over4}
+{(b+as)^2\over{}a^2\alpha^2}
-{2b^3\over{}a^4\alpha^4}
\left(b-as-\sqrt{(b-as)^2-a^2\alpha^2}\right)
\right],
}\\
\lefteqn{
{\cal{F}}_{40}^{1}(n,a,b,s)=
{\pi\alpha^5\over{}a}
\left[
{2b^4\over{}a^5\alpha^5}
\left(b-as-\sqrt{(b-as)^2-a^2\alpha^2}\right)
-{b\over4a\alpha}
-{b^3\over{}a^3\alpha^3}
\right.}\nonumber\\ &&\left.\!
-{s\over\alpha}\left({3\over4}+{(b+as)^2\over{}a^2\alpha^2}\right)
\right].
\label{eq:F40s}\end{aligned}$$ Using Eqs. (\[eq:F00s\])–(\[eq:F40s\]) together with Eqs. (\[eq:Fpq2\]) and (\[eq:Fpq3\]) when they are needed, the solution to the integrals appearing in Eqs. (\[eq:SigmaAA\])–(\[eq:XiAG\]), (\[eq:C\]), (\[eq:D\]), (\[eq:N\]), and (\[eq:A-xx\])–(\[eq:A-zz\]) are obtained in a straightforward manner. They can be found on detailed form in Appendix \[app:C\].
Analytic solution to the integrals over $\vec{\kappa}_{\|}$ appearing in the conductivity tensors when scattering takes place in the $x$-$z$-plane {#app:C}
==================================================================================================================================================
Taking a close look at the expressions for the different nonlinear conductivity tensor parts in Eqs. (\[eq:XiAA\])–(\[eq:XiAG\]) we observe by insertion of the transition current density given by Eq. (\[eq:Jm->n||\]) that they can be separated into two independent parts. One of these parts depends solely on $\vec{\kappa}_{\|}$ and the other part depends only on $z$. In the remaining of this Appendix they will be denoted by $\xi$ and ${\cal{Z}}$, respectively. The quantity $\xi$ can be solved according to the solution scheme given in Appendix \[ch:Solve-Q\], and the explicit solution will therefore in the following be given in terms of the functions solved in Appendix \[ch:Solve-Q\]. Furthermore it is possible to split the $z$-dependent part into independent functions of each $z$-coordinate. Since the $z$-dependence involves only the wave functions, we define the following three new quantities $$\begin{aligned}
{\cal{Z}}_{nm}^{x}(z)&=&{\cal{Z}}_{nm}^{\,y}(z)=\psi_m^*(z)\psi_n(z),
\label{eq:Z-nm-xy}\\
{\cal{Z}}_{nm}^{z}(z)&=&\psi_m^*(z){\partial\psi_n(z)\over\partial{}z}
-\psi_n(z){\partial\psi_m^*(z)\over\partial{}z},
\label{eq:Z-nm-z}\end{aligned}$$ in order to reduce the expressions in the following.
As the first step in preparing the solutions to the integrals over $\vec{\kappa}_{\|}$ we identify the transition frequencies occuring in the nonlinear conductivity tensor parts. Each of these transition frequencies can on general form be written $$\begin{aligned}
{\omega}_{nm}(\vec{\kappa}_{\|}+\vec{\beta},\vec{\kappa}_{\|}+\vec{\gamma})=
{1\over\hbar}\left[\varepsilon_n-\varepsilon_m+{\hbar^2\over2m_e}
\left(2\kappa_x(\beta_x-\gamma_x)+\beta_x^2-\gamma_x^2\right)\right].
\label{eq:transit-nm}\end{aligned}$$ Looking at Eqs. (\[eq:XiAA\])–(\[eq:XiAG\]), we observe that the following transition frequencies appear in the nonlinear conductivity tensor: $$\begin{aligned}
{\omega}_{nm}(\vec{\kappa}_{\|},\vec{\kappa}_{\|})&=&
{\omega}_{nm}(\vec{\kappa}_{\|}+\vec{q}_{\|},\vec{\kappa}_{\|}+\vec{q}_{\|})=
{1\over\hbar}\left[\varepsilon_n-\varepsilon_m\right],
\label{eq:omega-nm-1}\\
\omega_{nm}(\vec{\kappa}_{\|}+\vec{k}_{\|},\vec{\kappa}_{\|})
&=&{1\over\hbar}\left[\varepsilon_n-\varepsilon_m+{\hbar^2k_x\over2m_e}
\left(2\kappa_x+k_x\right)\right],
\label{eq:omega-nm-2}\\
\omega_{nm}(\vec{\kappa}_{\|},\vec{\kappa}_{\|}+\vec{k}_{\|})
&=&{1\over\hbar}\left[\varepsilon_n-\varepsilon_m-{\hbar^2k_x\over2m_e}
\left(2\kappa_x+k_x\right)\right],
\label{eq:omega-nm-3}\\
{\omega}_{nm}(\vec{\kappa}_{\|}-\vec{k}_{\|},\vec{\kappa}_{\|})&=&
{1\over\hbar}\left[\varepsilon_n-\varepsilon_m+{\hbar^2k_x\over2m_e}
\left(k_x-2\kappa_x\right)\right],
\label{eq:omega-nm-4}\\
{\omega}_{nm}(\vec{\kappa}_{\|},\vec{\kappa}_{\|}-\vec{k}_{\|})&=&
{1\over\hbar}\left[\varepsilon_n-\varepsilon_m+{\hbar^2k_x\over2m_e}
\left(2\kappa_x-k_x\right)\right],
\label{eq:omega-nm-5}\\
{\omega}_{nm}(\vec{\kappa}_{\|}+\vec{q}_{\|},\vec{\kappa}_{\|})&=&
{1\over\hbar}\left[\varepsilon_n-\varepsilon_m+{\hbar^2q_x\over2m_e}
\left(2\kappa_x+q_x\right)\right],
\label{eq:omega-nm-6}\\
\omega_{nm}(\vec{\kappa}_{\|}+\vec{q}_{\|}
,\vec{\kappa}_{\|}+\vec{k}_{\|})&=&
{1\over\hbar}\left[\varepsilon_n-\varepsilon_m+{\hbar^2\over2m_e}
\left(2\kappa_x(q_x-k_x)+q_x^2-k_x^2\right)\right],
\label{eq:omega-nm-7}\\
{\omega}_{nm}(\vec{\kappa}_{\|}+\vec{q}_{\|},
\vec{\kappa}_{\|}-\vec{k}_{\|})&=&
{1\over\hbar}\left[\varepsilon_n-\varepsilon_m+{\hbar^2\over2m_e}
\left(2\kappa_x(q_x+k_x)+q_x^2-k_x^2\right)\right],
\label{eq:omega-nm-8}\\
{\omega}_{nm}(\vec{\kappa}_{\|}-\vec{k}_{\|}+\vec{q}_{\|},
\vec{\kappa}_{\|})&=&
{1\over\hbar}\left[\varepsilon_n-\varepsilon_m+{\hbar^2\over2m_e}(q_x-k_x)
\left(2\kappa_x+(q_x-k_x)\right)\right],
\label{eq:omega-nm-9}\\
\omega_{nm}(\vec{\kappa}_{\|}+\vec{k}_{\|}+\vec{q}_{\|}
,\vec{\kappa}_{\|})&=&
{1\over\hbar}\left[\varepsilon_n-\varepsilon_m+{\hbar^2\over2m_e}
(q_x+k_x)\left(2\kappa_x+(q_x+k_x)\right)\right],
\label{eq:omega-nm-10}\\
\omega_{nm}(\vec{\kappa}_{\|}+\vec{k}_{\|}+\vec{q}_{\|}
,\vec{\kappa}_{\|}+\vec{k}_{\|})&=&
{1\over\hbar}\left[\varepsilon_n-\varepsilon_m+{\hbar^2q_x\over2m_e}
\left(2\kappa_x+q_x+2k_x\right)\right],
\label{eq:omega-nm-11}\\
\omega_{nm}(\vec{\kappa}_{\|}+\vec{k}_{\|}+\vec{q}_{\|}
,\vec{\kappa}_{\|}+\vec{q}_{\|})&=&
{1\over\hbar}\left[\varepsilon_n-\varepsilon_m+{\hbar^2k_x\over2m_e}
\left(2\kappa_x+k_x+2q_x\right)\right],
\label{eq:omega-nm-12}\\
{\omega}_{nm}(\vec{\kappa}_{\|}+\vec{q}_{\|},
\vec{\kappa}_{\|}-\vec{k}_{\|}+\vec{q}_{\|})&=&
{1\over\hbar}\left[\varepsilon_n-\varepsilon_m+{\hbar^2k_x\over2m_e}
\left(2\kappa_x+2q_x-k_x\right)\right],
\label{eq:omega-nm-13}\\
{\omega}_{nm}(\vec{\kappa}_{\|}-\vec{k}_{\|}+\vec{q}_{\|},
\vec{\kappa}_{\|}-\vec{k}_{\|})&=&
{1\over\hbar}\left[\varepsilon_n-\varepsilon_m+{\hbar^2q_x\over2m_e}
\left(2\kappa_x+q_x-2k_x\right)\right],
\label{eq:omega-nm-14}\\
{\omega}_{nm}(\vec{\kappa}_{\|}-\vec{k}_{\|}+\vec{q}_{\|},
\vec{\kappa}_{\|}+\vec{q}_{\|})&=&
{1\over\hbar}\left[\varepsilon_n-\varepsilon_m+{\hbar^2k_x\over2m_e}
\left(k_x-2q_x-2\kappa_x\right)\right].
\label{eq:omega-nm-15}\end{aligned}$$ We observe from Eqs. (\[eq:XiAA\])–(\[eq:XiAG\]) that these transition frequencies gives rise to a number of different $a$ and $b$ coefficients, which we will use in the later sections of this Appendix. It turns out that there are a total of four different $a$’s and thirteen different $b$’s. To present an overview and for the sake of easy reference they are all listed together below, viz. $$\begin{aligned}
a_{1}&=&{\hbar{}k_x\over{}m_e},
\\
a_{2}&=&{\hbar{}q_x\over{}m_e},
\\
a_{3}&=&{\hbar\over{}m_e}(q_x+k_x),
\\
a_{4}&=&{\hbar\over{}m_e}(q_x-k_x),
\\
b_{nm}^{1}&=&{1\over\hbar}(\varepsilon_n-\varepsilon_m)
+{\hbar{}k_x^2\over2m_e}-\omega-{\rm{i}}\tau_{nm}^{-1},
\\
b_{nm}^{2}&=&{1\over\hbar}(\varepsilon_n-\varepsilon_m)
-{\hbar{}k_x^2\over2m_e}-\omega-{\rm{i}}\tau_{nm}^{-1},
\\
b_{nm}^{3}&=&{1\over\hbar}(\varepsilon_n-\varepsilon_m)
+{\hbar{}q_x^2\over2m_e}+\omega-{\rm{i}}\tau_{nm}^{-1},
\\
b_{nm}^{4}&=&{1\over\hbar}(\varepsilon_n-\varepsilon_m)
+{\hbar{}q_x^2\over2m_e}-\omega-{\rm{i}}\tau_{nm}^{-1},
\\
b_{nm}^{5}&=&{1\over\hbar}(\varepsilon_n-\varepsilon_m)
+{\hbar\over2m_e}(q_x-k_x)^2-{\rm{i}}\tau_{nm}^{-1},
\\
b_{nm}^{6}&=&{1\over\hbar}(\varepsilon_n-\varepsilon_m)
-{\hbar\over2m_e}(q_x-k_x)^2-{\rm{i}}\tau_{nm}^{-1},
\\
b_{nm}^{7}&=&{1\over\hbar}(\varepsilon_n-\varepsilon_m)
+{\hbar\over2m_e}(q_x+k_x)^2-{\rm{i}}\tau_{nm}^{-1},
\\
b_{nm}^{8}&=&{1\over\hbar}(\varepsilon_n-\varepsilon_m)
+{\hbar\over2m_e}(q_x^2-k_x^2)-{\rm{i}}\tau_{nm}^{-1},
\\
b_{nm}^{9}&=&{1\over\hbar}(\varepsilon_n-\varepsilon_m)
+{\hbar{}q_x\over2m_e}(q_x+2k_x)+\omega-{\rm{i}}\tau_{nm}^{-1},
\\
b_{nm}^{10}&=&{1\over\hbar}(\varepsilon_n-\varepsilon_m)
+{\hbar{}k_x\over2m_e}(k_x+2q_x)-\omega-{\rm{i}}\tau_{nm}^{-1},
\\
b_{nm}^{11}&=&{1\over\hbar}(\varepsilon_n-\varepsilon_m)
+{\hbar{}q_x\over2m_e}(q_x-2k_x)+\omega-{\rm{i}}\tau_{nm}^{-1},
\\
b_{nm}^{12}&=&{1\over\hbar}(\varepsilon_n-\varepsilon_m)
+{\hbar{}k_x\over2m_e}(2q_x-k_x)-\omega-{\rm{i}}\tau_{nm}^{-1},
\\
b_{nm}^{13}&=&{1\over\hbar}(\varepsilon_n-\varepsilon_m)
+{\hbar{}k_x\over2m_e}(k_x-2q_x)-\omega-{\rm{i}}\tau_{nm}^{-1}.\end{aligned}$$
Nonlinear process A
-------------------
Starting with the pure interband term in Eq. (\[eq:XiAA\]) we separate the $z$-dependent and the $\vec{\kappa}_{\|}$-dependent parts in the following way: $$\begin{aligned}
{\Xi}_{xxxx}^{\rm{A}}(z,z',z'',z''';\vec{q}_{\|},\vec{k}_{\|})=
{2\over({\rm{i}}\omega)^3}
\sum_{nm}{\cal{Z}}_{nm}^{\rm{A}}(z,z',z'',z''')
\xi_{nm}^{\rm{A}},\end{aligned}$$ where the indices on the quantities ${\cal{Z}}$ and $\xi$ follows the indices of the quantum numbers in the sum. The $z$-dependent part above is $${\cal{Z}}_{nm}^{\rm{A}}(z,z',z'',z''')=
{\cal{Z}}_{mn}^{x}(z''){\cal{Z}}_{nm}^{x}(z)
\delta(z-z')\delta(z''-z'''),$$ and the solution to the integral over $\vec{\kappa}_{\|}$ in the low-temperature limit $$\xi_{nm}^{\rm{A}}=
-{e^4\over32\pi{}m_e^2\hbar}
{\alpha_n^2-\alpha_m^2\over(\varepsilon_n-\varepsilon_m)/\hbar
-2\omega-{\rm{i}}\tau_{nm}^{-1}}$$ readily appears by use of Eq. (\[eq:Solve-Q-Beta=0\]), since only the Fermi-Dirac distribution functions depend on $\vec{\kappa}_{\|}$.
Nonlinear process B
-------------------
In order to solve the integral over $\kappa_{\|}$ in Eq. (\[eq:XiAB\]), we rewrite it into $$\begin{aligned}
{\Xi}_{xxkh}^{\rm{B}}(z,z',z'',z''';\vec{q}_{\|},\vec{k}_{\|})=
{2\over({\rm{i}}\omega)^3}\left({e\hbar\over2m_e}\right)^2\sum_{nmv}
{\cal{Z}}_{kh,nmv}^{\rm{B}}(z,z',z'',z''')
\xi_{kh,nmv}^{\rm{B}}(\vec{q}_{\|},\vec{k}_{\|}),
\nonumber\\
\label{eq:XiAB-M}\end{aligned}$$ in which both quantities ${\cal{Z}}$ and $\xi$ are indexed according to their dependence on the two Cartesian indices of $\Xi$ and the quantum numbers in the sum. Above, the $z$-dependent part in general is given by $$\begin{aligned}
{\cal{Z}}_{kh,nmv}^{\rm{B}}(z,z',z'',z''')=
{\cal{Z}}_{v{}n}^{h}(z'''){\cal{Z}}_{mv}^{k}(z'')
\delta(z-z'){\cal{Z}}_{nm}^{x}(z),\end{aligned}$$ in terms of Eqs. (\[eq:Z-nm-xy\]) and (\[eq:Z-nm-z\]). Of these, the two with Cartesian indices $xx$ and $yy$ are equal. The solution to the other quantity above, $\xi_{kh,nmv}^{\rm{B}}(\vec{q}_{\|},\vec{k}_{\|})$, in terms of the $a$’s and $b$’s and the functions solved in Appendix \[ch:Solve-Q\], is written $$\begin{aligned}
\lefteqn{
\xi_{kh,nmv}^{\rm{B}}(\vec{q}_{\|},\vec{k}_{\|})=
-{e^2\over4m_e\hbar^2}{1\over(2\pi)^2}
{1\over(\varepsilon_n-\varepsilon_m)/\hbar-2\omega-{\rm{i}}\tau_{nm}^{-1}}
\left\{
{\cal{F}}_{kh}^{\rm{B}}(m,a_1,b_{v{}m}^{1},0)
\right.}\nonumber\\ &&\left.
-{\cal{F}}_{kh}^{\rm{B}}(v,a_1,b_{v{}m}^{1},k_x)
+{\cal{F}}_{kh}^{\rm{B}}(n,-a_1,b_{nv}^{2},0)
-{\cal{F}}_{kh}^{\rm{B}}(v,-a_1,b_{nv}^{2},k_x)
\right\},\end{aligned}$$ which is written in terms of a set of functions ${\cal{F}}$ that vary from element to element. These functions are of the order $\beta=1$ because only one transition frequency appears inside each integral. They are determined from the $\vec{\kappa}_{\|}$-dependent parts of the microscopic current densities, and they become $$\begin{aligned}
{\cal{F}}_{zz}^{\rm{B}}&=&{\cal{F}}_{00}^{1},
\\
{\cal{F}}_{yy}^{\rm{B}}&=&4{\cal{F}}_{02}^{1},
\\
{\cal{F}}_{xz}^{\rm{B}}&=&{\cal{F}}_{zx}^{\rm{B}}=
2{\cal{F}}_{10,n}^{1}+k_x{\cal{F}}_{00}^{1},
\\
{\cal{F}}_{xx}^{\rm{B}}&=&4{\cal{F}}_{20}^{1}
+4k_x{\cal{F}}_{10}^{1}+k_x^2{\cal{F}}_{00}^{1},\end{aligned}$$ in short notation, since the functions at the right side of these equations take the same arguments as the functions to the left.
Nonlinear process C
-------------------
Separating Eq. (\[eq:XiAC\]) into its $z$-dependent and $\vec{\kappa}_{\|}$-dependent parts, we write $$\begin{aligned}
{\Xi}_{xxxx}^{\rm{C}}(z,z',z'',z''';\vec{q}_{\|},\vec{k}_{\|})=
{2\over({\rm{i}}\omega)^3}
\sum_{nm}{\cal{Z}}_{nm}^{\rm{C}}(z,z',z'',z''')
\xi_{nm}^{\rm{C}}(\vec{q}_{\|},\vec{k}_{\|})
\label{eq:XiAC-M}\end{aligned}$$ where the indices on the new quantities follows the quantum numbers in the sum. The $z$-independent part in this part of the conductivity tensor is $${\cal{Z}}_{nm}^{\rm{C}}(z,z',z'',z''')=
{\cal{Z}}_{mn}^{x}(z'){\cal{Z}}_{nm}^{x}(z)
\delta(z'-z''')\delta(z-z''),$$ in terms of Eq. (\[eq:Z-nm-xy\]). The solution to the quantity $\xi_{nm}^{\rm{C}}(\vec{q}_{\|},\vec{k}_{\|})$ then appears as $$\begin{aligned}
\xi_{nm}^{\rm{C}}(\vec{q}_{\|},\vec{k}_{\|})=
-{e^4\over16\pi^2m_e^2\hbar}
\left\{
{\cal{F}}_{00}^{1}(n,a_4,b_{nm}^{5},q_x-k_x)
-{\cal{F}}_{00}^{1}(m,a_4,b_{nm}^{5},0)
\right\}\end{aligned}$$ in terms of the $a$’s and $b$’s and the functions solved in Appendix \[ch:Solve-Q\].
Nonlinear process D
-------------------
Performing an adequate separation of variables in Eq. (\[eq:XiAD\]), it is written $$\begin{aligned}
\lefteqn{
{\Xi}_{xjkx}^{\rm{D}}(z,z',z'',z''';\vec{q}_{\|},\vec{k}_{\|})=
{2\over({\rm{i}}\omega)^3}\left({e\hbar\over2m_e}\right)^2
\sum_{nmv}
\left\{
{\cal{Z}}_{jk,nmv}^{\rm{Da}}(z,z',z'',z''')
\xi_{jk,nmv}^{\rm{Da}}(\vec{q}_{\|},\vec{k}_{\|})
\right.}\nonumber\\ &\quad&\left.
+
{\cal{Z}}_{jk,nmv}^{\rm{Db}}(z,z',z'',z''')
\xi_{jk,nmv}^{\rm{Db}}(\vec{q}_{\|},\vec{k}_{\|})
\right\},
\label{eq:XiAD-M}\end{aligned}$$ where the four new quantities are indexed according to the varying Cartesian coordinates of $\Xi$ and the quantum numbers in the sum. The $z$-dependent terms in Eq. (\[eq:XiAD-M\]) are $$\begin{aligned}
{\cal{Z}}_{jk,nmv}^{\rm{Da}}(z,z',z'',z''')&=&\delta(z-z''')
{\cal{Z}}_{mv}^{k}(z''){\cal{Z}}_{v{}n}^{j}(z'){\cal{Z}}_{nm}^{x}(z),
\\
{\cal{Z}}_{jk,nmv}^{\rm{Db}}(z,z',z'',z''')&=&\delta(z-z''')
{\cal{Z}}_{v{}n}^{k}(z''){\cal{Z}}_{mv}^{j}(z'){\cal{Z}}_{nm}^{x}(z)\end{aligned}$$ in terms of Eqs. (\[eq:Z-nm-xy\]) and (\[eq:Z-nm-z\]). In both equations above, the $xx$ and $yy$ permutations are the same. The solution to the $\xi$ quantities we write in terms of the $a$’s, $b$’s, and the functions solved in Appendix \[ch:Solve-Q\], the result being $$\begin{aligned}
\lefteqn{
\xi_{jk,nmv}^{\rm{Da}}(\vec{q}_{\|},\vec{k}_{\|})=
-{e^2\over16\pi^2m_e\hbar^2}
\left\{
{\cal{F}}_{jk}^{\rm{Da}}(m,\{a_3,a_1\},\{b_{nm}^{7},b_{v{}m}^{1}\},0)
\right.}\nonumber\\ &\quad&
-{\cal{F}}_{jk}^{\rm{Da}}(v,\{a_3,a_1\},\{b_{nm}^{7},b_{v{}m}^{1}\},k_x)
+{\cal{F}}_{jk}^{\rm{Da}}(n,\{a_3,a_2\},\{b_{nm}^{7},b_{nv}^{9}\},q_x+k_x)
\nonumber\\ &&\left.
-{\cal{F}}_{jk}^{\rm{Da}}(v,\{a_3,a_2\},\{b_{nm}^{7},b_{nv}^{9}\},k_x)
\right\},
\\
\lefteqn{
\xi_{jk,nmv}^{\rm{Db}}(\vec{q}_{\|},\vec{k}_{\|})=
-{e^2\over16\pi^2m_e\hbar^2}
\left\{
{\cal{F}}_{jk}^{\rm{Db}}(m,\{a_3,a_2\},\{b_{nm}^{7},b_{v{}m}^{3}\},0)
\right.}\nonumber\\ &&
-{\cal{F}}_{jk}^{\rm{Db}}(v,\{a_3,a_2\},\{b_{nm}^{7},b_{v{}m}^{3}\},q_x)
+{\cal{F}}_{jk}^{\rm{Db}}(n,\{a_3,a_1\},\{b_{nm}^{7},b_{nv}^{10}\},q_x+k_x)
\nonumber\\ &&\left.
-{\cal{F}}_{jk}^{\rm{Db}}(v,\{a_3,a_1\},\{b_{nm}^{7},b_{nv}^{10}\},q_x)
\right\},\end{aligned}$$ which have been written in terms of a set of functions ${\cal{F}}$ that vary from element to element. These functions are again determined from the $\vec{\kappa}_{\|}$-dependent part of the microscopic current densities appearing in ${\Xi}_{xjkx}^{\rm{D}}(z,z',z'',z''';\vec{q}_{\|},\vec{k}_{\|})$. They are $$\begin{aligned}
{\cal{F}}_{zz}^{\rm{Da}}&=&
{\cal{F}}_{zz}^{\rm{Db}}=
{\cal{F}}_{00}^{2},
\\
{\cal{F}}_{yy}^{\rm{Da}}&=&
{\cal{F}}_{yy}^{\rm{Db}}=
4{\cal{F}}_{02}^{2},
\\
{\cal{F}}_{xz}^{\rm{Da}}&=&2{\cal{F}}_{10}^{2}+(2k_x+q_x){\cal{F}}_{00,n}^{2},
\\
{\cal{F}}_{xz}^{\rm{Db}}&=&2{\cal{F}}_{10}^{2}+q_x{\cal{F}}_{00}^{2},
\\
{\cal{F}}_{zx}^{\rm{Da}}&=&2{\cal{F}}_{10}^{2}+k_x{\cal{F}}_{00}^{2},
\\
{\cal{F}}_{zx}^{\rm{Db}}&=&2{\cal{F}}_{10}^{2}+(k_x+2q_x){\cal{F}}_{00}^{2},
\\
{\cal{F}}_{xx}^{\rm{Da}}&=&4{\cal{F}}_{20}^{2}+2(q_x+3k_x){\cal{F}}_{10}^{2}
+(2k_x^2+q_xk_x){\cal{F}}_{00}^{2},
\\
{\cal{F}}_{xx}^{\rm{Db}}&=&4{\cal{F}}_{20}^{2}+2(3q_x+k_x){\cal{F}}_{10}^{2}
+(2q_x^2+q_xk_x){\cal{F}}_{00}^{2},\end{aligned}$$ again in short notation, and for the same reason as before.
Nonlinear process E
-------------------
Separation of the $z$-dependent part and the $\vec{\kappa}_{\|}$-dependent part in Eq. (\[eq:XiAE\]) yields $$\begin{aligned}
\lefteqn{
{\Xi}_{ijxx}^{\rm{E}}(z,z',z'',z''';\vec{q}_{\|},\vec{k}_{\|})=
{2\over({\rm{i}}\omega)^3}\left({e\hbar\over2m_e}\right)^2\sum_{nmv}
\left\{
{\cal{Z}}_{ij,nmv}^{\rm{Ea}}(z,z',z'',z''')
\xi_{ij,nmv}^{\rm{Ea}}(\vec{q}_{\|},\vec{k}_{\|})
\right.}\nonumber\\ &\quad&\left.
+
{\cal{Z}}_{ij,nmv}^{\rm{Eb}}(z,z',z'',z''')
\xi_{ij,nmv}^{\rm{Eb}}(\vec{q}_{\|},\vec{k}_{\|})
\right\},
\label{eq:XiAE-M}\end{aligned}$$ the new quantities being indexed according to their dependence on the varying Cartesian coordinates in ${\Xi}_{ijxx}^{\rm{E}}(z,z',z'',z''';\vec{q}_{\|},\vec{k}_{\|})$ and the quantum numbers in the sum. The $z$-dependent quantities are again written in terms of Eqs. (\[eq:Z-nm-xy\])–(\[eq:Z-nm-z\]), with the result $$\begin{aligned}
{\cal{Z}}_{ij,nmv}^{\rm{Ea}}(z,z',z'',z''')&=&\delta(z''-z''')
{\cal{Z}}_{mv}^{x}(z''){\cal{Z}}_{v{}n}^{j}(z'){\cal{Z}}_{nm}^{i}(z),
\\
{\cal{Z}}_{ij,nmv}^{\rm{Eb}}(z,z',z'',z''')&=&\delta(z''-z''')
{\cal{Z}}_{v{}n}^{x}(z''){\cal{Z}}_{mv}^{j}(z'){\cal{Z}}_{nm}^{i}(z),\end{aligned}$$ and again it appears, the quantities with Cartesian indices $xx$ and $yy$ are equal in each of the above equations. The solutions to the two $\xi$ quantities are obtained in terms of the $a$’s and $b$’s and the functions solved in Appendix \[ch:Solve-Q\], and they become $$\begin{aligned}
\lefteqn{
\xi_{ij,nmv}^{\rm{Ea}}(\vec{q}_{\|},\vec{k}_{\|})=
-{e^2\over2^6\pi^2m_e\hbar^2}
\left\{
{{\cal{F}}_{ij}^{\rm{E}1}(m,a_2,b_{nm}^{4},0)
-{\cal{F}}_{ij}^{\rm{E}1}(v,a_2,b_{nm}^{4},0)
\over(\varepsilon_{v}-\varepsilon_m)/\hbar-2\omega-{\rm{i}}\tau_{v{}m}^{-1}}
\right.}\nonumber\\ &\quad&
+{\cal{F}}_{ij}^{\rm{E}2}(n,\{a_2,a_2\},\{b_{nm}^{4},b_{nv}^{3}\},q_x)
-{\cal{F}}_{ij}^{\rm{E}2}(v,\{a_2,a_2\},\{b_{nm}^{4},b_{nv}^{3}\},0)
\Bigr\},
\\
\lefteqn{
\xi_{ij,nmv}^{\rm{Eb}}(\vec{q}_{\|},\vec{k}_{\|})=
-{e^2\over2^6\pi^2m_e\hbar^2}
\left\{
{{\cal{F}}_{ij}^{\rm{E}1}(n,a_2,b_{nm}^{4},q_x)
-{\cal{F}}_{ij}^{\rm{E}1}(v,a_2,b_{nm}^{4},q_x)
\over(\varepsilon_n-\varepsilon_{v})/\hbar-2\omega-{\rm{i}}\tau_{nv}^{-1}}
\right.}\nonumber\\ &&
+{\cal{F}}_{ij}^{\rm{E}2}(m,\{a_2,a_2\},\{b_{nm}^{4},b_{v{}m}^{3}\},0)
-{\cal{F}}_{ij}^{\rm{E}2}(v,\{a_2,a_2\},\{b_{nm}^{4},b_{v{}m}^{3}\},q_x)
\Bigr\},\end{aligned}$$ which is written in terms of a set of functions ${\cal{F}}$ that vary from element to element. Their structure is as before determined from the $\vec{\kappa}_{\|}$-dependent parts of the transition current densities, and we find for the pure interband transitions functions of order $\beta=1$, since only one transition frequency occurs in each integral. They are $$\begin{aligned}
{\cal{F}}_{zz}^{\rm{E}1}&=&{\cal{F}}_{00}^{1},
\\
{\cal{F}}_{yy}^{\rm{E}1}&=&4{\cal{F}}_{02}^{1},
\\
{\cal{F}}_{xz}^{\rm{E}1}&=&{\cal{F}}_{zx}^{\rm{E}1}=
2{\cal{F}}_{10}^{1}+q_x{\cal{F}}_{00}^{1},
\\
{\cal{F}}_{xx}^{\rm{E}1}&=&4{\cal{F}}_{20}^{1}+2q_x{\cal{F}}_{10}^{1}
+2q_x^2{\cal{F}}_{00}^{1},\end{aligned}$$ in short notation, and for the mixed interband/intraband transitions functions of order $\beta=2$ because two transition frequencies occur in each integral. They are $$\begin{aligned}
{\cal{F}}_{zz}^{\rm{E}2}&=&{\cal{F}}_{00}^{2},
\\
{\cal{F}}_{yy}^{\rm{E}2}&=&4{\cal{F}}_{02}^{2},
\\
{\cal{F}}_{xz}^{\rm{E}2}&=&{\cal{F}}_{zx}^{\rm{E}2}=
2{\cal{F}}_{10}^{2}+q_x{\cal{F}}_{00}^{2},
\\
{\cal{F}}_{xx}^{\rm{E}2}&=&4{\cal{F}}_{20}^{2}
+2q_x{\cal{F}}_{10}^{2}+2q_x^2{\cal{F}}_{00}^{2},\end{aligned}$$ again in short notation, since all arguments to the functions are of the same type.
Nonlinear process F
-------------------
The separation of variables of Eq. (\[eq:XiAF\]) into $z$-independent and $\vec{\kappa}_{\|}$-independent terms gives $$\begin{aligned}
\lefteqn{ {\Xi}_{ixxh}^{\rm{F}}(z,z',z'',z''';\vec{q}_{\|},\vec{k}_{\|})=
{2\over({\rm{i}}\omega)^3}\left({e\hbar\over2m_e}\right)^2
\sum_{nmv}
\left\{
{\cal{Z}}_{ih,nmv}^{\rm{Fa}}(z,z',z'',z''')
\xi_{ih,nmv}^{\rm{Fa}}(\vec{q}_{\|},\vec{k}_{\|})
\right.}\nonumber\\ &\quad&\left.
+
{\cal{Z}}_{ih,nmv}^{\rm{Fb}}(z,z',z'',z''')
\xi_{ih,nmv}^{\rm{Fb}}(\vec{q}_{\|},\vec{k}_{\|})
\right\},
\label{eq:XiAF-M}\end{aligned}$$ where the new quantities have been indexed according to their dependence on the Cartesian indices of $\Xi$ and the quantum numbers in the sum. In Eq. (\[eq:XiAF-M\]) above, the $\vec{\kappa}_{\|}$-independent terms are $$\begin{aligned}
{\cal{Z}}_{ih,nmv}^{\rm{Fa}}(z,z',z'',z''')&=&\delta(z'-z''')
{\cal{Z}}_{v{}n}^{h}(z''){\cal{Z}}_{mv}^{x}(z'){\cal{Z}}_{nm}^{i}(z),
\\
{\cal{Z}}_{ih,nmv}^{\rm{Fb}}(z,z',z'',z''')&=&\delta(z'-z''')
{\cal{Z}}_{mv}^{h}(z''){\cal{Z}}_{v{}n}^{x}(z'){\cal{Z}}_{nm}^{i}(z).\end{aligned}$$ in terms of the three quantities defined in Eqs. (\[eq:Z-nm-xy\])–(\[eq:Z-nm-z\]), and again the $xx$ and $yy$ elements in each of the two above quantities are equal. The solutions to the $z$-independent terms appear in terms of the $a$’s and $b$’s and the functions solved in Appendix \[ch:Solve-Q\] as $$\begin{aligned}
\lefteqn{
\xi_{ih,nmv}^{\rm{Fa}}(\vec{q}_{\|},\vec{k}_{\|})=
-{e^2\over2^5\pi^2m_e\hbar^2}
\Bigl\{
{\cal{F}}_{ih}^{\rm{Fa}}(m,\{a_2,a_4\},\{b_{nm}^{4},b_{v{}m}^{5}\},0)
}\nonumber\\ &\quad&
-{\cal{F}}_{ih}^{\rm{Fa}}(v,\{a_2,a_4\},\{b_{nm}^{4},b_{v{}m}^{5}\},q_x-k_x)
+{\cal{F}}_{ih}^{\rm{Fa}}(n,\{a_2,a_1\},\{b_{nm}^{4},b_{nv}^{12}\},q_x)
\nonumber\\ &&
-{\cal{F}}_{ih}^{\rm{Fa}}(v,\{a_2,a_1\},\{b_{nm}^{4},b_{nv}^{12}\},q_x-k_x)
\Bigr\},
\\
\lefteqn{
\xi_{ih,nmv}^{\rm{Fb}}(\vec{q}_{\|},\vec{k}_{\|})=
-{e^2\over2^5\pi^2m_e\hbar^2}
\Bigl\{
{\cal{F}}_{ih}^{\rm{Fb}}(m,\{a_2,a_1\},\{b_{nm}^{4},b_{v{}m}^{1}\},0)
}\nonumber\\ &&
-{\cal{F}}_{ih}^{\rm{Fb}}(v,\{a_2,a_1\},\{b_{nm}^{4},b_{v{}m}^{1}\},k_x)
+{\cal{F}}_{ih}^{\rm{Fb}}(n,\{a_2,a_4\},\{b_{nm}^{4},b_{nv}^{8}\},q_x)
\nonumber\\ &&
-{\cal{F}}_{ih}^{\rm{Fb}}(v,\{a_2,a_4\},\{b_{nm}^{4},b_{nv}^{8}\},k_x)
\Bigr\},\end{aligned}$$ which are written in terms of a set of functions ${\cal{F}}$ that vary from element to element. They are again determined from the $\vec{\kappa}_{\|}$-dependent parts of the transition current densities appearing, and thus they become $$\begin{aligned}
{\cal{F}}_{zz}^{\rm{Fa}}&=&{\cal{F}}_{zz}^{\rm{Fb}}={\cal{F}}_{00}^{2},
\\
{\cal{F}}_{yy}^{\rm{Fa}}&=&{\cal{F}}_{yy}^{\rm{Fb}}=4{\cal{F}}_{02}^{2},
\\
{\cal{F}}_{xz}^{\rm{Fa}}&=&{\cal{F}}_{xz}^{\rm{Fb}}=2{\cal{F}}_{00}^{2}
+q_x{\cal{F}}_{00}^{2},
\\
{\cal{F}}_{zx}^{\rm{Fa}}&=&2{\cal{F}}_{10}^{2}+(q_x-k_x){\cal{F}}_{00}^{2},
\\
{\cal{F}}_{zx}^{\rm{Fb}}&=&2{\cal{F}}_{10}^{2}+k_x{\cal{F}}_{00}^{2},
\\
{\cal{F}}_{xx}^{\rm{Fa}}&=&4{\cal{F}}_{20}^{2}
+2(2q_x-k_x){\cal{F}}_{00}^{2}+(2q_x^2-q_xk_x){\cal{F}}_{00}^{2},
\\
{\cal{F}}_{xx}^{\rm{Fb}}&=&4{\cal{F}}_{20}^{2}
+2(q_x+k_x){\cal{F}}_{10}^{2}+q_xk_x{\cal{F}}_{00}^{2},\end{aligned}$$ again in the abbreviated notation, where the functions in general take arguments of the type $(n,\{a_1,a_2\},\{b_1,b_2\},s)$.
Nonlinear process G
-------------------
Finally, Eq. (\[eq:XiAG\]) becomes in terms of $z$-independent and $\vec{\kappa}_{\|}$-independent terms $$\begin{aligned}
\lefteqn{{\Xi}_{ijkh}^{\rm{G}}(z,z',z'',z''';\vec{q}_{\|},\vec{k}_{\|})=
{2\over({\rm{i}}\omega)^3}\left({e\hbar\over2m_e}\right)^4
\sum_{nmvl}
\left\{
{\cal{Z}}_{ijkh,nmvl}^{\rm{Ga}}(z,z',z'',z''')
\xi_{ijkh,nmvl}^{\rm{Ga}}(\vec{q}_{\|},\vec{k}_{\|})
\right.}\nonumber\\ &\quad&\left.
+{\cal{Z}}_{ijkh,nmvl}^{\rm{Gb}}(z,z',z'',z''')
\xi_{ijkh,nmvl}^{\rm{Gb}}(\vec{q}_{\|},\vec{k}_{\|})
+{\cal{Z}}_{ijkh,nmvl}^{\rm{Gc}}(z,z',z'',z''')
\xi_{ijkh,nmvl}^{\rm{Gc}}(\vec{q}_{\|},\vec{k}_{\|})
\!\right\},
\nonumber\\
\label{eq:XiAG-M}\end{aligned}$$ where again the new quantities have been indexed according to their dependence on the various Cartesian indices of $\Xi$ and the quantum numbers of the sum. The $z$-dependent terms in Eq. (\[eq:XiAG-M\]) are on general form $$\begin{aligned}
{\cal{Z}}_{ijkh,nmvl}^{\rm{Ga}}(z,z',z'',z''')=
{\cal{Z}}_{ml}^{h}(z'''){\cal{Z}}_{lv}^{k}(z'')
{\cal{Z}}_{v{}n}^{j}(z'){\cal{Z}}_{nm}^{i}(z),
\label{eq:Z-ijkh,nmvl-Ga}\\
{\cal{Z}}_{ijkh,nmvl}^{\rm{Gb}}(z,z',z'',z''')=
{\cal{Z}}_{ml}^{h}(z'''){\cal{Z}}_{v{}n}^{k}(z'')
{\cal{Z}}_{lv}^{j}(z'){\cal{Z}}_{nm}^{i}(z),
\label{eq:Z-ijkh,nmvl-Gb}\\
{\cal{Z}}_{ijkh,nmvl}^{\rm{Gc}}(z,z',z'',z''')=
{\cal{Z}}_{lv}^{h}(z'''){\cal{Z}}_{v{}n}^{k}(z'')
{\cal{Z}}_{ml}^{j}(z'){\cal{Z}}_{nm}^{i}(z),
\label{eq:Z-ijkh,nmvl-Gc}
\label{eq:Z-ijkh,nmvl-Gf}\end{aligned}$$ in terms of the quantities defined in Eqs. (\[eq:Z-nm-xy\])–(\[eq:Z-nm-z\]), and as in the previous cases we may observe that any element with a Cartesian index $x$ is equal to the element with the Cartesian index $y$ on the same place, the other Cartesian indices unchanged. The $z$-independent terms we write using the $a$’s and $b$’s and the functions solved in Appendix \[ch:Solve-Q\], as before. They finally become $$\begin{aligned}
\lefteqn{
\xi_{ijkh,nmvl}^{\rm{Ga}}(\vec{q}_{\|},\vec{k}_{\|})=
-{1\over8\hbar^3}{1\over(2\pi)^2}
\left\{
{1\over(\varepsilon_{v}-\varepsilon_m)/\hbar-2\omega-{\rm{i}}\tau_{v{}m}^{-1}}
\right.}\nonumber\\ &\quad&\times
\Bigl[
{\cal{F}}_{ijkh}^{\rm{Ga}1}
\left(l,\{a_2,-a_1\},\{b_{nm}^{4},b_{{l}m}^{1}\},-k_x\right)
-{\cal{F}}_{ijkh}^{\rm{Ga}1}
\left(m,\{a_2,-a_1\},\{b_{nm}^{4},b_{{l}m}^{1}\},0\right)
\nonumber\\ &&
+{\cal{F}}_{ijkh}^{\rm{Ga}1}
\left(l,\{a_2,a_1\},\{b_{nm}^{4},b_{vl}^{2}\},-k_x\right)
-{\cal{F}}_{ijkh}^{\rm{Ga}1}
\left(v,\{a_2,a_1\},\{b_{nm}^{4},b_{vl}^{2}\},0\right)
\Bigr]
\nonumber\\ &&
+{\cal{F}}_{ijkh}^{\rm{Ga}2}
\left(l,\{a_2,a_1,a_3\},\{b_{nm}^{4},b_{vl}^{2},b_{nl}^{8}\},-k_x
\right)
\nonumber\\ &&
-{\cal{F}}_{ijkh}^{\rm{Ga}2}
\left(v,\{a_2,a_1,a_3\},\{b_{nm}^{4},b_{vl}^{2},b_{nl}^{8}\},0
\right)
\nonumber\\ &&
+{\cal{F}}_{ijkh}^{\rm{Ga}2}
\left(n,\{a_2,a_2,a_3\},\{b_{nm}^{4},b_{nv}^{3},b_{nl}^{8}\},q_x\right)
\nonumber\\ &&
-{\cal{F}}_{ijkh}^{\rm{Ga}2}
\left(v,\{a_2,a_2,a_3\},\{b_{nm}^{4},b_{nv}^{3},b_{nl}^{8}\},0\right)
\Bigr\}
\\
\lefteqn{
\xi_{ijkh,nmvl}^{\rm{Gb}}(\vec{q}_{\|},\vec{k}_{\|})=
-{1\over8\hbar^3}{1\over(2\pi)^2}
\Bigl\{
{\cal{F}}_{ijkh}^{\rm{Gb}}
\left(l,\{a_2,-a_1,a_4\},\{b_{nm}^{4},b_{{l}m}^{1},b_{{v}m}^{5}\},
-k_x\right)
}\nonumber\\ &&
-{\cal{F}}_{ijkh}^{\rm{Gb}}
\left(m,\{a_2,-a_1,a_4\},\{b_{nm}^{4},b_{{l}m}^{1},b_{{v}m}^{5}\},0
\right)
\nonumber\\ &&
+{\cal{F}}_{ijkh}^{\rm{Gb}}
\left(l,\{a_2,a_2,a_4\},\{b_{nm}^{4},b_{vl}^{11},b_{{v}m}^{5}\},
-k_x\right)
\nonumber\\ &&
-{\cal{F}}_{ijkh}^{\rm{Gb}}
\left(v,\{a_2,a_2,a_4\},\{b_{nm}^{4},b_{vl}^{11},b_{{v}m}^{5}\},
q_x-k_x\right)
\nonumber\\ &&
+{\cal{F}}_{ijkh}^{\rm{Gb}}
\left(l,\{a_2,a_2,a_3\},\{b_{nm}^{4},b_{vl}^{11},b_{nl}^{8}\},
-k_x\right)
\nonumber\\ &&
-{\cal{F}}_{ijkh}^{\rm{Gb}}
\left(v,\{a_2,a_2,a_3\},\{b_{nm}^{4},b_{vl}^{11},b_{nl}^{8}\},
q_x-k_x\right)
\nonumber\\ &&
+{\cal{F}}_{ijkh}^{\rm{Gb}}
\left(n,\{a_2,a_1,a_3\},\{b_{nm}^{4},b_{nv}^{12},b_{nl}^{8}\},q_x\right)
\nonumber\\ &&
-{\cal{F}}_{ijkh}^{\rm{Gb}}
\left(v,\{a_2,a_1,a_3\},\{b_{nm}^{4},b_{nv}^{12},b_{nl}^{8}\},
q_x-k_x\right)
\Bigr\}
\\
\lefteqn{
\xi_{ijkh,nmvl}^{\rm{Gc}}(\vec{q}_{\|},\vec{k}_{\|})=
-{1\over8\hbar^3}{1\over(2\pi)^2}
\Bigl\{
{\cal{F}}_{ijkh}^{\rm{Gc}1}
\left(l,\{a_2,a_2,a_4\},\{b_{nm}^{4},b_{{l}m}^{3},b_{{v}m}^{5}\},q_x
\right)
}\nonumber\\ &&
-{\cal{F}}_{ijkh}^{\rm{Gc}1}
\left(m,\{a_2,a_2,a_4\},\{b_{nm}^{4},b_{{l}m}^{3},b_{{v}m}^{5}\},0
\right)
\nonumber\\ &&
+{\cal{F}}_{ijkh}^{\rm{Gc}1}
\left(l,\{a_2,-a_1,a_4\},\{b_{nm}^{4},b_{vl}^{13},b_{{v}m}^{5}\},
q_x\right)
\nonumber\\ &&
-{\cal{F}}_{ijkh}^{\rm{Gc}1}
\left(v,\{a_2,-a_1,a_4\},\{b_{nm}^{4},b_{vl}^{13},b_{{v}m}^{5}\},
q_x-k_x\right)
+{1\over(\varepsilon_n-\varepsilon_{l})/\hbar-2\omega-{\rm{i}}\tau_{nl}^{-1}}
\nonumber\\ &&\times
\Bigl[
{\cal{F}}_{ijkh}^{\rm{Gc}2}
\left(l,\{a_2,-a_1\},\{b_{nm}^{4},b_{vl}^{13}\},q_x\right)
-{\cal{F}}_{ijkh}^{\rm{Gc}2}
\left(v,\{a_2,-a_1\},\{b_{nm}^{4},b_{vl}^{13}\},q_x-k_x\right)
\nonumber\\ &&
+{\cal{F}}_{ijkh}^{\rm{Gc}2}
\left(n,\{a_2,a_1\},\{b_{nm}^{4},b_{nv}^{12}\},q_x\right)
-{\cal{F}}_{ijkh}^{\rm{Gc}2}
\left(v,\{a_2,a_1\},\{b_{nm}^{4},b_{nv}^{12}\},q_x-k_x\right)
\Bigr]\Bigr\},
\nonumber\\
\label{eq:GenericXi}\end{aligned}$$ and again they are written in terms of a set of functions ${\cal{F}}$ that vary from element to element. As was the case in the previous sections, these functions are determined from the $z$-independent parts of the transition current densities appearing in $\Xi$.
In passing we should notice that parts $(Ga1)$ and $(Gc2)$ has $\beta=2$ because of the pure interband transition appearing in one of their denominators, while parts $(Ga2)$, $(Gb)$, and $(Gc1)$ has $\beta=3$ since all their transitions are mixed interband/intraband transitions, we observe that a lot of ${\cal{F}}$ functions are equal. In the simplest case, we observe $$\begin{aligned}
{\cal{F}}_{zzzz}^{\rm{Ga}1}&=&{\cal{F}}_{zzzz}^{\rm{Gc}2}
={\cal{F}}_{00}^{2}
\\
{\cal{F}}_{zzzz}^{\rm{Gb}}&=&{\cal{F}}_{zzzz}^{\rm{Gc}1}=
{\cal{F}}_{zzzz}^{\rm{Ga}2}
={\cal{F}}_{00}^{3}.\end{aligned}$$ At the second level of complexity we find $$\begin{aligned}
\lefteqn{
{\cal{F}}_{yyzz}^{\rm{Ga}1}={\cal{F}}_{yyzz}^{\rm{Gc}2}=
{\cal{F}}_{yzyz}^{\rm{Ga}1}={\cal{F}}_{yzyz}^{\rm{Gc}2}=
{\cal{F}}_{yzzy}^{\rm{Ga}1}={\cal{F}}_{yzzy}^{\rm{Gc}2}=
{\cal{F}}_{zyyz}^{\rm{Ga}1}={\cal{F}}_{zyyz}^{\rm{Gc}2}=
{\cal{F}}_{zyzy}^{\rm{Ga}1}={\cal{F}}_{zyzy}^{\rm{Gc}2}=
}\nonumber\\ &\quad&
{\cal{F}}_{zzyy}^{\rm{Ga}1}={\cal{F}}_{zzyy}^{\rm{Gc}2}
=4{\cal{F}}_{02}^{2},
\\
\lefteqn{
{\cal{F}}_{yyzz}^{\rm{Gb}}={\cal{F}}_{yyzz}^{\rm{Gc}1}=
{\cal{F}}_{yyzz}^{\rm{Ga}2}={\cal{F}}_{yzyz}^{\rm{Gb}}=
{\cal{F}}_{yzyz}^{\rm{Gc}1}={\cal{F}}_{yzyz}^{\rm{Ga}2}=
{\cal{F}}_{yzzy}^{\rm{Gb}}={\cal{F}}_{yzzy}^{\rm{Gc}1}=
{\cal{F}}_{yzzy}^{\rm{Ga}2}={\cal{F}}_{zyyz}^{\rm{Gb}}=
}\nonumber\\ &&
{\cal{F}}_{zyyz}^{\rm{Gc}1}={\cal{F}}_{zyyz}^{\rm{Ga}2}=
{\cal{F}}_{zyzy}^{\rm{Gb}}={\cal{F}}_{zyzy}^{\rm{Gc}1}=
{\cal{F}}_{zyzy}^{\rm{Ga}2}={\cal{F}}_{zzyy}^{\rm{Gb}}=
{\cal{F}}_{zzyy}^{\rm{Gc}1}={\cal{F}}_{zzyy}^{\rm{Ga}2}
=4{\cal{F}}_{02}^{3}.\end{aligned}$$ The third level of complexity gives $$\begin{aligned}
{\cal{F}}_{yyyy}^{\rm{Ga}1}&=&{\cal{F}}_{yyyy}^{\rm{Gc}2}
=16{\cal{F}}_{04}^{2},
\\
{\cal{F}}_{yyyy}^{\rm{Gb}}&=&{\cal{F}}_{yyyy}^{\rm{Gc}1}=
{\cal{F}}_{yyyy}^{\rm{Ga}2}
=16{\cal{F}}_{04}^{3}.\end{aligned}$$ At the fourth level of complexity we observe $$\begin{aligned}
{\cal{F}}_{xzzz}^{\rm{Ga}1}&=&{\cal{F}}_{xzzz}^{\rm{Gc}2}=
{\cal{F}}_{zxzz}^{\rm{Ga}1}={\cal{F}}_{zxzz}^{\rm{Gc}2}
=2{\rm{i}}{\cal{F}}_{10}^{2}+2{\rm{i}}q_x{\cal{F}}_{00}^{2},
\\
{\cal{F}}_{xzzz}^{\rm{Gb}}&=&{\cal{F}}_{xzzz}^{\rm{Gc}1}=
{\cal{F}}_{xzzz}^{\rm{Ga}2}={\cal{F}}_{zxzz}^{\rm{Gc}1}=
{\cal{F}}_{zxzz}^{\rm{Ga}2}
=2{\rm{i}}{\cal{F}}_{10}^{3}+2{\rm{i}}q_x{\cal{F}}_{00}^{3},
\\
{\cal{F}}_{zzxz}^{\rm{Ga}1}&=&{\cal{F}}_{zzzx}^{\rm{Ga}1}
=2{\rm{i}}{\cal{F}}_{10}^{2}-2{\rm{i}}k_x{\cal{F}}_{00}^{2},
\\
{\cal{F}}_{zzxz}^{\rm{Ga}2}&=&{\cal{F}}_{zzzx}^{\rm{Gb}}=
{\cal{F}}_{zzzx}^{\rm{Ga}2}
=2{\rm{i}}{\cal{F}}_{10}^{3}-{\rm{i}}k_x{\cal{F}}_{00}^{3},
\\
{\cal{F}}_{zzxz}^{\rm{Gc}2}&=&{\cal{F}}_{zzzx}^{\rm{Gc}2}
=2{\rm{i}}{\cal{F}}_{10}^{2}+{\rm{i}}(2q_x-k_x){\cal{F}}_{00}^{2},
\\
{\cal{F}}_{zzxz}^{\rm{Gb}}&=&{\cal{F}}_{zzxz}^{\rm{Gc}1}=
{\cal{F}}_{zzzx}^{\rm{Gc}1}
=2{\rm{i}}{\cal{F}}_{10}^{3}+{\rm{i}}(2q_x-k_x){\cal{F}}_{00}^{3},\end{aligned}$$ and the independent element $${\cal{F}}_{zxzz}^{\rm{Gb}}
=2{\rm{i}}{\cal{F}}_{10}^{3}+2{\rm{i}}(q_x-2k_x){\cal{F}}_{00}^{3}.$$ In the fifth case we find $$\begin{aligned}
\lefteqn{
{\cal{F}}_{xyyz}^{\rm{Ga}1}={\cal{F}}_{xyyz}^{\rm{Gc}2}=
{\cal{F}}_{xyzy}^{\rm{Ga}1}={\cal{F}}_{xyzy}^{\rm{Gc}2}=
{\cal{F}}_{xzyy}^{\rm{Ga}1}={\cal{F}}_{xzyy}^{\rm{Gc}2}=
{\cal{F}}_{yxyz}^{\rm{Ga}1}={\cal{F}}_{yxyz}^{\rm{Gc}2}=
{\cal{F}}_{yxzy}^{\rm{Ga}1}={\cal{F}}_{yxzy}^{\rm{Gc}2}=
}\nonumber\\ &\quad&
{\cal{F}}_{zxyy}^{\rm{Ga}1}={\cal{F}}_{zxyy}^{\rm{Gc}2}
=-8{\rm{i}}{\cal{F}}_{12}^{2}-4{\rm{i}}q_x{\cal{F}}_{02}^{2},
\\
\lefteqn{
{\cal{F}}_{xyyz}^{\rm{Gb}}={\cal{F}}_{xyyz}^{\rm{Gc}1}=
{\cal{F}}_{xyyz}^{\rm{Ga}2}={\cal{F}}_{xyzy}^{\rm{Gb}}=
{\cal{F}}_{xyzy}^{\rm{Gc}1}={\cal{F}}_{xyzy}^{\rm{Ga}2}=
{\cal{F}}_{xzyy}^{\rm{Gb}}={\cal{F}}_{xzyy}^{\rm{Gc}1}=
{\cal{F}}_{xzyy}^{\rm{Ga}2}={\cal{F}}_{yxyz}^{\rm{Gc}1}=
}\nonumber\\ &&
{\cal{F}}_{yxyz}^{\rm{Ga}2}={\cal{F}}_{yxzy}^{\rm{Gc}1}=
{\cal{F}}_{yxzy}^{\rm{Ga}2}={\cal{F}}_{zxyy}^{\rm{Gc}1}=
{\cal{F}}_{zxyy}^{\rm{Ga}2}
=-8{\rm{i}}{\cal{F}}_{12}^{3}-4{\rm{i}}q_x{\cal{F}}_{02}^{3},
\\
\lefteqn{
{\cal{F}}_{yxyz}^{\rm{Gb}}={\cal{F}}_{yxzy}^{\rm{Gb}}=
{\cal{F}}_{zxyy}^{\rm{Gb}}
=-8{\rm{i}}{\cal{F}}_{12}^{3}-4{\rm{i}}(q_x-2k_x){\cal{F}}_{02}^{3},
}\\
\lefteqn{
{\cal{F}}_{yyxz}^{\rm{Ga}1}={\cal{F}}_{yzxy}^{\rm{Ga}1}=
{\cal{F}}_{zyxy}^{\rm{Ga}1}={\cal{F}}_{yyzx}^{\rm{Ga}1}=
{\cal{F}}_{yzyx}^{\rm{Ga}1}={\cal{F}}_{zyyx}^{\rm{Ga}1}
=-8{\rm{i}}{\cal{F}}_{12}^{2}+4{\rm{i}}k_x{\cal{F}}_{02}^{2},
}\\
\lefteqn{
{\cal{F}}_{yyxz}^{\rm{Ga}2}={\cal{F}}_{yzxy}^{\rm{Ga}2}=
{\cal{F}}_{zyxy}^{\rm{Ga}2}={\cal{F}}_{yyzx}^{\rm{Gb}}=
{\cal{F}}_{yyzx}^{\rm{Ga}2}={\cal{F}}_{yzyx}^{\rm{Gb}}=
{\cal{F}}_{yzyx}^{\rm{Ga}2}={\cal{F}}_{zyyx}^{\rm{Gb}}=
{\cal{F}}_{zyyx}^{\rm{Ga}2}=
}\nonumber\\ &&
-8{\rm{i}}{\cal{F}}_{12}^{3}+4{\rm{i}}k_x{\cal{F}}_{02}^{3},
\\
\lefteqn{
{\cal{F}}_{yyxz}^{\rm{Gc}2}={\cal{F}}_{yzxy}^{\rm{Gc}2}=
{\cal{F}}_{zyxy}^{\rm{Gc}2}={\cal{F}}_{yyzx}^{\rm{Gc}2}=
{\cal{F}}_{yzyx}^{\rm{Gc}2}={\cal{F}}_{zyyx}^{\rm{Gc}2}
=-8{\rm{i}}{\cal{F}}_{12}^{2}+4{\rm{i}}(k_x-2q_x){\cal{F}}_{02}^{2},
}\\
\lefteqn{
{\cal{F}}_{yyxz}^{\rm{Gb}}={\cal{F}}_{yyxz}^{\rm{Gc}1}=
{\cal{F}}_{yzxy}^{\rm{Gb}}={\cal{F}}_{yzxy}^{\rm{Gc}1}=
{\cal{F}}_{zyxy}^{\rm{Gb}}={\cal{F}}_{zyxy}^{\rm{Gc}1}=
{\cal{F}}_{yyzx}^{\rm{Gc}1}={\cal{F}}_{yzyx}^{\rm{Gc}1}=
{\cal{F}}_{zyyx}^{\rm{Gc}1}=
}\nonumber\\ &&
-8{\rm{i}}{\cal{F}}_{12}^{3}+4{\rm{i}}(k_x-2q_x){\cal{F}}_{02}^{3},\end{aligned}$$ In the sixth case we observe the related functions $$\begin{aligned}
\lefteqn{
{\cal{F}}_{xxzz}^{\rm{Ga}1}={\cal{F}}_{xxzz}^{\rm{Gc}2}
=-4{\cal{F}}_{20}^{2}-4q_x{\cal{F}}_{10}^{2}-q_x^2{\cal{F}}_{00}^{2},
}\\
\lefteqn{
{\cal{F}}_{xxzz}^{\rm{Gc}1}={\cal{F}}_{xxzz}^{\rm{Ga}2}
=-4{\cal{F}}_{20}^{3}-4q_x{\cal{F}}_{10}^{3}-q_x^2{\cal{F}}_{00}^{3},
}\\
\lefteqn{
{\cal{F}}_{xzxz}^{\rm{Ga}1}={\cal{F}}_{xzzx}^{\rm{Ga}1}=
{\cal{F}}_{zxxz}^{\rm{Ga}1}={\cal{F}}_{zxzx}^{\rm{Ga}1}
=-4{\cal{F}}_{20}^{2}-2(q_x-k_x){\cal{F}}_{10}^{2}
+q_xk_x{\cal{F}}_{00}^{2},
}\\
\lefteqn{
{\cal{F}}_{xzxz}^{\rm{Ga}2}={\cal{F}}_{xzzx}^{\rm{Gb}}=
{\cal{F}}_{xzzx}^{\rm{Ga}2}={\cal{F}}_{zxxz}^{\rm{Ga}2}=
{\cal{F}}_{zxzx}^{\rm{Ga}2}
=-4{\cal{F}}_{20}^{3}-4(q_x-k_x){\cal{F}}_{10}^{3}
+q_xk_x{\cal{F}}_{00}^{3},
}\\
\lefteqn{
{\cal{F}}_{xzxz}^{\rm{Gb}}={\cal{F}}_{xzxz}^{\rm{Gc}1}=
{\cal{F}}_{xzzx}^{\rm{Gc}1}={\cal{F}}_{zxxz}^{\rm{Gc}1}=
{\cal{F}}_{zxzx}^{\rm{Gc}1}=
}\nonumber\\ &\quad&
-4{\cal{F}}_{20}^{3}-2(3q_x-k_x){\cal{F}}_{10}^{3}
-q_x(2q_x-k_x){\cal{F}}_{00}^{3},
\\
\lefteqn{
{\cal{F}}_{xzxz}^{\rm{Gc}2}={\cal{F}}_{xzzx}^{\rm{Gc}2}=
{\cal{F}}_{zxxz}^{\rm{Gc}2}={\cal{F}}_{zxzx}^{\rm{Gc}2}
=-4{\cal{F}}_{20}^{2}-2(3q_x-k_x){\cal{F}}_{10}^{2}
-q_x(2q_x-k_x){\cal{F}}_{00}^{2},
}\end{aligned}$$ and the eight independent functions $$\begin{aligned}
{\cal{F}}_{xxzz}^{\rm{Gb}}&=&
-4{\cal{F}}_{20}^{3}-4(q_x-k_x){\cal{F}}_{10}^{3}
-q_x(q_x-2k_x){\cal{F}}_{00}^{3},
\\
{\cal{F}}_{zxxz}^{\rm{Gb}}&=&
-4{\cal{F}}_{20}^{3}-6(q_x-k_x){\cal{F}}_{10}^{3}
-(2q_x^2+2k_x^2-5q_xk_x){\cal{F}}_{00}^{3},
\\
{\cal{F}}_{zxzx}^{\rm{Gb}}&=&
-4{\cal{F}}_{20}^{3}-2(q_x-3k_x){\cal{F}}_{10}^{3}
-k_x(2k_x-q_x){\cal{F}}_{00}^{3},
\\
{\cal{F}}_{zzxx}^{\rm{Ga}1}&=&
-4{\cal{F}}_{20}^{2}+4k_x{\cal{F}}_{10}^{2}-k_x^2{\cal{F}}_{00}^{2},
\\
{\cal{F}}_{zzxx}^{\rm{Ga}2}&=&
-4{\cal{F}}_{20}^{3}+4k_x{\cal{F}}_{10}^{3}-k_x^2{\cal{F}}_{00}^{3},
\\
{\cal{F}}_{zzxx}^{\rm{Gb}}&=&
-4{\cal{F}}_{20}^{3}-4(q_x-k_x){\cal{F}}_{10}^{3}
-k_x(k_x-2q_x){\cal{F}}_{00}^{3},
\\
{\cal{F}}_{zzxx}^{\rm{Gc}1}&=&
-4{\cal{F}}_{20}^{3}-2(2q_x-k_x){\cal{F}}_{10}^{3}
-(2q_x-k_x)^2{\cal{F}}_{00}^{3},
\\
{\cal{F}}_{zzxx}^{\rm{Gc}2}&=&
-4{\cal{F}}_{20}^{2}-2(2q_x-k_x){\cal{F}}_{10}^{2}
-(2q_x-k_x)^2{\cal{F}}_{00}^{2}.\end{aligned}$$ The seventh case has the following related functions $$\begin{aligned}
\lefteqn{
{\cal{F}}_{xxyy}^{\rm{Ga}1}={\cal{F}}_{xxyy}^{\rm{Gc}2}
=16{\cal{F}}_{22}^{2}+16q_x{\cal{F}}_{12}^{2}+4q_x^2{\cal{F}}_{02}^{2},
}\\
\lefteqn{
{\cal{F}}_{xxyy}^{\rm{Gc}1}={\cal{F}}_{xxyy}^{\rm{Ga}2}
=16{\cal{F}}_{22}^{3}+16q_x{\cal{F}}_{12}^{3}+4q_x^2{\cal{F}}_{02}^{3},
}\\
\lefteqn{
{\cal{F}}_{xyxy}^{\rm{Ga}1}={\cal{F}}_{xyyx}^{\rm{Ga}1}=
{\cal{F}}_{yxxy}^{\rm{Ga}1}={\cal{F}}_{yxyx}^{\rm{Ga}1}
=16{\cal{F}}_{22}^{2}+8(q_x-k_x){\cal{F}}_{12}^{2}-4q_xk_x{\cal{F}}_{02}^{2},
}\\
\lefteqn{
{\cal{F}}_{xyxy}^{\rm{Ga}2}={\cal{F}}_{xyyx}^{\rm{Gb}}=
{\cal{F}}_{xyyx}^{\rm{Ga}2}={\cal{F}}_{yxxy}^{\rm{Ga}2}=
{\cal{F}}_{yxyx}^{\rm{Ga}2}
=16{\cal{F}}_{22}^{3}+8(q_x-k_x){\cal{F}}_{12}^{3}-4q_xk_x{\cal{F}}_{02}^{3},
}\\
\lefteqn{
{\cal{F}}_{xyxy}^{\rm{Gb}}={\cal{F}}_{xyxy}^{\rm{Gc}1}=
{\cal{F}}_{xyyx}^{\rm{Gc}1}={\cal{F}}_{yxxy}^{\rm{Gc}1}=
{\cal{F}}_{yxyx}^{\rm{Gc}1}=
}\nonumber\\ &\quad&
16{\cal{F}}_{22}^{3}+8(3q_x-k_x){\cal{F}}_{12}^{3}
+4q_x(2q_x-k_x){\cal{F}}_{02}^{3},
\\
\lefteqn{
{\cal{F}}_{xyxy}^{\rm{Gc}2}={\cal{F}}_{xyyx}^{\rm{Gc}2}=
{\cal{F}}_{yxxy}^{\rm{Gc}2}={\cal{F}}_{yxyx}^{\rm{Gc}2}
=16{\cal{F}}_{22}^{2}+8(3q_x-k_x){\cal{F}}_{12}^{2}
+4q_x(2q_x-k_x){\cal{F}}_{02}^{2},
}\end{aligned}$$ and the eight independent functions $$\begin{aligned}
{\cal{F}}_{xxyy}^{\rm{Gb}}&=&
16{\cal{F}}_{22}^{3}+16(q_x-k_x){\cal{F}}_{12}^{3}
+4q_x(q_x-2k_x){\cal{F}}_{02}^{3},
\\
{\cal{F}}_{yxxy}^{\rm{Gb}}&=&
16{\cal{F}}_{22}^{3}+24(q_x-k_x){\cal{F}}_{12}^{3}
+4(2q_x^2+2k_x^2-5q_xk_x){\cal{F}}_{02}^{3},
\\
{\cal{F}}_{yxyx}^{\rm{Gb}}&=&
16{\cal{F}}_{22}^{3}+8(q_x-3k_x){\cal{F}}_{12}^{3}
+4k_x(2k_x-q_x){\cal{F}}_{02}^{3},
\\
{\cal{F}}_{yyxx}^{\rm{Ga}1}&=&
16{\cal{F}}_{22}^{2}-16k_x{\cal{F}}_{12}^{2}+4k_x^2{\cal{F}}_{02}^{2},
\\
{\cal{F}}_{yyxx}^{\rm{Ga}2}&=&
16{\cal{F}}_{22}^{3}-16k_x{\cal{F}}_{12}^{3}+4k_x^2{\cal{F}}_{02}^{3},
\\
{\cal{F}}_{yyxx}^{\rm{Gb}}&=&
16{\cal{F}}_{22}^{3}+16(q_x-k_x){\cal{F}}_{12}^{3}
+4k_x(k_x-2q_x){\cal{F}}_{02}^{3},
\\
{\cal{F}}_{yyxx}^{\rm{Gc}1}&=&
16{\cal{F}}_{22}^{3}+16(2q_x-k_x){\cal{F}}_{12}^{3}
+4(2q_x-k_x)^2{\cal{F}}_{02}^{3},
\\
{\cal{F}}_{yyxx}^{\rm{Gc}2}&=&
16{\cal{F}}_{22}^{2}+16(2q_x-k_x){\cal{F}}_{12}^{2}
+4(2q_x-k_x)^2{\cal{F}}_{02}^{2}.\end{aligned}$$ The eighth case gives $$\begin{aligned}
\lefteqn{
{\cal{F}}_{xxxz}^{\rm{Ga}1}={\cal{F}}_{xxzx}^{\rm{Ga}1}
=-8{\rm{i}}{\cal{F}}_{30}^{2}-4{\rm{i}}(2q_x-k_x){\cal{F}}_{20}^{2}
-2{\rm{i}}q_x(q_x-2k_x){\cal{F}}_{10}^{2}+{\rm{i}}q_x^2k_x{\cal{F}}_{00}^{2},
}\\
\lefteqn{
{\cal{F}}_{xxxz}^{\rm{Ga}2}={\cal{F}}_{xxzx}^{\rm{Ga}2}
=-8{\rm{i}}{\cal{F}}_{30}^{3}-4{\rm{i}}(2q_x-k_x){\cal{F}}_{20}^{3}
-2{\rm{i}}q_x(q_x-2k_x){\cal{F}}_{10}^{3}+{\rm{i}}q_x^2k_x{\cal{F}}_{00}^{3},
}\\
\lefteqn{
{\cal{F}}_{xxxz}^{\rm{Gc}1}={\cal{F}}_{xxzx}^{\rm{Gc}1}
=-8{\rm{i}}{\cal{F}}_{30}^{3}-4{\rm{i}}(4q_x-k_x){\cal{F}}_{20}^{3}
-2{\rm{i}}q_x(5q_x-2k_x){\cal{F}}_{10}^{3}
}\nonumber\\ &\quad&
-{\rm{i}}q_x^2(2q_x-k_x){\cal{F}}_{00}^{3},
\\
\lefteqn{
{\cal{F}}_{xxxz}^{\rm{Gc}2}={\cal{F}}_{xxzx}^{\rm{Gc}2}
=-8{\rm{i}}{\cal{F}}_{30}^{2}-4{\rm{i}}(4q_x-k_x){\cal{F}}_{20}^{2}
-2{\rm{i}}q_x(5q_x-2k_x){\cal{F}}_{10}^{2}
}\nonumber\\ &&
-{\rm{i}}q_x^2(2q_x-k_x){\cal{F}}_{00}^{2},
\\
\lefteqn{
{\cal{F}}_{xzxx}^{\rm{Ga}1}={\cal{F}}_{zxxx}^{\rm{Ga}1}
=-8{\rm{i}}{\cal{F}}_{30}^{2}-4{\rm{i}}(q_x-2k_x){\cal{F}}_{20}^{2}
-2{\rm{i}}k_x(k_x-2q_x){\cal{F}}_{10}^{2}-{\rm{i}}q_xk_x^2{\cal{F}}_{00}^{2},
}\\
\lefteqn{
{\cal{F}}_{xzxx}^{\rm{Ga}2}={\cal{F}}_{zxxx}^{\rm{Ga}2}
=-8{\rm{i}}{\cal{F}}_{30}^{3}-4{\rm{i}}(q_x-2k_x){\cal{F}}_{20}^{3}
-2{\rm{i}}k_x(k_x-2q_x){\cal{F}}_{10}^{3}-{\rm{i}}q_xk_x^2{\cal{F}}_{00}^{3},
}\\
\lefteqn{
{\cal{F}}_{xzxx}^{\rm{Gc}1}={\cal{F}}_{zxxx}^{\rm{Gc}1}
=-8{\rm{i}}{\cal{F}}_{30}^{3}-4{\rm{i}}(5q_x-2k_x){\cal{F}}_{20}^{3}
-2{\rm{i}}(2q_x-k_x)(4q_x-k_x){\cal{F}}_{10}^{3}
}\nonumber\\ &&
-{\rm{i}}q_x(2q_x-k_x)^2{\cal{F}}_{00}^{3},
\\
\lefteqn{
{\cal{F}}_{xzxx}^{\rm{Gc}2}={\cal{F}}_{zxxx}^{\rm{Gc}2}
=-8{\rm{i}}{\cal{F}}_{30}^{2}-4{\rm{i}}(5q_x-2k_x){\cal{F}}_{20}^{2}
-2{\rm{i}}(2q_x-k_x)(4q_x-k_x){\cal{F}}_{10}^{2}
}\nonumber\\ &&
-{\rm{i}}q_x(2q_x-k_x)^2{\cal{F}}_{00}^{2},\end{aligned}$$ and the four independent elements $$\begin{aligned}
\lefteqn{
{\cal{F}}_{xxxz}^{\rm{Gb}}=
-8{\rm{i}}{\cal{F}}_{30}^{3}-4{\rm{i}}(4q_x-3k_x){\cal{F}}_{20}^{3}
-2{\rm{i}}(5q_x^2+2k_x^2-8q_xk_x){\cal{F}}_{10}^{3}
}\nonumber\\ &\quad&
-{\rm{i}}q_x(2q_x^2+2k_x^2-5q_xk_x){\cal{F}}_{00}^{3},
\\
\lefteqn{
{\cal{F}}_{xxzx}^{\rm{Gb}}=
-8{\rm{i}}{\cal{F}}_{30}^{3}-4{\rm{i}}(2q_x-3k_x){\cal{F}}_{20}^{3}
-2{\rm{i}}(q_x^2+2k_x^2-4q_xk_x){\cal{F}}_{10}^{3}
}\nonumber\\ &&
+{\rm{i}}q_xk_x(q_x-2k_x){\cal{F}}_{00}^{3},
\\
\lefteqn{
{\cal{F}}_{xzxx}^{\rm{Gb}}=
-8{\rm{i}}{\cal{F}}_{30}^{3}-4{\rm{i}}(3q_x-2k_x){\cal{F}}_{20}^{3}
-2{\rm{i}}(2q_x^2+k_x^2-4q_xk_x){\cal{F}}_{10}^{3}
}\nonumber\\ &&
+{\rm{i}}q_xk_x(2q_x-k_x){\cal{F}}_{00}^{3},
\\
\lefteqn{
{\cal{F}}_{zxxx}^{\rm{Gb}}=
-8{\rm{i}}{\cal{F}}_{30}^{3}-4{\rm{i}}(3q_x-4k_x){\cal{F}}_{20}^{3}
-2{\rm{i}}(2q_x^2+5k_x^2-8q_xk_x){\cal{F}}_{10}^{3}
}\nonumber\\ &&
+{\rm{i}}k_x(2q_x^2+2k_x^2-5q_xk_x){\cal{F}}_{00}^{3}.\end{aligned}$$ The most complex solution group of Cartesian indices gives the five independent functions $$\begin{aligned}
\lefteqn{
{\cal{F}}_{xxxx}^{\rm{Ga}1}=
16{\cal{F}}_{40}^{2}+16(q_x-k_x){\cal{F}}_{30}^{2}
+4(q_x^2+k_x^2-4q_xk_x){\cal{F}}_{20}^{2}
+4q_xk_x(k_x-q_x){\cal{F}}_{10}^{2}
}\nonumber\\ &\quad&
+(q_xk_x)^2{\cal{F}}_{00}^{2},
\\
\lefteqn{
{\cal{F}}_{xxxx}^{\rm{Ga}2}=
16{\cal{F}}_{40}^{3}+16(q_x-k_x){\cal{F}}_{30}^{3}
+4(q_x^2+k_x^2-4q_xk_x){\cal{F}}_{20}^{3}
+4q_xk_x(k_x-q_x){\cal{F}}_{10}^{3}
}\nonumber\\ &&
+(q_xk_x)^2{\cal{F}}_{00}^{3},
\\
\lefteqn{
{\cal{F}}_{xxxx}^{\rm{Gb}}=
16{\cal{F}}_{40}^{3}+32(q_x-k_x){\cal{F}}_{30}^{3}
+4(5q_x^2+5k_x^2-12q_xk_x){\cal{F}}_{20}^{3}
}\nonumber\\ &&
+2(q_x-k_x)(2q_x^2+2k_x^2-8q_xk_x){\cal{F}}_{10}^{3}
-q_xk_x(2q_x^2+2k_x^2-5q_xk_x){\cal{F}}_{00}^{3},
\\
\lefteqn{
{\cal{F}}_{xxxx}^{\rm{Gc}1}=
16{\cal{F}}_{40}^{3}+16(3q_x-k_x){\cal{F}}_{30}^{3}
+4((2q_x-k_x)^2+q_x(9q_x-4k_x)){\cal{F}}_{20}^{3}
}\nonumber\\ &&
+4q_x(2q_x-k_x)(3q_x-k_x){\cal{F}}_{10}^{3}
+q_x^2(2q_x-k_x)^2{\cal{F}}_{00}^{3},
\\
\lefteqn{
{\cal{F}}_{xxxx}^{\rm{Gc}2}=
16{\cal{F}}_{40}^{2}+16(3q_x-k_x){\cal{F}}_{30}^{2}
+4((2q_x-k_x)^2+q_x(9q_x-4k_x)){\cal{F}}_{20}^{2}
}\nonumber\\ &&
+4q_x(2q_x-k_x)(3q_x-k_x){\cal{F}}_{10}^{2}
+q_x^2(2q_x-k_x)^2{\cal{F}}_{00}^{2}.\end{aligned}$$ The immediate conclusion of these observations is that only $65$ of the original $246$ possible functions are independent.
The ${\cal{Z}}$ coefficients with uniform pump field amplitudes {#sec:Z}
---------------------------------------------------------------
If the pump fields have uniform amplitudes along the $z$-axis, we take the local limit in the two coordinates $z'''$ and $z''$ in the ${\cal{Z}}$ coefficients, such that in general we may write, $${\cal{Z}}(z,z')=\iint{\cal{Z}}(z,z',z'',z''')dz'''dz'',$$ and thus $$\tensor{\Xi}(z,z';\vec{q}_{\|},\vec{k}_{\|})=
\iint\tensor{\Xi}(z,z',z'',z''';\vec{q}_{\|},\vec{k}_{\|})dz'''dz''.$$ Using an orthogonal set of wave functions, parity teaches that the integrals over $x$- and $y$-components gives $$\int{\cal{Z}}_{nm}^{x}(z)dz=\int{\cal{Z}}_{nm}^{\,y}(z)dz=\delta_{nm},$$ where $\delta_{nm}$ is the Kronecker delta. The only question left is the $z$-components, which may be determined as soon as the wave functions for the system is known. Then part $A$ of the nonlinear conductivity tensor does not contribute to the phase conjugated response because of the result of integration over $z''$. For the same reason, part E vanish, since the pure interband terms vanish by themselves, and the rest of part Ea becomes equal in magnitude to the rest of part Eb, but with the opposite sign. All other terms still contribute to the response.
### Infinite barrier quantum well
If we choose a quantum well within the infinite barrier model with boundaries at $0$ and $-d$ as the source, then we find $${\cal{Z}}(z,z')=\int_{-d}^{0}\int_{-d}^{0}{\cal{Z}}(z,z',z'',z''')dz'''dz'',$$ and thus $$\tensor{\Xi}(z,z';\vec{q}_{\|},\vec{k}_{\|})=
\int_{-d}^{0}\int_{-d}^{0}
\tensor{\Xi}(z,z',z'',z''';\vec{q}_{\|},\vec{k}_{\|})dz'''dz'',$$ in general. Since the individual ${\cal{Z}}_{nm}^{x,y,z}(z)$ are independent, the result is written as a product of these in the coordinates $z$, $z'$, $z''$, and $z'''$. Then the integrals over $z$-components gives $$\int_{-d}^{0}{\cal{Z}}_{nm}^{z}(z)dz=
{4nm[1-(-1)^{n+m}]\over(n^2-m^2)d}.$$
### Probe with a single wavevector
If we take the probe field as $\vec{E}(z';\vec{q}_{\|})=\vec{E}e^{{\rm{i}}q_{\perp}z'}$ and thus $\vec{E}^{*}(z';\vec{q}_{\|})=\vec{E}^{*}e^{-{\rm{i}}q_{\perp}z'}$, then we may further reduce the $z$-dependence, if we define $${\mathfrak{Z}}(z)=\int_{-d}^{0}{\cal{Z}}(z,z')e^{-{\rm{i}}q_{\perp}z'}dz',$$ where we have indicated the modification before integration of the ${\cal{Z}}(z,z')$ quantity by shifting the symbol from calligraphic style to fraktur. Then, from the two integrals $$\begin{aligned}
\lefteqn{
\int_{-d}^{0}{\cal{Z}}_{nm}^{x}(z)e^{{\pm}{\rm{i}}q_{\perp}z}dz=
\int_{-d}^{0}{\cal{Z}}_{nm}^{\,y}(z)e^{{\pm}{\rm{i}}q_{\perp}z}dz=
}\nonumber\\ &\quad&
\mp{4\pi^2nm{\rm{i}}q_{\perp}d[e^{{\mp}{\rm{i}}q_{\perp}d}(-1)^{n+m}-1]
\over[({\rm{i}}q_{\perp}d)^2+\pi^2(n+m)^2][({\rm{i}}q_{\perp}d)^2+\pi^2(n-m)^2]},
\\
\lefteqn{
\int_{-d}^{0}{\cal{Z}}_{nm}^{z}(z)e^{{\pm}{\rm{i}}q_{\perp}z}dz=
{4\pi^4(n^2-m^2)nm[e^{{\mp}{\rm{i}}q_{\perp}d}(-1)^{n+m}-1]
\over{}d[({\rm{i}}q_{\perp}d)^2+\pi^2(n+m)^2][({\rm{i}}q_{\perp}d)^2+\pi^2(n-m)^2]},
}\end{aligned}$$ we may deduce the reduced quantities.
### Combining ${\mathfrak{Z}}$ with $\Omega$
If we combine the ${\mathfrak{Z}}$-quantities with the two different exponentials appearing in the $\Omega$-quantities, we get new quantities that are independent of $z$. Defining $${\mathfrak{Z}}^{\pm}=\int_{-d}^{0}{\mathfrak{Z}}(z)e^{\pm{}{\rm{i}}q_{\perp}z}dz,$$ we may now deduce the $z$-independent quantities that will finally appear in the expressions for the $\Omega$-quantities. Even though we have modified the ${\mathfrak{Z}}$ coefficients again before integration, we keep the symbol, since the main purpose of selecting another symbol is to indicate that it is modified from the original ${\cal{Z}}$ quantities.
The nonlinear conductivity tensor in the $\Omega$ coefficients {#sec:Ps}
--------------------------------------------------------------
If we concatenate the definitions made in this compliment until now, we may define the new quantity to be considered in the $\Omega$ coefficients \[Eqs. (\[eq:14.44\])–(\[eq:14.46\])\] $${\tensor{{\mathfrak{X}}}}^{\pm}(\vec{q}_{\|},\vec{k}_{\|})=
\int_{-d}^{0}\int_{-d}^{0}\int_{-d}^{0}\int_{-d}^{0}
{\tensor{\Xi}}(z,z',z'',z''';\vec{q}_{\|},\vec{k}_{\|})
e^{-{\rm{i}}q_{\perp}z'}e^{\pm{}{\rm{i}}q_{\perp}z}
dz'''dz''dz'dz.$$ As we did with the ${\cal{Z}}$ coefficients, the modification before integration have led us to define the new quantity in fraktur rather as $\Xi$.
Then the nonlinear conductivity tensors combined with the $z$-dependent parts of the probe and the $\Omega$ coefficient can be written $$\begin{aligned}
\lefteqn{
{{\mathfrak{X}}}_{xxxx}^{\rm{A}\pm}(\vec{q}_{\|},\vec{k}_{\|})=
{2\over({\rm{i}}\omega)^3}\sum_{nm}{\mathfrak{Z}}_{nm}^{\rm{A}\pm}\xi_{nm}^{\rm{A}},
}\\
\lefteqn{
{{\mathfrak{X}}}_{xxkh}^{\rm{B}\pm}(\vec{q}_{\|},\vec{k}_{\|})=
{2\over({\rm{i}}\omega)^3}\left({e\hbar\over2m_e}\right)^2\sum_{nmv}
{\mathfrak{Z}}_{kh,nmv}^{\rm{B}\pm}\xi_{kh,nmv}^{\rm{B}}(\vec{q}_{\|},\vec{k}_{\|}),
}\\
\lefteqn{
{{\mathfrak{X}}}_{xxxx}^{\rm{C}\pm}(\vec{q}_{\|},\vec{k}_{\|})=
{2\over({\rm{i}}\omega)^3}
\sum_{nm}{\mathfrak{Z}}_{nm}^{\rm{C}\pm}\xi_{nm}^{\rm{C}}(\vec{q}_{\|},\vec{k}_{\|}),
}\\
\lefteqn{
{{\mathfrak{X}}}_{xjkx}^{\rm{D}\pm}(\vec{q}_{\|},\vec{k}_{\|})=
{2\over({\rm{i}}\omega)^3}\left({e\hbar\over2m_e}\right)^2\sum_{nmv}
\left\{
{\mathfrak{Z}}_{jk,nmv}^{\rm{Da}\pm}
\xi_{jk,nmv}^{\rm{Da}}(\vec{q}_{\|},\vec{k}_{\|})
+{\mathfrak{Z}}_{jk,nmv}^{\rm{Db}\pm}
\xi_{jk,nmv}^{\rm{Db}}(\vec{q}_{\|},\vec{k}_{\|})
\right\},
}\nonumber\\ \\
\lefteqn{
{{\mathfrak{X}}}_{ijxx}^{\rm{E}\pm}(\vec{q}_{\|},\vec{k}_{\|})=
{2\over({\rm{i}}\omega)^3}\left({e\hbar\over2m_e}\right)^2\sum_{nmv}
\left\{
{\mathfrak{Z}}_{ij,nmv}^{\rm{Ea}\pm}
\xi_{ij,nmv}^{\rm{Ea}}(\vec{q}_{\|},\vec{k}_{\|})
+{\mathfrak{Z}}_{ij,nmv}^{\rm{Eb}\pm}
\xi_{ij,nmv}^{\rm{Eb}}(\vec{q}_{\|},\vec{k}_{\|})
\right\},
}\nonumber\\ \\
\lefteqn{
{{\mathfrak{X}}}_{ixxh}^{\rm{F}\pm}(\vec{q}_{\|},\vec{k}_{\|})=
{2\over({\rm{i}}\omega)^3}\left({e\hbar\over2m_e}\right)^2\sum_{nmv}
\left\{
{\mathfrak{Z}}_{ih,nmv}^{\rm{Fa}\pm}
\xi_{ih,nmv}^{\rm{Fa}}(\vec{q}_{\|},\vec{k}_{\|})
+{\mathfrak{Z}}_{ih,nmv}^{\rm{Fb}\pm}
\xi_{ih,nmv}^{\rm{Fb}}(\vec{q}_{\|},\vec{k}_{\|})
\right\},
}\nonumber\\ \\
\lefteqn{
{{\mathfrak{X}}}_{ijkh}^{\rm{G}\pm}(\vec{q}_{\|},\vec{k}_{\|})=
{2\over({\rm{i}}\omega)^3}\left({e\hbar\over2m_e}\right)^4\sum_{nmvl}
\left\{
{\mathfrak{Z}}_{ijkh,nmvl}^{\rm{Ga}\pm}
\xi_{ijkh,nmvl}^{\rm{Ga}}(\vec{q}_{\|},\vec{k}_{\|})
\right.}\nonumber\\ &\quad&\left.
+{\mathfrak{Z}}_{ijkh,nmvl}^{\rm{Gb}\pm}
\xi_{ijkh,nmvl}^{\rm{Gb}}(\vec{q}_{\|},\vec{k}_{\|})
+{\mathfrak{Z}}_{ijkh,nmvl}^{\rm{Gc}\pm}
\xi_{ijkh,nmvl}^{\rm{Gc}}(\vec{q}_{\|},\vec{k}_{\|})
\right\},\end{aligned}$$ in terms of the ${\mathfrak{Z}}$ coefficients just calculated.
The local limit in three coordinates
------------------------------------
In the local limit, the vector potentials (and electric fields) are independent of their $z$-coordinate, and thus the nonlinear conductivity tensor alone has to be integrated over the $z'''$, $z''$, and $z'$ coordinates, viz. $$\tensor{\Xi}(z;\vec{q}_{\|},\vec{k}_{\|})=
\iiint\tensor{\Xi}(z,z',z'',z''';\vec{q}_{\|},\vec{k}_{\|})dz'''dz''dz'.$$
Analytical expressions for ${\cal{C}}$, ${\cal{D}}$, and ${\cal{N}}$ {#sec:CDN}
--------------------------------------------------------------------
In the quantity ${\cal{C}}(q_{\|}-k_{\|})$, given by Eq. (\[eq:C\]), $\beta=1$ in Eq. (\[eq:FpqB-indef\]) and $p=0$, and in the quantity ${\cal{D}}(q_{\|},k_{\|})$ \[Eq. (\[eq:D\])\], $\beta=p=2$. In the quantity ${\cal{N}}$ \[Eq. (\[eq:N\])\], $p=a_k=0$ and $b_k=1$, and it can be solved immediately, with the result $${\cal{N}}={4\over(2\pi)^2}\int_{0}^{{\alpha}}\int_{0}^{\pi}rd\theta dr
={{\alpha}^2\over2\pi}.
\label{eq:Solve-Q-Beta=0}$$ Using the analysis in Appendix \[app:B\], the quantities ${\cal{C}}(q_{\|}\pm{}k_{\|})$ and ${\cal{D}}(q_{\|},k_{\|})$ can be expressed as $${\cal{C}}(q_{\|}\pm{}k_{\|})=
2\left[{\cal{F}}_{0}^{1}(a_1,b_1,q_{\|}\pm{}k_{\|})
-{\cal{F}}_{0}^{1}(a_1,b_1,0)\right],
\label{eq:C-solved}$$ and $$\begin{aligned}
\lefteqn{
{\cal{D}}(q_{\|},k_{\|})=2\left[{\cal{F}}_{2}^{2}(a_2,a_3,b_2,b_3,0)
-{\cal{F}}_{2}^{2}(a_2,a_3,b_2,b_3,k_{\|})
\right.}\nonumber\\ &\quad&
+{\cal{F}}_{2}^{2}(a_2,a_4,b_2,b_4,k_{\|}+q_{\|})
-{\cal{F}}_{2}^{2}(a_2,a_4,b_2,b_4,k_{\|})
\nonumber\\ &&
+{\cal{F}}_{2}^{2}(a_2,a_4,b_2,b_5,0)
-{\cal{F}}_{2}^{2}(a_2,a_4,b_2,b_5,q_{\|})
\nonumber\\ &&\left.
+{\cal{F}}_{2}^{2}(a_2,a_3,b_2,b_6,k_{\|}+q_{\|})
-{\cal{F}}_{2}^{2}(a_2,a_3,b_2,b_6,q_{\|})\right],
\nonumber\\
\label{eq:D-solved}\end{aligned}$$ where $$\begin{aligned}
a_1&=&\hbar(q_{\|}-k_{\|})/m_e, \\
a_2&=&\hbar(q_{\|}+k_{\|})/m_e, \\
a_3&=&\hbar{}k_{\|}/m_e, \\
a_4&=&\hbar{}q_{\|}/m_e, \\
b_1&=&\hbar(q_{\|}-k_{\|})^2/(2m_e)-i/\tau, \\
b_2&=&\hbar(q_{\|}+k_{\|})^2/(2m_e)-i/\tau, \\
b_3&=&\hbar{}k_{\|}^2/(2m_e)-i/\tau-\omega, \\
b_4&=&\hbar{}q_{\|}(q_{\|}+2k_{\|})/(2m_e)-i/\tau+\omega, \\
b_5&=&\hbar{}q_{\|}^2/(2m_e)-i/\tau+\omega, \\
b_6&=&\hbar{}k_{\|}(k_{\|}+2q_{\|})/(2m_e)-i/\tau-\omega.\end{aligned}$$
Analytic expressions for the ${\cal{Q}}$ quantities {#sec:Q}
---------------------------------------------------
Let us finish this appendix by giving the solutions to Eqs. (\[eq:A-xx\])–(\[eq:A-zz\]). They are $$\begin{aligned}
\lefteqn{ {\cal{Q}}_{\,\,\,nm}^{xx}(\vec{q}_{\|},\omega)={2i\hbar\over(2\pi)^2}
\left({e\hbar\over2m_{e}}\right)^2
}\nonumber\\ &\quad&\times
\left[
4{\cal{F}}_{20,n}^{2}(a_1,a_2,b_{1,nm},b_{2,nm},q_x)
+4q_x{\cal{F}}_{10,n}^{2}(a_1,a_2,b_{1,nm},b_{2,nm},q_x)
\right.\nonumber\\ &&
+q_x^2{\cal{F}}_{00,n}^{2}(a_1,a_2,b_{1,nm},b_{2,nm},q_x)
-4{\cal{F}}_{20,m}^{2}(a_1,a_2,b_{1,nm},b_{2,nm},0)
\nonumber\\ &&\left.\!
-4q_x{\cal{F}}_{10,m}^{2}(a_1,a_2,b_{1,nm},b_{2,nm},0)
-q_x^2{\cal{F}}_{00,m}^{2}(a_1,a_2,b_{1,nm},b_{2,nm},0)
\right],
\label{eq:Q1}
\\
\lefteqn{ {\cal{Q}}_{\,\,\,nm}^{xz}(\vec{q}_{\|},\omega)={2i\hbar\over(2\pi)^2}
\left({e\hbar\over2m_{e}}\right)^2
}\nonumber\\ &&\times
\left[
2{\cal{F}}_{10,n}^{2}(a_1,a_2,b_{1,nm},b_{2,nm},q_x)
+q_x{\cal{F}}_{00,n}^{2}(a_1,a_2,b_{1,nm},b_{2,nm},q_x)
\right.\nonumber\\ &&\left.
-2{\cal{F}}_{10,m}^{2}(a_1,a_2,b_{1,nm},b_{2,nm},0)
-q_x{\cal{F}}_{00,m}^{2}(a_1,a_2,b_{1,nm},b_{2,nm},0)
\right],
\\
\lefteqn{
{\cal{Q}}_{\,\,\,nm}^{yy}(\vec{q}_{\|},\omega)={2i\hbar\over(2\pi)^2}
\left({e\hbar\over2m_{e}}\right)^2
}\nonumber\\ &&\times
4\left[
{\cal{F}}_{02,n}^{2}(a_1,a_2,b_{1,nm},b_{2,nm},q_x)
-{\cal{F}}_{02,m}^{2}(a_1,a_2,b_{1,nm},b_{2,nm},0)
\right],
\\
\lefteqn{
{\cal{Q}}_{\,\,\,nm}^{zz}(\vec{q}_{\|},\omega)={2i\hbar\over(2\pi)^2}
\left({e\hbar\over2m_{e}}\right)^2
}\nonumber\\ &&\times
\left[
{\cal{F}}_{00,n}^{2}(a_1,a_2,b_{1,nm},b_{2,nm},q_x)
-{\cal{F}}_{00,m}^{2}(a_1,a_2,b_{1,nm},b_{2,nm},0)
\right],
\label{eq:Q4}\end{aligned}$$ according to the treatment of these types of integrals given in Complement \[ch:Solve-Q\]. Above we have used $$\begin{aligned}
a_1&=&a_2={\hbar{}q_x\over{}m_{e}},
\\
b_{1,nm}&=&
\varepsilon_n-\varepsilon_m+{\hbar{}q_x^2\over2m_{e}}-{i\hbar\over\tau_{nm}},
\\
b_{2,nm}&=&
\varepsilon_n-\varepsilon_m+{\hbar{}q_x^2\over2m_{e}}-{i\hbar\over\tau_{nm}}
-\hbar\omega.\end{aligned}$$
Fermi energy, quantum well thickness, and $\alpha(n)$ {#app:D}
=====================================================
The number of electrons $n(\vec{r})$ in a system where the spin energies are degenerate can be written $$n(\vec{r})=2\sum_{N}|\Psi_{N}(\vec{r})|^2f_{N},
\label{eq:Number-r}$$ where the number $2$ represents the degeneracy of the spin energies, and the sum runs over all electron states in the system multiplied by the probability of finding an electron in that state. This probability is given as a Fermi-Dirac distribution $$f_{N}={1\over1+\exp(({\cal{E}}_{N}-\mu)/(k_BT))},$$ where ${\cal{E}}_{N}$ is the energy of the electron in state $N$, $\mu$ is the chemical potential, $k_B$ is Boltzmann’s constant and $T$ is the absolute temperature.
We will now look at the case where we have two-dimensional translational invariance along the $x$-$y$-plane. In this case the wave function gives plane-wave solutions in the direction of the plane, $$\Psi_{N}(\vec{r})={1\over2\pi}\psi_n(z)e^{i\vec{\kappa}_{\|}\cdot\vec{r}}$$ and the corresponding energy is $${\cal{E}}_{N}=\varepsilon_{n}+{\hbar^2\kappa_{\|}^2\over2m_e},$$ where $\kappa_{\|}=|\kappa_{\|}|$. By insertion into Eq. (\[eq:Number-r\]), it is converted into $$n(z)=2\sum_{n}|\psi_{n}(z)|^2\int_{-\infty}^{\infty}\left({1+\exp\left[
{\varepsilon_n+(\hbar^2\kappa_{\|}^2)/(2m_e)-\mu\over{}k_BT}
\right]}\right)^{-1}
{d^2\kappa_{\|}\over(2\pi)^2},$$ taking into account that the sum over the plane-wave expansion parallel to the surface can be converted into an integral, and the notation $n(z)\equiv{}n(\vec{r})$ is introduced for consistency. Solving the integral, we get $$n(z)={2m\over\hbar^2}\sum_{n}G_n|\psi_{n}(z)|^2,$$ with $$G_n={k_{B}T\over2\pi}
\ln\left[1+\exp\left({\mu-\varepsilon_n\over{}k_BT}\right)\right]$$ as the number of electrons in the quantum well for any temperature $T$.
Fermi energy in the low temperature limit
-----------------------------------------
In the low temperature limit, the chemical potential obeys, $$\lim_{T\rightarrow0}\mu={\cal{E}}_{F},$$ where ${\cal{E}}_{F}$ is the Fermi energy. Then $$G_n=\left\{
\begin{array}{lll}
0 & \mbox{for} & \varepsilon_n>{\cal{E}}_{F}, \\
({\cal{E}}_{F}-\varepsilon_n)/(2\pi)&\mbox{for}&\varepsilon_n<{\cal{E}}_{F},
\end{array}\right.$$ for $T\rightarrow0$, and thus $$n(z)|_{T\rightarrow0}={m\over\pi\hbar^2}\sum_{n}
({\cal{E}}_{F}-\varepsilon_n)\Theta({\cal{E}}_{F}-\varepsilon_n)
|\psi_n(z)|^2$$ is the number of electrons (negative charges) in the system.
Additionally, the global neutrality condition teaches that if the net electric charge should be zero, the number of positive charges should be equal to the number of negative charges, that is $$\begin{aligned}
ZN_+d&=&\int{}n(z)dz
\nonumber\\ &=&
{m\over\pi\hbar^2}\sum_{n}
({\cal{E}}_{F}-\varepsilon_n)\Theta({\cal{E}}_{F}-\varepsilon_n)
\int|\psi_n(z)|^2dz
\nonumber\\ &=&
{m\over\pi\hbar^2}\sum_{n}
({\cal{E}}_{F}-\varepsilon_n)\Theta({\cal{E}}_{F}-\varepsilon_n)\end{aligned}$$ where $N_+$ is the number of positive ions per unit volume and $Z$ is the valence of each of these ions.
Defining the quantity $N_F$ as the index of the highest occupied level, this may be rewritten into $$ZN_+d={m\over\pi\hbar^2}\sum_{n=1}^{N_F}
\left({\cal{E}}_{F}-\varepsilon_n\right),$$ from which the Fermi energy easily is extracted as $${\cal{E}}_{F}={1\over{}N_F}\left[{\pi\hbar^2\over{}m_e}ZN_+d
+\sum_{n=1}^{N_F}\varepsilon_n\right].
\label{eq:EF-T=0}$$
Infinite barrier quantum well
-----------------------------
In the infinite barrier model for a quantum well extending from $0$ to $-d$ in the $z$-direction and infinitely in the $x$-$y$-plane, we have $$\varepsilon_n={\pi^2\hbar^2n^2\over2m_ed^2},$$ which inserted into Eq. (\[eq:EF-T=0\]) gives $$\begin{aligned}
{\cal{E}}_{F}&=&{1\over{}N_F}\left[{\pi\hbar^2\over{}m_e}ZN_+d
+{\pi^2\hbar^2\over2m_ed^2}\sum_{n=1}^{N_F}n^2\right]
\nonumber\\ &=&
{\pi\hbar^2\over{}N_Fm_e}\left[ZN_+d+{\pi\over2d^2}{N_F(N_F+1)(2N_F+1)\over6}
\right].
\label{eq:EF-IB}\end{aligned}$$ From this equation, the limits on the thickness of the quantum well can be determined if we know the number of bound states we want below the Fermi level, the minimal thickness for the quantum well to have $n$ levels being determined from the simple relation ${\cal{E}}_{F}=\varepsilon_n$, and thus the maximal thickness can be determined from ${\cal{E}}_{F}=\varepsilon_{n+1}$, since it has the same limit value as the minimal thickness to obtain $n+1$ bound states. Thus, for $n$ bound states below the Fermi level, $${\pi\hbar^2\over{}nm_e}\left[ZN_+d+{\pi\over2d^2}{n(n+1)(2n+1)\over6}
\right]={\pi^2\hbar^2n^2\over2m_ed^2},$$ which gives the related minimal and maximal thicknesses to have these $n$ bound states $$\begin{aligned}
d_{\rm{min}}^{n}&=&
\sqrt[3]{{\pi{}n\over2ZN_+}\left[n^2-{(n+1)(2n+1)\over6}\right]},
\nonumber\\
d_{\rm{max}}^{n}&=&
\sqrt[3]{{\pi{}(n+1)\over2ZN_+}\left[(n+1)^2-{(n+2)(2n+3)\over6}\right]}.\end{aligned}$$ For a quantum well with only a single bound state we thus get $$\begin{aligned}
d_{\rm{min}}^{(1)}&=&0,
\nonumber\\
d_{\rm{max}}^{(1)}&=&\sqrt[3]{3\pi/2ZN_+},\end{aligned}$$ for two bound states $$\begin{aligned}
d_{\rm{min}}^{(2)}&=&\sqrt[3]{3\pi/2ZN_+},
\nonumber\\
d_{\rm{max}}^{(2)}&=&\sqrt[3]{39\pi/6ZN_+},\end{aligned}$$ and so on.
Since $\hbar^2k_F^2=2m_e{\cal{E}}_{F}$, the radius of the two-dimensional Fermi circle for state $n$, $\alpha(n)$, used as integration boundary in Appendix \[ch:Solve-Q\] can be found using Eq. (\[eq:EF-IB\]). It is $$\alpha(n)=\sqrt{{\pi{}ZN_+d\over{}N_F}+{\pi^2\over2d}{(N_F+1)(2N_F+1)\over6}
-{n^2\pi^2\over{}d^2}}.
\label{eq:alpha(n)}$$
Solution to integrals over $z$ in Chapter \[ch:14\] {#app:E}
===================================================
In this appendix we give some intermediate steps of the solution to integrals in the quantity $\tensor{K}(\vec{q}_{\|},\omega)$ appering in the description of the multilevel quantum well in Chapter \[ch:14\].
Inserting the expressions for the different $F$ quantities \[Eqs. (\[eq:F-xx\])–(\[eq:F-zz\])\] into Eqs. (\[eq:Kxx\])–(\[eq:Kzz\]), we get $$\begin{aligned}
\lefteqn{K_{xx,mn}^{vl}(\vec{q}_{\|},\omega)=
-{\rm{i}}\mu_0\omega
\left\{
{\cal{Q}}_{\,\,\,lv}^{xx}(\vec{q}_{\|},\omega)
\iint{\cal{Z}}^{x}_{mn}(z)G_{xx}(z,z'';\vec{q}_{\|},\omega){\cal{Z}}^{x}_{lv}(z'')dz''dz
\right.}\nonumber\\ &\quad&\left.\!
-i{\cal{Q}}_{\,\,\,lv}^{xz}(\vec{q}_{\|},\omega)
\iint{\cal{Z}}^{x}_{mn}(z)G_{xz}(z,z'';\vec{q}_{\|},\omega){\cal{Z}}^{z}_{lv}(z'')dz''dz
\right\},
\label{eq:K-xx-i}
\\
\lefteqn{K_{xz,mn}^{vl}(\vec{q}_{\|},\omega)=
{\rm{i}}\mu_0\omega
\left\{
i{\cal{Q}}_{\,\,\,lv}^{xz}(\vec{q}_{\|},\omega)
\iint{\cal{Z}}^{x}_{mn}(z)G_{xx}(z,z'';\vec{q}_{\|},\omega){\cal{Z}}^{x}_{lv}(z'')dz''dz
\right.}\nonumber\\ &&\left.\!
+{\cal{Q}}_{\,\,\,lv}^{zz}(\vec{q}_{\|},\omega)
\iint{\cal{Z}}^{x}_{mn}(z)G_{xz}(z,z'';\vec{q}_{\|},\omega){\cal{Z}}^{z}_{lv}(z'')dz''dz
\right\},
\\
\lefteqn{K_{yy,mn}^{vl}(\vec{q}_{\|},\omega)=
-{\rm{i}}\mu_0\omega {\cal{Q}}_{\,\,\,lv}^{yy}(\vec{q}_{\|},\omega)
\iint{\cal{Z}}^{x}_{mn}(z)G_{yy}(z,z'';\vec{q}_{\|},\omega){\cal{Z}}^{x}_{lv}(z'')dz''dz,
}\\
\lefteqn{K_{zx,mn}^{vl}(\vec{q}_{\|},\omega)=
-{\rm{i}}\mu_0\omega
\left\{
{\cal{Q}}_{\,\,\,lv}^{xx}(\vec{q}_{\|},\omega){q_{\|}\over{}q_{\perp}}
\iint{\cal{Z}}^{z}_{mn}(z)G_{xx}(z,z'';\vec{q}_{\|},\omega){\cal{Z}}^{x}_{lv}(z'')dz''dz
\right.}\nonumber\\ &&\left.\!
-i{\cal{Q}}_{\,\,\,lv}^{xz}(\vec{q}_{\|},\omega)
{q_{\|}\over{}q_{\perp}}
\iint{\cal{Z}}^{z}_{mn}(z)G_{xz}(z,z'';\vec{q}_{\|},\omega){\cal{Z}}^{z}_{lv}(z'')dz''dz
\right\},
\\
\lefteqn{K_{zz,mn}^{vl}(\vec{q}_{\|},\omega)=
{\rm{i}}\mu_0\omega
\left\{
i{\cal{Q}}_{\,\,\,lv}^{xz}(\vec{q}_{\|},\omega){q_{\|}\over{}q_{\perp}}
\iint{\cal{Z}}^{z}_{mn}(z)G_{xx}(z,z'';\vec{q}_{\|},\omega){\cal{Z}}^{x}_{lv}(z'')dz''dz
\right.}\nonumber\\ &&\left.\!
+{\cal{Q}}_{\,\,\,lv}^{zz}(\vec{q}_{\|},\omega)
{q_{\|}\over{}q_{\perp}}
\iint{\cal{Z}}^{z}_{mn}(z)G_{xz}(z,z'';\vec{q}_{\|},\omega){\cal{Z}}^{z}_{lv}(z'')dz''dz
\right\}.
\label{eq:K-zz-i}\end{aligned}$$ Using an infinite barrier potential along the $z$-direction of the quantum well, the ${\cal{Z}}(z)$ quantities are described in Eqs. (\[eq:ZxIB\]) and (\[eq:ZzIB\]). Then by use of , Eqs. 2.663.1 and 2.663.3, $$\begin{aligned}
\int e^{ax}\sin(bx)dx&=&{e^{ax}[a\sin(bx)-b\cos(bx)]\over{}a^2+b^2},
\\
\int e^{ax}\cos(bx)dx&=&{e^{ax}[a\cos(bx)+b\sin(bx)]\over{}a^2+b^2},\end{aligned}$$ we find that integrals over the source region takes the form $$\begin{aligned}
\int_{-d}^{0}e^{ax}\sin\left({b\pi{}x\over{}d}\right)dx&=&
{\pi{}bd[e^{-ad}(-1)^b-1]\over{}a^2d^2+\pi^2b^2},
\\
\int_{-d}^{0}e^{ax}\cos\left({b\pi{}x\over{}d}\right)dx&=&
{ad^2[1-e^{-ad}(-1)^b]\over{}a^2d^2+\pi^2b^2},\end{aligned}$$ in which $b$ is an integer. This result leads to $$\begin{aligned}
\lefteqn{
\int G_{xx}(z,z';\vec{q}_{\|},\omega){\cal{Z}}^{x}_{nm}(z')dz'=
}\nonumber\\ &\quad&
{2\pi^2nmc_0^2q_{\perp}^2d\left[
1+r^p-\left(e^{-iq_{\perp}d}+r^pe^{iq_{\perp}d}\right)(-1)^{n+m}\right]
\over\omega^2[(iq_{\perp}d)^2+\pi^2(n-m)^2][(iq_{\perp}d)^2+\pi^2(n+m)^2]}
e^{-iq_{\perp}z},
\\
\lefteqn{
\int G_{xz}(z,z';\vec{q}_{\|},\omega){\cal{Z}}^{x}_{nm}(z')dz'=
}\nonumber\\ &\quad&
{2\pi^2nmc_0^2q_{\|}q_{\perp}d\left[
1-r^p-\left(e^{-iq_{\perp}d}-r^pe^{iq_{\perp}d}\right)(-1)^{n+m}\right]
\over\omega^2[(iq_{\perp}d)^2+\pi^2(n-m)^2][(iq_{\perp}d)^2+\pi^2(n+m)^2]}
e^{-iq_{\perp}z},
\\
\lefteqn{
\int G_{yy}(z,z';\vec{q}_{\|},\omega){\cal{Z}}^{x}_{nm}(z')dz'=
}\nonumber\\ &\quad&
{2\pi^2nmd\left[
1-r^s-\left(e^{-iq_{\perp}d}-r^se^{iq_{\perp}d}\right)(-1)^{n+m}\right]
\over[(iq_{\perp}d)^2+\pi^2(n-m)^2][(iq_{\perp}d)^2+\pi^2(n+m)^2]}
e^{-iq_{\perp}z},
\\
\lefteqn{
\int G_{xz}(z,z';\vec{q}_{\|},\omega){\cal{Z}}^{z}_{nm}(z')dz'=
}\nonumber\\ &\quad&
{2\pi^4nm(n^2-m^2)c_0^2q_{\|}
[(e^{-iq_{\perp}d}+r^pe^{iq_{\perp}d})(-1)^{n+m}-1-r^p]
\over{}i\omega^2d
[(iq_{\perp}d)^2+\pi^2(n-m)^2][(iq_{\perp}d)^2+\pi^2(n+m)^2]}
e^{-iq_{\perp}z},\end{aligned}$$ and $$\begin{aligned}
\int {\cal{Z}}^{x}_{mn}(z)e^{-iq_{\perp}z}dz&=&
{4\pi^2nmiq_{\perp}d[e^{iq_{\perp}d}(-1)^{n+m}-1]
\over[(iq_{\perp}d)^2+\pi^2(n-m)^2][(iq_{\perp}d)^2+\pi^2(n+m)^2]},
\\
\int {\cal{Z}}^{z}_{mn}(z)e^{-iq_{\perp}z}dz&=&
{4\pi^4nm(n^2-m^2)[e^{iq_{\perp}d}(-1)^{n+m}-1]
\over{}d[(iq_{\perp}d)^2+\pi^2(n-m)^2][(iq_{\perp}d)^2+\pi^2(n+m)^2]},\end{aligned}$$ since $(-1)^{n-m}=(-1)^{n+m}$ for $n$ and $m$ integers. Then $$\begin{aligned}
\lefteqn{ F_{nm}^{xx}(z;\vec{q}_{\|},\omega)=
-{2\pi^2inm\over\epsilon_0\omega}
e^{-iq_{\perp}z}
\left\{
{\cal{Q}}_{\,\,\,nm}^{xx}(\vec{q}_{\|},\omega)
{q_{\perp}^2d}
+{\cal{Q}}_{\,\,\,nm}^{xz}(\vec{q}_{\|},\omega)
{\pi^2(n^2-m^2)q_{\|} \over{}d}
\right\}
}\nonumber\\ &\quad&\times
{1+r^p-\left(e^{-iq_{\perp}d}+r^pe^{iq_{\perp}d}\right)(-1)^{n+m}
\over[(iq_{\perp}d)^2+\pi^2(n-m)^2][(iq_{\perp}d)^2+\pi^2(n+m)^2]}
\label{eq:F-xx-i}\\
\lefteqn{ F_{nm}^{xz}(z;\vec{q}_{\|},\omega)=
{2\pi^2inm\over\epsilon_0\omega}
e^{-iq_{\perp}z}
\left\{
i{\cal{Q}}_{\,\,\,nm}^{xz}(\vec{q}_{\|},\omega)
q_{\perp}^2d
-{\cal{Q}}_{\,\,\,nm}^{zz}(\vec{q}_{\|},\omega)
{\pi^2(n^2-m^2)q_{\|}\over{}id}
\right\}
}\nonumber\\ &&\times
{1+r^p-\left(e^{-iq_{\perp}d}+r^pe^{iq_{\perp}d}\right)(-1)^{n+m}
\over[(iq_{\perp}d)^2+\pi^2(n-m)^2][(iq_{\perp}d)^2+\pi^2(n+m)^2]},
\label{eq:F-xz-i}\\
\lefteqn{ F_{nm}^{yy}(z;\vec{q}_{\|},\omega)=
-2\pi^2{\rm{i}}\mu_0\omega nmd
e^{-iq_{\perp}z}
{\cal{Q}}_{\,\,\,nm}^{yy}(\vec{q}_{\|},\omega)
}\nonumber\\ &&\times
{1-r^s-\left(e^{-iq_{\perp}d}-r^se^{iq_{\perp}d}\right)(-1)^{n+m}
\over[(iq_{\perp}d)^2+\pi^2(n-m)^2][(iq_{\perp}d)^2+\pi^2(n+m)^2]},
\label{eq:F-yy-i}\end{aligned}$$ If we now insert Eqs. (\[eq:F-xx-i\])–(\[eq:F-yy-i\]), (\[eq:F-zx\]) and (\[eq:F-zz\]) into Eqs. (\[eq:K-xx-i\])–(\[eq:K-zz-i\]) and perform the remaining integration, we get Eqs. (\[eq:Kxx\])–(\[eq:Kzz\]).
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[**Dansk resum[é]{}**]{}
For at studere den mulige fasekonjugation af optiske n[æ]{}rfelter er det n[ø]{}dvendigt at g[å]{} ud over den langsomtvarierende indhyldningskurve approksimation samt den elektriske dipoltiln[æ]{}rmelse, der normalt anvendes i fasekonjugationsstudier hvor rumligt ud[æ]{}mpede (eller i det mindste svagt d[æ]{}mpede) svingninger blandes. I den foreliggende afhandling pr[æ]{}senteres en vilk[å]{}rlig-fase tiln[æ]{}rmet beregning af den uline[æ]{}re og ikke-lokale optiske responstensor der beskriver den uliner[æ]{}re str[ø]{}mt[æ]{}thed af tredie orden, som genereres af fireb[ø]{}lgeblanding i en uensartet elektrongas. Beskrivelsen er baseret p[å]{} en halvklassisk model, hvori det elektromagnetiske felt antages at v[æ]{}re en klassisk st[ø]{}rrelse og udgangspunktet er bev[æ]{}gelsesligningen for t[æ]{}thedsmatrix-operatoren. Vekselvirknings Hamilton-operatoren anvendes i dens minimale koblingsform, og den indeholder det led i str[ø]{}mt[æ]{}thedsoperatoren, der er proportionalt med det p[å]{}trykte vektorpotential. Ved brug af denne formalisme er den rumlige struktur af systemets optiske respons beskrevet ved hj[æ]{}lp af mikroskopiske overgangsstr[ø]{}mt[æ]{}theder. Beregningen inkluderer derfor b[å]{}de bidrag fra $\vec{p}\cdot\vec{A}$ og $\vec{A}\cdot\vec{A}$ leddene i vekselvirknings-Hamilton-operatoren. Det er vist at der introduceres nogle vigtige f[æ]{}nomener, som er begrebsm[æ]{}ssigt forskellige fra de der har deres oprindelse i $\vec{p}\cdot\vec{A}$-leddet, ved at inkludere $\vec{A}\cdot\vec{A}$ leddet i vekselvirknings Hamilton-operatoren. For at fremh[æ]{}ve den fysiske mening af de forskellige processer er koblingerne mellem observationspunkter for feltet og str[ø]{}mt[æ]{}theden pr[æ]{}senteret i form af diagrammer. Resultatet af en analyse af tensorsymmetrierne, der er tilknyttet $\vec{p}\cdot\vec{A}$ og $\vec{A}\cdot\vec{A}$ vekselvirkningerne er summeret i form af symmetriskemaer for fasekonjugationsprocessen. Den teoretiske model efterf[ø]{}lges af en beregning af det fasekonjugerede respons fra en et-niveau metallisk kvantebr[ø]{}nd. Et-niveau kvantebr[ø]{}nden repr[æ]{}senterer den simplest mulige konfiguration en kvantebr[ø]{}nds-fasekonjugator kan have. Ydermere er den et interessant objekt, idet dens optiske respons ikke indeholder noget dipol-led. Diskussionen af responset er baseret p[å]{} stimulering af processen ved brug af lys, der er polariseret enten i spredningsplanet eller vinkelret p[å]{} spredningsplanet. Det vises, at fasekonjugationsprocessen er ekstremt effektiv i det d[æ]{}mpede omr[å]{}de af b[ø]{}lgevektor-spektret. Dern[æ]{}st anskues problemet med at generere plane b[ø]{}lger til excitation i den h[ø]{}je ende af det d[æ]{}mpede spektrum, og vi diskuterer brugen af en bredb[å]{}ndskilde (i vinkelspektret) til at stimulere processen. En s[å]{}dan bredb[å]{}ndskilde kan v[æ]{}re en kvantetr[å]{}d, og det fasekonjugerede vinkelspektrum fra en kvantetr[å]{}d pr[æ]{}senteres og diskuteres. Kvantetr[å]{}dens subb[ø]{}lgel[æ]{}ngde st[ø]{}rrelse g[ø]{}r den en mulig kandidat til en diskussion af den mulige rumlige komprimering af lys, og rumlig begr[æ]{}nsning af lys foran en et-niveau metallisk kvantebr[ø]{}ndsfasekonjugator er diskuteret i to dimensioner. Det retf[æ]{}rdigg[ø]{}res at man ved et passende valg af str[ø]{}mt[æ]{}thedens orientering i kvantetr[å]{}den kan opn[å]{} en feltkomprimering, der er v[æ]{}sentligt p[å]{} den anden side af Rayleighs gr[æ]{}nsev[æ]{}rdi. Afhandlingen afsluttes med en kort beskrivelse af det mere generelle tilf[æ]{}lde, hvor kvantebr[ø]{}nden tillades at have mere end en energi-egentilstand. Numeriske resultater, der viser responset hvis en to-niveau kvantebr[ø]{}nd anvendes som fasekonjugerende medium, er pr[æ]{}senteret og diskuteret.
to2cm[ ]{}
(0,0)(141,204) (0,204)
(0,0)\[tl\]
In order to study the possible phase conjugation of optical near-fields, it is necessary to go beyond the slowly varying envelope- and electric dipole approximations that are normally applied in phase conjugation studies where spatially non-decaying (or at least slowly decaying) modes are mixed. In the present dissertation a random-phase-approximation calculation of the nonlocal nonlinear optical response tensor describing the third order nonlinear current density generated by degenerate four-wave mixing in an inhomogeneous electron gas is established. The description is based on a semi-classical approach, in which the electromagnetic field is considered as a classical quantity, and the starting point is the equation of motion for the density matrix operator. The interaction Hamiltonian is taken in its minimal coupling form, and it includes the term in the current density operator which is proportional to the prevailing vector potential. Using this formalism the spatial structure of the optical response of the system is described in terms of the microscopic transition current densities. The calculation thus includes contributions originating from both the $\vec{p}\cdot\vec{A}$ and $\vec{A}\cdot\vec{A}$ terms in the interaction Hamiltonian. It is demonstrated that inclusion of the $\vec{A}\cdot\vec{A}$ term in the interaction Hamiltonian introduces some important phenomena that are conceptually different from those originating in the $\vec{p}\cdot\vec{A}$ part. To emphasize the physical meaning of the various processes, the couplings between observation points for the field and the current density is presented in a diagrammatic form. The result of an analysis of the tensor symmetries associated with the $\vec{p}\cdot\vec{A}$ and $\vec{A}\cdot\vec{A}$ interactions are summarized in terms of symmetry schemes for the phase conjugation process. The theoretical model is followed by a calculation of the phase conjugated response from a single-level metallic quantum well. The single-level quantum well represents the simplest possible configuration of a quantum-well phase conjugator. Furthermore, it is an interesting object, since its optical response contains no dipole terms. The discussion of the response is based on the use of light that is polarized either in the scattering plane or perpendicular to the scattering plane to excite the process. It is demonstrated that the phase conjugation process is extremely efficient in the evanescent regime of the wavevector spectrum. We address the problem of plane-wave excitation in the high wavenumber end of the evanescent regime and discuss the use of a broadband source to excite the process. One possible broad angular band source is a quantum wire, and the phase conjugated angular spectrum from a quantum wire is presented and discussed. The subwavelength size of the quantum wire makes it a possible candidate for discussion of confinement of light, and the confinement of light in two dimensions in front of a single-level metallic quantum-well phase conjugator is discussed. It is justified that by a proper choice of orientation of the current in the quantum wire a field compression substantially beyond the Rayleigh limit is obtained. The thesis is concluded with a short description of the more general case where the quantum well is allowed to have more than one energy eigenstate, and numerical results showing the response from a two-level quantum well as the phase conjugating medium are presented and discussed.
(0,0)[(0,0)\[bl\][Med et resum[é]{} p[å]{} dansk.]{}]{} (141,0)[(0,0)\[br\][ISBN 87-89195-16-7]{}]{}
[^1]: With respect to the result published in the last sum in Eq. (\[eq:J3-3\]) above is written in a more compact form than in Eq. (22) of . The compact form in Eq. (\[eq:J3-3\]) is obtained by exchanging indices $v$ and $l$ in the last three terms of , Eq. (22). As a consequence of this, the same difference occur between Eqs. (\[eq:XiAGrz\]) and (\[eq:XiAG\]) and Eqs. (34) and (51) of , respectively.
|
---
author:
- 'S. Issaoun'
- 'M. D. Johnson, L. Blackburn'
- 'M. Mo[ś]{}cibrodzka'
- 'A. Chael'
- 'H. Falcke'
bibliography:
- 'rgauss.bib'
nocite:
- '[@PaperI]'
- '[@PaperII]'
- '[@PaperIII]'
- '[@PaperIV]'
- '[@PaperV]'
- '[@PaperVI]'
title: VLBI imaging of black holes via second moment regularization
---
Introduction {#sec:intro}
============
Very-long-baseline interferometry (VLBI) is a technique able to achieve high angular resolution imaging through the use of widely separated antennas. Unfortunately, as the observing frequency is increased, the availability of suitable sites on Earth is greatly reduced, leading to sparse arrays with a high angular resolution but a low spatial dynamic range. In particular, a simple inverse Fourier transform of the visibilities measured by an interferometer, or ‘dirty image’, is dominated by artifacts introduced by sparse sampling of the Fourier plane. Short baselines are particularly important in imaging, as they anchor the flux distribution and provide a crucial link between high-resolution small-scale features and the large-scale extent and morphology of the target. The sparser the array, the more challenging it is to reconstruct images from interferometric measurements. Additionally, weather and technical issues at sites that provide short/mid-range baselines can greatly degrade the ability to image a given data set.
Array sparsity and station-based errors can have dramatic effects on reconstructed images. Thus, the imaging process requires further information and assumptions beyond the visibility measurements from the interferometer. The choice of imaging method imposes additional constraints on the reconstructed image. Here, we will focus on extending the method of regularized maximum likelihood (RML) that performs well under sparse sampling conditions and does not involve direct inverse Fourier transforms of the data in the imaging process.
In this paper we present an algorithmic contingency to array sparsity and site issues in the form of a second moment regularization function. That is, the compactness of the source can be expressed as the second moment of the source brightness distribution [@moffet_1962; @burn_1976], which can be constrained to match, for example, confident source size measurements from short baselines of previous experiments or epochs. Enforcing this source size constraint supplements limited short-baseline information while fitting to long-baseline smaller scale structure from newer observations.
The Event Horizon Telescope (EHT), observing at a frequency of 230GHz [@PaperI; @PaperII], is a prime example of a high-frequency VLBI imaging experiment with image uncertainties dominated by the effects of sparse coverage. The EHT currently has only a single short/mid-range VLBI baseline, joining the Large Millimeter Telescope Alfonso Serrano (LMT) in Mexico to the Submillimeter Telescope (SMT) in Arizona. Recent observations with the EHT have shown that the LMT is difficult to calibrate, giving baselines with large measurement uncertainties dominated by uncharacterized station behavior in 2017 [@PaperIII; @PaperIV].
Although the EHT observes a number of non-horizon-scale sources in conjunction with the Atacama Large Millimeter/submillimeter Array (ALMA), its primary targets are the two supermassive black hole candidates in the Galactic Center, Sagittarius A\* (), and at the center of the radio galaxy M87. At the frequency of the EHT, these two sources are very compact, with sizes on the sky historically measured with three stations, in California, Arizona, and Hawaii, in early EHT observations, and are thus ideal imaging targets for second moment regularization [@Doeleman_2008; @Fish_2011; @Doeleman_2012; @Akiyama_2015; @Johnson_2015; @Lu_2018]. Near-zero closure phases on the California–Arizona–Hawaii triangle are indicative of the source compactness and symmetry on scales of a few tens of $\mu$as [@Akiyama_2015; @Fish_2016]. The California–Arizona baseline provided the short-baseline measurements needed to constrain the compactness and size of the sources in the visibility domain. Recent observations of M87 in 2017 also found a source size of $\sim40\,\mu$as consistent with previous measurements [@PaperI; @PaperII; @PaperIII; @PaperIV; @PaperV; @PaperVI].
For , the source size is also well-constrained at lower frequencies due to its compactness and dominant diffractive scattering [@Shen_2005; @Bower_2006; @Lu_2011; @Johnson_2018]. VLBI observations at 86GHz taken one month apart give fitted Gaussian source sizes for the scattered image of with $<10\%$ difference [@Ortiz_2016; @Brinkerink_2019]. At this frequency, while the small scale structure is expected to vary, the large-scale information, dominated by the size of the scattering kernel, should be stable from epoch to epoch [@Johnson_2018].
Second moment regularization merges the benefits of model-fitting with the flexibility of imaging: compared to self-calibration to a known model, it does not actually modify the measured visibilities used for the imaging process or enforce a model-dependent solution, but instead provides additional information to improve image quality. The regularization constrains the spread of flux density to a motivated region in the image, discouraging non-physical morphology driven by fits to long-baseline data and accelerating convergence toward a plausible image. It is a natural extension of imaging tools that add source information in the imaging process in RML methods: a total flux constraint is in fact the zeroth moment of the image; an image centroid specification corresponds to the first moment of the image; and a short-baseline source size completes the picture by constraining the image second moment. The implementation of second moment regularization can be done in conjunction with other tools and constraints in RML, for both static and movie reconstructions. Furthermore, as the constraint function acts on the image itself and does not modify the visibility data, it can be used with any choice of data product, including minimally-calibrated closure phases and amplitudes.
The paper is structured as follows. We present the mathematical background to motivate the regularization in Sect. \[sec:background\]. We outline the method, assumptions, and physical motivation in Sect. \[sec:method\]. In Sect. \[sec:demo\] we demonstrate the improvements in image quality and fidelity using the regularization with or without a priori knowledge of the source size. We present possible applications of the second moment regularization to more sophisticated imaging techniques for scattering mitigation and movie reconstructions in Sect. \[sec:examples\]. A summary is given in Sect. \[sec:summary\].
Background {#sec:background}
==========
By the van Cittert-Zernike theorem, an interferometer samples complex visibilities corresponding to Fourier components of an image [@vancittert; @zernike]. Consequently, $n^{\text{th}}$ moments of an image correspond to $n^{\text{th}}$ derivatives of the visibility function at the origin. Specifically, an interferometric visibility $V(\mathbf{u})$ on a baseline $\mathbf{u}$ can be written as [e.g., @TMS] $$\begin{aligned}
V(\mathbf{u}) &= \int d^2\mathbf{x}\, I(\mathbf{x}) e^{-2\pi i \mathbf{u} \cdot \mathbf{x}}, \label{eq:cittert-zernicke}\end{aligned}$$ where $I(\mathbf{x})$ is the brightness distribution on the sky, and $\mathbf{x}$ is an angular unit.
From this expression, $V(\mathbf{0}) = \int d^2\mathbf{x}\, I(\mathbf{x}) \in \mathbb{R}$ gives the total flux density of the image (the $0^{\text{th}}$ moment). Likewise, the phase gradient of the visibility function at zero baseline gives a vector proportional to the centroid of the image, $$\begin{aligned}
\nonumber \left. \nabla V(\mathbf{u}) \right\rfloor_{\mathbf{u}=\mathbf{0}} &= -2\pi i \int d^2\mathbf{x}\, \mathbf{x} I(\mathbf{x}) \\
&= -2\pi i V(\mathbf{0}) \boldsymbol{\mu},\end{aligned}$$ where $\boldsymbol{\mu}$ is the image centroid (the normalized $1^{\text{st}}$ moment): $$\begin{aligned}
\boldsymbol{\mu} = ({\ensuremath{\bar{x}}}\mathbf{\hat{x}}, {\ensuremath{\bar{y}}}\mathbf{\hat{y}}) = \frac{\int d^2\mathbf{x} I(\mathbf{x}) \mathbf{x}}{\int d^2\mathbf{x}\, I(\mathbf{x}) }.\end{aligned}$$ Because the image is real, the gradient $\left. \nabla V(\mathbf{u}) \right\rfloor_{\mathbf{u}=\mathbf{0}}$ is purely imaginary. For images that are positive (e.g., images in total intensity), the visibility function must take its maximum amplitude at the origin. More generally, the visibility function is Hermitian; thus, its amplitude must always have a vanishing gradient at the origin because of the conjugation symmetry $V(\mathbf{u}) = V^\ast(-\mathbf{u})$.
The second derivative, or Hessian, of the visibility amplitude function at zero baseline gives a matrix (see Appendix \[sec:vis\_der\]): $$\begin{aligned}
\nonumber \left. \nabla\nabla ^\intercal |V(\mathbf{u})| \right\rfloor_{\mathbf{u}=\mathbf{0}} &= -4\pi^2 \int d^2\mathbf{x}\, I(\mathbf{x})(\mathbf{x}-\boldsymbol{\mu}) (\mathbf{x}-\boldsymbol{\mu})^\intercal \\
&= -4\pi^2 V(\mathbf{0}) \boldsymbol{\Sigma}, \label{eq:2mom}\end{aligned}$$ where $\boldsymbol{\Sigma}$ is the normalized second central moment (or covariance matrix) of the image. We show in Appendix \[sec:vis\_der\] that this expression is equivalent to the curvature of the centered complex visibility function [see also @moffet_1962; @burn_1976]. The visibility amplitude function is a more natural data product to use for observations with non-astrometric VLBI arrays such as the EHT, where there is no absolute phase information due to strong differential atmospheric propagation effects between sources, and thus no directly measured full complex visibilities. Therefore it is useful for us to determine image moments directly from the visibility amplitude function, which is measured.
Physical Analog
------------------- ------------------------------------ --------------------------------------------------------------------------------------- --------------------- --------------------------------------------------------------------------------------------------------------------
Mass Total Flux $\int I(\mathbf{x}) d^2\mathbf{x}$ Peak Visibility $V(\mathbf{0})$
Center of Mass Centroid ($\boldsymbol{\mu}$) $V(\mathbf{0})^{-1} \int \mathbf{x} I(\mathbf{x}) d^2\mathbf{x}$ Phase Gradient $(2\pi i V(\mathbf{0}))^{-1} \left. \nabla V(\mathbf{u}) \right\rfloor_{\mathbf{u}=\mathbf{0}}$
Moment of Inertia Covariance ($\boldsymbol{\Sigma}$) $V(\mathbf{0})^{-1} \int \mathbf{x} \mathbf{x}^\intercal I(\mathbf{x}) d^2\mathbf{x}$ Amplitude Curvature $(-4\pi^2 V(\mathbf{0}))^{-1} \left. \nabla\nabla ^\intercal V(\mathbf{u}) \right\rfloor_{\mathbf{u}=\mathbf{0}} $
\[tab:inertia\]
The image covariance matrix $\boldsymbol{\Sigma}$ can be more intuitively expressed in terms of its principal axes, corresponding to the perpendicular axes about which the second moment reaches its maximum [@Hu_1962]. The matrix has two eigenvalues ${\ensuremath{\lambda_\mathrm{min}}}$ and ${\ensuremath{\lambda_\mathrm{maj}}}$, and can be diagonalized as follows: $$\begin{aligned}
\mathbf{\Sigma}
= \mathbf{R_\phi} \begin{pmatrix}
{\ensuremath{\lambda_\mathrm{min}}}& 0 \\
0 & {\ensuremath{\lambda_\mathrm{maj}}}\end{pmatrix} \mathbf{R_\phi^{\intercal}}, \end{aligned}$$ where $\mathbf{R_\phi}$ is the rotation matrix based on the position angle east of north $\phi$ of the major principal axis (Appendix \[sec:vis\_axes\]). The eigenvalues of the covariance matrix are the variances of the normalized image projected along the principal (major and minor) axes. The correspondence between [$\lambda_\mathrm{maj}$]{}, [$\lambda_\mathrm{min}$]{}, $\phi$ and the individual terms of $\boldsymbol{\Sigma}$ is given in Appendix \[sec:vis\_axes\].
Following Equation \[eq:cittert-zernicke\], we can fully express the visibility function as a Taylor expansion in its derivatives. Each $n+1^\text{th}$ term of the Taylor expansion is proportional to the $n^\text{th}$ moment of the visibility function (see Table \[tab:inertia\]). At zero baseline, only the zeroth moment remains. We choose the coordinate system such that the centroid of the image is at the origin, and the first moment of the visibility function (the second term of the Taylor expansion) vanishes. At short baseline, the centered complex visibility function is therefore dominated by the quadratic term. The Taylor expansion of the visibility function at short baseline becomes: $$\begin{aligned}
\nonumber V(\mathbf{u}) &\simeq V(\mathbf{0}) - 2\pi^2 \int d^2\mathbf{x}\, (\mathbf{u}\cdot \mathbf{x})^2 I(\mathbf{x}) \\
&\simeq V(\mathbf{0}) - 2\pi^2 V(\mathbf{0})\mathbf{u}^\intercal \boldsymbol{\Sigma} \mathbf{u} . \label{eq:gen-short}\end{aligned}$$
Equation \[eq:gen-short\] describes the visibility function behavior on short baselines entirely in terms of the total flux $V(\mathbf{0})$ and the second moment covariance matrix $\boldsymbol{\Sigma}$ projected along the baseline direction. These parameters also describe a unique visibility function of a Gaussian source with total flux $V(\mathbf{0})$, and major/minor axes sizes and orientation prescribed by the same covariance matrix. We show this by comparing the general complex visibility function to that for a Gaussian source. For the simplest case of an isotropic Gaussian source of standard deviation $\sigma$ with the same total flux $V(\mathbf{0})$, we have the following intensity pattern on the sky and corresponding visibility function: $$\begin{aligned}
I_\mathrm{gauss}(\mathbf{x}) &= \frac{V(\mathbf{0})}{2\pi\sigma^2} e^{-|\mathbf{x}|^2/2\sigma^2}, \\
V_\mathrm{gauss}(\mathbf{u}) &= V(\mathbf{0})e^{-2\pi^2|\mathbf{u}|^2\sigma^2} .\end{aligned}$$ More generally, an anisotropic Gaussian with a covariance matrix $\boldsymbol{\Sigma}$ gives: $$\begin{aligned}
I_\mathrm{gauss}(\mathbf{x}) &= \frac{V(\mathbf{0})}{2\pi\sqrt{|\boldsymbol{\Sigma}|}} e^{-\mathbf{x}^\intercal \boldsymbol{\Sigma}^{-1}\mathbf{x}}, \\
V_\mathrm{gauss}(\mathbf{u}) &= V(\mathbf{0})e^{-2\pi^2\mathbf{u}^\intercal \boldsymbol{\Sigma}\mathbf{u}}.\end{aligned}$$ Taking the Taylor expansion of the anisotropic Gaussian visibility function at short baselines, the first two terms dominate: $$\begin{aligned}
V_\mathrm{gauss}(\mathbf{u}) \simeq V(\mathbf{0}) - 2\pi^2V(\mathbf{0}) \mathbf{u}^\intercal \boldsymbol{\Sigma}\mathbf{u} .\label{eq:gauss-short}\end{aligned}$$ We thus obtain an equivalence of the behavior of the general visibility function (Equation \[eq:gen-short\]) and the Gaussian visibility function (Equation \[eq:gauss-short\]) at short baselines. This relation allows us to translate the second moment covariance matrix of the general visibility function to the covariance matrix of an anisotropic Gaussian, which provides a simple parametrization to describe the second moment in terms of the characteristic source extent. The sizes of the major and minor axes [$\theta_\mathrm{maj}$]{} and [$\theta_\mathrm{min}$]{} are simply the full widths at half-maximum (FWHMs) of the equivalent Gaussian derived from the variances projected along each principal axis: $$\begin{aligned}
{\ensuremath{\theta_\mathrm{maj}}}&= \sqrt{8\ln(2){\ensuremath{\lambda_\mathrm{maj}}}}, \\
{\ensuremath{\theta_\mathrm{min}}}&= \sqrt{8\ln(2){\ensuremath{\lambda_\mathrm{min}}}}.\end{aligned}$$
![Three images with equal extent along their respective major axis, from left to right: a Gaussian; a crescent model; a ray-traced image from a general relativistic magneto-hydrodynamics (GRMHD) simulation of a black hole shadow and accretion disk. Model visibility amplitudes along the major axis of each source as a function of $(u,v)$ distance, after flux and size normalization, show identical behavior at short baseline length but diverge at longer baseline length: the Gaussian in black; the crescent in blue; and the GRMHD simulation in red. []{data-label="fig:behavior"}](figures/multimodel_radplot.pdf){width="\linewidth"}
The equivalence to the Gaussian also gives a natural break-off point where the characteristic source size constraint from the second moment ceases to be a good approximation to the full visibility function: the $1/e$ point determining the resolvability of a Gaussian translates to the baseline length at which the visibility amplitude reaches $V(\mathbf{0})/e$. Baseline lengths longer than the $1/e$ point will lead to higher order terms of the Taylor expansion dominating the behavior and sampling finer structure in the image. We employ the $1/e$ point as a conceptual and visual limit for the source size constraint applied via the second moment regularization. It is not a hard cut-off enforced by the imaging process.
In Fig. \[fig:behavior\], we demonstrate the behavior of the normalized visibility amplitudes sampled along the source major axis as a function of projected baseline length for three images with distinctly different structure but an identical second moment. The behavior on short baselines aligns well for all three images, the amplitudes start to diverge at longer baselines. We denote the $1/e$ limit, corresponding to the resolvability of the Gaussian image, with a magenta vertical line. On baselines past this line, the amplitudes show very different behavior, dominated by the smaller-scale features in each image (or lack thereof). We can thus express the visibility amplitude function behavior on short baselines via the second moment of the image, defined by the total flux and just three Gaussian parameters: the principal axes FWHMs [$\theta_\mathrm{maj}$]{} and [$\theta_\mathrm{min}$]{} and the position angle $\phi$ of the major axis east of north. In the RML imaging process, there is an additional fifth input parameter, governing the weight of the second moment regularization, or hyperparameter $\beta_{\textrm{R}}$, following Equation \[eq::objfunc\].
Method {#sec:method}
======
RML focuses on pixelized reconstructions of the image, iteratively maximizing an “objective function”, which is analogous to a log posterior probability function. This function is a weighted (via “hyperparameters”) sum of both $\chi^2_\textrm{D}$ goodness-of-fit data terms, and regularization functions $S_{\textrm{R}}$, or “regularizers”, governing specific image properties. In this paper, we use the RML method implemented in the `eht-imaging` Python library [@Chael_2016; @Chael_2018], where the objective function $J(I)$ is minimized via gradient descent, and can be written as: $$\begin{aligned}
\label{eq::objfunc}
J(I) = {\mathlarger{\sum}}_{\mathclap{\text{data terms}}} \alpha_{\textrm{D}} \chi^2_{\textrm{D}}\left(I\right) - {\mathlarger{\sum}}_{\mathclap{\text{regularizers}}} \beta_{\textrm{R}} S_{\textrm{R}}\left(I\right),\end{aligned}$$ where $\alpha_{\textrm{D}}$ and $\beta_{\textrm{R}}$ are the input hyperparameters.
Using only five input parameters to the regularization ($V(\mathbf{0})$, [$\theta_\mathrm{maj}$]{}, [$\theta_\mathrm{min}$]{}, $\phi$ and $\beta_{\textrm{R}}$) we can now constrain the second moment of the reconstructed image to match the size constraint provided by the user for RML imaging. In Sect. \[sec:2mom\] we present our implementation of the second moment regularization function within the `eht-imaging` library minimization framework. In Sect. \[sec:assumptions\] we describe the assumptions and physical motivation for second moment regularization using historical observational measurements, known source properties and theoretical expectations.
Second moment regularization {#sec:2mom}
----------------------------
Regularization functions in imaging enforce constraints on particular properties of the image, such as image entropy [e.g., @Narayan_Nityananda_1986], smoothness [@Bouman_2016; @Chael_2016; @Kuramochi_2018] and/or sparsity [@Wiaux_2009a; @Wiaux_2009b; @Honma_2014; @Akiyama_2017b; @Akiyama_2017a]. Simple constraints, such as image positivity, image total flux (zeroth moment) or image centering (first moment), are often applied to the image, utilizing known information on the behavior of the total intensity distribution of the source imaged. The implementation of a second moment regularization, constraining the size of the source, is thus a natural extension of common imaging tools that add source information to the imaging process.
We define a regularization function that is minimized when the covariance matrix of the reconstructed image $\boldsymbol{\Sigma}$ matches a user-specified covariance matrix $\boldsymbol{\Sigma}'$. In practice, this latter matrix is computed using user-specified principal axes FWHMs and position angle. We utilize the Frobenius norm to determine a penalty function that quantifies the difference between the user-specified and reconstructed covariance matrices: $$\begin{aligned}
\mathcal{R}_{\boldsymbol \Sigma} &\equiv \mathrm{Tr}\left[\left({\boldsymbol \Sigma} - {\boldsymbol \Sigma'}\right)^\intercal \left({\boldsymbol \Sigma} - {\boldsymbol \Sigma'}\right) \right]\end{aligned}$$ This regularizer is, by definition, simply the minimization of the difference between two covariance matrices. The procedure for the regularizer implementation in the [eht-imaging]{} library via gradient descent is presented in Appendix \[sec:implementation\].
Assumptions {#sec:assumptions}
-----------
The second moment regularization operates under a few key assumptions on the properties of the source observed. The main assumption of this method is the compactness of the source. In order to get a quadratic fall-off in the visibility function, as shown in Sect. \[sec:background\], the source must be compact and resolved on longer baselines of the interferometer. This method would break down for point sources or sources with complex morphology and diffuse flux on large scales.
Another assumption concerns the stability of the source size across multiple epochs. The input axis sizes and position angle for the regularization will only be valid if the source does not vary significantly in size between observations. The source size input is typically derived from observations where weather conditions, coverage, and station performance on short baselines were adequate for higher precision model fitting. The source size can then be used for data sets with larger uncertainties to improve the fidelity and convergence of the imaging process. This assumption is well-motivated for the compact sources observed with the EHT:
- at 86GHz, has been model-fitted with varying precision over two decades, with little variation in the obtained source size parameters, [@Rogers_1994; @Krichbaum_1998; @Doeleman_2001; @Shen_2005; @Lu_2011; @Ortiz_2016; @Brinkerink_2019]
- at 230GHz has been measured to be compact and stable in size between 2007 and 2013 [@Doeleman_2008; @Lu_2018; @Johnson_2018],
- M87 at 230GHz has been measured to be compact and stable in size over a decade [@Doeleman_2012; @Akiyama_2015; @PaperI; @PaperII; @PaperIII; @PaperIV; @PaperV; @PaperVI].
It is worth noting that this assumption breaks down for sources with multiple bright components moving relative to each other, as is common for multi-epoch images of bright jet sources. An overall size measurement from a single epoch would not translate to other observations due to components appearing or moving outward, changing the source morphology significantly between observations. The quadratic fall-off approximation until the $1/e$ point would also not hold for two separated point sources, which do show a quadratic fall-off in the visibility amplitudes but the amplitudes would quickly evolve to more complex structure on longer baselines that could be identified as the behavior of two point sources interfering. The method is most effective whenever the emission is confined within a single compact region or on multiple scales that are substantially separated, and particularly if the scale of the emission in the image is comparable to the resolution of the array.
![Principal axes FWHMs as a function of time for the simulation of in Fig. \[fig:grmhd\_im\] [@Moscibrodzka_2018]. The solid lines show sizes for the simulation, the dotted lines show sizes for the simulation including the effects of interstellar scattering [@Johnson_2016; @Johnson_2018]. The scattering major axis is aligned with the source minor axis, and thus the scattering kernel slightly dominates the minor axis size, which stabilizes the minor axis FWHM time series. The sizes were obtained from measurements of the image second moment per frame. For all four size trends, the deviation about the mean size is $<10\%$.[]{data-label="fig:grmhd_var"}](figures/monika_example_true_image.pdf "fig:"){width="0.515\linewidth"} ![Principal axes FWHMs as a function of time for the simulation of in Fig. \[fig:grmhd\_im\] [@Moscibrodzka_2018]. The solid lines show sizes for the simulation, the dotted lines show sizes for the simulation including the effects of interstellar scattering [@Johnson_2016; @Johnson_2018]. The scattering major axis is aligned with the source minor axis, and thus the scattering kernel slightly dominates the minor axis size, which stabilizes the minor axis FWHM time series. The sizes were obtained from measurements of the image second moment per frame. For all four size trends, the deviation about the mean size is $<10\%$.[]{data-label="fig:grmhd_var"}](figures/monika_example_scattered_image.pdf "fig:"){width="0.51\linewidth"}
![Principal axes FWHMs as a function of time for the simulation of in Fig. \[fig:grmhd\_im\] [@Moscibrodzka_2018]. The solid lines show sizes for the simulation, the dotted lines show sizes for the simulation including the effects of interstellar scattering [@Johnson_2016; @Johnson_2018]. The scattering major axis is aligned with the source minor axis, and thus the scattering kernel slightly dominates the minor axis size, which stabilizes the minor axis FWHM time series. The sizes were obtained from measurements of the image second moment per frame. For all four size trends, the deviation about the mean size is $<10\%$.[]{data-label="fig:grmhd_var"}](figures/LO_fwhm_timeseries.pdf){width="0.95\linewidth"}
![Geometric mean FWHM of principal axes as a function of frequency for the ray-traced simulation of in Fig. \[fig:grmhd\_im\] [@Moscibrodzka_2018]. The blue curve shows size evolution for the simulation, the red curve shows size evolution for the simulation including the effects of interstellar scattering [@Johnson_2016; @Johnson_2018]. The sizes were obtained from measurements of the image second moment per frequency bin of 20GHz. The change in size with increasing frequency becomes greatly reduced at frequencies above 300GHz, where the size of the source is dominated by the achromatic black hole shadow and the Doppler boosted features [@Falcke_2000].[]{data-label="fig:grmhd_freq"}](figures/size_freqseries.pdf){width="0.95\linewidth"}
We also assume that the extent of the source does not significantly vary within a single epoch. For static imaging of slow-varying sources, it suffices to assume that the average size of the source matches the input, but this has further implications on reconstructions of variable sources within a single epoch. The structural variability on short timescales should be contained within the region constrained by the second moment. This is an issue particularly for imaging , as the source is known to vary on timescales of minutes, much shorter than the length of a single observing epoch. We assess the degree of variability of the source extent in quiescent (non-flaring) models of using general relativistic magnetohydrodynamic (GRMHD) simulations of variable emission on horizon scales [Fig. \[fig:grmhd\_im\]; @Moscibrodzka_2018]. In Fig. \[fig:grmhd\_var\], we show the variation in the principal axes FWHMs for a typical GRMHD simulation of the accretion flow of at 230GHz, both excluding and including the effects of scattering due to the interstellar medium in our line of sight [@Johnson_2016; @Johnson_2018]. Although the simulation shows structural changes in the source morphology, deviations about the mean FWHM remain below 10% for both the model and scattered simulation principal axes.
Furthermore, the emitting gas around supermassive black holes in low-luminosity active galactic nuclei becomes optically thin as we increase the observing frequency. The source extent is therefore dominated by the black hole shadow and Doppler-boosted features at higher frequencies [@Falcke_2000]. This behavior is shown in Fig. \[fig:grmhd\_freq\] for the GRMHD simulation of the quiescent accretion flow of observed at frequencies from 80GHz to 1THz [@Moscibrodzka_2018]. At frequencies of $\sim$300GHz and above, the source size changes very little with increasing frequency. These achromatic properties motivate the extrapolation of a source size from lower-frequency observations with short baselines, such as the EHT at 230GHz, to higher-frequency imaging experiments such as the upcoming EHT at 345GHz [@PaperII; @Doeleman_astro2020].
![Visibility amplitudes for a model image of a semi-analytic advection-dominated accretion flow (ADAF) model of [@Broderick_2011] with a FWHM of $\sim60\,\mu$as as a function of $(u,v)$ distance sampled by the EHT in 2017 with and without the LMT (affecting mid-range baselines). The regularizer governs the visibility amplitude behavior at short baselines until the $1/e$ point. This allows us to constrain and correct limitations and uncertainties in LMT calibration based on the expected behavior of the LMT–SMT mid-range baseline.[]{data-label="fig:avery_radplot"}](figures/coverage_rsig.pdf){width="\linewidth"}
![Visibility amplitudes for a model image of a semi-analytic advection-dominated accretion flow (ADAF) model of [@Broderick_2011] with a FWHM of $\sim60\,\mu$as as a function of $(u,v)$ distance sampled by the EHT in 2017 with and without the LMT (affecting mid-range baselines). The regularizer governs the visibility amplitude behavior at short baselines until the $1/e$ point. This allows us to constrain and correct limitations and uncertainties in LMT calibration based on the expected behavior of the LMT–SMT mid-range baseline.[]{data-label="fig:avery_radplot"}](figures/avery_example_radplot.pdf){width="\linewidth"}
Demonstration {#sec:demo}
=============
The second moment regularization can be used with informed size constraints from previous experiments, GRMHD simulations, or achromatic features from other observing frequencies. In this section, we demonstrate how the second moment regularization adds information to the imaging process if the data set to be imaged has no short baselines. For all following tests, we use a high $\beta_{\textrm{R}}=10^5$, such that the input source size is strongly constrained in the imaging process. To put this value into perspective, $\beta_{\textrm{R}}=10^5$ would cause a ${\sim}10\%$ difference between the input and reconstructed source sizes to be penalized equivalently to a change in reduced $\chi^2$ of ${\sim}1$ in our imaging procedure. This regularization weight tends to drive the second moment of reconstructed images to be within 20% of the input values, therefore allowing some flexibility for the imaging process to deviate from the input second moment toward morphology favored by the available data.
In Sect. \[sec:demo1\] we show improvements to the reconstructions when the source size is known. In Sect. \[sec:demo2\] we study the image quality and fidelity dependence on the assumed size in the regularization. Finally in Sect. \[sec:demo3\] we demonstrate that high fidelity images can be obtained without a priori knowledge of the source extent via input parameter searches.
Imaging with complementary size constraints {#sec:demo1}
-------------------------------------------
In Fig. \[fig:coverage\], we illustrate the domain in which the second moment regularization () operates. The $(u,v)$ coverage is that of a typical observation of with the EHT at 230GHz. Assuming a source extent of 60$\mu$as from previous observations [@Johnson_2018], the $1/e$ boundary of the visibility function for a source with that characteristic size is shown as a disk on the $(u,v)$ coverage. The only EHT baselines that lie within the disk are intra-site baselines and the LMT–SMT short VLBI baseline. A single short VLBI baseline is very limited in constraining the overall extent of the source even assuming optimal performance of the telescopes.
We selected a ray-traced image of a semi-analytic advection-dominated accretion flow (ADAF) model of [@Broderick_2011] with a similar characteristic size to the observations to assess the performance of the regularizer and to test the robustness of the imaging process as a function of the input parameters [$\theta_\mathrm{maj}$]{}, [$\theta_\mathrm{min}$]{}, and $\phi$. We sample the image with EHT 2017 coverage (Fig. \[fig:coverage\]), where we have total flux density estimates from intra-site baselines and a valuable mid-range baseline (SMT–LMT) describing the extent of the source on the sky, as shown in Fig. \[fig:avery\_radplot\]. We chose to discard all LMT baselines to limit the coverage and remove the constraining mid-range baseline for the regularizer tests. The extent of the source will then solely be enforced by the user-defined [$\theta_\mathrm{maj}$]{}, [$\theta_\mathrm{min}$]{}, and $\phi$ input parameters for in the imaging process. It should be noted that imaging without the LMT not only removes short-baseline information on source extent but also long-baseline information on finer features, creating further differences in reconstructed images. The LMT, due to its size and central location, holds a strong weight in triggering decisions, while the SMT is a smaller and well-exercised station and is fairly flexible to various observing conditions. The choice to discard the LMT is thus mainly motivated by the known difficulties, to date, for the station to observe in a wide range of observing conditions and obtain adequate calibration information [@PaperIII; @PaperIV]. Removing the SMT instead, for the purposes of these tests, would give similar results due to the lack of short-baseline information.
{width="0.59\linewidth"}
In Fig. \[fig:avery\_demo\], we show the model crescent image in the left panel, and example reconstructions for four different scenarios in the right panel. The first scenario is a reconstruction of the full EHT observations of the crescent, using closure quantities and visibility amplitudes, and maximizing simple image entropy. In that case, we obtain a good fit to the visibility amplitudes, and we recover an image very similar to the model image. Then, we reconstruct the same observations constraining the image to match the true second moment, as measured on the true image. With this method, we obtain a marginally improved fit to the amplitudes, but visibly less diffuse flux outside the crescent due to the constraint of . Once we remove the LMT however, the simple imaging with maximum entropy is not able to reconstruct the morphology of the source, although some compact features are reconstructed that enable a decent fit to the visibility amplitudes. When adding to the process, the second moment constraint is able to offset the absence of short baselines and reconstructs an image of improved quality in terms of both image morphology and goodness-of-fit to the amplitudes. This demonstration shows that successfully adds additional information to reconstruct a more physically plausible image even when mid-range baselines are lacking in the underlying data set. The improvement in the amplitude $\chi^2$ also shows that is a useful tool to aid convergence in imaging.
![Quality of the images obtained with different input FWHM (major and minor axes equal, position angle is zero). The image quality is measured in three ways: (1) the normalized cross-correlation against the true image, or NXCORR; (2) the normalized root-mean-square error against the true image, or NRMSE, shown in the top panel; and (3) reduced $\chi^2$ goodness-of-fits to the three data products used in the reconstructions (visibility amplitudes, closure amplitudes and phases) shown in the bottom panel. NRMSE is more sensitive to subtle differences in the images than NXCORR due to the higher weight associated with large pixel-by-pixel errors and is minimized in a comparable range of input FWHMs to the reduced data $\chi^2$. The narrow range of FWHMs encompasses the true mean source FWHM (magenta vertical line). []{data-label="fig:avery_metrics"}](figures/image_quality_diameter.pdf "fig:"){width="0.98\linewidth"} ![Quality of the images obtained with different input FWHM (major and minor axes equal, position angle is zero). The image quality is measured in three ways: (1) the normalized cross-correlation against the true image, or NXCORR; (2) the normalized root-mean-square error against the true image, or NRMSE, shown in the top panel; and (3) reduced $\chi^2$ goodness-of-fits to the three data products used in the reconstructions (visibility amplitudes, closure amplitudes and phases) shown in the bottom panel. NRMSE is more sensitive to subtle differences in the images than NXCORR due to the higher weight associated with large pixel-by-pixel errors and is minimized in a comparable range of input FWHMs to the reduced data $\chi^2$. The narrow range of FWHMs encompasses the true mean source FWHM (magenta vertical line). []{data-label="fig:avery_metrics"}](figures/image_ch-squares_diameter.pdf "fig:"){width="0.98\linewidth"}
Dependence of reconstructed images on assumed size {#sec:demo2}
--------------------------------------------------
In the demonstration of we constrained the second moment to the true size of the source, to enable an accurate reconstruction of the image. However, in practice, the true size of the source is unknown, and is instead approximated from Gaussian model fitting to closure quantities and/or short-baseline visibility amplitudes and extrapolated from historical measurements. We therefore investigate the robustness of the image reconstructions when the input Gaussian parameters are strongly enforced in the imaging process, corresponding to a strong weight of the hyperparameter, while changing input principal axes FWHMs. We demonstrate this dependence by imaging the data set of the crescent model sampled by the EHT 2017 coverage without the LMT, such that the extent of the source is only enforced by the varying inputs to . For simplicity, we use a single common imaging script varying only the input principal axes FWHMs. We assume an isotropic source size such that [$\theta_\mathrm{maj}$]{}= [$\theta_\mathrm{min}$]{} and $\phi=0^\circ$, and a range of input FWHMs of $5-90\,\mu$as.
We utilize two metrics to compare the quality of the reconstructed image to the true model image. The normalized root-mean-square error (NRMSE) of each image is given by: $$\begin{aligned}
{\rm NRMSE} = \sqrt{\frac{{\ensuremath{\sum\limits_\mathrm{k}}}({\ensuremath{I_\mathrm{i}}}- {\ensuremath{I_\mathrm{i}}}')^2}{{\ensuremath{\sum\limits_\mathrm{k}}}{\ensuremath{I_\mathrm{i}}}^2}}\, , \end{aligned}$$ where $I'$ is the intensity of the reconstructed image and $I$ is that of the true image [e.g., @Chael_2018]. If the reconstructed image is identical to the true image, the NRMSE is zero. Therefore, the input FWHM for the reconstruction resulting in the minimum NRMSE in comparison to the true image gives the best fit.
The normalized cross-correlation (NXCORR) is a sliding inner-product of two normalized functions. For fast numerical computation, we determine the cross-correlation of the Fourier transforms of the normalized intensity patterns of the true image $I_\mathrm{norm}$ and the reconstructed image $I'_\mathrm{norm}$ at different relative shifts $\boldsymbol{\delta}$ across the extent of the images. For each pixel i in the image, we normalize the intensity via: $$\begin{aligned}
I_{\mathrm{norm,i}} = \frac{{\ensuremath{I_\mathrm{i}}}- \mu_I}{\sigma_I}\, , \end{aligned}$$ where $\mu_I$ and $\sigma_I$ are the mean and standard deviation of the intensity distribution in the image. The cross-correlation for a given shift $\boldsymbol{\delta}$ is then given by: $$\begin{aligned}
{\rm NXCORR}(\boldsymbol{\delta}) = |\mathcal{F}^{-1}\{\mathcal{F}\{I_\mathrm{norm}^\ast(\mathbf{x})\} \cdot \mathcal{F}\{I_\mathrm{norm}'(\mathbf{x}+\boldsymbol{\delta})\}\}|.\end{aligned}$$ The shift at which the cross-correlation is maximized is then used to output the final NXCORR value for the two images. This method is less sensitive to individual features in the reconstructed image than NRMSE as it compares the bulks of each intensity pattern as opposed to the NRMSE pixel-to-pixel comparison. The $\chi^2$ statistics follow the equations presented in Sect. 2.1 of @PaperIV.
![Cross-comparisons of reconstructed images with varying isotropic input FWHMs using symmetrically normalized root-mean-square error (SNRMSE). The SNRMSE grid shows a greater correspondence of images with input FWHMs near the true mean FWHM of $55\,\mu$as, marked by the dashed black lines. The reconstructed images with varying input size (5–90$\mu$as) are all compared to each other, where image 1 and image 2 are the two images to be compared ($I'_1$ and $I'_2$ respectively in Equation \[eq:snrmse\]). The diagonal is each image compared to itself. The SNRMSE grid gives a range of plausible input FWHMs for that result in high fidelity images even when the true source size is unknown.[]{data-label="fig:nrmse_avery"}](figures/NRMSE_gridsearch.pdf){width="\linewidth"}
In Fig. \[fig:avery\_metrics\], we show the NRMSE and NXCORR metric scores for the reconstructed images compared against the true image (left panel of Fig. \[fig:avery\_demo\]), and the reduced data $\chi^2$ goodness-of-fits to the imaged data set (Fig. \[fig:avery\_radplot\], no LMT). The NXCORR is maximized at an input FWHM of $55\,\mu$as, and the NRMSE is minimized at the same input FWHM. This value corresponds to the mean FWHM (average of [$\theta_\mathrm{maj}$]{}= $58\,\mu$as and [$\theta_\mathrm{min}$]{}= $52\,\mu$as) of the true image. With this test, we find an excellent correspondence between the reconstructed image with the highest quality (highest NXCORR, lowest NRMSE, and lowest reduced data $\chi^2$) and the image with the input FWHM closest to the true value. Images with input FWHMs close to the optimal value are of similarly good quality. We thus show a good performance of in the imaging process even with input sizes inaccurate to within 20% of the true size. The reduction in data $\chi^2$ values as we approach the true source size also indicates that gives a convergence boost toward a higher fidelity image. This behavior is caused by rapidly reducing the favored set of images to only those that constrain flux within a given region. The region limits that best represent the flux distribution in the true image allow the minimizing process to focus more quickly on the data terms and achieve better reduced $\chi^2$ values within the given imaging conditions. This property also allows us to survey the response of the imaging process and goodness-of-fits to the available data via parameter searches over different favored second moments (and thus favored flux regions) and determine optimal parameters that best represent properties of the data set.
Imaging without complementary size constraints {#sec:demo3}
----------------------------------------------
The NRMSE metric proves to be more sensitive to differences in the image structure than NXCORR, as shown in Fig. \[fig:avery\_metrics\], due to the higher weight associated with large errors in the computation of the NRMSE. For that reason, we have selected NRMSE to score comparisons between the reconstructed images themselves. For this test, we assume that the true image and true FWHM are unknown, as is the case for real experiments. We instead focus on the morphological characteristics that appear in the images based on the underlying data, and how the inputs to affect the correspondence between reconstructed images. We restructure the metric into a symmetrically-normalized root-mean-square error [SNRMSE; @hanna1985; @Mentaschi_2013] to render the NRMSE independent of the input and comparison image choice: $$\begin{aligned}
{\rm SNRMSE} = \sqrt{\frac{{\ensuremath{\sum\limits_\mathrm{k}}}(I'_{1,i} - I'_{2,i})^2}{{\ensuremath{\sum\limits_\mathrm{k}}}I'_{1,i} I'_{2,i}}}. \label{eq:snrmse}\end{aligned}$$ Here $I'_1$ and $I'_2$ are the two reconstructed images to be compared. In Fig. \[fig:nrmse\_avery\], we show an SNRMSE grid comparing each reconstructed image to all others, where the diagonal squares correspond to each image compared with itself. We have marked with dashed lines where the mean FWHM of the true image lies. We find that images with input FWHMs near the true FWHM of the source have a better SNRMSE with each other than all other combinations of images. This test enables the user to find a range of characteristic sizes minimizing SNRMSE via a size parameter search. For compact sources that are distinctly elliptical, a one-dimensional size parameter search is useful to quickly sweep through a wide range of sizes and determine a range of plausible sizes for the source extent. A search within that range, varying parameters in two dimensions (${\ensuremath{\theta_\mathrm{maj}}}$, ${\ensuremath{\theta_\mathrm{min}}}$, and $\phi$), can then be carried out to refine the source size estimate for the imaging process.
We find that the use of the regularizer improves the quality of the resulting image even if the input parameters deviate by 20% from the true values. We also find that the strong use of the regularization, when combined with a size parameter search, is able to converge toward the true FWHM values, even when the true source dimensions are unknown. The use of SNRMSE and $\chi^2$ statistics serve well to score individual images and parameters without a priori knowledge of the source extent.
{width="0.33\linewidth"} {width="0.33\linewidth"} {width="0.33\linewidth"}
Applications {#sec:examples}
============
In addition to simple static imaging, second moment regularization can easily be coupled to more sophisticated and complex imaging techniques. In Sect. \[sec:stochastic\] we present an example of the use of second moment regularization for scattering mitigation imaging of at longer wavelengths. In Sect. \[sec:movie\] we demonstrate how second moment regularization in individual sparse snapshots improves the quality of dynamical reconstructions of variable sources, such as a movie of an orbiting “hot spot” in ’s accretion flow.
Scattering mitigation {#sec:stochastic}
---------------------
The second moment constraint in imaging can both be used for data sets where short baselines are lacking, as demonstrated in Sect. \[sec:demo\], and for data sets where short-baseline measurements have large uncertainties due to difficult observing conditions. An example of the latter case is presented in @Issaoun_2019, where observations of at 86GHz with the Global Millimeter VLBI Array and ALMA (project code MB007) yielded high signal-to-noise (SNR) detections on long baselines but bad weather at select Very Long Baseline Array (VLBA) stations led to poorly constrained short-baseline measurements. Imaging of the source with RML would not have been feasible with these measurements alone, as the large uncertainties in the short-baseline measurements caused flux to spread nonphysically across the reconstructed images. Since the size of on the sky is well studied and known to be affected by anisotropic scatter-broadening from the interstellar medium [@Davies_1976; @vanLangevelde_1992; @Frail_1994; @Bower_2004; @Shen_2005; @Bower_2006; @Psaltis_2018; @Johnson_2018], previous size measurements [@Ortiz_2016; @Brinkerink_2019] were used to constrain the extent of in the imaging process with . In this manner, we obtained an image that was able to fit new long-baseline detections to ALMA, likely refractive noise from scattering substructure.
The second moment regularization was also implemented in the scattering mitigation code [*stochastic optics*]{} developed by @Johnson_2016. Stochastic optics aims to mitigate the effects of scattering to derive an intrinsic (unscattered) image of the source. The code solves for the unscattered image by separating and mitigating the two main components of the scattering screen: the diffractive scattering that causes the image to become a convolution of the true image and the scattering kernel; and the refractive scattering that introduces stochastic ripples that further distort the image. The stochastic optics framework therefore simultaneously solves for the unscattered image and the scattering screen assuming a given model for the diffractive blurring kernel and the time-averaged refractive properties. The model assumed here is the @Johnson_2018 scattering model, the best-fitting model to observations to date [@Issaoun_2019].
The implementation of in stochastic optics only constrains the size of the scattered source ( as we see it on the sky) based on historical measurements from model fitting, such that the technique can more accurately mitigate the effects of interstellar scattering to obtain a physically motivated intrinsic image of the accretion flow of [for further details, see @Issaoun_2019]. The intrinsic image itself is not directly constrained by the second moment regularization, but is derived from the combination of the constrained scattered image and knowledge of the interstellar scattering.
Here we illustrate the use of within stochastic optics using a lower frequency data set. Observations of at 22GHz with the VLBA+GBT (project code BG221A) showed clear long-baseline detections of refractive noise from interstellar scattering [@Gwinn_2014; @Johnson_2018]. These long-baseline detections should translate to substructure in the image, distorting the intensity pattern seen for away from the scatter-broadened smooth elongated Gaussian-like morphology. While the scattering substructure is very apparent in the data set, it is a non-trivial task to successfully show its effects on the image itself and obtain an intrinsic image of the source. This is due to the imaging process being driven predominantly by the abundance of intra-VLBA short-baseline measurements in comparison to the few VLBA–GBT long-baseline detections. We therefore test the addition of on this data set, using the source dimensions in Table 1 of @Johnson_2018 from elliptical Gaussian model fitting.
In Fig. \[fig:22ghz\_im\], we show three separate reconstructions of the 22GHz data set. A standard RML reconstruction of the data set (Fig. \[fig:22ghz\_im\] left panel) shows some distortions in the scattered image, but the morphology remains fairly smooth and elongated. Standard RML imaging cannot solve for the scattering properties, therefore the procedure is solely focused on obtaining the highest fidelity scattered image possible from the data set. We will thus treat this image as our comparison image for this data set. When using stochastic optics however, the imaging process is more complex, as it is simultaneously imaging the scattered source and solving for the scattering properties to disentangle scattering from intrinsic source structure. This process derives a scattered image that is not well-constrained in the north–south direction due to the configuration of the VLBA+GBT, resulting in a large source image that is not fully converged to the image obtained from standard RML (Fig. \[fig:22ghz\_im\] center panel). Since the scattered image does not match our expectations of the physical morphology of the source, the derived intrinsic image should also not be trusted. The challenge is then to improve the convergence of the imaging component of stochastic optics to quickly obtain a physically motivated scattered image and therefore undergo a higher-fidelity separation of the scattering and intrinsic structure. When using , where the scattered image is constrained to remain within the size obtained by @Johnson_2018 using elliptical Gaussian model fitting, the resulting scattered image is more elongated in the east–west direction (Fig. \[fig:22ghz\_im\] right panel) and showing distortions similar to those of the standard RML reconstruction This shows that the use of helps the convergence of the scattered image through stochastic optics to a more physically motivated reconstruction, and thus will give a more realistic underlying unscattered image of the source.
{width="1.\linewidth"}
Dynamical imaging {#sec:movie}
-----------------
There are additional applications for the second moment regularization in movie reconstructions of variable sources where single snapshots have very sparse coverage. We can test the robustness of movie reconstructions with the loss of short baselines using a simulated flare [model B of @Doeleman_2009b] with an orbiting period of 27minutes around the same crescent model as in Sect. \[sec:demo\]. We reconstruct movies of the orbiting “hot spot” using [*dynamical imaging*]{}, enforcing temporal continuity between individual frames [for further details, see @Johnson_2017]. We reconstruct a movie of the orbiting hot spot for four different scenarios: (1) we use the EHT 2017 array without the LMT, no short baselines are present in the individual snapshots to constrain the source extent; (2) we use the data set without the LMT, but constrain the extent of the source (the dimensions of the crescent model) with , (3) we use the full EHT 2017 to reconstruct the orbit; (4) we use the full EHT 2017 and to reconstruct the orbit. In Fig. \[fig:hotpspot\], we show individual frames of the true simulation and of the reconstructed movies for the four different scenarios. The reconstructions without either yield unphysical source structure dominated by the dirty image (due to the lack of information without LMT) or contain imaging artifacts from flux spreading due to the sparse coverage of individual snapshots. In particular, even with the full EHT2017 array, dynamical imaging without shows north–east and south–west artifacts from the dirty image that persist due to the sparse snapshot coverage. The reconstructions with , even without the LMT, are significantly cleaner and more accurately reconstruct the motion and morphology of the simulation, as shown by NXCORR results when compared to the truth simulated images.
Summary {#sec:summary}
=======
In summary, we have developed a regularization function , for use in a regularized maximum likelihood framework for interferometric imaging, that constrains the spread of flux in reconstructed images to match input parameters defined by the user. The second moment regularization is a natural extension of common imaging tools, such as image total flux and image centroid constraints (zeroth and first moment respectively), that help to mitigate the missing information problem in high frequency VLBI. The regularization assumes that the source is compact, with a stable size, and is resolved on longer baselines of the interferometer. The validity of these assumptions for the EHT’s primary targets, and M87, are well-motivated by state-of-the-art GRMHD simulations and long-term observational studies. For well-studied sources, this method allows for contingency against weather, a major deterrent for high frequency VLBI, and gives more flexibility for triggering decisions if key short baselines yield poorly constrained measurements or become unavailable during or between observations.
We have shown that successfully informs the source behavior on short baselines and is defined only by three Gaussian parameters and the regularization hyperparameter. Imaging with is able to reconstruct high fidelity images fitting to the data products even if the input source dimensions deviate from the true values by up to 20%. The regularization therefore gives a larger flexibility than needed to account for changes in size from, for example, GRMHD simulations of highly variable sources such as . We have also shown that parameter searches over a range of isotropic FWHMs using in conjunction with goodness-of-fit statistics to data products and symmetrically-normalized root-mean-square error of image comparisons help determine high-fidelity source extent even if the exact size and morphology are unknown.
The regularization can be used to image with any choice of data products and any choice of feature-driven regularizers within the framework of the [eht-imaging]{} library [@Chael_2016; @Chael_2018] and is easily transferable to other tools or other RML imaging packages [e.g., `SMILI`; @Akiyama_2017b; @Akiyama_2017a]. We have shown that the implementation complements other techniques tackling source properties that add difficulty and complexity to the imaging process, such as time variability [via dynamical imaging; @Johnson_2017; @Bouman_2017] and interstellar scattering [@Johnson_2016; @Issaoun_2019]. Source parameter inputs can either be obtained from model fitting to abundant short-baseline measurements, historical measurements from observations with short baselines present, extrapolated from other wavelengths based on achromatic features, or estimated via parameter searches. The second moment regularization could prove particularly useful in future work with the EHT, both for dynamical reconstructions of variable sources such as and for upcoming imaging observations at 345GHz [@PaperII; @Doeleman_astro2020].
We thank John Wardle for his helpful comments and careful review. This work is supported by the ERC Synergy Grant “BlackHoleCam: Imaging the Event Horizon of Black Holes”, Grant 610058. We thank the National Science Foundation (AST-1440254, AST-1716536, AST-1312651) and the Gordon and Betty Moore Foundation (GBMF-5278) for financial support of this work. This work was supported in part by the Black Hole Initiative at Harvard University, which is supported by a grant from the John Templeton Foundation.
Properties of the visibility function {#sec:vis_prop}
=====================================
Visibility derivatives and image moments {#sec:vis_der}
----------------------------------------
Non-astrometric VLBI experiments such as the EHT measure visibility amplitudes directly but do not provide absolute phase information. Nevertheless, the zeroth and second image moments are determined from visibility amplitudes alone [i.e., they do not depend on the measured phase; @moffet_1962; @burn_1976]. For instance, the total flux density $\int I(\mathbf{x}) d^2\mathbf{x} = V(\mathbf{0}) = |V(\mathbf{0})|$ because the zero-baseline visibility is real and positive, and therefore equal to its modulus.
More generally, we can express the visibility function as a Taylor expansion of its derivatives: $$\begin{aligned}
V(\mathbf{u}) &= \int d^2\mathbf{x}\ I(\mathbf{x}) \left[ 1 - 2i\pi\mathbf{u}\cdot \mathbf{x} - \frac{(2\pi\mathbf{u}\cdot \mathbf{x})^2}{2} \right. \nonumber \\
& \qquad \left. + \frac{i(2\pi\mathbf{u}\cdot \mathbf{x})^3}{6}
+ \frac{(2\pi\mathbf{u}\cdot \mathbf{x})^4}{24} + \cdots \right].\end{aligned}$$ The visibility amplitude function is image-translation invariant. To obtain a Taylor expansion for visibility amplitudes, we choose the image centroid to be at the origin. The first derivative of the visibility function (thus the second term of the Taylor expansion) then vanishes, giving $$\begin{aligned}
\nonumber V(\mathbf{u}) &\simeq \int d^2\mathbf{x}\ I(\mathbf{x}) \left[ 1 - \frac{(2\pi\mathbf{u}\cdot \mathbf{x})^2}{2} \right] \\
&\simeq V(\mathbf{0}) - 2\pi^2 \int d^2\mathbf{x}\ (\mathbf{u}\cdot \mathbf{x})^2 I(\mathbf{x}).\end{aligned}$$ On short baselines (i.e., those with $\mathbf{u}\cdot \mathbf{x} \ll 1$), the visibility function is then positive and real, so $\left| V(\mathbf{u}) \right| \simeq V(\mathbf{u})$. Since $\mathbf{u} =\begin{pmatrix} u \\ v \end{pmatrix} $ and $\mathbf{x} =\begin{pmatrix} x \\ y \end{pmatrix} $, we can expand the inner product of the two vectors: $$\begin{aligned}
\nonumber (\mathbf{u}\cdot \mathbf{x})^2 &= u^2x^2 + v^2y^2 + 2uvxy \\
&= \begin{pmatrix} u & v \end{pmatrix} \begin{pmatrix} x^2 & xy \\ xy & y^2 \end{pmatrix} \begin{pmatrix} u \\ v \end{pmatrix}.\end{aligned}$$ Combining these results with the definition of the covariance matrix $\boldsymbol{\Sigma}$ (see Appendix \[sec:vis\_axes\]), we obtain: $$\begin{aligned}
\nonumber |V(\mathbf{u})| &\simeq V(\mathbf{0}) - 2\pi^2 \int d^2\mathbf{x}\, (\mathbf{u}\cdot \mathbf{x})^2 I(\mathbf{x}) \\
\nonumber &\simeq V(\mathbf{0}) - 2\pi^2 \begin{pmatrix} u & v \end{pmatrix} \int d^2\mathbf{x}\,I(\mathbf{x})\begin{pmatrix} x^2 & xy \\ xy & y^2 \end{pmatrix} \begin{pmatrix} u \\ v \end{pmatrix} \\
&\simeq V(\mathbf{0}) - 2\pi^2 V(\mathbf{0})\mathbf{u}^\intercal \boldsymbol{\Sigma} \mathbf{u} . \label{eq:ap_gen-short}\end{aligned}$$ The downward curvature of the amplitude function at zero baseline is thus related to the image covariance by: $$\begin{aligned}
\left. \nabla\nabla ^\intercal |V(\mathbf{u})| \right\rfloor_{\mathbf{u}=\mathbf{0}} = \left. \nabla\nabla^\intercal V(\mathbf{u}) \right\rfloor_{\mathbf{u}=\mathbf{0}} = -4\pi^2 V(\mathbf{0}) \boldsymbol{\Sigma}. \label{eq:ap_2mom}\end{aligned}$$
Image principal axes and visibility curvature {#sec:vis_axes}
---------------------------------------------
From Equation \[eq:ap\_2mom\], the curvature of the visibility function on short baselines is proportional to the second central moment of the image projected along the baseline direction. The second central moment of the image is naturally expressed as a covariance matrix: $$\begin{aligned}
\mathbf{\Sigma} &\equiv \frac{\int d^2\mathbf{x} I(\mathbf{x}) (\mathbf{x}-\boldsymbol{\mu})(\mathbf{x}-\boldsymbol{\mu})^\intercal}{\int d^2\mathbf{x}\, I(\mathbf{x}) } =
\begin{pmatrix}
{\ensuremath{\Sigma_\mathrm{xx}}}& {\ensuremath{\Sigma_\mathrm{xy}}}\\
{\ensuremath{\Sigma_\mathrm{yx}}}& {\ensuremath{\Sigma_\mathrm{yy}}}\end{pmatrix}, \label{eq:ap_matrix} \\
\nonumber {\ensuremath{\Sigma_\mathrm{xx}}}&= \frac{\int d^2\mathbf{x}\, I(\mathbf{x}) (x-{\ensuremath{\bar{x}}})^2}{\int d^2\mathbf{x}\, I(\mathbf{x})},\\
\nonumber {\ensuremath{\Sigma_\mathrm{yy}}}&= \frac{\int d^2\mathbf{x}\, I(\mathbf{x}) (y-{\ensuremath{\bar{y}}})^2}{\int d^2\mathbf{x}\, I(\mathbf{x})},\\
\nonumber {\ensuremath{\Sigma_\mathrm{xy}}}&= \frac{\int d^2\mathbf{x}\, I(\mathbf{x}) (x-{\ensuremath{\bar{x}}})(y-{\ensuremath{\bar{y}}})}{\int d^2\mathbf{x}\, I(\mathbf{x})} = {\ensuremath{\Sigma_\mathrm{yx}}}.\end{aligned}$$ To put the covariance matrix in a more intuitive form, we express it in terms of its principal axes. The image covariance matrix has two eigenvalues, and can be diagonalized as follows: $$\begin{aligned}
\mathbf{\Sigma}
= \mathbf{R_\phi} \begin{pmatrix}
{\ensuremath{\lambda_\mathrm{min}}}& 0 \\
0 & {\ensuremath{\lambda_\mathrm{maj}}}\end{pmatrix} \mathbf{R_\phi^{\intercal}}, \end{aligned}$$ where the rotation matrix $\mathbf{R_\phi}$, based on the position angle $\phi$ (East of North) of the major principal axis, is given by: $$\begin{aligned}
\mathbf{R_\phi} = \begin{pmatrix}
\cos(\phi) & \sin(\phi) \\
-\sin(\phi) & \cos(\phi)
\end{pmatrix} . \end{aligned}$$ The eigenvalues are derived from the quadratic equation: $$\begin{aligned}
{\ensuremath{\lambda_\mathrm{maj}}}&= \frac{{\ensuremath{\Sigma_\mathrm{xx}}}+{\ensuremath{\Sigma_\mathrm{yy}}}}{2} + \frac{\sqrt{4({\ensuremath{\Sigma_\mathrm{xy}}})^2 + ({\ensuremath{\Sigma_\mathrm{xx}}}- {\ensuremath{\Sigma_\mathrm{yy}}})^2}}{2}, \\
{\ensuremath{\lambda_\mathrm{min}}}&= \frac{{\ensuremath{\Sigma_\mathrm{xx}}}+{\ensuremath{\Sigma_\mathrm{yy}}}}{2} - \frac{\sqrt{4({\ensuremath{\Sigma_\mathrm{xy}}})^2 + ({\ensuremath{\Sigma_\mathrm{xx}}}- {\ensuremath{\Sigma_\mathrm{yy}}})^2}}{2}. \end{aligned}$$ We can also express each term of the covariance matrix in terms of the eigenvalues and position angle $\phi$: $$\begin{aligned}
{\ensuremath{\Sigma_\mathrm{xx}}}&= \cos^2(\phi){\ensuremath{\lambda_\mathrm{min}}}+ \sin^2(\phi) {\ensuremath{\lambda_\mathrm{maj}}}, \\
{\ensuremath{\Sigma_\mathrm{yy}}}&= \sin^2(\phi) {\ensuremath{\lambda_\mathrm{min}}}+ \cos^2(\phi) {\ensuremath{\lambda_\mathrm{maj}}}, \\
{\ensuremath{\Sigma_\mathrm{xy}}}&= ({\ensuremath{\lambda_\mathrm{maj}}}- {\ensuremath{\lambda_\mathrm{min}}})\cos(\phi) \sin(\phi).\end{aligned}$$ The eigenvalues of the covariance matrix are the variances along the principal axes (major and minor axes).
Implementation via gradient descent {#sec:implementation}
===================================
Pixel-based derivation of principal axes
----------------------------------------
Here we present the computation of the covariance matrix for the pixel-based reconstructions from RML. The centroid of an $n \times n$ pixel-based image is given by the following parameters: $$\begin{aligned}
{{\ensuremath{\bar{x}}}} = \frac{{\ensuremath{\sum\limits_\mathrm{k}}}{\ensuremath{x_\mathrm{i}}}{\ensuremath{I_\mathrm{i}}}}{{\ensuremath{\sum\limits_\mathrm{k}}}{\ensuremath{I_\mathrm{i}}}} \mbox{ and }
{{\ensuremath{\bar{y}}}} = \frac{{\ensuremath{\sum\limits_\mathrm{k}}}{\ensuremath{y_\mathrm{i}}}{\ensuremath{I_\mathrm{i}}}}{{\ensuremath{\sum\limits_\mathrm{k}}}{\ensuremath{I_\mathrm{i}}}},\end{aligned}$$ where i is the pixel number (from 1 to k), [$I_\mathrm{i}$]{} is the intensity at that pixel, [$x_\mathrm{i}$]{} is the x-position and [$y_\mathrm{i}$]{} is the y-position of the pixel in the image. The second moment of the image is given by the covariance matrix $$\begin{aligned}
\mathbf{\Sigma} &=
\begin{pmatrix}
{\ensuremath{\Sigma_\mathrm{xx}}}& {\ensuremath{\Sigma_\mathrm{xy}}}\\
{\ensuremath{\Sigma_\mathrm{xy}}}& {\ensuremath{\Sigma_\mathrm{yy}}}\end{pmatrix} , \\
\mbox{ where }\\
{\ensuremath{\Sigma_\mathrm{xx}}}&= \frac{{\ensuremath{\sum\limits_\mathrm{k}}}({\ensuremath{x_\mathrm{i}}}- {\ensuremath{\bar{x}}})^2 {\ensuremath{I_\mathrm{i}}}}{{\ensuremath{\sum\limits_\mathrm{k}}}{\ensuremath{I_\mathrm{i}}}},\\
{\ensuremath{\Sigma_\mathrm{yy}}}&= \frac{{\ensuremath{\sum\limits_\mathrm{k}}}({\ensuremath{y_\mathrm{i}}}- {\ensuremath{\bar{y}}})^2 {\ensuremath{I_\mathrm{i}}}}{{\ensuremath{\sum\limits_\mathrm{k}}}{\ensuremath{I_\mathrm{i}}}}, \\
{\ensuremath{\Sigma_\mathrm{xy}}}&= \frac{{\ensuremath{\sum\limits_\mathrm{k}}}({\ensuremath{x_\mathrm{i}}}- {\ensuremath{\bar{x}}})({\ensuremath{y_\mathrm{i}}}-{\ensuremath{\bar{y}}}) {\ensuremath{I_\mathrm{i}}}}{{\ensuremath{\sum\limits_\mathrm{k}}}{\ensuremath{I_\mathrm{i}}}}.\end{aligned}$$ As in Appendix \[sec:vis\_axes\], the image covariance matrix has two eigenvalues and can be diagonalized to obtain the principal axes FWHMs.
Gradient Descent Implementation
-------------------------------
We have defined our regularization function via the Frobenius norm: $$\begin{aligned}
\mathcal{R}_\mathbf{\Sigma} = ({\ensuremath{\Sigma_\mathrm{xx}}}- {\ensuremath{\Sigma_\mathrm{xx}}}')^2 + ({\ensuremath{\Sigma_\mathrm{yy}}}- {\ensuremath{\Sigma_\mathrm{yy}}}')^2 + 2({\ensuremath{\Sigma_\mathrm{xy}}}- {\ensuremath{\Sigma_\mathrm{xy}}}')^2 .\end{aligned}$$ Within the framework of the `eht-imaging` library, the objective function is minimized via gradient descent. Therefore, the regularization functions must also individually be minimized via gradient descent. The gradients for the quantities describing the properties of the image introduced thus far, for a given pixel j, are given below: $$\begin{aligned}
\nonumber \frac{\delta {\ensuremath{\bar{x}}}}{\delta {\ensuremath{I_\mathrm{j}}}} &= \frac{{\ensuremath{x_\mathrm{j}}}{\ensuremath{\sum\limits_\mathrm{k}}}{\ensuremath{I_\mathrm{i}}}- {\ensuremath{\sum\limits_\mathrm{k}}}({\ensuremath{x_\mathrm{i}}}{\ensuremath{I_\mathrm{i}}})}{\big({\ensuremath{\sum\limits_\mathrm{k}}}{\ensuremath{I_\mathrm{i}}}\big)^2} = \frac{{\ensuremath{x_\mathrm{j}}}- {\ensuremath{\bar{x}}}}{\big({\ensuremath{\sum\limits_\mathrm{k}}}{\ensuremath{I_\mathrm{i}}}\big)},\\
\frac{\delta {\ensuremath{\bar{y}}}}{\delta {\ensuremath{I_\mathrm{j}}}} &= \frac{{\ensuremath{y_\mathrm{j}}}{\ensuremath{\sum\limits_\mathrm{k}}}{\ensuremath{I_\mathrm{i}}}- {\ensuremath{\sum\limits_\mathrm{k}}}({\ensuremath{y_\mathrm{i}}}{\ensuremath{I_\mathrm{i}}})}{\big({\ensuremath{\sum\limits_\mathrm{k}}}{\ensuremath{I_\mathrm{i}}}\big)^2} = \frac{{\ensuremath{y_\mathrm{j}}}- {\ensuremath{\bar{y}}}}{\big({\ensuremath{\sum\limits_\mathrm{k}}}{\ensuremath{I_\mathrm{i}}}\big)},\end{aligned}$$
$$\begin{aligned}
\nonumber \frac{\delta {\ensuremath{\Sigma_\mathrm{xx}}}}{\delta {\ensuremath{I_\mathrm{j}}}} &=
\frac{[({\ensuremath{x_\mathrm{j}}}- {\ensuremath{\bar{x}}})^2 - 2({\ensuremath{x_\mathrm{j}}}- {\ensuremath{\bar{x}}})\frac{\delta {\ensuremath{\bar{x}}}}{\delta {\ensuremath{I_\mathrm{j}}}} {\ensuremath{I_\mathrm{j}}}] {\ensuremath{\sum\limits_\mathrm{k}}}{\ensuremath{I_\mathrm{i}}}- {\ensuremath{\sum\limits_\mathrm{k}}}[({\ensuremath{x_\mathrm{i}}}- {\ensuremath{\bar{x}}})^2 {\ensuremath{I_\mathrm{i}}}]}{\big({\ensuremath{\sum\limits_\mathrm{k}}}{\ensuremath{I_\mathrm{i}}}\big)^2} \\
&= \frac{[({\ensuremath{x_\mathrm{j}}}- {\ensuremath{\bar{x}}})^2 - 2({\ensuremath{x_\mathrm{j}}}- {\ensuremath{\bar{x}}})\frac{\delta {\ensuremath{\bar{x}}}}{\delta {\ensuremath{I_\mathrm{j}}}} {\ensuremath{I_\mathrm{j}}}]- {\ensuremath{\Sigma_\mathrm{xx}}}}{{\ensuremath{\sum\limits_\mathrm{k}}}{\ensuremath{I_\mathrm{i}}}},\end{aligned}$$
$$\begin{aligned}
\nonumber \frac{\delta {\ensuremath{\Sigma_\mathrm{yy}}}}{\delta {\ensuremath{I_\mathrm{j}}}} &=
\frac{[({\ensuremath{y_\mathrm{j}}}- {\ensuremath{\bar{y}}})^2 - 2({\ensuremath{y_\mathrm{j}}}- {\ensuremath{\bar{y}}})\frac{\delta {\ensuremath{\bar{y}}}}{\delta {\ensuremath{I_\mathrm{j}}}} {\ensuremath{I_\mathrm{j}}}] {\ensuremath{\sum\limits_\mathrm{k}}}{\ensuremath{I_\mathrm{i}}}- {\ensuremath{\sum\limits_\mathrm{k}}}[({\ensuremath{y_\mathrm{i}}}- {\ensuremath{\bar{y}}})^2 {\ensuremath{I_\mathrm{i}}}]}{\big({\ensuremath{\sum\limits_\mathrm{k}}}{\ensuremath{I_\mathrm{i}}}\big)^2} \\
&= \frac{[({\ensuremath{y_\mathrm{j}}}- {\ensuremath{\bar{y}}})^2 - 2({\ensuremath{y_\mathrm{j}}}- {\ensuremath{\bar{y}}})\frac{\delta {\ensuremath{\bar{y}}}}{\delta {\ensuremath{I_\mathrm{j}}}} {\ensuremath{I_\mathrm{j}}}] - {\ensuremath{\Sigma_\mathrm{yy}}}}{{\ensuremath{\sum\limits_\mathrm{k}}}{\ensuremath{I_\mathrm{i}}}},\end{aligned}$$
$$\begin{aligned}
\nonumber &\frac{\delta {\ensuremath{\Sigma_\mathrm{xy}}}}{\delta {\ensuremath{I_\mathrm{j}}}} = \frac{[({\ensuremath{x_\mathrm{j}}}- {\ensuremath{\bar{x}}})({\ensuremath{y_\mathrm{j}}}- {\ensuremath{\bar{y}}}) - ({\ensuremath{y_\mathrm{j}}}- {\ensuremath{\bar{y}}})\frac{\delta {\ensuremath{\bar{x}}}}{\delta {\ensuremath{I_\mathrm{j}}}} {\ensuremath{I_\mathrm{j}}}] {\ensuremath{\sum\limits_\mathrm{k}}}{\ensuremath{I_\mathrm{i}}}}{\big({\ensuremath{\sum\limits_\mathrm{k}}}{\ensuremath{I_\mathrm{i}}}\big)^{2}} \\
\nonumber &- \frac{[ ({\ensuremath{x_\mathrm{j}}}- {\ensuremath{\bar{x}}})\frac{\delta {\ensuremath{\bar{y}}}}{\delta {\ensuremath{I_\mathrm{j}}}} {\ensuremath{I_\mathrm{j}}}] {\ensuremath{\sum\limits_\mathrm{k}}}{\ensuremath{I_\mathrm{i}}}}{\big({\ensuremath{\sum\limits_\mathrm{k}}}{\ensuremath{I_\mathrm{i}}}\big)^{2}}- \frac{ {\ensuremath{\sum\limits_\mathrm{k}}}[({\ensuremath{x_\mathrm{i}}}- {\ensuremath{\bar{x}}})({\ensuremath{y_\mathrm{i}}}- {\ensuremath{\bar{y}}}) {\ensuremath{I_\mathrm{i}}}]}{\big({\ensuremath{\sum\limits_\mathrm{k}}}{\ensuremath{I_\mathrm{i}}}\big)^{2}} \\
&= \frac{[({\ensuremath{x_\mathrm{j}}}- {\ensuremath{\bar{x}}})({\ensuremath{y_\mathrm{j}}}- {\ensuremath{\bar{y}}}) - ({\ensuremath{y_\mathrm{j}}}- {\ensuremath{\bar{y}}})\frac{\delta {\ensuremath{\bar{x}}}}{\delta {\ensuremath{I_\mathrm{j}}}} {\ensuremath{I_\mathrm{j}}}- ({\ensuremath{x_\mathrm{j}}}- {\ensuremath{\bar{x}}})\frac{\delta {\ensuremath{\bar{y}}}}{\delta {\ensuremath{I_\mathrm{j}}}} {\ensuremath{I_\mathrm{j}}}]- {\ensuremath{\Sigma_\mathrm{xy}}}}{{\ensuremath{\sum\limits_\mathrm{k}}}{\ensuremath{I_\mathrm{i}}}} \end{aligned}$$
We can now compute the gradient of the second moment regularization within the minimization framework of the `eht-imaging` library: $$\begin{aligned}
\nonumber \frac{\delta \mathcal{R}_\Sigma}{\delta {\ensuremath{I_\mathrm{j}}}} = 2({\ensuremath{\Sigma_\mathrm{xx}}}- {\ensuremath{\Sigma_\mathrm{xx}}}')\frac{\delta {\ensuremath{\Sigma_\mathrm{xx}}}}{\delta {\ensuremath{I_\mathrm{j}}}} + 2({\ensuremath{\Sigma_\mathrm{yy}}}- {\ensuremath{\Sigma_\mathrm{yy}}}')\frac{\delta {\ensuremath{\Sigma_\mathrm{yy}}}}{\delta {\ensuremath{I_\mathrm{j}}}} \\
+ 4({\ensuremath{\Sigma_\mathrm{xy}}}- {\ensuremath{\Sigma_\mathrm{xy}}}')\frac{\delta {\ensuremath{\Sigma_\mathrm{xy}}}}{\delta {\ensuremath{I_\mathrm{j}}}} .\end{aligned}$$
Note that these equations correspond to regularization of the normalized second central moment of an image. In cases where the total flux density of an image is constrained or regularized, it would be advantageous to instead regularize the unnormalized second central moment, giving a substantially simplified and convex optimization problem.
|
---
abstract: 'In this paper we prove the Leibniz analogue of Whitehead’s vanishing theorem for the Chevalley-Eilenberg cohomology of Lie algebras. As a consequence, we obtain the second Whitehead lemma for Leibniz algebras. Moreover, we compute the cohomology of several Leibniz algebras with adjoint or irreducible coefficients. Our main tool is a Leibniz analogue of the Hochschild-Serre spectral sequence, which is an extension of (the dual of) a spectral sequence of Pirashvili for Leibniz homology from symmetric bimodules to arbitrary bimodules.'
address:
- 'Department of Mathematics and Statistics, University of South Alabama, Mobile, AL 36688-0002, USA'
- 'Laboratoire de mathématiques Jean Leray, UMR 6629 du CNRS, Université de Nantes, 2, rue de la Houssinière, F-44322 Nantes Cedex 3, France'
author:
- Jörg Feldvoss
- Friedrich Wagemann
date: 'October 27, 2019'
title: On Leibniz cohomology
---
Introduction {#introduction .unnumbered}
============
In [@ALO], the authors study the cohomology of semi-simple Leibniz algebras, i.e., the cohomology of finite-dimensional Leibniz algebras ${\mathfrak{L}}$ with an ideal of squares ${\mathrm{Leib}}({\mathfrak{L}})$ such that the corresponding canonical Lie algebra ${\mathfrak{L}}_{\mathrm{Lie}}:=
{\mathfrak{L}}/{\mathrm{Leib}}({\mathfrak{L}})$ is semi-simple, and conjecture that ${\mathrm{HL}}^2({\mathfrak{L}},{\mathfrak{L}}_{\mathrm{ad}})=0$. In [@FMM], the authors determine the deviation of the second Leibniz cohomology of a complex Lie algebra with adjoint or trivial coefficients from the corresponding Chevalley-Eilenberg cohomology. With these motivations in mind, we systematically transpose Pirashvili’s results and tools from homology (see [@P]) to cohomology, generalize one of Pirashvili’s spectral sequences from symmetric bimodules to arbitrary bimodules, and prove the conjecture mentioned above.
Obtaining this kind of vanishing results would be easy with a strong analogue of the Hochschild-Serre spectral sequence for Leibniz cohomology. Recall that the Hochschild-Serre spectral sequence for a Lie algebra extension $0\to{\mathfrak{k}}\to{\mathfrak{g}}\to{\mathfrak{q}}\to 0$ arises from a filtration of the standard cochain complex of ${\mathfrak{g}}$ by cochains which vanish in case one inserts for a certain fixed number $q$ elements of the ideal ${\mathfrak{k}}$ in $q$ arguments of the cochain (see [@HS Sections 2 and 3]). When trying to generalize this filtration from Lie algebras to Leibniz algebras, one needs to choose whether to filter from the left or from the right. Another difficulty is that the arising spectral sequence does not converge to the cohomology of the Leibniz algebra, but rather to the cohomology of some quotient complex. Furthermore, one must impose that the ideal acts trivially from the left (right) on the left (right) Leibniz algebra. This last issue excludes the application of the spectral sequence to many interesting ideals in the Leibniz algebra. Pirashvili [@P Theorem C] has constructed an analogue of the Hochschild-Serre spectral sequence using the filtration from the right for right Leibniz algebras and indicated how to use it together with a long exact sequence in order to extract cohomology. We use Pirashvili’s framework and extend his spectral sequence from symmetric bimodules to arbitrary bimodules (see Theorem \[hs1\]). The two main changes of perspective with respect to [@P] are the systematic use of arbitrary bimodules and computations in which we consider ground fields of all characteristics. We hope that this might be useful for further applications in the future.
The main application of Theorem \[hs1\] is Theorem \[vansemsim\] in which we compute the cohomology of a finite-dimensional semi-simple Leibniz algebra over a field of characteristic zero with coefficients in an arbitrary finite-dimensional bimodule. The case $n=2$ of Theorem \[vansemsim\] is the second Whitehead lemma for Leibniz algebras. But note that contrary to Chevalley-Eilenberg cohomology, Leibniz cohomology vanishes in any degree $n\ge 2$. This is one of several instances that we found by our computations in this paper which indicates that Leibniz cohomology behaves more uniformly than Chevalley-Eilenberg cohomology. We also show by examples that the theorem fails in prime characteristic or for infinite-dimensional modules (see Examples E and F, respectively).
As an immediate consequence of Theorem \[vansemsim\] we obtain the rigidity of finite-dimensional semi-simple Leibniz algebras in characteristic zero (see Corollary \[rigid\]). More generally, we obtain a complete description of the cohomology of a finite-dimensional semi-simple left Leibniz algebra with coefficients in the adjoint bimodule and its (anti-)symmetric counterparts (see Theorem \[adj\]). In particular, we deduce that a finite-dimensional semi-simple non-Lie Leibniz algebra in characteristic zero always possesses outer derivations (see Corollary \[outder\]) which might be somewhat surprising as this shows that derivations of non-Lie Leibniz algebras are more complicated than derivations of Lie algebras.
In addition to the results just mentioned, we dualize another spectral sequence obtained by Pirashvili for Leibniz homology (see [@P Theorem A]) that relates the Leibniz cohomology of a Lie algebra to its Chevalley-Eilenberg cohomology (see Theorems \[theorem\_A\] and \[vanlie\]). As an application we generalize some known results on rigidity to complete Lie algebras (see Corollary \[complet\] and Corollary \[witt\]) and to parabolic subalgebras of finite-dimensional semi-simple Lie algebras (see Proposition \[borel\]). Moreover, we compute the Leibniz cohomology for the non-abelian two-dimensional Lie algebra (see Example A) and the three-dimensional Heisenberg algebra (see Example B) with coefficients in irreducible bimodules. The authors believe that Leibniz cohomology is an important invariant of a Lie algebra that behaves more uniformly than Chevalley-Eilenberg cohomology. The motivation for including so many details in Section \[cel\] was to provide the reader with a solid foundation for computing this invariant in arbitrary characteristics.
The subject of Leibniz algebras, and especially its (co)homology theory, owes a great deal to Jean-Louis Loday and Teimuraz Pirashvili (see [@CH], [@L], [@LP], [@LP2], and [@P]). Many fundamental definitions and tools are due to them. While Loday and Pirashvili work with right Leibniz algebras, we work with left Leibniz algebras. Obviously, results in left Leibniz algebras are equivalent to the corresponding results in right Leibniz algebras. For the convenience of the reader we shall indicate where to find the corresponding formulae for left Leibniz algebras, even when they have been invented in the framework of right Leibniz algebras and are due to Loday and Pirashvili.
In this paper we will follow the notation used in [@F]. All tensor products are over the relevant ground field and will be denoted by $\otimes$. For a subset $X$ of a vector space $V$ over a field ${\mathbb{F}}$ we let $\langle X\rangle_{\mathbb{F}}$ be the subspace of $V$ spanned by $X$. We will denote the space of linear transformations from an ${\mathbb{F}}$-vector space $V$ to an ${\mathbb{F}}$-vector space $W$ by ${\mathrm{Hom}}_{\mathbb{F}}(V,W)$. In particular, $V^*:={\mathrm{Hom}}_{\mathbb{F}}(V,{\mathbb{F}})$ will be the space of linear forms on a vector space $V$ over a field ${\mathbb{F}}$. Moreover, $S^2(V)$ will denote the symmetric square of a vector space $V$. Finally, the identity function on a set $X$ will be denoted by ${\mathrm{id}}_X$, and the set $\{0,1,2,\dots\}$ of non-negative integers will be denoted by ${\mathbb{N}}_0$.
Preliminaries {#prelim}
=============
In this section we recall some definitions and collect several results that will be useful in the remainder of the paper.
A [*left Leibniz algebra*]{} is an algebra ${\mathfrak{L}}$ such that every left multiplication operator $L_x:{\mathfrak{L}}\to{\mathfrak{L}}$, $y\mapsto xy$ is a derivation. This is equivalent to the identity $$\label{LLI}
x(yz)=(xy)z+y(xz)$$ for all $x,y,z\in{\mathfrak{L}}$, which in turn is equivalent to the identity $$\label{RLLI}
(xy)z=x(yz)-y(xz)$$ for all $x,y,z\in{\mathfrak{L}}$. We will call both identities the [*left Leibniz identity*]{}. There is a similar definition of a [*right Leibniz algebra*]{} but in this paper we will only consider left Leibniz algebras.
Every left Leibniz algebra has an important ideal, its Leibniz kernel, that measures how much the Leibniz algebra deviates from being a Lie algebra. Namely, let ${\mathfrak{L}}$ be a left Leibniz algebra over a field ${\mathbb{F}}$. Then $${\mathrm{Leib}}({\mathfrak{L}}):=\langle x^2\mid x\in{\mathfrak{L}}\rangle_\mathbb{F}$$ is called the [*Leibniz kernel*]{} of ${\mathfrak{L}}$. The Leibniz kernel ${\mathrm{Leib}}({\mathfrak{L}})$ is an abelian ideal of ${\mathfrak{L}}$, and ${\mathrm{Leib}}({\mathfrak{L}})\ne{\mathfrak{L}}$ when ${\mathfrak{L}}\ne 0$ (see [@F Proposition 2.20]). Moreover, ${\mathfrak{L}}$ is a Lie algebra if, and only if, ${\mathrm{Leib}}({\mathfrak{L}})=0$. It follows from the left Leibniz identity (\[RLLI\]) that ${\mathrm{Leib}}({\mathfrak{L}})\subseteq C_\ell({\mathfrak{L}})$, where $C_\ell({\mathfrak{L}}):=\{c\in{\mathfrak{L}}\mid\forall\,x\in{\mathfrak{L}}:cx=0\}$ denotes the [*left center*]{} of ${\mathfrak{L}}$.
By definition of the Leibniz kernel, ${\mathfrak{L}}_{\mathrm{Lie}}:={\mathfrak{L}}/{\mathrm{Leib}}({\mathfrak{L}})$ is a Lie algebra which we call the [*canonical Lie algebra*]{} associated to ${\mathfrak{L}}$. In fact, the Leibniz kernel is the smallest ideal such that the corresponding factor algebra is a Lie algebra (see [@F Proposition 2.22]).
Next, we will briefly discuss left modules and bimodules of left Leibniz algebras. Let ${\mathfrak{L}}$ be a left Leibniz algebra over a field ${\mathbb{F}}$. A [*left ${\mathfrak{L}}$-module*]{} is a vector space $M$ over ${\mathbb{F}}$ with an ${\mathbb{F}}$-bilinear left ${\mathfrak{L}}$-action ${\mathfrak{L}}\times M\to M$, $(x,m)\mapsto x\cdot m$ such that $$\label{LLM}
(xy)\cdot m=x\cdot(y\cdot m)-y\cdot(x\cdot m)$$ is satisfied for every $m\in M$ and all $x,y\in{\mathfrak{L}}$.
By virtue of [@F Lemma 3.3], every left ${\mathfrak{L}}$-module is an ${\mathfrak{L}}_{\mathrm{Lie}}$-module, and vice versa. Therefore left Leibniz modules are sometimes called Lie modules. Consequently, many properties of left Leibniz modules follow from the corresponding properties of modules for the canonical Lie algebra.
The correct concept of a module for a left Leibniz algebra ${\mathfrak{L}}$ is the notion of a Leibniz bimodule. An [*${\mathfrak{L}}$-bimodule*]{} is a left ${\mathfrak{L}}$-module $M$ with an ${\mathbb{F}}$-bilinear right ${\mathfrak{L}}$-action $M\times{\mathfrak{L}}\to M$, $(m,x)\mapsto m\cdot x$ such that $$\label{LML}
(x\cdot m)\cdot y=x\cdot(m\cdot y)-m\cdot(xy)$$ and $$\label{MLL}
(m\cdot x)\cdot y=m\cdot(xy)-x\cdot(m\cdot y)$$ are satisfied for every $m\in M$ and all $x,y\in{\mathfrak{L}}$ (see [@F Section 3] for the motivation behind this definition of a bimodule for a left Leibniz algebra).
The usual definitions of the notions of [*sub-(bi)module*]{}, [*irreducibility*]{}, [*complete reducibility*]{}, [*composition series*]{}, [*homomorphism*]{}, [*isomorphism*]{}, etc., hold for left Leibniz modules and Leibniz bimodules.
Let ${\mathfrak{L}}$ be a left Leibniz algebra over a field ${\mathbb{F}}$, and let $M$ be an ${\mathfrak{L}}$-bimodule. Then $M$ is said to be [*symmetric*]{} if $m\cdot x=-x\cdot m$ for every $x\in{\mathfrak{L}}$ and every $m\in M$, and $M$ is said to be [*anti-symmetric*]{} if $m\cdot x=0$ for every $x\in{\mathfrak{L}}$ and every $m\in M$. We call $$M_0:=\langle x\cdot m+m\cdot x\mid x\in{\mathfrak{L}},m\in M
\rangle_{\mathbb{F}}$$ the [*anti-symmetric kernel*]{} of $M$. It is known that $M_0$ is an anti-symmetric ${\mathfrak{L}}$-sub-bimodule of $M$ (see [@F Proposition 3.12]) such that $M_{\mathrm{sym}}:=M/M_0$ is symmetric (see [@F Proposition 3.13]).
Moreover, for any ${\mathfrak{L}}$-bimodule $M$ we will need its [*space of right ${\mathfrak{L}}$-invariants*]{} $$M^{\mathfrak{L}}:=\{m\in M\mid\forall\,x\in{\mathfrak{L}}:m\cdot x=0\}$$ and the [*annihilator*]{} $${\mathrm{Ann}}_{\mathfrak{L}}^{\mathrm{bi}}(M):=\{x\in{\mathfrak{L}}\mid\forall\,m\in M:x\cdot m=0=m\cdot x\}\,.$$
Our first result will be useful in the proof of Theorem \[whitehead\].
\[sym\] Let ${\mathfrak{L}}$ be a left Leibniz algebra, and let $M$ be an ${\mathfrak{L}}$-bimodule such that $M^{\mathfrak{L}}=0$. Then $M$ is symmetric. In particular, ${\mathrm{Leib}}({\mathfrak{L}})\subseteq{\mathrm{Ann}}_{\mathfrak{L}}^{\mathrm{bi}}(M)$.
Since $M_0$ is anti-symmetric, it follows from the hypothesis that $$M_0=M_0^{\mathfrak{L}}\subseteq
M^{\mathfrak{L}}=0\,.$$ Hence we obtain from the definition of $M_0$ that $M$ is symmetric. The second part is then an immediate consequence of [@F Lemma 3.10].
It is clear from the definition of $M^{\mathfrak{L}}$ that an ${\mathfrak{L}}$-bimodule $M$ is anti-symmetric if, and only if, $M^{\mathfrak{L}}=M$. We will use Lemma \[sym\] to show that the symmetry of non-trivial irreducible Leibniz bimodules can also be characterized by the behavior of their spaces of right invariants. As a preparation for this, we need to know that the latter space is a sub-bimodule.
\[inv\] Let ${\mathfrak{L}}$ be a left Leibniz algebra, and let $M$ be an ${\mathfrak{L}}$-bimodule. Then $M^{\mathfrak{L}}$ is a sub-bimodule of $M$.
It follows from (\[LML\]) that $M^{\mathfrak{L}}$ is invariant under the left action on $M$, and it follows from (\[MLL\]) that $M^{\mathfrak{L}}$ is invariant under the right action on $M$.
Now we can characterize the symmetry of a non-trivial irreducible Leibniz bimodule by the vanishing of its space of right invariants. In particular, for non-trivial irreducible Leibniz bimodules we obtain the converse of Lemma \[sym\]. (Recall that an irreducible bimodule $M$ is a bimodule that has exactly two sub-bimodules, namely, $0$ and $M$. In particular, an irreducible bimodule is by definition a non-zero vector space.)
\[irr\] Let ${\mathfrak{L}}$ be a left Leibniz algebra, and let $M$ be an irreducible ${\mathfrak{L}}$-bimodule. Then $M$ is symmetric with non-trivial ${\mathfrak{L}}$-action if, and only if, $M^{\mathfrak{L}}=0$.
Since $M$ is irreducible, we obtain from Lemma \[inv\] that $M^{\mathfrak{L}}=0$ or $M^{\mathfrak{L}}=M$. Suppose first that $M$ is symmetric with non-trivial ${\mathfrak{L}}$-action. Then we have that $M^{\mathfrak{L}}=0$. On the other hand, the converse follows immediately from Lemma \[sym\].
Recall that every left ${\mathfrak{L}}$-module $M$ of a left Leibniz algebra ${\mathfrak{L}}$ determines a unique symmetric ${\mathfrak{L}}$-bimodule structure on $M$ by defining $m\cdot x:=-x\cdot m$ for every element $m\in M$ and every element $x\in{\mathfrak{L}}$ (see [@F Proposition 3.15(b)]). We will denote this symmetric ${\mathfrak{L}}$-bimodule by $M_s$. Similarly, every left ${\mathfrak{L}}$-module $M$ with trivial right action is an anti-symmetric ${\mathfrak{L}}$-bimodule (see [@F Proposition 3.15(a)]). We will denote this module by $M_a$. Note that for any irreducible left ${\mathfrak{L}}$-module $M$ the ${\mathfrak{L}}$-bimodules $M_s$ and $M_a$ are irreducible, and every irreducible ${\mathfrak{L}}$-bimodule arises in this way from an irreducible left ${\mathfrak{L}}$-module (see [@LP2]).
Similar to the boundary map in [@CH] for the homology of a right Leibniz algebra with coefficients in a right module one can also introduce a coboundary map ${\widetilde{\mathrm{d}}}^\bullet$ for the cohomology of a left Leibniz algebra with coefficients in a left module as follows.
Let ${\mathfrak{L}}$ be a left Leibniz algebra over a field ${\mathbb{F}}$, and let $M$ be a left ${\mathfrak{L}}$-module. For any non-negative integer $n$ set ${\mathrm{CL}}^n({\mathfrak{L}},M):={\mathrm{Hom}}_{\mathbb{F}}({\mathfrak{L}}^{\otimes n},M)$ and consider the linear transformation ${\widetilde{\mathrm{d}}}^n:{\mathrm{CL}}^n({\mathfrak{L}},M)
\to{\mathrm{CL}}^{n+1}({\mathfrak{L}},M)$ defined by $$\begin{aligned}
({\widetilde{\mathrm{d}}}^nf)(x_1,\dots,x_{n+1}) & := & \sum_{i=1}^{n+1}(-1)^{i+1}x_i\cdot f(x_1,
\dots,\hat{x}_i,\dots,x_{n+1})\\
& + & \sum_{1\le i<j\le n+1}(-1)^if(x_1,\dots,\hat{x}_i,\dots,x_ix_j,\dots,x_{n+1})\end{aligned}$$ for any $f\in{\mathrm{CL}}^n({\mathfrak{L}},M)$ and all elements $x_1,\dots,x_{n+1}\in{\mathfrak{L}}$. (Here and in the remainder of the paper we identify the tensor power ${\mathfrak{L}}^{\otimes n}$ with the corresponding Cartesian power.)
Now let $M$ be an ${\mathfrak{L}}$-bimodule and for any non-negative integer $n$ consider the linear transformation ${\mathrm{d}}^n:{\mathrm{CL}}^n({\mathfrak{L}},M)\to{\mathrm{CL}}^{n+1}({\mathfrak{L}},M)$ defined by $$\begin{aligned}
({\mathrm{d}}^nf)(x_1,\dots,x_{n+1}) & := & \sum_{i=1}^n(-1)^{i+1}x_i\cdot f(x_1,\dots,\hat{x}_i,
\dots,x_{n+1})\\
& + & (-1)^{n+1}f(x_1,\dots,x_n)\cdot x_{n+1}\\
& + & \sum_{1\le i<j\le n+1}(-1)^if(x_1,\dots,\hat{x}_i,\dots,x_ix_j,\dots,x_{n+1})\end{aligned}$$ for any $f\in{\mathrm{CL}}^n({\mathfrak{L}},M)$ and all elements $x_1,\dots,x_{n+1}\in{\mathfrak{L}}$.
It is proved in [@C Lemma 1.3.1] that ${\mathrm{CL}}^\bullet({\mathfrak{L}},M):=({\mathrm{CL}}^n({\mathfrak{L}},M),{\mathrm{d}}^n)_{n\in
{\mathbb{N}}_0}$is a cochain complex, i.e., ${\mathrm{d}}^{n+1}\circ{\mathrm{d}}^n=0$ for every non-negative integer $n$. Of course, the original idea of defining Leibniz cohomology as the cohomology of such a cochain complex for right Leibniz algebras is due to Loday and Pirashvili [@LP Section 1.8]. Hence one can define the [*cohomology of ${\mathfrak{L}}$ with coefficients in an ${\mathfrak{L}}$-bimodule*]{} $M$ by $${\mathrm{HL}}^n({\mathfrak{L}},M):={\mathrm{H}}^n({\mathrm{CL}}^\bullet({\mathfrak{L}},M)):={\mathrm{Ker}}({\mathrm{d}}^n)/{\mathrm{Im}}({\mathrm{d}}^{n-1})$$ for every non-negative integer $n$. (Note that ${\mathrm{d}}^{-1}:=0$.)
If $M$ is a symmetric ${\mathfrak{L}}$-bimodule, then we have the identity ${\widetilde{\mathrm{d}}}^n={\mathrm{d}}^n$ for any non-negative integer $n$. Namely, $$\begin{aligned}
({\widetilde{\mathrm{d}}}^nf)(x_1,\dots,x_{n+1}) & = & \sum_{i=1}^n(-1)^{i+1}x_i\cdot f(x_1,\dots,\hat{x}_i,
\dots,x_{n+1})\\
& + & (-1)^{n+2}x_{n+1}\cdot f(x_1,\dots,x_n)\\
& + & \sum_{1\le i<j\le n+1}(-1)^if(x_1,\dots,\hat{x}_i,\dots,x_ix_j,\dots,x_{n+1})\\
& = & ({\mathrm{d}}^nf)(x_1,\dots,x_{n+1})\end{aligned}$$ for any $f\in{\mathrm{CL}}^n({\mathfrak{L}},M)$ and all elements $x_1,\dots,x_{n+1}\in{\mathfrak{L}}$. In particular, as mentioned earlier, any left ${\mathfrak{L}}$-module $M$ can be turned into a symmetric ${\mathfrak{L}}$-module $M_s$, and the fact that ${\mathrm{d}}^\bullet$ is a coboundary map for ${\mathrm{CL}}^\bullet({\mathfrak{L}},M_s)$ shows that ${\mathrm{\widetilde{CL}}}^\bullet({\mathfrak{L}},M):=({\mathrm{CL}}^n({\mathfrak{L}},M),{\widetilde{\mathrm{d}}}^n)_{n\in{\mathbb{N}}_0}$ is a cochain complex, i.e., ${\widetilde{\mathrm{d}}}^{n+1}\circ{\widetilde{\mathrm{d}}}^n=0$ for every non-negative integer $n$. Hence one can define the [*cohomology of ${\mathfrak{L}}$ with coefficients in a left ${\mathfrak{L}}$-module*]{} $M$ by $${\mathrm{\widetilde{HL}}}^n
({\mathfrak{L}},M):={\mathrm{H}}^n({\mathrm{\widetilde{CL}}}^\bullet({\mathfrak{L}},M)):={\mathrm{Ker}}({\widetilde{\mathrm{d}}}^n)/{\mathrm{Im}}({\widetilde{\mathrm{d}}}^{n-1})$$ for every non-negative integer $n$. (Note that as before ${\widetilde{\mathrm{d}}}^{-1}:=0$.)
Now we are ready to state the next result (see [@P Lemma 2.2] for the analogous result in Leibniz homology) whose second part generalizes [@F Corollary 4.4(b)] to arbitrary degrees and which will be crucial in Section \[semsim\]. (Note that the first part has already been obtained in [@C Section 1.3.4]). For the convenience of the reader we include a detailed proof.
\[antisym\] Let ${\mathfrak{L}}$ be a left Leibniz algebra over a field ${\mathbb{F}}$, and let $M$ be a left ${\mathfrak{L}}$-module. Then the following statements hold:
1. If $M$ is considered as a symmetric ${\mathfrak{L}}$-bimodule $M_s$, then $${\mathrm{HL}}^n({\mathfrak{L}},M_s)=
{\mathrm{\widetilde{HL}}}^n({\mathfrak{L}},M)$$ for every integer $n\ge 0$.
2. If $M$ is considered as an anti-symmetric ${\mathfrak{L}}$-bimodule $M_a$, then $${\mathrm{HL}}^0
({\mathfrak{L}},M_a)=M$$ and $${\mathrm{HL}}^n({\mathfrak{L}},M_a)\cong{\mathrm{\widetilde{HL}}}^{n-1}({\mathfrak{L}},{\mathrm{Hom}}_{\mathbb{F}}({\mathfrak{L}},M))={\mathrm{HL}}^{n-1}
({\mathfrak{L}},{\mathrm{Hom}}_{\mathbb{F}}({\mathfrak{L}},M)_s)$$ for every integer $n\ge 1$, where ${\mathrm{Hom}}_{\mathbb{F}}({\mathfrak{L}},M)$ is a left ${\mathfrak{L}}$-module via $$(x\cdot f)(y):=x\cdot f(y)-f(xy)$$ for every $f\in{\mathrm{Hom}}_{\mathbb{F}}({\mathfrak{L}},M)$ and any elements $x,y\in{\mathfrak{L}}$.
By virtue of the computation before Lemma \[antisym\], we only need to prove part (b). Note that the first part of (b) is just [@F Corollary 4.2(b)].
First, we show that ${\mathrm{Hom}}_{\mathbb{F}}({\mathfrak{L}},M)$ is a left ${\mathfrak{L}}$-module via the given action. Let $f\in{\mathrm{Hom}}_{\mathbb{F}}({\mathfrak{L}},M)$ and $x,y,z\in{\mathfrak{L}}$ be arbitrary. Then we obtain from the defining identity of a left Leibniz module (\[LLM\]) and the left Leibniz identity (\[RLLI\]) that $$\begin{aligned}
((xy)\cdot f)(z) & = & (xy)\cdot f(z)-f((xy)z)\\
& = & x\cdot(y\cdot f(z))-y\cdot(x\cdot f(z))-f(x(yz))+f(y(xz))\,,\end{aligned}$$ and $$\begin{aligned}
(x\cdot(y\cdot f))(z) & = & x\cdot(y\cdot f)(z)-(y\cdot f)(xz)\\
& = & x\cdot(y\cdot f(z))-x\cdot f(yz)-y\cdot f(xz)+f(y(xz))\,,\end{aligned}$$ as well as $$\begin{aligned}
(y\cdot(x\cdot f))(z) & = & y\cdot(x\cdot f)(z)-(x\cdot f)(yz)\\
& = & y\cdot(x\cdot f(z))-y\cdot f(xz)-x\cdot f(yz)+f(x(yz))\,.\end{aligned}$$ Hence $((xy)\cdot f)(z)=(x\cdot(y\cdot f)(z)-(y\cdot(x\cdot f)(z)$ for every $z\in{\mathfrak{L}}$, or equivalently, $(xy)\cdot f=x\cdot(y\cdot f)-y\cdot(x\cdot f)$.
Now we will prove the second part of (b). Let $n$ be any positive integer. Consider the linear transformations $\varphi^n:{\mathrm{CL}}^n({\mathfrak{L}},M)\to{\mathrm{CL}}^{n-1}({\mathfrak{L}},{\mathrm{Hom}}_{\mathbb{F}}({\mathfrak{L}},M))$ defined by $\varphi^n(f)(x_1,
\dots,x_{n-1})(x):=f(x_1,\dots,x_{n-1},x)$ for any elements $x_1,\dots,x_{n-1},x\in{\mathfrak{L}}$ and $\psi^n:{\mathrm{CL}}^{n-1}({\mathfrak{L}},{\mathrm{Hom}}_{\mathbb{F}}({\mathfrak{L}},M))\to{\mathrm{CL}}^n({\mathfrak{L}},M)$ defined by $\psi^n(g)(x_1,\dots,
x_{n-1},x_n)$ $:=g(x_1,\dots,x_{n-1})(x_n)$ for any elements $x_1,\dots,x_{n-1},x_n
\in{\mathfrak{L}}$. Then $\varphi^n$ and $\psi^n$ are inverses of each other.
Next, we will show that ${\widetilde{\mathrm{d}}}^{n-1}\circ\varphi^n=\varphi^{n+1}\circ{\mathrm{d}}^n$. Compute $$\begin{aligned}
({\widetilde{\mathrm{d}}}^{n-1}\circ\varphi^n)(f)(x_1,\dots,x_n)(x) & = & {\widetilde{\mathrm{d}}}^{n-1}(\varphi^n(f))(x_1,\dots,x_n)(x)\\
& = & \sum_{i=1}^n(-1)^{i+1}(x_i\cdot\varphi^n(f))(x_1,\dots,\hat{x_i},\dots,x_n)(x)\\
& + & \sum_{1\le i<j\le n}(-1)^i\varphi^n(f)(x_1,\dots,\hat{x_i},\dots,x_ix_j,\dots,x_n)(x)\\
& = & \sum_{i=1}^n(-1)^{i+1}x_i\cdot\varphi^n(f)(x_1,\dots,\hat{x_i},\dots,x_n)(x)\\
& - & \sum_{i=1}^n(-1)^{i+1}\varphi^n(f)(x_1,\dots,\hat{x_i},\dots,x_n)(x_ix)\\
& + & \sum_{i\le i<j\le n}(-1)^if(x_1,\dots,\hat{x_i},\dots,x_ix_j,\dots,x_n,x)\\
& = & \sum_{i=1}^n(-1)^{i+1}x_i\cdot f(x_1,\dots,\hat{x_i},\dots,x_n,x)\\
& + & \sum_{i=1}^n(-1)^if(x_1,\dots,\hat{x_i},\dots,x_n,x_ix)\\
& + & \sum_{1\le i<j\le n}(-1)^if(x_1,\dots,\hat{x_i},\dots,x_ix_j,\dots,x_n,x)\end{aligned}$$ and $$\begin{aligned}
(\varphi^{n+1}\circ{\mathrm{d}}^n)(f)(x_1,\dots,x_n)(x) & = & \varphi^{n+1}({\mathrm{d}}^n(f))(x_1,\dots,x_n)(x)\\
& = & {\mathrm{d}}^n(f)(x_1,\dots,x_n,x)\\
& = & \sum_{i=1}^n(-1)^{i+1}x_i\cdot f(x_1,\dots,\hat{x_i},\dots,x_n,x)\\
& + & (-1)^{n+1}f(x_1,\dots,x_n)\cdot x\\
& + & \sum_{1\le i<j\le n}(-1)^if(x_1,\dots,\hat{x_i},\dots,x_ix_j,\dots,x_n,x)\\
& + & \sum_{i=1}^n(-1)^if(x_1,\dots,\hat{x_i},\dots,x_n,x_ix)\end{aligned}$$ for any elements $x_1,\dots,x_n,x\in{\mathfrak{L}}$. Since $M$ is anti-symmetric, the second of the last four summands vanishes, and thus the two compositions are equal. From the identity ${\widetilde{\mathrm{d}}}^{n-1}\circ\varphi^n=\varphi^{n+1}\circ{\mathrm{d}}^n$ for every integer $n\ge 1$ we obtain that $$\varphi^n({\mathrm{Ker}}({\mathrm{d}}^n))\subseteq{\mathrm{Ker}}({\widetilde{\mathrm{d}}}^{n-1})$$ and $$\varphi^n({\mathrm{Im}}({\mathrm{d}}^{n-1}))
\subseteq{\mathrm{Im}}({\widetilde{\mathrm{d}}}^{n-2})$$ for every integer $n\ge 1$. Hence $\varphi^n$ induces an isomorphism of vector spaces between ${\mathrm{HL}}^n({\mathfrak{L}},M)$ and ${\mathrm{\widetilde{HL}}}^{n-1}({\mathfrak{L}},{\mathrm{Hom}}_{\mathbb{F}}({\mathfrak{L}},M))$ for every integer $n\ge 1$. In order to see the remainder of the assertion, apply part (a).
In the special case of the trivial one-dimensional Leibniz bimodule we obtain from Lemma \[antisym\] the following result which will be needed in Section \[semsim\] (see [@CH Exercise E.10.6.1] for the analogous result in Leibniz homology).
\[coadj\] Let ${\mathfrak{L}}$ be a left Leibniz algebra over a field ${\mathbb{F}}$. Then for every integer $n\ge 1$ there are isomorphisms $${\mathrm{HL}}^n({\mathfrak{L}},{\mathbb{F}})\cong{\mathrm{\widetilde{HL}}}^{n-1}({\mathfrak{L}},{\mathfrak{L}}^*)={\mathrm{HL}}^{n-1}({\mathfrak{L}},({\mathfrak{L}}^*)_s)$$ of vector spaces, where ${\mathfrak{L}}^*$ is a left ${\mathfrak{L}}$-module via $(x\cdot f)(y):=-f(xy)$ for every linear form $f\in{\mathfrak{L}}^*$ and any elements $x,y\in{\mathfrak{L}}$.
[**Remark.**]{} Note that [@HPL Theorem 3.5] is an immediate consequence of the case $n=2$ of Corollary \[coadj\] and [@F Corollary 4.4(a)].
A relation between Chevalley-Eilenberg cohomology and Leibniz cohomology for Lie algebras {#cel}
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Let ${\mathfrak{g}}$ be a Lie algebra, and let $M$ be a left ${\mathfrak{g}}$-module, also viewed as a symmetric ${\mathfrak{g}}$-bimodule $M_s$. In this section, we will investigate how the Chevalley-Eilenberg cohomology ${\mathrm{H}}^\bullet({\mathfrak{g}},M)$ and the Leibniz cohomology ${\mathrm{HL}}^\bullet({\mathfrak{g}},M_s)$ are related. The tools set forth in this section have been developed by Pirashvili, and we follow the analogous treatment for homology given in [@P] very closely.
The Chevalley-Eilenberg cohomology of a Lie algebra ${\mathfrak{g}}$ with trivial coefficients is not isomorphic (up to a degree shift) to the Chevalley-Eilenberg cohomology of ${\mathfrak{g}}$ with coadjoint coefficients as it is the case for Leibniz cohomology (see Corollary \[coadj\]). Instead these cohomologies are only related by a long exact sequence (see Proposition \[lescoadj\]). The cohomology measuring the deviation from such an isomorphism will appear in a spectral sequence (see Theorem \[theorem\_A\]) which can be used to relate the Leibniz cohomology of a Lie algebra to its Chevalley-Eilenberg cohomology (see Proposition \[lesrelcoh\]).
The exterior product map $m:\Lambda^n{\mathfrak{g}}\otimes {\mathfrak{g}}\to\Lambda^{n+1}{\mathfrak{g}}$ given on homogeneous tensors by $x_1\wedge\ldots\wedge x_n\otimes x\mapsto x_1\wedge\ldots
\wedge x_n\wedge x$ induces a monomorphism$$m^\bullet:\overline{{\mathrm{C}}}^\bullet({\mathfrak{g}},{\mathbb{F}})
[-1]\hookrightarrow{\mathrm{C}}^\bullet({\mathfrak{g}},{\mathfrak{g}}^*),$$ where $\overline{{\mathrm{C}}}^\bullet({\mathfrak{g}},{\mathbb{F}})$ is the truncated cochain complex $$\overline{{\mathrm{C}}}^0({\mathfrak{g}},{\mathbb{F}}):=0\,\mbox{ and }\,\overline{{\mathrm{C}}}^n
({\mathfrak{g}},{\mathbb{F}}):={\mathrm{C}}^n({\mathfrak{g}},{\mathbb{F}})\,\,\,\mbox{ for every integer}\,\,\,n>0\,.$$ The complex ${\mathrm{CR}}^\bullet({\mathfrak{g}})$ is defined by ${\mathrm{CR}}^\bullet({\mathfrak{g}}):={\mathrm{Coker}}(m^\bullet)[-1]$. Observe that classes in ${\mathrm{CR}}^n({\mathfrak{g}})$ are represented by cochains of degree $n+1$ with values in ${\mathfrak{g}}^*$, i.e., they have $n+2$ arguments. From the short exact sequence $$0\to
\overline{{\mathrm{C}}}^\bullet({\mathfrak{g}},{\mathbb{F}})[-1]\to{\mathrm{C}}^\bullet({\mathfrak{g}},{\mathfrak{g}}^*)\to{\mathrm{CR}}^\bullet({\mathfrak{g}})[1]\to
0$$ of cochain complexes we obtain the following long exact sequence:
\[lescoadj\] For every Lie algebra ${\mathfrak{g}}$ over a field ${\mathbb{F}}$ there is a long exact sequence $$\begin{aligned}
0 & \to & {\mathrm{H}}^2({\mathfrak{g}},{\mathbb{F}})\to{\mathrm{H}}^1({\mathfrak{g}},{\mathfrak{g}}^*)\to{\mathrm{HR}}^0({\mathfrak{g}})\\
& \to & {\mathrm{H}}^3({\mathfrak{g}},{\mathbb{F}})\to{\mathrm{H}}^2({\mathfrak{g}},{\mathfrak{g}}^*)\to{\mathrm{HR}}^1({\mathfrak{g}})\to\cdots\end{aligned}$$ and an isomorphism ${\mathrm{H}}^1({\mathfrak{g}},{\mathbb{F}})\cong{\mathrm{H}}^0({\mathfrak{g}},{\mathfrak{g}}^*)$.
[**Remark.**]{} If we assume that the characteristic of the ground field ${\mathbb{F}}$ is not 2, then ${\mathrm{HR}}^0({\mathfrak{g}})\cong[S^2({\mathfrak{g}})^*]^{\mathfrak{g}}$ is the space of invariant symmetric bilinear forms on ${\mathfrak{g}}$ (see [@P p. 403]). As a consequence, we obtain from Proposition \[lescoadj\] in the case that ${\mathrm{char}}({\mathbb{F}})\ne 2$ the five-term exact sequence $$0\to{\mathrm{H}}^2({\mathfrak{g}},{\mathbb{F}})\to{\mathrm{H}}^1
({\mathfrak{g}},{\mathfrak{g}}^*)\to[S^2({\mathfrak{g}})^*]^{\mathfrak{g}}\to{\mathrm{H}}^3({\mathfrak{g}},{\mathbb{F}})\to{\mathrm{H}}^2({\mathfrak{g}},{\mathfrak{g}}^*)\,,$$ which generalizes [@Fa Proposition 1.3(1) & (3)]. Note that the map $[S^2({\mathfrak{g}})^*]^{\mathfrak{g}}\to{\mathrm{H}}^3({\mathfrak{g}},{\mathbb{F}})$ is the classical Cartan-Koszul map defined by $\omega\mapsto\overline{\omega}+{\mathrm{B}}^3({\mathfrak{g}},{\mathbb{F}})$, where $\overline{\omega}(x\wedge y\wedge z):=\omega(xy,z)$ for any elements $x,y,z\in{\mathfrak{g}}$ (see [@P p. 403]).
For a Lie algebra ${\mathfrak{g}}$ and a left ${\mathfrak{g}}$-module $M$ viewed as a symmetric Leibniz ${\mathfrak{g}}$-bimodule $M_s$, we have a natural monomorphism $${\mathrm{C}}^\bullet({\mathfrak{g}},M)\hookrightarrow{\mathrm{CL}}^\bullet({\mathfrak{g}},
M_s)\,.$$ The cokernel of this morphism is by definition (up to a shift in degree) the cochain complex ${\mathrm{C}}_{\mathrm{rel}}^\bullet({\mathfrak{g}},M)$: $${\mathrm{C}}_{\mathrm{rel}}^\bullet({\mathfrak{g}},M):={\mathrm{Coker}}({\mathrm{C}}^\bullet({\mathfrak{g}},M)
\to{\mathrm{CL}}^\bullet({\mathfrak{g}},M_s))[-2]\,.$$ We therefore have another long exact sequence. (For the isomorphisms in degrees 0 and 1 see [@F Corollary 4.2(a)] and [@F Corollary 4.4(a)], respectively.)
\[lesrelcoh\] Let ${\mathfrak{g}}$ be a Lie algebra, and let $M$ be a left ${\mathfrak{g}}$-module considered as a symmetric Leibniz ${\mathfrak{g}}$-bimodule $M_s$. Then there are a long exact sequence $$\begin{aligned}
0 & \to & {\mathrm{H}}^2({\mathfrak{g}},M)\to{\mathrm{HL}}^2({\mathfrak{g}},M_s)\to{\mathrm{H}}_{\mathrm{rel}}^0({\mathfrak{g}},M)\\
& \to & {\mathrm{H}}^3({\mathfrak{g}},M)\to{\mathrm{HL}}^3({\mathfrak{g}},M_s)\to{\mathrm{H}}_{\mathrm{rel}}^1({\mathfrak{g}},M)\to\cdots\end{aligned}$$ and isomorphisms $${\mathrm{HL}}^0({\mathfrak{g}},M_s)\cong{\mathrm{H}}^0({\mathfrak{g}},M),\,\,\,{\mathrm{HL}}^1({\mathfrak{g}},M_s)\cong{\mathrm{H}}^1({\mathfrak{g}},M)\,.$$
[**Remark.**]{} If we again assume that the characteristic of the ground field ${\mathbb{F}}$ is not 2, it follows from Theorem \[theorem\_A\] below in conjunction with the remark after Proposition \[lescoadj\] that ${\mathrm{H}}_{\mathrm{rel}}^0({\mathfrak{g}},{\mathbb{F}})\cong{\mathrm{HR}}^0({\mathfrak{g}})\cong[S^2({\mathfrak{g}})^*]^{\mathfrak{g}}$ is the space of invariant symmetric bilinear forms on ${\mathfrak{g}}$. So when ${\mathrm{char}}({\mathbb{F}})\ne 2$, we obtain the five-term exact sequence $$0\to{\mathrm{H}}^2({\mathfrak{g}},{\mathbb{F}})\to{\mathrm{HL}}^2({\mathfrak{g}},{\mathbb{F}})\to[S^2({\mathfrak{g}})^*]^{\mathfrak{g}}\to{\mathrm{H}}^3({\mathfrak{g}},{\mathbb{F}})\to{\mathrm{HL}}^3({\mathfrak{g}},{\mathbb{F}})$$ as a special case of Proposition \[lesrelcoh\] (cf.[@HPL Proposition 3.2] for fields of characteristic zero). Note that Corollary \[coadj\] implies that the second terms of the five-term exact sequences in Proposition \[lescoadj\] and Proposition \[lesrelcoh\] are isomorphic, but Example A below shows that the fifth terms are not always isomorphic.
Observe that as for ${\mathrm{CR}}^n({\mathfrak{g}})$, representatives of classes in ${\mathrm{C}}_{\mathrm{rel}}^n({\mathfrak{g}},M)$ have $n+2$ arguments.
On the quotient complex ${\mathrm{C}}_{\mathrm{rel}}^\bullet({\mathfrak{g}},M)$ there is the following filtration $${\mathcal F}^p{\mathrm{C}}_{\mathrm{rel}}^{n}({\mathfrak{g}},M)=\{[c]\in{\mathrm{C}}_{\mathrm{rel}}^{n}({\mathfrak{g}},M)\mid c(x_1,\ldots,
x_{n+2})=0\mbox{ if }\exists\,j\leq p+1:\,x_{j-1}=x_j\}\,.$$ Note that the condition is independent of the representative $c$ of the class $[c]$. This defines a finite decreasing filtration $${\mathcal F}^0{\mathrm{C}}_{\mathrm{rel}}^{n}({\mathfrak{g}},M)={\mathrm{C}}_{\mathrm{rel}}^{n}({\mathfrak{g}},M)\supset{\mathcal
F}^1{\mathrm{C}}_{\mathrm{rel}}^{n}({\mathfrak{g}},M)\supset\cdots\supset{\mathcal F}^{n+1}{\mathrm{C}}_{\mathrm{rel}}^{n}({\mathfrak{g}},M)
=\{0\}\,,$$ whose associated spectral sequence converges thus in the strong (i.e., finite) sense to ${\mathrm{H}}_{\mathrm{rel}}^n({\mathfrak{g}},M)$ thanks to the following lemma:
\[lemma\_filtration\_compatible\] This filtration is compatible with the Leibniz coboundary map ${\mathrm{d}}^\bullet$.
The Leibniz coboundary map, acting on a cochain $c$, is an alternating sum of operators ${\mathrm{d}}_{ij}
(c)$, $\delta_i(c)$ and $\partial(c)$, where ${\mathrm{d}}_{ij}(c)$ is the term involving the product of the $i$-th and the $j$-th element, $\delta_i(c)$ is the term involving the left action of the $i$-th element, and $\partial(c)$ is the term involving the right action of the $(n+1)$-th element. As the module is symmetric, the term involving the right action can be counted among the terms involving the left actions.
We have to show that ${\mathrm{d}}^\bullet(\,{\mathcal F}^p{\mathrm{C}}_{\mathrm{rel}}^{n}({\mathfrak{g}},M))\subseteq{\mathcal F}^p
{\mathrm{C}}_{\mathrm{rel}}^{n+1}({\mathfrak{g}},M)$. We thus consider the different terms of ${\mathrm{d}}^\bullet(c)$ with two equal elements as arguments in the first $p+1$ slots and have to show that all terms are zero. For ${\mathrm{d}}_{ij}
(c)$ with $i,j\leq p+1$, the assertion is clear because either the two equal elements do not occur in the product, and then it is correct, or at least one of them occurs, and then the product generates out of the sum of ${\mathrm{d}}_{ij}$ and ${\mathrm{d}}_{ij+1}$ (or ${\mathrm{d}}_{ij-1}$) an element $x_ix\otimes x+x\otimes
x_ix$, which is a sum of symmetric elements thanks to $$x_ix\otimes x+x\otimes x_ix=(x_ix+x)
\otimes(x_ix+x)-x_ix\otimes x_ix-x\otimes x\,.$$
Even more elementary, the assertion holds for ${\mathrm{d}}_{ij}(c)$ with $i,j\geq p+1$. For ${\mathrm{d}}_{ij}
(c)$ with $i\leq p+1$ and $j\geq p+2$, the assertion is clear in case $x_i$ is not one of the equal elements. In case it is, the two terms corresponding to the product action of the two equal elements cancel (as they are equal and have different sign).
For the action terms $\delta_i(c)$ the reasoning is similar. In case $i\leq p+1$, either the two equal elements do not occur and the assertion holds, or both occur and cancel each other because of the alternating sign. For $\delta_i(c)$ with $i\geq p+2$, the assertion is clear in any case.
The next step is then to compute the $0$-th term of the corresponding spectral sequence, i.e., the associated graded vector space of the filtration: $$E_0^{p,q}:= {\mathcal F}^p
{\mathrm{C}}_{\mathrm{rel}}^{p+q}({\mathfrak{g}},M)\,/\,{\mathcal F}^{p+1}{\mathrm{C}}_{\mathrm{rel}}^{p+q}({\mathfrak{g}},M)\,.$$
Observe that the filtration can be expressed as $${\mathcal F}^p{\mathrm{C}}_{\mathrm{rel}}^{p+q}({\mathfrak{g}},M)
=\{[c]\in{\mathrm{C}}_{\mathrm{rel}}^{n}({\mathfrak{g}},M)\,|\,c|_{({\mathrm{Ker}}(\otimes^{p+1}{\mathfrak{g}}\to\Lambda^{p+1}{\mathfrak{g}}))\otimes
(\otimes^{q+1}{\mathfrak{g}})}=0\}.$$ This is useful, because by elementary linear algebra, we have $$F^{\perp}/\,G^{\perp}={\mathrm{Hom}}_{\mathbb{F}}(G/F,M)\,,$$ where $F^{\perp}:=\{f:E\to M\,|\,f|_F=
0\}$ and $G^{\perp}:=\{f:E\to M\,|\,f|_G=0\}$ for $F\subseteq G\subseteq E$.
In order to be able to find $E_0^{p,q}$, we therefore have to compute $$({\mathrm{Ker}}(\otimes^{p+2}{\mathfrak{g}}\to\Lambda^{p+2}{\mathfrak{g}})\otimes(\otimes^{q}{\mathfrak{g}}))\,/\,({\mathrm{Ker}}(\otimes^{p+1}{\mathfrak{g}}\to\Lambda^{p+1}{\mathfrak{g}})\otimes(\otimes^{q+1}{\mathfrak{g}})).$$ Using the isomorphism (see the proof of Theorem A in [@P]) $${\mathrm{Ker}}(\otimes^{p+2}{\mathfrak{g}}\to
\Lambda^{p+2}{\mathfrak{g}})\,/\,({\mathrm{Ker}}(\otimes^{p+1}{\mathfrak{g}}\to\Lambda^{p+1}{\mathfrak{g}})\otimes{\mathfrak{g}})
\cong{\mathrm{Ker}}(\Lambda^{p+1}{\mathfrak{g}}\otimes{\mathfrak{g}}\to\Lambda^{p+2}{\mathfrak{g}})\,.$$ we obtain $$\begin{gathered}
E_0^{p,q}=\{c:{\mathrm{Ker}}(\Lambda^{p+1}{\mathfrak{g}}\otimes{\mathfrak{g}}\to\Lambda^{p+2}{\mathfrak{g}})\otimes{\mathrm{CL}}_{q}
({\mathfrak{g}})\to M\}\,/\,{\mathrm{C}}^{p+q+2}({\mathfrak{g}},M)\\
={\mathrm{Ker}}(\Lambda^{p+1}{\mathfrak{g}}\otimes{\mathfrak{g}}\to\Lambda^{p+2}{\mathfrak{g}})^*\otimes{\mathrm{CL}}^{q}({\mathfrak{g}},M)\,/\,
{\mathrm{C}}^{p+q+2}({\mathfrak{g}},M)\,.\end{gathered}$$ It remains to observe that the first tensor factor is the kernel of the exterior multiplication map $m$, and thus $${\mathrm{Ker}}(\Lambda^{p+1}{\mathfrak{g}}\otimes{\mathfrak{g}}\to\Lambda^{p+2}{\mathfrak{g}})^*={\mathrm{Ker}}(m)^*=
{\mathrm{Coker}}(m^*)={\mathrm{CR}}^p({\mathfrak{g}})\,.$$ Therefore the term $E_0^{p,q}$ takes the form $$E_0^{p,q}
={\mathrm{CR}}^p({\mathfrak{g}})\otimes{\mathrm{CL}}^{q}({\mathfrak{g}},M)\,.$$
Next, let us find the differential on $E_0^{p,q}$.
The differential on $E_0^{\bullet,\bullet}$ induced from ${\mathrm{d}}^\bullet$ on ${\mathrm{C}}_{\mathrm{rel}}^\bullet
({\mathfrak{g}},M)$ identifies with ${\mathrm{id}}_{{\mathrm{CR}}^p({\mathfrak{g}})}^{\otimes(p+2)}\otimes{\mathrm{d}}^q_{{\mathrm{CL}}^{q}({\mathfrak{g}},M)}$.
By definition, the differential ${\mathrm{d}}_0$ of the spectral sequence is the differential which is induced from the Leibniz coboundary map ${\mathrm{d}}^\bullet$ on the associated graded quotients $${\mathrm{d}}_0:
{\mathcal F}^p{\mathrm{C}}_{\mathrm{rel}}^{p+q}({\mathfrak{g}},M)\,/\,{\mathcal F}^{p+1}{\mathrm{C}}_{\mathrm{rel}}^{p+q}({\mathfrak{g}},M)\to
{\mathcal F}^p{\mathrm{C}}_{\mathrm{rel}}^{p+q+1}({\mathfrak{g}},M)\,/\,{\mathcal F}^{p+1}{\mathrm{C}}_{\mathrm{rel}}^{p+q+1}({\mathfrak{g}},M)\,.$$ In order to examine which terms ${\mathrm{d}}_{ij}(c)$, $\delta_i(c)$ and $\partial(c)$ are zero for a cochain $c\in{\mathcal F}^p{\mathrm{C}}_{\mathrm{rel}}^{p+q}({\mathfrak{g}},M)$, we have to insert two consecutive equal elements in the arguments of $c$ within the first $p+2$ arguments.
Now, by the same reasoning as in the proof of Lemma \[lemma\_filtration\_compatible\], the terms ${\mathrm{d}}_{ij}(c)$ vanish in case $i,j\leq p+2$, because in case the equal elements are not involved, the formula for ${\mathrm{d}}_{ij}(c)$ diminishes the number of arguments by one, and as $c$ is of degree $p$ in the filtration, this gives then zero. In case the elements occur, they create once again a symmetric element of the form $x_ix\otimes x+x\otimes x_ix$. Also for ${\mathrm{d}}_{ij}(c)$ with $i\leq p+2$ and $j\geq p+3$, the terms are zero when the equal elements are not involved and are zero in addition with ${\mathrm{d}}_{ij+1}(c)$ (or ${\mathrm{d}}_{ij-1}(c)$), in case of multiplying with one of the equal elements. The terms $\delta_i(c)$ for $i\leq p+1$ vanish as the corresponding formula diminishes the number of arguments by one in case the equal elements do not occur and annihilate each other in case they occur.
There remain thus the terms ${\mathrm{d}}_{ij}(c)$ with $i,j\geq p+3$, $\delta_i(c)$ with $i\geq p+3$, and $\partial(c)$, which form together the coboundary map of the cochain complex ${\mathrm{CL}}^\bullet({\mathfrak{g}},M)$.
In conclusion, we have as second term of the spectral sequence $$E_2^{p,q}={\mathrm{HR}}^p({\mathfrak{g}})\otimes{\mathrm{HL}}^{q}({\mathfrak{g}},M_s)\,,$$ because the differential ${\mathrm{d}}_1$ is again induced by ${\mathrm{d}}^\bullet$, but all terms involving terms of ${\mathrm{HL}}^\bullet({\mathfrak{g}},M)$ vanish, because a Leibniz algebra acts trivially on its cohomology. This follows from the Cartan relations for Leibniz cohomology (due to Loday-Pirashvili [@LP Proposition 3.1]). For the reader interested in left Leibniz algebras, a proof of the formulae adapted to this case can be found in [@C Proposition 1.3.2].
This discussion proves the following result which is (up to dualization) Theorem A in [@P] and is due to Pirashvili. In the case of trivial coefficients (and possibly topological Fréchet Lie algebras) Theorem \[theorem\_A\] has also been obtained by Lodder (see [@L Theorem 2.10]).
\[theorem\_A\] Let ${\mathfrak{g}}$ be a Lie algebra, and let $M$ be a left ${\mathfrak{g}}$-module considered as a symmetric Leibniz ${\mathfrak{g}}$-bimodule $M_s$. Then there is a spectral sequence converging to ${\mathrm{H}}^\bullet_{\mathrm{rel}}({\mathfrak{g}},M)$ with second term $$E_2^{p,q}={\mathrm{HR}}^p({\mathfrak{g}})\otimes {\mathrm{HL}}^{q}({\mathfrak{g}},M_s)\,.$$
[**Remark.**]{} As the spectral sequence is the spectral sequence of a filtered complex, the higher differentials in the spectral sequence are again induced by the total Leibniz cohomology differential ${\mathrm{d}}^{\bullet}$. We will see in Example B below an instance of a concrete computation of ${\mathrm{d}}_2$.
Our main application of the spectral sequence will be the next theorem which is a refinement of the cohomological analogue of [@P Corollary 1.3]:
\[vanlie\] Let ${\mathfrak{g}}$ be a Lie algebra, let $M$ be a left ${\mathfrak{g}}$-module considered as a symmetric Leibniz ${\mathfrak{g}}$-bimodule $M_s$, and let $n$ be a non-negative integer. If ${\mathrm{H}}^k({\mathfrak{g}},M)=0$ for every integer $k$ with $0\le k\le n$, then ${\mathrm{HL}}^k({\mathfrak{g}},M_s)=0$ for every integer $k$ with $0\le k\le
n$ and ${\mathrm{HL}}^{n+1}({\mathfrak{g}},M_s)\cong{\mathrm{H}}^{n+1}({\mathfrak{g}},M)$ as well as ${\mathrm{HL}}^{n+2}({\mathfrak{g}},M_s)\cong
{\mathrm{H}}^{n+2}({\mathfrak{g}},M)$. In particular, ${\mathrm{H}}^\bullet({\mathfrak{g}},M)=0$ implies that ${\mathrm{HL}}^\bullet({\mathfrak{g}},M_s)
=0$.
The proof follows the proof of Corollary 1.3 in [@P] very closely.
According to Proposition \[lesrelcoh\], it suffices to prove that ${\mathrm{H}}^k({\mathfrak{g}},M)=0$ for every integer $k$ with $0\le k\le n$ implies that ${\mathrm{H}}_{\mathrm{rel}}^n ({\mathfrak{g}},M)=0$ for every integer $k$ with $0\le k\le n$. We proceed by induction on $n$. In the case $n=0$, the hypothesis yields that $E_2^{0,0}=0$ for the second term of the spectral sequence of Theorem \[theorem\_A\], and therefore we obtain from the convergence of the spectral sequence that ${\mathrm{H}}_{\mathrm{rel}}^0
({\mathfrak{g}},M)=0$ initializing the induction.
So suppose now that $n\ge 1$ and ${\mathrm{H}}^k({\mathfrak{g}},M_s)=0$ for every integer $k$ with $0\le
k\le n+1$. By induction hypothesis, we obtain that ${\mathrm{H}}_{\mathrm{rel}}^n ({\mathfrak{g}},M)=0$ for every integer $k$ with $0\le k\le n$. Hence it follows from Proposition \[lesrelcoh\] that ${\mathrm{HL}}^k({\mathfrak{g}},M_s)=0$ for every integer $k$ with $0\le k\le n$ and ${\mathrm{HL}}^{n+1}({\mathfrak{g}},M_s)
\cong{\mathrm{H}}^{n+1}({\mathfrak{g}},M)=0$. Consequently, the second term $E_2^{p,q}$ of the spectral sequence in Theorem \[theorem\_A\] is zero for $p+q\leq n+1$, and therefore ${\mathrm{H}}_{\mathrm{rel}}^{n+1}({\mathfrak{g}},M)=0$.
Finally, the isomorphisms in degree $n+1$ and $n+2$, respectively, are an immediate consequence of Proposition \[lesrelcoh\].
[**Remark.**]{} Note that the converse of Theorem \[vanlie\] is also true, namely, ${\mathrm{H}}^k({\mathfrak{g}},M)=0$ for every integer $k$ with $0\le k\le n$ if, and only if, ${\mathrm{HL}}^k({\mathfrak{g}},M_s)=0$ for every integer $k$ with $0\le k\le n$. In particular, ${\mathrm{H}}^\bullet({\mathfrak{g}},M)=0$ if, and only if, ${\mathrm{HL}}^\bullet({\mathfrak{g}},M_s)=0$.
Next, we illustrate the use of the spectral sequence of Theorem \[theorem\_A\] and the associated long exact sequences (see Propositions \[lescoadj\] and \[lesrelcoh\]) by two examples. We begin by computing the Leibniz cohomology of the smallest non-nilpotent Lie algebra with coefficients in an arbitrary irreducible bimodule (see also [@P Example 1.4i)] for trivial coeffcients in characteristic $\ne 2$). Note that for a ground field of characteristic 2 the Leibniz cohomology of this Lie algebra is far more complicated than for a field of characteristic $\ne 2$.
[**Example A.**]{} Let ${\mathbb{F}}$ denote an arbitrary field, and let ${\mathfrak{a}}:={\mathbb{F}}h\oplus{\mathbb{F}}e$ be the non-abelian two-dimensional Lie algebra over ${\mathbb{F}}$ with multiplication determined by $he=e=-eh$. It follows from Theorem \[vanlie\] in conjunction with [@B1 Theorem 3] that the Leibniz cohomology of a finite-dimensional supersolvable Lie algebra with coefficients in an irreducible bimodule of dimension $\ne 1$ vanishes. So in the following we will only consider one-dimensional bimodules.
For any scalar $\lambda\in{\mathbb{F}}$ one can define a one-dimensional left ${\mathfrak{a}}$-module $F_\lambda:={\mathbb{F}}1_\lambda$ with ${\mathfrak{a}}$-action defined by $h\cdot 1_\lambda:=\lambda
1_\lambda$ and $e\cdot 1_\lambda:=0$. Then the Chevalley-Eilenberg cohomology of ${\mathfrak{a}}$ with coefficients in $F_\lambda$ is as follows: $$\begin{aligned}
{\mathrm{H}}^n({\mathfrak{a}},F_\lambda)\cong
\left\{
\begin{array}{cl}
{\mathbb{F}}& \mbox{if }\lambda=0\mbox{ and }n=0,1\mbox{ or }\lambda=1\mbox{ and }n=1,2\\
0 & \hspace{2.5cm}\mbox{otherwise}\,.
\end{array}
\right.\end{aligned}$$ In particular, if $\lambda\ne 0,1$, then ${\mathrm{H}}^\bullet({\mathfrak{a}},F_\lambda)=0$.
First, let us consider $F_\lambda$ as a symmetric ${\mathfrak{a}}$-bimodule $(F_\lambda)_s$. Then it follows from Theorem \[vanlie\] that ${\mathrm{HL}}^\bullet({\mathfrak{a}},(F_\lambda)_s)=0$ for $\lambda\ne
0,1$.
In order to be able to compute the Leibniz cohomology for $\lambda=0,1$, and for the anti-symmetric ${\mathfrak{a}}$-bimodules $(F_\lambda)_a$, let $M$ be an arbitrary left ${\mathfrak{a}}$-module considered as a symmetric ${\mathfrak{a}}$-bimodule $M_s$. Since ${\mathrm{H}}^n({\mathfrak{a}},M)=0$ for every integer $n\ge 3$, we obtain from Proposition \[lesrelcoh\] the short exact sequence $$\label{leibnizses}
0\to{\mathrm{H}}^2({\mathfrak{a}},M)\to{\mathrm{HL}}^2({\mathfrak{a}},M_s)\to{\mathrm{H}}_{\mathrm{rel}}^0({\mathfrak{a}},M)\to 0$$ and the isomorphisms $$\label{leibnizreduc}
{\mathrm{HL}}^n({\mathfrak{a}},M_s)\cong{\mathrm{H}}_{\mathrm{rel}}^{n-2}({\mathfrak{a}},M)$$ for every integer $n\ge 3$. Moreover, we have that ${\mathrm{HL}}^0({\mathfrak{a}},M_s)\cong M^{\mathfrak{a}}$ and ${\mathrm{HL}}^1({\mathfrak{a}},M_s)\cong{\mathrm{H}}^1({\mathfrak{a}},M)$.
For the computation of the relative cohomology spaces ${\mathrm{H}}_{\mathrm{rel}}^n({\mathfrak{a}},M)$ we need the coadjoint Chevalley-Eilenberg cohomology of ${\mathfrak{a}}$. It is easy to verify that $$\dim_{\mathbb{F}}{\mathrm{H}}^0({\mathfrak{a}},{\mathfrak{a}}^*)=1\,,$$ $$\begin{aligned}
\dim_{\mathbb{F}}{\mathrm{H}}^1({\mathfrak{a}},{\mathfrak{a}}^*)=
\left\{
\begin{array}{cl}
2 & \mbox{if }{\mathrm{char}}({\mathbb{F}})=2\\
1 & \mbox{if }{\mathrm{char}}({\mathbb{F}})\ne 2\,,
\end{array}
\right.\end{aligned}$$ and $$\begin{aligned}
\dim_{\mathbb{F}}{\mathrm{H}}^2({\mathfrak{a}},{\mathfrak{a}}^*)=
\left\{
\begin{array}{cl}
1 & \mbox{if }{\mathrm{char}}({\mathbb{F}})=2\\
0 & \mbox{if }{\mathrm{char}}({\mathbb{F}})\ne 2\,.
\end{array}
\right.\end{aligned}$$ Consequently, we have to consider the cases ${\mathrm{char}}({\mathbb{F}})=2$ and ${\mathrm{char}}({\mathbb{F}})\ne 2$ differently.
Let us first assume that ${\mathrm{char}}({\mathbb{F}})\ne 2$. Then it follows from Proposition \[lescoadj\] that ${\mathrm{HR}}^0({\mathfrak{a}})\cong{\mathrm{H}}^1({\mathfrak{a}},{\mathfrak{a}}^*)\cong{\mathbb{F}}$ and ${\mathrm{HR}}^n({\mathfrak{a}})=0$ for every integer $n\ge 1$. Hence the spectral sequence of Theorem \[theorem\_A\] implies that ${\mathrm{H}}_{\mathrm{rel}}^n({\mathfrak{a}},M)\cong{\mathrm{HL}}^n({\mathfrak{a}},M_s)$ for every non-negative integer $n$. In conclusion, we obtain that $$\label{leibniz2}
{\mathrm{HL}}^2({\mathfrak{a}},M_s)\cong M^{\mathfrak{a}}\oplus{\mathrm{H}}^2({\mathfrak{a}},M)$$ and $$\label{leibnizhigh}
{\mathrm{HL}}^n({\mathfrak{a}},M_s)\cong{\mathrm{HL}}^{n-2}({\mathfrak{a}},M_s)\hspace{.1cm}\mbox{for every integer }n\ge 3\,.$$
As an immediate consequence, we deduce when ${\mathrm{char}}({\mathbb{F}})\ne 2$ that $\dim_{\mathbb{F}}{\mathrm{HL}}^n({\mathfrak{a}},{\mathbb{F}})
=1$ for every non-negative integer $n$ and $$\begin{aligned}
\dim_{\mathbb{F}}{\mathrm{HL}}^n({\mathfrak{a}},(F_1)_s)=
\left\{
\begin{array}{cl}
0 & \mbox{if }n=0\\
1 & \mbox{if }n>0\,.
\end{array}
\right.\end{aligned}$$
In summary, we have for the Leibniz cohomology of ${\mathfrak{a}}$ over a field ${\mathbb{F}}$ of characteristic $\ne 2$ with coefficients in a one-dimensional symmetric bimodule that $$\begin{aligned}
\dim_{\mathbb{F}}{\mathrm{HL}}^n({\mathfrak{a}},(F_\lambda)_s)=
\left\{
\begin{array}{cl}
1 & \mbox{if }\lambda=0\mbox{ and }n\mbox{ is arbitrary or if }\lambda=1\mbox{ and }n>0\\
0 & \hspace{3cm}\mbox{otherwise}\,.
\end{array}
\right.\end{aligned}$$
Next, let us assume that ${\mathrm{char}}({\mathbb{F}})=2$. Then it follows from Proposition \[lescoadj\] that ${\mathrm{HR}}^0({\mathfrak{a}})\cong{\mathrm{H}}^1({\mathfrak{a}},{\mathfrak{a}}^*)\cong{\mathbb{F}}^2$, ${\mathrm{HR}}^1({\mathfrak{a}})\cong{\mathrm{H}}^2({\mathfrak{a}},{\mathfrak{a}}^*)
\cong{\mathbb{F}}$, and ${\mathrm{HR}}^n({\mathfrak{a}})=0$ for every integer $n\ge 2$. Hence in the spectral sequence of Theorem \[theorem\_A\], we have only two non-zero columns, namely the $p=0$ and the $p=1$ column. In the $p=0$ column, we have spaces ${\mathbb{F}}^2\otimes{\mathrm{HL}}^q({\mathfrak{a}},M_s)
\cong{\mathrm{HL}}^q({\mathfrak{a}},M_s)\oplus{\mathrm{HL}}^q({\mathfrak{a}},M_s)$, while in the $p=1$ column, we have just ${\mathrm{HL}}^q({\mathfrak{a}},M_s)$ for every integer $q\geq 0$. Therefore, the spectral sequence degenerates at the term $E_2$, and for every integer $n\geq 1$ we obtain that $$\label{leibnizrel}
{\mathrm{H}}^n_{\mathrm{rel}}({\mathfrak{a}},M)\cong{\mathrm{HL}}^n({\mathfrak{a}},M_s)\oplus{\mathrm{HL}}^n({\mathfrak{a}},M_s)\oplus{\mathrm{HL}}^{n-1}({\mathfrak{a}},M_s)\,,$$ and $$\label{leibnizrel0}
{\mathrm{H}}^0_{\mathrm{rel}}({\mathfrak{a}},M)\cong E_2^{0,0}\cong{\mathrm{HL}}^0({\mathfrak{a}},M_s)\oplus{\mathrm{HL}}^0({\mathfrak{a}},M_s)\cong M^{\mathfrak{a}}\oplus M^{\mathfrak{a}}\,.$$ This, together with (\[leibnizses\]), (\[leibnizreduc\]), and induction yields the recursive relation $$\label{fibonacci}
{\mathrm{HL}}^n({\mathfrak{a}},M_s)\cong{\mathrm{HL}}^{n-1}({\mathfrak{a}},M_s)\oplus{\mathrm{HL}}^{n-2}({\mathfrak{a}},M_s)$$ for every integer $n\geq 2$.
As a consequence, we obtain for the Leibniz cohomology of ${\mathfrak{a}}$ over a field ${\mathbb{F}}$ of characteristic $2$ with coefficients in a one-dimensional symmetric bimodule that $$\begin{aligned}
\dim_{\mathbb{F}}{\mathrm{HL}}^n({\mathfrak{a}},(F_\lambda)_s)=
\left\{
\begin{array}{cl}
f_{n+1} & \mbox{if }\lambda=0\\
f_n & \mbox{if }\lambda=1\\
0 & \mbox{otherwise}
\end{array}
\right.\end{aligned}$$ for every non-negative integer $n$, where $f_n$ denotes the $n^{\rm th}$ term of the standard Fibonacci sequence given by $f_0:=0$, $f_1:=1$, and $f_n:=f_{n-1}+f_{n-2}$ for every integer $n\geq 2$. In particular, we have that $${\mathrm{HL}}^n({\mathfrak{a}},(F_1)_s)\cong
{\mathrm{HL}}^{n-1}({\mathfrak{a}},{\mathbb{F}})$$ for every integer $n\geq 1$.
Next, let us consider $F_\lambda$ as an anti-symmetric Leibniz ${\mathfrak{a}}$-bimodule $(F_\lambda)_a$ with the same left ${\mathfrak{a}}$-action as above and with the trivial right ${\mathfrak{a}}$-action (see Section \[prelim\]). Then we conclude from Lemma \[antisym\](b) that $$\dim_{\mathbb{F}}{\mathrm{HL}}^0({\mathfrak{a}},
(F_\lambda)_a)=1\mbox{ for every }\lambda\in{\mathbb{F}}\,.$$
Let us now compute ${\mathrm{HL}}^n({\mathfrak{a}},(F_\lambda)_a)$ for any integer $n\ge 1$. It follows from Lemma \[antisym\](b) that $$\label{leibnizalt}
{\mathrm{HL}}^n({\mathfrak{a}},(F_\lambda)_a)\cong{\mathrm{HL}}^{n-1}({\mathfrak{a}},{\mathrm{Hom}}_{\mathbb{F}}({\mathfrak{a}},F_\lambda)_s)\cong{\mathrm{HL}}^{n-1}
({\mathfrak{a}},({\mathfrak{a}}^*\otimes F_\lambda)_s)\,.$$ A straightforward computation shows that $$0\to F_\lambda\to{\mathfrak{a}}^*\otimes F_\lambda\to
F_{\lambda-1}\to 0$$ is a short exact sequence of left ${\mathfrak{a}}$-modules. Then we obtain from the long exact cohomology sequence and another straightforward computation in the case $\lambda=1$: $$\begin{aligned}
\dim_{\mathbb{F}}({\mathfrak{a}}^*\otimes F_\lambda)^{\mathfrak{a}}=
\left\{
\begin{array}{cl}
1 & \mbox{if }\lambda=0\\
0 & \mbox{otherwise}\,,
\end{array}
\right.\end{aligned}$$ $$\begin{aligned}
\dim_{\mathbb{F}}{\mathrm{H}}^1({\mathfrak{a}},{\mathfrak{a}}^*\otimes F_\lambda)=
\left\{
\begin{array}{cl}
2 & \mbox{if }\lambda=0\mbox{ and }{\mathrm{char}}({\mathbb{F}})=2\\
1 & \mbox{if }\lambda=0,2\mbox{ and }{\mathrm{char}}({\mathbb{F}})\ne 2\,,\\
0 & \hspace{1cm}\mbox{otherwise}
\end{array}
\right.\end{aligned}$$ and $$\begin{aligned}
\dim_{\mathbb{F}}{\mathrm{H}}^2({\mathfrak{a}},{\mathfrak{a}}^*\otimes F_\lambda)=
\left\{
\begin{array}{cl}
1 & \mbox{if }\lambda=0\mbox{ and }{\mathrm{char}}({\mathbb{F}})=2\mbox{ or }\lambda=2
\mbox{ and }{\mathrm{char}}({\mathbb{F}})\ne 2\\
0 & \hspace{3cm}\mbox{otherwise}\,.
\end{array}
\right.\end{aligned}$$
If ${\mathrm{char}}({\mathbb{F}})\ne 2$, we conclude by applying (\[leibniz2\]) and (\[leibnizalt\]) to the symmetric ${\mathfrak{a}}$-bimodule $M_s:=({\mathfrak{a}}^*\otimes F_\lambda)_s$ that $$\begin{aligned}
\dim_{\mathbb{F}}{\mathrm{HL}}^1({\mathfrak{a}},(F_\lambda)_a)=
\left\{
\begin{array}{cl}
1 & \mbox{if }\lambda=0\\
0 & \mbox{otherwise}\,,
\end{array}
\right.\end{aligned}$$ and $$\begin{aligned}
\dim_{\mathbb{F}}{\mathrm{HL}}^3({\mathfrak{a}},(F_\lambda)_a)=\dim_{\mathbb{F}}{\mathrm{HL}}^2({\mathfrak{a}},(F_\lambda)_a)=
\left\{
\begin{array}{cl}
1 & \mbox{if }\lambda=0,2\\
0 & \mbox{otherwise}\,,
\end{array}
\right.\end{aligned}$$ Finally, we use (\[leibnizhigh\]) to deduce for every integer $n\ge 2$: $$\begin{aligned}
\dim_{\mathbb{F}}{\mathrm{HL}}^n({\mathfrak{a}},(F_\lambda)_a)=
\left\{
\begin{array}{cl}
1 & \mbox{if }\lambda=0,2\\
0 & \mbox{otherwise}\,.
\end{array}
\right.\end{aligned}$$
In summary, we have for the Leibniz cohomology of ${\mathfrak{a}}$ over a field ${\mathbb{F}}$ of characteristic $\ne 2$ with coefficients in a one-dimensional anti-symmetric bimodule that $$\begin{aligned}
\dim_{\mathbb{F}}{\mathrm{HL}}^n({\mathfrak{a}},(F_\lambda)_a)=
\left\{
\begin{array}{cl}
1 & \mbox{if }\lambda=0\mbox{ and }n\mbox{ is arbitrary or }\lambda=2\mbox{ and }n\ge 2\\
& \mbox{or }n=0\mbox{ and }\lambda\mbox{ is arbitrary}\\
0 & \hspace{3cm}\mbox{otherwise}\,.
\end{array}
\right.\end{aligned}$$
If ${\mathrm{char}}({\mathbb{F}})=2$, we obtain by applying (\[fibonacci\]) and (\[leibnizalt\]): $$\begin{aligned}
\dim_{\mathbb{F}}{\mathrm{HL}}^n({\mathfrak{a}},(F_\lambda)_a)=
\left\{
\begin{array}{cl}
1 & \mbox{if }n=0\mbox{ and }\lambda\mbox{ is arbitrary}\\
f_{n+1} & \mbox{if }\lambda=0\mbox{ and }n\mbox{ is arbitrary}\\
0 & \hspace{1cm}\mbox{otherwise}\,.
\end{array}
\right.\end{aligned}$$
[**Remark.**]{} Since every invariant symmetric bilinear form on ${\mathfrak{a}}$ is a multiple of the Killing form, we have that $[S^2({\mathfrak{a}})^*]^{\mathfrak{a}}\cong{\mathbb{F}}$. On the other hand, from the computations in Example A we obtain that $$\begin{aligned}
{\mathrm{H}}_{\mathrm{rel}}^0({\mathfrak{a}},{\mathbb{F}})\cong{\mathrm{HR}}^0({\mathfrak{a}})\cong{\mathrm{H}}^1({\mathfrak{a}},{\mathfrak{a}}^*)\cong
\left\{
\begin{array}{cl}
{\mathbb{F}}^2 & \mbox{if }{\mathrm{char}}({\mathbb{F}})=2\\
{\mathbb{F}}& \mbox{if }{\mathrm{char}}({\mathbb{F}})\ne 2\,,
\end{array}
\right.\end{aligned}$$ This shows that, in general, ${\mathrm{HR}}^0({\mathfrak{a}})\not\cong[S^2({\mathfrak{a}})^*]^{\mathfrak{a}}$ and ${\mathrm{H}}_{\mathrm{rel}}^0({\mathfrak{a}},{\mathbb{F}})
\not\cong[S^2({\mathfrak{a}})^*]^{\mathfrak{a}}$ when ${\mathrm{char}}({\mathbb{F}})=2$.
Since the Leibniz cohomology of an abelian Lie algebra with trivial coefficients is known, in Example B we compute this cohomology for the smallest non-abelian nilpotent Lie algebra. Note that in [@P Example 1.4.iv)] the corresponding Leibniz homology has been computed. In fact, homology and cohomology of a finite-dimensional Leibniz algebra ${\mathfrak{L}}$ with [*trivial*]{} coefficients are isomorphic, as we have the duality isomorphism ${\mathrm{CL}}_{\bullet}
({\mathfrak{L}},{\mathbb{F}})^*\cong{\mathrm{CL}}^{\bullet}({\mathfrak{L}},{\mathbb{F}})$ already on the level of cochain complexes. Therefore our results coincide with those of Pirashvili. We furthermore compute in Example D the Leibniz cohomology of the smallest nilpotent non-Lie Leibniz algebra with coefficients in the trivial bimodule. Note that it follows from Theorem \[vanlie\] in conjunction with [@B1 Lemma 3] (see also [@D Théorème 1] for infinite fields) that the Leibniz cohomology of a finite-dimensional nilpotent Lie algebra with coefficients in a non-trivial irreducible bimodule vanishes.
[**Example B.**]{} Let ${\mathbb{F}}$ denote an arbitrary field of characteristic $\ne 2$, and let ${\mathfrak{h}}:={\mathbb{F}}x\oplus{\mathbb{F}}y\oplus{\mathbb{F}}z$ be the three-dimensional Heisenberg algebra over ${\mathbb{F}}$ with multiplication determined by $xy=z=-yx$. Then the Chevalley-Eilenberg cohomology of ${\mathfrak{h}}$ with coefficients in the trivial module ${\mathbb{F}}$ is well-known: $$\begin{aligned}
\dim_{\mathbb{F}}{\mathrm{H}}^n({\mathfrak{h}},{\mathbb{F}})=
\left\{
\begin{array}{cl}
1 & \mbox{if }n=0,3\\
2 & \mbox{if }n=1,2\,.\\
0 & \mbox{if }n\ge 4
\end{array}
\right.\end{aligned}$$ Consequently, we have that $\dim_{\mathbb{F}}{\mathrm{HL}}^0({\mathfrak{h}},{\mathbb{F}})=1$ and $\dim_{\mathbb{F}}{\mathrm{HL}}^1({\mathfrak{h}},{\mathbb{F}})=2$.
As ${\mathrm{H}}^n({\mathfrak{h}},{\mathbb{F}})=0$ for every integer $n\ge 4$, we obtain from Proposition \[lesrelcoh\] the following six-term exact sequence: $$0\to{\mathrm{H}}^2({\mathfrak{h}},{\mathbb{F}})\to{\mathrm{HL}}^2({\mathfrak{h}},{\mathbb{F}})\to{\mathrm{H}}_{\mathrm{rel}}^0({\mathfrak{h}},{\mathbb{F}})\to{\mathrm{H}}^3({\mathfrak{h}},{\mathbb{F}})\to
{\mathrm{HL}}^3({\mathfrak{h}},{\mathbb{F}})\to{\mathrm{H}}_{\mathrm{rel}}^1({\mathfrak{h}},{\mathbb{F}})\to 0$$ and $${\mathrm{HL}}^n({\mathfrak{h}},{\mathbb{F}})\cong{\mathrm{H}}_{\mathrm{rel}}^{n-2}
({\mathfrak{h}},{\mathbb{F}})$$ for every integer $n\ge 4$.
Since we assume that ${\mathrm{char}}({\mathbb{F}})\ne 2$, it follows from the remark after Proposition \[lesrelcoh\] that we can identify ${\mathrm{H}}_{\mathrm{rel}}^0({\mathfrak{h}},{\mathbb{F}})$ with the space of invariant symmetric bilinear forms on ${\mathfrak{h}}$ and the map ${\mathrm{H}}_{\mathrm{rel}}^0({\mathfrak{h}},{\mathbb{F}})\to{\mathrm{H}}^3({\mathfrak{h}},{\mathbb{F}})$ with the classical Cartan-Koszul map. It is easy to see that the latter map is zero for the Heisenberg algebra, which yields the surjectivity of the map ${\mathrm{HL}}^2({\mathfrak{h}},{\mathbb{F}})\to{\mathrm{H}}_{\mathrm{rel}}^0({\mathfrak{h}},{\mathbb{F}})$ and the injectivity of the map ${\mathrm{H}}^3({\mathfrak{h}},{\mathbb{F}})\to{\mathrm{HL}}^3({\mathfrak{h}},{\mathbb{F}})$. As a consequence, we obtain the following two short exact sequences: $$0\to{\mathrm{H}}^2({\mathfrak{h}},{\mathbb{F}})\to{\mathrm{HL}}^2({\mathfrak{h}},{\mathbb{F}})\to
{\mathrm{H}}_{\mathrm{rel}}^0({\mathfrak{h}},{\mathbb{F}})\to 0\,,$$ $$0\to{\mathrm{H}}^3({\mathfrak{h}},{\mathbb{F}})\to{\mathrm{HL}}^3({\mathfrak{h}},{\mathbb{F}})\to{\mathrm{H}}_{\mathrm{rel}}^1
({\mathfrak{h}},{\mathbb{F}})\to 0\,.$$
In order to compute ${\mathrm{H}}_{\mathrm{rel}}^0({\mathfrak{h}},{\mathbb{F}})$ and ${\mathrm{H}}_{\mathrm{rel}}^1({\mathfrak{h}},{\mathbb{F}})$, we need the coadjoint Chevalley-Eilenberg cohomology of ${\mathfrak{h}}$. We have that $\dim_{\mathbb{F}}{\mathrm{H}}^0
({\mathfrak{h}},{\mathfrak{h}}^*)=2$, $\dim_{\mathbb{F}}{\mathrm{H}}^1({\mathfrak{h}},{\mathfrak{h}}^*)=5$, $\dim_{\mathbb{F}}{\mathrm{H}}^2({\mathfrak{h}},{\mathfrak{h}}^*)=4$, and ${\mathrm{H}}^3({\mathfrak{h}},{\mathfrak{h}}^*)=1$. (This can be computed directly but for the complex numbers as a ground field it also follows from the main result of [@M] in conjunction with [@W Theorem 3.4].)
Similar to the discussion of the consequences of Proposition \[lesrelcoh\] above, we obtain from Proposition \[lescoadj\] the two short exact sequences $$0\to{\mathrm{H}}^2
({\mathfrak{h}},{\mathbb{F}})\to{\mathrm{H}}^1({\mathfrak{h}},{\mathfrak{h}}^*)\to{\mathrm{HR}}^0({\mathfrak{h}})\to 0\,,$$ $$0\to{\mathrm{H}}^3({\mathfrak{h}},{\mathbb{F}})\to{\mathrm{H}}^2
({\mathfrak{h}},{\mathfrak{h}}^*)\to{\mathrm{HR}}^1({\mathfrak{h}})\to 0\,,$$ the isomorphism ${\mathrm{HR}}^2({\mathfrak{h}})\cong{\mathrm{H}}^3({\mathfrak{h}},
{\mathfrak{h}}^*)$, and ${\mathrm{HR}}^n({\mathfrak{h}})=0$ for every integer $n\ge 3$.
From these two short exact sequences and the isomorphism we derive that $$\dim_{\mathbb{F}}{\mathrm{HR}}^0({\mathfrak{h}})=\dim_{\mathbb{F}}{\mathrm{H}}^1({\mathfrak{h}},{\mathfrak{h}}^*)-\dim_{\mathbb{F}}{\mathrm{H}}^2({\mathfrak{h}},{\mathbb{F}})=5-2=3\,,$$ $$\dim_{\mathbb{F}}{\mathrm{HR}}^1({\mathfrak{h}})=\dim_{\mathbb{F}}{\mathrm{H}}^2({\mathfrak{h}},{\mathfrak{h}}^*)-\dim_{\mathbb{F}}{\mathrm{H}}^3({\mathfrak{h}},{\mathbb{F}})=4-1=3\,,$$ and $$\dim_{\mathbb{F}}{\mathrm{HR}}^2({\mathfrak{h}})=\dim_{\mathbb{F}}{\mathrm{H}}^3({\mathfrak{h}},{\mathfrak{h}}^*)=1\,,$$ respectively. Therefore we obtain from ${\mathrm{H}}_{\mathrm{rel}}^0({\mathfrak{h}},{\mathbb{F}})\cong{\mathrm{HR}}^0({\mathfrak{h}})$ that $$\dim_{\mathbb{F}}{\mathrm{HL}}^2({\mathfrak{h}},{\mathbb{F}})=
\dim_{\mathbb{F}}{\mathrm{H}}^2({\mathfrak{h}},{\mathbb{F}})+\dim_{\mathbb{F}}{\mathrm{H}}_{\mathrm{rel}}^0({\mathfrak{h}},{\mathbb{F}})=2+3=5\,.$$
Now we want to apply the spectral sequence of Theorem \[theorem\_A\]. For this let us compute the differential $${\mathrm{d}}_2^{0,1}:E_2^{0,1}={\mathrm{HR}}^0({\mathfrak{h}})\otimes{\mathrm{HL}}^1({\mathfrak{h}},{\mathbb{F}})\to
E_2^{2,0}={\mathrm{HR}}^2({\mathfrak{h}})\otimes{\mathrm{HL}}^0({\mathfrak{h}},{\mathbb{F}})\,.$$ In characteristic $\not=2$, an element of ${\mathrm{HR}}^0({\mathfrak{h}})$ is an invariant symmetric bilinear form $\omega$. It is considered as a $1$-cochain with values in ${\mathfrak{h}}^*$ and, as it is a representative of an element of a quotient cochain complex, it is zero in case it is skew-symmetric in all entries. Take furthermore a cocycle $c\in {\mathrm{CL}}^1({\mathfrak{h}},{\mathbb{F}})$ and compute for three elements $r,s,t\in{\mathfrak{h}}$: $$\begin{aligned}
{\mathrm{d}}^1(\omega\otimes c)(r,s,t)&=&\omega(rs,-)c(t)+\omega(s,-)c(rt)-\omega(r,-)c(st)+\\
&+&\omega(s,r-)c(t)-\omega(r,s-)c(t)+\omega(r,t-)c(s)\end{aligned}$$ Now as $c$ is a cocycle with trivial coefficients, $c$ vanishes on products, thus the second and third terms are zero. Furthermore, the first and fourth term cancel by the invariance of the form and skew-symmetry of the Lie product. We are left with the two last terms $-\omega(r,s-)c(t)+\omega(r,t-)c(s)$, which are skew-symmetric in the three entries of the element in ${\mathrm{HR}}^2({\mathfrak{h}})$ and vanish therefore as well. In conclusion, the differential ${\mathrm{d}}_2^{0,1}$ is zero, and we have that $${\mathrm{H}}^1_{\rm rel}({\mathfrak{h}})={\mathrm{HR}}^0({\mathfrak{h}})\otimes{\mathrm{HL}}^1
({\mathfrak{h}},{\mathbb{F}})\oplus{\mathrm{HR}}^1({\mathfrak{h}})\otimes{\mathrm{HL}}^0({\mathfrak{h}},{\mathbb{F}})\,.$$ This implies in turn $$\dim_{\mathbb{F}}{\mathrm{HL}}^3({\mathfrak{h}},{\mathbb{F}})=\dim_{\mathbb{F}}{\mathrm{H}}^3({\mathfrak{h}},{\mathbb{F}})+\dim_{\mathbb{F}}{\mathrm{H}}_{\mathrm{rel}}^1({\mathfrak{h}},{\mathbb{F}})=1+9=10\,.$$
It seems that all differentials ${\mathrm{d}}_2$ are zero and thus that this scheme persists to yield the dimensions of the higher ${\mathrm{H}}_{\mathrm{rel}}^n({\mathfrak{h}},{\mathbb{F}})$ and thus of ${\mathrm{HL}}^n({\mathfrak{h}},{\mathbb{F}})$ (see the dimension formula in [@P Example 1.4iv)]).
[**Remark.**]{} As a by-product of the above computations we obtain that the space $[S^2({\mathfrak{h}})^*]^{\mathfrak{h}}$ of invariant symmetric bilinear forms on ${\mathfrak{h}}$ is three-dimensional when ${\mathrm{char}}({\mathbb{F}})\ne 2$.
We proceed by proving an extension of a result by Fialowski, Magnin, and Mandal (see Corollary 2 in [@FMM]), namely, the fact that the vanishing of the center $C({\mathfrak{g}})$ of a Lie algebra ${\mathfrak{g}}$ implies ${\mathrm{HL}}^2({\mathfrak{g}},{\mathfrak{g}}_{\mathrm{ad}})={\mathrm{H}}^2({\mathfrak{g}},{\mathfrak{g}})$, where ${\mathfrak{g}}_{\mathrm{ad}}$ denotes the adjoint Leibniz ${\mathfrak{g}}$-bimodule induced by the left and right multiplication operator. Observe that for Lie algebras, this bimodule is indeed symmetric. The spectral sequence of Theorem \[theorem\_A\] with adjoint coefficients has as its $E_2$-term $$E_2^{p,q}={\mathrm{HR}}^p({\mathfrak{g}})
\otimes{\mathrm{HL}}^q({\mathfrak{g}},{\mathfrak{g}}_{\mathrm{ad}})\,.$$ We note that ${\mathrm{HL}}^0({\mathfrak{g}},{\mathfrak{g}}_{\mathrm{ad}})=C({\mathfrak{g}})$. Therefore it is an immediate consequence of the case $n=0$ of Theorem \[vanlie\] that the vanishing of the center implies that ${\mathrm{HL}}^2({\mathfrak{g}},{\mathfrak{g}}_{\mathrm{ad}})={\mathrm{H}}^2({\mathfrak{g}},{\mathfrak{g}})$. By the same token for $n=1$, we can extend this to complete Lie algebras, i.e., to those Lie algebras ${\mathfrak{g}}$ for which ${\mathrm{H}}^0({\mathfrak{g}},{\mathfrak{g}})={\mathrm{H}}^1({\mathfrak{g}},{\mathfrak{g}})=0$:
\[complet\] Let ${\mathfrak{g}}$ be a complete Lie algebra. Then $${\mathrm{HL}}^2({\mathfrak{g}},{\mathfrak{g}}_{\mathrm{ad}})\cong{\mathrm{H}}^2({\mathfrak{g}},{\mathfrak{g}})\,\,\,{\rm
and}\,\,\,{\mathrm{HL}}^3({\mathfrak{g}},{\mathfrak{g}}_{\mathrm{ad}})\cong{\mathrm{H}}^3({\mathfrak{g}},{\mathfrak{g}})\,.$$
A class of examples of complete Lie algebras over an algebraically closed field ${\mathbb{F}}$ of characteristic zero consists of those finite-dimensional Lie algebras ${\mathfrak{g}}$ for which ${\mathfrak{g}}$ has the same dimension as its Lie algebra of derivations and $\dim_{\mathbb{F}}{\mathfrak{g}}/{\mathfrak{g}}^2>1$ (see [@Car Proposition 3.1]). Another example is the two-sided Witt algebra over a field of characteristic zero. Indeed, this infinite-dimensional simple Lie algebra is complete (see [@ES Theorem A.1.1]). Hence we obtain from [@S Theorem 3.1] and [@ES Theorem 4.1] in conjunction with the case $n=3$ of Theorem \[vanlie\] the following result:
\[witt\] Let ${\mathcal{W}}:={\mathrm{Der}}({\mathbb{F}}[t,t^{-1}])$ be the two-sided Witt algebra over a field ${\mathbb{F}}$ of characteristic zero. Then ${\mathrm{HL}}^2({\mathcal{W}},{\mathcal{W}}_{\mathrm{ad}})=0$ and ${\mathrm{HL}}^3({\mathcal{W}},{\mathcal{W}}_{\mathrm{ad}})=0$. Moreover, $${\mathrm{HL}}^4({\mathcal{W}},{\mathcal{W}}_{\mathrm{ad}})
\cong{\mathrm{H}}^4({\mathcal{W}},{\mathcal{W}})\,\,\,{\rm and}\,\,\,{\mathrm{HL}}^5({\mathcal{W}},{\mathcal{W}}_{\mathrm{ad}})\cong{\mathrm{H}}^5({\mathcal{W}},{\mathcal{W}})\,.$$
[**Remark.**]{} Very recently, Camacho, Omirov, and Kurbanbaev also proved that the second adjoint Leibniz cohomology of ${\mathcal{W}}$ vanishes (see [@COK Theorem 4]) by explicitly showing that every adjoint Leibniz 2-cocycle (resp. Leibniz 2-coboundary) is an adjoint Chevalley-Eilenberg 2-cocycle (resp. Chevalley-Eilenberg 2-coboundary) for ${\mathcal{W}}$.
We conclude this section with another application of Theorem \[vanlie\]. Let ${\mathbb{F}}$ be an algebraically closed field of characteristic zero, let $n$ be a non-negative integer, and let $L_n({\mathbb{F}})\subseteq{\mathbb{F}}^{n^3}$ denote the affine variety of structure constants of the $n$-dimensional left Leibniz algebras over ${\mathbb{F}}$ with respect to a fixed basis of ${\mathbb{F}}^n$. Then the general linear group ${\mathrm{GL}}_n({\mathbb{F}})$ acts on $L_n({\mathbb{F}})$, and a point (= Leibniz multiplication law) $\phi\in L_n({\mathbb{F}})$ is called [*rigid*]{} if the orbit ${\mathrm{GL}}_n({\mathbb{F}})\cdot\phi$ is open in $L_n
({\mathbb{F}})$. It follows from Corollary \[witt\] in conjunction with [@Ba Théorème 3] that the infinite-dimensional two-sided Witt algebra over an algebraically closed field of characteristic zero is rigid as a Leibniz algebra.
It is well known that the Chevalley-Eilenberg cohomology of the non-abelian two-dimensional Lie algebra with coefficients in the adjoint module vanishes. According to Theorem \[vanlie\], this implies that the corresponding Leibniz cohomology vanishes as well. Similarly, by applying Theorem \[vanlie\] in conjunction with [@T Theorem 1] (see also [@LL2 Section 1]) one obtains the following more general result in characteristic zero (cf. also [@P Proposition 2.3] for the rigidity of parabolic subalgebras). Recall that a subalgebra of a semi-simple Lie algebra ${\mathfrak{g}}$ is called [*parabolic*]{} if it contains a maximal solvable (= Borel) subalgebra of ${\mathfrak{g}}$.
\[borel\] Let ${\mathfrak{p}}$ be a parabolic subalgebra of a finite-dimensional semi-simple Lie algebra over a field of characteristic zero. Then ${\mathrm{HL}}^n({\mathfrak{p}},{\mathfrak{p}}_{\mathrm{ad}})=0$ for every non-negative integer $n$. In particular, parabolic subalgebras of a finite-dimensional semi-simple Lie algebra over an algebraically closed field of characteristic zero are rigid as Leibniz algebras.
[**Remark.**]{} It would be interesting to know whether Proposition \[borel\] remains valid in prime characteristic.
A Hochschild-Serre type spectral sequence for Leibniz cohomology {#specseq}
================================================================
In this section we consider a Leibniz analogue of the Hochschild-Serre spectral sequence for the Chevalley-Eilenberg cohomology of Lie algebras that converges to some relative cohomology. It will play a predominant role in Section \[semsim\]. The (homology version of the) spectral sequence (with values in symmetric bimodules) is due to Pirashvili (see [@P Theorem C]). Our arguments follow Pirashvili very closely but we include all the details as it turns out that the spectral sequence holds for arbitrary bimodules.
Let $\pi:{\mathfrak{L}}\to{\mathfrak{Q}}$ be an epimorphism of left Leibniz algebras, and let $M$ be a ${\mathfrak{Q}}$-bimodule. Then $M$ is also an ${\mathfrak{L}}$-bimodule via $\pi$. Moreover, the epimorphisms $\pi^{\otimes n}:
{\mathfrak{L}}^{\otimes n}\to{\mathfrak{Q}}^{\otimes n}$ induce a monomorphism ${\mathrm{CL}}^\bullet({\mathfrak{Q}},M)\to{\mathrm{CL}}^\bullet
({\mathfrak{L}},M)$ of cochain complexes. Now set $${\mathrm{CL}}^\bullet({\mathfrak{L}}\vert{\mathfrak{Q}},M):={\mathrm{Coker}}({\mathrm{CL}}^\bullet
({\mathfrak{Q}},M)\to{\mathrm{CL}}^\bullet({\mathfrak{L}},M))[-1]$$ and $${\mathrm{HL}}^\bullet({\mathfrak{L}}\vert{\mathfrak{Q}},M):={\mathrm{H}}^\bullet({\mathrm{CL}}^\bullet
({\mathfrak{L}}\vert{\mathfrak{Q}},M))\,.$$ Then by applying the long exact cohomology sequence to the short exact sequence $$0\to{\mathrm{CL}}^\bullet({\mathfrak{Q}},M)\to{\mathrm{CL}}^\bullet({\mathfrak{L}},M)\to{\mathrm{CL}}^\bullet({\mathfrak{L}}\vert{\mathfrak{Q}},M)[1]\to 0$$ of cochain complexes one obtains the following result (see also [@P Proposition 4.1] for the corresponding result on Leibniz homology).
\[les\] For every epimorphism $\pi:{\mathfrak{L}}\to{\mathfrak{Q}}$ of left Leibniz algebras and every ${\mathfrak{Q}}$-bimodule $M$ there exists a long exact sequence $$\begin{aligned}
0 & \to & {\mathrm{HL}}^1({\mathfrak{Q}},M)\to{\mathrm{HL}}^1({\mathfrak{L}},M)\to{\mathrm{HL}}^0({\mathfrak{L}}\vert{\mathfrak{Q}},M)\\
& \to & {\mathrm{HL}}^2({\mathfrak{Q}},M)\to{\mathrm{HL}}^2({\mathfrak{L}},M)\to{\mathrm{HL}}^1({\mathfrak{L}}\vert{\mathfrak{Q}},M)\to\cdots\,.\end{aligned}$$
Let us now derive Pirashvili’s analogue of the Hochschild-Serre spectral sequence for Leibniz cohomology (see [@P Theorem C] for the homology version). While Pirashvili considers only symmetric bimodules, we extend the dual of his spectral sequence to arbitrary bimodules.
\[hs1\] Let $0\to{\mathfrak{I}}\to{\mathfrak{L}}\stackrel{\pi}\to{\mathfrak{Q}}\to 0$ be a short exact sequence of left Leibniz algebras such that ${\mathfrak{I}}\subseteq C_\ell({\mathfrak{L}})$. Then ${\mathfrak{I}}$ is a ${\mathfrak{Q}}$-bimodule via $x
\cdot y:=\pi^{-1}(x)y$ and $y\cdot x:=y\pi^{-1}(x)$ for every element $x\in{\mathfrak{Q}}$ and every element $y\in{\mathfrak{I}}$, and there is a similar bimodule structure for its dual ${\mathfrak{I}}^*$, where ${\mathfrak{I}}^*$ is a left ${\mathfrak{L}}$-module via $(x\cdot f)(y):=-f(xy)$ for every linear form $f\in{\mathfrak{I}}^*$ and any elements $x\in{\mathfrak{L}},y\in{\mathfrak{I}}$. Moreover, there is a spectral sequence converging to ${\mathrm{HL}}^\bullet({\mathfrak{L}}\vert{\mathfrak{Q}},M)$ with second term $$\begin{aligned}
E_2^{p,q}=
\left\{
\begin{array}{cl}
{\mathrm{HL}}^{p}({\mathfrak{Q}},{\mathrm{Hom}}_{\mathbb{F}}({\mathfrak{I}},{\mathrm{HL}}^0({\mathfrak{L}},M))_s) & \mbox{if }p\ge 0,q=0\\
{\mathrm{HL}}^{p}({\mathfrak{Q}},({\mathfrak{I}}^*)_s)\otimes{\mathrm{HL}}^q({\mathfrak{L}},M)& \mbox{if }p\ge 0,q\ge 1
\end{array}
\right.\end{aligned}$$ for every ${\mathfrak{Q}}$-bimodule $M$.
\[pirashvili\] If in the above situation the ${\mathfrak{Q}}$-bimodule $M$ is symmetric, then for any integers $p,q\ge 0$ the $E_2$-term of the spectral sequence simply reads $$E_2^{p,q}={\mathrm{HL}}^{p}({\mathfrak{Q}},({\mathfrak{I}}^*)_s)\otimes{\mathrm{HL}}^q({\mathfrak{L}},M)\,.$$
[**Remarks.**]{}
- According to [@F Proposition 2.13], Theorem \[hs1\] applies to ${\mathfrak{I}}:={\mathrm{Leib}}({\mathfrak{L}})$ and ${\mathfrak{Q}}:={\mathfrak{L}}_{\mathrm{Lie}}$ (see [@P Remark 4.2] for the analogous statement for Leibniz homology). Note that in the cohomology space ${\mathrm{HL}}^p({\mathfrak{Q}},({\mathfrak{I}}^*)_s)$, the left ${\mathfrak{Q}}$-module ${\mathfrak{I}}^*$ is here viewed as a [*symmetric*]{} bimodule (while it is naturally an anti-symmetric ${\mathfrak{Q}}$-bimodule).
- The higher differentials in the spectral sequence are again induced by the the total Leibniz cohomology differential ${\mathrm{d}}^{\bullet}$. Observe that the spectral sequence of Corollary \[pirashvili\] is isomorphic to the spectral sequence of the bicomplex ${\mathrm{CL}}^{\bullet}({\mathfrak{Q}},{\mathfrak{I}}^*)\otimes{\mathrm{CL}}^{\bullet}({\mathfrak{L}},M)$. Therefore the description of the higher differentials can be adapted from [@Hue] (see, in particular, Remark 3.2 therein). For example, it is clear that in case one of the two differentials in the bicomplex is zero, all higher differentials vanish. We will see an instance of this case in Example D below.
We consider the following filtration on the complex ${\mathrm{CL}}^\bullet({\mathfrak{L}},M)[-1]$. $${\mathcal F}^p{\mathrm{CL}}^n({\mathfrak{L}},M)[-1]:=\{c\in{\mathrm{CL}}^{n+1}({\mathfrak{L}},M)\mid c(x_1,\ldots,x_{n+1})=0\mbox{ if }
\exists\,i\leq p:\,x_i\in {\mathfrak{I}}\}\,.$$ This defines a finite decreasing filtration $$\begin{gathered}
{\mathcal F}^0{\mathrm{CL}}^n({\mathfrak{L}},M)[-1]={\mathrm{CL}}^n({\mathfrak{L}},M)[-1]\supset{\mathcal F}^1{\mathrm{CL}}^n({\mathfrak{L}},M)[-1]\supset\cdots\\
\cdots\supset{\mathcal F}^{n+1}{\mathrm{CL}}^n({\mathfrak{L}},M)[-1]={\mathrm{CL}}^n({\mathfrak{Q}},M)[-1],\end{gathered}$$ whose associated spectral sequence converges in the strong (i.e., finite) sense to the cohomology ${\mathrm{HL}}^n({\mathfrak{L}}|{\mathfrak{Q}},M)$ of the quotient complex ${\mathrm{CL}}^n({\mathfrak{L}},M)[-1]\,/\,{\mathrm{CL}}^n({\mathfrak{Q}},M)[-1]$ thanks to the following result:
The filtration is compatible with the Leibniz coboundary map ${\mathrm{d}}^\bullet$.
We have to prove that ${\mathrm{d}}^\bullet({\mathcal F}^p{\mathrm{CL}}^n({\mathfrak{L}},M)[-1])\subseteq{\mathcal F}^p
{\mathrm{CL}}^{n+1}({\mathfrak{L}},M)[-1]$. For this, we consider the different terms ${\mathrm{d}}_{ij}(c)$, $\delta_i(c)$, and $\partial(c)$, which constitute the differential ${\mathrm{d}}_0(c)$, where we have inserted an element of ${\mathfrak{I}}$ within the first $p$ arguments. The vanishing is clear for the terms ${\mathrm{d}}_{ij}(c)$ with $i,j
\leq p$, because even if the element of ${\mathfrak{I}}$ occurs in the product, the product will again be in the ideal ${\mathfrak{I}}$. The vanishing is also clear for the terms ${\mathrm{d}}_{ij}(c)$ with $i,j\geq p+1$. Concerning the terms ${\mathrm{d}}_{ij}(c)$ with $i\leq p$ and $j\geq p+1$, we use the condition ${\mathfrak{I}}\subseteq C_\ell({\mathfrak{L}})$ to conclude that these are zero.
The action terms follow a similar pattern. The terms $\delta_i(c)$ with $i\leq p$ vanish, because either the element of ${\mathfrak{I}}$ occurs in the arguments, or it acts on $M$, which is zero by assumption. The terms $\delta_i(c)$ with $i\geq p+1$ are zero for elementary reasons, as is the term $\partial(c)$.
We get for the 0-th term of the spectral sequence $$E_0^{p,q}={\mathrm{Hom}}_{\mathbb{F}}({\mathfrak{Q}}^p\otimes{\mathfrak{L}}^{q+1},M)\,/\,{\mathrm{Hom}}_{\mathbb{F}}({\mathfrak{Q}}^{p+1}\otimes{\mathfrak{L}}^q,M)\,
\cong\,{\mathrm{Hom}}_{\mathbb{F}}({\mathfrak{Q}}^p\otimes{\mathfrak{I}}\otimes{\mathfrak{L}}^q,M)\,,$$ where the isomorphism is induced by the inclusion ${\mathfrak{I}}\hookrightarrow{\mathfrak{L}}$.
The differential ${\mathrm{d}}_0$ on $E_0^{p,q}$ reads ${\mathrm{id}}_{{\mathrm{CL}}^{p+1}({\mathfrak{Q}},{\mathfrak{I}}^*)}^{\otimes{p+1}}
\otimes{\mathrm{d}}^q|_{{\mathrm{CL}}^q({\mathfrak{L}},M)}$.
The differential $$\begin{gathered}
{\mathrm{d}}_0:{\mathcal F}^p{\mathrm{CL}}^n({\mathfrak{L}},M)[-1]\,/\,{\mathcal F}^{p+1}{\mathrm{CL}}^n({\mathfrak{L}},M)[-1]\to\\
\to{\mathcal F}^p{\mathrm{CL}}^{n+1}({\mathfrak{L}},M)[-1]\,/\,{\mathcal F}^{p+1}{\mathrm{CL}}^{n+1}({\mathfrak{L}},M)[-1]\end{gathered}$$ is the differential induced by ${\mathrm{d}}^\bullet$. Thus we have to examine which terms ${\mathrm{d}}_{ij}(c)$, $\delta_i(c)$, and $\partial(c)$ composing the differential ${\mathrm{d}}_0(c)$ are non-zero in case we put an element of ${\mathfrak{I}}$ within the first $p+1$ entries.
It is clear that ${\mathrm{d}}_{ij}(c)=0$ for $i,j\leq p+1$, because this is true if the element of ${\mathfrak{I}}$ is not involved in the product as the number of elements is diminished by one, and it is also true if the element of ${\mathfrak{I}}$ is in the product as ${\mathfrak{I}}$ is an ideal. We then have ${\mathrm{d}}_{ij}(c)=0$ for $i\leq p+1$ and $j\geq p+2$ because in case the element of ${\mathfrak{I}}$ acts in the product, it acts trivially by ${\mathfrak{I}}\subseteq C_\ell({\mathfrak{L}})$. Furthermore, the terms $\delta_i(c)$ for $i\leq p+1$ are zero, because if the element of ${\mathfrak{I}}$ acts on $M$, it acts trivially.
Note that in all action terms on ${\mathfrak{I}}^*$ the action is from the left, thus, in order to make appear the Leibniz coboundary operator with values in ${\mathfrak{I}}^*$, we have to switch around the last action term. This is the reason why we view ${\mathfrak{I}}^*$ here as a symmetric ${\mathfrak{Q}}$-bimodule. We therefore are left with the terms composing the differential ${\mathrm{d}}_0|_{{\mathrm{CL}}^q({\mathfrak{L}},M)}$.
The first term of the spectral sequence is thus $$E_1^{p,q}={\mathrm{Hom}}_{\mathbb{F}}({\mathfrak{Q}}^p\otimes{\mathfrak{I}},{\mathbb{F}})\otimes{\mathrm{HL}}^q({\mathfrak{L}},M)\,,$$ and the second term reads for $q>0$ $$E_2^{p,q}={\mathrm{HL}}^p({\mathfrak{Q}},({\mathfrak{I}}^*)_s)\otimes{\mathrm{HL}}^q({\mathfrak{L}},M)\,,$$ because a Leibniz algebra acts trivially on its cohomology. This follows again from the Cartan relations for Leibniz cohomology (see [@LP Proposition 3.1] for the case of right Leibniz algebras and [@C Proposition 1.3.2] for the case of left Leibniz algebras), but one needs to be careful since the Cartan relations do only hold for $q\ge 1$. Therefore, for an arbitrary bimodule $M$, ${\mathfrak{Q}}$ will act non-trivially on ${\mathrm{HL}}^0({\mathfrak{L}},M)$. In case the bimodule $M$ is symmetric, however, the action is indeed trivial on ${\mathrm{HL}}^0({\mathfrak{L}},M)$. This ends the proof of Theorem \[hs1\].
[**Remark.**]{} One might wonder what one gets when one uses the filtration by the [*last*]{} $p$ arguments instead of the [*first*]{} $p$ arguments. It turns out that this spectral sequence has an $E_2$-term which is more difficult to describe (and which we stated erroneously in a first version of this article), because one takes in the $E_2$-term the cohomology of a complex which appears as [*coefficients*]{} in the Leibniz cohomology that constitutes the $E_1$-term.
As in the previous section, we illustrate the use of the spectral sequence of Theorem \[hs1\] and the associated long exact sequence (see Proposition \[les\]) by two examples.
In the first example we compute the Leibniz cohomology of the smallest non-nilpotent non-Lie left Leibniz algebra with coefficients in one-dimensional bimodules. Note that contrary to the semidirect product of two one-dimensional Lie algebras in Example A the Leibniz algebra in Example C is the hemi-semidirect product of two one-dimensional Lie algebras. It turns out that this somewhat simplifies matters.
[**Example C.**]{} Let ${\mathbb{F}}$ denote an arbitrary field, and let ${\mathfrak{A}}:={\mathbb{F}}h\oplus{\mathbb{F}}e$ be the two-dimensional supersolvable left Leibniz algebra over ${\mathbb{F}}$ with multiplication determined by $he=e$. For any scalar $\lambda\in{\mathbb{F}}$ one can define a one-dimensional left ${\mathfrak{A}}$-module $F_\lambda:={\mathbb{F}}1_\lambda$ with ${\mathfrak{A}}$-action defined by $h\cdot 1_\lambda:=\lambda 1_\lambda$ and $e\cdot 1_\lambda:=0$. Note that ${\mathrm{Leib}}({\mathfrak{A}})={\mathbb{F}}e$, and thus ${\mathfrak{A}}_{\mathrm{Lie}}$ is a one-dimensional abelian Lie algebra. Then we obtain from [@B1 Lemma 1] and Theorem \[vanlie\] that $$\begin{aligned}
\dim_{\mathbb{F}}{\mathrm{HL}}^n({\mathfrak{A}}_{\mathrm{Lie}},(F_\lambda)_s)=
\left\{
\begin{array}{cl}
1 & \mbox{if }\lambda=0\mbox{ and }n\mbox{ is arbitrary}\\
0 & \hspace{1cm}\mbox{otherwise}\,.
\end{array}
\right.\end{aligned}$$ We deduce from Lemma \[antisym\](b) that $$\begin{aligned}
{\mathrm{HL}}^n({\mathfrak{A}}_{\mathrm{Lie}},(F_\lambda)_a)&\cong&{\mathrm{HL}}^{n-1}({\mathfrak{A}}_{\mathrm{Lie}},{\mathrm{Hom}}_{\mathbb{F}}({\mathfrak{A}}_{\mathrm{Lie}},F_\lambda)_s)\\
&\cong&{\mathrm{HL}}^{n-1}({\mathfrak{A}}_{\mathrm{Lie}},(F_\lambda)_s)\end{aligned}$$ for every integer $n\ge 1$, and therefore $$\begin{aligned}
\dim_{\mathbb{F}}{\mathrm{HL}}^n({\mathfrak{A}}_{\mathrm{Lie}},(F_\lambda)_a)=
\left\{
\begin{array}{cl}
1 & \mbox{if }\lambda=0\mbox{ and }n\mbox{ is arbitrary or if }\lambda\ne 0\mbox{ and }n=0\\
0 & \hspace{3cm}\mbox{otherwise}\,.
\end{array}
\right.\end{aligned}$$ In order to be able to apply the spectral sequence of Theorem \[hs1\], we first compute ${\mathrm{HL}}^\bullet({\mathfrak{A}}_{\mathrm{Lie}},[{\mathrm{Leib}}({\mathfrak{A}})^*]_s)$. Observe that the module ${\mathrm{Leib}}({\mathfrak{A}})^*={\mathbb{F}}e^*\cong
F_{-1}$ is non-trivial irreducible and furthermore viewed as a symmetric ${\mathfrak{A}}_{\mathrm{Lie}}$-bimodule. Hence from the above it follows that ${\mathrm{HL}}^n({\mathfrak{A}}_{\mathrm{Lie}},[{\mathrm{Leib}}({\mathfrak{A}})^*]_s)=0$ for every non-negative integer $n$. This implies in turn that the spectral sequence of Theorem \[hs1\] collapses at the $E_2$-term and that $$\begin{aligned}
{\mathrm{HL}}^n({\mathfrak{A}}\vert{\mathfrak{A}}_{\mathrm{Lie}},(F_\lambda)_a)&=&{\mathrm{HL}}^n({\mathfrak{A}}_{\mathrm{Lie}},{\mathrm{Hom}}_{\mathbb{F}}({\mathrm{Leib}}({\mathfrak{A}}),{\mathrm{HL}}^0({\mathfrak{A}},
(F_\lambda)_a))_s)\\
&=&{\mathrm{HL}}^n({\mathfrak{A}}_{\mathrm{Lie}},{\mathrm{Hom}}_{\mathbb{F}}({\mathrm{Leib}}({\mathfrak{A}}),F_\lambda)_s)\end{aligned}$$ for all non-negative integers $n$, while ${\mathrm{HL}}^n({\mathfrak{A}}\vert{\mathfrak{A}}_{\mathrm{Lie}},(F_\lambda)_s)=0$ for all $n\ge 0$ by Corollary \[pirashvili\]. Notice that as an ${\mathfrak{A}}$-bimodule ${\mathrm{Hom}}_{\mathbb{F}}({\mathrm{Leib}}({\mathfrak{A}}),
F_\lambda)_s\cong[F_{\lambda-1}]_s$. We have already remarked that the long exact sequence of Proposition \[les\] splits, and therefore we conclude from Proposition \[les\] that $${\mathrm{HL}}^n({\mathfrak{A}},(F_{\lambda})_a)\cong{\mathrm{HL}}^n({\mathfrak{A}}_{\mathrm{Lie}},[F_{\lambda}]_a)\oplus{\mathrm{HL}}^n({\mathfrak{A}}_{\mathrm{Lie}},
[F_{\lambda-1}]_s)$$ for all $\lambda$ and all non-negative integers $n$. Consequently, we obtain that $$\begin{aligned}
\dim_{\mathbb{F}}{\mathrm{HL}}^n({\mathfrak{A}},(F_\lambda)_s)=
\left\{
\begin{array}{cl}
1 & \mbox{if }\lambda=0\mbox{ and }n\mbox{ is arbitrary}\\
0 & \hspace{1cm}\mbox{otherwise}\,,
\end{array}
\right.\end{aligned}$$ and $$\begin{aligned}
\dim_{\mathbb{F}}{\mathrm{HL}}^n({\mathfrak{A}},(F_\lambda)_a)=
\left\{
\begin{array}{cl}
1 & \mbox{if }\lambda=0,1\mbox{ and }n\mbox{ is arbitrary or if }\lambda\ne 0,1\mbox{ and }n=0\\
0 & \hspace{3cm}\mbox{otherwise}\,.
\end{array}
\right.\end{aligned}$$
[**Remark.**]{} In particular, we have that $\dim_{\mathbb{F}}{\mathrm{HL}}^n({\mathfrak{A}},{\mathbb{F}})=1$ for every non-negative integer $n$. Note that this follows as well from the scheme of proof of Proposition 4.3 in [@P] (using also the isomorphism between Leibniz homology and cohomology with trivial coefficients). Indeed, the characteristic element ${\mathrm{ch}}({\mathfrak{A}})\in{\mathrm{HL}}^2({\mathfrak{A}}_{\mathrm{Lie}},{\mathrm{Leib}}({\mathfrak{A}}))$ of ${\mathfrak{A}}$ is zero as ${\mathrm{Leib}}({\mathfrak{A}})
={\mathbb{F}}e\cong F_1$. Since also ${\mathrm{HL}}^\bullet({\mathfrak{A}}_{\mathrm{Lie}},[{\rm Leib}({\mathfrak{A}})^*]_s)$ is zero, we can reason in the same way as Pirashvili does.
In the second example we compute the Leibniz cohomology of the smallest nilpotent non-Lie left Leibniz algebra with trivial coefficients.
[**Example D.**]{} Let ${\mathbb{F}}$ denote an arbitrary field, and let ${\mathfrak{N}}:={\mathbb{F}}e\oplus
{\mathbb{F}}f$ be the two-dimensional nilpotent left (and right) Leibniz algebra over ${\mathbb{F}}$ with multiplication determined by $ff=e$. Then ${\mathrm{Leib}}({\mathfrak{N}})={\mathbb{F}}e$, and thus ${\mathfrak{N}}_{\mathrm{Lie}}$ is a one-dimensional abelian Lie algebra. Hence ${\mathrm{HL}}^n({\mathfrak{N}}_{\mathrm{Lie}},{\mathbb{F}})\cong{\mathbb{F}}$ for every non-negative integer $n$. We have that $\dim_{\mathbb{F}}{\mathrm{HL}}^0({\mathfrak{N}},{\mathbb{F}})=1$ and $\dim_{\mathbb{F}}{\mathrm{HL}}^1({\mathfrak{N}},{\mathbb{F}})=1$.
Next, we compute the higher cohomology with the help of the spectral sequence of Corollary \[pirashvili\]. As observed in the remark after the proof of Theorem \[hs1\], all higher differentials are zero in our case, because the differential of the abelian Lie algebra with values in the trivial module vanishes. With the input data $\dim_{\mathbb{F}}{\mathrm{HL}}^0({\mathfrak{N}},{\mathbb{F}})
=1$ and $\dim_{\mathbb{F}}{\mathrm{HL}}^1({\mathfrak{N}},{\mathbb{F}})=1$, we therefore get from the spectral sequence $$\dim_{\mathbb{F}}{\mathrm{HL}}^0({\mathfrak{N}}|{\mathfrak{N}}_{\mathrm{Lie}},{\mathbb{F}})=1\,\,\,{\rm and}\,\,\,\dim_{\mathbb{F}}{\mathrm{HL}}^1({\mathfrak{N}}|{\mathfrak{N}}_{\mathrm{Lie}},{\mathbb{F}})
=2\,.$$ In order to apply now the long exact sequence from Proposition \[les\] and deduce the dimensions of the cohomology spaces from here, we want to argue that the sequence is split. In fact, it is split, because the connecting homomorphism is surjective. This comes from the fact that the cochain complex ${\mathrm{CL}}^{\bullet}({\mathfrak{N}}_{\mathrm{Lie}},
{\mathbb{F}})$ is one-dimensional in each degree and a generator can be hit via the connecting homomorphism which is easy to see directly (take a cochain in ${\mathrm{CL}}^n({\mathfrak{N}}|{\mathfrak{N}}_{\mathrm{Lie}},{\mathbb{F}})$ represented by an element in ${\mathrm{CL}}^{n+1}({\mathfrak{N}},{\mathbb{F}})$ with exactly one slot in $e^*$ at the first place: the Leibniz product in this slot gives the only non-zero contribution). The long exact sequence from Proposition \[les\] splits into short exact sequences $$0\to{\mathrm{HL}}^n({\mathfrak{N}},{\mathbb{F}})\to{\mathrm{HL}}^{n-1}({\mathfrak{N}}|{\mathfrak{N}}_{\mathrm{Lie}},{\mathbb{F}})\to{\mathrm{HL}}^{n+1}({\mathfrak{N}}_{\mathrm{Lie}},{\mathbb{F}})\to 0\,,$$ starting from $n=2$, where the right-hand term is one-dimensional. These short exact sequences, together with the spectral sequence where all differentials are zero, permit to determine all relative and absolute cohomology spaces. For example, we obtain $\dim_{\mathbb{F}}{\mathrm{HL}}^2({\mathfrak{N}},{\mathbb{F}})=1$, and then $\dim_{\mathbb{F}}{\mathrm{HL}}^2({\mathfrak{N}}|{\mathfrak{N}}_{\mathrm{Lie}},{\mathbb{F}})=3$, $\dim_{\mathbb{F}}{\mathrm{HL}}^3({\mathfrak{N}},{\mathbb{F}})=2$, and then $\dim_{\mathbb{F}}{\mathrm{HL}}^3({\mathfrak{N}}|{\mathfrak{N}}_{\mathrm{Lie}},{\mathbb{F}})=5$, and so on. In general, we obtain by induction that $\dim_{\mathbb{F}}{\mathrm{HL}}^n({\mathfrak{N}},{\mathbb{F}})=2^{n-2} $ for every integer $n\ge
2$ and $\dim_{\mathbb{F}}{\mathrm{HL}}^n({\mathfrak{N}}|{\mathfrak{N}}_{\mathrm{Lie}},{\mathbb{F}})=2^{n-1}+1$ for every integer $n\ge 1$.
Cohomology of semi-simple Leibniz algebras {#semsim}
==========================================
Recall that a left Leibniz algebra ${\mathfrak{L}}$ is called [*semi-simple*]{} if ${\mathrm{Leib}}({\mathfrak{L}})$ contains every solvable ideal of ${\mathfrak{L}}$ (see [@F Section 7]). In particular, a finite-dimensional left Leibniz algebra ${\mathfrak{L}}$ is semi-simple if, and only if, ${\mathrm{Leib}}({\mathfrak{L}})={\mathrm{Rad}}({\mathfrak{L}})$, where ${\mathrm{Rad}}({\mathfrak{L}})$ denotes the largest solvable ideal of ${\mathfrak{L}}$ (see [@F Proposition 7.4]).
\[leibnizinv\] Let ${\mathfrak{L}}$ be a finite-dimensional semi-simple left Leibniz algebra over a field of characteristic zero. Then $[{\mathrm{Leib}}({\mathfrak{L}})^*]_s^{{\mathfrak{L}}_{\mathrm{Lie}}}=0$, where ${\mathrm{Leib}}({\mathfrak{L}})^*$ is a left ${\mathfrak{L}}$-module, and thus a left ${\mathfrak{L}}_{\mathrm{Lie}}$-module, via $(x\cdot f)(y):=-f(xy)$ for every linear form $f\in{\mathrm{Leib}}({\mathfrak{L}})^*$ and any elements $x,y\in{\mathfrak{L}}$.
It follows from Levi’s theorem for Leibniz algebras (see [@P Proposition 2.4] and [@B2 Theorem 1]) that there exists a semi-simple Lie subalgebra ${\mathfrak{s}}$ of ${\mathfrak{L}}$ such that ${\mathfrak{L}}={\mathfrak{s}}\oplus{\mathrm{Leib}}({\mathfrak{L}})$ (see [@FM Corollary 2.14]). Note that then ${\mathfrak{L}}_{\mathrm{Lie}}\cong{\mathfrak{s}}$. Since ${\mathfrak{s}}$ is a Lie algebra and ${\mathrm{Leib}}({\mathfrak{L}})$ is abelian, we obtain that $(s+x)(s+x)=s\cdot x$ for any elements $s\in{\mathfrak{s}}$ and $x\in{\mathrm{Leib}}({\mathfrak{L}})$. This shows that ${\mathrm{Leib}}({\mathfrak{L}})={\mathfrak{s}}{\mathrm{Leib}}({\mathfrak{L}})$. Now let $\varphi\in[{\mathrm{Leib}}({\mathfrak{L}})^*]_s^{\mathfrak{s}}$ be arbitrary. Since $(s\cdot\varphi)(x)=-\varphi(sx)$ for any $\varphi\in{\mathrm{Leib}}({\mathfrak{L}})^*$, $s\in{\mathfrak{s}}$, and $x\in{\mathrm{Leib}}({\mathfrak{L}})$, we conclude that $\varphi[{\mathrm{Leib}}({\mathfrak{L}})]=\varphi[{\mathfrak{s}}{\mathrm{Leib}}({\mathfrak{L}})]=0$, which proves the assertion.
The first main result in this section is the Leibniz analogue of Whitehead’s vanishing theorem for the Chevalley-Eilenberg cohomology of finite-dimensional semi-simple Lie algebras over a field of characteristic zero (see [@CE Theorem 24.1] or [@HS Theorem 10]). Note that in the special case of a Lie algebra, Theorem \[whitehead\] is an immediate consequence of Whitehead’s classical vanishing theorem and Theorem \[vanlie\].
\[whitehead\] Let ${\mathfrak{L}}$ be a finite-dimensional semi-simple left Leibniz algebra over a field of characteristic zero. If $M$ is a finite-dimensional ${\mathfrak{L}}$-bimodule such that $M^{\mathfrak{L}}=0$, then ${\mathrm{HL}}^n({\mathfrak{L}},M)=0$ for every non-negative integer $n$.
According to Lemma \[sym\], the hypothesis $M^{\mathfrak{L}}=0$ implies that $M$ is symmetric. We can therefore use the spectral sequence of Corollary \[pirashvili\] with ${\mathfrak{I}}:={\mathrm{Leib}}({\mathfrak{L}})$ and ${\mathfrak{Q}}:={\mathfrak{L}}_{\mathrm{Lie}}$. The $E_2$-term reads $$E_2^{p,q}={\mathrm{HL}}^p({\mathfrak{Q}},({\mathfrak{I}}^*)_s)
\otimes{\mathrm{HL}}^q({\mathfrak{L}},M)\,.$$ It follows from [@F Proposition 7.8] and the Ntolo-Pirashvili vanishing theorem for the Leibniz cohomology of a finite-dimensional semi-simple Lie algebra over a field of characteristic zero (see [@N Théorème 2.6] and the sentence after the proof of Lemma 2.2 in [@P]) that ${\mathrm{HL}}^p({\mathfrak{Q}},({\mathfrak{I}}^*)_s)=0$ for every positive integer $p$. Hence the spectral sequence collapses, and we deduce $${\mathrm{HL}}^n({\mathfrak{L}}|{\mathfrak{Q}},M)
=({\mathfrak{I}}^*)_s^{\mathfrak{Q}}\otimes{\mathrm{HL}}^n({\mathfrak{L}},M)\,.$$ By virtue of Lemma \[leibnizinv\], the relative cohomology ${\mathrm{HL}}^n({\mathfrak{L}}\vert{\mathfrak{Q}},M)$ vanishes for every non-negative integer $n$, and thus we obtain from Proposition \[les\] in conjunction with [@F Proposition 4.1] and the Ntolo-Pirashvili vanishing theorem that ${\mathrm{HL}}^n({\mathfrak{L}},M)\cong{\mathrm{HL}}^n({\mathfrak{Q}},M)=0$ for every non-negative integer $n$.
[**Remark.**]{} It is possible to prove Theorem \[whitehead\] without using the Ntolo-Pirashvili vanishing theorem. Namely, the first time the Ntolo-Pirashvili vanishing theorem is used in the above proof, one can instead use Lemma \[leibnizinv\], Whitehead’s classical vanishing theorem, and Theorem \[vanlie\], and the second time, by hypothesis, it is enough to apply just the last two results. As a consequence, the proof of Theorem \[vansemsim\] gives another proof of the Ntolo-Pirashvili vanishing theorem.
Next, we generalize the Ntolo-Pirashvili vanishing theorem from Lie algebras to arbitrary Leibniz algebras. The main tools in the proof are Theorem \[whitehead\], Lemma \[antisym\], and Corollary \[coadj\], where the first result and its use in this proof seems to be new.
\[vansemsim\] Let ${\mathfrak{L}}$ be a finite-dimensional semi-simple left Leibniz algebra over a field of characteristic zero, and let $M$ be a finite-dimensional ${\mathfrak{L}}$-bimodule. Then ${\mathrm{HL}}^n({\mathfrak{L}},M)=0$ for every integer $n\ge 2$, and there is a five-term exact sequence $$0\to M_0\to{\mathrm{HL}}^0({\mathfrak{L}},M)
\to M_{\mathrm{sym}}^{{\mathfrak{L}}_{\mathrm{Lie}}}\to{\mathrm{Hom}}_{\mathfrak{L}}({\mathfrak{L}}_{{\mathrm{ad}},\ell},M_0)\to{\mathrm{HL}}^1({\mathfrak{L}},M)\to 0\,.$$ Moreover, if $M$ is symmetric, then ${\mathrm{HL}}^n({\mathfrak{L}},M)=0$ for every integer $n\ge 1$.
The proof is divided into three steps. First, we will prove the assertion for symmetric ${\mathfrak{L}}$-bimodules. So suppose that $M$ is symmetric. Since $M$ is finite-dimensional, it has a composition series. It is clear that sub-bimodules and homomorphic images of a symmetric bimodule are again symmetric. By using the long exact cohomology sequence, it is therefore enough to prove the second part of the theorem for finite-dimensional irreducible symmetric ${\mathfrak{L}}$-bimodules. So suppose now in addition that $M$ is irreducible and non-trivial. Then we obtain from Corollary \[irr\] that $M^{\mathfrak{L}}=0$, and thus Theorem \[whitehead\] yields that ${\mathrm{HL}}^n({\mathfrak{L}},M)=0$ for every non-negative integer $n$. Finally, suppose that $M={\mathbb{F}}$ is the trivial irreducible ${\mathfrak{L}}$-bimodule. In this case it follows from Corollary \[coadj\] that ${\mathrm{HL}}^n({\mathfrak{L}},{\mathbb{F}})\cong{\mathrm{HL}}^{n-1}({\mathfrak{L}},({\mathfrak{L}}^*)_s)$ for every integer $n\ge 1$. Since ${\mathfrak{L}}_{\mathrm{Lie}}$ is perfect, we obtain from Corollary \[coadj\] that $$({\mathfrak{L}}^*)_s^{\mathfrak{L}}\cong{\mathrm{HL}}^0({\mathfrak{L}},({\mathfrak{L}}^*)_s)\cong{\mathrm{HL}}^1({\mathfrak{L}},{\mathbb{F}})\cong{\mathrm{H}}^1({\mathfrak{L}}_{\mathrm{Lie}},
{\mathbb{F}})=0\,.$$ Therefore another application of Theorem \[whitehead\] yields that $${\mathrm{HL}}^n({\mathfrak{L}},{\mathbb{F}})\cong{\mathrm{HL}}^{n-1}({\mathfrak{L}},({\mathfrak{L}}^*)_s)=0$$ for every integer $n\ge 1$. This finishes the proof for symmetric ${\mathfrak{L}}$-bimodules.
If $M$ is anti-symmetric, then we obtain the assertion from Lemma \[antisym\](b) and the statement for symmetric bimodules. Finally, if $M$ is arbitrary, then in the short exact sequence $0\to M_0\to M\to M_{\mathrm{sym}}\to 0$ the first term is anti-symmetric and the third term is symmetric. Hence another application of the long exact cohomology sequence in conjunction with the statement for the anti-symmetric and the symmetric case yields ${\mathrm{HL}}^n({\mathfrak{L}},M)=0$ for every integer $n\ge 2$. Now we deduce the five-term exact sequence from the long exact cohomology sequence together with [@F Corollary 4.2], [@F Corollary 4.4(b)], and the symmetric case.
Note that Theorem \[vansemsim\] contains [@F Theorem 7.15] as the special case $n=1$ and the second Whitehead lemma for Leibniz algebras as the special case $n=2$. But contrary to Chevalley-Eilenberg cohomology, Leibniz cohomology vanishes in any degree $n\ge 2$.
The following example shows that the Ntolo-Pirashvili vanishing theorem (and therefore also Theorem \[vansemsim\]) does not hold over fields of prime characteristic.
[**Example E.**]{} Let ${\mathfrak{g}}:={\mathfrak{sl}}_2({\mathbb{F}})$ be the three-dimensional simple Lie algebra of traceless $2\times 2$ matrices over a field ${\mathbb{F}}$ of characteristic $p>2$. Moreover, let ${\mathbb{F}}_p$ denote the field with $p$ elements, and let $L(n)$ ($n\in{\mathbb{F}}_p$) denote the irreducible restricted ${\mathfrak{g}}$-module of heighest weight $n$. (If the ground field ${\mathbb{F}}$ is algebraically closed, these modules represent all isomorphism classes of restricted irreducible ${\mathfrak{g}}$-modules.) It is well known (see [@Dz Theorem 4]) that ${\mathrm{H}}^1({\mathfrak{g}},L(p-2))\cong{\mathbb{F}}^2\cong{\mathrm{H}}^2({\mathfrak{g}},L(p-2))$. (Note that by virtue of [@Dz Theorem 2], ${\mathrm{H}}^\bullet({\mathfrak{g}},M)=0$ for every non-restricted irreducible ${\mathfrak{g}}$-module. In fact, $L(p-2)$ is the only irreducible ${\mathfrak{g}}$-module $M$ such that ${\mathrm{H}}^1({\mathfrak{g}},M)\ne 0$ or ${\mathrm{H}}^2({\mathfrak{g}},M)\ne 0$.)
We obtain from Proposition \[lesrelcoh\] that $${\mathrm{HL}}^1({\mathfrak{g}},L(p-2)_s)\cong{\mathrm{H}}^1({\mathfrak{g}},L(p-2))
\cong{\mathbb{F}}^2\ne 0$$ and $$0\ne{\mathbb{F}}^2\cong{\mathrm{H}}^2({\mathfrak{g}},L(p-2))\hookrightarrow{\mathrm{HL}}^2({\mathfrak{g}},L(p-2)_s)\,.$$ In particular, this shows that the Ntolo-Pirashvili vanishing theorem (and therefore also Theorem \[vansemsim\]) is not true over fields of prime characteristic.
[**Remark.**]{} By using more sophisticated tools one can also say something about the Leibniz cohomology of anti-symmetric irreducible ${\mathfrak{g}}$-bimodules, where again ${\mathfrak{g}}:=
{\mathfrak{sl}}_2({\mathbb{F}})$. We obtain from Lemma \[antisym\](b) that $${\mathrm{HL}}^1({\mathfrak{g}},L(n)_a)\cong{\mathrm{HL}}^0
({\mathfrak{g}},{\mathrm{Hom}}_{\mathbb{F}}({\mathfrak{g}},L(n))_s)\cong{\mathrm{Hom}}_{\mathbb{F}}({\mathfrak{g}},L(n))^{\mathfrak{g}}$$ and $${\mathrm{HL}}^2({\mathfrak{g}},L(n)_a)\cong{\mathrm{HL}}^1
({\mathfrak{g}},{\mathrm{Hom}}_{\mathbb{F}}({\mathfrak{g}},L(n))_s)\cong{\mathrm{H}}^1({\mathfrak{g}},{\mathrm{Hom}}_{\mathbb{F}}({\mathfrak{g}},L(n)))\,.$$ Since ${\mathfrak{g}}\cong L(2)$ is a self-dual ${\mathfrak{g}}$-module, we have the following isomorphisms of ${\mathfrak{g}}$-modules: $${\mathrm{Hom}}_{\mathbb{F}}({\mathfrak{g}},L(n))\cong L(2)\otimes L(n)\,.$$ Let us first consider the case $p>3$. Then we obtain from the modular Clebsch-Gordan rule (see [@BO Theorem 1.11(a)] or Satz a) in Chapter 5 of [@Fi]) that $$L(2)\otimes L(2)\cong L(4)\oplus L(2)\oplus L(0)$$ and $$\begin{aligned}
L(2)\otimes L(p-4)\cong
\left\{
\begin{array}{cl}
L(3)\oplus L(1) & \mbox{\rm if }p=5\\
L(p-2)\oplus L(p-4)\oplus L(p-6) & \mbox{\rm if }p\ge 7\,.
\end{array}
\right.\end{aligned}$$ Hence we conclude for $p>3$ that $${\mathrm{HL}}^1({\mathfrak{g}},L(2)_a)\cong(L(2)\otimes L(2))^{\mathfrak{g}}\cong L(0)^{\mathfrak{g}}\cong{\mathbb{F}}\ne 0$$ and $${\mathrm{HL}}^2({\mathfrak{g}},L(p-4)_a)\cong{\mathrm{H}}^1({\mathfrak{g}},L(2)\otimes L(p-4))\cong{\mathrm{H}}^1({\mathfrak{g}},
L(p-2))\cong{\mathbb{F}}^2\ne 0\,.$$
Let us now consider $p=3$. Note that in this case $L(2)$ is the [*Steinberg module*]{}, i.e., $L(2)$ is the unique projective irreducible restricted ${\mathfrak{g}}$-module. This implies that $L(2)\otimes
L(n)$ is also projective for every highest weight $n\in{\mathbb{F}}_3$. Then we obtain from the modular Clebsch-Gordan rule (cf. [@BO Theorem 1.11(b) and (c)] or Satz b) and c) in Chapter 5 of [@Fi]) for $p=3$ that $$\begin{aligned}
L(2)\otimes L(n)\cong
\left\{
\begin{array}{cl}
L(2) & \mbox{\rm if }n\equiv 0\,\,(\mathrm{mod}\,3)\\
P(1) & \mbox{\rm if }n\equiv 1\,\,(\mathrm{mod}\,3)\,,\\
P(0)\oplus L(2) & \mbox{\rm if }n\equiv 2\,\,(\mathrm{mod}\,3)
\end{array}
\right.\end{aligned}$$ where $P(n)$ denotes the projective cover (and at the same time also the injective hull) of $L(n)$. As a consequence, we have that $$\begin{aligned}
(L(2)\otimes L(n))^{\mathfrak{g}}\cong
\left\{
\begin{array}{cl}
{\mathbb{F}}& \mbox{\rm if }n\equiv 2\,\,(\mathrm{mod}\,3)\\
0 & \mbox{\rm if }n\not\equiv 2\,\,(\mathrm{mod}\,3)\,.
\end{array}
\right.\end{aligned}$$ Therefore, we obtain that $${\mathrm{HL}}^1({\mathfrak{g}},L(2)_a)\cong(L(2)\otimes L(2))^{\mathfrak{g}}\cong P(0)^{\mathfrak{g}}\cong{\mathbb{F}}\ne 0\,.$$ Moreover, by using the six-tem exact sequence relating Hochschild’s cohomology of a restricted Lie algebra to its Chevalley-Eilenberg cohomology (see [@Ho p.575]), we also conclude that $${\mathrm{HL}}^2({\mathfrak{g}},L(2)_a)\cong{\mathrm{H}}^1({\mathfrak{g}},L(2)\otimes L(2))
\cong{\mathbb{F}}^3\ne 0\,.$$
The next example shows that the Ntolo-Pirashvili vanishing theorem (and therefore also Theorem \[vansemsim\]) does not hold for infinite-dimensional modules.
[**Example F.**]{} Let ${\mathfrak{g}}:={\mathfrak{sl}}_2({\mathbb{C}})$ be the three-dimensional simple complex Lie algebra of traceless $2\times 2$ matrices, and let $V(\lambda)$ ($\lambda\in{\mathbb{C}}$) denote the [*Verma module*]{} of highest weight $\lambda$. (Here we identify every complex multiple of the unique fundamental weight with its coefficient.) Verma modules are infinite-dimensional indecomposable ${\mathfrak{g}}$-modules (see, for example, [@H Theorem 20.2(e)]). Furthermore, it is well known (see [@H Exercise 7(c) in Section 7.2]) that $V(\lambda)$ is irreducible if, and only if, $\lambda$ is not a dominant integral weight (i.e., with our identification, $\lambda$ is not a non-negative integer). Moreover, it follows from [@W Theorem 4.19] that $$\begin{aligned}
{\mathrm{H}}^n({\mathfrak{g}},V(\lambda))\cong
\left\{
\begin{array}{cl}
{\mathbb{C}}& \mbox{\rm if }\lambda=-2\mbox{ and }n=1,2\\
0 & \hspace{1cm}\mbox{\rm otherwise}\,.
\end{array}
\right.\end{aligned}$$ This in conjunction with Proposition \[lesrelcoh\] yields that $${\mathrm{HL}}^1({\mathfrak{g}},V(-2)_s)\cong
{\mathrm{H}}^1({\mathfrak{g}},V(-2))\cong{\mathbb{C}}\ne 0$$ and $$0\ne{\mathbb{C}}\cong{\mathrm{H}}^2({\mathfrak{g}},V(-2))\hookrightarrow
{\mathrm{HL}}^2({\mathfrak{g}},V(-2)_s)\,.$$ In particular, the Ntolo-Pirashvili vanishing theorem (and therefore also Theorem \[vansemsim\]) is not true for infinite-dimensional modules.
We obtain as an immediate consequence of Theorem \[vansemsim\] the following generalization of [@F Corollary 7.9].
\[triv\] If ${\mathfrak{L}}$ is a finite-dimensional semi-simple left Leibniz algebra over a field of characteristic zero, then ${\mathrm{HL}}^n({\mathfrak{L}},{\mathbb{F}})=0$ for every integer $n\ge 1$.
[**Remark.**]{} It is well known that the analogue of Corollary \[triv\] does not hold for the Chevalley-Eilenberg cohomology of Lie algebras as ${\mathrm{H}}^3({\mathfrak{g}},{\mathbb{F}})\ne 0$ for any finite-dimensional semi-simple Lie algebra ${\mathfrak{g}}$ over a field ${\mathbb{F}}$ of characteristic zero (see [@CE Theorem 21.1]).
Next, we apply Theorem \[vansemsim\] to compute the cohomology of a finite-dimensional semi-simple left Leibniz algebra over a field of characteristic zero with coefficients in its adjoint bimodule and in its (anti-)symmetric counterparts.
\[adj\] For every finite-dimensional semi-simple left Leibniz algebra ${\mathfrak{L}}$ over a field of characteristic zero the following statements hold:
1. $$\begin{aligned}
{\mathrm{HL}}^n({\mathfrak{L}},{\mathfrak{L}}_s)=
\left\{
\begin{array}{cl}
{\mathrm{Leib}}({\mathfrak{L}}) & \mbox{\rm if }n=0\\
0 & \mbox{\rm if }n\ge 1\,.
\end{array}
\right.\end{aligned}$$
2. $$\begin{aligned}
{\mathrm{HL}}^n({\mathfrak{L}},{\mathfrak{L}}_a)=
\left\{
\begin{array}{cl}
{\mathfrak{L}}& \mbox{\rm if }n=0\\
{\mathrm{End}}_{\mathfrak{L}}({\mathfrak{L}}_{{\mathrm{ad}},\ell}) & \mbox{\rm if }n=1\,,\\
0 & \mbox{\rm if }n\ge 2
\end{array}
\right.\end{aligned}$$
where ${\mathrm{End}}_{\mathfrak{L}}({\mathfrak{L}}_{{\mathrm{ad}},\ell})$ denotes the vector space of endomorphisms of the left adjoint ${\mathfrak{L}}$-module ${\mathfrak{L}}_{{\mathrm{ad}},\ell}$.
3. $$\begin{aligned}
{\mathrm{HL}}^n({\mathfrak{L}},{\mathfrak{L}}_{\mathrm{ad}})=
\left\{
\begin{array}{cl}
{\mathrm{Leib}}({\mathfrak{L}}) & \mbox{\rm if }n=0\\
{\mathrm{Hom}}_{\mathfrak{L}}({\mathfrak{L}}_{{\mathrm{ad}},\ell},{\mathrm{Leib}}({\mathfrak{L}})) & \mbox{\rm if }n=1\,,\\
0 & \mbox{\rm if }n\ge 2
\end{array}
\right.\end{aligned}$$
where ${\mathrm{Hom}}_{\mathfrak{L}}({\mathfrak{L}}_{{\mathrm{ad}},\ell},{\mathrm{Leib}}({\mathfrak{L}}))$ denotes the vector space of homomorphisms from the left adjoint ${\mathfrak{L}}$-module ${\mathfrak{L}}_{{\mathrm{ad}},\ell}$ to the Leibniz kernel ${\mathrm{Leib}}({\mathfrak{L}})$ considered as a left ${\mathfrak{L}}$-module.
(a): According to [@F Proposition 4.1] and [@F Proposition 7.5] we have that that ${\mathrm{HL}}^0({\mathfrak{L}},{\mathfrak{L}}_s)=({\mathfrak{L}}_s)^{\mathfrak{L}}=C_\ell({\mathfrak{L}})={\mathrm{Leib}}({\mathfrak{L}})$. Moreover, we obtain the statement for degree $n\ge 1$ from the second part of Theorem \[vansemsim\].
(b): It follows from [@F Corollary 4.2(b)] that ${\mathrm{HL}}^0({\mathfrak{L}},{\mathfrak{L}}_a)={\mathfrak{L}}$, and it follows from [@F Corollary 4.4(b)] that ${\mathrm{HL}}^1({\mathfrak{L}},{\mathfrak{L}}_a)={\mathrm{End}}_{\mathfrak{L}}({\mathfrak{L}}_{{\mathrm{ad}},\ell})$. The remainder of the assertion is an immediate consequence of the first part of Theorem \[vansemsim\].
(c): As for the symmetric adjoint bimodule, we obtain from [@F Proposition 4.1] and [@F Proposition 7.5] that ${\mathrm{HL}}^0({\mathfrak{L}},{\mathfrak{L}}_{\mathrm{ad}})=({\mathfrak{L}}_{\mathrm{ad}})^{\mathfrak{L}}=C_\ell({\mathfrak{L}})={\mathrm{Leib}}({\mathfrak{L}})$. Next, by applying the five-term exact sequence of Theorem \[vansemsim\] to the adjoint ${\mathfrak{L}}$-bimodule $M:={\mathfrak{L}}_{\mathrm{ad}}$, we deduce that $${\mathrm{HL}}^1({\mathfrak{L}},{\mathfrak{L}}_{\mathrm{ad}})\cong{\mathrm{Hom}}_{\mathfrak{L}}({\mathfrak{L}}_{{\mathrm{ad}},
\ell},{\mathrm{Leib}}({\mathfrak{L}}))\,,$$ as the third term is ${\mathfrak{L}}_{\mathrm{Lie}}^{{\mathfrak{L}}_{\mathrm{Lie}}}=C({\mathfrak{L}}_{\mathrm{Lie}})=0$. Finally, the assertion for degree $n\ge 2$ is again an immediate consequence of the first part of Theorem \[vansemsim\].
[**Remark.**]{} Note that the vanishing part of Theorem \[adj\](c) confirms a generalization of the conjecture at the end of [@ALO]. Moreover, parts (a) and (b) of Theorem \[adj\] show that the statements in Theorem \[vansemsim\] are best possible.
In particular, one can derive from Theorem \[adj\](c) that finite-dimensional semi-simple non-Lie Leibniz algebras over a field of characteristic zero have outer derivations. In this respect non-Lie Leibniz algebras behave differently than Lie algebras (see, for example, [@H Theorem 5.3]).
\[outder\] Every finite-dimensional semi-simple non-Lie Leibniz algebra over a field of characteristic zero has derivations that are not inner.
If one applies the contravariant functor ${\mathrm{Hom}}_{\mathbb{F}}(-,{\mathrm{Leib}}({\mathfrak{L}}))$ to the short exact sequence $$0\to{\mathrm{Leib}}({\mathfrak{L}})\to{\mathfrak{L}}_{\mathrm{ad}}\to{\mathfrak{L}}_{\mathrm{Lie}}\to 0$$ considered as a short exact sequence of left ${\mathfrak{L}}$-modules, one obtains the short exact sequence $$0\to{\mathrm{Hom}}_{\mathbb{F}}({\mathfrak{L}}_{\mathrm{Lie}},{\mathrm{Leib}}({\mathfrak{L}}))\to
{\mathrm{Hom}}_{\mathbb{F}}({\mathfrak{L}}_{{\mathrm{ad}},\ell},{\mathrm{Leib}}({\mathfrak{L}}))\to{\mathrm{Hom}}_{\mathbb{F}}({\mathrm{Leib}}({\mathfrak{L}}),{\mathrm{Leib}}({\mathfrak{L}}))\to 0$$ of left ${\mathfrak{L}}$-modules. Then the long exact cohomology sequence in conjunction with Lemma \[antisym\](a) yields the long exact sequence $$\begin{aligned}
0 & \to & {\mathrm{Hom}}_{\mathfrak{L}}({\mathfrak{L}}_{\mathrm{Lie}},{\mathrm{Leib}}({\mathfrak{L}}))\to{\mathrm{Hom}}_{\mathfrak{L}}({\mathfrak{L}}_{{\mathrm{ad}},\ell},{\mathrm{Leib}}({\mathfrak{L}}))\to{\mathrm{Hom}}_{\mathfrak{L}}({\mathrm{Leib}}({\mathfrak{L}}),{\mathrm{Leib}}({\mathfrak{L}}))\\
& \to & {\mathrm{\widetilde{HL}}}^1({\mathfrak{L}},{\mathrm{Hom}}_{\mathbb{F}}({\mathfrak{L}}_{\mathrm{Lie}},{\mathrm{Leib}}({\mathfrak{L}})))={\mathrm{HL}}^1({\mathfrak{L}},{\mathrm{Hom}}_{\mathbb{F}}({\mathfrak{L}}_{\mathrm{Lie}},{\mathrm{Leib}}({\mathfrak{L}}))_s)\,.\end{aligned}$$ According to the second part of Theorem \[vansemsim\], the fourth term is zero. Since the third term contains the identity map, it is non-zero as by hypothesis ${\mathfrak{L}}$ is a not a Lie algebra. Hence in this case the second term is non-zero, and we obtain from Theorem \[adj\](c) that ${\mathrm{HL}}^1({\mathfrak{L}},
{\mathfrak{L}}_{\mathrm{ad}})\cong{\mathrm{Hom}}_{\mathfrak{L}}({\mathfrak{L}}_{{\mathrm{ad}},\ell},{\mathrm{Leib}}({\mathfrak{L}}))\ne 0$.
In the same way as at the end of Section \[cel\] for the infinite-dimensional two-sided Witt algebra, by using [@Ba Théorème 3] in conjunction with Theorem \[adj\](c), one obtains the rigidity of any finite-dimensional semi-simple Lie algebra as a Leibniz algebra.
\[rigid\] Every finite-dimensional semi-simple left Leibniz algebra over an algebraically closed field of characteristic zero is rigid as a Leibniz algebra.
[**Acknowledgments.**]{} Most of this paper was written during a sabbatical leave of the first author in the Fall semester 2018. He is very grateful to the University of South Alabama for giving him this opportunity.
The first author would also like to thank the Laboratoire de mathématiques Jean Leray at the Université de Nantes for the hospitality and the financial support in the framework of the program DéfiMaths during his visit in August and September 2018. Moreover, the first author wishes to thank Henning Krause and the BIREP group at Bielefeld University for the hospitality and the financial support during his visit in October and November 2018 when large portions of the paper were written.
Finally, both authors thank Bakhrom Omirov for useful discussions. We also thank Teimuraz Pirashvili for spotting a mistake in a previous version of the manuscript and Geoffrey Powell for his help in understanding it.
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|
---
address: |
Arnold Sommerfeld Institute for Mathematical Physics, Technical University Clausthal, Leibnizstr. 10, D-38678 Clausthal, GERMANY\
E-mail: [email protected]
author:
- 'W. Lücke'
title: '.-25mm Nonlocality in Nonlinear Quantum Mechanics[^1]'
---
Nonlinear Modifications of Quantum Mechanics
============================================
Many authors[^2] considered nonlinear Schrödinger equations of the form $$\label{NLSE}
i\hbar\frac{\partial}{\partial t} \Psi_t(\xv) =
\left(-\frac{\hbar^2}{2m}\Delta+V(\xv,t)\right) \Psi_t(\xv) +
R[\Psi_t](\xv) \Psi_t(\xv)\,,$$ where $F[\Psi] = R[\Psi] \Psi$ is a local[^3] nonlinear Functional of $\Psi\,$. The essential point is that (\[NLSE\]) is not interpreted as an equation for some field operator but as a classical evolution equation for the quantum mechanical wave function: $$\label{StInt}
\int_{{\cal B}} \modulus{\Psi_t(\xv)}^2\,{\rm d}\xv =
\parbox{6cm}{probability for location within ${\cal {\mbox{$\cal B$}}}$}$$ (at time $t$ for normalized $\Psi_t$). Special cases were even tested experimentally.[@tests]
All these efforts seemed useless according to N. Gisin’s claim[@gis1; @gis2]: $$\label{gis-cl}
\mbox{``All deterministic nonlinear Schr\"odinger equations are irrelevant.''}$$ By this Gisin meant the following:
> Consider a Bell-like situation as sketched in Figure \[fig:gisin\].
>
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>
> Then, if the source produces entangled 2-particle states, there is always a physical observable for the particle sent to the right,[^4] the probability distribution of which is instantaneously (substantially) changed by suitable measurements (involving only low energy transfer) on the other particle ‘behind the moon’.
Actually, Gisin[@gis1] assumed the following:[^5]
> - If the observable $A$ resp. $B$ of the particle ‘behind the moon’ is measured at time $t=0$ the partial state of the other particle at times $t\geq0$ is given by a density matrix $\rho_{\rm right}(t)$ of the form[^6] $$\sum_{\alpha} x_\alpha\, P_\Psi(\alpha,t)\quad
> \mbox{resp.}\quad \sum_{\beta} x_\beta\,
> P_\Psi(\beta,t)\,,$$ where the $P_\Psi(\alpha,t)$ resp. $P_\Psi(\beta,t)$ are pure states evolving according to the corresponding 1-particle equation.
>
> - [**All**]{} self-adjoint operators correspond to observables.
>
That Gisin’s claim (\[gis-cl\]) is wrong since Gisin’s assumption (G1), the projection postulate, is unjustified in nonlinear quantum mechanics has already been pointed out by Polchinski[@pol], who determined conditions which are sufficient for the absence of essential nonlocality.[^7]
Accepting Gisin’s assumption (G2), Polchinski concluded that his conditions – violated for prominent examples of nonlinear Schrödinger equations – are also necessary to avoid essential nonlocality. However, as explained already in [@lu1], also assumption (G2) is unjustified in nonlinear quantum mechanics and definitely wrong for the situation reconsidered in the next section. This is why valid proofs of essential nonlocality have to be more involved.
What we can Learn from Nonlinear Gauge Transformations {#section:CE}
======================================================
Consider the well-defined special case $$\left(\Op N_{\! D}(\Psi)\right)(\xv) \stackrel{\rm def}{=}
e^{i\frac{2mD}{\hbar}\ln\modulus{\Psi(\xv)}}\Psi(\xv)\;,\quad D\in\bbbr\,.$$ of the ‘nonlinear gauge transformations’ exploited by H.-D. Doebner et al.[@nlgt] If $\Psi_t'(\xv)$ is a solution of (\[NLSE\]) for $R=0$ then straightforward calculation shows that $$\Psi_t(\xv) = \left(\Op N_{\! D}(\Psi'_t)\right)(\xv)$$ is a solution of (\[NLSE\]) for $$\label{GDGE}
R[\Psi_t] = \hbar D \Bigl(\frac{i}{2}
\frac{\Delta\rho_t}{\rho_t}\Psi_t + c_1 \frac{\Nv\cdot \vec
{J}_t}{\rho_t} + c_2 \frac{\Delta\rho_t}{\rho_t} + c_3
\frac{\vec{J}_t}{\rho_t}+ +c_4 \frac{\vec{J}_t\cdot
\Nv\rho_t}{(\rho_t)^2} + c_5 \frac{(\Nv\rho_t)^2}{(\rho_t)^2}\Bigr)$$ with[^8] $$\label{el}
c_1=1\;, \quad c_2=-2c_5=-mD/\hbar\;,\quad c_3=0\;,\quad c_4=-1\,,$$ where $$\rho_t\stackrel{\rm def}{=}\modulus{\Psi_t}^2\,,\; \vec{J}_t
\stackrel{\rm def}{=} \frac{1}{2i} \left( \overline{\Psi_t} \Nv \Psi_t
- \Psi_t \Nv \overline{\Psi_t} \right)\,.$$ This way we get a deterministic nonlinear Schrödinger equation which, interpreted by (\[StInt\]), describes the same physics as the corresponding ($R=0$) linear Schrödinger equation, since $$\modulus{\Psi_t(\xv)} = \modulus{\Psi'_t(\xv)}\,.$$ That, contrary to Gisin’s as well as Polchinski’s claim, there is no real problem with locality is no surprise since now nonlinear projection operators $$\Op E = \Op N_D \circ \Op E' \circ \Op N_D^{-1}\,,$$ instead of linear projectors $\Op E'\,$, have to be used[@lu1] to get the correct probabilities $$\norm{\Op E(\Psi_t)}^2 = \norm{\Op E'(\Psi'_t)}^2 = \left\langle
\Psi'_t \mid \Op E'\,\Psi'_t \right\rangle\,.$$ Hence assumption (G2) is obviously wrong in nonlinear quantum mechanics. Moreover, w.r.t. the nonlinear $\Op E\,$, density matrices are inadequate for the description of classical mixtures: $$\sum_{\alpha} x_\alpha \norm{\Op E(\Psi_\alpha)}^2 = {\rm
trace}\left(\Op E'\, \sum_\alpha x_\alpha \Op P_{\Psi'_\alpha}\right)
\stackrel{\rm i.g.}{\ne} {\rm trace}\left(\Op E\, \sum_\alpha x_\alpha
\Op P_{\Psi_\alpha}\right)\,.$$ Therefore also assumption (G1) turns out to be quite inadequate for nonlinear modifications of quantum mechanics.
The simple example (\[NLSE\])/(\[GDGE\])/(\[el\]) tells us that essential nonlocality should to be checked by using nothing else than the evolution equation together with its basic interpretation (\[StInt\]).
The Doebner-Goldin Equation Interpreted as 2-Particle Equation {#section:dg}
==============================================================
Let us interpret (\[NLSE\]) as a two-particle equation: $$\xv=(\xv_1,\xv_2)\;,\quad \xv_j = \mbox{position of particle } j\,.$$ Moreover assume that the potential is of the form $$V(\xv) = V_2(\xv_2-\xv_0)\;,\quad \xv_0 \mbox{ fixed}\,,$$ and that $\Psi_t(\xv)$ is the solution of (\[NLSE\]) fulfilling the initial condition $$\Psi_0(\xv_1,\xv_2) = f(\xv_1,\xv_2-\xv_0)\,.$$ Then, obviously, we have an unacceptable nonlocality, if the position probability density[^9] $$\label{pp1}
\rho_1(\xv_1,t) \stackrel{\rm def}{=}
\int \modulus{\Psi_t(\xv_1,\xv_2)}^2{\rm d}\xv_2$$ for particle 1 depends on $V_2\,$.
That the latter happens for certain cases of the general Doebner-Goldin equation, if interpreted as 2-particle equation, was first proved by Werner.[@werner1] Inspired by E. Nelson[@nelson] he considered pairs of 1-dimensional particles, i.e. $\xv=(x_1,x_2)\in\bbbr^2\,$, with $$\label{osz}
V_2(x_2) = \lambda\,(x_2)^2\;,\quad \lambda\in\bbbr\,,$$ and entangled initial conditions of Gaussian type. The corresponding solutions are of the form $$\Psi_t(x_1,x_2)= e^{\gamma(t)
-\sum_{j,k=1}^2 C_{jk}(t)x_jx_k/2}\;,\quad C_{jk}=C_{kj}\,,$$ where the $C_{jk}(t)$ fulfill a simple system of first order ordinary differential equations that can be used to determine their time derivatives at $t=0$ and thus $$\label{test1}
\left({\partial_t}^n\int (x_1)^2 \rho_1(x_1,t)\,{\rm
d}x_1\right)_{|_{t=0}}$$ for arbitrary $n\in\bbbz_+\,$. Werner found that (\[test1\]) depends on $\lambda$ for $n=3$ and suitable $C_{jk}(0)$ unless $$\label{gi}
c_3=c_1+c_4=0\,.$$ In principle, using (\[NLSE\]) directly,[@luna] one may calculate $$\label{test2}
\partial_\lambda \left(\partial_t^n \int\modulus{\Psi_t(x,y)}^2\,{\rm
d}y\right)_{|_{t=0}}$$ as a functional of $\Psi_0$ and $V$ for given $R\,$. This shows that (\[test1\]) varies also with strictly localized changes of $V_2\,$. However, for $n>3$ the calculation becomes too involved and could not even managed by use of computer algebra. On the other hand, Werner’s Ansatz turned out to be too special to uncover essential nonlocality of the Galilei covariant cases of the Doebner-Goldin 2-particle equation.[^10] This is why $$\label{test3}
\left({\partial_t}^n\int x_1 \rho_1(x_1,t)\,{\rm
d}x_1\right)_{|_{t=0}}$$ was checked for $n=4$ in [@lu2] showing that the Doebner-Goldin equation is essentially nonlocal, when interpreted as a 2-particle equation, unless the parameters $c_\nu$ are chosen such that (\[NLSE\]) is (formally) linearizable by some nonlinear gauge transformation.
Inconsistency of Bialynicki-Birula’s and Mycielski’s Theory {#section:bbm}
===========================================================
For the nonlinear Schrödinger equation of Bialynicki-Birula and Mycielski, given by[^11] $$\label{bbm}
\left(R[\Psi]\right)(\xv) = -2b\,\ln\modulus{\Psi(\xv)}\;,\quad b\in\bbbr\,,$$ testing (\[test3\]) is of no use, since the Ehrenfest relations hold.[@bbm] In such cases one should check $$\label{test4}
S_{k,n}[\Psi_0,V_2] \stackrel{\rm def}{=} \partial_\lambda
\left(\partial_t^n \int e^{ikx}\,\modulus{\Psi_t(x,y)}^2 {\rm d}x\,{\rm
d}y\right)_{|_{t=0}}$$ for $k\in\bbbr$ and $n\in\bbbz_+\,$. In principle this is equivalent to testing (\[test2\]) but has the advantage of testing (\[test3\]): The integral over $x$ allows for partial integrations which simplify the resulting expressions considerably.
For simplicity, let $\frac{\hbar^2}{2m}=1$ and assume $R$ to be real valued,[^12] as in (\[bbm\]): $$\label{rR}
\left(R[\Psi]\right)(\xv) = \overline{\left(R[\Psi]\right)(\xv)}\,.$$ Then, defining $$\begin{array}[c]{rcl}
T_{k,\nu}(t) &\stackrel{\rm def}{=}&\displaystyle \int
e^{ikx}\,\overline{\Psi_t}\,\partial_x^\nu \Psi_t\,{\rm d}x\,{\rm d}y\,,\\
\displaystyle D_{k,\mu,\nu}(t) &\stackrel{\rm def}{=}&\displaystyle
\int e^{ikx}\,\left(\partial_x^\mu R[\Psi_t]\right)\overline{\Psi_t}\,
\partial_x^\nu \Psi_t\,{\rm d}x\,{\rm d}y\,,
\end{array}$$ we get from (\[NLSE\]) by partial integration $$i\frac{{\rm d}}{{\rm d} t} T_{k,\nu}= -k^2\, T_{k,\nu} + 2ik\, T_{k,\nu+1}
+ \sum_{\mu=1}^{\nu}{\nu\choose \mu} D_{k,\mu,\nu-\mu}$$ for $\xv=(x_1,x_2)\in\bbbr^2$ and $V(\xv)=V_2(x_2)\,$. Iteration of this gives $$\begin{array}[c]{rcl}
\displaystyle \left(i \frac{{\rm d}}{{\rm d} t}\right)^3 T_{k,0}
&=& -k^6\, T_{k,0} + 6ik^5\, T_{k,1} + 12k^4 T_{k,2} - 8 ik^3 T_{k,3}\\
&& -\left(4ik^3\,D_{k,1,0} +8k^2 D_{k,1,1} + 4k^2 D_{k,2,0} +
2k\partial_t D_{k,1,0}\right)\,.
\end{array}$$ Hence, e.g., $$\begin{array}[c]{rcl}
\displaystyle \left(\partial_t^{n+3}\!\int
x^2\modulus{\Psi_t(x,y)}^2{\rm d}x\,{\rm d}y \right)_{|_{t=0}}
&=&\displaystyle -\left(\partial_t^{n+3}\,(\partial_k)^2
T_{k,0}(t)\right)_{|_{t=k=0}}\\[4mm]
&=& 8i\,\partial_t^n\left(2\,D_{0,1,1} + D_{0,2,0}\right).
\end{array}$$ For the special case $$m=1\;,\quad V_2(y) = \lambda\,y^2\;,\quad \Psi_0(x,y) = \frac{e^{-x^2 -y^2
-xy}}{\int e^{-2x^2 -2y^2 -2xy}\, {\rm d}x\,{\rm d}y}\,,$$ running a simple computer algebra program (see appendix) shows that (\[NLSE\]) and (\[bbm\]) imply[^13] $$\partial_\lambda\left((i\partial_t)^3\left(2\,D_{0,1,1}(t) +
D_{0,2,0}(t)\right)\right)_{|_{t=0}} = 32b\,.$$ This means that (\[NLSE\])/(\[bbm\]) is essential nonlocal — against the basic philosophy of Bialynicki-Birula’s and J. Mycielski’s theory.[@bbm]
Identical Particles
===================
Up to now we tacitly assumed that the two particles (with equal masses) considered in Figure \[fig:gisin\] can be distinguished. Therefore one might still hope that Bialynicki-Birula’s and J. Mycielski’s theory is consistent for identical particles. However, even for 2-particles states which are symmetric or antisymmetric w.r.t. exchange of the particles essential nonlocality is unavoidable. To show this denote by $\Psi_t^{g,U}$ the solution of $$i\partial_t \Psi_t^{g,U} = \left(-\Delta +\lambda U - 2b
\ln\modulus{\Psi_t^{g,U}}\right)\Psi_t^{g,U}$$ fulfilling the initial condition $$\Psi_0^{g,U}(x,y) = g(x,y)\,.$$ For fixed $f(x,y)$ and $\sigma\in\left\{+1,-1\right\}$ define $$\begin{array}[c]{c}
\begin{array}[c]{rcl}
U^{(d)}(x,y) &\stackrel{\rm def}{=}& V(y-d) + V(x-d)\,,\\
\Psi_0^{(d)}(x,y) &\stackrel{\rm def}{=}& f(x,y-d) +\sigma f(y,x-d)\,,
\end{array}\\
\chi_0^{(d)}(x,y) \stackrel{\rm def}{=} f(x,y-d)\;,\quad
\phi_0^{(d)}(x,y) \stackrel{\rm def}{=} f(y,x-d)\,.
\end{array}$$ and $$\chi_t^{(d)} \stackrel{\rm def}{=} \Psi_t^{\chi_0^{(d)},U^{(d)}}\;,\quad
\phi_t^{(d)} \stackrel{\rm def}{=} \Psi_t^{\phi_0^{(d)},U^{(d)}}\;,\quad
\Psi_t^{(d)} \stackrel{\rm def}{=} \Psi_t^{\Psi_0^{(d)},U^{(d)}}\,.$$ Obviously, if $V\in\testspace D(\bbbr)$ and $f\in\testspace S(\bbbr^2)\,$, $$\partial_\lambda\left(\partial_t^6 \int_{\cal G}\left(
\modulus{\Psi^{(d)}_t(x,y)}^2 - \modulus{\chi^{(d)}_t(x,y)}^2 -
\modulus{\phi^{(d)}_t(x,y)}^2\right){\rm d}x{\rm d}y\right)_{|_{t=0}}
\stackscript{\longrightarrow}_{d \to \infty} 0$$ holds for every region ${\cal G}\subset\bbbr^3\times\bbbr^3\,$.
Moreover, $$\begin{array}[c]{l}
\displaystyle \lim_{d\to\infty} \partial_\lambda\left( \partial_t^6
\int_{\cal G} \modulus{\phi^{(d)}_t(x,y)}^2\,{\rm d}x{\rm
d}y\right)_{|t=0}\\[8mm]
=\displaystyle \lim_{d\to\infty} \partial_\lambda\left( \partial_t^6
\int_{\cal G} \modulus{\chi^{(d)}_t(x,y)}^2\,{\rm d}x{\rm
d}y\right)_{|t=0}\\[8mm]
= c^{f,U}_{\lambda} \stackrel{\rm def}{=}\displaystyle \partial_\lambda\left(
\partial_t^6 \int_{x\in\cal O} \modulus{\Psi_t^{f,U}(x,y)}^2 {\rm
d}x\,{\rm d}y\right)_{|t=0}
\end{array}$$ holds for $${\mbox{$\cal G$}}=\left\{(x,y)\in\bbbr^2: x\in{\mbox{$\cal O$}}\,\lor\,y\in{\mbox{$\cal O$}}\right\}\;,\quad {\mbox{$\cal O$}} \mbox{ bounded}\;,\quad U(x,y)=V(y)\,.$$ Therefore, under these conditions, $$\partial_\lambda\left(\partial_t^6 \int_{\cal G}\left(
\modulus{\Psi^{(d)}_t(x,y)}^2\,{\rm d}x\,{\rm d}y\right)
\right)_{|_{t=0}} \stackscript{\longrightarrow}_{d \to \infty} =
c^{f,U}_\lambda\,.$$ Since, as shown in Section \[section:bbm\], $c^{f,U}_{\lambda}$ can be arranged to be nonzero we conclude:
> The postulate of symmetry or antisymmetry of the wave function w.r.t. to exchange of particles does not prevent essential nonlocality.
Acknowledgments {#acknowledgments .unnumbered}
===============
I am grateful to M. Czachor, H.-D. Doebner, G.A. Goldin, and R. Werner for stimulating discussions.
Appendix: Maple V (Release 4) Session {#appendix-maple-v-release-4-session .unnumbered}
=====================================
PROCEDURES:
> del := proc(f)
> global x,y,t;
> option operator;
> unapply( diff(f(x,y,t), x$2) + diff(f(x,y,t), y$2), (x,y,t));
> end:
> pot := proc(f)
> global x,y,t;
> option operator;
> unapply(V(y) * f(x,y,t),(x,y,t));
> end:
> Idot := proc(f)
> global x,y,t;
> option operator;
> unapply(simplify(subs(diff(P(x,y,t),t)=(-del(P)(x,y,t)+ pot(P)(x,y,t)),
diff(PB(x,y,t),t)=(del(PB)(x,y,t)-pot(PB)(x,y,t)), diff(f(x,y,t), t))), (x,y,t));
> end:
EVALUATION:
> term0 := (x,y,t) -> 2 * diff(ln( PB(x,y,t) * P(x,y,t)), x) * PB(x,y,t) *
diff(P(x,y,t), x) + diff(ln( PB(x,y,t) * P(x,y,t)), x$2) * PB(x,y,t) * P(x,y,t):
> term1 := (x,y,t) -> simplify(Idot(term0)(x,y,t)):
> term2 := (x,y,t) -> simplify(Idot(term1)(x,y,t)):
> term3 := (x,y,t) -> simplify(Idot(term2)(x,y,t)):
SPECIAL CASE:
> spec := proc(f)
> global x,y,t;
> option operator;
> unapply( subs(V(y)=lambda * y^2, P(x,y,t)=exp(-x^2 -y^2 -x*y),
PB(x,y,t)=exp(-x^2 -y^2 -x*y), f(x,y,t)), (x,y,t));
> end:
RESULT:
> int(int(simplify(diff(spec(term3)(x,y,t),lambda)), x=-infinity..infinity),
y=-infinity..infinity);
1/2
- 32/3 Pi 3
> int(int(exp(-2*x^2 -2*y^2 -2*x*y), x=-infinity..infinity),
y=-infinity..infinity);
1/2
1/3 Pi 3
[99]{}
A. Ashtekar and T.A. Schilling, “Geometrical Formulation of Quantum Mechanics”, gr-qc/9706069.
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T.W.B. Kibble, “Relativistic models of nonlinear quantum mechanics”, .
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C.G. Shull, D.K. Atwood, J. Arthur, and M.A. Horne: .
R. Gaehler, A.G. Klein, and A. Zeilinger: .
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N. Gisin, .
N. Gisin, .
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M. Czachor, .
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H.-D. Doebner, G.A. Goldin, and P. Nattermann, “Gauge Transformations in Quantum Mechanics and the Unification of Nonlinear Schrödinger Equations”, quant-ph/9709036.
W. Lücke, [*Gisin Nonlocality of the Doebner-Goldin 2-Particle Equation*]{}, quant-ph/9710033; not to be published.
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W.L. and P. Nattermann, “Nonlinear Quantum Mechanics and Locality”, in [*Symmetry in Science X*]{}, eds. B. Gruber and M. Ramek (Plenum Press, 1998), pp. 197–205.
[^1]: Based on a talk presented at “New Insights in Quantum Mechanics – Fundamentals, Experimental Results, Theoretical Directions”, Goslar, August 31 – September 3, 1998.
[^2]: Let us mention just some of them: T.W.B. Kibble[@Kibble], A. Ashtekar and T.A. Schilling[@Ashtekar], P. Bona[@Bona], R. Haag and U. Bannier[@Haag-Bannier], Mielnik[@Mielnik], S. Weinberg[@Weinberg], G. Auberson and P.C. Sabatier[@Sabatier], M.D. Kostin[@Kostin], H.D. Doebner and Goldin[@dg1; @dg2], I. Bialynicki-Birula and J. Mycielski[@bbm].
[^3]: Here locality means $\Psi(\xv)=0\Longrightarrow \left(F[\Psi]\right)(\xv) =0$ in the distribution theoretical sense.
[^4]: Note that, contrary to Bell nonlocality,[@cl-sh] the observable effect does not refer to the resulting correlations between both particles.
[^5]: His justification: “So far we have only used linear quantum mechanics”.
[^6]: This especially implies $\sum_{\alpha} x_\alpha\,
P_\Psi(\alpha,t) = \sum_{\beta} x_\beta\, P_\Psi(\beta,t)$ for $t=0$ but — as realized by Gisin — not generally for $t>0\,$.
[^7]: See also [@czach1] for examples.
[^8]: The [*general Doebner-Goldin equation*]{} is given by (\[NLSE\]) and (\[GDGE\]) without the restriction (\[el\]) on the parameters $c_\nu\in\bbbr\,$.
[^9]: Note that (\[pp1\]) does not depend on $\xv_0\,$. Therefore the effect on particle 1, if any, can be produced by acting on particle 2 as far away as one likes.
[^10]: Note that (\[gi\]) is equivalent to Galilei covariance of the Doebner Goldin equation.[@dg2]
[^11]: Equation (\[bbm\]) was already considered by H. Košťál[@kostal] with $b<0\,$.
[^12]: For the Doebner-Goldin case (\[GDGE\]) the latter can always be achieved[@NaDip] by some nonlinear gauge transformation of the type considered in Section \[section:CE\].
[^13]: This result was confirmed by R. Werner[@werner2] using his method described in Section \[section:dg\].
|
---
abstract: 'AMANDA-II is the largest neutrino telescope collecting data at the moment, and its main goal is to search for sources of high energy extra-terrestrial neutrinos. The detection of such sources could give non-controversial evidence for the acceleration of charged hadrons in cosmic objects like Supernova Remnants, Micro-quasars, Active Galactic Nuclei or Gamma Ray Bursts. No significant excess has been found in searching for neutrinos from both point-like and non-localized sources. However AMANDA-II has significantly improved analysis techniques for better signal-to-noise optimization. The km$^3$-scale IceCube telescope will enlarge the observable energy range and improve the sensitivities of high energy neutrino searches due to its 30 times larger effective area.'
author:
- 'Paolo Desiati, for the IceCube Collaboration[^1]'
title: |
Neutrino Astronomy at the South Pole:\
latest Results from AMANDA-II
---
[ address=[University of Wisconsin, Madison, WI, U.S.A.]{} ]{}
Introduction {#s:intro}
============
The detection of extra-terrestrial multi-TeV neutrinos would significantly contribute to the understanding of the origin, acceleration and propagation of high energy cosmic rays. It would also influence the interpretation of results of the most recent gamma ray experiment results, like those of H.E.S.S. [@hess].
Neutrinos can propagate through the Universe without being affected by interactions or magnetic fields. Therefore they are ideal high energy cosmic messengers and, for the same reason, unfortunately very difficult to detect. Even the so called “guaranteed” neutrino fluxes, i.e. those associated with known cosmic accelerators with an identified pion production target, typically require a $\sim$ km$^3$ scale detector. AMANDA-II [@amanda; @amanda2], with its $\sim 0.016$ km$^3$ instrumented volume, is currently the largest operating neutrino telescope. Its sensitivity is improving because of the larger data sample collected, better apparatus understanding, and more efficient background rejection techniques.
The main background for neutrino telescopes is the intense flux of downward penetrating cosmic muons which can mimic the upward neutrino-induced event signature. A total rejection factor of $10^6$ is required to isolate muons arising from neutrinos created in atmospheric cosmic ray showers. AMANDA-II has improved reconstruction algorithms [@reco] and analysis techniques and currently it is able to select about four upward going muons per day [@icrc05]. After rejecting the downward muon background, these atmospheric neutrinos remain as an irreducible background in searches for extraterrestrial neutrino fluxes.
The following sections describe the searches for localized neutrino sources and for a diffuse neutrino flux from many weak sources, respectively.
Neutrinos from point-like sources {#s:point}
=================================
Since the main background consists of downward-going cosmic ray muons, the search for neutrinos is restricted to the northern hemisphere. A point source in the sky would be identified as a localized significant excess of events, above the irreducible measured atmospheric muon neutrino background. The event selection is independently performed on different declination bands, for assumed spectra with $E^{-2}$ to $E^{-3}$ energy dependence. The band width depends on the angular resolution at the given declination. The optimization of event selection is done by requiring the best sensitivity, i.e. the best 90% CL expected average upper limit in the case of no signal [@mrf], at each declination band. The final sample contains 3329 events in the first 807 days of lifetime of AMANDA-II (see Fig \[fig:point\]), which has an estimated cosmic ray muon background contamination of less than 10% and corresponds to a median angular resolution of $\sim 1.5^{\circ} - 2.5^{\circ}$, depending of declination. The sensitivity, for an assumed energy spectrum of $E^{-2}$, is $6\times 10^{-8}$ GeVcm$^{-2}$s$^{-1}$, and it is almost independent on declination [@amanda; @icrc05]. No excess with respect to the expected background has been observed so far [@point1; @point2].
![Left: sky map of the selected 3329 neutrino events from the year 2000-03. Right: significance map of the search for clusters of events in the northern hemisphere, based on the selected events (gray scale is in sigma).[]{data-label="fig:point"}](Bernardini_Fig1.epsi "fig:"){height=".12\textheight"} ![Left: sky map of the selected 3329 neutrino events from the year 2000-03. Right: significance map of the search for clusters of events in the northern hemisphere, based on the selected events (gray scale is in sigma).[]{data-label="fig:point"}](Bernardini_Fig2.epsi "fig:"){height=".11\textheight"}
The absence of a significant excess could simply be caused by the small size of the AMANDA-II apparatus and faint neutrino fluxes from the point sources. Besides increasing the collected data sample, a possibility for improving the signal-to-noise ratio is to search for events from a pre-selected list of source candidates [@amanda2; @point1; @point2], or for events in coincidence with known periods of enhanced electromagnetic emission of selected objects [@amanda2]. Another possibility is to perform a stacking source search using well defined classes of Active Galactic Nuclei, under the assumption that neutrino emission is similar to the electromagnetic one [@stacking]. With respect to the single point source candidate search the stacking analysis achieves $\sim$ 2-3 times better sensitivity for some of the selected source classes.
Searches for neutrinos in coincidence with Gamma Ray Bursts were also performed [@mike], where the time constraint significantly reduces the background contamination, but no signal was found.
Neutrinos from non-localized sources {#s:diffuse}
====================================
The search for high energy neutrinos from all the possible sources in the Universe emitted throughout its evolution could enhance a very faint signal not detectable from single sources. The search for a diffuse neutrino flux relies on the simulation of the background and the signal-to-noise optimization is performed on the energy estimation of the selected events. It is expected that neutrinos have a hard energy spectrum proportional to $E^{-2}$, whereas the atmospheric neutrinos have a steeper spectrum ($\sim E^{-3.7}$). Therefore, after having assured the quality of selected events, an energy cut is optimized to have the best sensitivity. The preliminary sensitivity for the search of diffuse muon neutrinos is $9.5\times 10^{-8}$ GeVcm$^{-2}$s$^{-1}$sr$^{-1}$ for 807 days of livetime, in the energy range between 13 TeV and 3.2 PeV [^2]. This preliminary search did not reveal any significant excess with respect to the background.
A search for all neutrino flavor events undergoing neutral current interaction as well was also performed. Reconstruction of the cascade position and energy allows to extend the search to the full sky (not only the northern hemisphere), since shower events could be discriminated from tracks. Even so, the major background consists of bremsstrahlung showers produced by cosmic muons [@cascades]. At Ultra High Energy ranges (i.e. above 10$^3$ TeV) the events are very extensive and full reconstruction is not necessary. At these energies the background of atmospheric events starts to become negligible and the search can be done by counting the fraction of optical sensors with more than one detected photon, which increases with energy [@uhe].
The absence of signal detection in AMANDA-II is an indication that the flux of extra-terrestrial neutrinos, if any, is very small. A $\sim$ km$^3$-scale experiment may open this new observation window by profiting not only of its larger size and more powerful analysis techniques being developed in AMANDA-II, but also of the full digitized response of IceCube array sensors [@spencer].
[9]{}
W. Benbow for the H.E.S.S. Collaboration, in these Proceedings.
P. Desiati for the IceCube Collaboration, in *International Journal of Modern Physics A* edited by O. Adriani et al., XIX ECRS Conference Proceedings Vol 20, No. 29, Florence, Italy 6533-7068 (2005).
E. Bernardini for the IceCube Collaboration, to appear in the Proceedings of 7th Workshop on *Towards a Network of Atmospheric Cherenkov Detectors 2005*, Palaiseau, France, 27-29, (2005).
E. Andrés et al., *Astropart. Phys.* **13**, No. 1, (2000).
The IceCube Collaboration, contributions to the XXIX ICRC, Pune, India, 2005, astro-ph/0509330.
G. C. Hill and K. Rawlins, *Astropart. Phys.* **19**, 393 (2003).
J. Ahrens et al., *Phys. Rev. Lett.* **92**, 071102 (2004).
M. Ackermann et al., *Phys. Rev. D* **71**, 077102 (2005).
M. Ackermann et al., submitted to Astropart. Phys..
M. Stamatikos et al., contribution to the XXIX ICRC, Pune, India, 2005, astro-ph/0510336.
M. Ackermann et al., *Astropart. Phys.* **22**, 127 (2004).
M. Ackermann et al., *Astropart. Phys.* **22**, 339 (2005).
S. Klein for the IceCube Collaboration, in these Proceedings.
[^1]: For the Collaboration list see [http://www.icecube.wisc.edu/pub\_and\_doc/conferences/panic05/]{}
[^2]: which contains 90% of the selected events
|
---
abstract: 'Relatively prime pairs of integers can be represented as nodes in three way branching trees. We construct trees of Bézout coefficients which correspond to the relatively prime pairs in the aforementioned trees. As one application, we compare the Bézout coefficients in these trees to those returned by the `gcd` function in <span style="font-variant:small-caps;">Matlab</span>. As another application, we use these trees to decrease the computation time required to create computer generated hyperbolic wallpaper designs.'
address:
- |
Department of Mathematics\
University of Wisconsin-Eau Claire
- |
Department of Mathematics\
University of Wisconsin-Eau Claire
author:
- 'EMILY GULLERUD AND JAMES S. WALKER'
title: GENERATING BÉZOUT TREES FOR PYTHAGOREAN PAIRS
---
Generating the trinary trees
============================
The pair of integers $(m,n)$ is defined to be a Pythagorean pair if they generate a Pythagorean triplet $(x,y,z)$ where $x^2+y^2=z^2$ for $x,y,z\in \mathbb{Z}$. These integers $m$ and $n$ are relatively prime and satisfy the equations $$x=m^2-n^2, \quad y=2mn, \quad z=m^2+n^2.$$ For example, the pair $(2,1)$ generates the Pythagorean triplet $(3,4,5)$. Not only does $(m,n)$ generate a Pythagorean triplet, but its *associated pairs* $(n,m)$ and $(m,-n)$ generate the triplets $(-x,y,z)$ and $(x,-y,z)$ respectively, which clearly still satisfy the equation $x^2+y^2=z^2$.
Using the fact that $(2m+n,m)$ is a Pythagorean pair given that $(m,n)$ is a Pythagorean pair, Randall and Saunders [@Randall1994] proved that every relatively prime pair of integers can be obtained through two trinary trees, one starting with the pair $(2,1)$ and another with the pair $(3,1)$. These are three way branching trees since they use associated pairs to obtain subsequent branches. The construction of these trees is summarized below.
\[thm:treenodes\] Let $(m,n)$ be a Pythagorean pair such that $m>n$ with associated pairs $(n,m)$ and $(m,-n)$. Define $f: \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z} \times \mathbb{Z}$ by $f(m,n) = (2m+n,m)$. Then $$\begin{aligned}
f(m,n) &= (2m+n,m),\\
f(n,m) &= (2n+m,n), \text{ and}\\
f(m,-n) &= (2m-n,m)\end{aligned}$$ are all Pythagorean pairs.
Starting with the Pythagorean pair $(m,n)$, applying $f$ as defined in Theorem \[thm:treenodes\] to $(m,n)$ and its associated pairs $(n,m)$ and $(m,-n)$ and then successively applying $f$ to the resultant Pythagorean pairs results in a trinary tree. We denote this the **trinary tree generated by $\mathbf{(m,n)}$**; see Figure \[fig:TrinaryTree\]. Randall and Saunders proved that the trinary tree produced from $(3,1)$ contains all pairs of relatively prime odd integers. Similarly, the trinary tree produced from $(2,1)$ contains all pairs of relatively prime integers of opposite parity. Hence using only two trinary trees we can produce all the pairs of relatively prime integers.
at (-3,0) (A)[$(m,n)$]{}; at (0,2) (B\_1)[$f(m,-n)$]{}; at (0,0) (B\_2)[$f(m,n)$]{}; at (0,-2) (B\_3)[$f(n,m)$]{}; at (3,2.65) (C\_1)[$\cdots$]{}; at (3,2) (C\_2)[$\cdots$]{}; at (3,1.35) (C\_3)[$\cdots$]{}; at (3,.65) (C\_4)[$\cdots$]{}; at (3,0) (C\_5)[$\cdots$]{}; at (3,-.65) (C\_6)[$\cdots$]{}; at (3,-1.35) (C\_7)[$\cdots$]{}; at (3,-2) (C\_8)[$\cdots$]{}; at (3,-2.65) (C\_9)[$\cdots$]{}; (A) – (B\_1); (A) – (B\_2); (A) – (B\_3); (B\_1) – (C\_1); (B\_1) – (C\_2); (B\_1) – (C\_3); (B\_2) – (C\_4); (B\_2) – (C\_5); (B\_2) – (C\_6); (B\_3) – (C\_7); (B\_3) – (C\_8); (B\_3) – (C\_9);
Since the Pythagorean pair $(m,n)$ is relatively prime, we can find a pair of integers $(u,v)$ such that $mu+nv=1$. We show that the pairs $(u,v)$ can be produced recursively in a similar fashion to the pairs $(m,n)$.
Let $(m,n)$ be a Pythagorean pair such that $m>n$ with associated pairs $(n,m)$ and $(m,-n)$. Let $f$ be defined as in Theorem \[thm:treenodes\] and let $mu+nv=1$ for some $u,v \in \mathbb{Z}$. Define the function $g: \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z} \times \mathbb{Z}$ by $g(u,v) = (v,u-2v)$. Then $$\begin{aligned}
g(u,v) &= (v,u-2v),\\
g(v,u) &= (u,v-2u),\text{ and}\\
g(u,-v) &= (-v,u+2v)\end{aligned}$$ yield the necessary $u'$ and $v'$ to satisfy $$\begin{aligned}
(2m+n)u' + mv' &= 1,\\
(2n+m)u' + nv' &= 1,\text{ and}\\
(2m-n)u' + mv' &= 1\end{aligned}$$ respectively.
Let $(m,n)$ be a Pythagorean pair with associated pairs $(n,m)$ and $(m,-n)$. Then by Theorem \[thm:treenodes\], we have the Pythagorean pairs
1. $f(m,n) = (2m+n,m)$
2. $f(n,m) = (2n+m,n)$
3. $f(m,-n) = (2m-n,m)$
Since $(m,n)$ is a Pythagorean pair, we know $\gcd{(m,n)} = 1$. So there exist $u,v \in \mathbb{Z}$ such that $mu + nv = 1$. We will consider each of three cases above separately.
**CASE 1:** Consider $f(m,n) = (2m+n,m)$. Since $(2m+n,m)$ is a Pythagorean pair, $\gcd{(2m+n,m)} = 1$. So there exist $u',v' \in \mathbb{Z}$ such that $(2m+n)u' + mv' = 1$. Then $$(2m+n)u' + mv' = 2mu' + nu' + mv' = (2u'+v')m + u'n = 1.$$ Since $mu + nv = 1$, we get $2u'+v' = u$ and $u' = v$. Hence $u' = v$ and $v' = u-2v$.
**CASE 2:** Consider $f(n,m) = (2n+m,n)$. Since $(2n+m,n)$ is a Pythagorean pair, $\gcd{(2n+m,n)} = 1$. So there exist $u',v' \in \mathbb{Z}$ such that $(2n+m)u' + nv' = 1$. Then $$(2n+m)u' + nv' = 2nu' + mu' + nv' = u'm + (2u'+v')n = 1.$$ Since $mu + nv = 1$, we get $u'= u$ and $2u'+v'=v$. Hence $u' = u$ and $v' = v-2u$.
**CASE 3:** Consider $f(m,-n) = (2m-n,m)$. Since $(2m-n,m)$ is a Pythagorean pair, $\gcd{(2m-n,m)} = 1$. So there exists $u',v' \in \mathbb{Z}$ such that $(2m-n)u' + mv' = 1$. Then $$(2m-n)u' + mv' = 2mu' - nu' + mv' = (2u'+v')m + (-u')n = 1.$$ Since $mu + nv = 1$, we get $2u'+v' = u$ and $-u' = v$. Hence $u' = -v$ and $v' = u+2v$.
Define $g: \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z} \times \mathbb{Z}$ by $g(u,v) = (v,u-2v)$. One can verify that this function satisfies the three cases above.
We call the trinary tree generated by $(u,v)$ for the corresponding Pythagorean pair $(m,n)$ the **Bézout tree of $\mathbf{(m,n)}$ generated by $\mathbf{(u,v)}$**. Notice that there is not a unique pair $(u,v)$ for each $(m,n)$; in fact, there are infinitely many such pairs. Thus there exist infinitely many Bézout trees of $(m,n)$, where each is dependent on the pair chosen to be the root of the tree.
\[ex:trees\] The trinary tree generated by $(3,1)$ is given below on the left and the Bézout tree of $(3,1)$ generated by $(0,1)$ is given below on the right both to a depth of two:
at (-5,0) (A)[$(3,1)$]{}; at (-3,2) (B\_1)[$(5,3)$]{}; at (-3,0) (B\_2)[$(7,3)$]{}; at (-3,-2) (B\_3)[$(5,1)$]{}; at (-1,2.65) (C\_1)[$(7,5)$]{}; at (-1,2) (C\_2)[$(13,5)$]{}; at (-1,1.35) (C\_3)[$(11,3)$]{}; at (-1,.65) (C\_4)[$(11,7)$]{}; at (-1,0) (C\_5)[$(17,7)$]{}; at (-1,-.65) (C\_6)[$(13,3)$]{}; at (-1,-1.35) (C\_7)[$(9,5)$]{}; at (-1,-2) (C\_8)[$(11,5)$]{}; at (-1,-2.65) (C\_9)[$(7,1)$]{}; (A) – (B\_1); (A) – (B\_2); (A) – (B\_3); (B\_1) – (C\_1); (B\_1) – (C\_2); (B\_1) – (C\_3); (B\_2) – (C\_4); (B\_2) – (C\_5); (B\_2) – (C\_6); (B\_3) – (C\_7); (B\_3) – (C\_8); (B\_3) – (C\_9); at (1,0) (X)[$(0,1)$]{}; at (3,2) (Y\_1)[$(-1,2)$]{}; at (3,0) (Y\_2)[$(1,-2)$]{}; at (3,-2) (Y\_3)[$(0,1)$]{}; at (5,2.65) (Z\_1)[$(-2,3)$]{}; at (5,2) (Z\_2)[$(2,-5)$]{}; at (5,1.35) (Z\_3)[$(-1,4)$]{}; at (5,.65) (Z\_4)[$(2,-3)$]{}; at (5,0) (Z\_5)[$(-2,5)$]{}; at (5,-.65) (Z\_6)[$(1,-4)$]{}; at (5,-1.35) (Z\_7)[$(-1,2)$]{}; at (5,-2) (Z\_8)[$(1,-2)$]{}; at (5,-2.65) (Z\_9)[$(0,1)$]{}; (X) – (Y\_1); (X) – (Y\_2); (X) – (Y\_3); (Y\_1) – (Z\_1); (Y\_1) – (Z\_2); (Y\_1) – (Z\_3); (Y\_2) – (Z\_4); (Y\_2) – (Z\_5); (Y\_2) – (Z\_6); (Y\_3) – (Z\_7); (Y\_3) – (Z\_8); (Y\_3) – (Z\_9);
Comparing the <span style="font-variant:small-caps;">Matlab</span> `gcd` function to trinary trees
==================================================================================================
The `gcd` function in <span style="font-variant:small-caps;">Matlab</span> [@Matlab] has the following syntax: $$\texttt{gcd}(A,B) = [G,U,V]$$ where $G$ is the greatest common divisor of $A$ and $B$, while $U$ and $V$ are the Bézout coefficients satisfying $A\cdot U+B\cdot V=G$. Even though there are infinitely many choices for $U$ and $V$, the `gcd` function will yield only one choice. This is done so using the extended Euclidean algorithm. We now consider the question of **when does the `gcd` function yield the same Bézout coefficients as the Bézout tree?**
Recall that the two trinary trees produced by $(2,1)$ and $(3,1)$ yield all the pairs of relatively prime integers. Hence the Bézout trees of $(2,1)$ and $(3,1)$ will produce one pair of Bézout coefficients for each pair of relatively prime integers. We have the freedom to choose any pair of Bézout coefficients which correspond to $(2,1)$ and $(3,1)$ to be the roots of our Bézout trees. In an attempt to match the Bézout coefficients produced by the `gcd` function to those in the Bézout trees, we select the following coefficients $(u_r,v_r)$ for our roots:
- The Bézout tree of $(2,1)$: $(u_r,v_r) = (0,1)$ since $\texttt{gcd}(2,1)=[1,0,1]$.
- The Bézout tree of $(3,1)$: $(u_r,v_r) = (0,1)$ since $\texttt{gcd}(3,1)=[1,0,1]$.
Notice that the Bézout trees of $(2,1)$ and $(3,1)$ defined as above are identical. Recall that we gave this tree to a depth of $2$ in Example \[ex:trees\].
We computed the trinary trees generated by $(2,1)$ and $(3,1)$ and their corresponding Bézout trees generated by $(0,1)$ to a depth of $13$. This yielded a total of $4782966$ relatively prime pairs and their corresponding Bézout coefficients. We then found the percentage of pairs $(u,v)$ in the Bézout trees which were equal to the pairs $(U,V)$ given by the `gcd` function. Our test showed that exactly one sixth of these pairs differed. Of the pairs which differed, all were generated in the same branch of the Bézout tree of $(2,1)$, specifically the branch corresponding to the relatively prime pair $(2,-1)$ found in the second level. In Figure \[fig:BezoutTreeGCD\] we show the Bézout tree of $(2,1)$ generated by $(0,1)$ with the differing pairs indicated; the pairs from the Bézout trees are denoted in blue and the pairs given by the `gcd` function are denoted in red. Pairs which are the same in both the tree and `gcd` function are given once in black.
at (-3,0) (A)[$(0,1)$]{}; at (0,2.5) (B\_1)[$(-1,2)$]{}; at (0,2.9) (D\_1)[$(1,-1)$]{}; at (0,0) (B\_2)[$(1,-2)$]{}; at (0,-2) (B\_3)[$(0,1)$]{}; at (3,3.5) (C\_1)[$(-2,3)$]{}; at (3,3.9) (D\_1)[$(1,-1)$]{}; at (3,2.5) (C\_2)[$(2,-5)$]{}; at (3,2.9) (D\_1)[$(1,-1)$]{}; at (3,1.5) (C\_3)[$(-1,4)$]{}; at (3,1.9) (D\_1)[$(1,-1)$]{}; at (3,.65) (C\_4)[$(2,-3)$]{}; at (3,0) (C\_5)[$(-2,5)$]{}; at (3,-.65) (C\_6)[$(1,-4)$]{}; at (3,-1.35) (C\_7)[$(-1,2)$]{}; at (3,-2) (C\_8)[$(1,-2)$]{}; at (3,-2.65) (C\_9)[$(0,1)$]{}; (A) – (-.7,2.6); (A) – (B\_2); (A) – (B\_3); (.7,2.85) – (2.4,3.7); (.7,2.7) – (2.4,2.7); (.7,2.55) – (2.4,1.7); (B\_2) – (C\_4); (B\_2) – (C\_5); (B\_2) – (C\_6); (B\_3) – (C\_7); (B\_3) – (C\_8); (B\_3) – (C\_9); (-.6,2.3) – (.6,2.3) – (.6,2.7) – (-.6,2.7) – cycle;
In our Bézout tree generated by $(0,1)$, if we change the pair $(-1,2)$ to the pair given by the `gcd` function, which is $(1,-1)$, then computation to a depth of $13$ shows that the Bézout tree will have the same exact pairs as the `gcd` function. This leads to the following conjecture, which we strongly believe to be true given our data.
\[conj:BezoutTreeGCD\] Consider the trinary trees generated by $(2,1)$ and $(3,1)$. Let $(u,v)$ be the pair in a Bézout tree corresponding to the relatively prime pair $(m,n)$ and $(U,V)$ be the pair given by the $\texttt{gcd}$ function for the same pair $(m,n)$. Then the following hold:
1. For all $(u,v)$ in the Bézout tree of $(3,1)$ generated by $(0,1)$, $(u,v) = (U,V)$.
2. One third of the $(u,v)$ in the Bézout tree of $(2,1)$ generated by $(0,1)$ are not equal to $(U,V)$. Changing the value of $g(0,-1)$ in the second level of the Bézout tree from $(-1,2)$ to $(1,-1)$ results in a tree in which $(u,v) = (U,V)$ for all $(u,v)$.
Application to mathematical art
===============================
Our purpose for exploring trinary trees of relatively prime pairs was to use them to create hyperbolic wallpaper in a computationally inexpensive manner. These hyperbolic wallpapers are plotted in the upper-half plane and are created using a domain coloring algorithm. We start with a color map and a blank image. Then we use a mapping function $f(z)$ to color the pixels in the blank image, where $f(z)$ maps pixel locations from the blank image to pixel locations in the color map. This mapping function $f(z)$ is given by $$f(z) = \sum_{\gcd(c,d)=1}f(\gamma_{c,d}(z))$$ where $\gamma_{c,d}(z) = \frac{az+b}{cz+d}$ and $ad-bc = 1$. For our purposes, this sum will be finite yet have hundreds of terms. Notice that we need relatively prime pairs $(c,d)$ in our computation. The restriction $ad-bc = 1$ requires us to have the Bézout coefficients $(a,b)$ as well. Since the `gcd` function in <span style="font-variant:small-caps;">Matlab</span> is computationally expensive, we instead generate the trinary trees and Bźout trees in a matter of seconds and pull values from these. More information on these wallpapers will be given in [@Handbook] upon publication.
[3]{}
T. Randall and R. Saunders, “The family tree of the Pythagorean triplets revisited.” *Math. Gaz.* **78**(482), pp. 190-193, 1994.
<span style="font-variant:small-caps;">Matlab</span>. A language and environment for technical computing. Product of MathWorks.
E. Gullerud and J. S. Walker, “Symmetry and Art.” <span style="font-variant:small-caps;">Handbook of Complex Analysis</span>, Ed. Steven G. Krantz, CRC Press, to be published in 2019.
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abstract: 'The two-color above-threshold [ionization]{} (ATI) of atoms and ions is investigated for a vortex [Bessel beam in the presence of]{} a strong near-infrared (NIR) light field. While the photoionization is caused by the photons from the [weak but extreme]{} ultra-violet (XUV) vortex Bessel beam, the energy [and angular distribution of]{} the photoelectrons [and their sideband structure]{} are affected by the plane-wave NIR field. We here explore the energy spectra and angular emission of the photoelectrons [in such two-color fields as a function]{} of the size and location of the target atoms with regard to the beam axis. [In addition,]{} analogue to the circular dichroism in typical two-color ATI [experiments]{} with circularly polarized [light, we define and discuss seven]{} different dichroism signals for such [vortex Bessel beams that arise from]{} the various combinations of the orbital and spin angular momenta of the two light fields. [For localized targets, it]{} is found that these dichroism signals [strongly depend on]{} the size and position [of the atoms]{} relative to the beam. For macroscopically extended targets, in contrast, three of these [dichroism]{} signals tend to zero, while the other four [just]{} coincide with the standard circular dichroism, [similar as for]{} Bessel beams with small opening angle. [Detailed computations of the dichroism are performed]{} and discussed for the $4s$ valence-shell photoionization of Ca$^+$ ions.'
author:
- 'D. Seipt'
- 'R. A. Müller'
- 'A. Surzhykov'
- 'S. Fritzsche'
title: 'Two-color above threshold ionization of atoms and ions in XUV Bessel beams [and combined with]{} intense laser light'
---
Introduction
============
[Studies on non-perturbative multiphoton processes in intense laser pulses have rapidly advanced during recent years and helped to explore the inner-atomic motion of electrons at femto- and attosecond timescales [@Corkum:NaturePhys2007; @Krausz:RevModPhys2009; @Pazourek:RevModPhys2015]. For example, such multiphoton ionization and inner-shell processes have not only been observed for noble gases [@Gilbertson:PRL2010; @Feist:PRL2009] but also for molecules [@Haessler:NaturePhys2010], surfaces and elsewhere [@Cavalieri:Nature2007; @Kruger:JPB2012]. Today, these studies enable one to generate quite routinely attosecond pulses by high-order harmonic generation [@Sansone:Science2006; @Popmintchev:Science2012; @Calegari:JPB2016], or to control the above-threshold ionization (ATI) [@Milosevic:JPB2006; @Wittmann:NaturePhys2009]. ]{}
In typical ATI experiments, an electron is released from an atom or molecule by absorbing one or several photons from a near-infrared (NIR) laser field *more* than required energetically in order to overcome the ionization threshold. The ATI energy spectra of the photoelectrons therefore exhibit a series of peaks, just separated by the NIR photon energy, while the relative strength of these peaks may depend significantly on the intensity and temporal structure of the incident laser pulses. These ATI spectra are thus quite in contrast to the photoelectron spectra as obtained by just weak high-frequency (XUV) radiation, where the absorption of a single photon is sufficient to eject a bound electron and where the single photoline (for each possible final state of the photoion) is usually well described by perturbation theory. In two-color ATI, such a XUV field is often combined with intense NIR laser pulses in order to investigate the ionization of sub-valence electrons: While, under these circumstances, the NIR field alone is not sufficient to ionize the atoms or molecules efficiently, it is intense enough to stimulate the absorption or emission of one or several additional NIR photons by the outgoing electron. This *non-perturbative* interaction of the electrons with both, the XUV *and* NIR fields then leads to the well-known ’sidebands’ that typically occur as satellites to the normal photolines [@Ehlotzky:PhysRep2001; @Radcliffe:NJP2012]. Such sidebands were first observed in the two-color ATI of noble gases as well as in laser-assisted Auger processes [@Muller:JPB1986; @Schins:PRL1994; @Glover:PRL1996; @Meyer:PRA2006; @Meyer:PRL2008; @Meyer:PRL2010]. More recently, these sidebands have been applied in pump-probe photoelectron spectroscopy [@Helml:NatPhoton2014] or in imaging the molecular orbitals of $\rm H_2O$, $\rm O_2$ and $\rm N_2$ [@Leitner:PRA2015].
[Apart from the temporal structure of the XUV and NIR pulses, the intensity of the sidebands depends of course also on the relative orientation and linear polarization of these fields [@OKeeffe:PRA2004; @Guyetand:JPB2005]. This orientation dependence has been explored especially by Meyer and coworkers [@Meyer:PRL2008] for the angular distribution of the sidebands in the photoionization of helium. Later, it was shown theoretically [@Kazansky:PRA2012] that the two-color ATI sideband spectra are rather sensitive also with regard to the circular polarization of both, the XUV and NIR light, and an *asymmetry* in the photoelectron spectra was found, if the circular polarization of one of the field is changed from same to the opposite direction, a phenomenon that is termed today as circular dichroism in two-color ATI. This circular dichroism, that is associated with some flip of the spin-angular momentum (SAM) of the incident light field, has recently been utilized, e.g., for *measuring* the polarization state of an ultra-violet free electron laser [@Kazansky:PRL2011; @Kazansky:PRA2012; @Mazza:NatCommun2014]. For molecules, in addition, the question has been raised how two-color ATI spectra are affected by the molecular symmetry and the polarization of the incident radiation [@Guyetand:JPB2008; @Lux:AngewChemie2012].]{}
In this work, we investigate the two-color ATI process if the usual plane-wave XUV field is *replaced* by a vortex (Bessel) beam, [also]{} known as ’twisted light’, that carries not only SAM but also orbital angular momentum (OAM). Indeed, the study of such OAM light has attracted much recent interest [for the manipulation of microparticles [@Molina:NatPhys2007], for investigating fundamental interactions [@Stock:PRA2015; @Quinteiro:PRA2015; @Surzhykov:PRA2015; @Schmiegelow:NatureCommun2016; @Radwell:PRL2015], and for multiplexing in telecommunication [@Bozinovic:Science2013; @Krenn:NJP2014], to name a few.]{} In particular, we here explore the energy spectra and angular emission of photoelectrons ejected by a vortex Bessel beam in the presence of strong NIR light, and how these photoelectron spectra depend on the size and location of the target (atoms) with regard to the axis of the vortex beam. We assume an XUV Bessel beam that is energetic enough to photoionize the atom, while the plane-wave NIR field affects the sideband structure as well as the energy and angular distribution of the photoelectrons. Analogously to the circular dichroism from above, we define and discuss moreover *seven* possible dichroism signals which arise from different combinations of the orbital and spin angular momenta of the two light fields involved. For localized targets, we find that these dichroism signals sensitively depend on the size and position of the atoms relative to the beam axis. For macroscopically extended targets, in contrast, three of these signals tend to zero, while the other four just approach the (standard) circular dichroism. To discuss these finding, detailed computations of the dichroisms are performed and discussed for the $4s$ valence-shell photoionization of Ca$^+$ ions.
[In the next section, we shall first evaluate the transition amplitude and photoionization probability for the two-color ATI within the strong-field approximation (SFA), if a vortex Bessel beam interacts with an atom in the presence of a strong, plane-wave NIR field. Note that atomic units are used throughout the paper. Details about the twisted XUV Bessel beam are given in Subsection \[sect:bessel\], while the choice of targets is explained in [Subsection \[sect:target\]. The]{} possible dichroism signals for such a two-color field are defined in Section \[sect:dichroism\] . Emphasize is placed here especially on the influence of a *localized *versus* macroscopically extended* target in exploring the sensitivity of the dichroism with regard to the target size. Detailed calculations of the photoelectron spectra and the various dichroism signals are presented and discussed later in Section \[sect:results\]. Finally, conclusions are given in Section \[sect:summary\].]{}
![[Scheme for]{} the two-color ATI of an atom by an XUV vortex beam (blue) and in the presence of a strong NIR field (orange). The atoms are assumed to be [either]{} a) localized with regard to the common beam axis or b) equally distributed over the cross section of the XUV Bessel beam.[]{data-label="fig:1"}](fig1.pdf){width="0.9\columnwidth"}
Theoretical Background {#sect:theory}
======================
Two-color ATI in Strong-Field Approximation {#subsect:system}
-------------------------------------------
We here explore the two-color ATI of atoms (and ions) if they interact with a weak XUV vortex beam and an intense plane-wave NIR field. In particular, we assume an (almost) monochromatic XUV vortex beam, as for instance generated by free-electron lasers (FEL), and which is energetic enough to ionize the target atom. Although quite strong, moreover, we suppose some NIR laser pulse with many optical cycles so that it can be described as a monochromatic plane-wave. Together, these two assumptions ensure that the ’sideband regime’ holds [@Radcliffe:NJP2012], in which photoelectrons are expected not only at the given photoline but also at energies that differ by one or several energy quanta of the the NIR field. Moreover, both fields are supposed to propagate along a common beam axis that is taken also as the quantization axis ($z$-axis). Finally, the atomic target is either a microscopic target of trapped atoms or ions, localized at some position $\vec b$ in the $xy$-plane perpendicular to the beam axis, cf. Ref. [@Schmiegelow:NatureCommun2016] and Fig. \[fig:1\]a, or as *macroscopic* and uniformly distributed target over the cross section of the twisted XUV beam (Fig. \[fig:1\]b).
[After their interaction with the two-color field, the photoelectrons leave the interaction region with the asymptotic (canonical) momentum $\vec p$ as measured at the detectors. Below, we shall analyze the angular and momentum distribution of these electrons as a function of the polar angle $\vartheta_p$ of the momentum, i.e. with regard to the common beams axis, as well as for different azimuth angles $\varphi_p$ \[as defined by the impact vector $\vec b \,=\, (b, \varphi_p = 0, z=0)$ of the target atom\]. [In particular, we aim]{} to understand how the photoelectron distributions are affected by the OAM of the XUV beam and by relative changes in either the SAM and/or OAM of the two-color fields.]{}
Within the SFA, the transition amplitude ${\mathcal T}$ for the two-color ATI of a single *active* electron, being initially in the bound state $ | \phi_0 \rangle$, reads as
$$\begin{aligned}
\label{eq:def:amplitude}
{\mathcal T}(\vec p) &= -i \int_{-\infty}^\infty \! { {\mathrm d} }t \,
\mem{ \Psi^{(V)}_{\vec q(t)} }
{ \, {\vec{\hat{p}}} \cdot \vec A_X(\vec r) \, \vphantom{\Psi^{(V)}_{\vec q(t)} } }
{ \phi_0 } \;
e^{i\, (E_B-\omega_X) t} \,,\end{aligned}$$
where $E_B$ is the binding energy of the active electron and $\vec A_X(\vec r)$ the vector potential of the XUV Bessel beam with frequency $\omega_X$. We shall describe the details of this vector potential in the next subsection. In the SFA, moreover, the final (continuum) state of the electron is typically described by a Volkov state $\bram{ \Psi^{(V)}_{\vec q(t)} } = \bram{ \vec q(t) }\, e^{-iS_V(t)}$ (in length gauge), which neglects the effect of the parent ion upon the motion of the liberated electron, and where $\bram{ \vec q(t) }$ describes [the plane-wave electron wave function]{} with the kinetic momentum $\vec q(t) = \vec p - \vec A_L(t)$. In the presence of an external NIR field $\vec A_L (t)$, this kinetic momentum is different from the *conserved* canonical momentum $ \vec p $ [which is]{} measured at the detector, eventually. Finally, the phase of the Volkov wavefunction is given by [@Volkov:1935] $$\begin{aligned}
S_V(t) &= \int_t^\infty \! { {\mathrm d} }t' \, \frac{\vec q(t)^2}{2}
= \frac{1}{2} \int_t^\infty \! { {\mathrm d} }t' \, \left[ \vec p - \vec A_L(t') \right]^2 \,.\end{aligned}$$ While we have employed the length gauge for the strong assisting NIR laser field $\vec A_L$, we still use the interaction operator with the high-frequency XUV vortex field $\vec A_X$ [in]{} the velocity gauge form because of the complex spatial structure of [$\vec A_X(\vec r)$]{}.
Characterization of twisted Bessel beams {#sect:bessel}
----------------------------------------
[So-called twisted or [vortex]{} (light) beams are known to carry, in addition their possible polarization or SAM, also an orbital angular momentum (OAM) owing to their helical phase fronts. Typically, [twisted]{} beams show a very characteristic annular intensity distribution with zero intensity on the beam axis. This zero intensity line is called the *vortex line* of the field and it embodies a phase singularity. In the XUV frequency region, twisted light beams have been generated recently by means of undulators [@Hemsing:NatPhys2013; @Bahrdt:PRL2013] or by using high-harmonic generation [@Zuerch:NatPhys2012; @Gariepy:PRL2014; @Hernandez-Garcia:PRL2013]. Experimentally, such twisted beams can be prepared in different modes with regard to the (components of the) angular momenta that are *conserved* for some given beam. In the further analysis, we shall assume Bessel beams that are obtained as non-paraxial solutions of the vector Helmholtz equation, and which are classified here by the wave vector components $k_z$ and $\varkappa$ \[with $\varkappa=|\vec k_\perp|$ and $\vec k \:\equiv\: ( \vec k_\perp, k_z)^T \:=\: (\varkappa, \varphi_k, k_z)$\], the topological charge $m$ as well as the helicity $\Lambda_X$. ]{}
Here, we shall restrict ourselves to the representation of the vector potential of these Bessel beams in terms of plane waves and how we can distinguish between their spin-angular (polarization) and orbital angular momenta. Following Refs. [@Jentschura:PRL2011; @Ivanov:PRA2011; @Matula:JPB2013], we can write the vector potential $$\begin{aligned}
\label{eq:def:vortex}
\vec A_X(\vec r) &= \int \! {\frac{\mathrm d^2 \vec k_\perp}{(2\pi)^2}}\: {a_{\varkappa m}(\vec k_\perp)}\:
e^{i \vec k \cdot \vec r } \, \vec \varepsilon_{\vec k \Lambda_X } \,,\end{aligned}$$ as a superposition of plane waves with wave vectors $\vec k = (\vec k_\perp,k_z)^T$ and the Fourier coefficients $$\begin{aligned}
\label{eq:akm}
{a_{\varkappa m}(\vec k_\perp)}&= \sqrt{\frac{2\pi}{\varkappa}} \: (-i)^m e^{i m \varphi_k} \: \delta( k_\perp - \varkappa ) \, ,\end{aligned}$$ and where $\vartheta_k = \arctan \varkappa /k_z$ is the so-called cone *opening* angle. Like in the atomic case, the (quantum) number $m$ determines the projection of the OAM or the so-called topological charge and ${\vec \varepsilon_{\vec k \Lambda_X}}$ the polarization (vector) of the plane-wave components. Obviously, the polarization vector must depend explicitly not only on the helicity $\Lambda_X$ of the plane-wave components, but also on the angles $\vartheta_k$ and $\varphi_k$, $$\begin{aligned}
\label{eq:xuvpol}
\vec \varepsilon_{\vec k \Lambda_X}
&= \frac{e^{i \Lambda_X \varphi_k}}{\sqrt{2}}
\left( \begin{matrix}
\cos \vartheta_k \cos \varphi_k - i \Lambda_X \sin \varphi_k \\
\cos \vartheta_k \sin \varphi_k + i \Lambda_X \cos \varphi_k \\
- \sin \vartheta_k
\end{matrix}
\right) \, ,\end{aligned}$$ due to the transversality condition $\vec k \cdot {\vec \varepsilon_{\vec k \Lambda_X}}\: = \: 0$. Indeed, this definition of the polarization vector ${\vec \varepsilon_{\vec k \Lambda_X}}$ ensures that, in the limit of small opening angles $\vartheta_k \to 0$, we obtain the usual polarization vectors for circularly polarized plane waves $$\begin{aligned}
\label{eq:xuvpol.plane}
{\vec \varepsilon_{\vec k \Lambda_X}}\stackrel{\vartheta_k\to0}{\to} \frac{1}{\sqrt2} \: (1,i\Lambda_X,0)^T \,.\end{aligned}$$ It can be shown [@Matula:JPB2013] that the polarization vector ${\vec \varepsilon_{\vec k \Lambda_X}}$ from Eq. is an eigenvector also of the $z$-component ${\hat{J}}_z \,=\, {\hat{L}}_z + {\hat{S}}_z$ of the *total* angular momentum operator with eigenvalues $m_J = \Lambda_X$: $\: {\hat{J}}_z \: {\vec \varepsilon_{\vec k \Lambda_X}}\,=\, \Lambda_X \: {\vec \varepsilon_{\vec k \Lambda_X}}$. With this definition of ${\vec \varepsilon_{\vec k \Lambda_X}}$, the Bessel beam is constructed as an eigenfunction of the *total* angular momentum projection $\hat J_z$ with eigenvalue $m_J = m + \Lambda_X$ [^1]: $$\hat J_z \vec A_X (\vec r) = (m + \Lambda_X) \vec A_X(\vec r)\,.$$
For the sake of completeness, let us write here the vector potential of the XUV Bessel beam explicitly in cylindrical co-ordinates $$\begin{gathered}
\label{eq:bessel}
\vec A_X(\vec r) = \sum_{m_s=-1,0,1} \vec \eta_{m_s} \:
\sqrt{ \frac{\varkappa}{2\pi} } \: c_{m_s} \: e^{i k_z z} \: i^{\Lambda_X-m_s} \\[0.1cm]
\times \: e^{i(m+\Lambda_X-m_s)\varphi} \: J_{m+\Lambda_X-m_s} (\varkappa r_\perp) \,.\end{gathered}$$ In this expression, $\vec \eta_0 = (0,0,1)$ and $\vec \eta_{\pm1} = (1,\pm i,0)/\sqrt{2}$ denote the (spherical) unit vectors, and the coefficients are $c_0 = - (\sin \vartheta_k)/\sqrt2$ and $c_{\pm1} = (\cos\vartheta_k \pm \Lambda_X )/2$, respectively. As seen from Eq. , a Bessel beam consist of three terms with topological charges $m + \Lambda_X$ and $m+\Lambda_X \pm 1$. The relative weight of these terms depend on the opening angle $\vartheta_k$, and only [the one]{} with the topological charge $m$ remains *non-zero* for paraxial beams, i.e. if $\vartheta_k \ll 1$ [^2].
Transition amplitude for a well-localized atom in a XUV Bessel beam
-------------------------------------------------------------------
We can use the vector potential of the XUV Bessel beam, Eq. , to evaluate the transition amplitude for the two-color ATI of atoms and ions and for the emission of photoelectrons with well-defined *asymptotic* momentum $\vec p$. To do so, we also need to specify the position of the atom with regard to the beam axis, i.e. in terms of an impact parameter vector $\vec b \:\equiv\: (b, \varphi_b = 0, b_z = 0)$. If $\vec r$ denotes the electronic coordinate with respect to the atomic nucleus, that is the center of the atomic potential, we have to replace $\vec r \,\to\, \vec b + \vec r$ in the electron-photon interaction operator. We therefore see that, for vortex beams, the transition amplitude generally depends on the location of the atom within the beam,
as indicated by the subscript $\vec b$ in the notation of the transition amplitude:
$$\begin{aligned}
\label{eq:twisted:amplitude:superposition}
{\mathcal T}_{\vec b}(\vec p) & = \int \! {\frac{\mathrm d^2 \vec k_\perp}{(2\pi)^2}}\: {a_{\varkappa m}(\vec k_\perp)}\: e^{i \vec k \cdot \vec b} \:
{\mathcal T}^{\rm pw}({\vec p,\vec k}) \,.\end{aligned}$$
It is readily expressed as a superposition of typical SFA plane-wave amplitudes
[ $$\begin{gathered}
\label{eq:amplitude:planewave}
{\mathcal T}^{\rm pw}(\vec p, \vec k) = -i \int \! { {\mathrm d} }t \, \vec q(t) \cdot {\vec \varepsilon_{\vec k \Lambda_X}}\, e^{i(E_B-\omega_X) t - i S_V(t)}\\
\times \, \mem{ \vec q(t) }{ \: e^{i\,\vec k \cdot \vec r} \:}{ \phi_0 } \, ,\end{gathered}$$ just weighted by the Fourier coefficients ${a_{\varkappa m}(\vec k_\perp)}$ of the Bessel beam and the given phase factor $e^{i\vec b\cdot \vec k}$. An analogue superposition of plane-wave amplitudes was found also for the single-photon ionization by light from a vortex beam [@Surzhykov:PRA2015], and this remains true when adding an assisting laser field, at least *within* the SFA. Let us mention finally that all the time-dependence resides in the plane-wave amplitudes ${\mathcal T}^{\rm pw}(\vec p,\vec k)$. ]{}
### Time-dependence of the Volkov phase: Sideband structure
To obtain and describe the (well-known) sideband structures in the energy spectrum of the emitted photoelectrons in the two-color ATI, we next need to specify the vector potential of the strong NIR laser pulse that enters the plane-wave amplitude ${\mathcal T}^{\rm pw}(\vec p,\vec k)$. For the strong laser pulse, we here apply the vector potential $$\begin{aligned}
\vec A_L(t) &= A_L \left( \begin{matrix}
\cos \omega_L t \\[0.1cm]
\Lambda_L \sin \omega_L t \\[0.1cm]
0
\end{matrix}
\right) \, ,\end{aligned}$$ of a plane wave with laser frequency $\omega_L$, helicity $\Lambda_L$ and field amplitude $A_L$. In the plane-wave amplitude of [Eq. ]{}, moreover, we can cast the Volkov phase factor into the form $$\begin{aligned}
\label{eq:jacobi:anger}
e^{-iS_V(t)}
= e^{i ( \frac{p^2}{2} + U_p ) t } \sum_{\ell={-\infty}}^\infty
J_\ell(\alpha_L) e^{-i \ell (\omega_L t - \Lambda_L \varphi_p)} \end{aligned}$$ by applying the Jacobi-Anger expansion [@book:Watson], and where $\alpha_L = ( A_L\, p \sin \vartheta_p ) / \omega_L $ just refers to the amplitude of the classical oscillation of an electron in the laser field, while $U_p \,=\, A_L^2/2$ is the ponderomotive potential.
With this reformulation of the Volkov phase factor in Eq. , the (strong-field) transition amplitude ${\mathcal T}^{\rm pw}(\mathbf p, \mathbf k)$ therefore becomes
$$\begin{aligned}
\label{eq:amplitude:jacobi}
{\mathcal T}^{\rm pw}(\mathbf p, \mathbf k)
&= - i \sum_\ell J_\ell(\alpha_L) \: e^{i \ell \Lambda_L \varphi_p} \,
\int \! { {\mathrm d} }t \: e^{i(\frac{p^2}{2}+U_p+E_B-\omega_X - \ell \omega_L) t } \:
\vec q(t) \cdot {\vec \varepsilon_{\vec k \Lambda_X}}\: \mem{\vec q(t) }{\, e^{i\vec k \cdot \vec r}\,}{ \phi_0 } \, .\end{aligned}$$
To further simplify this amplitude we next have to analyze the scalar product between the kinetic momentum $\vec q(t)$ of the electron and the polarization vector of the twisted light ${\vec \varepsilon_{\vec k \Lambda_X}}$ in the following subsection.
Before we shall continue, let us note, that the summations in Eqs. and formally runs from $\ell = - \infty \ldots \infty$. In practice, however, just a finite number of sidebands, $\ell_\mathrm{min} \leq \ell \leq \ell_\mathrm{max}$, can be resolved experimentally, while the magnitude of these sidebands decays exponentially beyond these cut-off values. These cut-off values can be determined by either a saddle point analysis of the Volkov phase [@Lewenstein:PRA1994; @Zhang:PRA2013; @Seipt:PRA2015; @Seipt:NJP2016] or by just making use of the properties of the Bessel functions [@book:Watson]. From the prior analysis, we have found these cut-off values as $$\begin{aligned}
\label{eq:cutoffs}
\ell_\mathrm{max/min}
= \frac{A_L^2\sin^2\vartheta_p}{\omega_L}
\pm \sqrt{ \frac{A_L^4 \sin^4 \vartheta_p}{\omega_L^2}
+ 2 \frac{A_L^2 \sin^2\vartheta_p}{\omega_L} \: (\omega_X- E_B - U_p)} \,,\end{aligned}$$
and where the upper/lower sign refer to the max/min values.
### Angular dependence of the photoelectron emission in the transition amplitude
The angular distribution of the photoelectrons emitted in the two-color ATI process is mainly determined by the scalar product $\vec q(t) \cdot {\vec \varepsilon_{\vec k \Lambda_X}}$ in the plane-wave amplitudes . This scalar product becomes maximum when the *kinetic* momentum of the photelectron $\vec q(t) \,=\, \vec p - \vec A_L(t)$ is, at the moment of the ionization, parallel to the polarization vector of the XUV field. We remember that this kinetic momentum $\vec q(t) $ differs from the conserved *canonical* momentum $\vec p$ [as long as the electron is inside the laser pulse.]{}
#### Plane Waves: {#subsubsect:theory:plane}
Here, let us first (re-)consider the scalar product $\vec q(t) \cdot {\vec \varepsilon_{\vec k \Lambda_X}}$ for the case of circularly polarized plane waves [@Kazansky:PRA2012]. If the XUV pulse propagates for instance along the $z$-direction, $\vec k = k \vec e_z$, we can apply the [plane-wave limit of the]{} XUV polarization vector from Eq. and readily obtain for the scalar product $$\begin{aligned}
\label{eq:scalarproduct:plane}
\vec q(t) \cdot {\vec \varepsilon_{\vec k \Lambda_X}}&= \frac{p}{\sqrt 2} \sin \vartheta_p e^{i\Lambda_X \varphi_p}
-\frac{ A_L}{\sqrt 2} e^{i\Lambda_X\Lambda_L \omega_L t} \,.\end{aligned}$$ Moreover, if we combine this expression with Eq. , the *plane-wave* transition amplitude reads as
$$\begin{aligned}
\label{eq:amplitude:plane}
{\mathcal T}^{\rm pw} (\vec p, \vec k = k \vec e_z)
&= -i \sum_\ell \mathcal F_\ell(\Lambda_L,\Lambda_X) \, e^{i ( \ell \Lambda_L +\Lambda_X ) \varphi_p } \:
\int \!{ {\mathrm d} }t \, e^{i(\frac{p^2}{2} + U_p + E_B-\omega_X - \ell \omega_L) t }
\mem{ \vec q(t) }{\, e^{i\vec k \cdot \vec r}\,}{ \phi_0 } \,,\end{aligned}$$
where the *sideband* amplitudes $$\begin{aligned}
\label{eq:Fl:plane}
\mathcal F_\ell(\Lambda_L,\Lambda_X)
&= \frac{1}{\sqrt{2}}
\Big( J_\ell (\alpha_L) p \sin \vartheta_p - A_L J_{\ell +\Lambda_L \Lambda_X} (\alpha_L) \Big) \,\end{aligned}$$ describe the strength and the angular distribution of the photoelectrons of the $\ell$-th sideband (ATI-peak). In order to arrive at Eqs. and , we have shifted the summation variable $\ell$ in the second term of Eq. .
From the sideband amplitude , we immediately find: (i) Only the central photoline ($\ell=0$) occurs with the typical $\mathbb P(\vartheta_p) \propto \sin^2 \vartheta_p$ angular dependence if the laser field vanishes, i.e. for $A_L\to 0$ and $\alpha_L\to 0$. Moreover, (ii) the second term of $\mathcal F_\ell$ in Eq. contains the product $\Lambda_L \, \Lambda_X$ of the spin angular momenta (helicities) of the XUV and the NIR laser fields in the *order* of the Bessel function $J$. Therefore, the angular distribution of the photoelectrons differ from each other if $\Lambda_X$ and $\Lambda_L$ have either equal or opposite signs. Indeed, it is the sign of $\Lambda_L \, \Lambda_X$ that leads to the circular dichroism in the two-color photoionization of atoms by plane-wave radiation [@Kazansky:PRA2012; @Mazza:NatCommun2014].
#### XUV Bessel beams:
Of course, the same scalar product in the plane-wave amplitude becomes much more complex for a vortex beam since it now depends explicitly on the direction of the momentum vector $\vec k \:=\: \vec k(\vartheta_k,\varphi_k)$ of the plane-wave components, forming a cone in momentum space. Using expression , this product can be evaluated as \[compare with Eq. \]
$$\begin{gathered}
\label{eq:scp:twisted}
\vec q(t) \cdot {\vec \varepsilon_{\vec k \Lambda_X}}= \frac{p}{\sqrt 2}\: \left[ \sin \vartheta_p e^{i\Lambda_X \varphi_p}
\:-\: 2 \sin \vartheta_p \sin^2 \frac{\vartheta_k}{2} \cos (\varphi_p
\:-\: \varphi_k) e^{i \Lambda_X \varphi_k}
\:-\: \cos \vartheta_p \sin \vartheta_k e^{i\Lambda_X \varphi_k}
\right] \\[0.15cm]
- \frac{A_L}{\sqrt 2}\: \left[ e^{i \Lambda_X \Lambda_L \omega_L t}
\:-\: \sin^2 \frac{\vartheta_k}{2}
\left( e^{i \omega_Lt } \, e^{i(\Lambda_X - \Lambda_L) \varphi_k}
+ e^{-i \omega_L t} \, e^{i(\Lambda_X +\Lambda_L) \varphi_k}
\right)
\right] \,.\end{gathered}$$
If we substitute this expression into Eq. , the transition amplitude for the two-color ATI of an atom at position $\vec b$ by a XUV Bessel beam can be written as the superposition of the plane-wave transition amplitudes $$\begin{aligned}
\label{eq:amplitude:pw:2}
{\mathcal T}^{\rm pw} (\vec p, \vec k)
&= - i \sum_\ell \mathcal F_\ell(\vartheta_k , \varphi_k;\Lambda_X,\Lambda_L) \:
e^{i(\ell \Lambda_L + \Lambda_X)\varphi_p} \:
\int \! { {\mathrm d} }t \: e^{i(\frac{p^2}{2} + U_p + E_B-\omega_X - \ell \omega_L ) t }
\, \mem{ \vec q(t)}{\: e^{i\vec k \cdot \vec r } \:}{\phi_0} \,,\end{aligned}$$ and with the *modified* sideband amplitudes $$\begin{gathered}
\label{eq:sideband:tw}
\mathcal F_\ell( \vartheta_k , \varphi_k ; \Lambda_X, \Lambda_L)
= J_\ell(\alpha_L) \frac{p}{\sqrt2}
\left[ \sin \vartheta_p \:-\: 2 \sin \vartheta_p \sin^2 \frac{\vartheta_k}{2} \cos (\varphi_k
\:-\: \varphi_p) e^{i \Lambda_X (\varphi_k - \varphi_p)}
\:-\: \cos \vartheta_p \sin \vartheta_k e^{i \Lambda_X ( \varphi_k - \varphi_p)}
\right] \\[0.15cm]
- \frac{A_L}{\sqrt{2}}
\left[ J_{\ell + \Lambda_X \Lambda_L}(\alpha_L)
\:-\: \sin^2 \frac{\vartheta_k}{2}
\left\{ J_{\ell+1} (\alpha_L) e^{i(\Lambda_X-\Lambda_L)(\varphi_k-\varphi_p)}
\:+\: J_{\ell-1}(\alpha_L) e^{i(\Lambda_X+\Lambda_L)(\varphi_k-\varphi_p)}
\right\}
\right] \, ,\end{gathered}$$ which now depends on the particular direction of the wave vector $\vec k$.
### Analytical time integration of the two-color ATI transition amplitude
The plane-wave transition amplitude still contains a time integration which cannot be performed in general. However, this time integral can be solved analytically if we assume a sufficiently weak assisting NIR laser field, $A_L \ll p$, so that the kinetic momentum $\vec q(t)$ of the photoelectron can be reasonably well approximated by the canonical momentum $\vec p$ in the atomic matrix. This then results also in time-independent atomic matrix elements [@Kazansky:PRA2010; @Kazansky:PRA2012]. In the dipole approximation, moreover, we can approximate these matrix elements by $$\begin{aligned}
\mem{\vec q(t) }{\, e^{i \vec k \cdot \vec r} \,}{ \phi_0 } \:\simeq\: {\langle \vec p | \phi_0 \rangle } \, ,
\label{eq:atomic}\end{aligned}$$ which is valid almost everywhere apart from the region close to the vortex line. For $\vec b \,=\, 0$, in contrast, the integral over the transverse momentum $\vec k_\perp$ in vanishes when the electric-dipole approximation is employed, and the leading contribution to the twisted-wave amplitude ${\mathcal T}_{\vec b \simeq 0} (\vec p)$ will then arise from higher-order multipoles [@Surzhykov:PRA2015; @Schmiegelow:NatureCommun2016].
With these assumptions about the NIR field, we can perform the time integration in the plane-wave amplitude $$\begin{gathered}
\label{eq:time:integral}
\int \! { {\mathrm d} }t \, e^{i t ( \frac{p^2}{2} + U_p + E_B - \omega_X - \ell \omega_L )} \\
= 2 \pi \, \delta( p^2/2 + U_p + E_B - \omega_X - \ell \omega_L ) \,.\end{gathered}$$ Here, the delta function ensures the energy conservation in this two-color interaction process and shows that the kinetic energy of the photoelectrons becomes discrete for sufficiently weak fields. In each of these *sidebands* of the main photo line (that arise from the ionization by the XUV pulse), the modulus of the electron momenta is constant, $|\vec p| = p \to p_\ell = \sqrt{ 2 (\omega_X + \ell \omega_L - E_B - U_p ) }$, while these electrons may still exhibit an (angular) distribution as function $\vartheta_p$ and $\varphi_p$. Using expression , the transition amplitude for two-color ATI of an atom at position $\vec b$ by a XUV Bessel beam now becomes $$\begin{aligned}
\label{eq:amplitude:final1}
{\mathcal T}_{\vec b}(\vec p)
&= 2 \pi \sum_\ell \: \delta(p^2/2 + U_p + E_B - \omega_X - \ell \omega_L) \: {\mathcal T}^{(\ell)} \, ,\end{aligned}$$ and where $$\begin{aligned}
\label{eq:amplitude:final2}
{\mathcal T}^{(\ell)}
&= {\langle \vec p_\ell | \phi_0 \rangle } \: \int \! {\frac{\mathrm d^2 \vec k_\perp}{(2\pi)^2}}\, e^{i \vec k \cdot \vec b} \, {a_{\varkappa m}(\vec k_\perp)}\mathcal F_\ell(\vartheta_k,\varphi_k;\Lambda_X,\Lambda_L) \end{aligned}$$ [are]{} often referred to as *partial amplitudes*. As seen from expression , the angular distribution of the photoelectrons is now determined by a convolution of the sideband amplitudes $\mathcal F_\ell$ with the Fourier coefficients of the vortex Bessel beam ${a_{\varkappa m}(\vec k_\perp)}$ from Eq. and a phase factor that just contains the impact vector $\vec b$. Indeed, the expressions and are one of our major results of this work, although they still describe the transition amplitude for a single atom at some (fixed) position $\vec b$ with regard to the beam axis.
Photoionization probability of localized and macroscopically extended targets {#sect:target}
-----------------------------------------------------------------------------
We can use the two-color ATI amplitude to express the photoionization probability (per unit time) for an atom at position $\vec b$ within a vortex beam by $$\begin{aligned}
\mathbb P_{\vec b}(\vec p) &= \frac{1}{T} \left| {\mathcal T}_{\vec b} (\vec p) \right|^2 \nonumber \\
&= 2\pi \sum_\ell \delta(p^2/2+U_p+E_B-\omega_X - \ell \omega_L)
\mathbb P^{(\ell)}_{\vec b} (\vec p) \, ,
\label{eq:probability:b2}\end{aligned}$$ if $T$ denotes here the interaction time of the atom with the two-color field, and if we make use of the usual interpretation of the delta function $\delta(0) = T/2\pi$ in the second line. Expression shows that the photoionization probability is a sum of partial probabilities $$\begin{aligned}
\label{eq:probability:partial}
\mathbb P^{(\ell)}_{\vec b}(\vec p) = \left|{\mathcal T}^{(\ell)} \right|^2\end{aligned}$$ that describe the individual sidebands in the photoelectron spectrum. The partial probabilities still refer, as before, to a single atom at impact vector $\vec b$ with regard to the beam axis. To further analyze and compare the photoelectron spectra and angular distribution with those obtained experimentally, we need to know (or assume) also the distribution of atoms in the overall cross section of the Bessel beam.
### Macroscopically extended targets
If, for example, the twisted Bessel beam interacts with a homogeneous and (infinitely in the cross section of the beam) *extended* target of atoms, we have to average the partial photoionization probabilities $\mathbb P^{(\ell)}_{\vec b}(\vec p)$ from Eq. incoherently over all impact vectors $\vec b$, $$\begin{aligned}
\label{eq:def:probability:infinite}
\mathbb P^{(\ell)} ( {\vec p} ) = \int \! { {\mathrm d} }^2 \vec b \;\: \mathbb P^{(\ell)}_{\vec{b}}(\vec p) \, ,\end{aligned}$$ in order to obtain the partial photoionization probabilities, and the same is true for the total photoionization probability . Using Eqs. and , we then obtain
$$\begin{aligned}
\mathbb P^{(\ell)} (\vec p) & = |{\langle \vec p_\ell | \phi_0 \rangle }|^2
\int \! {\frac{\mathrm d^2 \vec k_\perp}{(2\pi)^2}}\, \frac{{ {\mathrm d} }^2 \vec k'_\perp}{(2\pi)^2} \, { {\mathrm d} }^2 \vec b \;\:
a^*_{\varkappa m}(\vec k'_\perp) \: {a_{\varkappa m}(\vec k_\perp)}\: e^{i \vec b\cdot (\vec k - \vec k')} \:
\mathcal F_\ell^*(\vartheta_{k'},\varphi_{k'} {;\Lambda_X,\Lambda_L} )
\: \mathcal F_\ell(\vartheta_k,\varphi_k {;\Lambda_X,\Lambda_L} ) \,.\end{aligned}$$
Since the impact vector occurs here only in the exponential, $e^{i \vec b \cdot (\vec k - \vec k') }$, the integration over $\vec b$ just gives rise to a delta function in momentum space, and the partial photoionization probability of sideband $\ell$ becomes $$\begin{aligned}
\mathbb P^{(\ell)}(\vec p)
&= |{\langle \vec p_\ell | \phi_0 \rangle }|^2 \,\int \frac{{ {\mathrm d} }\varphi_k}{2\pi} \:
|\mathcal F_\ell( \vartheta_k, \varphi_k; \Lambda_X , \Lambda_L ) |^2 \, .\end{aligned}$$ [Employing the expression for]{} the sideband amplitude , we can now, furthermore, perform the integral over the azimuthal angle $\varphi_k$ and finally obtain for the partial photoionization probability $$\begin{gathered}
\label{eq:probability:infinite}
\mathbb P^{(\ell)}(\vec p)
= |{\langle \vec p_\ell | \phi_0 \rangle }|^2
\left\{ \frac{p^2_\ell}{2} J_\ell^2
\left[ \sin^2 \vartheta_p \left( 1 - 2 \sin^2 \frac{\vartheta_k}{2} + 2 \sin^4 \frac{\vartheta_k}{2} \right)
\:+\: \cos^2 \vartheta_p \sin^2 \vartheta_k
\right] \right. \\[0.15cm]
\;-\; p_\ell \sin \vartheta_p A_L J_\ell \:
\left[ J_{\ell + \Lambda_X \Lambda_L} \left( 1 - 2 \sin^2 \frac{\vartheta_k}{2} \right)
\:+\: (J_{\ell+1}+J_{\ell-1}) \sin^4 \frac{\vartheta_k}{2}
\right] \\[0.15cm]
\left. + \frac{A_L^2}{2}
\left[ J_{\ell +\Lambda_X \Lambda_L}^2 \left( 1 - 2 \sin^2 \frac{\vartheta_k}{2} \right)
\:+\: (J_{\ell+1}^2+J_{\ell-1}^2) \sin^4 \frac{\vartheta_k}{2}
\right]
\right\} \; .\end{gathered}$$
Obviously, this probability depends on the cone opening angle $\vartheta_k$ of the (vortex) Bessel beam, while it is independent of the topological charge $m$ for a macroscopically extended target.
![Illustration of a localized atomic target of size $\sigma_b$ that is displaced by the impact vector $b_0$ with regard to the center of the vortex beam.[]{data-label="fig:beam:target"}](fig2.pdf){width="0.89\columnwidth"}
### Localized targets
Another (localized) target refers to a small cloud of atoms that is centered around the impact vector $\vec b_0$ in a plane perpendicular to the beam axis. We here assume a normalized Gaussian distribution of target atoms
$$\begin{aligned}
\rho(\vec b) = \frac{1}{\sqrt{ 2\pi } \sigma_b } \exp \left\{ -\frac{(\vec b - \vec b_0)^2}{2 \sigma_b^2} \right\} \, \end{aligned}$$
where $\sigma_b$ denotes the (r.m.s.) size of the target, cf. Fig. \[fig:beam:target\]. Without loss of generality, moreover, we may assume that the impact vector $\vec b_0 = b_0 \vec e_x$ defines the $x$-axis and, hence, the angle $\varphi_p$ in the angular distribution of the photoelectrons below (and with the $z$-axis along the beam). For such a localized target with distribution [$\rho(\vec b)$]{}, the partial photoionization probability becomes $$\begin{aligned}
\label{eq:probability:cloud}
\mathbb P^{(\ell)}_\rho(\vec p)
&= \int \! {{ {\mathrm d} }^2 \vec b} \, \rho(\vec b) \: \mathbb P_{\vec b}^{(\ell)} (\vec p) \, ,\end{aligned}$$ and where the integration over the target distribution below will be performed numerically.
Dichroism in two-color fields {#sect:dichroism}
-----------------------------
In the previous section, we saw how the partial photoionization [probabilities , and describe]{} the yield of photoelectrons for a given sideband, as a function of the two emission angles $\vartheta_p$ and $\varphi_p$ [for different kinds of targets]{}. Of course, these probabilities also depend on the spin- and orbital angular momenta of the incident XUV and assisting NIR laser fields. To further understand how the coupling of these angular momenta affects the relative photoionization probabilities, we may resort to different kinds of *dichroism* signals as often used in describing the interaction of light with atoms, molecules and solids [@Kazansky:PRL2011; @Mazza:NatCommun2014; @Lux:AngewChemie2012].
### Circular dichroism for plane-waves
Let us start from the (atomic) *circular* dichroism which has been frequently used in characterizing the photoelectron emission if both, the XUV and the assisting NIR fields are described by plane waves. For two plane waves, as shown in Sect. \[subsubsect:theory:plane\], the two-color ATI amplitude and, hence, the corresponding photoionization probability only depends on the product of the two spin angular momentum (SAM) projections, i.e. the helicities of the XUV and laser photons. While there are four possible combinations of these helicities, only two $\Lambda_X\Lambda_L \:=\: \pm1$ are distinguishable from each other. For the interaction of atoms with two plane waves, we can therefore define just *one* dichroism signal, $$\begin{aligned}
\label{eq:def:CD}
\mathrm{CD}
= \frac{ \mathbb P({\vec p};\Lambda_X , \Lambda_L)
\;-\; \mathbb P (\vec p; \Lambda_X , -\Lambda_L )
}{ \mathbb P(\vec p; \Lambda_X,\Lambda_L ) \:+\: \mathbb P (\vec p; \Lambda_X, -\Lambda_L )} \,,\end{aligned}$$ commonly known also as circular dichroism [@Kazansky:PRA2012], and which is a function of the photoelectron emission angles $\vartheta_p$ and $\varphi_p$, respectively. This circular dichroism can be defined uniquely for each sideband as long as the incident light beams are sufficiently monochromatic.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- ------------------------------------------------------ ----- ------------------------------------------------------------------------------------------------------------------------------
\[-0.25cm\]
\[-0.25cm\] Dichroism due to a flip of … Definition
\[0.1cm\]
\[-0.25cm\] the helicity of the assisting NIR laser field. $\displaystyle {\mathcal D^{(\Lambda_L)}}$ $=$ [$\frac{\mathbb P(\vec p, m, \Lambda_X,\Lambda_L) - \mathbb P(\vec p, m, \Lambda_X,-\Lambda_L)
}{\mathbb P(\vec p, m, \Lambda_X,\Lambda_L) + \mathbb P(\vec p, m, \Lambda_X,-\Lambda_L) }$ ]{}
\[0.3cm\] the helicity of the XUV photons. $\displaystyle {\mathcal D^{(\Lambda_X)}}$ $=$ [$ \frac{\mathbb P(\vec p, m, \Lambda_X,\Lambda_L) - \mathbb P(\vec p, m, -\Lambda_X,\Lambda_L)
}{\mathbb P(\vec p, m, \Lambda_X,\Lambda_L) + \mathbb P(\vec p, m, -\Lambda_X,\Lambda_L) }$ ]{}
\[0.3cm\] the projection of the orbital angular momentum. $\displaystyle {\mathcal D^{(m)}}$ $=$ [$ \frac{\mathbb P(\vec p, m, \Lambda_X,\Lambda_L) - \mathbb P(\vec p, -m, \Lambda_X,\Lambda_L)
}{\mathbb P(\vec p, m, \Lambda_X,\Lambda_L) + \mathbb P(\vec p, -m, \Lambda_X,\Lambda_L) }$ ]{}
\[0.3cm\] the helicity and the orbital angular momentum of the XUV Bessel beam. This is equivalent to just a flip of the projection of the total angular momentum. $\displaystyle {\mathcal D^{(m\Lambda_X)}}$ $=$ [$ \frac{\mathbb P(\vec p, m, \Lambda_X,\Lambda_L) - \mathbb P(\vec p,- m, - \Lambda_X,\Lambda_L)
}{\mathbb P(\vec p, m, \Lambda_X,\Lambda_L) + \mathbb P(\vec p,- m, - \Lambda_X,\Lambda_L) }$ ]{}
\[0.3cm\] the helicities of both the laser and XUV photons. For two plane waves this dichroism signal is always zero because of the symmetry. $\displaystyle {\mathcal D^{(\Lambda_X\Lambda_L)}}$ $=$ [$ \frac{\mathbb P(\vec p, m, \Lambda_X,\Lambda_L) - \mathbb P(\vec p, m, - \Lambda_X,-\Lambda_L)
}{\mathbb P(\vec p, m, \Lambda_X,\Lambda_L) + \mathbb P(\vec p, m, - \Lambda_X,-\Lambda_L) }$ ]{}
\[0.3cm\] the projection of the orbital angular momentum of the Bessel beam and of the helicity of the laser field. $\displaystyle {\mathcal D^{(m\Lambda_L)}}$ $=$ [$ \frac{\mathbb P(\vec p, m, \Lambda_X,\Lambda_L) - \mathbb P(\vec p,- m, \Lambda_X,-\Lambda_L)
}{\mathbb P(\vec p, m, \Lambda_X,\Lambda_L) + \mathbb P(\vec p,- m, \Lambda_X,-\Lambda_L) }$ ]{}
\[0.3cm\] all three projections of the angular momenta simultaneously. $\displaystyle {\mathcal D^{(m\Lambda_X\Lambda_L)}}$ $=$ [$ \frac{\mathbb P(\vec p, m, \Lambda_X,\Lambda_L) - \mathbb P(\vec p,- m, -\Lambda_X,-\Lambda_L)
}{\mathbb P(\vec p, m, \Lambda_X,\Lambda_L) + \mathbb P(\vec p,- m, -\Lambda_X,-\Lambda_L) } $ ]{}
\[0.3cm\]
-------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- ------------------------------------------------------ ----- ------------------------------------------------------------------------------------------------------------------------------
### Dichroism signals for (vortex) Bessel beams
For vortex Bessel beams, the photoionization probability depends not only on the SAM of the XUV ($\Lambda_X=\pm1$) and laser beams ($\Lambda_L=\pm1$) but also on the *orbital* angular momentum $\pm m$ of the XUV photons. With three angular momenta, we can form eight combinations of $(m,\Lambda_X,\Lambda_L)$ by just changing the sign of one or more of these quantum numbers. This enables us to define *seven* different dichroism signals for the two-color ionization of atoms by a vortex and a plane-wave beam since one of the combinations, $\mathbb P (\vec p,|m|,\Lambda_X=+1,\Lambda_L=+1)$, should occur as reference. For example, the dichroism that is associated with a flip of the projection of the orbital angular momentum $\pm m$ is easily defined by $$\begin{aligned}
{\mathcal D^{(m)}}
&= \frac{\mathbb P(\vec p,m,\Lambda_X,\Lambda_L) \:-\: \mathbb P (\vec p,-m, \Lambda_X,\Lambda_L)
}{\mathbb P (\vec p,m,\Lambda_X,\Lambda_L) \:+\: \mathbb P(\vec p,-m ,\Lambda_X,\Lambda_L)} \,.\end{aligned}$$ Very similarly, we can define all the other dichroism signals as associated with some flip in the helicity and/or OAM quantum numbers, and which are displayed explicitly in Table \[tab:1\] . As for the circular dichroism, all these (seven) dichroism signals generally depend for a localized target on the photoelectron emission angles $\vartheta_p$ and $\varphi_p$ as well as on the particular sideband $\ell$.
For sufficiently extended targets ($\sigma_b \varkappa \gg 1$), however, the photoionization probability and, hence, the dichroism signals above become independent of the (projection of the) orbital angular momentum or topological charge, $m$. This can be seen for instance from the analytical expression for the photoionization probability for infinitely extended targets , which is independent of $m$. For large targets, therefore, all the dichroism signals will depend just on the product $\Lambda_X\:\Lambda_L$ of the two helicities and, thus, all signals with $\Lambda_X\:\Lambda_L\:=\: +1$ must vanish in this limit, ${\mathcal D^{(\Lambda_X\Lambda_L)}} = {\mathcal D^{(m \Lambda_X \Lambda_L)}} = {\mathcal D^{(m)}} =0$. Moreover, all other signals with $\Lambda_X\:\Lambda_L\:=\: -1$ then coincide with the usual circular dichroism, ${\mathcal D^{(\Lambda_L)}} = {\mathcal D^{(\Lambda_X)}} = {\mathcal D^{(m \Lambda_X)}} = {\mathcal D^{(m\Lambda_L)}} $, cf. Eq. . For extended targets, a nonzero dichroism signal can be observed only if just *one* of the helicities $\Lambda_X$ or $\Lambda_L$ is changed.
{width="0.99\columnwidth"}
{width="95.00000%"}
{width="95.00000%"}
Results and Discussion {#sect:results}
======================
In the last section, we have analyzed the transition amplitude and photoionization probability for the two-color ATI of atoms by a vortex (Bessel) beam and combined with an intense plane-wave (NIR) laser field. Emphasize was placed here on the evaluation of this amplitude and the sideband structure of the [central photoline]{} due to the interaction of the emitted electrons with the NIR field. We also introduced various dichroism signals by flipping the projections of the spin and orbital angular momenta of the involved fields in order to quantify the dependence of the photoionization probability upon the angular momentum properties of the incident light beams.
As discussed above, the two-color ATI probability crucially depends for Bessel beams also on the size of the atomic target. To better understand the influence of this target size, detailed computations were performed for the photoionization of the $4s$ valence electron of Ca$^+$ ions with binding energy $E_B = \unit{11.7}{\electronvolt} = {0.43}\,{\rm a.u.}$. Simple core-Hartree wave functions in a screened potential have been applied to calculate all the necessary (one-electron) atomic matrix elements [@Fritzsche:CPC2012], cf. .
[In Fig. \[fig:probability\], we display the two-color ATI probability $\mathbb P^{(\ell)}(\vartheta_p)$ as function of the emission angle $\vartheta_p$ (horizontal axis) and the sideband number $\ell$ (vertical axis). The sideband $\ell$ gives directly the net number of laser photons from the NIR field that are either absorbed or emitted by the outgoing photoelectrons. The ATI probabilities are encoded by colors and are shown for an infinitely extended target. Results are compared for the two-color ATI by a plane-wave XUV beam (upper panel) as well as for a XUV Bessel beam with cone opening angles $\vartheta_k=0.2$ rad (middle panel) and $\vartheta_k=0.5$ rad (lower panel), respectively. In these computations, we applied a XUV beam with the rather high frequency $\omega_X = 3\,{\rm a.u.} = \unit{81.6}{\electronvolt}$ and for a NIR laser field with $\omega_L = {0.05}\,{\rm a.u.} = \unit{1.36}{\electronvolt}$ and amplitude $A_L=0.1$. As seen from Fig. \[fig:probability\], the photoelectron distributions [exhibit]{} an almond-like shape for which the largest number of sidebands occurs at $\vartheta_p \,\simeq\, \unit{90}{\degree}$, while only the main photoline ($\ell=0$) is seen along the beam axis, i.e. for $\vartheta=\unit{0}{\degree}$ and $\unit{180}{\degree}$. This shape of the photoelectron distributions is well predicted also by the semiclassical cutoffs, Eq. , as indicated by the white dotted curves in the figures.]{}
[For the two-color ATI by a plane-wave XUV beam, the calculated photoelectron distribution agrees qualitatively well with the calculations by Kazansky and coworkers [@Kazansky:PRA2012]. While no photoelectrons are seen in this case along the axis for plane-waves \[cf. Fig. \[fig:probability\] a\], this changes in the case of a twisted Bessel XUV beams in Figs. \[fig:probability\] b,c. For such Bessel beams, the photoionization probabilities along the beam axis increases with the cone opening angle $\vartheta_k$. We note that the plane-wave result is of course recovered in the paraxial approximation for $\vartheta_k\to 0$.]{}
[To analyze the localization effects of the target, we use the different dichroism signals as defined in Section \[sect:dichroism\] and Table \[tab:1\]. Figure \[fig:polar:dichroism\], for example, shows these dichroism signals as function of the emission angle $\vartheta_p$ and sideband number $\ell$ of the emitted electrons, and with the magnitude of the signals encoded by colors in the (seven) rows of the figure. We here applied a Bessel beam of the same frequency $\omega_X = \unit{81.6}{\electronvolt}$ and opening angle $\vartheta_k=0.2$ as in Fig. \[fig:probability\], and with the projection of the angular momentum $m=4$. Detailed computations are carried out for the three target sizes $\sigma_b=\unit{1}{\nano\metre}$ (left column), $\sigma_b=\unit{10}{\nano\metre}$ (middle column) and $\sigma_b=\unit{100}{\nano\metre}$ (right column), and for photoelectrons that are observed at the azimuthal angle $\varphi_p=\unit{90}{\degree}$ with regard to the impact vector $\vec b_0 = b_0 \, \vec e_x$ with $b_0=\unit{30}{\nano\metre}$ as the center of the target. While all the dichroism signals are quite different from each other for a small target (left column) and, hence, sensitive to the particular localization of the target, these differences become less pronounced as the target size increases. For target sizes (much) larger than the typical width of the rings in the Bessel beam, moreover, the dichroism signals approach the two limits: They either vanish identically if the product of the helicities of the two-color field is positive, $\Lambda_X\:\Lambda_L\:=\: +1$ [\[cf. the right panels of rows 3, 5 and 7\]]{}, or these signals coincide with the known circular dichroism for $\Lambda_X\:\Lambda_L\:=\: -1$ [\[cf. the right panels of rows 1, 2, 4, and 6\]]{}. Let us note also that the (usual) circular dichroism signal in row 1 appears to be rather insensitive to the size of the target. In fact, these dichroism signals do not depend much on the details of the applied matrix elements as, in the electric-dipole approximation, the prefactor in Eq. cancel in the ratio that is formed by any dichroism. Finally, the black dotted curves indicate the cut-off values of the number of sidebands as given analytically by Eq. .]{} [That means, while we can calculate a dichroism signal also outside the almond-shaped area, its measurement might be challenging since the photoionization probability is very small in these regions, cf. Fig. \[fig:probability\].]{}
[Due to the phase of the XUV Bessel beam, a localization of the target affects not only the (polar) angular emission of the photoelectrons but may results also in a non-trivial azimuthal distribution. Therefore, Fig. \[fig:azimuthal:dichroism\] shows the same as Fig. \[fig:polar:dichroism\] but here as function of the azimuthal angle $\varphi_p$ and for a Bessel beams with slightly higher photon energy $\omega_X = {5}\,{\rm a.u.} = \unit{136}{\electronvolt}$ and for a target centered at $b_0 = \unit{25}{\nano\metre}$. In this figure, the photoelectrons are assumed to be observed under the polar angle $\vartheta_p=\unit{45}{\degree}$ with regard to the beam axis. An azimuthal anisotropy of the ATI probabilities is found for the localized targets as it was obtained before for the azimuthal distribution of photoelectrons [@Matula:JPB2013]. This anisotropy of the ionization probabilities occurs of course also in the dichroism signals, while no azimuthal dependence appears for the usual circular dichroism (first row). As for the polar-angle dependence in Fig. \[fig:polar:dichroism\], all dichroism signals become either zero or simply approach the circular dichroism for sufficiently large targets. ]{}
Summary and Conclusions {#sect:summary}
=======================
[In this work, we investigated the two-color ATI of atoms and ions if light from a weak XUV Bessel beam is combined with a strong NIR laser field. While the emission of photoelectron occurs due to the weak XUV beam, the energy and angular distribution of the photoelectrons is affected by the plane-wave NIR field due to a net absorption or emission of one or several laser photons. Thus, the interaction of the atoms with such a two-color field results in sidebands to the normal photoline which are affected not only by the intensity and temporal structure of the NIR field but also by the location and extent of the atomic target as well as by the spin and orbital angular momenta of the two fields involved.]{}
[Emphasis in our analysis has been placed upon the energy spectra and angular emission of the photoelectrons as well as on the asymmetry in the photoelectron spectra if some of the SAM or OAM components of the fields are flipped relative to each other. For a XUV Bessel beam and a plane-wave NIR field, seven different dichroism signals can be defined. These signals differ for localized target but become either zero or coincide with the usual circular dichroism for macroscopically extended targets, similar as for Bessel beams with small opening angle.]{} Our investigation of two-color strong field ATI with XUV vortex Bessel beams and the discussion of the seven different dichroism signals opens up avenues for future investigations of the interaction of atomic and molecular targets with twisted light in the high-intensity regime.
Acknowledgments
===============
This work was supported by the DFG priority programme 1840, “Quantum Dynamics in Tailored Intense Fields”.
[^1]: Note the different definition of the vector ${\vec \varepsilon_{\vec k \Lambda_X}}$ in [@Matula:JPB2013] which furnishes a different interpretation of the quantum number $m = m_J$ as the TAM eigenvalue of the Bessel beam, but without [any]{} physical consequences.
[^2]: Strictly speaking, the orbital quantum number $m$ here refers to the dominant component of the three possible projections of the orbital angular momentum, and it coincides with the true OAM only in the paraxial limit $\vartheta_k\ll 1 $.
|
---
abstract: 'We study nonparametric Bayesian inference with location mixtures of the Laplace density and a Dirichlet process prior on the mixing distribution. We derive a contraction rate of the corresponding posterior distribution, both for the mixing distribution relative to the Wasserstein metric and for the mixed density relative to the Hellinger and $L_q$ metrics.'
address: |
Mathematical Institute\
Leiden University\
Niels Bohrweg 1\
2333 CA Leiden, Netherlands.\
author:
-
-
bibliography:
- 'nguyen.bib'
title: 'Posterior contraction rates for deconvolution of Dirichlet-Laplace mixtures'
---
Introduction
============
Consider statistical inference using the following hierarchical Bayesian model for observations $X_1,\ldots, X_n$:
1. A probability distribution $G$ on ${\mathbb{R}}$ is generated from the Dirichlet process prior $\operatorname{DP}(\alpha)$ with base measure $\alpha$.
2. An i.i.d. sample $Z_1,\ldots, Z_n$ is generated from $G$.
3. An i.i.d. sample $e_1,\ldots, e_n$ is generated from a known density $f$, independent of the other samples.
4. The observations are $X_i=Z_i+e_i$, for $i=1,\ldots, n$.
In this setting the conditional density of the data $X_1,\ldots, X_n$ given $G$ is a sample from the convolution $$p_G = f \ast G$$ of the density $f$ and the measure $G$. The scheme defines a conditional distribution of $G$ given the data $X_1,\ldots, X_n$, the *posterior distribution* of $G$, and consequently also posterior distributions for quantities that derive from $G$, including the convolution density $p_G$. We are interested in whether this posterior distribution can recover a “true" mixing distribution $G_0$ if the observations $X_1,\ldots, X_n$ are in reality a sample from the mixed distribution $p_{G_0}$, for some given probability distribution $G_0$.
The main contribution of this paper is for the case that $f$ is the Laplace density $f(x)=e^{-|x|}/2$. For distributions on the full line Laplace mixtures seem the second most popular class next to mixtures of the normal distribution, with applications in for instance speech recognition or astronomy ([@Kotzetal]) and clustering problem in genetics ([@bailey1994fitting]). For the present theoretical investigation the Laplace kernel is interesting as a test case of a non-supersmooth kernel.
We consider two notions of recovery. The first notion measures the distance between the posterior of $G$ and $G_0$ through the *Wasserstein metric* $$W_{k} (G, G') = \inf_{\gamma \in \Gamma (G, G')}
\Bigl(\int |x- y|^k \, d\gamma (x, y)\Bigr)^{1/k},$$ where $\Gamma(G, G')$ is the collection of all *couplings* $\gamma$ of $G$ and $G'$ into a bivariate measure with marginals $G$ and $G'$ (i.e. if $(x,y)\sim \gamma$, then $x \sim G $ and $y \sim G'$), and $k\ge 1$. The Wasserstein metric is a classical metric on probability distributions, which is well suited for use in obtaining rates of estimation of measures. It is weaker than the total variation distance (which is more natural as a distance on densities), can be interpreted through transportation of measure (see [@villani_optimal_2009]), and has also been used in applications such as as comparing the color histograms of digital images. Recovery of the posterior distribution relative to the Wasserstein metric was considered by [@nguyen_convergence_2013], within a general mixing framework. We refer to this paper for further motivation of the Wasserstein metric for mixtures, and to [@villani_optimal_2009] for general background on the Wasserstein metric. In the present paper we improve the upper bound on posterior contraction rates given in [@nguyen_convergence_2013], at least in the case of the Laplace mixtures, obtaining a rate of nearly $n^{-1/8}$ for $W_1$ (and slower rates for $k>1$). Apparently the minimax rate of contraction for Laplace mixtures relative to the Wasserstein metric is currently unknown. Recent work on recovery of a mixing distribution by non-Bayesian methods is given in [@zhang_cunhui_1990]. It is not clear from our result whether the upper bound $n^{-1/8}$ is sharp.
The second notion of recovery measures the distance of the posterior of $G$ to $G_0$ indirectly through the Hellinger or $L_q$-distances between the mixed densities $p_G$ and $p_{G_0}$. This is equivalent to studying the estimation of the true density $p_{G_0}$ of the observations through the density $p_G$ under the posterior distribution. As the Laplace kernel $f$ has Fourier transform $$\tilde f(\lambda) = \frac{1}{ 1 + \lambda^2},$$ it follows that the mixed densities $p_G$ have Fourier transforms satisfying $$|\tilde p_G(\lambda)|\le \frac{1}{ 1 + \lambda^2}.$$ Estimation of a density with a polynomially decaying Fourier transform was first considered in [@watson_estimation_1963]. According to their Theorem in Section $3$A, a suitable kernel estimator possesses a root mean square error of $n^{-3/8}$ with respect to the $L_2$-norm for estimating a density with Fourier transform that decays exactly at the order 2. This rate is the “usual rate” $n^{-\alpha/(2\alpha+1)}$ of nonparametric estimation for smoothness $\alpha=3/2$. This is understandable as $|\tilde p(\lambda)|\lesssim 1/(1+|\lambda|^2)$ implies that $\int (1+|\lambda|^2)^{\alpha}|\tilde p(\lambda)|^2\,d\lambda<\infty$, for every $\alpha<3/2$, so that a density with Fourier transform decaying at square rate belongs to any Sobolev class of regularity $\alpha<3/2$. Indeed in [@golubev1992nonparametric], the rate $n^{-\alpha/(2\alpha+1)}$ is shown to be minimax for estimating a density in a Sobolev ball of functions on the line. In the present paper we show that the posterior distribution of Laplace mixtures $p_G$ contracts to $p_{G_0}$ at the rate $n^{-3/8}$ up to a logarithm factor, relative to the $L_2$-norm and Hellinger distance, and also establish rates for other $L_q$-metrics. Thus the Dirichlet posterior (nearly) attains the minimax rate for estimating a density in a Sobolev ball of order $3/2$. It may be noted that the Laplace density itself is Hölder of exactly order 1, which implies that Laplace mixtures are Hölder smooth of at least the same order. This insight would suggest a rate $n^{-1/3}$ (the usual nonparametric rate for $\alpha=1$), which is slower than $n^{-3/8}$, and hence this insight is misleading.
Besides recovery relative to the Wasserstein metric and the induced metrics on $p_G$, one might consider recovery relative to a metric on the distribution function on $G$. Frequentist recovery rates for this problem were obtained in [@fan_optimal_1991] under some restrictions. There is no simple relation between these rates and rates for the other metrics. The same is true for the rates for deconvolution of densities, as in [@fan_optimal_1991]. In fact, the Dirichlet prior and posterior considered here are well known to concentrate on discrete distributions, and hence are useless as priors for recovering a density of $G$.
Contraction rates for Dirichlet mixtures of the normal kernel were considered in [@ghosal_entropies_2001; @ghosal_posterior_2007; @kruijer; @shen_adaptive_2011; @scricciolo]. The results in these papers are driven by the smoothness of the Gaussian kernel, whence the same approach will fail for the Laplace kernel. Nevertheless we borrow the idea of approximating the true mixed density by a finite mixture, albeit that the approximation is constructed in a different manner. Because more support points than in the Gaussian case are needed to obtain a given quality of approximation, higher entropy and lower prior mass concentration result, leading to a slower rate of posterior contraction. To obtain the contraction rate for the Wasserstein metrics we further derive a relationship of these metrics with a power of the Hellinger distance, and next apply a variant of the contraction theorem in [@ghosal_convergence_2000], which is included in the appendix of the paper. Contraction rates of mixtures with other priors than the Dirichlet were considered in [@scricciolo]. Recovery of the mixing distribution is a deconvolution problem and as such can be considered an inverse problem. A general approach to posterior contraction rates in inverse problems can be found in [@knapik2014general], and results specific to deconvolution can be found in [@donnet2014posterior]. These authors are interested in deconvolving a (smooth) mixing density rather than a mixing distribution, and hence their results are not directly comparable to the results in the present paper.
The paper is organized as follows. In the next section we state the main results of the paper, which are proved in the subsequent sections. In Section \[sec-approximation\] we establish suitable finite approximations relative to the $L_q$- and Hellinger distances. The $L_q$-approximations also apply to other kernels than the Laplace, and are in terms of the tail decay of the kernel’s characteristic function. In Sections \[sec-entropy-mixture\] and \[sec-prior-mass\] we apply these approximations to obtain bounds on the entropy of the mixtures relative to the $L_q$, Hellinger and Wasserstein metrics, and a lower bound on the prior mass in a neighbourhood of the true density. Sections \[SectionProofsMain\] and \[SectionProofsMainTwo\] contain the proofs of the main results.
Notation and preliminaries
--------------------------
Throughout the paper integrals given without limits are considered to be integrals over the real line $\mathbb{R}$. The $L_q$-norm is denoted $${\|{g}\|}_q = \left(\int |g(x)|^q\, dx \right)^{1/q},$$ with ${\|{\cdot}\|}_{\infty}$ being the uniform norm. The *Hellinger distance* on the space of densities is given by $$h(f,g) = \left(\int (f^{1/2}(x) - g^{1/2}(x))^2\, dx \right)^{1/2}.$$ It is easy to see that $h^2(f, g) \le {\|{f - g}\|}_1 \le 2 h(f,g)$, for any two probability densities $f$ and $g$. Furthermore, if the densities $f$ and $g$ are uniformly bounded by a constant $M$, then ${\|{f -g }\|}_2 \le 2 \sqrt{M} h(f,g)$. The Kullback-Leiber discrepancy and corresponding variance are denoted by $$K(p_0, p) = \int \log (p_0/p)\ dP_0, \qquad K_2(p_0, p) = \int (\log (p_0/p))^2\, dP_0$$ with $P_0$ the measure corresponding to the density $p_0$.
We are primarily interested in the Laplace kernel, but a number of results are true for general kernels $f$. The Fourier transform of a function $f$ and the inverse Fourier transform of a function $\tilde f$ are given by $$\tilde{f}(\lambda) = \int e^{\imath \lambda x} f(x) dx, \qquad
f(x) = \frac{1}{2 \pi} \int e^{-\imath \lambda x} \tilde{f}(\lambda) d\lambda.$$ For $\frac{1}{p} + \frac{1}{q} =1$ and $1 \le p\le 2$, *Hausdorff-Young’s inequality* gives that ${\|{f}\|}_q \le (2\pi)^{-1/p} {\|{\tilde{f}}\|}_p$. The covering number $N(\varepsilon, \Theta, \rho)$ of a metric space $(\Theta, \rho)$ is the minimum number of $\varepsilon$-balls needed to cover the entire space $\Theta$. $D(\varepsilon, \Theta, \rho)$ is the packing number of $(\Theta, \rho)$, which is the maximum number of points that are pairwise at least $\varepsilon$-separated. It is known that $
N(\varepsilon, \Theta, \rho) \le D(\varepsilon, \Theta, \rho) \le N(\varepsilon/2, \Theta, \rho)
$.
Throughout the paper $\lesssim$ denotes inequality up to a constant multiple, where the constant is universal or fixed within the context. Furthermore $a_n \asymp b_n$ means $c \le\liminf_{n \rightarrow \infty} a_n/b_n \le \limsup_{n \rightarrow \infty} a_n/b_n \le C$, for some positive constants $c$ and $C$.
We denote by $\mathcal{M}[-a,a]$ the set of all probability measures on a given interval $[-a, a]$.
Main Results
============
Let $\Pi_n(\cdot | X_1,\ldots, X_n)$ be the posterior distribution for $G$ in the scheme (i)-(iv) introduced at the beginning of the paper. We study this random distribution under the assumption that $X_1,\ldots,X_n$ are an i.i.d. sample from the mixture density $p_{G_0}=f\ast G_0$, for a given probability distribution $G_0$. We assume that $G_0$ is supported in a compact interval $[-a,a]$, and that the base measure $\alpha$ of the Dirichlet prior in (i) is concentrated on this interval with a Lebesgue density bounded away from 0 and $\infty$.
\[thm-rate-wasserstein\] If $G_0$ is supported on $[-a, a]$ with $f$ being Laplace kernel and $\alpha$ has support $[-a,a]$ with Lebesgue density bounded away from 0 and $\infty$, then for every $k\ge 1$, there exists a constant $M$ such that $$\label{eqn-main-results-rate-wasserstein}
\Pi\bigl( G: W_k(G, G_0) \ge M n^{-3/(8k+16)}(\log n)^{(k+7/8)/(k+2)}| X_1,\ldots, X_n) \rightarrow 0,$$ in $P_{G_0}$-probability.
The rate for the Wasserstein metric $W_k$ given in the theorem deteriorates with increasing $k$, which is perhaps not unreasonable as the Wasserstein metrics increase with $k$. The fastest rate is $n^{-1/8}(\log n)^{5/8}$, and is obtained for $W_1$.
\[thm-rate-mixture\] If $G_0$ is supported on $[-a, a]$ with $f$ being Laplace kernel and $\alpha$ has support $[-a,a]$ with Lebesgue density bounded away from 0 and $\infty$, then there exists a constant $M$ such that $$\Pi_n\bigl( G: h(p_G , p_{G_0}) \ge M (\log n/n)^{3/8}| X_1,\ldots, X_n\bigr) \rightarrow 0,
\label{eqn-main-results-rate-mixture-h}$$ in $P_{G_0}$-probability. Furthermore, for every $q\in[ 2,\infty)$ there exists $M_q$ such that $$\Pi_n\bigl( G: \|p_G -p_{G_0}\|_q \ge M_q (\log n/n)^{(q+1)/(q(q+2))}| X_1,\ldots, X_n\bigr) \rightarrow 0,
\label{eqn-main-results-rate-mixture}$$ in $P_{G_0}$-probability.
The rate for the $L_q$-distance given in deteriorates with increasing $q$. For $q=2$ it is the same as the rate $(\log n/n)^{3/8}$ for the Hellinger distance.
In both theorems the mixing distributions are assumed to be supported on a fixed compact set. Without a restriction on the tails of the mixing distributions, no rate is possible. The assumption of a compact support ensures that the rate is fully determined by the complexity of the mixtures, and not their tail behaviour.
Finite Approximation {#sec-approximation}
====================
In this section we show that a general mixture $p_G$ can be approximated by a mixture with finitely many components, where the number of components depends on the accuracy of the approximation, the distance used, and the kernel $f$. We first consider approximation with respect to the $L_q$-norm, which applies to mixtures $p_G=f\ast G$, for a general kernel $f$, and next approximation with respect to the Hellinger distance for the case that $f$ is the Laplace kernel. The first result generalizes the result of [@ghosal_entropies_2001] for normal mixtures. Also see [@scricciolo].
The result splits in two cases, depending on the tail behaviour of the Fourier transform $\tilde f$ of $f$:
1. $\limsup_{|\lambda|\rightarrow\infty}
\bigl|\tilde{f}(\lambda)\bigr| |\lambda|^\beta<\infty$, for some $\beta>1/2$.
2. $\limsup_{|\lambda|\rightarrow\infty} \bigl|\tilde{f} (\lambda) \bigr|
e^{|\lambda|^\beta} <\infty$, for some $\beta>0$.
\[lemma-find-discrete-measure\] Let $\varepsilon<1$ be sufficiently small and fixed. For a probability measure $G$ on an interval $[-a, a]$ and $2 \le q \le \infty$, there exists a discrete measure $G'$ on $[-a, a]$ with at most $N$ support points in $[-a, a]$ such that $${\|{p_G - p_{G'}}\|}_q \lesssim \varepsilon,$$ where
1. $N \lesssim \varepsilon^{- (\beta - p^{-1})^{-1}}$ if $f$ is ordinary smooth of order $\beta$, for $p$ and $q$ being conjugate ($p^{-1} + q^{-1} =1$).
2. $N \lesssim (\log \varepsilon^{-1} )^{\max(1,\beta^{-1})}$ if $f$ is supersmooth of order $\beta$.
The Fourier transform of $p_G$ is given by $ \tilde{f}\tilde{G} $, for $\tilde{G}(\lambda) = \int e^{\imath\lambda z}\,dG(z)$. Determine $G'$ so that it possesses the same moments as $G$ up to order $k-1$, i.e. $$\int z^j d(G - G')(z) = 0, \quad \forall\ 0\le j \le k-1.$$ By Lemma A.1 in [@ghosal_entropies_2001] $G'$ can be chosen to have at most $k$ support points.
Then for $G$ and $G'$ supported on $[-a, a]$, we have $$\begin{aligned}
|\tilde{G}(\lambda) - \tilde{G'} (\lambda) |
& = \left|\int\Big( e^{\imath \lambda z} - \sum_{j=0}^{k-1}\frac{(\imath \lambda z)^j}{ j!}\Big)\, d(G-G')(z)\right|\\
& \le \int \frac{| \imath \lambda z|^k}{ k!} d(G + G')(z) \le \Big( \frac{a e |\lambda|}{ k} \Big)^k.
\end{aligned}$$ The inequality comes from $| e^{iy} - \sum_{j=0}^{k-1} {(iy)^j}/{j!} | \le {|y|^k}/{ k!} \le{ (e|y|)^k}/{ k^k}$, for every $y\in{\mathbb{R}}$.
Therefore, by Hausdorff-Young’s inequality, $$\begin{aligned}
{\|{p_G - p_{G'}}\|}_q^p
& \le \frac{1}{ 2 \pi} \int |\tilde{f}(\lambda) |^p |\tilde{G} (\lambda) - \tilde{G'} (\lambda) |^p d\lambda \\
& \lesssim \int_{|\lambda| >M} |\tilde{f}(\lambda)|^p d\lambda + \int_{|\lambda| \le M} \Big( \frac{ea|\lambda|}{k} \Big)^{pk}\ d\lambda.
\end{aligned}$$ We denote the first term in the preceding display by $I_1$ and the second term by $I_2$. It is easy to bound $I_2$ as: $$I_2 \asymp \left( \frac{ea}{k} \right)^{kp} \frac{M^{kp+1}}{kp +1}
\lesssim \Bigl(\frac{eaM}{k}\Bigr)^{kp+1}\frac 1p.$$ For $I_1$ we separately consider the supersmooth and ordinary smooth cases.
In the supersmooth case with parameter $\beta$, we note that the function $(t^{\beta^{-1} -1})/e^{\delta t}$ is monotonely decreasing for $t\ge pM^{\beta}$, when $\delta \ge (\beta^{-1} -1 ) / (pM^{\beta})$. Thus, for large $M$, $$\begin{aligned}
I_1 & \lesssim \int_{|\lambda|>M } e^{-p|\lambda|^{\beta}} d\lambda
= \frac{2}{\beta p^{\beta^{-1}}} \int_{t>pM^{\beta}} e^{-t} t^{\beta^{-1} -1} dt \\
& \le\frac{2}{\beta p^{\beta^{-1}}} \int_{t>pM^{\beta}}e^{-(1-\delta)t}\,dt \frac{(pM^\beta)^{\beta^{-1}-1}}{ e^{\delta p M^{\beta}}}
= \frac{2}{ 1- \delta} \frac{1}{ \beta p} e^{-p M^{\beta}} M^{1- \beta},
\end{aligned}$$ where the bound is sharper if $\delta$ is smaller. Choosing the minimal value of $\delta$, we obtain $$\begin{aligned}
I_1 & \lesssim \frac{1}{ 1 - (\beta^{-1} -1)/(p M^{\beta})} \frac{1}{\beta p} e^{-p M^{\beta}} M^{1- \beta}
\lesssim M^{1-\beta} e^{-pM^{\beta}},
\end{aligned}$$ for $M$ sufficiently large. We next choose $ M = 2 \left(\log ({1}/{\varepsilon}) \right)^{\frac{1}{\beta}}$ in order to ensure that $I_1 \le \varepsilon^p$. Then $I_2\lesssim\varepsilon^p$ if $k\ge 2ea M$ and $2^{-kp}\le \varepsilon^p$. This is satisfied if $k = 2( \log \varepsilon^{-1})^{\max(\beta^{-1},1)} $.
In the ordinary smooth case with smoothness parameter $\beta$, we have the bound $$I_1 \lesssim \int_{\lambda > M} |\lambda|^{-\beta p} d\lambda \lesssim \left(\frac{1}{M}\right)^{\beta p -1}.$$ We choose $M = ( {1}/{\varepsilon})^{-(\beta -1/p)^{-1}}$ to render the right side equal to $\varepsilon^p$. Then $I_2 \lesssim \varepsilon^p$ if $k = 2 \varepsilon^{- (\beta-1/p)^{-1}}$.
The number of support points in the preceding lemma is increasing in $q$ and decreasing in $\beta$. For approximation in the $L_2$-norm ($q=2$), the number of support points is of order $\varepsilon^{-1/(\beta-1/2)}$, and this reduces to $\varepsilon^{-2/3}$ for the Laplace kernel (ordinary smooth with $\beta=2$). The exponent $\beta - 1/2$ can be interpreted as (almost) the Sobolev smoothness of $p_G$, since, for $\alpha < \beta - 1/2$, $$\int (1 + |\lambda|^2)^\alpha |\tilde{p}_G(\lambda)|^2 d \lambda \lesssim \int (1 + |\lambda|^2)^\alpha |\tilde{f}(\lambda)|^2 d \lambda <\infty.$$ We do not have a compelling intuition for this correspondence.
The Hellinger distance is more sensitive to areas where the densities are close to zero. This causes that the approach in the preceding lemma does not give sharp results. The following lemma does, but is restricted to the Laplace kernel.
\[lemma-bound-hellinger-discrete\] For a probability measure $G$ supported on $[-a, a]$ there exists a discrete measure $G'$ with at most $N \asymp \varepsilon^{- 2/3}$ support points such that for $p_{G} = f \ast G$ and $f$ the Laplace density $$h(p_{G}, p_{G'}) \le \varepsilon.$$
Since $p_{G}(x) \ge f(|x|+a)=e^{-a}e^{-|x|}/2$, for every $x$ and probability measure $G$ supported on $[-a,a]$, the Hellinger distance between Laplace mixtures satisfies $$\begin{aligned}
h^2(p_{G}, p_{G'}) & \le \int \frac{ (p_{G} - p_{G'})^2 }{ p_{G}+ p_{G'}}(x)\, dx
\le e^a \int (p_{G'}(x) -p_{G}(x))^2 e^{|x|}\, dx.
\end{aligned}$$ If we write $q_{G}(x) = p_{G}(x) e^{|x|/2}$, and $\tilde q_G$ for the corresponding Fourier transform, then the integral in the right side is equal to $(1/ 2\pi) \int |\tilde{q}_{G'} - \tilde{q}_{G}|^2(\lambda)\, d\lambda$, by Plancherel’s theorem. By an explicit computation we obtain $$\begin{aligned}
\tilde{q}_{G} (\lambda) & = \frac{1}{2} \int \int e^{\imath \lambda x} e^{-|x-z| + |x|/2}\, dx\, dG(z)
= \frac{1}{2} \int r(\lambda,z)\, d{G}(z),
\end{aligned}$$ where $r(\lambda,z)$ is given by $$\begin{aligned}
r(\lambda, z) & = \frac{e^{-z}}{\imath \lambda + 1/2} + e^{-z} \frac{e^{(\imath \lambda + 3/2)z} -1 }{\imath \lambda + 3/2} - \frac{e^{(\imath \lambda +1/2)z}}{\imath \lambda - 1/2} \label{eqn-approx-kl-explicit-r} \nonumber \\
& = \frac{e^{-z}}{(\imath\lambda+1/2)(\imath\lambda+3/2)}
- \frac{2 e^{\imath \lambda z} e^{z/2}}{(\imath\lambda + 3/2) (\imath \lambda - 1/2)}.
\end{aligned}$$ Now let $G'$ be a discrete measure on $[-a,a]$ such that $$\begin{aligned}
& \int e^{-z}\, d({G'}-G)(z) = 0,\qquad
\int e^{z/2} z^j\,d({G'}- G)(z) = 0, \quad \forall\ 0\le j \le k-1 .
\end{aligned}$$ By Lemma A.1 in [@ghosal_entropies_2001] $G'$ can be chosen to have at most $k+1$ support points.
By the choice of $G'$ the first term of $r(\lambda,z)$ gives no contribution to the difference $\int r(\lambda, z)\, d ({G'} - G)(z)$. As the second term of $r(\lambda,z)$ is for large $|\lambda|$ bounded in absolute value by a multiple of $|\lambda|^{-2}$, it follows that $$\begin{aligned}
I_2:=\int_{|\lambda|>M} \left| \int r(\lambda, z)\, d ({G'} - G)(z) \right|^2\, d \lambda
\lesssim \int_{\lambda >M} \lambda^{-4} d\lambda \asymp M^{-3}.\end{aligned}$$ By the choice of $G'$ in the second term of $r(\lambda,z)$ we can replace $e^{i\lambda z}$ by $e^{\imath\lambda z}-\sum_{j=0}^k (\imath\lambda z)^j/j!$ again without changing the integral $\int r(\lambda, z)\, d ({G'} - G)(z)$. It follows that $$\begin{aligned}
I_1&:= \int_{|\lambda|\le M} \left| \int r(\lambda, z)\, d ({G'} - G)(z) \right|^2\, d \lambda\\
& \le \int_{|\lambda| \le M} \left|\frac{2}{(\imath\lambda+1/2)(\imath\lambda+3/2)}\right|^2 \left| \int e^{z/2}\Big[e^{\imath \lambda z} - \sum_{j=0}^k (\imath \lambda z)^j\Bigr]\, d({G'}-G)(z)\right|^2 d \lambda \\
& \qquad \qquad \qquad \lesssim \int_{0}^{M} \frac{ (z\lambda)^{2k}}{(k!)^2}\, d\lambda
\lesssim \frac{(aeM)^{2k+1}}{k^{2k+1}}.\end{aligned}$$ It follows, by a similar argument as in the proof of Lemma \[lemma-find-discrete-measure\], that we can reduce both $I_1$ and $I_2$ to $\varepsilon^2$ by choosing and $M\asymp\varepsilon^{-2/3}$ and $k =2aeM$.
Entropy {#sec-entropy-mixture}
=======
We study the covering numbers of the class of mixtures $p_G=f\ast G$, where $G$ ranges over the collection $\mathcal{M}[-a,a]$ of all probability measures on $[-a, a]$. We present a bound for any $L_q$-norm and general kernels $f$, and a bound for the Hellinger distance that is specific to the Laplace kernel.
\[lemma-entropy-estimate-lq\] If both ${\|{f}\|}_q$ and ${\|{f'}\|}_q$ are finite and $\tilde{f}$ has ordinary smoothness $\beta$, then, for $p_G=f\ast G$, and any $q\ge 2$, $$\log N\bigl(\varepsilon, \{p_G: G\in\mathcal{M}[-a,a]\}, {\|{\cdot}\|}_q\bigr)
\lesssim \left( \frac{1}{\varepsilon}\right)^{\frac{1}{ \beta - 1 + 1/q}} \log \Big( \frac{1}{\varepsilon} \Big).
\label{eqn-entropy-estimate-lq}$$
Consider an $\varepsilon$-net of ${\mathcal{P}}_a=\{p_G: G\in\mathcal{M}[-a,a]\}$ by constructing $\mathcal{I}$ the collection of all $p_G$’s such that the mixing measure $G \in \mathcal{M}[-a, a]$ is discrete and has at most $N
\le D\varepsilon^{-(\beta -1 + q^{-1})^{-1}}$ support points for some proper constant $D$.
In light of the approximation Lemma \[lemma-find-discrete-measure\], the set of all mixtures $p_G$ with $G$ a discrete probability measure with $N\lesssim \varepsilon^{-(\beta -1 + q^{-1})^{-1}}$ support points forms an $\varepsilon$-net over the set of all mixtures $p_G$ as in the lemma. It suffices to construct an $\varepsilon$-net of the given cardinality over this set of discrete mixtures.
By Jensen’s inequality and Fubini’s theorem, $$\begin{aligned}
{\|{f(\cdot- \theta ) - f }\|}_q
& = \left( \int \left| \theta \int_0^{1} f'(x- \theta s)\, ds \right|^q\, dx \right)^{1/q}
\le {\|{f'}\|}_q \theta.\end{aligned}$$ Furthermore, for any probability vectors $p$ and $p'$ and locations $\theta_i$, $$\begin{aligned}
\left\|\sum_{i=1}^N p_i f(\cdot-\theta_i) - \sum_{i=1}^N p'_i f(\cdot-\theta_i)\right\|_q
& \le \sum_{i=1}^N |p_i - p'_i| {\|{f(\cdot-\theta_i)}\|}_q
= {\|{f}\|}_q \|p-p'\|_1. \end{aligned}$$ Combining these inequalities, we see that for two discrete probability measures $G = \sum_{i=1}^N p_i \delta_{\theta_i}$ and $G' =\sum_{i=1}^N p_i' \delta_{\theta'_i}$, $$\begin{aligned}
{\|{p_G - p_{G'}}\|}_q
& \le {\|{f'}\|}_q \max_i|\theta_i-\theta_i'|+ {\|{f}\|}_q\|p-p'\|_1.
\label{eqn-bound-mixtures-by-relocating}\end{aligned}$$ Thus we can construct an $\varepsilon$-net over the discrete mixtures by relocating the support points $(\theta_i)_{i=1}^N$ to the nearest points $(\theta'_i)_{i=1}^N$ in a $\varepsilon$-net on $[-a,a] $, and relocating the weights $p$ to the nearest point $p'$ in an $\varepsilon$-net for the $l_1$-norm over the $N$-dimensional $l_1$-unit simplex. This gives a set of at most $$\left( \frac{2a}{ \varepsilon} \right)^N \left( \frac{5}{\varepsilon} \right)^N
\sim \left( \frac{10a}{ \varepsilon^2 } \right)^N$$ measures $p_G$ (cf. Lemma A.4 of [@ghosal_posterior_2007] for the entropy of the $l_1$-unit simplex). This gives the bound of the lemma.
\[LemmaEntropyHellinger\] For $f$ the Laplace kernel and $p_G=f\ast G$, $$\log N\bigl(\varepsilon, \{p_G: G\in \mathcal{M}[-a,a]\}, h\bigr)
\lesssim \varepsilon^{-3/8}\log (1/ \varepsilon).
\label{eqn-entropy-estimate-h}$$
Because the function $\sqrt f$ is absolutely continuous with derivative $x\mapsto -2^{-3/2}e^{-|x|/2}{\mathop{\rm sgn}\nolimits}(x)$, we have by Jensen’s inequality and Fubini’s theorem that $$\begin{aligned}
h^2\bigl(f, f(\cdot-\theta)\bigr)
& =\int \Bigl(\theta \int_0^1 -2^{-3/2}e^{-|x-\theta s|/2}{\mathop{\rm sgn}\nolimits}(x-\theta s)\, ds\Bigr)^2 \, dx \\
& \le \theta^2 \int_0^1 \int e^{-|x-\theta s|}\, dx\, ds=2\theta^2.
\end{aligned}$$ It follows that $h\bigl(f, f(\cdot-\theta) \bigr) \lesssim \theta$.
By convexity of the map $(u,v)\mapsto (\sqrt u-\sqrt v)^2$, we have $$\Bigl|\sqrt{\sum_{i}p_if(\cdot-\theta_i)}-\sqrt{\sum_{i}p_if(\cdot-\theta_i')}\Bigr|^2
\le \sum_{i}p_i\bigl[\sqrt{f(\cdot-\theta_i)}-\sqrt{f(\cdot-\theta_i')}\bigr]^2.$$ By integrating this inequality we see that the densities $p_G$ and $p_{G'}$ with mixing distributions $G = \sum_{i=1}^N p_i \delta_{\theta_i}$ and $G' = \sum_{i=1}^N p_i
\delta_{\theta'_i}$ satisfy $h^2(p_G, p_{G'})\lesssim \sum p_i|\theta_i-\theta_i'|^2\le
\|\theta-\theta'\|_\infty^2$.
Furthermore, for distributions $G = \sum_{i=1}^N p_i \delta_{\theta_i}$ and $G' =\sum_{i=1}^N p'_i
\delta_{\theta_i}$ with the same support points, but different weights, we have $$\begin{aligned}
h^2(p_G, p_{G'}) &
\le \int \frac{\big( \sum_{i=1}^N (p_i - p'_i) f(x- \theta_i) \big)^2}{ \sum_{i=1}^N (p_i + p'_i) f(x- \theta_i)} dx\\
& \le \int \big( \sum_{i=1}^N |p_i - p'_i| \big)^2\frac{ f^2(|x| -a) }{2 f(|x| +a) } dx
\lesssim {\|{p- p'}\|}_{1}^2.
\end{aligned}$$ Therefore the bound follows by arguments similar as in the proof of Proposition \[lemma-entropy-estimate-lq\], where presently we use Lemma \[lemma-bound-hellinger-discrete\] to determine suitable finite approximations.
The map $G\mapsto p_G=f\ast G$ is one-to-one as soon as the characteristic function of $f$ is never zero. Under this condition we can also view the Wasserstein distance on the mixing distribution as a distance on the mixtures. Obviously the covering numbers are then free of the kernel.
\[PropositionEntropyEstimateW\] For any $k\ge 1$, and any sufficiently small $\varepsilon>0$, $$\log N\bigl(\varepsilon, \mathcal{M}[-a,a], W_k\bigr)
\lesssim \Bigl(\frac1\varepsilon\Bigr)\log (1/ \varepsilon).
\label{eqn-entropy-estimate-W}$$
The proposition is a consequence Lemma \[LemmaEntropyWGeneral\], below, which applies to the set of all Borel probability measures on a general metric space $(\Theta,\rho)$ (cf. [@nguyen_convergence_2013]).
\[lemma:countable-partitions\] For any probability measure $G$ concentrated on countably many disjoint sets $\Theta_1,\Theta_2, \ldots$ and probability measure $G'$ concentrated on disjoint sets $\Theta_1',\Theta'_2,\ldots$, $$W_k(G,G') \le \sup_i \sup_{\theta_i \in \Theta_i, \theta'_i \in \Theta'_i} \rho(\theta_i, \theta'_i) + \operatorname{diam}(\Theta) \Big( \sum_i |G(\Theta_i) - G'(\Theta'_i)| \Big)^{1/k}.$$ In particular, $$W_k \Big(\sum_i p_i \delta_{\theta_i}, \sum_i p'_i \delta_{\theta'_i} \Big) \le \max_i \rho(\theta_i, \theta'_i) + \operatorname{diam}(\Theta) {\|{p-p'}\|}_1^{1/k}.$$
For $p_i = G(\Theta_i)$ and $p'_i = G'(\Theta'_i)$ divide the interval $[0, \sum_i p_i \wedge
p'_i]$ into disjoint intervals $I_i$ of lengths $p_i \wedge p'_i$. We couple variables $\bar{\theta}$ and $\bar{\theta}'$ by an auxiliary uniform variable $U$. If $U \in I_i$, then generate $\bar{\theta} \sim G(\cdot|\Theta_i)$ and $\bar{\theta}' \sim G'(\cdot|\Theta'_i)$. Divide the remaining interval $[\sum_i p_i \wedge p_i', 1] $ into intervals $J_i$ of lengths $p_i -
p_i \wedge p'_i$ and, separately, intervals $J'_i$ of length $p'_i - p_i\wedge p'_i$. If $U\in
J_i$, then generate $\bar{\theta}\sim G(\cdot| \Theta_i) $ and if $U\in J'_i$, then generate $\bar{\theta}' \sim G'(\cdot| \Theta'_i)$. Then $\bar\theta$ and $\bar\theta'$ have marginal distributions $G$ and $G'$, and $${\mathbb{E}}\rho^k ( \bar{\theta}, \bar{\theta}') \le {\mathbb{E}}\big[ \rho^k(
\bar{\theta}, \bar{\theta}') \text{1}_{U\le \sum_i p_i \wedge p'_i} \big] +
\operatorname{diam}(\Theta)^k {\mathbb{P}}\big( U > \sum_i p_i \wedge p'_i \big).$$ The first term is bounded by the $k$-th power of the first term of the lemma, while the probability in the second term is equal to $1- \sum_i p_i \wedge p'_i = \sum_i |p_i - p'_i|/2$.
\[LemmaEntropyWGeneral\] For the set $\mathcal{M}(\Theta)$ of all Borel probability measures on a metric space $(\Theta,\rho)$, any $k\ge 1$, and $0< \varepsilon< \min\{ 2/3, \operatorname{diam}(\Theta)\}$, $$N\bigl(\varepsilon, \mathcal{M}(\Theta), W_k\bigr)
\le \Big( \frac{ 4 \operatorname{diam}(\Theta)}{\varepsilon}\Big)^{kN(\varepsilon, \Theta, \rho)}.$$ \[lemma:bound-packing-numbers\]
For a minimal $\varepsilon$-net over $\Theta$ of $N=N(\varepsilon,\Theta,\rho)$ points, let $\Theta = \cup_i \Theta_i$ be the partition obtained by assigning each $\theta$ to a closest point. For any $G$ let $G_\varepsilon = \sum_i G(\Theta_i)\delta_{\theta_i}$, for arbitrary but fixed $\theta_i\in \Theta_i$. Since $W_k(G,G_\varepsilon) \le \varepsilon$ by Lemma \[lemma:countable-partitions\], we have $N(2\varepsilon, \mathcal{M}(\Theta), W_k) \le N(\varepsilon, \mathcal{M}_\varepsilon, W_k) $, for $\mathcal{M}_\varepsilon$ the set of all $G_\varepsilon$. We next form the measures $G_{\varepsilon,p} = \sum_i p_i \delta_{\theta_i}$ for $(p_1,\ldots, p_N)$ ranging over an $(\varepsilon/\operatorname{diam}(\Theta))^k$-net for the $l_1$-distance over the $N$-dimensional unit simplex. By Lemma \[lemma:countable-partitions\] every $G_\varepsilon$ is within $W_k$-distance of some $G_{\varepsilon,p}$. Thus $N(\varepsilon, \mathcal{M}_\varepsilon, W_k)$ is bounded from above by the number of points $p$, which is bounded by $(4\operatorname{diam}(\Theta)/\varepsilon)^{k N}$ (cf. Lemma A.4 in [@ghosal_convergence_2000]).
Prior mass {#sec-prior-mass}
==========
This main result of this section is the following proposition, which gives a lower bound on the prior mass of the prior (i)-(iv) in a neighbourhood of a mixture $p_{G_0}$.
\[PropositionPriorMass\] If $\Pi$ is the Dirichlet process $\operatorname{DP}(\alpha)$ with base measure $\alpha$ that has a Lebesgue density bounded away from 0 and $\infty$ on its support $[-a,a]$, and $f$ is the Laplace kernel, then for every sufficiently small $\varepsilon>0$ and every probability measure $G_0$ on $[-a,a]$, $$\log \Pi \Big(G: K(p_G, p_{G_0}) \le \varepsilon^2, K_2(p_G, p_{G_0}) \le \varepsilon^2 \Big)
\gtrsim \Big( \frac{1}{\varepsilon} \Big)^{2/3} \log\Big( \frac{1}{\varepsilon}\Big).$$
By Lemma \[lemma-bound-hellinger-discrete\] there exists a discrete measure $G_1$ with $N\lesssim \varepsilon^{-2/3}$ support points such that $h (p_{G_0} , p_{G_1}) \le \varepsilon$. We may assume that the support points of $G_1$ are at least $2\varepsilon^2$-separated. If not, we take a maximal $2\varepsilon^2$-separated set in the support points of $G_1$, and replace $G_1$ by the discrete measure obtained by relocating the masses of $G_1$ to the nearest points in the $2\varepsilon^2$-net. Then $h(p_{G_1}, p_{G'_1}) \lesssim \varepsilon^2$, as seen in the proof of Proposition \[LemmaEntropyHellinger\].
Now by Lemmas \[lemma-bound-h-with-L2\] and \[lemma-L1-discrete\], if $G_1 = \sum_{i=1}^N p_j \delta_{z_j}$, with the support points $z_j$ at least $2 \varepsilon^2$-separated, $$\begin{aligned}
\bigl\{G: \max( K, K_2) (p_{G_0},p_G)< d_1\varepsilon^2 \bigr\}
& \supset \bigl\{G: h(p_{G_0}, p_G) \le 2 \varepsilon \bigr\} \\
& \supset \bigl\{G: h(p_{G_1}, p_G) \le \varepsilon \bigr\} \\
& \supset \bigl\{ G: {\|{p_G - p_{G_1}}\|}_{1} \le d_2 \varepsilon^2 \bigr\} \\
& \supset \bigl\{G: \sum_{j=1}^N \big|G[z_j - \varepsilon^2, z_j + \varepsilon^2] - p_j\big| \le \varepsilon^2 \bigr\}.\end{aligned}$$ By Lemma A.$2$ of [@ghosal_entropies_2001], since the base measure $\alpha$ has density bounded away from zero and infinity on $[-a, a]$ by assumption, we have $$\log \Pi \left(G: \sum_{j=1}^N \big| G[z_j - \varepsilon^2, z_j + \varepsilon^2] - p_j\big| \le
\varepsilon^2 \right)
\gtrsim -N\log\Big( \frac{1}{\varepsilon} \Big)$$ The lemma follows upon combining the preceding.
\[lemma-L1-discrete\] If $G'= \sum_{j=1}^{N} p_i \delta_{z_j}$ is a probability measure supported on points $z_1,\ldots, z_N$ in ${\mathbb{R}}$ with $|z_j-z_k| > 2\varepsilon$ for $j \neq k$, then for any probability measure $G$ on $\mathbb{R}$ and kernel $f$ , $${\|{p_G-p_{G'}}\|}_1 \le 2\|f'\|_1 \varepsilon
+2 \sum_{j=1}^N \big| G[z_j-\varepsilon, z_j+\varepsilon] - p_j\big|.$$
First we claim, for $\delta$ sufficiently small, $$\int |f(x) - f(x+\delta) | dx \lesssim \delta.$$ By Fubini’s theorem $$\begin{split}
\int |f(x) - f(x+\delta) | dx & \le \int \int_{x}^{x+\delta} |f'(s)|\, ds\, dx \\
& = \delta \int |f'(s)| ds \lesssim \delta.
\end{split}
\label{eqn-bound-L1-f-x+delta-f}$$
We bound $|p_G - p_{G'}|$ as follows $$\begin{aligned}
|(p_G- p_{G'})(x)| & = \left| \int f(x-z) dG(z) - \sum_{j=1}^N f(x-z_j) p_j \right| \\
& \le \left| \sum_{j=1}^N \int_{|z-z_j|\le \varepsilon} f(x-z) d G(z) - \sum_{j=1}^N f(x-z_j)p_j \right| \\
& \quad + \left| \int_{z: |z-z_j|>\varepsilon\forall j} f(x-z) dG(z) \right|\\
& \le \left| \sum_{j=1}^N \int_{|z-z_j|\le \varepsilon} \big(f(x-z) - f(x-z_j) \big) dG(z) \right| \\
& \quad + \left| \sum_{j=1}^N f(x-z_j) G[z_j-\varepsilon, z_j+\varepsilon] - \sum_{j=1}^N f(x-z_j) p_j \right| \\
& \quad + \int_{z: |z-z_j|>\varepsilon\forall j} f(x-z) dG(z) \\
& \le \sum_{j=1}^N \int_{|z-z_j|\le \varepsilon} |f(x-z) - f(x-z_j)| dG(z) \\
& \quad + \sum_{j=1}^N f(x-z_j) \big| p_j - G[z_j-\varepsilon, z_j+\varepsilon] \big| \\
& \quad + \int_{z: |z-z_j|>\varepsilon\forall j} f(x-z) d G(z).
\end{aligned}$$ Denote the three terms in the preceding display as $I_1(x)$, $I_2(x)$ and $I_3(x)$.
Fix $z$ and $z_j$, by (\[eqn-bound-L1-f-x+delta-f\]) we have $$\begin{aligned}
\int | f(x-z) - f(x- z_j) | dx \lesssim |z-z_j|.
\end{aligned}$$ By Fubini’s theorem, we have $$\begin{aligned}
\int I_1(x) dx & = \sum_{j=1}^N \int_{|z-z_j| \le \varepsilon} \int \big| f(x-z) - f(x-z_j) \big| dx dG(z) \\
& \lesssim \sum_{j=1}^N \int_{|z-z_j| \le \varepsilon} |z-z_j| d G(z) \le \varepsilon.
\end{aligned}$$ Note that $f(x)$ integrates to $1$, $$\int I_2(x) dx \le \sum_{j=1}^N \big|p_j - G[z_j-\varepsilon, z_j+\varepsilon]\big|.$$ noticing $(p_j)_j$ sums up to $1$, $$\begin{aligned}
\int I_3(x)dx & \le G(z: |z-z_j|>\varepsilon \forall j) \\
& = 1-\sum_{j=1}^N G[z_j-\varepsilon, z_j+\varepsilon] \\
& \le \sum_{j=1}^N \big|G[z_j-\varepsilon, z_j+\varepsilon] - p_j\big|.
\end{aligned}$$ Combing the above displays, we have the desired result.
\[lemma-bound-h-with-L2\] If $G$ and $G'$ are probability measures on $[-a,a]$, and $f$ is the Laplace kernel, then $$\begin{aligned}
\label{eqn-bound-hellinger-l2}
h^2(p_G, p_{G'})& \lesssim {\|{p_G - p_{G'}}\|}_{2},\\
\max \big(K(p_{G}, p_{G'}), K_2(p_{G}, p_{G'})\big) & \lesssim h^2(p_{G}, p_{G'}).
\label{lemma-bound-kl-hellinger}\end{aligned}$$
The first lemma is a generalization of Lemma 4 in [@ghosal_posterior_2007] from normal to general kernels, and is proved in the same manner.
In view of the shape of the Laplace kernel, it is easy to see that for $G$ compactly supported on $[-a,a]$, $$f(|x| +a) \le p_G(x) \le f(|x| - a),$$ We bound the squared Hellinger distance as follows: $$\begin{aligned}
h^2(p_G, p_{G'}) & \le \int \frac{(p_G - p_{G'})^2}{ p_G + p_{G'}} dx\\
& \le \int_{|x| \le A} e^{A+a} (p_G - p_{G'} )^2 dx + \int_{|x| > A} (p_G + p_{G'}) dx \\
& \lesssim e^a {\|{p_G - p_{G'}}\|}_2^2 e^A + e^{-A}.
\end{aligned}$$ By the elementary inequality $ t + \frac{u}{t } \ge 2 \sqrt{u}$, for $u, t>0$, we obtain upon choosing $A = \min(a, \log {\|{p_G - p_{G'}}\|}_2^{-1}- a/2) $.
For the proof of the second assertion we first note that, if both $G$ and $G'$ are compactly supported on $[-a, a]$, $$\frac{p_{G}(x)}{p_{G'}(x)} \le \frac{ f(|x| -a )}{ f (|x| +a) } \le e^{2a}.$$ Therefore ${\|{p_{G}/p_{G'}}\|}_{\infty} \le e^{2a}$, and follows by Lemma 8 in [@ghosal_posterior_2007].
Proof of Theorem \[thm-rate-wasserstein\] {#SectionProofsMain}
=========================================
The proof is based on the following comparison between the Wasserstein and Hellinger metrics. The lemma improves and generalizes Theorem $2$ in [@nguyen_convergence_2013]. Let $C_k$ be a constant such that the map $\varepsilon\mapsto \varepsilon[\log (C_k/\varepsilon)]^{k+1/2}$ is monotone on $(0,2]$.
\[LemmaCompareWh\] For probability measures $G$ and $G'$ supported on $[-a,a]$, and $p_G=f\ast G$ for a probability density $f$ with $\inf_{\lambda}(1+|\lambda|^\beta)|\tilde f(\lambda)|>0$, and any $k\ge 1$, $$W_k(G,G')\lesssim h(p_G,p_{G'})^{1/(k+\beta)}\Bigl(\log \frac{C_k} {h(p_G,p_{G'})}\Bigr)^{(k+1/2)/(k+\beta)}.$$
By Theorem 6.15 in [@villani_optimal_2009] the Wasserstein distance $W_k(G,G')$ is bounded above by a multiple of the $k$th root of $\int |x|^k\,d|G-G'|(x)$, where $|G-G'|$ is the total variation measure of the difference $G-G'$. We apply this to the convolutions of $G$ and $G'$ with the normal distribution $\Phi_\delta$ with mean 0 and variance $\delta^2$, to find, for every $M>0$, $$\begin{aligned}
W_k(G\ast \Phi_\delta,&G'\ast\Phi_\delta)^k
\lesssim \int |x|^k\,\bigl|(G-G')\ast \phi_\delta(x)\bigr|\,dx\\
&\le\Bigl(\int_{-M}^M x^{2k}\,dx\,\int_{-M}^M \bigl|(G-G')\ast \phi_\delta(x)\bigr|^2\,dx\Bigr)^{1/2}\\
&\qquad\qquad+ e^{-M}\int_{|x|>M}|x|^ke^{|x|}\bigl|(G-G')\ast \phi_\delta(x)\bigr|\,dx\\
&\lesssim M^{k+1/2}\bigl\|(G-G')\ast \phi_\delta\bigr\|_2+e^{-M}e^{2|a|}{\mathord{\rm E}}e^{2|\delta Z|},\end{aligned}$$ where $Z$ is a standard normal variable. The number $K_\delta:=e^{2|a|}{\mathord{\rm E}}e^{2|\delta Z|}$ is uniformly bounded by if $\delta\le \delta_k$, for some fixed $\delta_k$.
By Plancherel’s theorem, $$\begin{aligned}
\bigl\|(G-G')\ast \phi_\delta\bigr\|_2^2
&=\int |\tilde G-\tilde G'|^2(\lambda)\tilde \phi_\delta^2(\lambda)\,d\lambda
=\int |\tilde f(\tilde G-\tilde G')|^2(\lambda) \frac{\tilde \phi_\delta^2}{|\tilde f|^2}(\lambda)\,d\lambda\\
&\lesssim \|p_G-p_{G'}\|_2^2\, \sup_\lambda \frac{\tilde \phi_\delta^2}{|\tilde f|^2}(\lambda)
\lesssim h^2(p_G,p_{G'}) \delta^{-2\beta},\end{aligned}$$ where we have again applied Plancherel’s theorem, used that the $L_2$-metric on uniformly bounded densities is bounded by the Hellinger distance, and the assumption on the Fourier transform of $f$, which shows that $(\tilde\phi_\delta/|\tilde f|)(\lambda)\lesssim
(1+|\lambda|^\beta)e^{-\delta^2\lambda^2/2}\lesssim \delta^{-\beta}$.
If $U\sim G$ is independent of $Z\sim N(0,1)$, then $(U,U+\delta Z)$ gives a coupling of $G$ and $G\ast \Phi_\delta$. Therefore the definition of the Wasserstein metric gives that $W_k(G,G\ast\Phi_\delta)^k\le {\mathord{\rm E}}|\delta Z|^k\lesssim \delta^k$.
Combining the preceding inequalities with the triangle inequality we see that, for $\delta\in (0,\delta_k]$ and any $M>0$, $$W_k(G,G')^k
\lesssim M^{k+1/2}h(p_G,p_{G'})\delta^{-\beta}+e^{-M}+\delta^k.$$ The lemma follows by optimizing this over $M$ and $\delta$. Specifically, for $\varepsilon=h(p_G,p_{G'})$, we choose $M=k/(k+\beta)\,\log(C_k/\varepsilon)$ and $\delta=(M^{k+1/2}\varepsilon)^{1/(k+\beta)}$. These are eligible choices for $$\delta_k=\sup_{\varepsilon\in(0,2]}\Bigl[\frac k{k+\beta}\log\frac{C_k}\varepsilon\Bigr]^{(k+1/2)/(k+\beta)}
\varepsilon^{1/(k+\beta)},$$ which is indeed a finite number. In fact the supremum is taken at $\varepsilon=2$, by the assumption on $C_k$.
For the Laplace kernel $f$ we choose $\beta=2$ in the preceding lemma, and then obtain that $d(p_G, p_{G'}) \le h(p_G, p_{G'})$, for the “discrepancy” $d=\gamma^{-1}(W_k)$, and $\gamma(\varepsilon)=D_k\varepsilon^{1/(k+\beta)}[\log (C_k/\varepsilon)]^{(k+1/2)/(k+\beta)}$ a multiple of the (monotone) transformation in the right side of the preceding lemma. For small values of $W_k(G_1,G_2)$ we have $$\label{def-connect-G-PG}
d(p_{G_1}, p_{G_2}) \asymp W_k^{k+2} ( G_1, G_2)\Bigl(\log \frac{1}{W_k(G_1,G_2)}\Bigr)^{-k-1/2}.$$ As $k+2>1$ the discrepancy $d$ may not satisfy the triangle inequality, but it does possess the properties (a)–(d) in the appendix, Section \[Appendix\]. The balls of the discrepancy $d$ are convex, as the Wasserstein metrics are convex (see [@villani_optimal_2009]).
It follows that Theorem \[thm-rate-semi-metric\] applies to obtain a rate of posterior contraction relative to $d$ and hence relative to $W_k\sim d^{1/(k+2)}\bigl(\log (1/d)\bigr){}^{(k+1/2)/(k+2)}$. We apply the theorem with $\mathcal{P}=\mathcal{P}_n$ equal to the set of mixtures $p_G=f\ast G$, as $G$ ranges over $\mathcal{M}[-a,a]$. Thus is trivially satisfied.
For the entropy condition we have, by Proposition \[PropositionEntropyEstimateW\], $$\begin{aligned}
\log N(\varepsilon, \mathcal{P}_n, d) & = \log N\Bigl(\varepsilon^{1/(k+2)}\Bigl(\log\frac1\varepsilon\Bigr)^{(k+1/2)/(k+2)},
\mathcal{M}[-a, a], W_k) \\
&\lesssim \Bigl(\frac{1}{\varepsilon}\Big)^{1/(k+2)} \Bigl(\log\frac{1}{\varepsilon}\Bigr)^{1+(k+1/2)/(k+2)}.\end{aligned}$$ Thus holds for the rate $ \varepsilon_n\gtrsim n^{-\gamma}$, for every $\gamma< (k+2)/(2k+5)$.
The prior mass condition is satisfied with the rate $\varepsilon_n\asymp(\log n / n)^{3/8}$, in view of Proposition \[PropositionPriorMass\].
Theorem \[thm-rate-semi-metric\] yields a rate of contraction relative to $d$ equal to the slower of the two rates, which is $(\log n/n)^{3/8}$. This translates into the rate for the Wasserstein distance as given in Theorem \[thm-rate-wasserstein\].
Proof of Theorem \[thm-rate-mixture\] {#SectionProofsMainTwo}
=====================================
We apply Theorem \[thm-rate-semi-metric\], with $\mathcal{P} = \mathcal{P}_n $ the set of all mixtures $p_G$ as $G$ ranges over $\mathcal{M}[-a,a]$. For $d = h$ the rate follows immediately by combining Propositions \[lemma-entropy-estimate-lq\] and \[PropositionPriorMass\].
Since the densities $p_G$ are uniformly bounded by $1/2$, the $L_q$ distance $\|p_G-p_{G'}\|_q$ is bounded above by a multiple of $h(p_G,p_{G'})^{2/q}$. We can therefore apply Theorem \[thm-rate-semi-metric\] with the discrepancy $d(p,p')=\|p-p'\|_q^{q/2}$. In view of Proposition \[lemma-entropy-estimate-lq\] $$\log N\bigl(\varepsilon,\mathcal{P}_n,d\bigr)\lesssim \varepsilon ^{-2/(q+1)}\log
(1/\varepsilon).$$ Therefore the entropy condition is satisfied with $\varepsilon_n\asymp (\log n/n )^{(q+1)/(2q+4)}$. By Proposition \[PropositionPriorMass\] the prior mass condition is satisfied for $\varepsilon_n\asymp(\log n/n)^{3/8}$. By Theorem \[thm-rate-semi-metric\] the rate of contraction relative to $d$ is the slower of these two rates, which is the first. The rate relative to the $L_q$-norm is the $(2/q)$th power of this rate.
In the case of $L_2$, we apply Proposition $2$ of [@golubev1992nonparametric] to show the rate is minimax. If we take $\sigma(\lambda) = (i\lambda)^{3/2 - \delta}$ for any $\delta >0$, we have $$\int |\sigma(\lambda)|^{2+ \rho} |\tilde{p}_{G_0}(\lambda)|^2 d\lambda \le \int
|\sigma(\lambda)|^{2+\rho} |\tilde{f}(\lambda)|^2 d\lambda < \infty$$ for some $\rho>0$. It is implicitly shown in [@golubev1992nonparametric] that the rate of square risk, i.e., the square of $L_2$-norm, is bounded below by $n^{-3/4}$. This concludes our statement that the rate $n^{-3/8}$, up to a logarithm factor, is optimal in the minimax sense.
Normal mixtures
===============
We reproduce the results on normal mixtures from [@ghosal_entropies_2001], but in $L_2$-norm. Note the normal kernel is supersmooth with $\beta=2$, by the approximation lemma, for any measure $G_1$ compactly supported on $[-a, a]$ we can always find a discrete measure $G_2$ with number of support points of order $N \asymp \log \varepsilon^{-1}$ such that ${\|{p_{G_1} - p_{G_2}}\|}_2 \le \varepsilon$. It is easy to establish $$h^2(p_{G_1}, p_{G_2}) \lesssim {\|{p_{G_1} - p_{G_2}}\|}_2.$$
Following the same procedure as before, assuming $G_0$ is the true measure, we obtain for prior mass condition $$\log \Pi \left(G: \max\Big( P_{G_0} \log \frac{p_{G_0}}{p_G}, P_{G_0} \Big( \log \frac{p_{G_0}}{p_G}\Big)^2 \Big) \le \varepsilon^2 \right) \gtrsim - \Big( \log \frac{1}{\varepsilon} \Big)^2,$$ Thus we obtain $\varepsilon_n = \log n / \sqrt{n}$.
By Lemma \[lemma-entropy-estimate-lq\], we have the following estimate for entropy condition $$\log N( \varepsilon, {\mathcal{P}}_a, \| \cdot \|_2 ) \lesssim \Big( \log \frac{1}{\varepsilon} \Big)^2,$$ this coincides with the estimate of prior mass condition, thus we obtain the rate of $\varepsilon_n = \log n / \sqrt{n}$ with respect to $L_2$-norm. This is the same with what is obtained in [@ghosal_entropies_2001], only in $L_2$-norm. However we lose a $\sqrt{\log n}$-factor comparing to [@watson_estimation_1963], which is $\sqrt{\log n / n}$.
Appendix: contraction rates relative to non-metrics {#Appendix}
===================================================
The basic theorem of [@ghosal_convergence_2000] gives a posterior contraction rate in terms of a metric on densities that is bounded above by the Hellinger distance. In the present situation we would like to apply this result to a power smaller than one of the Wasserstein metric, which is not a metric. In this appendix we establish a rate of contraction which is valid for more general discrepancies.
We consider a general “discrepancy measure” $d$, which is a map $d: \mathcal{P}\times\mathcal{P}\to{\mathbb{R}}$ on the product of the set of densities on a given measurable space and itself, which has the properties, for some constant $C>0$:
- $d(x,y)\ge 0$;
- $d(x,y) = 0$ if and only if $x=y$;
- $d(x,y) = d(y,x)$;
- $d(x,y) \le C \big(d(x,z) + d(y,z)\big)$.
Thus $d$ is a metric except that the triangle inequality is replaced with a weaker condition that incorporates a constant $C$, possibly bigger than 1. Call a set of the form $\{x: d(x,y)<c\}$ a $d$-ball, and define covering numbers $N(\varepsilon,\mathcal{P},d)$ relative to $d$ as usual.
Let $\Pi_n(\cdot|X_1,\ldots, X_n)$ be the posterior distribution of $p$ given an i.i.d. sample $X_1,\ldots, X_n$ from a density $p$ that is equipped with a prior probability distribution $\Pi$.
\[thm-rate-semi-metric\] Suppose $d$ has the properties as given, the sets $\{p: d(p,p')<\delta\}$ are convex, and satisfies $d(p_0,p)\le h(p_0,p)$, for every $p\in\mathcal{P}$. Then $\Pi_n\bigl(d(p,p_0)>M \varepsilon_n | X_1,\ldots, X_n\bigr)\rightarrow 0 $ in $P_0^n$-probability for any $\varepsilon_n$ such that $n \varepsilon_n^2 \rightarrow \infty$ and such that, for positive constants $c_1$, $c_2$ and sets $\mathcal{P}_n \subset \mathcal{P}$, $$\begin{gathered}
\log N(\varepsilon_n, \mathcal{P}_n, d) \le c_1 n \varepsilon_n^2, \label{eqn-semi-metric-entropy} \\
\Pi_n(p: K(p_0, p) < \varepsilon_n^2, K_2(p_0, p) < \varepsilon_n^2 ) \ge e^{- c_2 n \varepsilon_n^2},
\label{eqn-semi-metric-prior} \\
\Pi_n(\mathcal{P}- \mathcal{P}_n) \le e^{- (c_2+4)n \varepsilon_n^2}. \label{eqn-semi-metric-sieve}\end{gathered}$$
For every $\varepsilon > 4 C \varepsilon_n$, we have $\log N( C^{-1} \varepsilon /4 , \mathcal{P}_n, d) \le \log N(\varepsilon_n, \mathcal{P}_n , d) \le c_1 n \varepsilon_n^2 $, take $N(\varepsilon) = \exp(c_1 n \varepsilon_n^2) $ and $\varepsilon = MC^{-1} \varepsilon_n$, $j =1 $ in Lemma \[prop-convex-test\], where $M > 4C$ is a large constant to be chosen later, there exist tests $\varphi_n$ with errors $$P^n_0 \varphi_n \le e^{c_1 n \varepsilon_n^2} \frac{e^{-n M^2 C^{-2} \varepsilon_n^2 /32}}{1 - e^{-n M^2 C^{-2} \varepsilon_n^2 /32}}, \quad
\sup_{p\in \mathcal{P}_n: d(p,p_0) > M\varepsilon_n} P^n (1 - \varphi_n) \le e^{-n M^2 C^{-2} \varepsilon_n^2 /32}.$$ Next the proof proceeds as in [@ghosal_convergence_2000]. All terms should tend to zero for $ M^2 / (32 C^{2} ) > c_1 $ and $M^2 / (32 C^{2} ) > 2 + c_2 $.
\[prop-convex-test\] Let $d$ be a discrepancy measure in the sense of (a)–(d) whose balls are convex and which is bounded from above by the Hellinger distance $h$. If $N(C^{-1}\varepsilon/4 , \mathcal{Q}, d) \le
N(\varepsilon)$ for any $\varepsilon>C\varepsilon_n >0$ and some non-increasing function $N:
(0,\infty) \rightarrow (0,\infty)$, then for every $\varepsilon > C\varepsilon_n$ and $n$, there exists a test $\varphi_n$ such that for all $j\in \mathbb{N}$, $$P^n \varphi_n \le N(\varepsilon) \frac{e^{-n\varepsilon^2/32}}{1-e^{-n\varepsilon^2/32}},
\quad \sup_{Q \in \mathcal{Q}, d(P,Q) > C j\varepsilon} Q^n (1-\varphi_n) \le e^{-n\varepsilon^2 j^2 /32}.$$
For a given $j \in \mathbb{N}$, choose a maximal set $Q_{j,1}, Q_{j,2},\dots, Q_{j, N_j}$ in the set $\mathcal{Q}_j = \{Q\in \mathcal{Q}: Cj \varepsilon < d(P,Q) < 2C j \varepsilon \}$ such that $d(Q_{j,k}, Q_{j,l}) \ge j\varepsilon/2$ for every $k \neq l$. By property (d) of the discrepancy every ball in a cover of $\mathcal{Q}_j$ by balls of radius $C^{-1}j\varepsilon/4$ contains at most one $Q_{j,k}$. Thus $N_j \le N(C^{-1}j\varepsilon/4, \mathcal{Q}_j, d) \le N(\varepsilon)$. Furthermore, the $N_j$ balls $ B_{j,l}$ of radius $j\varepsilon/2$ around $Q_{j,l}$ cover $\mathcal{Q}_j$, as otherwise the set of $Q_{j,l}$ would not be maximal. For any point $Q$ in each $B_{j,l}$, we have $$d(P, Q) \ge C^{-1} d(P,Q_{j,l}) - d(Q, Q_{j,l}) \ge j\varepsilon/2.$$ Since the Hellinger distance bounds $d$ from above, also $h(P, B_{j,l}) \ge j\varepsilon/2$. By Lemma \[tests-convex-sets\], there exist a test $\varphi_{j,l}$ of $P$ versus $B_{j,l}$ with error probabilities bounded from above by $e^{-nj^2\varepsilon^2 /32}$. Let $\varphi_n$ be the supremum of all the tests $\varphi_{j,l}$ obtained in this way, for $j = 1, 2, \dots$, and $l =
1, 2, \dots, N_j$. Then, $$\begin{aligned}
P^n \varphi &\le \sum_{j=1}^{\infty}\sum_{l=1}{N_j} e^{-n j^2 \varepsilon^2 /32} \le \sum_{j=1}^{\infty} N(C^{-1} j \varepsilon/4, \mathcal{Q}_j, d) e^{-nj^2 \varepsilon^2/32} \\
&\le N(\varepsilon) \frac{e^{-n\varepsilon^2/32}}{1- e^{-n\varepsilon^2/32}},\end{aligned}$$ and for every $j\in \mathbb{N}$, $$\sup_{Q\in \cup_{l>j} \mathcal{Q}_l} Q^n (1 - \varphi_n) \le \sup_{l>j} e^{-nl^2 \varepsilon^2/32} \le e^{-nj^2 \varepsilon^2/32},$$ by the construction of $\varphi_n$.
The following lemma comes from the general results of [@birge1984MR764150] and [@le_cam_asymptotic_1986].
\[tests-convex-sets\] For any probability measure $P$ and dominated, convex set of probability measures $\mathcal{Q}$ with $h(p,q) > \varepsilon$ for any $q \in \mathcal{Q}$ and any $n\in \mathbb{N}$, there exists a test $\phi_n$ such that $$P^n \phi_n \le e^{-n\varepsilon^2 /8}, \quad \sup_{Q \in \mathcal{Q}} Q^n(1-\phi_n) \le e^{-n\varepsilon^2/8}$$
|
---
author:
- 'Guy Fayolle [^1]'
- Cyril Furtlehner
bibliography:
- 'refer.bib'
date: January 2006
title: 'Stochastic Dynamics of Discrete Curves and Multi-Type Exclusion Processes'
---
Introduction {#INTRO}
============
Interplay between discrete and continuous description is a recurrent question in statistical physics, which in some cases can be addressed quite rigorously via probabilistic methods. In the context of reaction-diffusion systems this amounts to studying fluid or hydrodynamic limits, and number of approaches have been proposed, inparticular in the framework of exclusion processes, see [@Li],[@MaPr] [@Sp], [@KiLa] and references therein. As far as the above limits are at stake, all these methods have in common to be limited to systems having stationary states given in closed product form, or at least to systems for which the invariant measure for finite $N$ is explicitly known. For instance, <span style="font-variant:small-caps;">asep</span> with open boundary are described in terms of matrix product forms (really a sort of non-commutative product form), and the continuous limits can be understood by means of Brownian bridges [@DeEnLe]. We propose to tackle these problems from a different view-point. The initial objects are discrete sample paths enduring stochastic deformations, and our primary concern is to understand the nature of the limit curves, when $N$ goes to infinity: how do they evolve in time, and which limiting process do they represent as $t$ goes to infinity: in other words, what are the equilibrium curves? Following [@FaFu] and [@FaFu2], we give here some partial answers to these questions.
In [@FaFu] a specific model was considered, namely paths on the square lattice, and we could reformulate the problem in terms of coupled exclusion processes, to understand the thermodynamic equilibrium and a phase transition point above which curves reach a deterministic profile, solution of a nonlinear dynamical system which was solved explicitly by means of elliptic functions. Two extensions of this system were introduced in [@FaFu2] :
- one which comprises multi-type exclusion particle systems encountered in another context (see e.g. [@EvFoGoMu; @EvKaKoMu]), including the $ABC$ model for which similar features occur [@ClDeEv];
- a tri-coupled exclusion process to represent the stochastic dynamics of curves in the three-dimensional space.
With this extended formulation, we provided a set of general conditions for reversibility, by analyzing cycles in the state space and the corresponding invariant measure.
This paper focuses on non-Gibbs states and transient regimes. In another work in progress [@FaFu3], we analyze the asymmetric simple exclusion process (<span style="font-variant:small-caps;">asep</span>) on a torus. Under suitable initial conditions, the usual sequence of empirical measures converges in probability to a deterministic measure, which is the unique weak solution of a Cauchy problem. The method presents some new features, and relies on the analysis of a family of parabolic differential operators, involving variational calculus. This approach let hope for a pretty large level of generalization, and we are working over its general conditions of validity.
Sections \[gibbs\] and \[nongibbs\] are devoted to the stationary regime, for which, from [@FaFu] and [@FaFu2], the limit curves are known to satisfy a differential system of Lotka-Volterra type which is the essence of the fluid limits in our context. Section \[gibbs\] solves the steady state regime in the reversible case. A geometric interpretation of the free energy is provided (involving the algebraic area enclosed by the curve), as well as an urn model description for the underlying dynamical system, leading precisely to a Lotka-Volterra system.
Non-Gibbs states are considered in section \[nongibbs\]. In [@FaFu2], necessary and sufficient conditions for reversibility where given, by identification of a family of independent cycles in the state graph, for which Kolmogorov’s criteria have to be fulfilled. We pursue this analysis by showing that irreversibility occurs as a result of particle currents attached to these cycles. A connection between recursion properties (originating matrix solutions) and particle cycles in the state-graph is found, with the introduction of loop currents, on the analogy with electric circuits. These recursions at discrete level connect together invariant measures of systems of size $N$ (the number of sites) and of size $N-1$, and they involve coefficients which are given a concrete meaning. Indeed, by means of a functional approach, we map explicitly these structure coefficients onto special constants which intervene in the Lotka-Volterra systems describing the fluid limit, as $N\to\infty$.
In the last section \[fluctuations\], we observe that local equilibrium takes place at a rapid time-scale, compared to the diffusion time which is the natural scale of the system. We extend the iterative scheme procedure initiated in [@FaFu] and developed in [@FaFu2], which originally concerned only the steady-state regime. In fact, this scheme allow us to express in transient regime particle-currents in terms of deterministic particle densities: this is a mere consequence of a law of large numbers. At least when the diffusion scale is identical for all particle species, local correlations are found to be absent at the hydrodynamical scale. Finally, in the spirit of the study made in [@BeLa], we obtain the Lagrangian describing the fluctuations of currents, and we analyze the related Hamilton-Jacobi equations.
Model definition
================
A stochastic clock model
------------------------
The system consists of an oriented path embedded in a bidimensional manifold, with $N$ steps of equal size, each one being chosen among a discrete set of $n$ possible orientations, drawn from the set of angles with some given origin $\{\frac{2k\pi}{n}, k=0,\ldots,n-1\}$. The stochastic dynamics in force consists in displacing one single point at a time without breaking the path, while keeping all links within the set of admissible orientations. In this operation, two links are simultaneously displaced. This constrains quite strongly the possible dynamical rules, which are given in terms of *reactions* between consecutive links.
For any $n$, we can define $$\label{exchange}
X^k X^l\ \operatorname*{\rightleftarrows}_{\lambda_{lk}}^{\lambda_{kl}}\ X^l X^k,\quad
k\in[1,\,n],\, k\ne l,$$ which in the sequel will be sometimes referred to as a local exchange process. It is necessary to discriminate between $n$ odd and $n$ even. Indeed, for $n=2p$, there is another set of possible stochastic rules: $$\label{evnmod}
\begin{cases}
\DD X^k X^l\ \operatorname*{\rightleftarrows}_{\lambda_{lk}}^{\lambda_{kl}}\ X^l X^k,\qquad
k=1,\ldots,n, \quad l\ne k+p,\\
\DD X^k X^{k+p}\ \operatorname*{\rightleftarrows}_{\delta^{k+1}}^{\gamma^k}
\ X^{k+1} X^{k+p+1},\qquad k=1,\ldots,n .
\end{cases}$$ The distinction is simply due to the presence, for even $n$, of *folds* (two consecutive links with opposite directions), which may undergo different transition rules, leading to a richer dynamics. The parameters $\{\lambda_{kl}\}$ represent the exchange rates between two consecutive links, while the $\gamma_{k}$’s and $\delta_k$’s correspond to the rotation of a fold to the right or to the left.
Examples {#abcdef}
--------
*1) The simple exclusion process*
The first elementary and most studied example is the simple exclusion process, which after mapping particles onto links corresponds to a one-dimensional fluctuating interface. In that case, we simply have a binary alphabet. Letting $X^1=\tau$ and $X^2=\bar\tau$, the reactions rewrite $$\tau\bar\tau \operatorname*{\leftrightarrows}_{\lambda^+}^{\lambda^-}\ \bar\tau\tau,$$ where $\lambda^\pm$ is the transition rate for the jump of a particle to the right or to the left.
*2) The triangular lattice and the ABC model*
Here the evolution of the random walk is restricted to the triangular lattice. A link (or step) of the walk is either $1$, $e^{2i\pi/3}$ or $e^{4i\pi/3}$, and quite naturally will be said to be of type A, B and C, respectively. This corresponds to the so-called *ABC model*, since there is a coding by a $3$-letter alphabet. The set of *transitions* (or reactions) is given by $$\begin{aligned}
AB\ \operatorname*{\leftrightarrows}_{\lambda_{ab}}^{\lambda_{ba}}\ BA, \qquad
BC\ \operatorname*{\leftrightarrows}_{\lambda_{bc}}^{\lambda_{cb}}\ CB, \qquad
CA\ \operatorname*{\leftrightarrows}_{\lambda_{ca}}^{\lambda_{ac}}\ AC, \qquad \end{aligned}$$ where the rates are arbitrary positive numbers. Also we impose *periodic boundary conditions* on the sample paths. This model was first introduced in [@EvFoGoMu] in the context of particles with exclusion, and, for some cases corresponding to reversibility, a Gibbs form has been found in [@EvKaKoMu].
*3) A coupled exclusion model in the square lattice*
This model was introduced in [@FaFu] to analyze stochastic distortions of a walk in the square lattice. Assuming links are counterclockwise oriented, the following transitions can take place. $$\begin{aligned}
AB\ \operatorname*{\rightleftarrows}_{\lambda_{ab}}^{\lambda_{ba}}\ BA, \qquad
BC\ \operatorname*{\rightleftarrows}_{\lambda_{bc}}^{\lambda_{cb}}\ CB, &\qquad&
CD\ \operatorname*{\rightleftarrows}_{\lambda_{cd}}^{\lambda_{dc}}\ DC, \qquad
DA\ \operatorname*{\rightleftarrows}_{\lambda_{da}}^{\lambda_{ad}}\ AD, \\[0.2cm]
AC\ \operatorname*{\rightleftarrows}_{\gamma_{ac}}^{\delta_{bd}}\ BD, \qquad
BD\ \operatorname*{\rightleftarrows}_{\gamma_{bd}}^{\delta_{ca}}\ CA, &\qquad&
CA\ \operatorname*{\rightleftarrows}_{\gamma_{ca}}^{\delta_{db}}\ DB, \qquad
DB\ \operatorname*{\rightleftarrows}_{\gamma_{db}}^{\delta_{ac}}\ AC. \end{aligned}$$ We studied a rotation invariant version of this model, namely when $$\label{symmetric}
\begin{cases}
\lambda^+ {\stackrel{\mbox{\upshape\tiny def}}{=}}\lambda_{ab} = \lambda_{bc} =
\lambda_{cd} = \lambda_{da}, \\
\lambda^- {\stackrel{\mbox{\upshape\tiny def}}{=}}\lambda_{ba} = \lambda_{cb} =
\lambda_{dc} = \lambda_{ad}, \\
\gamma^+ {\stackrel{\mbox{\upshape\tiny def}}{=}}\gamma_{ac} = \gamma_{bd}
= \gamma_{ca} = \gamma_{db}. \\
\gamma^- {\stackrel{\mbox{\upshape\tiny def}}{=}}\delta_{ac} = \delta_{bd}
= \delta_{ca} = \delta_{db}.
\end{cases}$$ Define the mapping $(A,B,C,D)\to(\tau^a,\tau^b)\in\{0,1\}^2$, such that $$\label{cpmapping}
\begin{cases}
A \to (0,0),\\
B \to (1,0),\\
C \to (1,1),\\
D \to (0,1).\\
\end{cases}$$ The dynamics can be formulated in terms of coupled exclusion processes. The evolution of the sample path is represented by a Markov process with state space the set of $2N$-tuples of binary random variables $\{\tau_i^a\}$ and $\{\tau_i^b\}$, $i=1,\ldots, N$, taking the value $1$ if a particle is present and $0$ otherwise. The jump rates to the right ($+$) or to the left ($-$) are then given by $$\label{taux}
\begin{cases}
\lambda_a^{\pm}(i)={\bar \tau_i^b}{\bar
\tau_{i+1}^b}\lambda^{\mp}+\tau_i^b
\tau_{i+1}^b\lambda^{\pm} + {\bar
\tau_i^b}\tau_{i+1}^b\gamma^{\mp} + \tau_i^b{\bar
\tau_{i+1}^b}\gamma^{\pm},\\[0.2cm] \lambda_b^{\pm}(i)={\bar
\tau_i^a}{\bar \tau_{i+1}^a}\lambda^{\pm}+\tau_i^a
\tau_{i+1}^a\lambda^{\mp} +
{\bar \tau_i^a}\tau_{i+1}^a\gamma^{\pm} +
\tau_i^a{\bar \tau_{i+1}^a}\gamma^{\mp}.
\end{cases}$$ Notably, one sees the jump rates of a given sequence are locally conditionally defined by the complementary sequence.
Stationary regime for reversible systems {#gibbs}
========================================
In this section, we quote the main characteristics of the steady state distribution when the processes at stake are reversible.
The general form of the invariant measure
-----------------------------------------
Up to a slight abuse in the notation, we let $X_i^k\in\{0,1\}$ denote the binary random variable representing the occupation of site $i$ by a letter of type $k$. The state of the system is represented by the array $\eta{\stackrel{\mbox{\upshape\tiny def}}{=}}\{X_i^k, i=1,\ldots,N;
k=1,\ldots,n\}$ of size $N\times\,n$. The invariant measure of the Markov process of interest is given by $$\label{inv}
\pi_\eta =
\frac{1}{Z}\exp\bigl[-\mathcal{H}\bigl(\eta\bigr)\bigr],$$ where $$\label{ansatz}
\mathcal{H}(\eta) = \frac{1}{N}
\sum_{i<j}\sum_{k,l} \alpha_{kl}{^{\scriptscriptstyle (N)}}X_i^k X_j^l,$$ with $\alpha_{kl}{^{\scriptscriptstyle (N)}}$ and $\alpha_{lk}{^{\scriptscriptstyle (N)}}$ two $N$-dependent coefficients related by $$\label{alphacoef}
\alpha_{kl}{^{\scriptscriptstyle (N)}}- \alpha_{lk}{^{\scriptscriptstyle (N)}}= N\log\frac{\lambda_{kl}}{\lambda_{lk}},$$ provided that some *balance* conditions hold (see e.g. [@Kel]). For example, in the clock model (\[exchange\]), these conditions take the simple form $$\label{oddinv}
\sum_{k\ne l} \bigl(\alpha_{kl}{^{\scriptscriptstyle (N)}}-\alpha_{lk}{^{\scriptscriptstyle (N)}}\bigr) N_k = 0,$$ and they follow indeed directly from Kolmogorov’s criteria (applied to a particle crossing the system), which is tantamount to detailed balance equations.
### An example in the square lattice
To show a concrete exploitation of the form (\[inv\]), we consider the square-lattice model introduced in [@FaFu]. It does illustrate the rules (\[evnmod\]). Instead of handling the problem directly with the natural set of four letters $\{A,B,C,D\}$, we found convenient to represent the degrees of freedom by pairs of binary components. In the symmetric version of the model defined by (\[symmetric\]), when cycles are absent ($N_a=N_b=1/2$ and $\gamma^+=\gamma^-$), we could derive the invariant measure $$\label{invform}
\pi_\eta = \frac{1}{Z}\exp\bigl[\beta\sum_{i<j}
(\tau_i^a\bar\tau_j^b - \tau_i^b\bar\tau_j^a)\bigr],$$ with $\eta = \{(\tau_i^a,\tau_i^b),i=1\ldots N\}$ with $\beta = \log\frac{\lambda^-}{\lambda^+}$. Let us see how this relates to the original formulation of the model in terms of the four letters $A, B, C$ and $D$.
Under the reversibility conditions imposed on the transitions rates $\{\lambda_{kl},\gamma^k,\delta^k,k=1\ldots4,l=1\ldots4\}$, the measure given by (\[inv\])and (\[ansatz\]) reduces to $$\begin{aligned}
\pi_\eta &=& \frac{1}{Z} \exp\biggl\{\frac{\beta}{2}\sum_{i<j}
B_iA_j - A_iB_j\ +\ A_iD_j - D_iA_j \nonumber \\[0.2cm]
& & \qquad \quad + \ C_iB_j - B_iC_j\ +\ D_iC_j - C_iD_j\biggr\},
\label{invariantform}\end{aligned}$$ and is equivalent to (\[invform\]).
The proof is not difficult, starting from (\[invform\]). It can also be achieved by a direct argument, i.e. without using (\[invform\]), from theorem 3.2 of [@FaFu2].
Free energy
-----------
We consider again the $ABC$ model as a typical example, and the extension to other models will be straightforward. Assume conditions (\[oddinv\]) hold, so that the invariant measure is given by $$\pi_\eta = \frac{1}{Z}\exp\bigl[\frac{1}{N}\sum_{i<j}^N\alpha_{ab}{^{\scriptscriptstyle (N)}}A_iB_j
+\alpha_{bc}{^{\scriptscriptstyle (N)}}B_iC_j+\alpha_{ca}{^{\scriptscriptstyle (N)}}C_iA_j\bigr],$$ where the constants $\alpha_{ab}{^{\scriptscriptstyle (N)}}$, $\alpha_{bc}{^{\scriptscriptstyle (N)}}$ and $\alpha_{ca}{^{\scriptscriptstyle (N)}}$ take the values $$\alpha_{ab}{^{\scriptscriptstyle (N)}}= N\log\frac{\lambda_{ab}}{\lambda_{ba}},\quad
\alpha_{bc}{^{\scriptscriptstyle (N)}}= N\log\frac{\lambda_{bc}}{\lambda_{cb}},\quad
\alpha_{ca}{^{\scriptscriptstyle (N)}}= N\log\frac{\lambda_{ca}}{\lambda_{ac}},$$ while $\alpha_{ba}{^{\scriptscriptstyle (N)}}$ $\alpha_{cb}{^{\scriptscriptstyle (N)}}$ and $\alpha_{ac}{^{\scriptscriptstyle (N)}}$ are set to zero, to be consistent with (\[alphacoef\]). The constraints (\[oddinv\]) now become $$\label{conditions}
\frac{N_A}{N_B} = \frac{\alpha_{bc}{^{\scriptscriptstyle (N)}}}{\alpha_{ca}{^{\scriptscriptstyle (N)}}},\quad
\frac{N_B}{N_C} = \frac{\alpha_{ca}{^{\scriptscriptstyle (N)}}}{\alpha_{ab}{^{\scriptscriptstyle (N)}}},\quad
\frac{N_C}{N_A} = \frac{\alpha_{ab}{^{\scriptscriptstyle (N)}}}{\alpha_{bc}{^{\scriptscriptstyle (N)}}}.$$ Following [@ClDeEv], we want to write a large deviation functional corresponding to the above Gibbs measure when $N\to\infty$. Set $x=\frac{i}{N}, \,J=\exp(2i\pi/3)$, and let $Z(x)$ denote the complex number given by $$Z(x) = \frac{1}{N} \sum_{i=1}^{[xN]}\Bigl(\frac{A_i}{\alpha}+
J\frac{B_i}{\beta}+J^2\frac{C_i}{\gamma} \Bigr),$$ where we have introduced the parameters $$\alpha {\stackrel{\mbox{\upshape\tiny def}}{=}}\lim_{N\to\infty} \alpha_{bc}{^{\scriptscriptstyle (N)}}, \qquad
\beta {\stackrel{\mbox{\upshape\tiny def}}{=}}\lim_{N\to\infty} \alpha_{ca}{^{\scriptscriptstyle (N)}}, \qquad
\gamma {\stackrel{\mbox{\upshape\tiny def}}{=}}\lim_{N\to\infty} \alpha_{ab}{^{\scriptscriptstyle (N)}}.$$ The sequence $\eta= \{(A_i,B_i,C_i),i=1\ldots N\}$ is thus represented by a discrete path $\Gamma$ in the complex plane, made of oriented links having only three possible directions $$\{\theta=0, \ \theta=2\pi/3,\ \theta=4\pi/3\},$$ depending on whether a particle A, B or C is present. The length of a link corresponding to A, B, or C is, respectively, $1/(N\alpha)$, $1/(N\beta)$ or $1/(N\gamma)$.
The equation of $\Gamma$ is given by a function $Z :{\stackrel{\mbox{\upshape\tiny def}}{=}}x\rightarrow Z(x), x\in \mathbb{C}$. Note that condition (\[conditions\]) ensures $\Gamma$ is closed, that is $$Z(1) = \frac{1}{\alpha+\beta+\gamma}(1+J+J^2) = 0.$$ The area $\mathcal {A}$ enclosed by $\Gamma$ is given by $$\mathcal {A} {\stackrel{\mbox{\upshape\tiny def}}{=}}\frac{1}{2i}\oint_\Gamma\bigl(\bar z dz - z d\bar z\bigr),$$ and, for large $N$, this coincide with $$\label{AIRE}
\mathcal{A} = \frac{\sqrt{3}}{N^2}
\sum_{l<k} \frac{A_l}{\alpha}\Bigl(\frac{B_k}{\beta}
-\frac{C_k}{\gamma}\Bigr) + \frac{B_l}{\beta}\Bigl(\frac{C_k}{\gamma} -
\frac{A_k}{\alpha}\Bigr) + \frac{C_l}{\gamma}\Bigl(\frac{A_k}{\alpha}
- \frac{B_k}{\beta}\Bigr)
+ o(1).$$ As a result, $${\cal H}(\{\eta\}) = \frac{N \alpha\beta\gamma}{2\sqrt 3}\ {\cal A} +
\frac{3N\alpha\beta\gamma}{(\alpha+\beta+\gamma)^2} + O(1).$$ The large deviation probability is easily obtained from the law of large numbers. It is given by $$\label{ldfunctional}
P_N(\rho_a,\rho_b,\rho_c) =
\frac{1}{Z}\exp\bigl(-N{\cal F}(\rho_a,\rho_b,\rho_c)\bigr),$$ with the free energy $$\label{freen}
{\cal F}(\rho_a,\rho_b,\rho_c) = \frac{\alpha\beta\gamma}{2\sqrt{3}}\
{\cal A}(\rho_a,\rho_b,\rho_c)
-{\cal S}(\rho_a,\rho_b,\rho_c),$$ where $$\mathcal{A}(\rho_a,\rho_b,\rho_c) {\stackrel{\mbox{\upshape\tiny def}}{=}}\sqrt{3}\int_0^1dx\int_x^1dy
\frac{\rho_a(x)}{\alpha}\Bigl(\frac{\rho_b(y)}{\beta}
-\frac{\rho_c(y)}{\gamma}\Bigr)+\frac{\rho_b(x)}{\beta}\Bigl(\frac{\rho_c(y)}{\gamma}
-\frac{\rho_a(y)}{\alpha}\Bigr)+\frac{\rho_c(x)}{\gamma}\Bigl(\frac{\rho_a(y)}{\alpha}
-\frac{\rho_b(y)}{\beta}\Bigr).$$ and where the entropy term comes from a multinomial combinatorial factor $\frac{n!}{n_a!n_b!n_c!}$, namely the way of arranging a box of n=\[N dx\] sites, with $3$ species of identical particles having respective populations $n_i=\rho_i(x) N dx$, $i\in\{a,b,c\}$. Stirling’s formula for large $N$ yields $${\cal S}(\rho_a,\rho_b,\rho_c) =
- \int_0^1 dx[\rho_a(x)\log\rho_a(x)+\rho_b(x)\log\rho_b(x)+\rho_c(x)\log\rho_c(x)].$$
Stable and metastable deterministic profiles correspond to local minima of the free-energy. According to (\[freen\]), an optimal profile is a compromise between a maximal entropy and a minimum of the enclosed algebraic area. Curves of maximal entropy are typically Brownian, and they have an area which scales like $1/N$; on the other hand, the opposite extreme configuration consisting of an equilateral triangle with negative orientation achieves the minimum algebraic area, but belongs to a class of profiles for which the entropy contribution is equal to zero (since $\rho\log\rho$ vanishes both for $\rho=0$ and $\rho=1$). Depending on the ratio $\alpha\beta\gamma/2\sqrt{3}$ of the two contributions, we obtain either Brownian (the degenerate point of the deterministic equations, see below) or deterministic profiles, both regimes being separated by a second order phase transition.
Lotka Volterra systems
----------------------
Under the scaling earlier defined, letting $N\to\infty$, we show on two examples that the limiting invariant measure is the solution of a non-linerar differential system of Lotka-Volterra type.
### Urn model
Consider three species, denoted by $\{A,B,C\}$, and let $N_a^{(N)}(t)$, $N_b^{(N)}(t)$ and $N_c^{(N)}(t)$ be the corresponding time-dependent populations. The system is closed, $N_a+N_b+N_c=N$. At random times taken as exponential events, individuals do meet and population transfer take place at rates $\alpha$, $\beta$, $\gamma$, associated with the reactions $$\begin{cases}
\DD AB \operatorname*{\rightarrow}_{\gamma} BB ,\\[0.2cm]
\DD BC \operatorname*{\rightarrow}_{\alpha} CC ,\\[0.2cm]
\DD CA \operatorname*{\rightarrow}_{\beta} AA .
\end{cases}\nonumber$$ This zero-range process is an urn-type model of Ehrenfest *Class*, as defined in [@GoLu], where indivivuals, rather than urns, are chosen at random. When $N$ increases to infinity, we rather consider concentrations instead of integer numbers: $$\rho_i(t) {\stackrel{\mbox{\upshape\tiny def}}{=}}\lim_{N\to\infty}\frac{N_i^{(N)}(t)}{N},$$ for $i=a,b,c$. After a proper scaling limit, the dynamics of the model is described by the following Lotka-Volterra system $$\begin{cases}
\DD \frac{\partial \rho_a}{\partial x} = \rho_a(\beta \rho_c - \gamma \rho_b), \\[0.2cm]
\DD \frac{\partial \rho_b}{\partial x} = \rho_b(\gamma \rho_a - \alpha \rho_c), \\[0.2cm]
\DD \frac{\partial \rho_c}{\partial x} = \rho_c(\alpha \rho_b - \beta \rho_a),
\end{cases}\nonumber$$ which, after replacing $x$ by $t$ and densities by concentrations, is nothing else but the differential system giving the invariant measure of the $(A,B,C)$ model, in the fluid limit at thermodynamical equilibrium [@ClDeEv].
### The square lattice model
From (\[invariantform\]), one can write down the large deviation functional ${\cal F}(\rho_A,\rho_B,\rho_C,\rho_D)$, \[as in (\[ldfunctional\])\], together with the conditions ensuring an optimal profile. This leads again to a differential system of Lotka-Volterra class $$\begin{aligned}
\label{LotkaII}
\frac{\partial\rho_A}{\partial x} = \eta\rho_A(\rho_B - \rho_D), &\qquad&
\frac{\partial\rho_B}{\partial x} = \eta\rho_B(\rho_C - \rho_A), \nonumber \\
\frac{\partial\rho_C}{\partial x} = \eta\rho_C(\rho_D - \rho_B), &\qquad&
\frac{\partial\rho_D}{\partial x} = \eta\rho_D(\rho_A - \rho_C), \end{aligned}$$ in which the last equation follows merely by summing up the three other ones. This system is structurally different from the one obtained in [@FaFu], which involved only two independent profiles $(\rho_a,\rho_b)$ corresponding to deterministic densities for the particles $\tau_a$ and $\tau_b$, while in the present case there are three ($\rho_A,\rho_B,\rho_C$ for example).
It is interesting to notice that, in both models, explicit level surfaces exist. Indeed, the above system satisfies $\rho_A\rho_B\rho_C\rho_D=cte$, in addition to constraint $\rho_A+\rho_B+\rho_C+\rho_D=1$. On the other hand, $\rho_a(1-\rho_a)\rho_b(1-\rho_b)$ is the level surface of the former system encountered in [@FaFu]. This can be explained by reversing the mapping (\[mapping\]), so that $$\begin{aligned}
A_i = \bar\tau_i^a\bar\tau_i^b,&\qquad& B_i = \tau_i^a\bar\tau_i^b,
\nonumber \\[0.2cm]
\label{nlmapping}C_i = \tau_i^a\tau_i^b,&\qquad& D_i = \bar\tau_i^a\tau_i^b. \end{aligned}$$ This indicates that the set of $4$-tuples $\{\tau_i^a,\bar\tau_i^a,\tau_i^b,\bar\tau_i^b\}$ constitutes the elementary blocks of the system, and that letters $A_i,B_i,C_i,D_i$ are composite variables encoding correlations of these building blocks. Therefore, in the continuous limit, we are left with two different descriptions of the same system, related in a non trivial manner. We propose now to explore more carefully this connection. In particular, while the linear mapping $$\label{mapping}
\begin{cases}
\tau_i^a = B_i+C_i , \qquad \bar\tau_i^a = A_i+D_i, \\
\tau_i^b = C_i+D_i , \qquad \bar\tau_i^a = A_i+B_i.
\end{cases}$$ still holds in the continuous limit, as a relation between expected values $$\label{lmap}
\begin{cases}
\rho_a = \rho_B+\rho_C, \\
\rho_b = \rho_C+\rho_D,
\end{cases}$$ the non-linear equations (\[nlmapping\]) are instead expected to bring a different form, since they involve correlations.
The differential system given by $$\label{stat}
\begin{cases}
\DD\frac{\partial}{\partial
x}\bigl[\log\frac{\rho^a(x)}{1-\rho^a(x)}\bigr] = 2\eta
(2\rho_b(x)-1),\\[0.3cm]
\DD\frac{\partial}{\partial
x}\bigl[\log\frac{\rho^b(x)}{1-\rho^b(x)}\bigr] = -2\eta
(2\rho_a(x)-1),
\end{cases}$$ is related to (\[LotkaII\]) through the invertible functional mapping given by $$\label{revmap}
\begin{cases}
\rho_A = \bar\rho_a\bar\rho_b+K, \qquad \rho_B = \rho_a\bar\rho_b-K, \\
\rho_C = \rho_a\rho_b+K, \qquad \rho_D = \bar\rho_a\rho_b-K ,
\end{cases}$$ where $K$ is a constant to be determined.
First, let $\{\rho_B,\rho_C,\rho_D\}$ be the set of independent variables in (\[LotkaII\]), and express them in terms of the new triple $\{\rho_a,\rho_b,\rho_C\}$ given by (\[lmap\]). This gives $$\begin{aligned}
\frac{\partial(\rho_a-\rho_C)}{\partial x} &=&
\eta(\rho_a - \rho_C)(\rho_a+\rho_b -1), \nonumber \\
\frac{\partial(\rho_b-\rho_C)}{\partial x} &=&
\eta(\rho_b - \rho_C)(1 - \rho_a+\rho_b), \nonumber \\
\frac{\partial \rho_C}{\partial x} \hspace{1cm}
&=& \eta\rho_C(\rho_b - \rho_a). \label{rhoC}\end{aligned}$$ Combining these equations yields $$\label{tmp}
\begin{cases}
\DD \frac{\partial\rho_a}{\partial x} =
\eta\rho_a(\rho_a + \rho_b -1) +\eta\rho_C(1-2\rho_a), \\[0.2cm]
\DD \frac{\partial\rho_b}{\partial x} =
\eta\rho_b(1-\rho_a - \rho_b) +\eta\rho_C(2\rho_b -1),
\end{cases}$$ which in turn allows to express $\rho_C$ as $$\rho_C = \frac{1}{\rho_a-\rho_b}\biggl(\rho_a\frac{\partial\rho_b}{\partial x}\
+\ \rho_b\frac{\partial\rho_a}{\partial x} \biggr).$$
Instantiating this last value of $\rho_C$ in (\[tmp\]) and in (\[rhoC\]), we obtain (\[stat\]), after immediate recombination, together with the relation $$\frac{\partial \rho_C}{\partial x} = \frac{\partial(\rho_a\rho_b)}{\partial x}.$$ This last equation has its counterpart for $\rho_A$, $\rho_B$ and $\rho_D$: after integration, we are left with four constants, which reduce to the one given in (\[revmap\]) only when compatibility with (\[lmap\]) is imposed.
Non-Gibbs steady state regime {#nongibbs}
=============================
We call non-Gibbs steady state regime, a regime for which the invariant measure is not described by means of a potential. This occurs when reversibility is broken, that is when there exists at least one cycle in the state graph for which the Kolmogorov criteria fails. A complete set of detailed balanced equations cannot be written in such a case, there exist at least two states $\eta$ and $\eta'$, connected by a single particle jump, with rate $\lambda_{\eta\eta'}$, $\lambda_{\eta'\eta}$ such that $$\label{eq:gcurrent}
\lambda_{\eta\eta'} \pi_\eta - \lambda_{\eta'\eta}\pi_{\eta'}
= \phi \ne 0,$$ if $\pi_\eta$ denotes the invariant measure. It is the second member of this equation we wish to study in this section. In the sequel we note $\cal S$ the state space, $\cal G$ the corresponding state graph, by assigning oriented edges between pair of nodes $(\alpha,\beta)\in{\cal S}^2$, when the rate $\lambda_{\alpha\beta}$ is non-zero, $\cal C$ will denote a cycle in $\cal G$ and we denote $\cal T$ the set of spanning trees on $\cal G$.
The tagged particle cycle
-------------------------
Cycles in the state graph for the $n$ odd model are important in the analysis of reversibility, and they are the ones for which at least one particle performs a complete round-trip. For example if a given particle makes $N-1$ successive jumps to the right, because of the circular geometry, the initial and final states are identical, up to a 1-step global shift to the left of the particles. As long as this particle is the only one in movement, the permutation order of the remaining other $N-1$ particles is kept frozen. The corresponding subsequence $\eta{^{\scriptscriptstyle (N-1)}}$ will in the sequel denote these specific cycles. Let us examine this one particle model, by tagging a specific particle which is given a new label $Y$, and by following its motion conditionally on $\eta^*=\{X_i^{k},i=1\ldots N, k\in\{1\ldots n\}\}$, the complementary frozen set of particles. This is equivalent to consider $Y$ moving in the inter-sites $\{i+1/2,i=0\ldots N-1\}$ of the $N-1$ frozen particles. The question is then to analyze the steady-state regime of a particle moving around a circular lattice in a random environment. To all allowed transitions which are jumps of $Y$ between sites $i-\frac{1}{2}$ and $i+\frac{1}{2}$, we let correspond the set of conditional transition rates given by $$\begin{cases}
\DD\lambda_y^{+}(i) = \sum_{k=1}^n \lambda_{yk}\ X_i^k,\\[0.2cm]
\DD\lambda_y^{-}(i) = \sum_{k=1}^n \lambda_{ky}\ X_i^k.
\end{cases}$$ Violation of condition (\[oddinv\]) corresponds to have, $$\label{def:det}
\det(\eta^*) {\stackrel{\mbox{\upshape\tiny def}}{=}}\prod_{i=0}^{N-1} \lambda_y^{+}(i) -
\prod_{i=0}^{N-1}\lambda_y^{-}(i) \ne 0,$$ This coefficient attached to the cycle $\eta^*$ is the determinant of the set of flux equations $$\label{eq:flux}
\lambda_y^{+}(i)\pi_{i-\frac{1}{2}} - \lambda_y^{-}(i)\pi_{i+\frac{1}{2}}
= \phi(\eta^*), \quad\qquad i=0,\ldots ,N-1,$$ giving the invariant measure $\pi_{i+\frac{1}{2}}$ which reads, $$\pi_{i+\frac{1}{2}} = \frac{1}{Z} \sum_{l=1}^N
\exp\biggl\{\sum_{m=1}^{n,N}\sum_{l+1<j<i} X_j^m\log\lambda_{ym} +
\sum_{i<j<l} X_j^m\log\lambda_{my}\biggr\}, \qquad i=0,\ldots, N-1,$$ where $Z$ is a normalization constant. A diagramatic representation of each term in the summation (over $l$) is given in Fig. \[loop\].b. Each term is in fact a spanning tree on the reduced tagged-particle state-graph, weighted by the transitions rates and rooted at the considerd point ($i+\frac{1}{2}$ for $\pi_{i+\frac{1}{2}}$). The constant $Z$ is therefore the sum of all spanning-trees on the reduced tagged-particle state-graph. The probability current between site $i-\frac{1}{2}$ and site $i+\frac{1}{2}$ reads $$\lambda_y^{+}(i)\pi_{i-\frac{1}{2}} -
\lambda_y^{-}(i)\pi_{i+\frac{1}{2}}
= \frac{1}{Z}\biggl[\exp\bigl(\sum_{m=1}^n N_m\log\lambda_{ym}\bigr)
\ -\ \exp\bigl(\sum_{m=1}^n N_m\log\lambda_{my}\bigr)\biggr],$$ with $N_m$ the number of particles of type $m$, a quantity independent of $i$. This shows that $\phi(\eta^*)$ is a quantity attached to the cycle $\eta^*$, which will be referred to as [*cycle current*]{} and reads $$\label{def:flux}
\phi(\eta^*) = \frac{1}{Z} \det(\eta^*).$$ Depending on the sign of $\det(\eta^*)$, the diffusion of particle $Y$ is biased in the right ($\det(\eta^*)>0$) or in the left ($\det(\eta^*)<0$) direction. Of course the reversible case is recovered when the determinant vanishes, which corresponds exactly to Kolmogorov’s criterion.
[**Case of open systems: example of ASEP**]{}
Consider the well studied asymmetric simple exclusion process <span style="font-variant:small-caps;">asep</span> with open boundary conditions, defined by $\alpha$ the rate of particle entering to the left side and $\beta$ the rate at which particles exit from the right side. The generalization to open systems of our definition of the tagged particle cycle (<span style="font-variant:small-caps;">tpc</span>) is depicted in figure \[asepcycles\].a. We adopt the convention for the cycle orientation that particles move positively to the right and holes to the left. Assume we give a tag to one of the particles. Let it perform sucessive jumps until reaching the right side; when it leaves the system it is in fact transformed into a hole; We keep the tag attached to the hole which performs successive jumps in the opposite direction until it reaches the left side; again it transformed back into a particle which in turn performs jumps to the right until the reaching of the initial position, to conclude the cycle.
Combinatorial formulas for invariant measure and currents
---------------------------------------------------------
We give here a combinatorial way of expressing the stationary measure on a connex finite state space $\cal S$ of size $\cal N$, the number of states. We consider a continuous-time irreducible Markov chain, with a set of transition rates $\lambda_{\alpha\beta}$ between states $\alpha$ and $\beta$ and define the corresponding state graph $\cal G$, based on $\cal S$, by assigning oriented edges between pair of nodes $(\alpha,\beta)$, when the corresponding rate $\lambda_{\alpha\beta}$ is non-zero.
\[combin1\] The invariant measure $\pi_\alpha$ is given by $$\label{invmeasure}
\pi_\alpha = \frac{\sum_{t\in{\cal T}_\alpha}w(t)}{\sum_{t\in{\cal T}}w(t)}$$ where $\cal T$ is the set of spanning tree over $\cal G$, ${\cal T}_\alpha$ is the set of spanning tree over $\cal G$ rooted in $\alpha$, and $w(t)$ the weight of a tree $t$ given by $$w(t) = \prod_{(\alpha,\beta)\in t} \lambda_{\alpha\beta}.$$
This follows from reexpressing the solution to the steady-state equation $$\pi_\alpha G_{\alpha\beta} = 0,\qquad\qquad \forall\beta\in{\cal S}$$ where G is the generator, and $G_{\alpha\beta} = -
\Bigl(\sum_\gamma \lambda_{\alpha\gamma}\Bigr)\delta_{\alpha\beta} +
\lambda_{\alpha\beta}$, using the Cramer relation. Indeed since $$\sum_{\beta=1}^{\cal N} G_{\alpha\beta} = 0,$$ the set of steady-state equations is of rank ${\cal N}-1$ and $\pi_\alpha$ can be written as the ratio of two determinants, namely the cofactor $\tilde G_{\alpha {\cal N}}$ of $G_{\alpha {\cal N}}$ and the determinant $|\tilde G|$ of the matrix obtained from $G$ by replacing $G_{\beta {\cal N}}$ by $1$ for $\beta=1,\ldots, {\cal N}$. $G$ has a structure of an admittance-matrix, as a result, expanding $\tilde G_{\alpha
\beta}$ and $|\tilde G|$ amounts to sum over spanning trees, $$\tilde G_{\alpha {\cal N}} = \sum_{t\in{\cal T}_\alpha}w(t),\qquad\qquad
|\tilde G| = \sum_{t\in{\cal T}}w(t),$$ which leads to formula (\[invmeasure\]).
From this observation we deduce a way to express the probability currents at steady-state, which generalizes formula (\[eq:flux\]) and (\[def:flux\]). First call $\det(C)$ a coefficient attached to each cycle $C$, $$\det(C) {\stackrel{\mbox{\upshape\tiny def}}{=}}\prod_{(\gamma,\delta)\in C}\lambda_{\gamma\delta}
- \prod_{(\gamma,\delta)\in C}\lambda_{\delta\gamma},$$ generalizing (\[def:det\]) and where the orientation of $C$ is prescribed by the orientation of $(\alpha,\beta)$ and the product over the set $(\gamma,\delta)\in C$, is understood according to this orientation. Let ${\cal C}_{\alpha\beta}$ the set of cycles in $\cal G$ containing the oriented edge $(\alpha,\beta)$. Let ${\cal T}_{\cal C}$ a set of subgraph of $\cal G$, s.t. when $\cal C$ is glued into a single node $\alpha_{\cal C}$, ${\cal T}_{\cal C}$ represents the set of spanning trees rooted in $\alpha_C$.
The steady state current between states $(\alpha,\beta)\in {\cal S}^2$ is given by $$\label{invcurrent}
\lambda_{\alpha\beta}\pi_\alpha -
\lambda_{\beta\alpha}\pi_\beta = \sum_{C\in{\cal C}_{\alpha\beta}}
\frac{\sum_{t\in{\cal T}_{\cal C}}w(t)}{\sum_{t\in{\cal T}}w(t)}\det(C)$$
When $\pi_\alpha$ is multiplied by $\lambda_{\alpha\beta}$, each spanning tree contribution is transformed by drawing an oriented edge between $\alpha$ and $\beta$. Since the spanning tree contains by construction of $\pi_\alpha$ a path going from $\beta$ to $\alpha$, the added edges contributes to the forming of a cycle which contains $\alpha$ and $\beta$. If each oriented edge in this cycle have a reversed counterpart, then in $\lambda_{\beta\alpha}\pi_\beta$ there is to be found a corresponding term with the same edges but with reversed orientation in the cycle (see Fig. \[state\_graph\]). In any case, $\det(C)$ factors out of an ensemble of contributions which consist in drawing trees spanning all the subgraph ${\cal G}$ with end-points on $C$, divided by the global normalization constant $\sum_{t\in{\cal T}}w(t)$. This complete the justification of formula (\[invcurrent\]).
Note that $\sum_{t\in{\cal T}_{\cal C}}w(t)$ in (\[invcurrent\]) represents the unormalized invariant measure of $\alpha_{\cal C}$ on the reduced graph ${\cal G}/{\cal C}$. This indicates that (\[invcurrent\]) bears recursive properties which could be used for asymptotic limits when the size of the system tends to infinity. Let us call $C$ a reversible \[resp. non-reversible\] cycle if $\det(C)=0$ \[resp. $\det(C)\ne0$\]. In the loop expansion of the currents provided by (\[invcurrent\]), only non-reversible cycles do contribute. For particle system, this distinction is embedded into a topological classification of cycles with respect to their corresponding determinant value $\det(C)$.\
[**Connection with the matrix ansatz for ASEP**]{}
For the <span style="font-variant:small-caps;">asep</span> model, a simple algorithm has been discovered [@DeEvHaPa] to obtain the steady-state probabilities of each individual state with the help of a matrix ansatz . In this representation, a given sequence $\eta = 1010\ldots 00$ is represented by a product of matrices $D$ (for $1$) and $E$ (for $0$), and the corresponding probability measure is obtained by taking the trace $$\pi_\eta = \frac{1}{Z}\text{Tr}\bigl(WDEDE\ldots EE\bigr),$$ where $W$ is an additional matrix which takes into account the boundary property. A sufficient condition for this to be the invariant measure is that $D,E,W$ satisfy $$\begin{aligned}
\DD \lambda_{10}DE - \lambda_{01}ED &=& D+E \label{algebra}\\[0.2cm]
\DD DW &=& \frac{1}{\beta}W \nonumber\\[0.2cm]
\DD WE &=& \frac{1}{\alpha}W .\nonumber \end{aligned}$$ If $\lambda_{01}=0$, the process is totally asymmetric (<span style="font-variant:small-caps;">tasep</span>), particles can jump only to the right. Consider the system with only 3 sites, which graph is depicted in figure \[cycleasep\]. Using these rules we find e.g. that $$\begin{aligned}
\pi_{000} &= \frac{1}{Z}\alpha^3\\[0.2cm]
\pi_{100} &= \frac{1}{Z}\Bigl(\bigl(
\frac{1}{\alpha}+\frac{1}{\beta}\bigr)\frac{1}{\lambda^2}+
\frac{1}{\alpha^2\lambda}\Bigr).\end{aligned}$$
Comparison with the spanning tree expansion is done by counting deletions. A spanning tree is obtained from the complete graph by the deleting of a certain number of edges, and each deletion is accounted for by dividing with respect to the corresponding transition rate. The set of spanning trees contributing to $\pi_{000}$ is given in figure \[cycleasep\].b. The brut result (without normalization is): $$\begin{aligned}
\pi_{000} &\propto \Bigl(\frac{1}{\alpha\beta}+
\frac{1}{\alpha\beta}+\frac{1}{\beta\lambda}
\Bigr)\alpha^3\\[0.2cm]
\pi_{100} &\propto \Bigl(\frac{1}{\alpha\beta}+
\frac{1}{\alpha\beta}+\frac{1}{\beta\lambda}
\Bigr)\Bigl(\bigl(
\frac{1}{\alpha}+\frac{1}{\beta}\bigr)\frac{1}{\lambda^2}+\frac{1}{\alpha^2\lambda}\Bigr).\end{aligned}$$ The factor $\Bigl(\frac{1}{\alpha\beta}+\frac{1}{\alpha\beta}+\frac{1}{\beta\lambda}
\Bigr)$ shows up for each state, and disapears after normalization. Nervertheless, it induces in this simple example a factor of 3 in the enumeration of terms, by comparison with the matrix ansatz. An underlying symmetry of the state graph is at the origin of this combinatorial factor. Indeed for the <span style="font-variant:small-caps;">asep</span> system, the steady-state probability current between two sequences $\eta$ and $\eta'$ separated by a single jump between site $i$ and $i+1$ reads, $$\label{asepcurrent}
\lambda_{10}\pi_{\eta}{^{\scriptscriptstyle (N)}}- \lambda_{10}\pi_{\eta'}{^{\scriptscriptstyle (N)}}=
\pi_{\eta_i^*}{^{\scriptscriptstyle (N-1)}}+\pi_{\eta_{i+1}^*}{^{\scriptscriptstyle (N-1)}},$$ as a consequence of (\[algebra\]), with the subsequence $\eta_i^*$ \[resp. $\eta_{i+1}^{*'}$\] of $\eta$ obtained by deleting bit $i$ \[resp. $i+1$\]. We have not been able yet to fill the gap between (\[invcurrent\]) and (\[asepcurrent\]). We believe that the combinatorial arrangement which occur is due to a hierarchical structure of the state-graph, revealed with the help of the tagged particle. The complete analysis of (\[invcurrent\]) is the subject of another work in progress. Beforehand, in the next sections, we simply propose a possible general form for the detailed current equation (\[eq:gcurrent\]), which leads (see section \[hydrodyn\]) to the correct form of the Lotka-Volterra equations describing the fluid limits at steady state.
Cycle currents
--------------
We interpret relation (\[asepcurrent\]) in terms of [*cycle currents*]{}. A transition taking place between two particles of different types, say $AB\rightarrow BA$, can be viewed either as a particle $A$ travelling to the right or, conversely, as a particle $B$ travelling to the left. In this exchange two joint <span style="font-variant:small-caps;">tpc</span> are involved. In the state-graph, each <span style="font-variant:small-caps;">tpc</span> defines a face, which we will identify with a subsequence $\eta^*$, obtained from $\eta$ by removing the tagged particle. Accordingly, we attach a set of variables $\{\phi(\eta^*)\}\in {\mathbb R}$ to each <span style="font-variant:small-caps;">tpc</span> face, while currents between states are variables attached to the edges of the graph. Conservation of probability currents at a given node is automatically fulfilled, provided that if one write (assuming a transition between site $i$ and $i+1$), see figure \[cycleabc\]), $$\label{balance}
\lambda_{ab}\pi_{\eta}-\lambda_{ba}\pi_{\eta'} = \phi_a\bigl(\eta_i^*\bigr) -
\phi_b\bigl(\eta_{i+1}^*\bigr),$$ which is tantamount to changing current variables into cycle variables.
The right-hand side members in (\[algebra\]) and (\[asepcurrent\]) is reminiscent of the second member of (\[balance\]). In fact we have $$\begin{aligned}
\phi_a(\eta_i^*) &= \text{Tr}\bigl(W\eta_i^*\bigr) \\[0.2cm]
\phi_b(\eta_{i+1}^*) &= - \text{Tr}\bigl(W\eta_{i+1}^*\bigr)\end{aligned}$$
With each edge of the state-graph, we associate such an extended detailed balance equation. Then, eliminating all $\phi$’s from this set of equations leads to the invariant measure equation. Consider the example given in figure \[cycleabc\]. The transition rules are $$AB \operatorname*{\rightarrow}^1 BA\qquad AC\operatorname*{\rightarrow}^1 CA \qquad BC\operatorname*{\rightleftarrows}_q^1 CB.$$ The various weights corresponding to each sequence and subsequence associated with cycles are given in the following table, for $q=0$ and $q=1$. Note that one should expect $\pi_{c1}=\frac{1}{3}$ and $\pi_{c2}=\frac{2}{3}$ from the subgraph of figure \[cycleabc\]. The correction results from the different degeneracy w.r.t circular permutation symmetry (4 for $C_1$ and 2 for $C_2$).
----------------- ---------------- ---------------- ---------------- ---------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- ------------------ ------------------- -------------------
$\pi_1$ $\pi_2$ $\pi_3$ $\pi_4$ $\pi_5$ $\pi_6$ $\pi_{a1}$ $\pi_{a2}$ $\pi_{a3}$ $\pi_{b1}$ $\pi_{b2}$ $\pi_{b3}$ $\pi_{c1}$ $\pi_{c2}$ $C_a$ $C_b$ $C_c$
\[0.2cm\] $q=0$ $\frac{1}{10}$ $\frac{1}{10}$ $\frac{3}{10}$ $\frac{1}{10}$ $\frac{2}{5}$ $\frac{2}{5}$ $\frac{1}{4}$ $\frac{1}{4}$ $\frac{1}{2}$ $\frac{1}{6}$ $\frac{1}{2}$ $\frac{1}{3}$ $\frac{1}{2}$ $\frac{1}{2}$ $\frac{2}{5}$ $0$ $-\frac{1}{5}$
\[0.2cm\] $q=1$ $\frac{1}{6}$ $\frac{1}{6}$ $\frac{1}{6}$ $\frac{1}{6}$ $\frac{1}{6}$ $\frac{1}{6}$ $\frac{1}{3}$ $\frac{1}{3}$ $\frac{1}{3}$ $\frac{1}{3}$ $\frac{1}{3}$ $\frac{1}{3}$ $\frac{1}{2}$ $\frac{1}{2}$ $\frac{32}{105}$ $-\frac{41}{210}$ $-\frac{41}{315}$
\[0.2cm\]
----------------- ---------------- ---------------- ---------------- ---------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- ------------------ ------------------- -------------------
In this two cases one has the decomposition of the dual variables $\phi$ of relation \[balance\] according to $$\label{eq:structure}
\phi_x(\eta^*) = C_x\pi_{\eta^*} \qquad\text{with}\ x\in\{a,b,c\}$$ with the value of the structure coefficient also given in the table. For example we have $$\pi_1 - q\pi_2 = C_b\pi_{b1} - C_c\pi_{c1}.$$ This decomposition is however not valid for arbitrary $q$. A certain number of compatibility constraint have to be imposed on the $\phi'$, because the <span style="font-variant:small-caps;">tpc</span> do not constitute a complete bases of cycles in the state graph. When considering the complete system (\[balance\]) of detailed currents, we have at hand $m$ equations, $m$ being the number of edges of the state-graph, and $n+\nu_\text{tpc}$ unknowns, where $n$ is the number of nodes and $\nu_\text{tpc}$ the number of <span style="font-variant:small-caps;">tpc</span>. In matrix form, this reads $$\label{currents_system}
M\Pi = \Phi,$$ where
- $M$ is a $m\times n$ matrix;
- $\Pi$ a column vector of size $n$, with the elements the invariant probability measure;
- $\Phi$ is a column vector of size $m$, where each component $l$ is the algebraic contribution of the (two in general) <span style="font-variant:small-caps;">tpc</span> having the edge corresponding to $l$ in common.
To fix the sign conventions, we agree that orientations of cycles are given by the natural orientation of the system, i.e. each particle travels positively from left to right. An exception is made for the simple exclusion system, since in this case holes travel positively to the left and there is only one type of <span style="font-variant:small-caps;">tpc</span>. $$\begin{aligned}
\lambda_{10}\pi_\eta - \lambda_{01}\pi_{\eta'} &=
\phi(\eta_i^*)+\phi(\eta_{i+1}^*) \qquad\ \ \text{for}\,\textsc{ asep,}\\[0.2cm]
\lambda_{ab}\pi_{\eta} - \lambda_{ba}\pi_{\eta'} &=
\phi_a(\eta_i^*)-\phi_b(\eta_{i+1}^*)\qquad \text{for multi-type systems,} \end{aligned}$$ (with $(i,i+1)$ the sites involved in the transition). From basic graph theory (see [@Berge]), the quantity giving the number of independent cycles in an arbitrary graph $\cal G$ is called the *cyclomatic number* $$\nu({\cal G}) = m - n + p,$$ where $n,m$ and $p$ are respectively the number of nodes, edges and components. In our cases, the system is irreducible, so $p=1$. Since $m$ is the number of equations and $n+\nu_\text{tpc}$ the number of unknown, the system is over-determined by a quantity $$m - (n+\nu_\text{tpc}) = \nu - \nu_\text{tpc} -1.$$ This over-determination is understood as follows. To each line of the matrix $M$ corresponds a transition between two states, so that a given cycle in the state-graph corresponds to some combination of lines of $M$ (namely the successive transitions taking part in the cycle), and the resulting sub-matrix is a square matrix of size the number of states visited by the cycle. The corresponding determinant vanishes for all trivial cycles. Hence the number of independent equations is $m-\nu+\nu_\text{tpc}$, which is equal to the number of unknown minus $1$, the remaining degree of freedom being related to the global normalization condition. However, a certain number of compatibility conditions have to be imposed on the $\phi$’s in order to eliminate safely all dependent equations of our system (\[currents\_system\]). These conditions are somehow related to the basic recurrence scheme which is at the origin of matrix-solutions obtained in the context of <span style="font-variant:small-caps;">asep</span>, but also for multi-type particle systems [@ArHeRi]. Let us see how the specific form (\[eq:structure\]) encountered precedingly do combine with these compatibility conditions.
The form $$\label{flux}
\phi_a{^{\scriptscriptstyle (N)}}(\eta^*) = C_a{^{\scriptscriptstyle (N)}}\pi{^{\scriptscriptstyle (N-1)}}_{\eta^*},$$ of the [*cycle currents*]{} fulfills the compatibility condition imposed by trivial cycles if and only if $$C_a{^{\scriptscriptstyle (N)}}C_b{^{\scriptscriptstyle (N-1)}}= C_b{^{\scriptscriptstyle (N)}}C_a{^{\scriptscriptstyle (N-1)}}\qquad\qquad\forall a,b\in\{1\ldots n\}.$$
Instead of proving this for an arbitrary trivial cycle, we do it for the one depicted in figure \[trivial\], the completion of the general case follows by recurrence, since any trivial cycle can be constructed as a combination of cycle of this type.
To fix some notation, let $\eta^1$, $\eta^2$, $\eta^3$ and $\eta^4$ be the states visited by the cycle, with $i$ the position of $A$ and $j$ the position of $C$ in $\eta_1$, so that $$\begin{aligned}
\eta^1 &= \ldots{\bf AB}\ldots{\bf CD}\ldots
\qquad \eta_i^{1*}=\ldots{\bf B}\ldots{\bf CD}\ldots
\qquad \eta_{i+1}^{1*}=\ldots{\bf A}\ldots{\bf CD}\ldots\\[0.2cm]
\eta^2 &= \ldots{\bf BA}\ldots{\bf CD}\ldots
\qquad \eta_j^{2*}=\ldots{\bf BA}\ldots{\bf D}\ldots
\qquad \eta_{j+1}^{2*}=\ldots{\bf BA}\ldots{\bf C}\ldots\\[0.2cm]
\eta^3 &= \ldots{\bf BA}\ldots{\bf DC}\ldots
\qquad \eta_i^{3*}=\ldots{\bf A}\ldots{\bf DC}\ldots
\qquad \eta_{i+1}^{3*}=\ldots{\bf B}\ldots{\bf DC}\ldots\\[0.2cm]
\eta^4 &= \ldots{\bf AB}\ldots{\bf DC}\ldots
\qquad \eta_j^{4*}=\ldots{\bf AB}\ldots{\bf C}\ldots
\qquad \eta_{j+1}^{4*}=\ldots{\bf AB}\ldots{\bf D}\ldots\end{aligned}$$ The sub-system of \[currents\_system\] corresponding to this cycle reads $$\begin{aligned}
\lambda_{ab}\pi_{\eta^1} - \lambda_{ba}\pi_{\eta^2}
&= \phi_a[\eta_i^{1*}] - \phi_b[\eta_{i+1}^{1*}],\qquad(a)\\[0.2cm]
\lambda_{cd}\pi_{\eta^2} - \lambda_{dc}\pi_{\eta^3}
&= \phi_c[\eta_j^{2*}] - \phi_d[\eta_{j+1}^{2*}],\qquad(b)\\[0.2cm]
\lambda_{ba}\pi_{\eta^3} - \lambda_{ab}\pi_{\eta^4}
&= \phi_b[\eta_i^{3*}] - \phi_a[\eta_{i+1}^{3*}],\qquad(c)\\[0.2cm]
\lambda_{dc}\pi_{\eta^4} - \lambda_{cd}\pi_{\eta^1}
&= \phi_d[\eta_j^{4*}] - \phi_c[\eta_{j+1}^{4*}].\qquad(d) \end{aligned}$$ As already noted, these equations are not independent. Hence the combination $\lambda_{cd}(a)+\lambda_{ba}(b)+\lambda_{dc}(c)+\lambda_{ab}(d)$ eliminates one equation, but with the resulting constraint on the $\phi$’s: $$\begin{aligned}
\label{subid}
&\lambda_{cd}\phi_a[\eta_i^{1*}]-\lambda_{dc}\phi_a[\eta_{i+1}^{3*}]\ +\
\lambda_{dc}\phi_b[\eta_i^{3*}]-\lambda_{cd}\phi_b[\eta_{i+1}^{1*}]\ +\nonumber\\[0.2cm]
&\lambda_{ba}\phi_c[\eta_j^{2*}]-\lambda_{ab}\phi_c[\eta_{j+1}^{4*}]\ +\
\lambda_{ab}\phi_d[\eta_j^{4*}]-\lambda_{ba}\phi_d[\eta_{j+1}^{2*}]\ =\ 0.\end{aligned}$$ $\eta_i^{1*}$ and $\eta_{i+1}^{3*}$ are in correspondence through the transition $CD\to DC$ at site $j$, $j+1$, as well as $\eta_j^{2*}$ and $\eta_{j+1}^{4*}$ with respect to the transition $AB\to BA$ at site $i,i+1$ …. From the hypothesis of the lemma, (\[subid\]) rewrites $$\begin{aligned}
&C_a{^{\scriptscriptstyle (N)}}\left(C_c{^{\scriptscriptstyle (N-1)}}\pi{^{\scriptscriptstyle (N-2)}}_{\eta_{i,j}^{1**}}\ -\
C_d{^{\scriptscriptstyle (N-1)}}\pi{^{\scriptscriptstyle (N-2)}}_{\eta_{i,j+1}^{1**}}\right)\ +\
C_b{^{\scriptscriptstyle (N)}}\left(C_d{^{\scriptscriptstyle (N-1)}}\pi{^{\scriptscriptstyle (N-2)}}_{\eta_{i,j}^{3**}}\ -\
C_c{^{\scriptscriptstyle (N-1)}}\pi{^{\scriptscriptstyle (N-2)}}_{\eta_{i,j+1}^{3**}}\right)\ +\ \\[0.2cm]
&C_c{^{\scriptscriptstyle (N)}}\left(C_b{^{\scriptscriptstyle (N-1)}}\pi{^{\scriptscriptstyle (N-2)}}_{\eta_{i,j}^{2**}}\ -\
C_a{^{\scriptscriptstyle (N-1)}}\pi{^{\scriptscriptstyle (N-2)}}_{\eta_{i+1,j}^{2**}}\right)\ +
C_d{^{\scriptscriptstyle (N)}}\left(C_a{^{\scriptscriptstyle (N-1)}}\pi{^{\scriptscriptstyle (N-2)}}_{\eta_{i,j}^{4**}}\ -\
C_b{^{\scriptscriptstyle (N-1)}}\pi{^{\scriptscriptstyle (N-2)}}_{\eta_{i+1,j}^{4**}}\right)\ =\ 0,\end{aligned}$$ where $\eta_{i,j}^{1**}$ is the sequence obtained from $\eta^1$ by suppressing letters at site $i$ and $j$.The elimination of letters in sequences is a commutative process, therefore this last equality holds because of the following identities: $$\eta_{i,j}^{1**}=\eta_{i+1,j}^{2**},\quad \eta_{i,j}^{3**}=\eta_{i+1,j}^{4**},\quad
\eta_{i,j}^{2**}=\eta_{i,j+1}^{3**}, \quad \eta_{i,j}^{4**}=\eta_{i,j+1}^{1**}.$$
The complete study to establishing the range of validity of the recurrence relation (\[balance\]) altogether with (\[flux\]) is the object of another work in progress. We expect that in general this relation to be valid only asymptotically for large $N$, which could be proved possibly by selecting the dominant terms in the expansion (\[invcurrent\]).
Fluid limits {#hydrodyn}
------------
In this section we examine how the microscopic coefficients $C_k{^{\scriptscriptstyle (N)}}$, whenever (\[flux\]) holds, can be transposed at macroscopic level and how they are related to important coefficients showing up in the Lotka-Volterre equations of the fluid limit. Using the preliminary study [@FaFu3], where a new functional method was introduced to handle the hydrodynamic limit of a simple exclusion process, we consider hereafter the $n$-type case.
### Functional approach
Let $\phi_k, k=1\ldots n$ a set of arbitrary functions in ${\bf C}^2[0,1]$, ${\mathbf{G}}{^{\scriptscriptstyle (N)}}{\stackrel{\mbox{\upshape\tiny def}}{=}}{\mathbb{Z}}/N{\mathbb{Z}}$ the discrete torus (circle). For $i\in{\mathbf{G}}{^{\scriptscriptstyle (N)}}$, $X_i^k(t)$ is a binary random variable and, at time $t$, the presence of a particle of type $k$ at site $i$ is equivalent to $X_i^k(t)=1$. The exclusion constraint reads $$\sum_{k=1}^n X_i^k(t) =1 , \quad\forall i\in{\bf G}.$$ The whole trajectory is represented by $\eta{^{\scriptscriptstyle (N)}}(t){\stackrel{\mbox{\upshape\tiny def}}{=}}\{X_i^k(t),i\in{\bf G}{^{\scriptscriptstyle (N)}},k=1\ldots n\}$ which is a Markov process. $\Omega{^{\scriptscriptstyle (N)}}$ will denote its generator and ${\cal F}_t{^{\scriptscriptstyle (N)}}= \sigma(\eta{^{\scriptscriptstyle (N)}}(s),s\le t)$ is the associated natural filtration.
Define the real-valued positive measure $$Z_t{^{\scriptscriptstyle (N)}}[\phi] {\stackrel{\mbox{\upshape\tiny def}}{=}}\exp\left[\frac{1}{N}\sum_{k=1,i\in{\bf G}{^{\scriptscriptstyle (N)}}}^n
\phi_k\bigl(\frac{i}{N}\bigr)X_i^k\right],$$ where $\phi$ denotes the set $\{\phi_k,k=1\ldots n\}$. In [@FaFu3] the convergence of this measure was analyzed for $n=2$. A functional integral operator was used to characterize limit points of this measure, these were shown to be indeed the unique weak solution of a partial differential equation of Cauchy type.
In what follows, we will be interested in the quantities $$\begin{cases}
f_t{^{\scriptscriptstyle (N)}}(\phi) {\stackrel{\mbox{\upshape\tiny def}}{=}}\Bigl[{\mathbb E}\bigl(Z_t{^{\scriptscriptstyle (N)}}[\phi]\bigr)\Bigr], \\[0.3cm]
g_t{^{\scriptscriptstyle (N)}}(\phi) {\stackrel{\mbox{\upshape\tiny def}}{=}}\log\Bigl[{\mathbb E}\bigl(Z_t{^{\scriptscriptstyle (N)}}[\phi]\bigr)\Bigr],
\end{cases}$$ respectively the moment and cumulant generating function. The idea of using $Z_t{^{\scriptscriptstyle (N)}}[\phi]$ is that the generator, when applied to $Z_t{^{\scriptscriptstyle (N)}}$, can be expressed as a differential operator with respect to the arbitrary functions $\phi$. Indeed, we have $$\Omega{^{\scriptscriptstyle (N)}}\bigl[Z_t{^{\scriptscriptstyle (N)}}\bigr] = L_t{^{\scriptscriptstyle (N)}}Z_t{^{\scriptscriptstyle (N)}},$$ with $$L_t{^{\scriptscriptstyle (N)}}= N^2\sum_{k\ne l,i\in{\cal G}{^{\scriptscriptstyle (N)}}}\tilde\lambda_{kl}
\frac{\partial^2}{\partial\phi_k(\frac{i}{N})\partial\phi_l(\frac{i+1}{N})},$$ after having set $$\begin{cases}
\DD \Delta\psi_{kl}\bigl(\frac{i}{N}\bigr) {\stackrel{\mbox{\upshape\tiny def}}{=}}\phi_k\bigl(\frac{i+1}{N}\bigr) - \phi_k\bigl(\frac{i}{N}\bigr)
+\phi_l\bigl(\frac{i}{N}\bigr) - \phi_l\bigl(\frac{i+1}{N}\bigr) , \\[0.4cm]
\DD \tilde\lambda_{kl}(i,N) {\stackrel{\mbox{\upshape\tiny def}}{=}}2\lambda_{kl}(N)
e^{\frac{\Delta\psi_{kl}(\frac{i}{N})}{2N}}
\sinh\Bigl(\frac{\Delta\psi_{kl}\bigl(\frac{i}{N}\bigr)}{2N}\Bigr).
\end{cases}$$ We introduce now the key quantities for hydrodynamic scalings, by assuming an asymptotic expansion of the form $$\lambda_{kl}(N) = D\bigl(N^2 + \frac{\alpha_{kl}}{2}N\bigr) +
{\cal O}(1),\qquad\forall k,l\ k\ne l,$$ where $\alpha_{kl}=-\alpha_{lk}$ are real constants. Here the system is assumed to be *equidiffusive*, which means there exists a constant $D$ such that, for all pairs $(k,l)$, $$\lim_{N\to\infty}\frac{\lambda_{kl}(N)}{N^2} = D.$$ From now on we will omit the argument of $\lambda_{kl}(N)$ and retain the initial notation $\lambda_{kl}$. The coefficients $\alpha_{kl}$ express the asymmetry between types $k$ and $l$. Now one can write $$\begin{aligned}
\label{geneq}
\frac{\partial f_t{^{\scriptscriptstyle (N)}}}{\partial t} =
N^2\sum_{k\ne l,i\in{\bf G}{^{\scriptscriptstyle (N)}}}^n
\tilde\lambda_{kl}(i,N)
\frac{\partial^2 f_t{^{\scriptscriptstyle (N)}}}{\partial\phi_k(\frac{i}{N})
\partial\phi_l(\frac{i+1}{N})} .\end{aligned}$$ To rearrange the sum in (\[geneq\]), in order to select dominant terms in the expansion with respect to $1/N$, we make use of the exclusion property, which is formally equivalent to $$\sum_{k=1}^n \frac{\partial}{\partial\phi_k(\frac{i}{N})} = \frac{1}{N}.$$ Since we are on the circle $i\in{\mathbf{G}}{^{\scriptscriptstyle (N)}}$, Abel’s summation formula does not produce any boundary term, so that, skipping details, (\[geneq\]) can be rewritten as $$\begin{aligned}
\frac{\partial f_t{^{\scriptscriptstyle (N)}}}{\partial t} =
D N^2 &\sum_{k=1,i\in{\bf G}{^{\scriptscriptstyle (N)}}}^n \Bigl[\phi_k\bigl(\frac{i+1}{N}\bigr)
-\phi_k\bigl(\frac{i}{N}\bigr)\Bigr]
\Bigl[\frac{\partial f_t{^{\scriptscriptstyle (N)}}}{\partial\phi_k(\frac{i}{N})}
-\frac{\partial f_t{^{\scriptscriptstyle (N)}}}{\partial\phi_k(\frac{i+1}{N})}
\nonumber\\[0.2cm]
&+\frac{1}{2}\sum_{l\ne k}\alpha_{kl}
\Bigl(\frac{\partial^2 f_t{^{\scriptscriptstyle (N)}}}{\partial\phi_k(\frac{i}{N})
\partial\phi_l(\frac{i+1}{N})}
+\frac{\partial^2 f_t{^{\scriptscriptstyle (N)}}}{\partial\phi_l(\frac{i+1}{N})
\partial\phi_k(\frac{i}{N})}\Bigr)\Bigr]+{\cal O}(N^{-1}).\end{aligned}$$ It is worth remarking that operators like $\DD\frac{\partial}{\partial\phi_k(\frac{i}{N})}$ and $\phi_k(\frac{i+1}{N})-\phi_k(\frac{i}{N})$ produce a scale factor $1/N$, while $\DD\frac{\partial}{\partial\phi_k(\frac{i}{N})} -
\frac{\partial}{\partial\phi_k(\frac{i+1}{N})}$ and $\DD\frac{\partial}{\partial\phi_k(\frac{i+1}{N})}
\frac{\partial}{\partial\phi_l(\frac{i}{N})}$ scale as $1/N^2$: this explains the selection of dominant terms in the above expansion.
Let $N\to\infty$ and assume the convergence of the sequence $f_0{^{\scriptscriptstyle (N)}}$. Then, from the tightness of the process, together with a zeste of variational and complex variable calculus, as in [@FaFu3], we claim \[the proof is omitted\] $f_t{^{\scriptscriptstyle (N)}}$ also converges, in a *good tempered* functional space, and its limit $f_t$ satisfies the functional integral equation $$\frac{\partial f_t}{\partial t} = D\int_0^1 dx \sum_{k=1}^n\phi_k(x)
\ \frac{\partial}{\partial x}\Bigl[\ \frac{\partial}{\partial x}
\frac{\partial f_t}{\partial\phi_k(x)}
-\sum_{l\ne k}\alpha_{kl}\Bigl(\frac{\partial^2 f_t}{\partial\phi_k(x)
\partial\phi_l(x)}\Bigr)\Bigr].$$ Similarly, the cumulant characteristic function is a solution of $$\begin{aligned}
\label{eqcar}
\frac{\partial g_t}{\partial t} = D\int_0^1 dx \sum_{k=1}^n\phi_k(x)
&\ \frac{\partial}{\partial x}\Bigl[\ \frac{\partial}{\partial x}
\frac{\partial g_t}{\partial\phi_k(x)}
\nonumber\\[0.2cm]
&-\sum_{l\ne k}\alpha_{kl}\Bigl(\frac{\partial g_t}{\partial\phi_k(x)}
\frac{\partial g_t}{\partial\phi_l(x)}
-\frac{\partial^2 g_t}{\partial\phi_k(x)
\partial\phi_l(x)}\Bigr)\Bigr].\end{aligned}$$ Assume at time $0$ the given initial profile $\rho_k(x,0)$ to be twice differentiable with repect to $x$. Then (\[eqcar\]) is given by $$g_t\bigl(\phi\bigr) = \int_0^1 dx \sum_{k=1}^n\rho_k(x,t)\phi_k(x),$$ where $\rho_k(x,t)$ satisfy the hydrodynamic system of coupled Burger’s equations $$\frac{\partial\rho_k}{\partial t} = D\Bigl[\frac{\partial^2\rho_k}{\partial x^2}
+ \frac{\partial}{\partial x}\Bigl(\sum_{l\ne k}\alpha_{lk}\rho_k\rho_l\Bigr)\Bigr]
,\qquad k=1,\ldots, n,$$ with a set of given initial conditions $\rho_k(x,0), k=1,\ldots,n$.
**Remark** It is important to note that, without differentiability conditions for the intial profiles $\rho_k(x,0)$, one can only assert the existence of *weak solutions* (in the sense of Schwartz’s distributions) of Burger’s system.
### Functional equation at steady-state
Consider a particle system of size $N$, with rules \[exchange\], with $n$ types of particles and periodic boundary conditions. Assume the detailed current equations holds, for any pair of particle types $k$ and $l$, $$\label{structure}
\lambda_{kl}{^{\scriptscriptstyle (N)}}\pi_\eta - \lambda_{lk}{^{\scriptscriptstyle (N)}}\pi_{\eta'} = C_k{^{\scriptscriptstyle (N)}}\pi{^{\scriptscriptstyle (N-1)}}_{\eta_i^*}
- C_l{^{\scriptscriptstyle (N)}}\pi{^{\scriptscriptstyle (N-1)}}_{\eta_{i+1}^*},\qquad k,l=1\ldots n,$$ Then the limit functional $\DD f_\infty[\phi] = \lim_{N\to\infty} f_\infty{^{\scriptscriptstyle (N)}}[\phi]$, where $$f_\infty{^{\scriptscriptstyle (N)}}[\phi] = \sum_{\{\eta\}} \pi_\eta\exp\Bigl(\frac{1}{N}\sum_{k=1,i=1}^{n,N}
X_i^k\phi_k(\frac{i}{N})\Bigr),$$ satisfies the equation $$\label{fluide}
\frac{\partial}{\partial x}\frac{\partial f_\infty}{\partial\phi_k(x)}
\ +\ \sum_{l\ne k}\alpha_{kl}
\frac{\partial^2 f_\infty}{\partial\phi_k(x)\partial\phi_l(x)}
\ =\
c_k f_\infty\ - v\frac{\partial f_\infty}{\partial\phi_k(x)},$$ under the *fundamental scaling* $$\lim_{N\to\infty} \log\frac{\lambda_{kl}{^{\scriptscriptstyle (N)}}}{\lambda_{lk}{^{\scriptscriptstyle (N)}}} =
\alpha_{kl}\quad \text{and}\quad\forall l\ne k,\
\lim_{N\to\infty} \frac{N^2 C_k{^{\scriptscriptstyle (N)}}}{\lambda_{kl}{^{\scriptscriptstyle (N)}}}= \lim_{N\to\infty}\frac{C_k{^{\scriptscriptstyle (N)}}}{D} = c_k,$$ with $$v {\stackrel{\mbox{\upshape\tiny def}}{=}}\sum_{l=1}^n c_k.$$
We use the notation of section \[hydrodyn\]. In order to extract additional information at steady state, we refine our preceding variational analysis by defining the functional $$\label{composite}
T{^{\scriptscriptstyle (N)}}\bigl(\{\phi,\partial_x\phi\}\bigr)
= \frac{N^2}{2}\Bigg[\sum_{k\ne l,i=1}^{n,N}
\tilde\lambda_{kl}{^{\scriptscriptstyle (N)}}\frac{\partial^2}{\partial\phi_l(\frac{i}{N})
\partial\phi_k(\frac{i+1}{N})}\\[0.2cm]
+\tilde\lambda_{lk}{^{\scriptscriptstyle (N)}}\frac{\partial^2}{\partial\phi_k(\frac{i+1}{N})
\partial\phi_l(\frac{i}{N})}\Bigg]
f_\infty{^{\scriptscriptstyle (N)}},$$ which corresponds to the second member of equation (\[geneq\]) at steady state, and where it is understood that the sets $\{\phi\}{\stackrel{\mbox{\upshape\tiny def}}{=}}\{\phi(\frac{i}{N}),i=1\ldots N\}$ and $\{\partial\phi\}{\stackrel{\mbox{\upshape\tiny def}}{=}}\{\frac{\partial\phi}{\partial x}(\frac{i}{N}),i=1\ldots N\}$ are taken as independant parameters. This functional can be writen in two different manners. Recalling the definitions $$\bigtriangleup\psi_{kl}(i) {\stackrel{\mbox{\upshape\tiny def}}{=}}\phi_k(i+1)-\phi_k(i)-\phi_l(i+1)+\phi_l(i)
{\stackrel{\mbox{\upshape\tiny def}}{=}}\bigtriangleup\psi_k(i) - \bigtriangleup\psi_l(i),$$ (\[composite\]) may be rewritten in the form $$\begin{aligned}
T{^{\scriptscriptstyle (N)}}\bigl(\{\phi,\partial_x\phi\}\bigr) =
N D &\sum_{k=1,i=1}^{n,N}\partial_x\phi_k(\frac{i}{N})
\Bigl[\frac{\partial f_\infty^{(N)}}{\partial\phi_k(\frac{i}{N})}
-\frac{\partial f_\infty^{(N)}}{\partial\phi_k(\frac{i+1}{N})} \nonumber
\\[0.2cm]
&+\sum_{l\ne k}\frac{\alpha_{kl}}{2}
\Bigl(\frac{\partial^2 f_\infty^{(N)}}{\partial\phi_k(\frac{i}{N})
\partial\phi_l(\frac{i+1}{N})}
+\frac{\partial^2 f_\infty^{(N)}}{\partial\phi_k(\frac{i+1}{N})
\partial\phi_l(\frac{i}{N})}\Bigr)\Bigr] +{\cal O}(\frac{1}{N}), \label{hydro1}\end{aligned}$$ On the other hand, combining the sums in (\[composite\]) yields $$\begin{aligned}
T{^{\scriptscriptstyle (N)}}\bigl(\{\phi,\partial_x\phi\}\bigr) = N^2\sum_{k,l=1,i=1}^{n,N}\sum_{\{\eta\}}
&e^{\frac{1}{N}\vec\phi.\vec\eta+\frac{1}{2N}\bigtriangleup\psi_{kl}(i)}
\sinh\frac{\bigtriangleup\psi_{kl}(i)}{2N} \nonumber\\[0.2cm]
&\times X_i^kX_{i+1}^l\Bigl[\lambda_{kl}{^{\scriptscriptstyle (N)}}\pi{^{\scriptscriptstyle (N)}}_\eta -\lambda_{lk}{^{\scriptscriptstyle (N)}}\pi{^{\scriptscriptstyle (N)}}_{T_i\eta} \Bigr],
\label{hydro2}\end{aligned}$$ where $\eta$ is a given configuration, $T_i\eta$ being the one obtained from $\eta$ by exchanging $i$ and $i+1$, and the shorthand notation $$\vec\phi.\vec\eta = \sum_{k=1,i=1}^{n,N} X_i^k\phi_k\bigl(\frac{i}{N}\bigr).$$ From the assumptions in the statement of the proposition, we can rewrite (\[hydro2\]) as $$\begin{aligned}
T{^{\scriptscriptstyle (N)}}\bigl(\{\phi,\partial_x\phi\}\bigr) = N^2\sum_{k,l=1,i=1}^{n,N}\sum_{\{\eta\}}
&e^{\frac{1}{N}\vec\phi.\vec\eta+\frac{1}{2N}\bigtriangleup\psi_{kl}(i)}
\sinh\frac{\bigtriangleup\psi_{kl}(i)}{2N} \nonumber\\[0.2cm]
&\times X_i^kX_{i+1}^l\Bigl[
C_k{^{\scriptscriptstyle (N)}}\pi{^{\scriptscriptstyle (N-1)}}_{\eta_i^*}
-C_l{^{\scriptscriptstyle (N)}}\pi{^{\scriptscriptstyle (N-1)}}_{\eta_{i+1}^*}
\Bigr], \label{hydro3}\end{aligned}$$ where $\eta_i^*$ is the sequence obtained from $\eta$ by removing the site $i$. We also have $$\sum_{\{\eta\}} X_i^k \pi_{\eta_i^*}
e^{\frac{1}{N}\vec\phi.\vec\eta} = f_\infty{^{\scriptscriptstyle (N-1)}}[\phi_i^*]
e^{\frac{1}{N}\phi_k(\frac{i}{N})},$$ where $f_\infty{^{\scriptscriptstyle (N-1)}}[\phi_i^*]$ means that $f_\infty{^{\scriptscriptstyle (N-1)}}$ is considered as a function of the $n(N-1)$ variables $\{\phi_k(\frac{j}{N}), k=1,\ldots, n; j=1,\ldots, N,j\ne i\}$. Using all these ingredients, expanding (\[hydro3\]) in powers of $\frac{1}{N}$ and keeping the dominant terms, we get $$\label{hydro4}
T{^{\scriptscriptstyle (N)}}\bigl(\{\phi,\partial_x\phi\}\bigr) = \frac{N^2}{2}\sum_{k,l=1,i=1}^{n,N}
\bigtriangleup\psi_{kl}(i)
\Bigl[
C_k{^{\scriptscriptstyle (N)}}\frac{\partial f_\infty{^{\scriptscriptstyle (N-1)}}}{\partial\phi_l(\frac{i+1}{N})}
-C_l{^{\scriptscriptstyle (N)}}\frac{\partial f_\infty{^{\scriptscriptstyle (N-1)}}}{\partial\phi_k(\frac{i}{N})}
\Bigr]+{\mathcal{O}}\Bigl(\frac{1}{N}\Bigr) .$$ Now, rearranging the summation, using the exclusion property $$\sum_{l=1}^n\frac{\partial}{\partial\phi_l(\frac{i}{N})}=\frac{1}{N},$$ comparing (\[hydro1\]) and (\[hydro4\]), we finally obtain $$\begin{aligned}
N^2\sum_{k=1,i=1}^{n,N}\partial_x\phi_k\bigl(\frac{i}{N}\bigr)
&\biggl[\frac{\partial f_\infty^{(N)}}{\partial\phi_k(\frac{i}{N})}
-\frac{\partial f_\infty^{(N)}}{\partial\phi_k(\frac{i+1}{N})}
+\frac{\alpha^{kl}}{2}
\Bigl(\frac{\partial^2 f_\infty^{(N)}}{\partial\phi_k(\frac{i}{N})
\partial\phi_l(\frac{i+1}{N})}
+\frac{\partial^2 f_\infty^{(N)}}{\partial\phi_k(\frac{i+1}{N})
\partial\phi_l(\frac{i}{N})}\Bigr)\biggr] \\[0.2cm]
&= N^2\sum_{k,i=1}^{n,N}\partial_x\phi_k\bigl(\frac{i}{N}\bigr)
\Bigl[\frac{C_k{^{\scriptscriptstyle (N)}}}{D} f_\infty{^{\scriptscriptstyle (N-1)}}-\sum_{l=1}^n \frac{C_l{^{\scriptscriptstyle (N)}}}{D}
\frac{\partial f_\infty{^{\scriptscriptstyle (N-1)}}}{\partial\phi_k(\frac{i}{N})}\Bigr] + {\mathcal{O}}\Bigl(\frac{1}{N}\Bigr).\end{aligned}$$ As the last equality holds for any $\partial_x\phi_k$, letting $N\to\infty$ implies easily (\[fluide\]), which was to be proved.
### Lotka-Volterra systems and out-of-equilibrium stationary states
Here we will make the link between the structure coefficients of the current equations (\[structure\]) and the fluid limit description of stationary states. A solution is sought of the form $$f_\infty(\phi) = \exp\Bigl(\int_0^1dx\sum_{k=1}^N\rho_k^\infty(x)\phi_k(x)\Bigr),$$ which, instantiated into (\[fluide\]), yields gives the following equations for the $\rho_k^\infty$’s . $$\frac{\partial\rho_k^\infty}{\partial x}-
\rho_k^\infty\sum_{l\ne k}\alpha^{kl}\rho_l^\infty
= c_k - v\rho_k^\infty,\qquad k=1\ldots n .$$ The interpretation of this system is now quite clear : it is exactly a particular stationary solution of the system formed by the coupled Burger’s equations $$\frac{\partial\rho_k}{\partial t} = \frac{\partial^2\rho_k}{\partial x^2}
-\frac{\partial}{\partial x}\Bigl(\rho_k\sum_{l\ne k}\alpha^{kl}\rho_l\Bigr),
\qquad k=1\ldots n,$$ where the functions $\rho_k$ are sought in the class $$\rho_k(x,t) {\stackrel{\mbox{\upshape\tiny def}}{=}}\rho_k^\infty(x-vt),$$ the variable $(x-vt)$ being taken \[modulo $1$\]. Hence, there is a frame rotating at velocity $v$, in which $\rho_k^\infty$ is periodic. Moreover, in this frame, the stationary currents do not vanish and have constant values $$J_k(x) =
\frac{\partial\rho_k^\infty}{\partial x}+
\rho_k^\infty\bigl(v-\sum_{l\ne k}\alpha^{kl}\rho_l^\infty\bigr) = c_k.$$ Therefore, while the macroscopic constants $\{c_k,k=1,\ldots ,n\}$ are in principle determined from the periodic boundary conditions constraints and from the fixed average values of each particle species, they can also be directly derived from the microscopic model.
Transient regime and fluctuations {#fluctuations}
=================================
The goal of this section is twofold : first, establish relationships between currents and particle densities at the deterministic level by means of the law of large numbers; secondly, compute the stochastic corrections to these relationships for large but finite systems by using central limit theorems and large deviations.
Time-scale for local equilibrium
--------------------------------
In keeping with our approach, we discuss the question of local equilibrium [@Sp] by means of the following functional $$Y_t{^{\scriptscriptstyle (N)}}{\stackrel{\mbox{\upshape\tiny def}}{=}}\exp\biggl[\frac{1}{N}\sum_{k,l=1,i=1}^{n,N}
\phi_{kl}(\frac{i}{N})X_i^kX_{i+1}^l \biggr].$$ Without entering into cumbersome technical details, let us just notice that the explicit computation of $L_t{^{\scriptscriptstyle (N)}}Y_t{^{\scriptscriptstyle (N)}}$ shows that $L_t{^{\scriptscriptstyle (N)}}Y_t{^{\scriptscriptstyle (N)}}$ scales like ${\cal O}(N)$ instead of ${\cal O}(1)$ as $L_t{^{\scriptscriptstyle (N)}}Z_t{^{\scriptscriptstyle (N)}}$. This fact can be interpreted as follows. The empirical measure $$\mu_t{^{\scriptscriptstyle (N)}}{\stackrel{\mbox{\upshape\tiny def}}{=}}\frac{1}{N}\sum_{k,l=1,i=1}^{n,N}
\phi_{kl}(\frac{i}{N})X_i^kX_{i+1}^l$$ is a convolution of the distribution of interfaces between particle domains with a set of arbitrary functions. To any given particle density distribution, drawn from the set of local hydrodynamic densities, there corresponds an arrangement of these interfaces which somehow characterizes the local correlations between particles. At steady-state, at least in the reversible case, it is easy to show that these correlations vanish. Moreover this scaling tells us that correlations vanish at a time-scale *faster* than the diffusion scale, by a factor of $N$. Therefore, even in transient regime, we expect correlations to be negligible for the family of diffusive processes under study. A more formal proof of this fact is postponed to the completion of the functional approach initiated in [@FaFu3].
Hydrodynamical currents {#hydrocurrents}
-----------------------
In our preceding studies, we devised a scheme to obtain a fluid limit at steady state, first for the reversible square-lattice model in [@FaFu], and also for the non-reversible <span style="font-variant:small-caps;">abc</span> model [@FaFu2]. Here we generalize this procedure to transient $n$-type particle systems, resting upon the hydrodynamic hypothesis, which will be precisely stated. The principle of the method is to reverse the relationship between particle and current variables in a suitable manner, in order to apply a law of large numbers.
### Diffusion models {#scheme2}
The system corresponds to rules (\[exchange\]). For any particle-type $k$, the rescaled discrete current reads $$\label{dcurrent}
J_k{^{\scriptscriptstyle (N)}}\bigl(\frac{i}{N}\bigr) {\stackrel{\mbox{\upshape\tiny def}}{=}}\lambda_k^+(i+1)X_i^k
- \lambda_k^-(i)X_{i+1}^k,\qquad i=1,\ldots, N,$$ with $$\begin{cases}
\DD \lambda_k^+(i)
{\stackrel{\mbox{\upshape\tiny def}}{=}}\sum_{l\ne k} \frac{\lambda_{kl}}{N}X_i^l +\Gamma_kX_i^k,\\[0.2cm]
\DD \lambda_k^-(i)
{\stackrel{\mbox{\upshape\tiny def}}{=}}\sum_{l\ne k} \frac{\lambda_{lk}}{N}X_i^l +\Gamma_kX_i^k,
\end{cases}$$ where arbitrary constants $\Gamma_k$ have been introduced (they not modify the value of $J_k$) to ensure that the $\lambda_k^\pm$’s never vanish. To be consistent with other scalings, $\Gamma_k$ is assumed to scale like $N
$. Our hypothesis is that $J_k$ has a limiting distribution, $J_k(x)$, such that, for any integrable complex-valued function $\alpha$, $$\label{hhydro}
\lim_{N\to\infty} \frac{1}{N}\sum_{i=1}^N\alpha\bigl(\frac{i}{N}\bigr)
J_k{^{\scriptscriptstyle (N)}}\bigl(\frac{i}{N}\bigr)
=\int_0^1\alpha(x)J_k(x)dx.$$ In addition, the system will be said [*equidiffusive*]{}, if there exists a single diffusion constant $D$, such that, for all pair of species $(k,l)$, $$\lim_{N\to\infty}\frac{\lambda_{kl}(N)}{N^2} = D \qquad \mathrm{[equidiffusion]} .$$ To simplify the notation, consider equation for $k=1$, writing $J_a{\stackrel{\mbox{\upshape\tiny def}}{=}}J_1$ and replacing $X_i^1$ by $A_i$. Then solving (\[dcurrent\]) as a linear system yields $$A_{i+1} = \frac{\lambda_a^+(i+1)A_i - J_a{^{\scriptscriptstyle (N)}}\bigl(\frac{i}{N}\bigr)}
{\lambda_a^-(i)}.$$ This relationship between $A_i$ and $A_{i+1}$ can be iterated, by means of a $2\times2$ matrix products. Indeed, introducing the pair of numbers $(u_i,v_i)$ such that $A_i = \frac{u_i}{v_i}$, the recursion becomes $$\left[\begin{matrix}
u_{i+1}\\[0.4cm]
v_{i+1}
\end{matrix}\right] =
\left[\begin{matrix}
\sqrt{\frac{\lambda_a^+(i+1)}{\lambda_a^-(i)}}
& - \frac{J_a{^{\scriptscriptstyle (N)}}\bigl(\frac{i}{N}\bigr)}{\sqrt{\lambda_a^+(i+1)\lambda_a^-(i)}}
\\[0.4cm]
0 & \sqrt{\frac{\lambda_a^-(i)}{\lambda_a^+(i+1)}}
\end{matrix}\right]
\left[\begin{matrix}
u_i\\[0.4cm]
v_i
\end{matrix}\right]
{\stackrel{\mbox{\upshape\tiny def}}{=}}M_i \left[\begin{matrix}
u_i\\[0.4cm]
v_i
\end{matrix}\right] ,$$ where for convenience we divided everything by the common factor $\sqrt{\lambda_a^-(i)\lambda_a^+(i+1)}$. Let us define the matrices ($p$ being a positive integer) $$\begin{aligned}
G^0\bigl(\frac{i+p}{N},\frac{i}{N}\bigr) &{\stackrel{\mbox{\upshape\tiny def}}{=}}& \prod_{j=i}^{i+p}
\left[\begin{matrix}
\sqrt{\frac{\lambda_a^+(j+1)}{\lambda_a^-(j)}}
& 0\\[0.4cm]
0
& \sqrt{\frac{\lambda_a^-(j)}{\lambda_a^+(j+1)}}
\end{matrix}\right] , \\[0.2cm]
G\bigl(\frac{i+p}{N},\frac{i}{N}\bigr) &{\stackrel{\mbox{\upshape\tiny def}}{=}}& \prod_{j=i}^{i+p}M_j, \\[0.2cm]
\Sigma\bigl(\frac{i}{N}\bigr) &{\stackrel{\mbox{\upshape\tiny def}}{=}}&
\left[\begin{matrix}
0
& - \frac{J_a{^{\scriptscriptstyle (N)}}\bigl(\frac{i}{N}\bigr)}{\sqrt{\lambda_a^+(i+1)\lambda_a^-(i)}}
\\[0.4cm]
0 & 0
\end{matrix}\right],\end{aligned}$$ (explicit references to the species $(a)$ and the size $N$ is omitted here, to lighten the notations). Because of the upper triangular structure of $\Sigma$, we may simply express $G$ as $$\begin{split}
G\bigl(\frac{i+p}{N},\frac{i}{N}\bigr) = G^0\bigl(\frac{i+p}{N},\frac{i}{N}\bigr)
+\sum_{j=0}^p G^0\bigl(\frac{i+p}{N},\frac{i+j+1}{N}\bigr)
\Sigma\bigl(\frac{i+j}{N}\bigr)
G^0\bigl(\frac{i+j-1}{N},\frac{i}{N}\bigr).
\end{split}$$ To handle this equation in the continuous limit, we need an additional transformation. Define $$L_i =
\left[\begin{matrix}
\sqrt{\frac{\Gamma_a}{\lambda_a^+(i)}} & 0\\[0.4cm]
0 & \sqrt{\frac{\lambda_a^+(i)}{\Gamma_a}}
\end{matrix}\right] , \qquad
R_i = \left[\begin{matrix}
\sqrt{\frac{\lambda_a^-(i)}{\Gamma_a}} &0 \\[0.4cm]
0 & \sqrt{\frac{\lambda_a^-(i)}{\Gamma_a}}
\end{matrix}\right],$$ together with $$\label{transform}
\begin{cases}
\DD\tilde G\bigl(\frac{i+p}{N},\frac{i}{N}\bigr) = L_{i+p+1}
G\bigl(\frac{i+p}{N},\frac{i}{N}\bigr) R_i\\[0.2cm]
\DD\tilde G^0\bigl(\frac{i+p}{N},\frac{i}{N}\bigr) = L_{i+p+1}
G^0\bigl(\frac{i+p}{N},\frac{i}{N}\bigr) R_i
\end{cases}.$$ Then $\tilde G$, $\tilde G^0$ and $\tilde\Sigma$ verify the same relation, $$\label{Dyson}
\begin{split}
\tilde G\bigl(\frac{i+p}{N},\frac{i}{N}\bigr) =
\tilde G^0\bigl(\frac{i+p}{N},\frac{i}{N}\bigr)
+ \sum_{j=0}^p \tilde G^0\bigl(\frac{i+p}{N},\frac{i+j+1}{N}\bigr)
\tilde\Sigma\bigl(\frac{i+j}{N}\bigr)
\tilde G^0\bigl(\frac{i+j}{N},\frac{i+1}{N}\bigr),
\end{split}$$ but $$\tilde\Sigma\bigl(\frac{i}{N}\bigr) =
\left[\begin{matrix}
0 & -\frac{\Gamma_a J_a{^{\scriptscriptstyle (N)}}(\frac{i}{N})}{\lambda_a^+(i+1)\lambda_a^-(i)}\\[0.4cm]
0 & 0
\end{matrix}\right].$$ Noting that $A_{i+p+1}\Gamma_a/\lambda_a^+(i+p+1) = A_{i+p+1}$ and $A_i\Gamma_a/\lambda_a^-(i) = A_i$, the iteration between $i$ and $i+p$ gives $$\label{iterp}
A_{i+p+1} = \frac{\tilde G_{11} \bigl(\frac{i+p}{N},\frac{i}{N}\bigr)
A_i + \tilde G_{12}\bigl(\frac{i+p}{N},\frac{i}{N}\bigr)}
{\tilde G_{22}\bigl(\frac{i+p}{N},\frac{i}{N}\bigr)}.$$ We can now take advantage of the law of large numbers in equation (\[Dyson\]). First of all, for $N$ large, and fixing $x=i/N$ and $y=p/N$, letting $\sigma = \left[\begin{matrix}1 & 0\\0&-1\end{matrix}\right]$, we have, $$\tilde G^0\bigl(\frac{i+p}{N},\frac{i}{N}\bigr)
= \exp\Bigl(\frac{\sigma}{2}\sum_{j=i+1,k=2}^{i+p,n}
\log\frac{\lambda_{ak}}{\lambda_{ka}}X_j^k\Bigr)
= \exp\Bigl(\frac{\sigma}{2}\int_x^{x+y}du\sum_{k=2}^n\alpha^{ak}\rho_k(u)
+o(1) \Bigr),$$ from the hydrodynamic hypothesis. To proceed further, we have to distinguish between two situations.
[**\[The equidiffusion case\]**]{}
Recalling that $\Gamma_a$ is a free parameter which scales like $N$, it is convenient in the [*equidiffusion*]{} case to impose the limit $$\lim_{N\to\infty} \frac{\Gamma_a(N)}{N} = D.$$ Then, expanding $\tilde\Sigma(i/N)$ with respect to $1/N$ yields $$\tilde\Sigma\bigl(\frac{i}{N}\bigr) =
\left[\begin{matrix}
0 & -\frac{J_a{^{\scriptscriptstyle (N)}}(\frac{i}{N})}{ND}\\[0.4cm]
0 & 0
\end{matrix}\right]
+ {\cal O}\bigl(N^{-2}\bigr),$$ and the limit $${\cal G}(x+y,x) {\stackrel{\mbox{\upshape\tiny def}}{=}}\lim_{N\to\infty} \tilde G\bigl(\frac{i+p}{N},\frac{i}{N}\bigr)$$ is provided by equation (\[Dyson\]). Hence $$\label{Cdyson}
{\cal G}(x+y,x) = {\cal G}^0(x+y,x) +\int_x^{x+y}du\
{\cal G}^0(x+y,x+u)\ \Xi(x+u)\
{\cal G}^0(x+u,x),$$ with $$\label{freemat}
{\cal G}^0(y,x) =
\exp\Bigl(\frac{\sigma}{2}\int_x^ydu\sum_{k=2}^n\alpha_{ak}\rho_k(u)
\Bigr),\qquad\text{and}\qquad \Xi(x) = \left[\begin{matrix}
0 & -\frac{J_a{^{\scriptscriptstyle (N)}}(x)}{D}\\[0.4cm]
0 & 0
\end{matrix}\right],$$ still by virtue of the hydrodynamic hypothesis (\[hhydro\]). Now it is possible to close the equations between densities and currents. Using again the hydrodynamic hypothesis with the fact that $\cal G$ is a smooth deterministic operator, (\[iterp\]) leads to, $$\rho_a(x+y) = \frac{{\cal G}_{11} (x+y,x)\rho_a(x)+ {\cal G}_{12}(x+y,x)}
{{\cal G}_{22}(x+y,x)}.$$ By differentiating this last relation w.r.t. $y$, altogether with (\[Cdyson\]) and (\[freemat\]), we obtain the final deterministic expression for the current $$\label{detercurrent}
J_a(x) = D\bigl(-\frac{\partial\rho_a}{\partial x}+
\sum_{k=2}^n\alpha_{ak}\rho_k\rho_a\Bigr),$$ which, combined with the continuity equation $$\frac{\partial \rho_a}{\partial t} +\frac{\partial J_a}{\partial x} = 0,$$ leads again to a Burger’s hydrodynamic equation.
Here, the limit (\[Dyson\]) is a bit more tricky. In fact, the expansion of $\tilde\Sigma$ involves correlations between currents and densities which already appear in the leading terms, and we expect an effective diffusion constant of the form $$D_a(\rho) = D\exp\Bigl(\sum_{k=2}^n \beta^{ak}\rho_k\Bigr) ,$$ with $$\begin{cases}
\DD D {\stackrel{\mbox{\upshape\tiny def}}{=}}\lim_{N\to\infty}\frac{1}{N^2}\exp\Bigl(\frac{1}{n-1}
\sum_{k=2}^n\log\lambda_{ak}(N)\Bigr) ,\\[0.2cm]
\DD\beta^{ak} {\stackrel{\mbox{\upshape\tiny def}}{=}}\lim_{N\to\infty}
\log\Bigl(\frac{\lambda_{ak}}{N^2 D}\Bigr).
\end{cases}$$ We pursue no further the study of this case, which presumably could be handled with block-estimates techniques (see [@Sp]).
### Diffusion with reaction
Here we treat the square-lattice model, a special case of (\[evnmod\]),where reactions take place, in addition to diffusion. The procedure follows the lines of the preceding subsection. Using the mapping (\[cpmapping\]), the model is formulated in terms of two coupled exclusion processes, and the current equations corresponding to both species have the form $$\begin{aligned}
J_a{^{\scriptscriptstyle (N)}}\bigl(\frac{i}{N}) &= \lambda_a^+(i) \tau_i^a\bar\tau_{i+1}^a -
\lambda_a^-(i) \bar\tau_i^a\tau_{i+1}^a , \\[0.2cm]
J_b{^{\scriptscriptstyle (N)}}\bigl(\frac{i}{N}) &= \lambda_b^+(i) \tau_i^b\bar\tau_{i+1}^b -
\lambda_b^-(i) \bar\tau_i^b\tau_{i+1}^b ,\end{aligned}$$ with the rates given by (\[taux\]), and we restrict the present analysis to the symmetric case (see relations (\[symmetric\])). Reversing for example the equation for $J_a$ leads to the homographic relationship $$\tau_{i+1}^a = \frac{\lambda_a^+(i)\tau_i^a - J_a{^{\scriptscriptstyle (N)}}\bigl(\frac{i}{N}\bigr)}
{(\lambda_a^+(i)-\lambda_a^-(i))\tau_i^a+\lambda_a^-(i)},$$ which again can be iterated by means of a $2\times 2$ matrix product, after defining $u_i^a$ and $v_i^a$ s.t. $\tau_i^a=u_i^a/v_i^a,$ $\forall i\in\{1\ldots N\}$. Define $$\lambda(N) {\stackrel{\mbox{\upshape\tiny def}}{=}}\frac{\lambda^+(N)+\lambda^-(N)}{2},\qquad\mu(N)
{\stackrel{\mbox{\upshape\tiny def}}{=}}\frac{\lambda^+(N)-\lambda^-(N)}{2},$$ and $$\gamma(N) {\stackrel{\mbox{\upshape\tiny def}}{=}}\frac{\gamma^+(N)+\gamma^-(N)}{2}.$$ Then the proper scalings for large $N$ are given by $$\lim_{N\to\infty}\frac{\lambda(N)}{N^2} = D , \qquad
\lim_{N\to\infty}\frac{\gamma(N)}{N^2} = \Gamma , \qquad
\lim_{N\to\infty}\frac{\mu(N)}{N} = \eta.$$ Letting now, $$\Sigma\bigl(\frac{i}{N}\bigr) =
\left[\begin{matrix}
0 & -\frac{J_a{^{\scriptscriptstyle (N)}}(\frac{i}{N})}
{\sqrt{\lambda_a^+(i)\lambda_a^-(i)}}\\[0.4cm]
\sqrt{\frac{\lambda_a^+(i)}{\lambda_a^-(i)}}
-\sqrt{\frac{\lambda_a^-(i)}{\lambda_a^i(i)}}& 0
\end{matrix}\right].$$ $G$ cannot be given explicitly, it is instead solution of the following combinatorial self-consistent equation $$\label{Dyson2}
G\bigl(\frac{i+p}{N},\frac{i}{N}\bigr) =
G^0\bigl(\frac{i+p}{N},\frac{i}{N}\bigr)
+\sum_{j=0}^p G^0\bigl(\frac{i+p}{N},\frac{i+j}{N}\bigr)
\Sigma\bigl(\frac{i+j}{N}\bigr)\ G\bigl(\frac{i+j}{N},\frac{i+1}{N}\bigr).$$ The iteration now reads, $$\left[\begin{matrix}
u_{i+p+1}\\[0.4cm]
v_{i+p+1}
\end{matrix}\right] = G\bigl(\frac{i+p}{N},\frac{i}{N}\bigr)
\left[\begin{matrix}
u_{i}\\[0.4cm]
v_{i}
\end{matrix}\right]$$ For the same reason as before, the limit $\cal G$ of $G$ when $N\to\infty$ does satisfy $$\label{dyson3}
{\cal G}(x+y,x) = {\cal G}^0(x+y,x) +\int_x^{x+y}du\ {\cal G}^0(x+y,x+u)\Sigma(x+u)
{\cal G}(x+u,x),$$ with $${\cal G}^0(y,x) =
\exp\Bigl(\eta\sigma\int_x^y (2\rho_b(u)-1)du
\Bigr),$$ by just applying the law of large numbers in the formal expansion of $G$ with respect to $\Sigma$. We leave aside the question concerning existence and analytic properties of a solution of (\[dyson3\]). We must again discriminate between two situations.
$$\Sigma(x) =
\left[\begin{matrix}
\eta(2\rho_b-1) & -\frac{J_a(x)}{D}\\[0.4cm]
2\eta(2\rho_b-1) & \eta(1-2\rho_b)
\end{matrix}\right],$$ which leads to the following differential system, $$\begin{aligned}
\frac{\partial u^a}{\partial x} &= \eta(2\rho_b-1)u^a
-\frac{1}{D}J_a(x)v^a , \\[0.2cm]
\frac{\partial v^a}{\partial x} &= 2\eta(2\rho_b-1)u^a + \eta(1-2\rho_b)v^a , \end{aligned}$$ after making use of the law of large numbers and the hydrodynamic hypothesis. Combining these last two equations to express $\rho_a'=(u_a'v_a-v_a'u_a)/v_a^2$, leads to the relation $$J_a(x) = -D\Bigl(\frac{\partial\rho_a}{\partial x}+
2\eta\rho_a(1-\rho_a)(1-2\rho_b)\Bigr).$$
[**\[Case $\gamma\ne\lambda$\]**]{}
Like in the [*hetero-diffusion*]{} case of the last section, the effective diffusion constant $D_a(\rho)$ involves correlations between $\tau_i^b$ and $\tau_{i+1}^b$ and $J_a(i/N)$ in the leading order term, and we expect a behavior of the form [@FaFu] $$D_a(\rho_b) = D\exp\Bigl[2\rho_b(1-\rho_b)\log\frac{\gamma}{\lambda}\Bigr],$$ as a result of a multiplicative process. This could be obtained through renormalization techniques applied directly to equation (\[Dyson2\]).
To conclude this section, we see that, for $\gamma=\lambda$, the differential system expressing, at steady state, the deterministic limit of the square lattice model with periodic boundary conditions finally reads, setting $\nu_{a,b} = 2\rho_{a,b}-1$, $$\label{nldiff}
\begin{cases}
\DD\frac{\partial\nu_a}{\partial x} = \eta(1-\nu_a^2)\nu_b\ +\
v\nu_a+\varphi^a, \\[0.2cm] \DD\frac{\partial\nu_b}{\partial x} = -
\eta(1-\nu_b^2)\nu_a\ +\ v\nu_b+\varphi^b,
\end{cases}$$ where $v$ is a possibly finite drift velocity and $\varphi^a=\varphi(\bar\nu_a,\bar\nu_b)$ and $\varphi^b(\bar\nu_a,\bar\nu_b)$ are two constant currents in the translating frame. These currents have to be determined in a self-consistent manner, after fixing the average densities $\bar\nu_a$ and $\bar\nu_b$ and the periodic boundary conditions. For $v=0$, the system (\[nldiff\]) is Hamiltonian with $$H = \frac{\eta}{2}\bigl[\nu_a^2\nu_b^2 - \nu_a^2-\nu_b^2\bigr]+
\varphi_b\nu_a-\varphi_a\nu_b.$$ Indeed, it is easy to observe that (\[nldiff\]) can be rewritten as $$\frac{\partial\nu_a}{\partial x} = -\frac{\partial H}{\partial\nu_b}, \qquad
\frac{\partial\nu_b}{\partial x} = \frac{\partial H}{\partial\nu_a}.$$ The degenerate fixed point $\nu_{a,b}(x)=\bar\nu_{a,b}$ is always a trivial solution and corresponds to the relations $$\varphi_a = \eta(\bar\nu_a^2 - 1)\bar\nu_b, \qquad \varphi_b =
\eta(1-\bar\nu_b^2)\bar\nu_a.$$
Microscopic currents
--------------------
### Particle currents
An important feature of our particle systems is that the number of particles is locally conserved. This property is reflected as $N\to\infty$ by a continuity equation, which relates local variations of particle density to inhomogeneous currents. In a discretized framework, conservation of particles is expressed according to the following
\[condprodform\] Let $\{J_i^k(t,\epsilon)\}\ i=1,\ldots, N$ be stochastic variables corresponding to the fluxes of particles of type $k\in\{1,\ldots, n\}$ between site $i$ and $i+1$, such that $$J_i^k(t,\epsilon) {\stackrel{\mbox{\upshape\tiny def}}{=}}\frac{1}{\epsilon}\sum_{l\ne k}
\Bigl( X_i^k(t) X_{i+1}^l(t)X_i^l(t+\epsilon) X_{i+1}^k(t+\epsilon)-
X_i^l(t) X_{i+1}^k(t)X_i^k(t+\epsilon) X_{i+1}^l(t+\epsilon)\Bigr)$$ with $\epsilon>0$. By definition $ J_i^{k}(t,\epsilon)$ are ternary variables in $ \{-\frac{1}{\epsilon},0,+\frac{1}{\epsilon}\}$. The following identity, equivalent to particle conservation, $$\label{Ward}
\lim_{\epsilon\to 0} \frac{X_i^k(t+\epsilon)-
X_i^k(t)}{\epsilon} +J_{i+1}^k(t,\epsilon) - J_i^k(t,\epsilon) = 0 \qquad a.s.,$$ holds for all $i\in\{1,\ldots, N\}$, $ \forall t\in{\mathbb R}^+$. In addition, letting $\eta{^{\scriptscriptstyle (N)}}(t)$ denote the sequence $\{X_i^k(t)\},\ i=1,\ldots, N; k=1,\ldots ,n\}$, then the variables $\{J_i^{k}(t,\epsilon)\},\ i=1,\ldots, N; k =1,\ldots, n\}$, have a joint conditional Laplace transform given by $$\begin{aligned}
\label{caract}
h_{t,\epsilon}{^{\scriptscriptstyle (N)}}(\phi) &{\stackrel{\mbox{\upshape\tiny def}}{=}}{\mathsf{E}}_t\left(\exp\Bigl(\frac{1}{N}\sum_{k<l\atop i=1}^{n,N}
\phi_k(\frac{i}{N})\epsilon
J_i^k(t,\epsilon)\Bigr)\Big\vert \eta(t)\right) = \nonumber \\[0.2cm]
&{\mathsf{E}}_t\biggl[\exp\Bigl(\epsilon\sum_{k\ne l\atop i=1}^{n,N}
\lambda_{kl}X_i^kX_{i+1}^l\bigl(e^{\frac{1}{N}\psi_{kl}(\frac{i}{N})}-1\bigr)
+\lambda_{lk}X_i^lX_{i+1}^k\bigl(e^{-\frac{1}{N}\psi_{kl}(\frac{i}{N})} -1\bigr)
\Bigr)\biggr]+\ o(\epsilon),\end{aligned}$$ where $\phi_k,\ k =1,\ldots, n$ is a set of $\mathbf{C^{\infty}}$ bounded functions, and $\psi_{kl} = \phi_k-\phi_l$.
The points are mere consequences of the Markovian feature of the process and of its generator. In particular, (\[Ward\]) results from the fact that, almost surely, at most one jump takes place in the time-interval $\epsilon$, when $\epsilon\to 0$, since all events are due to independent Poisson processes. In addition, on the time interval $[t,t+\epsilon]$, the occurrence of a particle exchange between sites $i$ and $i+1$, corresponding to $\epsilon J_i^k(t,\epsilon) = 1$ is only conditioned by the presence of a pair $(k,l)$ at $(i,i+1)$, with a transition rate given by $\lambda_{kl}X_i^kX_{i+1}^l$. Therefore $$h_{t,\epsilon}{^{\scriptscriptstyle (N)}}(\phi) = {\mathbb E}_t
\biggl(\prod_{k\ne l\atop i=1}^{n,N}\Bigl[1+\epsilon\lambda_{kl}X_i^kX_{i+1}^l
\bigl(e^{\frac{1}{N}\psi_{kl}(\frac{i}{N})}-1\bigr)\Bigr]\biggr),$$ which, after a first order expansion with respect to $\epsilon$, leads to (\[caract\]) .
### An iterative numerical scheme
Given a sample path $\eta{^{\scriptscriptstyle (N)}}(t)$ at time $t$, we may generate a current sequence $\{J_i^k(t,\epsilon)\}$ according to the local product form encountered earlier. In turn, once the set $\{J_i^k(t,\epsilon)\}$ is known, the sequence $\eta(t+\epsilon)$ is almost surely determined, as $\epsilon\to 0$, by the identity (\[Ward\]), expressing conservation law of particles. We therefore have at hand an explicit stochastic numerical scheme to generate the sequence $\eta(t)$ step by step.
For any $\epsilon>0$, $N\in {\mathbb N}$, the iterative scheme given by $$Q_{n+1}(\eta) = \sum_{\eta'} P_\epsilon(\eta \vert \eta') Q_n(\eta),$$ where $P_\epsilon(\eta \vert \eta')$ is defined according to (\[Ward\]) and (\[caract\]), converges when $\epsilon\to 0$ to the original probability measure $P_{t=n\epsilon}(\eta)$ corresponds to the original process.
There is only one thing to show: $\forall T>0$, the probability $p_\epsilon$ that $\exists\,t\in[0,T]$, such that two adjacent transitions occur within the same time-interval $[t,t+\epsilon]$, tends to $0$ when $\epsilon\to 0$. This is warranted by the fact that the total number of transitions for $t<T$ is almost certainly finite. Indeed, we have $$p_\epsilon\le 1- \bigl(1-(\max_{kl}\lambda_{kl})^2
\epsilon^2\bigr)^{\frac{NT}{\epsilon}}
\operatorname*{\rightarrow}_{\epsilon\to 0} 0.$$ For the hydrodynamic limit the rates $\lambda_{kl}$ scale like $N^2$ for large $N$. Thus, it will be convenient to take a single limit $\epsilon {\stackrel{\mbox{\upshape\tiny def}}{=}}\epsilon(N)\to 0 $ as $N \to \infty$, since the condition for the scheme to be meaningful writes $$N\epsilon(N) (\max_{kl}\lambda_{kl})^2 = o(1),$$ so that we get a scaling of $\epsilon(N) = o(N^{-5})$ to meet our needs. This will allow us, in the sequel, to make use of the approximation $$\sum_{i=1}^N \alpha_i^k\Bigl(X_i^k(t+\epsilon)-X_i^k(t)
-\sum_l\bigl(J_{i-1}^k-J_i^k\bigr)\epsilon\Bigr) =
o(\epsilon),$$ for any set of bounded complex numbers $\{\alpha_i^k\}$.
### Central limit theorem for currents {#fluctu}
We are in position to exploit the conditional product form (\[caract\]) to perform a mapping, in the spirit of Lemma 4.1 of [@FaFu] , allowing to obtain a dynamical description of the system, in terms of some external free random process. To this end we assume, as a basic point, the hydrodynamic limit holds and we rest on the following lemma.
Suppose the existence of a set of density functions $\rho_k$, such that $${\mathbb E}\Bigl[\exp\bigl(\frac{1}{N}\sum_{k=1\atop i=1}^{n,N}
X_i^k\phi_k\bigl(\frac{i}{N}\bigr)\bigr)\Bigr]
= \exp\Bigl(\sum_{k=1\atop i=1}^{n,N}\log\Bigl[1+\rho_k\bigl(\frac{i}{N}\bigr)
\bigl(e^{\phi_k\bigl(\frac{i}{N}\bigr)}-1\bigr)\Bigr]+ o(N^{-2})\Bigr),$$ for any given bounded complex function $\phi_k$, and let $\DD \phi = \sup_{k\in\{1\ldots n\}\atop x\in[0,1]}(\phi_k(x))$. Then, $${\mathbb E}\Bigl[\exp\bigl(\frac{1}{N}\sum_{k<l\atop i=1}^N
\phi_k(\frac{i}{N})\phi_l(\frac{i}{N})X_i^kX_i^l\bigr)\Bigr] =
\exp\Bigl(\frac{1}{N}\sum_{k<l\atop i=1}^N \phi_k(\frac{i}{N})\phi_l(\frac{i}{N})
\rho_k\bigl(\frac{i}{N}\bigr)\rho_l\bigl(\frac{i}{N}\bigr)
\ +\ o\bigl(\frac{\phi}{N}\bigr)\Bigr).$$
From this we deduce the following identity, $$\begin{aligned}
h_{t,\epsilon}{^{\scriptscriptstyle (N)}}(\phi) = \exp\Bigl(
\epsilon\sum_{k<l\atop i=1}^{n,N}
\lambda_{kl}\rho^k\bigl(\frac{i}{N}\bigr)
\rho^l\bigl(\frac{i+1}{N}\bigr)\bigl(e^{\frac{1}{N}
\psi_{kl}(\frac{i}{N})}-1\bigr)
+\lambda_{lk}\rho^l\bigl(\frac{i}{N}\bigr)
\rho^k\bigl(\frac{i+1}{N}\bigr)\bigl(e^{-\frac{1}{N}\psi_{kl}(\frac{i}{N})}-1\bigr)
\Bigr)+o(\epsilon)\Bigg),\end{aligned}$$ which leads to recover (in our specific context) a formulation of the general result of [@BeLa] concerning fluctuation laws of currents for diffusive systems.
Keeping up to quadratic terms w.r.t. to functions $\phi$’s its argument, $h_{t,\epsilon}{^{\scriptscriptstyle (N)}}(\phi)$ reads, $$\label{heps}
h_{t,\epsilon}{^{\scriptscriptstyle (N)}}(\phi) = \exp\Bigl(\epsilon\sum_{k=1\atop i=1}^{n,N}
\phi_k(\frac{i}{N}){\cal J}^k(\rho(\frac{i}{N})) +
\frac{D\epsilon}{N^2}\sum_{k,l=1}^n
\phi_k\bigl(\frac{i}{N}\bigr) Q_{kl}\bigl(\frac{i}{N}\bigl)\phi_l\bigl(\frac{i}{N}\bigr)
+o(\frac{\phi^2}{N^2})\Bigr)$$ where ${\cal J}^k$ are deterministic currents expressed, in terms of densities, by $${\cal J}^k(\{\rho_l,l=1\ldots n\}) {\stackrel{\mbox{\upshape\tiny def}}{=}}-D\Bigl(\frac{\partial\rho_k}{\partial x}
+\sum_{l\ne k}\alpha_{kl}\rho_k\rho_l\Bigr),$$ and $Q$ is a $n\times n$ symmetric matrix $$\begin{cases}
Q_{ij} = -\rho_i\rho_j,\qquad i\ne j, \\[0.2cm]
Q_{ii} = \rho_i(1-\rho_i).
\end{cases}$$ $Q$ is of rank $n-1$, because due to the exclusion constraint, currents are not independants, $$\sum_{k=1}^n J_i^k(t,\epsilon) = 0, \qquad\forall i\in\{1,\ldots,N\}.$$ Let $M$ the reduced matrix obtained from $Q$ by deleting last row and last column. Its determinant is $\prod_{k=1}^n\rho_k$, so that it $M$ invertible if none of the $\rho_k$ vanishes, with $$\label{minverse}
\begin{cases}
\DD M^{-1}_{ij} = \frac{1}{\rho_i}+\frac{1}{\rho_n},\qquad i\ne j , \\[0.2cm]
\DD M^{-1}_{ii} = \frac{1}{\rho_n},
\end{cases}$$ after having taken into account the exclusion condition $\sum_{k=1}^n \rho_k=1$. Since every line $k$ or column $k$ sums to $\rho_k\rho_n >0$, all the eigenvalues are strictly positive, and hence $M(\rho)$ ows a real square-root matrix $M^{\frac{1}{2}}(\rho)$.
Let $\phi_k, k=1,\ldots, n-1$ denote a set of $C^\infty$ bounded functions of the real variable $x\in [0,1]$, $\{w_i^k,k=1,\ldots, n-1\}$ a set of independent identically distributed Bernoulli random variables with parameters $1/2$, taking at time $t$ values in $\{-1/2,1/2\}$. Then there exists a probability space, such that $$\label{currentmapping}
\frac{1}{N} \sum_{k=1\atop i=1}^{n,N}\ \phi_k(\frac{i}{N}) J_i^k\epsilon =
\frac{1}{N} \sum_{k=1\atop i=1}^{n-1,N}\ \psi_{kn}(\frac{i}{N})\Bigl
[{\cal J}^k(\rho(\frac{i}{N}))\epsilon +\sqrt{2D\epsilon} \sum_{l=1}^{n-1}
M_{kl}^\frac{1}{2}\Bigl(\rho\bigl(\frac{i}{N}\bigr)\Bigr)\ w_i^l\Bigr]
+{\cal O}(N^{-2}),\ a.s.,$$
The lines of arguments bare some features in common with the ones proposed in [@FaFu] (to study fluctuations at steady state). Recall, by law of large numbers, that correlations are negligible and do not affect the expression of the deterministic currents (\[detercurrent\]). This justifies the mapping (\[currentmapping\]). On the other hand, the calculation of coefficents $M_{ij}^{\frac{1}{2}}$ is done by comparing $h_{t,\epsilon}{^{\scriptscriptstyle (N)}}$ in (\[heps\]) with $${\mathbb E}\Bigl[\exp\bigl(\frac{1}{N}\sum_{k=1\atop i=1}^{n-1,N}
\psi_{kn}\bigl(\frac{i}{N}\bigr) \sqrt{2D\epsilon}
M^\frac{1}{2}_{kl}(\frac{i}{N})w_i^l\bigr)\Bigr]
=
\exp\Bigl(\frac{D\epsilon}{N^2}\sum_{kl}^n
\phi_k\bigl(\frac{i}{N}\bigr) Q_{kl}\bigl(\frac{i}{N}\bigl)\phi_l\bigl(\frac{i}{N}\bigr)
+o(\epsilon)\Bigr),$$ because $M^\frac{1}{2}$ is symmetric and $$\sum_{kl}^{n-1} \psi_{kn}\bigl(\frac{i}{N}\bigr)M_{kl}\psi_{ln}\bigl(\frac{i}{N}\bigr)
= \sum_{kl}^n
\phi_k\bigl(\frac{i}{N}\bigr) Q_{kl}\bigl(\frac{i}{N}\bigl)\phi_l.$$ Setting, for $k=1,\ldots,n-1$, $$Y_k{^{\scriptscriptstyle (N)}}(x,t) {\stackrel{\mbox{\upshape\tiny def}}{=}}\frac{1}{\sqrt N}\sum_{i=1}^{[xN]} w_i^k,$$ the corresponding space time white noise processes $$W^k(x,t) = \lim_{N\to\infty} \frac{d Y_k{^{\scriptscriptstyle (N)}}}{dx}(x,t),$$ describe current fluctuations in the continuous limit.
Macroscopic fluctuations
------------------------
Two main quantities with be explored in this section: the Lagrangian and the large deviation functional.
### The Lagrangian
The preceding section provides us with all coefficients required to achieve an informal derivation of the Lagrangian [@BeLa] describing the current fluctuations. Given the empirical measure $$\rho_k{^{\scriptscriptstyle (N)}}(x,t) {\stackrel{\mbox{\upshape\tiny def}}{=}}\frac{1}{N}\sum_{i=1}^n X_i^k(t)
\delta\bigl(x-\frac{i}{N}\bigr),$$ and assuming the system admits a hydrodynamical description in terms of a density field $\rho_k(x,t)$, the statement in [@BeLa] says that there is a large deviation principle for the stationary measure. In other words, the probability that the measure $\rho_k{^{\scriptscriptstyle (N)}}$ deviates from the hydrodynamic density profile $\rho_k$ is exponentially small and given by $$P\bigl\{\rho{^{\scriptscriptstyle (N)}}(t)\simeq\hat\rho(t),t\in[t_1,t_2]\bigr\}
\simeq e^{-NI_{[t_1,t_2]}(\hat\rho)},$$ where $$I_{[t_1,t_2]}(\hat\rho) = \int_{t_1}^{t_2}{\cal L}(\hat\rho(t),\partial_t\hat\rho(t)) dt .$$ Here the deviation from hydrodynamic solutions is due to current fluctuations.\
Writing $\bigtriangledown^{-1}{\stackrel{\mbox{\upshape\tiny def}}{=}}\int_0^x$, the quantity $\bigtriangledown^{-1}\frac{\partial\hat\rho_k{^{\scriptscriptstyle (N)}}}{\partial t} + {\cal J}^k(\hat\rho)$, represents the fluctuations of the current $J^k$. Reversing the relationship between current fluctuations and white noise process leads formally to $$\label{noise}
W^l(x,t) \simeq \sqrt\frac{\epsilon}{2DN}
\sum_{k=1}^{n-1} M^{-\frac{1}{2}}_{lk}\Bigl(
\bigtriangledown^{-1}\frac{\partial\hat\rho_k}{\partial t}
+ {\cal J}_k(\hat\rho)\Bigr) , \quad l=1,\ldots, n-1.$$ Then, replacing (\[noise\]) in the joint distribution of $\{W^k(x,t); x\in[0,1],k=1,\ldots,n-1\}$ , we obtain $$\begin{aligned}
{\cal L}(\hat\rho(t),\partial_t\hat\rho(t))dt
&= \frac{1}{2}\int_0^1 dxdt \sum_{k=1}^{n-1} \Bigl(W^k(x,t)\Bigr)^2 \\[0.2cm]
&= \frac{1}{4D}\int_0^1 dxdt \sum_{k=1}^{n-1} \Bigl(\sum_{l=1}^{n-1} M^{-\frac{1}{2}}_
{lk}
\bigtriangledown^{-1}\frac{\partial\hat\rho_k}{\partial t}
+ {\cal J}_k(\hat\rho)\Bigr)^2,\end{aligned}$$ where $\epsilon$ has been identified with $dt$ and $dx$ with $1/N$. Then, the symmetry of $M^{-\frac{1}{2}}$, the form (\[minverse\]) of $M^{-1}$ and the exclusion constraint $$\sum_{k=0}^n \bigtriangledown^{-1}\frac{\partial\hat\rho_k}{\partial t}
+ {\cal J}_k(\hat\rho) = 0,$$ lead to the final compact form $$\mathcal{L}(\hat\rho,\partial_t\hat\rho)
=\frac{1}{4D}\int_0^1 dx \sum_{k=1}^n \frac{
\Bigr(\bigtriangledown^{-1}\frac{\partial\hat\rho_k}{\partial t}
+ {\cal J}_k(\hat\rho)\Bigl)^2}{\hat\rho_k} .$$
### Hamilton-Jacobi equation and large deviation functional
Here we proceed as in [@BeLa]. Let $\pi_k$, the conjugate variable of $\rho_k$, $$\pi_k(x,t) {\stackrel{\mbox{\upshape\tiny def}}{=}}\frac{\partial\mathcal{L}(\rho,\partial_t\rho)}
{\partial\partial_t\rho_k(x,t)}.$$ The Hamiltonian is then given by $${\cal H}(\{\rho_k,\pi_k\})
{\stackrel{\mbox{\upshape\tiny def}}{=}}\int_0^1 dx\sum_{k=1}^n\pi_k(x,t)\partial_t\rho_k(x,t) - {\cal L} .$$ Algebraic manipulations lead to the expression $${\cal H}(\{\rho_k,\pi_k\}) = \int_0^1 dx
\Bigl[\partial_x\pi_k {\cal J}_k(\rho) +D\rho_k
\Bigl(\partial_x\pi_k\Bigr)^2 \Bigr].$$ Then the large deviation functional $\mathcal{F}$, satisfying $$P(\rho{^{\scriptscriptstyle (N)}}\simeq\rho) \simeq e^{-N \mathcal{F}(\rho)},$$ might be derived as in [@BeLa], from the following regular variational principle $$\mathcal{F}(\rho) = \inf_{\hat\rho}I_{[-\infty,0]}(\hat\rho),$$ where the minimum is taken over all trajectories $\hat\rho$ connecting the stationary deterministic equilibrium profiles $\bar\rho_k$ to $\rho$. This means that $\mathcal{F}$ and the action functional $I$ must satisfy the related Hamilton-Jacobi equation $${\cal H}\Bigl(\{\rho_k,\frac{\partial{\cal F}}{\partial\rho_k}\}\Bigr) = 0.$$ In addition, one can check the relation $${\cal F} = {\cal U} - {\cal S},$$ where $$\begin{cases}
\DD {\cal U} = \int_0^1dx\int_0^x\sum_{k\ne l}
\alpha_{kl}\rho_k(x)\rho_l(y)dy,\\[0.3cm]
\DD {\cal S} = -\int_0^1 dx\sum_{k=1}^n\rho_k\log\rho_k,
\end{cases}$$ a form already encountered in the reversible case, see equation (\[freen\]). Indeed, when the process is reversible, $\cal U$ is translation invariant (i.e. independent of the initial integration point, here set to zero), and so $$\partial_x\frac{\partial {\cal F}}{\partial\rho_k(x)} = -\frac{{\cal J}_k}{D\rho_k}.$$ This approach could be used to analyse the non-reversible case.
Concluding Remarks
==================
In this report we strove to put forward some techniques, and to extend methods to tackle the problem of mapping discrete model to continuous equations. Even in the context of a very specific model, namely stochastic distortions of discrete curves, some difficult questions remain.
- The determination of the invariant measure in the general case, at the discrete level, which would generalize the totally asymmetric case [@FeMa; @MaMaRa].
- The analysis of Hamilton-Jacobi equations to obtain a kind of continuous counterpart of the invariant measures, namely large deviation functionals.
With regard to hydrodynamic limits, there is a puzzling issue, namely when particle-species diffuse at various speeds, in what we called the *heterodiffusive* case. For many one-dimensional models, it is well known that a single slow particle may considerably modify the macroscopic behavior of the system (see e.g. [@Ma]). For the time being, our approach is restricted to diffusive one-dimensional systems. Yet, other scalings (like Euler), as well as processes in higher dimension, are definitely worth being studied. in particular, it could be interesting to deal with more realistic exclusion processes, for instance those encountered in the field of traffic modelling. Besides, the analysis of irreversible invariant states in terms of cycles in a state-graph might well be extended to tackle <span style="font-variant:small-caps;">asep</span> on closed networks.
[^1]: INRIA - Domaine de Voluceau, Rocquencourt BP 105 - 78153 Le Chesnay Cedex - France. Contact: `[email protected], [email protected]`
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---
abstract: |
The free energy that is dissipated in a magnetic reconnection process of a solar flare, generally accompanied by a coronal mass ejection (CME), has been considered as the ultimate energy source of the global energy budget of solar flares in previous statistical studies. Here we explore the effects of the aerodynamic drag force on CMEs, which supplies additional energy from the slow solar wind to a CME event, besides the magnetic energy supply. For this purpose we fit the analytical aerodynamic drag model of Cargill (2004) and Vrsnak et al. (2013) to the height-time profiles $r(t)$ of LASCO/SOHO data in 14,316 CME events observed during the first 8 years (2010-2017) of the SDO era (ensuring EUV coverage with AIA). Our main findings are: (i) a mean solar wind speed of $w=472 \pm 414$ km s$^{-1}$, (ii) a maximum drag-accelerated CME energy of $E_{drag} \lapprox
2 \times 10^{32}$ erg, (iii) a maximum flare-accelerated CME energy of $E_{flare} \lapprox 1.5 \times 10^{33}$ erg; (iv) the ratio of the summed kinetic energies of all flare-accelerated CMEs to the drag-accelerated CMEs amounts to a factor of 4; (v) the inclusion of the drag force slightly lowers the overall energy budget of CME kinetic energies in flares from $\approx 7\%$ to $\approx 4\%$; and (vi) the arrival times of CMEs at Earth can be predicted with an accuracy of $\approx 23\%$.
author:
- 'Markus J. Aschwanden$^1$'
- Nat Gopalswamy$^2$
title: 'Global Energetics of Solar Flares: VII. Aerodynamic Drag in Coronal Mass Ejections'
---
INTRODUCTION
==============
The motivation for this study is the determination of the energy budget of [*coronal mass ejections (CMEs)*]{} in the overall global energetics and energy partitioning of solar flare/CME events. Previous statistical work on flare energies has been pioneered by Emslie et al. (2004, 2005, 2012), and has been focused to the dissipation of magnetic energies (Aschwanden et al. 2014), thermal energies (Aschwanden et al. 2015), non-thermal energies (Aschwanden et al. 2016), CMEs (Aschwanden 2016, 2017), and the global energy closure between these various forms of energies (Aschwanden et al. 2017). It goes without saying that we cannot claim to understand the physics of flares and CMEs if we cannot pin down the relative amounts of energies in such a way that we obtain closure in the total energy budget. In the big picture we assumed that the magnetic free energy (which is defined as the difference between the non-potential and potential magnetic energy) provides the ultimate source and upper limit of energy that can be dissipated during a flare/CME event, most likely to be driven by a magnetic reconnection process. Consequently, the potential gravitational force and the kinetic energy of a CME have to be supplied entirely by the magnetic free energy and the associated Lorentz forces, in addition to the energy needed for the acceleration of particles and direct heating of the flare plasma. In the meantime it became clear that additional energy (besides the dissipated magnetic energy) can supply part of the CMEs kinematics, in form of the aerodynamic drag force that is exerted onto CMEs from the ambient slow solar wind (Vrsnak and Gopalswamy 2002; Cargill 2004; Vrsnak et al. 2008, 2010, 2013). For a brief review see Aschwanden (2019, Section 15.5). The main focus of this study is therefore the question to what extent the presence of the aerodynamic drag force affects the energy partition ratios of flare/CME events, compared with previous studies where this effect was not taken into account.
The role of the aerodynamic drag force on coronal mass ejections (CMEs) and interplanetary coronal mass ejections (ICME) has been brought to recent attention (Cargill 2004; Chen 1997; Gopalswamy et al. 2001a). Cargill (2004) demonstrated that tenuous ICMEs rapidly equalize in velocity due to the very effective drag force, while ICMEs that are denser than the ambient solar wind are less affected by the aerodynamic drag, although the drag coefficient is approximately independent of the propagation distance. An anti-correlation between the CME acceleration and velocity was established from LASCO/SOHO data (Gopalswamy et al. 2000, 2001a), which confirms that massive CMEs are less affected by the aerodynamic drag (Vrsnak et al. 2008). Massive CMEs have been found to be accelerated for masses of $m_{cme} > 3\times 10^{14}$ g, while less massive CMEs are generally decelerated (Michalek 2012). The shortest transit times and hence the fastest velocities have been identified in narrow and massive ICMEs (i.e., high-density eruptions) propagating in high-speed solar wind streams (Gopalswamy et al. 2000, 2001a; Vrsnak et al. 2010). Extremely short transit times of 14 hours (Gopalswamy et al. 2005a) and 21 hours have been observed (Temmer and Nitta 2005), with maximum speeds of $v \approx 2600$ km s$^{-1}$, but agreement with the aerodynamic drag model requires a decrease of the solar wind density near 1 AU (Temmer and Nitta 2015), but see Gopalswamy et al. (2016) for an alternative interpretation. Fast CMEs were found to show a linear dependence for the velocity difference between CMEs and solar wind, while slow CMEs show a quadratic dependence (Maloney and Gallagher 2010). A quadratic dependence is expected in a collisionless environment, where drag is caused primarily by emission of magnetohydrodynamic (MHD) waves (Vrsnak et al. 2013). A distinction between the aerodynamic drag force and the hydrodynamic Stokes drag force has been suggested (Iju et al. 2014), but was found to be equivalent in other cases (Gopalswamy et al. 2001b). Analytical models for the drag coefficient include the viscosity in the turbulent solar wind (Subramanian et al. 2012). The aerodynamic drag model has been used increasingly as the preferred physical model to quantify the propagation of ICMEs and to forecast their arrival times at Earth, and this way it became a key player in space weather predictions (Michalek et al. 2004; Vrsnak et al. 2010; Song 2010; Shen et al. 2012; Kilpua et al. 2012; Lugaz and Kintner 2013; Hess and Zhang 2014 ; Tucker-Hood et al. 2014; Mittal and Narain 2015; Zic et al. 2015; Sachdeva et al. 2015; Dumbovic et al. 2018; Verbeke et al. 2019). Arrival times at Earth inferred from the “drag-based model” have been compared with the numerical “WSA-ENLIL+Cone model” (Wang-Sheeley-Arge), which enables early space-weather forecast 2-4 days before the arrival of the disturbance at Earth (Vrsnak et al. 2014; Dumbovic et al. 2018). The Stokes form was the basis for the empirical shock arrival model, whose prediction is comparable to that of the ENLIL+cone model (Gopalswamy et al. 2005b, 2013). New models, such as the [*Forecasting a CMEs Altered Trajectory (ForeCAT)*]{} deal also with CME reflections based on magnetic forces and non-radial drag coefficients (Kay et al. 2015). Geometric models, such as the [*Graduated Cylindrical Shell (GCS)*]{} model are fitted to LASCO and STEREO data, finding that the Lorentz forces generally peak at $(1.65-2.45) R_{\odot}$, and become negligible compared with the aerodynamic drag already at distances of $(3.5-4.0) R_{\odot}$, but only at $(12-50) R_{\odot}$ for slow CME events (Sachdeva et al. 2017).
In this paper we are fitting the aerodynamic drag model to all CMEs observed with LASCO/SOHO during the SDO era (2010-1017), which yields the physical parameters that are necessary to determine the kinetic energies, the energy ratios of flare-associated and drag-accelerated CMEs, as well as their arrival times near Earth. We present the analytical description of the constant-acceleration and aerodynamic drag model in Section 2, the data analysis of forward-fitting the analytical models to LASCO data and the related results in Section 3, a discussion of some relevant issues in Section 4 and conclusions in Section 5.
THEORY AND METHODS
====================
The Constant-Acceleration Model
---------------------------------
The simplest model of the kinematics of a coronal mass ejection (CME) has a minimum number of three free parameters, which includes a constant (time-averaged) acceleration $a_0$, an initial height $r(t=t_0)=r_0$, and a starting time at a reference time $t=t_0$. A slightly more general model (with 4 free parameters) allows also for a non-zero velocity $v_0=v(t=t_0)$ at the starting time $t=t_0$, which constitutes four free model parameters $[a_0, v_0, r_0, t_0]$, defining the time dependence of the acceleration $a(t)$, $$a(t)=a_0 \ ,$$ the velocity $v(t)$ of the CME leading edge, $$v(t)=\int_{t_0}^t a(t) dt = v_0 + a_0 (t-t_0) \ ,$$ and the radial distance $r(t)$ from Sun center, $$r(t)=\int_{t_0}^t v(t) \ dt =
r_0 + v_0 (t-t_0) + {a_0 \over 2} (t-t_0)^2 \ .$$ The radial distance $r(t)$, which is directly obtained from the observations, can be fitted with a simple second-order polynomial, $$r(t) = c_0 + c_1 t + c_2 t^2 \ ,$$ where the free parameters as functions of the coefficients $c_0, c_1, c_2$ follow directly from Eqs. (3) and (4), $$a_0 = 2 c_2 \ ,$$ $$t_0 = (v_0 - c_1)/a_0 \ ,$$ $$r_0 = c_0 + v_0 t_0 - {a_0 \over 2} t_0^2 \ .$$ A practical example of the height-time profile $r(t)$, the velocity profile $v(t)$, and the acceleration profile $a(t)$ of the constant-acceleration CME kinematic model is shown for an event in Fig. 1 (left), where the observed datapoints $r_i=r(t=t_i)$ are marked with crosses (top left panel), and the fitted model is rendered with thick curves, covering the fitted time range $[t_1, t_2]$. For the fitting of an acceleration model to LASCO data we have to be aware that CMEs are observed at a heliocentric distance of $\gapprox 2.5 R_{\odot}$, by which time most CMEs have finished acceleration (Bein et al. 2011) and we are observing a residual acceleration only, combined with gravity and drag.
For the calculation of the CME starting time $t_s$ we extrapolate the model $r(t)$, which is observed in the time range $[t_1, t_2]$, to an expanded range $[t_0,t_2]$ with double length (with lower boundary $t_0=t_1-(t_2-t_1)$). The actual starting time $t_s$ of the CME launch can now be derived from the height-time profile $r(t)$ within the expanded time range $[t_0, t_2]$, where two possible cases can occur. One case is when the extrapolated minimum height at the start of the CME is lower than the solar limb, in which case the solution $r(t)$ can simple be extrapolated to the nominal height $r_s = 1 R_{\odot}$, as it is shown for the case depicted in Fig. 1 (left), for the CME event on 2011 September 24, 18:36 UT.
The other case is when the minimum height $r_s=min[r(t)]=r(t=t_s)$ is higher than the solar limb $(r_{min} > R_{\odot})$, in which case the starting time $t_s$ coincides with the height minimum, where the velocity is zero, i.e., $v_s=v(t=t_s)=0$. An example of the second case is shown in Fig. 2 (left), where the starting height is estimated to $r_s=1.963 R_{\odot}$ for the event of 2010 January 3, 05:30 UT. Note that the starting time, defined by the extrapolated zero velocity $v_s(t=t_s)=0$, is dependent on the model, estimated at $t_s=2.375$ hrs for the constant-acceleration model, and $t_s=6.287$ hrs for the aerodynamic drag model. So there is an uncertainty of the order of $\approx 4$ hrs for the start of this particular event.
The Aerodynamic Drag Model
----------------------------
We define now, besides the constant-acceleration model, a second model that is based on a physical mechanism. The interaction of a coronal mass ejection (CME) (or an interplanetary coronal mass ejection (ICME)) with the solar wind leads to an adjustment or equalization of their velocities at heliocentric distances from a few solar radii out out to one astronomical unit. When an ICME has initially a higher velocity (or take-off speed) than the solar wind, it is then slowed down to a lower value that is closer to the solar wind speed (Fig. 1). Vice versa, ICMEs with slower speeds than the ambient solar wind speed become accelerated to about the solar wind speed (Fig. 2). There are also cases where the CME accelerates to high speeds, but quickly slows down even before the solar wind formation (Gopalswamy et al. 2012; 2017). Following the physical model of aerodynamic drag formulated by Cargill (2004) and the analytical solution of Vrsnak et al. (2013), we can describe the velocity time profile $v(t)$, $$v(t) = \left( { dr(t) \over dt } \right)
= {(v_s - w) \over
1 \pm \gamma (v_s - w) (t - t_s) } + w \ ,$$ where $v_s$ is the CME velocity at an initial start time $t_s$ (also called “take-off” velocity), $w$ is the (constant) solar wind speed, $\gamma \approx 1 \times 10^{-7}$ cm$^{-1}$ is the drag parameter (in units of inverse length), and $r_s$ is the initial height at the starting time $t=t_s$. The drag parameter $\gamma$ has been defined as $$\gamma = {c_d A \rho_w \over M + M_v} \ ,$$ where $c_d$ is the dimensionless drag coefficient (Cargill 2004), $A$ is the ICME cross-sectional area, $\rho_s$ is the ambient solar-wind density, and $M$ is the ICME mass. The so called virtual mass, $M_v$, can be expressed approximately as $M_v \approx \rho_w V/2$, where $V$ is the ICME volume. Here we assumed a constant solar wind speed $w$ and a constant $\gamma$, which is justified to some extent by MHD simulations (Cargill 2004), which show the constant drag coefficient $c_d$ varies slowly between the Sun and 1 AU, and is of order unity. When the ICME and solar wind densities are similar, $c_d$ becomes larger, but remains approximately constant with radial distance. For ICMEs denser than the ambient solar wind, $\gamma$ is approximately independent of radius, while $\gamma$ falls off linearly with distance for tenuous ICMEs (Cargill 2004). Regarding the variability of the solar wind speed $w(r)$ as a function of the distance $r$, the largest deviation from a constant value $w(r)$ is expected in the corona, where the solar wind transitions from subsonic to supersonic speed at a distance of a few solar radii from Sun center, but this coronal zone is also the place where flare-associated acceleration of CMEs occurs and aerodynamic drag is less dominant, which alleviates the influence of the (non-constant) solar wind.
Vrsnak et al. (2003) integrated the velocity dependence $v(t)$ (Eq. 8) to obtain an analytical function for the height-time profile $r(t)$ explicitly, $$r(t) = \pm {1 \over \gamma}
\ln{ [ 1 \pm \gamma (v_s - w) (t - t_s) ] }
+ w (t - t_s) + r_s \ .$$ Differentiating the speed $v(t)$, we obtain an analytical expression for the acceleration time profile $a(t)$, $$a(t) = \left({ dv(t) \over dt } \right) =
{ \mp \gamma (v_s - w)^2 \over
[1 \pm \gamma (v_s - w) (t - t_s)]^2} \ .$$ We see that this model has 5 free parameters $[t_s, v_s, r_s, w, \gamma]$. The two regimes of $\pm$ correspond to the deceleration/acceleration regime, i.e., it is plus for $v_s > w$, and minus for $v_s < w$. Comparing with the constant-acceleration model, we see that three parameters are equivalent, i.e., $[t_s, r_s, v_s]=
[t_0, r_0, v_0]$, while the acceleration $a_0$ is constrained by the drag coefficient $\gamma$ and the solar wind speed $w$.
Two examples of CME kinematic models with the aerodynamic drag model are shown in Figs. 1 and 2 (right-hand panels). The case shown in Fig. 1 (middle right panel) reveals deceleration from and initial value of $v_s = 801$ km s$^{-1}$ towards the solar wind speed of $w = 405$ km s$^{-1}$, while the other case shows acceleration from $v_s = 0$ km s$^{-1}$ to $w = 421$ km s$^{-1}$, according to the aerodynamic drag model (Fig.2, middle right panel).
For the calculation of the free parameters we define a fitting time range $[t_s,t_2]$ that is bound by the starting time $t_s$ of the CME (inferred from the constant-acceleration model) and the last observed time $t_2$ of the LASCO/SOHO data. The remaining four free parameters $[r_s, v_s, w, \gamma]$ are optimized by forward-fitting of the height-time profile $r(t)$ (Eq. 10) to the observed heights $r_i=r(t_i), i=0,...,n_t$ of the LASCO/SOHO data, using the [*Direction Set (Powell’s)*]{} methods in multidimensions (Press et al. 1986). A robust performance of this optimization algorithm is achieved by optimizing the parameters \[$ln(w)$, $\ln(v_s/w)$, $\ln{(\gamma)}$, $r_s/R_{\odot}$\]. The iteration of logarithmic parameters avoids (unphysical) negative values for the velocities and the drag parameter, \[$v_s, w, \gamma$\].
OBSERVATIONS AND DATA ANALYSIS RESULTS
========================================
LASCO/SOHO Data
----------------
In the following we describe the observations from LASCO/SOHO and characterize the statistical results of our data analysis. We make use of the SOHO/LASCO CME catalog that is publicly available at [*https://cdaw.gsfc.nasa.gov/CME$\_$list*]{}, based on visually selected CME events, created and maintained by Seiji Yashiro and Nat Gopalswamy (Yashiro et al. 2008; Gopalswamy et al. 2009a, 2010). A brief description of the algorithm of measuring height-time profiles $r(t)$ is given on the same website. From the LASCO/SOHO data archive, only C2 and C3 data have been used for uniformity, because LASCO/C1 has been disabled in June 1998. We downloaded the time sequences of height time profiles, $r_i = r(t=t_i), i=1,...,n_t$, that are available for every CME detected with LASCO/SOHO during the first 8 years (2010-2017) of the [*Solar Dynamics Observatory (SDO)*]{} mission. This data set comprises 14,316 events, covering almost a full solar cycle.
Fitting of CME Kinematic Models
---------------------------------
The forward-fitting of both the constant-acceleration model (Section 2.1) and the aerodynamic drag model (Section 2.2) to the LASCO height-time profiles yields dynamical parameters that are important for extrapolating the CME kinematics from the LASCO-covered distance range of $r \approx (3 - 32) R_{\odot}$ to the lower corona at $r \lapprox 1.5 R_{\odot}$ (for identification of simultaneous flare events), and extrapolating out into the heliosphere to $r \approx 1$ AU (for forecasting of the CME arrival time at Earth).
We fitted the constant-acceleration model (Section 2.1) to the LASCO/SOHO CME height-time profiles in the same way as the second-order polynomial fits have been carried out in the LASCO CME catalog, and we verified consistency between our fits and those listed in the CDAW LASCO CME catalog. We found that the forward-fitting of both models is fairly robust. Unsatisfactory fits have been found in very few cases, identified by a low fitting accuracy ($\sigma \lapprox 5\%$, in 6% of the cases for the constant-acceleration model, and in 7% of the cases for the aerodynamic drag model), or a low drag coefficient ($\gamma \le 10^{-8}$ cm$^{-1}$, in 10% of the cases).
The fitting quality of the two (analytical) theoretical models used here is defined as follows. We calculate the average ratios of the fitted (modeled) distances $r_i^{model}$ and compare them with the observed distances, $r_i^{obs}=r(t_i), i=1,...,n_t$, $$q = {1 \over n_{t}} \sum_{i=1}^{n_t}
\left( {r_i^{model} \over r_i^{obs}} \right)
\approx 1.0 \pm \sigma \ .$$ This measure of the accuracy has been found to be very suitable, yielding a standard deviation of $\sigma_{CA}=2.7\%\pm2.7\%$ for the constant-acceleration model, and a very similar value of $\sigma_{AD}=2.9\%\pm2.5\%$ for the aerodynamic drag model. The accuracies were calculated for all 14,316 events, based on an average of $n_t \approx 23$ distance measurements per event. The fact that both models fit the data with equal accuracy suggests that either model is suitable. The example shown in Fig. 1 reveals an equal accuracy of $\sigma_{AC}=\sigma_{AD}=4.4\%$ for both models. The example shown in Fig. 2 yields a better performance for the constant-acceleration model ($\sigma_{CA}=3.3\%$), versus $\sigma_{AD}=5.4\%$ for the aerodynamic drag model. However, the aerodynamic drag model represents a physical model and yields the five parameters $[t_s, r_s, v_s, w, \gamma]$, while the constant-acceleration model requires four parameters $[t_0, r_0, v_0, a_0]$. A fundamental difference between the two models is that the acceleration $a_0$ is not time-dependent in the constant-acceleration model, while it is variable in the aerodynamic drag model, and the CME speed asymptotically approaches the solar wind speed constraining the solar wind speed $w$ and the aerodynamic drag coefficient $\gamma$.
Eruptive and Failed CMEs
--------------------------
A distinction is generally made by the dynamical characteristics of CME events, which defines the type of [*eruptive flares or CMEs*]{} when the final CME speed exceeds the gravitational escape velocity ($v_2 \ge v_{esc}$), and alternatively the type [*failed eruptions*]{}, when the escape velocity is not reached ($v_s < v_{esc}$). Failed eruptions have mass motions, but do not escape (Gopalswamy et al. 2009b). The escape speed depends only on the radial distance from the Sun center, $$v_{esc}(r) = \sqrt{ {2 G M_{\odot} \over r} }
\approx 618\
\left( {r \over R_{\odot}} \right)^{-1/2}
{\rm km\ s}^{-1} \ ,$$ where $G$ is the gravitational constant, $M_{\odot}$ the solar mass, and $R_{\odot}$ the solar radius. Examples of the escape speed dependence on the radial distance are shown in Figs. 1 and 2 (middle panels), where the escape speed is indicated with dotted curves. In the first event, the CME speed exceeds the escape velocity all the times (Fig. 1 middle right), while the second case reaches escape speed at $t = 12.1$ hrs (Fig. 2, middle right), which is reached at a distance of $r = 7.2 R_{\odot}$ (Fig. 2 top right). So, both events are eruptive CMEs.
We determined the time $t_{esc}$ and distance $r_{esc}$ where the CME gained sufficient speed to overcome the combined Lorentz force, gravitational force, and drag force, as a function of the time (Fig. 3a) and as a function of the distance (Fig. 3b), for all analyzed 14,316 CME events. The start time $t_s$ has been extrapolated from the aerodynamic drag model to a starting height of $r_s \approx 1 R_{\odot}$. We see that $\approx 10\%$ of the CMEs reach escape velocity below the solar limb (Fig. 3b), that $\approx 30\%$ of the CMEs reach escape velocity at $r_1 \lapprox 3 R_{\odot}$, at the location of the LASCO first detection, and that 100% reach escape speed at a distance of $r_2 \lapprox 10 R_{\odot}$, at the location of the LASCO last detection. Conversely, all CMEs reach escape velocity at $\lapprox 25$ hrs after launch (Fig. 3a). This confirms the selection criterion of the CDAW CME list, where eruptive CME events have been measured only, by definition. No confined flare is contained in the CDAW CME list, but we will encounter such events when we compare the association of soft X-ray flare events with CME detections, using GOES flare data (see Section 3.5).
Statistical Results of LASCO Fitting
--------------------------------------
We summarize the statistical results of our fitting of the two CME kinematic models (Eqs. 3 and 10) in Table 1 and in the Figures 4 to 7. In Fig. 4 we show the near-final speed $v_2$ (that is measured from the last detection in LASCO data), versus the ambient slow solar wind speed $w$ for all CME events. Although the last detection with LASCO yields a wide range of final speeds $v_2 \approx 100-1000$ km s$^{-1}$ (Fig. 4), the slow slow solar wind is mostly concentrated in the velocity range of $w \approx 200-500$ km s$^{-1}$. This implies that the aerodynamic drag model accelerates CMEs with $v_s < w$ and decelerates CMEs with $v_s > w$ towards the near-final speed of $v_2 \approx 200-500$ km s$^{-1}$, as indicated with the concentration of data along the vertical ridge of $w \approx 400$ km s$^{-1}$ (Fig. 4). This agrees also with the conclusion obtained from the empirical acceleration formula derived by Gopalswamy et al. (2001b), i.e., $a=-0.0054 (v_{cme} - 406)$.
This confirms that the solar wind speed $w$ is reliably retrieved from forward-fitting of the kinematic model (Eq. 8) to the LASCO data, regardless what the value of the CME speed $v_2$ is. The most frequent starting height is in the lower corona, at a median distance of $r_s \lapprox 1.2 R_{\odot}$ from Sun center ($h_S \lapprox 140,000$ km) (Fig. 5a, Table 1). The first detection with LASCO occurs at a mean distance of $r_1 = (3.0 \pm 0.8) R_{\odot}$ (Fig. 5b), while the last detection with LASCO is around $r_2 = (10.3 \pm 6.4) R_{\odot}$ (Fig. 5c, Table 1).
Statistics of the velocities are particularly interesting here because the propagation of most CMEs depends on their relative speed to the slow solar wind speed. The distribution of starting speeds $v_s$ has two peaks (Fig. 6a), one near $v_s \approx 0$, and a second peak at $v_s \approx 200$ km s$^{-1}$. This bimodality depends on the time resolution, which is typically 0.2 hrs or 12 minutes (Table 1) for LASCO data. If the initial acceleration of CMEs in the lower solar corona peaks before 12 minutes, we do not resolve the initial speed increase from $v_s=0$ to $v_1 \approx 200$ km s$^{-1}$, while a peak acceleration later than 12 minutes after the starting time $t=t_s$ will reveal an initial value of $v_s \approx 0$ (e.g., Fig. 2). This explains the large range of obtained starting velocities $v_s=482 \pm 1294$ km s$^{-1}$, which has a higher mean than that at the first detection with LASCO, $v_1 = 320 \pm 283$ km s$^{-1}$ (Fig. 6b), or at the last detection with LASCO, $v_2 = 368 \pm 198$ km s$^{-1}$ (Fig. 6c), due to the aerodynamic drag that streamlines the CME velocities.
The most interesting statistical result is the distribution of slow solar wind speeds, which have an average of $w=472\pm414$ km s$^{-1}$ or a median of $w=405$ km s$^{-1}$ (Fig. 6d), obtained at heliocentric distances in the range of $r_2 \approx 3-30 R_{\odot}$, according to the last LASCO detection (shown in Fig. 5c). Note that these forward-fitting results of the aerodynamic drag model, based on 14,306 LASCO CME events, represent one of the largest statistical measurements of the slow solar wind speeds.
We present the statistical time scales related to the LASCO detection delay $(t_1-t_s)$ and propagation duration in the LASCO field-of-view in Fig. 7 and Table 1. The average detection delay is $(t_1-t_s)=1.0 \pm 1.3$ hrs (Fig. 7a). The mean duration of CME propagation in the LASCO observed zone is $(t_2-t_1)=4.3 \pm 3.7$ hrs (Fig. 7b).
CME Start Times and GOES Flare Times
--------------------------------------
While the previously described results make exclusively use of LASCO/SOHO data, we compare now these measurements with other data sets based on HMI/SDO, AIA/SDO, and GOES data. In particular we focus on a subset of 576 M and X-class GOES flare events that have been observed during the first 7 years of the SDO mission (2010-2016), for which measurements of temporal, spatial, and energetic parameters were published previously (Aschwanden 2016, 2017).
In order to identify LASCO CME events that are associated with each of the 576 M and X-class GOES flares we use the GOES flare start reference times $t_s^{GOES}$ issued by NOAA, and find the CME events (of the entire LASCO catalog of 14,316 events during 2010-2017) that have their first LASCO detection time $t_1^{LASCO}$ closest to the starting time $t_s^{GOES}$ of the GOES flares. The relative time difference can be significantly improved by extrapolating the LASCO height-time plot $r(t)$ to the LASCO starting time $t_s^{LASCO}$ at the initial height of $r_s=r(t=t_s)$, which yields a time difference between the GOES and LASCO starting times, $$\Delta t = t_s^{LASCO} - t_s^{GOES} \ .$$ A histogram of the time differences between the GOES start times $t_s^{GOES}$ and the extrapolated LASCO starting times $t_s^{LASCO}$ is shown in Fig. 8. Out of the 576 events we find a total of 480 events (83%) with relative time delays within a time window of $\pm 4$ hrs. We calculate a Gaussian fit to the core distribution within a time window of $\pm0.7$ hrs, which encompasses 231 events (40%) that may be considered as a lower limit of events with good time coincidence. The association rate of a CME with a flare increases from 20% for C-flares to 100% in large X-class events (Yashiro et al. 2005). Therefore, the flare-associated fraction of LASCO CME events may vary between the limits of 20% and 100%. On the other hand, the complementary fraction of GOES flare events that have no CME detected with LASCO may vary in the range of 17% to 60%. These CME-less events that are not associated with a $>$M.1 GOES class flare may consist of confined flares or weak non-detected CME events.
The distribution of starting delays is $\tau = t_s^{LASCO}-t_s^{GOES} = 0.07 \pm 0.28$ hrs (Fig. 8), evaluated with a Gaussian fit at the peak of the distribution. This is consistent with previous measurements of 275 flare/CME events, where also no significant delay was found, i.e., $\tau = 0.02 \pm 0.77$ hrs (Fig. 17c in Aschwanden 2016). In this relative timing analysis we neglected the difference of the heliographic position of CME source locations, since the propagation time difference from disk center to the limb, i.e., $\Delta t_{prop}=R_{\odot}/v_{cme} \approx 0.2$ hrs (for $v_{cme} \approx 1000$ km s$^{-1}$) is smaller than the bin width of the histogram shown in Fig. 8.
Four examples of flare/CME events are shown in form of GOES flux time profiles and CME height-time plots (Fig. 9), which illustrate different uncertainties in the time coincidence. The first example (Fig. 9a) exhibits a simple single-peak GOES time profile, where the GOES and LASCO starting times coincide within 0.174 hrs (or 10 minutes). The second example (Fig. 9b) has an extremely impulsive peak, but the extrapolated LASCO starting time has due to the low initial speed ($v_1=71$ km s$^{-1}$ at the first detection with LASCO) a large uncertainty, so that the coincidence is within 0.542 hrs or 33 minutes). The third case (Fig. 9c) exhibits a substantial uncertainty in the GOES starting time, so that the coincidence is within 0.40 hrs (or 24 minutes). The fourth example (Fig. 9d) shows an X-class flare with a well-defined starting time, but the initial CME speed is so low that it implies a large negative delay of -1.56 hrs (or 94 minutes). In summary, the accuracy of the relative time coincidence between soft X-ray emission (detected with GOES) and the starting height of CMEs (detected with LASCO) depends on the definition of the flaring time (starting, peak, or end time) and the uncertainties of the CME speed extrapolation, particularly in the case of slow CMEs.
Another time marker of CME starting times is the EUV dimming, which has been measured with large statistics using AIA data. A significant delay has been observed between the AIA dimming and the GOES starting time, with a mean of $\tau = (t_s^{AIA} - t_s^{GOES}) = 0.35 \pm 0.40$ hrs (or $21 \pm 24$ minutes) (Fig. 17d in Aschwanden 2016).
Energetics of CMEs
--------------------
Our main interest of this study is how much the aerodynamic drag force affects the global energetics of flare/CME events. In order to evaluate this effect quantitatively, we have to discriminate between the flare-associated acceleration in the lower corona and the solar wind-associated acceleration in the heliosphere. Given a cadence of $\Delta t=12$ minutes for LASCO data, and assuming a minimal CME velocity of $v_{min} \approx 100$ km $s^{-1}$, the altitude range of flare-associated acceleration is estimated to be $h_{min} = v_{min} \Delta t \gapprox 72,000$ km or $0.1 R_{\odot}$, which corresponds to a radial distance of $r \le 1.1 R_{\odot}$. Since the velocity corresponds to the product of the acceleration $a$ and the acceleration time interval $\Delta t$, i.e., $v = a \Delta t$, the absolute value of the unresolved acceleration $a$ cannot be determined from LASCO observations alone, but only the product. Using EUV dimming data in addition, however, the acceleration can be resolved, as shown from AIA/SDO data (Aschwanden 2017), where a median acceleration rate of $a=0.8$ km s$^{-1}$, median acceleration times of 500 s (or 7 minutes), and an acceleration height of $h_{acc}=0.75 R_{\odot}$ have been determined (Table 1 in Aschwanden 2017). These measurements justify the assumption that the flare-associated acceleration occurs at low coronal heights of $r \lapprox 1.5 R_{\odot}$ (Gopalswamy et al. 2009; Bein et al. 2011), even for ground-level enhancement (GLE) events (Gopalswamy et al. 2013). For clarification, we emphasize that the term acceleration refers to the combination of the Lorentz force, the gravitational force, and the drag force.
A clear indication of dominant flare-associated acceleration is given when the CME velocity profile shows the maximum velocity at the starting time $t_s$, i.e., $v_s=v(t=t_s)$, while the velocity decreases during the outward propagation, which can be observed from the velocity difference between the first ($v_1$) and last ($v_2$) LASCO detection, i.e., when $v_2 < v_1$, as used in Gopalswamy et al. (2017), $$E_{flare} = {1 \over 2}\ m_{cme} \left\{
\begin{array}{cc}
v_s^2 & \mbox{for $v_1 > v_2$} \\
0 & \mbox{for $v_1 < v_2$}
\end{array}
\right.$$ On the other hand, the aerodynamic drag acceleration becomes progressively more important after the first detection of LASCO (at velocity $v_1$), while the last detection with LASCO (at velocity $v_2$) approaches the final CME speed, often close to the slow solar wind $w$, $$E_{drag} = {1 \over 2}\ m_{cme} \left\{
\begin{array}{cc}
0 & \mbox{for $v_1 > v_2$} \\
w^2 & \mbox{for $v_1 < v_2$}
\end{array}
\right.$$
Using these criteria we find $N_{flare}=313$ that show flare-associated acceleration (Fig. 10a), and $N_{drag}=263$ that exhibit aerodynamic drag acceleration (Fig. 10b), out of the total number of 576 cases (Fig. 10c). We show the (logarithmic) size distributions of these three data sets, which reveal that flare-associated acceleration processes produce the largest CME energies (Fig. 10a), while aerodynamic drag acceleration appears to have an upper limit of $E_{drag}
\lapprox 2 \times 10^{32}$ erg (Fig. 10b, vertical dashed line). When we integrate the CME kinetic energies over all events of each subgroup, we find the flare-associated acceleration processes make up for a fraction of $E_{flare}/E_{all}=80.5\%$, while aerodynamic drag acceleration accounts for the remainder $E_{drag}/E_{all}=19.5\%$, which obviously is dominated by the largest events (or the most energetic CMEs). This result is consistent with the expectation that the fastest and most energetic CMEs are less influenced by aerodynamic drag, because they have higher masses and higher velocities, (although the drag fraction is larger for faster CMEs before the solar wind takes over).
For comparison, we show also the distribution of CME energies in Fig. 10c (histogram with thin line style) from a previous study (Aschwanden 2017), which has the same number of 576 events, but contains about 1.5 times the total energy, which appears to be produced by a factor of 1.25 higher velocities for the largest events at energies $E_{max} \approx 2 \times 10^{32}$ erg.
In summary, the energy ratio of the flare-accelerated CMEs to the drag-accelerated CMEs is a factor of 4 for the CME kinetic energies. Since the CME kinetic energy accounts for 7% of the total flare energy budget (Aschwanden et al. 2017), the inclusion of the aerodynamic drag effect lowers the CME contribution from 7% to $7\% \times 0.8 = 5.6\%$. In addition, the absolute value of the CME energies is a factor of 1/1.5=0.67 lower in this study, which lowers the CME contribution to $7\% \times 0.8 \times 0.67 \approx 3.8\%$, causing an overall change of 7%-3.8%=3.2% in the global flare/CME energy budget.
The kinetic energies of CMEs shown in Fig. (10) have been derived from the AIA dataset of $\approx 576$ M and X-class flares, and thus are all associated with flares. If we ask whether flare-less CME events have a different distribution of kinetic energies, because they are all accelerated by the aerodynamic drag force, we would need a data set of LASCO-detected CMEs that have no associated flares, but heliographic flare locations are unfortunately not provided in the LASCO CME catalog, and thus we are not able to derive kinetic energies of events that are not associated with flares. However, since the association rate is near 100% for X-class flares, we do not expect that the size distributions shown in Fig.10 change at the upper end. For C-class flares, however, where the flare-association rate is 20%, we would expect a lot of smaller CME events without flares, which would steepen the size distributions of kinetic energies at the lower end (Fig. 10).
Extrapolated CME-Earth Arrival Times
--------------------------------------
We may ask whether the LASCO/SOHO data (providing height-time series of the leading edge of propagating CMEs in a distance range of $\approx (3-20) R_{\odot}$) are sufficient to predict the arrival times of [*Interplanetary Coronal Mass Ejections (ICME)*]{} near Earth (e.g., Gopalswamy et al. 2013). Using data acquired with the instruments onboard WIND, ACE, SOHO/CELIAS/MTOF/PM, we obtain timing information for the arrival times at Earth from the ICME catalog http://www.srl.caltech.edu/ACE/ASC/DATA/level3/icmetable2.html (produced by I. Richardson and H. Cane), which contains ICME observations during 1996-2018. During the SDO+LASCO era (2010-2017), which is of primary interest here, information on the LASCO (or GOES) starting time are available for 78 ICME events, whereof 19 events are associated with GOES $>M1.0$ class flares. Eliminating events that have insufficient data points ($n_t < 5$) or have extremely low drag coefficient values ($\gamma
\lapprox 10^{-8}$ cm$^{-1}$), we are left with 11 events, which are listed in Table 2. In Table 2 we list the GOES flare starting times (which are good proxies for the CME starting times $t_s$) and the ICME arrival times at Earth, based on the time of the associated geomagnetic storm sudden commencement, which is typically related to the arrival of a shock at Earth (see footnotes in ICME catalog by Richardson and Cane). The resulting observed ICME propagation time delay $\tau_{obs}$ ranges from 35 to 87 hours (Fig. 11). For the predicted delay we assume radial propagation of CMEs, which corresponds to an interplanetary path length $L_{path}$ of one astronomical unit $d_{AU}$, $$L_{path}=d_{AU} \ ,$$ and the mean CME speed is approximated by the last LASCO detection $v_2$, which yields a predicted propagation delay $\tau_{pred}$ of $$\tau_{pred} = {L_{path} \over v_2}
= {d_{AU}\ q_{corr} \over v_2} \ ,$$ where the correction factor $q_{corr}$ includes various effects that have to be determined empirically, such as velocity corrections due to projection effects (since all CME velocities are measured in the plane-of-sky and may underestimate the true 3-D velocity, by factors up to 2), the temporal variability of the CME speed, velocity changes due to CME-CME interactions, and the temporal evolution of the solar wind speed. We find that an empirical value of $q_{corr}=0.81$ provides the optimum correction factor. The resulting ratio of the theoretically predicted to the observationally measured ICME propagation delays has then a mean and standard deviation of $\tau_{pred}/\tau_{obs}=1.00\pm0.23$ (Fig. 11a), which implies that we can predict the arrival times at Earth with an accuracy of $\approx 23\%$.
Comparing our result with the empirical formula of transit times $\tau_{pred}$ as a function of the CME speed $v$ fitted in Gopalswamy et al. 2005a (Fig. 8 therein) for 4 ICME events, i.e., $\tau_{pred} = a b^v + c$ (with the best-fitting coefficients a=151.02, b=0.998625, c=11.5981), we find an almost identical result, with a mean and standard deviation of $\tau_{pred}/\tau_{obs}=1.01\pm0.23$ (Fig. 11b).
DISCUSSION
============
Eruptive and Failed CMEs
--------------------------
This classification into eruptive and failed CMEs is not trivial, because both the CME speed and the escape speed are spatially and temporally varying. In previous studies, the kinetic energy $E_{kin}(r)$ and the CME gravitational energy $E_{grav}(r)$ were calculated as a function of the distance $r$ from the Sun (Vourlidas et al. 2000; Aschwanden 2016). In the study of Vourlidas et al. (2000) it was concluded that the potential (gravitational) energy is larger than the kinetic energy of the CMEs for relatively slow CMEs (which is expected for failed eruptions), while the kinetic energy was found to exceed the gravitational energy for a relatively fast CME (as expected for eruptive CMEs). In the study of Aschwanden (2016) the gravitational energy was found to make up a fraction of $E_{grav}/E_{cme}=0.75\pm0.28$ of the total energy $E_{cme}
=E_{grav}+E_{kin}$, so that the most energetic CMEs (of GOES M and X-class flares) have a kinetic energy larger than the gravitational energy, which was the case in 22% of the events. This low fraction is most likely caused by the neglect of the solar wind drag force. Emslie et al. (2012) estimated the CME kinetic energy in the rest frame of the solar wind by subtracting 400 km s$^{-1}$ from the measured CME speed, which lowers the energy demand to overcome the flare-associated Lorentz force and thus increases the percentage of eruptive CME events (compared with the percentage of failed eruptions). In the study of Aschwanden (2017), the deceleration due to the gravitational force was included in the dynamical model of initial CME acceleration, leading to a very small fraction of $\approx 2.3\%$ for failed eruptions. These are relatively low values compared with the study of Cheng et al. (2010), who found a fraction of 43% for confined flare events. The lowest values of $\approx 2.3\%$ for failed CME eruptions may be a consequence of dynamic models that over-estimate the CME velocity (Aschwanden 2017). In the present study we estimate a fraction of 40%-83% CME events to be associated with ($>$M1.0 class) flares, depending on the chosen uncertainty of the time overlap ($\Delta t \approx 0.7-4.0$ hrs; see Fig. 8), but is largely consistent with earlier results, i.e., 43% (Cheng et al. 2010) and 22% (Aschwanden 2016).
Aerodynamic Drag and Global Flare Energetics
----------------------------------------------
How does the phenomenon of the aerodynamic drag force, which we neglected in this series of statistical studies so far, effect the global energy budget of a flare/CME event? In the study of Emslie et al. (2012), the CME is estimated to dissipate 19% of the magnetic flare energy in the statistical average. The total primary dissipated energy (by acceleration of nonthermal electrons and ions, as well as the kinetic energy of CMEs) amount only to 25% of the magnetic energy in the study of Emslie et al (2012), while the CME kinetic energy (with the slow solar wind energy subtracted) is estimated to consume 19% of the available magnetic energy. Since the effects of the slow solar wind has already been corrected, no additional correction is needed to account for the aerodynamic drag force, and thus the discrepancy in energy closure does not change, mostly caused by a massive over-estimate of the magnetic energy, which was estimated ad hoc to 30% of the potential energy.
In the study of Aschwanden (2017), energy closure is almost reached (87%$\pm$18%), where the CMEs are estimated to dissipate 7% of the available magnetic free energy. Including the energy supply by aerodynamic drag, the CME energy budget changes from 7% to 4% of the total flare energy budget, and thus it just drops slightly in the energy closure from 87% to 83%. Hence there is no dramatic change in the global flare energetics.
Coincidence of Flare and CME Starting Times
---------------------------------------------
The association of flares and CMEs is fairly well established by observing the initial rise of soft X-ray emission (such as from a GOES light curve) and identifying a near-simultaneous EUV dimming, because these two time markers are produced cospatially. It is more difficult to find the corresponding flare-associated CME event from white-light observations (such as with a height-time profile of the CME leading edge, because the two associated phenomena are not cospatial. The time delay between the GOES flare starting time and the first detection with LASCO at $r \approx 3.0 R_{\odot}$ is $(t_1^{LASCO}-t_s^{GOES}) = 1.0 \pm 1.3$ hrs (Fig. 7a), during which multiple flares can occur. One way to improve the simultaneity is to extrapolate the LASCO height-time profile to the initial starting height $r_s$, which indeed improves the coincidence to $(t_s^{LASCO}-t_s^{GOES}) = 0.07 \pm 0.28$ hrs = $4 \pm 15$ minutes (Fig. 8a). The extrapolation from the first detection at $t_1^{LASCO}$ to the expected CME starting time $t_s^{LASCO}$, however, is model-dependent, and hence the timing uncertainties can amount from $(t_s^{LASCO}-t_s^{GOES}) = 0.174$ hrs (Fig. 9a) to $(t_s^{LASCO}-t_s^{GOES}) = -1.659$ hrs (Fig. 9d). More accurate starting time measurements could be achieved by using occulter disks closer to the solar surface, such as LASCO/C1, which unfortunately was disabled on June 1998.
Estimating CME arrival times at Earth
---------------------------------------
An important parameter for space weather predictions is the estimated propagation time from the solar CME site to the Earth at a distance of 1 AU. In our study we compare the observed travel time (Fig. 11, x-axis) with the predicted travel time (Fig. 11, y-axis) based on the velocity profile $v(t)$ obtained from fitting the aerodynamic drag model, which essentially is close to the travel time one obtains from the slow solar wind speed of $w \approx 400$ km s$^{-1}$. The comparison demonstrates that an accuracy of $\pm 23\%$ of the observed travel time can be achieved, which translates for a range of travel times ($\approx 35-87$) hrs to an uncertainty of $\approx 8-20$ hrs.
Our results compare favorably with other measurements. Tucker-Hood et al. (2014) report an average error of 22 hrs in the predicted transit time, which exceeds the largest uncertainty of our measurements. Kim et al. (2007) compared 91 predictions of shocks made with the empirical shock arrival model and found that 60% of the predicted travel times were within $\pm 12$ hrs. McKenna-Lawlor et al. (2006) found only 40% of the cases within $\pm 12$ hrs. One advantage of our method is that the solar wind speed $w$ is measured from fitting the aerodynamic drag model, so that no assumptions need to be made about the time-dependent variation of the slow solar wind.
CONCLUSIONS
=============
Our motivation for this study is the role of the aerodynamic drag force on the acceleration of CMEs, in the context of global energetics of flares and CMEs. In previous studies on the energy closure and partition in solar flares and CMEs we neglected this effect. Here we investigate three data sets: one CME set that covers all (14,316) LASCO CME detections during the SDO era (2010-2017), one flare data set with (576) GOES M- and X-class flares, and one set with (11) interplanetary CMEs with known arrival times at Earth. We obtain the following results:
1. [We apply two different forward-fitting models: (i) A second-order polynomial fit based on the assumption of constant acceleration during the propagation across the LASCO/C2 and C3 coronagraph, and (ii) the aerodynamic drag model of Cargill (2004) and Vrsnak et al. (2013). Both are analytical models that can be fitted to the observed height-time profiles $r(t)$ from LASCO and yield either the acceleration constant $a$, or the ambient slow solar wind speed $w$ and the drag coefficient $\gamma$. Both models fit the data with an accuracy of $\approx 3\%$ in the ratio of modeled to observed distances $r$. Both models can be applied to extrapolate the starting time $t_s$ to the CME at a coronal base level $r_s = 1 R_{\odot}$ and to predict the arrival time of a CME at Earth.]{}
2. [The extrapolated starting times $t_s^{LASCO}$ are found to coincide with the flare starting time $t_s^{GOES}$ in soft X-rays within $\pm 4$ hrs in 83%, or within $\pm 0.7$ hrs in 40%, which implies that a fraction of 17%-60% of flare events have no GOES $>$1 M class counterpart in LASCO-detected CMEs, possibly representing failed eruptions or confined flare events. All LASCO-detected CMEs were found to develop final speeds above the gravitational escape velocity, latest after a distance of $r \gapprox 10 R_{\odot}$ or a travel time of $t \gapprox 25$ hrs.]{}
3. [The LASCO-detected CME events can be subdivided into two classes, (i) one with dominant flare-associated acceleration in the lower corona at heights of $r \lapprox 1.5 R_{\odot}$, inferred in 313 out of the 576 events, and (ii) one with dominant aerodynamic drag acceleration in the upper corona of $r \approx (1.5-10.0) R_{\odot}$, identified in 263 out of the 576 cases. The aerodynamic drag acceleration appears to have an upper limit of CMEs kinetic energies at $E_{drag}\lapprox
2 \times 10^{32}$ erg, while the flare-associated acceleration can produce CME kinetic energies up to $E_{flare} \approx 1.5 \times 10^{33}$ erg. The ratio of the summed kinetic energies for the two acceleration processes is $E_{flare}/E_{all} \approx 80\%$ for flare acceleration, and $E_{drag}/E_{all} \approx 20\%$ for the aerodynamic drag model, so that $E_{flare}/E_{drag} \approx 4$.]{}
4. [The aerodynamic drag model predicts the velocity $v(t)$ of the CME leading edges from the locations of LASCO detection all the way to Earth, approaching asymptotically the solar wind speed at a distance of $r \gapprox 10 R_{\odot}$. For a subset of 11 events, for which the arrival times $\tau_{obs}$ at Earth are known, we predict the arrival times $\tau^{pred}$ within an accuracy of $\approx 23\%$, which translates into an uncertainty of 8-20 hrs.]{}
5. [For the global energetics of flare/CME events we found that CMEs contribute in the average $\approx 7\%$ to the total energy budget, for which we reached closure within $87\%\pm18\%$ (Aschwanden et al. 2017). Including the effects of the aerodynamic drag, which boosts the CME kinetic energies in addition to the dissipated magnetic energies, we find a correction of the estimated total energy by $\approx -4\%$, which modifies energy closure from 87% slightly downward to 83%.]{}
In summary, neglecting the aerodynamic drag does not modify the overall energy budget by a large amount, i.e., the total dissipated magnetic energy is reduced from a closure value of 87% to 83%, the fraction of CME energies reduces from 7% to $\approx 4\%$, but the kinetic energies in flare-accelerated CMEs are a factor of 4 higher than the total kinetic energies transferred from the slow solar wind aerodynamic drag to the final CME kinetic energies. This preponderance of flare-accelerated CME energies results from the inability of the aerodynamic drag to accelerate CMEs to larger kinetic energies than $\lapprox 2 \times 10^{32}$ erg, while flares can produce CME kinetic energies that are up to an order of magnitude higher.
We acknowledge helpful discussions with Ian Richardson and Nariaki Nitta. This CME catalog is generated and maintained at the CDAW Data Center by NASA and The Catholic University of America in cooperation with the Naval Research Laboratory. SOHO is a project of international cooperation between ESA and NASA. Support for the CDAW catalog is provided by NASA/LWS and by the Air Force Office of Scientific Research (AFOSR). This work was partially supported by NASA contracts NNX11A099G, 80NSSC18K0028, NNX16AF92G, and NNG04EA00C (SDO/AIA).
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Parameter Mean and standard dev. Median
------------------------------------------------------- ---------------------------------------- --------------------------------
Starting height $r_s$ $1.7\pm1.4$ $R_{\odot}$ 1.2 $R_{\odot}$
Height of first LASCO detection $r_1$ $3.0\pm0.8$ $R_{\odot}$ 2.7 $R_{\odot}$
Height of last LASCO detection $r_2$ $10.3\pm6.4$ $R_{\odot}$ 8.1 $R_{\odot}$
Starting velocity $v_s$ $482\pm1294$ km/s 202 km/s
Velocity at first LASCO detection $v_1$ $320\pm283$ km/s 284 km/s
Velocity at last LASCO detection $v_2$ $368\pm198$ km/s 326 km/s
Slow solar wind speed $w$ $472\pm414$ km/s 405 km/s
Acceleration $|a|$ $0.013\pm0.029$ km/s$^{-2}$ $0.005$ km/s$^{-2}$
LASCO detection delay $t_1-t_s$ $1.0\pm1.3$ hrs 0.9 hrs
LASCO detection duration $t_2-t_1$ $4.3\pm3.7$ hrs 3.2 hrs
Aerodynamic drag coefficient $\gamma$ $(2.6\pm3.3) \times 10^{-7}$ cm$^{-1}$ $1.3 \times 10^{-7}$ cm$^{-1}$
Accuracy of constant-acceleration model $\sigma_{CA}$ $2.7\%\pm2.7\%$ 2.4%
Accuracy of aerodynamic drag model $\sigma_{AD}$ $2.9\%\pm2.5\%$ 2.5%
Number of observed LASCO images $n_t$ $23\pm17$ 19
Average cadence $0.21 \pm 0.07$ hrs 0.20 hrs = 12 min
Starting time delay $(t_s^{LASCO}-t_s^{GOES})$ 0.07$\pm$0.27 hrs -0.06 hrs
: Statistics of CME parameters (mean, standard deviation, median) for 14,316 eruptive CME events detected with LASCO/SOHO during 2010-2017.
------ --------------------- --------------------- ---------- ------------ ---------- ----------- ---------------
\# Start time Arrival time Velocity Solar wind Observed Predicted Ratio
GOES ICME at Earth $v_2$ speed $w$ delay delay
(UT) (UT) (km/s) (km/s) (hrs) (hrs)
12 2011-02-15T01:44:00 2011-02-18T01:30:00 581 436 71 69 0.961
54 2011-08-02T05:19:00 2011-08-04T21:53:00 611 438 64 67 1.038
58 2011-08-04T03:41:00 2011-08-05T17:51:00 1110 467 38 47 1.231
66 2011-09-06T22:12:00 2011-09-09T12:42:00 565 425 62 64 1.024
98 2011-10-02T00:37:00 2011-10-05T07:36:00 264 389 78 103 1.304
115 2011-11-09T13:04:00 2011-11-12T05:59:00 789 448 64 55 0.847
147 2012-03-07T00:02:00 2012-03-08T11:03:00 2405 487 35 22 0.628
273 2013-04-11T06:55:00 2013-04-13T22:54:00 775 439 63 57 0.891
409 2014-02-04T01:16:00 2014-02-07T17:05:00 488 419 87 74 0.843
421 2014-02-12T06:54:00 2014-02-15T13:16:00 432 818 78 64 0.817
504 2014-09-10T17:21:00 2014-09-12T15:53:00 955 403 46 64 1.375
mean 1.00$\pm$0.23
------ --------------------- --------------------- ---------- ------------ ---------- ----------- ---------------
: Observed and predicted arrival time at Earth for 14 eruptive CME events, (data extracted from Interplanetary Coronal Mass Ejection web site, provided by Ian Richardson and Hillary Cane), and based on an empirical correction factor of $q_{corr}=0.81$ due to velocity projection effects.
![The height-time profile $r(t)$ (top panels), the velocity profile $v(t)$ (middle panels), and the acceleration profile $a(t)$ (bottom panels) of two CME kinematic models, the constant-acceleration model (left panels), and the aerodynamic drag model (right panels), showing an example of CME decelaration ($v_2 < v_1$). The observed data points (crosses in top panels) are detected during the time interval $[t_1, t_2]$, while the approximate starting time $t_s$ is constrained by the initial height $r_s=r(t=t_s)$.](f1.eps){width="100.00000%"}
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|
---
abstract: |
The paper deals about Hardy-type inequalities associated with the following higher order Poincaré inequality:
$$\left( \frac{N-1}{2} \right)^{2(k -l)} := \inf_{ u \in C_{c}^{\infty} \setminus \{0\}} \frac{\int_{{\mathbb{H}^{N}}} |\nabla_{{\mathbb{H}^{N}}}^{k} u|^2 \ dv_{{\mathbb{H}^{N}}}}{\int_{{\mathbb{H}^{N}}} |\nabla_{{\mathbb{H}^{N}}}^{l} u|^2 \ dv_{{\mathbb{H}^{N}}} }\,,$$ where $0 \leq l < k$ are integers and ${\mathbb{H}^{N}}$ denotes the hyperbolic space. More precisely, we improve the Poincaré inequality associated with the above ratio by showing the existence of $k$ Hardy-type remainder terms. Furthermore, when $k = 2$ and $l = 1$ the existence of further remainder terms are provided and the sharpness of some constants is also discussed. As an application, we derive improved Rellich type inequalities on upper half space of the Euclidean space with non-standard remainder terms.
address:
-
-
author:
- Elvise BERCHIO
- Debdip GANGULY
date:
-
-
title: |
Improved higher order poincaré inequalities\
on the hyperbolic space via Hardy-type remainder terms
---
Introduction
============
Let ${\mathbb{H}^{N}}$ denote the hyperbolic space and let $k, l$ be non-negative integers such that $l < k$. The following higher order Poincaré inequality [@SD Lemma 2.4] holds $$\label{high}
\int_{\mathbb{H}^{N}} |\nabla_{\mathbb{H}^{N}}^{k} u|^{2} \ dv_{\mathbb{H}^{N}} \geq \left( \frac{N-1}{2} \right)^{2(k - l)}
\int_{\mathbb{H}^{N}} |\nabla_{\mathbb{H}^{N}}^{l} u|^2 \ dv_{\mathbb{H}^{N}},$$ for all $u \in H^{k}(\mathbb{H}^{N})$, where $$\nabla_{\mathbb{H}^{N}}^{j}:=
\left \{\begin{array}{ll}
\Delta_{{\mathbb{H}^{N}}}^{j/2} & \text{if $j$ is an even integer}\,,\\
\nabla_{{\mathbb{H}^{N}}} \Delta_{{\mathbb{H}^{N}}}^{(j-1)/2} & \text{if $j$ is an odd integer}
\end{array}\right.$$ and $\nabla_{\mathbb{H}^{N}}$ denotes the Riemannian gradient while $\Delta_{{\mathbb{H}^{N}}}^j$ denotes the $j-$th iterated Laplace-Beltrami operator. The present paper takes the origin from the basic observation that the inequality in is strict for $u\neq 0$, namely the following infimum is never achieved $$\label{inf}
\left( \frac{N-1}{2} \right)^{2(k-l)} = \inf_{u \in H^{k}(\mathbb{H}^{N}) \setminus \{ 0 \} }
\frac{\int_{\mathbb{H}^{N}} |\nabla_{\mathbb{H}^{N}}^{k} u|^{2} \ dv_{\mathbb{H}^{N}}}{\int_{\mathbb{H}^{N}} |\nabla_{\mathbb{H}^{N}}^{l} u|^2 \ dv_{\mathbb{H}^{N}}}\,.$$ It becomes then a natural problem to look for possible remainder terms for . In this direction, when $k=1$ and $l=0$, a remainder term of Sobolev type has been determined in [@mancini]. The aim of our study is to deal with Hardy remainder terms, namely to determine improved Hardy inequalities for higher order operators, where the improvement is meant with respect to the higher order Poincaré inequality . More precisely, settled $r:=\varrho(x,x_0)$, where $\varrho$ denotes the geodesic distance and $x_0\in{\mathbb H}^N$ denotes the pole, we wish to answer the question\
*Does there exist positive constants $C$ and $\gamma$ such that the following Poincaré-Hardy inequality $$\label{motivation}
\int_{\mathbb{H}^{N}} |\nabla_{\mathbb{H}^{N}}^{k} u|^{2} \ dv_{\mathbb{H}^{N}} - \left( \frac{N-1}{2} \right)^{2(k - l)}
\int_{\mathbb{H}^{N}} |\nabla_{\mathbb{H}^{N}}^{l} u|^2 \ dv_{\mathbb{H}^{N}}\ge C \int_{{\mathbb H}^N}\,\frac{u^2}{r^{\gamma}}\,{\rm d}v_{{\mathbb{H}^{N}}}\,$$ holds for all $u\in H^k({\mathbb{H}^{N}})$?\
The literature on improved Hardy and Rellich inequalities in the Euclidean setting dates back to the seminal works of Brezis-Vazquez [@Brezis] and Brezis-Marcus [@BrezisM]. Without claiming of completeness, we also recall [@A; @GFT; @BFT2; @BT; @DH; @FT; @FTT; @gaz; @GM; @MMP; @R; @TZ] and references therein. The reason of such a great interest is surely do to the fact that Hardy inequalities and their improved versions have various applications in the theory of partial differential equations and nonlinear analysis, see for istance [@Brezis; @vaz; @VZ]. Further generalizations to Riemannian manifolds are quite recent and a subject of intense research after the work of Carron [@Carron]. We enlist few important recent works [@Mitidieri1; @Mitidieri2; @Dambrosio; @pinch; @Kombe1; @Kombe2; @LW; @Mitidieri; @Yang] and references therein. Most of these works deals with classical Hardy inequalities and their improvement on Riemannian manifolds. Namely, differently from , the optimal Hardy constant is taken as fixed and one looks for bounds of the constant in front of other remainder terms. The main motivation of our study initiated in [@BGG] on improved Poincaré inequalities comes from a paper of Devyver-Fraas-Pinchover [@pinch], which deals with optimal Hardy inequalities for general second order operators. In particular, the existence of at least one Hardy-type remainder term for with $k=1$ and $l=0$ follows as an application of their results. Nevertheless, their weight is given in terms of the Green’s function of the associated operator and does not imply the validity of an inequality like . See [@BGG] for further details. The same can be said for the inequality in [@BMR Example 5.3] where $N=3$. The above mentioned goal was achieved in [@BGG] where, developing a suitable construction of super solution, the following inequality was shown*
[$\bullet$ **[Case $k = 1$ and $l=0$.]{}**]{} For $N>2$ and for all $ u \in C^{\infty}_{0}(\mathbb{H}^{N} )$ there holds $$\label{poincareeq}
\int_{\mathbb{H}^{N}} |\nabla_{{\mathbb{H}^{N}}} u|^2 \ dv_{{\mathbb{H}^{N}}}- \left( \frac{N-1}{2} \right)^{2} \int_{\mathbb{H}^{N}} u^2 \ dv_{{\mathbb{H}^{N}}}\,
\geq \frac{1}{4} \int_{\mathbb{H}^{N}} \frac{u^2}{r^2} \ dv_{{\mathbb{H}^{N}}}\,,$$ where the constants $\left( \frac{N-1}{2} \right)^{2}$ and $\frac{1}{4}$ are sharp.
Unfortunately, the super solution construction applied in the proof of seems not applicable to the higher order case. Nevertheless, by exploiting a completely different technique based on spherical harmonics, in [@BGG] the following second order analogue of was obtained
[**[$\bullet$ Case $k = 2$ and $l=0$.]{}**]{} For $N>4$ and for all $ u \in C^{\infty}_{0}(\mathbb{H}^{N})$ there holds
$$\begin{aligned}
\label{PR}
\int_{\mathbb{H}^{N}} (\Delta_{{\mathbb{H}^{N}}} u)^2 \ dv_{{\mathbb{H}^{N}}} - \left( \frac{N-1}{2} \right)^{4}\int_{{\mathbb{H}^{N}}} u^2 \ dv_{{\mathbb{H}^{N}}}
\geq \frac{(N-1)^2}{8} \int_{\mathbb{H}^{N}} \frac{u^2}{r^2} \ dv_{{\mathbb{H}^{N}}} + \frac{9}{16} \int_{\mathbb{H}^{N}} \frac{u^2}{r^4} \ dv_{{\mathbb{H}^{N}}} \,,\end{aligned}$$
where the constant $\left( \frac{N-1}{2} \right)^{4}$ and $\frac{(N-1)^2}{8}$ are sharp.
It is clear that and do not give a complete proof of . The aim of the present paper is either to generalize to the higher order and and to investigate all the remaining cases when $l\neq 0$. A first step in this direction is represented by the proof of the validity of when $k = 2$ and $ l= 1$. This case is not covered by and and its proof requires some effort. A clever transformation which uncovers the Poincaré term and spherical harmonics technique are the main tools applied, see Sections \[21\] and \[proof21\]. Also we note that when $k = 2$ and $ l= 1$ further singular remainder terms, involving hyperbolic functions, are provided and some optimality issues are proved. Namely, we have
\[main\_intro0\] [ **[(Case $k = 2$ and $l=1$)]{}**]{} Let $ N >4$. For all $ u \in C^{\infty}_{0}(\mathbb{H}^{N})$ there holds
$$\int_{\mathbb{H}^{N}} (\Delta_{{\mathbb{H}^{N}}} u)^2 \ dv_{{\mathbb{H}^{N}}} - \left( \frac{N-1}{2} \right)^{2} \int_{\mathbb{H}^{N}} |\nabla_{{\mathbb{H}^{N}}} u|^2 \ dv_{{\mathbb{H}^{N}}} \geq \frac{(N-1)^2}{16} \int_{\mathbb{H}^{N}} \frac{u^2}{r^2} \ dv_{{\mathbb{H}^{N}}}
+ \frac{9}{16} \int_{\mathbb{H}^{N}} \frac{u^2}{r^4} \ dv_{{\mathbb{H}^{N}}}$$ $$+\frac{(N-1)(N-3)(N^2 -2N - 7)}{16} \int_{{\mathbb{H}^{N}}} \frac{u^2}{\sinh^2 r} \ dv_{{\mathbb{H}^{N}}} +\frac{(N-1)(N-3)(N^2-4N-3)}{16} \int_{{\mathbb{H}^{N}}} \frac{u^2}{\sinh^4 r} \ dv_{{\mathbb{H}^{N}}}.$$ The constant $\left( \frac{N-1}{2} \right)^{2}$ is sharp by construction and sharpness of the other constants is discussed in Section 2.
Theorem \[main\_intro0\] turns out to be one of the key ingredients in our strategy to get the arbitrary case, i.e. inequality for every $l <k$. Furthermore, from Theorem \[main\_intro0\] we derive improved Rellich type inequalities on upper half space of the Euclidean space having their own interest. See Corollary \[cor2\] for the details. The technique adopted relies on the so-called Conformal Transformation" to the Euclidean space.
As concerns the general case $l <k$, a fine combination of the previous results and some technical inequalities allow us to finally derive the following family of inequalities
\[main\_intro\] [ **[(Case $0\leq l < k$)]{}**]{} Let $k,l$ be integers such that $0\leq l < k$ and let $ N> 2k$. There exist $k$ *positive* constants $\alpha_{k,l}^j=\alpha_{k,l}^j(N)$ such that the following inequality holds
$$\int_{\mathbb{H}^{N}} |\nabla_{\mathbb{H}^{N}}^{k} u|^{2} \ dv_{\mathbb{H}^{N}} -\left( \frac{N-1}{2} \right)^{2(k - l)}
\int_{\mathbb{H}^{N}} | \nabla^{l}_{\mathbb{H}^{N}} u|^{2} \ dv_{\mathbb{H}^{N}}\ge \sum_{j = 1}^{k} \alpha_{k,l}^j \int_{\mathbb{H}^{N}} \frac{u^2}{r^{2j}} \ dv_{\mathbb{H}^{N}}$$ for all $u \in C^{\infty}_{0}(\mathbb{H}^{N})$. Furthermore, the constant $\left( \frac{N-1}{2} \right)^{2(k - l)} $ is sharp and the leading terms as $r\rightarrow 0$ and $r\rightarrow +\infty$, namely $\alpha_{k,l}^{1}$ and $\alpha_{k,l}^{k}$, are given explicitly in Theorems \[mainhigher0\] and \[mainmain\] below.
In view of possible applications to differential equations, we point out that the strategy of our proofs basically allows to determine explicitly all the constants $\alpha_{k,l}^j$ in Theorem \[main\_intro\]. Nevertheless, for the sake of simplicity, we prefer to focus on the leading terms $\alpha_{k,l}^{1}$ and $\alpha_{k,l}^{k}$. This choice is also justified by the fact that our interest is devoted to the non-Euclidean behavior of inequalities and the constant highlighting this aspect is exactly $\alpha_{k,l}^{1}$, i.e. the constant in front of the leading term as $r\rightarrow +\infty$. As a matter of example, here below we specify our family of inequalities for some particular choices of $k$ and $l$.
\[mainhigher0\] [ **[(Case $0= l < k$)]{}**]{} Let $k$ be a positive integer and let $ N> 2k$.
If $k=2m$ for some positive integer $m$, there holds $$\int_{\mathbb{H}^{N}} (\Delta_{{\mathbb{H}^{N}}}^{m} u)^2 \ dv_{\mathbb{H}^{N}} -\left( \frac{N-1}{2} \right)^{4m }
\int_{\mathbb{H}^{N}} u^2 \ dv_{\mathbb{H}^{N}}\geq$$ $$\sum_{j = 1}^{ m} \frac{(N-1)^{4m-2j}}{2^{4m-1}} \int_{\mathbb{H}^{N}} \frac{u^2}{r^{2}} \ dv_{\mathbb{H}^{N}}+ \frac{9}{2^{4m}} \prod_{j = 1}^{ m -1} (N+ 4j)^2(N-4j-4)^2 \int_{\mathbb{H}^{N}} \frac{u^2}{r^{4m}} \ dv_{\mathbb{H}^{N}}$$ for all $u \in C^{\infty}_{0}(\mathbb{H}^{N})$, where we use the convention $\prod_{j = 1}^{ 0}=1$.
If $k=2m+1$ for some positive integer $m$, there holds $$\int_{\mathbb{H}^{N}} |\nabla_{{\mathbb{H}^{N}}} (\Delta_{{\mathbb{H}^{N}}}^{m} u)|^{2} \ dv_{\mathbb{H}^{N}} -\left( \frac{N-1}{2} \right)^{4m+2 }
\int_{\mathbb{H}^{N}} u^2 \ dv_{\mathbb{H}^{N}}\geq$$ $$\left[ \sum_{j = 1}^{ m} \frac{(N-1)^{4m-2j+2}}{2^{4m+1}}+ \frac{(N-1)^{2m}}{2^{4m+2}} \right]\int_{\mathbb{H}^{N}} \frac{u^2}{r^{2}}
\ dv_{\mathbb{H}^{N}}+ \frac{1}{2^{4m+2}} \prod_{j = 1}^{ m} (N+ 4j - 2)^2(N- 4j - 2)^2 \int_{\mathbb{H}^{N}} \frac{u^2}{r^{4m+2}} \ dv_{\mathbb{H}^{N}}$$ for all $u \in C^{\infty}_{0}(\mathbb{H}^{N})$.
[ **[(Case $k-1= l < k$)]{}**]{} Let $k$ be a positive integer and let $ N> 2k$.
If $k=2m$ for some positive integer $m$, there holds $$\int_{\mathbb{H}^{N}} (\Delta_{{\mathbb{H}^{N}}}^{m} u)^2 \ dv_{\mathbb{H}^{N}} - \left( \frac{N-1}{2} \right)^{2} \int_{\mathbb{H}^{N}} |\nabla_{{\mathbb{H}^{N}}} (\Delta_{{\mathbb{H}^{N}}}^{m-1} u)|^{2} \ dv_{\mathbb{H}^{N}} \geq$$ $$\frac{(N-1)^{2m}}{2^{4m}} \int_{\mathbb{H}^{N}} \frac{u^2}{r^{2}} \ dv_{\mathbb{H}^{N}} + \frac{9}{2^{4m}} \prod_{j = 1}^{ m-1} ((N+ 4j )(N- 4j -4))^2 \int_{\mathbb{H}^{N}} \frac{u^2}{r^{4m}} \ dv_{\mathbb{H}^{N}}$$ for all $u \in C^{\infty}_{0}(\mathbb{H}^{N})$, where we use the convention $\prod_{j = 1}^{ 0}=1$.
If $k=2m+1$ for some positive integer $m$, there holds $$\int_{\mathbb{H}^{N}} |\nabla_{{\mathbb{H}^{N}}} (\Delta_{{\mathbb{H}^{N}}}^{m} u)|^{2} \ dv_{\mathbb{H}^{N}} - \left( \frac{N-1}{2} \right)^{2} \int_{\mathbb{H}^{N}} (\Delta_{{\mathbb{H}^{N}}}^{m} u)^{2} \ dv_{\mathbb{H}^{N}} \geq$$ $$\frac{(N-1)^{2m}}{2^{4m+2}} \int_{\mathbb{H}^{N}} \frac{u^2}{r^{2}} \ dv_{\mathbb{H}^{N}} + \frac{1}{2^{4m+2}} \prod_{j = 1}^{m}(N+ 4j-2)^2(N - 4j- 2)^2 \int_{\mathbb{H}^{N}} \frac{u^2}{r^{4m+2}} \ dv_{\mathbb{H}^{N}}$$ for all $u \in C^{\infty}_{0}(\mathbb{H}^{N})$.
The article is organized as follows. Section \[21\] is devoted to the precise statement and discussion of results for the case $k = 2$ and $ l = 1.$ The complete proof of the results discussed in Section \[21\] is postponed to Section \[proof21\]. Section \[l0\] and Section \[karbitrary\] are devoted to discussions and proofs of the results for $0= l < k$ and for $0\neq l < k$. The statements of the results given in these sections will contain the precise constants for the leading terms mentioned in the statement of Theorem \[main\_intro\].
Case $k=2$ and $l=1$ {#21}
====================
We start by restating Theorem \[main\_intro0\] in its complete form. The proof of the results given in this section will be postponed to Section \[proof21\].
\[PRHinequality\] Let $ N >4$. For all $ u \in C^{\infty}_{0}(\mathbb{H}^{N})$ there holds
$$\begin{aligned}
\label{npoincare}
\int_{\mathbb{H}^{N}} (\Delta_{{\mathbb{H}^{N}}} u)^{2} \ dv_{{\mathbb{H}^{N}}} - \left( \frac{N-1}{2} \right)^{2} \int_{\mathbb{H}^{N}} |\nabla_{{\mathbb{H}^{N}}} u|^2 \ dv_{{\mathbb{H}^{N}}}
&\geq \frac{(N-1)^2}{16} \int_{\mathbb{H}^{N}} \frac{u^2}{r^2} \ dv_{{\mathbb{H}^{N}}}
+ \frac{9}{16} \int_{\mathbb{H}^{N}} \frac{u^2}{r^4} \ dv_{{\mathbb{H}^{N}}} \notag \\
& +\frac{(N-1)(N-3)(N^2 -2N - 7)}{16} \int_{{\mathbb{H}^{N}}} \frac{u^2}{\sinh^2 r} \ dv_{{\mathbb{H}^{N}}} \notag \\ &+\frac{(N-1)(N-3)(N^2-4N-3)}{16} \int_{{\mathbb{H}^{N}}} \frac{u^2}{\sinh^4 r} \ dv_{{\mathbb{H}^{N}}}\,.\end{aligned}$$
The constant $\left( \frac{N-1}{2} \right)^{2}$ is sharp by construction, namely cannot be replaced by a larger one. Furthermore, the constant $\frac{(N-1)^2}{16}$ is sharp in the sense that no inequality of the form $$\int_{{\mathbb{H}^{N}}} (\Delta_{{\mathbb{H}^{N}}} u)^{2} \ dv_{{\mathbb{H}^{N}}} - \left( \frac{N-1}{2} \right)^{2} \int_{\mathbb{H}^{N}} |\nabla_{{\mathbb{H}^{N}}} u|^2 \ dv_{{\mathbb{H}^{N}}}
\geq c\, \int_{{\mathbb{H}^{N}}} \frac{u^2}{r^2} \ dv_{{\mathbb{H}^{N}}}$$ holds for all $ u \in C^{\infty}_{c}(\mathbb{H}^{N})$ when $c>\frac{(N-1)^2}{16}$.
Inequality does not follow directly from and but requires an independent proof which is achieved by means of a suitable modification of the proof of as given in [@BGG]. As already remarked in the Introduction, the main tools exploited are a suitable transformation which uncovers the Poincaré term and spherical harmonic analysis. Recently, spherical harmonics technique has been successfully exploited in the context of Weighted Calderón–-Zygmund and Rellich inequalities [@MSS].
As already explained in the introduction, the leading term of inequality is the one in front of $1/r^2$ for functions supported outside a large ball. Hence, it is particularly important to determine the sharp constant in front of such a term to highlight the non-Euclidean behavior of the inequality. Nevertheless, as happens for inequality , the problem of finding the best constant in front of the term $1/r^4$ is still open. See also [@BGG Remark 6.1].
It’s worth noting that, as happens for inequality , the constants appearing in front of the terms $1/r^4$ and $1/\sinh^4 r$ are jointly sharp. In the sense that the inequality $$\begin{aligned}
\int_{\mathbb{H}^{N}} (\Delta_{{\mathbb{H}^{N}}} u)^{2}dv_{{\mathbb{H}^{N}}} - \left( \frac{N-1}{2} \right)^{2} \int_{\mathbb{H}^{N}} |\nabla_{{\mathbb{H}^{N}}} u|^2 \ dv_{{\mathbb{H}^{N}}} \ge a \int_{{\mathbb{H}^{N}}} \frac{u^2}{r^4} \ dv_{{\mathbb{H}^{N}}}
+b \int_{{\mathbb{H}^{N}}} \frac{u^2}{\sinh^4 r} \ dv_{{\mathbb{H}^{N}}}
\end{aligned}$$ cannot hold for all $ u \in C^{\infty}_{c}(\mathbb{H}^{N})$, or even for for all $ u \in C^{\infty}_{c}(B_\varepsilon)$ given any $\varepsilon>0$, if $$\begin{aligned}
&a=\frac9{16},\ \ \ b>\frac{(N-1)(N-3)(N^2-4N-3)}{16}\ \ \ \textrm{or}\\
&a>\frac9{16},\ \ \ b=\frac{(N-1)(N-3)(N^2-4N-3)}{16}.
\end{aligned}$$ This follows by noting that $\sinh r\sim r$ as $r\to0$ and that yields $$\begin{aligned}
\int_{\mathbb{H}^{N}} (\Delta_{{\mathbb{H}^{N}}} u)^{2} \ dv_{{\mathbb{H}^{N}}} \geq
\frac{9}{16} \int_{\mathbb{H}^{N}} \frac{u^2}{r^4} \ dv_{{\mathbb{H}^{N}}} +\frac{(N-1)(N-3)(N^2-4N-3)}{16} \int_{{\mathbb{H}^{N}}} \frac{u^2}{\sinh^4 r} \ dv_{{\mathbb{H}^{N}}}
\end{aligned}$$ where $$\frac9{16}+\frac{(N-1)(N-3)(N^2-4N-3)}{16}=\frac{N^2(N-4)^2}{16}\,$$ and $\frac{N^2(N-4)^2}{16}$ is the best constant (namely, the larger) for the standard $N$ dimensional Euclidean Rellich inequality, both on the whole ${\mathbb R}^N$ or in any open set containing the origin.
Consider the upper half space model for ${\mathbb{H}^{N}}$, namely $\mathbb{R}^{N}_{+} = \{ (x, y) \in \mathbb{R}^{N-1} \times \mathbb{R}^{+} \} $ endowed with the Riemannian metric $\frac{\delta_{ij}}{y^2}$. We set $$\label{dd}
d:=d((x,y),(0,1) ): = \cosh^{-1} \left( 1 + \frac{(|y| - 1)^2 + |x|^2}{2 |y|} \right)\,.$$ It is readily seen that $d \sim \log(1/y)$ as $y \rightarrow 0$. By exploiting the transformation $$v(x,y) : = y^{\alpha} u(x,y), \quad x \in \mathbb{R}^{N-1}, y \in \mathbb{R}^{+} ,$$ with $\alpha=-\frac{N-2}{2}$ or $\alpha=-\frac{N-4}{2}$, from we derive the following statements
\[cor2\] Let $N >4$ and $d$ as defined in . For all $v \in C_c^{\infty}(\mathbb{R}^{N}_{+})$ the following inequalities hold
$$\begin{aligned}
\label{HALFrellich1}
& \int_{\mathbb{R}^{+}} \int_{\mathbb{R}^{N-1}} \left( y^2 (\Delta v)^2 + \frac{(N^2 - 2N -1 )}{4} |\nabla v|^2 \right) \ dx \ dy
\geq \frac{N(N-2)}{16} \int_{\mathbb{R}^{+}} \int_{\mathbb{R}^{N-1}} \frac{v^2}{y^2} \ dx \ dy \notag \\
& + \frac{(N-1)^2}{16} \int_{\mathbb{R}^{+}} \int_{\mathbb{R}^{N-1}}
\frac{v^2}{y^2 d^2} \ dx \ dy + \frac{9}{16} \int_{\mathbb{R}^{+}} \int_{\mathbb{R}^{N-1}} \frac{v^2}{y^2 d^4} \ dx \ dy\end{aligned}$$
*and*
$$\begin{aligned}
\label{HALFrellich2}
& \int_{\mathbb{R}^{+}} \int_{\mathbb{R}^{N-1}} \left( (\Delta v)^2 + \frac{(N^2 - 2N -9 )}{4} \frac{|\nabla v|^2}{y^2} \right) \ dx \ dy
\geq \frac{9}{16} (N+2)(N - 4) \int_{\mathbb{R}^{+}} \int_{\mathbb{R}^{N-1}} \frac{ v^2}{y^4} \ dx \ dy \notag \\
& + \frac{(N-1)^2}{16} \int_{\mathbb{R}^{+}} \int_{\mathbb{R}^{N-1}}
\frac{ v^2}{y^4 d^2} \ dx \ dy + \frac{9}{16} \int_{\mathbb{R}^{+}} \int_{\mathbb{R}^{N-1}} \frac{ v^2}{y^4 d^4} \ dx \ dy.\end{aligned}$$
Furthermore, we have:
$\bullet$ no inequality of the form $$\int_{\mathbb{R}^{+}} \int_{\mathbb{R}^{N-1}} \left( y^2 (\Delta v)^2 +
c |\nabla v|^2 \right) \ dx \ dy
\geq \frac{N(N-2)}{16} \int_{\mathbb{R}^{+}} \int_{\mathbb{R}^{N-1}} \frac{v^2}{y^2} \ dx \ dy$$ holds for all $ v \in C^{\infty}_{c}(\mathbb{H}^{N})$ when $c< \frac{( N^2 - 2N -1)}{4}$;
$\bullet$ no inequality of the form $$\int_{\mathbb{R}^{+}} \int_{\mathbb{R}^{N-1}} \left( y^2 (\Delta v)^2 + \frac{(N^2 - 2N -1)}{4} |\nabla v|^2 \right) \ dx \ dy
\geq c \int_{\mathbb{R}^{+}} \int_{\mathbb{R}^{N-1}} \frac{v^2}{y^2} \ dx \ dy$$ holds for all $ v \in C^{\infty}_{c}(\mathbb{H}^{N})$ when $c> \frac{N(N-2)}{16} $;
$\bullet$ no inequality of the form $$\begin{aligned}
& \int_{\mathbb{R}^{+}} \int_{\mathbb{R}^{N-1}} \left( y^2 (\Delta v)^2 + \frac{(N^2 - 2N -1)}{4} |\nabla v|^2 \right) \ dx \ dy
\geq \frac{N(N-2)}{16} \int_{\mathbb{R}^{+}} \int_{\mathbb{R}^{N-1}} \frac{v^2}{y^2} \ dx \ dy \notag \\
& + c \int_{\mathbb{R}^{+}} \int_{\mathbb{R}^{N-1}}
\frac{v^2}{y^2 d^2} \ dx \ dy\end{aligned}$$ holds for all $ v \in C^{\infty}_{c}(\mathbb{H}^{N})$ when $c> \frac{(N-1)^2}{16} $.
Similar conclusions hold for the constants $ \frac{(N^2 - 2N - 9)}{2},$ $\frac{9}{16}(N+2)(N-4)$ and $\frac{(N-1)^2}{16}$ in .
Case $k$ arbitrary and $l=0$ {#l0}
============================
In this section we restate and prove Theorem \[main\_intro\] for $0=l<k$.
\[mainhigher0\] Let $k$ be a positive integer and $ N> 2k$. There exist $k$ *positive* constants $c_{k}^i=c_{k}^i(N)$ such that the following inequality holds $$\label{global}
\int_{\mathbb{H}^{N}} |\nabla_{\mathbb{H}^{N}}^{k} u|^{2} \ dv_{\mathbb{H}^{N}} - \left( \frac{N-1}{2} \right)^{2k}
\int_{\mathbb{H}^{N}} u^2 \ dv_{\mathbb{H}^{N}}\ge \sum_{i= 1}^{k} c_{k}^i\int_{\mathbb{H}^{N}} \frac{u^2}{r^{2i}} \ dv_{\mathbb{H}^{N}}\,,$$ for all $u \in C^{\infty}_{0}(\mathbb{H}^{N})$. Furthermore, the leading terms a $r\rightarrow 0$ and $r \rightarrow\infty$ are explicitly given by $$c_{k}^1=d_k:=
\left \{\begin{array}{ll}
\sum_{j = 1}^{ m} \frac{(N-1)^{4m-2j}}{2^{4m-1}}& \text{if $k=2m$}\,,\\
\sum_{j = 1}^{ m} \frac{(N-1)^{4m-2j+2}}{2^{4m+1}}+ \frac{(N-1)^{2m}}{2^{4m+2}} & \text{if $k=2m+1$\,,}
\end{array}\right.$$ $$c_{k}^k=e_k:=
\left \{\begin{array}{ll}
\frac{9}{2^{4m}} \prod_{j = 1}^{ m -1} (N+ 4j)^2(N-4j-4)^2 & \text{if $k=2m$}\,,\\
\frac{1}{2^{4m+2}} \prod_{j = 1}^{ m} (N+ 4j - 2)^2(N- 4j - 2)^2 & \text{if $k=2m+1$\,,}
\end{array}\right.$$ where we use the conventions: $\sum_{j = 1}^{ 0}=0$ and $\prod_{j = 1}^{ 0}=1$.
Here and after, for shortness we will write $\Delta_{\mathbb{H}^{N}}=\Delta$. In the proof we will repeatedly exploit the following inequality from [@Yang Theorem 4.4]: $$\begin{aligned}
\label{yangsukong}
\int_{\mathbb{H}^{N}} \frac{(\Delta u)^2}{r^{\beta}} \ dv_{\mathbb{H}^{N}}\geq &\frac{(N+\beta)^2(N-\beta-4)^2}{16} \int_{\mathbb{H}^{N}} \frac{u^2}{r^{4+\beta}} \ dv_{\mathbb{H}^{N}} \notag \\
& +\frac{(N-2-\beta)(N-2+\beta)(N-1)}{8} \int_{\mathbb{H}^{N}} \frac{u^2}{r^{2+\beta}} \ dv_{\mathbb{H}^{N}}+\frac{(N-1)^2}{16} \int_{\mathbb{H}^{N}} \frac{u^2}{r^{\beta}} \ dv_{\mathbb{H}^{N}} , \end{aligned}$$ for all $u \in C^{\infty}_{0}(\mathbb{H}^{N})$ and $0\leq \beta<N-4$.
We prove separately the case $k$ even and $k$ odd. First we assume $k=2m$ even. If $m = 1$, follows directly from . When $m = 2$, by and with $N>8$, we have $$\begin{aligned}
\int_{\mathbb{H}^{N}} (\Delta (\Delta u))^2 \ dv_{\mathbb{H}^{N}} & \geq \frac{(N-1)^4}{16} \int_{\mathbb{H}^{N}} (\Delta u)^2 \ dv_{\mathbb{H}^{N}}
+ \frac{(N-1)^2}{8} \int_{\mathbb{H}^{N}} \frac{(\Delta u)^2}{r^2} \ dv_{\mathbb{H}^{N}} + \frac{9}{16} \int_{\mathbb{H}^{N}} \frac{(\Delta u)^2}{r^4} \ dv_{\mathbb{H}^{N}} \notag\\
& \geq \frac{(N-1)^4}{16} \left[ \frac{(N-1)^4}{16} \int_{\mathbb{H}^{N}} u^2 \ dv_{\mathbb{H}^{N}} + \frac{(N-1)^2}{8} \int_{\mathbb{H}^{N}} \frac{u^2}{r^2} \ dv_{\mathbb{H}^{N}}
+ \frac{9}{16} \int_{\mathbb{H}^{N}} \frac{u^2}{r^4} \ dv_{\mathbb{H}^{N}} \right] \notag \\
& + \frac{(N-1)^2}{8} \left[ \frac{(N + 2)^2 (N - 6)^2}{16} \int_{\mathbb{H}^{N}} \frac{ u^2}{r^6} \ dv_{\mathbb{H}^{N}}
+ \frac{N(N-1)(N-4)}{8} \int_{\mathbb{H}^{N}} \frac{u^2}{r^4} \ dv_{\mathbb{H}^{N}} \right. \notag \\
& \left. + \frac{(N-1)^2}{16} \int_{\mathbb{H}^{N}} \frac{ u^2}{r^2} \ dv_{\mathbb{H}^{N}} \right] + \frac{9}{16} \left[ \frac{(N + 4)^2(N - 8)^2}{16} \int_{\mathbb{H}^{N}}
\frac{u^2}{r^8} \ dv_{\mathbb{H}^{N}} \right. \notag \\
& \left. + \frac{(N + 2)(N - 1) (N - 6)}{8} \int_{\mathbb{H}^{N}} \frac{u^2}{r^6} \ dv_{\mathbb{H}^{N}} + \frac{(N - 1)^2}{16} \int_{\mathbb{H}^{N}}
\frac{u^2}{r^4} \ dv_{\mathbb{H}^{N}} \right] \notag \\
& = \left( \frac{N-1}{2} \right)^8 \int_{\mathbb{H}^{N}} u^2 \ dv_{\mathbb{H}^{N}} + \sum_{i = 1}^{4} c_{4}^i \int_{\mathbb{H}^{N}} \frac{u^2}{r^{2i}},\end{aligned}$$ where $c_{4}^1 = \frac{(N - 1)^6}{2^7} + \frac{(N - 1)^4}{2^7},$ $c_{4}^4 = \frac{9}{2^8} {(N + 4)^2 (N - 8)^2}$. Hence, is proved for $k= 4.$\
Next we assume holds for $k=2m$ with $m > 2,$ namely $$\begin{aligned}
\label{higher1}
\int_{\mathbb{H}^{N}} (\Delta^{m} u)^2 \ dv_{\mathbb{H}^{N}} - \left( \frac{N-1}{2} \right)^{4m} \int_{\mathbb{H}^{N}} u^{2} \ dv_{\mathbb{H}^{N}} & \geq \sum_{j = 1}^{ m} \frac{(N-1)^{4m-2j}}{2^{4m-1}} \int_{\mathbb{H}^{N}} \frac{u^2}{r^{2}} \ dv_{\mathbb{H}^{N}}+\sum_{i = 2}^{2m - 1} c_{2m}^i \int_{\mathbb{H}^{N}} \frac{u^2}{r^{2i}} \ dv_{\mathbb{H}^{N}} \notag \\
&+\frac{9}{2^{4m}} \prod_{j = 1}^{ m -1} (N+ 4j)^2(N-4(j+1))^2 \int_{\mathbb{H}^{N}} \frac{u^2}{r^{4m}} \ dv_{\mathbb{H}^{N}}, \end{aligned}$$ where, for $2\leq i\leq 2m-1$, the $c_{2m}^i$ are suitable positive constants and $N>4m$. Inequality yields $$\begin{aligned}
\int_{\mathbb{H}^{N}} (\Delta^{m+1} u)^2 = \int_{\mathbb{H}^{N}} (\Delta^{m} (\Delta u))^2\geq
\left( \frac{N-1}{2} \right)^{4m} \int_{\mathbb{H}^{N}} (\Delta u)^2 \ dv_{\mathbb{H}^{N}} +\sum_{j = 1}^{ m} \frac{(N-1)^{4m-2j}}{2^{4m-1}} \int_{\mathbb{H}^{N}} \frac{(\Delta u)^2}{r^{2}} \ dv_{\mathbb{H}^{N}}\notag \\
+ \sum_{ i = 2}^{2m - 1} c_{2m}^i \int_{\mathbb{H}^{N}} \frac{(\Delta u)^2}{r^{2i}} \ dv_{\mathbb{H}^{N}}+ \frac{9}{2^{4m}} \prod_{i = 1}^{m -1} (N + 4j)^2(N-4(j + 1))^2
\int_{\mathbb{H}^{N}} \frac{(\Delta u)^2}{r^{4m}} \ dv_{\mathbb{H}^{N}} \,.\end{aligned}$$ Next, by , for $N > 4m + 4$ we have $$\begin{aligned}
\int_{\mathbb{H}^{N}} \frac{(\Delta u)^2}{r^{2}} \ dv_{\mathbb{H}^{N}} &\geq\frac{(N+2)^2(N-6)^2}{16} \int_{\mathbb{H}^{N}} \frac{u^2}{r^{6}} \ dv_{\mathbb{H}^{N}} \notag \\ &+\frac{(N-4)(N)(N-1)}{8} \int_{\mathbb{H}^{N}} \frac{u^2}{r^{4}} \ dv_{\mathbb{H}^{N}}+\frac{(N-1)^2}{16} \int_{\mathbb{H}^{N}} \frac{u^2}{r^{2}} \ dv_{\mathbb{H}^{N}}\,,\end{aligned}$$ $$\begin{aligned}
\sum_{ i = 2}^{2m - 1} c_{2m}^i \int_{\mathbb{H}^{N}} \frac{(\Delta u)^2}{r^{2i}} \ dv_{\mathbb{H}^{N}}\geq \sum_{ i = 2}^{2m+1} \bar c_{2m}^i \int_{\mathbb{H}^{N}} \frac{u^2}{r^{2i}} \ dv_{\mathbb{H}^{N}}\,,\end{aligned}$$ $$\begin{aligned}
\int_{\mathbb{H}^{N}} \frac{(\Delta u)^2}{r^{4m}} \ dv_{\mathbb{H}^{N}}&\geq \frac{(N+4m)^2(N-4m-4)^2}{16} \int_{\mathbb{H}^{N}} \frac{u^2}{r^{4+4m}} \ dv_{\mathbb{H}^{N}}\notag \\ &+\frac{(N-2-4m)(N-2+4m)(N-1)}{8} \int_{\mathbb{H}^{N}} \frac{u^2}{r^{2+4m}} \ dv_{\mathbb{H}^{N}}+\frac{(N-1)^2}{16} \int_{\mathbb{H}^{N}} \frac{u^2}{r^{4m}} \ dv_{\mathbb{H}^{N}}\,,\end{aligned}$$ where, for $2\leq i\leq 2m+1$, $\bar c_{2m}^i$ are suitable positive constants. The above inequalities and , finally yield $$\begin{aligned}
\int_{\mathbb{H}^{N}} (\Delta^{m+1} u)^2 - \left( \frac{N-1}{2} \right)^{4(m + 1)} \int_{\mathbb{H}^{N}} u^2 \ dv_{\mathbb{H}^{N}} \geq
\left(\frac{(N-1)^{4m+2}}{2^{4m+3}}+\sum_{j = 1}^{ m} \frac{(N-1)^{4m-2j+2}}{2^{4m+3}} \right)\int_{\mathbb{H}^{N}} \frac{u^2}{r^{2}} \ dv_{\mathbb{H}^{N}}
\notag\\ + \sum_{ i = 2}^{2m+1} \hat c_{2m}^i \int_{\mathbb{H}^{N}} \frac{u^2}{r^{2i}} \ dv_{\mathbb{H}^{N}}+
\notag\\
+ \frac{9}{2^{4m}} \prod_{j = 1}^{m-1} (N + 4j)^2(N-4(j + 1))^2
\frac{(N+4m)^2(N-4m-4)^2}{16} \int_{\mathbb{H}^{N}} \frac{u^2}{r^{4+4m}} \ dv_{\mathbb{H}^{N}}\\
=\sum_{j = 1}^{ m+1} \frac{(N-1)^{4(m+1)-2j}}{2^{4(m+1)-1}} \int_{\mathbb{H}^{N}} \frac{u^2}{r^{2}} \ dv_{\mathbb{H}^{N}}+ \sum_{ l = 2}^{2(m+1)-1} \hat c_{2m}^i \int_{\mathbb{H}^{N}} \frac{u^2}{r^{2i}} \ dv_{\mathbb{H}^{N}}
\notag\\
+ \frac{9}{2^{4m+4}} \prod_{j = 1}^{m } (N + 4j)^2(N-4(j + 1))^2 \int_{\mathbb{H}^{N}} \frac{u^2}{r^{4+4m}} \ dv_{\mathbb{H}^{N}}\end{aligned}$$ where, for $2\leq i\leq 2m+1$, $\hat c_{2m}^i$ are suitable positive constants. By induction, this completes the proof of for $k$ even.\
Next we turn to the case $k=2m+1$ odd. For $m = 1$ and $N > 6$, by , and , we deduce $$\begin{aligned}
\int_{\mathbb{H}^{N}} |\nabla (\Delta u)|^2 \ dv_{\mathbb{H}^{N}} & \geq \left( \frac{N-1}{2} \right)^2 \int_{\mathbb{H}^{N}} (\Delta u)^2 \ dv_{\mathbb{H}^{N}}
+ \frac{1}{4} \int_{\mathbb{H}^{N}} \frac{(\Delta u)^2}{r^2} \ dv_{\mathbb{H}^{N}} \notag \\
& \geq \left( \frac{N-1}{2} \right)^2 \left[ \left( \frac{N-1}{2} \right)^2\int_{\mathbb{H}^{N}} u^2 \ dv_{\mathbb{H}^{N}} + \frac{(N-1)^2}{8} \int_{\mathbb{H}^{N}} \frac{u^2}{r^2} \ dv_{\mathbb{H}^{N}}
+ \frac{9}{16} \int_{\mathbb{H}^{N}} \frac{u^2}{r^4} \ dv_{\mathbb{H}^{N}} \right] \notag \\
& + \frac{1}{4} \left[ \frac{(N + 2)^2 (N-6)^2}{16} \int_{\mathbb{H}^{N}} \frac{u^2}{r^6} \ dv_{\mathbb{H}^{N}} + \frac{N(N-1)(N-4)}{8} \int_{\mathbb{H}^{N}} \frac{u^2}{r^4} \ dv_{\mathbb{H}^{N}} \right. \notag \\
& \left. + \frac{(N-1)^2}{16} \int_{\mathbb{H}^{N}} \frac{u^2}{r^2} \ dv_{\mathbb{H}^{N}} \right]. \end{aligned}$$ Hence we obtain, $$\int_{\mathbb{H}^{N}} |\nabla (\Delta u)|^2 \ dv_{\mathbb{H}^{N}} - \left( \frac{N-1}{2} \right)^6 \int_{\mathbb{H}^{N}} u^2 \ dv_{\mathbb{H}^{N}} \geq \sum_{l = 1}^{3} c_{3}^i \int_{\mathbb{H}^{N}} \frac{u^2}{r^{2i}},$$ where $c_{3}^1 = \left( \frac{(N-1)^4}{2^5} + \frac{(N-1)^2}{2^6} \right)$ and $c_{3}^3=\frac{1}{4^3} (N+2)^2(N-6)^2$ hence, with $m=1$ is verified. For general $ m + 1$ the proof follows very similar to the case $k$ even, we skip the details for brevity. This completes the proof.
Case $k>l>0$ arbitrary {#karbitrary}
======================
In this section we restate and prove Theorem \[main\_intro\] for $l>0$, the case $l=0$ has already been dealt with in Section \[l0\].
\[mainmain\] Let $k>l$ be positive integers and $ N> 2k$. There exist $k$ *positive* constants $\alpha_{k,l}^i=\alpha_{k,l}^i(N)$ such that the following inequality holds $$\int_{\mathbb{H}^{N}} |\nabla_{\mathbb{H}^{N}}^{k} u|^{2} \ dv_{\mathbb{H}^{N}} -\left( \frac{N-1}{2} \right)^{2(k - l)}
\int_{\mathbb{H}^{N}} |\nabla_{\mathbb{H}^{N}}^{l} u|^2 \ dv_{\mathbb{H}^{N}}\ge \sum_{i = 1}^{k} \alpha_{k,l}^i \int_{\mathbb{H}^{N}} \frac{u^2}{r^{2i}} \ dv_{\mathbb{H}^{N}}\,,$$ for all $u \in C^{\infty}_{0}(\mathbb{H}^{N})$. Furthermore, the leading terms a $r\rightarrow 0$ and $r \rightarrow\infty$, namely $\alpha_{k,l}^1 $ and $\alpha_{k,l}^k$ are explicitly given as follows $$\alpha_{k,l}^1:=
\left \{\begin{array}{lllll}
d_{2(m-h)} \,a_{h} & \text{if $k=2m$ and $l=2h$}\,,\\
2^{4(m-h)-2}\,a_{2(m-h)}\,a_{h}+d_{2(m-h-1)} \,a_{h+1} & \text{if $k=2m$ and $l=2h+1$, $h\neq m-1$}\,,\\
a_1\,a_{m-1} & \text{if $k=2m$ and $l=2m-1$}\,,\\
4a_1\, a_{h} d_{2(m-h)} + \frac{1}{4} a_{m} & \text{if $k=2m+1$ and $l=2h$}\,,\\
\frac{1}{4} a_{m} + 4^{2(m-h)}\, a_{2(m-h)}\,a_{h}\,a_1+4\,d_{2(m - h -1)} a_{h + 1} a_1 & \text{if $k=2m+1$ and $l=2h+1$, $h\neq m-1$}\,,\\
\frac{1}{4} a_{m} + 4\,a_2\, a_{m - 1} & \text{if $k=2m+1$ and $l=2m-1$}\,,
\end{array}\right.$$
$$\alpha_{k,l}^k:=
\left \{\begin{array}{lllll}
e_{2(m-h)}\, b_{h,4(m-h)}& \text{if $k=2m$ and $l=2h$}\,,\\
e_{2(m-h-1)}\, b_{h+1,4(m-h-1)} & \text{if $k=2m$ and $l=2h+1$, $h\neq m-1$}\,,\\
\frac{9}{16}\, b_{m-1,4} & \text{if $k=2m$ and $l=2m-1$}\,,\\
\frac{1}{4} b_{m,2} & \text{if $k=2m+1$ and $l<k$}\,,
\end{array}\right.$$
where $a_0=1$, $d_0=0$ and, for any $\gamma$ and $\beta$ positive integers, $a_{\gamma}=\frac{(N-1)^{2\gamma}}{2^{4\gamma}}$, $b_{\gamma,\beta} = \prod_{j = 0}^{\gamma-1}\frac{(N+(\beta+4j))^2(N-(\beta+4j)-4)^2}{16}$, $d_{\gamma}$ and $e_{\gamma}$ are the constants defined in Theorem \[mainhigher0\].
The proof is achieved by considering separately four cases. In each proof we will exploit the following technical lemma whose proof can be obtained by induction, iterating . Notice that, except for the main statements, for shortness we will always write $\Delta_{\mathbb{H}^{N}}=\Delta$.
Let $\gamma$ be a positive integer. For all $u \in C^{\infty}_{0}(\mathbb{H}^{N})$ and $0\leq \beta<N-4\gamma$, the following inequality holds $$\begin{aligned}
\label{yangxtended}
\int_{\mathbb{H}^{N}} \frac{(\Delta^{\gamma} u)^2}{r^{\beta}} \ dv_{\mathbb{H}^{N}}\geq a_{\gamma}\int_{\mathbb{H}^{N}} \frac{u^2}{r^{\beta}} \ dv_{\mathbb{H}^{N}} +\sum_{ j= 1}^{2\gamma-1} a_{\gamma,\beta}^j \int_{\mathbb{H}^{N}} \frac{u^2}{r^{2j+\beta}} \ dv_{\mathbb{H}^{N}}+b_{\gamma,\beta} \int_{\mathbb{H}^{N}} \frac{u^2}{r^{4\gamma+\beta}} \ dv_{\mathbb{H}^{N}} \,, \end{aligned}$$ where $a_{\gamma}=\frac{(N-1)^{2\gamma}}{2^{4\gamma}}$, $a_{m,\beta}^j$ are suitable positive constants and $b_{\gamma,\beta} = \prod_{j = 0}^{\gamma-1}\frac{(N+(\beta+4j))^2(N-(\beta+4j)-4)^2}{16}$.
Case $k=2m$ even and $l=2h$ even
--------------------------------
\[mainhigheree\] Let $m,h$ be integers such that $0<h< m$ and $ N> 4m$. There exist $2m$ *positive* constants $\alpha^i=\alpha^i(N,m,h)$ such that the following inequality holds $$\begin{aligned}
\label{highereven1}
\int_{\mathbb{H}^{N}} (\Delta_{{\mathbb{H}^{N}}}^{m} u)^2 \ dv_{\mathbb{H}^{N}} - \left( \frac{N-1}{2} \right)^{4(m-h)} \int_{\mathbb{H}^{N}} (\Delta_{{\mathbb{H}^{N}}}^{h} u)^{2} \ dv_{\mathbb{H}^{N}} \geq
\sum_{i = 1}^{2m} \alpha^{i} \int_{\mathbb{H}^{N}} \frac{u^2}{r^{2i}} \ dv_{\mathbb{H}^{N}},\end{aligned}$$ for all $u \in C^{\infty}_{0}(\mathbb{H}^{N})$. Furthermore, the leading terms a $r\rightarrow 0$ and $r \rightarrow\infty$ are explicitly given by $$\alpha^1:=d_{2(m-h)}\,a_h \quad \text{and}\quad \alpha^{2m}:=e_{2(m-h)}\, b_{h,4(m-h)} \,,$$ where, for any $\gamma$ and $\beta$ positive integers, $a_{\gamma}=\frac{(N-1)^{2\gamma}}{2^{4\gamma}}$, $b_{\gamma,\beta} = \prod_{j = 0}^{\gamma-1}\frac{(N+(\beta+4j))^2(N-(\beta+4j)-4)^2}{16}$, $d_{\gamma}$ and $e_{\gamma}$ are the constants defined in Theorem \[mainhigher0\].
By applying with $k=2(m-h)$ and with $\gamma=h$ and $\beta=2i$ we deduce $$\int_{\mathbb{H}^{N}} (\Delta^{m} u)^2 \ dv_{\mathbb{H}^{N}} =\int_{\mathbb{H}^{N}} (\Delta^{m-h}(\Delta^h u))^2 \ dv_{\mathbb{H}^{N}}$$ $$\geq \left( \frac{N-1}{2} \right)^{4(m-h)}
\int_{\mathbb{H}^{N}} (\Delta^h u)^2 \ dv_{\mathbb{H}^{N}}+ \sum_{i= 1}^{2(m-h)} c_{2(m-h)}^i\int_{\mathbb{H}^{N}} \frac{(\Delta^h u)^2}{r^{2i}} \ dv_{\mathbb{H}^{N}}\geq \left( \frac{N-1}{2} \right)^{4(m-h)}
\int_{\mathbb{H}^{N}} (\Delta^h u)^2 \ dv_{\mathbb{H}^{N}}$$ $$+ \sum_{i= 1}^{2(m-h)} c_{2(m-h)}^i \left(a_{h}\int_{\mathbb{H}^{N}} \frac{u^2}{r^{2i}} \ dv_{\mathbb{H}^{N}} +\sum_{ j= 1}^{2h-1} a_{h,2i}^j \int_{\mathbb{H}^{N}} \frac{u^2}{r^{2j+2i}} \ dv_{\mathbb{H}^{N}}+b_{h,2i} \int_{\mathbb{H}^{N}} \frac{u^2}{r^{4h+2i}} \ dv_{\mathbb{H}^{N}} \right)\,,$$ where all the constants are positive. Settled $g(j,i):=2j+2i$, for $0\leq j\leq 2h$ and $1\leq i\leq 2(m-h)$ it is readily verified that $g$ has a unique global minimum $g(0,1)=1$ and a unique global maximum $g(2h,2(m-h))=4m$. Furthermore, by the fact that $g(j,1)$ goes monotonically from $2$ to $4h+2$ and $g(2h,i)$ goes monotonically from $4h+2$ to $4m$, we deduce the existence of $2m$ positive constants $\alpha^i=\alpha^i(N,m,h)$ such that holds. Moreover, $$\alpha^1=d_{2(m-h)} \,a_{h} \quad \text{and} \quad \alpha^k:=e_{2(m-h)}\, b_{h,4(m-h)}\,.$$
Case $k=2m$ even and $l=2h+1$ odd
---------------------------------
\[mainhighereo\] Let $m,h$ be integers such that $0\leq h< m$ and $ N> 4m$. There exist $2m$ *positive* constants $\bar\alpha^i=\bar \alpha^i(N,m,h)$ such that the following inequality holds $$\begin{aligned}
\label{highereven2}
\int_{\mathbb{H}^{N}} (\Delta_{{\mathbb{H}^{N}}}^{m} u)^2 \ dv_{\mathbb{H}^{N}} - \left( \frac{N-1}{2} \right)^{4(m-h)-2} \int_{\mathbb{H}^{N}} |\nabla_{{\mathbb{H}^{N}}} \Delta_{{\mathbb{H}^{N}}}^{h} u|^{2} \ dv_{\mathbb{H}^{N}} \geq
\sum_{i = 1}^{2m}\bar \alpha^{i} \int_{\mathbb{H}^{N}} \frac{u^2}{r^{2i}} \ dv_{\mathbb{H}^{N}}\,,\end{aligned}$$ for all $u \in C^{\infty}_{0}(\mathbb{H}^{N})$. Furthermore, the leading terms a $r\rightarrow 0$ and $r \rightarrow\infty$ are explicitly given by:
if $0\leq h<m-1$ $$\bar \alpha^1:=2^{4(m-h)-2}\,a_{2(m-h)}\,a_{h}+d_{2(m-h-1)} \,a_{h+1} \quad \text{and} \quad \bar\alpha^k:=e_{2(m-h-1)}\, b_{h+1,4(m-h-1)}\,,$$ if $h=m-1$ $$\bar \alpha^1:=a_1\,a_{m-1} \quad \text{and} \quad \bar\alpha^k:=\frac{9}{16}\, b_{m-1,4}\,.$$ where $a_0=1$ and, for any $\gamma$ and $\beta$ positive integers, $a_{\gamma}=\frac{(N-1)^{2\gamma}}{2^{4\gamma}}$, $b_{\gamma,\beta} = \prod_{j = 0}^{\gamma-1}\frac{(N+(\beta+4j))^2(N-(\beta+4j)-4)^2}{16}$, $d_{\gamma}$ and $e_{\gamma}$ are the constants defined in Theorem \[mainhigher0\].
Let $0< h<m-1$, by applying first with $k=2(m-h-1)$, then and finally with $\gamma=h,h+1$ and $\beta=2,4,2i$, we deduce $$\int_{\mathbb{H}^{N}} (\Delta^{m} u)^2 \ dv_{\mathbb{H}^{N}} =\int_{\mathbb{H}^{N}} (\Delta^{m-h-1}(\Delta^{h+1} u))^2 \ dv_{\mathbb{H}^{N}}$$ $$\geq \left( \frac{N-1}{2} \right)^{4(m-h-1)} \int_{\mathbb{H}^{N}} (\Delta^{h+1} u)^2 \ dv_{\mathbb{H}^{N}}+ \sum_{i= 1}^{2(m-h-1)} c_{2(m-h-1)}^i\int_{\mathbb{H}^{N}} \frac{(\Delta^{h+1} u)^2}{r^{2i}} \ dv_{\mathbb{H}^{N}}$$ $$\geq \left( \frac{N-1}{2} \right)^{4(m-h)-2} \int_{\mathbb{H}^{N}} |\nabla \Delta^{h} u|^2 \ dv_{\mathbb{H}^{N}}
+\frac{1}{4}\left( \frac{N-1}{2} \right)^{4(m-h)-2} \int_{\mathbb{H}^{N}} \frac{(\Delta^{h} u)^2}{r^{2}} \ dv_{\mathbb{H}^{N}}$$ $$+ \frac{9}{16} \left( \frac{N-1}{2} \right)^{4(m-h-1)} \int_{\mathbb{H}^{N}} \frac{(\Delta^{h} u)^2}{r^{4}} \ dv_{\mathbb{H}^{N}}
+ \sum_{i= 1}^{2(m-h-1)} c_{2(m-h-1)}^i\int_{\mathbb{H}^{N}} \frac{(\Delta^{h+1} u)^2}{r^{2i}} \ dv_{\mathbb{H}^{N}}$$ $$\geq \left( \frac{N-1}{2} \right)^{4(m-h)-2} \int_{\mathbb{H}^{N}} |\nabla \Delta^{h} u|^2 \ dv_{\mathbb{H}^{N}}$$ $$+\frac{1}{4}\left( \frac{N-1}{2} \right)^{4(m-h)-2} \left( a_{h}\int_{\mathbb{H}^{N}} \frac{u^2}{r^{2}} \ dv_{\mathbb{H}^{N}} +\sum_{ j= 1}^{2h-1} a_{h,2}^j \int_{\mathbb{H}^{N}} \frac{u^2}{r^{2j+2}} \ dv_{\mathbb{H}^{N}}+b_{h,2} \int_{\mathbb{H}^{N}} \frac{u^2}{r^{4h+2}} \ dv_{\mathbb{H}^{N}} \right)$$ $$+\frac{9}{16} \left( \frac{N-1}{2} \right)^{4(m-h-1)} \left( a_{h}\int_{\mathbb{H}^{N}} \frac{u^2}{r^{4}} \ dv_{\mathbb{H}^{N}} +\sum_{ j= 1}^{2h-1} a_{h,4}^j \int_{\mathbb{H}^{N}} \frac{u^2}{r^{2j+4}} \ dv_{\mathbb{H}^{N}}+b_{h,4} \int_{\mathbb{H}^{N}} \frac{u^2}{r^{4h+4}} \ dv_{\mathbb{H}^{N}} \right)$$ $$+\sum_{i= 1}^{2(m-h-1)} c_{2(m-h-1)}^i \left(a_{h+1}\int_{\mathbb{H}^{N}} \frac{u^2}{r^{2i}} \ dv_{\mathbb{H}^{N}} +\sum_{ j= 1}^{2h+1} a_{h+1,2i}^j \int_{\mathbb{H}^{N}} \frac{u^2}{r^{2j+2i}} \ dv_{\mathbb{H}^{N}}+b_{h+1,2i} \int_{\mathbb{H}^{N}} \frac{u^2}{r^{4h+4+2i}} \ dv_{\mathbb{H}^{N}} \right)\,.$$
Hence, with an argument similar to that applied in the proof of Theorem \[mainhigheree\], it’s readily deduced the existence of $2m$ positive constants $\bar\alpha^i=\alpha^i(N,m,h)$ such that holds. Furthermore, $$\bar \alpha^1=\frac{1}{4}\left( \frac{N-1}{2} \right)^{4(m-h)}\,a_{h}+d_{2(m-h-1)} \,a_{h+1} \quad \text{and} \quad \bar\alpha^k=e_{2(m-h-1)}\, b_{h+1,4(m-h-1)}\,.$$ When $h=0$ the above computations may be slightly modified to show the validity of . Furthermore, by setting $a_0=0$, the leading terms are still given as above.
When $h=m-1$, the existence of $2m$ positive constants $\bar\alpha^i=\alpha^i(N,m,h)$ such that holds follows similarly by applying first and then , with $\gamma=m-1$ and $\beta=2,4$. Finally, the leading terms turn out to be $$\bar\alpha^1= \frac{(N-1)^2}{16}\,a_{m-1} \quad \text{and} \quad \bar \alpha^k=\frac{9}{16}\, b_{m-1,4}\,.$$
Case $k=2m + 1$ odd and $l=2h$ even
-----------------------------------
Let $m,h$ be integers such that $0< h\leq m$ and $ N> 4m + 2$. There exist $2m+1$ *positive* constants $\delta^i=\delta^i(N,m,h)$ such that the following inequality holds
$$\begin{aligned}
\label{higherodd1}
\int_{\mathbb{H}^{N}} |\nabla_{{\mathbb{H}^{N}}} (\Delta_{{\mathbb{H}^{N}}}^{m} u) |^2 \ dv_{\mathbb{H}^{N}} - \left( \frac{N-1}{2} \right)^{4(m-h) + 2} \int_{\mathbb{H}^{N}} ( \Delta_{{\mathbb{H}^{N}}}^{h} u)^{2} \ dv_{\mathbb{H}^{N}} \geq
\sum_{i = 1}^{2m + 1} \delta^{i} \int_{\mathbb{H}^{N}} \frac{u^2}{r^{2i}} \ dv_{\mathbb{H}^{N}},\end{aligned}$$
for all $u \in C^{\infty}_{0}(\mathbb{H}^{N})$. Furthermore, the leading terms as $r\rightarrow 0$ and $r \rightarrow\infty$ are explicitly given by:
$$\delta^1:= 4a_1\, a_{h} d_{2(m-h)} + \frac{1}{4} a_{m}\quad \text{and} \quad \delta^{2m +1} := \frac{1}{4} b_{m,2}$$
where $d_0=0$ and, for any $\gamma$ and $\beta$ positive integers, $a_{\gamma}=\frac{(N-1)^{2\gamma}}{2^{4\gamma}}$, $b_{\gamma,\beta} = \prod_{j = 0}^{\gamma-1}\frac{(N+(\beta+4j))^2(N-(\beta+4j)-4)^2}{16}$, $d_{\gamma}$ and $e_{\gamma}$ are the constants defined in Theorem \[mainhigher0\].
From we know
$$\begin{aligned}
\int_{{\mathbb{H}^{N}}} |\nabla (\Delta^m u)|^2 \ dv_{{\mathbb{H}^{N}}} & \geq \left( \frac{N-1}{2} \right)^{2} \int_{{\mathbb{H}^{N}}} (\Delta^{m} u)^2 \ dv_{{\mathbb{H}^{N}}} + \frac{1}{4} \int_{{\mathbb{H}^{N}}} \frac{(\Delta^m u)^2}{r^2} \ dv_{{\mathbb{H}^{N}}} \,.
\end{aligned}$$
If $0< h< m$, from and we readily get $$\int_{{\mathbb{H}^{N}}} |\nabla (\Delta^m u)|^2 \ dv_{{\mathbb{H}^{N}}} \geq \left( \frac{N-1}{2} \right)^{2} \left(\left( \frac{N-1}{2} \right)^{4(m-h)} \int_{\mathbb{H}^{N}} (\Delta_{{\mathbb{H}^{N}}}^{h} u)^{2} \ dv_{\mathbb{H}^{N}} +
\sum_{i = 1}^{2m} \alpha^{i} \int_{\mathbb{H}^{N}} \frac{u^2}{r^{2i}} \ dv_{\mathbb{H}^{N}} \right)$$ $$+ \frac{1}{4} \left( a_{m} \int_{{\mathbb{H}^{N}}} \frac{u^2}{r^2} \ dv_{{\mathbb{H}^{N}}} + \sum_{j = 1}^{2m-1} a^{j}_{m,2} \int_{{\mathbb{H}^{N}}} \frac{u^2}{r^{2j + 2}} \ dv_{{\mathbb{H}^{N}}}
+ b_{m,2} \int_{{\mathbb{H}^{N}}} \frac{u^2}{r^{4m + 2}} \right)$$ by which the existence of $2m+1$ positive constants $\delta^i=\delta^i(N,m,h)$ such that holds follows. Furthermore, one has $$\delta^1:= \left( \frac{N-1}{2} \right)^{2} a_{h} d_{2(m-h)} + \frac{1}{4} a_{m} \quad \text{and} \quad \delta^{2m +1} := \frac{1}{4} b_{m,2}\,.$$ When $h=m$ the same proof may be adopted without applying . In this case, the leading terms are defined as above by assuming $d_0=0$.
Case $k=2m + 1$ odd and $l=2h + 1$ odd
--------------------------------------
Let $m,h$ be integers such that $0\leq h< m$ and $ N> 4m + 2$. There exist $2m+1$ *positive* constants $\bar \delta^i=\bar\delta^i(N,m,h)$ such that the following inequality holds $$\begin{aligned}
\label{higherodd2}
\int_{\mathbb{H}^{N}} |\nabla_{{\mathbb{H}^{N}}} (\Delta_{{\mathbb{H}^{N}}}^{m} u) |^2 \ dv_{\mathbb{H}^{N}} - \left( \frac{N-1}{2} \right)^{4(m-h)} \int_{\mathbb{H}^{N}} | \nabla_{{\mathbb{H}^{N}}} (\Delta_{{\mathbb{H}^{N}}}^{h} u)|^{2} \ dv_{\mathbb{H}^{N}} \geq
\sum_{i = 1}^{2m + 1}\bar \delta^{i} \int_{\mathbb{H}^{N}} \frac{u^2}{r^{2i}} \ dv_{\mathbb{H}^{N}},\end{aligned}$$ for all $u \in C^{\infty}_{0}(\mathbb{H}^{N})$. Furthermore, the leading terms as $r\rightarrow 0$ and $r \rightarrow\infty$ are explicitly given by:
if $0\leq h<m-1$ $$\bar\delta^1:= \frac{1}{4} a_{m} + 4^{2(m-h)}\, a_{2(m-h)}\,a_{h}\,a_1+4\,d_{2(m - h -1)} a_{h + 1} a_1
\mbox{ and } \quad \bar\delta^{2m + 1} := \frac{1}{4} b_{m, 2},$$ if $h=m-1$ $$\bar\delta^1:= \frac{1}{4} a_{m} + 4\,a_2\, a_{m - 1} \quad \text{ and } \quad \bar \delta^{2m+1} = \frac{1}{4} b_{m,2},$$ where $a_0=1$ and, for any $\gamma$ and $\beta$ positive integers, $a_{\gamma}=\frac{(N-1)^{2\gamma}}{2^{4\gamma}}$, $b_{\gamma,\beta} = \prod_{j = 0}^{\gamma-1}\frac{(N+(\beta+4j))^2(N-(\beta+4j)-4)^2}{16}$, $d_{\gamma}$ and $e_{\gamma}$ are the constants defined in Theorem \[mainhigher0\].
From we know
$$\begin{aligned}
\int_{{\mathbb{H}^{N}}} |\nabla (\Delta^m u)|^2 \ dv_{{\mathbb{H}^{N}}} & \geq \left( \frac{N-1}{2} \right)^{2} \int_{{\mathbb{H}^{N}}} (\Delta^{m} u)^2 \ dv_{{\mathbb{H}^{N}}} + \frac{1}{4} \int_{{\mathbb{H}^{N}}} \frac{(\Delta^m u)^2}{r^2} \ dv_{{\mathbb{H}^{N}}} \,.
\end{aligned}$$
Now, by applying and we deduce $$\int_{{\mathbb{H}^{N}}} |\nabla (\Delta^m u)|^2 \ dv_{{\mathbb{H}^{N}}} \geq \left( \frac{N-1}{2} \right)^{2}\left(\left( \frac{N-1}{2} \right)^{4(m-h)-2} \int_{\mathbb{H}^{N}} |\nabla \Delta_{{\mathbb{H}^{N}}}^{h} u|^{2} \ dv_{\mathbb{H}^{N}} +
\sum_{i = 1}^{2m}\bar \alpha^{i} \int_{\mathbb{H}^{N}} \frac{u^2}{r^{2i}} \ dv_{\mathbb{H}^{N}} \right)$$ $$+ \frac{1}{4} \left( a_{m} \int_{{\mathbb{H}^{N}}} \frac{u^2}{r^2} \ dv_{{\mathbb{H}^{N}}} + \sum_{j = 1}^{2m -1} a^{j}_{m, 2} \int_{{\mathbb{H}^{N}}} \frac{u^2}{r^{2j + 2}} \ dv_{{\mathbb{H}^{N}}} + b_{m, 2} \int_{{\mathbb{H}^{N}}} \frac{u^2}{r^{4m + 2}} \right)\,,$$ by which the existence of $2m+1$ positive constants $\bar \delta^i=\delta^i(N,m,h)$ such that holds follows. Furthermore, one has $$\bar \delta^1:=\left( \frac{N-1}{2} \right)^{2}\bar \alpha_1+\frac{1}{4} a_{m} \quad \text{and} \quad \bar\delta^{2m +1} :=\frac{1}{4} b_{m, 2}$$ and the thesis follows by recalling the definition of $\bar \alpha_1$ in Theorem \[mainhighereo\] for $h=m-1$ and $h\neq m-1$.
Proof of Theorem \[PRHinequality\] and Corollary \[cor2\] {#proof21}
=========================================================
This section is devoted to the proofs of Theorem \[PRHinequality\] and Corollary \[cor2\]. The proof of Theorem \[PRHinequality\] mainly relies on the transformation $u \rightarrow (\sinh r)^{\frac{(N-1)}{2}} u$, which uncovers the Poincaré term, and spherical harmonics technique. Before entering the proof we recall some facts on spherical harmonics.
The Laplace-Beltrami operator on hyperbolic space in spherical coordinates is given by $$\Delta_{{\mathbb{H}^{N}}} = \frac{\partial^2}{\partial r^2} + (N-1)\, \coth r
\frac{\partial}{\partial r} + \frac{1}{\sinh ^2 r} \Delta_{\mathbb{S}^{N-1}},$$ where $\Delta_{\mathbb{S}^{N-1}}$ is the Laplace-Beltrami operator on the unit sphere $\mathbb{S}^{N-1}.$ If we write $u(x) = u(r,\sigma) \in C_c^{\infty}({\mathbb{H}^{N}}),$ $r \in [0, \infty), \sigma \in \mathbb{S}^{N-1},$ then by [@ES Ch.4, Lemma 2.18] we have that $$u(x): = u(r, \sigma) = \sum_{n= 0}^{\infty} d_{n}(r) P_{n}(\sigma)$$ in $L^2({{\mathbb{H}^{N}}}),$ where $\{ P_{n}\}$ is a complete orthonormal system of spherical harmonics and $$d_{n}(r) = \int_{\mathbb{S}^{N-1}} u(r, \sigma) P_{n}(\sigma) \ d\sigma.$$ We note that the spherical harmonic $P_{n}$ of order $n$ is the restriction to $\mathbb{S}^{N-1}$ of a homogeneous harmonic polynomial of degree $n.$ Now we recall the following
\[arm\] [@MSS Lemma 2.1] Let $P_{n}$ be a spherical harmonic of order $n$ on $\mathbb{S}^{N-1}.$ Then for every $n \in \mathbb{N}_{0}$
$$\Delta_{\mathbb{S}^{N-1}} P_{n} = -(n^2 + (N-2)n)P_{n}.$$ The values $\lambda_{n} := n^2 + (N-2)n$ are the eigenvalues of the Laplace-Beltrami operator $- \Delta_{\mathbb{S}^{N-1}}$ on $\mathbb{S}^{N-1}$ and enjoy the property $\lambda_{n} \geq 0$ and $\lambda_{0} = 0.$ The corresponding eigenspace consists of all the spherical harmonics of order $n$ and has dimension $d_{n}$ where $d_{0} = 1,$ $d_{1} = N$ and $$d_{n} = \left(\begin{array}{c}
N+n-1\\
n
\end{array} \right) - \left(\begin{array}{c}
N+n-3\\
n-2\
\end{array}\right),$$ for $n\geq 2.$
From Lemma \[arm\] it is easy to see that $$\Delta_{{\mathbb{H}^{N}}} u(r, \sigma) = \sum_{n= 0}^{\infty} \left( d_{n}^{\prime \prime}(r) +
(N-1) \coth r d_{n} (r) - \frac{\lambda_{n} d_{n}(r)}{\sinh^2 r} \right) P_{n}(\sigma).$$
In the sequel we will also exploit the following 1-dimensional Hardy-type inequality from [@BGG]:
\[hardytype\]
For all $ u \in C^{\infty}_c(0,\infty)$ there holds
$$\int_{0}^{\infty} \frac{u^{\prime 2}}{\sinh^2 r} \ dr \geq \frac{9}{4} \int_{0}^{\infty} \frac{u^2}{\sinh^4 r} \ dr
+ \int_{0}^{\infty} \frac{u^2}{\sinh^2 r} \ dr.$$
[**[Proof of Theorem \[PRHinequality\].]{}**]{}
The proof is divided in several steps.
[**[Step 1.]{}**]{} For $u \in C_{c}^{\infty}(\mathbb{H}^{N})$ we define $$v(x) = (\sinh r)^{\frac{N-1}{2}} u(x), \ \mbox{where} \ r = \rho(x,x_0)$$ Then the following relation holds for $x = (r, \sigma) \in (0, \infty) \times \mathbb{S}^{N-1},$ $$\label{ph1}
|\nabla_{\mathbb{H}^{N}} u|^2 = (\sinh r)^{-(N-1)} \left( |\nabla_{\mathbb{H}^{N}} v|^2 +
\frac{(N-1)^2}{4} \coth^2 r v^2 - (N-1) \coth r v \frac{\partial v}{\partial r} \right).$$\
[**[Proof of Step 1.]{}**]{} A straightforward computation gives $$\begin{aligned}
|\nabla_{\mathbb{H}^{N}} v|^2& = \left(\frac{\partial v}{\partial r}\right)^2+\frac{1}{\sinh^2 r}|\nabla_{\mathbb{S}^{N}}v|^2
\notag\\
& = \frac{(N-1)^2}{4} (\sinh r)^{N-1} \coth^2 r u^2 +
|\nabla_{\mathbb{H}^{N}} u|^2 (\sinh r)^{N-1} \notag \\
& + (N-1) (\sinh r)^{N-1} \coth r (\sinh r)^{\frac{-(N-1)}{2}} v \frac{\partial u}{\partial r} \notag \\
& = \frac{(N-1)^2}{4} \coth^2 r v^2 +
|\nabla_{\mathbb{H}^{N}} u|^2 (\sinh r)^{N-1} - \frac{(N-1)^2}{2} \coth^2 r v^2 \notag \\
& + (N-1) \coth r v \frac{\partial v}{\partial r}.\end{aligned}$$ Now by rearranging the terms above we conclude the proof of Step 1.
[**[Step 2.]{}**]{} In this step we compute, $$\label{transformation}
\Delta_{{\mathbb{H}^{N}}} v = \left( \frac{\partial^2}{\partial r^2} + (N-1) \coth r \frac{\partial}{\partial r} +
\frac{1}{\sinh^2} \Delta_{\mathbb{S}^{N-1}}\right) (\sinh r)^{\frac{N-1}{2}} u.$$
$$\begin{aligned}
\Delta_{{\mathbb{H}^{N}}} v& = \frac{(N-1)(N-3)}{4} (\sinh r)^{\frac{N-1}{2}} \coth^2 r \ u + (N-1) (\sinh r)^{\frac{N-3}{2}} \cosh r
\frac{\partial u}{\partial r} \\
& + \frac{(N-1)}{2} (\sinh r)^{\frac{N-1}{2}} u + (\sinh r)^{\frac{N-1}{2}} \frac{\partial^2 u}{\partial r^2} \\
& + \frac{(N-1)^2}{2} \coth^2 r (\sinh r)^\frac{N-1}{2} u + (N-1) \coth r (\sinh r)^{\frac{N-1}{2}} \frac{\partial u}{\partial r} \\
& + (\sinh r)^{\frac{N-1}{2}} \frac{1}{\sinh^2 r} \Delta_{\mathbb{S}^{N-1}} u\\
& = (\sinh r)^{\frac{N-1}{2}} \left[ \frac{\partial^2 u}{\partial r^2} + (N-1) \coth r \frac{\partial u}{\partial r}
+ \frac{1}{\sinh^2 r} \Delta_{\mathbb{S}^{N-1}} u \right] \\
& + \left[ \frac{(N-1)(N-3)}{4} + \frac{(N-1)^2}{2} \right] \coth^2 r (\sinh r)^{\frac{N-1}{2}} u +
\frac{(N-1)}{2} (\sinh r)^{\frac{N-1}{2}} u \\
&+ (N-1) (\sinh r)^{\frac{N-3}{2}} \cosh r \left[ \frac{\partial}{\partial r} ((\sinh r)^{- \frac{(N-1)}{2}} v) \right]\\
& = (\sinh r)^{\frac{N-1}{2}} (\Delta_{{\mathbb{H}^{N}}} u) + \left[ \frac{(N-1)(N-3)}{4} + \frac{(N-1)^2}{2} \right] \coth^2 r v \\
& + \frac{(N-1)}{2} v - \frac{(N-1)^2}{2} \coth^2 r v + (N-1) \coth r \frac{\partial v}{\partial r} \\
& = (\sinh r)^{\frac{N-1}{2}} (\Delta_{{\mathbb{H}^{N}}} u) + \frac{(N-1)(N-3)}{4} \coth^2 r v + \frac{(N-1)}{2} v + (N-1) \coth r
\frac{\partial v}{\partial r}\,.\end{aligned}$$
Hence, we get $$\begin{aligned}\label{trans1}
\Delta_{{\mathbb{H}^{N}}} u &= \frac1{(\sinh r)^{\frac{(N-1)}{2}}} \left[ \Delta_{{\mathbb{H}^{N}}} v -
\left( \frac{(N-1)(N-3)}{4} \coth^2 r + \frac{(N-1)}{2} \right) v \right. \left. -(N-1)\coth r \frac{\partial v}{\partial r} \right] \\
&= \frac1{(\sinh r)^{\frac{(N-1)}{2}}} \left[ \frac{\partial^2 v}{\partial r^2} - \left( \frac{(N-1)(N-3)}{4} \coth^2 r
+ \frac{(N-1)}{2} \right) v \right. \left. + \frac{1}{\sinh^2 r} \Delta_{\mathbb{S}^{N-1}} v \right].
\end{aligned}$$
[**[ Step 3.]{}**]{} Expanding $v$ in spherical harmonics $$v(x) : = v(r, \sigma ) = \sum_{n=0}^{\infty} d_{n}(r) P_{n}(\sigma)$$ and observing that $$\int_{\mathbb{H}^{N}} (\sinh r)^{-(N-1)} |\nabla_{\mathbb{H}^{N}} v|^2 \ dv_{\mathbb{H}^{N}} =
\sum_{ n = 0}^{\infty} \int_{0}^{\infty} \left( (d_{n}^{\prime}(r))^2 + \lambda_{n} \frac{d_{n}^2(r)}{\sinh^2 r} \right) \ dr,$$ and substituting in , we infer $$\begin{aligned}
\label{ph4}
\int_{\mathbb{H}^{N}} |\nabla_{\mathbb{H}^{N}} u|^2 \ dv_{\mathbb{H}^{N}}
= &\notag \\\sum_{n = 0 }^{\infty} \int_{0}^{\infty} & \left( (d_{n}^{\prime}(r))^2 +
\lambda_{n} \frac{d_{n}^2(r)}{\sinh^2 r} + \frac{(N-1)^2}{4} \coth^2 r d_{n}^2(r)- (N-1) \coth r d_{n}(r) d_{n}^{\prime}(r) \right) dr.\end{aligned}$$
Further expanding in spherical harmonics and putting this in , we have $$\begin{aligned}
\int_{\mathbb{H}^{N}} |\Delta_{{\mathbb{H}^{N}}} u|^2 \ dv_{\mathbb{H}^{N}}
=& \notag \\ \sum_{n=0}^{\infty} \int_{0}^{\infty} &\left( d_{n}^{\prime \prime}(r) - \frac{(N-1)(N-3)}{4} \coth^2 r d_{n}(r) - \frac{(N-1)}{2} d_{n}(r) - \frac{\lambda_{n}}{\sinh^2 r} d_{n}(r) \right)^2 dr, \end{aligned}$$ where the eigenvalues $\lambda_{n}$ are repeated according to their multiplicity. Let us write using and ,
$$\begin{aligned}
& \int_{\mathbb{H}^{N}} (\Delta_{\mathbb{H}^{N}} u)^2 \ dv_{\mathbb{H}^{N}} - \left( \frac{N-1}{2} \right)^2 \int_{\mathbb{H}^{N}} |\nabla_{\mathbb{H}^{N}} u|^2
\ dv_{\mathbb{H}^{N}} = \sum_{n=0}^{\infty} \left[ \int_{0}^{\infty} \left( d_{n}^{\prime \prime}(r) - \frac{(N-1)(N-3)}{4} \coth^2 r d_{n}(r) \right. \right. \notag \\
& \left. \left. - \frac{(N-1)}{2} d_{n}(r) - \frac{\lambda_{n}}{\sinh^2 r} d_{n}(r) \right)^2 \ dr - \left( \frac{N-1}{2} \right)^2 \int_{0}^{\infty}
\left( (d_{n}^{\prime}(r))^2 + \lambda_{n} \frac{d_{n}^2(r)}{\sinh^2 r} + \frac{(N-1)^2}{4} \coth^2 r d_{n}^2(r) \right. \right. \notag \\
& \left. \left. - (N-1) \coth r d_{n}(r) d_{n}^{\prime}(r) \right) \right] \ dr.\end{aligned}$$
Considering each term separately and simplifying further, for detail see the proof of [@BGG Theorem 3.1]), we get
$$\begin{aligned}
\notag \label{estimate1}
&\int_{0}^{\infty} \left( d_{n}^{\prime \prime}(r) - \frac{(N-1)(N-3)}{4} \coth^2 r d_{n}(r)
- \frac{(N-1)}{2} d_{n}(r) - \frac{\lambda_{n}}{\sinh^2 r} d_{n}(r) \right)^2 \ dr \\ \notag
& = \int_{0}^{\infty} (d_{n}^{\prime \prime}(r))^2 \ dr + \frac{(N-1)^2}{2} \int_{0}^{\infty} (d_{n}^{\prime}(r))^2 \ dr + \left( \frac{(N-1)(N-3)}{2} +
2 \lambda_{n} \right) \int_{0}^{\infty} \frac{1}{\sinh^ 2 r} (d_{n}^{\prime}(r))^2 \ dr \notag \\
& \frac{(N-1)^4}{16} \int_{0}^{\infty} (d_{n}(r))^2 \ dr + \left( \lambda_{n}^2 + \frac{(N-1)(N-3)}{2} \lambda_{n} - 6 \lambda_{n} + \frac{(N-1)^2 (N-3)^2}{16} \right. \notag \\
& \left. - \frac{3}{2} (N-1)(N-3) \right) \int_{0}^{\infty} \frac{1}{\sinh^4 r} (d_{n}(r))^2 \ dr + \left( \frac{(N-1)^2(N-3)^2}{8} + \frac{(N-1)^2(N-3)}{4} \right. \notag \\
& \left. + \frac{(N-1)(N-3)}{2} \lambda_{n} + (N-5) \lambda_{n} - (N-1)(N-3) \right) \int_{0}^{\infty} \frac{1}{\sinh^2 r} (d_{n}(r))^2 \ dr \notag \\\end{aligned}$$
and
$$\begin{aligned}
\label{estimate2}
\left(\frac{N-1}{2} \right)^2 \int_{\mathbb{H}^{N}} |\nabla_{\mathbb{H}^{N}} u|^2 \ dv_{\mathbb{H}^{N}} & = \left( \frac{N-1}{2} \right)^2 \int_{0}^{\infty} (d_{n}^{\prime}(r))^2 \ dr - \frac{(N-1)^4}{16} \int_{0}^{\infty} (d_{n}(r))^2 \ dr \notag \\
& - \frac{(N-1)^3 (N - 3)}{16} \int_{0}^{\infty}
\frac{1}{\sinh^2 r} (d_{n}(r))^2 \ dr.\end{aligned}$$
By and , using Lemma \[hardytype\], the 1-dimensional Hardy inequality: $$\int_{0}^{\infty} (d_{n}^{\prime}(r))^2 \ dr \geq \frac{1}{4} \int_{0}^{\infty} \frac{d_{n}^{2}(r)}{r^2} \ dr$$ and the 1-dimensional Rellich inequality: $$\int_{0}^{\infty} (d_{n}^{\prime \prime}(r))^2 \ dr \geq \frac{9}{16} \int_{0}^{\infty} \frac{d_{n}^2 (r)}{r^4} \ dr ,$$ we obtain $$\begin{aligned}
& \int_{\mathbb{H}^{N}} (\Delta_{\mathbb{H}^{N}} u)^2 \ dv_{\mathbb{H}^{N}} - \left( \frac{N-1}{2} \right)^2 \int_{\mathbb{H}^{N}} |\nabla_{\mathbb{H}^{N}} u|^2
\ dv_{\mathbb{H}^{N}} \notag \\
& \geq \frac{9}{16} \int_{0}^{\infty} \frac{d_{n}^{2}(r)}{r^4} \ dr + \frac{(N-1)^2}{16} \int_{0}^{\infty} \frac{d_{n}^{2}(r)}{r^2 } \ dr + A_{N} \int_{0}^{\infty} \frac{d_{n}^{2}(r)}{\sinh^4 r} \ dr +
B_{N} \int_{0}^{\infty} \frac{d_{n}^{2}(r)}{\sinh^4 r} \ dr,\end{aligned}$$ where $$A_{n} = \left[ \lambda_{n}^2 + \frac{N(N-4)}{2} \lambda_{n} +
\frac{((N-1)(N-3))^2}{16} - \frac{3}{8} (N-1)(N-3) \right]$$ and $$B_{n} = \left[ \frac{(N^2 - 2N - 5)}{4} \lambda_{n} + \frac{(N-1)^2(N-3)}{4} + \frac{(N-1)^2(N-3)(N-5)}{16} - \frac{(N-1)(N-3)}{2} \right].$$ We note that $$\min_{n \in \mathbb{N}_{0}} A_{n}=\frac{(N-1)(N-3)(N^2-4N-3)}{16} \quad \mbox{and} \quad \min_{n \in \mathbb{N}_{0}} B_{n}= \frac{(N-1)(N-3)(N^2 - 2N - 7)}{16}$$ for $N \geq 5$ and hence they are both positive. Also we have $$\int_{{\mathbb{H}^{N}}} u^2 \ dv_{{\mathbb{H}^{N}}} = \int_{{\mathbb{H}^{N}}} v^2 (\sinh r)^{-(N-1)} \ dv_{{\mathbb{H}^{N}}} = \sum_{n= 0}^{\infty} \int_{0}^{\infty} d_{n}^2(r) \ dr,$$ similarly $$\int_{{\mathbb{H}^{N}}} \frac{u^2}{r^2} \ dv_{{\mathbb{H}^{N}}} = \sum_{n=0}^{\infty} \int_{0}^{\infty} \frac{d_{n}^2(r)}{r^2} \ dr$$ and so on. Now, using all these facts, we obtain $$\begin{aligned}
\int_{{\mathbb{H}^{N}}} (\Delta_{{\mathbb{H}^{N}}} u)^2 \ dv_{{\mathbb{H}^{N}}} - \left(\frac{N-1}{2} \right)^2 \int_{{\mathbb{H}^{N}}} |\nabla_{{\mathbb{H}^{N}}} u|^2 \ dv_{{\mathbb{H}^{N}}}
\geq \frac{9}{16} \int_{{\mathbb{H}^{N}}} \frac{u^2}{r^4} \ dv_{{\mathbb{H}^{N}}} + \frac{(N-1)^2}{16} \int_{{\mathbb{H}^{N}}} \frac{u^2}{r^2} \ dv_{{\mathbb{H}^{N}}}\\
\frac{(N-1)(N-3)(N^2-4N-3)}{16} \int_{0}^{\infty} \frac{u^2}{\sinh^4 r} \ dv_{{\mathbb{H}^{N}}} +\frac{(N-1)(N-3)(N^2 -2N -7)}{16} \int_{0}^{\infty} \frac{u^2}{\sinh^2 r} \ dv_{{\mathbb{H}^{N}}},\end{aligned}$$ and hence the proof of inequality .
[**[Step 4.]{}**]{} Next we show the optimality of the constant $\frac{(N-1)^2}{16}.$ Let us suppose that the $\frac{(N-1)^2}{16}$ is not optimal, i.e., there exist $ C > \frac{(N-1)^2}{16}$ such that there holds, $$\begin{aligned}
\int_{{\mathbb{H}^{N}}} (\Delta_{{\mathbb{H}^{N}}} u)^2 \ dv_{{\mathbb{H}^{N}}} - \left(\frac{N-1}{2} \right)^2 \int_{{\mathbb{H}^{N}}} |\nabla_{{\mathbb{H}^{N}}} u|^2 \ dv_{{\mathbb{H}^{N}}}
\geq C \int_{{\mathbb{H}^{N}}} \frac{u^2}{r^2} \ dv_{{\mathbb{H}^{N}}},
\end{aligned}$$ using [@BGG Theorem 2.1]) and above we obtain
$$\begin{aligned}
\label{estimate3}
\int_{\mathbb{H}^{N}} (\Delta_{\mathbb{H}^{N}} u)^2 \ dv_{\mathbb{H}^{N}} & \geq C \int_{\mathbb{H}^{N}} \frac{u^2}{r^2} \ dv_{\mathbb{H}^{N}}
+ \frac{(N-1)^2}{4} \left[ \frac{(N-1)^2}{4} \int_{\mathbb{H}^{N}} u^2 \ dv_{{\mathbb{H}^{N}}} + \frac{1}{4} \int_{{\mathbb{H}^{N}}} \frac{u^2}{r^2} \ dv_{{\mathbb{H}^{N}}} \right] \notag \\
& =\left( C + \frac{(N-1)^2}{16} \right) \int_{\mathbb{H}^{N}} \frac{u^2}{r^2} \ dv_{\mathbb{H}^{N}}
+ \frac{(N-1)^4}{16} \int_{{\mathbb{H}^{N}}} u^2 \ dv_{\mathbb{H}^{N}},
\end{aligned}$$
comparing with [@BGG Theorem 3.1], we conclude that $C \leq \frac{(N-1)^2}{16}$ which gives a contradiction and hence $\frac{(N-1)^2}{16}$ is the best constant. $\Box$
[**[Proof of Corollary 2.2. ]{}**]{} By considering the upper half space model $\mathbb{R}^{N}_{+} $ for ${\mathbb{H}^{N}}$ and using the explicit expression of the gradient in these coordinates, namely $\nabla_{\mathbb{H}^{N}} = y^2 \nabla,$ we obtain
$$\begin{aligned}
\label{pf1}
\int_{{\mathbb{H}^{N}}} |\nabla_{{\mathbb{H}^{N}}} u(x,y)|^2 \ dv_{{\mathbb{H}^{N}}} & = \int_{\mathbb{R}^{+}} \int_{\mathbb{R}^{N-1}} y^{2\alpha + 2 - N} |\nabla v|^2 \ dx \ dy
+ \alpha^2 \int_{\mathbb{R}^{+}} \int_{\mathbb{R}^{N-1}} y^{2 \alpha - N} v^2 \ dx \ dy \notag \\
& - \alpha (2 \alpha + 1 - N) \int_{\mathbb{R}^{+}} \int_{\mathbb{R}^{N-1}} y^{2 \alpha - N} v^2 \ dx \ dy\end{aligned}$$
and also using Laplacian expression, $\Delta_{\mathbb{H}^{N}} = y^2 \Delta - (N-2) y$, we get
$$\label{pf2}
\Delta_{\mathbb{H}^{N}} u = y^{\alpha + 2} \Delta v + (2 \alpha - (N-2)) y^{\alpha} \frac{\partial v}{\partial y} + \alpha (\alpha - (N-1)) y^{\alpha} v,$$
where and $u (x,y) : = y^{\alpha} v(x,y).$ With $\alpha = \frac{N-2}{2}, $ we get
$$\begin{aligned}
\int_{{\mathbb{H}^{N}}} (\Delta_{{\mathbb{H}^{N}}} u(x,y))^2 \ dv_{{\mathbb{H}^{N}}} & = \int_{\mathbb{R}^{+} \times \mathbb{R}^{N-1}} \left( y^{N+1} (\Delta v)^2 + \frac{N^2(N-2)^2}{16} y^{N-2} v^2
- \frac{N(N-2)}{2} y^{N} v \Delta v \right) \frac{dx \ dy}{y^{N}} \\
& = \int_{\mathbb{R}^{N-1} \times \mathbb{R}^{+}} \left( y^{2} (\Delta v)^2 + \frac{N^2(N-2)^2}{16} \frac{v^2}{y^2}
+ \frac{N(N-2)}{2} y^{N} |\nabla v|^2 \right) dy \ dx.\end{aligned}$$
Similarly with $\alpha = \frac{N-4}{2}, $ and by denoting $\frac{\partial v}{\partial y} := v_{y},$ we get
$$\begin{aligned}
\int_{{\mathbb{H}^{N}}} (\Delta_{{\mathbb{H}^{N}}} u(x,y))^2 \ dv_{{\mathbb{H}^{N}}} & = \int_{\mathbb{R}^{+} \times \mathbb{R}^{N-1}} \left( (\Delta v)^2 + 4 \frac{v_{y}^2}{y^2}
+ \frac{(N-4)^2 (N +2)^2}{16} \frac{v^2}{y^4} \right. \\
& \left. - 4 \frac{v_{y} \Delta v}{y} - \frac{(N-4)(N + 2)}{2} \frac{v \Delta v}{y^2} + (N-4)(N+2) \frac{v v_{y}}{y^3} \right) dy \ dx \\
& = \int_{\mathbb{R}^{+} \times \mathbb{R}^{N-1}} \left( (\Delta v)^2 + 4 \frac{v_{y}^2}{y^2} + \frac{(N-4)^2 (N+2)^2}{16} \frac{v^2}{y^4} \right. \\
& \left. - 4 \frac{v_{y}^2}{y^2} + 2 \frac{|\nabla v|^2}{y^2} + \frac{(N-4)(N+2)}{2} \frac{|\nabla v|^2}{y^2} \right) dy \ dx \\
& = \int_{\mathbb{R}^{+} \times \mathbb{R}^{N-1}} \left( (\Delta v)^2 + \frac{(N-4)^2 (N+2)^2}{16} \frac{v^2}{y^4}
+ 2 \frac{|\nabla v|^2}{y^2} \right. \\
& \left. + \frac{(N-4)(N+2)}{2} \frac{|\nabla v|^2}{y^2} \right) dy \ dx
\end{aligned}$$
Now the proof follows by inserting , and the above computations in with $\alpha = \frac{N-2}{2}$ and $\alpha = \frac{N-4}{2}$ successively.\
Next we turn to the optimality issues. Assume by contradiction that the following inequality holds $$\int_{\mathbb{R}^{+}} \int_{\mathbb{R}^{N-1}} \left( y^2 (\Delta v)^2 +
c |\nabla v|^2 \right) \ dx \ dy
\geq \frac{N(N-2)}{16} \int_{\mathbb{R}^{+}} \int_{\mathbb{R}^{N-1}} \frac{v^2}{y^2} \ dx \ dy$$ for all $ u \in C^{\infty}_{c}(\mathbb{H}^{N})$ with $c< \frac{N^2 -2N -1}{4}$. The above inequality, jointly with with $k = 1,$ $l = 0$ and Hardy-Maz’ya inequality:
$$\int_{\mathbb{R}^{+}} \int_{\mathbb{R}^{N-1}} |\nabla v|^2 \ dx \ dy \geq \frac{1}{4} \int_{\mathbb{R}^{+}} \int_{\mathbb{R}^{N-1}} \frac{v^2}{y^2} \ dx \ dy,$$ where $\frac{1}{4}$ is the best constant, see [@Ma] and [@FMT; @FTT], yields $$\int_{\mathbb{H}^{N}} |\Delta_{\mathbb{H}^{N}} u|^2 \ dv_{\mathbb{H}^{N}}\geq\frac{(N -1)^2}{4} \int_{\mathbb{H}^{N}} |\nabla_{{\mathbb{H}^{N}}}u|^2 \ dv_{\mathbb{H}^{N}}+\left(\frac{N^2-2N-1}{4}-c\right) \int_{\mathbb{R}^{+}} \int_{\mathbb{R}^{N-1}} |\nabla v|^2 \ dx \ dy$$ $$\geq\frac{(N -1)^4}{16} \int_{\mathbb{H}^{N}} u^2 \ dv_{\mathbb{H}^{N}}+\frac{1}{4} \left(\frac{N^2-2N-1}{4}-c\right) \int_{\mathbb{R}^{+}} \int_{\mathbb{R}^{N-1}} \frac{v^2}{y^2} \ dx \ dy$$ $$=\left(\frac{(N -1)^4}{16}+\frac{1}{4} \left(\frac{N^2-2N-1}{4}-c\right) \right) \int_{\mathbb{H}^{N}} u^2 \ dv_{\mathbb{H}^{N}}\,,$$ a contradiction with with $k = 2$ and $l = 0$. The optimality of the other constants follows straightforwardly from what remarked above. $\Box$
**Acknowledgments.** We are grateful to G. Grillo for fruitful discussions during the preparation of the manuscript. The authors are partially supported by the Research Project FIR (Futuro in Ricerca) 2013 *Geometrical and qualitative aspects of PDE’s*. The first author is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
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|
---
author:
- 'H. Peter, Y.-M. Huang, L. P. Chitta, and P. R. Young'
date: 'Received 2 May 2019 / Accepted 1 July 2019'
title: 'Plasmoid-mediated reconnection in solar UV bursts'
---
[Ultraviolet bursts are transients in the solar atmosphere with an increased impulsive emission in the extreme UV lasting for one to several tens of minutes. They often show spectral profiles indicative of a bi-directional outflow in response to magnetic reconnection. ]{} [To understand UV bursts, we study how motions of magnetic elements at the surface can drive the self-consistent formation of a current sheet resulting in plasmoid-mediated reconnection. In particular, we want to study the role of the height of the reconnection in the atmosphere. ]{} [We conducted numerical experiments solving the 2D magnetohydrodynamic equations from the solar surface to the upper atmosphere. Motivated by observations, we drove a small magnetic patch embedded in a larger system of magnetic field of opposite polarity. This type of configuration creates an X-type neutral point in the initial potential field. The models are characterized by the (average) plasma-$\beta$ at the height of this X point. ]{}
The driving at the surface stretches the X-point into a thin current sheet, where plasmoids appear, accelerating the reconnection, and a bi-directional jet forms. This is consistent with what is expected for UV bursts or explosive events, and we provide a self-consistent model of the formation of the reconnection region in such events. The gravitational stratification gives a natural explanation for why explosive events are restricted to a temperature range around a few 0.1MK, and the presence of plasmoids in the reconnection process provides an understanding of the observed variability during the transient events on a timescale of minutes.
[Our numerical experiments provide a comprehensive understanding of UV bursts and explosive events, in particular of how the atmospheric response changes if the reconnection happens at different plasma-$\beta$, that is, at different heights in the atmosphere. This analysis also gives new insight into how UV bursts might be related to the photospheric Ellerman bombs. ]{}
Introduction\[S:intro\]
=======================
The solar atmosphere is in a highly dynamic state. Since the early days of space-based spectroscopy at extreme ultraviolet (EUV) and X-ray wavelengths, observations have hinted at small-scale dynamics that are either barely or not resolved. Using data from rocket flights, [@1983ApJ...272..329B] isolated what they originally called turbulent events and what were later termed explosive events. They stood out as excessive broadening of spectral lines interpreted as nonresolved motions in response to heating of the plasma. The broadened spectral features often appeared with additional components in the the red and blue wings of the line profile about 50kms$^{-1}$ to 100kms$^{-1}$ away from line center [@1989SoPh..123...41D]. The spatial location of bi-directional line profiles has been connected to places of flux emergence, cancelation, and therefore reconnection [@1991JGR....96.9399D]. Given the limited amount of observing time of the experiments on rockets and a Space Shuttle flight, these explosive events have been mostly seen in the quiet Sun where a clear connection could be established to the bright network structures and the magnetic field [@1991ApJ...370..775P]. With the era of the Solar and Heliospheric Observatory [SOHO; @1995SoPh..162.....F], observations became much more abundant. This allowed [@1997Natur.386..811I] to show that the misalignment of the line of sight with the axis of the reconnection outflow explains the asymmetry of the line profiles and to confirm the reconnection scenario. This is consistent with the finding that explosive events occur mostly at locations where opposite polarities come into close contact and cancel [@1998ApJ...497L.109C]. For a review of the zoo of transient events in the UV, see [@2018SSRv..214..120Y].
In active regions, in particular while magnetic flux is still emerging, energetic events are seen more frequently. Forming in the photosphere, Ellerman bombs are visible through enhancements in the wings of H$\alpha$ [@1917ApJ....46..298E]. Observations give clear indications that these too are forming through reconnection [e.g., @2002ApJ...575..506G]. Recent observations in the UV reveal a new type of event that can be characterized by very strong enhancements in chromospheric and transition region lines (e.g., , ). Here spectral absorption features of chromospheric lines can be seen in the strongly broadened transition region line profiles [@2014Sci...346C.315P]. These UV bursts seem to originate in the chromosphere, thus from higher up (at lower densities) than Ellerman bombs. While the plasma in UV bursts is heated sufficiently to give rise to emission in, for example, , the plasma temperature probably does not reach coronal temperature of the order of 1MK.
Explosive events have been seen abundantly in the quiet Sun in emission lines forming (under ionization equilibrium conditions) at around 0.1MK to 0.4MK, for example, in , or . They are one of the classical transition-region phenomena. They seem not to appear at higher temperatures (in the corona) in that they have no counterparts in the upper transition region or low corona, for example in forming at about 0.7MK or at about 1MK . This poses the question of why the mechanism that works in the transition region would not be efficient in the corona.
After it had been established observationally that explosive events are caused by reconnection, 2D magnetohydrodynamics (MHD) models were constructed with the aim to recreate the observed bi-directional reconnection outflows. Most of the models so far have used a Harris-type current sheet [@Harris:1962] as an initial setup. For example, [@1999SoPh..185..127I] used a magnetic field that changes as $\tanh$ across the current sheet. To initiate reconnection, these early models used an anomalously enhanced resistivity at a single location along the current sheet. The resulting reconnection outflow speeds and temperatures are consistent with observations [@1999SoPh..185..127I] in that they provide an explanation for the enhanced emission in the line wings often showing up as separate spectral components . When also considering gravitational stratification, were able to find asymmetric line profiles , but they could still not recover the increase of the intensity in the line core that is associated with most (but not all) transient events.
This shortcoming of the early explosive event models has been worked out clearly by [@2015ApJ...813...86I] and they suggested that the presence of plasmoids in the current sheet provides a solution. Direct observational evidence for plasmoids has been found in the solar atmosphere, at least in currents sheets forming following a coronal mass ejection [@2003ApJ...594.1068K; @2005ApJ...622.1251L], in flares [@2004ApJ...605L..77A; @2012ApJ...745L...6T], in the corona [e.g., @2016NatPh..12..847L], and recently also in the chromosphere [@2017ApJ...851L...6R]. The key importance of the formation of plasmoids for reconnection was established early on [@1986PhFl...29.1520B]. For large Lundquist numbers, thin current sheets will undergo a super-Alfvénic tearing instability [e.g., @2007PhPl...14j0703L; @2009PhPl...16k2102B] which results in a rich variety of possible reconnection dynamics [e.g., @2013PhPl...20e5702H]. The importance of plasmoid-mediated reconnection for the solar chromosphere and corona became evident quickly [e.g., @2012ApJ...758...20N; @2015ApJ...799...79N; @2012ApJ...760..109L]. Using models of plasmoid-mediated reconnection, [@2015ApJ...813...86I] showed that this indeed can explain the strong enhancement of the line cores along with the increased emission in the line wings. This is because the plasmoid instability provides not only faster reconnection (outflows), but also more plasma at slow speeds in the forming plasmoids. One main reason why the early explosive event models did not show the plasmoid instability is a lack of spatial resolution.
The 2D reconnection modeling is now also applied to UV bursts and Ellerman bombs by placing the reconnection region in places of different plasma-$\beta$, where $\beta$ is the ratio of magnetic to thermal energy density, or magnetic to gas pressure. If $\beta$ is lower, that is, higher in the atmosphere, the plasma can be heated to higher temperatures, and in the low chromosphere gravity can hinder the formation of plasmoids [@2016ApJ...832..195N]. However, the situation changes if one accounts for the presence of neutrals in the chromosphere and the reconnection in the low chromosphere can result in peak temperatures of about 30kK [@2018ApJ...852...95N; @2018ApJ...868..144N]. During reconnection, the 2D models can produce turbulent small-scale structures in plasmoids that cover a range of temperatures from $10^4$K to $10^5$K, that is, covering the temperatures expected in Ellerman bombs and UV bursts. Still, whether or not the observational signatures expected from those models are consistent with observations remains unknown. Observations show that there is a relation between Ellerman bombs and UV bursts [@2015ApJ...812...11V]. In a significant fraction of cases (10% to 20%) Ellerman bombs and UV bursts appear at the same time and location [@2016ApJ...824...96T].
The first 3D MHD models to reproduce the radiative signatures of Ellerman bombs were presented in and [@2017ApJ...839...22H], and the latter model was also able to reproduce UV burst emission from , but unrelated to the Ellerman bomb at a different location. In a very recent 3D model, find a UV burst flashing at the same time and location as the Ellerman bomb. While these models provide important insight into the evolution and driving of these events, the resolution is (naturally) not comparable to the 2D models and thus they barely resolve the process of plasmoid-mediated reconnection, if at all.
From the above discussion we can summarize some of the major shortcomings and open questions for understanding explosive events and UV bursts. (1) Current 2D models assume the existence of a Harris-type current sheet as the initial setup of the model. How is this current sheet produced in the first place? (2) How is the response of the atmosphere changing when the site of the reconnection is located at sites of different plasma-$\beta$? (3) Why are explosive events seen only at transition region temperatures and not above approximately 0.6MK in the (low) corona? (4) How are explosive events, UV bursts, and Ellerman bombs connected in terms of magnetic configurations and models? Are they based on the same mechanism, despite the fact that they seem to appear in regions where plasma-$\beta$ is above and below unity?
Our study addresses questions (1) to (3). Concerning question (4), we can only provide an educated guess. We address these questions using a 2D model with sufficient resolution to resolve the plasmoids that form in the current sheet. The main improvements over existing 2D models are as follows: we use a magnetic field setup that is motivated by solar observations; the simulation is driven by a horizontal motion at the bottom boundary, that is, the solar surface; and this driving is inspired by observations. Our setup is guided by the work of who found a small magnetic patch of one magnetic polarity moving between two large regions of opposite polarity. A magnetic field extrapolation of the observation revealed an X-type neutral point above the small magnetic patch and the spectroscopic EUV observations showed the continued presence of a UV burst above that patch. We capture this magnetic setup in our model, drive the small patch into the larger region of magnetic field, and find that the X-point stretches into a current sheet. We therefore create the thin current sheet in a self-consistent way. With the plasmoid instability setting in, we get significant energy conversion that we can relate to the observed UV burst. Running numerical experiments with different plasma-$\beta$ we investigate how the atmosphere reacts in these cases and check the peak temperatures during the reconnection events.
Magnetohydrodynamic model setup\[S:model\]
==========================================
The idealized 2D setup of the model is inspired by observations of UV bursts. Therefore, we first give a general figurative explanation of the setup of the magnetic field and how it is driven from the surface ([Sect.\[S:setup\]]{}). Subsequently, we present the MHD equations solved here together with the initial and boundary conditions ([Sects.\[S:model.eqs\]]{} to \[S:boundary\]) and discuss the range of parameters for our numerical experiments ([Sect.\[S:experiments\]]{}).
General setup\[S:setup\]
------------------------
The magnetic setup is inspired by a UV burst observation near a site of flux cancelation by . In their study they related the emission from the UV burst to the magnetic field extrapolated into the upper atmosphere. They found the UV burst to be located at an X-type magnetic null point. Essentially, this X-point was located some 500km above a (minor) parasitic polarity that moved in a region of opposite polarity .
In our 2D model, the initial magnetic field at the lower boundary reflects this setup (cf. [Fig.\[F:setup\]]{}a). The whole computational domain stretches over 12Mm in the horizontal ($x$) and 2.4Mm in the vertical ($z$) direction. To mimic the magnetic setup from , we choose a magnetic field as defined later in [Eq.(\[E:B.initial\])]{} where at the bottom boundary the main (negative) polarity covers the region outside $|x|\gtrsim1.2$Mm and the small patch of parasitic (positive) polarity is in the center of the box inside $|x|\lesssim1.2$Mm. This is illustrated by the field lines and the vertical component of the magnetic field at the bottom boundary shown in [Fig.\[F:setup\]]{}a,b. Clearly, this magnetic field configuration has a magnetic null point in the form of an X-point above the parasitic polarity at a height of about 550km above the bottom boundary as can be seen from [Fig.\[F:setup\]]{}a and detailed in [Eq.(\[E:null\])]{}. This would correspond to a location in the chromosphere on the real Sun.
![Model setup. In panel (a) we show the initial condition of the magnetic field through the field lines in the 2D computational domain following [Eq.(\[E:B.initial\])]{}. Panel (b) displays the vertical component of the magnetic field at the bottom boundary at $z{=}0$. The magnetic field is driven in the positive $x$-direction with a Gaussian-type velocity profile $v_0(x)$ as defined in [Eq.(\[E:v.space\])]{} and depicted in panel (c). The peak value of the driving velocity $v_d(t)$, given through [Eq.(\[E:v.time\])]{}, is shown in panel (d). While panel (a) shows the full computational domain, the red dashed rectangle indicates the region of interest displayed in [Fig.\[F:evolution\]]{}. See [Sect.\[S:model\]]{} for details. \[F:setup\]](f1_initial){width="75mm"}
In their observations, reported that the small parasitic polarity moved from one main polarity into the other main polarity. In our model we mimic this by applying a horizontal motion in the $+x$ direction. The normalized spatial profile $v_0(x)$ of the velocity driver is shown in [Fig.\[F:setup\]]{}c and detailed later in [Eq.(\[E:v.space\])]{}. We restrict the driving to the middle part of the computational domain for the simple reason that on the real Sun the opposite polarity would be pushed only so far into the main polarity before it gets completely canceled, while the main polarity would remain (roughly at the same location). Furthermore, we restrict the peak value of the driving velocity, $v_d(t)$, in time for some 20 minutes, as illustrated in [Fig.\[F:setup\]]{}d and defined in [Eq.(\[E:v.time\])]{}. Starting with the stable initial potential magnetic field setup, we slowly ramp up the velocity for about two minutes at the bottom boundary and similarly ramp down to reduce the velocity at the end of the driving period to zero. Driving with a maximum speed of just below 2kms$^{-1}$ (which is compatible with the observations) will move the parasitic polarity by about 2Mm during the period of driving.
As expected, when running the numerical experiments, the driving of the magnetic field at the bottom boundary moves the patch of opposite polarity in the $+x$ direction. Consequently, the null point above the opposite polarity also moves in the same direction . The X-point then stretches into a current sheet and plasmoid-mediated reconnection begins (cf. [Fig.\[F:evolution\]]{}).
Model equations\[S:model.eqs\]
------------------------------
The governing equations for our simulations are the 2D resistive MHD equations in a vertical $x$-$z$ plane. Here we solve for the magnetic potential $\psi$ instead of the magnetic field itself, meaning that $\mathbf B=\nabla\psi\times\hat{y}$, where $\hat{y}$ is the unit vector in the $y$ direction. The induction equation then reads $$\label{E:induction}
\partial_{t}\psi+\mathbf{v}\cdot\nabla\psi=\eta\nabla^{2}\psi,$$ where $\mathbf{v}$ denotes the velocity vector and $\eta$ is the (constant) magnetic resistivity. In the dimensionless units we chose $\eta$ to be $10^{-5}$, which in real units corresponds to $4.4\times10^5$m$^2$s$^{-1}$. This value is about a factor of 300 larger than the value according to classical transport theory [e.g., @Boyd+Sanderson:2003] for temperatures at about $10^4$K, that is, at the typical background temperature of our models (cf. [Sect.\[S:init\]]{}). We chose this anomalously high value to ensure that the current sheets are dissipated at the grid scale (the vertical grid spacing $\delta{z}$ is just above 1km; see [Sect.\[S:init\]]{}). For the velocities of about $U\approx1$kms$^{-1}$ driving the system (cf. [Sect.\[S:boundary\]]{}), $U\,\delta{z}$ would be of the same order of magnitude as $\eta$, and consequently the magnetic Reynolds number on the grid scale would be about unity. This ensures efficient dissipation at the grid scale and avoids numerical dissipation.
The continuity and momentum equations are given by $$\label{E:mass}
\partial_{t}\rho+\nabla\cdot\left(\rho\mathbf{v}\right)=0,$$ $$\label{E:momentum}
\partial_{t}(\rho\mathbf{v})+\nabla\cdot\left(\rho\mathbf{vv}\right)=-\nabla p-{\rho}g\hat{z}-\nabla\psi\nabla^{2}\psi+\nabla\cdot(\rho\nu\mbox{\boldmath$E$\unboldmath}),$$ where we assume the pressure of an ideal gas, $p=2n{k_{\rm{B}}}T$, with number density $n{=}\rho/m_{\rm{p}}$, Boltzmann’s constant $k_{\rm{B}}$, and temperature $T$. The mass density is denoted by $\rho$ and the proton mass by $m_{\rm{p}}$. The gravitational acceleration $g=274$ms$^{-2}$ operates in the negative $z$ direction ($\hat{z}$ is the unit vector in the $+z$ direction). The Lorentz force is given through $\nabla\psi\nabla^{2}\psi$. The effects of viscosity are included through the strain rate tensor $\mbox{\boldmath$E$\unboldmath}=(\nabla\mathbf{v}+\nabla\mathbf{v}^{T})/2$ and the kinematic viscosity $\nu$. To choose the value for the viscosity $\nu$, we use the same reasoning as above for the resistivity $\eta$. Consequently they have the same values, $\nu{=}\eta$.
Finally, the energy balance is described by $$\label{E:energy}
\partial_{t}p+\nabla\cdot(p\mathbf v)=(\gamma-1)\left(-p\nabla\cdot\mathbf{v}-\nabla\cdot\mathbf{q}+H_{\eta}+H_{\nu}\right).$$ Heat conduction parallel to the magnetic field is through the heat flux $\mathbf{q}=-\kappa_{\parallel}\mathbf{bb}\cdot\nabla T$, with $\mathbf{b}$ being the unit vector along the magnetic field. The coefficient of the thermal conductivity is given by $\kappa_{\parallel}=c_{\rm{v}}\rho\chi$, with the specific heat at constant volume $c_{\rm{v}}$, and the thermal diffusivity $\chi$. For simplicity, we assume a constant diffusivity of $\chi=4.4\times10^9$m$^2$s$^{-1}$. According to classical transport theory, $\kappa_{\parallel}$ should depend on temperature as ${\propto}T^{5/2}$. Considering this temperature dependence, the typical values of $\chi$ at the top of the chromosphere, the transition region and the corona would be of the order of $10^5$m$^2$s$^{-1}$, $10^8$m$^2$s$^{-1}$, and $10^{11}$m$^2$s$^{-1}$, respectively.[^1] Thus the value of $\chi$ we use in our model is higher than expected for the top of the chromosphere or the transition region, that is, the regions where the UV bursts or explosive events we intend to model are located. The higher efficiency of heat conduction in our model implies that the temperatures we find during the reconnection events will be a lower limit.
Energy is added to the plasma through Ohmic dissipation, $H_{\eta}=\eta j^{2}$ (with currents $j{=}|\nabla{\times}B|)$, and viscous dissipation $H_{\nu}=n\nu\,\nabla\mathbf{v}\,{:}\,\mbox{\boldmath$E$\unboldmath}$. The ratio of specific heats is denoted by $\gamma{=}5/3$. We neglected optically thin radiative losses, because the timescale of the events is short compared to the (coronal) radiative cooling times.
The code that is used to solve the above equations under the boundary conditions discussed below has been described in [@1993PhFlB...5.3712G]. More details on the code as well as on testing and validation are given there.
Setup and initial conditions\[S:init\]
--------------------------------------
The simulation stretches in the horizontal direction over $x\in[-L_{x},L_{x}]$ and in the vertical direction over $z\in[0,L_{z}]$. We chose $L_x=6$Mm and $L_z=2.4$Mm motivated by the observations of ; see [Sect.\[S:setup\]]{}. The computational domain is covered by 2400 grid points horizontally and 2000 grid points vertically. Consequently the (equidistant) grid is spaced by $5\,{\rm{km}}\times1.2\,{\rm{km}}$.
The analytical form of the initial magnetic field, $\mathbf B_0=\nabla\psi_0\times\hat{y}$, is defined through $$\label{E:B.initial}
\psi_{0}(x,z;t{=}0)=\left[f_b\frac{L_{x}}{\pi}\sin\left(\frac{\pi x}{L_{x}}\right)\exp\left(\frac{-\pi z}{L_{x}}\right)-x \right] \overline{B}.$$ This represents a potential magnetic field, that is, one can easily show that $\nabla\times\mathbf{B}_0=0$, and of course, $\nabla\cdot\mathbf{B}_0=0$. Most importantly, this magnetic field structure as depicted in [Fig.\[F:setup\]]{}a roughly matches the field geometry motivated by the UV burst observations of . In particular it has an X-type null point at $$\label{E:null}
x_{\rm{null}}=0 \quad , \quad z_{\rm{null}}=\left(L_{x}/\pi\right)~\ln f_b.$$ Considering the horizontal extent of our box, $2L_x=12$Mm, we choose the free parameter $f_b=4/3$ so that the height of the null point is $z_{\rm{null}}\approx550$km. This then closely matches the height of the null point derived by . In the form of [Eq.(\[E:B.initial\])]{}, the average vertical magnetic field in the model region is $\overline{B}$. Here we choose $\overline{B}=75$G, which is typical for a plage or enhanced network area, where UV bursts are seen frequently.
We choose the temperature at the beginning of the simulation to be constant. This is motivated by our goal to simulate a UV burst that happens at the base of the corona. Embedded in an comparatively cool environment the plasma should be heated to transition region temperatures, as expected for a UV burst. Consequently our constant initial temperature is set to $T_0=10^4$K. The initial state is at rest, that is, all velocities are zero.
For the density we choose an initial state with a vertical (barometric) stratification in the atmosphere with the initially constant temperature, $$\label{E:init.density}
\rho\,(x,z;t{=}0) =\rho_{0}\exp\left(-z/H_g\right),$$ where $H_g=(2k_{\rm{B}}T_0)/(m_{\rm{p}}\,g)$ is the barometric scale height. In our setup this is about $H_g\approx600$km. The density at the bottom boundary, $\rho_0$, is a free parameter and is chosen differently in the numerical experiments conducted here (see. [Sect.\[S:experiments\]]{}).
Boundary conditions\[S:boundary\]
---------------------------------
To drive the system, we apply a spatially and temporally variable velocity profile at the bottom boundary. This velocity is always in the positive $x$ direction. The spatially variable part is defined as $$\label{E:v.space}
v_0(x) = \left(\frac{1+\cos\left(\pi x/L_{x}\right)}{2}\right)^{\!\!10}.$$ This closely resembles a Gaussian profile with a full width at half maximum of 2Mm for our choice of $L_x=6$Mm (cf. [Fig.\[F:setup\]]{}c). In contrast to a Gaussian, this form in [Eq.(\[E:v.space\])]{} is strictly periodic and therefore satisfies our horizontal boundary conditions. We keep driving for a timescale of $t_d$. For a smooth transition we ramp the velocity up and down over the timescale $t_r$, $$\label{E:v.time}
v_d(t)=\hat{v} \times
\left\{
\begin{array}{l@{~~~~~~}r@{~}c@{~}l}
\sin\left(0.5\pi t/t_{r}\right) & 0 \le&t&\le t_{r} \\
1 & t_{r} \le &t&\le t_{d}-t_{r} \\
\sin\left(0.5\pi(t_{d}-t)/t_{r}\right) & t_{d}-t_{r} \le &t&\le t_{d} \\
0 & & t&\ge t_d
\end{array}
\right.
.$$ Loosely guided by the observational study of , we choose the driving time to be $t_d=1320$s, or about 20 minutes, and apply a ramping time of one tenth of that, $t_r=132$s, to ensure a smooth transition. The peak driving velocity is $\hat{v}=1.8$kms$^{-1}$, and therefore during the major part of the driving the velocity $v_{\rm{d}}$ is slightly faster than in . This makes up for the slightly shorter period of driving in our numerical experiments. The time profile of the driving as defined in [Eq.(\[E:v.time\])]{} is depicted in [Fig.\[F:setup\]]{}d. Combining the spatial and temporal variation in [Eq.(\[E:v.space\])]{} and [Eq.(\[E:v.time\])]{} yields the final horizontal velocity imposed at the bottom boundary, $$\label{E.v}
v_x\,(x,z{=}0;t) = v_d(t) ~\, v_0(x).$$
The horizontal velocity at the top boundary and the vertical velocities at the top and bottom boundaries are set to zero. In a real system, the plasma outflow from the reconnection region should be free to escape along the spine through the top boundary. In the numerical model, we set the flow to zero at the top boundary for convenience and place a frictional layer near the top boundary to absorb the flow. This mimics the outflow boundary condition and ensures that the plasma outflow does not bounce back from the top boundary and feedback to the reconnection site. The magnetic field at the bottom and top boundaries is line-tied to the flow field that drives the system. The temperatures at the top and the bottom boundaries are fixed at their initial values. The density above and below the top and bottom boundaries, respectively, is extrapolated by a local gravitational stratification, ${\partial}p/{\partial}z=-\rho\,g$ (to fill the ghost cells). The boundary conditions in the horizontal direction are periodic.
Range of numerical experiments\[S:experiments\]
-----------------------------------------------
In our study we also want to investigate the effects that plasma-$\beta$, that is, the ratio of gas pressure to magnetic pressure, has on the UV burst resulting from the driving. To characterize $\beta$ for the different numerical experiments, we use the average $\langle\beta\rangle$ in the dome region in the initial condition. This dome region we define as a horizontal section in $x\in[-2.4,+2.4]$Mm at the height where the X-type null point as defined in [Eq.(\[E:null\])]{} is located. In our model setup the average field strength in the dome region is $\sqrt{\langle{B}^2\rangle}\approx 52$G (we first average $B^2$, because we are interested in the average magnetic pressure to calculate the average $\beta$).
To change $\beta$ in our idealized setup we could change the average magnetic field $\overline{B}$ (and thus $\sqrt{\langle{B}^2\rangle}$), the density at the bottom boundary, $\rho_0$, or a combination of both. Here we opt to change $\rho_0$. In our models we choose $\rho_0$ so that the density in the initial condition at the X-type null point has values from $4.9\times10^{-10}\,{\rm{kg}}\,{\rm{m}}^{-3}$ to $2.4\times10^{-8}\,{\rm{kg}}\,{\rm{m}}^{-3}$. These are typical chromospheric densities [cf. e.g., @1981ApJS...45..635V]. Essentially, with the choice of $\rho_0$ we select whether the X-type neutral point is in the low, middle, or upper chromosphere.
This combination of initial density and magnetic field in the dome region together with the initial temperature of $10^4$K gives a range of $\langle\beta\rangle$ from $0.015$ to $0.735$ in the dome region (even though we also ran more models with lower and higher $\beta$). The respective $\langle\beta\rangle$ values will be given with the figures and the discussion of the results. These averages of $\beta$ in the dome region around the X-type null point apply only for the initial condition. When the numerical experiments evolve, the density, temperature, and magnetic field changes self-consistently and the location of the X-point moves (and eventually stretches into a current sheet). Still, the initial value of $\langle\beta\rangle$ is a good measure to characterize the (average) plasma-$\beta$ in and around the reconnection region.
In our numerical experiments we change the density in order to change plasma-$\beta$ while we keep the magnetic field fixed. What can we expect if we keep the density fixed and change the magnetic field to run models for the same (average) plasma-$\beta$? For a case with the same $\beta$ but higher magnetic field, the magnetic energy density (per volume) would be increased by the same factor as the density. This is because the energy density is ${\propto}B^2$ and $\beta{\propto}B^2/\rho$. Furthermore, we can expect the energy dissipation (or conversion) to increase with the magnetic energy density. Consequently, the heating per particle (i.e., heat input divided by density) will be the same in two models with the same $\beta$, even if the magnetic field and the density are higher. According to the energy equation, the (change of) temperature goes with the heating per particle. Therefore, we can expect to have models with different $\beta $ but the same peak temperatures in the reconnection region, even if the magnetic field is stronger. However, this does not hold if radiative cooling and effects of partial ionization are also included. Increasing the temperature would then become more difficult if the X-point were in a region at higher density (and higher magnetic field so that $\beta$ is the same). This has been shown in the multi-fluid models of [@2018ApJ...868..144N].
Results\[S:results\]
====================
General evolution of the magnetic structure and formation of plasmoids\[S:res.general\]
---------------------------------------------------------------------------------------
![Overall evolution of magnetic field and kinetic energy during reconnection for $\beta{=}0.147$. Panel (a) shows the temporal evolution of the kinetic energy integrated in a rectangle around the reconnection region (dashed lines in lower panels). Panels (b) to (d) show snapshots of the kinetic energy density along with magnetic field lines in part of the computational domain (cf. dashed rectangle in [Fig.\[F:setup\]]{}a). These are taken sequentially at the times indicated in the top panel by the vertical blue lines. The current sheet forms roughly along the diagonal of the rectangle. The compact near-circular enhancements of the kinetic energy indicate the location of the plasmoids that form. The blue double-arrow in panel (a) indicates the time of the driving at the bottom boundary. The plasmoids and the overall evolution are best seen in the animation attached to this figure (also available at <http://www2.mps.mpg.de/data/outgoing/peter/papers/2019-bursts/f2.mp4>); see [Sect.\[S:res.general\]]{}. \[F:evolution\] ](f2_3frames_Ek){width="75mm"}
To describe the overall evolution of the system we first concentrate on one of the numerical experiments, and choose the case where $\beta{\approx}0.15$ at the height of the null point in the initial condition. To illustrate the evolution of the system, we show in [Fig.\[F:evolution\]]{} the kinetic energy together with the changing magnetic connectivity. A movie attached to the figure shows the full temporal evolution. The lower panels (b) to (d) show part of the computational domain. On the background with the kinetic energy density we plot the changing magnetic field lines. The top panel (a) displays the kinetic energy integrated in a rectangle around the reconnection region as outlined by the dashed red lines in panels (b) to (d).
When driving the small opposite magnetic polarity in the positive $x$-direction (right), the magnetic field gets distorted and the X-type neutral point is stretched into a current sheet. The current sheet is visible here through the kinetic energy that is enhanced due to the reconnection outflow that roughly follows the diagonal of the dashed rectangles in [Fig.\[F:evolution\]]{} (we discuss the currents later in [Sect.\[S:res.beta\]]{} and [Fig.\[F:current.sheets\]]{}). We see a clear increase of the kinetic energy around the reconnection region due to the reconnection outflow associated with the motion of plasmoids ([Fig.\[F:evolution\]]{}a). After an initial phase, this enhancement of the kinetic energy essentially lasts as long as we drive the system (i.e., up to time $t{\approx}1300$s; cf. [Fig.\[F:setup\]]{}d). Details of the temporal evolution are presented in [Sect.\[S:res.evolution\]]{}.
![Zoom into reconnection region showing plasmoids and temperature for $\beta{=}0.147$. This is similar to [Fig.\[F:evolution\]]{}, but we now show the temporal evolution of the maximum temperature around the reconnection region in panel (a) and a snapshot of the temperature in panel (b) that is taken at the time indicated by the blue vertical line in panel (a). At this snapshot there are two (resolved) plasmoids present in the reconnection region clearly seen as near-circular regions of enhanced temperature in the middle of panel (b). The field of view corresponds to the rectangle indicated in [Fig.\[F:evolution\]]{}b-d. The blue double-arrow indicates the time of the driving at the bottom boundary; see [Sect.\[S:res.general\]]{}. \[F:zoom\] ](f3_T_crop){width="75mm"}
The plasmoids form continuously and are regions of enhanced temperature and density. This is underlined by [Fig.\[F:zoom\]]{} where we show a zoom into the reconnection region. In the snapshot we show here, two plasmoids are clearly visible as compact, near-circular enhancements of the temperature (see [Sect.\[S:disc.beta\]]{} for a discussion of the size of plasmoids and resolution). Also, the thin current sheet is visible as a thread of enhanced temperature running diagonally through the two plasmoids. The plasmoids are hotter for a couple of reasons. Ohmic heating is stronger in the current sheets, that is, in regions between plasmoids rather than in the plasmoids themselves. The heated plasma is ejected into the plasmoids and trapped there. The plasmoids then retain the heat because they are essentially thermally insulated. This is because they are magnetic islands and thermal conduction is parallel to the field lines. Furthermore, the plasma is heated by adiabatic compression due to the magnetic tension force in plasmoids. This latter effect also causes the higher density within the plasmoids. Consequently, the peak temperature ([Fig.\[F:zoom\]]{}a) as well as the kinetic energy (cf. [Fig.\[F:evolution\]]{}) are closely connected to the presence of plasmoids: whenever plasmoids are present there will be (local) maxima in temperature and in kinetic energy. We discuss the resulting temporal substructure in [Sect.\[S:disc.beta\]]{} and compare it to observations.
In the outflow regions (top left and bottom right regions in [Fig.\[F:zoom\]]{}b), that is, in the region that is fed by the reconnection outflow, we see a clear enhancement of the temperature as compared to the inflow regions. The case for the kinetic energy shown in [Fig.\[F:evolution\]]{} is the same (but less clear). As expected, the reconnection energizes a region much larger than just the immediate surrounding of the current sheet.
Temporal evolution of the reconnection and plasma-$\beta$\[S:res.evolution\]
----------------------------------------------------------------------------
The different experiments we conduct are distinguished by the value of the average plasma-$\beta$ at the height of the reconnection region (at the initial condition; cf. [Sect.\[S:experiments\]]{}). To characterize how the reconnection evolves in the different cases, we consider a sub-region of size 2Mm $\times$ 1Mm that fully encloses the current sheet, the plasmoids, and the reconnection outflows (indicated by the dashed boxes in [Fig.\[F:evolution\]]{}b-d). In this region we track changes of three characteristic quantities as a function of time, namely the maximum current, $J_{\rm{max}}$, the maximum velocity, $V_{\rm{max}}$, and the integrated kinetic energy, $E_{\rm{kin}}$.
The driving at the bottom boundary is the same for all the cases studied here (only the density stratification changes; see [Sect.\[S:experiments\]]{}). Therefore the difference in for example the energy deposition in the reconnection processes in the different cases is solely due to the different evolution of the magnetic field within the computational domain. The experiments with values of plasma-$\beta$ from ${\approx}0.75$ to ${\approx}0.01$ range from (almost) plasma-dominated to magnetic-field-dominated in the vicinity of the reconnection region. This then determines how efficiently reconnection can operate.
The overall evolution of the maximum current, maximum velocity, and kinetic energy for all the cases is similar ([Fig.\[F:evolution.beta\]]{}). These quantities increase, reach a peak while the driving at the footpoints is active, and then decline; they also show a substructure in the temporal evolution that looks almost like noise, but is real and is related to the presence of plasmoids. This is discussed further when relating our models to observations in [Sect.\[S:disc.beta\]]{}. The high-$\beta$ cases are different in these quantities ($J_{\rm{max}}$, $V_{\rm{max}}$, $E_{\rm{kin}}$) in that they increase only later or do not reach a peak at all while driving.
### Current sheet and plasmoids {#S:curr.sheet.plasmoids}
For the maximum current, $J_{\rm{max}}$, we see a clear ordering of the timing of its sharp rise with plasma-$\beta$ ([Fig.\[F:evolution.beta\]]{}a). This is mainly due to the different Alfvén crossing times from the surface to the X-type neutral point, that will be longer for the larger $\beta$ values. For lower $\beta$ values the Alfvén speed is higher and consequently the disturbance induced by the surface motions will reach the neutral point earlier and stretch it into a current sheet much sooner and quicker.
For small $\beta$, below about 0.1, the current sheet forms essentially at the same time as the driving begins. As a result, the current sheet will be shorter as compared to the experiments with higher $\beta$. This is evident from a comparison of the snapshots of the currents in panels (a) and (b) of [Fig.\[F:current.sheets\]]{} that show the cases for $\beta$ at about 0.015 and 0.15, respectively.
{width="184mm"}
In the cases of high $\beta$, above about 0.4, the current sheet forms later not only because of the longer Alfvén crossing time, but also because the inertia of the plasma around the X-type neutral point can keep the magnetic field from forming a current sheet, at least for a while. The resulting current sheets will be shorter at these higher values of $\beta$ (cf. [Fig.\[F:current.sheets\]]{}c), because the stressing of the magnetic field has to work against the inertia of the plasma. This also causes longer-lasting significant magnetic gradients, that is, high currents lasting for longer (see case $\beta{=}0.735$ in [Fig.\[F:evolution.beta\]]{}a).
In all models we find that plasmoids develop irrespective of the value for plasma-$\beta$ at the (initial) location of the X-type neutral point or the current sheet. They can be seen in all quantities; in the current density ([Fig.\[F:current.sheets\]]{}) of course, but also in the kinetic energy ([Fig.\[F:evolution\]]{}) and in the temperature ([Fig.\[F:zoom\]]{}a). As expected, plasmoids will move away from the center of the current sheet and finally collide with the ambient medium and thermalize (see the animation attached to [Fig.\[F:evolution\]]{} to see the motion of the plasmoids).
### Maximum velocity
The reconnection outflow will be driven by the Lorentz force in the current sheet, with higher currents implying stronger forcing. In the case of low $\beta$ the density in the vicinity of the current sheets will be smaller, and thus the Lorentz force can accelerate the gas to higher speeds. Thus the resulting maximum velocity, $V_{\rm{max}}$, shows a comparable time variation as the maximum currents at the same $\beta$, but the highest $V_{\rm{max}}$ in each $\beta$ case drops with increasing $\beta$. This is the simple consequence of the Lorentz force being applied to plasma at lower density.
### Kinetic energy\[S:res.kinetic\]
The time variation of the kinetic energy, $E_{\rm{kin}}$, integrated in the vicinity of the reconnection region shows a two-part evolution: an increase at the beginning of the driving and then an evolution consistent with the maximum current and velocity. In the initial phase of the driving, the horizontal motion at the bottom boundary simply carries the whole system in the positive $x$-direction and thus causes the initial bump of $E_{\rm{kin}}$ at $t{\approx}200$s ([Fig.\[F:evolution.beta\]]{}c). The speed of this motion is slow, of the order of less than 1 kms$^{-1}$ and therefore does not show up in the maximum velocity ([Fig.\[F:evolution.beta\]]{}b), but because $E_{\rm{kin}}$ is integrated in a rather large volume, the bump is visible. This small peak during the first 200s in $E_{\rm{kin}}$ is lower if $\beta$ is lower (and hence density is also lower). This simply reflects the scaling of the kinetic energy with the density. While not being clear in the plot in [Fig.\[F:evolution.beta\]]{}c, closer inspection shows that this early bump (at much smaller amplitude) is also present for the very small $\beta$ cases. With the system finding some type of driving equilibrium, this increase of the kinetic energy vanishes again. This initial bump of $E_{\rm{kin}}$, most prominent at high $\beta$, is an artifact of our driving and we do not discuss it further.
The most important part of the evolution of the kinetic energy, $E_{\rm{kin}}$, in the reconnection region is the main part that is closely related to the maximum velocity, $V_{\rm{max}}$. The envelope of $V_{\rm{max}}$ reaches its highest values for small $\beta$ ([Fig.\[F:evolution.beta\]]{}b). In contrast, $E_{\rm{kin}}$ reaches only small values for very low $\beta$ ([Fig.\[F:evolution.beta\]]{}c). This is again because of the lower density for the low-$\beta$ cases. Thus, with increasing $\beta$ also the maximum $E_{\rm{kin}}$ increases, but only up to values of $\beta$ of around 0.1. For higher values of $\beta$, the inertia of the plasma can work strong enough against the magnetic driving, so that the outflow velocities drops faster with $\beta$ than the density increases associated with higher $\beta$. Consequently, for values of $\beta$ significantly above 0.1 we find only a small increase of $E_{\rm{kin}}$ during the reconnection event.
In addition to this evolution of the envelope of $E_{\rm{kin}}$ (and also $J_{\rm{max}}$ and $V_{\rm{max}}$), we find a substructure in the temporal variation with a timescale of about one minute ([Fig.\[F:evolution.beta\]]{}) that looks almost like noise. This substructure is clearly associated with the presence of plasmoids that form in the current sheet naturally through the plasmoid instability. This can be seen upon close inspection of [Fig.\[F:evolution\]]{}. The snapshots in panels (b) and (d) are taken when there are no (or small) plasmoids in the current sheet. At these times $E_{\rm{kin}}$ has local minima ([Fig.\[F:evolution\]]{}a). When there is a plasmoid (in the snapshot in [Fig.\[F:evolution\]]{}c), there is a local maximum of $E_{\rm{kin}}$ in [Fig.\[F:evolution\]]{}a. This relation between plasmoids and local peaks of the kinetic energy is very clear when following the animation that is associated with [Fig.\[F:evolution\]]{}. Likewise, there is a tight relation between the presence of plasmoids and $J_{\rm{max}}$ and $V_{\rm{max}}$ as well as the maximum temperature in the reconnection region, $T_{\rm{max}}$. The timescale of the order of one minute for this substructure is related to the time a plasmoid moves along the current sheet, bounded by the Alfvén speed. This is discussed further in [Sect.\[S:disc.beta\]]{}.
Reconnection, plasma-$\beta$, and temperature\[S:res.beta\]
-----------------------------------------------------------
We now turn to the peak temperatures during the reconnection event for the runs with different plasma-$\beta$. In [Fig.\[F:T.plasma.b\]]{}a we show the maximum temperature in the reconnection region as a function of time. Here for each time we evaluated the maximum value of the temperature in the vicinity of the current sheet (i.e., in the rectangles shown in [Fig.\[F:evolution\]]{}b-d, and the full region in [Fig.\[F:zoom\]]{}b). We see a clear trend that higher maximum temperatures are reached for lower plasma-$\beta$.
To quantify this, we estimate the peak temperature $T_{\rm{peak}}$ during the evolution of the maximum temperature in [Fig.\[F:T.plasma.b\]]{}a for each run. The maximum temperature in the reconnection region, $T_{\rm{max}}$, varies rapidly because of the presence of plasmoids ([Sect.\[S:curr.sheet.plasmoids\]]{}). Therefore we employ the following procedure to avoid spurious results due to individual peaks. We evaluate $T_{\rm{max}}$ during the time period after it first becomes larger than half of the overall maximum value, until it finally drops below half of the overall maximum value and never increases back again. We define $T_{\rm{peak}}$ as the average of $T_{\rm{max}}$ over this period. These values of $T_{\rm{peak}}$ are shown in [Fig.\[F:T.plasma.b\]]{}b for the models with plasma-$\beta$ ranging from 0.007 to above unity.
We find that the peak temperature $T_{\rm{peak}}$ and plasma-$\beta$ are connected (roughly) by a power law, $T_{\rm{peak}}\propto\beta^{\,-0.3}$ (cf. [Fig.\[F:T.plasma.b\]]{}b). For high plasma-$\beta$ we find no significant enhancement of the temperature, simply because the density is too high, meaning that the deposited energy can lead only to a modest increase in temperature. The smaller plasma-$\beta$ , the higher the temperature enhancement can be, because the added energy is now distributed over many fewer particles. In our models we can find the temperature rising to about 50000K.
Discussion\[S:discussion\]
==========================
Self-consistent formation of an inclined current sheet\[S:disc.current.sheet\]
------------------------------------------------------------------------------
One of the main goals of this study is to answer the question of whether or not footpoint motions at the solar surface can create a current sheet higher up in the atmosphere in which reconnection would lead to a bi-directional outflow like in an explosive event or UV burst. The short answer is yes.
There have been other models of reconnection events in the upper atmosphere for explosive events or UV bursts. But these are idealized in so far as they assume that oppositely directed magnetic fields already exist in the initial configuration of the numerical experiment, for example in the form of a Harris-type current sheet (see [Sect.\[S:intro\]]{}).
The initial condition in our model with the X-type neutral point would be stable if we did not drive the system. Only after the footpoint driving starts, is the X-point stretched into a current sheet. Thus the current sheet in our system forms self-consistently, which is a major difference as compared to earlier models for explosive events (as outlined in [Sect.\[S:intro\]]{}). Essentially, this process first leads to a (slow) Sweet-Parker-type reconnection, after which the plasmoid instability begins to take effect and greatly enhances the efficiency of the energy conversion process.
Considering the geometry of the magnetic setup ([Figs.\[F:setup\]]{} and [Fig.\[F:evolution\]]{}b-d), it is natural that the resulting current sheet will be inclined to the vertical. Earlier models with a gravitational stratification were experimenting with cases where the current sheet was horizontal or vertical [e.g., @2016ApJ...832..195N], but they simply assumed an orientation. The magnetic setup we use in our model matches not only the observations by but also the more general observation that explosive events typically happen at locations where opposite polarities come into close contact and cancel [@1998ApJ...497L.109C]. Therefore, we can assume that our finding of an inclined current sheet in explosive events is quite general. Consequently, the (bi-directional) outflow from the reconnection site will also be inclined, as has been inferred by the spectroscopic observations of [@1997Natur.386..811I]. Such inclined outflows have also been suggested for UV bursts [see Sect.S3 and Fig.S7 in the supplemental material of @2014Sci...346C.315P].
We also see that the orientation (or inclination) of the current sheet in our model is consistent with the observations of . In our model, the upward-directed part of the outflow is pointing away from the main polarity that the small opposite polarity is moving into. In the observations of , the blueshifted, i.e., upflowing, part of the UV bursts is located on the side of the UV bursts pointing away from the pore, that is, the main polarity. This is yet another nice match between model and observations, even though the analysis of more observational cases is needed to finally confirm this.
Ultraviolet bursts in a low-$\beta$ environment: temporal evolution and plasmoids\[S:disc.beta\]
------------------------------------------------------------------------------------------------
The reconnection in our model is driven by the motion at the bottom boundary (i.e., the solar surface) moving the small opposite-polarity magnetic patch into the main polarity. As such, the timescale of the driving determines the overall evolution of the system. Still, independent of the driving, the dynamics in the reconnection region lead to a temporal variability on shorter timescales (${\approx}1$min) controlled by plasmoid formation. This is consistent with observations of UV bursts.
We find that the conversion of energy in the reconnection region lasts for about 1000s ([Fig.\[F:evolution.beta\]]{}, [Sect.\[S:res.evolution\]]{}). Once the driving stretched the current sheet sufficiently, the plasmoid-mediated reconnection begins and lasts as long as we drive ([Sect.\[S:curr.sheet.plasmoids\]]{}). While the duration of the reconnection event follows from our choice of the model parameters, our choice for the duration of the driving is not completely free. Instead, the driving time we choose is governed by observations, or more precisely by the (horizontal) magneto-convective motions on the surface. In the UV burst discussed by the timescale of driving the small magnetic patch into the pore is about 10 to 20min; hence our choice of about 20min for the driving time. In a more general context, this timescale corresponds to a motion with about 1kms$^{-1}$ over a distance of about 1Mm. These velocities and length scales are typical for granular magneto-convection near the surface. Thus we can also consider our assumption for the driving time to be typical in cases where magnetic elements are pushed into (opposite) magnetic patches, for example with explosive events in the network [@1998ApJ...497L.109C].
During the time of the reconnection we see a clear bi-directional outflow from the reconnection region. This is consistent with observations of enhanced wings or a bi-modal shape of line profiles that is seen in explosive events [e.g., @1989SoPh..123...41D] or in part of the UV bursts [e.g., @2014Sci...346C.315P]. At the same time, our model also shows plasmoids forming along the current sheets that can explain the enhanced emission observed in the line core, similar to the model of [@2015ApJ...813...86I].
{width="150mm"}
What we find in our study is that the plasmoids are key to understanding the substructure seen in the temporal evolution of UV bursts, that is, their light curve. In [Fig.\[F:obs.burst\]]{} we provide an example for the intensity variability during a UV burst. This shows that the event lasts for about one half to three quarters of an hour with a large number of local peaks in intensity, each lasting one or a few minutes. Another example is shown in Fig.3 of [@2018SSRv..214..120Y] with substructures spaced by some 30s. Yet another case is discussed in [@2015ApJ...809...82G] who found a flickering with a period from 30 to 90s. In our models we see a similar sub-structure on timescales of 1 to 2min in all quantities derived from the reconnection region ([Figs.\[F:zoom\]]{}a, \[F:evolution.beta\], \[F:T.plasma.b\]a), and this substructure can be clearly related to the presence of plasmoids ([Sect.\[S:res.general\]]{}). Assuming that the UV bursts are driven by reconnection, it stands to reason that the observed flickering in UV bursts is also caused by plasmoids, with the enhanced temperature and density in the plasmoids leading to the enhancements in intensity (see also [Sect.\[S:disc.int\]]{}).
The plasmoids move along the current sheet reaching (almost) the local Alfvén speed. In our setup the Alfvén crossing time along the current sheet is of the order of 10 to 20s. However, this can only be a lower limit for the crossing of the plasmoids. When formed, the plasmoid first has to accelerate and only when almost at the end of the current sheet hitting the ambient media does it reach the Alfvén speed. Typically the time it takes the plasmoid to cross from the middle of the current sheet to its end is about 1 min. The density and magnetic field in our model are motivated by a UV burst observation, and hence we expect that the (local) Alfvén speed in a UV burst is comparable to what we find in our model. We therefore conclude that the plasmoids are indeed responsible for the 30 to 90s flickering seen in UV bursts.
In our model, at any given time we see one or two (large) plasmoids in the current sheet (e.g., [Fig.\[F:zoom\]]{}b). This would also hold if we were to increase the spatial resolution. For higher resolution, one expects that there are still large plasmoids, typically one in each current sheet at any given time, but these are now accompanied by smaller ones in a self-similar fashion [@2012PhRvL.109z5002H]. Essentially, at (much) higher resolution we would still see the clear peaks in all quantities of about one minute in length due to the (large) plasmoids, similar to the observed sub-structure in UV bursts. But now a noise-like pattern on much shorter timescales would be superimposed by plasmoids over a wide range of smaller scales.
Plasmoids triggering fast-mode waves\[S:disc.wave\]
---------------------------------------------------
The plasmoids that appear in a quasi-periodic pattern also launch fast magneto-acoustic waves. A more thorough analysis of this phenomenon would deviate from the main goal of this study. Therefore, here we only refer to the movie attached to [Fig.\[F:evolution\]]{} that shows the field lines shaking back and forth in a fashion expected for a fast-mode wave propagating away from the reconnection site. Recently, such a phenomenon was observed in the corona above the limb by [@2018ApJ...868L..33L], albeit at larger scales. These latter authors found a fast-mode wave with a period of about 4 min lasting for many hours. In their observation they find evidence for reconnection lasting all that time, so we can speculate that plasmoids forming in the process could give rise to the excitation of fast-mode waves over such a long time in a coherent fashion.
Intensity from plasmoids compared to UV bursts\[S:disc.int\]
------------------------------------------------------------
The reconnection events we find in our model are energetic enough to power a typical UV burst. At least, the intensity emerging from a plasmoid is consistent with the intensity observed in solar UV bursts.
In the plasmoids of our run with $\beta{=}0.007$ we typically find a peak temperature of about 50000K and densities of about $5{\times}10^{11}$cm$^{-3}$. Assuming a line-of-sight length of 0.2Mm (representing the size of the plasmoid; cf. [Fig.\[F:current.sheets\]]{}), we calculate an intensity radiating from the structure of about $10^{5}$ergcm$^{-2}$s$^{-1}$sr$^{-1}$. For this estimate we use the Chianti database and the procedure as described in [@2006ApJ...638.1086P]. This fits well with the value of $5{\times}10^{5}$ergcm$^{-2}$s$^{-1}$sr$^{-1}$ for the UV burst observations quoted by [@2018SSRv..214..120Y] in their Sect.4. We therefore conclude that our model can also give a quantitative explanation for the UV burst intensity.
Why are explosive events restricted to temperatures below a few 10$^{\,5}$K?\[S:disc.temp\]
-------------------------------------------------------------------------------------------
As mentioned in the introduction, the question remains open as to why explosive events are observed only in spectral lines forming below about 0.4MK. Here, our results on how the peak temperature in the reconnection region depends on plasma-$\beta$ are instructive ([Sect.\[S:res.beta\]]{}, [Fig.\[F:T.plasma.b\]]{}b) and can provide an answer.
Clearly, we find a monotonic increase of the peak temperature for smaller $\beta$. In short, if plasma-$\beta$ is smaller in the reconnection region, there are less particles to be heated and thus the temperature can reach higher values. While the upper solar atmosphere is a low-$\beta$ plasma, $\beta$ cannot be arbitrarily small and thus there is an upper limit for the temperature to be reached (realistically) during an explosive event. Of course, this is particular for this model, and in a different setup, for example with continued flux emergence and higher driving speeds supported for longer times, one might reach coronal temperatures in the event [@2018ApJ...864..165W].
From a self-consistent 3D MHD model of the upper solar atmosphere one can derive plasma-$\beta$. In their model, [@2006ApJ...638.1086P] showed that the smallest values of $\beta$ are found in the (low) transition region at temperatures from $10^4$K to $10^5$K. Typically, there $\beta$ is larger than $10^{-3}$, and essentially never below $10^{-4}$ [Fig.12c of @2006ApJ...638.1086P]. Therefore, we can consider $10^{-4}$ as a lower limit for plasma-$\beta$. In particular, this lower limit is also applied in the source region of typical transition region lines such as [Fig.12a of @2006ApJ...638.1086P] and .
We can now extrapolate the peak temperatures down to the smallest plasma-$\beta$ values to be expected and use that temperature as an upper limit for the temperatures to be expected in explosive events and UV bursts. The lowest plasma-$\beta$ case we have in our numerical experiments is about 0.007. Extrapolating using the power law based on [Fig.\[F:T.plasma.b\]]{}b, we find values of the peak temperature of just below 0.1MK for $\beta{=}10^{-3}$ and 0.2MK for $\beta{=}10^{-4}$. Consequently, we would not expect temperatures in UV bursts and explosive events to reach values much higher than 0.2 MK. On the real Sun this upper limit might be slightly higher, because we most likely underestimate the temperatures in the reconnection region as discussed in [Sect.\[S:model.eqs\]]{} following [Eq.(\[E:energy\])]{}. Therefore we consider our finding of a maximum reachable temperature in UV bursts and explosive events to be consistent with the observations.
Of course, this result applies only to situations for a driving as assumed in our model, that is, if one small patch of one magnetic polarity is moving into and canceling a larger patch of opposite polarity. Under other circumstances the driving of the magnetic field might be located closer to the reconnection location, as expected for Ellerman bombs or some UV bursts [see cartoons in @2002ApJ...575..506G; @2014Sci...346C.315P]. A model might therefore produce much higher temperatures. Still, such hot plasma, even if reaching coronal temperatures, might not show up in extreme UV emission (e.g., in the widely used 171[Å]{} band showing mostly coronal plasma in ). This is because of absorption by overlying cooler material in the Lyman-continua of hydrogen and helium, [cf. suppl. material SM2 of @2014Sci...346C.315P].
Relation to Ellerman bombs and high-$\beta$ reconnection\[S:disc.EB\]
---------------------------------------------------------------------
The particular configuration we use here cannot explain more violent bursts at deeper atmospheric layers, that is, events that originate where plasma-$\beta$ is close to unity or even larger. For the particular driving mechanism we employ here, we do not find any significant increase of the kinetic energy or of the temperature during the reconnection event for values of $\beta$ larger than about 0.5 (cf. [Figs.\[F:evolution.beta\]]{} and \[F:T.plasma.b\]). When driving the magnetic configuration from the footpoints, the inertia of the plasma can hinder the propagation of the changes of the magnetic field into the upper atmosphere ([Sect.\[S:res.kinetic\]]{}). Consequently, there is only a small amount of energy conversion and hence no observational consequences are expected if plasma-$\beta$ is close to or above unity.
Of course, reconnection can happen also in a high-$\beta$ plasma [e.g., @2010PhPl...17f2104H]. However, in this situation the reconnection has to be driven by a flow of plasma converging in the reconnection region. This is exactly the scenario that recent 3D models suggest for Ellerman bombs originating in the photosphere, that is, in a region where $\beta$ is above unity on average****. Here the converging horizontal flows arising from the granular convective motions push together opposite magnetic field which then drives magnetic reconnection .
In a configuration like the one we use here to study explosive events and UV bursts we cannot explain Ellerman bombs. This is because the (horizontal) motions in the high-$\beta$ regions that push together oppositely directed magnetic field are not present. On the other hand, in a scenario in which an Ellerman bomb is initiated, one might well expect this to also induce effects at higher temperatures resembling UV bursts; this has indeed been suggested recently based on observations by [@2019ApJ...875L..30C]. Therefore, one might speculate that a higher fraction of Ellerman bombs is accompanied by UV bursts, rather than the other way round. However, more modeling and observational efforts are needed to conclude on the processes connecting these transient events.
Applicability of the model to UV bursts\[S:limitation\]
-------------------------------------------------------
One major limitation of the model is the incompleteness of the physical processes in the chromosphere. Observations indicate that UV bursts are launched somewhere between the temperature minimum and the middle chromosphere . In response to the reconnection, the local plasma gets heated and radiates in emission lines such as . Therefore, during the onset of the reconnection, we have to expect the plasma to be only partially ionized. This would require consideration of multi-fluid effects, as was the case in recent numerical reconnection experiments [@2018ApJ...852...95N; @2018PhPl...25d2903N; @2018ApJ...868..144N]. These models showed that nonequilibrium (partial) ionization alters the radiative cooling and thus can significantly affect the temperature evolution in the reconnection region. In consequence, the temperature increase would be lower than what we find in our numerical experiments that do not consider multi-fluid effects.
If we included multi-fluid effects and set off the reconnection near the temperature minimum, i.e., below 5000K, the peak temperature would be lower than what we find in our study. However, not all transients seen in the extreme UV will originate at the bottom of the chromosphere. For example, explosive events seen in the quiet Sun are generally thought to be reconnection events starting at lower densities (i.e., higher temperatures) than what is found in the chromosphere. For those events, as we discuss in [Sect.\[S:disc.temp\]]{}, our setup with an initial temperature of $10^4$K, that is, well above the temperature minimum, might provide a more realistic temperature estimate.
Conclusions\[S:conclusions\]
============================
In our study we investigate a reconnection model around an X-type neutral point suitable to understand the dynamics in an explosive event or UV burst. The magnetic setup is motivated by observations showing that such events usually take place in situations where a (small) patch of one magnetic polarity is moving into a (main) patch of opposite polarity. In particular, we select our simulation parameters based on the observations of .
The driving stretches the X-point into a current sheet and quickly the plasmoid instability takes effect. The enhancement of for example kinetic energy and temperature in the reconnection region is very closely related to the presence of plasmoids. In contrast to earlier studies, in our model the current sheet forms self-consistently in response to the driving of the (small) magnetic patch at the bottom boundary of our computational box, that is, close to the solar surface ([Sect.\[S:disc.current.sheet\]]{}).
The enhancement of the energy deposition essentially lasts as long as we drive the system. However, we find a fine structure in the temporal evolution with short busts lasting for around one minute. This intermittency is governed by the plasmoids, and the timescale corresponds to the Alfvén crossing time along the current sheet. This corresponds very well to the observed sub-structure (or flickering) of the light curve observed in UV bursts ([Sect.\[S:disc.beta\]]{}). Estimating the radiative losses from our reconnection experiments, we find that these roughly match the observations of UV bursts ([Sect.\[S:disc.int\]]{}).
One main goal of our study was to investigate the effect of plasma-$\beta$ on the reconnection. For high values of $\beta$, that is, if the thermal energy dominates, we find that this reconnection process driven from the surface is not efficient. In this case, the inertia of the plasma can hinder the changes of the magnetic field in reaching the X-type neutral point. For low $\beta$, that is, if the magnetic field dominates, the driving has an increasing effect and hence the resulting temperatures and the velocities found in the reconnection region increase with decreasing $\beta$. However, we cannot expect this process to reach arbitrarily high temperatures because generally, plasma-$\beta$ will never drop below $10^{-4}$ in the solar atmosphere. Therefore, practically, we find that the temperatures in the reconnection region should not reach values significantly above 10$^5$K. This is in accordance with observations that show that explosive events are essentially limited to a narrow temperature range of about 10$^5$K. Still, under certain circumstances in UV bursts, with different driving from that applied here, one might expect coronal temperatures ([Sect.\[S:disc.temp\]]{}).
Our reconnection models provide new insight into the physics of explosive events and UV bursts. These can be driven by motions of magnetic patches at the solar surface, self-consistently resulting in current sheets and plasmoids. Using this approach we reproduce key features such as duration, temporal sub-structure of the light curve (flickering), and the preferential temperatures of explosive events around 10$^5$K.
We gratefully acknowledge the constructive comments from the referee. L.P.C. received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sk[ł]{}odowska-Curie grant agreement No. 707837. This work was supported by the Max Planck/Princeton Center for Plasma Physics sponsoring trips of H.P., L.P.C, and Y.-M.H. The simulations were performed using facilities of the National Energy Research Scientific Computing Center. H.P. acknowledges the hospitality at the Princeton Plasma Physics Laboratory during his stay. P.R.Y. acknowledges funding from NASA grant NNX15AF48G, and thanks the Max Planck Institute in Göttingen for kind hospitality during visits in 2015 and 2016. Y.-M.H. acknowledges funding from NSF grant AGS-1460169, DOE grant DE-SC0016470, and NASA grant 80NSSC18K1285, and thanks the Max Planck Institute in Göttingen for kind hospitality during visits in 2016 and 2017. This research has made use of NASA’s Astrophysics Data System.
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[^1]: Assuming temperatures of $10^4$K, $10^5$K, and $10^6$K in an isobaric atmosphere with a typical coronal density of $10^{9}$cm$^{-3}$ ($\hat{=}10^{-12}$kgm$^{-3}$) at $10^6$K and using $\kappa_{\parallel}=10^{-11} (T[{\rm{K}}])^{5/2}$Wm$^{-1}$K$^{-1}$ [e.g., @Priest:1982].
|
---
abstract: 'The presence of flavor symmetries in the lepton sector may have several consequences for the generation of the baryon asymmetry of the Universe via leptogenesis. We review the mechanism in general type-I, type-II and type-III seesaw models. We then turn to the discussion of the cases when the asymmetry is generated in the context of seesaw models extended with flavor symmetries, before or after flavor symmetry breaking. Finally we explain how the interplay between type-I and type-II seesaws can (or not) lead to viable models for leptogenesis even when there is an exact mixing pattern enforced by the flavor symmetry.'
bibliography:
- 'refs.bib'
---
**The role of lepton flavor symmetries**\
**in leptogenesis**\
D. Aristizabal Sierra$^{a,}$[^1], I. de Medeiros Varzielas$^{b,}$[^2]\
$^a$[[*IFPA, Dep. AGO, Universite de Liege, Bat B5,\
*Sart Tilman B-4000 Liege 1, Belgium**]{}]{}.\
$^b$[[*Facultät für Physik, Technische Universität Dortmund\
D-44221 Dortmund, Germany*]{}]{}.\
Introduction {#sec:intro}
============
Baryogenesis via leptogenesis is a scenario in which the baryon asymmetry of the Universe is first generated in leptons and partially reprocessed—via standard model sphaleron processes—into a baryon asymmetry. From a general point of view three conditions (Sakharov conditions [@Sakharov:1967dj]) have to be satisfied in order for leptogenesis to take place at some stage during the evolution of the expanding Universe, namely there must be ($i$) interactions that break lepton number; ($ii$) CP violation; ($iii$) departure from thermodynamical equilibrium. In principle any framework in which these conditions can be satisfied can be regarded as a playground for leptogenesis.
In models for Majorana neutrino masses lepton number is broken, so they provide intrinsic frameworks for leptogenesis. The most well studied scenarios for leptogenesis correspond to the type-I [@Minkowski:1977sc; @Yanagida:1979as; @Glashow:1979nm; @GellMann:1980vs; @Mohapatra:1979ia; @Schechter:1980gr], type-II [@Schechter:1980gr; @Lazarides:1980nt; @Mohapatra:1980yp; @Wetterich:1981bx] and type-III [@Foot:1988aq] seesaw models (tree-level realizations of the dimension five effective operator ${\cal O}_5\sim \ell\ell HH$ [@Weinberg:1980bf]). In these cases the generation of a $B-L$ asymmetry proceeds via the decay of heavy fermion right-handed electroweak singlets (RH neutrinos for brevity) (see e.g. [@Davidson:2008bu]) or triplets [@Hambye:2003rt; @Hambye:2005tk] (type-I or type-III) or scalar $SU(2)$ triplets [@Hambye:2003ka; @Antusch:2004xy; @Hambye:2005tk]. Due to the different electroweak charges of these states their thermodynamical behavior is different and so is the way in which leptogenesis takes place.
The idea of flavor symmetries dates back to the late 1970’s [@Froggatt:1978nt]. Initially, due to the lack of experimental data in the lepton sector, flavor symmetries were used to explain quark masses and mixing patterns, but with the advent of neutrino data [@Schwetz:2011zk; @GonzalezGarcia:2010er; @Fogli:2011qn] the idea was increasingly extended to the lepton sector. In particular, in recent years it has been shown that lepton mixing is well described by non-Abelian flavor symmetries (see e.g. [@Altarelli:2010gt]). In association with these developments, the issue of leptogenesis in flavor models has attracted a great deal of attention.
In this short review we describe the relationship between flavor symmetries and leptogenesis. In section \[sec:lepto-gen\] we provide a brief review of general aspects of leptogenesis, covering leptogenesis in type-I seesaw, and also leptogenesis in type-II and III seesaw models. We take some care in establishing the notation to be used in the other sections. The connection with flavor symmetries has been covered in several works: [@AristizabalSierra:2007ur; @AristizabalSierra:2009bh; @Sierra:2011vk] cover the flavor symmetric phase and are reviewed in section \[sec:sym\]. In section \[sec:single\] we review the results of [@Jenkins:2008rb; @Hagedorn:2009jy; @Bertuzzo:2009im; @AristizabalSierra:2009ex; @Felipe:2009rr; @Choubey:2010vs], addressing the case where only type-I seesaw takes place and identify cases where the presence of the symmetries can lead to strong predictions for the viability of leptogenesis. In section \[sec:both\] we summarize the results from [@AristizabalSierra:2011ab], where the scenario considered has both type-I and II seesaws taking place—here too, in certain circumstances conclusions about leptogenesis can be derived due to the presence of the flavor symmetry. Other papers studying leptogenesis in the context of a flavor symmetry include [@Mohapatra:2005ra; @Antusch:2006cw; @Adhikary:2008au; @Lin:2009ic; @Branco:2009by; @Altarelli:2009kr; @Riva:2010jm] although here we do not review their results explicitly.
For consistency we employ the same notation throughout the review. The notation is mostly based on what was used in [@AristizabalSierra:2009ex] and parts of the notation used in [@AristizabalSierra:2011ab] (in particular where type-II seesaw is discussed, such as in section \[sec:both\]). In general we consider the basis where the charged lepton mass matrix is diagonal. For type-I seesaw we also take the basis where the RH neutrino mass matrix is diagonal unless otherwise stated. Matrices that appear with a hat are in the basis where that matrix is diagonal (e.g. $\pmb{\hat{m}_D}$) and we denote matrices in boldface.
Leptogenesis: generalities {#sec:lepto-gen}
==========================
We now discuss the general framework of leptogenesis in more detail: in a hot plasma with $N$ lepton number and CP violating states $S_1,\cdots, S_N$, assuming they all have tree-level couplings with the standard model leptons (only with electroweak doublets to simplify the discussion), their out-of-equilibrium tree-level decays will produce a net $B-L$ asymmetry. The determination of the exact amount of $B-L$ asymmetry depends on the dynamics of the $S_\alpha$ states and requires—in general—an analysis based on kinetic equations accounting for the evolution of the $S_\alpha$ densities and the $B-L$ asymmetry density itself. For the evolution of the $B-L$ asymmetry one can write $$\label{eq:total-lepton-asymm}
\dot Y_{\Delta_{B-L}}(z)=\sum_{\alpha=1}^N\;
\dot Y_{\Delta_{B-L}}^{(S_\alpha)}(z)\,,$$ where, following ref. [@Nardi:2007jp], we are using the notation $s(z)H(z)z \,d\,Y_X(z)/dz\equiv \dot Y_X(z)$. Here $z=M_1/T$ ($M_1$ being the mass of the lightest state), $Y_{\Delta_X}=n_X-n_{\bar X}/s$ with $n_X$ ($n_{\bar X}$) the number density of particles (antiparticles), $s$ the entropy density and $H(z)$ the expansion rate of the Universe (the expressions for these functions are given in appendix \[sec:conventions\]). $\dot Y_{\Delta_{B-L}}^{(S_\alpha)}(z)$ is the asymmetry generated by each of the states $S_\alpha$. Note that we have written the dimensionless inverse temperature of the remaining states as $z_\alpha=M_\alpha/M_1\, z$.
The evolution of the asymmetries generated by each $S_\alpha$ is in turn determined by the “competition” between source ($S_{S_\alpha}$) and washout ($W_{S_\alpha}$) terms. The size of the source terms is fixed by how much the $S_\alpha$’s deviate from thermodynamical equilibrium when decaying, by the strength of the decays and by the amount of CP violation. The size of the washout terms is, instead, dictated by $S_\alpha$ processes that tend to diminish the lepton asymmetry created via the source terms like e.g. inverse decays and lepton number breaking scatterings.
As discussed in the introduction, models for Majorana neutrino masses are intrinsic scenarios for leptogenesis to take place and indeed from this approach it turns out that there is a link between two in principle unrelated problems: the origin of neutrino masses and the origin of the baryon asymmetry of the Universe. It is well known that Majorana neutrino masses can be generated in a model independent way by adding to the standard model Lagrangian an effective dimension five operator ${\cal O}_5\sim\ell
\ell H H$, that generates the corresponding Majorana masses after electroweak symmetry breaking [@Weinberg:1980bf]. The tree-level realizations of this operator give rise to type-I, II and III seesaws which constitute the usual frameworks for almost all the studies of leptogenesis.
Leptogenesis in type-I seesaw {#sec:leptogenesis-type-I}
-----------------------------
In type-I seesaw the states $S_\alpha$ correspond to RH neutrinos $N_{\alpha}$. In a general basis the interactions of these states are given by $$\label{eq:seesaw-lag}
-{\cal L}^{(I)}=
\text{i}\bar N\,\gamma^\mu\partial_\mu\,N
+ \bar \ell\,\pmb{\lambda}^*\,N \tilde H
+ \frac{1}{2} N^T\,C\,\pmb{M_N}\,N + \mbox{h.c.}\,,$$ Here $\tilde H = \text{i}\tau_2 H^*$, $C$ is the charge conjugation operator, and for 3 $N_{\alpha}$, $\pmb{\lambda}^*$ is a $3\times 3$ Yukawa coupling matrix in flavor space and $\pmb{M_N}$ is the $3\times 3$ Majorana mass matrix. At energy scales well below the RH neutrino masses, the light neutrinos masses are determined by the effective matrix $$\label{eq:eff-mass-matrix-type-I}
\pmb{m_\nu^\text{eff}}=\pmb{m_\nu^I}
=-\pmb{m_D}\pmb{M_{N}}^{-1}\pmb{m_D}^T
=-\sum_{\alpha=1,2,3}M_{N_\alpha}^{-1}
\pmb{m_{D_\alpha}}\otimes\pmb{m_{D_\alpha}}\,,$$ where in order to facilitate the discussion in section \[sec:both\] we have expressed the matrix in terms of the parameter space vectors $\pmb{m_{D_\alpha}}^T=(m_{D_{\alpha 1}},m_{D_{\alpha 2}},m_{D_{\alpha
3}})$, with $\pmb{m_D}=v\,\pmb{\lambda}$ and $v=\langle
H\rangle\simeq 174$ GeV. Diagonalization of (\[eq:eff-mass-matrix-type-I\]) by means of the PMNS mixing matrix $\pmb{U}$ leads to the light neutrino mass spectrum: $$\label{eq:diagonalization-type-I}
\pmb{U}^T\pmb{m_\nu^\text{eff}}\pmb{U}=\pmb{\hat m_\nu}\,,$$ with $\pmb{U}=\pmb{V}\,\pmb{D}$ (with $\pmb{V}$ the part of the PMNS matrix having a CKM-like form and $\pmb{D}=\mbox{diag}(e^{\text{i}\phi_1},e^{\text{i}\phi_2},1)$ containing the Majorana CP phases).
The $3\times 3$ Dirac mass matrix $\pmb{m_D}$, being a general complex matrix, contains 18 parameters (9 moduli and 9 phases) of which 3 phases can be removed by rotation of the lepton doublets in (\[eq:seesaw-lag\]). The number of physical parameters defining $\pmb{m_D}$ is therefore 15. A very useful parametrization in which this is explicitly taken into account is the Casas-Ibarra parametrization [@Casas:2001sr], in which the Dirac mass matrix is expressed in terms of low-energy neutrino observables and a general complex orthogonal matrix $\pmb{R}$, namely $$\label{eq:casas-ibarra}
\pmb{m_D}=\pmb{U}^*\,\pmb{\hat m_\nu}^{1/2}\pmb{R}\,\pmb{\hat M_N}^{1/2}\,.$$
In the conventional thermal leptogenesis scenario the RH neutrino mass spectrum is taken to be hierarchical, $M_{N_\alpha}\ll M_{N_\beta}$ for $\alpha<\beta$ (for a throughout review see [@Davidson:2008bu]). Under this simplification—well justified as far as $T_\text{Reheat}<M_{N_{2,3}}$—the effects of $N_{2,3}$ can be neglected and thus the asymmetry is entirely produced by the dynamics of $N_1$. The kinetic equations that describe the evolution of the asymmetry involve $N_1$ decays, $\Delta L=1$ and $\Delta L=2$ scatterings, and depending on the temperature regimen at which leptogenesis takes place ($T\sim M_{N_1}$) should include the lepton flavor degrees of freedom [@Barbieri:1999ma; @Endoh:2003mz; @Fujihara:2005pv; @Nardi:2006fx; @Abada:2006fw; @Abada:2006ea]. At the leading order in the Yukawa couplings, however, the kinetic equations are determined by the decays and the off-shell pieces of the $\Delta L=2$ scattering processes $\ell_j H\leftrightarrow \ell_i H$ and $\bar\ell_j H^\dagger\leftrightarrow \ell_i H$. For $T\gtrsim
10^{12}$ GeV (or otherwise neglecting flavor effects) they read as follows $$\begin{aligned}
\label{eq:BEQ-type-I}
\dot Y_{N_1}&=-(y_{N_1}-1)\gamma_{D_{N_1}}\,,\nonumber\\
\dot Y_{\Delta_{B-L}}&=S_{N_1} + W_{N_1}\nonumber\\
&=-\left[
(y_{N_1}-1)\epsilon_{N_1}
+ \frac{y_{\Delta_{B-L}}}{2}
\right]\gamma_{D_{N_1}}\,,\end{aligned}$$ where we are using the notation $y_X\equiv Y_X/Y_X^\text{Eq}$ and $y_{\Delta_{B-L}}\equiv Y_{\Delta_{B-L}}/Y_\ell^\text{Eq}$ (the expressions for the equilibrium densities are given in appendix \[sec:conventions\]). The strength of the decays $$\label{eq:mtilde-type-I}
\tilde m_1=\frac{v^2}{M_{N_1}}\,
\left(
\pmb{\lambda}^\dagger\pmb{\lambda}
\right)_{11}$$ determines the size of the reaction density $\gamma_{D_{N_1}}$ appearing in the source term $S_{N_1}$ as well as in the washout term $W_{N_1}$, namely $$\label{eq:reaction-dens-type-I}
\gamma_{D_{N_1}}=\frac{1}{8\pi^3}\frac{M_{N_1}^5}{v^2}
\frac{K_1(z)}{z}\,\tilde m_1\,.$$ Here $K_1(z)$ is the first-order modified Bessel function of the second-type. The amount of CP violation in $N_1$ decays is given by the CP violating asymmetry $\epsilon_{N_1}=\sum_{i=e,\mu,\tau}\epsilon_{N_1}^{\ell_i}$. At the leading order this quantity arises through the interference between the $N_1$ tree-level decay and the one-loop vertex and wave function corrections [@Covi:1996wh]. The flavored CP violating asymmetries arising from the diagrams depicted in figure \[fig:cp-asymm-type-I\] read $$\begin{aligned}
\label{eq:CPV-asymmetries-type-I}
\epsilon_{N_1}^{\ell_i\text{(V)}}&=\frac{1}{8\pi}\sum_{\beta\neq 1}
\frac{\mathbb{I}\mbox{m}[\sqrt{\omega_\beta}
(\pmb{\lambda}^\dagger\pmb{\lambda})_{\beta 1}
\lambda_{i\beta}^*\lambda_{i1}]}
{(\pmb{\lambda}^\dagger\pmb{\lambda})_{11}}f(\omega_\beta)
\;,\nonumber\\
\epsilon_{N_1}^{\ell_i\text{(W)}}&=
-\frac{1}{8\pi}\sum_{\beta\neq 1}
\frac{
\mathbb{I}\mbox{m}
\{
[(\pmb{\lambda}^\dagger\pmb{\lambda})_{1\beta}
+
\sqrt{\omega_\beta}(\pmb{\lambda}^\dagger\pmb{\lambda})_{\beta 1}]
\lambda_{i\beta}^*\lambda_{i1}
\}}
{(\pmb{\lambda}^\dagger\pmb{\lambda})_{11}}g(\omega_\beta)\;,\end{aligned}$$ with obvious generalization if the decaying state is $N_\alpha$, where $\omega_\beta=M_{N_\beta}^2/M_{N_\alpha}^2$ and the loop functions are $$\begin{aligned}
\label{eq:loop-functions-type-I}
f(\omega_\beta)&=(1+\omega_\beta)\ln
\left(
\frac{\omega_\beta + 1}{\omega_\beta}
\right) - 1\;,\nonumber\\
g(\omega_\beta)&=\frac{1}{\omega_\beta-1}\;.\end{aligned}$$ Since $\omega_\beta^{-1}\ll 1$ the loop functions can be expanded in powers of $\omega_\beta^{-1}$ dropping the subleading terms. At leading order in these parameters the flavored CP violating asymmetries can be expressed as $$\label{eq:flavored-cpv-asymmetry-leading-type-I}
\epsilon_{N_1}^{\ell_i}=
-\frac{1}{8\pi(\pmb{\lambda}^\dagger\pmb{\lambda})_{11}}
\sum_{\beta\neq 1}
\mathbb{I}\mbox{m}
\left\{
\left[
\frac{(\pmb{\lambda}^\dagger\pmb{\lambda})_{1\beta}}{\omega_\beta}
+
\frac{3(\pmb{\lambda}^\dagger\pmb{\lambda})_{\beta 1}}{2\sqrt{\omega_\beta}}
\right]
\lambda_{i\beta}^*\lambda_{i 1}
\right\}\,.$$ The total CP asymmetry then is obtained from $$\label{eq:total-CP-type-I}
\epsilon_{N_1}=\sum_{i=e,\mu,\tau}\epsilon_{N_1}^{\ell_i}=
-\frac{3}{16\pi v^2}\sum_\beta\frac{1}{\sqrt{\omega_\beta}}
\frac{\mathbb{I}\mbox{m}[(\pmb{m_D}^\dagger\pmb{m_D})_{\beta 1}^2]}
{(\pmb{m_D}^\dagger\pmb{m_D})_{11}}\;,$$ where anticipating the discussions of sections \[sec:single\] and \[sec:both\] we have rewritten the CP violating asymmetry in terms of the Dirac mass matrix (in the basis where the RH neutrino mass matrix is diagonal).
![Feynman diagrams generating the CP violating asymmetry $\epsilon_{N_1}^{\ell_i}$ in type-I seesaw.[]{data-label="fig:cp-asymm-type-I"}](cp-asymm-plots.pdf){width="9cm" height="2cm"}
Formal integration of eqs. (\[eq:BEQ-type-I\]), using the integrating factor technique and assuming a vanishing primordial $B-L$ asymmetry, gives $$\label{eq:BmL-astmm-type-I}
Y_{\Delta_{B-L}}(z)=-\epsilon_{N_1}\,Y_{N_1}^\text{Eq}(z\to 0)\,\eta(z)\,.$$ Here $\eta(z)$ is the efficiency function that determines the evolution of $Y_{\Delta_{B-L}}(z)$ and its final value at $z\to\infty$ (see appendix \[sec:conventions\] for details). At ${\cal
O}(\pmb{\lambda}^2)$ (leading order) the problem of determining the final $Y_{\Delta_{B-L}}$ is a two parameters problem, $\tilde m_1$ and $\epsilon_{N_1}$, and requires numerical solutions of eqs. (\[eq:BEQ-type-I\]) [^3]. Figure \[fig:efficiency-type-I\] shows the efficiency factor $\eta\equiv\eta(z\to \infty)$ as a function of the parameter $\tilde
m_1$.
![Efficiency factor as a function of the parameter $\tilde
m_1$ in type-I seesaw. The efficiency has been calculated from leading order Boltzmann equations in the one-flavor approximation.[]{data-label="fig:efficiency-type-I"}](eff-mtilde-type-I.pdf){width="9cm" height="7cm"}
Leptogenesis in type-II seesaw {#sec:leptogenesis-type-II}
------------------------------
Consistent models for leptogenesis involving scalar electroweak triplets require going beyond a single scalar triplet. There are several ways in which this can be done, namely adding at least another triplet [@Ma:1998dx], adding RH neutrinos [@Hambye:2003ka] or adding fermionic triplets. Here we will discuss scenarios that include RH neutrinos, as these schemes will be further analyzed in the context of flavor symmetries in section \[sec:both\]. In this case, depending on the triplet and RH mass spectrum, the states $S_\alpha$ can be identified with the RH neutrinos, the triplet or both.
The interactions of the RH neutrinos are given by the Lagrangian in (\[eq:seesaw-lag\]) whereas the interaction of the triplet are determined by $$\label{eq:Lag-type-II}
-{\cal L}^{(II)}=\ell^T C \pmb{Y}\text{i}\tau_2\pmb{\tau}\ell
\pmb{\Delta}
+M^2_\Delta \mbox{Tr}\pmb{\Delta}^\dagger\pmb{\Delta}
-\mu H^T \text{i}\tau_2 \pmb{\tau}H \pmb{\Delta}+ \mbox{h.c.}\,.$$ Here $\pmb{Y}$ is a $3\times 3$ matrix in flavor space and $\pmb{\Delta}$, the $SU(2)$ scalar electroweak triplet has hypercharge +1 (to the lepton doublets -1/2) and is given by $$\label{eq:triplet}
\pmb{\Delta}=
\begin{pmatrix}
\Delta^{++} & \Delta^{+}/\sqrt{2}\\
\Delta^{+}/\sqrt{2} & \Delta^{0}
\end{pmatrix}\,.$$ After electroweak symmetry breaking the light neutrino mass matrix receives the contributions from the dimension five effective operators of the RH neutrino and triplet $$\label{eq:eff-mass-matrix}
\pmb{m_\nu^{\text{eff}}}=\pmb{m_\nu^I} + \pmb{m_\nu^{II}}\quad\text{with}\quad
\pmb{m_\nu^{II}}=2\,v_\Delta\,\pmb{Y}\,.$$ The first term is the contribution from the RH neutrinos (eq. (\[eq:eff-mass-matrix-type-I\])) whereas the second one is the contribution from the triplet, with the triplet vacuum expectation value fixed by $\langle\Delta^0\rangle=v_\Delta=\mu^*\,v^2/M_\Delta^2$.
As already mentioned the generation of $B-L$ asymmetry depends on the heavy mass spectrum. One can define three possible scenarios:
- $M_{N_1}\ll M_\Delta$: the effects of $\pmb{\Delta}$ are decoupled and the lepton asymmetry is generated via $N_1$ dynamics. This case resembles leptogenesis in type-I seesaw.
- $M_{N_1}\gg M_\Delta$: the lepton asymmetry is entirely produced by the dynamics of $\pmb{\Delta}$ [@Hambye:2005tk].
- $M_\Delta\sim M_{N_1}$: both the triplet and the lightest RH neutrino generate the asymmetry [@AristizabalSierra:2011ab].
Here we will discuss the third possibility in the regimen $M_{N_1,\Delta}>10^{12}$ GeV. We will closely follow the presentation in [@AristizabalSierra:2011ab].
Since in this case the scalar triplet, carrying non-trivial $SU(2)$ quantum numbers, couples to the standard model electroweak gauge bosons, and the number of degrees of freedom participating in the generation of the lepton asymmetry is larger, the Boltzmann equations are more involved. At leading order in the Yukawa couplings $\pmb{\lambda}$ and $\pmb{Y}$ the kinetic equation for the lepton asymmetry, $Y_{\Delta_L}$[^4], involve the RH neutrino and triplet decays and inverse decays $N_1\leftrightarrow \ell \tilde
H^\dagger$ and $\pmb{\Delta}\leftrightarrow \bar \ell\bar \ell$ and the off-shell Yukawa generated scattering reactions $\ell \tilde
H^\dagger\leftrightarrow \ell \tilde H^\dagger$ and $H^\dagger
H^\dagger \leftrightarrow \ell\ell$. In addition to the evolution of the $Y_{\Delta_L}$ asymmetry the full network of Boltzmann equations should include the equations accounting for the evolution of the RH neutrino and triplet number densities and the triplet and Higgs asymmetries[^5]. The resulting system of five coupled differential equations can be reduced to four by using the constraint imposed by hypercharge neutrality [@Hambye:2005tk]: $$\label{eq:hypercharge-neutrality}
2\,Y_{\Delta_\Delta} + Y_{\Delta H} - Y_{\Delta_L}=0\,.$$
The resulting kinetic equations can thus be written as $$\begin{aligned}
\label{eq:BEQ-type-II}
\dot Y_{N_1}&=-(y_{N_1}-1)\,\gamma_{D_{N_1}}\,,
\nonumber\\
\dot Y_\Sigma&=-(y_\Sigma - 1)\,\gamma_{D_\Delta}
- 2(y_\Sigma^2 - 1)\,\gamma_A\,,\nonumber\\
\dot Y_{\Delta_L}&=\left[(y_{N_1}-1)\,\epsilon_{N_1}^\text{tot} -
\left(y_{\Delta_L} + y_{\Delta_\Delta}^H\right)\right]\,\gamma_{D_{N_1}}
+\left[(y_\Sigma - 1)\,\epsilon_\Delta -
2K_\ell\, (y_{\Delta_L} + y_{\Delta_\Delta})\right]\,\gamma_{D_\Delta}\,,
\nonumber\\
\dot Y_{\Delta_\Delta}&=-\left[y_{\Delta_\Delta} + (K_\ell - K_H)\,y_{\Delta_L}
+ 2K_H \,y_{\Delta_\Delta}^H\right]\,,\end{aligned}$$ where $\Sigma\equiv\Delta + \Delta^\dagger$ and $y_{\Delta_{\Delta}}^H\equiv
Y_{\Delta_\Delta}/Y_H^\text{Eq}$ and the rest of the variables in the equations follow the conventions introduced in the previous section when writing the eqs. in (\[eq:BEQ-type-I\]). The reaction densities involving the triplet are given by $$\label{eq:reaction-densities-triplet}
\gamma_{D_\Delta}=\frac{1}{8\,\pi^3}\frac{M_\Delta^5}{v^2}
\frac{K_1(z)}{z}\,
\left(
\tilde m_\Delta^\ell
+
\frac{\tilde m_\Delta^2}{4\tilde m_\Delta^\ell}
\right)\,,
\quad
\gamma_A(z)=\frac{M_\Delta^4}{64\,\pi^4}\,\int_4^\infty
dx\sqrt{x}\frac{K_1(zx)}{z}\,\widehat\sigma_A(x)\,,$$ with $x=s/M_\Delta^2$. The reduced cross section $\widehat\sigma_A(x)=2\,x\,\lambda(1,x^{-1},0)$ (where we have $\lambda(a,b,c)=(a-b-c)^2-4bc$) can be found in appendix \[sec:conventions\]. The factors $K_{\ell,H}$ resemble the flavor projectors defined in standard flavored leptogenesis [@Nardi:2006fx; @Abada:2006ea] as they project triplet decays into either the Higgs or the lepton doublet directions. They are defined as follows $$\label{eq:projectors}
K_\ell=\frac{\tilde m_\Delta^\ell}{\tilde m_\Delta^\ell + \frac{\tilde m_\Delta^2}
{4\,\tilde m_\Delta^\ell}}\,
\qquad\mbox{and}\qquad
K_H=\frac{\tilde m_\Delta^2}{4\,\tilde m_\Delta^\ell\left(
\tilde m_\Delta^\ell + \frac{\tilde m_\Delta^2}
{4\,\tilde m_\Delta^\ell}\right)}\,,$$ where the parameters $\tilde m_\Delta^\ell$ and $\tilde m^2_\Delta$ are given by $$\label{eq:definition-mtildes}
\tilde m_\Delta^\ell=\frac{v^2\,|\pmb{Y}|^2}{M_\Delta}
\qquad\mbox{and}\qquad
\tilde m^2_\Delta=\mbox{Tr}[\pmb{m_\nu^{II}}\pmb{m_\nu^{II}}^\dagger]\,,$$ with $|\pmb{Y}|^2=\mbox{Tr}[\pmb{Y}\,\pmb{Y}^\dagger]$. In these definitions we have replaced the trilinear coupling $\mu$ by the contribution of the type-II sector to the effective light neutrino mass matrix, encoded in $\tilde m^2_\Delta$. In principle this is just a matter of choice, but it proves to be quite convenient given that in contrast to $\mu$ the parameter $\tilde m_\Delta$ is (partially) constrained by experimental neutrino data.
The CP asymmetry for the RH neutrino arises as in type-I but, due to the trilinear scalar coupling in (\[eq:Lag-type-II\]), there is an additional contribution coming from a vertex correction involving the triplet, as shown in fig. \[fig:cp-asymm-type-II\] (left-hand side). The interference between the tree-level decay $N_\alpha\to
\ell\,\tilde H^\dagger$ and this 1-loop vertex diagram yields [@Hambye:2003ka; @Antusch:2004xy][^6] $$\label{eq:cp-asymm-triplet}
\epsilon_{N_1}^\Delta=-\frac{3}{2\,\pi\,M_\Delta}
\frac{1}{
\left(
\pmb{m_D}\,\pmb{m_D}^\dagger
\right)_{11}}
\mathfrak{I}\mbox{m}
\left[
\left(\pmb{m_D}\,\pmb{Y}^*\pmb{m_D}^T
\right)_{11}\,\mu
\right]\,h(\sigma_1)\,.$$ The function $h(\sigma_1)$, with $\sigma_\alpha=M_\Delta^2/M_{N_\alpha}^2$, is given by $$\label{eq:triplet-cp-asymm-loop-function}
h(\sigma_1)=\sqrt{\sigma_1}
\left[
1 - \sigma_1\log
\left(\frac{1+\sigma_1}{\sigma_1}\right)
\right]\,.$$ The total CP violating asymmetry in $N_1$ decays therefore reads $$\label{eq:total-CP-N1-type-II}
\epsilon_{N_1}^\text{tot}=\epsilon_{N_1}+\epsilon_{N_1}^\Delta\,,$$ where, for the scenario considered, $\epsilon_{N_1}$ is determined by eqs. (\[eq:flavored-cpv-asymmetry-leading-type-I\]) and (\[eq:total-CP-type-I\]).
![Left-hand side: Vertex loop correction involving the triplet and contributing to $\epsilon_{N_1}^\Delta$. Right-hand side: Vertex loop correction for triplet decays [@Hambye:2003ka].[]{data-label="fig:cp-asymm-type-II"}](cp-asymm-plots-triplet.pdf){width="7cm" height="2.4cm"}
The CP violating asymmetry in triplet decays arises from the interference between the tree-level $\pmb{\Delta}\to \ell\ell$ process and the interference with the 1-loop vertex diagram shown in figure \[fig:cp-asymm-type-II\] (right-hand side). The result reads [@Hambye:2003ka] $$\label{eq:triplet-CPasymmetry}
\epsilon_\Delta=-\frac{1}{8\,\pi\,v^2}\frac{1}{M_\Delta}
\frac{
\sum_\beta\mathfrak{I}\mbox{m}
\left[
\left(
\pmb{m_D}\,\pmb{Y}^*\,\pmb{m_D}^T
\right)_{\beta\beta}\,\mu
\right]}{\mbox{Tr}\left[\pmb{Y}\,\pmb{Y}^\dagger\right] + \mu^2/M_\Delta^2}\,
H(\sigma_\beta)\,,$$ where the loop function in this case is given by $$\label{eq:loop-function-triplet-decay}
H(\sigma_\beta)=\frac{1}{\sqrt{\sigma_\beta}}
\log\left(1 + \sigma_\beta \right)\,.$$
As in the type-I case the kinetic equation for the lepton asymmetry can be formally integrated. The resulting asymmetry, assuming a zero primordial asymmetry, can be expressed in two different ways [@AristizabalSierra:2011ab] $$\label{eq:efficiencies}
Y_{\Delta_L}(z)=-\epsilon_{N_1}^\text{tot}\,Y^\text{Eq}_\text{tot}\,\eta^I(z)
\quad\mbox{or}\quad
Y_{\Delta_L}(z)=-\epsilon_\Delta\,Y^\text{Eq}_\text{tot}\,\eta^{II}(z)\,.$$ The functions $\eta^{I,II}(z)$ are defined in such a way that in the limit in which the triplet (RH neutrino) interactions are absent $\eta^{I}$ ($\eta^{II}$) corresponds to the efficiency function of standard leptogenesis (pure triplet leptogenesis), see appendix \[sec:conventions\] for details. As in the type-I case the final $L$ asymmetry is obtained from these functions in the limit $z\to \infty$.
A precise determination of the lepton asymmetry generated in $N_1$ and $\pmb{\Delta}$ decays requires solving the network of equations in (\[eq:BEQ-type-II\]). Taking $z=M_\Delta/T$ and $z_N=r z$, with $r=\sigma_1^{-1/2} = M_{N_1}/M_\Delta$, and once the CP asymmetries $\epsilon_{N_1}^\text{tot}$ and $\epsilon_\Delta$ are fixed, the problem of studying the evolution of the lepton asymmetry is entirely determined by five parameters: $\tilde m_1$, $\tilde m_\Delta$, $\tilde m_\Delta^\ell$, $M_\Delta$ and $r$ [^7].
As pointed out in [@AristizabalSierra:2011ab], in models featuring a mild hierarchy between $M_\Delta$ and $M_{N_1}$ three scenarios can be defined:
I. \
The relevant parameters follow the hierarchy $\tilde m_1\ll
\tilde m_\Delta^\ell, \tilde m_\Delta$. The $L$ asymmetry is generated through the processes $\pmb{\Delta}\to \bar\ell\bar\ell$ or $\pmb{\Delta}\to H H$ and the details strongly depend on whether $\tilde m_\Delta^\ell\gg \tilde m_\Delta$, $\tilde m_\Delta^\ell\ll
\tilde m_\Delta$ or $\tilde m_\Delta^\ell\sim \tilde m_\Delta$. Interestingly, when $\tilde m_\Delta^\ell\gg \tilde m_\Delta$ the Higgs asymmetry—being weakly washed out—turns out to be large and implies a large lepton asymmetry.
II. \
These scenarios are defined according to $\tilde m_1\gg \tilde
m_\Delta^\ell, \tilde m_\Delta$ thus leptogenesis is mainly determined by $N_1$ dynamics. The relative difference between the parameters $\tilde m_\Delta^\ell$ and $\tilde m_\Delta$ determines whether either the Higgs asymmetry or the $L$ asymmetry are strongly or weakly washed out, thus three cases can be distinguished: $\tilde
m_\Delta^\ell\gg \tilde m_\Delta$, $\tilde m_\Delta^\ell\ll \tilde
m_\Delta$ or $\tilde m_\Delta^\ell\sim \tilde m_\Delta$. Each of them exhibit different features.
III. \
In these models the parameters controlling the gauge reaction densities strengths are all of the same order i.e. $\tilde
m_1\sim \tilde m_\Delta^\ell\sim \tilde m_\Delta$.
For the sake of illustration in figure \[fig:models-case1-case2\] we show two numerical examples for scenarios I and II. They were obtained with the parameter space points $P_I$=($\tilde m_1$, $\tilde
m_\Delta$, $\tilde m_\Delta^\ell$,$M_\Delta$,$r$) =($10^{-4}$ eV, $10^{-2}$ eV, $10^{-1}$ eV, $10^{10}$ GeV,2) and $P_{II}$=($\tilde
m_1$, $\tilde m_\Delta$, $\tilde m_\Delta^\ell$,$M_\Delta$,$r$) =($10^{-2}$ eV, $10^{-4}$ eV, $10^{-3}$ eV, $10^{10}$ GeV,2) for fixed $\epsilon_\Delta=10^{-6}$ and $\epsilon_{N_1}=10^{-5}$ and assuming initial vanishing asymmetries.
![[*Reaction densities for triplet and RH neutrino processes (left-panels) and evolution of the different densities (right-panels) entering in the kinetic equations for the scenarios of purely triplet leptogenesis (upper panels) and singlet dominated models (lower panels) [@AristizabalSierra:2011ab]. See the text for more details.*]{}[]{data-label="fig:models-case1-case2"}](readens-ptc.pdf "fig:"){width="7.4cm" height="6cm"} ![[*Reaction densities for triplet and RH neutrino processes (left-panels) and evolution of the different densities (right-panels) entering in the kinetic equations for the scenarios of purely triplet leptogenesis (upper panels) and singlet dominated models (lower panels) [@AristizabalSierra:2011ab]. See the text for more details.*]{}[]{data-label="fig:models-case1-case2"}](pure-trip-case.pdf "fig:"){width="7.4cm" height="6cm"} ![[*Reaction densities for triplet and RH neutrino processes (left-panels) and evolution of the different densities (right-panels) entering in the kinetic equations for the scenarios of purely triplet leptogenesis (upper panels) and singlet dominated models (lower panels) [@AristizabalSierra:2011ab]. See the text for more details.*]{}[]{data-label="fig:models-case1-case2"}](readens-sdl.pdf "fig:"){width="7.4cm" height="6cm"} ![[*Reaction densities for triplet and RH neutrino processes (left-panels) and evolution of the different densities (right-panels) entering in the kinetic equations for the scenarios of purely triplet leptogenesis (upper panels) and singlet dominated models (lower panels) [@AristizabalSierra:2011ab]. See the text for more details.*]{}[]{data-label="fig:models-case1-case2"}](sdl-case.pdf "fig:"){width="7.4cm" height="6cm"}
Leptogenesis in type-III seesaw {#sec:leptogenesis-type-III}
-------------------------------
In type-III [@Foot:1988aq] seesaw the states $S_\alpha$ correspond to fermion electroweak triplets (here we consider 3 for definiteness) with vanishing hypercharge. In a general basis the interactions of these states are given by the following Lagrangian $$\label{eq:lagrangian-type-III}
-{\cal L}^{(III)}=-\overline{\pmb{T}}_\alpha \slashed{D}\,\pmb{T}_\alpha
+ \overline{\ell}\,\pmb{h}^*\pmb{T} \widetilde H
+ \frac{1}{2}
\overline{\pmb{T}}_\alpha^\dagger C \pmb{M_{T}} \pmb{T}_\alpha
+ \mbox{h.c.}\;,$$ where the fermion triplets can be written as a matrix $$\label{eq:electric-triplet}
\pmb{T}_\alpha = \pmb{\tau} \cdot \pmb{T}_\alpha=
\begin{pmatrix}
T_\alpha^0 & \sqrt{2}T_\alpha^+\\
\sqrt{2}T_\alpha^- & -T_\alpha^0
\end{pmatrix}\,,$$ with $T^0 = T^3$, $T^\pm =(T^1\mp \text{i} T^2)/\sqrt{2}$. In this notation, the covariant derivative is defined as $D_\mu = \partial_\mu - \text{i} g
\tau^a W_\mu^a / 2$ ($a$ being $SU(2)$ indices). Lepton number is broken by the Majorana triplet mass terms and the effective light neutrino mass matrix has the same structure than in type-I seesaw, eq. (\[eq:eff-mass-matrix-type-I\]), with the right-handed neutrino mass matrix replaced by that of the triplets and $\pmb{m_D}=v\,\pmb{h}$: $$\label{eq:light-neutrino-mm-typeIII}
\pmb{m_\nu^\text{eff}}=\pmb{m_\nu^{III}}=
\sum_{\alpha=1,2,3}M_{T_\alpha}^{-1}
\pmb{m_{D_\alpha}}\otimes\pmb{m_{D_\alpha}}\,,$$ where we are using the same conventions used in the type-I case discussion.
In what concerns leptogenesis, in several aspects, these models resemble models based on type-I seesaw. For example assuming a hierarchical triplet mass spectrum $M_{T_\alpha}<M_{T_\beta}$ ($\alpha<\beta$) the $B-L$ asymmetry is completely produced by $T_1$ decays. There is, however, a significant difference arising from the fact that the triplets couple to the standard model electroweak gauge bosons. Thus, at high temperatures the triplet distribution is thermalized by gauge reactions, and only when these reactions are frozen a net $B-L$ asymmetry can be built [@Hambye:2003rt; @Strumia:2008cf].
As done in sections \[sec:leptogenesis-type-I\] and \[sec:leptogenesis-type-II\], in what follows, we will discuss the generation of the $B-L$ asymmetry in these models in the one-flavor approximation (the effects of flavor have been considered in [@AristizabalSierra:2010mv]). At ${\cal O}(\pmb{h}^2)$, the leading order in the couplings $\pmb{h}$, the kinetic equations consist of $T_1$ decays and off-shell $\Delta L=2$ processes. The main difference with the conventional leptogenesis scenario is the inclusion of the couplings of $T_1$ with gauge bosons. The Boltzmann equations in this case read $$\begin{aligned}
\label{eq:BEQs-type-III}
\dot Y_{T_1}&=-\left(y_{T_1}-1\right)\gamma_{D_{T_1}}
-\left(y_{T_1}^2-1\right)\gamma_A\,,\nonumber\\
\dot Y_{\Delta_{B-L}}&=-
\left[
\left(y_{T_1}-1\right)\epsilon_{T_1}
+\frac{y_{\Delta_{B-L}}}{2}
\right]\gamma_{D_{T_1}}\,.\end{aligned}$$ The Yukawa reaction density $\gamma_{D_{T_1}}$ is given by eq. (\[eq:reaction-dens-type-I\]), changing $\pmb{\lambda}\to
\pmb{h}$ and $M_{N_1}\to M_{T_1}$ in the definition of $\tilde m_1$ (eq. (\[eq:mtilde-type-I\])) whereas the gauge reaction density by (\[eq:reaction-densities-triplet\]) using, of course, the corresponding fermion triplet reduced cross section (see appendix \[sec:conventions\]). The CP violating asymmetry is a factor of three smaller than in type-I seesaw due to contractions of the $SU(2)$ indices in the Yukawa interaction terms entering in the 1-loop corrections, thus $$\label{eq:CP-asym-type-III}
\epsilon_{T_1}=\sum_{i=e,\mu,\tau}\epsilon_{T_1}^{\ell_i}=
-\frac{1}{16\pi v^2}\sum_\beta\frac{1}{\sqrt{\omega_\beta}}
\frac{\mathbb{I}\mbox{m}[(\pmb{m_D}^\dagger\pmb{m_D})_{\beta 1}^2]}
{(\pmb{m_D}^\dagger\pmb{m_D})_{11}}\;.$$
From the formal integration of the $B-L$ asymmetry kinetic equation in (\[eq:BEQs-type-III\]) the asymmetry can be written as $$\label{eq:BmL-astmm-type-III}
Y_{\Delta_{B-L}}(z)=-3 \epsilon_{T_1}\,Y_{T_1}^\text{Eq}(z\to 0)
\,\eta(z)\,.$$ The expression is similar to the one obtained in the type-I case but the efficiency is different, as it now includes the gauge reaction density. The factor of $3$ comes from the $SU(2)$ degrees of freedom of $T_1$.
A precise determination of the $B-L$ asymmetry relies on numerical solutions of the kinetic equations, which in this case—even at the leading order in the couplings—requires $\epsilon_{T_1}$, $\tilde m_1$ and also the triplet mass $M_{T_1}$ to be specified. The results for the efficiency factor are shown in fig. \[fig:eff-mtilde-aligned-type-III\] (left panel) where a strong dependence with $M_{T_1}$ can be seen. This dependence, introduced by the gauge reactions, diminishes as $\tilde m_1$ increases and disappears at certain $\widetilde m_1^{\text{min}}$. This implies that above this value $T_1$ leptogenesis proceeds as in type-I seesaw [^8]. Thus, as highlighted in [@AristizabalSierra:2010mv], in this type of models the generation of the $B-L$ asymmetry can proceed either in in a region determined by the condition $\widetilde
m_1<\widetilde m_1^{\text{min}}$ (“gauge region”) or conversely in a region defined by $\widetilde m_1>\widetilde m_1^{\text{min}}$ (“Yukawa region”). These regions are displayed in figure \[fig:eff-mtilde-aligned-type-III\] (right panel).
![Efficiency factor as a function of $\widetilde m_1$ in the flavor aligned case (left panel) and regions for which gauge interactions freeze out after (lower region) and before (upper region) Yukawa reaction decoupling (right panel) in type-III leptogenesis.[]{data-label="fig:eff-mtilde-aligned-type-III"}](YDeltaBmLvsmt.pdf "fig:"){height="6.4cm" width="7.4cm"} ![Efficiency factor as a function of $\widetilde m_1$ in the flavor aligned case (left panel) and regions for which gauge interactions freeze out after (lower region) and before (upper region) Yukawa reaction decoupling (right panel) in type-III leptogenesis.[]{data-label="fig:eff-mtilde-aligned-type-III"}](diff-reg.pdf "fig:"){height="6.4cm" width="7.4cm"}
Leptogenesis in the flavor symmetric phase {#sec:sym}
==========================================
We now turn to the discussion of the implications of the presence of lepton flavor symmetries for leptogenesis in models based on type-I seesaw. In general in these models four energy scales can be distinguished: a cutoff scale $\Lambda$ (typically a scale of heavy matter), the lepton number breaking scale $M_N$, the flavons scale $M_\phi$—determined by the scale of the fields that trigger flavor symmetry breaking—and the scale at which the flavor symmetry is broken, denoted hereafter by $v_F$. The scale of heavy matter is constrained to be the largest one, the remaining three scales, being free parameters, can follow any hierarchy. In principle six possible hierarchical patterns can be considered; however, since lepton number is an intrinsic feature of seesaw models these possibilities can be split in two generic scenarios:
1. \[scenarioI-fsl\] The flavor symmetry related scales $M_\phi$ and $v_F$ are larger than the number breaking scale.
2. \[scenarioII-fsl\] The flavor symmetry related scales $M_\phi$ and $v_F$ are smaller than the number breaking scale.
The scale at which leptogenesis takes place is intimately related with the lepton number violating scale. Accordingly in scenarios \[scenarioI-fsl\] leptogenesis proceeds once the flavor symmetry is already broken whereas in scenarios \[scenarioII-fsl\] leptogenesis takes place when the Lagrangian and the vacuum are still flavor invariant i.e in the flavor symmetric phase. The former is considered in sections \[sec:single\] and \[sec:both\], the latter cases are the subject of this section.
From now on we will assume the Lagrangian and the vacuum to be invariant under a flavor group $G_F$. The standard model leptons and RH neutrinos, thus, belong to $G_F$ representations $R_a^{(X)}\sim
(X_1,\dots,X_m)$ (with $X=N,\ell,e$ and $a,b,c\dots$ denoting $G_F$ indices) in such a way that all the terms in (\[eq:seesaw-lag\]) are $G_F$ singlets. As can be seen in (\[eq:BmL-astmm-type-I\]) a vanishing $\epsilon_{N_1}$ implies in turn a vanishing $B-L$ asymmetry. Two conditions have to be satisfied in order to get $\epsilon_{N_1}\neq 0$: ($i$) Mass splittings among the RH states, otherwise the loop integrals arising from the vertex and wave function corrections do not acquire an imaginary part; ($ii$) the matrix $\pmb{m_D}^\dagger\pmb{m_D}$ must have non-zero and imaginary off-diagonal elements. The first condition is satisfied if the RH neutrinos belong to different $G_F$ representations (RH neutrinos belonging to the same representation have a common universal mass). But the second condition can never be achieved in the flavor symmetric phase: recovering the correct kinetic terms for the RH neutrinos and lepton doublets requires $R_a^{(N)*}R_b^{(N)}=\delta_{ab}$ and $R_a^{(\ell)}R_b^{(\ell)*}=\delta_{ab}$, for the lepton doublets transforming according to $\bar \ell\sim R^{(\ell)}_a$. Taking the scalar electroweak doublet as a $G_F$ singlet, the Yukawa coupling matrix $\pmb{\lambda}$ is determined by the Clebsch-Gordan coefficients arising from the contraction $R^{(\ell)}_aR^{(N)}_b$, thus implying that the matrix $\pmb{\lambda}^\dagger\pmb{\lambda}$ arises from the contractions $R^{(N)*}_aR^{(\ell)}_bR^{(\ell)*}_cR^{(N)}_d=\delta_{ad}\delta_{bc}$ [@Sierra:2011vk].
A non-vanishing $B-L$ asymmetry is possible only if new contributions to the CP violating asymmetry exist ($\epsilon_{N_1}^\text{New}$) i.e. if the flavons play a role, which they can do as propagating states or virtually via loop corrections. In both cases the kinematical constraint $M_\phi<M_N$ (where $M_N$ is the mass parameter of the $R_a^{(N)}$ representation) must be guaranteed, as otherwise either RH neutrino decays to flavons are kinematical forbidden or the loop integral in which the flavons intervene can not acquire an imaginary part. The flavor models one can envisage can be described by a Lagrangian involving effective operators or models with ultraviolet completions, regardless of the approach the presence of new energy scales, different from that of lepton number violation, can have an impact in the way leptogenesis takes place. In [@Sierra:2011vk], where the conditions for leptogenesis in the flavor symmetric phase were established, an $A_4$ inspired model involving effective operators was analyzed in full detail. In contrast, [@AristizabalSierra:2007ur; @AristizabalSierra:2009bh] discussed an ultraviolet completed flavor toy model that we now discuss with the purpose of illustrating the previous statements.
We will consider a setup inspired by $U(1)_X$ flavor models [*à la*]{} Froggatt-Nielsen [^9]. Thus, in addition to the standard model fields and RH neutrinos, the setup also contains vectorlike fermion fields $F$ and a complex scalar field $S$ (flavon), all of them being electroweak singlets. With the horizontal charge assignment $X(\ell,F)=+1$, $X(S)=-1$ and $X(H,N)=0$ the following Lagrangian can be written $$\label{eq:U1-Lag}
-{\cal L}=\bar \ell\,\pmb{h}\,F\,H + \bar N\,\pmb{\lambda}\,F\,S
+\frac{1}{2}\bar N^T\,C\,\pmb{M_N}\,N + \bar F\,\pmb{M_F}\,F\,.$$ Here the Yukawa coupling matrices $\pmb{h}$ and $\pmb{\lambda}$ are $3\times 3$ matrices in flavor space. The $U(1)_X$ symmetry is spontaneously broken by the vacuum expectation value of the complex scalar field, $\langle S\rangle=v_F$. In addition to the $U(1)_X$ symmetry the terms in the Lagrangian (\[eq:U1-Lag\]) preserve a global $U(1)$ symmetry with charge assignments $L(\ell,F,N)=+1$ and $L(H,S)=0$. This symmetry is only broken by the RH Majorana mass term and thus can be identified with lepton number. In this setup the scale $\Lambda$ corresponds to $M_F$, and $M_\phi$ to $M_S$. Since leptogenesis in the flavor symmetric phase requires $M_N>v_F,M_S$ the following hierarchies follow $M_F>M_N>v_F,M_S$. With $G_F$ being Abelian the standard contribution to the CP asymmetry does not vanish, but due to the absence of the tree-level coupling $\bar \ell N \tilde H$—enforced by the flavor charge assignments—$\epsilon_{N_1}$ arises at the second loop-order, rendering its value far below the one needed for successful leptogenesis ($\epsilon_{N_1}\gtrsim 10^{-6}$). Therefore, leptogenesis is viable only if new contributions to the CP violating asymmetry are present.
![Tree-level and one-loop self-energy diagrams determining $\epsilon_{N_1}^{\text{(New)}\,\ell_i}$.[]{data-label="fig:purely-flavored-lepto"}](purely-flv.pdf){width="9cm" height="2.5cm"}
With the couplings in (\[eq:U1-Lag\]), and due to the kinematical constraint $M_F>M_N, M_S$, RH neutrinos have three body decay modes, $N_\alpha\to SH\ell_i$. So in this case the flavon $S$ intervenes in the generation of the $B-L$ asymmetry as a propagating state. The interference between the tree-level decay and the one-loop self-energy correction diagrams shown in fig. \[fig:purely-flavored-lepto\] determine the new contribution to the flavored CP violating asymmetry, which at leading order in the mass ratio $r_A=M_{N_1}^2/M_{F_A}^2$ reads [@AristizabalSierra:2007ur] $$\label{eq:new-contribution-CP}
\epsilon^{(\text{New})\,\ell_i}_{N_1}=\frac{3}{128\pi}
\frac{1}{(\pmb{\tilde\lambda}\pmb{\tilde\lambda})_{11}}
\sum_j\mathbb{I}\text{m}
\left[
\left(
\pmb{h}\,\pmb{\hat r}^2\pmb{h}^\dagger
\right)_{ji}
\tilde\lambda_{1j}\tilde\lambda_{1i}^*
\right]\,,$$ where $\pmb{\hat
r}=\mbox{diag}(M_{N_1}^2/M_{F_1}^2,M_{N_1}^2/M_{F_2}^2,M_{N_1}^2/M_{F_3}^2)$ and the effective couplings $\pmb{\tilde \lambda}$ are defined as $$\label{eq:effective-couplings}
\pmb{\tilde \lambda}=v_F\pmb{\lambda}\,\pmb{\hat M_F}^{-1}\,\pmb{h}^\dagger\,.$$ The total CP violating asymmetry obtained from (\[eq:new-contribution-CP\]) by summing over the flavor indices vanishes $$\label{eq:new-contribution-CP-total}
\epsilon_{N_1}^\text{New}=\sum_{i=e,\mu,\tau}
\epsilon^{(\text{New})\,\ell_i}_{N_1}
=\frac{3}{128\pi}
\frac{1}{(\pmb{\tilde\lambda}\pmb{\tilde\lambda})_{11}}
\sum_j\mathbb{I}\text{m}
\left[
\pmb{\tilde\lambda}\,
\pmb{h}\,\pmb{\hat r}^2\pmb{h}^\dagger\,
\pmb{\tilde\lambda}^\dagger
\right]_{11}=0\,.$$ Accordingly in the resulting scheme leptogenesis becomes possible only via flavor dynamics and in that sense it is a purely flavored leptogenesis realization [@AristizabalSierra:2009bh; @GonzalezGarcia:2009qd]. Note that since in this simple case $N_{2,3}$ are not involved in the loop corrections the RH neutrino mass splittings are not relevant. Even if they were relevant a mass splitting could always be accommodated due to the Abelian nature of $G_F$. When $G_F$ is non-Abelian and the RH neutrinos are placed in multiplets, as already stressed the mass splittings can only be achieved if they belong to different multiplets.
Leptogenesis with flavor symmetries: type-I seesaw {#sec:single}
==================================================
The connection between flavor symmetry enforced Tribimaximal (TB) mixing and leptogenesis was investigated by [@Jenkins:2008rb], considering models based on $A_4$ and $Z_7 \rtimes Z_3$. The conclusion derived was that due to the specific construction those models implement, the relevant quantity $\pmb{\cal M}\equiv \pmb{m_D}^{\dagger}
\;\pmb{m_D}$ is proportional to the identity matrix $\pmb{\mathbb{I}}$ and therefore the CP asymmetry must vanish at leading order (LO) ${\cal O}(\eta^0)$, with $\eta\equiv V/\Lambda$ and $V$ a generic flavon vacuum expectation value $\langle\phi\rangle=V$. Importantly, it was noted that there was a difference between having TB at low energy accidentally (which allows leptogenesis to be viable) and TB being enforced by a symmetry. Deviations from the exact mixing limit were also considered and the magnitude of the CP asymmetry was estimated as being connected to the magnitude of the next-to-leading order (NLO), ${\cal O}(\eta^1)$, deviations of the mixing angles. These conclusions were illustrated by considering the SUSY model $A_4 \times Z_3$ of [@Altarelli:2005yx]. In [@Hagedorn:2009jy], two specific $A_4$ models were carefully studied (taking into account washout effects) in order to derive the correlations between the deviation from the exact mixing limit and the generation of leptonic asymmetries. The existing collection of particular cases was generalized into two model-independent results in [@Bertuzzo:2009im; @AristizabalSierra:2009ex]. Although the conclusions of both generalizations are to some extent equivalent, they rely on different assumptions and it is worth considering both in detail. While [@Bertuzzo:2009im] is based on group theoretical arguments, [@AristizabalSierra:2009ex] is based on general arguments hinging explicitly on the absence of unnatural fine-tuning.
The group theoretical proof [@Bertuzzo:2009im] starts by assuming invariance of the Lagrangian in (\[eq:seesaw-lag\]) under a generic flavor group $G_F$ in the limit $v_F=0$. Under this assumption the Dirac and RH neutrino mass matrices must remain invariant under $G_F$ transformations of $\ell$ and $N$, namely $$\label{eq:field-transformations}
X\to \Omega_X(g)\,X\quad (\mbox{with}\;X=\ell,N)\,,$$ where $\Omega_X(g)$ corresponds to unitary representations of the group $G_F$ for the generic group element $g$. Different conclusions can be derived depending on whether the representations are irreducible or not:
- If 3 RH neutrinos are in a 3-dimensional irreducible representation the CP asymmetry vanishes at LO. Invariance of the Lagrangian implies the following equality $$\label{eq:case1-irr-rep}
\pmb{\cal M}= \Omega_N(g)^\dagger \pmb{\cal M} \Omega_N(g)\,.$$
Since the irreducible representation is 3-dimensional $\Omega_N(g)$ is in general a non-diagonal matrix. Therefore as a direct consequence of this $\pmb{\cal M}$ is proportional to $\mathbb{I}$ so that the equality can be verified for any group element $g$. In general all the parameters in (\[eq:seesaw-lag\]) receive NLO corrections, from higher dimensional effective operators, and so do the total and flavored CP asymmetries. At ${\cal O}(\eta^1)$ two cases can be identified:
- The loop-functions $f(\omega_\alpha)$ and $g(\omega_\alpha)$ are independent of $\eta$: The flavored CP asymmetries $\epsilon_{N_\alpha}^{\ell_i}$ arise at ${\cal O}(\eta)$ as they have only one power of $\pmb{\cal M}$, and $(m_{D_{i
\alpha}}^{\ast}\,m_{D_{i \beta}})$ needs not depend on $\eta$ (the combination has flavor indices so the transformation properties of lepton doublets can be relevant). With the sum over the lepton flavor index $i$ taken, the total CP asymmetry $\epsilon_{N_\alpha}$ depends on the square of $\pmb{\cal M}$ and it arises only at order $\eta^2$ (this is in agreement with the results of [@Jenkins:2008rb]).
- The non-Abelian symmetry produces degeneracies in the RH neutrino mass spectrum: there is an enhancement of one order in both asymmetries ($\epsilon_{N_\alpha}^{\ell_i} \sim {\cal
O}(\eta^0)$, $\epsilon_{N_\alpha} \sim {\cal O}(\eta)$) due to the loop functions $f(\omega_\alpha)$ and $g(\omega_\alpha)$ having $\eta^{-1}$ dependence.
- If the RH neutrinos are in a reducible representation the conclusions do not follow so straightforwardly, but if the LO matrices $\pmb{M_N}$ and $\pmb{\cal M}$ are simultaneously diagonalizable then the same conclusions as in the case with irreducible representations apply. As a particular case, if the symmetry is Abelian its 1-dimensional representations are in general unable to make the asymmetry vanish, with the requirement that $\pmb{\cal M}$ is diagonal simultaneously with $\pmb{M_N}$ typically not being fulfilled.
The authors also investigated thoroughly a particular based in the $A_4 \times Z_3 \times U(1)_{FN}$ model ([@Altarelli:2005yx]).
The general argument proof in [@AristizabalSierra:2009ex] starts from an exact mixing scheme (in the form-diagonalizable sense [@Low:2003dz]). The exact mixing is the outcome of a symmetry, not accidental. The proof relies fundamentally on the assumption that the resulting effective light neutrino mass matrix can be diagonalized by a special unitary matrix that does not depend on relationships between the parameters that govern the masses. For definiteness the TB mixing was considered: $$\label{eq:diagonalization}
\pmb{\hat{m}_\nu}=\pmb{D}\,\pmb{U_{\text{TB}}}^T\,
\pmb{m_\nu^\text{eff}}\,\pmb{U_{\text{TB}}}\,\pmb{D}\,,$$ $\pmb{D}$, defined in section \[sec:leptogenesis-type-I\], has the low-energy Majorana phases and $\pmb{U_{\text{TB}}}$ is the PMNS matrix with the corresponding TB values for the mixing angles. The Dirac and RH neutrino mass matrix are diagonalized according to $$\label{def}
\begin{array}{rcl}
\pmb{\hat{m}_D}&=&\pmb{U_L}^\dagger \, \pmb{m_D}\, \pmb{U_R} \,,\\
\pmb{\hat{M}_N}&=&\pmb{V_R}^T \, \pmb{M_N}\, \pmb{V_R} \,,
\end{array}$$ with $\pmb{U_{L,R}}$ and $\pmb{V_R}$ unitary matrices. Then, from the seesaw formula we can write: $$\label{ss}
\pmb{m_\nu^\text{eff}}= - \pmb{U_L}\,\pmb{\hat{m}_D} \,
\left(\pmb{U_R}^\dagger\,\pmb{V_R}\right)\,
\pmb{\hat{M}_N}^{-1}\,
\left(\pmb{V_R}^T \pmb{U_R}^*\right)\,
\pmb{\hat{m}_D}\,\pmb{U_L}^T\,.$$ We assume $\pmb{m_\nu^{\text{eff}}}$ is diagonalized by the mixing scheme without special relationships between masses: $$\pmb{\hat{m}_\nu}= - \pmb{D}\,
\left(\pmb{U_{\text{TB}}}^T \pmb{U_L}\right)\,
\pmb{\hat{m}_D} \, \left(\pmb{U_R}^\dagger\,\pmb{V_R}\right)\,
\pmb{\hat{M}_N}^{-1}\,
\left(\pmb{V_R}^T \pmb{U_R}^*\right)\,
\pmb{\hat{m}_D}\,
\left(\pmb{U_L}^T \pmb{U_{\text{TB}}}\right)\,\pmb{D}\,,$$ therefore the matrix on the left-hand side is diagonal (denoted by the hat), which then implies that the combinations of matrices appearing on the right-hand side of the equation, $\left(\pmb{U_{\text{TB}}}^T
\pmb{U_L}\right)$, $\left(\pmb{U_R}^\dagger\,\pmb{V_R}\right)$ and conversely $\left(\pmb{V_R}^T \pmb{U_R}^*\right)$, $\left(\pmb{U_L}^T
\pmb{U_{\text{TB}}}\right)$ should also be diagonal (up to orthogonal rotations in case of degenerate eigenvalues, but this does not alter the conclusion). Consider for simplicity a case without degeneracies, and evaluate off-diagonal elements of the expression on the right hand side: if the matrix combinations identified above were not diagonal, then the off-diagonal elements of the right-hand side will depend on combinations of the masses of RH neutrinos and Yukawa couplings $\pmb{\pmb{\lambda}}$, which could only vanish for very specific relations between them—which explicitly violates form-diagonalizability. Therefore $\left(\pmb{U_{\text{TB}}}^T
\pmb{U_L}\right)$, $\left(\pmb{U_R}^\dagger\,\pmb{V_R}\right)$ should indeed be diagonal up to orthogonal rotations of degenerate eigenvalues. Assuming no degeneracies this implies $$\label{eq:UL-and-UR}
\pmb{U_L}=\pmb{U_{\text{TB}}}\,\pmb{\hat P_L}\,,
\qquad
\pmb{U_R}^\dagger=\pmb{\hat P_R}\,\pmb{V_R}^\dagger\,,$$ with $\pmb{\hat
P_{L,R}}=\mbox{diag}(e^{\text{i}\alpha_1^{L,R}},e^{\text{i}\alpha_2^{L,R}},
e^{\text{i}\alpha_3^{L,R}})$. These relations allow to fix the structure of the Dirac mass matrix as $$\label{eq:Dirac-mass-matrix-secTBM}
\pmb{m_D}=\pmb{U_{\text{TB}}}\,\pmb{\hat D}^*\pmb{\hat m_D}\,,$$ that when compared with the Casas-Ibarra parametrization in (\[eq:casas-ibarra\]) leads to $\pmb{R}=\pmb{\hat
m_\nu}^{-1/2}\,\pmb{\hat m_D}\,\pmb{\hat M_N}^{-1/2}$, showing that $\pmb{R}$ is diagonal and real. As the total CP asymmetry can be expressed as: $$\label{eq:cp-asymm-CI}
\epsilon_{N_\alpha} = -\frac{3 M_{N_\alpha}}{8 \pi v^2}
\frac{{\mathbb I}\mbox{m}
\left[\sum_i m_{\nu_i}^2 R_{i \alpha}^2\right]}
{\sum_i m_{\nu_i} |R_{i \alpha}|^2}\,,$$ then the asymmetry must vanish. Alternatively one can consider the following: $(\pmb{U_R}^\dagger\,\pmb{V_R})$ is diagonal from our assumption, this means that the basis where $\pmb{m_D}$ is diagonal and the basis where $\pmb{M_N}$ is diagonal have a special relationship (this is often denoted as form-dominance [@Chen:2009um] and is essentially also the requirement outlined in the group theoretical approach of [@Bertuzzo:2009im] in the case of reducible representations). We can simply start with the diagonal basis of $\pmb{m_D}$, use $\pmb{U_R}$ to bring it to the general basis, $\pmb{V_R}$ to bring it to the basis of diagonal $\pmb{M_N}$ and see that in that basis $\pmb{m_D}$ is essentially $\pmb{U_{\text{TB}}}\;\pmb{\hat{m}_D}$—its columns are the eigenvectors of the mixing scheme. Naturally when $\pmb{m_D}^{\dagger} \pmb{m_D}$ is taken the mixing cancels out and the relevant quantity for the asymmetry is diagonal (consistently with [@Jenkins:2008rb; @Bertuzzo:2009im]). Although the example uses TB mixing for definiteness, it should be stressed that any exact mixing scheme enforced by a symmetry leads to the same conclusion. The paper also looked into several particular cases of TB mixing, dividing them into classes of models according to the structure of $\pmb{m_D}$ and $\pmb{M_N}$. The structure of NLO contributions was considered explicitly with expansions around the the LO values, leading to: $$\begin{aligned}
\label{NLO-corrections}
\pmb{m_D}^{\prime\dagger} \pmb{m_D}^{\prime}&=
\pmb{m_D}^{\dagger} \pmb{m_D} +
\pmb{m_D}^{\dag}
\left(
\pmb{U_\ell^{(1)}}^\dag\pmb{m_D} +
\pmb{U_L}\pmb{U_L^{(1)}}\pmb{\hat{m}_D} \pmb{U_R}^{\dag}\pmb{V_R} +
\pmb{U_L}\pmb{\hat{m}_D}^\prime \pmb{U_R}^\dag \pmb{V_R} +
\right.\nonumber\\
& + \left.
\pmb{U_L}\pmb{\hat{m}_D}\pmb{U_R^{(1)}}^\dag\pmb{U_R}^\dag\pmb{V_R} +
\pmb{m_D} \pmb{V_R^{(1)}}\right) + \mathrm{h.c.}\,.\end{aligned}$$ The superscript (1) refers to those quantities corrected by NLO contributions and $\pmb{U_\ell}$ diagonalizes the charged lepton mass matrix (we started on the basis where it is diagonal at LO, but it becomes non-diagonal after NLO corrections are introduced). The $A_4$ model of [@Lin:2009ic] was used to illustrate the conclusions and to highlight how it can be possible to link low and high-energy CP violation parameters. Finally it was noted that with added degrees of freedom (such as from having type-II seesaw) it would be possible to generate an asymmetry even while remaining in the exact mixing limit.
Not long after these two important generalizations, further results were presented by [@Felipe:2009rr] and [@Choubey:2010vs], clarifying some points that we summarize very briefly here. Assuming that the symmetries of the mass matrices involved in type-I are residual symmetries of the Lagrangian, [@Felipe:2009rr] shows that $\pmb{{\cal M}}$ is diagonal and therefore the asymmetry vanishes. They also consider the exact mixing schemes so characteristic of models with flavor symmetries and connect that requirement with their assumption: if the effective neutrino mass matrix has nonzero determinant, then the Lagrangian contains the maximal residual symmetry (that of the mass matrices) and so leptogenesis can not proceed at LO and in fact even when the determinant vanishes, $\epsilon_{N_\alpha}$ is still zero at LO. The implication of form-dominance [@Chen:2009um] on the Casas-Ibarra matrix $\pmb{R}$ is considered in detail in [@Choubey:2010vs]: the vanishing CP asymmetry is not particular to TB. Rather, exact mixing schemes enforced by symmetries are a particular case of form-dominance [@Chen:2009um]. The main conclusions are that form-dominance by itself is sufficient to make the CP asymmetry vanish and that it is possible to violate form-dominance softly without perturbing the mixing. The cases considered earlier in [@Antusch:2006cw] were summarised, and they exemplify very clearly the separation between TB and form-dominance.
Before concluding this section it is important to stress that corrections to the exact mixing scheme are typically expected at NLO as is explicitly considered in the literature (see e.g. [@Altarelli:2010gt]). An exception which does preserve exact mixing is one of the renormalizable UV complete models in [@Varzielas:2010mp] (due to the lack of certain messengers). Furthermore, in [@Cooper:2011rh] it is shown that RG corrections can also provide the deviations necessary to lift the vanishing CP asymmetries.
Leptogenesis with flavor symmetries: type-I and II seesaws {#sec:both}
==========================================================
Recently, a model-independent analysis in the style of [@AristizabalSierra:2009ex] considered cases with both type-I and type-II seesaw [@AristizabalSierra:2011ab]. Flavor models limit themselves to type-I and/or II seesaws with few exceptions (e.g. [@Bazzocchi:2009qg]). As noted in [@AristizabalSierra:2009ex], in general the CP asymmetries involving the additional degrees of freedom can be non-vanishing even in the exact mixing limit and [@AristizabalSierra:2011ab] considered the framework with both seesaw types in detail. It was shown that non-vanishing CP asymmetries depend on the existence of repeated eigenvectors across the seesaw types. The main point is the following: leptogenesis can become viable through the CP asymmetries in which the triplets intervene i.e. $\epsilon_{N_\alpha}^\Delta$ or $\epsilon_\Delta$ (see eqs. (\[eq:cp-asymm-triplet\]) and (\[eq:triplet-CPasymmetry\])), depending—of course—on whether it proceeds via RH neutrino or scalar triplet dynamics (or both as in the case treated in sec. \[sec:leptogenesis-type-II\]). Both CP asymmetries depend on the imaginary parts of $$\label{eq:matrix-involved-type-i-ii}
\pmb{{\cal Y}}\,\mu=\pmb{m_D}\,\pmb{Y}^*\,\pmb{m_D}^T\,\mu\,.$$ Since the parameter $\mu$ is in general complex, and the presence of $G_F$ does not allow a definitive statement about its phase, the CP asymmetries are non-vanishing even if the matrix $\pmb{\cal Y}$ turns out to be real. Vanishing $\pmb{\cal Y}$, however, implies $\epsilon_{N_\alpha}^\Delta,\epsilon_\Delta=0$. In that sense the quantity to be analyzed is $\pmb{\cal Y}$.
Definitive conclusions about this matrix can be made by writing the effective light neutrino mass matrix as an outer product of the eigenvectors of the assumed mixing scheme [^10] and its mass eigenvalues: $$\label{eq:mass-matrix-eff-i-plus-ii}
\pmb{m_\nu^\text{eff}}=\sum_{i=e,\mu,\tau}=m_{\nu_i}\,
\pmb{v_i}\otimes\pmb{v_i}\,.$$ According to (\[eq:eff-mass-matrix\]) the eigenvectors come from the contributions of type-I and/or type-II seesaws: $$\label{eq:sesaw-mm-outer}
\pmb{m_\nu^X} = \sum_{i=1,2,3}m_{\nu_i}^X
\pmb{v_i}\otimes\pmb{v_i}\qquad (X=I,II)\,.$$ This decomposition is based on the assumption that both $\pmb{m_\nu^I}$ and $\pmb{m_\nu^{II}}$ are diagonalized by the PMNS matrix fixed by the assumed mixing scheme. This needs not be the case but if it is not then somehow a contribution that is incompatible with the mixing scheme is present in both seesaw types in just the correct quantities to cancel each other out (which amounts to unrealistic fine-tuning given the separate physical degrees of freedom involved). With the decomposition in (\[eq:sesaw-mm-outer\]) we classify the possible models:
A) \[g-models\] : The eigenvectors $\pmb{v_i}$, defining the effective light neutrino mass matrix, stem from both type-I and type-II contributions. Note that in this case in addition to the pieces involving the eigenvectors $\pmb{v_i}$ each (or only one) seesaw contribution may involve also the identity matrix $\mathbb{I}$.
B) \[int-models\] : The eigenvectors $\pmb{v_i}$ entirely arise from either type-I or type-II contributions.
C) \[min-models\] : Two eigenvectors $\pmb{v_i}$ stem from the type-I (type-II) contributions and the third one $\pmb{v_k}$ (with $\pmb{v_i}\cdot\pmb{v_k}=0$) from type-II (type-I).
Note that being able to parametrize each seesaw contribution with these eigenvectors does not mean they are all explicitly present. A common scenario can be the explicit presence of only a single eigenvector in a given seesaw type in either cases \[int-models\] or \[min-models\] (with at least one more eigenvector present in the other seesaw). Another relevant observation is that the underlying symmetry may be arranging structures which can be reparametrized in terms of the eigenvectors, meaning one does not necessarily need separate physical degrees of freedom to have more than one eigenvector represented—see e.g. [@deMedeirosVarzielas:2011tp] where the $\mu-\tau$ structure $$P=\begin{pmatrix}
1 & 0 & 0 \\
0 & 0 & 1 \\
0 & 1 & 0
\end{pmatrix}$$ arises directly from specific discrete groups—as a TB-compatible contribution it can be expressed in terms of the TB eigenvectors as explicitly seen with $b=2a$ and $c=-3a$ in the parametrization: $$\label{eq:tbm-light-mm}
\pmb{m_\nu^{\text{eff}}}=\pmb{m_\nu^I} + \pmb{m_\nu^{II}}
=
\begin{pmatrix}
4a + b & -2a +b & -2a + b\\
\cdot & a+b+c & a+b-c\\
\cdot & \cdot & a+b+c
\end{pmatrix}\,.$$
With the models classified according to the eigenvectors of their mixing scheme, we can determine the structures of $\pmb{m_D}$, $\pmb{Y}$ and then also $\pmb{{\cal
Y}}$ (see [@AristizabalSierra:2011ab] for details). However, even without determining explicitly these structures, it can be realized from the definitions in \[g-models\], \[int-models\] and \[min-models\] that vanishing $\pmb{{\cal Y}}$ occurs only when $\pmb{m_D}$ and $\pmb{Y}$ are orthogonal, and in principle this happens only in models of type \[min-models\]: in the other cases, the presence of the scalar triplet degrees of freedom allows the generation of the baryon asymmetry via leptogenesis even in the limit of an exact mixing pattern (in agreement with what was suggested in [@AristizabalSierra:2009ex]).
Having identified models where leptogenesis becomes viable in the exact mixing limit, the obvious step is to study those in which the constraints enforced by $G_F$ allow the CP asymmetry to be constrained by the low-energy data. There are in general 6 observables: 3 light neutrino masses and 3 CP phases (only 2 Majorana phases in TB mixing). Thus, models involving more than 6 parameters barely allow to make any statement about the asymmetry. The most general models in \[g-models\] are within that class, with 8 complex parameters. Models in which the asymmetries $\epsilon_{N_\alpha}^\Delta,\epsilon_\Delta$ can be constrained by the low-energy data fall within classes \[int-models\] or \[min-models\]. One can add a contribution proportional to $\mathbb{I}$ to either (or both) seesaw types, and any such contribution counts as all (and any) 3 eigenvectors, so the quantity $\mathfrak{I}\mbox{m}[\pmb{\cal
Y}]$ can be expressed in terms of the combinations of parameters defining the quantities $\pmb{m_{\nu_i}^{I,II}}$. Denoting them as $a_0^X$ and $a_i^X$ ($X=I,II$) for $\mathbb{I}$ and the eigenvectors contributions respectively, it turns out that $$\label{eq:explicit}
\mathfrak{I}\mbox{m}\left[\pmb{\cal Y}\right]=
\mathfrak{I}\mbox{m}
\left[ (a_0^I a_0^{II \star})
+ \sum_i \left( a_i^I a_i^{II \star} \right)
+ \left(\sum_i a_i^I \right) a_0^{II \star}
+ a_0^I \left(\sum_i a_i^{II \star}\right) \right]\,.$$ In particular for the class of models discussed in item \[min-models\] with only two eigenvectors $\pmb{v_i}$ stemming from type-I (type-II) and $\mathbb{I}$ from type-II (type-I) we have $$\label{eq:explicit-particular}
\mathfrak{I}\mbox{m}\left[\pmb{\cal Y}\right]=
\mathfrak{I}\mbox{m}
\Bigg[(a_0^X)^\star \sum_{\substack{i<j\\j=2,3}} a_i^Y\Bigg]\,,$$ with $X=I$ and $Y=II$ or vice versa. The parameters of these models are only 3 and can be well restricted by means of the solar and atmospheric squared mass differences [@Schwetz:2011zk] yielding tight constraints on the CP asymmetries. Figure \[fig1\] shows $\epsilon_\Delta$ in models for which two eigenvectors originate from type-I and the contribution from type-II is proportional to $\mathbb{I}$, assuming a TB mixing pattern. For comparison we have also included the results for the general cases discussed in \[g-models\] involving contributions proportional to the $\mathbb{I}$ in type-I and II. The scatter plot was obtained by randomly scanning the parameters defining the neutrino masses and selecting those points that lead to solar and atmospheric squared mass differences within the experimental range. Figure \[fig1\] shows that in general models, even in the limit of an exact mixing pattern no statement about the CP asymmetry can be established. In contrast, in the simplified model considered, specific values of the CP asymmetry require somehow specific ranges for the triplet mass.
![*$\epsilon_\Delta$ as function of $M_\Delta$. Red squares and orange crosses for normal and inverted hierarchy of a specific 3-parameter predictive case. Green squares and blue crosses for normal and inverted hierarchy of the general 8-parameter case [@AristizabalSierra:2011ab].*[]{data-label="fig1"}](Greps.jpg){width="10.5cm" height="7cm"}
Conclusions {#sec:conclusions}
===========
From a general perspective the problem of studying leptogenesis in the presence of flavor symmetries $G_F$ depends on whether the lepton number breaking scale $\Lambda_L$ is above or below the scales involved in $G_F$ (flavor breaking and flavon scales, denoted generically $v_F$ and $M_\phi$). In the flavor symmetric phase, defined as a scenario in which $\Lambda_L>v_F,M_\phi$, as described in section \[sec:sym\], flavons must play a relevant role in the generation of the lepton asymmetry either as propagating or loop intermediate states. Indeed it turns out that the presence of these states apart from rendering leptogenesis viable can change the conventional picture by e.g. leading to models in which leptogenesis proceeds entirely via lepton flavor effects[^11].
In the case of type-I seesaw in the flavor broken phase,defined as a scenario where $v_F,M_\phi>\Lambda_L$, the model-independent conclusion reviewed in section \[sec:single\] is that CP asymmetries vanish in the exact mixing limit enforced by flavor symmetries. This is not an intrinsic feature of the exact mixing, and this result can be attributed to the property of form dominance in the neutrino mass matrices. Within the scenario of type-I seesaw with symmetry enforced mixing, interesting correlations between low energy observables (mixing angles and CP phases) and high-energy parameters (CP asymmetries) can be present when there is departure from the exact mixing limit. It is possible even in a model-independent context to identify rather generally the order of magnitudes associated with a small parameter responsible for the mixing deviations.
When other degrees of freedom that can contribute to leptogenesis are added, such as those associated with type-II seesaw, the above conclusions need not apply. Section \[sec:both\] considers specifically the interplay between type-I and II, where it is possible to conclude that the associated asymmetries still vanish in special cases. Classifying these hybrid scenarios according to the eigenvectors of the exact mixing is helpful, and even without departure from exact mixing leptogenesis can occur whenever eigenvectors are repeated across the two seesaw types—with contributions proportional to $\mathbb{I}$ counting as any and all eigenvectors.
Finally, we note that in accordance with section \[sec:sym\], in the flavor symmetric phase it is possible to have non-vanishing asymmetry originating just from type-I seesaw while the type-II asymmetries vanish due to orthogonality of the eigenvectors. For this to occur there must be a specific hierarchy of scales so that the RH neutrinos decay in the flavor symmetric phase, avoiding the results described in section \[sec:single\], while $\Delta$ decays in the broken phase with vanishing contributions as described in section \[sec:both\].
Acknowledgement {#acknowledgement .unnumbered}
===============
We specially thank Federica Bazzocchi for helpful discussions. DAS also wants to acknowledge Marta Losada, Luis Alfredo Muñoz, Jernej Kamenik and Miha Nemevšek for the enjoyable collaboration on the subjects discussed here. Special thanks to Enrico Nardi for the always enlightening leptogenesis discussions. DAS is supported by a Belgian FNRS postdoctoral fellowship. IdMV is supported by DFG grant PA 803/6-1 and partially through PTDC/FIS/098188/2008.
Conventions and notation {#sec:conventions}
========================
In this appendix we collect the equations used in the calculations discussed in section \[sec:lepto-gen\]. We start by specifying well known statistical and cosmological quantities.
Equilibrium distributions and Cosmological quantities {#sec:eq-dis-cosmo}
-----------------------------------------------------
All the results presented in this short review were done using Maxwell-Boltzmann distribution functions. For type-I and type-III seesaws the equilibrium number densities read $$\label{eq:equilibrim-num-densities}
n^{\text{Eq}}_{\ell,H}(z)=\frac{2M^3}{\pi^2\;z^3}\,,
\quad
n^{\text{Eq}}_{N_1}(z) =\frac{M^3_{N_1}}{\pi^2}\frac{K_2(z)}{z}\,,
\quad
n^{\text{Eq}}_{X}(z) =\frac{3M^3_{X}}{2\pi^2}\frac{K_2(z)}{z}
\quad(\mbox{with}\;\; X=\Delta,T_\alpha)\,.$$ Here $K_2(z)$ is the second-order modified Bessel function of the second-type and $z\equiv M/T$ where $M$ can refer to $M_{N_{1},\Delta,T_\alpha}$ depending on the considered case (this also applies for $n^{\text{Eq}}_{\ell,H}(z)$). For the type-II scenario discussed in \[sec:leptogenesis-type-II\] the $N_1$ equilibrium number density is given by $$\label{eq:N1-equilibrium-number-dens-type-II}
n^{\text{Eq}}_{N_1}(z) =\frac{M^3_\Delta}{\pi^2}r^2\frac{K_2(rz)}{z}\,,$$ with $r=M_{N_1}/M_\Delta$. The energy density $\rho(z)$ and pressure $p(z)$ become $$\label{eq:energy-dens}
\rho(z)=\frac{3 M^4}{z^4\pi^2} g_*\,,\quad
p(z)=\frac{M^4}{z^4\pi^2} g_*$$ where $g_*=\sum_{i=\text{All species}} g_i$ is the number of standard model relativistic degrees of freedom (118 for $T\gg$ 300 GeV). Accordingly, the expansion rate of the Universe and entropy density can be written as $$\label{eq:HandS}
H(z)=\sqrt{\frac{8g_*}{\pi}}\frac{M^2}{M_{\text{Planck}}}
\frac{1}{z^2}\,,
\quad
s(z)=\frac{4 M^3}{z^3\pi^2} g_*\,.$$
Formal solutions of the kinetic equations {#sec:formal-sol}
-----------------------------------------
In the type-I and III seesaw cases the integration of the differential equations accounting for the evolution of the $B-L$ asymmetry leads to $$\label{eq:BmL-appendix-typeI-III}
Y_{\Delta_{B-L}}(z)=-n\times\epsilon_X\,Y_X^\text{Eq}(z_0)\eta(z)\,,$$ where $X=N_1,T_1$ and $n=1,3$ depending on whether the decaying state is the singlet or the triplet. Assuming a vanishing initial asymmetry ($Y_{\Delta_{B-L}}^\text{(In)}=0$) the efficiency function can be written as $$\label{eq:eff-type-I-III}
\eta(z)=\frac{1}{Y_{T_\alpha}^{\text{Eq}}(z_0)}
\int_{z_0}^{z}\;Q_X(z')\frac{dY_{X}(z')}{dz'}
e^{-\int_{z'}^z dz'' P_{X}(z'')}\;,$$ with the functions $Q_X(z),P_X(z)$ given by $$\label{eq:integrand-efficiencyappendix}
Q_{N_1}(z)=1\,,\quad
Q_{T_1}(z)=\frac{\gamma_{D_{T_1}}}{\gamma_{D_{T_1}}+2\gamma_{A}}\,,\quad
P_{N_1,T_1}(z)=\frac{1}{2 Y^{\text{Eq}}_\ell(z)}
\frac{\gamma_{D_{N_1,T_1}}(z)}{s(z) H(z) z}\,.$$ Freeze-out of the asymmetry is at $z=z_f$ with $z_0\ll
z_f$. The efficiency factor is determined by $\eta=\eta(z_f)$.
The case for type-II is more involved but the kinetic equation for the $L$ asymmetry in (\[eq:BEQ-type-II\]) can still be formally integrated [@AristizabalSierra:2011ab]. Again, assuming an initial vanishing $L$ asymmetry, we get $$\label{eq:BmL-formal-sol}
Y_{\Delta_L}(z)=\int_{z_i}^z\,dz'\,Q(z')\,e^{-\int_{z'}^z\,dz''\,P(z'')}\,,$$ with the functions $Q(z)=Q^I(z) + Q^{II}(z)$ and $P(z)$ given by $$\begin{aligned}
\label{eq:PandQ-functions1}
Q^I(z)&=\frac{1}{s(z)H(z)z}
\left\{
\left[
(y_{N_1}(z)-1)\epsilon_{N_1}^\text{tot}-y_{\Delta_\Delta}^H(z)
\right]\gamma_{D_{N_1}}(z)
\right\}\,,\\
\label{eq:PandQ-functions2}
Q^{II}(z)&=\frac{3}{s(z)H(z)z}
\left\{
\left[
(y_\Sigma(z) - 1)\epsilon_\Delta - 2 K_\ell\,y_{\Delta_\Delta}(z)
\right]\gamma_{D_\Delta}(z)
\right\}\,,\\
\label{eq:PandQ-functions3}
P(z)&=\frac{1}{s(z)H(z)z}
\left[
\frac{1}{Y^\text{Eq}_\ell}
\left(
\gamma_{D_{N_1}}(z) + 2\,K_\ell\,\gamma_{D_\Delta}(z)
\right)
\right]\,.\end{aligned}$$ Note that in $Q^{II}(z)$ we have included a factor of 3 coming from the $SU(2)$ physical degrees of freedom of the triplet. By factorizing either $\epsilon_{N_1}^\text{tot}$ or $\epsilon_\Delta$ from the functions $Q^{I,II}(z)$ and normalizing to $Y^\text{Eq}_\text{tot}\equiv
Y^\text{Eq}_\text{tot}(z\to0)=
Y^\text{Eq}_{N_1}(z)+Y^\text{Eq}_\Sigma(z)|_{z\to 0}$ the $L$ asymmetry in (\[eq:BmL-formal-sol\]) can be written in terms of efficiency functions that depend on the dynamics of the scalar triplet and the fermionic singlet as done in eq. (\[eq:efficiencies\]).
Reduced cross sections for triplet scalar and fermion {#sec:reduced-cross-secion}
-----------------------------------------------------
The reduced cross section for the scalar electroweak triplet involves the $s$-channel processes $\pmb{\Delta}\pmb{\Delta}\to F\bar F, AA,
HH$ ($F$ and $A$ stand for standard model fermions and $SU(2)\times
U(1)$ gauge bosons respectively), $t$ and $u$ channel triplet mediated processes $\pmb{\Delta}\pmb{\Delta}\to AA$ and the “quartic” process $\pmb{\Delta}\pmb{\Delta}\to AA$. In powers of the kinematic factor $\omega(x)=\sqrt{1-4/x}$ (with $x=M_\Delta^2/s$) it can be split in three pieces [@Hambye:2005tk]: $$\begin{aligned}
\label{eq:reduced-cross-section}
\widehat \sigma_1(x)&=\frac{1}{\pi}
\left[
g^4\left(5 + \frac{34}{x}\right)
+
\frac{3}{2}g'^4\left(1 + \frac{4}{x}\right)
\right]\omega(x)\,,
\nonumber\\
\widehat \sigma_2(x)&=\frac{1}{8\pi}
\left(
25 g^4 + \frac{41}{2}g'^4
\right)\omega(x)^3\,,
\nonumber\\
\widehat \sigma_3(x)&=\frac{6}{\pi x^2}
\left[
4 g^4 (x-1) + g'^4 (x-2)
\right]\ln\left[
\frac{1+\omega(x)}{1-\omega(x)}
\right]
\,,\end{aligned}$$ with $\widehat\sigma_A(x)=\sum_{i=1}^3\widehat \sigma_i(x)$ and $g,
g'$ the $SU(2)$ and $U(1)$ gauge couplings.
For the fermion $SU(2)$ triplet the reduced cross sections involves the gauge boson mediated $s$-channel processes $T_\alpha
T_\alpha\leftrightarrow \ell\bar\ell$ and $T_\alpha
T_\alpha\leftrightarrow q\bar q$ and the $t$ and $u$-channel triplet mediated process $T_\alpha T_\alpha\leftrightarrow A_\mu A^\mu$. The full result where now $x=M_{T_1}^2/s$, reads [@Hambye:2003rt]: $$\label{eq:red-cross-sec}
\widehat \sigma_A(x)=
\frac{6g^4}{\pi}\left(1+\frac{2}{x}\right)\omega(x)
+
\frac{2g^4}{\pi}\left[
3\left(
1 + \frac{4}{x} - \frac{4}{x^2}
\right)
\log
\left(
\frac{1+\omega(x)}{1-\omega(x)}
\right)
-\left(
4 + \frac{17}{x}
\right)\omega(x)
\right]\,.$$
[^1]: e-mail address: [[email protected]]{}
[^2]: e-mail address: [[email protected]]{}
[^3]: Analytically the problem has been addressed yielding quite accurate expressions for the efficiency [@Buchmuller:2004nz; @Blanchet:2006dq].
[^4]: In contrast to the previous section, in this case we do not include the change in the lepton densities due to sphaleron processes, and thus study only the evolution of the $L$ asymmetry.
[^5]: These asymmetries are a consequence of these fields not being self-conjugate.
[^6]: This equation follows from [@Antusch:2004xy] which differs from [@Hambye:2003ka] by a factor of $3/2$.
[^7]: This is to be compared with the pure triplet leptogenesis scenario [@Hambye:2005tk] where the generation of the $L$ asymmetry is entirely determined by only three parameters: $\tilde m_\Delta$, $\tilde m_\Delta^\ell$, $M_\Delta$.
[^8]: In standard leptogenesis at ${\cal O}(\pmb{\lambda}^2)$ the efficiency does not depend on the RH neutrino mass.
[^9]: Leptogenesis in models based on the Froggatt-Nielsen mechanism have been studied in [@Cerdeno:2006ha].
[^10]: These eigenvectors are determined by the column vectors of the PMNS matrix for a fixed mixing pattern.
[^11]: The viability of these models depends on whether lepton flavor equilibrating effects can be circumvented [@AristizabalSierra:2009mq].
|
---
abstract: 'Topological quantum phases underpin many concepts of modern physics. While the existence of disorder-immune topological edge states of electrons usually requires magnetic fields, direct effects of magnetic field on light are very weak. As a result, demonstrations of topological states of photons employ synthetic fields engineered in special complex structures or external time-dependent modulations. Here, we reveal that the quantum Hall phase with topological edge states, spectral Landau levels and Hofstadter butterfly can emerge in a simple quantum system, where topological order arises solely from interactions without any fine-tuning. Such systems, arrays of two-level atoms (qubits) coupled to light being described by the classical Dicke model, have recently been realized in experiments with cold atoms and superconducting qubits. We believe that our finding will open new horizons in several disciplines including quantum physics, many-body physics, and nonlinear topological photonics, and it will set an important reference point for experiments on qubit arrays and quantum simulators.'
author:
- 'Alexander V. Poshakinskiy'
- Janet Zhong
- Yongguan Ke
- 'Nikita A. Olekhno'
- Chaohong Lee
- 'Yuri S. Kivshar'
- 'Alexander N. Poddubny'
title: Quantum Hall phase emerging in an array of atoms interacting with photons
---
The study of electrons propagating in magnetic fields has been driving many problems of physics since the discoveries of the Landau levels [@ezawa2013quantum] and a self-similar structure of the energy spectrum in crystals subjected to ultra-high magnetic fields [@Azbel1964; @Hofstadter1976]. Quantum Hall effect [@TKNN1982] and topological insulators [@Bernevig2006; @Hasan2010] brought the concepts of topological phases to condensed matter physics. However, many effects predicted long time ago including the Hofstadter butterfly spectrum have been realized only recently [@Dean2013].
These developments inspired a rapid progress in [*topological photonics*]{} aiming at creating robust edge states of light immune to disorder [@Ozawa2019; @wang2009; @Haldane2008a; @Chang2020]. Since the effects of magnetic fields on light are weak, the realisation of topological concepts in photonics requires artificial structures and metamaterials [@Khanikaev2017]. Alternative approaches rely on time modulation of structure parameters [@hauke2012; @Sounas2017; @Roushan2017; @Dutt2019] or engineered nonlinearities [@Hadad2018; @smirnova2019nonlinear]. These approaches allow creating effective gauge fields in real or synthetic dimensions, and mimick the effects of magnetic fields or spin-orbit couplings for photons.
{width="70.00000%"}
Here, we uncover that the hallmarks of the quantum Hall phases, including Landau energy levels, topological edge states, and Hofstadter butterfly spectrum, can appear in a simple quantum system: an array of closely spaced two-level atoms (qubits) coupled to photons in a waveguide, see Fig. \[fig:1\]a. In this system, photons become strongly coupled to atoms and create [*polaritons*]{}. These polaritons are not independent but strongly interacting, because one atom cannot absorb two photons simultaneously [@Birnbaum2005]. While the considered model is paradigmatic for quantum optics [@Roy2017; @KimbleRMP2018; @FriskKockum2019], its two-particle Hilbert space was not analyzed until recently. As shown in Figs. \[fig:1\](c,d), when the polariton wave vector is comparable with that of light, a collective atomic state is easily excited optically, and it generally gets “darker" for larger wave vectors. In a two-particle “bright" state, the wave vectors of both excitations are small, which corresponds to the Dicke superradiance [@Dicke1954]. Novel two-particle dark states, where both wave vectors are large, were predicted only last year, and they originate from fermionization of strongly interacting polaritons [@Molmer2019]. It has also been suggested that interactions in the corner regions [@Ke2019] of the diagram of Fig. \[fig:1\]d can localize one of the two polaritons in the center of the array [@Zhong2020].
In this paper, we predict novel types of topological edge states driven by polariton-polariton interactions in the regions indicated by butterflies in Fig. \[fig:1\]d. Here, one polariton forms a standing wave with multiple nodes and a periodic potential for the other indistinguishable polariton, see Figs. \[fig:1\](b,e). As a result, the interaction is described by the [*self-induced*]{} Aubry-André-Harper [@aubrey] model that is mathematically equivalent to the quantum Hall problem on a lattice [@Kraus2012; @Poshakinskiy2014].
The striking novelty of our prediction is that the quantum Hall phase can emerge for interacting indistinguishable particles without any special fine-tuning. The periodic modulation is an intrinsic feature that arises naturally due to the polariton-polariton interactions, in a sharp contrast to previous studies [@Baboux2017; @Ke2020], where one had to impose the modulation deliberately, either by engineering the lattice [@verbin2013; @Ke2020] or applying external fields [@Roushan2017; @Chang2020]. The full Hofstadter butterfly-like spectrum could be obtained in a single shot from just one fixed atomic array, eliminating the need to continuously tune an external magnetic field in a conventional setup [@Roushan2017; @Dean2013]. Our results apply to the experiments with cold atoms [@Corzo2019] or superconducting qubits coupled to a waveguide [@Astafiev2010; @Wang2019; @Haroche2020; @Browaeys2020; @Blais2020; @Clerk2020; @Carusotto2020] and emerging quantum simulators based on excitonic polaritons [@Ghosh2020]. This offers new possibilities to understand quantum many-body topological phases of interacting matter and protect them against decoherence.
{width="\textwidth"}
Two-polariton states {#two-polariton-states .unnumbered}
====================
![[**Self-induced Hofstadter butterfly.**]{} [**(a)**]{} Energy spectrum for the two-polariton states in the cluster corresponding to $j=11$, calculated from the approximate Eq. and by the exact diagonalization of the two-particle Hamiltonian Eq. . [**(b)**]{} Butterfly energy spectrum obtained by the exact diagonalization as a function of cluster index $j$, determining the effective magnetic field. Localization degree is determined as the inverse participation ratio of the vector $\chi$ in Eq. and is shown by color. Thin horizontal line in (b) indicates magnetic field $j/N=11/200$, corresponding to panel (a). Calculation has been performed for $N=200$ and $\varphi=0.02$, energy is measured in the units of $\Gamma_0$. []{data-label="fig:3"}](Fig3){width="45.00000%"}
We consider a periodic array of two-level atoms (qubits) coupled to light, described by an effective Dicke-type Hamiltonian [@Molmer2019; @Ke2019; @Albrecht2019] $$\label{eq:H}
H=\sum_n\omega_0 \sigma_n^\dag\sigma_n^{\vphantom{\dag}}-{{\rm i}}\Gamma_0\sum\limits_{n,m}{{\rm e}}^{{{\rm i}}\omega_0 d|n-m|/c}\sigma_n^\dag\sigma_m^{\vphantom{\dag}}\:,$$ where $\sigma_n^\dag$ is the operator creating excitation of the atom $n$ with the resonance frequency $\omega_0$, $(\sigma_n^{\dag})^2=0$ and $\Gamma_0$ is the radiative decay rate of a single atom. While for $d=0$ the Hamiltonian is equivalent to the conventional Dicke model [@Dicke1954], even small interatomic spacings $0<d\ll 2\pi c/\omega_0$ make the model considerably richer.
Single-particle eigenstates of Eq. are polaritons with the energy dispersion ${\varepsilon}(k)=\omega_0+\Gamma_0/[\cos kd-\cos( \omega_0d/c)]$ [@Ivchenko1991; @Albrecht2019], schematically shown in Fig. \[fig:1\]e. The dispersion consists of two polaritonic branches, resulting from the avoided crossing of light with the atomic resonance and we focus on the lower branch. In the finite array of $N$ atoms, the wave vectors are quantized $k_jd=\pi j/N$, $j=1,2,\ldots N$ [@vladimirova1998ru], and the eigenstates are standing waves, see Figs. \[fig:2\][a–d]{}. Negative imaginary part of the energies in Fig. \[fig:2\]a characterizes radiative decay into the waveguide. Crucially, the spectrum in Figs. \[fig:2\]a condenses near the resonance ${\varepsilon}=\omega_0$, where the group velocity of polaritons decreases.
Next, we proceed to the double-excited states $\Psi=\sum_{n,m}\psi_{nm}\sigma_n^\dag\sigma_m^\dag|0\rangle$. Their spectrum, obtained from the Schrödinger equation $H\Psi=2{\varepsilon}\Psi$, is shown in Figs. \[fig:2\]e–h in different energy scales and demonstrates a distinct clustered structure. Each cluster resembles the single-particle spectrum in Fig. \[fig:2\]a and is formed by a polariton with a certain wave vector $k_j$ interacting with polaritons with larger wave vectors. Therefore, most of the spectrum in Figs. \[fig:2\][e–g]{} could be described by ${\varepsilon}\approx ({\varepsilon}_j+{\varepsilon}_i)/2$, where ${\varepsilon}_j$ and ${\varepsilon}_i$ are the single-particle energies from Fig. \[fig:2\]a. However, the dense part of the cluster, which corresponds to ${\varepsilon}_i \to \omega_0$ (see red arrows in Fig. \[fig:2\]f), is drastically transformed by the interaction. Three characteristic states from the cluster with $j=7$ are presented in Figs. \[fig:2\]i–k. While the state in Fig. \[fig:2\]i is just a symmetrized product of two standing waves, weakly modified by interaction, the role of the interaction dramatically increases for ${\mathop\mathrm{Re}\nolimits}{\varepsilon}-\omega_0>-0.66\Gamma_0$ in Fig. \[fig:2\]h. The spectrum is split by interaction into relatively delocalized states with smaller radiative decay rate (yellow ellipse in Fig. \[fig:2\]h and Fig. \[fig:2\]k) and the states with larger radiative losses, where one of the two polaritons is localized at the edge of the structure (blue ellipse in Fig. \[fig:2\]h and Fig. \[fig:2\]j).
This interaction-induced transformation of the two-polariton spectrum is our central result. The delocalized states are almost $(j-1)$-fold degenerate, where $j$ is the cluster number and correspond to the Landau levels in the effective magnetic field. The states in Fig. \[fig:2\]j come in degenerate pairs corresponding to topological edge states localized at the opposite sides of the array.
Landau levels, topological edge states, and Hofstadter butterfly {#landau-levels-topological-edge-states-and-hofstadter-butterfly .unnumbered}
=================================================================
We now present an analytical model explaining the topological origin behind the interaction-induced edge states in Fig. \[fig:2\]j. In the basis $|x\rangle=\frac1{\sqrt{N}}\sum_{n=1}^N\exp({{\rm i}}\omega_0d|x-n|/c)\sigma_n^\dag|0\rangle$, $x=1,2, \ldots N$ [@Zhong2020; @Poddubny2019quasiflat], the following ansatz can be used for the two-polariton state $$\label{eq:psi}
\psi_{xy} = \psi^{(j)}_{y} \chi_{x}+\psi^{(j)}_{x}\chi_{y}\:,\quad x,y=1\ldots N$$ where $\psi^{(j)}_{x}$ and $\chi_{x}$ are the wave functions of the first and second polaritons. The former is assumed known and corresponds to the standing wave $\psi^{(j)}_{x} = \cos k_j(x-\tfrac{1}{2})$. To determine the latter, we derive the Schrödinger equation, that accounts for interaction between the polaritons and reads (see Supplementary Materials for more details) $$\begin{aligned}
\label{eq:main}
&\chi_{x+1}+\chi_{x-1}-2\chi_x\\ \nonumber
&+\left\{\tfrac{\omega_0+\omega_j-2{\varepsilon}}{2\varphi \Gamma_0}+\tfrac{4}{Nk_j^2}\cos^2[ k_j (x-\tfrac12)]\right\}^{-1}
\chi_x=0\:,\end{aligned}$$ where $\omega_j\approx \omega_0 -2\varphi \Gamma_0/k_j^2$ is the real part of the eigenfrequency of the single-polaritonic state $\psi^{(j)}$ and $\varphi = \omega_0d/c$. Equation describes a motion of a particle on a lattice in an external potential of a standing wave with the period $N/j$. It has a striking similarity to the Harper equation for an electron moving in a square lattice subjected to the perpendicular magnetic field [@Harper1955]: $$\label{eq:Harper}
\chi_{x+1}+\chi_{x-1}+2\cos (2\pi x\alpha-k_y)\chi_x={\varepsilon}\chi_x\:.$$ Here, $\alpha$ is the magnetic flux through the unit cell and $k_y$ is the wave vector in the perpendicular direction. For small magnetic fields $\alpha \ll 1$, the energy spectrum of Eq. is a ladder of degenerate Landau levels for electrons moving along quantized cyclotron orbits. In the finite structure, the edge states of topological nature arise in the gaps between the Landau levels. Such states correspond to electrons moving along skipping orbits at the structure edge, and are the origin for the quantum Hall effect [@TKNN1982].
In our system, the ratio $j/N$ of the cluster index to the total number of atoms in Eq. plays the same role as the magnetic field flux in Eq. . The spectrum also consists of degenerate Landau levels and the topological edge states in the gaps between them, see Fig. \[fig:3\]a. The result of exact numerical diagonalization of the two-polariton Hamiltonian Eq. \[bold symbols in Fig. \[fig:3\]a\] agrees quantitatively with the solution of Eq. \[open symbols in Fig. \[fig:3\]a\]. Thus, the states Eq. acquire a peculiar internal structure, with nontrivial topology induced by interaction for each of the two indistinguishable polaritons.
The energy spectrum of the Harper Eq. becomes very rich when the magnetic flux $\alpha$ increases. The Landau levels split and transform into a celebrated Hofstadter butterfly [@Hofstadter1976], shown also in Supplementary Fig. S2. The butterfly has a self-similar structure with $q$ allowed energy bands at the rational fluxes $\alpha=p/q$ [@Azbel1964] and a Cantor-set spectrum for irrational fluxes. Even though in our case the effective magnetic flux $j/N$ is rational, we can still extract an analogue of the Hofstadter butterfly from the two-polariton spectrum in Figs. \[fig:2\][e–g]{}. We separate the clusters in Figs. \[fig:2\][e–h]{} formed by different standing waves (i.e. different effective magnetic fields) and align them horizontally, the details are presented in Methods and Supplementary Fig. S3. The resulting butterfly is shown in Fig. \[fig:3\]b and it qualitatively resembles the Hofstadter butterfly \[Fig. S2\].
In accordance with Fig. \[fig:3\]a and Fig. \[fig:2\], for small magnetic fluxes $j/N$ the butterfly in Fig. \[fig:3\]b features distinct Landau levels with edge states in the gaps between them. These edge states correspond to red points in Fig. \[fig:3\]b. At high magnetic fields the Landau levels split, but the spectrum still retains a surprisingly delicate structure.
{width="\textwidth"}
Polariton-polariton entanglement {#polariton-polariton-entanglement .unnumbered}
================================
The internal structure of the two-polariton states is represented by their entanglement entropy [@Eisert2010], $S=-\sum\lambda_\nu \ln \lambda_\nu$, obtained from the Schmidt expansion $\psi_{nm}=\sum_{\nu=1}^N \sqrt{\lambda_\nu} \psi^\nu_n \psi^\nu_m\:,$ $(\sum \lambda_\nu=1)$. The result, presented in Fig. \[fig:4\]b, demonstrates a rich variety of eigenstates with different localization degrees, indicated by the color. Characteristic examples of wave functions are shown in Fig. \[fig:4\]a. The entropy of entanglement tells us the number of distinct single-particle states in a given two-body state, so it is low for the scattering states, where two polaritons are quasi-independent. The topological states Eq. also have an intrinsically low entropy, being just a product of a standing wave and a localized or an edge state. However, the states Eq. can mix with each other resulting in larger entanglement entropy. This entangled mixing becomes especially prominent for the Landau level states, cf. points LL, ClS and ES in Fig. \[fig:4\]b. When the real part of energy approaches $\omega_0$ from the negative side, the mixing between different standing waves increases since the spectrum gets denser, and the states become chaotic-like, see also the top right corner of Fig. \[fig:1\]d. At very small negative energies ${\varepsilon}-\omega_0\sim-\varphi \Gamma_0$ the single-particle dispersion changes from ${\varepsilon}\propto -1/k^2$ to ${\varepsilon}\propto -(k-\pi/d)^2$ because both polaritons get closer to the Brillouin zone edge, and the fermionic correlations [@Molmer2019] emerge from chaos. The dense cluster of two-polariton states in the right panel of Fig. \[fig:4\]b, where ${\mathop\mathrm{Re}\nolimits}\varepsilon>0$, is formed by the interaction with the quasi-superradiant mode with ${\mathop\mathrm{Re}\nolimits}{\varepsilon}-\omega_0\approx 71\Gamma_0$ in Fig. \[fig:2\]a. It is one of the states of this cluster that has the entanglement entropy even higher than that of the chaotic-like states, see the point MES at ${\mathop\mathrm{Re}\nolimits}{\varepsilon}-\omega_0\approx 35 \Gamma_0$.
In summary, we have discovered a novel interaction-induced internal topological order for the two-polariton states in a light-coupled atomic array. We have revealed that the underlying Dicke-type model demonstrates an incredible diversity of quantum states with different topologies, lifetimes, and entanglement in a strikingly simple system. While its importance for quantum optics [@Roy2017; @KimbleRMP2018] is already well understood, it could rightfully take its place also in the many-body physics, along with such celebrated examples as a Heisenberg model, Bose-Hubbard model, or a Luttinger liquid. The waveguide-mediated long-ranged couplings are quite uncharacteristic for traditional quantum systems and there is much more to expect. For example, we have focused here only on the regime of extremely subwavelength distances between the atoms, where two-polariton bound states [@Zhang2019arXiv; @Poddubny2019quasiflat] play no role. Polariton-polariton interactions could be even more interesting in the Bragg-spaced lattices, where the non-Markovian effects are drastically enhanced [@Ivchenko1994; @Hubner1999; @Goldberg2009]. The ultra-strong coupling regime [@FriskKockum2019] is also unexamined for the quantum waveguides to the best of our knowledge. On the more practical side, it is promising to explore recently proposed high-quality states [@Koshelev2020] in the many-body domain to increase the quantum coherence. The waveguide-based setups could be used to route and manipulate signals [@Leung2012], propagating on future quantum chips [@Arute2019] and our results open new possibilities to engineer the quantum entanglement.
[**Author contributions.**]{} ANP and AVP conceived the idea and developed an analytical model. JZ, ANP, NAO and YKe performed the numerical calculations. ANP, CL, and YSK supervised the project. All authors contributed to discussion of the results and writing the manuscript.
Methods {#methods .unnumbered}
=======
Calculation of the energy spectrum of the Hamiltonian Eq. , shown in Fig. \[fig:2\], is relatively straightforward. The spectrum is found by standard linear algebra techniques, see also Supplementary Materials. However, it is more challenging to extract the butterfly spectrum in Fig. \[fig:3\]b from the spectrum in Fig. \[fig:2\]e. This task requires careful separation of the clusters corresponding to different single-polariton states. We start by performing the Schmidt decomposition of the two-polariton state $$\label{eq:psi1b}
\psi_{xy}=\sum\limits_{\nu=1}^N \sqrt{\lambda_\nu} \psi^\nu_x \psi^\nu_y\:$$ for all the states that have ${\mathop\mathrm{Re}\nolimits}\varepsilon<\omega_0$. Our analysis of the Schmidt decomposition confirms, that for most of the states it is governed by two largest singular values $\lambda_1$ and $\lambda_2$, that have close absolute values. Keeping only these two terms, we obtain new linear combinations of the wave functions $ \psi^1_x$ and $ \psi^2_y$ as $u^{\pm}_n=\lambda_1^{-1/4}\psi^1_n\pm \lambda_2^{-1/4}\psi^{2}_n$. After that, the two-polariton state can be approximately presented as $\psi_{xy}\propto u^{+}_{x} u^{-}_{y}+u^{-}_{x} u^{+}_{y}\:.$ Next, we select one of the two states $u^{+}_{x}$, $u^{-}_{y}$ that has lower inverse participation ratio, $\sum |u_x|^4/[\sum |u_x|^2]^2$, which means it is less localized in space. We designate this state as $u^{(\rm free)}$ and the more localized one as $u^{(\rm loc)}$, perform the discrete Fourier transform $$u^{(\rm free)}(k)=\sum\limits_{x=1}^N{{\rm e}}^{-{{\rm i}}kx}u^{(\rm free)}_x$$ and calculate the wave vector $k_{\rm max}$, corresponding to the maximum of the Fourier decomposition. The number of the cluster can be then determined from the quantization rule $$\label{eq:cluster1}
j\approx \left[\frac{k_{\rm max}N}{\pi}\right],$$ where square brackets indicate the rounding to the nearest integer. In order to improve the precision in Eq. for large $j$, we also characterize the vectors $\chi$ by their mirror symmetry. Then we apply Eq. separately for odd and even states with odd and even $j$, respectively. The results of Fourier transform for $N=200$ atoms are shown in Figs. S3, S4 of the Supplementary Materials. Except for very large $j$ close to $N$, the spectrum is clearly separated into well-defined steps of alternating parity. Each step is assigned to a different cluster of eigenvalues. Next, we align the clusters with respect to each other. This is done by subtracting the energy with the largest (smallest negative) real part from the energies of the states of each cluster, ${\varepsilon}_{j,\rm max}$. In order to keep the points with the highest energy on the semilogarithmic plot after this subtraction, we also add a small value of $1.1\times 10^{-4}\Gamma_0$ to all the energies. The result is the butterfly spectrum, shown in Fig. \[fig:3\]b.
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**Supplementary Materials**
Analytical model for polariton-polariton interactions
=====================================================
In this section, we start from the Hamiltonian (1) in the main text [@Ke2019; @Albrecht2019] $$\begin{aligned}
\label{eq:HM}
&H=\sum_n\omega_0 \sigma_n^\dag\sigma_n^{\vphantom{\dag}}+\sum\limits_{n,m}
\sigma_n^\dag\sigma_m^{\vphantom{\dag}}\mathcal H_{nm}\:,\\
&\mathcal H_{nm}= -{{\rm i}}\Gamma_0 {{\rm e}}^{{{\rm i}}\varphi|n-m|},\quad \varphi=\frac{\omega_0d}{c}\:,
\end{aligned}$$ and proceed to derive Eq. (3) that describes an interaction of two polaritons. Substituting the ansatz $|\Psi\rangle=\sum\psi_{mn} \sigma_n^\dag \sigma_m^\dag |0\rangle$ into the Schrödinger equation $ H \Psi =2{\varepsilon}\Psi$ we obtain the two-polariton Schrödinger equation in the form [@Ke2019; @Zhong2020] $$\begin{gathered}
\label{eq:S1}
\mathcal H_{mn'}\psi_{n'n}+\psi_{mn'}\mathcal H_{n'n}-2\delta_{mn}\mathcal H_{nn'}\psi_{n'n}\\=2({\varepsilon}-\omega_0) \psi_{mn}\:,\end{gathered}$$ or, in a matrix form, $$\label{eq:S1b}
\mathcal H\psi+\psi \mathcal H-2\:{\rm diag\:}[{\rm diag\:} \mathcal H\psi]=2({\varepsilon}-\omega_0) \psi\:.$$ Here, the “diag” operator transforms a given matrix to the column-vector filled by the diagonal entries of this matrix, and vice versa.
This system is readily solved numerically after the wavefunction $\psi$ is rewritten in the basis of $N(N-1)/2$ localized states of the type $$[\widetilde\psi]_{mn}=[\widetilde\psi]_{nm}=\frac{1}{\sqrt{2}},\quad n\ne m\:.$$
Our next goal is to go beyond Refs. [@Ke2019; @Zhong2020; @Poddubny2019quasiflat] and obtain Eq. (3). To this end, we notice that [@Zhong2020; @Poddubny2019quasiflat] $$K\equiv \mathcal H^{-1}\approx \frac{1}{2\varphi \Gamma_0}\partial^2,\text{ where } \partial^{2}\equiv \left(\begin{smallmatrix}
-1&1&0&\ldots\\
1&-2&1&\ldots\\
&&\ddots&\\
\ldots&1&-2&1\\
\ldots &0& 1&-1
\end{smallmatrix}\right)\:.\label{eq:ih}$$ Here, the matrix $\partial^{2}$ represents the one-dimensional discrete Laplacian (or the operator of discrete second-order derivative) This means that for a vector $\psi_n$ with a smooth dependence on $n$ one has $$[\partial^{2}\psi]_n=\psi_{n+1}+\psi_{n-1}-2\psi_n\approx\frac{{{\rm d}}^2 \psi_n}{{{\rm d}}n^2}\:.$$ Thus, for a short-period array with $\varphi\ll 1$ the operator $K$ reduces to the second derivative operator. The inverted Hamiltonian $K$ Eq. is a sparse matrix with only nearest-neighbor couplings. This fact inspires us to perform the transformation $$\label{eq:trans}
\psi=K \psi'K$$ that means change of the basis to $$\label{eq:basis}
|x\rangle=\frac1{\sqrt{N}}\sum_{n=1}^N{{\rm e}}^{{{\rm i}}\omega_0d|x-n|/c}\sigma_n^\dag|0\rangle\:,
\text{where } x=1,2, \ldots N\:.$$ This basis inherits the distribution of electric field emitted by a given atom. Indeed, $\exp({{\rm i}}\omega_0d|x-n|/c)$ is just the photon Green function in one dimension. Since the wave equations for electric field are local, the transformed two-polariton Schrödinger equation will be local as well, i.e. it will involve only sparse matrices. Substituting Eq. into Eq. we find [@Zhong2020; @Poddubny2019quasiflat] $$K\psi'+\psi'K-2\:{\rm diag\:}[{\rm diag\:} \psi' K]=2({\varepsilon}-\omega_0) K\psi'K\:.\label{eq:Sh3}$$ Next, we look for the solution of the transformed equation in the form $$\label{eq:ansatz}
\psi'_{xy} = \psi^{(j)}_{y} \chi_{x}+\psi^{(j)}_{x}\chi_{y}\:,\quad x,y=1\ldots N\:,$$ corresponding to Eq. (2) in the main text. Here, one of the two excitations is a single-particle eigenstate of the matrix $\mathcal H$ with the eigenfrequency $\omega_j$. Using the definition $K\equiv \mathcal H^{-1}$,e we find $$\label{eq:u}
K\psi^{(j)}=\frac{1}{\omega_j} \psi^{(j)}\:.$$ The state is normalized as $ \sum_x [\psi_x^{(j)}]^2=1$. The normalization does not involve complex conjugation, because the original matrix $\mathcal H$ is not Hermitian but symmetric. As such, its eigenvectors $\psi^{(j)}$ satisfy the non-conjugated orthogonality condition $$\langle j|j'\rangle\equiv \sum\limits_{x=1}^N \psi_x^{(j)}\psi_x^{(j')}=\delta_{jj'}\:.$$ Due to the translational symmetry the vector $ \psi^{(j)}$ is just a standing wave [@Ivchenko1991]: $$\label{eq:ansatz1}
\psi^{(j)}_x\approx \sqrt{\frac{2}{N}}\cos \frac{\pi j(x-1/2)}{N}\:.$$ We note, that the ansatz and where the eigenstate $\psi^{(j)}$ does not take into account the interaction effects, works only for the transformed Schrödinger equation . This ansatz does not adequately describe the solutions of the original equation because the wavefunction $\psi'$ does not turn to zero for $n=m$.
{width="80.00000%"}
Substituting Eq. into Eq. , we obtain $$\begin{gathered}
\frac{\psi_n^{(j)}}{\omega_j} \chi_m+(K\chi)_n \psi^{(j)}_m+
\frac{\psi_m^{(j)}}{\omega_j} \chi_n+(K\chi)_m \psi^{(j)}_n\\-
2\delta_{mn}\left[\frac{\psi^{(j)}_n}{\omega_j} \chi_n+(K\chi)_n \psi^{(j)}_n\right]\\=\frac{2({\varepsilon}-\omega_0)}{\omega_j} \left[ \psi^{(j)}_n (K\chi)_m+\psi^{(j)}_m (K\chi)_n\right]\:.\end{gathered}$$ Next, we multiply this equation by $\psi^{(j)}_m$ and sum over $m$: $$\begin{gathered}
\label{eq:sum1}
\frac{1}{\omega_j}\chi_n+(K\chi)_n+ \psi^{(j)}_n \psi^{(j)}_m (K\chi)_m+\frac{\psi^{(j)}_n}{\omega_j}\chi_m \psi^{(j)}_m
\\-\frac{2}{\omega_j} \psi^{(j),2}_n \chi_n-2\psi^{(j),2}_n (K\chi)_n\\=\frac{2({\varepsilon}-\omega_0)}{\omega_j} \left[ \psi^{(j)}_n \psi^{(j)}_m(K\chi)_m+ (K\chi)_n\right]\:.\end{gathered}$$ For the sake of brevity, the summation over the dummy index $m$ is assumed but not indicated explicitly. Using the fact that $\psi^{(j)}$ is the eigenstate of $K$, we simplify Eq. to $$\begin{gathered}
\label{eq:sum2}
\left[\frac{\chi_n}{\omega_j}+K_{nm}\chi_m+\frac{2}{\omega_j} \psi^{(j)}_n \psi^{(j)}_m \chi_m\right]\\-2 \psi^{(j),2}_n \left[\frac1{\omega_j} \chi_n+K_{nm}\chi_m\right]\\=\frac{2({\varepsilon}-\omega_0)}{\omega_j}\left(K_{nm}\chi_m+\frac{1}{\omega_j} \psi^{(j)}_n \psi^{(j)}_m \chi_m\right)\:.\end{gathered}$$ We are going to consider strongly localized eigenstates that are orthogonal to the standing wave $\psi^{(j)}$. Thus, the terms $\propto \psi^{(j)}_m \chi_m$ in Eq. can be omitted to find our main result: $$\begin{gathered}
\label{eq:main2}
\left[\frac{1}{\omega_j}+K\right]\chi-\frac{2}{\omega_j}{\mathop{\mathrm{diag}}\nolimits}[\psi^{(j),2}]\chi-{\mathop{\mathrm{diag}}\nolimits}[(\psi^{(j),2})K]\chi\\=\frac{2({\varepsilon}-\omega_0)}{\omega_j} K \chi\:.\end{gathered}$$ Moreover, for relatively small $j$ the function $\chi$ changes with $n$ much faster than $\psi^{(j)}$. Hence, the term, proportional to ${\mathop{\mathrm{diag}}\nolimits}[(\psi^{(j),2})K]\chi$, is larger than the terms ${\mathop{\mathrm{diag}}\nolimits}[\psi^{(j),2}]\chi$. Neglecting the terms $\propto \psi^{(j),2}_n \chi_n$, we obtain $$\label{eq:main2}
\left[\frac{1}{\omega_j}+K\right]\chi-2{\mathop{\mathrm{diag}}\nolimits}[(\psi^{(j),2})K]\chi=\frac{2({\varepsilon}-\omega_0)}{\omega_j} K \chi\:.$$ Taking Eq. into account, we get Eq. (3) from the main text, i.e. $$\begin{aligned}
\label{eq:mainb}
&\chi_{x+1}+\chi_{x-1}-2\chi_x\\ \nonumber
&+\left\{\tfrac{\omega_0+\omega_j-2{\varepsilon}}{2\varphi \Gamma_0}+\tfrac{4}{Nk_j^2}\cos^2[ k_j (x-\tfrac12)]\right\}^{-1}
\chi_x=0\:.\end{aligned}$$ The butterfly spectrum, calculated from Eq. , is shown in Fig. \[fig:anbut\]. In the calculation we have neglected the imaginary part of $\omega_j$, replaced the exact operator $K=\mathcal H^{-1}$ by its approximation Eq. and used the analytical expression Eq. for the wavefunction $\psi^{(j)}$. In order to better resolve the small band gaps at large $j$ we have colored the points by the density of states, with red meaning more dense spectrum. Namely, the color scale corresponds to the logarithm of the fourth derivative of the spectrum $\varepsilon_\nu$, $\nu=1\ldots N$, obtained separately for each value of $j$. For low $j$, the result in Fig. \[fig:anbut\] is generally similar to the numerically obtained butterfly, shown in Fig. 3b of the main text. To our surprise, the exact numerical spectrum in Fig. 3b of the main text is actually richer than the semi-analytical one in Fig. \[fig:anbut\].
![Hofstadter butterfly obtained from solution of Eq. . Calculation has been performed for an array with $N=200$ sites, $k_y=0$, open boundary conditions, and the flux $\alpha$ changing from $0$ to $1$ with the step $1/(4N)$. Color of the points corresponds to their inverse participation ratio (IPR), higher IPR corresponds to topological edge states.[]{data-label="fig:OrigBut"}](FigS2){width="45.00000%"}
For comparison, we also present in Fig. \[fig:OrigBut\] the original Hofstadter butterfly [@Hofstadter1976]. It is obtained by studying the dependence of the spectrum of the Harper equation [@Harper1955] (Eq. (4) in the main text) on the magnetic flux $\alpha$ $$\label{eq:Harper1}
\chi_{x+1}+\chi_{x-1}+2\cos (2\pi x\alpha-k_y)\psi_x={\varepsilon}\chi_x\:.$$ Three butterfly spectra, calculated from full two-polariton Hamiltonian Eq. , semi-analytical model Eq. and the Harper equation Eq. , are shown in Fig. \[fig:all\] (corresponding to Fig. 3b of the main text), Fig. \[fig:anbut\] and Fig. \[fig:OrigBut\], respectively. These spectra have both similarities and differences. At low magnetic fields they all demonstrate distinct degenerate Landau levels separated by the band gaps with topological edge states. However, their behavior diverges at higher magnetic fields. The difference between the semi-analytical butterfly in Fig. \[fig:anbut\] and the original Hofstadter butterfly in Fig. \[fig:OrigBut\] could be attributed to the peculiar nature of Eq. . Namely, in contrast to the conventional eigenvalue problem Eq. where the energy ${\varepsilon}$ enters the term in the right-hand side, Eq. contains the energy ${\varepsilon}$ in the denominator of the last term, i.e. it is a generalized eigenvalue problem. This is related to the long-ranged photon-mediated couplings between the atoms. Namely, the price for the transformation from the original Hamiltonian Eq. to the model Eq. with nearest-neighbor couplings was the appearance of the $K$ operators in the right hand side of Eq. , i.e. the Eq. became a generalized eigenvalue problem. It is not fully clear at the moment, which of the conclusions derived from the original Harper model Eq. should be valid for the generalized Aubry-André-Harper model Eq. .
As mentioned above, the spectrum of the full Hamiltonian Eq. is even richer than that of the generalized Aubry-André-Harper model Eq. . The reason for this might be the mixing between the standing waves with different orders $j$ that is not accounted in the ansatz Eq. . This mixing becomes prominent for high values of $j$, as is also indicated by the calculation of the entanglement entropy in Fig. 4 of the main text. It is important, that the butterfly has a fine structure, band gaps and edge states even for high values of effective magnetic flux $j/N\sim 0.3$, see in particular the top state in the left column of Fig. \[fig:all\].
Fourier analysis of the eigenstates
===================================
We present in Fig. \[fig:fourier\] the result of the Fourier analysis of the eigenstates, described in the Methods section of the main text. Figure \[fig:all\] shows the examples of singular value decomposition for several characteristic two-polariton states.
![Fourier analysis of the eigenstates for $N=200$ atoms, $\varphi=0.02$. Red and blue colors correspond to even and odd eigenstates, respectively. Inset shows the spectrum in a larger scale for small $k_{\rm max}$. Energy is measured in the units of $\Gamma_0$.[]{data-label="fig:fourier"}](FigS3){width="45.00000%"}
{width="80.00000%"}
|
---
abstract: 'We demonstrate a multipartite protocol to securely distribute and reconstruct a quantum state. A secret quantum state is encoded into a tripartite entangled state and distributed to three players. By collaborating, any two of the three players can reconstruct the state, whilst individual players obtain nothing. We characterize this $(2,3)$ threshold quantum state sharing scheme in terms of fidelity, signal transfer and reconstruction noise. We demonstrate a fidelity averaged over all reconstruction permutations of $0.73 \pm 0.04$, a level achievable only using quantum resources.'
author:
- 'Andrew M. Lance'
- Thomas Symul
- 'Warwick P. Bowen'
- 'Barry C. Sanders'
- Ping Koy Lam
title: Tripartite Quantum State Sharing
---
Secret sharing[@Sha79] is a powerful technique in computer science, which enables secure and robust communication in information networks, such as the internet, telecommunication systems and distributed computers. The security of these networks can be enhanced using quantum resources to protect the information. Such schemes have been termed [*quantum secret sharing*]{}[@Hil99]. Many applications in quantum information science, however, require the distribution of quantum states. One such example are quantum information networks, which are expected to consist of nodes where quantum states are created, processed and stored, connected by quantum channels[@Cir97]. It is of paramount importance that the quantum channels in these networks allow the robust and secure distribution of quantum states between nodes. Cleve [*et al.*]{}[@Cle99] proposed the secret sharing of quantum states as a protocol that provides these capabilities, overcoming failures or conspiracies by nodes. We term this [*quantum state sharing*]{} to differentiate from the quantum secret sharing of classical information. In $(k,n)$ threshold quantum state sharing[@Cle99], the “dealer” node encodes a secret state into an $n$-party entangled state and distributes it to $n$ “player” nodes. Any $k$ players (the access structure) can collaborate to retrieve the quantum state, whereas the remaining $n-k$ players (the adversary structure), even when conspiring, acquire nothing. This scheme provides quantum information networks with a secure framework for distributed quantum computation and quantum communication.
The original quantum state sharing scheme by Cleve [*et al.*]{} was formulated for discrete states and requires the control and coupling of qudits ($d$-dimensional extensions of qubits), which is extremely experimentally challenging. In the continuous variable regime, however, quantum state sharing is feasible utilizing Einstein-Podolsky-Rosen (EPR) entanglement[@Tyc02], an experimentally accessible quantum resource[@Ou92; @Bow03]. We demonstrate $(2,3)$ threshold quantum state sharing in this regime. In our scheme, a secret coherent state is encoded into a tripartite entangled state and distributed to three players. We demonstrate that any two of the three players can form an access structure to reconstruct the state. The state reconstruction is characterized in terms of fidelity, signal transfer, and reconstruction noise. These measures show a direct verification of our tripartite continuous variable entanglement. As coherent states form an over-complete basis for all quantum states, arbitrary states can be shared by this scheme.
![ Schematic of the $(2,3)$ quantum state sharing scheme. $\psi_{\rm in}$: secret quantum state, OPA: optical parametric amplifier, $G$: electronic gain, AM: amplitude modulator, LO: optical local oscillator. ${\rm
x\!\!:\!\!y}$: beam splitter with reflectivity ${\rm x}/\!({\rm x}\!+\!{\rm y})$ and transmitivity ${\rm y}/\!({\rm x}\!+\!{\rm y})$.[]{data-label="fig:QSSexperiment"}](Figure1_Lance_PRL.EPS){width="9cm"}
The quantum states of interest in this paper reside at the frequency sidebands of an electromagnetic field. In the Heisenberg picture of quantum mechanics, a quantum state can be represented by the field annihilation operator $\hat{a}\!=\!(\hat X^{+}\!+\!i\hat
X^{-})/2$, where $\hat X^{\pm}\!=\!\langle\hat
X^{\pm}\rangle\!+\!\delta\hat X^{\pm}$ are the amplitude (+) and phase (-) quadratures, with variances of $V^{\pm}\!=\!\langle(\hat X^{\pm})^{2}\rangle$. In our dealer protocol, the dealer interferes the secret state $\hat{a}_{{\rm in}}$ with one of a pair of EPR entangled beams $\hat{a}_{\rm EPR1}$ on a 1:1 beam splitter (Fig. \[fig:QSSexperiment\]). The two output fields and the second entangled beam $\hat{a}_{\rm EPR2}$ form the three shares which are distributed to the players. The entangled state ensures that the secret is protected from each player individually and is generated by interfering a pair of amplitude squeezed beams $\hat{a}_{\rm sqz1}$ and $\hat{a}_{\rm sqz2}$[@Bow03]. The dealer can further enhance the security of the scheme by displacing the coherent amplitudes of the shares with correlated Gaussian white noise[@Lan03]. Choosing the Gaussian noise to have the same correlations as the quadrature entanglement, the shares can be expressed as $$\begin{aligned}
\hat a_1 &=& (\hat a_{{\rm in}}\!+\!\hat a_{\rm EPR1}\!+\!\delta\mathcal{N}
)/\sqrt2 \\
\hat a_2 &=& (\hat a_{{\rm in}}\!-\!\hat a_{\rm EPR1}\!-\!\delta\mathcal{N}
)/\sqrt2 \\
\hat a_3 &=& \hat a_{\rm EPR2}\!+\!\delta\mathcal{N}^{\ast}\end{aligned}$$ where $\delta\mathcal{N}\!=\!(\delta{N}^{+}\!+\!i\delta{N}^{-})/2$ represents the Gaussian noise with mean $\langle\delta{N}^{\pm}\rangle\!=\!0$ and variance $\langle(\delta{N}^{\pm})^{2}\rangle\!=\!V_{N}$, and $^\ast$ denotes the complex conjugate.
The reconstruction protocol used for the $(2,3)$ quantum state sharing scheme is dependent on the corresponding access structure (Fig. \[fig:QSSexperiment\]). The access structure formed when players 1 and 2 collaborate, henceforth denoted {1,2}, reconstructs the secret quantum state by completing a Mach-Zehnder interferometer using a 1:1 beam splitter[@Tyc02]. The access structures {2,3} and {1,3} reconstruct the quantum state by utilizing a 2:1 beam splitter and an electro-optic feedforward loop[@Lan03]. In the latter protocol, combining the shares on the beam splitter with appropriate relative phase reconstructs the phase quadrature of the secret state on one of the beam splitter outputs. In contrast, the amplitude quadrature has additional noise as a result of this process. This noise, however, is correlated with the amplitude quadrature of the other beam splitter output, which is detected. The resulting photocurrent is fedforward to displace the amplitude quadrature of the first output. Assuming no losses, the quadratures of the reconstructed secret can then be expressed as [@Lan03] $$\begin{aligned}
\label{eqn:23reconstruction}
\delta\hat X^+_{{\rm out}} &=& \!g^{+}\delta\hat{X}^+_{{\rm
in}}\!+\!\frac{\sqrt{3}}{2}(1\!-\!\sqrt{3}g^{+})(\delta\hat{X}^+_{\rm
sqz_1}\!+\!\delta\hat{X}^+_{\rm sqz_2})+\nonumber\\
& & \!\!\frac{1}{2}(g^{+}\!\!-\!\sqrt{3})(\delta\hat X^-_{\rm
sqz_1}\!\!-\!\delta\hat X^-_{\rm
sqz_2})\!+\!(\!\sqrt{3}\!-\!g^{+}\!)\delta N^+\\
\delta\hat X^-_{{\rm out}} &=& \!\frac{1}{\sqrt{3}} (\delta\hat
X^-_{{\rm in}}\!+\!\delta\hat{X}^+_{\rm sqz_1}\!-\!\delta\hat
{X}^+_{\rm sqz_2})\end{aligned}$$ where $g^{\pm}\!=\!\langle\hat X^{\pm}_{{\rm
out}}\rangle /\langle\hat X^{\pm}_{{\rm in}}\rangle$ are the optical quadrature gains. The phase quadrature gain $g^{-}\!=\!1/\sqrt{3}$ is set by the 2:1 beam splitter, whilst the amplitude quadrature gain $g^{+}\!=\!(1/\sqrt{3}\!+\!G/\sqrt{6})$ has an additional term which is a function of the electronic feedforward gain $G$. We refer to the specific gain of $g^{+}g^{-}\!=\!1$ as the *unitary gain point*. At unitary gain and in the limit of perfect squeezing, the quadratures of the reconstructed state are given by $\delta\hat{X}^{\pm}_{{\rm out}}\!=\! (\sqrt{3})^{\pm
1}\delta\hat{X}^{\pm}_{{\rm in}}$. This state is directly related to the secret state via a local unitary parametric operation. Although not in the same form as the secret state, such a reconstructed state is only achievable using entanglement. On the other hand, the unitary parametric operation is local and requires no entanglement. Therefore, the essence of the reconstruction protocol is contained within the feedforward scheme.
![Experimental results for the {1,2} access structure. (a) and (b) show the spectra of the amplitude and phase quadrature variances for the secret (input, blue/dark grey) and reconstructed (output, green/light grey) quantum states. $\Delta{f}$ is the offset from the signal frequency at 6.12 MHz. Resolution Bandwidth = 1 kHz, Video Bandwidth = 30 Hz. (c) Standard deviation contours of Wigner functions of the secret (blue/dark grey) and reconstructed (green/light grey) quantum states. (d) Measured fidelity as a function of gain deviation $r^{2}\!=\!(\langle\!\hat{X}^{+}_{\rm out} \! \rangle
\! - \! \langle \! \hat{X}^{+}_{\rm in} \! \rangle)^{2} \! +
\! (\langle \! \hat{X}^{-}_{\rm out} \! \rangle \! - \! \langle
\! \hat{X}^{-}_{\rm in} \! \rangle)^{2}$. (d) Grey area highlights the accessible fidelity region. Points plotted are from six different experimental runs.[]{data-label="fig:Fig_spectra3"}](Figure2_Lance_PRL.EPS){width="8.5cm"}
![Experimental results for the {2,3} access structure. (a) and (b) show the spectra of the amplitude and phase quadrature variances for the secret (input, blue/dark grey) and reconstructed (output, green/light grey) quantum states. (c) Standard deviation contours of Wigner functions of the secret (blue/dark grey) and reconstructed (green/light grey) quantum states. The dashed circle represents the quantum state $\delta\hat{X}^{\pm}_{\rm para} \!\! =
\!\! (\sqrt{3})^{\mp 1}\delta\hat{X}^{\pm}_{\rm out}$ after the a posteriori local unitary parametric operation.[]{data-label="fig:Fig_spectra2"}](Figure3_Lance_PRL.EPS){width="8.5cm"}
In our experiment we use a Nd:YAG laser producing a coherent field at 1064nm to provide a shared time frame between all parties; to yield the dealer secret state, produced by displacing the sideband vacuum state of the laser field using an amplitude and a phase modulator at 6.12MHz; and to produce two amplitude squeezed states generated in hemilithic MgO:LiNbO$_{3}$ optical parametric amplifiers (OPAs) pumped with 532nm light. The output fields of each OPA are squeezed $4.5\pm{0.2}$dB below the quantum noise limit. These squeezed beams are interfered on a 1:1 beam splitter with an observed visibility of $99.1\pm{0.2}\%$. The beam splitter outputs are EPR entangled and satisfy the wave-function inseparability criterion $\langle(\delta\hat{X}^{+}_{\rm EPR1}\!+\!\delta\hat{X}^{+}_{\rm
EPR2})^{2}\rangle\langle(\delta\hat{X}^{-}_{\rm
EPR1}\!-\!\delta\hat{X}^{-}_{\rm
EPR2})^{2}\rangle/4\!=\!0.44\pm{0.02}\!<\!1$[@Dua00; @Bow03]. To enhance the security of the secret state against the adversaries, the coherent quadrature amplitudes of the entangled beams are displaced with Gaussian noise of variance $V_{N}=3.5\pm{0.1}$dB. Experimentally, this noise can be actively applied using electro-optic modulation techniques, but in our case it appears naturally as a result of de-coherence[^1]. A homodyne detector is used to characterize the secret, adversary and reconstructed quantum states using a configuration of removable mirrors. To ensure accurate results, the total homodyne detection efficiency, $\eta_{\rm
hom}\!=\!0.89\!\pm\!{0.01}$, is inferred out of each measurement.
![ Experimental fidelity for the {2,3} access structure as a function of the product of $g^{+}g^{-}$. Dashed line: calculated theoretical curve with squeezing of $-4.5$ dB, added noise of $+3.5$ dB, electronic noise of $-13$ dB with respect to the quantum noise limit, and feed forward detector efficiency of $0.93$. Solid line and dotted lines: experimental fidelity for the adversary structure and error bar. Grey area highlights the classical boundary for the access structure.[]{data-label="fig:QSS_Fidelity"}](Figure4_Lance_PRL.EPS){width="8.5cm"}
![Experimental signal transfer ($\mathcal{T}$) and additional noise ($\mathcal{V}$) for the {2,3} access structure (green/light grey circles), and the adversary structure (red/dark grey diamonds). Solid line: calculated theoretical curve for varying gain with same parameters as in Figure \[fig:QSS\_Fidelity\]. Triangle symbol: unitary gain point for the {2,3} access structure. Square symbol: calculated theoretical point for the adversary structure. Grey area: the classical region for the {2,3} access structure. (inset) Experimental $\mathcal{T}$ and $\mathcal{V}$ for the {1,2} access structure (green/light grey circles) and the theoretical point (black circles). []{data-label="fig:QSS_TV"}](Figure5_Lance_PRL.EPS){width="8.5cm"}
We characterize the quality of the state reconstruction for the access and adversary structures using fidelity $\mathcal{F}\!=\!\langle \psi_{\rm in} |\rho_{\rm out}|\psi_{\rm
in}\rangle$, which measures the overlap between the secret and reconstructed quantum states[@Shu95]. Assuming that all fields involved have Gaussian statistics and that the secret is a coherent state, the fidelity can be expressed in terms of experimentally measurable parameters as $$\mathcal{F}=2e^{-(k^{+}\!+\!k^{-})/4}\Big/ \sqrt{(1\!+\!V_{{\rm
out}}^{+})(1\!+\!V_{{\rm out}}^{-})}$$ where $k^{\pm}\!=\!\langle X_{{\rm
in}}^{\pm}\rangle^{2}(1\!-\!g^{\pm})^{2}/ (1\!+\!V_{{\rm
out}}^{\pm})$. In our experiment, the fidelity for {1,2} can be determined directly; however, for {2,3} and {1,3} a unitary parametric transformation must be applied before a meaningful fidelity is obtained. This unitary transform can be applied either optically or a posteriori. The final state is then $\delta\hat{X}^{\pm}_{{\rm para}}\!=\! (\sqrt{3})^{\mp
1}\delta\hat{X}^{\pm}_{{\rm out}}$, so in the ideal case $\delta\hat{X}^{\pm}_{{\rm para}}\!=\!\delta\hat{X}^{\pm}_{{\rm
in}}$. Under ideal conditions and at unitary gain, any one of the access structures can achieve $\mathcal{F}\!=\!1$ corresponding to perfect reconstruction of the secret quantum state, whilst the corresponding adversary structure obtains $\mathcal{F}\!=\!0$.
When no entanglement is used, the maximum fidelity achievable by {2,3} and {1,3} is $\mathcal{F}^{\rm clas}_{\{2,3\}}\!=\!\mathcal{F}^{\rm
clas}_{\{1,3\}}\!=\!1/2$, whilst {1,2} can still achieve $\mathcal{F}^{\rm clas}_{\{1,2\}}\!=\!1$, so the average fidelity achieved by all permutations of the access structure cannot exceed $\mathcal{F}^{\rm
clas}_{\rm avg}\!=\!2/3$. This defines the classical boundary for $(2,3)$ threshold quantum state sharing. Note that for general $(k,n)$ threshold quantum state sharing of a coherent state, independent of the dealer protocol, the maximum average fidelity achievable without entanglement resources is $\mathcal{F}^{\rm clas}_{\rm
avg}\!=\!k/n$.
With the $\{1,3\}$ and $\{2,3\}$ protocols being equivalent, our $(2,3)$ threshold quantum state sharing scheme is demonstrated through the implementations of the $\{1,2\}$ and $\{2,3\}$ reconstruction protocols. Figure \[fig:Fig\_spectra3\] shows the noise spectra and corresponding inferred Wigner function standard deviation contours for the secret and reconstructed state for the {1,2} protocol. The fidelity obtained from these noise spectra is $\mathcal{F}_{\{1,2\}}\!=\!0.93\!\pm\!{0.03}$ with $g^{+}\!=\!0.94\!\pm\!{0.01}$ and $g^{-}\!=\!0.97\!\pm\!{0.01}$. The corresponding adversary structure {3} gets a fidelity of $\mathcal{F}_{\{3\}}\!=\!0$ since the share contains no component of the secret state. Figure \[fig:Fig\_spectra3\] (d) shows several measured fidelity points as a function of phase space distance, $r$, between the coherent amplitudes of the secret and reconstructed states. Each fidelity point has a non-zero distance due to mode mismatch, optical losses and imperfect phase locking.
Similarly, Figure \[fig:Fig\_spectra2\] shows an example of the secret and reconstructed state for the {2,3} protocol. In this case, to allow a direct comparison between the secret and reconstructed states, the inferred Wigner function standard deviation contour of the reconstructed state after the a posteriori local unitary parametric operation is also shown. Figure \[fig:QSS\_Fidelity\] shows the measured fidelity for a range gains. Around the unitary gain point, we achieve a fidelity of $\mathcal{F}_{\{2,3\}}\!=\!0.63\!\pm\!{0.02}$ with $g^{+}g^{-}\!=\!1.02\pm{0.03}$. The corresponding adversary structure {1} achieves a fidelity of only $\mathcal{F}_{\{1\}}\!=\!0.03\!\pm\!{0.01}$. The quantum nature of our protocol is demonstrated by the fidelity averaged over all the access structures $\mathcal{F}_{\rm avg}\!=\!0.74\!\pm\!{0.04}$, which exceeds the classical limit $\mathcal{F}^{\rm clas}_{\rm avg}=2/3$.
Fidelity is a single state dependent measure of the efficacy of quantum information protocols. There are alternative measures which provide complementary information about these processes. One obvious technique is to measure the signal transfer to ($\mathcal{T}$), and the additional noise on ($\mathcal{V}$), the reconstructed state[@Ral98]. Such analysis has been used to characterize quantum non-demolition[@Poi94] and quantum teleportation experiments[@Bow03b]. Unlike the fidelity measure, both $\mathcal{T}$ and $\mathcal{V}$ are invariant to unitary transformations of the reconstructed state. Therefore, for the $\mathcal{T}$ and $\mathcal{V}$ analysis, an a posteriori unitary transform is not required. The signal transfer is given by $\mathcal{T}\!=\!T^{+}\!+\!T^{-}$, where $T^{\pm}\!=\!{\rm SNR}^{\pm}_{\rm out}/{\rm SNR}^{\pm}_{\rm in}$ are the quadrature signal transfer coefficients, with ${\rm SNR}^{\pm}$ being the standard signal-to-noise ratios. The additional noise is given by $\mathcal{V}\!=\!V^{+}_{\rm cv}V^{-}_{\rm cv}$, where $V^{\pm}_{\rm cv}\!=\!V^{\pm}_{\rm out}\!-\!|\langle\delta\hat{X}^{+}_{\rm in}
\delta\hat{X}^{+}_{\rm out}\rangle |^{2}/V^{\pm}_{\rm out}$ are the conditional variances. Experimentally, the signal-to-noise ratios that define $\mathcal{T}$ can be obtained from our measured noise spectra (Fig. \[fig:Fig\_spectra3\] and \[fig:Fig\_spectra2\]), whilst $V^{\pm}_{\rm cv}$ can be determined from the output quadrature variance and the optical quadrature gains $V^{\pm}_{\rm cv}\!=\!V^{\pm}_{\rm out}\!-\!(g^{\pm})^{2}$. In the ideal case, any one of the access structures can achieve perfect state reconstruction with $\mathcal{T}\!=\!2$ and $\mathcal{V}\!=\!0$, whilst the corresponding adversary structure obtains no information with $\mathcal{T}\!=\!0$ and $\mathcal{V}\!=\!\infty$.
Figure \[fig:QSS\_TV\] (inset) shows the experimental $\mathcal{T}$ and $\mathcal{V}$ points for the {1,2} protocol. We measure a best state reconstruction of $\mathcal{T}_{\{1,2\}}\!=\!1.77\pm{0.05}$ and $\mathcal{V}_{\{1,2\}}\!=\!0.01\pm{0.01}$. Both of these values are close to optimal, being degraded only by optical losses and experimental inefficiencies. Similarly, Figure \[fig:QSS\_TV\] shows the points for the {2,3} protocol for a range of gains, together with the corresponding adversary structure {1}. The majority of the experimental points are in agreement with the theoretical prediction, with the discrepancies accountable for by drifts in our control system. The accessible region for the {2,3} protocol without entanglement is illustrated by the shaded region. The quantum nature of the state reconstruction is demonstrated by the experimental points which exceed this classical region. For the {2,3} protocol we measure a lowest reconstruction noise of $\mathcal{V}_{\{2,3\}}\!=\!0.46\pm{0.08}$ and a largest signal transfer of $\mathcal{T}_{\{2,3\}}\!=\!1.03\pm{0.05}$. Points with $\mathcal{T}\!>\!1$ exceed the information cloning limit[@Bow03b] and demonstrate that the {2,3} protocol has better access to information encoded on the secret state than any other parties. The adversary structure obtains a mean $\mathcal{T}_{\{1\}}\!=\!0.41\pm{0.01}$ and $\mathcal{V}_{\{1\}}\!=\!3.70\pm{0.08}$. The separation of the adversary structure $\mathcal{T}$ and $\mathcal{V}$ points from that of the {2,3} protocol in Figure \[fig:QSS\_TV\] illustrates that in such a protocol the access structure performs far better than any adversary structure.
Our experimental demonstration of $(2,3)$ threshold quantum state sharing is the first application of continuous variable tripartite entanglement. Furthermore, it is extendable to an arbitrary $(k,n)$ scheme, without a corresponding scale-up of the required quantum resources[@Tyc03]. The implementation of quantum state sharing broadens the scope of quantum information networks allowing quantum communication between multiple nodes, whilst providing security against malicious parties in the network as well as node and channel failures.
We wish to thank Timothy Ralph, Tomáš Tyc, Roman Schnabel and Hans Bachor for useful discussions and the support of the Australian Research Council and iCORE.
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[^1]: This aspect of de-coherence will be described in detail in a later publication.
|
**Numerical Analysis of a Contact Problem with Wear**
Danfu Han[^1], Weimin Han[^2], Michal Jureczka[^3]and Anna Ochal[^4]
> [**Abstract**]{}. This paper represents a sequel to [@JO18] where numerical solution of a quasistatic contact problem is considered for an elastic body in frictional contact with a moving foundation. The model takes into account wear of the contact surface of the body caused by the friction. Some preliminary error analysis for a fully discrete approximation of the contact problem was provided in [@JO18]. In this paper, we consider a more general fully discrete numerical scheme for the contact problem, derive optimal order error bounds and present computer simulation results showing that the numerical convergence orders match the theoretical predictions.
>
> [**Keywords**]{}. Quasistatic contact problem, variational inequality, numerical methods, optimal order error estimate.
>
> [**AMS Classification.**]{} 65N30
Introduction
============
0
Contact phenomenon is common in engineering applications. Mathematical studies and numerical analysis of contact problems are most suitably carried out within the frameworks of variational inequalities or hemivariational inequalities, which have attracted the attention of many researchers. The related mathematical literature grows rapidly. Some representative comprehensive references in this area are [@DL1976; @KO1988; @HHNL1988; @HS2002; @SST2004; @EJK2005; @SHS2006; @HR2013] in the context of variational inequalities and [@P1993; @NP1995; @HMP1999; @MOS2013; @SM2018] in the context of hemivariational inequalities.
For a contact problem, the mathematical model is constructed based on considerations of various aspects of the contact process. Factors to be taken into account include the type of the contact process (static, quasi-static, or dynamic), constitutive relations of the deformable bodies, contact conditions of various application-specific forms. In certain applications, it is important to consider heating or thermo effects ([@OJ18]), or piezoelectricity effects ([@Sz17]). Since the contact process inevitably causes material wear or even damage, it is not surprising that the wear effect has been built into mathematical models for a variety of contact processes, cf. [@CHS00; @HSS01; @RTS01; @KS02; @Bar06; @VRS13; @GOS16]. In a recent paper [@SPS17], a mathematical model is proposed and studied for contact with wear described by Archard’s law of surface wear. In this model, the friction between a deformable body and the foundation leads to wear of the contact surface of the body over time. Solution existence and uniqueness for the model are provided in [@SPS17]. Numerical approximation of the contact problem is the subject of [@JO18] where some error bounds are derived for a fully discrete scheme. In this paper, we take a further step by considering a more general fully discrete numerical scheme for the contact problem that allows an arbitrary partition of the time interval, providing optimal order error estimates of the fully discrete scheme to solve the contact problem. Moreover, we present numerical results showing deformation of the contact body and numerical convergence orders of the fully discrete solutions that confirm the theoretical error bounds.
The remainder of this paper is organized as follows. In Section \[sec2\] we introduce the contact problem and its variational formulation. In Section \[sec4\], we study a fully discrete numerical scheme and derive optimal order error bounds. In Section \[sec5\], we present computational simulation results for numerical convergence orders that match the theoretical predictions.
The contact problem and its variational formulation {#sec2}
===================================================
0
First, we describe the physical setting of the contact problem. Consider a deformable body that occupies a domain $\Omega\subset\mathbb{R}^d$, $d = 2, 3$ in application. The body is subject to the action of volume forces with a total density ${\mbox{\boldmath{$f$}}}_0$. The boundary $\Gamma$ of the domain $\Omega$ is assumed to be Lipschitz continuous and is divided into three disjoint measurable parts $\Gamma_D$, $\Gamma_N$ and $\Gamma_C$, with ${\rm meas}\,(\Gamma_D)>0$. Denote by ${\mbox{\boldmath{$\nu$}}}$ the unit outward normal vector on $\Gamma$ that is defined a.e. on $\Gamma$. The body is clamped on $\Gamma_D$, i.e., the displacement is equal to ${\mbox{\boldmath{$0$}}}$ on $\Gamma_D$. Surface transactions of a total density ${\mbox{\boldmath{$f$}}}_N$ act on the boundary $\Gamma_N$. The contact boundary is $\Gamma_C$ where the contact is modeled by a normal compliance condition with a unilateral constraint and Coulomb’s law of dry friction. Following [@SPS17], we assume that the body is elastic, in contact with a moving obstacle (foundation) made of a hard perfectly rigid material, and assume that the contact surface of the body $\Gamma_C$ is covered by a layer of soft material. This layer is deformable and the foundation may penetrate it. Frictional contact with the foundation may cause this layer to wear over time.
We assume that the acceleration of the body is negligible and so the problem is quasistatic. In our model, the framework of the small strain theory is employed. We are interested in the body displacement and foundation wear in a time interval $[0, T]$, with $T > 0$. We denote by “$\cdot$” and $\|\cdot\|$ the scalar product and the Euclidean norm in $\mathbb{R}^d$ or $\mathbb{S}^d$, respectively, where $\mathbb{S}^d$ is the space of symmetric matrices of order $d$. The indices $i$ and $j$ run from $1$ to $d$ and the index after a comma represents the partial derivative with respect to the corresponding component of the independent variable. Summation convention over repeated indices is adopted. We denote the divergence operator by ${\rm Div}{\mbox{\boldmath{$\sigma$}}}= (\sigma_{ij,j})$ for an $\mathbb{S}^d$-valued field ${\mbox{\boldmath{$\sigma$}}}$. Standard Lebesgue and Sobolev spaces will be used, such as $L^2(\Omega)^d = L^2(\Omega; \mathbb{R}^d)$ and $H^1(\Omega)^d =H^1(\Omega; \mathbb{R}^d)$. Recall that the linearized strain tensor of a displacement field ${\mbox{\boldmath{$u$}}}\in H^1(\Omega)^d$ is $${\mbox{\boldmath{$\varepsilon$}}}({\mbox{\boldmath{$u$}}})=(\varepsilon_{ij}({\mbox{\boldmath{$u$}}})),\quad\varepsilon_{ij}({\mbox{\boldmath{$u$}}})=\frac{1}{2}\left(u_{i,j}+u_{j,i}\right).$$
Let $u_\nu = {\mbox{\boldmath{$u$}}}\cdot{\mbox{\boldmath{$\nu$}}}$ and $\sigma_\nu ={\mbox{\boldmath{$\sigma$}}}{\mbox{\boldmath{$\nu$}}}\cdot{\mbox{\boldmath{$\nu$}}}$ be the normal components of ${\mbox{\boldmath{$u$}}}$ and ${\mbox{\boldmath{$\sigma$}}}$, respectively, and let ${\mbox{\boldmath{$u$}}}_\tau = {\mbox{\boldmath{$u$}}}-u_\nu {\mbox{\boldmath{$\nu$}}}$ and ${\mbox{\boldmath{$\sigma$}}}_\tau= {\mbox{\boldmath{$\sigma$}}}{\mbox{\boldmath{$\nu$}}}-\sigma_\nu {\mbox{\boldmath{$\nu$}}}$ be their tangential components, respectively. To simplify the notation, we will usually not indicate explicitly the dependence of various functions on the spatial variable ${\mbox{\boldmath{$x$}}}$.
Denote by ${\mbox{\boldmath{$v$}}}^*(t)\not={\mbox{\boldmath{$0$}}}$ the velocity of the foundation. Let $${\mbox{\boldmath{$n$}}}^*(t)=-{\mbox{\boldmath{$v$}}}^*(t)/\|{\mbox{\boldmath{$v$}}}^*(t)\|, \quad \alpha(t)=\kappa\,\|{\mbox{\boldmath{$v$}}}^*(t)\|,
\label{bn}$$ where $\kappa$ represents the wear coefficient, and let $\mu$ be the friction coefficient. The classical formulation of the contact problem with wear is as follows.
\[P1\] Find a displacement field ${\mbox{\boldmath{$u$}}}\colon\Omega\times [0,T]\to \mathbb{R}^d$, a stress field ${\mbox{\boldmath{$\sigma$}}}\colon\Omega\times [0,T]\to \mathbb{S}^d$, and a wear function $w\colon\Gamma_C\times[0, T] \to \mathbb{R}_{+}=[0,\infty)$ such that for all $t \in [0, T]$, $$\begin{aligned}
{\mbox{\boldmath{$\sigma$}}}(t)={\mathcal F}{\mbox{\boldmath{$\varepsilon$}}}({\mbox{\boldmath{$u$}}}(t))\quad&{\rm in}\ \Omega,\label{eq1}\\[1mm]
{\rm Div}\,{\mbox{\boldmath{$\sigma$}}}(t)+{\mbox{\boldmath{$f$}}}_0(t)={\mbox{\boldmath{$0$}}}\quad&{\rm in}\ \Omega,\label{eq2}\\[1mm]
{\mbox{\boldmath{$u$}}}(t)={\mbox{\boldmath{$0$}}}\quad &{\rm on}\ \Gamma_D,\label{eq3}\\[1mm]
{\mbox{\boldmath{$\sigma$}}}(t){\mbox{\boldmath{$\nu$}}}={\mbox{\boldmath{$f$}}}_N(t)\quad&{\rm on}\ \Gamma_N,\label{eq4}\\[1mm]
\left. \begin{array}{ll}
u_\nu(t)\le g,\ \sigma_\nu(t)+p(u_\nu(t)-w(t))\le 0,\\
(u_\nu(t)-g)(\sigma_\nu(t)+p(u_\nu(t)-w(t)))=0
\end{array}\right\} \quad&{\rm on}\ \Gamma_C,\label{eq5}\\[1mm]
-{\mbox{\boldmath{$\sigma$}}}_\tau(t)=\mu\,p(u_\nu(t)-w(t))\,{\mbox{\boldmath{$n$}}}^*(t)\quad&{\rm on}\ \Gamma_C,\label{eq6}\\
w^\prime(t)=\alpha(t)\,p(u_\nu(t)-w(t))\quad&{\rm on}\ \Gamma_C,\label{eq7}\\
w(0)=0\quad&{\rm on}\ \Gamma_C. \label{eq8}\end{aligned}$$
In Problem \[P1\], equation represents an elastic constitutive law with an elasticity operator ${\cal F}$. Equation is the equilibrium equation. The equality describes the fact that body is clamped on $\Gamma_D$ and represents external forces acting on $\Gamma_N$. The relations in describe the damping response of the foundation, $g > 0$ being the thickness of a soft layer covering $\Gamma_C$. The friction is modeled by equation . Here, the size of ${\mbox{\boldmath{$v$}}}^*$ is assumed to be significantly larger than that of the tangential body velocity ${\mbox{\boldmath{$u$}}}^\prime_\tau$. Equations and govern the evolution of the wear function. Detailed derivation of this model is presented in [@SPS17].
The contact problem will be studied in its variational formulation. For this purpose, we introduce function spaces and hypotheses on the problem data. We recall that for a normed space $X$, $C([0,T];X)$ is the space of continuous functions from $[0, T]$ to $X$. We will use the following Hilbert spaces: $${\cal H} = L^2(\Omega;\mathbb{S}^d),\quad V=\{{\mbox{\boldmath{$v$}}}\in H^1(\Omega)^d\mid {\mbox{\boldmath{$v$}}}={\mbox{\boldmath{$0$}}}\ {\rm on}\ \Gamma_D\}$$ endowed with the inner scalar products $$({\mbox{\boldmath{$\sigma$}}},{\mbox{\boldmath{$\tau$}}})_{\cal H}=\int_\Omega \sigma_{ij}\tau_{ij} dx,\quad
({\mbox{\boldmath{$u$}}},{\mbox{\boldmath{$v$}}})_V=({\mbox{\boldmath{$\varepsilon$}}}({\mbox{\boldmath{$u$}}}),{\mbox{\boldmath{$\varepsilon$}}}({\mbox{\boldmath{$v$}}}))_{\cal H}$$ with the corresponding norms. Denote by $\langle\cdot,\cdot\rangle_{V^*\times V}$ the duality pairing between a dual space $V^*$ and $V$. The set of admissible displacements is $$U = \{{\mbox{\boldmath{$v$}}}\in V \mid v_\nu\le g \ {\rm on}\ \Gamma_C \}.$$
For a function ${\mbox{\boldmath{$v$}}}\in V$, we use the same symbol ${\mbox{\boldmath{$v$}}}$ for its trace on the boundary $\Gamma$. By the Sobolev trace theorem, there exists a constant $c_0 > 0$ depending only on $\Omega$, $\Gamma_D$ and $\Gamma_C$ such that $$\|{\mbox{\boldmath{$v$}}}\|_{L^2(\Gamma_C)^d}\le c_0\|{\mbox{\boldmath{$v$}}}\|_V\quad\forall\,{\mbox{\boldmath{$v$}}}\in V.
\label{eq9}$$
Now we introduce the hypotheses on the data needed in the study of Problem \[P1\].
: For the elasticity operator ${\cal F}\colon\Omega\times\mathbb{S}^d\to\mathbb{S}^d$,\
(a) ${\cal F}(\cdot,{\mbox{\boldmath{$\varepsilon$}}})$ is measurable on $\Omega$ for all ${\mbox{\boldmath{$\varepsilon$}}}\in \mathbb{S}^d$, ${\cal F}(\cdot,{\mbox{\boldmath{$0$}}})\in {\cal H}$;\
(b) $\exists$ $L_{\cal F} > 0$ s.t. $\|{\cal F}({\mbox{\boldmath{$x$}}},{\mbox{\boldmath{$\varepsilon$}}}_1)-{\cal F}({\mbox{\boldmath{$x$}}},{\mbox{\boldmath{$\varepsilon$}}}_2)\|
\le L_{\cal F}\|{\mbox{\boldmath{$\varepsilon$}}}_1- {\mbox{\boldmath{$\varepsilon$}}}_2\|$ $\forall\,{\mbox{\boldmath{$\varepsilon$}}}_1,{\mbox{\boldmath{$\varepsilon$}}}_2\in \mathbb{S}^d$, a.e. ${\mbox{\boldmath{$x$}}}\in\Omega$;\
(c) $\exists$ $m_{\cal F} > 0$ s.t. $({\cal F}({\mbox{\boldmath{$x$}}},{\mbox{\boldmath{$\varepsilon$}}}_1)- {\cal F}({\mbox{\boldmath{$x$}}},{\mbox{\boldmath{$\varepsilon$}}}_2))
\cdot({\mbox{\boldmath{$\varepsilon$}}}_1- {\mbox{\boldmath{$\varepsilon$}}}_2)\ge m_{\cal F}\|{\mbox{\boldmath{$\varepsilon$}}}_1- {\mbox{\boldmath{$\varepsilon$}}}_2\|^2$ $\forall\,{\mbox{\boldmath{$\varepsilon$}}}_1,{\mbox{\boldmath{$\varepsilon$}}}_2\in \mathbb{S}^d$, a.e. ${\mbox{\boldmath{$x$}}}\in\Omega$.
: For the normal compliance function $p\colon\Gamma_C\times\mathbb{R}\to\mathbb{R}_+$,\
(a) $p(\cdot, r)$ is measurable on $\Gamma_C$ $\forall\,r \in\mathbb{R}$;\
(b) $\exists$ $L_p>0$ s.t. $|p({\mbox{\boldmath{$x$}}},r_1)-p({\mbox{\boldmath{$x$}}},r_2)|\le L_p|r_1-r_2|$ $\forall\,r_1,r_2\in \mathbb{R}$, a.e. ${\mbox{\boldmath{$x$}}}\in\Gamma_C$;\
(c) $(p({\mbox{\boldmath{$x$}}}, r_1)-p({\mbox{\boldmath{$x$}}}, r_2))(r_1-r_2)\ge 0$ $\forall\,r_1, r_2\in \mathbb{R}$, a.e. ${\mbox{\boldmath{$x$}}}\in\Gamma_C$;\
(d) $p({\mbox{\boldmath{$x$}}}, r) = 0$ $\forall\, r\le 0$, a.e. ${\mbox{\boldmath{$x$}}}\in\Gamma_C$.
Note that (b) and (d) imply $$|p({\mbox{\boldmath{$x$}}},r)|\le L_p|r|\quad \forall\,r\in\mathbb{R},\ {\rm a.e.}\ {\mbox{\boldmath{$x$}}}\in\Gamma_C.
\label{eq2.9a}$$
: For the densities of body and traction forces, $${\mbox{\boldmath{$f$}}}_0 \in C([0, T]; L^2(\Omega)^d ),\quad {\mbox{\boldmath{$f$}}}_N\in C([0, T]; L^2(\Gamma_N)^d).$$
: For the friction and wear coefficients, and the foundation velocity,\
(a) $\mu\in L^\infty(\Gamma_C)$, $\mu({\mbox{\boldmath{$x$}}})\ge 0$ a.e. ${\mbox{\boldmath{$x$}}}\in \Gamma_C$;\
(b) $\kappa \in L^\infty(\Gamma_C)$, $\kappa({\mbox{\boldmath{$x$}}})\ge 0$ a.e. ${\mbox{\boldmath{$x$}}}\in \Gamma_C$;\
(c) ${\mbox{\boldmath{$v$}}}^* \in C([0, T ];\mathbb{R}^d)$, $\|{\mbox{\boldmath{$v$}}}^*(t)\|\ge v_0 > 0$ $\forall\, t \in [0, T]$.
We notice that hypotheses implies the following regularities: $${\mbox{\boldmath{$n$}}}^*\in C([0, T];\mathbb{R}^d),\quad \alpha\in C([0, T ]; L^\infty(\Gamma_C)),
\label{eq10}$$ where ${\mbox{\boldmath{$n$}}}^*$ and $\alpha$ are defined in .
Finally, we will need a smallness assumption on the combined effect of the Lipschitz constant of the normal compliance function $p$ and the friction coefficient $\mu$. Recall that $c_0$ is the constant in the inequality .
: $c^2_0 L_p\|\mu\|_{L^\infty(\Gamma_C)}< m_{\cal F}$.
Now we define some operators and functions needed in the variational formulation of Problem \[P1\]. Let $F\colon V\to V^*$, ${\mbox{\boldmath{$f$}}}\colon[0, T ]\to V^*$ and $\varphi\colon [0,T]\times L^2(\Gamma_C)\times
V \times V\to \mathbb{R}$ be defined for all ${\mbox{\boldmath{$u$}}},{\mbox{\boldmath{$v$}}}\in V$, $w \in L^2(\Gamma_C)$, $t\in[0,T]$ as follows: $$\begin{aligned}
&\langle F{\mbox{\boldmath{$u$}}},{\mbox{\boldmath{$v$}}}\rangle_{V^*\times V}=({\cal F}({\mbox{\boldmath{$\varepsilon$}}}({\mbox{\boldmath{$u$}}})),{\mbox{\boldmath{$\varepsilon$}}}({\mbox{\boldmath{$v$}}}))_{\cal H},\\
&\langle {\mbox{\boldmath{$f$}}}(t),{\mbox{\boldmath{$v$}}}\rangle_{V^*\times V}=\int_\Omega{\mbox{\boldmath{$f$}}}_0(t)\cdot{\mbox{\boldmath{$v$}}}\,dx+\int_{\Gamma_N}{\mbox{\boldmath{$f$}}}_N(t)\cdot{\mbox{\boldmath{$v$}}}\,da,\\
&\varphi(t,w,{\mbox{\boldmath{$u$}}},{\mbox{\boldmath{$v$}}})=\int_{\Gamma_C} p(u_\nu-w) \left[v_\nu+\mu\,{\mbox{\boldmath{$n$}}}^*(t)\cdot{\mbox{\boldmath{$v$}}}_\tau\right] da.\end{aligned}$$
Let $W=L^2(\Gamma_C)$ be the space for the wear variable $w$. Using the standard procedures in the mathematical theory of contact mechanics, we obtain the week formulation of Problem \[P1\].
\[P2\] Find ${\mbox{\boldmath{$u$}}}\colon [0,T]\to U$ and $w\colon [0,T] \to W$ such that for all $t\in[0,T]$, $$\begin{aligned}
& \langle F{\mbox{\boldmath{$u$}}}(t),{\mbox{\boldmath{$v$}}}-{\mbox{\boldmath{$u$}}}(t)\rangle_{V^*\times V}+\varphi(t,w(t),{\mbox{\boldmath{$u$}}}(t),{\mbox{\boldmath{$v$}}})-\varphi(t,w(t),{\mbox{\boldmath{$u$}}}(t),{\mbox{\boldmath{$u$}}}(t))
\nonumber\\
&\qquad \qquad\qquad {}\ge \langle {\mbox{\boldmath{$f$}}}(t),{\mbox{\boldmath{$v$}}}-{\mbox{\boldmath{$u$}}}(t)\rangle_{V^*\times V}\quad\forall\,{\mbox{\boldmath{$v$}}}\in U,\label{eq11}\\
& w(t)=\int_0^t \alpha(s)\,p(u_\nu(s)-w(s))\,ds. \label{eq12}\end{aligned}$$
We recall the following existence and uniqueness result for Problem \[P2\] from [@SPS17].
\[thm1\] Assume , , , and . Then Problem \[P2\] has a unique solution with the regularity $${\mbox{\boldmath{$u$}}}\in C([0, T ]; V), \quad {\mbox{\boldmath{$\sigma$}}}\in C([0, T ];{\cal H}),\quad w \in C^1([0,T];W).$$ In addition, $w(t)\ge 0$ for all $t\in [0,T]$, a.e. on $\Gamma_C$.
Numerical analysis {#sec4}
==================
0
We turn to the numerical solution of Problem \[P2\]. Let $V^h \subset V$ and $W^h\subset W$ be two families of finite dimensional subspaces with a discretization parameter $h > 0$. Then define $U^h = U \cap V^h$. Let $0=t_0<t_1<\cdots<t_N=T$ be a partition of the time interval $[0,T]$. Denote $k_n=t_{n+1}-t_n$, $0\le n\le N-1$, and $k=\max_{0\le n\le N-1} k_n$ for the time step size. For a function $z$ continuous in $t$, we write $z_n = z(t_n)$.
We make the following additional assumptions on the solution ${\mbox{\boldmath{$u$}}}$ to Problem \[P2\] and the velocity of the foundation ${\mbox{\boldmath{$v$}}}^*$.
: ${\mbox{\boldmath{$u$}}}\in H^1(0, T ; V)$, ${\mbox{\boldmath{$v$}}}^*\in W^{1,\infty}(0,T;\mathbb{R}^d)$.
Note that assumptions and (b) imply that $$\alpha\in W^{1,\infty}(0,T;L^\infty(\Gamma_C)).
\label{eq13}$$
Consider the following fully discrete scheme for solving Problem \[P2\].
\[P3\] Find ${\mbox{\boldmath{$u$}}}^{hk}=\{{\mbox{\boldmath{$u$}}}^{hk}_n\}_{n=0}^N\subset U^h$ and $w^{hk}=\{w^{hk}_n\}_{n=0}^N\subset W^h$, $w^{hk}_0=0$, such that for $0\le n\le N$, $$\begin{aligned}
&\langle F{\mbox{\boldmath{$u$}}}^{hk}_n,{\mbox{\boldmath{$v$}}}^h-{\mbox{\boldmath{$u$}}}^{hk}_n\rangle_{V^*\times V}+\varphi(t_n,w^{hk}_n,{\mbox{\boldmath{$u$}}}^{hk}_n,{\mbox{\boldmath{$v$}}}^h)
-\varphi(t_n,w^{hk}_n,{\mbox{\boldmath{$u$}}}^{hk}_n,{\mbox{\boldmath{$u$}}}^{hk}_n)\nonumber\\
&\qquad\qquad\qquad\ge \langle {\mbox{\boldmath{$f$}}}_n,{\mbox{\boldmath{$v$}}}^h-{\mbox{\boldmath{$u$}}}^{hk}_n\rangle_{V^*\times V}\quad\forall\,{\mbox{\boldmath{$v$}}}^h\in U^h,
\label{eq14}\end{aligned}$$ and for $1\le n\le N$, $$w^{hk}_n=\sum_{j=0}^{n-1} k_j \alpha_j p(u^{hk}_{j,\nu}-w^{hk}_j). \label{eq15}$$
We remark that existence of a unique solution to Problem \[P3\] follows from an application of discrete version of Theorem \[thm1\]. We also remark that the numerical scheme considered in [@JO18] is a special case of Problem \[P3\] where a uniform partition of the time interval $[0,T]$ is used. For a uniform partition of $[0,T]$ into $N$ equal size sub-intervals, we let $k = T/N$ be the time step and $t_n = n\,k$, $0\le n\le N$, the node points.
We will make use of the following discrete Gronwall inequality ([@HS2002 Lemma 7.25]).
\[lem3\] Assume $\{g_n\}_{n=1}^N$ and $\{e_n\}_{n=1}^N$ are two sequences of non-negative numbers satisfying $$e_n\le c\,g_n+c\,\sum_{j=1}^{n-1} k_j e_j,\quad n=1,\dots,N.$$ Then $$e_n\le c\,\Big(g_n+\sum_{j=1}^{n-1} k_j g_j\Big),\quad n=1,\dots,N.$$ Therefore, $$\max_{1 \le n \le N} e_n \le c\,\max_{1 \le n \le N} g_n.$$
We have Ceá’s inequality useful for error estimation.
\[thm4\] Under the assumptions stated in Theorem \[thm1\] and the additional hypothesis , there exists a constant $c > 0$ such that for any ${\mbox{\boldmath{$v$}}}^h_n\in U^h$, $1\le n\le N$, $$\begin{aligned}
\max_{1\le n\le N}\left(\|{\mbox{\boldmath{$u$}}}_n-{\mbox{\boldmath{$u$}}}^{hk}_n\|_V^2+\|w_n-w^{hk}_n\|_W^2\right)
& \le c\,k^2+ c\,k\,\|{\mbox{\boldmath{$u$}}}_0-{\mbox{\boldmath{$u$}}}^{hk}_0\|_V^2\nonumber\\
&\quad{} +c\,\max_{1\le n\le N}\left(\|{\mbox{\boldmath{$u$}}}_n-{\mbox{\boldmath{$v$}}}^h_n\|_V^2+|R_n(w_n,{\mbox{\boldmath{$u$}}}_n,{\mbox{\boldmath{$v$}}}^h_n)|\right)
\label{eq17}\end{aligned}$$ where $$\begin{aligned}
R_n(w_n,{\mbox{\boldmath{$u$}}}_n,{\mbox{\boldmath{$v$}}}^h_n)& =\langle F{\mbox{\boldmath{$u$}}}_n,{\mbox{\boldmath{$v$}}}^h_n-{\mbox{\boldmath{$u$}}}_n\rangle_{V^*\times V}
+\varphi(t_n,w_n,{\mbox{\boldmath{$u$}}}_n,{\mbox{\boldmath{$v$}}}^h_n)-\varphi(t_n,w_n,{\mbox{\boldmath{$u$}}}_n,{\mbox{\boldmath{$u$}}}_n)\nonumber\\
&\quad{} -\langle {\mbox{\boldmath{$f$}}}_n,{\mbox{\boldmath{$v$}}}^h_n-{\mbox{\boldmath{$u$}}}_n\rangle_{V^*\times V}.
\label{eq18}\end{aligned}$$
By modifying the proof of Theorem 4 in [@JO18], we can establish the inequality $$\begin{aligned}
\|{\mbox{\boldmath{$u$}}}_n-{\mbox{\boldmath{$u$}}}^{hk}_n\|_V^2+\|w_n-w^{hk}_n\|_W^2 &\le c\,\|{\mbox{\boldmath{$u$}}}_n-{\mbox{\boldmath{$v$}}}^h_n\|_V^2+\left|R_n(w_n,{\mbox{\boldmath{$u$}}}_n,{\mbox{\boldmath{$v$}}}^h_n)\right|
+c\,k^2+c\,k\,\|{\mbox{\boldmath{$u$}}}_0-{\mbox{\boldmath{$u$}}}^{hk}_0\|_V^2\nonumber\\
&\quad{}+ c\sum_{j=1}^{n-1} k_j \left(\|{\mbox{\boldmath{$u$}}}_j-{\mbox{\boldmath{$u$}}}^{hk}_j\|_V^2+\|w_j-w^{hk}_j\|_W^2\right).
\label{eq28}\end{aligned}$$ Applying Lemma \[lem3\] on , we get the inequality .
Note that from and , we have (cf. [@JO18 (27)]), for $t\in [0,T]$, $$\begin{aligned}
&\varphi(t,w_1,{\mbox{\boldmath{$u$}}}_1,{\mbox{\boldmath{$v$}}}_2)+\varphi(t,w_2,{\mbox{\boldmath{$u$}}}_2,{\mbox{\boldmath{$v$}}}_1)-\varphi(t,w_1,{\mbox{\boldmath{$u$}}}_1,{\mbox{\boldmath{$v$}}}_1)-\varphi(t,w_2,{\mbox{\boldmath{$u$}}}_2,{\mbox{\boldmath{$v$}}}_2)
\nonumber\\
&\qquad{}\le L_p\left(c_0\|{\mbox{\boldmath{$u$}}}_1-{\mbox{\boldmath{$u$}}}_2\|_V+\|w_1-w_2\|_W\right)
\left(c_0\|\mu\|_{L^\infty(\Gamma_C)}\|{\mbox{\boldmath{$v$}}}_1-{\mbox{\boldmath{$v$}}}_2\|_V+\|w_1-w_2\|_W\right)\nonumber\\
&\qquad{}\qquad\quad\forall\,{\mbox{\boldmath{$u$}}}_1,{\mbox{\boldmath{$u$}}}_2,{\mbox{\boldmath{$v$}}}_1,{\mbox{\boldmath{$v$}}}_2\in V,\,w_1,w_2\in W.
\label{eq49}\end{aligned}$$ The inequality is the starting point for further error estimation. For simplicity, we assume $\Omega$ is a polygonal/polyhedral domain. Then $\Gamma_D$, $\Gamma_N$ and $\Gamma_C$ can be expressed as unions of flat components (line segments for $d=2$ and polygons for $d=3$) that have pairwise disjoint interiors. In particular, we write $\overline{\Gamma_C}=\cup_{i=1}^{i_0}\Gamma_{C,i}$, where each component $\Gamma_{C,i}$ is a line segment if $d=2$ or a polygon if $d=3$. Consider a regular family of finite element partitions $\{{\cal T}^h\}$ of the domain $\overline{\Omega}$ into triangular or tetrahedral elements such that if the intersection of one side/face of an element with one flat component of the boundary has a positive relative measure, then the side/face lies entirely in that flat component. Corresponding to ${\cal T}^h$, we define the linear element space $$V^h=\left\{{\mbox{\boldmath{$v$}}}^h\in C(\overline{\Omega})^d \mid {\mbox{\boldmath{$v$}}}^h|_T\in \mathbb{P}_1(T)^d,\ T\in {\cal T}^h,\
{\mbox{\boldmath{$v$}}}^h={\mbox{\boldmath{$0$}}}\ {\rm on\ }\Gamma_D\right\}.
\label{Vh}$$ Then we define the discrete admissible finite element set $$U^h=\left\{{\mbox{\boldmath{$v$}}}^h\in V^h\mid v^h_\nu\le g\ {\rm at\ all\ nodes\ on\ }\Gamma_C\right\}.
\label{Uh}$$ We assume $g$ is a concave function. Then, $U^h=V^h\cap U\subset U$. We proceed to derive an optimal order error estimate for the finite element solution defined by Problem \[P3\].
\[thm5\] Keep the assumptions stated in Theorem \[thm4\]. Assume further the solution regularities $$\begin{aligned}
&{\mbox{\boldmath{$u$}}}\in C([0,T];H^2(\Omega)^d),\quad {\mbox{\boldmath{$u$}}}|_{\Gamma_{C,i}}\in C([0,T];H^2(\Gamma_{C,i})^d),\quad 1\le i\le i_0,
\label{eq29}\\
& {\mbox{\boldmath{$\sigma$}}}{\mbox{\boldmath{$\nu$}}}|_{\Gamma}\in C([0,T];L^2(\Gamma)^d). \label{eq30}\end{aligned}$$ Then we have the optimal order error estimate $$\max_{1\le n\le N}\left(\|{\mbox{\boldmath{$u$}}}_n-{\mbox{\boldmath{$u$}}}^{hk}_n\|_V^2+\|w_n-w^{hk}_n\|_W^2\right)
\le c\left(k^2+h^2\right). \label{eq31}$$
By following the arguments presented in [@HS2002 Section 8.1], it can be shown that under the stated regularity assumptions, the solution of Problem \[P2\] satisfies, for $t\in [0,T]$, $$\begin{aligned}
{\rm Div}\,{\mbox{\boldmath{$\sigma$}}}(t)+{\mbox{\boldmath{$f$}}}_0(t)={\mbox{\boldmath{$0$}}}\quad&{\rm a.e.\ in}\ \Omega,\\[1mm]
{\mbox{\boldmath{$\sigma$}}}(t){\mbox{\boldmath{$\nu$}}}={\mbox{\boldmath{$f$}}}_N(t)\quad&{\rm a.e.\ on}\ \Gamma_N,\end{aligned}$$ where $${\mbox{\boldmath{$\sigma$}}}(t)={\mathcal F}{\mbox{\boldmath{$\varepsilon$}}}({\mbox{\boldmath{$u$}}}(t)).$$ Using these relations we find that $$R_n(w_n,{\mbox{\boldmath{$u$}}}_n,{\mbox{\boldmath{$v$}}}^h_n)=\int_{\Gamma_C}\left\{{\mbox{\boldmath{$\sigma$}}}_n{\mbox{\boldmath{$\nu$}}}{\cdot}({\mbox{\boldmath{$v$}}}^h_n-{\mbox{\boldmath{$u$}}}_n)
+p(u^{hk}_{n,\nu}-w_n)\left[v^h_{n,\nu}-u_{n,\nu}
+\mu\,{\mbox{\boldmath{$n$}}}^*_n\cdot({\mbox{\boldmath{$v$}}}_{n,\tau}^h-{\mbox{\boldmath{$u$}}}_{n,\tau})\right]\right\}da.$$ Thus, $$\left|R_n(w_n,{\mbox{\boldmath{$u$}}}_n,{\mbox{\boldmath{$v$}}}^h_n)\right|\le c\,\|{\mbox{\boldmath{$u$}}}_n-{\mbox{\boldmath{$v$}}}_n^h\|_{L^2(\Gamma_C)^d}.
\label{eq32}$$ This provides an upper bound for the term $\left|R_n(w_n,{\mbox{\boldmath{$u$}}}_n,{\mbox{\boldmath{$v$}}}^h_n)\right|$ on the right hand side of .
Now we bound the error $\|{\mbox{\boldmath{$u$}}}_0-{\mbox{\boldmath{$u$}}}^{hk}_0\|_V$. For simplicity, we denote $$\varphi_0({\mbox{\boldmath{$u$}}},{\mbox{\boldmath{$v$}}}):=\varphi(0,0,{\mbox{\boldmath{$u$}}},{\mbox{\boldmath{$v$}}}).$$ Write $$\begin{aligned}
\langle F{\mbox{\boldmath{$u$}}}_0-F{\mbox{\boldmath{$u$}}}^{hk}_0,{\mbox{\boldmath{$u$}}}_0-{\mbox{\boldmath{$u$}}}^{hk}_0\rangle_{V^*\times V}
& = \langle F{\mbox{\boldmath{$u$}}}_0-F{\mbox{\boldmath{$u$}}}^{hk}_0,{\mbox{\boldmath{$u$}}}_0-{\mbox{\boldmath{$v$}}}^h_0\rangle_{V^*\times V}
+\langle F{\mbox{\boldmath{$u$}}}_0,{\mbox{\boldmath{$v$}}}^h_0-{\mbox{\boldmath{$u$}}}_0\rangle_{V^*\times V} \nonumber\\
&\quad{} +\langle F{\mbox{\boldmath{$u$}}}_0,{\mbox{\boldmath{$u$}}}_0-{\mbox{\boldmath{$u$}}}^{hk}_0\rangle_{V^*\times V}
-\langle F{\mbox{\boldmath{$u$}}}^{hk}_0,{\mbox{\boldmath{$v$}}}^h_0-{\mbox{\boldmath{$u$}}}^{hk}_0\rangle_{V^*\times V}.
\label{eq53}\end{aligned}$$ From with $t=0$, $$\langle F{\mbox{\boldmath{$u$}}}_0,{\mbox{\boldmath{$v$}}}-{\mbox{\boldmath{$u$}}}_0\rangle_{V^*\times V}+\varphi_0({\mbox{\boldmath{$u$}}}_0,{\mbox{\boldmath{$v$}}})-\varphi_0({\mbox{\boldmath{$u$}}}_0,{\mbox{\boldmath{$u$}}}_0)
\ge \langle {\mbox{\boldmath{$f$}}}_0,{\mbox{\boldmath{$v$}}}-{\mbox{\boldmath{$u$}}}_0\rangle_{V^*\times V}\quad\forall\,{\mbox{\boldmath{$v$}}}\in U.
\label{eq51}$$ From with $n=0$, $$\langle F{\mbox{\boldmath{$u$}}}^{hk}_0,{\mbox{\boldmath{$v$}}}^h_0-{\mbox{\boldmath{$u$}}}^{hk}_0\rangle_{V^*\times V}+\varphi_0({\mbox{\boldmath{$u$}}}^{hk}_0,{\mbox{\boldmath{$v$}}}^h_0)
-\varphi_0({\mbox{\boldmath{$u$}}}^{hk}_0,{\mbox{\boldmath{$u$}}}^{hk}_0)\ge \langle {\mbox{\boldmath{$f$}}}_0,{\mbox{\boldmath{$v$}}}^h_0-{\mbox{\boldmath{$u$}}}^{hk}_0\rangle_{V^*\times V}
\quad\forall\,{\mbox{\boldmath{$v$}}}^h_0\in U^h.
\label{eq52}$$ Take ${\mbox{\boldmath{$v$}}}={\mbox{\boldmath{$u$}}}^{hk}_0$ in , and use the resulting inequality and the inequality in to obtain $$\begin{aligned}
\langle F{\mbox{\boldmath{$u$}}}_0-F{\mbox{\boldmath{$u$}}}^{hk}_0,{\mbox{\boldmath{$u$}}}_0-{\mbox{\boldmath{$u$}}}^{hk}_0\rangle_{V^*\times V}
& \le \langle F{\mbox{\boldmath{$u$}}}_0-F{\mbox{\boldmath{$u$}}}^{hk}_0,{\mbox{\boldmath{$u$}}}_0-{\mbox{\boldmath{$v$}}}^h_0\rangle_{V^*\times V}+R_0(0,{\mbox{\boldmath{$u$}}}_0,{\mbox{\boldmath{$v$}}}^h_0)\nonumber\\
&\quad{} +\varphi_0({\mbox{\boldmath{$u$}}}_0,{\mbox{\boldmath{$u$}}}^{hk}_0)+\varphi_0({\mbox{\boldmath{$u$}}}^{hk}_0,{\mbox{\boldmath{$v$}}}_0^h)-\varphi_0({\mbox{\boldmath{$u$}}}_0,{\mbox{\boldmath{$v$}}}_0^h)
-\varphi_0({\mbox{\boldmath{$u$}}}^{hk}_0,{\mbox{\boldmath{$u$}}}^{hk}_0).
\label{eq54}\end{aligned}$$ By (c), $$m_{\cal F}\|{\mbox{\boldmath{$u$}}}_0-{\mbox{\boldmath{$u$}}}^{hk}_0\|_V^2\le \langle F{\mbox{\boldmath{$u$}}}_0-F{\mbox{\boldmath{$u$}}}^{hk}_0,{\mbox{\boldmath{$u$}}}_0-{\mbox{\boldmath{$u$}}}^{hk}_0\rangle_{V^*\times V}.$$ By (b), $$\langle F{\mbox{\boldmath{$u$}}}_0-F{\mbox{\boldmath{$u$}}}^{hk}_0,{\mbox{\boldmath{$u$}}}_0-{\mbox{\boldmath{$v$}}}^h_0\rangle_{V^*\times V}
\le L_{\cal F}\|{\mbox{\boldmath{$u$}}}_0-{\mbox{\boldmath{$u$}}}^{hk}_0\|_V\|{\mbox{\boldmath{$u$}}}_0-{\mbox{\boldmath{$v$}}}^h_0\|_V.$$ Then, for an arbitrarily small $\epsilon>0$, there is a constant $c$ depending on $\epsilon$ such that $$\langle F{\mbox{\boldmath{$u$}}}_0-F{\mbox{\boldmath{$u$}}}^{hk}_0,{\mbox{\boldmath{$u$}}}_0-{\mbox{\boldmath{$v$}}}^h_0\rangle_{V^*\times V}
\le \epsilon\,\|{\mbox{\boldmath{$u$}}}_0-{\mbox{\boldmath{$u$}}}^{hk}_0\|_V^2+c\,\|{\mbox{\boldmath{$u$}}}_0-{\mbox{\boldmath{$v$}}}^h_0\|_V^2.$$ By , $$R_0(0,{\mbox{\boldmath{$u$}}}_0,{\mbox{\boldmath{$v$}}}^h_0)\le c\,\|{\mbox{\boldmath{$u$}}}_0-{\mbox{\boldmath{$v$}}}^h_0\|_{L^2(\Gamma_C)^d}.$$ By , $$\begin{aligned}
& \varphi_0({\mbox{\boldmath{$u$}}}_0,{\mbox{\boldmath{$u$}}}^{hk}_0)+\varphi_0({\mbox{\boldmath{$u$}}}^{hk}_0,{\mbox{\boldmath{$v$}}}_0^h)-\varphi_0({\mbox{\boldmath{$u$}}}_0,{\mbox{\boldmath{$v$}}}_0^h)
-\varphi_0({\mbox{\boldmath{$u$}}}^{hk}_0,{\mbox{\boldmath{$u$}}}^{hk}_0)\\
&\qquad{}\le c^2_0 L_p\|\mu\|_{L^\infty(\Gamma_C)}\|{\mbox{\boldmath{$u$}}}_0-{\mbox{\boldmath{$u$}}}^{hk}_0\|_V\|{\mbox{\boldmath{$u$}}}^{hk}_0-{\mbox{\boldmath{$v$}}}^h_0\|_V.\end{aligned}$$ Since $$\|{\mbox{\boldmath{$u$}}}^{hk}_0-{\mbox{\boldmath{$v$}}}^h_0\|_V\le \|{\mbox{\boldmath{$u$}}}_0-{\mbox{\boldmath{$u$}}}^{hk}_0\|_V+ \|{\mbox{\boldmath{$u$}}}_0-{\mbox{\boldmath{$v$}}}^h_0\|_V,$$ for the arbitrarily small $\epsilon>0$, there is a constant $c$ depending on $\epsilon$ such that $$\begin{aligned}
& \varphi_0({\mbox{\boldmath{$u$}}}_0,{\mbox{\boldmath{$u$}}}^{hk}_0)+\varphi_0({\mbox{\boldmath{$u$}}}^{hk}_0,{\mbox{\boldmath{$v$}}}_0^h)-\varphi_0({\mbox{\boldmath{$u$}}}_0,{\mbox{\boldmath{$v$}}}_0^h)
-\varphi_0({\mbox{\boldmath{$u$}}}^{hk}_0,{\mbox{\boldmath{$u$}}}^{hk}_0)\\
&\qquad{}\le \left(c^2_0 L_p\|\mu\|_{L^\infty(\Gamma_C)}+\epsilon\right)\|{\mbox{\boldmath{$u$}}}_0-{\mbox{\boldmath{$u$}}}^{hk}_0\|_V^2
+c\,\|{\mbox{\boldmath{$u$}}}_0-{\mbox{\boldmath{$v$}}}^h_0\|_V^2.\end{aligned}$$ Using these relations in , we obtain $$\left(m_{\cal F}-c^2_0 L_p\|\mu\|_{L^\infty(\Gamma_C)}-2\,\epsilon\right)\|{\mbox{\boldmath{$u$}}}_0-{\mbox{\boldmath{$u$}}}^{hk}_0\|_V^2\le
c\left(\|{\mbox{\boldmath{$u$}}}_0-{\mbox{\boldmath{$v$}}}^h_0\|_V^2+\|{\mbox{\boldmath{$u$}}}_0-{\mbox{\boldmath{$v$}}}^h_0\|_{L^2(\Gamma_C)^d}\right).$$ Recall the condition ; choosing $\epsilon=\left(m_{\cal F}-c^2_0 L_p\|\mu\|_{L^\infty(\Gamma_C)}\right)/4$ we obtain from the above inequality that $$\|{\mbox{\boldmath{$u$}}}_0-{\mbox{\boldmath{$u$}}}^{hk}_0\|_V^2\le c\left(\|{\mbox{\boldmath{$u$}}}_0-{\mbox{\boldmath{$v$}}}^h_0\|_V^2+\|{\mbox{\boldmath{$u$}}}_0-{\mbox{\boldmath{$v$}}}^h_0\|_{L^2(\Gamma_C)^d}\right).
\label{eq55}$$ Using and in , we have $$\begin{aligned}
\max_{0\le n\le N}\left(\|{\mbox{\boldmath{$u$}}}_n-{\mbox{\boldmath{$u$}}}^{hk}_n\|_V^2+\|w_n-w^{hk}_n\|_W^2\right)
& \le c\,k^2+ c\,k\left(\|{\mbox{\boldmath{$u$}}}_0-{\mbox{\boldmath{$v$}}}^h_0\|_V^2+\|{\mbox{\boldmath{$u$}}}_0-{\mbox{\boldmath{$v$}}}^h_0\|_{L^2(\Gamma_C)^d}\right)\nonumber\\
&\quad{} +c\,\max_{1\le n\le N}\left(\|{\mbox{\boldmath{$u$}}}_n-{\mbox{\boldmath{$v$}}}^h_n\|_V^2+\|{\mbox{\boldmath{$u$}}}_n-{\mbox{\boldmath{$v$}}}_n^h\|_{L^2(\Gamma_C)^d}\right)
\label{eq56}\end{aligned}$$ for any ${\mbox{\boldmath{$v$}}}^h_n\in U^h$.
Thus, by applying the finite element interpolation theory (e.g., [@AH2009; @Ciar1978]), we have the optimal order error bound from , under the solution regularities and .
We comment that if ${\cal F}({\mbox{\boldmath{$x$}}},{\mbox{\boldmath{$\varepsilon$}}})$ is a smooth function of ${\mbox{\boldmath{$x$}}}$, in particular if ${\cal F}({\mbox{\boldmath{$x$}}},{\mbox{\boldmath{$\varepsilon$}}})$ does not depend on ${\mbox{\boldmath{$x$}}}$, then follows from and thus there is no need to assume .
Numerical results {#sec5}
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0
In this section, we report computer simulation results on a numerical example. Let $d=2$ and consider a square-shaped set $\Omega=(0,1)\times(0,1)$ with the following partition of the boundary $$\Gamma_D=\{0\}\times[0,1],\quad\Gamma_N=([0,1]\times\{1\})\cup(\{1\}\times[0,1]),
\quad\Gamma_C=[0,1]\times\{0\}.$$
The linear elasticity operator $\mathcal{F}$ is defined by $$\mathcal{F}(\bm{\tau}) = 2\eta\bm{\tau} + \lambda \mbox{tr}(\bm{\tau})I,\qquad \bm{\tau} \in \mathbb{S}^2.$$ Here $I$ denotes the identity matrix, $\mbox{tr}$ denotes the trace of the matrix, $\lambda>0$ and $\eta>0$ are the Lame coefficients. In our simulations, we choose $\lambda = \eta = 4$, $T=1$ and take the following data $$\begin{aligned}
&\bm{u}_{0}(\bm{x}) = (0,0), \quad \bm{x} \in \Omega,\\
&p(r) = \left \{ \begin{array}{ll}
100 \, r, \quad r \in [0, \infty), \\
0, \quad r \in (-\infty, 0), \\
\end{array} \right.\\
&\bm{f}_N(\bm{x},t) = (0,0), \quad \bm{x} \in \Omega,\ t \in [0,T],\\
&\bm{f}_0(\bm{x},t) = (0,-2), \quad \bm{x} \in \Omega,\ t \in [0,T],\\
&g = 0.1.\end{aligned}$$
We use the linear finite element space $V^h$ defined in and its subset $U^h$ defined in , based on uniform triangular partitions of $\overline{\Omega}$. We use the uniform partition of the time interval $[0,1]$ with the time step size $k=1/N$ for a positive integer $N$.
We first demonstrate the effect of some input data on the deformation of the body. In all cases, we show the shape of the body at final time $t=1$, and the numerical solutions correspond to the time step size $1/16$ and where the boundary $\Gamma_C$ of the body is divided into $16$ equal parts.
In Figure \[figOne\] we show the deformed configuration for $\mu(\bm{x}) = 0.3$, $\kappa(\bm{x})= 0.04$ and $\bm{v^*}(\bm{x},t)= (1,0)$. We push the body down towards the moving foundation with a force $\bm{f}_0$, and as a result of friction, the soft layer of material covering $\Gamma_C$ wears out allowing the body to move downward. We observe that in this case coefficient $\kappa$, governing the rate of wear, is not big enough to cause the body to touch the foundation. Because of the friction, the body moves in the same direction as the foundation, i.e. to the right.
We then increase the wear coefficient $\kappa$ to $\kappa(\bm{x}) = 0.08$. The deformed configuration is shown in Figure \[figTwo\]. We observe that the layer of soft material on part of the boundary $\Gamma_C$ completely wears out, allowing the body to rest on the rigid foundation as it cannot penetrate it further.
In Figure \[figThree\], we show the deformed configuration for $\mu(\bm{x}) =1$, $\kappa(\bm{x})= 0.04$ and $\bm{v^*}(\bm{x},t)= (1,0)$. We observe that the body moves further to the right, which is a result of increased friction between soft layer of material covering $\Gamma_C$ and the rigid foundation.
The result in Figure \[figFour\] corresponds to $\mu(\bm{x}) = 0.3$, $\kappa(\bm{x})= 0.02$ and $\bm{v^*}(\bm{x},t)= (-1,0)$. Note that the direction of the motion of the foundation is reversed. As a result, the lower part of the body squeezes to the left and we observe that the boundary $\Gamma_C$ is slightly curled. We conclude that all those modifications lead to results that can be expected.
![Deformed configuration at $t=1$, $\mu= 0.3$, $\kappa= 0.08$, $\bm{v^*}= (1,0)$[]{data-label="figTwo"}](A.jpg){width="0.9\linewidth"}
![Deformed configuration at $t=1$, $\mu= 0.3$, $\kappa= 0.08$, $\bm{v^*}= (1,0)$[]{data-label="figTwo"}](B.jpg){width="0.9\linewidth"}
![Deformed configuration at $t=1$, $\mu= 0.3$, $\kappa= 0.02$, $\bm{v^*}= (-1,0)$[]{data-label="figFour"}](C.jpg){width="0.9\linewidth"}
![Deformed configuration at $t=1$, $\mu= 0.3$, $\kappa= 0.02$, $\bm{v^*}= (-1,0)$[]{data-label="figFour"}](D.jpg){width="0.9\linewidth"}
$h+k$ $1$ $1/2$ $1/4$ $1/8$ $1/16$
----------------------------------------- ---------------- ---------------- ---------------- ---------------- ----------------
$\|\bm{u}-\bm{u}^{hk}\|_V/\|\bm{u}\|_V$ $4.1698e^{-1}$ $2.6840e^{-1}$ $1.4360e^{-1}$ $7.3979e^{-2}$ $3.4882e^{-2}$
[Convergence order]{} 0.6355 0.9022 0.9569 1.0846
$\|w-w^{hk}\|_{W}/\|w\|_W$ $2.9009e^{-1}$ $1.0328e^{-1}$ $3.8385e^{-2}$ $1.4694e^{-2}$ $5.0891e^{-3}$
[Convergence order]{} 1.4898 1.4280 1.3853 1.5297
: Numerical errors[]{data-label="tabOne"}
![Error estimate $\|w - w^{hk}\|_{W} / \|w\|_{W}$[]{data-label="figSix"}](errorsU.jpg){width="0.9\linewidth"}
![Error estimate $\|w - w^{hk}\|_{W} / \|w\|_{W}$[]{data-label="figSix"}](errorsW.jpg){width="0.9\linewidth"}
Finally, we explore the numerical convergence orders of the numerical method on the model problem with $\bm{f}_N(\bm{x},t)=(-0.5,-0.5)$, $\bm{f}_0(\bm{x},t) = (-0.5,-2)$, $\mu(\bm{x}) = 1.0$, $\kappa(\bm{x})= 0.05$, and $\bm{v^*}(\bm{x},t)= (1,0)$. We present a comparison of numerical errors $\|\bm{u} - \bm{u}^{hk}\|_V$ and $\|w - w^{hk}\|_{W}$ computed for a sequence of solutions to discretized problems. We use a uniform discretization of the problem domain and time interval according to the spatial discretization parameter $h$ and time step size $k$, respectively. The boundary $\Gamma_C$ of $\Omega$ is divided into $1/h$ equal parts. We start with $h = 1/2$ and $k = 1/2$, which are successively halved. The numerical solution corresponding to $h = 1/64$ and $k = 1/64$ is taken as the “exact” solution $\bm{u}$ and $w$ with $\|\bm{u}\|_{V}\doteq 0.19131$ and $\|w\|_{W}\doteq 0.08192$. The results are presented in Table \[tabOne\] and Figures \[figFive\] and \[figSix\], where the dependence of the relative error estimates $\|\bm{u} - \bm{u}^{hk}\|_V / \|\bm{u}\|_V$ and $\|w - w^{hk}\|_{W} / \|w\|_{W}$ with respect to $h+k$ are plotted on a log-log scale. A first order convergence is clearly observed for the numerical solutions of the displacement. The numerical convergence orders for the numerical solutions of the wear function are somewhat higher than 1.
[**Acknowledgments**]{}\
The project leading to this application has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement no. 823731 CONMECH.
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A. Ochal and M. Jureczka, Numerical treatment of contact problems with thermal effect, *Discrete Contin. Dyn. Syst. Ser. B* [**23**]{} (2018), 387–400.
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[^1]: Department of Mathematics, Hangzhou Normal University, Hangzhou, China. Email: [[email protected]]{}
[^2]: Program in Applied Mathematical and Computational Sciences (AMCS) & Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA. Email: [[email protected]]{}
[^3]: Jagiellonian University in Krakow, Faculty of Mathematics and Computer Science, Lojasiewicza 6, 30-348 Krakow, Poland. Email: [[email protected]]{}
[^4]: Jagiellonian University in Krakow, Faculty of Mathematics and Computer Science, Lojasiewicza 6, 30-348 Krakow, Poland. Email: [[email protected]]{}
|
---
author:
- Bokun Kong
bibliography:
- 'report.bib'
title: Analysing Russian Trolls via NLP tools
---
|
---
abstract: 'We have measured the clustering of $0.30<z<0.90$ red galaxies and constrained models of the evolution of large-scale structure using the initial $1.2\Box^\circ$ data release of the NOAO Deep Wide-Field Survey (NDWFS). The area and $B_WRI$ passbands of the NDWFS allow samples of $\gtrsim 10^3$ galaxies to be selected as a function of spectral type, absolute magnitude, and photometric redshift. Spectral synthesis models can be used to predict the colors and luminosities of a galaxy population as a function of redshift. We have used PEGASE2 models, with exponentially declining star formation rates, to estimate the observed colors and luminosity evolution of galaxies and to connect, as an evolutionary sequence, related populations of galaxies at different redshifts. A red galaxy sample, with present-day rest-frame Vega colors of $B_W-R>1.44$, was chosen to allow comparisons with the 2dF Galaxy Redshift Survey and Sloan Digital Sky Survey. We find the spatial clustering of red galaxies to be a strong function of luminosity, with $r_0$ increasing from $4.4\pm 0.4 h^{-1} {\rm Mpc}$ at $M_R-5{\rm log}h\approx -20.0$ to $11.2\pm 1.0 h^{-1} {\rm Mpc}$ at $M_R-5{\rm log}h\approx -22.0$. Clustering evolution measurements using samples where the rest-frame selection criteria vary with redshift, including all deep single-band magnitude limited samples, are biased due to the correlation of clustering with rest-frame color and luminosity. The clustering of $-21.5<M_R-5{\rm log}h<-20.5$, $B_W-R>1.44$ galaxies exhibits no significant evolution over the redshift range observed with $r_0= 6.3\pm 0.5 h^{-1} {\rm Mpc}$ in comoving coordinates. This is consistent with recent $\Lambda$CDM models where the bias of $L^*$ galaxies undergoes rapid evolution and $r_0$ evolves very slowly at $z<2$.'
author:
- 'Michael J. I. Brown, Arjun Dey, Buell T. Jannuzi, Tod R. Lauer,'
- 'Glenn P. Tiede, and Valerie J. Mikles'
title: 'Red Galaxy Clustering in the NOAO Deep Wide-Field Survey'
---
Introduction {#sec:intro}
============
The strength and evolution of the spatial clustering of galaxies is a function of cosmology, galaxy mass, and galaxy formation scenarios in Cold Dark Matter models [e.g., @pea97; @kau99; @ben01; @som01]. The predicted linear and quasi-linear growth of density perturbations results in the spatial clustering of dark matter undergoing rapid evolution at $z<2$ [e.g., @pea97]. However, CDM semi-analytic models and simulations predict the distribution of $M_R<-20$ galaxies to be highly biased with respect to the underlying dark matter distribution [e.g., @col89; @kau99; @ben01; @som01]. CDM models for an $\Omega_m=0.3$, $\Lambda=0.7$ cosmology ($\Lambda$CDM) predict little evolution of the early-type galaxy spatial correlation function (measured in comoving coordinates) at $z<2$.
At low redshift, strong constraints on galaxy clustering as a function of spectral type and absolute magnitude are provided by the 2dF Galaxy Redshift Survey (2dFGRS) and Sloan Digital Sky Survey (SDSS) wide-field spectroscopic surveys [@nor02; @zeh02]. At $z>0.3$, the comoving volume and galaxy counts of spectroscopic surveys are orders of magnitude less than the 2dFGRS and SDSS and the resulting estimates of the spatial correlation function suffer from small number statistics and biases from unrepresentative sample volumes [@hog00; @gia01]. Only wide-field surveys, with large samples ($>10^3$) of galaxies and volumes of $\gtrsim10^6 h^{-3} {\rm Mpc^3}$, are able to provide robust estimates of $r_0$ (the spatial scale where the correlation function equals 1) with errors of $<10\%$.
It is possible to obtain constraints on the clustering of high-redshift galaxies using the two-point angular correlation function and deep wide-field imaging. While survey volumes at high redshift have increased with the advent of large-format CCDs, significant issues remain in the interpretation of angular correlation functions derived from these samples. Estimates of the spatial correlation function derived from imaging rely on models of the redshift distribution which differ significantly from each other. In addition, the distribution of galaxy types and luminosities can be a strong function of apparent magnitude and redshift. As the clustering of galaxies is a function of spectral type and luminosity [e.g., @dav76; @lov95; @nor02; @zeh02], the apparent evolution of the correlation function can be dominated by selection effects rather than evolution of large-scale structure [@efs91].
The advent of reliable photometric redshifts allows improved constraints on the redshift distribution of faint galaxies in imaging surveys [e.g., Brunner, Szalay & Connolly 2000; @bro01; @tep01; @fir02]. However, as discussed in §\[sec:cor\], the resulting model redshift distributions and spatial correlation functions strongly depend on the uncertainties of photometric redshifts. If rest-frame color criteria are also applied, samples of galaxies containing a comparable range of spectral types can be selected over a broad range of redshifts [Brown et al. 2001, @fir02; @wil03], thus reducing the selection effect which dominates clustering evolution estimates derived from single-band imaging. Absolute magnitude selection criteria can also be used, though this requires accurate photometric redshifts.
To measure the evolution of galaxy clustering, we have used optical images from the NDWFS [@jan99 Jannuzi et al., in preparation] to select comparable samples of red galaxies at multiple epochs. We have used photometric redshifts and the PEGASE2 galaxy spectral evolution models [@fio97] with exponentially declining star formation rates to estimate the redshifts, luminosities, and rest-frame colors of all galaxies detected in the $B_W$, $R$, and $I$-bands. In addition, we used the best-fit PEGASE2 models to estimate the spectral evolution and the present-day rest-frame colors and luminosities. We selected a red galaxy sample, with present-day rest-frame colors of $B_W-R>1.44$, to allow direct comparison of the NDWFS clustering measurements with the low redshift early-type galaxy clustering measurements from the 2dFGRS. We also selected a $B_W-R>1.77$ subsample to allow a measurement of galaxy clustering as a function rest-frame color.
The outline of the paper is as follows. We provide a brief description of the NDWFS imaging data and catalogs in §\[sec:ndwfs\]. We describe the estimation of photometric redshifts and the galaxy spectral evolution models in §\[sec:photoz\]. In §\[sec:red\], we discuss the motivation for studying red galaxies and the selection of these objects. We discuss the measurement of the angular and spatial two-point correlation functions in §\[sec:cor\]. In §\[sec:clust\], we present the clustering of $0.30<z<0.90$ red galaxies as a function of absolute magnitude and redshift. We discuss the implications of our results in §\[sec:dis\] and summarize the paper in §\[sec:sum\].
The NOAO Deep Wide-Field Survey {#sec:ndwfs}
===============================
The NDWFS is a multiband ($B_W,R,I,J,H,K$) survey of two $\approx 9\Box^\circ$ high Galactic latitude fields with the CTIO $4{\rm m}$, KPNO $4 {\rm m}$, and KPNO $2.1 {\rm m}$ telescopes [@jan99]. A thorough description of the observing strategy and data reduction will be provided by Jannuzi et al. (in preparation). This paper utilizes the first $1.2\Box^\circ$ $B_WRI$ public data release of four adjacent KPNO $4 {\rm m}$ MOSAIC subfields in the Boötes field. The coordinates, depth, and image quality of the four subfields are provided in Table \[table:fields\]. This dataset, along with explanatory material, is available from the NOAO Science Archive on the World Wide Web[^1]. We generated object catalogs using SExtractor $2.2.2$ [@ber96] run in single-image mode with the minimum detection area, convolution filter, and signal above sky threshold optimized to provide the deepest catalogs possible as a function of the seeing. Detections in the different bands were then matched using criteria based on the distance between the image centroids. At faint magnitudes ($R>22$), images with centroids within $1^{\prime\prime}$ of each other in different bands were matched. At bright magnitudes, images were matched if the probability of the centroids being within a given distance of each other by random chance was $<0.005$. This probability was determined using the number of objects as a function of magnitude per unit area measured from the NDWFS. Where multiple matches were found, the closest centroids were matched. In practice, such criteria work well for both faint objects and bright objects including saturated stars and well resolved galaxies.
We have used SExtractor MAG\_AUTO magnitudes [@ber96], which are similar to Kron total magnitudes [@kro80], due to their low uncertainties and small systematic errors at faint magnitudes. The $1\sigma$ uncertainty of the MAG\_AUTO photometry decreases from $\simeq 0.3$ magnitudes at the $50\%$ completeness limit to $\simeq 0.1$ magnitudes at 2 magnitudes brighter than the completeness limit. For unsaturated objects, the MAG\_AUTO photometry has systematic errors which are significantly less than the $1\sigma$ uncertainties. The measured correlation functions do not strongly depend on the technique used to measure object fluxes. Correlation functions using galaxies selected with MAG\_AUTO photometry and $5^{\prime\prime}$ aperture photometry differed by $\lesssim 1 \sigma$. Throughout the remainder of the paper, all the results and conclusions are derived from NDWFS samples using MAG\_AUTO photometry.
We determined the completeness as a function of magnitude by adding artificial stellar objects to copies of the data and recovering them with SExtractor. The $50\%$ completeness limits vary between the 4 subfields in the ranges of $26.2<B_W<26.8$, $24.8<R<25.3$, and $23.6<I<25.6$. We used SExtractor’s star-galaxy classifier to remove objects from the galaxy catalog which had a stellarity of $>0.7$ in 2 or more bands brighter than $B_W<23.8$, $R<22.8$, and $I<21.4$. Tests with artificial stars indicate $\lesssim 5 \%$ of stars are being misclassified as galaxies at the classification limit in all four subfields.
Regions surrounding saturated stars were removed from the catalog to exclude clustered spurious objects detected in the wings of the point spread function of bright stars. Unclustered contamination reduces the amplitude of the correlation function by $(1-f)^2$ where $f$ is the fractional contamination of the sample. We assume that $1\%$ of bright stars are misclassified as galaxies, a rate which is consistent with tests with artificial stars which are a magnitude brighter than the classification limits. While this is a crude assumption, misclassified bright stars would only significantly alter the correlation function results if they were misclassified at a rate of more than $10\%$.
The number of objects classified as stars rapidly decreases at magnitudes fainter than $R\approx 22$. To estimate the number counts of faint red stars, we assumed stars fainter than the classification limits have the same color distribution as stars within a magnitude of the classification limits. We then modeled the star counts with a power law given by $$\frac{dN}{dm} \propto 10^{\alpha m}.$$ This relationship was normalized so the counts agreed with the measured star counts at the classification limits. For this work, we assumed $\alpha=0.07$. While this is only an approximation, the increase in the counts is comparable to the observed star counts in the Medium Deep Survey [@san96]. It is also comparable to the power-law index for the counts of $19<R<21$ red stars in the NDWFS. Tests with $\alpha$ between $0.0$ and $0.15$ did not alter the amplitude of the angular correlation function significantly. In addition, the amplitude of the correlation function (after correcting for contamination) does not significantly change if the star-galaxy classification magnitude limits are decreased. For all the correlation function bins, the estimated rate of stellar contamination was less than $12\%$.
Photometric redshifts {#sec:photoz}
=====================
Photometric redshifts were determined for all objects with $B_W$, $R$, and $I$-band detections. Model spectra as a function of redshift were generated using PEGASE2 spectral synthesis code [@fio97]. Models with $z=0$ Solar metallicity, Miller-Scalo initial mass functions, $z=0$ ages of $12 ~{\rm Gyr}$ (formation $z \approx 4.1$), and exponentially decreasing star formation rates with $e$-folding times between $0.6~{\rm Gyr}$ and $15~{\rm Gyr}$ ($\tau$-models) for an $\Omega=0.3$ $\Lambda=0.7$ $H_0=70~{\rm km~s}^{-1} ~{\rm Mpc}^{-1}$ cosmology[^2] were used to estimate galaxy colors, $k$-corrections and spectral evolution corrections. At $z>3.9$, $\tau$-models with ages of $100~{\rm Myr}$ were used to model the spectra of high redshift galaxies. The model spectra were multiplied by intrinsic dust extinction with $E(B-V)=0.04$ and $R_V=3.1$, comparable to estimates for $0<z<1$ early-type galaxies [@fal99; @kau03]. We multiplied the $B_W$, $R$, and $I$ filter transmission curves with the MOSAIC CCD quantum efficiency as a function of wavelength, the mirror reflectivity and a measurement of the KPNO atmospheric extinction to improve the accuracy of the model galaxy colors. Galaxy photometry was corrected for Galactic dust extinction using the dust maps of [@sch98], though it should be noted that the maximum estimate of $E(B-V)$ was only $0.014$ in the four subfields. We used the Vega spectral energy distribution of [@hay85] to zeropoint the model galaxy colors. Model galaxy colors zeropointed with the [@hay85] spectrum or the [@cas94] model of Vega differ by $B_W-R\simeq 0.03$ from model galaxy colors zeropointed with the frequently used [@kur79] model of Vega. Interpolation between the $\tau$-model was used to fill the color-space occupied by galaxies observed in the NDWFS. The uncertainties of the photometric redshifts would be underestimated if the color-space was not filled. This is particularly true if color-redshift degeneracies present in the data are not reproduced by the models.
Photometric redshifts were estimated by finding the minimum value of $\chi^2$ as a function of redshift, spectral type ($\tau$), and luminosity. To reduce the CPU time required to evaluate the photometric redshifts, the interpolated models were only evaluated when they differed from neighboring models sufficiently to significantly alter the photometric redshifts. As the model spectra do not account for the observed width of the galaxy locus, we increased the photometric uncertainties for the galaxies by $0.05$ magnitudes (added in quadrature). This results in the photometric redshift code producing $1\sigma$ errors which are consistent with the observed scatter between the photometric and spectroscopic redshifts in Figure \[fig:photoz\].
To further improve the accuracy of the photometric redshifts and their uncertainties, the estimated redshift distribution of galaxies as a function of spectral type and apparent magnitude was introduced as a prior. This approach, rather than a Bayesian prior [@kod99; @ben00], was undertaken, as the number of objects as a function of $\tau$ is poorly known at present. The 2dFGRS $z=0$ luminosity functions for different spectral types [@mad02], with luminosity evolution given by the $\tau$-models, were used to produce estimates of the redshift distributions. At $M_R-5{\rm log}h<-23$, where the luminosity function of red galaxies is poorly fitted by a Schechter function [@mad02], the luminosity function was approximated by a power-law. To obtain the correspondence between the 2dFGRS principal component $\eta$ parameter and the $\tau$ parameter, we fitted PEGASE2 models to the 2dFGRS principal component spectra between 3900Å and 4100Å. We did not fit for the entire 2dFGRS spectrum as there may be errors in the 2dFGRS continuum calibration [@mad02]. The prior has little effect on the best-fit estimates of the photometric redshifts but does significantly alter the photometric redshift uncertainties.
To confirm the reliability of the photometric redshifts, spectral types, and absolute magnitudes, simulated data were generated using the $\tau$ spectral evolution models and the 2dFGRS luminosity functions. The simulated data consisted of $\tau \leq 15~{\rm Gyr}$ galaxies with the redshift range $0<z\leq5$ and luminosity range $0.01<L^*\leq 100$. The simulated object photometry was scattered using the estimated uncertainties (including the 0.05 component discussed earlier) as a function of apparent magnitude, thus mimicking what would be present in the real catalogs. In addition to the simulated data, the accuracy of the photometric redshifts was confirmed with spectroscopic redshifts and $B_WRI$ photometry for selected objects in the NDWFS Boötes, NDWFS Cetus, and Lockman Hole fields.
A comparison of the photometric and spectroscopic redshifts for $-22.5<M_R-5{\rm log}h<-19.5$ red galaxies is shown in right-hand panel of Figure \[fig:photoz\]. The observed colors of the galaxies with spectroscopic redshifts are shown in Figure \[fig:col\]. We discuss the selection criteria for the red galaxies in §\[sec:red\]. The photometric redshifts have a $5\%$ systematic error and $\pm5\%$ $1\sigma$ uncertainty. Only two of the 29 galaxies have photometric redshifts with errors of $>30\%$. One of the outliers has strong \[OII\] emission and Balmer absorption. The other is a blended object consisting of an emission line galaxy and an M-star.
Increasing or decreasing the internal dust extinction increases the offset between the photometric and spectroscopic redshifts and increases the random scatter. As varying the dust extinction does not improve the accuracy of the photometric redshifts, we use the constant value of $E(B-V)=0.04$. Photometric redshifts using the GISSEL01 [@bru93; @liu00] $\tau$-model spectral energy distributions produce a slightly larger residual offset and more random scatter, so we use the PEGASE2 models throughout the paper. However, there are no large differences between correlation functions determined using galaxies with PEGASE2 and GISSEL01 photometric redshifts. Throughout the remainder of the paper, photometric redshifts and absolute magnitudes have been corrected for the $5\%$ systematic error shown in Figure \[fig:photoz\].
The red galaxy sample {#sec:red}
=====================
If the PEGASE2 $\tau$-models are a good approximation for the spectral evolution of $z<0.90$ galaxies, the spectral model fits by the photometric redshift code can be used to measure the spectral types and luminosity evolution of galaxies. This assumption is consistent with the $B_WRI$ colors of the galaxy locus (Dey et al., in preparation) and comparisons of photometric and spectroscopic redshifts discussed in §\[sec:photoz\]. We were therefore able to select comparable populations of galaxies at multiple epochs.
If a galaxy sample is going to be used to measure the evolution of clustering, an evolutionary sequence of related galaxy populations must be selected at different redshifts. As the clustering of galaxies is known to be a function of absolute magnitude at low redshift [e.g., @nor02; @zeh02], accurate photometric redshifts are required so objects with similar luminosities can be selected at multiple epochs, enabling unbiased studies of the evolution of clustering. Galaxy types with low rates of spectral evolution are advantageous as selection criteria relying on rapidly evolving models will be very sensitive to errors in the model spectral energy distributions. Red galaxies have accurate photometric redshifts and low rates of spectral evolution. At $z<1$, the 4000Å break is moving through the optical so accurate photometric redshifts can be obtained with a limited number of optical passbands. The PEGASE2 $\tau\sim 1~{\rm Gyr}$ models and observations [@jor99; @sch99; @im02] indicate $z=0.9$ red galaxies are only $\approx 0.9$ magnitudes brighter in rest-frame $R$-band than $z=0$ red galaxies.
For this work, we selected a sample of red galaxies to be comparable to the early-type sample of the 2dFGRS [@nor02]. Galaxies fitted with $\tau<4.5 {\rm ~Gyr}$ were chosen so their $z=0$ spectra matched the 2dFGRS principal component selection criterion. The rest-frame color of the $12~{\rm Gyr}$ old $\tau=4.5 {\rm ~Gyr}$ model with $E(B-V)=0.04$ intrinsic dust extinction is $B_W-R=1.44$. This is approximately the same color as the Sab template of [@fuk95]. The $\tau<4.5 {\rm ~Gyr}$ selection criterion is $\simeq 0.15$ magnitudes redder in $B_W-R$ than the $u^*-r^*>1.8$ (AB) color cut for SDSS red galaxies in [@zeh02]. A subsample, selected with $\tau<2.0 {\rm ~Gyr}$, was chosen to allow estimates of clustering as a function of rest-frame color and comparisons with redder samples, including $z>0.8$ EROs. The rest-frame color of the $12~{\rm Gyr}$ old $\tau=2.0 {\rm ~Gyr}$ model with $E(B-V)=0.04$ intrinsic dust extinction is $B_W-R=1.77$, which is only $0.07$ magnitudes bluer in $B_W-R$ than the $12~{\rm Gyr}$ old $\tau=0.6 {\rm ~Gyr}$ model. The observed colors of the red galaxy sample, along with the $\tau$-models, are shown in Figure \[fig:col\]. For comparison, the rest-frame colors of the [@col80] E and Sbc templates are $B_W-R=1.71$ and $B_W-R=1.14$. Objects with colors which differ significantly from the models can contaminate the sample. To reduce this contamination, only $0.5<R-I<1.6$ galaxies with photometric redshift fits with $\chi^2<3$ (including the prior) were included in the final sample.
The conclusions of this paper rely on the accuracy of the galaxy spectral classifications and absolute magnitudes. Significant contamination by late-type spirals or $R>21$ faint blue galaxies will dramatically decrease angular and spatial two-point correlation functions [@efs91]. To minimize contamination, the photometric redshift range was restricted to $0.30<z<0.90$, where the $\tau$-models are not degenerate for $B_WRI$ photometry. The bulk of the spectra used in Figure \[fig:photoz\] have relatively low signal-to-noise ratios making it difficult to verify the selection criteria spectroscopically with currently available datasets. Instead, we used the simulated data and compared the values of $\tau$ and $M_R$ used to generate the model object with the output $\tau$ and $M_R$ values from the photometric redshift code. As shown in Figure \[fig:tau\], there is good agreement between the input and output values of $\tau$ and $M_R$ for red galaxies. Few blue galaxies are scattered into the red sample while a small fraction of red galaxies are scattered out of the sample. For the remainder of the paper, we restrict the magnitude range to $R\leq 23.25$, where tests with simulated galaxies indicate $\tau$ is being reliably measured. While some contamination is inevitable, it is reasonable to assume that weakly clustered faint blue galaxies [e.g., @efs91] are not dominating the measured angular correlation function. The final red galaxy sample contains 5325 objects, which is 14% of all 39316, $R\leq23.25$ galaxies in the $1.2\Box^\circ$ sample area.
The correlation function {#sec:cor}
========================
We determined the angular correlation function using the [@lan93] estimator: $$\hat\omega(\theta)=\frac{DD-2DR+RR}{RR}$$ where $DD$, $DR$, and $RR$ are the number of galaxy-galaxy, galaxy-random and random-random pairs at angular separation $\theta\pm\delta\theta/2$. The pair counts were determined in logarithmically spaced bins between $10^{\prime\prime}$ and $1^\circ$.
The random objects consist of copies of real galaxies which have had their positions changed to mimic objects that are randomly distributed across the sky. This does not result in a perfectly uniform surface density of objects across the field due to the completeness variations in the 4 subfields. When the “random” objects were distributed across the field, the probability of each object being detected in a given subfield was estimated and the product of this and the subfield area was used when determining which subfield the object would be placed. To decrease the contribution of the random objects to the shot-noise, 100 random object catalogs were generated and $DR$ and $RR$ were renormalized accordingly.
The estimator of the correlation function is subject to the integral constraint $$\int \int \hat \omega (\theta_{12}) d\Omega_1 d\Omega_2 \simeq 0$$ [@gro77] which results in a systematic underestimate of the clustering. To remove this bias, the term $$\omega_\Omega =
\frac{1}{\Omega^2} \int \int \omega (\theta_{12}) d\Omega_1 d\Omega_2$$ was added to $\hat \omega (\theta)$ where $\Omega$ is the survey area. The value of $\bar{n}^2\omega_\Omega$, where $\bar{n}$ is the mean number of galaxies per area $\Omega$, is the contribution of clustering to the variance of the galaxy number counts [@gro77; @efs91]. The angular correlation function was assumed to be a power-law given by $$\omega(\theta) = \omega(1^\prime) \left( \frac{\theta}{1^\prime} \right)^{1-\gamma}$$ where $\gamma$ is a constant. This is a good approximation of the observed spatial correlation function from the 2dFGRS and SDSS surveys on scales of $\lesssim 10 h^{-1} {\rm Mpc}$ [@nor01; @zeh02]. For a $\gamma=1.87$ power-law, the integral constraint for this study was approximately $6\%$ of the amplitude of the correlation function at $1^\prime$.
We determined the covariance matrix of $\omega(\theta)$ using the approximation of [@eis01]: $$C_\omega(\theta_i, \theta_j) = \frac{1}{\pi \Omega^2}
\int_0^\infty K P^2_2(K) J_0(K\theta_i) J_0(K\theta_j) dK
\label{eq:ez}$$ where $J_0$ is a Bessel function and $P^2_2(K)$ is the angular power spectrum, $$P_2(K) = 2\pi \int_0^\infty w(\theta) J_0(K\theta)\theta d\theta.$$ This approximation is best suited to correlation functions where $\omega(\theta)\ll 1$ and underestimates the covariance of very strongly clustered objects. For the evaluation of $P_2(K)$, we truncated the power-law form of the correlation function at $2^\circ$ as the $z\sim 0$ galaxy correlation function is $\approx 0$ on scales of $>20 h^{-1} {\rm Mpc}$ [e.g., @mad96; @con02]. However, the power-law fits to the data are only marginally affected by the value of $P_2(K)$ on large scales. If the angular correlation function bins have significant width, Equation \[eq:ez\] is modified to $$C_\omega(\theta_i, \theta_j) =
\left(\frac{2}{\theta_{i,2}^2-\theta_{i,1}^2}\right)\left(\frac{2}{\theta_{j,2}^2-\theta_{j,1}^2}\right)
\int^{\theta_{i,2}}_{\theta_{i,1}}\int^{\theta_{j,2}}_{\theta_{j,1}}\theta \theta^\prime
C_\omega(\theta,\theta^\prime) d\theta d\theta^\prime$$ (D. Eisenstein 2003, private communication) where $\theta_1$ and $\theta_2$ are the inner and outer radii of the bins. This can be rewritten as the single integral $$\begin{aligned}
C_\omega(\theta_i, \theta_j)
= \frac{4}{(\theta_{i,2}^2-\theta_{i,1}^2)(\theta_{j,2}^2-\theta_{j,1}^2)\pi\Omega^2} \times \nonumber \\
\int^\infty_0 P_2^2(K) \left[ \theta_{i,2}J_1(K\theta_{i,2})-\theta_{i,1}J_1
(K\theta_{i,1})\right] \left[\theta_{j,2}J_1(K\theta_{j,2})-\theta_{j,1}J_1(K\theta_{j,1})\right] \frac{dK}{K}.\end{aligned}$$ The contribution of shot noise to the estimate of the covariance was included by adding the reciprocal of the sky surface density of galaxies (per steradian) to $P_2(k)$. However, the shot noise only dominates the covariance on scales of $\lesssim 1^\prime$ for the red galaxy sample.
The spatial correlation function was obtained using Limber’s (1954) equation: $$\omega(\theta)= \int^\infty_0 \frac{dN}{dz}
\left[ \int^\infty_0 \xi (r(\theta,z,z^\prime),z) \frac{dN}{dz^\prime} dz^\prime \right] dz
\left/
\left( \int^\infty_0 \frac{dN}{dz} dz \right)^2 \right.$$ where $\frac{dN}{dz}$ is the redshift distribution, $\xi$ is the spatial correlation function and $r(\theta,z,z^\prime)$ is the comoving distance between two objects at redshifts $z$ and $z^\prime$ separated by angle $\theta$ on the sky. The spatial correlation function was assumed to be a power law given by $$\xi (r,z) = [ r/r_0(z)]^{-\gamma}
\label{eq:spa}$$ where $$r_0(z) = r_0(0) [1+z] ^{-(3+\epsilon-\gamma)/\gamma}
\label{eq:ep}$$ and $\epsilon$ is a constant [@gro77]. Clustering is fixed in physical or comoving coordinates if $\epsilon=0$ or $\epsilon=\gamma-3$ respectively.
We estimated the redshift distribution for Limber’s equation by summing the redshift likelihood distributions of the individual galaxies in each subsample. As shown in Figure \[fig:pz\], the redshift distributions of individual galaxies can not be modeled with Gaussians and estimates derived from the photometric redshift code $\chi^2$ as a function of redshift must be used instead. Model redshift distributions for several subsamples selected by luminosity and photometric redshift are shown in Figure \[fig:dndz\].
The clustering of red galaxies {#sec:clust}
==============================
The angular and spatial correlation functions
---------------------------------------------
The angular correlation function was determined for red galaxies in a series of photometric redshift bins between $z=0.30$ and $z=0.90$, and absolute magnitude bins between $M_R-5{\rm log}h=-22.5$ and $M_R-5{\rm log}h=-19.5$. All bins are volume limited samples containing galaxies brighter than $R=23.25$. A power-law of the form $\omega (\theta) = \omega(1^\prime) (\theta/1^\prime)^{1-\gamma}$ was fitted to the $\theta<0.25^\circ$ data with $\gamma$ fixed to $1.87$, the value for $z<0.15$ early-type galaxies [@nor02; @zeh02]. We use $\omega(1^\prime)$ rather than $\omega(1^\circ)$ for the power-law fits as it depends less on the assumed value of $\gamma$. The amplitude of the two-point angular correlation functions for these subsamples are summarized in Tables \[table:r0mag\] and \[table:r0z\]. Angular correlation functions for $-21.5<M_R-5{\rm log}h<20.5$, $\tau<4.5~{\rm Gyr}$ galaxies are also plotted in Figure \[fig:ang\].
A summary of the spatial clustering (parameterized by $r_0$) as a function of spectral type, absolute magnitude, and redshift is presented in Tables \[table:r0mag\] and \[table:r0z\]. The estimates of $r_0$ in the narrow redshift bins are consistent with the values in the widest redshift bins. While this is expected, it is not always the case as the largest redshift bin contains object pairs which are not present in the smallest bin. The width of the redshift distribution for the smallest bin strongly depends on the uncertainties of the photometric redshifts as these uncertainties are comparable to the bin width. In contrast, the shape of the redshift distribution of the largest bin depends mostly on the bin width as the uncertainties of the photometric redshifts are smaller than the width of the bin. If the uncertainties are systematically underestimated, Limber’s equation will overestimate the number of close object pairs in the narrowest bin and underestimate the value of $r_0$. If the uncertainties in the redshift distribution were not included, $r_0$ would vary between $3.9\pm 0.4 h^{-1} {\rm Mpc}$ and $5.5 \pm 0.4 h^{-1} {\rm Mpc}$ for the narrowest and widest redshift bins for $-21.0<M_R-5{\rm log}h<-20.0$, $\tau<4.5~{\rm Gyr}$ galaxies. The measured evolution of $r_0$ should be independent of the width of the redshift bins and confirming this is an extremely useful internal consistency check which should always be applied to correlation functions using photometric redshifts.
Clustering as a function of absolute magnitude {#sec:abs}
----------------------------------------------
Figure \[fig:r0mag\] and Table \[table:r0mag\] present estimates of the spatial clustering of red galaxies as a function of evolution corrected absolute magnitude. Absolute magnitude bins containing galaxies brighter than $M_R-5{\rm log}h=-20.5$ include the entire $0.30<z<0.90$ photometric redshift range while fainter bins have truncated redshift ranges which are listed in Table \[table:r0mag\]. While redder galaxies are more strongly clustered than bluer galaxies, the striking correlation is between $r_0$ and absolute magnitude. While there is a mild correlation with luminosity at $M_R-5{\rm log}h\sim -20.0$, the value of $r_0$ rapidly increases from $r_0=6.3 h^{-1} {\rm Mpc}$ at $M_R-5{\rm log}h=-21$ to $r_0=11.2 h^{-1} {\rm Mpc}$ at $M_R-5{\rm log}h=-22$. Similar behavior is seen for at $z<0.15$ in the SSRS2 and 2dFGRS [@wil98; @nor02], and, with lower significance, in CNOC2 at $z<0.4$ [@she01]. [@hog03] observe similar trends in the SDSS at $0.05<z<0.22$ by measuring density of galaxy neighbors within $8 h^{-1} {\rm Mpc}$ spheres as a function of galaxy color and luminosity. [@wil03] measures $r_0=4.02\pm 0.22 h^{-1} {\rm Mpc}$ for $0.2<z<1.0$, $M_B<-18.61$ red galaxies, but her model redshift distribution does not include the uncertainties of her $V-I$ photometric redshifts, so her value of $r_0$ is a lower limit. A summary of previous measurements of red galaxy correlation functions is provided in Table \[table:prev\].
While the $r_0$ values of $M_R-5{\rm log} h>-21.5$ galaxies from the NDWFS and 2dFGRS show only marginal differences, the $r_0$ values of luminous galaxies in the NDWFS are higher than those of the 2dFGRS with $\approx 2\sigma$ significance. Clustering evolution would be expected to produce decreasing $r_0$ values with increasing redshift rather than the opposite trend seen in Figure \[fig:r0mag\]. However, as luminous galaxies are more strongly clustered than $L^*$ galaxies, estimates of their clustering are also more susceptible to cosmic variance. The distribution of galaxies on the plane of the sky, which is plotted in Figure \[fig:dist\], clearly shows that luminous galaxies are in structures comparable in size to the field-of-view. While it is plausible that selection effects could produce the observed structures, stars selected with the same selection criteria do not show similar large-scale structure. In addition, in Figure \[fig:dndz\], the model redshift distribution of the most luminous red galaxies shows evidence of individual structures. We therefore assume that the $2\sigma$ difference between the clustering of luminous galaxies in the NDWFS and 2dFGRS is due cosmic variance. Even if cosmic variance were not an issue, it is difficult to measure clustering evolution with galaxies in the absolute magnitude range where $r_0$ is strongly correlated with luminosity as small luminosity errors can translate into large errors in $r_0$. It is therefore preferable to measure clustering evolution with $M_R-5{\rm log}h>-21.5$ galaxies, as we have done in §\[sec:z\].
As the correlation between $r_0$ and spectral type is relatively weak in the NDWFS and 2dFGRS, our measurements of the evolution of clustering are not sensitive to small errors in the estimates of the spectral types. The strong correlation between $r_0$ and absolute magnitude for luminous galaxies is a prediction of recent large volume $\Lambda$CDM simulations [e.g., @ben01]. As shown in Figure \[fig:r0mag\], a $141^3 h^{-3} {\rm Mpc^3}$ $\Lambda$CDM simulation with galaxy selection criteria similar to the $\tau<4.5 ~{\rm Gyr}$ sample [@ben01 A. Benson 2002, private communication] is a good approximation of the clustering of $M_R\sim -21$ red galaxies.
Evolution of the spatial correlation function {#sec:z}
---------------------------------------------
The evolution of spatial clustering galaxies was studied with $-21.5<M_R-5{\rm log}h<-20.5$ red galaxies. Luminous galaxies were excluded as their spatial clustering is strongly correlated with absolute magnitude and small redshift or spectral evolution errors could produce large changes in the measured spatial clustering. The faint limit was chosen to allow the same range of absolute magnitudes to be studied over a broad redshift range.
As shown in Figure \[fig:r0\] and Table \[table:r0z\], no significant evolution of $r_0$ (comoving) occurs over the redshift range studied. Two models of the clustering of red galaxies, derived from the GIF $\Lambda$CDM simulations [@jen98], are plotted in Figure \[fig:r0\] and provide good approximations to the measured clustering from the NDWFS, SDSS and 2dFGRS. The [@kau99] model predicts the clustering of early-type galaxies with stellar masses of $>3\times 10^{10}M_{\sun}$ while the Benson (2002, private communication) simulation models the clustering of $B-R>1.24$, $-20.5<M_R-5{\rm log}h<-21.5$ galaxies. If the evolution of the underlying dark matter distribution at $z<1$ is well described by the linear or quasi-linear growth of density perturbations, the bias of red galaxies must be rapidly evolving with redshift.
The evolution of $r_0$ was empirically measured by estimating the clustering evolution parameter $\epsilon$ (from Equation \[eq:ep\]). The $0.30<z<0.50$ and $0.70<z<0.90$ bins were assumed to be independent estimates of $r_0$ at the median redshifts of their model redshift distributions. The 2dFGRS and SDSS clustering estimates were also included to provide additional constraints on $\epsilon$. The selection criteria for the NDWFS $\tau<4.5~{\rm Gyr}$ sample allow direct comparison with the 2dFGRS early-type sample of [@nor02]. The present-day colors of the NDWFS $\tau<4.5~{\rm Gyr}$ sample are only $\simeq0.15$ magnitudes redder in $B_W-R$ than the SDSS $u^*-r^*>1.8$ sample of [@zeh02]. The best-fit estimates of $\epsilon$ are summarized in Table \[table:ep\] and plotted in Figure \[fig:epev\]. Models with $\epsilon=0$ are rejected with $>2.5\sigma$ confidence when constraints from the 2dFGRS are included with the NDWFS data. The evolution of $r_0$ is consistent with clustering fixed in comoving coordinates ($\epsilon=\gamma-3$) at $z<0.90$.
Discussion {#sec:dis}
==========
The clustering of red galaxies undergoes little or no evolution from $z\sim 0.9$ to the present-day. At $z>0.9$, the only published clustering study using comparable template selection criteria is that of [@fir02] which used galaxies fitted with the evolving E and Sbc templates of the HYPERZ photometric redshift code [@bol00]. Their estimate of $r_0 = 7.0 \pm 1.9 h^{-1}{\rm Mpc}$, with $\gamma$ assumed to be $1.8$, is consistent with $r_0$ remaining fixed in comoving coordinates to $z\approx 1.5$. However, to differentiate between currently plausible $\Lambda$CDM models larger survey areas, on the order of the complete NDWFS ($\approx 18 \Box^\circ$), will be required.
Further constraints on the clustering of $1<z<2$ galaxies are available from samples of QSOs [@cro01] and $R-K>5$ extremely red objects [EROs; @dad01; @fir02; @roc02]. The QSO spatial correlation function, measured in redshift space ($s_0$ replacing $r_0$), marginally increases from $s_0=5.28\pm^{0.72}_{0.89} h^{-1} {\rm Mpc}$ with $\gamma=1.72\pm^{0.23}_{0.22}$ at $z<0.95$ to $s_0=6.93\pm^{1.32}_{1.64} h^{-1} {\rm Mpc}$ with $\gamma=1.64\pm^{0.29}_{0.27}$ at $2.10<z<2.90$ [@cro01]. However, as pointed out by [@cro01], interpretation of QSO clustering relies on poorly constrained models of QSOs lifetimes and host populations as a function of redshift.
EROs are easier to relate to low redshift populations than QSOs as the majority appear to be the progenitors of early-type galaxies with the remainder being dusty starbursts [@dey99; @lid00; @mor00]. EROs could therefore extend the redshift range of early-type spatial correlation function evolution estimates to $z\approx 1.5$. With the exception of $H<20.5$ EROs in the Las Campanas Infrared Survey [@fir02], all EROs clustering studies measure $r_0\geq10 h^{-1} {\rm Mpc}$ [@dad01; @fir02; @roc02]. This is comparable to the clustering of the most luminous red and early-type galaxies at $z<1$.
If $K>19$ EROs correspond to $\lesssim L^*$ galaxies [@dad01; @roc02], the ERO clustering results are difficult to reconcile with $z<1$ studies and $\Lambda$CDM theory. The clustering of red $L^*$ galaxies in the NDWFS can be increased to $r_0\sim 9 h^{-1} {\rm Mpc}$ only by increasing the photometric redshift $1\sigma$ uncertainties to $\sim 40\%$ of the photometric redshifts or increasing the stellar contamination to $\sim 40\%$. These scenarios are inconsistent with the redshift comparisons in Figure \[fig:photoz\] and the lack of a stellar locus in Figure \[fig:col\]. The discrepancy between the ERO $r_0$ values and $z<1$ studies could be due to errors in the ERO model redshift distributions used to deproject the angular correlation function. [@fir02], who measure $r_0=7.7\pm 2.4 h^{-1} {\rm Mpc}$ for $H<20.5$ EROs, have few $z\gtrsim 1.5$ objects while the [@dad01] and [@roc02] models have a significant fraction of EROs at $z\approx 2$. Improved constraints on ERO clustering should be provided with model redshift distributions constrained with photometric redshifts which have been verified with spectroscopic samples. The clustering of EROs, as measured from the NDWFS, will be described in a future paper (Brown et al., in preparation) using a $B_WRIK$ dataset.
Summary {#sec:sum}
=======
We have used the NOAO Deep Wide-Field Survey to measure the clustering of $0.30 \lesssim z \lesssim 0.90$ red galaxies. The wide-field and $B_WRI$ bands allow large galaxy samples to be selected as a function of spectral type and absolute magnitude using photometric redshifts. PEGASE2 spectral evolution models with exponentially decreasing star formation rates have been used to select an evolutionary sequence of related galaxies as a function of redshift. The red sample, with present-day rest-frame colors of $B_W-R>1.44$, was chosen to allow direct comparison with the low redshift early-type sample from the 2dFGRS. The clustering of red galaxies is strongly correlated with luminosity, with $r_0$ increasing from $4.4\pm 0.4 h^{-1} {\rm Mpc}$ at $M_R-5{\rm log}h\approx -20.0$ to $11.2\pm 1.0 h^{-1} {\rm Mpc}$ at $M_R-5{\rm log}h\approx -22.0$. Clustering evolution measurements with samples where the distribution of spectral types and luminosities are a function of redshift will be dominated by selection effects. The strength of $r_0$ (comoving) as a function of absolute magnitude in our sample is comparable to estimates at $z<0.15$ from the 2dFGRS, with differences at high luminosity being attributable to structures of sizes comparable to the field-of-view. No significant evolution of $r_0$ was detected in comparisons of the NDWFS with the 2dFGRS and SDSS. For $0.30<z<0.90$, $-21.5<M_R-5{\rm log}h<-20.5$, $B_W-R>1.44$ galaxies, the largest sample studied, the value of $r_0$ is $6.3\pm 0.5 h^{-1} {\rm Mpc}$ with $\gamma$ fixed at $1.87$. The strong clustering and lack of detectable evolution appears consistent with recent $\Lambda$CDM models where the bias undergoes rapid evolution and $r_0$ undergoes little evolution at $z<2$.
This research was supported by the National Optical Astronomy Observatory which is operated by the Association of Universities for Research in Astronomy (AURA), Inc. under a cooperative agreement with the National Science Foundation. We thank our colleagues on the NDWFS team and the KPNO and CTIO observing support staff. We thank Frank Valdes, Lindsey Davis and the IRAF team for the MSCRED and astrometry packages used to reduce the mosaic imaging data. S. Croom, S. Dawson, R. Green, S. Malhotra, J. Rhoads, P. Smith, H. Spinrad, D. Stern and S. Warren kindly provided spectroscopic redshifts for galaxies in the Lockman Hole and NDWFS prior to publication. We thank Daniel Eisenstein for productive discussions concerning the uncertainties of correlation functions. Andrew Benson kindly provided estimates of the clustering of red galaxies derived from the GIF $\Lambda$CDM simulation. Valerie Mikles’ research was supported by the NOAO/KPNO REU Program, funded by the National Science Foundation. This research has made use of the NASA/IPAC Extragalactic Database, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.
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[cccccccccccc]{} NDWFS J1426+3531 & 14 26 00.8 & +35 31 32 & 1.2 & 1.6 & 1.3 & 2.1 & 1.7 & 3.0 & 26.8 & 24.8 & 24.7\
NDWFS J1426+3456 & 14 26 01.4 & +34 56 32 & 1.3 & 1.2 & 1.0 & 2.3 & 1.2 & 2.1 & 26.5 & 25.3 & 24.9\
NDWFS J1428+3531 & 14 28 52.8 & +35 31 39 & 1.5 & 1.3 & 0.7 & 2.3 & 1.7 & 2.2 & 26.4 & 25.0 & 25.6\
NDWFS J1428+3456 & 14 28 52.2 & +34 56 39 & 1.6 & 1.3 & 1.3 & 2.3 & 1.7 & 2.6 & 26.2 & 25.0 & 23.6\
[ccccccccc]{} $\tau<4.5 ~{\rm Gyr}$ & 0.30-0.90 & $-22.50<M_R<-21.50$ & $18.18\leq R\leq 22.43$ & 660 & 0.009 & $0.41 \pm 0.07$ & 0.64 & $11.2 \pm 1.0$\
$\tau<4.5 ~{\rm Gyr}$ & 0.30-0.90 & $-22.00<M_R<-21.00$ & $18.30\leq R\leq 23.01$ & 1677 & 0.005 & $0.24 \pm 0.03$ & 0.61 & $8.5 \pm 0.6$\
$\tau<4.5 ~{\rm Gyr}$ & 0.30-0.90 & $-21.50<M_R<-20.50$ & $18.76\leq R\leq 23.25$ & 2651 & 0.023 & $0.14 \pm 0.02$ & 0.60 & $6.3 \pm 0.5$\
$\tau<4.5 ~{\rm Gyr}$ & 0.30-0.75 & $-21.00<M_R<-20.00$ & $19.17\leq R\leq 23.17$ & 2429 & 0.009 & $0.12 \pm 0.02$ & 0.55 & $5.0 \pm 0.4$\
$\tau<4.5 ~{\rm Gyr}$ & 0.30-0.65 & $-20.50<M_R<-19.50$ & $19.72\leq R\leq 23.08$ & 1756 & 0.025 & $0.12 \pm 0.02$ & 0.51 & $4.4 \pm 0.4$\
\
$\tau<2.0 ~{\rm Gyr}$ & 0.30-0.90 & $-22.50<M_R<-21.50$ & $18.18\leq R\leq 22.43$ & 488 & 0.011 & $0.52 \pm 0.10$ & 0.66 & $12.4 \pm 1.2$\
$\tau<2.0 ~{\rm Gyr}$ & 0.30-0.90 & $-22.00<M_R<-21.00$ & $18.30\leq R\leq 23.01$ & 1122 & 0.006 & $0.29 \pm 0.05$ & 0.64 & $9.2 \pm 0.8$\
$\tau<2.0 ~{\rm Gyr}$ & 0.30-0.90 & $-21.50<M_R<-20.50$ & $18.76\leq R\leq 23.25$ & 1592 & 0.037 & $0.20 \pm 0.03$ & 0.62 & $7.5 \pm 0.6$\
$\tau<2.0 ~{\rm Gyr}$ & 0.30-0.75 & $-21.00<M_R<-20.00$ & $19.25\leq R\leq 23.17$ & 1337 & 0.016 & $0.16 \pm 0.04$ & 0.57 & $5.7 \pm 0.7$\
$\tau<2.0 ~{\rm Gyr}$ & 0.30-0.65 & $-20.50<M_R<-19.50$ & $19.87\leq R\leq 23.08$ & 929 & 0.043 & $0.14 \pm 0.04$ & 0.53 & $4.7 \pm 0.7$\
[ccccccccc]{} $\tau<4.5 ~{\rm Gyr}$ & 0.30-0.50 & $-21.50<M_R<-20.50$ & $18.76\leq R\leq 21.25$ & 853 & 0.003 & $0.25 \pm 0.05$ & 0.42 & $5.3 \pm 0.6$\
$\tau<4.5 ~{\rm Gyr}$ & 0.50-0.70 & $-21.50<M_R<-20.50$ & $20.05\leq R\leq 22.50$ & 1023 & 0.002 & $0.24 \pm 0.05$ & 0.60 & $5.5 \pm 0.6$\
$\tau<4.5 ~{\rm Gyr}$ & 0.70-0.90 & $-21.50<M_R<-20.50$ & $20.99\leq R\leq 23.25$ & 775 & 0.071 & $0.20 \pm 0.05$ & 0.85 & $6.7 \pm 0.8$\
$\tau<4.5 ~{\rm Gyr}$ & 0.30-0.70 & $-21.50<M_R<-20.50$ & $18.76\leq R\leq 22.50$ & 1876 & 0.003 & $0.19 \pm 0.03$ & 0.52 & $5.9 \pm 0.5$\
$\tau<4.5 ~{\rm Gyr}$ & 0.50-0.90 & $-21.50<M_R<-20.50$ & $20.05\leq R\leq 23.25$ & 1798 & 0.032 & $0.17 \pm 0.03$ & 0.69 & $6.3 \pm 0.5$\
$\tau<4.5 ~{\rm Gyr}$ & 0.30-0.90 & $-21.50<M_R<-20.50$ & $18.76\leq R\leq 23.25$ & 2651 & 0.023 & $0.14 \pm 0.02$ & 0.60 & $6.3 \pm 0.5$\
\
$\tau<2.0 ~{\rm Gyr}$ & 0.30-0.50 & $-21.50<M_R<-20.50$ & $18.76\leq R\leq 21.25$ & 435 & 0.005 & $0.31 \pm 0.08$ & 0.44 & $5.5 \pm 0.8$\
$\tau<2.0 ~{\rm Gyr}$ & 0.50-0.70 & $-21.50<M_R<-20.50$ & $20.12\leq R\leq 22.50$ & 665 & 0.004 & $0.29 \pm 0.07$ & 0.61 & $5.9 \pm 0.8$\
$\tau<2.0 ~{\rm Gyr}$ & 0.70-0.90 & $-21.50<M_R<-20.50$ & $21.20\leq R\leq 23.25$ & 492 & 0.111 & $0.31 \pm 0.07$ & 0.85 & $8.4 \pm 1.0$\
$\tau<2.0 ~{\rm Gyr}$ & 0.30-0.70 & $-21.50<M_R<-20.50$ & $18.76\leq R\leq 22.50$ & 1100 & 0.004 & $0.20 \pm 0.04$ & 0.55 & $5.8 \pm 0.7$\
$\tau<2.0 ~{\rm Gyr}$ & 0.50-0.90 & $-21.50<M_R<-20.50$ & $20.12\leq R\leq 23.25$ & 1157 & 0.050 & $0.22 \pm 0.04$ & 0.69 & $7.1 \pm 0.7$\
$\tau<2.0 ~{\rm Gyr}$ & 0.30-0.90 & $-21.50<M_R<-20.50$ & $18.76\leq R\leq 23.25$ & 1592 & 0.037 & $0.20 \pm 0.03$ & 0.62 & $7.5 \pm 0.6$\
[ccccccc]{} NDWFS & $0.30<z<0.90 $ & 2651 & $-21.5<M_R<-20.5$ & $\tau<4.5~{\rm Gyr}$ & $6.3\pm 0.5$ & $1.87$\
NDWFS & $0.30<z<0.90 $ & 1592 & $-21.5<M_R<-20.5$ & $\tau<2.0~{\rm Gyr}$ & $7.5\pm 0.6$ & $1.87$\
\
Perseus-Pisces & $z \leq 0.04 $ & 278 & $M_{Z_W}<-19.5$ & Morphology & $8.35\pm 0.75$ & $2.05^{+0.10}_{-0.08}$\
SSRS2 & $z \leq 0.020 $ & 395 & $M_B<-19.4$ & Morphology & $5.10\pm 0.38$ & $1.91\pm 0.26$\
SSRS2 & $z \leq 0.027 $ & 418 & $M_B<-20.0$ & Morphology & $5.27\pm 0.46$ & $1.86\pm 0.29$\
SSRS2 & $z \leq 0.033 $ & 372 & $M_B<-20.5$ & Morphology & $5.73\pm 0.56$ & $2.30\pm 0.46$\
SSRS2 & $z \leq 0.040 $ & 272 & $M_B<-20.9$ & Morphology & $8.60\pm 1.44$ & $2.45\pm 0.71$\
APM & $z \lesssim 0.1 $ & 336 & $-22<M_{B_J}<-15 $ & Morphology & $5.9\pm 0.7$ & $1.85\pm 0.13$\
2dFGRS & $0.016 \leq z \leq 0.071 $ & 1909 & $-18.5<M_{B_J}<-17.5 $ & Red SED ($\eta<-1.4$) & $8.33\pm 1.82$ & $1.87\pm 0.23$\
2dFGRS & $0.020 \leq z \leq 0.086 $ & 3717 & $-19.0<M_{B_J}<-18.0 $ & Red SED ($\eta<-1.4$) & $6.28\pm 1.46$ & $1.98\pm 0.11$\
2dFGRS & $0.031 \leq z \leq 0.125 $ & 10,135 & $-20.0<M_{B_J}<-19.0 $ & Red SED ($\eta<-1.4$) & $5.71\pm 0.57$ & $1.87\pm 0.09$\
2dFGRS & $0.048 \leq z \leq 0.150 $ & 6434 & $-21.0<M_{B_J}<-20.0 $ & Red SED ($\eta<-1.4$) & $6.10\pm 0.72$ & $1.80\pm 0.12$\
2dFGRS & $0.072 \leq z \leq 0.150 $ & 686 & $-22.0<M_{B_J}<-21.0 $ & Red SED ($\eta<-1.4$) & $9.74\pm 1.16$ & $1.95\pm 0.37$\
SDSS & $0.019\leq z\leq 0.13 $ & 19,603 & $14.5<r^*< 17.6$ & Rest-frame $u^*-r^*>1.8$ & $6.78\pm 0.23$ & $1.86\pm 0.03$\
\
PDF & $z \lesssim 0.5 $ & 22,359 & $B_J <22.5$ & Redder than CWW Sbc & $6.0\pm 0.3$ & 1.9\
K20 & $0.5 \lesssim z \lesssim 2.0 $ & 400 & $K <19.2$ & $R-K>5$ & $12\pm 3$ & 1.8\
CNOC2 & $0.120 < z < 0.270 $ & 248 & $R_C<21.5$, $M_R<-20$ & Red CWW templates & $5.35\pm 0.20$ & $2.05\pm 0.08$\
CNOC2 & $0.270 < z < 0.382 $ & 234 & $R_C<21.5$, $M_R<-20$ & Red CWW templates & $6.55\pm 1.16$ & $2.05\pm 0.08$\
CNOC2 & $0.382 < z < 0.510 $ & 238 & $R_C<21.5$, $M_R<-20$ & Red CWW templates & $6.99\pm 0.57$ & $2.10\pm 0.11$\
CNOC2 & $0.12 < z < 0.40 $ & 254 & $-20.00<M_R<-19.25$ & Red CWW templates & $5.82\pm 0.81$ & $1.85\pm 0.08$\
CNOC2 & $0.12 < z < 0.40 $ & 276 & $-20.65<M_R<-20.00$ & Red CWW templates & $5.40\pm 0.19$ & $1.89\pm 0.06$\
CNOC2 & $0.12 < z < 0.40 $ & 278 & $-22.52<M_R<-20.65$ & Red CWW templates & $6.71\pm 0.53$ & $2.14\pm 0.02$\
UH8K & $0.20<z<0.90$ & 3382 & $M_B\lesssim-18.61$ & Redder than CWW Sbc & $4.02\pm 0.22$ & 1.8\
LCIRS & $0.3 \lesssim z \lesssim 0.8 $ & 272 & $H <20.5$ & Evolving E & Sbc & $7.0\pm 1.6$ & 1.8\
LCIRS & $0.8 \lesssim z \lesssim 1.5 $ & 355 & $H <20.5$ & Evolving E & Sbc & $7.0\pm 1.9$ & 1.8\
LCIRS & $0.7 \lesssim z \lesssim 1.5 $ & 337 & $H <20.0$ & $R-H>4$ & $11.1\pm 2.0$ & 1.8\
LCIRS & $0.7 \lesssim z \lesssim 1.5 $ & 312 & $H <20.5$ & $R-H>4$ & $7.7\pm 2.4$ & 1.8\
ELAIS N2 & $0.5 \lesssim z \lesssim 2.0 $ & 166 & $K <21.0$ & $R-K>5$ & 10-13 & 1.8\
[ccccc]{} $\tau<2.0 ~{\rm Gyr}$ & none & $3.1\pm_{1.0}^{1.3}$ & $-4.2\pm_{1.3}^{1.3}$\
$\tau<4.5 ~{\rm Gyr}$ & none & $3.8\pm_{1.0}^{1.3}$ & $-2.9\pm_{1.1}^{1.2}$\
$\tau<4.5 ~{\rm Gyr}$ & SDSS $u^*-r^*>1.8$, $-22.0<M_{r^*}<-19.0$ & $6.9\pm_{0.3}^{0.3}$ & $-0.5\pm_{0.4}^{0.5}$\
$\tau<4.5 ~{\rm Gyr}$ & 2dFGRS $\eta<-1.4$, $-20.0<M_{B_J}<-19.0$ & $5.3\pm_{0.6}^{0.7}$ & $-1.6\pm_{0.6}^{0.6}$\
[^1]: http://www.archive.noao.edu/ndwfs/
[^2]: Throughout this paper $h \equiv \frac{H_0}{100 {\rm km~s}^{-1} ~{\rm Mpc}^{-1}}$, $\Omega_m=0.3$, and $\Lambda=0.7$.
|
---
abstract: 'We consider a spherically symmetric (magnetic) $SU(2)$ Yang-Mills field propagating on the exterior of the extremal Reissner-Nordström black hole. Taking advantage of the conformal symmetry, we reduce the problem to the study of the Yang-Mills equation in a geodesically complete spacetime with two asymptotically flat ends. We prove the existence of infinitely many static solutions (two of which are found in closed form) and determine the spectrum of their linear perturbations and quasinormal modes. Finally, using the hyperboloidal approach to the initial value problem, we describe the process of relaxation to the static endstates of evolution for various initial data.'
address:
- |
Institute of Physics, Jagiellonian University, Kraków, Poland\
and Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Potsdam, Germany
- 'Institute of Physics, Jagiellonian University, Kraków, Poland'
author:
- Piotr Bizoń
- Michał Kahl
title: 'A Yang-Mills field on the extremal Reissner-Nordström black hole'
---
Introduction and setup
======================
The global dynamics of a Yang-Mills field propagating in a four-dimensional Minkowski spacetime is well understood: all solutions starting from smooth initial data at $t=0$ remain smooth for all times [@em] and decay to zero as $t\rightarrow \pm\infty$ [@ch; @bcr]. The global-in-time regularity holds true in any globally hyperbolic four-dimensional curved spacetime [@cs], however the phase portrait can be richer due to existence of nontrivial stationary solutions which play the role unstable attractors, as was shown for the Schwarzschild background [@brz].
In this paper we consider the evolution of a Yang-Mills field on the exterior of the extremal Reissner-Nordström black hole. Our study is motivated by the expectation that an interplay between the conformal structure of the extremal Reissner-Nordström black hole and the conformal invariance of the Yang-Mills equations may lead to an interesting behaviour of solutions. This is a continuation of our studies of how the dissipation-by-dispersion phenomena, responsible for the relaxation to a stationary equilibrium in extended Hamiltonian systems, depend on the geometry of the underlying spacetime [@bk; @bm].
The exterior (i.e., the domain of outer communication) of the extremal Reissner-Nordström black hole is a globally hyperbolic static spacetime $(\mathcal{M},\hat g)$ whose metric, in coordinates $t\in \mathbb{R}, r>M,(\vartheta,\varphi)\in S^2$, reads $$\label{rn}
\hat g=-\left(1-\frac{M}{r}\right)^{2} dt^2+\left(1-\frac{M}{r}\right)^{-2} dr^2 + r^2 (d\vartheta^2+\sin^2{\!\vartheta}\, d\varphi^2)\,,$$ where $M$ is positive constant. The metric $\hat g$ is a spherically symmetric solution of the Einstein-Maxwell equations with mass $M$ and charge $Q=\sqrt{M}$. In order to better see the global properties of this spacetime, it is convenient to use dimensionless variables $(\tau,x)\in \mathbb{R}^2$: $$\tau=\frac{t}{4M}, \quad \mbox{and} \quad x=\ln\left(\frac{r}{M}-1\right)\,,$$ in terms of which the metric takes the form $$\label{rnx}
\hat g=\frac{16 M^2 }{(1+e^{-x})^2} \; g\,,$$ where $$\label{sigma}
g= -d\tau^2 + C^4 \, (dx^2 + d\vartheta^2+\sin^2{\!\vartheta}\, d\varphi^2)\,.$$ Hereafter, for typographical convenience we use the abbreviation $C=C(x)=\cosh{\frac{x}{2}}$. We note that the Ricci scalar vanishes both for $\hat g$ and $g$. In contrast to $(\mathcal{M}, \hat g)$, the spacetime $(\mathcal{M}, g)$ is geodesically complete. It has two asymptotically flat ends at $x=\pm \infty$ (see Fig. 1). Asymptotic flatness is easily seen in terms of the coordinate $\rho=C^2(x)$ for which we have $$\label{RNr}
g= -d\tau^2 + \left(1-\frac{1}{\rho}\right)^{-1} d\rho^2 + \rho^2 (d\vartheta^2+\sin^2{\!\vartheta}\, d\varphi^2)\,.$$
![[Penrose diagram for $(\mathcal{M},g)$.]{}[]{data-label="fig1"}](fig1.pdf){width="35.00000%"}
Note that the reflection $x\mapsto -x$ is the isometry of the metric $g$ but only the conformal isometry for the metric $\hat g$ [@ct]. On $(\mathcal{M},g)$ the reflection interchanges the ’left’ and the ’right’ future null infinities, ${\mathcal J}_L^+$ and ${\mathcal J}_R^+$, while on $(\mathcal{M},\hat g)$ it interchanges the event horizon and ${\mathcal J}_R^+$.
We consider an $SU(2)$ Yang-Mills field propagating in the spacetime $(\mathcal{M},g)$. The gauge potential $A_{\mu}=A_{\mu}^a \tau_a$ takes values in the Lie algebra $su(2)$, where the generators $\tau_a$ satisfy $[\tau_a,\tau_b]=i \epsilon_{abc} \tau_c$. In terms of the Yang-Mills field strength $F_{\mu\nu}=\nabla_{\mu} A_{\nu} - \nabla_{\nu} A_{\mu} +[A_{\mu}, A_{\nu}]$, the lagrangian density reads $$\label{action}
\mathcal{L} = \mathrm{Tr}\left(F_{\alpha\beta} F_{\mu\nu} g^{\alpha\mu} g^{\beta\nu}\right)\, \sqrt{-\text{det}(g_{\mu\nu})}.$$ In four dimensions the quantity $g^{\alpha\mu} g^{\beta\nu}\, \sqrt{-\text{det}(g_{\mu\nu})}$ is invariant under a conformal transformation $g_{\mu\nu} \mapsto\Omega^2 g_{\mu\nu}$, hence if $A_{\mu}$ solves the Yang-Mills equations in one metric, so does it in any conformally related metric. Taking advantage of this conformal invariance of the Yang-Mills equations, in the following we consider only the spacetime $(\mathcal{M},g)$.
For the Yang-Mills potential we assume the spherically symmetric purely magnetic ansatz $$\label{A}
A = W(\tau,x)\, \omega +\tau_3 \cos{\vartheta} d\varphi\,,\quad\mbox{where} \quad \omega=\tau_1 d\vartheta+\tau_2 \sin{\vartheta} \,d\varphi\,,$$ which gives $$\label{F}
F=\partial_{\tau} W d\tau\wedge \omega + \partial_x W dx\wedge\omega -(1-W^2)\, \tau_3 \,d\vartheta \wedge \sin{\vartheta} \,d\varphi\,.$$ Note that the vacuum is given by $W=\pm 1$, while $W=0$ corresponds to the magnetic monopole with unit charge. Inserting the ansatz into we get the reduced lagrangian density $$\label{red_action}
\mathcal{L}=-\frac{1}{2} \left(\partial_{\tau} W\right)^2 C^2+\frac{1}{2} \left(\partial_{x} W\right)^2 C^{-2}+\frac{1}{4} \left(1-W^2\right)^2 C^{-2}\,,$$ and the corresponding Euler-Lagrange equation $$\label{ymx}
\partial_{\tau\tau} W= C^{-2} \partial_x \left(C^{-2} \partial_x W\right) + C^{-4} W (1-W^2)\,.$$ We know from Chruściel and Shatah [@cs] that solutions of Eq. starting at $\tau=0$ from smooth initial data remain smooth for all future times. The goal of this paper is to describe their asymptotic behaviour for $\tau\rightarrow \infty$. For physical reasons our analysis is restricted to solutions with finite (conserved) energy $$\label{energy}
E=\frac{1}{2} \int_{-\infty}^{\infty} \left[C^{2} (\partial_{\tau} W)^2
+ C^{-2}\left( (\partial_x W)^2+\frac{(1-W^2)^2}{2}\right)\right] dx < \infty\,.$$ Due to dissipation of energy by dispersion, such solutions are expected to settle down to critical points of the potential energy, i.e. static solutions of Eq..
Static solutions
================
Time-independent solutions $W=W(x)$ of Eq. satisfy the ordinary differential equation $$\label{ode}
W''-\tanh\left(\frac{x}{2}\right) W'+W (1-W^2)=0\,.$$ We claim that, besides the constant solution $W_0=1$, Eq. has a countable family of smooth finite energy solutions $W_n(x)$ ($n\in \mathbb{N})$ with the following properties (which, not very surprisingly, bear remarkable similarities to the Bartnik-McKinnon solutions of Einstein-Yang-Mills equations [@bmk]):
- $W_n(x)$ has $n$ zeros,
- $|W_n(x)|<1$ for all finite $x$ and $\lim_{|x|\rightarrow\infty} |W_n(x)| = 1$,
- $W_n(x)$ is an even (resp. odd) function for even (resp. odd) $n$,
- As $n\rightarrow\infty$, $W_n(x)$ tend pointwise to $W_{\infty}=0$ for any finite $x$.
The proof of existence of solutions $W_n$ and their properties is a straightforward adaptation of the proof given in [@hm] in the case of harmonic maps between 3-spheres (which satisfy the same equation as with the nonlinearity $\sin(2W)$ instead of $W(1-W^2)$). Key to the proof is the fact that Eq. is asymptotically autonomous with the limiting equations for $x\rightarrow\pm\infty$, $ W''\mp W'+W (1-W^2)=0$, having saddle points at $W=\pm 1$ and a spiral at $W=0$ (stable at $-\infty$ and unstable at $+\infty$). Using a shooting method one can show that there exist infinitely many homoclinic and heteroclinic orbits connecting the saddle points. The solutions are parametrized by the coefficients of the stable directions of the saddle points $$\label{expinfty}
W_n(x)=1-a_n e^{-x}+\mathcal{O}(e^{-3x}).$$ We refer the interested reader to [@hm] for the details. Note that due to the reflection symmetry $W\mapsto -W$, each solution $W_n$ has a copy $-W_n$. We adopt the convention that $W_n(\infty)=1$. The parameters and energies of the first few solutions are given in Table 1 and their profiles are depicted in Fig. 2.
----------------- --------------- ------------------ ------------------- ------------------- ------------------- --------------------
$\,\,$n$\,\,$ $1$ $2$ $3$ $4$ $5$ $6$
$\,\,a_{n}\,\,$ $\,\,2.0\,\,$ $\,\,15.798\,\,$ $\,\,101.108\,\,$ $\,\,624.538\,\,$ $\,\,3835.14\,\,$ $\,\,23528\,\,$
$\,\,E_{n}\,\,$ $\,\,0.8\,\,$ $\,\,0.9664\,\,$ $\,\,0.9945\,\,$ $\,\,0.9991\,\,$ $\,\,0.99985\,\,$ $\,\,0.999976\,\,$
\[1ex\]
----------------- --------------- ------------------ ------------------- ------------------- ------------------- --------------------
: [The parameters of the first few static solutions $W_n(x)$.]{}[]{data-label="tab:static.params"}
0.2cm
![[The first four odd and even static solutions.]{}[]{data-label="fig2"}](fig2a.pdf "fig:"){width="49.00000%"} ![[The first four odd and even static solutions.]{}[]{data-label="fig2"}](fig2b.pdf "fig:"){width="49.00000%"}
Experimenting with Maple, we found the first two nontrivial solutions in closed form $$\label{exact}
W_1(x)=\tanh\left(\frac{x}{2}\right),\qquad W_2(x)=\frac{2\cosh{x}-2-\sqrt{6}}{2\cosh{x}+4+3\sqrt{6}}\,.$$ 0.3cm We turn now to the linear stability analysis of the static solutions $W_n(x)$. This analysis is essential in understanding the role these solutions may play in the evolution. Following the standard procedure, we substitute $W(\tau,x)=W_n(x)+\delta W(\tau,x)$ into Eq., linearize and separate the time dependence $\delta W(\tau,x)=e^{\lambda \tau} v(x)$. This yields the eigenvalue problem[^1] $$\label{pert}
L_n\, v:= -\frac{1}{C^2}\,\frac{d}{dx}\left(\frac{1}{C^2}\,\frac{d v}{dx}\right)+\frac{3 W_n^2(x)-1}{C^4(x)}\,v=-\lambda^2\,v \,.$$ Note that due to the reflection symmetry $x\rightarrow -x$, the eigenfunctions are alternately even and odd. We denote them by $v^{(n)}_k(x)$ and the corresponding eigenvalues by $-(\lambda_k^{(n)})^2$ ($k\in \mathbb{N}$). We claim that the operator $L_n$ has exactly $n$ negative eigenvalues. For $n=0$ this is evident because the potential in is everywhere positive. For $n=\infty$ this follows from the fact that the $\lambda=0$ solution is oscillating at infinity. For any finite $n\geq 1$, one can obtain a lower bound as follows. Consider the function $u_n(x):=W'_n(x)$. Differentiating Eq., one finds that $$\label{U}
\tilde L_n\, u_n=0\,, \quad\mbox{where}\quad \tilde L_n=L_n + \frac{1}{2 C^6(x)}\,.$$ By construction $u_n(x)$ has $(n-1)$ zeros, hence from the Sturm oscillation theorem it follows that the operator $\tilde L_n$ has exactly $(n-1)$ negative eigenvalues. Consequently, the operator $L_n$, which is the exponentially localized negative perturbation of $\tilde L_n$, has a least $(n-1)+1=n$ negative eigenvalues (where ’$+1$’ stands for the zero eigenvalue of $\tilde L_n$ going negative). Numerics (see Table 2) shows that this lower bound is sharp but it seems hard to prove this fact rigorously for general $n$. In the case $n=1$, non-existence of the second negative eigenvalue follows by the Sturm oscillation theorem from an easy to prove fact that the odd solution solution of Eq. with $\lambda=0$ is monotone.
------------------ -------------------- -------------------- -------------------- ---------------------
$\,\,$n$\,\,$ $\lambda^{(n)}_1$ $\lambda^{(n)}_2$ $\lambda^{(n)}_3$ $\lambda^{(n)}_4$
$\,\,$1$\,\,$ $\,\,0.54089 \,\,$ $\,\, \,\,$ $\,\,\,\,$ $\,\,\,\,$
$\,\,$2$\,\,$ $\,\,0.69937 \,\,$ $\,\,0.17161 \,\,$ $\,\,\,\,$ $\,\,\,\,$
$\,\,$3$\,\,$ $\,\,0.72553 \,\,$ $\,\,0.21010 \,\,$ $\,\,0.033792\,\,$ $\,\,\,\,$
$\,\,$4$\,\,$ $\,\,0.72934 \,\,$ $\,\,0.21772 \,\,$ $\,\,0.040983\,\,$ $\,\,0.0057005\,\,$
$\,\,\infty\,\,$ $\,\,0.73015 \,\,$ $\,\,0.21884 \,\,$ $\,\,0.042775\,\,$ $\,\,0.0072103\,\,$
\[1ex\]
------------------ -------------------- -------------------- -------------------- ---------------------
: [Lyapunov exponents of the unstable modes of $W_n$.]{}[]{data-label="tab:unstable"}
Hyperboloidal formulation
=========================
We will use the method of hyperboloidal foliations and Scri-fixing as developed by Zenginoğlu [@anil1] on the basis of concepts introduced by Friedrich [@f]. To implement this method we define a new time coordinate $$\label{s}
s=\tau - \frac{1}{2} \left(\cosh{x}+\ln(2\cosh{x})\right)$$ and foliate the spacetime by hyperboloidal hypersurfaces $\Sigma_s$ of constant $s$. These are spacelike hypersurfaces that approach the ’left’ future null infinity along outgoing null cones of constant advanced time $v=\tau + \frac{1}{2} (\sinh{x}+x)$ and the ’right’ future null infinity along outgoing null cones of constant retarded time $u=\tau - \frac{1}{2} (\sinh{x}+x)$. To see this, note that $$\label{n}
(\partial_{\alpha} s)(\partial_{\beta} s) g^{\alpha\beta} = -\frac{1}{\cosh^2{\!x}} \rightarrow 0 \quad\mbox{as}\quad |x|\rightarrow \infty\,,$$ and $$\begin{aligned}
\label{u}
v_{\vert_{\Sigma_s}}&=&s+\frac{1}{2} (\cosh{x}+\ln(2\cosh{x}))+ \frac{1}{2} (\sinh{x}+x)\rightarrow s \quad \mbox{as} \quad x\rightarrow-\infty\,,\\
u_{\vert_{\Sigma_s}}&=&s+\frac{1}{2} (\cosh{x}+\ln(2\cosh{x}))- \frac{1}{2} (\sinh{x}+x)\rightarrow s \quad \mbox{as} \quad x\rightarrow\infty\,.\end{aligned}$$ There is plenty of freedom in choosing a hyperboloidal foliation; the particular choice is motivated by computational convenience. In terms of the coordinates $(s,x)$, Eq. takes the form[^2] $$\label{eqs}
\frac{C^2}{\cosh^2{\!x}}\partial_{ss} W +2 \tanh{x}\, \partial_{x s} W + \frac{1}{\cosh^{2}{\!x}} \,\partial_s W = \partial_x\left(C^{-2}\partial_x W\right) + C^{-2} W (1-W^2)\,.$$ Next, we compactify the real line $-\infty<x<\infty$ to the interval $[-1,1]$ by the coordinate transformation $z=\tanh{\frac{x}{2}}$. This fixes ${\mathcal J}_L^+$ at $z=-1$ and ${\mathcal J}_R^+$ at $z=1$. Eq. now becomes $$\label{eqz}
\frac{1}{(1+z^2)^2}\, \partial_{ss} W +\frac{2z}{1+z^2} \,\partial_{z s} W + \frac{1-z^2}{(1+z^2)^2} \,\partial_s W=\partial_z\left(\frac{(1-z^2)^2}{4}\, \partial_z W\right) + W (1-W^2)\,.$$ Multiplying this equation by $\partial_s W$ we obtain the local conservation law $$\label{cons}
\partial_s e +\partial_z f=0\,,$$ where $$\begin{aligned}
\label{cons2}
e(s,z)&=& \frac{1}{(1+z^2)^2}\, (\partial_s W)^2 + \frac{(1-z^2)^2}{4}\,(\partial_z W)^2
+\frac{(1-W^2)^2}{2}\,, \\
f(s,z)&=&\frac{(1-z^2)^2}{2}\,\partial_s W \partial_z W-\frac{2z}{1+z^2}\,(\partial_s W)^2\,.\end{aligned}$$ Integrating over a hypersurface $\Sigma_s$ we get the energy balance $$\label{flux}
\frac{d\mathcal{E}}{ds} =-(\partial_s W(s,1))^2-(\partial_s W(s,-1))^2\,,$$ where $$\label{bondi}
\mathcal{E}(s)=\int_{-1}^{1} e(s,z) dz$$ is the Bondi-type energy. The formula expresses the radiative loss of energy through the future null infinities. Since the energy $\mathcal{E}(s)$ is positive and monotone decreasing, it has a nonnegative limit for $s\rightarrow \infty$. For this reason the hyperboloidal formulation is very natural in analyzing relaxation processes that are due to the dispersive dissipation of energy. In the remainder of the paper we describe in detail the convergence to one of the static solutions $W_n(z)$. We focus our attention on the first two static solutions $W_0$ and $W_1$ because, as follows from the linear stability analysis in section 2, only these solutions may participate in the evolution of generic and codimension-one initial data.
We return now to the linear perturbation analysis and compute the quasinormal modes for $W_0$ and $W_1$. Substituting $$W(s,z)=W_n(z)+(1+z^2)^{\lambda/2} e^{\lambda s} u(z)$$ into and linearizing, we obtain the quadratic eigenvalue problem (the purpose of the factor $(1+z^2)^{\lambda/2}$ is to simplify the resulting equation) $$\label{pert2}
(1-z^2)^2 u''+2 z \left(\lambda(z^2-3)+2z^2+2\right) u'+ \left(\lambda^2 (z^2-4)+3\lambda (z^2-1) + 4-12 W_n^2\right) u=0\,.$$ We shall compute the quasinormal modes for the first two static solutions $W_0=1$ and $W_1=z$ using Leaver’s method[^3] [@leaver]. We seek solutions in the form of a power series around the ordinary point $z=0$ $$\label{series}
u(z)=\sum_{j=0} a_j z^j\,.$$ This series converges for $|z|<1$. The discrete values of $\lambda$ for which the function defined by the series is *analytic* at $z=\pm 1$ correspond to the eigenvalues (for real $\lambda>0$) and quasinormal modes (for $\Re(\lambda)<0$). To find those values we need to determine the asymptotics of the coefficients $a_j$ for large $j$. To this end we substitute the series into Eq. for $n=0,1$ and get the three-term recurrence relation $$\begin{aligned}
\label{rec}
& \alpha_0\, a_2 + \beta_0\, a_0=0\,, \nonumber\\
& \alpha_1\, a_3 + \beta_1\, a_1=0\,,\nonumber \\
& \alpha_j\, a_{j+2} +\beta_j\, a_{j} +\gamma_j\, a_{j-2}=0\,, \quad j\geq 2\end{aligned}$$ with $$\begin{aligned}
\alpha_j&=&j^2+3j+2\,,\\
\beta_j&=&-2 j^2 -(6\lambda+2) j -4 \lambda^2 - 3\lambda +4-12 b_0^2\,,\\
\gamma_j&=& j^2+(2\lambda-1) j +\lambda^2-\lambda-2-12b_1^2\,,\end{aligned}$$ where $b_0$ and $b_1$ are the first two coefficients of the Taylor expansion of $W_n(z)$ around $z=0$. For $n=0$ we have $b_0=1, b_1=0$ and for $n=1$ we have $b_0=0,b_1=1$. Even modes satisfy the boundary condition $u(0)=1, u'(0)=0$, hence $a_0=1$ and $a_1=0$, while for odd modes we have $u(0)=0, u'(0)=1$, hence $a_0=0$ and $a_1=1$.
The recurrence relation has two linearly independent asymptotic solutions for $j\rightarrow \infty$ (in the theory of finite difference equations, such solutions are called Birkhoff’s solutions, see Chapter 8.6 in [@pin]) $$\label{birk}
a_j^{\pm} \sim j^{\beta} \exp(\pm 2 \sqrt{\lambda}\sqrt{j})\,,\quad\mbox{where}\quad \beta=\frac{\lambda}{2}-\frac{3}{4}\,,$$ (the leading order behaviour does not depend on $n$), thus asymptotically $$\label{c12}
a_j \sim c_{+}(\lambda) a_j^{+} + c_{-}(\lambda) a_j^{-}\,.$$ The series converges at $z=\pm 1$ if and only if the coefficient of an exponentially growing term in vanishes. The corresponding solution is then called a minimal solution of the recurrence relation and denoted $a_j^{min}$. To find the minimal solution we define $$\label{ab}
A_j=\frac{\beta_j}{\alpha_j},\qquad B_j=\frac{\gamma_j}{\alpha_j},\qquad r_j=\frac{a_{j}}{a_{j-2}}\,,$$ and rewrite in the form $$\label{cf0}
r_j=-\frac{B_j}{A_j+r_{j+2}}\,.$$ Iterating this formula we get the continued fraction representation $$\label{cf}
r_j=-\frac{B_j}{A_j-}\,\frac{B_{j+2}}{A_{j+2}-}\,\frac{B_{j+4}}{A_{j+4}-}\,\dots\,.$$ According to Pincherle’s theorem [@pin], this continued fraction converges if and only if the recurrence relation has a minimal solution $a_j^{min}$ and then $r_j=a_{j}^{min}/a_{j-2}^{min}$ for each $j\geq 2$. Using Pincherle’s theorem we obtain the following quantization conditions $$\label{quant0}
\frac{a_2}{a_0}=-\frac{B_0}{A_0-}\,\frac{B_{2}}{A_{2}-}\,\frac{B_{4}}{A_{4}-}\,\dots\ \quad\mbox{for even modes}\,,$$ $$\label{quant1}
\frac{a_3}{a_1}=-\frac{B_1}{A_1-}\,\frac{B_{3}}{A_{3}-}\,\frac{B_{5}}{A_{5}-}\,\dots\ \quad\mbox{for odd modes}\,.$$ The continued fractions above can be computed to any desired precision by downward recursion starting from some large $j_{max}$ and an arbitrary initial value $r_{j_{max}}$. Finally, the roots of the quantization conditions and are determined numerically (see Table 3).
----------------- ------------------------------------ ------------------------------------ ------------------------------------
$\,\,W_{0}\,\,$ $\,\,-0.33756 \pm 1.34121 i \,\,$ $\,\, -1.04619 \pm 1.21202 i \,\,$ $\,\, -1.85831 \pm 0.99066 i \,\,$
$\,\,W_{1}\,\,$ $\,\, -0.39316 \pm 0.08844 i \,\,$ $\,\, -0.26656 \pm 0.45639 i \,\,$ $\,\, -1.39277 \pm 0.15802 i \,\,$
\[1ex\]
----------------- ------------------------------------ ------------------------------------ ------------------------------------
: [The three least damped quasinormal frequencies for $\!W_0\!$ and $\!W_1\!$ (the corresponding modes are alternately even and odd).]{}[]{data-label="tab:qnm"}
Numerical results
=================
Following [@anil2] and [@bk], we define the auxiliary variables $$\label{aux}
\Psi=\frac{1}{2} \partial_z W\,,\quad\mbox{and}\quad \Pi=\frac{1}{(1+z^2)^2}\, \partial_s W + \frac{z}{1+z^2}\, \partial_z W\,,$$ and rewrite Eq. as the first order symmetric hyperbolic system $$\begin{aligned}
\label{symhyp}
\partial_s W &=& (1+z^2)^2\, \Pi - 2z(1+z^2)\,\Psi\,,\\
\partial_s \Psi &=& \partial_z \left(\frac{1}{2} (1+z^2)^2\,\Pi - z (1+z^2)\,\Psi\right)\,,\\
\partial_s \Pi &=& \partial_z\left(\frac{1}{2} (1+z^2)^2\,\Psi - z (1+z^2)\,\Pi\right)+W(1-W^2)\,.
\end{aligned}$$ We solve this system numericaly using the method of lines with an 8th-order finite difference scheme in space and 4th-order Runge-Kutta integration in time. At the boundaries we use one-sided stencils. Note that there are no ingoing characteristics at the boundaries, hence no boundary conditions need, or can, be imposed.
As expected, for generic intial data the solution tends to one of the vacuum solutions $\pm W_0$. For intermediate times we observe ringdown along the fundamental quasinormal mode, while for later times the polynomial tail is seen to dominate (see Fig. 3).
![[Evolution of time-symmetric initial data $W(0,z)=1+\frac{1}{2}\exp(-2 z^2)$ at a sample point $z=\frac{1}{2}$. The ringdown to $W_0$ for intermediate times is governed by the quasinormal mode with frequency $\lambda \approx -0.337+1.341 i$. For late times $|\partial_s W(s,\frac{1}{2})| \sim s^{-5}$. ]{}[]{data-label="fig3"}](fig3.pdf){width="75.00000%"}
If initial data are close to an unstable static solution, then for early times we observe an exponentially fast departure from this solution along its principal unstable mode. This behaviour, in the case of a small perturbation of $W_{\infty}$, is shown in Fig. 4.
![[Evolution of time-symmetric initial data $W(0,z)=10^{-8}\,\exp(-16 \text{atanh}^2z)$ at a sample point $z=\frac{1}{2}$. For early times we see the exponential growth along the principal unstable mode around $W_{\infty}$ with the Lyapunov exponent $\lambda^{(\infty)}_1\approx 0.73$. Afterwards, the evolution proceeds as in Fig. 3.]{}[]{data-label="fig4"}](fig4.pdf){width="75.00000%"}
Since parity is preserved in evolution, solutions starting from odd initial data cannot tend to $W_0$. Generically, odd solutions converge to $W_1$ (whose single unstable mode is even and therefore not excited). If we add a small even admixture of size $\varepsilon$ to odd initial data, then $W_1$ appears as an intermediate attractor with lifetime $\sim \frac{1}{\lambda^{(1)}_1} \log(\frac{1}{\varepsilon})$, where $\lambda^{(1)}_1\approx 0.54$ is the Lyapunov exponent of the unstable mode of $W_1$ (see Fig. 5).
![[Evolution of time-symmetric initial data $W(0,z)=(0.01\tan(z)-10^{-24})\,\exp(-16 \text{atanh}^2z)$ (black) and $W(0,z)=0.01\tan(z)\,\exp(-16 \text{atanh}^2z)$ (red) at a sample point $z=\frac{1}{2}$. Initially, both solutions evolve together and ring down to $W_1$ along its least damped odd quasinormal mode with frequency $\lambda\approx -0.266+0.456i$. Later, the solution with the small even admixture departs towards $W_0$ along the unstable mode of $W_1$ with the Lyapunov exponent $\lambda^{(1)}_1 \approx 0.54$.]{}[]{data-label="fig5"}](fig5.pdf){width="75.00000%"}
In all cases, the dynamics in the vicinity of static solutions is in excellent quantitative agreement with the results of linear stability analysis displayed in Tables 2 and 3.
Finally, we discuss the late time polynomial tails. In contrast to the ringdown, which is sensitive to the interior structure of the spacetime and depends on the final attractor, the tails for the Yang-Mills field on an asymptotically flat spacetime are universal. They decay as $s^{-4}$ in the interior ($|z|<1$) and as $s^{-2}$ along future null infinities ($z=\pm 1$) [@ch; @bcr]. Moreover, the spatial profiles of tails also appear to be universal. This can be seen by the following heuristic argument. Consider a solution tending asymptotically to $W_0$. Substituting $$\label{profil-ansatz}
W(s,z)=1+s^{-2} f(y), \quad \mbox{where}\quad y=s(1-z),$$ into Eq. and linearizing, we get $$\label{profil-eq}
y(y+1) f'' + (2y-1) f' - 2f =0.$$ Discarding the growing solution, we obtain $$\label{f}
f(y)=\frac{A_R}{(1+y)^2}\,,$$ hence for $0<z\leq 1$ and large $s$ $$\label{prof_R}
W(s,z)-1 \sim \frac{1}{s^2} \, \frac{A_R}{(1+s(1-z))^2}\,,$$ where the amplitude $A_R$ is the only trace of initial data. By an analogous argument, for $-1\leq z<0$ and large $s$ $$\label{prof_L}
W(s,z)-1 \sim \frac{1}{s^2} \, \frac{A_L}{(1+s(1+z))^2}\,.$$ In general, $A_R\neq A_L$. It follows from that for large $s$ $$\label{deriv}
\frac{\partial^n W}{\partial z^n} \bigg|_{z=1} \sim A_R (n+1)!\, s^{n-2},$$ hence higher derivatives grow in time. This result can also be derived by inserting the power series expansion near ${\mathcal J}_R^+$, $$W(s,z)=1+\sum_{n=0}c_n(s) (1- z)^n,$$ into Eq. and solving the resulting system of ordinary differential equations iteratively starting from $c_0(s)=A_R s^{-2}$. The formula reflects the fact that the decay of the Yang-Mills field is not uniform in space (because the decay along ${\mathcal J}^+$ is slower than the decay in the interior) . By the conformal invariance of the problem at hand, the formula holds for the derivatives of Yang-Mills field at the future horizon of the extremal Reissner-Nordström black hole. This type of instability for solutions of the wave equation on the extremal Reissner-Nordström black hole was discovered by Aretakis [@a1; @a2] (see also [@bf] and [@lmrt] for a discussion of the relationship between the behavior of fields near the future (degenerate) horizon and the future null infinity).
The numerical evidence confirming the profile and the growth of higher derivatives along the future null infinity is shown in Figs. 6 and 7.
![[Snaphots of the solution shown in Fig. 3. The late time profiles are seen to converge to the universal profile $f(y)$ (depicted in red) given by formula with $A_R\approx 0.1319$.]{}[]{data-label="fig7"}](fig7.pdf){width="75.00000%"}
![[For the solution shown in Fig. 3 we depict (in the log-log scale) the transversal derivatives at ${\mathcal J}^+_R$. The late time behaviour ($s>50$) agrees with formula for $A_R\approx 0.1319$.]{}[]{data-label="fig6"}](fig6.pdf){width="75.00000%"}
### Acknowledgments. {#acknowledgments. .unnumbered}
We thank Paweł Biernat and Maciej Maliborski for helpful discussions. This work was supported in part by the Polish National Science Centre grant no. DEC-2012/06/A/ST2/00397. P.B. also gratefully acknowledges the support of the Alexander von Humboldt Foundation.
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[^1]: We point out in passing that in terms of the variable $y=\frac{1}{2} (\sinh{x}+x)$, defined by $dy/dx=C^2(x)$, Eq. takes the form of the one-dimensional Schrödinger equation $-\frac{d^2 v}{dy^2}+V_n(x(y)) v(y) =-\lambda^2 v(y)$, however this form is not very helpful because the function $x(y)$ is given only implicitly.
[^2]: We slightly abuse notation and use the same letter for the function $W(\tau,x)$ and any other function obtained from it by changing the variables.
[^3]: An alternative (black box) method is to exploit the fact that Eq. for $n=0,1$ has the form of a double confluent Heun equation and get Maple to do the rest of the job [@bk; @fs], however we prefer a more transparent approach.
|
---
abstract: 'The Linked Data community is focused on integrating Resource Description Framework (RDF) data sets into a single unified representation known as the Web of Data. The Web of Data can be traversed by both man and machine and shows promise as the *de facto* standard for integrating data world wide much like the World Wide Web is the *de facto* standard for integrating documents. On February 27$^\text{th}$ of 2009, an updated Linked Data cloud visualization was made publicly available. This visualization represents the various RDF data sets currently in the Linked Data cloud and their interlinking relationships. For the purposes of this article, this visual representation was manually transformed into a directed graph and analyzed.'
author:
- 'Marko A. Rodriguez'
title: A Graph Analysis of the Linked Data Cloud
---
=1
Introduction
============
The World Wide Web is a distributed document and media repository [@lee94]. Hyper-Text Markup Language (HTML) documents reference other HTML documents and media (e.g. images, audio, etc.) by means of an [href]{} citation. The resulting document citation graph has been the object of scholastic research [@bowtie:huberman1999; @bowtie:broder] as well as a component utilized in web page ranking [@anatom:brin1998]. Similarly, the Semantic Web is a distributed resource identifier repository [@pubsem:lee2001]. The Resource Description Framework (RDF) serves as one of the primary standards of the Semantic Web [@rdfintro:miller1998]. RDF provides the means by which Uniform Resource Identifiers (URI) [@uri:berners2005] are interrelated to form a multi-relational or edge labeled graph. If $U$ is the set of all URIs, $L$ is the set of all literals, and $B$ is the set of all blank (or anonymous) nodes, the the Semantic Web RDF graph is defined as the set of triples $$G \subseteq (U \cup B) \times U \times (U \cup L \cup B).$$ Given that the URI is the foundational standard of both the World Wide Web and the Semantic Web, the Semantic Web serves as an extension to the World Wide Web in that it provides a semantically-rich graph overlay for URIs. Thus, the Semantic Web moves the Web beyond the simplistic [href]{} citation into a rich relational structure that can be utilized for numerous end user applications.
The Linked Data community is actively focused on integrating RDF data sets into a single connected data set [@berners:ldata2006]. The Linked Data model allows
> “\[any man or machine\] to start with one data source and then move through a potentially endless Web of data sources connected by RDF links. Just as the traditional document Web can be crawled by following hypertext links, the Web of Data can be crawled by following RDF links. Working on the crawled data, search engines can provide sophisticated query capabilities, similar to those provided by conventional relational databases. Because the query results themselves are structured data, not just links to HTML pages, they can be immediately processed, thus enabling a new class of applications based on the Web of Data." [@linkeddata:bizer2008]
While the Linked Data community has focused on providing a distributed data structure, they have not focused on providing a distributed process infrastructure [@rodriguez:distributed2008]. Unfortunately, if only a data structure is provided, then processing that data structure will lead to what has occurred with the World Wide Web: a commercial industry focused on downloading, indexing, and providing search capabilities to that data. For the problem space of keyword search, this model suffices. However, the RDF data model is much richer than the World Wide Web citation data model. If data must be downloaded to a remote machine for processing, then only so much of the Web of Data can be processed in a reasonable amount of time. This ultimately limits the sophistication of the algorithms that can be executed on the Web of Data. The RDF data model is rich enough to conveniently support the representation of relational objects [@activerdf:oren2008] and their computational instructions [@rodriguez:gpsemnet2007]. Moreover, with respect to searching, the RDF data model requires a new degree of sophistication in graph analysis algorithms [@semrank:boan2005]. For one, the typical PageRank centrality calculation is nearly meaningless on an edge labeled graph [@grammar:rodriguez2007]. To leave this algorithmic requirement to a small set of search engines will ultimately yield a limited set of algorithms and not a flourishing democracy of collaborative development. As a remedy to this situation, a distributed process infrastructure (analogous in many ways to the Grid [@grid:foster2004]) may be a necessary requirement to ensure the accelerated, grass roots use of the Web of Data, where processes are migrated to the data, not data to the processes. In such a model, computational clock cycles are as open as the data upon which they operate.
With respect to the Web of Data as a distributed RDF data structure, this article presents a graph analysis of the March 2009 Linked Data cloud visualization that was published on February 27, 2009 by Chris Bizer.[^1] The remainder of this article is organized as follows. §\[sec:construction\] articulates how the Linked Data cloud graph was constructed from the February 27$^\text{th}$ Linked Data cloud visualization. §\[sec:statistics\] provides a collection of standard graph statistics for the constructed Linked Data cloud graph. Finally §\[sec:structural\] provides a more in-depth analysis of the structural properties of the graph.
Constructing the Linked Data Cloud Graph\[sec:construction\]
============================================================
The current Linked Data cloud visualization was published by Chris Bizer on February 27, 2009. This visualization is provided in Figure \[fig:lod-cloud\].
![The Linked Data cloud visualization as provided by the Linked Data community. This version is dated February 27, 2009. The author was not responsible for the creation of this visualization. This is only provided in order to better elucidate the means by which the Linked Data cloud graph was created.\[fig:lod-cloud\]](images/lod-cloud){width="45.00000%"}
The Linked Data cloud visualization represents various data sets as vertices (i.e. nodes) and their interlinking relationships as directed unlabeled edges (i.e. links). Moreover, it is assumed that vertex size denotes the number of triples in the data set and edge thickness denotes the extent to which one data set interlinks with another. Data set $A$ links to data set $B$ if data set $B$ has a URI that is maintained (according to namespace) by data set $A$. In this way, by resolving a data set $B$ URI within data set $A$, the man or machine is able to traverse to data set $B$ from $A$.
A manual process was undertaken to turn the Linked Data cloud visualization into a Linked Data cloud graph denoted $G = (V, E)$, where $V$ is the set of vertices (i.e. data sets), $E$ is the set of unlabeled edges (i.e data set links), and $E \subseteq (V \times V)$. The link weights and the node sizes in the original visualization were ignored. A new visualization of the manually generated Linked Data cloud graph is represented in Figure \[fig:lod-graph\]. The properties of this visualization are discussed throughout the remainder of this article.
Standard Graph Statistics\[sec:statistics\]
===========================================
Given the constructed Linked Data cloud graph visualized in Figure \[fig:lod-graph\], it is possible to calculate various graph statistics. A collection of standard graph statistics are provided in Table \[tab:graphstats\].
statistic statistic value
--------------------- -----------------
number of vertices $86$
number of edges $274$
weakly connected true
strongly connected false
diameter $10$
average path length $3.916$
: \[tab:graphstats\] A collection of standard graph statistics for the Linked Data cloud graph represented in Figure \[fig:lod-graph\].
Strongly Connected Components
-----------------------------
The Linked Data graph is not strongly connected. This means that there does not exist a path from every data set to every other data set. Therefore, a walk along the graph can lead to an “island" of data sets that can not be returned from. The number of strongly connected components is $31$ with $26$ of those components only maintaining a single data set (that is, they are either the source of a path or the sink of a path). The size of the remaining strongly connected components is $37$, $15$, $4$, $2$, and $2$. The largest component (with size of $37$) is the “DBpedia component". The second largest (with size of $15$) is the “DBLP RKB Explorer component".
Given the large diameter and average path length, the Linked Data cloud graph can be seen as a two weakly connected components: the larger DBpedia component and the smaller DBLP RKB Explorer component. However, as will be seen later, other communities in the larger DBpedia component exist such as biological and medical communities.
Degree Distributions
--------------------
The in- and out-degree distributions of the graph are plotted in Figure \[fig:in-degree\] and Figure \[fig:out-degree\] on a log-log plot, respectively. These plots show the number (frequency) of data sets that have a particular in- or out-degree. The top 11 in- and out-degree data sets are presented in Table \[tab:in-top\] and Table \[tab:out-top\], respectively. It is interesting to note that the two leaders (DBpedia and DBLP RKB Explorer) are also the leaders of the two largest strongly connected components identified previously.
![The in-degree distribution of the Linked Data cloud graph on a log-log plot.\[fig:in-degree\]](images/in-degree){width="40.00000%"}
![The out-degree distribution of the Linked Data cloud graph on a log-log plot.\[fig:out-degree\]](images/out-degree){width="40.00000%"}
data set in-degree
------------------- -----------
DBpedia $14$
DBLP RKB Explorer $13$
ACM $10$
GeneID $10$
GeoNames $10$
CiteSeer $9$
ePrints $9$
UniProt $9$
ECS Southampton $8$
FOAF Profiles $7$
RAE 2001 $7$
: \[tab:in-top\] The top $11$ Linked Data data sets with the highest in-degree.
data set out-degree
------------------- ------------
DBpedia $17$
DBLP RKB Explorer $14$
ACM $10$
CiteSeer $9$
EPrints $9$
GeneID $8$
UniProt $8$
DrugBank $7$
ECS Southampton $7$
FOAF Profiles $7$
RAE 2001 $7$
: \[tab:out-top\] The top $11$ Linked Data data sets with the highest out-degree.
While the number of data points is small, a power-law fit is provided according to a distribution that is defined as $p(x) \sim x^{-\alpha}$, where $p(x)$ is the probability of seeing a data set with a degree of $x$. A power-law fit to the total degree distribution (i.e. ignoring edge directionality) yields an exponent of $\alpha = 1.496$. In other words, the larger the degree, the fewer number of data sets.
Degree Correlations
-------------------
The correlation between the in- and out-degrees of the vertices yields a Spearman $\rho = 0.6753$ with a significant $p < 9.85^{-13}$. Similarly, the Kendall $\tau = 0.5640$ with a significant $p < 7.27^{-12}$. In other words, data sets that frequently link to other data sets tend to get linked to frequently.
If a graph is degree assortative then vertices with high degree are connected to other vertices with high degree. Likewise, vertices with low degree connect to vertices with low degree. Assortativity is calculated by creating two vectors of length $|E|$. One vector maintains the degree of the vertices at the head of each edge and the other vector maintains the degree of the vertices at the tail of each edge. These two vectors are then correlated. The popular assortative mixing value [@newman:assort] is calculated with a Pearson correlation over the two vectors as $$r = \frac{|E| \sum_{i} j_i k_i - \sum_i j_i \sum_i k_i}{\sqrt{\left[|E| \sum_i j^2_i - \left(\sum_i j_i\right)^2\right]\left[|E| \sum_i k^2_i -\left(\sum_i k_i\right)^2\right]}},$$ where $j_i$ is the degree of the vertex on the tail of edge $i$, and $k_i$ is the degree of the vertex on the head of edge $i$. The correlation coefficient $r$ is in $[-1,1]$, where $-1$ represents a fully disassortative graph, $0$ represents an uncorrelated graph, and $1$ represents a fully assortative graph. Given that the degree distribution is non-parametric, a non-parametric assortativity correlation is also provided using both Spearman $\rho$ and Kendall $\tau$. All of these assortativity correlations are presented in Table \[tab:assort\], where the only significant values are from the standard Pearson correlation and all the in-degree correlations.
method in-degree out-degree total-degree
---------- ------------------ ------------------ ------------------
pearson -0.1911 (0.0015) -0.1728 (0.0042) -0.1868 (0.0019)
spearman -0.1319 (0.0292) -0.0311(0.6089) -0.0629 (0.2998)
kendall -0.0933 (0.0346) -0.0193 (0.6626) -0.0364 (0.3982)
: \[tab:assort\] Various degree assortativity correlations for the Linked Data cloud graph. The first number is the correlation and the second number in parentheses is the $p$-value. A significant $p$-value is less than $0.05$.
These results demonstrate that Linked Data data sets tend to connect to data sets with differing degrees. That is, for instance, high degree data sets connect to low degree data sets. This is made apparent when looking at DBpedia which has a total-degree of $32$. DBpedia’s neighbors in the graph have the following total-degrees: $1$, $1$, $2$, $3$, $3$, $3$, $3$, $3$, $4$, $4$, $4$, $6$, $8$, $9$, $11$, $12$, $12$, and $18$. However, in general, the degree assortativity correlation is weak and for the non-parametric correlations, mostly insignificant.
Structural Analysis\[sec:structural\]
=====================================
This section presents an analysis of the community structures that exist within the Linked Data cloud graph. A community is loosely defined as a set of vertices that have a high number of intra-connections and low number of inter-connections. In other words, vertices in the same community tend to link to vertices in the same community as opposed to vertices in other communities. In order to compare the algorithmically determined structural communities to the metadata properties of the vertices that compose those communities, two metadata properties were gathered:
1. a string label denoting the type of content maintained in the data set
2. an integer value denoting the number of triples contained in the data set.
The content labels were determined manually. The set of labels used were: biology, business, computer science, general, government, images, library, location, media, medicine, movie, music, reference, and social. Note that many data sets could have been labeled with more than one label. However, only one label was chosen. Moreover, these labels were determined by reviewing the websites of the data sets and not by looking at the structure of the graph.
The data set triple counts were taken from the “Linking Open Data on the Semantic Web" web page.[^2] Of the $86$ data sets in the Linked Data cloud, only $31$ of those data sets have published triple counts.
Labeled Structural Communities
------------------------------
The graph analysis method for comparing nominal vertex metadata with structural communities as originally presented in [@onthe:rodriguez2008] was used to compare the content labels of the data sets to their structural communities. The purpose of this analysis is to determine the semantics of the structural communities. The hypothesis is that structural communities denote shared content. That is, data sets in the same structural community maintain the same type of content data (e.g. biology, medicine, computer science, etc.).
A contingency table was created that denotes the number of vertices that have a particular content label and are in a particular structural community. An example contingency table that has community values that were determined using the leading eigenvector community detection algorithm [@newman-eigen] is presented in Table \[tab:conting\].
\[tab:chisquare\]
[ccccccccccc]{}
content/community
& 0 &1 & 2 & 3 & 4 & 5& 6& 7& 8 & 9\
biology & 2 & 0 & 4 & 1 & 0 & 0 & 0& 0 & 3 &10\
business & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\
computer science & 1& 12 & 0 & 0 & 0 & 0 & 0 & 2 & 0& 0\
general & 4 & 3 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\
government & 3 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\
images & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\
library & 2 &0& 0& 0& 0 & 1 & 0 & 1 & 0 & 0\
location & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0\
media & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\
medicine & 0 & 1& 0 & 4 & 0 & 0 & 0 & 0 & 1 & 1\
movie & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0& 0\
music & 5 & 0 & 0& 0& 0 & 1& 1 & 1 & 0 & 0\
reference & 2 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0\
social & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 5 & 0 & 1\
The contingency table is subjected to a $\chi^2$ analysis in order to determine if the manually generated content labels are statistically related to the algorithmically determined structural communities. Four community detection algorithms (and thus, four individual contingency tables) were used for this analysis and the $\chi^2$ $p$-values are presented in Table \[tab:chisquare\].
\[tab:chisquare\]
community algorithm $\chi^2$ $p$-value
--------------------- --------------------
Leading Eigenvector $6.6^{-12}$
WalkTrap $2.2^{-16}$
Edge Betweenness $0.0323$
Spinglass $2.4^{-16}$
: \[tab:chisquare\] The $p$-values for four $\chi^2$ tests using four structural community detection algorithms: leading eigenvector [@newman-eigen], walktrap [@latapy], edge betweenness [@girvan-2002], and spinglass [@spinglass:reichardt2006].
The analysis demonstrates that data sets that maintain similar content tend to exist in the same structural areas of the graph. This is made salient by a qualitative analysis of various subsets of the graph (see Figure \[fig:lod-graph\] where the vertex colors denote their structural community). Moreover, this makes sense intuitively. Data sets that share the same content labels are more than likely to reference to the same resources. For example, it is true that medical data sets tend to be connected to other medical data sets and not to music data sets. Table \[tab:comstruct\] provides a review of 15 randomly chosen Linked Data data sets, their structural community values according to the leading eigenvector community detection algorithm, and their manually determined content labels.
data set community content label
----------------- ----------- ------------------ --
SurgeRadio 0 music
MusicBrainz 0 music
DBpedia 0 general
Riese 5 government
LinkedCT 3 medicine
World Fact Book 5 government
OpenCyc 0 general
Yago 0 general
DrugBank 3 medicine
DailyMed 3 medicine
UniParc 2 biology
Reactome 9 biology
ACM 1 computer science
CiteSeer 1 computer science
IEEE 1 computer science
: \[tab:comstruct\]A sample of 15 Linked Data data sets, their leading eigenvector structual community value, and their manually determined content label.
Data Set Triple Counts
----------------------
Of the $86$ data sets in the Linked Data cloud, only $31$ of those data sets have triple counts that were published on the “Linking Open Data on the Semantic Web" web page. Given the statistically significant, positive correlation between the in-degree and out-degree of the vertices, it is hypothesized that those data sets that are more central in the graph will have a larger triple count. The centrality of all $86$ vertices was determined using the PageRank centrality algorithm with a $\delta = 0.85$ [@page98pagerank]. For those $31$ data sets that have triple counts, their triple count value was correlated with their PageRank centrality value. The Spearman $\rho = 0.6274$ with a significant $p < 0.00016$. Similarly, the Kendall $\tau = 0.4566$ with a significant $p < 0.00039$. Thus, those data sets that have the most RDF triples tend to be centrally located in the Linked Data cloud.
Finally, an assortative mixing calculation over data set triple counts was performed. Given that only $31$ data sets have triple count values, a $31$ vertex subgraph was created. This $31$ vertex graph has $56$ edges. These $56$ edges were used to determine the assortative triple count correlation. Thus, two vectors of length $56$ were created where one vector maintained the triple count of the data sets on the head of each edge and the other vector maintained the triple count of the data sets on the tail of each edge. Table \[tab:assortsize\] provides three assortativity correlations. Note that the triple count data distribution is non-parametric. From these results, only the non-parametric Kendall correlation is statistically significant with a correlation that demonstrates that the data sets are loosely disassortative according. This means that small data sets tend to connect to large data sets and large data sets tend to connect to small data sets. Again, this correlation is relatively weak.
method size assortativity
---------- --------------------
pearson 0.0682 (0.3230)
spearman -0.2546 (0.0559)
kendall -0.2064 (0.0302)
: \[tab:assortsize\] Data set triple count assortativity correlations for the Linked Data cloud graph. Given that only $31$ data sets have published triple counts, these assortativity values are determined according to this $31$ data set subgraph. The first number is the correlation and the second number in parentheses is the $p$-value. A significant $p$-value is less than $0.05$.
Data Set Centrality
-------------------
The PageRank centrality (with $\delta = 0.85$) of each of the $86$ data sets in the Linked Data cloud graph was calculated. Table \[tab:pagerank\] provides the top $15$ central data sets. From this analysis, and assuming that centrality denotes “importance", it appears that content in computer science and biology are of major import to the current instantiation of the Linked Data cloud.
data set page rank content label
------------------- ----------- ------------------
DBLP Berlin 0.0484 computer science
DBLP Hannover 0.0464 computer science
DBpedia 0.0384 general
KEGG 0.0370 biology
UniProt 0.0357 biology
GeneID 0.0346 biology
DBLP RKB Explorer 0.0341 computer science
GeoNames 0.0294 location
ACM 0.0257 computer science
Pfam 0.0254 biology
Prosite 0.0233 biology
ePrints 0.0218 computer science
CiteSeer 0.0218 computer science
PDB 0.0209 biology
: \[tab:pagerank\] The top 15 PageRank central data sets in the Linked Data cloud graph.
Conclusion
==========
The Linked Data initiative is focused on unifying RDF data sets into a single global data set that can be utilized by both man and machine. This initiative is providing a fundamental shift in the way in which data is maintained, exposed, and interrelated. This shift is both technologically and culturally different from the relational database paradigm. For one, the address space of the Web of Data is the URI address space, which is inherently distributed and infinite. Second, the graph data structure is becoming a more accepted, flexible representational medium and as such, may soon displace the linked table data structure of the relational database model. Finally, with respects to culture, the Web of Data maintains publicly available interrelated data. In the relational database world, rarely are database ports made publicly available for harvesting and rarely are relational schemas published for reuse. The Semantic Web, the Linked Data community, and the Web of Data are truly emerging as a radical rethinking of the way in which data is managed and distributed in the modern world.
{width="90.00000%"}
[10]{} \[1\][\#1]{} url@samestyle \[2\][\#2]{} \[2\][[l@\#1=l@\#1\#2]{}]{}
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[^2]: Linking Open Data on the Semantic Web is available at: [http://tinyurl.com/5fcmzm]{}.
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[\
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hep-th/9906106\
AEI-1999-8
[**On Hidden Symmetries in $d=11$ Supergravity\
and Beyond**]{}[^1]\
[H. Nicolai]{}\
[*Max-Planck-Institut für Gravitationsphysik*]{}\
[*Am Mühlenberg, Haus 5*]{}\
[*D-14476 Golm, Germany*]{}\
This talk is about hidden symmetries in eleven dimensions, but it is equally a tribute to a scientist and friend, who is eminently visible in four space-time dimensions: Fran[ç]{}ois Englert, in whose honor this meeting is being held. Therefore, before entering [*dans le vif du sujet*]{} I would like to express my gratitude for having had the opportunity and privilege to learn from him and to work with him, and for all the fun we have had — involving, amongst other things, dinosaurs within dinosaurs [@englert1] (the ancestor of all modern inflationary theories) and their eleven-dimensional avatars [@englert2], as well as higher states of consciousness [@englert3] and monster strings [@englert4].
These days, many of us who have not yet attained the wisdom that comes with an [*émeritat*]{}, but who share Fran[ç]{}ois’ enthusiasm for all of physics’ mysteries, are participants in the hunt for a still elusive theory, called “M Theory”, which is to unify all known consistent string theories and to relate them through a web of non-perturbative dualities [@M; @T]. This theory would also accommodate $d=11$ supergravity [@CJS] as a strong coupling limit via the relations $$R_{11} = \ell_s g_s \qquad \ell_P^3 = \ell_s^3 g_s$$ where $\ell_P$ is the $d=11$ Planck length, $\ell_s$ the string length, $g_s$ the string coupling constant, and $R_{11}$ the radius of the circle on which $d=11$ supergravity is compactified to ten dimensions (the limit is taken in such a way that $\ell_P$ stays finite while $g_s\rightarrow\infty$ and $\ell_s\rightarrow 0$, hence $R_{11}\rightarrow\infty$). It is clear from these relations that present knowledge covers only the “boundary” of M Theory (where either the massive string modes or the $d=11$ Kaluza Klein modes are sent to infinity), but tells us almost nothing about its “bulk” — the true domain of quantum gravity. Still, we can probably anticipate that it will be a pregeometrical theory in the sense that space time as we know it will emerge as a derived concept and that it should possess a huge symmetry group involving new types of Lie algebras (such as hyperbolic Kac Moody algebras) and perhaps even more general structures such as quantum groups.
According to the currently most popular proposal, M Theory “is” the $N\rightarrow\infty$ limit of the maximally supersymmetric quantum mechanical $SU(N)$ matrix model (see e.g. [@dW] for reviews and many references). This model had already appeared in an earlier study of the $d=11$ supermembrane in a flat background in the light cone gauge, and for any finite $N$, it can alternatively be obtained by dimensional reduction of the maximally extended supersymmetric Yang Mills theory in $d=10$ with gauge group $SU(N)$ to one (time) dimension. However, while matrix theory is pregeometrical in the sense that the target space coordinates are replaced by matrices, thus implying a kind of non-commutative geometry, the symmetries of dimensionally reduced supergravities that we are concerned with here, are hard to come by.
In this contribution, I will briefly describe some recent work done in collaboration with S. Melosch [@NM], and with K. Koepsell and H. Samtleben [@KNS2], which was motivated by recent advances in string theory (as well as the possible existence of an Ashtekar-type canonical formulation of $d=11$ supergravity). Although, at first sight, this work may seem to have little to do with the issues raised above, it could actually be relevant in the context of M Theory, assuming (as we do) that further progress will crucially depend on the identification of the underlying symmetries, and that the hidden exceptional symmetries of maximal supergravity theories discovered long ago [@CJ; @Julia1] may provide important clues as to where we should be looking. Support for this strategy derives from the fact that some local symmetries of the dimensionally reduced theories can be “lifted" back to eleven dimensions. More precisely, it was shown in [@dewnic1; @nic1] that there exist new versions of $d=11$ supergravity with local ${ SO(1,3) \times SU(8)}$ and ${ SO(1,2) \times SO(16)}$ tangent space symmetry, respectively. In both versions the supersymmetry variations acquire a polynomial form from which the corresponding formulas for the maximal supergravities in four and three dimensions can be read off directly and without the need for complicated duality redefinitions. This reformulation can thus be regarded as a step towards the complete fusion of the bosonic degrees of freedom of $d=11$ supergravity (i.e. the elfbein and the antisymmetric tensor $A_{MNP}$) in a way which is in harmony with the hidden symmetries of the dimensionally reduced theories.
The existence of alternative versions of $d=11$ supergravity, which, though equivalent on-shell to the original version of [@CJS], differ from it off-shell, strongly suggests the existence of a novel kind of “exceptional geometry” for $d=11$ supergravity, and thus the bigger theory containing it. This new geometry would be intimately tied to the special properties of the exceptional groups, and would be characterized by relations which have no analog in ordinary Riemannian geometry. Much of the ongoing work centers on the role of extended objects (such as 2- and 5-branes in eleven dimensions), which couple to the antisymmetric tensor fields present in $d=11$ and $d=10$ supergravities. Since these antisymmetric tensors are here “dualized away”, our formulation might open new vistas on a unified description of the basic “objects” of M Theory.
We will here concentrate on the ${ SO(1,2) \times SO(16)}$ invariant version of $d=11$ supergravity [@nic1]. To derive it from the original formulation of $d=11$ supergravity, one first breaks the original tangent space symmetry SO(1,10) to its subgroup $SO(1,2)\times SO(8)$ through a partial choice of gauge for the elfbein, and subsequently enlarges it again to $SO(1,2) \times SO(16)$ by introducing new gauge degrees of freedom. The construction thus requires a 3+8 split of the $d=11$ coordinates and indices, implying a similar split for all tensors of the theory. The symmetry enhancement of the transverse (helicity) group $SO(9) \subset SO(1,10)$ to $SO(16)$ requires suitable redefinitions of the bosonic and fermionic fields, or, more succinctly, their combination into tensors w.r.t. the new tangent space symmetry. It is important, however, that the dependence on all eleven coordinates is retained throughout.
In the bosonic sector, the elfbein and the three-index photon are combined into new objects covariant w.r.t. to $d=3$ coordinate reparametrizations and the new tangent space symmetry $SO(1,2)\times SO(16)$ (similar redefinitions must be made for the fermionic fields, but we will not give explicit formulas here for lack of space). In a special Lorentz gauge the elfbein takes the form $$E_M^{~A} = \left(\begin{array}{cc}
\Delta^{-1}e_\mu^{~a} & B_\mu^{~m} e_m^{~a}\\
0& e_m^{~a} \end{array} \right)
\label{11bein}$$ where curved $d=11$ indices are decomposed as $M=(\mu ,m)$ with $\mu =0,1,2$ and $m= 3,...,10$ (with a similar decomposition of the flat indices), and $\Delta := {\rm det} \, e_m^{~a}$. It thus contains the (Weyl rescaled) dreibein and the Kaluza Klein vectors $B_\mu{}^{m}$, all of which are left untouched. By contrast, we will trade the internal achtbein $e_m^{~a}$ for a rectangular 248-bein $e^m_{{\mathcal{A}}}\equiv (e^{m}_{IJ},e^{m}_{A})$ containing the remaining “matter-like” degrees of freedom, where the index ${{\mathcal{A}}}\equiv ([IJ],A)$ labels the 248-dimensional adjoint representation of $E_{8(8)}$ in the $SO(16)$ decomposition ${\bf 248} \rightarrow {\bf 120}\oplus {\bf 128}$. This 248-bein, which in the reduction to three dimensions contains all the propagating bosonic matter degrees of freedom of $d=3,N=16$ supergravity, is defined in a special $SO(16)$ gauge by $$(e^m_{IJ},e^m_A ) := \left\{ \begin{array}{ll}
\Delta^{-1} e_a^{~m} \Gamma^a_{\alpha \dot \beta}
& \mbox{if $[IJ]$ or $A = (\alpha \dot \beta)$}\\
0 & \mbox{otherwise}
\end{array} \right.$$ where the $SO(16)$ indices $IJ$ or $A$ are decomposed w.r.t. the diagonal subgroup $ SO(8)\equiv (SO(8)\times SO(8))_{diag}$ of $SO(16)$ (see [@nic1] for details). Being the inverse densitized internal achtbein contracted with an SO(8) $\Gamma$-matrix, this object is similar to the inverse densitized triad in Ashtekar’s reformulation of Einstein’s theory [@A].
In addition we need composite fields ${{\mathcal{Q}}}_\mu^{{\mathcal{A}}}\equiv (Q_{\mu}^{IJ}, P_{\mu}^{A})$ and ${{\mathcal{Q}}}_m^{{\mathcal{A}}}\equiv(Q_{m}^{IJ}, P_{m}^{A})$, which together make up an $E_{8(8)}$ connection again [*in eleven dimensions*]{}. Their explicit expressions in terms of the $d=11$ coefficients of anholonomity and the four-index field strength $F_{MNPQ}$ are, however, too lengthy to reproduce here [@nic1].
The new geometry is encoded into constraints between the vielbein components, which rely in an essential way on special properties of the exceptional group $E_{8(8)}$. With the $E_{8(8)}$ indices ${\cal A},{\cal B},\dots (=1,\dots,248)$, we have $$({\cal P}_j)_{\cal A B}{}^{\cal C D} e^m_{\cal C} e^n_{\cal D} = 0$$ where ${\cal P}_j$ are the projectors onto the $j = {\bf 1}\, ,\,
{\bf 248}$ and $\bf 3875$ representations of $E_{8(8)}$. (Note that the projectors onto the $j= {\bf 27000}$ and $\bf 30380$ representations do [*not*]{} vanish.) In addition, the 248-bein and the new connection fields are subject to a “vielbein postulate" similar to the usual vielbein postulate, which states the covariant constancy of the 248-bein w.r.t. to an $E_{8(8)}$ covariant derivative involving the $E_{8(8)}$ connection ${{\mathcal{Q}}}_M^{{\mathcal{A}}}$. For instance, for $M=m$ we have $$\partial_m e^n_{{\mathcal{A}}}+ f_{{{\mathcal{A}}}{{\mathcal{B}}}}{}^{{\mathcal{C}}}{{\mathcal{Q}}}_m^{{\mathcal{B}}}e^n_{{\mathcal{C}}}= 0$$ where $f^{{{\mathcal{A}}}{{\mathcal{B}}}{{\mathcal{C}}}}$ are the $E_{8(8)}$ structure constants. (The relations with $M=\mu$ involve the Kaluza Klein vectors $B_\mu{}^m$ and are slightly more complicated). The supersymmetry variations of $d=11$ supergravity can now be re-expressed entirely in terms of these new variables and their fermionic partners [@nic1; @NM].
Despite the “$E_{8(8)}$ covariance” of these relations, it must be stressed, however, that the full theory does not respect $E_{8(8)}$ invariance, as is already obvious from the fact that the fermions do not fit into representations of $E_{8(8)}$. However, the algebraic relations given above can be exploited to show [@KNS2] that there exists an $E_{8(8)}$ matrix ${{\mathcal{V}}}$ [*in eleven dimensions*]{} such that $$e^m_{{\mathcal{A}}}= \frac1{60} {\rm Tr} \Big( Z^m {{\mathcal{V}}}X_{{\mathcal{A}}}{{\mathcal{V}}}^{-1}\Big)$$ where the $X_{{\mathcal{A}}}$ are the generators of $E_{8(8)}$, and the $Z^m$ span an eight-dimensional nilpotent subalgebra of $E_{8(8)}$ (there are altogether $36=8+28$ such nilpotent generators, whose role in relating the various dualized forms of dimensionally reduced supergravity has been explained in [@CJPL]). Because the fundamental and the adjoint representations of $E_{8(8)}$ are the same, we have ${{\mathcal{V}}}X_{{\mathcal{A}}}{{\mathcal{V}}}^{-1} = X_{{\mathcal{B}}}{{\mathcal{V}}}^{{\mathcal{B}}}{}_{{\mathcal{A}}}$ and can thus rewrite this relation in the form $$e^m_{{\mathcal{A}}}= {{\mathcal{V}}}^m{}_{{\mathcal{A}}}$$ This means that the (inverse densitized) achtbein, which itself is part of the elfbein of $d=11$ supergravity, has become part of an $E_{8(8)}$ matrix ${{\mathcal{V}}}$ in eleven dimensions! Furthermore, it then follows from the generalized vielbein postulate stated above that the $M=m$ part of the $E_{8(8)}$ connection ${{\mathcal{Q}}}_M^{{\mathcal{A}}}$ can be simply expressed in terms of this matrix via $${{\mathcal{Q}}}_m = {{\mathcal{V}}}^{-1} \partial_m {{\mathcal{V}}}$$ This simple formula, however, does not work for the low dimensional components ${{\mathcal{Q}}}_\mu^{{\mathcal{A}}}$.
The results obtained so far suggest further extensions incorporating infinite dimensional symmetries. More specifically, the fact that the construction outlined above works with a 4+7 and 3+8 split of the indices suggests that we should be able to construct versions of $d=11$ supergravity with infinite dimensional tangent space symmetries, which would be based on a 2+9 or even a 1+10 split of the indices. This would also be desirable in view of the fact that the new versions are “simple” only in their internal sectors, as put in evidence by the above formula for ${{\mathcal{Q}}}_m^{{\mathcal{A}}}$. The general strategy would thus be to further enlarge the internal sector by absorbing more and more degrees of freedom into it, such that in the final step, only an einbein would be left in the low dimensional sector. However, it is also clear that the elaboration of these ideas will not be an easy task. After all, it took a considerable effort extending over many years to show that the general pattern continues when one descends to $d=2$ and that the hidden symmetries become infinite dimensional, generalizing the Geroch group of general relativity [@Geroch].
There is some reason to believe that a generalization along these lines will take us beyond $d=11$ supergravity. The fundamental object of the theory could then turn out to be an infinite generalization of the vierbein of general relativity, which would be acted upon from one side by a vast extension of the Lorentz group, containing not only space-time, but also internal symmetries, and perhaps even local supersymmetries. For the left action, one would have to appeal to some kind of generalized covariance principle, which would involve the $E_{11-d}$ symmetries.
To put these ideas into perspective, let us recall some facts about dimensionally reduced maximal supergravity to two dimensions. Following the empirical rules of dimensional reduction one is led to predict $E_9 = E_8^{(1)}$ as a symmetry for the dimensional reduction of $d=11$ supergravity to two dimensions [@Julia1; @Julia2]. This expectation is borne out by the existence of a linear system for maximal $N=16$ supergravity in two dimensions [@nic2] (see [@BM] for the bosonic theory, and [@JN1] for a more recent summary). As is usually the case for integrable systems, the linear system requires the introduction of an extra spectral parameter $t$, and the extension of the $\sigma$-model matrix ${{\mathcal{V}}}(x)$ to a matrix ${{\widehat{\mathcal{V}}}}(x;t)$ depending on this extra parameter $t$. An unusual feature is that, due to the presence of gravitational degrees of freedom, this parameter becomes coordinate dependent, i.e. we have $t=t(x;w)$, where $w$ is an integration constant, sometimes referred to as the “constant spectral parameter” whereas $t$ itself is called the “variable spectral parameter”.
The (finite dimensional) coset structure of the higher dimensional theories has a natural continuation in two dimensions, with the only difference that the symmetry groups are infinite dimensional. This property is manifest from the transformation properties of the linear system matrix ${{\widehat{\mathcal{V}}}}$, with a global affine symmetry acting from the left, and a local symmetry corresponding to some “maximal compact” subgroup acting from the right: $${{\widehat{\mathcal{V}}}}(x;t) \longrightarrow g(w) {{\widehat{\mathcal{V}}}}(x;t) h(x;t)$$ Here $g(w)\in E_{9(9)}$ with affine parameter $w$, and the subgroup to which $h(x;t)$ belongs is defined as follows [@Julia2; @BM; @JN1]. Let $\tau$ be the involution characterizing the coset space $E_{8(8)}/SO(16)$: then $h(t)\in{ SO(16)^\infty}$ is defined to consist of all ${\tau^\infty}$ invariant elements of $E_{9(9)}$, where the extended involution ${\tau^\infty}$ is defined by ${\tau^\infty}(h(t)):= \tau h({\varepsilon}t^{-1})$, with ${\varepsilon}=+1$ (or $-1$) for a Lorentzian (Euclidean) worldsheet. Observe that ${ SO(16)^\infty}$ is different from the affine extension of ${ SO(16)}$ for either choice of sign.
Introducing a suitable triangular gauge and taking into account the compensating ${ SO(16)^\infty}$ transformations to re-establish the chosen gauge where necessary, one finds that these symmetries are realized in a non-linear and non-local fashion on the basic physical fields. Moreover, they act as duality transformations in the sense that they mix scalar fields with their duals. At the linear level, a scalar field ${\varphi}$ and its dual $\tilde{\varphi}$ in two dimensions are related by $$\partial_\mu \tilde{\varphi}= \epsilon_{\mu\nu} \partial^\nu {\varphi}$$ If we were just dealing with free fields (as in conformal field theory), there would not be much more to duality than this simple equation, since a second dualization obviously brings us back to the original field (up to an integration constant). The crucial difference here is that, as a consequence of the non-linearity of the field equations, there are [*infinitely many*]{} dual potentials because each dualization now produces a new (i.e. higher order) dual potential. It is basically this non-linearity inherited from Einstein’s equations which explains why the group of duality transformations becomes infinite dimensional in two dimensions. Remarkably, however, already the free field relation above (with ${\varphi}$ replaced by any target space coordinate) is central to modern string duality — for instance implying the emergence of D(irichlet) branes through the interchange of Neumann and Dirichlet boundary conditions for open strings [@Pol]. It is furthermore well known that the integration constant arising in the dualization of a compactified string target space coordinate is associated with string winding modes, and that duality interchanges Kaluza Klein and winding modes. Since we here get infinitely many such integration constants (i.e. one for every dualization), we are led to predict the existence of an infinite tower of novel “winding modes” over and above the ones seen so far seen in string theory. These could be related to the mysterious states found in [@EGKR] that cannot be accounted for by the standard counting arguments.
By representing the “moduli space of solutions” ${{\mathcal{M}}}$ of the bosonic equations of motion of $d=11$ supergravity with nine commuting space-like Killing vectors as $${{\mathcal{M}}}= \frac{{\rm solutions \, of \, field \, equations}} {{\rm diffeomorphisms}}
= \frac{E_{9(9)}}{{ SO(16)^\infty}} \label{coset1}$$ one has managed to endow this space, which a priori is very complicated, with a group theoretic structure that makes it much easier to handle. In particular, the integrability of the system is directly linked to the fact that ${{\mathcal{M}}}$ possesses an infinite dimensional “isometry group” $E_{9(9)}$. The introduction of infinitely many gauge degrees of freedom embodied in the subgroup ${ SO(16)^\infty}$ linearizes and localizes the action of this isometry group on the space of solutions. Of course, in making such statements, one should keep in mind that a mathematically rigorous construction of such spaces is a thorny problem. We can ignore these subleties here, not least because these spaces ultimately will have to be “quantized” anyway.
Elevating the local symmetries of maximal supergravity in two dimensions to eleven dimensions would thus require the existence of yet another extension of the theory, for which the Lorentz group $SO(1,10)$ is replaced by ${ SO(1,1) \times SO(16)^\infty}$ (the subgroup ${ SO(16)^\infty}$ can be interpreted as an extension of the transverse group $SO(9)$ in eleven dimensions). Accordingly, we would now decompose the elfbein into a zweibein and nine Kaluza Klein vectors $B_\mu^{~m}$ (with $m=2,...,10$). The remaining internal neunbein would have to be replaced by an “Unendlichbein” (or “$\infty$-bein”, for short) $e^m_{{{\mathcal{A}}}}(x;t)$. The parameter $t$ is necessary in order to parametrize the infinite dimensional extension of the symmetry group; whether it would still be a “spectral parameter” in the conventional sense of the word for the “lifted” theory, remains to be seen. One important difference with the dimensionally reduced theory is, however, clear: in eleven dimensions, there is no anolog of the dualization mechanism, which would ensure that despite the existence of infinitely many dual potentials, there are only finitely many physical degrees of freedom. This means that the construction will almost certainly take us beyond $d=11$ supergravity.
Some information can be deduced from the requirement that in the dimensional reduction to $d=2$, there should exist a formula relating $e^m_{{\mathcal{A}}}(x;t)$ to the linear system matrix ${{\widehat{\mathcal{V}}}}(x;t)$, analogous to the one relating $e^m_{{\mathcal{A}}}(x)$ to the $E_{8(8)}$ matrix ${{\mathcal{V}}}(x)$ before. For this purpose, we would need a ninth nilpotent generator to complement the $Z^m$’s; an obvious candidate is the central charge generator $c$, since it obeys $\langle c|c \rangle =
\langle c| Z^m \rangle = 0$ for all $m=3,...,10$. The parameter $t$, introduced somewhat ad hoc for the parametrization of the $\infty$-bein, must coincide in the dimensional reduction with the spectral parameter of the $d=2$ theory. Furthermore, the generalized “$\infty$-bein postulate” should reduce to the linear system of $d=2$ supergravity in this reduction.
One difference with the previous situation, where the tangent space symmetry was still finite, is that the Lie algebra of ${ SO(16)^\infty}$ also involves the non-compact $E_{8(8)}$ generators, but in such a way that the generalized Cartan Killing form on $E_{9(9)}$ is still positive on all these generators. This follows from consideration of the $t$ dependence of the linear system of the dimensionally reduced theory and shows that the new connections would constitute an ${ SO(16)^\infty}$ rather than an $E_{9(9)}$ gauge connection. This means that the covariantizations in the generalized vielbein postulate would be in precise correpondence with the local symmetries, in contrast with the previous relations which looked $E_{8(8)}$ covariant, whereas the full theory was actually invariant only under ${ SO(16)}$. Another curious feature is the following: in two dimensions, the linear system matrix contains all degrees of freedom, including the fermionic ones, and the local $N=16$ supersymmetry can be bosonized into a local ${ SO(16)^\infty}$ gauge transformation [@NW]. This could mean that there is a bosonization of fermions in the sense that $e^m_{{\mathcal{A}}}(x;t)$ would describe bosonic and fermionic degress of freedom.
What has been said here could be summarized as follows: in searching for a possible candidate M Theory, one should not only concentrate on dimensionally reduced maximally extended [*rigidly*]{} supersymmetric theories (= supersymmetric Yang Mills theories), but also consider the dimensionally reduced maximally extended [*locally*]{} supersymmetric theory. The idea (already proposed in [@nic2]) is that a third quantized version of maximal supergravity in two dimensions would give rise via a kind of bootstrap to a theory beyond $d=11$ supergravity that would contain the latter in the same way as superstring theories contain $d=10$ supergravity and $d=10$ super-Yang-Mills theories as special limits. However, it is not clear how (and if) this idea fits with presently accepted points of view.
[**Acknowledgements**]{}: I am very grateful to the organizers for inviting me to this splendid event. I would also like to thank K. Koepsell, S. Melosch and H.A.J. Samtleben for the enjoyable collaboration on which the new and as yet unpublished results reported here are based, and B. de Wit for discussions.
[99]{}
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[^1]: Invited talk at the Conference “Fundamental Interactions: From Symmetries to Black Holes”, 24 - 27 March, Université Libre de Bruxelles, Belgium.
|
****
Decays of degeneracies in ${\cal PT}-$symmetric ring-shaped lattices
[**Miloslav Znojil**]{}
Nuclear Physics Institute ASCR,
250 68 Řež, Czech Republic
[e-mail: [email protected]]{}
Abstract {#abstract .unnumbered}
========
Many apparently non-Hermitian ring-shaped discrete lattices share the appeal with their more popular linear predecessors. Irrespectively of the change of the topology, their dynamics controlled by the nearest-neighbor interaction is shown to remain phenomenologically interesting. For the perturbation-caused removals of spectral degeneracy at exceptional points, in particular, alternative scenarios are illustrated via solvable examples.
Introduction
============
The concept of ${\cal PT}-$symmetry [@Carl] is an inspiring source of experimental activities in several branches of physics like optics [@Makris] or solid-state physics [@Joglekar; @Joglekarbe] or magnetohydrodynamics [@Eva]. The idea itself originates in quantum mechanics. In a way illustrated by numerous toy models [@BB; @Geza] the interest in ${\cal
PT}-$symmetry has been motivated there by a counterintuitive contrast between the manifestly non-Hermitian form of a given ${\cal
PT}-$symmetric interaction Hamiltonian $H$ and the reality of the spectrum of the related energies inside a certain domain ${\cal D}$ of optional parameters.
The mathematical key to the puzzling existence of a non-empty domain ${\cal D}$ of parameters giving real spectra has been found in Ref. [@Geyer]. The authors emphasized there the compatibility of the non-Hermiticity of $H $ in a given Hilbert space (say, in ${\cal
H}^{(friendly)}:=L^2(\mathbb{R})$ where $H \neq H^\dagger$) with the [*simultaneous*]{} Hermiticity of the same operator in another Hilbert space. In particular, the new space ${\cal
H}^{(sophisticated)}$ may be allowed to coincide with ${\cal
H}^{(friendly)}$ as a vector space, being just endowed with another, nontrivial, [*ad hoc*]{} inner product.
The main phenomenological appeal of such a situation lies, paradoxically, in the possibility of a [*loss*]{} of the reality of the spectrum. This happens, naturally, at the boundary $\partial{\cal D}$, i.e., at a certain value of parameter(s) called, according to Kato [@Kato], exceptional point(s) (EP). They specify, in effect, the strength of a critical non-Hermiticity at which the [*ad hoc*]{} inner product ceases to exist (cf. [@ali] and [@SIGMA])
One of the weak points of the recipe (which the authors of Ref. [@Geyer] could have called “quasi-Hermitian quantum mechanics", QHQM) lies in the ambiguity of the definition of the [*ad hoc*]{} inner product mediated by the so called metric operator $\Theta=\Theta^\dagger>0$, |\^[(S)]{}:=||\^[(F)]{}. A way has been found in the so called ${\cal PT}-$symmetric quantum mechanics (PTSQM) as reviewed in Ref. [@Carl]. The essence of PTSQM lies in the introduction of an additional postulate $
{\cal PT} H = H {\cal PT} $ (called ${\cal PT}-$symmetry of the Hamiltonian for a suitable operator $
{\cal PT}$) and in its combination with another, [*ad hoc*]{} requirement of factorization $\Theta^{(PT)}={\cal PC}$ where operator ${\cal C}$ is a charge (cf. Ref. [@Carl] for more details). The nontriviality of the [*additional*]{} requirement of ${\cal PT}-$symmetry proved more than compensated by its heuristic efficiency, with ${\cal P}$ chosen, most often, as the operator of parity.
A disappointing failure of the extension of the PTSQM formalism to the dynamical regime of scattering [@Jones] has been discussed in Ref. [@fund]. The obstacle has been found removable via a return to the QHQM recipe, with the emphasis shifted from the charge in $\Theta^{(PT)}$ to an alternative treatment of the ambiguity of the metric.
The perceivable increase of the technical difficulty of the required explicit construction of the metrics $\Theta\neq \Theta^{(PT)}$ has been softened by the discretization of the coordinates, $x \to x_n$ (cf. also [@smear]). The common differential toy-model Hamiltonians $H=-\triangle+V(\vec{x})$ were replaced by their difference-operator analogues. Typically, the role of the kinetic energy $-\triangle$ was taken by a tridiagonal matrix (i.e., say, by the well known Runge-Kutta approximation of the Laplacean). Simultaneously, the diagonal potential-energy matrix $V(x_n)$ was generalized into a tridiagonal matrix which was not necessarily Hermitian in ${\cal H}^{(friendly)}$.
The resulting $N$ by $N$ Hamiltonians of the generic form \^[(N)]{}= \[kitielor\] may be interpreted as representing an open-end $N$-site nearest-neighbor-interaction quantum lattice [@Joglekar].
In what follows we intend to complement the existing studies of various open-end versions of the $N$-site quantum lattice (\[kitielor\]) (cf., e.g., Refs. [@chain] or [@laguerre], with further references) by the next-step study of its generalization \^[(N)]{}= , N = 2J. \[kitiel\] This Hamiltonian matrix may be read as representing a circular, ring-shaped discrete lattice (cf. its graphical representation in Figs. \[fi3\] and \[sefi\]).
The motivation of such a project is twofold. Firstly, the introduction of the “anomalous" matrix elements $c_N$ enables us to study the spectral consequences of the tunable input interaction of a long-range character. Section \[pertur\] will pay particular attention to the effects of this long-range perturbation on a [*maximally degenerate*]{} EP singularity at $N=6$. We shall sample alternative scenarios of the removal of this degeneracy.
Secondly, we feel motivated by the simplicity-preserving character of the transition from Eq. (\[kitielor\]) to Eq. (\[kitiel\]). In section \[nepertur\] a few exactly solvable benchmark models will be presented, demonstrating an enrichment of the variability of the spectrum in non-perturbative regime. Our models will exhibit [*multiple complexifications*]{} of the energies at the EP boundary $\partial{\cal D}$.
Our observations and proposals will be finally summarized in section \[VIs\].
The decays of a multiple degeneracy \[pertur\]
==============================================
In Ref. [@chain] we proposed a family of multi-parametric $N-$dimensional matrices (\[kitielor\]) for which one is able to construct certain parts of the boundary $\partial{\cal D}$ in closed form, non-numerically. In an $N=6$ illustration of such a toy-model scenario let us recall the multiple-degeneracy-generating Hamiltonian matrix $H_{(MDG)}=$ =. \[sixsit\] The corresponding spectrum of energies stays unobservable (complex) at the negative “times" $t<0$ while becoming, suddenly, completely degenerate at $t=0$ and strictly real and fully non-degenerate at all the positive $t>0$. This fine-tuned multiple-degeneracy behavior of the spectrum (sampled here in Fig. \[fiebb\]) may immediately be extended to any dimension $N$ [@chaindva].
We shall check what may happen when the linear chain of Eq. (\[kitielor\]) is being replaced by the ring-shaped chain represented by Eq. (\[kitiel\]).
${\cal PT}-$symmetry in lattices \[Vs\]
---------------------------------------
It has recently been hinted in the literature that the open-end non-Hermitian quantum lattices are exceptional “since periodic boundary conditions are incompatible with the ${\cal PT}-$symmetry" [@Joglekar]. For several reasons, such a comment attracted our attention. First of all, we felt puzzled by the potential physical consequences of such a statement since it contradicted our older empirical experience with the existence of strong parallels between the open- and coupled-end quantum lattices [@aindva].
During our subsequent analysis of the problem we imagined that its very core is entirely artificial. The source of the misunderstanding may simply be traced back to certain ambiguity of the current terminology. In an introductory remark let us, therefore, recall a few relevant definitions while restricting our present attention, for the sake of simplicity, just to Hamiltonians (\[kitielor\]) or (\[kitiel\]) with [*real*]{} matrix elements.
In the first step let us slightly simplify the complex-Hamiltonian conventions as accepted in Ref. [@Joglekar]. Thus, our present interpretation of the ${\cal PT}-$symmetry property $H{\cal
PT}={\cal PT}H$ of the real Hamiltonians will employ the time-reversal ${\cal T}$ represented just by the operator of matrix transposition. Secondly, the specific choice of the parity operator = \[parita\] will be assumed fixed in advance. Thirdly, let us clearly distinguish between the concepts of the so called unbroken and broken ${\cal PT}-$symmetry where, by definition, the whole spectrum of $H$ is real or not real, respectively.
In a more explicit concise explanation involving, for the sake of brevity, just the non-Hermitian ${\cal PT}-$symmetric Hamiltonians (i.e., matrices $H\neq H^\dagger$ such that $H^\dagger{\cal P} =
{\cal P}H$) with non-degenerate (though, in general, complex) spectra, one must distinguish between the right-eigenvector Schrödinger equation $H|R_n{\rangle}= E_n|R_n{\rangle}$ and the left-eigenvector Schrödinger equation ${\langle}L_n|H = {\langle}L_n|\,E_n$ (or, equivalently, $K\, |L_n{\rangle}= F_n |L_n{\rangle}$ where $K
\equiv H^\dagger \neq H$ and $F_n\equiv E_n^*$).
In this notation one easily separates the real-energy case “A" (in which $F_n=E_n$) and the complex-energy case “B" (in which $F_n=E_n^* = E_m$ at a subscript $m =m(n)\neq n$). As long as we can always write ${\cal P}H|R_n{\rangle}= H^\dagger \left ({\cal
P}|R_n{\rangle}\right ) = E_n \left ({\cal P}|R_n{\rangle}\right )$ in both of these cases, it is easy to conclude that |R\_n= const |L\_n [iff]{} E\_n=E\_n\^\*. \[symwf\] In the major part of the current literature on ${\cal PT}-$symmetric Hamiltonians, the validity of proportionality (\[symwf\]) between the left and right eigenvectors [*at all*]{} $n=1,2,\ldots,N$ is, conveniently, called the unbroken ${\cal PT}-$symmetry of the quantum system in question. Thus, one must be a bit careful when reading Ref. [@Joglekar] where the models with unbroken ${\cal
PT}-$symmetry are called “models in ${\cal PT}-$symmetric phase".
This being explained, we believe that there is no true reason for taking the circular lattices (i.e., in our case, systems with Hamiltonians (\[kitiel\]) mimicking the periodic boundary conditions and representing the circular lattices of the shape sampled in Fig. \[sefi\]) as a perceivably more complicated option. We might even conjecture that as long as the circular shape of the lattice may be perceived as an elementary exemplification of a topologically nontrivial quantum graph of a non-tree shape, the presence of the end-point bonds might be interpreted, in the spirit of Ref. [@anomal], as a hidden source of potentially interesting anomalies in the spectrum.
We shall restrict our attention to the even$-N$ subset of models (\[kitiel\]). The main reason is that under this restriction our Hamiltonians will exhibit more parallels with their differential-equation predecessors. In particular, we shall always employ just the manifestly coupling-independent operator ${\cal P}$ of Eq. (\[parita\]) which strongly resembles the standard parity with its equal number of positive and negative eigenvalues.
A destabilization via a coupling between ends\[prece\]
------------------------------------------------------
For a sampling of the effects of the periodicity-simulating lattice-ends couplings $\pm c_N$ let us return to the six-site open-end-lattice spectrum of Fig. \[fiebb\] and let us treat the bonding matrix elements $\pm c_N$ as a small perturbation to this unperturbed form of the ${\cal PT}-$symmetric Hamiltonian.
Let us consider the first sample of such a perturbed Hamiltonian $H^{(6)}_1(t)=H_{(MDG)}+W_1$, with the perturbation specified as follows, W\_1= , w =. The resulting modified $t-$dependence of the spectrum is displayed in Fig. \[fiedeeb\]. We see that the original collapse of the whole spectrum gets split. A twin partial collapse is shifted to the right (i.e., to $t\to 0. 159^+$) while the ultimate complete complexification moved to the left (i.e., to negative $t\to-0.2818^+$).
Competing agents of destabilization
-----------------------------------
From the observations made in paragraph \[prece\] one can conclude that both the maximal degeneracy (at $t=0$) and the Big-Bang-like behavior (at the small $t>0$) of the spectrum as sampled by Fig. \[fiebb\] are in fact very sensitive to perturbations. The form of this perturbation is highly relevant for the guarantee of the stability of the perturbed quantum system, i.e., of the reality of its bound-state energies.
The determination of the boundary $\partial{\cal D}$ of the domain ${\cal D}$ of the parameters which are compatible with the stability is important. Certain qualitative features of this boundary (or “horizon") may even be guessed [*a priori*]{}. Typically, one may expect that the system gets less stable, i.e., the size of the domain ${\cal D}$ will shrink in the strong-coupling regime. In parallel, the destabilizing effect of the growth of the end-bond $c_6=a$ (cf. Fig. \[sefi\]) may be compensated by the decrease (or, alternatively, enhanced by the further increase) of some other non-Hermiticity measure (i.e., for the sake of definiteness, of $c_3=d$ in Fig. \[sefi\]).
The consequences are both interesting and hardly predictable. Figures \[fiedeece\] and \[fiedeeef\] offer an illustration of the effect. These pictures illustrate a scenario of destabilization in which, in spite of the perceivable growth of $a$ (i.e., in spite of our making the ring better bound), the dominant role is still played by the smaller perturbation of the close-to-critical $d$.
This guess may be assisted by the toy-model Hamiltonian $H^{(6)}_2(t)=H_{(MDG)}+W_2$ with W\_2= , w =. \[fifi\] The mechanism of the dominance of the enhancement of the central coupling is demonstrated by Figs. \[fiedeece\] and \[fiedeeef\]. We see there the remarkable pattern of complexification in which the rightmost exceptional point $t_{+} \approx 0.3$ of the loss of the crypto-Hermiticity (i.e., of the reality of the whole spectrum) is determined by $d$ while the next coordinate $t_{-} \approx 0.1$ of the remaining two mergers and subsequent complexifications already reflects the combined effect of the whole perturbation.
A qualitative novelty may be seen in the re-emergence of a small island of the reality of as many as four central energies in a very small vicinity of $t=0$. This phenomenon is a close analogue of the similar spectral “reality-island" anomaly encountered, in Ref. [@anomal], in the strong-coupling regime of another topologically nontrivial model.
Decays of separate degeneracies \[nepertur\]
============================================
The explicit evaluation of the periodic-lattice spectra remains a more or less purely numerical problem in the tight-binding regime, especially at the larger $N$. One still encounters exactly solvable secular equations at the smallest even $N=2J$. [*In extremis*]{}, many generic features of the periodic-lattice spectra may be understood even via their first nontrivial four-site-lattice realization.
An exactly solvable model
-------------------------
Once we pick up the following one-parametric four-site toy model with equal couplings, H\_[(EC)]{}= \[away\] we reveal that the whole spectrum remains real up to the strong-coupling dynamical regime, i.e., even beyond $|t|=1$ (cf. the graphical representation of this spectrum in Fig. \[fiedeero\]). For our present purposes it is also useful that the spectrum is in fact available in closed form, E\_0=- , E\_1=-1, E\_2=1, E\_3=. \[twoe\] One localizes, precisely, the unavoided-crossing points $t=t_{\pm}^{(UC)} = \pm \sqrt{2}$ as well as the points of the ultimate complexification $t=t_{\pm}^{(C)} = \pm {3/2}$. All of these four values are the exceptional points in the Kato’s sense [@Kato]. In order to see this, one may recall Eq. (\[twoe\]) and pick up, say, eigenvalues $E_2$ and $E_3$ of our Hamiltonian (\[away\]). Then, the two respective eigenvectors, viz., the four-component vector $\vec{\psi}_2=[0, t, 2, t]$ and its partner $\vec{\psi}_3$ with components , \_4= will strictly coincide in either of the limits of $t \to
t_{\pm}^{(UC)}$. The Hamiltonian will only remain diagonalizable (i.e., crypto-Hermitian) inside the three separate intervals of $t$, viz, inside domain ${\cal D}^{(H)}_0= (-\sqrt{2},\sqrt{2})$ (combining the weak- and strong-coupling dynamical regimes) or inside ${\cal D}^{(H)}_-= (-3/2,-\sqrt{2})$ or ${\cal D}^{(H)}_+=
(\sqrt{2},3/2)$ (= the two perceivably smaller domains of anomalously strong couplings).
Inside the non-anomalous domain ${\cal D} = (-\sqrt{2},\sqrt{2})$ of $t$ our choice of the first nontrivial $N=4$ enables us to find a [*complete*]{} family of all of the candidates for the metric in principle. A sample of the necessary linear algebra reconstructing, basically, the metric from the relation H\^= H\^[-1]{} \[dieu\] may be found, say, in Ref. [@fund]. An entirely exhaustive explicit construction of the metrics $\Theta$ has been performed there for a certain extremely elementary one-parametric model. Here, we shall only pick up a single, characteristic solution = \[metra2\] as a candidate for the metric in ${\cal H}^{(sophisticated)}$. In order to confirm its eligibility we must prove that it is positive definite. Such a proof is easy since the two eigenvalues of our candidate matrix happen to possess the elementary form $$3+{t}^{2}+t \pm \sqrt {13\,{t}^{2}+{t}^{4}+6\,{t}^{3}}\,.$$ The other two eigenvalues are obtained by the replacement $t \to
-t$. This implies the positivity of our matrix (\[metra2\]) inside the interval of t (-,):=[D]{}\_(-1.225,1.225). We see that our choice of special metric is satisfactory since this interval covers more than 86 % of the whole crypto-Hermiticity domain of the Hamiltonian itself.
The strengthened bond
----------------------
Let us now replace the fine-tuned one-parametric four-site model (\[away\]) by its perturbation H= H\_[(EC)]{}+ . \[awayab\] Figure \[fiedeese\] illustrates the changes. The decay of the two original point-like unavoided crossings at $t_{\pm}^{(UC)}$ spreads over the respective two small non-empty intervals of $t$ in which all of the four eigenenergies become complex, acquiring a non-vanishing imaginary part.
A more detailed inspection of Fig. \[fiedeese\] reveals that the central domain shrinks a bit in comparison, ${\cal D}^{(H)}_0
\approx (-1.13137,1.13137)$. The ultimate onset of the large$-t$ survival of the two real energies moves from the points $t_{\pm}^{(C)}=3/2$ slightly closer to the origin.
Figure \[fiedeese\] seems to indicate the complete disappearance of the separate anomalous domains ${\cal D}^{(H)}_\pm$. The magnified version of the same picture (cf. Fig. \[fiedeesese\]) reveals that it is not so. The two very small but still non-empty intervals ${\cal D}^{(H)}_+ \approx (1.37199,1.37228)$ and ${\cal
D}^{(H)}_- \approx (-1.37228,-1.37199)$ survive and yield the whole quadruplet of the energies purely real again, in a truly fine-tuned manner.
A challenging task emerges in connection with an appropriate modification of the metric. In place of Eq. (\[metra2\]) the present would-be positive-definite solution of Eq. (\[dieu\]) reads $$\left[ \begin {array}{cccc} 3+{t}^{2}&{\frac { \left( 3+{t}^{2}
\right) t \left( 13\,{t}^{2}-96 \right) }{17\,{t}^{2}+96}}&24\,{
\frac { \left( 3+{t}^{2} \right) {t}^{2}}{17\,{t}^{2}+96}}&1/2\,{
\frac { \left( 3+{t}^{2} \right) t \left( {t}^{2}+96 \right)
}{17\,{t} ^{2}+96}}\\\noalign{\medskip}{\frac { \left( 3+{t}^{2}
\right) t
\left( 13\,{t}^{2}-96 \right) }{17\,{t}^{2}+96}}&3+{t}^{2}&{\frac {
\left( 3+{t}^{2} \right) t \left( 7\,{t}^{2}-96 \right) }{17\,{t}^{2}
+96}}&24\,{\frac { \left( 3+{t}^{2} \right)
{t}^{2}}{17\,{t}^{2}+96}}
\\\noalign{\medskip}24\,{\frac { \left( 3+{t}^{2} \right) {t}^{2}}{17
\,{t}^{2}+96}}&{\frac { \left( 3+{t}^{2} \right) t \left(
7\,{t}^{2}- 96 \right) }{17\,{t}^{2}+96}}&3+{t}^{2}&{\frac { \left(
3+{t}^{2}
\right) t \left( 13\,{t}^{2}-96 \right) }{17\,{t}^{2}+96}}
\\\noalign{\medskip}1/2\,{\frac { \left( 3+{t}^{2} \right) t \left( {t
}^{2}+96 \right) }{17\,{t}^{2}+96}}&24\,{\frac { \left( 3+{t}^{2}
\right) {t}^{2}}{17\,{t}^{2}+96}}&{\frac { \left( 3+{t}^{2} \right) t
\left( 13\,{t}^{2}-96 \right) }{17\,{t}^{2}+96}}&3+{t}^{2}
\end {array} \right]\,.$$ In spite of its perceivably more complicated structure this matrix may still be shown, by the same techniques as above, to be safely positive definite (i.e., to become eligible as a metric) for t \_ (-1.082854389,1.082854389) i.e., in a slightly diminished range of the “time" parameter.
A recoupled regime
------------------
In a way complementing Eq. (\[awayab\]) let us consider H= H\_[(EC)]{}+. \[awayaber\] The fine-tuned model (\[away\]) looks now perturbed in an opposite direction. The perceivable weakening of the matrix element in the corner (i.e., of the periodicity-guaranteeing bond) enhances the parallels with the open-end systems. By itself, this change should lead to a completely complex spectrum at the larger $|t|$s.
This expectation is confirmed by Fig. \[fiedeeze\]. The picture shows that the choice of the stronger central attraction in Eq. (\[awayaber\]) works in the fragility-enhancing direction, diminishing the central crypto-Hermiticity domain, \^[(H)]{}\_0:= ( , ) (-0.9658391622, 0.9658391622). The presence of the inflexions in the outer energy loop finds its origin in the highly unstable unavoided-crossing points of model (\[away\]). The relevance of the inflexion points as emphasized in Ref. [@Joglekar] might be recalled.
The construction of the metric preserves the algebraic structure shown above. From an updated formula (using just different numerical coefficients) we derive the boundary points of the interval ${\cal
D}_\Theta$. The definition of these boundaries is provided by the minimal root of the expression ( -[t]{}\^[3]{}+631[t]{}\^[2]{}-720t+1152-1/24) ( 3+[t]{}\^[2]{} ) ( 631[t]{}\^[2]{}+1152 ) \^[-1]{}, V=[2772145[t]{}\^[6]{}-50595840[t]{}\^[5]{}+348491520[t]{}\^[4]{}-955514880 [t]{}\^[3]{}+1147944960[t]{}\^[2]{}]{} giving t \_ (-0.9658391622,0.9658391622). The size of this interval of the positivity of $\Theta$ is again comparable with the size of the interval of the crypto-Hermiticity of the Hamiltonian.
Outlook \[VIs\]
===============
The recent growth of interest in ${\cal PT}-$symmetric quantum lattices offers a natural motivation of the transition to the loop-shaped lattices. We found that a core of the consistency of such a transition (which could suffer from its potential “fragility" in general [@fragile]) is similar to the suppression of the fragility in open-ends models.
Via a thorough description of a few not too complicated examples we illustrated that in [*both*]{} the open-end and matched-ends models, the stability and the robust nature (i.e., non-fragile nature) of the models results from the absence of the degeneracy of spectra of the zero-coupling versions of the Hamiltonians in question.
The role of the matching matrix elements of the Hamiltonian remains, in the phenomenological perspective, slightly counterintuitive. Still, the decisive conceptual parallels between periodic lattices and the mathematically more friendly open-end lattices were noticed. They involve not only the encouraging emergence of the parallel structures in the spectra of energies but also in the shapes of the domains of the reality of the energies or of the positivity of the matrix candidates for the metrics.
An important merit of our specific models may be seen in the feasibility of calculations. This resulted from the preservation of multiple parallels between our present matched-ends models and their open-ends predecessors (let us mention just the up-down symmetry, equidistance of the unperturbed spectra or the reality of the interaction matrices). Nevertheless, even beyond the level of the low-dimensional solvable examples the more general questions of consistency of the underlying quantum theory were addressed. Our constructive study of the chains defined along discrete loops appeared more friendly than expected.
Our detailed analyses covered the extensive dynamical domain, far beyond the mere weak-coupling subdomain. Our periodicity-simulating bonds proved connected to the emergence of unexpected spectral phenomena (like the strong-coupling-related islands of stability) which will certainly deserve a further study in the future.
Acknowledgments {#acknowledgments .unnumbered}
---------------
Work supported by the GAČR grant Nr. P203/11/1433, by the Institutional Research Plan AV0Z10480505 and by the MŠMT “Doppler Institute" project Nr. LC06002.
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|
---
abstract: |
FCC046 is a Fornax Cluster dwarf elliptical galaxy. Optical observations have shown that this galaxy, besides an old and metal-poor stellar population, also contains a very young centrally concentrated population and is actively forming stars, albeit at a very low level.
Here, we report on 21 cm observations of FCC046 with the Australia Telescope Compact Array (ATCA) which we conducted in the course of a small survey of Fornax Cluster early-type dwarf galaxies. We have discovered a $\sim 10^7$ M$_\odot$ H[i]{} cloud surrounding FCC046. We show that the presence of this significant gas reservoir offers a concise explanation for this galaxy’s optical morphological and kinematical properties.
Surprisingly, the H[i]{} gas, as evidenced by its morphology and its rotational motion around the galaxy’s optical major axis, is kinematically decoupled from the galaxy’s stellar body. This is the first time such a ring of gaseous material in minor-axis rotation is discovered around a dwarf galaxy.
author:
- 'S. De Rijcke, P. Buyle, M. Koleva'
title: 'FCC046: a candidate gaseous polar ring dwarf elliptical galaxy in the Fornax Cluster '
---
Introduction {#sect:exdat}
============
FCC046 was included in the Fornax Cluster Catalog by @fe89. It was classified as a dE4, i.e. as a rather flattened dwarf elliptical galaxy, with a total B-band apparent magnitude $m_B=15.99$ mag and a half-light radius of $R_e = 6.7''$. We adopt a distance of 20.3 Mpc [@je03] for the distance to the Fornax Cluster, yielding a total absolute magnitude of $M_B=-15.55$ mag and a half-light radius of $R_e
= 660$ parsec for FCC046 [@de05]. Using VLT photometry and spectroscopy, @dd04 showed that FCC046, despite its pronounced flattening, has zero net stellar rotation about neither the projected major or the minor axis. It also has a very pronounced nucleus which is resolved from the ground. The nucleus’ most striking feature is that, surprisingly, it is off-center with respect to the galaxy’s outer isophotes by almost 1 arcsec. This displacement has been interpreted as a consequence of the counter-rotation instability.
Existing optical observations
=============================
Observations with the FLAIR-II spectrograph on the UK Schmidt Telescope by @dr01 have shown that FCC046 has H$\alpha$ emission, with an equivalent width of 2.1 [Å]{} over a fibre 6.7 arcsec accross centered on the galaxy. The presence of ionized gas was interpreted as evidence for star formation, casting doubt on its classification as a “true” dwarf elliptical.
This has later been confirmed by @de03, who showed, based on VLT photometry, that FCC046 has strong positive color gradients and that its nucleus is surrounded by 6 sources of H$\alpha$ emission. The largest of the emission sources have properties (diameters and H$\alpha$+\[N[ii]{}\] luminosities) similar to those of supernova remnants. The smaller ones could be H[ii]{} regions or nebulae around Wolf-Rayet stars. Blindly applying the heuristic relation between the star-formation rate and the H$\alpha$ emission line’s equivalent width [@ke92] leads to a star-formation rate of $\sim
10^{-3}$ M$_\odot$/year. The colour gradient also suggests the occurrence of centrally concentrated recent star formation in FCC046.
Applying the full-spectrum fitting code ULySS to an optical VLT spectrum of FCC046, @ko09 showed that the SSP-equivalent age of the stellar population increases from less than 1 Gyr near the center to $\sim 5$ Gyr beyond 1 kpc (see fig. 2 of [@ko09]). At the same time, the mean luminosity weighted metallicity, quantified by the SSP-equivalent \[Fe/H\], increases very slightly from a central value of \[Fe/H\]$\sim -1.0$ dex to \[Fe/H\]$\sim -0.8$ dex at 1 $R_e$.
By fitting a weighted sum of several SSPs to the spectra one can reconstruct a star-formation history [@lb04]. The number of age bins used in this fit was steadily increased until the goodness-of-fit stopped increasing significantly. The bounds of the bins were varied to ensure that the age and metallicity of each SSP were well within their respective bounds (e.g. if the age of an SSP coincided with the lower age bound of its bin, this bound was lowered).
In Table \[tab1\], we present the results from fits to spectra summed within the inner seeing disk (1 arcsec accross) and within one $R_e$ (excluding the inner seeing disk). Within the inner 1 arcsecond, young stars make up 59 % of the light; within the inner 1 R$_e$, they constitute 47 % of the light, with about 10 % of newborn stars. Apparently, star formation picked up about 300 Myr ago in the body of the galaxy while the increase in the inner 1 arcsecond happened only $\sim 70~$Myr ago. Clearly, the positive age gradient observed in FCC046 is caused by the presence of centralised ongoing star formation, resulting in a very young stellar population whose fractional light contribution decreases with radius.
Dividing the stellar mass in each age bin by its age width yields the mean star-formation rate in each bin, shown in Fig. \[fig:SFH\]. The central 1 arcsecond wide disk has a B-band luminosity of $L_B(1'')=1.0
\times 10^7~L_{B,\odot}$; the inner 1 R$_e$ (excluding the inner disk) has a luminosity $L_B(1~R_e)=1.2 \times 10^8~L_{B,\odot}$. The corresponding masses for the stellar populations inside the inner 1 arcsec and inside 1 $R_e$, derived with the Pegase.HR evolutionary code for a Salpeter IMF using the SFH reconstruction of ULySS, are M$(1'')=1.5 \times 10^7$ M$_{\odot}$ and M$(1~R_e)=3.3 \times
10^8$ M$_{\odot}$. We normalize the SFR such that the time-integral over the star-formation history in each radial bin yields the correct total mass. The recent star-formation event appears to have been very pronounced in the inner regions. There, the star-formation rate increased by over a factor of 5 compared with the mean rate over the previous $\sim 4$ Gyr, although the time resolution of the SFH does not allow us to pinpoint other similar bursts older than $\sim 1$ Gyr.
The oldest stars (age $>3$ Gyr) have metallicities in the range \[Fe/H\]$\sim -2$ to $-1$ throughout the whole galaxy. Judging from Table \[tab1\], the young and intermediate-age stellar populations have metallicities consistent with solar while the 12 Gyr old population has a higher metallicity in the outer radial bin (\[Fe/H\]$\sim -1.5$ versus \[Fe/H\]$\sim -2.3$ inside the inner 1 arcsec). This appears to be the cause for the slight radial increase of the SSP-equivalent metallicity. The metallicity of the very youngest stellar populations is rather uncertain since their massive stars, which dominate their optical light output, have no conspicuous metallicity-sensitive absorption features. However, there appears to be a tendency for the very youngest stellar populations to have metalicities below solar.
[|c|c|c|]{}\
Age & light fraction & \[Fe/H\]\
$67 \pm 5$ Myr & 59% & $-0.11 \pm 0.07$\
$1176 \pm 352$ Myr & 14% & $-0.01 \pm 0.27$\
$12000$ Myr (fixed) & 27% & $-2.27 \pm 0.13$\
\
Age & light fraction & \[Fe/H\]\
$34 \pm 10$ Myr & 11% & $-0.80 \pm 0.45$\
$309 \pm 51$ Myr & 36% & $0.16 \pm 0.12$\
$3613 \pm 1857$ Myr & 8% & $0.27 \pm 0.38$\
$12000$ Myr (fixed) & 45% & $-1.46 \pm 0.16$\
Within the inner 1 arcsecond, the nucleus contributes $\sim 80$ % of the light while $\sim 20$ % comes from the galaxy’s main body, adopting the latter’s Sérsic profile parameters from @ko09. If we assume that the light fraction of 11 % contributed by very young stars is typical for the galaxy’s body (see Table \[tab1\]) then the 59 % of light from the central 1 arcsecond coming from very young stars, with ages below 100 Myr, translates into a nucleus whose youngest stellar populations contribute $\sim 70$ % of its light. The mass of the nucleus is $M_{\sf nuc} \approx 8 \times
10^6$ M$_\odot$, 7 % of which consists of very young stars. For this estimate, we adopted a $M/L \approx 0.1$ in solar units for a 70 Myr old population from PEGASE-HR [@lb04]. Therefore, most of the mass of the nucleus was in place before the most recent star-formation event.
With such a massive nucleus, box orbits are scattered onto tube and chaotic orbits; conservation of angular momentum then requires that direct and retrograde loop orbits be equally populated [@ec94; @mq98; @va10]. If this scenario holds, FCC046 would have developed an axially symmetric counter-rotating stellar body, prone to the counter-rotation instability, over the course of only a few crossing times [@dd04].
New H[i]{} observations {#sect:HI}
=======================
We used the Australia Telescope Compact Array (ATCA) during February 2006 to search for 21 cm emission in a small sample of Fornax Cluster dEs. The observations took place during daytime but were not disturbed by solar radio-frequency interference. The ATCA was positioned in the large EW367 configuration with baselines ranging from 46 m up to 4408 m. Due to the typical small velocity widths of dwarf galaxies, we opted for a correlator setup with 256 channels of width 31.25 kHz, giving a total baseband of 8 MHz or roughly 1700 km/s at the distance of FCC046. No on-line Hanning smoothing was applied, resulting in a velocity resolution of 6.6 km/s. The observation was initiated with a calibration on the source 1934-638 which acted as primary calibrator. After the initial calibration, this source was observed for 15 minutes, followed by alternating observations of the phase calibrator 0332-403 (integration of 5 minutes) and of the targeted dwarf galaxy (integration of 40 minutes). The total integration time on each target (including calibration) was 12h.
The standard data reduction steps (phase, amplitude and bandpass calibration) were performed with the MIRIAD package [@sa95], the standard ATCA data analysis program. We subtracted the continuum by performing a first-order fit to the visibilities over the line-free channels that were not affected by the edge effects of the band (300 km/s on each edge). The data cubes were created by using natural weighting. A minimal cleaning of 500 iterations was performed. Our final data cube had a synthesized beam of $147 \times 78$ arcsec$^2$ and a noise of 3 mJy/beam. These steps resulted in a detection of FCC046 in at least 3 subsequent 6.6 km/s channels in the centre of our baseband at a velocity of 2209 km/s, very close to the optical velocity of FCC046 at 2220 km/s. Channel maps of FCC046, rebinned to 13.2 km/s wide channels, are presented in Fig. \[fig:velocity.eps\].
To derive a spectrum we summed the flux within a $4 \times
4$ arcmin$^2$ box centred on the central radio position of FCC046, which was derived from the total H[i]{} intensity map (see Fig. \[fig:histo.eps\]). The global H[i]{} profile shows a double horned spectrum suggesting rotation. A single Gaussian was fitted to the H[i]{} profile, giving a velocity linewidth of 52 km/s at the 20 per cent level and 34 km/s at the 50 per cent level. After correction for broadening and random motions (method of @vs01) we find $W_{20}=43$ km/s and $W_{50}=32$ km/s.
After summing the channels that contain emission of the galaxy, we found a total velocity-integrated H[i]{} flux density of 0.15 Jy km/s, resulting in a total estimated H[i]{} mass of $1.45 \times
10^7$ M$_\odot$.
Discussion {#sect:disc}
==========
The most striking feature of the 21 cm emission of FCC046 is that it is elongated in a direction almost perpendicular to the galaxy’s stellar body. This is shown clearly in Fig. \[fig:moment0.eps\] in which a zeroth-order map of the 21 cm emission of FCC046 is presented. While the beam size is, admittedly, quite significant, the emission region is clearly resolved and, at FWHM, is almost 1.5 times as large as the beam in the north-south direction. Its east-west extent is not resolved. The first-order moment map shows evidence for a velocity progression from the galaxy’s southern side towards the north-west (the velocity increase eastwards of the galaxy is due to a $2\sigma$ peak in a single channel and is likely to be spurious). If, allowing for the noise level and resolution of our observations, we interpret this as ordered rotation this means that the H[i]{} gas surrounding FCC046 is rotating essentially around the optical galaxy’s major axis.
Polar rings, i.e. rings of stars and/or gas and dust orbiting in a plane perpendicular to their host galaxy’s equatorial plane, are known in a small fraction of low-mass early-type galaxies [@whi90; @co06]. Cosmological simulations [@mac06] suggest that these galaxies acquire this material through cold accretion of gas filaments. A similar scenario could apply to FCC046. The position of FCC046, at $3.5^\circ$ from the cluster center [@bu05], places it far outside the X-ray halo of the Fornax Cluster, which does not extend significantly beyond the NGC1399/NGC1404/NGC1387 central region [@jo97]. This means that any gas accreted onto FCC046 will not immediately be removed by ram-pressure stripping, keeping it available for star formation. It then stands to reason that FCC046, an “ordinary” dE with a stellar population with a mean age over $\sim
3$ Gyr, has accreted a few $10^7$ M$_\odot$ of gas orbiting roughly perpendicularly to its stellar body. The slightly distorted morphology of the gas may be due to the orbital movement of FCC046 through the Fornax cluster, combined with its weak gravitational field, or by warping. Moreover, if FCC046 has a strongly oblate or prolate gravitational potential, the polar ring gas is expected to rapidly flow inwards where it can be used for star formation [@ch92]. This inflow, flooding the ring’s central hole with gas, would occur on a timescale corresponding to only a few rotation periods. In the case of FCC046, this would amount to a few 100 Myrs which is in line with the observed upturn of the star-formation rate.
Assuming we are indeed observing a gas ring, the observed velocity gradient will be much lower than what could be expected from the galaxy’s rotation curve. In the central hole, it is precisely the gas on orbits tangent to the line of sight that is missing. The gas within the ring (back and front side) has much less favorable projections of its rotation velocity onto the line of sight and therefore results in small projected velocities. Only along sufficiently distant lines of sight, those that avoid the central hole, is it possible to observe the full rotation of the gas. However, given the theoretically predicted ongoing warping and disruption of the ring, even there gas is most likely not moving at the local rotation velocity. Along with the severe beam smearing, this helps explains the observed small velocity gradient in the right panel of Fig. \[fig:moment0.eps\].
Clearly, the hypothesis that FCC046 has accreted a fresh reservoir of gas that triggered a recent star-formation event offers a concise explanation for most of this galaxy’s observed properties. Other explanations work far less well. For instance, the observed H[i]{} gas could have been expelled by the centralized star-formation event, rendering it an outflow rather than an infall. One might expect that, in a flattened dwarf galaxy, such outflows are preferably aligned along the minor axis, as is sometimes observed [@mf99]. However, the recent star-formation rate measured in FCC046 is orders of magnitudes smaller than what is observed in e.g. the star-bursting bipolar outflow dwarf M82, which has a central star-formation rate upwards of $1$ M$_\odot$/year [@fs03]. This makes it unlikely that FCC046 could produce the $G{\rm M}_{\rm gal}{\rm M}_{\rm
gas}/{\rm 1~kpc}\sim 10^{54}$ erg, the equivalent of over $\sim
1000$ combined supernova explosions [@cl12], needed to lift M$_{\rm gas}= 1.45 \times 10^7$ M$_\odot$ of gas out to a distance of 1 kpc in a M$_{\rm gal} \sim 10^9$ M$_\odot$ galaxy. Moreover, realistic simulations of flattened dwarf galaxies fail to produce such collimated outflows if the star-formation activity is not very centrally concentrated [@sc11]. Another possibility is that we are seeing gas being removed from FCC046 by ram-pressure stripping [@mb00; @ma03; @ma06]. However, FCC046 is very far outside the X-ray halo of the Fornax Cluster, making the stripping scenario highly unlikely.
Conclusions {#sect:conc}
===========
Based on optical observations, FCC046, a dwarf galaxy in the Fornax Cluster, has properties that make it stand out from the general dwarf elliptical galaxy population. All evidence supports the conclusion that $\sim 300$ Myr ago, the star-formation rate in FCC046 increased dramatically. The strength of this recent star-formation event decreases with radius since these young stars contribue $\sim 60$ % of the light within the inner arcsecond (a region largely coinciding with the galaxy’s nucleus) while they make up only 47 % of the light within one half-light radius, excluding the central arcsecond. Another odd feature is this galaxy’s relatively massive, off-center nucleus. Given the fact that FCC046, despite its significant flattening, has zero net rotation it is possible that the well-known counter-rotation instability is responsible for driving the nucleus off-center.
In this paper, we have presented H[i]{} observations with the ATCA of FCC046 that may offer a concise explanation for its optical properties. We have discovered a $\sim 10^7$ M$_\odot$ H[i]{} cloud surrounding FCC046 which, as evidenced by its morphology and its rotational motion around the galaxy’s optical major axis, is kinematically decoupled from FCC046’s stellar body. It seems plausible that this gas reservoir has been accreted by FCC046 on a highly inclined orbit and, as it flows to the center of the gravitational well, fuels centrally concentrated star formation.
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|
---
abstract: 'We performed an intensive accretion disk reverberation mapping campaign on the high accretion rate active galactic nucleus Mrk 142 in early 2019. Mrk 142 was monitored with the [*Neil Gehrels Swift Observatory*]{} for 4 months in X-rays and 6 UV/optical filters. Ground-based photometric monitoring was obtained from the Las Cumbres Observatory, Liverpool Telescope and Dan Zowada Memorial Observatory in [*ugriz*]{} filters and the Yunnan Astronomical Observatory in [*V*]{}. Mrk 142 was highly variable throughout, displaying correlated variability across all wavelengths. We measure significant time lags between the different wavelength light curves, finding that through the UV and optical the wavelength-dependent lags, $\tau(\lambda)$, generally follow the relation $\tau(\lambda) \propto \lambda^{4/3}$, as expected for the $T\propto R^{-3/4}$ profile of a steady-state optically-thick, geometrically-thin accretion disk, though can also be fit by $\tau(\lambda) \propto \lambda^{2}$, as expected for a slim disk. The exceptions are the [*u*]{} and [*U*]{} band, where an excess lag is observed, as has been observed in other AGN and attributed to continuum emission arising in the broad-line region. Furthermore, we perform a flux-flux analysis to separate the constant and variable components of the spectral energy distribution, finding that the flux-dependence of the variable component is consistent with the $f_\nu\propto\nu^{1/3}$ spectrum expected for a geometrically-thin accretion disk. Moreover, the X-ray to UV lag is significantly offset from an extrapolation of the UV/optical trend, with the X-rays showing a poorer correlation with the UV than the UV does with the optical. The magnitude of the UV/optical lags is consistent with a highly super-Eddington accretion rate.'
author:
- 'Edward M. Cackett'
- Jonathan Gelbord
- 'Yan-Rong Li'
- Keith Horne
- 'Jian-Min Wang'
- 'Aaron J. Barth'
- 'Jin-Ming Bai'
- 'Wei-Hao Bian'
- 'Russell W. Carroll'
- Pu Du
- Rick Edelson
- 'Michael R. Goad'
- 'Luis C. Ho'
- Chen Hu
- 'Viraja C. Khatu'
- Bin Luo
- Jake Miller
- 'Ye-Fei Yuan'
bibliography:
- 'agn.bib'
title: |
Supermassive black holes with high accretion rates in active galactic nuclei.\
XI. Accretion disk reverberation mapping of Mrk 142
---
Introduction {#sec:intro}
============
At typical mass accretion rates onto supermassive black holes in Seyfert galaxies (a few percent of the Eddington limit), accretion is expected to take place via a geometrically thin ($H/R \ll 1$), optically thick accretion disk [@shakurasunyaev]. However, once mass accretion rates exceed the Eddington limit, then radiation pressure becomes important, and is expected to change the structure of the accretion flow. In the ‘slim disk’ class of models, at super-Eddington rates, radiation pressure dominates the accretion flow at most radii, and the disk becomes slim (rather than thin), with $H \lesssim R$ [e.g. @abramowicz88]. Slim disks are characterized by sub-Keplerian rotation and transonic radial motion. The fast radial transportation in slim disks means that most photons are trapped by optically thick Thomson scattering and advected into the black hole before escaping. Within this inner photon-trapping region the disk increases significantly in scale height, which can cast a shadow on the outer disk [e.g. @wang14_shadow]. Alternatively, @begelman02 proposes that through the photon bubble instability, the disks may remain thin even above the Eddington limit. However, observational tests of the nature of super-Eddington accretion flows in Seyfert galaxies are rare.
One way to observationally test the accretion flow and nearby broad-line region (BLR) is to use reverberation mapping [RM; @blandmckee82; @peterson14]. In RM, time lags between light curves at different wavelengths (either between the continuum and emission lines or the continuum at different wavelengths) can be used to determine the size-scale of the emitting region. Applying RM to Active Galactic Nuclei (AGN) thought to be accreting at high rates is therefore a way to observationally test super-Eddington accretion. One class of AGN thought to be accreting at high rates are Narrow-Line Seyfert 1 (NLS1) galaxies. NLS1s are characterized by relatively narrow broad emission lines, strong lines, weak \[\] lines and steep 2–10 keV spectra [e.g., @boller96; @veroncetty01].
Over the last 7 years or so the Super-Eddington Accreting Massive Black Holes (SEAMBH) collaboration has been performing extensive optical monitoring of super-Eddington AGN candidates that show these characteristics of strong optical Fe [ii]{} and weak \[O [iii]{}\] emission lines [e.g., @Du14; @Du16; @du18; @Hu15]. These observations show that the BLR structure in these super-Eddington objects differs significantly from more typical sub-Eddington Seyferts [@Du16; @du18]. One of the main findings is that these super-Eddington AGN lie below the well known relation between the radius of the H$\beta$-emitting region and the optical luminosity [the $R-L$ relation, e.g., @kaspi00; @bentz13]. Hence, this suggests that the BLR size depends on more than just luminosity, i.e. for objects of the same luminosity, those with lower mass and thus higher Eddington ratio show more compact BLRs. This can be understood as the inner part of the slim disk acting as an optically thick torus, creating a self-shadowing effect that lowers the ionizing flux seen by the BLR [@wang14_shadow].
In order to test the accretion disk structure in a super-Eddington AGN, we carried out the first accretion disk RM campaign on a super-Eddington AGN, Mrk 142 (PG 1022+519, $z = 0.045$). Accretion disk RM uses time lags between the continuum at different wavelengths to probe the size and temperature of the accretion disk [e.g. @cackett07]. In the lamp-post reprocessing picture, high energy X-ray/EUV photons from a central corona irradiate the accretion disk, driving variability at longer wavelengths. The hotter, inner disk will respond to variability in the irradiating photons before the cooler, outer disk. This then leads to correlated continuum lightcurves with longer wavelengths lagging shorter wavelengths. Measuring the wavelength-dependence of the lag therefore gives both the size-scale of the disk and its temperature profile for an assumed disk geometry. For instance, for an optically thick, geometrically thin accretion disk [@shakurasunyaev] the temperature profile goes like $T(R) \propto R^{-3/4}$. Since $\tau \sim R/c$ and $\lambda \propto 1/T$ (from Wien’s law), such a temperature profile leads to wavelength-dependent lags following $\tau(\lambda) \propto \lambda^{4/3}$. On the other hand, since a slim disk has a temperature profile following $T(R) \propto R^{-1/2}$ within the photon-trapping region [@wang99], the wavelength-dependent lags should follow $\tau(\lambda) \propto \lambda^{2}$ instead.
Recently, advances in accretion disk RM have come from intensive (better than daily) monitoring with the [*Neil Gehrels Swift Observatory*]{} (hereafter, [*Swift*]{}) on four Seyferts: NGC 5548 [@edelson15; @fausnaugh16], NGC 4151 [@edelson17], NGC 4593 [@cackett18; @mchardy18] and Mrk 509 [@edelson19]. See @edelson19 for a comparison of all four [*Swift*]{} datasets. These campaigns have shown three main results. Firstly, the time lag, $\tau$ generally follows $\lambda^{4/3}$, as expected for a standard thin disk, however, the magnitude of the lags is larger than expected by a factor of 2 – 3. Secondly, the lag in the $u$ band (3465Å) consistently lies above this $\tau \propto \lambda^{4/3}$ relation, which indicates significant continuum emission from the BLR [@koristagoad01; @koristagoad19; @lawther18]. This is further highlighted by [*Hubble Space Telescope*]{} monitoring of NGC 4593 which spectroscopically resolved the “lag spectrum” in finding a discontinuity at the Balmer jump, as expected if BLR continuum emission is important [@cackett18]. Finally, the X-ray to UV correlation is significantly weaker than the UV to optical correlation [@edelson19], which raises the question of whether the X-rays drive the variability at longer wavelength. In the case of NGC 5548 the shape of the X-ray lightcurve is not consistent with driving the UV/optical variability [@starkey17; @gardnerdone17], while in NGC 4593 it is [@mchardy18].
These four Seyferts all accrete at rates significantly lower than the Eddington limit, and so serve as a good comparison for wavelength-dependent lags measured in objects accreting at much higher rates. In this paper, we present an intensive accretion disk RM campaign on Mrk 142, using [*Swift*]{} along with ground-based monitoring. Mrk 142 has a black hole mass of $\log(M/M_\odot) = 6.23^{+0.26}_{-0.45}$, and a dimensionless accretion rate of $\dot{\mathscr{M}} = \dot{m}c^2/L_{\rm Edd} = 250$ [@li18]. The paper is organized as follows. In Section \[sec:obs\] we describe the observations, and in Section \[sec:data\] we detail the data reduction. The time series analysis and results are presented in Section \[sec:timeseries\], while in Section \[sec:spec\] we use variability to isolate the spectral energy distribution of the disk. Finally, in Section \[sec:discuss\] we discuss the implications.
Observations {#sec:obs}
============
A large, coordinated monitoring campaign on Mrk 142 took place from December 2018 – June 2019. The core of the campaign was centered around X-ray and UV/optical observations taken with [*Swift*]{}. In addition, we obtained further X-ray observations with [*NICER*]{}, and supporting ground-based photometric and spectroscopic monitoring from multiple telescope sites. Details and results from the spectroscopic monitoring and [*NICER*]{} X-ray analysis will be presented in future follow-up papers. Here, we focus only on the [*Swift*]{} and ground-based photometric data. The ground-based monitoring involved Las Cumbres Observatory (LCO), Liverpool Telescope, Dan Zowada Memorial Observatory (hereafter Zowada Observatory) and the Yunnan Astronomical Observatory. Further details about observations from each telescope are given below. A summary of the observations used is given in Tab. \[tab:summary\].
Swift
-----
Mrk 142 was monitored by [*Swift*]{} from 1 January 2019 to 30 April 2019 through Cycle 14 proposal 1417139 (PI: E. M. Cackett). Initially, [*Swift*]{} observations were obtained twice per day. However, following a successful request for Director’s Discretionary Time to extend the campaign by 1 month, the cadence of observations became once per day from 20 March 2019 onwards. In total, 184 epochs of observations were obtained. The typical visit duration was 1000s, with the exact length varying depending on scheduling. X-ray observations were taken in Photon Counting mode. UVOT exposures were taken in 0X30ED mode, which gives the bluer filters longer exposure times. For a typical 1000s visit, this gives exposures of approximately 333s for [*UVW2*]{}, 250s for [*UVM2*]{}, 167s for [*UVW1*]{}, and 83s for [*U*]{}, [*B*]{} and [*V*]{}.
Las Cumbres Observatory
-----------------------
LCO is a global network of robotic telescopes. As part of an LCO Key Project (KEY-2018B-001, PI: R. Edelson) monitoring was obtained in Sloan [*u, g, r, i*]{}, and PanSTARRS [*z*]{} filters from both the 2-m Faulkes Telescope North at the Haleakala Observatory (OGG), and the 1-m telescope at the McDonald Observatory (ELP). Since most of the ground-based data comes from the two LCO telescopes we adopt the effective wavelengths of the LCO filters[^1] (see Tab. \[tab:lags\]) for the subsequent analysis. On OGG we use the Spectral camera which has a $10.5\arcmin \times 10.5$ field of view, while at ELP we use the Sinistro camera with a $26.5\arcmin\times26.5$ field of view.
Exposures were taken in pairs with individual exposure times being initially 300s for [*u*]{}, 60s for [*g, r*]{} and [*i*]{}, and 120s for [*z*]{} for OGG. After analysis of early data the exposure time in the z filter was increased to 240s. For ELP, the initial exposure times were 300s for [*u*]{}, 60s for [*g, r*]{}, and [*i*]{} and 120s for [*z*]{}. These exposure times were increased to 600s for $u$, 180s for $g$, $r$, and $i$ and 360s for $z$ after inspection of early data. Observations with LCO took place between 15 December 2018 and 19 June 2019.
Liverpool Telescope
-------------------
Photometric monitoring was obtained with the robotic 2-m Liverpool Telescope located on La Palma, Spain through program PL19A01 (PI: M. Goad). Observations were taken using the IO:O instrument in $u$, $g$, $r$, $i$, and $z$ filters. IO:O has $4096 \times 4112$ pixels with a pixel scale of 015 per pixel. Pairs of exposures were taken during each epoch, with individual exposure times of 90s for $u$, 10s for $g$ and $r$, 15s for $i$ and 20s for $z$. Observations took place between 3 January 2019 and 22 April 2019.
Zowada Observatory
------------------
The Zowada Observatory is a robotic 20-inch f/6.8 PlaneWave telescope located near Rodeo, New Mexico and owned and operated by Wayne State University. During the monitoring campaign two different detectors were used. Prior to 19 January 2019 a FLI Proline 16803 CCD with 4096 $\times$ 4096 pixels was used. On 19 January 2019 a back-illuminated FLI Proline 230-42-1-MB CCD with 2048 $\times$ 2048 pixels was installed. The pixel size for this detector is 15 microns, leading to a plate scale of 09 per pixel.
Observations began on 31 October 2018 and continued daily (when possible) until 30 May 2019. Images were obtained using [*u, g, r, i*]{}, and $z$ filters. Individual exposure times were 300s for $u$, 200s for $z$ and 100s for [*g, r*]{}, and $i$. Multiple exposures per filter were obtained on each night (typically 5 per filter, but it varied depending on weather).
Yunnan Astronomical Observatory
-------------------------------
Observations at the Lijiang Station of the Yunnan Observatories, Chinese Academy of Sciences, were obtained with the 2.4-m telescope. The telescope is equipped with the Yunnan Faint Object Spectrograph and Camera (YFOSC), which is a versatile instrument usable both for photometry and spectroscopy. An e2v back-illuminated $2048 \times 4608$ pixels CCD is mounted in YFOSC and covers a field of view of $10\arcmin \times 10\arcmin$ (with a pixel size of 0283pixel$^{-1}$) in the imaging mode. While the Lijiang telescope was primarily used for spectroscopy, images in the $V$ filter were also obtained as part of the program. Observations span from 22 October 2018 to 21 June 2019. The typical exposure time is 120 – 150s (three 40 – 50s consecutive exposures in each of the nights).
Data Reduction {#sec:data}
==============
Swift
-----
The X-ray lightcurve is produced using the [*Swift*]{}/XRT data products generator[^2] [@evans07; @evans09]. We used this to extract the background-subtracted count rate of Mrk 142 in the 0.3 – 10 keV energy range for each [*Swift*]{} snapshot during the campaign.
The *Swift*/UVOT [@poole08] data analysis largely follows the same procedure detailed in @edelson15 [@edelson17; @edelson19] and is only described briefly here, focusing on details that differ. The data were processed using HEASOFT v6.24. In the present study, field stars from the GAIA DR2 catalog [@GAIA_DR2] are used to refine the astrometry of each exposure before making photometric measurements. For each epoch and filter, fluxes are measured using the tool UVOTSOURCE. Source extractions are measured using a circular region with a radius of 5, while the background is measured in an annulus from 40–90 from which small circular regions centered on background stars are excluded (consequently, the background region resembles a ring of Swiss cheese, with holes of radius 12 centered on sources from the GAIA DR2 catalog that lie within 102 of Mrk 142).
The standard pipeline processing includes a correction for the gradual decline in UVOT sensitivity; the correction applied to the Mrk 142 data is an updated version that has been approved by the instrument team but has not yet (as of Summer 2019) been released in the CALDB (Alice Breeveld, private communication). The data are then screened to identify which measurements are likely to be affected by detector regions with reduced sensitivity, applying the updated masks presented in Hern[á]{}ndez Santisteban et al. (submitted). Any observation where Mrk 142 is identified as falling within the detector mask is then removed from the light curve. Note that the masks used here do not include a correction for the UVOT shift-and-add processing, as the impact of this was not recognized until after the Mrk 142 data were analyzed. The result is that the masks of Hern[á]{}ndez Santisteban et al. are effectively smoothed by a blur on the scale of 7–10, which causes a few false positive and false negative errors when screening the Mrk 142 measurements.
Ground-based optical photometry
-------------------------------
The optical lightcurves were obtained using relative photometry. For each exposure, the count rate within a circular aperture was obtained for Mrk 142 and a number of comparison stars. Aperture radii for each telescope were: 11 pixels (33) for OGG; 13 pixels (51) for ELP; 7 pixels (21) for the Liverpool Telescope and 5 pixels (45) for the Zowada Observatory. Background rates were extracted from an annulus with the following inner and outer radii: 40 to 60 pixels for OGG and ELP; 15 pixels to 20 pixels for the Liverpool Telescope and 20 to 30 pixels for Zowada Observatory. These were optimized to maximize the signal to noise ratio (S/N) for each telescope. Any differences in host galaxy contribution with changes in aperture size are corrected for by the inter-calibration of the lightcurves (described later).
For a given telescope, the fluxes for all exposures taken within 3 hours of each other were averaged, to improve S/N. At each epoch, relative photometry is performed by dividing the observed rates by the sum of count rates from the chosen comparison stars. We assess the reliability of the relative photometry through looking at the fractional standard deviation of the comparison stars used. We experimented with the choice of how many comparison stars, and which ones to use as well as the aperture size. We find that the choice of comparison stars does not affect the overall shape of the AGN light curve, but does have an important impact on the S/N. The comparison stars used to produce the final lightcurves were chosen to give the lowest fractional standard deviation. The comparison stars are between a factor of 2 – 4 brighter than the AGN in the [*g, r, i*]{} and $z$ bands. We add, in quadrature, the largest fractional standard deviation from the selected comparison stars to the statistical uncertainty in the relative AGN flux (though note that the fractional standard deviation is comparable for each of the comparison stars). For the [*g, r, i*]{}, and $z$ filters we use the same 4 comparison stars for all detectors. However, since the throughput in the $u$ band is much lower, and most stars are typically redder, we found more reliable photometry from choosing a different set of comparison stars for this band. Despite this, the field of view of the OGG detector is significantly smaller than the other telescopes, forcing us to use a different set of comparison stars for the $u$ band. Again, we find this does not change the shape of the lightcurve, and only effects the S/N. The systematic and statistical uncertainties are of the same order, and on average we get better than 1% photometry in the [*g, r*]{}, and $i$ filters for all telescopes/detectors. In the $u$ filter the mean uncertainty is 2.4%. In the $z$ filter it varies by telescope/detector (see more below).
The Lijiang 2-m data were analyzed separately, but also using relative photometry. That analysis made use of 3 comparison stars, with a circular aperture of radius 99. The background rate was extracted from an annulus from 113 to 141.
![$g$-band light curves of Mrk 142 from each of the ground-based telescopes. Panels (a) and (b) show the LCO light curves from the OGG (purple) and ELP (red) telescopes, panel (c) shows the Liverpool Telescope (blue) light curve, while panel (d) shows the Zowada Observatory (gray) light curve. Panel (e) shows the combined light curve from all telescopes,[]{data-label="fig:scalelc"}](lc_g_all.pdf){width="0.99\columnwidth"}
Combining the lightcurves from each telescope and detector requires adding small shifts and scaling of the individual light curves to account for differences in bandpass and sensitivity for each combination of telescope and detector for a given filter. In order to perform this intercalibration of the light curves we use the Bayesian method described by @li14[^3]. This method fits a damped-random walk model to all datasets simultaneously, allowing for a shift and scaling of each dataset in order to optimize the intercalibration. It also takes into account the uncertainties on the best-fitting shift and scale parameters, increasing the uncertainties on the data points accordingly. We show an example of the separate and combined light curves for the $g$-band only in Fig. \[fig:scalelc\].
The mean uncertainty in the $z$ band is typically 1.5%, however, this longest-wavelength band is also where the variability amplitude is lowest and on par with the flux uncertainty. When performing the time-lag analysis (see Section \[sec:timeseries\]) we find that including all telescopes/bands combined gives significant scatter and leads to a poorly constrained lag measurement. We explore the lags from the lightcurves from each telescope separately, finding that they are all consistent within 1$\sigma$, however, all are poorly constrained aside from the OGG $z$-band lightcurve. We therefore opt to use only the OGG $z$-band lightcurve in all subsequent analysis since it is the highest quality (0.6% mean photometric uncertainty) and thus provides the best-constrained lag measurement alone. The light curves for all wavebands can be seen in Fig. \[fig:lc\]. All lightcurves are given in Tab. \[tab:lc\].
[ccccc]{} \[tab:summary\] X-ray & [*Swift*]{} & 58484.3 – 58603.9 & 185 & 1.55\
[*UVW2*]{} & [*Swift*]{} & 58484.3 – 58603.9 & 149 & 1.25\
[*UVM2*]{} & [*Swift*]{} & 58484.4 – 58602.5 & 146 & 1.24\
[*UVW1*]{} & [*Swift*]{} & 58484.3 – 58603.9 & 151 & 1.26\
$U$ & [*Swift*]{} & 58484.3 – 58603.9 & 154 & 1.29\
$B$ & [*Swift*]{} & 58484.3 – 58603.9 & 168 & 1.40\
$V$ & [*Swift*]{} & 58484.4 – 58602.5 & 159 & 1.35\
$V$ & Lijiang & 58413.9 – 58655.6 & 62 & 0.26\
$u$ & Liverpool & 58486.0 – 58595.9 & 60 & 0.55\
& LCO, OGG & 58483.5 – 58653.3 & 78 & 0.46\
& LCO, ELP & 58467.3 – 58555.4 & 42 & 0.48\
& Zowada & 58422.4 – 58636.2 & 107 & 0.50\
& All & 58422.4 – 58653.3 & 287 & 1.24\
$g$ & Liverpool & 58486.0 – 58595.9 & 63 & 0.57\
& LCO, OGG & 58483.5 – 58653.3 & 84 & 0.49\
& LCO, ELP & 58467.3 – 58653.1 & 80 & 0.43\
& Zowada & 58422.5 – 58633.2 & 134 & 0.64\
& All & 58422.5 – 58653.3 & 361 & 1.56\
$r$ & Liverpool & 58486.0 – 58595.9 & 62 & 0.56\
& LCO, OGG & 58483.5 – 58653.3 & 84 & 0.49\
& LCO, ELP & 58467.3 – 58653.1 & 78 & 0.42\
& Zowada & 58424.5 – 58633.2 & 134 & 0.64\
& All & 58424.5 – 58653.3 & 358 & 1.56\
$i$ & Liverpool & 58486.0 – 58595.9 & 63 & 0.57\
& LCO, OGG & 58483.5 – 58653.3 & 84 & 0.49\
& LCO, ELP & 58467.3 – 58653.1 & 77 & 0.41\
& Zowada & 58424.5 – 58633.2 & 131 & 0.63\
& All & 58424.5 – 58653.3 & 355 & 1.55\
$z$ & LCO, OGG & 58483.5 – 58653.3 & 81 & 0.48\
{width="85.00000%"}
[ccccc]{} \[tab:lc\] 58484.348 & X-ray & 0.096 & 0.011 & Swift\
58484.814 & X-ray & 0.111 & 0.012 & Swift\
58485.139 & X-ray & 0.241 & 0.017 & Swift\
58485.934 & X-ray & 0.134 & 0.014 & Swift\
58486.065 & X-ray & 0.212 & 0.015 & Swift\
Time Series Analysis {#sec:timeseries}
====================
The lightcurves all show significant variability and are correlated – the same prominent structures (peaks and troughs) can generally be seen at all wavelengths. We therefore proceed to measure the time lags between the different wavebands. We measure all time lags with respect to the [*Swift*]{}/[*UVW2*]{} light curve. As the shortest-wavelength UV/optical band, with the highest variability amplitude (see variability amplitudes, $F_{\rm var}$, in Tab. \[tab:lags\]), [*UVW2*]{} is the natural choice for the reference band. There are two methods typically used to measure lags between UV/optical AGN lightcurves: the interpolated cross-correlation function (ICCF) combined with flux randomization and random subset sampling [FR/RSS; as implemented by @petersonetal04] and the [Javelin]{} analysis package [@zu11; @zu13]. Several recent works noted that uncertainties determined by ICCF were approximately two times larger than those determined by [Javelin]{} [e.g. @edelson19]. This motivated @yu19, who performed a detailed comparison of the two methods through extensive simulations. Their conclusion was that [Javelin]{} generally produces a more realistic estimate of the uncertainties than the ICCF method. Here, we present lags and uncertainties determined via both methods, which we briefly describe in more detail.
For the ICCF method, we create many realizations of each of the lightcurves following the flux randomization and random subset sampling approach. The data points in the lightcurve are randomly selected with replacement, meaning that some points are selected multiple times while others are not selected at all. Error bars for data points are scaled appropriately for the number of times they are selected. Gaussian noise is then added to the data, with a mean equal to the observed flux, and standard deviation equal to the error bar. The cross-correlation function of the realization is then calculated by linearly interpolating one lightcurve, then the other, and averaging the two CCFs. The peak and centroid of the CCF is then determined. The centroid is calculated using CCF values higher than 80% of the peak value. This process is repeated $N=10,000$ times, leading to CCF centroid and peak distributions from which the median and uncertainties are determined (using the 16% and 84% quantiles).
[Javelin]{} models the variability of the lightcurves assuming a damped random walk prior constrained by the observed fluxes, and that the responding lightcurve is a delayed, blurred version of the reference lightcurve. The transfer function connecting the two lightcurves is assumed to be a top-hat function. [Javelin]{} fits the lightcurves using a Markov Chain Monte Carlo algorithm, recovering the probability density distribution for the lightcurve and transfer function parameters. We limit the lags to be within $-10$ to $+10$ days but otherwise run [Javelin]{} with the default parameters.
The lags measured from both methods are given in Tab. \[tab:lags\] and are quoted in the observed frame. While we quote both the peak and centroid lags from the ICCF method, we only use the centroid lags in the following analysis. We also give the top-hat width from the [Javelin]{} fits. The right-hand panels of Fig. \[fig:lc\] show the CCFs as well as the lag distributions determined from both methods. Note that we determine the uncertainty in the reference band by calculating the lag of the [*UVW2*]{} light curve with respect to itself. Tab. \[tab:lags\] also gives the fractional variability amplitude, $F_{\rm var}$ [@vaughan03], and the maximum correlation coefficient, $R_{\rm max}$, between the lightcurve of interest and the reference [*UVW2*]{} lightcurve. The lags generally increase with wavelength rising from $<$1 days in the UV bands to $\sim$1.7–2.4 days in the $i/z$ bands (discussed more below). We note that the UVM2 lag is slightly negative (though consistent with zero within 1$\sigma$). The expected lag there is very small given the closeness in wavelength of the two filters, but we also note that the UVW2 filter has a larger red wing [@poole08] than the UVM2 filter, and thus may be more contaminated by longer wavelength light. The UVM2 lag (with respect to UVW2) is always observed to be consistent with zero within 1$\sigma$ [e.g., see Tab. 3 in @edelson19].
The UV/optical lightcurves (aside from the [*Swift/V*]{} band) are well-correlated with the [*UVW2*]{} band (with $R_{\rm max} > 0.73$), and the X-ray lightcurve is the least well correlated of all the bands with $R_{\rm max} = 0.54$. The [*Swift/V*]{} correlation is poor because the lightcurve is noisy – the uncertainties on the data points are approximately the same size as the variability amplitude. On the other hand, the X-ray lightcurve is poorly correlated with the [*UVW2*]{} because the well-measured rapid and large amplitude variations in the X-rays are absent from the [*UVW2*]{} lightcurve.
We explore this further by smoothing the X-ray lightcurve using a boxcar average and then recalculating the CCF with respect to the UVW2 lightcurve. We vary the width of the boxcar from 1 to 10 days and re-evaluate $R_{\rm max}$ and the lag. We find that smoothing significantly increases $R_{\rm max}$, with the strongest correlation of $R_{\rm max} = 0.74$ occurring with a boxcar width of 5 days. However, the smoothing does not significantly alter the UVW2 to X-ray lag, with the lag remaining consistent within 1$\sigma$.
Finally, we test splitting the [*Swift*]{} X-ray lightcurve up into soft (0.3 – 1.5 keV) and hard (1.5 – 10 keV) energies, and find that the [*UVW2*]{} lags of the two bands are consistent within 1$\sigma$, and therefore we do not explore these separate energy bands any further.
[lcCCCCcc]{} \[tab:lags\] X-ray & 0.3 – 10 keV & -2.09\_[-0.40]{}\^[+0.34]{} & -2.2\_[-0.6]{}\^[+0.9]{} & -1.74\_[-0.23]{}\^[+0.27]{} & 4.93\_[-0.53]{}\^[+0.50]{} & $0.533\pm0.007$ & 0.54\
[*UVW2*]{} & 1928Å& 0.000.20 & 0.00.1 & 0.000.01 & 0.07\_[-0.05]{}\^[+0.08]{} & $0.101\pm0.002$ & 1.00\
[*UVM2*]{} & 2236Å& -0.150.24 & -0.2\_[-0.1]{}\^[+0.4]{} & -0.150.12 & 1.57\_[-0.54]{}\^[+0.51]{} & $0.097\pm0.003$ & 0.95\
[*UVW1*]{} & 2600Å& 0.180.29 & 0.3\_[-0.5]{}\^[+0.1]{} & 0.160.13 & 1.29\_[-0.84]{}\^[+0.94]{} & $0.084\pm0.003$ & 0.92\
Swift, $U$ & 3467Å& 1.090.39 & 0.7\_[-0.3]{}\^[+0.1]{} & 0.590.11 & 0.24\_[-0.17]{}\^[+0.34]{} & $0.073\pm0.003$ & 0.88\
$u$ & 3540Å& 0.870.30 & 0.8\_[-0.6]{}\^[+0.7]{} &0.69\_[-0.12]{}\^[+0.13]{} & 3.58\_[-0.45]{}\^[+0.48]{} & $0.062 \pm 0.002$ & 0.87\
Swift, $B$ & 4392Å& 0.92\_[-0.53]{}\^[+0.50]{} & 0.3\_[-0.5]{}\^[+0.9]{} & 0.62\_[-0.19]{}\^[+0.24]{} & 1.32\_[-0.88]{}\^[+1.44]{} & $0.051\pm0.003$ & 0.80\
$g$ & 4770Å& 0.75\_[-0.20]{}\^[+0.23]{} & 0.6\_[-0.2]{}\^[+0.4]{} & 0.58\_[-0.09]{}\^[+0.11]{} & 1.70\_[-0.28]{}\^[+0.48]{} & $0.044 \pm 0.001$ & 0.91\
$V$ & 5383Å& 0.32\_[-0.54]{}\^[+0.44]{} & 0.6\_[-0.4]{}\^[+0.3]{} & 0.66\_[-0.20]{}\^[+0.24]{} & 0.32\_[-0.20]{}\^[+0.44]{} & $0.025\pm0.001$ & 0.90\
Swift, $V$ & 5468Å& 1.40\_[-1.20]{}\^[+1.50]{} & 0.9\_[-0.7]{}\^[+1.5]{} & 1.14\_[-0.61]{}\^[+0.74]{} & 2.99\_[-2.22]{}\^[+4.02]{} & $0.021\pm0.008$ & 0.59\
$r$ & 6215Å& 1.380.27 & 1.2\_[-0.6]{}\^[+0.5]{} & 0.97\_[-0.06]{}\^[+0.18]{} & 0.009\_[-0.006]{}\^[+0.03]{} & $0.028\pm0.001$ & 0.87\
$i$ & 7545Å& 1.85\_[-0.29]{}\^[+0.33]{} & 1.7\_[-0.4]{}\^[+0.6]{} & 1.69\_[-0.17]{}\^[+0.13]{} & 0.05\_[-0.04]{}\^[+5.97]{} & $0.024\pm0.001$ & 0.84\
$z$ & 8700Å& 2.08\_[-0.73]{}\^[+0.64]{} & 2.0\_[-1.0]{}\^[+0.8]{} & 2.42\_[-0.19]{}\^[+0.15]{} & 0.31\_[-0.26]{}\^[+6.20]{} & $0.019\pm0.001$ & 0.73\
Lag-wavelength relation
-----------------------
For a standard thin accretion disk the lags are expected to follow $\tau \propto \lambda^{4/3}$. We therefore fit this relation to the observed lags in Mrk 142 using the following form: $$\label{eq:lag}
\tau = \tau_0 \left[ \left(\lambda/\lambda_0\right)^\beta - 1.0 \right]$$ where $\lambda_0 = 1928$Å (the wavelength of the UVW2 band). We fit the relation both with $\beta = 4/3$ for a standard thin disk, $\beta = 2$ for a slim disk, and allowing $\beta$ to be a free parameter. Initial fits show that the X-ray to [*UVW2*]{} lag is significantly offset from the best-fitting trend through the UV/optical, and therefore we remove the X-ray point from the fits. Moreover, we find that the $u/U$ band lags also sit above the best-fitting relations (as has been seen in other objects), and therefore we also remove those points from the fits. The best-fitting parameters are given in Tab. \[tab:lagwave\], and are shown in Fig. \[fig:lagwave\].
The best-fitting slope is consistent with both $\beta = 4/3$ and $\beta=2$ for the ICCF lags, however, for the [Javelin]{} lags a better fit is achieved with $\beta = 2$ than with $\beta = 4/3$. The lag normalization parameter $\tau_0$ ranges from $\tau_0 = 0.07$ to $\tau_0 = 0.34$ days with the lower value from the [Javelin]{} lags, which are systematically shorter than the ICCF lags (aside from the Lijiang [*V*]{} band, and the [*z*]{} band). Noticing these lower [Javelin]{} lags we also investigate the width of the top-hat function, finding that in some cases the width is large. For instance, for the $u$ band we find a width of $\sim$4 days, meaning that a significant portion of the response has negative lags. We experiment with modifying the [Javelin]{} code to force positive time lags. For the $u$ band this increases the median lag from 0.59 days to 0.88 days, closer to the ICCF value. We do not pursue this further here.
The difference in slope determined by the ICCF and [Javelin]{} lags seems to be dominated by the $z$ band lag. The best-fit to the [Javelin]{} lags with the slope fixed at $\beta=4/3$ goes significantly below the $z$ band lag. While generally the [Javelin]{} lags are smaller than the ICCF lags, the $z$ band lag is larger. The combination of these leads to a larger slope when fitting the [Javelin]{} lags. Excluding the $z$-band lag the fit with $\beta=4/3$ (fixed) significantly improves, giving an acceptable fit with $\chi^2 = 8.5$ for 8 degrees of freedom. Given the strong dependence of the fits on the $z$ band lag – the light curve with the lowest variability amplitude, and a lower number of data points – we do not put too much weight on the implied larger slope from the [Javelin]{} fits.
Previous studies have found that de-trending the light curves can reduce/remove the X-ray offset [e.g., @mchardy14; @mchardy18]. We therefore explored de-trending the X-ray and UVW2 lightcurves using a linear fit, a quadratic fit, and a boxcar average (of various widths). We find that none of the de-trending methods significantly change the X-ray to UVW2 lag.
[lCcc]{} \[tab:lagwave\] ICCF & 0.340.04 & 4/3 (fixed) & 3.46 (9)\
ICCF & 0.130.01 & 2 (fixed) & 3.67 (9)\
ICCF & 0.230.23 & $1.60\pm0.68$ & 3.30 (8)\
[Javelin]{} & 0.310.01 & 4/3 (fixed) & 18.3 (9)\
[Javelin]{} & 0.120.01 & 2 (fixed) & 6.91 (9)\
[Javelin]{} & 0.070.03 & $2.36\pm0.28$ & 5.30 (8)\
{width="\textwidth"}
Spectral Analysis {#sec:spec}
=================
The structure of the disk can also be tested through an analysis of the spectrum of the variable component of the light curves. To perform spectral modeling we first flux-calibrated the ground-based light curves using the magnitudes of the comparison stars from the AAVSO Photometric All-Sky Survey DR10 [@henden18] for the [*u, g, r*]{} and $i$ bands, and the SDSS catalog for the $z$ band. The flux-calibrated lightcurves (Tab. \[tab:lc\]) were then corrected for Galactic absorption assuming $E(B-V) = 0.0136$ and the extinction law of @cardelli89, and shifted to the rest-frame flux.
We perform a modified version of the flux-flux analysis to separate the constant (galaxy) and variable (AGN) components [e.g. @cackett07; @starkey17; @mchardy18]. We fit the light curves using the following linear model:
$$f_\lambda (\lambda, t) = A_\lambda(\lambda) + R_\lambda(\lambda)X(t)
\label{eq:flfl}$$
Here $X(t)$ is a dimensionless light curve with a mean of 0 and standard deviation of 1. $A_\lambda(\lambda)$ is a constant for each light curve, while $R_\lambda(\lambda)$ is the rms spectrum. While this is a simplified model that does not take account of any time lags, the time lags only act to add scatter around the linear flux-flux relations. We estimate the minimum host galaxy contribution in each band by extrapolating the best-fitting relations to where the first band crosses $f_\lambda = 0$. In this case both the [*UVW2*]{} and [*UVM2*]{} bands cross $f_\lambda = 0$ at essentially the same value of $X$, which we denote $X_g$. The host-galaxy components in the other bands are then the best-fitting relation evaluated at $X = X_g$. Fig. \[fig:fluxflux\] shows the flux-flux relation ($X(t)$ vs $f_\lambda$) for each band. Note that the linear relation in Eq. \[eq:flfl\] provides a good fit over the full range of observed fluxes in Fig. \[fig:fluxflux\]. The absence of curvature here validates the assumption of a constant spectral shape for the variable light, and shows that any ‘bluer-when-brighter’ effect in Mrk 142 is entirely due to a relatively blue spectrum of the variable light being diluted by a relatively red non-variable spectrum.
In Fig. \[fig:sed\] we show the resulting spectral energy distributions. The maximum and minimum spectra are determined from evaluating the best-fitting relations at the brightest and faintest values of $X = X_B$ and $X_F$ respectively. The average spectrum is evaluated at $X = 0$. The host galaxy components are shown as $f_{\rm gal}$ and generally increase with wavelength, as expected for an old stellar population. The variable spectrum is plotted as both the maximum-minimum spectrum and the rms spectrum. These variable spectra decrease with wavelength, and are well represented by the $\lambda f_\lambda \propto \lambda^{-4/3}$ relation expected for a standard thin disk (dotted lines in Fig. \[fig:sed\]). If we allow the index to be a free parameter, we find $-1.36\pm0.01$, very close to the expected thin disk value of $-4/3$. Note that a slim disk should have a spectrum of $\lambda f_\lambda =$ constant [@wang99b], inconsistent with what we see here. Given the excess lags in the $U/u$ bands, we exclude those points from the spectral fits. The $U/u$ fluxes lie 12 and 17% above the best-fitting disk spectrum, putting additional constraints on contribution to the variable flux from the diffuse BLR.
![Flux-flux analysis: $f_\lambda$ vs $X(t)$ for each of the light curves. The best-fitting relations are shown as solid lines. $X_F$ and $X_B$ indicate the faint and bright values of $X(t)$ during the campaign. $X_g$ indicates the value of $X(t)$ where $f_\lambda = 0$ for the UVW2 band. The flux-flux relations for other bands evaluated at $X_g$ give the minimum galaxy contribution.[]{data-label="fig:fluxflux"}](Xt_flux.pdf){width="\columnwidth"}
![The spectral energy distribution of Mrk 142 from the flux-flux analysis. The maximum and minimum spectra are derived from the best-fitting relation evaluated at $X(t) = X_B$ and $X_F$, respectively, while the average spectrum comes from $X(t) = 0$. The galaxy spectrum ($f_{\rm gal}$, orange squares) is derived from the best-fitting relation at $X(t) = X_g$. The rms spectrum comes from the slope of the best-fitting relation. The dotted lines indicate the best-fitting thin disk spectrum, $\lambda f_\lambda \propto \lambda^{-4/3}$, excluding the $U/u$ bands. The variable spectrum of Mrk 142 is consistent with a standard thin accretion disk.[]{data-label="fig:sed"}](fluxflux_sed.pdf){width="\columnwidth"}
Discussion {#sec:discuss}
==========
We have monitored the super-Eddington AGN Mrk 142 for 4 months with [*Swift*]{} with an average sampling rate of better than once per day. Moreover, we obtained ground-based photometric monitoring in the Sloan [*ugriz*]{} filters over approximately 230 days, overlapping with the [*Swift*]{} monitoring. By combining light curves from multiple telescopes around the globe (LCO, Liverpool, and Zowada) we obtain an average sampling rate of 1.6 observations per day in the $g$ band – comparable to that obtained with [*Swift*]{}.
Mrk 142 was highly variable, with a variability amplitude as high as 53% in the X-ray, dropping to 10% in the [*Swift*]{}/[*UVW2*]{} band (1928Å) and 1.9% in $z$ ($\sim$9000Å). All the UV/optical bands are highly correlated with the [*UVW2*]{} light curve, with their maximum correlation coefficients all above 0.75 (aside from the noisy [*Swift/V*]{} band). However, while the 0.3 – 10 keV X-ray band shows a number of features that are apparent in the longer wavelength lightcurves, it shows much more variability on shorter timescales ($\sim$few days) that is not apparent at longer wavelengths, and has a significantly lower peak correlation coefficient of 0.54 (with respect to the [*UVW2*]{} band). Smoothing the X-ray lightcurve with a boxcar average of width 5 days removes the short timescale variability and leads to an increased peak correlation coefficient of 0.74 without affecting the lag measurement. While a significantly higher correlation, it remains weaker than correlations between the UVW2 and the highest quality UV/optical lightcurves.
The goal of the intensive photometric monitoring campaign was to perform the first continuum reverberation mapping of a super-Eddington AGN to test whether the accretion disk structure is notably different from that of previously-studied sub-Eddington AGN. We therefore determine time lags between the [*Swift*]{}/UVW2 and other light curves. We find that the lags increase with wavelength, approximately following $\tau \propto \lambda^{4/3}$, though they can also be fit with $\tau \propto \lambda^{2}$ (but we caution that this is dependent on the $z$ band lag). There are noticeable outliers, with the X-ray to UV time delay significantly longer than an extrapolation of the best-fit through the UV and optical. Moreover, the $u/U$ lags are also significantly offset from the general trend with wavelength. Rather surprisingly given the significantly higher mass accretion rate of Mrk 142, the main observational results – that approximately $\tau \propto \lambda^{4/3}$; the X-ray offset and poor correlation with the UV/optical; and the enhanced $u/U$ lags – are seen in all the other high-cadence [*Swift*]{} monitoring campaigns to date on sub-Eddington AGN [@edelson15; @edelson17; @edelson19; @fausnaugh16; @cackett18; @mchardy18].
The origin of the poor X-ray/UV correlation remains unclear. In comparing NGC 5548, NGC 4151, NGC 4593 and Mrk 509, @edelson19 noted that they all show a poorer correlation between the X-rays and the UV than the UV and the optical. This is hard to reconcile with a picture where the X-rays directly irradiate the UV/optical part of the accretion disk driving the variability. Even more puzzling are objects such as Mrk 817, where no correlation at all is seen between the X-rays and the UV [@morales19], despite this being quite a typical object where broad emission line reverberation is observed. @gardnerdone17 explain the poor X-ray correlation in NGC 5548 through a vertically extended inner Comptonizing region that prevents the X-rays directly irradiating the disk. Even then, light travel time from this inner Comptonizing region is too short to explain the lags, and thus they suggest the lags may instead be a dynamical timescale for the outer disk to respond to changing FUV illumination. @edelson17 invoke a similar inner torus to explain the long X-ray to UV lags in NGC 4151. Here, in Mrk 142 this may also be a natural explanation – given the high mass accretion rate, the inner disk is expected to be slim (not thin) within the photon trapping radius and vertically extended. This might act in the way envisaged by @gardnerdone17. However, while such an inner-disk structure is expected for slim-disk models for super-Eddington AGN it remains a puzzle as to why the sub-Eddington AGN exhibit the same phenomenon.
The large X-ray to UV lag has important implications for the size of the BLR too. If the X-rays are a good proxy for the driving light curve, then the 2 day lag between X-rays and [*UVW2*]{} variations would imply that the BLR size in @Du16 is underestimated by $\sim$40%. @Du16 measure a H$\beta$ lag with respect to the 5100Å continuum of approximately 8 days. Since the [*UVW2*]{} to 5100Å lag is approximately 1 day, and the X-ray to [*UVW2*]{} lag is about 2 days, the ‘true’ H$\beta$ lag would be 11 days. If, however, the [*UVW2*]{} band is closer to the driving continuum, then the BLR size is only underestimated by $\sim$10% (8 vs. 9 days). Therefore, it is important to develop a better understanding of what band is driving the optical variability, since it has implications for BLR size and hence black hole mass estimates.
The excess lag in the $U/u$ bands is thought to be due to continuum emission arising in the BLR. This emission, which contributes to the observed continuum over a broad range in wavelengths from the UV to the near-IR, has a significant discontinuity at the Balmer jump (3646Å) [@koristagoad01; @lawther18; @koristagoad19] and therefore leads to an increase in the lags particularly around that wavelength. $U/u$ band excesses have been seen in NGC 5548 [@edelson15; @fausnaugh16], NGC 4151 [@edelson17], NGC 4593 [@cackett18; @mchardy18], and Mrk 509 [@edelson19]. The UV spectroscopic observations of NGC 4593 were particularly powerful in highlighting this, showing a broad excess in the lags around the Balmer jump [@cackett18], rather than from just a single broadband photometric filter. The $U/u$ band excess observed here in Mrk 142 likely has the same origin due to continuum emission from the BLR. @edelson19 compared the magnitude of the $U$ excesses in four objects, finding that on average the excess was a factor of 2.2 larger than expected from the best-fitting lag-wavelength relation, with values ranging from 1.6 to 2.9. For Mrk 142 we find that the $U/u$ lags are on average a factor of 2.4 larger than the best-fitting lag-wavelength relation, consistent with the results of @edelson19. Properly assessing the impact of the BLR continuum on the lags requires careful spectral deconvolution and light curve simulations [e.g., @koristagoad19], which should be possible for Mrk 142 once analysis of the optical spectra from this campaign is completed in the future.
Another consideration is how the normalization of the lag-wavelength relation compares to the expectations from assuming a standard Shakura-Sunyaev disk [@shakurasunyaev] with temperature profile $T \propto R^{-3/4}$. To do this, we use Eq. 12 from @fausnaugh16 for the normalization of the lag-wavelength relation, $\tau_0$, which we reproduce here: $$\tau_0 = \frac{1}{c}\left( X \frac{k \lambda_0}{hc} \right)^{4/3} \left[ \left(\frac{GM}{8 \pi \sigma}\right)\left(\frac{L_{\rm Edd}}{\eta c^2}\right)(3 + \kappa)\dot{m}_{\rm E}\right]^{1/3} \; .
\label{eq:faus}$$ In this equation $\eta$ is the accretion efficiency, $X$ is a factor for converting from $\lambda$ to $T$ for a given radius, $\kappa$ is the local ratio of external to internal heating, and $\dot{m}_{\rm E} = L_{\rm bol}/L_{\rm Edd}$. Here, we assume a flux-weighted value for $X = 2.49$ (though note response-weighted values will be larger), $\kappa = 1$ and a black hole mass of $M = 1.7\times10^6 $ M$_\odot$ [@li18]. To compare with the observations we first consider several estimates for the bolometric luminosity, $L_{\rm bol}$, given that bolometric corrections can sometimes be highly uncertain. First, we determine $L_{\rm bol}$ using $L_{\rm bol} = 9\lambda L_\lambda$ (5100Å) [@kaspi00]. Since we do not directly measure the 5100Å flux, we use the [*Swift*]{} $V$-band flux as an estimate. From the flux-flux analysis we determine an average host-galaxy subtracted rest-frame flux of $8.3\times10^{-16}$ erg cm$^{-2}$ s$^{-1}$ Å$^{-1}$. This leads to $L_{\rm bol} = 1.85\times10^{44}$ erg s$^{-1}$ for a luminosity distance of $D_L = 201.5$ Mpc, and $L_{\rm bol}/L_{\rm Edd} = 0.86$ (for a black hole mass of $M = 1.7\times10^6 $ M$_\odot$). Alternatively, we can use the observed 2 – 10 keV X-ray flux and the bolometric correction of @marconi04. Using the average X-ray spectrum from [*Swift*]{} we measure a 2 – 10 keV flux of $1.9\times10^{-12}$ erg s$^{-1}$ cm$^{-2}$, which in turn leads to $L_{\rm bol} = 1.6\times10^{44}$ erg s$^{-1}$ and $L_{\rm bol}/L_{\rm Edd} = 0.74$. Finally, we can use the observed host galaxy-subtracted 5100Å luminosity combined with the Shakura-Sunyaev disk model itself to estimate the dimensionless mass accretion rate $\dot{\mathscr{M}}$ following Eq. 2 in @Du15. $\dot{\mathscr{M}}$ relates to the Eddington ratio via $L_{\rm bol}/L_{\rm Edd} = \eta \dot{ \mathscr{M}}$. We get $\dot{\mathscr{M}} = 100$ during this campaign. To convert to an Eddington ratio we must assume some accretion efficiency, but, this is expected to drop with increasing mass accretion rate for slim disk models [@wang99; @mineshige00; @sadowski11]. Using the formulation of @mineshige00 we determine $\eta = 0.034$ and $L_{\rm bol}/L_{\rm Edd} = 3.4$ for $\dot{\mathscr{M}} = 100$. Thus, the three estimates give a range of 0.74 to 3.4 for $L_{\rm bol}/L_{\rm Edd}$.
Using these estimates for $L_{\rm bol}/L_{\rm Edd}$ and $\eta$ we can now compare the observed and predicted values for $\tau_0$. Under the assumptions above, taking $L_{\rm bol}/L_{\rm Edd} = 3.4$, and $\eta = 0.034$ we predict $\tau_0 = 0.1$ days. In other words, the observed $\tau_0$ (assuming $\beta = 4/3$; see Tab. \[tab:lagwave\]) is a factor of 3.1 to 3.4 larger than predicted from the standard disk model and our largest estimate of $L_{\rm bol}/L_{\rm Edd}$. This discrepancy between observed and predicted disk size is comparable to what is seen in other objects, [e.g., @edelson19 and references therein]. Either a significantly higher Eddington ratio, or lower accretion efficiency would be needed to reconcile the model lags, in other words, the magnitude of the lags is consistent with a highly super-Eddington accretion rate. However, we recognize that since Eq. \[eq:faus\] is for a standard sub-Eddington disk it would no longer be applicable.
The discrepancy between the observed and predicted disk size is similar to the issue in sub-Eddington objects – for reasonable accretion rates, the magnitude of the predicted lags is a factor of a few smaller than observed [e.g., @edelson15; @edelson19; @cackett18; @mchardy14; @mchardy18]. Solutions that have been proposed for sub-Eddington objects include inhomogeneous accretion disks [@dexter11], a tilted inner disk [@starkey17], that the lags are due to a dynamical timescale for the outer disk to respond to changing FUV illumination [@gardnerdone17], that the X-ray source is located higher above the disk than usually assumed [@kammoun19], or that the lags are due to disk turbulence [@cai20]. Continuum emission from the BLR will also contribute, or even dominate, the observed lag [@koristagoad01; @koristagoad19; @lawther18; @chelouche19]. Those same solutions could work here also. Alternatively, for higher-mass accretion rate objects there may be other solutions. For instance, the model used for the lag-wavelength relation assumes a standard optically-thick geometrically-thin accretion disk, and so will be not applicable if the disk is instead a slim disk. At high mass accretion rates the inner region of a slim disk is expected to be geometrically thick and will create an anisotropic radiation field that is not taken into account here.
Additonal tests of the disk structure can be performed through analysis of the variable spectrum. Thus, we also performed a flux-flux analysis to decompose the observed spectrum into constant and variable components. We found that the spectrum of the variable component is well-represented by $\lambda f_\lambda \propto \lambda^{-4/3}$, as expected for a standard thin disk. The variable spectrum is not consistent with a slim disk. The constant component increases with wavelength, as expected for stellar population. The $U/u$ band fluxes are enhanced by a little over 10% with respect to the best-fitting $\lambda^{-4/3}$ relation, which can be used to constrain any flux due to continuum emission from the BLR.
Since the variable spectrum is consistent with a thin disk (and rules out a slim disk in the UV/optical), and given that the UV/optical lags can be fit by $\tau \propto \lambda^{4/3}$, this has implications for the accretion disk structure at such high Eddington ratios. In the slim disk model [e.g., @abramowicz88] the accretion disk increases in scale height within the photon trapping radius. Observations of the BLR support this, with higher mass accretion rate objects falling significantly below the radius-luminosity relation, as would be expected if an inflated inner disk was shadowing it [@Du15; @Du16; @du18]. Broad-line reverberation of Mrk 142 shows that it also falls below the radius-luminosity relation [@Du16], suggesting it contains a slim disk. If the accretion disk in Mrk 142 is a slim disk, then our spectral analysis shows that since the optical/UV emitting part looks like a thin disk, then the inflated inner disk must be well within the region producing the UV emission we observe with the [*Swift*]{}/[*UVW2*]{} (1928Å). We can therefore put observational constraints on the size of the photon-trapping radius by assuming that the extrapolation of the UV/optical lags ($\tau_0$) sets the maximum extent of the photon-trapping region. For our largest $\tau_0$ estimate of 0.34 days this corresponds to the light travel time for a distance of $1\times10^{13}$m, or for a black hole mass of $M = 1.7\times10^6 $M$_\odot$, it corresponds to approximately $4\times10^3$$R_{\rm g}$ (where $R_{\rm g} = GM/c^2)$.
From a theoretical perspective, according to the self-similar solution [@wang99], the trapping radius is given by $$\frac{R_{\rm tr}}{R_{\rm g}}=4.5\times 10^{2}\left(\frac{\dot{\mathscr{M}}}{250}\right),$$ and the effective temperature is $$T_{\rm eff}=9.4\times 10^{5}\left(\frac{M}{10^{6}M_{\odot}}\right)^{-1/4}
\left(\frac{R}{R_{\rm g}}\right)^{-1/2},$$ Using Wien’s law to go from temperature to wavelength we have $$\frac{R}{R_{\rm g}}=1.1\times 10^{3}\left(\frac{M}{10^{6}M_{\odot}}\right)^{-1/2}
\left(\frac{\lambda}{1000{\rm \AA}}\right)^{2},$$ giving the trapping radius in terms of wavelength as: $$\lambda_{\rm tr}=0.65\times 10^{3}\left(\frac{M}{10^{6}M_{\odot}}\right)^{1/4}
\left(\frac{\dot{\mathscr{M}}}{250}\right)^{1/2}\,{\rm \AA}.$$ These scaling relations show that $\tau\propto \lambda^{2}$ in the trapping region, which is steeper than the standard disk model. While this is consistent with the lags we observe in Mrk 142, the spectrum of the variable component is far from the flat spectrum expected for a slim disk, and very well-fit by a standard thin disk spectrum. Moreover, for Mrk 142 with $M=1.7\times 10^{6}$M$_{\odot}$ and $\dot{\mathscr{M}}\approx 100$ (from our estimate above), the optical and UV photons are not trapped ($\lambda_{\rm tr} = 462$Å), but the soft X-ray photons should be trapped. This appears to be consistent with the X-ray offset and poor X-ray/UV correlation. However, as noted above, while an inner geometrically-thick region is expected for slim disks, it is not expected in standard geometrically-thin disks in sub-Eddington sources and thus it remains a puzzle as to why those sources also show an X-ray offset and poor X-ray/UV correlation.
In summary, the high cadence, multi-wavelength photometric monitoring of Mrk 142 has provided a rare opportunity to place observational constraints on the accretion flow at super-Eddington rates.
EMC and JM gratefully acknowledge support for the [*Swift*]{} analysis from NASA through grant 80NSSC19K0150, and support for analysis of the ground-based data from the NSF through grant AST-1909199. EMC and Wayne State University deeply thank the 419 Foundation for donating the Dan Zowada Memorial Observatory. We are also grateful to Terry Friedrichsen, Philip Moores, Nick Paizis, and Dennis Recla for help with maintenance at the Zowada Observatory. Without them we would not be able to operate as many nights as we do.
JMG and RE gratefully acknowledge support from NASA under the ADAP award 80NSSC17K0126. KH acknowledges support from STFC grant ST/R000824/1. Research by AJB is supported by NSF grant AST-1907290. LCH was supported by the National Science Foundation of China (11721303, 11991052) and the National Key R&D Program of China (2016YFA0400702). JMW acknowledges financial support from the the National Science Foundation of China (11833008 and 11991054), from the National Key R&D Program of China (2016YFA0400701), from the Key Research Program of Frontier Sciences of the Chinese Academy of Sciences (CAS; QYZDJ-SSW-SLH007), and from the CAS Key Research Program (KJZD-EW-M06). PD acknowledges financial support from the the National Science Foundation of China (11873048 and 11991051) and from the Strategic Priority Research Program of the CAS (XDB23010400). CH acknowledges financial support from the the National Science Foundation of China (11773029). YRL acknowledges financial support from the the National Science Foundation of China (11922304),from the Strategic Priority Research Program of the CAS (XDB23000000), and from the Youth Innovation Promotion Association CAS. BL acknowledges financial support from the National Science Foundation of China grant 11991053. YFY is supported by National Natural Science Foundation of China (Grant No. 11725312, 11421303). This work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester.
[^1]: <https://lco.global/observatory/instruments/filters/>
[^2]: <https://www.swift.ac.uk/user_objects/>
[^3]: The code is publicly available here: <https://github.com/LiyrAstroph/CALI>
|
---
abstract: 'We describe a chip-scale, telecommunications-band frequency conversion interface designed for low-noise operation at wavelengths desirable for common single photon emitters. Four-wave-mixing Bragg scattering in silicon nitride waveguides is used to demonstrate frequency upconversion and downconversion between the 980 nm and 1550 nm wavelength regions, with signal-to-background levels $>10$ and conversion efficiency of $\approx -60$ dB at low continuous wave input pump powers ($<50$ mW). Finite element simulations and the split-step Fourier method indicate that increased input powers of $\approx$10 W (produced by amplified nanosecond pulses, for example) will result in a conversion efficiency $>25~\%$ in existing geometries. Finally, we present waveguide designs that can be used to connect shorter wavelength (637 nm to 852 nm) quantum emitters with 1550 nm.'
address: |
$^1$Center for Nanoscale Science and Technology, National Institute of Standards and Technology, Gaithersburg, MD 20899\
$^2$Maryland NanoCenter, University of Maryland, College Park, MD 20742\
$^3$Department of Applied Physics, California Institute of Technology, Pasadena, CA 91125\
$^{4}$These authors contributed equally.\
$^{5}$[[email protected]]{}\
$^{\ast}$[[email protected]]{}
author:
- 'Imad Agha,$^{1,2,4,5}$ Serkan Ates,$^{1,2,4}$ Marcelo Davanço,$^{1,3}$ and Kartik Srinivasan$^{1,\ast}$'
title: 'A chip-scale, telecommunications-band frequency conversion interface for quantum emitters '
---
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Introduction
============
Interfaces that connect photonic quantum systems operating at different frequencies constitute an important resource in the development of a hybrid architecture that leverages the key advantages of its component entities. Such hybrid architectures may be at the heart of modern quantum networks, which must be able to generate, send, manipulate, and store quantum information with high fidelity and low loss. For example, single photon sources (SPSs) based on single quantum emitters like InAs quantum dots [@ref:Michler_book_2009], nitrogen vacancy centers in diamond [@ref:Kurtsiefer], and neutral alkali atoms [@ref:McKeever] all exhibit desirable features, such as on-demand generation with the potential for high single photon purity and indistinguishability. These properties are crucial in both quantum cryptography [@ref:Grangier_QKD] and quantum computing [@ref:Knill; @ref:OBrien_Furusawa_Vuckovic] applications. Unfortunately, for these common quantum emitters, emission occurs at wavelengths below 1000 nm, where long-distance transmission through optical fibers is not optimal. Similarly, many promising quantum memories [@ref:Simon_Qmemory_review] operate in the visible or near-visible, whereas silicon-based systems that have been developed for applications in nanophotonics, CMOS electronics, and microelectromechanics are opaque at such wavelengths. Thus, interfaces that can connect disparate wavelength regions without otherwise disturbing the relevant properties of the quantum state (such as photon statistics and coherence time) are an important resource for future photonic quantum information processing systems. Such quantum frequency conversion [@ref:Kumar_OL; @ref:Raymer_Srinivasan_PT] has been demonstrated with single photon Fock states generated by single semiconductor quantum dots [@ref:Rakher_NPhot_2010; @ref:Zaske_Becher_downconversion; @ref:Ates_Srinivasan_PRL; @ref:de_Greve_Yamamoto_Nature], where three-wave-mixing in periodically-poled lithium niobate waveguides was used for both upconversion [@ref:Rakher_NPhot_2010; @ref:Ates_Srinivasan_PRL] and downconversion [@ref:Zaske_Becher_downconversion; @ref:de_Greve_Yamamoto_Nature].
Future applications may benefit from the development of such frequency conversion technology within silicon-based material systems, for which complex and highly scalable fabrication technology at a level beyond lithium niobate has been demonstrated. To that end, frequency conversion through four-wave-mixing Bragg scattering (FWM-BS) [@ref:McKinstrie], a process used in quantum frequency conversion experiments in optical fibers [@ref:McGuinnes_PRL10], was recently demonstrated in silicon nitride (Si$_3$N$_4$) waveguides [@ref:Agha_OL], where Si$_3$N$_4$ was chosen due to its wide optical transparency window (including visible and near-visible wavelengths of importance to many quantum optical systems), its relatively high nonlinear refractive index, and low linear and nonlinear absorption. In Ref. [@ref:Agha_OL], narrowband conversion over a few nanometers was measured, and the low-noise nature was inferred by measuring the pump-induced noise. Here, we show the flexibility of FWM-BS in these Si$_3$N$_4$ devices by demonstrating wideband downconversion and upconversion between the 980 nm and 1550 nm bands, with a signal-to-noise ratio $\gtrsim10$ as determined via photon counting measurements. Through calculations of waveguide dispersion and simulations using the split-step Fourier method, we further show that simple variations in the waveguide geometry should extend this technique to enable quantum emitters in the 637 nm, 780 nm, and 850 nm wavelength bands to be connected to the 1550 nm band. Finally, as the demonstrated conversion efficiencies are low ($\approx-60$ dB) due to relatively low pump power used ($<50$ mW), we use simulations to establish the pump powers required to achieve conversion efficiencies high enough for realistic experiments with single photon states of light. We find that pump powers of $\approx$10 W, achievable through amplified nanosecond pulses, for example, should enable conversion efficiencies $>25~\%$ to be achieved.
Principle of operation
======================
Four-wave-mixing Bragg scattering is a nonlinear mixing process involving four non-degenerate fields, whereby two pumps $\omega_1$ and $\omega_2$ mediate the scattering from a signal $\omega_s$ to an idler $\omega_i$. In this paper, we consider a process in which the idler frequencies are given by $\omega_{i}^{\pm}=\omega_2 \pm (\omega_s - \omega_1)$. In the slowly-varying envelope approximation, and ignoring the effects of pump-depletion, the conversion efficiency from the signal to the idler at a propagation distance $z$ along the waveguide is given by:
$$\eta (z) = \frac{4\gamma_1 \gamma_2 P_1 P_2}{g^2} \sin^2(gz)$$
where $\gamma_{1}$ and $\gamma_{2}$ are the waveguide’s effective nonlinearity parameter at the two pump wavelengths, $P_1$ and $P_2$ are the powers in pumps 1 and 2, respectively, and $g$ is the four-wave-mixing gain that depends both on the pump powers (through nonlinear dispersion) and the linear phase mismatch [@ref:Uesaka_Kazovksy]. A typical estimated value for the effective nonlinearity parameter is $\gamma\approx6$ W$^{-1}$m$^{-1}$ (depending on the specific waveguide cross-section and wavelength), which is determined assuming a nonlinear refractive index $n_{2}=2.5\times10^{-19}$ m$^2$W$^{-1}$[@ref:Ikeda_Fainman_silicon_nitride_nonlinear_OE]. By tailoring the dispersion appropriately, and by having the two pumps widely separated in wavelength, this process allows ultra-wide band frequency conversion. Moreover, due to the fact that this is a wavelength exchange process with no net amplification of either signal or idler, the conversion is theoretically noiseless and is suitable for quantum state translation [@ref:McKinstrie]. In reality, the noise limit may come either from poor filtering of the pumps or, in certain materials, from Raman scattering or other four-wave-mixing processes.
In this work, our main motivation is to develop an interface for frequency conversion from 980 nm to 1550 nm (InAs quantum dot SPSs to telecom wavelength) in a silicon nitride waveguide (Fig. \[fig:waveguide and schematics\](a)). Silicon nitride is chosen here due to CMOS fabrication compatibility [@ref:Levy_Lipson_comb], simple growth techniques, large transparency window, high optical nonlinearity (x10 that of SiO$_2$) [@ref:Tan_Fainman], and low nonlinear loss (absence of two-photon absorption and free-carrier generation). Details on the fabrication technique and dispersion calculations (Fig. \[fig:waveguide and schematics\](b)) can be found in Ref. [@ref:Agha_OL]. Unlike this earlier work, whereby we employed two telecom-band pumps to convert a signal to its idler in the 980 nm band, here we place one of pumps near 1550 nm and the other around 980 nm (Fig. \[fig:waveguide and schematics\](c,d)), allowing for downconversion (upconversion) over a $\approx 600$ nm range, a significantly larger wavelength translation range compared to wideband conversion recently demonstrated in optical fibers through FWM-BS [@ref:McGuinness_PTL]. As the nonlinear phase-mismatch scales as $\Delta\beta_{nl} = \gamma_1P_1-\gamma_2P_2$, where $\gamma_{1}$ ($\gamma_{2}$) is the nonlinear coefficient at $\omega_{1}$ ($\omega_{2}$) and $\Delta\beta_{nl}\approx\gamma(P_1-P_2)$ for two pumps in the same band [@ref:Uesaka_Kazovksy], we use the same design parameters as [@ref:Agha_OL] in our device fabrication. We produce 18 mm long, 550 nm x 1200 nm (height x width) Si$_3$N$_4$-on-SiO$_2$ waveguides with an air top cladding, and can adjust the ratio of pump powers to eliminate the nonlinear phase mismatch. In this experiment, a straight waveguide was employed, however, mature fabrication techniques (both in writing and etching the structure) and the index contrast between Si$_3$N$_4$ and SiO$_2$ allow for integrating the same length in a much smaller ($\approx 1$ mm) footprint via a spiral layout.
Experimental results - Wideband Frequency Downconversion
========================================================
The experimental setup for frequency downconversion of a 980 nm band signal to a telecom–band idler is shown in Fig. \[fig:downconversion\_data\](a). A strong (550 mW) pump consisting of a 974 nm distributed feedback laser is combined with a weak, tunable signal on a 90:10 coupler, and the combined output is sent to the 980 nm port of a 980 nm/1550 nm wavelength-division multiplexer (WDM). The signal power at the input of the waveguide is in the range of a few $\mu W$. The second pump is a 1550 nm laser that is amplified via an erbium-doped amplifier (EDFA) and sent to the other port of the WDM. The combined pumps and signal are coupled to the waveguide via a lensed fiber, and the output of the waveguide is filtered to reject both the residual (unconverted) signal and the pumps. This output is either sent to a spectrometer equipped with a cooled InGaAs photodiode array for spectral analysis, or is further bandpass-filtered and sent to an InGaAs single photon avalanche diode (SPAD) for signal-to-noise analysis. Data is synchronously recorded on the spectrometer while the 980 nm band signal laser is tuned from 965 nm to 985 nm in 0.2 nm steps and monitored on an optical spectrum analyzer (OSA).
![a) Experimental setup for frequency downconversion from 980 nm to 1550 nm: A tunable 980 nm signal laser is combined with a strong 974 nm distributed feedback (DFB) pump laser and then multiplexed (through a fiber 980 nm/1550 nm wavelength-division multiplexer (WDM)) with a 1550 nm laser that is amplified via an erbium-doped fiber amplifier (EDFA). Fiber polarization controllers (FPCs) ensure that the signal and pumps are co-polarized and launched into the desired waveguide mode. An optical spectrum analyzer (OSA) monitors the 980 nm wavelengths, while the output of the waveguide is first filtered through a notch filter (NF) to remove the residual 1550 nm pump, and then either sent to an InGaAs spectrometer or through a bandpass filter (BPF) to an InGaAs single photon avalanche diode (SPAD). b) Sample spectrum of the generated idlers together with an OSA trace of the 980 nm-band pump+signal. The position of the 1550 nm pump, which has been notch filtered, is denoted by a dashed line. c) Contour plot of the generated 1550 nm idler spectra (in units of detuning from the 1550 nm pump) for differing levels of 980 nm band pump-input signal detuning (taken with 0.2 nm tuning steps). d) Normalized idler power as a function of its wavelength for fixed pumps and tunable 980 nm input signal. The gray central region denotes the bandwidth of the 1550 nm pump rejection notch filter. e) Conversion efficiency (bottom) and converted idler and background count rate (top) as a function of the 1550 nm pump power inside the waveguide at its input. The error bars (often smaller than the data point size) are due to the variation in the detected SPAD count rates, and are one standard deviation values. The signal-to-background level for the converted idler increases from 5.6$~\pm~$2.5 at the lowest pump power to 9.6$~\pm~$2.4 at the highest pump power.[]{data-label="fig:downconversion_data"}](Figure2.eps){width="\linewidth"}
Figure \[fig:downconversion\_data\](b) shows the generated idlers in the 1550 nm band for a specific set of pump and input signal wavelengths in the 980 nm band (the 1550 nm pump has been removed by a notch filter after the WG chip, as shown in Fig. \[fig:downconversion\_data\](a)). Tuning the signal away from the 974 nm pump causes the idlers to tune in accordance with energy conservation $\omega_i=\omega_2 \pm (\omega_s - \omega_1)$, as shown in the image plot in Fig. \[fig:downconversion\_data\](c). Here, the x-axis is in units of detuning between the generated idlers and the fixed 1550 nm pump, while the y-axis is in units of detuning between the input 980 nm band signal and fixed 974 nm pump.
Figure \[fig:downconversion\_data\](d) shows the normalized conversion efficiency as a function of the idler wavelength (for fixed pump wavelengths). This plot shows both the phase matching bandwidth of our device as well as oscillations which are likely due to its expected sinc$^2$ character (Eq. 1). We next use an InGaAs SPAD to determine both the conversion efficiency as well as the signal-to-background ratio for the converted idler as a function of the 1550 nm pump power. The signal-to-background is determined by measuring the counts in the $\omega_{i}^{+}$ idler band with the input 980 nm signal on and off, and dividing the two values (after subtraction of the detector dark count rate of $\approx 150$ s$^{-1}$). As the pump power inside the waveguide and at its input is increased from 5 mW to 43 mW, the internal conversion efficiency (taking into account 7.5 dB input and output coupling losses) increases from -74 dB to -64 dB while the signal-to-background increases from $\approx 5$ to $\approx 10$. While nearly background-free FWM-BS conversion has been demonstrated in situations for which the detuning between pumps and signal and idler is large [@ref:McGuinnes_PRL10], for smaller detuning levels, high signal-to-background levels are generally harder to achieve. This can be due to technical reasons, such as incomplete suppression of the pump at the converted idler wavelength, or more fundamental reasons related to the waveguide system itself, such as spontaneous Raman scattering in SiO$_2$ [@ref:clark2013] or modulation instability when the strong pumps are situated in regions of anomalous dispersion. Assuming these effects contribute to the measured noise level, the relatively high signal-to-background levels observed suggest that the Si$_3$N$_4$ system is less susceptible to noise from Raman scattering than systems such as silica optical fibers. If these signal-to-background levels can be maintained at higher conversion efficiency levels, they would be adequate for initial quantum frequency conversion experiments with true single photon sources (signal-to-background ratios between 2 and 7 were used in Ref. [@ref:Rakher_NPhot_2010]).
In moving to such experiments, of principal concern is the very low conversion efficiencies thus far demonstrated. However, this experiment was done under continuous-wave and relatively low pump power conditions, with maximum powers of $P_1=13$ mW and $P_2=43$ mW at the input inside the waveguide. These results are similar to those in Ref. [@ref:Agha_OL] achieved under similar power conditions. In that work, by using amplified pulses, the conversion efficiency improved to $\approx 5~\%$ for peak powers of a few Watts coupled into the chip. With similar pump amplification (for example, using nanosecond pulses and an EDFA to generate the telecom-band high peak-power pump and a tapered amplifier for the 980 nm band pump), and by reducing the input coupling losses via optimized geometries (for example, inverse tapers in symmetrically clad coupling regions) - already demonstrated for Si$_3$N$_4$ waveguides [@ref:Ferdous_Weiner] - we expect the conversion efficiency to improve to over $25~\%$, as confirmed by split-step Fourier method simulations [@ref:Agha_OL]. This conversion efficiency level would render the interface suitable for quantum frequency conversion with single photon emitters [@ref:Rakher_NPhot_2010; @ref:McGuinnes_PRL10].
Experimental results - Wideband Frequency Upconversion
======================================================
The flexibility of the two pump FWM-BS process implies that by switching the roles of the signal and idler, we can perform frequency upconversion within the same device. To ensure the capability of single photon counting in the 980 nm band, the 974 nm pump laser is heavily filtered via volume Bragg gratings to yield better than 140 dB of suppression. The 1550 nm fixed pump laser is combined with a tunable telecommunications-band weak signal via a 90:10 coupler, and the combined pumps+signal are multiplexed via a WDM and sent to the waveguide chip (Fig. \[fig:upconversion\_data\](a)). The output of the chip is notch-filtered (to get rid of the strong 974 nm laser) and sent to either a visible-wavelength spectrometer or silicon SPAD for signal-to-background measurements.
Figure \[fig:upconversion\_data\](b) shows the generated 980 nm band idlers for a specific set of pump and input signal wavelengths in the 1550 nm band (the 980 nm band pump has been removed by a notch filter after the chip, as shown in Fig. \[fig:upconversion\_data\](b)). Combining a series of upconversion spectra as the 1550 nm band input signal is swept yields the contour plot shown in Fig. \[fig:upconversion\_data\](c), which confirms a tuning of the generated idlers that matches the energy conservation condition, $\omega_i=\omega_s \pm (\omega_1 - \omega_2)$. By integrating the power within each idler peak, we can deduce the normalized conversion efficiency and hence the phase-matching bandwidth, which, as expected, is similar in behavior to that observed in downconversion (Fig. \[fig:downconversion\_data\](d)). As the 1550 nm pump power is increased up to 50 mW, the internal conversion efficiency improves from -77 dB to -62 dB, while the signal-to-background ratio for the $\omega_{i}^{+}$ idler reaches a maximum of $\approx 20$ (Fig. 3(e)). As in the case of downconversion, the signal-to-background ratio is determined by measuring the counts in the $\omega_{i}^{+}$ band with the input 980 nm signal turned on and off, and dividing the two values (after subtraction of the detector dark count rate of $\approx 170$ s$^{-1}$). The higher signal-to-background we measure here for upconversion can most likely be attributed to the improved filtering of the pumps both at the input and the output of the waveguide via volume Bragg gratings that provide better extinction than fiber-based WDMs. This also provides some indication that the process may not be limited by noise contributions due to Raman scattering. In comparison to SiO$_2$, we note that Raman scattering in silicon nitride has not been observed to be comparable to four-wave-mixing, even at high power conditions [@ref:Levy_Lipson_comb; @ref:Ferdous_Weiner; @ref:Foster_Levy; @ref:Okawachi]. Another potential reason for the higher signal-to-background levels is that the strong 980 nm pump is situated deep within the normal dispersion region (Fig. \[fig:waveguide and schematics\](b)), so that modulation instability is completely suppressed. In comparison, the 1550 nm pump, though nominally also in a region of normal dispersion, is much closer to the dispersion zero. As a result, deviations in the waveguide geometry with respect to the simulated structure (e.g., due to fabrication tolerances or a non-uniform waveguide cross-section along its length) may lead to modulation instability induced noise.
Experimental results - Narrowband Frequency Conversion
======================================================
As noted previously, FWM-BS in the case of pumps that are far-detuned from the input signal and generated idlers has likely advantages from a signal-to-background perspective. To some extent, it can thus provide a reference point for the best achievable signal-to-background levels within the system. To verify this, we revert to the narrowband conversion configuration (Fig. \[Fig:narrowband\_SNR\_data\]) we studied recently in Ref. [@ref:Agha_OL]. Here, two telecommunications-wavelengths pumps $\omega_1$ and $\omega_2$ convert a 980 nm band input signal $\omega_s$ to an idler $\omega_i$ in the same band, according to the rule $\omega_i=\omega_s \pm (\omega_2 -
\omega_1)$. At optimal conversion conditions, the pumps are separated by $\approx 10$ nm, while the signal/idler are separated by $\approx 5$ nm (Fig. \[Fig:narrowband\_SNR\_data\](b)). We bandpass filter the generated idler $\omega_{i}^{+}$ and use a Si SPAD to measure the power in this wavelength band with the input signal turned on and off (Fig. \[Fig:narrowband\_SNR\_data\](c)). When the input is turned off, we measure no excess background counts above the SPAD dark count rate ($\approx 400$ s$^{-1}$) for the majority of the 1550 nm pump power range sent into the waveguide (so that after SPAD dark count subtraction, the process is noise-free). Even at the very highest pump powers, for which we observe significant excess noise in our EDFA, we measure signal-to-background levels of $\approx 50$.
Discussion
==========
We have demonstrated the first steps towards a quantum interface for ultra-wide band frequency conversion, and shown how it is experimentally possible to translate photons from 980 nm to 1550 nm and back via FWM-BS in a CMOS-compatible platform. The goal of this work is to pave the way towards both high-efficiency devices that render on-chip frequency conversion of single photon emitters practical, as well as to provide design guidelines for down(up)conversion of photons from various quantum emitters via FWM-BS. While our devices targeted the 980 nm band which is the wavelength range of InAs quantum dot SPSs that we have employed in recent experiments [@ref:Ates_Srinivasan_PRL; @ref:Ates_SR], the ability to engineer the dispersion, coupled with the transparency of silicon nitride, allows for devices that in principle can target most common single photon emitters.
![a) Design parameters for Si$_3$N$_4$-on-SiO$_2$ waveguides targeting up(down)conversion between the telecommunications band and various single photon sources (SPSs) (Nitrogen vacancy center in diamond, Rubidium, Cesium, and InAs QD in descending order). The table specifies, for each SPS wavelength ($\lambda$), the waveguide width (W) and the necessary input peak power (P$_{in}$) for maximum conversion. The waveguide thickness has been fixed at 550 nm in all cases. b) The corresponding dispersion parameter $D$ for each waveguide geometry. (c) Split-step Fourier simulation for the case of downconversion between 780 nm and 1550 nm. Conversion efficiency is plotted as a function of pump power, where the power in both pumps are assumed to be equal.[]{data-label="fig:Dispersion_table"}](Figure5.eps){width="\linewidth"}
Figure \[fig:Dispersion\_table\](a) is a table of common SPS emission wavelengths, with the required geometry of the waveguide to achieve phase matching for downconversion towards the 1550 nm telecommunications band. Along with linear phase-matching, of key importance is for the strong pump fields to be in regions of normal dispersion, to avoid the onset of modulation instability. This has been achieved here by simply tuning the waveguide width, while keeping the thickness fixed at 550 nm. Figure \[fig:Dispersion\_table\](b) shows the resulting dispersion parameter $D=\frac{-2\pi c}{\lambda^2}\frac{d^2\beta}{d\omega^2}$ for the different optimized geometries, where the zero dispersion point, to a first approximation, is close to the average of the two pumps’ wavelengths [@ref:Uesaka_Kazovksy]. Finally, Fig. \[fig:Dispersion\_table\](c) shows an example of the results from a split-step Fourier simulation [@ref:Agrawal_NFO] in which we take into account the calculated dispersion to assess the efficiency of the frequency conversion process without the approximations that go into the analytic coupled-mode theory model (e.g., no pump depletion, no pump mixing, and no degenerate four-wave-mixing). The figure plots the conversion efficiency as a function of pump power (assumed to be equal for the two pumps) in the case of downconversion from 780 nm to 1550 nm. We see that a conversion efficiency of $\approx25~\%$ can be achieved for approximately 9 W of power in each pump. Simulations reveal that conversion efficiencies $\gtrsim25~\%$ can be achieved for the other sets of wavelengths considered, and the corresponding pump powers are listed in the table in Fig. \[fig:Dispersion\_table\](a). It should be noted that while 9 W is a relatively high power when considering continuous-wave sources, it is readily achievable by amplifying nanosecond pulses in saturated erbium-doped or semiconductor tapered amplifiers, making our chip-scale interface a practical prospect for conversion of single photon states of light from quantum emitters (particularly systems like semiconductor quantum dots, which have characteristic timescales on the order of 1 ns). Moreover, by implementing the structure in Si$_3$N$_4$, peak power is not limited by phenomena such as two-photon absorption, and hence such powers can be coupled to a Si$_3$N$_4$ waveguide without major problems, as observed in experiment for few Watt level pumps in Ref. [@ref:Agha_OL]. On the other hand, implementation of the four-wave-mixing Bragg scattering process in crystalline silicon waveguides could be of interest, in part due to its higher nonlinear refractive index than Si$_3$N$_4$ [@ref:Foster_Gaeta; @ref:Lin_Painter_Agrawal]. Such work would be limited to wavelengths $>1000$ nm (as silicon is opaque at shorter wavelengths), and may require long wavelength ($>2000$ nm) pumps to avoid significant nonlinear absorption [@ref:Liu_Green_NatPhoton; @ref:Zlatanovic_Radic_NatPhoton]. Alternately, materials like amorphous silicon, which combine high optical nonlinearity and a broader optical transparency window than silicon (though less than Si$_3$N$_4$) may ultimately prove to be an attractive candidate for four-wave-mixing Bragg scattering within certain wavelength regions [@ref:Wang_Foster_amorphous_Si].
In conclusion, we have demonstrated a CMOS-compatible chip-scale interface for ultra-wide band frequency conversion via four-wave-mixing Bragg scattering in a silicon nitride waveguide, and characterized both its tuning bandwidth and noise properties. We performed proof-of-principle experiments demonstrating both downconversion and upconversion between the 980 nm and 1550 nm bands, rendering our chip a promising interface for connecting self-assembled InAs quantum dots to the telecommunications band. With improved input coupling and higher input pump peak powers, we expect our interface to be ready for frequency conversion of true single photon sources, which will be the target of future work.
Acknowledgments
===============
The authors thank Yoshitomo Okawachi for helpful comments on this work. I.A and S.A. acknowledge support under the Cooperative Research Agreement between the University of Maryland and NIST-CNST, Award 70NANB10H193. The authors also acknowledge the DARPA MESO program for partial support.
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abstract: 'A second-gradient elastic (SGE) material is identified as the homogeneous solid equivalent to a periodic planar lattice characterized by a hexagonal unit cell, which is made up of three different linear elastic bars ordered in a way that the hexagonal symmetry is preserved and hinged at each node, so that the lattice bars are subject to pure axial strain while bending is excluded. Closed form-expressions for the identified non-local constitutive parameters are obtained by imposing the elastic energy equivalence between the lattice and the continuum solid, under remote displacement conditions having a dominant quadratic component. In order to generate equilibrated stresses, in the absence of body forces, the applied remote displacement has to be constrained, thus leading to the identification in a condensed’ form of a higher-order solid, so that imposition of further constraints becomes necessary to fully quantify the equivalent continuum. The identified SGE material reduces to an equivalent Cauchy material only in the limit of vanishing side length of hexagonal unit cell. The analysis of positive definiteness and symmetry of the equivalent constitutive tensors, the derivation of the second-gradient elastic properties from those of the higher-order solid in the condensed’ definition, and a numerical validation of the identification scheme are deferred to Part II of this study.'
author:
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G. Rizzi, F. Dal Corso, D. Veber, and D. Bigoni$^1$\
DICAM, University of Trento\
via Mesiano 77, I-38123 Trento, Italy
bibliography:
- 'Biblio1.bib'
title: 'Identification of second-gradient elastic materials from planar hexagonal lattices. Part I: Analytical derivation of equivalent constitutive tensors'
---
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i
Keywords: Strain gradient elasticity; non-local material; non-centrosymmetric material; internal length; homogenization
Introduction
============
Research on the equivalence between spring networks and continuous bodies was initiated by Cauchy [@cauchy1828] and later continued by Born [@born1954dynamical], with the purpose of determining the overall elastic properties of crystalline materials subject to small strain. Considering a linear interaction between atoms, a material is modelled as a three-dimensional linear elastic lattice, with elements only subject to axial deformation. This is the so-called Cauchy-Born rule’, which yields the rari-constant’ theory of elasticity, relating the elastic property of a solid to the interactions between its atoms or molecules.
Over the years, the approach has been extended to evaluate mechanical characteristics such as Young modulus, Poisson’s ratio and normal modes of vibration for a number of geometrically different networks [@Genoese2018; @keating1966effect; @kirkwood1939skeletal; @latture; @neumann1975equations]. With reference to a hexagonal lattice, composed of linearly elastic bars pinned to each other (so that bending effects are excluded) and characterized by three different values of stiffness, as reported in Fig. \[fig:intro\_lattice\], Day et al. [@day1992elastic; @snyder1992elastic] have shown that the overall behaviour of this lattice may be modelled through an equivalent isotropic Cauchy linear elastic solid defined by the elastic bulk $K$ and shear $\mu$ moduli given by $$K =\dfrac{\overline{k} + \widehat{k} + \widetilde{k}}{\sqrt{12}}, ~~~~ \mu = \sqrt{\dfrac{27}{16}}\left( \dfrac{1}{\overline{k}} + \dfrac{1}{\widehat{k}} + \dfrac{1}{\widetilde{k}} \right)^{-1},
\label{eq:const_day}$$ where $\overline{k}$, $\widehat{k}$ and $\widetilde{k}$ are the three in-plane bars’ stiffnesses (so that their dimension is a force per unit out-of-plane thickness divided by a length) defining the hexagonal lattice.
![(Left) A planar lattice obtained as the periodic repetition of a hexagonal unit cell (with side length $\ell$) made up of linear elastic bars, characterized by three stiffnesses $\overline{k}$ (red bars), $\widehat{k}$ (green bars), and $\widetilde{k}$ (blue bars). The bars are connected through hinge joints, so that only axial strain is present and bending is excluded. Reference systems are also reported. (Right) Explosion of the hexagonal lattice displaying the cell and node nomenclature and highlighting how the perimeter nodes are shared among adjacent cells.[]{data-label="fig:intro_lattice"}](Figures/intro_lattice){width="100.00000%"}
The goal of the present research is to extend the theory developed by Day et al. [@day1992elastic; @snyder1992elastic] towards a higher-order approximation for the elastic material equivalent to the hexagonal lattice, showing nonlocal effects related to the four parameters defining the lattice properties at the micro-scale, the hexagon side length $\ell$ and the stiffnesses $\overline{k}$, $\widehat{k}$ and $\widetilde{k}$.
Phenomenological constitutive theories, used to model materials of engineering relevance, were traditionally assumed to be local, or, in other words, did not comprise any internal characteristic length. Recently, experimental observations at the micro- and nano-scale have evidenced size-effects [@BEVERIDGE2013246; @buechner2003size; @lakes1986experimental; @WASEEM2013148], which cannot be described with local constitutive models. Therefore, an enhanced modelling has been introduced, which becomes particularly useful when large strain gradient are involved, as in contact mechanics [@gourgiotis2016analysis; @zisis2015contact] indentation processes [@begley1998mechanics; @danas2012size], fracture [@gourgiotis2014steady; @piccolroaz2012mode], and shear band formation [@dal2011stability; @sluys1993wave].
Several authors [@Seppecher2018; @askar1968structural; @bacigalupo2012computational; @bacigalupo2014second2; @le2013homogenization; @ostoja2002lattice; @Shi1995; @SPADONI2012156; @warren2002three] have proposed non-classical continuum models to treat lattice structures involving beam-type interactions. For these lattices, non-local effects emerge as the response to non simple interactions between material points, generated, for example, when rotational springs are used [@suiker2001comparison].
The primary goal of the present study is the determination of the non-local response of lattices (having elements only subject to axial forces), which has been scarcely considered so far (an example is the case of pantographic trusses [@Seppecher2011]). In particular, it will be shown that a hexagonal lattice structure with axially-deformable bars can be identified with a form I’ Mindlin elastic material, a special type of second-gradient elastic law [@mindlin1964micro].
The present article is organized as follows. After the kinematics and the equilibrium of the hexagonal lattice (Fig. \[fig:intro\_lattice\]) is introduced (Sect. \[hexa\]), the quadratic remote displacement conditions, plus the additional terms needed to enforce equilibrium, are presented in Sect. \[qqua\]. The homogeneous Second Gradient Elastic ($\mathsf{SGE}$) solid equivalent to lattice is identified in Sect. \[identi\]. In particular, by imposing an elastic energy matching, closed-form expressions for the higher-order tensors are derived. As a consequence of the fact that the energy matching is imposed under the condition that the applied displacement field generates equilibrated stress states, only a condensed’ form of the constitutive equations is determined for the $\mathsf{SGE}$ solid. As a conclusion, it is shown that the elastic second-gradient solid equivalent to the lattice structure exhibits non-locality, anisotropy, and non-centro-symmetry (despite the fact that the equivalent Cauchy material, derived on linear displacement fields, is local, isotropic, and centro-symmetric). Important issues related to: the analysis of (i.) positive definiteness and (ii.) symmetry of the equivalent material, (iii.) the derivation of the full $\mathsf{SGE}$ solid from the properties of the condensed’ one, and (iv.) the validation of the derived second-gradient model are deferred to Part II [@rizzipt2] of this study.
The hexagonal lattice {#hexa}
=====================
Preliminaries: the periodic structure and its elastic equilibrium
-----------------------------------------------------------------
An infinite periodic lattice (Fig. \[fig:intro\_lattice\], left), defined in the plane containing the orthonormal basis $\be_1$–$\be_2$, is considered as the repetition of a hexagonal unit cell, which will eventually be identified with a representative volume element (RVE) of an equivalent continuum. The hexagonal cell is regular and has side of length $\ell$, it is characterized by linear elastic bars with three different values of axial stiffnesses, namely, $\overline{k}$,$ \widehat{k}$, and $\widetilde{k}$, distributed according to the scheme reported in Fig. \[fig:intro\_lattice\], which preserves the hexagonal symmetry. Therefore, a total of six bars (two groups of three bars having the same stiffness) converge at each hinge node of the lattice.
Among the three tessellations equivalent for the realization of the periodic lattice, the one is chosen for which the unit cell has its center defined by the convergence of the bars of stiffness $\widehat{k}$ and $\widetilde{k}$, while the other bars of stiffness $\overline{k}$ define the hexagon perimeter. Each node of the cell is denoted by the index $i=\left\{0,1,2,3,4,5,6\right\}$ and each cell is singled out by the integers $\left\{m,n\right\}\in\mathbb{Z}$, which determine the cell position with reference respectively to the non-orthogonal directions $\be_1$ and $\be_{\pi/3}=1/2\be_1+\sqrt{3}/2\be_2$, see Fig. \[fig:intro\_lattice\] . It follows that the position $\bx^{(m,n|i)}$ of the $i$-th node of the $\left\{m,n\right\}$ cell can be described with reference to the central node ($i=0$) position $\textbf{x}^{(m,n|0)}$ through the following expression $$\bx^{(m,n|i)} = \bx^{(m,n|0)} + \ell \bg^{(i)},
\label{eq:Posizio1}$$ where $\bg^{(i)} $ defines the direction spanning from the central node to the $i$-th node, $$\bg^{(i)} = (1-\delta_{i0})\left\{-\sin\left[\frac{\pi(i-1)}{3} \right]\be_1+\cos\left[\frac{\pi(i-1)}{3}\right] \be_2\right\},
\label{eq:unitevectorG}$$ in which the index $i$ is not summed and the Kronecker delta $\delta_{i0}$ is defined to include the null index value, so that $\delta_{00}=1$ while $\delta_{i0}=0$ for every $i\neq 0$. From the definition expressed by Eq. (\[eq:unitevectorG\]), it follows that the vector $\bg^{(i)}$ has unit modulus for every $i\neq0$, while it vanishes when $i=0$ (central node), $$\begin{array}{lll}
\bg^{(0)}=\b0, \quad|\bg^{(i)}|=1, \qquad \mbox{for} \qquad i=1,2,...,6.
\end{array}$$ Furthermore, due to the RVE symmetry, the unit vectors $\bg^{(i)} $ satisfy the following property $$\bg^{(i)} = -\bg^{(i+3)},\qquad i=1,2,3,$$ and the following combination of the unit vectors $\bg^{(1)}$, $\bg^{(5)}$, and $\bg^{(6)}$ provides the unit vectors $\be_1$ and $\be_{\pi/3}$ $$\be_1=\frac{\bg^{(5)}+\bg^{(6)}}{\sqrt{3}},\qquad
\be_{\pi/3}=\frac{\bg^{(1)}+\bg^{(6)}}{\sqrt{3}}.$$
Considering the definition of the unit vector $\bg^{(i)}$, Eq. (\[eq:unitevectorG\]), the position $\textbf{x}^{(m,n|0)}$ of the central node of the cell $\left\{m,n\right\}$ can be expressed with reference to the position $\bx^{(0,0|0)}$ of the central node of the cell $\left\{m,n\right\}=\left\{0,0\right\}$ as $$\bx^{(m,n|0)}=\bx^{(0,0|0)}+ \ell\,\left[m \, \left(\bg^{(5)}+\bg^{(6)}\right) +n\, \left(\bg^{(1)}+\bg^{(6)}\right)\right],
\label{eq:36}$$ so that the position $\bx^{(m,n|i)}$ of each node $i$ of every $\left\{m,n\right\}$ cell, expressed by Eq.(\[eq:Posizio1\]), can be finally reduced to $$\bx^{(m,n|i)}=\bx^{(0,0|0)}+ \ell\left[\textbf{g}^{(i)} + m \, \left(\bg^{(5)}+\bg^{(6)}\right) +n\, \left(\bg^{(1)}+\bg^{(6)}\right) \right].
\label{eq:37}$$
All the perimeter nodes ($i=\left\{1,2,...,6\right\}$) join three adjacent hexagonal cells, Fig. \[fig:intro\_lattice\] (right), so that the following identities hold $$\begin{array}{llllll}
&\bx^{(m,n|1)} = \bx^{(m,n+1|3)} = \bx^{(m-1,n+1|5)},\qquad
&\bx^{(m,n|2)} = \bx^{(m-1,n+1|4)} = \bx^{(m-1,n|6)}, \\[3mm]
&\bx^{(m,n|3)} = \bx^{(m-1,n|5)} = \bx^{(m,n-1|1)},\qquad
&\bx^{(m,n|4)} = \bx^{(m,n-1|6)} = \bx^{(m+1,n-1|2)}, \\[3mm]
&\bx^{(m,n|5)} = \bx^{(m+1,n-1|1)} = \bx^{(m+1,n|3)},\qquad
&\bx^{(m,n|6)} = \bx^{(m+1,n|2)} = \bx^{(m,n+1|4)}.
\end{array}
\label{eq:38}$$ Introducing $\bu^{(m,n|i)}$ as the (small) displacement of the $i$-th node belonging to the cell $\left\{m,n\right\}$, which according to Eq. Eq. (\[eq:38\]) satisfies $$\begin{array}{llllll}
&\bu^{(m,n|1)} = \bu^{(m,n+1|3)} = \bu^{(m-1,n+1|5)},\qquad
&\bu^{(m,n|2)} = \bu^{(m-1,n+1|4)} = \bu^{(m-1,n|6)}, \\[3mm]
&\bu^{(m,n|3)} = \bu^{(m-1,n|5)} = \bu^{(m,n-1|1)},\qquad
&\bu^{(m,n|4)} = \bu^{(m,n-1|6)} = \bu^{(m+1,n-1|2)}, \\[3mm]
&\bu^{(m,n|5)} = \bu^{(m+1,n-1|1)} = \bu^{(m+1,n|3)},\qquad
&\bu^{(m,n|6)} = \bu^{(m+1,n|2)} = \bu^{(m,n+1|4)},
\end{array}
\label{eq:39}$$ the elongation $E^{(m,n|i,j)}$ of the bar connecting the nodes $i$ and $j$ (with $i\neq j$) is given by $$E^{(m,n|i,j)}=\left(\bu^{(m,n|i)} - \bu^{(m,n|j)}\right)\scalp
\left(\bg^{(i)} - \bg^{(j)}\right), \qquad i\neq j,
\label{eq:ElonGeneral}$$ which is insensitive to a permutation of the node indexes $i$ and $j$, $$E^{(m,n|i,j)}=E^{(m,n|j,i)}.$$ Considering that the bars have a linear elastic response, the force $\bF^{(m,n|i,j)}$ (positive if tensile and negative if compressive) acting on the $i$-th node of the cell $\left\{m,n\right\}$ and generated by the elongation $E^{(m,n|i,j)}$ of the bar with stiffness $k^{(i,j)}$ is given by $$\bF^{(m,n|i,j)}=-k^{(i,j)}\,E^{(m,n|i,j)}\left(\bg^{(i)} - \bg^{(j)}\right),
\label{eq:ForceGeneral}$$ which, according to the second Newton’s law, is also the opposite of that acting at the $j$-th node and due to the elongation $E^{(m,n|i,j)}$ of the same bar $$\bF^{(m,n|j,i)}=-\bF^{(m,n|i,j)}.$$
Independently of the cell indexes $\{m,n\}$, the stiffness $k^{(i,j)}$ related to the bar connecting the nodes $i$ and $j$ is defined as (Fig. \[fig:intro\_lattice\], left) $$k^{(i,j)}=
\left\{\begin{array}{lllll}
\overline{k},\qquad \qquad &i\neq 0 \,\,\mbox{and}\,\, j\neq 0,\\
\widetilde{k},\qquad \qquad &i= 0\,\, \mbox{and}\,\, j \,\,\mbox{even}\qquad
&\mbox{or}
\qquad
&i\,\,\mbox{even}\, \mbox{and}\, j = 0,\\
\widehat{k},\qquad \qquad &i= 0\,\, \mbox{and}\,\, j \,\,\mbox{odd}\qquad
&\mbox{or}
\qquad
&i\,\,\mbox{odd}\,\, \mbox{and}\,\, j = 0.
\end{array}
\right.$$
The sum of all the forces $\bF^{(m,n|i,j)}$, acting on the node $i$ (belonging to the cell $\left\{m,n\right\}$) and generated by the elongation of all the bars jointed at that node, provides the resultant $\bR^{(m,n|i)}$, Fig. \[fig:CorrEForceNodal\] (left). Considering the properties expressed by Eq. (\[eq:39\]), the resultant forces at all of the lattice nodes are given through the three primary resultants $\bR^{(m,n|0)}$, $\bR^{(m,n|1)}$, $\bR^{(m,n|2)}$ as $$\begin{array}{lll}
\bR^{(m,n|0)}=\sum\limits_{j=1}^6 \bF^{(m,n|0,j)},
\\[6mm]
\bR^{(m,n|1)}=\bF^{(m,n|1,0)}+\bF^{(m,n|1,2)}+\bF^{(m,n|1,6)}+\bF^{(m,n+1|3,0)}+\bF^{(m-1,n+1|5,6)}+\bF^{(m-1,n+1|5,0)},
\\[6mm]
\bR^{(m,n|2)}=\bF^{(m,n|2,0)}+\bF^{(m,n|2,1)}+\bF^{(m,n|2,3)}+\bF^{(m-1,n|6,0)}+\bF^{(m-1,n+1|4,3)}+\bF^{(m-1,n+1|4,0)}.
\end{array}
\label{eq:ResultantGeneral}$$
![Resultant forces $\bR^{(m,n|i)}$ (left) and additional displacements $\bDelta\bu^{(m,n|i)}$ (right) associated with the node $i$ ($i=0,...,6$) belonging to the cell $\{m,n\}$ within the lattice drawn in its undeformed configuration.[]{data-label="fig:CorrEForceNodal"}](Figures/Fig_2b.pdf){width="0.7\linewidth"}
Assuming quasi-static conditions, from property (\[eq:39\]) the equilibrium of the whole lattice is attained when the three primary resultants $\bR^{(m,n|0)}$, $\bR^{(m,n|1)}$, and $\bR^{(m,n|2)}$ vanish for every cell $\left\{m,n\right\}$ $$\label{equilibriobrio}
\bR^{(m,n|0)}=\bR^{(m,n|1)}=\bR^{(m,n|2)}=\b0,\qquad \forall \left\{m,n\right\}.$$ The elastic energy $\mathsf{U}_{lat}^{(m,n)}$ stored within the cell $\left\{m,n\right\}$ (instrumental to later identify the energetically equivalent microstructured solid) is provided by $$\mathsf{U}_{lat}^{(m,n)} =\dfrac{1}{2}\sum_{i=1}^{6}
k^{(i,0)} \left[E^{(m,n|i,0)}\right]^2
+
\dfrac{1}{4}\sum_{i=1}^{6}
k^{(i,i+1-6\delta_{i6})} \left[E^{(m,n|i,i+1-6\delta_{i6})}\right]^2 ,
\label{eq:EnergyGeneral}$$ where only one half of the energy stored within the bars along the hexagon perimeter has been considered, so that the total energy of the infinite lattice is obtained by summing the energy of each cell $$\mathsf{U}_{lat} =\sum_{m,n\in \mathbb{Z}}\mathsf{U}_{lat}^{(m,n)}.$$
Definition of an average operator for the displacement gradient in the lattice structure {#secAvarage}
----------------------------------------------------------------------------------------
With reference to a generic field $\textbf{f}(x_1,x_2)$ over a domain $\Omega$ of a continuous body, its gradient and the related average are respectively given by $$\label{strnz0}
f_{j,k}(x_1, x_2)=\frac{\partial f_j(x_1, x_2)}{\partial x_k}, \qquad
\langle f_{j,k}\rangle = \dfrac{1}{||\Omega||}\int_{\Omega}f_{j,k} \mbox{d}\Omega,$$ where $||\Omega||$ is the measure of $\Omega$. By means of the divergence theorem, the gradient average can be rewritten as $$\label{strnz}
\langle f_{j,k}\rangle = \dfrac{1}{||\Omega||}\int_{\partial\Omega} f_j n_k \mbox{ds},$$ where only the evaluation of the field $\textbf{f}(x_1,x_2)$ along the cell perimeter is needed. In order to compute the displacement gradient average, the displacement field along the cell perimeter can be linearly interpolated as $$\bu(s;m,n,i) =\bu^{(m,n|i)} + \left( \bu^{(m,n|i+1-6\delta_{i6})} - \bu^{(m,n|i)}\right) \frac{s}{\ell}, \qquad
i=1,...,6,
\label{eq:MeanDisp}$$ where $s$ is the curvilinear coordinate along the bar of the cell $\{m,n\}$ connecting the node $i$ to node $i+1-6\delta_{i6}$ and measuring the distance from the former ($i$=1,...,6). Considering this interpolating field $\bu(s;m,n,i)$ and identifying $\Omega$ with the hexagonal domain, the average of the displacement gradient for the lattice structure (identified with the subscript ${\textsf{lat}}$’ to highlight its relation with the lattice, and not with the continuum) can be obtained by substituting Eq. (\[eq:MeanDisp\]) into Eq. (\[strnz\]) as $$\langle{u}_{j,k}\rangle^{(m,n)}_{\textsf{lat}} =
\dfrac{1}{3\sqrt{3} \ell}\sum_{i=1}^{6} \left(u^{(m,n|i)}_j + u^{(m,n|i+1-6\delta_{i6})}_j \right)n^{(i)}_{k},
\label{eq:GradMedia}$$ which, when the normal vectors $n^{(i)}_{k}$ are expressed with respect to the unit vectors $g^{(i)}_{k}$, reduces to $$\langle{u}_{j,k}\rangle^{(m,n)}_{\textsf{lat}} =
\dfrac{1}{9\ell}\sum_{i=1}^{6} \left(u^{(m,n|i)}_j + u^{(m,n|i+1-6\delta_{i6})}_j \right)\left(g^{(i)}_{k}+g^{(i+1-6\delta_{i6})}_{k}\right).
\label{eq:GradMediaRVE}$$ More specifically, the four components of $\langle{u}_{j,k}\rangle^{(m,n)}_{\textsf{lat}}$ can be expressed in the reference system $\be_1$–$\be_2$ as $$\begin{split}
\langle{\nabla \bu(\bx)}\rangle^{(m,n)}_{\textsf{lat}}
=\dfrac{1}{\ell}
\resizebox{0.8\textwidth}{!}{$
\begin{bmatrix}
\dfrac{-u_{1}^{(m,n|2)}-u_{1}^{(m,n|3)}+u_{1}^{(m,n|5)}+u_{1}^{(m,n|6)}}{2 \sqrt{3}} &
\dfrac{2 u_{1}^{(m,n|1)}+u_{1}^{(m,n|2)}-u_{1}^{(m,n|3)}-2 u_{1}^{(m,n|4)}-u_{1}^{(m,n|5)}+u_{1}^{(m,n|6)}}{6}\\[3mm]
\dfrac{-u_{2}^{(m,n|2)}-u_{2}^{(m,n|3)}+u_{2}^{(m,n|5)}+u_{2}^{(m,n|6)}}{2 \sqrt{3}} &
\dfrac{2 u_{2}^{(m,n|1)}+u_{2}^{(m,n|2)}-u_{2}^{(m,n|3)}-2u_{2}^{(m,n|4)}-u_{2}^{(m,n|5)}+u_{2}^{(m,n|6)}}{6}\\
\end{bmatrix}
$}
.
\label{eq:mean3}
\end{split}$$
An alternative but equivalent way for deriving the average of the displacement gradient, Eq. (\[eq:mean3\]), can be obtained with reference to the piecewise description of the displacement field along each one of the six equilateral triangles, subdomains of the hexagonal cells and enclosed by the three different bars. Such a piecewise description of the field $\bu^{(m,n,j)}(\bx)$ follows from the linear interpolation of the displacements of the central node and the two consecutive perimeter nodes $j$ and $j+1-6\delta_{j6}$ (with $j=1,...,6$), corresponding to the three vertices of the $j$-th triangle composing the $\{m,n\}$ hexagonal cell, as $$\bu^{(m,n,j)}(\bx)=\bA^{(m,n,j)}\bx+\bc^{(m,n,j)} \qquad\mbox{with}\qquad j=1,...,6 \qquad m,n\in\mathbb{Z}
\label{eq:sposttriangular}$$ where matrix $\bA^{(m,n,j)}$ and the vector $\bc^{(m,n,j)}$ are $$\resizebox{0.98\textwidth}{!}{$
\begin{array}{lll}
&A^{(m,n,j)}_{11}=\dfrac{2 \cos \left(\frac{\pi j}{3}\right) (u_{1}^{(m,n|j)}-u_{1}^{(m,n|0)})+2 \cos \left(\frac{\pi (j-1)}{3} \right) (u_{1}^{(m,n|0)}-u_{1}^{(m,n|j+1)})}{\sqrt{3} \ell},
&A^{(m,n,j)}_{12}=\dfrac{2 \cos \left(\frac{\pi j}{3}\right) (u_{2}^{(m,n|j)}-u_{2}^{(m,n|0)})+2 \cos \left(\frac{\pi (j-1)}{3} \right) (u_{2}^{(m,n|0)}-u_{2}^{(m,n|j+1)})}{\sqrt{3} \ell},\\[3mm]
&A^{(m,n,j)}_{21}=\dfrac{2 \sin \left(\frac{\pi j}{3}\right) (u_{1}^{(m,n|j)}-u_{1}^{(m,n|0)})+2 \sin \left(\frac{\pi (i-1)}{3} \right) (u_{1}^{(m,n|0)}-u_{1}^{(m,n|j+1)})}{\sqrt{3} \ell},&A^{(m,n,j)}_{22}=\dfrac{2 \sin \left(\frac{\pi j}{3}\right) (u_{2}^{(m,n|j)}-u_{2}^{(m,n|0)})+2 \sin \left(\frac{\pi (j-1)}{3} \right) (u_{2}^{(m,n|0)}-u_{2}^{(m,n|j+1)})}{\sqrt{3} \ell},\\[3mm]
&c^{(m,n,j)}_{1}=u_{1}^{(m,n|0)},&c^{(m,n,j)}_{2}=u_{2}^{(m,n|0)}.
\end{array}
$}
\label{eq:CoeffMedie2}$$ The average of the displacement gradient within the unit cell $\{m,n\}$ follows from Eq. (\[strnz0\]) as $$\langle{\nabla \bu(\bx)}\rangle^{(m,n)}_{\textsf{lat}}=\frac{1}{||\Omega||}\sum_{j=1}^{6}\int_{\Omega^{(m,n,j)}} \nabla \bu^{(m,n,j)}(\bx) \mbox{d} \Omega .
\label{eq:mean2}$$ which, considering the piecewise description of displacement (\[eq:sposttriangular\]), Eq. (\[eq:mean2\]) can be rewritten as $$\begin{split}
&\langle{\nabla \bu(\bx)}\rangle^{(m,n)}_{\textsf{lat}}
=\frac{1}{6}\sum\limits_{j=0}^{6}
\begin{bmatrix}
A^{(m,n,j)}_{11} & A^{(m,n,j)}_{12} \\[5mm]
A^{(m,n,j)}_{21} & A^{(m,n,j)}_{22}
\end{bmatrix}
,
\end{split}$$ and that, recalling Eq. (\[eq:CoeffMedie2\]), reduces to the same expression given by Eq. (\[eq:GradMediaRVE\]).
Second-order displacement boundary condition {#qqua}
============================================
The key for the identification procedure performed in the next Section is the imposition to the infinite lattice of a linear and a quadratic nodal displacement fields (as in [@bacca2013mindlin], [@mattia2013mindlin], [@BACIGALUPO2017], [@bigoni2007analytical]), together with an additional field’ $\bDelta \bu^{(m,n|i)}$, namely, $$u_r^{(m,n|i)}= \alpha_{sr} x_s^{(m,n|i)} + \beta_{str} x_s^{(m,n|i)} x_t^{(m,n|i)} + \Delta u_r^{(m,n|i)},\qquad\mbox{with}\qquad r,s,t=1,2
\label{eq:campospost}$$ where $\alpha_{sr}$ and $\beta_{str}$ are tensors defining the displacement amplitudes and satisfying the symmetry properties $\alpha_{sr}=\alpha_{rs}$ and $\beta_{str}= \beta_{tsr}$, so that they have in general three and six independent components, respectively. The presence of the additional term $\Delta u_r^{(m,n|i)}$ is necessary, as shown further on, for attaining the quasi-static equilibrium for every $\alpha_{sr}$ and $\beta_{str}$ as defined by Eq. (\[equilibriobrio\]). The displacement field expressed through Eq. (\[eq:campospost\]) can equivalently be written as $$\bu^{(m,n|i)} = \balpha \bx^{(m,n|i)} + \left(\bx^{(m,n|i)} \otimes \bx^{(m,n|i)}\right) \scalpp \bbeta
+\bDelta \bu^{(m,n|i)},
\label{eq:campospost2}$$ where the second-order tensor $\balpha$ and the third-order tensor $\bbeta$ have components $\alpha_{sr}=(\balpha)_{sr}$ and $\beta_{str}=(\bbeta)_{str}$. In Eq. (\[eq:campospost2\]), the dyadic product $\otimes$ and double scalar product $\scalpp$ are introduced, respectively defined as $\left(\ba \otimes \bb\right)_{st}=a_s b_t$ and $\left(\bA \scalpp \boldsymbol{\mathcal{B}}\right)_{r}=A_{st} \mathcal{B}_{str}$. Considering the displacement field (\[eq:campospost2\]), the elongation of the bars can be computed from Eq. (\[eq:ElonGeneral\]) as $$\begin{split}
E^{(m,n|i,j)} =& \ell \left\{ \balpha \left(\bg^{(i)} - \bg^{(j)}\right)
+ 2 \left[\bx^{(m,n|0)} \otimes \left(\bg^{(i)} - \bg^{(j)}\right)\right] \scalpp \bbeta
\right.\\
& \left.+ \ell \left[\left(\bg^{(i)} + \bg^{(j)}\right) \otimes \left(\bg^{(i)} - \bg^{(j)}\right)\right] \scalpp \bbeta \right\}\scalp
\left(\bg^{(i)} - \bg^{(j)}\right)+\Delta E^{(m,n|i,j)}, \qquad i\neq j,
\end{split}
\label{eq:44}$$ so that the corresponding force at the $i$-th node can be evaluated from Eq. (\[eq:ForceGeneral\]) as $$\begin{split}
\bF^{(m,n|i,j)} =& -k^{(i,j)}\ell~\bG^{(i,j)}
\left\{ \balpha \left(\bg^{(i)} - \bg^{(j)}\right)
+ 2 \left[\bx^{(m,n|0)} \otimes \left(\bg^{(i)} - \bg^{(j)}\right)\right] \scalpp \bbeta
\right.\\
& \left.+ \ell \left[\left(\bg^{(i)} + \bg^{(j)}\right) \otimes \left(\bg^{(i)} - \bg^{(j)}\right)\right]\scalpp \bbeta \right\} +\bDelta \bF^{(m,n|i,j)}, \qquad i\neq j,
\end{split}
\label{eq:Forces}$$ where $$\label{eq:ForcCorretti}
\begin{array}{lll}
\Delta E^{(m,n|i,j)}=
\left(\bDelta\bu^{(m,n|i)} - \bDelta\bu^{(m,n|j)}\right)\scalp
\left(\bg^{(i)} - \bg^{(j)}\right), \\[4mm]
\bDelta \bF^{(m,n|i,j)}=-
k^{(i,j)}\,\Delta E^{(m,n|i,j)}\left(\bg^{(i)} - \bg^{(j)}\right),
{\end{array}}\qquad i\neq j,$$ and $$\bG^{(i,j)}=\left(\bg^{(i)} - \bg^{(j)}\right)\otimes \left(\bg^{(i)} - \bg^{(j)}\right).$$ In combination with Eqs. (\[eq:Forces\]) and (\[eq:ForcCorretti\])$_2$, the three primary resultants $\bR^{(m,n|0)}$, $\bR^{(m,n|1)}$, $\bR^{(m,n|2)}$, Eqs. (\[eq:ResultantGeneral\]), reduce to $$\begin{split}
\bR^{(m,n|0)}=&\left ( \widehat{k} - \widetilde{k} \right ) \ell \sum_{i=1,3,5} \left(\bg^{(i)} \scalp \balpha \bg^{(i)}\right) \bg^{(i)}
+ \left ( \widehat{k} + \widetilde{k} \right ) \ell^2 \sum_{i=1,3,5} \left [ \left( \bg^{(i)} \otimes \bg^{(i)}\right) \scalpp \bbeta \scalp \bg^{(i)} \right ] \bg^{(i)} \\
& + 2 \left ( \widehat{k} - \widetilde{k} \right ) \ell \sum_{i=1,3,5} \left [ \left(\bx^{(m,n|0)} \otimes \bg^{(i)} \right) \scalpp \bbeta \scalp \bg^{(i)} \right ] \bg^{(i)}
+ \sum_{j=1}^{6} k^{(0,j)} \bG^{(0,j)} \left(\bDelta \bu^{(m,n|0)} - \bDelta \bu^{(m,n|j)}\right),
\end{split}
\label{eqEq0Big}$$ $$\begin{split}
\bR^{(m,n|1)}=&\left ( \overline{k} - \widehat{k} \right ) \ell \sum_{i=1,3,5} \left(\bg^{(i)} \scalp \balpha \bg^{(i)}\right) \bg^{(i)}
+ \left ( \overline{k} + \widehat{k} \right ) \ell^2 \sum_{i=1,3,5} \left [ \left( \bg^{(i)} \otimes \bg^{(i)}\right) \scalpp \bbeta \scalp \bg^{(i)} \right ] \bg^{(i)} \\
&+ 2 \left ( \overline{k} - \widehat{k} \right ) \ell \sum_{i=1,3,5} \left [ \left(\bx^{(m,n|0)} \otimes \bg^{(i)} \right) \scalpp \bbeta \scalp \bg^{(i)}\right ]\bg^{(i)}
+ 2 \left(\overline{k} - \widehat{k} \right ) \ell^2 \sum_{i=1,3,5} \left [ \left(\bg^{(1)} \otimes \bg^{(i)} \right) \scalpp \bbeta \scalp \bg^{(i)} \right ] \bg^{(i)}\\
&+\overline{k}
\left[\bG^{(1,0)}
\left(\bDelta \bu^{(m,n|1)} - \bDelta \bu^{(m,n|0)}\right)
+ \bG^{(5,0)}
\left(\bDelta \bu^{(m,n|1)} - \bDelta \bu^{(m-1,n+1|0)}\right)\right.\\
&+\left. \bG^{(3,0)} \left(\bDelta \bu^{(m,n|1)} - \bDelta \bu^{(m,n+1|0)}\right)\right]-\widehat{k} \left[\bG^{(3,0)} \left(\bDelta \bu^{(m,n|1)} - \bDelta \bu^{(m,n|2)}\right) \right.\\
&+ \left. \bG^{(5,0)} \left(\bDelta \bu^{(m,n|1)} - \bDelta \bu^{(m,n|6)}\right) + \bG^{(1,0)} \left(\bDelta \bu^{(m,n|1)} - \bDelta \bu^{(m,n+1|2)}\right)\right],
\end{split}
\label{eqEq1Big}$$ $$\begin{split}
\bR^{(m,n|2)}=&\left ( \widetilde{k} - \overline{k} \right ) \ell \sum_{i=1,3,5} \left(\bg^{(i)} \scalp \balpha \bg^{(i)}\right) \bg^{(i)}
+ \left ( \widetilde{k} + \overline{k} \right ) \ell^2 \sum_{i=1,3,5} \left [ \left( \bg^{(i)} \otimes \bg^{(i)}\right) \scalpp \bbeta \scalp \bg^{(i)} \right ] \bg^{(i)} \\
& + 2 \left ( \widetilde{k} - \overline{k} \right ) \ell \sum_{i=1,3,5} \left [ \left(\bx^{(m,n|0)} \otimes \bg^{(i)} \right) \scalpp \bbeta \scalp \bg^{(i)}\right ] \bg^{(i)}
+ 2 \left ( \widetilde{k} - \overline{k} \right ) \ell^2 \sum_{i=1,3,5} \left [ \left(\bg^{(2)} \otimes \bg^{(i)} \right) \scalpp \bbeta \scalp \bg^{(i)}\right ] \bg^{(i)}\\
&+
\widetilde{k} \left[
\bG^{(2,0)} \left(\bDelta \bu^{(m,n|2)} - \bDelta \bu^{(m,n|0)}\right)
+ \bG^{(4,0)} \left(\bDelta \bu^{(m,n|2)} - \bDelta \bu^{(m-1,n+1|0)}\right)\right.\\
&+ \left.
\bG^{(6,0)} \left(\bDelta \bu^{(m,n|2)} - \bDelta \bu^{(m-1,n|0)}\right)\right]-
\overline{k}
\left[\bG^{(4,0)} \left(\bDelta \bu^{(m,n|2)} - \bDelta \bu^{(m,n|3)}\right)\right.\\
&\left.
+ \bG^{(6,0)} \left(\bDelta \bu^{(m,n|2)} - \bDelta \bu^{(m,n|1)}\right)
+ \bG^{(2,0)} \left(\bDelta \bu^{(m,n|2)} - \bDelta \bu^{(m-1,n|1)}\right
)\right].
\end{split}
\label{eqEq2Big}$$
It follows from the above that all of the resultant forces $\bR^{(m,n|i)}$ may be annihilated only when the additional field $\bDelta \bu^{(m,n|i)}$ assumes a linear expression which, under the constraint given by equations (\[eq:39\]), is provided in the following general form (Fig. \[fig:CorrEForceNodal\], right) $$\bDelta \bu^{(m,n|i)}=
\left\{\begin{array}{lllll}
\bDelta \ba^{(m,n|0)}=\bZ\bx^{(m,n|0)} + \bz, \\[3mm]
\bDelta \bb^{(m,n|i)}=\bV \bx^{(m,n|i)} + \bv,\qquad \qquad &i\,\, \mbox{odd},\\[3mm]
\bDelta \bc^{(m,n|i)}=\bW \bx^{(m,n|i)} + \bw,\qquad \qquad &i\neq 0\,\, \mbox{and}\,\, \mbox{even},
\end{array}
\right.
\label{eq:EspreCampCorr}$$ which implies that the average of the displacement gradient (\[eq:GradMediaRVE\]) in the lattice is $$\langle{\bnabla\bu}\rangle^{(m,n)}_{\textsf{lat}}=\balpha+\ell \bbeta \scalp
\left[
\begin{array}{ccc}
\sqrt{3}(2m+n)\\
3n
\end{array}
\right]
+\dfrac{\bV + \bW }{2}.
\label{eq:MeanGradCorr}$$
Considering the additional field , Eq. (\[eq:EspreCampCorr\]), the three primary resultants $\bR^{(m,n|0)}$, $\bR^{(m,n|1)}$, $\bR^{(m,n|2)}$, Eqs. (\[eqEq0Big\])–(\[eqEq2Big\]), reduce to $$\begin{split}
\bR^{(m,n|0)}=&\left ( \widehat{k} - \widetilde{k} \right ) \ell \sum_{i=1,3,5} \left(\bg^{(i)} \scalp \balpha \bg^{(i)}\right) \bg^{(i)}
+ \left ( \widehat{k} + \widetilde{k} \right ) \ell^2 \sum_{i=1,3,5} \left [ \left( \bg^{(i)} \otimes \bg^{(i)}\right) \scalpp \bbeta \scalp \bg^{(i)} \right ] \bg^{(i)} \\
& + 2 \left ( \widehat{k} - \widetilde{k} \right ) \ell \sum_{i=1,3,5} \left [ \left(\bx^{(m,n|0)} \otimes \bg^{(i)} \right) \scalpp \bbeta \scalp \bg^{(i)} \right ] \bg^{(i)}+\\
&+\sum_{i=1,3,5} \left(\bg^{(i)} \otimes \bg^{(i)} \right) \left[ \widehat{k} \left( \ell \bV \bg^{(i)} + \left(\bV - \bZ \right) \bx^{(m,n|0)} + \bv - \bz \right) \right.\\
& \left. + \widetilde{k} \left( - \ell \bW \bg^{(i)} + \left(\bW - \bZ \right) \bx^{(m,n|0)} + \bw - \bz \right) \right] ,
\end{split}
\label{eqEq0BigCorr}$$ $$\begin{split}
\bR^{(m,n|1)}=&\left ( \overline{k} - \widehat{k} \right ) \ell \sum_{i=1,3,5} \left(\bg^{(i)} \scalp \balpha \bg^{(i)}\right) \bg^{(i)}
+ \left ( \overline{k} + \widehat{k} \right ) \ell^2 \sum_{i=1,3,5} \left [ \left( \bg^{(i)} \otimes \bg^{(i)}\right) \scalpp \bbeta \scalp \bg^{(i)} \right ] \bg^{(i)} \\
&+ 2 \left ( \overline{k} - \widehat{k} \right ) \ell \sum_{i=1,3,5} \left [ \left(\bx^{(m,n|0)} \otimes \bg^{(i)} \right) \scalpp \bbeta \scalp \bg^{(i)}\right ]\bg^{(i)} \\
&+ 2 \left ( \overline{k} - \widehat{k} \right ) \ell^2 \sum_{i=1,3,5} \left [ \left(\bg^{(1)} \otimes \bg^{(i)} \right) \scalpp \bbeta \scalp \bg^{(i)} \right ] \bg^{(i)}+\\
&+\sum_{i=1,3,5} \left(\bg^{(i)} \otimes \bg^{(i)} \right) \left[ \widehat{k} \left( - \ell\bV \bg^{(1)} + \left(\bZ - \bV\right) \bx^{(m,n|0)} + \ell\bZ \left(\bg^{(1)} - \bg^{(i)}\right) + \bz - \bv \right) \right.\\
& \left. + \overline{k} \left( - \ell\bV \bg^{(1)} + \left(\bW - \bV \right)\bx^{(m,n|0)} + \ell\bW \left(\bg^{(i)} + \bg^{(1)}\right) + \bw - \bv \right) \right] ,
\end{split}
\label{eqEq1BigCorr}$$ $$\begin{split}
\bR^{(m,n|2)}=&\left ( \widetilde{k} - \overline{k} \right ) \ell \sum_{i=1,3,5} \left(\bg^{(i)} \scalp \balpha \bg^{(i)}\right) \bg^{(i)}
+ \left ( \widetilde{k} + \overline{k} \right ) \ell^2 \sum_{i=1,3,5} \left [ \left( \bg^{(i)} \otimes \bg^{(i)}\right) \scalpp \bbeta \scalp \bg^{(i)} \right ] \bg^{(i)} \\
& + 2 \left ( \widetilde{k} - \overline{k} \right ) \ell \sum_{i=1,3,5} \left [ \left(\bx^{(m,n|0)} \otimes \bg^{(i)} \right) \scalpp \bbeta \scalp \bg^{(i)}\right ] \bg^{(i)} \\
& + 2 \left ( \widetilde{k} - \overline{k} \right ) \ell^2 \sum_{i=1,3,5} \left [ \left(\bg^{(2)} \otimes \bg^{(i)} \right) \scalpp \bbeta \scalp \bg^{(i)}\right ] \bg^{(i)}+\\
&+\sum_{i=1,3,5} \left(\bg^{(i)} \otimes \bg^{(i)} \right) \left[ \widetilde{k} \left( - \ell\bW \bg^{(2)} + \left(\bZ - \bW\right) \bx^{(m,n|0)} + \ell\bZ \left(\bg^{(i)} + \bg^{(2)}\right) + \bz - \bw \right) \right.\\
& \left. + \overline{k} \left( -\ell\bW \bg^{(2)} + \left(\bV - \bW \right)\bx^{(m,n|0)} - \ell\bV \left(\bg^{(i)} - \bg^{(2)}\right) + \bv - \bw \right) \right] .
\end{split}
\label{eqEq2BigCorr}$$
The annihilation of the three resultant forces $\textbf{R}^{(m,n|0)}$, $\textbf{R}^{(m,n|1)}$, and $\textbf{R}^{(m,n|2)}$ for every unit cell $\{m,n\}$ is equivalent to a system of 30 linear equations in the 18 unknown components of the vectors $\textbf{v}$, $\textbf{w}$, and $\textbf{z}$, and of the matrices $\textbf{V}$, $\textbf{W}$, and $\textbf{Z}$ (Eqs. (\[eq:EspreCampCorr\])). Solving this system leads to two results, namely, (i.) the determination of 12 out of the 18 additional field components, which depend on the components of $\textbf{z}$ and $\textbf{Z}$ assumed as free parameters as $$\begin{array}{ccc}
\bv
=
\mathcal{K}^{[1]}
\left\{
\begin{bmatrix}
\alpha_{12} \\
\dfrac{\alpha_{11}-\alpha_{22}}{2}
\end{bmatrix}
+
\begin{bmatrix}
\dfrac{Z_{12} + Z_{12}}{2} \\
\dfrac{Z_{11}-Z_{22}}{2}
\end{bmatrix}
\right\}\ell
+
\left\{
\mathcal{K}^{[3]}
\begin{bmatrix}
\beta_{111}+\beta_{122} \\
\beta_{222}+\beta_{211}
\end{bmatrix}
+
\mathcal{K}^{[5]}
\begin{bmatrix}
\beta_{221}-\beta_{122} \\
\beta_{112}-\beta_{211}
\end{bmatrix}
\right\}\ell^2
+\bz,
\vspace{4mm}
\\
\bw
=
\mathcal{K}^{[2]}
\left\{
\begin{bmatrix}
\alpha_{12} \\
\dfrac{\alpha_{11}-\alpha_{22}}{2}
\end{bmatrix}
+
\begin{bmatrix}
\dfrac{Z_{12} + Z_{12}}{2} \\
\dfrac{Z_{11}-Z_{22}}{2}
\end{bmatrix}
\right\}\ell
+
\left\{
\mathcal{K}^{[4]}
\begin{bmatrix}
\beta_{111}+\beta_{122} \\
\beta_{222}+\beta_{211}
\end{bmatrix}
+
\mathcal{K}^{[6]}
\begin{bmatrix}
\beta_{221}-\beta_{122} \\
\beta_{112}-\beta_{211}
\end{bmatrix}
\right\}\ell^2
+\bz ,
\vspace{4mm}
\\
\bV =
\mathcal{K}^{[1]}
\begin{bmatrix}
\beta_{112} + \beta_{211} & \beta_{122} + \beta_{221}\\
\\
\beta_{111} - \beta_{122} & \beta_{211} - \beta_{222}
\end{bmatrix}\ell+\bZ
,
\qquad
\bW =
\mathcal{K}^{[2]}
\begin{bmatrix}
\beta_{112} + \beta_{211} & \beta_{122} + \beta_{221}\\
\\
\beta_{111} - \beta_{122} & \beta_{211} - \beta_{222}
\end{bmatrix}\ell+\bZ,
\label{eq:CorrVWvw}
\end{array}$$ and (ii.) two linear equations for the six components of $\bbeta$.
It follows from these two equations that tensor $\bbeta$ is constrained to have only four independent components and will be henceforth referred as $\bbeta^{\textsf{lat}}$, a symbol defining the set of generic quadratic amplitude tensors $\bbeta$, for which the lattice structure is in equilibrium in the absence of external nodal forces. Considering $\beta_{111}$, $\beta_{221}$, $\beta_{112}$, $\beta_{222}$ as the four independent components, tensor $\bbeta^{\textsf{lat}}$ is defined by the six components $\beta_{111}, \beta_{221}, \beta_{112}, \beta_{222}, \beta_{211}^{\textsf{lat}},$ and $
\beta_{122}^{\textsf{lat}}$, where the last two are $$\left[
\begin{array}{cc}
\beta_{211}^{\textsf{lat}}\\
\beta_{122}^{\textsf{lat}}
\end{array}
\right]=
-\left(1+\dfrac{9I_{[3]}}{2I_{[1]}I_{[2]}}\right)
\left[
\begin{array}{cc}
\beta_{222}\\
\beta_{111}
\end{array}
\right]
-\dfrac{9I_{[3]}}{2I_{[1]}I_{[2]}}
\left[
\begin{array}{cc}
\beta_{112}\\
\beta_{221}
\end{array}
\right].
\label{eq:EquiConstrainBeta}$$ In Eqs. (\[eq:CorrVWvw\]) and (\[eq:EquiConstrainBeta\]), the coefficients $I_{[j]}$ ($j=1,2,3$) are the three invariants of the diagonal matrix $\bK$ $$\bK = \left[
\begin{array}{cccc}
\overline{k} & 0 & 0 \\
0 & \widehat{k} & 0 \\
0 & 0 & \widetilde{k}
\end{array}\right],
\label{eq:59}$$ so that $$I_{[1]} = \text{tr} \bK= \overline{k} + \widehat{k} +\widetilde{k}, \qquad
I_{[2]} = \dfrac{1}{2}\left[\left(\text{tr}\bK\right)^2-\text{tr}\bK^2 \right]=
\overline{k} \, \widehat{k}\,+\overline{k} \, \widetilde{k}+ \widehat{k}\, \widetilde{k},\qquad
I_{[3]} = \det(\bK)=\overline{k} \, \widehat{k}\, \widetilde{k},$$ while the coefficients $\mathcal{K}^{[j]}$ ($j=1,...,6$) are given by $$\begin{array}{cccc}
&\mathcal{K}^{[1]}=\dfrac{\overline{k} (\widehat{k}-\widetilde{k})+\widetilde{k} (\widehat{k}-\overline{k})}{I_{[2]}}, \qquad &\mathcal{K}^{[2]}=\dfrac{\widehat{k} (\overline{k}-\widetilde{k})+\overline{k} (\widehat{k}-\widetilde{k})}{I_{[2]}}, \qquad &\mathcal{K}^{[3]}=\dfrac{3 \widehat{k} (\overline{k}+\widetilde{k})+4 \overline{k} (\overline{k}+2 \widetilde{k})}{4 I_{[2]}},
\vspace{4mm}\\
&\mathcal{K}^{[4]}=\dfrac{3 \widetilde{k} (k+\widehat{k})+4 \overline{k} (k+2 \widehat{k})}{4 I_{[2]}}, \qquad &\mathcal{K}^{[5]}=\dfrac{I_{[2]}+3 \overline{k} \widetilde{k}}{4 I_{[2]}}, \qquad &\mathcal{K}^{[6]}=\dfrac{I_{[2]}+3 \overline{k} \widehat{k}}{4 I_{[2]}}.
\end{array}
\label{eq:CoeffCampCorr}$$ Imposing that the additional field $\bDelta \bu^{(m,n|i)}$ does not affect the mean value of the displacement gradient $\langle\nabla\bu\rangle^{(m,n)}_{\textsf{lat}}$, Eq. (\[eq:MeanGradCorr\]), leads to the condition $$\bV + \bW=\b0,
\label{eq:MeanGradCorr2}$$ which, considering Eq. (\[eq:CorrVWvw\]), implies the following expression for $\bZ$ $$\bZ =
-\frac{\mathcal{K}^{[1]}+\mathcal{K}^{[2]}}{2}
\normalsize{\begin{bmatrix}
\beta_{112} + \beta_{211}^{\mathsf{lat}} & \beta_{122}^{\mathsf{lat}} + \beta_{221}\\
\\
\beta_{111} - \beta_{122}^{\mathsf{lat}} & \beta_{211}^{\mathsf{lat}} - \beta_{222}
\end{bmatrix}}\ell,
\label{eq:CorrZ}$$ while the vector $\bz$ appearing in Eqs. (\[eq:CorrVWvw\]) remains indeterminate because it only produces a rigid-body translation.
It is worth noting that:
- in the case of bars with same stiffness ($\overline{k}=\widetilde{k}=\widehat{k}$), enforcing Eqs.(\[eq:EquiConstrainBeta\]) automatically provides the equilibrium Eqs. (\[eqEq0BigCorr\])–(\[eqEq2BigCorr\]) for the generic purely quadratic displacement field augmented by a rigid translation $\bz$, $$\bar{k}=\widetilde{k}=\widehat{k}
\quad
\Longrightarrow
\quad
\begin{cases}
\bv=\bw=\bz,\\
\bV=\bW=\bZ=\b0,
\end{cases}$$ so that the additional field reduces to a rigid-body translation, $\bDelta \bu=\bz$;
- in the case when $\bbeta=\b0$, it follows that $\bV=\bW=\bZ=\b0$ but the additional field is in general non-null when two over the three stiffnesses are different from each other. Indeed, the additional field is annihilated only when $\bg^{(1)} \scalp \balpha \bg^{(1)}=\bg^{(3)} \scalp \balpha \bg^{(3)}=\bg^{(5)} \scalp \balpha \bg^{(5)}$ (or equivalently, $\alpha_{11}=\alpha_{22}$ and $\alpha_{12}=0$), except in the particular case of bars having same stiffness ($\overline{k}=\widetilde{k}=\widehat{k}$), in which case the additional field is always null;
- the second-order tensors $\bV$, $\bW$, and $\bZ$ of the additional field display the following permutation properties $$\bV\left(\kappa_1, \kappa_2, \kappa_3\right) = \bV\left(\kappa_1, \kappa_3, \kappa_2\right), \qquad
\bW\left(\kappa_1, \kappa_2, \kappa_3\right) = \bW\left(\kappa_1, \kappa_3, \kappa_2\right), \qquad
\bZ\left(\kappa_1, \kappa_2, \kappa_3\right) = - \bZ\left(\kappa_1, \kappa_3, \kappa_2\right).$$
In the case $\bbeta=\b0$, the above equations are also complemented by following properties for the vectors $\bv$, $\bw$ of the additional field $$\bv\left(\kappa_1, \kappa_2, \kappa_3\right) = \bv\left(\kappa_3,\kappa_2, \kappa_1\right),\qquad
\bw\left(\kappa_1, \kappa_2, \kappa_3\right) = \bw\left(\kappa_2, \kappa_1, \kappa_3\right), \qquad \mbox{when}\,\,\bbeta=\b0.$$
At this stage, the additional field $\bDelta \bu^{(m,n|i)}$, Eq. (\[eq:EspreCampCorr\]), results completely defined through Eqs. (\[eq:CorrVWvw\]), (\[eq:EquiConstrainBeta\]), and (\[eq:CorrZ\]). With the purpose of highlighting the contribution of the additional field $\bDelta\bu$ to the considered second-order displacement, Eq. (\[eq:campospost\]), three deformed configurations of the lattice are shown in Fig. \[fig:DefoLinQuadCorr\].
Looking to the upper row of the figure, the first image on the left shows the displacement produced by a purely linear ($\bbeta=\b0$) didplacement, while the second image depicts the corresponding additional field only. Finally the image on the right is the composition of the two. The lower row shows respectively a purely quadratic ($\balpha=\b0$) displacement, its additional field $\bDelta \bu^{(m,n|i)}$, and the composition of the two. In the figure, the following stiffnesses of the lattice have been considered: $\overline{k}=\widehat{k}=10\widetilde{k}$.
![(Upper part) Deformed configurations for a lattice with bars of stiffness $\overline{k}=\widehat{k}=10\widetilde{k}$ subject to (left) a purely linear displacement condition with $\{\alpha_{11},\alpha_{22},\alpha_{12}\}=\{0,0,1/5\}$, (center) its additional field, and (right) the sum of these two. (Lower part) As in the upper part, but for a purely quadratic displacement condition with $\{\beta_{111},\beta_{221},\beta_{112},\beta_{222},\beta_{211}^{\mathsf{lat}},\beta_{122}^{\mathsf{lat}}\}=\{-1,1,1,-1,1,1\}1/(80\ell)$.[]{data-label="fig:DefoLinQuadCorr"}](Figures/Fig_3.pdf){width="0.8\linewidth"}
Identification of the higher-order solid equivalent to the lattice structure {#identi}
============================================================================
Considering the second-order displacement field Eq. (\[eq:campospost\]) defined by the tensors $\balpha$ and $\bbeta^{\mathsf{lat}}$ Eqs. (\[eq:EquiConstrainBeta\]) and by the additional field’ $\bDelta \bu^{(m,n|i)}$, Eqs. (\[eq:CorrVWvw\]) and Eq. (\[eq:CorrZ\]), the elastic energy stored within the lattice cell $\{m,n\}$ is computed. This elastic energy is shown to display the same mathematical structure of the elastic energy stored within a unit cell made up of a homogeneous elastic second-gradient solid ($\mathsf{SGE}$) when subject to a quadratic displacement field, defined by the tensors $\balpha$ and $\bbeta^{\mathsf{SGE}}$ (note that $\bbeta^{\mathsf{SGE}}$ defines the coefficients of all quadratic fields which generate equilibrated stresses in a second-gradient elastic material without body forces). Therefore, imposing the elastic energy matching between the lattice and the $\mathsf{SGE}$ solid allows for the identification of the constitutive parameters of the latter and shows that the self-equilibrium condition provides the same constrained boundary condition for the two materials, so that $\bbeta^{\mathsf{lat}}=\bbeta^{\mathsf{SGE}}$.
It is instrumental to represent the components of the tensors $\balpha$ and $\bbeta^{(\boldsymbol{\cdot})}$ (where the superscript $(\boldsymbol{\cdot})$ denotes either $(\mathsf{lat})$ or $(\mathsf{SGE})$) using a vectorial notation through the vectors $\boldsymbol{\mathsf{a}}$ and $\boldsymbol{\mathsf{b}}^{(\boldsymbol{\cdot})}$ as $$\boldsymbol{\mathsf{a}}
=\begin{bmatrix}
\alpha_{11}\\
\alpha_{22}\\
2\alpha_{12}
\end{bmatrix}, \qquad
\boldsymbol{\mathsf{b}}^{(\boldsymbol{\cdot})}
=\begin{bmatrix}
\beta_{111}\\
\beta_{221}\\
\beta_{112}\\
\beta_{222}\\
2\beta_{211}\\
2\beta_{122}
\end{bmatrix},
\label{eq:contrazioBD}$$ and to collect the four components of $\bbeta^{(\boldsymbol{\cdot})}$ not constrained by the equilibrium Eq. (\[eq:EquiConstrainBeta\]) in the vector $\boldsymbol{\mathsf{b}}^{*}$ $$\boldsymbol{\mathsf{b}}^{*}
=\begin{bmatrix}
\beta_{111}\\
\beta_{221}\\
\beta_{112}\\
\beta_{222}
\end{bmatrix},$$ so that vector $\boldsymbol{\mathsf{b}}^{(\boldsymbol{\cdot})}$ can be obtained as $$\boldsymbol{\mathsf{b}}^{(\boldsymbol{\cdot})}=
\boldsymbol{\mathsf{T}}^{(\boldsymbol{\cdot})}
\boldsymbol{\mathsf{b}}^{*}$$ where the matrix $\boldsymbol{\mathsf{T}}^{(\boldsymbol{\cdot})}$ is the transformation matrix enforcing the equilibrium conditions in the lattice (in which case it will be denoted as $\boldsymbol{\mathsf{T}}^{\mathsf{lat}}$) or in the second-gradient elastic solid (in which case it will be denoted as $\boldsymbol{\mathsf{T}}^{\mathsf{SGE}}$).
Energy stored within the lattice structure
------------------------------------------
Considering the second-order displacement field Eq. (\[eq:campospost\]) defined by the tensors $\balpha$ and $\bbeta^{\textsf{lat}}$ under the equilibrium constraint Eqs. (\[eq:EquiConstrainBeta\]) and with the additional displacement given by Eqs. (\[eq:CorrVWvw\]) and (\[eq:CorrZ\]), the elastic strain energy $\mathsf{U}_{\mathsf{lat}}^{(m,n)}\left(\balpha,\bbeta^{\textsf{lat}}\right)$, stored within the lattice unit cell $\{m,n\}$ can be written in terms of vectors $\boldsymbol{\mathsf{a}}$ and $\boldsymbol{\mathsf{b}}^{\mathsf{lat}}$, as $$\mathsf{U}_{\mathsf{lat}}^{(m,n)}\left(\boldsymbol{\mathsf{a}}, \boldsymbol{\mathsf{b}}^{\mathsf{lat}}\right)=
\mathsf{U}_{\mathsf{lat}}^{(m,n)}\left(\boldsymbol{\mathsf{a}}, \boldsymbol{\mathsf{T}}^{\mathsf{lat}}\boldsymbol{\mathsf{b}}^{*}\right),$$ so that $\boldsymbol{\mathsf{b}}^{\mathsf{lat}}=
\boldsymbol{\mathsf{T}}^{\mathsf{lat}}
\boldsymbol{\mathsf{b}}^{*}$ with the definition $$\boldsymbol{\mathsf{T}}^{\mathsf{lat}}
=
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & -\frac{9 I_{[3]}}{I_{[1]} I_{[2]}} & -\frac{9 I_{[3]}}{I_{[1]} I_{[2]}}-2 \\
-\frac{9 I_{[3]}}{I_{[1]} I_{[2]}}-2 & -\frac{9 I_{[3]}}{I_{[1]} I_{[2]}} & 0 & 0 \\
\end{array}
\right).
\label{eq:passaggioAStarMic}$$
Therefore, from Eq. (\[eq:EnergyGeneral\]), the elastic energy of the lattice can be expressed as $\mathsf{U}_{\mathsf{lat}}^{(m,n)}\left(\boldsymbol{\mathsf{a}}, \boldsymbol{\mathsf{b}}^{*}\right)$ and therefore can be represented as the following quadratic form in $\boldsymbol{\mathsf{a}}$ and $\boldsymbol{\mathsf{b}}^*$ $$\begin{split}
\mathsf{U}_{\mathsf{lat}}^{(m,n)}\left(\boldsymbol{\mathsf{a}}, \boldsymbol{\mathsf{b}}^{*}\right)=
\ell^2&\left\{ \boldsymbol{\mathsf{a}} \scalp \boldsymbol{\mathsf{H}}^{[1]}\kern-0.3em\left(\overline{k},\widehat{k},\widetilde{k}\right)\,\boldsymbol{\mathsf{a}}
+ 2\ell \boldsymbol{\mathsf{a}}\scalp\left[ m \boldsymbol{\mathsf{H}}^{[2]}\kern-0.3em\left(\overline{k},\widehat{k},\widetilde{k}\right)\, + n \boldsymbol{\mathsf{H}}^{[3]}\kern-0.3em\left(\overline{k},\widehat{k},\widetilde{k}\right)\, + \boldsymbol{\mathsf{H}}^{[4]}\kern-0.3em\left(\overline{k},\widehat{k},\widetilde{k}\right)\,\right]\boldsymbol{\mathsf{b}}^{*}\right.\\
&\left. + \ell^2 \boldsymbol{\mathsf{b}}^{*} \scalp\left[m^2 \boldsymbol{\mathsf{H}}^{[5]}\kern-0.3em\left(\overline{k},\widehat{k},\widetilde{k}\right)\, + n^2 \boldsymbol{\mathsf{H}}^{[6]}\kern-0.3em\left(\overline{k},\widehat{k},\widetilde{k}\right)\, + m\,n \boldsymbol{\mathsf{H}}^{[7]}\kern-0.3em\left(\overline{k},\widehat{k},\widetilde{k}\right)\, + m \boldsymbol{\mathsf{H}}^{[8]}\kern-0.3em\left(\overline{k},\widehat{k},\widetilde{k}\right)\,+ \right.\right.\\
&\left.\left. + n \boldsymbol{\mathsf{H}}^{[9]}\kern-0.3em\left(\overline{k},\widehat{k},\widetilde{k}\right)\, + \boldsymbol{\mathsf{H}}^{[10]}\kern-0.3em\left(\overline{k},\widehat{k},\widetilde{k}\right)\, \right] \boldsymbol{\mathsf{b}}^{*}\right\},
\end{split}
\label{eq:Ene2ordRVE}$$ where the matrices $\boldsymbol{\mathsf{H}}^{[r]}$ ($r=1,...,10$) depend on the values of the three stiffnesses $\overline{k},\widehat{k}$, and $\widetilde{k}$. These matrices have different dimensions ($3\times 3$ for $r=1$, $3\times 4$ for $r=2,3,4$, and $4 \times 4$ in the other cases) and their components $\mathsf{H}_{ij}^{[r]}$ are reported in Appendix A. From Eq. (\[eq:Ene2ordRVE\]) it is evident that the strain energy depends on the cell position whenever $\boldsymbol{\mathsf{b}}^*\neq0$, so that it becomes independent of indexes $m$ and $n$ only when $\boldsymbol{\mathsf{b}}^*=0$, a condition corresponding to $\boldsymbol{\mathsf{b}}^{\mathsf{lat}}=0$ and also implying $\bbeta^{\mathsf{lat}}= 0$.
Energy stored within a second-gradient elastic solid {#Sec:SecGradCont}
----------------------------------------------------
With reference to the form I’ elastic material introduced by Mindlin [@mindlin1964micro], a second-gradient elastic ($\textsf{SGE}$) solid has a quadratic strain energy density $\mathcal{U}_{\mathbb{SGE}}$ function of the strain $\bepsilon$ and the curvature $\bchi$, which can be derived from the displacement field $\bu$ as $$\epsilon_{ij}=\frac{u_{i,j}+u_{j,i}}{2},
\qquad
\chi_{ijk}=u_{k,ij},$$ displaying the symmetry properties $\epsilon_{ij}=\epsilon_{ji}$ and $\chi_{ijk}=\chi_{jik}$. The quadratic strain energy density $\mathcal{U}_{\mathbb{SGE}}$ can be decomposed as $$\mathcal{U}_{\mathbb{SGE}}\left(\bepsilon,\bchi\right)= \mathcal{U}_{\mathbb{C}}\left(\bepsilon\right) + \mathcal{U}_{\mathbb{M}}\left(\bepsilon,\bchi\right) + \mathcal{U}_{\mathbb{A}}\left(\bchi\right) ,
\label{eq:EneCont1}$$ where $\mathcal{U}_{\mathbb{C}}\left(\bepsilon\right)$ is a purely local’ (Cauchy) energy term and $\mathcal{U}_{\mathbb{A}}\left(\bchi\right)$ a completely non-local’ energy term, while the mutual energy term $\mathcal{U}_{\mathbb{M}}\left(\bepsilon,\bchi\right)$ expresses the coupling between strain and curvature, $$\mathcal{U}_{\mathbb{C}}\left(\bepsilon\right)=\frac{1}{2}\mathbb{C}_{ijkl}\epsilon_{ij}\epsilon_{kl}, \qquad
\mathcal{U}_{\mathbb{M}}\left(\bepsilon,\bchi\right)=\mathbb{M}_{ijklm}\chi_{ijk}\epsilon_{lm},
\qquad
\mathcal{U}_{\mathbb{A}}\left(\bchi\right)= \frac{1}{2}\mathbb{A}_{ijklmn}\chi_{ijk}\chi_{lmn},
\label{eq:EneCont2}$$ being $\boldmath{\mathbb{C}}$, $\boldmath{\mathbb{M}}$, and $\boldmath{\mathbb{A}}$ the fourth-, fifth-, and sixth-order constitutive tensors, respectively, possessing the following symmetries $$\begin{array}{cc}
\mathbb{C}_{ijkl}=\mathbb{C}_{jikl}=\mathbb{C}_{ijlk}=\mathbb{C}_{klij}, \qquad
\mathbb{M}_{ijklm}=\mathbb{M}_{ijkml}=\mathbb{M}_{jiklm}, \qquad
\mathbb{A}_{ijklmn}=\mathbb{A}_{jiklmn}=\mathbb{A}_{ijkmln}=\mathbb{A}_{lmnijk}.
\end{array}$$
The tensors work-conjugate to the fundamental kinematic fields $\bepsilon$ and $\bchi$ are respectively the stress $\bsigma$ and double stress $\btau$, defined as $$\sigma_{ij} = \mathbb{C}_{ijlm}\epsilon_{lm} + \mathbb{M}_{ijlmn}\chi_{lmn},\qquad
\tau_{kji} = \mathbb{A}_{kjilmn}\chi_{lmn} + \mathbb{M}_{lmkji}\epsilon_{lm},
\label{eq:23}$$ which are restricted to satisfy the equilibrium equations, that in the absence of body-forces are expressed by $$\sigma_{ij,j} - \tau_{kji,jk} = 0.
\label{eq:selfequiEqui2}$$ The vectorial representations for the strain $\bepsilon$ and the curvature $\bchi$ are introduced through the strain $\boldsymbol{\mathsf{p}}$ and curvature $\boldsymbol{\mathsf{q}}$ vectors as $$\boldsymbol{\mathsf{p}}
=
\begin{bmatrix}
\epsilon_{11}\\
\epsilon_{22}\\
2\epsilon_{12}
\end{bmatrix},
\qquad
\boldsymbol{\mathsf{q}}
=\begin{bmatrix}
\chi_{111}\\
\chi_{221}\\
\chi_{112}\\
\chi_{222}\\
2\chi_{211}\\
2\chi_{122}
\end{bmatrix},
\label{eq:curv_def_vett1}$$ so that the elastic energy densities (\[eq:EneCont2\]) can be rewritten as $$\mathcal{U}_{\mathbb{C}}\left(\bepsilon\right)= \mathcal{U}_{\boldsymbol{\mathsf{C}}}\left(\boldsymbol{\mathsf{p}}\right),
\quad
\mathcal{U}_{\mathbb{M}}\left(\bepsilon,\bchi\right)=\mathcal{U}_{\boldsymbol{\mathsf{M}}}\left(\boldsymbol{\mathsf{p}},\boldsymbol{\mathsf{q}}\right),
\quad
\mathcal{U}_{\mathbb{A}}\left(\bchi\right)=\mathcal{U}_{\boldsymbol{\mathsf{A}}}\left(\boldsymbol{\mathsf{q}}\right),
\label{eq:EneCont3}$$ where $$\mathcal{U}_{\boldsymbol{\mathsf{C}}}\left(\boldsymbol{\mathsf{p}}\right)=\frac{1}{2}\mathsf{C}_{ij}\mathsf{p}_{i}\mathsf{p}_{j},
\quad
\mathcal{U}_{\boldsymbol{\mathsf{M}}}\left(\boldsymbol{\mathsf{p}},\boldsymbol{\mathsf{q}}\right)=\mathsf{M}_{jk} \mathsf{p}_{j}\mathsf{q}_k,
\quad
\mathcal{U}_{\boldsymbol{\mathsf{A}}}\left(\boldsymbol{\mathsf{q}}\right)=\frac{1}{2} \mathsf{A}_{kl}\mathsf{q}_k\mathsf{q}_l ,\quad i,j=1,2,3 \quad k,l=1,...,6,
\label{eq:EneCont4}$$ with the matrices $\mathsf{C}_{ij}$, $\mathsf{M}_{jk}$, and $\mathsf{A}_{kl}$ respectively representing the constitutive tensors $\boldmath{\mathbb{C}}$, $\boldmath{\mathbb{M}}$, and $\boldmath{\mathbb{A}}$ in the Voigt notation. Note that matrices $\mathsf{C}_{ij}$ and $\mathsf{A}_{jk}$ are square and symmetric (the former of order 3 and the latter of order 6), while $\mathsf{M}_{jk}$ is a rectangular (3 $\times$ 6) matrix. Considering this notation, the strain energy density $\mathcal{U}_{\mathsf{SGE}}\left(\boldsymbol{\mathsf{p}},\boldsymbol{\mathsf{q}}\right)$ can be introduced as $$\mathcal{U}_{\mathsf{SGE}}\left(\boldsymbol{\mathsf{p}},\boldsymbol{\mathsf{q}}\right)=\mathcal{U}_{\boldsymbol{\mathsf{C}}}\left(\boldsymbol{\mathsf{p}}\right)+\mathcal{U}_{\boldsymbol{\mathsf{M}}}\left(\boldsymbol{\mathsf{p}},\boldsymbol{\mathsf{q}}\right)+\mathcal{U}_{\boldsymbol{\mathsf{A}}}\left(\boldsymbol{\mathsf{q}}\right),$$ representing the strain energy density $\mathcal{U}_{\mathbb{SGE}}\left(\bepsilon,\bchi\right)$ in the Voigt notation, so that $$\mathcal{U}_{\mathbb{SGE}}\left(\bepsilon,\bchi\right)=\mathcal{U}_{\mathsf{SGE}}\left(\boldsymbol{\mathsf{p}}\left(\bepsilon\right),\boldsymbol{\mathsf{q}}\left(\bchi\right)\right).
\label{eq:EquivaEner}$$
It is assumed now that the second-gradient elastic material is subject to remote quadratic displacement boundary conditions provided by the second-order displacement field, Eq. (\[eq:campospost\]), in the absence of the additional field ($\bDelta \bu^{(m,n|i)}~=~\b0$, see also Sect. \[Homogenization\]), $$\bu (\bx) = \balpha \bx + \left(\bx \otimes \bx\right) \scalpp \bbeta.
\label{eq:campospostCont}$$ The quadratic displacement field (\[eq:campospostCont\]) is restricted, at first order, by equilibrium, $$\mathbb{C}_{ljkh}~\beta_{jkh}=0,
\label{eq:selfequiEqui}$$ an equation which introduces two relationships between the six coefficients $\beta_{ijk}$, so that two of them are dependent on the remaining four. Therefore, the coefficients $\beta_{ijk}$ are re-assembled in the vector $\bbeta^{\mathsf{SGE}}$, so that $$\bu (\bx) = \balpha \bx + \left(\bx \otimes \bx\right) \scalpp \bbeta^{\mathsf{SGE}} ,
\label{eq:spostselfequiEqui}$$ where $$\begin{array}{c}
\beta^{\mathsf{SGE}}_{111} = \beta_{111},\quad
\beta^{\mathsf{SGE}}_{221} = \beta_{221},\quad
\beta^{\mathsf{SGE}}_{112} = \beta_{112},\quad
\beta^{\mathsf{SGE}}_{222} = \beta_{222}\\ [5 mm]
\beta^{\mathsf{SGE}}_{211}=\beta_{111} \mathcal{D}_{1}+\beta_{221} \mathcal{D}_{2}+\beta_{112} \mathcal{D}_{3}+\beta_{222} \mathcal{D}_{4}, ~~~~~
\beta^{\mathsf{SGE}}_{122}=\beta_{111} \mathcal{D}_{5}+\beta_{221} \mathcal{D}_{6}+\beta_{112} \mathcal{D}_{7}+\beta_{222} \mathcal{D}_{8},
\end{array}
\label{eq:selfequiResults}$$ in which $$\begin{array}{ccc}
\mathcal{D}_{1} = \dfrac{2 \mathsf{C}_{13}^2-\mathsf{C}_{11} \left(\mathsf{C}_{12}+\mathsf{C}_{33}\right)}{\left(\mathsf{C}_{12}+\mathsf{C}_{33}\right)^2-4 \mathsf{C}_{13} \mathsf{C}_{23}}, ~~~~~
\mathcal{D}_{2} = \dfrac{2 \mathsf{C}_{13} \mathsf{C}_{23}-\mathsf{C}_{33} \left(\mathsf{C}_{12}+\mathsf{C}_{33}\right)}{\left(\mathsf{C}_{12}+\mathsf{C}_{33}\right)^2-4 \mathsf{C}_{13} \mathsf{C}_{23}}, ~~~~~
\mathcal{D}_{3} = \dfrac{\mathsf{C}_{13} \left(\mathsf{C}_{33}-\mathsf{C}_{12}\right)}{\left(\mathsf{C}_{12}+\mathsf{C}_{33}\right)^2-4 \mathsf{C}_{13} \mathsf{C}_{23}} ,
\vspace{4mm}\\
\mathcal{D}_{4} = \dfrac{2 \mathsf{C}_{13} \mathsf{C}_{22}-\mathsf{C}_{23} \left(\mathsf{C}_{12}+\mathsf{C}_{33}\right)}{\left(\mathsf{C}_{12}+\mathsf{C}_{33}\right)^2-4 \mathsf{C}_{13} \mathsf{C}_{23}}, ~~~~~
\mathcal{D}_{5} = \dfrac{2 \mathsf{C}_{11} \mathsf{C}_{23}-\mathsf{C}_{13} \left(\mathsf{C}_{12}+\mathsf{C}_{33}\right)}{\left(\mathsf{C}_{12}+\mathsf{C}_{33}\right)^2-4 \mathsf{C}_{13} \mathsf{C}_{23}}, ~~~~~
\mathcal{D}_{6} = \dfrac{\mathsf{C}_{23} \left(\mathsf{C}_{12}-\mathsf{C}_{33}\right)}{4 \mathsf{C}_{13} \mathsf{C}_{23}-\left(\mathsf{C}_{12}+\mathsf{C}_{33}\right)^2} ,
\vspace{4mm}\\
\mathcal{D}_{7} = \dfrac{\mathsf{C}_{33} \left(\mathsf{C}_{12}+\mathsf{C}_{33}\right)-2 \mathsf{C}_{13} \mathsf{C}_{23}}{4 \mathsf{C}_{13} \mathsf{C}_{23}-\left(\mathsf{C}_{12}+\mathsf{C}_{33}\right)^2}, ~~~~~
\mathcal{D}_{8} = \dfrac{2 \mathsf{C}_{23}^2-\mathsf{C}_{22} \left(\mathsf{C}_{12}+\mathsf{C}_{33}\right)}{\left(\mathsf{C}_{12}+\mathsf{C}_{33}\right)^2-4 \mathsf{C}_{13} \mathsf{C}_{23}}.
\end{array}
\label{eq:coeffD}$$ From now on the constrained tensor $\bbeta$, due to Eqs.(\[eq:selfequiResults\]), will be denoted by $\bbeta^{\mathsf{SGE}}$, so that the strain $\boldsymbol{\mathsf{p}}$ and the curvature $\boldsymbol{\mathsf{q}}$ vectors can be rewritten as $$\boldsymbol{\mathsf{p}}^{\mathsf{SGE}}=
\boldsymbol{\mathsf{p}}\left(\boldsymbol{\mathsf{a}},\boldsymbol{\mathsf{T}}^{\mathsf{SGE}}\boldsymbol{\mathsf{b}}^{*}\right),\qquad
\boldsymbol{\mathsf{q}}^{\mathsf{SGE}}=
2\boldsymbol{\mathsf{T}}^{\mathsf{SGE}}\boldsymbol{\mathsf{b}}^{*},$$ where $$\boldsymbol{\mathsf{T}}^{\mathsf{SGE}}=
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
2 \mathcal{D}_{5} & 2 \mathcal{D}_{6} & 2 \mathcal{D}_{7} & 2 \mathcal{D}_{8} \\
2 \mathcal{D}_{1} & 2 \mathcal{D}_{2} & 2 \mathcal{D}_{3} & 2 \mathcal{D}_{4} \\
\end{array}
\right),$$ and $\boldsymbol{\mathsf{p}}^{\mathsf{SGE}}$ can also be expressed as $$\boldsymbol{\mathsf{p}}^{\mathsf{SGE}}\left(\boldsymbol{\mathsf{a}},\boldsymbol{\mathsf{b}}^{*}\right)=
\boldsymbol{\mathsf{a}}+2\left(\boldsymbol{\mathsf{P}}^{[1]}x_1^{(m,n)}+\boldsymbol{\mathsf{P}}^{[2]}x_2^{(m,n)}\right)\boldsymbol{\mathsf{b}}^{*}
\label{eq:curv_def_vett3}$$ with $$\boldsymbol{\mathsf{P}}^{[1]}=
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
\mathcal{D}_{1} & \mathcal{D}_{2} & \mathcal{D}_{3} & \mathcal{D}_{4} \\
\mathcal{D}_{5} & \mathcal{D}_{6} & \mathcal{D}_{7}+1 & \mathcal{D}_{8} \\
\end{array}
\right),
\qquad
\boldsymbol{\mathsf{P}}^{[2]}=
\left(
\begin{array}{cccc}
\mathcal{D}_{5} & \mathcal{D}_{6} & \mathcal{D}_{7} & \mathcal{D}_{8} \\
0 & 0 & 0 & 1 \\
\mathcal{D}_{1} & \mathcal{D}_{2}+1 & \mathcal{D}_{3} & \mathcal{D}_{4} \\
\end{array}
\right).
\label{eq:passaggioAStarSGM}$$
From Eqs. (\[eq:selfequiResults\]), the energy densities, Eqs.(\[eq:EneCont4\]), become $$\mathcal{U}_{\boldsymbol{\mathsf{C}}}\left(\boldsymbol{\mathsf{p}}^{\mathsf{SGE}}\right)=\frac{1}{2}\mathsf{C}_{ij}\mathsf{p}^{\mathsf{SGE}}_{i}\mathsf{p}^{\mathsf{SGE}}_{j},
\quad
\mathcal{U}_{\boldsymbol{\mathsf{M}}}\left(\boldsymbol{\mathsf{p}}^{\mathsf{SGE}},2\boldsymbol{\mathsf{T}}^{\mathsf{SGE}}\boldsymbol{\mathsf{b}}^{*}\right)=\mathsf{M}^{*}_{jk} \mathsf{p}^{\mathsf{SGE}}_{j}\mathsf{q}^{*}_k,
\quad
\mathcal{U}_{\boldsymbol{\mathsf{A}}}\left(2\boldsymbol{\mathsf{T}}^{\mathsf{SGE}}\boldsymbol{\mathsf{b}}^{*}\right)=\frac{1}{2} \mathsf{A}^{*}_{kl}\mathsf{q}^{*}_k\mathsf{q}^{*}_l ,\quad
\begin{array}{ll}
i,j=1,2,3, \\
k,l=1,2,3,4,
\end{array}
\label{eq:EneCont5}$$ where $\boldsymbol{\mathsf{q}}^{*}=2\boldsymbol{\mathsf{b}}^{*}$ and $$\boldsymbol{\mathsf{M}}^{*}=\boldsymbol{\mathsf{M}}\boldsymbol{\mathsf{T}}^{\mathsf{SGE}},
\qquad
\boldsymbol{\mathsf{A}}^{*}=\left(\boldsymbol{\mathsf{T}}^{\mathsf{SGE}}\right)^T\boldsymbol{\mathsf{A}}\boldsymbol{\mathsf{T}}^{\mathsf{SGE}}.$$
Matrices $\boldsymbol{\mathsf{M}}^{*}$ and $\boldsymbol{\mathsf{A}}^{*}$ have reduced dimensions, so that the former is a rectangular 3 $\times$ 4 matrix and the latter a symmetric square matrix of order 4. They define the *condensed representation* for the constitutive matrices $\boldsymbol{\mathsf{M}}$ and $\boldsymbol{\mathsf{A}}$, so that the strain energy density of the second-gradient elastic material can be seen as a function of $\boldsymbol{\mathsf{a}}$ and $\boldsymbol{\mathsf{b}}^{*}$, namely, $\mathcal{U}_{\mathsf{SGE}}\left(\boldsymbol{\mathsf{a}},\boldsymbol{\mathsf{b}}^{*}\right)$.
The elastic energy stored in a hexagonal domain $\Omega^{(m,n)}$ made up of a second-gradient elastic continuum (with the same shape and location of the lattice’s unit cell $\{m,n\}$) is obtained through volume integration of the strain energy density $$\begin{split}
\mathsf{U}_{\mathsf{SGE}}^{(m,n)}\left(\boldsymbol{\mathsf{a}}, \boldsymbol{\mathsf{b}}^{*}\right)&= \int_{\Omega^{(m,n)}}\mathcal{U}_{\mathsf{SGE}}\left(\boldsymbol{\mathsf{a}}, \boldsymbol{\mathsf{b}}^{*}\right) d \Omega ,
\end{split}$$ which is evaluated as $$\begin{split}
\mathsf{U}_{\mathsf{SGE}}^{(m,n)}\left(\boldsymbol{\mathsf{a}}, \boldsymbol{\mathsf{b}}^{*}\right)&= \ell^2\left\{ \boldsymbol{\mathsf{a}} \scalp \boldsymbol{\mathsf{G}}^{[1]}\kern-0.3em\left(\mathsf{C}_{ij}\right)\boldsymbol{\mathsf{a}}
+ 2\ell \boldsymbol{\mathsf{a}}\scalp\left[ m \boldsymbol{\mathsf{G}}^{[2]}\kern-0.3em\left(\mathsf{C}_{ij}\right) + n \boldsymbol{\mathsf{G}}^{[3]}\kern-0.3em\left(\mathsf{C}_{ij}\right) + \boldsymbol{\mathsf{G}}^{[4]}\kern-0.3em\left(\mathsf{M}^{*}_{ij}\right)\right]\boldsymbol{\mathsf{b}}^{*}\right.\\
&\left. + \ell^2 \boldsymbol{\mathsf{b}}^{*} \scalp\left[m^2 \boldsymbol{\mathsf{G}}^{[5]}\kern-0.3em\left(\mathsf{C}_{ij}\right) + n^2 \boldsymbol{\mathsf{G}}^{[6]}\kern-0.3em\left(\mathsf{C}_{ij}\right) + m\,n \boldsymbol{\mathsf{G}}^{[7]}\kern-0.3em\left(\mathsf{C}_{ij}\right) + m\boldsymbol{\mathsf{G}}^{[8]}\kern-0.3em\left(\mathsf{M}^{*}_{ij}\right) \right.\right. \\
&\left.\left. + n \boldsymbol{\mathsf{G}}^{[9]}\kern-0.3em\left(\mathsf{M}^{*}_{ij}\right) + \boldsymbol{\mathsf{G}}^{[10]}\kern-0.3em\left(\mathsf{C}_{ij},\mathsf{A}^{*}_{ij}\right) \right] \boldsymbol{\mathsf{b}}^{*}\right\},
\end{split}
\label{eq:Ene2ordCont2}$$ where the coefficients of the matrices $\boldsymbol{\mathsf{G}}^{[r]}$ ($r=1,...,10$) are reported in Appendix A.
Identification of the condensed’ second-gradient material equivalent to the lattice structure {#Homogenization}
---------------------------------------------------------------------------------------------
By imposing the elastic energy matching between the lattice, Eq. (\[eq:Ene2ordRVE\]), and for the moment unknown effective second-gradient material in the condensed form’, Eq. (\[eq:Ene2ordCont2\]), to hold for every unit cell $\{m,n\}$ and every pair of vectors $\boldsymbol{\mathsf{a}}$ and $\boldsymbol{\mathsf{b}}^{*}$ $$\mathsf{U}_{\mathsf{lat}}^{(m,n)}\left(\boldsymbol{\mathsf{a}}, \boldsymbol{\mathsf{b}}^{*}\right) = \mathsf{U}_{\mathsf{SGE}}^{(m,n)}\left(\boldsymbol{\mathsf{a}}, \boldsymbol{\mathsf{b}}^{*}\right), \qquad \forall \,\, m,n, \boldsymbol{\mathsf{a}}, \boldsymbol{\mathsf{b}}^{*},
\label{eq:EnergyEqui}$$ the following identities are obtained $$\boldsymbol{\mathsf{G}}^{[r]}=\boldsymbol{\mathsf{H}}^{[r]}
\qquad\forall\,r\in[1,10].
\label{eq:DiffeRela}$$
It is highlighted that imposing the energy equivalence, Eq.(\[eq:EnergyEqui\]), at first-order ($\bbeta=\b0$ and therefore $\boldsymbol{\mathsf{b}}^{*}=\b0$) implies $$\boldsymbol{\mathsf{G}}^{[1]}=\boldsymbol{\mathsf{H}}^{[1]},
\label{eq:DiffeRela2}$$ providing all the coefficients of the matrix $\boldsymbol{\mathsf{C}}$ as $$\begin{array}{ccc}
\mathsf{C}_{11} = \mathsf{C}_{22} = \dfrac{2I_{[1]}I_{[2]} + 9I_{[3]}}{4 \sqrt{3} I_{[2]}},\quad
\mathsf{C}_{12} = \dfrac{2I_{[1]}I_{[2]} - 9I_{[3]}}{4 \sqrt{3} I_{[2]}},\quad
\mathsf{C}_{13} = \mathsf{C}_{23} = 0,\quad
\mathsf{C}_{33} = \dfrac{\mathsf{C}_{11}-\mathsf{C}_{12}}{2} = \dfrac{9I_{[3]}}{4 \sqrt{3} I_{[2]}},
\end{array}
\label{eq:1stcostant}$$ which coincide with the corresponding constants obtained in [@day1992elastic] through a different identification technique. From the first-order result, Eq. (\[eq:1stcostant\]), it follows that the two transformation matrices are the same for both the lattice and the equivalent material, namely, $$\boldsymbol{\mathsf{T}}^{\mathsf{lat}}=\boldsymbol{\mathsf{T}}^{\mathsf{SGE}},$$ so that $\boldsymbol{\mathsf{b}}^{\mathsf{lat}} = \boldsymbol{\mathsf{b}}^{\mathsf{SGE}}$ and therefore $\bbeta^{\mathsf{lat}}\left(\boldsymbol{\mathsf{b}}^{*}\right)=\bbeta^{\mathsf{SGE}}\left(\boldsymbol{\mathsf{b}}^{*}\right)$, meaning that the linear and quadratic components of the displacement field imposed to both the solid and the lattice coincide.
The non-local properties can now be identified from Eq. (\[eq:DiffeRela\]) for $r=2,...,10$. In particular, the ten components of the matrix $\boldsymbol{\mathsf{A}}^{*}$ are identified as $$\begin{split}
&\hspace{2mm} \mathsf{A}^{*}_{13} = 0, \quad
\mathsf{A}^{*}_{14} = 0, \quad
\mathsf{A}^{*}_{23} = 0, \quad
\mathsf{A}^{*}_{24} = 0
,\\
&\begin{array}{lcl}
\mathsf{A}^{*}_{11} &=& \dfrac{\sqrt{3} I_{[3]} \ell^2}{64 I_{[1]}^2 I_{[2]}^4}\left[-50 \overline{k}^5 \left(\widehat{k}+\widetilde{k}\right)^3-\overline{k}^4 \left(\widehat{k}+\widetilde{k}\right)^2 \left(100 \widehat{k}^2+359 \widehat{k} \widetilde{k}+100 \widetilde{k}^2\right) + \right.\\
&&\left. - \overline{k}^3 \left(\widehat{k}+\widetilde{k}\right) \left(50 \widehat{k}^4+419 \widehat{k}^3 \widetilde{k}+339 \widehat{k}^2 \widetilde{k}^2+419 \widehat{k} \widetilde{k}^3+50 \widetilde{k}^4\right) + \right.\\
&&\left. + 2 \overline{k}^2 \widehat{k} \widetilde{k} \left(24 \widehat{k}^4+459 \widehat{k}^3 \widetilde{k}+1853 \widehat{k}^2 \widetilde{k}^2+459 \widehat{k} \widetilde{k}^3+24 \widetilde{k}^4\right) + \right.\\
&&\left. + \overline{k} \widehat{k}^2 \widetilde{k}^2 \left(\widehat{k}+\widetilde{k}\right) \left(219 \widehat{k}^2+1283 \widehat{k} \widetilde{k}+219 \widetilde{k}^2\right)+121 \widehat{k}^3 \widetilde{k}^3 \left(\widehat{k}+\widetilde{k}\right)^2\right]
,\\
\mathsf{A}^{*}_{12} &=& \dfrac{\sqrt{3} I_{[3]} \ell^2}{64 I_{[1]}^2 I_{[2]}^4}\left[10 \overline{k}^5 \left(\widehat{k}+\widetilde{k}\right)^3+5 \overline{k}^4 \left(\widehat{k}+\widetilde{k}\right)^2 \left(4 \widehat{k}^2+5 \widehat{k} \widetilde{k}+4 \widetilde{k}^2\right) + \right.\\
&&\left. + \overline{k}^3 \left(\widehat{k}+\widetilde{k}\right) \left(10 \widehat{k}^4-71 \widehat{k}^3 \widetilde{k}-303 \widehat{k}^2 \widetilde{k}^2-71 \widehat{k} \widetilde{k}^3+10 \widetilde{k}^4\right) + \right.\\
&&\left. + 2 \overline{k}^2 \widehat{k} \widetilde{k} \left(6 \widehat{k}^4-9 \widehat{k}^3 \widetilde{k}+641 \widehat{k}^2 \widetilde{k}^2-9 \widehat{k} \widetilde{k}^3+6 \widetilde{k}^4\right) + \right.\\
&&\left. - \overline{k} \widehat{k}^2 \widetilde{k}^2 \left(\widehat{k}+\widetilde{k}\right) \left(33 \widehat{k}^2+\widehat{k} \widetilde{k}+33 \widetilde{k}^2\right)-35 \widehat{k}^3 \widetilde{k}^3 \left(\widehat{k}+\widetilde{k}\right)^2\right]
,\\
\mathsf{A}^{*}_{22} &=& \dfrac{\sqrt{3} I_{[3]} \ell^2}{64 I_{[1]}^2 I_{[2]}^4}\left[-10 \overline{k}^5 \left(\widehat{k}+\widetilde{k}\right)^3-\overline{k}^4 \left(\widehat{k}+\widetilde{k}\right)^2 \left(20 \widehat{k}^2-137 \widehat{k} \widetilde{k}+20 \widetilde{k}^2\right) + \right.\\
&&\left. - \overline{k}^3 \left(\widehat{k}+\widetilde{k}\right) \left(10 \widehat{k}^4-53 \widehat{k}^3 \widetilde{k}+219 \widehat{k}^2 \widetilde{k}^2-53 \widehat{k} \widetilde{k}^3+10 \widetilde{k}^4\right) + \right.\\
&&\left. + 2 \overline{k}^2 \widehat{k} \widetilde{k} \left(12 \widehat{k}^4-45 \widehat{k}^3 \widetilde{k}+349 \widehat{k}^2 \widetilde{k}^2-45 \widehat{k} \widetilde{k}^3+12 \widetilde{k}^4\right) + \right.\\
&&\left. + \overline{k} \widehat{k}^2 \widetilde{k}^2 \left(\widehat{k}+\widetilde{k}\right) \left(51 \widehat{k}^2-197 \widehat{k} \widetilde{k}+51 \widetilde{k}^2\right)+17 \widehat{k}^3 \widetilde{k}^3 \left(\widehat{k}+\widetilde{k}\right)^2\right],\\
\mathsf{A}^{*}_{33} &=& \dfrac{\sqrt{3} \ell^2}{192 I_{[1]}^2 I_{[2]}^4}\left[2 \overline{k}^6 \left(\widehat{k}+\widetilde{k}\right)^3 \left(4 \widehat{k}^2-7 \widehat{k} \widetilde{k}+4 \widetilde{k}^2\right) + \right.\\
&&\left. + \overline{k}^5 \left(\widehat{k}+\widetilde{k}\right)^2 \left(16 \widehat{k}^4-132 \widehat{k}^3 \widetilde{k}+181 \widehat{k}^2 \widetilde{k}^2-132 \widehat{k} \widetilde{k}^3+16 \widetilde{k}^4\right) + \right.\\
&&\left. + \overline{k}^4 \left(\widehat{k}+\widetilde{k}\right) \left(8 \widehat{k}^6-110 \widehat{k}^5 \widetilde{k}+301 \widehat{k}^4 \widetilde{k}^2+667 \widehat{k}^3 \widetilde{k}^3+301 \widehat{k}^2 \widetilde{k}^4-110 \widehat{k} \widetilde{k}^5+8 \widetilde{k}^6\right) + \right.\\
&&\left. + 2 \overline{k}^3 \widehat{k} \widetilde{k} \left(4 \widehat{k}^6+27 \widehat{k}^5 \widetilde{k}-101 \widehat{k}^4 \widetilde{k}^2-587 \widehat{k}^3 \widetilde{k}^3-101 \widehat{k}^2 \widetilde{k}^4+27 \widehat{k} \widetilde{k}^5+4 \widetilde{k}^6\right) + \right.\\
&&\left. - \overline{k}^2 \widehat{k}^2 \widetilde{k}^2 \left(\widehat{k}+\widetilde{k}\right) \left(6 \widehat{k}^4-121 \widehat{k}^3 \widetilde{k}-349 \widehat{k}^2 \widetilde{k}^2-121 \widehat{k} \widetilde{k}^3+6 \widetilde{k}^4\right) + \right.\\
&&\left. - \overline{k} \widehat{k}^3 \widetilde{k}^3 \left(\widehat{k}+\widetilde{k}\right)^2 \left(4 \widehat{k}^2+43 \widehat{k} \widetilde{k}+4 \widetilde{k}^2\right)+2 \widehat{k}^4 \widetilde{k}^4 \left(\widehat{k}+\widetilde{k}\right)^3\right]
\end{array}
\end{split}$$ $$\begin{array}{lcl}
\mathsf{A}^{*}_{34} &=& \dfrac{\sqrt{3} \ell^2}{64 I_{[1]}^2 I_{[2]}^4}\left[-2 \overline{k}^6 \left(\widehat{k}+\widetilde{k}\right)^3 \left(4 \widehat{k}^2+3 \widehat{k} \widetilde{k}+4 \widetilde{k}^2\right) + \right.\\
&&\left. - \overline{k}^5 \left(\widehat{k}+\widetilde{k}\right)^2 \left(16 \widehat{k}^4+4 \widehat{k}^3 \widetilde{k}-63 \widehat{k}^2 \widetilde{k}^2+4 \widehat{k} \widetilde{k}^3+16 \widetilde{k}^4\right) + \right.\\
&&\left. - \overline{k}^4 \left(\widehat{k}+\widetilde{k}\right) \left(8 \widehat{k}^6+6 \widehat{k}^5 \widetilde{k}-267 \widehat{k}^4 \widetilde{k}^2-173 \widehat{k}^3 \widetilde{k}^3-267 \widehat{k}^2 \widetilde{k}^4+6 \widehat{k} \widetilde{k}^5+8 \widetilde{k}^6\right) + \right.\\
&&\left. - 2 \overline{k}^3 \widehat{k} \widetilde{k} \left(4 \widehat{k}^6-15 \widehat{k}^5 \widetilde{k}+115 \widehat{k}^4 \widetilde{k}^2+461 \widehat{k}^3 \widetilde{k}^3+115 \widehat{k}^2 \widetilde{k}^4-15 \widehat{k} \widetilde{k}^5+4 \widetilde{k}^6\right) + \right.\\
&&\left. + \overline{k}^2 \widehat{k}^2 \widetilde{k}^2 \left(\widehat{k}+\widetilde{k}\right) \left(6 \widehat{k}^4+71 \widehat{k}^3 \widetilde{k}+43 \widehat{k}^2 \widetilde{k}^2+71 \widehat{k} \widetilde{k}^3+6 \widetilde{k}^4\right) + \right.\\
&&\left. + \overline{k} \widehat{k}^3 \widetilde{k}^3 \left(\widehat{k}+\widetilde{k}\right)^2 \left(4 \widehat{k}^2+35 \widehat{k} \widetilde{k}+4 \widetilde{k}^2\right)-2 \widehat{k}^4 \widetilde{k}^4 \left(\widehat{k}+\widetilde{k}\right)^3\right]
,\\
\mathsf{A}^{*}_{44} &=& \dfrac{\sqrt{3} \ell^2}{64 I_{[1]}^2 I_{[2]}^4}\left[2 \overline{k}^6 \left(\widehat{k}+\widetilde{k}\right)^3 \left(12 \widehat{k}^2-\widehat{k} \widetilde{k}+12 \widetilde{k}^2\right) + \right.\\
&&\left. + \overline{k}^5 \left(\widehat{k}+\widetilde{k}\right)^2 \left(48 \widehat{k}^4+260 \widehat{k}^3 \widetilde{k}+103 \widehat{k}^2 \widetilde{k}^2+260 \widehat{k} \widetilde{k}^3+48 \widetilde{k}^4\right) + \right.\\
&&\left. + \overline{k}^4 \left(\widehat{k}+\widetilde{k}\right) \left(24 \widehat{k}^6+286 \widehat{k}^5 \widetilde{k}+583 \widehat{k}^4 \widetilde{k}^2-255 \widehat{k}^3 \widetilde{k}^3+583 \widehat{k}^2 \widetilde{k}^4+286 \widehat{k} \widetilde{k}^5+24 \widetilde{k}^6\right) + \right.\\
&&\left. + 2 \overline{k}^3 \widehat{k} \widetilde{k} \left(12 \widehat{k}^6-3 \widehat{k}^5 \widetilde{k}-735 \widehat{k}^4 \widetilde{k}^2-1753 \widehat{k}^3 \widetilde{k}^3-735 \widehat{k}^2 \widetilde{k}^4-3 \widehat{k} \widetilde{k}^5+12 \widetilde{k}^6\right) + \right.\\
&&\left. - \overline{k}^2 \widehat{k}^2 \widetilde{k}^2 \left(\widehat{k}+\widetilde{k}\right) \left(18 \widehat{k}^4+309 \widehat{k}^3 \widetilde{k}+937 \widehat{k}^2 \widetilde{k}^2+309 \widehat{k} \widetilde{k}^3+18 \widetilde{k}^4\right) + \right.\\
&&\left. - \overline{k} \widehat{k}^3 \widetilde{k}^3 \left(\widehat{k}+\widetilde{k}\right)^2 \left(12 \widehat{k}^2+17 \widehat{k} \widetilde{k}+12 \widetilde{k}^2\right)+6 \widehat{k}^4 \widetilde{k}^4 \left(\widehat{k}+\widetilde{k}\right)^3\right],
\end{array}
\label{eq:2stcostant}$$ while the twelve components of the matrix $\boldsymbol{\mathsf{M}}^{*}$ as $$\begin{array}{ccc}
\mathsf{M}^{*}_{11} = \mathsf{M}^{*}_{12} = \mathsf{M}^{*}_{21} = \mathsf{M}^{*}_{22} = \mathsf{M}^{*}_{31} = \mathsf{M}^{*}_{32} = \mathsf{M}^{*}_{33} = \mathsf{M}^{*}_{34} =0,
\vspace{4mm}\\
\mathsf{M}^{*}_{13} = \mathsf{M}^{*}_{23} = \dfrac{\left(\widehat{k}-\widetilde{k}\right) \left(I_{[1]} I_{[2]}-9 I_{[3]}\right) \left(\widehat{k} \widetilde{k}-2 \overline{k} \left(\widehat{k}+\widetilde{k}\right) \right) }{8 \sqrt{3} I_{[1]} I_{[2]}^2}\ell,
\vspace{4mm}\\
\mathsf{M}^{*}_{14} = \mathsf{M}^{*}_{24} = -\dfrac{3 \left(\widehat{k}-\widetilde{k}\right) \left(I_{[1]} I_{[2]}+3 I_{[3]}\right) \left(\widehat{k} \widetilde{k}-2 \overline{k} \left(\widehat{k}+\widetilde{k}\right)\right)}{8 \sqrt{3} I_{[1]} I_{[2]}^2}\ell.
\end{array}
\label{eq:3stcostant}$$ It is worth to note that the result provided by Eqs. (\[eq:1stcostant\])–(\[eq:3stcostant\]) shows that the constitutive matrices are invariant with respect the following permutations of $\{\overline{k},\widehat{k},\widetilde{k}\}$: $$\begin{array}{ccc}
\boldsymbol{\mathsf{C}}\left(\kappa_1, \kappa_2, \kappa_3\right) = \boldsymbol{\mathsf{C}}\left(\kappa_1, \kappa_3, \kappa_2\right) =
\boldsymbol{\mathsf{C}}\left(\kappa_2, \kappa_1, \kappa_3\right) = \boldsymbol{\mathsf{C}}\left(\kappa_2, \kappa_3, \kappa_1\right) =
\boldsymbol{\mathsf{C}}\left(\kappa_3, \kappa_1, \kappa_2\right) = \boldsymbol{\mathsf{C}}\left(\kappa_3, \kappa_2, \kappa_1\right)\\
\boldsymbol{\mathsf{A}}^{*}\left(\kappa_1, \kappa_2, \kappa_3\right) = \boldsymbol{\mathsf{A}}^{*}\left(\kappa_1, \kappa_3, \kappa_2\right),\qquad
\boldsymbol{\mathsf{M}}^{*}\left(\kappa_1, \kappa_2, \kappa_3\right) = -\boldsymbol{\mathsf{M}}^{*}\left(\kappa_1, \kappa_3, \kappa_2\right).
\end{array}$$
It can be therefore concluded that
> *[ the effective response approaches that of a Cauchy elastic material only in the limit of vanishing length of the lattice’s bars, $\ell \longrightarrow 0$, a condition for which $\mathsf{M}^{*}_{ij}=\mathsf{A}^{*}_{ij}=0$. ]{}*
Finally, from Eqs. (\[eq:1stcostant\])–(\[eq:3stcostant\]) it is evident that the stiffness ratio between the bars may have a dramatic effect on the equivalent solid response, as further discussed in second part of this article [@rizzipt2].
Influence of the additional field $\bDelta \bu^{\{m,n|i\}}$
-----------------------------------------------------------
It is remarked that, although $\bbeta^{\mathsf{SGE}}=\bbeta^{\textsf{lat}}
$, the displacement fields imposed to the lattice differs from that imposed to the equivalent solid due to the presence of the additional field $\bDelta\bu^{(m,n|i)}$ in the former. From the practical point of view, however, the amplitude of such an additional field does not play an important role when compared to the amplitude of the quadratic part, so that the deformed configuration of the solid very well represents that of the lattice, even if in the latter the additional field is present.
To analyze the influence of the additional field on the kinematics of the lattice and of the equivalent solid, a rectangular domain (having sides $25\sqrt{3}\ell$ $\times$ $37\ell$) is considered, occupied in one case by the lattice, which is shown on the left in Fig. \[fig:CompaContDisc\], (625 hexagonal unit cells, namely, 25 along each axis of the rectangle) and in the other case by the equivalent continuum with its boundary reported on the left in Fig. \[fig:CompaContDisc\]. The solid is subject to a displacement field characterized by tensors $\balpha$ and $\bbeta^{\mathsf{SGE}}$, while the lattice is subject to the same $\balpha$ and to $\bbeta^{\textsf{lat}}=\bbeta^{\mathsf{SGE}}$ plus the additional field $\bDelta \bu^{\{m,n|i\}}$. In particular, the following values have been selected to produce the figure $\alpha_{11}=0.0\overline{18}$, $\alpha_{22}=0.02$, $\alpha_{12}=0.0\overline{2}$ and $\beta_{111}^{\mathsf{SGE}}=\beta_{111}^{\textsf{lat}}=0.0029$, $\beta_{221}^{\mathsf{SGE}}=\beta_{221}^{\textsf{lat}}=0.0028\overline{6}$, $\beta_{112}^{\mathsf{SGE}}=\beta_{112}^{\textsf{lat}}=0.00\overline{3}$, $\beta_{222}^{\mathsf{SGE}}=\beta_{222}^{\textsf{lat}}=0.004$. Moreover, having selected $\widehat{k}/\overline{k}=2$ and $\widetilde{k}/\overline{k}=3$ as bars’ stiffness ratios, the remaining two components of $\bbeta^{\mathsf{SGE}}$ result from Eq. (\[eq:EquiConstrainBeta\]) and (\[eq:selfequiResults\]) as $\beta_{211}^{\mathsf{SGE}}=\beta_{211}^{\textsf{lat}}=-0.007$ and $\beta_{122}^{\mathsf{SGE}}=\beta_{122}^{\textsf{lat}}=-0.0052$. The additional field $\bDelta \bu^{\{m,n|i\}}$ applied to the lattice has been calculated with the given values of $\balpha$ and $\bbeta^{\textsf{lat}}$ through Eq. (\[eq:CorrVWvw\]) and (\[eq:CorrZ\]).
The undeformed and deformed configurations (visible as lines for the equivalent solid and as spots for the lattice) are reported in Fig. \[fig:CompaContDisc\]. The positions of the undeformed lattice’s nodes were chosen to lie on the undeformed lines of the continuum. The fact that, after deformation, the dots overlap the deformed lines demonstrates that the additional field (needed to enforce equilibrium in the lattice) affects only marginally the overall displacement of the lattice, in which the linear and quadratic displacement fields prevail.
![(Left) Rectangular domain (having sides $25\sqrt{3}\ell$ $\times$ $37\ell$) occupied in one case by the lattice (625 hexagonal unit cells) and in the other case by the equivalent continuum (only its boundary is reported). (Right) Undeformed and deformed configurations for initially straight lines of the equivalent continuum, when subject to a linear and quadratic displacement field. The same displacement, plus the additional field $\bDelta \bu^{\{m,n|i\}}$, are applied to the lattice, of which the nodal positions are reported as spots. The additional field $\bDelta \bu^{\{m,n|i\}}$ is observed to play only a marginal role in the overall deformation of the lattice.[]{data-label="fig:CompaContDisc"}](Figures/Fig_4.pdf){width="80.00000%"}
Discussion
==========
An infinite hexagonal lattice of bars (only subject to axial forces and characterized by three different elastic stiffnesses) has been considered and solved, when loaded at infinity with a quadratic displacement field, enhanced with an additional displacement to comply with the periodicity constraint of the lattice. Its elastic energy has been shown to match with that of a second-gradient (form I’ Mindlin) elastic material, subject to the same quadratic field. In this way, a homogeneous continuum, enriched with an internal length, has been derived, which is equivalent to the discrete lattice. However, this continuum was only identified in a condensed’ form, so that not all constitutive parameters have been identified. For those appearing in the condensed version, closed form expressions have been given, showing the influence of the lattice properties (the hexagon side length $\ell$ and the bars stiffness $\overline{k}$, $\widehat{k}$, $\widetilde{k}$). As an example, the higher-order constitutive parameters $\mathsf{M}^{*}_{13}$ and $\mathsf{M}^{*}_{14}$ ruling the non-centrosymmetric behaviour (and made dimensionless through division by $\overline{k} \ell$) are portrayed in Fig. \[fig:m13\] where two stiffness ratios $\widehat{k}/\overline{k}$ and $\widetilde{k}/\overline{k}$ are varied. The red lines highlight the condition for which both parameters vanish, so that, correspondingly, centrosymmetric response is retrieved, while in all the other cases non-centrosymmetry characterizes the mechanical behaviour of the equivalent material.
![Nonlocal constitutive parameters $\mathsf{M}^{*}_{13}$ (left) and $\mathsf{M}^{*}_{14}$ (right) as functions of the bar stiffness ratios $\widehat{k}/\overline{k}$ and $\widetilde{k}/\overline{k}$. The red lines represent the stiffness ratios pairs for which a centrosymmetric response is attained, while in all the other cases the solid equivalent to the hexagonal bars’ lattice displays a non-centrosymmetric mechanical behaviour.[]{data-label="fig:m13"}](Figures/Fig_5.pdf){width="\textwidth"}
The fact that the equivalent material is only defined in a condensed’ form is a consequence of the fact that the elastic energy equivalence between the solid and the lattice has been so far restricted to self-equilibrated displacement fields. This means, in other words, that the mechanical tests applied both to the lattice and to the continuum are not enough in number to completely characterize the latter. Nevertheless, the presented results allow already to conclude that even a simple hexagonal lattice, which corresponds to an equivalent isotropic, local, and centrosymmetric material at a first-order of approximation, at a higher approximation displays anisotropic, non-local, and non-centrosymmetric effects. Therefore, the presented results provide a tool for advanced mechanical design of microstructured solids. The complete derivation of all material constants of the second-gradient equivalent elastic solids is deferred to Part II of this study, together with the analysis of positive definitess and symmetry of the equivalent material and with an assessment of the validity of the second-gradient model.
#### Acknowledgements.
G.R., D.V., F.D.C. gratefully acknowledge financial support from the grant ERC Advanced Grant Instabilities and nonlocal multiscale modelling of materials’ ERC-2013-ADG-340561-INSTABILITIES. D.B. gratefully acknowledges financial support from PRIN 2015 Multi-scale mechanical models for the design and optimization of micro-structured smart materials and metamaterials’ 2015LYYXA8-006.
Appendix A - Components of the matrices $\boldsymbol{\mathsf{H}}^{[r]}$ and $\boldsymbol{\mathsf{G}}^{[r]}$ {#appendix-a---components-of-the-matrices-boldsymbolmathsfhr-and-boldsymbolmathsfgr .unnumbered}
===========================================================================================================
The coefficients of the matrices $\boldsymbol{\mathsf{H}}^{[r]}$ ($r=1,...,10$) are $$\begin{array}{cccc}
&\mathsf{H}_{11}^{[1]}=\dfrac{3}{16} \left(2 I_{[1]}+\dfrac{9 I_{[3]}}{I_{[2]}}\right),\quad
\mathsf{H}_{12}^{[1]}=\dfrac{3}{16} \left(2 I_{[1]}-\dfrac{9 I_{[3]}}{I_{[2]}}\right),\quad
\mathsf{H}_{13}^{[1]}=0,\\[5mm]
&\mathsf{H}_{22}^{[1]}=\dfrac{3}{16} \left(2 I_{[1]}+\dfrac{9 I_{[3]}}{I_{[2]}}\right),\quad
\mathsf{H}_{23}^{[1]}=0,\quad
\mathsf{H}_{33}^{[1]}=\dfrac{27 I_{[3]}}{16 I_{[2]}}
\end{array}$$ $$\begin{array}{ccc}
&\mathsf{H}_{11}^{[2]}=\dfrac{27 \sqrt{3} I_{[3]} \left(2 I_{[1]} I_{[2]}+9 I_{[3]}\right)}{16 I_{[1]} I_{[2]}^2},\quad
\mathsf{H}_{12}^{[2]}=\dfrac{27 \sqrt{3} I_{[3]} (9 I_{[3]}-2 I_{[1]} I_{[2]})}{16 I_{[1]} I_{[2]}^2},\quad
\mathsf{H}_{13}^{[2]}=0,\\[5mm]
& \mathsf{H}_{14}^{[2]}=0,\quad
\mathsf{H}_{21}^{[2]}=-\dfrac{81 \sqrt{3} I_{[3]} \left(2 I_{[1]} I_{[2]}+3 I_{[3]}\right)}{16 I_{[1]} I_{[2]}^2},\quad
\mathsf{H}_{22}^{[2]}=-\dfrac{27 \sqrt{3} I_{[3]} \left(2 I_{[1]} I_{[2]}+9 I_{[3]}\right)}{16 I_{[1]} I_{[2]}^2},\\[5mm]
& \mathsf{H}_{23}^{[2]}=0,\quad
\mathsf{H}_{24}^{[2]}=0,\quad
\mathsf{H}_{31}^{[2]}=0,\\[5mm]
& \mathsf{H}_{32}^{[2]}=0,\quad
\mathsf{H}_{33}^{[2]}=\dfrac{27 \sqrt{3} I_{[3]} \left(2 I_{[1]} I_{[2]}-9 I_{[3]}\right)}{16 I_{[1]} I_{[2]}^2},\quad
\mathsf{H}_{34}^{[2]}=-\dfrac{27 \sqrt{3} I_{[3]} \left(2 I_{[1]} I_{[2]}+9 I_{[3]}\right)}{16 I_{[1]} I_{[2]}^2}
\end{array}$$ $$\begin{array}{ccc}
& \mathsf{H}_{11}^{[3]}=\dfrac{27 \sqrt{3} I_{[3]} \left(2 I_{[1]} I_{[2]}+9 I_{[3]}\right)}{32 I_{[1]} I_{[2]}^2},\quad
\mathsf{H}_{12}^{[3]}=\dfrac{27 \sqrt{3} I_{[3]} \left(9 I_{[3]}-2 I_{[1]} I_{[2]}\right)}{32 I_{[1]} I_{[2]}^2},\quad
\mathsf{H}_{13}^{[3]}=-\dfrac{81 I_{[3]} \left(2 I_{[1]} I_{[2]}+9 I_{[3]}\right)}{32 I_{[1]} I_{[2]}^2},\\[5mm]
& \mathsf{H}_{14}^{[3]}=-\dfrac{243 I_{[3]} \left(2 I_{[1]} I_{[2]}+3 I_{[3]}\right)}{32 I_{[1]} I_{[2]}^2},\quad
\mathsf{H}_{21}^{[3]}=-\dfrac{81 \sqrt{3} I_{[3]} \left(2 I_{[1]} I_{[2]}+3 I_{[3]}\right)}{32 I_{[1]} I_{[2]}^2},\quad
\mathsf{H}_{22}^{[3]}=-\dfrac{27 \sqrt{3} I_{[3]} \left(2 I_{[1]} I_{[2]}+9 I_{[3]}\right)}{32 I_{[1]} I_{[2]}^2},\\[5mm]
& \mathsf{H}_{23}^{[3]}=\dfrac{81 I_{[3]} \left(9 I_{[3]}-2 I_{[1]} I_{[2]}\right)}{32 I_{[1]} I_{[2]}^2},\quad
\mathsf{H}_{24}^{[3]}=\dfrac{81 I_{[3]} \left(2 I_{[1]} I_{[2]}+9 I_{[3]}\right)}{32 I_{[1]} I_{[2]}^2},\quad
\mathsf{H}_{31}^{[3]}=-\dfrac{81 I_{[3]} \left(2 I_{[1]} I_{[2]}+9 I_{[3]}\right)}{32 I_{[1]} I_{[2]}^2},\\[5mm]
& \mathsf{H}_{32}^{[3]}=\dfrac{81 I_{[3]} \left(2 I_{[1]} I_{[2]}-9 I_{[3]}\right)}{32 I_{[1]} I_{[2]}^2},\quad
\mathsf{H}_{33}^{[3]}=\dfrac{27 \sqrt{3} I_{[3]} \left(2 I_{[1]} I_{[2]}-9 I_{[3]}\right)}{32 I_{[1]} I_{[2]}^2},\quad
\mathsf{H}_{34}^{[3]}=-\dfrac{27 \sqrt{3} I_{[3]} \left(2 I_{[1]} I_{[2]}+9 I_{[3]}\right)}{32 I_{[1]} I_{[2]}^2}
\end{array}$$ $$\begin{array}{ccc}
& \mathsf{H}_{11}^{[4]}=0,\quad
\mathsf{H}_{12}^{[4]}=0,\quad
\mathsf{H}_{13}^{[4]}=\dfrac{3 \left(\widehat{k}-\widetilde{k}\right) \left(I_{[1]} I_{[2]}-9 I_{[3]}\right) \left(\widehat{k} \widetilde{k}-2 k \left(\widehat{k}+\widetilde{k}\right)\right)}{16 I_{[1]} I_{[2]}^2},\\[5mm]
& \mathsf{H}_{14}^{[4]}=-\dfrac{9 \left(\widehat{k}-\widetilde{k}\right) \left(I_{[1]} I_{[2]}+3 I_{[3]}\right) \left(\widehat{k} \widetilde{k}-2 k \left(\widehat{k}+\widetilde{k}\right)\right)}{16 I_{[1]} I_{[2]}^2},\quad
\mathsf{H}_{21}^{[4]}=0,\quad
\mathsf{H}_{22}^{[4]}=0,\\[5mm]
& \mathsf{H}_{23}^{[4]}=\dfrac{3 \left(\widehat{k}-\widetilde{k}\right) \left(I_{[1]} I_{[2]}-9 I_{[3]}\right) \left(\widehat{k} \widetilde{k}-2 k \left(\widehat{k}+\widetilde{k}\right)\right)}{16 I_{[1]} I_{[2]}^2},\quad
\mathsf{H}_{24}^{[4]}=-\dfrac{9 \left(\widehat{k}-\widetilde{k}\right) \left(I_{[1]} I_{[2]}+3 I_{[3]}\right) \left(\widehat{k} \widetilde{k}-2 k \left(\widehat{k}+\widetilde{k}\right)\right)}{16 I_{[1]} I_{[2]}^2},\\[5mm]
&
\mathsf{H}_{31}^{[4]}=0,\quad
\mathsf{H}_{32}^{[4]}=0,\quad
\mathsf{H}_{33}^{[4]}=0,\quad
\mathsf{H}_{34}^{[4]}=0
\end{array}$$ $$\begin{array}{ccc}
& \mathsf{H}_{11}^{[5]}=\dfrac{81 I_{[3]} \left(2 I_{[1]} I_{[2]}+9 I_{[3]}\right) \left(8 I_{[1]} I_{[2]}+9 I_{[3]}\right)}{16 I_{[1]}^2 I_{[2]}^3},\quad
\mathsf{H}_{12}^{[5]}=\dfrac{2187 I_{[3]}^2 \left(2 I_{[1]} I_{[2]}+3 I_{[3]}\right)}{16 I_{[1]}^2 I_{[2]}^3},\quad
\mathsf{H}_{13}^{[5]}=0,\\[5mm]
& \mathsf{H}_{14}^{[5]}=0,\quad
\mathsf{H}_{22}^{[5]}=\dfrac{729 I_{[3]}^2 \left(2 I_{[1]} I_{[2]}+9 I_{[3]}\right)}{16 I_{[1]}^2 I_{[2]}^3},\quad
\mathsf{H}_{23}^{[5]}=0,\\[5mm]
& \mathsf{H}_{24}^{[5]}=0,\quad
\mathsf{H}_{33}^{[5]}=\dfrac{81 I_{[3]} \left(2 I_{[1]} I_{[2]}-9 I_{[3]}\right)^2}{16 I_{[1]}^2 I_{[2]}^3},\quad
\mathsf{H}_{34}^{[5]}=\dfrac{81 I_{[3]} \left(\dfrac{81 I_{[3]}^2}{I_{[1]}^2}-4 I_{[2]}^2\right)}{16 I_{[2]}^3},\\[5mm]
& \mathsf{H}_{44}^{[5]}=\dfrac{81 I_{[3]} \left(2 I_{[1]} I_{[2]}+9 I_{[3]}\right)^2}{16 I_{[1]}^2 I_{[2]}^3}
\end{array}$$ $$\begin{array}{ccc}
& \mathsf{H}_{11}^{[6]}=\dfrac{81 I_{[3]} \left(2 I_{[1]} I_{[2]}+9 I_{[3]}\right) \left(7 I_{[1]} I_{[2]}+18 I_{[3]}\right)}{32 I_{[1]}^2 I_{[2]}^3},\quad
\mathsf{H}_{12}^{[6]}=\dfrac{243 I_{[3]} \left(-2 I_{[1]}^2 I_{[2]}^2+9 I_{[1]} I_{[2]} I_{[3]}+54 I_{[3]}^2\right)}{32 I_{[1]}^2 I_{[2]}^3},\\[5mm]
& \mathsf{H}_{13}^{[6]}=-\dfrac{81 \sqrt{3} I_{[3]} \left(2 I_{[1]} I_{[2]}+9 I_{[3]}\right)}{32 I_{[1]} I_{[2]}^2},\quad
\mathsf{H}_{14}^{[6]}=-\dfrac{243 \sqrt{3} I_{[3]} \left(2 I_{[1]} I_{[2]}+3 I_{[3]}\right)}{32 I_{[1]} I_{[2]}^2},\\[5mm]
& \mathsf{H}_{22}^{[6]}=\dfrac{243 I_{[3]} \left(2 I_{[1]}^2 I_{[2]}^2-15 I_{[1]} I_{[2]} I_{[3]}+54 I_{[3]}^2\right)}{32 I_{[1]}^2 I_{[2]}^3},\quad
\mathsf{H}_{23}^{[6]}=\dfrac{81 \sqrt{3} I_{[3]} \left(2 I_{[1]} I_{[2]}-9 I_{[3]}\right)}{32 I_{[1]} I_{[2]}^2},\\[5mm]
& \mathsf{H}_{24}^{[6]}=-\dfrac{81 \sqrt{3} I_{[3]} \left(2 I_{[1]} I_{[2]}+9 I_{[3]}\right)}{32 I_{[1]} I_{[2]}^2},\quad
\mathsf{H}_{33}^{[6]}=\dfrac{81 I_{[3]} \left(2 I_{[1]}^2 I_{[2]}^2+9 I_{[1]} I_{[2]} I_{[3]}+162 I_{[3]}^2\right)}{32 I_{[1]}^2 I_{[2]}^3},\\[5mm]
& \mathsf{H}_{34}^{[6]}=\dfrac{81 I_{[3]} \left(-2 I_{[1]}^2 I_{[2]}^2+81 I_{[1]} I_{[2]} I_{[3]}+162 I_{[3]}^2\right)}{32 I_{[1]}^2 I_{[2]}^3},\quad
\mathsf{H}_{44}^{[6]}=\dfrac{81 I_{[3]} \left(2 I_{[1]} I_{[2]}+9 I_{[3]}\right) \left(13 I_{[1]} I_{[2]}+18 I_{[3]}\right)}{32 I_{[1]}^2 I_{[2]}^3}
\end{array}$$ $$\begin{array}{ccc}
& \mathsf{H}_{11}^{[7]}=\dfrac{81I_{[3]} \left(2I_{[1]}I_{[2]}+9I_{[3]}\right) \left(8I_{[1]}I_{[2]}+9I_{[3]}\right)}{16I_{[1]}^2I_{[2]}^3},\quad
&\mathsf{H}_{12}^{[7]}=\dfrac{2187I_{[3]}^2 \left(2I_{[1]}I_{[2]}+3I_{[3]}\right)}{16I_{[1]}^2I_{[2]}^3},\\[5mm]
&\mathsf{H}_{13}^{[7]}=-\dfrac{81 \sqrt{3}I_{[3]} \left(2I_{[1]}I_{[2]}+9I_{[3]}\right)}{16I_{[1]}I_{[2]}^2},\quad
& \mathsf{H}_{14}^{[7]}=-\dfrac{243 \sqrt{3}I_{[3]} \left(2I_{[1]}I_{[2]}+3I_{[3]}\right)}{16I_{[1]}I_{[2]}^2},
\\[5mm]
&\mathsf{H}_{22}^{[7]}=\dfrac{729I_{[3]}^2 \left(2I_{[1]}I_{[2]}+9I_{[3]}\right)}{16I_{[1]}^2I_{[2]}^3},\quad
&\mathsf{H}_{23}^{[7]}=\dfrac{81 \sqrt{3}I_{[3]} \left(2I_{[1]}I_{[2]}-9I_{[3]}\right)}{16I_{[1]}I_{[2]}^2},\\[5mm]
& \mathsf{H}_{24}^{[7]}=-\dfrac{81 \sqrt{3}I_{[3]} \left(2I_{[1]}I_{[2]}+9I_{[3]}\right)}{16I_{[1]}I_{[2]}^2},\quad
&\mathsf{H}_{33}^{[7]}=\dfrac{81I_{[3]} \left(2I_{[1]}I_{[2]}-9I_{[3]}\right)^2}{16I_{[1]}^2I_{[2]}^3},\\[5mm]
& \mathsf{H}_{34}^{[7]}=\dfrac{81I_{[3]} \left(\dfrac{81I_{[3]}^2}{\text{I1}^2}-4I_{[2]}^2\right)}{16I_{[2]}^3},\quad
& \mathsf{H}_{44}^{[7]}=\dfrac{81I_{[3]} \left(2I_{[1]}I_{[2]}+9I_{[3]}\right)^2}{16I_{[1]}^2I_{[2]}^3}
\end{array}$$ $$\begin{array}{lcl}
& \mathsf{H}_{11}^{[8]}=0,\quad
\mathsf{H}_{12}^{[8]}=0,\quad
\mathsf{H}_{13}^{[8]}=-\dfrac{27 \sqrt{3} I_{[3]} \left(\widehat{k}-\widetilde{k}\right) \left(I_{[1]} I_{[2]}-9 I_{[3]}\right)\left(\widehat{k} \widetilde{k}-2 k \left(\widehat{k}+\widetilde{k}\right)\right)}{16 I_{[1]}^2 I_{[2]}^3},\\[5mm]
& \mathsf{H}_{14}^{[8]}=\dfrac{81 \sqrt{3} I_{[3]} \left(\widehat{k}-\widetilde{k}\right) \left(I_{[1]} I_{[2]}+3 I_{[3]}\right) \left(\widehat{k} \widetilde{k}-2 k \left(\widehat{k}+\widetilde{k}\right)\right)}{16 I_{[1]}^2 I_{[2]}^3},\quad
\mathsf{H}_{22}^{[8]}=0,\\[5mm]
& \mathsf{H}_{23}^{[8]}=-\dfrac{27 \sqrt{3} I_{[3]} \left(\widehat{k}-\widetilde{k}\right) \left(I_{[1]} I_{[2]}-9 I_{[3]}\right) \left(\widehat{k} \widetilde{k}-2 k \left(\widehat{k}+\widetilde{k}\right)\right)}{16 I_{[1]}^2 I_{[2]}^3},\\
&\mathsf{H}_{24}^{[8]}=\dfrac{81 \sqrt{3} I_{[3]} \left(\widehat{k}-\widetilde{k}\right) \left(I_{[1]} I_{[2]}+3 I_{[3]}\right) \left(\widehat{k} \widetilde{k}-2 k \left(\widehat{k}+\widetilde{k}\right)\right)}{16 I_{[1]}^2 I_{[2]}^3},\\[5mm]
& \mathsf{H}_{33}^{[8]}=0,\quad
\mathsf{H}_{34}^{[8]}=0,\quad
\mathsf{H}_{44}^{[8]}=0
\end{array}$$ $$\begin{array}{ccc}
& \mathsf{H}_{11}^{[9]}=0,\quad
\mathsf{H}_{12}^{[9]}=0,\quad
\mathsf{H}_{13}^{[9]}=-\dfrac{27 \sqrt{3} I_{[3]} \left(\widehat{k}-\widetilde{k}\right) \left(I_{[1]} I_{[2]}-9 I_{[3]}\right) \left(\widehat{k} \widetilde{k}-2 k \left(\widehat{k}+\widetilde{k}\right)\right)}{32 I_{[1]}^2 I_{[2]}^3},\\[5mm]
& \mathsf{H}_{14}^{[9]}=\dfrac{81 \sqrt{3} I_{[3]} \left(\widehat{k}-\widetilde{k}\right) \left(I_{[1]} I_{[2]}+3 I_{[3]}\right) \left(\widehat{k} \widetilde{k}-2 k \left(\widehat{k}+\widetilde{k}\right)\right)}{32 I_{[1]}^2 I_{[2]}^3},\quad
\mathsf{H}_{22}^{[9]}=0,\\[5mm]
& \mathsf{H}_{23}^{[9]}=-\dfrac{27 \sqrt{3} I_{[3]} \left(\widehat{k}-\widetilde{k}\right) \left(I_{[1]} I_{[2]}-9 I_{[3]}\right) \left(\widehat{k} \widetilde{k}-2 k \left(\widehat{k}+\widetilde{k}\right)\right)}{32 I_{[1]}^2 I_{[2]}^3},\\
& \mathsf{H}_{24}^{[9]}=\dfrac{81 \sqrt{3} I_{[3]} \left(\widehat{k}-\widetilde{k}\right) \left(I_{[1]} I_{[2]}+3 I_{[3]}\right) \left(\widehat{k} \widetilde{k}-2 k \left(\widehat{k}+\widetilde{k}\right)\right)}{32 I_{[1]}^2 I_{[2]}^3},\\[5mm]
& \mathsf{H}_{33}^{[9]}=-\dfrac{81 I_{[3]} \left(\widehat{k}-\widetilde{k}\right) \left(I_{[1]} I_{[2]}-9 I_{[3]}\right) \left(\widehat{k} \widetilde{k}-2 k \left(\widehat{k}+\widetilde{k}\right)\right)}{16 I_{[1]}^2 I_{[2]}^3},\\
&
\mathsf{H}_{34}^{[9]}=\dfrac{81 I_{[3]} \left(\widehat{k}-\widetilde{k}\right) \left(I_{[1]} I_{[2]}+9 I_{[3]}\right) \left(\widehat{k} \widetilde{k}-2 k \left(\widehat{k}+\widetilde{k}\right)\right)}{16 I_{[1]}^2 I_{[2]}^3},\\[5mm]
& \mathsf{H}_{44}^{[9]}=\dfrac{243 I_{[3]} \left(\widehat{k}-\widetilde{k}\right) \left(I_{[1]} I_{[2]}+3 I_{[3]}\right) \left(\widehat{k} \widetilde{k}-2 k \left(\widehat{k}+\widetilde{k}\right)\right)}{16 I_{[1]}^2 I_{[2]}^3}
\end{array}$$
$$\mathsf{H}_{13}^{[10]}= \mathsf{H}_{14}^{[10]} = \mathsf{H}_{23}^{[10]}= \mathsf{H}_{24}^{[10]}=0$$
$$\begin{array}{rcl}
\mathsf{H}_{11}^{[10]}&=&\dfrac{9 I_{[3]}}{64 I_{[1]}^2 I_{[2]}^4} \left[\left(-50 \left(\widehat{k}+\widetilde{k}\right)^3 \overline{k}^5-\left(\widehat{k}+\widetilde{k}\right)^2 \left(100 \widehat{k}^2+359 \widetilde{k} \widehat{k}+100 \widetilde{k}^2\right) \overline{k}^4 + \right.\right. \\[5mm]
&&\left.\left. - \left(\widehat{k}+\widetilde{k}\right) \left(50 \widehat{k}^4+419 \widetilde{k} \widehat{k}^3+339 \widetilde{k}^2 \widehat{k}^2+419 \widetilde{k}^3 \widehat{k}+50 \widetilde{k}^4\right) \overline{k}^3 + \right.\right.\\[5mm]
&&\left.\left. + 2 \widehat{k} \widetilde{k} \left(24 \widehat{k}^4+459 \widetilde{k} \widehat{k}^3+1853 \widetilde{k}^2 \widehat{k}^2+459 \widetilde{k}^3 \widehat{k}+24 \widetilde{k}^4\right) \overline{k}^2 + \right.\right.\\[5mm]
&&\left.\left. + \widehat{k}^2 \widetilde{k}^2 \left(\widehat{k}+\widetilde{k}\right) \left(219 \widehat{k}^2+1283 \widetilde{k} \widehat{k}+219 \widetilde{k}^2\right) \overline{k}+121 \widehat{k}^3 \widetilde{k}^3 \left(\widehat{k}+\widetilde{k}\right)^2\right) \ell^2\right] + \\[5mm]
&& + \left[45 I_{[3]} \left(2 I_{[1]} I_{[2]}+9 I_{[3]}\right) \left(5 I_{[1]} I_{[2]}+9 I_{[3]}\right)\right]/\left(64 I_{[1]}^2 I_{[2]}^3\right),\\[5mm]
\mathsf{H}_{12}^{[10]}&=&\dfrac{9 I_{[3]}}{64 I_{[1]}^2 I_{[2]}^4} \left[\left(10 \left(\widehat{k}+\widetilde{k}\right)^3 \overline{k}^5+5 \left(\widehat{k}+\widetilde{k}\right)^2 \left(4 \widehat{k}^2+5 \widetilde{k} \widehat{k}+4 \widetilde{k}^2\right) \overline{k}^4 + \right.\right. \\[5mm]
&&\left.\left. + \left(\widehat{k}+\widetilde{k}\right) \left(10 \widehat{k}^4-71 \widetilde{k} \widehat{k}^3-303 \widetilde{k}^2 \widehat{k}^2-71 \widetilde{k}^3 \widehat{k}+10 \widetilde{k}^4\right) \overline{k}^3 + \right.\right. \\[5mm]
&&\left.\left. + 2 \widehat{k} \widetilde{k} \left(6 \widehat{k}^4-9 \widetilde{k} \widehat{k}^3+641 \widetilde{k}^2 \widehat{k}^2-9 \widetilde{k}^3 \widehat{k}+6 \widetilde{k}^4\right) \overline{k}^2 + \right.\right. \\[5mm]
&&\left.\left. - \widehat{k}^2 \widetilde{k}^2 \left(\widehat{k}+\widetilde{k}\right) \left(33 \widehat{k}^2+\widetilde{k} \widehat{k}+33 \widetilde{k}^2\right) \overline{k}-35 \widehat{k}^3 \widetilde{k}^3 \left(\widehat{k}+\widetilde{k}\right)^2\right) \ell^2 \right]+ \\[5mm]
&& + \left[(45 I_{[3]} \left(-2 I_{[1]}^2 I_{[2]}^2+27 I_{[1]} I_{[2]} I_{[3]}+81 I_{[3]}^2\right)\right]/\left(64 I_{[1]}^2 I_{[2]}^3\right),\\[5mm]
\mathsf{H}_{22}^{[10]}&=&\dfrac{9 I_{[3]}}{64 I_{[1]}^2 I_{[2]}^4} \left[\left(-10 \left(\widehat{k}+\widetilde{k}\right)^3 \overline{k}^5-\left(\widehat{k}+\widetilde{k}\right)^2 \left(20 \widehat{k}^2-137 \widetilde{k} \widehat{k}+20 \widetilde{k}^2\right) \overline{k}^4 + \right.\right. \\[5mm]
&&\left.\left. - \left(\widehat{k}+\widetilde{k}\right) \left(10 \widehat{k}^4-53 \widetilde{k} \widehat{k}^3+219 \widetilde{k}^2 \widehat{k}^2-53 \widetilde{k}^3 \widehat{k}+10 \widetilde{k}^4\right) \overline{k}^3 + \right.\right.\\[5mm]
&&\left.\left. + 2 \widehat{k} \widetilde{k} \left(12 \widehat{k}^4-45 \widetilde{k} \widehat{k}^3+349 \widetilde{k}^2 \widehat{k}^2-45 \widetilde{k}^3 \widehat{k}+12 \widetilde{k}^4\right) \overline{k}^2 + \right.\right.\\[5mm]
&&\left.\left. + \widehat{k}^2 \widetilde{k}^2 \left(\widehat{k}+\widetilde{k}\right) \left(51 \widehat{k}^2-197 \widetilde{k} \widehat{k}+51 \widetilde{k}^2\right) \overline{k}+17 \widehat{k}^3 \widetilde{k}^3 \left(\widehat{k}+\widetilde{k}\right)^2\right) \ell^2 \right]+ \\[5mm]
&& + \left[45 I_{[3]} \left(2 I_{[1]}^2 I_{[2]}^2-9 I_{[1]} I_{[2]} I_{[3]}+81 I_{[3]}^2\right)\right]/\left(64 I_{[1]}^2 I_{[2]}^3\right),\\[5mm]
\mathsf{H}_{33}^{[10]}&=&\dfrac{3}{64 I_{[1]}^2 I_{[2]}^4} \left[\left(2 \left(\widehat{k}+\widetilde{k}\right)^3 \left(4 \widehat{k}^2-7 \widetilde{k} \widehat{k}+4 \widetilde{k}^2\right) \overline{k}^6 + \right.\right.\\[5mm]
&&\left.\left. + \left(\widehat{k}+\widetilde{k}\right)^2 \left(16 \widehat{k}^4-132 \widetilde{k} \widehat{k}^3+181 \widetilde{k}^2 \widehat{k}^2-132 \widetilde{k}^3 \widehat{k}+16 \widetilde{k}^4\right) \overline{k}^5 + \right.\right.\\[5mm]
&&\left.\left. + \left(\widehat{k}+\widetilde{k}\right) \left(8 \widehat{k}^6-110 \widetilde{k} \widehat{k}^5+301 \widetilde{k}^2 \widehat{k}^4+667 \widetilde{k}^3 \widehat{k}^3+301 \widetilde{k}^4 \widehat{k}^2-110 \widetilde{k}^5 \widehat{k}+8 \widetilde{k}^6\right) \overline{k}^4 + \right.\right.\\[5mm]
&&\left.\left. + 2 \widehat{k} \widetilde{k} \left(4 \widehat{k}^6+27 \widetilde{k} \widehat{k}^5-101 \widetilde{k}^2 \widehat{k}^4-587 \widetilde{k}^3 \widehat{k}^3-101 \widetilde{k}^4 \widehat{k}^2+27 \widetilde{k}^5 \widehat{k}+4 \widetilde{k}^6\right) \overline{k}^3 + \right.\right.\\[5mm]
&&\left.\left. - \widehat{k}^2 \widetilde{k}^2 \left(\widehat{k}+\widetilde{k}\right) \left(6 \widehat{k}^4-121 \widetilde{k} \widehat{k}^3-349 \widetilde{k}^2 \widehat{k}^2-121 \widetilde{k}^3 \widehat{k}+6 \widetilde{k}^4\right) \overline{k}^2 + \right.\right.\\[5mm]
&&\left.\left. - \widehat{k}^3 \widetilde{k}^3 \left(\widehat{k}+\widetilde{k}\right)^2 \left(4 \widehat{k}^2+43 \widetilde{k} \widehat{k}+4 \widetilde{k}^2\right) \overline{k}+2 \widehat{k}^4 \widetilde{k}^4 \left(\widehat{k}+\widetilde{k}\right)^3\right) \ell^2 \right]+ \\[5mm]
&& + \left[45 I_{[3]} \left(2 I_{[1]}^2 I_{[2]}^2-9 I_{[1]} I_{[2]} I_{[3]}+81 I_{[3]}^2\right)\right]/\left(64 I_{[1]}^2 I_{[2]}^3\right),
\end{array}$$
$$\begin{array}{rcl}
\mathsf{H}_{34}^{[10]}&=&-\dfrac{9}{64 I_{[1]}^2 I_{[2]}^4} \left[\left(2 \left(\widehat{k}+\widetilde{k}\right)^3 \left(4 \widehat{k}^2+3 \widetilde{k} \widehat{k}+4 \widetilde{k}^2\right) \overline{k}^6 + \right.\right.\\[5mm]
&&\left.\left. + \left(\widehat{k}+\widetilde{k}\right)^2 \left(16 \widehat{k}^4+4 \widetilde{k} \widehat{k}^3-63 \widetilde{k}^2 \widehat{k}^2+4 \widetilde{k}^3 \widehat{k}+16 \widetilde{k}^4\right) \overline{k}^5 + \right.\right.\\[5mm]
&&\left.\left. + \left(\widehat{k}+\widetilde{k}\right) \left(8 \widehat{k}^6+6 \widetilde{k} \widehat{k}^5-267 \widetilde{k}^2 \widehat{k}^4-173 \widetilde{k}^3 \widehat{k}^3-267 \widetilde{k}^4 \widehat{k}^2+6 \widetilde{k}^5 \widehat{k}+8 \widetilde{k}^6\right) \overline{k}^4 + \right.\right.\\[5mm]
&&\left.\left. + 2 \widehat{k} \widetilde{k} \left(4 \widehat{k}^6-15 \widetilde{k} \widehat{k}^5+115 \widetilde{k}^2 \widehat{k}^4+461 \widetilde{k}^3 \widehat{k}^3+115 \widetilde{k}^4 \widehat{k}^2-15 \widetilde{k}^5 \widehat{k}+4 \widetilde{k}^6\right) \overline{k}^3 + \right.\right.\\[5mm]
&&\left.\left. - \widehat{k}^2 \widetilde{k}^2 \left(\widehat{k}+\widetilde{k}\right) \left(6 \widehat{k}^4+71 \widetilde{k} \widehat{k}^3+43 \widetilde{k}^2 \widehat{k}^2+71 \widetilde{k}^3 \widehat{k}+6 \widetilde{k}^4\right) \overline{k}^2 + \right.\right.\\[5mm]
&&\left.\left. - \widehat{k}^3 \widetilde{k}^3 \left(\widehat{k}+\widetilde{k}\right)^2 \left(4 \widehat{k}^2+35 \widetilde{k} \widehat{k}+4 \widetilde{k}^2\right) \overline{k} + 2 \widehat{k}^4 \widetilde{k}^4 \left(\widehat{k}+\widetilde{k}\right)^3\right) \ell^2 \right]+ \\[5mm]
&& + \left[45 I_{[3]} \left(-2 I_{[1]}^2 I_{[2]}^2+27 I_{[1]} I_{[2]} I_{[3]}+81 I_{[3]}^2\right)\right]/\left(64 I_{[1]}^2 I_{[2]}^3\right),\\[5mm]
\mathsf{H}_{44}^{[10]}&=&\dfrac{9}{64 I_{[1]}^2 I_{[2]}^4} \left[\left(2 \left(\widehat{k}+\widetilde{k}\right)^3 \left(12 \widehat{k}^2-\widetilde{k} \widehat{k}+12 \widetilde{k}^2\right) \overline{k}^6 + \right.\right.\\[5mm]
&&\left.\left. + \left(\widehat{k}+\widetilde{k}\right)^2 \left(48 \widehat{k}^4+260 \widetilde{k} \widehat{k}^3+103 \widetilde{k}^2 \widehat{k}^2+260 \widetilde{k}^3 \widehat{k}+48 \widetilde{k}^4\right) \overline{k}^5 + \right.\right.\\[5mm]
&&\left.\left. + \left(\widehat{k}+\widetilde{k}\right) \left(24 \widehat{k}^6+286 \widetilde{k} \widehat{k}^5+583 \widetilde{k}^2 \widehat{k}^4-255 \widetilde{k}^3 \widehat{k}^3+583 \widetilde{k}^4 \widehat{k}^2+286 \widetilde{k}^5 \widehat{k}+24 \widetilde{k}^6\right) \overline{k}^4 + \right.\right.\\[5mm]
&&\left.\left. + 2 \widehat{k} \widetilde{k} \left(12 \widehat{k}^6-3 \widetilde{k} \widehat{k}^5-735 \widetilde{k}^2 \widehat{k}^4-1753 \widetilde{k}^3 \widehat{k}^3-735 \widetilde{k}^4 \widehat{k}^2-3 \widetilde{k}^5 \widehat{k}+12 \widetilde{k}^6\right) \overline{k}^3 + \right.\right.\\[5mm]
&&\left.\left. - \widehat{k}^2 \widetilde{k}^2 \left(\widehat{k}+\widetilde{k}\right) \left(18 \widehat{k}^4+309 \widetilde{k} \widehat{k}^3+937 \widetilde{k}^2 \widehat{k}^2+309 \widetilde{k}^3 \widehat{k}+18 \widetilde{k}^4\right) \overline{k}^2 + \right.\right.\\[5mm]
&&\left.\left. - \widehat{k}^3 \widetilde{k}^3 \left(\widehat{k}+\widetilde{k}\right)^2 \left(12 \widehat{k}^2+17 \widetilde{k} \widehat{k}+12 \widetilde{k}^2\right) \overline{k}+6 \widehat{k}^4 \widetilde{k}^4 \left(\widehat{k}+\widetilde{k}\right)^3\right) \ell^2 \right]+ \\[5mm]
&& + \left[45 I_{[3]} \left(2 I_{[1]} I_{[2]}+9 I_{[3]}\right) \left(5 I_{[1]} I_{[2]}+9 I_{[3]}\right)\right]/\left(64 I_{[1]}^2 I_{[2]}^3\right)
\end{array}$$
The coefficients of the matrices $\boldsymbol{\mathsf{G}}^{[r]}$ ($r=1,...,10$) are $$\mathsf{G}_{ij}^{[1]}=\dfrac{3}{4} \sqrt{3}\mathsf{C}_{ij}$$ $$\begin{array}{ccc}
\mathsf{G}_{11}^{[2]}=\dfrac{9}{2}\left(\mathsf{C}_{11}+\mathsf{C}_{12} \mathcal{D}_{1}+\mathsf{C}_{13} \mathcal{D}_{5}\right),\quad
\mathsf{G}_{12}^{[2]}=\dfrac{9}{2}\left(\mathsf{C}_{12} \mathcal{D}_{2}+\mathsf{C}_{13} \mathcal{D}_{6}\right),\quad
\mathsf{G}_{13}^{[2]}=\dfrac{9}{2}\left(\mathsf{C}_{12} \mathcal{D}_{3}+\mathsf{C}_{13} \mathcal{D}_{7}+\mathsf{C}_{13}\right),\\[5mm]
\mathsf{G}_{14}^{[2]}=\dfrac{9}{2}\left(\mathsf{C}_{12} \mathcal{D}_{4}+\mathsf{C}_{13} \mathcal{D}_{8}\right),\quad
\mathsf{G}_{21}^{[2]}=\dfrac{9}{2}\left(\mathsf{C}_{12}+\mathsf{C}_{22} \mathcal{D}_{1}+\mathsf{C}_{23}\mathcal{D}_{5}\right),\quad
\mathsf{G}_{22}^{[2]}=\dfrac{9}{2}\left(\mathsf{C}_{22} \mathcal{D}_{2}+\mathsf{C}_{23} \mathcal{D}_{6}\right),\\[5mm]
\mathsf{G}_{23}^{[2]}=\dfrac{9}{2}\left(\mathsf{C}_{22} \mathcal{D}_{3}+\mathsf{C}_{23} \mathcal{D}_{7}+\mathsf{C}_{23}\right),\quad
\mathsf{G}_{24}^{[2]}=\dfrac{9}{2}\left(\mathsf{C}_{22} \mathcal{D}_{4}+\mathsf{C}_{23} \mathcal{D}_{8}\right),\quad
\mathsf{G}_{31}^{[2]}=\dfrac{9}{2}\left(\mathsf{C}_{13}+\mathsf{C}_{23} \mathcal{D}_{1}+\mathsf{C}_{33} \mathcal{D}_{5}\right),\\[5mm]
\mathsf{G}_{32}^{[2]}=\dfrac{9}{2}\left(\mathsf{C}_{23} \mathcal{D}_{2}+\mathsf{C}_{33}
\mathcal{D}_{6}\right),\quad
\mathsf{G}_{33}^{[2]}=\dfrac{9}{2}\left(\mathsf{C}_{23} \mathcal{D}_{3}+\mathsf{C}_{33} \mathcal{D}_{7}+\mathsf{C}_{33}\right),\quad
\mathsf{G}_{34}^{[2]}=\dfrac{9}{2}\left(\mathsf{C}_{23} \mathcal{D}_{4}+\mathsf{C}_{33} \mathcal{D}_{8}\right).
\end{array}$$ $$\begin{array}{ccc}
\mathsf{G}_{11}^{[3]}=\dfrac{9}{4}\left(\sqrt{3} \mathsf{C}_{11} \mathcal{D}_{5}+\mathsf{C}_{11}+\mathsf{C}_{12} \mathcal{D}_{1}+\mathsf{C}_{13} \left(\sqrt{3} \mathcal{D}_{1}+\mathcal{D}_{5}\right)\right),\quad
\mathsf{G}_{12}^{[3]}=\dfrac{9}{4}\left(\sqrt{3} \mathsf{C}_{11} \mathcal{D}_{6}+\mathsf{C}_{12} \mathcal{D}_{2}+\mathsf{C}_{13} \left(\sqrt{3} \mathcal{D}_{2}+\mathcal{D}_{6}+\sqrt{3}\right)\right),\\[5mm]
\mathsf{G}_{13}^{[3]}=\dfrac{9}{4}\left(\sqrt{3}\mathsf{C}_{11} \mathcal{D}_{7}+\mathsf{C}_{12} \mathcal{D}_{3}+\mathsf{C}_{13} \left(\sqrt{3} \mathcal{D}_{3}+\mathcal{D}_{7}+1\right)\right),\quad
\mathsf{G}_{14}^{[3]}=\dfrac{9}{4}\left(\sqrt{3} \mathsf{C}_{11} \mathcal{D}_{8}+\mathsf{C}_{12} \left(\mathcal{D}_{4}+\sqrt{3}\right)+\mathsf{C}_{13} \left(\sqrt{3} \mathcal{D}_{4}+\mathcal{D}_{8}\right)\right),\\[5mm]
\mathsf{G}_{21}^{[3]}=\dfrac{9}{4}\left(\sqrt{3} \mathsf{C}_{12}\mathcal{D}_{5}+\mathsf{C}_{12}+\mathsf{C}_{22} \mathcal{D}_{1}+\mathsf{C}_{23} \left(\sqrt{3} \mathcal{D}_{1}+\mathcal{D}_{5}\right)\right),\quad
\mathsf{G}_{22}^{[3]}=\dfrac{9}{4}\left(\sqrt{3} \mathsf{C}_{12} \mathcal{D}_{6}+\mathsf{C}_{22} \mathcal{D}_{2}+\mathsf{C}_{23} \left(\sqrt{3} \mathcal{D}_{2}+\mathcal{D}_{6}+\sqrt{3}\right)\right),\\[5mm]
\mathsf{G}_{23}^{[3]}=\dfrac{9}{4}\left(\sqrt{3} \mathsf{C}_{12}\mathcal{D}_{7}+\mathsf{C}_{22} \mathcal{D}_{3}+\mathsf{C}_{23} \left(\sqrt{3} \mathcal{D}_{3}+\mathcal{D}_{7}+1\right)\right),\quad
\mathsf{G}_{24}^{[3]}=\dfrac{9}{4}\left(\sqrt{3} \mathsf{C}_{12} \mathcal{D}_{8}+\mathsf{C}_{22} \left(\mathcal{D}_{4}+\sqrt{3}\right)+\mathsf{C}_{23} \left(\sqrt{3} \mathcal{D}_{4}+\mathcal{D}_{8}\right)\right),\\[5mm]
\mathsf{G}_{31}^{[3]}=\dfrac{9}{4}\left(\sqrt{3} \mathsf{C}_{13}\mathcal{D}_{5}+\mathsf{C}_{13}+\mathsf{C}_{23} \mathcal{D}_{1}+\mathsf{C}_{33} \left(\sqrt{3} \mathcal{D}_{1}+\mathcal{D}_{5}\right)\right),\quad
\mathsf{G}_{32}^{[3]}=\dfrac{9}{4}\left(\sqrt{3} \mathsf{C}_{13} \mathcal{D}_{6}+\mathsf{C}_{23} \mathcal{D}_{2}+\mathsf{C}_{33} \left(\sqrt{3} \mathcal{D}_{2}+\mathcal{D}_{6}+\sqrt{3}\right)\right),\\[5mm]
\mathsf{G}_{33}^{[3]}=\dfrac{9}{4}\left(\sqrt{3} \mathsf{C}_{13}\mathcal{D}_{7}+\mathsf{C}_{23} \mathcal{D}_{3}+\mathsf{C}_{33} \left(\sqrt{3} \mathcal{D}_{3}+\mathcal{D}_{7}+1\right)\right),\quad
\mathsf{G}_{34}^{[3]}=\dfrac{9}{4}\left(\sqrt{3} \mathsf{C}_{13} \mathcal{D}_{8}+\mathsf{C}_{23} \left(\mathcal{D}_{4}+\sqrt{3}\right)+\mathsf{C}_{33} \left(\sqrt{3} \mathcal{D}_{4}+\mathcal{D}_{8}\right)\right).
\end{array}$$ $$\mathsf{G}_{ij}^{[4]}=\dfrac{3}{2\ell} \sqrt{3}\mathsf{M}^{*}_{ij}$$ $$\begin{array}{ccc}
\mathsf{G}_{11}^{[5]}=9 \sqrt{3} \left(\mathsf{C}_{11}+2 \mathsf{C}_{12} \mathcal{D}_{1}+2 \mathcal{D}_{5} \left(\mathsf{C}_{13}+\mathsf{C}_{23} \mathcal{D}_{1}\right)+\mathsf{C}_{22} \mathcal{D}_{1}^2+\mathsf{C}_{33} \mathcal{D}_{5}^2\right),\\[5mm]
\mathsf{G}_{12}^{[5]}=9 \sqrt{3} \left(\mathcal{D}_{2} \left(\mathsf{C}_{12}+\mathsf{C}_{22} \mathcal{D}_{1}+\mathsf{C}_{23} \mathcal{D}_{5}\right)+\mathcal{D}_{6}\left(\mathsf{C}_{13}+\mathsf{C}_{23} \mathcal{D}_{1}+\mathsf{C}_{33} \mathcal{D}_{5}\right)\right),\\[5mm]
\mathsf{G}_{13}^{[5]}=9 \sqrt{3} \left(\mathcal{D}_{3} \left(\mathsf{C}_{12}+\mathsf{C}_{22} \mathcal{D}_{1}\right)+\mathsf{C}_{13} \left(\mathcal{D}_{7}+1\right)+\mathsf{C}_{23} \left(\mathcal{D}_{1} \mathcal{D}_{7}+\mathcal{D}_{1}+\mathcal{D}_{3} \mathcal{D}_{5}\right)+\mathsf{C}_{33}\mathcal{D}_{5} \left(\mathcal{D}_{7}+1\right)\right),\\[5mm]
\mathsf{G}_{14}^{[5]}=9\sqrt{3} \left(\mathcal{D}_{4} \left(\mathsf{C}_{12}+\mathsf{C}_{22} \mathcal{D}_{1}+\mathsf{C}_{23} \mathcal{D}_{5}\right)+\mathcal{D}_{8} \left(\mathsf{C}_{13}+\mathsf{C}_{23} \mathcal{D}_{1}+\mathsf{C}_{33} \mathcal{D}_{5}\right)\right),\quad
\mathsf{G}_{22}^{[5]}=9 \sqrt{3} \left(\mathsf{C}_{22} \mathcal{D}_{2}^2+\mathcal{D}_{6} \left(2 \mathsf{C}_{23} \mathcal{D}_{2}+\mathsf{C}_{33}\mathcal{D}_{6}\right)\right),\\[5mm]
\mathsf{G}_{23}^{[5]}=9 \sqrt{3} \left(\mathsf{C}_{22} \mathcal{D}_{2} \mathcal{D}_{3}+\mathsf{C}_{23} \left(\mathcal{D}_{2} \mathcal{D}_{7}+\mathcal{D}_{2}+\mathcal{D}_{3} \mathcal{D}_{6}\right)+\mathsf{C}_{33} \mathcal{D}_{6} \left(\mathcal{D}_{7}+1\right)\right),\\[5mm]
\mathsf{G}_{24}^{[5]}=9 \sqrt{3} \left(\mathsf{C}_{22} \mathcal{D}_{2} \mathcal{D}_{4}+\mathsf{C}_{23} \mathcal{D}_{2} \mathcal{D}_{8}+\mathsf{C}_{23} \mathcal{D}_{4} \mathcal{D}_{6}+\mathsf{C}_{33} \mathcal{D}_{6} \mathcal{D}_{8}\right),\quad
\mathsf{G}_{33}^{[5]}=9 \sqrt{3} \left(\mathcal{D}_{3} \left(\mathsf{C}_{22} \mathcal{D}_{3}+2 \mathsf{C}_{23} \left(\mathcal{D}_{7}+1\right)\right)+\mathsf{C}_{33} \left(\mathcal{D}_{7}+1\right)^2\right),\\[5mm]
\mathsf{G}_{34}^{[5]}=9 \sqrt{3} \left(\mathsf{C}_{22} \mathcal{D}_{3} \mathcal{D}_{4}+\mathsf{C}_{23} \left(\mathcal{D}_{3} \mathcal{D}_{8}+\mathcal{D}_{4} \mathcal{D}_{7}+\mathcal{D}_{4}\right)+\mathsf{C}_{33} \left(\mathcal{D}_{7}+1\right) \mathcal{D}_{8}\right),\quad
\mathsf{G}_{44}^{[5]}=9 \sqrt{3} \left(\mathsf{C}_{22} \mathcal{D}_{4}^2+\mathcal{D}_{8} \left(2 \mathsf{C}_{23} \mathcal{D}_{4}+\mathsf{C}_{33} \mathcal{D}_{8}\right)\right).
\end{array}$$ $$\begin{array}{rcl}
\mathsf{G}_{11}^{[6]}&=&\dfrac{9}{4} \left(\left(3 \sqrt{3} \mathsf{C}_{11}+6 \mathsf{C}_{13}+\sqrt{3} \mathsf{C}_{33}\right) \mathcal{D}_{5}^2+2 \left(3 \mathsf{C}_{11}+\left(3 \mathsf{C}_{12}+\sqrt{3} \mathsf{C}_{23}+3 \mathsf{C}_{33}\right) \mathcal{D}_{1}+\sqrt{3} \mathsf{C}_{13} \left(3 \mathcal{D}_{1}+1\right)\right)
\mathcal{D}_{5} + \right.\\[5mm]
&&\left. + \sqrt{3} \mathsf{C}_{11}+\mathcal{D}_{1} \left(2 \sqrt{3} \mathsf{C}_{12}+6 \mathsf{C}_{13}+\left(\sqrt{3} \mathsf{C}_{22}+6 \mathsf{C}_{23}+3 \sqrt{3} \mathsf{C}_{33}\right) \mathcal{D}_{1}\right)\right),\\[5mm]
\mathsf{G}_{12}^{[6]}&=&\dfrac{9}{4} \left(\sqrt{3} \mathsf{C}_{12} \mathcal{D}_{2}+3 \mathsf{C}_{11}\mathcal{D}_{6}+\mathsf{C}_{13} \left(3 \mathcal{D}_{2} \left(\sqrt{3} \mathcal{D}_{5}+1\right)+\sqrt{3} (3 \mathcal{D}_{1}+1) \mathcal{D}_{6}+3 \mathcal{D}_{5} \left(2 \mathcal{D}_{6}+\sqrt{3}\right)+3\right) + \right.\\[5mm]
&&\left. + \mathcal{D}_{1} \left(\sqrt{3} \mathsf{C}_{22} \mathcal{D}_{2}+3 \mathsf{C}_{12} \mathcal{D}_{6}+3 \mathsf{C}_{33}\left(\sqrt{3} \mathcal{D}_{2}+\mathcal{D}_{6}+\sqrt{3}\right)+\mathsf{C}_{23} \left(6 \mathcal{D}_{2}+\sqrt{3} \mathcal{D}_{6}+3\right)\right) + \right.\\[5mm]
&&\left. + \mathcal{D}_{5} \left(3 \mathsf{C}_{12} \mathcal{D}_{2}+\mathsf{C}_{33} \left(3 \mathcal{D}_{2}+\sqrt{3} \mathcal{D}_{6}+3\right)+\sqrt{3} (\mathsf{C}_{23} \mathcal{D}_{2}+3\mathsf{C}_{11} \mathcal{D}_{6})\right)\right),
\\[5mm]
\mathsf{G}_{13}^{[6]}&=&\dfrac{9}{4} \left(\sqrt{3} \mathsf{C}_{12} \mathcal{D}_{3}+3 \mathsf{C}_{11} \mathcal{D}_{7}+\mathcal{D}_{5} \left(\left(3 \mathsf{C}_{12}+\sqrt{3} \mathsf{C}_{23}+3 \mathsf{C}_{33}\right) \mathcal{D}_{3}+\sqrt{3} (\mathcal{D}_{7} \mathsf{C}_{33}+\mathsf{C}_{33}+3\mathsf{C}_{11} \mathcal{D}_{7})\right) + \right.\\[5mm]
&&\left. + \mathsf{C}_{13} \left(3 \mathcal{D}_{5}+\sqrt{3} \left(3 \mathcal{D}_{1} \mathcal{D}_{7}+\mathcal{D}_{7}+1\right)+3 \left(\sqrt{3} \mathcal{D}_{5} \mathcal{D}_{3}+\mathcal{D}_{3}+2 \mathcal{D}_{5} \mathcal{D}_{7}\right)\right) + \right.\\[5mm]
&&\left. + \mathcal{D}_{1} \left(\sqrt{3} \mathsf{C}_{22} \mathcal{D}_{3}+3 \mathsf{C}_{12} \mathcal{D}_{7}+3 \mathsf{C}_{33} \left(\sqrt{3} \mathcal{D}_{3}+\mathcal{D}_{7}+1\right)+\mathsf{C}_{23} \left(6 \mathcal{D}_{3}+\sqrt{3} (\mathcal{D}_{7}+1)\right)\right)\right),
\\[5mm]
\mathsf{G}_{14}^{[6]}&=&\dfrac{9}{4} \left(\mathsf{C}_{22} \mathcal{D}_{1} \left(\sqrt{3} \mathcal{D}_{4}+3\right)+\mathsf{C}_{12} \left(3 \mathcal{D}_{5}\mathcal{D}_{4}+\sqrt{3} \mathcal{D}_{4}+3 \sqrt{3} \mathcal{D}_{5}+3 \mathcal{D}_{1} \mathcal{D}_{8}+3\right)+3 \mathsf{C}_{11} \left(\sqrt{3} \mathcal{D}_{5} \mathcal{D}_{8}+\mathcal{D}_{8}\right) + \right.\\[5mm]
&&\left. + \mathsf{C}_{33} \left(3 \mathcal{D}_{4} \mathcal{D}_{5}+\sqrt{3} \mathcal{D}_{8} \mathcal{D}_{5}+3 \mathcal{D}_{1} \left(\sqrt{3}\mathcal{D}_{4}+\mathcal{D}_{8}\right)\right)+\mathsf{C}_{13} \left(\sqrt{3} \left(3 \mathcal{D}_{1}+1\right) \mathcal{D}_{8}+3 \left(\sqrt{3} \mathcal{D}_{5} \mathcal{D}_{4}+\mathcal{D}_{4}+2 \mathcal{D}_{5} \mathcal{D}_{8}\right)\right) + \right.\\[5mm]
&&\left. + \mathsf{C}_{23} \left(\left(\sqrt{3} \mathcal{D}_{4}+3\right) \mathcal{D}_{5}+\mathcal{D}_{1}\left(6 \mathcal{D}_{4}+\sqrt{3} (\mathcal{D}_{8}+3)\right)\right)\right),\\[5mm]
\mathsf{G}_{22}^{[6]}&=&\dfrac{9}{4} \left(\sqrt{3} \mathsf{C}_{22} \mathcal{D}_{2}^2+2 \mathsf{C}_{23} \left(3 \mathcal{D}_{2}+\sqrt{3} \mathcal{D}_{6}+3\right) \mathcal{D}_{2}+3 \mathcal{D}_{6} \left(2 \mathsf{C}_{12} \mathcal{D}_{2}+\sqrt{3} \mathsf{C}_{11}\mathcal{D}_{6}+2 \mathsf{C}_{13} \left(\sqrt{3} \mathcal{D}_{2}+\mathcal{D}_{6}+\sqrt{3}\right)\right) + \right.\\[5mm]
&&\left. + \mathsf{C}_{33} \left(3 \sqrt{3} \mathcal{D}_{2}^2+6 \left(\mathcal{D}_{6}+\sqrt{3}\right) \mathcal{D}_{2}+\mathcal{D}_{6} \left(\sqrt{3} \mathcal{D}_{6}+6\right)+3\sqrt{3}\right)\right),\\[5mm]
\mathsf{G}_{23}^{[6]}&=&\dfrac{9}{4} \left(\sqrt{3} \mathsf{C}_{22} \mathcal{D}_{2} \mathcal{D}_{3}+3 \left(\mathsf{C}_{12} \mathcal{D}_{2}+\sqrt{3} \mathsf{C}_{13} \left(\mathcal{D}_{2}+1\right)\right) \mathcal{D}_{7}+3 \mathcal{D}_{6} \left(\sqrt{3} \mathcal{D}_{3} \mathsf{C}_{13}+2 \mathcal{D}_{7}\mathsf{C}_{13}+\mathsf{C}_{13}+\mathsf{C}_{12} \mathcal{D}_{3}+\sqrt{3} \mathsf{C}_{11} \mathcal{D}_{7}\right) + \right.\\[5mm]
&&\left. + \mathsf{C}_{23} \left(\mathcal{D}_{3} \left(6 \mathcal{D}_{2}+\sqrt{3} \mathcal{D}_{6}+3\right)+\sqrt{3} \mathcal{D}_{2} \left(\mathcal{D}_{7}+1\right)\right)+\mathsf{C}_{33} \left(3 \mathcal{D}_{3} \left(\sqrt{3}\mathcal{D}_{2}+\mathcal{D}_{6}+\sqrt{3}\right)+\left(3 \mathcal{D}_{2}+\sqrt{3} \mathcal{D}_{6}+3\right) \left(\mathcal{D}_{7}+1\right)\right)\right),\\[5mm]
\mathsf{G}_{24}^{[6]}&=&\dfrac{9}{4} \left(\mathsf{C}_{22} \mathcal{D}_{2} \left(\sqrt{3} \mathcal{D}_{4}+3\right)+3 \sqrt{3} \mathsf{C}_{12} \mathcal{D}_{6}+3 \left(\mathsf{C}_{12}\mathcal{D}_{2}+\sqrt{3} \mathsf{C}_{13} \left(\mathcal{D}_{2}+1\right)\right) \mathcal{D}_{8} + \right.\\[5mm]
&&\left. + 3 \mathcal{D}_{6} \left(\mathsf{C}_{12} \mathcal{D}_{4}+\sqrt{3} \mathsf{C}_{13} \mathcal{D}_{4}+\sqrt{3} \mathsf{C}_{11} \mathcal{D}_{8}+2 \mathsf{C}_{13} \mathcal{D}_{8}\right)+\mathsf{C}_{33} \left(3 \mathcal{D}_{4} \left(\sqrt{3}
\mathcal{D}_{2}+\mathcal{D}_{6}+\sqrt{3}\right)+\left(3\mathcal{D}_{2}+\sqrt{3}\mathcal{D}_{6}+3\right)\mathcal{D}_{8}\right) + \right.\\[5mm]
&&\left. + \mathsf{C}_{23}\left(\sqrt{3}\mathcal{D}_{4}\mathcal{D}_{6}+3\left(\mathcal{D}_{4}+\mathcal{D}_{6}+\sqrt{3}\right)+\mathcal{D}_{2} \left(6 \mathcal{D}_{4}+\sqrt{3}(\mathcal{D}_{8}+3)\right)\right)\right),\\[5mm]
\end{array}$$ $$\begin{array}{rcl}
\mathsf{G}_{33}^{[6]}&=&\dfrac{9}{4} \left(\sqrt{3} \mathsf{C}_{22} \mathcal{D}_{3}^2+2 \mathsf{C}_{23} \left(3 \mathcal{D}_{3}+\sqrt{3} (\mathcal{D}_{7}+1)\right) \mathcal{D}_{3}+\mathsf{C}_{33} \left(3 \sqrt{3} \mathcal{D}_{3}^2+6 (\mathcal{D}_{7}+1) \mathcal{D}_{3}+\sqrt{3}\left(\mathcal{D}_{7}+1\right)^2\right) + \right.\\[5mm]
&&\left. + 3\mathcal{D}_{7}\left(2\mathsf{C}_{12}\mathcal{D}_{3}+\sqrt{3}\mathsf{C}_{11}\mathcal{D}_{7}+2\mathsf{C}_{13}\left(\sqrt{3}\mathcal{D}_{3}+\mathcal{D}_{7}+1\right)\right)\right),\\[5mm]
\mathsf{G}_{34}^{[6]}&=&\dfrac{9}{4} \left(\mathsf{C}_{22} \mathcal{D}_{3} \left(\sqrt{3} \mathcal{D}_{4}+3\right)+3\sqrt{3} \mathsf{C}_{12} \mathcal{D}_{7}+3 \mathcal{D}_{4} \left(\left(\mathsf{C}_{12}+\sqrt{3} \mathsf{C}_{13}\right) \mathcal{D}_{7}+\mathsf{C}_{33} \left(\sqrt{3} \mathcal{D}_{3}+\mathcal{D}_{7}+1\right)\right) + \right.\\[5mm]
&&\left. +\left(3 (\mathsf{C}_{12}+\mathsf{C}_{33}) \mathcal{D}_{3}+3 \mathsf{C}_{13} \left(\sqrt{3}\mathcal{D}_{3}+2 \mathcal{D}_{7}+1\right)+\sqrt{3} (\mathcal{D}_{7} \mathsf{C}_{33}+\mathsf{C}_{33}+3 \mathsf{C}_{11} \mathcal{D}_{7})\right) \mathcal{D}_{8} + \right.\\[5mm]
&&\left. + \mathsf{C}_{23} \left(\left(\sqrt{3} \mathcal{D}_{4}+3\right)\left(\mathcal{D}_{7}+1\right)+\mathcal{D}_{3} \left(6\mathcal{D}_{4}+\sqrt{3}(\mathcal{D}_{8}+3)\right)\right)\right),\\[5mm]
\mathsf{G}_{44}^{[6]}&=&\dfrac{9}{4} \left(3 \sqrt{3} \mathsf{C}_{33} \mathcal{D}_{4}^2+6 \left(\mathsf{C}_{12}+\sqrt{3} \mathsf{C}_{13}+\mathsf{C}_{33}\right) \mathcal{D}_{8} \mathcal{D}_{4}+\left(3 \sqrt{3} \mathsf{C}_{11}+6 \mathsf{C}_{13}+\sqrt{3} \mathsf{C}_{33}\right)\mathcal{D}_{8}^2 + \right.\\[5mm]
&&\left. + \mathsf{C}_{22} \left(\mathcal{D}_{4} \left(\sqrt{3} \mathcal{D}_{4}+6\right)+3 \sqrt{3}\right)+6 \sqrt{3} \mathsf{C}_{12} \mathcal{D}_{8}+2 \mathsf{C}_{23} \left(3 \mathcal{D}_{4}^2+\sqrt{3} \left(\mathcal{D}_{8}+3\right) \mathcal{D}_{4}+3 \mathcal{D}_{8}\right)\right).
\end{array}$$ $$\begin{array}{rcl}
\mathsf{G}_{11}^{[7]}&=&9 \left(\mathcal{D}_{1} \left(2 \sqrt{3} \mathsf{C}_{12}+3 \mathsf{C}_{13}+\sqrt{3} \mathsf{C}_{22} \mathcal{D}_{1}+3 \mathsf{C}_{23} \mathcal{D}_{1}\right)+\mathsf{C}_{11} \left(3 \mathcal{D}_{5}+\sqrt{3}\right) + \right.\\[5mm]
&&\left. + \mathcal{D}_{5} \left(3 \mathcal{D}_{5} \mathsf{C}_{13}+2 \sqrt{3} \mathsf{C}_{13}+3 \mathsf{C}_{12} \mathcal{D}_{1}+2 \sqrt{3} \mathsf{C}_{23} \mathcal{D}_{1}+3 \mathsf{C}_{33} \mathcal{D}_{1}+\sqrt{3} \mathsf{C}_{33} \mathcal{D}_{5}\right)\right),\\[5mm]
\mathsf{G}_{12}^{[7]}&=&\dfrac{9}{2} \left(2 \sqrt{3} \mathsf{C}_{12} \mathcal{D}_{2}+3 \mathsf{C}_{33} \mathcal{D}_{5}+3 \mathsf{C}_{11} \mathcal{D}_{6}+\mathcal{D}_{5} \left(3 \mathsf{C}_{12}
\mathcal{D}_{2}+2 \sqrt{3} \mathsf{C}_{23} \mathcal{D}_{2}+3 \mathsf{C}_{33} \mathcal{D}_{2}+2 \sqrt{3} \mathsf{C}_{33} \mathcal{D}_{6}\right) + \right.\\[5mm]
&&\left. + \mathsf{C}_{13} \left(3 \mathcal{D}_{2}+6 \mathcal{D}_{5} \mathcal{D}_{6}+2 \sqrt{3} \mathcal{D}_{6}+3\right)+\mathcal{D}_{1} \left(2 \sqrt{3} \mathsf{C}_{22} \mathcal{D}_{2}+3
\left(\mathsf{C}_{12}+\mathsf{C}_{33}\right) \mathcal{D}_{6}+\mathsf{C}_{23} \left(6 \mathcal{D}_{2}+2 \sqrt{3} \mathcal{D}_{6}+3\right)\right)\right),\\[5mm]
\mathsf{G}_{13}^{[7]}&=&\dfrac{9}{2} \left(2 \sqrt{3} \mathsf{C}_{12} \mathcal{D}_{3}+3 \mathsf{C}_{11} \mathcal{D}_{7}+\mathcal{D}_{1} \left(2 \sqrt{3} \mathsf{C}_{22} \mathcal{D}_{3}+3 \mathsf{C}_{12}
\mathcal{D}_{7}+3 \mathsf{C}_{33} (\mathcal{D}_{7}+1)\right) + \right.\\[5mm]
&&\left. + \mathcal{D}_{5} \left(3 \left(\mathsf{C}_{12}+\mathsf{C}_{33}\right) \mathcal{D}_{3}+2 \sqrt{3} \mathsf{C}_{33} \left(\mathcal{D}_{7}+1\right)\right)+2 \mathsf{C}_{23} \left(\mathcal{D}_{3} \left(3 \mathcal{D}_{1}+\sqrt{3} \mathcal{D}_{5}\right)+\sqrt{3} \mathcal{D}_{1}(\mathcal{D}_{7}+1)\right) + \right.\\[5mm]
&&\left. + \mathsf{C}_{13} \left(3 \mathcal{D}_{3}+2 \sqrt{3} \left(\mathcal{D}_{7}+1\right)+\mathcal{D}_{5} \left(6 \mathcal{D}_{7}+3\right)\right)\right),\\[5mm]
\mathsf{G}_{14}^{[7]}&=&\dfrac{9}{2} \left(3 \mathsf{C}_{13} \mathcal{D}_{4}+6 \mathsf{C}_{23} \mathcal{D}_{1} \mathcal{D}_{4}+2 \sqrt{3} \mathsf{C}_{23} \mathcal{D}_{5} \mathcal{D}_{4}+3 \mathsf{C}_{33}
\mathcal{D}_{5} \mathcal{D}_{4}+\mathsf{C}_{22} \mathcal{D}_{1} \left(2 \sqrt{3} \mathcal{D}_{4}+3\right) + \right.\\[5mm]
&&\left. + 3 \mathsf{C}_{23} \mathcal{D}_{5}+\left(3 \mathsf{C}_{11}+2 \sqrt{3} \mathsf{C}_{13}+2 \sqrt{3} \mathsf{C}_{23} \mathcal{D}_{1}+3 \mathsf{C}_{33} \mathcal{D}_{1}+6 \mathsf{C}_{13} \mathcal{D}_{5}+2 \sqrt{3} \mathsf{C}_{33}
\mathcal{D}_{5}\right) \mathcal{D}_{8} + \right.\\[5mm]
&&\left. + \mathsf{C}_{12} \left(3 \mathcal{D}_{5} \mathcal{D}_{4}+2 \sqrt{3} \mathcal{D}_{4}+3 \mathcal{D}_{1} \mathcal{D}_{8}+3\right)\right),\\[5mm]
\mathsf{G}_{22}^{[7]}&=&9 \left(\sqrt{3} \mathsf{C}_{22} \mathcal{D}_{2}^2+\mathsf{C}_{23} \left(3 \mathcal{D}_{2}+2 \sqrt{3} \mathcal{D}_{6}+3\right) \mathcal{D}_{2}+\mathcal{D}_{6}
\left(\sqrt{3} \mathcal{D}_{6} \mathsf{C}_{33}+3 \mathsf{C}_{33}+3 \left(\mathsf{C}_{12}+\mathsf{C}_{33}\right) \mathcal{D}_{2}+3 \mathsf{C}_{13} \mathcal{D}_{6}\right)\right),
\\[5mm]\mathsf{G}_{23}^{[7]}&=&\dfrac{9}{2} \left(2 \sqrt{3} \mathsf{C}_{22} \mathcal{D}_{2} \mathcal{D}_{3}+3 \mathsf{C}_{12} \mathcal{D}_{6} \mathcal{D}_{3}+3 \mathsf{C}_{13} \mathcal{D}_{6}+3
\mathsf{C}_{12} \mathcal{D}_{2} \mathcal{D}_{7}+6 \mathsf{C}_{13} \mathcal{D}_{6} \mathcal{D}_{7}+\mathsf{C}_{23} \left(\mathcal{D}_{3} \left(6 \mathcal{D}_{2}+2 \sqrt{3} \mathcal{D}_{6}+3\right) + \right.\right.\\[5mm]
&&\left.\left. + 2 \sqrt{3} \mathcal{D}_{2} \left(\mathcal{D}_{7}+1\right)\right)+\mathsf{C}_{33} \left(3 \left(\mathcal{D}_{2}+1\right) \left(\mathcal{D}_{7}+1\right)+\mathcal{D}_{6}
\left(3 \mathcal{D}_{3}+2 \sqrt{3} \left(\mathcal{D}_{7}+1\right)\right)\right)\right),\\[5mm]
\mathsf{G}_{24}^{[7]}&=&\dfrac{9}{2} \left(\mathsf{C}_{22} \mathcal{D}_{2} \left(2 \sqrt{3} \mathcal{D}_{4}+3\right)+3 \left(\mathsf{C}_{12}+\mathsf{C}_{33}\right) \mathcal{D}_{4} \mathcal{D}_{6}+\left(3 \mathsf{C}_{12} \mathcal{D}_{2}+6 \mathsf{C}_{13}
\mathcal{D}_{6}+\mathsf{C}_{33} \left(3 \mathcal{D}_{2}+2 \sqrt{3} \mathcal{D}_{6}+3\right)\right) \mathcal{D}_{8} + \right.\\[5mm]
&&\left. + \mathsf{C}_{23} \left(3 \mathcal{D}_{6}+\mathcal{D}_{4} \left(6 \mathcal{D}_{2}+2 \sqrt{3} \mathcal{D}_{6}+3\right)+2 \sqrt{3} \mathcal{D}_{2} \mathcal{D}_{8}\right)\right), \\[5mm]
\mathsf{G}_{33}^{[7]}&=&9 \left(\sqrt{3}
\mathsf{C}_{22} \mathcal{D}_{3}^2+\mathsf{C}_{23} \left(3 \mathcal{D}_{3}+2 \sqrt{3} \left(\mathcal{D}_{7}+1\right)\right) \mathcal{D}_{3}+3 \mathcal{D}_{7} \left(\mathcal{D}_{7} \mathsf{C}_{13}+\mathsf{C}_{13}+\mathsf{C}_{12} \mathcal{D}_{3}\right) + \right.\\[5mm]
&&\left. + \mathsf{C}_{33} \left(\mathcal{D}_{7}+1\right) \left(3 \mathcal{D}_{3}+\sqrt{3}
\left(\mathcal{D}_{7}+1\right)\right)\right),\\[5mm]
\end{array}$$ $$\begin{array}{rcl}
\mathsf{G}_{34}^{[7]}&=&\dfrac{9}{2} \left(\mathsf{C}_{22} \mathcal{D}_{3} \left(2 \sqrt{3} \mathcal{D}_{4}+3\right)+3 \mathcal{D}_{4} \left(\mathsf{C}_{33}+\left(\mathsf{C}_{12}+\mathsf{C}_{33}\right) \mathcal{D}_{7}\right)+3 \mathsf{C}_{13} \mathcal{D}_{8} + \right.\\[5mm]
&&\left. + \left(3 \mathsf{C}_{12} \mathcal{D}_{3}+6 \mathsf{C}_{13}
\mathcal{D}_{7}+\mathsf{C}_{33} \left(3 \mathcal{D}_{3}+2 \sqrt{3} (\mathcal{D}_{7}+1)\right)\right) \mathcal{D}_{8} + \right.\\[5mm]
&&\left. + \mathsf{C}_{23} \left(3 \mathcal{D}_{7}+2 \mathcal{D}_{4} \left(3 \mathcal{D}_{3}+\sqrt{3} \left(\mathcal{D}_{7}+1\right)\right)+2 \sqrt{3} \mathcal{D}_{3} \mathcal{D}_{8}+3\right)\right),\\[5mm]
\mathsf{G}_{44}^{[7]}&=&9 \left(\mathsf{C}_{22}
\mathcal{D}_{4} \left(\sqrt{3} \mathcal{D}_{4}+3\right)+\mathcal{D}_{8} \left(3 \left(\mathsf{C}_{12}+\mathsf{C}_{33}\right) \mathcal{D}_{4}+3 \mathsf{C}_{13} \mathcal{D}_{8}+\sqrt{3} \mathsf{C}_{33} \mathcal{D}_{8}\right) + \right.\\[5mm]
&&\left. + \mathsf{C}_{23} \left(3 \mathcal{D}_{4}^2+2 \sqrt{3} \mathcal{D}_{8} \mathcal{D}_{4}+3\mathcal{D}_{8}\right)\right).
\end{array}$$
$$\begin{array}{ccc}
\mathsf{G}_{11}^{[8]}=\dfrac{18}{\ell} \left(\mathcal{D}_{1} \mathsf{M}^{*}_{21}+\mathcal{D}_{5} \mathsf{M}^{*}_{31}+\mathsf{M}^{*}_{11}\right),\quad
\mathsf{G}_{12}^{[8]}=\dfrac{9}{\ell} \left(\mathcal{D}_{1} \mathsf{M}^{*}_{22}+\mathcal{D}_{2} \mathsf{M}^{*}_{21}+\mathcal{D}_{5} \mathsf{M}^{*}_{32}+\mathcal{D}_{6} \mathsf{M}^{*}_{31}+\mathsf{M}^{*}_{12}\right),\\[5mm]
\mathsf{G}_{13}^{[8]}=\dfrac{9}{\ell} \left(\mathcal{D}_{1} \mathsf{M}^{*}_{23}+\mathcal{D}_{3} \mathsf{M}^{*}_{21}+\mathcal{D}_{5} \mathsf{M}^{*}_{33}+\mathcal{D}_{7} \mathsf{M}^{*}_{31}+\mathsf{M}^{*}_{13}+\mathsf{M}^{*}_{31}\right),\quad
\mathsf{G}_{14}^{[8]}=\dfrac{9}{\ell} \left(\mathcal{D}_{1} \mathsf{M}^{*}_{24}+\mathcal{D}_{4} \mathsf{M}^{*}_{21}+\mathcal{D}_{5} \mathsf{M}^{*}_{34}+\mathcal{D}_{8} \mathsf{M}^{*}_{31}+\mathsf{M}^{*}_{14}\right),\\[5mm]
\mathsf{G}_{22}^{[8]}=\dfrac{18}{\ell} \left(\mathcal{D}_{2} \mathsf{M}^{*}_{22}+\mathcal{D}_{6} \mathsf{M}^{*}_{32}\right),\quad
\mathsf{G}_{23}^{[8]}=\dfrac{9}{\ell} \left(\mathcal{D}_{2} \mathsf{M}^{*}_{23}+\mathcal{D}_{3} \mathsf{M}^{*}_{22}+\mathcal{D}_{6} \mathsf{M}^{*}_{33}+\mathcal{D}_{7} \mathsf{M}^{*}_{32}+\mathsf{M}^{*}_{32}\right),\\[5mm]
\mathsf{G}_{24}^{[8]}=\dfrac{9}{\ell} \left(\mathcal{D}_{2} \mathsf{M}^{*}_{24}+\mathcal{D}_{4} \mathsf{M}^{*}_{22}+\mathcal{D}_{6} \mathsf{M}^{*}_{34}+\mathcal{D}_{8} \mathsf{M}^{*}_{32}\right),\quad
\mathsf{G}_{33}^{[8]}=\dfrac{18}{\ell} \left(\mathcal{D}_{3} \mathsf{M}^{*}_{23}+\mathcal{D}_{7} \mathsf{M}^{*}_{33}+\mathsf{M}^{*}_{33}\right),\\[5mm]
\mathsf{G}_{34}^{[8]}=\dfrac{9}{\ell} \left(\mathcal{D}_{3} \mathsf{M}^{*}_{24}+\mathcal{D}_{4} \mathsf{M}^{*}_{23}+\mathcal{D}_{7} \mathsf{M}^{*}_{34}+\mathcal{D}_{8} \mathsf{M}^{*}_{33}+\mathsf{M}^{*}_{34}\right),\quad
\mathsf{G}_{44}^{[8]}=\dfrac{18}{\ell} \left(\mathcal{D}_{4} \mathsf{M}^{*}_{24}+\mathcal{D}_{8} \mathsf{M}^{*}_{34}\right)
\end{array}$$
$$\begin{array}{ccc}
\mathsf{G}_{11}^{[9]}=\dfrac{9}{\ell} \left(\mathcal{D}_{1} \left(\mathsf{M}^{*}_{21}+\sqrt{3} \mathsf{M}^{*}_{31}\right)+\sqrt{3} \mathcal{D}_{5} \mathsf{M}^{*}_{11}+\mathcal{D}_{5} \mathsf{M}^{*}_{31}+\mathsf{M}^{*}_{11}\right),\\[5mm]
\mathsf{G}_{12}^{[9]}=\dfrac{9}{2} \left(\mathcal{D}_{1} \mathsf{M}^{*}_{22}+\sqrt{3} \left(\mathcal{D}_{1} \mathsf{M}^{*}_{32}+\mathsf{M}^{*}_{31}\right)+\mathcal{D}_{2}
\left(\mathsf{M}^{*}_{21}+\sqrt{3} \mathsf{M}^{*}_{31}\right)+\sqrt{3} \mathcal{D}_{5} \mathsf{M}^{*}_{12}+\mathcal{D}_{5} \mathsf{M}^{*}_{32}+\mathcal{D}_{6} \left(\sqrt{3} \mathsf{M}^{*}_{11}+\mathsf{M}^{*}_{31}\right)+\mathsf{M}^{*}_{12}\right),\\[5mm]
\mathsf{G}_{13}^{[9]}=\dfrac{9}{2\ell} \left(\mathcal{D}_{1} \mathsf{M}^{*}_{23}+\sqrt{3} \mathcal{D}_{1}
\mathsf{M}^{*}_{33}+\mathcal{D}_{3} \mathsf{M}^{*}_{21}+\sqrt{3} \mathcal{D}_{3} \mathsf{M}^{*}_{31}+\sqrt{3} \mathcal{D}_{5} \mathsf{M}^{*}_{13}+\mathcal{D}_{5} \mathsf{M}^{*}_{33}+\mathcal{D}_{7} \left(\sqrt{3} \mathsf{M}^{*}_{11}+\mathsf{M}^{*}_{31}\right)+\mathsf{M}^{*}_{13}+\mathsf{M}^{*}_{31}\right),\\[5mm]
\mathsf{G}_{14}^{[9]}=\dfrac{9}{2\ell} \left(\sqrt{3} \left(\mathcal{D}_{1}
\mathsf{M}^{*}_{34}+\mathcal{D}_{4} \mathsf{M}^{*}_{31}\right)+\mathcal{D}_{1} \mathsf{M}^{*}_{24}+\left(\mathcal{D}_{4}+\sqrt{3}\right) \mathsf{M}^{*}_{21}+\sqrt{3} \mathcal{D}_{5} \mathsf{M}^{*}_{14}+\mathcal{D}_{5} \mathsf{M}^{*}_{34}+\mathcal{D}_{8} \left(\sqrt{3} \mathsf{M}^{*}_{11}+\mathsf{M}^{*}_{31}\right)+\mathsf{M}^{*}_{14}\right),\\[5mm]
\mathsf{G}_{22}^{[9]}=\dfrac{9}{\ell} \left(\mathcal{D}_{2}
\mathsf{M}^{*}_{22}+\sqrt{3} \left(\mathcal{D}_{2}+1\right) \mathsf{M}^{*}_{32}+\mathcal{D}_{6} \left(\sqrt{3} \mathsf{M}^{*}_{12}+\mathsf{M}^{*}_{32}\right)\right),\\[5mm]
\mathsf{G}_{23}^{[9]}=\dfrac{9}{2\ell} \left(\mathcal{D}_{2} \mathsf{M}^{*}_{23}+\sqrt{3} \mathcal{D}_{2} \mathsf{M}^{*}_{33}+\mathcal{D}_{3} \mathsf{M}^{*}_{22}+\sqrt{3} \mathcal{D}_{3} \mathsf{M}^{*}_{32}+\mathcal{D}_{6}
\left(\sqrt{3} \mathsf{M}^{*}_{13}+\mathsf{M}^{*}_{33}\right)+\mathcal{D}_{7} \left(\sqrt{3} \mathsf{M}^{*}_{12}+\mathsf{M}^{*}_{32}\right)+\mathsf{M}^{*}_{32}+\sqrt{3} \mathsf{M}^{*}_{33}\right),\\[5mm]
\mathsf{G}_{24}^{[9]}=\dfrac{9}{2\ell} \left(\sqrt{3} \left(\mathcal{D}_{2} \mathsf{M}^{*}_{34}+\mathcal{D}_{4} \mathsf{M}^{*}_{32}+\mathsf{M}^{*}_{34}\right)+\mathcal{D}_{2}
\mathsf{M}^{*}_{24}+\left(\mathcal{D}_{4}+\sqrt{3}\right) \mathsf{M}^{*}_{22}+\mathcal{D}_{6} \left(\sqrt{3} \mathsf{M}^{*}_{14}+\mathsf{M}^{*}_{34}\right)+\mathcal{D}_{8} \left(\sqrt{3} \mathsf{M}^{*}_{12}+\mathsf{M}^{*}_{32}\right)\right),\\[5mm]
\mathsf{G}_{33}^{[9]}=\dfrac{9}{\ell} \left(\mathcal{D}_{3} \left(\mathsf{M}^{*}_{23}+\sqrt{3} \mathsf{M}^{*}_{33}\right)+\mathcal{D}_{7}
\left(\sqrt{3} \mathsf{M}^{*}_{13}+\mathsf{M}^{*}_{33}\right)+\mathsf{M}^{*}_{33}\right),\\[5mm]
\mathsf{G}_{34}^{[9]}=\dfrac{9}{2\ell} \left(\mathcal{D}_{3} \mathsf{M}^{*}_{24}+\sqrt{3} \mathcal{D}_{3} \mathsf{M}^{*}_{34}+\left(\mathcal{D}_{4}+\sqrt{3}\right) \mathsf{M}^{*}_{23}+\sqrt{3} \mathcal{D}_{4} \mathsf{M}^{*}_{33}+\mathcal{D}_{7} \left(\sqrt{3}
\mathsf{M}^{*}_{14}+\mathsf{M}^{*}_{34}\right)+\mathcal{D}_{8} \left(\sqrt{3} \mathsf{M}^{*}_{13}+\mathsf{M}^{*}_{33}\right)+\mathsf{M}^{*}_{34}\right),\\[5mm]
\mathsf{G}_{44}^{[9]}=\dfrac{9}{\ell} \left(\left(\mathcal{D}_{4}+\sqrt{3}\right) \mathsf{M}^{*}_{24}+\sqrt{3} \mathcal{D}_{4} \mathsf{M}^{*}_{34}+\mathcal{D}_{8} \left(\sqrt{3}
\mathsf{M}^{*}_{14}+\mathsf{M}^{*}_{34}\right)\right)
\end{array}$$
$$\begin{array}{rcl}
\mathsf{G}_{11}^{[10]}&=&\dfrac{3 \sqrt{3}}{\ell^{2}}\mathsf{A}^{*}_{11} + \dfrac{5}{8} \sqrt{3} \left(\mathcal{D}_{5}^2 \left(\mathsf{C}_{11}+\mathsf{C}_{33}\right)+\mathsf{C}_{11}+2 \mathsf{C}_{12} \mathcal{D}_{1}+2 \mathcal{D}_{5} \left(\mathcal{D}_{1} \left(\mathsf{C}_{13}+\mathsf{C}_{23}\right)+\mathsf{C}_{13}\right)+\mathcal{D}_{1}^2 \left(\mathsf{C}_{22}+\mathsf{C}_{33}\right)\right),\\[5mm]
\mathsf{G}_{12}^{[10]}&=&\dfrac{3 \sqrt{3}}{\ell^{2}}\mathsf{A}^{*}_{12} + \dfrac{5}{8} \sqrt{3} \left(\mathcal{D}_{6}
\left(\mathsf{C}_{11} \mathcal{D}_{5}+\mathcal{D}_{1} \left(\mathsf{C}_{13}+\mathsf{C}_{23}\right)+\mathsf{C}_{13}\right)+\mathcal{D}_{2} \left(\mathsf{C}_{12}+\mathcal{D}_{5} \left(\mathsf{C}_{13}+\mathsf{C}_{23}\right)+\mathsf{C}_{22} \mathcal{D}_{1}\right)+\mathsf{C}_{13} \mathcal{D}_{5} + \right.\\[5mm]
&&\left. + \mathsf{C}_{33} \left(\mathcal{D}_{1} \mathcal{D}_{2}+\mathcal{D}_{1}+\mathcal{D}_{5} \mathcal{D}_{6}\right)\right),\\[5mm]
\mathsf{G}_{13}^{[10]}&=&\dfrac{3 \sqrt{3}}{\ell^{2}}\mathsf{A}^{*}_{13} + \dfrac{5}{8}\sqrt{3} \left(\mathcal{D}_{7} \left(\mathcal{D}_{5} \left(\mathsf{C}_{11}+\mathsf{C}_{33}\right)+\mathcal{D}_{1} \left(\mathsf{C}_{13}+\mathsf{C}_{23}\right)+\mathsf{C}_{13}\right)+\mathsf{C}_{12} \mathcal{D}_{3}+\mathsf{C}_{13} \mathcal{D}_{3} \mathcal{D}_{5}+\mathsf{C}_{13} + \right.\\[5mm]
&&\left. +\mathcal{D}_{1} \mathcal{D}_{3} \left(\mathsf{C}_{22}+\mathsf{C}_{33}\right)+ \mathsf{C}_{23} \mathcal{D}_{1}+\mathsf{C}_{23}
\mathcal{D}_{3} \mathcal{D}_{5}+\mathsf{C}_{33} \mathcal{D}_{5}\right),\\[5mm]
\mathsf{G}_{14}^{[10]}&=&\dfrac{3 \sqrt{3}}{\ell^{2}}\mathsf{A}^{*}_{14} + \dfrac{5}{8} \sqrt{3} \left(\mathcal{D}_{8} \left(\mathsf{C}_{11} \mathcal{D}_{5}+\mathsf{C}_{13} \mathcal{D}_{1}+\mathsf{C}_{13}+\mathsf{C}_{33} \mathcal{D}_{5}\right)+\mathsf{C}_{12} \left(\mathcal{D}_{4}+\mathcal{D}_{5}\right)+\mathsf{C}_{13} \mathcal{D}_{4} \mathcal{D}_{5}+\mathcal{D}_{1} \mathcal{D}_{4}
\left(\mathsf{C}_{22}+\mathsf{C}_{33}\right) + \right.\\[5mm]
&&\left. + \mathsf{C}_{23} \left(\mathcal{D}_{1} \mathcal{D}_{8}+\mathcal{D}_{1}+\mathcal{D}_{4} \mathcal{D}_{5}\right)\right),\\[5mm]
\mathsf{G}_{22}^{[10]}&=&\dfrac{3 \sqrt{3}}{\ell^{2}}\mathsf{A}^{*}_{22} + \dfrac{5}{8} \sqrt{3} \left(\mathcal{D}_{6} \left(\mathsf{C}_{11} \mathcal{D}_{6}+2 \left(\mathcal{D}_{2} \left(\mathsf{C}_{13}+\mathsf{C}_{23}\right)+\mathsf{C}_{13}\right)\right)+\mathsf{C}_{22} \mathcal{D}_{2}^2+\mathsf{C}_{33}
\left(\left(\mathcal{D}_{2}+1\right)^2+\mathcal{D}_{6}^2\right)\right),\\[5mm]
\mathsf{G}_{23}^{[10]}&=&\dfrac{3 \sqrt{3}}{\ell^{2}}\mathsf{A}^{*}_{23} + \dfrac{5}{8} \sqrt{3} \left(\mathsf{C}_{11} \mathcal{D}_{6} \mathcal{D}_{7}+\mathsf{C}_{13} \left(\mathcal{D}_{2} \mathcal{D}_{7}+\mathcal{D}_{3} \mathcal{D}_{6}+\mathcal{D}_{7}\right)+\mathsf{C}_{22} \mathcal{D}_{2} \mathcal{D}_{3}+\mathsf{C}_{23} \left(\mathcal{D}_{2}
\mathcal{D}_{7}+\mathcal{D}_{2}+\mathcal{D}_{3} \mathcal{D}_{6}\right) + \right.\\[5mm]
&&\left. + \mathsf{C}_{33} \left(\mathcal{D}_{2} \mathcal{D}_{3}+\mathcal{D}_{3}+\mathcal{D}_{6} \mathcal{D}_{7}+\mathcal{D}_{6}\right)\right),\\[5mm]
\mathsf{G}_{24}^{[10]}&=&\dfrac{3 \sqrt{3}}{\ell^{2}}\mathsf{A}^{*}_{24} + \dfrac{5}{8} \sqrt{3} \left(\mathcal{D}_{8} \left(\mathcal{D}_{6} \left(\mathsf{C}_{11}+\mathsf{C}_{33}\right)+\mathsf{C}_{13} \left(\mathcal{D}_{2}+1\right)\right)+\mathsf{C}_{12} \mathcal{D}_{6}+\mathcal{D}_{4}
\left(\mathsf{C}_{13} \mathcal{D}_{6}+\mathsf{C}_{22} \mathcal{D}_{2}+\mathsf{C}_{33} \mathcal{D}_{2}+\mathsf{C}_{33}\right) + \right.\\[5mm]
&&\left. + \mathsf{C}_{23} \left(\mathcal{D}_{2} \mathcal{D}_{8}+\mathcal{D}_{2}+\mathcal{D}_{4} \mathcal{D}_{6}+1\right)\right),\\[5mm]
\mathsf{G}_{33}^{[10]}&=&\dfrac{3 \sqrt{3}}{\ell^{2}}\mathsf{A}^{*}_{33} + \dfrac{5}{8} \sqrt{3} \left(\mathsf{C}_{11} \mathcal{D}_{7}^2+2 \mathcal{D}_{3} \mathcal{D}_{7}
\left(\mathsf{C}_{13}+\mathsf{C}_{23}\right)+\mathcal{D}_{3} \left(\mathsf{C}_{22} \mathcal{D}_{3}+2 \mathsf{C}_{23}\right)+\mathsf{C}_{33} \left(\mathcal{D}_{3}^2+\left(\mathcal{D}_{7}+1\right)^2\right)\right),\\[5mm]
\mathsf{G}_{34}^{[10]}&=&\dfrac{3 \sqrt{3}}{\ell^{2}}\mathsf{A}^{*}_{34} + \dfrac{5}{8} \sqrt{3} \left(\mathcal{D}_{8} \left(\mathcal{D}_{7} \left(\mathsf{C}_{11}+\mathsf{C}_{33}\right)+\mathsf{C}_{13} \mathcal{D}_{3}+\mathsf{C}_{33}\right)+\mathsf{C}_{12}
\mathcal{D}_{7}+\mathsf{C}_{13} \mathcal{D}_{4} \mathcal{D}_{7}+\mathcal{D}_{3} \mathcal{D}_{4} \left(\mathsf{C}_{22}+\mathsf{C}_{33}\right) + \right.\\[5mm]
&&\left. + \mathsf{C}_{23} \left(\mathcal{D}_{3} \mathcal{D}_{8}+\mathcal{D}_{3}+\mathcal{D}_{4} \mathcal{D}_{7}+\mathcal{D}_{4}\right)\right),\\[5mm]
\mathsf{G}_{44}^{[10]}&=&\dfrac{3 \sqrt{3}}{\ell^{2}}\mathsf{A}^{*}_{44} + \dfrac{5}{8} \sqrt{3} \left(\mathcal{D}_{8}^2 \left(\mathsf{C}_{11}+\mathsf{C}_{33}\right)+2 \mathcal{D}_{8}
\left(\mathsf{C}_{12}+\mathcal{D}_{4} \left(\mathsf{C}_{13}+\mathsf{C}_{23}\right)\right)+\mathsf{C}_{22} \mathcal{D}_{4}^2+\mathsf{C}_{22}+2 \mathsf{C}_{23} \mathcal{D}_{4}+\mathsf{C}_{33} \mathcal{D}_{4}^2\right).
\end{array}$$
|
---
author:
- 'Shari Moskow [^1]'
title: 'Nonlinear eigenvalue approximation for compact operators.'
---
Introduction
============
Frequently spectral perturbation problems arise in the study of electromagnetics and composite materials where the dependence on the spectral parameter is nonlinear. Here we state and prove some results which can be applied to both asymptotic and numerical approximations in those contexts. The general problem is as follows. Assume we have a set of compact linear operators $T_h(\lambda)$ where $\lambda$ is a spectral parameter and $h$ is some (w.l.o.g. small) perturbative parameter, and that we want to find a nontrivial pair $( u_h,\lambda_h)$ such that $$\label{nolineig} \lambda_h T_h(\lambda_h) u_h=u_h.$$ Such a pair is what we refer to as a nonlinear eigenpair. We assume we have also some limiting problem $$\label{nolineiglim} \lambda_0 T_0(\lambda_0) u_0= u_0$$ whose solutions are much simpler. The limiting problem could, for example, correspond to a background problem where solutions are known, or a lower dimensional or homogeneous problem whose solutions are far easier to compute. To understand the behavior of $\lambda_h$, it is useful to have an expansion $$\label{eigasymp}\lambda_h = \lambda_0 + h\lambda^{(1)} +o(h) ,$$ where the expression for $\lambda^{(1)}$ is as explicit as possible, and depends only on solutions to (\[nolineiglim\]). We refer to $\lambda^{(1)}$ as the eigenvalue correction.
Note that in this class of nonlinear spectral problem, we are looking for $\lambda$ such that $ S(\lambda ) =\lambda T(\lambda)- I$ has a nontrivial null space, where unlike standard eigenvalue problems, $S$ depends nonlinearly on $\lambda$. The nonlinearity is only in $\lambda$, that is, we assume here that $T(\lambda)$ is itself a linear operator. The study of nonlinear eigenvalue problems, quadratic eigenvalue problems and operator pencils is not new, and we refer to [@Markus],[@chanillo],[@Ne], as examples of a vast literature on the subject. However, this author is unaware of results on formulas for first order corrections to nonlinear eigenvalues for general Banach space operator perturbations.
Background: Linear theory
=========================
There is an established and expansive theory for linear eigenvalue perturbations of compact operators, in particular if they are self adjoint [@Ka]. To derive the correction for nonlinear eigenvalues, we begin with a [*linear*]{} eigenvalue correction theorem which is a restatement of Theorem 3 of [@Os]. This theorem does not require that the operators be self-adjoint, and within the proof provides an explicit formula for the correction (noticed in [@Vo]), which we state here. Suppose $X$ is a Banach space and $K_n: X\rightarrow X$ is a sequence of compact linear operators such that $K_n\rightarrow K$ pointwise (i.e. $\forall f\in X$, $K_n f\rightarrow Kf$ in norm). Assume also that the sequence $\{ K_n \}$ is collectively compact, meaning that the set $\{ K_n f\ | \ \| f\|\leq 1, n=1,2,\ldots \}$ has compact closure. We also suppose that $K_n^*\rightarrow K^*$ pointwise and $\{ K_n^*\} $ is also collectively compact. (Note that all of the above conditions are met when $K_n\rightarrow K$ in the operator norm.) Let $\mu$ be a nonzero eigenvalue of $K$ of algebraic multiplicity $m$. It is well known that for $n$ large enough, there exist $m$ eigenvalues of $K_n$, $\mu_1^n,\ldots \mu_m^n$ (counted according to algebraic multiplicity) such that $\mu_j^n\rightarrow\mu$ as $n\rightarrow\infty$, for each $1\leq j\leq m$. Let $E$ be the spectral projection onto the generalized eigenspace of $T$ corresponding to eigenvalue $\mu$. The space $X$ can be decomposed in terms of the range and null space of $E$: $X=R(E)\oplus N(E)$. Elements in $\phi^*\in R(E)^*$ therefore can be extended to act on all of $X$ by initial projection onto $R(E)$, that is, $\phi^* f=\phi^* Ef $.
\[osthm\] Let $\phi_1,\phi_2,\ldots \phi_m$ be a normalized basis for $R(E)$, and let $\phi_1^*,\ldots \phi_m^* $ denote the corresponding dual basis for $R(E^*)$. Then there exists a constant $C$ such that $$\left| \mu-{1\over{m}}\sum_{j=1}^m\mu_j^n -{1\over{m}}\sum_{j=1}^m \langle
(K-K_n)\phi_j,\phi^*_j\rangle \right| \leq C\|(K-K_n)|_{R(E)} \|\cdot \|
(K^*-K_n^*)|_{R(E^*)}\| .$$
Now suppose the operators depend on a continuous parameter $h$, and $K_h\rightarrow K$ in norm for example, and they differ by order $h$. Note that the right hand side is of higher order and the correction term for the average of the perturbed eigenvalues is merely $${1\over{m}}\sum_{j=1}^m \langle (K_h-K)\phi_j,\phi^*_j\rangle$$ so that if one has $K_h\approx K + h K^{(1)} $ for $K^{(1)}$ the operator correction, this yields the formula ${1\over{m}}\sum_{j=1}^m\mu_j^h \approx \mu +h\mu^{(1)}$ where $\mu^{(1)}= {1\over{m}}\sum_{j=1}^m \langle K^{(1)} \phi_j,\phi^*_j\rangle.$ Of course for a simple eigenpair $(\mu,\phi)$ this means that $$\mu_h = \mu + h \langle K^{(1)} \phi,\phi^*\rangle +O(h^2).$$
Convergence of nonlinear eigenvalues
====================================
One expects convergence of the nonlinear eigenvalues due to the analytic Fredholm theory. Here we include a statement of convergence and its proof for completeness. Define the modified resolvent type operator valued functions on $\mathbb{C}$ $$R_h(\lambda)=(I-\lambda T_h(\lambda))^{-1}$$ and $$R_0(\lambda)=(I-\lambda T_0(\lambda))^{-1}.$$ An important note is that if $R_h(\lambda)$ does not exist as a bounded linear operator from $X$ to itself, then $\lambda$ is a nonlinear eigenvalue of $T_h$. This is because if $R_h(\lambda)$ does not exist, then ${1/\lambda }$ is in the spectrum of the compact operator $T_h(\lambda)$. Hence since $1/\lambda$ is nonzero, it must be an eigenvalue and $(I-\lambda T_h(\lambda))$ must have nontrivial and finite dimensional null space. The same argument holds for the limiting operator $R_0(\lambda)$.
In the following proposition, we show that the nonlinear eigenvalues converge to those of the unperturbed operator.
\[convergeprop\] Assume that $\lambda_0$ is a nonlinear eigenvalue of $T_0$, and that $R_0$ and $R_h$ are meromorphic in some region $U$ of $\mathbb{C}$ containing $\lambda_0$. Assume also that for any $\lambda\in U$, $T_h(\lambda)\rightarrow T_0(\lambda)$ in norm. Then for any ball $B$ around $\lambda_0$, there exists $h_0 >0$ such that $T_h$ has a nonlinear eigenvalue in $B$ for all $h<h_0$. Conversely, if $\{ \lambda_h\}$ is a sequence of nonlinear eigenvalues of $T_h$ that converges as $h\rightarrow 0$, the limit is a nonlinear eigenvalue of $T_0$.
[*Proof*]{} Since $U$ is open and $R_0$ meromorphic, we can choose $B$, a ball around $\lambda_0$ such that $T_0$ has no other nonlinear eigenvalues in $\overline{B}$. We will use a well known result about the inverses of perturbed operators, see for example [@Ka] p. 31: If $S-T=A$ and $T^{-1}$ exists, then for $\| A\| < {1\over{\| T^{-1}\| }}$, $S^{-1}$ exists and $$\label{inversebound} \| S^{-1}-T^{-1}\| \leq {\| A\| \| T^{-1}\|^2 \over{1-\| A\| \| T^{-1}\| }}.$$ Apply this, with $$S=I-\lambda T_h(\lambda)$$ $$T=I-\lambda T_0(\lambda)$$ to get $$\label{resolventerror}\| R_h(\lambda)-R_0(\lambda)\| \leq {\lambda \| T_0(\lambda)-T_h(\lambda)\| \| R_0(\lambda)\|^2\over{1-\lambda\| T_0(\lambda)-T_h(\lambda)\| \| R_0(\lambda)\|}}.$$ Let $\Gamma={\partial}B$, positively oriented. By the choice of $B$, $\Gamma$ does not intersect with any poles of $R_0$, and $\lambda_0$ is the only pole of $R_0$ in the closed disk. Then $R_0(\lambda)$ is continuous with respect to $\lambda$ on $\Gamma$, hence $\| R_0(\lambda)\|$ is uniformly bounded for $\lambda$ on $\Gamma$. Using (\[resolventerror\]), we have that $$R_h(\lambda)\rightarrow R_0(\lambda)$$ in norm as $h\rightarrow 0$, uniformly for $\lambda\in \Gamma$. This implies that the operator valued integral $${1\over{2\pi i}}\int_\Gamma R_h(\lambda)d\lambda\rightarrow {1\over{2\pi i}}\int_\Gamma R_0(\lambda)d\lambda$$ in norm as $h\rightarrow 0$. More generally, the integrals $${1\over{2\pi i}}\int_\Gamma (\lambda-\lambda_0)^\alpha R_h(\lambda)d\lambda\rightarrow {1\over{2\pi i}}\int_\Gamma (\lambda-\lambda_0)^\alpha R_0(\lambda)d\lambda$$ also converge for $\alpha$ a positive integer. From the residue theorem, the integral $${1\over{2\pi i}}\int_\Gamma (\lambda-\lambda_0)^\alpha R_0(\lambda)d\lambda$$ gives us the coefficient of the $(\lambda-\lambda_0)^{-(\alpha+1)}$ term in the Laurent series expansion for $R_0(\lambda)$. Since $R_0$ has a pole at $\lambda_0$ and is meromorphic, this must be nonzero for some finite integer $\alpha\geq 0$. Hence for that $\alpha$, the integrals $${1\over{2\pi i}}\int_\Gamma (\lambda-\lambda_0)^\alpha R_h(\lambda)d\lambda$$ must all be nonzero for $h$ small enough. This means that all $R_h$ must have at least one pole in $B$ for $h$ small enough. That is, for $h$ small enough, all $T_h$ have a nonlinear eigenvalue in $B$. This proves the first part of the statement of the proposition. For the converse, if $\lambda_0$ is not a nonlinear eigenvalue of $T_0$, then $R_0(\lambda)$ exists in some neighborhood of $\lambda_0$. The formula (\[resolventerror\]) implies that $R_h(\lambda)$ also exists in that neighborhood for $h$ small enough. Hence the nonlinear eigenvalues of $T_h$ are bounded away from $\lambda_0$ for $h$ small enough.
Some remarks about the assumptions in this theorem:
- If the operator functions $I-\lambda T_h(\lambda)$ and $I-\lambda T_0(\lambda)$ are analytic in some region $U$, then this combined with the fact that the $T$’s are compact, means that the inverses are meromorphic.
- If $\lambda_0$ is a nonlinear eigenvalue of $T_0$, then the classical resolvent of $T_0(\lambda_0)$, given by $ (zI- T_0(\lambda_0))^{-1}$, automatically has nonzero residue at $z={1\over{\lambda_0}}$; its residue is the projection onto the generalized eigenspace [@Ka]. Using arguments as above, one can show that the coefficient operators in the Laurent series expansions for $R_h$ must converge to those of $R_0$. However, it is not clear how these coefficients relate to the nonlinear eigenspaces.
Nonlinear eigenvalue corrections
================================
Now assume we have a series of problems of the form (\[nolineig\]),(\[nolineiglim\]). For the case of resonances for the Helmholtz equation we were able to extend and modify the linear eigenvalue correction theorem of Osborn to apply such a situation [@GoMoSa]. Using the ideas there, we will state and prove a general nonlinear eigenvalue correction theorem that works for simple eigenvalues, or for higher multiplicity if the perturbed nonlinear eigenvalue has multiplicity just as high. We remark that the following theorem does not apply at all for multiple eigenvalues in the general case.
If $\lambda_0$ is a nonlinear eigenvalue of $T_0$, then ${1\over{\lambda_0}}$ is a standard eigenvalue of $T_0(\lambda_0)$, with algebraic multiplicity $m$ and $E$ the projection onto the corresponding generalized eigenspace. We will say in this case that $\lambda_0$ has multiplicity $m$. As in the linear case, let $\{ \phi_j \}_{j=1,\ldots m},\{ \phi^*_j \}_{j=1,\ldots m} $ be normalized bases of the generalized eigenspace $R(E)$ and its dual space $R(E)^*$ respectively. Again $X=R(E)\oplus N(E)$, and elements in $\phi^*\in R(E)^*$ can be extended to act on all of $X$ by initial projection onto $R(E)$, that is, $\phi^* f=\phi^* Ef $. Here $X$ is again a Banach space, and $\langle f,g^* \rangle=g^*(f)$ represents the duality pairing for $f\in X, g^*\in X^*$.
\[nonlinearosthm\] Let $\{ T_h(\lambda):X\rightarrow X \} $ be a set of compact linear operator valued functions of $\lambda$ which are analytic in a region $U$ of the complex plane, collectively compact for any $\lambda\in U$. Assume that $T_h(\lambda)\rightarrow T_0(\lambda)$ pointwise as $h\rightarrow 0$, uniformly for $\lambda\in U$. Assume also that $T^*_h(\lambda)\rightarrow T^*_0(\lambda)$ pointwise as $h\rightarrow 0$, and that $\{ T^*_h(\lambda)\} $ are collectively compact, uniformly for $\lambda\in U$. Let $\lambda_0\neq 0, \lambda_0\in U$ be a nonlinear eigenvalue (\[nolineiglim\]) of $T_0$ , of algebraic multiplicity $m$. Assume that there exists $h_0$ such that for $h< h_0$ there exists $\{ \lambda_h\} $ a nonlinear eigenvalue of $T_h$ of multiplicity m, such that $\lambda_h\rightarrow \lambda_0$. Assume $B\subset U$ is a ball around $\lambda_0$ containing all $\lambda_h$ for $h<h_0$. Let $DT_0(\lambda_0)$ be the derivative of $T_0$ with respect to $\lambda$ evaluated at $\lambda_0$. Then if $$\label{DTcond}{ \lambda_0^2\over{m}} \sum_{j=1}^m \langle DT_0(\lambda_0) \phi_j,\phi_j^*\rangle\neq -1 ,$$ we have the following formula $$\begin{gathered}
\label{nonlinearcor}
\lambda_h=\lambda_0 + {{\lambda_0^2\over{m}}\sum_{j=1}^m\langle (T_0(\lambda_0)-T_h(\lambda_0))\phi_j, \phi_j^*\rangle \over {1+{\lambda_0^2\over{m}}\sum_{j=1}^m\langle DT_0 (\lambda_0) \phi_j,\phi_j^* \rangle}} \\ + O\left( \sup_{\lambda\in B} \| (T_h(\lambda)- T_0(\lambda))|_{R(E)} \| \| (T^*_h(\lambda)- T^*_0(\lambda))|_{R(E)^*}\| \right).\end{gathered}$$
[*Proof*]{} Note that $$\lambda_h T_h(\lambda_h) u_h=u_h$$ and $$\lambda_0 T_0(\lambda_0) u_0=u_0,$$ that is, ${1\over{\lambda_h}}$ is an eigenvalue of $T_h(\lambda_h )$ and ${1\over{\lambda_0}}$ is an eigenvalue of $T_0(\lambda_0)$. Also, by assumption we know that $$T_h(\lambda_h)\rightarrow T_0(\lambda_0)$$ pointwise, and the sequence is collectively compact; likewise for the adjoints. So, what we have are the eigenvalues of a convergent sequence of compact operators, $\{ {1\over{\lambda_h}}, T_h(\lambda_h)\} $ converging to $\{ {1\over{\lambda_0}}, T_0(\lambda_0) \} $, so we now apply Theorem \[osthm\]. Since ${1\over{\lambda_h}}$ has multiplicity $m$, all of the eigenvalues of $T_h(\lambda_h)$ are equal to ${1\over{\lambda_h}}$ if $h$ is small enough. (We remark that if this is not the case, the desired nonlinear eigenvalue must get averaged with the other [*linear*]{} eigenvalues of $T_h(\lambda_h)$, which are not of interest to us here.) Hence this theorem yields $$\begin{gathered}
\left| {1\over{\lambda_0}}-{1\over{\lambda_h}}
- {1\over{m}}\sum_{j=1}^m \langle
(T_0(\lambda_0)-T_h(\lambda_h))\phi_j,\phi_j^*\rangle \right|
\\ \leq \| (T_h(\lambda_h)- T_0(\lambda_0))|_{R(E)} \| \| (T^*_h(\lambda_h)- T^*_0(\lambda_0))|_{R(E)^*} \| . \label{osborn2}\end{gathered}$$ Since $R(E)$ is finite dimensional, $$\| (T_h(\lambda)- T_0(\lambda))|_{R(E)} \| \leq c(h)$$ where $c(h)\rightarrow 0$ as $h\rightarrow 0$ and is independent of $\lambda\in U$. Similarly, we have $$\| (T^*_h(\lambda)- T^*_0(\lambda))|_{R(E)^*} \| \leq c^*(h)$$ where $c^*(h)\rightarrow 0$ as $h\rightarrow 0$. (Note that these rates need not necessarily be the same.) From the regularity of $T_0$ with respect to $\lambda$ and the assumptions of this theorem, there exists $C$ independent of $h$ and $\lambda\in U$ such that $$\| T_0(\lambda_0)-T_h(\lambda_h)|_{R(E)}\| \leq C( c(h) + |\lambda_h-\lambda_0| )$$ and likewise $$\|
T^*_0(\lambda_0)-T_h^*(\lambda_h)|_{R(E)^*} \|\leq C( c^*(h) + |\lambda_h-\lambda_0| ) .$$ Inserting this into (\[osborn2\]) we have $$\begin{gathered}
\left| {1\over{\lambda_0}}-{1\over{\lambda_h}}
- {1\over{m}}\sum_{j=1}^m \langle
(T_0(\lambda_0)-T_h(\lambda_h))\phi_j,\phi_j^*\rangle \right|
\\ \leq C( c(h) + |\lambda_h-\lambda_0| )\cdot( c^*(h) + |\lambda_h-\lambda_0| ) . \label{osborn3}\end{gathered}$$ If we multiply everything by $\lambda_0\lambda_h$, $$\left|\lambda_h-\lambda_0-{\lambda_0\lambda_h\over{m}} \sum_{j=1}^m\langle
(T_0(\lambda_0)-T_h(\lambda_h))\phi_j,\phi_j^*\rangle\right|\leq C( c(h) + |\lambda_h-\lambda_0| )\cdot( c^*(h) + |\lambda_h-\lambda_0| )\nonumber$$ which we manipulate to get $$\begin{gathered}
\nonumber \lambda_h=\lambda_0 +{\lambda_0^2\over{m}}\sum_{j=1}^m\langle
(T_0(\lambda_0)-T_h(\lambda_h))\phi_j,\phi_j^*\rangle
+{\lambda_0\over{m}}(\lambda_h-\lambda_0)\sum_{j=1}^m\langle
(T_0(\lambda_0)-T_h(\lambda_h))\phi_j,\phi_j^* \rangle \\ +O\left(
(c(h)+|\lambda_h-\lambda_0|)\cdot (c^*(h)+|\lambda_h-\lambda_0|)\right). \nonumber \end{gathered}$$ Now, we again see that the third term on the right hand side is bounded by the error term, and hence $$\begin{gathered}
\lambda_h=\lambda_0 +{\lambda_0^2\over{m}}\sum_{j=1}^m\langle
(T_0(\lambda_0)-T_h(\lambda_h))\phi_j,\phi_j^*\rangle
+O\left(
(c(h)+|\lambda_h-\lambda_0|)\cdot (c^*(h)+|\lambda_h-\lambda_0|)\right). \label{eexpand} \end{gathered}$$ Now, since the correction term above depends on $\lambda_h$, we need to expand the term further. We can write $$T_0(\lambda_0) -T_h(\lambda_h)= (T_0(\lambda_0)
-T_h(\lambda_0))+(T_h(\lambda_0) -T_h(\lambda_h)) \label{Tdivide2}$$ and compute using the regularity with respect to $\lambda$, $$\langle (T_h(\lambda_0) -T_h(\lambda_h))\phi_j,\phi_j^*\rangle =(\lambda_0-\lambda_h) \langle DT_h (\lambda_0)\phi_j,\phi_j^*\rangle +O(|\lambda_0-\lambda_h|^2) \nonumber$$ where $DT_h(\lambda_0)$ is the derivative with respect to $\lambda$ of $T_h$ evaluated at $\lambda_0$. Since the pointwise convergence of $T_h$ is uniform with respect to $\lambda$, we have that $$\langle (DT_h (\lambda_0)-DT_0(\lambda_0) )\phi_j,\phi_j^*\rangle \leq C c(h)$$ which yields $$\begin{gathered}
\langle (T_h(\lambda_0) -T_h(\lambda_h))\phi_j,\phi_j^*\rangle =(\lambda_0-\lambda_h) \langle DT_0 (\lambda_0)\phi_j,\phi_j^*\rangle \\ +O\left(
(c(h)+|\lambda_h-\lambda_0|)\cdot (c^*(h)+|\lambda_h-\lambda_0|)\right) \label{dest}\end{gathered}$$
Combining (\[eexpand\]), (\[Tdivide2\]) , and (\[dest\]), we obtain $$\begin{gathered}
\lambda_h=\lambda_0+{\lambda_0^2\over{m}}\sum_{j=1}^m \langle
(T_0(\lambda_0)-T_h(\lambda_0))\phi_j,\phi_j^*\rangle
-{\lambda_0^2\over{m}} (\lambda_h-\lambda_0)\sum_{j=1}^m\langle DT_0(\lambda_0)\phi_j,\phi_j^* \rangle \\ +
O\left((c(h)+|\lambda_h-\lambda_0|)\cdot (c^*(h)+|\lambda_h-\lambda_0|) \right) \nonumber .\end{gathered}$$ We now collect terms for $(\lambda_h-\lambda_0)$ so that $$\begin{gathered}
(\lambda_h-\lambda_0)\left(1+{\lambda_0^2\over{m}}\sum_{j=1}^m\langle DT_0(\lambda_0)\phi_j,\phi_j^* \rangle \right) ={\lambda_0^2\over{m}}\sum_{j=1}^m \langle
(T_0(\lambda_0)-T_h(\lambda_0))\phi_j,\phi_j^*\rangle \\ +
O\left((c(h)+|\lambda_h-\lambda_0|)\cdot (c^*(h)+|\lambda_h-\lambda_0|) \right). \end{gathered}$$ At this point we need to use the assumption (\[DTcond\]) to obtain $$\begin{gathered}
\label{finalest} \lambda_h=\lambda_0 + {{\lambda_0^2\over{m}}\sum_{j=1}^m\langle (T_0(\lambda_0)-T_h(\lambda_0))\phi_j, \phi_j^*\rangle \over {1+{\lambda_0^2\over{m}}\sum_{j=1}^m\langle DT_0 (\lambda_0) \phi_j,\phi_j^* \rangle}} \\ + O\left((c(h)+|\lambda_h-\lambda_0|)\cdot (c^*(h)+|\lambda_h-\lambda_0|) \right) \end{gathered}$$ By the looking at either the operators or their adjoints, we must have that $$\langle (T_0(\lambda_0)-T_h(\lambda_0))\phi_j,\phi_j^*\rangle \leq \min{\{ c(h),c^*(h)\} } ,$$ which implies that $$\lambda_h-\lambda_0=O(\min{\{ c(h),c^*(h)\} } ) +O\left((c(h)+|\lambda_h-\lambda_0|)\cdot (c^*(h)+|\lambda_h-\lambda_0|) \right) .$$ Since we assume that $\lambda_h-\lambda_0
\rightarrow 0$, this can only hold if $$\lambda_h-\lambda_0=O(\min{\{ c(h),c^*(h)\} }).$$ Inserting this into (\[finalest\]) completes the proof.
The following simplified version of the above theorem is applicable in many situations.
\[nonlinearcor\] Let $\{ T_h(\lambda):X\rightarrow X \} $ be a set of compact linear operator valued functions of $\lambda$ which are analytic in a region $U$ of the complex plane, such that $T_h(\lambda)\rightarrow T_0(\lambda)$ in norm as $h\rightarrow 0$ uniformly for $\lambda\in U$. Let $\lambda_0\neq 0, \lambda_0\in U $ be a simple nonlinear eigenvalue (\[nolineiglim\]) of $T_0$, define $DT_0(\lambda_0)$ to be the derivative of $T_0$ with respect to $\lambda$ evaluated at $\lambda_0$, and let $\phi$ be the normalized eigenfunction and $\phi^*$ its dual. Then for any $h$ small enough there exists $\lambda_h$ a simple nonlinear eigenvalue of $T_h$ , such that if $$\label{DTcond}{ \lambda_0^2} \langle DT_0(\lambda_0) \phi,\phi^*\rangle\neq -1 ,$$ we have the following formula $$\begin{gathered}
\label{nonlinearcorform}
\lambda_h=\lambda_0 + {{\lambda_0^2}\langle (T_0(\lambda_0)-T_h(\lambda_0))\phi, \phi^*\rangle \over {1+{\lambda_0^2}\langle DT_0 (\lambda_0) \phi,\phi^* \rangle}} \\ + O\left(\sup_{\lambda\in U} \| (T_h(\lambda)- T_0(\lambda))|_{R(E)} \| \| (T^*_h(\lambda)- T^*_0(\lambda))|_{R(E)^*}\| \right).\end{gathered}$$
Examples
========
Generalized eigenvalue problems {#gensub}
-------------------------------
If one has a sequence of generalized eigenvalue problems of the form $$Au=\lambda B u,$$ then if either $A$ or $B$ is invertible and the other compact, this case reduces to the linear case and one can use Theorem \[osthm\], assuming all conditions are met. However, if instead one has the form $$(A +K)u=\lambda Bu$$ where $A$ is invertible and $K$,$B$ are compact, this does not reduce to a standard eigenvalue problem unless one knows that $(A+K)$ is also invertible. However, one can multiply by $A^{-1}$ to have $$u=( -A^{-1}K+\lambda A^{-1} B)u ,$$ or $$u= \lambda ( -{1\over{\lambda}} A^{-1}K+ A^{-1} B) u.$$ Hence for perturbations of this problem, one can potentially apply Theorem \[nonlinearosthm\] with $$T(\lambda) = -{1\over{\lambda}} A^{-1}K+ A^{-1} B .$$
Transmission eigenvalues
------------------------
In classical scattering problems, the far field scattering operator is a measure of the difference between the free space solution and solution of the equation modeling the presence of a (bounded here) scatterer. Wave numbers for which the scattering operator has a nontrivial kernel yield solutions to what is called the interior transmission eigenvalue problem for a given scatterer. Since these transmission eigenvalues are so closely related to non-scattering incident waves [@CakHad2], [@sp], it is clear they play an important role in inversion [@ccm07], [@haddex]. There has been quite a bit of progress made in these problems, including existence proofs [@cakginhad] [@paisyl] [@coltpaisyl]. See also the survey paper [@CakHad2].
Let $D\subset {\mathbb R}^d$, $d\geq 2$ be a bounded connected region with smooth boundary $\partial D$ and let $\nu$ denote the unit normal vector oriented outward to $D$. We consider a real valued function $n(x)$ defined in $D$, such that and $n(x)\geq n_0>0$. The transmission eigenvalue problem associated with $D$, $n$ are the values of $k$ for which the interior transmission problem $$\begin{aligned}
&\Delta v+k^2v=0&\qquad \mbox{in}\; {D}\label{iit1}\\
&\Delta w+k^2n(x)w=0&\qquad \mbox{in}\; {D} \label{ids2}\\
&w=v \qquad
\displaystyle{\frac{\partial w}{\partial \nu}=\frac{\partial v}{\partial \nu}} &\qquad \mbox{on} \;\partial D\label{iit4}\end{aligned}$$ has a nontrivial solution pair $(v, w)$. It was shown in [@cakginhad] under a fixed sign assumption on $n-1$ one can write the equivalent eigenvalue problem for $u=v-w\in H^2_0(D)$: $$\label{gr}
\left(\Delta+k^2n\right)\frac{1}{n-1}\left(\Delta +k^2\right)u=0.$$ Note that this eigenvalue problem is not self adjoint and it is no longer linear, it is quadratic in $\lambda=k^2$. So its analysis is not covered by standard elliptic eigenvalue theory. This can be multiplied out and written in the operator form [@cakginhad] $${{\mathbb A}}u+\, \lambda {{\mathbb B}}u+ \lambda^2 {{\mathbb C}}u=0$$ where ${{\mathbb A}}, {{\mathbb B}}, {{\mathbb C}}:\!H^2_0(D)\to H^2_0(D)$ are defined using Riesz representation $$\left({{\mathbb A}}u,v\right)_{H_0^2(D)}=\int_D\frac{1}{n -1}\Delta u \, \Delta\overline{v} \, \textrm{d}x,$$ $$\left({{\mathbb B}}u,v\right)_{H_0^2(D)}=\int_D\frac{1}{n -1}\left(\Delta u\,\overline v + n u\, \Delta\overline v \right)\,dx$$ $$\left({{\mathbb C}}u, v\right)_{H_0^2(D)} = \int_D\frac{n}{n-1} u \, \overline{v} \, dx.$$ Here the $H^2_0$ inner product is the $L^2$ inner product of the Laplacians. Note that ${{\mathbb A}}$ is invertible and ${{\mathbb B}}, {{\mathbb C}}$ are compact. Imagine we are interested in how these transmission eigenvalues are perturbed by material perturbations (i.e. perturbations in $n$ or $D$), or by numerical approximations (i.e. discretizations of ${{\mathbb A}}, {{\mathbb B}}, {{\mathbb C}}$). Since quadratic eigenvalue problems can be converted to 2$\times$2 linear problems, Osborn’s Theorem \[osthm\] may be applied directly, and this approach was indeed used in [@CaMo]. However, depending on the type of perturbation, the use of a system may be quite inconvenient, or may not work at all (e.g. in [@CaMo] the formula does not work for complex eigenvalues). The ability to use a nonlinear formulation leaves us a number of options. The simplest may be to invert ${{\mathbb A}}$, and use $$T(\lambda)= -{{\mathbb A}}^{-1} {{\mathbb B}}-\lambda {{\mathbb A}}^{-1}{{\mathbb C}}.$$ If we have a simple nonlinear eigenvalue $\lambda_0$ and some perturbation indexed by $h$, then Corollary \[nonlinearcor\] (assuming all hypotheses are met) yields $$\begin{gathered}
\label{nonlinearcorform}
\lambda_h=\lambda_0 + {{\lambda_0^2}\langle ({{\mathbb A}}_h^{-1} {{\mathbb B}}_h +\lambda_0 {{\mathbb A}}_h^{-1}{{\mathbb C}}_h - ( {{\mathbb A}}^{-1} {{\mathbb B}}+\lambda_0 {{\mathbb A}}^{-1}{{\mathbb C}}))\phi, \phi^*\rangle \over {1+{\lambda_0^2}\langle {{\mathbb A}}^{-1}{{\mathbb C}}\phi,\phi^* \rangle}} \\ + O( \| (T_h(\lambda)- T_0(\lambda))|_{R(E)} \| \| (T^*_h(\lambda)- T^*_0(\lambda))|_{R(E)^*}\| . )\end{gathered}$$ where one expects the square of the norms to be asymptotically smaller than the correction term. Furthermore, when $n-1$ is sign changing, one will need to use the formulation in [@sy], a generalized eigenvalue problem which for some situations can be converted to an compact nonlinear problem in the manner presented in subsection \[gensub\].
Resonances
----------
Assume we are interested in the propagation of scalar waves in free space, $x\in\mathbb{R}^d$, with the presence of a scattering obstacle. The fields may obey the Helmholtz equation: $$\Delta u(x) + k^2 (1 +\eta(x) ) u=0$$ where $u=u_{\bf i} + u_{\bf s}$ is divided into a given incident wave and scatterered field, and where the scattered field $u_{\bf s}$ satisfies the appropriate Sommerfeld radiation conditions at infinity. Here the scatterer is modeled by $\eta(x)$, which has compact support contained in the compact set $D$. Standard integration by parts yields the equivalent Lippmann-Schwinger formulation for the total field $u$, $$u(x)=u_{\bf i}(x) + k^2 \int_{D} {\eta}(y) G(x,y) u({y}) d{y}$$ where, in dimension three for example, $$G(x,y) = {1\over{4\pi}} {e^{ik|x-y|}\over{|x-y|}}$$ is the Helmholtz fundamental solution. Unlike waves in a bounded domain, operators in free space with an obstacle scatterer do not have real eigenvalues and corresponding standing waves/modes. However, if one allows $\lambda= k^2$ to extend into the complex plane, you can have poles, or values of $\lambda$ where the above equation has nontrivial solutions for no incident wave. Although such solutions are nonphysical, if their imaginary part is small they are observed as resonances. In the time domain these solutions correspond to very slowly decaying modes. That is, the resonances are values of $\lambda$ for which $$u(x)= \lambda \int_{D} {\eta}(y) {1\over{4\pi}} {e^{i\sqrt{\lambda}|x-y|}\over{|x-y|}} u({y}) d{y}$$ has nontrivial solutions $u$. This is a nonlinear eigenvalue problem where the operator $$( T(\lambda)u)(x) = \int_{D} {\eta}(y) {1\over{4\pi}} {e^{i\sqrt{\lambda}|x-y|}\over{|x-y|}} u({y}) d{y}$$ is analytic with respect to $\lambda$ away from the negative real axis. Perturbations in $\eta$ which correspond to material defects, or numerical discretizations of such an operator, can be handled by the theory presented here for the case of simple eigenvalues. Since the eigenvalue problem is not polynomial, converting to a linear system would require the use of an infinite system and would potentially be far more complicated.
Acknowledgements {#acknowledgements .unnumbered}
================
The author was supported by the National Science Foundation under grant DMS-1108858.
[^1]: Department of Mathematics, Drexel University, Philadelphia, PA 19104, USA, ([email protected]).
|
---
title: 'Constraining the Doublet Left-Right Model'
---
One of the most puzzling features of the Standard Model (SM) consists in the different treatment of left and right chiralities of fermions, as shown by the violation of parity by weak interactions. In order to restore this symmetry at high energies, Left-Right Models (LRM) have been introduced in the 70’s [@History] and they assume a symmetry between left- and right-handed fermions broken spontaneously, implying that left- and right-handed currents behave differently at low energies.
Historically, LRM have been considered with doublets in order to break the left-right symmetry spontaneously [@History]. Later the focus was on triplet models, due to their ability to generate both Dirac and Majorana masses for neutrinos. The triplet models have the advantage of potentially introducing a see-saw mechanism, though it is difficult to reconcile the very light masses of neutrinos with a TeV scale LRM without fine tuning [@Deshpande]. Moreover, combined constraints coming from meson oscillations, among other observables, tend to push the mass scale of the new scalar particles to a few TeV [@Buras] or beyond [@Mohapatra]. The new vector particles must not be far away from this scale, otherwise the couplings present in the Higgs potential would become non-perturbative. Much effort has been done to avoid these constraints, but stringent lower bounds persist [@Basecq].
Our aim is to reconsider the breaking of the left-right gauge group, via doublet rather than triplet fields, and see the constraints set on the scale and pattern of symmetry breaking. We also want to determine whether experimental data can be accommodated only through this spontaneous breakdown or if it requires also an explicit breaking of parity through different couplings in the left and right sectors.
Doublet LRM
===========
The gauge group of LRM is $ SU(3)_{c} \times SU(2)_{L} \times SU(2)_{R} \times U(1)_{B-L} $, where $ B $ is the baryon number and $ L $ is the lepton number. We call $ g_{B-L} $ the gauge coupling of $ U(1)_{B-L} $, and $ g_{L} $ and $ g_{R} $ the ones of $ SU(2)_{L} $ and $ SU(2)_{R} $, respectively – the case $ g_{L} \neq g_{R} $ explicitly violates parity. The LR symmetry is spontaneously broken into the EW symmetry, $ SU(3)_{c} \times SU(2)_{L} \times U(1)_{Y} $, where $ Y $ is the hypercharge, given by $ Y = T^{3}_{R} + \frac{B-L}{2} $. This breaking can be triggered by a scalar in any representation whose Vacuum Expectation Value (VEV) does not preserve the LR symmetry, but preserves the EW symmetry. One considers here a doublet representation $ \chi_{R} = \begin{pmatrix}
\chi^{\pm}_{R} , \chi^{0}_{R}
\end{pmatrix} $, with quantum numbers under the gauge group $ (0,0,1/2,1) $, whose VEV is $ \langle \chi_{R} \rangle = \begin{pmatrix}
0 , \frac{1}{\sqrt{2}} \kappa_{R}
\end{pmatrix} $. Since LRM is assumed to be valid at energies much higher than the scale $ \kappa $ of the EW Symmetry Breaking (EWSB), it follows that $ \kappa_{R} \gg \kappa $. The Higgs mechanism then leads to new heavy gauge bosons, called $ {W'}^{\pm} $ and $ {Z'}^{0} $, whose masses are of order $ \kappa_{R} $, coupled predominantly to right-handed fermions.
To discuss the physics occurring at the EWSB, one introduces a bidoublet $ \phi = \begin{pmatrix}
\varphi^{0}_{1} & \varphi^{+}_{2} \\
\varphi^{-}_{1} & \varphi^{0}_{2} \\
\end{pmatrix} $, $ (0,1/2,1/2,0) $, whose VEV, $ \langle \phi \rangle = \frac{\operatorname{diag} \left( \kappa_{1}, \kappa_{2} \right)}{\sqrt{2}} $, is not invariant under the SM gauge group and breaks it spontaneously into $ SU(3)_{c} \times U(1)_{\operatorname{EM}} $. Even though a second doublet $ \chi_{L} $, with quantum numbers $ (0,1/2,0,1) $, is not necessary from the point-of-view of EWSB, it is introduced in order to preserve the structural symmetry between left and right sectors before symmetry breaking. Its VEV is $ \langle \chi_{L} \rangle = \begin{pmatrix}
0 , \frac{1}{\sqrt{2}} \kappa_{L}
\end{pmatrix} $, with $ \kappa_{L} $ of the order of the EWSB scale at most, and thus it also triggers the EWSB. Since $ \chi_{L} $ is a doublet, its VEV does not contribute to the $ \rho $ parameter at tree-level, and must be constrained by other observables, in particular EW Precision Observables (EWPO). The scale of EWSB is set by $ \kappa^{2} \equiv \kappa^{2}_{1} + \kappa^{2}_{2} + \kappa^{2}_{L} $ and the SM Higgs is given by a combination of the real degrees of freedom of $ \phi $ and $ \chi_{L} $. The gauge bosons $ {W}^{\pm} $ and $ {Z}^{0} $ acquire masses of order $ \kappa $ by the Higgs mechanism, and they couple predominantly to left-handed fermions. For simplicity, we take $ \kappa_{1,2} $ and $ \kappa_{L,R} $ to be real and positive. We also consider that there is no complex phase in the Higgs potential, so that no new CP-violation terms are generated by the extended Higgs sector.
In the LRM, right-handed (left-handed) fermions come into doublets (singlets) of $ SU(2)_{R} $ and singlets (doublets) of $ SU(2)_{L} $, denoted $ Q_{R} $ ($ Q_{L} $). The mechanism responsible for giving them a mass is the Yukawa coupling $ \overline{Q}_{L} \left( Y \phi + \tilde{Y} \sigma_{2} \phi^{*} \sigma_{2} \right) Q_{R} + h.c. $, where generation indices are not shown. As in the SM, one introduces the mixing matrices $ V^{L,R} $, where $ V^{L} $ is the equivalent of the SM-CKM matrix and $ V^{R} $ is a new mixing matrix for right-handed quarks. Discrete symmetries can be imposed to relate L and R sectors, implying relations between $ V^{L} $ and $ V^{R} $ [@Maiezza]. The more general case where $ V^{L,R} $ are independent corresponds to an explicit violation of parity.
The spectrum of physical scalars of the Doublet LRM is composed of one light neutral Higgs $ h^{0} $, five heavy neutral Higgses $ H^{0}_{1,2,3} $ (CP-even) and $ A^{0}_{1,2} $ (CP-odd), and four heavy charged Higgses $ H^{\pm}_{1,2} $. All of the heavy Higgses have masses of the order of $ \kappa_{R} $. The neutral Higgses $ H^{0}_{1,2} $ and $ A^{0}_{1,2} $ couple to quarks with a strength proportional to the Yukawa couplings and VEV’s and induce Flavor Changing Neutral Currents (FCNC), whereas charged Higgses induce left- and right-handed Flavor Changing Charged Currents (FCCC).
Some differences/advantages of the doublet model compared to the triplet case are the following: (1) A VEV, whose size (of the order of the EWSB or less) is less constrained than in the triplet model, modifies the structure of FCNC couplings between Higgses and quarks; (2) This impacts the analysis of neutral-meson mixing (mainly corrected by neutral Higgs scalars in this class of models); (3) There are no doubly-charged Higgses in the theory; (4) There is no particular mechanism of mass generation for neutrinos, leaving the smallness of their masses unexplained.
The observables constraining the model will be the following: (a) EWPO (constraining VEVs and gauge couplings); (b) meson mixing and (semi-)leptonic decays (for $ V^{L} $, $ V^{R} $, Higgs masses, etc.); (c) and finally other observables as $ b \rightarrow s \gamma $ and the relation among the masses of up- and down-type quarks. The firs two categories will be discussed in the following sections.
EWPO
====
Among the observables one can use to constrain models of New Physics (NP), EWPO are of particular importance due to the accuracy reached by SM computations and experiments [@PDG]. The SM global fit of these observables shows good agreement between them, but some tension is present specially between the Forward-Backward asymmetry $ A_{FB} (b) $ and the Left-Right asymmetry $ A_{LR} (e) $. The observables we consider here are (a) Z-lineshape and asymmetries: $ A_{LR} (f) $, for $ f = e, \mu, \tau, c, b $, $ A_{FB} (f) $, for $ f = e, \mu, \tau, c, b $, the hadronic cross section at the Z-pole ($ \sigma^{0}_{\operatorname{had}} $), ratios of partial widths ($ R_{\ell} $, for $ \ell = e, \mu, \tau $, and $ R_{q} $, for $ q = c, b $) and the total width ($ \Gamma_{Z} $); (b) Mass ($ M_{W} $) and total width ($ \Gamma_{W} $) of the W; (c) Atomic parity violation of cesium and thallium.
In the SM, one usually parameterizes EWPO in terms of $ \mathcal{S} \equiv \{ m_{h}, m_{t}, \alpha_{s} (M_{Z}), \Delta \alpha, M_{Z} \} $:
$$\mathcal{X} = c_{0} + c_{1} \cdot L_{H} + c_{2} \cdot \Delta_{t} + c_{3} \cdot \Delta_{\alpha_{s}} + c_{4} \cdot \Delta^{2}_{\alpha_{s}} + c_{5} \cdot \Delta_{\alpha_{s}} \Delta_{t} + c_{6} \cdot \Delta_{\alpha} + c_{7} \cdot \Delta_{Z} ,$$
where $ \Delta_{t} = \left( \frac{m_{t}}{173.2 \, \operatorname{GeV}} \right)^{2} - 1 $, $ \Delta_{\alpha_{s}} = \frac{\alpha_{s} (M_{Z})}{0.1184} - 1 $, $ \Delta_{\alpha} = \frac{\Delta \alpha}{0.059} - 1 $, $ \Delta_{Z} = \frac{M_{Z}}{91.1876 \, \operatorname{GeV}} - 1 $ and $ {L_{H} = \log \frac{m_{h}}{125.7 \, \operatorname{GeV}}} $. The values of the coefficients $ c_{i} $ for some observables ($ \Gamma_{Z} $, $ \sigma^{0}_{\operatorname{had}} $, $ R_{b, c} $), including 2-loop fermionic EW corrections, are given by [@Freitas]. Using the numerical program [@Zfitter], one can determine the values of the coefficients $ c_{i} $ of the other observables with a level of accuracy somewhat lower. The fundamental parameters $ \mathcal{R} \equiv \{ \epsilon^2 \equiv \frac{\kappa^{2}}{\kappa^{2}_{R}}, c^{2}_R \equiv 1 - \frac{s^{2}_{W}}{1 - s^{2}_{W}} \left( \frac{g_{L}}{g_{R}} \right)^{2}, r \equiv \frac{\kappa_{2}}{\kappa_{1}}, w \equiv \frac{\kappa_{L}}{\kappa_{1}} \} $ are also present in the LRM, where $ s_{W} $ is the sine of the weak angle. The EWPO are given by $ \mathcal{X}_{LR} = \mathcal{X} + \delta \mathcal{X} $, where $ \delta \mathcal{X} $ is the Leading-Order (LO) correction to the SM and $ \frac{\delta \mathcal{X}}{\mathcal{X}} = \mathcal{O} (\epsilon^{2}) $. A similar treatment of EWPO can be found in [@Yuan].
 \[fig:fig1\]
$ w $ $ \epsilon^{2} $ $ c_{R} $ $ g_{R} $ $ g_{B-L} $ $ M_{Z'} $\[TeV\] $ \chi^{2}_{min} $
------- ------------------ ----------- ----------- ------------- ------------------- --------------------
0 0.88 0.11 0.36 3.57 13.1 26.12
1 1.04 0.40 0.39 0.90 3.8 25.14
2 1.43 0.63 0.46 0.56 2.4 24.06
\[tab:changingw\]
In order to combine the different EWPO and constrain the parameters $ \mathcal{S} \cup \mathcal{R} $, we use the [@CKMfitter] frequentist framework (with *Range fit* treatment of systematic uncertainties). The correlated constraint among the scale of LRM, $ \epsilon $, and the size of $ c_R $ is seen in Figure \[fig:fig1\]. We do not use bounds on masses coming from direct searches for the $ W' $ boson, as the latter are tied to specific assumptions on the structure of the LRM couplings [@directsearches] and the analysis should be adapted to the more general framework considered here.
The constraints are not powerful enough to constrain $ c_{R} $, $ r $ and $ w $ independently at $ 1 \sigma $. The global fit of LRM is similar to the SM one: $ \chi^{2}_{\min} \vert_{SM} = 22.24 $ and $ \chi^{2}_{\min} \vert_{LRM} = 22.19 $. The agreement is improved for some observables (e.g. $ \sigma^{0}_{\operatorname{had}} $) at the expense of others (e.g. $ \Gamma_{Z} $), [@Luiz]. Though not constrained at $ 1 \sigma $, $ w $ has an impact on the fit, as seen in Table \[tab:changingw\]. The fit prefers $ w > 0 $, though $ \chi^{2}_{\min} $ does not change by large amounts. Moreover, when $ w = 0 $, $ g_{B-L} $ reaches its perturbativity limit, $ g^{2}_{B-L} = 4 \pi $. The fact that $ w $ is pushed towards non-vanishing values is an interesting feature of EWPO, but it remains to be seen if the other sectors of the theory agree with this tendency.
Neutral-meson mixing
====================
In order to further test the LRM, in particular the scale of the masses of the Higgses and the general structure of the $ V^R $ mixing matrix, we consider meson oscillation observables. The SM calculation of these observables consists of $ {W}^{\pm} {W}^{\pm} $, $ {W}^{\pm} {G}^{\pm} $ and $ {G}^{\pm} {G}^{\pm} $ box diagrams ($ {G}^{\pm} $ is the Goldstone associated to $ {W}^{\pm} $), and it is corrected at order $ \epsilon^2 $ by (a) new boxes $ {W}^{\pm} {W'}^{\pm} $ and $ {G}^{\pm} {W'}^{\pm} $; (b) $ W $ gauge boson/charged scalar boxes, $ {W}^{\pm} {H}^{\pm}_{1,2} $ and $ {G}^{\pm} {H}^{\pm}_{1,2} $; (c) FCNC introduced by $ H^{0}_{1,2} $ and $ A^{0}_{1,2} $ at tree-level; and (d) self-energy and vertex corrections to the FCNC, necessary for gauge invariance of the $ {W}^{\pm} {W'}^{\pm} $ box [@Pal]. Usually, the tree-level Higgs exchanges dominate over the other new contributions. In the triplet case (which is similar to the limit $ \kappa_{L} = 0 $), only the pair $ H^{0}_{1} $, $ A^{0}_{1} $ contributes. In the doublet case, the presence of contributions from other Higgses, $ H^{0}_{2} $ and $ A^{0}_{2} $, and different FCNC couplings when $ \kappa_{L} \neq 0 $, as suggested by the analysis of EWPO, means that the constraint from neutral meson mixing is less stringent, in particular on the mass of FCNC Higgses.
The general structure of the neutral meson mixing observables is $ \Delta m = \sum_{i} C^{q_{1} q_{2}}_{i} \eta^{q_{1} q_{2}}_{i} \langle O_{i} \rangle $, where $ i $ runs over the number of operators, and $ q_{1}, q_{2} $ are the flavors of the up-type quarks in the box, also related to the FCCC mixing matrices arising in $ C_{i} $. The Wilson coefficients $ C_{i} $ can be computed perturbatively by matching the low energy EFT (effective Hamiltonian) and the underlying theory (SM here), whereas the matrix element $ \langle O_{i} \rangle $ can be determined from lattice QCD; $ \eta $, collecting the short-distance QCD corrections are precisely known (up to NNLO) in the SM through the use of EFT. There is also a simplified method for computing the $ \eta $’s, described by Vysotskii [@Vysotskii]. Consider the LO diagrams (for instance, the $ W W $ box) with the addition of a gluon (corresponding to a two-loop integral on a gluon momentum and a quark momentum). Vysotskii’s simplified method aims at extracting the main contributions to the short-distance coefficients by determining, at a first stage, the range of momenta of the gluon contributing to the leading-order QCD corrections. One then improves the result with the help of Renormalization Group Equations (RGE), resumming the gluon corrections thanks to the running of four-fermion operators. Finally, the quark momentum of the 2-loop integral is determined from the range of energies dominating the Inami-Lim functions (LO computation). The values of the short-distance QCD corrections in the SM for the $ K $ system using this method are given in Table \[tab:SMetas\], and they reproduce the values calculated from a systematic use of EFT [@Burasetas].
For the LR operators, only calculations of the $ \eta $ for top-top box below $ \mathcal{O} (m_{t}) $ are known [@Burasgammas]. To derive constraints from meson mixing, we compute the remaining $ \eta $’s applying the procedure described by Vysotskii, extending what was done by [@Ecker]. We give our preliminary results in Table \[tab:LRetas\]. The same approach was also employed by [@Bertolini]. More details of our calculations at LO, a possible NLO extension and the corresponding constraints on the parameters of the Doublet LRM from meson mixing will be given in ref. [@Luiz].
$ K \overline{K} $, LO $ \eta_{tt} $ $ \eta_{cc} $ $ \eta_{ct} $
----------------------------- --------------- --------------- ---------------
Vysotskii [@Vysotskii] 0.60 0.92 0.34
systematic EFT [@Burasetas] 0.612 1.12 0.35
: SM short-distance QCD corrections at LO showing a comparison between Vysotskii’s prescription and a systematic use of Effective Field Theory (EFT). Flavor thresholds are taken into account.[]{data-label="tab:SMetas"}
LO $ \overline{\eta}^{K \overline{K}}_{tt} $ $ \overline{\eta}^{K \overline{K}}_{cc} $ $ \overline{\eta}^{K \overline{K}}_{ct} $ $ \overline{\eta}^{B \overline{B}}_{tt} $
--------------------------------------------- ------------------------------------------- ------------------------------------------- ------------------------------------------- -------------------------------------------
$ W^{\pm} {W'}^{\pm} $, $ W^{\pm} H^{\pm} $ 2.89 0.78 1.50 2.19
$ G^{\pm} {W'}^{\pm} $ 2.89 0.92 1.50 2.19
: Preliminary results for the short-distance QCD corrections at LO to the LRM, using the Vysotskii’s procedure described briefly in the text. Flavor thresholds are taken into account. The $ \overline{\eta} $’s are dependent on the hadronisation scale: $ \mu_{\operatorname{had}} = 2 $ GeV is taken for the $ K $ system and $ \mu_{\operatorname{had}} = 4 $ GeV for the $ B $ systems.[]{data-label="tab:LRetas"}
LO $ \overline{\eta}^{K \overline{K}}_{tt} $ $ \overline{\eta}^{K \overline{K}}_{cc} $ $ \overline{\eta}^{K \overline{K}}_{ct} $ $ \overline{\eta}^{B \overline{B}}_{tt} $
--------------------- ------------------------------------------- ------------------------------------------- ------------------------------------------- -------------------------------------------
$ G^{\pm} H^{\pm} $ 2.89 0.31 0.41 2.18
tree-level FCNC 2.15 0.58 1.12 1.63
: Preliminary results for the short-distance QCD corrections at LO to the LRM, using the Vysotskii’s procedure described briefly in the text. Flavor thresholds are taken into account. The $ \overline{\eta} $’s are dependent on the hadronisation scale: $ \mu_{\operatorname{had}} = 2 $ GeV is taken for the $ K $ system and $ \mu_{\operatorname{had}} = 4 $ GeV for the $ B $ systems.[]{data-label="tab:LRetas"}
Outlook
=======
We reanalyze a version of LRM where the spontaneous breakdown of the LR gauge group is triggered by doublet rather than triplet representations. This changes the structure of the model and introduces a new degree of freedom, the VEV $ \kappa_{L} $ of a $ SU(2)_{L} $ doublet field. Our first concern was to set constraints in this model using EWPO. Our analysis shows that these observables impose a correlation between the scale of the left-right symmetry breaking, which occurs at the scale of several TeV scale, and the size of the couplings.
We are presently analyzing meson mixing observables in order to constrain the remaining parameters of the model. These observables are of great impact because LRM introduces FCNC through new heavy scalars. The structure of the corrections from LRM is different, in the doublet and triplet cases, and they require a good knowledge of short-distance QCD corrections, which can be large, as seen in Table \[tab:LRetas\]. We aim at investigating the consequences of this new scalar sector, especially for FCNC, which are sensitive to $ \kappa_{L} $. A joint fit of EWPO and meson mixing observables, together with leptonic and semileptonic decays, is in progress [@Luiz].
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---
abstract: 'We prove that the Drinfeld double of an arbitrary finite group scheme has finitely generated cohomology. That is to say, for $G$ any finite group scheme, and $D(G)$ the Drinfeld double of the group ring $kG$, we show that the self-extension algebra of the trivial representation for $D(G)$ is a finitely generated algebra, and that for each $D(G)$-representation $V$ the extensions from the trivial representation to $V$ form a finitely generated module over the aforementioned algebra. As a corollary, we find that all categories $\operatorname{rep}(G)^\ast_{\mathscr{M}}$ dual to $\operatorname{rep}(G)$ are of also of finite type (i.e. have finitely generated cohomology), and we provide a uniform bound on their Krull dimensions. This paper completes earlier work of E. M. Friedlander and the author.'
address: 'Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599'
author:
- Cris Negron
title: Finite generation of cohomology for Drinfeld doubles of finite group schemes
---
Introduction {#sect:intro}
============
Fix $k$ an arbitrary field of finite characteristic. Let us recall some terminology [@negronplavnik]: A finite $k$-linear tensor category ${\mathscr{C}}$ is said to be of [*finite type (over $k$)*]{} if the self-extensions of the unit object $\operatorname{Ext}^\ast_{\mathscr{C}}({\mathbf{1}},{\mathbf{1}})$ are a finitely generated $k$-algebra, and for any object $V$ in ${\mathscr{C}}$ the extensions $\operatorname{Ext}^\ast_{\mathscr{C}}({\mathbf{1}},V)$ are a finitely generated module over this algebra. In this case, the [*Krull dimension*]{} $\operatorname{Kdim}{\mathscr{C}}$ of ${\mathscr{C}}$ is the Krull dimension of the extension algebra of the unit. One is free to think of ${\mathscr{C}}$ here as the representation category $\operatorname{rep}(A)$ of a finite-dimensional Hopf algebra $A$, with monoidal structure induced by the comultiplication, and unit ${\mathbf{1}}=k$ provided by the trivial representation.
It has been conjectured [@etingofostrik04 Conjecture 2.18] [@friedlandersuslin97] that any finite tensor category, over an arbitrary base field, is of finite type. Here we consider the category of representations for the Drinfeld double $D(G)$ of a finite group scheme $G$, which is identified with the so-called Drinfeld center $Z(\operatorname{rep}(G))$ of the category of finite $G$-representations [@muger03; @egno15]. The Drinfeld double $D(G)$ is the smash product ${\mathscr{O}}(G)\rtimes kG$ of the algebra of global functions on $G$ with the group ring $kG$, under the adjoint action. So, one can think of $Z(\operatorname{rep}(G))$, alternatively, as the category of coherent $G$-equivariant sheaves on $G$ under the adjoint action $$Z(\operatorname{rep}(G))=\operatorname{rep}(D(G))=\operatorname{Coh}(G)^G.$$
In the present work we prove the following.
For any finite group scheme $G$, the Drinfeld center $Z(\operatorname{rep}(G))$ is of finite type and of Krull dimension $$\operatorname{Kdim}Z(\operatorname{rep}(G))\leq \operatorname{Kdim}\operatorname{rep}(G)+\operatorname{embed.dim}(G).$$
Here $\operatorname{embed.dim}(G)$ denotes the minimal dimension of a smooth (affine) algebraic group in which $G$ embeds as a closed subgroup. The above theorem was proved for $G={\mathbb{G}}_{(r)}$ a Frobenius kernel in a smooth algebraic groups ${\mathbb{G}}$ in work of E. M. Friedlander and the author [@friedlandernegron18]. Thus Theorem \[thm:ZG\] completes, in a sense, the project of [@friedlandernegron18].
One can apply Theorem \[thm:ZG\], and results of J. Plavnik and the author [@negronplavnik], to obtain an additional finite generation result for all dual tensor categories $\operatorname{rep}(G)^\ast_{\mathscr{M}}(:=\operatorname{End}_{\operatorname{rep}(G)}({\mathscr{M}}))$, calculated relative to an exact $\operatorname{rep}(G)$-module category ${\mathscr{M}}$ [@etingofostrik04 Section 3.3].
\[cor:ZGM\] Let $G$ be a finite group scheme, and ${\mathscr{M}}$ be an arbitrary exact $\operatorname{rep}(G)$-module category. Then the dual category $\operatorname{rep}(G)^\ast_{\mathscr{M}}$ is of finite type and of uniformly bounded Krull dimension $$\operatorname{Kdim}\operatorname{rep}(G)^\ast_{\mathscr{M}}\leq \operatorname{Kdim}\operatorname{rep}(G)+\operatorname{embed.dim}(G).$$
Immediate from Theorem \[thm:ZG\] and [@negronplavnik Corollary 4.1].
We view Theorem \[thm:ZG\], and Corollary \[cor:ZGM\], as occurring in a continuum of now very rich studies of cohomology for finite group schemes, e.g. [@friedlanderparshall86II; @friedlandersuslin97; @suslinfriedlanderbendel97; @friedlanderpevtsova07; @touzevanderkallen10; @drupieski16; @bikp19].
Exact $\operatorname{rep}(G)$-module categories have been classified by Gelaki [@gelaki15], and correspond to pairs $(H,\psi)$ of a subgroup $H\subset G$ and certain $3$-cocycle $\psi$ which introduces an associativity constraint for the action of $\operatorname{rep}(G)$ on $\operatorname{rep}(H)$.
For an analysis of support theory for Drinfeld doubles of some solvable height $1$ group schemes, one can see [@negronpevtsova; @negronpevtsovaII]. The problem of understanding support for general doubles $D(G)$ is, at this point, completely open.
Approach via equivariant deformation theory
-------------------------------------------
In [@friedlandernegron18], where the Frobenius kernel ${\mathbb{G}}_{(r)}$ in a smooth algebraic group ${\mathbb{G}}$ is considered, we basically use the fact that ambient group ${\mathbb{G}}$ provides a smooth, equivariant, deformation of ${\mathbb{G}}_{(r)}$ parametrized by the quotient ${\mathbb{G}}/{\mathbb{G}}_{(r)}\cong {\mathbb{G}}^{(r)}$ in order to gain a foothold in our analysis of cohomology. In particular, the adjoint action of ${\mathbb{G}}_{(r)}$ on ${\mathbb{G}}$ descends to a trivial action on the twist ${\mathbb{G}}^{(r)}$, so that the Frobenius map ${\mathbb{G}}\to {\mathbb{G}}^{(r)}$ can be viewed as smoothly varying family of ${\mathbb{G}}_{(r)}$-algebras which deforms the algebra of functions ${\mathscr{O}}({\mathbb{G}}_{(r)})$. Such a deformation situation provides “deformation classes" in degree $2$, $$\{\text{deformation classes}\}=T_1{\mathbb{G}}^{(r)}\subset \operatorname{Ext}^2_{\operatorname{Coh}({\mathbb{G}}_{(r)})^{{\mathbb{G}}_{(r)}}}({\mathbf{1}},{\mathbf{1}})=\operatorname{Ext}^2_{D({\mathbb{G}}_{(r)})}({\mathbf{1}},{\mathbf{1}}).$$ One uses these deformation classes, in conjunction with work of Friedlander and Suslin [@friedlandersuslin97], to find a finite set of generators for extensions.
For a general finite group scheme $G$, we can try to pursue a similar deformation approach, where we embed $G$ into a smooth algebraic group ${\mathcal{H}}$, and consider ${\mathcal{H}}$ as a deformation of $G$ parametrized by the quotient ${\mathcal{H}}/G$. However, a general finite group scheme may not admit any *normal* embedding into a smooth algebraic group. (This is the case for certain non-connected finite group schemes, and should also be the case for restricted enveloping algebras $kG=u^{\rm res}({\mathfrak{g}})$ of Cartan type simple Lie algebras, for example). So, in general, one accepts that $G$ acts nontrivially on the parametrization space ${\mathcal{H}}/G$, and that the fibers in the family ${\mathcal{H}}$ are permuted by the action of $G$ here. Thus we do not obtain a smoothly varying family of $G$-algebras deforming ${\mathscr{O}}(G)$ in this manner.
One can, however, consider a *type* of equivariant deformation theory where the group $G$ is allowed to act nontrivially on the parametrization space, and attempt to obtain *higher* deformation classes in this instance $$\{\text{higher deformation classes}\}\subset \operatorname{Ext}^{\geq 2}_{\operatorname{Coh}(G)^G}({\mathbf{1}},{\mathbf{1}})=\operatorname{Ext}_{D(G)}^{\geq 2}({\mathbf{1}},{\mathbf{1}}).$$ We show in Sections \[sect:equiv1\] and \[sect:equiv2\] that this equivariant deformation picture can indeed be formalized, and that–when considered in conjunction with work of Touzé and Van der Kallen [@touzevanderkallen10]–it can be used to obtain the desired finite generation results for cohomology (see in particular Theorems \[thm:equiv\_fg\] and \[thm:tv2\]).
From a geometric perspective, one can interpret our main theorem as a finite generation result for the cohomology of non-tame stacky local complete intersections. (Formally speaking, we only deal with the maximal codimension case here, but the general situation is similar.) One can compare with works of Gulliksen [@gulliksen74], Eisenbud [@eisenbud80], and many others regarding the homological algebra of complete intersections.
Acknowledgements
----------------
Thanks to Ben Briggs, Christopher Drupieski, Eric Friedlander, Julia Pevtsova, Antoine Touzé, and Sarah Witherspoon for helpful conversations. The proofs of Lemmas \[lem:231\] and \[lem:well\_def\] are due to Ben Briggs and Ragnar Buchweitz (with any errors in their reproduction due to myself). This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140, while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2020 semester.
Differential generalities
=========================
Throughout $k$ is a field of finite characteristic, which is not necessarily algebraically closed. Schemes and algebras are $k$-schemes and $k$-algebras, and ${\otimes}={\otimes}_k$. All group schemes are *affine* group schemes which are of finite type over $k$, and throughout $G$ denotes an (affine) group scheme.
Commutative algebras and modules
--------------------------------
A *finite type* commutative algebra over a field $k$ is a finitely generated $k$-algebra. A *coherent* module over a commutative Noetherian algebra is a finitely generated module. We adopt this language, at times, to distinguish clearly between these two notions of finite generation.
$G$-equivariant dg algebras
---------------------------
Consider $G$ an affine group scheme. We let $\operatorname{rep}(G)$ denote the category of finite-dimensional $G$-representations, $\operatorname{Rep}(G)$ denote the category of integrable, i.e. locally finite, representations, and $\operatorname{Ch}(\operatorname{Rep}(G))$ denote the category of cochain complexes over $\operatorname{Rep}(G)$. Each of these categories is considered along with its standard monoidal structure.
By a $G$-algebra we mean an algebra object in $\operatorname{Rep}(G)$, and by a dg $G$-algebra we mean an algebra object in $\operatorname{Ch}(\operatorname{Rep}(G))$. For $T$ any commutative $G$-algebra, by a *$G$-equivariant dg $T$-algebra* $S$ we mean a $T$-algebra in $\operatorname{Ch}(\operatorname{Rep}(G))$. Note that, for such a dg algebra $S$, the associated sheaf $S^\sim$ on $\operatorname{Spec}(T)$ is an equivariant sheaf of dg algebras, and vice versa. Note also that a dg $G$-algebra is the same thing as an equivariant dg algebra over $T=k$.
DG modules and resolutions
--------------------------
For $S$ a dg $G$-algebra, we let $S\text{-dgmod}^G$ and $D(S)^G$ denote the category of $G$-equivariant dg modules over $S$ and its corresponding derived category $D(S)^G=(S\text{-dgmod}^G)[{\rm quis}^{-1}]$. (Of course, by an equivariant dg module we mean an $S$-module in the category of cochains over $G$.) If we specify some commutative Noetherian graded $G$-algebra $T$, and equivariant $T$-algebra structure on cohomology $T\to H^\ast(S)$, then we take $$D_{coh}(S)^G:=\left\{\begin{array}{c}
\text{The full subcategory in $D(S)^G$ consisting of dg modules}\\
\text{$M$ with finitely generated cohomology over $T$}
\end{array}\right\}.$$ When $T=k$ we take $D_{fin}(S)^G=D_{coh}(S)^G$.
A (non-equivariant) *free* dg $S$-module is an $S$-module of the form $\oplus_{j\in J}\Sigma^{n_j}S$, where $J$ is some indexing set. A semi-projective resolution of a (non-equivariant) dg $S$-module $M$ is a quasi-isomorphism $F\to M$ from a dg module $F$ equipped with a filtration $F=\cup_{i\geq 0} F(i)$ by dg submodules such that each subquotient $F(i)/F(i-1)$ is a summand of a free $S$-module. An *equivariant semi-projective resolution* of an equivariant dg module $M$ is a $G$-linear quasi-isomorphism $F\to M$ from an equivariant dg module $F$ which is non-equivariantly semi-projective. The usual shenanigans, e.g. [@drinfeld04 Lemma 13.3], shows that equivariant semi-projective resolutions always exist.
Homotopy isomorphisms {#sect:dg_htop}
---------------------
Consider $S$ and $A$ dg $G$-algebras, over some given group scheme $G$. By an *(equivariant) homotopy isomorphism* $f:S\to A$ we mean a zig-zag of $G$-linear dg algebra quasi-isomorphism $$\label{eq:207}
S\overset{\sim}\leftarrow S_1\overset{\sim}\to S_2\cdots\overset{\sim}\leftarrow S_{N-1}\overset{\sim}\to A.$$
We note that we use the term *homotopy* informally here, as we do not propose any particular model structure on the category of dg $G$-algebras (cf. [@tabuada05; @toen07]). Throughout the text, when we speak of homotopy isomorphisms between dg $G$-algebras we always mean equivariant homotopy isomorphisms.
A homotopy isomorphism $f:S\to A$ as in specifies a triangulated equivalence between the corresponding derived categories of dg modules $$\label{eq:211}
f_\ast:D(S)^G\overset{\sim}\to D(A)^G,$$ via successive application of base change and restriction along the maps to/from the $S_i$. To elaborate, an equivariant quasi-isomorphism $f:S_1\to S_2$ specifies mutually inverse equivalences $S_2{\otimes}_{S_1}^{\rm L}-:D(S_1)^G\to D(S_2)^G$ and $\operatorname{res}_f:D(S_2)^G\to D(S_1)$. So for a homotopy isomorphism $f:S\to A$, compositions of restriction and base change produce the equivalence .
Note that, on cohomology, such a homotopy isomorphism $f:S\to A$ induces an actual isomorphism of algebras $H^\ast(f):H^\ast(S)\to H^\ast(A)$, and one can check that for a dg module $M$ over $S$ we have $$H^\ast(f_\ast M)\cong H^\ast(A){\otimes}_{H^\ast(S)}H^\ast(M)\cong\operatorname{res}_{H^\ast(f)^{-1}}H^\ast(M).$$ So, in particular, if $H^\ast(S)$ and $H^\ast(A)$ are $T$-algebras, for some commutative Noetherian $T$, and $H^\ast(f)$ is $T$-linear, then the equivalence restricts to an equivalence $$f_\ast:D_{coh}(S)^G\overset{\sim}\to D_{coh}(A)^G$$ between the corresponding equivariant, coherent, derived categories.
We say a dg $G$-algebra $S$ is *equivariantly formal* if $S$ is equivariantly homotopy isomorphic to its cohomology $H^\ast(S)$.
Derived maps and derived endomorphisms
--------------------------------------
Fix $S$ a dg $G$-algebra, over a group scheme $G$. For such $S$, the dg $\operatorname{Hom}$ functor $\operatorname{Hom}_S$ on $S\text{-dgmod}^G$ naturally takes values in $\operatorname{Ch}(\operatorname{Rep}(G))$. Namely, for $x$ in the group ring $kG={\mathscr{O}}(G)^\ast$, we act on functions $f\in \operatorname{Hom}_S(M,N)$ according to the formula $$(x\cdot f)(m):=x_1 f(S(x_2)m).$$ With these actions each $\operatorname{Hom}_S(M,N)$ is a dg $G$-representation, and composition $$\operatorname{Hom}_S(N,L){\otimes}\operatorname{Hom}_S(M,N)\to \operatorname{Hom}_S(M,L)$$ is a map of dg $G$-representations. In particular, $\operatorname{End}_S(M)$ is a dg $G$-algebra for any equivariant dg module $M$ over $S$.
One needs to use cocommutativity of $kG$ here to see that $x\cdot f$ is in fact $S$-linear for $S$-linear $f$.
We derive the functor $\operatorname{Hom}_S$ to $\operatorname{Ch}(\operatorname{Rep}(G))$ by taking $$\operatorname{RHom}_S(M,N):=\operatorname{Hom}_S(M',N),$$ where $M'\to M$ is any equivariant semi-projective resolution of $M$. One can apply their favorite arguments to see that $\operatorname{RHom}_S(M,N)$ is well-defined as an object in $D(\operatorname{Rep}(G))$, or refer to the following lemma.
\[lem:231\] For any two equivariant resolutions $M_1\to M$ and $M_2\to M$ there is an equivariant semi-projective dg module $F$ which admits two surjective, equivariant, quasi-isomorphisms $F\to M_1$ and $F\to M_2$.
By adding on acyclic semi-projective summands we may assume that the given maps $f_i:M_i\to M$ are surjective. For example, one can take a surjective resolution $N\to M$, consider the mapping cone $\operatorname{cone}(id_N)$, then replaces the $M_i$ with $(\Sigma^{-1}\operatorname{cone}(id_N))\oplus M_i$ . So, let us assume that the $f_i$ here are surjective.
We consider now the fiber product $F_0$ of the maps $f_1$ and $f_2$ to $M$. Note that the structure maps $F_0\to M_i$ are surjective, since the $f_i$ are surjective. We have the exact sequence $$0\to F_0\to M_1\oplus M_2\overset{[f_1\ -f_2]^T}\to M\to 0$$ and by considering the long exact sequence on cohomology find that we have also an exact sequence $$0\to H^\ast(F_0)\to H^\ast(M_1)\oplus H^\ast(M_2)\to H^\ast(M)\to 0,$$ with the map from $H^\ast(M_1)\oplus H^\ast(M_2)$ the sum of isomorphisms $\pm H^\ast(f_i)$. It follows that the composites $H^\ast(F_0)\to H^\ast(M_1)\oplus H^\ast(M_2)\to H^\ast(M_i)$ are both isomorphisms, and hence that the maps $F_0\to M_1$ and $F_0\to M_2$ are quasi-isomorphisms. One considers $F\to F_0$ any surjective, equivariant, semi-projective resolution to obtain the claimed result.
For $M$ in $D(S)^G$ we take $\operatorname{REnd}_S(M)=\operatorname{End}_S(M')$, for $M'\to M$ any equivariant semi-projective resolution. The following result should be known to experts. The proof we offer is due to Benjamin Briggs and Ragnar Buchweitz. I thank Briggs for communicating the proof to me, and allowing me to repeat it here.
\[lem:well\_def\] $\operatorname{REnd}_S(M)$ is well-defined, as a dg $G$-algebra, up to homotopy isomorphism. Furthermore, if $M$ and $N$ are isomorphic in $D(S)^G$, then $\operatorname{REnd}_S(M)$ and $\operatorname{REnd}_S(N)$ are homotopy isomorphic as well.
Given an explicit isomorphism $\xi:M\to N$ in $D(S)^G$, the homotopy isomorphism $\operatorname{RHom}_S(M)\to \operatorname{RHom}_S(N)$ can in particular be chosen to lift the canonical isomorphism $\operatorname{Ad}_\xi:\operatorname{Ext}^\ast_S(M,M)\to \operatorname{Ext}^\ast_S(N,N)$ on cohomology.
Consider two equivariant semi-projective resolutions $M_1\to M$ and $M_2\to M$. By Lemma \[lem:231\] we may assume that the map $M_1\to M$ lifts to a surjective, equivariant, quasi-isomorphism $f:M_1\to M_2$. In this case we have the two quasi-isomorphisms $f_\ast$ and $f^\ast$ of $\operatorname{Hom}$ complexes, and consider the fiber product $$\label{eq:261}
\xymatrixrowsep{3mm}
\xymatrix{
& B\ar@{-->}[dl]\ar@{-->}[dr] & \\
\operatorname{End}_S(M_1)\ar[dr]_{f_\ast} & & \operatorname{End}_S(M_2)\ar[dl]^{f^\ast}\\
&\operatorname{Hom}_S(M_1,M_2)
}$$ As $f_\ast$ and $f^\ast$ are maps of dg $G$-representations, $B$ is a dg $G$-representation. Furthermore, one checks directly that $B$ is a dg algebra, or more precisely a dg subalgebra in the product $\operatorname{End}(M_1)\times \operatorname{End}(M_2)$. So the top portion of is a diagram of maps of dg $G$-algebras.
As $M_1$ is projective, as a non-dg module, the map $f_\ast$ is a *surjective* quasi-isomorphism. One can therefore argue as in the proof of Lemma \[lem:231\] to see that the structure maps from $B$ to the $\operatorname{End}_S(M_i)$ are quasi-isomorphisms. So we have the explicit homotopy isomorphism $$\operatorname{End}_S(M_1)\overset{\sim}\leftarrow B\overset{\sim}\to \operatorname{End}_S(M_2).$$
Now, if $M$ is isomorphic to $N$ in $D(S)^G$, then there is a third equivariant dg module $\Omega$ with quasi-isomorphisms $M\overset{\sim}\leftarrow \Omega\overset{\sim}\to N$. Any resolution $F\overset{\sim}\to \Omega$ therefore provides a simultaneous resolution of $M$ and $N$, and we may take $\operatorname{REnd}_S(M)=\operatorname{End}_S(F)=\operatorname{REnd}_S(N)$.
Equivariant deformations and Koszul resolutions {#sect:equiv1}
===============================================
In Sections \[sect:equiv1\] and \[sect:equiv2\] we develop the basic homological algebra associated with equivariant deformations. Our main aim here is to provide equivariant versions of results of Bezrukavnikov and Ginzburg [@bezrukavnikovginzburg07], and Pevtsova and the author [@negronpevtsova §4] (cf. [@eisenbud80; @arkhipovbezrukavnikovginzburg04]).
Equivariant deformations
------------------------
We recall that a deformation of an algebra $R$, parametrized by an augmented commutative algebra $Z$, is a choice of flat $Z$-algebra $Q$ along with an algebra map $Q\to R$ which reduces to an isomorphism $k{\otimes}_Z Q\cong R$. We call such a deformation $Q\to R$ an *equivariant deformation* if all of the algebras present are $G$-algebras, and all of the structure maps $Z\to Q$, $Z\to k$, and $Q\to R$ are maps of $G$-algebras.
The interesting point here, and the point of deviation with other interpretations of equivariant deformation theory, is that we allow $G$ to act nontrivially on the parametrization space $\operatorname{Spec}(Z)$ (or $\operatorname{Spf}(Z)$ in the formal setting).
An equivariant Koszul resolution {#sect:KZ}
--------------------------------
We fix a group scheme $G$, and equivariant deformation $Q\to R$ of a $G$-algebra $R$ with formally smooth parametrization space space $\operatorname{Spf}(Z)$. We require specifically that $Z$ is isomorphic to a power series $k{[\!\hspace{.1mm}[{x_1,\dots,x_n}]\!\hspace{.1mm}]}$ in finitely many variables. As the distinguished point $1\in \operatorname{Spf}(Z)$ is a fixed point for the $G$-action, the cotangent space $T_1\operatorname{Spf}(Z)=m_Z/m_Z^2$ admits a natural $G$-action, and so does the graded algebra $$\operatorname{Sym}(\Sigma m_Z/m_Z^2)=\wedge^\ast(m_Z/m_Z^2),$$ which we view as a dg $G$-algebra with vanishing differential.
\[lem:equiv\_kos\] One can associate to the parametrization algebra $Z$ a commutative equivariant dg $Z$-algebra ${\mathcal{K}}_Z$ such that
1. ${\mathcal{K}}_Z$ is finite and flat over $Z$, and
2. ${\mathcal{K}}_Z$ admits quasi-isomorphisms ${\mathcal{K}}_Z\overset{\sim}\to k$ and $k{\otimes}_Z {\mathcal{K}}_Z\overset{\sim}\to \operatorname{Sym}(\Sigma m_Z/m_Z^2)$ of equivariant dg algebras.
We first construct an unbounded dg resolution ${\mathcal{K}}'$ of $k$, as in [@ciocankapranov01 Section 2.6], then truncate to obtain ${\mathcal{K}}$. We construct ${\mathcal{K}}'$ as a union ${\mathcal{K}}'=\varinjlim_{i\geq 0} {\mathcal{K}}(i)$ of dg subalgebras ${\mathcal{K}}(i)$ over $Z$. We define the ${\mathcal{K}}(i)$ inductively as follows: Take ${\mathcal{K}}(0)=Z$ and, for $V_1$ a finite-dimensional $G$-subspace generating the maximal ideal $m_Z$ in $Z$, we take ${\mathcal{K}}(1)=\operatorname{Sym}_Z(Z{\otimes}\Sigma V_1)$ with differential $d(\Sigma v)=v$, $v\in V_1$.
Suppose now that we have ${\mathcal{K}}(i)$ an equivariant dg algebra which is finite and flat over $Z$ in each degree, and has (unique) augmentation ${\mathcal{K}}(i)\to k$ which is a quasi-isomorphism in degrees $> -i$. Let $V_i$ be an equivariant subspace of cocycles in ${\mathcal{K}}(i)^{-i}$ which generates $H^{-i}({\mathcal{K}}(i))$, as a $Z$-module. Define $${\mathcal{K}}(i+1)=\operatorname{Sym}_Z(Z{\otimes}\Sigma V_i){\otimes}_Z {\mathcal{K}}(i),\ \text{with extended differential $d(\Sigma v)=v$ for }v\in V_i.$$ We then have the directed system of dg algebras ${\mathcal{K}}(0)\to {\mathcal{K}}(1)\to \dots$ with colimit ${\mathcal{K}}'=\varinjlim_i {\mathcal{K}}(i)$. By construction ${\mathcal{K}}'$ is finite and flat over $Z$ in each degree, and has cohomology $H^\ast({\mathcal{K}}')=k$.
Since $Z$ is of finite flat dimension, say $n$, the quotient $$({\mathcal{K}}_Z:=){\mathcal{K}}={\mathcal{K}}'/(({\mathcal{K}}')^{<-n}+B^{-n}({\mathcal{K}}'))$$ is finite and flat over $Z$ in all degrees. Furthermore, ${\mathcal{K}}$ inherits a $G$-action so that the quotient map ${\mathcal{K}}'\to {\mathcal{K}}$ is an equivariant quasi-isomorphism. So we have produced a finite flat dg $Z$-algebra ${\mathcal{K}}$ with equivariant quasi-isomorphism ${\mathcal{K}}\overset{\sim}\to k$.
We consider a section $m_Z/m_Z^2\to V_1$ of the projection $V_1\to m_Z/m_Z^2$, and let $\bar{S}_1\subset V_1$ denote the image of this section. Take $S=\operatorname{Sym}_Z(Z{\otimes}\Sigma \bar{S}_1)$ with differential specified by $d(\Sigma v)=v$ for $v\in \bar{S}_1$. Then $S$ the the standard Koszul resolution for $k$, and the inclusion $S\to {\mathcal{K}}$ is a (non-equivariant) dg algebra quasi-isomorphism. Since ${\mathcal{K}}$ and $S$ are bounded above and flat over $Z$ in each degree, the reduction $k{\otimes}_Z S\to k{\otimes}_Z {\mathcal{K}}$ remains a quasi-isomorphism and we have an isomorphism of algebras $$\operatorname{Sym}(\Sigma m_Z/m_Z^2)\cong H^\ast(k{\otimes}_ZS)\overset{\cong}\to H^\ast(k{\otimes}_Z {\mathcal{K}}).$$
Note that the dg subalgebra $\operatorname{Sym}(\Sigma V_1)\subset k{\otimes}_Z {\mathcal{K}}$ consists entirely of cocycles, and furthermore $Z^{-1}(k{\otimes}_Z {\mathcal{K}})=\Sigma V_1$. We see also that the intersection $V_1\cap m_Z^2$ consists entirely of coboundaries, as such vectors $v$ lift to cocycles in the acyclic complex ${\mathcal{K}}$ which are of the form $v+m_Z{\otimes}V_1$. A dimension count now implies that the projection $$V_1=Z^{-1}(k{\otimes}_Z {\mathcal{K}})\to H^{-1}(k{\otimes}_Z {\mathcal{K}})$$ reduces to an isomorphism $V_1/(m_Z^2\cap V_1)=H^1(k{\otimes}_Z K)$. Hence, for the degree $-1$ coboundaries in $k{\otimes}_Z{\mathcal{K}}$, we have $B^{-1}=V_1\cap m_Z^2$. One now consults the diagram $$\xymatrix{
\operatorname{Sym}(\Sigma m_Z/m_Z^2)\ar[d]_\cong\ar[r]^{\rm incl} & \operatorname{Sym}(\Sigma V_1)\ar[d]\ar[r]^(.25){\rm proj} & \operatorname{Sym}(\Sigma V_1)/(B^{-1})\cong \operatorname{Sym}(\Sigma m_Z/m_Z^2)\ar[dl]\\
H^\ast(k{\otimes}_Z S)\ar[r]^\cong & H^\ast(k{\otimes}_Z {\mathcal{K}}),
}$$ to see that the intersection $B^\ast(k{\otimes}_Z{\mathcal{K}})\cap \operatorname{Sym}(\Sigma V_1)$ is necessarily the ideal $(B^1)$ generated by the degree $-1$ coboundaries. So we find that the projection $$f:k{\otimes}_Z {\mathcal{K}}\to \operatorname{Sym}(\Sigma V_1)/(B^1)\cong \operatorname{Sym}(\Sigma m_Z/m_Z^2)$$ which annihilates (the images of) all cells $\Sigma V_i$ with $i>1$ is an equivariant dg algebra map, and furthermore an equivariant dg algebra quasi-isomorphism.
In the following $Z$ a commutative $G$-algebra which is isomorphic to a power series in finitely many variables, as above.
An *equivariant Koszul resolution* of $k$ over $Z$ is a $G$-equivariant dg $Z$-algebra ${\mathcal{K}}_Z$ which is finite and flat over $Z$, comes equipped with an equivariant dg algebra quasi-isomorphism $\epsilon:{\mathcal{K}}_Z\overset{\sim}\to k$, and also comes equipped with an equivariant dg map $\pi:{\mathcal{K}}_Z\to \operatorname{Sym}(\Sigma m_Z/m_Z^2)$ which reduces to a quasi-isomorphism $k{\otimes}_Z{\mathcal{K}}_Z\overset{\sim}\to \operatorname{Sym}(\Sigma m_Z/m_Z^2)$ along the augmentation $Z\to k$.
Lemma \[lem:equiv\_kos\] says that equivariant Koszul resolutions of $k$, over such $Z$, always exists.
The Koszul resolution associated to an equivariant deformation {#sect:KQ}
--------------------------------------------------------------
Consider $Q\to R$ an equivariant deformation, parameterized by a formally smooth space $\operatorname{Spf}(Z)$, as in Section \[sect:KZ\]. For any equivariant Koszul resolution ${\mathcal{K}}_Z\overset{\sim}\to k$ over $Z$, the product $$\label{eq:205}
{\mathcal{K}}_Q:=Q{\otimes}_Z {\mathcal{K}}_Z$$ is naturally a dg $G$-algebra which is a finite and flat extension of $Q$. Since finite flat modules over $Z$ are in fact free, ${\mathcal{K}}_Q$ is more specifically *free* over $Q$ in each degree. Flatness of $Q$ over $Z$ implies that the projection $$id_Q{\otimes}_Z\epsilon:{\mathcal{K}}_Q\overset{\sim}\to Q{\otimes}_Z k=R$$ is a quasi-isomorphism of dg $G$-algebras (cf. [@arkhipovbezrukavnikovginzburg04 Section 5.2], [@bezrukavnikovginzburg07 Section 3], [@avramovsun98 Section 2]). We call the dg algebra , deduced from a particular choice of equivariant Koszul resolution for $Z$, the (or *a*) Koszul resolution of $R$ associated to the equivariant deformation $Q\to R$.
Deformations associated to group embeddings
===========================================
Consider now $G$ a *finite* group scheme, and a closed embedding of $G$ into a smooth affine algebraic group ${\mathcal{H}}$. (We mean specifically a map of group schemes $G\to {\mathcal{H}}$ which is, in addition, a closed embedding.) We explain in this section how such an embedding $G\to {\mathcal{H}}$ determines an equivariant deformation ${\mathscr{O}}\to {\mathscr{O}}(G)$ which fits into the general framework of Section \[sect:equiv1\].
Note that such closed embeddings $G\to {\mathcal{H}}$ always exists for finite $G$. For example, if we choose a faithful $G$-representation $V$ then the corresponding action map $G\to \operatorname{GL}(V)$ is a closed embedding of $G$ into the associated general linear group.
The quotient space
------------------
For any embedding $G\to {\mathcal{H}}$ of $G$ into smooth ${\mathcal{H}}$ we consider the quotient space ${\mathcal{H}}/G$. The associated quotient map ${\mathcal{H}}\to{\mathcal{H}}/G$ is $G$-equivariant, where we act on ${\mathcal{H}}$ via the adjoint action and on ${\mathcal{H}}/G$ via translation.
This is all clear geometrically, but let us consider this situation algebraically. Functions on the quotient ${\mathscr{O}}({\mathcal{H}}/G)$ are the right $G$-invariants ${\mathscr{O}}({\mathcal{H}})^G$ in ${\mathscr{O}}({\mathcal{H}})$, or rather the left ${\mathscr{O}}(G)$-coinvariants. Then ${\mathscr{O}}({\mathcal{H}}/G)$ is a right ${\mathscr{O}}({\mathcal{H}})$-coideal subalgebra in ${\mathscr{O}}({\mathcal{H}})$, in the sense that the comultiplication on ${\mathscr{O}}({\mathcal{H}})$ restricts to a coaction $$\rho:{\mathscr{O}}({\mathcal{H}}/G)\to {\mathscr{O}}({\mathcal{H}}/G){\otimes}{\mathscr{O}}({\mathcal{H}}).$$ We project along ${\mathscr{O}}({\mathcal{H}})\to {\mathscr{O}}(G)$ to obtain the translation coaction of ${\mathscr{O}}(G)$ on ${\mathscr{O}}({\mathcal{H}}/G)$. The left translation coaction of ${\mathscr{O}}(G)$ on ${\mathscr{O}}({\mathcal{H}})$ restricts to a trivial coaction on ${\mathscr{O}}({\mathcal{H}}/G)$. So, ${\mathscr{O}}({\mathcal{H}}/G)$ is a sub ${\mathscr{O}}(G)$-bicomodule in ${\mathscr{O}}({\mathcal{H}})$.
We consider the dual *action* of the group ring $kG={\mathscr{O}}(G)^\ast$ on ${\mathscr{O}}({\mathcal{H}})$, and find that the inclusion ${\mathscr{O}}({\mathcal{H}}/G)\to {\mathscr{O}}({\mathcal{H}})$ is an inclusion of $G$-algebras, where we act on ${\mathscr{O}}({\mathcal{H}})$ via the adjoint action and on ${\mathscr{O}}({\mathcal{H}}/G)$ by translation. We have the following classical result, which can be found in [@milne17 Proposition 5.25 and Corollary 5.26].
Consider a closed embedding $G\to {\mathcal{H}}$ of a finite group scheme into a smooth algebraic group ${\mathcal{H}}$. The algebra of functions ${\mathscr{O}}({\mathcal{H}})$ is finite and flat over ${\mathscr{O}}({\mathcal{H}}/G)$, and ${\mathscr{O}}({\mathcal{H}}/G)$ is a smooth $k$-algebra.
The associated equivariant deformation sequence {#sect:embed_def}
-----------------------------------------------
Consider $G\to {\mathcal{H}}$ as above and let $1\in {\mathcal{H}}/G$ denote the image of the identity in ${\mathcal{H}}$, by abuse of notation. We complete the inclusion ${\mathscr{O}}({\mathcal{H}}/G)\to {\mathscr{O}}({\mathcal{H}})$ at the ideal of definition for $G$ to get a finite flat extension ${\widehat}{{\mathscr{O}}}_{{\mathcal{H}}/G}\to{\widehat}{{\mathscr{O}}}_{\mathcal{H}}$. Take $$Z={\widehat}{{\mathscr{O}}}_{{\mathcal{H}}/G}\ \ \text{and}\ \ {\mathscr{O}}={\widehat}{{\mathscr{O}}}_{\mathcal{H}}.$$ So we have the deformation ${\mathscr{O}}\to {\mathscr{O}}(G)$, with formally smooth parametrizing algebra $Z$. A proof of the following Lemma can be found at [@negronpevtsova Lemma 2.10].
The completion ${\mathscr{O}}={\widehat}{{\mathscr{O}}}_{\mathcal{H}}$ is Noetherian and of finite global dimension.
Note that the ideal of definition for $G$ is the ideal ${\mathfrak{m}}{\mathscr{O}}(G)$, where ${\mathfrak{m}}\subset {\mathscr{O}}({\mathcal{H}}/G)$ is associated to the closed point $1\in {\mathcal{H}}/G$.
\[prop:eq\_def\_seq\] Consider a closed embedding $G\to {\mathcal{H}}$ of a finite group scheme into a smooth algebraic group ${\mathcal{H}}$. Take ${\mathscr{O}}={\widehat}{{\mathscr{O}}}_{{\mathcal{H}}}$ and $Z={\widehat}{{\mathscr{O}}}_{{\mathcal{H}}/G}$, where we complete at the augmentation ideal ${\mathfrak{m}}$ in ${\mathscr{O}}({\mathcal{H}}/G)$. Then
1. the quotients ${\mathscr{O}}({\mathcal{H}}/G)/{\mathfrak{m}}^n$ and ${\mathscr{O}}({\mathcal{H}})/{\mathfrak{m}}^n{\mathscr{O}}({\mathcal{H}})$ inherit $G$-algebra structures from ${\mathscr{O}}({\mathcal{H}}/G)$ and ${\mathscr{O}}({\mathcal{H}})$ respectively.
2. The completions $Z$ and ${\mathscr{O}}$ inherit unique continuous $G$-actions so that the inclusions ${\mathscr{O}}({\mathcal{H}}/G)\to Z$ and ${\mathscr{O}}({\mathcal{H}})\to {\mathscr{O}}$ are $G$-linear.
3. Under the actions of [(b)]{}, the projection ${\mathscr{O}}\to {\mathscr{O}}(G)$ is an equivariant deformation of ${\mathscr{O}}(G)$ parametrized by $\operatorname{Spf}(Z)=({\mathcal{H}}/G)^{\wedge}_1$.
All of (a)–(c) will follow if we can simply show that ${\mathfrak{m}}\subset {\mathscr{O}}({\mathcal{H}}/G)$ is stable under the translation action of $kG$. This is clear geometrically, and certainly well-known, but let us provide an argument for completeness. If we let $\ker(\epsilon)\subset {\mathscr{O}}({\mathcal{H}})$ denote the augmentation ideal, we have ${\mathfrak{m}}=\ker(\epsilon)\cap{\mathscr{O}}({\mathcal{H}}/G)$.
For the adjoint coaction $\rho_{\rm ad}:f\mapsto f_2{\otimes}S(f_1)f_3$ of ${\mathscr{O}}({\mathcal{H}})$ on itself, and $f\in \ker(\epsilon)$, we have $$\begin{array}{l}
(\epsilon{\otimes}1)\circ\rho_{\rm ad}(f)=\epsilon(f_2)S(f_1)f_3\\
\hspace{2cm}=S(f_1)(\epsilon(f_2)f_3)=S(f_1)f_2=\epsilon(f)=0.
\end{array}$$ So we see that under the adjoint coaction $\rho_{\rm ad}(\ker(\epsilon))\subset \ker(\epsilon){\otimes}{\mathscr{O}}({\mathcal{H}})$. It follows that $\ker(\epsilon)$ is preserved under the adjoint coaction of ${\mathscr{O}}(G)$, and hence the adjoint action of $kG$, as well. So, the intersection ${\mathfrak{m}}={\mathscr{O}}({\mathcal{H}}/G)\cap \ker(\epsilon)$ is an intersection of $G$-subrepresentations in ${\mathscr{O}}({\mathcal{H}})$, and hence ${\mathfrak{m}}$ is stable under the action of $kG$.
Equivariant formality results and deformation classes {#sect:equiv2}
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We observe cohomological implications of the existence of a (smooth) equivariant deformation, for a given finite-dimensional $G$-algebra $R$. The main results of this section can been seen as particular equivariantizations of [@bezrukavnikovginzburg07 Theorem 1.2.3] and [@negronpevtsova Corollary 4.7], as well as of classical results of Gulliksen [@gulliksen74 Theorem 3.1].
We fix an equivariant deformation {#sect:fix}
---------------------------------
We fix a $G$-equivariant deformation $Z\to Q\to R$, with $Z$ isomorphic to a power series in finitely many variables. Fix also a choice of equivariant Koszul resolution $${\mathcal{K}}:={\mathcal{K}}_Z,\ \ \text{with}\ \epsilon:{\mathcal{K}}\overset{\sim}\to k\ \text{and}\ \pi:{\mathcal{K}}\to \operatorname{Sym}(\Sigma m_Z/m_Z^2).$$ Recall the associated dg resolution $K_Q\overset{\sim}\to R$, with ${\mathcal{K}}_Q=Q{\otimes}_Z{\mathcal{K}}$. Via general phenomena (Section \[sect:dg\_htop\]) we observe
\[lem:433\] Restriction provides a derived equivalence $D_{fin}(R)^G\overset{\sim}\to D_{coh}({\mathcal{K}}_Q)^G$.
Following the notation of [@negronpevtsova], we fix $$\label{eq:a_z}
A_Z:=\operatorname{Sym}(\Sigma^{-2}T_1\operatorname{Spf}(Z))=\operatorname{Sym}(\Sigma^{-2}(m_Z/m_Z^2)^\ast).$$
Equivariant formality and deformation classes
---------------------------------------------
\[lem:formal\] Consider ${\mathcal{K}}$ the regular dg ${\mathcal{K}}$-bimodule. There is a ($G$-)equivariant homotopy isomorphism $$\operatorname{REnd}_{{\mathcal{K}}{\otimes}_Z {\mathcal{K}}}({\mathcal{K}})\overset{\sim}\to A_Z.$$ In particular, $\operatorname{REnd}_{{\mathcal{K}}{\otimes}_Z{\mathcal{K}}}({\mathcal{K}})$ is equivariantly formal.
Consider our algebra $A=A_Z$ from and take $B=\operatorname{Sym}(\Sigma m_Z/m_Z^2)$. Let $F\to k$ be the standard resolution of the trivial module over $B$. The resolution $F$ is of the form $B{\otimes}A^\ast$, as a graded space, with differential given by right multiplication by the identity element $\sum_i x_i{\otimes}x^i$ in $B^{-1}{\otimes}A^2$, and so $F$ admits a natural dg $(B,A)$-bimodule structure. The action map for $A$ now provides an equivariant quasi-isomorphism $A\overset{\sim}\to \operatorname{End}_{B}(F)=\operatorname{REnd}_{B}(k)$.
For the Koszul resolution ${\mathcal{K}}$ over $Z$, we have the equivariant quasi-isomorphism $\pi{\otimes}_Z\epsilon:{\mathcal{K}}{\otimes}_Z {\mathcal{K}}\overset{\sim}\to B$ and corresponding restriction and base change equivalences $D({\mathcal{K}}{\otimes}_Z{\mathcal{K}})^G\leftrightarrows D(B)^G$, which are mutually inverse. Restriction sends the trivial representation $k$ over $B$ to the regular ${\mathcal{K}}$-bimodule $k\cong {\mathcal{K}}$. Hence the base change $B{\otimes}^{\rm L}_{{\mathcal{K}}{\otimes}_Z{\mathcal{K}}}{\mathcal{K}}$ is isomorphic to $k$. We then get then an equivariant quasi-isomorphism $$B{\otimes}^{\rm L}_{{\mathcal{K}}{\otimes}_Z{\mathcal{K}}}-:\operatorname{REnd}_{{\mathcal{K}}{\otimes}_Z{\mathcal{K}}}({\mathcal{K}})\overset{\sim}\to \operatorname{REnd}_{B}(B{\otimes}^{\rm L}_{{\mathcal{K}}{\otimes}_Z{\mathcal{K}}}{\mathcal{K}}),$$ with the latter algebra homotopy isomorphic to $\operatorname{REnd}_{B}(k)\cong A$ by Lemma \[lem:well\_def\].
In odd characteristic, one can replace the quasi-isomorphism $\pi{\otimes}_Z\epsilon:{\mathcal{K}}{\otimes}_Z{\mathcal{K}}\to B$ with the more symmetric map $$mult(\frac{1}{2}\pi{\otimes}_Z\frac{-1}{2}\pi):{\mathcal{K}}{\otimes}_Z{\mathcal{K}}\to B.$$ The point is to provide an equivariant quasi-isomorphism which is a retract of the non-equivariant quasi-isomorphism $B\to {\mathcal{K}}{\otimes}_Z{\mathcal{K}}$ implicit in [@bezrukavnikovginzburg07 Lemma 2.4.2].
Recall that we are considering an equivariant deformation $Q\to R$, with associated dg resolution $K_Q\overset{\sim}\to R$, as in Section \[sect:KQ\]. We have the natural action of $A_Z$ on $D_{coh}(K_Q)$ [@negronpevtsova §3.4], which is expressed via the algebra map $$\label{eq:500}
A_Z=\operatorname{End}^\ast_{D({\mathcal{K}}{\otimes}_Z{\mathcal{K}})}({\mathcal{K}})\to Z(D_{coh}({\mathcal{K}}_Q))$$ to the center of the derived category $Z(D_{coh}({\mathcal{K}}_Q))=\oplus_i\operatorname{Hom}_{\text{Fun}}(id,\Sigma^i)$. Specifically, for any endomorphism $f:{\mathcal{K}}\to \Sigma^n{\mathcal{K}}$ in the derived category of $Z$-central bimodules, and $M$ in $D_{coh}(K_Q)$, we have the induced endomorphism $$f{\otimes}_{{\mathcal{K}}}^{\rm L}M:M\to \Sigma^n M.$$
Suppose, for convenience, that $Q$ is of finite global dimension. We lift the maps $$\label{eq:ast}
-{\otimes}^{\rm L}_{\mathcal{K}}M:\operatorname{End}^\ast_{D({\mathcal{K}}{\otimes}_Z{\mathcal{K}})}({\mathcal{K}})\to \operatorname{End}^\ast_{D({\mathcal{K}}_Q)}(M)$$ to a dg level, for *equivariant* $M$, as follows [@bezrukavnikovginzburg07]: Fix an equivariant semi-projective resolution $F\to {\mathcal{K}}$ over ${\mathcal{K}}{\otimes}_Z{\mathcal{K}}$ and, at each $M$, chose an equivariant quasi-isomorphism $M'\to M$ from a dg ${\mathcal{K}}_Q$-module which is bounded and projective over $Q$ in each degree. (Such a resolution exists since $Q$ is of finite global dimension.) Then $F{\otimes}_{{\mathcal{K}}}M'\to M$ is an equivariant semi-projective resolution of $M$ over ${\mathcal{K}}_Q$ [@negronpevtsova Lemma 4.4]. We now have the lift $$-{\otimes}_{\mathcal{K}}M':\operatorname{End}_{{\mathcal{K}}{\otimes}_Z{\mathcal{K}}}(F)\to \operatorname{End}_{{\mathcal{K}}_Q}(F{\otimes}_{\mathcal{K}}M')$$ of , and we write this lift simply as $${\mathfrak{def}}^G_M:\operatorname{REnd}_{{\mathcal{K}}{\otimes}_Z{\mathcal{K}}}({\mathcal{K}})\to \operatorname{REnd}_{{\mathcal{K}}_Q}(M).$$ Direct calculation verifies that ${\mathfrak{def}}^G_M$, constructed in this manner, is in fact $G$-linear. The following result is an equivariantization of [@negronpevtsova Corollary 4.7].
\[thm:equiv\_fg\] Consider a $G$-equivariant deformation $Q\to R$, with $R$ finite-dimensional, $Q$ of finite global dimension, and parametrization algebra $Z$ isomorphic to a power series in finitely many variables. Let ${\mathscr{R}}$ denote the formal dg algebra $\operatorname{REnd}_{{\mathcal{K}}{\otimes}_Z{\mathcal{K}}}({\mathcal{K}})$ (Lemma \[lem:formal\]).
For any $M$ in $D_{coh}({\mathcal{K}}_Q)^G$, the equivariant dg algebra map ${\mathfrak{def}}^G_M:{\mathscr{R}}\to \operatorname{REnd}_{{\mathcal{K}}_Q}(M)$ defined above has the following properties:
1. The induced map on cohomology $H^\ast({\mathfrak{def}}^G_M):A_Z\to \operatorname{End}^\ast_{D({\mathcal{K}}_Q)}(M)$ is a finite morphism of graded $G$-algebras.
2. For any $N$ in $D_{coh}({\mathcal{K}}_Q)^G$, the induced action of ${\mathscr{R}}$ on $\operatorname{RHom}_{{\mathcal{K}}_Q}(M,N)$ is such that $$\operatorname{RHom}_{{\mathcal{K}}_Q}(M,N)\in D_{coh}({\mathscr{R}})^G.$$
By $D_{coh}({\mathscr{R}})^G$ we mean the category of $G$-equivariant dg modules over ${\mathscr{R}}$ with finitely generated cohomology over $A_Z=H^\ast({\mathscr{R}})$.
The map ${\mathfrak{def}}^G_M$ was already constructed above. We just need to verify the implications for cohomology, which actually have nothing to do with the $G$-action. We note that the cohomology $H^\ast({\mathfrak{def}}^G_M)$ is, by construction, obtained by evaluating the functor $$-{\otimes}^{\rm L}_{{\mathcal{K}}}M:D({\mathcal{K}}{\otimes}_Z{\mathcal{K}})\to D({\mathcal{K}}_Q)$$ at the object ${\mathcal{K}}$. (Again, we forget about equivariance here.) We can factor this functor through the category of ${\mathcal{K}}_Q$-bimodules $$D({\mathcal{K}}{\otimes}_Z{\mathcal{K}})\overset{-{\otimes}_Z^{\rm L}Q}\longrightarrow D(K_Q{\otimes}K_Q)\overset{-{\otimes}^{\rm{L}}_{{\mathcal{K}}_Q}M}\longrightarrow D({\mathcal{K}}_Q)$$ to see that the corresponding map to the center agrees with that of [@bezrukavnikovginzburg07 (3.1.5)] [@negronpevtsova Section 3.4]. So the finiteness claims of (1) and (2) follow from [@negronpevtsova Corollary 4.7].
Via Lemma \[lem:formal\] we may replace $D({\mathscr{R}})^G$ with $D(A_Z)^G$, and view $\operatorname{RHom}_{K_Q}$, or equivalently $\operatorname{RHom}_R$, as a functor to $D(A_Z)^G$. Alternatively, we could work with the dg scheme (shifted affine space) ${\mathcal{T}}^\ast=T_1^\ast\operatorname{Spf}(Z)=\operatorname{Spec}(A_Z)$, and view $\operatorname{RHom}_R$ as a functor taking values in the derived category of equivariant dg sheaves on ${\mathcal{T}}^\ast$.
From this perspective, Theorem \[thm:equiv\_fg\] tells us that $\operatorname{RHom}_R$ has image in the subcategory of dg sheaves on ${\mathcal{T}}^\ast$ with coherent cohomology, $$\operatorname{RHom}_R:(D_{fin}(R)^G)^{op}\times D_{fin}(R)^G\to D_{coh}(A_Z)^G\cong D_{coh}({\mathcal{T}}^\ast)^G.$$
We only use the finiteness claims of Theorem \[thm:equiv\_fg\] in the case in which all of $Z$, $Q$, and $R$ are commutative. In this case in particular, claims (1) and (2) of Theorem \[thm:equiv\_fg\] should be obtainable directly from Gulliksen [@gulliksen74 Theorem 3.1].
One may compare the above analyses with the formality arguments of [@arkhipovbezrukavnikovginzburg04 Sections 5.4–5.8].
Touzé-Van der Kallen and derived invariants
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We recall some results of Touzé and Van der Kallen [@touzevanderkallen10]. Our aim is to take derived invariants of Theorem \[thm:equiv\_fg\] to obtain a finite generation result for equivariant extensions $\operatorname{Hom}_{D(R)^G}^\ast$. We successfully realize this aim via an invocation of [@touzevanderkallen10]. Throughout this section $G$ is a *finite* group scheme.
Basics and notations
--------------------
For $V$ any $G$-representation we have the standard group cohomology $H^\ast(G,V)=\operatorname{Ext}^\ast_G({\mathbf{1}},V)$. For more general objects in $D(\operatorname{Rep}(G))$ we adopt a hypercohomological notation.
We let $(-)^{{\mathrm{R}G}}:D(\operatorname{Rep}(G))\to D(Vect)$ denote the derived invariants functor, $(-)^{{\mathrm{R}G}}=\operatorname{RHom}_G({\mathbf{1}},-)$. For $M$ in $D(\operatorname{Rep}(G))$ we take $${\mathbb{H}}^\ast(G,M):=H^\ast(M^{{\mathrm{R}G}}).$$
We note that the hypercohomology ${\mathbb{H}}^\ast(G,M)$ is still identified with morphisms $\operatorname{Hom}^\ast_{D(\operatorname{Rep}(G))}({\mathbf{1}},M)$ in the derived category. Since $G$ is assumed to be finite, we are free to employ an explicit identification $$(-)^{{\mathrm{R}G}}=\operatorname{Hom}_G(Bar_G,-),$$ where $Bar_G$ is the standard Bar resolution. For any dg $G$-algebra $S$ the derived invariants $S^{{\mathrm{R}G}}$ are naturally a dg algebra in $Vect$, and for any equivariant dg $S$-module $M$, $M^{{\mathrm{R}G}}$ is a dg module over $S^{{\mathrm{R}G}}$. (Under our explicit expression of derived invariants in terms of the bar resolution, these multiplicative structures are induced by a dg coalgebra structure on $Bar_G$, see e.g. [@sanada93 §2.2].) We therefore obtain at any dg $G$-algebra a functor $$\label{eq:RG}
(-)^{{\mathrm{R}G}}:D(S)^G\to D(S^{{\mathrm{R}G}}).$$
The following well-known fact can be proved by considering the hypercohomology ${\mathbb{H}}^\ast(G,S)$ as maps ${\mathbf{1}}\to \Sigma^nS$ in the derived category.
If $A$ is a commutative dg $G$-algebra, then the hypercohomology $\mathbb{H}^\ast(G,A)$ is a also commutative.
Derived invariants and coherence of dg modules
----------------------------------------------
We have the following result of Touzé and Van der Kalen.
\[thm:tv1\] Consider $G$ a finite group scheme, and $A$ a commutative $G$-algebra which is of finite type over $k$. Then the cohomology $H^\ast(G,A)$ is also of finite type, and for any finitely generated equivariant $A$-module $M$, the cohomology $H^\ast(G,M)$ is a finite module over $H^\ast(G,A)$.
One can derive this results to obtain
\[thm:tv2\] Consider $G$ a finite group scheme, and $S$ a dg $G$-algebra which is equivariantly formal and has commutative, finite type, cohomology. Suppose additionally that the cohomology of $S$ is bounded below. Then the derived invariants functor restricts to a functor $$(-)^{{\mathrm{R}G}}:D_{coh}(S)^G\to D_{coh}(S^{{\mathrm{R}G}}).$$ Equivalently, for any equivariant dg $S$-module $M$ with finitely generated cohomology over $H^\ast(S)$, the hypercohomology ${\mathbb{H}}^\ast(G,M)$ is finite over ${\mathbb{H}}^\ast(G,S)$.
Take $A=H^\ast(S)$. We are free to view, momentarily, $A$ as a non-dg object. We have that $A$ is finite over its even subalgebra $A^{ev}$, which is a commutative algebra in the classical sense, so that Theorem \[thm:tv1\] implies that cohomology $H^\ast(G,-)$ sends $A$ to a finite extension of $H^\ast(G,A^{ev})$, and any finitely generated $A$-module to a finitely generated $H^\ast(G,A^{ev})$-module. Hence $H^\ast(G,A)$ is of finite type over $k$, and $H^\ast(G,N)$ is finite over $H^\ast(G,A)$ for any finitely generated, equivariant, non-dg, $A$-module $N$.
Since $G$ is a finite group scheme, $A$ is also a finite module over its (usual) invariant subalgebra $A^G$, and any $A$-module is finitely generated over $A$ if and only if it is finitely generated over $A^G$. Theorem \[thm:tv1\] then tells us that, for any finitely generated $A$-module $N$, the cohomology $H^\ast(G,N)$ is finitely generated over $H^\ast(G,A^G)=H^\ast(G,{\mathbf{1}}){\otimes}A^G$, where $H^\ast(G,A^G)$ acts through the algebra map $$H^\ast(G,{\rm incl}):H^\ast(G,A^G)\to H^\ast(G,A).$$
Consider now any dg module $M$ in $D_{coh}(S)^G$. Formality implies an algebra isomorphism $S\cong A$ in $D(\operatorname{Rep}(G))$ and so identifies ${\mathbb{H}}^\ast(G,S)$ with ${\mathbb{H}}^\ast(G,A)=H^\ast(G,A)$. We want to show that, for such a dg module $M$, the hypercohomology ${\mathbb{H}}^\ast(G,M)$ is a finitely generated module over ${\mathbb{H}}^\ast(G,S)\cong H^\ast(G,A)$. It suffices to show that ${\mathbb{H}}^\ast(G,M)$ is finite over $H^\ast(G,A^G)=H^\ast(G,{\mathbf{1}}){\otimes}A^G$. We have the first quadrant spectral sequence (via our bounded below assumption) $$E_2^{\ast,\ast}=H^\ast(G,H^\ast(M))\ \Rightarrow\ {\mathbb{H}}^\ast(G,M),$$ and the $E_2$-page is finite over $H^\ast(G,A^G)$ by the arguments given above. Since $H^\ast(G,A^G)$ is Noetherian, it follows that the associated graded module $E_\infty^{\ast,\ast}=\operatorname{gr}{\mathbb{H}}^\ast(G,M)$ is finite over $H^\ast(G,A^G)$, and since the filtration on ${\mathbb{H}}^\ast(G,M)$ is bounded in each cohomological degree it follows that the hypercohomology ${\mathbb{H}}^\ast(G,M)$ is indeed finite over $H^\ast(G,A^G)\subset {\mathbb{H}}^\ast(G,S)$ [@friedlandersuslin97 Lemma 1.6].
Finite generation of cohomology for Drinfeld doubles
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Consider $G$ a finite group scheme. Fix a closed embedding $G\to {\mathcal{H}}$ into a smooth algebraic group ${\mathcal{H}}$, and fix also the associated $G$-equivariant deformation $$Z\to {\mathscr{O}}\to {\mathscr{O}}(G),\ \ Z={\widehat}{{\mathscr{O}}}_{{\mathcal{H}}/G},\ {\mathscr{O}}={\widehat}{{\mathscr{O}}}_{{\mathcal{H}}},$$ as in Section \[sect:embed\_def\]. Here $kG$ acts on ${\mathscr{O}}(G)$ and ${\mathscr{O}}$ via the adjoint action, and this adjoint action restricts to a translation action on $Z$. We recall that the embedding dimension of $G$ is the minimal dimension of such smooth ${\mathcal{H}}$ admitting a closed embedding $G\to {\mathcal{H}}$.
We consider the tensor category $$Z(\operatorname{rep}(G))\cong \operatorname{rep}(D(G))\cong \operatorname{Coh}(G)^G$$ of representations over the Drinfeld double of $G$, aka the Drinfeld center of $\operatorname{rep}(G)$. We prove the following below.
\[thm:ZG\] For any finite group scheme $G$, the Drinfeld center $Z(\operatorname{rep}(G))$ is of finite type and of bounded Krull dimension $$\operatorname{Kdim}Z(\operatorname{rep}(G))\leq \operatorname{Kdim}\operatorname{rep}(G)+\operatorname{embed.dim}(G).$$
One can recall our definition of a finite type tensor category, and of the Krull dimension of such a category, from the introduction. For ${\mathcal{T}}^\ast$ the cotangent space $T^\ast_1\operatorname{Spf}(Z)$, considered as a variety with a linear $G$-action, we show in particular that there is a finite map of schemes $\operatorname{Spec}\operatorname{Ext}^\ast_{Z(\operatorname{rep}(G))}({\mathbf{1}},{\mathbf{1}})\to (G\setminus {\mathcal{T}}^\ast)\times \operatorname{Spec}H^\ast(G,{\mathbf{1}})$.
Preliminaries for Theorem \[thm:ZG\]: Derived maps in $Z(\operatorname{rep}(G))$
--------------------------------------------------------------------------------
We let $G$ act on itself via the adjoint action, and have $\operatorname{Coh}(G)^G=\operatorname{rep}({\mathscr{O}}(G))^G$. The unit object ${\mathbf{1}}\in \operatorname{Coh}(G)^G$ is the residue field of the fixed point $1:\operatorname{Spec}(k)\to G$. We have $$\operatorname{REnd}_{\operatorname{Coh}(G)^G}({\mathbf{1}})=\operatorname{REnd}_{\operatorname{Coh}(G)}({\mathbf{1}})^{{\mathrm{R}G}},$$ as an algebra, and for any $V$ in $\operatorname{Coh}(G)^G$ we have $$\operatorname{RHom}_{\operatorname{Coh}(G)^G}({\mathbf{1}},V)=\operatorname{RHom}_{\operatorname{Coh}(G)}({\mathbf{1}},V)^{{\mathrm{R}G}},$$ as a dg $\operatorname{REnd}_{\operatorname{Coh}(G)^G}({\mathbf{1}})$-module.
One can observe these identifications essentially directly, by noting that for the projective generator ${\mathscr{O}}(G)\rtimes kG$ we have an identification of $G$-representations $$\operatorname{Hom}_{\operatorname{Coh}(G)}({\mathscr{O}}(G)\rtimes kG,V)=\operatorname{Hom}_k(kG,V)={\mathscr{O}}(G){\otimes}V,$$ and ${\mathscr{O}}(G){\otimes}V$ is an injective over $kG$ for any $V$. Hence the functor $\operatorname{Hom}_{\operatorname{Coh}(G)}(-,V)$ sends projectives objects in $\operatorname{Coh}(G)^G$ to injectives in $\operatorname{Rep}(G)$, and for a projective resolution $F\to {\mathbf{1}}$ we have identifications in the derived category of vector spaces $$\begin{array}{rl}
\operatorname{RHom}_{\operatorname{Coh}(G)^G}({\mathbf{1}},V)&=\operatorname{Hom}_{\operatorname{Coh}(G)^G}(F,V)\\
&=\operatorname{Hom}_{\operatorname{Coh}(G)}(F,V)^G\\
&\cong\operatorname{Hom}_{\operatorname{Coh}(G)}(F,V)^{{\mathrm{R}G}}=\operatorname{RHom}_{\operatorname{Coh}(G)}({\mathbf{1}},V)^{{\mathrm{R}G}}
\end{array}$$ and $$\operatorname{REnd}_{\operatorname{Coh}(G)^G}({\mathbf{1}},{\mathbf{1}})=\operatorname{End}_{\operatorname{Coh}(G)}(F)^G\cong\operatorname{End}_{\operatorname{Coh}(G)}(F)^{{\mathrm{R}G}}=\operatorname{REnd}_{\operatorname{Coh}(G)}({\mathbf{1}})^{{\mathrm{R}G}}.$$ The middle identification for derived endomorphisms comes from the diagram $$\xymatrix{
\operatorname{End}(F)^G\ar[r]\ar[d]_\sim & \operatorname{End}(F)^{{\mathrm{R}G}}\ar[d]^\sim\\
\operatorname{Hom}(F,{\mathbf{1}})^G\ar[r]_\sim & \operatorname{Hom}(F,{\mathbf{1}})^{{\mathrm{R}G}}.
}$$
Proof of Theorem \[thm:ZG\]
---------------------------
Fix an embedding $G\to {\mathcal{H}}$ and associated equivariant deformation ${\mathscr{O}}\to {\mathscr{O}}(G)$ as above, and take $A=A_Z=\operatorname{Sym}(\Sigma^{-2}(m_Z/m_Z^2)^\ast)$, as in . Take also ${\mathscr{R}}$ the dg $G$-algebra $\operatorname{REnd}_{{\mathcal{K}}_Z{\otimes}_Z{\mathcal{K}}_Z}({\mathcal{K}}_Z)$. We recall from Lemma \[lem:formal\] that ${\mathscr{R}}$ is equivariantly formal, and so homotopy isomorphic to $A$. We adopt the abbreviated notations $\operatorname{RHom}=\operatorname{RHom}_{\operatorname{Coh}(G)}$ and $\operatorname{REnd}=\operatorname{REnd}_{\operatorname{Coh}(G)}$ when convenient.
We consider the equivariant dg algebra map $${\mathfrak{def}}^G_{\mathbf{1}}:{\mathscr{R}}\to \operatorname{REnd}_{\operatorname{Coh}(G)}({\mathbf{1}})$$ of Theorem \[thm:equiv\_fg\], and the action of ${\mathscr{R}}$ on each $\operatorname{REnd}_{\operatorname{Coh}(G)}({\mathbf{1}},V)$ through ${\mathfrak{def}}^G_{\mathbf{1}}$. By Theorems \[thm:equiv\_fg\] and \[thm:tv2\], the hypercohomology ${\mathbb{H}}^\ast(G,\operatorname{REnd}({\mathbf{1}}))$ is a finite algebra extension of ${\mathbb{H}}^\ast(G,{\mathscr{R}})$, and ${\mathbb{H}}^\ast(G,\operatorname{RHom}({\mathbf{1}},V))$ is a finitely generated module over ${\mathbb{H}}^\ast(G,{\mathscr{R}})$ for any $V$ in $\operatorname{Coh}(G)^G$. In particular, ${\mathbb{H}}^\ast(G,\operatorname{RHom}({\mathbf{1}},V))$ is finite over ${\mathbb{H}}^\ast(G,\operatorname{REnd}({\mathbf{1}}))$.
Since ${\mathbb{H}}^\ast(G,{\mathscr{R}})\cong {\mathbb{H}}^\ast(G,A)$ is of finite type over $k$, by Touzé-Van der Kallen (Theorem \[thm:tv2\]), the above arguments imply that $${\mathbb{H}}^\ast(G,\operatorname{REnd}_{\operatorname{Coh}(G)}({\mathbf{1}}))=\operatorname{Ext}^\ast_{\operatorname{Coh}(G)^G}({\mathbf{1}},{\mathbf{1}})$$ is a finite type $k$-algebra, and that each $${\mathbb{H}}^\ast(G,\operatorname{RHom}_{\operatorname{Coh}(G)}({\mathbf{1}},V))=\operatorname{Ext}^\ast_{\operatorname{Coh}(G)^G}({\mathbf{1}},V)$$ is a finitely generated module over this algebra, for $V$ in $\operatorname{Coh}(G)^G$. That is to say, the tensor category $Z(\operatorname{rep}(G))\cong\operatorname{Coh}(G)^G$ is of finite type over $k$.
As for the Krull dimension, ${\mathbb{H}}^\ast(G,A)$ is finite over $H^\ast(G,A^G)=H^\ast(G,{\mathbf{1}}){\otimes}A^G$, by Touzé-Van der Kallen, so that $$\begin{array}{rl}
\operatorname{Kdim}Z(\operatorname{rep}(G))&=\operatorname{Kdim}\operatorname{Ext}^\ast_{Z(\operatorname{rep}(G))}({\mathbf{1}},{\mathbf{1}})\\
&\leq \operatorname{Kdim}H^\ast(G,k){\otimes}A^G\\
&\hspace{.5cm}=\operatorname{Kdim}H^\ast(G,k){\otimes}A\\
&\hspace{.5cm}=\operatorname{Kdim}\operatorname{rep}(G)+\dim {\mathcal{H}}/G=\operatorname{Kdim}\operatorname{rep}(G)+\dim{\mathcal{H}}.
\end{array}$$ When ${\mathcal{H}}$ is taken to be of minimal possible dimension we find the proposed bound, $$\operatorname{Kdim}(Z(\operatorname{rep}(G)))\leq \operatorname{Kdim}\operatorname{rep}(G)+\operatorname{embed.dim}(G).$$
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abstract: '[If $f$ is an idempotent in a ring $\Lambda$, then we find sufficient conditions which imply that the cohomology rings $\oplus_{n\ge 0}{\operatorname{Ext}\nolimits}^n_{\Lambda}(\Lambda/{{{\bf r}}},\Lambda/{{{\bf r}}})$ and $\oplus_{n\ge 0}{\operatorname{Ext}\nolimits}^n_{f\Lambda f}(f\Lambda f/f{{{\bf r}}} f,f\Lambda f/f{{{\bf r}}} f)$ are eventually isomorphic. This result allows us to compare finite generation and GK dimension of the cohomology rings $\Lambda$ and $f\Lambda f$. We are also able to compare the global dimensions of $\Lambda$ and $f\Lambda f$. ]{}'
address:
- |
Department of Mathematics\
Virginia Tech\
Blacksburg, VA 24061\
USA
- |
Dag Oskar Madsen\
University of Nordland\
Faculty of Professional Studies
- |
Eduardo Marcos\
Departmento Matemática\
Universidade de São Paulo\
Brasil
author:
- 'Edward L. Green'
- Dag Oskar Madsen
- Eduardo Marcos
title: Cohomological Comparison Theorem
---
\[section\] \[prop\][Theorem]{} \[prop\][Lemma]{} \[prop\][Corollary]{} \[prop\][Question]{}
\[prop\][Example]{} \[prop\][Examples]{} \[prop\][Remark]{}
[^1]
Introduction
============
If $M$ is a $\Lambda$-module for some ring $\Lambda$, knowledge of the cohomology ring of $M$, $\oplus_{n\ge 0}{\operatorname{Ext}\nolimits}_{\Lambda}^n(M,M)$, is useful in the study of the representation theory of $\Lambda$-modules. In view of this, connecting the cohomology rings of two modules over different rings can provide helpful information. The main goal of this paper is to find sufficient conditions so that the cohomology rings $\bigoplus_{n\ge 0}{\operatorname{Ext}\nolimits}^n_{\Lambda}(\Lambda/{{{\bf r}}},\Lambda/{{{\bf r}}})$ and $\bigoplus_{n\ge 0}{\operatorname{Ext}\nolimits}^n_{f\Lambda f}(f\Lambda f/f{{{\bf r}}} f,f\Lambda f/f{{{\bf r}}} f)$ are eventually isomorphic, where $f$ is an idempotent in the ring $\Lambda$ and ${{\bf r}}$, denotes the Jacobson radical of $\Lambda$. Our results are stated in the more general setting of graded rings. In [@dk], conditions are found implying the full cohomology rings are isomorphic and [@pos] contains results that are related to ours.
To properly summarize the contents of this paper, we introduce some definitions and notation. Let $G$ be a group and let $\Lambda=\oplus_{g\in G}\Lambda_{g}$ be a $G$-graded ring; that is, if $g,h\in G$, then $\Lambda_g\cdot \Lambda_{h}
\subseteq \Lambda_{gh}$. We denote the identity of $G$ by $\mathfrak{e}$, the graded Jacobson radical of $\Lambda$ by ${{\bf r}}$ and set ${{\bf r}}_{\mathfrak e}=\Lambda_{\mathfrak e}\cap {{\bf r}}$. A $G$-grading on $\Lambda$ will be called a *proper $G$-grading* when it satisfies the following conditions: if $g\ne \mathfrak e$ then $\Lambda_g\cdot
\Lambda_{g^{-1}}\subseteq {{\bf r}}_{\mathfrak e}$ and $\Lambda_{\mathfrak e}/{{\bf r}}_{\mathfrak e}$ is a semisimple Artin algebra over a commutative Artin ring $C$.
For a proper $G$ grading it is easy to see that that ${{\bf r}}={{\bf r}}_{\mathfrak e}\oplus(\oplus_{g\in G}\Lambda_g)$ and that ${{\bf r}}_{\mathfrak e}$ is the intersection of the left maximal graded ideals in $\Lambda_{\mathfrak e}$ that contain $\sum_{g\in G\setminus\{\mathfrak e\}}\Lambda_g\cdot \Lambda_{g^{-1}}$. We view $\Lambda$ as a $G$-graded $\Lambda$-module with $\Lambda_g$ living in degree $g$.
We also fix the following notation: ${\operatorname{\mathbf{Mod}}\nolimits}(\Lambda)$ will denote the category of left $\Lambda$-modules, ${\operatorname{\mathbf{Gr}}\nolimits}(\Lambda)$ the category of graded $\Lambda$-modules, and ${\operatorname{\mathbf{gr}}\nolimits}(\Lambda)$ the full subcategory of ${\operatorname{\mathbf{Gr}}\nolimits}(\Lambda)$ consisting of the finitely generated graded modules. Given a $\Lambda$ module $X$, we let ${\operatorname{pd}\nolimits}_{\Lambda}(X)$ and ${\operatorname{id}\nolimits}_{\Lambda}(X)$ denote the projective dimension and the injective dimension of $X$ over $\Lambda$ respectively. If $${{\mathcal P}}:\cdots \to P^2{\xrightarrow}{\delta^2}P^1{\xrightarrow}{\delta^1}P^1{\xrightarrow}{\delta^1}P^0\to X\to
0$$ is a graded projective $\Lambda$-resolution of a graded module $X$, then ${{\mathcal P}}^{>c}$ will denote the resolution of the $\Omega^n(X) ={\mbox{Im}}(P^n\to P^{n-1})$, obtained from ${{\mathcal P}}$. The main comparison is Theorem \[grG-to-ss-thm\] which we state below, omitting some technicalities.
Let $G$ be a group and $\Lambda=\oplus_{g\in G}\Lambda_g$ be a properly $G$-graded ring. Suppose that $e$ is an idempotent in $\Lambda$, $f=1-e$ and set $\Lambda^*$ be the ring $f\Lambda
f$ and ${{\bf r}}^*=f{{\bf r}}f$. Assume that ${\operatorname{pd}\nolimits}_{\Lambda^*}(f\Lambda e)=c<\infty$, ${\operatorname{pd}\nolimits}_{\Lambda}((\Lambda/{{\bf r}}) e)=a<\infty$, and that ${\operatorname{id}\nolimits}_{\Lambda}((\Lambda/{{\bf r}}) e)=b<\infty$. Then, for $n>\max\{a,b+c+2\}$, there are isomorphisms ${\operatorname{Ext}\nolimits}^n_{\Lambda}(\Lambda/{{\bf r}},\Lambda/{{\bf r}})\cong
{\operatorname{Ext}\nolimits}^n_{\Lambda^*}(\Lambda^*/{{\bf r}}^*,\Lambda^*/{{\bf r}}^*)$ such that the induced isomorphism $$\bigoplus_{n>\max\{a,b+c+2\}}{\operatorname{Ext}\nolimits}^n_{\Lambda}(\Lambda/{{\bf r}},\Lambda/{{\bf r}})\cong
\bigoplus_{n>\max\{a,b+c+2\}}{\operatorname{Ext}\nolimits}^n_{\Lambda^*}(\Lambda^*/{{\bf r}}^*,\Lambda^*/{{\bf r}}^*)$$
We also obtain the following applications; see Theorem \[bigthm\]. To simplify notation, we write $E(\Lambda)$ for the cohomology ring $\oplus_{n\ge 0}{\operatorname{Ext}\nolimits}^n_{\Lambda}(\Lambda/{{\bf r}},\Lambda/{{\bf r}}) $.
Keeping the hypotheses of the above Theorem, the following hold.
1. Assume that $f\Lambda e$ has a finitely generated minimal graded projective $\Lambda^*$-resolution. The cohomology ring $E(\Lambda)$ is finitely generated over${\operatorname{Ext}\nolimits}^0_{\Lambda}(\Lambda/{{\bf r}},\Lambda/{{\bf r}})\cong {\mbox{Hom}}_{\Lambda}(\Lambda/{{\bf r}},\Lambda/{{\bf r}})
\cong (\Lambda/{{\bf r}})^{op}$ if and only if the cohomology ring $E(\Lambda^*)$ is finitely generated as a $(\Lambda^*/{{\bf r}}^*)^{op}$-algebra.
2. Assume that $\Lambda$ is $K$-algebra, where $K$ is a field and that $\Lambda/{{\bf r}}$ is a finite dimensional $K$-algebra. Assume further that both $E(\Lambda)$ and $E(\Lambda^*)$ are finitely generated $K$-algebras. Then ${\operatorname{GKdim}\nolimits}(E(\Lambda))
={\operatorname{GKdim}\nolimits}(E(\Lambda^*))$.
3. We have that ${\operatorname{pd}\nolimits}_{\Lambda}(S)<\infty$, for all graded simple $\Lambda$-modules $S$ if and only if ${\operatorname{pd}\nolimits}_{\Lambda^*}(S^*)<\infty$, for all graded simple $\Lambda^*$-modules $S^*$.
Comparison theorem {#main result}
==================
Let $G$ be a group and let $\Lambda=\oplus_{g\in G}\Lambda_{g}$ be a $G$-graded ring; in particular, if $g,h\in G$, then $\Lambda_g\cdot \Lambda_{h}
\subseteq \Lambda_{gh}$. We denote the identity of $G$ by $\mathfrak{e}$, the graded Jacobson radical of $\Lambda$ by ${{\bf r}}$ and set ${{\bf r}}_{\mathfrak e}=\Lambda_{\mathfrak e}\cap {{\bf r}}$. We assume throughout this section that if $g\ne \mathfrak e$ then $\Lambda_g\cdot
\Lambda_{g^{-1}}\subseteq {{\bf r}}_{\mathfrak e}$ and that $\Lambda_{\mathfrak e}/{{\bf r}}_{\mathfrak e}$ is a semisimple Artin algebra over a commutative Artin ring $C$. We call such a $G$-grading on $\Lambda$ a *proper $G$-grading*. It is easy to see that ${{\bf r}}={{\bf r}}_{\mathfrak e}\oplus(\oplus_{g\in G}\Lambda_g)$ and that ${{\bf r}}_{\mathfrak e}$ is the intersection of the left maximal [graded]{} ideals in $\Lambda_{\mathfrak e}$ that contain $\sum_{g\in G\setminus\{\mathfrak e\}}\Lambda_g\cdot \Lambda_{g^{-1}}$. We view $\Lambda$ as a $G$-graded $\Lambda$-module with $\Lambda_g$ living in degree $g$.
We denote the category of $G$-graded $\Lambda$-modules and degree $\mathfrak e$ maps by ${\operatorname{\mathbf{Gr}}\nolimits}(\Lambda)$. We let ${\operatorname{\mathbf{gr}}\nolimits}(\Lambda)$ denote the full subcategory of finitely generated $G$-graded $\Lambda$-modules and let $\Lambda=\oplus_{g\in G}\Lambda_g$ be a properly $G$-graded ring. The shifts by elements of $G$ induce a group of endofunctors on ${\operatorname{\mathbf{Gr}}\nolimits}(\Lambda)$. More precisely, the shift functor associated to an element $h\in G$ is defined as follows: if $X=\oplus_{g\in G}X_g$ is a graded $\Lambda$-module, we let $X[h]=\oplus_{g\in G}Y_g$, where $Y_g=X_{hg}$. Let $\Phi\colon {\operatorname{\mathbf{Gr}}\nolimits}(\Lambda)\to {\operatorname{\mathbf{Mod}}\nolimits}(\Lambda)$ denote the forgetful functor.
If $X\in {\operatorname{\mathbf{gr}}\nolimits}(\Lambda)$ and $Y$ is a graded $\Lambda$-module then $${\mbox{Hom}}_{\Lambda}(\Phi(X),\Phi(Y))={\mbox{Hom}}_{{\operatorname{\mathbf{Gr}}\nolimits}(\Lambda)}(X,\oplus_{g\in G}Y[g])\cong
\bigoplus_{g\in G}{\mbox{Hom}}_{{\operatorname{\mathbf{Gr}}\nolimits}(\Lambda)}(X,Y[g]).$$
We need one further assumption; namely, if $\overline \epsilon$ is an idempotent element in $\Lambda_{\mathfrak e}/{{\bf r}}_{\mathfrak e}$, then there is an idempotent $\epsilon\in
\Lambda_{\mathfrak e}$ such that $\pi(\epsilon)=\overline{\epsilon}$, where $\pi\colon \Lambda_{\mathfrak e}
\to \Lambda_{\mathfrak e}/{{\bf r}}_{\mathfrak e}$ is the canonical surjection. If a graded ring $\Lambda$ has this property, we say *graded idempotents lift*. Assume that graded idempotents lift in $\Lambda$. It follows that if $S$ is simple graded $\Lambda$-module, then $S\cong (\Lambda_{\mathfrak e}/{{\bf r}}_{\mathfrak e})
\epsilon[g]$, for some primitive idempotent $\epsilon\in \Lambda_{\mathfrak e}$ and some $g\in G$. We also see that the canonical surjection $\Lambda \epsilon[g]\to (\Lambda_{\mathfrak e}/{{\bf r}}_{\mathfrak e})\epsilon[g]$ is a projective cover. The next three examples provide important classes of graded rings satisfying our assumptions. 2
\[Gfinite\]
Let $K$ be a field, ${{\mathcal Q}}$ a finite quiver, $G$ a group, and $W\colon {{\mathcal Q}}_1\to G\setminus\{\mathfrak e\}$. We call $W$ a *weight function*; see [@gr]. Setting $W(v)=\mathfrak e$ for all vertices $v$ in ${{\mathcal Q}}$, and, if $p=a_1\cdots a_n$ is a path of length $n\ge 1$, with the $a_i\in{{\mathcal Q}}_1$, then set $W(p)=W(a_n)W(a_{n-1})\cdots W(a_1)$. In this case, we say $p$ has *weight $W(p)$*. We $G$-grade the path algebra $K{{\mathcal Q}}$ by defining $K{{\mathcal Q}}_g$ to be the $K$-span of paths $p$ of weight $g$. Let $I$ be an ideal in $K{{\mathcal Q}}$ such that $I$ can be generated by elements $x_i$, such that, for each $i$, the paths occuring in $x_i$ are all of length at least 2 and all have the same weight. Let $\Lambda= K{{\mathcal Q}}/I$. The $G$-grading on $K{{\mathcal Q}}$ induces a $G$-grading on $\Lambda$.
Note that if $a\in{{\mathcal Q}}_1$ with $W(a)=g$, then $a+I$ is a nonzero element in $\Lambda_g$. Using that $g\ne\mathfrak e$, one can show that ${{\bf r}}$ is the ideal generated by $\{a+I\mid
a\in{{\mathcal Q}}_1\}$. It follows that $\Lambda/{{\bf r}}$ is the semisimple ring $\prod_{v\in{{\mathcal Q}}_0}K$, which is a semisimple Artin algebra over $K$. Furthermore, one may check that ${{\bf r}}_{\mathfrak e}$ is the ideal in $\Lambda_{\mathfrak e}$ generated by the elements of the form $p+I$, where $p$ is a path of length $\ge 1$ in ${{\mathcal Q}}$ of weight $\mathfrak e$. Thus the $G$-grading on $\Lambda$ is a proper $G$-grading. It is also clear that graded idempotents lift.
\[pos-Z\]
Let $G=\mathbb Z$ and let $\Lambda =\Lambda_0\oplus \Lambda_1\oplus \Lambda _2\oplus \cdots$ be a positively $\mathbb Z$-graded ring such that $\Lambda_0$ is an Artin algebra. It is immediate that $\Lambda$ is a properly $\mathbb Z$-graded ring in which graded idempotents lift.
\[artin-alg\]
Let $\Lambda$ be an Artin algebra over a commutative Artin ring $C$. Let $G$ be any group and $\Lambda_{\mathfrak e}=\Lambda$, and, for $g\in G\setminus\{\mathfrak e\}$, $\Lambda_g=0$. We see that $\Lambda$, as a $G$-graded ring, is properly $G$-graded and graded idempotents lift. One choice for $G$ is the trivial group $\{\mathfrak e\}$.
We recall some known results about graded projective resolutions over properly graded rings in which graded idempotents lift. We leave the proof to the reader.
\[proj-cov\] Let $\Gamma=\oplus_{g\in G}\Gamma_g$ be a properly $G$-graded ring in which graded idempotents lift and let ${{\bf r}}_{\Gamma}$ denote the graded Jacobson radical of $\Gamma$. Suppose $X$ is a finitely generated graded $\Gamma$-module and $X/{{\bf r}}_{\Gamma}X\cong\oplus_{i=1}^nS_i$, where each $S_i$ is a graded simple $\Gamma$-module. Let $P_i{\xrightarrow}{\alpha_i}S_i$ be graded projective covers for each $i$ and let $P=\oplus_{i=1}^nP_i$. Then
1. For each $i=1,\dots,n$, $P_i\cong \Gamma \epsilon_i[g]$, for some idempotent $\epsilon_i\in \Gamma$ and $g\in G$.
2. The map $P{\xrightarrow}{\oplus_{i=1}^n\alpha_i}\oplus_{i=1}^nS_i$ is a graded projective cover.
3. If $P{\xrightarrow}{\beta}X$ is a graded map such that the following diagram commutes $$\xymatrix{ X\ar^{\pi\phantom{xxx}}[r]&X/{{\bf r}}_{\Gamma}X\\
P\ar^{\beta}[u]\ar^{\oplus_i\alpha_i\phantom{xx}}[r]&\oplus_{i=1}^nS_i,\ar@{=}[u],
}$$ where $\pi$ is the canonical surjection, then $\beta\colon P\to X$ is a graded projective cover. Moreover, $\ker(\beta)\subseteq
{{\bf r}}_{\Gamma}P$.
4. If $0\to K{\xrightarrow}{\sigma} P{\xrightarrow}{\beta} X\to 0$ is a short exact sequence in ${\operatorname{\mathbf{Gr}}\nolimits}(\Gamma)$ with $P$ finitely generated, such that $\sigma(K)\subseteq {{\bf r}}_{\Gamma}P$, then $\beta$ is a graded projective cover.
5. Suppose that $${{\mathcal P}}:\cdots \to P^2{\xrightarrow}{\delta^2}P^1{\xrightarrow}{\delta^1}P^1{\xrightarrow}{\delta^1}P^0\to X\to
0$$ is a graded projective $\Gamma$-resolution of $X$ with each $P^n$ finitely generated. Then ${{\mathcal P}}$ is minimal if and only if, for $n\ge 1$, $\delta^n(P^n)\subseteq {{\bf r}}_{\Gamma}P^{n-1}$.
6. If $P$ and $Q$ are finitely generated graded projective $\Gamma$-modules and $\alpha\colon P \to Q$ is a map in ${\operatorname{\mathbf{gr}}\nolimits}(\Gamma)$, then there are primitive idempotents $\epsilon_i$ and $\epsilon'_{j}$ and elements $g_i$ and $h_i$ of $G$, $i=1,\dots,m$ and $j=1,\dots,n$, for some integers $m$ and $n$ such that
1. $P\cong\oplus_{i=1}^m\Gamma \epsilon_i[g_i]$,
2. $Q\cong\oplus_{j=1}^n\Gamma \epsilon_j[h_j]$, and
3. viewing (a) and (b) as identifications, $\alpha$ is given by an $m\times n$ matrix $(\gamma_{i,j})$, where $\gamma_{i,j}\in
\epsilon_i\Gamma_{h_jg_i^{-1}}\epsilon_j$.
7. Keeping the notation and assumptions of part (5), we see that ${{\mathcal P}}$ is a minimal graded projective resolution of $X$ if and only if the matrices that give the $\delta^n$, $n\ge 0$, all have entries in ${{\bf r}}_{\Gamma}$.
8. The forgetful functor $\Phi$ is exact, preserves direct sums, and, if $Y$ is a graded $\Gamma$-module, $\Phi(Y)$ is a projective $\Gamma$-module if and only if $Y$ is a graded projective $\Gamma$-module. Thus, $\Phi$ takes graded projective $\Gamma$-resolutions to projective $\Gamma$-resolutions.
Let $e$ be an idempotent in $\Lambda_{\mathfrak e}$. We say that $(e,f)$ is a *suitable idempotent pair* if $f=1-e$ and $f\Lambda e\subseteq
{{\bf r}}$. Note that if $(e,f)$ is a suitable idempotent pair, then, since $e$ and $1$ are homogeneous of degree $\mathfrak e$, so is $f$. Furthermore, if $(e,f)$ is a suitable idempotent pair, then ${\mbox{Hom}}_{\Lambda}(
(\Lambda/{{\bf r}})e,(\Lambda/{{\bf r}})f)= {\mbox{Hom}}_{\Lambda}(
(\Lambda/{{\bf r}})f,(\Lambda/{{\bf r}})e)= 0$. Note that if $(e,f)$ is a suitable idempotent pair, then $(f,e)$ is also a suitable idempotent pair..
For the remainder of this section, we fix a suitable idempotent pair $(e,f)$. Let $\Lambda^*=f\Lambda f$ and ${{\bf r}}^*=f{{\bf r}}f$. The $G$-grading of $\Lambda$ induces a $G$-grading on $\Lambda^*$ and it is not hard to show that ${{\bf r}}^*$ is the graded Jacobson radical of $\Lambda^*$.
The main tool in this section is the functor $F\colon {\operatorname{\mathbf{Gr}}\nolimits}(\Lambda)\to {\operatorname{\mathbf{Gr}}\nolimits}(\Lambda^*)$ given by $F(X)=f\Lambda\otimes_{\Lambda}X$. Let $H\colon {\operatorname{\mathbf{Gr}}\nolimits}(\Lambda^*)\to {\operatorname{\mathbf{Gr}}\nolimits}(\Lambda)$ be given by $H(X)={\mbox{Hom}}_{\Lambda^*}(f\Lambda ,X)$. Note that both $F(X)$ and $H(X)$ have induced $G$-gradings obtained from the gradings of $X$ and $\Lambda f$. The next result is well-known.
\[FGbasic\] Keeping the above notation, we have that
1. the functor $F$ is exact,
2. $(F,H)$ is an adjoint pair, and
3. $f\Lambda\cong \Lambda^*\oplus f\Lambda e$, as left $\Lambda^*$-modules.
The functor $H$ is exact if and only if $f\Lambda$ is a left projective $\Lambda^*$-module, and, by Proposition \[FGbasic\](3), $H$ is exact if and only if $f\Lambda e$ is a left projective $\Lambda^*$-module. Note that $F(\Lambda e)\cong f\Lambda e$ does not, in general, have finite projective dimension as a left $\Lambda^*$-module, as the example below demonstrates.
[Let ${{\mathcal Q}}$ be the quiver $$\xymatrix{
\stackrel{u}{\circ}\ar_a[r]&\stackrel{v}{\circ}\ar@(r,u)[]^b
}$$ Let $I$ be the ideal generated by $ba$ and $b^2$ and let $\Lambda={{\mathcal Q}}/I$. Taking $e=u$ and $f=v$, we see that $f\Lambda e$ has infinite projective dimension viewed as a left $\Lambda^*$-module where $\Lambda^*=f\Lambda f$. ]{}
We note that if $X$ is a graded $\Lambda$-module, then $F(\Phi(X))\cong \Phi(F(X))$ and if ${{\mathcal P}}:\cdots \to P^2{\xrightarrow}{\delta^2}P^1{\xrightarrow}{\delta^1}P^0\to X\to
0$ is a graded projective resolution with syzygies $\Omega^n_{\Lambda}(X)$ then $\Phi({{\mathcal P}})$ is projective resolution of $\Phi(X)$, $$F(\Phi(\Omega^n_{\Lambda}(X))\cong \Phi(\Omega^n_{\Lambda}(F(X)),$$ where $\Omega^n_{\Lambda}(F(X))$ denotes the $n$-th syzygy of $F({{\mathcal P}})$.
We abuse notation by denoting the forgetful functor from ${\operatorname{\mathbf{Gr}}\nolimits}(\Lambda^*)$ to $-(\Lambda^*)$ also by $\Phi$. We also use $F$ to denote the functor $f\Lambda \otimes_{\Lambda}-$ from ${\operatorname{\mathbf{Mod}}\nolimits}(\Lambda)$ to ${\operatorname{\mathbf{Mod}}\nolimits}(\Lambda^*)$. The meaning of both $F$ and $\Phi$ will be clear from the context.
The next result is quite general and will allow us to apply the functor $F$ and keep control of the cohomology if ${\operatorname{pd}\nolimits}_{\Lambda^*}(f\Lambda e)<\infty$. One does not need that the $G$-grading is proper.
\[grapplyF\] Let $G$ be a group with identity element $\mathfrak e$ and $\Lambda$ be a $G$-graded ring and let $(e,f)$ be a suitable idempotent pair in $\Lambda$. Set $\Lambda^*=f\Lambda f$. Suppose that ${\operatorname{pd}\nolimits}_{\Lambda^*}(f\Lambda e)=c<\infty$. Let $X$ be a graded left $\Lambda$-module and $\Omega^i_{\Lambda}(X)$ (respectively, $\Omega^i_{\Lambda^*}(F(X))$) denote the $i$-th sygyzy of $X$ (resp., $F(X)$) in a graded projective $\Lambda$-resolution of $X$ (resp., a graded projective $\Lambda^*$-resolution of $F(X)$). Then, for $t>c+1$ and $n\ge 0$, $${\operatorname{Ext}\nolimits}_{\Lambda^*}^t(\Phi(F(\Omega^n_{\Lambda}(X))),-)\cong
{\operatorname{Ext}\nolimits}_{\Lambda^*}^t(\Phi(\Omega^n_{\Lambda^*}(F(X))),-).$$
For $n=0$ the result is clear and hence we assume $n\ge 1$. Without loss of generality, we may start with a graded $\Lambda$-resolution of $X$ in which each graded projective module is a direct sum of copies of graded projective modules of the form $\Lambda[g]$, for $g\in G$. Since $1=e+f$, this resolution has the form: $$\cdots\to P^2\oplus Q^2\to P^1\oplus Q^1\to P^0\oplus Q^0\to X\to
0,$$ where $P^i$ is a direct sum of copies of graded modules of the form $\Lambda f[g]$ and $Q^i$ is a direct sum of copies of graded modules of the form $\Lambda e[g]$, for $i\ge 0$. Setting $F(P^i)=L^i$ and $F(Q^i)=M^i$, we note that $L^i$ is a graded projective $\Lambda^*$-module and $M^i$ is a direct sum of copies of graded modules of the form $(f\Lambda e)[g]$. Applying the exact functor $F$ to the resolution above, we obtain an exact sequence of graded $\Lambda^*$-modules $$\cdots\to L^2\oplus M^2\to L^1\oplus M^1\to L^0\oplus M^0\to F(X)\to
0.$$ For $i\ge 1$, note $F(\Omega^i_{\Lambda}(X))={\mbox{Im}}(L^i\oplus
M^i\to L^{i-1}\oplus M^{i-1})$ and $L^i$ is a graded left projective $\Lambda^*$-module. For ease of notation, we let $Z_i=F(\Omega^i_{\Lambda}(X))$, for $i\ge 1$ and $Z_0=F(X)$.
For $n\ge 1$, we have a short exact sequence of graded $\Lambda^*$-modules $$0\to Z_n\to L^{n-1}\oplus M^{n-1}\to Z_{n-1}\to 0.$$
Let $ P(M^{n-1})\to M^{n-1}\to 0$ be exact sequence of graded $\Lambda^*$-modules with $ P(M^{n-1})$ a graded projective module. Then we obtain the following exact commutative diagram: $$\xymatrix{ &0&0\\
0\ar[r]&Z_n\ar[u]\ar[r]&L^{n-1}\oplus
M^{n-1}\ar[u]\ar[r]\ar[u]&Z_{n-1}\ar[r] &0\\
0\ar[r]&\Omega^1_{\Lambda^*}(Z_{n-1})\ar[u]\ar[r]&L^{n-1}\oplus
P(M^{n-1})\ar[u]\ar[r]&Z_{n-1}\ar[r]\ar[u]^= &0\\
0\ar[r]&\Omega^1_{\Lambda^*}(M^{n-1})\ar[u]\ar[r]^= &
\Omega^1_{\Lambda^*}(M^{n-1})\ar[u]\\
&0\ar[u]&0\ar[u]
}$$ The first column yields the short exact sequence $$0\to
\Omega^1_{\Lambda^*}(M^{n-1}) \to \Omega^1_{\Lambda^*}(Z_{n-1})\to
Z_n\to 0.$$ Taking graded projective $\Lambda^*$-resolutions of the two end modules, applying the Horseshoe lemma, and taking syzygies, we obtain short exact sequences $$\quad 0\to \Omega^{j+1}_{\Lambda^*}(M^{n-1}) \to
\Omega^{j+1}_{\Lambda^*}(Z_{n-1})\to \Omega_{\Lambda^*}^j(Z_n)\to
0,$$ for $j\ge 0$. Hence we obtain short exact sequences of $\Lambda^*$-modules $$\quad 0\to \Phi( \Omega^{j+1}_{\Lambda^*}(M^{n-1})) \to
\Phi(\Omega^{j+1}_{\Lambda^*}(Z_{n-1}))\to \Phi( \Omega_{\Lambda^*}^j(Z_n))\to
0.$$
Note that $\Phi(\Omega^{j+1}_{\Lambda^*}(M^{n-1})))$ a projective $\Lambda^*$-module if $j\ge c$ since $c\ge{\operatorname{pd}\nolimits}_{\Lambda^*}(\Phi(M^{n-1}))$. It follows that, for $j\ge c$ and $t\ge 2$, $${\operatorname{Ext}\nolimits}^{t+j}_{\Lambda^*}(\Phi(Z_n),-)\cong
{\operatorname{Ext}\nolimits}^{t}_{\Lambda^*}(\Phi(\Omega_{\Lambda^*}^j(Z_n)),-)\cong
{\operatorname{Ext}\nolimits}^{t}_{\Lambda^*}(\Phi(\Omega_{\Lambda^*}^{j+1}(Z_{n-1})),-) \cong$$$$\cong{\operatorname{Ext}\nolimits}^{t}_{\Lambda^*}(\Phi(\Omega_{\Lambda^*}^{j+2}(Z_{n-2})),-)\cong
\cdots\cong {\operatorname{Ext}\nolimits}^{t}_{\Lambda^*}(\Phi(\Omega_{\Lambda^*}^{j+n}(Z_{0})),-)\cong$$$$\cong {\operatorname{Ext}\nolimits}^{t +j}_{\Lambda^*}(\Phi(\Omega_{\Lambda^*}^{n}(Z_{0})),-) .$$ Finally, we note that $Z_n=F(\Omega^n_{\Lambda}(X))$ and $Z_0= F(X)$ and the result follows.
The next result is immediate and we only provide a sketch of the proof.
\[grExt-G-S\]Let $G$ be a group and $\Lambda$ a properly $G$-graded ring with graded Jacobson radical ${{\bf r}}$ and suitable idempotent pair $(e,f)$. We have that ${\operatorname{id}\nolimits}_{\Lambda}((\Lambda/{{\bf r}})e)\le a<\infty$ if and only if, for every graded $\Lambda$-module $X$, $$\bigoplus_{n>a}{\operatorname{Ext}\nolimits}^n_{\Lambda}(\Phi(X),(\Lambda/{{\bf r}})f)
\cong
\bigoplus_{n>a}{\operatorname{Ext}\nolimits}^n_{\Lambda}(\Phi(X),\Lambda/{{\bf r}})$$ as $\mathbb Z\times G$-graded modules over the $\mathbb Z\times G$-graded ring $\oplus{\operatorname{Ext}\nolimits}^n_{\Lambda}(\Lambda/{{\bf r}},\Lambda/{{\bf r}})$. Furthermore, ${\operatorname{pd}\nolimits}_{\Lambda}((\Lambda/{{\bf r}})e)\le a<\infty$, ${\operatorname{id}\nolimits}_{\Lambda}((\Lambda/{{\bf r}})e)\le a<\infty$ if and only if $$\bigoplus_{n>a}{\operatorname{Ext}\nolimits}^n_{\Lambda}((\Lambda/{{\bf r}})f,(\Lambda/{{\bf r}})f)
\cong
\bigoplus_{n>a}{\operatorname{Ext}\nolimits}^n_{\Lambda}(\Lambda/{{\bf r}},\Lambda/{{\bf r}})$$ as $\mathbb Z\times G$-graded rings without identity.
Since $\Lambda/{{\bf r}}=\Lambda_0\cong \Lambda_0e\oplus \Lambda_0f$, $${\operatorname{Ext}\nolimits}^i_{\Lambda}(X,\Lambda/{{\bf r}})=
{\operatorname{Ext}\nolimits}^i_{\Lambda}(X,(\Lambda/{{\bf r}})e)\oplus{\operatorname{Ext}\nolimits}^i_{\Lambda}(X,(\Lambda/{{\bf r}})f)$$ and
$${\operatorname{Ext}\nolimits}^i_{\Lambda}(\Lambda_0,\Lambda_0)=
{\operatorname{Ext}\nolimits}^i_{\Lambda}(\Lambda_0e,\Lambda_0e)\oplus{\operatorname{Ext}\nolimits}^i_{\Lambda}(\Lambda_0e,\Lambda_0f)\oplus{\operatorname{Ext}\nolimits}^i_{\Lambda}(\Lambda_0f,\Lambda_0e)
\oplus{\operatorname{Ext}\nolimits}^i_{\Lambda}(\Lambda_0f,\Lambda_0f)$$ the result follows.
If $X$ is a graded $\Lambda$-module and ${{\mathcal P}}:\cdots \to P^2{\xrightarrow}{\delta^2}P^1{\xrightarrow}{\delta^1}P^1{\xrightarrow}{\delta^1}P^0\to X\to
0$ is a graded projective $\Lambda$-resolution of $X$, then we say that *${{\mathcal P}}$ is finitely generated* if $P^n$ is a finitely generated graded $\Lambda$-module for $n\ge 0$.
Let $\epsilon$ be an idempotent element of $\Lambda_{\mathfrak e}$. We say a graded simple module $S$ *belongs to $\epsilon$* if $\epsilon S\ne 0$. Equivalently, $S$ belongs to $\epsilon$ if $S$ is isomorphic to a summand of $(\Lambda/{{\bf r}})\epsilon [g]$, for some $g\in G$. We say a graded projective $\Lambda$-module $P$ *belongs to $\epsilon$*, if $P/{{\bf r}}P$ is a direct sum of graded simple $\Lambda$-modules with each summand belonging to $\epsilon$. We now state a useful result.
\[grFresol\] Let $X$ be a graded $\Lambda$-module and assume that ${{\mathcal P}}^{\bullet}:\cdots\stackrel{d^3}{\to} P^2\stackrel{d^2}{\to}
P^1\stackrel{d^1}{\to} P^0\stackrel{d^0}{\to}X\to 0$ is a graded projective $\Lambda$-resolution of $X$ such that, for $n>c$, $P^n$ belongs to $f$. Then
1. $F({{\mathcal P}}^{>c})$ is a projective $\Lambda^*$-resolution of $F(\Omega^{c+1}X)$, where $(\Omega^{c+1}X)$ is $(c+1)$-st syzygy in ${{\mathcal P}}^{\bullet}$.
2. If ${{\mathcal P}}^{\bullet}$ is a finitely generated minimal graded projective $\Lambda$-resolution of $X$, then $F({{\mathcal P}}^{\ge c+1})$ is a finitely generated minimal graded projective $\Lambda^*$-resolution of $F(\Omega^{c+1}X)$.
The functor $F$ is exact. We need to show that if $P$ belongs to $f$, then $F(P)$ is a projective $\Lambda^*$-module. Since $P$ belongs to $f$, $P$ is a direct sum of indecomposable projective modules, each of which is a summand of $(\Lambda f)[g]$, for some $g\in G$. Thus, it suffices to show that, for $g\in G$, $F((\Lambda f)[g])$ is a graded projective $\Lambda^*$-module. But $F((\Lambda f)[g])=
(f\Lambda\otimes_{\Lambda}\Lambda f)[g]\cong (f\Lambda f)[g]=\Lambda^*[g]$ and part (1) follows.
By minimality and our assumptions, the maps $F(d^i)$, viewed as matrices (as in Proposition \[proj-cov\]), have entries in $f{{\bf r}}f$. But $f{{\bf r}}f={{\bf r}}^*$, the graded Jacobson radical of $\Lambda^*$, and (2) follows.
The following is an immediate consequence of the above Lemma.
\[grF-to-resol\] Assume that ${\operatorname{id}\nolimits}_{\Lambda}((\Lambda/{{\bf r}})e)=b<\infty$. Suppose that $X$ is a graded $\Lambda$-module and let $${{\mathcal P}}^{\bullet}:\cdots{\to} P^2\stackrel{d^2}{\to}
P^1\stackrel{d^1}{\to} P^0\stackrel{d^0}{\to}X\to 0$$ be a minimal graded projective $\Lambda$-resolution of $X$. Then, for $n>b$, $P^n$ belongs to $f$ and $F({{\mathcal P}}^{>b})$ is a minimal graded projective $\Lambda^*$-resolution of $F(\Omega^{b+1}_{\Lambda}(X))$.
Let $n>b$ and consider $P^n$. If there is an indecomposable summand of $ P^n$ belonging to $e$, then ${\operatorname{Ext}\nolimits}^n_{\Lambda}(X,(\Lambda/{{\bf r}})e)\ne 0$, contradicting ${\operatorname{id}\nolimits}_{\Lambda}((\Lambda/{{\bf r}})e)=b$. Hence, $P^n$ belongs to $f$ and the result follows.
Using the above result we have one of the main results of this section.
\[grfirst-half\] Let $G$ be a group and $\Lambda=\oplus_{g\in G}\Lambda_g$ be a properly $G$-graded ring in which graded idempotents lift. Let ${{\bf r}}$ denote the graded Jacobson radical of $\Lambda$ and $(e,f)$ be a suitable idempotent pair. Set $\Lambda^*$ be the ring $f\Lambda
f$ and ${{\bf r}}^*=f{{\bf r}}f$. Assume that ${\operatorname{pd}\nolimits}_{\Lambda^*}f\Lambda e=c<\infty$, and that ${\operatorname{id}\nolimits}_{\Lambda}(\Lambda/{{\bf r}}e)=b<\infty$. Then, for a graded $\Lambda$-module $X$ having finitely generated projective resolutions and for $n>b+c+2$, the functor $F=f\Lambda\otimes_{\Lambda}-\colon{\operatorname{\mathbf{Gr}}\nolimits}(\Lambda)\to
{\operatorname{\mathbf{Gr}}\nolimits}(\Lambda^*)$ induces isomorphisms $${\operatorname{Ext}\nolimits}^n_{\Lambda}(\Phi(X),(\Lambda/{{\bf r}})f)\cong
{\operatorname{Ext}\nolimits}^n_{\Lambda^*}(\Phi(F(X)),\Lambda^*/{{\bf r}}^*).$$ Moreover, assuming that every graded simple $\Lambda$-module belonging to $f$ has a finitely generated minimal graded projective resolution, then the induced isomorphism $$\bigoplus_{n>
b+c+2}{\operatorname{Ext}\nolimits}^n_{\Lambda}((\Lambda/{{\bf r}}) f,(\Lambda/{{\bf r}})f)\cong
\bigoplus_{n>
b+c+2}{\operatorname{Ext}\nolimits}^n_{\Lambda^*}(\Lambda^*/{{\bf r}}^*,\Lambda^*/{{\bf r}}^*)$$ is an isomorphism of $\mathbb Z\times G$-graded rings without identity. Furthermore, identifying $\oplus_{n>
b+c+2}{\operatorname{Ext}\nolimits}^n_{\Lambda}((\Lambda/{{\bf r}}) f,(\Lambda/{{\bf r}})f)$ and $\oplus_{n>
b+c+2}{\operatorname{Ext}\nolimits}^n_{\Lambda^*}(\Lambda^*/{{\bf r}}^*,\Lambda^*/{{\bf r}}^*)$ and denoting this ring by $\Delta$, $\oplus_{n>b+c+2}{\operatorname{Ext}\nolimits}^n_{\Lambda}(\Phi(X),(\Lambda/{{\bf r}})f)$ and $\oplus_{n>b+c+2}{\operatorname{Ext}\nolimits}^n_{\Lambda^*}(\Phi(F(X)),\Lambda^*/{{\bf r}}^*)$ are isomorphic as graded $\Delta$-modules.
Let $X$ be a graded $\Lambda$-module and let $${{\mathcal P}}^{\bullet}:\cdots{\to} P^2\stackrel{\delta^2}{\to}
P^1\stackrel{\delta^1}{\to} P^0\stackrel{\delta^0}{\to}X\to 0$$ be a minimal graded projective $\Lambda$-resolution of the graded module $X$. By our assumption that $id_{\Lambda}((\Lambda/{{\bf r}})e)=b$, for $n>b$, $P^n$ belongs to $f$. Hence, applying the functor $F$ and Proposition \[grF-to-resol\], we see that $$F({{\mathcal P}}^{\ge n}):\cdots{\to} F(P^{n+2})\stackrel{F(\delta^{n+2})}{\to}
F(P^{n+1})\stackrel{F(\delta^{n+1})}{\to}
F(P^n)\stackrel{F(\delta^n)}{\to}F(\Omega^n(X))\to 0$$ is a minimal graded projective $\Lambda^*$-resolution of $F(\Omega^n(S))$. By Theorem \[grapplyF\], $$(**)\quad\quad
{\operatorname{Ext}\nolimits}_{\Lambda^*}^t(\Phi(F(\Omega^n_{\Lambda}(X))),-)\cong
{\operatorname{Ext}\nolimits}_{\Lambda^*}^t(\Omega^n_{\Lambda^*}(\Phi(F(X))),-),$$ for $t>c$. By (\*\*) , dimension shift, and since $\Phi$ commutes with $F$ and $\Omega$, $${\operatorname{Ext}\nolimits}_{\Lambda^*}^t(\Phi(F(\Omega^n_{\Lambda}(X))),-)\cong
{\operatorname{Ext}\nolimits}_{\Lambda^*}^{t+n}(\Phi(F(X)),-).$$ Let $S$ be a simple graded $\Lambda$-module belonging to $f$ and let $S^*=F(S)$. First we show that, using the above isomorphisms, if $n>b+c+2$, then $F$ induces a monomorphism $${\operatorname{Ext}\nolimits}^n_{\Lambda}(\Phi(X),\Phi(S))\to
{\operatorname{Ext}\nolimits}^n_{\Lambda^*}(\Phi(F(X)),\Phi(S^*)).$$ We recall that ${\operatorname{Ext}\nolimits}^n_{\Lambda}(\Phi(X),\Phi(S)) \cong {\operatorname{Ext}\nolimits}^n_{{\operatorname{\mathbf{Gr}}\nolimits}(\Lambda)}(X,\oplus_{g\in G}S[g])$ and that ${\operatorname{Ext}\nolimits}^n_{\Lambda^*}(\Phi(F(X)),\Phi(S^*)\cong{\operatorname{Ext}\nolimits}^n_{{\operatorname{\mathbf{Gr}}\nolimits}(\Lambda)}(F(X),\oplus_{g\in G}S^*[g])$. Suppose $\alpha:P^n\to S[g]$ represents a nonzero element in ${\operatorname{Ext}\nolimits}^n_{{\operatorname{\mathbf{Gr}}\nolimits}(\Lambda)}(X,S[g])$. It is now easy to see that $F(\alpha)\colon F(P^n)\to S^*[g]$ is nonzero. By minimality of $F({{\mathcal P}}^{\bullet})$ from $b+1$ on, $F(\alpha)$ represents a nonzero element of ${\operatorname{Ext}\nolimits}^n_{{\operatorname{\mathbf{Gr}}\nolimits}(\Lambda)}(F(X),S^*[g])$.
Having shown that if $n>b+c+2$, then $F$ induces an monomorphism $${\operatorname{Ext}\nolimits}^n_{\Lambda}(X,S)\to
{\operatorname{Ext}\nolimits}^n_{\Lambda^*}(F(X),S^*),$$ we now show that $F$ induces an epimorphism. Since $${\operatorname{Ext}\nolimits}^n_{\Lambda}(\Phi(X),\Phi(S))\cong {\mbox{Hom}}_{\Lambda}(\Phi(P^n),\Phi(S))\cong {\mbox{Hom}}_{\Lambda_0}(\Phi(P^n/{{\bf r}}P^n),\Phi(S))$$ and $${\operatorname{Ext}\nolimits}^n_{\Lambda^*}(\Phi(F(X)),\Phi(S^*))\cong {\mbox{Hom}}_{\Lambda^*}(\Phi(F(P^n)),\Phi(S^*))$$$$\cong {\mbox{Hom}}_{\Lambda^*_0}(\Phi(F(P^n)/{{\bf r}}^* F( P^n)),\Phi(S^*)),$$ we conclude that the lengths of ${\operatorname{Ext}\nolimits}^n_{\Lambda}(\Phi(X),\Phi(S))$ and ${\operatorname{Ext}\nolimits}^n_{\Lambda^*}(\Phi(F(X),\Phi(S^*))$ are equal as modules over the commutative Artin ring $C$, over which both $\Lambda/{{\bf r}}$ and $\Lambda^*/{{\bf r}}^*$ are both finite length modules. Since $F$ induces a monomorphism, we conclude that $F$ is an isomorphism.
By taking direct sums over simple modules belonging to $f$, the isomorphisms ${\operatorname{Ext}\nolimits}^n_{\Lambda}(\Phi(X),\Phi(S))\to
{\operatorname{Ext}\nolimits}^n_{\Lambda^*}(\Phi(F(X)),\Phi(S^*))$ induced by $F$ extends to an isomorphism $${\operatorname{Ext}\nolimits}^n_{\Lambda}(X,\Lambda/{{\bf r}}f)\cong
{\operatorname{Ext}\nolimits}^n_{\Lambda^*}(F(X),\Lambda^*/{{\bf r}}^*),$$ Taking $X=(\Lambda/{{\bf r}})f$ we obtain the isomorphism $${\operatorname{Ext}\nolimits}^n_{\Lambda}((\Lambda/{{\bf r}})f,(\Lambda/{{\bf r}}) f)\cong
{\operatorname{Ext}\nolimits}^n_{\Lambda^*}(\Lambda^*/{{\bf r}}^*,\Lambda^*/{{\bf r}}^*),$$ Since $F$ is an exact functor, the induced isomorphism $$\bigoplus_{n\ge b+c+2}{\operatorname{Ext}\nolimits}^n_{\Lambda}(\Lambda/{{\bf r}}f,\Lambda/{{\bf r}}f)\cong
\bigoplus_{n\ge b+c+2}{\operatorname{Ext}\nolimits}^n_{\Lambda^*}(\Lambda^*/{{\bf r}}^*,\Lambda^*/{{\bf r}}^*),$$ is an isomorphism of $\mathbb Z\times G$-graded rings (without identity) the assertion about $\oplus_{n>b+c+2}{\operatorname{Ext}\nolimits}^n_{\Lambda}((X,(\Lambda/{{\bf r}}) f)\cong
\oplus_{n>b+c+2}{\operatorname{Ext}\nolimits}^n_{\Lambda^*}(F(X),\Lambda^*/{{\bf r}}^*)$ being a graded module isomorphism follows.
We have the following consequence of the above proof.
\[pd-simple\] Keeping the notation and hypothesis of Theorem \[grfirst-half\], let $X$ be a graded $\Lambda$-module. Then ${\operatorname{pd}\nolimits}_{\Lambda}(\Phi(X)) <\infty$ if and only if ${\operatorname{pd}\nolimits}_{\Lambda^*}(\Phi(F(X)) <\infty$.
From the proof of Theorem \[grfirst-half\], we see that for every graded simple $\Lambda$-module $S$ belonging to $f$, $${\operatorname{Ext}\nolimits}_{\Lambda}^n(\Phi(X),\Phi(S))\cong {\operatorname{Ext}\nolimits}_{\Lambda^*}^n(\Phi(F(X)),\Phi(F(S))),$$ for $n>b +c+2$. But by our assumption on finitely generated resolutions, and that ${\operatorname{id}\nolimits}_{\Lambda}(\Lambda e)=b$, we see that ${\operatorname{Ext}\nolimits}_{\Lambda}^n(\Phi(X),-)=0$ if and only if ${\operatorname{Ext}\nolimits}_{\Lambda}^n(\Phi(X),\Phi(S)) =0$ for all graded simple modules $S$ belonging to $f$. Finally, ${\operatorname{Ext}\nolimits}_{\Lambda}^n(\Phi(X),-)=0$ if and only if ${\operatorname{pd}\nolimits}_{\Lambda}(X)\le n-1$.
By combining Proposition \[grExt-G-S\] and Theorem \[grfirst-half\], we obtain the desired result.
\[grG-to-ss-thm\] Let $G$ be a group with identity element $\mathfrak e$ and $\Lambda=\oplus_{g\in G}\Lambda_g$ be a properly $G$-graded ring in which graded idempotents lift. Assume that every graded simple $\Lambda$-module has a finitely generated minimal graded projective $\Lambda$-resolution. Let ${{\bf r}}$ denote the graded Jacobson radical of $\Lambda$. Suppose that $(e,f)$ is a suitable idempotent pair in $\Lambda_{\mathfrak e}$ and set $\Lambda^*$ be the ring $f\Lambda
f$ and ${{\bf r}}^*=f{{\bf r}}f$. Assume that ${\operatorname{pd}\nolimits}_{\Lambda^*}(f\Lambda e)=c<\infty$, ${\operatorname{pd}\nolimits}_{\Lambda}((\Lambda/{{\bf r}}) e)=a<\infty$, and that ${\operatorname{id}\nolimits}_{\Lambda}((\Lambda/{{\bf r}}) e)=b<\infty$. Then, for $n>\max\{a,b+c+2\}$, there are isomorphisms ${\operatorname{Ext}\nolimits}^n_{\Lambda}(\Lambda/{{\bf r}},\Lambda/{{\bf r}})\cong
{\operatorname{Ext}\nolimits}^n_{\Lambda^*}(\Lambda^*/{{\bf r}}^*,\Lambda^*/{{\bf r}}^*)$ such that the induced isomorphism $$\bigoplus_{n>\max\{a,b+c+2\}}{\operatorname{Ext}\nolimits}^n_{\Lambda}(\Lambda/{{\bf r}},\Lambda/{{\bf r}})\cong
\bigoplus_{n>\max\{a,b+c+2\}}{\operatorname{Ext}\nolimits}^n_{\Lambda^*}(\Lambda^*/{{\bf r}}^*,\Lambda^*/{{\bf r}}^*)$$
Letting $\Delta = \oplus_{n>\max\{a,b+c+2\}}{\operatorname{Ext}\nolimits}^n_{\Lambda}(\Lambda/{{\bf r}},\Lambda/{{\bf r}})$, if $X$ is a graded $\Lambda$-module having a finitely generated projective resolution, then $$\bigoplus_{n>\max\{a,b+c+2\}}{\operatorname{Ext}\nolimits}^n_{\Lambda}(\Phi(X),\Lambda/{{\bf r}})\text{ and }
\bigoplus_{n>\max\{a,b+c+2\}}{\operatorname{Ext}\nolimits}^n_{\Lambda^*}(\Phi(F(X)),\Lambda^*/{{\bf r}}^*)$$ are isomorphic as graded $\Delta$-modules.
Applications
============
We begin this section with a well-known result whose proof we include for completeness.
\[large-gen\]Let $R=R_0\oplus R_1\oplus
R_2\oplus\cdots$ be a finitely generated positively $\mathbb Z$-graded $C$-algebra where $C$ is a commutative ring. Let $N$ be a fixed positive integer. Then there is a positive integer $D$ with $N< D$ such that the following holds.
1. If $j> D$ and $r\in R_j$, then $r=\sum_ic_iu_{i,1}u_{i,2}\cdots u_{i,m_i}$, where $c_i\in C$ and each $u_{i,k}\in
R_{\ell}$, with $N\le \ell< D$.
Assume that $R$ can be generated over $C$ by homogeneous elements $x_1,\dots,x_n$ with each $x_i$ having degree at least $0$ and $L=\max\{\deg x_i\mid 1\le i\le n\}$. Set $D=2LN$ and suppose $r\in R_j$ with $j>D$. Then, by finite generation, $r=\sum_ic_iy_{i,1}\cdots y_{i,t_i}$ where, for all $i,k$, $c_i\in C$, $y_{i,k}\in\{x_1,\dots,x_n\}$ and $\sum_{k=1}^{t_i}\deg(y_{i,k})=j$, for each $i$. Fix $i$ and write $y_{j}$ instead of $y_{i,j}$ and set $t=t_i$. We see that $$D=2NL<j=\sum_{k=1}^{t}\deg(y_{k})\le Lt.$$ Hence $2N<t$. Write $t=AN+S$, where $0\le S<N$. For $i=1,\dots,A-1$, set $u_i=y_{(i-1)N+1}y_{(i-1)N+2}\cdots y_{iN}$ and $u_A=y_{(A-1)N+1} \cdots y_{AN}\cdot y_{AN+1}\cdots y_{t}$. It is immediate that for $1\le i\le A$, $N\le \deg(u_i)< 2NL=D$. This completes the proof.
We have some immediate consequences.
\[eventual\]Let $R=R_0\oplus R_1\oplus
R_2\oplus\cdots$ be a positively $\mathbb Z$-graded ring such that, $R_0$ is an Artin algebra over a commutative Artin ring $C$, and, for $i\ge 0$, $R_i$ has finite length over $R_0$. Let $N$ be a fixed positive integer. Then $R$ is finitely generated as ring over $C$ if and only if $T=\oplus_{i\ge N}R_i$ is finitely generated as a ring (without identity) over $C$.
Note that $R_0\oplus R_1\oplus \cdots\oplus R_{N-1}$ is of finite length over $C$. If $T$ is finitely generated over $C$, adding a finite numbers generators of $R_0\oplus R_1\oplus \cdots\oplus R_{N-1}$ over $C$ to a set of generators $T$ yields a finite generating set for $R$.
If $R$ is finitely generated as a ring over $C$, the proof of Proposition \[large-gen\] implies that $T$ is finitely generated, by taking as a generating set, all products of the form $$y_1y_2\cdots y_t,$$ where the $y_i$’s are elements of a finite generating set of $R$ and $N\le t <2NL$, with $L$ being the maximum degree of the generators of $R$.
Before stating the main theorem of the section, we consider low terms in resolutions of simple $\Lambda$- and $\Lambda^*$-modules. More precisely, suppose that $G$ is a group and that $\Lambda$ is a properly $G$-graded ring in which graded idempotents lift. Let $(e, f)$ be a suitable idempotent pair in $\Lambda$ and let $\Lambda^*=f\Lambda f$, ${{\bf r}}$ and ${{\bf r}}^*$ the graded Jacobson radicals of $\Lambda$ and $\Lambda^*$ respectively. Assume all the conditions of Theorem \[grG-to-ss-thm\]. Let $S$ be a graded simple $\Lambda$-module and $S^*=f\Lambda\otimes_{\Lambda}S$, viewed as a graded $\Lambda^*$-module. Example \[ex41\] shows that even if $S$ has a finitely generated graded projective $\Lambda$-resolution, $S^*$ need not have a finitely generated graded projective $\Lambda^*$-resolution. To remedy this situation, we have the following result and its collorary.
\[gen-fp\] Let $G$ be a group and $R=\oplus_{g\in G}R_g$ be a $G$-graded ring. Let $\cdots \to X^2{\xrightarrow}{d^2}X^1{\xrightarrow}{d^1}X^0{\xrightarrow}{d^0}M\to 0$ be an exact sequence of graded $R$-modules. If, for all $n\ge 0$, $X^n$ has a finitely generated graded projective $R$-resolution, then $M$ has a finitely generated graded projective $R$-resolution.
For $j\ge 0$, let $X^{0,j}= X^j$ and $Y^{0,j}= {\mbox{Im}}(d^j)$. Note that $Y^{0,0}=M$. For each $j\ge 0$, let $$\cdots \to P^{2,j}{\xrightarrow}{\delta^{2,j}}P^{1,j}{\xrightarrow}{\delta^{1,j}}P^{0,j}{\xrightarrow}{\delta^{0,j}
}X^{0,j}\to 0$$ be a finitely generated graded projective $R$-resolution of $X^{0,j}$. For $i\ge 0$, define $X^{i,j}= {\mbox{Im}}(\delta^{i,j})$. Thus, for each $i\ge 0$, we have short exact sequences $$0\to X^{i+1,j}\to P^{i,j}\to X^{i,j}\to 0.$$
We inductively construct graded $R$-modules $Y^{i,j}$ and finitely generated graded projective $R$-modules $Q^{i,j}$ such that
1. for each $i,j\ge 0$, there is a short exact sequence $0\to Y^{i+1,j}\to Q^{i,j}\to Y^{i,j}\to 0$,
2. for $i\ge 0$ and $j\ge 1$, there is a short exact sequence $0\to X^{i+1,j-1}\to Y^{i+1,j-1}\to Y^{i,j}\to 0$ and,
3. for $i\ge 1$ and $j\ge 0$, $Q^{i,j}= Q^{i-1,j+1}\oplus P^{i,j}$.
Once this is accomplished, splicing together the short exact sequences $0\to Y^{i+1,0}\to Q^{i,0}\to Y^{i,0}\to 0$ we obtain a long exact sequence $$\cdots\to Q^{2,0}\to Q^{1,0}\to Q^{0,0}\to Y^{0,0}\to 0.$$ But $Y^{0,0}=M$ and the result follows.
We have defined $Y^{0,s}$ and $P^{0,s}$ for all $s\ge 0$. Set $Q^{0,i}=P^{0,i}$, for all $i\ge 0$. We have exact sequences $$0\to Y^{0,s+1}\to X^{0,s}\to Y^{0,s}\to 0,$$ for all $s\ge $. We also have exact sequences $0\to X^{1,s}\to P^{0,s}\to X^{0,s}\to 0$ for all $s\ge 0$. From these exact sequences we obtain the following commutative diagram.
$$\xymatrix{
&&0&0\\
0\ar[r]&Y^{0,s+1}\ar[r]&X^{0,s}\ar[u]\ar[r]&Y^{0,s}\ar[r]\ar[u]&0\\
&0\ar[r]&P^{0,s}\ar@{=}[r]\ar[u]& P^{0,s}\ar[r]\ar[u]&0\\
&0\ar[r]&X^{1,s}\ar[r]\ar[u]& Y^{1,s}\ar[r]\ar[u]& Y^{0,s+1}\ar[r]&0\\
&&0\ar[u]&0\ar[u],
}$$ where $Y^{1,s}$ is defined to be the kernel of the surjection $P^{0,s}$ to $Y^{0,s}$. Thus, we have defined $Y^{1,j}$ such that (1) holds for all $i=0$ and $j\ge 0$ and (2) holds for all $j\ge 1$ and $i=0$. For $i=0$, (3) vacuously holds.
Now consider $0\to X^{1,s}\to Y^{1,s}\to Y^{0,s+1}\to 0$ Using the exact sequences $0\to Y^{2,s}\to Q^{1,s}\to Y^{1,s}\to 0$ and $0\to X^{1,s+1}\to P^{0,s+1}\to X^{0,s+1}\to 0$ and using the Horseshoe Lemma, we obtain the following commutative diagram $$\xymatrix{
&0&0&0\\
0\ar[r]&X^{1,s}\ar[u]\ar[r]&Y^{1,s}\ar[r]\ar[u]&Y^{0,s+1}\ar[u]\ar[r]&0\\
0\ar[r]&P^{1,s}\ar[r]\ar[u]& P^{1,s}\oplus Q^{0,s+1}\ar[r]\ar[u]&Q^{0,s+1}\ar[r]\ar[u]&0\\
0\ar[r]&X^{2,s}\ar[r]\ar[u]& Y^{2,s}\ar[r]\ar[u]& Y^{1,s+1}\ar[u]\ar[r]&0\\
&0\ar[u]&0\ar[u]&0\ar[u],
}$$ where $Y^{2,s}$ is the kernel of $ P^{1,s}\oplus Q^{0,s+1}\to Y^{1,s}$. Let $Q^{1,s}= P^{1,s}\oplus Q^{0,s+1}$. It is immediate that (1) holds for all $j\ge 0$, (2) holds for all $j\ge 1$ and $i=1$, and (3) holds for $i=1$ and all $j\ge 0$.
Continuing in this fashion, we define the $X^{i,j}$ and $P^{i,j}$ for all $i,j\ge 0$ satisfying (1), (2), and (3).
\[fg-resol\] Let $G$ be a group with identity element $\mathfrak e$ and $\Lambda=\oplus_{g\in G}\Lambda_g$ be a properly $G$-graded ring in which graded idempotents lift. Suppose that $(e,f)$ is suitable idempotent pair and set $\Lambda^*$ be the ring $f\Lambda
f$. Assume that, as a left $\Lambda^*$-module, $f\Lambda e$ has a finitely generated graded projective resolution. Let $M$ be a graded $\Lambda$-module having a finitely generated graded projective $\Lambda$-resolution. Then $F(M)$ has a finitely generated graded projective $\Lambda^*$-resolution.
Let $M$ be a graded $\Lambda$-module and let ${{\mathcal P}}: \cdots\to P^1\to P^0\to M\to 0$ be a finitely generated graded projective $\Lambda$-resolution of $M$. Applying the exact functor $F$, we get an exact sequence graded $\Lambda^*$-modules $F({{\mathcal P}}): \cdots\to F(P^1)\to F(P^0)\to F(M)\to 0$. The result will follow if we show that each $F(P^n)$ has a finitely generated graded projective $\Lambda^*$-resolution. For each $n\ge 0$, set $P^n=P^n_e\oplus P^n_f$, where $P^n_e$ belongs to $e$ and $P^n_f$ belongs to $f$. By our hypothesis, $F(P^n_e)$ has a finitely generated graded projective $\Lambda^*$-resolution. Since $F(P^n_f)$ is a graded projective $\Lambda^*$ module and since $F(P^n)=F(P^n_e)
\oplus F(P^n_f)$ we are done.
We can state the main theorem of this section. If $\Lambda$ is a ring, then let ${\operatorname{GKdim}\nolimits}(\Lambda)$ denote the Gelfand-Krillov dimension of $\Lambda$ and ${\operatorname{gl.dim}\nolimits}(\Lambda)$ denote the (left) global dimension of $\Lambda$.
\[bigthm\] Let $G$ be a group with identity element $\mathfrak e$ and $\Lambda=\oplus_{g\in G}\Lambda_g$ be a properly $G$-graded ring in which graded idempotents lift. Assume that every graded simple $\Lambda$-module has a finitely generated minimal graded projective $\Lambda$-resolution. Let ${{\bf r}}$ denote the graded Jacobson radical of $\Lambda$. Suppose that $(e,f)$ is a suitable idempotent pair and set $\Lambda^*$ be the ring $f\Lambda
f$ and ${{\bf r}}^*=f{{\bf r}}f$. Suppose that ${\operatorname{pd}\nolimits}_{\Lambda^*}(f\Lambda e)<\infty$, ${\operatorname{pd}\nolimits}_{\Lambda}((\Lambda/{{\bf r}}) e)<\infty$, and that ${\operatorname{id}\nolimits}_{\Lambda}((\Lambda/{{\bf r}}) e)<\infty$. Then the following hold.
1. Assume that $f\Lambda e$ has a finitely generated minimal graded projective $\Lambda^*$-resolution. The cohomology ring $E(\Lambda)$ is finitely generated over${\operatorname{Ext}\nolimits}^0_{\Lambda}(\Lambda/{{\bf r}},\Lambda/{{\bf r}})\cong {\mbox{Hom}}_{\Lambda}(\Lambda/{{\bf r}},\Lambda/{{\bf r}})
\cong (\Lambda/{{\bf r}})^{op}$ if and only if the cohomology ring $E(\Lambda^*)$ is finitely generated as a $(\Lambda^*/{{\bf r}}^*)^{op}$-algebra.
2. Assume that $\Lambda$ is $K$-algebra, where $K$ is a field and that $\Lambda/{{\bf r}}$ is a finite dimensional $K$-algebra. Assume further that both $E(\Lambda)$ and $E(\Lambda^*)$ are finitely generated $K$-algebras. Then ${\operatorname{GKdim}\nolimits}(E(\Lambda))
={\operatorname{GKdim}\nolimits}(E(\Lambda^*))$.
3. We have that ${\operatorname{pd}\nolimits}_{\Lambda}(S)<\infty$, for all graded simple $\Lambda$-modules $S$ if and only if ${\operatorname{pd}\nolimits}_{\Lambda^*}(S^*)<\infty$, for all graded simple $\Lambda^*$-modules $S^*$.
Suppose $C$ is a commutative Artin algebra over which $\Lambda/{{\bf r}}$ has finite length. Note that if $S^*$ is a graded simple $\Lambda^*$-module, then there exists a graded simple $\Lambda$-module $S$ such that $S^*\cong F(S)$. By Corollary \[fg-resol\] and our assumptions, it follows every graded simple $\Lambda^*$-module has a finitely generated graded projective $\Lambda^*$-resolution. In particular, for $n\ge 0$, ${\operatorname{Ext}\nolimits}_{\Lambda^*}^n(
\Lambda^*/{{\bf r}}^*,\Lambda^*/{{\bf r}}^*)$ has finite length over $C$. Part (1) follows from Theorems \[grfirst-half\] and \[grG-to-ss-thm\], Proposition \[large-gen\], and Corollary \[eventual\]. Part (2) follows from the definition of Gelfand-Kirillov dimension and Theorem \[grG-to-ss-thm\]. For Part (3), the proof is basically given in the proof of Proposition \[pd-simple\].
Applying these results the Artin algebra case we get the following Corollary.
\[bigcor\] Let $\Lambda$ be an Artin algebra. Let ${{\bf r}}$ denote the graded Jacobson radical of $\Lambda$. Suppose that $(e,f)$ is a suitable idempotent pair in $\Lambda$ and set $\Lambda^*$ be the ring $f\Lambda
f$ and ${{\bf r}}^*=f{{\bf r}}f$. Suppose that ${\operatorname{pd}\nolimits}_{\Lambda^*}(f\Lambda e)<\infty$, ${\operatorname{pd}\nolimits}_{\Lambda}((\Lambda/{{\bf r}}) e)<\infty$, and that ${\operatorname{id}\nolimits}_{\Lambda}((\Lambda/{{\bf r}}) e)<\infty$. Then the following hold.
1. The cohomology ring $E(\Lambda)$ is finitely generated over ${\operatorname{Ext}\nolimits}^0_{\Lambda}(\Lambda/{{\bf r}},\Lambda/{{\bf r}})\cong {\mbox{Hom}}_{\Lambda}(\Lambda/{{\bf r}},\Lambda/{{\bf r}})
\cong (\Lambda/{{\bf r}})^{op}$ if and only if the cohomology ring $E(\Lambda^*)$ is finitely generated as a $(\Lambda^*/{{\bf r}}^*)^{op}$-algebra.
2. Assume that $\Lambda$ is a finite dimensional $K$-algebra, where $K$ is a field. Assume further that both $E(\Lambda)$ and $E(\Lambda^*)$ are finitely generated rings. Then ${\operatorname{GKdim}\nolimits}(E(\Lambda))
={\operatorname{GKdim}\nolimits}(E(\Lambda^*)$.
3. We have that ${\operatorname{gl.dim}\nolimits}(\Lambda)$ is finite if and only if ${\operatorname{gl.dim}\nolimits}(\Lambda^*)$ is finite.
We take $G$ to be the trivial group and view $\Lambda$ as a graded algebra. Then the grading is proper and graded idempotents lift. Every (graded) simple $\Lambda$-module and (graded) simple $\Lambda^*$-module has a finitely generated projective resolution, as does $f\Lambda e$. The result is now a direct consequence of Theorem \[bigthm\].
Concluding remarks and examples {#rmk-ex}
===============================
We begin this section with a discussion of the construction of $\Lambda^*=f\Lambda f$ in case $\Lambda$ is a quotient of a path algebra. We keep the notation of Example \[Gfinite\]; namely, let $K$ be a field, ${{\mathcal Q}}$ be a finite quiver, $G$ a group, $W\colon {{\mathcal Q}}_1\to G\setminus
\{\mathfrak e\}$ be a weight function, and $I$ a graded ideal in the path algebra $K{{\mathcal Q}}$ generated by weight homogeneous elements. We also assume that $I$ is contained in the ideal of $K{{\mathcal Q}}$ generated by the arrows of ${{\mathcal Q}}$. Setting $\Lambda=K{{\mathcal Q}}/I$, the $G$-grading on $K{{\mathcal Q}}$ obtained from $W$ induces a proper $G$-grading on $\Lambda$ such that graded idempotents lift.
To simplify notation, if $x\in K{{\mathcal Q}}$, then we denote the element $x +I$ of $\Lambda$ by $\overline{x}$. We wish to describe $\Lambda^*=f\Lambda f$, where $f =\sum_{v\in X}\overline{v}$ and $X$ is a subset of the vertex set ${{\mathcal Q}}$. Set $e=\sum_{v\in {{\mathcal Q}}_0\setminus X}\overline{v}$. We keep the notation that ${{\bf r}}$ is the graded Jacobson radical of $\Lambda$ and ${{\bf r}}^*=f{{\bf r}}f$. Note that ${{\bf r}}$ is generated all elements of the form $\overline a$, for $a\in{{\mathcal Q}}_1$, $\Lambda^*$ has an $G$-grading induced from the $G$-grading on $\Lambda$, and ${{\bf r}}^*$ is the graded Jacobson radical of $\Lambda^*$. Furthermore, $\Lambda/{{\bf r}}\cong \prod_{v\in {{\mathcal Q}}_0}K$ and $\Lambda^*/{{\bf r}}^*\cong \prod_{v\in X}K$
We define the quiver ${{\mathcal Q}}^*$ as follows. Let ${{\mathcal Q}}^*_0=X$. To define the set of arrows ${{\mathcal Q}}^*_1$ consider set of paths ${{\mathcal M}}$ in ${{\mathcal Q}}$ such that $p\in {{\mathcal M}}$ if $p$ is a path of length $n$, $n\ge 1$ in ${{\mathcal Q}}$ such that $p=u_1\stackrel{a_1}{\to}u_2\stackrel{a_2}{\to}u_3\to\cdots\to
u_{n}\stackrel{a_{n}}{\to}u_{n+1}$, with $u_i$ belonging to $e$ for $i=2,\dots,n$ and $u_1$ and $u_{n+1}$ belonging to $f$. Note that a vertex $u$ belongs to $f$ (resp., to $e$) just means $u\in X$ (resp., $u\not\in X$). Then $
{{\mathcal Q}}^*_1=\{a_p \mid p\in {{\mathcal M}}, p \text{ is a path from } u_1\text{ to }
u_{n+1}\}$. A path $p\in {{\mathcal M}}$ is called a *minimal $f$-path* and the arrow $a_p$ in ${{\mathcal Q}}^*$ is called the *arrow in ${{\mathcal Q}}^*$ associated to minimal $f$-path $p$*. We note that if $a\colon u\to v$ is an arrow with $u$ and $v$ belonging to $f$, then $a$ is a minimal $f$-path. It is also easy to see that if $p$ is a path in $Q$ from vertex $u$ to vertex $v$ with $u$ and $v$ belonging to $f$, then $p$ can be uniquely written as a product of paths $p_1\cdots p_m$, where each $p_i$ is a minimal $f$-path.
We now turn our attention to relations. Let $I^*$ be the ideal in $K{{\mathcal Q}}^*$ generated as follows. If $r\in I$ is an element with $r=urv$, where $u$ and $v$ are vertices belonging to $f$ and $r=\sum_ic_ip_i$, where $c_i\in K$ and $p_i$ is a path from $u$ to $v$, then we set $r^*$ to be $\sum_ic_ip_i^*$ where $p_i^*$ is the path in ${{\mathcal Q}}^*$ obtained from $p_i$ by replacing each minimal $f$-subpath $p$ in $p_i$ by $a_p$. Note that if a minimal $f$ is in $I$, then associated arrow is in $I^*$. We also note that although ${{\mathcal Q}}$ is a finite quiver and ${{\mathcal Q}}^*_0$ is a finite set, ${{\mathcal Q}}^*$ may have an infinite number of arrows. The next example demonstrates this and that even if $I$ is an ideal in $K{{\mathcal Q}}$, finitely generated by homogeneous elements, $I^*$ need not be finitely generated.
\[ex41\]
Let ${{\mathcal Q}}$ be the quiver $$\xymatrix{
\stackrel{u}{\circ}\ar_a[r]&\stackrel{v}{\circ}\ar@(r,u)[]^b\ar_c[r]&\stackrel{w}{\circ}
}$$ Take $e=v$ and $f=u+w$. It is not hard to see that each path of the form $cb^na$, $n\ge 0$ is a minimal $f$-path and that these are the only minimal $f$-paths. Hence, ${{\mathcal Q}}^*$ is the quiver with two vertices $u$ and $w$, and a countable number of arrows $a_{ca}, a_{cba}, a_{cb^2a},\dots$, each starting at $u$ and ending at $w$.
Let $W\colon {{\mathcal Q}}_1\to \mathbb Z_{> 0}$ by $W(a)=W(b)=W(c)=1$ and $I$ be the ideal in $K{{\mathcal Q}}$ generated by $b^2$. Set $\Lambda=K{{\mathcal Q}}/I$ and $\Lambda^*=K{{\mathcal Q}}^*/I^*$. Then $\Lambda^*=f \Lambda f$ and $I^*=fIf$. We have $I^*$ is generated by $\{a_{cb^na}\mid n\ge 2\}$. Note that both $\Lambda$ and $\Lambda^*$ are Artin algebras. Note that ${\operatorname{pd}\nolimits}_{\Lambda}((\Lambda/{{\bf r}}) e)= {\operatorname{id}\nolimits}_{\Lambda}((\Lambda/{{\bf r}}) e)
=\infty$, where ${{\bf r}}$ is the graded Jacobson radical of $\Lambda$. Moreover, ${\operatorname{gl.dim}\nolimits}(\Lambda^*)=1$. This example shows that the finiteness of the projective or injective dimension of $(\Lambda/{{\bf r}})e$ is necessary in Colloray \[bigcor\].
If we take $I=(0)=I^*$ above, then both $\Lambda =K{{\mathcal Q}}$ and $\Lambda^*=K{{\mathcal Q}}^*$ are hereditary algebras. Hence Theorem \[grG-to-ss-thm\] holds; in fact, the ${\operatorname{Ext}\nolimits}^n$’s are 0 for $n\ge 2$. But ${\operatorname{Ext}\nolimits}^1_{\Lambda^*}(\Lambda^*/{{\bf r}}^*,\Lambda^*/{{\bf r}}^*)$ is infinite dimensional and hence Theorem \[bigthm\] fails. Note that $f\Lambda e$ does not have a finitely generated graded projective $\Lambda^*$-resolution.
We leave the proof of the following result to the reader.
\[newquiver\] Keeping the notation above, $\Lambda^*\cong K{{\mathcal Q}}^*/I^*$.
In the quiver case where $e$ is a idempotent associated to a vertex, the next result gives a sufficient condition for exactness of the functor $H$ where $H\colon {\operatorname{\mathbf{Gr}}\nolimits}(\Lambda^*)\to{\operatorname{\mathbf{Gr}}\nolimits}(\Lambda)$ given by $H(X)={\mbox{Hom}}_{\Lambda^*}(f\Lambda, X)$ (see Section \[main result\]).
\[Gexact\] Let $K{{\mathcal Q}}/I$ be a finite dimensional $K$-algebra where $K$ is field and $I$ is an admissible ideal in the path algebra $K{{\mathcal Q}}$; that is, for some $n\ge 2$, $J^n\subseteq I\subseteq
J^2$ where $J$ is the ideal generated by the arrows of ${{\mathcal Q}}$. Assume that $K{{\mathcal Q}}$ is $G$-graded with the grading coming from a weight function $W\colon {{\mathcal Q}}_1\to G$ and that $I$ can be generated by homogeneous elements. Let $e$ be an idempotent element of $K{{\mathcal Q}}$ associated to a vertex $v$. If ${\operatorname{pd}\nolimits}_{\Lambda}((\Lambda/{{\bf r}})e)\le 1$, then ${\operatorname{pd}\nolimits}_{\Lambda^*}(f\Lambda e)\le 1$ and $H$ is exact.
By [@ILP], there is no loop at $v$. Let $e$ be the idempotent in $K{{\mathcal Q}}$ associated to the vertex $v$ and let $f=1-e$. Consider the short exact sequence $0\to {{\bf r}}e\to \Lambda
e\to (\Lambda/{{\bf r}})e\to 0$. Applying the functor $F$, we obtain $$\xymatrix{
0\ar[r]&f\Lambda\otimes_{\Lambda}{{\bf r}}e\ar[r]\ar[d]^{\cong}&
f\Lambda\otimes_{\Lambda}\Lambda
e\ar[r]\ar[d]^{\cong}&f\Lambda\otimes_{\Lambda}(\Lambda/{{\bf r}})e\ar[r]\ar[d]^{\cong}&0\\
&f{{\bf r}}e&f\Lambda e&0 }$$ It follows that $f\Lambda e\cong f{{\bf r}}e$. Since ${\operatorname{pd}\nolimits}_{\Lambda}(\Lambda/{{\bf r}})e\le 1$, ${{\bf r}}e\cong \oplus \Lambda
w$ where the direct sum runs over the arrows $v\to w$ in ${{\mathcal Q}}$ and $w$ belongs to $f$, and where $\Lambda
w$ is the projective $\Lambda$-module associated to the vertex $w$. Since each $w$ belongs to $f$, it follows that $f\Lambda w =f\Lambda fw=\Lambda^*w$, which is a projective $\Lambda^*$-module. Thus $f\Lambda e)$ is a projective $\Lambda^*$-module and by the remark after Proposition \[FGbasic\], $H$ is exact.
Let $K{{\mathcal Q}}/I$ be a finite dimensional $K$-algebra where $K$ is field and $I$ is an admissible ideal in the path algebra $K{{\mathcal Q}}$. Assume that $K{{\mathcal Q}}$ is $G$-graded with the grading coming from a weight function $W\colon {{\mathcal Q}}_1\to G$ and that $I$ can be generated by homogeneous elements. Let $e$ be an idempotent element of $K{{\mathcal Q}}$ associated to a vertex $v$. As usual let $f=1-e$. It is well known that ${\operatorname{pd}\nolimits}_{\Lambda}(\Lambda/{{\bf r}})e\le 1$ if and only if there exists a uniform set $\rho$ of generators of $I$ such that $gv=0$ for all $g\in\rho$, where an element $r\in K{{\mathcal Q}}$ is *uniform* if there exist vertices $u$ and $v$ in ${{\mathcal Q}}$ such that $r=urv$. Thus, if there exists a uniform set $\rho$ of generators of $I$ such that $g v=0$ for all $g\in \rho$ and if ${\operatorname{id}\nolimits}_{\Lambda}(\Lambda/{{\bf r}}e)<\infty$ , then Theorems \[grG-to-ss-thm\] and \[bigthm\] hold.
The next result gives sufficient conditions so that $pd_{\Lambda^*}(f\Lambda e)<\infty$.
\[pd-finite\] Let $G$ be a group and $\Lambda=\oplus_{g\in G}\Lambda_g$ be a properly $G$-graded ring in which graded idempotents lift. Suppose that $(e,f)$ is a suitable idempotent pair and set $\Lambda^*$ be the ring $f\Lambda
f$. Suppose that $0\to P^n\to\cdots\to P^1\to P^0\to(\Lambda/{{\bf r}})e\to 0$ is a minimal graded projective $\Lambda$-resolution of $(\Lambda/{{\bf r}})e$ and that each $P^i$, for $i\ge 1$, is a direct sum of indecomposable projective $\Lambda$-modules of the form $\Lambda w$ with $w$ a vertex belonging to $f$. Then ${\operatorname{pd}\nolimits}_{\Lambda^*}(f\Lambda e)<\infty$.
Note that $f\Lambda\otimes_{\Lambda}(\Lambda/{{\bf r}})e=0$ and that $P^0\cong \Lambda e$; so that $f\Lambda\otimes_{\Lambda}\Lambda e \cong f\Lambda e$. We see that the result follows by tensoring the projective resolution of $(\Lambda/{{\bf r}})e$ with $f\Lambda\otimes_{\Lambda}-$.
We end with a nontrivial class of examples where the hypothesis of main theorems of the paper hold. Let $K$ be a field and $\Delta$ and $\Sigma$ be finite dimensional $K$-algebras. Suppose that $A$ is $K$-$\Sigma$-bimodule, $B$ is $\Delta$-$K$-bimodule and $C$ is a $\Delta$-$\Sigma$-bimodule. Let $$\Lambda =\left(\begin{array}{ccc}K&A&0\\
0&\Sigma&0\\
B&C&\Delta
\end{array}\right),$$ where the ring operations are given by matrix addition and multiplication. Set $e=\left(\begin{array}{ccc}1&0&0\\0&0&0\\0
&0&0\end{array}\right)$ and $f=\left(\begin{array}{ccc}0&0&0\\0&1&0\\0
&0&1\end{array}\right).$ Note that $f\Lambda f=
\left(\begin{array}{cc}
\Sigma&0\\
C&\Delta
\end{array}\right)$.
The reader may verify that if ${\operatorname{pd}\nolimits}_{\Delta}B<\infty$ and ${\operatorname{id}\nolimits}_{\Sigma}(A)<\infty$, then ${\operatorname{pd}\nolimits}_{\Lambda}(\Lambda/{{\bf r}})e <\infty$, ${\operatorname{id}\nolimits}_{\Lambda}(\Lambda/{{\bf r}})e <\infty$, and ${\operatorname{pd}\nolimits}_{f\Lambda f}(f\Lambda
e)<\infty$. Note that $C$ is an arbitrary finite dimensional bimodule. Thus if ${\operatorname{pd}\nolimits}_{\Delta}B<\infty$ and ${\operatorname{id}\nolimits}_{\Sigma}(A)<\infty$, then, taking $G=\{\mathfrak e\}$ to be the trivial group, Theorem \[grG-to-ss-thm\] and Corollary \[bigcor\] hold.
[99]{} Diracca, Luca; Koenig, Steffen *Cohomological reduction by split pairs*. J. Pure Appl. Algebra 212 (2008), no. 3, 471–485. Green, Edward L. *Graphs with relations, coverings and group-graded algebras*. Trans. Amer. Math. Soc. 279 (1983), no. 1, 297–310. Igusa, Kiyoshi; Liu, Shiping; Paquette, Charles *A proof of the strong no loop conjecture*. Adv. Math. 228 (2011), no. 5, 2731–2742. Psaroudakis, Chrysostomos; Skartsæterhagen, Øystein Ingmar; Solberg, Øyvind; *Gorenstein categories, singular equivalences and finite generation of cohomology rings in recollements*, Preprint 2014
[^1]: The third autor was partially supported by a research grant of CNPq-Brazil, (bolsa de pesquisa), and also from a tematic grand from Fapesp- São Paulo, Brazil\
This work was mostly done during some visits to Virginia Tech
|
---
abstract: 'Hindman proved that, whenever the set $\mathbb {N}$ of naturals is finitely colored, there must exist non-constant monochromatic solution of the equation $a+b=cd$. In this paper we extend this result for dense subsemigroups of $((0, \infty), +)$ to near zero.'
address:
- 'Department of Mathematics, Ramakrishna Mission Vidyamandira, Belur Math, Howrah-711202, West Bengal, India'
- 'Department of Mathematics, Maharaja Manindra Chandra College, 20, Ramkanto Bose Street, Kolkata-700 003, West Bengal, India'
author:
- Sourav Kanti Patra and Md Moid Shaikh
title: MONOCHROMATIC SUMS EQUAL TO PRODUCTS NEAR ZERO
---
introduction
============
In Ref. [@csikvari], P. Csikvári, A. Sárközy, and K. Gyarmati asked whether whenever the set $\mathbb {N}$ of naturals is finitely colored, there must exist monochromatic $a,b,c$ and $d$ with $a\neq b$ such that $a+b=cd$. In Ref. [@Hindman1], Hindman answered this question affirmatively by showing in addition that one can demand that $a,b, c, d$ are all distinct and the color of $a+b$ is that same as that of $a,b,c$ and $d$. In fact he proved considerably more stronger result using the algebraic structure of $\beta \mathbb {N}$, the Stone-$\breve{C}$ech compactification of $\mathbb {N}$.
Let $(S,\cdot)$ be an infinite discrete semigroup. Now the points of $\beta S$ are taken to be the ultrafilters on $S$, the principal ultrafilters being identified with the points of $S$. Given $A\subseteq S$, let us set $\bar{A}=\{p\in\beta S:A\in p\}$. Then the set $\{\bar{A}:A\subseteq S\}$ will become a basis for a topology on $\beta S$. The operation $\cdot$ on $S$ can be extended to the Stone-$\breve{C}$ech compactification $\beta S$ of $S$ so that $(\beta S,\cdot)$ is a compact right topological semigroup (meaning that for any $p\in \beta S$, the function $\rho_p:\beta S \rightarrow \beta S$ defined by $\rho_p(q)=q\cdot
p$ is continuous) with $S$ contained in its topological center (meaning that for any $x\in S$, the function $\lambda_x:\beta S
\rightarrow \beta S$ defined by $\lambda_x(q)=x\cdot q$ is continuous). Given $p,q\in\beta S$ and $A\subseteq S$, $A\in p\cdot
q$ if and only if $\{x\in
S:x^{-1}A\in q\}\in p$, where $x^{-1}A=\{y\in S:x\cdot y\in A\}$.\
A nonempty subset $I$ of a semigroup $(T,\cdot)$ is called a left ideal of $T$ if $T\cdot I\subseteq I$, a right ideal if $I.T\subseteq I$, and a two-sided ideal (or simply an ideal) if it is both a left and a right ideal. A minimal left ideal is a left ideal that does not contain any proper left ideal. Similarly, we can define minimal right ideal and smallest ideal. Any compact Hausdorff right topological semigroup $(T,\cdot)$ has the unique smallest two-sided ideal $$\begin{array}{ccc}
K(T) & = & \bigcup\{L:L \text{ is a minimal left ideal of } T\} \\
& = & \,\,\,\,\,\bigcup\{R:R \text{ is a minimal right ideal of } T\}\\
\end{array}$$
Given a minimal left ideal $L$ and a minimal right ideal $R$ of $T$, $L\cap R$ is a group, and in particular $K(T)$ contains an idempotent. An idempotent that belongs to $K(T)$ is called a minimal idempotent.\
In Ref. [@Hindman3], authors introduced different notations of largeness for arbitrary semigroup and studied their combinatorial properties. From now we use $\mathcal{P}_{f}(X)$ to denote the set of all finite subsets of a set $X$.
**Definition 1.1** [@Hindman3 Definition 3.1]: Let $(S,\cdot)$ be a semigroup.
\(a) A set $A\subseteq S$ is syndetic if only if there exists $G \in \mathcal{P}_{f}(S)$ with
$S\subseteq \bigcup_{t\in G} t^{-1}A$.
\(b) A set $A\subseteq S$ is piecewise syndetic if only if there exists $G \in \mathcal{P}_{f}(S)$ such that $\{y^{-1}(\bigcup_{t\in G} t^{-1}A) :y\in S\}$ has the finite intersection property.
\(c) A family $\mathcal{A}\subseteq \mathcal{P}(S)$ is collectionwise piecewise syndetic if and only if there exists a function $G: \mathcal{P}_{f}(\mathcal{A})\longrightarrow\mathcal{P}_{f}(S)$ such that
$\{y^{-1}(G(\mathcal{F}))^{-1}(\cap\mathcal{F}): y\in S$ and $\mathcal{F}\in \mathcal{P}_{f}(\mathcal{A}) \}$ has the finite intersection property.
**Theorem 1.2**: Let $(S,\cdot)$ be a semigroup and let $\mathcal{A}\subseteq \mathcal{P}(S)$. Then there exists $p\in K(\beta S)$ with $\mathcal{A}\subseteq p$ if and only if $\mathcal{A}$ is collectionwise piecewise syndetic. In particular, given $A\subseteq S$, $K(\beta S)\cap cl A\neq\O{}$ if and only if $A$ is piecewise syndetic.
**Proof**: See the proof [@Hindman3 Theorem 3.2].\
Furstenberg introduced a class of large sets in terms of notion from topological dynamics. There is a nice characterization of such sets in terms of algebraic structure of $\beta\mathbb{N}$.
We now recall the following definitions [@Hindman4 Definitions 4.42 and 15.3]
**Definition 1.3**: Let $(S,\cdot)$ be a semigroup and let $A\subseteq S$.
\(a) $A$ is called Central in $(S,\cdot)$ if there is some idempotent $p\in K(\beta S,\cdot) $ such that $A\in p$.
\(b) $A$ is called Central\* in $(S,\cdot)$ if $A\cap B \neq\O{}$ for every central set $B$ in $(S,\cdot)$.
In Ref. [@Bergelson1], authors investigated the interplay between additive and multiplicative largeness. We need the following definition [@Hindman1 Definition 1] to state such results.
**Definition 1.4**: Let $\langle{x_n}\rangle_{n=1}^{\infty}$ be an infinite sequence of positive real numbers, let $m\in\mathbb{N}$ and let $\langle{y_n}\rangle_{n=1}^{m}$be a finite sequence of positive real numbers. Then\
(a) $FS(\langle{x_n}\rangle_{n=1}^{\infty})=\{\sum_{n\in F}x_n:F\in\mathcal{P}_f(\mathbb N)\}$ and\
$FP(\langle{x_n}\rangle_{n=1}^{\infty})=\{\prod_{n\in F}x_n:F\in\mathcal{P}_f(\mathbb N)\}$.\
(b) $FS(\langle{y_n}\rangle_{n=1}^{m})=\{\sum_{n\in F}y_n:\O{}\neq F\subseteq\{1,2,.....,m\}\}$ and\
$FP(\langle{y_n}\rangle_{n=1}^{m})=\{\prod_{n\in F}y_n:\O{}\neq F\subseteq\{1,2,.....,m\}\}$.\
(c) The sequence $\langle{x_n}\rangle_{n=1}^{\infty}$ has distinct finite sums if and only if whenever
$F, G\in\mathcal{P}_f(\mathbb N)$ and $F\neq G$ , one has $\sum_{n\in F}x_n\neq \sum_{n\in G}x_n$. The analogous definition applies to $\langle{y_n}\rangle_{n=1}^{m}$.
\(d) The sequence $\langle{x_n}\rangle_{n=1}^{\infty}$ has distinct finite products if and only if whenever $F, G\in\mathcal{P}_f(\mathbb N)$ and $F\neq G$ , one has $\prod_{n\in F}x_n\neq \prod_{n\in G}x_n$. The analogous definition applies to $\langle{y_n}\rangle_{n=1}^{m}$.
\(e) The sequence $\langle{x_n}\rangle_{n=1}^{\infty}$ is strongly increasing if and only if for each $n\in \mathbb{N}$, $\sum_{t=1}^nx_{t}< x_{n+1}$.
Clearly if $\langle{x_n}\rangle_{n=1}^{\infty}$ is strongly increasing, then it has distinct finite sums. We also need to recall the following definition from Ref. [@Bergelson1].
**Definition 1.5**: Let $A\subseteq \mathbb N$. Then $A$ is said to be an $IP_0$-set in $(\mathbb{N}, +)$ if and only if for each $m\in\mathbb N$, there exists a finite sequence $\langle{y_n}\rangle_{n=1}^{m}$ such that $FS(\langle{y_n}\rangle_{n=1}^{m})\subseteq A$.
**Theorem 1.6**: For all $A\subseteq \mathbb N$, if $A$ is syndetic in $(\mathbb{N},\cdot)$ then $A$ is central in $(\mathbb{N},+)$.\
**Proof**: See the proof [@Bergelson2 Lemma 5.11].
**Theorem 1.7**: For all $A\subseteq \mathbb N$, if $A$ is central in $(\mathbb{N},\cdot)$ then $A$ is an $IP_0$-set in $(\mathbb{N},+)$.\
**Proof**: See the proof [@Bergelson3 Theorem 3.5].\
In the present paper we want to extend the Theorem $5$ of Ref. [@Hindman1] in section 3 for a dense subsemigroup of $((0, \infty), +)$. Let $S$ be a dense subsemigroup of $((0, \infty), +)$, then one can define\
$ 0^+(S)= \{p\in \beta S_d:$ for all $(\epsilon>0)((0, \epsilon), \in p)\}$, ($S_d$ is the set $S$ with the discrete topology).\
**Theorem 1.8**: Let $S$ be a dense subsemigroup of $((0, \infty), +)$ such that $S\cap(0,1)$ is a subsemigroup of $((0, 1), \cdot)$ and assume that for each $y\in S\cap(0,1)$ and each $x\in S$, $x/y$ and $yx\in S$. Then $ 0^+(S)$ is a two-sided ideal of $\beta(S\cap(0,1),\cdot)$.\
**Proof**: See the proof[@Hindman4 Lemma 13.29.(f)]
Additive and Multiplicative Largeness Near Zero
===============================================
Hindman and Leader introduced different notions of large sets near zero. In this section we study the interplay between additive and multiplicative large sets near zero. We now recall some notions from Ref. [@Hindman2].\
**Definition 2.1**: Let $S$ be a dense subsemigroup of $((0, \infty), +)$.\
(a) [@Hindman2 Definition 3.2]: A set $A\subseteq S$ is syndetic near zero if only if for every $\epsilon>0$ there exist some $F \in \mathcal{P}_{f}((0, \epsilon)\cap S)$ and some $\delta>0$ such that $S\cap (0, \delta)\subseteq \bigcup_{t\in F} (-t+A)$.\
(b) [@Hindman2 Definition 3.4]: A subset $A$ of $S$ is piecewise syndetic near zero if and only if there exist sequences $\langle{F_n}\rangle_{n=1}^{\infty}$ and $\langle{\delta_n}\rangle_{n=1}^{\infty}$ such that\
(1) for each $n\in\mathbb{N}$, $F_n\in\mathcal{P}_{f}((0, 1 /n)\cap S)$ and $\delta_n\in (0, 1 /n)$ and\
(2) for all $G\in\mathcal{P}_{f}(S)$ and all $\mu>0$ there is some $x\in(0, \mu)\cap S$ such that for all $n\in\mathbb{N}$, $(G\cap (0, \delta_{n}))+x\subseteq \bigcup_{t\in F_n} (-t+A)$.\
**Theorem 2.2** [@Hindman2 Theorem 3.5]: Let $S$ be a dense subsemigroup of $((0, \infty), +)$ and let $A\subseteq S$. Then $K(0^+(S))\cap\bar{A} \neq\O{}$ if and only if $A$ is piecewise syndetic near zero.\
**Definition 2.3**: Let $S$ be a dense subsemigroup of $((0, \infty), +)$. A subset $A$ of $ S$ is said to be an $IP$-set near zero if and only if for each $\epsilon>0$ there exists some sequence $\langle{x_n}\rangle_{n=1}^{\infty}$ in $S$ such that $\sum_{n=1}^{\infty}x_{n}$ converges and $FS(\langle{x_n}\rangle_{n=1}^{\infty})\subseteq A\cap (0, \epsilon)$.\
**Theorem 2.4**: Let $S$ be a dense subsemigroup of $((0, \infty), +)$ and let $A\subseteq S$. Then $A$ is an $IP$-set near zero if and only if there is some idempotent $p$ in $0^+(S)$ such that $A\in p$.\
**Proof**: See the proof [@Hindman2 Theorem 3.1].\
**Definition 2.5**: Let $S$ be a dense subsemigroup of $((0, \infty), +)$. Define $\varGamma_0(S)=${$p\in \beta S:$ if $A\in p$ then $A$ is an $IP$-set near zero}\
**Lemma 2.6**: Let $S$ be a dense subsemigroup of $((0, \infty), +)$ such that $S\cap(0,1)$ is a subsemigroup of $((0, 1), \cdot)$ and assume that for each $y\in S\cap(0,1)$ and each $x\in S$, $x/y$ and $yx\in S$. Then $\varGamma_0(S)$ is a left ideal of $\beta(S\cap(0,1),\cdot)$.\
**Proof**: Note that $\varGamma_0(S)= cl\{p\in 0^+(S): p+p=p\}$, and therefore is non-empty. Let $p\in \varGamma_0$, and $q\in \beta(S\cap(0,1),\cdot)$, and let $A\in q\cdot p$. Then $\{y\in S\cap(0,1):y^{-1}A\in p\}\in q$, so pick $y\in S\cap(0,1)$ such that $y^{-1}A\in p$. Thus $y^{-1}A$ is an $IP$-set near zero. Take some sequence $\langle{x_n}\rangle_{n=1}^{\infty}$ in $S$ such that $FS(\langle{x_n}\rangle_{n=1}^{\infty})\subseteq (y^{-1} A)\cap (0, \epsilon)$ and $\sum_{n=1}^{\infty}x_{n}$ converges.\
Let $z_n=yx_n$ for all $n\in \mathbb{N}$. Then $FS(\langle{z_n}\rangle_{n=1}^{\infty})\subseteq A\cap (0, \epsilon)$ and $\sum_{n=1}^{\infty}z_{n}$ be a convergent series in $S$. Therefore $p\cdot q\in \varGamma_0(S)$ which completes the proof.\
**Theorem 2.7**: Let $S$ be a dense subsemigroup of $((0, \infty), +)$ such that $S\cap(0,1)$ is a subsemigroup of $((0, 1), \cdot)$ and assume that for each $y\in S\cap(0,1)$ and each $x\in S$, $x/y$ and $yx\in S$. Let $r\in \mathbb{N}$ and let $S\cap(0,1) = \bigcup_{i=i}^{r}B_{i}$. Then there is some $i\in \{1,2,.....,r\}$ such that $B_{i}$ is an $IP$-set near zero and $B_{i}$ is central in $(S\cap(0,1),\cdot)$.\
**Proof**: By Lemma 2.6, $\varGamma_0(S)$ is a left ideal of $\beta(S\cap(0,1),\cdot)$. Pick a minimal left ideal $L$ of $\beta(S\cap(0,1),\cdot)$ with $L\subseteq \varGamma_0(S)$ and pick $p=p\cdot p\in L$. Note that $p\in K(\beta(S\cap(0,1),\cdot))$ with $p=p\cdot p$ so all of its member are central in ($S\cap(0,1),\cdot))$ Pick $i\in \{1,2,.....,r\}$ such that $B_{i}\in p$. Then $B_{i}$ is central in $(S\cap(0,1),\cdot)$ and, since $p\in \varGamma_0(S)$, $ B_{i}$ is an $IP$-set near zero.\
**Definition 2.8**: Let $S$ be a dense subsemigroup of $((0, \infty), +)$. Define $M_0(S)=\{p\in \beta S: $ if $A\in p$ then $A$ is a central set near zero}.\
**Lemma 2.9**: Let $S$ be a dense subsemigroup of $((0, \infty), +)$ such that $S\cap(0,1)$ is a subsemigroup of $((0, 1), \cdot)$ and assume that for each $y\in S\cap(0,1)$ and each $x\in S$, $x/y$ and $yx\in S$. Then $M_0(S)$ is a left ideal of $\beta(S\cap(0,1),\cdot)$.\
**Proof**: See the proof [@Hindman2 Theorem 5.6].\
**Theorem 2.10**: Let $S$ be a dense subsemigroup of $((0, \infty), +)$ such that $S\cap(0,1)$ is a subsemigroup of $((0, 1), \cdot)$ and assume that for each $y\in S\cap(0,1)$ and each $x\in S$, $x/y$ and $yx\in S$. Let $r\in \mathbb{N}$ and let $S\cap(0,1) = \bigcup_{i=i}^{r}B_{i}$. Then there is some $i\in \{1,2,.....,r\}$ such that $B_{i}$ is central near zero and $B_{i}$ is central in $(S\cap(0,1),\cdot)$.\
**Proof**: See the proof [@Hindman2 Theorem 5.6].\
**Theorem 2.11**: Let $S$ be a dense subsemigroup of $((0, \infty), +)$ such that $S\cap(0,1)$ is a subsemigroup of $((0, 1), \cdot)$ and assume that for each $y\in S\cap(0,1)$ and each $x\in S$, $x/y$ and $yx\in S$. If $A$ is syndetic in $(S\cap (0, 1), \cdot)$ then $A$ is central near zero.\
**Proof**: Since $A$ is syndetic in $(S\cap (0, 1), \cdot)$, there exists $G \in \mathcal{P}_{f}((0, 1)\cap S)$ such that $S\cap(0,1)=\bigcup_ {t\in G}t^{-1}A$. Now take an idempotent $p$ in $K(0^+(S))$. Choose $t\in G$ such that $t^{-1}A\in p$. Thus $t^{-1}A$ is central near zero. So by [@Hindman2 Lemma 4.8], $A$ is central near zero.\
**Theorem 2.12**: Let $S$ be a dense subsemigroup of $((0, \infty), +)$ such that $S\cap(0,1)$ is a subsemigroup of $((0, 1), \cdot)$ and assume that for each $y\in S\cap(0,1)$ and each $x\in S$, $x/y$ and $yx\in S$. If $A$ is piecewise syndetic in $(S\cap (0, 1), \cdot)$ then for each $\epsilon>0$, $l\in \mathbb{N}$ there exist $a, d\in S$ such that $\{a, a+d, .....,a+(l-1)d\}\subseteq A\cap (0,\epsilon)$.\
**Proof**: Le $I=\{p\in \beta S:$ if $A\in p$ then for each $l\in \mathbb{N}$, there exist $a, d\in S$ such that $\{a, a+d, .....,a+(l-1)d\}\subseteq A$}. By [@Hindman2 Theorem 4.11], $E(K(0^+(S))\subseteq I$ (where for a semigroup $S$, $E(S)=\{x\in S: x$ is an idempotent}). Let $I_0=I\cap 0^+(S)$ . Then clearly $I_0\neq \O{}$. By theorem 1.8, $0^+(S)$ is a two-sided ideal of $\beta(S\cap(0,1),\cdot)$. To show that $I_0$ is a two-sided ideal of $\beta(S\cap(0,1), \cdot)$, it is enough to prove that $I$ is a two-sided ideal of $\beta(S\cap(0,1), \cdot)$. To this end let $p\in I$ and $q\in \beta(S\cap(0,1), \cdot)$ . Suppose $A\in q\cdot p$ then $\{x\in S: x^{-1}A\in p\}\in q$. Choose $x\in S\cap(0,1)$ such that $x^{-1}A\in p$. Pick $a, d\in S$ such that $\{a, a+d, .....,a+(l-1)d\}\subseteq x^{-1}A$. Then $\{ax, ax+dx, .....,ax+(l-1)dx\}\subseteq A$. Thus $q\cdot p\in I$. Also let $A\in p\cdot q$. Then $B=\{x\in S: x^{-1}A\in q\}\in p$. Pick $a, d\in S$ such that $\{a, a+d, .....,a+(l-1)d\}\subseteq B$. Then $\bigcap_{i=0}^{l-1}(a+id)^{-1}A\in q$. Take $s\in S\cap(0,1)$ such that $s\in \bigcap_{i=0}^{l-1}(a+id)^{-1}A$. Then $\{as, as+ds, .....,as+(l-1)ds\}\subseteq A$. Thus $p\cdot q\in I$. Therefore $I$ is a two-sided ideal of $\beta(S\cap(0,1), \cdot)$.\
Let $A$ be a piecewise syndetic set in $(S\cap (0, 1), \cdot)$. Now by Theorem 1.2 $K(\beta (S\cap(0,1), \cdot))\cap cl A\neq\O{}$. Since $I_0$ is a two-sided ideal of $\beta (S\cap(0,1), \cdot))$, $K(\beta (S\cap(0,1), \cdot))\subseteq I_0$ and hence $clA\cap I_0\neq\O{}$. Now Choose $p\in I_0$ such that $A\in p$. Therefore for each $\epsilon>0$, $l\in \mathbb{N}$ there exist $a, d\in S$ such that $\{a, a+d, .....,a+(l-1)d\}\subseteq A\cap (0,\epsilon)$.\
We can define $IP_0$-set near zero for a dense subsemigroup of $((0, \infty), +)$ as is defined on $(\mathbb{N}, +)$\
**Definition 2.13**: Let $S$ be a dense subsemigroup of $((0, \infty), +)$ and $A\subseteq S$. Then $A$ is siad to be an $IP_0$-set near zero if for each $m\in \mathbb{N}$ and $\epsilon>0$ there exists a finite sequence $\langle{y_n}\rangle_{n=1}^{m}$ of positive reals such that $FS(\langle{y_n}\rangle_{n=1}^{m})\subseteq A\cap (0,\epsilon)$.\
**Theorem 2.14**: Let $S$ be a dense subsemigroup of $((0, \infty), +)$ such that $S\cap(0,1)$ is a subsemigroup of $((0, 1), \cdot)$ and assume that for each $y\in S\cap(0,1)$ and each $x\in S$, $x/y$ and $yx\in S$. If $A$ is central in $(S\cap (0, 1), \cdot)$ then $A$ is an $IP_0$-set near zero.\
**Proof**: Let $J_0=\{p\in \beta S: $ if $A\in p$ then $A$ is an $IP_0$-set near zero}.\
Notice that $\varGamma_0(S)\subseteq J_0$ and therefore $ J_0\neq\O{}$. We now claim that $J_0$ is a two-sided ideal of $\beta (S\cap(0,1), \cdot)$. To this end let $p\in J_0$ and $q\in \beta (S\cap(0,1), \cdot)$. Suppose $A\in q\cdot p$. Then $B=\{x\in S: x^{-1}A\in p\}\in q$. Choose $x\in S\cap(0,1)$ such that $x^{-1}A\in p$. Then for each $m\in \mathbb{N}$ and $\epsilon>0$ there exists a finite sequence $\langle{y_n}\rangle_{n=1}^{m}$ of positive reals such that $FS(\langle{y_n}\rangle_{n=1}^{m})\subseteq x^{-1}A\cap (0,\epsilon)$.\
Thus $FS(\langle{xy_n}\rangle_{n=1}^{m})\subseteq A\cap (0,\epsilon)$ and therefore $q\cdot p\in J_0$. Also let $A\in p\cdot q$. Then $B=\{x\in S: x^{-1}A\in q\}\in p$. So for each $m\in \mathbb{N}$ and $\epsilon>0$ there exists a finite sequence $\langle{y_n}\rangle_{n=1}^{m}$ of positive reals such that $F=FS(\langle{y_n}\rangle_{n=1}^{m})\subseteq B$. Then $\bigcap_{y\in F}y^{-1}A\in q$. Choose $x\in \bigcap_{y\in F}y^{-1}A\cap (0, 1)$. Then $FS(\langle{xy_n}\rangle_{n=1}^{m})\subseteq A\cap (0,\epsilon)$. Therefore $p\cdot q\in J_0$. Thus $J_0$ is a two-sided ideal of $\beta (S\cap(0,1), \cdot)$. Let $A$ be central in $(S\cap(0,1), \cdot)$. Then there exists an idempotent $p\in K(\beta (S\cap(0,1), \cdot))$ such that $A\in p$. Also since $J_0$ is a two-sided ideal of $\beta (S\cap(0,1), \cdot)$, $p\in J_0$. Therefore $A$ is an $IP_0$-set near zero.\
Monochromatic Solution To $\sum_{t=1}^nx_{t}=\prod_{t=1}^ny_t$ Near Zero
========================================================================
In Ref. [@Hindman1], author generalized the affirmative answer to the question of P. Csikvári, A. Sárközy, and K. Gyarmati regarding the monochromatic solution of $a+b=cd $, whenever the set of naturals $\mathbb N$ is finitely colored.\
**Theorem 3.1**: Let $r\in \mathbb{N}$ and let $\mathbb{N}=\bigcup_{i=i}^{r}A_{i}$.There exists $i\in \{1,2,.....,r\}$ such that for each $m\in \mathbb{N}$,\
(1) there exists an increasing sequence $\langle{y_n}\rangle_{n=1}^{\infty}$ with distinct finite products such that $FP(\langle{y_n}\rangle_{n=1}^{\infty})\subseteq A_{i}$ and whenever $F \in \mathcal{P}_{f}(\mathbb{N})$, there exists a strongly increasing sequence $\langle{x_n}\rangle_{n=1}^{m}$ such that $FS(\langle{x_n}\rangle_{n=1}^{m})\subseteq A_{i}$ and $\sum_{n=1}^mx_{n}=\prod_{n\in F}y_n$ and\
(2) there exists a strongly increasing sequence $\langle{x_n}\rangle_{n=1}^{\infty}$ such that\
$FS(\langle{x_n}\rangle_{n=1}^{\infty})\subseteq A_{i}$ and whenever $F \in \mathcal{P}_{f}(\mathbb{N})$, there exists an increasing sequence $\langle{y_n}\rangle_{n=1}^{m}$ with distinct finite products such that $FP(\langle{y_n}\rangle_{n=1}^{m})\subseteq A_{i}$ and $\prod_{n=1}^{m}y_n =\sum_{n\in F}x_{n}$.\
**Proof**: See the proof [@Hindman1 Theorem 5].\
In this section we extend this result for a dense subsemigroup of $((0, \infty), +)$ to near zero.\
To establish the main result we need the following lemma.\
**Lemma 3.2**: Let $S$ be a dense subsemigroup of $((0, \infty), +)$ such that $S\cap(0,1)$ is a subsemigroup of $((0, 1), \cdot)$ and assume that for each $y\in S\cap(0,1)$ and each $x\in S$, $x/y$ and $yx\in S$. Let $\langle{w_t}\rangle_{t=1}^{\infty}$ be a sequence in $S\cap(0,1)$ such that $\sum_{t=1}^{\infty}w_{t}$ converges. Then there exist sequences $\langle{x_t}\rangle_{t=1}^{\infty}$ and $\langle{y_t}\rangle_{t=1}^{\infty}$ such that $\langle{x_t}\rangle_{t=1}^{\infty}$ is strictly decreasing (and therefore it is distinct), $\langle{y_t}\rangle_{t=1}^{\infty}$ is decreasing and has distinct finite products, $FS(\langle{x_t}\rangle_{t=1}^{\infty})\subseteq FS(\langle{w_t}\rangle_{t=1}^{\infty})$ and $FP\langle{y_t}\rangle_{t=1}^{\infty}\subseteq FP(\langle{w_t}\rangle_{t=1}^{\infty})$
**Proof**: We construct the sequence $\langle{x_t}\rangle_{t=1}^{\infty}$ inductively. Since $\sum_{t=1}^{\infty}w_{t}$ converges, $\langle{w_t}\rangle_{t=1}^{\infty}$ converges to $0$. Hence we can construct a strictly decreasing subsequence of $\langle{w_t}\rangle_{t=1}^{\infty}$ in the following way. Let $x_1=w_1$. Then clearly $(0, w_1/2)$ contains infinitely many points of the sequence $\langle{w_t}\rangle_{t=2}^{\infty}$. Let us take $x_2=w_l$ such that $w_l\in (0, w_1/2)$. Then clearly $x_1>x_2$. Inductively, let $\langle{x_t}\rangle_{t=1}^{k}$ be a strictly decreasing sequence such that for each $t$, $x_t=w_{p_t}$ and $x_t\in (0, x_{t-1}/2)$ for some $p_t \in \mathbb{N}$ and $x_t>x_{t+1}$ for $t\in \{1,2,.....,k-1\}$. Clearly $FS(\langle{x_t}\rangle_{t=1}^{k})\subseteq FS(\langle{w_t}\rangle_{t=1}^{\infty})$. Now $(0, x_k/2)$ contains infinitely many points of the sequence $\langle{w_t}\rangle_{t=1}^{\infty}$ hence we can choose min$\{t\in \mathbb{N}: w_t\in (0, x_k/2)\}=q$. Let $x_{k+1}=w_q$.\
By similar arguments, we construct the sequence $\langle{y_t}\rangle_{t=1}^{\infty}$ inductively. Let $\langle{y_k}\rangle_{k=1}^{n}$ be a strictly decreasing subsequence of $\langle{w_t}\rangle_{t=1}^{\infty}$ such that $y_k=w_{t_k}$ with $t_1<t_2.....<t_n$. Let $E=FP(\langle{y_k}\rangle_{k=1}^{n})$ also let $\mu=$ min $E\cup \{u^{-1}v: u,v\in E\}$. Now $(0,\mu)$ contains infinitely many points of $\langle{w_t}\rangle_{t={t_n}+1}^{\infty}$. Let min$\{t\in \mathbb{N}:t\geq {t_n}+1, w_t\in (0, \mu)\}=r$. Let $y_{n+1}=w_r$. Therefore, $FP\langle{y_t}\rangle_{t=1}^{\infty}\subseteq FP(\langle{w_t}\rangle_{t=1}^{\infty})$. This completes the proof.\
**Theorem 3.3**: Let $S$ be a dense subsemigroup of $((0, \infty), +)$ such that $S\cap(0,1)$ is a subsemigroup of $((0, 1), \cdot)$ and assume that for each $y\in S\cap(0,1)$ and each $x\in S$, $x/y$ and $yx\in S$. Let $r\in \mathbb{N}$, $\epsilon>0$ and let $S\cap (0,1)=\bigcup_{i=i}^{r}A_{i}$. There exists $i\in \{1,2,.....,r\}$ such that for each $m\in \mathbb{N}$,\
(1) there exists a decreasing sequence $\langle{y_n}\rangle_{n=1}^{\infty}$ with distinct finite products such that $FP(\langle{y_n}\rangle_{n=1}^{\infty})\subseteq A_{i}\cap (0, \epsilon)$ and whenever $F \in \mathcal{P}_{f}(\mathbb{N})$, there exists a strictly decreasing sequence $\langle{x_n}\rangle_{n=1}^{m}$ such that $FS(\langle{x_n}\rangle_{n=1}^{m})\subseteq A_{i}\cap (0, \epsilon)$ and $\sum_{n=1}^mx_{n}=\prod_{n\in F}y_n$ and\
(2) there exists a strictly decreasing sequence $\langle{x_n}\rangle_{n=1}^{\infty}$ such that\
$FS(\langle{x_n}\rangle_{n=1}^{\infty})\subseteq A_{i}\cap (0, \epsilon)$ and whenever $F \in \mathcal{P}_{f}(\mathbb{N})$, there exists a decreasing sequence $\langle{y_n}\rangle_{n=1}^{m}$ with distinct finite products such that $FP(\langle{y_n}\rangle_{n=1}^{m})\subseteq A_{i}\cap (0, \epsilon)$ and $\prod_{n=1}^{m}y_n =\sum_{n\in F}x_{n}$.\
**Proof**: Pick $p\in \beta (S\cap(0,1), \cdot)$ such that, for every $A\in p$ there exist sequences $\langle{x_n}\rangle_{n=1}^{\infty}$ and $\langle{y_n}\rangle_{n=1}^{\infty}$ with $FS(\langle{x_n}\rangle_{n=1}^{\infty})\subseteq A\cap (0, \epsilon)$ and $FP(\langle{y_n}\rangle_{n=1}^{\infty})\subseteq A\cap (0, \epsilon)$ (In section 2 we have proved the existence of such a $p$). Pick $i\in \{1,2,.....,r\}$ such that $A_{i}\cap (0, \epsilon)\in p$.\
Let $m\in \mathbb{N}$ be given. Let $B_0=\{z\in A_{i}\cap (0, \epsilon):$ there exists a strictly decreasing sequence $\langle{x_n}\rangle_{n=1}^{m}$ such that $FS(\langle{x_n}\rangle_{n=1}^{m})\subseteq A_{i}\cap (0, \epsilon)$ and $z=\sum_{n=1}^mx_{n}$}. Let $C_0= \{z\in A_{i}\cap (0, \epsilon):$ there exists a decreasing sequence $\langle{y_n}\rangle_{n=1}^{m}$ with distinct finite products such that $FP(\langle{y_n}\rangle_{n=1}^{m})\subseteq A_{i}\cap (0, \epsilon)$ and $z==\prod_{n=1}^my_{n}$}\
We claim that $B_0\in p$. If possible, let $B_0\notin p$, in which case\
$(A_{i}\cap (0, \epsilon))\backslash B_0\in p$. Pick a sequence $\langle{x_n}\rangle_{n=1}^{\infty}$ with\
$FS(\langle{x_n}\rangle_{n=1}^{\infty})\subseteq (A_{i}\cap (0, \epsilon))\backslash B_0$. By Lemma 3.2 we may assume that $\langle{x_n}\rangle_{n=1}^{\infty}$ is strictly decreasing. But then $\sum_{n=1}^mx_{n}\in B_0$, a contradiction. Similarly $C_0\in p$.\
For conclusion (1) pick a decreasing sequence $\langle{y_n}\rangle_{n=1}^{\infty}$ with distinct finite products such that $FP(\langle{y_n}\rangle_{n=1}^{\infty})\subseteq B_0$. For conclusion (2) pick a strictly decreasing sequence $\langle{x_n}\rangle_{n=1}^{\infty}$ with distinct finite products such that $FS(\langle{x_n}\rangle_{n=1}^{\infty})\subseteq C_0$.\
**Corollary 3.4**: Let $S$ be a dense subsemigroup of $((0, \infty), +)$ such that $S\cap(0,1)$ is a subsemigroup of $((0, 1), \cdot)$ and assume that for each $y\in S\cap(0,1)$ and each $x\in S$, $x/y$ and $yx\in S$. Let $r\in \mathbb{N}$ and let $\mathbb{N}=\bigcup_{i=i}^{r}A_{i}$. There exists $i\in \{1,2,.....,r\}$ and $a$, $b$, $c$, and $d$ in $S$ such that $\{a, b, c, d\}\subseteq A_{i}\cap (0, \epsilon)$ with $a+b= cd$ and $a\neq b$.\
**Proof**: It is the special case of the above theorem when $n=2$
[3]{}
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P. Csikvári, A. Sárközy, and K. Gyarmati, Density and Ramsey type results on algebraic equations with restricted solution sets, Manuscript.
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author:
- 'Ning-An Lai, Nico Michele Schiavone and Hiroyuki Takamura'
title: 'Wave-like blow-up for semilinear wave equations with scattering damping and negative mass term'
---
Introduction
============
In this paper, we consider the Cauchy problem for semilinear wave equations with scattering damping and negative mass term $$\label{eq:main_problem}
\left\{
\begin{aligned}
& u_{tt} - \Delta u + \frac{\mu_1}{(1+t)^\beta} u_t - \frac{\mu_2}{(1+t)^{\alpha+1}} u = |u|^p, \quad\text{in $\R^n\times[0, T)$}, \\
& u(x,0)=\e f(x), \quad u_t(x,0)=\e g(x), \quad x\in\R^n,
\end{aligned}
\right.$$ where $\mu_1, \mu_2 > 0$, $\alpha>1$, $\beta >1$, $n\in\N$ and $\e>0$ is a “small” parameter.
We call the term $\mu_1u_t/(1+t)^\beta\,(\beta>1)$ scattering damping, due to the reason that the solution of the following Cauchy problem $$\label{linear}
\left\{
\begin{aligned}
& u^0_{tt}-\Delta u^0+\frac{\mu}{(1+t)^\beta}u^0_t=0,
\quad \text{in $\R^n \times[0,\infty)$},\\
& u^0(x,0)=u_1(x), \quad u^0_t(x,0)=u_2(x), \quad x\in\R^n,
\end{aligned}
\right.$$ scatters to that of the free wave equation when $\beta>1$ and $t\rightarrow \infty$. In fact, according to the works of Wirth [@Wir1; @Wir2; @Wir3], we may classify the damping for different values of $\beta$ into four cases, as shown in the next table.
[p[3cm]{}p[4.5cm]{}]{} Range of $\beta$ & Classification\
$\beta\in(-\infty,-1)$ & overdamping\
$\beta\in[-1,1)$ & effective\
& scaling invariant\
& if $\mu\in(0,1)\Rightarrow$ non-effective\
$\beta\in(1,\infty)$ & scattering\
If we come to the nonlinear problem with power nonlinearity, thus $$\label{nonlinear}
\left\{
\begin{aligned}
& u_{tt}-\Delta u+\frac{\mu}{(1+t)^\beta}u_t=|u|^p,
\quad \text{in $R^n \times[0,\infty)$},\\
& u(x,0)=u_1(x), \quad u_t(x,0)=u_2(x), \quad x\in\R^n,
\end{aligned}
\right.$$ we want to determine the long time behaviour of the solution according to the different value of $p$, $n$ and even $\mu$. Ikeda and Wakasugi [@IWnew] proved global existence for for all $p>1$ when $\beta<-1$. For $\beta\in [-1, 1)$, due to the work [@DLR13; @LNZ12; @N1; @WY17; @IO; @FIW; @II], we know that problem admits a critical power $p_F(n):=1+2/n$ (Fujita power), which means that for $p\in (1, p_F(n)]$ the solution will blow up in a finite time, while for $p\in \left(p_F(n), \infty\right)$ we have global existence. Obviously, in this case the critical is exactly the same as that of the Cauchy problem of semilinear heat equation $$u_t-\Delta u=u^p,$$ and so we call it admits heat-like" behaviour.
For the case $\beta=1$ in , we say that the damping is scale invariant, due to the reason that the equation in the corresponding linear problem is invariant under the following scaling transformation $$\wt{u^0}(x,t):=u^0(\sigma x, \sigma(1+t)-1),\ \sigma>0.$$ It is a bit sophisticated for the scale invariant nonlinear problem , since the size of the positive constant $\mu$ will also have an effect on the long time behaviour of the solution. Generally speaking, according to the known results ([@DABI; @DL1; @DLR15; @WY14_scale; @LTW; @IS; @TL1; @TL2]), it is believed that if $\mu$ is large enough, then the critical power is related to the Fujita power, while if $\mu$ is relatively small, then the critical power is related to the Strauss power, i.e. $p_S(n)$, which is denoted to be the positive root of the following quadratic equation $$\gamma(p,n):=2+(n+1)p-(n-1)p^2=0,$$ and which is also the critical power of the small data Cauchy problem of the semilinear wave equation $$u_{tt}-\Delta u=|u|^p.$$ It means that for relatively small $\mu$ we have wave-like“ behaviour. Unfortunately, we are not clear of the exact threshold determined by the value $\mu$ between the heat-like” and wave-like" phenomenon till now.
For the scattering case $(\beta>1)$, one expects that problem admits the long time behaviour as that of the corresponding problem without damping. In [@LT], Lai and Takamura obtained the blow-up results for $$1<p<
\begin{cases}
p_S(n) &\text{for $n\ge2$},\\
\infty &\text{for $n=1$}
\end{cases}$$ and the upper bound of the lifespan estimate $$T\leq C\varepsilon^{-2p(p-1)/\gamma(p, n)}.$$ What is more, when $n=1, 2$ and $$\int_{\R^n}g(x)dx\neq 0,$$ they established an improved upper bound of the lifespan for $1<p<2$, $n=2$ and $p>1$, $n=1$. However, it remains to determine the exact critical power for with $\beta>1$.
Recently, the small data Cauchy problem for semilinear wave equation with scale-invariant damping and mass and power non-linearity, i.e., $$\label{nonlinearmass}
\left\{
\begin{aligned}
& u_{tt} - \Delta u + \frac{\mu_1}{1+t} u_t + \frac{\mu_2^2}{(1+t)^{2}} u = |u|^p,
\quad \text{in $\R^n \times[0,\infty)$},\\
& u(x,0)=u_1(x), \quad u_t(x,0)=u_2(x), \quad x\in\R^n,
\end{aligned}
\right.$$ attracts more and more attention. Denote $$\label{eq:def_delta}
\delta:=(\mu_1-1)^2-4\mu_2^2.$$ Then in [@NPR] and [@Pal] a blow-up result was established for $$1<p\leq p_{F}\left(n+\frac{\mu_1-1-\sqrt{\delta}}{2}\right)$$ assuming $\delta\geq 0$, by using two different approaches. Furthermore, in [@NPR] they improved the result for $\delta=1$ to $$1<p\leq \max\left\{p_S(n+\mu_1),\ p_F\left(n+\frac{\mu_1}{2}-1\right)\right\}.$$ Recently, Palmieri and Reissig [@P-R] generalized the blow-up result for $n\geq 1$ and $\delta\in (0, 1]$ to the following power: $$\begin{cases}
p<p_{\mu_1, \mu_2}(n):=\max\left\{p_S(n+\mu_1),\ p_F\left(n+\frac{\mu_1}{2}-\frac{\sqrt{\delta}}{2}\right)\right\},\\
p=p_{\mu_1, \mu_2}(n)=p_F\left(n+\frac{\mu_1}{2}-\frac{\sqrt{\delta}}{2}\right),\\
p=p_{\mu_1, \mu_2}(n)=p_S(n+\mu_1), \quad\text{for $n=2$}.
\end{cases}$$ We note that a transform by $v:=(1+t)^{\mu_1/2}u$ changes the equation in (\[nonlinearmass\]) into $$v_{tt}-\Delta v+\frac{1-\delta}{4(1+t)^2}v=\frac{|v|^p}{(1+t)^{\mu_1(p-1)/2}},$$ so that the assumption of $\delta\in(0,1]$ implies the non-negativeness of the mass term in this equation.
In this paper, we are going to study the small data Cauchy problem of semilinear wave equations with power nonlinearity, scattering damping and mass term with negative sign, thus, problem . Blow-up results and lifespan estimates will be established for $1<p<p_S(n)$, which are the same as that in the work [@LT]. We could say that we experience a double phenomenon of scattering, due to the damping term and the mass term. For the proof, we will borrow the idea from [@LT], by introducing a key multiplier to absorb the damping term and establishing an iteration frame. However, we have to deal with the mass term. Due to the negative sign, we use a comparison argument to eliminate the effect from the mass term. Although the calculations in this work hold for any mass exponent $\alpha\in\R$, we suppose that it satisfies $\alpha>1$ because otherwise we have shorter lifespan estimates due to the effect of the negative mass term. This analysis will appear in our forthcoming paper.
Main Result
===========
Before the statement of our main results, we first denote the energy and weak solutions of problem .
We say that $u$ is an energy solution of over $[0,T)$ if $$u \in C([0,T), H^1(\R^n)) \cap C^1([0,T),L^2(\R^n)) \cap C((0,T), L^p_{loc}(\R^n))$$ satisfies $u(x,0)=\e f(x)$ in $H^1(\R^n)$ and $u_t(x,0)=\e g(x)$ in $L^2(\R^n)$, and $$\label{eq:energy_solution}
\begin{split}
&\int_{\R^n}u_t(x,t)\phi(x,t)dx-\int_{\R^n}\e g(x)\phi(x,0)dx\\
&+\int_0^tds\int_{\R^n}\left\{-u_t(x,s)\phi_t(x,s)+\nabla u(x,s)\cdot\nabla\phi(x,s)\right\}dx \\
&+\int_0^tds\int_{\R^n}\frac{\mu_1 }{(1+s)^{\beta}}u_t(x,s) \phi(x,s)dx - \int_0^t ds \ints \frac{\mu_2}{(1+s)^{\alpha+1}}u(x,s)\phi(x,s) \\
= & \int_0^tds\int_{\R^n}|u(x,s)|^p\phi(x,s)dx
\end{split}$$ with any test function $\phi \in C_0^\infty(\R^n \times [0,T))$ and for any $t\in[0,T)$.
Employing the integration by part in the above equality and letting $t\to T$, we got the definition of the weak solution of , that is $$\begin{split}
&\int_{\R^n\times[0,T)}
u(x,s)\bigg\{\phi_{tt}(x,s)-\Delta \phi(x,s)
-\frac{\p}{\p s} \left(\frac{\mu_1}{(1+s)^{\beta}}\phi(x,s)\right)\\
& - \frac{\mu_2}{(1+s)^{\alpha+1}} \phi(x,s) \bigg\}dxds\\
=&\ \int_{\R^n}\mu_1 \e f(x)\phi(x,0)dx -\int_{\R^n}\e f(x)\phi_t(x,0)dx +\int_{\R^n}\e g(x)\phi(x,0)dx\\ &+\int_{\R^n\times[0,T)}|u(x,s)|^p\phi(x,s)dxds.
\end{split}$$
As in the introduction, set $$\gamma(p,n) := 2+(n+1)p - (n-1)p^2$$ and, for $n\ge2$, define $p_S(n)$ the positive root of the quadratic equation $\gamma(p,n)=0$, the so-called Strauss exponent, that is $$p_S(n) = \frac{n+1+\sqrt{n^2+10n-7}}{2(n-1)}.$$ Note that if $n=1$, then $\gamma(p,1)=2+2p$ and we can set $p_S(1):=+\infty$.
Now we announce our main results.
\[thm1\] Let $n=1$ and $p>1$, or $n \geq 2$ and $1<p<p_S(n)$. Assume that both $f \in H^1(R^n)$ and $g\in L^2(\R^n)$ are non-negative, and at least one of them does not vanish identically. Suppose that $u$ is an energy solution of on $[0,T)$ that satisfies $$\label{support}
\operatorname*{supp}u \subset\{(x,t)\in\R^n\times[0,\infty) \colon |x|\le t+R\}$$ with some $R\ge1$. Then, there exists a constant $\e_0=\e_0(f,g,n,p,\mu_1, \beta, R)>0$ which is independent of $\mu_2$, such that $T$ has to satisfy $$\label{eq:1}
T\leq C\varepsilon^{-2p(p-1)/\gamma(p, n)}$$ for $0<\e\le\e_0$, where $C$ is a positive constant independent of $\e$.
In low dimensions $(n=1, 2)$, with some additional hypothesis, we may have improvements on the lifespan estimates as follows.
\[thm2\] Let $n=2$ and $1<p<2$. Assume that both $f \in H^1(R^2)$ and $g\in L^2(\R^2)$ are non-negative and that $g$ does not vanish identically. Then the lifespan estimate is replaced by $$\label{1a}
T\leq C\varepsilon^{-(p-1)/(3-p)}.$$
\[thm3\] Let $n=1$ and $p>1$. Assume that both $f \in H^1(R^1)$ and $g\in L^2(\R^1)$ are non-negative and that $g$ does not vanish identically. Then the lifespan estimate is replaced by $$\label{1b}
T\leq C\varepsilon^{-(p-1)/2}.$$
\[thm4\] Let $n=p=2$. Suppose that $\alpha\le\beta$ and $$\label{eq:cond_thm4}
\mu_2 \ge
\begin{cases}
\frac{\beta\mu_1}{2} & \text{if $\alpha=\beta$}, \\
\frac{\beta\mu_1}{2}\frac{\beta-1}{2\beta-\alpha-1}\left(4\frac{\mu_1^2}{\mu_2}\frac{\beta-\alpha}{\beta-1} \right)^{\frac{\beta-\alpha}{2\beta-\alpha-1}} & \text{if $\alpha<\beta$}.
\end{cases}$$ Assume that $f \equiv 0$ and $g \in C^2(\R^2)$ is non-negative and does not vanish identically. Suppose also that $u$ is a classical solution of on $[0,T)$ with the support property . Then, $T$ satisfies $$\label{1c}
T \leq C a(\e)$$ where $a=a(\e)$ is a number satisfying $$\label{def:a}
a^2 \e^2 \log(1+a)=1.$$
In Theorem \[thm1\], we require that at least one of the initial data does not vanish identically, which is weaker than that in the corresponding result (Theorem 2.1) in [@LT].
Observe that:
- is stronger than by the fact that $1<p<2$ is equivalent to $$\frac{p-1}{3-p}<\frac{2p(p-1)}{\gamma(p,2)};$$
- is stronger than by the fact that $p>1$ is equivalent to $$\frac{p-1}{2}<\frac{2p(p-1)}{\gamma(p,1)};$$
- is stronger than by the fact that when $n=p=2$ $$a(\e) < \e^{-1} =\e^{-2\cdot2(2-1)/\gamma(2,2)}$$ for sufficiently small $\e$.
Lower bound for derivative of the functional
============================================
Following the idea in [@LT], we introduce the multiplier $$\label{eq:m}
m(t):= \exp\left(\mu_1 \frac{(1+t)^{1-\beta}}{1-\beta}\right).$$ Clearly $$\label{eq:m_bound}
1 \geq m(t) \geq m(0) >0 \quad\text{for $t \geq 0$}.$$ Moreover, let us define the functional $$F_0(t) := \int_{\R^n} u(x,t)dx,$$ then $$F_0(0) = \e\int_{\R^n} f(x)dx, \quad F'_0(0) = \e\int_{\R^n}g(x)dx$$ are non-negative due to the hypothesis of positiveness on the initial data. Our final target is to establish a lower bound for $F_0(t)$.
Let us start finding the lower bound of the derivative of the functional, i.e., $F'_0(t)$. Due to , choosing the test function $\phi=\phi(x,s)$ in to satisfy $\phi\equiv 1$ in $\{(x,s)\in \R^n\times[0,t]:|x|\le s+R\}$, we get $$\begin{split}
&\int_{\R^n}u_t(x,t)dx-\int_{\R^n}u_t(x,0)dx+\int_0^tds\int_{\R^n}\frac{\mu_1}{(1+s)^\beta}u_t(x,s)dx \\
=& \int_0^t \int_{\R^n} \frac{\mu_2}{(1+s)^{\alpha+1}} u(x,s) dx
+\int_0^tds\int_{\R^n}|u(x,s)|^pdx,
\end{split}$$ which yields by taking derivative with respect to $t$ $$\label{2}
F_0''(t)+\frac{\mu_1}{(1+t)^\beta}F_0'(t)
= \frac{\mu_2}{(1+t)^{\alpha+1}}F_0(t) + \int_{\R^n}|u(x,t)|^pdx.$$ Here we note that can be established by regularity assumption on the solution. Multiplying both sides of with $m(t)$ yields $$\label{3}
\left\{m(t)F'_0(t)\right\}'
= m(t)\frac{\mu_2}{(1+t)^{\alpha+1}} \, F_0(t) + m(t)\int_{\R^n}|u(x,t)|^pdx.$$ Integrating the above equality over $[0, t]$ we get $$\label{4}
\begin{split}
F'_0(t) =&\ \frac{m(0)}{m(t)}F'_0(0) + \frac{1}{m(t)} \int_0^t m(s)\frac{\mu_2}{(1+s)^{\alpha+1}}F_0(s) ds\\
&+ \frac{1}{m(t)} \int_0^t m(s)ds\int_{\R^n}|u(x,s)|^pdx.
\end{split}$$
To get the lower bound for $F'_0$, we need the positiveness of $F_0$, and this can be obtained by a comparison argument. However, since we assume that at least one of the initial data does not vanish identically, we have to consider the following two cases.
**Case 1: $f\geq 0 (\not\equiv 0)$, $g \geq 0$.** This means that $F_0(0)>0$, $F'_0(0) \geq 0$. By the continuity of $F_0$, it is positive at least for small time. Suppose that $t_0$ is the smallest zero point of $F_0$, such that $F_0>0$ in $[0,t_0)$. Then, integrating over this interval we have $$\begin{split}
0 = F_0(t_0) =&\ F_0(0) + m(0)F'_0(0) \int_0^{t_0} \frac{ds}{m(s)} \\
&+ \int_{0}^{t_0} \frac{ds}{m(s)} \int_0^s m(r)\frac{\mu_2}{(1+r)^{\alpha+1}}F_0(r) dr\\
&+ \int_{0}^{t_0}\frac{ds}{m(s)} \int_0^s m(r)dr\int_{\R^n}|u(x,r)|^pdx >0,
\end{split}$$ which leads to a contradiction, and hence $F(t)$ is positive all the time.
**Case 2: $f\geq 0$, $g \geq 0 (\not\equiv 0)$.** This imply that $F_0(0) \geq 0$, $F'_0(0)>0$. We apply the same argument as in the first case to $F'_0$. Suppose that $t_0$ is the smallest zero point of $F'_0$, such that $F'_0$ is positive on the interval $[0,t_0)$. Therefore $F_0$ is strictly monotone increasing on the same interval, and hence positive due to $F_0(0) \geq 0$. Letting $t=t_0$ in , we again come to a contradiction. Therefore $F'_0$ is always strictly positive, and hence $F_0(t)>0$ holds for all $t>0$.
Coming back to , using the positivity of $F_0$, the boundedness of $m(t)$ and that $F'_0(0) \geq 0$, we obtain the lower bound for $F'_0$ as $$\label{eq:bound_F'_0}
F'_0(t) \geq m(0) \int_0^t \int_{\R^n} |u(x,s)|^p dxds \quad\text{for $t \geq 0$}.$$
Lower bound for the weighted functional
=======================================
Set $$F_1(t):=\int_{\R^n}u(x,t)\psi_1(x,t)dx,$$ where $\psi_1$ is the test function introduced by Yordanov and Zhang [@YZ06] $$\psi_1(x,t):=e^{-t}\phi_1(x),
\quad
\phi_1(x):=
\begin{cases}
\d\int_{S^{n-1}}e^{x\cdot\omega}dS_\omega & \text{for $n\ge2$},\\
e^x+e^{-x} & \text{for $n=1$}.
\end{cases}$$
\[lem1\] $$\label{14}
\int_{|x|\leq t+R}\left[\psi_1(x,t)\right]^{p/(p-1)}dx
\leq C(1+t)^{(n-1)\{1-p/(2(p-1))\}},$$ where $C_1=C_1(n,p,R)>0$.
Next we aim to establish the lower bound for $F_1$. From the definition of energy solution , we have that $$\begin{split}
&\frac{d}{dt}\int_{\R^n}u_t(x,t)\phi(x,t)dx +\int_{\R^n}\left\{-u_t(x,t)\phi_t(x,t)-u(x,t)\Delta\phi(x,t)\right\}dx\\
&+\int_{\R^n}\frac{\mu_1 }{(1+t)^\beta}u_t(x,t)\phi(x,t)dx
- \int_{\R^n} \frac{\mu_2}{(1+t)^{\alpha+1}}u(x,t)\phi(x,t)dx\\
=&\ \int_{\R^n}|u(x,t)|^p\phi(x,t)dx.
\end{split}$$ Multiplying both sides of the above equality with $m(t)$ yields $$\begin{split}
&\frac{d}{dt}\left\{m(t)
\int_{\R^n}u_t(x,t)\phi(x,t)dx\right\}\\
&+m(t)\int_{\R^n}\left\{-u_t(x,t)\phi_t(x,t)-u(x,t)\Delta\phi(x,t)\right\}dx\\
=&\ m(t)\int_{\R^n} \frac{\mu_2}{(1+t)^{\alpha+1}}u(x,t)\phi(x,t)dx + m(t)\int_{\R^n}|u(x,t)|^p\phi(x,t)dx,
\end{split}$$ integrating which over $[0,t]$ yields $$\begin{split}
&m(t)
\int_{\R^n}u_t(x,t)\phi(x,t)dx
-m(0)\e\int_{\R^n}g(x)\phi(x,0)dx\\
&-\int_0^tds\int_{\R^n}m(s)u_t(x,s)\phi_t(x,s)dx - \int_0^tds\int_{\R^n} m(s)
u(x,s)\Delta\phi(x,s) \\
=&\ \int_0^tds\int_{\R^n}m(s) \frac{\mu_2}{(1+s)^{\alpha+1}}u(x,s)\phi(x,s)dx\\
&+ \int_0^tds\int_{\R^n} m(s)|u(x,s)|^p\phi(x,s)dx.
\end{split}$$ Integrating by parts the first term in the second line of the above equality, we have $$\label{eqforF_1}
\begin{split}
&m(t)
\int_{\R^n}u_t(x,t)\phi(x,t)dx
-m(0)\e\int_{\R^n}g(x)\phi(x,0)dx\\
&-m(t)\int_{\R^n} u(x,t)\phi_t(x,t)dx + m(0)\e \int_{\R^n} f(x)\phi_t(x,0)dx\\
&+ \int_0^t ds \int_{\R^n} m(s)\frac{\mu_1}{(1+s)^\beta}u(x,s)\phi_t(x,s)dx
\\
& + \int_0^t ds \int_{\R^n} m(s)u(x,s)\phi_{tt}(x,s)dx - \int_0^tds\int_{\R^n} m(s)
u(x,s)\Delta\phi(x,s) \\
=&\ \int_0^tds\int_{\R^n}m(s) \frac{\mu_2}{(1+s)^{\alpha+1}}u(x,s)\phi(x,s)dx\\
&+ \int_0^tds\int_{\R^n} m(s)|u(x,s)|^p\phi(x,s)dx.
\end{split}$$
Setting $$\phi(x,t)=\psi_1(x,t)=e^{-t}\phi_1(x)
\quad\text{on $\operatorname*{supp}u$},$$ then we have $$\phi_t=-\phi,\quad \phi_{tt}=\Delta\phi \quad\text{on $\operatorname*{supp}u$}.$$ Hence we obtain from $$\begin{split}
m(t)\{F_1'(t)+2F_1(t)\}
=& \ m(0)\e\int_{\R^n}\left\{ f(x)+g(x)\right\}\phi_1(x)dx \\
&+ \int_0^t m(s) \left\{\frac{\mu_1}{(1+s)^\beta} + \frac{\mu_2}{(1+s)^{\alpha+1}} \right\} F_1(s)ds\\
&+\int_0^tds\int_{\R^n}m(s)|u(x,s)|^pdx,
\end{split}$$ which implies $$\label{eq:F'_1+2F_1}
\begin{split}
F'_1(t)+2F_1(t) &\ge\frac{m(0)}{m(t)}
C_{f,g}\e+\frac{1}{m(t)}\int_0^tm(s)
\left\{\frac{\mu_1}{(1+s)^\beta} + \frac{\mu_2}{(1+s)^{\alpha+1}} \right\}F_1(s)ds\\
&\ge m(0)C_{f,g}\e+\int_0^tm(s)
\left\{\frac{\mu_1}{(1+s)^\beta} + \frac{\mu_2}{(1+s)^{\alpha+1}} \right\}F_1(s)ds,
\end{split}$$ where $$C_{f,g}:=\int_{\R^n}\left\{f(x)+g(x)\right\}\phi_1(x)dx >0.$$ Integrating the above inequality over $[0,t]$ after a multiplication with $e^{2t}$, we get $$\label{6}
\begin{split}
e^{2t}F_1(t)
\ge & \ F_1(0)+m(0)C_{f,g}\e \int_0^t e^{2s}ds \\
&+\int_0^t e^{2s}ds
\int_0^sm(r) \left\{\frac{\mu_1}{(1+r)^\beta} + \frac{\mu_2}{(1+r)^{\alpha+1}} \right\} F_1(r)dr.
\end{split}$$ Applying a comparison argument, we have that $F_1(t)>0$ for $t>0$. Again, we should consider two cases due to the hypothesis on the data.
**Case 1: $f\geq0 (\not\equiv0)$, $g\geq0$.** In this case $F_1(0)=C_{f,0}\e>0$. The continuity of $F_1$ yields that $F_1(t)>0$ for small $t>0$. If there is the nearest zero point $t_0$ to $t=0$ of $F_1$, then gives a contradiction at $t_0$.
**Case 2: $f\geq0$, $g\geq0(\not\equiv0)$.** If $f\not\equiv0$, we are in the previous case. If $f\equiv0$, then $F_1(0)=0$, $F'_1(0)= C_{0,g}\e >0$. By the continuity of $F'_1$, we have that $F'_1$ is strictly positive for small $t$, hence there exists some $t_1>0$ such that $F'_1 >0$ over $[0,t_1]$. Then $F_1$ is strictly monotone increasing on this interval, and then strictly positive on $(0,t_1]$. Now, suppose by contradiction that $t_2(>t_1)$ is the smallest zero point of $F_1$, and so $F_1>0$ on $(0, t_2)$. Then we claim that $F'_1(t_2) \leq 0$. If not, by continuity, $F'_1$ is strictly positive in a small interval $(t_3,t_2]$ for some time $t_3$ satisfying $0<t_3<t_2$. This implies that $F_1$ is strictly monotone increasing on $(t_3,t_2]$ and then negative due to the fact that $F_1(t_2)=0$, a contradiction. We then verify the claim $(F'_1(t_2) \leq 0)$. Letting $t=t_2$ in the inequality , noting the fact that $F_1(t_2)=0$, $F'_1(t_2) \leq 0$ and $F_1 \geq 0$ on $[0,t_2]$, we come to a contradiction. And we show that $F_1 >0$ for $t>0$ also in this case.
Therefore, coming back to , we may ignore the last term, and then we have $$e^{2t}F_1(t) \ge F_1(0)+m(0)C_{f,g}\e\int_0^te^{2s}ds \ge \frac{1}{2}m(0)C_{f,g}\e (e^{2t}-1),$$ from which, finally, we get the lower bound of $F_1(t)$ in the form $$\label{eq:bound_F_1}
F_1(t)>\frac{1-e^{-2}}{2}m(0)C_{f,g}\e
\quad\text{for $t\ge1$}.$$
Note that we have to cut off the time because $f$ can vanish and so $F_1(0)$ can be equal to 0, due to our assumption on the data. If $f$ is not identically equal to zero, then the lower bound of $F_1$, i.e. , holds for all $t\geq 0$.
Lower bound for the functional
==============================
By Hölder inequality and using the compact support of the solution , we have $$\label{7}
\int_{\R^n}|u(x,t)|^pdx\ge C_2(1+t)^{-n(p-1)}|F_0(t)|^p
\quad\text{for $t\ge0$},$$ where $C_2=C_2(n,p,R)>0$. Plugging this inequality into and then integrating it over $[0,t]$, we have $$\label{18}
F_0(t) \geq C_3\int_0^tds
\int_0^s(1+r)^{-n(p-1)}F_0(r)^pdr
\quad\text{for $t\ge0$},$$ where $C_3:=C_2m(0)>0.$
Moreover, by Hölder inequality, Lemma \[lem1\] and estimate , we get $$\begin{split}
\int_{\R^n} |u(x,t)|^p dx
&\geq \left( \int_{\R^n} |\psi_1(x,t)|^{p/(p-1)}\right)^{1-p} |F_1(t)|^p\\
&\geq C_1^{1-p} \left(\frac{1-e^{-2}}{2}m(0)C_{f,g}\right)^p \e^p (1+t)^{(n-1)(1-p/2)} \quad\text{for $t\ge1$}.
\end{split}$$ Plugging this inequality into we have $$\label{19}
F'_0(t)\ge C_4\e^p\int_{1}^t(1+s)^{(n-1)(1-p/2)}ds
\quad\text{for $t\ge1$},$$ where $$C_4:=m(0)C_1^{1-p} \left(\frac{1-e^{-2}}{2}m(0)C_{f,g}\right)^p >0.$$ Integrating over $[1, t]$, we obtain $$\label{20}
\begin{split}
F_0(t)
& \ge C_4\e^p\int_{1}^tds
\int_{1}^s(1+r)^{(n-1)(1-p/2)}dr\\
& \ge C_4\e^p(1+t)^{-(n-1)p/2}\int_{1}^tds
\int_{1}^s (r-1)^{n-1}dr\\
& =\frac{C_4}{n(n+1)}\e^p(1+t)^{-(n-1)p/2} (t-1)^{n+1}
\quad\text{for $t\ge1$}.
\end{split}$$
Iteration argument
==================
Now we come to the iteration argument to get the upper bound of the lifespan estimates. First we make the ansatz that $F_0(t)$ satisfies $$\label{21}
F_0(t) \ge D_j(1+t)^{-a_j}(t-1)^{b_j} \quad\text{for $t\ge 1,\quad j=1,2,3,\dots$}$$ with positive constants $D_j,a_j,b_j$, which will be determined later. Due to , note that is true when $j=1$ with $$\label{22}
D_1=\frac{C_4}{n(n+1)}\e^p,
\quad a_1=(n-1)\frac{p}{2},
\quad b_1=n+1.$$ Plugging into , we have $$\begin{split}
F_0(t)
& \ge C_3D_j^p\int_{1}^tds
\int_{1}^s(1+r)^{-n(p-1)-pa_j}(r-1)^{pb_j}dr\\
& \ge C_3D_j^p(1+t)^{-n(p-1)-pa_j}\int_{1}^tds
\int_{1}^s(r-1)^{pb_j}dr\\
& \ge \frac{C_3D_j^p}{(pb_j+2)^2}(1+t)^{-n(p-1)-pa_j}(t-1)^{pb_j+2} \quad\text{for $t\ge1$}.
\end{split}$$ So we can define the sequences $\{D_j\}_{j\in\N}$, $\{a_j\}_{j\in\N}$, $\{b_j\}_{j\in\N}$ by $$\label{23}
D_{j+1}\ge\frac{C_3D_j^p}{(pb_j+2)^2},
\quad
a_{j+1}=pa_j+n(p-1),
\quad
b_{j+1}=pb_j+2$$ to establish $$F_0(t)\ge D_{j+1}(1+t)^{-a_{j+1}}(t-1)^{b_{j+1}}\quad\text{for $t\ge1$}.$$ It follows from and that for $j=1,2,3,\dots$ $$a_j=p^{j-1}\left((n-1)\frac{p}{2}+n\right)-n,
\quad
b_j=p^{j-1}\left(n+1+\frac{2}{p-1}\right)-\frac{2}{p-1}.$$ Employing the inequality $$b_{j+1}=pb_j+2\le p^j\left(n+1+\frac{2}{p-1}\right)$$ in , we have $$\label{24a}
D_{j+1}\ge C_5\frac{D_j^p}{p^{2j}},$$ where $$C_5:=\frac{C_3}{\left(n+1+\d\frac{2}{p-1}\right)^2} >0.$$ From it holds that $$\begin{split}
\log D_j&\geq p\log D_{j-1}-2(j-1)\log p+\log C_5\\
&\geq p^2\log D_{j-2}-2\big(p(j-2)+(j-1)\big)\log p+(p+1)\log C_5\\
&\geq \cdots\\
&\geq p^{j-1} \log D_1 - \sum_{k=1}^{j-1} 2 p^{k-1}(j-k) \log p + \sum_{k=1}^{j-1} p^{k-1} \log C_5\\
&= p^{j-1} \left(\log D_1-\sum_{k=1}^{j-1}\frac{2k\log p-\log C_5}{p^k}\right),
\end{split}$$ which yields that $$D_j\ge\exp\left\{p^{j-1}\left(\log D_1-S_p(j)\right)\right\},$$ where $$S_p(j):=\sum_{k=1}^{j-1}\frac{2k\log p-\log C_5}{p^k}.$$ We know that $\sum_{k=0}^{\infty} x^k = 1/(1-x)$ and $\sum_{k=1}^{\infty} kx^k= x/(1-x)^2$ when $|x|<1$. Then $$S_p(\infty) := \lim_{j \to \infty} S_p(j) = \log\{ C_5^{p/(1-p)} p^{2p/(1-p)^2}\}.$$ Moreover $S_p(j)$ is a sequence definitively increasing with $j$. Hence we obtain that $$D_j\ge\exp\left\{p^{j-1}\left(\log D_1-S_p(\infty)\right)\right\},~~~j\ge 2.$$ Turning back to , we have $$\label{27}
\begin{aligned}
F_0(t)\ge(1+t)^n(t-1)^{-2/(p-1)}\exp\left(p^{j-1}J(t)\right)
\quad\mbox{for}\ t\ge 1,
\end{aligned}$$ where $$\begin{split}
J(t)=& -\bigg((n-1)\frac{p}{2}+n\bigg)\log(1+t)
+\bigg(n+1+\frac{2}{p-1}\bigg)\log(t-1)\\
&+\log D_1-S_p(\infty).
\end{split}$$ For $t\geq 2$, by the definition of $J(t)$, we have $$\begin{split}
J(t)\geq& -\bigg((n-1)\frac{p}{2}+n\bigg)\log(2t)+\bigg(n+1+\frac{2}{p-1}\bigg)\log \bigg(\frac{t}{2}\bigg)\\
&+\log D_1-S_p(\infty)\\
=&\ \frac{\gamma(p, n)}{2(p-1)}\log t+\log D_1-\bigg((n-1)\frac{p}{2}+2n+1+\frac{2}{p-1}\bigg)\log 2-S_p(\infty)\\
=&\ \log \big(t^{\gamma(p, n)/\{2(p-1)\}}D_1\big)-C_6,\\
\end{split}$$ where $$C_6:=\bigg((n-1)\frac{p}{2}+2n+1+\frac{2}{p-1}\bigg)\log 2+S_p(\infty).$$
Thus, if $$\begin{aligned}
t>C_7\varepsilon^{-2p(p-1)/\gamma(p, n)}
\end{aligned}$$ with $$C_7:=\Big(\frac{n(n+1)e^{C_6+1}}{C_4}\Big)^{2(p-1)/\gamma(p, n)}>0,$$ we then get $J(t)>1$, and this in turn gives that $F_0(t)\rightarrow \infty$ by letting $j\rightarrow \infty$ in . Since we assume that $t\geq2$ in the above iteration argument, we require $$0<\e\leq \e_0 := \left(\frac{C_7}{2} \right)^{\frac{\gamma(p,n)}{2p(p-1)}}.$$ Therefore we get the desired upper bound, $$\begin{aligned}
T\leq C_7\varepsilon^{-2p(p-1)/\gamma(p, n)}
\end{aligned}$$ for $0<\e\leq \e_0$, and hence we finish the proof of Theorem \[thm1\].
Proof for Theorem \[thm2\] and Theorem \[thm3\]
===============================================
To prove the theorems in low dimensions, we proceed similarly as for Theorem \[thm1\], but we change the first step of the iteration argument to get the desired improvement.
From , using and noting that $F_0$ is positive, we have $$\begin{aligned}
F_0'(t)\geq \frac{m(0)}{m(t)}F_0'(0)\geq C_8\e,
\end{aligned}$$ where $$C_8:=m(0)\int_{\R^n}g(x)dx > 0$$ due to the assumption on $g$. The above inequality implies that $$\label{81}
F_0(t)\geq C_8\e t \quad\text{for $t\geq 0$}.$$
By and , we have $$\label{81a}
\int_{\R^2}|u(x, t)|^pdx\geq C_{9}\e^p(1+t)^{-n(p-1)}t^p,$$ with $C_{9}:=C_2C_8^p>0$. Plugging into and integrating it over $[0, t]$ we come to $$\label{83}
\begin{split}
F_0(t)&\geq m(0)C_{9}\e^p\int_0^tds\int_0^s(1+r)^{-n(p-1)}r^p dr\\
&\geq m(0)C_{9}\e^p(1+t)^{-n(p-1)}\int_0^tds\int_0^s r^pdr\\
&=C_{10}\e^p(1+t)^{-n(p-1)}t^{p+2}
\quad\text{for $t\ge0$}
\end{split}$$ with $$C_{10}:=\frac{m(0)C_{9}}{(p+1)(p+2)}>0.$$
Note that the inequality improves the lower bound of for $n=2$ and $1<p<2$, and for $n=1$ and $p>1$. Hence we may establish the improved lifespan estimate as stated in Theorem \[thm2\] and Theorem \[thm3\].
In a similar way as in the last section, we define our iteration sequences, $\{\widetilde{D}_j\}, \{\widetilde{a}_j\}, \{\widetilde{b}_j\}$, such that $$\label{85}
\begin{aligned}
F_0(t)\geq \widetilde{D}_j(1+t)^{-\widetilde{a}_j}t^{\widetilde{b}_j}
\quad\text{for $t\geq 0$ and $j=1, 2, 3, \dots$}
\end{aligned}$$ with positive constants, $\widetilde{D}_j, \widetilde{a}_j, \widetilde{b}_j$, and $$\nonumber
\begin{aligned}
\widetilde{D}_1=C_{10}\e^p, \quad \widetilde{a}_1=n(p-1), \quad \widetilde{b}_1=p+2.
\end{aligned}$$ Combining and , we have $$\begin{split}
F_0(t)&\geq C_3\widetilde{D}_j^p\int_0^tds\int_0^s(1+r)^{-n(p-1)-p\widetilde{a}_j}r^{p\widetilde{b}_j}dr\\
&\geq \frac{C_3\widetilde{D}_j^p}{(p\widetilde{b}_j+2)^2}(1+t)^{-n(p-1)-p\widetilde{a}_j}t^{p\widetilde{b}_j+2}
\quad\text{for $t\geq 0$}.\\
\end{split}$$ So the sequences satisfy $$\begin{aligned}
\widetilde{a}_{j+1}&=p\widetilde{a}_j+n(p-1),\\
\widetilde{b}_{j+1}&=p\widetilde{b}_j+2,\\
\widetilde{D}_{j+1}&\geq \frac{C_3\widetilde{D}_j^p}{(p\widetilde{b}_j+2)^2},
\end{aligned}$$ which means that $$\nonumber
\begin{aligned}
\widetilde{a}_{j}&=np^j-n,\\
\widetilde{b}_{j}&=\frac{p+1}{p-1}p^{j}-\frac{2}{p-1},\\
\widetilde{D}_{j+1} &\ge C_{11} \frac{\widetilde{D}_j^p}{p^{2j}},\\
\end{aligned}$$ where $C_{11}:= C_3(p-1)^2/[p(p+1)]^2$, from which we get $$\log\widetilde{D}_{j}\geq p^{j-1}\left(\log\widetilde{D}_1-\sum_{k=1}^{j-1}\frac{2k\log p-\log C_{11}}{p^k}\right).$$ Then proceeding as above we have $$\begin{aligned}
F_0(t)&\geq\widetilde{D}_j(1+t)^{n-np^j}
t^{p^j(p+1)/(p-1)-2/(p-1)}\\
&\geq (1+t)^n t^{-2/(p-1)}\exp\big(p^{j-1}\widetilde{J}(t)\big),
\end{aligned}$$ where $$\widetilde{J}(t):=-np\log(1+t)+\left(p\,\frac{p+1}{p-1}\right)\log t+\log \widetilde{D}_1-\widetilde{S}_p(\infty)$$ and $$\widetilde{S}_p(\infty) = \log\{ C_{11}^{p/(1-p)} p^{2p/(1-p)^2}\}.$$ Estimating $\wt{J}(t)$ for $t\ge1$ we get $$\begin{split}
\wt{J}(t)
&\geq -np\log(2t)+\left(p\,\frac{p+1}{p-1}\right)\log t+\log \widetilde{D}_1-\widetilde{S}_p(\infty)\\
&=\frac{\gamma(p,n)-2}{p-1} \log t+\log \widetilde{D}_1-\widetilde{S}_p(\infty)-np\log 2,
\end{split}$$ and then we obtain that $$\wt{J}(t)\ge\log\Big(t^{(\gamma(p,n)-2)/(p-1)}\widetilde{D}_1\Big)-C_{12}
\quad\text{for $t\geq 1$},$$ where $C_{12}:=\widetilde{S}_p(\infty)+np\log 2$. In particular, $$\gamma(p,n)-2=
\begin{cases}
p(3-p) &\text{if $n=2$}, \\
2p &\text{if $n=1$}.
\end{cases}$$ By the definition of $\widetilde{D}_1$, proceeding in the same way as that in the previous section, we get the lifespan estimate in Theorem \[thm2\] when $n=2$, and the lifespan estimate in Theorem \[thm3\] when $n=1$.
Proof for Theorem \[thm4\]
==========================
Let us come back to our initial equation , with $n=p=2$. In this case we introduce another multiplier $$\label{2problem_trans}
\l(t):=\exp\left(\frac{\mu_1}{2}\frac{(1+t)^{1-\beta}}{1-\beta}\right),$$ which yields $$\l'(t)=\frac{\mu_1}{2(1+t)^{\beta}}\l(t)$$ and $$\l''(t)=\left( \frac{\mu_1^2}{4(1+t)^{2\beta}} - \frac{\beta \mu_1}{2(1+t)^{\beta+1}}\right)\l(t).$$ Introducing a new unknown function by $$w(x,t):=\l(t)u(x,t),$$ then it is easy to get $$w_t=\frac{\mu_1}{2(1+t)^\beta}\l u+\l u_t$$ and $$w_{tt}=\frac{\mu_1^2}{4(1+t)^{2\beta}} \l u
-\frac{\beta\mu_1}{2(1+t)^{\beta+1}} \l u
+\frac{\mu_1}{(1+t)^\beta} \l u_t+ \l u_{tt}.$$ With this in hand the equation can be transformed to $$\label{eq:main_problem_n=p=2}
\left\{
\begin{aligned}
& w_{tt}-\Delta w = Q w + \l^{-1} |w|^2 \\
& w(x,0)=0, \quad w_t(x,0)= \l(0) \e g(x)
\end{aligned}
\right.$$ where $$Q=Q(t):= \frac{\mu_1^2}{4(1+t)^{2\beta}} - \frac{\beta \mu_1}{2(1+t)^{\beta+1}} + \frac{\mu_2}{(1+t)^{\alpha+1}}.$$
A key property of the function $Q$ is its positivity. Indeed, we can write this function as $Q=\widetilde{Q}/(1+t)^{\beta+1}$, where $$\widetilde{Q}=\widetilde{Q}(t):= \frac{\mu_1^2}{4(1+t)^{\beta-1}} - \frac{\beta \mu_1}{2} + \frac{\mu_2}{(1+t)^{\alpha-\beta}},$$ and so it is enough to check the positivity of $\widetilde{Q}$. If $\alpha=\beta$, then $\widetilde{Q}$ is strictly decreasing to $\mu_2-\beta\mu_1/2$, that is positive by our assumption. If $\alpha<\beta$, than we can easily find the minimum $t_{0}$ of $\widetilde{Q}$, that is $$t_0 = -1+\left( \frac{\mu_1^2(\beta-1)}{4\mu_2(\beta-\alpha)} \right)^{\frac{1}{2\beta-\alpha-1}},$$ and verify that the condition in is equivalent to $\widetilde{Q}(t_0)\ge0$.
Observe that:
- when $\alpha<\beta$, the condition can be replaced by the more strong but easier condition $$\mu_2 \ge \frac{\mu_1^2}{4}\frac{\beta-1}{\beta-\alpha},$$ that is equivalent to ask that $t_0 \le 0$, so that $\widetilde{Q}$ is increasing and positive for $t>0$;
- when $\alpha>\beta$, $\widetilde{Q}$ is strictly decreasing to $-\beta\mu_1/2 <0$, and then we have no chance to achieve the positivity of this function for all the time.
We can rewrite the function $Q$ also as $$Q(t)=\frac{1}{4(1+t)^2}\left[\left(\frac{\mu_1}{(1+t)^{\beta-1}}-\beta\right)^2 + \frac{4\mu_2}{(1+t)^{\alpha-1}} - \beta^2\right],$$ which implies some connection with the definition of $\delta$ in the scale invariant case ($\beta=1$) with positive mass and $\alpha=1$.
Now, it is well-known that our integral equation is of the form $$\label{eq:w_int}
\begin{split}
w(x,t) =&\ \frac{\l(0)\e}{2\pi} \int_{|x-y|\le t} \frac{g(y)}{\sqrt{t^2-|x-y|^2}}dy\\
&+\frac{1}{2\pi}\int_0^td\tau \int_{|x-y|\le t-\tau} \frac{Q(\tau)w(y,\tau)+\l^{-1}(\tau)|w(y,\tau)|^2}{\sqrt{(t-\tau)^2-|x-y|^2}}dy.
\end{split}$$ Before we can move forward, we need the positivity of the solution.
\[lem:w\_pos\] Under the assumption of Theorem \[thm4\], the solution $w$ of is positive.
Let $\wt{w}=\wt{w}(x,t)$ be the classical solution of the Cauchy problem $$\left\{
\begin{aligned}
& \wt{w}_{tt}-\Delta \wt{w} = Q |\wt{w}| + \l^{-1} |\wt{w}|^2, \quad \mbox{in}\ \R^n\times[0,\infty),\\
& \wt{w}(x,0)=0, \quad \wt{w}_t(x,0)= \l(0) \e g(x), \quad x \in \R^n.
\end{aligned}
\right.$$ It is clear from the analogous of for $\wt{w}$ that this function is positive, and then satisfies the system . But $u$ is the unique solution of , and so $w=\lambda u$ is the unique solution of . Then $w \equiv \wt{w} \ge 0$.
By Lemma \[lem:w\_pos\], we can neglect the second term on the right-hand side of . Using the relation $|y|\le R, |x|\le t+R$ due to the support property in the first term on the right-hand side, from which the inequalities $$\begin{gathered}
t-|x-y| \le t-||x|-|y|| \le t-|x|+R \quad\mbox{for}\ |x|\ge R,\\
t+|x-y|\le t+|x|+R \le 2(t+R),\end{gathered}$$ we obtain that $$w(x,t) \ge \frac{\l(0)\e}{2\sqrt{2}\pi\sqrt{t+R}\sqrt{t-|x|+R}}\int_{|x-y|\le t} g(y)dy \quad\text{for $|x|\ge R$}.$$ If we assume $|x|+R\le t$, which implies $|x-y| \le t$ for $|y| \le R$, we get $$\int_{|x-y|\le t} g(y)dy = {\left\lVertg\right\rVert}_{L^1(\R^2)},$$ and then we obtain $$\label{37}
w(x,t) \ge \frac{\l(0){\left\lVertg\right\rVert}_{L^1(\R^2)}}{2\sqrt{2}\pi\sqrt{t+R}\sqrt{t-|x|+R}} \e \quad\text{for $R \le |x| \le t-R$}.$$
Defining the functional $$W(t) := \int_{\R^2} w(x,t)dx,$$ we reach to $$W''(t) = Q(t)W(t) +\l^{-1}(t) \int_{\R^2} |w(x,t)|^2dx.$$ Noting that $W$ is also positive by Lemma \[lem:w\_pos\] (or by the fact that $W=\lambda F$), then we have $$W''(t) \ge \l^{-1}(t) \int_{\R^2} |w(x,t)|^2dx \ge \int_{R \le |x| \le t-R} |w(x,t)|^2dx \quad\text{for $t \ge 2R$},$$ where we used the fact that $\l^{-1}(t) >1$. Plugging into the right-hand side of the above inequality, we have $$W''(t) \ge \frac{\l(0)^2{\left\lVertg\right\rVert}^2_{L^1(\R^2)}}{8\pi^2(t+R)}\e^2 \int_{R \le |x| \le t-R} \frac{1}{t-|x|+R}dx,$$ which yields $$W''(t) \ge \frac{\l(0)^2 {\left\lVertg\right\rVert}^2_{L^1(\R^2)}}{4\pi(t+R)}\e^2 \int_{R}^{t-R} \frac{r}{t-r+R}dr \quad\text{for $t \ge 2R$}.$$ Then, the rest of the demonstration is exactly the same as that of Theorem 4.1 in [@T], and we omit the details here.
We want to emphasize that the results stated in our four Theorems are still true if we have no damping term, that is if $\mu_1=0$. In fact, a key point in our proofs was to introduce multipliers to absorb this term. If $\mu_1=0$, then $m\equiv\lambda\equiv1$ and the demonstrations proceed analogously. In this case we do not need any additional condition on $\mu_2$ in Theorem \[thm4\], but it is enough to ask $\mu_2>0$.
Acknowledgement {#acknowledgement .unnumbered}
===============
The first author is partially supported by Zhejiang Province Science Foundation (LY18A010008), NSFC (11501273, 11726612), Chinese Postdoctoral Science Foundation (2017M620128, 2018T110332), CSC(201708330548), the Scientific Research Foundation of the First-Class Discipline of Zhejiang Province (B)(201601). The second author is partially supported by the Global Thesis study award in 2016–2017, University of Bari. And he is also grateful to Future University Hakodate for hearty hospitality during his stay there, 12/01/2018–04/04/2018. The third author is partially supported by the Grant-in-Aid for Scientific Research (C) (No.15K04964) and (B)(No.18H01132), Japan Society for the Promotion of Science, and Special Research Expenses in FY2017, General Topics (No.B21), Future University Hakodate. This work started when the third author was working in Future University Hakodate.
|
---
abstract: 'High resolution FUV echelle spectra showing absorption features arising from and CO gas in the [$\beta$ Pictoris]{} circumstellar (CS) disk were obtained on 1997 December 6 and 19 using the Space Telescope Imaging Spectrograph (STIS). An unsaturated spin-forbidden line of at 1613.376 Å not previously seen in spectra of [$\beta$ Pictoris]{} was detected, allowing for an improved determination of the column density of at zero velocity relative to the star (the stable component), [$N(^{3}\rm{P})$]{}$~=~(2 - 4) \times 10^{16}$ cm$^{-2}$. Variable components with multiple velocities, which are the signatures of infalling bodies in the [$\beta$ Pictoris]{} CS disk, are observed in the $\lambda\lambda$1561 and 1657 multiplets. Also seen for the first time were two lines arising from the metastable $^{1}$D level of carbon, at 1931 Å and 1463 Å. The results of analysis of the CO $A-X$ (0-0), (1-0), and (2-0) bands are presented, including the bands arising from $^{13}$CO, with much better precision than has previously been possible due to the very high resolution provided by the STIS echelle gratings. Only stable CO gas is observed, with a column density $N(\rm{CO})~=~(6.3 \pm 0.3) \times 10^{14} \ \rm{cm}^{-2}$. An unusual ratio of the column densities of $^{12}$CO to $^{13}$CO is found ($R$ = $15 \pm 2$). The large difference between the column densities of and CO indicates that photodissociation of CO is not the primary source of gas in the disk, contrary to previous suggestion.'
author:
- 'A. Roberge and P. D. Feldman'
- 'A. M. Lagrange'
- 'A. Vidal-Madjar and R. Ferlet'
- 'A. Jolly, J. L. Lemaire, and F. Rostas'
title: |
High resolution HST STIS spectra of and CO in the\
circumstellar disk
---
Introduction
============
[$\beta$ Pictoris]{} is the most extensively studied of the young planetary systems discovered in the last decade and a half. It is a bright Southern hemisphere star (type A5 V), located about 19.3 pc distant from the Sun, with a systemic radial velocity of 20 km s$^{-1}$ (for a review of the [$\beta$ Pictoris]{} system, see @vid98). It was observed in 1983 by the IRAS satellite to have a large excess of emission at infrared wavelengths. This was referred to as the Vega-like phenomenon and was identified as arising from an edge-on circumstellar (hereafter CS) dust disk, presumed to be associated with planetary formation [@smi84]. It was soon determined through absorption spectroscopy that there was CS gas associated with the dust as well.
A large body of evidence has accumulated indicating that there are comet-like bodies present in the [$\beta$ Pictoris]{} CS disk. Collisions between dust particles are expected to produce submicron fragments which should be expelled from the system by radiation pressure on time scales much shorter than any plausible stellar age. Thus there must be a secondary source of particles; one model for the production of dust and gas in the CS disk focusses on evaporating comets and is called the Orbiting Evaporating Bodies model (OEB). In this picture, the comets orbit the star at several tens of AU, and thus, the [$\beta$ Pictoris]{} CS disk is a kind of “gigantic multi-cometary tail with its natural constituents: gas and dust” [@lec96]. Spectra of [$\beta$ Pictoris]{} show variable redshifted absorption features arising from gas infalling toward the star at high velocities (and infrequently, blueshifted features as well); these features are best attributed to the evaporation of star-grazing comets, called the Falling Evaporating Bodies scenario (FEB) [@beu90]. Also, gas at close to 20 km s$^{-1}$ (zero radial velocity relative to the star) is identified in all observations and is called the stable gas component. This gas is difficult to understand, as modeling indicates that it should be expelled from the system by radiation pressure; a continuous source for this gas is required.
Neutral carbon and carbon monoxide have been observed in HST-GHRS UV absorption spectra of [$\beta$ Pictoris]{}; carbon monoxide is the only molecule detected in the CS disk to date [@vid94; @jol98]. Since CO and can be dissociated and ionized by interstellar UV photons on time scales of the order of 200 years, both must be continuously replenished. Carbon monoxide, in particular, is difficult to reform after dissociation in the [$\beta$ Pictoris]{} environment. Thus, the presence of these species indicates that a secondary source for this gas should exist, just as for the CS dust [@vid94]. @jol98 found the column densities of and CO to be comparable, around $10^{15}$ cm$^{-2}$; since their rates of destruction are also comparable, this was taken as evidence that the is produced by photodissociation of CO, which evaporates from comets orbiting at various distances and velocities. However, the column density was determined from a heavily saturated multiplet and is therefore quite uncertain. In the hopes of further constraining the characteristics of and CO in the [$\beta$ Pictoris]{} system, we have reinvestigated the transitions observed in the GHRS data, as well as some that were not seen due to the relatively low spectral resolution of GHRS compared with that of the STIS high resolution echelle.
Observations
============
HST STIS high resolution echelle spectra of [$\beta$ Pictoris]{} were obtained on 1997 December 6 and 1997 December 19, covering the wavelength range from 1459 Å to 2888 Å in six exposures each day. Table 1 shows the log of observations. All the absorption features discussed in this paper appear in either the first or second data set listed for each day (o4g001010/o4g002010 or o4g001020/o4g002020).The data were initially reduced and calibrated using the STScI IRAF package *calstis v1.8*. Spectra with a signal-to-noise ratio of around 10 were achieved. Examination of the errors in the flux values showed that the error propagation calculation had not been performed correctly and that thestated errors were too small. Therefore the data were re-calibrated using *calstis v1.9a*, correcting the underestimate of the measurement errors.
The particular advantage of this data set over previous comparable ones is the very high spectral resolution achieved. The instrumental line spread function using the E140H grating is well described by a Gaussian with a FWHM of 1 pixel, corresponding to FWHM = $\lambda$/220,000 [@sah99]. However, since the detector undersamples the line spread function, the effective resolution to separate two adjacent lines with this grating is only $R$ = 110,000. Using the E230H grating, the FWHM is about 2 pixels, corresponding to FWHM = $\lambda$/110,000 [@sah99].
Analysis
========
\] $\lambda$1613 line
---------------------
This previously undetected spin-forbidden transition $(^{3}$P$_{0}\:
\rightarrow \:^{1}$P$_{1})$ at 1613.376 Å was clearly observed on both days. It is unsaturated, which allows for an improved determination of the column density of . As there was no change in the line between the two days of observation, the spectra were averaged together to improve the S/N; the result is shown in Figure 1. Unfortunately, the other two fine structure lines in the multiplet were not reliably detected. Atomic data were taken from @mor91 and the continuum around the line was fit with a sixth-degree polynomial. Voigt line profiles were used to generate transmission functions, which were then convolved with a Gaussian instrumental line spread function with FWHM = $\lambda$/220,000 to create model spectra. The $\chi^{2}$ statistic between the model and the data was then minimized to determine the best velocity centroid, $v$, column density of in the ground level, $N(^{3}\rm{P}_{0})$, and Doppler broadening parameter, $b$; error bars for these parameters were determined from the contours of $\chi^{2}$. Since the line was unsaturated, the column density was also determined from the equivalent width of the line and $b$ and $v$ were determined from a simple Gaussian fit to the line, as a check on the results of the $\chi^{2}$ minimization.
[($^{3}$P)]{} $\lambda\lambda$1561 and 1657 multiplets
------------------------------------------------------
The central portion of the 1561 Å and 1657 Å multiplets, arising from the $^{3}$P ground term, were heavily saturated in the STIS data. However, this allowed us to examine the smaller variable red and blueshifted features in the spectra which are the signatures of infalling comets in the CS disk. In Figures 2 and 3, one can easily see absorption lying at higher redshift in the December 6 data that is not visible in the December 19 data. A model of the multiplet which contains only one velocity component is unable to reproduce the absorption dip at 1561.5 Å or the absorption at higher redshift in the December 6 data.
Due to the fact that the multiplets were extremely saturated and that the multiple velocity components were blended together, we were not able to perform $\chi^{2}$ minimizations to find unique best values for the model parameters. absorption features in our data set were analyzed and the velocities of their multiple velocity components found. Models of the $\lambda$1561 multiplet were constructed, containing the same number of components as were found in the features, with roughly the same velocities. The parameters of the stable component at $v$ = 20 km s$^{-1}$ were set to those determined from the analysis of the unsaturated 1613.376 Å line. The remaining parameters were then adjusted to find the best model by eye. This model was compared to the 1657 Å multiplet to confirm that the values found in this way were reasonable. A small constant value ($\approx 2 \times 10^{-13}$ erg s$^{-1}$ cm$^{-2}$ Å$^{-1}$) was subtracted from the 1657 Å data to bring the baselines of the totally saturated features to zero.
[($^{1}$D)]{} $\lambda\lambda$1931 and 1463 lines
-------------------------------------------------
These lines arise from the excited, metastable $^{1}$D state of the ground configuration and have not previously been observed in spectra of [$\beta$ Pictoris]{}. The $^{1}$D level has a lifetime of about 4000 s, lying about 10,000 cm$^{-1}$ above the ground level; C atoms in this state may be produced during the photodissociation of CO in solar system comet comae [@toz98]. The 1931 Å lines were analyzed using the same model generating and $\chi^{2}$ minimization procedures described in § 3.1; the atomic data used were from @hib93. Although there may be some reason to suspect that the strong central absorption features near 1931.05 Å contain multiple velocity components, since this behavior is seen in lines arising from excited levels of other atomic species, the $\chi^{2}$ minimizations indicated that stable unique solutions containing more than one component in the central absorption features could not be found. Thus, best models were found using only one velocity component. However, the difficulty involved in modeling a saturated, blended line is such that there may well be other undetected components present, so models with two components in the central absorption feature were compared to the data by eye. The 1463 Å spectra, shown in Figure 5, were so noisy that they served only to roughly confirm the values found from the 1931 Å line. A small constant flux value ($\approx 1 \times 10^{-13}$ erg s$^{-1}$ cm$^{-2}$ Å$^{-1}$) was subtracted from the 1463 Å data to bring the baselines of the saturated lines to zero.
CO Fourth Positive band system
------------------------------
The (0-0), (1-0), and (2-0) bands of the Fourth Positive system of CO ($A \: ^{1}\Pi \:-\: X \:^{1}\Sigma ^{+}$) did not vary between the two days of observation. The spectra were therefore averaged together to improve the S/N; the result appears in Figure 6. Only a single velocity component was observed. Note the detection of bands arising from $^{13}$CO, the perturbation band $e \: ^{3}\Sigma^{-} \:-\: X \:^{1}\Sigma ^{+}$ (1-0), not previously observed in spectra of [$\beta$ Pictoris]{}, and the strong perturbation band $d \: ^{3}\Delta \:-\: X \:^{1}\Sigma ^{+}$ (5-0).
Models were generated using wavelengths and oscillator strengths from @mor94. Energies of the ground state levels were calculated using the Dunham coefficients from @far91 and LTE assumed in order to determine the population of the rotational levels. The parameters of the model are the rotational excitation temperature, $T$, the Doppler broadening parameter, $b$, the column density of $^{12}$CO in the ground vibrational state, $N(^{12}\rm{CO})$, the column density of $^{13}$CO in the ground vibrational state, $N(^{13}\rm{CO})$, and the velocity centroid, $v$. $\chi^{2}$ minimization was then performed on all three bands simultaneously.
Results
=======
[($^{3}$P)]{} stable component
------------------------------
The \] $\lambda$1613.376 line showed only one velocity component at 20 km s$^{-1}$, the systemic velocity of the star; the nominal uncertainty in the absolute wavelength calibration of STIS leads to an error in velocity determinations of about 1 km s$^{-1}$ [@sah99]. This velocity and the fact that the line did not change between the two days of observation identifies the line as arising from stable gas. The results of the $\chi^{2}$ minimizations for all the features analyzed appear in Table 2. The column density for stable carbon in the ground level, $^{3}$P$_{0}$, determined from $\chi^{2}$ minimization was found to agree with that determined from the equivalent width of the line. Similarly the $v$ and $b$ determined from $\chi^{2}$ minimization agreed with the values found from fitting a simple Gaussian to the unsaturated line.
Using a 3-$\sigma$ upper limit on the column density of in the $^{3}$P$_{2}$ level (from the non-detection of the fine structure line at 1614.5068 Å) and assuming LTE, a firm upper limit of 100 K on the excitation temperature of $^{3}$P carbon in the stable component was found. From analysis of the 1561 Å multiplet, discussed below, it was found that the excitation temperature of the stable component must be greater than about 50 K, or the multiplet could not be reasonably modeled. This range in temperature allows us to determine that the total column density of stable in the $^{3}$P ground term is $(2~-~4)~\times~10^{16}$ cm$^{-2}$.
This column density is more than an order of magnitude larger than the total $^{3}$P column density found by @jol98 from GHRS data taken in November 1994, [$N(^{3}\rm{P})$]{}$~=~2 \times 10^{15}$ cm$^{-2}$. Either the abundance of stable has varied over the three years between observations or, as is more likely, the difficult modeling of the heavily saturated $\lambda$1561 multiplet in the GHRS data led to an inaccurate column density. Models with column densities of $10^{16}$ cm$^{-2}$ in the stable component were compared to the 1994 GHRS spectrum of the 1561 Å multiplet and found to fit the data equally well as models with the lower column density. We found the Doppler broadening parameter of the stable [($^{3}$P)]{} to be $1.3 \pm 0.5 \ \rm{km} \ \rm{s}^{-1}$. The previous work on the GHRS data found a Doppler broadening parameter of 4.2 km s$^{-1}$, which was much greater than the parameter found for the other atomic species and CO [@jol98]. Our smaller $b$ value is equal to that found for CO in our data; we thus do not see any excess kinetic energy in the motions of the atoms.
[($^{3}$P)]{} multiple velocity components
------------------------------------------
The difficulty in modeling a saturated, blended multiplet is formidable, and the values determined for the multiple velocity components of [($^{3}$P)]{} from the 1561 Å multiplet are not reliable. However, this analysis did confirm that models with column densities of $10^{16}$ cm$^{-2}$ in the stable component could reasonably fit the 1561 Å spectra. We can also conclude that the total column density of [($^{3}$P)]{} in the variable components is about $10^{14}$ cm$^{-2}$ on December 6 and about $10^{15}$ cm$^{-2}$ on December 19, to within an order of magnitude or so. It is also clear that the variable features are generally better fit with higher excitation temperatures and larger $b$ parameters, but these values are very poorly determined.
[($^{1}$D)]{}
-------------
Using models with only one velocity component in the strong central absorption features, two components arising from the excited $^{1}$D level of were found in the December 6 data, only one in the December 19 data. The best models are overplotted on the spectra in Figures 4 and 5. Note that the central absorption features do not appear at the same velocity on the two days, that they also do not appear at the systemic velocity of the star, and that the column density of [($^{1}$D)]{} changes significantly between the two days of observation. This would seem to indicate that there is no stable component for [($^{1}$D)]{}. Models with two velocity components in the central absorption feature, which were compared to the data by eye, indicate that there could be a “stable” component which has the same column density on both days. However, it must be at about 22-23 km s$^{-1}$, which is significantly different from the systemic velocity of [$\beta$ Pictoris]{}. These data are not able to conclusively determine the velocity structure present in the [($^{1}$D)]{} gas; observation of unsaturated lines at similarly high resolution will be necessary.
CO
--
The CO bands showed no multiple velocity component structure and no change between the two days of observation; the absorbing gas is thus entirely associated with the stable component at the systemic velocity of the star. The best model is shown overplotted on the data in Figure 6. The rotational excitation temperature and the ratio of $N(^{12}\rm{CO})$ to $N(^{13}\rm{CO})$ show no significant change from the values found by @jol98. $R(^{12}\rm{CO}/^{13}\rm{CO})$ = $15 \pm 2$ is quite small compared to typical values found in the ISM, e.g. $R$ = $150 \pm 27$ for the diffuse clouds toward $\zeta$ Ophiuchi [@she92]. However, chemical fractionation at low kinetic temperatures can explain this unusual ratio, as discussed below. The column densities of both $^{12}$CO and $^{13}$CO were found to be about 1/3 the values found from the GHRS data. Note that the column density of $^{12}$CO is about 50 times smaller than the column density of stable [($^{3}$P)]{}.
Discussion
==========
Although CO in solar system comet comae is photodissociated by solar FUV photons, the CO in the [$\beta$ Pictoris]{} CS disk is primarily destroyed by interstellar photons. The dissociation energy for CO is 11.1 eV; thus CO may only be dissociated by photons with wavelengths shortward of $\sim$ 1100 Å. The type A5 star [$\beta$ Pictoris]{} lacks the strong FUV emission lines created in the Sun’s chromosphere; thus it emits very little flux in the FUV. The only source for CO-dissociating photons at [$\beta$ Pictoris]{} is therefore the interstellar UV radiation field. Since CO photodissociates primarily through discrete line absorptions, self-shielding can have a strong effect on the abundance of CO in interstellar clouds [@van88]. However, this is unlikely to occur in the [$\beta$ Pictoris]{} CS disk because of the very small transverse dimension of the disk. Thus, the photodissociation rate for $^{12}$CO in the [$\beta$ Pictoris]{} disk should be equal to the unshielded value and should also be equal to the value for $^{13}$CO. Since $^{13}$CO is not selectively dissociated in this situation, its high abundance was explained by the reaction $$^{13}\rm{C}^{+} + ^{12}\rm{CO} \;
\rightleftharpoons \; ^{13}\rm{CO} + ^{12}\rm{C}^{+} + 35 \ \rm{K}$$ which favors the production of $^{13}$CO at gas kinetic temperatures below 35 K [@jol98]. Assuming chemical equilibrium and that isotopic exchange is much more important than photodissociation for both $^{12}$CO and $^{13}$CO [@she92], $$\frac{n(^{12}\rm{CO})}{n(^{13}\rm{CO})} = \exp
\left(\frac{-35}{T_{kin}}\right) \left(\frac{^{12}\rm{C}}{^{13}\rm{C}}\right) =
15 \pm 2.$$ The rotational excitation temperature of CO often does not accurately describe the gas kinetic temperature in diffuse environments [@wan97]. However, assuming a $(^{12}\rm{C}/^{13}\rm{C})$ ratio, we may use the above expression to estimate the true gas kinetic temperature of the CO.
The average carbon isotopic ratio in the local ISM is $\sim \ 60-70$ [@lan93] and the typical solar system value found in comets is $(^{12}\rm{C}/^{13}\rm{C}) \ = \ 89$ [@jew97]. Using the range $(^{12}\rm{C}/^{13}\rm{C}) \ = \ 89-60$, we find that the gas kinetic temperature of the carbon monoxide is 20 K $ - $ 25 K, indicating that the CO gas is indeed colder than the stable gas. This suggests that the and CO are not located in the same regions of the disk. The assumption that isotopic exchange is more important than photodissociation should be reasonable. For this assumption to apply, $$\Gamma \; \ll \; k^{f} \exp
\left(\frac{-35}{T_{kin}}\right) \ n(^{12}\rm{C}^{+}),$$ where $\Gamma$ is the unshielded photodissociation rate, $2 \times 10^{-10} \ \rm{s}^{-1}$ [@van88], k$^{f}$ is the forward reaction rate, $6.8 \times 10^{-10} \
\rm{cm}^{3} \ \rm{s}^{-1}$ at 80 K [@smi80], and n$(^{12}\rm{C}^{+})$ is the volume density of $^{12}\rm{C}^{+}$. The average volume density of C, $n(^{12}\rm{C}) \ = \ \it{N}(\rm{^{3}P})/\it{r} \ \simeq \ \rm{20 \ cm}^{-3}$, assuming that the carbon extends over a distance $r$ = 100 AU. Further assuming that $n(^{12}\rm{C}^{+}) \geq \it{n}(\rm{^{12}C})$, as is likely, the right hand side of the above expression is greater than or equal to about $2 \times 10^{-9}
\ \rm{s}^{-1}$, an order of magnitude larger than $\Gamma$. However, a more complete treatment of the relationship between $(^{12}\rm{CO}/^{13}\rm{CO})$ and $(^{12}\rm{C}/^{13}\rm{C})$ takes into account photodissociation but requires exact knowledge of the density of $^{12}\rm{C}^{+}$ in the CS disk [@she92].
The much larger column density of [($^{3}$P)]{} compared to that of CO leads us to believe that photodissociation of CO cannot be the only source of stable in the [$\beta$ Pictoris]{} CS disk. Since there is no evidence that CO is produced by infalling comets (no variability or red and blueshifted features), it has been postulated that the CO gas slowly evaporates from the OEBs at several tens of AU from the star and is photodissociated to produce the stable [@lec98]. Obviously some portion of the gas must be produced directly from the FEBs (the portion giving rise to the variable red and blueshifted absorption); perhaps this gas is decelerated somehow and accumulates in the stable component before being destroyed by photoionization. In this scenario, the equilibrium column density of is $$N(^{3}\rm{P}) =
\frac{\it{n} \times N_{FEB}}{\Gamma}$$ where $n$ is the mean number of infalling comets per year, $\sim 10^{2}$ per year [@vid98], $N_{FEB}$ is the total column density of gas produced by an infalling comet, and $\Gamma$ is the photoionization rate for , 0.004 yr$^{-1}$. This expression, which assumes that atoms are lost only through photoionization by interstellar UV photons, gives $N_{FEB} \sim 10^{11}$ cm$^{-2}$, which should easily be produced by infalling comets like the ones giving rise to the variable [($^{3}$P)]{} components in our data. However, this is a rough treatment and the number of infalling comets per year varies significantly over time scales of a few years [@vid98]. Also, this treatment does not take into account reformation of by radiative recombination; we cannot take this process into account properly without knowledge of the amount of in the CS disk and the electron density.
Considering the [($^{1}$D)]{}, when such atoms are produced by photodissociation of CO, ($^{1}$D) atoms must be produced also to conserve spin. The minimum total photon energy needed for this dissociation is 14.33 eV, corresponding to a threshold wavelength of 865 Å. Since this threshold is below the Lyman limit, there are virtually no interstellar UV photons capable of producing [($^{1}$D)]{} by photodissociation of CO. Thus, the [($^{1}$D)]{} atoms in the [$\beta$ Pictoris]{} CS disk cannot be produced by photodissociation of CO by stellar or interstellar photons and must be produced by a collisional process involving ground state carbon atoms. Since the energy of the $^{1}$D state relative to the ground state is high, the collisional process must be a very energetic one and therefore is likely to be closely associated with the infalling bodies. Consequently, with a short [($^{1}$D)]{} lifetime, there would be no stable component in this gas; this behavior has not been previously seen in any constituent of the [$\beta$ Pictoris]{} CS disk. But this result is tentavive; examination of an unsaturated line at high resolution is needed to determine the velocity structure of the [($^{1}$D)]{} gas.
Concluding Remarks
==================
The very high resolution, low scattered light contamination, and good order separation of this STIS echelle data set has provided some clear advantages over previous observations of [$\beta$ Pictoris]{}. The rotational lines of CO have been resolved, allowing for a much more precise determination of the physical parameters of the gas. The column density of CO is $N(\rm{CO})~=~(6.3 \pm 0.3) \times 10^{14}
\ \rm{cm}^{-2}$ and the ratio $R(^{12}\rm{CO}/^{13}\rm{CO})~=~15 \pm 2$ is found. The absence of transient red or blueshifted components in the high resolution CO spectra supports the suggestion that this gas evaporates from cometary bodies orbiting far (several tens of AU) from the star. But the fact that the column density of CO is only about 2% of the total column density of in the $^{3}$P ground term implies that photodissociation of this CO is not the primary source for gas. It could perhaps be produced directly from infalling comets close to the star, but the mechanism by which it comes to zero velocity relative to the star and accumulates before being photoionized is unclear (although see @lag98). The [($^{1}$D)]{} gas may not have a stable component at 20 km s$^{-1}$; this unique species could prove to be a valuable tracer of FEB activity in the [$\beta$ Pictoris]{} CS disk.
Despite the advantages of this data set, our lack of success in modeling the heavily saturated multiplets indicates that in order to really determine the characteristics of the variable components of the gas, we need to observe an unsaturated line or multiplet, with an oscillator strength between that of the 1561 Å multiplet and that of the spin-forbidden 1613.376 Å line. A number of suitable multiplets and lines lie in the FUV, shortward of $\sim$ 1300 Å. Also, an unsaturated line arising from the $^{1}$D level would allow us to confirm the velocity structure in this gas and to determine if the velocities of the $^{1}$D gas components correspond with any of the velocities of the variable components in the $^{3}$P gas. Three likely lines lie between 1311 Å and 1359 Å. Measurement of the densities of and would greatly help to unravel the carbon chemistry of the [$\beta$ Pictoris]{} disk. Again, potentially useful multiplets of these species lie in the FUV below 1340 Å. Thus, although this data set has vastly increased our knowledge about the important species and CO in the [$\beta$ Pictoris]{} disk, our understanding would probably benefit greatly from investigation of [$\beta$ Pictoris]{} at shorter ultraviolet wavelengths.
We thank Jason McPhate for his work on our absorption line profile codes and John Debes for his work on the lines. We also thank B-G Andersson and our reviewer, X. Tielens, for their fruitful comments. This work is based on observations with the National Aeronautics and Space Administration – European Space Agency HST obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555. Support for this work at JHU was provided by grant GO-07512.01-96A from the Space Telescope Science Institute.
Beust, H., Lagrange-Henri, A. M., Vidal-Madjar, A., & Ferlet, R. 1990, , 236, 202 Draine, B. T. 1978, , 36, 595 Farrenq, R., Guelachvili, G., Sauval, A. J., Grevesse, N., & Farmer, C. B. 1991, J. Molec. Spectrosc., 149, 375 Hibbert, A., Biemont, E., Godefroid, M., & Vaeck, N. 1993, , 99, 179 Jewitt, D. C., Matthews, H. E., Owen, T., & Meier, R. 1997, Science, 278, 90 Jolly, A. et al. 1998, , 329, 1028 Lagrange, A. M. et al. 1998, , 330, 1091 Langer, W. D., & Penzias, A. A. 1993, , 408, 539 Lecavelier des Etangs, A., Vidal-Madjar, A., & Ferlet, R. 1996, , 307, 542 Lecavelier des Etangs, A. 1998, , 337, 501 Morton, D. C. 1991, , 77, 119 Morton, D. C. & Noreau, L. 1994, , 95, 301 Sahu, K. et al. 1999, STIS Instrument Handbook v3.0 (Baltimore: STScI) Sheffer, Y., Federman, S. R., Lambert, D. L., & Cardelli, J. A. 1992, , 397, 482 Smith, B. A., & Terrile, R. J. 1984, Science, 226, 1421 Smith, D., & Adams, N. G. 1980, , 242, 424 Tozzi, G. P., Feldman, P. D., & Festou, M. C. 1998, , 330, 753 Van Dishoeck, E. F. & Black, J. H. 1988, , 334, 771 Vidal-Madjar, A. et al. 1994, , 290, 245 Vidal-Madjar, A., Lecavelier des Etangs, A. L., & Ferlet, R. 1998, , 46, 629 Wannier, P., Penprase, B. E., & Andersson, B-G 1997, , 487, L165
[cccccc]{} o4g001010 & E140H & $0.2 \times 0.09$ & 1461 – 1663 & 07:17:58 & 900.0\
o4g001020 & E230H & $0.1 \times 0.03$ & 1874 – 2146 & 07:40:26 & 80.0\
o4g001030 & E230H & $0.1 \times 0.09$ & 1624 – 1896 & 07:48:20 & 678.8\
o4g001040 & E230H & $31 \times 0.05$ & 2624 – 2895 & 08:48:28 & 360.0\
o4g001050 & E230H & $31 \times 0.05$ & 2374 – 2646 & 09:02:20 & 360.0\
o4g001060 & E230H & $31 \times 0.05$ & 2124 – 2396 & 09:16:12 & 288.0\
o4g002010 & E140H & $0.2 \times 0.09$ & 1461 – 1663 & 19:51:29 & 900.0\
o4g002020 & E230H & $0.1 \times 0.03$ & 1874 – 2146 & 20:13:57 & 80.0\
o4g002030 & E230H & $0.1 \times 0.09$ & 1624 – 1896 & 20:21:51 & 678.8\
o4g002040 & E230H & $31 \times 0.05$ & 2624 – 2895 & 21:25:45 & 360.0\
o4g002050 & E230H & $31 \times 0.05$ & 2374 – 2646 & 21:39:37 & 360.0\
o4g002060 & E230H & $31 \times 0.05$ & 2124 – 2396 & 21:53:29 & 288.0\
[lcccc]{} & $20 \pm 1$ & $(6 \pm 1) \times 10^{15}$ & $1.3 \pm 0.5$ &\
December 6 & $20 \pm 1$ & $(2 - 4) \times 10^{16}$ & $1.3 \pm 0.5$ & 50 – 100\
& 26 & $10^{14}$ & 4 & $10^{2}$\
& 41 & $10^{14}$ & 10 & $10^{3}$\
& 57 & $10^{13}$ & 12 & $10^{3}$\
& & & &\
December 19 & 11 & $10^{13}$ & 5 & $10^{1}$\
& $20 \pm 1$ & $(2 - 4) \times 10^{16}$ & $1.3 \pm 0.5$ & 50 – 100\
& 27 & $10^{15}$ & 5 & $10^{2}$\
December 6 & $23 \pm 1$ & $(4 \pm 3) \times 10^{13}$ & $(4 \pm
1)$ &\
& $49 \pm 2$ & $(4 \pm 2) \times 10^{12}$ & $(4 \pm 2)$ &\
& & & &\
December 6 & $22$ & $4 \times 10^{13}$ & $2$ &\
& $28$ & $8 \times 10^{12}$ & $3$ &\
& $48$ & $6 \times 10^{12}$ & $3$ &\
& & & &\
December 19 & $26 \pm 1$ & $(1.2 \pm 0.7) \times 10^{14}$ & $4.4 \pm 0.8$ &\
& & & &\
December 19 & $23$ & $4 \times 10^{13}$ & $2$ &\
& $29$ & $3 \times 10^{13}$ & $3$ &\
& $20 \pm 1$ & $(6.3 \pm 0.3) \times 10^{14}$ & $1.3 \pm 0.1$ & $15.8 \pm 0.6$\
& $20 \pm 1$ & $(4.3 \pm 0.4) \times 10^{13}$ & $1.3 \pm 0.1$ & $15.8 \pm 0.6$\
|
---
abstract: |
The dependence on the structure functions and Z, N numbers of the nuclear binding energy is investigated within the inverse problem approach. This approach allows us to infer the underlying model parameters from experimental observation, rather than to predict the observations from the model parameters. The inverse problem was formulated for the numerical generalization of the semi-empirical mass formula of Bethe and von Weizsäcker. It was solved in step by step way based on the AME2012 nuclear database. The established parametrization describes the measured nuclear masses of 2564 isotopes with a maximum deviation less than 2.6 MeV, starting from the number of protons and number of neutrons equal to 1.
The set of parameters $\{a_{i}\}$, $i=1,\dots, {\mathcal{N}}_{\rm{param}}$ of our fit represent the solution of an overdetermined system of nonlinear equations, which represent equalities between the binding energy $E_{B,j}^{\rm{Expt}}(A,Z)$ and its model $E_{B,j}^{\rm{Th}}(A,Z,\{a_{i}\})$, where $j$ is the index of the given isotope. The solution of the overdetermined system of nonlinear equations has been obtained with the help of the Aleksandrov’s auto-regularization method of Gauss-Newton type for ill-posed problems. The efficiency of the above methods was checked by comparing relevant results with the results obtained independently. The explicit form of unknown functions was discovered in a step-by-step way using the modified least $\chi^{2}$ procedure, that realized in the algorithms which were developed by Aleksandrov to solve nonlinear systems of equations via the Gauss-Newton method, lets us to choose between two functions with same $\chi^{2}$ the better one.
In the obtained generalized model the corrections to the binding energy depend on nine proton (2, 8, 14, 20, 28, 50, 82, 108, 124) and ten neutron (2, 8, 14, 20, 28, 50, 82, 124, 152, 202) magic numbers as well on the asymptotic boundaries of their influence.
address:
- '$^{a}$The Institute for Nuclear Research and Nuclear Energy, BAS, Sofia, Bulgaria'
- '$^{b}$Department of High Energy Nuclear Physics, Institute of Modern Physics, CAS, Lanzhou, China'
-
author:
- 'S. Cht. Mavrodiev$^{a}$ and M.A. Deliyergiyev$^{b}$'
title: 'Numerical Generalization of the Bethe-Weizsäcker Mass Formula'
---
Bethe-Weizsäcker mass formula, magic numbers, binding energy, Wigner term, inverse problem 27.30+t ,21.10.Dr ,32.10.Bi ,21.60.Ev ,21.60.Cs
Introduction
============
Semi-empirical mass formulas have been a staple of nuclear physics since shortly after the identification of the neutron as a constituent of the nucleus, done by Chadwick [@Chadwick692], which was followed by the proposal of the composed atomic model given by D. Ivanenko [@Ivanenko:1932], in 1932, and the first nuclear shell model [@IvanenkoGapon:1932]. The venerable formula of Bethe and von Weizsäcker, which was described in detail in [@Weizsacker:1935; @RevModPhys.8.82], has been extended and modified frequently over the years.
The principal domain of low-energy nuclear physics is the table of the nuclides, shown in Fig.\[fig:Map\_of\_bound\_EvEv\_nuclei\]. There are several hundred stable nuclei, 288 (black squares), of the several thousand nuclides, or isotopes, that inhabit the nuclear landscape are either stable or practically stable (that is, have half-lives longer than the expected life of the Solar System) [@Erler_nature:2012]. Exactly these 288 nuclei form the so-called valley of stability, see Fig.\[fig:Map\_of\_bound\_EvEv\_nuclei\]. However, the total number of nuclides is unknown, (see the white gap between known nuclei and the two-neutron drip line), with theoretical estimates suggesting approximately seven thousand in total [@Erler_nature:2012]. The recent dramatic expansion of experimental nuclear physics facilities, allowing the measurement of nuclear properties very far from that valley (Fig.\[fig:Map\_of\_bound\_EvEv\_nuclei\]), has triggered, among others, a renewed interest in nuclear astrophysics. Only in 2011, 100 new nuclides were discovered [@Thoennessen_nature:2011]. They joined the approximately 3,000 stable and radioactive nuclides that either occur naturally on Earth or are synthesized in the laboratory [@ENSDF:2006; @Thoennessen:2004; @Wapstra2003129; @Audi2003337]. We hope that many new unstable nuclei (‘rare isotopes’) will be created and studied in new and planned experimental facilities around the world. By moving away from this valley, by adding nucleons, nuclear physics enters the vast territory of short-lived radioactive nuclei, which disintegrate by emitting $\beta$- and $\alpha$-particles or split into smaller parts through spontaneous fission.
As known nuclear existence ends at the drip lines, where there is no longer enough binding energy to prevent the last nucleons from escaping the nucleus. The placement of this line for the heavier elements is based on uncertain hypothetical predictions. However, it is not known how uncertain it is or how many protons and neutrons can be bound in a nucleus. As indicated in Fig.\[fig:Map\_of\_bound\_EvEv\_nuclei\], the proton-rich border of the nuclear territory has been experimentally delineated up to protactinium [@ENSDF:2006] (proton number, $Z=91$). The neutron-rich boundary is known only up to oxygen ($Z=8$) because of the long distance separating the valley of stability from the neutron drip line [@Thoennessen:2004]. The superheavy nucleus with $Z=118$ and $A=294$ [@PhysRevC.74.044602] marks the current limit of nuclear charge and mass. The borders of the superheavy region are unknown and difficult to predict because competition between Coulomb and shell effects can cause voids and form exotic topologies (see Section 4 of Ref.[@Nazarewicz2002165]).
Although the masses of many very exotic nuclei are now amenable to measurement [@PhysRevC.74.044602; @0954-3899-34-4-R01; @PhysRevC.76.011601], many more nuclei of great interest in astrophysics remain beyond the capabilities of modern mass measurements [@Erler_nature:2012]. This has conferred increased urgency on the development of reliable extrapolations of measured masses [@Moller1995185], one of the classic applications of mass formulas. However, there is significance in the numerical values of the coefficients of the semi-empirical mass formula, influencing, for instance, the deduction of the binding energy per nucleon of infinite nuclear matter, or the symmetry energy and surface tension of nuclear matter. These are not easily accessed by the macroscopic-microscopic mass formulas [@NILSSON19691; @RevModPhys.44.320; @PhysRevC.5.1050] that have been elaborated in order to reproduce the irregularity of the masses as functions of $A$ and $Z$, partly due to shell closings and proton and neutron number parity. Mainly the finite range liquid drop model and the finite-range droplet model [@Moller1995185]. All these works represent an on-going challenge for low-energy nuclear theory, which is to describe the structure and reactions of all nuclei, whether measured or not.
The main goal of our studies was to determine how well the existing data, and only data, determines the mapping from the proton and neutron numbers to the mass of the nuclear ground state. Another is to find presumed regularities by analysis of observed nuclei masses [@AleksandrovGadjokov1971]. And in addition is to provide reliable predictive model that can be used to forecast mass values away from the valley of stability. The results suggest that with further development this approach may provide a valuable complement to conventional global models.
A set of experimental nuclear masses from AME2012, the most recent evaluation database, that was published in December 2012 in [@Ami2012_ChinPhysC; @NUBASE2012_ChinPhysC], constitutes the raw material for this work. Only measured nuclei are included into our consideration. The masses extrapolated from systematics and marked with the symbol $\#$ in the error column are not taken into account here. Therefore we use only 2564 experimental nuclear masses, including the Hydrogen atom, to provide a deep understanding of the mutual influence of terms in the semi-empirical formula previously investigated in [@Kirson200829]. Those experimental values play a crucial role in our study of the separated influence of the Wigner term, the curvature energy and different powers of the relative neutron excess $I=(N-Z)/A$. The improvement of the generalized liquid drop model formula (GLDM) previously proposed in [@Royer1985477], whose coefficients were recently determined once again in [@Chowdhury:2004jr] and [@PhysRevC.73.067302], is also the focus of this paper.
In present work we demonstrate the applicability of the inverse problem approach for solution of such nuclear physics problems. First we formalize the nuclear mass problem in the framework of the inverse problem. Second we propose the generalized form of the Bethe-Weizsäker (BW) mass formula, which helps us to discover the latent regularities in the nuclear masses from AME2012. And afterwards we provide a solution of the formalized inverse problem that has been obtained with the help of the Alexandrov dynamic auto-regularization method of the Gauss-Newton type for ill-posed problems (REGN-Dubna) [@Alexandrov:1970; @Alexandrov:1973; @Alexandrov:1983; @Aleksandrov197146; @Alexandrov:1982; @AleksandrovPriv1997; @Alexandrov66:1977], which is a constructive development of the Tikhonov regularization method [@Tikhonov:1963; @Tikhonov:1983; @Tikhonov:1986; @Tikhonov:1995] for ill-posed problems.
The formalism of the applied approach is given in Sec. \[Theory\_and\_Method\]. The basis of the classic BW mass formula is sketched in Sec. \[Nuclear Binding Energy\] of this paper. The numerical generalization of the BW mass formula is described in Sec. \[Parametrization\_of\_the\_BWmass\_formulae\]. The main conclusions of the paper are drawn in Sec. \[Results\], which also includes a discussion of the principal results, the resulting rms deviations and parameter values. The description of experimental nuclear masses from the AME2012 database is presented in Table \[tab:Results\_output\] in \[Resulting table\]. Predicted binding energies, nuclear and atomic masses, and mass excess for the recently discovered nuclei at the DGFRS separator based on fusion reactions of $^{48}\rm{Ca}$ with $^{238}\rm{U}$-$^{249}\rm{Cf}$ target nuclei [@Oganessian201562] are shown in \[Predicted binding energies for super heave nuclei\]. The Fortran source code of the generalized BW mass formula is shown in \[Fortran CODE\].
Theory and Method {#Theory_and_Method}
=================
This section begins with the concept of latent regularities in the experimental data. An attempt is made to provide a general mathematical formulation of this problem and to connect its occurrence to an effective research strategy. It explains regularization technique within Aleksandrov method. Furthermore it describes how to construct solution of non-regular problems, based on the iteration processes for the solution of nonlinear equations. The example of such a process is given.
The problem of analyzing the latent regularities {#AleksandrovMethod_LatentRegul}
------------------------------------------------
The aforementioned goals stimulated us to try to clarify the features and to find hidden regularities of the well known semi-empirical mass formula of Bethe-Weizsäker, based exclusively on experimental data. This idea motivate us to apply the inverse problem approach to solve this task, which will be briefly discussed in this and the next section.
The conventional approach to physical data analysis consists of constructing different physical models and comparing the theoretical predictions computed for these models with the observed data. Numerical modeling of physical data for given model parameters is usually called a $forward$ problem. The forward problem solution makes it possible to predict physical observation for specific physical areas.
Usually we approximate real physics by a more or less simple model and try to determine the model parameters from the data. One may call this problem an $inverse$ problem. The success of physical interpretation depends on our ability to approximate real physical dependencies by reasonable models, and to solve the corresponding inverse problems effectively.
A wide variety of nonlinear inverse problems exist in science and engineering, and many examples can be found in the monographs and surveys by Tikhonov and Arsenin [@Tikhonov:1995; @TikhonovArsenin:1977], Hofmann [@BHofmann:1986], Banks and Kunisch [@Banks:1989], Engl [@Engl:1993], Groetsch [@Groetsch:1993], and Vasin-Ageev [@Vasin:1995]. Such problems are based on the comparison of the theoretical and experimental data by solving the system of the nonlinear operator equations on the field of real numbers $\bm{R}$ of following type: $$\begin{split}
&F_{j}(x_{1}, x_{2},\dots, x_{n};p_{1}, p_{2},\dots, p_{\nu}) = y_{j}, \\
&j=1,2,\dots,m; ~~1 \leq n \leq m;
\label{eq:fxy}
\end{split}$$ where $x\in D_{f} \subseteq \bm{R}^{n}$ and $y \in D_{f} \subseteq \bm{R}^{m}$, $\bm{R}^{n}$ and $\bm{R}^{m}$ are the $n$ and $m$ dimensional real Cartesian canonical space correspondingly, $D_{f}$ is the convex unbounded domain, the given nonlinear function $$%f_{j}(x_{1}, x_{2},\dots, x_{n};p_{1}, p_{2},\dots, p_{\nu}) \in C^{2}(D_{f}), ~~j=1,2,\dots,m,
F_{j}(x_{1}, x_{2},\dots, x_{n};p_{1}, p_{2},\dots, p_{\nu}) \in C^{2}(D_{f}), ~~j=1,2,\dots,m,
\label{eq:GivenNonlinearFunc}$$ where $$\begin{split}
&D_{f}=(x_{1}^{\prime}, x_{1}^{\prime\prime})\times \dots \times (x_{n}^{\prime}, x_{n}^{\prime\prime}),\\
&x_{i}^{\prime}, x_{i}^{\prime\prime} \in \bm{R}^{1}; p_{1}, p_{2}, \dots,p_{\nu} \in \bm{R}^{1}, ~~\nu \geq 0
\label{eq:GivenSpaceForNonlinearFunc}
\end{split}$$ are the given parameters.
The solution of the system of Eqs.\[eq:fxy\] can be found with respect to the unknowns $x_{i}\in(x_{i}^{\prime}, x_{i}^{\prime\prime})$, where $i=1,2,\dots, n$. Taking into account that $\bm{R}^{n}$ and $\bm{R}^{m}$ are the $n$ and $m$ dimensional real coordinate space, that corresponds to the $x$ and $y$ respectively and using the following notation $$\begin{split}
&x=(x_{1},x_{2},\dots,x_{n})^{T} \in D_{f} \subset \bm{R}^{n}\\
&y=(y_{1},y_{2},\dots,y_{m})^{T} \in \bm{R}^{m}
\label{eq:XYvariables}
\end{split}$$ the system of Eqs.\[eq:fxy\] can be written in a vector form: $$%&fx = \bar{y},
%&F(x) = \bar{y},
F(x) = y.
%&f\bm{x} = \bar{\bm{y}},
\label{eq:fxy_VectorForm}$$ Suppose that observed function describing some property of some physical phenomenon may be denoted by $f(t)$. Note, that in our case the role of this function will be played by the BW mass formula, we come to this in later sections. Let us assume that $f(t)$ is determined on the closed segment $S = [t_{1}, t_{2}]$, $-\infty \textless t_{1} \textless t_{2} \textless \infty$. For sake of simplicity the independent variable $t$ and the function $f(t)$ will be considered scalar quantities. Let a set $L$ of scalar functions $g_{i}(i = 1, 2,\dots, n)$, depending on the parameters $p_{ij}(i=1,2,\dots,n;j=1,2,\dots,s_{i})$ and on the independent variable $x\in S$, be given: $$%L=\{g_{1}(p_{11},\dots, p_{1n_{1}};t),\dots,g_{\eta}(p_{\eta 1},\dots, p_{\eta n_{\eta}};t)\}.
L=\{g_{1}(p_{11},\dots, p_{1n_{1}};t),\dots,g_{n}(p_{n 1},\dots, p_{n s_{n}};t)\}.
\label{eq:L_scalarFunc}$$ Further a certain degree of smoothness of the function $g_{i}$ with respect to the parameters $p_{ij}$ is assumed, namely, that continuous second-order derivatives exist.
Based on the previous knowledge of the observed functions or on some other conceptions (theory, analogy, phenomenology, etc.), the following basic hypothesis is made: the observed function $f(t)$ may be approximated by a linear combination of the elements of the set $L$ as: $$f(t)=\sum_{i=1}^{n}{\rm{a}}_{i} g_{i}(p_{i1},\dots, p_{is_{i}};t)=
{\rm{a}}_{0}+\sum_{i=1}^{n}{\rm{a}}_{i}{\rm{e}}^{-h_{i}t},
\label{eq:Comb_Elements_Lf}$$ where ${\rm{a}}_{i}\geq 1$ and the number of exponents in the general case is unknown. Due to high sensitivity of the solution to the infinitesimal changes of values $f(t)$ one has to use special steady solutions of the systems Eq.\[eq:Comb\_Elements\_Lf\].
Let’s demonstrate the high sensitivity of the solution of this system. Suppose that $f(u)$ will define the transformation of the right hand side of Eq.\[eq:Comb\_Elements\_Lf\], that transforms the points: $$%u=({\rm{a}}_{0}, {\rm{a}}_{1},\ldots, {\rm{a}}_{n}, h_{1},\dotsc, h_{n} ) \in R_{2n+1}
u=({\rm{a}}_{0}, {\rm{a}}_{1},\ldots, {\rm{a}}_{n}, h_{1},\dotsc, h_{n} ) \in R_{2n+1}
\label{eq:TransformFunction}$$ into the points $\upsilon=(y_{1},y_{2},\dotsc,y_{m}) \in R_{n_{i}}$. Let $\Omega$ be the union of the all subspaces $${\rm{a}}_{0}=0, ~~{\rm{a}}_{i}=0, ~~ h_{p}=h_{q}, ~~~i,p,q=1,2,\ldots,n, ~~~p\neq q,
%{\rm{a}}_{0}=0, ~~{\rm{a}}_{i}=0, ~~ h_{p}=h_{q}, ~~~i,p,q=1,2,\ldots,\eta, ~~~p\neq q,
\label{eq:H_Subspaces}$$ and $U=\{ u/u\in R_{2n+1}, {\rm{det}}J(u)\neq 0 \}$ at $2n+1=m$, where $J(u)$ is the Jacobi matrix of the function $f(u)$.
Thus we have $U\cap \Omega=\O$. One may easily check, that once $u\in f(U)$, then the non-linear problem Eq.\[eq:Comb\_Elements\_Lf\] is correct [@Tikhonov:1963; @Tikhonov:1995].
In the case where the point $u$ is located close enough to the manifold $\Omega$ one may note a sensitivity feature of the solution Eq.\[eq:Comb\_Elements\_Lf\] with respect to small deviations of the point $\upsilon$.
Now we back to the made hypothesis, according to it the set $L$ is related to the function $f(t)$. In this case $L$ will be called “characteristic function set" and denoted as $L_f$. In addition we can make the following definition [@AleksandrovGadjokov1971].
[**[Definition 1]{}**]{}: The problem of finding the set $L_f$, the parameters $p_{ij}(i=1,2,\dots,n;j=1,2,\dots,s_{i})$ and the amplitudes ${\rm{a}}_{i}(i=1,2,\dots,n)$ is defined as the full problem of analysis of the latent regularities.
Let’s us analyze the series of observations of the function $\{f(t_{k})\}$[^1], where $k = 1,2,\dots,m$, and a set $L_f$ with a known number $n$ of presumed relationships be given.
Based on relation Eq.\[eq:Comb\_Elements\_Lf\], the unknown quantities $p_{ij}$ and ${\rm{a}}_{i}$ can be found by solving the following non-linear (in general, over-determined) system $$%y(x_{j})={\rm{a}}_{0}+\sum_{i=1}^{n}{\rm{a}}_{i}{\rm{e}}^{-h_{i}x_{j}}, ~~j=1,2,\ldots, m,
f(t_{k})={\rm{a}}_{0}+\sum_{i=1}^{n}{\rm{a}}_{i}{\rm{e}}^{-h_{i}t_{k}}, ~~k=1,2,\ldots, m,
\label{eq:NonLinearSystem}$$ The following vector denotations are introduced: $$\begin{split}
&x=( {\rm{a}}_{1},\dots,{\rm{a}}_{n},p_{11},\dots, p_{1s_{1}},\dots, p_{n s_{1}},\dots, p_{n s_{n}} ) \in \bm{R}^{l(n)}\\
&l(n)=n+\sum_{i=1}^{n}s_{i},\\
&F(x)=[F_{1}(x),\dots,F_{m}(x)]\in \bm{R}^{m},\\
\end{split}
\label{eq:VectorDenotation}$$ where $$%F_{k}=\left({\rm{a}}_{0}+\sum_{i=1}^{n}{\rm{a}}_{i}{\rm{e}}^{-h_{i}x_{k}} \right) -f(x_{k}), ~~k=1,2,\ldots, m.
F(x_{k})=\left({\rm{a}}_{0}+\sum_{i=1}^{n}{\rm{a}}_{i}{\rm{e}}^{-h_{i}x_{k}} \right) -f(x_{k}), ~~k=1,2,\ldots, m.
\label{eq:VectorDenotation}$$ The non-linear system Eqs.\[eq:NonLinearSystem\] may be expressed in the vectorial form as written in Eq.\[eq:fxy\_VectorForm\].
Let us assume that the operator $F$ transforms the limited convex subset $X \subseteq \bm{R}^{l(n)}$ into $\bm{R}^{m}$. The Jacobian matrix of the operator $F$ with respect to $x$ is denoted as $F^{\prime}$. When different relationships have no common parameter the Jacobi matrix can be written in following form [@AleksandrovGadjokov1971]: $$F^{\prime}(x)=\\
\begin{bmatrix}
g_{1}(t_{1}),..,g_{n}(t_{1}) &{\rm{a}}_{1}\frac{\partial g_{1}(t_{1})}{\partial p_{11}}
&...~{\rm{a}}_{1}\frac{\partial g_{1}(t_{1})}{\partial p_{1s_1}}
%&... {\rm{a}}_{b}\frac{\partial g_{\eta}(t_{1})}{\partial p_{\eta 1}}
&...~{\rm{a}}_{n}\frac{\partial g_{n}(t_{1})}{\partial p_{n s_{n}}}\\
g_{1}(t_{2}),..,g_{n}(t_{2}) &{\rm{a}}_{1}\frac{\partial g_{1}(t_{2})}{\partial p_{11}}
&...~{\rm{a}}_{1}\frac{\partial g_{1}(t_{2})}{\partial p_{1s_1}}
%&... {\rm{a}}_{b}\frac{\partial g_{\eta}(t_{2})}{\partial p_{\eta 1}}
&...~{\rm{a}}_{n}\frac{\partial g_{n}(t_{2})}{\partial p_{n s_{n}}}\\
\dots\dots\dots\dots
&\dots\dots\dots
&\dots\dots\dots
&\dots\dots\dots\\
g_{1}(t_{m}),..,g_{n}(t_{m}) &{\rm{a}}_{1}\frac{\partial g_{1}(t_{m})}{\partial p_{11}}
&...~{\rm{a}}_{1}\frac{\partial g_{1}(t_{m})}{\partial p_{1s_1}}
%&... {\rm{a}}_{\eta}\frac{\partial g_{\eta}(t_{n})}{\partial p_{\eta 1}}
&...~{\rm{a}}_{n}\frac{\partial g_{n}(t_{m})}{\partial p_{n s_{n}}}
\end{bmatrix}
\label{eq:JacobiMatrix_of_F}
%\end{split}$$ If two or more relationships exist which have a common parameter (or parameters) the modifications in matrix $F^{\prime}(x)$ are not essential for further considerations.
The operator $H(x) = F^{\prime T}(x)F^{\prime}(x)$ will be considered regular. Let us consider its condition number [@GeorgeForsythe:1967]: $${\rm{cond}}H(x)=\Arrowvert H^{-1}(x) \Arrowvert, ~~ \Arrowvert H(x) \Arrowvert =\frac{\mu_{1}}{\mu_2}\geq 1,
\label{eq:condition}$$ where $\mu_{1}$ and $\mu_{2}$ are its greatest and smallest singular numbers [@GeorgeForsythe:1967], respectively. The operator $H(x)$ is only then irregular when $\mu_2=0$. In this case, if $\mu_1 \neq 0 $, the condition number in Eq.\[eq:condition\] will be considered as infinitely large.
Let’s define a real number $r$, such as $1\textless r \textless \infty$. Then
[**[Definition 2]{}**]{}: If the non-equality $${\rm{cond}}H(x)~\textless~ r
\label{eq:Nonequality}$$ is fulfilled for all $x\in X$, the solution of Eq.\[eq:fxy\_VectorForm\] is defined as “the regular problem with the modulus $r$".
It can be easily verified that the regularity of problem Eq.\[eq:fxy\_VectorForm\] depends on the closeness of the relationships or on their smallness with regard to the modulus, and also on the smallness of the amplitudes ${\rm{a}}_{i}$ or on the smallness with regard to the modulus of the derivatives of the relationships.
If one will assume that in Eq.\[eq:fxy\_VectorForm\] $\mu_{1}(x)\neq 0$ for all $x\in X$. Then, if $r$ is fixed, there will always exist a sufficiently small real number $\delta \textgreater 0$ with which, if at least one of the non-equalities $$\begin{cases}
&\sum_{k=1}^{m}\left( g_{i_{1}}(t_{k})-g_{i_{2}}(t_{k}) \right)^{2}\leq\delta; i_{1},i_{2}=1,2,\dots,n; i_{1}\neq i_{2}\\
&{\rm{a}}_{i}~\leq~\delta\\
&\vert g_{i}(t_{k}) \vert ~\leq~\delta;~~k=1,2,\dots,m\\
&\vert \frac{\partial g_{i}(t_{k})}{\partial p_{ij}}\vert~\leq~\delta;~~k=1,2,\dots,m \\
\end{cases}
\label{eq:System_Of_Nonequalities}$$ is fulfilled, the problem Eq.\[eq:fxy\_VectorForm\] becomes non-regular.
It should be particularly stressed that perturbations of the observed function $\{f(t_k)\}$ (and the presence of perturbations is very characteristic for the problems concerning latent regularities) may also infringe on the regularity of problem Eq.\[eq:fxy\_VectorForm\], if at least one of the non-equalities Eq.\[eq:System\_Of\_Nonequalities\] is fulfilled with a sufficiently small value of $\delta$.
When the problem Eq.\[eq:fxy\_VectorForm\] is non-regular, general calculation algorithms ensuring its solution can be constructed, based on the regularized iteration processes for the solution of non-linear equations [@Alexandrov5137:1970; @Alexandrov:1970; @Alexandrov:1973].
Aleksandrov method for Ill-posed nonlinear systems of equations {#AleksandrovMethod_REGN}
---------------------------------------------------------------
When it is not feasible to give a definite indication of the type and number of regularities (i.e. to define exactly the set $L_f$), the full analysis of latent regularities becomes much more difficult. However, even in this case it is possible to find an approximate solution of problem Eq.\[eq:fxy\_VectorForm\] based on regularized or an auto-regularized iteration processes[^2] for solving non-linear problems.
The solution of the inverse problem, Eq.\[eq:fxy\_VectorForm\], consists in determining such a model $x$ (predicted model) that generates predicted data $y^{\delta}$, that fit well the observed data $y$, this solution is generally obtained by deducing system parameters from observations of system behavior.
Such kind of problems, Eq.\[eq:fxy\] or Eq.\[eq:fxy\_VectorForm\], typically involve the estimation of certain quantities based on indirect measurements of these quantities. The estimation process is often ill-posed in the sense that noise in the data may give rise to significant errors in the estimate. In other words, the problem Eq.\[eq:fxy\_VectorForm\] is ill-posed [@Tikhonov:1986] if its solution does not depend continuously on the right hand side $y$, which is often obtained by measurement and hence contains errors: $$\Arrowvert y^{\delta}-y \Arrowvert ~\leq\delta.
\label{eq:ySigma}$$ Here we denote the measured perturbed data by $y^{\delta}$, $\delta$ is the experimental uncertainty. The $y$ in Eqs.\[eq:ySigma\] and \[eq:fxy\_VectorForm\] one may interpret as a data set, which can also be characterized as a function of the observation point (in the case of continuous observations), or as a vector (in the case of discrete observations). Operator $F$ in Eq.\[eq:fxy\_VectorForm\] is a forward modeling nonlinear operator, that transforms any model $x$ into the corresponding data $y$. The inverse problem is formulated as the solution of the operator Eq.\[eq:fxy\_VectorForm\]. Therefore, Eqs.\[eq:fxy\_VectorForm\] connect the unknown parameters of the model with some given quantities (variables) describing the model, in our case atomic mass number, $A$, and proton mass number, $Z$. These quantities take the form of the so-called input data. Generally, the input data as well as the unknown parameters are the elements of certain metric spaces, in particular, Banach or Hilbert spaces, with the operator of the model acting from the solution space to the data space.
Techniques known as regularization methods [@Tikhonov:1995; @TikhonovArsenin:1977; @Alexandrov:1970; @Alexandrov:1973; @Alexandrov:1983; @Aleksandrov197146; @Alexandrov:1982; @AleksandrovPriv1997] have been developed to deal with this ill-posedness, to get stable approximations of solutions of Eq.\[eq:fxy\_VectorForm\]. Thus on one hand the classic solution to ill-posed problems is to regularize the unknowns, e.g. by penalizing deviations from a prior model. It consists in approximating a solution of Eq.\[eq:fxy\] by a minimizer $x_{\alpha}^{\delta}$ of the functional $$x\mapsto \Arrowvert F(x)-y^{\delta}\Arrowvert^{2} + \alpha\Arrowvert x - x_{0}\Arrowvert^{2},
\label{eq:TikhonovRegularization}$$ where $x_{0}$ typically unifies all available $a~priori$ information on the solution, or the initial input data (approximation), $\alpha$ is a positive number. On another hand one may use the iterative approaches which are attractive alternatives to regularization, since for solution of the ill-posed problems it is usually necessary to approximate the initial, often infinite-dimensional, problem by a finite-dimensional problem, for which numerical algorithms and computer programs have been devised, for instance, the steepest descent method [@ScherzerOmar:1996], the Landweber iteration [@Hank:1995], and the Landweber-Kaczmarz method, also called algebraic reconstruction techniques (ART) [@Herman:1980]. But in the current work we apply the auto-regularization method of Gauss-Newton type for ill-posed problems (REGN) [@Alexandrov:1970; @Alexandrov:1973; @Alexandrov:1983; @Aleksandrov197146; @Alexandrov:1982; @AleksandrovPriv1997; @Alexandrov66:1977], which we found an appropriate choice for our inverse problem that will be formulated in the next section.
The ill-posedness degree provides indications on how badly the deviations from the experimental data are echoed on the quality of the solution. For severely ill-posed problems, the effect of that noise is highly damaging to the solution and has a major impact on the numerical methods. That is why savvy practitioners use a safe approximation of it. A relevant computed solution cannot be achieved without sophisticated regularization strategies. Regarding these issues, we refer to [@Alexandrov:1982; @Tikhonov:1986; @PerChristian:1991; @RainerKress:1999] for how to numerically handle severely ill-posed problems, in general, and to [@0266-5611-22-4-012; @FrankWubbeling:1993], for the Cauchy problem, specifically.
One question, which arises when solving an inverse task, is whether the results obtained are proven. This question can be answered by comparing the results obtained by independent methods. To illustrate such a comparison, the binding energies and atomic masses were used as an example, see next section.
Alexandrov in his works [@Alexandrov:1970; @Alexandrov:1973; @Alexandrov:1983; @Aleksandrov197146; @Alexandrov:1982; @AleksandrovPriv1997; @Alexandrov66:1977] propose to search for the solution of Eq.\[eq:fxy\], or Eq.\[eq:fxy\_VectorForm\], for two formulations $$y\in \bm{R}_{f}\equiv FD_{f},\\
\label{eq:Formulations1}$$ $$y\in \bm{R}^{m} \setminus \bm{R}_{f},
\label{eq:Formulations2}$$ which are covered by transition to the “$\varphi$-transformed" [@Alexandrov:1973] problem $$F(x_{k}) \equiv \varphi(x) (F(x_{k})-y)=0,
\label{eq:PhiTransformedProblem}$$ where $$\varphi(x):\bm{R}_{\varphi} ( \equiv {\bm{R}}_{f}\setminus \{ y \} ) \rightarrow D_{f}
\label{eq:varphi_x}$$ is the given linear operator function. Since $\varphi(x)$ as the linear operator transform $\bm{R}_{\varphi}$ to $D_{f}$, and as the nonlinear function of the $x$ transform $D_{f}$ into the space of the confined real matrices, ${\mathcal{L}}(\bm{R}^{n},\bm{R}^{m})$. The space of the confined real matrices is characterized by the norm $$\Arrowvert A \Arrowvert_{\infty} =\max_{i} \left( g_{i} \sum_{i=1}^{n} \frac{\vert a_{ij}\vert}{g_{i}} \right),
\label{eq:RealMatrixNorm}$$ where $A=\{ a_{ij}\} \in {\mathcal{L}}(\bm{R}^{n},\bm{R}^{m})$.
The transformation function, $\varphi(x)$ in Eq.\[eq:PhiTransformedProblem\] in the basic case (second Gauss transformation) can be written as: $$\varphi(x)=(F^{\prime}(x))^{T}=\Bigg\{ \frac{ \partial F_{j}(x)}{ \partial x_{i}} \Bigg\}^{T}_{{i=1,2,\dots,n;}{j=1,2,\dots,m;}}
\label{eq:PhiTransfFunction}$$ Therefore, the Eq.\[eq:PhiTransformedProblem\] for both formulations, Eqs.\[eq:Formulations1\]-\[eq:Formulations2\] in accordance to the Alexandrov proposal [@Alexandrov:1970; @Alexandrov:1973; @Alexandrov:1983; @Aleksandrov197146; @Alexandrov:1982; @AleksandrovPriv1997; @Alexandrov66:1977] reduced to search for the solution of the “$F^{\prime T}(x)W^{1/2}$-transformed" [@Alexandrov66:1977] and the $\alpha$-regularized [@Tikhonov:1986] problem $$F(x_{k}) \equiv F^{\prime T}(x_{k})W(F(x_{k})-y)+\alpha x_{k}=0, ~~\alpha\textgreater 0,
\label{eq:RegularizedChi2}$$ where $$W={\rm{diag}}(\omega_{1},\omega_{2},\dots, \omega_{m}), ~~\omega_{m}\textgreater 0,
\label{eq:WeightMatrix}$$ is the weighting matrix, $\omega_{m}\in (0, {\infty})$ are the given constants, and $\alpha: D_{f}\rightarrow D_{f}$ is the given nonlinear operator, $F$ is a continuous and Frechet differential nonlinear operator.
In this case when the $\bm{R}^{n}$ is the Banach space, then it is normalized by the vector norm $$\Arrowvert x \Arrowvert_{\infty} =\max_{i}(g_{i} \vert x_{i}\vert).
\label{eq:VectorNorm}$$ where $g_{i}~\textgreater~0 ~(i=1,2,\dots, n)$ are the given weights. When the $\bm{R}^{n}$ is the Euclidean space then it is characterized by the Euclidean norm: $$\Arrowvert x \Arrowvert_{2} = \left( \sum_{i=1}^{n} x_{i}^{2} \right)^{1/2},
\label{eq:ChebyshevNorm}$$ Further by applying the iteratively regularized Gauss-Newton method within double regularization we can build the following regularized process: $$\begin{split}
R_{\varepsilon_{k}}: x_{0}, x_{k+1} & = x_{k}-\left[ F^{\prime T}(x_{k})F^{\prime}(x_{k})+(\alpha + \varepsilon_{k}) I_{\bm{R}^{n}} \right]^{-1} \\
&\times \left[ F^{\prime T}(x_{k}) (F(x_{k})- y) + \alpha x_{k} \right], %default Gauss-Newton
\label{eq:GaussNewton}
\end{split}$$ which can be used to find an approximate solution of Eq.\[eq:fxy\_VectorForm\]. Where $x_{0}$ and $\varepsilon_{0}$ are the initial input data (approximation), in other words is an initial guess which may incorporate $a~priori$ knowledge on an exact solution, in our case it is the generalized BW mass formula. The $x_{k}$ denotes the iterative solution, $k=1,2,\dots,k^{\ast}\textless \infty$, where $k^{\ast}$ is the pseudo solution $x_{k^{\ast}}$ (i.e. the solution in the best iteration), $\alpha$ it is an adjustable from iteration to iteration the Tikhonov‘s regularizator. $F^{\prime}(x)$ is the Frechet derivative of $F$ at $x\in D_{f}$. The process, $R_{\varepsilon_{k}}$ is converge according to the theorem stated in Ref.[@Alexandrov:1970].
Therefore, as one may note the key idea of any Newton type method consists in repeatedly linearizing the operator Eq.\[eq:fxy\_VectorForm\] around some approximate solution $x_{k}$, and then solving the linearized problem Eq.\[eq:GaussNewton\] for $x_{k+1}$. However, usually these linearized problems are also ill-posed if the nonlinear problem is ill-posed and, therefore, they have to be regularized.
The linear Eq.\[eq:GaussNewton\] can be solved with the help of one of the three methods [@Alexandrov:1970]: Cholesky Decomposition [@GeneGolub:1996; @WilliamHPress], Gauss-Jordan Elimination [@Steven:102307] or Singular Value Decomposition [@WilliamHPress_C:1992]. Aleksandrov in turn propose [@Alexandrov:1973; @Alexandrov:1983; @Aleksandrov197146; @Alexandrov66:1977] the following regularizator for the process Eq.\[eq:GaussNewton\]: $$\begin{split}
&r_{k}:\varepsilon_{k}=\frac{\alpha_{2}}{2} \left[ \left( \tau_{k}^{2} + 4 N_{0}(\varrho_{k})^{p_{r}} \right)^{1/2} - \tau_{k}\right],
\label{eq:AleksandrovReg}
\end{split}$$ where $$\begin{split}
\varrho_{k}& = \Arrowvert F(x_{k})\Arrowvert_{\infty} ~~~{\rm{or}}~~~\varrho_{2k} = \Arrowvert F(x_{k})\Arrowvert_{2},\\
\chi^{2}_{k}& = \Arrowvert W^{1/2}(F(x_{k})-y) \Arrowvert_{2}^{2}.
\label{eq:AleksandrovVar}
\end{split}$$ Here the $\varrho_{k}$ and $\chi^{2}_{k}$ are both the iteration process criteria, namely the “target criteria". The $F(x_{k})$ for $\varrho_{k}$ defined from Eq.\[eq:RegularizedChi2\]. The functionals $\tau_{k}$ and $N_{0}$ in Eq.\[eq:AleksandrovReg\] are the iteration process behavior criteria, that defined as: $$\begin{split}
\tau_{k} &=\Arrowvert F^{\prime T}(x_{k})WF^{\prime}(x_{k}) +\alpha I_{\bm{R}^{n}} \Arrowvert_{\infty}, \\
N_{0} & = \frac{\alpha_{1}}{\varrho_{0}} \left(\tau_{0}\varepsilon_{0} + \varepsilon_{0}^{2}\right) , ~~\varepsilon_{0}\geq 0.
\label{eq:IteraProcBehaviorCriteria}
\end{split}$$ where $\varrho_{0}\neq 0$, or $N_{0}$ is a given positive number. The $\alpha_{1}$, $\alpha_{2}$ and $p_{r}$ are the internal coefficient of the auto-regularized process. By default they are equal to unity, but in more general case $\alpha_{1},\alpha_{2}\in [0,1]$ and $p_{r}\geq 1$.
In the case when the auto-regularization parameter of Eq.\[eq:AleksandrovReg\] is taken in the form of: $$\varepsilon_{k}=\frac{1}{2} \left( (\lambda_{\rm{min}}^{k})^{2} + 4 N\varrho_{k} - \lambda_{\rm{min}}^{k} \right),
\label{eq:AleksandrovAutoReg}$$ where $\lambda_{\rm{min}}^{k}$ is the minimal eigenvalue of the matrix $F^{\prime T}(x_{k})F^{\prime}(x_{k})$, and $$\begin{split}
%N_{0} & = \frac{ \left(\varepsilon_{0}^{2}+ \lambda_{\rm{min}}^{0}\varepsilon_{0} \right)} {\varrho_{0}},\\
N_{0} & = \frac{1} {\varrho_{0}} \left( \lambda_{\rm{min}}^{0}\varepsilon_{0} + \varepsilon_{0}^{2} \right),\\
\varrho_{k}& = \Arrowvert F(x_{k}) \Arrowvert_{2},
\label{eq:AleksandrovAutoRegVar}
\end{split}$$ where, as noted previously, $\varrho_{0}\neq 0$, or $N_{0}$ is a given positive number.
The weighting matrix in this case looks like: $$W={\rm{diag}}(\sigma_{1}^{-1},\sigma_{2}^{-1},\dots, \sigma_{m}^{-1}),
\label{eq:WeightMatrixAutoRegEuclNorm}$$ where $\sigma_{m}$ is the standard deviation. When the mathematical formulation of the problem is clear, but the experimental uncertainties $$((\Delta y)_{1},(\Delta y)_{2},\dots, (\Delta y)_{n})^{T}
\label{eq:ExpUncertaities}$$ are very rough or unknown, then one may use robust methods to determine the weights [@AleksandrovMavrodiev:1976; @Aleksandrov1980520] or the LCH-weighting procedure[^3] [@AleksandrovMavrodiev:1976; @AlexandrovLCH:1976; @Aleksandrov2004519; @LCH:1999]. The meaning of this procedure is based on the preliminary solution of the problem Eq.\[eq:RegularizedChi2\] with the help of the weight matrix $W=I_{\bm{R}^{n}}={\rm{diag}}(1,1,\dots, 1)$ one finds the intermediate pseudo solution $x^{\ast}$ of the system Eq.\[eq:RegularizedChi2\] (or Eq.\[eq:GaussNewton\]) and then we solve it again, with the help of the weight matrix $W$ [@AleksandrovMavrodiev:1976; @Aleksandrov1980520]: $$W={\rm{diag}} \left( \frac{1}{ \vert F_{1}(x^{\ast})-y_{1}\vert }, \frac{1}{\vert F_{2}(x^{\ast})-y_{2}\vert } \dots, \frac{1}{\vert F_{n}(x^{\ast})-y_{n}\vert } \right),
\label{eq:W2}$$ To be sure that the true solution is converged one may repeat this process several times.
The essential difference between Aleksandrov method and similar methods [@Bjork:1996; @Nguyen_UMJ:2005; @ARGYROS20151318; @YANG20151339] based on especially effective ideology of regularization of the inverse problem, is that at each step (iteration) not only the decision, but also the uncertainty of the solution is controlled, which is important. Aleksandrov regularization stabilizes the solution of our ill-posed operator equations by minimizing the weighted sum of a strictly convex regularization term. The regularization term stabilizes the solution of the problem at the expense of biasing the solution. Note, that the regularizator proposed by Aleksandrov to solve nonlinear ill-posed problems with the help of the iterative approach is the constructive development of the Tikhonov-Glasko regularizator [@Tikhonov196593]: $$\alpha_{k}=\left( \frac{1}{2} \right)^{k-1} x_{k}
\label{eq:TikhonovGlasko}$$ and the regularization parameter proposed by Ramm [@Ramm:2000] $$\alpha_{k}=\left( \frac{1}{\rm{e}} \right)^{k-1} x_{k}.
\label{eq:Ramm}$$ Therefore the purpose of such functions, Eq.\[eq:AleksandrovReg\], \[eq:TikhonovGlasko\] and Eq.\[eq:Ramm\], to regularize the solution by filtering out the contributions of the noise in $y$ to the computed solution.
The processing of the experimental data of different nature often leads to the automatized solution of one-type systems, connected with the analysis of hidden (latent) regularities, i.e. hidden exponents, hidden Gauss- or Breit-Wigner functions and so on [@AleksandrovGadjokov1971; @Alexandrov:19701285; @AleksandrovHRegul:1972]. The number of hidden regularities involved in each problem is unknown in the general case. In the next section we show how these regularities have been discovered in the description of the nuclear binding energies.
Note, that with the help of the developed by Collatz method of justification of the “enhanced Newton’s method" [@LotharCollatz:1964] Alexandrov manage to obtain the general theory of convergence and attraction, which cover the iterative regularized Newton-Kantorovich and Gauss-Newton processes. Prior to work [@Alexandrov:1973] the general theory of convergence and attraction for the ordinary methods of the Newton-Kantorovich and Gauss-Newton type did not exist.
The Bethe-Weizsäcker mass formula and binding energy of the nucleus {#Nuclear Binding Energy}
===================================================================
During the first few decades of nuclear physics, the term ‘nuclear forces’ was often used as being synonymous with nuclear physics as a whole [@Rosenfeld:1948; @Eder:1965]. Nevertheless, even today with the onset of quantum chromodynamics, in any first approach towards a nuclear structure problem, one assumes the nucleons to be elementary particles.
In the theory of the liquid drop model proposed by George Gamow [@GGamow:1931], the binding energy per nucleon is given by formula $$\begin{split}
E_{B}(A,Z)&=\alpha_{vol}+\alpha_{surf}\frac{1}{A^{1/3}}+\alpha_{comb}\frac{Z(Z-1)}{A^{4/3}}\\
&+\alpha_{sym}\frac{(N-Z)^{2}}{A^{2}}+\alpha_{pairing}\frac{\delta(A,Z)}{A}+\alpha_{\rm{Wigner}}\frac{1}{A},
\label{eq:LiquidDrop_BW}
\end{split}$$ when $\alpha_{\rm{Wigner}} = 0$, Eq. \[eq:LiquidDrop\_BW\] reduces to the usual BW formula.
Due to historical reasons the BW mass formula, Eq.\[eq:LiquidDrop\_BW\], provides the baseline fit, relative to which all refinements can be judged. That is why we decide to use this formula in our formulation of the inverse problem, which will be discussed later in the paper.
One may note, that the leading linear behavior of the binding energy as a function of the number of nucleons is a manifestation of the saturation of nuclear forces, see Fig.\[fig:BindingEnergyA\]. Another manifestation of saturation, following Bethe [@RevModPhys.8.82], the proportionality of both the total binding energy and the nuclear volume at first approximation to the number of nucleons. Nuclei have diffuse surfaces, so the notion of volume is somewhat fuzzy, but an indicator is provided by the measured rootmean-squared (rms) radius, $\langle r^{2} \rangle^{1/2}$, which sets a linear scale of the nuclear size. The volume term, $\alpha_{vol}$, in Eq.\[eq:LiquidDrop\_BW\] is expected to be positive, the other three $\alpha$’s negative.
Another an obvious feature in the BW is the effect of the repulsive Coulomb interaction between protons, that information is encoded in the third term of Eq.\[eq:LiquidDrop\_BW\]. On the basis of simple electrostatics and dimensional arguments, the total Coulomb energy of the nucleus is expected to be proportional to the square of its charge and inversely proportional to its length scale, $E_{C}\propto Z^{2}e^{2}/A^{1/3}$ (called also the proton form-factor correction to the Coulomb energy in [@Moller1995185]), where $e^{2} = 1.44$ MeV fm in convenient units.
The surface energy term, second term of Eq.\[eq:LiquidDrop\_BW\], takes into account the deficit of binding energy of the nucleons at the nuclear surface and corresponds to semi-infinite nuclear matter.
The symmetry term, the fourth term of Eq.\[eq:LiquidDrop\_BW\], has another motivation in the BW mass formula, it produces a rough balance between the number of neutrons $N$ and number of protons $Z$ among the $A = N + Z$ nucleons. In the absence of this term, the semi-empirical mass formula would suggest maximum binding for a nucleus consisting only of neutrons. The symmetry energy has two contributing factors $\textendash$ a kinetic energy term and a potential energy term [@Angeli2004185], that have the same form and proportional to $(N-Z)^{2}/A$. Since the isoscalar part of the interaction does not distinguish between protons and neutrons, a qualitative estimate of the potential energy contribution is obtained by considering the isovector part of the nucleon-nucleon interaction. It has the same form as the kinetic energy, which is usually estimated in a Fermi gas model and proportional to $(N-Z)^{2}/A$. Note the symmetry term, recalling binding energy connection to the isospin $I$, which was introduced by Myers and Swiatecki [@WDMyers:1966]. It should be noted that a more sophisticated treatment of the nuclear interaction (as in the seniority shell model or the Wigner supermultiplet theory [@Talmni:1993]) would produce symmetry energies of the form $I(I+1)$ or $I(I+4)$, respectively, where the nuclear isospin $I=\vert N-Z\vert/2$.
Despite the widespread use of formula Eq.\[eq:LiquidDrop\_BW\] in nuclear physics it was found that it is severely inadequate for light-mass nuclei, especially away from the line of stability [@Heyde:1999], after the discovery of neutron rich light nuclei. The unusual stability of nuclei with preferred nucleon numbers, commonly referred to as magic numbers, can be clearly delineated by comparing the experimental binding energies with predictions of the BW mass formula. However, this comparison does not indicate the recently observed features like the disappearance of some traditional magic numbers [@GuillemaudMueller198437; @Motobayashi19959; @PhysRevLett.83.496; @Keller:1994ZurPhysA; @PhysRevLett.84.5493] and extra stability for some newly observed nuclei.
These issues motivate the recent phenomenological searches for a more generalized BW formula [@PhysRevC.65.037301; @ADHIKARI:2004], which can explain the gross features of the shapes of the binding energy versus neutron number curves of all the elements from $Li$ to $Bi$. Nevertheless in order to understand and appreciate the relevance and reliability of the semi-empirical mass formula, it is necessary to study not only the effect of adding specific terms to the original BW formula, but also the interplay between the different terms. At this point we would like to reduce the excessive inflation of discussion about different terms of the BW, the interested reader is referred to this Ref. [@Kirson200829] and references therein to familiarize oneself with a rich history proposals of additional terms for inclusion in the formula.
-- -- -- -- --
-- -- -- -- --
: List of the Bethe-Weizsäcker mass formula coefficients (in MeV) obtained using different fitting procedures and the corresponding rms deviations (in keV). The sign “-" in the $\alpha_{pairing}$ column corresponds to even-even nuclei, and the “+" to odd-odd nuclei. Macroscopic parameters of different mass formulas based on Hartree-Fock-Bogoliubov theories [@Samyn2002142; @GORIELY2001311; @PhysRevC.66.024326] and the FRDM [@Moller1995185; @Moller1988213; @MYERS1974186] are shown in the center of the table and are underlined by a horizontal line. The quantities in parentheses for the FRDM denote the values used in the microscopic part of the model. Results of the current work shown in the bottom line. []{data-label="tab:BW_CoeffFits"}
However, from the recent literature one may note that there is still lack of knowledge about the binding energy of new nuclides both in the super-heavy element region and the regions close to the proton and neutron drip lines, and the different model predictions do not completely agree. The existence of the heaviest atomic nuclei bound against immediate disintegration depends on the detailed arrangement of protons, $Z$, and neutrons, $N$, in shells that provide a second effect on nuclear stability, which is analogous to the way electrons fill atomic orbitals; see, for instance, [@0954-3899-34-4-R01; @Cwiok:2005; @GRYZINSKI1973131; @Gryzinski1976180] and references therein. Some of the effective interactions used in the shell model approach [@PhysRev.75.1969; @PhysRev.75.1766.2] can be traced back to the nucleon-nucleon Brueckner $G$-matrix [@PhysRev.95.217; @PhysRev.96.508; @PhysRev.97.1344]. Closed shells are energetically favorable, and this modulates the smooth trend of the binding energy predicted by the liquid drop model. It has proven very successful, that few valence nucleons (outside closed shells) can describe the properties of nuclei [@Talmni:1993], including energy levels, magnetic and quadrupole moments, electromagnetic transition probabilities, beta-decay rates and reaction cross-sections. Moreover this approach has also been used as the theoretical basis for several algebraic nuclear models [@DaGeng:1997; @Cseh2015213]. Therefore, the shell model became the standard model for describing the systematics observed in the spectra and transition intensities of $p-$ [@COHEN19651], $sd-$ [@Wildenthal19845; @Brown1988191; @BrownAnnRev:1988] and lower $fp$-shell [@French:1969; @RICHTER1991325; @PhysRevC.50.225; @PhysRevC.55.187] nuclei. Since the size of the model space increases rapidly with the number of valence nucleons and/or orbits, full major shell calculations were limited to nuclei with $A \textless 49$ [@PhysRevC.50.225; @PhysRevC.55.187].
Therefore continuous improvements are still needed to determine of the nuclear masses and their structure.
Parametrization of the Bethe-Weizsäcker mass formula {#Parametrization_of_the_BWmass_formulae}
====================================================
So far due to absence of rigorous theory of the nucleus the role of the various correction terms in the BW formula remain unclear [@PhysRevC.65.037301; @ADHIKARI:2004; @Kirson200829]. As far as their empirical basis is concerned, the introduction of correction terms into the equation for the binding energy, despite the addition of new parameters, does not lead to any appreciable improvement in the agreement with experiment. Nevertheless, the introduction of purely empirical corrections to the parity and the effects of the shells considerably improves the agreement with experiment [@Cameron:1957; @KUMMEL1966129; @Strutinskii:1966].
The present paper attempts to provide a general mathematical formulation of this problem and to relate its occurrence to a new research strategy. A general calculational approach to the analysis of latent regularities is suggested, based on auto-regularized iteration processes for solving non-linear problems, see Sec.\[AleksandrovMethod\_REGN\].
The aim of our study was to investigate the BW mass formula correction terms correlation for the whole set of nucleus masses that have been measured [@Ami2012_ChinPhysC] and provide information on latent regularities that may affect the nuclei formation. Recent fits to the AME2012 atomic mass evaluation [@Wapstra2003129; @Ami2012_ChinPhysC] are given in Table \[tab:BW\_CoeffFits\]. One may consider the AME database as a good playground for developing and testing different nuclear models.
In this section we analyze fluctuations of the nuclei and atomic masses as a many-body quantum system, we rewrite the BW mass formula Eq.\[eq:LiquidDrop\_BW\] in the functional analytical framework of an operator equation. In particular we discuss the existence and origin of a chaotic part in the binding energy and its possible implications on the accuracy of theoretical mass predictions in atomic nuclei. A careful analysis of the previous attempts of calculation nucleus masses reveals that all models can reproduce experimental/empirical trends on the average [@Moller1995185; @Wapstra2003129; @Chowdhury:2004jr; @Kirson200829; @WDMyers:1966; @Myers1996141; @Samanta:2004et; @Rohlf:1994; @NIX19651; @Wapstra:1958; @PhysRev.89.1102; @PhysRevC.33.2039; @PhysRevC.67.044316; @Royer20131]. In order to overcome this issue we consider a parametrized nonlinear dynamical system of equations for determining nuclei and atomic masses from the experimental bound-state energies, which can be written using matrix notation of Eq.\[eq:fxy\_VectorForm\] as: $$%FE_{B}^{\rm{Th}}(A_{j},Z_{j},\{a_{i}\}) =E_{B}^{\rm{Expt}}(A_{j},Z_{j}),
FE_{B,j}^{\rm{Th}}(A,Z,\{a_{i}\}) =E_{B,j}^{\rm{Expt}}(A,Z),
\label{eq:BE_j}$$ $$%FM_{a.m.}^{\rm{Th}}(A_{j},Z_{j},\{a_{i}\}) = M_{a.m.}^{\rm{Expt}}(A_{j},Z_{j}),
FM_{a.m.,j}^{\rm{Th}}(A,Z,\{a_{i}\}) = M_{a.m.,j}^{\rm{Expt}}(A,Z),
\label{eq:Mam_j}$$ $$%FM_{n.m.}^{\rm{Th}}(A_{j},Z_{j},\{a_{i}\}) = M_{n.m.}^{\rm{Expt}}(A_{j},Z_{j}),
FM_{n.m.,j}^{\rm{Th}}(A,Z,\{a_{i}\}) = M_{n.m.,j}^{\rm{Expt}}(A,Z),
\label{eq:Mnm_j}$$ $$%F\Delta m^{\rm{Th}}(A_{j},Z_{j},\{a_{i}\}) = \Delta m^{\rm{Expt}}(A_{j},Z_{j}),
F\Delta m_{j}^{\rm{Th}}(A,Z,\{a_{i}\}) = \Delta m_{j}^{\rm{Expt}}(A,Z),
\label{eq:MExess_j}$$ where $F$ is a rectangular $d\times m$ matrix representing the forward operator $F$, $\{d=1,\dots,{\mathcal{N}}_{\rm{data}}; m=1,\dots,{\mathcal{N}}_{\rm{param}}\}$, $j$ is the index of the nonlinear equation in the system. Each of the systems, Eqs.\[eq:BE\_j\] - \[eq:MExess\_j\], contain 2564 equations, that correspond to the number of the experimental data-points. Therefore, these systems are overdetermined because the number of considered equations exceeds the number of parameters used in the fit. The right hand side of these equations represented by the vector of observed experimental data.
The solution of an overdetermined system of Eqs.\[eq:BE\_j\] for experimental data for binding energy and its model is given by the real values of the parameters $a_{i}$, where $i=1,\dots, {\mathcal{N}}_{\rm{param}}$ is the number of unknown parameters. Due to ill-posedness of the formulated problem we apply the iterative approach, which is based on the auto-regularized iteration process of Gauss-Newton type that was described in the previous sections (Sec.\[Theory\_and\_Method\]), to obtain a true solution of it. For more details we recommend reader the following references [@Alexandrov:1970; @Aleksandrov197146; @Alexandrov:19701285].
The main idea of any regularization algorithm is to consider, instead of one ill-posed inverse problem Eq.\[eq:BE\_j\], a family of well-posed problems, $$%F_{\varepsilon}E_{B}^{\rm{Th}}(A_{j},Z_{j},\{a_{i}\}) =E_{B}^{\rm{Expt}}(A_{j},Z_{j}),
F_{\varepsilon}E_{B,j}^{\rm{Th}}(A,Z,\{a_{i}\}) =E_{B,j}^{\rm{Expt}}(A,Z),
\label{eq:BE_WellposedFamily}$$ which approximate the original inverse problem in some sense. The scalar parameter $\varepsilon > 0$ is called a regularization parameter, and it was defined in Eq.\[eq:AleksandrovReg\]. We require also that $$%E_{B}^{\rm{Th}}(A_{j},Z_{j},\{a_{i}\}) \rightarrow E_{B}^{\rm{true}}(A_{j},Z_{j}),~{\rm{if}}~ \varepsilon \rightarrow 0,
E_{B,j}^{\rm{Th}}(A,Z,\{a_{i}\}) \rightarrow E_{B,j}^{\rm{true}}(A,Z),~{\rm{if}}~ \varepsilon \rightarrow 0,
\label{eq:BE_true}$$ where $E_{B,j}^{\rm{Th}}(A,Z,\{a_{i}\})$ is the solution of the inverse problem Eq.\[eq:BE\_WellposedFamily\], and $E_{B,j}^{\rm{true}}(A,Z)$ is the true solution of the original problem Eq.\[eq:BE\_j\]. Thus, we replace the solution of one ill-posed inverse problem by the solutions of the family of well-posed problems, assuming that these solutions, $E_{B,j}^{\rm{Th}}(A,Z,\{a_{i}\})$, tend asymptotically to the true solution, as $\varepsilon$ tends to zero.
In other words, the applied regularization algorithm, as in any other, is based on the approximation of the noncontinuous inverse operator $F^{-1}$ by the family of continuous inverse operators $F^{-1}_{\varepsilon} E_{B,j}^{\rm{Expt}}(A,Z)$ that depend on the regularization parameter $\varepsilon$. The regularization must be such that, as $\varepsilon$ vanishes, the operators in the family should approach the exact inverse operator $F^{-1}$.
The regularized solution of the ill-posed inverse problem, Eq.\[eq:BE\_j\], is provided by minimization of the corresponding parametric functional $$\begin{split}
&{\mathcal{J}}_{\varepsilon}(E_{B}^{\rm{Th}}(A,Z,\{a_{i}\})) =\\
&{\rm{min}}~\Bigg\{ \Arrowvert F E_{B}^{\rm{Th}}(A,Z,\{a_{i}\}) - E_{B}^{\rm{Expt}}(A,Z)\Arrowvert^{2} + (\alpha+\varepsilon) \Arrowvert {\mathcal{W}} E_{B}^{\rm{Th}}(A,Z,\{a_{i}\})\Arrowvert^{2} \Bigg\}
%& ={\rm{min}},
\label{eq:Parametric_functional}
\end{split}$$ here the $j$ and $k$ indices are omitted for brevity, $\alpha$ is the Tikhonov regularization parameter, ${\varepsilon}$ is the Aleksandrov regularization parameter compromising the $a~priori$ information with data fitting, ${\mathcal{W}}\in \bm{R}^{p\times {\mathcal{N}}_{\rm{param}}}$ is a positively determined linear continuous operator in the space of model parameters that imposes smoothness. We assume that ${\rm{ker}}F \cap {\rm{ker}}{\mathcal{W}} = \{ 0 \}$, which guarantees the uniqueness of the minimizer, and the rank of the matrix ${\mathcal{W}}$ is ${\mathcal{N}}_{\rm{param}}$.
The solution of the penalized least-squares problem Eq.\[eq:Parametric\_functional\] is given by Eq.\[eq:GaussNewton\], which can be rewritten as $$\begin{split}
&E_{B}(A,Z,\{a_{i}\})_{k+1} = E_{B}(A,Z,\{a_{i}\})_{k}\\
&- \left[ F^{\prime T}(E_{B}(A,Z,\{a_{i}\})_{k} ) F^{\prime}(E_{B}(A,Z,\{a_{i}\})_{k})+(\alpha + \varepsilon_{k}) I_{\bm{R}^{n}} \right]^{-1} \\
&\times \left[ F^{\prime T}(E_{B}(A,Z,\{a_{i}\})_{k}) \left(F(E_{B}(A,Z,\{a_{i}\})_{k})- E_{B}^{\rm{Expt}}(A,Z)_{k} \right) \right.\\
&\left. + \alpha E_{B}(A,Z,\{a_{i}\})_{k}^{}\right].
\label{eq:SolutionPenalizedLeastSquaresPr}
\end{split}$$ We assume that the misfit between the observed data and the true data is less than the given level of the experimental uncertainties. We will just comment here that in this statement we just follow the exposition given by Tykhonov and Arsenin [@Tikhonov:1986] and Lavrenti’ev [@Lavrentiev:1986]. For more details please see Ref.[@Zhdanov:2002]. So, in this situation, it is natural to search for a solution in the set of the models such that $$\begin{split}
%\Arrowvert F E_{B}^{\rm{Th}}(A,Z,\{a_{i}\}) - E_{B}^{\rm{Expt}}(A,Z)\Arrowvert^{2}~\le~\delta
\Arrowvert F E_{B,j}^{\rm{Th}}(A,Z,\{a_{i}\}) - E_{B,j}^{\rm{Expt}}(A,Z)\Arrowvert^{2}~\le~\delta
\label{eq:CondOfMinimization}
\end{split}$$ Note, that here we describe the basic idea of the applied method focusing only on the binding energy, the same method have been used for the rest of the systems, Eqs.\[eq:Mam\_j\] -\[eq:MExess\_j\]. In order to proceed further one have to choose the initial model for description of the binding energy. The BW mass formula was chosen for this role, the motivation for this choice has been already explained in the previous sections. It provides the baseline fit for all the rest models [@Moller1995185; @Wapstra2003129; @Chowdhury:2004jr; @Kirson200829; @WDMyers:1966; @Myers1996141; @Samanta:2004et; @Rohlf:1994; @NIX19651; @Wapstra:1958; @PhysRev.89.1102; @PhysRevC.33.2039; @PhysRevC.67.044316; @Royer20131]. The generalization of the BW mass formula, Eq.\[eq:LiquidDrop\_BW\], for the binding energy per nucleon in our approach has the following form: $$\begin{split}
%&-------------------------
& E_{B}(A,Z,\{a_{i}\}) = \alpha_{vol}(A,Z,\{a_{i}\}_{1})-\alpha_{surf}(A,Z,\{a_{i}\}_{2}) \frac{1}{A^{p_{1}(A,Z, \{a_{i}\}_{6})}}\\
&-\alpha_{comb}(A,Z,\{a_{i}\}_{3})\frac{Z(Z-1)}{A^{p_{2}(A, Z, \{a_{i}\}_{7})}} -\alpha_{sym}(A,Z, \{a_{i}\}_{4} )\frac{(N-Z)^{2}}{A^{p_{3}(A,Z, \{a_{i}\}_{8} )}}\\
&+\alpha_{Wigner}(A,Z,\{a_{i}\}_{5})\frac{\delta(A,Z)}{A^{p_{4}(A, Z, \{a_{i}\}_{1}) }}+K_{MN}(A,Z,\{a_{i}\}_{10}).\\
\end{split}
\label{eq:BWparametrization_Wigner_CorMN}$$ The term $\delta(A,Z)$ is expressed in the following way: $$\delta(A,Z) =
\begin{cases}
+1, & \mbox{for even}~N,Z\\
-1, & \mbox{for odd}~N,Z\\
0, & \mbox{for odd}~(Z+N)\rightarrow \mbox{for odd}~A.\\
\end{cases}
\label{eq:pairingTerm}$$ Here we have to clarify the important point in our approach. To perform the generalization of the BW mass formula we assume, that all the BW constant terms that are obtained as results of the different fitting procedure (see Table \[tab:BW\_CoeffFits\]) in our framework are unknown functions of $A$, $Z$, and $N$, namely structures, and the set of the unknown parameters $\{a_{i}\}$. In addition we involve the Wigner term, and the correction term instead of the pairing term, which are sensitive to the magicity of the given atomic mass number. Since the Bethe-Weizsäcker mass formula has no shell effect incorporated and thus when the shell effect in a nucleus quenches, by including this term, the theoretical mass should come closer to the experimental ones. Also we implement the original Coulomb term in order to have the possibility to describe the light nuclei. Therefore we propose that all structures corresponding to the appropriate type of energy, namely, volume, surface, Coulomb, asymmetry and pairing, as well as the power factors $A^{-1/3}$, $A^{-4/3}$, $A^{-2}$, $A^{-3/2}$ are all determined with the help of the function of $A$, $Z$ and unknown number of the parameters $a_{i}$, where $i=1,\dotsc,{\mathcal{N}}_{\rm{param}}$.
Given that we determine the binding energy of the nuclei with the help of Eq.\[eq:BWparametrization\_Wigner\_CorMN\], one may describe the atomic masses that are related to the nuclear mass $M_{n.m}(A,Z,\{a_{i}\})$ by $$M_{a.m}(A,Z,\{a_{i}\})=Zm_{p}+Nm_{n}-AE_{B}(A,Z,\{a_{i}\}),
\label{eq:AtomicMassFunc}$$ $M_{a.m}(A,Z,\{a_{i}\})$ may thus be connected to the experimental atomic masses given in [@Ami2012_ChinPhysC] since: $$M_{n.m}(A,Z, \{a_{i}\})=M_{a.m}(A,Z,\{a_{i}\})-Zm_{e}+B_{e}(Z).
\label{eq:Mnm}$$ The cloud of electrons that defines atomic dynamical [@PhysRevLett.24.45], chemical and mechanical properties generates its own binding field, driven by the strong force that exists between all of its constituent nucleons (neutrons and protons). As such, nuclei have a much less well-defined ‘centre’. Thus the $B_{e}(Z)$ in Eq.\[eq:Mnm\] is the binding energy of all removed electrons that is fitted by a power law accordingly to [@PhysRev.83.397; @RevModPhys.75.1021] as $$%B_{e}(Z)=a_{el}Z^{2.39}+b_{el}Z^{5.35},
B_{e}(Z)=a_{el}Z^{a_{240}}+b_{el}Z^{a_{241}},
\label{eq:Electrons_BindEn}$$ with $a_{el}=1.44381\times10^{-5}$ MeV and $b_{el}=1.55468\times10^{-12}$ MeV.
The mass excess of a nuclide is $$\Delta m(A,Z,\{a_{i}\})=M_{a.m}(A,Z,\{a_{i}\})-A,
\label{eq:Mnm_Strashemir}$$ where $m_{H} = 938.782303(0.084)$ MeV, $m_{n} = 939.56538(4.56)$ MeV, $m_{p} = 938.272046(21)$ MeV, $m_{el} = 0.510998928(11)$ MeV [@Ami2012_ChinPhysC; @PhysRevD.86.010001] and $u = 931.494061$ MeV [@Ami2012_ChinPhysC; @PhysRevD.86.010001]. The $1-\sigma$ uncertainties in the last digits are given in parentheses after the values.
Therefore, in order to solve all systems Eqs.\[eq:BE\_j\]-\[eq:MExess\_j\] we adopt the following procedure. First, we searched for the appropriate solution for the binding energy for the given $A$ and $Z$. The obtained solution was used as an approximation (‘fake solution’) to find the atomic masses, the solution for atomic masses was in turn used to find nuclear mass. Then, the fit is performed again, allowing the binding energy to vary for the given $A$ and $Z$. The resultant binding energy is then taken to be the new seed, and this procedure is repeated iteratively until convergence is reached for all systems. Indeed the system converge due to applied iterative process, that was discussed in Sec.\[AleksandrovMethod\_REGN\].
In Figs.\[fig:BindingEnergyA\], we plot the result binding energy and the mass excess. In the bottom pads, we show subtracted the true solution from the experimental data for each of the isotopes. There is no significant deviation for the binding energy, while the maximum absolute deviation for the mass excess is less than 2.6 MeV.
The unknown structure functions of Eq.\[eq:BWparametrization\_Wigner\_CorMN\] with exponential nonlinear terms are expressed as follows: $$\begin{split}
&\alpha_{vol}(A, Z, \{a_{i}\}_{1})={\rm{exp}} \left(a_{1} + P\left(\upsilon, \{a_{i}\}_{1} \right) \right),\\
&\alpha_{surf}(A, Z, \{a_{i}\}_{2})={\rm{exp}} \left(a_{2}+P\left(\upsilon, \{a_{i}\}_{2} \right)\right),\\
&\alpha_{comb}(A, Z, \{a_{i}\}_{3})={\rm{exp}} \left(a_{3}+P\left(\upsilon, \{a_{i}\}_{3} \right)\right),\\
&\alpha_{sym}(A, Z, \{a_{i}\}_{4})={\rm{exp}}\left(a_{4} +P\left(\upsilon, \{a_{i}\}_{4} \right) \right),\\
%&\alpha_{Wigner}(A, Z, \{a_{i}\}_{5})={\rm{exp}}\left(a_{5} +P\left(\upsilon, \{a_{i}\}_{5}\right) \right) +\omega_{W}(A,Z,\{a_{i}\}_{{\mathcal{N}}_{\rm{param}}-4} )\\
&\alpha_{Wigner}(A, Z, \{a_{i}\}_{5})={\rm{exp}}\left(a_{5} +P\left(\upsilon, \{a_{i}\}_{5}\right) \right) +\omega_{W}(A,Z,\{a_{i}\} ).\\
\end{split}
\label{eq:BW_coeff}$$ The explicit form of the functions in Eq.\[eq:BW\_coeff\] has been obtained by implementation of the LCH procedure [@Aleksandrov2004519; @LCH:1999] in the iteratively regularized method [@Alexandrov:1970] for inverse problems, see Eq.\[eq:BE\_j\], of discovering the explicit form of the unknown functions, which is realized in the Dubna REGN (Regularized Gauss-Newton iteration method) program [@Alexandrov:1973; @Alexandrov:1983; @Alexandrov:1982], and allows us to choose the better one of two functions with the same $\chi^{2}$. Therefore, by solving the overdetermined nonlinear systems with condition Eqs.\[eq:BE\_j\]-\[eq:MExess\_j\], using step-by-step different models for unknown functions $\alpha_{vol}(A, Z, \{a_{i}\})$, $\alpha_{vol}(A, Z, \{a_{i}\})$,\
$\alpha_{surf}(A, Z, \{a_{i}\})$, $\alpha_{comb}(A, Z, \{a_{i}\})$, $\alpha_{sym}(A, Z, \{a_{i}\})$, $\alpha_{Wigner}(A, Z, \{a_{i}\})$, we obtain their explicit form, which was written in Eq.\[eq:BW\_coeff\].
The parametrized powers that were implemented in Eq.\[eq:BWparametrization\_Wigner\_CorMN\], have been obtained using the same LCH procedure, that was used for the structure functions, and defined as: $$\begin{split}
& p_{1}(Z,A, \{a_{i}\}_{6})={\rm{exp}} \left(a_{6}+P(\upsilon, \{a_{i}\}_{6} )\right),\\
& p_{2}(Z,A, \{a_{i}\}_{7})={\rm{exp}} \left(a_{7}+P(\upsilon, \{a_{i}\}_{7} )\right),\\
& p_{3}(Z,A, \{a_{i}\}_{8})={\rm{exp}} \left(a_{8}+P(\upsilon, \{a_{i}\}_{8} )\right),\\
& p_{4}(Z,A, \{a_{i}\}_{9})={\rm{exp}} \left(a_{9}+P(\upsilon, \{a_{i}\}_{9} )\right).
\end{split}
\label{eq:BW_Powers_coeff}$$ The function, $P(\upsilon, \{a_{i}\})$, that determines corrections to the structure terms in Eqs.\[eq:BW\_coeff\], and also implies corrections to the power factors of Eq.\[eq:BW\_Powers\_coeff\] with respect to the different isotopes in order to obtain the appropriate binding energy, defined using the exponential distributions as: $$\begin{split}
%&P(\upsilon, a, i)= {\rm{exp}}\left( -\left(\sum_{k=1}^{4} c_{k}(\upsilon, a, i) \right)^{2} \right),
&P(\upsilon, \{a_{i}\}_{j})= {\rm{exp}}\left( -\left(\sum_{k=1}^{4} c_{k}(\upsilon, \{a_{i}\}_{j}) \right)^{2} \right),
\end{split}
\label{eq:func_CorPow}$$ where the four functions $c_{k}(\upsilon, \{a_{i}\}_{j})$ defined through linear series of the variables $\upsilon_{k}$, where $k=1,..,6$ and parameters $a_{i}$ as $$\begin{split}
&c_{1}(\upsilon, \{a_{i}\}_{j} )=a_{i_1}\upsilon_{1}+a_{i_2}\upsilon_{2}+a_{i_3}\upsilon_{3}+a_{i_4}\upsilon_{4},\\
&c_{2}(\upsilon, \{a_{i}\}_{j})=a_{i_5}(\upsilon_{1})^{2}+a_{i_6}(\upsilon_{2})^{2}+a_{i_7}(\upsilon_{3})^{2}+a_{i_8}(\upsilon_{4})^{2},\\
&c_{3}(\upsilon, \{a_{i}\}_{j})=a_{i_9}(\upsilon_{1})^{3}+a_{i_{10}}(\upsilon_{2})^{3}+a_{i_{11}}(\upsilon_{3})^{3}+a_{i_{12}}(\upsilon_{4})^{3},\\
&c_{4}(\upsilon, \{a_{i}\}_{j})=a_{i_{13}}\upsilon_{6}+a_{i_{14}}\upsilon_{5}.
\end{split}
\label{eq:func_CorGam_coeff}$$ Here $\{a_{i}\}_{j}$ represents the subset of parameters from the whole set of parameters that is used for the particular structure function. The consecutive index $j$ defines a mapping for the chosen subset in the full set of free parameters used in our model. For $j=1,\dots,9$ the mapping is following: $$\begin{split}
& \{a_{i}\}_{1}\rightarrow a_{10},a_{11},\dots,a_{23},\\
& \{a_{i}\}_{2}\rightarrow a_{24},a_{25},\dots,a_{37},\\
& \cdots\\
& \{a_{i}\}_{9}\rightarrow a_{122},a_{123},\dots,a_{135}.
\end{split}
\label{eq:ParamMappring_j19}$$ For reasons of explicity, let us examine the case when the consecutive index $j=2$ in Eq.\[eq:func\_CorPow\]. Therefore, we have $c_{k}(\upsilon, \{a_{i}\}_{2})$, whose sequence of the parameters, taken into account Eq.\[eq:ParamMappring\_j19\], can be written in the following way $a_{i_{1}}\rightarrow a_{24}, a_{i_2}\rightarrow a_{25},\cdots, a_{i_{14}}\rightarrow a_{37}$. Thus, for each of the functions $P(\upsilon, \{a_{i}\})$ we assign the 14 parameters from the whole set $\{a_{i}\}$.
The presented approach enables calculations of nuclear properties in large model spaces. The concrete subjects of investigation are the nucleus masses and the structure behavior of the BW mass formula, whose bound states are interpreted as highly stable elements $\textendash$ nuclei with closures shells, magic numbers. In order to obtain such kind of dependencies from magic numbers and boundaries between them, which define their influence, different inverse problems were formulated using different values of proton and neutron magic numbers. The LCH analysis of solutions of these additional problems helps us to establish the explicit form of the correction, which depends on nine proton and ten neutron magic numbers, see Fig.\[fig:MagNumberList\] and Table \[tab:ZN\_magnumbers\]. Therefore, the function that determines the correction energy with respect to the magic numbers is: $$\begin{split}
&K_{MN}(Z,A, \{a_{i}\}_{10} )\\
&=W_{\rm{BrWigner}}\left(Z, Z_{MN}^{\rm{nearest}}, w_{Z}, N, N_{MN}^{\rm{nearest}}, w_{N}, \{a_{i}\}_{10} \right)\\
&+\omega_{MN}(A,Z, \{a_{i}\}),\\
\end{split}
\label{eq:CorMN_func}$$ where $w_{Z}$ and $w_{N}$ are the mean values of the two closest proton magic numbers and the two neutron magic numbers respectively. For this function we use subset of $\{a_{i}\}_{10}$ which now includes 15 parameters, starting from $a_{136}$ parameter in the sequence. This correction function provides an indicator about the deviation in the level structure of nuclei away from closed shells. Also it includes the pairing effects, which can also provide reliable information about magicity of the different proton and neutron numbers. The maximum values of the correction energy corresponds to shell closure at certain nucleon number, as will be shown later in the paper.
The $\omega_{W}$ function from the Eq.\[eq:BW\_coeff\] can be written as: $$\begin{split}
&\omega_{W}(A,Z,\{a_{i}\})=
{\rm{exp}}\left( a_{237}\frac{\upsilon_{7}}{A}+a_{238}\frac{\upsilon_{8}}{Z}+a_{239}\frac{\upsilon_{9}}{N+1}\right).
\end{split}
\label{eq:omegaW_Functions}$$ The $\omega_{MN}$ function in Eq.\[eq:CorMN\_func\] defined in the similar way as $\omega_{W}$, the only difference it depends on the different subset of the parameters $a_{i}$ as: $$\begin{split}
&\omega_{MN}(A,Z, \{a_{i}\})
={\rm{exp}} \left( a_{234}\frac{\upsilon_{7}}{A}+a_{235}\frac{\upsilon_{8}}{Z}+a_{236}\frac{\upsilon_{9}}{N+1} \right),\\
\end{split}
\label{eq:omegaMN_Functions}$$ where the variables $\upsilon_{7}$, $\upsilon_{8}$ and $\upsilon_{9}$ that used in these functions will be defined later in the paper.
The Breit-Wigner function, $W_{\rm{BrWigner}}$, in Eq.\[eq:CorMN\_func\] was constructed in the following form: $$\begin{split}
&W_{\rm{BrWigner}}(Z,Z_{MN}^{\rm{nearest}},w_{Z}, N, N_{MN}^{\rm{nearest}}, w_{N}, \{a_{i}\}_{10} )\\
&=W_{\rm{BrWigner}}^{Z}(Z,Z_{MN}^{\rm{nearest}},w_{Z}, \{a_{i}\})+W_{\rm{BrWigner}}^{N}(N,N_{MN}^{\rm{nearest}},w_{N}, \{a_{i}\}),\\
\end{split}
\label{eq:BrWirnerDistribution}$$ where $$\begin{split}
&W_{\rm{BrWigner}}^{Z}(Z,Z_{MN}^{\rm{nearest}},w_{Z}, \{a_{i}\})={\mathcal{A}}_{Z}\frac{ {\rm{exp}} \left( \frac{- \left(Z-Z_{MN}^{\rm{nearest}}\right)^{2} } {2Q(\upsilon, \{a_{i}\}_{12} ) } \right)} {\left(Z-Z_{MN}^{\rm{nearest}}\right)^{2}+Q(\upsilon, \{a_{i}\}_{12} )},\\
&W_{\rm{BrWigner}}^{N}(N,N_{MN}^{\rm{nearest}},w_{N}, \{a_{i}\})={\mathcal{A}}_{N}\frac{ {\rm{exp} }\left( \frac{- \left(N-N_{MN}^{\rm{nearest}}\right)^{2} } {2Q(\upsilon, \{a_{i}\}_{13}) } \right) }{\left(N-N_{MN}^{\rm{nearest}}\right)^{2}+Q(\upsilon, \{a_{i}\}_{13} )},\\
\end{split}
\label{eq:func_CorGam}$$ Here $Z$ is the proton number, $N$ is neutron number, $w_{Z}$ and $w_{N}$ are half-width of the two closest magic numbers for the given proton and neutron numbers respectively. One may note that these widths are representing the arguments of the generalized BW mass function, which is defined as the product of the two distributions, the Gaussian and the Breit-Wigner. Given that prescription one may conclude that the nearest proton or neutron number is the location parameter, specifying the location of the peak of the binding energy distribution.
The amplitudes in Eq.\[eq:func\_CorGam\] are written as: $$\begin{split}
&{\mathcal{A}}_{Z}(\upsilon,\{a_{i}\}_{10})=w_{Z}+Q(\upsilon, \{a_{i}\}_{10} ),\\
&{\mathcal{A}}_{N}(\upsilon,\{a_{i}\}_{11})=w_{N}+Q(\upsilon, \{a_{i}\}_{11} ).\\
\end{split}
\label{eq:func_CorGam_Amplitude}$$ The scale functions, $Q(\upsilon, \{a_{i}\}_{j})$ in Eq.\[eq:func\_CorGam\], which specify the half-width at half-maximum, are given by this relation: $$\begin{split}
&Q(\upsilon,\{a_{i}\}_{j} )={\rm{exp}}\left(a_{i+15}- \left(\sum_{k=1}^{4} c_{k}(\upsilon, \{a_{i}\}_{j} ) \right)^{2} \right).\\
\end{split}
\label{eq:func_CorGam_Gam_ex}$$ As one may note, the Eq.\[eq:func\_CorGam\_Gam\_ex\] is similar to the $P(\upsilon, \{a_{i}\}_{j})$ function that was defined in the Eq.\[eq:func\_CorPow\], but here we have an extra term, $a_{i+15}$, due to this term each of the function in the expressions Eq.\[eq:func\_CorGam\] and Eq.\[eq:func\_CorGam\_Amplitude\] the $j$ index is now mapping 15 parameters: $$\begin{split}
& \{a_{i}\}_{10}\rightarrow a_{136},a_{137},\dots,a_{150},\\
& \{a_{i}\}_{11}\rightarrow a_{151},a_{152},\dots,a_{165},\\
& \cdots\\
& \{a_{i}\}_{13}\rightarrow a_{181},a_{182},\dots,a_{195},
\end{split}
\label{eq:ParamMappring_j1013}$$ Since experimental data has systematic uncertainties we apply robust procedures in order to estimate the weights of the explicit form of the Breit-Wigner distributions, which has been obtained with the help of the LCH-weights method [@Aleksandrov2004519; @LCH:1999].
In Fig.\[fig:BetheWeizsackerCorMNA\] we plot the distribution functions of Eq.\[eq:func\_CorGam\]. In the top pad, we show for comparison the spectrum of the shell correction energy, Eq.\[eq:CorMN\_func\], that reflects effect of the magicity of the given atomic mass number. From this figure, it is clear how the adopted Breit-Wigner functions correlates with the shell correction energy function.
In the given formalization, see Eqs.\[eq:BE\_j\] and \[eq:BWparametrization\_Wigner\_CorMN\] the problem of finding the unknown structure functions in Eq.\[eq:BW\_coeff\] becomes similar to the problem that was stressed in the Sec.\[Theory\_and\_Method\], namely the problem of allocating the exponents with unknown decrement $h_{i}$ and amplitudes ($\rm{a}_{0}$, $\rm{a}_{i}$) on the finite values of their sum, $f(t_{k})$, which are known approximately [@Alexandrov:19701285]. If we do not know the exact number of the exponents, then this problem can be resolved by assuming that number $n$ in Eq.\[eq:NonLinearSystem\] is very big, bigger than the expected number of the unknown exponents. However, to obtain the correct dependences of the decrements on the level of the $A$, $Z$ numbers in Eq.\[eq:BW\_coeff\], it is necessary to carry out extensive and time-consuming computations of the amplitude of the attenuating oscillations for subsequent calculations of the different values of the decrements.
Further, in this paper we apply the iteration process, see Sec.\[AleksandrovMethod\_REGN\], for the constructed systems of Eqs.\[eq:BE\_j\]-\[eq:MExess\_j\]. The Tikhonov regularization method within Aleksandrov regularizator [@Alexandrov:1973; @Alexandrov:1983; @Aleksandrov197146; @Alexandrov66:1977] has been applied for each iteration to the solution of the not-well defined systems of the nonlinear equations [@Tikhonov:1986; @Tikhonov:1995]. In the developed iteration scheme, the problem of choosing the initial approximations ($a~priori$ knowledge) is in some sense solved, by using values from [@Rohlf:1994], see the fifth row in Table \[tab:BW\_CoeffFits\], as initial values for structure functions. The choise of $p_{i}$ is arbitrary, and in present application we suppose, that initial values correspond to the liquid-drop model, hence we start from the power factors $p_{1}=1/3$, $p_{2}=4/3$, $p_{3}=2/3$ and $p_{4}=3/2$. Of course, in standard regularization $a~priori$ knowledge leads to restriction on the solution space. But we choose this approximation not because we think that it is a good description of the binding energy, but rather because it is a necessary first step towards the required formalism. Indeed as will be shown further, this approximation is a useful simplification of the parametrized powers, see Eq.\[eq:BW\_Powers\_coeff\]. The result power factors we plot in Fig.\[fig:PowerFactors\_AZN\].
For next step forward one may need to define the number and the form of the independent variables. Thus the next problem is to choose the independent variables of the functions $a_{vol}$, $a_{surf}$, $a_{comb}$, $a_{sym}$, $a_{Wigner}$, and $p_{j}$, where $j=1,2,3,4$. The hypothesis that they are linearly independent permits one to choose them as follow: $$\upsilon_{1}=\frac{Z}{A}, ~\upsilon_{2}=\frac{N}{A}, ~\upsilon_{3}=\frac{N-Z}{A}, ~\upsilon_{4}=\frac{Z}{N+1},
\label{eq:IndependentVariable_14}$$ where $Z$ and $N$ are the numbers of protons and neutrons in nuclei, and $A=Z+N$. The next two the linearly independent variables are $$\begin{split}
& \upsilon_{5}={\rm{log}}(A+1), ~\upsilon_{6}=\frac{1}{\upsilon_{5}},\\
\end{split}
\label{eq:IndependentVariable_56}$$ Plus we have the corresponding four square and the four cubic terms of Eq.\[eq:IndependentVariable\_14\] that used in Eq.\[eq:func\_CorGam\_coeff\]. In addition to those parameters we also have an extra three spin variables $\upsilon_{7}$, $\upsilon_{8}$, $\upsilon_{9}$, that were implemented in order to estimate the impact of valence proton and neutron in the nuclei to the binding energy, as well as to obtain dependencies of the binding energy with respect to the odd-odd, even-even, odd-even nuclei. They are defined in the following way $$\begin{split}
&\upsilon_{7}=
\begin{cases}
0,~~\mbox{for odd}~A\\
1,~~\mbox{otherwise}~\\
\end{cases}\\
&\upsilon_{8}=
\begin{cases}
0,~~\mbox{for odd}~Z\\
1,~~\mbox{otherwise}~\\
\end{cases}\\
&\upsilon_{9}=
\begin{cases}
0,~~\mbox{for odd}~N\\
1,~~\mbox{otherwise}~\\
\end{cases}
\label{eq:AAA}
\end{split}$$
Thus, totally we will use 17 linearly independent variables.
Summing up all the above let us compute the number of unknown parameters in our parametrization. The parameters that we found in order to describe the whole set of the experimental data are: $$\begin{split}
%& n_{str}=5, ~~n_{p}=4, ~~n_{Pow}=14,\\
& n_{str}=5, ~~n_{p}=4, ~~n_{map}=14,\\
\end{split}
\label{eq:n_BWpowers}$$ where $n_{str}$ is the number of the structures, see Eq.\[eq:BW\_coeff\] namely, volume, surface, Coulomb, asymmetry, Wigner and magic number terms; $n_{p}$ is the number of the power factors that used in our parametrization, see Eq.\[eq:BW\_Powers\_coeff\], while the total number of the structures and power factors defined as $n_{SP}=n_{str}+n_{p}$. In addition we define the number of parameters that we use for each of the unknown functions as $n_{map}$, see Eq.\[eq:func\_CorGam\_coeff\]. Thus only for the structure functions in the BW mass formula we need $$n_{SP}+n_{SP}n_{map}=135.
\label{eq:BW_paramNr}$$ The number of parameters for the function that determines the correction energy with respect to the magic numbers namely, the Breit-Wigner function in our fit is equal to $$n_{\rm{BrWign}}=2n_{\rm{width}}+2n_{\rm{amp}}=60,
\label{eq:nBW}$$ where we use 15 parameters to define the width, $n_{\rm{width}}$, and 15 parameters to define the amplitude, $n_{\rm{amp}}$, of the Breit-Wigner function.
The number of unknown parameters in our mass function before the correction of magic numbers starts affecting the final solution defined as $n_{0}$ and written as: $$\begin{split}
& n_{0}=n_{SP}+n_{SP}n_{map}+n_{\rm{BrWign}} = 195.\\
\end{split}
\label{eq:n_BWpowers}$$ Therefore, totally we have: $$\begin{split}
& {\mathcal{N}}_{\rm{param}}=n_{0}+2n_{mn,Z}+2n_{mn,N}+8 = 241,
\end{split}
\label{eq:Ntotal}$$ where the total number of the proton magic numbers is $n_{mn,Z}=9$, and for the neutron $\textendash$ $n_{mn,N}=10$. The factor of 2 in the front of the magic numbers appears due to fact that need take into account the effect of the boundaries between two closest magic numbers with respect to the examined nuclei. ${\mathcal{N}}_{\rm{param}}$ is the total number of unknown free parameters that describe the semi-empirical mass formula Eq.\[eq:BWparametrization\_Wigner\_CorMN\], and in the current work it is equal to 241. Here we have to emphasize that many inverse problems include nuisance parameters, such as variance parameters, regularization parameters, application-specific tuning parameters, or degrees of freedom parameters, that are not of primary interest but can have significant influence on the estimation of primary parameters. Such kind of issues arise in a great variety of applications, including seismic inverse problems [@Tarantola:1984], dynamic systems [@Fahrmeir:2010], optimal experimental design [@Horesh:2011], uncertainty quantification [@Flath:2011] and pharmacokinetic modeling [@Bradley:1996]. Nevertheless one may note that this is quite a big number with respect to the 31 independent mass-related parameters used in the finite-range droplet model (FRDM) [@RevModPhys.75.1021], or, for instance, 19 parameters used in the Hartree-Fock BCS-method, the HFBCS-1 mass formula, [@GORIELY2001311], and in Hartree-Fock-Bogoliubov HBF-1 mass formulas [@Samyn2002142]. But this number is less than 477 parameters required in the mass relations used in [@RevModPhys.41.S1], or by factor of three less than 928 parameters used in the mass predictions based on the Garvey-Kelson mass relations [@Janecke1988265]. Indeed we have a big number of parameters, because we do not know the exact number of exponents in our parametrization. However, in our case it is not an ultimate number, since most of these parameters are strongly correlated with each other, those in further computations, one can drop some of them off from the final solution in order to keep just uncorrelated ones. From our estimation the total number of unknown parameters can be reduced by a factor of $2$ or $4$. Unfortunately the search for this kind of correlations between unknown parameters requires a lot of computation resources due to nonlinearity of the problem, which is currently not available for us. In light of this issue we decide stick to this number in the current work.
Results {#Results}
=======
A major program in computational physics is to calculate nuclear properties from underlying realistic nuclear forces. A careful analysis of the AME2012 [@Ami2012_ChinPhysC; @NUBASE2012_ChinPhysC] database using the dynamic autoregularization method or, more precisely, the REGN program [@Alexandrov:19701285; @Alexandrov:1973; @Alexandrov:1982; @Alexandrov:1983], which is the constructive development of Tikhonov regularization method [@Tikhonov:1986; @Tikhonov:1995; @Tikhonov:1983], for solving the nonlinear system of equations Eq.\[eq:BWparametrization\_Wigner\_CorMN\], reveals phenomenological functional features of previously unknown parameters of the Bethe-Weizsäcker mass formula.
The search and selection of the best solution has been done with the help of the conventional statistical methods, such as assessments of the $\chi^{2}$ test. The LCH-weighting procedure [@Aleksandrov2004519; @LCH:1999] of the REGN program helps us to choose the better function out of two functions with the same $\chi^{2}$. The essential difference of the Alexandrov method [@Alexandrov:1973; @Alexandrov:1982; @Alexandrov:1983; @Alexandrov:19701285] from other similar methods is extremely effective ideology regularization of inverse problem solution, which on each iteration step controls not only the actual decision, but, very importantly, uncertainty of the solution. At the same time, it should be noted that the transition from the mathematical theory of the auto-regularized iterative processes, which is based on meaningful L. Aleksandrov’s theorems of convergence [@Alexandrov:1970], to Fortran codes, for example, REGN-Dubna [@Alexandrov:1983], FXY-Sofia-Dubna [@AleksandrovPriv1997] is a complicated, but technically clear work.
The presented parameterization of the binding energy allows us to solve the formulated inverse problem, see Eqs.\[eq:BE\_j\] - \[eq:MExess\_j\], with the help of the Aleksandrov’s auto-regularized method. This solution provides us description of the 2564 nucleus masses and their binding energies starting from $A=2$ with relative error $-1.1924\times10^{-6}$ and $3.2197\times10^{-4}$ for atomic mass and binding energy respectively. Where the relative error for the binding energies was computed as: $$\bar{\epsilon}_{rel}=\frac{1}{N}\sum_{j=1}^{N=2564} \frac{E_{B,j}^{\rm{Expt}}(A, Z) - E_{B,j}^{\rm{Th}}(A, Z, \{a_{i}\})}{E_{B,j}^{\rm{Expt}}(A, Z)}.
\label{eq:MeanAbsoluteError}$$ In the same way was computed the relative error for the atomic masses.
Mean absolute error $\bar{\epsilon}_{abs}$ is $-0.0509$ and $7.8488\times10^{-4}$ for atomic mass and binding energy respectively. The maximum absolute deviation is less than 2.6 MeV for the atomic mass and less than 0.82 MeV for the binding energy. In addition, the various characteristics have been obtained, see Fig.\[fig:PowerFactors\_AZN\] and Fig.\[fig:Structure\_constants\]. Here we would like to note, that one may reach a deviation less than 1.9 MeV for atomic masses by dropping off the Hydrogen atom from the data-point list using the same method.
The model standard deviation computed using the Möller and Nix formula: $$\begin{split}
&\sigma_{\rm{mod}}^{2} =\\
& \frac{1}{\sum w_{j}} \sum_{j=1}^{2564} w_{j}\left[ \left( M_{a.m.,j}^{\rm{Expt}}(A,Z) - M_{a.m.,j}^{\rm{Th}}(A,Z, \{a_{i}\}) \right)^{2} - (\sigma_{j}^{\rm{Expt}})^{2} \right],
\label{eq:MollerFit_quality}
\end{split}$$ where $M_{a.m.,j}^{\rm{Th}}(A,Z, \{a_{i}\})$ is the model prediction for the data point $i$, $\sigma_{j}^{\rm{Expt}}=\sigma_{j}^{\rm{Expt}}(A,Z)$ is the experimental error with which each of the data points $M_{a.m.,j}^{\rm{Expt}}(A,Z)$ is measured, assuming that the inherent errors of the model follow a Gaussian distribution, see for details [@Moller1988213]. The same formula has been used to compute the rms values for the binding energy. The results are shown in Table \[tab:RMS\_error\]. We assume that such big mean errors for the atomic mass appear due to the fact that we include all light atoms starting from Hydrogen isotope, $^{2}_{1}\rm{H}$.
The mean model error is $$\begin{split}
& \bar{\epsilon}_{\rm{mod}} =\\
& \frac{1}{\sum w_{j}} \sum_{j=1}^{2564} w_{j}\left[ \left( M_{a.m.,j}^{\rm{Expt}}(A,Z) - M_{a.m.,j}^{\rm{Th}}(A,Z, \{a_{i}\}) \right) \right],
\label{eq:MollerFit_meanModelError}
\end{split}$$ The weight factor is $$w_{j}= \left( \left(\sigma_{j}^{\rm{Expt}}(A,Z)\right)^{2} + \left(\sigma_{j}^{\rm{Th}}(A,Z)\right)^{2}\right)^{-k}. \\
\label{eq:SigmaRMSweight_factor}$$ The benefit of such weighting in the form given by Eq.\[eq:MollerFit\_quality\] is that one may ask if $k = 2$ is the only possibility, or if other values of $k$ are possible. We shall not discuss this question in depth here but instead will comment that if all $(\sigma_{j}^{\rm{Expt}})^{2}$ are equal, then all values of $k$ yield the same equation. However, the more interesting case is how $k$ should be chosen if the $(\sigma_{j}^{\rm{Expt}})^{2}$ have different values, for details see [@Moller1988213]. To gain some additional insight into the properties of the applied method we compare the root-mean-square deviation of our calculated masses from the AME2012 for three different choices of $k$ factor, namely 1, 1.5 and 2. In Table \[tab:RMS\_error\] we provide our $\sigma_{\rm{mod}}$ for the different $k$’s.
The mean modified error, Eq.\[eq:MollerFit\_meanModelError\], of our solution is of 1.232(1%) $\textendash$ 1.231(5%) MeV, the $\sigma_{\rm{mod}}$ of 1.11(1%) $\textendash$ 1.565(5%) MeV (assuming that the theoretical uncertainty varies between 1% and 5%, see Eq.\[eq:MeanError\], and $k=2$) for the atomic mass. While for the binding energy, using the same assumption for band of the theoretical uncertainties, we obtain $\bar{\epsilon}_{\rm{mod}}$, of fit of 0.044(1%) $\textendash$ -0.0057(5%) MeV, the $\sigma_{\rm{mod}}$ of 0.209(1%) $\textendash$ 0.223(5%) MeV, which can be compared with the latest fits for all modern mass formulas [@PhysRevLett.16.197; @Janecke1988265; @Liran1976431; @Nayak1999213; @Koura200047; @PhysRevC.52.R23], see Table \[tab:RMS\_error\].
The mean error of the model is computed, assuming that the theoretical uncertainty is modeled through a multi-band set of 3% and 5%, in the following way: $$\begin{split}
&\bar{\epsilon}_{\%}=\frac{1}{N}\sum_{j=1}^{N=2564} E_{B,j}^{\rm{Th}}(A, Z, \{a_{i}\})\times s_{2}\\
\label{eq:MeanError}
\end{split}$$ The hypothesis testing has been done to determine what significance there is with this value of $\chi^{2}$ using the formula (see Eq.6 in [@Ami2012_ChinPhysC]): $$\chi^{2}=\sum_{j=1}^{N=2564}\left( \frac{E_{B,j}^{\rm{Expt}}(A, Z)-E_{B,j}^{\rm{Th}}(A, Z, \{a_{i}\})}{\sigma_{j}(A, Z)} \right),
\label{eq:Chi2_test}$$ where $$\sigma_{j}(A, Z)=s_{1} \sigma_{j}^{\rm{Expt}}(A, Z)+s_{2}E_{B}^{\rm{Expt}}(A, Z).
\label{eq:Sigma_treat}$$ Here $\sigma_{j}^{\rm{Expt}}(A, Z)$ is the uncertainty of a nucleus as it has been reported in AME2012, $s_{1}$ and $s_{2}$ are the nuisance parameters, where $s_{2}$ is the percentage of the given experimental value. Eq.\[eq:Chi2\_test\] has been applied to all considered nuclei in this work. Table \[tab:Chi2Sigma\_treatment\] illustrates the quality of the descriptions of the binding energy, the nuclear and atomic masses, and the mass excess assuming a different hypothesis for the nuisance parameters $s_{1}$ and $s_{2}$. Note, that some masses of nuclei are measured with very high precision, which be can noticed from the mass excess column in AME2012, but due to artificial cutoff of the significant digits the uncertainties for these nuclei are given as zero uncertainty. Since we do not know the exact numbers we treat uncertainties for these nuclei as 1% of the given experimental value.
We used such analyses in order to be able to allocate questionable uncertainties in the 2012 Audi-Wapstra tables. In Table \[tab:Chi2Sigma\_treatment\] we report the value of corrections that have been made.
In addition one can define as above the normalized chi, $\chi_{n}$ (or ’consistency factor’ or ’Birge ratio): $$%\chi_{n}=\sqrt{\frac{\chi^{2}}{N-{\mathcal{N}}_{\rm{total}}}}
\chi_{n}=\sqrt{\frac{\chi^{2}}{N-{\mathcal{N}}_{\rm{param}}}}
\label{eq:NormalizedChi}$$ where $N-{\mathcal{N}}_{\rm{param}}$ is the number of degrees of freedom.
---------------------------------------------- ---------------------------------------------- -- -- --
$~~~~1 ~\leq~ 2 ~\textless~ 5$ $1 ~\leq~ 2 ~\textless~ 5$
$~~~~5 ~\leq~ 8 ~\textless~ 11$ $5 ~\leq~ 8 ~\textless~ 11$
$~~11 ~\leq~ 14 ~\textless~ 17$ $11 ~\leq~ 14 ~\textless~ 17$
$~~17 ~\leq~ 20 ~\textless~ 24$ $17 ~\leq~ 20 ~\textless~ 24$
$~~24 ~\leq~ 28 ~\textless~ 39$ $24 ~\leq~ 28 ~\textless~ 39$
$~~39 ~\leq~ 50 ~\textless~ 66$ $39 ~\leq~ 50 ~\textless~ 66$
$~~66 ~\leq~ 82 ~\textless~ 95$ $66 ~\leq~ 82 ~\textless~ 103$
$~~95 ~\leq~ 108 ~\textless~ 116$ $103 ~\leq~ 124 ~\textless~ 138$
$116 ~\leq~ 124 ~\textless~ {\rm{unknown}}$ $138 ~\leq~ 152 ~\textless~ 177$
$177 ~\leq~ 202 ~\textless~ {\rm{unknown}}$
---------------------------------------------- ---------------------------------------------- -- -- --
: The range of influence of proton and neutron magics numbers obtained in this work. []{data-label="tab:ZN_magnumbers"}
As shown in Fig.\[fig:Structure\_constants\], the products of the structure functions with the corresponding terms in the BW formula are not independent of each other but rather strongly correlated, see Eq.\[eq:BWparametrization\_Wigner\_CorMN\]. From this figure, it is clear how the effect of the binding energy saturation reflected on each of the structure functions. The mean value of the structure functions are shown in Table.\[tab:BW\_CoeffFits\].
The mean value of the power factors in the parametrized BW mass formula do not deviate too much from the initial approximation, while $p_{1}$ decreases by a factor of two with respect to the initial approximation, $p_{1}$ = 1/3. The mean values are $\langle p_{1}(A,Z,\{a_{i}\}) \rangle$ = 0.164398, $\langle p_{2}(A,Z,\{a_{i}\}) \rangle$ = 1.311083, $\langle p_{3}(A,Z,\{a_{i}\}) \rangle$ = 1.88221, $\langle p_{4}(A,Z,\{a_{i}\}) \rangle$ = 1.847739. The dependencies of the power factors with respect to the atomic mass number, $Z$ and $N$ are shown in Fig.\[fig:PowerFactors\_AZN\].
The correction function Eq.\[eq:CorMN\_func\] in our work plays an important role in the parametrization procedure. Since one has to take into account the fact that there is an especially high binding energy for nuclei with double shell closures, that is, where the neutron and proton shells are both closed. The shell model does a good job of explaining the stability of nuclei with neutron and proton shell closures at 2, 8, 20, 28, 50, and 82, and when the neutron shell closes at 126. For instance, Helium-4, which has a double shell closure with 2 neutrons and 2 protons, is an example of a highly stable element. But where the closed shells are located in the heavy element region is less certain. Various models predict that it should occur at $Z=114,120$, or 126 for the protons and $N=150, 164, 172$ and 184 for the neutrons [@Zhang2005106; @Satpathy2003C24; @Sorlin2008602; @Rydin2011238; @Rydin20112356]. The defined correction function in Eq.\[eq:CorMN\_func\] provides an indicator about the deviation in the level structure of nuclei away from closed shells. Also it includes the pairing effects, which can also provide reliable information about the magic number. Fig.\[fig:CorMagicNumbers\] shows the correction energies, see Eq.\[eq:CorMN\_func\], as a function of mass numbers $Z, N$ and $A$. The peaks on Fig.\[fig:CorMagicNumbers\] represent the possible magic proton and neutron numbers. A peak is enhanced if a magic proton number exists, the boundaries of the discovered magic numbers are shown in Table \[tab:ZN\_magnumbers\]. Therefore the analysis of the latent regularities allows us to rediscover the already known set of the magic numbers ($2, 8, 20, 28, 50, 82$), and obtain the additional magic numbers for proton and neutrons. Also as a consequence of our parameterized solution we obtain the total number of magic numbers. We have compared our magic number candidates with results of previous works, this result is illustrated in Fig.\[fig:MagNumberList\].
The obtained results and validation of our approach on the big set of the experimental data allows us to probe the outer boundary of the island of enhanced nuclear stability. So, we make a prediction for the binding energy, nuclear and atomic mass, and mass excess of the recently discovery nuclei at the DGFRS separator [@Oganessian201562], see Table \[tab:Utyonkov\_data\].
-- -- -- -- -- -- -- --
-- -- -- -- -- -- -- --
: The calculated $\chi^{2}$ and $\chi_{n}$ for the simplest choice of $s_{1}$ and $s_{2}$ values, where $s_{2}$ is given in percent. Our case is the top subsection of the table, see Eq.\[eq:Sigma\_treat\], and the case without uncertainty turning is the bottom subsection. []{data-label="tab:Chi2Sigma_treatment"}
-- -- -- -- -- -- -- -- -- -- -- -- -- -- -- --
-- -- -- -- -- -- -- -- -- -- -- -- -- -- -- --
The nuclear landscape, illustrated in Fig.\[fig:Map\_of\_bound\_EvEv\_nuclei\], is visualized by a plot of all known nuclei (green squares) and the valley of stability (black squares), with neutron number on the $x$-axis and proton number on the $y$- axis. The point where some of the nucleons, protons or neutrons, will be completely unbound is known as a nuclear drip line, because it is as if the extra nucleons drip right off the nucleus. To find where these drip lines are on the nuclear landscape, the separation energy is needed. In finding both the proton and neutron drip lines, even-even nuclei were used. Since only even-even nuclei were considered, two neutron ($S_{2n}$) and two proton ($S_{2p}$) separation energies were used. These energies required to remove two neutrons and two protons $$S_{2n}(N,Z)=(Z+N)E_{B}(N, Z, \{a_{i}\}) - (Z+(N-2))E_{B}(N-2, Z, \{a_{i}\}),
\label{eq:2NeutronSeparationEnergy}$$ in the case of neutron shells, and the two-proton separation energy $$S_{2p}(N,Z)=(Z+N)E_{B}(N, Z, \{a_{i}\}) - ((Z-2)+N)E_{B}(N, Z-2, \{a_{i}\}),
\label{eq:2ProtonSeparationEnergy}$$ in the case of proton shells. The two-neutron and two proton drip lines are reached when $S_{2n}\approx 0$ and $S_{2p}\approx 0$, respectively. In the real math we compute the separation energies until they do change the sign to minus, then we took the previous even-even nuclei for which the values of $S_{2n}$ and $S_{2p}$ are very close to zero as our drip-line limit. Since by convention, our values for the binding energy are per nucleon, that is, we put the additional factors in front of the binding energy. Results of our calculation of the borders of the drip lines are shown in Fig.\[fig:DripLines\_Predictions\]. We observe that the drip lines reach some asymptotic limit, which occurs due to the asymptotic behavior of the variables $\upsilon_{1}$, $\upsilon_{2}$, $\upsilon_{3}$, see Fig.\[fig:FitParam\_Asimptotic\], which were introduced in Eq.\[eq:IndependentVariable\_14\], and represent the discovered arguments of the separation energies.
![ The behavior of variables $\upsilon_{1}$, $\upsilon_{2}$, $\upsilon_{3}$, see Eq.\[eq:IndependentVariable\_14\], which are the discovered arguments of drip-lines of $S_{2p}(Z,N)$ and $S_{2n}(Z,N)$ presented in Fig.\[fig:DripLines\_Predictions\]. []{data-label="fig:FitParam_Asimptotic"}](S2nS2p_Asymtotic)
Summary {#Summary}
=======
Summarizing the above results, we may say that based on this solution, the signatures of the traditional magic numbers 2, 8, 20, 28, 50, 82, which are clearly seen to be close to the stable nuclei, there is indication for some other shell closures at $Z = 14, 108, 124$ and $N = 14, 124, 152, 202$, that are supported by the correction energies, see Eq.\[eq:CorMN\_func\]. But as one may note, the peaks for these magic numbers looks more smeared than those from the general set. The function Eq.\[eq:CorMN\_func\] used for the correction energy, representing the overall behavior of the single-particle spectra, may be a good candidate to identify the magicity of nuclei. As seen from Fig.\[fig:MagNumberList\] the obtained magic proton number candidates have been proposed in the modified Lucas’ geometrical packing scheme [@Rydin2011238], while the magic neutron number candidates are new and do not match any models.
We find that that results of the verification of the inverse problem in the framework of the Bethe-Weizsäcker mass formula greatly improve the agreement between the experimental masses and the calculated ones and thus predicts the drip-lines more accurately than calculated earlier [@PhysRevC.65.037301; @ADHIKARI:2004; @BASU:2004] with the modified Bethe-Weizsäcker mass formula alone.
We adopt a procedure in which we compute all the structure functions from scratch, which allows us to provide some predictions regarding the magic numbers. A correlation between the surface, Coulomb, asymmetry and Wigner terms has been found which allows us to avoid the introduction of different radius constants for the nuclear and the spin-orbit potentials.
The volume and surface symmetry terms are shown, see Fig.\[fig:Structure\_constants\], to be largely independent of the other terms in the formula, while the Coulomb diffuseness correction $Z(Z-1)/A^{p_{1}}$ or the charge exchange correction $(N-Z)^{2}/A^{p_{3}}$, asymmetry and Wigner correction terms are of critical importance in determining the nucleon binding energy and play the main role to improve the accuracy of the mass formula. The Wigner term and the curvature energy can also be used separately for the same purpose. The interplay between different terms is found to be important.
We were able to calculate the borders of the nuclear landscape (drip lines) and show their limit. This finding together with the asymptotic behavior, see Fig.\[fig:FitParam\_Asimptotic\], and accurate predictions of the binding energies of all known isotopes allows us to obtain quite precise predictions for the location of the proton and neutron drip lines, and claim the exact number of bound nuclei in the nuclear landscape. We are of course aware that the super-heavy isotopes in Fig.\[fig:DripLines\_Predictions\_limit\] cannot be produced at present and may be even at future facilities and that, so far, only theoretical studies may be carried out in such region of the nuclear chart.
The concept of ill-posed problems and the associated regularization theories seem to provide a satisfactory framework to solve nuclear physics problems. This new perspective can be used for testing the applications of the liquid model in different areas, for forecasting the mass values away from the valley of stability, for calculation of kinetic and total energy of nuclear proton, alpha, cluster decays and spontaneous fission, for preliminary research of island stability problem and the possibility of creating new super-heavy elements in the future.
[**[ACKNOWLEDGMENTS:]{}**]{} We are grateful to [**[Lubomir Aleksandrov]{}**]{} for the provided REGN program. Strachimir Cht. Mavrodiev would like to thank [**[Alexey Sissakian]{}**]{} and [**[Lubomir Aleksandrov]{}**]{} for many years of a very constructive collaboration and friendship. Unfortunately, they passed away, but they made enormous contributions to the REGN development and application of it for solving different inverse problems, especially for the discovering latent regularities.
This work was partially supported by the Chinese Academy of Sciences President’s International Fellowship Initiative under Grant No. 2016PM043.
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Binding energy per nucleon. Results {#BindEn.Results}
===================================
Resulting table {#Resulting table}
===============
=10.2in
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-- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- --
: Atomic mass table, where $Z$ is the number of protons, $A$ is the mass number $A = N +Z$, $\Delta m(A,Z)$ is the mass excess ($M$(in u)- $A$), in MeV, $E_{B}(A,Z)$ is the binding energy per nucleon (in MeV). $M_{a.m.}(A,Z)$ and $M_{n.m.}(A,Z)$ are the atomic mass and nuclear mass respectively. The theoretical energies and masses are compared with the recent experimental data taken from AME2012 [@Ami2012_ChinPhysC; @NUBASE2012_ChinPhysC]. []{data-label="tab:Results_output"}
Predicted binding energies for super-heave nuclei {#Predicted binding energies for super heave nuclei}
=================================================
=6.4in
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: Predicted binding energies, nuclear and atomic masses, and mass excess for the recently discovery nuclei at the DGFRS separator based on fusion reactions of $^{48}\rm{Ca}$ with $^{238}\rm{U} \textendash ^{249}\rm{Cf}$ target nuclei [@Oganessian201562]. []{data-label="tab:Utyonkov_data"}
Fortran program {#Fortran CODE}
===============
FUNCTION aNuclMass(Proton,aNeutron,BiEnTh,AtMassTh,aMassExcTh) IMPLICIT DOUBLE PRECISION(A-H,O-Z) COMMON/bwigner/BWZ,BWN,CoMn,BetheWeizsacker COMMON/power/pow1,pow2,pow3,pow4 COMMON/variables/AA,a1,a12,a13,a2,a22,a23,a3,a32,a33,a4,a42,a43,a5,a6 COMMON/nhelp/lexpt,iSP,nPow,nBw,nBWA,NBWG,MnZ,MnN,N0 DIMENSION A(241),X(2) DATA A / 0.29505599354845E+01, 0.28884885427112E+01, -0.66184782828700E+00, 0.25150441215832E+01, 0.38509313661189E+01, & -0.18331934846462E+01, 0.24989124476310E+00, 0.63500611595309E+00, 0.48206457697962E+00, 0.34826035147447E+01, & -0.31103437659677E+01, -0.64381051730034E+01, -0.23155639610509E+02, 0.28851770119732E+01, -0.37527758838446E+01, & -0.22983221671284E+01, 0.11230399866911E+02, 0.12708557258698E+01, -0.27651630317775E+01, 0.37119527742796E+01, & -0.24543376748414E+01, 0.16342477503052E+02, 0.36312110857756E+01, -0.83862938211403E+01, -0.78428293213680E+01, & 0.63939594939676E+00, -0.10167253104189E+02, -0.33709122569709E+01, -0.24839428509568E+01, 0.48767531463510E+01, & 0.11307735166797E+02, 0.74983916267884E+00, -0.12768492445712E+01, -0.91597752397203E+01, -0.38462880390659E+01, & 0.17113149664896E+02, 0.26439968153163E+01, -0.28309363766461E+03, -0.12248619773181E+03, 0.16048760233604E+03, & 0.12618436699407E+03, -0.21641100550244E+03, -0.55841073428324E+02, -0.14144144148520E+03, 0.59840812966271E+02, & -0.16261647736746E+03, -0.44453159819871E+02, -0.80928759939479E+01, -0.17259749460069E+02, 0.37983411980331E+03, & 0.11511955345053E+02, -0.55704502557592E+03, -0.52208256546162E+03, 0.36614671474719E+02, -0.37920857630041E+03, & -0.35894196931396E+03, -0.32295786429717E+03, -0.28569090839531E+03, 0.37869438907179E+03, -0.34824916607032E+03, & -0.13510730319705E+03, 0.74531833068162E+03, -0.10572087255554E+03, 0.21957071856905E+04, 0.87615178825115E+02, & -0.45426079677873E+02, -0.34843594165462E+02, 0.10678192226568E+02, -0.73468632009813E+01, -0.24757219474247E+02, & -0.14175971396252E+02, 0.23036868379919E+01, 0.10416791886258E+02, -0.29438951117593E+02, 0.11175114172202E+02, & 0.13081097472932E+03, 0.30726096131228E+01, 0.18430280306748E+02, 0.16793269642199E+02, -0.54883855518417E+02, & -0.41572899927593E+02, 0.13422808331938E+02, 0.48117868978840E+02, -0.22805258133787E+02, -0.97982985905013E+01, & 0.29875653157394E+02, -0.21637491716558E+02, -0.22206795928125E+01, 0.78360508224329E+00, -0.26643314137233E+02, & 0.30786471664485E+01, 0.44606410603121E+02, 0.48943480993327E+01, 0.23175960969827E+03, 0.13384388614033E+03, & -0.96316601833645E+02, -0.36517953724213E+03, 0.11914093371423E+03, 0.22310080634133E+02, -0.81435194761971E+02, & 0.18742258840656E+03, 0.44518432506530E+02, -0.14474469151351E+02, 0.52738598916777E+02, -0.39421556315459E+02, & -0.69170640239893E+02, 0.28607325916225E+01, -0.17021026463960E+03, -0.83285400234162E+02, 0.87001322827306E+02, & -0.47840918501569E+02, -0.10547234055459E+03, -0.18562558435311E+02, 0.53175256249661E+01, -0.13786219193877E+02, & -0.56994393682383E+02, -0.23108007530520E+01, -0.41906445426294E+02, 0.72367864288666E+02, 0.15176817608918E+03, & 0.41912886275550E+02, 0.11660615773449E+03, 0.74081651688635E+02, -0.42591900422841E+02, -0.88858615003034E+02, & 0.56855355569979E+02, 0.14400150358518E+02, -0.47852021665788E+02, -0.62285437441053E+02, 0.16838009044068E+02, & -0.44567706561424E+01, 0.43388797669805E+02, 0.79702652742833E+02, -0.69623025342584E+02, -0.34321990034314E+01, & -0.27845148929212E+03, -0.16392804879787E+03, 0.11438381234392E+03, 0.33248848979741E+03, -0.14197361405611E+03, & -0.27369894185035E+02, 0.10508253599696E+03, -0.16046679875033E+03, -0.21156242607124E+02, -0.10985329253354E+02, & -0.30400569131227E+03, 0.35683887062437E+02, 0.16119088166446E+03, 0.60519035315825E+01, -0.22782713519741E+00, & 0.72867306411600E+02, 0.58798345726480E+02, -0.13328363454432E+02, -0.12490771209866E+03, -0.15788829395717E+00, & -0.13577978108087E+02, -0.15944853119039E+03, 0.46570790262682E+02, -0.63653244345954E+02, -0.22704694679172E+02, & 0.20183131549482E+03, 0.12678044481597E+02, 0.26882086307558E+02, 0.17969301081794E+01, 0.15754081461258E+01, & 0.76951394324102E+02, -0.26095265769828E+02, -0.10457474359541E+03, -0.15176753331014E+03, 0.13236826469319E+03, & 0.28448033511240E+02, 0.27195688271866E+03, 0.22863870940513E+02, 0.20494813624508E+03, 0.11029556386425E+02, & -0.46321012800313E+03, -0.15108244966139E+02, 0.86090121856582E+02, 0.47165434267532E+01, 0.25058099088327E+02, & 0.74582866992219E+02, 0.14364809106642E+03, -0.60134184531319E+02, -0.74842009892629E+02, -0.61478778858071E+02, & -0.59617510413661E+02, 0.45435491920105E+02, 0.45216678216006E+02, -0.40806254905721E+02, -0.60642713547269E+02, & 0.17443374760096E+02, -0.10568046708119E+02, -0.22151726981108E+02, -0.74298493081817E+00, 0.32987693228959E+03, & 0.20000000000000E+01, 0.80000008605930E+01, 0.14000000000000E+02, 0.20000000000000E+02, 0.28000000000000E+02, & 0.50000000000000E+02, 0.82000000000000E+02, 0.10800000000000E+03, 0.12400000000000E+03, 0.10000000000000E+01, & 0.50000000000000E+01, 0.11000000000000E+02, 0.17000000000000E+02, 0.24000000000000E+02, 0.39000000000000E+02, & 0.66000000000000E+02, 0.95000000000000E+02, 0.11600000000000E+03, 0.20000000000000E+01, 0.80000000000000E+01, & 0.14000000000000E+02, 0.20000000000000E+02, 0.28000000000000E+02, 0.50000000000000E+02, 0.82000000000000E+02, & 0.12400000000000E+03, 0.15200000000000E+03, 0.20200000000000E+03, 0.10000000000000E+01, 0.50000000000000E+01, & 0.11000000000000E+02, 0.17000000000000E+02, 0.24000000000000E+02, 0.39000000000000E+02, 0.66000000000000E+02, & 0.10300000000000E+03, 0.13800000000000E+03, 0.17700000000000E+03, -0.40082290184326E+02, 0.17303948799355E+01, & 0.17306328131782E+01, 0.76071091945279E-01, 0.34470508756088E+00, 0.41051624843330E+00, 0.23900000000000E+01, & 0.53500000000000E+01 /
pi2 = 2.D0\*0.3141592653589793D1 HAtomMass = 938.782303D0 !2.072E-20 \[MeV\] aNeutronMass = 939.565381D0 !2.072\*10\^(-14) \[MeV\] ProtonMass = 938.270462D0 ElectronMass = 0.51099892800000000 u = 931.494061D0 Ael = 1.44381E-05 Bel = 1.554680E-12
iStr = 5; iPow = 4; iSP= iStr + iPow nPow = 14; nBWA = 15; nBWG = 15; nBW = 2\*nBWA + 2\*nBWG MnZ = 9; MnN = 10; N0 = iSP + iSP\*nPow + nBW N = N0 + 2\*MnZ + 2\*MnN + 8
x(1) = Proton x(2) = aNeutron AA = Proton+aNeutron a1 = x(1)/AA; a12 = a1\*\*2; a13 = a1\*\*3 a2 = x(2)/AA; a22 = a2\*\*2; a23 = a2\*\*3 a3 = (x(2)-x(1))/AA; a32 = a3\*\*2; a33 = a3\*\*3 a4 = x(1)/(x(2)+1); a42 = a4\*\*2; a43 = a4\*\*3 a5 = dlog(AA+1.D0); a6 = 1.D0/a5 Z0 = FunZ(N, A, x(1),WZ) a02Z = (x(1)- Z0)\*\*2 aN0 = FunN(N, A, x(2),WN) a02N = (x(2)-aN0)\*\*2 PRINT \*,AA,x(1),X(2),Z0,aN0; PAUSE
AAA=1.D0 IF(int(AA/2)\*2.ne.AA) AAA=0.D0 ZZZ=1.D0 IF(int(X(1)/2)\*2.ne.X(1)) ZZZ=0.D0 ANN=1.D0 IF(int(X(2)/2)\*2.ne.X(2)) ANN=0.D0
CorMN = BrWig(a02Z,WZ,a02N,WN, a,n,iSP+9\*nPow)+eexp(a(n-4)\*AAA/AA+a(n-3)\*ZZZ/x(1)+a(n-2)\*ANN/(x(2)+1))
c1 = eexp(a(1) + CorPow(a,n,isp)) c2 = eexp(a(2) + CorPow(a,n,isp + nPow)) c3 = eexp(a(3) + CorPow(a,n,isp + 2\*nPow)) c4 = eexp(a(4) + CorPow(a,n,isp + 3\*nPow)) c5 = eexp(a(5) + CorPow(a,n,isp + 4\*nPow)) + a(n-7)\*AAA + a(n-6)\*ZZZ + a(n-5)\*ANN pow1 = eexp(a(6) + CorPow(a,n,isp + 5\*nPow)) pow2 = eexp(a(7) + CorPow(a,n,isp + 6\*nPow)) pow3 = eexp(a(8) + CorPow(a,n,isp + 7\*nPow)) pow4 = eexp(a(9) + CorPow(a,n,isp + 8\*nPow))
Vol = c1 Sur = c2/AA\*\*Pow1 Cha = c3\*x(1)\*(x(1)-1.D0)/AA\*\*Pow2 Sym = c4\*(x(2)-x(1))\*\*2/AA\*\*Pow3
IF(int(AA/2)\*2.ne.AA) WigE=0.D0 IF(int(x(1)/2)\*2.ne.x(1).and. int(x(2)/2)\*2.ne.x(2)) WigE =-1.D0 IF(int(x(1)/2)\*2.eq.x(1).and. int(x(2)/2)\*2.eq.x(2)) WigE = 1.D0 Wig = c5\*WigE/AA\*\*Pow4
BetheWeizsacker = Vol - Sur - Cha - Sym + Wig BiEnTh = BetheWeizsacker + CorMN AtMassTh = x(1)\*HAtomMass + x(2)\*aNeutronMass - AA\*BiEnTh aNuclMass = AtMassTh - (ElectronMass\*x(1) + Ael\*x(1)\*\*a(n-1) + Bel\*x(1)\*\*a(n)) aMassExcTh = AtMassTh - AA\*u RETURN; END !\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*
FUNCTION CorPow(a,n,i) IMPLICIT DOUBLE PRECISION(A-H,O-Z) COMMON/variables/AA,a1,a12,a13,a2,a22,a23,a3,a32,a33,a4,a42,a43,a5,a6 DIMENSION a(n)
c1 = a(i+1)\*a1 + a(i+2)\*a2 + a(i+3)\*a3 + a(i+4)\*a4 c2 = a(i+5)\*a12 + a(i+6)\*a22 + a(i+7)\*a32 + a(i+8)\*a42 c3 = a(i+9)\*a13 + a(i+10)\*a23 + a(i+11)\*a33 + a(i+12)\*a43 c4 = a(i+13)\*a6 + a(i+14)\*a5 CorPow = eexp(-(c1+c2+c3+c4)\*\*2) RETURN; END !\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*
FUNCTION BrWig(a02Z,WZ,a02N,WN,a,n,i) IMPLICIT DOUBLE PRECISION(A-H,O-Z) COMMON/bwigner/BWZ,BWN,CorMn,BetheWeizsacker COMMON/nhelp/lexpt,iSP,nPow,nBw,nBWA,NBWG,MnZ,MnN,N0 DIMENSION a(n)
AmpZ = CorAmp(a,n,i) AmpN = CorAmp(a,n,i + nBWA) GamZ = WZ + CorAmp(a,n,i + 2\*nBWA ) GamN = WN + CorAmp(a,n,i + 2\*nBWA + nBWG ) BWZ = AmpZ\*eexp(-a02Z/(GamZ\*2.D0))/(a02Z+GamZ) BWN = AmpN\*eexp(-a02N/(GamN\*2.D0))/(a02N+GamN) BrWig = BWZ + BWN RETURN; END !\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*
FUNCTION CorAmp(a,n,i) IMPLICIT DOUBLE PRECISION(A-H,O-Z) COMMON/variables/AA,a1,a12,a13,a2,a22,a23,a3,a32,a33,a4,a42,a43,a5,a6 DIMENSION a(n)
c1 = a(i+1)\*a1 + a(i+2)\*a2 + a(i+3)\*a3 + a(i+4)\*a4 c2 = a(i+5)\*a12 + a(i+6)\*a22 + a(i+7)\*a32 + a(i+8)\*a42 c3 = a(i+9)\*a13 + a(i+10)\*a23 + a(i+11)\*a33 + a(i+12)\*a43 c4 = a(i+13)\*a6 + a(i+14)\*a5 CorAmp = eexp(a(i+15) - (c1+c2+c3+c4)\*\*2) RETURN; END !\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*
FUNCTION FunZ(N,A,x,WZ) IMPLICIT DOUBLE PRECISION (A-H,O-Z) ! Proton Magic numbers 2,8,20,28,50,82,108,124 COMMON/nhelp/lexpt,iSP,nPow,nBw,nBWA,NBWG,MnZ,MnN,N0 DIMENSION aMn(9),aB(9),A(N)
DO i=1, MnZ; aMn(i)=Int(a(N0+i)); aB(i)=int(a(N0 + MnZ + i)) !PRINT \*,i,aMn(i),aB(i) ENDDO !PAUSE DO i=1, MnZ-1 IF(x.ge.aB(i).and.x.lt.aB(i+1)) THEN FunZ = aMn(i) WZ = (aMn(i+1) - aMn(i))/2 RETURN; ENDIF ENDDO
IF(x.ge.aB(MnZ)) FunZ = aMn(MnZ) IF(x.ge.aB(MnZ)) WZ = (aMn(MnZ) - aMn(MnZ-1))/2 RETURN; END !\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*
FUNCTION FunN(N,A,x,WN) IMPLICIT DOUBLE PRECISION (A-H,O-Z) ! Neutron Magic numbers 2,8,20,28,50,82,124,152,202 COMMON/nhelp/lexpt,iSP,nPow,nBw,nBWA,NBWG,MnZ,MnN,N0 DIMENSION aMn(10),aB(10),A(N)
DO i=1, MnN; aMn(i)=int(a(N0 + 2\*MnZ + i)); aB(i)=int(a(N0 + 2\*MnZ + MnN + i)); ENDDO
DO i=1,MnN-1 IF(x.ge.aB(i).and.x.lt.aB(i+1)) THEN FunN = aMn(i) WN = (aMn(i+1) - aMn(i))/2 RETURN; ENDIF ENDDO
IF(x.ge.aB(MnN)) FunN = aMn(MnN) IF(x.ge.aB(MnN)) WN = (aMn(MnN) - aMn(MnN-1))/2 RETURN; END
[^1]: By the series of observation we will consider the list of the observed nuclei masses, that is given by the AME2012 database
[^2]: The subgroup of the regularized Newton-Kantorovich type processes (called “automatically controlled" or “auto-regularized"), for which the regularization in each iteration (with number $n$) of the process is performed with respect to value of “defect" norm of equation ($\vert\vert F(x_{n-1}) \vert\vert$), which was obtained basing on the previous iteration (with number $n-1$).
[^3]: LHC is the modified least $\chi^{2}$ procedure, which investigates the possibility of the existence of the minimal $\chi^{2}$ from the parameters that were excluded from the considered model due to strong correlation.
|
[**The general homothetic equations**]{}
[**John D. Steele**]{}
School of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia.
email: j.steele@unsw[.]{}edu[.]{}au
[**Abstract**]{}
In an earlier paper \[6\] the author wrote the homothetic equations for vacuum solutions in a first order formalism allowing for arbitrary alignment of the dyad. This paper generalises that method to homothetic equations in non-vacuum spaces and also provides useful second integrability conditions. An application to the well-known Petrov type O pure radiation solutions is given.
PACS number 0420J
Mathematics Subject Classification: 83C15, 83C20
Recap
=====
In \[6\] I gave the homothetic Killing equations in vacuum, written out in a first order form without the assumption that the spinor dyad used was aligned to either the symmetry or the curvature in any way, and indeed allowing for the dyad to be non-normalised. In this paper I will generalise to non-vacuum. Conventions and notation will follow Penrose and Rindler \[7\].
A homothetic vector $\xi^a$ by definition satisfies the equation $$\xi_{a;b}= F_{ab} + \psi g_{ab}\label{eq:Killing}.$$ Here $\psi$, the [**divergence**]{}, is a constant, and $F_{ab}$ will be called the [**homothetic bivector**]{}.
Let $\{\o^A,\i^A\}$ be a spinor dyad, with $\o_A\i^A=\chi$. A complex null tetrad is related to this dyad in the standard way: $$\ell^a=\o^A\o^{A'};\quad n^a=\i^A\i^{A'};\quad m^a=\o^A\i^{A'};\quad
\mb^a=\i^A\o^{A'},$$ (\[7\], (4.5.19)), and $\ell_an^a = -m_a\mb^a = \chi\bar\chi$. As in \[6\], we define components of the homothety: $$\xi_a =\Hn\ell_a + \Hl n_a - \Hmb m_a - \Hm \mb_a,$$ with $\{\ell^a,n^a,m^a,\mb^a\}$ a Newman-Penrose tetrad. Thus, for example, $\chi\bar\chi\Hl=\xi_a\ell^a$.
For the homothetic bivector $F_{ab}$ we define its anti-self dual by $$\begin{aligned}
\sfdual{F}_{ab}& =\frac{1}{2}\left(F_{ab}+i{}F^*_{ab}\right)\end{aligned}$$ and then $$\sfdual{F}_{ab} =(\chi\bar\chi)^{-1}\left( 2\phi_{00}\,\ell_{[a}m_{b]}
+ 2\phi_{01}\,(\ell_{[a}n_{b]} - m_{[a}\mb_{b]})
- 2\phi_{11}\,n_{[a}\mb_{b]}\right),$$ where $$\begin{aligned}
\phi_{11} & = (\chi\bar\chi)^{-1}F_{ab}\ell^am^b \\
\phi_{01} & = \frac{1}{2}(\chi\bar\chi)^{-1}\left(
F_{ab}n^a\ell^b - F_{ab}\mb^am^b\right) \\
\phi_{00} & = (\chi\bar\chi)^{-1}F_{ab}\mb^a n^b\end{aligned}$$
Most of the equations in this paper will be given using the compacted GHP-formalism, see \[3,7,8\]. In this formalism, we concentrate on those spin coefficients of good weight, that is, those that transform homogeneously under a spin-boost transformation of the dyad: if $$\o^A\mapsto \lambda \o^A \qquad \i^A\mapsto \mu\i^A$$ a [**weighted**]{} quantity $\eta$ of [**type**]{} $\{r',r;t',t\}$ undergoes a transformation $$\eta \mapsto \lambda^{r'}\bar\lambda^{t'} \mu^{r}\bar\mu^{t}\eta.$$ These weights will be referred to as the [**Penrose-Rindler (PR)**]{} weights, to distinguish them from the more familiar GHP-weights $(p,q)$ for a normalised dyad in e.g. \[3,7\]. The two sets of weights are related by $p=r'-r$ and $q=t'-t$.
General Equations
=================
The Killing equations themselves, (\[eq:Killing\]), are unaffected by the curvature and so are the same as in \[6\]. $$\begin{aligned}
\thorn\Hl & = -\bar\kappa\Hm - \kappa\Hmb; \tag{\thecompkilleq$a$}\\
\thorn'\Hl & = -\bar\tau\Hm -\tau\Hmb
-(\phi_{01}+\bar\phi_{01})+\psi; \tag{\thecompkilleq$b$} \\
\eth\Hl &= -\bar\rho\Hm-\sigma\Hmb +\phi_{11}; \tag{\thecompkilleq $c$}\\
\thorn\Hn &= -\tau'\Hm-\bar\tau'\Hmb
+(\phi_{01}+\bar\phi_{01})+\psi; \tag{\thecompkilleq $d$} \\
\thorn'\Hn &= - \kappa'\Hm -\bar\kappa'\Hmb ;
\tag{\thecompkilleq$e$}\\
\eth\Hn &= -\rho'\Hm-\bar\sigma'\Hmb -\bar\phi_{00};
\tag{\thecompkilleq$f$}\\
\thorn\Hm & = -\bar\tau'\Hl - \kappa\Hn -\phi_{11};\tag{\thecompkilleq$g$}\\
\thorn'\Hm & = -\bar\kappa'\Hl - \tau\Hn +\bar\phi_{00};
\tag{\thecompkilleq$h$}\\
\eth\Hm &= -\bar\sigma'\Hl -\sigma\Hn ;
\tag{\thecompkilleq$i$}\\
\eth'\Hm & = -\bar\rho'\Hl-\rho\Hn
+(\phi_{01}-\bar\phi_{01})- \psi; \tag{\thecompkilleq$j$}\end{aligned}$$
The spin-boost weights $(r',r,t',t)$ of the components of $\xi^a$ and $F_{ab}$ are given in Table I (correcting a minor typo in \[6\]).
$$\vbox{
\tabskip=0pt\offinterlineskip
\halign {\strut#&\vrule # \tabskip=.75em plus 2em minus 0.5em
&\hfil $#$\hfil&\vrule# &\hfil $#$\hfil&\vrule#
&\hfil $#$\hfil&\vrule# &\hfil $#$\hfil&\vrule#
&\hfil $#$\hfil&\vrule# &\hfil $#$\hfil&\vrule#
&\hfil $#$\hfil&\vrule# &\hfil $#$\hfil&\vrule# \tabskip=0pt\cr
\noalign{\hrule}
\omit & height 3pt && height 3pt && height 3pt && height 3pt && height 3pt && height 3pt && height 3pt && height 3pt
&& height 3pt\cr
&& && \Hl && \Hn && \Hm && \Hmb && \phi_{00} && \phi_{01} && \phi_{11} &\cr
\noalign{\hrule}
\omit & height 3pt && height 3pt && height 3pt && height 3pt && height 3pt && height 3pt && height 3pt && height 3pt
&& height 3pt\cr
&& r' && 0 && -1 && 1 && 0 && -1 && 0 && 1&\cr
\noalign{\hrule}
\omit & height 3pt && height 3pt && height 3pt && height 3pt && height 3pt && height 3pt && height 3pt && height 3pt
&& height 3pt\cr
&& r && -1 && 0 && -1 && 0 && 1 && 0 && -1&\cr
\noalign{\hrule}
\omit & height 3pt && height 3pt && height 3pt && height 3pt && height 3pt && height 3pt && height 3pt && height 3pt
&& height 3pt\cr
&& t' && 0 && -1 && -1 && 0 && 0 && 0 && 0&\cr
\noalign{\hrule}
\omit & height 3pt && height 3pt && height 3pt && height 3pt && height 3pt && height 3pt && height 3pt && height 3pt
&& height 3pt\cr
&& t && -1 && 0 && 0 && -1 && 0 && 0 && 0&\cr
\noalign{\hrule}
}}$$
The Ricci identity for $\xi^a$ implies $F_{cd;b}=R_{abcd}\xi^a$, from which the algebraic Bianchi identities lead to equations for the derivatives of the $\phi_{ij}$. The anti-self-dual of the above equation takes the spinor form $$\nabla_{CC'}\phi_{AB} = \left(\Psi_{ABDC}\epsilon_{D'C'} +
\Phi_{ABD'C'}\epsilon_{DC}\right)\xi^{DD'} -
\Lambda\left(\epsilon_{BC}\xi_{AC'}+\epsilon_{AC}\xi_{BC'}\right)
\label{eq:spinorintcond}$$ Here $\Psi_{ABCD}$ is the (totally symmetric) Weyl spinor, $\Phi_{ABA'B'}$ the Ricci spinor and $24\Lambda=R$, the Ricci scalar (see \[7\]).
The components of the Weyl and Ricci spinors are given in \[7\] (4.11.6) and (4.11.8) respectively, and then resolving equation (\[eq:spinorintcond\]) we get the (first) integrability conditions $$\begin{aligned}
\Psi_1\Hl-\Psi_0\Hmb +\Phi_{01}\Hl -\Phi_{00}\Hm &
=2\kappa\phi_{01} -\thorn\phi_{11};
\tag{\theintcons$a$}\\[1pt]
\Psi_1\Hm-\Psi_0\Hn +\Phi_{02}\Hl -\Phi_{01}\Hm &
= 2\sigma\phi_{01} -\eth\phi_{11};
\tag{\theintcons$b$}\\[1pt]
\Psi_2\Hl-\Psi_1\Hmb +\Phi_{01}\Hmb -\Phi_{00}\Hn +2\Pi\Hl &
= 2\rho\phi_{01} -\eth'\phi_{11};
\tag{\theintcons$c$}\\[1pt]
\Psi_2\Hm-\Psi_1\Hn +\Phi_{02}\Hmb -\Phi_{01}\Hn +2\Pi\Hm &
= 2\tau\phi_{01} -\thorn'\phi_{11};
\tag{\theintcons$d$}\\[1pt]
\Psi_3\Hl -\Psi_2\Hmb +\Phi_{21}\Hl -\Phi_{20}\Hm -2\Pi\Hmb &
= 2\tau'\phi_{01} -\thorn\phi_{00};
\tag{\theintcons$e$}\\[1pt]
\Psi_3\Hm-\Psi_2\Hn +\Phi_{22}\Hl -\Phi_{21}\Hm -2\Pi\Hn &
= 2\rho'\phi_{01} -\eth\phi_{00};
\tag{\theintcons$f$}\\[1pt]
\Psi_4\Hl -\Psi_3\Hmb +\Phi_{21}\Hmb -\Phi_{20}\Hn &
= 2\sigma'\phi_{01} -\eth'\phi_{00};
\tag{\theintcons$g$}\\[1pt]
\Psi_4\Hm -\Psi_3\Hn +\Phi_{22}\Hmb -\Phi_{21}\Hn &
= 2\kappa'\phi_{01} -\thorn'\phi_{00}
\tag{\theintcons$h$}\\[1pt]
\Psi_2\Hl-\Psi_1\Hmb +\Phi_{11}\Hl -\Phi_{10}\Hm -\Pi\Hl &
= \thorn\phi_{01}-\tau'\phi_{11}-\kappa\phi_{00};
\tag{\theintcons$i$}\\[1pt]
\Psi_2\Hm-\Psi_1\Hn +\Phi_{12}\Hl -\Phi_{11}\Hm -\Pi\Hm &
= \eth\phi_{01}-\rho'\phi_{11}-\sigma\phi_{00};
\tag{\theintcons$j$}\\[1pt]
\Psi_3\Hl -\Psi_2\Hmb +\Phi_{11}\Hmb -\Phi_{10}\Hn +\Pi\Hmb &
= \eth'\phi_{01}-\rho\phi_{00}-\sigma'\phi_{11};
\tag{\theintcons$k$}\\[1pt]
\Psi_3\Hm-\Psi_2\Hn +\Phi_{12}\Hmb -\Phi_{11}\Hn +\Pi\Hn &
= \thorn'\phi_{01}-\tau\phi_{00}-\kappa'\phi_{11}.
\tag{\theintcons$l$}\end{aligned}$$ where $\Pi=\chi\bar\chi\Lambda$. These are equivalent to the equations (20)–(22) in \[5\].
Note that there are four pairs of equations with the same Weyl curvature terms ($c$/$i$; $d$/$j$; $e$/$k$ and $f$/$l$). We can eliminate the Weyl curvature terms between these pairs to give equations equivalent to (23) in \[5\]: $$\begin{aligned}
\thorn\phi_{01} +\eth'\phi_{11} & - \kappa\phi_{00} -2\rho \phi_{01}
-\tau'\phi_{11}\notag\\
&= \left(\Phi_{11}-3\Lambda\right)\Hl +\Phi_{00}\Hn-\Phi_{10}\Hm-\Phi_{01}\Hmb
\tag{\themaxintcons$a$}\\[1pt]
\eth\phi_{01} +\thorn'\phi_{11} & - \sigma\phi_{00} -
2\tau\phi_{01} - \rho'\phi_{11}\notag\\
&=\Phi_{12}\Hl+\Phi_{01}\Hn -\left(\Phi_{11}+3\Lambda\right)\Hm -\Phi_{02}\Hmb
&\tag{\themaxintcons$b$}\\[1pt]
\eth'\phi_{01} +\thorn\phi_{00} &- \rho\phi_{00} -
2\tau'\phi_{01} - \sigma'\phi_{11}\notag\\
&= -\Phi_{21}\Hl-\Phi_{10}\Hn+\Phi_{20}\Hm+\left(\Phi_{11}+3\Lambda\right)\Hmb
&\tag{\themaxintcons$c$}\\[1pt]
\thorn'\phi_{01}+\eth\phi_{00}& - \tau\phi_{00} -
2\rho'\phi_{01} -\kappa'\phi_{11}\notag\\
& =-\Phi_{22}\Hl -\left(\Phi_{11}-3\Lambda\right)\Hn+\Phi_{21}\Hm+\Phi_{12}\Hmb
\tag{\themaxintcons$d$}\end{aligned}$$
Note that all these equations are consistent as far as spin and boost weight are concerned, and all reduce to the equations of \[6\] in vacuum.
Second integrability conditions
===============================
Since a homothetic transformation preserves connection and hence curvature, we have ${\mathcal L}_\xi R^a{}_{bcd}=0$, and resolving the spinor version of this equation and using ($\theintcons$) to eliminate first derivatives of the $\phi_{ij}$ leads to equations I will refer to as second integrability conditions, although they are not strictly integrability conditions in the case of a homothety. The same equations arises from applying the commutators to the components of the homothetic bivector of course. Using the Bianchi identities and the GHP-notation these equations can be reduced to a very compact form. Firstly, define the zero weight derivative operator $$\GHPL_\xi= \Hn\thorn+\Hl\thorn'-\Hm\eth'-\Hmb\eth,$$ and let $$X_{00}=\phi_{00}-\kappa'\Hl-\tau'\Hn+\sigma'\Hm+\rho'\Hmb\qquad
X_{11}=\phi_{11}+\kappa\Hn+\tau\Hl-\sigma\Hmb-\rho\Hm.$$ (Note that under the Sachs $*$ operation, $X_{11}$ and $X_{00}$ are unchanged but $\bar{X}_{11}^*=\bar{X}_{00}$ and $\bar{X}_{00}^*=\bar{X}_{11}$). Then we find that $$\begin{aligned}
\GHPL_\xi \Psi_i + 2\psi\Psi_i &= iX_{00}\Psi_{i-1} -p\,\phi_{01}\Psi_i +
(i-4)X_{11}\Psi_{i+1}\label{eq:Psitwo}\\
\GHPL_\xi \Phi_{ab} + 2\psi\Phi_{ab} &=
aX_{00}\Phi_{(a-1)b} + b\bar X_{00}\Phi_{a(b-1)} -(p\,\phi_{01} +
q\,\bar\phi_{01})\Phi_{ab} \notag \\
&\qquad {}+ (a-2)X_{11}\Phi_{(a+1)b} + (b-2)\bar
X_{11}\Phi_{a(b+1)}\label{eq:Phitwo} \\
\GHPL_\xi\Pi +2\psi\Pi & =0,\end{aligned}$$ where $p=r'-r$ and $q=t'-t$ are the GHP weights \[7,8\]. Note that $\Psi_i$ has PR-weight $[3-i,i-1,1,1]$ and $\Phi_{ab}$ PR-weight $[2-a,a,2-b,b]$. Equations (\[eq:Psitwo\]) are equivalent to Collinson and French’s equations ($2.2$) \[1\] and Kolassis and Ludwig’s equations (43)–(45) \[4\]; equations (\[eq:Phitwo\]) are equivalent to \[4\] equations (47)–(49). In these references the tetrad is assumed normalised. A comparison with \[4\] shows that equations (\[eq:Psitwo\]) are the same for a general conformal vector in a normalised tetrad: this is as expected since these equations actually arise from the derivative of the Weyl tensor part of the curvature.
Note that if $\ell^a$ is a Debever-Penrose direction then $\Psi_0=0$ and (\[eq:Psitwo\]) implies $X_{11}=0$, or $$\label{eq:DP}
\phi_{11}=-\kappa\Hn-\tau\Hl+\sigma\Hmb+\rho\Hm$$ correcting the error in equation (11) of \[6\]. Similarly, if $n^a$ is a Debever-Penrose direction then $X_{00}=0$.
Type O pure radiation metrics
=============================
Possibly the simplest non vacuum metric to consider would be that of conformally flat pure radiation solutions. In \[2\] Edgar and Ludwig performed the integration of this case, which I here repeat with the extra assumption of the existence of a homothetic vector, when the calculations can be pushed to completion in the sense that no free functions remain. The main difference here is that I use the homothety to choose coordinate candidates.
We begin by assuming that the dyad is normalised ($\chi=1$) and aligned to the Ricci tensor, so $R_{ab}=\Phi_{22}\ell_a\ell_b$, leaving complete four parameter null rotation freedom in choosing the dyad.
The Bianchi identities quickly tell us that $\kappa=\sigma=0$, and the remaining Bianchi identities (see \[7\]) are $$\eth'\Phi_{22}=\bar\tau\Phi_{22},\qquad
\thorn\Phi_{22}=\bar\rho\Phi_{22}\qquad
\thorn\Phi_{22}=(\rho+\bar\rho)\Phi_{22}$$ So $\rho=0$ and we are in Kundt’s class. Now if $\tau=0$ we have plane waves, a case which has been much studied and we will ignore. So from henceforth, $\tau\not=0$.
Of the second integrability equations only equation (\[eq:Phitwo\]) for $(a,b)$ either $(1,2)$ or $(2,2)$ are non-trivial, and the first of these gives $\phi_{11}=-\tau\Hl$. Suppose $\phi_{11}$, and hence $\Hl$, vanishes. Then equation ($d$) and $\tau\not=0$ implies $\phi_{01}$ vanishes and then by ($l$), $\phi_{00}=0$. But now ($d$) gives $\Hmb=0$, and the Killing equation ($h$) means that $\Hn=0$ and the homothetic vector vanishes.
Thus $\Hl$ is not identically zero: no symmetry vector is orthogonal to $\ell^a$. This means $\phi_{11}\not=0$ and we can perform a proper null rotation about $\ell^a$ to set $\phi_{01}$ to be identically zero: we will be left with boost and rotation freedom, which is what we would want for a GHP integration procedure.
With $\phi_{01}=0$, the integrability equations involving the derivatives of $\phi_{01}$, ($i$) – ($l$) show that $\sigma'=\rho'=\tau'=0$ and $\phi_{00}=\kappa'\Hl$.
The remaining integrability conditions () and (\[eq:Phitwo\]) allow us to find all directional derivatives of all the remaining scalars. We have $$\begin{aligned}
{5}
\thorn\tau &=0 & \qquad\quad \thorn\kappa' &=0 &\qquad\quad \thorn\Phi_{22} &
= 0 \\
\eth\tau &=\tau^2 &\qquad\quad \eth \kappa' &=\tau\kappa'-\Phi_{22} &\eth
\Phi_{22} & = \tau\Phi_{22} \\
\eth'\tau &=\tau\bar\tau & \eth' \kappa' &=\kappa'\bar\tau &\eth' \Phi_{22} &
= \bar\tau\Phi_{22} \\
\Hl\thorn'\tau &=\tau (W-\psi) & \qquad \quad
\Hl\thorn'\kappa' &=-\Hmb\Phi_{22} +\kappa'(W-\psi)
&\qquad
\Hl\thorn'\Phi_{22} & = \Phi_{22}(W-2\psi)\end{aligned}$$ where $W=\bar\tau\Hm+\tau\Hmb$ has weight $(0,0)$. The homothetic equations reduce to $$\begin{aligned}
{5}
\thorn\Hl&=0 &\qquad\quad \thorn\Hn & = \psi&\qquad\quad \thorn\Hm& =\tau\Hl
\\
\eth\Hl&=-\tau\Hl &\qquad\quad \eth\Hn & = -\bar\kappa'\Hl&\qquad\quad \eth\Hm&
=0 \\
\eth'\Hl&=-\bar\tau\Hl&\qquad\quad \eth'\Hn & =-\kappa'\Hl &\qquad\quad
\eth'\Hm& = -\psi \\
\thorn'\Hl&=-W+\psi &\qquad\quad \thorn'\Hn & =-\kappa'\Hm-\bar\kappa'\Hmb
&\qquad\quad \thorn'\Hm& = -\tau\Hn\end{aligned}$$ In \[2\], Edgar and Ludwig used the tetrad freedom to set $\tau'=\sigma'=\rho'=0$, $\Phi_{22}-\tau\kappa'-\bar\tau\bar\kappa'=0$ and $\thorn'(\tau/\bar\tau)=0$ as a preparation to performing the integration. In our approach, we have obtained $\tau'=\sigma'=\rho'=0$ without the need to solve a system of differential equations, and it is not difficult to check that we also have $\thorn'(\tau/\bar\tau)=0$. As for Edgar and Ludwig’s other term, we note that $\Phi_{22}-\tau\kappa'-\bar\tau\bar\kappa'$ is of weight $(-2,-2)$. Define the real scalar $Z=\Hl^2(\Phi_{22}-\tau\kappa'-\bar\tau\bar\kappa')$ of weight $(0,0)$. Then we can easily check that $Z$ is annihilated by all the derivative operators and is hence constant.
We will not attempt to show $Z$ is zero, as we wish to pick a set of coordinate candidates (real weight $(0,0)$ scalars) more attuned to the homothetic or Killing vector than those of Edgar and Ludwig.
Firstly, as in \[2\], define the convenient scalars $P
=\sqrt{\tau/2\bar\tau}$, complex of weight $(1,-1)$ and $A=(2\tau\bar\tau)^{-1/2}$, real of weight $(0,0)$. The scalar $P$ has the happy property of being annhilated by all the GHP operators, whereas $$\thorn A =0 ,\qquad \thorn' A = \frac1{\Hl}(A\psi -u),\qquad
\eth A= -P.\label{eq:QDEss}$$ Our coordinate candidates are $w$, $x$ and $y$ where $$w = \Re\left(\frac{\Hm}{P}\right),\qquad
z=x+iy = -2\frac{\kappa'P}{\Phi_{22}},$$ and also a real scalar $u$ of weight $(0,0)$ satisfying the equations (cf. \[2\]) $$\thorn u=0,\qquad \eth u=0,\qquad \thorn'u = 1/\Hl.$$ These latter equations are consistant, as can be checked by verifying the commutators are satisfied. The commutators acting on weight $(0,0)$ scalars here simplify to $$\left[\thorn,\thorn'\right] = A^{-1}\left(
\frac{1}{\bar P}\eth+\frac{1}{P}\eth'\right),\qquad
\left[\thorn,\eth\right] = \left[\eth,\eth'\right]=0,\qquad
\left[\thorn',\eth\right]=-\frac{1}{AP}\thorn'-\kappa'\thorn$$ and their conjugates.
Our table of deriviatives reads $$\begin{aligned}
{5}
\thorn u & = 0 & \qquad
\thorn'u & = 1/\Hl & \qquad
\eth u & = 0 \\
\thorn w & = \Hl/A & \qquad
\thorn'w & = -B/\Hl & \qquad
\eth w & = -\psi P \\
\thorn x & = 0 & \qquad
\thorn' x & = \frac{1}{\Hl}(\psi x+w) & \qquad
\eth x& = P\\
\thorn y & = 0 & \qquad
\thorn'y & = -\frac{1}{\Hl}(\psi y -v) & \qquad
\eth y & = -i P\end{aligned}$$ where $B=\Hn\Hl/A$ and $v=\Im(\Hm/P)$ are both real of weight $(0,0)$. With the coordinates in the order $(u,w,x,y)$, the tetrad is thus $$\begin{aligned}
\ell^a&=\Hl\left(0,A^{-1},0,0\right)\qquad
n^a = {\Hl}^{-1}\left(1,-B,\psi x+w,\psi y-v,\right)\qquad
m^a = P\left(0,-\psi,1,-i\right)\end{aligned}$$ The $\Hl$ and $P$ terms are a manifestation of the residual boost and spin freedom of course.
The other homothetic vector components are $\Hn=AB/\Hl$ and $\Hm=(w+iv)P$, and the homothetic vector simplifies to $$\xi^a = \left(1,\psi w,\psi x,\psi y\right)$$ so that in the Killing case $u$ is a cyclic coordinate.
To complete the integration we need to find $A$, $B$ and $v$. Since they are all weight (0,0) this is straightforward: $$v=\psi y -c_3 e^{\psi u},\qquad
A=c_2e^{\psi u}-x,\qquad
B = \psi w +\psi^2 x +(x^2+y^2)c_1e^{-\psi u}+c_4e^{\psi u},$$ where the $c_i$ are constant. Note that in the Killing case, $v$ is constant.
The metric is then $$ds^2 = H\,du^2 +2A\,du\,dw+ 2(w+c_2\psi e^{\psi u})du\,dx+
2c_3e^{\psi u}du\,dy - dx^2-dy^2$$ where $$H = 2A\left(c_1(x^2+y^2)e^{-\psi u}+c_4e^{\psi u} )\right) -
c_3^2e^{2\psi u} -w^2 -\psi^2x^2-2\psi w x.$$
Larger algebra
--------------
This leaves us to consider the solutions with further homothetic or Killing vectors. Since the bracket of two homothetic vectors is Killing, we begin by assuming the first symmetry vector is Killing, and then look for a second (possibly homothetic) vector $\xi^a$. So the metric is $$ds^2 = H\,du^2 +2(c_2-x)\,du\,dw+ 2w\,du\,dx+
2c_3du\,dy - dx^2-dy^2$$ where $$H = 2\left((x^2+y^2)c_2+c_4\right)(c_2-x) -
w^2 - c_3^2 $$ and the Killing vector is $K^a=\partial_u$. For the tetrad we perform a rotation to make $\tau$ (and hence $P$) real and also a boost to allow us to integrate more easily: so $$\ell^a=\left(0,1,0,0\right)\qquad
n^a = \frac{1}{c_2-x}\left(1,-c_4-c_1(x^2+y^2),w,c_3\right)\qquad
m^a = \frac1{\sqrt{2}}\left(0,0,1,-i\right).$$ The advantage of this tetrad is that the improperly weighted spin coefficients $\varepsilon$, $\alpha$, $\beta$ and $\gamma$ all vanish, leaving only $$\tau = \frac1{\sqrt{2}(c_2-x)}\quad \text{and}\quad
\kappa'=-\sqrt{2}c_1\,\frac{x+iy}{c_2-x}.$$ So the four GHP operators reduce to the basic Newman-Penrose operators $D$, $\delta$, $\delta'$ and $D'$ (see \[7,8\]) for all scalars. Furthermore, the same argument as used in the previous section tells us that $\Hl$ is non-zero and $\phi_{11}=-\tau\Hl$.
Integrating ($a$) and ($b$) and using the reality of $\Hl$ gives $\Hl=F_1(u)(c_2-x)$ for some $F_1(u)$. Next we integrate ($g$), ($i$) and ($j$) to get $$\Hm = \frac1{\sqrt{2}}\left(F_1(u)w- \psi(x-iy)\right) +F_2(u)$$ for some complex $F_2(u)$. However, ($d$) and the Killing equations imply that $F_2(u)=\frac1{\sqrt{2}}\,c_2\psi+i b(u)$ for real $b$. And now equation (\[eq:Phitwo\]) for $(a,b)=(2,2)$ gives $\phi_{01}=\frac14\psi$ and so ($l$) gives $\phi_{00}=\kappa'\Hl$.
The remaining equation for $\Hl$, ($b$), shows that $F_1=a_1-\frac12\psi u$ where $a_1$ is a constant, so $\Hl=\xi^a\ell_a=(a_1-\frac12\psi u)(c_2-x)$. But $K^a\ell_a=c_2-x$, which tells us that we cannot have a second Killing vector ($\psi=0$), as then a linear combination of $\xi^a$ and $K^a$ would be orthogonal to $\ell^a$, which we saw is not possible. Hence
Type O pure radiation metrics can admit at most a 2 parameter group of homothetic motions, and if the dimension is 2 it is a proper homothetic group.
We also see that for the second symmetry vector (the proper homothety), we can take $$\Hl=-\frac12\psi u(c_2-x).$$ The imginary part of ($h$) can now be solved for $b_2(u)$ to give $b_2(u) = -\frac1{\sqrt{2}}\psi c_3 u + a_2$ for constant $a_2$. The real part of ($h$) now gives us $$\Hn = -\frac12\psi\left( c_1(x^2+y^2)u+c_4u-w\right).$$ The only remaining Killing or integrability equation is ($e$) for $D'\Hn$, and this tells us that $c_2=c_3=c_4=a_2=0$ and thus $$\xi^a = \frac12\psi\left(-u,3w,x,y\right).$$ The remaining constant $c_1$ is non-zero (or the metric is flat) and can be absorbed in the coordinates. The type O pure radiation solution with the largest possible homothetic symmetry group is thus $$ds^2 =-\left(2x(x^2+y^2)+w^2\right)\,du^2 -2x\,du\,dw+ 2w\,du\,dx
- dx^2-dy^2$$ with a non-abelian homothetic algebra generated by $\{\partial_u, -u\partial_u+3w\partial_w+x\partial_x+y\partial_y\}$.
Acknowledgements
================
The author is indebted to John Carminati of Deakin Universtity for the idea of the second integrability equations given here. Calculations were carried out using Maple, and in particular the GHPII package of Vu and Carminati \[9\]. Maple is a registered trademark of Waterloo Maple Inc.
References
==========
1. Collinson CD and French DC [*J. Math. Phys.*]{} [**8**]{} (1967) p. 701
2. Edgar SB and Ludwig G [*Gen. Rel. Grav.*]{} [**29**]{} (1997) p. 1309
3. Geroch R, Held A and Penrose R [*J. Math. Phys.*]{} [**14**]{} (1973) p. 874
4. Kollasis C and Ludwig G [*Gen. Rel. Grav.*]{} [**25**]{} (1993) p. 625
5. Ludwig G [*Class. Quantum Grav.*]{} [**19**]{} (2002) p. 3799
6. Steele JD 2002 [*Class. Quantum Grav.*]{} [**19**]{} p. 259
7. Penrose R and Rindler W 1984 [*Spinors and Space-Time*]{} vol 1 (Cambridge: Cambridge University Press)
8. Stephani H, Kramer D, MacCallum M, Hoenselaers C and Herlt E 2003 [*Exact Solutions of Einstein’s Field Equations, 2nd Edition*]{} (Cambridge: Cambridge University Press)
9. Vu K and Carminati J [*Gen. Rel. Grav.*]{} [**35**]{} (2003) p. 263
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---
abstract: 'The motion of dark striations across bright filaments in a sunspot penumbra has become an important new diagnostic of convective gas flows in penumbral filaments. The nature of these striations has, however, remained unclear. Here we present an analysis of small scale motions in penumbral filaments in both simulations and observations. The simulations, when viewed from above, show fine structure with dark lanes running outwards from the dark core of the penumbral filaments. The dark lanes either occur preferentially on one side or alternate between both sides of the filament. We identify this fine structure with transverse (kink) oscillations of the filament, corresponding to a sideways swaying of the filament. These oscillations have periods in the range of 5-7 min and propagate outward and downward along the filament. Similar features are found in observed G-band intensity time series of penumbral filaments in a sunspot located near disk center obtained by the Broadband Filter Imager (BFI) on board [*Hinode*]{}. We also find that some filaments show dark striations moving to both sides of the filaments. Based on the agreement between simulations and observations we conclude that the motions of these striations are caused by transverse oscillations of the underlying bright filaments.'
author:
- 'L. Bharti$^{1}$, R.H. Cameron$^{1}$, M. Rempel$^2$, J. Hirzberger$^{1}$ and S. K. Solanki$^{1,3}$'
bibliography:
- 'MURaM.bib'
title: Waves as the source of apparent twisting motions in sunspot penumbrae
---
=1
Introduction
============
High resolution Hinode and Swedish Solar Telescope observations of sunspots located away from disk center have revealed fine structure in penumbral filaments: filaments which are nearly perpendicular to the solar disk radius vector display a “twisting motion” indicated by dark striations moving across the filaments (i.e. perpendicular to the filament’s axis) always directed from the limb-side to the center-side of the filament [@ichimoto_07]. These striations can be used as tracers of the flow, and have observationally established the presence of overturning convection in the filaments [@zakharov08; @bharti10a], an idea which is consistent with theory and recent simulations [@heinemann07; @Rempel2009b; @Rempel2009]. The direct measurement of the velocity is more difficult: [@almeida07] found a local correlation between upflows and bright structures as well as between downflows and dark structures in a penumbra. Such correlations are suggestive of convective energy transport in the penumbra. The clear signal of an upflow along the central axis of a bright filament has been reported by [@franz09], [@bellot10] and [@ichimoto10], downflows at its sides are more challenging to observe. Line syntheses from sunspot simulations (cf. [@bharti11]) suggest that these downflows are partly hidden in observations due to both limited spatial resolution, and the fact that commonly used lines form above the heights where the downflows are strongest. Moreover, the Evershed flow affects line of sight velocities, thus the detection of downflows at edges of filaments also depends on the location of the sunspot on the solar disk. Recently [@Joshi11] and [@Scharmer11] found downflows in dark regions at the edges of the penumbral filaments in the C I 5380Å line formed deep in the photosphere. The results of these publications support the prediction made by [@bharti11]. [@Scharmer12] reported that downflows at the edges of filaments detected in the C I 5380Å line ([@Scharmer11]) are also present in the wing of Fe I 6301.5 Å with reduced amplitude.
The striations have the advantage that they potentially allow horizontal motions to be followed, if detected close to disk center, which would make them of unique diagnostic value for the velocity field we have. In previous studies by [@ichimoto_07], [@zakharov08], [@spruit10] and [@bharti10a], the apparent twist of the striations was analyzed only in sunspots away from the disk center where the twisting motions is seen in filaments perpendicular to the line of symmetry (i.e. in filaments directed parallel to the nearest portion of the limb). The twists are always directed toward the center-side. This was interpreted as a geometrical effect by [@zakharov08] – these “twisting” motions were exclusively interpreted in terms of overturning convective flows perpendicular to the filament’s major axis [@ichimoto_07; @zakharov08; @scharmer09; @spruit10]. [@spruit10] proposed that these striations originate from the “corrugation of the boundary between an overturning convective flow inside the filament and the magnetic field wrapping around it”. Based on their modeling results, they also argue that the striations are not compatible with a horizontal field along the axis of filaments in excess of 300 G. However, it is not certain if they really trace convective flows or not. This is an important point to establish, since the motions of the striations have been employed by [@zakharov08] to conclude that convective motions transport sufficient energy to explain the brightness of the penumbra (cf. [@bharti10a]) . In this paper we use the “realistic” numerical radiative MHD simulations of [@Rempel2009b; @Rempel2009; @Rempel2011] to investigate the causes of the observed brightness striations. In addition, we analyze such striations in penumbral structures observed at disk center and compare their properties with those found in the numerical simulations.
The paper is organized as follows: in Section 2 we describe the numerical simulation and present an analysis of the fine structure we find there, in Section 3 we describe the observations and compare the simulations with the observations. We then present our conclusions in Section 4.
Numerical simulations
=====================
The simulations analyzed here were carried out with the MURaM code [@Voegler05]. The code includes the effects of partial ionization on the equation of state, and non-grey radiative transfer. For details of the code and the equations see [@Voegler05] and for recent modifications, essential for the sunspot simulations presented here, see [@Rempel2009]. This code has been used extensively to treat problems both in the quiet Sun [@Keller04; @Khomenko05; @Voegler07; @Pietarila_Graham09; @Graham10; @Danilovic10b; @Danilovic10a] as well as flux concentrations reaching from pores to entire active regions [@Cameron07; @Cheung08; @Cheung10; @Yelle_Chaouche09; @Schuessler06; @Rempel2009b; @Rempel2009; @bharti10b; @Rempel2010; @Rempel2011].
Here we present results from two different simulation runs. The first simulation uses a setup in ’slab’ geometry, in which only a narrow slice through the center of a sunspot is simulated. The geometry and size of this simulation make it ideal for studying the detailed 3 dimensional evolution of individual penumbral filaments, albeit in a somewhat artificial geometry. The second simulation uses a setup with a pair of opposite polarity sunspots, leading to more extended penumbrae with a more realistic geometry.
Slab Geometry
-------------
A snapshot from the sunspot simulation in slab geometry described by [@Rempel2009] was used as the initial condition for the calculations presented here. The simulation domain is periodic in both horizontal ($x$ and $y$) directions, with dimensions of 4.6 Mm $\times$ 36.864 Mm, and has a dimension of 6.144 Mm in the vertical ($z$) direction. The average value of the $\tau_{\mathrm{Ross}}=1$ ( $\tau_{\mathrm{Ross}}=1$ levels refer to optical depth computed from the grey opacities (which is Rossland mean opacity, an “average” opacity such that if we assume that the opacity at all frequencies is this average.))height in the quiet Sun is used to define $z=0$, and $z$ is defined to be positive above this height and negative in the interior of the Sun. The vertical boundary conditions are unchanged from [@Rempel2009]: the top boundary is closed and the magnetic field above it is assumed to be potential. The bottom boundary is open as described in [@Voegler05].
From this initial condition, the simulation was continued for 133 minutes of solar time, with snapshots saved every 34.5s.
### Simulation analysis
The bolometric intensity of the entire simulation domain viewed from above at $t=71.9$ min is shown in Fig. \[fig:sim\_ref\]. Several penumbral filaments with central dark cores can be seen. In the following we consider the filament in the lower right region of the penumbra ($21.1 \le y \le 28.1$ Mm and $0 \le y \le 1.44$ Mm). A blow up of this region is shown in Fig. \[fig:I\_times\]. There we see that, at the height of the cut $z=-384$ km (i.e. 384 km below the average $\tau_{\mathrm{Ross}}=1$ height of the quiet Sun) there is a continuous upflow along the central part of the filament and downflows along the borders. This cut lies below the local $\tau_{\mathrm{Ross}}=1$ surface, so that the shown up and downflows are not directly observable. The filament exhibits fine structure in the form of ’wiggles’ in the bolometric intensity as well as horizontal and vertical velocities along its entire length. The wavelength in the $x$ direction of the ’wiggles’, at this fixed time, is approximately 700 km. Figure \[fig:I\_inc\] shows that, when viewed at an angle, inclined striations which propagate away from the umbra appear, somewhat similar to what is seen in the high resolution observations by [@ichimoto_07], [@zakharov08], [@spruit10] and [@bharti10a].
In order to study the time evolution of the fine structure, we first focus on cuts across the filament at $x=25.7$ Mm. The time evolution of the cuts for various quantities are shown in Fig. \[fig:I\_time\_slice\]. The range of the intensity image has been restricted in order to better reveal the fine structure between $t=50$ and $t=90$ min. Between $t=0$ and $t=30$ min there are clear variations in the intensity producing an asymmetric fishbone pattern, as dark lanes propagate first to one edge of the filament, then to the other. Somewhat weaker oscillations occur between 50 and 80 minutes, and these are followed by larger amplitude oscillations from $t\approx 100$ min onwards.
These bolometric intensity variations correspond to variations in the vertical velocity, with the minima in intensity corresponding to stronger downflows. These fluctuations are accompanied by variations in the $y$ component of the velocity, corresponding to the tube at this height swaying first in the negative $y$ direction, and then in the positive $y$ direction. The second, weaker burst of oscillations occurs between $t=50$ and $t=90$ min. The intensity fluctuations associated with this second set of oscillations are more pronounced for $y<700$ km. The weak velocity fluctuations, again best seen in $v_y$, indicate a swaying of the tube in the $y$ direction, consistent with magnetic field strength variations which are asymmetric. The period of the oscillations in both phases is about 8 min. We emphasize that the perturbations occur across the entire inhomogeneous filament, despite the tube having very different velocities at different locations.
The vertical structure of $v_y$, as a function of time at the three points indicated by stars in Fig. \[fig:I\_times\] is shown in Fig. \[fig:vz\_vert\_slice\]. At $x=25.7$ Mm, $y=510$ km (top frame of Fig. \[fig:vz\_vert\_slice\]), we see that mostly $v_y<0$, corresponding to an outflow away from the central axis of the filament. At $x=25.7$ Mm, $y=660$ km (middle panel), we see that along the center of the filament neither flows in the positive or negative $y$ direction dominate and the clearest signature is of oscillatory motions. The situation at $x=24.5$ Mm, $y=810$ km (bottom panel of Fig. 5) is conceptually similar to that at $y=510$ km, except that on this side of the filament $v_y>0$ mostly dominates which again corresponds to a lateral outflow from the filament. Oscillatory motions can be seen, and are in phase at $y=510$ km and $y=660$ km, and especially at early times at $y=810$ km. The oscillations are mainly propagating downwards (towards lower $z$ at later times). We measured the wavelength to be $\approx 730$ km.
As well as studying the $x$ dependence of the oscillations at a particular time as in Fig. \[fig:I\_times\], we also took a space-time cut along the violet line in Fig. \[fig:I\_times\]. This cut is shown in Fig. \[fig:iisl\]. The red line is placed at 25.7 Mm, corresponding to the red line in Fig. \[fig:I\_times\]. Oscillations can be seen near the red line between $t=0$ and $t=30$ min and between $t=50$ and $t=90$ min. They appear as light and dark ridges running from the umbral end of the filament towards the granulation. The apparent $x$ component of the wavelength is on the order of 1Mm.
The phase speed of the oscillations $\omega/\sqrt{k_x^2+k_z^2}$ is then approximately 2 km/s. The wavevector, $(k_x,0,k_z)$, is inclined by approximately 45$^{\circ}$ to the vertical, directed downwards and away from the umbra. The latter is consistent with the fact that the striations, when observed near the limb, appear to propagate only away from the umbra.
To visualize the waves, Fig. \[fig:T\_cuts\] shows the temperature in vertical cuts through the filament at the location indicated by the red line in Fig \[fig:I\_times\] at $t=11$ and $t=16.5$ min, corresponding to two nearly opposite phases of the oscillations. The differences in the temperature structure at the two phases is large in the top 200 km of the filament, indicating that the oscillations are outside the linear regime.
We comment that the mode is global with respect to the penumbral filament, although the filament has strong velocity, temperature and field gradients. For this reason we think it is dangerous to interpret the associated intensity fluctuations, such as those plotted in the top left of Fig. \[fig:iisl\], as simple tracers of the velocity field.
There are numerous physical forces and processes which affect the oscillations. The period of 8 minutes (similar to the lifetime of granules) is long enough to make radiative processes important, the magnetic field is strong and structured, the flows are a significant fraction of the local sound speed and highly structured, whilst the density varies strongly across the filament and with depth. We leave the difficult task of disentangling the various waves and instabilities which could play a role to a future study.
Round sunspots
--------------
To see if the oscillations found above are specific to penumbral filaments in the slab geometry, we looked for similar features in the sunspot simulation described in [@Rempel2009b; @Rempel2011]. The simulation domain is periodic in both horizontal directions ($x$ and $y$), with dimensions of 98.304 Mm $\times$ 49.152 Mm, and has a vertical ($z$) extent of 6.144 Mm. The boundary conditions are identical to the slab simulation described above. The setup of this simulation contains a pair of opposite polarity sunspots with about $1.6\cdot 10^{22}$ Mx each and maximum field strengths of about $3$ and $4$ kG, respectively. This setup leads to an extended penumbra in between both spots and we focus here our investigation on a part of the penumbra belonging to the (left side) for the $4$ kG sunspot (see, e.g., Figs. 1 of @Rempel2009b [@Rempel2011 see, e.g., Figs. 1 of]). This part of the penumbra is the most extended displaying long, well formed fibrils, which are shown in Fig. \[fig:MS1\]. A detailed analysis of this region was recently performed by @Rempel2011.
The spatial location of the space-time slices plotted in Fig. \[fig:MS2\] is marked by the horizontal line in Fig. \[fig:MS1\]. The time slices displayed in Fig. \[fig:MS2\] are computed using a constant geometrical height about $300$ km beneath the quiet Sun $\tau=1$ level, which is below the local $\tau=1$ in bright filaments and above it in the dark filaments. The presence of oscillations in this simulation is suggested by the typical asymmetric fishbone pattern, earlier detected in the slab simulation, most clearly visible in the filament at the position $x=9$ Mm. This suspicion is heightened by the clear oscillatory signal in the velocity perpendicular to the filament. A more detailed view of the filaments located between $x=2.25$ and $x=5.25$ Mm as well as $x=8.1$ and $x=9.9$ Mm is shown in Figs. \[fig:MS3\] and \[fig:MS4\], respectively. Clearly, this simulation also displays oscillations in several filaments in the innermost parts of the penumbra with periods around 7-8 minutes. The most significant difference compared to the oscillations in the slab geometry is a substantially smaller horizontal displacement. While the slab simulation presented in Fig. \[fig:I\_time\_slice\] shows a lateral displacement of the filament with an amplitude comparable to the width of the filament in all plotted variables, the oscillations here occur within otherwise mostly unaffected flow channels. The oscillation is most prominent in the intensity and in the velocity component lateral to the filament. Variations in the magnetic field strength are mostly restricted to a narrow boundary layer characterized by enhanced horizontal and reduced vertical field strength. This boundary layer coincides with the region of convective downflows at the edge of the filaments, as has been shown in @Rempel2011 [see Fig. 17 therein]. The vertical flow velocity shows moderate changes in the central upflows, but a rather intermittent behavior in the lateral downflows. The average outflow velocity from filament has its largest amplitudes near the outer edge of the filaments, where we also find the strongest flow variations. They remain small compared to the mean flow velocity, however. Both, the radial magnetic field and radial outflow originate mostly from a thin boundary layer just below the $\tau=1$ level [see Fig. 17 of @Rempel2011], where strong horizontal field is induced and the Evershed flow is driven. The horizontal field and accelerated fluid are transported downward by the overturning convection within the filament, leading in deeper layers to a filament structure with enhancements of both radial magnetic field strength and flow velocity at the lateral boundaries of the filaments. The most significant difference between this simulation and the previously discussed slab simulation is in terms of a substantially stronger Evershed flow that is accompanied with stronger horizontal field within flow channels: the flow velocity along filaments reaches in most filaments 3 km/s, while the horizontal field strength remains of the order of 1.5 kG. The horizontal field enhances the stiffness in the direction along filaments and thus suppresses lateral deformations. This could be taken as an indication that the lateral displacement of the flow channel in Fig. \[fig:I\_time\_slice\] is more a consequence than a cause of the oscillation mode. Furthermore the substantial difference in flow velocity and field strength does not seem to affect the oscillation period compared to the example shown in Fig. \[fig:I\_time\_slice\].
Observations
============
For the present study we selected a time series of G-band (4305Å) images of a sunspot located almost at disk center and recorded by the Broadband Filter Imager (BFI) of the Solar Optical Telescope (SOT) onboard [*Hinode*]{} [@tsuneta07]. Such a location differs significantly from all previous studies of moving dark stripes in penumbral filaments, which were all carried out closer to the limb. The time series, recorded on January 5, 2007, consists of 432 images at 30s cadence and contains a sunspot (NOAA 10933) located at disk center. The spatial resolution in the G-band is approximately 022. The image scale is 0054 per pixel. The Solar Soft pipelines for the Hinode SOT/BFI were used for flat field and dark current corrections. The images are reconstructed for the instrumental PSF, applying a Wiener filter [@sobotka93] and assuming diffraction on an ideal circular 50 cm aperture. Finally, a subsonic filter [@title89] with a cut-off velocity of 6kms$^{-1}$ is used to filter out contributions of five-minute oscillations. The first and last 14 images from the time series of January 5, 2007 have been omitted due to the apodizing window used in the subsonic filtering. The mean intensity of quiet regions (i.e., regions containing undisturbed granulation and an absence of bright points) close to the observed sunspot were used for intensity normalization in all images.
Observational analysis
----------------------
Figure 12 shows a G-band image of the sunspot at the beginning of the filtered time series. A central umbra and a fully developed penumbra can be seen. Filaments oriented in all azimuthal directions clearly exhibit central dark cores. An animation of the analyzed time series reveals a lateral motion of dark lanes in the filaments from the axis of a particular filament toward either one of its edges, or toward both of its edges. These lateral motions are found in all azimuthal directions from the center of the sunspot. They are strongest at the inner ends of the filaments, in particular in filaments that protrude into the umbra.
Two representative space-time diagrams are shown in Figs. 13 and 14, respectively. they correspond to the lines ’S1’ and ’S2’ lying on opposite sides of the penumbra in Fig. 12. The motion of thin dark stripes across some of the filaments is apparent as inclined dark stripes in Figs. 13 and 14.
Figure 13 shows the space time diagram for the horizontal slit ’S1’. The white boxes indicate locations of filaments where inclined stripes or “twists” can be clearly seen. Only some of the filaments show such stripes and only for a part of the time. In the panels on the right, each box is displayed two times, once simply to highlight the stripes, the second time with white lines overplotted on the dark stripes for better visibility. Filament ’A’, located at $x=19$5, shows a central dark core and dark stripes, from 0 to 20 min, pointing toward both edges. The dark stripes in opposite directions occur alternatively, recalling from the spacing between asymmetric fishbone appearance already seen in the left panel of Fig. 4. This pattern can also be seen for filament ’B’ at $x=23$5 from 22 to 38 min. Filament ’C’ at $x=27$0 shows stripes in one direction only. Filament ’D’ located at $x=23$0 shows dark stripes from 100 to 135 min also just in one direction, but opposite to ’C’. Such “one-sided” stripes may correspond to the weaker, asymmetric oscillations seen between 50 and 90 min in the slab simulations (Fig. 3) and the leftmost filament located between 2.5 to 3.0 Mm in Fig. 9 for the round spot.
Figure 14 is a space-time plot along slit ’S2’ depicted in Fig. 12. Filament ’E’, located at $x=27$0, shows stripes in one direction from 30 to 75 min. Filament ’F’ at $x=29$5 shows an asymmetric fishbone pattern from 56 to 70 min. Figure 14 demonstrates that such stripes are not restricted to a single location and that, because the sunspot is nearly at disk center, opposite sides of the penumbra display essentially the same behavior (in a statistical sense). From these two images we estimate that the period of the observed oscillations is in the range of 3 to 7 minutes.
Space-time diagrams along further lines (slits) cutting across filaments slits oriented perpendicular to S1 and S2 (e.g. at solar $x=-22$ and at +4 arc sec) were also produced (not shown). Similar stripes and patterns as shown in Figs. 13 and 14 can also be recognized there, although slightly less clearly.
It is also instructive to consider space-time diagrams running along the filaments (i.e. along slit S3 in Fig. 12), the observational analogy to that shown for the simulation in Fig. \[fig:iisl\]. The oscillations we saw in Figs. 13 and 14 are also visible in Fig. 15. They can be clearly seen between $y=-35$ and $y=-34$ arsec between $t=0$ and $t=30$ as a stack of dark and white stripes, whose inclination to the horizontal indicates that the waves are propagating outwards from the umbra. However, it is also clear that oscillations and propagating waves are ubiquitous. In particular, on the opposite side of the umbra waves (now propagating in the opposite direction, i.e. still away from the umbra) are visible at many locations. From this figure we can also determine that the wavelength along the filament is about 500 km.
Not only are moving stripes visible also at disk center and not just at the limb, but they also move with roughly the same phase-speed (1.2-2.8kms$^{-1}$) and display roughly the same periodicity (3-7 min) at disk center as they do closer to the limb. This observed velocity is consistent with the apparent phase velocity inferred from the simulation, e.g. 2kms$^{-1}$ for the slab geometry (see Section 2.1.1), although the period is somewhat larger in the simulations (possibly related to the subsonic filtering we applied to the observations), e.g. 8 min in the slab geometry and 7-8 min for the round sunspots.
Conclusion
==========
We analyzed striations moving across bright penumbral filaments in both numerical simulations and in a time series of seeing-free G-band images obtained with [*Hinode*]{}. In both cases we find that these moving striations, which give the filaments a twisting appearance, are also visible at disk center and appear relatively similar in simulations and observations. Interestingly, at disc center the two halves of a filament on either side of its centeral dark core can display “twists” in opposite directions, with the stripes on the two sides being out of phase.
Two different numerical simulations indicate that the striations are oscillations propagating along the penumbral filaments directed away from the umbra and downward. The periods of the oscillations one found to be 3-7 min from observations and 7-8 min in the simulations. The explanation for the striations suggested by the simulations is quite different from that proposed by [@spruit10] and, in particular does not support their conclusion that the striations imply the absence of a dynamically significant horizontal magnetic field strength in bright penumbral filaments. The MHD simulations reproduce the data rather well, although they have a rather dynamically significant horizontal field in the filaments. Although the Simulations with a stronger horizontal field (1500 G) in the filaments produce oscillations with a smaller amplitude, they are still consistent with the observations. These oscillations are potentially a new seismic diagnostic which can be used to better understand penumbral filaments. This would, however, require a physical understanding of the underlying oscillatory mode, which itself will require further study.
![Bolometic intensity from the simulation in slab geometry at $t=71.9$min. The red box indicates the filament we chose for further study.[]{data-label="fig:sim_ref"}](Figure1.pdf)
![Intensity map and horizontal cuts through the simulated filament in the red box in Fig \[fig:sim\_ref\]. From top to bottom, the panels show: the bolometric intensity; the component of the velocity in the $y$ direction saturated at $-500$ m/s (red) and $+500$ m/s (blue) at a height $z=-384$ km below the average height of $\tau_{\mathrm Ross}=1$ in the quiet Sun ; and the component of the velocity in the $z$ direction saturated at $-2$ km/s (red) and $+2$ km/s (blue) at the same height. The red line at $x=25.6$ Mm, the red crosses and the blue line at $y=0.51$ Mm show the locations at which slices were taken for further analysis (black in lower frames).[]{data-label="fig:I_times"}](Figure2a.pdf "fig:"){width="14cm"} ![Intensity map and horizontal cuts through the simulated filament in the red box in Fig \[fig:sim\_ref\]. From top to bottom, the panels show: the bolometric intensity; the component of the velocity in the $y$ direction saturated at $-500$ m/s (red) and $+500$ m/s (blue) at a height $z=-384$ km below the average height of $\tau_{\mathrm Ross}=1$ in the quiet Sun ; and the component of the velocity in the $z$ direction saturated at $-2$ km/s (red) and $+2$ km/s (blue) at the same height. The red line at $x=25.6$ Mm, the red crosses and the blue line at $y=0.51$ Mm show the locations at which slices were taken for further analysis (black in lower frames).[]{data-label="fig:I_times"}](Figure2b.pdf "fig:"){width="14cm"} ![Intensity map and horizontal cuts through the simulated filament in the red box in Fig \[fig:sim\_ref\]. From top to bottom, the panels show: the bolometric intensity; the component of the velocity in the $y$ direction saturated at $-500$ m/s (red) and $+500$ m/s (blue) at a height $z=-384$ km below the average height of $\tau_{\mathrm Ross}=1$ in the quiet Sun ; and the component of the velocity in the $z$ direction saturated at $-2$ km/s (red) and $+2$ km/s (blue) at the same height. The red line at $x=25.6$ Mm, the red crosses and the blue line at $y=0.51$ Mm show the locations at which slices were taken for further analysis (black in lower frames).[]{data-label="fig:I_times"}](Figure2c.pdf "fig:"){width="14cm"}
![Snapshots of the simulated penumbral filament seen at an angle of $50^{\circ}$ from disk centre at times (a) $t=57$ min, (b) 60 min, (c) 63 min, (d) 66 min and (e) 69 min. The umbra is situated at the bottom of the figure and the quiet-Sun above the top of the figure. At this viewing angle, inclined striations, indicated by the blue arrows, can be seen propagating away from the umbra. []{data-label="fig:I_inc"}](Figure3.pdf)
. Shown are (a) the bolometric intensity, (b) the $x$ component of the magnetic field, (c) the vertical component of the magnetic field, (d) the $x$ component of the velocity, (e) the $y$ component of the velocity, and (f) the vertical component of the velocity. The last five quantities are sampled at a constant geometric height of $z=-384$ km below the average $\tau_{\mathrm Ross}$ height of the quiet Sun.[]{data-label="fig:I_time_slice"}](Figure4a.pdf "fig:"){width="4cm"}. Shown are (a) the bolometric intensity, (b) the $x$ component of the magnetic field, (c) the vertical component of the magnetic field, (d) the $x$ component of the velocity, (e) the $y$ component of the velocity, and (f) the vertical component of the velocity. The last five quantities are sampled at a constant geometric height of $z=-384$ km below the average $\tau_{\mathrm Ross}$ height of the quiet Sun.[]{data-label="fig:I_time_slice"}](Figure4b.pdf "fig:"){width="4cm"}. Shown are (a) the bolometric intensity, (b) the $x$ component of the magnetic field, (c) the vertical component of the magnetic field, (d) the $x$ component of the velocity, (e) the $y$ component of the velocity, and (f) the vertical component of the velocity. The last five quantities are sampled at a constant geometric height of $z=-384$ km below the average $\tau_{\mathrm Ross}$ height of the quiet Sun.[]{data-label="fig:I_time_slice"}](Figure4c.pdf "fig:"){width="4cm"}
. Shown are (a) the bolometric intensity, (b) the $x$ component of the magnetic field, (c) the vertical component of the magnetic field, (d) the $x$ component of the velocity, (e) the $y$ component of the velocity, and (f) the vertical component of the velocity. The last five quantities are sampled at a constant geometric height of $z=-384$ km below the average $\tau_{\mathrm Ross}$ height of the quiet Sun.[]{data-label="fig:I_time_slice"}](Figure4d.pdf "fig:"){width="4cm"}. Shown are (a) the bolometric intensity, (b) the $x$ component of the magnetic field, (c) the vertical component of the magnetic field, (d) the $x$ component of the velocity, (e) the $y$ component of the velocity, and (f) the vertical component of the velocity. The last five quantities are sampled at a constant geometric height of $z=-384$ km below the average $\tau_{\mathrm Ross}$ height of the quiet Sun.[]{data-label="fig:I_time_slice"}](Figure4e.pdf "fig:"){width="4cm"}. Shown are (a) the bolometric intensity, (b) the $x$ component of the magnetic field, (c) the vertical component of the magnetic field, (d) the $x$ component of the velocity, (e) the $y$ component of the velocity, and (f) the vertical component of the velocity. The last five quantities are sampled at a constant geometric height of $z=-384$ km below the average $\tau_{\mathrm Ross}$ height of the quiet Sun.[]{data-label="fig:I_time_slice"}](Figure4f.pdf "fig:"){width="4cm"}
![Space-time plots from the MHD simulations at the three locations marked by crosses along the slit marked in Fig. \[fig:I\_times\], i.e. from top to bottom at $y=510$ km, $y=660$ km and $y=810$ km, respectively. The spatial dimension corresponds to height, with $z=0$ corresponding to the average value of $\tau_{\mathrm{Ross}}=1$ in the qiet Sun. Positive values of $z$ correspond to heights above the quiet-Sun $\tau=1$ level. The $y$ component of the velocity, saturated between $\pm 500$ m/s, is shown. The black liness show where $v_y=0$ at $y=510$ km, they are intended to provide a reference for comparing the structure of the velocity field between all three sub-images. []{data-label="fig:vz_vert_slice"}](Figure5.png){width="17cm"}
![Space-time plots of the normalized bolometric intensity along the blue line shown in Fig. \[fig:I\_times\] extended into the ’quiet-Sun’ granulation. This cut is along the penumbral filament, but offset from the centre of the filament. The red line shows the $x$ value used to make Figures \[fig:I\_time\_slice\], and \[fig:vz\_vert\_slice\]. To give an impression of how the signal discussed here differs from that of the umbra and quiet Sun, we have used a bigger box than that used in Fig. \[fig:I\_times\], with the green line showing the extent of the box shown in Fig. \[fig:I\_times\].[]{data-label="fig:iisl"}](Figure6.pdf){width="15cm"}
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---
abstract: 'Real-time monitoring of functional tissue parameters, such as local blood oxygenation, based on optical imaging could provide groundbreaking advances in the diagnosis and interventional therapy of various diseases. While photoacoustic (PA) imaging is a novel modality with great potential to measure optical absorption deep inside tissue, quantification of the measurements remains a major challenge. In this paper, we introduce the first machine learning based approach to quantitative PA imaging (qPAI), which relies on learning the fluence in a voxel to deduce the corresponding optical absorption. The method encodes relevant information of the measured signal and the characteristics of the imaging system in voxel-based feature vectors, which allow the generation of thousands of training samples from a single simulated PA image. Comprehensive *in silico* experiments suggest that context encoding (CE)-qPAI enables highly accurate and robust quantification of the local fluence and thereby the optical absorption from PA images.'
author:
- '[^1]'
-
- 'Lena Maier-Hein$^*$'
date: 'June 12th, 2017'
title: Context encoding enables machine learning based quantitative photoacoustics
---
**Keywords:** photoacoustics, quantification, multispectral imaging, machine learning
Introduction {#introduction .unnumbered}
============
Photoacoustic (PA) imaging is a novel imaging concept with a high potential for real-time monitoring of functional tissue parameters such as blood oxygenation deep inside tissue. It measures the acoustic waves arising from the stress-confined thermal response of optical absorption in tissue [@Wang2016-nb]. More specifically, a photoacoustic signal $S(\bm{v})$ in a location $\bm{v}$ is a pressure response to the locally absorbed energy $H(\bm{v})$, which, in turn, is a product of the absorption coefficient $\mu_\textrm{a}(\bm{v})$, the Grueneisen coefficient $\Gamma(\bm{v})$ and the light fluence $\phi(\bm{v})$. $$S(\bm{v}) \propto H(\bm{v}) = \mu_\textrm{a}(\bm{v}) \cdot \Gamma(\bm{v}) \cdot \phi(\bm{v})
\label{eqSignal}$$ Given that the local light fluence not only depends on the imaging setup but is also highly dependent on the optical properties of the surrounding tissue, quantification of optical absorption based on the measured PA signal is a major challenge [@Wang2012-ms; @Cox2009-vn]. So far, the field of quantitative PA imaging (qPAI) has focussed on model-based iterative optimization approaches to infer optical tissue parameters from measured signals (cf. e.g. [@Cox2009-vn; @Iftimia2000-bb; @Cox2005-tl; @Cox2006-hw; @Yuan2006-qx; @Laufer2007-va; @Malone2016-ew; @Haltmeier2015-eq; @Cox2012-ao; @Banerjee2008-rj]). While these methods are well-suited for tomographic devices with high image quality (cf. e.g. [@Wang2009-op; @Xia2014-hs; @Tzoumas2016-yt]) as used in small animal imaging, translational PA research with clinical ultrasound transducers or similar handheld devices (cf. e.g. [@Niederhauser2005-wk; @Zackrisson2014-pb; @Wang2016-nb; @Upputuri2017-fz; @Gamelin2008-we; @Song2008-rr; @Kim2010-cu; @Garcia-Uribe2015-oi]) focusses on qualitative image analysis.
As an initial step towards clinical qPAI, we introduce a novel machine learning based approach to quantifying PA measurements. The approach features high robustness to noise while being computationally efficient. In contrast to all other approaches proposed to date, our method relies on learning the light fluence on a voxel level to deduce the corresponding optical absorption. Our core contribution is the development of a voxel-based *context image* (CI) that encodes relevant information of the measured signal voxel together with characteristics of the imaging system in a single feature vector. This enables us to tackle the challenge of fluence estimation as a machine learning problem that we can solve in a fast and robust manner. Comprehensive *in silico* experiments indicate high accuracy, speed, and robustness of the proposed context encoding (CE)-qPAI approach. This is demonstrated for estimation of (1) fluence and optical absorption from PA images, as well as (2) blood oxygen saturation as an example of functional imaging using multispectral PA images.
Materials and Methods {#materials-and-methods .unnumbered}
=====================
A common challenge when applying machine learning methods to biomedical imaging problems is the lack of labeled training data. In the context of PAI, a major issue is the strong dependence of the signal on the surrounding tissue. This renders separation of voxels from their context - as in surface optical imaging [@Wirkert2016-yz] - impossible or highly inaccurate. Simulation of a sufficient number of training volumes covering a large range of tissue parameter variations, on the other hand, is computationally not feasible given the generally long runtime of Monte Carlo methods which are currently the gold standard for the simulation of light transportation in tissue [@Cox2012-ao].
Inspired by an approach to shape matching, where the shape context is encoded in a so-called *spin image* specifically for each node in a mesh [@Johnson1999-ct], we encode the voxel-specific context in so-called context images (CIs). This allows us to train machine learning algorithms on a voxel level rather than image level and thus require orders of magnitude fewer simulated training volumes. CIs encode relevant information of the measured signal as well as characteristics of the imaging system (represented by so-called voxel-specific fluence contribution maps (FCMs)). The CIs serve as a feature vector for said machine learning algorithm which are trained to estimate fluence in a voxel. The entire quantification method is illustrated in Figure\[figMethod\] which serves as an overview with details given in the following subsections.
![**Machine learning approach to fluence estimation with context images.** Context images (CIs) are generated individually for each voxel and encode both (1) relevant information on the measured signal extracted from the PAI signal volume and (2) prior knowledge on the characteristics of the imaging system represented by fluence contribution maps (FCMs). During algorithm training, a regressor is presented tuples of context images and corresponding ground truth fluence values for each voxel in the training data. For estimation of optical absorption in voxels of a previously unseen image, the voxel-specific context image is generated and used to infer the local fluence using the trained regressor.[]{data-label="figMethod"}](figure1jbo.png "fig:")\
Fluence Contribution Map {#fluence-contribution-map .unnumbered}
------------------------
An important prerequisite for computing the CI for a voxel $\bm{v}$ is the computation of the corresponding FCM, referred to as $\textrm{FCM}[\bm{v}]$. $\textrm{FCM}[\bm{v}](\bm{v'})$ represents a measure for the likelihood that a photon arriving in voxel $\bm{v}$ has passed $\bm{v'}$. In other words, a FCM reflects the impact of a PA signal in $\bm{v'}$ on the drop in fluence in voxel $\bm{v}$. An illustration of a FCM corresponding to a typical handheld PA setup is shown in Figure1(2). The $\textrm{FCM}[\bm{v}]$ is dependent on how the PA excitation light pulse propagates through homogeneous tissue to arrive in $\bm{v}$ given a chosen hardware setup. The $x \times y$ FCMs per imaging plane are generated once for each new hardware setup and each voxel in the imaging plane.
In this first implementation of the CE-qPAI concept, FCMs are simulated with the same resolution as the input data assuming a background absorption coefficient of 0.1cm$^{-1}$ and a constant reduced scattering coefficient of 15cm$^{-1}$ [@Jacques2013-pm]. The number of photons is varied to achieve a consistent photon count in the target voxel. The FCMs are generated with the widely used Monte Carlo simulation tool *mcxyz* [@Jacques2014-ia]. We integrated mcxyz into the open-source Medical Image Interaction Toolkit MITK [@Wolf2004-jw] as mitkMcxyz and modified it to work in a multi-threaded environment. Sample FCMs for different voxels are illustrated in Figure\[figCIandFCM\], which also shows the generation of context images.
Context Image {#context-image .unnumbered}
-------------
The CI for a voxel $\bm{v}$ in a PA volume is essentially a 2D histogram composed of (1) the measured PA signal S in the tissue surrounding $\bm{v}$ and (2) the corresponding $\textrm{FCM}[\bm{v}]$. More specifically, it is constructed from the tuples $\{(\textrm{S}(\bm{v'}), \textrm{FCM}[\bm{v}](\bm{v'})) | \bm{v'} \in N(\bm{v})\}$ where $N(\bm{v})$ is defined as $N(\bm{v}) = \{\bm{v'} | \textrm{FCM}[\bm{v}](\bm{v'}) > \epsilon\}$. This constraint is set to exclude voxels with a negligible contribution to the fluence in $\bm{v}$. The tuples are arranged by magnitude of $\textrm{S}(\bm{v'})$ and $\textrm{FCM}[\bm{v}](\bm{v'})$ into a 2D histogram and thereby encode the relevant context information in a compact form. In our prototype implementation of the CE-qPAI concept, the fluence contribution and signal axes of the histogram are discretized in 12 bins and scaled logarithmically to better represent the predominantly low signal and fluence contribution components. The ranges of the axes are set as $0 < \log(S) < \log(\textrm{255})$ and $\log(\epsilon) < \log(\textrm{FCM}) < -1$. Signals and fluence contributions larger than the upper boundary are included in the highest bin while smaller signals and fluence contributions are not. Figure\[figCIandFCM\] illustrates the generation of CIs from FCMs and PA signals. Labelled CIs are used for training a regressor that can later estimate fluence, which, in turn, is used to reconstruct absorption (Eq.\[eqSignal\]).
![**Generation of context images** using FCMs and PA signal volumes. The voxel-specific FCMs serve as a representation of how the PA excitation light pulse propagates through homogeneous tissue to arrive in a target voxel given a chosen hardware setup. Signal volume and FCM are superimposed to generate 2D histograms from tuples of measured signal and corresponding fluence contribution.[]{data-label="figCIandFCM"}](figure2jbo.png)
Machine learning based regression for fluence estimation {#machine-learning-based-regression-for-fluence-estimation .unnumbered}
--------------------------------------------------------
During the training phase, a regressor is presented tuples $(\textrm{CI}(\bm{v}), \phi(\bm{v}))$ of context images $\textrm{CI}(\bm{v})$ and corresponding ground truth fluence values $\phi(\bm{v})$ for each voxel $\bm{v}$ in a set of PAI volumes. For estimation of optical absorption in a voxel $\bm{v}_\textrm{u}$ of a previously unseen image, the voxel-specific CI is generated and used to infer fluence $\hat{\phi}(\bm{v}_\textrm{u})$ using the trained algorithm.
In our prototype implementation of the CE-qPAI method we use a random forest regressor. With voxel-based CIs, thousands of training samples can be extracted from a single slice of a simulated PA training volume. Ground truth training data generation is performed using a dedicated software plugin integrated into MITK and simulating the fluence with mitkMcxyz. It should be noted that the simulated images consist mainly of background voxels and not of vessel structures which are our regions of interest (ROI). This leads to an imbalance in the training set. To avoid poor estimation for underrepresented classes [@Estabrooks2004-yg], we undersample background voxels in the training process to ensure a 1:1 ROI / background sample ratio. The parameters of the random forest are set to the defaults of sklearn 0.18 using python 2.7, except for the tree count which was set to $n_\textrm{regressors}$ = 100. CIs are used as feature vectors and labeled with the optical property to be estimated (e.g. fluence or oxygenation). The parameters were chosen based on a grid search on a separate dataset not used in the experiments of this work.
Hardware setup {#hardware-setup .unnumbered}
--------------
We assume a typical linear probe hardware setup [@Kirchner2016-rp] where the ultrasound detector array and the light source move together and the illumination geometry is the same for each image recorded. This is also the case for other typical tomographic devices [@Neuschmelting2016-we; @Needles2013-vy]. All simulation were performed on high-end CPUs (Intel i7-5960X).
Experiments and Results {#experiments-and-results .unnumbered}
=======================
In the following validation experiments, we quantify the fluence up to an imaging depth of 28mm in unseen test images for each dataset. With our implementation and setup, all images comprise 3008 training samples, which results in an average simulation time of 2 seconds per training sample. This allows us to generate enough training samples in a feasible amount of time, to train a regressor that enables fluence estimation in a previously unseen image in near real-time. The measured computational time for quantifying fluence in a single $64\times47$ voxel image slice is 0.9s $\pm$ 0.1s.
In the following, we present the experimental design and results of the validation of CE-qPAI. First we will validate the estimation of absorption from PAI volumes acquired at a fixed wavelength and then estimate blood oxygenation from multispectral PAI volumes.
Monospectral absorption estimation {#monospectral-absorption-estimation .unnumbered}
----------------------------------
### Experiment {#experiment .unnumbered}
To assess the performance of CE-qPAI in PA images of blood vessels, we designed five experimental datasets with varying complexity as listed in Table\[tabSetsIni\]. With the exception of DS$_\textrm{multi}$, each of the five experimental datasets is composed of 150 training items, 25 validation items and 25 test items, where each item comprises a 3D simulated PA image of dimensions $64 \times 47 \times 62$ and 0.6mm equal spacing as well as a corresponding (ground truth) fluence map.
---------------------------------- ------------- ------------ -----------
Dataset
(lr)[1-4]{} DS$_\textrm{base}$ 3 4.7 1
(lr)[1-4]{} DS$_\textrm{radius}$ **0.5 - 6** 4.7 1
DS$_\textrm{absorb}$ 3 **1 - 12** 1
DS$_\textrm{vessel}$ 3 4.7 **1 - 7**
(lr)[1-4]{} DS$_\textrm{multi}$ 0.5 - 6 1 - 12 1 - 7
---------------------------------- ------------- ------------ -----------
: **Dataset design.** The design parameters of the datasets (DS)[]{data-label="tabSetsIni"}
As labels of the generated CIs we used a fluence correction $\phi_\textrm{c}(\bm{v'}) = \phi(\bm{v'}) / \phi_\textrm{h}(\bm{v'})$, where $\phi_\textrm{h}(\bm{v'})$ is a fluence simulation based on a homogeneous background tissue assumption. We used 5 equidistant slices out of each volume, resulting in a generation of a total of 2,256,000; 376,000 and 376,000 context images for each dataset - for training, parameter optimization and testing respectively. To account for the high complexity of DS$_\textrm{multi}$, we increased the number of training volumes for that set from 150 to 400. The baseline dataset DS$_\textrm{base}$ represents simulations of a transcutaneously scanned simplified model of a blood vessel of constant radius (3mm) and constant absorption (vessel: 4.73cm$^{-1}$, background: 0.1cm$^{-1}$) and reduced scattering coefficient (15cm$^{-1}$). To approximate partial volume effects, the absorption coefficients in the ground truth images were Gaussian blurred with a sigma of 0.6mm. Single slices were simulated using 10$^8$ photons and then compounded in a fully scanned volume. Different shapes and poses of the vessel were generated by a random walk defined with steps $\bm{r}$ defined as $$\bf{r}_\textrm{i} = \bf{r}_\textrm{i-1} + \eta \cdot \bf{a}$$ where $\eta$ is a free parameter constant in each vessel with an inter-vessel variation within a uniform distribution $( 0 < \eta < 0.2)$ and $\bm{a}$ is varied for each of its components in each step within a uniform distribution $( - 0.2\,\textrm{mm} < a_\textrm{i} < 0.2\,\textrm{mm})$. To investigate how variations in geometry and optical properties impact the performance of our method, we designed further experimental datasets in which the number of vessels (DS$_\textrm{vessel}$), the radii of the vessels (DS$_\textrm{radius}$), the optical absorption coefficients within the vessels (DS$_\textrm{absor.}$) as well as all of the above (DS$_\textrm{multi}$) were varied. We tested the robustness of CE-qPAI to this range of scenarios without retuning CI or random forest parameters.
While most studies assess the performance of a method in the entire image (cf. e.g. [@Cox2006-hw; @Tarvainen2013-cc; @Zemp2010-tr]), it must be pointed out that the accuracy of signal quantification is often most relevant in a defined region of interest - such as in vessels or in regions which provide a meaningful PA signal. These are typically also the regions where quantification is particularly challenging due to the strongest signals originating from boundaries with discontinuous tissue properties. To address this important aspect we validated our method, not only on the entire image, but also in the region of interest (ROI), which we define for our datasets as voxels representing a vessel and at the same time having a contrast to noise ratio (CNR) of larger than 2, to only include significant signal in the ROI. We define CNR following Walvaert and Rosseel [@Welvaert2013-vd] in a voxel $\bm{v}$ as
$$\textrm{CNR} = \dfrac{S(\bm{v}) - avg(b)}{std(b)}$$
where the $avg(b)$ and $std(b)$ are the average and standard deviation of the background signal $b$ over a simulated image slice with a background absorption coefficient of 0.1cm$^{-1}$ and no other structures. Using such an image without application of a noise model, we simulated an intrinsic background noise of $(4.2 \pm 2.8)$a.u.
To investigate the robustness of CE-qPAI to noise we added the following noise models to each dataset. The noise models consist of an additive Gaussian noise term applied on the signal volumes followed by a multiplicative white Gaussian noise term, similar to noise assumptions used in prior work [@Cox2006-hw; @Tarvainen2013-cc]. We examined three noise levels to compare against the simulation-intrinsic noise case:
(1)
: [ 2% multiplicative and $(0.125 \pm 0.125)$a.u. additive component]{}
(2)
: [ 10% multiplicative and $(0.625 \pm 0.625)$a.u. additive component]{}
(3)
: [ 20% multiplicative and $(1.25 \pm 1.25)$a.u. additive component]{}
The additive and multiplicative noise components follow an estimation of noise components on a custom PA system [@Kirchner2016-rp]. For each experimental dataset introduced in Table\[tabSetsIni\] and each noise set, we applied the following validation procedure separately. Following common research practice, we used the training data subset for training of the random forest and the validation data subset to ensure the convergence of the training process, as well as to set suitable parameters for the random forest and ROI, whereas we only evaluated the test data subset to report the final results (as described in [@Ripley2007-qt]). As an error metric we report the *relative fluence estimation error* $e_\textrm{r}$ $$e_\textrm{r}(\bm{v}) = \dfrac{|\hat{\phi}(\bm{v}) - \phi(\bm{v})|}{\phi(\bm{v})}$$ rather than an absorption estimation error, to separate the error in estimating fluence with CE-qPAI from errors introduced through simulation-intrinsic or added noise on the signal which will affect the quantification regardless of fluence estimation.
### Results {#results .unnumbered}
![ **Absorption reconstruction results after fluence estimation.** For the slices with the **a** lowest, **b** median, and **c** highest median fluence estimation error $e_\textrm{r}$ within the region of interest (ROI) of DS$_\textrm{base}$. We show (from left to right) (1) the estimated fluence, (2) the corresponding signal images, (3) the resulting estimation of the absorption coefficient, and (4) the ground truth optical absorption, for reference. **d** shows a histogram of the relative absorption estimation over all ROI voxels ($n=5347$) in DS$_\textrm{base}$ illustrating absorption estimation accuracy rather than fluence estimation accuracy measured by $e_\textrm{r}$. Precorrecting the signal with the fluence of a homogeneous tissue assumption underestimates the absorption and is considerably outperformed by CE-qPAI in the ROI. The CE-qPAI plot omits 5 outliers larger 2.[]{data-label="figReconstruction"}](figure2a.png "fig:") ![ **Absorption reconstruction results after fluence estimation.** For the slices with the **a** lowest, **b** median, and **c** highest median fluence estimation error $e_\textrm{r}$ within the region of interest (ROI) of DS$_\textrm{base}$. We show (from left to right) (1) the estimated fluence, (2) the corresponding signal images, (3) the resulting estimation of the absorption coefficient, and (4) the ground truth optical absorption, for reference. **d** shows a histogram of the relative absorption estimation over all ROI voxels ($n=5347$) in DS$_\textrm{base}$ illustrating absorption estimation accuracy rather than fluence estimation accuracy measured by $e_\textrm{r}$. Precorrecting the signal with the fluence of a homogeneous tissue assumption underestimates the absorption and is considerably outperformed by CE-qPAI in the ROI. The CE-qPAI plot omits 5 outliers larger 2.[]{data-label="figReconstruction"}](figure2b.png "fig:") ![ **Absorption reconstruction results after fluence estimation.** For the slices with the **a** lowest, **b** median, and **c** highest median fluence estimation error $e_\textrm{r}$ within the region of interest (ROI) of DS$_\textrm{base}$. We show (from left to right) (1) the estimated fluence, (2) the corresponding signal images, (3) the resulting estimation of the absorption coefficient, and (4) the ground truth optical absorption, for reference. **d** shows a histogram of the relative absorption estimation over all ROI voxels ($n=5347$) in DS$_\textrm{base}$ illustrating absorption estimation accuracy rather than fluence estimation accuracy measured by $e_\textrm{r}$. Precorrecting the signal with the fluence of a homogeneous tissue assumption underestimates the absorption and is considerably outperformed by CE-qPAI in the ROI. The CE-qPAI plot omits 5 outliers larger 2.[]{data-label="figReconstruction"}](figure2c.png "fig:")\
\
![ **Absorption reconstruction results after fluence estimation.** For the slices with the **a** lowest, **b** median, and **c** highest median fluence estimation error $e_\textrm{r}$ within the region of interest (ROI) of DS$_\textrm{base}$. We show (from left to right) (1) the estimated fluence, (2) the corresponding signal images, (3) the resulting estimation of the absorption coefficient, and (4) the ground truth optical absorption, for reference. **d** shows a histogram of the relative absorption estimation over all ROI voxels ($n=5347$) in DS$_\textrm{base}$ illustrating absorption estimation accuracy rather than fluence estimation accuracy measured by $e_\textrm{r}$. Precorrecting the signal with the fluence of a homogeneous tissue assumption underestimates the absorption and is considerably outperformed by CE-qPAI in the ROI. The CE-qPAI plot omits 5 outliers larger 2.[]{data-label="figReconstruction"}](figure2dB.png "fig:")
Figure\[figReconstruction\]a-c show representative examples of the previously unseen 125 simulated test images from the baseline dataset DS$_\textrm{base}$, with their corresponding fluence estimation results. The optical absorption is reconstructed using the fluence estimation. A histogram illustrating absorption estimation accuracy in region of interest (ROI) voxels of DS$_\textrm{base}$ is shown in Figure\[figReconstruction\]d and compared with a static fluence correction approach.
--------------------------------- -------- ------------- -------- --------------
(lr)[2-5]{}
(lr)[2-3]{} (lr)[4-5]{} Dataset Median Median
(lr)[1-5]{} DS$_\textrm{base}$ 0.5% (0.2, 1.0)% 3.9% (1.9, 7.3)%
DS$_\textrm{radius}$ 0.8% (0.3, 2.7)% 5.8% (2.6, 11.4)%
DS$_\textrm{absorb}$ 0.7% (0.3, 1.9)% 14.0% (5.0, 31.5)%
DS$_\textrm{vessel}$ 1.6% (0.5, 5.9)% 6.8% (2.9, 13.4)%
DS$_\textrm{multi}$ 1.2% (0.4, 5.4)% 20.1% (7.3, 49.0)%
--------------------------------- -------- ------------- -------- --------------
: **Descriptive statistics of fluence estimation results.** The median and interquartile range (IQR) of the relative fluence estimation error $e_\textrm{r}$ for the five validation datasets used for the single wavelength experiments. The median error and IQR are provided (1) for all voxels in the respective test set as well as (2) for the voxels in the region of interest (ROI) only.[]{data-label="tabSets"}
![**Robustness of the fluence estimation against noise.** Median relative fluence estimation errors $e_\textrm{r}$ with interquartile range over all datasets for, **a** all test voxels, and **b** in region of interest test voxels. The whiskers in this plot show the first and third quartile.[]{data-label="figRobustness"}](figure4.png)
Table\[tabSets\] summarizes the descriptive statistics of the relative fluence estimation errors $e_\textrm{r}$ for the experiments on absorption estimation using single wavelength PA images. The relative fluence estimation error $e_\textrm{r}$ does not follow a normal distribution due to large outliers especially in complex datasets, which is why we report median $e_\textrm{r}$ with interquartile ranges (IQR) for all datasets. Even for the most complex dataset DS$_\textrm{multi}$ with variations of multiple parameters, specifically, number of vessels, vessel absorption coefficient and vessel radii CE-qPAI yields a median overall relative fluence estimation error $e_\textrm{r}$ below 2%. Errors are higher in the ROI, especially in datasets with high variations of absorption.
Previously proposed qPAI approaches reveal high drops in estimation performance when dealing with noisy data (cf. e.g. [@Beretta2016-qc]). To remedy this, methods have been proposed to incorporate more accurate noise representations into model based reconstruction algorithms [@Tarvainen2013-cc; @Tarvainen2016-yq]. When validating the robustness of CE-qPAI to noise, it yields high accuracy even under unrealistically high noise levels of up to 20% (cf. Figure\[figRobustness\]). Regardless of the noise level applied, the highest median errors occur in the ROIs of datasets which are characterized by high absorption and inhomogeneous tissue properties.
Multispectral blood oxygenation estimation {#multispectral-blood-oxygenation-estimation .unnumbered}
------------------------------------------
The concept of context encoding cannot only be used to estimate fluence and absorption but also derived functional parameters such as blood oxygenation. To this end, the estimated absorption in a voxel for multiple wavelengths can be applied to resolve oxygenation via linear spectral unmixing. Alternatively, a regressor can be trained using the CIs labeled with ground truth oxygenation.
### Experiment {#experiment-1 .unnumbered}
To investigate the performance of CE-qPAI for blood oxygenation (sO$_2$) estimation we designed an additional multispectral simulated dataset DS$_\textrm{oxy}$ using the wavelengths 750nm, 800nm and 850nm. It consists of 240 multispectral training volumes and 11 multispectral test volumes, each featuring homogeneous oxygenation and one vessel with a radius of 2.3 to 4mm - modeled after a carotid artery [@Krejza2006-wj]. For each image slice and at each wavelength, $10^7$ photons were used for simulation. Oxygenation values for the training images were drawn randomly from a uniform sO$_2$ distribution $U(0\,\%, 100\,\%)$. For testing, we simulated 11 multispectral volumes at 3 wavelengths and 11 blood oxygenation levels (sO$_2 \in \{ 0\,\%, 10\,\%, 20\,\%, \dots , 100\,\% \}$). The optical absorption was adjusted by wavelength and oxygenation, as described by Jacques [@Jacques2013-pm]. Hemoglobin concentration was assumed to be 150g/liter [@Jacques2013-pm]. The blood volume fraction was set to 0.5% in the background tissue and to 100% in the blood vessels. The reduced scattering coefficient was again set to 15cm$^{-1}$. We estimated the oxygenation using three methods:
\(1) *Linear spectral unmixing on the signal images as a baseline [@Keshava2002-wv].* For this, we applied a non-negative constrained least squares approach as also used in [@Tzoumas2016-yt] that minimizes $||A\bm{x}-\bm{b}|| = 0$, where $A$ is the matrix containing the reference spectra, $\bm{b}$ is the measurement vector, and $\bm{x}$ is the unmixing result. Specifically, we used the python scipy.optimize.minimize function with the Sequential Least SQuares Programming (SLSQP) method and added a non-negativity inequality constraint. We evaluated the unmixing results of this method on all voxels in the ROI as well as exclusively on those voxels with the maximum intensity projection (MIP) along image x-axis at wavelength 800nm to account for nonlinear fluence effects deep inside the vessels.
\(2) *Linear spectral unmixing of the signal after quantification of the three input images with CE-qPAI.* After correcting the raw signal images for nonlinear fluence effects using CE-qPAI, we applied the same method as described in (1) and evaluated on the same voxels that were used in (1) to ensure comparability of the results.
\(3) *Direct estimation of oxygenation using a functional adaptation of CE-qPAI.* For functional CE-qPAI (fCE-qPAI), triples of CIs for the three chosen wavelengths were concatenated into one feature vector and labeled with the ground truth oxygenation.
### Results {#results-1 .unnumbered}
![**Oxygenation estimation.** **a** shows the median oxygen estimation with the interquartile range (IQR) on the maximum intensity projection (MIP) voxels using linear spectral unmixing of (blue) the uncorrected signal, (green) the signal corrected by CE-qPAI, and (red) direct estimation by functional CE-qPAI (fCE-qPAI). **b** shows the oxygenation estimation for a representative patch of signal showing a vessel in 15mm depth and with 3mm radius. The signal for one of the measurement wavelengths is shown for reference together with the oxygen estimation results for 0%, 50%, and 100% ground truth homogeneous oxygenation and the three examined methods.[]{data-label="figOxygenation"}](figure5a.png "fig:") ![**Oxygenation estimation.** **a** shows the median oxygen estimation with the interquartile range (IQR) on the maximum intensity projection (MIP) voxels using linear spectral unmixing of (blue) the uncorrected signal, (green) the signal corrected by CE-qPAI, and (red) direct estimation by functional CE-qPAI (fCE-qPAI). **b** shows the oxygenation estimation for a representative patch of signal showing a vessel in 15mm depth and with 3mm radius. The signal for one of the measurement wavelengths is shown for reference together with the oxygen estimation results for 0%, 50%, and 100% ground truth homogeneous oxygenation and the three examined methods.[]{data-label="figOxygenation"}](figure5b.png "fig:")
Estimation of local blood oxygen saturation (sO$_2$) is one of the main qPAI applications and is only possible with multispectral measurements. As such, the presented approaches were validated together with the baseline method on the dataset DS$_\textrm{oxy}$. As shown in Figure\[figOxygenation\]a, the estimation results for both methods are in very close agreement with the ground truth. In fact, the median absolute oxygen estimation error was 3.1% with IQR (1.1%, 6.4%) for CE-qPAI and 0.8% with IQR (0.3%, 1.8%) for the fCE-qPAI adaptation. Furthermore, our methodology outperforms a baseline approach based on linear spectral unmixing of the raw signal (as also compared to in [@Tzoumas2016-yt]). By means of example Figure\[figOxygenation\]b shows that linear spectral unmixing of the ROI on the uncorrected signal fails deep inside the ROI where the fluence varies strongly for different wavelengths. To compensate for this effect when comparing the approach to our method, we validate all methods only on the maximum intensity projection along the depth axis (as also used in [@Dean-Ben2014-uu]) in Figure\[figOxygenation\]a.
Discussion {#discussion .unnumbered}
==========
This paper addresses one of the most important challenges related to photoacoustic imaging, namely the quantification of optical absorption based on the measured signal. In contrast to all other approaches proposed to qPAI to date (cf. e.g. [@Cox2009-vn; @Iftimia2000-bb; @Cox2005-tl; @Cox2006-hw; @Yuan2006-qx; @Laufer2007-va; @Malone2016-ew; @Haltmeier2015-eq; @Cox2012-ao; @Banerjee2008-rj]), our method relies on *learning* the light fluence in a voxel to deduce the corresponding optical absorption. Comprehensive *in silico* experiments presented in this manuscript show the high potential of this entirely novel approach to estimate optical absorption as well as derived functional properties, such as oxygenation, even in the presence of high noise.
Although machine learning methods have recently been applied to PAI related problems (cf. e.g. [@reiter2017-ml; @hauptmann2017-mb; @antholzer2017-dl]), these have mainly focused on image reconstruction but not signal quantification. We attribute this to the fact that training generation for machine learning based qPAI is not at all straightforward given the lack of reference methods for estimating optical absorption in depth and the long simulation times of Monte Carlo based methods. Note also that commonly applied methods of data augmentation (i.e. methods that may be used to automatically enlarge training data sets as discussed in [@Dosovitskiy2014]) cannot be applied to PA images due to the interdependence of fluence and signal. There have been recent developments, however, that could speed up training data generation using hybrid diffusion approximation and Monte Carlo methods [@zhu2012-hm]. With our contribution, we have addressed the challenge by introducing the concept of context images, which allow us to generate one training case from each *voxel* rather than from each image.
As an important contribution with high potential impact, we adapted CE-qPAI to estimate functional tissue properties from multi wavelength data. Both variants - linear spectral unmixing of the fluence corrected signal, as well as direct estimation of oxygenation from multi wavelength CIs, yielded accurate results that outperformed a baseline approach based on linear spectral unmixing of the raw PA signal. It should be noted that linear spectral unmixing of the signal for sO$_2$ estimation is usually performed on a wider range of wavelengths to increase accuracy. However, even this increase in number of wavelengths cannot fully account for nonlinear fluence effects [@Cox2009-vn]. Combined with the separately established robustness to noise, multi wavelength applications of CE-qPAI are very promising.
In our first prototype implementation of CE-qPAI we used random forests regressors with standard parameters. It should be noted, however, that fluence estimation from the proposed CI can in principle be performed by any other machine learning method in a straightforward manner. Given the recent breakthrough successes of convolutional neural networks [@He2016-vv], we expect even better performance of our approach when applying deep learning algorithms or using data augmentation in the training data generation process.
By relating the measured signals $\textrm{S}(\bm{v'})$ in the neighborhood of $\bm{v}$ to the corresponding fluence contributions $\textrm{FCM}[\bm{v}](\bm{v'})$ we relate the absorbed energy in $\bm{v'}$, to the fluence contribution of $\bm{v'}$ to $\bm{v}$. In this context it has to be noted that the fluence contribution $\textrm{FCM}[\bm{v}](\bm{v'})$ is only an approximation of the true likelihood that a photon passing $\bm{v}$ has previously passed $\bm{v'}$, because $\textrm{FCM}[\bm{v}]$ is generated independently of the scene under observation assuming constant background absorption and scattering. Nevertheless due to the generally low variance of scattering in tissue it serves as a reliable input for the proposed machine learning based quantification.
A limitation of our study can be seen in the fact that we performed the validation *in silico*. To apply CE-qPAI *in vivo*, further research will have to be conducted in two main areas. Firstly, the acoustical inverse problem for specific scanners must be integrated into the quantification algorithm to enable quantification of images acquired with common PAI probes such as clinical linear transducers. Secondly, training data has to be generated as close to reality as possible - considering for example imaging artifacts.
In contrast to prior work (cf. e.g. [@Cox2006-hw; @Yuan2006-qx; @Zemp2010-tr; @Tarvainen2013-cc; @Naetar2014-zu]) our initial validation handles the whole range of near infrared absorption in whole blood at physiological hemoglobin concentrations and demonstrates high robustness to noise. The impact of variations of scattering still needs investigation although these should be small in the near infrared.
Long-term goal of our work is the transfer of CE-qPAI to *clinical* data. In this context, run-time of the algorithm will play an important role. While our current implementation can estimate absorption on single slices within a second, this might not be sufficient for interventional clinical estimation of whole tissue volumes and at higher resolutions. An efficient GPU implementation of the time intensive CI generation should enable real-time quantification.
In summary, CE-qPAI is the first machine learning based approach to quantification of PA signals. The results of this work suggest that quantitative real-time functional PA imaging deep inside tissue is feasible.
Disclosures {#disclosures .unnumbered}
===========
The authors have no relevant financial interests in this article and no potential conflicts of interest to disclose.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to acknowledge support from the European Union through the ERC starting grant COMBIOSCOPY under the New Horizon Framework Programme grant agreement ERC-2015-StG-37960.
We would like to thank the ITCF of the DKFZ for the provision of their computing cluster, C.Feldmann for her support with figure design. A.M.Franz, A.Seitel, F.Sattler, S.Wirkert and A.Vemuri for reading the manuscript.
Author Contributions {#author-contributions .unnumbered}
====================
T.K., J.G. and L.M. conceived of the research, analyzed the results and wrote the manuscript. T.K. and J.G. wrote the software and performed the experiments. L.M. supervised the project.
Code and Data Availability {#code-and-data-availability .unnumbered}
==========================
The code for the method as well as the experiments was written in C++ and python 2.7 and is partially open source and available at https://phabricator.mitk.org/source/mitk.git. Additional code and all raw and processed data generated in this work is available from the corresponding authors on reasonable request.
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[^1]: Please address all correspondence to [email protected] or [email protected]
|
---
author:
- |
[**Radoslaw Martin Cichy$^{1,*}$ ([email protected]), Gemma Roig$^{2}$ ([email protected]), Alex**]{}\
[**Andonian$^{3}$ ([email protected]), Kshitij Dwivedi$^{2}$ (kshitij\[email protected]), Benjamin** ]{}\
[**Lahner$^{3}$ ([email protected]), Alex Lascelles$^{3}$ ([email protected]), Yalda Mohsenzadeh$^{3}$**]{}\
[**([email protected]), Kandan Ramakrishnan$^{3}$ ([email protected]), Aude Oliva$^{3}$ ([email protected])**]{}\
$^{1}$Department of Education and Psychology, Freie Universit[ä]{}t Berlin, Berlin, Germany\
$^{2}$Information Systems Technology and Design, Singapore University Technology and Design, Singapore\
$^{3}$Computer Science and Artificial Intelligence Laboratory, MIT, Cambridge, USA\
$^*$ corresponding author
bibliography:
- 'ccn\_style.bib'
title: |
The Algonauts Project: A Platform for Communication between\
the Sciences of Biological and Artificial Intelligence
---
Abstract
========
[ **In the last decade, artificial intelligence (AI) models inspired by the brain have made unprecedented progress in performing real-world perceptual tasks like object classification and speech recognition. Recently, researchers of natural intelligence have begun using those AI models to explore how the brain performs such tasks. These developments suggest that future progress will benefit from increased interaction between disciplines. Here we introduce the Algonauts Project as a structured and quantitative communication channel for interdisciplinary interaction between natural and artificial intelligence researchers. The project’s core is an open challenge with a quantitative benchmark whose goal is to account for brain data through computational models. This project has the potential to provide better models of natural intelligence and to gather findings that advance AI. The 2019 Algonauts Project focuses on benchmarking computational models predicting human brain activity when people look at pictures of objects. The 2019 edition of the Algonauts Project is available online: <http://algonauts.csail.mit.edu/>.** ]{}
> **Keywords:** human neuroscience; vision; object recognition; prediction; challenge; competition; benchmark
Introduction
============
The quest to understand the nature of human intelligence and engineer advanced forms of artificial intelligence (AI) are increasingly intertwined [@Hassabis17; @Kriegeskorte15; @Yamins16]. To explain human intelligence, we require computational models that can handle the complexity of real-world tasks. To engineer artificial intelligence, biological systems can provide inspiration and guidance of how to solve the task efficiently.
With this algorithmic exploration paradigm for explaining the brain, it is becoming essential to have standardized benchmarks for comparing how well different algorithms account for neural data. Open challenges are a particular form of standardized benchmark that foster fast-paced advance in a collaborative and transparent manner.
Open challenges have helped science to thrive in many times and fields. As early as 1900, Hilbert proposed 23 problems as challenges in mathematics to be solved. More recently, benchmarks for open competition have emerged in other disciplines such as robotics (e.g. the DARPA robotics challenge) and computer science on a diverse sets of topics including visual recognition [@Everingham15; @ILSVRC15; @PlacesChallenge]), reasoning [@CLEVR] and natural language understanding [@GLUE]. Those challenges are well accepted in the their scientific communities and suggest standardized benchmarks as fruitful platforms for collaboration.
Inspired by these approaches, we propose a challenge platform with standardized benchmarks for the artificial and biological sciences. At the core of the platform is an open competition with the goal of accounting for brain activity through computational models and algorithms. We coin the platform the Algonauts Project. Inspired by the astronauts (i.e. sailors of the stars) who launched into space to explore a new frontier, the algonauts (i.e. sailors of algorithms) set out to relate brains and computer algorithms in an exploratory way.
We believe that the Algonauts Project will facilitate the interaction between biological and artificial intelligence researchers, allowing the communities to exchange ideas and advance both fields rapidly and in a transparent way.
The 2019 Edition of the Algonauts Project: Explaining the Human Visual Brain
============================================================================
The 2019 edition is the first edition of the Algonauts Project’s challenge and workshop. It is titled “Explaining the Human Visual Brain”, and its specific target is to determine which computational model best accounts for human visual brain activity.
We focus on visual object recognition as it is an essential cognitive capacity of systems embedded in the real world. Visual object recognition has long fascinated neuroscientists and computer scientists alike, and it is here that the recent advances in AI and their adoption into neurosciences have taken place most prominently. Currently, particular deep neural networks trained with the engineering goal to recognize objects in images do best in accounting for brain activity during visual object recognition [@Schrimpf18; @Bashivan19]. However, a large portion of the signal measured in the brain remains unexplained. This is so because we do not have models that capture the mechanisms of the human brain well enough. Thus, what is needed are advances in computational modelling to better explain brain activity.
### Related challenges in neuroscience.
The 2019 edition “Explaining the Human Visual Brain” relates to initiatives such as the “The neural prediction challenge” (<http://neuralprediction.berkeley.edu/>) and “brain-score” (<http://www.brain-score.org/>) [@Schrimpf18] that provide benchmarks and leaderboards. The Algonauts Project emphasizes human brain data, and an automated submission procedure with immediate assessment. It couples neural prediction benchmarks to a challenge limited in time, and adds educational and collaborative components through the accompanying workshop.
Materials and Methods
=====================
The target of the 2019 challenge is to account for activity in the human visual brain responsible for object recognition. This is the so-called ventral visual stream [@Grill04], a hierarchically ordered set of brain regions in which neural activity unfolds across regions in space and time when human beings see an object. It starts with early visual cortex (EVC) and continues in inferior temporal (IT) cortex. Neurons in EVC respond preferentially to simple visual features such as oriented edges, whereas neurons in IT respond to more complex and larger features such as object parts. Consistent with their position in the processing hierarchy, neurons in EVC have been found to respond to visual stimulation earlier in time than neurons in IT. Stages of brain processing can thus be identified both in space (different regions) and in time (early and late). Correspondingly we have two challenge tracks.
[**Track 1**]{} aims to account for brain data in space, providing data from the start and later point of the ventral visual stream: early visual cortex (EVC) and inferior temporal cortex (IT), respectively (Fig. \[fig1\]a). We provide brain data measured with functional magnetic resonance imaging (fMRI[^1]), a technique with high spatial resolution (millimeters) that measures blood flow changes associated with neural activity.
[**Track 2**]{} aims to account for brain data in time, providing data recorded early and late in visual processing (Fig. \[fig1\]a). For this we provide brain data measured with magnetoencephalography (MEG) at time points identified to correspond to processing in EVC and IT. MEG is a technique with very high temporal resolution (milliseconds) that measures the magnetic fields accompanying electrical activity in the brain.
### Comparison metric from brain activity and models to challenge score.
Comparing human brains and models is challenging because of the numerous differences between them (e.g. in-silico vs. biological, number of units). Different approaches have been proposed [@Diedrichsen17; @Wu06], and here we make use of a technique called representational similarity analysis (RSA) [@Kriegeskorte08; @Kriegeskorte13]. RSA has low computational demands and is straightforward to implement. The idea behind RSA is that models and brains are similar if they treat the same images as similar (or equivalently dissimilar). RSA is a two-step procedure. In a first step (Fig. \[fig1\]b), we abstract from the incommensurate signal spaces into similarity space by calculating pairwise dissimilarities between signals for all conditions (images) and order them in so-called representational dissimilarity matrices (RDMs) indexed in rows and columns by the conditions compared. RDMs for the different signal spaces have the same dimensions and are directly comparable. We relate RDMs in a second step (Fig. \[fig1\]c) by calculating their similarity (Spearman’ R). Finally, we square the result to R$^2$ to indicate the amount of variance explained, and display results in the leaderboard (Fig. \[fig1\]d).
![*Procedure of the Algonauts 2019 edition challenge.* [**a)**]{} In two tracks, the goal is to account for human brain activity measured during object perception in space and time. [**b)**]{} RSA makes models and brain activity comparable, yielding [**c)**]{} percent variance explained relative to the noisiness of the data. [**d)**]{} Models are ranked in a leaderboard (i.e. for Track 2).[]{data-label="fig1"}](FigCNN_v2.pdf){width="\linewidth"}
### Noise ceiling.
The noise ceiling is the expected RDM correlation achieved by the (unknown) ideal model, given the noise in the data. The noise ceiling is computed by the assumption that the subject-averaged RDM is the best estimate of the ideal model RDM, i.e. by averaging the correlation of each subject’s RDM with the subject-averaged RDM. We use the noise ceiling to normalize R$^2$ values to noise-normalized variance explained. Thus, any model can explain from 0 to 100% of the explainable variance.
![*Training and Testing Material.* [**a)**]{} There are two sets of training data, each consisting of an image set and brain activity in RDM format (for fMRI and MEG). Training set 1 has 92 silhouette object images, and training set 2 has 118 object images with natural backgrounds. [**b)**]{} Testing data consists of 78 images of objects on natural backgrounds. Associated brain data is held back and used to evaluate models online for the leaderboard.[]{data-label="fig2"}](Fig2CCN.pdf){width="\linewidth"}
### Training Data.
Participants can submit their models out of the box to determine how well they predict brain activity in each track. We also provide training data that can help optimizing models for predicting brain data. We provide two sets of training data published previously (Fig. \[fig2\]a) [@Cichy14; @Cichy16]. Each set consists of a set of images (92 silhouette object images and 118 images of objects on natural background), and brain data recorded with fMRI (EVC and IT) and MEG (early and late in time) in response to viewing those images (by 15 participants). Participants differ across training sets but are the same across imaging modalities (MEG and fMRI).
### Testing Data and Procedure.
The testing set consists of 78 images and the respective brain activity recorded with fMRI and MEG (Fig. \[fig2\]b). Participants in the challenge receive only the images, and the brain data is held back. On the basis of the image test set participants calculate model RDMs as predictions of human brain activity. Participants submit the RDMs which are compared against the held-out brain data using RSA as described above. This results in a challenge score and determines the relative place in the leaderboard.
### Rules.
To encourage broad participation the challenge consists of a simple submission process. Participants can use any model trained on any type of data, however we explicitly forbid the use of human brain responses to the test image set. We request participants to submit a short report to a preprint server describing their final submitted model.
### Development Kit.
The development kit contains the aforementioned training and testing data. In addition, we provide example extraction code (matlab and python) to extract activation values from models into RDMs and evaluation code that compares model RDMs with brain RDMs, calculating the noise-normalized score for a model.
### Baseline Model.
Deep neural networks trained on object classification are currently the model class best performing in predicting visual brain activity. We used *AlexNet* [@Krizhevsky12] as an example often used in neuroscientific studies as baseline model. *AlexNet* is a feedforward deep neural network, trained on object categorization, with 5 convolutional and 3 fully connected layers. In Track 1 (fMRI), *AlexNet* accounts for 6.58% (layer 2) and 8.22% (layer 8) of noise-normalized variance in EVC and IT. In track 2 (MEG), it accounts for 5.82% (layer 2) and 22.93% (layer 4) noise-normalized variance in early and late visual processing.
Discussion
==========
### Challenges as scientific instruments in cognitive science.
Open challenges at the intersection of natural and artificial intelligence sciences hold promise for both sides. The natural intelligence sciences, in particular neuroscience and psychology, might benefit in two ways. For one, open challenges provide the incentive structure to promote and ensure transparency and openness. These are values recognized to promote replicability of results [@Nosek15; @Poldrack17]. Second, challenges provide a clear and quantitatively concise metric for success. They can thus play an important role in guiding research by differentiating between theories: predictive success is a necessary property of a good explanatory model [@Kriegeskorte15]. The sciences creating artificial intelligence in turn might benefit, too, in several ways. Biological systems can provide insight into how a cognitive problem might be solved mechanistically. More specifically, neuroscience can provide constraints on the infinite number of free parameters when engineering a model from scratch.
### Prediction vs. explanation.
Challenges like the Algonauts Project provide one measure of success: predictive power. Having an artifact that even perfectly predicts a phenomenon does not by itself explain the phenomenon. However, prediction and explanation are related goals [@Cichy19]. For one, successful explanations ultimately must also provide successful predictions [@Breiman01; @Yarkoni17]. Second, the ordering of models on a challenge benchmark can help scientist to concentrate future research efforts in creating explanations based on the most successful models. Further, bringing success rate in connection with the models’ properties can reveal what it is about those models that is responsible for the success. It can thus generate hypotheses and guide the next engineering steps.
### Limitations of the current approach.
Constitutive for a challenge are the choice of a particular data set and analysis steps. We readily assert that we could have structured the challenge differently (e.g. which data to provide, in which format, how to relate brain data and models). The choices we made were motivated by providing a low threshold to participation and a low computational load. Future challenges that make use of other data sets (e.g. large-scale) will invite a different type of data format and analytic treatment. We will invite an open discussion on those issues during the workshop.
### The future of the project.
We hope that the 2019 edition of the Algonauts Project will inspire other researchers to initiate open challenges and collaborate with the Algonauts Project. We see potential in tackling problems that become increasingly interesting to both natural and artificial intelligence communities. In the context of perception, future challenges might put the focus on action recognition or involve other sensory modalities such as audition or the tactile sense, or focus on other cognitive functions such as learning and memory.
Acknowledgments
===============
This research was funded by DFG (CI-241/1-1 CI-241/1-3) and an ERC grant (ERC-2018-StG 803370) to R.M.C; NSF award (1532591) in Neural and Cognitive Systems and the Vannevar Bush Faculty Fellowship program funded by the ONR (N00014-16-1-3116) to A.O. We thank our sponsors: the MIT Quest for Intelligence and the MIT-IBM Watson AI Lab.
[^1]: For more details on MEG and fMRI see <http://algonauts.csail.mit.edu/fmri_and_meg.html>
|
INJE-TP-02-06\
hep-th/0210300
[**Holography and Entropy Bounds in Gauss-Bonnet Gravity**]{}
[Rong-Gen Cai[^1]$^1$ and Yun Soo Myung[^2]$^2$]{}\
[**[Abstract]{}**]{}
We discuss the holography and entropy bounds in Gauss-Bonnet gravity theory. By applying a Geroch process to an arbitrary spherically symmetric black hole, we show that the Bekenstein entropy bound always keeps its form as $S_{\rm B}=2\pi E R$, independent of gravity theories. As a result, the Bekenstein-Verlinde bound also remains unchanged. Along the Verlinde’s approach, we obtain the Bekenstein-Hawking bound and Hubble bound, which are different from those in Einstein gravity. Furthermore, we note that when $HR=1$, the three cosmological entropy bounds become identical as in the case of Einstein gravity. But, the Friedmann equation in Gauss-Bonnet gravity can no longer be cast to the form of cosmological Cardy formula.
Introduction
============
According to the holographic principle [@HP], within a given volume $V$ the number of degrees of freedom is bounded by a quantity proportional to the surface area $A$ of the volume. This is obtained from the idea that the maximal entropy inside the volume is given by the largest black hole that just fits inside the volume, while the entropy of the latter obeys the Bekenstein-Hawking entropy formula $A/4G$, where $G$ is the Newton constant. Thus, the holographic principle gives an entropy bound on matter inside the volume $$\label{1eq1}
S \le \frac{A}{4G},$$ which is called the holographic bound.
Fischler and Susskind [@FS] were the first to consider entropy bound in the cosmological setting. In a closed universe, the holographic bound in its naive form (\[1eq1\]) is not applicable because there is no boundary in the closed universe. On the other hand, the argument leading to (\[1eq1\]) assumes that it is possible to form a black hole filling the whole volume. This is no longer valid in the universe since the expansion rate $H$ of the universe and the total energy in the universe restrict the maximal size of black hole [@Verl]. Following Fischler and Susskind, it was argued that the maximal entropy inside the universe is produced by black holes with size of Hubble horizon [@Hubb]. The usual holographic arguments lead to the result that the total entropy should be less than or equal to the Bekenstein-Hawking entropy of a Hubble horizon-sized black hole times the number of Hubble regions in the universe. That is, one has $S \le \beta HV/G$, where $V$ represents the volume of the universe and $\beta$ is a pure coefficient. This coefficient is fixed by Verlinde [@Verl] by using a local version of holographic bound [@FS; @Bous]. This bound is called the Hubble entropy bound, which has the form $$\label{1eq2}
S_{\rm H}= (n-1) \frac{HV}{4G},$$ where $n$ stands for spatial dimensions of the universe. The Hubble bound is valid for a strongly self-gravitating universe ($HR\ge 1$). Except for the Hubble bound, Verlinde introduced other two entropy bounds [@Verl]: $$\begin{aligned}
\label{1eq3}
{\rm Bekenstein-Verlinde\ bound}:&& S_{\rm BV}=\frac{2\pi}{n}ER
\nonumber \\
{\rm Bekenstein-Hawking\ bound}:&& S_{\rm BH}=(n-1)\frac{V}{4G_{n+1}R}.
\end{aligned}$$ Here $E$ is the total energy of the matter filling the universe and $R$ is the scale factor. The Bekenstein-Verlinde bound $S_{\rm BV}$ is the counterpart of the Bekenstein entropy bound [@Beke] in the cosmological setting [@CMO], which is believed to hold for a weakly self-gravitating universe ($HR \le 1$). The Bekenstein-Hawking entropy bound does not serve as an entropy bound, but acts as a criterion whether the universe is in a weakly self-gravitating phase ($HR \le 1$) or in a strongly self-gravitating phase ($HR \ge 1$) [@Verl]. The Friedmann equation of a ($n+1$)-dimensional, closed Friedmann-Robertson-Walker (FRW) universe is $$\label{1eq4}
H^2 = \frac{16\pi G}{n(n-1)} \frac{E}{V} -\frac{1}{R^2},$$ from which one can see that $S_{\rm BV} \le S_{\rm BH}$ for $HR \le 1$, while $S_{\rm BV} \ge S_{\rm BH}$ for $HR \ge 1$. Clearly one has $S_{\rm BV}=S_{\rm BH}
=S_{\rm H}$ at the critical point $HR=1$. Furthermore, Verlinde found that with the three cosmological entropy bounds, the Friedmann equation (\[1eq4\]) can be cast to $$\label{1eq5}
S_{\rm H}=\sqrt{S_{\rm BH} (2 S_{\rm BV}-S_{\rm BH})},$$ the cosmological Cardy formula. This formula (\[1eq5\]) has a close relation to the Cardy-Verlinde formula describing the entropy of conformal field theories. For more discussions, see [@Verl].
We note that those discussions on the entropy bounds crucially depend on the area entropy formula of black holes (\[1eq1\]). However, it is well-known that the area entropy formula of black holes holds only in Einstein gravity. If some higher derivative curvature terms appear, for example, one has to include some additional terms to the area entropy formula of black holes [@TM]. Therefore it would be interesting to see how those entropy bounds get modified in higher derivative gravity theories. In this note we will discuss entropy bounds in the Gauss-Bonnet gravity, which belongs to a special class of higher derivative gravity theories in the sense that the equation of motion for the Gauss-Bonnet gravity contains no more than second derivatives of metric.
Bekenstein bound and Bekenstein-Verlinde bound
==============================================
Bekenstein was the first to consider the issue of maximal entropy for a macroscopic system. He argued that for a closed system with total energy $E$, which fits in a sphere with radius $R$ in three spatial dimensions, there exists an upper bound on the entropy of the system $$\label{2eq1}
S \le S_{\rm B} =2 \pi ER,$$ which is called the Bekenstein entropy bound. This bound is believed to be valid for a system with limited self-gravity, which means that the gravitational self-energy is negligibly small compared to its total energy. However, it is interesting to note that this bound gets saturated for a ($3+1$) dimensional Schwarzschild black hole, which of course is a strongly self-gravitating object. Furthermore it was found that the form (\[2eq1\]) is independent of spatial dimensionality. That is, the form (\[2eq1\]) keeps unchanged for any dimensional object. This is obtained by considering a Geroch process in an arbitrary dimensional Schwarzschild black hole and the generalized second law of black hole thermodynamics [@Bousso]. It is easy to show that for a higher ($n+1>4$) dimensional Schwarzschild black hole, the Bekenstein entropy bound still holds, but it is not saturated.
In deriving the Bekenstein entropy bound [@Beke; @Bousso], black hole thermodynamics is used. And the thermodynamics of black holes is dependent of gravity theories under consideration. In fact, we show here that the Bekenstein entropy bound is independent of gravity theories. As a result, the Bekenstein-Verlinde bound has also the same feature of independence of gravity theories. Consider an arbitrary, ($n+1$)-dimensional spherically symmetric black hole solution $$\label{2eq2}
ds^2 =- e^{2\delta (r)}\left (1-\frac{2m(r)}{r^{n-2}}\right) dt^2
+\left( 1-\frac{2 m(r)}{r^{n-2}}\right)^{-1}dr^2 +r^2d\Omega_{n-1}^2,$$ where $\delta $ and $m$ are two continuous functions of $r$, It is assumed that $e^{2\delta (r)} \ne 0$ in the whole spacetime. The black hole horizon $r_+$ is determined by equation $1 -2m(r_+)/r^{n-2}_+=0$. The Hawking temperature $T$ associated with the horizon is $$\label{2eq3}
T=\frac{e^{\delta(r_+)}}{4\pi}\left(\frac{n-2}{r_+}-
\frac{2m'(r_+)}{r_+^{n-2}}\right),$$ where a prime denotes derivative with respect to $r$. We denote by $M$ the mass of the black hole. According to the first law of black hole thermodynamics, which always holds because a black hole behaves as a thermodynamic system, one has the entropy variation $\triangle S$ when the mass gets increase by a small amount $\triangle M$, $$\label{2eq4}
\triangle S = T^{-1}\triangle M.$$ Let us consider a Geroch process in the black hole background (\[2eq2\]). Suppose that one has a thermodynamic system with energy $E$ and $R$ being the radius of the smallest ($n-1$)-sphere circumscribing the system. Now move this system from infinity to a place just outside the horizon of the black hole (\[2eq2\]), and drop the matter into the black hole.
The mass added to the black hole is given by the energy $E$ of the system, which gets redshifted according to the position of the center of mass at the drop-off point, at which the circumscribing sphere almost touches the horizon. The center of mass can be brought to within a proper distance $R$ from the horizon, while all parts of the system still remain outside the horizon. Thus one needs to calculate the redshift factor at a proper distance $R$ from the horizon [@Bousso].
Let $x$ be the radial coordinate distance from the horizon $x=r-r_+$. The redshift factor near the horizon is given by $$\label{2eq5}
\chi ^2(x)= e^{2\delta(r_+)}\left( \frac{n-2}{r_+}-\frac{2m'(r_+)}{r_+^{n-2}}
\right) x$$ up to the leading order of $x$. Near the horizon, the proper distance $R$ has a relation to the coordinate distance $x$, $$R=2 \sqrt{\frac{x}{(n-2)/r_+ -2m'(r_+)/r_+^{n-2}}}.$$ Hence the absorbed mass is $\triangle M = E\chi(x)$. Substituting this into (\[2eq4\]), we find that the increased entropy of the black hole is $$\label{2eq7}
\triangle S = T^{-1} E \chi (x) = 2\pi ER.$$ According to the generalized second law of black hole thermodynamics [@Beke2], which says that the total entropy of black hole and matter outside the black hole never decreases in any physical process, we can immediately obtain the maximal entropy of the system, $$\label{2eq8}
S_m \le 2\pi ER.$$ This is just the Bekenstein entropy bound (\[2eq1\]). From the above one can see that we have neither specified what the black hole solution (\[2eq2\]) is, nor in which gravity theory it is. Hence the resulting conclusion (\[2eq8\]) is independent of gravity theories. This is an expected result since as stated above, the Bekenstein bound is valid only for systems with limited self-gravity, which implies that gravity effect is negligible. In addition, the Bekenstein-Verlinde bound $S_{\rm BV}$ in (\[1eq3\]) is the counterpart of the Bekenstein bound in the cosmological setting. Therefore we conclude that the Bekenstein-Verlinde bound is also independent of gravity theories.
Hubble bound and Bekenstein-Hawking bound in Gauss-Bonnet gravity
=================================================================
Now we consider the so-called Gauss-Bonnet gravity theory by adding the Gauss-Bonnet term to the Einstein-Hilbert action, $$\label{3eq1}
{\cal S} =\frac{1}{16\pi G}\int d^{n+1}x\sqrt{-g} \left ({\cal R}
+\alpha ({\cal R}_{\mu\nu\gamma\sigma}
{\cal R}^{\mu\nu\gamma\sigma} -4 {\cal R}_{\mu\nu}{\cal R}^{\mu\nu} +{\cal R}^2)\right ),$$ where $\alpha $ is a constant. Here we exclude the case of $n=3$ since in that case the Gauss-Bonnet term is a topological term. The static spherically symmetric black hole solutions in (\[3eq1\]) have been found in [@Deser; @Whee]. The entropy of the black holes has the expression [@Myers; @Cai] $$\label{3eq2}
S =\frac{A}{4G}\left ( 1 +\frac{n-1}{n-3}\frac{2\tilde \alpha }{R^2}\right),$$ where $\tilde \alpha =(n-2)(n-3)\alpha $, $A$ represents the horizon area of the black hole and $R$ the horizon radius. Following Verlinde [@Verl], in this section we “derive" the Hubble bound for a closed FRW universe in the Gauss-Bonnet theory.
In [@Verl] Verlinde used a version of holographic bound proposed by Fischler and Susskind [@FS] and subsequently developed by Bousso [@Bous], which gives a restriction of entropy flow $S$ through a contracting light sheet: the entropy flow $S$ is less than or equal to $A/4G$, where $A$ is the area of the surface from which the light sheet originates. The infinitesimal version of the holographic bound plays a crucial role in the “derivation" by Verlinde. According to the infinitesimal version, for every ($n-1)$-dimensional surface at time $t +dt$ with area $A+dA$, one has $dS \le dA/4G$. Here $dS$ represents the entropy flow through the infinitesimal light sheets originating at the surface at $t +dt$ and extending back to time $t$, and $dA$ denotes the increase in area between $t$ and $t +dt$. Obviously the holographic bound is based on the area entropy formula of black holes. In our case, the black hole entropy is given by (\[3eq2\]). The infinitesimal version is then changed to $$\label{3eq3}
dS \le \frac{1}{4G}d\left[ A \left( 1 +\frac{n-1}{n-3}\frac{2\tilde \alpha }{R^2}\right)
\right].$$ In a ($n+1$) dimensional closed FRW universe, for a surface which is fixed in comoving coordinates, the area $A$ changes as a result of the expansion of the universe by an amount $$\label{3eq4}
dA =(n-1)HA dt,$$ where the relation $A \sim R^{n-1}$ has been used. Choose one of two past light sheets that originate at the surface: the inward or the outward going. The entropy flow through this light sheet between $t$ and $t +dt$ is given by the entropy density $s=S/V$ times the infinitesimal volume $A dt$ swept out by the light sheet. That is, one has $$\label{3eq5}
dS =\frac{S}{V} A dt.$$ Applying $A \sim R^{n-1}$ to (\[3eq3\]), and then substituting (\[3eq3\]) and (\[3eq4\]) into (\[3eq5\]), we obtain $$\label{3eq6}
S \le S_{\rm H} =(n-1) \frac{HV}{4G}\left (1+\frac{2\tilde \alpha}{R^2}\right),$$ which is the Hubble entropy bound in the Gauss-Bonnet gravity. This is the main result of ours. If a cosmological constant is added to the action (\[3eq1\]), the entropy of black holes still has the expression (\[3eq2\]) [@Cai]. Hence the Hubble entropy bound still takes the form (\[3eq6\]) even if a cosmological constant is present in the Gauss-Bonnet gravity.
On the other hand, the Friedmann equation for the Gauss-Bonnet gravity (\[3eq1\]) is $$\label{3eq7}
H^2 +\frac{1}{R^2} +\tilde \alpha \left (H^2 +\frac{1}{R^2}\right)^2
= \frac{16\pi G}{n(n-1)}\frac{E}{V},$$ from which we see that when $HR=1$, the Bekenstein-Verlinde bound $S_{\rm BV}=
2\pi E R/n$ equals the Hubble bound $S_{\rm H}$ given by (\[3eq6\]). This is a good check for our “derivation" of the Hubble bound. Furthermore, from the Friedmann equation (\[3eq7\]), we find that the Bekenstein-Hawking bound has the form $$\label{3eq8}
S_{\rm BH} =(n-1)\frac{V}{4GR}\left( 1 +\frac{2\tilde \alpha}{R^2}\right).$$ As in the case of Einstein theory [@Verl], this Bekenstein-Hawking bound (\[3eq8\]) was obtained by identifying $S_{\rm BH}=S_{\rm BV}$ via the Friedmann equation (\[3eq7\]) at the critical point $HR=1$. Hence at the critical point the property that three cosmological entropy bounds become identical in Einstein theory persists in the Gauss-Bonnet gravity. Inspecting the Friedmann equation (\[3eq8\]), however, we find that it can no longer be rewritten in the form (\[1eq5\]), which might be related to that the black hole entropy in higher derivative theories cannot be cast to the Cardy-Verlinde formula [@Cai2].
Conclusion
==========
In summary this paper has initiated the study of holography in gravity theories with higher derivative curvature terms. As a concrete model, we have considered the Gauss-Bonnet theory. We have shown that as expected, the Bekenstein bound and the Bekenstein-Verlinde bound keep the same forms as in Einstein theory, while the Hubble bound and Bekenstein-Hawking bound get modified. Along the Verlinde’s approach, we have obtained expressions of the Hubble bound and Bekenstein-Hawking bound. When the universe undergoes a transition from a weakly self-gravitating phase ($HR \le 1$) to a strongly self-gravitating phase ($HR\ge 1$), the three cosmological entropy bounds get matched at the critical point $HR=1$, as in the case of Einstein gravity. However, the Friedmann equation of the Gauss-Bonnet gravity cannot be rewritten in the form of the cosmological Cardy formula.
Acknowledgment {#acknowledgment .unnumbered}
==============
We thank H.W. Lee for useful discussions. The work of R.G.C. was supported in part by a grant from Chinese Academy of Sciences and a grant from Ministry of Education, PRC. Y.S.M. acknowledges partial support from the KOSEF, Project Number: R02-2002-000-00028-0. R.G.C. is grateful to Relativity Research Center and School of Computer Aided Science, Inje University for warm hospitality during his visit.
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[^1]: e-mail address: [email protected]
[^2]: e-mail address: [email protected]
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