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[
"Mammogram Edge Detection Using Hybrid Soft Computing Methods",
"Mammogram Edge Detection Using Hybrid Soft Computing Methods"
]
| [
"I Laurence Aroquiaraj ",
"K Thangavel "
]
| []
| []
| Image segmentation is a crucial step in a wide range of method image processing systems. It is useful in visualization of the different objects present in the image. In spite of the several methods available in the literature, image segmentation still a challenging problem in most of image processing applications. The challenge comes from the fuzziness of image objects and the overlapping of the different regions. Detection of edges in an image is a very important step towards understanding image features. There are large numbers of edge detection operators available, each designed to be sensitive to certain types of edges. The Quality of edge detection can be measured from several criteria objectively. Some criteria are proposed in terms of mathematical measurement, some of them are based on application and implementation requirements. Since edges often occur at image locations representing object boundaries, edge detection is extensively used in image segmentation when images are divided into areas corresponding to different objects. This can be used specifically for enhancing the tumor area in mammographic images. Different methods are available for edge detection like Roberts, Sobel, Prewitt, Canny, Log edge operators. In this paper a novel algorithms for edge detection has been proposed for mammographic images. Breast boundary, pectoral region and tumor location can be seen clearly by using this method. For comparison purpose Roberts, Sobel, Prewitt, Canny, Log edge operators are used and their results are displayed. Experimental results demonstrate the effectiveness of the proposed approach. | null | [
"https://arxiv.org/pdf/1307.4516v1.pdf"
]
| 14,021,910 | 1307.4516 | a58a8649b3f611d91d64612fe4e3a3302baad691 |
Mammogram Edge Detection Using Hybrid Soft Computing Methods
I Laurence Aroquiaraj
K Thangavel
Mammogram Edge Detection Using Hybrid Soft Computing Methods
Index Terms-Fuzzy LogicFuzzy Relative PixelStandard Deviation with GradientDigital Mammogram
Image segmentation is a crucial step in a wide range of method image processing systems. It is useful in visualization of the different objects present in the image. In spite of the several methods available in the literature, image segmentation still a challenging problem in most of image processing applications. The challenge comes from the fuzziness of image objects and the overlapping of the different regions. Detection of edges in an image is a very important step towards understanding image features. There are large numbers of edge detection operators available, each designed to be sensitive to certain types of edges. The Quality of edge detection can be measured from several criteria objectively. Some criteria are proposed in terms of mathematical measurement, some of them are based on application and implementation requirements. Since edges often occur at image locations representing object boundaries, edge detection is extensively used in image segmentation when images are divided into areas corresponding to different objects. This can be used specifically for enhancing the tumor area in mammographic images. Different methods are available for edge detection like Roberts, Sobel, Prewitt, Canny, Log edge operators. In this paper a novel algorithms for edge detection has been proposed for mammographic images. Breast boundary, pectoral region and tumor location can be seen clearly by using this method. For comparison purpose Roberts, Sobel, Prewitt, Canny, Log edge operators are used and their results are displayed. Experimental results demonstrate the effectiveness of the proposed approach.
INTRODUCTION
reast cancer has been one of the major causes of death among w omen since the last decades and it has become an emergency for the healthcare systems of industriali zed countries. This disease became a commonest cancer among w omen. If the cancer can be detected early, the options of treatment and the chances of total recovery w ill increase. Intra-operative diagnosis of the disease has steadily become more important w ith respect to the recent introduction of sentinel lymph node biopsy. Image segmentation is referred to as the procedure in w hich the input image is divided into meaningful regions in such a w ay that the output image w ill consist of a set of labeled region describing the input image.
The potency of digital mammography for detecting of breast cancer is currently under Investigation. In medical science, mammography image is to be a cornerstone for examining breast cancer in human. Even if image output obtained in scanning process w ith X ray is frequent blueprint and unclearness for edge of image. Screen-film mammography has limited detection ability for low contrast lesions in dense breasts. This limitation poses a problem for the estimated 40% of w omen w ith dense breast w ho undergo mammography [1]. Thus, to be required a process for improving the contrast of mammography images w as a digital image processing. The researchers have been done several methods for improving the capability of mammography image to detect the abnormality. Enhancement of contrast of the mammograms has been done by the researchers in the past. Edge detection is a critical element in image processing, since edges contain a major function of image information. The function of edge detection is to identify the boundaries of homogeneous regions in an image based on properties such as intensity and texture. M any edge detection algorithms have been developed based on computation of the intensity gradient vector, w hich, in general, is sensitive to noise in the image.
The rest of this paper is organized as follow s. Section (2) review s edge detection operators. Section (3) classifies the edge detection algorithms. Section (4) discusses proposed edge detection algorithm. Section, (5) the experimental result of many edge detections obtained and finally in section (7) the discussion and conclusion are given.
Review of Previous Work
In the past tw o decades several algorithms w ere developed to extract the contour of homogeneous regions w ithin digital mammogram image. A lot of the attention is focused to edge detection, being a crucial part in most of the algorithms.
Classically, the first stage of edge detection (e.g. the gradient operator, Robert operator, the Sobel operator, the Prew itt operator) is the evaluation of derivatives of the image intensity. Smoothing filter and surface fitting are used as regularization techniques to make differentiation more immune to noise. Raman M aini and J. S. Sobel [2] evaluated the performance of the Prew itt edge detector for noisy image and demonstrated that the Prew itt edge detector w orks quite w ell for digital image corrupted w ith Poisson noise w hereas its performance decreases sharply for other kind of noise. Ferrari et al. (2004) [3] has proposed a new method using Gabor w avelets for the identification of the pectoral muscle in medio-lateral oblique (M LO) mammograms based upon a multiresolution technique. The magnitude value of each pixel w as propagated in the direction of the phase after computing the magnitude and phase images using a vector-summation procedure. The resulting image w as then used to detect the relevant edges and true pectoral muscle edge. H ow ever, in many cases, cancer is not easily detected by the eyes.
Bellotti et al. [4] characterized ROI by means of textural features computed from the gray level co-occurrence matrix (GLCM ), also know n as spatial gray level dependence (SGLD) matrix. Varela et al. [5] used features based on the iris filter output, toget her w ith gray level, texture, contour-rel ated and morphological features.Yuan et al . [6] used three groups of features in thei r study. The first group incl uded features characterizing spiculation, margin, sh ape and contrast of the lesion. Sahiner et al. [7] developed an algorithm for extracting spiculation feature and circumsribed margin feature. Both features had high accuracy for characterizing mass margins according to BI-RADS descriptors.
Timp and Karssemei jer [8] proposed temporal feature set consisted of complete set of single view features together w ith temporal features. Timp et al . [9] designed two kinds of temporal features: differ ence features and similarity features. Difference features measured changes in feature values between corresponding regions in the prior and the current view . Similarity features measured w hether two regions ar e comparable in appearance. Fauci et al . [10] extracted 12 features from segmented masses. Some features gave the geomet rical information, others provided sh ape parameters. The criterion for feature selection w as based on morphological differ ences betw een pathological and healthy regions. Rangayyan et al. [11] proposed methods to obtain shape features from the turning angle functions of contours. Features are useful in the analysis of contours of breast masses and tumors because of their ability to capture diagnostically important details of shape related to spicules and lobulations.
Davis, L. S. [12] has suggested Gaussian pre convolution for this purpose. H ow ever, all the Gaussian and Gaussian-like smoothing filters, w hile smoothing out the noise, also remove genuine high frequency edge features, degrade localization and degrade the detection of low -contrast edges.
Zhao Yu-qian et al. [13] proposed a novel mathematic morphological algorithm to detect lungs CT medical image edge. They show ed that this algorithm is more efficient for medical image denoising and edge detecting than the usually used template-based edge detection algorithms such as Laplacian of Gaussian operator and Sobel edge detector, and general morphological edge detection algorithm such as morphological gradient operation and dilation residue edge detector. Fesharaki, M .N .and H ellestrand, G.R [14] presented a new edge detection algorithm based on a statistical approach using the student t-test. They selected a 5x5 w indow and partitioned into eight different orientations in order to detect edges.
H ow ever there is no such loss in the fuzzy based method described here. Research has clearly demonstrated that methods involving Gaussian filtering suffer from problems such an edge displacement, vanishing edges and false edges [15]. A nother problem faced by few methods like the anisotropic diffusion lies in obtaining the locations of semantically meaningful edges at coarse scales generated by convoluting images w ith Gaussian kernels [16]. M ethods that involve simple scan line approach are not able to detect all the edges due to limitation of the methodology to trace only the horizontal and vertical neighbours [17] of a point. Fuzzy logic is a pow erful problem-solving methodology w ith a myriad of applications in embedded control and information processing [18]. Fuzzy provides a remarkably simple w ay to draw definite conclusions from vague, ambiguous or imprecise information. In a sense, fuzzy logic resembles human decision making w ith its ability to w ork from approximate data and find precise solutions.
The fuzzy relative pixel value algorithm has been developed w ith the know ledge of vision analysis w ith low or no illumination [19], thus making this method optimized for application requiring such methods. The method helps us to detect edges in an image in all cases due to subjection of pixel values to an algorithm involving host of fuzzy conditions for edges associated w ith an image. The purpose of this paper is to present a new methodology for image edge detection w hich is undoubtedly one of the most important operations related to low level computer vision, in particular w ithin area of feature extraction w ith plethora of techniques, each based on a new methodology, having been published. The method described here uses a fuzzy based logic model w ith the help of w hich high performance is achieved along w ith simplicity in resulting model [20]. Fuzzy logic helps to deal w ith problems w ith imprecise and vague information and thus helps to create a model for image edge detection as presented here [21] displaying the accuracy of fuzzy methods in digital image processing [22].
Edge Detection Techniques 3.1 Robert Edge Detector
The calculation of the gradient magnitude of an image is obtained by the partial derivatives G x and G y at every pixel location. The simplest w ay to implement the first order partial derivative is by using the Roberts cross gradient operator. Therefore
) 1 , 1 ( ) , ( j i f j i f G x (1) ) 1 , ( ) , 1 ( j i f j i f G y (2)
Prewitt Edge detector
The Prew itt edge detector is a much better operator than Roberts's operator. This operator having a 3 x 3 masks deals better w ith the effect of noise. A n approach using the masks of size 3 x 3 is given below , the arrangement of pixels about the pixels[i, j]. The partial derivatives of the Prew itt operator are calculated as
) 2 1 0 ( ) 4 5 6 ( a ca a a ca a G x (3) ) 6 7 0 ( ) 4 3 2 ( a ca a a ca a G y (4)
The constant c implies the emphasis given to pixels closer to the centre of the mask. G x and G y are the approximation at [i, j]. Setting c=1, the Prew itt operator is obtained. Therefore the Prew itt masks are as follow s these masks have longer support. They differentiate in one direction and average in the other direction, so the edge detector is less vulnerable to noise.
Sobel Edge Detector
The Sobel edge detector is very much similar to the Prew itt edge detector. The difference betw een the both is that the w eight of the centre coefficient is 2 in the Sobel operator. The partial derivatives of the Sobel operator are calculated as
) 2 1 2 0 ( ) 4 5 2 6 ( a a a a a a G x (5) ) 6 7 2 0 ( ) 4 3 2 2 ( a a a a a a G y (6)
A lthough the Prew itt masks are easier to implement than Sobel masks, the later has better noise suppression characteristics.
Laplacian of Gradient (LoG) Operator Edge Detector
The Laplacian of an image ) ,
( y x f is a second order derivative defined as:
2 2 2 2 2 y f x f f (7)
The Laplacian of Gradient (LoG) is usually used to establish w hether a pixel is on the dark or light side of an edge.
Canny Edge Detector
Canny technique is very important method to find edges by isolating noise from the image before find edges of image, w ithout affecting the features of the edges in the image and then applying the tendency to find the edges and the critical
) , ( c r f ) 6 , , ( * ) , ( ) , ( c r G c r f c r f (8)
Step 2: A pply first difference gradient operator to compute edge strength then edge magnitude and direction are obtain as before.
Step 3: A pply non-maximal or critical suppression to the gradient magnitude.
Step 4: A pply threshold to the non-maximal suppression image.
The Proposed Edge Detection algorithms
Fuzzy Canny Edge Detector
The hybrid of Fuzzy and Canny edge detection technique is very important method to find edges by isolating noise from the image before find edges of image, w ithout affecting the features of the edges in the image and then applying the tendency to find the edges and the critical value for threshold. The algorithmic steps for canny edge detection technique are Fuzzy provides a remarkably simple w ay to draw definite conclusions from vague, ambiguous or imprecise information. In a sense, fuzzy logic resembles human decision making w ith its ability to w ork from approximate data and find precise solutions. , ( c r f
Step 3: A pply first difference gradient operator to compute edge strength then edge magnitude and direction are obtain as before.
Step 4: A pply non-maximal or critical suppression to the gradient magnitude.
Step 5: A pply threshold to the non-maximal suppression image.
Fuzzy Relative Pixel Edge Detector
The A lgorithm begins w ith reading an M xN image. The first set of nine pixels of a 3x3 w i ndow is chosen w ith central pixel having values (2,2). A fter the initialization, the pixel values are subjected to the fuzzy conditions for edge existence show n in Fig.2.(a-i). A fter the subjection of the pixel values to the fuzzy conditions the algorithm generates an intermediate image. It is checked w hether all pixels have been checked or now , if not then first the horizontal coordinate pixels are checked. If all horizontal pixels have been checked the vertical pixels are checked else the horizontal pixel is incremented to retrieve the next set of pixels of a w indow . In this manner the w indow shifts and checks all the pixels in one horizontal line then increments to check the next vertical location.
A fter edge highlighting image is subjected to another set of condition w ith the help of w hich the unw anted parts of the output image of type show n in Fig.2.(a-b) are removed to generate an image w hich contains only the edges associated w ith the input image. Let us now consider the case of the fuzzy condition displayed in Fig.1. (g). for an input image A and an output image B of size M xN pixels respectively w e have the follow ing set of conditions that are implemented to detect the edges pixel values.
Fuzzy Edge Detection Based on Pixel's Gradient and Standard Deviation Values (SDGD Edge Detector)
The gradient and standard deviation of pixels value, edges are separately extracted and then based on fuzzy logic, final decision about w hether each pixel is edge or not is made.
Problematic results could be gained if each of the methods be used solely. It may causes on identifying of edge pixels as non-edge pixels and vice versa. , ( c r f gray level standard deviation to compute over adjacent neighbourhood pixels.
Step 3: Similarly pixels with standard deviation values is greater than a threshold value are edge.
Step 4: A pply threshold to the non-maximal suppression image using 3x3 w indow size.
Results and Discussion
Obtaining real mammogram images (322 images) for carrying out research is highly difficult due to privacy issues, legal issues and technical hurdles. H ence the M ammography Image A nalysis Society (M IAS) database (ftp:/ / peipa.essex.ac.uk) is used in this paper to study the efficiency of the proposed image segmentation and evaluated using mammography images.
The proposed system w as tested w ith different mammogram images, its performance being compared to that of the other edge detection operators.
The objective methods used to measure the performance of edge detectors using signal to noise ratio and mean square error between the edge detectors images and the original one. The objective methods borrow ed from digital signal processing and information theory, and provide us with equations that can be used to measure the amount of error in a processed image by comparison to know n image. Although the objective methods are w idely used, are not necessarily correlated w ith our perception of image quality. For instance, an image w ith a low error as determined by an objective measure may actually look much w orse than an image with a high error metric. Table 1 explains the objective measures are the root-mean-square error, e RM S , the root-meansquare signal-to-noise ratio, SN R RM S , and the peak signal-to-noise ratio, SN R PEA K and Contrast Improved Index (CII).
pixels.
If the value of e RM S is low and the values of SN R RM S and SN R PEA K are larger than the enhancement approach is better. Fig. 3 Different Edge Detection M ethods and Proposed Fuzzy based Edge Detection M ethods using mdb002, mdb067, mdb171, mdb240 and mdb320 mammogram images.
It is observed from the Table 2 that according to edge detectors provide the performance rates for overall 322 mammogram images are equal except values of M SE, e RM S, SN R RM S and SN R PEA K . The proposed Fuzzy based Edge detectors provides the best performance rate. Fig. 4 show s the performance analysis of different edge detectors.
It is observed from the Table 3 that according to edge detectors provide the percentage of total number of w hite pixels in selected mammogram images for different edge detectors. The proposed Fuzzy based Edge detectors provides the best percentage of total number of w hite pixels. Fig. 5 show s the performance analysis of different edge detectors.
CONCLUSION
In this paper, better algorithm has been proposed to improve the detection of edges by using fuzzy rules. This algorithm is adaptable to various environments. The w eights associated w ith each fuzzy rule w ere tuned to allow good results to be obtained w hile extracting edges of the image, w here contrast varies a lot from one region to another. During the performance tests, how ever, all parameters w ere kept constant. Experimental results show the higher quality and superiority of the extracted edges compared to the other methods in the literature such as Sobel, Robert, and Prew itt. To achieve good result, some parameters and thresholds are experimentally set. Improving fuzzy system performance by the w ays such as using different kind of input, rough fuzzy hybridization techniques also need to be investigated in future w orks.
A lgorithm: Fuzzy based Gradient and Standard Deviation Value Edge Detection Step 1: Convolve image
Fig
Fig 1(a-i). Fuzzy conditions have been displayed
value for threshold. The algorithmic steps for canny edge detection technique are follow s: A lgorithm: Canny Edge Detection Step1: Convolve image)
,
( c
r
f
w ith a Gaussian function
to get smooth image
TABLE 1 OBJECTIVE
1METHOD MEASURES
TABLE 2 PERFORMANCE
2RATES OF EDGE DETECTORS FOR 322 MAMMOGRAM IMAGES Fig. 4 Performance analysis of different Edge Detectors
TABLE 3 PERCENTAGE
3OF WHITE PIXELS VALUES FOR EDGE DETECTORS Fig. 5 Performance analysis of different Edge Detectors using Percentage of White Pixels
ACKNOWLEDGEMENTThe first author of this paper acknowledges UGC-Minor Research Project (No.F.41-1361/2012(SR)) for the financial support.[22] Chen Y., 2002.A region-based fuzzy feature matching approach to content based image retrieval. In proceedings of IEEE Transactions on pattern analysis and machine intelligence, VOL 24, N O. 9. and mdb320 mammogram imagesEdge
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| []
|
[
"Angular Quantization in CFT",
"Angular Quantization in CFT"
]
| [
"Nicholas Agia \nCenter for the Fundamental Laws of Nature\nHarvard University\nCambridgeMAUSA\n",
"Daniel L Jafferis \nCenter for the Fundamental Laws of Nature\nHarvard University\nCambridgeMAUSA\n"
]
| [
"Center for the Fundamental Laws of Nature\nHarvard University\nCambridgeMAUSA",
"Center for the Fundamental Laws of Nature\nHarvard University\nCambridgeMAUSA"
]
| []
| The most common quantization scheme in which to study a conformal field theory is radial quantization, wherein a Hilbert space of states is defined on a sphere, whose Hamiltonian when mapped to the plane is the dilatation generator and which boasts a state/operator correspondence. In this paper, we consider an alternative quantization scheme for 2d CFTs in which the plane is foliated by constant-angle slices, as opposed to concentric circles, whose Hamiltonian is the rotation generator. In this angular quantization, there is no state/operator correspondence but instead an "asymptotics/operator correspondence". A central feature is that the quantization slice ends on two operators, and a regulator must be chosen by excising holes around each operator and imposing suitable boundary conditions such that the holes shrink to the desired local operators. This angular quantization may be viewed as constructing CFTs in Minkowski space, or separately as studying non-conformal (but local) boundary conditions in CFTs. We provide explicit Fock space constructions for various free 2d CFTs. In addition to the motivation of applying angular quantization in string theory situations where traditional radial quantization is insufficient, we comment on its relation to modular Hamiltonian approaches to entanglement entropy. | null | [
"https://arxiv.org/pdf/2204.11872v1.pdf"
]
| 248,392,022 | 2204.11872 | f2d5ee21298454a3c7dcfba37653048493ac20ad |
Angular Quantization in CFT
25 Apr 2022
Nicholas Agia
Center for the Fundamental Laws of Nature
Harvard University
CambridgeMAUSA
Daniel L Jafferis
Center for the Fundamental Laws of Nature
Harvard University
CambridgeMAUSA
Angular Quantization in CFT
25 Apr 2022
The most common quantization scheme in which to study a conformal field theory is radial quantization, wherein a Hilbert space of states is defined on a sphere, whose Hamiltonian when mapped to the plane is the dilatation generator and which boasts a state/operator correspondence. In this paper, we consider an alternative quantization scheme for 2d CFTs in which the plane is foliated by constant-angle slices, as opposed to concentric circles, whose Hamiltonian is the rotation generator. In this angular quantization, there is no state/operator correspondence but instead an "asymptotics/operator correspondence". A central feature is that the quantization slice ends on two operators, and a regulator must be chosen by excising holes around each operator and imposing suitable boundary conditions such that the holes shrink to the desired local operators. This angular quantization may be viewed as constructing CFTs in Minkowski space, or separately as studying non-conformal (but local) boundary conditions in CFTs. We provide explicit Fock space constructions for various free 2d CFTs. In addition to the motivation of applying angular quantization in string theory situations where traditional radial quantization is insufficient, we comment on its relation to modular Hamiltonian approaches to entanglement entropy.
Introduction and Motivation
A conformal field theory is defined, in part, by its spectrum of local operators together with their correlators on a conformal equivalence class of manifolds. However, to make contact with fundamental principles of QFT such as unitarity and locality, one must construct a Hilbert space of the CFT by foliating spacetime in spatial slices, allowing one to discuss the states of the CFT and an associated Hamiltonian. The most common choice of quantization scheme in the literature for a d-dimensional CFT is radial quantization, where the spatial slices are the spheres of the Lorentzian cylinder R × S d−1 , which upon analytic continuation to Euclidean signature becomes the plane R d foliated in concentric spheres via the usual exponential map. A great deal is known about CFT Hilbert spaces in radial quantization, such as possessing a unique ground state with a gapped spectrum, primarily due to the famous state/operator correspondence providing an isomorphism between the local operators and the Hilbert space of states on the sphere.
Of course, one is free to choose any foliation to define a quantization scheme, and neither the local operators nor their correlators can depend on this choice, as the latter are intrinsic to the CFT. When two foliations differ by a continuous deformation, the Hilbert spaces so constructed are isometrically isomorphic, and in particular each state in one Hilbert space has a unique and convergent expansion in a basis of the other. This isometry is clear from the point of view of a path integral, where the expansion coefficients are given by the path integral over the bulk region between the two slices with fixed boundary wavefunctionals.
However, the story is far more subtle for Euclidean foliations which are not deformable into each other. The simplest example is that of Euclidean R d foliated in concentric spheres S d−1 versus Euclidean R d foliated in Cartesian planes R d−1 . The former upon analytically continuing the radial direction becomes the CFT on the Lorentzian cylinder R × S d−1 , while the latter upon analytically continuing the Cartesian direction becomes the CFT on Minkowski space R d−1,1 . Far less is known about the structure of general CFTs on Minkowski space, whose Hilbert space is not isomorphic to the one in radial quantization and indeed is not even gapped. Of course, that the Lorentzian cylinder Hilbert space is gapped is not surprising because the cylinder has some finite radius R and the physical energy of a state is E = ∆ R , where ∆ is the scaling dimension of the associated operator under the state/operator correspondence. One way to obtain the CFT on Minkowski space is to take the R → ∞ limit of the theory on the Lorentzian cylinder, upon which the energy spectrum generically becomes a gapless continuum, and it becomes possible for a moduli space of vacua to arise 1 . This limit is necessarily singular, and the relation between the radially-quantized Hilbert space and the flat Minkowski space Hilbert space is far from obvious.
In this paper we shall describe in detail yet another quantization scheme in the specific context of two-dimensional CFTs wherein the Euclidean plane is foliated by constant-angle rays extending from the origin to infinity, which we generically call "angular quantization". In the sphere conformal frame, one considers an operator O i at the origin and an operator O j at its antipodal point (one or both of which may be the identity), and the goal is to construct the angularly-quantized Hilbert space H ij associated to the spatial slice ending on those two operators and its associated Hamiltonian H R , which we shall call the "Rindler Hamiltonian". This procedure requires regularization near the endpoints where the two operators sit, so we shall excise holes of radius ε and place suitable boundary conditions which shrink to the appropriate operators, constructing the desired Hilbert space as the ε → 0 limit of the regulated one. This construction is performed very explicitly for the free field examples given below, though its generalization to arbitrary CFTs is straightforward, albeit abstract.
The concept of angular quantization is hardly new. It is the analytic continuation of the Rindler quantization performed by a uniformly accelerating observer in Minkowski space, which is why we call H R the Rindler Hamiltonian despite being in Euclidean signature. Furthermore, the description of the modular Hamiltonian in the context of entanglement entropy [15,16,17] is essentially a special case of angular quantization. This latter construction defines a modular Hamiltonian K from a reduced density matrix ρ via ρ = e −2πK , and the von Neumann entropy of the reduced density matrix is computed from the n th Rényi entropy Tr[e −2πnK ] analytically continued in n. The connection to the entanglement entropy literature is commented on at the end. It should be emphasized that the angular quantization constructed here is equivalent to considering the CFT on Minkowski space, which is carefully constructed from the Hilbert space on a very wide strip in the limit that the width of the strip becomes infinite. As such, it can be a useful stepping stone to considering Minkowski space CFTs more generally.
Finally, there is another important application of the angular quantization formalism which is actually our main motivation. In conventional string theory, asymptotic scattering states are described via BRST-invariant worldsheet vertex operators in radial quantization [1]. This suffices to describe, for example, the perturbative S-matrix of excitations above the Minkowski vacuum, but is not sufficient to capture all the on-shell states in more interesting scenarios involving black hole backgrounds [2]; even string theory in Rindler space cannot be described by vertex operators in radial quantization alone [3]. This can be seen in a lack of unitarity of the standard scattering amplitudes, indicating the existence of additional physical string states. In contrast, quantum field theory in such backgrounds is perfectly unitarity when all modes, on both sides of any horizon, are included. The distinction is because strings are extended objects that can be bisected by horizons.
Spacetime states in string theory should be defined via a Keldysh field contour for the complexified time boson X 0 , and the aforementioned cases where radial quantization may be insufficient arise when the Keldysh contour is chosen such that X 0 is compact in the Euclidean section, whence on-shell Euclidean time-winding operators may arise. One place such operators are prominent is in the work [2], which presented a Lorentzian description of the FZZ duality between the SL(2, R)/U(1) cigar black hole [4] and sine-Liouville theory [5], as well as threedimensional uplifts of this duality. There, the dual sine-Liouville description of the cigar black hole involves a condensate of Euclidean time-winding operators.
The need for angular quantization for such worldsheet CFTs with BRST-invariant vertex operators that wind Euclidean time was already described in [2], so here we shall only briefly summarize the logic based on the Witten iε prescription. In order for a string theory to make sense, there must be a well-defined prescription for computing observables from correlators, which must involve a treatment of potential divergences in moduli space integrals. In [6], Witten showed that the way to regulate a region in Euclidean moduli space where one scattering state vertex operator approaches another that results in causal propagation in spacetime is to cut out a disk of radius ε around the coincident point and glue in a Lorentzian cylinder. With this prescription, the worldsheet time must be analytically continued from Euclidean to Lorentzian signature so that as one vertex operator approaches another, it continues along an infinite Lorentzian tube rather than reaching the singular point.
For ordinary scattering states, Witten's iε contour prescription thus dictates using radial quantization, or any continuous deformation thereof, since the codimension-one gluing locus between the Lorentzian cylinder section and the remaining Euclidean worldsheet is topologically a circle, and the gluing slice is the fixed locus of a local Z 2 time-reflection symmetry. However, this iε prescription fails for time-winding vertex operators because the modified contour with the Lorentzian tube does not avoid the multi-valued cut in the OPE due to the winding of X 0 . In order for the moduli space integrand to be finite and single-valued, the vicinity of the timewinding vertex operator must be replaced with a Lorentzian Rindler wedge, on which the OPE is indeed single-valued and finite. Therefore, the correct Witten iε prescription for time-winding vertex operators dictates using angular quantization, as the gluing locus between the Euclidean worldsheet and the Lorentzian Rindler wedge is topologically a semi-infinite ray extending away from the would-be singularity. We may always locally choose the gauge such that the gluing locus in this Witten contour is at constant angle θ so that the constant-time slices are indeed the constant-θ slices of angular quantization. Hence describing string states corresponding to time-winding vertex operators in angular quantization is a formal requirement of the Witten iε prescription in order that the string theory make sense in the first place 2 .
As described in [2], the spacetime string states associated to Euclidean time-winding operators are not scattering states in the usual sense, and the ones appearing in that work (and also in [7]) are anchored at infinity but extend to finite locations in the bulk; we shall present a detailed analysis of the Hilbert space of the resulting "long strings" in [8]. The present paper can be viewed as a self-contained reference for the CFT techniques and results which we shall subsequently use in the string theory context. The structure of this paper is as follows. In Section 2, we provide a brief general outline of the construction of angular quantization on the plane/sphere and its main characteristics. In Section 3, we study the simplest example of a noncompact free boson in detail, including different non-conformal boundary conditions that shrink to exponential vertex operators which are either "Neumann-like" or "Dirichlet-like". In Section 4, we present the main results for other free CFTs for which the angular quantization admits an explicit Fock space description, in particular for the linear dilaton theory, the compact boson and the bc ghost system. From these results, one may apply the formalism to BRST quantization in limits of NLσM worldsheet string theory containing free or linear dilaton target space directions. Lastly, in Section 5 we mention the connection to entanglement entropy as a special case of the formalism; while the work in this paper does not provide any new results in that application, it does allow one to have a more systematic operator method of computing certain entanglement entropies which could potentially be useful even in non-conformal theories. More importantly, the rationale behind obtaining local operators by shrinking boundary conditions allows one to study more carefully the admissibility of boundary conditions necessary to define entanglement entropy in continuum quantum field theory. For the entirety of this paper we work in Euclidean signature. A collection of some operator algebra calculations is contained in two appendices. 2 One might try to gain intuition on the need for angular quantization near Euclidean time-winding operators using logic from effective string theory by fixing static gauge in order to define the spacetime string states. The gauge choice fixing the radial direction away from such an operator to be X 0 is inconsistent with the shift in X 0 around a Euclidean time-winding operator. However angular gauge, fixing X 0 in terms of the angular coordinate, is consistent and leads to a real condition in the combined Lorentzian continuation. Such static gauges are incompatible with having a flat worldsheet metric unless the equation of motion for X 0 is satisfied, so one would either have to work with a dynamical metric mode which complicates the BRST quantization or to confine oneself to classical string theory.
O i O j ←→ z = e −iy
General Construction on the Sphere
Angular quantization is most naturally discussed on the genus zero Riemann sphere; for correlators at higher genera, one may use a standard pair-of-pants decomposition to express everything in terms of genus zero quantities. So consider a sphere correlator arranged via a global PSL(2, C) transformation to have an operator O i inserted at the origin z = 0 and an operator O j inserted at z = ∞, the latter of which is understood as being at the origin of the inverted patch with transition function z ′ = 1/z. Either operator could be chosen to be the identity so that the procedure works for all n-point correlators including the partition function. We would like to define a Hilbert space H ij of states defined on the line between the two operators which is time-evolved with a Rindler Hamiltonian H R in angles around the origin, resulting in all sphere correlators being computed as "thermal traces" over this angularly-quantized Hilbert space. We shall often freely pass between the sphere and cylinder conformal frames, related by z = e −iy , where we write the cylinder complex coordinate as y = y 1 + iy 2 with y 1 ∼ y 1 + 2π the circle direction; see Figure 1.
The immediate problem is that the picture in Figure 1 does not define a trace over a Hilbert space because of the fixed points of the angular evolution at z = 0 and z = ∞, and so a regularization prescription must be given. A natural regulator is provided by excising a hole of some radius ε around an endpoint operator O and imposing a boundary condition B such that the boundary "shrinks" back to the operator O in the limit ε → 0; we are of course free to consider independently-sized holes cut around the operator at the origin and the one at infinity, but it is easiest to take the symmetric regulator where the excised hole around infinity has the same radius ε in the z ′ = 1/z patch. That is, in lieu of the operators O i (0) and O j (∞), we shall place a boundary condition B i at |z| = ε and a boundary condition B j at |z| = 1 ε , as shown on the left of Figure 2, performing all calculations at finite ε and then taking the "shrinking limit" ε → 0 at the end. Then, if we choose B i and B j to be local boundary conditions, they allow us to define the regulated Hilbert space H B i B j ij of a very wide strip between the two boundary conditions, and the Rindler Hamiltonian H
B i B j R
is just the generator of time translations on this strip, as shown on the right of Figure 2. The angular quantization Hilbert space H ij with associated Rindler Hamiltonian H R is then extracted from the limit ε → 0 by constructing the Figure 2: Left: The finite-regulator set-up for sphere correlators obtained by excising holes of radius ε around the endpoint operators and imposing suitable boundary conditions B such that the operators are reproduced in the shrinking limit ε → 0; at finite regulator it is manifest that correlators are computed as traces over some Hilbert space after evolving 2π in angular time. Right: The appropriate Hilbert space H B i B j ij and its Hamiltonian H
B i B j ε 1 ε × × × × B i y 2 = ln ε B j y 2 = − ln ε H B i B j ij H B i B j RB i B j R
as defined on the very wide strip with the local boundary condition B i on the bottom and the local boundary condition B j on the top. The Hilbert space H ij of angular quantization is obtained in the ε → 0 limit of H
B i B j ij .
equivalence classes of normalizable, finite-energy states 3 . To avoid some notational clutter, we shall usually leave the dependence on ε implicit, as everything in this paper is at strictly finite regulator. Now, the procedure to compute correlators is clear from Figure 2; one computes the trace over the evolution operator e −2πH B i B j R on the two-holed sphere in the Hilbert space H B i B j ij and takes the limit ε → 0 at the end. That is,
lim ε→0 Tr H B i B j ij O · · · Oe −2πH B i B j R S 2 ∝ O i (0)O · · · OO j (∞) S 2 ,(2.1)
where O · · · O represents any number of bulk operator insertions, and we explicitly use an S 2 subscript to remind us of the conformal frame. Of course, the correlator on the right side of this equation vanishes simply because O i and O j are infinitely far separated in this conformal frame. Once we write more careful expressions below to extract the finite quantities, it will be important to determine the (finite) normalization, which depends only on B i and B j .
Shrinking of Boundary Conditions
The first step in this construction is to determine the set of admissible boundary conditions B i that will shrink to a given operator O i . The shrinking condition may be characterized as 3 The Minkowski space Hamiltonian will generally differ from limε→0 H
B i B j R
by an infinite constant, so one really means that the states have finite energy above some vacuum. However, the identification of the physical states in the shrinking limit is still subtle due to their gapless nature; we shall be more explicit about this limit in [8]. Figure 3: The procedure of determining which operator is obtained upon shrinking a given boundary condition. One path integrates from |z| = ε to some fixed larger radius r * , takes the limit ε → 0 in the resulting state (for example by taking ε → λε followed by λ → 0) and then back-evolves this state to the origin using the dilatation generator.
B i ε =⇒ |Ψ B i (r * , ε) r * =⇒ Ψ B i r * λ , ε ε → λε =⇒ ε ∆ i O i (0) ×
follows -the boundary condition B i placed at |z| = ε shrinks to the operator O i if and only if
lim ε→0 O · · · O B i (ǫ) = N B i O i (0)O · · · O (2.2)
for all bulk operator insertions O · · · O outside the hole at all steps, where the correlator on the left is that in the presence of the hole of radius ε with boundary condition B i and that on the right is in the original theory; N B i is a normalization constant which depends on the boundary condition B i chosen. Abstractly, the necessary and sufficient condition which B i must satisfy is depicted in Figure 3, which follows a similar logic as for the state/operator correspondence in radial quantization. For simplicity, suppose one constructs an abstract path integral description of the CFT. Inside any correlator with the boundary condition B i at radius ε, one may perform the partial path integral over the annular region ε |z| r * for any r * smaller than the radius of the nearest bulk operator insertion. The result of this path integral produces a state |Ψ B i (r * , ε) in the wavefunctional basis (in radial quantization), in the sense that it is a linear functional which maps the rest of the configuration to the number obtained by the path integral over all |z| r * with the initial condition set by Ψ B i . Let |Ψ O (r * ) likewise denote the state in the wavefunctional basis produced by the path integral in the original theory over the disk |z| < r * with the operator O(0) inserted at the origin. We may always expand the state |Ψ B i in the wavefunctional basis {|Ψ O }, i.e.
|Ψ B i (r * , ε) = O c i O (r * , ε)|Ψ O (r * ) , (2.3)
where the sum is over all independent local operators in the theory, which we may take to have definite scaling dimensions (both primaries and descendants being included explicitly).
We would now like to find the state |Ψ B i (r * ) obtained when shrinking the excised hole away, i.e. when taking ε → 0, and so we need to know how the coefficients c i O (r * , ε) depend on ε. To do so, consider shrinking ε → λε for a finite constant 0 < λ < 1. Since the state obtained at |z| = r * with the transformed boundary condition at |z| = λε is equivalent to the state obtained at |z| = r * λ with the boundary condition at |z| = ε unchanged 4 , the fact that |Ψ B i is a state 4 It is implicitly understood that the conformal transformation of the boundary state also involves transforming the boundary condition itself so that the resulting overlaps with the bras that represent the rest of the correlator are identical before and after the transformation. If this transformation of the boundary condition is nontrivial, one says that it breaks conformal symmetry because the set of conformal transformations preserving the boundary in the original CFT and so evolves with the plane dilatation generator r −D diagonalized by {|Ψ O } tells us that
|Ψ B i (r * , λε) = Ψ B i r * λ , ε = r * /λ r * −D |Ψ B i (r * , ε) = O c i O (r * , ε)λ ∆ O |Ψ O (r * ) , (2.4) where ∆ O = h O + h O is the conformal dimension of O, with h and h the conformal weights. Hence we see that c i O (r * , ε) = b i O (r * )ε ∆ O , where the coefficients b i O (r * ) are independent of ε. Since λ ∆ O |Ψ O (r * ) = |Ψ O ( r * λ )
, the above equation also says that b i O (r * ) is in fact independent of r * , a totally obvious fact from quantum mechanics that time-evolving a Schrödinger-picture state amounts to time-evolving each basis state in an expansion starting with fixed coefficients. Thus, the wavefunctional state defined by Figure 3 is
|Ψ B i (r * , ε) = O b i O ε ∆ O |Ψ O (r * ) ,(2.5)
where b i O are pure constants. Then, in the limit ε → 0, the leading contribution in |Ψ B i (r * , ε) is just from the state with the lowest scaling dimension, and therefore, up to normalization, the boundary condition B i shrinks to the lowest-dimension operator O min that appears in (2.5); this last step is identical to the state/operator correspondence in radial quantization, wherein one simply back-evolves |Ψ O min (r * ) to the origin via the dilatation generator; since |Ψ O min (r * ) was defined as the state obtained by path integrating over O min (0), note that (2.5) says that the coefficient when shrinking the boundary condition to the origin of the plane decays as ε ∆ O min . Of course, if there are multiple independent states of lowest scaling dimension in the above expansion, then the boundary condition would shrink to the linear combination of those operators that appears in the expansion. It should be noted that, in the order of limits described above, it is important that ε be sent to zero at fixed and finite r * first.
Generically, boundary conditions are specified by the limiting behavior of bulk correlators as an operator approaches the boundary. Since the state |Ψ B i (r * , ε) above is the linear functional on configurations of bulk operators which by definition gives the respective correlator with the boundary condition B i imposed at |z| = ε, the abstract boundary condition B i is uniquely encoded in the set of coefficients b i O appearing in (2.5). Thus, the abstract procedure for enumerating all boundary conditions which shrink to a multiple of an operator O i , which we assume to be the only operator of dimension ∆ i for simplicity, is very simple
-set O min = O i (so that b i O = 0 for all ∆ O < ∆ i and b i O i is some finite number) and set b i O for all ∆ O > ∆ i
to be anything one wants; by the above discussion, the boundary condition so constructed shrinks to the operator
b i O i ε ∆ i O i .
Thus there are always infinitely many boundary conditions which shrink to a given operator, since the shrinking procedure kills all contributions except that from b i O i itself. "Most" boundary conditions in a unitary CFT will simply shrink to the identity operator, as b 1 would have to be tuned to zero for this not to occur. In order to have an angular quantization Hilbert space as defined in Figure 2, we should also demand an appropriate notion of locality; we shall not attempt to provide a systematic discussion of the condition of locality here, as the explicit boundary conditions described in the following sections are manifestly local. It should also be emphasized that these finite-regulator boundary conditions are generically nonconformal. Indeed, a condition that a 2d conformally-invariant boundary condition must satisfy is that the integral of v α T αβ n β around the boundary vanish (when inserted into correlators), where T αβ is the stress-energy tensor and v α is any vector field whose diffeomorphism flow condition does not leave the correlators invariant. Note also that we slightly abuse notation by directly comparing boundary states in Hilbert spaces H S 1 of different radii because we have an isometric isomorphism between them provided by the dilatation generator.
preserves the boundary with outward normal vector n β . Acting with this integral on (2.5) will not generically annihilate it because there is no relation between the coefficients b i O for a primary and its descendants. This shrinking procedure may therefore be regarded as a generalization of Cardy's construction of conformally-invariant boundary states out of Ishibashi states [9,10]. In particular, the Ishibashi states are the very special states of the form (2.5) where the b i O are only nonvanishing for the conformal family of a single scalar primary with the descendant coefficients determined by the primary coefficient so that the boundary integral of v α T αβ n β does annihilate it, and the conformally-invariant boundary states are the linear combinations of Ishibashi states which further satisfy the Cardy condition of "modular invariance" between the open and closed string channels. For the free CFT examples discussed below, we shall not need to make contact with this abstract prescription since boundary conditions can be defined on the fields themselves; the connection to and generalization of Cardy states could be useful for performing angular quantization calculations in other CFTs, for example in Liouville CFT, by setting up and solving bootstrap-like equations in the shrinking limit.
Since there are infinitely many ways to choose the coefficients b i O so that the boundary condition shrinks to the desired operator O i , (2.5) is not necessarily the most useful characterization of the shrinking condition. Consider the case of a primary operator O i . Taking an orthonormal basis of two-point functions, it suffices to demonstrate the original shrinking condition (2.2) for just a single bulk primary operator insertion; specifically, one need only show that the limit of O(z, z) B i (ǫ) is proportional to the nonvanishing two-point function when O = O i is the conjugate primary of O i and is zero when O is any other primary. Then (2.2) will be satisfied for all insertions of two bulk operators by an application of the OPE, and so on. In the free theory examples considered in the following sections, checking these one-point functions is a simple way to prove the boundary conditions there indeed shrink to the primaries claimed.
Matching Conditions and Asymptotics/Operator Correspondence
Having discussed the shrinking of a single boundary condition to a local operator, the set-up of angular quantization as depicted in Figure 2 follows from imposing boundary condition B i at |z| = ε which shrinks to O i (0) and simultaneously imposing boundary condition B j at |z| = 1 ε which shrinks to O j (∞). The usual two-and three-point functions of conformal primaries on the sphere are
O i (z i , z i )O j (z j , z j ) S 2 = δ ij z 2h ij z 2 h ij (2.6) O i (z i , z i )O k (z k , z k )O j (z j , z j ) S 2 = C ikj z h i +h k −h j ik z h k +h j −h i kj z h j +h i −h k ji z h i + h k − h j ik z h k + h j − h i kj z h j + h i − h k ji ,(2.7)
where C ijk are the three-point coefficients.
The thermal trace on the two-holed sphere over H
B i B j ij
as depicted on the left of Figure 2 (with no bulk insertions) should reproduce in the shrinking limit the two-point function above with z i = 0 and z j = ∞; the thermal trace with O k (z k , z k ) inserted is likewise supposed to reproduce in the shrinking limit the three-point function above with z i = 0 and z j = ∞. However, we must be careful to keep track of the various powers of ε which cause various quantities to decay or diverge. For example, both correlators vanish since z ji = ∞, which is purely an artifact of working in a conformal frame with two operators at infinite coordinate separation; finite quantities are obtained by simply stripping off this zero. Furthermore, as elucidated above, there are powers of ε incurred by shrinking the boundary conditions themselves. In particular, B i shrinks to (a finite constant) times ε ∆ i O i (0); for aesthetic reasons explained below, we shall separate out a spin factor of i −s i to write the shrinking limit as
b O i i −s i ε ∆ i O i (0), where b O i is a finite constant.
For the other endpoint, consider placing the operator O j at large but finite position z j = 1 ε and transition to the inverted patch
z ′ j = 1/z j in which O ′ j (ε) = (−1) s j ε −2∆ j O j ( 1 ε ), where s j = h j − h j is the operator's spin. Then, shrinking B ′ j at |z ′ | = ε is identical to shrinking B j at |z| = ε, and hence the original B j at |z| = 1 ε shrinks to b O j i s j ε −∆ j O j (∞)
. That is, angular quantization traces on the sphere shrink according to
Tr H B i B j ij O 1 (z 1 , z 1 ) · · · O n (z n , z n )e −2πH B i B j R S 2 ε→0 −→ b O i b O j i s j −s i ε ∆ i −∆ j O i (0)O 1 (z 1 , z 1 ) · · · O n (z n , z n )O j (∞) S 2 . (2.8) Finally, it is O ′ j (0)
which has a finite two-point function with O i (0) and a finite three-point function with O i (0)O k (z, z), which are extracted via further multiplication by ε −2∆ j to strip off the infinite-separation zero. Altogether, therefore, the matching conditions which must be satisfied in angular quantization are
lim ε→0 i s i −s j ε −∆ i −∆ j Tr H B i B j ij e −2πH B i B j R S 2 = N B i B j ij δ ij (2.9) lim ε→0 i s i −s j ε −∆ i −∆ j Tr H B i B j ij O k (z, z)e −2πH B i B j R S 2 = N B i B j ij C ikj z h i +h k −h j z h i + h k − h j , (2.10) where N B i B j ij
is a finite normalization constant that depends on which boundary conditions we use to shrink to the two operators (but not on ε) which factorizes between B i and B j because the excision regulators are local and independent.
It may seem peculiar that we are defining these matching conditions from the limit of sphere correlators involving O j (z j , z j ) evaluated at z j = 1 ε even though the sphere correlators have no relation to the artificial excision regulator ε. Doing so simply defines the normalization and phase convention of the operators in angular quantization. The statement (2.9) taken by itself is very weak because it cannot disentangle the Hamiltonian H
B i B j R from the Hilbert space H B i B j ij .
That is, the "matching condition" (2.9) is really just a partial definition of the numbers N B i B j ij , which we demand to be finite without loss of generality. With this understanding, the bona fide matching condition (2.10) becomes a highly nontrivial constraint on the asymptotic spectrum of the Rindler Hamiltonian and the matrix elements of all primaries in the states of angular quantization. Note also that (2.9) says that the limit of the thermal trace over H B i B j ij must vanish if the endpoint operators do not lie in conjugate Verma modules, and this vanishing cannot come from an overall positive power of ε without contradicting (2.10), meaning that the Rindler Hamiltonian H
B i B j R
is not generally Hermitian. It may seem odd that a non-Hermitian Hamiltonian describes a unitary theory, but we shall see in our examples below that its non-Hermiticity is of a rather mild form, essentially that the only non-Hermitian part 5
of H B i B j R
is what provides the orthogonality δ ij needed in (2.9). Despite our notation and language, it is an assumption for the moment that there exist choices of boundary conditions B i such that H
B i B j ij
is an honest Hilbert space; one is always free to define a vector space H B i B j ij and scalar operator H
B i B j R
based on reproducing the finite-regulator thermal partition function and onepoint functions, but there is no guarantee a priori that there exists a Lorentzian continuation thereof such that H
B i B j ij
is a Hilbert space of normalizable states with a local operator algebra obeying microcausality.
It suffices to consider only primary endpoint operators. That is, the map O i × O j → H ij is actually well-defined on the full Verma module product. Indeed, if one of the endpoint operators is a descendant, we rewrite it as the appropriate contour integrals surrounding a primary and then excise the primary with a boundary condition contained inside the stress-energy contour integrals. Thus, the angular quantization Hilbert space for endpoint descendants is the same as the Hilbert space for their primaries, and the entire effect of the descendants is described by stress-energy insertions in the thermal traces which may be reabsorbed into a new Rindler evolution operator.
Finally, we emphasize that there is no state/operator correspondence in angular quantization. In radial quantization, the state/operator correspondence provides an isomorphism between the Hilbert space of states on S 1 and local operators. The construction of angular quantization that we have described in this section gives an entirely different map from all pairs of local operators to all angularly-quantized Hilbert spaces, O i × O j → H ij . Notably, this correspondence requires two operators instead of one, and they are associated with an entire Hilbert space and not just a single state within a Hilbert space. We shall call the map O i × O j → H ij the "asymptotics/operator correspondence" instead. From the abstract definition provided in this section, the "asymptotics" part of this correspondence is given by the Hilbert space with suitable boundary conditions on a very wide strip in the limit that the width of the strip tends to infinity. After the shrinking limit, the states in the angularly-quantized Hilbert space H ij live on an infinite line and obey an asymptotic fall-off condition; the latter is most easily seen in the wavefunctional basis, where "shrinking" the finite-regulator boundary conditions has the effect of determining the asymptotic form of the wavefunctionals Ψ[O(y 1 , y 2 )], where O stands for all local operators of the theory, as y 2 → ±∞. While the map O i × O j → H ij described above requires a pair of endpoint operators, the asymptotic fall-off condition in the wavefunctional basis holds for the endpoint operators individually. In the free CFT examples described in detail below, the asymptotic fall-off conditions associated to given endpoint operators will be even more explicitly written in terms of the free field description. This asymptotics/operator correspondence is central to angular quantization applications to the NLσM limit of string theory, as we shall describe in [8].
The Weyl Anomaly
It is important to note that the thermal trace depicted on the left of Figure 2 computes correlators in the sphere conformal frame after shrinking ε → 0 whereas the actual definition of the angular quantization Hilbert space on the right of Figure 2 is on the very wide strip which becomes the cylinder conformal frame after identifying y 1 ∼ y 1 + 2π. As such, we must be careful of the frame dependence of the Hamiltonian. Despite the fact that the boundary conditions involved are non-conformal, the only contribution to the Weyl transformation of the angular quantization traces here is from the usual bulk Weyl anomaly.
First recall that, if a 2d CFT on a Riemann surface Σ with metric g αβ and boundary embedding f : ∂Σ ֒→ Σ has partition function Z[g], then the partition function under the local Weyl transformation g αβ → g ′ αβ = e −2ω g αβ is given by 6
Z[g ′ ] = e −S W Z[g] (2.11) S W = c 24π Σ d 2 σ √ g g αβ ∂ α ω∂ β ω + ωR + c 12π ∂Σ dℓ f * g ωK, (2.12)
where R is the scalar curvature associated to g αβ and K the (trace of the) extrinsic curvature of the boundary whose invariant line element is dℓ √ f * g; the Euler characteristic of Σ of genus g with b boundaries is
χ = 1 4π Σ d 2 σ √ g R + 1 2π ∂Σ dℓ f * g K = 2 − 2g − b. (2.13)
We shall always discuss CFTs for which the holomorphic and antiholomorphic central charges agree, c = c, so that there is no gravitational anomaly. On the very wide cylinder with complex coordinate y = y 1 + iy 2 , where ln ε y 2 − ln ε and y 1 ∼ y 1 + 2π, we simply take the flat metric g αβ = δ αβ with scalar curvature R = 0 everywhere and boundary extrinsic curvature K = 0 at y 2 = ∓ ln ε (where d 2 σ √ g = dy 1 dy 2 and dℓ √ f * g = dy 1 ). The two-holed sphere at finite regulator with complex coordinates (z, z), where ε |z| 1 ε , is related by the exponential map z = e −iy . For the latter we wish to take the flat plane metric ds 2 = dz ⊗ s dz = e 2y 2 dy ⊗ s dy which has scalar curvature R = 0 and boundary extrinsic curvature K = ± 1 |z| at |z| = ε ∓1 (where now χ = 0 is due to this opposite orientation of the two boundaries). Hence the flat plane metric g ′ αβ = δ αβ is obtained from the flat cylinder metric via the Weyl transformation ω = −y 2 , for which the Weyl anomaly action above is
S W = c 24π − ln ε ln ε dy 2 2π 0 dy 1 = − c 6 ln ε.
(2.14)
So the Weyl anomaly says that the finite-regulator partition functions in the two conformal frames are related by Z flat plane = ε c/6 Z flat cylinder .
(2.15)
The fact that (2.15) is the only factor in the Weyl transformation between the flat finite cylinder and the flat annulus requires some explanation. The thermal traces in angular quantization are just a different way of slicing up and computing correlators in any conformal frame. In particular, the thermal trace over the evolution operator is always equal to the partition function in that frame. Moreover, since the underlying correlators are those of a CFT, a conformal transformation acts on both the local operators and the boundary conditions so that, up to the Weyl anomaly, the final number (or distribution) computed from a given configuration does not change; see footnote 4 above.
In path integral language, the boundary condition B i that shrinks to the operator O i is imposed via a boundary action S O i [g, B i ]; the dependence on B i means that the boundary action depends on a set of boundary condition parameters which should be thought of as background fields in the path integral. When performing a Weyl transformation, the change in 6 Recall that this result comes from insertions of the stress-energy tensor in correlators being written as
T αβ = − 4π √ g δW δg αβ , where Z[g] = e −W [g] , whose trace is T α α = − 2π √ g δW δω .
The usual Weyl anomaly action for a 2d CFT then follows from the curved-space one-point function T α α g = − c 12 R − c 6 δ(Σ − ∂Σ)K, where δ(Σ − ∂Σ) is the covariant delta function localizing to the boundary, together with the Weyl transformation of the scalar curvature being R ′ = e 2ω (R + 2∇ 2 ω); i.e. SW is the unique diffeomorphism-invariant action whose Weyl variation is exactly that dictated by the trace of the stress-energy tensor. the boundary action due to the metric variation is exactly compensated by the variation of the parameters in the boundary condition, so that schematically
δS O i [g, B i ] δω = 2g αβ δS O i [g, B i ] δg αβ + δS O i δB i ∂B i ∂ω = 0. (2.16)
Therefore, the Weyl transformation for the path integral whose action is
S CFT 0 + S O i [g, B i ] is
identical to that of the path integral whose action is S CFT 0 , i.e. it is given by (2.15). Note that the stress-energy tensor is defined as the Noether current for translations so that the Hamiltonian is constructed from it in the usual way; that is, T αβ is computed from varying the metric with fixed boundary conditions, and S O i [g, B i ] does give a boundary stress-energy contribution which makes T α α generically nonzero even in flat space with flat boundaries. Including the other endpoint operator, the full path integral action is
S CFT 0 [g] + S O i [g, B i ] + S O j [g, B j ]
, and the Weyl invariance of the boundary actions again results in the relation
Z[g ′ ] = e −S W Z[g]
, where S W is the same Weyl anomaly action as before. Despite the fact that, barring the Weyl anomaly, the partition functions in any two conformal frames are equal, the boundary conditions B i and B j are not generically conformally invariant, which would have required equality of the partition functions when the boundary conditions are held fixed.
While angular quantization traces are equal to the appropriate partition functions on any surface, the flat cylinder frame plays a distinguished role. On the finite cylinder, the spatial and temporal directions are flat and perpendicular, so the equality Z S 1 ×R = Tr H [e −2πH R ] S 1 ×R (together with its generalizations with operator insertions) describes the thermal trace at inverse temperature β = 2π. Due to this distinguished role, we define the normalization in angular quantization so that the shrinking limit on the cylinder produces the two endpoint operators with no powers of ε. Specifically,
Tr H B i B j ij O 1 (y 1,1 , y 1,2 ) · · · O n (y n,1 , y n,2 )e −2πH B i B j R S 1 ×R ε→0 −→ b O i b O j O i (0, ln ε)O 1 (y 1,1 , y 1,2 ) · · · O n (y n,1 , y n,2 )O j (0, − ln ε) S 1 ×R ; (2.17)
the ε → 0 limit of the correlator of course contains powers of ε, but the point is that the relation between the trace and the correlator does not. Then, the shrinking of the boundary conditions in any other conformal frame is obtained from the above cylinder limit simply by applying conformal transformations to O i (0, ln ε) and O j (0, − ln ε). To reiterate, the partition functions are always the same (modulo the Weyl anomaly), but the relation between the thermal traces and the correlators changes between frames. Indeed, mapping the endpoint operators from the cylinder to the sphere is precisely why shrinking the boundary conditions on the plane acquires factors of i −s ε ±∆ , as we used to obtain the matching conditions (2.9) and (2.10) above.
In summary, the angular quantization thermal traces in the regulated (flat) sphere and in the regulated (flat) cylinder conformal frames are related by
Tr H B i B j ij O · · · Oe −2πH B i B j R S 2 = ε c/6 Tr H B i B j ij O ′ · · · O ′ e −2πH B i B j R S 1 ×R , (2.18)
where any bulk operator insertions are also transformed in the usual way, and the Rindler Hamiltonian is the same on both sides. We are free to interpret the factor from this Weyl transformation as the shift in the Rindler Hamiltonian between the conformal frames, which is a local and covariant shift corresponding to a bulk cosmological constant. Unlike in radial quantization where the shift in energy is finite, the shift in energy in angular quantization tends to infinity; the difference, of course, is because the states in radial quantization live on a finite circle whereas angular quantization is really constructing the states in Minkowski space with potential sources at infinity 7 . This precise infinite shift will be important in the explicit calculations presented in the following sections. The ensuing derivations make it clear that the care required to take the shrinking limit properly manifests, in part, as an extreme sensitivity of angular quantization to factors of 2 and minus signs!
Warm-Up: Noncompact Free Boson
We now illustrate the procedure outlined above for the simplest example, that of the noncompact free boson X(z, z) with central charge c = c = 1. Strictly speaking, the noncompact free boson is singular as a CFT, but nevertheless it is an instructive pedagogical exercise, and the pathology-free example of the compact free boson will be considered in the following section. We shall explicitly perform the angular quantization for endpoint exponential primary operators O k ≡ :e ikX :, which have conformal weights h = h = k 2 4 , with normalization
O k 1 (z 1 )O k 2 (z 2 ) = 2πδ(k 1 + k 2 ) |z 12 | k 2 1 (3.1)
and hence three-point coefficient
C k 1 k 2 k 3 = 2πδ(k 1 + k 2 + k 3 ). (3.2)
Since the theory is free, we shall use canonical quantization to provide a Fock space construction of H k 1 k 2 . In particular, we perform the excision regularization for two particularly simple classes of boundary conditions -one which is "Neumann-like" and one which is "Dirichlet-like". As we shall show, pure Neumann and pure (indefinite) Dirichlet boundary conditions both shrink to the identity operator, but small modifications thereof will shrink to nontrivial exponential primaries. It is perhaps non-intuitive that Neumann and Dirichlet boundary conditions are equivalent in angular quantization, given how different they seem at finite regulator.
Boundary Conditions for Exponential Primaries
We must first find appropriate boundary conditions B k 1 at |z| = ε and B k 2 at |z| = 1 ε which shrink to a multiple of O k 1 (0) and O k 2 (∞), respectively; since this is a free theory, we may impose the boundary condition directly on X(z, z) itself. We first present a heuristic derivation of the relevant boundary conditions; that the angular quantization constructed out of these boundary conditions correctly reproduces the matching equations (2.9) and (2.10) may be viewed as a posteriori proof that they indeed shrink to the exponential primaries claimed. Nevertheless, it is useful pedagogically to derive rigorously that the boundary conditions shrink to the exponential primaries, as this property is logically independent from the construction of angular quantization, so we then provide this derivation for the Neumann-like boundary conditions.
Heuristic Derivation from the OPE
The easiest way to find such appropriate boundary conditions is via the OPE, which tells us for instance how X(z, z) must behave in the vicinity of :e ikX(0) :. Our normalization convention is such that the free-field OPE at separated points on the plane is 7 We should clarify that this shift is not a Casimir energy which, while still present, is proportional to 1 ln ε and so vanishes in the shrinking limit. It is an example of a far more dramatic effect of an infinite energy shift between the Minkowski vacuum and the Rindler vacuum.
X(z, z)X(0) = − 1 2 ln |z| 2 + :X(z, z)X(0): . (3.3)
Since X(z, z) is not a well-defined local operator in the CFT, we really should consider its derivatives ∂X(z) and ∂X(z) instead. Their OPEs with an exponential primary are The fact that the singular parts of these OPEs contain the same operator with which we started at the origin says that the operators z∂X(z) and z∂X(z) both behave like constants (in fact the same constant − ik 2 ) near the :e ikX(0) : insertion; it is understood that X(z, z) is analytically extended to take values in a curve in one complex dimension 8 . However, since we would like to perform canonical quantization, for which X(z, z) and its conjugate momentum Π(z, z) obey the usual canonical commutation relations at equal angle θ, it is inconsistent to impose constancy of z∂X(z) and of z∂X(z) on the boundary separately since ∂X and ∂X will not commute on the quantization slice. We could take one of these conditions or the other, or any linear combination of the two, but the most natural choices are to take their sum or to take their difference, which say that
r∂ r X(z, z):e ikX(0) : = −ik:e ikX(0) : + :r∂ r X(z, z)e ikX(0) : (3.6) ∂ θ X(z, z):e ikX(0) : = :∂ θ X(z, z)e ikX(0) :, (3.7)
where r∂ r X = z∂X + z∂X and ∂ θ X = i(z∂X − z∂X) are the radial and angular derivatives on the plane. For k = k 1 , we may then impose the constancy of r∂ r X or of ∂ θ X at |z| = ε; for k = k 2 , the above OPEs hold in the inverted z ′ = 1/z patch, so the boundary conditions at |z| = 1 ε receive a minus sign from the reversed orientation, i.e. from
r ′ ∂ r ′ X ′ (z ′ , z ′ ) = −r∂ r X(z, z) and ∂ θ ′ X ′ (z ′ , z ′ ) = −∂ θ X(z, z)
. Therefore, the two simplest boundary conditions on the plane that shrink to a multiple of :e ik 1 X(0) : and :e ik 2 X(∞) : are Neumann-like:
r∂ r X |z|=ε = −ik 1 (3.8) r∂ r X |z|= 1 ε = +ik 2 (3.9)
and Dirichlet-like:
∂ θ X |z|=ε = ∂ θ X |z|= 1 ε = 0. (3.10)
We call the former "Neumann-like" because it fixes the derivative normal to the boundary (along the spatial slice) and the latter "Dirichlet-like" because it fixes the derivative parallel to the boundary (along the time direction). Another easy way to see that the Neumann-like boundary condition shrinks to an exponential primary is by noting that the global translation current is j α = i∂ α X under which :e ikX : has charge k; since the translation charge of the operator at the origin can be computed as the constant-|z| integral of 1 2π j r , the boundary condition j r = k/ε at |z| = ε simply ensures that the boundary shrinks to an operator of charge k, and the lowest-dimension operator of charge k is the primary :e ikX :.
The Dirichlet-like boundary condition written above, on the other hand, seems insufficient to shrink to :e ikX :, as it does not seem to depend on k at all. Indeed, all this boundary condition says by itself is that X(z, z) is single-valued around the origin, so it only guarantees that the boundary condition shrink to a local operator (i.e. not contain a bare field X(z, z) that is neither differentiated nor exponentiated). Therefore, that the charge of the resulting operator be exactly k must be fixed dynamically for this Dirichlet-like boundary condition, which will determine what the actual action for this regulator is. We shall show this dynamical fixing below, but roughly speaking it arises from the equation of motion for the boundary Lagrange multiplier which must be introduced to impose the boundary condition.
It may seem like we have ignored the simplest boundary condition consistent with the OPE, which would just set X = −ik ln ε at |z| = ε, fixing the actual value of the free boson itself more akin to what would usually be called a Dirichlet boundary condition in the literature. We have opted not to write this boundary condition because it is misleading for the compact boson discussed in the following section due to the need to split X(z, z) into chiral and antichiral halves. The chiral splitting for the compact boson is pivotal to angular quantization and is in fact determined by the theory unlike in radial quantization where one usually says it amounts to a choice of branch cut which disappears at the end of the day for correlators of truly local operators. There is also a technical reason the boundary condition X = −ik ln ε is inappropriate for the noncompact boson -due to the continuous spectrum, fixing the value of the field does not shrink to a well-defined primary. This latter fact holds even for the pure Dirichlet boundary condition X = 0, which does not quite shrink to the identity operator as may be verified by applying the framework below. This difficulty is of course due to the pathological continuous spectrum of the noncompact theory and hence does not persist in the compact theory. Nevertheless, it is the Dirichlet-like boundary condition described above and below which most easily generalizes to one which shrinks to winding operators for the compact boson. We shall revisit this point later.
Since we wish to perform the canonical quantization on the very wide (but finite) strip, where ln ε y 2 − ln ε and y 1 ∈ R is not compactified, we merely need to transform the above boundary conditions via z = e −iy . Since X(z, z) is a scalar, the relevant boundary conditions on the strip are simply Neumann-like:
∂ y 2 X y 2 =+ ln ε = −ik 1 (3.11) ∂ y 2 X y 2 =− ln ε = +ik 2
and Dirichlet-like:
∂ y 1 X y 2 =± ln ε = 0. (3.12)
Rigorous Derivation from the Path Integral
Here we provide a rigorous derivation that the Neumann-like boundary condition r∂ r X = −ik at |z| = ε shrinks to (a constant times) the operator :e ikX(0) : as ε → 0; this subsection is independent from the rest of the paper and included for completeness, so there is no loss of continuity for the reader wishing to jump ahead to the angular quantization Hilbert spaces and Rindler Hamiltonians in the following subsection. The simplest way to prove the desired shrinking is to derive the one-point function limit lim ε→0 :e ik ′ X(z,z) :
N(k,ε) = 2πbε k 2 2 δ(k + k ′ ) |z| k 2 , (3.13)
where · · · N(k,ε) denotes the correlator of the free boson theory on the plane with the disk |z| < ε cut out and the Neumann-like boundary condition r∂ r X = −ik imposed at |z| = ε, and b is an ε-independent constant. The normalization constant b is not relevant to prove that we obtain the correct operator in the shrinking limit, so we need not compute the one-loop determinant in the path integral; the exact normalization relevant to angular quantization will be obtained in the following subsection. The bare path integral on the excised plane Σ(ε) = C \ D 2 (ε) subject to the Neumann-like boundary condition at |z| = ε is
Z[0] = DX exp − 1 2π Σ(ε) d 2 z ∂X∂X + ik 2π 0 dθ 2π X(|z| = ε) . (3.14)
The boundary term is precisely how the boundary condition r∂ r X = −ik at |z| = ε is imposed via the path integral, because the variation of the action is
δS = − 1 π Σ(ε) d 2 z ∂∂XδX + 2π 0 dθ 2π f * g n α ∂ α X − ik δX |z|=ε , (3.15)
where the line element on the boundary is dθ √ f * g = εdθ and the outward unit normal vector is n = n α ∂ α = −∂ r , with the minus sign due to the chosen orientation of the boundary. The vanishing of the action variation on the boundary then indeed correctly imposes r∂ r X = −ik at |z| = ε. Before computing the propagator, we must first address the fact that the noncompact free boson has a non-normalizable zero-mode which causes the bare path integral to be illdefined. The zero-mode here, of course, takes the form X = x 0 for constant x 0 ; without the boundary term, the bare path integral is infinity, while with the boundary term the bare path integral vanishes unless k = 0 (when it is also infinity). When fermionic path integrals have zero-modes, the correct procedure is to compare correlators not with the bare path integral but with the path integral with the correct number of zero-modes inserted in order to render the result finite. Morally speaking, that should also be the correct prescription for non-normalizable bosonic zero-modes, namely we must separate out the zero-mode contribution from the rest of the path integral that has the zero-mode absorbed by a delta function. Then, if we schematically write O = O 0 O ′ for any operator as its zero-mode part times its nonzero-mode part (which is appropriate for the exponential operators, for instance), we compute correlators as
O 1 O 2 · · · O n N(k,ε) = O 1 O 2 · · · O n 0 O 1 O 2 · · · O n ′ N(k,ε) ,(3.16)
where
O 1 O 2 · · · O n 0 ≡ ∞ −∞ dx 0 (O 1 O 2 · · · O n ) 0 e ikx 0 (3.17)
is the zero-mode contribution, and
O 1 O 2 · · · O n ′ N(k,ε) = DX ′ O ′ 1 O ′ 2 · · · O ′ n exp − 1 2π Σ(ε) d 2 z ∂X ′ ∂X ′ + ik 2π 0 dθ 2π X ′ (|z| = ε) (3.18
) is the part of the path integral orthogonal to the zero-mode. Note that the part of the path integral with the zero-mode removed has well-defined normalization, but the zero-mode part may be given arbitrary (ε-independent) normalization as usual for a continuous spectrum. For the exponential operators, we have O i = :e ik i X : = e ik i x 0 :e ik i X : ′ so the zero-mode part of the path integral immediately gives O 1 · · · O n 0 = 2πδ(k + k 1 + . . . + k n ) in the conventional continuousmomentum state normalization, and after the zero-mode has been removed the correlator of normal-ordered exponentials may be computed via the series expansion of the exponentials. In particular, the one-point function we wish to compute is given by
:e ik ′ X(z,z) : N(k,ε) = 2πbδ(k + k ′ ) ∞ n=0 (ik ′ ) n n! :X n (z, z): ′ N(k,ε) ,(3.19)
where b is the normalization from the one-loop fluctuation determinant, which we do not need to compute here, and hence the remaining correlators · · · ′ N(k,ε) are computed with the normalization 1 ′ N(k,ε) = 1. The last ingredients we need are the nonzero-mode one-point function X(z, z) ′ N(k,ε) and propagator X(z 1 , z 1 )X(z 2 , z 2 ) ′ N(k,ε) . These may be computed by evaluating the sourced partition function Z ′ [J] in the usual way by shifting the variable of integration so as to complete the square and hence decouple X from its source J. It is easier, however, to directly solve the free Schwinger-Dyson equation subject to the boundary condition, which for the two-point function read
∂ 1 ∂ 1 X(z 1 , z 1 )X(z 2 , z 2 ) ′ N(k,ε) = −πδ 2 (z 12 ) (3.20) z 1 ∂ 1 + z 1 ∂ 1 X(z 1 , z 1 )X(z 2 , z 2 ) ′ N(k,ε) |z 1 |=ε = −ik X(z 2 , z 2 ) ′ N(k,ε) ,(3.21)
where ∂ 1 and ∂ 1 denote the holomorphic and antiholomorphic derivatives with respect to (z 1 , z 1 ), and which for the one-point function read
∂∂X(z, z) ′ N(k,ε) = 0 (3.22) z∂ + z∂ X(z, z) ′ N(k,ε) |z|=ε = −ik, (3.23)
where we have normalized 1 ′ N(k,ε) = 1. The one-point function equations are uniquely solved by
X(z, z) ′ N(k,ε) = − ik 2 ln z ε 2 + γ, (3.24)
where γ is an arbitrary constant independent of ε which reflects the shift ambiguity in the free boson; since γ is shifted by the zero-mode which has been removed and integrated over separately, we may simply set γ = 0. Note that the bulk equation of motion is indeed satisfied because ∂∂(ln |z| 2 ) = 2πδ 2 (z) which has no support on Σ(ε) and so equals the zero distribution; it is also important to realize that the ε dependence of the correlator is fixed by the invariance under the simultaneous z → λz and ε → λε manifest in the action, which is why we know that the constant ambiguity γ cannot in fact depend on ε. With this one-point function (with γ = 0), the two-point function Schwinger-Dyson equation with boundary condition is uniquely solved by
X(z 1 , z 1 )X(z 2 , z 2 ) ′ Σ(ε) = − 1 2 ln z 12 ε 2 − 1 4 ln ε z 1 − z 2 ε 2 − 1 4 ln z 1 ε − ε z 2 2 + 1 4 ln z 1 z 2 ε 2 2 + − ik 2 ln z 1 ε 2 − ik 2 ln z 2 ε 2 . (3.25)
It is worthwhile to note that taking separate holomorphic derivatives of the two-point function gives
∂X(z 1 )∂X(z 2 ) ′ N(k,ε) = − 1 2z 2 12 − k 2 4z 1 z 2 , (3.26)
and hence the nonzero-mode correlator of the stress-energy tensor T (z) = −:∂X(z)∂X(z): is
T (z) ′ N(k,ε) = k 2 4z 2 . (3.27)
This result, together with its antiholomorphic counterpart, says that the boundary condition must shrink to a scalar (primary) operator with weights h = h = k 2 4 , for which the only candidates are :e ±ikX :. It might seem that the one-point function of the stress-energy tensor is nonvanishing in the limit ε → 0 for k = 0 since it does not even depend on ε, but we must remember that the full one-point function includes a δ(k) from the zero-mode and hence gives T (z) N(k,ε) = 0 as one would expect. It is very nice that, by separating out the zero-mode contribution, we can determine the weights of the operator to which the boundary condition shrinks before even computing the one-point functions of the exponential primaries. Finally we may compute the one-point functions :X n (z, z): ′ N(k,ε) needed in (3.19). To do so, we note that the one-and two-point functions computed above immediately tell us that the nonzero-mode sourced partition function, in which the integral over iJ(z, z)X(z, z) is added to the exponential, is
Z ′ [J] Z ′ [0] = exp − 1 2 Σ(ε) d 2 z Σ(ε) d 2 z ′ J(z, z)K(z, z; z ′ , z ′ )J(z ′ , z ′ ) + i Σ(ε) d 2 z X(z, z) ′ N(k,ε) J(z, z) ,
(3.28) where the integral kernel K(z 1 , z 1 ; z 2 , z 2 ) is the first line in (3.25). In addition to taking n functional derivatives with respect to J, one combinatorically subtracts off all divergences using Wick's Theorem before taking the multi-coincident limit to obtain the one-point function of :X n (z, z):. Since the combinatorics of Wick's Theorem exponentiate [1], the normal-ordered n-point function at distinct points is obtained by replacing the bulk kernel K(z, z; z ′ , z ′ ) above with K(z, z; z ′ , z ′ )+ 1 2 ln |z −z ′ | 2 , and the subsequent multi-coincident limit allows one to express the desired one-point function as
:X n (z, z): ′ N(k,ε) = (−i) n δ n δJ(z, z) n exp − 1 2 Σ(ε) d 2 z ′ √ AJ(z ′ , z ′ ) 2 + Σ(ε) d 2 z ′ BJ(z ′ , z ′ ) J=0 , (3.29) where the two integral kernels are A(z ′ , z ′ ) = ln ε − ln 1 − ε 2 |z ′ | 2 (3.30) B(z ′ , z ′ ) = k 2 ln z ′ ε 2 . (3.31)
Then, the sum of these one-point functions is
∞ n=0 (ik ′ ) n n! :X n (z, z): = ∞ n=0 ⌊n/2⌋ m=0 k ′n 2 m m!(n − 2m)! [−A(z, z)] m B(z, z) n−2m (3.32) = ∞ m=0 k ′2m 2 m m! ln m 1 ε 1 − ε 2 |z| 2 ∞ n=0 (k ′ k) n 2 n n! ln n z ε 2 (3.33) = ε −k ′2 /2 1 − ε 2 |z| 2 k ′2 /2 z ε k ′ k . (3.34)
Therefore, including the prefactor in (3.19), we have derived the general exponential primary one-point function in the presence of the Neumann-like boundary condition at |z| = ε to be :e ik ′ X(z,z) :
N(k,ε) = 2πbε k 2 2 δ(k + k ′ ) |z| k 2 1 − ε 2 |z| 2 k 2 /2 , (3.35)
whose limit as ε → 0 is indeed given by (3.13). This result proves that the Neumann-like boundary condition shrinks to the operator bε k 2 /2 :e ikX(0) : on the plane as claimed. The shrinking of the other boundary conditions written in this paper may be proven directly with only minimal modifications to these computations, so we shall not explicitly write these derivations.
Hilbert Space and Rindler Hamiltonian
Having determined boundary conditions that shrink to exponential primaries, it is now straightforward to construct the angular quantization Hilbert spaces and associated Rindler Hamiltonians via canonical quantization on the very wide strip with complex coordinate y = y 1 + iy 2 . First, one determines the correct action which imposes the given boundary conditions; since we are working in Euclidean signature with y 1 the strip time direction, the canonical momentum conjugate to X in terms of the Lagrangian is Π = −i δL δ∂y 1 X . Then, one mode expands X and Π at a fixed time y 1 and imposes the usual equal-time canonical commutation relations
[X(y 1 , y 2 ), Π(y 1 , y ′ 2 )] = iδ(y 2 − y ′ 2 ) (3.36) [X(y 1 , y 2 ), X(y 1 , y ′ 2 )] = [Π(y 1 , y 2 ), Π(y 1 , y ′ 2 )] = 0; (3.37)
for this step it is useful to note the distributional identities
δ(y 2 − y ′ 2 ) = − 1 2 ln ε n∈Z cos πn(y 2 − ln ε) 2 ln ε cos πn(y ′ 2 − ln ε) 2 ln ε (3.38) = − 1 2 ln ε n∈Z sin πn(y 2 − ln ε) 2 ln ε sin πn(y ′ 2 − ln ε) 2 ln ε , (3.39)
valid on the domain ln ε y 2 − ln ε. Next, the Rindler Hamiltonian H R must be computed via 9
H R (y 1 ) = 1 2π − ln ε ln ε dy 2 T y 1 y 1 (y 1 , y 2 ); (3.40)
since the mode expansions obey the equation of motion and boundary conditions, this Hamiltonian will be time independent, and hence the Euclidean strip evolution operator will simply be 10 e ∆y 1 H R . However, it is crucial that the finite-regulator stress-energy tensor is not traceless due to the explicit breaking of conformal symmetry by the boundary conditions. As such, the component appearing in the Hamiltonian is T y 1 y 1 (y 1 , y 2 ) = T yy (y, y) + T yy (y, y) + 2T yy (y, y),
(3.41)
where T yy and T yy are now not holomorphic and antiholomorphic, respectively. However, the trace 4T yy as well as the non-(anti)holomorphic parts of T yy (T yy ) are localized to the boundary because local conformal symmetry is still preserved in the bulk. We may thus write each stressenergy component as a bulk piece plus a boundary piece, and we still have T bulk yy = 0 as well as T bulk yy ≡ T (y) being holomorphic and T bulk yy ≡ T (y) antiholomorphic. As such, we may still use the usual conformal transformations
T (y) = −z(y) 2 T z(y) + c 24 (3.42) T (y) = −z(y) 2 T z(y) + c 24 (3.43) 9
The 1/2π normalization is due to the stringy convention of normalizing all currents with an extra 2π and subsequently dividing all Noether charges by 2π. 10 We are admittedly using an unfortunate definition for the direction of Euclidean time. Having defined angular quantization to evolve in angles counterclockwise in the plane, the positive direction of Euclidean time on the strip is in decreasing y1 coordinate. The formula (3.40) remains the same because the usual right-handed orientation corresponding to spatial normal vector n = −∂y 1 gives the spatial tangent vector t = −∂y 2 . Then, evolution in Euclidean time always corresponds to ∆y1 being negative, so that e ∆y 1 H R is indeed a decaying operator. For example, the thermal trace corresponds to evolving from y1 = 0 to y1 = −2π and gives the usual operator e −2πH R . This convention is also why the canonical momentum is Π = i δL δ∂τ X = −i δL δ∂y 1 X .
of the bulk stress-energy tensor components between the cylinder and the plane; the boundary contributions provide important boundary terms in the Rindler Hamiltonian which are necessary for the consistency of the theory. The angular quantization Hilbert space at finite regulator is thus constructed as the Fock space built up from a vacuum by the positive-energy mode operators of the canonical quantization, whose energies are immediately read off from the mode expression for the Rindler Hamiltonian. Finally, one computes the mode expansions of the exponential primaries themselves and evaluates the thermal trace over the Fock space as well as the thermal one-point functions of the bulk primaries, showing that the matching conditions (2.9) and (2.10) are indeed satisfied with specific normalization constants.
Neumann-like Boundary Conditions
Consider the Neumann-like boundary conditions (3.11) on the very wide strip, which will result in the endpoint operators :e ik 1 X(0) : and :e ik 2 X(∞) : in the shrinking limit on the sphere. The free boson action which imposes these boundary conditions is given by
S = 1 4π ∞ −∞ dy 1 − ln ε ln ε dy 2 (∂ y 1 X) 2 + (∂ y 2 X) 2 − ik 1 2π ∞ −∞ dy 1 X y 2 =ln ε − ik 2 2π ∞ −∞ dy 1 X y 2 =− ln ε .
(3.44) For fixed k 1 and k 2 , the field X is analytically continued to be defined on an appropriate complexified contourà la Keldysh in the target space, so strictly speaking it is only a real field when k 1 and k 2 are pure imaginary. The stress-energy tensor for this theory is
T αβ = −: ∂ α X∂ β X − 1 2 δ αβ ∂ γ X∂ γ X : − t α t β X [ik 1 δ(y 2 − ln ε) + ik 2 δ(y 2 + ln ε)] ,(3.45)
where δ αβ is the flat strip metric and t α is the unit vector tangent to the boundaries. We immediately see that the classical trace of this stress-energy tensor is 11 T α α (y 1 , y 2 ) = −X(y 1 , y 2 ) [ik 1 δ(y 2 − ln ε) + ik 2 δ(y 2 + ln ε)] ; are the usual bulk holomorphic and antiholomorphic stress-energy components, so that the integrand appearing in the Rindler Hamiltonian in this case is T y 1 y 1 = T (y) + T (y) + T α α . The mode expansions of X and its canonical momentum conjugate Π = − i 2π ∂ y 1 X obeying the equation of motion (∂ 2 y 1 + ∂ 2 y 2 )X = 0 and the boundary conditions are
X(y 1 , y 2 ) = x + i(k 1 + k 2 ) 12 ln ε − πip ln ε y 1 − i(k 1 − k 2 ) 2 y 2 + i(k 1 + k 2 ) 4 ln ε y 2 1 − y 2 2 (3.49) + i √ 2 n∈Z n =0 α n n e − πn 2 ln ε y 1 cos πn(y 2 − ln ε) 2 ln ε Π(y 1 , y 2 ) = − p 2 ln ε + k 1 + k 2 4π ln ε y 1 − 1 2 √ 2 ln ε n∈Z n =0
α n e − πn 2 ln ε y 1 cos πn(y 2 − ln ε) 2 ln ε , (3.50) 11 In covariant notation, this classical trace is T α α = −Xn α ∂αXδ(Σ − ∂Σ), where n α is the outward-pointing unit vector normal to the boundaries, which equals the classical trace written here on account of the boundary conditions. 12 By convention, the variable of a holomorphic or antiholomorphic derivative is implicitly given by its argument, so that for example ∂X(y) means ∂yX(y) and not ∂zX z(y) .
where the constant contribution i 12 (k 1 + k 2 ) ln ε in the expansion of X appears so that the target-space NLσM center-of-mass variables are
x = − 1 2 ln ε − ln ε ln ε dy 2 X(0, y 2 ) (3.51) p = − ln ε ln ε dy 2 Π(0, y 2 ). (3.52)
As a classical expression, (3.49) is indeed a saddle-point of the Euclidean path integral on the very wide strip for all values of its parameters, but note that it is not a saddle-point of the Euclidean path integral on the very wide cylinder (where y 1 ∼ y 1 − 2π is now periodic) for any values of its parameters unless k 1 + k 2 = 0 (for which a periodic saddle-point is given by p = α n = 0). The lack of a saddle on the cylinder means that the path integral on the two-holed plane vanishes. Thus, even at the level of the classical expansion we see that the path integral on the two-holed plane at finite regulator will already provide the δ(k 1 + k 2 ) appearing in the two-point function of :e ik 1 X : and :e ik 2 X :. Now, imposing the canonical commutation relations fixes the commutators between the operators x, p and α n to be
[x, p] = i (3.53) [α n , α m ] = nδ n,−m ,(3.54)
with all other commutators vanishing.
Next we want to compute the Hamiltonian and evolution operator so that the angular quantization Hilbert space can be identified with the Fock space of positive-energy mode operators acting on a Fock vacuum annihilated by all negative-energy mode operators. We have already seen that T y 1 y 1 = T (y)+ T (y)+T α α , so the Rindler Hamiltonian on the very wide strip associated to Neumann-like boundary conditions on both ends is computed by
H NN R (y 1 ) = 1 2π − ln ε ln ε dy 2 T (y) + T (y) − 1 2π
[ik 1 X(y 1 , y 2 = ln ε) + ik 2 X(y 1 , y 2 = − ln ε)] .
(3.55) It remains to compute the bulk stress-energy tensors T (y) and T (y). Below we will see that the usual relations α † n = α −n and [α n , α −n ] = n identify α n<0 as creation operators and α n>0 as annihilation operators. So define creation/annihilation normal ordering • •O • • in canonical quantization as the mode expansion where the operators x and α −n (n > 0) are all placed to the left of the operators p and α n (n > 0); it is of course irrelevant whether we group x with the creation operators and p with the annihilation operators or vice versa, so we make this aesthetic choice as we shall usually take a momentum basis of the non-oscillator Hilbert space. Then, the bulk stress-energy tensors on the strip are
T (y) = (k 1 + k 2 ) 2 16 ln 2 ε y 2 − π(k 1 + k 2 ) 4 √ 2 ln 2 ε y n∈Z α n e πin 2 e − πny 2 ln ε (3.56) + π 2 8 ln 2 ε n,m∈Z • •α n α m • •e πi(n+m) 2 e − π(n+m)y 2 ln ε − π 2 96 ln 2 ε T (y) = (k 1 + k 2 ) 2 16 ln 2 ε y 2 − π(k 1 + k 2 ) 4 √ 2 ln 2 ε y n∈Z α n e − πin 2 e − πny 2 ln ε (3.57) + π 2 8 ln 2 ε n,m∈Z • • α n α m • •e − πi(n+m) 2 e − π(n+m)y 2 ln ε − π 2 96 ln 2 ε ,
where we have introduced the oscillator zero-modes
α 0 ≡ √ 2 p − i(k 1 − k 2 ) 2π ln ε (3.58) α 0 ≡ √ 2 p + i(k 1 − k 2 ) 2π ln ε (3.59)
and defined α n = α −n for all n = 0. The finite-regulator Rindler Hamiltonian (3.55) computed from the above mode expansions is where we split the exponentials using the Zassenhaus form of the Baker-Campbell-Hausdorff formula.
H NN R = − πp 2 2 ln ε − i(k 1 + k 2 ) 2π x − π 2 ln ε ∞ n=1 α −n α n − (k 1 − k 2 ) 2 8π ln ε − (k 1 + k 2 ) 2 24π ln ε + π 48 ln ε .
Note that x appears in the Hamiltonian itself and is hence not a zero-mode in the strict sense of the word; more accurately it is a zero-momentum mode, so we shall refrain from using the conflicting term "zero-mode" as much as possible. Recall that in radial quantization of a unitary theory, real operators obey reflection positivity; see, for instance, [12]. Here, however, the free boson is defined on a complexified contour, so the appropriate notion of Hermiticity in angular quantization is
X(y 1 , y 2 ) † k 1 ,k 2 = X(−y 1 , y 2 ) −k 1 ,−k 2 ,(3.62)
where we have labeled its dependence on the boundary conditions explicitly, from which it follows that x † = x, p † = p and α † n = α −n as expected. The Rindler Hamiltonian (3.60) then also obeys
H NN R (k 1 , k 2 ) † = H NN R (−k 1 , −k 2 ). (3.63)
One may formally work with purely imaginary momenta and say that the free boson and the Rindler Hamiltonian are ordinary Hermitian operators in this analytically continued theory. Note that this is actually the situation that will arise in later applications to winding operators in certain theories with a linear dilaton. For real Lorentzian momenta, note that the vertex operator :e ikX : is itself non-Hermitian. Moreover, the only non-Hermitian part of the Hamiltonian is the constant-mode term − i(k 1 +k 2 ) 2π
x, whose role is to provide the correct momentum-conserving delta functions when computing thermal traces, as apparent from the appearance of e i(k 1 +k 2 )x in the evolution operator (3.61). The non-invariance of the Hamiltonian under the shift symmetry of the free boson is not problematic and in fact is required, as it encapsulates the effects of endpoint operators which do transform nontrivially under this symmetry.
The finite-regulator angular quantization Hilbert space H NN k 1 k 2 is defined as the Fock space built on top of the oscillator vacuum states |p; {0} defined by α n |p; {0} = 0 for all n > 0 and p|p; {0} = p|p; {0} ; since we shall always work in the momentum basis for the center-of-mass modes, we abuse notation and do not make a distinction between the momentum operator and its eigenvalue. Explicitly, the states of H NN k 1 k 2 may be written as
|p; {N n } ≡ ∞ n=1 1 n Nn N n ! α Nn −n |p; {0} (3.64)
where N n 0 is the integer occupation number of the n th mode, which are Dirac orthonormalized in the sense
p ′ ; {N ′ n }|p; {N n } = 2πδ {N ′ n },{Nn} δ(p ′ − p). (3.65)
This basis of H NN k 1 k 2 is not an eigenbasis of the Rindler Hamiltonian (3.60) due to the fact that x is not a zero-mode for k 1 + k 2 = 0, but we only ever need to compute traces over the angular quantization Hilbert space, which may be performed in any Dirac-orthonormal basis.
The finite-regulator cylinder thermal trace over (3.61) is straightforwardly computed as
Tr H NN k 1 k 2 e −2πH NN R S 1 ×R = δ(k 1 + k 2 ) √ 2 η − 2i ln ε π e ln ε 4 (k 1 −k 2 ) 2 + ln ε 12 (k 1 +k 2 ) 2 + π 2 12 ln ε (k 1 +k 2 ) 2 ,(3.66)
where η(τ ) is the Dedekind eta function (see Appendix A for relevant formulas) and the delta function arose from p|e i(k 1 +k 2 )x |p = p − k 1 − k 2 |p = 2πδ(k 1 + k 2 ). The dependence in the exponential on (k 1 + k 2 ) 2 then disappears on the support of the delta function. The result in the sphere conformal frame is simply obtained by multiplying by the Weyl transformation factor ε 1/6 from (2.18). Therefore, since the dimensions of the endpoint operators are ∆ 1 = k 2 1 2 and ∆ 2 = k 2 2 2 and we may write 1 4 (k 1 − k 2 ) 2 = ∆ 1 + ∆ 2 on the support of the delta function, the finite-regulator thermal trace on the sphere is
Tr H NN k 1 k 2 e −2πH NN R S 2 = δ(k 1 + k 2 ) √ 2 ε ∆ 1 +∆ 2 ε −1/6 η − 2i ln ε π .
(3.67)
We have written the result in this way because of the limit
lim ε→0 ε −1/6 η − 2i ln ε π = 1. (3.68)
Hence, we have shown that the angular quantization two-point function matching condition (2.9) is indeed satisfied, namely
lim ε→0 ε −∆ 1 −∆ 2 Tr H NN k 1 k 2 e −2πH NN R S 2 = 2πN NN k 1 k 2 δ(k 1 + k 2 ), (3.69) with the normalization constant N NN k 1 k 2 = 1 2 √ 2π . (3.70)
In principle, the normalization constant appearing in (2.10) could be this result times f (k 1 +k 2 ), where f (0) = 1; the factorization of N NN k 1 k 2 still allows a nontrivial function f (k 1 + k 2 ) given, for instance, by an exponential. The exact constant N NN k 1 k 2 as a function of k 1 and k 2 is only uniquely determined by matching the thermal one-point functions computed in the following subsection; we shall find that (3.70) is the full answer.
Dirichlet-like Boundary Conditions
Consider the Dirichlet-like boundary conditions (3.12) on the very wide strip. These boundary conditions by themselves shrink to the identity operator. To obtain the endpoint operators :e ik 1 X(0) : and :e ik 2 X(∞) : we add appropriate factors of ikX to the boundary action, which will dynamically fix the operator charges.
The free boson action which imposes the Dirichlet-like boundary conditions that shrink to :e ik 1 X(0) : and :e ik 2 X(∞) : is then given by
S = 1 4π dy 1 dy 2 (∂ y 1 X) 2 + (∂ y 2 X) 2 − i 2π y 2 =ln ε dy 1 (k 1 X − λ 1 ∂ y 1 X) − i 2π y 2 =− ln ε dy 1 (k 2 X + λ 2 ∂ y 1 X) , (3.71)
where λ 1 (y 1 ) and λ 2 (y 1 ) are boundary Lagrange multipliers imposing the vanishing of ∂ y 1 X there. In addition to the bulk equation of motion (∂ 2 y 1 + ∂ 2 y 2 )X = 0 and boundary conditions ∂ y 1 X(y 2 = ∓ ln ε) = 0, the equations of motion for the boundary Lagrange multipliers are
∂ y 1 λ 1 = i∂ y 2 X y 2 =+ ln ε − k 1 (3.72) ∂ y 1 λ 2 = i∂ y 2 X y 2 =− ln ε + k 2 . (3.73)
The stress-energy tensor for this theory is
T αβ = −: ∂ α X∂ β X − 1 2 δ αβ ∂ γ X∂ γ X : − t α t β X [ik 1 δ(y 2 − ln ε) + ik 2 δ(y 2 + ln ε)] − t (α n β) ∂ y 2 X [iλ 1 δ(y 2 − ln ε) + iλ 2 δ(y 2 + ln ε)] , (3.74)
where t α and n α are the unit tangent and normal vectors to the boundary, respectively. Despite having different boundary conditions and a different form of the boundary stress-energy tensor, the Dirichlet-like theory has the same classical trace as the Neumann-like theory, namely T α α (y 1 , y 2 ) = −X(y 1 , y 2 ) [ik 1 δ(y 2 − ln ε) + ik 2 δ(y 2 + ln ε)] , (3.75) as well as the same expression which computes the Rindler Hamiltonian, namely
H DD R (y 1 ) = 1 2π − ln ε ln ε dy 2 T (y) + T (y) − 1 2π
[ik 1 X(y 1 , y 2 = ln ε) + ik 2 X(y 1 , y 2 = − ln ε)] .
(3.76) Now, however, the momentum density canonically conjugate to X is
Π(y 1 , y 2 ) = − i 2π ∂ y 1 X(y 1 , y 2 ) + 1 2π [λ 1 (y 1 )δ(y 2 − ln ε) − λ 2 (y 1 )δ(y 2 + ln ε)] . (3.77)
The Dirichlet-like free boson X has two continuous degrees of freedom, a spatially constant mode and a spatially linear mode, and the Lagrange multipliers λ 1 and λ 2 each have a continuous constant mode. We shall write the non-oscillator part of the X expansion as x+(x s − i(k 1 −k 2 ) 2 )y 2 , calling x the constant mode and x s the "slope" mode; the definition of the slope mode with the constant shift − i(k 1 −k 2 ) 2 is only for later convenience, but its presence mimics the same term in the Neumann-like expansion (3.49). Then, we see from the above form of Π that the constant modes of λ 1 and λ 2 should be linear combinations of the conjugate momenta of x and x s . Specifically, mode expanding X, λ 1 and λ 2 on the strip and imposing the equal-time canonical commutation relations 14 between X and Π results in
X(y 1 , y 2 ) = x + x s − i(k 1 − k 2 ) 2 y 2 − √ 2 n∈Z n =0
α n n e − πny 1 2 ln ε sin πn(y 2 − ln ε) 2 ln ε (3.78) As usual, we are free to define the operators in the mode expansion of X however we wish, and we may make any operator redefinitions that preserve the commutators. With these definitions, the Heisenberg time evolutions of the operators p and p s are p(y 1 ) = p − k 1 +k 2 2π y 1 and p s (y 1 ) = p s + ixs π y 1 ln ε, which are indeed the total target space momentum and first moment thereof,
λ 1 (y 1 ) = +π p + p s ln ε + i x s + i(k 1 + k 2 ) 2 y 1 + i √ 2 n∈Z n =0 α n n e − πny 1 2 ln ε (3.79) λ 2 (y 2 ) = −π p − p s ln ε + i x s − i(k 1 + k 2 ) 2 y 1 + i √ 2 n∈Z n =0 α n n (−1) n e − πny 1 2 ln ε ,(3.− ln ε ln ε dy 2 Π(y 1 , y 2 ) = p − k 1 + k 2 2π y 1 (3.83) − ln ε ln ε dy 2 y 2 Π(y 1 , y 2 ) = p s + ix s π y 1 ln ε. (3.84)
The constant mode x, on the other hand, is no longer the target space center-of-mass, which now involves the oscillators as well. There is no point separating out this oscillator contribution, as the constant mode x will become the center-of-mass in the shrinking limit ε → 0. In the special case k 1 = k 2 = 0, (−2 ln ε)x s is the distance between the string endpoints, so − 1 2 ln ε p s is the momentum of one endpoint relative to the other.
Hermitian conjugation works as before, where now in addition λ 1 (y 1 ) † k 1 ,k 2 = λ 1 (−y 1 ) −k 1 ,−k 2 and λ 2 (y 1 ) † k 1 ,k 2 = λ 2 (−y 1 ) −k 1 ,−k 2 ; formally, one may again consider the case of pure imaginary k 1 and k 2 for which the target space contour is real and Hermiticity behaves normally. Therefore, x, p, x s and p s are all Hermitian, and α † n = α −n . Note moreover that there exists a Euclidean saddle point on the cylinder if and only if k 1 + k 2 = 0, where the classical saddle point obeys x s = α n = 0 with p and p s arbitrary; we have defined the slope mode of X with the shift − i(k 1 −k 2 ) 2 y 2 precisely so that the classical saddle point occurs at x s = 0.
14 Strictly speaking, since λ1 and λ2 do not have canonical conjugates, one should more properly perform Dirac quantization with the constraints that Π λ 1 and Π λ 2 both vanish and that λ1 and λ2 are proportional to ΠX(y2 = ln ε) and ΠX (y2 = − ln ε), respectively. One may explicitly check that the results of Dirac quantization are identical to those of canonical quantization here, essentially because the constraints are all compatible with a truncation of the original Poisson algebra. 15 To obtain this result, the Fourier expansions
1 ± y2 ln ε = 1 ± 1 if y2 = ln ε ∓ 2 π n =0 (±1) n n sin πn(y 2 −ln ε) 2 ln ε if ln ε < y2 < − ln ε 1 ∓ 1 if y2 = − ln ε are useful.
The bulk stress-energy tensors on the strip are T (y) = π 2 8 ln 2 ε n,m∈Z
• •α n α m • •e πi(n+m) 2 e − π(n+m)y 2 ln ε − π 2 96 ln 2 ε (3.85) T (y) = π 2 8 ln 2 ε n,m∈Z • •α n α m • •e − πi(n+m) 2 e − π(n+m)y 2 ln ε − π 2 96 ln 2 ε ,(3.86)
where we have defined
α 0 ≡ − √ 2 π x s − i(k 1 − k 2 ) 2 ln ε; (3.87)
by convention we shall also group x and p s with the creation operators and p and x s with the annihilation operators. Therefore, the Rindler Hamiltonian on the very wide strip associated to Dirichlet-like boundary conditions on both ends, as computed by (3.76), is
H DD R = − i(k 1 + k 2 ) 2π x − x 2 s 2π ln ε − π 2 ln ε ∞ n=1 α −n α n − (k 1 − k 2 ) 2 8π ln ε + π 48 ln ε . (3.88)
This Hamiltonian is time independent, and moreover the evolution operator e ∆y 1 H DD R is trivially split as the product of the individual exponentials since all the terms in (3.88) mutually commute. The finite-regulator angular quantization Hilbert space H DD k 1 k 2 on the strip thus consists of the tensor product of the oscillator Fock space and the two continuum modes; we write the general state as |p, x s ; {N n } , taking the momentum basis for (x, p) and the position basis for (x s , p s ), with conventional normalization [e −2πH DD R ] is computing. The cylinder action (3.71) has two independent symmetries corresponding to constant shifts of the Lagrange multipliers, since ∂ y 1 X integrates along the boundary to zero. These shift symmetries would cause the path integral to diverge if left untreated. However, it is a familiar fact that any path integral with a noncompact symmetry must be divided by the volume of the symmetry group to render the partition function welldefined and finite 16 . Note that this divergence is not present for the path integral on the strip, as the integrals over the zero-modes of λ 1 and λ 2 in that case just produce delta functions equating the boundary values of X in the infinite future and the infinite past. Thus the treatment of the path integral divergence on the cylinder should be interpreted as how to define the trace in angular quantization. As such, the zero-mode sector of the trace in angular quantization must always be divided by the zero-mode volume factor
p ′ , x ′ s ; {N ′ n }|p, x s ; {N n } = 2πδ {N ′ n },{Nn} δ(p ′ − p)δ(x ′ s − x s ).Vol(λ 1,0 , λ 2,0 ) = ∞ −∞ dλ 1,0 ∞ −∞ dλ 2,0 = − 8π 4 ln ε ∞ −∞ dp 2π ∞ −∞ dp s 2π . (3.90)
This infinite division also makes sense from the point of view of the angular quantization Hamiltonian (3.88), where it is obvious that p and p s are true continuous zero-modes, making all traces proportional to an overall unphysical infinity 17 . For example, the zero-mode part of the trace of the cylinder evolution operator is
Tr (p,xs) e −2πH DD R = 1 Vol(λ 1,0 , λ 2,0 ) ∞ −∞ dp 2π p|e i(k 1 +k 2 )x |p ∞ −∞ dx s x s |e x 2 s + (k 1 −k 2 ) 2 4 ln ε |x s (3.91) = − ln ε 4π 3 δ(k 1 + k 2 ) ∞ −∞ dp 2π ∞ −∞ dp 2π x s |x s ∞ −∞ dps 2π − π ln ε e (k 1 −k 2 ) 2 4 ln ε (3.92) = δ(k 1 + k 2 ) 4π 3 √ −π ln ε ε (k 1 −k 2 ) 2 /4 . (3.93)
The full finite-regulator cylinder thermal trace is therefore
Tr H DD k 1 k 2 e −2πH DD R S 1 ×R = δ(k 1 + k 2 ) 4 √ 2π 2 ε k 2 2 η − 2i ln ε π .
(3.94)
Transforming back to the sphere again requires the Weyl transformation factor ε 1/6 from (2.18), and therefore the finite-regulator thermal trace on the sphere is
Tr H DD k 1 k 2 e −2πH DD R S 2 = δ(k 1 + k 2 ) 4 √ 2π 2 ε ∆ 1 +∆ 2 ε −1/6 η − 2i ln ε π .
(3.95)
Thus, the angular quantization two-point function matching condition (2.9) is also satisfied with these boundary conditions, namely
lim ε→0 ε −∆ 1 −∆ 2 Tr H DD k 1 k 2 e −2πH DD R S 2 = 2πN DD k 1 k 2 δ(k 1 + k 2 ),(3.96)
with the normalization constant
N DD k 1 k 2 = √ 2π (2π) 4 .
(3.97)
As before, matching the two-point function is not sensitive to an overall function f (k 1 +k 2 ) with f (0) = 1 multiplying this normalization, which can only be completely determined by matching the thermal one-point functions; we shall do this computation in the following subsection and find that (3.97) is the full answer.
Thermal One-Point Function
The last necessary computation is the matching condition were derived explicitly in the previous subsection, so it remains only to compute the mode expansions of the exponential primaries O k 3 (z, z) = :e ik 3 X(z,z) : in order to evaluate the traces. The mode expansions are most easily obtained by relating the point-split normal-ordered product :X(z 1 , z 1 )X(z 2 , z 2 ): to the creation/annihilation normal-ordered product • •X (z 1 , z 1 )X(z 2 , z 2 ) • •, taking the coincident limit and using Wick's Theorem to exponentiate this result. We perform these calculations directly on the two-holed sphere, since we already know that the evolution operator on the sphere differs from that on the cylinder by the Weyl factor ε 1/6 in both cases.
Neumann-like Boundary Conditions
Since X transforms with weights (h, h) = (0, 0), the mode expansion X(z, z) of the free boson on the two-holed sphere is immediately obtained from the mode expansion (3.49) by setting y = i ln z. The relation between conformal and creation/annihilation normal ordering on the plane is then straightforwardly computed as
:X(z 1 , z 1 )X(z 2 , z 2 ): = • •X (z 1 , z 1 )X(z 2 , z 2 ) • • − 1 2 ln z −πi/2 ln ε 1 − z −πi/2 ln ε 2 (z 1 z 2 ) −πi/2 ln ε z 12 1 z −πi/2 ln ε 1 + z −πi/2 ln ε 2 2 . (3.98)
Taking the coincident limit and using Wick's Theorem, the relation between the conformal and creation/annihilation normal-ordered exponential operator on the plane is :e ik 3 X(z,z) : = π 2 ln 2 ε z z πi/ ln ε 1 |z| 2 cos 2 π ln |z| 2 ln ε
k 2 3 /4 • •e ik 3 X(z,z)• •,(3.99)
where the creation/annihilation normal-ordered exponential is
• •e ik 3 X(z,z)• • = exp − (k 1 + k 2 )k 3 12 ln ε + (k 1 − k 2 )k 3 2 ln |z| + (k 1 + k 2 )k 3 8 ln ε ln 2 z + ln 2 z (3.100) × exp ik 3 x + √ 2k 3 ∞ n=1 α −n n z z πin/4 ln ε cos πn 2 ln ε ln z ε × exp πik 3 p 2 ln ε ln z z − √ 2k 3 ∞ m=1 α m m z z −πim/4 ln ε cos πm 2 ln ε ln z ε .
It is now a straightforward, albeit slightly tedious, exercise to compute the finite-regulator thermal trace with this mode expansion inserted, using the cylinder evolution operator (3.61) and the Weyl transformation back to the sphere. The (x, p) sector in the trace is still simple, and the details of the oscillator calculation are provided in Appendix A; the final result is
Tr H NN k 1 k 2 O k 3 (z, z)e −2πH NN R S 2 = δ(k 1 + k 2 + k 3 ) √ 2ε − 1 6 η − 2i ln ε π ε (k 2 1 +k 2 2 )/2 |z| (k 2 1 +k 2 3 −k 2 2 )/2 ε − 1 6 η − 2i ln ε π ϑ 4 − i ln |z| π − 2i ln ε π k 2 3 /2 , (3.101)
where ϑ 4 (ν|τ ) is the fourth Jacobi elliptic theta function, obeying lim τ →i∞ ϑ 4 (ν|τ ) = 1. Since the three-point coefficient of exponential primaries is C k 1 k 2 k 3 = 2πδ(k 1 + k 2 + k 3 ), we indeed find the desired matching condition (2.10), namely
lim ε→0 ε − k 2 1 2 − k 2 2 2 Tr H NN k 1 k 2 O k 3 (z, z)e −2πH NN R S 2 = N NN k 1 k 2 C k 1 k 3 k 2 |z| (k 2 1 +k 2 3 −k 2 2 )/2 ,(3.102)
with the same normalization constant (3.70).
Dirichlet-like Boundary Conditions
k 2 3 /2 • •e ik 3 X(z,z)• •, (3.104) where • •e ik 3 X(z,z)• • = e ik 3 x |z| ik 3 xs+ (k 1 −k 2 )k 3 2 exp −i √ 2k 3 ∞ n=1 α −n n z z πin/4 ln ε sin πn 2 ln ε ln z ε × exp −i √ 2k 3 ∞ m=1 α m m z z −πim/4 ln ε sin πm 2 ln ε ln z ε . (3.105)
The trace is computed similarly as before, remembering to divide by the volume of the Lagrange multiplier zero-modes. The result is
Tr H DD k 1 k 2 O k 3 (z, z)e −2πH DD R S 2 = δ(k 1 + k 2 + k 3 ) 4 √ 2π 2 ε − 1 6 η − 2i ln ε π ε (k 2 1 +k 2 2 )/2 |z| (k 2 1 +k 2 3 −k 2 2 )/2 ε − 1 2 η 3 − 2i ln ε π ϑ 4 − i ln |z| π − 2i ln ε π k 2 3 /2 .
(3.106) We again find the desired matching condition (2.10), namely
lim ε→0 ε − k 2 1 2 − k 2 2 2 Tr H DD k 1 k 2 O k 3 (z, z)e −2πH DD R S 2 = N DD k 1 k 2 C k 1 k 3 k 2 |z| (k 2 1 +k 2 3 −k 2 2 )/2 ,(3.107)
with the same normalization constant (3.97) as before.
As a final comment, it may not be surprising that the normalization constants N NN k 1 k 2 and N DD k 1 k 2 are essentially those appearing in the Cardy states associated to the conformally-invariant Neumann and Dirichlet boundary conditions. Recall that the latter are non-normalizable states determined by the Cardy condition [9], i.e. they are constructed from the usual Hilbert space on S 1 so that their annulus matrix elements reproduce the cylinder traces over the strip Hilbert spaces (with conventional spatial length of π) with Neumann or Dirichlet boundary conditions at the endpoints. The usual Cardy states associated to ∂ n X = 0 and to X = x 0 are then determined to be
|N = 1 (8π 2 ) 1/4 e − ∞N NN k 1 k 2 = N 2 |N (3.110) N DD k 1 k 2 = N |D (2π) 2 2 , (3.111)
which is to be expected since the oscillator parts of the thermal trace matching computations above are nearly identical to those appearing in the Cardy condition 18 . We should also point out that the conformally-invariant boundary condition on which the Dirichlet-like approach here is based is ∂ t X = 0, whose associated boundary state is
|D = ∞ −∞ dx 0 |D, x 0 = (2π 2 ) 1/4 e ∞ n=1 1 n α −n α −n |0; {0}, {0} . (3.112)
It is this boundary state which shrinks to the identity operator. Even though the lowest dimension state appearing in the expression (3.109) is that corresponding to the identity, |D, x 0 also contains a continuum of states with infinitesimal dimension all with the same coefficient, and the contribution from the identity operator cannot be separated from this tail of exponential primaries. It is in this sense that pure Dirichlet boundary conditions are pathological for the noncompact free boson, a problem which is obviously rectified in the compact theory.
Free CFT Examples
Having presented the angular quantization of the noncompact free boson for the two most natural boundary conditions in great detail, in this section we present the results of angular quantization for three other free CFTs -the linear dilaton, the compact free boson and the bc ghost system. As these CFTs are free, we again analyze them in canonical quantization, leading to explicit Fock space constructions of the associated Hilbert spaces. Many of the details are only slight modifications of those presented for the noncompact free boson in Section 3, so after deriving the relevant boundary conditions we shall focus primarily on presenting the results and noting where they differ; some computational details are again relegated to an Appendix.
Linear Dilaton
Even though the linear dilaton CFT is also somewhat pathological -having a non-closed OPE 19 and being logarithmic -its appearance in string theory makes it relevant for our later applications in [8]. The linear dilaton φ with background charge Q has central charges c = c = 1 + 6Q 2 ; we normalize the linear dilaton on the plane in the same way as the free boson, so that its OPE with itself is φ(z 1 , z 1 )φ(z 2 , z 2 ) = − 1 2 ln |z 12 | 2 + :φ(z 1 , z 1 )φ(z 2 , z 2 ):. (4.1)
The exponential primaries are O α ≡ :e 2αφ : with conformal weights h α = h α = α(Q − α). We shall perform the angular quantization for the linear dilaton with exponential endpoint operators analogously to the Neumann-like approach for the noncompact free boson, but an important distinction is that the linear dilaton field φ(z, z) has different conformal transformations. Under a conformal transformation associated to z → z ′ (z), the field transforms as
φ(z, z) → φ ′ (z, z), where φ ′ (z ′ , z ′ ) = φ(z, z) − Q 2 ln ∂ z z ′ 2 . (4.2)
The anomalous holomorphic current ∂φ(z) does transform as a weight-(1, 0) operator but is not a primary; its anomalous conformal transformation is instead given by
(∂φ) ′ (z ′ ) = (∂ z z ′ ) −1 ∂φ(z) − Q 2 ∂ 2 z z ′ (∂ z z ′ ) 2 . (4.3)
The most important conformal transformation for us is that associated with the map z = e −iy between the flat plane and the flat cylinder; for ease of notation, we shall drop the prime on 19 Alternatively, one could take its closure under the OPE at the cost of unitarity.
the transformation of the operator itself, instead letting the argument determine whether the operator is the one on the plane with coordinates (z, z) or on the strip with coordinates (y, y) or equivalently (y 1 , y 2 ). Then, the transformations of the linear dilaton and its derivatives from the plane to the cylinder are φ(y, y) = φ z(y), z(y) + Qy 2 (4.4)
∂φ(y) = −iz(y)∂φ z(y) − iQ 2 (4.5)
∂φ(y) = +iz(y)∂φ z(y) + iQ 2 .
(4.6)
There will thus be a shift in the boundary conditions shrinking to exponential operators on the plane versus on the cylinder. We now wish to write Neumann-like boundary conditions on the plane at |z| = ε and at |z| = 1 ε which shrink to O α 1 = :e 2α 1 φ(0) : and to O α 2 = :e 2α 2 φ(∞) :, respectively. Naïvely, the appropriate boundary conditions might seem to be obtained from the free boson ones with the replacement k → −2iα, but this is not quite true. Recall that the covariant linear dilaton action on a closed Riemann surface Σ with metric g αβ is
S = 1 4π Σ d 2 σ √ g g αβ ∂ α φ∂ β φ + QRφ , (4.7)
with equation of motion ∇ 2 φ = 1 2 QR. Since the sphere has Euler characteristic χ S 2 = 2, working in the "flat" sphere conformal frame really means using the flat metric in the z-patch stereographic plane with a curvature singularity at infinity such that R = 8π √ g δ 2 (z − ∞). Therefore, when inserting the operators :e 2α 1 φ(0) : and :e 2α 2 φ(∞) : into the path integral on the sphere, the effective action (in the sense of absorbing the contributions of the operator insertions into the action up to some renormalization terms 20 ) in the flat z-coordinates is
S = 1 2π d 2 z ∂φ∂φ − 2α 1 φ(|z| = 0) + 2(Q − α 2 )φ(|z| = ∞). (4.8)
Thus the Neumann-like boundary conditions are obtained from those of the free boson via the replacements k 1 → −2iα 1 and k 2 → −2i(α 2 − Q). That is, on the two-holed sphere we impose the boundary conditions
r∂ r φ |z|=ε = −2α 1 (4.9) r∂ r φ |z|= 1 ε = +2(α 2 − Q). (4.10)
Another way to obtain the latter result is to apply the inversion z → z ′ = 1/z using the parity-reversed version of the anomalous transformation law (4.2), which says that φ ′ ( 1 z , 1 z ) = φ(z, z) + 2Q ln |z|. Since the inverted operator obeys the boundary condition near the origin r ′ ∂ r ′ φ ′ (r ′ = ε) = −2α 2 without subtleties, the boundary condition in the original z-frame is
r∂ r φ(r = 1 ε ) = −r ′ ∂ r ′ φ ′ (r ′ = ε) − 2Q = 2(α 2 − Q)
. These statements are equivalent to the familiar fact that the linear dilaton theory on the stereographically-projected sphere behaves like a free boson with a "background charge operator" :e −2Qφ : placed at infinity. Using (4.4), the appropriate boundary conditions on the very wide strip are
∂ y 2 φ y 2 =+ ln ε = −2α 1 + Q (4.11) ∂ y 2 φ y 2 =− ln ε = +2α 2 − Q. (4.12)
The boundary conditions on the strip are more symmetric, corresponding to the fact that the infinite cylinder without boundaries is typically described in the flat y-patch frame with curvature singularities equally distributed between y 2 = ±∞. Note also that there can only be a Euclidean saddle on the two-holed sphere or on the very wide cylinder if the Neumannlike boundary conditions at ln |z| = y 2 = ± ln ε are equal, meaning that the corresponding Euclidean path integrals will only be nonvanishing when α 1 + α 2 = Q, reproducing the wellknown anomalous conservation law of the linear dilaton translation current. The strip action imposing the boundary conditions (4.11) and (4.12) is
S = 1 4π dy 1 dy 2 (∂ y 1 φ) 2 + (∂ y 2 φ) 2 − 2α 1 −Q 2π y 2 =ln ε dy 1 φ − 2α 2 −Q 2π y 2 =− ln ε dy 1 φ. (4.13)
One must be careful in computing the stress-energy tensor, however, as even the "pure" linear dilaton on a Riemann surface with boundary has a boundary contribution to the stress-energy tensor, which is not often written in the literature since one typically works with a boundary condition in a frame where this term vanishes. It is useful first to write the fully covariant stressenergy tensor for the "pure" linear dilaton theory on a manifold with boundary, i.e. without including the extra boundary terms yet which will impose our choice of boundary conditions. The covariant pure action is
S pure = 1 4π Σ d 2 σ √ g g αβ ∂ α φ∂ β φ + QRφ + 1
2π ∂Σ dℓ f * g QKφ, (4.14) whose covariant stress-energy tensor is
T pure αβ = −: ∇ α φ∇ β φ− 1 2 g αβ ∇ γ φ∇ γ φ−Q ∇ α ∇ β φ−g αβ ∇ 2 φ :+ Qδ(Σ − ∂Σ)(g αβ − n α n β )n γ ∇ γ φ.
(4.15) The boundary contribution to T pure αβ only vanishes identically if one chooses exactly Neumann boundary conditions n α ∇ α φ ∂Σ = 0, which is not the case here (and is not even the conformallyinvariant choice). The full boundary action in a general frame is the strip boundary action in (4.13) plus the extrinsic curvature term in (4.14). Including the boundary contribution from the "pure" stress-energy tensor with the normal derivatives (4.11) and (4.12) as well as the boundary terms from the additional boundary action in (4.13) on the strip (for which R = 0 and K = 0), the correct stress-energy tensor for this theory in the flat cylinder frame is
T αβ (y 1 , y 2 ) = −: ∂ α φ∂ β φ − 1 2 δ αβ ∂ γ φ∂ γ φ − Q∂ α ∂ β φ : − t α t β (φ − Q) [(2α 1 − Q)δ(y 2 − ln ε) + (2α 2 − Q)δ(y 2 + ln ε)] . (4.16)
The Rindler Hamiltonian with these boundary conditions is therefore computed by
H LD R = 1 2π − ln ε ln ε dy 2 T (y) + T (y) − 1 2π (2α 1 − Q)(φ − Q) y 2 =ln ε + (2α 2 − Q)(φ − Q) y 2 =− ln ε ,(4.17)
where T (y) = −:∂φ(y)∂φ(y): + Q∂ 2 φ(y) (4.18)
T (y) = −:∂φ(y)∂φ(y): + Q∂ 2 φ(y) (4.19) are the usual bulk holomorphic and antiholomorphic components of the stress-energy tensor. Before proceeding, let us briefly comment on conformal invariance in boundary linear dilaton theory. A linear boundary condition for a linear dilaton is frame dependent since φ does not have definite weights. The boundary condition imposed by the pure linear dilaton action on its own is n α ∇ α φ = −QK, which is exactly Neumann only in frames where the boundary has no extrinsic curvature. The presence of the boundary stress-energy tensor even in the pure theory is a consequence of the fact that the background charge Q already gives rise to a "classical central charge". In particular, using the bulk equation of motion ∇ 2 φ = 1 2 QR and boundary condition n α ∇ α φ = −QK arising from the action S pure with no additional boundary terms, the classical trace of the pure stress-energy tensor is (T α α ) pure = − 1 2 Q 2 R−Q 2 δ(Σ−∂Σ)K, which is indeed the form of the trace dictated by diffeomorphism invariance for "classical central charge" c cl. = 6Q 2 . The difference between the actual quantum stress-energy tensor and the classical one is identical to that of the free boson, since such effects arise from coincident operator renormalization which is due only to the quadratic kinetic term in the action. The action S pure is hence conformally invariant with central charge c = 1 + 6Q 2 because the trace of the stress-energy tensor is exactly zero on a flat space with flat boundaries. Thus, the correct conformally-invariant boundary condition for the linear dilaton is actually n α ∇ α φ = −QK, which we still call "Neumann" even though the normal derivative only vanishes at a flat boundary, corresponding to the construction here on the strip/cylinder with α 1 = α 2 = Q 2 . Therefore, the conformally-invariant linear dilaton Neumann boundary condition actually shrinks to the scalar primary :e Qφ : of dimension ∆ = Q 2 2 on the sphere. This is a simple example of a conformally-invariant local boundary condition that does not shrink to the identity.
Back to the action (4.13), the strip mode expansions of the dilaton and its conjugate momentum Π φ = − i 2π ∂ y 1 φ satisfying the bulk equation of motion and the boundary conditions are
φ(y 1 , y 2 ) = ϕ + α 1 + α 2 − Q 6 ln ε − πip ln ε y 1 − (α 1 − α 2 )y 2 + α 1 + α 2 − Q 2 ln ε y 2 1 − y 2 2 (4.20) + i √ 2 n∈Z n =0 φ n n e − πn 2 ln ε y 1 cos πn(y 2 − ln ε) 2 ln ε Π φ (y 1 , y 2 ) = − p ϕ 2 ln ε − i(α 1 +α 2 −Q) 2π ln ε y 1 − 1 2 √ 2 ln ε n∈Z n =0
φ n e − πn 2 ln ε y 1 cos πn(y 2 − ln ε) 2 ln ε ,
T (y) = − (α 1 + α 2 − Q) 2 4 ln 2 ε y 2 + πi(α 1 + α 2 − Q) 2 √ 2 ln 2 ε y n∈Z φ n e πin 2 e − πny 2 ln ε (4.22) + π 2 8 ln 2 ε n,m∈Z • •φ n φ m • •e πi(n+m) 2 e − π(n+m)y 2 ln ε − π 2 96 ln 2 ε + Q∂ 2 φ(y) + iQ∂φ(y) − Q 2 4 T (y) = − (α 1 + α 2 − Q) 2 4 ln 2 ε y 2 + πi(α 1 + α 2 − Q) 2 √ 2 ln 2 ε y n∈Z φ n e − πin 2 e − πny 2 ln ε (4.23) + π 2 8 ln 2 ε n,m∈Z
The thermal trace and bulk thermal one-point function on the two-holed sphere are
Tr H LD α 1 α 2 e −2πH LD R S 2 = δ(α 1 +α 2 −Q) 2 √ 2 ε 2α 1 (Q−α 1 )+2α 2 (Q−α 2 ) ε −1/6 η − 2i ln ε π (4.28) Tr H LD α 1 α 2 O α 3 (z, z)e −2πH LD R S 2 = δ(α 1 +α 2 +α 3 −Q) 2 √ 2 ε 2α 1 (Q−α 1 )+2α 2 (Q−α 2 ) |z| 2[α 1 (Q−α 1 )+α 3 (Q−α 3 )−α 2 (Q−α 2 )] (4.29) × 1 ε − 1 6 η − 2i lnε π ε − 1 6 η − 2i lnε π ϑ 4 − i ln|z| π − 2i lnε π −2α 2 3 .
Since the canonically normalized linear dilaton sphere two-and three-point coefficients are
O α 1 (0)O α 2 (1) S 2 = πδ(α 1 + α 2 − Q) and C α 1 α 2 α 3 = πδ(α 1 + α 2 + α 3 − Q)
, we again find the required matching conditions (2.9) and (2.10), namely
lim ε→0 ε −2α 1 (Q−α 1 )−2α 2 (Q−α 2 ) Tr H LD α 1 α 2 e −2πH LD R S 2 = πN LD α 1 α 2 δ(α 1 + α 2 − Q) (4.30) lim ε→0 ε −2α 1 (Q−α 1 )−2α 2 (Q−α 2 ) Tr H LD α 1 α 2 O α 3 (z, z)e −2πH LD R S 2 = πN LD α 1 α 2 δ(α 1 + α 2 + α 3 − Q) |z| 2[α 1 (Q−α 1 )+α 3 (Q−α 3 )−α 2 (Q−α 2 )] ,(4.31)
where the normalization N LD
α 1 α 2 = 1 2 √ 2π (4.32)
is the same as in the Neumann-like angular quantization of the noncompact free boson.
Compact Free Boson
Now we turn to the free boson CFT compactified at radius R, so that X ∼ X + 2πR, rendering the spectrum discrete. The exponential primaries O n,w are characterized by integer momentum n and integer winding w; as usual on the plane we write O n,w (z, z) ≡ :e i[k L X L (z)+k R X R (z)] :, where k L ≡ n R + wR, k R ≡ n R − wR and X(z, z) = X L (z) + X R (z). The momentum and winding exponential primary O n,w has conformal weights (h, h) = ( k 2 L 4 , k 2 R 4 ), i.e. scaling dimension ∆ = 1 2 ( n 2 R 2 + w 2 R 2 ) and spin s = nw, but the expression :e i(k L X L +k R X R ) : is really only a slogan until one defines the chiral splitting more precisely, as it leaves the zero-mode ambiguous. In radial quantization, one often just says that the choice of zero-mode splitting is tantamount to a choice of branch cut, and at the end of the day the choice will be irrelevant for correlators of well-defined local operators. In more modern parlance, we would say that X L and X R are chiral defect operators and the monodromy from a branch cut is just the effect of passing an operator through the defect line [13]. When mode expanding the compact boson in radial quantization, the zero-mode ambiguity is thus left unfixed and requires appending 2-cocycles to the oscillator exponentials in order to recover the correct operator algebra [1] (just like with bosonization). In angular quantization, on the other hand, we shall find that the zero-mode splitting is actually determined by construction and moreover automatically incorporates the 2-cocycles that seemed ad hoc in radial quantization.
The example of most importance for us in future applications is the angular quantization associated to Euclidean time-winding operators. Using the results laid out here, in [8] we shall perform the BRST quantization associated to pairs of worldsheet vertex operators which wind Euclidean target-space time, describing a distinct class of string states which are neither open nor closed.
The chiral OPEs of the compact boson are X L (z 1 )X L (z 2 ) = − 1 2 ln z 12 + :X L (z 1 )X L (z 2 ): (4.33)
X R (z 1 )X R (z 2 ) = − 1 2 ln z 12 + :X R (z 1 )X R (z 2 ):, (4.34) which are well-defined as soon as we give a proper definition of the chiral splitting below. The OPEs of the radial and angular derivatives in the presence of an exponential primary are
z∂X L (z) + z∂X R (z) :e i[k L X L (0)+k R X R (0)] : = − in R :e i[k L X L (0)+k R X R (0)] : + . . . (4.35) iz∂X L (z) − iz∂X R (z) :e i[k L X L (0)+k R X R (0)] : = wR:e i[k L X L (0)+k R X R (0)] : + . . . ; (4.36)
the first OPE suggests a boundary condition r∂ r X = − in R to fix the momentum charge, while the second OPE suggests a boundary condition ∂ θ X = wR to fix the winding charge. However, it is incompatible with the equal-time canonical commutation relation to impose both of these boundary conditions, as one fixes the coordinate and the other the conjugate momentum. Instead we may use the boundary condition to fix one of the charges and construct the action so that the other charge is fixed dynamically; this is really the same situation that we saw for the Dirichlet-like approach for the noncompact free boson, so we follow this Dirichlet-like approach again here 21 . That is, for endpoint operators O n 1 ,w 1 (0) and O n 2 ,w 2 (∞), we take the boundary conditions on the plane to be
∂ θ X |z|=ε = +w 1 R (4.37) ∂ θ X |z|= 1 ε = −w 2 R,(4.38)
with the relative minus sign in the latter again due to the reversed orientation of the boundary near infinity 22 . We therefore wish to impose the very wide strip boundary conditions
∂ y 1 X y 2 =+ ln ε = −w 1 R (4.39) ∂ y 1 X y 2 =− ln ε = +w 2 R (4.40)
via Lagrange multipliers and impose the momentum conservation dynamically as before. Therefore, the strip action is
S = 1 4π dy 1 dy 2 (∂ y 1 X) 2 + (∂ y 2 X) 2 − i 2π y 2 =ln ε dy 1 n 1 R X − λ 1 (∂ y 1 X + w 1 R) − i 2π y 2 =− ln ε dy 1 n 2 R X + λ 2 (∂ y 1 X − w 2 R) , (4.41)
whose equations of motion in addition to the usual bulk (∂ 2 y 1 + ∂ 2 y 2 )X = 0 together with the boundary conditions are
∂ y 1 λ 1 (y 1 ) = i∂ y 2 X y 2 =+ ln ε − n 1 R (4.42) ∂ y 1 λ 2 (y 1 ) = i∂ y 2 X y 2 =− ln ε + n 2 R . (4.43)
It should be noted that the action (4.41) might look strange because it contains an undifferentiated X which is circle-valued. Nevertheless, the quantity that actually needs to be a well-defined functional is e −S on the cylinder, which it is because shifting X by 2πR shifts the cylinder action by −2πi(n 1 + n 2 ). On the very wide strip where the quantization is actually performed, there is also no issue because X, while circle-valued, is forced to be single-valued since the strip has trivial homology, and again a global shift of 2πR does not change e −S . Furthermore, the boundary Lagrange multipliers are defined to be noncompact. Another way to view the compact boson CFT is from the orbifold R/2πRZ, in which non-single-valued operators in the compact theory are viewed as defect operators in the noncompact theory; in this way functionals such as (4.41) are always well-defined in the covering theory.
Proceeding with the canonical quantization on the very wide strip to determine the Hilbert space and Rindler Hamiltonian is nearly identical to the Dirichlet-like approach to the noncompact free boson; we just need to put in the winding dependence. The general solution to the equations of motion and boundary conditions on the very wide strip is
X(y 1 , y 2 ) = x + x s − i(n 1 − n 2 ) 2R y 2 − (w 1 − w 2 )R 2 y 1 − (w 1 + w 2 )R 2 ln ε y 1 y 2 (4.44) − √ 2 n∈Z n =0 α n n e − πn 2 ln ε y 1 sin πn(y 2 − ln ε) 2 ln ε λ 1 (y 1 ) = λ 1,0 + i x s + i(n 1 +n 2 ) 2R y 1 − i(w 1 +w 2 )R 4 ln ε y 2 1 + i √ 2 n∈Z n =0
α n n e − πn 2 ln ε y 1 (4.45)
λ 2 (y 1 ) = λ 2,0 + i x s − i(n 1 +n 2 ) 2R y 1 − i(w 1 +w 2 )R 4 ln ε y 2 1 + i √ 2 n∈Z n =0
α n n (−1) n e − πn 2 ln ε y 1 , (4.46)
where canonical quantization determines that λ 1,0 equals π(p + p s / ln ε) plus a constant and that λ 2,0 equals −π(p − p s / ln ε) plus a constant. As before, the slope mode x s is defined with the same shift − i(n 1 −n 2 ) 2R so that the classical Euclidean saddle point exists when n 1 + n 2 = w 1 + w 2 = 0 with x s = 0, and we define the momenta p(y 1 ) and p s (y 1 ) canonically conjugate to x(y 1 ) = x − (w 1 −w 2 )R 2 y 1 and x s (y 1 ) = x s − (w 1 +w 2 )R 2 ln ε y 1 again as the total momentum and first moment thereof. As Π has the same relation to X, λ 1 and λ 2 as in the noncompact case, these integrals are trivially computed as
− ln ε ln ε dy 2 Π(y 1 , y 2 ) = − i(w 1 − w 2 )R 2π ln ε + λ 1,0 − λ 2,0 2π − n 1 + n 2 2πR y 1 (4.47) ! = p(y 1 ) = p − n 1 + n 2 2πR y 1 (4.48) − ln ε ln ε dy 2 y 2 Π(y 1 , y 2 ) = − i(w 1 +w 2 )R 6π ln 2 ε + ix s π y 1 ln ε − i(w 1 +w 2 )R 4π y 2 1 + λ 1,0 +λ 2,0 2π ln ε (4.49) ! = p s (y 1 ) = p s + ix s π y 1 ln ε − i(w 1 + w 2 )R 4π y 2 1 ,(4.50)
which determines the precise relations
λ 1,0 = +π p + p s ln ε + i 2 w 1 + w 2 3 + (w 1 − w 2 ) R ln ε (4.51) λ 2,0 = −π p − p s ln ε + i 2 w 1 + w 2 3 − (w 1 − w 2 ) R ln ε. (4.52)
We emphasize that one is free to redefine the mode operators with any normalization and constant offsets preserving the commutators. The definition here that p is the total momentum conjugate to the compact constant mode x means that the spectrum of this operator will be 1 R Z; one of course will obtain the same results if one alternatively defined λ 1,0 − λ 2,0 = 2πp, but would instead have to work with the spectrum of p being i(w 1 −w 2 )R 2π ln ε + 1 R Z. The offset of p s does not change its continuous spectrum, but it is still a convenient calculational choice in the same spirit of the chosen offset of x s .
The stress-energy tensor associated to the action (4.41) is
T αβ = −: ∂ α X∂ β X − 1 2 δ αβ ∂ γ X∂ γ X : − t α t β i n 1 R X + w 1 Rλ 1 δ(y 2 − ln ε) + i n 2 R X + w 2 Rλ 2 δ(y 2 + ln ε) − t (α n β) ∂ y 2 X [iλ 1 δ(y 2 − ln ε) + iλ 2 δ(y 2 + ln ε)] , (4.53)
so that the Rindler Hamiltonian is computed to be
H DD R = 1 2π − ln ε ln ε dy 2 T (y)+ T (y) − 1 2π i n 1 R X+w 1 Rλ 1 y 2 =ln ε +i n 2 R X+w 2 Rλ 2 y 2 =−ln ε (4.54) = − i(n 1 + n 2 ) 2πR x + i(w 1 − w 2 )R 2 p + i(w 1 + w 2 )R 2 ln ε p s − x 2 s 2π ln ε − π 2 ln ε ∞ n=1 α −n α n (4.55) − (n 1 − n 2 ) 2 8πR 2 ln ε − (w 1 + w 2 ) 2 R 2 24π ln ε − (w 1 − w 2 ) 2 R 2 8π ln ε + π 48 ln ε .
The finite-regulator angular quantization Hilbert space H DD n 1 w 1 ;n 2 w 2 is spanned by the states |m, x s ; {N n } , where p = m R with m ∈ Z are the discrete momentum eigenvalues. The explicit mode expansions of the stress-energy tensors and the resulting cylinder evolution operator are written in Appendix A.
Note that the generic finite-regulator Hilbert space H DD n 1 w 1 ;n 2 w 2 contains no true zero-modes because all operators and their conjugates now appear in (4.55), unlike in the noncompact case (3.88), though x and p s both become zero-modes in the special case n 1 + n 2 = w 1 + w 2 = 0 for which the two endpoint operators are BPZ conjugate. Nevertheless, the path integral on the cylinder still has the shift symmetries in the Lagrange multiplier constant modes 23 , so the angular quantization prescription to compute the sphere correlators still involves taking the thermal trace divided by the volume of this symmetry group. The only difference from the noncompact case is that p = m R is now discrete, so the relevant volume factor is
Vol(λ 1,0 , λ 2,0 ) = − 4π 3 R ln ε m∈Z 1 ∞ −∞ dp s 2π . (4.56)
The last ingredient we need before computing the thermal trace and thermal one-point function is the chiral splitting of the mode expansion X(y 1 , y 2 ) into X L (y) and X R (y). The chiral expansions are immediately written as
X L (y) = x L − i 2 x s − i(n 1 −n 2 ) 2R − i(w 1 −w 2 )R 2 y+ i(w 1 +w 2 )R 8 ln ε y 2 − i √ 2 n∈Z n =0
α n n e πin 2 e − πny 2 ln ε (4.57)
X R (y) = x R + i 2 x s − i(n 1 −n 2 ) 2R + i(w 1 −w 2 )R 2 y− i(w 1 +w 2 )R 8 ln ε y 2 + i √ 2 n∈Z n =0 α n n e − πin 2 e − πny 2 ln ε ,(4.58)
where we must determine x L and x R . Their sum is of course x L + x R = x, so it remains to determine their difference, which may be inferred directly from the constant modes in the path integral boundary terms, as these must capture the corresponding constant modes of the endpoint operators O n 1 ,w 1 (0) and O n 2 ,w 2 (∞) in the shrinking limit. This derivation is performed in Appendix B, with the result being
x L = x 2 − π 2 p + p s ln ε + i(w 1 + w 2 )R 24 ln ε (4.59) x R = x 2 + π 2 p + p s ln ε − i(w 1 + w 2 )R 24 ln ε,(4.60)
so that the exponential primary operator expression O n 3 ,w 3 ≡ :e i( n 3
R +w 3 R)X L +i( n 3 R −w 3 R)X R :
is finally well-defined; all the freedom in redefining the mode operators by multiplicative and additive constants is gone, so the chiral splitting is uniquely determined by the consistency of the definition :e i( n 3 R +w 3 R)X L +i( n 3 R −w 3 R)X R : with the path integral. It is very important that bulk winding operators therefore do actually depend on the momenta p and p s , which is necessary to obtain conservation of winding in traces with bulk operator insertions. It might seem strange that a winding operator, which is a perfectly well-defined bulk local operator, could depend on the Lagrange multipliers which are defined only on the boundary. However, this construction of the compact boson theory really is an orbifold of the noncompact theory, so the action (4.41) and all mode expansions are really constructed in the covering theory before the quotient is taken. In the noncompact theory, winding operators are defect operators and hence not local; its attached topological defect line can run all the way out to the boundary and give a dependence on λ.
The rest of the story proceeds without surprises. Let k L = n 3 R + w 3 R and k R = n 3 R − w 3 R denote the chiral charges of a bulk exponential primary. The mode expansion of this primary on the plane is
:e i[k L X L (z)+k R X R (z)] : = − π ln ε k 2 L +k 2 R 4 e − πi 4 (k 2 L −k 2 R ) |z| − πi 8 ln ε (k 2 L −k 2 R ) z z − πi 16 ln ε (k L −k R ) 2 2 (k L +k R ) 2 4 z k 2 L 4 z k 2 R 4 cos k L k R 2 π ln |z| 2 ln ε • •e i[k L X L (z)+k R X R (z)]• •,(4.61)
where It is worth pointing out that these exponential operators :e i(k L X L +k R X R ) : automatically commute at equal times, a property which in radial quantization is only achieved by appending 2-cocycles by hand. This commutativity is also derived in Appendix B, and it holds because angular quantization determines the chiral splitting for us, essentially accounting for the defect lines attached to X L and X R . The thermal trace on the sphere is then computed to be
Tr H DD n 1 w 1 ;n 2 w 2 e −2πH DD R S 2 = Rδ n 1 +n 2 ,0 δ w 1 +w 2 ,0 4 √ 2π 2 e − πi 2 (n 1 w 1 −n 2 w 2 ) ε 1 2 n 2 1 R 2 +w 2 1 R 2 + 1 2 n 2 2 R 2 +w 2 2 R 2 ε −1/6 η − 2i ln ε π ,
(4.63) and the thermal one-point function of an exponential primary is computed to be
Tr H DD n 1 w 1 ;n 2 w 2 O n 3 ,w 3 (z, z)e −2πH DD R S 2 = Rδ n 1 +n 2 +n 3 ,0 δ w 1 +w 2 +w 3 ,0 4 √ 2π 2 e − πi 2 (n 1 w 1 −n 2 w 2 ) ε h 1 + h 1 +h 2 + h 2 z h 1 +h 3 −h 2 z h 1 + h 3 − h 2 × e πi 2 (n 1 w 2 −n 2 w 1 ) ε −1/6 η − 2i ln ε π [ε −1/6 η − 2i ln ε π ] 1 2 ( 3n 2 3 R 2 +w 2 3 R 2 ) ϑ 4 − i ln |z| π − 2i ln ε π 1 2 ( n 2 3 R 2 −w 2 3 R 2 )
. (4.64)
Recalling that the usual normalization of the compact boson exponential operators gives the two-and three-point coefficients as 24 O n 1 ,w 1 (0)O n 2 ,w 2 (1) S 2 = 2πRδ n 1 +n 2 ,0 δ w 1 +w 2 ,0 (4.65)
C n 1 w 1 ;n 2 w 2 ;n 3 w 3 = 2πRe πi 2 (n 1 w 2 −n 2 w 1 ) δ n 1 +n 2 +n 3 ,0 δ w 1 +w 2 +w 3 ,0 , (4.66)
we see that the requisite matching conditions (2.9) and (2.10) are again satisfied. Explicitly, we have found
lim ε→0 e πi 2 (s 1 −s 2 ) ε −∆ 1 −∆ 2 Tr H DD 1;2 e −2πH DD R S 2 = N DD n 1 w 1 ;n 2 w 2 O n 1 ,w 1 (0)O n 2 ,w 2 (1) S 2 (4.67) lim ε→0 e πi 2 (s 1 −s 2 ) ε −∆ 1 −∆ 2 Tr H DD 1;2 O n 3 ,w 3 (z, z)e −2πH DD R S 2 = N DD n 1 w 1 ;n 2 w 2 C n 1 w 1 ;n 3 w 3 ;n 2 w 2 z h 1 +h 3 −h 2 z h 1 + h 3 − h 2 (4.68)
with normalization constant
N DD n 1 w 1 ;n 2 w 2 = √ 2π (2π) 4 , (4.69)
which is the same as before. Note that no phase has been generated in the normalization constant for spinning operators precisely because the spin-dependent factors in the matching conditions (2.9) and (2.10) have been included. Obviously it does not matter where these spin factors are included, but our choice of having no phases in the shrinking on the cylinder is the natural choice for which the normalization constant N DD is always positive.
bc Ghost System
As a final explicit free-field example, we perform the angular quantization associated to the bc ghost system. As our original motivation is to describe the angular quantization associated to certain BRST-invariant vertex operators in string theory, we want to consider the case where both endpoints operators are c c. Even though the ghost fields b αβ and c α are free, the fact that they are Grassmann variables with a first-order action makes finding a simple boundary condition shrinking to c c problematic. Indeed, the previous free field boundary conditions were all imposing the charge under a current, so an appropriate boundary condition here would impose the correct charge under the total ghost number current j α = −:b αβ c β :. Since this current is bilinear in the fields, there will not be a linear and local boundary action which can fix it at finite regulator. The easiest way to proceed is to bosonize the bc ghost system to a compact linear dilaton [14]. In this language, the angular quantization of the ) and ghost numbers (n gh , n gh ). The ghost number anomaly coincides with the linear dilaton shift anomaly, so the ghost number anomalous conservation law (e.g. that nonzero sphere correlators have n gh = n gh = 3) is simply reproduced by the linear dilaton anomalous conservation law (e.g. that nonzero sphere
correlators have − i √ 2 n gh = − i √ 2
n gh = Q gh ). We do not use momentum and winding notations since they are frame-dependent quantities due to the background charge. The only place where the "radius" of the linear dilaton H matters is in the periodic identification of its constant mode h, whose minimal choice 26 is h ∼ h + 2 √ 2π; the bosonized field H should not be thought of as a target space dimension compactified at radius R = √ 2. Since the linear dilaton is compact, we choose the Dirichlet-like boundary conditions so that the compact boson results may be borrowed directly, even though the endpoint operators c c ∼ = :e −i √ 2H : themselves have no "winding" (but c and c individually do). That is, we take the very wide strip boundary conditions to be
∂ y 1 H y 2 =± ln ε = 0 (4.75)
and impose the ghost number charges dynamically via the action
S = 1 4π dy 1 dy 2 (∂ y 1 H) 2 + (∂ y 2 H) 2 − i 2π y 2 =ln ε dy 1 1 √ 2 H − λ 1 ∂ y 1 H − i 2π y 2 =− ln ε dy 1 1 √ 2 H + λ 2 ∂ y 1 H , (4.76)
where the boundary action terms linear in H are obtained from the linear dilaton result (4.13) by setting
α 1 = α 2 = − i √ 2 and Q = Q gh = − 3i √ 2 .
When going back to the plane, we must remember the anomalous transformation H(z, z) = H y(z), y(z) −Q gh ln |z| and its derivatives.
The mode expansions of H(y 1 , y 2 ) = H L (y) + H R (y) and of the Lagrange multipliers on the very wide strip are
H L (y) = h 2 − π(p h + ps ln ε ) 2 − ih s 2 y − i √ 2 n∈Z n =0 h n n e πin 2 e − πny 2 ln ε (4.77) H R (y) = h 2 + π(p h + ps ln ε ) 2 + ih s 2 y + i √ 2 n∈Z n =0
h n n e − πin 2 e − πny 2 ln ε (4.78)
λ 1 (y 1 ) = +π p h + p s ln ε + i h s + i √ 2 y 1 + i √ 2 n∈Z n =0
h n n e − πn 2 ln ε y 1 (4.79)
λ 2 (y 1 ) = −π p h − p s ln ε + i h s − i √ 2 y 1 + i √ 2 n∈Z n =0
h n n (−1) n e − πn 2 ln ε y 1 .
(4.80)
The rest of the cylinder computations are identical for that of the compact boson, so we may directly import the Rindler Hamiltonian as (n gh + n gh )h , which is single-valued only if h is identified with a shift of 2 √ 2π times some integer. It is irrelevant whether we choose 2 √ 2π as the period or any integer multiple of it, because the momentum eigenvalues will be divided by that integer but the Hamiltonian does not depend on this momentum at all, so we might as well make the minimal choice.
H gh R = − i √ 2π h − h 2 s 2π ln ε − π 2 ln ε ∞ n=1 h −n h n + π 48 ln ε ,(4.
traces are defined in this Dirac orthonormalized basis together with dividing by the Lagrange multiplier zero-mode volume
Vol(λ 1,0 , λ 2,0 ) = − 2 √ 2π 3 ln ε m h ∈Z 1 ∞ −∞ dp s 2π
. which should vanish (at least in the shrinking limit) because cc c c S 2 = 0. The first nonvanishing correlator on the sphere must come from the thermal two-point function of c(z 1 ) c(z 2 ) and must reproduce the ε → 0 limit of the correlator
(c c)(ε)c(z 1 ) c(z 2 )(c c)(ε −1 ) S 2 ε→0 −→ 1 ε 4 z 1 z 2 ; (4.84)
we define the bc ghost system so that this first nonzero correlator has this normalization and sign. Of course, thanks to holomorphic factorization the thermal two-point function here is equivalent to a thermal one-point function via
:e −i √ 2H L (z 1 ) ::e −i √ 2H R (z 2 ) : = e −z 12 ∂ z 1 :e −i √ 2H(z 1 ,z 1 ) :. (4.85) with normalization N gh c c = 1 (2π) 2 .
(4.90)
Note that this differs from the compact boson normalization constant because the correlators of the bosonized exponential primaries are not normalized to have two-point coefficient 2πR, since there is no target space interpretation. Even though we did not need the explicit mode expansions of the general exponential primaries O n gh , n gh = :e −i √ 2(n gh H L + n gh H R ) : here, we should point out that they also automatically obey the correct operator algebra, meaning that they anticommute at equal times if and only if both bc primaries are Grassman odd and commute at equal times otherwise. This derivation is also performed in Appendix B, with the result given in (B.21).
It should be clear that the procedure laid out in this section immediately generalizes to the free fermion CFT as well as to the βγ superghost system of the RNS formalism, as the former can be written as a twisted bc ghost system (where the twist ensures the correct weights and central charges) and the latter may be bosonized into another twisted bc ghost system plus linear dilaton (the latter being the picture field).
Connection to Entanglement Entropy
Finally, we mention the connection to the computation of 2d entanglement entropy. Most of the set-up described below is well-known in the literature; for a review, see for example [15] and references therein. The idea that the simplest CFT entanglement scenarios are those for which there exists a map to the annulus was expounded in [16]. Using this idea, the computation of vacuum entanglement entropy of an interval for the 2d free boson with different boundary conditions via oscillator methods was already performed in [17] by utilizing the properties of the modular Hamiltonian. However, the interpretation of the methodology here is different. In particular, the role of angular quantization in the general set-up is manifest, and the incorporation of states other than the vacuum is straightforward. This construction also emphasizes that the associated "entanglement spectrum" is not merely a mathematical tool but is used to define a Minkowski Hilbert space, and it highlights the need to enumerate the full class of boundary condition regulators for which the physical part of the entanglement entropy is actually independent of boundary condition.
Consider the CFT on the plane in a state ρ, where we wish to compute the entanglement entropy between an interval A of length L and its complement A, which is
S EE "=" − Tr H A [ρ A ln ρ A ] ,(5.1)
where ρ A = Tr H A ρ is the reduced density matrix on the interval A. The equality sign is in quotes because the expression on the right side is not well-defined, due to the lack of a tensor product decomposition of the Hilbert space of states in continuum quantum field theory,
H = H A ⊗ H A .
In the context of algebraic quantum field theory, a rigorously defined quantity is the relative entropy, from which mutual information between subregions can be defined. However, there are applications in which various finite derivatives of the entanglement entropy itself play an important role, and in general those are a distinct quantity from mutual informations. For example, in (2+1)-dimensional theories that flow to a gapped phase described by Chern-Simons theory, entanglement entropies are nontrivial, while mutual informations vanish [18]. It remains an open problem to give a precise definition of entanglement entropy and its unambiguous, regulator-independent content in QFT.
It is of course straightforward to define entanglement entropy in finite-dimensional lattice Hilbert spaces, and one may introduce a UV regulator in a QFT in order to compute a regulated reduced density matrix and the von Neumann entropy thereof; the real challenge is finding the necessary and sufficient conditions guaranteeing that the physical quantities extracted from the entanglement entropy are actually finite and independent of regulator.
Of particular interest is the study of RG monotones, beginning with the famous Zamolodchikov c-theorem in 2d [19] and its later generalizations to the 3d F -theorem and the 4d atheorem, which have been proposed to be constructed in d dimensions from certain entanglement entropy quantities [20]. While there has been strong evidence in support of the generalized d-dimensional F -theorem, there is still work to be done in proving the universality of its entanglement entropy construction.
One notable roadblock in proceeding beyond formal arguments is the computational difficulty even for free theories. The most widely-adopted prescription is the replica trick, wherein one defines the n th Rényi entropy
S n = − 1 n − 1 ln Tr H A ρ n A (Tr H A ρ A ) n ,(5.2)
analytically continues in n (assuming some boundedness properties) and constructs the entanglement entropy as
S EE = lim n→1 S n = − ∂ ∂n ln Tr H A ρ n A (Tr H A ρ A ) n n=1 . (5.3)
To proceed, one must define ρ A and its integer powers ρ n A by introducing a UV regulator ε and defining a map which splits the Hilbert space into a tensor product. A natural methodology is supplied by the Euclidean path integral [21]; we consider only the simplest case where A is a single line segment of length L. A state in the Hilbert space H at any fixed Im z < 0 in the plane is prepared by performing the Euclidean path integral up from Im z = −∞ with appropriate initial conditions. One then performs the path integral all the way up to Im z = 0 except for excised half-disks of radius ε around the endpoints of A, on which one places some fixed boundary condition B as depicted on the left in Figure 4.
This procedure defines a regularized splitting map
ı B (ε) : H ֒−→ H B A ⊗ H B A ,(5.4)
where H B A is the Hilbert space of states on a finite line segment of length L − 2ε flanked by boundary conditions B; we must demand that in the ε → 0 limit, the splitting map is isometric, meaning that lim ε→0 ı B (ε) † ı B (ε) = 1 H . The split state ρ is then defined so that its matrix element between any |ψ A ⊗ |ψ A ∈ H B A ⊗ H B A and (the CPT conjugate of) any |ψ ′ A ⊗ |ψ ′ A ∈ H B A ⊗ H B A equals the Euclidean path integral over the entire two-holed plane obtained by gluing the lower-half-plane with future boundary conditions (ψ A , ψ A ) at Im z = 0 to the reversed upper-half-plane with past boundary conditions (ψ ′ A , ψ ′ A ) at Im z = 0. The finiteregulator reduced density matrix ρ A is obtained by tracing over H B A , which in the path integral language just produces the completeness relation on A at Im z = 0 and is hence equivalent to erasing the boundary conditions on A. That is, the matrix elements ψ ′ A |ρ A |ψ A of the reduced density matrix are computed by the path integral shown on the right in Figure 4; the trace Tr H A ρ A is therefore just the partition function on the two-holed plane, which we keep explicit here instead of normalizing it to unity.
The integer powers ρ n A are then defined by inserting completeness relations between each factor of ρ A so that the matrix element ψ ′ A |ρ n A |ψ A is given by the path integral on the n-fold copy of the diagram on the right in Figure 4 with the boundary condition on the upper branch on the i th copy identified and summed over with the boundary condition on the lower branch of the (i + 1) th copy for 1 i n − 1. The trace Tr H A ρ n A is therefore just the partition function on the n-fold cover of the two-holed plane. Note that this n-fold cover does not have a conical singularity; topologically, the two-holed plane is a cylinder, and the n-fold cover of a cylinder is again a cylinder whose compact direction is n times as large.
So far our discussion has been completely standard. The main open question is -what are the admissible boundary conditions B such that one may extract a universal finite quantity from the entanglement entropy which is independent of B? The most obvious condition is that B must be a boundary condition which shrinks to the identity operator, since the right side of Figure 4 is supposed to be the regularized version of the plane branched at the two endpoints of the interval, and there are no nontrivial operator insertions at these branching points; if B were to shrink to a nontrivial operator, then the splitting map would not be isometric in the limit.
As we have discussed in Section 2, this shrinkability condition is easy to satisfy in a 2d CFT since most boundary conditions will shrink to the identity operator. However, shrinkability is far less obvious for non-conformal 2d QFTs as well as for higher-dimensional field theories. In the latter, one regularizes the reduced density matrix by excising a tubular neighborhood of the boundary subregion, which will shrink to a codimension-2 surface operator. Lack of shrinkability is the culprit behind earlier results in [22] for 4d free Maxwell theory seemingly not agreeing with the general result from the a-anomaly. This issue was later rectified in [23] which obtained the correct entanglement entropy in agreement with the a-anomaly. In the latter, the discrepancy was found to be from so-called "edge modes" which had to be present for the boundary condition to be admissible. However, we emphasize that all that matters is shrinkability of the boundary condition, which is the common ground which must unite all different approaches.
It is also natural to demand that an admissible B be a local boundary condition. Locality in Euclidean time is required to express the path integral in the Hilbert space formulation and give a Lorentzian entropy interpretation. Furthermore, one might require the boundary conditions to respect various symmetries of the problem, such as local Lorentz covariance.
Angular quantization applied to non-conformal 2d QFTs can provide oscillator methods of computing entanglement entropy quantities that may prove useful. Here we just comment on the well-understood conformal case; the hope is that more general situations are amenable to similar methods. The regulated prescription of computing Tr H A ρ n A above is simply angular quantization after performing an appropriate special conformal transformation. Since PSL(2, C) transformations map circles to circles but do not map their centers to each other, we slightly modify the boundary locations drawn in Figure 4 so that after this transformation the boundary circles are symmetrically placed at |z ′ | = ε ′ and |z ′ | = 1 ε ′ like before. Taking the original interval A to have endpoints at z = 0 and z = L, we define the boundary Figure 5: The PSL(2, C) transformation mapping the two-holed plane used in computing entanglement entropy to the symmetric two-holed plane used to construct angular quantization. The point at infinity is mapped to z ′ = 1.
L √ L 2 + 4ε 2 × =⇒ PSL(2, C) |z ′ | = ε ′ |z ′ | = 1 ε ′ z ′ = 1
circles to have radius ε but centered at the points z = 1 2 (L ± √ L 2 + 4ε 2 ). Then, the special conformal transformation
z −→ z ′ = z z − L (5.5)
maps the boundaries to the circles at ln |z ′ | = ± ln ε ′ , where
ε ′ = √ L 2 + 4ε 2 − L 2ε ; (5.6)
see Figure 5. The Rényi entropy traces Tr H A ρ n A are then computed as sphere thermal traces over e −2πnH R in the appropriate angular quantization Hilbert space. For a chosen shrinkable boundary condition, the latter depends only on the regulator ε ′ , which itself depends only on the ratio ε L ; at very small ε, the relation is ε ′ = ε L + . . . . The only difference between the trace needed for the replica trick and the associated sphere thermal trace over 2πn angular evolution is the Weyl anomaly induced by the transformation (5.5). Fortunately, this Weyl anomaly is almost inconsequential as it cancels in the Rényi entropy definition (5.2) because the curvatures and Weyl transformation factor are all single-valued functions and so the Weyl anomaly for Tr H A ρ n A is exactly the n th power of the Weyl anomaly for Tr H A ρ A .
One must be slightly careful because the flat z ′ -plane metric is obtained from the flat z-plane metric by applying the Weyl transformation g αβ → g ′ αβ = e −2ω g αβ with ω(z ′ , z ′ ) = ln( L |z ′ −1| 2 ), which diverges at z ′ = 1. One should really cut out a hole of some small radius δ ′ centered around z ′ = 1 and place an appropriate boundary condition there. The Weyl anomaly still cancels in (5.2) for the same reason as before, and now the new boundary condition on the zplane fixes the asymptotic path integral behavior near the point at infinity and hence defines the total state ρ in which we wish to compute the entanglement entropy. For example, if we place a shrinkable boundary condition there, then the thermal trace over the z ′ -plane will compute the entanglement entropy of an interval in the vacuum on S 1 . Similarly, the entanglement entropy of an interval in any pure state ρ = |Ψ O Ψ O | simply corresponds to the thermal trace over the z ′ -plane with the operator O inserted at z ′ = 1 by the usual state/operator correspondence; in particular, Tr H A ρ n A involves n insertions of O, one at each pre-image of z ′ = 1 in the n-fold cover. Finally, the z ′ thermal traces are easily computed by transforming to the y ′ cylinder frame, where the Weyl anomaly action is again simply multiplied by n for the n-fold cover. The n-fold cover cylinder trace is then evaluated by changing coordinates to y ′′ = y ′ /n. In these new coordinates, the cylinder is covered exactly once with the n insertions of O(y ′′ 1 , y ′′ 2 ) at y ′′ 1 = 0 separated by angle ∆y ′′ 2 = 2π n ; the only change is that the boundaries are now located at y ′′ 2 = ± ln ε ′ n . Therefore, the end result is that the traces involved in the regularized Rényi entropy (5.2) of an interval in any pure state ρ = |Ψ O Ψ O | are computed in angular quantization by
Tr H A ρ n A = e −nS W (ε ′1/n ) Tr H 1 O(0, 0)O 0, 2π n · · · O 0, 2π(n−1) n e −2πH 1 R (ε ′1/n ) S 1 ×R , (5.7)
where S W is the Weyl anomaly action to go from the cylinder frame to the sphere frame, H 1 is the angular quantization Hilbert space associated to two local boundary conditions which shrink to the identity with associated Rindler Hamiltonian H 1 R , and we have explicitly indicated that all thermal traces are to be evaluated at regulator ε ′1/n where ε ′ is given by (5.6). The benefit of performing the calculation at finite regulator here is that the replica direction is an isometry (i.e. non-contractible) and hence there is no conical deficit in the resulting geometry.
Using the above methodology and the explicit expressions given in Section 3, it is simple to obtain the exact finite-regulator vacuum entanglement entropy for, say, the noncompact free boson by setting k 1 = k 2 = 0 and writing V = 2πδ(k = 0) for the volume of target space. The boundary conditions considered in Section 3 then correspond to ordinary Neumann and (indefinite) Dirichlet boundary conditions, which are the familiar conformally-invariant ones. For Neumann boundary conditions, the trace over ρ n A is therefore NN:
Tr H A ρ n A = V 2 √ 2π ε ′ 1 6 (n− 1 n ) ε ′− 1 6n η − 2i ln ε ′ πn ,(5.8)
from which it follows that the exact finite-regulator vacuum entanglement entropy of an interval of length L is
S NN EE (L, ε) = ln L 2ε + 1+ L 2ε 2 6 E 2 2i π ln L 2ε + 1+ L 2ε 2 −lnη 2i π ln L 2ε + 1+ L 2ε 2 +ln V 2 √
2π , (5.9) where E 2 (τ ) is the quasi-modular Eisenstein series of weight 2 defined in (A.6). For (indefinite) Dirichlet boundary conditions, the trace over ρ n A is DD:
Tr H A ρ n A = V 8 √ 2π 3 ε ′ 1 6 (n− 1 n ) ε ′− 1 6n η − 2i ln ε ′ πn ,(5.10)
and hence the entanglement entropy from Dirichlet boundary conditions differs from that from Neumann boundary conditions by a pure constant, namely
S NN EE (L, ε) = S DD EE (L, ε) + ln(2π) 2 . (5.11)
Indeed, it is a familiar fact in 2d CFTs that the constant part in the entanglement entropy as computed from the modular Hamiltonian with different conformally-invariant boundary conditions differs precisely by (the logarithm of) the ratio of their corresponding boundary g-functions [24]. From the explicit form of the Neumann and Dirichlet Cardy states in (3.108) and (3.109), the boundary g-functions obey 27 N 2 |D /N 2 |N = (2π) 2 . However, (5.11) expresses the much stronger fact that the constant logarithm of the ratio of the boundary g-functions is the only 27 The shift in (5.11) actually has the wrong sign compared to the logarithm of the ratio of the Neumann and Dirichlet boundary g-functions, but this is due to the extra factor of 1/(2π) 4 in N DD k 1 k 2 compared to N 2 |D .
difference between the exact entanglement entropies even at finite regulator. Of course, we are free to choose other boundary conditions that shrink to the identity which do differ nontrivially at finite regulator. The leading expansion of (5.9), subtracting the contribution from the boundary g-function, as ε → 0 is
S EE (L, ε) − ln(N V ) ε→0 −→ 1 3 − 1 2 ε L 4 + . . . ln L ε + 1 3 ε L 2 + 1 2 ε L 4 + . . . ,(5.12)
which holds for the Dirichlet result as well, agreeing with the famous result that the universal part of the vacuum entanglement entropy of an interval of length L in a 2d CFT is
lim ε→0 ∂S EE (L, ε) ∂ ln L = c 3 ; (5.13)
see, for example, [25] and references therein. Indeed, the angular quantization methodology also immediately produces this result in general. By the matching condition (2.9), the leading contribution to Tr H A ρ A is the normalization constant N , and hence the leading contribution to
Tr H 1 [e −2πH R (ε ′ ) ] S 1 ×R is ε ′−c/3
. Therefore, the leading term in the n th Rényi entropy is S n = − 1 n − 1 ln ε ′(1−n)c/3 N ε ′−c/3n N n ε ′−nc/3 + . . . from which (5.13) immediately follows due to ε ′ = ε L + . . . . Of course, relating the trace to the sphere partition function via the Weyl anomaly is one way (5.13) is derived in general. What we gain here is not new in terms of this result, but provides a solid foundation for the independence of boundary conditions given shrinkability. For example, this perspective makes it clear why pure Dirichlet boundary conditions, X = x 0 for fixed x 0 , are actually inadmissible for the noncompact free boson and so would not compute the correct entanglement entropy quantities. It is clear from the general construction in Section 2 that pure Dirichlet boundary conditions fail the shrinkability requirement because the finite-regulator state |D, x 0 contains infinitely many primaries of arbitrarily small dimension with equal coefficients, as mentioned before. Pure Dirichlet boundary conditions do however shrink to the identity operator for the compact boson because of the discrete spectrum of exponential primaries, indeed leading to the correct entanglement entropy as shown in [17].
Summary
In this paper we have described the framework of angular quantization as it applies to 2d CFTs. While our primary motivation is to use these results to describe string theory in the presence of Euclidean time-winding operators, there are other applications of interest where this formalism can be useful. At the computational level, angular quantization can be considered as a study of non-conformal boundary conditions in bulk CFTs, thus being a generalization of the familiar Cardy state construction for conformally-invariant boundary conditions. In this paper we have focused on explicit examples of free CFTs, namely the noncompact and compact free bosons, the linear dilaton and the bc ghost system. For a generic CFT, one may consider the bootstrap equations for the angular quantization Hilbert spaces associated to slicing correlators in different ways; these conditions encode locality of the boundary conditions, but it is difficult to provide explicit realizations of the Hilbert spaces because the lack of a state/operator correspondence means that the states are not organized into conformal primaries and descendants.
From a more practical point of view, analytically continuing angular quantization on the sphere results in Rindler quantization, thus providing an explicit construction of the CFT on Minkowski space R 1,1 . While a great deal is known about the structure of CFTs living on a circle, far less is known about the nonperturbative structure of CFTs living on an infinite line. Angular quantization as described here provides a connection between the two and thus can help learn more about Minkowski space CFTs. In particular, a Hilbert space of a CFT living on a line R is labeled by two asymptotic conditions which in the Euclidean language determine the endpoint operators to which the associated boundary conditions shrink. That the Hilbert space should be labeled by these asymptotics conditions is perhaps not so surprising, as it is by now a common viewpoint that theories in asymptotically flat or asymptotically AdS spacetimes should have the chosen asymptotic conditions as part of their intrinsic data, since the Hilbert space consists of normalizable modes in the interior none of which can alter the asymptotics.
Lastly, we briefly described the connection to entanglement entropy; although we have not obtained any new results in Section 5, the perspective of angular quantization allows us to be more careful about the shrinkability of boundary conditions. A general construction of entanglement entropy in QFT that is well-defined and independent of regulator for an appropriate class of admissible boundary conditions is still lacking. It is clear that a necessary requirement for admissibility is that the boundary condition placed on a tubular neighborhood around the subregion boundary must shrink to the trivial surface operator (the identity local operator in 2d). Demonstrating the equivalence of entanglement entropy for any boundary condition which shrinks to the identity remains a formidable task even for the free massive scalar. We hope to be able to say more about this matter in the future.
where the elliptic nome is q ≡ e 2πiτ with τ living in the upper-half-plane. They obey the modular-S transformations
η − 1 τ = √ −iτ η(τ ) (A.4) ϑ 2 (ν|τ ) = i τ e −πiν 2 /τ ϑ 4 ν τ − 1 τ . (A.5)
The Eisenstein series E 2n (τ ), defined to be 28 the sum of 1/λ 2n for all lattice vectors λ ∈ Λ(τ ) associated to the torus T 2 (τ ) ∼ = C/Λ(τ ), is a holomorphic modular form of weight 2n for any integer n 2. For n = 1, the Eisenstein series E 2 (τ ) is instead a quasi-modular form of weight 2 and is given explicitly by (A.14)
The oscillator traces that appear in the thermal one-point functions are only slightly more annoying. We perform the calculation here explicitly for the compact boson, as the other traces are obtained from it as special cases. Define the quantity where L N (x) is the Laguerre polynomial of degree N , whose generating function is
∞ N =0 L N (x)t N = e −tx/(1−t) 1 − t . (A.21)
The first infinite product is the same as above, given by the Dedekind eta function. For the second infinite product, we use The Euclidean path integral is performed by decomposing the field into eigenmodes of the Laplacian on the surface in question, separating out any "zero-modes" which appear only linearly or not at all in the action. For the path integral on the sphere, such terms are necessarily zero-modes of the Laplacian and hence satisfy the bulk equation of motion. For the compact boson path integral with action (4.41), one decomposes dX = dX 0 + 2πwRω into the singlevalued part dX 0 and the winding part where ω generates H 1 (Σ; Z) and subsequently sums over winding sectors. Only the single-valued part of X can have a zero-mode since the bulk kinetic term involves a term proportional to w 2 . This zero-mode is thus obtained by setting X = x 0 to a constant, for which the exponentiated boundary action at |z| = ε reads
exp i 2π |z|=ε dθ n 1 R X + λ 1 (∂ θ X − w 1 R) zero-mode = e in 1 R x 0 −iw 1 Rλ 1,0 , (B.1)
since the remaining integral over λ 1 just yields its constant Fourier mode. The regulated path integral is constructed to reproduce the path integral with the insertions O n 1 ,w 1 (ε)O n 2 ,w 2 (ε −1 ) in the ε → 0 limit, so in particular the above zero-mode must match that of the mode expansion O n 1 ,w 1 = :e i[ n 1 R X+w 1 R(X L −X R )] :. From the chiral mode expansions (4.57) and (4.58) with x s = α n = 0, we evaluate
in 1 R X(ε) + iw 1 R[X L (ε) − X R (ε)] zero-mode = in 1 R x + n 1 (n 1 − n 2 ) 2R 2 ln ε + iw 1 R(x L − x R ) + w 1 (w 1 − w 2 )R 2 2 ln ε + w 1 (w 1 + w 2 )R 2 4 ln ε. (B.2)
Matching of these two zero-modes using (4.51) then requires
x L − x R = −λ 1,0 + i(w 1 − w 2 )R 2 ln ε + i(w 1 + w 2 )R 4 ln ε (B.3)
= −π p + p s ln ε + i(w 1 + w 2 )R 12 ln ε, (B.4) as claimed in the text. Matching the zero-modes near infinity produces the same condition after mapping it to the origin via an inversion, which flips chirality and hence replaces p s with −p s . Finally, we may compute the equal-time commutator of two exponential primaries from their mode expansions. From (4.61), the mode expansion of O n,w (y 1 , y 2 ) on the strip is :e i( n R +wR)X L (y)+i( n R −wR)X R (y) : = A(n, w, y 1 , y 2 )e in R x e −πiwRp e in R y 2 +wRy 1 xs e − πiwR ln ε ps (B.5) e 2F O n 4 ,w 4 (y 1 ,y ′ 2 )O n 3 ,w 3 (y 1 ,y 2 ) (B.13) = O n 4 ,w 4 (y 1 , y ′ 2 )O n 3 ,w 3 (y 1 , y 2 ). (B.14)
× e i √ 2 ∞ ℓ=1 α −ℓ ℓ f −ℓ (n,w) e −i √ 2 ∞
Therefore the mode expansions of the exponential primaries in angular quantization automatically obey the correct operator algebra, as claimed.
For the bosonized bc ghost system, the story is nearly identical. The bosonized exponential primary O n gh , n gh has strip mode expansion O n gh , n gh = B(n gh , n gh ,y 1 ,y 2 )e where the oscillator commutator in the exponential is F = − πi 2 (n gh + n gh ) n ′ gh − n ′ gh θ(y 2 −y ′ 2 )+ y 2 −lnε 2 ln ε −(n gh − n gh ) n ′ gh + n ′ gh θ(y ′ 2 −y 2 )+ y ′ 2 −lnε 2 ln ε .
(B.19) As before, the (y 2 − ln ε)/2 ln ε and (y ′ 2 − ln ε)/2 ln ε terms in F exactly cancel the commutator contributions from the non-oscillator terms. Now, however, the θ(y 2 − y ′ 2 ) and θ(y ′ 2 − y 2 ) terms in F do not simply give unity because of the extra 1/2 in front. Instead, we may write, for example, − πi 2 (n gh + n gh ) n ′ gh − n ′ gh θ(y 2 − y ′ 2 ) − (n gh − n gh ) n ′ gh + n ′ gh θ(y ′ 2 − y 2 ) = − πi 2 (n gh + n gh ) n ′ gh − n ′ gh + πi n gh n ′ gh − n gh n ′ gh θ(y ′ 2 − y 2 ), (B.20)
where the second term is always an integer times πi, and hence contributes unity, but the first term is not. The total equal-time commutator is hence O n gh , n gh (y 1 ,y 2 )O n ′ gh , n ′ gh (y 1 ,y ′ 2 ) = (−1) (n gh + n gh )(n ′ gh + n ′ gh ) O n ′ gh , n ′ gh (y 1 ,y ′ 2 )O n gh , n gh (y 1 ,y 2 ), (B.21)
where we freely replaced (n gh + n gh )(n ′ gh − n ′ gh ) with (n gh + n gh )(n ′ gh + n ′ gh ) since their difference is an even integer. The integer n gh + n gh is the total ghost number of O n gh , n gh , and the ghost number modulo 2 is the "fermion" number for the bc system. Therefore, the result (B.21) says that the mode expansions of two bc ghost system primaries anticommute if and only if both their total ghost numbers are odd, otherwise they commute. Therefore, these mode expansions also obey the correct operator algebra, showing that angular quantization automatically takes into account the 2-cocycles in bosonization that must be added by hand in radial quantization.
Figure 1 :
1The angular quantization slicing on the sphere and on the infinite cylinder, related by the exponential map. The red arrows denote Euclidean time evolution via the Rindler Hamiltonian in the appropriate conformal frame.
T yy = T (y) + 1 4 T α α and T yy = T (y) + 1 4 T α α , where 12 T (y) = −:∂X(y)∂X(y): (3.47) T (y) = −:∂X(y)∂X(y): (3.48)
is manifestly independent of time because the original expansion (3.49) was chosen to obey the equation of motion, i.e. the time-dependent operators x(y1 ) ≡ x− πip(y 1 ) ln ε y 1 − i(k 1 +k 2 ) 4 ln ε y 2 1 , p(y 1 ) ≡ p − k 1 +k 22π y 1 and α n (y 1 ) ≡ α n e − πn 2 ln ε y 1 automatically obey the Heisenberg equation13 O = −[H NN R , O]; the boundary terms arising from the classical trace of the stress-energy tensor were needed to obtain this simple result. Therefore, the evolution operator around one full y 1 ∼ y 1 − 2π cycle on the cylinder is the ordinary exponential given by
80) where the nonvanishing commutators are 15 [x, p] = [x s , p s ] = i (3.81) [α n , α m ] = nδ n,−m . (3.82)
p and p s are true zero-modes of the Hamiltonian (3.88), reflecting the conservation of x and x s . The Hermitian conjugate again obeys H DD R(k 1 , k 2 ) † = H DD R (−k 1 , −k 2 ). The next step is to compute the thermal trace and thermal one-point functions on the cylinder and transform them back to the sphere. However, we must pause to consider the path integral on the cylinder with the action (3.71), which is what the angular quantization trace Tr H DD k 1 k 2
( 2 .
210) for the thermal one-point functions of bulk primary insertions. The angular quantization Hilbert spaces H NN k 1 k 2 and H DD k 1 k 2 with respective Rindler Hamiltonians H NN R and H DD R
From the mode expansion(3.78), the relation between conformal and creation/annihilation normal ordering on the plane is :X(z 1 , z 1 )X(z 2 , z 2 ): = • •X (z 1 , z 1 )X(z 2 , z 2 ) • • − 1 2 ln 1 − (z 1 /z 2 ) −πi/2 ln ε [1 + (z 1 z 2 ) −πi/2 ln ε ]
−n α −n |k; {0}, {0} , (3.109) where |k; {0}, {0} is the oscillator vacuum in H S 1 with target space momentum k. The normalizations N |N = 1 (8π 2 ) 1/4 and N |D = (2π 2 ) 1/4 of the above Cardy states are related to the angular quantization normalizations here by
( 4 .
421) where ϕ and p ϕ are again the center-of-mass variables; the nontrivial commutators are the usual [ϕ, p ϕ ] = i and [φ n , φ m ] = nδ n,−m . The mode expansions of the bulk holomorphic and antiholomorphic stress-energy tensor components are
bc ghost system is nearly identical to the previous examples, essentially combining the technique of the compact boson with the background charge effects of the linear dilaton. The bosonization of the ghost system amounts to the identifications b(z) ∼ = :e i √ 2H L (z) : b ∼ = :H = H L + H R is a canonically normalized linear dilaton (with chiral splitting defined below) with background chargeQ gh = − 3i √ 2 ,(4.72) which reproduces the bulk bc stress-energy tensor with central charges c = c = −26. The holomorphic and antiholomorphic ghost number currents bosonize to −:b(z)c(z): ∼ = bosonized current is linear is why this description possesses simple linear local boundary conditions shrinking to the primaries. The bosonized bc primaries generically take the form of the linear dilaton exponentials :e −i √ 2(n gh H L + n gh H R ) : for a pair of integers (n gh , n gh ) ∈ Z 2 , with conformal weights (h, h) = ( n gh (n gh +3) 2, n gh ( n gh +3) 2
81)and we take the basis of the finite-regulator angular quantization Hilbert space H gh c c to consist of the states |m h , h s ; {N n } with eigenvalues h s ∈ R and p h = m h √ 2 , m h ∈ Z; as usual, the thermal 26 The identification must be such that :e −i √ 2(n gh H L + n gh H R ) : is a well-defined local operator for all pairs (n gh , n gh ) ∈ Z 2 . The part of this operator involving h is e
0
0the trace over the evolution operator e −2πH gh R , the fact that m h |e i√ 2h |m h = m h − 2|m h =
Figure 4 :
4Left: The regularized splitting map ı B (ε) : H ֒→ H B A ⊗H B A as defined by the Euclidean path integral over the shaded region. Right: The regularized reduced density matrix element ψ ′ A |ρ A |ψ A as defined by the Euclidean path integral over everywhere outside the two excised holes.
N ε ′−c/3n + . . . ,(5.16)
trace needed for the thermal one-point function of O n 3 ,w 3 for the compact boson is Tr osc.
Re
(n,w) ,where A(n, w, y 1 , y 2 ) is a c-number multiplicative coefficient not important for this calculation, and f m (n 3 , w 3 ) ≡ f m z(y), z(y) is the quantity defined in (A.15). Now consider swapping the order of the exponentials in the equal-time product O n 3 ,w 3 (y 1 , y 2 )O n 4 ,w 4 (y 1 , y ′ 2 ). The nonoscillator exponentials result ine in 3 R x e −πiw 3 Rp e in 4 R x e −πiw 4 Rp = e πi(n 3 w 4 −n 4 w 3 ) y 2 +w 3 Ry 1 xs e − πiw 3 R ln ε ps e in 4 R y ′ 2 +w 4 Ry 1 xs e − πiw 4 R ln ε ps = e πi ln ε (n 3 w 4 y 2 −n 4 w 3 y ′ 2 ) e in 4 R y ′ 2 +w 4 Ry 1 xs e − πiw 4 R ln ε ps e in 3 R y 2 +w 3 Ry 1 xs e − πiw 3 R ln ε ps . (B.2F = e πi(n 3 w 4 −n 4 w 3 ) e − πi ln ε (n 3 w 4 y 2 −n 4 w 3 y ′ 2 ) , (B.12) so that the total equal-time commutator is O n 3 ,w 3 (y 1 ,y 2 )O n 4 ,w 4 (y 1 ,y ′ 2 ) = e πi(n 3 w 4 −n 4 w 3 ) e πi ln ε (n 3 w 4 y 2 −n 4 w 3 y ′ 2 )
−ℓ (n gh , n gh ) (n gh , n gh ) , (B.16) where f m (n gh , n gh ) is the same as the expression (A.15) with the replacements n R = − n gh + n gh √ 2and wR = − n gh − n gh √ 2. Therefore, the equal-time commutator isO n gh , n gh (y 1 ,y 2 )O n ′ gh , n ′ gh (y 1 ,y ′ 2 ) = O n ′ gh , n ′ gh (y 1 ,y ′ 2 )O n gh , n gh (y 1 ,y 2 ) (B.17) × e −πi(n gh n ′ gh − n gh n ′ gh ) e πi 2 ln ε [(n gh + n gh )(n ′ gh − n ′ gh )y 2 −(n gh − n gh )(n ′ gh + n ′ gh )y ′ 2 ]e 2F , (B.18)
It is commonly believed that a moduli space of vacua can only exist in free or supersymmetric theories.
For quantum field theory in asymptotically-flat or -AdS space, specifications of the asymptotic behavior of correlators is part of the definition of the theory, as it cannot be changed by any finite-energy excitations in the bulk. The Rindler Hamiltonian here is only non-Hermitian under a naïve application of Hermitian conjugation descended from radial quantization, which can change the asymptotic conditions. One can modify the definition of Hermitian conjugation so that it does not mix asymptotic conditions, under which the Rindler Hamiltonian is perfectly Hermitian.
Equivalently, one may stick to a real field and consider the analytic continuation to complex values of k. While the OPE of two exponential operators with non-real k is no longer unitary, none of the analyticity properties relevant to the angular quantization procedure detailed here change for complex k.
As a reminder, the Euclidean Heisenberg equation is ∂τ O = [H, O], and the sign change is due to our time conventionȮ ≡ ∂y 1 O = −∂τ O.
The most notable exception is for the constant mode of the noncompact free boson itself, which is usually left in so as to keep track of the spacetime volume dependence of quantities. Nevertheless, even for the free boson the constant mode of the path integral must be factored out, and correlators are normalized by dividing by the partition function with the zero-mode absorbed, as we have done previously.
Note that the Neumann-like Hamiltonian (3.60) did not share this property, as in that case x is a true zeromode if and only if k1 + k2 = 0, which is why all those traces are proportional to 2πδ(k1 + k2), supplying the infinity only for the Hilbert space with the actual zero-mode.
The extra factor of (2π) 2 accompanying the Dirichlet normalization is simply due to the fact that the normalization of a continuous basis is arbitrary, and the normalization we chose here differs slightly from that conventionally used in the Cardy states.
Or, more precisely, one may consider separating out the zero-mode from the exponential operators like we did in the path integral derivation for the shrinking of the free boson Neumann-like boundary condition. Then just the zero-modes are absorbed into the action, with the bulk insertions of the nonzero-mode exponentials still present. This is what one would do to explicitly evaluate the path integral by expanding the exponentials, for example.
One may try to work out the Neumann-like approach here directly, though it is more awkward to impose the winding condition dynamically on the strip. It it easier to first perform the Dirichlet-like quantization here, but with momentum w and winding n, and then T-dualize everything.22 This can be seen in two equivalent ways. One may define the operator at infinity by first transitioning to the z ′ = 1/z coordinate patch, for which ∂ θ ′ = −∂ θ . If one instead maps the operator at infinity to the origin by an actual conformal inversion via z → z ′ = 1/z, one has ∂ θ ′ = ∂ θ but also kL ⇌ kR because the inversion flips chirality.
Technically the path integral on the cylinder sums over all winding sectors of the free boson around the nontrivial homology cycle, and it is only one of these winding sectors that has the Lagrange multiplier shift symmetries. However, not coincidentally, the single winding sector with this symmetry is also the single winding sector that gives a nonvanishing contribution to the path integral (i.e. it is the one which satisfies winding conservation), so the net effect is that the symmetry is still present.
The phase factor in the three-point coefficient is easily seen to be what is required for the three-point function to be symmetric under operator exchange. There is no problem for this three-point coefficient in a unitary theory to be complex, because On,w is a complex observable; it is only in the "good" self-conjugate basis of operators with a positive two-point coefficient that the three-point coefficient must be real, which does indeed happen for the sine and cosine basis.
We apologize for the horrible notation of using HR to denote the antichiral linear dilaton for the bosonized ghost, despite the same symbol denoting the Rindler Hamiltonian. The latter will always be denoted H gh R , so there should not be any confusion.
This sum does define the Eisenstein series, usually denoted G2n(τ ), which conventionally equals 2ζ(2n) times E2n(τ ). We shall stick to calling E2n(τ ) the Eisenstein series, for the painfully obvious reason that it starts with 'E'.
AcknowledgmentsWe are grateful to Thomas Dumitrescu, Elliot Schneider and Xi Yin for discussions. This work is supported in part by DOE grant DE-SC0007870.where we have definedas well as φ n = φ n for all n = 0. Using these expansions in(4.17)results in the Rindler Hamiltonian on the very wide strip asln ε+ (α 1 +α 2 −Q) 2 6π ln ε+ π 48 ln ε .(4.26) The finite-regulator angular quantization Hilbert space H LD α 1 α 2 is isomorphic as a vector space to that of the noncompact free boson with Neumann-like boundary conditions, being spanned by |p ϕ ; {N n } . The mode expansion of the bulk exponential primary O α 3 (z, z) = :e 2α 3 φ(z,z) : on the plane isIt is no more difficult to compute the mode expansion and thermal one-point function of the general exponential ::, but we already know that the linear dilaton anomalous conservation law will immediately cause all thermal one-point functions to vanish except that of c c with n gh = n gh = 1. So here we only record the explicit mode expansions of the c ghosts themselves on the plane, which areEven though these expansions individually are not needed here, they are required for example in the computation of the worldsheet BRST charge in angular quantization, which we present in[8]. The finite-regulator thermal two-point function on the sphere is then easily shown to beA Oscillator TracesThe Dedekind eta function and Jacobi elliptic theta functions appearing in this paper are given byThe quasi-modularity here means that it is E 2 (τ ) − 3 π Im τ which is the actual modular form, with the non-holomorphic offset needed to cancel the part of the modular transformation where ∂ τ acts on √ −iτ . Let us also record here the mode expansions of the stress-energy tensors and evolution operator for the compact boson angular quantization with endpoint operators O n 1 ,w 1 (0) and O n 2 ,w 2 (∞). The stress-energy tensors on the plane arewhere we have defined the zero-momentum modeswith α n =0 = α n =0 . The cylinder evolution operator determined by the Hamiltonian (4.55) isTherefore, the original oscillator trace isThe noncompact Neumann-like free boson trace is obtained by setting n 3 R = 0 and w 3 R = −k 3 , that for the noncompact Dirichlet-like free boson by setting n 3 R = k 3 and w 3 R = 0, that for the linear dilaton by setting n 3 R = 0 and w 3 R = 2iα 3 and finally that for the bosonized bc ghost system by setting n 3 R = − √ 2 and w 3 R = 0 (for the thermal one-point function of c c). The elliptic functions are all evaluated at the nome q = ε 4 , from which it is obvious that ε −1/6 η(τ ) and ϑ 4 (ν|τ ) both approach unity as ε → 0 for any fixed ν.B Chiral Splitting and the Operator AlgebraIn this appendix, we obtain the chiral splitting of the free compact boson X(y, y) = X L (y) + X L (y) in angular quantization as dictated by the path integral and demonstrate that the resulting mode expansions of the exponential primaries O n,w automatically commute at equal times.
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| []
|
[
"A Robust Information Source Estimator with Sparse Observations",
"A Robust Information Source Estimator with Sparse Observations",
"A Robust Information Source Estimator with Sparse Observations",
"A Robust Information Source Estimator with Sparse Observations"
]
| [
"Kai Zhu \nSchool of Electrical\nComputer and Energy Engineering\nArizona State University Tempe\n85287AZUnited States\n",
"Lei Ying [email protected] \nSchool of Electrical\nComputer and Energy Engineering\nArizona State University Tempe\n85287AZUnited States\n",
"Kai Zhu \nSchool of Electrical\nComputer and Energy Engineering\nArizona State University Tempe\n85287AZUnited States\n",
"Lei Ying [email protected] \nSchool of Electrical\nComputer and Energy Engineering\nArizona State University Tempe\n85287AZUnited States\n"
]
| [
"School of Electrical\nComputer and Energy Engineering\nArizona State University Tempe\n85287AZUnited States",
"School of Electrical\nComputer and Energy Engineering\nArizona State University Tempe\n85287AZUnited States",
"School of Electrical\nComputer and Energy Engineering\nArizona State University Tempe\n85287AZUnited States",
"School of Electrical\nComputer and Energy Engineering\nArizona State University Tempe\n85287AZUnited States"
]
| []
| In this paper, we consider the problem of locating the information source with sparse observations. We assume that a piece of information spreads in a network following a heterogeneous susceptible-infected-recovered (SIR) model and that a small subset of infected nodes are reported, from which we need to find the source of the information. We adopt the sample path based estimator developed in[1], and prove that on infinite trees, the sample path based estimator is a Jordan infection center with respect to the set of observed infected nodes. In other words, the sample path based estimator minimizes the maximum distance to observed infected nodes. We further prove that the distance between the estimator and the actual source is upper bounded by a constant independent of the number of infected nodes with a high probability on infinite trees. Our simulations on tree networks and real world networks show that the sample path based estimator is closer to the actual source than several other algorithms. | 10.1186/s40649-014-0003-2 | [
"https://arxiv.org/pdf/1309.4846v1.pdf"
]
| 7,886,671 | 1309.4846 | bcdee769cc356f2ffb0d5426485ec4506bbd21d1 |
A Robust Information Source Estimator with Sparse Observations
Kai Zhu
School of Electrical
Computer and Energy Engineering
Arizona State University Tempe
85287AZUnited States
Lei Ying [email protected]
School of Electrical
Computer and Energy Engineering
Arizona State University Tempe
85287AZUnited States
A Robust Information Source Estimator with Sparse Observations
In this paper, we consider the problem of locating the information source with sparse observations. We assume that a piece of information spreads in a network following a heterogeneous susceptible-infected-recovered (SIR) model and that a small subset of infected nodes are reported, from which we need to find the source of the information. We adopt the sample path based estimator developed in[1], and prove that on infinite trees, the sample path based estimator is a Jordan infection center with respect to the set of observed infected nodes. In other words, the sample path based estimator minimizes the maximum distance to observed infected nodes. We further prove that the distance between the estimator and the actual source is upper bounded by a constant independent of the number of infected nodes with a high probability on infinite trees. Our simulations on tree networks and real world networks show that the sample path based estimator is closer to the actual source than several other algorithms.
I. INTRODUCTION
In this paper, we are interested in locating the source of information that spreads in a network by using sparse observations. The solution to this problem has important applications such as locating the sources of epidemics, news/rumors in social networks or online computer virus. The problem has been studied in [2], [3], [4], [5] under a homogeneous susceptible-infection (SI) model and in [1] under a homogeneous susceptible-infection-recover (SIR) model, assuming that a complete snapshot of the network is given.
While [2], [3], [4], [5], [1] answered some fundamental questions about information source detection in large-scale networks, a complete snapshot of a real world network, which may have hundreds of millions of nodes, is expensive to obtain. Furthermore, these works assume homogeneous infection across links and homogeneous recovery across nodes, but in reality, most networks are heterogeneous. For example, people close to each other are more likely to share rumors and epidemics are more infectious in the regions with poor medical care systems. Therefore, it is important to take sparse observations and network heterogeneity into account when locating information sources. In this paper, we consider a heterogeneous SIR model and assume only a small subset of infected nodes are reported to us. The goal is to identify the information source in a heterogeneous network by using sparse observations. We use the sample path based approach developed in [1] for locating the information source with sparse observations. Surprisingly, we find that the sample path based estimator is robust to network heterogeneity and the number of observed infected nodes. In particular, our results show that even under a heterogeneous SIR model and with sparse observations, the sample path based estimator remains to be a Jordan infection center in infinite trees, where the Jordan infection center with a partial observation is the node that minimizes the maximum distance to observed infected nodes. We further show that in an infinite tree, the distance between a Jordan infection center and the actual source can be bounded by a value independent of the size of infected subnetwork with a high probability, where the infected subnetwork is the subnetwork that consists of nodes are either infected or recovered and is a connected component. Assume the size of the infected subnetwork is n, the result says that a Jordan infection center is a distance of O(1) from the actual source.
We remark that the locations of the Jordan centers only depend on the network topology and are independent of the infection and recovery probabilities, so the sample path based estimators (or the Jordan infection centers) are also robust to the information diffusion models, which makes it very appealing in practice since the accurate knowledge of the SIR parameters can be difficult to measure in reality.
A. Related Works
Other than [2], [3], [4], [5], [1], there are several related works in this area including: (1) detecting the first adopter of an innovation based on game theory [6], in which the maximum likelihood estimator is derived but the computational complexity of finding the estimator is exponential in the number of nodes; (2) distinguishing epidemic infection from random infection under the SI model [7]; (3) geospatial abduction which deals with reasoning certain locations in a two-dimensional geographical area that can explain observed phenomena [8], [9]. A recent paper [10] also proposed a dynamic message passing algorithm (DMP) to detect the information source under a general SIR model with complete or partial observations. However, the algorithm needs the complete information of infection and recovery probabilities. In addition, the complexity of DMP is very high under partial observations since almost all nodes in the network are candidates of the source, and the calculation needs to be repeated for every possible candidate. In the simulations, we will show that our algorithm significantly outperforms DMP in terms of both accuracy and speed. We will see that our algorithm is 400× faster even when we limit the DMP algorithm to a subnetwork.
II. A HETEROGENEOUS SIR MODEL
In this section, we introduce the heterogeneous SIR model for information propagation. Different from the homogeneous SIR model in which infection and recovery probabilities are both homogeneous [1], the heterogeneous SIR model we consider allows different infection probabilities at different links and different recovery probabilities at different nodes.
Consider an undirected graph G = {V, E}, where V is the set of nodes and E is the set of edges. Denote by (u, v) ∈ E the edge between node u and node v. Each node v ∈ V has three states: susceptible (S), infected (I), and recovered (R). Time is slotted. At the beginning of each time slot, each infected node attempts to contact all its susceptible neighbors. A contact from node u to node v succeeds with probability q uv . A susceptible node becomes infected after being successfully contacted by one of its infected neighbors. At the middle of each time slot, an infected node, if it is infected before the current time slot, recovers with probability p v . A recovered node cannot be infected again. We assume that contacts succeed independently across links and time slots; and nodes recover independently across nodes and time slots.
Consider a network shown in Figure 1, where node e is in the susceptible state, nodes a and c are in the infected state and nodes b and d are in the recovered state. Then at the next time slot, node e becomes infected with probability
1 − (1 − q ae )(1 − q ce ),
and nodes a and c recover with probability p a and p c , respectively. In this section, we formally define the problem of information source detection. Adopting the notation in [1], we define X v (t) to be the states of node v at the end of time slot t such that
X v (t) = S, if v is in state S at time t; I, if v is in state I at time t; R, if v is in state R at time t.
Let X(t) = {X v (t) : ∀v ∈ V} denote the state of all nodes in the network at time t.
In this paper, we assume that we only have one partial snapshot of the network, which is a subset of the infected nodes. This observation can be sparse, and details will be given in the next section. We assume that the states of other nodes are unknown. We let Y v denote the state of node v in the snapshot such that
Y v = 1, if node v is observed to be infected; 0, otherwise. Let Y = {Y v : ∀v ∈ V}.
We denote by v * the information source. The problem of information source detection is to locate v * based on the partial observation Y and the network topology G. Due to recovery and partial observations, all nodes in the network are potential candidates of the information source. The maximum likelihood estimator of the problem is therefore computationally expensive to find as pointed out in [1]. In this paper, we follow the sample path based approach proposed in [1] to find an estimator of v * .
Since X(t) is the state of the network at time t, the sequence {X(τ )} 0≤τ ≤t specifies the complete infection process. Therefore, we call X[0, t] = {X(τ ) : 0 ≤ τ ≤ t} a sample path. We further define a function F (·) such that
F (X v (t)) = 1, if X v (t) = I and v is observed; 0, otherwise.
This function maps the actual state of a node to the observed state of the node. F(X(t)) = Y if and only if F (X v (t)) = Y v , ∀v ∈ V. The optimal sample path X * [0, t * ] is defined to be the most likely sample path that results in the observed snapshot, i.e., it solves the following optimization problem:
X * [0, t * ] = arg max t,X[0,t]∈X (t) Pr(X[0, t]),(1)
where X (t) = {X[0, t]|F(X(t)) = Y}. The source associates with X * [0, t * ] is called the sample path based estimator. It is proved in [1] that the sample path based estimator on an infinite tree is a Jordan infection center under the homogeneous SIR model with a complete snapshot. The focus of this paper is to identify the sample path based estimator under the heterogeneous SIR model with sparse observations.
IV. MAIN RESULTS
In this section, we summarize the main results of this paper.
A. Main result 1: The Jordan infection centers as the sample path based estimators
In our theoretical analysis, we consider tree networks with infinitely many levels (or called infinite trees) to derive the sample path based estimator under the heterogeneous SIR model with a partial snapshot. Let I Y denote the set of observed infected nodes. We define the observed infection eccentricityẽ(v, I Y ) of node v to be the maximum distance between v and any observed infected node where the distance is defined to be the shortest distance between two nodes. The Jordan infection centers of the partial snapshot are then defined to be the nodes with the minimum observed infection eccentricity. The following theorem states that on an infinite tree, the sample path based estimator is a Jordan infection center of the partial snapshot. Theorem 1. Consider an infinite tree and assume the partial snapshot Y contains at least one infected node. The sample path based estimator, denoted by v † , is a Jordan infection
center, i.e., v † ∈ arg max v∈Vẽ (v, I Y ).
The proof of this theorem consists of the following key steps.
1) In the first step, we focus on the sample paths originated from node v (i.e., we assume node v is the source). We consider two groups of sample paths: X v (t) and X v (t + 1), where X v (t) is the set of the sample paths that are originated from v, have time duration t, and are consistent with the partial snapshot, i.e., F(X(t)) = Y for any X[0, t] ∈ X v (t).
The set X v (t + 1) is similarly defined. We show that for any t ≥ẽ(v, I Y ), the sample path with the highest probability in X v (t) occurs more likely than the one in X v (t + 1). In other words, Pr(X[0, t + 1]).
As a consequence of this result, we conclude that the sample path that has the highest probability among those originated from node v has a duration ofẽ(v, I Y ) (the observed infection eccentricity of node v). This result will be proved in Lemma 3 in Section VI.
2) In the second step, we consider two neighboring nodes, say nodes u and v; and assume node v has a smaller observed infection eccentricity than node u. Based on Lemma 3, we will prove that the optimal sample path associated with node v occurs with a higher probability than that of node u. The key idea is to construct a sample path originated from node v based on the optimal sample path originated from node u and show that it occurs with a higher probability. This result will be proved in Lemma 4 in Section VI.
3) We will finally prove that starting from any node, there exists a path from the node to a Jordan infection center such that the observed infection eccentricity strictly decreases along the path. Consider an example in Figure 2. Nodes b and f are two observed infected nodes. So node a is a Jordan infection center with observed infection eccentricity 1. The path from node e to node a is
e → d → c → b → a,
along which the observed infection eccentricity decreases as
5 → 4 → 3 → 2 → 1.
By repeatedly using Lemma 4, it can be shown that the optimal sample path originated from a Jordan infection center occurs with a higher probability than the optimal sample path originated from a node which is not a Jordan infection center, which implies the sample path based estimator must be a Jordan infection center. Unlike the maximum likelihood estimator, the sample path estimator is not guaranteed that the estimator is the node that most likely leads to the observation. It has been shown in [1] that on tree networks and under the homogeneous SIR model, the distance between the estimator and the actual source is a constant with a high probability. It is easy to see that with a partial observation, the distance between the estimator and the actual source cannot be bounded if the observed infection nodes are arbitrarily chosen. In this paper, we consider a class of fairly general sampling algorithms that generate the partial observation (and maybe sparse). The sampling algorithms have the following property: for any set of M infected nodes, the probability that at least one node in the set is reported approaches to one as M goes to infinity. We call such a sampling algorithm unbiased, in other words, any subset of infected nodes is likely to contain an observed infected node when the size of the subset is large enough. Note that if an infected node is reported with probability at least δ for some δ > 0, independent of other nodes, then it satisfies the property above. Our second main result is that the sample path estimator is within a constant distance from the actual source independent of the size of the infected subnetwork if the sampling algorithm is unbiased. We also emphasize that the observation generated by an unbiased sampling algorithm can be very sparse since we only require one observed infected node is reported with a high probability among M nodes when M is sufficiently large.
Theorem 2. Consider an infinite tree. Let g min be the lower bound on the number of children, and q min > 0 be the lower bound on q. Assume g min > 1, g min q min > 1, and the observed infection topology Y contains at least one infected node and is generated by an unbiased sampling algorithm. Then given > 0, the distance between the sample path estimator and the actual source is d with probability 1 − , where d is independent of the size of the infected subnetwork. In other words, the distance is O(1) with a high probability. . The idea of the proof is illustrated using Figure 3, which consists of the following key steps: 1) We first define a one-time-slot infection subtree to be a subtree of the infected subnetwork such that each node on the subtree is infected in the next time slot after parent is infected, except the source node. Note that the depth of a one-time-slot infection subtree grows by one deterministically until it terminates. We further say a node survives at time t if it is the root of a one-time-slot infection subtree which has not terminated by time t.
2) In the first step, we will prove that there exist at least two survived nodes within a distance L from the information source. In Figure 3, node a is the information source, and nodes b and c are two survived nodes.
3) In the second step, we will show that with a high probability, at least one infected node at the bottom of a one-time-slot infection subtree, which has not terminated, is observed under an unbiased sampling algorithm. In Figure 3, nodes d and f are two sampled nodes corresponding to the two one-time-slot infection subtrees starting from nodes b and c, respectively. 4) Since a one-time-slot infection subtree grows by one deterministically at each time slot, the depth of a one-timeslot infection subtree is t − t I k , where k is the root node of the one-time-slot infection subtree. Recall that the Jordan infection centers minimize the maximum distance to observed infected nodes, so a Jordan infection center must be within a O(1) distance from the two survived nodes (nodes b and c). Considering Figure 3, we know that the actual source (node a) has an infection eccentricity ≤ t since the information can propagate at most t hops at time t. So the infection eccentricity of the Jordan infection centers is no more than t according to the definition. Assume node e in Figure 3 is a Jordan infection center, then it is within a distance of O(t) from nodes d and f, so is within a distance of O(1) from nodes b and c. Since nodes b and c are no more than L hops from the actual source a, we can conclude that the distance between the actual source a and the estimator e is O(1).
C. Reverse Infection Algorithm
The Jordan infection centers for general graphs can be identified by the reverse infection algorithm proposed in [1]. In the algorithm, each observed infected node broadcasts its identity (ID) to its neighbors. All nodes in the network record the distinct IDs they received. When a node receives a new distinct ID, it records it and then broadcasts it to its neighbors. This process stops when there is a node who receives the IDs from all observed infected nodes. It is easy to verify the set of nodes who first receive all infected IDs is the set of Jordan infection centers. When there are multiple Jordan infection centers in the graph, we select the one with the maximum infection closeness centrality as the information center. The infection closeness centrality is defined as the inverse of the sum of the distances from one node to all observed infected nodes.
D. Discussion: Robustness
According to the two main results above, we know that the sample path based estimator remains to be a Jordan infection center. This is a somewhat surprising result since the locations of the Jordan infection centers are determined by the topology of the network, and are independent of the parameters of the heterogeneous SIR model. In other words, the locations of the Jordan infection centers remain the same for different SIR processes as long as the set of observed infected nodes is the same. This property suggests that the sample path based estimator is a robust estimator and can be used in the case when the parameters of the SIR model are unknown, which is a very desirable property since knowing these parameters can be difficult in practice.
In the simulations, we also consider a weighted graph with the link weights chosen proportionally according to the SIR parameters and use the weighted Jordan infection centers as the estimator. Interestingly, we will see that the performance is worse than the unweighted Jordan infection centers, which again demonstrates the robustness of the sample path based estimator.
Furthermore, the main results hold as long as the sampling algorithm is unbiased and are independent of the number of samples. So the results are valid for sparse observations and are robust to the number of observations.
V. SIMULATIONS
In this section, we evaluate the performance of the reverse infection algorithm for the heterogeneous SIR model on different networks including tree networks and real world networks.
We first describe the heterogeneous SIR model we used in the simulation. Each edge e ∈ E is assigned with a weight q e which is uniformly distributed over (0, 1). The infection time over each edge e ∈ E is geometrically distributed with mean 1/q e . Similarly, each node v ∈ V is assigned with a weight p v generated by an uniform distribution over (0, 1) and the recovery time is geometrically distributed with mean 1/p v . The information source is randomly selected. The total number of infected and recovered nodes in each infection We briefly introduce the three main algorithms which were used to compare with the reverse infection algorithm (RI).
1) Closeness centrality algorithm (CC): The closeness centrality algorithm selects the node with the maximum infection closeness as the information source.
2) Weighted reverse infection algorithm (wRI): The weighted reverse infection algorithm selects the node with the minimum weighted infection eccentricity as the information source where the weighted infection eccentricity is similar to the infection eccentricity except that the length of a path is defined to be the sum of the link weights instead of the number of hops, and the link weight is the average time it takes to spread the information over the link, i.e., 1/q e on edge e.
3) Weighted closeness centrality algorithm (wCC): The weighted closeness centrality algorithm selects the node with the maximum weighted infection closeness as the information source.
A. Tree Networks
We first evaluated the performance of the RI algorithm on tree networks. 1) Regular Trees: A g-regular tree is a tree where each node has g neighbors. We set the degree g = 5 in our simulations.
We varied the sample probability σ from 0.01 to 0.1. The simulation results are summarized in Figure 4a, which shows the average distance between the estimator and the actual information source versus the sampling probability. When the sample probability increases, the performance of all algorithms improve. When the sample probability is larger than 6%, the average distance becomes stable which means a small number of infected nodes is enough to obtain a good estimator. We also notice that the average distance of RI is smaller than all other algorithms and is less than one hop when σ ≥ 0.04. wRI has a similar performance with RI when the sample probability is small (=0.01) but becomes much worse when the sample probability increases.
2) Binomial Trees: We further evaluated the performance of RI and other algorithms on binomial trees T (ξ, β) where the number of children of each node follows a binomial distribution such that ξ is the number of trials and β is the success probability of each trial. In the simulations, we selected ξ = 10 and β = 0.4. Again, we varied σ from 0.01 to 0.1. The results are shown in Figure 4b. Similar to the regular trees, the performance of RI dominates CC, wRI and wCC, and the difference in terms of the average number of hops is approximately one when σ ≥ 0.03.
B. Real World Networks
In this section, we conducted experiments on two real world networks: the Internet Autonomous Systems network (IAS) 1 , and the power grid network (PG) 2 .
1) The Power Grid Network: The power grid network has 4,941 nodes and 6,594 edges. On average, each node has 1.33 edges. So the power grid network is a sparse network. The simulation results are shown in Figure 4c. In the power grid network, we can see that RI and wRI have similar performance, and both outperform CC and wCC by at least one hop when σ ≥ 0.04.
2) The Internet Autonomous Systems Network: The Internet Autonomous Systems network is the data collected on March, 31st, 2001. There are 10,670 nodes and 22,002 edges in the network. The simulation results are shown in Figure 4d. wRI and wCC always perform worse than RI. Although RI and CC have similar performance when the sample probability is large, RI outperforms CC when σ ≤ 0.03.
C. RI versus DMP
We finally compared the performance of RI and DMP with sparse observations. We conducted the simulation on the power grid network and fixed the sample probability to be 10%. Under this setting, the complexity of DMP is very high since the DMP computation needs to be repeated for every node in the network. Since nodes far away from the observed infected nodes are not likely to be the information source, we ran DMP over a subset of nodes close to the Jordan infection centers to reduce the complexity of the algorithm.
We tested the speed of RI and DMP on a machine with 1.8 GB memory, 4 cores 2.4 GHz Intel i5 CPU and Ubuntu 12. 10. The algorithms are implemented in Python 2.7. On average, it took RI 0.57 seconds to locate the estimator for one snapshot and took DMP 229.12 seconds. So RI is much faster than DMP. Figure 5 shows the CDF of the distance from the estimator to the actual source under DMP and RI. We can see that RI dominates DMP, in particular, 71% of the estimators under RI are no more than 7 hops from the actual source comparing to 57% under DMP. Therefore, RI outperforms DMP in terms of both speed and accuracy. We remark that we did not compare the performance of RI and DMP on the Internet Autonomous System (IAS) network because the complexity of running DMP on a large size network like the IAS network is prohibitively high.
VI. PROOFS
In this section, we present the proofs of the main results. i.e., it is the duration of the optimal sample path with node v as the information source.
Lemma 3. (Time Inequality)
Consider an infinite tree rooted at v r . Assume that v r is the information source and the observed snapshot Y contains at least one infected node. If e(v r , I Y ) ≤ t 1 < t 2 , the following inequality holds,
max X[0,t1]∈X (t1) Pr(X[0, t 1 ]) > max X[0,t2]∈X (t2) Pr(X[0, t 2 ]), whereX (t) = {X[0, t]|Y = F(X(t))}. In addition, t * vr =ẽ(v r , I Y ) = max u∈I Y d(v r , u),
i.e., t * vr is equal to the observed infection eccentricity of v r with respect to I Y .
Proof. We adopt the notations defined in [1], which are listed below:
• C(v) is the set of children of v. • φ(v) is the parent of node v.
• Y k is the set of infection topologies where the maximum distance from v r to an infected node is k. All possible infection topologies are then partitioned into countable subsets {Y k }.
• T v is the tree rooted in v. • T −u v
is the tree rooted in v without the branch from its neighbor u.
• X([0, t], T −u v ) is the sample path restricted to topology T −u v .
Considering the case where the time difference of two sample paths is one, we will show that
max X[0,t]∈X (t) Pr(X[0, t]) > max X[0,t+1]∈X (t+1)
Pr(X[0, t + 1]).
Next, we use induction over Y k .
Step 1 k = 0 v r is the only observed infected node in this case. Given a sample path X[0, t + 1] ∈X (t + 1), the probability of the sample path can be written as
Pr (X[0, t + 1]) = Pr (X[0, t]) Pr(X(t + 1)|X[0, t])
Since v r is the only observed infected node and all other nodes' states are unknown, we assign X [0, t] ∈X (t) to be same as the first t time slots in X[0, t + 1], i.e., X [0, t] = X[0, t]. Hence, we obtain that
Pr (X [0, t]) = Pr (X[0, t]) > Pr (X[0, t + 1])
Therefore, the case k = 0 is proved.
Step 2 Assume the inequality holds for k ≤ n, and consider k = n + 1, i.e., Y ∈ Y n+1 . Clearly, t ≥ n + 1 ≥ 1 for each X[0, t]. Furthermore, the set of subtrees T = {T −vr u |u ∈ C(v r )} are divided into two subsets:
T h = {T −vr u |u ∈ C(v r ), T −vr u ∩ I Y = ∅}, and T i = T \T h .
Given t R vr , the infection processes on the sub-trees are mutually independent. We construct X [0, t] which occurs more likely than X * [0, t + 1] according to the following steps, where
X * [0, t + 1] = max X[0,t+1]∈X (t+1) Pr(X[0, t + 1]).
Part 1 T i . For a subtree in T i , the proof follows Step 2.b and Step 2.c of Lemma 1 in [1]. The intuition is as follows: Consider a subtree and a sample path on it with duration t+1. If u is not infected at the first time slot, we can construct a sample path with duration t by moving the events one time slot earlier. The new sample path (with duration t) has a higher probability to occur than the original one. If u is infected in the first time slot, we can invoke the induction assumption to the subtree rooted at u, which belongs to Y n .
Part 2 v r . In this part, we have the freedom to assign the unobserved node as infected or healthy. In Part 1, the infection time of each root u in subtrees T i of X [0, t] is either the same as or one time slot earlier than its infection time in X * [0, t+1]. Therefore, if t R vr ≤ t, the recovery time of the source v r in X [0, t] can be assigned the same as that in X * [0, t + 1].
If t R vr = t + 1, the source v r recovers at time slot t + 1 which means v r is not observed since the observation set only contains infected nodes. Therefore, in X [0, t] we assign the source to be in state I at time t, which is the same as the state of v r at time t in X * [0, t + 1].
If t R vr > t + 1, v r remains infected in the sample path X * [0, t + 1]. We assign the source to be in state I in X [0, t].
As a summary, according to the assignment above, the states of the source v r in X [0, t] are the same as those of the first t time slots in X * [0, t + 1].
Part 3 T h . Based on the conclusion of Part 2, the subtrees belonging to T h in X [0, t] mimic the behaviors of the first t time slots in X * [0, t + 1].
Since X * [0, t+1] has one extra time slot during which some extra events occur, X [0, t] occurs with a higher probability on the subtrees in T h .
According to the discussion above, we conclude that time inequality holds for k = n + 1, hence for any k according to the principle of induction. Therefore, the lemma holds.
, v ∈ V such that (u, v) ∈ E, if t * u > t * v Pr(X * u [0, t * u ]) < Pr(X * v [0, t * v ]), where X * u [0, t * u ]
is the optimal sample path associated with root u.
Proof. The proof of the lemma follows the proof of Lemma 2 in [1]. The key idea is to construct a sample path rooted at v, which has a higher probability than the optimal sample path rooted at u. It is not hard to see that t * u = t * v + 1 based on the definition of the infection eccentricity. The graph is partitioned into T −u v and T −v u which are mutually independent after the infection of v and u. With this observation,
we constructX v [0, t * v ]) which infects u at the first time slot.X v ([0, t * v ], T −u v ) then mimics the behavior of X * u ([0, t * u ], T −u v ) andX v ([0, t * v − 1], T −v u ) has a higher probability than X * u ([0, t * u ], T −v u )
based on Lemma 3. The adjacent nodes inequality results in partial orders in the tree and makes it possible to compare the likelihood of optimal sample paths associated with adjacent nodes without knowing the actual probability of the optimal sample path. Following the proof of Theorem 4 in [1], it can be shown that in tree networks, from any node, there exists a path from the node to a Jordan infection center such that the observed infection eccentricity strictly decreases along the path. By repeatedly using Lemma 4, we can then prove that the source of the optimal sample path must be a Jordan infection center.
B. Proof of Theorem 2
In this subsection, we present the proof that shows that the sample path estimator is within a constant distance from the actual source independent of the size of the infected subnetwork. Given a tree rooted in v * where the information starts from v * following the general SIR model, we define the following three branching processes.
1) Z l (T v * ) denotes the set of nodes which are in infected or recovered states at level l on tree T v * . Let Z l (T v * ) denote the cardinality of Z l (T v * ). Note that Z 0 (T v * ) = {v * }. We call this process the original infection process.
2) Z τ l (T v * ) denotes the set of infected and recovered nodes at level l whose parents are in set Z τ l−1 (T v * ) and who were infected within τ time slots after their parents were infected. This process adds a deadline τ on infection. If a node is not infected within τ time slots after its parent is infected, it is not included in this branching process. This process is called
τ −deadline infection process. From the definition, if u, v ∈ Z τ l (T v * ), then |t I u − t I v | ≤ l(τ − 1). For τ = 1, we call Z 1 l (T v * )
the one time slot infection process. The extinction probability of a branching process is the probability that there is no offspring at certain level of the branching process, i.e., Z 1 l (T v * ) = 0 for some l. Denote by ρ v the extinction probability of Z 1
l (T −φ(v) v )
. 3) We define the binomial branching process as a branching process whose offspring distribution follows binomial distribution B(g, ϕ) where g is the number of trials and ϕ is the success probability. Denote by ρ the extinction probability of the binomial branching process.
The following notations will be used in later analysis.
• v † denotes the optimal sample path estimator.
• g min is the lower bound on the number of children, i.e.,
min v |C(v)| ≥ g min , ∀v ∈ V.
• q min is the lower bound on the infection probability, i.e., q min = min e q e , ∀e ∈ E.
• σ τ v is the probability that a node v infects at least one of its children within τ time slot after v is infected.
Given n 0 > 0 and τ > 0, define l † = min l where Z τ l (T v * ) > n 0 , i.e., l † is the first level where the τ -deadline infection process has more than n 0 offsprings.
Given τ and level L ≥ 2, we consider the following two events:
Event 1: Z L (T v * ) = 0. Event 2: l † ≤ L and at least two one time slot infection processes starting from level l † survive, i.e., ∃u,
v ∈ Z τ l † (T v * ) such that ∀l, Z 1 l (T −φ(u) u ) = 0 and Z 1 l (T −φ(v) v ) = 0.
In addition, at least one infected node at the bottom of each survived one time slot infection process is observed.
For event 1, no node at level L gets infected and the infection process terminates at level L − 1. So the infection eccentricity of v * is at most L − 1, and the minimum infection eccentricity of the network is at most L − 1. Therefore, the distance between v * and v † is no more than 2(L − 1).
Considering event 2, we assume the information propagates for t time slots. The deadline property of the τ -deadline infection process indicates t I u1 ≤ τ l † and t I u2 ≤ τ l † . Given a nodeṽ at level (τ + 1)
l † − 1 whereṽ ∈ T −φ(u2) u2 and a node v ∈ T −φ(u1) u1
which is an observed infected node at the bottom of the infection tree, from Figure 6, we obtain
d(ṽ, v ) = t − t I u1 + τ l † + 1 ≥ t + 1.
Note that ∀u ∈ I,
d(v * , u) ≤ t < d(ṽ, v ).
Since l † ≤ L, any node at or below level L(τ + 1) − 1 has an infection eccentricity larger than that of v * . Hence, v † cannot be at or below level L(τ + 1) − 1. Therefore,
d(v † , v * ) < (τ + 1)L − 1.
Next, we prove the probability that either event 1 or event 2 happens goes asymptotically to 1. Denote by K l † the number of one time slot infection processes which start from level l † and survive. Denote by E the event that a survived one time slot infection process has at least one observed infected node at its lowest level. According to the discussion above, the probability that the distance between the estimator and the actual source is no more than (τ + 1)L − 1 is at least
Pr(Z L (T v * ) = 0) + Pr(K l † ≥ 2, l † ≤ L) Pr(E) 2 ≥ Pr(Z L (T v * ) = 0)+Pr l † ≤ L Pr K l † ≥ 2 l † ≤ L Pr(E) 2 = Pr(Z L (T v * ) = 0) + Pr L i=1 Z τ i > n 0 × Pr(K l † ≥ 2|l † ≤ L) Pr(E) 2 = 1−Pr L i=1 0 < Z τ i (T v * ) ≤ n 0 −Pr L i=1 Z τ i (T v * ) = 0 × Pr(K l † ≥ 2|l † ≤ L) Pr(E) 2 + Pr(Z L (T v * ) = 0).
In addition, we have
Pr(K l † ≥ 2|l † ≤ L) (2) = L l=1 Pr(K l † ≥ 2, l † = l|l † ≤ L) (3) = L l=1 Pr(K l † ≥ 2|l † = l) Pr(l † = l|l † ≤ L).(4)
In Lemma 5, we prove that the extinction probability of each branching process from level l † is upper bounded by the exitinction probability ρ of the binomial infection process B(g min , q min ). Therefore, at level l † we have n 0 i.i.d one time infection processes whose extinction probabilities are upper bounded by ρ. The probability that at least two of them survive goes asymptotic to 1 when n 0 increases. Therefore, ∀ 1 > 0, we have enough large n 0 , such that
Pr(K l † ≥ 2|l † = l) ≥ 1 − 1 .
Therefore, equation (4) becomes
Pr(K l † ≥ 2|l † ≤ L) ≥ (1 − 1 ) L l=1
Pr(l † = l|l † ≤ L)
= (1 − 1 ).
We show in Lemma 7 that Pr(E) ≥ 1 − 2 given 2 > 0. If n 0 and t are sufficiently large, we have
Pr(K l † ≥ 2|l † ≤ L) Pr(E) 2 ≥ (1 − 1 )(1 − 2 ) 2 .
Therefore,
Pr(Z L (T v * ) = 0) + Pr(K l † ≥ 2, l † ≤ L) Pr(E) 2 (5) ≥ 1 − Pr L i=1 0 < Z τ i (T v * ) ≤ n 0 (6) × (1 − 1 )(1 − 2 ) 2 (7) − Pr L i=1 Z τ i (T v * ) = 0 + Pr(Z L (T v * ) = 0) (8) = 1 − Pr L i=1 0 < Z τ i (T v * ) ≤ n 0 Part 1 (9) × (1 − 1 )(1 − 2 ) 2 (10) + Pr(Z L (T v * ) = 0) − Pr(Z τ L (T v * ) = 0) Part 2 ,(11)
where equation (11)
holds since Z τ l (T v * ) = 0 implies that Z τ L (T v * ) = 0 for l ≤ L.
For part 1 in equation (11), we prove in Lemma 6, given 3 > 0, when τ and L are sufficiently large,
1 − Pr L i=1 0 < Z τ i (T v * ) ≤ n 0 > 1 − 3 .
For part 2 in equation (11), we have Therefore, given 4 > 0, when τ is sufficiently large,
Pr(Z L (T v * ) = 0) − Pr(Z τ L (T v * ) = 0) ≥ − 4 .
Hence, we have
Pr(Z L (T v * ) = 0) + Pr(K l † ≥ 2, l † ≤ L) Pr(E) 2 ≥ (1 − 1 )(1 − 2 ) 2 (1 − 3 ) − 4 .
Now choosing 1 = 2 = 3 = 4 = 5 /5 for some 4 > 0, we have
Pr(Z L (T v * ) = 0) + Pr(K l † ≥ 2, l † ≤ L) Pr(E) 2 ≥ 1 − 5 .
Now let |Y| denote the number of infected nodes in the observation Y. Define events E 1 = {Z L = 0} and E 2 = {K l ≥ 2 for some l ≤ L} and E 3 is the event that two of the survived one time slot infection processes have at least one observed infected node each at their bottoms. We have
Pr(E 1 ||Y| ≥ 1) + Pr (E 2 ∩ E 3 ||Y| ≥ 1) = 1 Pr(|Y| ≥ 1) (Pr(E 1 ∩ {|Y| ≥ 1}) +Pr (E 2 ∩ E 3 ∩ {|Y| ≥ 1})) .
Since E 2 ∩ E 3 implies that |Y| ≥ 1, we have
Pr(E 1 ||Y| ≥ 1) + Pr (E 2 ∩ E 3 ||Y| ≥ 1) = 1 Pr(|Y| ≥ 1) (Pr(E 1 ∩ {|Y| ≥ 1}) + Pr (E 2 ∩ E 3 )) = 1 Pr(|Y| ≥ 1) (Pr(E 1 ) − Pr(E 1 ∩ {|Y| = 0}) +Pr (E 2 ∩ E 3 )) ≥ 1 Pr(|Y| ≥ 1) (Pr(E 1 ) − Pr({|Y| = 0}) + Pr (E 2 ∩ E 3 )) ≥ 1 Pr(|Y| ≥ 1) (Pr({|Y| ≥ 1}) − 5 ) =1 − 5 Pr(|Y| ≥ 1) .
Note that Pr(|Y| ≥ 1) is a positive constant since the one time slot infection process starting from the information source survives with non-zero probability. The theorem holds by choosing 5 = Pr(|Y| ≥ 1).
Lemma 5. The extinction probability of an one time slot infection process is smaller than the extinction probability of a binomial branching process B(g min , q min ), i.e., ∀v ∈ V,
ρ v < ρ.
Proof. As shown in Figure 7, we construct a virtual source process Z
(vs) l (T −φ(v) v
) and a min-infection process Z In the min-infection process, infection spreads over edges with probability q min . In the virtual source process, the probability that a node gets infected is
Pr(Y (vs) v = 1) = Pr(Y (mi) v = 1)+Pr(Y (mi) v = 0)· q uv − q min 1 − q min = q uv ,
i.e., for each node u ∈ C(v), v tries to infects u with probability q min . If v fails to infect u, a virtual source v tries to infect u with probability qvu−qmin 1−qmin . Therefore, the virtual source process has the same distribution with the one time slot infection process.
We now couple the min-infection process and the virtual source infection process as follows:
• If Y (mi) v = 1, then Y (vs) v = 1. • If Y (mi) v = 0, then Y (vs) v
= 1 with probability quv−qmin 1−qmin . Since a node is more likely to get infected in the virtual source infection process, we obtain
ρ (vs) v ≤ ρ (mi) v .
Recalling the one time slot infection process has the same distribution with the virtual source branching process, we obtain ρ v ≤ ρ (mi) v , ∀v. In addition, the min-infection process has more children than the binomial branching process with the same infection probability for each children. It is obvious that the binomial branching process is more likely to die out, i.e., ρ (mi) v < ρ. As a summary, we prove ρ v < ρ. Lemma 6. Assume ∃ξ > 0 such that σ τ v < 1 − ξ, ∀v ∈ V. Given any > 0, there exists a constant L such that for any L ≥ L ,
Pr L i=1 0 < Z τ i (T v * ) ≤ n 0 ≤
Proof. Follows the same argument of Lemma 7 in [1], by choosing L = log log (1 − ξ n0 ) ,
we obtain for any L ≥ L , > 0
Pr L i=1 0 < Z τ i (T v * ) ≤ n 0 ≤ .
Lemma 7. For any > 0, there exists a sufficiently large t such that Pr(E) ≥ 1 − Proof. Note the binomial branching process B(g min , q min ) is a Galton-Watson (GW) process [11] which requires each node has an i.i.d offspring distribution. The previous result about the instability of the Galton-Watson process in Theorem 6.2 in [11] proves that the GW process either goes to infinity or goes to 0. If the GW process survives, the number of offsprings goes to infinity as the level increases. Therefore, for sufficiently long time, the survived binomial branching process will have a sufficiently large number of offsprings at the lowest level. Since the one time slot infection process always has at least the same number of children as the binomial branching process, the survived one time slot infection process will have enough number of infected nodes at the lowest level as time increases. According to the unbiased property of the partial observation, after sufficiently long time, the probability that at least one infected node in the lowest level is observed goes to 1 asymptotically, i.e.,
Pr(E) ≥ 1 − .
VII. CONCLUSION
In this paper, we studied the problem of detecting the information source in a heterogeneous SIR model with sparse observations. We proved that the optimal sample path estimator on an infinite tree is a node with the minimum infection eccentricity with partial observations. With a fairly general condition, we proved that the estimator is within constant distance from the actual information source with a high probability with a sparse observation. Extensive simulation results showed our estimator outperforms other algorithms significantly.
Fig. 1 :
1An example for illustrating the heterogeneous SIR model III. PROBLEM FORMULATION
Fig. 2 :
2The key intuition behind Theorem 1 B. Main result 2: An O(1) bound on the distance between a Jordan infection center and the actual information source
Fig. 3 :
3The key intuition behind Theorem 2
Fig. 4 :
4The Performance of RI, CC, wRI and wCC on Different Graphs graph is within the range of [100, 300]. Each infected node v in the infection graph reports with probability σ, independently. The snapshots used in the simulations have at least one infected node. We changed σ and evaluated the performance on different networks.
Fig. 5 :
5Denote by I Y = {v|Y v = 1} the set of observed infected nodes and H Y = {v|Y v = 0} the set of unobserved nodes. Given a node v, define the optimal time t * v to be t * v arg t max t,X[0,t]∈X (t) Pr (X[0, t]|v is information source) , 1 Available at http://snap.stanford.edu/data/index.html 2 Available at http://www-personal.umich.edu/ ∼ mejn/netdata/ The CDF of RI and DMP on the Power Grid Network
Lemma 4. (Adjacent Nodes Inequality)Consider an infinite tree with partial observation Y which contains at least one infected node. For u
Fig. 6: A Pictorial Description of the Distance Relations in Theorem 2
T v * ) = 0) = Pr(Z L (T v * ) = 0).
Fig. 7: A Pictorial Description of the Two Auxiliary Processes in Lemma 5
Information source detection in the SIR model: A sample path based approach. K Zhu, L Ying, arXiv:1206.5421Arxiv preprintK. Zhu and L. Ying, "Information source detection in the SIR model: A sample path based approach," Arxiv preprint arXiv:1206.5421, 2012.
Detecting sources of computer viruses in networks: Theory and experiment. D Shah, T Zaman, Proc. Ann. ACM SIGMETRICS Conf. Ann. ACM SIGMETRICS ConfNew York, NYD. Shah and T. Zaman, "Detecting sources of computer viruses in networks: Theory and experiment," in Proc. Ann. ACM SIGMETRICS Conf., New York, NY, 2010, pp. 203-214.
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Rumor centrality: a universal source detector. Proc. Ann. ACM SIGMETRICS Conf. Ann. ACM SIGMETRICS ConfLondon, England, UK--, "Rumor centrality: a universal source detector," in Proc. Ann. ACM SIGMETRICS Conf., London, England, UK, 2012, pp. 199-210.
Identifying infection sources and regions in large networks. W Luo, W P Tay, M Leng, arXiv:1204.0354Arxiv preprintW. Luo, W. P. Tay, and M. Leng, "Identifying infection sources and regions in large networks," Arxiv preprint arXiv:1204.0354, 2012.
Spotting trendsetters: Inference for network games. V G Subramanian, R Berry, Proc. Annu. Allerton Conf. Communication, Control and Computing. Annu. Allerton Conf. Communication, Control and ComputingV. G. Subramanian and R. Berry, "Spotting trendsetters: Inference for network games," in Proc. Annu. Allerton Conf. Communication, Control and Computing, 2012.
Network forensics: Random infection vs spreading epidemic. C Milling, C Caramanis, S Mannor, S Shakkottai, Proc. Ann. ACM SIGMETRICS Conf. Ann. ACM SIGMETRICS ConfC. Milling, C. Caramanis, S. Mannor, and S. Shakkottai, "Network forensics: Random infection vs spreading epidemic," in Proc. Ann. ACM SIGMETRICS Conf., 2012, pp. 223-234.
GAPs: Geospatial abduction problems. P Shakarian, V S Subrahmanian, M L Sapino, ACM Trans. Intell. Syst. Technol. 31P. Shakarian, V. S. Subrahmanian, and M. L. Sapino, "GAPs: Geospatial abduction problems," ACM Trans. Intell. Syst. Technol., vol. 3, no. 1, pp. 1-27, Oct. 2011.
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Inferring the origin of an epidemy with dynamic message-passing algorithm. A Y Lokhov, M Mezard, H Ohta, L Zdeborova, arXiv:1303.5315arXiv preprintA. Y. Lokhov, M. Mezard, H. Ohta, and L. Zdeborova, "Inferring the origin of an epidemy with dynamic message-passing algorithm," arXiv preprint arXiv:1303.5315, 2013.
The Theory of Branching Processes. T E Harris, Dover PubnsT. E. Harris, The Theory of Branching Processes. Dover Pubns, 1963.
| []
|
[
"Hypothesis Set Stability and Generalization Spark Wave",
"Hypothesis Set Stability and Generalization Spark Wave"
]
| [
"Dylan J Foster ",
"Spencer Greenberg ",
"Satyen Kale [email protected] ",
"Google Research ",
"Haipeng Luo [email protected] ",
"Mehryar Mohri [email protected] ",
"Karthik Sridharan [email protected] ",
"\nMIT Institute for Foundations of Data Science\[email protected]\nUniversity of Southern\nCalifornia\n",
"\nInstitute of Mathematical Sciences\nGoogle Research and Courant\nCornell University\nNew York\n"
]
| [
"MIT Institute for Foundations of Data Science\[email protected]\nUniversity of Southern\nCalifornia",
"Institute of Mathematical Sciences\nGoogle Research and Courant\nCornell University\nNew York"
]
| []
| We present an extensive study of generalization for data-dependent hypothesis sets. We give a general learning guarantee for data-dependent hypothesis sets based on a notion of transductive Rademacher complexity. Our main results are two generalization bounds for data-dependent hypothesis sets expressed in terms of a notion of hypothesis set stability and a notion of Rademacher complexity for data-dependent hypothesis sets that we introduce. These bounds admit as special cases both standard Rademacher complexity bounds and algorithm-dependent uniform stability bounds. We also illustrate the use of these learning bounds in the analysis of several scenarios. | null | [
"https://arxiv.org/pdf/1904.04755v1.pdf"
]
| 104,291,883 | 1904.04755 | 5ce431736ace5c5215686521f8dddcc4b158b5cc |
Hypothesis Set Stability and Generalization Spark Wave
Dylan J Foster
Spencer Greenberg
Satyen Kale [email protected]
Google Research
Haipeng Luo [email protected]
Mehryar Mohri [email protected]
Karthik Sridharan [email protected]
MIT Institute for Foundations of Data Science
[email protected]
University of Southern
California
Institute of Mathematical Sciences
Google Research and Courant
Cornell University
New York
Hypothesis Set Stability and Generalization Spark Wave
We present an extensive study of generalization for data-dependent hypothesis sets. We give a general learning guarantee for data-dependent hypothesis sets based on a notion of transductive Rademacher complexity. Our main results are two generalization bounds for data-dependent hypothesis sets expressed in terms of a notion of hypothesis set stability and a notion of Rademacher complexity for data-dependent hypothesis sets that we introduce. These bounds admit as special cases both standard Rademacher complexity bounds and algorithm-dependent uniform stability bounds. We also illustrate the use of these learning bounds in the analysis of several scenarios.
Introduction
Most generalization bounds in learning theory hold for a fixed hypothesis set, selected before receiving a sample. This includes learning bounds based on covering numbers, VC-dimension, pseudo-dimension, Rademacher complexity, local Rademacher complexity, and other complexity measures (Pollard, 1984;Zhang, 2002;Vapnik, 1998;Koltchinskii and Panchenko, 2002;Bartlett et al., 2002). Some alternative guarantees have also been derived for specific algorithms. Among them, the most general family is that of uniform stability bounds given by Bousquet and Elisseeff (2002). These bounds were recently significantly improved by Feldman and Vondrak (2018), who proved guarantees that are informative, even when the stability parameter β is only in o(1), as opposed to o(1 √ m). New bounds for a restricted class of algorithms were also recently presented by Maurer (2017), under a number of assumptions on the smoothness of the loss function. Appendix A gives more background on stability.
In practice, machine learning engineers commonly resort to hypothesis sets depending on the same sample as the one used for training. This includes instances where a regularization, a feature transformation, or a data normalization is selected using the training sample, or other instances where the family of predictors is restricted to a smaller class based on the sample received. In other instances, as is common in deep learning, the data representation and the predictor are learned using the same sample. In ensemble learning, the sample used to train models sometimes coincides with the one used to determine their aggregation weights. However, standard generalization bounds cannot be used to provide guarantees for these scenarios since they assume a fixed hypothesis set. This paper studies generalization in a broad setting that admits as special cases both that of standard learning bounds for fixed hypothesis sets based on some complexity measure, and that of algorithm-dependent uniform stability bounds. We present an extensive study of generalization for sample-dependent hypothesis sets, that is for learning with a hypothesis set H S selected after receiving the training sample S. This defines two stages for the learning algorithm: a first stage where H S is chosen after receiving S, and a second stage where a hypothesis h S is selected from H S . Standard generalization bounds correspond to the case where H S is equal to some fixed H independent of S. Algorithm-dependent analyses, such as uniform stability bounds, coincide with the case where H S is chosen to be a singleton H S = {h S }. Thus, the scenario we study covers both existing settings and, additionally, includes many other intermediate scenarios. Figure 1 illustrates our general scenario.
We present a series of results for generalization with data-dependent hypothesis sets. We first present general learning bounds for data-dependent hypothesis sets using a notion of transductive Rademacher complexity (Section 3). These bounds hold for arbitrary bounded losses and improve upon previous guarantees given by Gat (2001) and Cannon et al. (2002) for the binary loss, which were expressed in terms of a notion of shattering coefficient adapted to the data-dependent case, and are more explicit than the guarantees presented by Philips (2005)[corollary 4.6 or theorem 4.7]. Nevertheless, such bounds may often not be sufficiently informative, since they ignore the relationship between hypothesis sets based on similar samples.
To derive a finer analysis, we introduce a key notion of hypothesis set stability, which admits algorithmic stability as a special case, when the hypotheses sets are reduced to singletons. We also introduce a new notion of Rademacher complexity for data-dependent hypothesis sets. Our main results are two generalization bounds for stable data-dependent hypothesis sets, both expressed in terms of the hypothesis set stability parameter, our notion Figure 1: Decomposition of the learning algorithm's hypothesis selection into two stages. In the first stage, the algorithm determines a hypothesis H S associated to the training sample S which may be a small subset of the set of all hypotheses that could be considered, say H = ⋃ S∈Z m H S . The second stage then consists of selecting a hypothesis h S out of H S .
S H S h S 2 H S H = [ S2Z m H S
of Rademacher complexity, and a notion of cross-validation stability that, in turn, can be upper-bounded by the diameter of the family of hypothesis sets. Our first learning bound (Section 4) is expressed in terms of a finer notion of diameter but admits a dependency in terms of the stability parameter β similar to that of uniform stability bounds of Bousquet and Elisseeff (2002). In Section 5, we use proof techniques from the differential privacy literature (Steinke and Ullman, 2017;Bassily et al., 2016;Feldman and Vondrak, 2018) to derive a learning bound expressed in terms of a somewhat coarser definition of diameter but with a more favorable dependency on β, matching the dependency of the recent more favorable bounds of Feldman and Vondrak (2018). Our learning bounds admit as special cases both standard Rademacher complexity bounds and algorithm-dependent uniform stability bounds. Shawe-Taylor et al. (1998) presented an analysis of structural risk minimization over datadependent hierarchies based on a concept of luckiness, which generalizes the notion of margin of linear classifiers. Their analysis can be viewed as an alternative study of datadependent hypothesis sets, using luckiness functions and ω-smallness (or ω-smoothness) conditions. A luckiness function helps decompose a hypothesis set into lucky sets, that is sets of functions luckier than a given function. The ω-smallness condition requires that the size of the family of loss functions corresponding to the lucky set of any function f with respect to a double-sample, measured by packing or covering numbers, be bounded with high probability by a function ω of the luckiness of f on the sample. The luckiness framework is attractive and the notion of luckiness, for example margin, can in fact be combined with our results. However, finding pairs of truly data-dependent luckiness and ω-smallness functions, other than those based on the margin and the empirical VC-dimension, is quite difficult, in particular because of the very technical ω-smallness condition (see Philips, 2005, p. 70). In contrast, our hypothesis set stability is simpler and often easier to bound. The notions of luckiness and ω-smallness have also been used by Herbrich and Williamson (2002) to derive algorithm-specific guarantees. The authors show a connection with algorithmic stability (not hypothesis set stability), at the price of a guarantee requiring the strong condition that the stability parameter be in o(1 m), where m is the sample size (see Herbrich and Williamson, 2002, pp. 189-190).
In section 6, we illustrate the generality and the benefits of our hypothesis set stability learning bounds by applying them to the analysis of several scenarios (see also Appendix K).
In Appendix J, we briefly discuss several extensions of our framework and results, including the extension to almost everywhere hypothesis set stability as in (Kutin and Niyogi, 2002). The next section introduces the definitions and properties used in our analysis.
Definitions and Properties
Let X be the input space and Y the output space. We denote by D the unknown distribution over X × Y according to which samples are drawn.
The hypotheses h we consider map X to a set Y ′ sometimes different from Y. For example, in binary classification, we may have Y = {−1, +1} and Y ′ = R. Thus, we denote by ∶ Y ′ × Y → [0, 1] a loss function defined on Y ′ × Y and taking non-negative real values bounded by one. We denote the loss of a hypothesis h∶ X → Y ′ at point z = (x, y) ∈ X × Y by L(h, z) = (h(x), y). We denote by R(h) the generalization error or expected loss of a hypothesis h ∈ H and byR S (h) its empirical loss over a sample S = (z 1 , . . . , z m ):
R(h) = E z∼D [L(h, z)]R S (h) = E z∼S [L(h, z)] = 1 m m i=1 L(h, z i ).
In the general framework we consider, a hypothesis set depends on the sample received. We will denote by H S the hypothesis set depending on the labeled sample S ∈ Z m of size m ≥ 1.
Definition 1 (Hypothesis set uniform stability) Fix m ≥ 1. We will say that a family of data-dependent hypothesis sets H = (H S ) S∈Z m is β-uniformly stable (or simply β-stable) for some β ≥ 0, if for any two samples S and S ′ of size m differing only by one point, the following holds:
∀h ∈ H S , ∃h ′ ∈ H S ′ ∶ ∀z ∈ Z, L(h, z) − L(h ′ , z) ≤ β.(1)
Thus, two hypothesis sets derived from samples differing by one element are close in the sense that any hypothesis in one admits a counterpart in the other set with β-similar losses.
Next, we define a notion of cross-validation stability for data-dependent hypothesis sets. The notion measures the maximal change in loss of a hypothesis on a training example and the loss of a hypothesis on the same training example, when the hypothesis is chosen from the hypothesis set corresponding to the a sample where the training example in question is replaced by a newly sampled example.
Definition 2 (Hypothesis set Cross-Validation (CV) stability) Fix m ≥ 1. We will say that a family of data-dependent hypothesis sets H = (H S ) S∈Z m has χ CV-stability for some χ ≥ 0, if the following holds (here, S z↔z ′ denotes the sample obtained by replacing z ∈ S by z ′ ):
∀S ∈ Z m ∶ E z ′ ∼D,z∼S sup h∈H S ,h ′ ∈H S z↔z ′ L(h ′ , z) − L(h, z) ≤ χ.(2)
We say that H hasχ average CV-stability for someχ ≥ 0 if the following holds:
E S∼D m z ′ ∼D,z∼S sup h∈H S ,h ′ ∈H S z↔z ′ L(h ′ , z) − L(h, z) ≤χ.(3)
We also define a notion of diameter of data-dependent hypothesis sets, which is useful in bounding CV-stability. In applications, we will typically bound the diameter, thereby the CV-stability.
Definition 3 (Diameter of data-dependent hypothesis sets) Fix m ≥ 1. We define the diameter ∆ and average diameter∆ of a family of data-dependent hypothesis sets H = (H S ) S∈Z m by
∆ = sup S∈Z m E z∼S sup h,h ′ ∈H S L(h ′ , z) − L(h, z) ∆ = E S∼D m z∼S sup h,h ′ ∈H S L(h ′ , z) − L(h, z) . (4)
Notice that, for consistent hypothesis sets, the diameter is reduced to zero since L(h, z) = 0 for any h ∈ H S and z ∈ S. As mentioned earlier, CV-stability of hypothesis sets can be bounded in terms of their stability and diameter:
Lemma 4 A family of data-dependent hypothesis sets H with β-uniform stability, diameter ∆, and average diameter∆ has (∆ + β)-CV-stability and (∆ + β)-average CV-stability.
Proof Let S ∈ Z m , z ∈ S, and z ′ ∈ Z. For any h ∈ H S and h ′ ∈ H S z↔z ′ , by the β-uniform stability of H, there exists h ′′ ∈ H S such that L(h ′ , z) − L(h ′′ , z) ≤ β. Thus,
L(h ′ , z)−L(h, z) = L(h ′ , z)−L(h ′′ , z)+L(h ′′ , z)−L(h, z) ≤ β + sup h ′′ , h∈H S L(h ′′ , z)−L(h, z).
This implies the inequality
sup h∈H S ,h ′ ∈H S z↔z ′ L(h ′ , z) − L(h, z) ≤ β + sup h ′′ , h∈H S L(h ′′ , z) − L(h, z),
and the lemma follows.
We also introduce a new notion of Rademacher complexity for data-dependent hypothesis sets. To introduce its definition, for any two samples S, T ∈ Z m and a vector of Rademacher variables σ, denote by S T,σ the sample derived from S by replacing its ith element with the ith element of T , for all i ∈ [m] = {1, 2, . . . , m} with σ i = −1. We will use H σ S,T to denote the hypothesis set H S T,σ .
Definition 5 (Rademacher complexity of data-dependent hypothesis sets) Fix m ≥ 1. The empirical Rademacher complexityR ◇ S,T (H) and the Rademacher complexity R ◇ m (H) of a family of data-dependent hypothesis sets H = (H S ) S∈Z m for two samples S = (z S 1 , . . . , z S m ) and T = (z T 1 , . . . , z T m ) in Z m are defined bŷ
R ◇ S,T (H) = 1 m E σ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ sup h∈H σ S,T m i=1 σ i h(z T i ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ R ◇ m (H) = 1 m E S,T ∼D m σ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ sup h∈H σ S,T m i=1 σ i h(z T i ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ .(5)
When the family of data-dependent hypothesis sets H is β-stable with β = O(1 m), the empirical Rademacher complexityR ◇ S,T (G) is sharply concentrated around its expectation R ◇ m (G), as with the standard empirical Rademacher complexity (see Lemma 13). Let H S,T denote the union of all hypothesis sets based on subsamples of S ∪ T of size m:
H S,T = ⋃U⊆(S∪T) U ∈Z m H U .
Since for any σ, we have H σ S,T ⊆ H S,T , the following simpler upper bound in terms of the standard empirical Rademacher complexity of H S,T can be used for our notion of empirical Rademacher complexity:
R ◇ m (H) ≤ 1 m E S,T ∼D m σ sup h∈H S,T m i=1 σ i h(z T i ) = E S,T ∼D m R T (H S,T ) ,
whereR T (H S,T ) is the standard empirical Rademacher complexity of H S,T for the sample T .
The Rademacher complexity of data-dependent hypothesis sets can be bounded by E S,T ∼D m R S (H S,T ) , as indicated previously. It can also be bounded directly, as illustrated by the following example of data-dependent hypothesis sets of linear predictors. For any sample S = (x S 1 , . . . , x S m ) ∈ R N , define the hypothesis set H S as follows:
H S = x ↦ w S ⋅ x∶ w S = m i=1 α i x S i , α 1 ≤ Λ 1 ,
where Λ 1 ≥ 0. Define r T and r S∪T as follows:
r T = ∑ m i=1 x T i 2 2 m
and r S∪T = max x∈S∪T x 2 . Then, it can be shown that the empirical Rademacher complexity of the family of datadependent hypothesis sets H = (H S ) S∈X m can be upper-bounded as follows (Lemma 11):
R ◇ S,T (H) ≤ r T r S∪T Λ 1 2 log(4m) m ≤ r 2 S∪T Λ 1 2 log(4m) m .
Notice that the bound on the Rademacher complexity is non-trivial since it depends on the samples S and T , while a standard Rademacher complexity for non-data-dependent hypothesis set containing H S would require taking a maximum over all samples S of size m. Other upper bounds are given in Appendix B.
Let G S denote the family of loss functions associated to H S :
G S = {z ↦ L(h, z)∶ h ∈ H S },(6)
and let G = (G S ) S∈Z m denote the family of hypothesis sets G s . Our main results will be expressed in terms of R ◇ m (G). When the loss function is µ-Lipschitz, by Talagrand's contraction lemma (Ledoux and Talagrand, 1991), in all our results, R ◇ m (G) can be replaced by µ E S,T ∼D m [R T (H S,T )].
General learning bound for data-dependent hypothesis sets
In this section, we present general learning bounds for data-dependent hypothesis sets that do not make use of the notion of hypothesis set stability.
One straightforward idea to derive such guarantees for data-dependent hypothesis sets is to replace the hypothesis set H S depending on the observed sample S by the union of all such hypothesis sets over all samples of size m, H m = ⋃ S∈Z m H S . However, in general, H m can be very rich, which can lead to uninformative learning bounds. A somewhat better alternative consists of considering the union of all such hypothesis sets for samples of size m included in some supersample U of size m + n, with n ≥ 1, H U,m = ⋃S∈Z m S⊆U H S . We will derive learning guarantees based on the maximum transductive Rademacher complexity of H U,m . There is a trade-off in the choice of n: smaller values lead to less complex sets H U,m , but they also lead to weaker dependencies on sample sizes. Our bounds are more refined guarantees than the shattering-coefficient bounds originally given for this problem by Gat (2001) in the case n = m, and later by Cannon et al. (2002) for any n ≥ 1. They also apply to arbitrary bounded loss functions and not just the binary loss. They are expressed in terms of the following notion of transductive Rademacher complexity for data-dependent hypothesis sets:
R ◇ U,m (G) = E σ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ sup h∈H U,m 1 m + n m+n i=1 σ i L(h, z U i ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ,
where U = (z U 1 , . . . , z U m+n ) ∈ Z m+n and where σ is a vector of (m + n) independent random variables taking value m+n n with probability n m+n , and − m+n m with probability m m+n . Our notion of transductive Rademacher complexity is simpler than that of El-Yaniv and Pechyony (2007) (in the data-independent case) and leads to simpler proofs and guarantees. A by-product of our analysis is learning guarantees for standard transductive learning in terms of this notion of transductive Rademacher complexity, which can be of independent interest.
Theorem 6 Let H = (H S ) S∈Z m be a family of data-dependent hypothesis sets. Then, for any > 0 with n 2 ≥ 2 and any n ≥ 1, the following inequality holds:
P sup h∈H S R(h)−R S (h) > ≤ exp ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ − 2 η mn m + n ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 2 − max U ∈Z m+nR ◇ U,m (G) − log(2e)(m + n) 3 2(mn) 2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ 2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ,
where η = m+n m+n− 1 2 1 1− 1 2 max{m,n} ≈ 1. For m = n, the inequality becomes:
P sup h∈H S R(h) −R S (h) > ≤ exp ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ − m η ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 2 − max U ∈Z m+nR ◇ U,m (G) − 2 log(2e) m ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ 2⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .
Proof We use the following symmetrization result, which holds for any > 0 with n 2 ≥ 2 for data-dependent hypothesis sets (Lemma 14, Appendix 14):
P S∼D m sup h∈H S R(h) −R S (h) > ≤ 2 P S∼D m T ∼D n sup h∈H SR T (h) −R S (h) > 2 .
To bound the right-hand side, we use an extension of McDiarmid's inequality to sampling without replacement (Cortes et al., 2008)
applied to Φ(S) = sup h∈H U,mR T (h) −R S (h).
Lemma 15 (Appendix E) is then used to bound E[Φ(S)] in terms of our notion of transductive Rademacher complexity. The full proof is given in Appendix C.
Learning bound for stable data-dependent hypothesis sets
In this section, we present generalization bounds for data-dependent hypothesis sets using the notion of Rademacher complexity defined in the previous section, as well as that of hypothesis set stability.
Theorem 7 Let H = (H S ) S∈Z m be a β-stable family of data-dependent hypothesis sets with χ average CV-stability. Let G be defined as in (6). Then, for any δ > 0, with probability at least 1 − δ over the draw of a sample S ∼ Z m , the following inequality holds for all h ∈ H S :
∀h ∈ H S , R(h) ≤R S (h) + min{2R ◇ m (G),χ} + [1 + 2βm] log 1 δ 2m .(7)
Proof For any two samples S, S ′ , define Ψ(S, S ′ ) as follows:
Ψ(S, S ′ ) = sup h∈H S R(h) −R S ′ (h).
The proof consists of applying McDiarmid's inequality to Ψ(S, S). The first stage consists of proving the ∆-sensitivity of Ψ(S, S), with ∆ = 1 m + 2β. The main part of the proof then consists of upper bounding the expectation E S∼D m [Ψ(S, S)] in terms of both our notion of Rademacher complexity, and in terms of our notion of cross-validation stability. The full proof is given in Appendix F.
The generalization bound of the theorem admits as a special case the standard Rademacher complexity bound for fixed hypothesis sets (Koltchinskii and Panchenko, 2002;Bartlett and Mendelson, 2002): in that case, we have H S = H for some H, thus R ◇ m (G) coincides with the standard Rademacher complexity R m (G); furthermore, the family of hypothesis sets is 0-stable, thus the bound holds with β = 0. It also admits as a special case the standard uniform stability bound (Bousquet and Elisseeff, 2002): in that case, H S is reduced to a singleton, H S = {h S }, and our notion of hypothesis set stability coincides with that of uniform stability of single hypotheses; furthermore, we haveχ ≤∆ + β = β, since∆ = 0. Thus, using min{2R ◇ m (G),χ} ≤ β in the right-hand side inequality, the expression of the learning bound matches that of a uniform stability bound for single hypotheses.
Differential privacy-based bound for stable data-dependent hypothesis sets
In this section, we use recent techniques introduced in the differential privacy literature to derive improved generalization guarantees for stable data-dependent hypothesis sets (Steinke and Ullman, 2017; Bassily et al., 2016) (see also (McSherry and Talwar, 2007)).
Our proofs also benefit from the recent improved stability results of Feldman and Vondrak (2018). We will make use of the following lemma due to Steinke and Ullman (2017, Lemma 1.2), which reduces the task of deriving a concentration inequality to that of upper bounding an expectation of a maximum.
Lemma 8 Fix p ≥ 1. Let X be a random variable with probability distribution D and X 1 , . . . , X p independent copies of X. Then, the following inequality holds:
P X∼D X ≥ 2 E X k ∼D max 0, X 1 , . . . , X p ≤ log 2 p .
We will also use the following result which, under a sensitivity assumption, further reduces the task of upper bounding the expectation of the maximum to that of bounding a more favorable expression. The sensitivity of a function 2∆ . Then, A is -differentially private and, for any S ∈ Z m , the following inequality holds:
f ∶ Z m → R is sup S,S ′ ∈Z m , S∩S ′ =m−1 f (S) − f (S ′ ) .max k∈[p] f k (S) ≤ E k=A(S) f k (S) + 2∆ log p.
Notice that, if we define f p+1 = 0, then, by the same result, the algorithm A returning the index k ∈ [p + 1] with probability proportional to e f k (S)1 k≠(p+1)
2∆
is -differentially private and the following inequality holds for any S ∈ Z m :
max 0, max k∈[p] f k (S) = max k∈[p+1] f k (S) ≤ E k=A(S) f k (S) + 2∆ log(p + 1).(8)
Theorem 10 Let H = (H S ) S∈Z m be a β-stable family of data-dependent hypothesis sets with χ CV-stability. Let G be defined as in (6). Then, for any δ ∈ (0, 1), with probability at least 1 − δ over the draw of a sample S ∼ Z m , the following inequality holds for all h ∈ H S :
R(h) ≤R S (h) + min ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 2R ◇ m (G) + [1 + 2βm] log 2 δ 2m , √ eχ + 4 1 m + 2β log 6 δ ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ .
Proof For any two samples S, S ′ of size m, define Ψ(S, S ′ ) as follows:
Ψ(S, S ′ ) = sup h∈H S R(h) −R S ′ (h).
The proof consists of deriving a high-probability bound for Ψ(S, S). To do so, by Lemma 8 applied to the random variable
X = Ψ(S, S), it suffices to bound E S∼D pm max 0, max k∈[p] Ψ(S k , S k ) , where S = (S 1 , . . . , S p ) with S k , k ∈ [p], independent samples of size m drawn from D m .
To bound that expectation, we use Lemma 9 and instead bound E S∼D pm
k=A(S) [Ψ(S k , S k )], where
A is an -differentially private algorithm. To apply Lemma 9, we first show that, for any
k ∈ [p], the function f k ∶ S → Ψ(S k , S k ) is ∆-sensitive with ∆ = 1 m + 2β.
Lemma 16 helps us express our upper bound in terms of the CV stability coefficient χ. The full proof is given in Appendix G.
The hypothesis set-stability bound of this theorem admits the same favorable dependency on the stability parameter β as the best existing bounds for uniform-stability recently presented by Feldman and Vondrak (2018). As with Theorem 7, the bound of Theorem 10 admits as special cases both standard Rademacher complexity bounds (H = H for some fixed H and β = 0) and uniform-stability bounds (H S = {h S }). In the latter case, our bound coincides with that of Feldman and Vondrak (2018) modulo constants that could be chosen to be the same for both results. 1 Notice that the current bounds for standard uniform stability may not be optimal since no matching lower bound is known yet (Feldman and Vondrak, 2018). It is very likely, however, that improved techniques used for deriving more refined algorithmic stability bounds could also be used to improve our hypothesis set stability guarantees. In Appendix H, we give an alternative version of Theorem 10 with a proof technique only making use of recent methods from the differential privacy literature, including to derive a Rademacher complexity bound. It might be possible to achieve a better dependency on β for the term in the bound containing the Rademacher complexity. In Appendix I, we initiate such an analysis by deriving a finer analysis on the expectation
E S∼D pm max 0, max k∈[p] Ψ(S k , S k ) .
1. The differences in constant terms are due to slightly difference choices of the parameters and a slightly different upper bound in our case where e multiplies the stability and the diameter, while the paper of Feldman and Vondrak (2018) does not seem to have that factor.
Applications
In this section, we discuss several applications of the learning guarantees presented in the previous sections. We discuss other applications in Appendix K. As already mentioned, both the standard setting of a fixed hypothesis set H S not varying with S, that is that of standard generalization bounds, and the uniform stability setting where H S = {h S }, are special cases benefitting from our learning guarantees.
Stochastic convex optimization
Here, we consider data-dependent hypothesis sets based on stochastic convex optimization algorithms. As shown by Shalev-Shwartz et al. (2010), uniform convergence bounds do not hold for the stochastic convex optimization problem in general. As a result, the datadependent hypothesis sets we will define cannot be analyzed using standard tools for deriving generalization bounds. However, using arguments based on our notion of hypothesis set stability, we can provide learning guarantees here.
Consider K stochastic optimization algorithms A j , each returning vectorŵ S j , after receiving sample S ∈ Z m , j ∈ [K]. We assume that the algorithms are all β-sensitive in norm, that is,
for all j ∈ [K], we have ŵ S j −ŵ S ′ j
≤ β if S and S ′ differ by one point. We will also assume that these vectors are bounded by some
D > 0 that is ŵ S j 2 ≤ D, for all j ∈ [K]
. This can be shown to be the case, for example, for algorithms based on empirical risk minimization with a strongly convex regularization term with β = O( 1 m ) (Shalev-Shwartz et al., 2010). Assume that the loss L(w, z) is µ-Lipschitz with respect to its first argument w. Let the data-dependent hypothesis set be defined as follows:
H S = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ K j=1 α jŵ S j ∶ α ∈ ∆ K ∩ B 1 (α 0 , r) ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ ,
where α 0 is in the simplex of distributions ∆ K and B 1 (α 0 , r) is the L 1 ball of radius r > 0 around α 0 . We choose r = 1 2µD √ m . A natural choice for α 0 would be the uniform mixture. Since the loss function is µ-Lipschitz, the family of hypotheses H S is µβ-stable. Additionally, for any α, α ′ ∈ ∆ K ∩ B 1 (α 0 , r) and any z ∈ Z, we have
L K j=1 α jŵ S j , z − L K j=1 α ′ jŵ S j , z ≤ µ K j=1 (α i − α ′ j )ŵ S j 2 ≤ µ [w S 1 ⋯ w S K ] 1,2 α − α ′ 1 ≤ 2µrD, where [w S 1 ⋯ w S K ] 1,2 is the subordinate norm of matrix [w S 1 ⋯ w S K ] defined by [w S 1 ⋯ w S K ] 1,2 = max x≠0 ∑ k j=1 x j w S j 2 x 1 = max i∈[K] w S i 2 ≤ D.
Thus, the average diameter admits the following upper bound:∆ ≤ 2µrD = 1 √ m . In view of that, by Theorem 10, for any δ > 0, with probability at least 1 − δ, the following holds for all α ∈ ∆ K ∩ B 1 (α 0 , r):
E z∼D L K j=1 α jŵ S j , z ≤ 1 m m i=1 L K j=1 α iŵ S j , z S i + e m + √ eµβ + 4 1 m + 2µβ log 6 δ .
The second stage of an algorithm in this context consists of choosing α, potentially using a non-stable algorithm. This application both illustrates the use of our learning bounds using the diameter and its application even in the absence of uniform convergence bounds.
∆-sensitive feature mappings
Consider the scenario where the training sample S ∈ Z m is used to learn a non-linear feature
mapping Φ S ∶ X → R N that is ∆-sensitive for some ∆ = O( 1 m )
. Φ S may be the feature mapping corresponding to some positive definite symmetric kernel or a mapping defined by the top layer of an artificial neural network trained on S, with a stability property.
The second stage may consist of selecting a hypothesis out of the family H S of linear hypotheses based on Φ S :
H S = x ↦ w ⋅ Φ S (x)∶ w ≤ γ .
Assume that the loss function is µ-Lipschitz with respect to its first argument. Then, for any hypothesis h∶ x ↦ w ⋅ Φ S (x) ∈ H S and any sample S ′ differing from S by one element, the hypothesis h ′ ∶ x ↦ w ⋅ Φ S ′ (x) ∈ H S ′ admits losses that are β-close to those of h, with β = µγ∆, since, for all (x, y) ∈ X × Y, by the Cauchy-Schwarz inequality, the following inequality holds:
(w ⋅ Φ S (x), y) − (w ⋅ Φ S ′ (x), y) ≤ µw ⋅ (Φ S (x) − Φ S ′ (x)) ≤ µ w Φ S (x) − Φ S ′ (x) ≤ µγ∆.
Thus, the family of hypothesis set H = (H S ) S∈Z m is uniformly β-stable with β = µγ∆ = O( 1 m ). In view that, by Theorem 7, for any δ > 0, with probability at least 1 − δ over the draw of a sample S ∼ D m , the following inequality holds for any h ∈ H S :
R(h) ≤R S (h) + 2R ◇ m (G) + [1 + 2µγ∆m] log 2 δ 2m .(9)
Notice that this bound applies even when the second stage of an algorithm, which consists of selecting a hypothesis h S in H S , is not stable. A standard uniform stability guarantee cannot be used in that case. The setting described here can be straightforwardly extended to the case of other norms for the definition of sensitivity and that of the norm used in the definition of H S . Figure 2: Illustration of the distillation hypothesis sets. Notice that the diameter of a hypothesis set H S may be large here.
f ⇤ S f ⇤ S 0 h 0 h H S H S 0
Distillation
Here, we consider distillation algorithms which, in the first stage, train a very complex model on the labeled sample. Let f * S ∶ X → R denote the resulting predictor for a training sample S of size m. We will assume that the training algorithm is β-sensitive, that is
f * S − f * S ′ ≤ β m = O
(1 m) for S and S ′ differing by one point. In the second stage, a distillation algorithm selects a hypothesis that is γ-close to f * S from a less complex family of predictors H. This defines the following sample-dependent hypothesis set:
H S = h ∈ H∶ (h − f * S ) ∞ ≤ γ .
Assume that the loss is µ-Lipschitz with respect to its first argument and that H is a subset of a vector space. Let S and S ′ be two samples differing by one point. Note, f * S may not be in H, but we will assume that f * Figure 2 illustrates the hypothesis sets. By the µ-Lipschitzness of the loss, for any z = (x, y) ∈ Z,
S ′ − f * S is in H. Let h be in H S , then the hypothesis h ′ = h + f * S ′ − f * S is in H S ′ since h ′ − f * S ′ ∞ = h − f * S ∞ ≤ γ.(h ′ (x), y) − (h(x), y) ≤ µ h ′ (x) − h(x) ∞ = µ f * S ′ − f * S ≤ µβ m .
Thus, the family of hypothesis sets H S is µβ m -stable.
In view that, by Theorem 7, for any δ > 0, with probability at least 1 − δ over the draw of a sample S ∼ D m , the following inequality holds for any h ∈ H S :
R(h) ≤R S (h) + 2R ◇ m (G) + [1 + 2µβ m m] log 2 δ 2m .
Notice that a standard uniform-stability argument would not necessarily apply here since H S could be relatively complex and the second stage not necessarily stable.
Bagging
Bagging (Breiman, 1996) is a prominent ensemble method used to improve the stability of learning algorithms. It consists of generating k new samples B 1 , B 2 , . . . , B k , each of size p, by sampling uniformly with replacement from the original sample S of size m. An algorithm A is then trained on each of these samples to generate k predictors A(B i ), i ∈ [k].
In regression, the predictors are combined by taking a convex combination ∑ k i=1 w i A(B i ). Here, we analyze a common instance of bagging to illustrate the application of our learning guarantees: we will assume a regression setting and a uniform sampling from S without replacement. 2 We will also assume that the loss function is µ-Lipschitz in the predictions and that the predictions are in the range [0, 1], and all the mixing weights w i are bounded by C k for some constant C ≥ 1, in order to ensure that no subsample B i is overly influential in the final regressor (in practice, a uniform mixture is typically used in bagging).
To analyze bagging in this setup, we cast it in our framework. First, to deal with the randomness in choosing the subsamples, we can equivalently imagine the process as choosing indices in [m] to form the subsamples rather than samples in S, and then once S is drawn, the subsamples are generated by filling in the samples at the corresponding indexes. Thus, for any index i ∈ [m], the chance that it is picked in any subsample is p m . Thus, by Chernoff's bound, with probability at least 1 − δ, no index in [m] appears in more than
t ∶= kp m + 2kp log( m δ ) m
subsamples. In the following, we condition on the random seed of the bagging algorithm so that this is indeed the case, and later use a union bound to control the chance that the chosen random seed does not satisfy this property, as elucidated in section J.2.
Define the data-dependent family of hypothesis sets H as
H S ∶= ∑ k i=1 w i A(B i )∶ w ∈ ∆ C k k , where ∆ C k k
denotes the simplex of distributions over k items with all weights w i ≤ C k . Next, we give upper bounds on the hypothesis set stability and the Rademacher complexity of H. Assume that algorithm A admits uniform stability β A (Bousquet and Elisseeff, 2002), i.e. for any two samples B and B ′ of size p that differ in exactly one data point and for all x ∈ X , we have A(B)(x) − A(B ′ )(x) ≤ β A . Now, let S and S ′ be two samples of size m differing by one point at the same index, z ∈ S and z ′ ∈ S ′ . Then, consider the subsets B ′ i of S ′ which are obtained from the B i 's by copying over all the elements except z, and replacing all instances of z by z ′ . For any B
i , if z ∉ B i , then A(B i ) = A(B ′ i ) and, if z ∈ B i , then A(B i )(x) − A(B ′ i )(x) ≤ β A for any x ∈ X .
We can bound now the hypothesis set uniform stability as follows: since L is µ-Lipschitz in the prediction, for any z ′′ ∈ Z, and any w ∈ ∆ C k k we have
L(∑ k i=1 w i A(B i ), z ′′ ) − L(∑ k i=1 w i A(B ′ i ), z ′′ ) ≤ p m + 2p log( 1 δ ) km ⋅ Cµβ A .
Bounding the Rademacher complexityR S (H S,T ) for S, T ∈ Z m is non-trivial. Instead, we can derive a reasonable upper bound by analyzing the Rademacher complexity of a larger function class. Specifically, for any z ∈ Z, define the d ∶= 2m p dimensional vector u z = 2. Sampling without replacement is only adopted to make the analysis more concise; its extension to sampling with replacement is straightforward. Thus, by Talagrand's inequality, we conclude thatR S (G S,T ) ≤ µ 2p log(4m) m . In view of that, by Theorem 10, for any δ > 0, with probability at least 1 − 2δ over the draws of a sample S ∼ D m and the randomness in the bagging algorithm, the following inequality holds for any h ∈ H S :
⟨A(B)(z)⟩ B⊆S∪T, B =p . Then the class of functions is
F S,T ∶= {z ↦ w ⊺ u z ∶ w ∈ R d , w 1 = 1}. Clearly H S,T ⊆ F S,T . Since u z ∞ ≤ 1,R(h) ≤R S (h) + 2µ 2p log(4m) m + ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 1 + 2 ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ p + 2pm log( 1 δ ) k ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⋅ Cµβ A ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ log 2 δ 2m
.
For p = o( √ m) and k = ω(p), the generalization gap goes to 0 as m → ∞, regardless of the stability of A. This gives a new generalization guarantee for bagging, similar (but incomparable) to the one derived by Elisseeff et al. (2005). Note however that unlike their bound, our bound allows for non-uniform averaging schemes.
As an aside, we note that the same analysis can be carried over to the stochastic convex optimization setting of section 6.1, by setting A to be a stochastic convex optimization algorithm which outputs a weight vectorŵ. This yields generalization bounds for aggregating over a larger set of mixing weights, albeit with the restriction that each algorithm uses only a small part of S.
Conclusion
We presented a broad study of generalization with data-dependent hypothesis sets, including general learning bounds using a notion of transductive Rademacher complexity and, more importantly, learning bounds for stable data-dependent hypothesis sets. We illustrated the applications of these guarantees to the analysis of several problems. Our framework is general and covers learning scenarios commonly arising in applications for which standard generalization bounds are not applicable. Our results can be further augmented and refined to include model selection bounds and local Rademacher complexity bounds for stable datadependent hypothesis sets (to be presented in a more extended version of this manuscript), and further extensions described in Appendix J. Our analysis can also be extended to the non-i.i.d. setting and other learning scenarios such as that of transduction.
Appendix A. Further background on stability
The study of stability dates back to early work on the analysis of k-neareast neighbor and other local discrimination rules (Rogers and Wagner, 1978;Devroye and Wagner, 1979). Stability has been critically used in the analysis of stochastic optimization (Shalev-Shwartz et al., 2010) and online-to-batch conversion (Cesa-Bianchi et al., 2001). Stability bounds have been generalized to the non-i.i.d. settings, including stationary (Mohri and Rostamizadeh, 2010) and non-stationary (Kuznetsov and Mohri, 2017) φ-mixing and βmixing processes. They have also been used to derive learning bounds for transductive inference (Cortes et al., 2008). Stability bounds were further extended to cover almost stable algorithms by Kutin and Niyogi (2002). These authors also discussed a number of alternative definitions of stability, see also (Kearns and Ron, 1997). An alternative notion of stability was also used by Kale et al. (2011) to analyze k-fold cross-validation for a number of stable algorithms.
Appendix B. Properties of data-dependent Rademacher complexity
In this section, we highlight several key properties of our notion of data-dependent Rademacher complexity.
B.1. Upper-bound on Rademacher complexity of data-dependent hypothesis sets
Lemma 11 For any sample S = (x S 1 , . . . , x S m ) ∈ R N , define the hypothesis set H S as follows:
H S = x ↦ w S ⋅ x∶ w S = m i=1 α i x S i , α 1 ≤ Λ 1 ,
where Λ 1 ≥ 0. Define r T and r S∪T as follows:
r T = ∑ m i=1 x T i 2 2 m
and r S∪T = max x∈S∪T x 2 . Then, the empirical Rademacher complexity of the family of data-dependent hypothesis sets H = (H S ) S∈X m can be upper-bounded as follows:
R ◇ S,T (H) ≤ r T r S∪T Λ 1 2 log(4m) m ≤ r 2 S∪T Λ 1 2 log(4m) m .
Proof The following inequalities hold:
R ◇ S,T (H) = 1 m E σ sup h∈H σ S,T m i=1 σ i h(x T i ) = 1 m E σ sup α 1 ≤Λ 1 m i=1 σ i m j=1 α j x S T,σ j ⋅ x T i = 1 m E σ sup α 1 ≤Λ 1 m j=1 α j x S T,σ j m i=1 σ i ⋅ x T i = Λ 1 m E σ max j∈[m] x S T,σ j ⋅ m i=1 σ i x T i ≤ Λ 1 m E σ max x ′ ∈S∪T σ ′ ∈{−1,+1} m i=1 σ i (σ ′ x ′ ⋅ x T i ) .
The norm of the vector z ′ ∈ R m with coordinates (σ ′ x ′ ⋅ x T i ) can be bounded as follows:
m i=1 (σ ′ x ′ ⋅ x T i ) 2 ≤ x ′ m i=1 x T i 2 ≤ r S∪T √ m r T .
Thus, by Massart's lemma, since S ∪ T ≤ 2m, the following inequality holds:
R ◇ S,T (H) ≤ r T r S∪T Λ 1 2 log(4m) m ≤ r 2 S∪T Λ 1 2 log(4m) m ,
which completes the proof.
Lemma 12 Suppose X = R N , and for every sample S ∈ Z m we associate a matrix A S ∈ R d×N for some d > 0, and let W S,Λ = {w ∈ R d ∶ A ⊺ S w 2 ≤ Λ} for some Λ > 0. Consider the hypothesis set H S ∶= x ↦ w ⊺ A S x∶ w ∈ W S,Λ . Then, the empirical Rademacher complexity of the family of data-dependent hypothesis sets H = (H S ) S∈Z m can be upperbounded as follows:
R ◇ S,T (H) ≤ Λ ∑ m i=1 x T i 2 2 m ≤ Λr √ m , where r = sup i∈[m] x T i 2 . Proof Let X T = [x T 1 ⋯ x T m ].
The following inequalities hold:
R ◇ S,T (H) = 1 m E σ sup h∈H σ S,T m i=1 σ i h(x T i ) = 1 m E σ sup w∶ A ⊺ S w 2 ≤Λ w ⊺ A S X T σ ≤ Λ m E σ X T σ 2 (Cauchy-Schwarz) ≤ Λ m E σ X T σ 2 2 (Jensen's ineq.) ≤ Λ m E σ m i,j=1 σ i σ j (x T i ⋅ x T j ) = Λ ∑ m i=1 x T i 2 2 m ,
which completes the proof.
B.2. Concentration
Lemma 13 Let H a family of β-stable data-dependent hypothesis sets. Then, for any δ > 0, with probability at least 1 − δ (over the draw of two samples S and T with size m), the following inequality holds:
R ◇ S,T (G) − R ◇ m (G) ≤ [(mβ + 1) 2 + m 2 β 2 ] log 2 δ 2m .
Proof Let T ′ be a sample differing from T only by point. Fix η > 0. For any σ, by definition of the supremum, there exists h ′ ∈ H σ S,T ′ such that:
m i=1 σ i L(h ′ , z T i ) ≥ sup h∈H σ S,T ′ m i=1 σ i L(h, z T ′ i ) − η.
By the β-stability of H, there exists h ∈ H σ S,T such that for any z ∈ Z, L(h ′ , z)−L(h, z) ≤ β. Thus, we have
sup h∈H σ S,T ′ m i=1 σ i L(h, z T ′ i ) ≤ m i=1 σ i L(h ′ , z T ′ i ) + η ≤ m i=1 [σ i (L(h, z T ′ i ) + β)] + η.
Since the inequality holds for all η > 0, we have
1 m sup h∈H σ S,T ′ m i=1 σ i L(h, z T ′ i ) ≤ 1 m m i=1 σ i (L(h, z T ′ i ) + β) ≤ 1 m sup h∈H σ S,T m i=1 σ i L(h, z T i ) + β + 1 m .
Thus, replacing T by T ′ affectsR ◇ S,T (G) by at most β + 1 m . By the same argument, changing sample S by one point modifiesR ◇ S,T (G) at most by β. Thus, by McDiarmid's inequality, for any δ > 0, with probability at least 1 − δ, the following inequality holds:
R ◇ S,T (G) − R ◇ m (G) ≤ [(mβ + 1) 2 + m 2 β 2 ] log 2 δ 2m .
This completes the proof.
Appendix C. Proof of Theorem 6
In this section, we present the proof of Theorem 6.
Theorem 6 Let H = (H S ) S∈Z m be a family of data-dependent hypothesis sets. Then, for any > 0 with n 2 ≥ 2 and any n ≥ 1, the following inequality holds:
P sup h∈H S R(h)−R S (h) > ≤ exp ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ − 2 η mn m + n ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 2 − max U ∈Z m+nR ◇ U,m (G) − log(2e)(m + n) 3 2(mn) 2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ 2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ,
where η = m+n m+n− 1 2 1 1− 1 2 max{m,n} ≈ 1. For m = n, the inequality becomes:
P sup h∈H S R(h) −R S (h) > ≤ exp ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ − m η ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 2 − max U ∈Z m+nR ◇ U,m (G) − 2 log(2e) m ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ 2⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .
Proof We will use the following symmetrization result, which holds for any > 0 with n 2 ≥ 2 for data-dependent hypothesis sets (Lemma 14, Appendix 14):
P S∼D m sup h∈H S R(h) −R S (h) > ≤ 2 P S∼D m T ∼D n sup h∈H SR T (h) −R S (h) > 2 .
Thus, we will seek to bound the right-hand side as follows, where we write (S, T ) ∼ U to indicate that the sample S of size m is drawn uniformly without replacement from U and that T is the remaining part of U , that is (S, T ) = U :
P S∼D m T ∼D n sup h∈H SR T (h) −R S (h) > 2 = E U ∼D m+n ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ P (S,T )∼U S =m, T =n sup h∈H SR T (h) −R S (h) > 2 U ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ≤ E U ∼D m+n ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ P (S,T )∼U S =m, T =n sup h∈H U,mR T (h) −R S (h) > 2 U ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ .
To upper bound the probability inside the expectation, we use an extension of McDiarmid's inequality to sampling without replacement (Cortes et al., 2008), which applies to symmetric functions. We can apply that extension to Φ(S) = sup h∈H U,mR T (h) −R S (h), for a fixed U , since Φ(S) is a symmetric function of the sample points z 1 , . . . , z m ) in S. Changing one point in S affects Φ(S) at most by 1 m + 1 m = m+u mu , thus, by the extension of McDiarmid's inequality to sampling without replacement, for a fixed U ∈ Z m+n , the following inequality holds:
P (S,T )∼U S =m, T =n sup h∈H U,mR T (h) −R S (h) > 2 ≤ exp − 2 η mn m + n 2 − E[Φ(S)]
Appendix D. Symmetrization lemma
In this section, we show that the standard symmetrization lemma holds for data-dependent hypothesis sets. This observation was already made by Gat (2001) (see also Lemma 2 in (Cannon et al., 2002)) for the symmetrization lemma of Vapnik (1998)[p. 139], used by the author in the case n = m. However, that symmetrization lemma of Vapnik (1998) holds only for random variables taking values in {0, 1} and its proof is not complete since the hypergeometric inequality is not proven.
Lemma 14 Let n ≥ 1 and fix > 0 such that n 2 ≥ 1. Then, the following inequality holds:
P S∼D m sup h∈H S R(h) −R S (h) > ≤ 2 P S∼D m T ∼D n sup h∈H SR T (h S ) −R S (h S ) > 2 .
Proof The proof is standard. Below, we are giving a concise version mainly for the purpose of verifying that the data-dependency of the hypothesis set does not affect its correctness.
Fix η > 0. By definition of the supremum, there exists h S ∈ H S such that
sup h∈H S R(h) −R S (h) − η ≤ R(h S ) −R S (h S ). SinceR T (h S ) −R S (h S ) =R T (h S ) − R(h S ) + R(h S ) −R S (h S ), we can write 1R T (h S )−R S (h S )> 2 ≥ 1R T (h S )−R(h S )>− 2 1 R(h S )−R S (h S )> = 1 R(h S )−R T (h S )< 2 1 R(h S )−R S (h S )> .
Thus, for any S ∈ Z m , taking the expectation of both sides with respect to T yields
P T ∼D n R T (h S ) −R S (h S ) > 2 ≥ P T ∼D n R(h S ) −R T (h S ) < 2 1 R(h S )−R S (h S )> = 1 − P T ∼D n R(h S ) −R T (h S ) ≥ 2 1 R(h S )−R S (h S )> ≤ 1 − 4 Var[L(h S , z)] n 2 1 R(h S )−R S (h S )> (Chebyshev's ineq.) ≥ 1 − 1 n 2 1 R(h S )−R S (h S )> ,
where the last inequality holds since L(h S , z) takes values in [0, 1]:
Var[L(h S , z)] = E z∼D [L 2 (h S , z)] − E z∼D [L(h S , z)] 2 ≤ E z∼D [L(h S , z)] − E z∼D [L(h S , z)] 2 = E z∼D [L(h S , z)](1 − E z∼D [L(h S , z)]) ≤ 1 4 .
Taking expectation with respect to S gives
P S∼D m T ∼D n R T (h S ) −R S (h S ) > 2 ≥ 1 − 1 n 2 P S∼D m R(h S ) −R S (h S ) > ≥ 1 2 P S∼D m R(h S ) −R S (h S ) > (n 2 ≥ 2) ≥ 1 2 P S∼D m sup h∈H S R(h) −R S (h) > + η .
Since the inequality holds for all η > 0, by the right-continuity of the cumulative distribution function, it implies
P S∼D m T ∼D n R T (h S ) −R S (h S ) > 2 ≥ 1 2 P S∼D m sup h∈H S R(h) −R S (h) > .
Since h S is in H S , by definition of the supremum, we have
P S∼D m T ∼D n sup h∈H SR T (h) −R S (h) > 2 ≥ P S∼D m T ∼D n R T (h S ) −R S (h S ) > 2 ,
which completes the proof.
Appendix E. Transductive Rademacher complexity bound
Lemma 15 Fix U ∈ Z m+n . Then, the following upper bound holds:
E (S,T )∼U S =m, T =n sup h∈H U,mR T (h) −R S (h) ≤R ◇ U,m (G) + log(2e)(m + n) 3 2(mn) 2 .
For m = n, the inequality becomes:
E (S,T )∼U S =m, T =n sup h∈H U,mR T (h) −R S (h) ≤R ◇ U,m (G) + 2 log(2e) m .
Proof The proof is an extension of the analysis of maximum discrepancy in ( Bartlett and Mendelson, 2002). Let σ denote ∑ m+n i=1 σ i and let I ⊆ − (m+n) 2 m , (m+n) 2 n denote the set of values σ can take. For any q ∈ I, define s(q) as follows:
s(q) = E σ sup h∈H U,m 1 m + n m+n i=1 σ i L(h, z U i ) σ = q .
Let σ + denote the number of positive σ i s, taking value m+n n , then σ can be expressed as follows:
σ = m+n i=1 σ i = σ + m + n n − (m + n − σ + ) m + n m = (m + n) 2 mn ( σ + − n).(10)
Thus, we have σ = 0 iff σ + = m, and the condition ( σ = 0) precisely corresponds to having the equality
1 m + n m+n i=1 σ i L(h, z U i ) =R T (h) −R S (h),
where S is the sample of size m defined by those z i s for which σ i takes value m+n n . In view of that, we have
E (S,T )∼U S =m, T =n sup h∈H U,mR T (h) −R S (h) = s(0).
Let q 1 , q 2 ∈ I, with q 1 = p 1 m+n n − (m + n − p 1 ) m+n m , q 2 = p 2 m+n n − (m + n − p 2 ) m+n m and q 1 ≤ q 2 . Then, we can write
s(q 1 ) = E sup g∈G p 1 i=1 1 n L(h, z i ) − m+n i=p 1 +1 1 m L(h, z i ) s(q 2 ) = E ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ sup g∈G p 1 i=1 1 n L(h, z i ) − m+n i=p 1 +1 1 m L(h, z i ) + p 2 i=p 1 +1 1 n + 1 m L(h, z i ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ .
Thus, we have the following Lipschitz property:
s(q 2 ) − s(q 1 ) ≤ p 2 − p 1 1 m + 1 n = (p 2 − n) − (p 1 − n) 1 m + 1 n (using (10)) = q 2 − q 1 mn (m + n) 2 1 m + 1 n = q 2 − q 1 m + n .
By this Lipschitz property, we can write
P s( σ ) − s(E[ σ ]) > ≤ P σ − E[ σ ] > (m + n) ≤ 2 exp − 2 (mn) 2 2 (m + n) 3 ,
since the range of each σ i is m+n n + m+n m = (m+n) 2 mn . We now use this inequality to bound the second moment of Z = s( σ ) − s(E[ σ ]) = s( σ ) − s(0), as follows, for any u ≥ 0:
E[Z 2 ] = +∞ 0 P[Z 2 > t] dt = u 0 P[Z 2 > t] dt + +∞ u P[Z 2 > t] dt ≤ u + 2 +∞ u exp − 2 (mn) 2 t (m + n) 3 dt ≤ u + (m + n) 3 (mn) 2 exp − 2 (mn) 2 t (m + n) 3 +∞ u = u + (m + n) 3 (mn) 2 exp − 2 (mn) 2 u (m + n) 3 .
Choosing u = 1 2 log(2)(m+n) 3 (mn) 2 to minimize the right-hand side gives E[Z 2 ] ≤ log(2e)(m+n) 3 2(mn) 2
. By
Jensen's inequality, this implies E[ Z ] ≤ log(2e)(m+n) 3
2(mn) 2
and therefore
E (S,T )∼U S =m, T =n sup h∈H U,mR T (h) −R S (h) = s(0) ≤ E[s( σ )] + log(2e)(m + n) 3 2(mn) 2 .
Since we have E[s( σ )] =R ◇ U,m (G), this completes the proof.
Appendix F. Proof of Theorem 7
In this section, we present the full proof of Theorem 7.
Theorem 7 Let H = (H S ) S∈Z m be a β-stable family of data-dependent hypothesis sets. Then, for any δ > 0, with probability at least 1 − δ over the draw of a sample S ∼ Z m , the following inequality holds for all h ∈ H S :
∀h ∈ H S , R(h) ≤R S (h) + min{2R ◇ m (G),χ} + [1 + 2βm] log 1 δ 2m .(11)
Proof For any two samples S, S ′ , define the Ψ(S, S ′ ) as follows:
Ψ(S, S ′ ) = sup h∈H S R(h) −R S ′ (h).
The proof consists of applying McDiarmid's inequality to Ψ(S, S). For any sample S ′ differing from S by one point, we can decompose Ψ(S, S) − Ψ(S ′ , S ′ ) as follows:
Ψ(S, S) − Ψ(S ′ , S ′ ) = Ψ(S, S) − Ψ(S, S ′ ) + Ψ(S, S ′ ) − Ψ(S ′ , S ′ ) .
Now, by the sub-additivity of the sup operation, the first term can be upper-bounded as follows:
Ψ(S, S) − Ψ(S, S ′ ) ≤ sup h∈H S R(h) −R S (h) − R(h) −R S ′ (h) ≤ sup h∈H S 1 m L(h, z) − L(h, z ′ ) ≤ 1 m ,
where we denoted by z and z ′ the labeled points differing in S and S ′ and used the 1boundedness of the loss function.
We now analyze the second term:
Ψ(S, S ′ ) − Ψ(S ′ , S ′ ) = sup h∈H S R(h) −R S ′ (h) − sup h∈H S ′ R(h) −R S ′ (h) .
By definition of the supremum, for any > 0, there exists h ∈ H S such that
sup h∈H S R(h) −R S ′ (h) − ≤ R(h) −R S ′ (h)
By the β-stability of (H S ) S∈Z m , there exists h ′ ∈ H S ′ such that for all z, L(h, z)−L(h ′ , z) ≤ β. In view of these inequalities, we can write
Ψ(S, S ′ ) − Ψ(S ′ , S ′ ) ≤ R(h) −R S ′ (h) + − sup h∈H S ′ R(h) −R S ′ (h) ≤ R(h) −R S ′ (h) + − R(h ′ ) −R S ′ (h ′ ) ≤ R(h) − R(h ′ ) + + R S ′ (h ′ ) −R S ′ (h) ≤ + 2β.
Since the inequality holds for any > 0, it implies that Ψ(S, S ′ ) − Ψ(S ′ , S ′ ) ≤ 2β. Summing up the bounds on the two terms shows the following:
Ψ(S, S) − Ψ(S ′ , S ′ ) ≤ 1 m + 2β.
Thus, by McDiarmid's inequality, for any δ > 0, with probability at least 1 − δ, we have
Ψ(S, S) ≤ E[Ψ(S, S)] + [1 + 2βm] log 1 δ 2m .(12)
We now seek a more explicit upper bound for the expectation appearing on the right-hand side, in terms of the Rademacher complexity. The following sequence of inequalities holds:
E S∼D m [Ψ(S, S)] = E S∼D m sup h∈H S R(h) −R S (h) = E S∼D m sup h∈H S E T ∼D m R T (h) −R S (h) (def. of R(h)) ≤ E S,T ∼D m sup h∈H SR T (h) −R S (h) (sub-additivity of sup) = E S,T ∼D m sup h∈H S 1 m m i=1 L(h, z T i ) − L(h, z S i ) = E S,T ∼D m E σ sup h∈H σ S,T 1 m m i=1 σ i L(h, z T i ) − L(h, z S i ) (symmetry) ≤ E S,T ∼D m σ sup h∈H σ S,T 1 m m i=1 σ i L(h, z T i ) + sup h∈H σ S,T 1 m m i=1 −σ i L(h, z S i ) (sub-additivity of sup) = E S,T ∼D m σ sup h∈H σ S,T 1 m m i=1 σ i L(h, z T i ) + sup h∈H −σ T,S 1 m m i=1 −σ i L(h, z S i ) (H σ S,T = H −σ T,S ) = E S,T ∼D m σ sup h∈H σ S,T 1 m m i=1 σ i L(h, z T i ) + sup h∈H σ T,S 1 m m i=1 σ i L(h, z S i ) (symmetry) = 2R ◇ m (G).
(linearity of expectation)
We can also show the following upper bound on the expectation: E S∼D m [Ψ(S, S)] ≤χ. To do so, first fix > 0. By definition of the supremum, for any S ∈ Z m , there exists h S such that the following inequality holds:
sup h∈H S R(h) −R S (h) − ≤ R(h S ) −R S (h S ).
Now, by definition of R(h S ), we can write
E S∼D m R(h S ) = E S∼D m E z∼D (L(h S , z) = E S∼D m z∼D L(h S , z) .
Then, by the linearity of expectation, we can also write
E S∼D m R S (h S ) = E S∼D m z∼S L(h S , z) = E S∼D m z ′ ∼D z∼S L(h S z↔z ′ , z ′ ) .
In view of these two equalities, we can now rewrite the upper bound as follows:
E S∼D m Ψ(S, S) ≤ E S∼D m R(h S ) −R S (h S ) + = E S∼D m z ′ ∼D L(h S , z ′ ) − E S∼D m z ′ ∼D z∼S L(h S z↔z ′ , z ′ ) + = E S∼D m z ′ ∼D z∼S L(h S , z ′ ) − L(h S z↔z ′ , z ′ ) + = E S∼D m z ′ ∼D z∼S L(h S z↔z ′ , z) − L(h S , z) + ≤χ + .
Since the inequality holds for all > 0, it implies E S∼D m Ψ(S, S) ≤χ. Plugging in these upper bounds on the expectation in the inequality (12) completes the proof.
Appendix G. Proof of Theorem 10
In this section, we present the proof of Theorem 10.
Theorem 10 Let H = (H S ) S∈Z m be a β-stable family of data-dependent hypothesis sets. Then, for any δ ∈ (0, 1), with probability at least 1 − δ over the draw of a sample S ∼ Z m , the following inequality holds for all h ∈ H S :
R(h) ≤R S (h) + min ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 2R ◇ m (G) + [1 + 2βm] log 2 δ 2m , √ eχ + 4 1 m + 2β log 6 δ ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ .
Proof For any two samples S, S ′ of size m, define Ψ(S, S ′ ) as follows:
Ψ(S, S ′ ) = sup h∈H S R(h) −R S ′ (h).
The proof consists of deriving a high-probability bound for Ψ(S, S). To do so, by Lemma 8 applied to the random variable X = Ψ(S, S), it suffices to bound E S∼D pm max 0, max k∈[p] Ψ(S k , S k ) , where S = (S 1 , . . . , S p ) with S k , k ∈ [p], independent samples of size m drawn from D m .
To bound that expectation, we can use Lemma 9 and instead bound E S∼D pm
k=A(S) [Ψ(S k , S k )],
where A is an -differentially private algorithm. Now, to apply Lemma 9, we first show that, for any k ∈ [p], the function f k ∶ S → Ψ(S k , S k ) is ∆-sensitive with ∆ = 1 m + 2β. Fix k ∈ [p]. Let S ′ = (S ′ 1 , . . . , S ′ p ) be in Z pm and assume that S ′ differs from S by one point. If they differ by a point not in S k (or S ′ k ), then f k (S) = f k (S ′ ). Otherwise, they differ only by a point in S k (or S ′ k ) and f k (S) − f k (S ′ ) = Ψ(S k , S k ) − Ψ(S ′ k , S ′ k ). We can decompose this term as follows:
Ψ(S k , S k ) − Ψ(S ′ k , S ′ k ) = Ψ(S k , S k ) − Ψ(S k , S ′ k ) + Ψ(S k , S ′ k ) − Ψ(S ′ k , S ′ k ) .
Now, by the sub-additivity of the sup operation, the first term can be upper-bounded as follows:
Ψ(S k , S k ) − Ψ(S k , S ′ k ) ≤ sup h∈H S k R(h) −R S k (h) − R(h) −R S ′ k (h) ≤ sup h∈H S k 1 m L(h, z) − L(h, z ′ ) ≤ 1 m ,
where we denoted by z and z ′ the labeled points differing in S k and S ′ k and used the 1-boundedness of the loss function.
We now analyze the second term:
Ψ(S k , S ′ k ) − Ψ(S ′ k , S ′ k ) = sup h∈H S k R(h) −R S ′ k (h) − sup h∈H S ′ k R(h) −R S ′ k (h) .
By definition of the supremum, for any η > 0, there exists h ∈ H S k such that
sup h∈H S k R(h) −R S ′ k (h) − η ≤ R(h) −R S ′ k (h)
By the β-stability of (H S ) S∈Z m , there exists h ′ ∈ H S ′ k such that for all z, L(h, z)−L(h ′ , z) ≤ β. In view of these inequalities, we can write
Ψ(S k , S ′ k ) − Ψ(S ′ k , S ′ k ) ≤ R(h) −R S ′ k (h) + η − sup h∈H S ′ k R(h) −R S ′ k (h) ≤ R(h) −R S ′ k (h) + η − R(h ′ ) −R S ′ k (h ′ ) ≤ R(h) − R(h ′ ) + η + R S ′ k (h ′ ) −R S ′ k (h) ≤ η + 2β.
Since the inequality holds for any η > 0, it implies that Ψ(S k , S ′ k ) − Ψ(S ′ k , S ′ k ) ≤ 2β. Summing up the bounds on the two terms shows the following:
Ψ(S k , S k ) − Ψ(S ′ k , S ′ k ) ≤ 1 m + 2β.
Having established the ∆-sensitivity of the functions f k , k ∈ [p], we can now apply Lemma 9. Fix > 0. Then, by Lemma 9 and (8), the algorithm A returning k ∈ [p + 1] with probability proportional to e Ψ(S k ,S k )1 k≠(p+1)
2∆
is -differentially private and, for any sample S ∈ Z pm , the following inequality holds:
max 0, max k∈[p] Ψ(S k , S k ) ≤ E k=A(S) Ψ(S k , S k ) + 2∆ log(p + 1).
Taking the expectation of both sides yields
E S∼D pm max 0, max k∈[p] Ψ(S k , S k ) ≤ E S∼D pm k=A(S) Ψ(S k , S k ) + 2∆ log(p + 1).(13)
We will show the following upper bound on the expectation:
E S∼D pm k=A(S) Ψ(S k , S k ) ≤ (e − 1) + e χ.
To do so, first fix η > 0. By definition of the supremum, for any S ∈ Z m , there exists h S ∈ H S such that the following inequality holds:
sup h∈H S R(h) −R S (h) − η ≤ R(h S ) −R S (h S ).
In what follows, we denote by S k,z↔z ′ ∈ Z pm the result of modifying S = (S 1 , . . . , S p ) ∈ Z pm by replacing z ∈ S k with z ′ . Now, by definition of the algorithm A, we can write:
E S∼D pm k=A(S) R(h S k ) = E S∼D pm k=A(S) E z ′ ∼D [L(h S k , z ′ )] (def. of R(h S k )) = E S∼D pm z ′ ∼D p k=1 P[A(S) = k] L(h S k , z ′ ) (def. of E k=A(S) ) = p k=1 E S∼D pm z ′ ∼D P[A(S) = k] L(h S k , z ′ ) (linearity of expect.) ≤ p k=1 E S∼D pm z ′ ∼D, z∼S k e P[A(S k,z↔z ′ ) = k] L(h S k , z ′ ) ( -diff. privacy of A) = p k=1 E S∼D pm z ′ ∼D, z∼S k e P[A(S) = k] L(h S z↔z ′ k , z) (swapping z ′ and z) ≤ p k=1 E S∼D pm z ′ ∼D, z∼S k e P[A(S) = k] L(h S k , z) + e χ. (By Lemma 16 below) Now, observe that E z∼S k [L(h S k , z)] coincides withR(h S k ), the empirical loss of h S k . Thus, we can write E S∼D pm k=A(S) R(h S k ) ≤ p k=1 E S∼D pm z∼S k e P[A(S) = k]R S k (h S k ) + e χ,
and therefore
E S∼D pm k=A(S) Ψ(S k , S k ) ≤ p k=1 E S∼D pm k=A(S) (e − 1)R S k (h S k ) + e χ + η ≤ (e − 1) + e χ + η.
Since the inequality holds for any η > 0, we have
E S∼D pm k=A(S) [Ψ(S k , S k )] ≤ (e − 1) + e χ.
Thus, by (13), the following inequality holds:
E S∼D pm max 0, max k∈[p] Ψ(S k , S k )
≤ (e − 1) + e χ + 2∆ log(p + 1).
For any δ ∈ (0, 1), choose p = log 2 δ , which implies log(p + 1) = log 2+δ δ ≤ log 3 δ . Then, by Lemma 8, with probability at least 1 − δ over the draw of a sample S ∼ D m , the following inequality holds for all h ∈ H S :
R(h) ≤R S (h) + (e − 1) + e χ + 2∆ log 3 δ .(15)
For ≤ 1 2 , the inequality (e − 1) ≤ 2 holds. Thus,
(e − 1) + e χ + 2∆ log 3 δ ≤ 2 + √ eχ + 2∆ log 3 δ Choosing = ∆ log 3 δ gives R(h) ≤R S (h) + √ eχ + 4 ∆ log 3 δ =R S (h) + √ eχ + 4 1 m + 2β log 3 δ .
Combining this inequality with the inequality of Theorem 7 related to the Rademacher complexity:
∀h ∈ H S , R(h) ≤R S (h) + 2R ◇ m (G) + [1 + 2βm] log 1 δ 2m ,(16)
and using the union bound complete the proof.
Lemma 16
The following upper bound in terms of the CV-stability coefficient χ holds:
p k=1 E S∼D pm z ′ ∼D, z∼S k e P[A(S) = k] [L(h S z↔z ′ k , z) − L(h S k , z)] ≤ e χ.
Proof Upper bounding the difference of losses by a supremum to make the CV-stability coefficient appear gives the following chain of inequalities:
p k=1 E S∼D pm z ′ ∼D, z∼S k e P[A(S) = k] [L(h S z↔z ′ k , z) − L(h S k , z)] ≤ p k=1 E S∼D pm z ′ ∼D, z∼S k e P[A(S) = k] sup h∈H S k , h ′ ∈H S z↔z ′ k [L(h ′ , z) − L(h, z)] = p k=1 E S∼D pm e P[A(S) = k] E z ′ ∼D, z∼S k sup h∈H S k , h ′ ∈H S z↔z ′ k [ L(h ′ , z) − L(h, z)] S ≤ p k=1 E S∼D pm e P[A(S) = k] χ = E S∼D pm p k=1 P[A(S) = k] ⋅ e χ = e χ,
which completes the proof.
We now analyze the second term:
Ψ(S k , S ′ k ) − Ψ(S ′ k , S ′ k ) = sup h∈H S k R(h) −R S ′ k (h) − sup h∈H S ′ k R(h) −R S ′ k (h) .
By definition of the supremum, for any η > 0, there exists h ∈ H S k such that
sup h∈H S k R(h) −R S ′ k (h) − η ≤ R(h) −R S ′ k (h)
By the β-stability of (H S ) S∈Z m , there exists h ′ ∈ H S ′ k such that for all z, L(h, z)−L(h ′ , z) ≤ β. In view of these inequalities, we can write
Ψ(S k , S ′ k ) − Ψ(S ′ k , S ′ k ) ≤ R(h) −R S ′ k (h) + η − sup h∈H S ′ k R(h) −R S ′ k (h) ≤ R(h) −R S ′ k (h) + η − R(h ′ ) −R S ′ k (h ′ ) ≤ R(h) − R(h ′ ) + η + R S ′ k (h ′ ) −R S ′ k (h) ≤ η + 2β.
Since the inequality holds for any η > 0, it implies that Ψ(S k , S ′ k ) − Ψ(S ′ k , S ′ k ) ≤ 2β. Summing up the bounds on the two terms shows the following:
Ψ(S k , S k ) − Ψ(S ′ k , S ′ k ) ≤ 1 m + 2β.
Having established the ∆-sensitivity of the functions f k , k ∈ [p], we can now apply Lemma 9. Fix > 0. Then, by Lemma 9 and (8), the algorithm A returning k ∈ [p + 1] with probability proportional to e Ψ(S k ,S k )1 k≠(p+1)
2∆
is -differentially private and, for any sample S ∈ Z pm , the following inequality holds: max 0, max k∈[p] Ψ(S k , S k ) ≤ E k=A(S) Ψ(S k , S k ) + 2∆ log(p + 1).
Taking the expectation of both sides yields
E S∼D pm max 0, max k∈[p] Ψ(S k , S k ) ≤ E S∼D pm k=A(S) Ψ(S k , S k ) + 2∆ log(p + 1).(17)
We will show the following upper bound on the expectation: E S∼D pm k=A(S) Ψ(S k , S k ) ≤ (e − 1) + e χ. To do so, first fix η > 0. By definition of the supremum, for any S ∈ Z m , there exists h S ∈ H S such that the following inequality holds:
sup h∈H S R(h) −R S (h) − η ≤ R(h S ) −R S (h S ).
In what follows, we denote by S k,z↔z ′ ∈ Z pm the result of modifying S = (S 1 , . . . , S p ) ∈ Z pm by replacing z ∈ S k with z ′ . Now, by definition of the algorithm A, we can write:
E S∼D pm k=A(S)
fixed family of hypotheses H. The second stage consists of using that prior p S to choose a hypothesis h S ∈ H, either deterministically or via a randomized algorithm. Our notion of hypothesis set stability could then be extended to that of stability of priors and lead to new learning bounds depending on that stability parameter.
Appendix K. Other applications K.1. Anti-distillation
A similar setup to distillation is that of anti-distillation where the predictor f * S in the first stage is chosen from a simpler family, say that of linear hypotheses, and where the sampledependent hypothesis set H S is the subset of a very rich family H. H S is defined as the set of predictors that are close to f * S :
H S = h ∈ H∶ ( (h − f * S ) ∞ ≤ γ) ∧ ( (h − f * S )1 S ∞ ≤ ∆ m ) , with ∆ m = O(1 √ m)
. Thus, the restriction to S of a hypothesis h ∈ H S is close to f * S in ∞ -norm. As shown in the previous section, the family of hypothesis sets H S is µβ mstable. However, here, the hypothesis sets H S could be very complex and the Rademacher complexity R ◇ m (H) not very favorable. Nevertheless, by Theorem 10, for any δ > 0, with probability at least 1 − δ over the draw of a sample S ∼ D m , the following inequality holds for any h ∈ H S : R(h) ≤R S (h) + √ eµ(∆ m + β m ) + 4 1 m + 2µβ m log 6 δ . Notice that a standard uniform-stability does not apply here since the (1 √ m)-closeness of the hypotheses to f * S on S does not imply their global (1 √ m)-closeness.
K.2. Principal Components Regression
Principal Components Regression is a very commonly used technique in data analysis. In this setting, X ⊆ R d and Y ⊆ R, with a loss function that is µ-Lipschitz in the prediction. Given a sample S = {(x i , y i ) ∈ X×Y∶ i ∈ [m]}, we learn a linear regressor on the data projected on the principal k-dimensional space of the data. Specifically, let Π S ∈ R d×d be the projection matrix giving the projection of R d onto the principal k-dimensional subspace of the data, i.e. the subspace spanned by the top k left singular vectors of the design matrix X S = [x 1 , x 2 , ⋯, x m ].
The hypothesis space H S is then defined as H S = {x ↦ w ⊺ Π S x∶ w ∈ R k , w ≤ γ}, where γ is a predefined bound on the norm of the weight vector for the linear regressor. Thus, this can be seen as an instance of the setting in section 6.2, where the feature mapping Φ S is defined as Φ S (x) = Π S x.
To prove generalization bounds for this setup, we need to show that these feature mappings are stable. To do that, we make the following assumptions:
1. For all x ∈ X, x ≤ r for some constant r ≥ 1.
2. The data covariance matrix E x [xx ⊺ ] has a gap of λ > 0 between the k-th and (k +1)-th largest eigenvalues.
The matrix concentration bound of Rudelson and Vershynin (2007) implies that with probability at least 1−δ over the choice of S, we have X S X ⊺ S −m E x [xx ⊺ ] ≤ cr 2 m log(m) log( 2 δ ) for some constant c > 0. Suppose m is large enough so that cr 2 m log(m) log( 2 δ ) ≤ λ 2 m. Then, the gap between the k-th and (k + 1)-th largest eigenvalues of X S X ⊺ S is at least λ 2 m. Now, consider changing one sample point (x, y) ∈ S to (x, y ′ ) to produce the sample set S ′ . Then, we have X S ′ X ⊺ S ′ = X S X ⊺ S − xx ⊺ + x ′ x ′ ⊺ . Since − xx ⊺ + x ′ x ′ ⊺ ≤ 2r 2 , by standard matrix perturbation theory bounds (Stewart, 1998), we have Π S − Π S ′ ≤ O( r 2 λm ). Thus,
Φ S (x) − Φ S ′ (x) ≤ Π S − Π S ′ x ≤ O( r 3 λm )
. Now, to apply the bound of (9), we need to compute a suitable bound on R ◇ m (H). For this, we apply Lemma 12. For any w ≤ γ, since Π S = 1, we have Π S w ≤ γ. So the hypothesis set H ′ S = {x ↦ w ⊺ Π S x∶ w ∈ R k , Π S w ≤ γ} contains H S . By Lemma 12, we have R ◇ m (H ′ ) ≤ γr √ m . Thus, by plugging the bounds obtained above in (9), we conclude that with probability at least 1 − 2δ over the choice of S, for any h ∈ H S , we have
R(h) ≤R S (h) + O ⎛ ⎝ µγ r 3 λ log 1 δ m ⎞ ⎠ .
Lemma 9 (
9(McSherry and Talwar, 2007; Bassily et al., 2016; Feldman and Vondrak, 2018)) Let f 1 , . . . , f p ∶ Z m → R be p scoring functions with sensitivity ∆. Let A be the algorithm that, given a dataset S ∈ Z m and a parameter > 0, returns the index k ∈ [p] with probability proportional to e f k (S)
a standard Rademacher complexity bound (seeTheorem 11.15 in (Mohri et al., 2018)) impliesR S (F S,
.
Plugging in the bound on E[Φ(S)] of Lemma 15 (Appendix E) completes the proof.
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Appendix H. Theorem 10 -Alternative proof techniqueIn this section, we give an alternative version of Theorem 10 with a proof technique only making use of recent methods from the differential privacy literature. In particular, the Rademacher complexity bound is obtained using only these techniques, as opposed to the standard use of McDiarmid's inequality. This can be of independent interest for future studies.Theorem 17 Let H = (H S ) S∈Z m be a β-stable family of data-dependent hypothesis sets. Then, for any δ ∈ (0, 1), with probability at least 1 − δ over the draw of a sample S ∼ Z m , the following inequality holds for all h ∈ H S :Proof For any two samples S, S ′ of size m, define Ψ(S, S ′ ) as follows:The proof consists of deriving a high-probability bound for Ψ(S, S). To do so, by Lemma 8 applied to the random variable X = Ψ(S, S), it suffices to bound E S∼D pm max 0, max k∈[p] Ψ(S k , S k ) , where S = (S 1 , . . . , S p ) with S k , k ∈ [p], independent samples of size m drawn from D m .To bound that expectation, we can use Lemma 9 and instead bound E S∼D pm k=A(S)where A is an -differentially private algorithm. Now, to apply Lemma 9, we first show that, for any k ∈ [p], the function. . , S ′ p ) be in Z pm and assume that S ′ differs from S by one point. If they differ by a point not inWe can decompose this term as follows:Now, by the sub-additivity of the sup operation, the first term can be upper-bounded as follows:where we denoted by z and z ′ the labeled points differing in S k and S ′ k and used the 1-boundedness of the loss function. δ . Then, by Lemma 8, with probability at least 1 − δ over the draw of a sample S ∼ D m , the following inequality holds for all h ∈ H S :For ≤ 1 2 , the inequality (e − 1) ≤ 2 holds. Thus,This completes the proof.Appendix J. ExtensionsWe briefly discuss here some extensions of the framework and results presented in the previous section.J.1. Almost everywhere hypothesis set stabilityAs for standard algorithmic uniform stability, our generalization bounds for hypothesis set stability can be extended to the case where hypothesis set stability holds only with high probability(Kutin and Niyogi, 2002).Definition 19 Fix m ≥ 1. We will say that a family of data-dependent hypothesis sets H = (H S ) S∈Z m is weakly (β, δ)-stable for some β ≥ 0 and δ > 0, if, with probability at least 1 − δ over the draw of a sample S ∈ Z m , for any sample S ′ of size m differing from S only by one point, the following holds:Notice that, in this definition, β and δ depend on the sample size m. In practice, we often have β = O( 1 m ) and δ = O(e −Ω(m) ). The learning bounds of Theorem 7 and Theorem 10 can be straightforwardly extended to guarantees for weakly (β, δ)-stable families of datadependent hypothesis sets, by using a union bound and the confidence parameter δ.J.2. Randomized algorithmsThe generalization bounds given in this paper assume that the data-dependent hypothesis set H S is deterministic conditioned on S. However, in some applications such as bagging, it is more natural to think of H S as being constructed by a randomized algorithm with access to an independent source of randomness in the form of a random seed s. Our generalization bounds can be extended in a straightforward manner for this setting if the following can be shown to hold: there is a good set of seeds, G, such that (a) P[s ∈ G] ≥ 1 − δ, where δ is the confidence parameter, and (b) conditioned on any s ∈ G, the family of data-dependent hypothesis sets H = (H S ) S∈Z m is β-uniformly stable. In that case, for any good set s ∈ G, Theorem 7 and 10 hold. Then taking a union bound, we conclude that with probability at least 1−2δ over both the choice of the random seed s and the sample set S, the generalization bounds hold. This can be further combined with almost-everywhere hypothesis stability as in section J.1 via another union bound if necessary.J.3. Data-dependent priorsAn alternative scenario extending our study is one where, in the first stage, instead of selecting a hypothesis set H S , the learner decides on a probability distribution p S on a
Rademacher and Gaussian complexities: Risk bounds and structural results. L Peter, Shahar Bartlett, Mendelson, Journal of Machine Learning. 3Peter L. Bartlett and Shahar Mendelson. Rademacher and Gaussian complexities: Risk bounds and structural results. Journal of Machine Learning, 3, 2002.
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"Yuichi Harikane \nInstitute for Cosmic Ray Research\nThe University of Tokyo\n5-1-5 Kashiwanoha277-8582KashiwaChibaJapan\n\nDepartment of Physics and Astronomy\nUniversity College London\nGower StreetWC1E 6BTLondonUK\n",
"Akio K Inoue \nDepartment of Physics\nSchool of Advanced Science and Engineering\nFaculty of Science and Engineering\nWaseda University\n3-4-1 Okubo169-8555ShinjukuTokyoJapan\n\nFaculty of Science and Engineering\nWaseda Research Institute for Science and Engineering\nWaseda University\n3-4-1 Okubo169-8555ShinjukuTokyoJapan\n",
"Ken Mawatari \nNational Astronomical Observatory of Japan\n2-21-1 Osawa181-8588MitakaTokyoJapan\n",
"Takuya Hashimoto \nFaculty of Pure and Applied Science\nTomonaga Center for the History of the Universe (TCHoU)\nUniversity of Tsukuba\n305-8571IbarakiJapan\n",
"Satoshi Yamanaka \nFaculty of Science and Engineering\nWaseda Research Institute for Science and Engineering\nWaseda University\n3-4-1 Okubo169-8555ShinjukuTokyoJapan\n\nGeneral Education Department\nNational Institute of Technology\nToba College\n1-1, Ikegami-cho517-8501TobaMieJapan\n\nResearch Center for Space and Cosmic Evolution\nEhime University\n2-5, Bunkyo-cho790-8577MatsuyamaEhimeJapan\n",
"Yoshinobu Fudamoto \nFaculty of Science and Engineering\nWaseda Research Institute for Science and Engineering\nWaseda University\n3-4-1 Okubo169-8555ShinjukuTokyoJapan\n\nNational Astronomical Observatory of Japan\n2-21-1 Osawa181-8588MitakaTokyoJapan\n",
"Hiroshi Matsuo \nNational Astronomical Observatory of Japan\n2-21-1 Osawa181-8588MitakaTokyoJapan\n\nDepartment of Astronomical Science\nThe Graduate University for Advanced Studies (SOKENDAI)\n2-21-1 Osawa181-8588MitakaTokyoJapan\n",
"Yoichi Tamura \nDivision of Particle and Astrophysical Science\nGraduate School of Science\nNagoya University\n464-8602NagoyaJapan\n",
"Pratika Dayal \nKapteyn Astronomical Institute\nUniversity of Groningen\nP.O. Box 8009700 AVGroningenthe Netherlands\n",
"L Y Aaron Yung \nAstrophysics Science Division\nNASA Goddard Space Flight Center\n20771GreenbeltMDUSA\n",
"Anne Hutter \nKapteyn Astronomical Institute\nUniversity of Groningen\nP.O. Box 8009700 AVGroningenthe Netherlands\n",
"Fabio Pacucci \nCenter for Astrophysics |\nHarvard & Smithsonian\n02138CambridgeMAUSA\n\nBlack Hole Initiative\nHarvard University\n02138CambridgeMAUSA\n",
"Yuma Sugahara \nFaculty of Science and Engineering\nWaseda Research Institute for Science and Engineering\nWaseda University\n3-4-1 Okubo169-8555ShinjukuTokyoJapan\n\nNational Astronomical Observatory of Japan\n2-21-1 Osawa181-8588MitakaTokyoJapan\n",
"Anton M Koekemoer \nSpace Telescope Science Institute\n3700 San Martin Dr21218BaltimoreMDUSA\n"
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"Institute for Cosmic Ray Research\nThe University of Tokyo\n5-1-5 Kashiwanoha277-8582KashiwaChibaJapan",
"Department of Physics and Astronomy\nUniversity College London\nGower StreetWC1E 6BTLondonUK",
"Department of Physics\nSchool of Advanced Science and Engineering\nFaculty of Science and Engineering\nWaseda University\n3-4-1 Okubo169-8555ShinjukuTokyoJapan",
"Faculty of Science and Engineering\nWaseda Research Institute for Science and Engineering\nWaseda University\n3-4-1 Okubo169-8555ShinjukuTokyoJapan",
"National Astronomical Observatory of Japan\n2-21-1 Osawa181-8588MitakaTokyoJapan",
"Faculty of Pure and Applied Science\nTomonaga Center for the History of the Universe (TCHoU)\nUniversity of Tsukuba\n305-8571IbarakiJapan",
"Faculty of Science and Engineering\nWaseda Research Institute for Science and Engineering\nWaseda University\n3-4-1 Okubo169-8555ShinjukuTokyoJapan",
"General Education Department\nNational Institute of Technology\nToba College\n1-1, Ikegami-cho517-8501TobaMieJapan",
"Research Center for Space and Cosmic Evolution\nEhime University\n2-5, Bunkyo-cho790-8577MatsuyamaEhimeJapan",
"Faculty of Science and Engineering\nWaseda Research Institute for Science and Engineering\nWaseda University\n3-4-1 Okubo169-8555ShinjukuTokyoJapan",
"National Astronomical Observatory of Japan\n2-21-1 Osawa181-8588MitakaTokyoJapan",
"National Astronomical Observatory of Japan\n2-21-1 Osawa181-8588MitakaTokyoJapan",
"Department of Astronomical Science\nThe Graduate University for Advanced Studies (SOKENDAI)\n2-21-1 Osawa181-8588MitakaTokyoJapan",
"Division of Particle and Astrophysical Science\nGraduate School of Science\nNagoya University\n464-8602NagoyaJapan",
"Kapteyn Astronomical Institute\nUniversity of Groningen\nP.O. Box 8009700 AVGroningenthe Netherlands",
"Astrophysics Science Division\nNASA Goddard Space Flight Center\n20771GreenbeltMDUSA",
"Kapteyn Astronomical Institute\nUniversity of Groningen\nP.O. Box 8009700 AVGroningenthe Netherlands",
"Center for Astrophysics |\nHarvard & Smithsonian\n02138CambridgeMAUSA",
"Black Hole Initiative\nHarvard University\n02138CambridgeMAUSA",
"Faculty of Science and Engineering\nWaseda Research Institute for Science and Engineering\nWaseda University\n3-4-1 Okubo169-8555ShinjukuTokyoJapan",
"National Astronomical Observatory of Japan\n2-21-1 Osawa181-8588MitakaTokyoJapan",
"Space Telescope Science Institute\n3700 San Martin Dr21218BaltimoreMDUSA"
]
| []
| We present two bright galaxy candidates at z ∼ 12 − 13 identified in our H-dropout Lyman break selection with 2.3 deg 2 near-infrared deep imaging data. These galaxy candidates, selected after careful screening of foreground interlopers, have spectral energy distributions showing a sharp discontinuity around 1.7 µm, a flat continuum at 2 − 5 µm, and non-detections at < 1.2 µm in the available photometric datasets, all of which are consistent with z > 12 galaxy. An ALMA program targeting one of the candidates shows a tentative 4σ [Oiii]88µm line at z = 13.27, in agreement with its photometric redshift estimate. The number density of the z ∼ 12 − 13 candidates is comparable to that of bright z ∼ 10 galaxies, and is consistent with a recently proposed double power-law luminosity function rather than the Schechter function, indicating little evolution in the abundance of bright galaxies from z ∼ 4 to 13. Comparisons with theoretical models show that the models cannot reproduce the bright end of rest-frame ultraviolet luminosity functions at z ∼ 10 − 13. Combined with recent studies reporting similarly bright galaxies at z ∼ 9 − 11 and mature stellar populations at z ∼ 6 − 9, our results indicate the existence of a number of star-forming galaxies at z > 10, which will be detected with upcoming space missions such as James Webb Space Telescope, Nancy Grace Roman Space Telescope, and GREX-PLUS. | null | [
"https://arxiv.org/pdf/2112.09141v3.pdf"
]
| 245,329,553 | 2112.09141 | 4357a236aa2e405736b83a40f1d0866f45b83817 |
Draft version February 10, 2022
Yuichi Harikane
Institute for Cosmic Ray Research
The University of Tokyo
5-1-5 Kashiwanoha277-8582KashiwaChibaJapan
Department of Physics and Astronomy
University College London
Gower StreetWC1E 6BTLondonUK
Akio K Inoue
Department of Physics
School of Advanced Science and Engineering
Faculty of Science and Engineering
Waseda University
3-4-1 Okubo169-8555ShinjukuTokyoJapan
Faculty of Science and Engineering
Waseda Research Institute for Science and Engineering
Waseda University
3-4-1 Okubo169-8555ShinjukuTokyoJapan
Ken Mawatari
National Astronomical Observatory of Japan
2-21-1 Osawa181-8588MitakaTokyoJapan
Takuya Hashimoto
Faculty of Pure and Applied Science
Tomonaga Center for the History of the Universe (TCHoU)
University of Tsukuba
305-8571IbarakiJapan
Satoshi Yamanaka
Faculty of Science and Engineering
Waseda Research Institute for Science and Engineering
Waseda University
3-4-1 Okubo169-8555ShinjukuTokyoJapan
General Education Department
National Institute of Technology
Toba College
1-1, Ikegami-cho517-8501TobaMieJapan
Research Center for Space and Cosmic Evolution
Ehime University
2-5, Bunkyo-cho790-8577MatsuyamaEhimeJapan
Yoshinobu Fudamoto
Faculty of Science and Engineering
Waseda Research Institute for Science and Engineering
Waseda University
3-4-1 Okubo169-8555ShinjukuTokyoJapan
National Astronomical Observatory of Japan
2-21-1 Osawa181-8588MitakaTokyoJapan
Hiroshi Matsuo
National Astronomical Observatory of Japan
2-21-1 Osawa181-8588MitakaTokyoJapan
Department of Astronomical Science
The Graduate University for Advanced Studies (SOKENDAI)
2-21-1 Osawa181-8588MitakaTokyoJapan
Yoichi Tamura
Division of Particle and Astrophysical Science
Graduate School of Science
Nagoya University
464-8602NagoyaJapan
Pratika Dayal
Kapteyn Astronomical Institute
University of Groningen
P.O. Box 8009700 AVGroningenthe Netherlands
L Y Aaron Yung
Astrophysics Science Division
NASA Goddard Space Flight Center
20771GreenbeltMDUSA
Anne Hutter
Kapteyn Astronomical Institute
University of Groningen
P.O. Box 8009700 AVGroningenthe Netherlands
Fabio Pacucci
Center for Astrophysics |
Harvard & Smithsonian
02138CambridgeMAUSA
Black Hole Initiative
Harvard University
02138CambridgeMAUSA
Yuma Sugahara
Faculty of Science and Engineering
Waseda Research Institute for Science and Engineering
Waseda University
3-4-1 Okubo169-8555ShinjukuTokyoJapan
National Astronomical Observatory of Japan
2-21-1 Osawa181-8588MitakaTokyoJapan
Anton M Koekemoer
Space Telescope Science Institute
3700 San Martin Dr21218BaltimoreMDUSA
Draft version February 10, 2022(Accepted 2022 February 8th)Typeset using L A T E X twocolumn style in AASTeX63 A Search for H-Dropout Lyman Break Galaxies at z ∼ 12 − 16galaxies: formation -galaxies: evolution -galaxies: high-redshift
We present two bright galaxy candidates at z ∼ 12 − 13 identified in our H-dropout Lyman break selection with 2.3 deg 2 near-infrared deep imaging data. These galaxy candidates, selected after careful screening of foreground interlopers, have spectral energy distributions showing a sharp discontinuity around 1.7 µm, a flat continuum at 2 − 5 µm, and non-detections at < 1.2 µm in the available photometric datasets, all of which are consistent with z > 12 galaxy. An ALMA program targeting one of the candidates shows a tentative 4σ [Oiii]88µm line at z = 13.27, in agreement with its photometric redshift estimate. The number density of the z ∼ 12 − 13 candidates is comparable to that of bright z ∼ 10 galaxies, and is consistent with a recently proposed double power-law luminosity function rather than the Schechter function, indicating little evolution in the abundance of bright galaxies from z ∼ 4 to 13. Comparisons with theoretical models show that the models cannot reproduce the bright end of rest-frame ultraviolet luminosity functions at z ∼ 10 − 13. Combined with recent studies reporting similarly bright galaxies at z ∼ 9 − 11 and mature stellar populations at z ∼ 6 − 9, our results indicate the existence of a number of star-forming galaxies at z > 10, which will be detected with upcoming space missions such as James Webb Space Telescope, Nancy Grace Roman Space Telescope, and GREX-PLUS.
INTRODUCTION
[email protected]
Observing the first galaxy formation is one of the main goals in the modern astronomy. One of the most straightforward approaches to achieve this goal is to observe forming galaxies directly in the early universe. arXiv:2112.09141v3 [astro-ph.GA] 9 Feb 2022 Large telescopes currently in operation have yielded the most distant objects so far. These highest redshift objects have posed various interesting questions for astronomy. For example, the most distant quasars at z > 7 raised a serious problem to form blackholes as massive as ∼ 10 9 M in the limited cosmic time (e.g., Mortlock et al. 2011;Bañados et al. 2018;Yang et al. 2020;Wang et al. 2021). Thus, searching for the most distant objects is not only the simplest frontier of the knowledge of human beings but also has a great power to reveal the formation physics of various objects in the early universe (e.g., see review by Stark 2016;Dayal & Ferrara 2018;Robertson 2021).
The current record of the highest redshift galaxy spectroscopically confirmed is GN-z11 at z ∼ 11 measured with detections of the Lyman break and rest-frame ultraviolet (UV) metal lines Jiang et al. 2021). A major surprise of GN-z11 is its remarkably high luminosity, M UV = −22.1 mag. Given that it is not gravitationally-lensed, GN-z11 is located in the brightest part of the rest-frame UV luminosity function. Although the narrow field-of-view (FoV) of Hubble Space Telescope (HST)/Wide Field Camera 3 (WFC3) in the near-infrared has limited the imaging survey areas to < 1 deg 2 , several studies using HST report very luminous Lyman break galaxies (LBGs) at z ∼ 9 − 10 more frequently than the expectation from a Schechtershape luminosity function (e.g., Morishita et al. 2018;Finkelstein et al. 2021a, see also Roberts-Borsani et al. 2021a). More statistically robust results have come from a few square-degree near-infrared imaging surveys with Visible and Infrared Survey Telescope for Astronomy (VISTA) and UK Infrared Telescope (UKIRT) such as UltraVISTA (McCracken et al. 2012), the UKIRT In-fraRed Deep Sky Surveys (UKIDSS, Lawrence et al. 2007), and the VISTA Deep Extragalactic Observation (VIDEO) Survey (Jarvis et al. 2013). These surveys have revealed that the UV luminosity functions at z ∼ 9 − 10 are more consistent with a double power-law than a standard Schechter function (Stefanon et al. 2017(Stefanon et al. , 2019Bowler et al. 2020). Previous studies also report similar number density excesses beyond the Schechter function at z ∼ 4 − 7 (Ono et al. 2018;Stevans et al. 2018;Adams et al. 2020;Harikane et al. 2021b), implying little evolution of the number density of bright galaxies at z ∼ 4 − 10 Harikane et al. 2021b). Although spectroscopic observations are required to confirm these results, the studies indicate that there are a larger number of luminous galaxies at z ∼ 9 − 11 than previously thought, which formed in the early universe of z > 10.
In addition to these observations of bright galaxies at z ∼ 9 − 11, several studies independently suggest the presence of star-forming galaxies in the early universe even at z ∼ 15. A candidate for a z ∼ 12 galaxy is photometrically identified in very deep HST/WFC3 images obtained in the Hubble Ultra Deep Field 2012 (UDF12) campaign (Ellis et al. 2013). Balmer breaks identified in z = 9 − 10 galaxies indicate mature stellar populations whose age is ∼ 300 − 500 Myr, implying early star formation at z ∼ 14 − 15 (Hashimoto et al. 2018, Laporte et al. 2021, see also Roberts-Borsani et al. 2020). An analysis of passive galaxy candidates at z ∼ 6 reports that their stellar population is dominated by old stars with ages of 700 Myr, consistent with star formation activity at z > 14 (Mawatari et al. 2020b).
Motivated by these recent works, we search for Hband dropout (H-dropout) LBGs whose plausible redshifts are z ∼ 12 − 16. 1 Given the observed number density of luminous galaxies at z ∼ 9 − 10 and its little redshift evolution from z ∼ 4 to 10, it is possible that one to several z ∼ 12 − 16 galaxies will be found in currently available datasets obtained by surveys with large ground-and space-based telescopes. This search for z ∼ 12 − 16 galaxies is important not only for understanding early galaxy formation, but also for designing survey strategies with upcoming space missions that will study the z > 10 universe such as James Webb Space Telescope (JWST).
This paper is organized as follows. We describe photometric datasets and a selection of z ∼ 12 − 16 galaxies in Section 2, and ALMA follow-up observations for one of our candidates in Section 3. Results of spectral energy distribution (SED) fitting and the UV luminosity function are presented in Sections 4 and 5, respectively. We discuss future prospects with space missions based on our results in Section 6, and summarize our findings in Section 7. Throughout this paper, we use the Planck cosmological parameter sets of the TT, TE, EE+lowP+lensing+ext result (Planck Collaboration et al. 2016): Ω m = 0.3089, Ω Λ = 0.6911, Ω b = 0.049, h = 0.6774, and σ 8 = 0.8159. All magnitudes are in the AB system (Oke & Gunn 1983).
PHOTOMETRIC DATASET AND SAMPLE SELECTION
Dataset
We use deep and wide photometric datasets available in the COSMOS (Scoville et al. 2007) and SXDS (Furu- Table 1. Since the COS-MOS field has the ultra-deep and deep stripes with different depths in the near-infrared images, we use the limiting magnitude of each stripe depending on the location of the source of interest.
Selection of H-dropout Galaxies
We construct multi-band photometric catalogs in the COSMOS and SXDS fields. We start from K s (K)-band detection catalogs made by the UltraVISTA (UKIDSS) team using SExtractor (Bertin & Arnouts 1996). We select sources detected in the K s (K) bands at > 5σ levels and not detected in the J-band at > 2σ levels in a 2 -diameter circular aperture. Then we measure magnitudes of these sources in the grizyJHK s (K) images using a 2 -diameter circular aperture centered at their coordinates in the catalogs. Since point-spread functions (PSFs) of the Spitzer/IRAC [3.6] and [4.5] images are relatively large (∼ 1. 7), source confusion and blending are significant for some sources. To remove the effects of the neighbor sources on the photometry, we first generate residual IRAC images where only the sources under analysis are left by using T-PHOT (Merlin et al. 2016), in the same manner as Harikane et al. (2018Harikane et al. ( , 2019. As high-resolution prior images in the T-PHOT run, we use HSC grizy stacked images whose PSF is ∼ 0. 7. Then we measure magnitudes in the IRAC images by using a 3 -diameter apertures in the same manner as Harikane et al. (2018). To account for the flux falling outside the aperture, we apply aperture corrections derived from samples of isolated point souces in Harikane et al. (2018).
To search for z ∼ 12−16 galaxies whose Lyman breaks are redshifted to ∼ 1.6 − 2.1 µm, we select H-dropout LBGs from the multi-band photometric catalogs constructed above. We adopt the following color selection criteria in the COSMOS and SXDS fields, respectively: COSMOS:
H − K s > 1.0,(1)K s − [3.6] < 0.1,(2)
SXDS:
H − K > 1.0,(3)K − [3.6] < 0.1.(4)
As shown in Figure 1, these color criteria can select sources at z > 12 while avoiding color tracks of z = 0−7 galaxies and stellar sources. To remove foreground interlopers, we exclude sources with detections at > 2σ levels in the grizyJ band images. Note that we use the same values for the color criteria in the COSMOS and SXDS fields. Since the filter response profiles are different in the VISTA JHK s filters for the COSMOS field and the UKIRT JHK filters for the SXDS field, the selection functions will not be identical. We will account for this difference by separately evaluating the selection functions in the COSMOS and SXDS fields in Section 5.1.
To remove foreground interlopers further, we conduct a photometric redshift analysis using BEAGLE (Chevallard & Charlot 2016). We adopt a constant star forma-
0.0 1.0 K s − [3.6] 0.0 1.0 H − K s Our z > 12 candidate z = 0 − 7 sources Galactic stars 0.0 1.0 K − [3.6] 0.0 1.0 H − K Our z > 12 candidate z = 0 − 7 sources
Galactic stars Figure 1. Two-color diagrams to select H-dropout galaxies. The left and right panels show two-color diagrams in the COSMOS and SXDS fields, respectively. The red lines indicate color criteria that we use to select H-dropout galaxies (Equations (1)-(4)), and the red squares are the selected candidates, HD1 and HD2. The black solid lines are colors of star-forming galaxies at z ≥ 12 calculated with BEAGLE (Chevallard & Charlot 2016) with τV = 0.0 and 0.4 (corresponding to UV spectral slopes of βUV −2.4 and −1.9, respectively) as a function of redshift. The circles on the line show their redshifts with an interval of ∆z = 0.1. The blue circles are z = 0 − 7 sources spectroscopically identified (Laigle et al. 2016;Mehta et al. 2018). The dotted, dashed, and dot-dashed lines are, respectively, typical spectra of elliptical, Sbc, and irregular galaxies (Coleman et al. 1980) redshifted from z = 0 to z = 7. The black stars indicate Galactic dwarf stars taken from Patten et al. (2006) and Kirkpatrick et al. (2011). tion history with the Chabrier (2003) initial mass function (IMF), stellar ages of 10 6 , 10 7 , and 10 8 yr, metallicities of 0.2 and 1 Z , and the Calzetti et al. (2000) dust attenuation law with the V -band optical depth of τ V = 0 − 2 (steps of 0.2) to account for very dusty low redshift interlopers. We select objects whose high redshift solution is more likely than the low redshift ones at a > 2σ level, corresponding to ∆χ 2 > 4.0, in the same manner as Bowler et al. (2020). Then we visually inspect images and SEDs of the selected sources to remove spurious sources, sources affected by bad residual features in the T-PHOT-made IRAC images, and extremely red sources (e.g., K s (K) − [4.5] 1) that are not likely to be z ∼ 12 − 16 galaxies.
After these careful screening processes, we finally identify two z ∼ 12 − 16 galaxy candidates, HD1 and HD2, in the COSMOS and SXDS fields, respectively. Figures 2 and 3 show images and SEDs of HD1 and HD2, and Table 2 summarizes their measured fluxes. HD1 and HD2 are spatially isolated from other nearby sources, ensuring the robustness of the photometry.
HD1 is also found in the COSMOS2020 catalog (Weaver et al. 2021). However, the photometric redshift of HD1 is 3.6 in the COSMOS2020 catalog. This
Flux density [µJy]
HD2 z = 12.3 (χ 2 = 4.2) z = 3.5 (χ 2 = 11.7) is due to the difference in the measured magnitudes in the IRAC images. Our measured magnitudes are 24.6 and 24.7 mag in the [3.6] and [4.5] images, respectively, while 24.2 and 23.9 mag (24.4 and 24.1 mag) are cataloged in the COSMOS2020 CLASSIC (FARMER) catalog with very small flux errors of 3 − 12 nJy. If we re-measure magnitudes by using a larger aperture in the original IRAC images before the T-PHOT run, the magnitudes become brighter due to a neighboring source located ∼ 3. 5 from HD1. We additionally test with deeper SMUVS images (Ashby et al. 2018). We carefully measure the magnitudes in the SMUVS images and still find that the best photometric redshift for HD1 is z > 12. Magnitudes in the other bands including the K s -band in the COSMOS2020 catalog are consistent with our measurements, although their flux errors are much smaller than ours. Thus in this study, we adopt our measured magnitudes for HD1. HD1 and HD2 will be observed in a JWST program (GO-1740, Harikane et al. 2021a). In addition to these two candidates, the program will target another source, Redshift Figure 4. Full ALMA spectrum of HD1. This spectrum is extracted from a 1. 0-radius circular aperture centered on the coordinate of HD1. No obvious emission line is identified at > 5σ, but there is a 4σ line-like feature at 237.8 GHz (the red arrow), where no severe atmospheric O3 absorption exists (the gray shades).
HD3 (R.A.=02:16:54.48, Decl.=-05:09:37.1), which is also a good candidate for a z ∼ 12 − 16 galaxy with its prominent break with H − K > 1.2 and the best photometric redshift of z phot = 14.6. However, due to its relatively red color with K − [3.6] = 0.2 ± 0.2, HD3 is not included in our final sample in this paper.
ALMA FOLLOW-UP OBSERVATION
We observed one of the candidates, HD1, in an ALMA Director's Discretionary Time (DDT) program (2019.A.00015.S, PI: A. K. Inoue). Following successful detections of [Oiii]88µm emission lines in high redshift galaxies (e.g., Inoue et al. 2016, Laporte et al. 2017, 2021, Carniani et al. 2017, Marrone et al. 2018, Hashimoto et al. 2018, Walter et al. 2018, Tamura et al. 2019, Harikane et al. 2020, see also Inoue et al. 2014b), we conducted a spectral scan targeting the [Oiii]88µm line using four tuning setups with the Band 6 covering the redshift range of 12.6 < z < 14.3. The antenna configurations were C43-2, C43-3, C43-4, and C43-5, and typical beam size is ∼ 0. 4 − 0. 7. We used four spectral windows with 1.875 GHz bandwidths in the Frequency Division Mode and the total bandwidth of 7.5 GHz in one tuning setup. The velocity resolution was set to ∼ 10 km s −1 . The data were reduced and calibrated using the Common Astronomy Software (CASA; McMullin et al. 2007) pipeline version 5.4.0 in the general manner with scripts provided by the ALMA observatory. Figure 4 shows the obtained spectrum for HD1 extracted from a 1. 0-radius circular aperture. Although there is no signal at a > 5σ level, we find a 4σ tentative line-like feature around 238 GHz. As shown in the top panel of Figure 5, this feature is at 237.8 GHz, and the significance level of the peak intensity is 3.8σ in the mo- The line luminosity is very small compared to the UV luminosity. Since the UV luminosity of HD1 is L UV = 4.8 × 10 11 L , the [Oiii]-to-UV luminosity ratio is L [OIII] /L UV ∼ 7 × 10 −4 . This ratio is the smallest among the galaxies observed in [Oiii]88µm emissions in the reionization epoch as well as in the local Universe so far (e.g., Inoue et al. 2016;Binggeli et al. 2021). Since the [Oiii]-to-UV ratio depends on the oxygen abundance (Harikane et al. 2020), this low ratio indicates a metallicity as low as ∼ 0.01 − 0.1 Z .
Another possibility is that the ALMA line scan just missed the true emission line and the redshift is out of the range of 12.6 < z < 14.3. As we will see in Section 5.1, the redshift selection function is as broad as 12 < z < 17. For HD1, the lower redshift case (z < 12.6) is not very favored by the SED fitting, but the higher redshift case (z > 14.3) is still equally likely, as we discuss later in Section 4. Therefore, additional spectroscopic data are highly desired to confirm redshifts of HD1 and HD2. We plan to conduct followup observations for the tentative signal in HD1 and to newly obtain spectroscopic data for HD2 in ALMA cycle 8 (2021.1.00207.S, PI: Y. Harikane). We will also observe these candidates with JWST (GO-1740, Harikane et al. 2021a), which allows us to examine a wider redshift range than ALMA.
The dust continuum of HD1 remains non-detection, which is consistent with the low metallicity interpretation from the low L [OIII] /L UV ratio discussed above. The obtained 1σ noise level is 8 µJy beam −1 . Assuming that HD1 is not resolved in this observation, we obtain the 3σ upper limit on the dust continuum of < 24 µJy.
SED FITTING
To examine the photometric redshifts of HD1 and HD2 more carefully, we perform a comprehensive SED fitting analysis from optical to sub-millimeter (sub-mm) wavelength using PANHIT (Mawatari et al. 2020a). PANHIT takes the energy conservation of the dust absorption in the rest-frame UV to near-infrared range and the emission in the far-infrared to sub-mm range into account. PANHIT deals with the upper limits for non-detection bands, following the probability distribution function formula proposed by Sawicki (2012). We adopt 1σ for the upper bound of the integral of the probability distribution. In addition to the fluxes in grizyJHK s (K)[3.6][4.5] measured in Section 2.2, we utilize far-infrared and sub-mm data of the Herschel survey (Oliver et al. 2012) of HD1 and HD2, and the ALMA data obtained for HD1 (see Section 3). Since dust continua of HD1 and HD2 are not detected in these data, we use the upper limits for the SED fitting.
We assume a delayed-τ model for the star formation history (SFH) covering a wide range of histories including a short time-scale burst, rising, declining, and almost constant cases (Speagle et al. 2014). It is important to include passive galaxy models because the red H−K s (K) color can be produced by the Balmer break as well as the Lyman break. This may be a major contamination case in our H-dropout selection. Template spectra include the BC03 stellar population synthesis model (Bruzual & Charlot 2003) with the Chabrier (2003) IMF of 0.1-100 M , the nebular continuum and line emission model (Inoue 2011), and the dust thermal emission with a modified black-body function. The dust temperature is assumed to be 30 K, 50 K, or 80 K to account for possibilities of dusty interlopers with various temperatures, and the dust emissivity index is fixed at β dust = 2.0. The effect of the cosmic microwave background on the dust emission (da Cunha et al. 2013) is also taken into account.
The considered fitting parameters are as follows; the SFH time-scale is τ SFH = 0.01, 0.03, 0.06, 0.1, 0.3, 0.6, 1, 3, 6, and 10 Gyr (10 cases), the metallicity is Z = 0.0001, 0.0004, 0.004, 0.008, 0.02(= Z ), and 0.05 (6 cases), the dust attenuation is A V = 0.01 to 10 with 20 logarithmic steps, the stellar population age is 7 cases in 1 Myr to 10 Myr, 8 cases in 10 Myr to 100 Myr, 15 cases in 100 Myr to 1 Gyr, and 8 cases in 1 Gyr to 15 Gyr, but limited by the cosmic age at the redshift of interest, and the redshift is 0.1 to 20.0 with a 0.1 step assuming a flat prior. Figure 2 shows the results of the SED fitting analyses, and Table 3 summarizes the results. The best photometric redshifts are always z > 12 for both HD1 and HD2 thanks to the sharp discontinuity between H and K s (K)-bands. The low redshift solutions are found at z ∼ 4 for both objects with larger χ 2 values than the z > 12 solutions. These are Balmer break galaxy so- lutions, and the dust temperature does not affect them because these solutions have very weak or no dust emission. Another type of possible solutions is dusty Hα emitters at z ∼ 2, although these solutions are not supported by the non-detections in the far-infrared and sub-mm bands. In these solutions, a strong Hα line boosts K s (K)-band and makes H − K s (K) color as red as z ∼ 12 − 16 galaxies. In the very high dust temperature case of 80 K, this solution gives a slightly smaller χ 2 than those of the Balmer break ones at z ∼ 4, while the lower, more normal dust temperature cases do not favor this type of the solution. Moreover, even the 80 K case is significantly less likely compared to the solutions at z > 12 (∆χ 2 > 4). For HD1, the best-fit redshift is z = 15.2 (χ 2 = 4.7), which is in fact out of the ALMA [O iii]88µm scan (Section 3). The case of z = 13.3, corresponding to the possible line feature at z = 13.27, gives χ 2 = 5.4. Since it is roughly equally likely (within the 1σ confidence range, see Table 3), we show this case in Figure 2. The physical properties are not well constrained, except for the dust attenuation that is A V < 0.08 (2σ). The stellar mass (M * ) is (1-100) × 10 9 M , depending on the stellar age that is not constrained. When the age is less than ∼ 10 Myr, the stellar mass and star formation rate (SFR) are estimated to be M * ∼ 1 × 10 9 M and SF R ∼ 10 2−3 M yr −1 , respectively. For an age of 10-100 Myr (> 100 Myr), M * ∼ (1 − 10) × 10 9 M and SF R ∼ 10 2 M yr −1 (M * ∼ (10 − 100) × 10 9 M and SF R < 10 2 M yr −1 ) are obtained. The SFH timescale also produces dependencies; for τ SFH > 100 Myr (a larger value is closer to a constant SFH), we obtain M * ∼ (1 − 10) × 10 9 M and SF R ∼ 10 2−3 M yr −1 , and for τ SFH < 100 Myr, M * and SF R show larger variations. The metallicity is not constrained at all.
For HD2, the best-fit redshift is z = 12.3 (χ 2 = 4.2) and the 1σ range (∆χ 2 < 1) is 12.0 < z < 12.7. We find two types of the high redshift solutions. One is a very young starburst: an age less than 10 Myr, M * ∼ 7 × 10 9 M , SF R ∼ 10 3−4 M yr −1 , and A V ∼ 0.8. τ SFH and metallicity are not constrained. The other case is a massive and relatively mature galaxy: an age greater than 100 Myr, M * ∼ 1×10 11 M , SF R < 10 2 M yr −1 , A V < 0.5, and τ SFH < 60 Myr. The metallicity is not constrained.
Although they are statistically less likely given the larger χ 2 values, the possible Balmer break solutions are as follows. For HD1, we obtain z ∼ 3.9, age of 0.3-1 Gyr, M * ∼ (6 − 10) × 10 9 M , SF R < 0.1 M yr −1 , A V < 0.5, τ SFH < 0.1 Gyr, and Z > 0.004. For HD2, we obtain z ∼ 3.5, age of 0.4-0.7 Gyr, M * ∼ 1 × 10 10 M , SF R ∼ 0 M yr −1 , A V < 0.1, τ SFH < 0.03 Gyr, and Z > 0.02. These stellar masses of ∼ 10 10 M are ∼ 10 times smaller than known passive galaxies at z ∼ 4 (Glazebrook et al. 2017;Tanaka et al. 2019;Valentino et al. 2020). Therefore, even these cases are also interesting to be examined further spectroscopically in future.
LUMINOSITY FUNCTION AND SFR DENSITY
Selection Completeness
To derive the rest-frame UV luminosity function of the z ∼ 12−16 galaxies, we estimate the selection completeness by conducting Monte Carlo simulations. We first make mock SEDs of galaxies at 9.0 < z < 19.0 (steps of 0.1) with UV spectral slopes of −3.0 < β UV < −1.0 (steps of 0.1). The IGM attenuation is taken into account by using a prescription of Inoue et al. (2014a), resulting almost zero flux densities at the wavelength bluer than the Lyα break. We then calculate fluxes in each band by integrating mock SEDs through our 10 filters (grizyJHK s (K)[3.6][4.5]), and scale to have apparent magnitudes of 23.0−25.0 mag in the K s (K)-band, whose central wavelength corresponds to ∼ 1500Å at z ∼ 13. We then perturb the calculated fluxes by adding photometric scatters based on a Gaussian distribution with a standard deviation equal to the flux uncertainties in each band. We generate 1000 mock galaxies at each redshift with UV spectral slopes following a Gaussian distribution with a mean of β UV = −2.0 and a scatter of ∆β UV = 0.2 (Rogers et al. 2014;Bowler et al. 2020). Finally, we select z ∼ 12 − 16 galaxy candidates with the same color selection criteria, and calculate the selection completeness as a function of the K s (K)-band magnitude and redshift, C(m, z), averaged over the UV spectral slope. Figure 6 shows the calculated selection completeness in the COSMOS and SXDS fields. Our selection criteria can select sources at 12 z 16. The mean redshift from the simulation is z = 14.3 and 14.6 in the COSMOS and SXDS fields, respectively, but in this paper we adopt z = 12.8 (z ∼ 13) that is the average of the nominal redshifts for HD1 and HD2, as the mean redshift of our H-dropout sample. The selection completeness is ∼ 70% even for very bright (23.0 mag) galaxies because our color criteria are very strict in order to remove foreground interlopers (see Figure 1) and miss some intrinsically-red (β UV −1.8) z 12 galaxies.
Based on the results of these selection completeness simulations, we estimate the survey volume per unit area as a function of the apparent magnitude (Steidel et al. 1999),
V eff (m) = C(m, z) dV (z) dz dz,(5)
where C(m, z) is the selection completeness, i.e., the probability that a galaxy with an apparent magnitude m at redshift z is detected and satisfies the selection criteria, and dV (z)/dz is the differential comoving volume as a function of redshift.
Contamination
The space number density of the z ∼ 13 galaxies that are corrected for incompleteness and contamination is calculated with the following equation:
ψ(m) = [1 − f cont ] n raw (m) V eff (m) ,(6)
where n raw (m) is the surface number density of selected galaxies in an apparent magnitude bin of m, and f cont is a contamination fraction. We estimate the contamination fraction of foreground sources by conducting Monte Carlo simulations. As discussed in Section 4, the most likely contaminants are z ∼ 4 passive galaxies whose Balmer breaks mimic the Lyman break at z ∼ 13. Stellar contaminations are not expected to be dominant, given observed colors of stellar sources (Figure 1). To investigate contamination from various z ∼ 4 passive galaxies, we prepare three types of mock SEDs at 3 ≤ z ≤ 5 based on 1) a classic spectrum of elliptical galaxies in Coleman et al. (1980), 2) model spectra with color distributions similar to real passive galaxies, and 3) the z ∼ 4 solutions from the SED fittings. In case 1, we use a spectrum of old elliptical galaxies in Coleman et al. (1980) as an input SED. In case 2, we first generate model spectra of galaxies by using PANHIT assuming a delayed-τ star formation history, τ = 0.01, 0.03, 0.1, 0.3, and 1 Gyr, the stellar age of 0.01 − 1.3 Gyr, metallicity of Z = 0.0001, 0.0004, 0.004, 0.008, 0.02, and 0.05, and dust attenuation of E(B − V ) = 0 − 1 (steps of 0.05). Then we calculate rest-frame N U V − r and r − J colors of the models, and compare these colors with those of observed passive galaxies in Davidzon et al. (2017). By selecting galaxies whose colors are consistent with the observed passive galaxies, we construct a set of passive galaxy SEDs that have realistic color distributions. In case 3, we use a spectrum of the z ∼ 4 passive galaxy solution in the SED fitting in Section 4. Since in this case we assume that all of the passive galaxies have the same SED as the z ∼ 4 solution, this case provides the most conservative estimate for the contamination fraction (i.e., the highest contamination fraction).
We then make mock SEDs redshifted to 3 ≤ z ≤ 5 (steps of 0.1) from the three types of the SEDs, calculate fluxes in each band, scale to have stellar masses of 10 9 ≤ M * /M ≤ 10 11 (steps of 0.1 dex), and perturb the calculated fluxes by adding photometric scatters in the same manner as Section 5.1. We generate ∼ 1000 mock galaxies at each redshift and stellar mass bin, and calculate the fraction of passive galaxies that satisfy our selection criteria in each bin. Finally, by integrating the product of the stellar mass function of passive galaxies in Davidzon et al. (2017) and the fraction of passive galaxies satisfying our selection criteria over the redshift and stellar mass, we calculate the number of passive galaxies at z ∼ 4 that are expected to be in our z ∼ 13 galaxy sample.
The expected numbers of passive galaxies in our sample are N cont = 0.00, 0.12, 1.36 in cases of 1), 2), and 3), respectively. We estimate the contamination fraction f cont by dividing the expected number of passive galaxies by the number of our z ∼ 13 candidates. The estimated contamination fractions are small in cases of 1 and 2 (f cont ∼ 0% and 6%, respectively), and f cont ∼ 70% in case 3, where we assume all of the passive galaxies have the same SED as the z ∼ 4 solution in the SED fitting as the extremely conservative case. Although the realistic simulation with the observed color distributions (i.e., case 2) indicates the very low contamination fraction, we adopt this very conservative estimate from case 3 for the UV luminosity function calculation. Note that even if we assume this conservative estimate as the prior, the z > 12 solutions for HD1 and HD2 in the SED fitting are still more likely than the z ∼ 4 solutions that give larger χ 2 values, as long as the true number density of z ∼ 13 galaxies is 10 −8 Mpc −3 (comparable with our estimate in Section 5.3). On the other hand, if the true number density is ∼ 10 −11 Mpc −3 at z ∼ 13 (comparable with model predictions in Section 5.4), the z ∼ 4 solutions are more likely due to the higher number density of z ∼ 4 passive galaxies compared to that of z ∼ 13 galaxies.
UV Luminosity Function
We convert the number density of z ∼ 13 galaxies as a function of apparent magnitude, ψ(m), into the UV luminosity functions, Φ[M UV (m)], which is the number densities of galaxies as a function of rest-frame UV absolute magnitude. We calculate the absolute UV magnitudes of galaxies from their apparent magnitudes in the K s (K)-band, whose central wavelength corresponds to ∼ 1500Å at z ∼ 13, assuming a flat rest-frame UV continuum, i.e., constant f ν , suggested by the SEDs of our galaxies. The 1σ uncertainty is calculated by taking into account the Poisson confidence limit (Gehrels 1986) on the expected number of galaxies at z ∼ 13 in our sample (N = 2 × (1 − f cont ) ∼ 1). Figure 7 shows the calculated UV luminosity function at z ∼ 13. The number density of our z ∼ 13 galaxies is (3.7 +8.4 (Bouwens et al. 2016), whereas the gray and red solid lines are the double power-law functions at z ∼ 10 and 13, respectively, whose parameters are determined by the extrapolation from lower redshifts in Bowler et al. (2020). galaxies at z ∼ 10 in Bowler et al. (2020), which is supported by the little evolution of the abundance of bright galaxies found by previous studies at z = 4 − 10 Harikane et al. 2021b). Indeed as shown in Figure 8, the number density of bright (M UV < −23 mag) galaxies do not show significant redshift evolution from z ∼ 4 to z ∼ 13. In Figure 7, we also plot the number density of z ∼ 10 galaxies estimated from GN-z11, (1.0 +2.2 −0.8 ) × 10 −6 Mpc −3 . 2 These results and the spectroscopic confirmation of GN-z11 by Oesch et al. (2016) and Jiang et al. (2021) indicate that the bright end of the luminosity function at high redshift cannot be explained by the Schechter function with the exponential cutoff, and is more consistent Number/mag/Mpc 3 z ∼ 4 z ∼ 5 z ∼ 6 z ∼ 7 z ∼ 8 z ∼ 9 z ∼ 10 z ∼ 13 Figure 8. Evolution of the rest-frame UV luminosity functions from z ∼ 4 to z ∼ 13. The red circle shows the number density of our z ∼ 13 galaxy candidates, and the grey, brown, purple, blue, green, yellow, and orange symbols show results at z ∼ 4, 5, 6, 7, 8, 9, and 10, respectively. The circles at z ∼ 4 − 7 are galaxy number densities from Harikane et al. (2021b), and those at z ∼ 8 − 10 are taken from Bowler et al. (2020). The squares show results taken from Bouwens et al. (2021) and Oesch et al. (2018) at z ∼ 4 − 9 and z ∼ 10, respectively. The diamond is a result in McLeod et al. (2016). The lines show double power-law functions in Harikane et al. (2021b) at z ∼ 4 − 7 and Bowler et al. (2020) at z ∼ 8 − 13. Note that the data point of Bowler et al. (2020) at z ∼ 10 is horizontally offset by −0.2 mag for clarity.
with the double power-law function. Indeed, the number density of our z ∼ 13 galaxies is consistent with the double power-law function with M * UV = −17.6 mag, φ * = 1.0 × 10 −4 Mpc −3 , α = −1.8, and β = −2.6 ( Figure 7), which are derived by extrapolating the redshift evolution of the parameters in Bowler et al. (2020) to z = 13. 3 Although spectroscopic confirmation is needed, these results indicate that upcoming surveys will detect a number of galaxies at z > 10, which will be discussed later in Section 6.
Comparison with Models
Both theoretical and empirical models predict the UV luminosity function of galaxies at z > 10 (e.g., Dayal et al. 2014, Mason et al. 2015, Tacchella et al. 2018, Behroozi et al. 2019, Yung et al. 2019 The extrapolated double power-law luminosity function from Bowler et al. (2020) predicts a ∼ 2 times higher number density at z ∼ 11 − 12 than those at z ∼ 10 and 13 in the magnitude regime of M UV −23.5 mag, but still consistent with our z ∼ 13 estimate within the errors. The detail of such a redshift evolution is beyond the scope of this paper. see also Hutter et al. 2021). We compare the number densities at z ∼ 10 and 13 with predictions from these models in Figure 9. At z ∼ 10, the predictions roughly agree with the observed number densities for relatively faint galaxies (M UV −21 mag), but the models underestimate the number densities of bright galaxies (M UV −22 mag) albeit with large uncertainties in the observations. Similarly at z ∼ 13, the models cannot reproduce the observed number density of our z ∼ 13 galaxy candidates. These discrepancies indicate that the current models do not account for the rapid mass growth within the short physical time since the Big Bang.
There are several possible physical processes to reconcile these discrepancies between the models and the observations at z ∼ 10 − 13. As discussed in Harikane et al. (2021b), less efficient mass quenching and/or lower dust obscuration than assumed in the models can explain the existence of these UV-bright galaxies. AGN activity may also boost the UV luminosity in these galaxies. Previous studies indicate that the AGN fraction starts to increase at M UV −22 mag (Ono et al. 2018;Stevans et al. 2018;Adams et al. 2020;Harikane et al. 2021b, see also Piana et al. 2021). If we assume that the UV luminosities of HD1 and HD2 are solely powered by black holes, the inferred black hole masses are ∼ 10 8 M , assuming accretion at the Eddington rate (Pacucci et al. 2022), in accordance with expectations for high redshift quasars (see, e.g., Haiman &Menou 2000 andWillott et al. 2010). In addition, note that a ∼ 10 8 M black hole at z ∼ 12 could be the progenitor of z ∼ 7 quasars, as the growth time to reach a mass of 10 9−10 M , typical of z > 6 quasars detected thus far, is shorter than the cosmic time between z = 12 − 13 and z = 7, for an Eddington-limited accretion. It is also possible that the observed bright source at z ∼ 10 − 13 are galaxies in a short-time starburst phase that is not captured in the models whose outputs are averaged over a time interval (see also Dayal et al. 2013). Finally, a top-heavier IMF would explain the discrepancies by producing more UV photons at the same stellar mass. It is possible that these bright galaxies (especially HD1 and HD2) are merging systems that are not resolved in the ground-based images. However, even if they are major mergers, the UV luminosity will decrease only by a factor of a few, which would not explain the discrepancy at z ∼ 13 (see also discussions in Harikane et al. 2021b andShibuya et al. 2021). In any case, if these bright z ∼ 10 − 13 galaxies are spectroscopically confirmed, the discrepancies will motivate the exploration of new physical processes that are responsible for driving the formation of these bright galaxies in the early universe. Number/mag/Mpc 3 Dayal+14,19 Yung+20 Behroozi+19 Mason+15 Figure 9. Comparison with predictions from theoretical and empirical models at z ∼ 10 (left) and z ∼ 13 (right). The black symbols and the gray shaded region are measurements at z ∼ 10 from the literature (symbols are the same as in Figure 7) and at z ∼ 13 from this study. The blue lines show predictions from models (solid: Dayal et al. 2014, dotted: Yung et al. 2019, dot-dashed: Behroozi et al. 2019, dashed: Mason et al. 2015.
Cosmic SFR Density
We calculate the cosmic SFR density at z ∼ 13 by integrating the rest-frame UV luminosity function. We use a double power-law luminosity function at z = 13 with M * UV = −17.6 mag, φ * = 1.0 × 10 −4 Mpc −3 α = −1.8, and β = −2.6, which is consistent with our number density measurement (see Section 5.3). We obtain the UV luminosity density by integrating the luminosity function down to −17 mag as previous studies (e.g., Bouwens et al. 2015Bouwens et al. , 2020Finkelstein et al. 2015;Oesch et al. 2018). We then convert the UV luminosity density to the SFR density by using the calibration used in Madau & Dickinson (2014) with the Salpeter (1955) IMF: SF R UV (M yr −1 ) = 1.15 × 10 −28 L UV (erg s −1 Hz −1 ).
(7) This SFR density estimation is true only if the restframe UV galaxy identification is complete with respect to the all galaxy populations at z ∼ 13 (but see Fudamoto et al. 2020). The uncertainty of the SFR density is scaled from that of the number density measurement.
The estimated SFR density is ρ SFR = (8.0 +18.4 −6.6 ) × 10 −5 M yr −1 Mpc −3 at z = 12.8. We compare the SFR density with previous results in Figure 10. The estimated SFR density at z ∼ 13 is consistent with the fitting function in Harikane et al. (2021b), which is calibrated with recent observations at z > 4 showing a more rapid decline (∝ 10 −0.5(1+z) ) than the extrapolation of the fitting function in Madau & Dickinson (2014) at z > 10. Furthermore, if we divide our sample into one galaxy at z ∼ 12 and another at z ∼ 13, given the low completeness at z = 12.3 (see Figure 6), the estimated SFR density would show a decrease from z ∼ 12 to 13, (7)). consistent with Harikane et al. (2021b). The estimate is also comparable to the edge of the range of the SFR density at z 13 expected from passive galaxies at z ∼ 6 in Mawatari et al. (2020b). Note that our estimated SFR density is dominated by relatively faint (M UV −20 mag) galaxies (see Roberts-Borsani et al. 2021a for discussions at z ∼ 8 − 10). In this calculation, we need to assume the shape of the UV luminosity function, be-cause there is no constraint on the number density of faint galaxies at z ∼ 13. Future JWST observations will measure the number density of faint z ∼ 13 galaxies and constrain the shape of the luminosity function combined with this work for bright galaxies, allowing a more robust measurement of the SFR density.
FUTURE PROSPECT
There are several upcoming space-based missions that can search for z > 10 galaxies such as JWST, Nancy Grace Roman Space Telescope (hereafter Roman), and Galaxy Reionization EXplorer and PLanetary Universe Spectrometer (GREX-PLUS). By taking advantage of the sensitivity of infrared observations in space, these missions are expected to detect galaxies at z > 10. In this section, we will discuss future prospects of these space missions based on our result of the z ∼ 12 − 16 galaxy search.
We here consider three missions, JWST, Roman, and GREX-PLUS, whose survey parameters are summarized in Table 4. Although Euclid has a remarkable capability to conduct wide-field surveys, it has a limited wavelength coverage to the H-band and can only select sources up to z ∼ 10 (J-dropout). We detail the three missions and survey plans below.
JWST is NASA's infrared space telescope that was launched on 2021 December 25th. Thanks to its large 6.5 m-diameter mirror, the sensitivity of JWST in the infrared is much higher than previous and current observational facilities. NIRCam is a near-infrared camera at 0.6 − 5.0 µm whose FoV is roughly 2 × 2.2 × 2.2 arcmin 2 (Rieke et al. 2005). In this paper we consider the following six surveys using NIRCam; the JWST Ad- (Jansen & Windhorst 2018).
Roman is NASA's optical to near-infrared space telescope whose launch is targeted around mid-2020s. Although the size of the mirror is comparable to that of HST, Roman is expected to conduct wide-field surveys in the near-infrared by taking advantage of the wide FoV of its camera (0.28 deg 2 ). The high latitude survey (HLS) will take images in the Y106, J129, H158, and F184bands over the ∼ 2000 deg 2 sky reaching ∼ 26.7 mag in F184. Two additional survey concepts are potentially possible; the Roman ultra-deep field survey (hereafter UltraDeep; Koekemoer et al. 2019), which will take very deep images (reaching ∼ 30 mag in the bluer filters, and 29.6 and 28.6 mag in F184 and K213, respectively) in a small area ( 1 deg 2 ), and the Roman cosmic dawn survey (hereafter Deep; see also Rhoads et al. 2018), which will conduct a relatively wide and deep survey (∼ 20 deg 2 , 27.5 and 27.2 mag in F184 and K213). 4 Both of these two possible surveys would cover a wavelength range up to 2.3 µm (the K213 filter), allowing us to select z ∼ 15 galaxies with the F184-dropout selection.
GREX-PLUS is a new 1.2-m class, cryogenic, widefield infrared space telescope mission concept proposed to ISAS/JAXA for its launch around mid-2030s. GREX-PLUS is planned to have a wide-field camera that will efficiently take wide and deep images at 2 − 10 µm, allowing us to select galaxies at z ∼ 10 − 17. Four surveys with different depths and areas (Ultra-Deep, Deep, Medium, and Wide) are now being planned.
For the surveys by these three missions, we calculate the expected number of detected galaxies at z ∼ 13, 15, and 17. For simplicity, we assume the survey volume of ∆z = 1 and 100% completeness. We calculate the number of galaxies based on two cases of the rest-frame UV luminosity functions (case A and case B). The one is the double power-law function with a redshift evolution suggested in Bowler et al. (2020, case A), which is consistent with the number density of our z ∼ 13 galaxies. The other is the Schechter function with a density evolution, whose parameters are M * UV = −20.5 mag, φ * = 44.7 × 10 −0.6(1+z) Mpc −3 , and α = −2.3 (case B).
15 − 0.5 1 − 0 0 − 0 CEERS 29.0/29.2 0.027 3 − 0 0.5 − 0 0 − 0 COSMOS-Web -/28.1 * 0.6 19 − 0.5 2 − 0 0 − 0 PRIMER ≤27.8/≤27.7 † 0.111 5 − 0 0.5 − 0 0 − 0 NGDEEP 30.6/30.7 0.0027 2 − 0 1 − 0 0 − 0 PANORAMIC 28.3/28.3 0.4 17 − 0.5 2 − 0 0 − 0 Roman UltraDeep 29.6/28.6 1 250 − 7 § 5 − 0 · · · Deep 27.5/27.2 20 270 − 3 § 9 − 0 · · · HLS 26.7/- 2000 8441 − 33 § · · · · · · GREX-PLUS UltraDeep 27.7 1 18 − 0 1 − 0 0 − 0 Deep 27.0 40 262 − 2 17 − 0 1 − 0 Medium 26.0 200 300 − 0 17 − 0 1 − 0 Wide 24.5 2000 322 − 0 14 − 0 0.5 − 0
Note-(1) Telescope.
(2) Planned survey.
(3) 5σ depth in the AB magnitude in the rest-frame UV band. In JWST, we quote depths of the F200W and F227W bands for the z ∼ 13 and z ∼ 15 − 17 galaxy selections, respectively. In Roman, depths of the F184 and K213 bands are quoted for the z ∼ 13 and z ∼ 15 galaxy selections, respectively. In GREX-PLUS, we quote depths in the F232 band for the z ∼ 13 − 17 galaxy selection. These depths are for point sources for simplicity and the actual depths for high redshift galaxies would be slightly shallower, except in PRIMER and NGDEEP, where resolved sources with typical sizes of high redshift galaxies are assumed. (4) Survey area in deg 2 . (5)-(7) Expected number of galaxies at z ∼ 13, 15, and 17 identified in the survey assuming ∆z = 1. Two values indicate the numbers in case A and case B (see text for details). Note that it may be difficult to select z ∼ 17 sources with Roman due to the lack of observational bands redder than the K213 band. Because the current plan of the Roman HLS does not include the K213 imaging, we do not calculate the expected number of z ∼ 15 galaxies. * The COSMOS-Web survey will take NIRCam F115W, F150W, F277W, and F444W images. We use the depth of the F277W band for the z ∼ 13 − 17 galaxy selections. † PRIMER is a "wedding cake" survey that is composed of several surveys with different depths and areas (27.8/27.7 mag for 400 arcmin 2 , 28.3/28.3 mag for 300 arcmin 2 , and 28.8/28.8 mag for 33 arcmin 2 ). § Although the expected number in the Roman HLS is larger than the other two Roman surveys, these three surveys will identify galaxies in different luminosities, and are complementary to each other. Please see Figure 11 for luminosity ranges that each survey covers.
We assume a relatively rapid decrease of the φ * parameter compared to the cosmic SFR density evolution at z > 10 (∝ 10 −0.5(1+z) , Harikane et al. 2021b), as a conservative estimate. These parameters are comparable to measurements for the z ∼ 10 UV luminosity function in Oesch et al. (2018). These two cases mostly cover the range of model predictions at z ∼ 13 as we show later in Figure 11. Given that these two cases are somewhat consistent with the model predictions and observations at z ∼ 10 and 13, we extrapolate these calculations to z ∼ 15 and 17 for reference. We integrate the luminosity functions down to the depths of detection bands pre-sented in Table 4, and calculate the number of detected galaxies at each redshift for each survey. Note that the 6σ depth is used for the z ∼ 15 galaxies identified with Roman to reduce the false positive rate in the one-band (K213) detection, while the 5σ depth is adopted for the others because multiple bands can be used. Figures 11, 12, and 13 show the expected UV luminosity functions in case A and case B and the depth and volume of each survey at z ∼ 13, 15, and 17, respectively, with predictions from models at z ∼ 13 and 15. Table 4 summarizes the expected number of detected galaxies in each survey. JWST will identify galaxies up . Predicted UV luminosity functions and depths and volumes of upcoming surveys. The red solid and dashed lines are the double power-law and Schechter functions (case A and case B in the text), respectively, whose parameters are determined by the extrapolation from lower redshifts (see text for details). The black, blue (left) and green (right) lines indicate expected depths and volumes of upcoming surveys with JWST, Roman, and GREX-PLUS, respectively. The gray lines show predictions from models (solid: Dayal et al. 2014, dotted: Yung et al. 2019, dot-dashed: Behroozi et al. 2019, dashed: Mason et al. 2015, same as the right panel of Figure 9). to z ∼ 15 in case A. In addition, JWST can conduct deep photometric and spectroscopic observations for relatively bright z > 10 galaxies identified in surveys with JWST and other telescopes, which will allow us to investigate physical properties (e.g., systemic redshift, stellar age, metallicity) in detail (e.g., Roberts-Borsani et al. 2021b). Roman will detect galaxies up to z ∼ 15 in case A, and identify galaxies at z ∼ 13 even in case B thanks to the wide survey areas. GREX-PLUS may be able to push the redshift frontier to z ∼ 17 in case A. The widearea surveys with Roman and GREX-PLUS can identify luminous z > 10 galaxies with 27 mag. These galaxies are bright enough to be followed up by spectroscopically with ALMA and JWST within a reasonable amount of observing time, to investigate the physical conditions of these galaxies in the early universe. didates, HD1 and HD2 ( Figure 2). SEDs of these candidates show a sharp discontinuity between H and K s (K)-bands, non-detections in the grizyJbands, and a flat continuum up to the [4.5]-band, all of which are consistent with a z ∼ 12 − 13 galaxy. Photometric redshift analyses based on these SEDs indicate that the most likely redshifts are z > 12 for both HD1 and HD2.
2. We calculate the number density of our galaxy candidates whose mean redshift is z ∼ 13 (Figure 7). The calculated number density at z ∼ 13 is comparable to that of bright galaxies at z ∼ 10 and consistent with the double power-law luminosity function extrapolated to z = 13 assuming the redshift evolution in Bowler et al. (2020). These results support little evolution of the abundance of bright galaxies to z ∼ 13 as suggested by previous studies at z ∼ 4 − 10. Comparisons with theoretical and empirical models show that these models underestimate the number densities of bright galaxies at z ∼ 10 − 13, although the uncertainties in observations are large (Figure 9). The inferred cosmic SFR density is consistent with the rapid decrease at z > 10 with ∝ 10 −0.5(1+z) suggested by Harikane et al. (2021b) (Figure 10).
3. We conducted ALMA follow-up observations targeting HD1. The obtained spectrum shows a ∼ 4σ tentative line-like feature at 237.8 GHz that is cospatial with the rest-frame UV emission, consistent with the [Oiii]88µm emission line at z = 13.27 ( Figure 5). Further spectroscopic efforts are needed to confirm the redshifts of HD1 and HD2.
Our results support the possibility that a number of bright galaxies exist at z > 10. If the UV luminosity function follows the double-power law function consistent with the number density of the bright galaxies at z ∼ 10−13, upcoming space missions such as JWST, Roman, and GREX-PLUS will detect more than ∼ 10000 galaxies at z ∼ 13 − 15 (Figures 11 and 12), and perhaps one to several at z ∼ 17 (Figure 13), allowing us to observe the first galaxy formation.
ACKNOWLEDGMENTS
We thank the anonymous referee for a careful reading and valuable comments that improved the clarity of the paper. We thank James Rhoads, Sangeeta Malhotra, Masami Ouchi, and the other members in the Roman cosmic dawn Science Investigation Team (SIT) for helpful discussions on the detectability of z > 10 galaxies with Roman. We are grateful to Caitlin Casey, James Dunlop, Steven Finkelstein, Christina Williams, and Rogier Windhorst for providing the expected depths in their JWST surveys, namely COSMOS-Web, PRIMER, NGDEEP, PANORAMIC, and the Webb Medium-Deep Field survey, respectively. We thank Takashiro Morishita for bringing an error in Figure 6 in the earlier manuscript to our attention. This work was partly supported by the joint research program of the Institute for Cosmic Ray Research (ICRR), University of Tokyo, JSPS KAKENHI Grant Numbers 17H06130, 19J01222, 20K22358, and 21K13953, the NAOJ ALMA Scientific Research Grant Codes 2018-09B and 2020-16B, and the Black Hole Initiative at Harvard University, which is funded by grants from the John Templeton Foundation and the Gordon and Betty Moore Foundation. T.H. was supported by Leading Initiative for Excellent Young Researchers, MEXT, Japan (HJH02007). P.D. and A. H. acknowledge support from the European Research Council's starting grant ERC StG-717001 ("DELPHI"). P.D. also acknowledges support from the NWO grant 016.VIDI.189.162 ("ODIN") and the European Commission's and University of Groningen's CO-FUND Rosalind Franklin program. A.Y. is supported by an appointment to the NASA Postdoctoral Program (NPP) at NASA Goddard Space Flight Center, administered by Oak Ridge Associated Universities under contract with NASA. F.P. acknowledges support from a Clay Fellowship administered by the Smithsonian Astrophysical Observatory.
This paper makes use of the following ALMA data: ADS/JAO. ALMA #2019.A.00015.S . ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST, and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO, and NAOJ.
This work is based on data products from observations made with ESO Telescopes at the La Silla Paranal Observatory under ESO programme ID 179.A-2005 and on data products produced by CALET and the Cambridge Astronomy Survey Unit on behalf of the UltraVISTA consortium
The HSC collaboration includes the astronomical communities of Japan and Taiwan, and Princeton University. The HSC instrumentation and software were
Figure 2 .
2Left: Optical-to-near-infrared SEDs of our H-dropout galaxy candidates, HD1 (top) and HD2 (bottom). The red symbols with error-bars are measured flux densities or the 2σ upper limits. The blue curve shows the best-fit model of an LBG at z > 12 in the SED fitting, and the gray curve shows a passive galaxy solution at z ∼ 4 (see Section 4). The upper panels show 10 × 10 images. The "[3.6] resid" and "[4.5] resid" are residual images after subtracting nearby objects with T-PHOT(Merlin et al. 2016). Middle: The same as the left panel but for the far-infrared-to-submm range. The curve shows the modified black-body function with the temperature of 50 K and the emissivity index of β dust = 2.0. Right: χ 2 value as a function of the redshift. The best-fit models are found at z > 12.
Figure 3 .
3Same as the left panels of Figure 2 but with fluxes plotted in linear scale to compare the observed fluxes with the models. The blue and grey open circles are fluxes of the models in each band.
Figure 5 .
5Top: ALMA spectrum showing the 4σ line-like feature at 237.8 GHz. This hints for the[Oiii]88µm line at z = 13.27. Bottom: Integrated intensity of the 4σ feature in HD1 overlaid on the VISTA Ks band image. This moment 0 map is made with the CASA task immoments, by integrating over 700 km s −1 covering most of the velocity range of the line emission (> 1.5×FWHM). The solid (dotted) lines show +1.5, +2.5, and +3.5σ (−1.5, −2.5, and −3.5σ) contours. The emission is co-spatial with the rest-frame UV emission in the Ks-band image. ment 0 map shown in the bottom panel. Although there are some other line-like features (e.g., 246.3 GHz), the feature at 237.8 GHz has the highest signal-to-noise ratio among the ones in the frequencies free from severe atmospheric O 3 absorption. If this feature is the [Oiii]88µm emission line, the redshift of HD1 is z = 13.27, in good agreement with the photometric redshift estimate. A relatively broad line width (∼ 400 km s −1 in a full width at half maximum; FWHM) is in fact comparable to similarly bright LBGs at z ∼ 6(Harikane et al. 2020). The emission feature is co-spatial with the rest-frame UV emission in the K s -band image (the bottom panel ofFigure 5). The integrated line flux is 0.24 ± 0.06 Jy km s −1 or (1.9 ± 0.5) × 10 −18 erg s −1 cm −2 , and the line luminosity is L [OIII] 3.3 × 10 8 L if z = 13.27 is assumed.
Figure 6 .
6Completeness estimated in our Monte Carlo simulations. The top and bottom panels are results for the COS-MOS and SXDS fields, respectively. The red, orange, yellow, green, blue, and purple lines show the completeness for Ks(K) = 23.0, 23.4, 23.8, 24.2, 24.6, and 25.0 mag sources, respectively.
Figure 7 .
7−3.0 ) × 10 −8 Mpc −3 mag −1 at M UV = −23.5 mag. This number density is comparable to that of bright Rest-frame UV luminosity functions at z ∼ 13 and z ∼ 10. The red circle shows the number density of our z ∼ 13 galaxy candidates. The black symbols and the gray shaded region are measurements at z ∼ 10 from the literature (diamond: McLeod et al. 2016, square: Oesch et al. 2018, pentagon: Morishita et al. 2018, circle: Bowler et al. 2020, shade: Finkelstein et al. 2021a). The green star is the number density of GN-z11 (see text). Note that the data point of Bowler et al. (2020) (GN-z11) is horizontally (vertically) offset by −0.2 mag (+0.03 dex) for clarity. The gray dashed line is the Schechter function fit
Figure 10 .
10Evolution of the cosmic SFR density. The red circle is our result at z ∼ 13 estimated by integrating the double power-law luminosity function down to −17 mag. The black circles are observed cosmic SFR densities taken from Madau & Dickinson (2014), Finkelstein et al. (2015), McLeod et al. (2016), and Bouwens et al. (2020). The blue and gray dashed curves represent the fits in Harikane et al. (2021b, their Equation (60)) and Madau & Dickinson (2014, extrapolated at z > 8), respectively. All results are converted to use the Salpeter (1955) IMF (Equation
vanced Deep Extragalactic Survey (JADES; Guaranteed Time Observation (GTO) program, Eisenstein et al. 2017a,b), the Cosmic Evolution Early Release Science (CEERS) survey (ERS-1345; Finkelstein et al. 2017), the COSMOS-Web survey (GO-1727; Kartaltepe et al. 2021), Public Release IMaging for Extragalactic Research (PRIMER; GO-1837; Dunlop et al. 2021), the Next Generation Deep Extragalactic Exploratory Public (NGDEEP) survey (GO-2079; Finkelstein et al. 2021b), and the Parallel wide-Area Nircam Observations to Reveal And Measure the Invisible Cosmos (PANORAMIC) survey (GO-2514; Williams et al. 2021). These surveys will take deep NIRCam imaging data at ∼ 1 − 5 µm in ∼ 10 − 2000 arcmin 2 survey fields. Other high redshift galaxy surveys are also planned in Cycle 1. For example, Through the Looking GLASS (ERS-1324, Treu et al. 2017) and the Ultra-deep NIRCam and NIRSpec Observations Before the Epoch of Reionization (UNCOVER; GO-2561; Labbe et al. 2021) will image the gravita-tional lensing cluster Abell 2744 with NIRCam. The Webb Medium-Deep Field survey (GTO-1176; Windhorst et al. 2017) will use 110 hours to observe 13 medium-deep (28.4 − 29.4 mag) fields, including the James Webb Space Telescope North Ecliptic Pole Timedomain Field
Figure 11
11Figure 11. Predicted UV luminosity functions and depths and volumes of upcoming surveys. The red solid and dashed lines are the double power-law and Schechter functions (case A and case B in the text), respectively, whose parameters are determined by the extrapolation from lower redshifts (see text for details). The black, blue (left) and green (right) lines indicate expected depths and volumes of upcoming surveys with JWST, Roman, and GREX-PLUS, respectively. The gray lines show predictions from models (solid: Dayal et al. 2014, 2019, dotted: Yung et al. 2019, 2020, dot-dashed: Behroozi et al. 2019, 2020, dashed: Mason et al. 2015, same as the right panel of Figure 9).
Figure 12 .
12Same asFigure 11but for z ∼ 15. The 6σ depth is used for Roman to reduce the false positive rate in the one-band (K213) detection.
Figure 13 .
13paper we have presented our search for Hdropout LBGs at z ∼ 12 − 16. We have used the multi-wavelength deep imaging data in the COSMOS and SXDS fields including Subaru/HSC grizy, VISTA JHK s , UKIRT JHK, and Spitzer/IRAC [3.6][4.5] images. Our major findings are summarized below:1. After the careful screening of foreground interlopers, we have identified two z ∼ 12 − 13 galaxy can-Same asFigure 11but for z ∼ 17. We only plot the survey depths and volumes of JWST and GREX-PLUS, which cover the wavelength redder than 2.3 µm (the Lyα break at z ∼ 17).
Table 1 .
15σ Limiting Magnitude of Imaging Data Used in This Study JHKs bands in the UD-COSMOS field represent limiting magnitudes in the ultra-deep and deep stripes.Subaru
VISTA/UKIRT
Spitzer
Table 2 .
2Photometry of Our H-Dropout Galaxy CandidatesName
R.A.
Decl.
Subaru
VISTA/UKIRT
Spitzer
g
r
i
z
y
J
H
Ks/K
[3.6]
[4.5]
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
HD1 10:01:51.31 02:32:50.0 < 27 < 32 < 46 < 63 < 157
< 107
< 145
510 ± 93
531 ± 108 494 ± 136
HD2 02:18:52.44 -05:08:36.1 < 25 < 33 < 46 < 68 < 133
< 95 296 ± 76 821 ± 63
888 ± 88 1252 ± 132
Note-(1) Name. (2) Right ascension. (3) Declination. (4)-(13) Flux densities in nJy or 2σ upper limits.
Table 3 .
3Physical Properties of Our H-Dropout Candidates SFR estimated from the UV magnitude by using Equation (7) in units of M yr −1 . (5) & (6) Stellar mass and dust attenuation suggested by the SED fitting in units of M and mag, respectively. See Section 4 for details. † z = 13.27 is suggested by the ALMA observations for HD1 (see Section 3), consistent with the photometric redshift estimate within 1σ. The absolute UV magnitude and SFR in this table are calculated based on the assumption of z = 13.27.Name
z phot
MUV
SF RUV
logM *
AV
(1)
(2)
(3)
(4)
(5)
(6)
HD1 15.2
+1.2(1.6)
−2.1(2.7)
†
−23.3 †
110 †
∼ 9 − 11
< 0.08
HD2
12.3
+0.4(0.7)
−0.3(0.7)
−23.8
170
∼ 9.8 − 11
0.8
Note-(1) Name. (2) The best photometric redshift with
1σ (2σ) errors. (3) Absolute UV magnitude in units of mag.
(4)
Table 4 .
4Expected Number of z 13 Galaxies Identified in Upcoming SurveysTelescope
Survey
m5σ
Asurvey N (z ∼ 13) N (z ∼ 15) N (z ∼ 17)
(1)
(2)
(3)
(4)
(5)
(6)
(7)
JWST
JADES/Deep
30.6/30.2
0.013
11 − 0.5
1 − 0
0 − 0
JADES/Medium
29.7/29.3
0.053
H-dropouts sources searched in this work are different from previously-studied dusty "H-dropouts" at z ∼ 3 − 6 (e.g.,Wang et al. 2019).
This number density is lower than that inBowler et al. (2020), because we adopt the number density estimate ofOesch et al. (2018). The UV magnitude of GN-z11 is estimated to be −22.1 and −21.6 mag inOesch et al. (2016) andOesch et al. (2018), respectively. We adopt their average value, −21.85 ± 0.25 mag, which is consistent with the recent estimate byTacchella et al. (2021).
The sensitivities of the survey are calculated based on the tables in the following website: https://roman.gsfc.nasa.gov/science/anticipated performance tables.html
developed by the National Astronomical Observatory of Japan (NAOJ), the Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU), the University of Tokyo, the High Energy Accelerator Research Organization (KEK), the Academia Sinica Institute for Astronomy and Astrophysics in Taiwan (ASIAA), and Princeton University. Funding was contributed by the FIRST program from the Japanese Cabinet Office, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), the Japan Society for the Promotion of Science (JSPS), Japan Science and Technology Agency (JST), the Toray Science Foundation, NAOJ, Kavli IPMU, KEK, ASIAA, and Princeton University.This paper makes use of software developed for the Large Synoptic Survey Telescope. We thank the LSST Project for making their code available as free software at http://dm.lsst.org This paper is based on data collected at the Subaru Telescope and retrieved from the HSC data archive system, which is operated by Subaru Telescope and Astronomy Data Center (ADC) at NAOJ. Data analysis was in part carried out with the cooperation of Center for Computational Astrophysics (CfCA), NAOJ.The Pan-STARRS1 Surveys (PS1) and the PS1 public science archive have been made possible through contributions by 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, the 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 grant No. AST-1238877, the University of Maryland, Eotvos Lorand University (ELTE), the Los Alamos National Laboratory, and the Gordon and Betty Moore Foundation.Software: BEAGLE(Chevallard & Charlot 2016), PAN-HIT(Mawatari et al. 2020a), SExtractor(Bertin & Arnouts 1996), T-PHOT(Merlin et al. 2016)
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| []
|
[
"On the DMT of TDD-SIMO Systems with Channel-Dependent Reverse Channel Training",
"On the DMT of TDD-SIMO Systems with Channel-Dependent Reverse Channel Training"
]
| [
"B N Bharath ",
"Senior Member, IEEEChandra R Murthy "
]
| []
| []
| This paper investigates the Diversity-Multiplexing gain Trade-off (DMT) of a training based reciprocal Single Input Multiple Output (SIMO) system, with (i) perfect Channel State Information (CSI) at the Receiver (CSIR) and noisy CSI at the Transmitter (CSIT), and (ii) noisy CSIR and noisy CSIT. In both the cases, the CSIT is acquired through Reverse Channel Training (RCT), i.e., by sending a training sequence from the receiver to the transmitter. A channel-dependent fixed-power training scheme is proposed for acquiring CSIT, along with a forward-link data transmit power control scheme. With perfect CSIR, the proposed scheme is shown to achieve a diversity order that is quadratically increasing with the number of receive antennas. This is in contrast with conventional orthogonal RCT schemes, where the diversity order is known to saturate as the number of receive antennas is increased, for a given channel coherence time. Moreover, the proposed scheme can achieve a larger DMT compared to the orthogonal training scheme. With noisy CSIR and noisy CSIT, a three-way training scheme is proposed and its DMT performance is analyzed. It is shown that nearly the same diversity order is achievable as in the perfect CSIR case. The time-overhead in the training schemes is explicitly accounted for in this work, and the results show that the proposed channel-dependent RCT and data power control schemes offer a significant improvement in terms of the DMT, compared to channel-agnostic orthogonal RCT schemes. The outage performance of the proposed scheme is illustrated through Monte-Carlo simulations.Index Terms-Diversity-multiplexing gain tradeoff, MMSE channel estimation, training sequence. | 10.1109/tcomm.2012.082712.110820 | [
"https://arxiv.org/pdf/1105.2375v2.pdf"
]
| 15,294,327 | 1105.2375 | 7fc8c176f702caef529ff42bac1ecc0b28cdd293 |
On the DMT of TDD-SIMO Systems with Channel-Dependent Reverse Channel Training
2 Nov 2012
B N Bharath
Senior Member, IEEEChandra R Murthy
On the DMT of TDD-SIMO Systems with Channel-Dependent Reverse Channel Training
2 Nov 20121Index Terms-Diversity-multiplexing gain tradeoffMMSE channel estimationtraining sequence
This paper investigates the Diversity-Multiplexing gain Trade-off (DMT) of a training based reciprocal Single Input Multiple Output (SIMO) system, with (i) perfect Channel State Information (CSI) at the Receiver (CSIR) and noisy CSI at the Transmitter (CSIT), and (ii) noisy CSIR and noisy CSIT. In both the cases, the CSIT is acquired through Reverse Channel Training (RCT), i.e., by sending a training sequence from the receiver to the transmitter. A channel-dependent fixed-power training scheme is proposed for acquiring CSIT, along with a forward-link data transmit power control scheme. With perfect CSIR, the proposed scheme is shown to achieve a diversity order that is quadratically increasing with the number of receive antennas. This is in contrast with conventional orthogonal RCT schemes, where the diversity order is known to saturate as the number of receive antennas is increased, for a given channel coherence time. Moreover, the proposed scheme can achieve a larger DMT compared to the orthogonal training scheme. With noisy CSIR and noisy CSIT, a three-way training scheme is proposed and its DMT performance is analyzed. It is shown that nearly the same diversity order is achievable as in the perfect CSIR case. The time-overhead in the training schemes is explicitly accounted for in this work, and the results show that the proposed channel-dependent RCT and data power control schemes offer a significant improvement in terms of the DMT, compared to channel-agnostic orthogonal RCT schemes. The outage performance of the proposed scheme is illustrated through Monte-Carlo simulations.Index Terms-Diversity-multiplexing gain tradeoff, MMSE channel estimation, training sequence.
I. INTRODUCTION
Reliability and system throughput are two fundamental parameters of interest in any wireless communication system, and the inherent tradeoff between the two at high SNR was elegantly captured by the Diversity Multiplexing gain Tradeoff (DMT) proposed in the seminal work of Zheng and Tse [2]. It is known that a significant improvement in the outage performance can be obtained if the Channel State Information (CSI) at the receiver (CSIR) and the transmitter (CSIT) are perfect [3], [4], while [2] considered perfect CSIR and no CSIT.
In a Time Division Duplex (TDD) system, CSI could be estimated at the transmitter and receiver by sending a known training sequence in the forward and reverse-link directions, respectively. This has two consequences. First, the estimation error results in incorrect data rate or power adaptation at the transmitter, in turn leading to higher outage rate. Second, training incurs a time overhead, which could be non-trivial when the training occupies a significant fraction of the channel coherence time, as it affects the pre-log term in the achievable data rate [5]. This paper therefore focuses on the important problem of analytically comparing the DMT performance of different channel estimation techniques and identifying training signals and data power control schemes that result in a good performance in terms of the achievable DMT. We start with a brief survey of related literature.
The impact of imperfect CSIT on the DMT of a multiple antenna system has been a popular area of research, and it is known that even with imperfect CSIR and CSIT, a significant improvement in DMT can be obtained, compared to the no-CSIT case (see, for example, [6]- [8]). The effect of imperfect CSIR on the DMT of a MIMO system was first studied in [9]. The DMT analysis of a multiple antenna system with perfect CSIR and when the CSIT is modeled as the CSI plus Gaussian noise whose variance decreases with training SNR was investigated in [10]- [12]. In a TDD setup, the achievable DMT improvement using power control based on noisy CSIT was shown in [12]- [14]. Other works that study the DMT performance with quantized feedback of CSI and/or target data rate control based on noisy CSIT include [6], [7], [11], [15]- [18]. In [17], [19], the DMT of two-way and multiround training schemes in a TDD system was derived. In these studies, the channel feedback signal on the reverse link is chosen to satisfy an average power constraint, rather than an instantaneous power constraint.
Most of the aforementioned studies of the DMT with imperfect CSI typically ignore the training duration overhead. Hence, they are primarily applicable to slowly varying channels, where the time overhead in training occupies an insignificant fraction of the channel coherence time. An exception is [13], where, taking the training overhead into account, the authors concluded that for nonzero multiplexing gain g m , the diversity order saturates as r increases, where r is the number of receive antennas. Hence, for fast varying channels, the authors suggest turning off receive antennas in order to achieve higher multiplexing gains. It is important to account for the training duration overhead in deriving the achievable DMT, because, as the SNR goes to infinity, although the estimation error goes to zero, the training duration overhead remains fixed and has a direct impact on the DMT. Also, by modeling the CSIT as the sum of the true CSI and an additive error, most of the past studies implicitly assume that a channel-agnostic orthogonal training signal is employed for channel estimation.
When the training signal is channel-dependent, the imperfect CSI can no longer be modeled as the sum of the true CSI and an additive noise. Due to this, the existing results cannot be directly extended to analyze the DMT performance of channeldependent training schemes.
When the channel is reciprocal and block-fading, e.g., in a TDD system, the receiver could exploit its channel knowledge (acquired through an initial forward-link training phase) in designing its reverse-training sequence, not only to reduce the channel estimation error at the transmitter, but also to reduce the required training duration overhead. Hence, the goals of this paper are two-fold: (a) to analyze the DMT performance of a channel dependent training scheme for acquiring CSIT and an associated power control mechanism for data transmission; and (b) to contrast the DMT performance of the proposed training and power control schemes with that achieved by conventional channel agnostic training schemes. Our study focuses on point-to-point Single Input Multiple Output (SIMO) systems. This is of practical importance, since it applies, for example, to the uplink of wireless networks where the base station has multiple antennas, the mobile users have a single antenna, and orthogonal access is used (e.g., OFDM/TDMA) as in WLANs and 4G/LTE systems. The channel dependent training sequence employed here was first proposed by us in [20] and [1] in a MIMO and SIMO context, respectively, and was independently explored in [21], although not in a DMT context.
In this paper, for analytical simplicity and clarity of presentation, we start by assuming that perfect CSI is available at the receiver, as in [10]- [12]. We propose a fixed-power RCT sequence, using which, the CSI can be estimated at the transmitter using a minimum duration of only one symbol, i.e., with a factor of r reduction in training duration compared to orthogonal RCT. For data transmission, we propose a modified truncated channel inversion-type power control scheme based on the noisy CSIT. For this system, we show that a diversity of d(g m ) = r s + 1 − gmLc Lc−LB,τ is achievable. Here, g m is the multiplexing gain, L c is the coherence time, L B,τ ≥ 1 is the reverse training duration, and 1 ≤ s < r is a parameter in the data power control scheme. (See Section III.)
Next, we consider the more practical case where noisy CSIR is acquired via a forward link training sequence, and propose a three-way training scheme followed by data transmission. We show that a DMT of d(g m ) = r(s + 1 − gmLc Lc−β ) is achievable, where β ≥ 3 is the total training overhead from all three training phases, which is again an improvement over conventional orthogonal training schemes. For example, a nonzero diversity order can be achieved with Lc−(r+2) Lc ≤ g m < Lc−3 Lc , which is not possible with orthogonal training schemes without switching off receive antennas and incurring an associated reduction in diversity order. (See Section IV.)
Note that although the perfect CSIR case is a special case of the three-way training scheme with infinite forward-link training power, we briefly present the perfect CSIR case also, as it provides insights into the impact of the reversetraining and data power control mechanisms on the DMT. Moreover, it is useful as an upper bound on the performance with imperfect CSIR. Also, we assume that power control is employed only at the transmitter and focus on fixed-power RCT in the sequel. Using power controlled RCT significantly changes the problem; we analyze this case in our follow up work [22].
An important implication of our work is that it shows that by exploiting the receiver's knowledge of the CSI in designing the reverse channel training (RCT) sequence and using our proposed data power control scheme, one can achieve a higher diversity order than conventional RCT for all values of g m . Somewhat surprisingly, we also demonstrate that although the DMT analysis corresponds to taking the SNR to infinity, it can nonetheless be used to discriminate between different training schemes both in terms of the estimation error as well as the training overhead. At finite SNR, this translates to an improvement in the outage probability performance and the achievable data rate, as will be illustrated through Monte-Carlo simulations in Section VI. We use the following notation. Bold face letters are used for vectors and normal font letters are used for scalars. We use E(·) to denote the expected value of (·). We use h 2 to represent the ℓ 2 norm of h. The transpose conjugate, absolute value, and real part are denoted by (·) H , | · | and ℜ{·}, respectively. We write f (P )
.
= 1 P k to mean − limP →∞ log f (P ) logP = k. Similarly, we define f (P ) 1 P k to mean − limP →∞ log f (P ) logP ≥ k.
II. SYSTEM MODEL
The system model consists of two communicating nodes, node A with a single antenna and node B with r antennas, with node A attempting to send data to node B over a wireless channel. The forward channel from node A to node B, denoted by h ∈ C r×1 , is modeled as a Rayleigh flat fading channel whose entries are i.i.d. Circularly Symmetric Complex Gaussian (CSCG) random variables with zero mean and unit variance, i.e., CN (0, 1). The channel is assumed to be blockfading, i.e., it remains constant for a duration of the coherence time L c , and evolve in an i.i.d. fashion across coherence times. We assume a TDD system with perfect reciprocity, and hence, taking the complex conjugate of the received signal at node A, the reverse link channel is h H . We let h = σv, where σ = h 2 is the singular value and v h h 2 is the singular vector of h. Since our goal is to study the achievable DMT performance with channel training, we first explain the twoway training protocol used for acquiring CSI at node B and node A. Later, in Sec. IV, an additional phase of forward link training is introduced, which is not presented here for simplicity.
1) Phase I (Forward-link training):
Here, the training sequence x A,τ = P L A,τ1 is transmitted from node A to node B, where L A,τ1 denotes the training duration andP is the training power 1 . Throughout this paper, we useP as the average power constraint during both training and data 1 Strictly speaking, x A,τ = √P is transmitted repeatedly L A,τ 1 times.
Mathematically, this is equivalent to using x A,τ = P L A,τ 1 for a duration of one unit.
transmission. The corresponding received training signal is given by,
y B,τ = h P L A,τ1 + w B,τ .(1)
The entries of w B,τ ∈ C r×1 are assumed to be distributed as i.i.d. CN (0, 1). From the received training signal y B,τ , node B computes an MMSE estimate of h, denotedĥ. The error in the estimate, denotedh h −ĥ, has i.i.d.
CN 0, 1/(1 +P L A,τ1 ) distributed entries.
In a TDD-SIMO system, node A only requires knowledge of σ to perform power control, which in turn improves the diversity order compared to the no-CSIT case. Therefore, in phase II, we estimate only σ at node A, using a channel dependent training sequence.
2) Phase II (Reverse-link training): Since node B has an estimate (say,v ĥ ĥ 2 ) of the channel, in this phase, it exploits its CSI to transmit the following training sequence [1], [20]:
x B,τ = P L B,τv ,(2)
where L B,τ is the reverse training duration. Using the corresponding received signal,
y A,τ h H x B,τ + w A,τ , where the AWGN w A,τ ∈ C is distributed as CN (0, 1)
, node A computes an estimate of the singular value as follows:
σ ℜ{y A,τ } P L B,τ = σℜ{v Hv } +w A,τ ,(3)wherew A,τ ℜ{wA,τ } √P LB,τ
. Note that the estimateσ could be negative; this is taken care of by the power control proposed in Sec. III, which usesσ only when it is greater than a positive threshold. Since a low or negativeσ is likely to be inaccurate, the thresholding technique helps to avoid the poor DMT performance due to such estimates. The RCT scheme employed above is different from existing channel agnostic methods in that the minimum training length in the proposed scheme is only 1 symbol. This represents a factor of r reduction compared to orthogonal RCT schemes, where the minimum training length increases linearly with r, and this difference in overhead could be significant when L c is small. Also, ifv is error-free, it is the optimal beamforming vector for estimating σ at node A.
3) Multiplexing Gain and Diversity Order:
We recall the definitions of the multiplexing gain, g m , and the diversity order d from [2]:
g m lim P →∞ RP logP , d − lim P →∞ log P out logP ,(4)
where RP is the target data rate when the average data power constraint isP , and P out is the corresponding outage probability, i.e., the probability that RP exceeds the channel capacity. In this work, the target data rate RP = g m logP is fixed and is independent of the CSIT; the extension of our proposed methods to joint rate and power adaptation is relegated to future work. The rate of data transmission RP is increased withP by increasing the cardinality of the signal set, keeping the symbol duration fixed. We ignore the effect of spectral leakage, and assume that the signal bandwidth remains fixed asP goes to infinity. Also, we use outage probability as a proxy for the probability of error at high SNR with finitelength codes; this is because the probability of error can be made to decrease as fast as the outage probability using finitelength approximately universal codes [23], [24].
In the next section, we assume perfect CSI at node B and derive the achievable DMT performance of our proposed training and data transmission schemes.
III. DMT ANALYSIS WITH PERFECT CSIR
When the CSIR is perfect, we havev = v, and in this case, it is easy to see that (2) is optimal for estimating σ given a power constraintP on the training signal. This is because, in general, the training signal can be expressed as the linear combination x B,τ = δv + βv ⊥ , where v ⊥ is orthogonal to v and δ and β are some constants. Then, the received training signal at node A is y A,τ = δσ + w A,τ , i.e., the power in v ⊥ does not help in estimating σ. From (3), an unbiased estimator of the singular value at node A is given bŷ
σ = σ +w A,τ .(5)
Note that since the channel is assumed to be Rayleigh fading, σ 2 is chi-square distributed with 2r degrees of freedom. Also, we employ this estimator primarily because we are interested in deriving the achievable DMT performance, and for this purpose, this simple unbiased estimator is sufficient.
A. Power-Controlled Data Transmission from Node A to Node B
Given the CSITσ in (5), node A uses a power P(σ) in the forward link data transmission phase, to avoid outages while satisfying the average data power constraintP . The corresponding data signal received at node B is given by,
y B,d = P(σ)hx A,d + w B,d ,(6)
where x A,d ∼ CN (0, 1), and with appropriate power normalization, the entries of the AWGN w B,d ∈ C r×1 are assumed to be i.i.d. CN (0, 1). Also, P(σ) is chosen independent of x A,d such that E{P(σ)} =P , where the expectation is with respect toσ given in (5), taken across all coherence blocks.
Since E{|x A,d | 2 } = 1 within a block, this ensures that the average data power constraint at node A is satisfied. We now present the data power control function P(σ) considered in this paper. Our proposed power control function is motivated as follows. The capacity of a fading channel with mismatched CSIT and CSIR is not known in closed form [25]. Since the outage probability computation requires a closed form expression for the capacity, we consider a genie-aided receiver as in [26], where node B is assumed to know P(σ). This is schematically illustrated in Fig. 1. Then, the achievable data rate conditioned on the knowledge of P(σ)h is given by [25] C L c − L B,τ L c log 1 + σ 2 P(σ) .
An outage occurs when RP , the target data rate, exceeds C. Its probability is upper bounded by Note that the exact outage probability is obtained by minimizing the right hand side above over all P(σ) satisfying E{P(σ)} =P . Hence, using our proposed data power control scheme leads to an upper bound on the outage probability, which is sufficient for obtaining the achievable DMT performance. If the CSIT is perfect (i.e.,σ 2 = σ 2 ), it is shown in [3] that the power control that minimizes the outage probability is given by
P out Pr L c − L B,τ L c log(1 + σ 2 P(σ)) < RP .(8Φ(σ 2 ) exp RP Lc Lc−LB,τ − 1 σ 2 .(9)
Note that since RP = g m logP and E 1
σ 2 = 1 r−1 , Φ(σ 2 ) satisfies E{Φ(σ 2 )} ≤P for large enoughP , provided g m ≤ (L c − L B,τ )/L c .
With inaccurate CSIT, due to the estimation error inσ, the natural extension of using a transmission power of Φ(σ 2 ) could result in allocating insufficient power or more power than required, which could lead to suboptimal performance. Also, inverting the channel for all values ofσ results in an infinite average power since the Gaussian noise can make the estimateσ arbitrarily small with a non-zero probability. One solution is to use a transmit power of Φ(σ 2 ) whenσ > θ 0 and a zero power otherwise, where θ 0 is chosen such that E[Φ(σ 2 )1σ >θ0 ] =P . The drawback of this method is that it results in an outage probability of 1 whenσ ≤ θ 0 , leading to a zero diversity order. To overcome this problem, we choose the threshold θ 0 such that θ 0 → 0 asP → ∞. Moreover, whenσ ≤ θ 0 , we do not necessarily want to use zero power, since the small value ofσ could be due to the estimation error. This motivates the following modified power control:
P(σ) P lσ ≤ θP , κP × Φ(σ 2s )σ > θP ,(10)
where s ≥ 1 is a parameter, and we use θP 1 P n , n > 0, for mathematical tractability. The parameters n, κP and l > 0 are chosen such that E[P(σ)] =P . Although similar power control schemes have been employed in the literature with perfect CSIT [3] or orthogonal RCT [12], [13], [19], the form in (10) is new. Specifically, the power control scheme in [3], [12], [13] can be obtained from (10) by setting s = 1, θP = 0 and l = −∞; while that in [19] can be obtained by setting s = r, θP = 0 and l = −∞.
Power constraint: The description of the power control would be complete if the parameters n, κP and l can be chosen such that E[P(σ)] =P , which is the essence of the following Lemma.
Lemma 1: Let θP 1 √P . For 1 ≤ s < r, there exists a κP . = 1 P gm α −1 , where α Lc−LB,τ Lc , such that E[P(σ)] =P , if 0 ≤ l ≤ r + 1. Proof : See Appendix B.
Due to Lemma 1, in the rest of this paper, we consider θP = 1/ √P . Also, in Sec. IV, we show that a minor modification of the above data power control scheme can be employed even with imperfect CSIR. The next subsection presents the achievable DMT of the proposed training and power control schemes.
B. Achievable DMT Analysis
Theorem 1: Given r receive antennas and L B,τ training symbols being used per coherence interval L c to estimate the CSIT in a SIMO system with perfect CSIR and a genieaided receiver, an achievable diversity order as a function of multiplexing gain g m is given by
d(g m ) = r min{l, s + 1} − g m α ,(11)
where 0 ≤ l ≤ r + 1, 1 ≤ s < r, 0 ≤ g m < α, and α Lc−LB,τ Lc represents the fractional data transmit duration. Proof: See Appendix C.
Remark: From a DMT perspective, it is clear from Theorem 1 that s → r, l = r+1 is superior to s = 1, l = 2. On the other hand, whenσ < 1, Φ(σ 2r ) could be much greater than Φ(σ 2 ). Thus, in practical systems with a peak power per transmitted codeword constraint, s = 1, l = 2 could be preferable over s → r, l = r + 1. In the sequel, for convenience, we associate l = 2 with s = 1 and l = r + 1 with s → r, and drop the explicit dependence of the diversity order on l. Further remarks and discussions on the result obtained here are deferred to Sec. V.
IV. THREE WAY TRAINING
In this section, we consider the more practical scenario where training is performed in both directions. We show that with fixed power training, one can achieve nearly the same DMT as derived in Sec. III for the perfect CSIR case. Unlike in the previous section, the analysis presented here is exact, in the sense that it does not require the assumption of a genie aided receiver, and hence, the DMT derived here is indeed achievable in practice. The transmission protocol now consists of four phases, as shown in Table I. The CSIR and CSIT are obtained by transmitting a fixed power training sequence in both directions, as explained in Sec. II. However, even a small mismatch in the CSI knowledge at node A and node B can potentially lead to a large mismatch in their estimate of the data transmit power [13]. Thus, it is essential to train node B about node A's knowledge of P(σ). This leads to a third phase of training, which is an additional power-controlled forward-link training phase. First, in the following subsection, we explain the power control scheme that is employed here.
A. Power Control Scheme
The power control scheme we propose to employ in this section is as given by (10), due to the following. Letĥ denote the MMSE estimate of the channel at node B, and considerσ in (3). We havê
σ ℜ{y A,τ } P L B,τ = ℜ{ĥ Hv } + ℜ{h Hv } + ℜ{w A,τ } P L B,τ = ĥ 2 +w ef f ,(12)wherew ef f ℜ{h Hv } + ℜ{wA,τ } √P LB,τ
. Note thatĥ andh are independent Gaussian random variables 2 . Sincev is uniformly distributed on the unit sphere and is independent ofh, ℜ{h Hv } is Gaussian distributed. This implies that the effective noise,w ef f , is Gaussian distributed with E|w ef f | 2 . = 1 P and independent ofĥ. Therefore, the estimate of the singular value at node A is statistically similar to the estimate given by (5) in the perfect CSIR case. Thus, we use a similar power control, P(σ) in (10), whereσ is given by (12). Also, with a slight abuse of notation, α
Lc−LB,τ −LA,τ 1 −LA,τ 2 Lc
, where L A,τ2 is the training duration in the third phase of training (phase III), which is in the forward-link direction.
In this section, without loss of generality, we move the power scaling √P into the data symbol transmitted by node A, so that E{P(σ)} = 1 (see (13) below), where the expectation is taken with respect to the distribution ofσ in (12). Now, in the proof of Lemma 1, using the probability density function (pdf) of ĥ 2 in place of the pdf of σ, and noting that the effective noise variance . = 1/P , we get κP . = 1 P gm /α and the constraint 0 ≤ l ≤ r to satisfy E{P(σ)} = 1 at high SNR. In the next subsection, we explain the third round of training that alleviates the mismatch in the knowledge of the data transmit power.
B. Phase III (Power-Controlled Forward Link Training)
In this phase, node A transmits the training sequence: x A,τ2 = P L A,τ2 P(σ), where L A,τ2 is the training duration. The corresponding received training signal at node B is given by,
y B,τ2 = P L A,τ2 P(σ)h + w B,τ2 ,(13)
where w B,τ2 ∈ C r×1 is the AWGN with CN (0, 1) entries. The goal at node B is to estimate the composite channel p c P(σ)h. Dividing (13) by P L A,τ2 , we get
y B,τ2 y B,τ2 P L A,τ2 = p c + w B,τ2 P L A,τ2 .(14)
2ĥ → h asP → ∞. Moreover, ĥ 2 is a chi distributed random variable.
From (14), node B computes an MMSE estimate of p c , denoted byp c . Letp c p c −p c . Although a closed form expression forp c is hard to find, the errorp c in the MMSE estimate has the following interesting property, which facilitates the calculation of the outage probability in Sec. IV-D. An analogous result has been shown in [27] for the scalar case.
Lemma 2: E p c 2z 2 1 P z for every z > 0. Proof: See Appendix D.
C. Phase IV (Data Transmission)
Using P(σ), node A sends the data signal x = P P(σ)x A,d , where x A,d is distributed as CN (0, 1) and is independent of P(σ). Note that E|x| 2 =P by construction, where the expectation is taken with respect to bothσ and x A,d . The corresponding signal received at node B is
y B,d = P P(σ)hx A,d + w B,d(15)= Pp c x A,d + Pp c x A,d + w B,d .(16)
Sincep c is an MMSE estimate, using the worst case noise theorem [5], we have the following lower bound on the mutual information,
I(x A,d ; y B,d |p c ) ≥ C AB , where C AB α log 1 +P p c 2 2 P r E[ p c 2 2 |ỹ B,τ2 ] + 1 ,(17)
and α
Lc−LB,τ −LA,τ 1 −LA,τ 2 Lc is the fractional data transmit duration after accounting for the time overheads in all three training phases.
D. DMT Analysis With Three-Way Training
Theorem 2: For a SIMO system with r receive antennas and three phases of training and the data transmission phase as described in Table I, an achievable DMT is given by
d(g m ) = r min{l, s} + 1 − g m α ,(18)
where 0 ≤ l ≤ r, 1 ≤ s < r, 0 ≤ g m < α, and α
Lc−LB,τ −LA,τ 1 −LA,τ 2 Lc . Proof: See Appendix E.
Remark: The above three way training scheme can be generalized to k training rounds to improve the diversity order, as in [17], [19]. However, this is mathematically cumbersome and out of the scope of our work. Fig. 2. The achievable DMT with the training and power control scheme proposed in Sec. III, compared with the performance with orthogonal RCT and the data power control proposed in [13], [19] (and appropriately accounting for the training duration overhead and switching off antennas to achieve higher values of gm). The plot corresponds to a SIMO system with r = 5 antennas, with coherence time Lc = 20 symbols, and reverse training duration of L B,τ = 1 symbol.
y A,τ = h H x B,τ + w A,τ III Power controlled training (Node A → Node B) y B,τ 2 = P L A,τ 2 P(σ)h + w B,τ 2 IV Power controlled data (Node A → Node B) y B,d = hx A,d + w B,d
V. DISCUSSION
Recall that with perfect CSIR and imperfect CSIT, with l ≥ s + 1, and for a genie aided channel, it was shown in Theorem 1 that the following DMT is achievable:
d(g m ) = r s + 1 − g m L c L c − L B,τ ,(19)
where 1 ≤ s < r, 0 ≤ g m ≤ L c − L B,τ L c . In contrast, for the same genie aided channel, it was shown in [26] that a diversity order of
d s (g m ) = r 2 − g m L c L c − rL B,τ , 0 ≤ g m ≤ L c − rL B,τ L c(
20) is achievable using orthogonal reverse channel training. Note that d s (g m ) saturates as r gets large, as opposed to (19), which is monotonically increasing in r. In order to achieve a g m > Lc−rLB,τ Lc , in [13], the authors suggest turning off one receive antenna at a time to reduce the training burden until r = 2. . This is in contrast to our result, which can accommodate a larger multiplexing gain, g m ≤ Lc−LB,τ Lc irrespective of r, while simultaneously achieving a higher diversity order at each g m . We note that for a SIMO channel, a diversity order of r(r + 1 − g m ) for 0 ≤ g m < 1 was obtained in [12], [19], using channel-independent training, and without accounting for the training duration overhead. This corresponds to taking L c → ∞ in (19). The performance of the proposed scheme is schematically contrasted with orthogonal RCT in Fig. 2 for a SIMO system with r = 5, L c = 20, and L B,τ = 1 symbol. The advantage of the proposed scheme at higher values of the multiplexing gain is clear from the plot. The proposed training scheme thus results in a factor r-reduction in the training duration, which, along with the proposed data power control scheme, translates to an increase in the range of achievable multiplexing gains, while simultaneously offering a better diversity order compared to orthogonal RCT schemes.
Comparing Theorems 1 and 2, we see that the DMT performance of a genie aided receiver with perfect CSIR is an upper bound on the performance of the system with imperfect CSIR and CSIT, as expected. Also, the performance of the two systems is similar, except that in the latter case, the factor α captures the loss in data transmission time due to all three training phases. Similar observations as the above regarding the improvement in DMT can be made for the three way training scheme compared to orthogonal RCT schemes.
VI. SIMULATION RESULTS
We now briefly present Monte-Carlo simulation results to illustrate the outage probability performance of our proposed RCT and forward-link data power control schemes. We consider a Rayleigh fading channel with three receive antennas. We calculate the outage probability by averaging over 10 8 i.i.d. channel and training noise instantiations. We set the channel coherence time and reverse training duration as L c = 40 and L B,τ = 1, respectively. Figures 3 and 4 show the outage probability of the proposed fixed-power training scheme and the data power control scheme in (10) with s = 1 and s = r = 3, respectively, as a function ofP , with g m = 0 and RP = 4 bits/channel use (1 and 1.5 bits/channel use in case of Fig. 4), and with g m = 0.8. Although the slopes of the curves do not match with the theoretical diversity order because the latter requires infinite SNR, the improved performance of the proposed schemes is clear from the graphs. Also, in Fig. 3, since the proposed scheme uses only L B,τ = 1 training symbol while the orthogonal RCT scheme uses rL B,τ = 3 training symbols, the former shows a higher outage than the latter at lower SNRs. Note that, we have not plotted the outage performance of the three-way training scheme in Sec. IV. This is because the outage probability is hard to compute, since a closed-form expression forp c is not available.
VII. CONCLUSIONS
This paper proposed reverse training and data power control schemes for a TDD-SIMO system with perfect/imperfect CSIR and investigated its DMT performance. It was shown that a diversity order of d(g m ) = r s + 1 − gm α is achievable for l ≥ s + 1, 1 ≤ s < r and 0 ≤ g m < α, where α represents the fractional data transmit duration. In contrast to channel agnostic orthogonal training schemes, the diversity order was shown to increase monotonically with r at nonzero multiplexing gain, which is a significant improvement. The DMT analysis was extended to a more practical situation where the training is done in both directions. In this case also, it was shown that the DMT performance can improve quadratically with the number of receive antennas, and nearly the same DMT can be achieved as that with perfect CSIR and a genie-aided receiver. In terms of system design for reciprocal SIMO systems, the key messages from this work are that it is important to (a) exploit the CSI at the receiver in designing the RCT and (b) use a modified channel-inversion type power control scheme that transmits data at some nonzero power even when the estimated singular value at the transmitter is poor. For fast varying channels, these ingredients can lead to a significant advantage in DMT performance, which, at finite SNR, can translate to a large improvement in outage probability performance compared to orthogonal training schemes. Future work could extend the DMT analysis to a time-selective block fading reciprocal channel, where the channel is correlated within a block [28].
APPENDIX
A. Useful Lemmas
Lemma 3: If the random variable σ 2 is a chi-square distributed with 2r degrees of freedom, then Pr{σ 2 < z} ≤ z r r! , z ≥ 0.
Proof : The result follows from
Pr{σ 2 < z} = 1 (r − 1)! z 0 e −x x r−1 dx (21) ≤ 1 (r − 1)! z 0 x r−1 dx = z r r! .(22)
Lemma 4: For the system in (3), |σ| ≤σ U , whereσ 2
U (σ + |w A,τ |) 2 , withw A,τ ℜ{wA,τ } √P LB,τ .
Proof : We upper bound the absolute value of (3) as follows:
|σ| (a) ≤ σ|ℜ{v Hv }| + | ℜ{w A,τ } P L B,τ | (b) ≤ σ + |w A,τ |,(23)
where (a) follows from the triangle inequality and (b) follows since |ℜ{v Hv }| ≤ 1.
B. Proof of Lemma 1
Consider the following constraint on the data power
E[P(σ)] = ∞ −∞ P(σ)fσ(σ;P )dσ =P ,(24)
where fσ(σ;P ) is the pdf ofσ. Substituting (10) in (24), we get
E[P(σ)] = κP exp L c RP L c − L B,τ − 1 F (P ) + IP ,(25)
where RP is the target data rate and the data transmit power isP ,
and showing that IP <P and that F (P ) is bounded for largē P when 0 ≤ l ≤ r + 1 and n = 1/2. From (26), IP = P l Pr{σ +w A,τ < θP } can be bounded as,
IP (a) ≤P l r! E(θP −w A,τ ) 2r (b) =P l r! E r j=0 θ 2(r−j) P 2r 2j w 2j A,τ(c)
. =P l max j∈{0,1,...,r}
1 P 2(r−j)n+j (d) . = 1 P r−l ,(28)
where (a) follows from Lemma 3 above, and the expectation is with respect to the distribution ofw A,τ , (b) follows from the binomial expansion and the fact that Ew i A,τ = 0 when i is odd, (c) follows from θP . = 1 P n and Ew 2j A,τ . = 1 P j , and (d) follows by substituting n = 1/2 in the left hand side. From (28), clearly, IP <P for largeP if l < r + 1 and n = 1/2. When l = r + 1 and n = 1/2, we have IP P , and therefore we can ensure that IP <P for largeP by scaling IP by an appropriately chosen constant scaling factor. Next, we show that F (P ) is bounded. Note that
F (P ) = 1 θP 1 x 2s fσ(x;P )dx + ∞ 1 1 x 2s fσ(x;P )dx. (29)
Now, it is sufficient to show that the first integral in (29) is bounded, since the second integral is clearly < 1. To this end, we need the distribution ofσ, i.e., Pr (σ +w A,τ ≤ x), wherē w A,τ ∼ N (0, σ 2 var ), and σ 2 var 1 2P LB,τ . Consider
G(x) Pr (σ +w A,τ ≤ x) (30) = ∞ 0 f σ (y) x−y −∞ 1 √ 2πσ var e −z 2 /2σ 2 var dzdy,(31)
where f σ (y) is the pdf of σ, which is chi distributed with 2r degrees of freedom. Taking the derivative of (30) with respect to x, we get
∂G(x) ∂x = J √ 2πσ var ∞ 0 y 2r−1 e − y 2 2 e − (x−y) 2 2σ 2 var dy (32) = Je −β3 √ 2πσ var ∞ 0 y 2r−1 e − (y−β 1 ) 2 2β 2 dy,(33)
where J is the constant term in the standard chi pdf, β 1
x 1+σ 2 var , β 2 σ 2 var 1+σ 2 var . = 1 P and β 3 β 2 x 2 /(2σ 2 var ). Let t = y−β1 √
β2 and using the binomial expansion, it can be shown that
∂G(x) ∂x = J exp(−β 3 ) √ 2πσ var 2r−1 j=0 2r − 1 j ( β 2 ) 2r−j × x j (1 + σ 2 var ) j ∞ −β1/ √ β2 t 2r−1−j e − t 2 2 dt. (34)
Now, using exp(−β 3 ) ≤ 1, we can upper bound the first term in (29) as
1 θP 1 x 2s ∂G(x) ∂x dx ≤ J √ 2πσ var 2r−1 j=0 2r − 1 j C j × ( √ β 2 ) 2r−j (1 + σ 2 var ) j 1 θP x j−2s dx,(35)
where s < r, and C j . = 1 is some constant that does not scale withP . Now, the behavior of the terms above withP is governed by
β r−j/2 2 σ var 1 θP x j−2s dx . = 1 j − 2s + 1 1 P a1 − 1 P a2 , (36)
where a 1 r − j/2 − 1/2, and a 2 (−2s + j + 1)n + r − j/2 − 1/2. The exponent corresponding to the first term above is r − j/2 − 1/2 ≥ 0 for all 0 ≤ j ≤ 2r − 1. Also, when n = 1/2, the exponent corresponding to the second term above is r − s > 0 for all 0 ≤ j ≤ 2r − 1, and hence the integral is bounded for 1 ≤ s < r.
Finally, let RP = g m log(P ). Since IP <P and F (P ) are bounded when 0 ≤ l ≤ r + 1, using exp LcRP Lc−LB,τ − 1
. = P gm α in (27), we get κP . = 1 P gm α −1 , where α Lc−LB,τ Lc
. This completes the proof of Lemma 1.
C. Proof of Theorem 1
Using the power control in (10), the outage probability in (8) can be written as
P out = Pr {σ≤θP } α log(1 +P l σ 2 ) < RP (37) + Pr {σ>θP } α log(1 + κP Φ(σ 2s )σ 2 ) < RP (38) ≤ Π 1 + Π 2 ,(39)
where Π 1 Pr α log(1 +P l σ 2 ) < RP , and Π 2 Pr α log(1 + κP Φ(σ 2s )σ 2 ) < RP . In the above, we have used Pr {A} {·} to mean Pr{· {A}}. Using RP = g m logP ,
we have Π 1 = Pr σ 2 < 1 P l− gm α for largeP and 0 ≤ l ≤ r + 1 from Lemma 1. From Lemma 3 in Appendix A, we have, Π 1 1 P ( l− gm α ) r . Next, substituting for Φ(σ 2s ) from (9), Π 2 can be written as, Π 2 = Pr σ 2 <σ 2s /κP . Usinĝ σ 2 ≤σ 2 U (σ + |w A,τ |) 2 from Lemma 4 in Appendix A witĥ σ 2 = σ 2 , we get
Π 2 ≤ Pr σ 2 < 1 κP (σ + |w A,τ |) 2s (40) ≤ Pr σ 2 < (2σ) 2s κP σ 2 > |w A,τ | 2 + Pr σ 2 < (2|w A,τ |) 2s κP σ 2 ≤ |w A,τ | 2 . (41)
It is straightforward to show that provided κP is strictly increasing withP , the first term in the above goes to zero exponentially withP for 1 ≤ s < r. This implies that g m < α, since κP . =P (1− gm α ) from Lemma 1. The second term in (41) is upper-bounded as
Pr σ 2 < |w A,τ | 2s 2 2s κP (a) ≤ 2 2sr E|w A,τ | 2sr κ rP r! (42) (b) . = 1 P r(s+1− gm α ) ,(43)
where (a) follows from Lemma 3 in Appendix A, and the . = in (b) uses the fact that κP . =P (1− gm α ) and E|w A,τ | 2sr . = 1/P sr . Hence, we have Pr σ 2 < |wA,τ | 2s 2 2s κP
1 P r ( s+1− gm α )
, which implies
Π 2 1 P r ( s+1− gm α )
. Using this and Π 1
1 P r ( l− gm α ) in (39), we have P out max 1 P r(l− gm α ) , 1 P r(s+1− gm α ) (44) = 1 P r(min{l,s+1}− gm α ) ,(45)
for 0 ≤ l ≤ r + 1, 1 ≤ s < r and 0 ≤ g m < α. This ends the proof of Theorem 1.
D. Proof of Lemma 2
Note thatp c can be written as
p c = p c −ỹ B,τ2 − E{p c −ỹ B,τ2 |ỹ B,τ2 } (46) = 1 P L A,τ2 [E{w B,τ2 |ỹ B,τ2 } − w B,τ2 ] . (47)
Now,
E p c 2z 2 = 1 P z L z A,τ2 E wB,τ 2 ,ỹB,τ 2 {A} (48) (a) ≤ 1 P z L z A,τ2 Eỹ B,τ 2 2 2z E{w B,τ2 |ỹ B,τ2 } 2z 2 + 2 2z E wB,τ 2 w B,τ2 2z 2 (49) (b) ≤ 2 2z+1 P z L z A,τ2 E w B,τ2 2z 2 . = 1 P z ,(50)
where A E{w B,τ2 |ỹ B,τ2 } − w B,τ2 2z 2 . In the above, (a) follows from the triangle inequality and using (a + b) n ≤ (2a) n + (2b) n for even n > 0, and (b) follows from the Jensen's inequality and the fact that E w B,τ2 2z 2 < ∞ as P → ∞. The subscripts on the expectation in the above denote the random variables over which the expectation is taken; and E{X|y} denotes the expectation of the random variable X conditioned on the instantiation Y = y. This completes the proof.
E. Proof of Theorem 2
Using the capacity lower bound in (17), the outage probability can be upper bounded as
P out ≤ Pr {C AB < RP } ,(51)
where RP g m logP is the target data rate. We choose η < 1, and arbitrarily close to 1. We split the event in the expression for P out as
P out ≤ Pr C AB < RP ∩ E[ p c 2 2 |ỹ B,τ2 ] ≤ 1 P η (52) + Pr C AB < RP ∩ E[ p c 2 2 |ỹ B,τ2 ] > 1 P η (a) ≤ Pr α log 1 +P p c 2 2 P (1−η) r + 1 < RP + Pr E[ p c 2 2 |ỹ B,τ2 ] > 1 P η ,(53)
where . Then, the first term in (53) can be written as:
Pr p c 2 2 <RP (a) ≤ Pr | p c 2 − p c 2 | < RP (54) ≤ Pr E 1 E 2 + Pr E 1 E c 2 ≤ Pr p c 2 > RP + Pr p c 2 2 < 4RP ,(55)
where E 1 { p c 2 < p c 2 + RP } and E 2 { p c 2 > RP }. In the above, (a) follows from the reverse triangle inequality, and the last two inequalities follow by ignoring one of the events in the intersection. The first term in (55) .
= e −Z ,(60)
where B {σ 2(s−1) > 1 2 2(s+1)RP }, and Z P η−gm /α s−1 . In the above, (a) follows by ignoring the constant factors and substituting for the chi-square pdf of σ 2 . Since 1/RP . = P (η−gm/α) when g m < ηα, and since the exponential term outside the summation dominates the polynomial terms inside the summation, we obtain (b). Note that the special case of s = 1 is trivial, since this corresponds to the probability that RP exceeds a constant, which becomes 0 for sufficiently largē P . The second term in (58) becomes:
Pr σ 2 < 2 2(s+1) |w A,τ | 2sRP ≤ 2 2r(s+1)RrP E{|w A,τ | 2sr } r! . = 1 P (η− gm α )rP rs (61)
= 1 P r(s+η−gm/α)(62)
for 0 ≤ g m < ηα. In the above, we have used Lemma 3 in Appendix A and E|w A,τ | 2s = 1 P s . Thus, in the good estimated channel case, we have , 0 ≤ g m < ηα.
(63) 2) Bad Estimated Channel {σ < θP }: Recall that when σ < θP , the composite channel is given by p c = √P l h. With this, the second term in (55) becomes
. = 1 P r(l+η− gm α ) ,(65)
where 0 ≤ l ≤ r. This completes the analysis of the first term in (53). Now, the second term in (53) can bounded as:
Pr (E[ p c 2 2 |ỹ B,τ2 ]) ζ > 1 P ζη (a) ≤ E(E[ p c 2 2 |ỹ B,τ2 ]) ζP ζη (b) ≤ E([ p c 2ζ 2 ])P ζη (67) (c) 1 P ζ(1−η) ,(68)
where ζ > 0 is an arbitrary number. In the above, (a) and (b) follow from the Markov inequality and Jensen's inequality, respectively, and (c) follows from Lemma 2. Since η < 1, and ζ can be chosen arbitrarily large, the second term in (53) goes to zero with an arbitrarily large exponent asP goes to infinity. Putting (57), (63), (66) and (68) together, a DMT of d(g m ) = r min{l, s} + η − gm α is achievable, for 0 ≤ l ≤ r, 1 ≤ s < r and 0 ≤ g m < ηα. Noting that η is arbitrarily close to 1 completes the proof of Theorem 2.
Fig. 3 .
3Outage probability versus the average data powerP for the fixedpower training scheme proposed in Sec. III, with the data power control scheme given by (10) with s = 1. Here, r = 3, Lc = 40 and L B,τ = 1. With gm = 0.8, the target data rate was set as RP = 4 + gm logP to facilitate the comparison of the curves.
Fig. 4 .
4Outage probability versus the average data powerP for the fixedpower training scheme proposed in Sec. III, with the data power control scheme given by (10) with s = r. Here, r = 3, Lc = 40 and L B,τ = 1.
F
(Lc−LB,τ − 1 F (P ) (P − IP ),
(a) follows by substituting 1/P η in place of E[ p c 2 2 |ỹ B,τ2 ] in the first term, and removing one of the events in the intersection. Definē RP
) Fig. 1. System model for reverse channel training with perfect CSIR used in Section III.1
r
Training
Data
Perfect CSI
Imperfect CSIT
Power Control
Power Control
Genie
Node A
Node B
TABLE I THREE
IWAY TRAINING IN A TDD-SIMO SYSTEMPhase
Description
Input-Output Equation
I
Fixed power training (Node A → Node B)
y B,τ = hx A,τ + w B,τ
II
Fixed power training (Node B → Node A)
where (a) follows from the Markov inequality and (b) follows from Lemma 2. Letting δ = r Pr p c 2 > RP 1In order to solve for the second term in (55), we need to handle two cases of the singular value estimate at node A separately; the good estimated channel case g {σ ≥ θP } and the bad estimated channel case b {σ < θP }. 1) Good Estimated Channel {σ ≥ θP }: Whenσ ≥ θP , substituting for p c P(σ)h and κP . =P − gm α , and defininĝ σ U (σ + |w A,τ |) as the upper bound onσ from Lemma 4 in Appendix A, the second term in (55) leads to:≤ Pr σ 2 < 2 2(s+1) σ 2sRP E 4 + Pr σ 2 < 2 2(s+1) |w A,τ | 2sRP E c 4 ≤ Pr σ 2(s−1) > 2 −2(s+1) RP + Pr σ 2 < 2 2(s+1) |w A,τ | 2sRP ,(58)< 4RP }, and E 4 {σ 2 > |w A,τ | 2 }. the above, we have used Pr {A} {·} to mean Pr{· {A}}, as before; and (a) follows by ignoring the event g. It can be shown that first term in (58) decreases exponentially with Pcan
be written as
Pr p c
2δ
2 >R δP
(a)
≤
E p c
2δ
2
R δP
(b) 1
P δ
1
P ( gm
α −η)δ
, (56)
1
gm
α −η+1 s + 1 − gm
α
> 0, we
have
P r(s+1− gm
α )
, 1 ≤ s < r. (57)
Pr
{σ≥θP }
{E 3 }
(a)
where E 3
{
h 2
2
σ 2s
U
In η−gm/α
s−1
, as follows:
Pr {B}
This work has appeared in part in[1].ACKNOWLEDGMENTSThe authors thank the anonymous reviewers for their detailed comments and suggestions, which have improved the quality of this paper.
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On the diversity gain in MIMO channels with joint rate and power control based on noisy CSITR. X J Zhang, Y Gong, K B Letaief, IEEE Trans. on Wireless Commun. 101X. J. Zhang, Y. Gong, and K. B. Letaief, "On the diversity gain in MIMO channels with joint rate and power control based on noisy CSITR," IEEE Trans. on Wireless Commun., vol. 10, no. 1, pp. 68-72, Jan. 2011.
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Two-way training: optimal power allocation for pilot and data transmission. X Zhou, T Lamahewa, P Sadeghi, S Durrani, IEEE Trans. on Wireless Commun. 92X. Zhou, T. Lamahewa, P. Sadeghi, and S. Durrani, "Two-way training: optimal power allocation for pilot and data transmission," IEEE Trans. on Wireless Commun., vol. 9, no. 2, pp. 564-569, Feb. 2010.
Power controlled reverse channel training achieves an infinite diversity order with perfect CSIR. B N Bharath, C R Murthy, Under PreparationB. N. Bharath and C. R. Murthy, "Power controlled reverse channel training achieves an infinite diversity order with perfect CSIR," Under Preparation, 2011.
Approximately universal codes over slowfading channels. S Tavildar, P Viswanath, IEEE Trans. Inf. Theory. 527S. Tavildar and P. Viswanath, "Approximately universal codes over slow- fading channels," IEEE Trans. Inf. Theory, vol. 52, no. 7, pp. 3233-3258, Jul. 2006.
Explicit space-time codes achieving the diversity-multiplexing gain tradeoff. P Elia, K R Kumar, S A Pawar, P V Kumar, H.-F Lu, IEEE Trans. Inf. Theory. 529P. Elia, K. R. Kumar, S. A. Pawar, P. V. Kumar, and H.-F. Lu, "Explicit space-time codes achieving the diversity-multiplexing gain tradeoff," IEEE Trans. Inf. Theory, vol. 52, no. 9, pp. 3869-3884, Sep. 2006.
On the capacity of some channels with channel state information. G Caire, S Shamai, IEEE Trans. Inf. Theory. 456G. Caire and S. Shamai, "On the capacity of some channels with channel state information," IEEE Trans. Inf. Theory, vol. 45, no. 6, pp. 2007- 2019, Sep. 1999.
The case for transmitter training. C Steger, A Khoshnevis, A Sabharwal, B Aazhang, IEEE Int. Symp. on Inf. Theory. Washington, U.S.AC. Steger, A. Khoshnevis, A. Sabharwal, and B. Aazhang, "The case for transmitter training," in IEEE Int. Symp. on Inf. Theory, Washington, U.S.A, Sep. 2007.
Estimation in Gaussian noise: Properties of the minimum mean-square error. D Guo, Y Wu, S Shamai, S Verdú, IEEE Trans. Inf. Theory. 574D. Guo, Y. Wu, S. Shamai, and S. Verdú, "Estimation in Gaussian noise: Properties of the minimum mean-square error," IEEE Trans. Inf. Theory, vol. 57, no. 4, pp. 2371-2385, Apr. 2011.
Capacity of noncoherent time-selective rayleigh-fading channels. Y Liang, V Veeravalli, IEEE Trans. Inf. Theory. 5012Y. Liang and V. Veeravalli, "Capacity of noncoherent time-selective rayleigh-fading channels," IEEE Trans. Inf. Theory, vol. 50, no. 12, pp. 3095-3110, 2004.
| []
|
[
"A Chameleon Primer",
"A Chameleon Primer"
]
| [
"Ph Brax ",
"C Van De Bruck ",
"A.-C Davis ",
"\nDepartment of Applied Mathematics\nService de Physique Théorique\nCEA/DSM/SPhT\nUnité de recherche associée au CNRS\nCEA-Saclay\nF-91191Gif/Yvette cedexFrance\n",
"\nDepartment of Applied Mathematics and Theoretical Physics\nUniversity of Sheffield\nHounsfield RoadS3 7RHSheffieldUnited Kingdom\n",
"\nUniversity of Cambridge\nClarkson RoadUK\n"
]
| [
"Department of Applied Mathematics\nService de Physique Théorique\nCEA/DSM/SPhT\nUnité de recherche associée au CNRS\nCEA-Saclay\nF-91191Gif/Yvette cedexFrance",
"Department of Applied Mathematics and Theoretical Physics\nUniversity of Sheffield\nHounsfield RoadS3 7RHSheffieldUnited Kingdom",
"University of Cambridge\nClarkson RoadUK"
]
| []
| We review some of the properties of chameleon theories. Chameleon fields are gravitationally coupled to matter and evade gravitational tests thanks to two fundamental properties. The first one is the density dependence of the chameleon mass. In most cases, in a dense environment, chameleons are massive enough to induce a short ranged fifth force. In other cases, non-linear effects imply the existence of a thin shell effect shielding compact bodies from each other and leading to an irrelevant fifth force. We also mention how a natural extension of chameleon theories can play a role to solve the PVLAS versus CAST discrepancy. | null | [
"https://arxiv.org/pdf/0706.1024v2.pdf"
]
| 15,296,928 | 0706.1024 | 97da658cce844bf6343d1a3f9b7dedf271294d5f |
A Chameleon Primer
13 Sep 2007
Ph Brax
C Van De Bruck
A.-C Davis
Department of Applied Mathematics
Service de Physique Théorique
CEA/DSM/SPhT
Unité de recherche associée au CNRS
CEA-Saclay
F-91191Gif/Yvette cedexFrance
Department of Applied Mathematics and Theoretical Physics
University of Sheffield
Hounsfield RoadS3 7RHSheffieldUnited Kingdom
University of Cambridge
Clarkson RoadUK
A Chameleon Primer
13 Sep 2007
We review some of the properties of chameleon theories. Chameleon fields are gravitationally coupled to matter and evade gravitational tests thanks to two fundamental properties. The first one is the density dependence of the chameleon mass. In most cases, in a dense environment, chameleons are massive enough to induce a short ranged fifth force. In other cases, non-linear effects imply the existence of a thin shell effect shielding compact bodies from each other and leading to an irrelevant fifth force. We also mention how a natural extension of chameleon theories can play a role to solve the PVLAS versus CAST discrepancy.
Introduction
Fifth force experiments such as the Cassini satellite experiment put stringent bounds on the gravitational coupling of nearly massless scalar particles. Future satellite tests of fifth forces and putative violations of the equivalence principle will even lead to stronger constraints. As no scalar field has ever been observed, these bounds would not be so dramatic if the existence of nearly massless fields was not suggested by the late time acceleration of the universe expansion 1,2 . In fact, these constraints have fundamental consequences for models of dark energy. Indeed, models of dark energy known as quintessence 3,4,5,6 require the existence of a runaway scalar field with a tiny mass now, of order H 0 ∼ 10 −43 GeV. The range of the interactions mediated by the quintessence scalar field is of order of the Hubble horizon size.
Hence, unless the quintessence field has a very small gravitational interaction with ordinary matter, fifth force experiments are not compatible with a quintessence scenario. For instance, embedding quintessence models in spontaneously broken supergravity proves to be extremely difficult as the gravitational couplings are generically large 7 . Within string theory, the dilaton has been argued to be a quintessence candidate provided the coupling to matter is universal and possesses a minimum playing the role of an attractor where all gravitational problems are evaded 8 . Of course, it would be extremely interesting to confirm this possibility explicitly. String moduli fields are also natural candidates for quintessence. Unfortunately, their gravitational coupling is generically of order one. On the other hand, there exists a well-motivated scalar field with a small gravitational field: the radion measuring the inter-brane distance in Randall-Sundrum scenarios. In this case, the gravitational coupling of the radion to matter on a warped brane is suppressed by the warp factor and becomes very small for a large radion.
Chameleon field theory combine both a quintessence-like behaviour leading to dark energy at late time and a gravitational coupling to matter which can be large 9,10 . So how come they are not definitely ruled out by fifth force experiments? In fact, it is useful to draw an analogy with photons. In some circumstances, photons do get a mass which alters their properties. This is notoriously the case in superconductors where the Meissner effect (the fact that the magnetic field is expelled from a superconductor) can be seen as the result of the Higgs mechanism with a mass given to the photons 11 . In less extreme situations, like in a crystal, photons are slowed down when interacting with matter. Similar phenomena can occur for scalar fields. Typically, scalar particles have an effective potential obtained as a combination of the bare potential appearing in the Lagrangian and a term proportional to the matter density. This effective potential may have a density dependent minimum. In this case, we call the field a chameleon as its mass depends on the environment. Chameleon fields are generically more massive in a dense environment. This is enough to evade the gravitational bounds in most cases. Indeed, the range of the chameleon mediated force becomes too small to be detected. Even when this is not the case, chameleon theories may enjoy another non-trivial property: the existence of thin shells. More precisely, the field created by a massive object may be essentially trapped inside the massive body. In this case, the interaction between massive bodies is essentially non-existent. Combining these two effects, one can build satisfying examples of chameleon theories. We will review their main properties here.
Recently, the PVLAS experiment has measured the dichroism of light propagating through a magnetic field 12 . This can be understood by coupling a scalar field to photons. In this case, one can use the environment dependent mass of chameleon fields to generate a large mass for the chameleon in the sun. Therefore, chameleons would not be produced by the Primakov effect and therefore the CAST experiment 13 would not see the photon regeneration by inverse Primakov effect. We will present some of these ideas very briefly.
Scalar-Tensor Theories
Coupling to matter
Chameleon fields appear in scalar-tensor theories of gravity 14 . We start with a discussion of these theories. We consider theories where a scalar field φ couples both to gravity and matter, generating a potential fifth force. The Lagrangian of such scalar-tensor theories reads
S = 1 2κ 2 4 d 4 x √ −g(R − (∂φ) 2 − 2κ 2 4 V (φ))(1)
Matter couples to both gravity and the scalar field according to
S m (ψ, A 2 (φ)g µν ),(2)
where ψ is a matter field and A is an arbitrary function of φ. The Klein-Gordon equation can be written in terms of an effective potential
V eff (φ) = V (φ) + ρ m A(φ).(3)
The effective potential depends on the environment through the matter energy density ρ m . We will assume that V (φ) is a runaway potential and for the models we consider A(φ) increases with φ. In that case the potential has a minimum whose location depends on ρ m , i.e. on the environment. Such a field has been called a chameleon field. The field φ acts on all types of matter and, in the Einstein frame, there is a new force associated with the scalar field
F φ = −κ 4 mα φ ∂φ ∂x µ ,(4)
where m is the mass of the test particle in the Einstein frame and
α φ = ∂ ln A ∂κ 4 φ (5)
The force F φ cannot be too large, otherwise experiments would have already detected it. For massless fields V (φ) ≡ 0 and a point-like matter source, the Klein-Gordon equation becomes
∆φ = −κ 4 mα φ | r=0 δ (3) (r)(6)
where m = A(φ)| r=0 m 0 is the Einstein frame mass and m 0 the bare mass of the source. The resulting field φ = −κ 4 α φ | r=0 /4πr leads to a force between bodies F φ = 2G N α φ|r=r 1 α φ|r=r 2 m 1 m 2 /r 12 where κ 2 4 = 8πG N . This produces a fifth force where
F φ = 2α 1 α 2 F Newton(7)
and α 1 = α φ|r=r 1 . The Cassini experiments impose that α 2 φ ≤ 5.10 −5 for a constant coupling. Hence massless particles (or nearly massless particles with a mass less than 10 −3 eV) must have a very small coupling to gravity. Chameleon field theories enable to overcome this obstacle.
The radion
A simple and interesting example of non-trivial coupling to gravity is provided by the Randall-Sundrum scenario where matter is confined on 4d hyperplanes embedded in an AdS 5 vacuum 15 . The two boundaries of space-time are called the UV and the IR brane reflecting the fact that the metric is warped. Distances on the IR branes are warped down compared to scales on the UV. Consider now matter on the UV brane of positive tension. The coupling of matter to gravity depends on the radion field φ (for a derivation of the following equations and references, see e.g. the review 16 )
A(φ) = cosh κ 4 φ √ 6 (8)
where the inter-brane distance is
d = −l ln(tanh κ 4 φ √ 6 )(9)
For small distances compared to the AdS curvature l, the coupling becomes
A(φ) = 1 2 e κ 4 φ/ √ 6(10)
The gravitational coupling is constant
α φ = 1 √ 6(11)
Of course, this is too big for the Cassini bound. However, in this case and in the large interbranedistance limit, the chameleon mechanism can be applied to hide the interaction mediated by the radion 17 by introducing a bare potential for the radion field.
Chameleon Cosmology
We concentrate on a particular model where A(φ) = e βφ and β = O(1). We consider the family of potential
V = M 4 f (( M φ ) n )(12)
where f leads to ordinary quintessence with a long time tracking solution. A typical example is provided by f (x) = e x . As φ ≫ M now, the potential is nothing but
V = M 4 + M 4+n φ n(13)
Cosmologically, it mimics a cosmological constant. For gravitational tests, only the Ratra-Peebles part of the potential matters. This model satisfies the chameleon property of having a ρ-dependent minimum. As β = 0(1), the coupling of matter to the chameleon field is large and may be in conflict with experiments. We will study the gravitational aspects in the next section. Here we concentrate on cosmological properties.
In a Friedmann-Robertson-Walker Universe, the (non)-conservation of matter equation readsρ + 3Hρ = α φφ ρ.
leading to
ρ = A(φ)ρ m , ρ m = ρ 0 a 3(1+wm)(15)
while the Klein-Gordon equation can be written in terms of an effective potential
V eff (φ) = V (φ) + ρ m (1 − 3w m )A(φ).(16)
Let us now go through the different cosmological eras 10 . During inflation the chameleon potential has an effective minimum which is time-independent. Moreover, as the mass of the chameleon field at the minimum is m ≫ H, the field oscillates rapidly and converges to the minimum extremely rapidly, behaving like a dust component. By the end of inflation, the field is stuck at the minimum. As inflation stops and the radiation era starts, the minimum is pushed far away (as it depends only of non-relativistic matter). The field is therefore in an overshooting regime where it becomes kinetically dominated, being far away to the right of the potential. The field overshoots before stopping at φ stop = φ in + 6Ω i φ m Pl where Ω i φ is the initial chameleon fraction density. After stopping the field is in an undershooting position. In that case, the field would remain still until either being caught up by the minimum or the beginning of the matter era. When caught by the minimum the field oscillate and converges to the minimum, which is a tracker solution. This follows from the fact that m ≫ H at the minimum throughout the history of the Universe. The field converges to the minimum faster than a −3 due to the time variation of the mass at the minimum.
In fact if the field is far away from the minimum after overshooting, it is sensitive to short bursts when relativistic particles become non-relativistic
φ + 3Hφ = β m Pl T µ µ(17)
as, during such periods, the trace of the energy momentum tensor T µ µ of the decoupling species is temporarily non-vanishing, resulting in a kick 18 of order of a fraction of β. Taking into account all these kicks, the field decreases by about ∆φ ∼ −βm Pl . By BBN, either the field is close to the minimum, in which case the electron kick which occurs during BBN does not lead to a large variation of φ during BBN, or the field is still far away from the minimum in which case the electron kick leads to large variations of φ and therefore of masses
| ∆m m | = β| ∆φ m Pl |(18)
the latter case being excluded. As a result, the initial value of φ cannot be larger that one and Ω i φ ≤ 1/6, a weaker bound than in quintessence. Once at the minimum by BBN, the field follows the attractor in the matter era. Once the vacuum energy dominates, the matter density decreases extremely fast. The chameleon field follows the minimum until m ∼ H where it starts lagging behind eventually having the same evolution as a quintessence field with no coupling to matter.
Gravitational Tests
The massive chameleon
The effective potential with f (x) = e x leads to a stabilisation of the scalar field for
φ = nΛ 4+n M ρ 1/(n+1) ,(19)
where ρ is the matter energy density . The mass at the bottom of the potential is given by
m 2 = n(n + 1) Λ n+4 φ n+2(20)
In the atmosphere, the mass of chameleons is larger than 10 −3 eV implying no consequence for Galileo's Pisa experiment and similar tests.
The thin shell
Let us now consider a situation where the gravitational experiments are performed on a body embedded in a surrounding medium. The body could be a small ball of metal in the atmosphere or a planet in the inter-planetary vacuum. The effective potential (16) is not the same inside the body and outside because ρ m is different. The effective potential can be approximated by
V eff ≃ 1 2 m 2 φ (φ − φ min ) 2 ,(21)
As already mentioned the minimum and the mass are different inside and outside the body. We denote by φ b and m b the minimum and the mass in the body and by φ ∞ and m ∞ the minimum and the mass of the effective potential outside the body. Then, the Klein-Gordon equation reads
d 2 φ dr 2 + 2 r dφ dr = ∂V eff ∂φ ,(22)
where r is a radial coordinate. Requiring that q remains bounded inside and outside the body and joining the interior and exterior solutions, one can determine the complete profile which can be expressed as
φ < (r) = φ b + R b (φ ∞ − φ b ) (1 + m ∞ R b ) sinh (m b R b ) [m ∞ R b + m b R b coth (m b R b )] sinh (m b r) r , r ≤ R b , φ > (r) = φ ∞ + R b (φ b − φ ∞ ) m b R b coth (m b R b ) − 1 [m ∞ R b + m b R b coth (m b R b )] e −m∞(r−R b ) r , r ≥ R b(23)
Assuming, as it is always the case in practise, that
m b ≫ m ∞ , m b R b ≫ 1, one has ∂φ > (r) ∂r ≃ − R b r 2 (φ ∞ − φ b ) ,(24)
from which we deduce that the acceleration felt by a test particle is given by
a = G N m b r 2 1 + α φ (φ ∞ − φ b ) Φ N ,(25)
where Φ N = G N m b /R b is the Newtonian potential at the surface of the body. Therefore, the theory is compatible with gravity tests if
α φ (φ ∞ − φ b ) Φ N ≪ 1 .(26)
Large compact bodies have a thin shell implying that no distortion of solar system planetary orbits are predicted. Lunar ranging experiments are not affected either.
Chameleon in a cavity
Gravitational experiments on earth and future satellite experiments involve vacuum chambers which can be modelled out as spherical cavities of radius R. Solving the chameleon equations in this situation, following the same method as in the previous subsection, we find that the mass of the chameleon field inside the cavity is determined by the resonance equation
sinh m 0 R m 0 R = n + 2(27)
Having determined m 0 , one can deduce the value of the field φ 0 inside the cavity. Notice that for most values of n we have
m 0 R = O(1)(28)
When β = O(1), the mass of the chameleon in gravitational experiments on earth is of order 1/R and is too large to evade gravitational tests (the range is given by R ∼ 1 m
PVLAS vs CAST
Recently, the coupling of a scalar field to photons have been invoked in order to explain the PVLAS results on dichroism 12 . The scalar field is required to have a mass of order 10 −12 GeV and a coupling strength suppressed by a scale of order M = 10 6 GeV. The coupling to photons is given by
− 1 4 d 4 xe φ/M F µν F µν(29)
The results of the PVLAS collaboration are in conflict with astrophysical bounds such as CAST 13 , which for the same mass for the scalar field, require much smaller couplings (M > 10 10 GeV).
The chameleon mechanism can help in explaining the PVLAS results and, at the same time, be in agreement with astrophysical bounds 21 . Our model is of the scalar-tensor type
S = d 4 x √ −g 1 2κ 2 4 R − g µν ∂ µ φ∂ ν φ − V (φ) − e φ/M 4 F 2 + S m (e φ/M g µν , ψ m )(30)
where S m is the matter action and the fields ψ m are the matter fields. The effective gravitational coupling is given by
β = m Pl M ,(31)
and therefore very large (β = 10 13 ) when assuming the results from the PLVAS experiment (M = 10 6 GeV). To prevent large deviations from Newton's law one must envisage non-linear effects shielding massive bodies from the scalar field. One natural possibility is that the scalar field φ coupled to photons has a runaway (quintessence)-potential leading to the chameleon effect. For exponential couplings, this is realised when
V (φ) = Λ 4 exp(Λ n /φ n ) ≈ Λ 4 + Λ 4+n φ n(32)
In the presence of matter, the dynamics of the scalar field is determined by an effective potential
V eff (φ) = Λ 4 exp(Λ n /φ n ) + e φ/M (ρ + B 2 2 )(33)
where ρ is the energy density of non-relativistic matter.
As already mentioned, the PVLAS experiment is in conflict with the CAST experiment on the detection of scalar particles emanating from the sun, as it requires M ≥ 10 10 GeV. However, this bound does not apply when the mass of the scalar field in the sun exceeds 10 −5 GeV. Let us evaluate the mass of the chameleon field inside the sun. Furthermore, from the effective potential one obtains m sun = m lab ρ sun ρ lab (n+2)/2(n+1)
.
Now ρ sun /ρ lab ≈ 10 14 and, with n = 0(1), one finds m sun ∼ 10 −2 GeV ≫ 10 −5 GeV
implying no production of chameleons in the sun. Hence, the CAST experiment is in agreement with the chameleon model due to the fact that the chameleon field is very massive in the sun.
Conclusion
We have given an brief overview of chameleon field theories. They provide exciting new mechanisms for both gravitation and cosmology. A scalar field coupled to matter can be problematic, since it mediates a new force. But if the field self-interacts in a non-linear way, as it is the case in chameleon field theories, the effect of the field can be hidden from current experiments. As we pointed out, future experiments will be able to search for such chameleon fields. We have speculated that the PVLAS anomaly finds a natural interpretation within these theories.
AcknowledgmentsWe would like to thank out friends and collaborators on various aspects of chameleon theories: A. Green, J. Khoury, D. Mota, D. Shaw and A. Weltman.
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| []
|
[
"Reading, writing and squeezing the entangled states of two nanomechanical resonators coupled to a SQUID",
"Reading, writing and squeezing the entangled states of two nanomechanical resonators coupled to a SQUID"
]
| [
"Guy Z Cohen \nDepartment of Physics\nUniversity of California\nSan Diego, La Jolla92093-0319California\n",
"Massimiliano Di Ventra \nDepartment of Physics\nUniversity of California\nSan Diego, La Jolla92093-0319California\n"
]
| [
"Department of Physics\nUniversity of California\nSan Diego, La Jolla92093-0319California",
"Department of Physics\nUniversity of California\nSan Diego, La Jolla92093-0319California"
]
| []
| We study a system of two nanomechanical resonators embedded in a dc SQUID. We show that the inductively-coupled resonators can be treated as two entangled quantum memory elements with states that can be read from, or written on by employing the SQUID as a displacement detector or switching additional external magnetic fields, respectively. We present a scheme to squeeze the even mode of the state of the resonators and consequently reduce the noise in the measurement of the magnetic flux threading the SQUID. We finally analyze the effect of dissipation on the squeezing using the quantum master equation, and show the qualitatively different behavior for the weak and strong damping regimes. Our predictions can be tested using current experimental capabilities. | 10.1103/physrevb.87.014513 | [
"https://arxiv.org/pdf/1211.1072v2.pdf"
]
| 59,438,249 | 1211.1072 | d1ecc359cd4ceb716c59958c52791685ea82a888 |
Reading, writing and squeezing the entangled states of two nanomechanical resonators coupled to a SQUID
7 Feb 2013
Guy Z Cohen
Department of Physics
University of California
San Diego, La Jolla92093-0319California
Massimiliano Di Ventra
Department of Physics
University of California
San Diego, La Jolla92093-0319California
Reading, writing and squeezing the entangled states of two nanomechanical resonators coupled to a SQUID
7 Feb 2013
We study a system of two nanomechanical resonators embedded in a dc SQUID. We show that the inductively-coupled resonators can be treated as two entangled quantum memory elements with states that can be read from, or written on by employing the SQUID as a displacement detector or switching additional external magnetic fields, respectively. We present a scheme to squeeze the even mode of the state of the resonators and consequently reduce the noise in the measurement of the magnetic flux threading the SQUID. We finally analyze the effect of dissipation on the squeezing using the quantum master equation, and show the qualitatively different behavior for the weak and strong damping regimes. Our predictions can be tested using current experimental capabilities.
I. INTRODUCTION
In recent years nanoelectromechanical systems 1,2 (NEMs), nanoscale mechanical oscillators coupled to electronic devices of comparable dimensions, have attracted substantial research effort. A major motivation for this effort is the ability to observe quantum behavior in a macroscopic system under realizable experimental conditions 3,4 . Indeed, NEMs today can be fabricated with vibrational mode frequencies of 1 MHz-10 GHz and quality factors in the range of 10 3 − 10 5 , allowing the quantum regime to be reached at milli-Kelvin temperatures for high frequency oscillators 5,6 . Possible quantum effects in NEMs under such conditions include quantized energy levels, superposition of states, entanglement and squeezing [7][8][9] . In addition, NEMs are applied to high-sensitive detection of mass [10][11][12] , force 13 and displacement 14 , electrometers 15 , and also to classical memory elements 16,17 .
Observing or changing the state of NEMs requires some type of transducer which couples to them. Optical coupling 18 can be performed, e.g., by a microwave cavity 19 , but is difficult to integrate in circuits and suffers from the diffraction limit and heating of the NEMS. Nonoptical coupling methods are therefore more common in experiments today. With magnetomotive coupling 20 , the magnetic force on a thin metallic layer on the NEMS is measured. Capacitive coupling can take many forms, one of which uses a normal or superconducting single electron transistor (SET) 8,21 . The NEMS changes the island charging energy in the SET and hence the tunneling rates, which can be read electronically. Other forms of capacitive coupling use Cooper pair boxes 22,23 , flux qubits 24 , quantum point contacts 25 and quantum dots 26 .
An inductive coupling scheme with a potential for displacement precision greater than the standard quantum limit is obtained by integrating a doubly clamped micron-scale beam within a superconducting quantum interference device (SQUID). In a dc SQUID the motion of the resonator changes the area of the SQUID loop and hence the magnetic flux and the current through it, which is then measured. This system was only recently implemented 27,28 . A more sophisticated design, where the dc SQUID, and hence the resonator, is coupled to a charge qubit, was also proposed 9 . For an rf SQUID it was found 29 that the change in the magnetic flux due to the motion of the beam affects the visibility of Rabi oscillations in the SQUID levels. The detection of discrete Fock states in a resonator integrated with an rf SQUID was suggested in another work 30 .
Squeezed states, originally introduced in quantum optics 31 , are defined as minimum-uncertainty states with less noise in one field quadrature than a coherent state 32 . Several methods to generate squeezing in NEMs were suggested. Coupling to a charge qubit 7,9 as means of generating squeezing was proposed, while another work described squeezing by periodic position measurement with a weakly-coupled detector 8 . Squeezing in nanoresonators can be applied to decrease the noise in force or displacement measurements to below the standard quantum limit, greatly improving the sensitivity of the device 7,33 .
In this work, we present a scheme to create quantum entanglement and squeezing in two nanoresonators integrated in a dc SQUID. A previous study 34 analyzed a similar system but introduced many approximations that are difficult to implement experimentally, whereas our present study is closer to an experimentally realizable system. For instance, we do not overlook the generally non-negligible self-inductance of the SQUID as done in previous work 34 , and we assume mega-Hertz frequency nanomechanical oscillators rather than giga-Hertz frequency resonators, which are difficult to integrate with a SQUID. Lastly, we do not require the SQUID to be prepared in a high-|α| coherent state in order to have squeezing, as the previous study does 34 , and require instead a thermal equilibrium state, which is easier to accomplish. Finally, we consider different aspects of the system and draw conclusions, e.g., on the reading and writing processes, that were not advanced in previous literature.
The paper is organized as follows. In Sec. II we present the system model and its classical Lagrangian and Hamiltonian formulations. We then proceed to quantize the Hamiltonian for the non-dissipative case and derive the effective Hamiltonian. Next, in Sec. III we treat the
I 1 I b I 3 I 4 FIG. 1.
Schematic of the device we study: Two nanomechanical beams oscillating in plane are embedded in a dc SQUID with area A when beams are at rest. Uniform magnetic field B threads the SQUID, and bias current I b is assumed. The two identical Josephson junctions on the SQUID have phase drops γi (i = 1, 2). The currents I3 and I4 create additional magnetic fields B1 and B2 at the resonators.
system as a quantum memory and explain how one can read its quantum state or write on it. In Sec. IV we put forward a scheme for generating quadrature-squeezed states of the nanomechanical beams when dissipation is neglected, whereas in Sec. V we use a quantum master equation approach to test the range of validity of our results in the the presence of dissipation. Lastly, we discuss our results and present the conclusions in Sec. VI.
II. SYSTEM MODEL AND HAMILTONIAN
Our system consists of a dc SQUID, shown schematically in Fig. 1, in which each arm includes a Josephson junction and an integrated doubly clamped beam of length l i and mass m i that can oscillate mechanically in the plane of the SQUID with an angular frequency ω i (i = 1, 2). The notation we use is similar to the one in Ref. 35. A uniform magnetic field B is applied perpendicularly to the plane of the loop, and a dc bias current I b flows through it after splitting to I 1 and I 2 in the lower and upper arms, respectively. Two currentcarrying wires with currents I 3 and I 4 create additional magnetic fields B 1 and B 2 at the positions of the first and second beams, respectively. The beam amplitudes are much smaller than the beam-wire distance, allowing these fields to be approximated as being spatially uniform. For simplicity, the two Josephson junctions on the SQUID arms are taken to be identical and their gaugeinvariant phase changes are denoted by γ i . The critical current and shunting capacitance of each junction are taken as I c and C, respectively, and are used to define the characteristic junction energy scales: the Josephson energy E J =hI c /2e and the charging energy E C = e 2 /2C. The plasma frequency ω pl = √ 2E J E C /h sets the typical time scale for the SQUID dynamics.
The area of the SQUID loop depends on the center of mass positions of the nanomechanical resonators, denoted by x i and defined as zero when the beam is at rest, and positive when it is inside the loop. Since the superconducting order parameter is single valued, we must have
γ 1 − γ 2 − 2πΦ Φ0 = 2πp,(1)Φ = BA − i (B + B i )l i x i + L(I 1 − I 2 )/2,(2)
where p is an integer, Φ is the total magnetic flux threading the loop, A is the loop area when the beams are at rest, Φ 0 = h/2e is the flux quantum, L is the selfinductance of the loop, and I i is the current in its ith arm. The first term in Eq. (2) comes from the external magnetic field and the second, responsible for the coupling of the mechanical and magnetic degrees of freedom, from the oscillation of the beams. The difference between l i x i and the actual area enclosed by the ith beam is negligible, being of third order in the ratio of the beam amplitude to its length. Lastly, the third term originates from the magnetic flux induced by the circulating current in the SQUID. The kinetic and potential energies of the system are functions of four dimensionless variables defined by γ =
(γ 1 + γ 2 )/2, φ = Φ/Φ 0 and ξ i = (B + B i )l i x i /Φ 0 . They are T = i (h 2 4EC 1 Ω 2 i 1 A 2 iξ 2 i ) +h 2 2ECγ 2 + π 2h2 2ECφ 2 ,(3)U = E J [−2 cos γ cos(πφ) − I b Ic γ + i ((−1) i π I b Ic ξ i + ξ 2 i 2A 2 i ) + 2π βL (φ − ξ 1 − ξ 2 − φ e ) 2 ],(4)
where mechanical dissipation was assumed to be negligible, and where we define the screening parameter β L = 2LI c /Φ 0 and external flux φ e = BA for the SQUID, while the dimensionless magnetic field
A i = E J m i (B + B i )l i ω i Φ 0(5)
and oscillation frequencies Ω i = ω i /ω pl are defined for each of the beams. The first term in Eq. (3) corresponds to the kinetic energy of the beams, while the second and third terms to the capacitive energy of the junctions. The first term in Eq. (4) relates to the Josephson junctions energy, while the second term is the washboard potential term 36 . The third term corresponds to the Lorentz force on the beams in the classical equations of motion (EOMs), and the fourth term to the beams' elastic potential, taken to be harmonic, as nonlinear terms are negligible at the amplitudes concerned 37 . Lastly, the fifth term corresponds to the inductive energy of the SQUID. The classical EOMs for the four variables γ, φ, ξ 1 and ξ 2 are the Euler-Lagrange equations for the system Lagrangian L = T − U . Before writing the Hamiltonian, we expand the potential in series about a minimum (φ, γ, ξ 1 , ξ 2 ), around which the system oscillates. Under current experimental conditions 27,28 such a minimum exists, as the Hessian matrix for U there, proportional to the one in Eq. 8, is positive definite. The well containing the minimum can accommodate ∼ 20 states in γ and ∼ 900 in φ. If we take these two parameters to be "frozen" at their respective ground states, as we will later assume, we find the well to be infinitely deep for the ξ i parameters. This assumption also allows us to neglect in the series expansion of U terms higher than quadratic ones in φ − φ and γ − γ. With these approximations in mind the Hamiltonian H reads
H = T + U = i E C 2h 2 p 2 i + i,j E J V ij q i q j ,(6)
where the coordinates q i are given by (7) (j = 1, 2), and the canonically conjugate momenta p i are
q 1 = γ−γ, q 2 = π(φ−φ), q 2+j = 1 √ 2Ω j A j (ξ j −ξ j )p i = (h 2 /E C )q i . In addition, V = r −s 0 0 −s r + 2 πβL − 2 √ 2Ω1A1 βL − 2 √ 2Ω2A2 βL 0 − 2 √ 2Ω1A1 βL Ω 2 1 (1 + 4πA 2 1 βL ) 4πΩ1A1Ω2A2 βL 0 − 2 √ 2Ω2A2 βL 4πΩ1A1Ω2A2 βL Ω 2 2 (1 + 4πA 2 2 βL ) ,(8)
where r = cos γ cos πφ and s = sin γ sin πφ were introduced. We see that the beam oscillations are coupled inductively via the V 34 term. This coupling can be used to generate squeezed states in the beams as we will show below.
The Hamiltonian is quantized in the standard way by converting the coordinates q i and their canonically conjugate momenta p i to operators and postulating the canonical commutation relation [ q i , p j ] = ihδ ij . In terms of creation and annihilation operators q i and p i arê
q i = 1 2 ( 2EC EJ Vii ) 1/4 (a † i + a i ),(9)p i = ih( EJ Vii 2EC ) 1/4 (a † i − a i ),(10)
and the quantized Hamiltonian is given by
H = ih ω i (a † i a i + 1 2 ) + 1 4h ω pl i =j ω pl √ ωiωj V ij (a i + a † i )(a j + a † j ),(11)
where ω i = ω pl √ V ii . We note the frequencies ω 3 and ω 4 are the same as the resonators frequencies, ω 1 and ω 2 , apart from each having a factor due to the magnetic field at the resonator. Thus we see that the Lagrangian classical memory variables ξ 1 and ξ 2 , in complete analogy with memory variables in electronic circuits 38 , become memory quanta in the Hamiltonian.
Taking the same experimental conditions 27,28 and tuning B to make r of order unity results inhω 1 ≫ k B T andhω 2 ≫ k B T . Consequently, the first and second harmonic oscillators are "frozen" at their respective ground states. Moreover, since ω 1 , ω 2 ≫ ω 3 , ω 4 , exciting the nanomechanical oscillators will not budge them from their ground state. Removing constant terms, we are then left with the effective Hamiltonian
H =hω 3 a † 3 a 3 +hω 4 a † 4 a 4 + V (a 3 + a † 3 )(a 4 + a † 4 ),(12)
where the interaction coefficient reads
V = 2πh √ ω 1 ω 2 A 1 A 2 β L (1 + 4πA 2 1 βL ) 1/4 (1 + 4πA 2 2 βL ) 1/4 .(13)
III. READING AND WRITING QUANTUM INFORMATION
We now wish to employ this system to create entangled nanomechanical quantum memory that can be read from and written on. We assume the beams are cooled to a temperature low enough so as to reduce the equilibrium state to the ground state for each of the beams. This is possible today, e.g. by coupling to a superconducting microwave resonator 39 , even if the environment of the beams, which includes the SQUID, has a higher temperature.
If the interaction term in Eq. (12) is small relative to the other two terms, perturbation theory gives firstorder energy corrections in V only when |ω 3 −ω 4 | ≪ V /h. Thus, we will henceforth assume the beams are identical. The Hamiltonian (12) is quadratic in the ladder operators and is thus amenable to an exact solution at all interaction strengths 40 . This solution is found by moving to the differential representation and then diagonalizing the quadratic form of the potential by a canonical transformation to even and odd coordinates,
x e,o = 1 √ 2 (x 1 ± x 2 ), p e,o = 1 √ 2 (p 1 ± p 2 ),(14)
where x i and p i are the position and momentum coordinates of the ith beam. Applying Eq. (14) on the ladder operators, we find
a 5 = 1 2 √ 2 ( ω 5 ω 3 + ω 3 ω 5 )(a 3 + a 4 ) + ( ω 5 ω 3 − ω 3 ω 5 )(a † 3 + a † 4 ) ,(15)a 6 = 1 2 √ 2 ( ω 6 ω 3 + ω 3 ω 6 )(a 3 − a 4 ) + ( ω 6 ω 3 − ω 3 ω 6 )(a † 3 − a † 4 ) ,(16)
where a † 5 corresponds to creation of an even mode quantum in which both beams oscillate in phase, and a † 6 to creation of an odd mode quantum, where the beams oscillate in anti-phase. The even and odd oscillation frequencies are given by
ω 5,6 = ω 2 3 ± 2 V ω 3 /h.(17)
Using this transformation and omitting constant terms, the Hamiltonian (12) is reduced to
H =hω 5 a † 5 a 5 +hω 6 a † 6 a 6 .(18)
We see that the even mode is decoupled from the odd mode in this Hamiltonian, which is thus separable to an even and an odd part. The energy spectrum of this Hamiltonian is given by
E nm = nhω 5 + mhω 6 ,(19)
while the eigenstates are
|nm = 1 √ n!m! (a † 5 ) n (a † 6 ) m |00 ,(20)
which, upon substitution of Eqs. (15) and (16), are seen to be highly entangled states of the two beams. The quantum state of the system is read by measuring the magnetic flux threading through the SQUID, which is done by a current measurement in the standard way 36 . The operator for this observable is
Φ = −(B + B 1 )l 1 x 1 − (B + B 2 )l 2 x 2 ,(21)
where the constant term BA was omitted. With the assumption of B 1 = B 2 , we have
Φ = − √ 2(B + B 1 )l 1 λ 3 (a 5 + a † 5 ),(22)
where the zero-point fluctuation, the resonator displacement uncertainty at the ground state, is defined as λ i = h/2m 1 ω i+2 with the definition extended also for i = 3, 4.
We thus see, as expected, that the measurement of the magnetic flux cannot detect the odd mode, since oscillations in this mode do not amount to a change in the area of the SQUID loop. We therefore set to read and write quantum information only in the n quantum number in the state |nm . Moreover, we note that for the eigenstates of the Hamiltonian we have Φ = 0, implying that a better observable would be the standard deviation ∆ Φ . This is indeed true, with the values of this observable on the eigenstates being
∆ Φ = √ 2(B + B 1 )l 1 λ 3 √ 1 + 2n,(23)
enabling us to measure the value of n.
Having established the reading process, we now set to describe how to write quantum information on this system. It would seem the best way to excite the system is via a resonant ac current of frequency ω 5 in the external wires that, according to the Hamiltonian (18), will pump the beams to their excited state. However, such a current will also pump the beams to even higher excited states, since the energy level difference is fixed in this system. A better method would be to use constant currents in the external wires. The addition to the potential (4) due to such currents, keeping only first-order terms in B 1 /B and B 2 /B, is (24) where constant terms were omitted and only linear terms in x i were kept, owing to the quadratic terms being smaller by several orders of magnitude.
H 1 = πE J l 1 I c β L Φ 0 {−B 1 x 1 [4I c (φ − φ e ) + I b β L ] +B 2 x 2 [I b β L − 4I c (φ − φ e )]}
Since reading can be done only for the even mode, and the Hamiltonian (18) is separable into odd and even components, we do not consider the odd part in Eq. (24), and by choosing also B 1 = B 2 , the even part of Eq. (24) is
H 1,e = −4 √ 2π E J l 1 λ 3 (φ − φ e ) β L Φ 0 B 1 (a 5 + a † 5 ) ≡ f (B 1 )(a 5 + a † 5 ),(25)
where constant terms were again omitted. The even part of the total Hamiltonian H + H 1 is therefore
H e =hω 5 a † 5 a 5 + f (B 1 )(a 5 + a † 5 ).(26)
The idea of writing is then the following: We create a constant magnetic field B 1 , which shifts the harmonic potential and then let the system relax to its new ground state. We then suddenly revert B 1 to zero thereby obtaining an excited state of the original system. The Hamiltonian in the differential representation is
H e = −h 2 2m 1 d 2 dx 2 + 1 2 m 1 ω 2 5 x 2 + λ −1 3 f (B 1 )x,(27)
where the mass m 1 is the beam mass. Apart from a constant, the Hamiltonian (27) is
H e = −h 2 2m 1 d 2 dx 2 + 1 2 m 1 ω 2 5 (x + ∆x) 2 ,(28)∆x = f (B 1 ) m 1 λ 3 ω 2 5 .(29)
According to the scheme described above, we wish to maximize the probability P nn = | ψ 0 (x + ∆x)|ψ n (x) | 2 , with ψ n (x) being the nth harmonic oscillator wavefunction, to obtain the desired state |n by tuning B 1 and with it ∆x. Using standard results of the quantum harmonic oscillator to write the integral ψ 0 (x + ∆x)|ψ n (x) and then using the generating function of the Hermite polynomials to find its value for every n, we find the maximum value of P nn is reached when ∆x = 2 √ nλ 3 with the probability then to measure k phonons after B 1 is removed given by
P kn = (k!) −1 e −n n k ,(30)
which is a Poisson distribution with mean n. The P nn function is plotted in Fig. 2. We see that the probability drops sharply for small n and evens out for larger values. The limit at n → ∞ is 0. Although the writing process does not create a pure number state |n , the standard deviation in the number of phonons, by the properties of the Poisson distribution, is √ n, which is reasonably low.
IV. SQUEEZED STATES
Having shown how to read and write quantum information in this system, we now wish to demonstrate the possibility of creating squeezed states. In an effort to mimic the Hamiltonian of a degenerate parametric amplifier from quantum optics 32 , we set the external wires magnetic fields to oscillate at double the frequency of the even mode, namely,
B 1 = B 1,0 e 2i ω5 t + c.c, B 2 = B 2,0 e 2i ω5 t + c.c,(31)
where ω 5 denotes the time average of ω 5 , and the oscillations of ω 5 about this average are small since B i ≪ B.
The non-interacting Hamiltonian is found from Eq. (18) to be H 0 =h ω 5 a † 5 a 5 +h ω 6 a † 6 a 6 .
Keeping only terms to first order in B i /B, the addition to the potential (4) due to the oscillating magnetic field in the rotating wave approximation (RWA), valid here due to the weak damping, is
H 1 = 4πE J l 2 1 B Φ 2 0 β L {(B 1 + B 2 )x 1 x 2 + B 1 x 2 1 + B 2 x 2 2 }. (33)
We write this addition in terms of ladder operators, take B 1,0 = B 2,0 to be pure imaginary and eliminate constant terms to find the interaction picture Hamiltonian in the RWA to be
H I = 8πi λ3 2 EJ l 2 1 B|B1,0| Φ 2 0 βL {a 2 5 − (a † 5 ) 2 },(34)
which is the squeezing Hamiltonian. In writing Eq. (34) we neglected the time-dependent term (ω 5 − ω 5 )a † 5 a 5 + (ω 6 − ω 6 )a † 6 a 6 , since it is negligible relative to the squeezing term under the assumed experimental conditions. It is interesting to note that squeezing of the odd mode is not possible using this scheme, even if ω 5 is replaced with ω 6 in Eq. (31). The fundamental reason for this is Lenz law, which makes the coefficient of x 1 x 2 in Eq. (33) positive and thus excludes terms proportional to a 2 6 in Eq. (34). The coefficient is positive because moving both beams in the positive direction costs energy, since both movements decrease the magnetic flux.
We now consider the effect of squeezing in this system and devise means to observe it. We assume dissipation is weak, and both beams are initially in the ground state. Conforming to standard notation, the squeezing parameter is
g = 16π λ 3 2 E J l 2 1 B|B 1,0 |t hΦ 2 0 β L ,(35)
which is real, and the squeezing operator reads
S(g) = exp[g(a 2 5 − (a † 5 ) 2 )/2].(36)
The time evolution of the a 5 operator in the interaction picture is given by
S † (g)a 5 S(g) = a 5 cosh g − a † 5 sinh g.(37)
In the rotating frame the uncertainty in the positions and momenta of the beams are
∆x i = λ 4 (1 + tanh(g + ln(ω 5 /ω 6 )/2)) −1/2 ,(38)∆p i =h 2 λ −1 4 (1 − tanh(g + ln(ω 5 /ω 6 )/2)) −1/2 ,(39)
where Eqs. (7), (9), (15), (16) and (37) were used. We see that we have limited squeezing to below the standard quantum limit in the positions and unlimited antisqueezing in the momenta, as the even mode is squeezed, while the odd mode is not. The interaction modifies the squeezing by adding the positive term of ln(ω 5 /ω 6 )/2 to the squeezing parameter. In addition, with the product of the uncertainties being ∆x i ∆p i =h 2 cosh(g + ln(ω 5 /ω 6 )/2), (40) we see that due to the interaction the minimum uncertainty is no longer attained before squeezing takes place. It is interesting to write the wavefunction for the beams in the differential representation in the presence of squeezing. This wavefunction can be found by using the relation between the two-photon coherent states and the squeezed states 32 to write for the squeezed state |g (cosh ga 5 + sinh ga † 5 )a 6 |g = 0.
After moving to the differential representation, Eq. (41) is solved to find a wavefunction of the form
ψ(x e , x o ) = C 1 e − m 1 ω 5 2h e 2g x 2 e e − m 1 ω 6 2h x 2 o ,(42)
with C 1 being a normalizing constant. When we transform this wavefunction to the beam coordinates via Eq. (14) we find that the new wavefunction is in a jointly Gaussian form, namely
ψ(x 1 , x 2 ) = C 1 exp(− x 2 1 σ 2 1 + x 2 2 σ 2 2 − 2rx1x2 σ1σ2 2(1 − r 2 ) ),(43)
where σ 1 = σ 2 = √ 2 ∆x 1 , with ∆x 1 given by Eq. (38), and the correlation coefficient given by
r = − tanh(g + ln(ω 5 /ω 6 )/2).(44)
We note that the factor of √ 2 in σ i comes from |ψ(x 1 , x 2 )| 2 , rather than ψ(x 1 , x 2 ), being the probability distribution. In addition, we see that, as before, the beam interaction results in an addition to the squeezing parameter, which gives negative correlation even at t = 0. The correlation due to the squeezing is negative, because the influence of the odd mode, which is not squeezed, increases with time, producing perfect anti-correlation when the squeezing parameter goes to infinity.
Lastly, we consider the effect of the squeezing on the measurement of the magnetic flux in the SQUID. Using Eq. (22), we find the standard deviation of Φ in the rotating frame to be
∆ Φ = √ 2(B + B 1 )l 1 λ 3 e −g ,(45)
which is fully squeezed, while in the lab frame we have
∆ Φ = √ 2(B + B 1 )l 1 λ 3 × cosh(2g) − sinh(2g) cos(2ω 5 t),(46)
which characteristically oscillates between fully squeezed values at t = (π/ω 5 )p, corresponding to Eq. (45), and fully anti-squeezed values at t = (π/ω 5 )(p + 1/2), where p is an integer. We conclude that the squeezing effect is measurable, and that the squeezing parameter can be found from the measurements.
V. EFFECT OF MECHANICAL DAMPING
In reality, the damping of the beam oscillations is weak but nonzero. With regard to reading and writing quantum information, this is not a problem, so long as the reading or writing is performed within a period much shorter than the characteristic decay time. The squeezed states, however, are measurably degraded even by very weak dissipation, as we show in this section.
Many models were devised for describing dissipation in quantum systems 41 . We choose here to work with the quantum master equation. In the interaction picture with the Hamiltonian (34), the quantum master equation takes the form 42
∂ ∂t ρ(t) = L S ρ(t) + L dis ρ(t),(47)L S ρ(t) = 1 2 ζ[a 2 − (a † ) 2 , ρ(t)],(48)L dis ρ(t) = − γ 2 (n cav + 1){[a † , aρ(t)] + [ρ(t)a † , a]} − γ 2 n cav {[a, a † ρ(t)] + [ρ(t)a, a † ]},(49)
where ζ = g/t is the squeezing rate and for brevity we write a instead of a 5 and ω instead of ω 5 . In Eqs. (47)(48)(49) ρ(t) is the statistical operator for the system, L S and L dis are the Liouville operators for the squeezing and dissipation, respectively, γ = ω/Q is the damping rate of the even mode, where Q is the beam quality factor, and n cav = (eh ω/kB T −1) −1 is the average phonon occupation in the even mode.
The system can be equivalently described by the Wigner quasi-probability distribution W (α, α * ) instead of by the statistical operator ρ(t), where we omit the explicit time dependence in W (α, α * ) to make the notation concise. The parameter α = X 1 + iX 2 is a complex number that is related to the phase space coordinates via α = 1 2λ x + iλ h p, where x and p are the even mode position and momentum coordinates, respectively, and λ = h/2m 1 ω as before. We convert 7,41 Eq. (47) to an equation for the Wigner distribution to find
∂W (X1,X2) ∂t = [ζ(X 1 ∂ ∂X1 − X 2 ∂ ∂X2 ) + γ 2 ( ∂ ∂X1 X 1 + ∂ ∂X2 X 2 ) + 1 4 γ(n cav + 1 2 )( ∂ 2 ∂X 2 1 + ∂ 2 ∂X 2 2 )]W (X 1 , X 2 ),(50)
where we note that the original Wigner function W (x, p) is related to the one used here by W (α, α * ) = W (X 1 , X 2 ) = 2hW (x, p). Equation (50) is seen to be a special case of the Fokker-Planck equation with W (u) corresponding to the probability distribution P (u; t), where u = (X 1 , X 2 ). Put in this form, the equation can be formally written as
∂W (u) ∂t = −∇ · [F(u)W (u)] + D 0 2 ∇ 2 W (u),(51)
where F = (−(ζ + γ 2 )X 1 , (ζ − γ 2 )X 2 ) and D 0 = 1 2 γ(n cav + 1 2 ) are the force and diffusion constant, respectively. Due to the form of the force in Eq. (51), we can use separation of variables to break this equation into two one-dimensional Fokker-Planck equations with solutions W 1 (X 1 ) and W 2 (X 2 ), where W (X 1 , X 2 ) = W 1 (X 1 )W 2 (X 2 ). These solutions are given by (i = 1, 2)
W i (X i ) = 1 √ 2πσ i (t) exp(− X 2 i 2σ 2 i (t) ),(52)σ i (t) = ( 1 4 − D 0 k i )e −kit + D 0 k i ,(53)k 1 = 2ζ + γ, (54) k 2 = γ − 2ζ,(55)
where k i are the decay rates. Equations (52-55) indicate that a steady-state solution always exists for W 1 (X 1 ) and is given by Eq. (52) with σ 1 (t) = D 0 /k 1 . This finite distribution width corresponds to a saturation in the squeezing in contrast with the dissipationless case, when the field quadrature X 1 is squeezed without limit. 32 For W 2 (X 2 ) on the other hand, we have a steady-state solution only at the strong damping regime, γ > 2ζ, and this solution exhibits σ 2 (t) = D 0 /k 2 . When the strong damping condition is not satisfied, k 2 is negative, there is no steady state, and X 2 is anti-squeezed as in the dissipationless case 32 , but at a slower pace since the leading behavior in ∆X 2 is e (ζ−γ/2)t instead of e ζt as in the dissipationless case.
The knowledge of the Wigner function in Eq. (52) enables us to calculate of system properties via the relation 41
{a r (a † ) s } sym = d 2 αα r (α * ) s W (α, α * ), (56)
where {·} sym indicates the average of all the permutations of the ladder operators, and d 2 α = dX 1 dX 2 . Working in the rotating frame, the resulting uncertainties in the positions and momenta of the beams read (i = 1, 2)
∆x i = √ 2λ 3 1 4 ω 5 ω 6 + σ 1 (t) 2 ,(57)∆p i = 1 √ 2h λ −1 3 1 4 ω 6 ω 5 + σ 2 (t) 2 ,(58)
which reduce to Eqs. (38)(39) when γ = 0. We see that the squeezing in the position coordinates, already limited to ω 3 /2ω 6 of the standard quantum limit, λ 1 , in the dissipationless case of Eq. (38), is limited here as well with the same limit, where we take n cav = 0 due to the previous assumption ofhω ≫ k B T . The momenta uncertainties, in comparison, are anti-squeezed only in the weak damping regime, γ < 2ζ, compared with being always anti-squeezed in Eq. (39), when there is no damping. As with the quadrature field X 2 , the momenta anti-squeezing in the weak damping regime has a slower rate relative to the dissipationless case with a leading behavior of e (ζ−γ/2)t vs. e ζt for the dissipationless case. The product of the position and momentum uncertainties in Eqs. (57-58) gives the lowest uncertainty at t = 0 and higher values afterwards.
As in Sec. IV, we wish to find here the effect of squeezing on the measurement of the magnetic flux. Using Eq. (22), we find the standard deviation in the rotating frame to be which is squeezed, though only to a finite extent unlike the dissipationless case in Eq. (45), where it is fully squeezed. In the lab frame we have
∆ Φ = 2 √ 2(B + B 1 )l 1 λ 3 σ 1 (t),(59)∆ Φ = 2 √ 2(B + B 1 )l 1 λ 3 × cos 2 (ωt)σ 2 1 (t) + sin 2 (ωt)σ 2 2 (t),(60)
which oscillates between squeezed values at t = (π/ω)p, corresponding to Eq. (59), and squeezed/anti-squeezed values, depending on the damping regime, at t = (π/ω)(p + 1/2), where p is an integer. In the dissipationless limit of γ = 0 Eq. (60) reduces to Eq. (46). Eq. (60) is plotted in Fig. 3 with normalized units for the nodamping, weak-damping and strong-damping cases. We conclude again that the squeezing effect is detectible and leads to reduced variation in the measured magnetic flux in the SQUID.
VI. CONCLUSIONS
In this work we have demonstrated that a system composed of two nanomechanical resonators embedded in a dc SQUID can be used as two units of quantum memory and that only the even mode in these two units is readable by the SQUID. We showed how the state of the beams can be altered, corresponding to writing quantum information, and proved the amplitude distribution of the number states in the resulting state is Poisson distributed. We then proposed a scheme to squeeze the even mode of the resonators and thus decrease the noise in the SQUID magnetic flux. Taking dissipation into account, we found a criterion that separates the weak damping regime, where a steady state exists only in one field quadrature, from the strong damping one, where both field quadratures exhibit steady states. We then predicted the form of the fluctuations in the magnetic flux in the SQUID, by which squeezing can be observed.
The approximations and assumptions made during our derivations hold well for reasonable experimental values. For instance, for two identical 8 MHz resonators of length 25 µm and quality factor Q = 2·10 4 , an external magnetic field of 10 T, beam temperature of 0.1 mK, SQUID temperature of 20 mK and other parameter values similar to the ones in Refs. 27 and 28, we find the energy level differences in the Hamiltonian (18) to be much larger than both k B T and the level widths. Moreover, for the reading process, Eq. (23) gives a required SQUID sensitivity of 1.3·10 −5 Φ0 √ Hz for n ∼ 1 and sensitivity of 1.3·10 −5 1 √ 2n Φ0 √ Hz for n ≫ 1. A typical SQUID with a flux sensitivity of 10 −6 Φ 0 / √ Hz satisfies these conditions for n < 80. Regarding the squeezing, a major question is whether substantial squeezing can be achieved within the decoherence time for the states. The decoherence time for the resonators here can be made to be at least 5 µs 19,43,44 , while substituting the parameters above in Eq. (35) gives a characteristic squeezing time of τ sq ∼ 2 µs. We therefore conclude that substantial squeezing is achievable within the dephasing time.
The experimental realization of this system will be an important demonstration of macroscopic quantum behavior and squeezing in a nanomechanical system. In addition it can be used for detecting the position of the embedded nanomechanical beams with accuracy higher than the standard quantum limit. Stacking such SQUIDS in series, with the upper arm of the lower SQUID being also the lower arm of the upper one, can form a quantum data bus 45,46 , lead to a multi-mode entangled state 47 , and possibly multi-mode squeezing 32 . Another application of this system or a close variant of it is that of a quantum gate 47 acting on the two states by means of currents in the external wires. A series of such quantum gates can form the basis of a nanomechanical quantum computer 48,49 . We leave the development of these ideas for future studies.
FIG. 2 .
2Graph of the maximum probability of the ground state of the shifted harmonic potential having a |n component in the unshifted potential. We have P00 = 1 and P11 = e −1 . The values for non-integer n were interpolated by n! = Γ(n + 1).
FIG. 3 .
3Normalized deviation in the SQUID magnetic flux ∆Φ(t)/∆Φ(t = 0) as a function of t/T1, where ncav = 0, T1 = 2π/ω and ζ = 0.1/T1. (a) No damping (γ = 0): Lower and upper limits are 0 and ∞. (b) Weak damping (γ = ζ): Lower and upper limits are 1/ √ 3 and ∞.
ACKNOWLEDGMENTSThis work has been partially funded by the NSF grant No. DMR-0802830. One of us (MD) is grateful to the Scuola Normale Superiore of Pisa for the hospitality during a visit where part of this work has been initiated, and to S. Pugnetti and R. Fazio for useful discussions.
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| []
|
[
"Sentiment Analysis of Covid-19 Tweets using Evolutionary Classification-Based LSTM Model",
"Sentiment Analysis of Covid-19 Tweets using Evolutionary Classification-Based LSTM Model"
]
| [
"Arunava Kumar Chakraborty [email protected] \nDept. of Computer Science & Engineering\nRCC Institute of Information Technology\nBeleghataKolkata -700015India\n",
"Sourav Das \nMaulana Abul Kalam Azad University of Technology\nSalt LakeKolkata -700064WBIndia\n",
"Anup Kumar Kolya \nDept. of Computer Science & Engineering\nRCC Institute of Information Technology\nBeleghataKolkata -700015India\n"
]
| [
"Dept. of Computer Science & Engineering\nRCC Institute of Information Technology\nBeleghataKolkata -700015India",
"Maulana Abul Kalam Azad University of Technology\nSalt LakeKolkata -700064WBIndia",
"Dept. of Computer Science & Engineering\nRCC Institute of Information Technology\nBeleghataKolkata -700015India"
]
| []
| As the Covid-19 outbreaks rapidly all over the world day by day and also affects the lives of million, a number of countries declared complete lockdown to check its intensity. During this lockdown period, social media platforms have played an important role to spread information about this pandemic across the world, as people used to express their feelings through the social networks. Considering this catastrophic situation, we developed an experimental approach to analyze the reactions of people on Twitter taking into account the popular words either directly or indirectly based on this pandemic. This paper represents the sentiment analysis on collected large number of tweets on Coronavirus or Covid-19. At first, we analyze the trend of public sentiment on the topics related to Covid-19 epidemic using an evolutionary classification followed by the n-gram analysis. Then we calculated the sentiment ratings on collected tweet based on their class. Finally, we trained the long-short term network using two types of rated tweets to predict sentiment on Covid-19 data and obtained an overall accuracy of 84.46%. | 10.1007/978-981-16-1543-6_7 | [
"https://arxiv.org/pdf/2106.06910v1.pdf"
]
| 235,421,792 | 2106.06910 | 4381e0336bedd09641f50bf357cb94d5aa7a4872 |
Sentiment Analysis of Covid-19 Tweets using Evolutionary Classification-Based LSTM Model
Arunava Kumar Chakraborty [email protected]
Dept. of Computer Science & Engineering
RCC Institute of Information Technology
BeleghataKolkata -700015India
Sourav Das
Maulana Abul Kalam Azad University of Technology
Salt LakeKolkata -700064WBIndia
Anup Kumar Kolya
Dept. of Computer Science & Engineering
RCC Institute of Information Technology
BeleghataKolkata -700015India
Sentiment Analysis of Covid-19 Tweets using Evolutionary Classification-Based LSTM Model
Covid-19Gram SelectionLSTMSentiment Analysis
As the Covid-19 outbreaks rapidly all over the world day by day and also affects the lives of million, a number of countries declared complete lockdown to check its intensity. During this lockdown period, social media platforms have played an important role to spread information about this pandemic across the world, as people used to express their feelings through the social networks. Considering this catastrophic situation, we developed an experimental approach to analyze the reactions of people on Twitter taking into account the popular words either directly or indirectly based on this pandemic. This paper represents the sentiment analysis on collected large number of tweets on Coronavirus or Covid-19. At first, we analyze the trend of public sentiment on the topics related to Covid-19 epidemic using an evolutionary classification followed by the n-gram analysis. Then we calculated the sentiment ratings on collected tweet based on their class. Finally, we trained the long-short term network using two types of rated tweets to predict sentiment on Covid-19 data and obtained an overall accuracy of 84.46%.
Introduction
On 31st December, 2019 the Covid-19 outbreak was first reported in the Wuhan, Hubei Province, China and it started spreading rapidly all over the world. Finally, WHO announced Covid-19 outbreak as pandemic on 11th March, 2020, when the virus continues to spread [1]. Starting from China, this virus infected and killed thousands of people from Italy, Spain, USA, UK, Brazil, Russia, and other many more countries as well. On 21st August 2020, more than 22.5 million cases of Covid-19 were reported in more than 188 countries and territories, yielding more than 7,92,000 deaths; although 14.4 million people have reported to be recovered 1 . While this pandemic has 1 https://gisanddata.maps.arcgis.com/apps/opsdashboard/index.html#/bda7594740fd40299 423467b48e9ecf6 continued to affect the lives of millions, many countries had enforced a strict lockdown for different periods to break the chain of this pandemic [1]. Since the Covid-19 vaccines are still yet to be discovered, therefore maintaining social distancing is the one and only one solution to check the spreading rate of this virus [2]. During the lockdown period a lot of people have chosen the Twitter to share their expression about this disease so we have been inspired to measure the human sensations about this epidemic by analyzing this huge Twitter data [3].
Initially, we have to face many challenges at the time of streaming the English tweets from the multilingual tweets all over the world as most of the peoples of foreign countries have used their native languages rather than English to express their feelings on social media [3]. However, we have developed our dataset considering the English tweets exclusively on Covid-19 of 160k tweets during April-May, 2020. We found the most popular words from the word corpus. Then we analyzed the trend of tweets using n-gram model. Further we assigned sentiment scores to our preprocessed tweets based on their sentiment polarity and classified our dataset on basis of their sentiment scores. Finally, we used those tweets and their sentiment ratings to train our LSTM model.
The following sections are furnished as follows: In Section 2 we have described some previous related research works. The architecture of our dataset and proposed pre-processing approach presented in Section 3. The Section 4 consists of Feature A for identifying the Covid-19 specified words based on the word lexicon. In Section 5, we have described Feature B as the trend of tweet words using n-gram model. The evolutionary classification on the sentiment-rated tweets based on their sentiment polarity has given in Section 6. In Section 7 we trained our LSTM model based on the classified tweets including their sentiment ratings. Whereas the Section 8 concludes the future prospects of our research work.
Related Works
A machine learning and cloud computing-based Covid-19 prediction model has been developed on May, 2020 to predict the future trend of this epidemic. They mainly used probabilistic distribution functions like Gaussian, Beta, Fisher-Tippet, and Log Normal functions to predict the trend [1]. A Covid-19 trend prediction model has introduced on June, 2020 for predicting the number of COVID-19 positive cases in different states of India. The researchers mainly developed a LSTM-based prediction model as LSTM model performs better for time series predictions. They tested different LSTM variants such as stacked, convolutional, and Bidirectional LSTM on the historical data, and based on the absolute error they found that Bi-LSTM gives more accurate results over other LSTM models for short-term prediction [4].
On July, 2020 the evolutionary K-means clustering on twitter data related to Covid-19 has been done by some researchers. They analyzed the tweet patterns using n-gram model. As the result they observed the difference between the occurrences of n-grams from the dataset [5].
Another research work describes a deep LSTM architecture for Message-level and Topic-based sentiment analysis. The authors used LSTM networks augmented with two kinds of attention mechanisms, on top of word embeddings pre-trained on a big collection of Twitter messages [7]. A group of researchers developed LSTM hyperparameter optimization for neural network-based Emotion Recognition framework. In their experiment they found that optimizing LSTM hyperparameters significantly improve the recognition rate of fourquadrant dimensional emotions with a 14% increase in accuracy and the model based on optimized LSTM classifier achieved 77.68% accuracy by using the Differential Evolution algorithm [8]. Fig. 1 we have represented an overview of our dataset.
Preparing Covid-19 Dataset
Data Pre-Processing
Data pre-processing is mainly used for cleaning the raw data by following certain steps to achieve the better result for further evaluations. We have done the preprocessing on our collected data by developing a user defined pre-processing function based on NLTK (Natural Language Toolkit, a Python library for NLP). Stemming helps to reduce inflected words to their word stem, base, or root form whereas by using Tokenization this function splits each of the sentence into smaller parts of word.
Feature A: Covid-19 Specified Words Identification
After pre-processing we have developed the Bag-of-Words (BOW) model using the frequently occurred words from the word lexicon and we obtained a list of most frequent Covid-19 exclusive words. We have represented a dense word-cloud in Fig. 2 of some of the mostly used words within the corpus.
Word Popularity
Several words within the generated corpus have been found at different times in different positions of the tweets. Here we have counted the recurrence of each word and presented the top 50 popular words along with their popularity in Fig. 3. After finding the word popularity, we have calculated the probability of repetition for each word on the basis of total 3,53,704 words from the corpus. Table 1 represents the popularity and probability scores of some most frequent words.
P(W i ) = count(W i ) ∑ count(W i=0 n ) n i=0
(1)
Feature b: Word Popularity Detection using N-gram
Lexical n-gram models are widely used in Natural Language Processing for statistical analysis & syntax feature mapping. We developed n-gram model to analyze our generated corpus consisting of tokenized words for finding the popularity of words or group of adjacent words. Here the probability of the occurrence of a sequence can be calculated using probability chain rule:
P(x 1 , x 2 , x 3 , ... x n ) = P(x 1 ) P(x 2 | x 1 ) P(x 3 | x 1 , x 2 ) … P(x n | x 1 , x 2 , x 3 ,… x (n-1) ) (2)
= ∏ P(x i | x 1 (i-1) ) n i=1(3)
For example, we can consider a sentence as "Still Covid-19 wave is running". Now as per the probability chain rule, P("Still Covid19 wave is running") = P("Still") x P("Covid19" | "Still") x P("wave" | "Still Covid19") x P("is" | "Still Covid19 wave") x P("running" | "Still Covid19 wave is"). The probabilities of words in each sentence after applying probability chain rule:
P(W 1 ,W 2 , W 3 , ... W n ) = ∏ P(W j | W 1 ,W 2 , W 3 , ... W (j-1) ) j (4) = ∏ P(W j | W 1 (j-1) ) n j=1(5)
The bigram model estimates the probability of a word by using only the conditional probability P(W i |W i-1 ) of one preceding word on given condition of all the previous words P(W i |W 1 i-1 ) [6].
P(W 1 , W 2 ) = ∏ P i=2 (W 2 | W 1 )(6)
The expression for the probability is - We have identified the most popular unigrams, bigrams, and trigrams within our corpus using the n-gram model. The graphical representations of 50 most popular unigrams, bigrams, and trigrams along with their popularity are presented in Fig. 4, Fig. 5 and Fig. 6 respectively. As the result of this analysis, we found that the popularity of trigrams is lesser than that of bigrams and the unigrams popularity is the highest according to this n-gram model.
P(W k | W (k-1) ) = count (W (k-1), W k ) count (W (k-1) )(7)
Sentiment Analysis
To measure the trend of public opinions we use Sentiment Analysis, a specific type of Data Mining through Natural Language Processing (NLP), computational linguistics and text analysis. The subjective information from the social media are analyzed and extracted to classify the text in multiple classes like positive, negative, and neutral.
Here we calculated the sentiment polarity of each cleaned and preprocessed tweet using the NLTK-based Sentiment Analyzer and get the sentiment scores for positive, negative and neutral classes to calculate the compound sentiment score for each tweet.
Sentiment Classification
We have classified the tweets on the basis of the compound sentiments into three different classes, i.e. Positive, Negative and Neutral. Then we have assigned the sentiment polarity rating for each tweet based on the algorithm presented in Table 2. Fig. 7, we have represented the sentiment classification with the overall percentage of each positive, negative, and neutral tweet found in the dataset. It can be visualized that the sentiment classes are naturally imbalanced as a large portion of social media users are either negative or neutral against the Covid-19 and the medical details associated with it.
Sentiment Modeling using Sequential LSTM
In traditional textual sentiment analysis, LSTM (Long-Short Term Model) network have already been proven to be performing better than the similar neural models [3]. We exploit a sequential LSTM model for sentiment evaluation of our Covid-19 dataset. We developed a new dataset consisting of the cleaned and preprocessed tweets along with their corresponding positive (1.0) and negative (0.0) sentiments. Then we created two sets X and y for the cleaned tweets and their sentiment scores, respectively and split the dataset into 80:20 ratio, i.e., 80% for training (X_train, y_train) and 20% for validation (X_test, y_test) purposes, respectively. A large number of Covid-19 exclusive words were generated by this model from the new dataset. Then we converted these words into word vectors using word2vec by setting the vector dimension as 200 for each collected n-grams within a sentence and developed new X_train, X_test sets consisting with the calculated word vectors for further processing. From updated training set the word vectors and the respective sentiment scores fed in the model as the first layer of inputs. In this experiment, we used TensorFlow framework and keras library to add Sequential LSTM model with Dense layers. We have trained our five-layered model for 30 epochs with two types of outline activation function with parameters, optimizer, loss, and accuracy. We have used ReLU (Rectified Linear Unit) activation function for the initial set of Dense layers with 128, 64, and 32 units, respectively and Sigmoid activation function for the outermost final Dense layer with 2 units. During the training we have used 32 batches and 2 verbose for our model. For some epochs Table 3 represents the training accuracy vs. loss and validation accuracy vs. loss, respectively. After completion of the training on our model, we have finally achieved 91.67% of overall training accuracy whereas the validation accuracy is 84.46% on the testing data. Table 4 and Table 5 are representing the confusion matrix and classification report to present the differences between predicted and the actual tweets along with the different classes. In Fig. 8, we have plotted the percentages of training accuracy vs. loss and validation accuracy vs. loss, achieved by the Sequential LSTM model during compilation. From the figure it is evident that there has been a significant loss difference between the training and testing epochs. This indicates a slight overfitting of the data which can be postulated from the several tweet collection parameters differing from time to time in the tweets streaming phase.
Conclusion & Future Scope
This experiment is mainly focused on Deep Learning-based Sentiment Analysis on Covid-19 tweets. We extracted the mostly popular words and analyzed the popularity of group of words using n-gram model as two main features of our dataset. However, later we developed a model to assign the sentiment ratings to the tweets based on their sentiment polarities calculated by sentiment analyzer and classify all tweets into positive and negative classes based on their assigned sentiment ratings. Then, using this classified dataset containing the cleaned and preprocessed tweets and their sentiment ratings, i.e., 1.0 for positive and 0.0 for negative, we trained our Deep Learning-based LSTM model. We divided the dataset into 80:20 ratio, i.e., 80% for training and 20% for testing purposes. After running 30 epochs on almost 93,474 parameters, we achieved validation accuracy as 84.46%.
For the future work, we want to develop a polarity-popularity model based on the features extracted during this experiment so that we can assign the refined sentiment ratings to the tweet based on the polarity of mostly recurred words [3]. With that data we will train the deep learning model to enhance the validation accuracy of our system.
Fig. 1 .
1Partial snapshot of the Covid-19 tweets corpus.
Fig. 2 .
2Some of the most popular Covid-19 related words from our corpus.
Fig. 4 .
4Graphical representation of the popularity for most frequent unigrams.
Fig. 5 .
5Graphical representation of the popularity for most frequent bigrams.
Fig. 6 .
6Graphical representation of the popularity for most frequent trigrams.
Fig. 7 .
7Sentiment distribution of three class polarity along with the percentage of Covid-19 tweets occurred from each class.
Fig. 8 .
8Performance metrics from the training loss vs. accuracy and validation loss vs. accuracy by the proposed model.
Table 1 .
1Popularity & probability of most frequent words.Fig. 3. Graphical representation of the popularity for most frequent Covid-19 exclusive words.Words
Popularity
Probability
0
Covid19
91794
0.259522
1
Test
11663
0.032974
2
New
11305
0.031962
3
People
10834
0.030630
4
Death
10783
0.030486
Table 2 .
2Algorithm used for sentiment classification of our Covid-19 tweets.Algorithm Sentiment Classification of Tweets (compound, sentiment):
1. for each k in range (0, len(tweet.index)):
2. if tweetk[compound] < 0:
3.
tweetk[sentiment] = 0.0 # assigned 0.0 for Negative Tweets
4. elif tweetk[compound] > 0:
5.
tweetk[sentiment] = 1.0 # assigned 1.0 for Positive Tweets
6. else:
7.
tweetk[sentiment] = 0.5 # assigned 0.5 for Neutral Tweets
8. end
In
Table 3 .
3Training accuracy vs. loss, validation accuracy vs. loss within 30 epochs.Epochs Train Loss
Train Accuracy
Val Loss
Val Accuracy
Initially
59.93%
67.63%
56.18%
70.45%
5 th
42.71%
79.27%
43.75%
78.74%
10 th
35.91%
83.38%
39.58%
81.71%
15 th
30.43%
86.36%
39.28%
82.81%
20 th
26.23%
88.40%
41.04%
83.47%
25 th
22.44%
90.15%
41.96%
84.24%
30 th
19.38%
91.67%
45.57%
84.46%
Table 4 .
4Confusion Matrix.Actual
Predicted
Positive
Negative
Positive
8298 (TP)
1941 (FP)
Negative
1946 (FN)
7924 (TN)
Table 5 .
5Classification Report.Precision
Recall
F1-Score
Support
Positive (1.0)
0.81
0.81
0.81
10239
Negative (0.0)
0.80
0.80
0.80
9870
Avg / Total
0.81
0.81
0.81
20109
Predicting the growth and trend of COVID-19 pandemic using machine learning and cloud computing. S Tuli, S Tuli, R Tuli, S S Gill, 100222Internet of ThingsTuli, S., Tuli, S., Tuli, R., Gill, S.S.:. Predicting the growth and trend of COVID-19 pan- demic using machine learning and cloud computing. Internet of Things, p. 100222 (2020)
Twitter sentiment analysis during COVID19 outbreak. A D Dubey, Available at SSRN 3572023. Dubey, A.D.: Twitter sentiment analysis during COVID19 outbreak. Available at SSRN 3572023 (2020)
S Das, D Das, A K Kolya, Sentiment classification with GST tweet data on LSTM based on polarity-popularity model. 45Das, S., Das, D., Kolya, A.K.: Sentiment classification with GST tweet data on LSTM based on polarity-popularity model. Sadhana. 45(1) (2020)
Prediction and analysis of COVID-19 positive cases using deep learning models: a descriptive case study of India. P Arora, H Kumar, B K Panigrahi, Chaos, Solitons Fractals. 139110017Arora, P., Kumar, H., Panigrahi, B.K.: Prediction and analysis of COVID-19 positive cases using deep learning models: a descriptive case study of India. Chaos, Solitons Fractals 139, 110017 (2020)
Analysis of twitter data using evolutionary clustering during the COVID-19 Pandemic. I Arpaci, S Alshehabi, M Al-Emran, M Khasawneh, I Mahariq, T Abdeljawad, A E Hassanien, CMC-Comput. Mat. Cont. 651Arpaci, I.,Alshehabi, S.,Al-Emran, M.,Khasawneh, M.,Mahariq, I.,Abdeljawad, T.,Hassanien, A.E.: Analysis of twitter data using evolutionary clustering during the COVID-19 Pandemic. CMC-Comput. Mat. Cont. 65(1), 193-203 (2020)
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Long short termmemory hyperparameter optimization for a neural network based emotion recognition framework. B Nakisa, M N Rastgoo, A Rakotonirainy, F Maire, V Chandran, IEEE Access. 6Nakisa, B., Rastgoo,M.N., Rakotonirainy, A.,Maire, F., Chandran, V.: Long short term- memory hyperparameter optimization for a neural network based emotion recognition framework. IEEE Access 6, 49325-49338 (2018)
| []
|
[
"Almost D-split sequences and derived equivalences",
"Almost D-split sequences and derived equivalences"
]
| [
"Wei Hu \nSchool of Mathematical Sciences\nLaboratory of Mathematics and Complex Systems\nBeijing Normal University\nMOE\n100875BeijingPeople's Republic of China\n",
"Changchang Xi [email protected]@163.com \nSchool of Mathematical Sciences\nLaboratory of Mathematics and Complex Systems\nBeijing Normal University\nMOE\n100875BeijingPeople's Republic of China\n"
]
| [
"School of Mathematical Sciences\nLaboratory of Mathematics and Complex Systems\nBeijing Normal University\nMOE\n100875BeijingPeople's Republic of China",
"School of Mathematical Sciences\nLaboratory of Mathematics and Complex Systems\nBeijing Normal University\nMOE\n100875BeijingPeople's Republic of China"
]
| []
| In this paper, we introduce almost D-split sequences and establish an elementary but somewhat surprising connection between derived equivalences and Auslander-Reiten sequences via BB-tilting modules. In particular, we obtain derived equivalences from Auslander-Reiten sequences (or n-almost split sequences), and Auslander-Reiten triangles. | null | [
"https://arxiv.org/pdf/0810.4757v1.pdf"
]
| 14,921,979 | 0810.4757 | 4a8edf5bc62e276bc83f807d73cadce65a0fc099 |
Almost D-split sequences and derived equivalences
27 Oct 2008
Wei Hu
School of Mathematical Sciences
Laboratory of Mathematics and Complex Systems
Beijing Normal University
MOE
100875BeijingPeople's Republic of China
Changchang Xi [email protected]@163.com
School of Mathematical Sciences
Laboratory of Mathematics and Complex Systems
Beijing Normal University
MOE
100875BeijingPeople's Republic of China
Almost D-split sequences and derived equivalences
27 Oct 2008
In this paper, we introduce almost D-split sequences and establish an elementary but somewhat surprising connection between derived equivalences and Auslander-Reiten sequences via BB-tilting modules. In particular, we obtain derived equivalences from Auslander-Reiten sequences (or n-almost split sequences), and Auslander-Reiten triangles.
Introduction
Derived equivalence and Auslander-Reiten sequence are two important objects in the modern representation theory of algebras and groups. On the one hand, derived equivalence preserves many significant invariants of groups and algebras; for example, the number of irreducible representations, Cartan determinants, Hochschild cohomology groups, algebraic K-theory and G-theory(see [7], [11] and [9]). One of the fundamental results on derived categories may be the Morita theory for derived categories established by Rickard in his several papers [20,21,22], which says that two rings A and B are derived-equivalent if and only if there is a tilting complex T of A-modules such that B is isomorphic to the endomorphism ring of T . Thus, starting with a ring A, we may construct theoretically all rings which are derived-equivalent to A by finding all tilting complexes of A-modules. However, in practice, it is not easy to show that two given rings are derivedequivalent by finding a suitable tilting complex, as is indicated by the famous unsolved Broue's abelian defect group conjecture, which states that the module categories of a block algebra A of a finite group algebra and its Brauer correspondent B should have equivalent derived categories if their defect groups are abelian (see [7]). On the other hand, as is well-known, Auslander-Reiten sequence is of significant importance in the modern representation theory of Artin algebras, it contains rich combinatorial information on the module category (see [3]). A natural and fundamental question is: Is there any relationship between Auslander-Reiten sequences and derived equivalences ? In other words, is it possible to construct derived equivalences from Auslander-Reiten sequences or n-almost split sequences or Auslander-Reiten triangles ?
In the present paper, we shall provide an affirmative answer to this question and construct derived equivalences by the so-called almost D-split sequences (see Definition 3.1 below). Such sequences include Auslander-Reiten sequences and occur very frequently in the representation theory of Artin algebras (see the examples in Section 3 below). Our result in this direction can be stated in the following general form: This result reveals a mysterious connection between Auslander-Reiten sequences and derived equivalences, namely we have the following corollary. (1) Suppose 0 −→ X i −→ M i −→ X i−1 −→ 0 is an Auslander-Reiten sequence of finitely generated A-modules for i = 1, 2, · · · , n. Let M = n i=1 M i . Then End A (M ⊕ X n ) and End A (M ⊕ X 0 ) are derivedequivalent via an n-BB-tilting module. In particular, if 0 −→ X −→ M −→ Y −→ 0 is an Auslander-Reiten sequence, then the endomorphism algebras End A (X ⊕ M ) and End A (M ⊕ Y ) are derived-equivalent via BBtilting module, and have the same Cartan determinant.
(2) If A is self-injective and X is an A-module, then the endomorphism algebra End (A ⊕ X) of A ⊕ X and the endomorphism algebra End A (A ⊕ Ω(X)) of A ⊕ Ω(X) are derived-equivalent, where Ω is the syzygy operator.
Thus, by Corollary 1.2 or more generally, by Proposition 3.13 in Section 3 below, one can produce a lot of derived equivalences from Auslander-Reiten sequences or n-almost split sequences. We stress that the algebra End A (X ⊕ M ) and the algebra End A (M ⊕ Y ) in Corollary 1.2 may be very different from each other (see the examples in Section 6), though the mesh diagram of the Auslander-Reiten sequence is somehow symmetric. Another result related to Corollary 1.2 is Proposition 5.1 in Section 5 below, which produces derived equivalences from Auslander-Reiten triangles in a triangulated category. In particular, we have The paper is organized as follows: In Section 2, we recall briefly some basic notions and a fundamental result of Rickard on derived categories. Our main results, Theorem 1.1, is proved in Section 3, where we also provide several generalizations of Corollary 1.2; among others is a formulation of Corollary 1.2(1) for n-almost split sequences. In section 4, we point out that if an almost D-split sequence is given by an Auslander-Reiten sequence then Theorem 1.1 can be viewed as a "generalized" version of a BB-tilting module. Thus an nalmost split sequence or concatenating n Auslander-Reiten sequences provides us a natural way to get an n-BB-tilting module (for definition, see Section 4). In Section 5, we discuss how to get derived equivalences from Auslander-Reiten triangles in a triangulated category. In particular, Corollary 1.3 is proved in this section. In the last section we present an example to illustrate our main result.
Preliminaries
In this section, we recall some basic definitions and results required in our proofs.
Let C be an additive category. For two morphisms f : X −→ Y and g : Y −→ Z in C, the composition of f with g is written as f g, which is a morphism from X to Z. But for two functors F : C −→ D and G : D −→ E of categories, their composition is denoted by GF . For an object X in C, we denote by add (X) the full subcategory of C consisting of all direct summands of finite sums of copies of X.
A complex X • over C is a sequence of morphisms d i X between objects X i in C: · · · → X i−1 d i−1 X −→ X i d i X −→ X i+1 d i+1 X −→ X i+2 → · · · , such that d i X d i+1 X = 0 for all i ∈ Z. We write X • = (X i , d i X ).
The category of all complexes over C with the usual complex maps of degree zero is denoted by C (C). The homotopy and derived categories of complexes over C are denoted by K (C) and D(C), respectively. The full subcategory of C (C) consisting of bounded complexes over C is denoted by C b (C). Similarly, K b (C) and D b (C) denote the full subcategories consisting of bounded complexes in K (C) and D(C), respectively.
An object X in a triangulated category C with a shift functor [1] is called self-orthogonal if Hom C (X, X[n]) = 0 for all integers n = 0.
Let A be a ring with identity. By A-module we shall mean a left A-module. We denote by A-Mod the category of all A-modules, by A-mod the category of all finitely presented A-modules, and by A-proj (respectively, A-inj) the category of finitely generated projective ( respectively, injective) A-modules. Let X be an A-module. If f : P −→ X is a projective cover of X with P projective, then the kernel of f is called a syzygy of X, denoted by Ω(X). Dually, if g : X −→ I is an injective envelope with I injective, then the cokernel of g is called a co-syzygy of X, denoted by Ω −1 (X). Note that a syzygy or a co-syzygy of an A-module X is determined, up to isomorphism, uniquely by X. Hence we may speak of the syzygy and the co-syzygy of a module.
It is well-known that K (A-Mod), K b (A-Mod), D(A-Mod) and D b (A-Mod) all are triangulated categories. Moreover, it is known that if
X ∈ K b (A-proj) or Y ∈ K b (A-inj), then Hom K b (A−Mod) (X, Z) ≃ Hom D b (A−Mod) (X, Z) and Hom K b (A−Mod) (Z, Y ) ≃ Hom D b (A-Mod) (Z, Y ) for all Z ∈ D b (A-Mod).
For further information on triangulated categories, we refer to [11]. In [20], Rickard proved the following theorem.
Theorem 2.1 For two rings A and B with identity, the following are equivalent:
(a) D b (A-Mod) and D b (B-Mod) are equivalent as triangulated categories;
(b) K b (A-proj) and K b (B-proj) are equivalent as triangulated categories; (c) B ≃ End K b (A-proj) (T • ), where T • is a complex in K b (A-proj) satisfying (1) T • is self-orthogonal in K b (A-proj), (2) add (T • ) generates K b (A-proj) as a triangulated category.
If two rings A and B satisfy the equivalent conditions of Theorem 2.1, then A and B are said to be derived- (1) and (2) in Theorem 2.1 is called a tilting complex over A. Given a derived equivalence F between A and B, there is a unique (up to isomorphism) tilting complex T • over A such that F T • = B. This complex T • is called a tilting complex associated to F .
equivalent. A complex T • in K b (A-proj) satisfying the conditions
To get derived equivalences, one may use tilting modules. Recall that a module T over a ring A is called a tilting module if (1) T has a finite projective resolution 0 −→ P n −→ · · · −→ P 0 −→ T −→ 0, where each P i is a finitely generated projective A-module;
(2) Ext i A (T, T ) = 0 for all i > 0, and
(3) there is an exact sequence 0 −→ A −→ T 0 −→ · · · −→ T m −→ 0 of A-modules with each T i in add (T ).
It is well-known that each tilting module supplies a derived equivalence. The following result in [8] is a generalization of a result in [11,Theorem 2.10].
Lemma 2.2 Let
A be a ring, A T a tilting A-module and B = End A (T ). Then A and B are derived-equivalent. In this case, we say that A and B are derived-equivalent via a tilting module.
In Theorem 2.1, if both A and B are left coherent rings, that is, rings for which the kernels of any homomorphisms between finitely generated projective modules are finitely generated, then A-mod and B-mod are abelian categories, and the equivalent conditions in Theorem 2.1 are further equivalent to the condition A special class of coherent rings is the class of Artin algebras. Recall that an Artin R-algebra over a commutative Artin ring R is an R-algebra A such that A is a finitely generated R-module. For the module category over an Artin algebra, there is the notion of Auslander-Reiten sequence, or equivalently, almost split sequence. It plays an important role in the modern representation theory of algebras and groups. Recall that a short exact sequence 0 −→ X f −→ Y g −→ Z −→ 0 in A-mod is called an Auslander-Reiten sequence if (1) the sequence does not split, (2) X and Z are indecomposable, (3) for any morphism h : V −→ Z in A-mod, which is not a split epimorphism, there is a homomorphism
f ′ : V −→ Y in A-mod such that h = f ′ f , and (4) for any morphism h : X −→ V in A-mod, which is not a split monomorphism, there is a homomorphism f ′ : Y −→ V in A-mod such that h = f f ′ .
For an introduction to Auslander-Reiten sequences and representations of Artin algebras, we refer the reader to the excellent book [3].
Almost D-split sequences and derived equivalences
In this section, we shall construct derived equivalences from Auslander-Reiten sequences. This builds a linkage between Auslander-Reiten sequences (or n-almost split sequences) and derived equivalences. We start first with a general setting by introducing the notion of almost D-split sequences, which is a slight generalization of Auslander-Reiten sequences, and then use these sequences to construct derived equivalences between the endomorphism rings of modules involved in almost D-split sequences. In Section 5, we shall consider the question of getting derived equivalences from Auslander-Reiten triangles. Now we recall some definitions from [4]. Let C be a category, and let D be a full subcategory of C, and X an object in C. A morphism f : D −→ X in C is called a right D-approximation of X if D ∈ D and the induced map Hom C (−, f ): Hom C (D ′ , D) −→ Hom C (D ′ , X) is surjective for every object D ′ ∈ D. A morphism f : X −→ Y in C is called right minimal if any morphism g : X −→ X with gf = f is an automorphism. A minimal right D-approximation of X is a right D-approximation of X, which is right minimal. Dually, there is the notion of a left D-approximation and a minimal left D-approximation. The subcategory D is called contravariantly (respectively, covariantly ) finite in C if every object in C has a right (respectively, left) D-approximation. The subcategory D is called functorially finite in C if D is both contravariantly and covariantly finite in C.
Let C be an additive category and e : X −→ X an idempotent morphism in C. We say that e splits if there are objects X ′ and X ′′ in C and an isomorphism ϕ : X ′ ⊕ X ′′ −→ X such that ϕe = πλϕ, where π : X ′ ⊕ X ′′ −→ X ′ and λ : X ′ −→ X ′ ⊕ X ′′ are the canonical morphisms. In an arbitrary additive category, all idempotents need not split, but of course, in the case where C is an abelian category, every idempotent splits. If all idempotents in C split, then so is every full subcategory D of C which is closed under direct summands. Moreover, for an additive category C such that every idempotent splits, we know that, for each object M in C, the functor Hom C (M, −) induces an equivalence between add (M ) and End C (M )-proj.
X f −→ M g −→ Y in C is called an almost D-split sequence if (1) M ∈ D;
(2) f is a left D-approximation of X, and g is a right D-approximation of Y ;
(3) f is a kernel of g, and g is a cokernel of f .
Recall that a morphism f : Y −→ X in an additive category C is a kernel of a morphism g : X −→ Z in C if f g = 0, and for any morphism h : V −→ X in C with hg = 0, there is a unique morphism h ′ : V −→ Y such that h = h ′ f . Note that if a morphism has a kernel in C then it is unique up to isomorphism. A cokernel of a given morphism in C is defined dually. If f : Y −→ X in C is a kernel of a morphism g :
X −→ Z in C, then f is a monomorphism, that is, if h i : U −→ Y is a morphism in C for i = 1, 2, such that h 1 f = h 2 f , then h 1 = h 2 . Similarly, if g : X −→ Z in C is a cokernel of a morphism f : Y −→ X in C, then g is an epimorphism, that is, if h i : Z −→ V is a morphism in C for i = 1, 2, such that gh 1 = gh 2 , then h 1 = h 2 .
Notice that an almost D-split sequence may split, whereas an Auslander-Reiten sequence never splits. Now we give some examples of almost D-split sequences.
Examples. (a) Let A be an Artin algebra and C = A-mod. Suppose D is the full subcategory of A-mod consisting of all projective-injective A-modules in C. If g : M −→ X is a surjective homomorphism in A-mod with M ∈ D, then the sequence 0 −→ ker(g) −→ M −→ X −→ 0 is an almost D-split sequence in C, where ker(g) stands for the kernel of the homomorphism g.
(b) Let A be an Artin algebra and
C = A-mod. Suppose 0 −→ X −→ M −→ Y −→ 0
is an Auslander-Reiten sequence. Let N be any module such that M ∈ add(N ), but neither X nor Y belongs to add(N ). If we take D = add(N ), then the Auslander-Reiten sequence is an almost D-split sequence in C.
(c) Let A be an Artin algebra and M ∈ A-mod. Recall that M is an almost complete tilting module if M is a partial tilting module (that is, M has finite projective dimension and Ext i A (M, M ) = 0 for all i > 0), and if the number of all non-isomorphic direct summands of M equals the number of non-isomorphic simple A-modules minus 1. An indecomposable A-module X ∈ A-mod is called a tilting complement to M if M ⊕X is a tilting A-module. If an almost complete tilting module M is faithful, then there is an exact (not necessarily infinite) sequence
0 −→ X 0 f1 −→ M 1 f2 −→ M 2 f3 −→ · · · of A-modules such that M i ∈ add(M ). Moreover, if we define X i = coker(f i ), the co-kernel of f i for i ≥ 1, then X i ≃ X j for i = j, proj.dim A (X i ) ≥ i for any i, and {X i | i ≥ 0}
is a complete set of nonisomorphic indecomposable tilting complements to M . In addition, each X i −→ M i+1 is a minimal left add(M )-approximation of X i and each M j −→ X j is a minimal right add(M )-approximation of X j . Thus
the sequence 0 −→ X i −→ M i+1 −→ X i+1 −→ 0 is an almost add(M )-split sequence in A-mod for all i ≥ 0.
For further information on almost complete tilting modules and relationship with the generalized Nakayama conjecture, we refer the reader to [6] and [13]. Now we consider some properties of an almost D-split sequence.
Proposition 3.2 Let C be an additive category and D a full subcategory of C.
(
1) Suppose D ′ is a full subcategory of D. If a sequence X −→ M −→ Y in C is an almost D-split sequence with M ∈ D ′ , then it is an almost D ′ -split sequence in C. (2) If X −→ M g −→ Y and X ′ −→ M ′ g ′ −→ Y ′ are almost D-split sequences in C such that both g and g ′ are right minimal, then Y ≃ Y ′ if and only if the two sequences are isomorphic. Similarly, If X f −→ M −→ Y and X ′ f ′ −→ M ′ −→ Y ′ are almost D-split sequences in C such that both f and f ′ are left minimal, then X ≃ X ′ if and only if the two sequences are isomorphic.
Proof. (1) is clear. We prove the first statement of (2). If the two sequences are isomorphic, then
X ≃ X ′ and Y ≃ Y ′ . Now assume that φ : Y −→ Y ′ is an isomorphism. Then gφ factors through g ′ since g ′ is a right D- approximation of Y ′ , and we may write gφ = hg ′ for some h : M −→ M ′ . Similarly, there is a homomorphism h ′ : M ′ −→ M such that g ′ φ −1 = h ′ g. Thus hh ′ g = hg ′ φ −1 = gφφ −1 = g and h ′ hg ′ = h ′ gφ = g ′ φ −1 φ = g ′ .
Since both g and g ′ are right minimal, the morphisms hh ′ and h ′ h are isomorphisms. It follows easily that h itself is an isomorphism. Since f ′ is a kernel of g ′ and since f is a kernel of g, there is a morphism k : X −→ X ′ and a morphism k ′ :
X ′ −→ X such that kf ′ = f h and k ′ f = f ′ h −1 . Thus kk ′ f = kf ′ h −1 = f hh −1 = f . It follows that kk ′ = 1 X since f is a monomorphism. Similarly, we have k ′ k = 1 X ′ .
Hence k is an isomorphism and the two sequences are isomorphic. Similarly, the other statements in (2) can be proved.
To get an almost D-split sequence, we may use the following proposition. First, we introduce some notations. Let D be a full subcategory of a category C. An object C in C is said to be generated (respectively,
co-generated ) by D if there is an epimorphism D −→ C (respectively, monomorphism C −→ D) with D ∈ D.
We denote by F (D) the full subcategory of C consisting of all objects C ∈ C generated by D, and by S (D) the full subcategory of C consisting of all objects C ∈ C co-generated by D.
Proposition 3.3 Suppose A is a ring with identity and
C = A-Mod. Let D be a full subcategory of C. We define X (D) = {X ∈ C | Ext 1 A (X, D) = 0} and Y (D) = {Y ∈ C | Ext 1 A (D, Y ) = 0}. (1) If D is contravariantly finite in C, then, for any A-module Y ∈ F (D) ∩ X (D), there is an almost D-split sequence 0 −→ X −→ D −→ Y −→ 0 in C. (2) If D is covariantly finite in C, then, for any A-module X ∈ S (D) ∩ Y (D), there is an almost D-split sequence 0 −→ X −→ D −→ Y −→ 0 in C. Proof. (1) Since Y is generated by D, there is a surjective right D-approximation of Y , say g : M −→ Y with M ∈ D. Let X be the kernel of g. Then it follows from the exact sequence 0 −→ X −→ M −→ Y −→ 0 that the sequence Hom A (M, D ′ ) −→ Hom A (X, D ′ ) −→ 0 is exact since Y ∈ X (D)
. This implies that the homomorphism X −→ M is a left D-approximation of X. Thus we get an almost D-split sequence in C. (2) can be proved analogously.
Our main purpose of introducing almost D-split sequences is to construct derived equivalences between the endomorphism algebras of objects appearing in almost D-split sequences. The following lemma is useful in our discussions.
Lemma 3.4 Let C be an additive category and M an object in
C. Suppose X f −→ M n −→ · · · −→ M 2 t −→ M 1 g −→ Y
is a (not necessarily exact) sequence of morphisms in C with M i ∈ add (M ) satisfying the following conditions:
(1) The morphism f : X −→ M n is a left add (M )-approximation of X, and the morphism g :
M 1 −→ Y is a right add (M )-approximation of Y ; (2) Put V = M ⊕ X and W = M ⊕ Y .
There are two induced exact sequences
0 −→ Hom C (V, X) f * −→ Hom C (V, M n ) → · · · → Hom C (V, M 1 ) g * −→ Hom C (V, Y ), 0 −→ Hom C (Y, W ) g * −→ Hom C (M 1 , W ) → · · · → Hom C (M n , W ) f * −→ Hom C (X, W ).
Then the endomorphism rings End C (M ⊕ X) and End C (M ⊕ Y ) are derived-equivalent via a tilting module of projective dimension at most n.
Proof. Let Λ be the endomorphism ring of V , and let T be the cokernel of the map [t 0] * : Hom C (V, M 2 ) −→ Hom C (V, M 1 ⊕ M ). Then, by (2), we have an exact sequence of Λ-modules:
0 → Hom C (V, X) → Hom C (V, M n ) → · · · → Hom C (V, M 2 ) → Hom C (V, M 1 ⊕ M ) → T → 0. ( * )
Note that all the Λ-modules appearing in the above exact sequence are finitely generated. Applying Hom Λ (−, Hom C (V, M )) to this sequence, we get a sequence which is isomorphic to the following sequence
0 −→ Hom Λ (T, Hom C (V, M )) −→ Hom C (M 1 ⊕ M, M ) −→ Hom C (M 2 , M ) −→ · · · −→ Hom C (M n , M ) f * −→ Hom C (X, M ) −→ 0.
By the second exact sequence in (2) and the fact that f is a left add (M )-approximation of X, we see that this sequence is exact. It follows that Ext i Λ (T, Hom C (V, M )) = 0 for all i > 0. Hence Ext i Λ (T, Hom C (V, M ′ )) = 0 for all i > 0 and M ′ ∈ add (M ). Thus, by applying Hom Λ (T, −) to the exact sequence ( * ), we get
Ext i Λ (T, T ) ≃ Ext i+n Λ (T, Hom C (V, X)) for all i > 0. But Ext i+n Λ (T, Hom C (V, X)) = 0 for all i > 0 since the projective dimension of T is at most n. Thus Ext i Λ (T, T ) = 0 for all i > 0.
Also, it follows from the exact sequence ( * ) that the following sequence
0 → Hom C (V, X ⊕ M )→Hom C (V, M n ⊕ M ) → · · · → Hom C (V, M 2 ) → Hom C (V, M 1 ⊕ M ) → T → 0 is exact, where Hom C (V, X ⊕ M )
is just Λ and the other terms are in add (T ). Thus T is a tilting Λ-module of projective dimension at most n.
Next, we show that End Λ (T ) and End C (W ) are isomorphic. If n = 1, we set V ′ = X and a = [f, 0] :
V ′ −→ M 1 ⊕ M . For n ≥ 2, we set V ′ = M 2 and a = [t, 0] : V ′ −→ M 1 ⊕ M . Let u : V ′ −→ V ′ and v : M 1 ⊕ M −→ M 1 ⊕ M be two morphisms in C. The morphism pair (u, v) is an endomorphism of the sequence V ′ −→ M 1 ⊕ M if ua = av. Let E be the endomorphism ring of the sequence V ′ −→ M 1 ⊕ M . Let I be the subset of E consisting of those endomorphisms (u, v) such that there exists h : M 1 ⊕ M −→ V ′ with ha = v. It is easy to check that I is an ideal of E. We shall show that End C (W ) is isomorphic to the quotient ring E/I. Let b be the morphism g 0 0 id : M 1 ⊕ M −→ W .
Then, by the second exact sequence of the condition (2), we have an exact sequence
0 −→ Hom C (W, W ) b * −→ Hom C (M 1 ⊕ M, W ) a * −→ Hom C (V ′ , W ). ( * * )
By considering the image of id W under the composition b * a * , we have ab = 0. Thus, for each (u, v) ∈ E, we have avb = uab = 0, which means that vb is in the kernel of a * . Therefore, there is a unique map q : W → W such that bq = vb. Now, we define η : E → End C (W ) by sending (u, v) to q. Then η is clearly a ring homomorphism. We claim that η is surjective. Indeed, since g is a right add (M )-approximation of Y , it is easy to check that the map b is a right add (M )-approximation of W . Let q be an endomorphism of W . Then there is a morphism v : M 1 ⊕ M −→ M 1 ⊕ M such that vb = bq. By the first exact sequence in (2), we have the following exact sequence:
Hom C (V ′ , V ′ ) a * −→ Hom C (V ′ , M 1 ⊕ M ) b * −→ Hom C (V ′ , W ).
It follows from avb = abq = 0 that av is in the kernel of b * and there is a map u :
V ′ −→ V ′ such that ua = av.
This implies that (u, v) is in E and η(u, v) = q. Hence η is surjective. Now, we determine the kernel of η. Note that, by the first exact sequence in (2), we have an exact sequence
Hom C (M 1 ⊕ M, V ′ ) a * −→ Hom C (M 1 ⊕ M, M 1 ⊕ M ) b * −→ Hom C (M 1 ⊕ M, W ).
Now, suppose (u, v) is in the kernel of η. Then vb = 0, which means that v is in the kernel of b * . Hence there is a map h :
M 1 ⊕ M −→ V ′ such that ha = v. This implies (u, v) ∈ I.
On the other hand, if (u, v) ∈ I and if η sends (u, v) to q, then bq = vb = hab = 0 and q is in the kernel of b * . By the exact sequence ( * * ), we have q = 0. Hence I is the kernel of η, and therefore E/I ≃ End C (W ). Let E be the endomorphism ring of the following complex of Λ-modules:
Hom C (V, V ′ ) a * −→ Hom C (V, M 1 ⊕ M ),
and I the ideal of E consisting of those (u, v) such that ha * = v for some h :
Hom C (V, M 1 ⊕ M ) −→ Hom C (V, V ′ )
. Similarly, we can show that End Λ (T ) is isomorphic to E/I. Finally, the natural map e : E −→ E, which sends (u, v) to (u * , v * ), is clearly an isomorphism of rings and induces an isomorphism from the ring E/I to the ring E/I. Thus End Λ (T ) and End C (W ) are isomorphic. The proof is completed.
Remarks.
(1) For an Auslander-Reiten sequence 0 → X → M → Y → 0 in A-mod with A an Artin algebra, the proof that End( A T ) of the tilting module T defined in Lemma 3.4 is isomorphic to End A (M ⊕ Y ) can be carried out very easily.
(2) From the proof of Lemma 3.4 we see that if we replace the second exact sequence in (2) by the following two exact sequences
0 −→ Hom C (Y, M ) g * −→ Hom C (M 1 , M ) → · · · → Hom C (M n , M ) f * −→ Hom C (X, M ), 0 −→ Hom C (Y, Y ) g * −→ Hom C (M 1 , Y ) t * −→ Hom C (M 2 , Y ),
then Lemma 3.4 still holds true. (Here M 2 = X if n = 1.) However, in most of cases that we are interested in, the second exact sequence in (2) does exist.
(3) A special case of Lemma 3.4 is the n-almost split sequences in a maximal (n−1)-orthogonal subcategory studied in [16]. Let A be a finite-dimensional algebra over a field. Suppose C is a functorially finite and full subcategory of A-mod. Recall that C is called a maximal (n− 1)-orthogonal subcategory if Ext i A (X, Y ) = 0 for all X, Y ∈ C and all 0 < i ≤ n − 1, and
C = C ∩ {X ∈ A-mod | Ext i A (C, X) = 0 for C ∈ C and 0 < i ≤ n − 1} = C ∩ {Y ∈ A-mod | Ext i A (Y, C) = 0 for C ∈ C and 0 < i ≤ n − 1}.
In [16]. It is shown that, for any non-projective indecomposable X in C (respectively, non-injective indecomposable Y in C), there is an exact sequence
( * ) 0 → Y fn −→ C n−1 fn−1 −→ · · · f1 −→ C 0 f0 −→ X → 0
with C j ∈ C and f j being radical maps such that the following induced sequences are exact on C:
0 −→ C(−, Y ) −→ C(−, C n−1 ) −→ · · · −→ C(−, C 0 ) −→ rad C (−, X) −→ 0, 0 −→ C(X, −) −→ C(C 0 , −) −→ · · · −→ C(C n−1 , −) −→ rad C (Y, −) −→ 0,
where rad C stands for the Jacobson radical of the category C. Note also that f 0 is a minimal right almost split morphism and that f n is a minimal left almost split morphism. The sequence ( * ) is called an n-almost split sequence in [16].
With Lemma 3.4 in mind, now we can show the significance of an almost D-split sequence for constructing derived equivalences by the following result.
( * ) 0 −→ Hom C (V, X) (−,f ) −→ Hom C (V, M ′ ) (−,g) −→ Hom C (V, Y ).
Since f is a monomorphism, the map (−, f ) is injective. Clearly, the image of the map (−, f ) is contained in the kernel of the map (−, g). Since f is a kernel of g, it is easy to see that the kernel of (−, g) is equal to the image of (−, f ). Thus ( * ) is exact. Similarly, we see that the sequence
0 −→ Hom C (Y, W ) (g,−) −→ Hom C (M ′ , W ) (f,−) −→ Hom C (X, W )
is exact. Thus Theorem 3.5 follows from Lemma 3.4 if we take n = 1.
In Theorem 3.5, the two rings End C (M ⊕X) and End C (M ⊕Y ) are linked by a tilting module of projective dimension at most 1. This is precisely the case of classic tilting module. Thus there is a nice linkage between the torsion theory defined by the tilting module in End C (M ⊕ X)-mod and the one in End C (M ⊕ Y )-mod. For more details we refer to [5] and [12].
In the following, we deduce some consequences of Theorem 3.5. Since an Auslander-Reiten sequence can be viewed as an almost D-split sequence, as explained in Example (b), we have the following corollary.
Corollary 3.6 Let A be an Artin algebra, and let 0 → X → M → Y → 0 be an Auslander-Reiten sequence in A-mod. Suppose N is an A-module in A-mod such that neither X nor Y belongs to add (N ). Then
End A (N ⊕M ⊕X) is derived-equivalent to End A (N ⊕M ⊕Y ). In particular, End A (M ⊕X) and End A (M ⊕Y ) are derived-equivalent.
As another consequence of Theorem 3.5, we have the following corollary.
Corollary 3.7 Let A be an Artin algebra and X a torsion-less A-module, that is, X is a submodule of a projective module in A-mod. If f : X → P is a left add ( A A)-approximation of X, then End A (A ⊕ X) and End ( A A ⊕ coker(f )) are derived-equivalent. In particular, if A is a self-injective Artin algebra, then, for any X in A-mod, the algebras End A (A ⊕ X) and End A (A ⊕ Ω(X)) are derived-equivalent via a tilting module.
Proof. Note that f is injective. Thus the short exact sequence
0 −→ X f −→ P −→ coker(f ) −→ 0
is an almost add( A A)-split sequence in A-mod. By Theorem 3.5, the corollary follows.
As a consequence of Corollary 3.7, we get the following corollary.
Corollary 3.8 Let A be a self-injective Artin algebra and X an A-module. Then the algebras End A (A ⊕ X)
and End A (A ⊕ τ X) are derived-equivalent, where τ stands for the Auslander-Reiten translation. Thus, for all n, the algebras End A (A ⊕ τ n X) are derived-equivalent.
Proof. Let ν be the Nakayama functor DHom A (−, A). It is known that if A is self-injective then τ ≃ νΩ 2 , ν(A) = A and the Nakayama functor is an equivalence from A-mod to itself. Since the algebra End A (A ⊕ τ X) is isomorphic to the algebra End A (A ⊕ Ω 2 (X)), the corollary follows immediately from Corollary 3.7.
Remark. If A is a finite-dimensional self-injective algebra, then, for any A-module X, it was shown in [19, Corollary 1.2] that the algebras End A (A ⊕ X), End A (A ⊕ Ω(X)) and End A (A ⊕ τ X) are stably equivalent of Morita type. Thus they are both derived-equivalent and stably equivalent of Morita type. For further information on stably equivalences of Morita type for general finite-dimensional algebras, we refer the reader to [17,18,19,24] and the references therein. Now, we point out the following consequence of Theorem 3.5: if 0 → X → M ′ → Y → 0 is an almost D-split sequence in A-mod with D = add(M ) for an A-module M , then X and Y have the same number of non-isomorphic indecomposable direct summands which are not in add(M ). This follows from the fact that a derived equivalence preserves the number of non-isomorphic simple modules.
Many other invariants of derived equivalences can be used to study the algebras End A (M ⊕ X) and End A (M ⊕ Y ); for example, End A (M ⊕ X) has finite global dimension if and only if End A (M ⊕ Y ) has finite global dimension. This follows from the fact that derived equivalence preserves the finiteness of global dimension. In fact, we have the following explicit formula by tilting theory (see [12] and [11, Proposition 3.4, p.116], for example):
If 0 → X → M ′ → Y → 0 is an almost D-split sequence in A-mod with D = add(M ) for an A-module M in A-mod, then gl.dim(End C (M ⊕ X)) − 1 ≤ gl.dim(End C (M ⊕ Y )) ≤ gl.dim(End C (M ⊕ X)) + 1,
where gl.dim(A) stands for the global dimension of A. Note that the global dimension of End C (M ⊕ X)) may be infinite (see Example 3 in Section 6). Concerning global dimensions and Auslander-Reiten sequences, there is a related result which can be found in [14].
Note that if a derived equivalence between two rings A and B is obtained from a tilting module A T , that is, there exists a tilting A-module A T such that B ≃ End A (T ), then the finitistic dimension of A is finite if and only if the finitistic dimension of B is finite (see [10]) a . Recall that the finitistic dimension of an Artin algebra a Recently, it is shown that the finiteness of finitistic dimension is invariant under an arbitrary derived equivalence.
A, denoted by fin.dim(A), is defined to be the supremum of the projective dimensions of finitely generated A-modules of finite projective dimension. The finitistic dimension conjecture states that fin.dim(A) should be finite for any Artin algebra A. This conjecture has closely been related to many other homological conjectures in the representation theory of algebras. For some advances and further information on the finitistic dimension conjecture, we may refer the reader to the recent paper [25] and the references therein.
Thus we have the following corollary.
Corollary 3.9 Let C be an additive category and M an object in C. Suppose If A is an Artin R-algebra over a commutative Artin ring R and M is an A-bimodule, then A ⋉ M , the trivial extension of A by M is the R-algebra whose underlying R-module is A ⊕ M , with multiplication given by
X f −→ M ′ g −→ Y(λ, m)(λ ′ , m ′ ) = (λλ ′ , λm ′ + mλ ′ )
for λ, λ ′ ∈ A, and m, m ′ ∈ M . It is shown in [21] that if A and B are finite-dimensional algebras over a field k that are derived-equivalent, then A ⋉ D(A) is derived-equivalent to B ⋉ D(B), where D = Hom k (−, k). Note that A ⋉ D(A) is a self-injective algebra and that a derived equivalence between two self-injective algebras implies a stable equivalence of Morita type between them by [21]. It is known in [23] that a stable equivalence of Morita type preserves representation dimension (see [2] for definition). Hence we have the following corollary. In the following, we consider several generalizations of Corollary 3.6, namely we deal with the case of a finite family of Auslander-Reiten sequences. Proof. First, we suppose X n ∈ add (M ). Then there is an M i such that X n is a direct summand of M i , and therefore there is an irreducible map from X i to X n . It follows that there is an irreducible map from X 0 = τ −i X i to X n−i = τ −i X n , where τ stands for the Auslander-Reiten translation. Thus X 0 is a direct summand of M n−i+1 , which implies X 0 ∈ add (M ). Hence add (M ⊕ X n ) = add (M ) = add (M ⊕ X 0 ). Consequently, the algebras End A (M ⊕ X n ) and End A (M ⊕ X 0 ) are Morita equivalent. Thus End A (M ⊕ X n ) and End A (M ⊕ X 0 ) are, of course, derived-equivalent via a (projective) tilting module.
Next, we assume X n ∈ add (M ). In this case, we claim that there is no integer i ∈ {0, 1, · · · , n} such that X i ∈ add (M ). If X 0 ∈ add (M ), then there is an M i , 1 ≤ i ≤ n, such that X 0 is a direct summand of M i . Thus there is an irreducible map from X i to X 0 . By applying the Auslander-Reiten translation, we see that there is an irreducible map from X n = τ n−i X i to X n−i = τ n−i X 0 . Hence X n is a direct summand of M n−i+1 , that is, X n is in add (M ). This is a contradiction and shows that X 0 does not belong to add (M ). Suppose X i ∈ add (M ) for some 0 < i < n. Then there is an integer j ∈ {1, 2, · · · , n} such that X i is a direct summand of M j . Clearly, i = j, and there is an irreducible map from X i to X j−1 . On the one hand, if i > j, then there is an irreducible map from X n = τ n−i X i to X n−i+j−1 = τ n−i X j−1 . This implies that X n is a direct summand of M n−i+j , which is a contradiction. On the other hand, if i < j, then there is an irreducible map from X 0 = τ −i X i to X j−1−i = τ −i X j−1 . It follows that X 0 is a direct summand of M j−i . This is again a contradiction. Hence there is no X i belonging to add (M ). Now let m be the minimal integer in {0, 1, · · · , n} such that X n ≃ X m . If m = 0, then add (M ⊕ X n ) = add (M ⊕ X 0 ). This means that the endomorphism algebras End A (M ⊕ X n ) and End A (M ⊕ X 0 ) are Morita equivalent. Now we assume m > 0. Then the A-modules X 0 , X 1 , · · · , X m are pairwise non-isomorphic. We consider the sequence
X m −→ M m −→ · · · −→ M 1 −→ X 0 .
Since X m ∈ add (M ), every homomorphism from X m to M factors through the map X m −→ M m in the Auslander-Reiten sequence starting at X m . This means that the map X m −→ M m is a left add (M )approximation of X m . Similarly, the map M 1 −→ X 0 is a right add (M )-approximation of X 0 . Let V = M ⊕ X m . Then X i ∈ add (V ) for all i = 0, 1, · · · , m − 1. It follows that we have exact sequences
0 −→ Hom A (V, X i ) −→ Hom A (V, M i ) −→ Hom A (V, X i−1 ) −→ 0
for i = 1, · · · , m. Connecting the above exact sequences, we get an exact sequence
0 −→ Hom A (V, X m ) −→ Hom A (V, M m ) −→ · · · −→ Hom A (V, M 1 ) −→ Hom A (V, X 0 ).
This gives the first exact sequence in Lemma 3.4 (2). The second exact sequence in Lemma 3.4(2) can be obtained similarly. Thus Corollary 3.11 follows immediately from Lemma 3.4.
Remark. In Corollary 3.11, if X n ∈ add (M ) and X 0 , X 1 , · · · , X n are pairwise non-isomorphic, then the tilting End (X ⊕ M )-module T defined in Lemma 3.4 has projective dimension n. Note that we always have
gl.dim(End A (X ⊕ M )) − n ≤ gl.dim(End A (M ⊕ Y )) ≤ gl.dim(End A (X ⊕ M )) + n.
The following is another type of generalization of Corollary 3.6.
Proposition 3.12 Let A be an Artin algebra.
(1) Suppose 0 −→ X i −→ M i −→ Y i −→ 0 is an Auslander-Reiten sequence for i = 1, 2, · · · , n. Let X = i X i , M = i M i and Y = i Y i . If add (X) ∩ add (M ) = 0 = add (M ) ∩ add (Y ), then End A (X ⊕ M ) and End A (M ⊕ Y ) are derived-equivalent. (2) Suppose 0 −→ X 1 −→ X 2 ⊕ M 1 −→ Y 1 −→ 0 and 0 −→ X 2 −→ Y 1 ⊕ M 2 −→ Y 2 −→ 0 are two Auslander-Reiten sequences such that neither X 2 is in add (M 1 ) nor Y 1 is in add (M 2 ). If X 1 ∈ add (Y 1 ⊕ M 2 )(or equivalently, Y 2 ∈ add (X 2 ⊕ M 1 )), then End A (X 1 ⊕ M 1 ⊕ M 2 ) and End A (M 1 ⊕ M 2 ⊕ Y 2 ) are derived- equivalent.
Proof. (1) Under our assumption, the exact sequence 0 −→ X −→ M −→ Y −→ 0 is an almost add(M )split sequence in A-mod. Therefore (1) follows from Theorem 3.5.
(2) There is an exact sequence
( * ) 0 −→ X 1 −→ M 1 ⊕ M 2 −→ Y 2 −→ 0,
which can be constructed by the given two Auslander-Reiten sequences. Clearly, X 1 ∈ add (X 2 ⊕ M 1 ) since Auslander-Reiten quiver has no loops. By assumption, we see X 1 ∈ add (M 1 ⊕ M 2 ). Hence we can verify that the morphism
X 1 −→ M 1 ⊕ M 2 in ( * ) is a left add (M 1 ⊕ M 2 )-approximation of X 1 . Similarly, we can see that the morphism M 1 ⊕ M 2 −→ Y 2 in ( * ) is a right add (M 1 ⊕ M 2 )-approximation of Y 2 .
Thus ( * ) is an almost add (M 1 ⊕ M 2 )-split sequence in A-mod, and therefore the conclusion (2) follows from Theorem 3.5.
Remark. Usually, given two Auslander-
Reiten sequences 0 → X i → M i → Y i → 0 (1 ≤ i ≤ 2), we cannot get a derived equivalence between End A (X 1 ⊕ X 2 ⊕ M 1 ⊕ M 2 ) and End A (M 1 ⊕ M 2 ⊕ Y 1 ⊕ Y 2 )
. For a counterexample, we refer the reader to Example 3 in the last section. Now, we mention that, for an n-almost split sequence studied in [16], we have a statement similar to Corollary 3.11. Proposition 3.13 Let C be a maximal (n − 1)-orthogonal subcategory of A-mod with A a finite-dimensional algebra over a field (n ≥ 1). Suppose X and Y are two indecomposable A-modules in C such that the sequence
0 −→ X f −→ M n tn −→ M n−1 −→ · · · −→ M 2 t2 −→ M 1 g −→ Y −→ 0 is an n-almost split sequence in C. Then End A (X ⊕ n i=1 M i ) and End A ( n i=1 M i ⊕Y ) are derived-equivalent. Proof. Let M := n i=1 M i . Suppose Y is a direct summand of some M i .
Then there is a canonical projection π : M i −→ Y . Let t 1 = g and t n+1 = f . We observe that all homomorphisms t 1 , · · · , t n+1 are radical maps by the definition of an n-almost split sequence. Hence the composition t i+1 π can not be a split epimorphism and consequently factors through t 1 = g, that is, t i+1 π = u 1 g for a homomorphism u 1 : M i+1 −→ M 1 . First, we assume that i = n. Then t i+2 u 1 g = t i+2 t i+1 π = 0. By [16, Theorem 2.5.3], we have t i+2 u 1 = u 2 t 2 for a homomorphism u 2 : M i+2 −→ M 2 . Similarly, we get a homomorphism u k : M i+k −→ M k such that t i+k u k−1 = u k t k for k = 2, 3, · · · , n − i. This allows us to form the following commutative diagram:
X f / / un−i+1
W ). Then the derived equivalence between Λ and Γ in Theorem 3.5 is given by a BB-tilting module. In particular, if the Auslander-Reiten sequence
0 −→ S −→ P ′ −→ τ −1 S −→ 0
defines an APR-tilting module T := P ⊕ τ −1 S, then the sequence is an almost add (P )-split sequence in A-mod and the derived equivalence between A and End A (T ) in Theorem 3.5 is given precisely by the APRtilting module T := P ⊕ τ −1 S.
Proof. From the Auslander-Reiten sequence we have the following exact sequence
0 → Hom A (V, X) → Hom A (V, M ) (−,g) −→ Hom A (V, Y ).
Let L be the image of the map (−, g). Then we have an exact sequence
( * ) 0 → Hom A (V, X) → Hom A (V, M ) (−,g) −→ L → 0.
(This is a minimal projective presentation of the Λ-module L). Let T := L ⊕ Hom A (V, M ). Then T is the tilting module which defines the derived equivalence in Theorem 3.5. We shall show that T is a BB-tilting Λ-module. To prove this, it is sufficient to show that L is of the form τ −1 S for a simple Λ-module S.
If we apply Hom Λ (−, Λ) to ( * ), then we get an exact sequence of right Λ-modules:
Hom Λ (Hom A (V, M ), Λ) −→ Hom Λ (Hom A (V, X), Λ) −→ Tr Λ (L) −→ 0,
which is isomorphic to the following exact sequence
Hom A (M, V ) (f,−) −→ Hom A (X, V ) −→ Tr Λ (L) −→ 0,
where Tr Λ stands for the transpose over Λ. Note that the image of the map (f, −) is the radical of the indecomposable projective right Λ-module Hom A (X, V ). Thus Tr Λ (L) is a simple right Λ-module, and consequently, τ Λ L is isomorphic to the socle S of the indecomposable injective Λ-module DHom A (X, V ). Hence L ≃ τ −1 Λ S. Since X is not a direct summand of M , we see that Ext 1 Λ (S, S) = 0. Thus T is a BB-tilting module. Note that if X ≃ Y then L ≃ Hom A (V, Y ). In case of an APR-tilting module, we can see that the given Auslander-Reiten sequence is an almost add (P )-split sequence. Thus Proposition 4.1 follows. Now, we introduce the notion of an n-BB-tilting module: Let A be an Artin algebra. Recall that we denote by Ω n the n-th syzygy operator, and by Ω −n the n-th co-syzygy operator. As usual, D is the duality of an Artin algebra. Suppose S is a simple A-module and n is a positive integer. If S satisfies (a) Ext j A (D(A), S) = 0 for all 0 ≤ j ≤ n − 1, and (b) Ext i A (S, S) = 0 for all 1 ≤ i ≤ n, we say that S defines an n-BB-tilting module, and that the module T := τ −1 Ω −n+1 (S) ⊕ P is an n-BB-tilting module, where P is the direct sum of all non-isomorphic indecomposable projective A-modules which are not isomorphic to P (S), the projective cover of S. Note that (a) implies that the injective dimension of S is at least n and that the case n = 1 is just the usual BB-tilting module. The terminology is adjudged by the following lemma. which is isomorphic to the following exact sequence
0 −→ 0 −→ P 0 −→ · · · −→ P n −→ L −→ 0.
This shows that L ≃ TrDΩ −n+1 A (S) and the projective dimension of L is at most n. Moreover, we have the following sequence:
( * ) 0 −→ Hom A (L, P ) −→ Hom A (P n , P ) −→ · · · −→ Hom A (P 0 , P ) −→ 0.
Since Hom A (νP j , νP ) ≃ Hom A (P j , P ), we see that ( * ) is isomorphic to the sequence 0 −→ Hom A (L, P ) −→ Hom A (νP n , νP ) −→ · · · −→ Hom A (νP 0 , νP ) −→ 0, which is exact because Hom A (−, νP ) is an exact functor. Note that Hom A (S, νP ) = 0 by the definition of P . This shows that Ext i A (L, P ) = 0 for all i > 0. Since Ext i A (S, S) = 0 for all 1 ≤ i ≤ n, this means that νP 0 is not a direct summand of νP i for 1 ≤ i ≤ n. Thus P (S) is not a direct summand of P i for 1 ≤ i ≤ n, that is, P i ∈ add (P ) for all 1 ≤ i ≤ n. Now, if we apply Hom A (L, −) to the projective resolution of L, we get Ext n+i A (L, P 0 ) ≃ Ext i A (L, L) for all i ≥ 1. Hence Ext i A (L, L) = 0 for all i ≥ 1. We note that P 0 = P (S) and there is an exact sequence
0 −→ A −→ P ⊕ P 1 −→ · · · −→ L −→ 0.
Altogether, we have shown that T is a tilting module of projective dimension at most n.
Proposition 4.3 (1) Suppose 0 → X i → M i → X i−1 → 0 is an Auslander-Reiten sequence in A-mod for i = 1, 2, · · · , n. Let M = n i=1 M i and V = M ⊕ X n . If X n ∈ add (M )
and if X 0 , X 1 , · · · , X n are pairwise non-isomorphic, then the tilting End A (V )-module T := Hom A (V, X 0 ) ⊕ Hom A (V, M ) is an n-BB-tilting module.
(2) Let C be a maximal (n − 1)-orthogonal subcategory of A-mod with A a finite-dimensional algebra over a field (n ≥ 1). Suppose X and Y are two indecomposable A-modules in C such that the sequence
0 −→ X f −→ M n tn −→ M n−1 −→ · · · −→ M 2 t2 −→ M 1 g −→ Y −→ 0
is an n-almost split sequence in C. We define V = n i=1 M i ⊕X, and L to be the image of the map Hom A (V, g).
If X ∈ add ( j M j ), then Hom A (V, M ) ⊕ L is an n-BB-tilting End A (V )-module.
right Σ-module, where S X is the top of the right Σ-module Hom A (X, U ). If we define ∆ = End (T Σ ), then Hom Σ (Hom A (V, U ) Σ , T Σ ) ⊕ Hom Σ (Hom A (Y, U ) Σ , T Σ ) is an (n + 1)-APR-tiling ∆-module, that is, it is an (n + 1)-BB-tilting ∆-module defined by the projective simple ∆-module Hom Σ (S X , T ). Note that ∆ is a one-point extension of End A (V ) because Hom Σ (S X , Σ) = 0.
Auslander-Reiten triangles and derived equivalences
By Corollary 3.6, one can get a derived equivalence from an Auslander-Reiten sequence. An analogue of an Auslander-Reiten sequence in a triangulated category is the notion of Auslander-Reiten triangle. Thus, a natural question rises: is it possible to get a derived equivalence from an Auslander-Reiten triangle in a triangulated category? In this section, we shall discuss this question. First, let us briefly recall some basic definitions concerning Auslander-Reiten triangles. For more details, we refer the reader to [11].
Let R be a commutative ring. Let C be a triangulated R-category such that Hom C (X, Y ) has finite length as an R-module for all X and Y in C. In this case, we say that C is a Hom-finite triangulated R-category. Suppose further that the category C is a Krull-Schmidt category. A triangle X
f −→ M g −→ Y w −→ X[1] in C is called an Auslander-Reiten triangle if (AR1) X and Y are indecomposable; (AR2) w = 0; (AR3) if t : U −→ Y
is not a split epimorphism, then tw = 0.
Note that neither f is a monomorphism nor g is an epimorphism in an Auslander-Reiten triangle. This is a difference of an Auslander-Reiten triangle from an almost D-split sequence. Thus, an Auslander-Reiten triangle in a triangulated category may not be an almost D-split sequence. Also, an Auslander-Reiten sequence in the module category of an Artin algebra in general may not give us an Auslander-Reiten triangle in its derived module category. For an Artin algebra, we even don't know whether its stable module category has a triangulated structure except that the Artin algebra is self-injective. In this case, an Auslander-Reiten sequence can be extended to an Auslander-Reiten triangle in the stable module category.
Recall that a morphism f : U −→ V in a category C is called a split monomorphism if there is a morphism g : V −→ U in C such that f g = id U ; a split epimorphism if gf = id V ; and an irreducible morphism if f is neither a split monomorphism nor a split epimorphism, and, for any factorization f = f 1 f 2 in C, either f 1 is a split monomorphism or f 2 is a split epimorphism.
Suppose
X f −→ M g −→ Y w −→ X[1]
is an Auslander-Reiten triangle in a triangulated category C. Then we have the following basic properties:
(1) f g = 0 and gw = 0. Moreover, both f and g are irreducible morphisms.
(2) If s : X → U is not a split monomorphism, then s factors through f . Similarly, if t : V → Y is not a split epimorphism, then t factors through g.
(3) Let V be an indecomposable object in C. Then V is a direct summand of M if and only if there is an irreducible map from V to Y if and only if there is an irreducible map from X to V .
We mention that in any triangulated category C the functors Hom C (V, −) and Hom C (−, V ) are cohomological functors for each object V ∈ C (see [11,Proposition 1.2,p.4]).
The following is an expected result for Auslander-Reiten triangles. Denote by Λ the endomorphism ring of V . Since X and Y are not in add(U ), we see that f is a left add(U )approximation of X and g is a right add(U )-approximation of Y . To see that the condition (2) in Lemma 3.4 is satisfied, we consider the exact sequence · · · → Hom C (V, M [−1]) δ −→ Hom C (V, Y [−1]) → Hom C (V, X) → Hom C (V, M ) → Hom C (V, Y ).
Proposition 5.1 Let C be a Hom-finite, Krull-Schmidt, triangulated R-category. Suppose X f −→ M g −→ Y w −→ X[1] is an Auslander-Reiten triangle in C such that X[1] ∈ add (M ⊕ Y ). If N is
We have to show that the map δ is surjective. By assumption, we have Y [−1] ∈ add (N ) and Y [−1] ≃ X since Y ≃ X [1]. If Y [−1] ∈ add (M ), then there is an irreducible map from X to Y [−1] by the property (3), and therefore there is an irreducible map from X[1] to Y . It follows that X[1] is a direct summand of M , which contradicts to our assumption that X[1] ∈ add (M ). This shows that Y [−1] ∈ add (M ). Thus any morphism from V to Y [−1] cannot be a split epimorphism. This implies that the map δ is surjective by the property (2) Proof. If A is a self-injective Artin algebra, then every Auslander-Reiten sequence 0 → X → M → Y → 0 in A-mod can be extended to an Auslander-Reiten triangle
X −→ M −→ Y −→ Ω −1 A X
in the triangulated category A-mod which is equivalent to D b (A)/K b (A) (for details, see [11]). Thus Corollary 5.2 follows.
Note that under the assumptions in Proposition 5.1 the corresponding statement of Proposition 4.1 holds true for an Auslander-Reiten triangle.
Finally, let us remark that Corollary 5.2 may fail if A is not self-injective; for example, if we take A to be the path algebra (over a field k) of the quiver 2 −→ 1 ←− 3, then there is an Auslander-Reiten sequence 0 −→ P (1) −→ P (2) ⊕ P (3) −→ I(1) −→ 0, where P (i) and I(i) stand for the projective and injective modules corresponding to the vertex i, respectively. Clearly, this is a desired counterexample.
An Example
In this section, we illustrate our results with an example. Example 1. Let k be a field, and let A = k[x, y]/(x 2 , y 2 ). If Y denotes the simple A-module, then there is an Auslander-Reiten sequence 0 −→ X −→ N ⊕ N −→ Y −→ 0 in A-mod. Note that X = Ω 2 A (Y ) and N is the radical of A. By Theorem 1.1 or Corollary 1.2, the two algebras End A (N ⊕ Y ) and End A (N ⊕ X) are derived-equivalent. Though the local diagram of the Auslander-Reiten sequence is reflectively symmetric, the two algebras End A (N ⊕ Y ) and End A (N ⊕ X) are very different. This can be seen by the following presentations of the two algebras given by quiver with relations: γ i α j = 0 = γ i β j , i = j, γ 1 β 1 = γ 2 β 2 , γ 1 α 1 = γ 2 α 2 , α 1 γ 2 = β 1 γ 1 , α 2 γ 2 = β 2 γ 1 .
Note that the algebra End A (N ⊕ Y ) is a 7-dimensional algebra of global dimension 2, while the algebra End A (N ⊕ X) is a 19-dimensional algebra of global dimension 3. Hence the two algebras are not stably equivalent of Morita type since global dimension is invariant under stable equivalences of Morita type (see [23]). A calculation shows that the Cartan determinants of the both algebras equal 1.
Recall that the Cartan matrix of an Artin algebra A is defined as follows: Let S 1 , · · · , S n be a complete list of non-isomorphic simple A-modules, and let P i be a projective cover of S i . We denote the multiplicity of S j in P i as a composition factor by [P i : S j ]. The Catan matrix of A is the n × n matrix ([P i : S j ]) 1≤i,j≤n , and its determinant is called the Cartan determinant of A. It is well-known that the Cartan determinant is invariant under derived equivalences.
Theorem 1. 1
1Let C be an additive category and M be an object in C. SupposeX −→ M ′ −→ Yis an almost add(M )-split sequence in C. Then the endomorphism ring End C (M ⊕ X) of M ⊕ X and the endomorphism ring End C (M ⊕Y ) of M ⊕Y are derived-equivalent via a tilting module. Moreover, the finitistic dimension of End C (M ⊕ X) is finite if and only if so is the finitistic dimension of End C (M ⊕ Y ).
Let A be an Artin algebra.
Corollary 1. 3
3Let A be a self-injective Artin algebra. Suppose 0 −→ X −→ M −→ Y −→ 0 is an Auslander-Reiten sequence such that Ω −1 (X) ∈ add (M ⊕ Y ). Then End A (M ⊕ X) and End A (M ⊕ Y ) are derivedequivalent,where End A (M ) denotes the stable endomorphism algebra of an A-module A M .
(d) D b (A-mod) and D b (B-mod) are equivalent as triangulated categories.
Definition 3. 1
1Let C be an additive category and D a full subcategory of C. A sequence
Theorem 3. 5
5Let C be an additive category and M an object in C. SupposeX f −→ M ′ g −→ Yis an almost add(M )-split sequence in C. Then the endomorphism ring End C (M ⊕ X) of M ⊕ X and the endomorphism ringEnd C (M ⊕ Y ) of M ⊕ Y are derived-equivalent. Proof. Let V = M ⊕ X and W = M ⊕ Y .We shall verify the conditions of Lemma 3.4 for n = 1. By the definition of an almost D-split sequence, we see immediately that the condition (1) in Lemma 3.4 is satisfied, while the condition (2) in Lemma 3.4 is implied by the condition (3) in Definition 3.1: In fact, by applying Hom C (V, −) to the above sequence, we get a complex of abelian groups
is an almost add(M )-split sequence in C. Then the finitistic dimension of End C (M ⊕ X) is finite if and only if the finitistic dimension of End C (M ⊕ Y ) is finite.
Corollary 3.10 Let Λ be a finite-dimensional algebra over a field k and M a Λ-module in Λ-mod. SupposeX f −→ M ′ g −→ Yis an almost add(M )-split sequence in Λ-mod, and let A = End Λ (X ⊕ M ) and B = End Λ (M ⊕ Y ). Then A ⋉ D(A) is derived-equivalent to B ⋉ D(B). In particular, the representation dimensions of A ⋉ D(A) and B ⋉ D(B) are equal.
Corollary 3.11 Let A be an Artin algebra, and let 0 −→ X i −→ M i −→ X i−1 −→ 0 be an Auslander-Reiten sequence in A-mod for i = 1, 2, · · · , n. Let M = n i=1 M i . Then End A (M ⊕ X n ) and End A (M ⊕ X 0 ) are derived-equivalent via a tilting module T of projective dimension at most n.
be an Auslander-Reiten sequence in A-mod. We define V := M ⊕ X, Λ = End A (V ), W = M ⊕ Y and Γ = End A (
Lemma 4. 2
2If S defines an n-BB-tilting A-module, then T := τ −1 Ω −n+1 S⊕P is a tilting module of projective dimension at most n.Proof. Let ν be the Nakayama functor DHomA (−, A A). Suppose the sequence 0 −→ S −→ νP 0 −→ νP 1 −→ · · · −→ νP n −→ · · ·is a minimal injective resolution of S with all P i projective. Since Exti A (D(A), S) = 0 for 0 ≤ i ≤ n − 1, we have the following exact sequence by applying Hom A (D(A), −) to the injective resolution: 0 −→ Hom A (D(A), S) −→ Hom A (D(A), νP 0 ) −→ · · · −→ Hom A (D(A), νP n ) −→ L −→ 0,
an object in C such that none of X, Y, X[1] and Y [−1] belongs to add (N ), then End C (N ⊕ M ⊕ X) and End C (N ⊕ M ⊕ Y ) are derived-equivalent via a tilting module. In particular, End C (M ⊕ X) and End C (M ⊕ Y ) are derived-equivalent via a tilting module. Proof. First, if X is a direct summand of M , then there is an irreducible map from X to Y . It follows from the property (3) of an Auslander-Reiten triangle that Y is a direct summand of M . Similarly, if Y is a direct summand of M , then so is X. Thus, if X or Y is in add (M ), then add (N ⊕ M ⊕ X) = add (N ⊕ M ⊕ Y ) = add (N ⊕ M ). In this case, both End C (N ⊕ M ⊕ X) and End C (N ⊕ M ⊕ Y ) are Morita equivalent to End C (N ⊕ M ), and therefore End C (N ⊕ M ⊕ X) and End C (N ⊕ M ⊕ Y ) are derived-equivalent. Now, we assume that neither X nor Y is in add (M ). For simplicity, we set U := N ⊕ M , V := U ⊕ X and W := U ⊕ Y .
of an Auslander-Reiten triangle since the triangle X[−1] −→ M [−1] −→ Y [−1] −→ X is also an Auslander-Reiten triangle. Hence we have a desired exact sequence0 −→ Hom C (V, X) −→ Hom C (V, M ) −→ Hom C (V, Y ).Similarly, we have an exact sequence0 −→ Hom C (Y, W ) −→ Hom C (M, W ) −→ Hom C (X, W ).Thus Proposition 5.1 follows from Lemma 3.4 by taking n = 1.From Proposition 5.1 we get the following corollary.
Corollary 5. 2
2Let A be a self-injective Artin algebra. Suppose 0 → X → M → Y → 0 is an Auslander-Reiten sequence such that Ω −1 (X) ∈ add (M ⊕ Y ). Then End A (M ⊕ X) and End A (M ⊕ Y ) are derived-equivalent,where End A (M ) stands for the quotient of End A (M ) by the ideal of those endomorphisms of M , which factor through a projective A-module.
−→ S −→ P ′ −→ τ −1 S −→ 0 in A-mod with P ′ projective.
Acknowledgements. The authors thank I. Reiten and M.C.R. Butler for comments, and Hongxing Chen at BNU for discussions on the first version of the manuscript. Also, C.C.Xi thanks NSFC (No.10731070) for partial support.Note that if i = n then the above diagram still makes sense. We claim that u n−i+1 is a split monomorphism. If this is not the case, then the map u n−i+1 factors through f . This means that there is some map h n : M n −→ M n−i+1 such that f h n = u n−i+1 . Then we have f (u n−i − h n t n−i+1 ) = f u n−i − u n−i+1 t n−i+1 = 0. By[16,Theorem 2.5.3], there is some homomorphism h n−1 :for V := X ⊕M and W := M ⊕Y . Thus the condition (2) in Lemma 3.4 is satisfied. Consequently, Proposition 3.13 follows from Lemma 3.4.Auslander-Reiten sequences and BB-tilting modulesIn this section, we point out that, when we restrict our consideration to Auslander-Reiten sequences, the tilting module defining the derived equivalence in Theorem 3.5 is of special form, namely it is a BB-tilting-module in the sense of Brenner and Butler[5]. This shows that the tilting theory and the Auslander-Reiten theory are so beautifully integrated with each other. We first recall the BB-tilting-module procedure in[5], and then give a generalization of a BB-tilting module, namely the notion of an n-BB-tilting module.Let A be an Artin algebra and S a non-injective simple A-module with the following two properties: (a) proj.dim A (τ −1 S) ≤ 1, and (b) Ext 1 A (S, S) = 0. Here τ −1 stands for the Auslander-Reiten translation TrD, and proj.dim A (S) means the projective dimension of S. We denote the projective cover of S by P (S), and assume that A A = P (S) ⊕ P such that there is not any direct summand of P isomorphic to P (S). Let T = τ −1 S ⊕ P . It is well-known that T is a tilting module. Such a tilting module is called a BB-tilting module. In particular, if S is a projective non-injective simple module, then T is automatically a BB-tilting module, this special case was first studied in[1], and the tilting module of this form is called an APR-tilting module in literature. Note that if S is a non-injective, projective simple A-module, then there is an Auslander-Reiten sequence Proof. The proof of (1) is similar to the one of Proposition 4.1. We leave it to the reader.(2) We shall show that L is isomorphic to τ −1 Ω −n+1 Λ (S) with S = τ Ω n−1 Λ (L) being a simple Λ-module. It is easy to see that D(S) = TrΩ n−1 Λ (L) is a simple right Λ-module. In fact, it is isomorphic to the top of the indecomposable right Λ-module Hom A (X, V ), and is not injective since X ∈ add ( j M j ). Further, it follows from X ∈ add ( i M i ) that we have an exact sequenceIf we apply Hom Λ (−, Λ) to this sequence, we can see that Ext (2) With the same method as in Proposition 4.3, we can prove the following fact: Let C be a maximal (n − 1)-orthogonal subcategory of A-mod with A a finite-dimensional algebra over a field (n ≥ 1). Suppose X and Y are two indecomposable A-modules in C such that the sequence 0 −→ X
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"Elena Zaninoni \nINAF Osservatorio Astronomico di Brera\nvia Bianchi 4623807MerateItaly\n\nUniversità di Padova\nDip. Astronomia, v. dell' Osservatorio 3135122PadovaItaly\n",
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"Margutti \nINAF Osservatorio Astronomico di Brera\nvia Bianchi 4623807MerateItaly\n\nHarvard-Smithsonian Center for Astrophysics\n60 Garden Street, MA02138 4 ICRANet, p.le della Repubblica 1065100Cambridge, PescaraItaly\n",
"Maria Grazia Bernardini \nINAF Osservatorio Astronomico di Brera\nvia Bianchi 4623807MerateItaly\n",
"Guido Chincarini \nINAF Osservatorio Astronomico di Brera\nvia Bianchi 4623807MerateItaly\n\nDip. Fisica G. Occhialini\nUniversità Milano Bicocca\nP.zza della Scienza 320126MilanoItaly\n",
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"INAF Osservatorio Astronomico di Brera\nvia Bianchi 4623807MerateItaly",
"Università di Padova\nDip. Astronomia, v. dell' Osservatorio 3135122PadovaItaly",
"INAF Osservatorio Astronomico di Brera\nvia Bianchi 4623807MerateItaly",
"Harvard-Smithsonian Center for Astrophysics\n60 Garden Street, MA02138 4 ICRANet, p.le della Repubblica 1065100Cambridge, PescaraItaly",
"INAF Osservatorio Astronomico di Brera\nvia Bianchi 4623807MerateItaly",
"INAF Osservatorio Astronomico di Brera\nvia Bianchi 4623807MerateItaly",
"Dip. Fisica G. Occhialini\nUniversità Milano Bicocca\nP.zza della Scienza 320126MilanoItaly"
]
| []
| We present the preliminary analysis of the GRB light curves obtained by Swift/XRT | null | [
"https://arxiv.org/pdf/1107.2870v1.pdf"
]
| 118,435,641 | 1107.2870 | 19d69660e23f4c13b450a8f58b7ac2044f01dec2 |
Poster presented in the "X-Ray Universe
2011. June 2011
Elena Zaninoni
INAF Osservatorio Astronomico di Brera
via Bianchi 4623807MerateItaly
Università di Padova
Dip. Astronomia, v. dell' Osservatorio 3135122PadovaItaly
Raffaella 2⋆⋆
Margutti
INAF Osservatorio Astronomico di Brera
via Bianchi 4623807MerateItaly
Harvard-Smithsonian Center for Astrophysics
60 Garden Street, MA02138 4 ICRANet, p.le della Repubblica 1065100Cambridge, PescaraItaly
Maria Grazia Bernardini
INAF Osservatorio Astronomico di Brera
via Bianchi 4623807MerateItaly
Guido Chincarini
INAF Osservatorio Astronomico di Brera
via Bianchi 4623807MerateItaly
Dip. Fisica G. Occhialini
Università Milano Bicocca
P.zza della Scienza 320126MilanoItaly
Al
Poster presented in the "X-Ray Universe
Berlin2011. June 2011between November 2004 and December 2010.arXiv:1107.2870v1 [astro-ph.HE]gamma-ray bursts: X-raySwiftafterglow
We present the preliminary analysis of the GRB light curves obtained by Swift/XRT
Introduction
The Swift satellite (Gehrels et al. 2004), launched on November 2004, during the past six years of operation detected and observed more than 600 Gamma-Ray Bursts (GRBs). The vast majority (∼67%) of these bursts were monitored in the soft X-ray band by the X-Ray Telescope (XRT, Burrows et al. 2005) starting as early as ∼80 s after the trigger. The standard model explains the X-ray afterglow of GRBs as synchrotron radiation arising from the deceleration of a relativistic blast wave into the external medium. The XRT follow up, therefore, is comprehensive of the tail of the prompt emission and of the afterglow. The XRT sample is now large enough to justify a statistical approach aimed at collecting, classifying and understanding the observational information of a wide and homogeneous sample of GRBs. This work will provide the most complete view of the X-ray properties of GRBs to date any existing theoretical model is asked to explain, while serving as a guide for future theoretical developments. The catalogue 1 is now under completion, here we report the status of the work.
Sample, Data Reduction and Analysis
We analysed all the GRBs detected until the end of 2010, for which the afterglow had been observed by XRT with enough photons to extract a measurable spectrum. The sample consists of 437 GRBs out of a total 658 GRBs detected by Swift of which 165 GRBs with redshift, 414 long GRBs (153 with z) and 23 short GRBs (12 with z). The original XRT data have been reduced with the method reported in Margutti et al. (2010). We extracted the XRT light curves in the 0.3-10 keV XRT band as well as in the 0.3-1 keV, 1-2 keV, 2-3 keV and 3-10 keV band. Our XRT archive contains light curves in count-rate, flux and luminosity (for the redshift subsample) and all the relevant parameters of interest. The light curves in flux units are calibrated accounting for spectral evolution and the spectra have been derived using the NH column density estimated in a time interval where no spectral evolution is apparent.
Data analysis
We fitted the light curves in flux units, using four functions:
-Single power-law:
F(t) = N t −α .
(1)
-Smoothed broken power-law:
F(t) = N t t b − α 1 s + t t b − α 2 s s .
(2)
-Sum of power-law and smoothed broken power-law:
F(t) = N 1 t −α 1 + N 2 t t b − α 2 s + t t b − α 3 s s .(3)
-Sum of two smoothed broken power-laws:
F(t) = N 1 t t b1 − α 1 s 1 + t t b1 − α 2 s 1 s 1 + N 2 t t b3 − α 3 s 2 + t t b3 − α 4 s 2 s 2 .
(4)
where α is the decay power-law index, t b the break time, s the smoothness parameter and N the normalization. The best fit parameters were determined using the IDL Levenberg-Marquard least-squares fit routine (MPFIT) supplied by Markwardt (2009). All variability and fluctuations superimposed on the basic underlying light curve have been subtracted by iteration following (Margutti et al. 2011). Using the fit parameters, we calculated the total fluence (and the energy for the subsample of GRBs with known z) of our light curves as well as the one of different parts of the light curves (E 1 , E 2 , E 3 , E 4 ), as shown in Fig. 1; moreover we calculate the fluence (energy when possible) of the excesses.
Classification and morphology
We classified the XRT light curves according to the fit function used, the presence of flares and the completeness of the curve (Fig. 2 and Tab. 1; see also Bernardini et al. 2011). For the different shapes (Fig. 1), we have:
-Type 0: no breaks, Eq. 1; -Type Ia: single break, Eq. 2 and s < 0; -Type Ib: single break, Eq. 2 and s > 0; -Type IIa: double broken power-law (steep-to-flat), Eq. 3; Fig. 1. Cartoon of the fit function used to classify and analyse the sample of XRT light curves. We also indicated some of the parameters derived and the time intervals for which we measured the emitted energy. A light curve is considered complete if the XRT re-pointing time is <300 s and the final count rate is comparable to the background. This ensures that the possible absence of the early steep decay is not due to an observational bias. See Tab. 1 and Fig. 1 Fig. 3. The Dainotti relation (left) and the histrogram of the total X-ray energy with errors calculated with the Monte Carlo method (right). In red the short GRBs.
Conclusion
The temporal and spectral properties of a homogenous sample of 437 GRBs have been extracted. For the first time, we have a large and homogeneous data set that allows us to perform a statistical study of the X-ray properties of GRB afterglow. Notably, the sample of GRBs with redshift information comprises 165 elements: for this sub-sample it is possible to study the intrinsic properties of these explosions. From Fig. 2 Chincarini et al. 2010).
As an example of the output of our analysis we plot the Dainotti relation (Dainotti et al. 2008(Dainotti et al. , 2010(Dainotti et al. , 2011, that is the relation between the end plateau luminosity and time, as we derive it from the row output and the distribution (histogram) of the total X-ray energy where the uncertainties for each bin have been computed by Monte Carlo simulations (Fig. 3).
Fig. 2 .
2Classification of the light curves in the sample, as illustrated in the Tab. 1. 0, I, II, III indicate the fit function used where the number refers to the numbers of observed breaks in the light curve; for instance Type 0 means that the light curve is a pure power law. "C" stands for complete and "U" for incomplete; "F" indicate the presence of flares; on the contrary, there are not flares if in the name is present a "N". For example, "ICN" means that the fitting function used was a smoothed broken power law with a complete set of the data and no flares.-Type IIb: double broken power-law (flat-to-steep), Eq. 3; -Type III: double broken power-law, Eq. 4.
Table 1. Distribution of the GRBs light curves in the different classes. For more details see Fig. 2., 2 for more details about our
sample.
Type UN UF CN
CF Total
0
65
4
28
17
114
I
29
7
86
31
153
II
9
4
84
51
148
III
0
0
12
10
22
Total 103 15 212 107
437
and Tab. 1, we notice that the probability to observe a Type I or Type II XRT light curve is higher if the data are complete (117/153 for Type I and 135/148 for Type II) evidencing an observational bias. The majority of incomplete light curves are Type 0 (69/118). In addition, almost all the incomplete light curves have not flares (103/118) and 33% of the complete light curves have flares (107/319; see
. M G Bernardini, SSRv. 120165Bernardini, M. G., et al. 2011, submitted Burrows, D. N., et al. 2005, SSRv, 120, 165
. G Chincarini, MNRAS. 4062113Chincarini, G., et al. 2010, MNRAS, 406, 2113
. M G Dainotti, ApJ. 730135Dainotti, M. G., et al. 2011, ApJ, 730, 135
. M G Dainotti, ApJ. 722215Dainotti, M. G., et al. 2010,ApJ, 722, L215
. M G Dainotti, MNRAS. 39179Dainotti, M. G., et al. 2008, MNRAS, 391, L79
. N Gehrels, ApJ. 6111005Gehrels, N., et al. 2004, ApJ, 611, 1005
. R Margutti, MNRAS. 4101064Margutti, R., et al. 2011, MNRAS, 410, 1064
. R Margutti, MNRAS. 40246Margutti, R., et al. 2010, MNRAS, 402, 46
C B Markwardt, Astronomical Data Analysis Software and Systems XVIII. 411251Markwardt, C. B. 2009, Astronomical Data Analysis Software and Systems XVIII, 411, 251
| []
|
[
"Influence of elastic strain on the thermodynamics and kinetics of lithium vacancy in bulk LiCoO 2",
"Influence of elastic strain on the thermodynamics and kinetics of lithium vacancy in bulk LiCoO 2"
]
| [
"Ashkan Moradabadi ",
"Payam Kaghazchi ",
"Jochen Rohrer ",
"Karsten Albe ",
"\nInstitut für Chemie und Biochemie\nInstitut für Materialwissenschaft\nFachgebiet Materialmodellierung\nFreie Universität Berlin\nTakustr. 314195BerlinGermany\n",
"\nInstitut für Chemie und Biochemie\nTechnische Universitat Darmstadt\nJovanka-Bontschits-Str. 264287DarmstadtGermany\n",
"\nInstitut für Materialwissenschaft\nFachgebiet Materialmodellierung\nFreie Universität Berlin\nTakustr. 314195BerlinGermany\n",
"\nTechnische Universitat Darmstadt\nJovanka-Bontschits-Str. 264287DarmstadtGermany\n"
]
| [
"Institut für Chemie und Biochemie\nInstitut für Materialwissenschaft\nFachgebiet Materialmodellierung\nFreie Universität Berlin\nTakustr. 314195BerlinGermany",
"Institut für Chemie und Biochemie\nTechnische Universitat Darmstadt\nJovanka-Bontschits-Str. 264287DarmstadtGermany",
"Institut für Materialwissenschaft\nFachgebiet Materialmodellierung\nFreie Universität Berlin\nTakustr. 314195BerlinGermany",
"Technische Universitat Darmstadt\nJovanka-Bontschits-Str. 264287DarmstadtGermany"
]
| []
| The influence of elastic strain on the lithium vacancy formation and migration in bulk LiCoO 2 is evaluated by means of first-principles calculations within density functional theory (DFT). Strain dependent energies are determined directly from defective cells and also within linear elasticity theory from the elastic dipole tensor (G i j ) for ground state and saddle point configurations. We analyze finite size-effects in the calculation of G i j , compare the predictions of the linear elastic model with those obtained from direct calculations of defective cells under strain and discuss the differences. Based on our data, we calculate the variations in vacancy concentration and mobility due to the presence of external strain in bulk LiCoO 2 cathodes. Our results reveal that elastic in-plane and out-of-plane strains can significantly change the ionic conductivity of bulk LiCoO 2 by an order of magnitude and thus strongly affect the performance of Li-secondary batteries. | 10.1103/physrevmaterials.2.015402 | [
"https://arxiv.org/pdf/1706.01709v1.pdf"
]
| 118,827,181 | 1706.01709 | 364c4eb8f80447642cd9ce6878e3558f821c5e41 |
Influence of elastic strain on the thermodynamics and kinetics of lithium vacancy in bulk LiCoO 2
Ashkan Moradabadi
Payam Kaghazchi
Jochen Rohrer
Karsten Albe
Institut für Chemie und Biochemie
Institut für Materialwissenschaft
Fachgebiet Materialmodellierung
Freie Universität Berlin
Takustr. 314195BerlinGermany
Institut für Chemie und Biochemie
Technische Universitat Darmstadt
Jovanka-Bontschits-Str. 264287DarmstadtGermany
Institut für Materialwissenschaft
Fachgebiet Materialmodellierung
Freie Universität Berlin
Takustr. 314195BerlinGermany
Technische Universitat Darmstadt
Jovanka-Bontschits-Str. 264287DarmstadtGermany
Influence of elastic strain on the thermodynamics and kinetics of lithium vacancy in bulk LiCoO 2
(Dated: March 5, 2018)Stress/straindefect thermodynamics and kineticselastic dipole tensorfinite-size-effectionic mobility
The influence of elastic strain on the lithium vacancy formation and migration in bulk LiCoO 2 is evaluated by means of first-principles calculations within density functional theory (DFT). Strain dependent energies are determined directly from defective cells and also within linear elasticity theory from the elastic dipole tensor (G i j ) for ground state and saddle point configurations. We analyze finite size-effects in the calculation of G i j , compare the predictions of the linear elastic model with those obtained from direct calculations of defective cells under strain and discuss the differences. Based on our data, we calculate the variations in vacancy concentration and mobility due to the presence of external strain in bulk LiCoO 2 cathodes. Our results reveal that elastic in-plane and out-of-plane strains can significantly change the ionic conductivity of bulk LiCoO 2 by an order of magnitude and thus strongly affect the performance of Li-secondary batteries.
I. INTRODUCTION
The presence of strain fields can significantly influence the efficiency and lifetime of functional materials such as semiconductors, solar cells and Li-ion batteries [1][2][3][4][5][6][7][8][9][10]. In principle, strain can be internally generated by structural defects within the bulk and/or interfaces or can be induced by external loads. These strains or corresponding stresses can cause lattice deformations and distortions and therefore also affect the formation and migration of point defects. As a result, the conductivity of an ionic conductor can be significantly changed [11][12][13][14][15]. This coupling is most relevant in Li-ion batteries, where due to charging/discharging processes, bulk and interfacial as well as thermal strains can occur. The fact that induced stresses, which raise during intercalation, could weaken the interface between electrode/electrolyte and finally degrade the battery performance has been extensively discussed in the past [16][17][18][19][20][21][22][23][24][25][26][27]. Much less is known, however, about the coupling of strain fields to the formation and migration of Li vacancies in the bulk part of the material.
As one of the first, Choi et al. [28] have examined the effect of intercalation-induced stress on Li migration in LiCoO 2 using experimental techniques such as electrochemical impedance spectroscopy together with cyclic voltammetry. They showed that the mismatch strain between intercalated and deintercalated states in LiCoO 2 is due to a phase transition between the α and β phases. This results in intercalationinduced stress in the phase boundary region. Since an inserted Li ion causes structural disorder in the host material, it intro-duces a strain field which affects the next intercalated Li and leads to a strain-induced elastic interaction between all intercalated Li ions. This elastic interaction has both short and long range effects and it was also found that as the particle size decreases, the stress gradient across LiCoO 2 particles increases.
Transmission electron microscopy (TEM), X-ray diffraction and electrochemical strain microscopy (ESM) have revealed that upon delithiation the c-axis in Li x CoO 2 (x=0.5) expands by about 2% [29,30] and rather large stresses occur during charging/discharging process at different charge/discharge rates (c-rates) [31][32][33][34]. Garcia et. al., for example, have shown based on a continuum model that large stress values (up to ± 200 MPa) are generated at particle contacts and these values increase with increasing c-rate of discharge [25]. Xiong et al. have shown in an ab-initio based study that during deintercalation of Li x CoO 2 , the c-axis increases up to 3.25% [35]. Li et al. also reported that compressive stress raises while the Li concentration is decreasing and at x=0.5, the measured strain at the O-Co-O octahedral slabs is 4.8% [36]. A mathematical model was developed by Renganathan et al. to reveal the mechanical stresses generated during the discharge process in carbon and LiCoO 2 [37]. Their findings show that at high discharge c-rates, stresses also increase and the stress caused by phase transformations is related to the amount of each phase present in the electrode. They also concluded that particle size and distribution can affect the generated stress [37]. Critical rates of charging and particle size below which fracture of LiCoO 2 occurs were pre-dicted by Zhao et al. using a kinetic and fracture mechanics model. They have shown that as the discharge rate increases the particle size should decrease so that fracture is prevented (for c-rates more than 5 C, particle sizes less than 200 nm) [38].
While all these studies point to the fact that significant stress and strain levels can occur in cathode materials of Lisecondary batteries, their influence on Li-ion diffusion has hardly been studied. In a recent theoretical work Ning et al. [23] have shown that by applying uniaxial tensile strain along the c-axis of bulk LiCoO 2 , the Li diffusion barrier decreases [23], but did not derive consider the case of a more complex tensorial strain field.
In this study, we calculate the elastic dipole tensor [39] in order to characterize the coupling of strain fields to the formation and migration energies of Li vacancy in bulk-LiCoO 2 . We obtain the components of the elastic dipole tensor for defect formation and also migration from total energy calculations within density functional theory. Similar calculations of the defect dipole tensor have been recently reported for defects in metals and silicon by Varvenne et al. [40] and in UO 2 by Goyal et al. [15]. In order to demonstrate the degree of coupling between lateral and longitudinal components of the stress tensor in LiCoO 2 , various supercell geometries are studied. Moreover, we compare the predictions from linear elasticity theory with directly calculated formation and migration energies of vacancy under strain and show that the elastic dipole-tensor allows us to quantify the influence of strain in a computationally efficient manner. Finally, we estimate the effect of lateral and longitudinal strains on ionic conductivity in bulk LiCoO 2 .
II. THEORY AND COMPUTATIONAL METHODS
Elastic dipole tensor
The insertion of a point defect into a crystal produces local elastic distortions. Moreover, there will be an interaction between this defect and a stress or strain field present in the crystal. This is similar to the interaction of an electric dipole with an applied electric field. Therefore, a defect inducing local distortions is called an elastic dipole, which is -contrary to the electric dipole-characterized by a second-rank tensor. This elastic dipole tensor, which is also called double force tensor [39], is the negative derivative of the defect formation energy E d with respect to an imposed bulk strain
G i j = − ∂E d ∂ i j ,
if, as usual, we ignore entropy contributions. Thus, G i j is relating the atomic structure of a point defect and its elastic field. In case of a purely dilatational strain the defect relaxation vol-
ume ∆V = 1 3B Tr{G i j },
where B is the bulk modulus, can be directly obtained from G i j . In principle, the concept of the elastic dipole tensor can be conveniently understood by ex-panding the free energy per volume in terms of the density of defects n d = N d /V and the strain tensor i j (both being intensive quantities) [41]. If entropy contributions are neglected the expansion of the energy density (T=0 K) reads as:
E(n d , ) = E o + i, j ∂E ∂ i j σ i j =0 i j + 1 2 i, j ∂ 2 E ∂ i j ∂ kl i j kl + ∂E ∂n d n d + i, j ∂ 2 E ∂n d ∂ i j i j n d + . . . = E o + 1 2 i, j C i jkl i j kl + n d E d + i, j ∂ 2 E ∂ i j ∂n d i j + . . . . = E o + 1 2 i, j C i jkl i j kl + n d E d + i, j ∂σ i j ∂n d i j + . . . . = E o + 1 2 i, j C i jkl i j kl + n d E d − i, j G i j i j + . . . .(1)
Here E o is the total energy of the non-strained defect-free system, n d the number of defects, respectively, E d is the formation energy of a defect, σ i j is the stress, C i jkl are the components of the stiffness tensor and G i j is the elastic dipole tensor. As such G i j describes the interaction of the defect with a strain field. The change in energy under strain that is exclusively due to the presence of a defect is given by
∆E = − i j G i j i j .(2)
Thus, for example, a positive lattice strain would lower the formation energy of a defect having a positive relaxation volume. The stress in a material under strain i j with a defect density n d is eventually given by
σ d i j ≡ ∂E(n d , i j ) ∂ i j = kl C i jkl kl − n d G i j = σ 0 i j − n d G i j ,(3)
if we only consider first-order components of the elastic dipole tensor. Note that -depending on the stress definition-different sign conventions have been proposed in literature. Here we stick to the original one used by Leibfried and Breuer [39]. With relation (3) it is straightforward to compute the components of the elastic dipole tensor numerically using atomistic methods. In a given periodic supercell, an individual defect is introduced. While the cell parameters are fixed, the atomic positions are relaxed and the induced stress is calculated. Then, the elastic dipole tensor can be obtained from the relation
G i j = − ∂E d ∂ i j = − ∂σ i j ∂n d i j = − 1 n d (σ d i j − σ 0 i j ) = −V 0 ∆σ i j ,(4)
where V 0 is the volume of the supercell containing one defect, σ d i j is the stress of the defective cell and σ 0 i j is the stress of the defect-free cell (which in principle should be or very close to zero).
In supercell calculations, there will be a certain unwanted contribution to the energy described in Eq. 1 from the interaction between the defects in their periodic images. It is therefore necessary, either to correct for this interaction, or to increase the size of the repeating unit to make it negligible. In the case of charged defects, another contribution comes from the Coulomb interaction between the defects, which must be corrected as shown by Leslie and Gillan [41]. Elastic contributions can usually be made small enough by increasing the cell size. If ab-initio methods are used, however, the accessible cell sizes are rather limited and elastic interactions can induce higher order effects, that are not covered by linear elasticity. As can be seen from Eq. 4, the variation of stress by a defect follows the relation ∆σ i j = −G i j /V o and thus goes to zero in the dilute limit (lim V o →∞ ∆σ i j = 0). Therefore, by plotting the components of ∆σ i j as function of the inverse volume, one can test the convergence behavior of the calculated stresses. For small supercell sizes higher order effects will affect the linearity of ∆σ i j with respect to the inverse volume. In practice, a polynomial fit including higher order terms are required. The coefficient of the linear term dominates at small values of inverse volume and thus still corresponds to the elastic dipole tensor.
Defect formation and migration energies under strain
The defect formation energy for a given homogeneous strain i j , considering Eq. 2, is then given by
E d ( i j ) = E d (0) − i j G i j i j .(5)
Note, that E d ( i j ) can also be directly calculated as shown later.
In the present work, we only consider a neutral Li vacancy. The reason is that we want to disentangle the electrostatic and elastic interactions. The conventional way to calculate the formation energy of defects in the neutral state and in the presence of strain is to use the following equation [42]
E d ( i j ) = E (Li−vacancy) tot ( i j ) − E p tot ( i j ) + µ Li .(6)
In this equation, the first term is the total energy of the strained system containing a single neutral Li vacancy, the second term is the total energy of the strained pristine system and the last term is the strain-free chemical potential of the Li reservoir.
This can be compared to the defect formation energy as a function of strain calculated from Eq. 5 for a specific concentration. It should be noted here that the vacancy formation energy depends on the chemical potential of lithium in the reservoir, which is taking up the removed Li atom. Thus, in principle one has to consider the fact that strain might also affect the reservoir. We now move to the coupling of defect migration and strain fields. The energy barrier that is required for an ion to jump between two sites is obtained by the energy difference between the saddle point and the initial configuration
E b ( ) = E S ( i j ) − E d ( i j ),(7)
where E S is the energy of the defective system in the saddle point (transition state) and E d the energy of the defective initial state.
For a system under strain, the energy barrier can be computed by applying a particular strain to the simulation cell and direct calculation of this energy difference. Alternatively, one can also calculate the defect dipole tensor G S i j at the saddle point configuration and obtain the strain dependent barrier from
E b ( i j ) = E b (0) − i j G S i j i j − G i j i j = E b (0) − i j ∆G i j i j ,(8)
where E b ( i j ) is the migration energy barrier in the strained material, E b (0) is the migration energy barrier in the unstrained material, while ∆G b i j is the change of the elastic dipole tensor by going from the initial to the transition state. For calculating the ∆G i j , we perform two single point calculations, one for the initial and one for the transition state in order to obtain ∆σ d i j . Afterwards, using Eq. 4, the elastic dipole tensor for the initial and transition state can be calculated. The atomic coordinations used in the single point calculations are obtained from the NEB calculation for the unstrained case. Therefore, by performing only one NEB calculation at i j = 0, the strain-dependent activation barriers for Li-diffusion can be investigated. Fig. 1 (a) shows the unit cell of bulk LiCoO 2 . Li atoms (violet color) are coordinated within the octahedrals of CoO −2 . During the delithiation, Li vacancies are formed and start to migrate. Considering the layered structure of LiCoO 2 , we expect a significant influence of strain on the formation and migration of these vacancies. The strain can either originate from the expansion/contraction during the deintercalation/intercalation, respectively, or can be applied externally (e.g. interfacial strains from SEI, solid electrolyte or binder). In Fig. 1(b), the direct mechanism of Li diffusion towards the single vacancy is indicated (or migration of vacancy towards Li). During this process, the Li ion must overcome an energy barrier E b by pushing the two nearby Li ions and passing through them. This mechanism is shown in Fig. 1(b) and (c) by gray arrows. Therefore, Li on the saddle point is under compressive stress (tensile on its neighbors). While there are also other Li migration mechanisms reported [16,43], in the following, we will focus on the strain dependence of the formation and migration of a neutral single Li vacancy moving on a direct pathway. Since we are only interested in the strain dependency, we deliberately study neutral cells in order to disentangle electrostatic image interactions of the charged defects from the elastic image interactions being also present.
Model structures
Computational details
The calculations were performed using the local atomicorbital DFT-code SeqQuest [44] with norm-conserving pseudopotentials and the generalized-gradient approximation of Perdew, Wang and Ernzerhof (PBE) [45] for exchange and correlation. Spin optimization is performed during all geometry relaxations. The energy convergence criterion for all calculations is 1 × 10 −5 eV. For smearing, we used the Gaussian method, together with a narrow width of smearing (0.005 eV) to make sure that the correct spin polarization can be achieved. Diffusion pathways are investigated using the nudge elastic band (NEB) method [46] (as implemented in the SeqQuest code). A 24×24×5 Monkhorst-Pack k-point mesh for the 1 × 1 × 1 unit cell is considered and for larger supercells, it is adjusted accordingly. For all calculations, grid spacing for the charge density integration is set to 0.16 Å. Convergence tests with respect to k-point sampling and grid spacing show that calculated stresses are well converged (less than 0.0002 GPa). For the charge density difference calculation, the relaxed structures from SeqQuest calculations were used in the DFT-code VASP [47]. The unit cells for all VASP calculations were optimized again while the energy-forces criteria and spin polarization were chosen similar to ones used in SeqQuest calculations.
RESULTS AND DISCUSSION
I. Size dependence of elastic dipole tensor (G i j in dilute limit)
In order to capture finite-size effects, we used supercells with different volumes and aspect ratios. We also investigated the degree of coupling between lateral and longitudinal components by studying various cell geometries. Figure 2 (left) shows the diagonal elements of the calculated stresses for isotropically repeated n × n × n (1 ≤ n ≤ 3) supercells containing a single (neutral) Li vacancy and those of nonisotropic replicas n × n × 1 (1 ≤ n ≤ 5, middle) and n × n × 2 (1 ≤ n ≤ 4, right) together with the second-order fit of the stresses and the diagonal components of the elastic dipole tensor. For the isotropic case ( Fig. 2 -left), due to computational limitations, the 4×4×4 supercell is not calculated. In all cases, the non-diagonal elements exhibit small values. This is due to the fact that a Jahn-Teller distortion of MO 2 octahedrals occurs [35,48], which can affect these non-diagonal elements of the elastic dipole tensor. The separation of defects along the z-axis between periodic images for 1 × 1 × 1 is equal to 13.96 Å, while across the xy-plane, it is equal to 2.85 Å. Therefore, we expect a stronger defect-defect interaction between the periodic images in the xy-direction rather than along the z-axis.
In all geometries, where the z-extension was varied (see Fig. 3), the xx and yy components of the elastic dipole tensor are comparable (about 1.2 eV), while the zz component is strongly varying and is even changing its sign in case of the 1 × 1 × n cells indicating a strong coupling between xx, yy and zz components.
We determined the components of the elastic dipole tensor by fitting the relation ∆σ i j = −G i j /V + β/V 2 to the calculated data. The second term is accounting for the fact that higher order contributions to the elastic dipole tensor might be significant. Using the mechanical definition of stress, we describe outward stress on the cell boundaries with positive and inward stress with negative signs. This means that the absence of positive (outward) stress would lead to cell contraction and vice versa (∆V = 1 3B Tr{G i j }). All data provide evidence for positive components of the dipole tensor in x-and y-direction and negative components in the z-direction. Therefore, the presence of a vacancy (during the deintercalation process) leads to a contraction in the xy-plane and expansion in the z-direction. These results are in agreement with the data reported by Xiong et al. [35]. The results for the 1 × 1 × 1 cells deviate from the expected scaling behavior, since due to the high defect concentration, non-linear contributions prevail. Thus, these data are not shown in the plots. It is also evident that the data for the non-isotropically replicated supercells exhibit significant nonlinear contributions for the xx-and yy-components of the defect elastic dipole tensor.
In Figure 4 we compare the values of the diagonal components of G i j using histogram plots. These plots show the scaling behavior from lateral (or longitudinal) extension alone towards isotropic one. It can be seen that the G xx /G yy and G zz components do not show a linear scaling behavior. This is due to the fact that in these relatively small cells image-image interactions are still strong (especially across the xy-plane). Figure 5 reveals the origin of the strong coupling between G xx -G yy and G zz components. For this, we have plotted the charge density differences (ρ defective − ρ pristine ) for two supercell dimensions, namely 3×3×1 and 3×3×2. For both supercells, similar isosurface value of 0.001 |e|/Å 3 is considered. Figure 5 indicates the charge distribution difference between defective and pristine structures as the lattice is increasing along z-axis. It can be seen that by lattice extension along the z-axis, the overlap of orbitals is decreasing which leads to the minimization of defect-defect interactions. However, due to the especial geometry of LiCoO 2 unit cell, despite the 3 times repeated cell along the xy-plane, the two defects between nearby images still show significant electrostatic interactions (although the overlap of orbitals along the xy-plane is also decreasing but with a less progress) which can be translated to the rather strong coupling of xy-plane stress components.
To sum up this part, our results reveal that size and geom- etry effects have a massive influence on the calculation of the elastic dipole tensor because of the coupling between stress components. We see that the fits to the data scale more linearly as the cell sizes are increasing homogeneously (n×n×n). Therefore, non-linear terms affect the result the least for the isotropically scaled cell.
II. Defect formation energy under strain
In order to investigate the effect of strain on the defect formation energy in LiCoO 2 , we used Eqs. 5 (dipole-tensor method) and 6 (direct method). With SeqQuest we calculated a value of 2.86 eV for the formation energy of a neutral Li vacancy in the 3×3×1 supercell as our reference for the unstrained case, which is in agreement with data previously reported by Hoang et al. [20]. Fig. 6 shows the change in defect formation energy as a function of external strain (laterally, in the xy−plane) with two FIG. 5. Charge density difference (ρ defective − ρ pristine ) for 3×3×1 and 3×3×2 supercells. Green and yellow colors show charge accumulation and depletion, respectively, with an isosurface value of 0.001 |e|/Å 3 for both supercells. The position of Li vacancy in the middle and orbitals separation are indicated with red circles and arrows, respectively. The supercells boundaries are also shown with dashed black lines. methods and two DFT codes. For all cases, the red points are obtained from the direct calculation in which the total energies of pristine and defective are strained. In this case, we can directly calculate the formation energies under strain. Since we need to calculate the total energies in each strain regime, this is computationally time-consuming. The second method is shown with the blue points which are obtained from Eq. 5. In this case, we only need the total energies of unstrained pristine and defective supercells, together with the alreadycalculated G i j for that specific supercell.
The directly calculated strain-dependent defect formation energies show a slightly non-linear behavior and do not follow the prediction of linear elastic theory. Moreover, this nonlinear behavior is not symmetric in case of 3×3×1 supercell using either DFT methods, as can be seen in Fig. 6.
A similar trend for other systems has also been previously reported. For example Zhu et al. investigated the effect of strain on the formation energy of Cu vacancy in Cu 2 ZnSn(S,Se) 4 system [49] in which they show how sym-FIG. 6. (color online) Top: SeqQuest-based calculation of defect formation energy as a function of strain (laterally, in the xy−plane) using direct and dipole-tensor methods for a neutral single Livacancy in a 3×3×1 supercell. Bottom: VASP-based calculation of defect formation energy as a function of strain (in the xy-plane) using direct and dipole-tensor methods for a neutral single Li-vacancy in a 3×3×1 supercell. By comparison between two plots, negligible difference between two methods of DFT calculations (atomic orbitals and plane waves basis sets and different pseudopotentials) is evident. metry breaking will lead to this deviation from linear elasticity. Similarly Aschauer et al. found a non-linear behavior for the strain dependence of oxygen formation in MnO [50]. The reason why linear elastic theory fails to predict this trend is first due to the fact that there are higher order terms relevant in the Taylor expansion given in Eq. 1. Another reason is that the elastic constants of the defective system are not the same as in pristine system. Regardless of the methods and supercell sizes, it can also be concluded that under the influence of lateral elastic strain, the formation energy of a single neutral vacancy in bulk LiCoO 2 varies by about 0.02 eV (with = 1%). We now focus on the effect of strain on the Li diffusion for the pathway indicated in Figs. 1 (b) and (c). In the first section of results and discussion, we calculated the elastic dipole tensor in the dilute limit and revealed that size and geometry of the supercell have a significant influence. This is why we first investigated the influence of cell size on the migration barrier for Li vacancy jumps using the NEB method. The results are shown in Fig. 7. The calculated values for the 3×3×1 and 4×4×1 deviate by about 10%, while the smaller cell shows a significantly smaller migration barrier. In order to reduced computational efforts, we have chosen the 3×3×1 supercell in order to investigate the strain effect on migration energy barrier. Figure 8 shows migration barriers for vacancy hopping as function of strain using the directly calculated data (red in the xy-plane) and those obtained from the elastic dipole tensor method (blue in the xy-plane and green along the z-axis) for 3×3×1 supercell using Eq. 8. For the direct calculations, four lateral strain states are introduced homogeneously on the lattice parameters of a and b. The calculation of the elastic dipole tensor in Eq. 8 was performed at the initial and saddle points for the 3×3×1 supercell. Since we already calculated G i j for the initial state (Fig. 2), we used the geometry of the saddle point from the NEB calculation for the unstrained case to obtain the G i j for the transition state. We obtain the following values for the elastic dipole tensor in the initial state and at the saddle point:
G initial = G saddle = −0.08 −0.81 1.48 −0.81 0.77 −2.30 1.60 −2.49 8.24
By considering the signs of diagonal components of elastic dipole tensor for the initial state, the direction of the gray arrows in Figs. 1-b and 1-c are justified. Fig. 8 shows the very good agreement between the directly calculated data and those obtained from elastic-dipole tensor in the xy-plane. Due to computational limitations, we do not compare the two methods for the longitudinal stain, however, we postulate that the good agreement observed for the xyplane strain is also established for the z-axis strain. It can also be seen that all curves behave linearly but with different slopes between lateral and longitudinal strains (between red-blue and green plots). By applying a positive strain, the energy barrier decreases and vice versa, which is also in agreement with previous results from Ning et. al. [23]. It can be seen that, as a result of tensile strain, there is more Li-Li separation and less Li-Li repulsion and therefore, moving Li atom can intercalate/deintercalate more easily. From another point of view, applying strain could disturb the octahedrals orientation and therefore affects the potential energy surface which directly influences the energy barrier.
Since we see a good agreement between two methods in Fig. 8 for the lateral strain, we also plotted the effect of longitudinal strain on the migration energy barrier which is indicated with green color. It is clear from Fig. 8 that the longitudinal strain has a much stronger effect on the energy barrier, compared to lateral strain. Therefore, this shows that instead of significant computational efforts due to performing NEB calculations at each strain regime, it is possible to use Eq. 8 and obtain the same results with much less computational resources.
A comparison between Figs. 6 and 8 also indicates that the effect of lateral (xy-plane) strain on the Li vacancy formation and migration are opposite of each other. This means that, while a lateral compressive strain decreases the energy barrier (due to above mentioned reasons), the same strain results in a larger Li vacancy formation energy. However, as can be seen from the plots, this effect on formation and migration energy is not equal. Therefore in the activation energy, which is the sum of formation and migration, at a lateral strain regime equals to +1%, for example, the migration term is decreased by 0.04 eV while formation term is increased by 0.02 eV. Thus, at this strain value, the overall trend is a decrease in the activation energy by 0.02 eV. Considering the relation of diffusivity,
D ≈ exp − ∆G A k B T ,
decreasing of 0.02 eV in activation energy leads to almost five times increase in the diffusivity and hence the conductivity. Therefore, the massive strain effect on ionic conductivity in bulk LiCoO 2 is evident. We again note that the effect of longitudinal strain is even more dominant than the lateral one. Particularly, only 1% strain along the z-axis can change the conductivity up to ten times compared to unstrained case. Moreover, according to theoretical findings [36], 4-5% and experimental evidences [29,30], 2-3% strain is expected during the lithiation/delithiation of LiCoO 2 to Li 0.5 CoO 2 . Therefore, strain fields can significantly influence the ionic conductivity in bulk LiCoO 2 which shows that it can be employed to tune the particle mobility in battery materials.
SUMMARY AND CONCLUSIONS
In summary, we have performed a detailed comparison for the variations of defect formation and migration energies with respect to strain in bulk LiCO 2 using (i) direct evaluation of strained supercells (at each strain regime separately) and (ii) classical elasticity theory by computing the elastic dipole tensor (G i j ). The latter method requires only three DFT calculations (pristine and defected cells for formation energies plus defected cell in transition state for energy barriers) for evaluation of both formation and migration energies to obtain the full variations as a function of any strain state.
We found that the calculated formation energies using the elastic dipole tensor method deviate slightly when they are compared with the direct method (by less then 1% for < 0.015). We note tha, however, deviation to some degree depends on particular basis sets and/or pseudopotentials. Moreover, the mentioned deviation largely cancels out for migration barriers and it makes this method even more error-free in case of strain-induced diffusion analysis. Therefore, using the elastic dipole tensor method for the analysis of strainedinduced formation and migration energies is computationally very efficient.
We also highlight that estimating the elastic dipole tensor in the dilute limit can be affected by finite-size effects and coupling between the stress components. Moreover, we found that the contribution of migration energy to the total activation energy when a lateral strain regime is applied, is more dominant than the contribution of formation energy. Finally, we can conclude that the effect of even small strains on ionic transport properties in bulk LiCoO 2 is very significant. We showed that only 1% in-plane strain can change the conductivity by a factor of 5, while the presence of out-of-plane strains can change the conductivity by an order of magnitude.
FIG. 1 .
1(color online) (a) 3D view of conventional 1×1×1 unit cell for bulk LiCoO 2 , (b) top view of Li diffusion through a singlevacancy mechanism on a direct pathway and (c) diffusing Li position in the saddle point while causing distortion for its nearest Li neighbors. The direction of the gray arrows determined according to the sign of G (which is equal to the opposite sign of stress). Li, O and Co are shown with violet, red and blue, respectively, while migrating Li and Li vacancy are shown with green and white, respectively.
FIG. 2 .
2(color online) Variations of the diagonal elements of the stress tensor for various supercell sizes and aspect ratios. The stress variation is fitted to a second-order polynomial in the range indicated by solid lines; the coefficients of the linear term which corresponds to the elastic dipole tensor in the dilute limit are given in the legend. Supercell sizes for the left figure are n × n × n (1 ≤ n ≤ 3), middle figure n × n × 1 (1 ≤ n ≤ 5) and right figure n × n × 2 (1 ≤ n ≤ 4). The (0, 0) point is included in all plots.FIG. 3. (color online) Variations of the diagonal elements of the stress tensor. The stress variation is fitted to a second-order polynomial in the range indicated by solid lines; the coefficients of the linear term which corresponds to the elastic dipole tensor in the dilute limit are given in the legend. Supercell sizes for left figure are 1 × 1 × n, middle figure is 2 × 2 × n and right figure is 3 × 3 × n (1 ≤ n ≤ 4). The (0, 0) point is included in all plots.
FIG. 4 .
4(color online) Histogram plot of the size effect analysis for the diagonal components of the elastic dipole tensor in bulk LiCoO 2 containing a single Li vacancy.
FIG. 7 .
7(color online) Calculated migration energy barriers of neutral Li vacancy jump in bulk LiCoO 2 . Given are the results for different supercell sizes.FIG. 8. (color online) Diffusion energy barrier vs.applied external strain (laterally, in the ab−plane) using direct and dipole-tensor methods for a neutral single Li-vacancy in a 3×3×1 supercell. The red points correspond to NEB results in which a plane strain regime is applied. A perfect agreement with the dipole-tensor method (blue points-line) for the similar strain regime can be seen. Green pointsline shows the dipole tensor method for the longitudinal strain (along c−axis) which indicates a stronger effect compared to strain in the ab−plane.
AcknowledgmentsAM and PK gratefully acknowledge support from the "Bundesministerium für Bildung und Forschung" (BMBF), the computing time granted on the Hessian high performance computer "LICHTENBERG" and Zentraleinrichtung für Datenverarbeitung (ZEDAT) at the Freie Universität Berlin. JR and KA acknowledge support through SPP 1473 of the German Research Foundation and project DFG-578/19-1.
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| []
|
[
"Bulk and Two-dimensional Silver and Copper Monohalides: A Unique Class of Materials with Modest Ionicity/Covalency and Ferroelasticity/Multiferroicity",
"Bulk and Two-dimensional Silver and Copper Monohalides: A Unique Class of Materials with Modest Ionicity/Covalency and Ferroelasticity/Multiferroicity"
]
| [
"Yaxin Gao \nSchool of Physics\nWuhan National High Magnetic Field Center\nHuazhong University of Science and Technology\n430074Hubei, WuhanChina\n",
"Menghao Wu \nSchool of Physics\nWuhan National High Magnetic Field Center\nHuazhong University of Science and Technology\n430074Hubei, WuhanChina\n",
"Xiao Cheng Zeng \nDepartment of Chemistry and Department of Mechanical & Materials Engineering\nUniversity of Nebraska-Lincoln\n68588LincolnNebraskaUnited States\n"
]
| [
"School of Physics\nWuhan National High Magnetic Field Center\nHuazhong University of Science and Technology\n430074Hubei, WuhanChina",
"School of Physics\nWuhan National High Magnetic Field Center\nHuazhong University of Science and Technology\n430074Hubei, WuhanChina",
"Department of Chemistry and Department of Mechanical & Materials Engineering\nUniversity of Nebraska-Lincoln\n68588LincolnNebraskaUnited States"
]
| []
| Silver and copper monohalides can be viewed as a class of compounds in the "neutral zone" between predominantly covalent and ionic compounds, thereby exhibiting neither strong ionicity nor strong covalency. We show ab initio calculation evidence that silver and copper monohalides entail relatively low transition barriers between the non-polar rock-salt phase and the polar zinc-blende phase, due largely to their unique chemical nature of modest iconicity/covalency. Notably, the low transition barriers endow both monohalides with novel mechanical and electronic properties, i.e., coupled ferroelasticity and ferroelectricity with large polarizations and relatively low switching barriers at ambient conditions. Several halides even possess very similar lattice constants and structures as the prevailing semiconductors such as silicon, thereby enabling epitaxial growth on silicon. Moreover, based on extensive structural search, we find that the most stable two-dimensional (2D) polymorphs of the monolayer halides are close or even greater in energy than their bulk counterparts, a feature not usually seen in the family of rock-salt or zinc-blende semiconductors. The low transition barrier between zincblende phase and 2D phase is predicted. Moreover, several 2D monolayer halides also exhibit multiferroicity with coupled ferroelasticity/ferroelectricity, thereby rendering their potential applications as high-density integrated memories for efficient data reading and writing. Their surfaces, covered by halides, also provide oxidation resistance and give low cleave energy from layered structure, suggesting high likelihood of experimental synthesis of these 2D polymorphs. | 10.1039/c9nh00172g | [
"https://arxiv.org/pdf/1912.05172v1.pdf"
]
| 146,470,135 | 1912.05172 | 75d85dbbdfd59d0d09a5cf5d433596db4632b59f |
Bulk and Two-dimensional Silver and Copper Monohalides: A Unique Class of Materials with Modest Ionicity/Covalency and Ferroelasticity/Multiferroicity
Yaxin Gao
School of Physics
Wuhan National High Magnetic Field Center
Huazhong University of Science and Technology
430074Hubei, WuhanChina
Menghao Wu
School of Physics
Wuhan National High Magnetic Field Center
Huazhong University of Science and Technology
430074Hubei, WuhanChina
Xiao Cheng Zeng
Department of Chemistry and Department of Mechanical & Materials Engineering
University of Nebraska-Lincoln
68588LincolnNebraskaUnited States
Bulk and Two-dimensional Silver and Copper Monohalides: A Unique Class of Materials with Modest Ionicity/Covalency and Ferroelasticity/Multiferroicity
1
Silver and copper monohalides can be viewed as a class of compounds in the "neutral zone" between predominantly covalent and ionic compounds, thereby exhibiting neither strong ionicity nor strong covalency. We show ab initio calculation evidence that silver and copper monohalides entail relatively low transition barriers between the non-polar rock-salt phase and the polar zinc-blende phase, due largely to their unique chemical nature of modest iconicity/covalency. Notably, the low transition barriers endow both monohalides with novel mechanical and electronic properties, i.e., coupled ferroelasticity and ferroelectricity with large polarizations and relatively low switching barriers at ambient conditions. Several halides even possess very similar lattice constants and structures as the prevailing semiconductors such as silicon, thereby enabling epitaxial growth on silicon. Moreover, based on extensive structural search, we find that the most stable two-dimensional (2D) polymorphs of the monolayer halides are close or even greater in energy than their bulk counterparts, a feature not usually seen in the family of rock-salt or zinc-blende semiconductors. The low transition barrier between zincblende phase and 2D phase is predicted. Moreover, several 2D monolayer halides also exhibit multiferroicity with coupled ferroelasticity/ferroelectricity, thereby rendering their potential applications as high-density integrated memories for efficient data reading and writing. Their surfaces, covered by halides, also provide oxidation resistance and give low cleave energy from layered structure, suggesting high likelihood of experimental synthesis of these 2D polymorphs.
Introduction
Silver halides like AgBr and AgI are well known as photosensitive materials for photographics 1 and photochemistry. 2 They exhibit excellent photocatalytic performance under visible-light irradiation. 3 Their structural, transport, and dynamic properties have also been studied extensively. [4][5][6] At ambient conditions, bulk silver chloride and silver bromide form a rock-salt (RS) structure, while silver iodide crystallizes in a mixed phase of wurtzite (WZ) and zinc-blende (ZB); it also becomes a super-ionic conductor beyond 148 °C, where the silver ions migrate into the interstitial sites of the bcc sub-lattice of the immobile iodine ions. 7 Due to the small energy difference, phase transition from WZ to ZB may occur upon a relatively modest pressure of 0.1 GPa, and the phase transition to RS structure can take place upon further compression, together with the associated increase in cation-anion coordination from tetrahedral to octahedral. [8][9][10][11] Generally, the RS structure tends to exhibit ionic bonding features, whereas the ZB structure tends to exhibit more covalent bonding features. 4 At ambient conditions, the compounds like silver monohalidescopper monohalides (e.g., CuCl, CuBr) -also incline to form the ZB structure. 12 Their ionicity approaches 0.7 according to the Philips scale, slightly lower than the critical value 0.785, marking the idealized boundary between predominantly ''covalent'' and ''ionic'' systems. 13 In this work, we show ab initio computation evidence, for the first time, of the possible ferroelectricity 14 and ferroelasticity [15][16][17][18][19] at ambient conditions for the silver and copper monohalides.
It is known that ferroic (ferromagnetic, ferroelectric, or ferroelastic) materials can be applied as nonvolatile access memories (RAMs) as these materials possess bi-stable states with degenerate in energy. 20 Indeed, these functional materials provide an alternative approach to overcome some major issues in current Si-based RAMs, e.g., the quantum-tunneling due to the inequivalent bi-states, and power dissipation due to volatile memories. Multiferroic materials with more than one ferroic order parameters may even have the merits of combining different ferroics for efficient data reading and writing. 20,21 Note that although many materials possess spontaneous polarization/strain, they are not necessarily ferroelectric/ferroelastic, unless a switching pathway entails a relatively low barrier. 22,23 For example, most WZ and ZB semiconductors like ZnO are polar, due to alternating planes of cations and anions, but are still non-ferroelectric materials due to high switching barrier. For many ZB silver and copper halides considered, both the ferroelectric and ferroelastic switching barriers are lower than 0.15 eV, thereby becoming multiferroic. Moreover, their ferroelectricity and ferroelasticity are coupled, which is required for efficient data reading and writing, since a 90 degree lattice rotation can be driven by either an electric field or a strain, equivalent to a ferroelastic switching and 90-degree ferroelectric switching. Note also that silver and copper halides may be integrated with Si-based circuits via epitaxial growth due to similar lattice constants and structures as Si. The small lattice mismatch may resolve a major issue in integrating traditional ferroelectrics into silicon-based circuits.
Finally, we show that these halides also possess stable two-dimensional (2D) van der Waals polymorphs, and their cohesive energy difference with respect to the bulk phases are within several tenth of meV/f.u.. These 2D monolayers are predicted to be 2D multiferroics as well, with coupled ferroelectricity and ferroelasticity and even lower switching barrier. Such 2D multiferroics with atomic thickness may be more demanded as high-density integrated memories since van der Waals interface does not require lattice matching.
Computational methods
Our ab initio calculations are performed within the framework of spin-unrestricted density-functionaltheory (DFT), implemented in the Vienna ab initio Simulation Package (VASP 5.4) 24 . The projector augmented wave (PAW) potentials 25 for the core, and either the local density approximation (LDA) or the generalized gradient approximation (GGA) in the Perdew-Burke-Ernzerhof (PBE) 26 form for the exchangecorrelation functional, is applied. The Monkhorst-Pack k-meshes are set to 14 × 14 × 10 in the Brillouin zone, and the electron wave function is expanded on a plane-wave basis set with a cutoff energy of 520 eV. All atoms are relaxed in each optimization cycle until atomic forces on each atom are less than 0.01 eV Å −1 and the energy variation between subsequent iterations falls below 10 −6 eV. The Berry phase method is adopted in computing the ferroelectric polarizations 27 , and a generalized solid-state elastic band (G-SSNEB) method is used to calculate the pathway of phase transition. 28 An unbiased swarmintelligence structural method implemented in the CALYPSO code 29,30 is employed to search for stable 2D silver/copper monohalides.
Results and discussion
I.
Phase transition
The RS and ZB structures of silver halide crystals are displayed in Fig. 1(a). Our DFT calculations based on the LDA functional shows that RS structures are the ground state for both AgCl and AgBr, consistent with previous experimental observations. Meanwhile, the ZB structure is energetically more favorable for AgI, as for CuCl and CuBr. Meanwhile, the ZB structure is also predicted to be the ground state for AgCl and AgBr based on the GGA functional, inconsistent with previous experimental reports. 4 Hence, the LDA computation appears to be more reasonable for predicting the ground state structure for this class of materials. In any case, the energy difference between the two phases, ΔE=E(ZB)-E(RS), is all within 0.09 eV/f.u. for the silver and copper monohalides MX (see Table 1). The absolute values are much smaller, compared with either more ionic compounds such as NaCl or more covalent compounds such as ZnS. Fig. 1(c) plotted ΔE vs Philips iconicity of the binary semiconductors listed in Table 1, revealing an approximately linear growth model, where the silver and copper monohalides are located within the transition zone marked by the circled region. Table 1.
The ZB structure can be obtained by simultaneously displacing all halide anions in the RS structure to (¼, ¼, ¼), and the calculated pathway based on SSNEB (see Fig. S1) shows a transition barrier of 0.17 eV/f.u.
for AgBr. However, as we rotate the lattice of RS and ZB structures by 45 degree and adopt a smaller unit cell illustrated by the grey area in Fig. 1(a), the ZB structure can be also obtained by buckling the linear -M-X-chains in the RS structure. In view of the distinct lattice parameters and the small energy difference of the two phases, the phase transformation from RS to ZB can be induced by a tensile strain along the z axis by prolonging the lattice constant along -z with the vertical displacement of halide ions and buckling of -M-X-chains, as displayed in Fig. 1(b). For example, the calculated transition barrier is only 0.10 eV/f.u.
for AgBr, where the ZB phase, only 0.07 eV/f.u. higher in energy, may become more stable upon a tensile strain. The ZB metastable phase might also be obtained via epitaxial growth on substrates with similar ZB structure and similar lattice constants, e.g., AgCl on Ge, or AgBr on InAs. However, compared with a tensile strain, which can be difficult to control uniformly, an external electric field may be a more feasible approach for inducing the phase transformation, noting that the ZB structure is polar (with alternating planes of cations and anions) while the RS structure is non-polar.
II. Ferroelasticity/Multiferroicity
The ZB structure is non-centrosymmetric and thus a polarization might be spontaneously formed. The ZB structure of AgI can be viewed as bundles of AgI zigzag chains with their polarization along the z axis, as marked by red circles in Fig. 2(a), and the polarization may be switchable due to a relatively low switching barrier for those chains. We note that the WZ ZnO also possesses a polar structure, but its polarization is not deemed as switchable due to a relatively high switching barrier of 0.25 eV/f.u.. 31 It only becomes ferroelectric in the form of ultra-thin multilayers as the associated barrier is lowered to 0.10 eV/f.u. due to the effect of depolarization field. 32 If this polarization can be reversed with the switching of AgI zigzag chains in a possible switching pathway from I to II (see Fig. 2 Similarly, for ZB CuCl, the computed switching pathway shows an even higher barrier of 0. 35eV/f.u., despite the calculated polarization is as high as 0.59 C/m 2 . However, this barrier is not necessarily the lowest one among all the possible switching pathways, which will be demonstrated later. Compared with AgI, the energy difference between ZB and RS state of CuCl is much larger. The switching pathway of ZB AgI, with a local minima and two transition states, indicates that RS is metastable, whereas for CuCl the RS state is the transition state.
Another possible switching pathway is also taken into consideration. As displayed in Fig. 3(a), the polar zigzag chains in the initial state I may also be switched from -z to the -x or -y axis upon a horizontal electricfield, leading to a 90 degree polarization switching with a lattice rotation at the final state III. Here, the zigzag chains are not be flat like in Fig. 2(a) According to our SSNEB calculation, a switching barrier of 0.13 eV/f.u. is obtained for the pathway of 90 degree switching for ZB AgI, which is slightly lower than the 180 degree switching barrier. The identified intermediate state II, however, is a titled metastable structure that is 0.11 eV higher in energy than that of the ground state. As the polarization is switched to -x/-y direction, the lattice constant in -z direction is also swapped with the lattice constant in -x/-y direction, so their ferroelectricity and ferroelasticity are coupled.
For the ferroelastic (90 degree ferroelectric) switching pathway of CuCl displayed in Fig. 3(b), the switching barrier of ~0.15 eV/f.u. is much lowered compared with 180 degree ferroelectric switching. This is plausible as the intermediate state II maintains tetra-coordinate for all atoms, which is energetically more favorable compared with RS state for CuCl. Here, the 180 degree polarization switching can be achieved by repeating 90 degree switching twice, and this pathway is more favorable with a much lower barrier compared with the pathway in Fig. 2(b). As a result, CuCl can be also multiferroic at ambient conditions with coupled ferroelectricity and ferroelasticity.
The epitaxial growth of traditional ferroelectrics like perovskites on silicon is still challenging due to issues such as lattice mismatch, which hinders substitution of silicon-based RAMs by ferroelectric RAMs to be integrated with silicon wafer utilizing a mature silicon process. Having demonstrated a series of ZB multiferroic semiconductors in this study, we illustrate that CuCl and silicon entail similar lattice constants (within 1% lattice mismatch) and structures, as in the case for CuBr and germanium, as well as in the case for ZB AgI and InSb. The lattice similarities have important implication for the epitaxial growth and combination with prevailing semiconductors like silicon, as shown in Fig. S2(a). According to previous experimental reports, both CuBr and AgI are wide-direct-gap semiconductors (band gap in the range of 2.9 -3.4 eV) with high mobility, while Ge and InSb are both narrow-gap semiconductors (0.2 -0.7 eV). 6,33 If the mixing alloy Cu1-xBr1-xGex or Ag1-xI1-xInxSbx can be achieved, in view of their similar lattice constants and structures, a chemical multi-junction, as shown in Fig. S2(b), with a broad range of bandgaps might be built, which in principle can overpass the Shockley-Queisser detailed-balance limit of photovoltaic conversion efficiency for mono-bandgap semiconductors. 34 The We also explore the low-energy monolayer structure of silver/copper monohalides. After an extensive structure search based on CALYPSO program, we obtained two stable structures, I and II, as displayed in Fig. 4(a) for each of CuCl, AgCl and AgBr. The structure I is composed of two buckled layer of honeycomb MX lattice, slightly lower in energy compared with the structure II which is composed of a buckled Cu/Ag monolayer covered by halides at two surfaces. Similar but still different polymorph structures I and II for AgI/CuBr are also obtained, where the structure I is composed of a buckled honeycomb Ag/Cu layer covered by I/Br atoms, with slightly lower energy than the more symmetrical structure II which is composed of a planar square Ag/Cu monolayer covered by I/Br atoms. As listed in Table 2 values for CuCl and CuBr based on GGA computation imply that some 2D polymorphs may be even more favorable in energy compared with ZB phase of bulk structure. Even for LDA results, the 2D polymorph II will become 0.016 eV/f.u. lower in energy compared with ZB phase when half of the Cu atoms in CuCl is substituted by Ni atoms, as shown in Fig. 4(d).
Synthesis of 2D monolayers is likely to be feasible, in view of the small energy difference shown in Table 2. As a comparison, the energy difference between bulk silicon and 2D silicene is notably much larger As shown in Fig. 4(a), both 2D structure I and II for CuCl/AgCl/AgBr possess spontaneous in-plane polarization and strain, which may be also multiferroic if they are switchable. Similarly, their polar direction initially along -x axis may be switched to -y axis upon an electric-field, as displayed in Fig. 5(a) and (b), leading to a 90 degree polarization and lattice rotation. This transformation may also be obtained by an in-plane strain along the -y direction, so ferroelastic switching and 90 degree ferroelectric switching are equivalent. According to our SSNEB calculation, a switching barrier of 0.08 eV/f.u. is predicted for the pathway of ferroelastic switching for 2D CuCl polymorph I (Fig. 5(c)), about a half of the barrier height for bulk CuCl, as shown in Fig. 3(b). The 180 degree ferroelectric switching can be obtained by repeating 90 degree switching twice with the same switching barrier, and the switchable polarization of 2.8 × 10 -10 C/m is even higher than 2D SnS and SnSe. As a result, 2D CuCl I can be also multiferroic at ambient conditions with coupled ferroelectricity and ferroelasticity. The switching barrier for polymorph II is even lower, as displayed in Fig. S4(a), and the switchable polarization is 1.5 × 10 -10 C/m. The band structures of both 2D polymorphs ( Fig. 5(d) and (e)) are highly anisotropic, with distinct effective mass along Γ-X and Γ-Y, so their ferroelastic switching (90 degree ferroelectric switching) can be electrically detected. One may even obtain a 2D triferroics 37 by doping 3d magnetic element. For example, the polymorph II of CuCl becomes ferromagnetic when some of Cu atoms are substituted by Ni atom, as shown in Fig. S4(b). Each Ni atom possesses 1 µB magnetic moment, while the easy axis of magnetism and electrical polarization are both aligned in the same in-plane direction, which can be also switched by 90 degree upon ferroelastic switching. As a result, their ferromagnetism, ferroelectricity and ferroelasticity are all coupled. In summary, through density-functional theory calculations we demonstrate that the modest barrier for the phase transition between bulk RS and ZB phase may give rise to ferroelectricity and ferroelasticity with relatively low switching barriers at ambient conditions. Some bulk halides even entail similar lattice constants and structures as prevailing semiconductors like silicon, thereby rendering epitaxial growth a high possibility to potentially resolving a major issue of integrating traditional ferroelectrics into siliconbased circuits. Moreover, their stable 2D polymorphs are close or even lower in energy compared with bulk phases, suggesting high likelihood for successful synthesis of these 2D monolayers in future. Some 2D monolayers are also predicted to be 2D multiferroics. As such, their coupled ferroelasticity and ferroelectricity together with anisotropic band structures are highly desired for efficient data reading and writing.
Electronic Supplementary Information is available
Notes
The authors declare no competing financial interest.
Figure 1 .
1Geometric structure of (a) RS and ZB silver halide crystals, and smaller unit cells marked by the dark area can be adopted. (b) Pathway of phase transition from RS (left) to ZB (right) structure. Blue, green, brown spheres denote Ag, Cl, Br/I atoms, and red arrows denote the strain direction. (c) Energy difference between RS and ZB phase (LDA computation) vs iconicity in
Figure 2 .
2An illustration of the ferroelectric switching pathway of (a) ZB and (b) WZ AgI, where the top panels show that the zigzag chains in red circles (lower panels) at the initial state (I) become flat at the intermediate state (II), and are finally reversed in state III. Thick/grey arrows in (a) denote the direction of polarizations. (c) Calculated energy profile of switching pathway for ZB AgI, WZ AgI and ZB CuCl by using the NEB method.
(a)), the symmetrical RS phase can be the intermediate state (II) as the zigzag AgI chains become flat with the halide anions moving along y direction. The switching pathway is computed by using the SSNEB method as shown in Fig. 2(a), suggesting a low switching barrier of 0.10 eV/f.u. The latter is much lower than the switching barrier of PbTiO3 (~0.20eV/f.u.). Meanwhile, a large polarization of 0.40 C/m 2 is formed in BZ-AgI, based on our Berry-phase calculation, a value much greater than that in BaTiO3 (~0.26 C/m 2 ). The WZ phase of AgI (0.2meV/f.u.) is also polar with a polarization of 0.20 C/m 2 , while our SSNEB calculation of the pathway inFig. 2(c)shows that the ferroelectric switching barrier can be as high as 0.27 eV/f.u., even higher than that for WZ ZnO (0.25 eV), making ferroelectric switching almost impossible at ambient condition.
Figure 3 .
3ferroelectric polarization may enhance the open-circuit voltage and the lifetime of excitons by hindering the recombination of electrons and holes. (a) The pathway of ferroelastic switching (90 degree ferroelectric switching) and (b) calculated energy profile based on SSNEB method, for AgI and CuCl. Here the zigzag chains at the initial state (I) are switched by 45 degree at the intermediate state (II), and then by 90 degree in state III.
, the energy difference between bulk phase and the 2D polymorphs is around a tenth of eV/f.u., much smaller than that (0.43/0.26 eV/f.u. based on LDA/GGA computation) of the highly ionic crystal of NaCl, or that (0.56/0.44 eV/f.u. based on LDA/GGA computation) of the covalent crystal of ZnS.Fig. 4(c)displays a possible pathway of phase transition from bulk ZB phase to multilayered polymorph II phase, which can be viewed as the rotation of zigzag chains in the red circles. A barrier lower than 0.10 eV/f.u. for such a phase transition of CuCl is predicted based on the SSNEB calculations and LDA method. The negative
( 0 .
076/0.65 eV/f.u. by LDA/GGA), and yet 2D silicene has been synthesized in the laboratory. Furthermore, the surfaces covered by halide anions also give rise to oxidation resistance and low cleave energy from layered structure: e.g., the cleavage energy from multilayer to monolayer structure for CuCl I and II are respectively 0.27 and 0.20 J/m 2 , even lower than that of graphite to graphene (0.3581 J/m 2 ). We have further verified dynamic stability of the 2D monolayers by computing their phonon dispersions, as shown in Figures S3. All the vibration spectra are free of soft modes associated with structural instabilities.
Figure 4 .
4(a,b) Top view (upper panels) and side view (lower panels) of 2D polymorphs, I and II, for (a) CuCl/AgCl/AgBr, and (b) AgI/CuBr. (c) Computed pathway (middle panel) of phase transition for CuCl from bulk ZB (left panel) to multilayered polymorph II (right panel) structure by using LDA method. (d) A comparison of bulk ZB (left panel) and multilayered polymorph II (right panel) structure of Cu0.5Ni0.5Cl.
Figure 5 .
5Ferroelastic switching (90 degree ferroelectric switching) of 2D polymorph (a) I and (b) II of CuCl. (c) Calculated pathway of ferroelastic switching for polymorph I. Band structure of polymorph (d) I and (e) II of CuCl are calculated based on the hybrid functional HSE06 38 .
Table 1 .
1Philips ionicity (from Ref. 14) and energy difference between RS and ZB phase calculated by using LDA/GGA (negative values reveal that ZB phase is more favorable in energy).AgCl
AgBr
AgI
CuCl
CuBr
NaCl
ZnS
ionicity
0.856
0.850
0.770
0.746
0.735
0.935
0.623
ΔE(eV/f.u.) LDA
0.09
0.07
-0.07
-0.31
-0.30
0.27
-0.54
GGA
-0.05
-0.08
-0.19
-0.35
-0.36
0.14
-0.63
, but can be switched by 90 degree in the intermediate state II.This transformation may also be obtained by a horizontal strain along the -x/-y direction with smaller lattice constants. Upon a strain or an electric field, the rotated and equivalent state III could be more favorable in energy. As a result, the transformation from I to III can be viewed as both ferroelastic switching and 90 degree ferroelectric switching, which may take place at ambient conditions if the switching barrier is within the desirable range (< 0.15 eV). The symmetrical RS state could still be an intermediate state in a possible switching pathway, and the switching barrier could be moderate.
Table 2 .
2Energy difference between bulk phase and 2D polymorph I/II (ΔI/ΔII), = E(2D)-E(bulk).
AgCl
AgBr
AgI
CuCl
CuBr
ΔI
(eV/f.u.)
LDA
0.17
0.12
0.05
0.11
0.04
GGA
0.07
0.05
0.035
0.02
-0.01
ΔII
(eV/f.u.)
LDA
0.24
0.15
0.08
0.13
0.07
GGA
0.13
0.08
0.05
-0.01
0.00
III.
Two-dimensional monolayer polymorphs
AcknowledgementThis work is supported by National Natural Science Foundation of China (Nos. 21573084). XCZ is supported by UNL Holland Computing Center.
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"A New Method for Evaluating the Effectiveness of Plastic Packaging Against Radon Penetration",
"A New Method for Evaluating the Effectiveness of Plastic Packaging Against Radon Penetration"
]
| [
"Yue Meng \nDepartment of Physics and Astronomy\nUniversity of Alabama\n35487TuscaloosaAL, US\n",
"Jerry Busenitz \nDepartment of Physics and Astronomy\nUniversity of Alabama\n35487TuscaloosaAL, US\n",
"Andreas Piepke \nDepartment of Physics and Astronomy\nUniversity of Alabama\n35487TuscaloosaAL, US\n"
]
| [
"Department of Physics and Astronomy\nUniversity of Alabama\n35487TuscaloosaAL, US",
"Department of Physics and Astronomy\nUniversity of Alabama\n35487TuscaloosaAL, US",
"Department of Physics and Astronomy\nUniversity of Alabama\n35487TuscaloosaAL, US"
]
| []
| Deposition of 222 Rn daughters onto detector materials pose a risk to ultra-low background experiments. To mitigate this risk, a common approach is to enclose detector components in sealed plastic bags made of films known to be effective barriers against radon. We describe a new method to evaluate radon barriers which is unique in that (a) it gauges not only the intrinsic resistance to radon penetration of a plastic film but also the integrity of bags fabricated from the film and sealed following some protocol, and (b) it employs gamma spectroscopy rather than alpha spectroscopy. We report the results of applying this method to sealed bags fabricated from polypropylene, Nylon, Mylar, metallized Mylar, FEP, and PFA. Evaluation of the fluoropolymers FEP and PFA as radon barriers are the first such measurements. | 10.1016/j.apradiso.2019.108963 | [
"https://arxiv.org/pdf/1903.02643v2.pdf"
]
| 119,344,830 | 1903.02643 | 45af284d45ef6c7d447d5e570d7d14f1815165f3 |
A New Method for Evaluating the Effectiveness of Plastic Packaging Against Radon Penetration
Yue Meng
Department of Physics and Astronomy
University of Alabama
35487TuscaloosaAL, US
Jerry Busenitz
Department of Physics and Astronomy
University of Alabama
35487TuscaloosaAL, US
Andreas Piepke
Department of Physics and Astronomy
University of Alabama
35487TuscaloosaAL, US
A New Method for Evaluating the Effectiveness of Plastic Packaging Against Radon Penetration
Radon PenetrationRadon PermeabilityHPGe DetectorPacking FilmsIntegrity of BagsDeposition of 222 Rn Daughters
Deposition of 222 Rn daughters onto detector materials pose a risk to ultra-low background experiments. To mitigate this risk, a common approach is to enclose detector components in sealed plastic bags made of films known to be effective barriers against radon. We describe a new method to evaluate radon barriers which is unique in that (a) it gauges not only the intrinsic resistance to radon penetration of a plastic film but also the integrity of bags fabricated from the film and sealed following some protocol, and (b) it employs gamma spectroscopy rather than alpha spectroscopy. We report the results of applying this method to sealed bags fabricated from polypropylene, Nylon, Mylar, metallized Mylar, FEP, and PFA. Evaluation of the fluoropolymers FEP and PFA as radon barriers are the first such measurements.
Introduction
For low background dark matter or neutrino experiments, 222 Rn daughter "plate-out" onto the surface of detector components is a potentially dangerous source of experiment backgrounds. The main process is neutron production via the (α, n) process induced by 210 P o decay. A common prevention method is to pack materials and parts inside sealed bags which are known to be effective barriers against radon diffusion. We apply a Pylon radon source (Pylon Model RN-1025), an air-tight purge box, bags fabricated from different materials, a steel can with a press-fit lid, and a high purity germanium detector to measure radon penetration through sealed bags. In addition to providing a quantitative measure of the effectiveness of a packaging method, the measurement results may be interpreted in terms of the radon diffusion constant and radon solubility for the bag material; we are providing such interpretation in Section 7.
Many studies of films as radon barriers have been carried out; see, for example, [1][2][3][4] and citations therein. What is unique to this method is that it employs gamma spectroscopy and evaluates not only the film but also sealed bags fabricated from these films.
Experimental Setup and Procedure
Radon penetration measurements through different sealed bags were conducted at room temperature. The purge system (Figure 1) consists of a carrier gas bottle, a flow controller, a flow meter, a radon source, an air-tight purge box and a "monitor can". The monitor can contains a sample of the radon loaded gas used to determine film penetration. At the end of the radon exposure it is counted and serves as normalization for the "sample can", described
below. An open steel can, equipped with a press-fit lid, is sealed (along with the lid) inside a test bag procured commercially. This test bag is made of the film to be studied. After placing the steel can and lid inside the bag, the open end is closed by making three adjacent seals with a hand-operated impulse heat sealer. This "sample can"-bag assembly is, in turn, placed inside an air-tight purge box where it is exposed to a radon loaded gas atmosphere for some period of time. Operation in gas purge mode assures a constant radon concentration.
The lateral dimensions of the sealed bag were measured to determine its area A.
The volume V of the bag interior was estimated, for some bags, by immersing representative samples in water and measuring the displacement. Radon was transferred to the purge box by means of nitrogen carrier gas, flowing through a Pylon (Model RN-2015) radon source, containing 246 kBq of 226 Ra. The radon concentration of the purge gas is given by the calibration certificate of the source. Multiple samples were exposed to this 222 Rn-rich atmosphere sufficiently long for radon concentrations to reach steady-state. The radon purge was then terminated by closing off and detaching the monitor can and quickly flushing the purge box with nitrogen gas. The bag -sample can combinations were then removed from the purge box. To seal the gas, contained inside the bag, into the sample can, the lid was pressed in, before opening and discarding the bag. The radon activity contained inside the, now sealed, "sample cans" was determined using HP Ge-detectors. This counting was performed in form a so-called time series, allowing a fit to the exponential decay, to determine the 214 P b and 214 Bi activities at the reference time, defined as the end of the radon purge.
Principle
The measured radon concentration inside the bag, relative to that outside the bag during exposure to the radon-rich atmosphere, provides a measure of the effectiveness of the sealed bag as a radon barrier. Assuming that the bag has been sealed properly, the effectiveness depends mainly on the diffusion of radon through the film, which is described in terms of the radon diffusion constant and the solubility of radon in the film material. In this section, we derive the relationships used to interpret the ratio of measured concentrations in terms of the diffusion constant and solubility. See Reference [5] for an alternate derivation with details.
The following assumptions are made. First, the time over which the bag is exposed to the radon-rich environment is sufficiently long for steady-state conditions to be achieved. Once the diffusion constant is determined, the time required to reach steady-state can be estimated and compared to the actual the typical film thickness is on the level of a few mils (∼100 microns) and the typical radius of curvature is on the level of a few cm. Third, the concentration of radon inside the bag is instantaneously uniform throughout the volume. This is effectively achieved at room temperature where the lineal dimension-about 10 cm-of the bag divided by root mean square speed of the radon atoms is short compared to the diffusion time through the film. Last, radon emanation from the bag or its contents is negligible. Radon diffusion through a planar barrier, where x is the distance into the barrier measured from the outside surface, obeys the diffusion equation, with the additional term λ · C, to account for the decay of the diffusing substance:
∂C ∂t = D · ∂ 2 C ∂x 2 − λ · C,(1)
where C is the radon number concentration (atoms per unit volume), D is the diffusion constant and λ is the radon decay constant. We emphasize that C is the radon number concentration within the barrier material itself.
At steady state
0 = D · ∂ 2 C ∂x 2 − λ · C(2)
Let C 0 ≡ C(0) be the radon number concentration at the outside edge of the film and C 1 ≡ C(d) be the radon number concentration at the inside edge of the film, where we have denoted the film thickness by d. The concentration of radon atoms outside the bag will not necessarily be the same as C 0 , owing to the fact that the solubility of radon in the bag material may be different than it is in the outside medium. For the same reason, C 1 will not necessarily be the same as the radon number concentration inside the bag. We therefore introduce the external and internal gas-space radon number concentrations, C 0 and C 1 , respectively. C 0 depends on the properties of the radon purge system and C 1 is measured. Finally, we link the boundary values of the radon number concentrations in the bag film with the radon number concentrations in the internal and external media by
C 0 C 0 = C 1 C 1 = S(3)
where we have assumed that the gases on the inside and outside of the bag are identical. S is the solubility of radon in the bag film relative to the gas. For our measurements, the outer medium is nitrogen gas and the inner medium initially air. One expects that the inner medium becomes dominantly nitrogen gas in steady state.
As already mentioned, C 0 is inferred from the properties of the radon purge system (radon source calibration and carrier gas flow rate) and C 1 is measured.
At steady-state, the number of radon atoms diffusing into the interior of the bag equals the decay rate inside the bag. φ C , the flux of radon atoms at depth
x in the film, is evaluated according using Fick's law:
φ C = −D · dC dx .(4)
Inside the sealed bag of total surface area A and volume V , the concentration C 1 obeys the equation
dC 1 dt = φ C1 · A V − λ · C 1 ,(5)
where φ C1 denotes the radon flux at the inner surface of the bag. The change in radon concentration inside the bag receives a contribution from diffusion into the bag and another one from Rn-decay. It is assumed that the bag itself is not a source of radon.
Once steady state has been achieved,
φ C1 = λ · V A · C 1(6)
Combining the above relationships, we obtain
α · β = S · A V 1 − β · cosh(α · d) sinh(α · d)(7)
where
β ≡ C 1 C 0 (8) α ≡ λ D(9)
This transcendental equation, which agrees with Reference The decay rate of 214 P b and 214 Bi, contained in the monitor and sample cans, can be inferred from the counting data, live-times, branching ratios and detection efficiencies [6]. This determines the 222 Rn activity contained in the cans. Example decay curves for sample cans sealed inside polypropylene and Nylon pouches are shown in Figure 3. An exponential function plus a constant background was used to fit the sequential decay rates in Figure 3. Excluding the first few hours after end of purge allows for secular equilibrium to be established. reference sample in each group. The results of this first series of measurements, in which the monitor can had not yet been implemented, are reported as a ratio R of the activity determined for the can sealed inside the bag material under study, divided by the activity of a can exposed in parallel but contained in a polypropylene bag. Polypropylene was chosen to serve as the reference material as it is known to be quite permeable to radon [2]. The R-values obtained this way are shown in Table 1 together with other relevant parameters.
Bag Integrity Testing
The data presented in Table 1 shows one relatively high R-value for a Nylon bag tested in Group 4. This reading seems to contradict the Nylon results obtained in Group 1. It should be noted that all Nylon bags were made from the same material stock. The high reading further seems to run counter previous permeability measurements made by others [7,8]. We take this result as evidence that this particular bag was not properly sealed. We carried out a 3-week radon exposure of three metallized Mylar bags and a polypropylene reference bag. Two of the metallized Mylar bags had pinholes ( Figure 4) bag dimensions are listed in Table 2. This data clearly identifies the punctured Mylar bags as such. This data shows that the method described here is capable of identifying even small breaches in radon enclosure films. Table 2: Radon concentrations and bag geometries for the "pinhole test" measurements.
Ratio of Concentrations Relative to Monitor Can
As described in Section 3, the method discussed here can be used to estimate the diffusion constant if the solubility of radon in the bag material is known.
This requires, however, that the ratio of the radon concentrations inside and outside of the bag are known. In order to interpret our radon permeation results in terms of terms of diffusion constants, we implemented a "monitor can" for absolute normalization, as mentioned earlier. With it we sample the radon-loaded gas directly. We denote the ratio of concentrations determined for a particular sample relative to the monitor can by R . A polypropylene can was used in this test as well. This measurement of the ratio of the polypropylene activity to the monitor can activity, denoted R pp allows one to re-normalize the R-data reported in Table 1 by the simple scaling relation:
R = R P P · R(10)
To determine R P P , 5 different polypropylene-sealed sample cans were analyzed.
The data resulting from these measurements is shown in Table 3. This data shows good consistency and reproducibility. An average variance-weighted correction factor of R P P = 0.856 ± 0.004 was obtained this way. Table 4 shows the R P P -corrected activity ratios for the films studied here. In case more than one measurement were available, the variance-weighted average was taken and renormalized by the correction factor R P P . surface area, and concentration ratio allow us to construct the ±1 σ confidence bands shown in Figure 5 (red: polypropylene, black: Nylon, blue: transparent Mylar).
Bag material
The red dashed lines in Figure 5 indicate the minimum and maximum values for the diffusion constant for polypropylene as reported in Reference [2]. The diffusion constants estimated from the data presented in this paper are, within a reasonable range of solubility, consistent with previous measurements.
Nylon seems to be an exception. A possible reason for this deviation could be uncontrolled experimental parameters. It was reported in reference [7] that the diffusion constant of Nylon depends on the water content of the material.
Uncontrolled environmental humidity may, thus, serve as an explanation. Note, however, that Reference [9] did not observe a dependence on relative humidity.
It is further not clear by how much the Nylon diffusion constant depends on the manufacturing details which we don't control. We therefore conclude that such measurements allow one to estimate the barrier effectiveness for various film materials while precise measurements require tight material specifications and controls. The latter was beyond the scope of the study presented here.
Conclusion
We describe a relatively simple method for evaluating the effectiveness of sealed bags as radon barriers. We show that this method is very sensitive for detecting even small breaches in the barrier. Results are reported for polypropylene, FEP, PFA, Nylon, transparent Mylar, and several types of metallized Mylar.
Acknowledgements
This work was motivated by the need of the LZ collaboration to effectively seal delicate detector components from environmental radon and the resulting plate-out of its daughters. We thank our LZ colleagues for encouragement and stimulating discussions. We are grateful to Devin Radloff for his contributions to the initial setup, measurements, and data analysis. This research was supported in part by the U.S. Department of Energy under DOE Grant de-sc0012447.
Figure 1 :
1The radon purge system. exposure time to check this assumption. Second, the film may be treated as a planar barrier, a good approximation if the characteristic radius of curvature of the bag surface is large compared to the film thickness. For our measurements,
[ 5 ]
5but uses different notation, can be solved for the diffusion constant if the ratio of concentrations and solubility are known. Alternately, if the ratio of concentrations and radon diffusion constant are known, one may determine the solubility. In order to determine both the radon diffusion constant and solubility, one must measure the concentrations as a function of time, not just at steady-state, as reported here. Therefore, our measurements determine the radon diffusion constant as a function of the solubility.
were exposed in each radon purge run, each sample was counted, in turn, with the counting time interval for each sample chosen based on the expected activity. For each daughter, only the most prominent gamma line, 352 keV for 214 P b and 609 keV for 214 Bi were used. Example fits to 214 P b and 214 Bi γ-peaks (Gaussian plus linear background) are shown in Figure 2.
Figure 2 :
2Left: 352 keV γ-peak of 214 P b. Right: 609 keV γ-peak of 214 Bi. Black dots are data and the red lines show the fits.
Figure 3 :
3214 P b and 214 Bi decay rates vs time. The red dots show the time dependence of the 352 keV γ-peak rates, resulting from 214 P b decay. The blue dots that of the 609 keV γ-peak of 214 Bi. Both are fitted (solid black line) with an exponential plus constant background.
1 :Figure 4 :
14Packaging properties and results for groups of samples. The left column gives the packaging material. (For FEP and PFA, bag fabrication and sealing was performed by American Durafilm Co., Holliston, MA, USA.) The bag area and volume are estimated from lateral dimensions and in some cases water displacement measurements. Column 4 lists the film thickness. R is the ratio of the radon concentration, determined for the sample can contained in the material of interest, divided by the concentration measured for a sample can packaged in polypropylene film. The errors given with the R-values are statistical only. *See the following section.All three metallized Mylar bags were sealed shut. The counting results and Meltallized Mylar bag with a pinhole, indicated by the red arrow. The diameter of the pinhole is less than 0.5 mm.
Figure 5 :
5Diffusion constants versus solubility of polypropylene (red band), Nylon (black band) and transparent Mylar (blue band), estimated from the data presented in this paper. Also shown (by points, lines, and hatched bands) are results from previous measurements.See the text for details.
punched into them to demonstrate the method presented here is capable of identifying them.Bag material
Bag volume
[L]
Bag area
[cm 2 ]
Film thickness
[µm]
R
Group 1
Polypropylene
0.7
697
101.6
1
Nylon
0.4
407
50.8
0.0104 ± 0.0003
Nylon
0.4
407
50.8
0.0076 ± 0.0003
Metallized Mylar-Type 1
0.5
503
63.5
0.0002 ± 0.0001
Metallized Mylar-Type 1
0.5
503
63.5
0.0005 ± 0.0001
Group 2
Polypropylene
0.8
929
101.6
1
Metallized Mylar-Type 2
0.5
542
101.6
0.0009 ± 0.0003
Metallized Mylar-Type 3
0.5
542
109.2
0.0012 ± 0.0003
Group 3
Polypropylene
0.7
668
101.6
1
PFA
0.4
387
50.8
1.17 ± 0.01
PFA
0.4
387
50.8
1.15 ± 0.01
FEP
0.4
439
127.0
0.114 ± 0.001
FEP
0.4
411
127.0
0.110 ± 0.002
Transparent Mylar
0.4
439
76.2
0.0006 ± 0.0002
Transparent Mylar
0.4
411
76.2
0.0009 ± 0.0003
Group 4
Polypropylene
0.4
397
101.6
1
Transparent Mylar
0.4
397
76.2
0.0001 ± 0.0004
Nylon*
0.4
397
50.8
0.99 ± 0.01
Metallized Mylar-Type 1
0.4
397
63.5
0.0011 ± 0.0002
Metallized Mylar-Type 2
0.4
397
101.6
0.0019 ± 0.0008
Metallized Mylar-Type 3
0.4
397
109.2
0.0027 ± 0.0003
Table
Table 3 :
3Determination of the scaling factor R P P (defined in the text) by means of repeated measurements. Uncertainties are statistical only.
Table 4 :
4Properly corrected measured ratios of radon concentrations inside and outside of bags made from the materials listed here. Diffusion constants for some of the materials investigated here have been previously measured. In order to demonstrate the validity of our method, we used our data to estimate the diffusion constant. This allows for a consistency check with previous measurements in the cases of polypropylene, Nylon, and transparent Mylar. Using Equation 7, we estimate the diffusion constant for each material as a function of the unknown solubility from our activity ratio, film thickness, bag area and bag enclosed volume data. The diffusion constant was calculated varying the solubility over a range of 1-20. Estimated uncertainties on volume,7. Determination of Diffusion Constants for Comparison with Previ-
ous Measurements
The dashed blue line shows the upper limit on the diffusion constant of transparent Mylar, as published in[1]. There are several previous measurements for Nylon, including ones in which the diffusion constant and solubility were simultaneously measured. The dotted black line denotes the value of the diffusion constant reported in[1]. The results of References[7] and[8] for dry Nylon are plotted as open-square points. Shown by the pink hatched band is the measurement of the dry Nylon diffusion constant reported by Reference[9].
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A theoretical approach to the measurement of radon diffusion and adsorption coefficients in radonproof membranes. P Fernández, L Quindós, C Sainz, J Gómez, 10.1016/j.nimb.2003.09.027Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms. 2171P. L Fernández, L. S Quindós, C. Sainz and J. Gómez, A theoretical approach to the measurement of radon diffusion and adsorption coefficients in radon- proof membranes, Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms 217 (1) (2004) 167 -176. doi:https://doi.org/10.1016/j.nimb.2003.09.027.
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| []
|
[
"Galaxy Evolution in all Five CANDELS Fields and IllustrisTNG: Morphological, Structural, and the Major Merger Evolution to z ∼ 3",
"Galaxy Evolution in all Five CANDELS Fields and IllustrisTNG: Morphological, Structural, and the Major Merger Evolution to z ∼ 3"
]
| [
"A Whitney \nSchool of Physics & Astronomy\nUniversity of Nottingham\nNG7 2RDNottinghamUK\n",
"L Ferreira \nSchool of Physics & Astronomy\nUniversity of Nottingham\nNG7 2RDNottinghamUK\n",
"C J Conselice \nSchool of Physics & Astronomy\nUniversity of Nottingham\nNG7 2RDNottinghamUK\n\nJodrell Bank Centre for Astrophysics\nUniversity of Manchester\nOxford RoadManchesterUK\n",
"K Duncan \nInstitute for Astronomy\nSUPA\nRoyal Observatory\nBlackford HillEH9 3HJEdinburghUK\n\nLeiden Observatory\nLeiden University\nPO Box 9513NL-2300 RALeidenThe Netherlands\n"
]
| [
"School of Physics & Astronomy\nUniversity of Nottingham\nNG7 2RDNottinghamUK",
"School of Physics & Astronomy\nUniversity of Nottingham\nNG7 2RDNottinghamUK",
"School of Physics & Astronomy\nUniversity of Nottingham\nNG7 2RDNottinghamUK",
"Jodrell Bank Centre for Astrophysics\nUniversity of Manchester\nOxford RoadManchesterUK",
"Institute for Astronomy\nSUPA\nRoyal Observatory\nBlackford HillEH9 3HJEdinburghUK",
"Leiden Observatory\nLeiden University\nPO Box 9513NL-2300 RALeidenThe Netherlands"
]
| []
| A fundamental feature of galaxies is their structure, yet we are just now understanding the evolution of structural properties in quantitative ways. As such, we explore the quantitative non-parametric structural evolution of 16,778 galaxies up to z ∼ 3 in all five CANDELS fields, the largest collection of high resolution images of distant galaxies to date. Our goal is to investigate how the structure, as opposed to size, surface brightness, or mass, changes with time. In particular, we investigate how the concentration and asymmetry of light evolve in the rest-frame optical. To interpret our galaxy structure measurements, we also run and analyse 300 simulation realisations from IllustrisTNG to determine the timescale of mergers for the CAS system. We measure that from z = 0 − 3, the median asymmetry merger timescale is 0.56 +0.23 −0.18 Gyr, and find it does not vary with redshift. Using this data, we find that galaxies become progressively asymmetric at a given mass at higher redshifts and we derive merger rates which scale as ∼ (1 + z) 1.87±0.04 Gyr −1 , which agrees well with recent machine learning and galaxy pair approaches, removing previous inconsistencies. We also show that far-infrared selected galaxies that are invisible to HST have a negligible effect on our measurements. We also find that galaxies are more concentrated at higher redshifts. We interpret this as a sign of how their formation occurs from a smaller initial galaxy that later grows into a larger one through mergers, consistent with the size growth of galaxies from 'inside-out', suggesting that the centres are the oldest parts of most galaxies. | 10.3847/1538-4357/ac1422 | [
"https://arxiv.org/pdf/2105.01675v1.pdf"
]
| 233,739,816 | 2105.01675 | d38a882140c3311cb5d00a1ad74306b8c9927984 |
Galaxy Evolution in all Five CANDELS Fields and IllustrisTNG: Morphological, Structural, and the Major Merger Evolution to z ∼ 3
May 6, 2021
A Whitney
School of Physics & Astronomy
University of Nottingham
NG7 2RDNottinghamUK
L Ferreira
School of Physics & Astronomy
University of Nottingham
NG7 2RDNottinghamUK
C J Conselice
School of Physics & Astronomy
University of Nottingham
NG7 2RDNottinghamUK
Jodrell Bank Centre for Astrophysics
University of Manchester
Oxford RoadManchesterUK
K Duncan
Institute for Astronomy
SUPA
Royal Observatory
Blackford HillEH9 3HJEdinburghUK
Leiden Observatory
Leiden University
PO Box 9513NL-2300 RALeidenThe Netherlands
Galaxy Evolution in all Five CANDELS Fields and IllustrisTNG: Morphological, Structural, and the Major Merger Evolution to z ∼ 3
May 6, 2021(Received XXX; Revised YYY; Accepted ZZZ)Draft version Typeset using L A T E X twocolumn style in AASTeX63
A fundamental feature of galaxies is their structure, yet we are just now understanding the evolution of structural properties in quantitative ways. As such, we explore the quantitative non-parametric structural evolution of 16,778 galaxies up to z ∼ 3 in all five CANDELS fields, the largest collection of high resolution images of distant galaxies to date. Our goal is to investigate how the structure, as opposed to size, surface brightness, or mass, changes with time. In particular, we investigate how the concentration and asymmetry of light evolve in the rest-frame optical. To interpret our galaxy structure measurements, we also run and analyse 300 simulation realisations from IllustrisTNG to determine the timescale of mergers for the CAS system. We measure that from z = 0 − 3, the median asymmetry merger timescale is 0.56 +0.23 −0.18 Gyr, and find it does not vary with redshift. Using this data, we find that galaxies become progressively asymmetric at a given mass at higher redshifts and we derive merger rates which scale as ∼ (1 + z) 1.87±0.04 Gyr −1 , which agrees well with recent machine learning and galaxy pair approaches, removing previous inconsistencies. We also show that far-infrared selected galaxies that are invisible to HST have a negligible effect on our measurements. We also find that galaxies are more concentrated at higher redshifts. We interpret this as a sign of how their formation occurs from a smaller initial galaxy that later grows into a larger one through mergers, consistent with the size growth of galaxies from 'inside-out', suggesting that the centres are the oldest parts of most galaxies.
INTRODUCTION
Throughout the history of galaxy studies, the most common way to derive galaxy evolution is through examining some property as a function of time. This famously includes the evolution of star formation, stellar mass, metallicity, and other properties. One of the most fundamental properties that we are still exploring is the morphological or structural evolution of galaxies, which is an integrated result of the many different galaxy properties and formation processes (e.g., Conselice et al. 2008;Mortlock et al. 2013;Huertas-Company et al. 2016).
There are many ways in which to trace the structural evolution of galaxies. The most simplistic and direct way is investigating the size evolution (e.g., Trujillo et al. 2007;Buitrago et al. 2008;Allen et al. 2017;Whitney et al. 2019), and the surface brightness evolution (e.g., Whitney et al. 2020), as well as simply the evolution of apparent morphology classified into Hubble types/peculiars (e.g., Conselice et al. 2005). Another way to examine the evolution of galaxy structure is to examine the bulge and disk components of galaxies and how these evolve together (Bruce et al. 2014;Margalef-Bentabol et al. 2018). What has not been carried out in any detail is the quantitative evolution of galaxies as measured with non-parametric parameters. These parameters, including concentration and asymmetry (Kent 1985;Conselice 2003), reveal the processes of galaxy assembly through the systematic change of galaxy light over time.
Throughout a galaxy's lifetime, it will undergo several processes that will alter its structure and its morphology. Within a cosmological context of Λ-CDM, this includes the formation of bulges and then disks. In the simplest paradigm galaxies collapse into small systems that grow through star formation and mergers with other galaxies. At some point gas accretion will also occur and this is a primary method by which spiral arms and disks are formed. This process includes the accumulation of gas forming into stars that will expand galaxies in their outer parts (Whitney et al. 2019). This also includes mergers that will lead to structural peculiarities, and eventually, for some, into more concentrated systems. Furthermore, within clusters of galaxies processes such as ram pressure stripping (Gunn & Gott 1972), starvation (Larson et al. 1980), and harassment (Moore et al. 1996) are all tied to galaxy star formation history and can strongly influence the physical and morphological properties of a galaxy.
A galaxy's morphology is traditionally defined as the point at which it lies on the 'tuning fork' diagram, first described by Edwin Hubble (Hubble 1926), whereby galaxies are defined as either spirals or ellipticals/S0s. Historically, these morphological classifications were done visually (e.g., de Vaucouleurs 1959;Sandage 1975;van den Bergh 1976;Lintott et al. 2008Lintott et al. , 2011, whereby an individual examines galaxy images and assigns labels to those images based in their visual appearance. However, this method gives rise to errors and biases and the sheer sample size of current and upcoming surveys mean this is quickly becoming inefficient even for large citizen-science projects such as Galaxy Zoo (Cheng et al. 2020). In order to remove some of these problems it is important to use a less subjective and more quantitative way of classifying galaxies. One such method is a non-parametric system that seeks to measure the concentration, asymmetry, clumpiness, Gini and M 20 (CAS parameters) of galaxies by using measured light distribution. This system is described in papers such as Conselice et al. (2000bConselice et al. ( , 2002; Lotz et al. (2004). Using this method, galaxies can be placed in parameter space and from this, it can be seen that all major classes of galaxies in various phases of evolution are easily differentiated (Conselice 2003). Furthermore, classical classifications of galaxies are unable to be used at higher redshifts, whereby most galaxies are not elliptical or spirals (e.g., Conselice et al. 2005;Mortlock et al. 2013).
An important stage of galaxy evolution that can be measured with these parameters is when a galaxy undergoes a merger. Mergers can be identified using methods such as pair fractions (Man et al. 2016;Mundy et al. 2017;Duncan et al. 2019;Ventou et al. 2019), deep learning models , and by using the CAS parameter space (Conselice 2003;Lotz et al. 2004;Conselice et al. 2008;López-Sanjuan et al. 2009). Major mergers lie in a specific areas of these parameter spaces and as such, they are a useful tool in determining whether a galaxy is a merger or not. The merger rate can then be calculated from this to determine the role of mergers in forming galaxies.
In this paper we investigate the general evolution of galaxy structure through cosmic time. We start with visual estimates of morphology and structure and then we examine the quantitative structural evolution of these systems. We use IllustrisTNG simulations to help us interpret these structures from which we derive the rates of galaxy formation processes such as those driven by mergers and how light concentration in a galaxy changes with time.
Throughout this paper we use AB magnitudes and assume a Λ-CDM cosmology with H 0 = 70 kms −1 Mpc −1 , Ω m = 0.3, and Ω Λ = 0.7.
DATA & METHODS
Data
We use a sample of 16,778 galaxies at redshifts in the range 0.5 < z < 3 with stellar masses from 10 9.5 M < M * < 10 12.2 M and only select galaxies with S/N > 10. This signal-to-noise cut removes only 340 galaxies from the initial sample. The mass and redshift distribution can be seen in Figure 1 where the yellow regions indicate a higher density of galaxies and purple indicates a lower density of galaxies. We select two samples from the 16,778 galaxies; a low mass sample with 10 9.5 M < M * < 10 10.5 M and a high mass sample with M * > 10 10.5 M . We later also consider how our results change if we used a constant co-moving number density selected sample. The mass limits for our massselected sample are shown as horizontal dashed lines on Figure 1. Masses and redshifts are determined using the method described in Duncan et al. (2019). A brief description of the process used to obtain the measurements is given in §2.4.
This study makes use of data from the Cosmic Assembly Near-infrared Deep Extragalactic Survey (CAN-DELS; Grogin et al. 2011;Koekemoer et al. 2011). This survey covers 800 arcmin 2 over five fields; GOODS-North, GOODS-South, COSMOS, UDS, and EGS. Our sample consists of galaxies from each of the five fields. CANDELS makes use of the Advanced Camera for Surveys (ACS) and the Wide Field Camera 3 (WFC3) on the Hubble Space Telescope (HST ). For this work we use imaging data from the F814W, F125W, and F160W fil- ters. This filters will be referred to as I 814 , J 125 , and H 160 from this point onward.
We measure all parameters in the optical rest-frame at approximately λ rest ∼ 4000Å. We do this by locating the observed filter where λ rest ∼ 4000Å falls as determined by its redshift. We then use that filter in our further analysis to have a consistent rest-frame optical view of our galaxies. The idea here is to measure everything at a constant optical rest-frame wavelength, to avoid structural biases produced by changes in restframe wavelength. The rest-frame wavelength and associated filter are given for each redshift bin in Table 1.
We use visual morphological classifications from Kartaltepe et al. (2015) who base the majority of their classifications on H-band images, however J-and V -band images are also used for some features such as clumpy light. For a summary of how these classifications are done and in which morphological classes they are placed, see Kartaltepe et al. (2015).
Morfometryka
We measure the non-parametric concentration, asymmetry, and clumpiness parameters using the Morfometryka code (Ferrari et al. 2015). These measurements are made within the Petrosian region (e.g., Conselice Note. Column 1 gives the midpoint of the redshift bin. Each redshift bin spans a redshift range of ∆z = 0.5. Column 2 gives the band corresponding to the optical rest-frame and column 3 gives the rest frame wavelength probed.
et al. 2000b
). This is defined as the area with the same axis ratio and position angle as the galaxy and with major axis equal to N R Petr R Petr , where N R Petr = 1.5 and R Petr is the Petrosian radius. Below we give a description of each index we use, although for more details see Conselice (2003).
Asymmetry
The asymmetry, A, is calculated in the same way as in Conselice (2003) whereby a galaxy image rotated by 180 • is subtracted from the original source image, and the absolute value of the total light in this selfsubtracted image is divided by the total light in the original image. Asymmetry through various tests has been shown to be one of the most robust non-parametric structural parameters to measure and use in analyses. The formula for calculating the asymmetry index (A) we use is outlined in detail in Conselice (2003) and is given by:
A = min Σ|I 0 − I 180 | Σ|I 0 | − A bkg(1)
where I 0 represents the original galaxy image, and I 180 is this image after rotating it by 180 deg from its centre. The asymmetry value is calculated through an iterative approach to find the centre of the rotation which is altered to find the one that gives the minimum asymmetry value. This minimum asymmetry which is used as the final asymmetry value (e.g., Conselice et al. 2000b) with a search radius typically a pixel or half pixel. A bkg is the background asymmetry of the image. This background term differs from the original application of the CAS parameter measurements whereby the background asymmetry is determined from a single region. Morfometryka initially did not include such a background correction. In this case we use the same method for determining A bkg as in Tohill et al. (2020) whereby a 10 × 10 grid is overlayed on the image in an area outside of the galaxy segmentation map, the asymmetry of the individual cells are measured and then the median of these values is taken for A bkg . This removes the bias present in the original method of measuring the asymmetry of a single background region and ensures the measurement is more robust.
Concentration
The concentration of a galaxy is used to distinguish between the different types of galaxies; for example, early type galaxies and their immediate progenitors tend to have a higher concentration than late type galaxies (e.g., Bershady et al. 2000;Conselice 2003). Concentration, C, is defined as the ratio between two circular radii containing certain fractions of the total flux of the galaxy (Kent 1985). We use R 20 , the radius containing 20% of the total light, and R 80 , the radius containing 80% of the total light. Therefore, the concentration is given as
C = 5 × log 10 R 80 R 20 .(2)
This calculation of the concentration is sensitive to seeing effects that are more prominent in the central regions and is therefore sensitive to the value of the inner radii R 20 (Ferrari et al. 2015). We however, investigate other forms of concentration indices, and fully explore the idea of light concentration and what it actually implies for our galaxies. We also consider the R 50 and R 90 radii to avoid problems with the inner parts of galaxies, as well as examine the Petrosian radii measures of light concentration.
Clumpiness
The clumpiness, S, is a measure of the small scale structure within a galaxy. A higher clumpiness indicates that there are clumps of material within a galaxy, for example spiral galaxies contain many star forming regions and therefore contain many clumps of material. Elliptical galaxies on the other hand are generally smooth and therefore have a clumpiness that is close to zero. S is calculated in the same way as in Conselice (2003) whereby the image is first smoothed by a filter and then subtracted from the original image. The flux contained within this residual is then divided by the flux contained with the original image. In this case, the filter used to smooth the original image is a Hamming window of size R Petr /4. Formally, the clumpiness can be described by the following
S = i,j |I(i, j) − I S (i, j)| |I(i, j)| − S bkg(3)
where I(i, j) is the original image, I S (i, j) is the smoothed image, and S bkg is the average smoothness of the background.
Correcting CAS Values for Redshift Effects
Galaxy structure will change for a given galaxy when that same galaxy is viewed at higher redshifts. To decouple evolution from this effect, we need to account for this. In order to correct for redshift effects, we artificially redshift a sample of galaxies at 0.5 < z < 1 in redshift intervals of 0.5 to a maximum redshift of z = 2.75. We take the galaxies in this initial bin from both GOODS fields and consider this our fiducial sample. We simulate these galaxies to higher redshifts and in the other CANDELS fields. We do this using the method described in Tohill et al. (2020). This method considers a number of effects. Firstly, it considers the rebinning factor b, which is the decrease in the apparent size of the galaxy when viewed at a higher redshift. This is done by following the method outlined in Conselice (2003) and de Albernaz Ferreira & Ferrari (2018). Luminosity evolution is also considered and this is implemented in the form found in Whitney et al. (2020) whereby the intrinsic surface brightness goes as
µ int ∝ (1 + z) −0.18(4)
for a mass-selected sample of galaxies. After applying the rebinning factor, cosmological dimming of the form (1 + z) −4 , as found by Tolman (1930), is applied. Finally, the image is convolved with the PSF corresponding to the rest-frame filter and inserted into an actual CANDELS background. For the first redshifting interval (z = 0.75 to z = 1.25) we do not convolve with the PSF as the filter corresponding to the optical rest-frame is the same for both redshifts. We then use Morfometryka to measure the CAS parameters at each redshift and compare these new values to the parameters measured at the original redshift. The differences are then applied to the real galaxies at the corresponding redshift. This change is dependent on the field due to the depth reached by each of the fields. The GOODS fields reach a greater depth than the COSMOS, EGS, and UDS fields and as such, we apply a different correction to each field. We also ensure that we are comparing the same galaxies at each redshift interval. To do this we only consider at all redshifts those galaxies that can be measured in the highest redshift as this ensures that we are comparing exactly the same galaxies at all redshifts to create a fair comparison and relevant correction factors. The left panel of Figure 2 shows the average asymmetry corrections applied to all CANDELS fields. The right panel of Figure 2 shows the average concentration corrections applied to the five CANDELS fields. Table 2 gives the values of these corrections for each redshift bin within each field.
Photometric Redshifts and Stellar Masses
The photometric redshifts of the galaxies within our sample are calculated using the method described in Duncan et al. (2019). The photometric redshift software eazy (Brammer et al. 2008) is used to determine the template-fitting estimates and three separate template sets are used and fit to all available photometric bands. The templates used include zero-point offsets to the input fluxes and additional wavelength-dependent errors. A Gaussian process code (GPz; Almosallam et al. (2016)) is then used to calculate further empirical estimates using a subset of the available photometric bands. Individual redshift posteriors are calibrated and the four estimates are combined in a statistical framework via a hierachical Bayesian combination to produce a final redshift estimate. For a more in-depth description of the process, see section 2.4 of Duncan et al. (2019).
The galaxy stellar masses we use are measured by using a modified version of the spectral energy distribution (SED) code described in Duncan et al. (2014). Instead of finding the best-fit mass for a fixed input redshift, the stellar mass is estimated at all redshifts in the photo-z fitting range. Also included in these estimates is a socalled 'template error function' (the method for this is described in Brammer et al. (2008)) to account for uncertainties introduced by the limited template set and any wavelength effects.
This mass-fitting technique uses Bruzual & Charlot (2003) templates and includes a wide range of stellar population parameters and assumes a Chabrier (2003) initial mass function. The assumed star formation histories follow exponential τ -models for both positive and negative values of τ . Characteristic timescales of |τ | = 0.25, 0.5, 1, 2.5, 5, and 10 are used, along with a short burst (τ = 0.05) and continuous star formation models (τ 1/H 0 ). We compare the mass measurements we make to the average of those determined by the several teams within the CANDELS collaboration (Santini et al. 2015). This is done in order to ensure that the stellar mass estimates do not suffer from systematic biases. There is some scatter between the two mass estimates, however our mass estimates are not affected by any significant biases compared to others. For further details on the method and models used, see §2.5 of Duncan et al. (2019) for an extensive discussion of the masses we use here.
IllustrisTNG Simulations
To interpret our results, we investigate how nonparametric morphologies change with redshift in a controlled manner with simulated galaxies. We do this as we want to disentangle real evolution effects from redshift effects. Additionally, by measuring these structural properties in simulated data we can estimate the observability timescales of galaxy mergers since it is possible to follow the time evolution of particular galaxies in this way (e.g., Lotz et al. 2008). With this goal in mind, we use data from the IllustrisTNG simulations Springel et al. 2018;Nelson et al. 2018;Naiman et al. 2018;Marinacci et al. 2018;Pillepich et al. 2019), which is a suite of cosmological, gravo-magnetohydrodynamical simulation runs, ranging within a diverse set of particle resolutions for three comoving simulation boxes of size, 50, 100, 300 Mpc h −1 , named TNG50, TNG100 and TNG300, respectively.
Here we use data from both TNG50-1 and TNG300-1. For estimating the merging timescales we focus in the largest simulation box TNG300-1, as it provides us with a more mass complete sample at higher redshifts. However, when doing direct comparisons between the morphology measured in CANDELS and in the simulations we use data from TNG50-1 as its mass and spatial resolution produce more realistic morphologies. This enable us to generate images of simulated galaxies that are embedded in a cosmological context.
To measure the observability timescales of pair galaxy mergers discussed in Section §3.4.3, we select galaxies using the TNG300-1 merger trees by searching and locating galaxies in the simulation that have had only one major merger event within redshifts z = 2 to z ∼ 0. This is to avoid contamination in the structure from past merger events. To balance any potential issues with the mass resolution of TNG300-1, we limit our analysis only to massive galaxies with M * > 10 10 M , which are in general represented by thousands of stellar particles. For these galaxies, we select all the snapshots and subhalos that are ± 2 Gyr of the snapshot from where the merger event takes place. We then narrow down our selection to 300 distinct galaxies, each with 35 different snapshots, resulting in a total of ∼ 10, 000 distinct objects. This ensures that we extensively probe not only around the merging event, but also the stages where the galaxies does not show signs of merging.
For a comparison between the non-parametric morphology and merger classifications of CANDELS galaxies and simulated galaxies we use the small box, high mass resolution TNG50-1 simulation, which is capable of producing output images in higher resolutions due to the high mass resolution and smaller gravitational soft- . Left: mean asymmetry correction applied to all five CANDELS fields. Right: mean concentration correction for all CANDELS fields. The error bars are equal to one standard deviation from the mean. In both panels, the GOODS-North correction is shown by a blue circle, GOODS-South by an orange diamond, COSMOS by a green triangle, EGS by a red square, and UDS by a purple inverted triangle. ening length. Here we do two different selections on TNG50-1 galaxies, which are described in §3.5.
The IllustrisTNG data contains the information from the stellar, gas, and dark matter particles for each source. However, to create mock broadband images from this information we post-process each stellar particle with a population synthesis process, as each particle represents a large region that can be described by a rich stellar population based on its age, mass, and metallicity. Instead of using the approach outlined in Ferreira et al. (2020), we follow the recipes from Trayford et al. (2017) and Vogelsberger et al. (2020) to post-process the simulation data with the Monte Carlo dusty radiative transfer code SKIRT (Camps & Baes 2015. We also include the resampling of the star-forming re-gions outlined in Camps et al. (2016) and Trayford et al. (2017), as this is particularly important to avoid problems with the coarse representation of star forming regions. For each source simulated with SKIRT we produce observations in four different orientations in the 3D volume containing the galaxy cutout. Three of those are aligned with the simulation box axis: xy, xz and yz. We also include a fourth orientation covering an octant of the 3D volume. This post-processing step is independent of the simulation run used, the only difference being the number of stellar and gas particles available and the output size of the datacube.
The output from SKIRT is a datacube with a spectral energy distribution (SED) for each pixel, which is then convolved with the filter response function for the HST CANDELS filters used in this work, namely I 814 , J 125 , and H 160 . The resulting broadband images are stored in two ways: without any observation effects and with HST -matched properties, including PSF, noise level, and sky background. The final images are then processed with Morfometryka to measure their CAS values. We later use these simulation outputs as a method for understanding our observational results.
RESULTS
In the following subsections we discuss the results of our study in terms of the morphological and structural evolution of galaxies to z = 3. We first describe the visual morphological classification evolution of our sample of galaxies, while later we discuss the quantitative evolution using the CAS parameters. Finally we use these results to derive the merger fraction and merger rate evolution for our sample and the resulting number of mergers and mass accretion from mergers over this cosmic time.
Visual Classifications
Using the visual classifications from the CANDELS survey from Kartaltepe et al. (2015), we examine the fraction of each galaxy type within our sample. We consider those galaxies that are classified as the following: spheroid, disk, peculiar, and other. A galaxy is considered to have a classification if the fraction of classifiers within Kartaltepe et al. (2015) that deem it to be that particular type of object is greater than 0.6, and the fraction of all other classifications is less than 0.6. For peculiar galaxies, we consider any galaxy where the fraction of classifiers that consider that galaxy to be irregular in shape is greater than 0.6; no other conditions are required for this group. The other group consists of galaxies for which no consensus could be reached as to its morphological type, galaxies that were deemed unclassifiable, or galaxies that had no classification given. Figure 3 shows the evolution of the fraction of galaxies in each of these groups within our sample across the redshift range 0.5 < z < 2.5. The spheroid galaxies are shown as red circles, disk galaxies as blue crosses, and peculiar galaxies as green triangles. The unclassifiable objects are shown as grey inverted triangles. We also fit power laws of the form α(1 + z) β to the spheroid (solid line), disk (dashed line), and peculiar (dotted line) categories. We find that the fractions evolve as:
f sp = (0.73 ± 0.09)(1 + z) −0.95±0.16(5)
for the spheroid galaxies,
f di = (0.48 ± 0.05)(1 + z) −0.60±0.13(6)
for the disk galaxies, and f pe = (0.11 ± 0.01)(1 + z) 1.04±0.10 (7)
for the peculiar galaxies. The fraction of both disk and spheroid galaxies increase with cosmic time, whereas the fraction of peculiar galaxies decreases as redshift decreases. Note that these fits are only valid down to z ∼ 0.5. It is clear that extrapolating these to z ∼ 0 would over-predict the number for each type. The fraction of 'other' galaxies evolves from ∼20% at z ∼ 2.25 and decreases to ∼6% at z ∼ 0.75 due to the fact that galaxy features will be more distinguishable at lower redshifts due to increased resolution and fewer effects such as surface brightness dimming. Figure 3. Evolution of the fractions of galaxies in our sample within a given visually morphological classification. Spheroid galaxies are shown as red red circles, disk galaxies as blue crosses, peculiar galaxies as green diamonds, and unclassifiable objects as grey inverted triangles. Also shown are the power law fits to the spheroid (solid line), disk (dashed line), and peculiar categories (dotted line). For the error bars on these fitted parameters see the text.
f sp = 0.73(1 + z) 0.95 f di = 0.48(1 + z) 0.60 f pe = 0.11(1 + z) 1.04
Asymmetry and Concentration
We next explore the data by comparing the measured asymmetry and concentration values with the visual classifications from Kartaltepe et al. (2015) as described in the previous section. In this section we examine those galaxies that are identified as disk, spheroid, or peculiar galaxies. We first examine the uncorrected CAS parameters here, and later investigate a way to correct these parameters for redshift effects. Essentially, the uncorrected values are comparable to each other at a given redshift (internally consistent), whereas the corrected values are comparable across all redshifts (externally and internally consistent).
In Figure 4, we show the evolution of the uncorrected concentration versus asymmetry for the galaxies within the sample that have one of our three main classifications. Red circles indicate a spheroid galaxy, blue crosses a disk galaxy, and green triangles a peculiar galaxy. Galaxies that fit none of these categories are shown as small grey circles. Also shown are the cuts between galaxy types taken from Bershady et al. (2000); the solid black line is the boundary between intermediate-and late-type galaxies whereby the galaxies that lay above this line are considered to be late-type galaxies, the dot dashed line is the boundary between early-and intermediate-type galaxies whereby galaxies below this line are considered to be early-type galaxies, and the dashed line is the boundary between mergers and non-mergers whereby galaxies that lie above this line are mergers.
At all redshifts, galaxies classified as spheroids tend to have a higher concentration and lower asymmetry, than the two other classifications. Whereas peculiar galaxies have a higher asymmetry but lower concentration. This is consistent with the results found by Conselice (2003) for nearby galaxies. The range of concentration values increases with time but the overall distribution of these values for the spheroid and disk galaxies remains roughly constant with redshift. The asymmetry of the peculiar galaxies is centred on the A > 0.35 merger condition, however, the mean concentration changes from C = 3.52 at 2.5 < z < 3.0 to C = 2.49 at 0.5 < z < 1.0. The mean values for spheroid galaxies are C = 2.67 and A = 0.16 at the highest redshift. These values change to C = 3.10 and A = 0.12 at the lowest redshift. Galaxies classified as being disks have average concentration and asymmetry values of C = 2.52 and A = 0.20 at 2.5 < z < 3.0 and these values do not change significantly within the lowest redshift bin of 0.5 < z < 1.0 where C = 2.50 and A = 0.23. Thus the disks change their value the least and the spheroids the most. These results also show that galaxy classification is consistent and can be carried out to z ∼ 3.
Disk/Bulge Dominated
Within the catalog of Kartaltepe et al. (2015) every image has flags based on a number of interesting structural features present. Two of these flags indicated whether the galaxy appears to be bulge or disk dominated. As with the visual morphological classifications, we assume a galaxy is bulge or disk dominated if the fraction of classifications that designate it to be such is greater than 0.6, and the other classification is less than 0.6. We show the evolution with redshift of the concentration-asymmetry plane for those galaxies classified as being bulge or disk dominated in Figure 5. Disk dominated galaxies are shown as blue points and bulge dominated galaxies are shown as red points. As with Figure 4, we also plot the boundaries between galaxy types. At higher redshifts there are fewer galaxies with either of the classifications in consideration here, likely due to greater noise and surface brightness dimming causing there to be less consensus among those classifying the images. This highlights the issues in classifying galaxy images and the need for a less subjective method of classifying galaxies.
Galaxies classified visually as disk-dominated predominantly lie in the late region of the concentrationasymmetry plane and the average values in the 0.5 < z < 1.0 bin are A = 0.22 and C = 2.68. These values remain roughly constant across redshift and become A = 0.20 and C = 2.69 at 2.5 < z < 3.0. At high redshifts, the bulge-dominated galaxies appear to lie in the late region but as redshift decreases, there are more galaxies in the intermediate and early regions of the plane. The average values of asymmetry and concentration of these bulge-dominated galaxies are A = 0.13 and C = 3.14 at 0.5 < z < 1.0. The asymmetry value remains roughly constant to higher redshifts and becomes A = 0.14 at 2.5 < z < 3.0 but the concentration decreases with redshift and becomes C = 2.80 at this redshift.
Clumpiness/Patchiness Matrix
Along with visual classifications of galaxies, classifiers involved in the work of Kartaltepe et al. (2015) also assigned flags based on how clumpy or patchy the distribution of the light within the galaxies is. Clumps are defined to be concentrated knots of light and patches are defined to be more diffuse structures of light. There are 9 flags associated with this method of identifying features, with each flag denoting a different combination of the clumpiness (C) and patchiness (P). For example, a galaxy that exhibits no clumpiness and no patchiness will be labelled as 0C0P. A galaxy that appears to be extremely clumpy and patchy will be labelled as 2C2P. The levels of clumpiness and patchiness range between . Evolution of the concentration-asymmetry plane with redshift, colour-coded by visual classification. Red circles are galaxies classified as spheroids, blue crosses as disks, and green triangles as peculiar galaxies. Any galaxies that do not fit within these three classifications are shown as small grey circles. The lines are boundaries between galaxy types derived from Bershady et al. (2000). Galaxies that are above the dashed line are considered to be mergers. The dot-dashed line is the boundary between early-and intermediate-type galaxies and the solid line is the boundary between late-and intermediate-type galaxies. The redshift indicated in the top right corner of each panel is the midpoint of the redshift bin whereby the upper and lower limits are 0.25 either side of this point. 0 and 2. As such we are able to organise the flags into a 3×3 grid with each square of the grid denoting the level of clumpiness or patchiness. Note, this clumpiness is not the same as the clumpiness, S, described in §2.2.3 and is a purely visual descriptor. A score of 0 indicates no clumpiness or patchiness and a score of 2 indicates a large amount of clumpiness or patchiness.
In Figure 6, we show this matrix of values and in each square of the grid, give the average concentration (top left, green), asymmetry (top right, red), and clumpiness (bottom left, blue). These average values are across all redshifts so any evolution in these parameters is not considered here. The lighter shades of colour denote a lower value of the CAS parameters while a darker shade denotes a higher value. In terms of concentration, the less clumpy or patchy a galaxy is, the higher its concentration on average. The opposite is true for the asymmetry and clumpiness; the more clumpy/patchy an image is, the higher the measured asymmetry and clumpiness.
CAS Parameter Evolution
We now investigate with our data the evolution of the CAS parameters. We ultimately use image simulations to determine how these parameters change with the effects of resolution and noise removed. Ultimately the evolution of these parameters will lead to a physical understanding of the driving forces behind galaxy formation over the epoch 0.5 < z < 3.
Concentration Evolution
The concentration of a galaxy tells us important information about how the light is distributed within a galaxy, relative to its centre. We show the evolution of the corrected concentration index defined in §2.2.2 in Figure 7 for two different mass ranges across the redshift range 0.5 < z < 3. The concentration values have been corrected using the method described in §2.3. The first includes galaxies in the mass range 10 9.5 M ≤ M * < 10 10.5 M and the evolution of the mean concentration for this mass bin is shown as red crosses (see Figure 7). The second mass bin includes galaxies that have a mass M * ≥ 10 10.5 M . This mass bin is plotted as the blue triangles. For both, the error bars represent one standard deviation from the mean. The lower mass galaxies have a lower concentration than the higher mass galaxies, but both samples on average exhibit a decrease in concentration with time. This is opposite to what we find when we examine the uncorrected concentration index evolution. This therefore deserves some attention to try to understand the origin of this, and whether it is in fact a real effect.
Asymmetry Evolution
The asymmetry of a galaxy is a useful indicator as to whether a galaxy is undergoing any interactions or mergers with other galaxies (Conselice et al. 2000a,b). We explore its use to define mergers in the following section, but first we examine the evolution of the corrected asymmetry for our sample of galaxies in the redshift range 0.5 < z < 3. Figure 8 shows this evolution for two different mass ranges. As in Figure 7, the lower mass bin is shown as red crosses, the higher mass bin is shown as blue triangles, and the error bars represent one standard deviation from the mean. On average, the asymmetry decreases with cosmic time for both mass bins, with the higher mass bin exhibiting a steeper decrease.
Concentration-Asymmetry Plane
In Figure 9, we show the evolution of the concentration-asymmetry plane for the corrected values with redshift. As with Figures 4 and 5 Figure 6. This figure shows the clumpiness (C) and patchiness (P) distribution visually vs. the CAS parameters. The values on the x-and y-axes are the visual estimates of clumpy galaxies, with high numbers denoting more visually looking clumpy systems. The clumpiness values on the y-axes are different to the clumpiness calculated by Morfometryka and are a purely visual score. Top left, green: mean concentration (C) values for each region of the clumpiness/patchiness matrix defined by Kartaltepe et al. (2015). Top right, red: mean asymmetry (A) values for each region of the clumpiness/patchiness matrix. Bottom left, blue: mean clumpiness (S) values for each region of the clumpiness/patchiness matrix. In each panel, a darker colour indicates a larger value of the relevant CAS parameter. Each three by three grid represents the clumpiness/patchiness matrix whereby a higher value on each axis represents a higher clumpiness or patchiness. The visual clumpiness/patchiness criteria appear to correlate well with the measured A and S values whilst concentrated galaxies are not visually clumpy or patchy. Figure 7, the lower mass bin is shown as a red cross and the higher mass bin is shown by the blue triangles. The error bars represent one standard deviation from the mean. Both mass bins show a decrease in asymmetry with decreasing redshift. the three visual classifications for each galaxy, along with the position of any galaxy that does not fit these three categories within this plane. Spheroid galaxies are shown as red circles, disks as blue crosses, and peculiar galaxies as green triangles. Any other galaxies are shown as small grey circles. The spread in distribution of both concentration and asymmetry increases at lower redshifts, leading to the evolution of the average of both these parameters as seen in Figures 7 and 8.
There are few galaxies that lie within the early region of the concentration-asymmetry plane and those that are are typically spheroid galaxies. This suggests that the galaxies within our sample are more asymmetric than is typical for spheroids. Spheroid galaxies that do not lie in this early region are mostly within the intermediate region. Within the lowest redshift bin (0.5 < z < 1.0), spheroids have a mean asymmetry of A = 0.14 and a mean concentration of C = 3.03. At the highest redshift (2.5 < z < 3.0), these values change to A = 0.25 and C = 3.77 with a steady increase in both A and C at intermediate redshifts. Disk galaxies are primarily located within the late region, as to be expected. The mean values are A = 0.21 and C = 2.52 at 0.5 < z < 1.0 and change to A = 0.28 and C = 3.62 at 2.5 < z < 3.0. The change in C is a steady increase however asymmetry remains at an approximately constant value until the highest redshift bin. Peculiar galaxies lie around the merger limit of A > 0.35 with 29.20 ± 0.01% of galaxies classified as being peculiar lying within this merger region. If these corrections are right, we can see that there is a diversity already at 2.5 < z < 3.0 in the galaxy population and that galaxies of all types are more asymmetric and more concentrated at higher redshifts.
Merger Fractions and Rates
This section of the paper describes our results of the merger history of galaxies within the five CANDELS fields using a CAS approach. The outline of this section is as follows: first we give a background description of the merger history of galaxies, include definitions of the merger process. We then describe in some detail how to measure the timescale for mergers and how these can then be used for measuring the merger rate of galaxies. Finally, from the merger rates we are able to say how many mergers these galaxies undergo on average and how much mass is added to galaxies through this process.
Merger Fractions
In this subsection we investigate the observational quantity that we can obtain directly from the data. This is the merger fraction, which in this paper we use the Figure 9. Evolution of the corrected concentration-asymmetry plane with redshift. As with Figure 4, the lines indicates the boundaries between galaxy types. Galaxies that lie above the dashed line are considered to be mergers. The dot-dashed line is the boundary between early-and intermediate-type galaxies and the solid line is the boundary between late-and intermediatetype galaxies. Spheroid galaxies are indicated by red circles, disks by blue crosses, and peculiar galaxies by green triangles. Galaxies that do not fall into any of the three categories are shown as small grey circles. The redshift indicated in the top right corner of each panel is the midpoint of the redshift bin whereby the upper and lower limits are 0.25 either side of this point.
CAS parameters described in §2.2 to define. We later compare the merger fractions and the merger rates we derive using merger timescales to other measurements of the merger history using machine learning methods ) and galaxies in pairs ). First we define what we mean by a merger fraction -that is, the fraction of galaxies within some galaxy sample which is undergoing a major merger. In our sample, a merger is defined within the CAS system as a galaxy that satisfies the following criteria:
(A > 0.35) & (A > S)(8)
where both A and S are the asymmetry and clumpiness values corrected for redshift effects. This method predominantly identifies only major mergers where the ratio of the stellar masses of the progenitors is at least 1:4 (Conselice 2003(Conselice , 2006Lotz et al. 2008). H-band images of examples of galaxies we consider to be a merger, based on this system, are shown in Figure 10. The left two columns show lower redshift (1 < z < 2) galaxies, and the right two columns show higher redshift (2 < z < 3) galaxies. The redshift and asymmetry of each example are indicated on each postage stamp. We calculate the merger fraction using this equation for each redshift bin, the evolution of which is shown in the left panel of Figure 12. The results for the low mass bin (10 9.5 M ≤ M * < 10 10.5 M ) are shown as red crosses and the results for the higher mass bin (M * ≥ 10 9.5 M ) are shown as blue triangles. We fit a power law to the merger fraction evolution given by the form:
f m (z) = f 0 (1 + z) β(9)
by applying a least squares fit to our data and setting f 0 to a fixed value of the merger fraction at z = 0. We take this z ∼ 0 value to be 0.0193 ± 0.0028 from Casteels et al. (2014). This point is shown as a grey inverted . Examples of galaxies within our sample that are classified as a merger using the criteria given in Equation 8. Lower redshift (1 < z < 2) galaxies are shown in the left two columns and higher redshift (2 < z < 3) galaxies are shown in the right two columns. The redshift and asymmetry are given for each example.
triangle. We also include both mass bins in this fit. We find that the merger fraction, f m evolves as f m = 0.0193 ± 0.004(1 + z) 1.87±0.04 .
This fit is shown in the left panel of Figure 12 Table 3 for both the lower and higher mass bins respectively.
IR Luminous Galaxies
One of the things that we need to consider in our approach is that some galaxies are invisible to optical/NIR light and are only detected in the far-IR, sub-mm or radio (e.g., Wang et al. 2019). These are sometimes called H-band drop-outs and are of interest as they may in fact be galaxies that are massive, within our limits, but are not considered part of the merger fraction as they would not be added into our H-band selected sample.
A simple argument can be made to show that this concern is insignificant for our merger fractions. The study of Wang et al. (2019) found 39 galaxies across the same fields we study, which are not detected in the Hband, and thus would likely not be within our sample. Wang et al. (2019) claim that these galaxies are typically massive ones at z > 2, with a median stellar mass of 10 10.6 M . Let us examine the results of assuming that all of these galaxies are mergers.
For our CANDELS sample, we have 6273 massive galaxies at z > 2, whilst the merger fraction we measure is f m = 0.183±0.005 for these galaxies. This gives in total 1135 mergers within our sample. Then if we consider all 39 Wang et al. (2019) galaxies to be mergers at z > 2 and within the mass range we are looking at, then the new merger fraction with the Wang et al. (2019) dropouts included would be (1135+39)/(6273+39) = 0.186. This gives a very small merger difference of ∼ 0.003. The Poisson error on our measurements is also 0.005, at the same level. If we consider that these galaxies have the same merger fraction as the bulk of our systems, at f m = 0.183, then the merger fraction would be: f m = 0.183, a difference of < 0.001 -almost unchanged and vastly lower than our counting errors.
Regardless, if we consider these FIR galaxies as mergers or that have a merger fraction higher than our measured one, it would only increase our values extremely slightly and well within our current measurement errors.
Merger Timescales from IllustrisTNG
In this section we investigate the merger timescales for a sample of galaxies in the IllustrisTNG 300-1 simulation. The process we use to generate mock images is outlined in §2.5 and here we use only HST -matched mocks. We measure the observability timescales with a process similar to the one developed in Lotz et al. (2008) and Nevin et al. (2019). The main difference here is that the galaxies used in this evaluation are generated as a result of a cosmological simulation instead of an isolated galaxy-galaxy merger simulation. Our sample is limited to massive M * > 10 10 M major-mergers. We briefly outline our steps to measure the timescales below.
First, we follow each galaxy in the simulation for a variety of snapshots spanning ± 2 Gyr around each merger event (limited to 0 < z < 2). The snapshot where the merger event happens is defined as the the central snapshot, S c . This information is extracted from the merger trees in the simulation and is originally defined by the SubLink friends-of-friends algorithm (Rodriguez-Gomez et al. 2015). The snapshots after S c are considered to be after the merger event, while the snapshot previous to S c are classified as before the merger event. This is the only distinction done here for particular merger stages, which is fundamentally different to the stages assigned in Lotz et al. (2008).
Secondly, we select only the broadband images that correspond to the rest-frame optical at those redshifts. For these images, we measure the asymmetry using Morfometryka. Thus, for each source simulated in §2.5 we have A for each snapshot in four different orientations. By averaging A by the viewing angle, we find where the asymmetries of each individual galaxy falls below the A > 0.35 threshold for both sides of the merger event.
Finally, by comparing the time difference between the redshift where the asymmetry threshold is no longer valid and the redshift of the central snapshot, we estimate an observability timescale for the asymmetry, both for the post-merger stage and the before merger stage, and, if combined, for the total merger event. By doing this for all our sample, we have a statistical estimation of the merger timescales for the asymmetry in Il-lustrisTNG, as shown in Figure 11. We find a τ = 0.56, lower than the values reported in Lotz et al. (2008), but higher than the ones found in Nevin et al. (2019) for major-mergers. We find an asymmetric distribution in the timescales that is broad but not deep; there is an asymmetric tail at higher timescales. The mean timescale of this distribution is ∼20% greater than the median at 0.67 Gyr so the tail does not significantly impact the timescale used. We consider the tail in our error in the timescale by finding the differences between the median of the full sample of simulations and the median of the sample both above and below 0.56 Gyr. From this, we yield a timescale of 0.56 +0.23 −0.18 Gyr. We find that this timescale does not vary with redshift unlike in the case of pair statistics (Snyder et al. 2017).
Merger Rates
Merger fractions are a solely observational quantity and can tell us little about the evolution of galaxy formation and evolution. The merger rate (time between mergers per galaxy) on the other hand is a fundamental parameter in galaxy evolution. We are able to convert from a merger fraction to a the merger rate (R) at each redshift using
R = f m τ (11)
where f m is the merger fraction and τ is the merger timescale. We calculate the merger rate evolution for our sample of galaxies using a timescale of τ = 0.56 +0.23 −0.18 Gyr found using mergers within the IllustrisTNG simulations as described above. The merger rate evolution is shown in the right panel of Figure 12. As with the merger fraction, we fit a power law to both the low-mass (red crosses) and high-mass (blue triangles) data of the form
R(z) = R 0 (1 + z) β Gyr −1(12)
where R 0 is the local merger rate calculated using the local merger fraction, f 0 , from Casteels et al. (2014) and the merger timescale we find using the IllustrisTNG simulations. This local merger rate is shown as a grey inverted triangle. We find that the merger rate evolves as:
R(z) = 0.03 ± 0.02(1 + z) 1.87±0.04 Gyr −1 .(13)
This fit is shown by a solid black line and one standard deviation from this fit is indicated by the grey shaded area in the right panel of Figure 12. As with the merger fractions, we also show the results of Duncan et al. (2019) and Ferreira et al. (2020). Due to the different methods used to identify mergers, different timescales must be used; Ferreira et al. (2020) use τ = 0.6 Gyr and Duncan et al. (2019) use a timescale that varies with redshift such that τ = 0.17 Gyr at 2.5 < z < 3.0 and τ = 0.77 Gyr at 0.5 < z < 1.0 in order to calculate their merger rates.
Despite using a timescale that differs from other merger identification methods, our results are largely consistent with these previous results. Our merger rates and errors for the low and high mass bins are given in Columns 3 and 5 of Table 3 respectively. We also compare our results to those found by Rodriguez-Gomez et al. (2015) for the Illustris simulation. This fit is shown by a dashed line here. The fit, given in Table 1 of Rodriguez-Gomez et al. (2015), is a complex function of redshift, the descendent galaxy stellar mass M * , and the progenitor stellar mass ratio µ * . We show the results for a stellar mass of 1×10 10 M and a stellar mass ratio of 1/4 (consistent with major mergers, the type of mergers we are able to probe with our asymmetry selection). The Illustris fit is steeper than the results we find in this paper and can be approximated as a power law of the form ∼ (1 + z) 2.7 . These simulations also predict a smaller merger rate, particularly at the lowest redshifts, with the local merger rate predicted to be ∼3 times smaller than the value we use here.
Number of Mergers Since z ∼ 3
From the merger rate we are able to calculate the number of mergers a galaxy at 0 < z < 3 undergoes by integrating the inverse of the characteristic time between mergers, Γ(z). First, we must convert the merger fraction, f m , to the galaxy merger fraction, f gm , which gives the number of galaxies merging as opposed to the number of mergers ). The galaxy merger fraction is given by
f gm = 2 × f m 1 + f m .(14)
We can then calculate Γ, which is essentially the time in between mergers, using this galaxy merger fraction:
Γ = τ f gm(15)
where τ is the merger timescale. We are able to fit a power law to Γ of the form Γ(z) = Γ 0 (1 + z) −m Gyr and find that Γ 0 = 14.4 ± 0.7 Gyr and m = 1.96 ± 0.13. . Left: evolution of the merger fraction. Right: evolution of the merger rate. The lower mass bin from this work is shown as a red cross and the higher mass bin from this work is shown as a blue triangle. The fit to both these points and a known local value from Casteels et al. (2014) (grey inverted triangle) is shown as a solid black line with the grey shaded area indicating one standard deviation from the fit. We find a fit of the form fm = 0.0193 ± 0.004(1 + z) 1.87±0.04 for the merger fractions. For the merger rates, we find a fit of the form R(z) = 0.03 ± 0.02(1 + z) 1.87±0.04 Gyr −1 . We compare our results to Duncan et al. (2019) (yellow circles) and Ferreira et al. (2020) (green diamonds). We also compare the merger rate to that found by Rodriguez-Gomez et al. (2015) for the Illustris simulation. This fit is given as a dashed line. Table 3. Merger fractions and rates for each redshift bin and each mass bin. We calculate the number of mergers a given galaxy will undergo between redshifts z 1 and z 2 , N m , using
N m = t2 t1 1 Γ(z) dt = z2 z1 1 Γ(z) t H 1 + z dz E(z)(16)
where t H is the Hubble time, and E(z) = [Ω m (1 + z) 3 + Ω k (1 + z) 2 + Ω Λ ] 1 2 = H(z) H0 . We assume Ω m = 0.3, Ω k = 0, and Ω Λ = 0.7.
The resulting number of mergers undergone between redshifts z ∼ 3 and z ∼ 0 using the time between mergers of Γ(z) = 14.4 ± 0.7(1 + z) −1.96±0.13 Gyr is shown in Figure 13 as a solid red line. We estimate that a galaxy with mass M * > 10 9.5 M will undergo 2.90 +0.50 −0.41 major mergers on average between z ∼ 3 and z ∼ 0, with 1.87 +0.39 −0.31 mergers occurring before z ∼ 1. This result lies between some previous results; Bluck et al. (2009) who find that massive (M * > 10 11 M ) galaxies undergo 1.7 ± 0.5 major mergers with a timescale of τ = 0.4 Gyr from z ∼ 3 to z ∼ 0. Conselice (2006) find that galaxies with M * > 10 10 M undergo 4.4 +1.6 −0.9 major mergers at z > 1 but undergo less than a single merger at z < 1. For comparison, we also plot the evolution of the number of mergers for a constant merger rate where we set Γ(z) to be that at z = 3 to show how many mergers would occur without the observed decrease in merger rate. This is shown as a blue dashed line. The grey shaded areas on both lines indicate the error on the number of mergers.
We are then able to determine how much mass is added to a galaxy due to these mergers. Major mergers as identified by the CAS parameters are defined as galaxy mergers where the mass ratio between the progenitors is µ = 1/4 or less and so the amount of mass added to a galaxy during a merger must be similar to the original galaxy's mass. We take the ratio between progenitor masses to be an average of 1:1.5, as in Conselice (2006), and we approximate the amount of stellar mass gained by a galaxy due to mergers by δM merger ∼ 1.65 Nm × M 0 , where N m is the number of mergers and M 0 is the mass of the initial galaxy (Conselice 2006). Therefore, a galaxy of mass M * = 10 9.5 M at z = 3 will gain a of mass M * = 1.3 × 10 10 M by z = 0, giving a relative mass increase from z = 3 to z = 0 of ∆M merger /M = 3.2. A galaxy of initial mass M * = 10 10.5 M will, on average, accumulate M * = 1.4 × 10 11 M by z = 0, giving a relative mass increase of ∆M merger /M = 4.1. The mass accretion rate, in terms of the amount of mass accreted from mergers per unit time, isṀ ∼ 10 9 M Gyr −1 at M * = 10 9.5 M , while for M * = 10 10.5 M the mass accretion rate is a factor of ten times high, on average, givingṀ ∼ 10 10 M Gyr −1 .
These are simply the average mass accretion rates per Gyr. Naturally, this process is not smooth and mass will be added in discrete jumps during the 2.9 mergers at z < 3. However, this mass accretion rate due to mergers per galaxy is an average value for all galaxies that can be applied to a mass selected population.
Comparison to Simulations
We compare our results to simulations within Illus-trisTNG. Instead of using TNG300-1 as we do to determine the merger timescale, we use the higher resolution TNG50-1. This is done to ensure that morphologies are not affected by mass or spatial resolution, especially in the central regions. For TNG300-1, the radius containing 20% of the light, R 20 , is close to the softening length of the simulation for some galaxies, and as a result could impact the concentration measurement. From TNG50-1 we randomly select galaxies in the same mass range as our mass-selected sample, generate the mocks following the description in §2.5 and measure the CAS parameters using Morfometryka. In Figure 14, we show a comparison of the concentration evolution for our results from CANDELS and the IllustrisTNG simulations. The CANDELS results are corrected for redshift effects, as described in §2.3. The IllustrisTNG simulations images are produced to have at least 3 times the resolution Figure 13. The total number of mergers since z ∼ 3, as a function of redshift. The solid red line shows this quantity for Γ = 14.4 ± 0.7(1 + z) −1.96±0.13 Gyr. We find that galaxies with M * > 10 9.5 M undergo ∼ 3 mergers on average since z ∼ 3. For comparison, also shown is the evolution of the number of mergers for a constant merger rate of Γ = 1.05 Gyr, which is equal to the merger rate at z = 3. This is indicated by a blue dashed line. The grey shaded regions indicate the error range on the number of mergers.
of the CANDELS images and have not been matched with HST realism as the concentration should be universal for all resolutions. The resulting images are 640 × 640 pixels and the spectra have been shifted to coincide with the same filters as in Table 1. As with Figure 7, the CANDELS low mass (10 9.5 M < M * < 10 10.5 M ) galaxies are shown as blue triangles and the CANDELS high mass (10 10.5.5 M < M * ) are shown as red crosses. The low mass IllustrisTNG simulations are shown as a dashed orange line and the high mass simulations are shown as a solid green line. We find that the concentration of the simulations show very little evolution with redshift for both mass bins. This suggests that the galaxies are growing in size with the inner and outer regions growing at a similar rate whereas the trend in the CANDELS data suggests the galaxies are growing from the inside out.
We also compare the measured merger fractions within the simulations to the CANDELS data. For this, we randomly select 5% of all available galaxies in TNG50-1 in this mass range. As this naturally selects more galaxies in the low mass bin than in the high mass bin, we further select more galaxies for the high mass bin Figure 14. Comparison of the evolution of the concentration for our results from CANDELS (corrected for redshift effects) and the galaxies within the IllustrisTNG simulations. As with Figure 7, the CANDELS low mass (10 9.5 M < M * < 10 10.5 M ) galaxies are shown as blue triangles and the CANDELS high mass (10 10.5.5 M < M * ) are shown as red crosses. The low mass IllustrisTNG simulations are shown as an dashed orange line and the high mass simulations are shown as a solid green line. The shaded regions indicate one standard deviation from the mean. The IllustrisTNG data is based upon high resolution images. until both bins have the same number of galaxies. The simulations in this case are calibrated on HST resolution. This comparison is shown in Figure 15. As for Figure 12, the CANDELS data is shown as red crosses (low mass bin) and blue triangles (high mass bin). The Illus-trisTNG merger fraction evolution determined using the same asymmetry condition as used for the CANDELS data for the low and high mass bins are shown as a solid orange line and a dashed green line respectively. We also determine the 'true' merger fraction within the simulation by using the merger labels produced based on the merger trees. We ensure we only select major mergers by selecting mergers that have a mass ratio greater than 1/4. We also select only those galaxies that are within 0.65 Gyr of their merging event, a time between the median and mean of our CAS merger time-scale. These 'true' merger fractions are shown as a purple dotted line and a dot-dashed yellow line for the low and high mass bins respectively. The shaded regions show the error on the merger fractions for the IllustrisTNG simulations. We find that both the 'true' and asymmetry defined mergers are consistent with the observations. When se-lecting galaxies within a shorter timescale than the one chosen here, the number of galaxies in each redshift bin are too few and thus the 'true' merger fractions decrease and do not agree very well with the asymmetry selected mergers from IllustrisTNG or the CANDELS data. This in part happens due to the time resolution between snapshots in IllustrisTNG, which is roughly ∆t ∼ 0.16 Gyr. IllustrisTNG -9.5 < log 10 (M * /M ) < 10.5 (True) IllustrisTNG -10.5 < log 10 (M * /M ) (True) IllustrisTNG -9.5 < log 10 (M * /M ) < 10.5 IllustrisTNG -10.5 < log 10 (M * /M ) CANDELS -9.5 < log 10 (M * /M ) < 10.5 CANDELS -10.5 < log 10 (M * /M ) Figure 15. Comparison of the evolution of the merger fraction for our results from CANDELS and the galaxies within the IllustrisTNG simulations. The IllustrisTNG data is based upon HST matched mocks. As with Figure 7, the CANDELS low mass (10 9.5 M < M * < 10 10.5 M ) galaxies are shown as blue triangles and the CANDELS high mass galaxies (10 10.5.5 M < M * ) are shown as red crosses. The merger fraction evolution for the simulations determined using the asymmetry condition is shown as a solid orange line and a dashed green line for the low and high mass bins respectively. Also shown are the 'true' merger fractions -the merger fraction determined based on the labels from the merger trees within the simulations and we select galaxies that are within 0.65 Gyr of a merging event -between the median and mean of the CAS time-scale distribution. These merger fractions are shown as a purple dotted line and a dot-dashed yellow line for the low and high mass bins respectively. Our results are consistent with both the asymmetry defined mergers and the 'true' mergers from the simulations.
We explore how many of galaxies within the Illustris simulation have A > 0.35 and also are considered to be a 'true' major merger (with a mass ratio greater than 0.25). In this case we consider galaxies that are within 0.45 Gyr of a merger event, or 0.9 Gyr in total, to be 'true' mergers in order to account for the time difference between snapshots within the simulation. We find that across all redshifts, an average of ∼ 37% of the 'true' major mergers lie above A > 0.35. This is slightly lower than the ∼ 50% found by Conselice (2006).
DISCUSSION
In this paper we have presented a number of observed features of galaxy evolution, as viewed through the structural changes in galaxies over the past 11 Gyr. We carry this out through an analysis of all five CAN-DELS fields, examining the change in light concentration and asymmetry for these systems. As opposed to other studies of galaxy evolution at these epochs we take a largely structurally evolutionary view of galaxies, rather than one based on stellar populations. In this viewing of galaxies we are interested in the build up of stars in galaxies and how that affects the distribution of light within galaxies, rather than examining the type and distribution of stars themselves.
Concentration
We find that galaxies are more concentrated at higher redshifts when probing galaxy structure in the optical rest-frame. This is an interesting result and is somewhat different from expectations, so it is important to examine the reasons for this. We test this result by examining the evolution of the concentration calculated using various measures of size. First, we correct the radii R 20 and R 80 (where R 20 is the radius containing 20% of the total light within a galaxy and R 80 is the radius containing 80% of the light) using the same method as described in §2.3 and use the mean of these corrected values to calculate the concentration using equation 2.
Examining the individual radii for our sample, we find that the evolution of the corrected R 80 values remain roughly constant with redshift whereas R 20 , on average, increases with redshift. Therefore, we see a lower concentration at lower redshifts. We show the evolution of the concentration calculated using the corrected values of R 20 and R 80 in Figure 16. As in Figure 7, the low mass bin is shown as red crosses and the high mass bin is shown as blue triangles. Also shown are power law fits to the concentrations, with the dotted line being for the low mass bin and the dashed line being for the high mass bin. For comparison, we show the evolution of the measured value of C for all masses as grey circles. This decrease in concentration is consistent with the corrected concentrations shown in Figure 7.
We further explore this by examining the Petrosian radius, R Petr (η). The Petrosian radius is defined to be the radius at which the surface brightness at a given radius is a specific fraction (η) of the surface brightness within that radius (e.g., Bershady et al. 2000;Conselice 2003;Whitney et al. 2019). η is given by: 4.00 C C lo = 2.28(1 + z) 0.22 C hi = 2.03(1 + z) 0.37 9.5 < log 10 (M * /M ) < 10.5 10.5 < log 10 (M * /M ) Measured C Figure 16. Evolution of the concentration as calculated from the corrected R20 and R80 growth radii. The low mass bin is shown as red crosses and the high mass bin is shown as blue triangles. The grey circles represent the evolution of the measured concentration across all masses.
η(r) = I(r) I(r)(17)
where I(r) is the surface brightness at radius r and I(r) is the mean surface brightness within that radius. Using this definition, η = 0 at large radii. We consider R Petr (η = 0.2) (the outer edge of a galaxy) and R Petr (η = 0.8) (the inner region of a galaxy). Whitney et al. (2019) show that R Petr (η = 0.2) grows more rapidly than R Petr (η = 0.8) for a mass-selected sample (where the masses lie in the range 10 9 M < M * < 10 10.5 M ) across a redshift range of 1 < z < 7 in the GOODS-North and GOODS-South CANDELS fields. So, one would expect the concentration at lower redshifts to be larger. In the same fashion as with the growth radii, we calculate the concentration of the galaxies in Whitney et al. (2019) with mass log 10 (M * /M ) > 9.5 using the Petrosian radii such that C P = 5 × log 10 R Petr (η = 0.2) R Petr (η = 0.8) .
The resulting concentration evolution is shown in Figure 17. We fit a power law to the entire redshift range from Whitney et al. (2019) (dotted line) and also the redshift range we consider in this paper (solid line). We find that C P goes as (1 + z) −0.09±0.06 for the redshift range of 1 < z < 6 and goes as (1 + z) 0.09±0.09 for the smaller redshift range of 1 < z < 3, which is within 2σ of being considered a flat relation. The vertical dashed line indicates the redshift limit of this work. Whilst on average over the redshift range of 1 < z < 6, the concentration decreases with redshift, when we consider the lowest redshift bins, the concentration appears to increase with redshift, which is consistent with the results we find and show in Figure 7. This is due to the fact that the inner radius appears to grow more rapidly compared to the outer radius within this small redshift range. We find the same result when considering galaxies with log 10 (M * /M ) > 10.5. for galaxies of mass log 10 (M * /M ) > 9.5 from Whitney et al. (2019). We fit a power law to the whole redshift range, shown as a dotted line, and also to the redshift range we consider here (solid line). We find that C P goes as (1 + z) −0.09±0.06 for the redshift range of 1 < z < 6 and as (1 + z) 0.09±0.09 for the smaller redshift range of 1 < z < 3. The vertical dashed line indicates the redshift limit of this work. On average, the concentration decreases at higher redshift however when considering the lowest redshifts, the concentration increases with redshift.
Alternatively, it might be the case that as we go to lower redshifts, we are sampling more and more galaxies using a constant mass limit (Mundy et al. 2015). The result of this is that intrinsically lower halo mass galaxies will be entering our sample and these may have lower concentrations. We test this by considering how the concentration changes for a number density selection sample. Doing this at number densities of 3 × 10 −5 Mpc −3 we find that the concentration decreases by a factor of 1.4 ± 0.3 from the highest redshift bin of 2.5 < z < 3 to the lowest bin of 0.5 < z < 1. This is comparable to the change seen in the mass-selected samples where the concentration decreases by a factor of 1.4 ± 0.3 for the lower mass bin and a factor of 1.2±0.2 for the higher mass bin over the same redshift range. A comparison of the evolution for the mass-selected and number density-selected samples are shown in Appendix A. This suggests that the evolution we see in the concentration is real, and not due to newer galaxies being introduced to the lower redshift bins.
Radius measurements of the inner regions can be inaccurate due to factors such as the size of the PSF. This could create inaccuracies in the concentration measurement. To ensure the concentration evolution we see is not an artefact of such issues, we also use R 50 and R 90 values in place of R 20 and R 80 in Equation 2. We find that the evolution of C 59 exhibits similar behaviour to C 28 , whereby the concentration increases by a factor of ∼1.5 over the redshift range 0.5 < z < 3. We can therefore conclude that the evolution in concentration we see is real and not as a result of problems with the inner radius measurement.
We interpret a decreasing concentration with lower redshift as an indication that a small initial galaxy will eventually grow in size as a result of galaxy mergers, as well as galaxy accretion. This growth is occurring in such a way that the inner regions of a galaxy grow more rapidly in comparison to the outer regions at z < 3.
Merger Consistency
Mergers can be identified in multiple ways such as pair statistics (e.g., Bundy et al. 2009;Man et al. 2016;Mundy et al. 2017;Duncan et al. 2019), CAS (e.g., Conselice et al. 2009;Bluck et al. 2012), Gini-M 20 (e.g., Lotz et al. 2008Lotz et al. , 2010, and Deep Learning (e.g., Ferreira et al. 2020) with each of these methods yielding differing merger counts, and therefore fractions due to the different criteria. Using a power law fit, we find that the merger fraction goes as (1 + z) 1.87±0.04 when fitting our data and setting the z ∼ 0 merger fraction to a fixed value. This is shallower than that found by Conselice et al. (2009) who derive merger fractions using the CAS parameters for a sample of galaxies within the Extended Groth Strip and Cosmic Evolution Survey at 0.2 < z < 1.2; they find a relation of (1 + z) 2.3±0.4 using their data and by using a fixed value of the merger fraction at z ∼ 0, the relation changes to (1 + z) 3.8±0.2 . Ferreira et al. (2020) use a convolution neural network trained on simulations from IllustrisTNG and applied to data from all five CANDELS fields and find an evolution of (1 + z) 2.82±0.46 , again slightly steeper than the results we find. Duncan et al. (2019) on the other hand find a shallower relation for their pair fractions of (1+z) 0.844 +0.216 −0.235 for galaxies with log 10 (M * /M ) > 10.3. In general, the morphologically defined mergers give a higher merger fraction than those derived through pairs ). However, this is not unexpected, as different methods of finding mergers will give different results for the fractions.
Consistency is however expected when converting to merger rates and comparing these values. Along with different methods giving different merger fractions, different selection criteria within the same sample also give varying results; Man et al. (2016) examine the pair fraction evolution up to z ∼ 3 using the UltraV-ISTA/COSMOS K S -band selected catalog along with 3DHST+CANDELS H-band imaging and explore the effect of using the flux ratio versus the mass ratio in the selection. When selecting via the stellar mass ratio, the merger fraction appears to increase with cosmic time in contradiction to the results we and others find. A selection via the H-band flux ratio however, the trend is similar to ours in that the merger fraction decreases.
As with merger fractions, merging timescales vary quite drastically depending on the selection method. For example, Lotz et al. (2008) use GADGET N-Body hydrodynamical simulations and the radiative transfer code SUNRISE to create a sample of galaxy mergers. They use these to determine merging timescales for multiple methods of identifying mergers. Using the same asymmetry condition we use here, they determine a timescale of 0.94 ± 0.13 Gyr, whilst using a Gini and M 20 condition, they find the timescale to be 0.26 ± 0.10 Gyr. Conselice et al. (2008) who examine the merger rate up to z ∼ 3 in the Hubble Ultra-Deep Field use a merger time-scale of 0.34 Gyr, determined using the asymmetry condition. These methods all assume a constant timescale with redshift, however using a close pair method of identifying mergers, Snyder et al. (2017) find that the timescale varies with redshift as τ ∝ (1 + z) −2 . These timescales all differ from the constant timescale of τ = 0.56 +0.23 −0.18 Gyr we find using the asymmetry criteria. Despite these differences in timescales and merger fractions for various methods of identifying mergers, merger rates are largely consistent with each other, as shown in Figure 12. The merger rates determined using the asymmetry condition to identify mergers as we have done here are, within the errors, similar to merger rates determined using other methods such as probabilistic pair-counts ) and finding mergers using Deep Learning ). This is despite using different timescales for a merging event, showing that while different methods identify different numbers of mergers, our CAS method gives a consistent merger history for galaxies. The merger rates presented here have greater errors associated with them than previous work due to the consideration of the error in the merger timescale; the IllustrisTNG simulations used here to determine the merger timescale have an asymmetric distribution in the timescale and so we therefore use an asymmetric error. Ferreira et al. (2020) use a timescale based on their selection and as such, do not have an associated error. Duncan et al. (2019) calculate the errors on their merger fractions using the bootstrap technique. The timescale taken from Snyder et al. (2017) does not have an associated statistical error so when determining the fit (by convolving the merger fraction fits with the redshift dependent timescale), the merger rates only consider the error in merger fraction. Therefore, we have found that using three independent methods that we are able to measure a consistent merger rate at z < 3, and therefore have a firm idea of what the history of galaxy major mergers are within this redshift range.
CONCLUSIONS
Galaxy structure is one of the most fundamental ways in which galaxies and their formation and evolution can be understood, yet there have been few systematic studies of this property at higher redshifts. In this paper we collate the high resolution HST imaging of galaxies from all five fields of the CANDELS survey to determine the time evolution of galaxy structure to derive the processes for galaxy evolution. We combine these 16,778 galaxy structures, both visually based and quantitatively based, with stellar mass measurements and photometric redshifts to derive the evolution of galaxy structure. To aid our investigation of these morphologies we also utilise Illustris TNG300 simulations to calibrate timescales for morphological features and Illustris TNG50 for direct morphological comparisons with observations. Our conclusions, based on a detailed non-parametric analysis of the structures of galaxies in all five CAN-DELS fields are as follows:
1. We find that galaxies are distributed in a concentration-asymmetry plane up to z ∼ 3 in a similar way as they do at lower redshifts. At these redshifts there is a continuum of parameters, suggesting unlike the colour-magnitude plane galaxies on this relation evolve continuously over time, rather than change dramatically.
2. Based on dynamical simulations with the Illus-trisTNG simulation, we show that the evolution of galaxies within the C − A plane over this time period is largely driven by galaxy formation pro-cesses, including the accretion of gas as well as galaxy mergers.
3. We find that galaxies are both more asymmetric and more concentrated at higher redshifts. This demonstrates that galaxies are assembled in a way where material is coming in through galaxy formation processes that are relatively changing the inner portions of galaxy light quicker than the outer radii at z < 3.
4. Using merger timescales from Illustris TNG300 we determine how the morphological timescales for mergers evolves with time. We find that for the CAS system of finding mergers, the mean merger timescale is 0.56 +0.23 −0.18 Gyr. We also find, that unlike in the galaxy pair situation, this merger timescale does not evolve with time.
5. Using these timescales we are able to determine the merger fraction evolution and the merger rate evolution up to z ∼ 3. We find a good agreement between our merger rate calculations in comparison with previous work using pairs of galaxies and finding mergers based on Deep Learning. This is therefore a consistent merger history for galaxies as measured in three different ways. We now have a firm and consistent measure of the galaxy major merger history at z < 3.
Our results are the best measurements of the merger history using HST imaging that will likely be possible unless a new large area survey in the near infrared is performed. When Euclid and JWST launch we will be able to expand these results in many ways. We will be able to measure the merger history in a similar way up to z ∼ 6, and at lower redshifts we will be able to probe down to lower masses to find the evolution of a fuller sampling of the galaxy population. Euclid will allow us to examine the merger history over 15,000 deg 2 , allowing detailed merger histories to be measured. The techniques and methods here can be used with both data sets, and others, to fully explore and measure this merger history, a principle part of the entire galaxy formation process.
ACKNOWLEDGEMENTS
This work is based on observations taken by the CANDELS Multi-Cycle Treasury Program with the NASA/ESA HST, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. We thank the rest of the CANDELS team for their heroic work making their products and data available. We acknowledge funding from the Science and Technology Facilities Council (STFC) to support this work. LF acknowledges funding from the Coordenaşão de Aperfeiçoamento de Pessoal de Nível Superior -Brazil (CAPES).
APPENDIX
A. NUMBER DENSITY-SELECTED SAMPLE
In order to test whether the mass limits of our sample are creating an artificial trend, we instead select galaxies using a constant number density of 3 × 10 −5 Mpc −3 . This yields galaxies in the mass range 10 10.7 M ≤ M * < 10 11.5 M . This method avoids problems with having a fixed stellar mass bin that galaxies can enter at lower redshift, compared with those that are at higher redshifts (e.g., Mundy et al. 2015;Ownsworth et al. 2016). This has an effect such that by z = 0 only ∼ 5 % of galaxies with a high mass selection would have been within the same mass selection at z > 2.5. However, by selecting through a constant co-moving number density we are able to remove this bias and always examine what are statistically the same number of massive galaxies at each redshift. Because there are so many 'new' galaxies in a mass-selected sample, it is possible that the relations we derive are biased by these new systems.
We determine the concentration evolution for this number density selected sample, and compare it to the evolution seen for the mass selected-sample. We find that the evolution for the number density-selected sample is very similar to that of the mass-selected sample, as shown in Figure 18. Green circles indicate the concentration for the number density-selected sample, red crosses indicate the lower mass bin, and blue triangles the higher mass bin. Therefore our finding of a reduction in concentration using the mass selected sample is not due to new galaxies entering the criteria.
As a test for another key result, we examine how the merger fraction evolution would occur by using a selection based on this number density selection of 3 × 10 −5 Mpc −3 . This is shown in Figure 19 where we find that the merger fraction is slightly lower for all but the final lowest redshift bin. However, the fitted merger fraction evolution and thus the corresponding merger rate is statistically very similar and does not give significantly different merger histories for 4.25 C Number Density-Selected 9.5 < log 10 (M * /M ) < 10.5 10.5 < log 10 (M * /M ) Figure 18. A comparison of the evolution of concentration for a number density-selected sample (green circles) and a massselected sample. The number density is kept constant at 3 × 10 −5 Mpc −3 . As for Figure 7, red crosses indicate the lower mass bin and blue triangles indicate the higher mass bin. We find no significant difference in the evolution of the C parameter using these different selection methods.
galaxies. Thus, our result using a constant mass selection is a viable way in which to trace the evolution of the massive galaxy population, and our results are robust to galaxy selection methods. Figure 19. A comparison of the evolution of the merger rate for a number density-selected (n = 3 × 10 −5 Mpc −3 ) sample of galaxies (green circles), low mass galaxies (red crosses), and higher mass galaxies (blue triangles). The fit to the number density-selected sample is shown by the solid line and the fit to the mass-selected sample is shown by the dashed line. The known local value of the merger fraction from Casteels et al. (2014) is from a mass-selected sample.
Figure 1 .
1Mass and redshift distribution of the sample of galaxies. Yellow indicates a higher density of galaxies and purple indicates the lowest density. White areas indicates there are no galaxies are present in this region of parameter space. The dashed horizontal lines at log10(M * /M ) = 9.5 and log10(M * /M ) = 10.5 indicate the boundaries between the two mass bins used in this study.
Figure 2
2Figure 2. Left: mean asymmetry correction applied to all five CANDELS fields. Right: mean concentration correction for all CANDELS fields. The error bars are equal to one standard deviation from the mean. In both panels, the GOODS-North correction is shown by a blue circle, GOODS-South by an orange diamond, COSMOS by a green triangle, EGS by a red square, and UDS by a purple inverted triangle.
Figure 4
4Figure 4. Evolution of the concentration-asymmetry plane with redshift, colour-coded by visual classification. Red circles are galaxies classified as spheroids, blue crosses as disks, and green triangles as peculiar galaxies. Any galaxies that do not fit within these three classifications are shown as small grey circles. The lines are boundaries between galaxy types derived from Bershady et al. (2000). Galaxies that are above the dashed line are considered to be mergers. The dot-dashed line is the boundary between early-and intermediate-type galaxies and the solid line is the boundary between late-and intermediate-type galaxies. The redshift indicated in the top right corner of each panel is the midpoint of the redshift bin whereby the upper and lower limits are 0.25 either side of this point.
Figure 5 .
5Evolution of the concentration-asymmetry plane with redshift for galaxies that have been given either a bulge dominated flag (red circles) or a disk dominated flag (blue crosses). Galaxies that have neither flag are not plotted. As withFigure 4, the lines are boundaries between galaxy types derived fromBershady et al. (2000). The redshift indicated in the top right corner of each panel is the midpoint of the redshift bin whereby the upper and lower limits are 0.25 either side of this point.
Figure 7 .Figure 8 .
78Evolution of the mean corrected concentration for both mass bins. The lower mass bin is shown as a red cross and the higher mass bin is shown by the blue triangles. The error bars represent one standard deviation from the mean. On average, the higher mass bin has a greater concentration than the lower mass bin at all redshifts, however both bins show a decrease in concentration with cosmic time. Evolution of the mean corrected asymmetry with redshift. As with
Figure 10
10Figure 10. Examples of galaxies within our sample that are classified as a merger using the criteria given in Equation 8. Lower redshift (1 < z < 2) galaxies are shown in the left two columns and higher redshift (2 < z < 3) galaxies are shown in the right two columns. The redshift and asymmetry are given for each example.
as a solid black line and one standard deviation from this fit is indicated by the grey shaded area. Also shown are the results of Duncan et al. (2019) (yellow circles) who identify mergers using pair statistics and fit a power law of the form (1 + z) 1.775 +0.205 −0.196 . We also compare to Ferreira et al. (2020) (green diamonds) who identify mergers using a convolutional neural network and find that the merger fraction evolves as (1 + z) 2.82±0.46 . Our merger fractions lie between these two examples. The merger fractions and their errors are given in Columns 2 and 4 of
Figure 11 .
11Distribution of the timescales for the total merger events within the IllustrisTNG sample. The dashed line indicates the median timescale used to determine the merger rates.
Figure 12
12Figure 12. Left: evolution of the merger fraction. Right: evolution of the merger rate. The lower mass bin from this work is shown as a red cross and the higher mass bin from this work is shown as a blue triangle. The fit to both these points and a known local value from Casteels et al. (2014) (grey inverted triangle) is shown as a solid black line with the grey shaded area indicating one standard deviation from the fit. We find a fit of the form fm = 0.0193 ± 0.004(1 + z) 1.87±0.04 for the merger fractions. For the merger rates, we find a fit of the form R(z) = 0.03 ± 0.02(1 + z) 1.87±0.04 Gyr −1 . We compare our results to Duncan et al. (2019) (yellow circles) and Ferreira et al. (2020) (green diamonds). We also compare the merger rate to that found by Rodriguez-Gomez et al. (2015) for the Illustris simulation. This fit is given as a dashed line.
Figure 17 .
17Evolution of the concentration calculated using the Petrosian radii at η = 0.2 and η = 0.8 as per Equation 18
Table 1 .
1The bands corresponding to the rest-frame optical at each redshift.z
O raw
i,j
λrest
0.75
I814
4650Å
1.25
I814
3620Å
1.75
J125
4550Å
2.25
J125
3850Å
2.75
H160
4270Å
Table 2 .
2The asymmetry and concentration corrections applied to each field at each redshift interval.Asymmetry
Concentration
z
GOODS-N GOODS-S COSMOS EGS
UDS GOODS-N GOODS-S COSMOS EGS UDS
0.75
0
0
0
0
0
0
0
0
0
0
1.25
-0.010
-0.010
-0.009
-0.004 -0.007
-0.27
-0.29
-0.30
-0.40 -0.20
1.75
0.008
-0.004
-0.008
-0.007 -0.023
-0.39
-0.43
-0.32
-0.46 -0.43
2.25
-0.032
-0.016
-0.036
-0.027 -0.060
-0.83
-0.87
-0.75
-0.85 -0.84
2.75
-0.084
-0.079
-0.104
-0.100 -0.118
-1.08
-1.10
-1.08
-1.09 -1.14
, we show the boundaries between the galaxy types. We also plot0
1
2
Patchiness
0
1
2
Clumpiness
2.759
2.439
2.285
2.639
2.466
2.383
2.573
2.484
2.356
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| []
|
[
"Stability of a nonlinear oscillator with random damping",
"Stability of a nonlinear oscillator with random damping"
]
| [
"N Leprovost [email protected] \nLaboratoire de Physique statistique de l'ENS\n24 rue Lhomond75231, cedex 05ParixFrance\n",
"S Aumaître \nLaboratoire de Physique statistique de l'ENS\n24 rue Lhomond75231, cedex 05ParixFrance\n",
"K Mallick \nService de Physique théorique, CEA Saclay\nF-91191Gif sur Yvette CedexFrance\n"
]
| [
"Laboratoire de Physique statistique de l'ENS\n24 rue Lhomond75231, cedex 05ParixFrance",
"Laboratoire de Physique statistique de l'ENS\n24 rue Lhomond75231, cedex 05ParixFrance",
"Service de Physique théorique, CEA Saclay\nF-91191Gif sur Yvette CedexFrance"
]
| []
| A noisy damping parameter in the equation of motion of a nonlinear oscillator renders the fixed point of the system unstable when the amplitude of the noise is sufficiently large. However, the stability diagram of the system can not be predicted from the analysis of the moments of the linearized equation.In the case of a white noise, an exact formula for the Lyapunov exponent of the system is derived. We then calculate the critical damping for which the nonlinear system becomes unstable. We also characterize the intermittent structure of the bifurcated state above threshold and address the effect of temporal correlations of the noise by considering an Ornstein-Uhlenbeck noise.PACS. 02.50.-r Probability theory, stochastic processes and statistics -05.40.-a Fluctuation phenomena, random process, noise and Brownian motion -05.10.Gg Stochastic analysis methods (Fokker-Planck, Langevin, etc.) | 10.1140/epjb/e2006-00089-9 | [
"https://arxiv.org/pdf/nlin/0510063v1.pdf"
]
| 18,905,337 | nlin/0510063 | a713697e90db85497bc6cc5463a77125172472af |
Stability of a nonlinear oscillator with random damping
24 Oct 2005
N Leprovost [email protected]
Laboratoire de Physique statistique de l'ENS
24 rue Lhomond75231, cedex 05ParixFrance
S Aumaître
Laboratoire de Physique statistique de l'ENS
24 rue Lhomond75231, cedex 05ParixFrance
K Mallick
Service de Physique théorique, CEA Saclay
F-91191Gif sur Yvette CedexFrance
Stability of a nonlinear oscillator with random damping
24 Oct 2005Received: date / Revised version: datearXiv:nlin/0510063v1 [nlin.CD] EPJ manuscript No. (will be inserted by the editor)
A noisy damping parameter in the equation of motion of a nonlinear oscillator renders the fixed point of the system unstable when the amplitude of the noise is sufficiently large. However, the stability diagram of the system can not be predicted from the analysis of the moments of the linearized equation.In the case of a white noise, an exact formula for the Lyapunov exponent of the system is derived. We then calculate the critical damping for which the nonlinear system becomes unstable. We also characterize the intermittent structure of the bifurcated state above threshold and address the effect of temporal correlations of the noise by considering an Ornstein-Uhlenbeck noise.PACS. 02.50.-r Probability theory, stochastic processes and statistics -05.40.-a Fluctuation phenomena, random process, noise and Brownian motion -05.10.Gg Stochastic analysis methods (Fokker-Planck, Langevin, etc.)
Introduction
It is a well-known fact that a multiplicative noise acting on a dynamical system can generate unexpected phenomena such as stabilization [1,2], stochastic transitions and patterns [3] or stochastic resonance [4]. As far as the stability of a random dynamical system is concerned, the exactly solvable example of a nonlinear first-order Langevin equation shows that the behaviour of the moments of the linearized equation can be misleading : higher moments seem to be always unstable although it is known that the critical value of the control parameter is the same as that of the deterministic system [5]. This apparent contradiction is due to the existence of long tails in the stationary probability distribution of the linearized equation which are suppressed when the nonlinearities are taken into account. The same type of conclusion can be drawn for a second-order system, namely a nonlinear oscillator with fluctuating frequency. The energetic stability analysis of the linearized system has been performed long ago [6,7] but it leads to an erroneous phase diagram. In fact, because of the nonlinearities, the noise can even enhance the stable phase [1,2]. Besides, a non-perturbative reentrant transition can occur : a noise of small amplitude stabilizes the system, but a strong noise destabilizes it. Similar features hold in spatially extended systems such as the Ginzburg-Landau and Swift-Hohenberg equations driven by a noisy control parameter [3,8]. Here again, the observed threshold shift can not be tackled by the analysis of the moments' behaviour alone.
In this paper, we study the nonlinear oscillator with random damping. This dynamical system, in which the noise acts on the velocity variable rather than on the position, provides another example of a bifurcation induced by a multiplicative noise. This model was introduced in the study of the generation of water waves by wind [9] where the turbulent fluctuations in the air flow is modeled by a noise. This model is also relevant in the study of dynamical systems with an advective term where the corresponding velocity fluctuates. This problem has been investigated in [10,11] via the stability of the moments of the linearized equation. Here, we demonstrate that, as in the case of the random frequency oscillator, the moments stability analysis has no relevance and can not be used to derive the instability threshold. We shall obtain the exact phase diagram of the system thanks to an instability criterion valid for general nonlinear stochastic dynamical systems [12,13].
The outline of this work is as follows. In section 2, we define the model and summarize the results obtained from the stability analysis of the moments. The main part of the paper (section 3) is devoted to the calculation of the Lyapunov exponent of the system from which the marginal stability curve is drawn. We then analyze the nature of the bifurcation and show that the bifurcated state is intermittent (section 4). In the last section, we consider an Ornstein-Uhlenbeck noise to investigate the effect of temporal correlations on the instability threshold.
Definition of the model and basic results
We consider the following stochastic dynamical system:
x + [r + ax 2 − ξ(t)]ẋ + αx = −bx 3 ,(1)
where ξ(t) is a white noise with intensity D and the pa- we observe that y satisfies:
y + (r − aα b + 2ax 0 y + ay 2 )ẏ − 2αy (2) =ẋξ(t) − b(3x 0 y 2 − y 3 ) .
The linear part of this equation is the same as that of equation (1) once the substitutions r ′ = r − aα/b and α ′ = −2α are made. The coefficient α ′ is now positive.
We thus take α > 0. Before proceeding, we write time and position in dimensionless units by multiplying t and
x by the factors 1/ √ α and a/r, respectively. Defining the following parameters:
γ = r √ α , ∆ = D √ α and k = br aα ,(3)
equation (1) becomes
x + γ(1 + x 2 )ẋ + x =ẋξ(t) − kx 3 ,(4)
where the autocorrelation of the white noise is given by We now summarize the results for the stability of the moments of the amplitude x, obtained by linearizing equation (4) around the fixed point x = 0. It can be shown that the first moment satisfies :
ξ(t)ξ(t ′ ) = ∆δ(t − t ′ ) (d 2 d t 2 x + (γ − ∆ 2 ) d d t x + x = 0 .(5)
The first moment remains bounded as t → ∞ provided and xẋ , and study the linear system that couples them.
These three quantities become zero in the long time limit provided γ > ∆. In this case, the system is said to be 'energetically' stable. The instability threshold thus depends on the moment which is considered, due to the interplay between noise and nonlinearities. Therefore, in contrast to the deterministic case, a naive linear stability analysis fails to lead to conclusive results. A more refined criterion is needed to determine stability.
The Lyapunov exponent
The correct criterion that determines the stability of the fixed point x = 0 is based upon the Lyapunov exponent Λ, defined as
Λ = 1 2 ∂ t ln[δx 2 ] = ∂ t (δx) x ,(6)
where δx satisfies equation (4) linearized around x = 0. It has been shown [12,13,14] that the sign of the Lyapunov exponent, calculated with the linear part of the equation, monitors the instability of the nonlinear oscillating system. When Λ is negative, the trajectories of the nonlinear system (4) almost surely decay to zero and in the long time limit, the oscillator becomes localized in its rest position x = 0. On the contrary, if Λ > 0, the fixed point x = 0 is unstable and the stationary probability density of the oscillator is extended.
We now calculate the Lyapunov exponent as a function of the parameters γ and ∆. Defining the variable z as z = ∂ t (δx)/δx, the equation satisfied by z is given bẏ
z = −1 − γz − z 2 + zξ(t) .(7)
This is a Langevin equation involving the variable z alone and coupled to a multiplicative noise (to be interpreted in the Stratonovich sense). By definition (6), we observe
that Λ(γ, ∆) = z .(8)
Using standard techniques [15,16], we obtain the Fokker-Planck equation for the probability density P (z, t) at time t. The stationary probability density P s (z) satisfies:
∆ 2 ∂ z zP s (z) + 1 + γz + z 2 z P s (z) = J z ,(9)
where J is the constant current of probability. Here, the current J does not vanish; intuitively, this is due to the fact that z plays the role of an angular variable in phase space and must be interpreted as a compact variable (see [17] for more details). Equation (9) can be solved by variation of constants method and we obtain
P s (z) = 2J ∆ z c | t z | 1+2γ/∆ 1 t 2 exp − 2 ∆ Φ(z) − Φ(t) dt + A|z| −1−2γ/∆ exp − 2Φ(z) ∆ ,(10)
where Φ(t) = t − 1/t; the constant A and the reference point c are not determined at this stage. We must impose A = 0 so that P s (z) is integrable at infinity (this implies J = 0). Furthermore, the function exp[Φ(t)] is exponentially divergent when t → 0 − ; this remark fixes the choice of the value of c in order to ensure that P s (z) is integrable at the origin (the cases z < 0 and z > 0 must be analyzed separately and two different reference points must be chosen). Finally, we obtain,
P s (z) = 1 N z −∞ H(z, t) dt if z < 0 1 N z 0 H(z, t) dt if z > 0(11)
where H(z, t) represents the function under the integral sign in (10). Using equation (9), we find that the probability density (11) satisfies
P s (z) ∼ J z 2 for z −→ ±∞ ,(12)P s (z) ∼ J for z −→ 0 .(13)
This quadratic decay insures that the probability density is indeed normalizable at infinity. The normalization constant N can calculated exactly [18] and we obtain
N = π 2 J 2 2γ/∆ 4 ∆ + Y 2 2γ/∆ 4 ∆ ,(14)
where J and Y are Bessel functions of the first kind and second kind, respectively. This formula was derived in [19] for the study of diffusion in a random medium.
According to equation (8), the Lyapunov exponent is equal to the mean value of z calculated with respect to P s (z); this quantity will be evaluated in the sense of principal parts
Λ = lim M→∞ +M −M zP s (z)dz .(15)
(Thanks to equation (12), the logarithmic divergencies at −∞ and +∞ cancel with each other by parity). An exact closed formula for the Lyapunov exponent can then be derived and is given by
Λ = 8 N +∞ 0 K 1 8 sinh x ∆ sinh 1 − 4γ ∆ x dx ,(16)
where K 1 is a modified Bessel function of the second kind and the normalization constant N is given in equation (14).
From this expression, the instability threshold is readily found. Indeed, because the Bessel function is always positive, the sign of the Lyapunov is the same as that of the argument of the hyperbolic sine. We conclude that the fixed point x = 0 is stable when
γ > ∆ 4 .(17)
We find, as usual, that the stable phase is wider than what moments stability predicts. Reverting to the initial variables, we observe that the instability threshold is given by stabilizing effect at small amplitudes : we notice that the curve for γ = 4 decreases for small values of ∆ and then increases. A closer inspection shows that this non-monotonic behaviour occurs when γ > 2. This feature seems to contradict the argument, presented in section 2, stating that a noise in the damping term always has a distabilizing effect [10]. This contradiction stems from the asymmetry of growth rate of the deterministic system around γ = 2.
Indeed, for γ < 2 the growth rate is a linear function of γ
Intermittency above the threshold
As shown in figure 2, the temporal series of the signal x(t) exhibits "on-off intermittency"above the instability threshold (i.e., for a positive Lyapunov exponent). In other words, the amplitude x is vanishingly small for the most of the time but exhibits sudden burst of activity. This behaviour has already been observed in chaotic systems [20,21] and in systems driven by multiplicative noise [22,23,24,25,26]. This intermittency can also be identified in the probability distribution density. In figure 3, we show the probability density of the energy of the oscillator defined as r = √ x 2 +ẋ 2 . Near the origin, we observe that the probability distribution of r is a power law which is generic of an intermittent signal. This power-law distribution for the energy of the oscillator can be derived as follows. From equation (4), we deduce that r satisfieṡ r = −(γ + ξ(t))r sin 2 θ − γ sin 2 θ cos 2 θ + k cos 3 θ sin θ r 3 ,
where θ = arctan(ẋ/x) = arctan(z). This equation, together with (7), forms a system of stochastic equations.
A Fokker-Planck equation for the joint density P (r, θ) or P (r, z) can be written but the resulting partial differential equation seems unlikely to be exactly solvable.
We therefore make an approximation along the lines of [13] and assume that, for small values of r, the probability density is separable and can be written as P (r, θ) = P s (r)P s (θ). An average over the angular variable yields an independent equation for P s (r) that can be exactly solved:
P s (r) = 1 N r r (B/A)−1 exp − Cr 2 2A ,(18)
where the coefficients A, B and C can all be expressed as mean values over the variable z (or, equivalently, over the angular variable θ) :
A = ∆ z 4 2(1 + z 2 ) 2 , B = z 2 ∆ − γ(1 + z 2 ) (1 + z 2 ) 2 , C = γz 2 + kz (1 + z 2 ) 2 .
Multiplying equation (9) by z 2 /(1 + z 2 ) and integrating the result over z, the following identity is obtained :
B = z = Λ .(19)
The coefficient A being a positive quantity, we remark that the function (18) (11). The agreement between the two curves is excellent as far as the power-law behaviour for small values of r is concerned. For higher values of r, a discrepancy appears : the assumption that the stationary distribution is separable is no more valid and a specific analysis for large r is needed [13]. In figure 4, we plot the probability density of the energy for a greater value of the noise : again the small r power law is very well described by equation (18).
Effect of a colored noise
In order to study the effect of the temporal correlations of the noise, we have performed numerical simulations of equation (4), taking the noise to be an Ornstein-Uhlenbeck process. In the stationary regime, the correlation of the Ornstein-Uhlenbeck process is exponential :
ξ(t)ξ(t ′ ) = ∆ 2τ exp − |t − t ′ | τ .(20)
Using the time series of the simulation results, we estimate the Lyapunov exponent (by computing ẋ/x ) and identify the bifurcation threshold. In figure 5, we plot the critical value of the noise intensity ∆ for the onset of instability as a function of γ, the correlation time τ being equal to 1 (in dimensionless units). We observe that the critical curve has the shape of a straight line. Performing this analysis
for different values of τ , we have found that the critical amplitude ∆ ⋆ of the noise is almost linear with γ; hence, to a good approximation, the critical curve is given by ∆ * (γ) = C(τ )γ. In figure 6, we plot C(τ ) as a function of τ ; again the evolution is well described by a straight line in the range 0 ≤ τ ≤ 1. These results indicate that the structure of the bifurcation diagram for the Ornstein-Uhlenbeck noise remains qualitatively the same as that for white noise.
Conclusion
In this paper, we have studied the noise-induced bifurcation of a nonlinear oscillator with a fluctuating damping term. We have shown that the instability threshold of the nonlinear oscillator can not be determined from a stability analysis of the moments of the linearized problem; the system is indeed more stable than the analysis of moments indicates. This feature, that has already been noticed in a variety of models [5,8,13], can be explained by the existence of long tails in the probability distribution of the linearized system. These long tails have a greater influence on higher moments of x which are therefore less and less stable. However, these long tails are suppressed for the nonlinear system, which has, therefore, a well defined stability threshold. For the problem considered here, an exact calculation of the Lyapunov exponent has allowed us to determine the exact stability diagram of the system.
Besides, we have shown that a good quantitative description of the system in the vicinity of the bifurcation can be obtained if the probability distribution is assumed to be separable in energy and angle variables. This Ansatz allows us to calculate the power law exponent of the stationary probability and to predict intermittency. Our theoretical findings agree with numerical simulations. Finally, we have studied numerically the effect of temporal correlation of the noise by coupling our dynamical system to an Ornstein-Uhlenbeck process, and have found that the bifurcation scenario remains qualitatively the same. For this latter problem, exact analytical calculations seem to be out of reach; some quantitative information may, however, be obtained either by using one of the various Markovian approximations for colored noise, or by considering a special type of random process, such as the dichotomic Poisson noise which leads, in the present case, to an exactly solvable problem.
We would like to thank Francois Petrelis and Stephan Fauve for interesting discussions and suggestions on this paper.
rameters a and b are taken to be positive in order to have a stabilizing effect. The mean value of the damping r is also positive whereas the coefficient α of the linear term can be either positive or negative. In fact, as far as the stability of the fixed point is concerned, the α < 0 case can be reduced to the α > 0 case as follows. For α < 0 and in the absence of noise, the oscillation around the x = 0 position is unstable and the new equilibrium position in the saturated regime is x 0 = ± −α/b. Defining y = x − x 0 ,
as usual, the brackets represent an average on the realizations of the noise). The aim of this work is to study the stability of the fixed point x = 0 of equation (4) as a function of the values of the different parameters γ, ∆ and k.
γ > ∆/ 2 .
2The effect of the noise is therefore to enhance the unstable phase. An intuitive argument of van Kampen[10] explains why this should be the case : positive and negative fluctuations are equiprobable, but because the noise is multiplied by the velocity, the negative fluctuations have a stronger effect because they tend to increase the velocity (whereas the positive fluctuations decrease the velocity and have a lesser impact). For the second order moment, one must consider simultaneously x 2 , ẋ 2
rFig. 1 .
1= D/4 which does not depend on the 'linear frequency' α of the oscillator. In the (r, D) plane, the critical curve in the stability diagram of the x = 0 fixed point is simply a straight line. We emphasize that this exact result is much simpler than that obtained for the Duffing oscillator with random frequency [13]. In figure 1, we plot the Lyapunov exponent of the nonlinear equation (4) for γ = 1 and γ = 4. We observe that, for Λ < 0, the numerical findings and the analytical prediction are in perfect agreement. For higher values of the noise, the nonlinear term can not be neglected : whereas the linear Lyapunov exponent, given by equation (16) continues to grow, the Lyapunov exponent of the nonlinear equation saturates to a value close to 0. This indicates that the nonlinear system has gone through a bifurcation : the fixed point x = 0 has become unstable and in the long time limit the system reaches an extended stationary state (in our case a noisy limit cycle). A remarkable feature of the curves displayed in figure 1, is that noise can have a Typical evolution of the Lyapunov exponent as a function of the noise amplitude ∆. The circles/crosses are obtained by numerical simulations of the nonlinear equation in the stationary regime for γ = 4 and γ = 1. The solid lines represent the theoretical values (16) for the linearized equation.
(σ = −γ/2), whereas it has a very different expression for γ > 2 (σ = −γ/2+ γ 2 − 4/2). The positive and negative fluctuations of the growth rate are therefore intrinsically asymmetric, in contrast to the implicit assumption in van Kampen's argument.
Fig. 2 .
2Numerical simulations of the system (4) with γ = 1, k = 1 and ∆ = 3 (top left panel) corresponding to an absorbing state and for ∆ = 4.5 (top right panel), corresponding to an intermittent state. The lower panels compares the probability distribution for z obtained by simulations (dots) with the exact result (11) (full lines).
Fig. 3 .
3Probability density (in log-log scale) of the variable r = √ x 2 +ẋ 2 for γ = 1, ∆ = 4.2 and k = 1. The dotted line corresponds to the approximation (18).
is normalizable if and only if B = Λ > 0. This provides a simple derivation of the stability criterion that we have used : when Λ < 0, the fixed point x = 0 is stable (the distribution (18) being not normalizable, the stationary distribution is the Dirac function at 0); when Λ > 0, the fixed point is unstable and the stationary distribution is extended. The distribution (18) is obviously a power law for small values of the variable r which is an indication for intermittency. In figure 3, numerical results (open circles) for the probability distribution of the variable r are compared with the analytical formula (18) (dashed line); the coefficients A, B and C have been calculated using
Fig. 4 .Fig. 5 .
45Probability density (in log-log scale) of the variable r = √ x 2 +ẋ 2 for γ = 1, ∆ = 6 and k = 1. Critical amplitude ∆⋆ of the noise versus γ for a time correlation τ = 1. The dotted line is a linear fit.
Fig. 6 .
6Evolution of the critical slope as a function of the correlation time. The dotted line is a linear fit.
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| []
|
[
"Externally Controlled Lotka-Volterra Dynamics in a Linearly Polarized Polariton Fluid",
"Externally Controlled Lotka-Volterra Dynamics in a Linearly Polarized Polariton Fluid"
]
| [
"Matthias Pukrop \nDepartment of Physics and CeOPP\nUniversität Paderborn\nWarburger Straße 10033098PaderbornGermany\n",
"Stefan Schumacher \nDepartment of Physics and CeOPP\nUniversität Paderborn\nWarburger Straße 10033098PaderbornGermany\n\nCollege of Optical Sciences\nUniversity of Arizona\n85721TucsonAZUSA\n"
]
| [
"Department of Physics and CeOPP\nUniversität Paderborn\nWarburger Straße 10033098PaderbornGermany",
"Department of Physics and CeOPP\nUniversität Paderborn\nWarburger Straße 10033098PaderbornGermany",
"College of Optical Sciences\nUniversity of Arizona\n85721TucsonAZUSA"
]
| []
| Spontaneous formation of transverse patterns is ubiquitous in nonlinear dynamical systems of all kinds. An aspect of particular interest is the active control of such patterns. In nonlinear optical systems this can be used for all-optical switching with transistor-like performance, for example realized with polaritons in a planar quantum-well semiconductor microcavity. Here we focus on a specific configuration which takes advantage of the intricate polarization dependencies in the interacting optically driven polariton system. Besides detailed numerical simulations of the coupled light-field exciton dynamics, in the present paper we focus on the derivation of a simplified population competition model giving detailed insight into the underlying mechanisms from a nonlinear dynamical systems perspective. We show that such a model takes the form of a generalized Lotka-Volterra system for two competing populations explicitly including a source term that enables external control. We present a comprehensive analysis both of the existence and stability of stationary states in the parameter space spanned by spatial anisotropy and external control strength. We also construct phase boundaries in non-trivial regions and characterize emerging bifurcations. The population competition model reproduces all key features of the switching observed in full numerical simulations of the rather complex semiconductor system and at the same time is simple enough for a fully analytical understanding of the system dynamics. arXiv:1903.12534v1 [cond-mat.other] | 10.1103/physreve.101.012207 | [
"https://arxiv.org/pdf/1903.12534v1.pdf"
]
| 88,516,903 | 1903.12534 | a331b5107ef2f1f051f4e3fe67814e88e05aa919 |
Externally Controlled Lotka-Volterra Dynamics in a Linearly Polarized Polariton Fluid
Matthias Pukrop
Department of Physics and CeOPP
Universität Paderborn
Warburger Straße 10033098PaderbornGermany
Stefan Schumacher
Department of Physics and CeOPP
Universität Paderborn
Warburger Straße 10033098PaderbornGermany
College of Optical Sciences
University of Arizona
85721TucsonAZUSA
Externally Controlled Lotka-Volterra Dynamics in a Linearly Polarized Polariton Fluid
(Dated: April 1, 2019)
Spontaneous formation of transverse patterns is ubiquitous in nonlinear dynamical systems of all kinds. An aspect of particular interest is the active control of such patterns. In nonlinear optical systems this can be used for all-optical switching with transistor-like performance, for example realized with polaritons in a planar quantum-well semiconductor microcavity. Here we focus on a specific configuration which takes advantage of the intricate polarization dependencies in the interacting optically driven polariton system. Besides detailed numerical simulations of the coupled light-field exciton dynamics, in the present paper we focus on the derivation of a simplified population competition model giving detailed insight into the underlying mechanisms from a nonlinear dynamical systems perspective. We show that such a model takes the form of a generalized Lotka-Volterra system for two competing populations explicitly including a source term that enables external control. We present a comprehensive analysis both of the existence and stability of stationary states in the parameter space spanned by spatial anisotropy and external control strength. We also construct phase boundaries in non-trivial regions and characterize emerging bifurcations. The population competition model reproduces all key features of the switching observed in full numerical simulations of the rather complex semiconductor system and at the same time is simple enough for a fully analytical understanding of the system dynamics. arXiv:1903.12534v1 [cond-mat.other]
I. INTRODUCTION
The demand for integrated optoelectronic devices in optical communication networks has resulted in an increase of research activities targeted at functional alloptical components. For example, a wide range of different approaches has been proposed to realize efficient all-optical switches exploiting the nonlinear optical properties of different material systems, including organic photonic crystals [1], rubidium atomic-vapor cells [2], or GaAs semiconductor microcavities as in Refs. [3][4][5] and Refs. [6][7][8]. The latter utilize the optical control of transverse optical patterns to achieve transistor-like switching performance [2,9]. Spontaneous formation of spatially extended patterns has been intensively studied during the past decades [10] with applications to different areas of science including formation of sand ripples and desert sand dunes [11], animal coat patterns such as zebra stripes [12,13], or geographic patterns in parasitic insect populations [14]. With applications to optical switching, however, the quest for efficient external control of these patterns naturally arises, but in many cases has not been explored in detail. In the present work we investigate a specific example of all-optical switching of polariton patterns in a semiconductor quantum-well microcavity system. In contrast to our previous work [9] we give a detailed analysis of the optical switching dynamics from a nonlinear dynamical system's perspective. To this end we derive a simplified mode competition model that governs the essentials of the system dynamics but is simple enough to fully characterize the possible stationary solutions and phase-space singularities in the relevant parameter space. This simplified model has the very gen-eral mathematical form of a generalized Lotka-Volterra system, with the addition of an inhomogeneity for external control. The solution space we obtain in dependence of spatial anisotropy and external control strength is of very general nature and may be similarly realized in other systems where external control of population competition is studied such as chemical reactions or in the life sciences.
As our specific example, here we study planar semiconductor microcavities with a strong coupling between the cavity field and the exciton polarization that gives rise to the formation of exciton polaritons [15]. These quasiparticles consist of a photonic and an excitonic part and are characterized by a normal-mode splitting in the dispersion relation, on the lower branch leading to long coherence times and strong nonlinear interactions. The latter are driven by four-wave mixing processes which can also be interpreted as polariton scattering. For certain excitation conditions, small spatially varying density fluctuations can experience huge growth in particular modes due to the intrinsic feedback mechanisms driven by four-wave mixing. This causes spatially homogeneous density distributions to become unstable such that the system's symmetry is spontaneously broken. This results in the formation of stationary patterns, directly observable in the far field emission from the microcavity. Figure 1(a) shows the excitation geometry used with the pump at normal incidence (zero in-plane momentum) and finite off-axis (k = 0) signals due to amplified fluctuations. Figure 1(b) shows the normal-mode splitting of the dispersion relation into a lower (LPB) and upper polariton branch (UPB) alongside the bare exciton and cavity dispersions. Phase-matching conditions determine 1. (a) Sketch of a planar quantum-well semiconductor microcavity with continuous-wave pump at normal incidence (on-axis) and optional off-axis control beam. (b) Corresponding polariton dispersion relation in the normal-mode splitting regime. The dominant four-wave mixing process is indicated, driving the pairwise scattering of pump-induced polaritons onto the lower polariton branch with opposite in-plane momenta for signal and idler modes.
the efficiencies of the different scattering processes, and therefore determine the possible pattern geometry. For scalar polariton fields hexagon patterns are favored [16][17][18]. Extending the model with polarization dependence a complex interplay of the TE/TM cavity-mode splitting in the linear regime and the spin-dependent excitonexciton interactions in the nonlinear regime arises. The resulting polarization-induced spatial anisotropy [19] determines the possible unstable modes and stable patterns formed. For linearly polarized pump excitation slightly above instability threshold, cross-linearly polarized resonant modes parallel and perpendicular to the pump's polarization plane constitute the basic instabilities [20], resulting in two-spot or four-spot patterns. Making use of this spatially anisotropic polariton scattering a reversible optical switching for orthogonal two-spot patterns can be realized that is triggered by a weak external control beam [9].
II. ORTHOGONAL SWITCHING OF TWO-SPOT PATTERNS
A detailed numerical investigation of the present system for the following excitation conditions was already discussed in Ref. [9]. The system is excited (driven) by an x-linearly polarized continuous wave pump at normal incidence with Gaussian shape in the QW plane and an intensity slightly above the off-axis instability threshold. In this case spontaneous breaking of spatial and polarization symmetry is observed. y-linearly polarized signals are formed by resonant polariton scattering onto the elastic circle defined by the polariton dispersion (cf. Fig. 1). As can also be understood based on a linear stability analysis [20,21], the scattering occurs predimonantly in four spatial directions oriented orthogonal and parallel to the pump polarization plane, respectively. In general, in any spatial direction signals can form either in the TE or in the TM mode as illustrated in Fig. 2. However, for the pump polaritons scattered to finite FIG. 2. Schematic illustration of the dominant contributions to the in-plane scattering of polaritons onto the elastic circles defined by the TE and TM modes of the lower polariton branch. The pump indicated in the center is x-polarized and slightly above off-axis instability threshold. Parallel and perpendicular to the pump polarization direction, the scattering can occur in either TE or TM mode (left panel). For pumping spectrally well below the exciton resonance, the scattering with polarization orthogonal to the pump's is preferred (center panel). With spatial anisotropy (which is partly induced by the linear pump polarization) scattering along the pump's polarization direction onto the TE mode can dominate (right panel).
k through Coulomb interaction, due to the underlying spin-dependent exciton-exciton interactions, the scattering probability is higher for a polarization state orthogonal to the pump polarization state [21]. Therefore, in the spatial direction parallel to the pump polarization state, the scattered signals preferentially form in the TE mode, and in the TM mode for scattering orthogonal to the pump polarization state (cf. Fig. 2). Out of these four signals, a stationary two-spot pattern (only one mode pair with opposite in-plane momenta) can be prepared by introducing some anisotropy favouring one direction over the other. Alongside a slight polarization induced spatial anisotropy that is due to a slightly higher density of states for the TE modes [20], an additional anisotropy can be introduced by tilting the pump beam slightly away from normal incidence (cf. Fig. 2). In the nonlinear system studied, at sufficiently high densities of the favored two-spot pattern, cross-saturation processes will lead to the extinction of the signals in the other orthogonal direction. This allows us to single out a two-spot pattern T 1 which will serve as the initial state in the switching process. Here, we start with the two-spot pattern oriented in the x-direction (direction of intrinsic anisotropy). This selection process is schematically visualized in Fig. 2.
If now a weak (compared to the pump intensity) ypolarized control beam with an in-plane momentum on the elastic circle and spatially orthogonal to the initial pattern is applied, stimulated scattering leads to population revival of the corresponding two-spot pattern. Due to the cross-saturation effect, the initial pattern is destabilized and finally switched off completely while the orthogonal pattern reaches a steady state T 2 (cf. Fig. 3). After the control beam has been switched off again, the anisotropy leads to re-emergence of the initial T 1 -pattern while the T 2 -state vanishes again. If the control beam is too weak so that complete switching may not be possible, the system will remain in a stationary four-spot pattern state F. Based on this scheme, in Ref. [9] transistor- like reversible switching was demonstrated including a systematic study of switching times, minimum control power needed, and achievable gain. This was done by numerical simulations of the nonlinear set of equations of motion governing the coherent coupled light field and exciton dynamics in the microcavity system in the twodimensional QW plane in real space,
i ∂ t E ± =(H c − iγ c )E ± + H ± E ∓ − Ωp ± + E ± pump i ∂ t p ± =( e 0 − iγ e )p ± − Ω(1 − α psf |p ± | 2 )E ± + T ++ |p ± | 2 p ± + T +− |p ∓ | 2 p ± .(1)
Here, the index ± denotes the different components in the circular polarization basis,
H c = ω 0 − 2 4 ( 1 mTM + 1 mTE )(∂ 2 x +∂ 2 y )
is the cavity Hamiltonian and e 0 the flat exciton dispersion. γ c and γ e represent the photon and exciton loss rates, Ω describes the photon-exciton coupling, and H ± =− 2 4 ( 1 mTM − 1 mTE )(∂ x ∓i∂ y ) 2 couples the two polarization components due to TE/TM-splitting. The cubic nonlinearities consist of a phase-space filling term α psf , repulsive interaction T ++ for excitons with parallel spins, and attractive interaction T +− for excitons with opposite spins.
Based on Eqs. (1) we perform a numerical simulation to demonstrate the basic switching process in the twodimensional plane of in-plane momenta. Figure 3 shows the photonic component, |E| 2 , in k-space without control beam such that the stationary pattern T 1 forms in panel (a) and with the control beam on, switching to the stationary pattern T 2 in panel (b). System parameters used and details of the calculations are given in Appendix A. Upon switching off the control beam the switching action is reversed and the pattern returns to its original state T 1 . Switching times in the range of ∼100 ps are achievable [9].
The focus of the present paper is not on the full numerical simulations of Eqs. (1) and resulting switching per-formance. Rather, we will derive a simplified population competition model for selected modes in k-space (details of the derivation are given in Appendix B) to provide further insight into the underlying phase space singularities that dictate the global behaviour and solution space of the nonlinear dynamical system studied. In a similar fashion this approach was previously applied to the switching between subsets of a hexagonal pattern [22]. We will systematically analyze the existence and stability properties of possible steady states in dependence of the strength of the different involved physical processes for a typical orthogonal switching setup comparable to the one introduced above. To this end, we will construct phase boundaries in representative regions of parameter space and characterize relevant bifurcations. This will lead to a general understanding of crucial parameter dependencies such as the ratio between the control beam strength and the anisotropy, and also of the coexistence of solutions in certain parameter regions and hysteresis behaviour. Based on the simplified model, we will also be able to show that the polariton dynamics and optical switching phenomena discussed in the present paper can mathematically be understood based on an generalized Lotka-Volterra model including an external control parameter.
III. POPULATION COMPETITION MODEL
The simplified population competition model discussed in the remainder of the present paper can be derived from the set of equations of motion in Eq. (1) as detailed in Appendix B. It reads
∂ t A 1 = α 1 A 1 − β 1 A 3 1 − θ 1 A 2 2 A 1 ∂ t A 2 = α 2 A 2 − β 2 A 3 2 − θ 2 A 2 1 A 2 + S.(2)
Here |A 1 | 2 and |A 2 | 2 denote the populations of the two elementary states of the system, i.e., the two orthogonal mode pairs in k-space. The six dimensionless realvalued positive parameters, α i , β i , θ i are directly related to the main (up to third order) scattering processes of the system as illustrated in Fig. 4. They can be calculated from the physical parameters of the full model (see Appendix C). These parameters are intrinsically different for A 1 and A 2 due to the polarization dependence and anisotropy. The linear process representing growth of the resonant modes is described by α i . Saturation processes are represented by the cubic terms which can be divided into self-saturation (β i ) and cross-saturation (θ i ). The external control is described by the inhomogeneity S. If we rewrite Eqs. R 2 ≥0 , i.e. the first quadrant of the (A 1 , A 2 )-plane. The linear term (α) leads to exponential growth of the corresponding mode pair and the self-saturation (β) has a stabilizing effect. For these two processes, the equations (2) are automatically decoupled and may be solved separately, resulting in a stable four-spot pattern. However, the cross-saturation terms (θ) couple the two modes and tend to suppress a particular mode pair, favoring the other mode pair, resulting in a two-spot pattern. Additionally, the external control acts as a source term for A 2 . The PC model thus describes the dynamical competition between the three types of possible stationary patterns: T 1 , T 2 and F. A phase in the PC model is defined as a set consisting of the number of steady states and their stability properties. These sets, i.e. the phase, are functions of the seven parameters and a phase boundary in parameter space indicates the change in the number of steady states and/or their stability. This can only happen at points where at least one eigenvalue of the corresponding Jacobian matrix J≡ ∂ Aj f i is zero [23], which is equivalent to the condition det J=0. We thus find phase boundaries in parameter space at points satisfying {f , det J}=0.
(2) as (∂ t A 1 , ∂ t A 2 ) T ≡ (f 1 , f 2 ) T = f ,
A. Homogeneous Case S = 0
The homogeneous case of Eq. (2) (S = 0) has the form of a generalized Lotka-Volterra (GLV) model [24] with cubic nonlinearities. The transformation A i → A 2 i ≡à i yields the usual Lotka-Volterra (LV) equations [25] with quadratic nonlinearities while conserving the (A 1 , A 2 ) phase space structure in the positive quadrant [26]. Hence, for S = 0 the phase portraits of (2) are topologically equivalent to those of the following well-known system
∂ tÃi =Ã i r i − 2 j=1 c ijÃj (3)
with i = 1, 2, growth rate vector r = 2(α 1 , α 2 ), and community matrix
C = 2 β 1 θ 1 θ 2 β 2 ,(4)
which describes self-and cross-interaction. This LV model (3) for interspecific competition has been studied in many different contexts, e.g. ecology [27], chemistry [refs], economics [28], physics [29] and it was shown [25,30] that its dynamical behavior is limited to three cases depending on the parameters: i) Coexistence regime for sgn(det C)=+1 (larger self-saturation), ii) bistability regime for sgn(det C)=−1 (larger cross-saturation), iii) dominance regime for sufficiently large/small growth rate ratios. In this regime long-time dynamics are independent of the interaction parameters and always result in extinction of one population. Although the LV system, Eq. (3), is well-known, we present a short discussion of the homogeneous case of the PC model, Eq. (2), and point out the importance for application to the polariton switching dynamics. The cubic nonlinearities imply an additional Z 2 × Z 2 symmetry leading to different bifurcations in comparison to the usual LV model, Eq. (3). Since for S = 0 the system is solvable analytically we obtain explicit expressions for all steady states and phase boundaries: i) T 1 :
A 1 = α1 β1 , A 2 =0, stable for α1 α2 > β1 θ2 , ii) T 2 : A 1 =0, A 2 = α2 β2 , stable for α1 α2 < θ1 β2 , iii) F: A 1 = α2θ1−α1β2 θ1θ2−β1β2 , A 2 = α1θ2−α2β1 θ1θ2−β1β2 , only exists for β1 θ2 ≶ α1 α2 ≶ θ1
β2 and is stable for θ 1 θ 2 <β 1 β 2 , iv) trivial solution: A 1 =A 2 =0. We already see that the four-spot solution does not exist in the entire S=0 plane in contrast to the two-spot pattern. The trivial solution also exists everywhere but is always unstable since the eigenvalues of its Jacobian α 1 , α 2 are positive. This solution will not be listed hereafter. For a systematic discussion we introduce a variable anisotropy parameter δα in the first equation of (2), α 1 → α 1 + δα, and set α 1 =α 2 =1. It favors A 1/2 for δα ≷ 0, respectively. The resulting phase boundaries are shown in Fig. 5 where the homogeneous case (S=0) is included in the extended region at the bottom of each plot. It shows the structure of the usual LV model. Green (red) letters mark stable (unstable) steady states. For large anisotropy |δα| only stable and unstable two-spot patterns are possible. This dominance regime does no longer depend on the interaction parameters β i and θ i . The middle region on the other hand is divided into two cases: A stable four-spot pattern for larger self-saturation (β 1 β 2 >θ 1 θ 2 , coexistence, upper panel) and two simultaneously stable two-spot patterns for larger cross-saturation (β 1 β 2 <θ 1 θ 2 , bistability, lower panel). The first four cases in Fig. 6 show the corresponding flow given by respresentative trajectories in state space for the homogeneous case. The system's dynamics are unambiguous in cases 1, 2, and 3a where only one attractor exists which determines the long time behavior. However, in case 3b two attractors exist and the dynamics now depend on the system's history leading to a hysteresis effect. The two basins of attraction are marked with different colors blue and orange. They are separated by the stable manifold of the saddle point F (often called separatrix ), defined by the set of points (A 1 , A 2 ) which satisfy (A 1 , A 2 ) → F for t → ∞. In contrast to this, the line connecting all three steady states is the unstable manifold of F, consisting of points which satisfy (A 1 , A 2 ) → F for t → −∞. Any initial point will stay in its region (blue or orange) and end up at the corresponding attractor (T 1 or T 2 ). The system thus shows hysteresis behavior which might prevent complete backswitching, and therefore should be avoided for switching purposes, for example by increasing the anisotropy.
The Hartman Grobman theorem [23] ensures that the behavior near hyperbolic equilibrium points is completely determined by its linearization. This no longer holds for non-hyperbolic fixed points which are characterized by the existence of at least one zero real-part eigenvalue of the corresponding Jacobian. Therefore, the behavior at the phase boundaries in parameter space can not be analyzed via linear stability analysis. Instead one can use center manifold theory and normal forms [23] in order to determine the occurring bifurcations. They are transcritical for the usual LV model but different in the case of cubic nonlinearities due to the additional Z 2 × Z 2 symmetry. For both two-spot solutions symmetric pitch-fork bifurcations occur at the phase boundaries when the anisotropy parameter is changed. They are supercritical for β 1 β 2 >θ 1 θ 2 leading to the coexistence regime with a stable F pattern and subcritical for β 1 β 2 <θ 1 θ 2 leading to the bistability regime with stable T 1 and T 2 pattern. Pitchfork bifurcations are typical for dynamical systems with inversion symmetry, here A i → −A i . Since we are only interested in positive solutions in the first quadrant of the phase plane, we either observe a transition from a stable T solution to an unstable T and a stable F solution (supercritical) or the same transition with reversed stability (subcritical).
In conclusion, all phase portraits of the homogeneous case can be completely reduced to the results of the wellknown Lotka-Volterra model for two competitive species. Due to the additional symmetry we obtain pitchfork bifurcations at the phase boundaries. The homogeneous case is crucial for the polariton switch because it describes the initial pattern formation and back-switching process in absence of the control beam. Both can only work reliably in regions where only the T 1 pattern exists as a single attractor. Therefore, a sufficient minimum anisotropy in favor of A 1 is needed. We have also seen that if the cross-saturation dominates over the selfsaturation, the coexistence of A 1 and A 2 is destabilized resulting in the extinction of A i and survival of A i+1 . A strong interspecific competition thus prevents the coexistence. Otherwise, if the self-saturation is stronger than the cross-saturation, the dominating intraspecific competition promotes coexistence.
B. Inhomogeneous Case S > 0
The inhomogeneous (S > 0) case of the PC model (2) describes the actual switching process induced by an external control beam. This term can also be motivated for other systems where the GLV model is commonly used, e.g. to include constant migration/harvesting in the description of ecological systems or constant influx in chemical reactions. Therefore, it can be interpreted as an extension of the generalized Lotka-Volterra dynamics and the following analysis is of very general interest but has not been investigated before. The influence of constant terms in the usual LV model with quadratic nonlinearities was investigated in Ref. [31] from a purely mathematical perspective, but in general inhomogeneous population competition models did not receive much attention in the past. Here, we analyze and apply the GLV model with a constant inhomogeneity to the case of the orthogonal switching of two-spot polariton patterns.
Also including inhomogeneity, all steady states and phase boundaries can still be determined analytically due to the system's simplicity. However, we note that in general solving multivariate polynomial equation systems is a difficult task which can be simplified using algebraic methods (e.g. Gröbner Basis [32,33]). The non-zero source term in (2) breaks the inversion symmetry for A 2 and prevents a T 1 solution. This leaves us with only two competing patterns (called T 2 & F 2 ) this time around. However, as we will see below it includes the possibility of an additional qualitatively different four-spot pattern solution F 1 . The control parameter S acts as a constant source term for the A 2 pattern, leading to linear growth. Supporting A 2 means simultaneously suppressing A 1 due to the cross-saturation effect, leading to a more asymmetric F 2 solution in favor of A 2 . A second four-spot pattern, F 1 , which is in favor of A 1 replaces T 1 . The phase boundaries are given explicitly by
S * 1 (δα) = β 2 1 + δα θ 1 3 − 1 + δα θ 1 ,(5)
and additionally, for the case of larger cross-saturation (β 1 β 2 <θ 1 θ 2 )
S * 2 (δα) = 6 3 [(1 + δα)θ 2 − β 1 ] 3 β 2 1 (θ 1 θ 2 − β 1 β 2 ) .(6)
The explicit expression for the steady states T 2 , F 1 , and F 2 are not given here due to their excessive length. The phase boundaries in the (δα, S) parameter space can be seen in Fig. 5 for both the case of higher self-and higher cross-saturation. Starting from the threshold values δα * 1,2 of the homogeneous case, we find continuously changing phase boundaries (S * 1,2 ) to higher anisotropy values for increasing control S. In the case of higher selfsaturation there is only one phase boundary S * 1 due to the absence of a T 1 state. For sufficiently strong control S there is only the stable steady state T 2 . This region represents the desired outcome of a successful switching process. Its phase boundary determines the minimum control strength required to achieve switching for a given anisotropy value. For lower values of S, a stable fourspot pattern arises and depending on the ratio θ1θ2 β1β2 a second unstable four-spot pattern occurs together with a stability transition of T 2 .
Again, we can draw the flow in the (A 1 , A 2 ) phase plane and mark the basins of attraction as shown in Fig. 6, cases 4, 5, and 6. For θ 1 θ 2 >β 1 β 2 a region with two attractors occurs similar to the S=0 case, but this time with a stable F 1 state instead of T 1 , which again implies hysteresis behavior. For increasing anisotropy, the unstable F 2 state will approach T 2 along the unstable manifold until they meet and an unstable T 2 and stable F 1 state emerge. This corresponds to the approach of the two phase boundaries S * 1 and S * 2 and the annihilation of S * 2 , which can only happen for parameter values 1< θ1θ2 β1β2 < 3 2 . Otherwise the two boundaries diverge. Thus, the case of higher cross saturation is divided into two subcases, defined by either the survival or the vanishing of the middle region with two attractors.
Another important effect of the inhomogeneity S is the explicit breaking of the Z 2 symmetry in the time evolution of A 2 . In the case of larger cross-saturation a cusp point arises in the two-dimensional bifurcation diagram in Fig. 5(b). The pitchfork bifurcation at S = 0 is replaced by a saddle-node bifurcation for S = 0, resulting in the creation/destruction of a stable-unstable pair of F-solutions while varying one of the parameters. In the cusp point two saddle-node phase boundaries, S * 2 and −S * 2 (not shown), meet tangentially. The other pitchfork bifurcation (δα * 1 ) remains (S * 1 ) in presence of the inhomogeneity since the Z 2 symmetry of A 1 is still conserved. This is the only bifurcation in the case of larger self-saturation in Fig. 5(a) for S = 0. Figure 7 shows the evolution of the equilibrium surfaces for A 1 and A 2 in the (δα, S) parameter space and the corresponding bifurcations. The pitchfork bifurcations are visible on the S=0 line. In the case of higher cross-saturation (lower panel) one of the pitchfork bifurcations unfolds into a saddlenode bifurcation with increasing S. In the case of higher self-saturation (upper panel) the supercritical pitchfork bifurcation remains stable with increasing S.
A 1 A 2 ‹Ş ‹Ş A 1 A 2 ‹Ş ‹Ş(
In conclusion, the main difference for the inhomogeneous case is the absence of a steady T 1 state and the existence of an additional stable F 1 state in the case of higher cross-saturation. This corresponds to the unfolding of one of the subcritical pitchfork bifurcations into a saddle-node bifurcation. The control term S thus prevents extinction of A 2 and promotes coexistence respectively. Furthermore, the F 1 -T 2 -bistability region van-ishes for high anisotropy values, if the parameter condition 1< θ1θ2 β1β2 < 3 2 is satisfied, resulting in a single remaining phase boundary S * 1 similar to the case of higher selfsaturation. The occurring saddle-node and subcritical pitchfork bifurcations are problematic since they both imply sudden vanishing of a stable fixed point, meaning the system undergoes an abrupt transition to another stable fixed point, i.e. hysteresis is possible. This can only happen in the bistability regime (larger crosssaturation). The numerical simulation of the switching process presented in section II takes places in the larger self-saturation regime (model parameters calculated in Appendix C), which is advantageous for the switching purpose, since no hysteresis can occur.
C. Remarks
So far, we have discussed steady states, their stability properties, and bifurcations occurring in the solution space of the population competition model in Eq. (2). From a dynamical perspective, we observe critical slowing down [34,35] near the phase boundaries due to the continuously vanishing real part of the Jacobian's eigenvalue responsible for the bifurcation. This corresponds to the divergence of switching times observed in the numerical simulations in Ref. [9]. For example, approaching S * 1,2 for β 1 β 2 ≷θ 1 θ 2 from above results in divergence of the switching time. Similarly, approaching δα * 2,1 for β 1 β 2 ≷θ 1 θ 2 from the left side results in divergence of the back-switching time. Hence, to achieve favourable performance, switching should be done for parameters sufficiently far away from the phase boundaries.
We note that in general a nonlinear dynamical system can have periodic solutions which are characterized by closed orbits in the phase plane. Here, we use Dulac's Criterion [36, p. 202] to rule out any periodic solutions. A simplified version reads: The existence of a function g(A 1 , A 2 ) with the property that ∇·(gf ) is sign definite in the entire considered state space, rules out any closed orbits in this area. If we choose g= 1 A1A2 , we obtain ∇ · (gf )= − 2(β 1
A1 A2 + β 2 A2 A1 + S A1A 2 2 )
, and therefore closed orbits in the positive quadrant are impossible. Another observation is that for θ 1 =θ 2 ≡θ (symmetric coupling) we can write the system in Eq. (2) as a gradient field f =∇V with the potential function
V = 2 i=1 α i 2 A 2 i − β i 4 A 4 i − θ 4 A 2 i A 2 i+1 + S i A i(7)
with A i+2 = A i , S 1 = 0, and S 2 = S. Closed orbits are impossible in gradient systems [36, p. 199], however, for the general case θ 1 =θ 2 (asymmetric coupling) this argument is no longer applicable. Also, assuming the gradient system defined by the potential function (7) allows us to use the language of catastrophe theory and observe that two of Thom's seven elementary catastrophies occur, namely folds (A 2 in Arnold's notation) and cusps (A 3 ). We further note that our detailed investigation above is limited to destabilizing linearity, stabilizing nonlinearities, and positive control parameter. These conditions match the numerical and experimental observations for the physical system under investigation here. For this case we are able to completely characterize all possible steady states, their corresponding bifurcations, and rule out any other bifurcations involving double zero and pure imaginary eigenvalues of the Jacobian.
IV. CONCLUSIONS
We have presented a detailed analysis of an all-optical switching concept based on transverse patterns in an interacting polariton system in a planar quantum-well semiconductor microcavity. From the relevant equations based on a microscopic semiconductor theory here we derived a simplified population competition (PC) model describing the system dynamics restricted to selected modes in k-space. Interestingly, the resulting rather simple PC model shows all key features of the system dynamics also observed in the full numerical simulations in the parameter range of interest here. In addition to what can be learned from the numerical simulations, the PC model enables us to systematically identify phase boundaries in parameter space and singularities governing the global dynamics of the nonlinear system. Interestingly the rather complicated original system of interacting microcavity polaritons, for the switching phenomenon studied here can be completely characterized by only seven flow portraits in a two-dimensional state space. The model derived has the very generic form of a generalized Lotka-Volterra (GLV) system extended with an inhomogeneity term to achieve external control. Such a system has not been investigated before. Considering the widespread use of GLV systems the understanding obtained in the present work is of very general nature and will be similarly applicable also to other fields where external control of population competition is studied such as chemical reactions or population competition in life-sciences.
V. ACKNOWLEDGMENTS
We greatfully acknowledge funding from the Deutsche Forschungsgemeinschaft through project SCHU 1980/5-2 and through the Heisenberg program (No. 270619725). We further acknowledge valuable discussions with Rolf Binder and a grant for computing time at the Paderborn Center for Parallel Computing (PC 2 ).
Appendix A: Numerical Details
We used the following parameters (appropriate for GaAs systems) to numerically simulate the switching presented in section II: mTE=1.05 · mTM=0.215 meVps 2 µm −2 , Ω=6.5 meV, γc=0.8 meV, γe=0.2 meV, α psf =5.188·10 −4 µm 2 ; T ++ = − 5T +− =5.69 · 10 −3 meVµm 2 , e 0 = 1.497 eV. Coherent excitation 2.5 meV above the lower polariton branch was used with flat top Gaussian profiles with intensities Ipump = 2.1 · 10 5 · I probe = 54 kWcm −2 . The equations (1) were solved on a finite 2D grid in real space using a 4th order Runge-Kutta method with variable step size. In section III we used parameters β1β2/θ1θ2 = 1.14 for the case of larger self-saturation and β1β2/θ1θ2 = 0.69 for the case of larger cross-saturation.
All bifurcations were determined analytically for the homogeneous case and numerically with the help of MATCONT, a matlab software package, for the inhomogeneous case.
Appendix B: Derivation of the PC Model
We follow qualitatively the derivation of the hexagon PC model [22], but here we are including polarization effects for linearly polarized excitation, and therefore considering a different reduced k-space. We start with the coupled equations of motion (1) for the exciton polarization and the cavity field and transform them into k-space and in the linear polarization basis. Introducing the reduced k-space consisting of modes {k0≡0, k1, k2, k3, k4} with relations k3= − k1 and k4= − k2 (see Figure 8) results in 20 equations. We consider an x- polarized pump. This leads to the following selection rules [19]:
• x y -polarized probe with k ex excites TM TE -mode • x y -polarized probe with k ⊥ ex excites TE TM -mode We assume that all dynamical quantities oscillate with pump frequency ω. Removing the phase factor e −iωt yields a shift of the dispersion by − ω. We consider all phase-matched scattering processes uo to third order within the reduced kspace. In agreement with the linear stability analysis of the corresponding system reporting a D2 ∼ = Z2 × Z2 symmetry [20], we assume an equal excitation of opposite modes in the reduced k-space, i.e. p k 1 =p k 3 ≡p1 and p k 2 =p k 4 ≡p2 (analogously for E). We further assume the pump induced densities E
x/y 0 and p x/y 0 at k = 0 to be constant. Using a cyclic definition for the two mode pairs pj=pj+2 with j=1, 2 results in the following phase-matched (q=k − k − k ) scattering processes (q, k , k ) for each mode pair in Eq. (1):
(j, 0, 0), (0, j, 0), (0, 0, j), 3×(j, j, j), 2×(j, j+1, j+1). (B1)
In our setup the pump is x-polarized. Therefore, terms ∝ p y 0 or ∝ E y 0 are omitted. Furthermore, we assume the co-linearly polarized off-axis modes to be very small as for these modes the instability threshold is not reached, i.e. p x j p y j and E x j E y j . Hence, we also neglect terms ∝ p x j or ∝ E x j . This leaves us with the following four equations for the y-polarized mode pairs:
i ∂tp y j = ( j − ω − iγe)p y j + 1 2 αPSFΩ[−p y * j p x 0 E x 0 + p x * 0 p y j E x 0 + |p x 0 | 2 E y j + 3p y * j p y j E y j + 2p y * j p y j+1 E y j+1 ] + 1 2 (T ++ + T +− )[3p y * j p y j p y j + 2p y * j p y2 j+1 ] − 1 2 (T ++ − T +− )[p y * j p x2 0 ] +T ++ |p x 0 | 2 p y j −ΩE y j (B2)
i ∂tE y j = ( ω y j − ω − iγc)E y j − Ωp y j + E y pump,j .
If we further assume that the time evolution of E follows adiabatically the evolution of p and set ∂tEj ≈ 0, we can write
Ej = Ω ωj − δj − ω − iγc pj ≡ 2λj αPSFΩ e iθ j pj(B4)
with j=0, 1, 2. The pump field was also set to zero Epump,j=0 and will be added later manually. We define θj as the phase between E y j and p y j and the ratio of their amplitudes is given by 2λ j α PSF Ω . Here, δj includes anisotropy effects due to higher density of states for the TE modes and tilting of the pump.
So the parameters are given by
λje iθ j = α PSF Ω 2 2( ω y j −δ j − ω−iγc)
for j = 1, 2 (B5)
λ0e iθ 0 = α PSF Ω 2 2( ω x 0 − ω−iγc)
for j = 0.
Finally, we obtain two equations for the two elementary states of the system. We also factorize the exciton field into phase and magnitude, i.e. p y j =p y j e iϕ j , and split the results into separate equations of motion for magnitude and phase:
If we define φj≡ϕj − ϕ0, the coefficients are then given by Lj = − γe + λ0p x2 o (sin(θ0) − sin(θ0 − 2φj)) + λjsin(θj)(p x2 0 − 2 α PSF ) − 1 2 (T ++ − T +− )p x2 0 sin(−2φj) C jk = 3λjsin(θj) , j = k 2λ k sin(θ k + 2φ k − 2φj) + (T ++ + T +− )sin(2φ k − 2φj)
, j = k
Kj = j − ω − T ++px2 0 (cos(θ0) − cos(θ0) − 2φj) − λjcos(θj)(p x2 0 − 2 α PSF ) − 1 2 (T ++ − T +− )p x2 0 cos(−2φj) D jk = −3λjcos(θj) − 3 2 (T ++ + T +− ) , j = k −2λ k cos(θ k + 2φ k − 2φj) − (T ++ + T +− )cos(2φ k − 2φj) , j = k .(B9)
Analogously to Ref. [22], we remove the phases as dynamical variables by assuming locked phases and linearization. We define the time-dependent phase as φj(t)≡δφj(t) + φ
k ) − D jk (φ (0) j ,φ (0) k )L j (φ (0) j ) K j (φ (0) j ) p y2 k p y j .(0)
We rewrite these two equations in a shorter form:
∂tp y 1 =α1p y 1 −β1p y3 1 −θ1p y2 2p y 1 ∂tp y 2 =α2p y 2 −β2p y3 2 −θ2p y2 1p y 2 (B11)
We replacep y j with a product of a dimensionless quantityp y j and a characteristic quantityp y j,c which carries the original dimension, i.e.p y j =p y j ·p y j,c . We do the same for the independent time variable t=t · tc, so that the derivative changes to The characteristic values are chosen in a way that the corresponding dimensionless quantities are of magnitude 1. With the definitionst ≡ t andp y j ≡ Aj, we can finally write down the population competition model as
∂tA1 = α1A1 − β1A 3 1 − θ1A 2 2 A1 ∂tA2 = α2A2 − β2A 3 2 − θ2A 2 1 A2 + S (B12)
where we have manually added a control parameter S for the A2 mode pair.
FIG.
FIG. 1. (a) Sketch of a planar quantum-well semiconductor microcavity with continuous-wave pump at normal incidence (on-axis) and optional off-axis control beam. (b) Corresponding polariton dispersion relation in the normal-mode splitting regime. The dominant four-wave mixing process is indicated, driving the pairwise scattering of pump-induced polaritons onto the lower polariton branch with opposite in-plane momenta for signal and idler modes.
FIG. 3 .
3(a) Stationary T1 and (b) T2 state of the polariton pattern switch as obtained from the numerical solution of Eq. (1). Shown is the stationary y-polarization component of |E| 2 in k-space in arbitrary units (a) before and (b) after switching on a continuous wave control beam. Switching off the control beam results in reversal of the switching process such that after sufficiently long times the system returns to the stationary T1-pattern.
FIG. 4 .
4Illustration of the different polariton scattering processes in k-space for the two elemetary mode-pairs A1 and A2 in Eq.(2). Green arrows mark incoming and red outgoing modes.
FIG. 5 .
5Phase boundaries in (δα, S) parameter space showing regions of different stable and unstable steady states in dependence of the anisotropy and external control. (a) Larger self-saturation β1β2>θ1θ2. (b) Larger cross-saturation β1β2<θ1θ2.
FIG. 6 .
6Phase plane flow for cases corresponding to Fig. 5. Black (white) dots mark stable (unstable) steady states. Black lines show representative orbits. The thick black line in 3b and 6 is the stable manifold of the saddle point which separates two basins of attraction.
FIG. 7 .
7Evolution of steady states for A1 and A2 in parameter space. Green/red surfaces belong to stable/unstable solutions. Black lines corresond to the phase boundaries projected on the (δα, S) plane (a) Larger self-saturation: one of the two pitchfork bifurcations on the S=0 line remains for S>0, whereas the other one vanishes completely. (b) Larger cross-saturation: one of the two pitchfork bifurcations on the S=0 line remains for S>0, the other one is replaced by a saddle-node bifurcation.
FIG. 8 .
8Definition of the reduced k-space.
=0 and δφj(t) is a small deviation. Expanding equations (B7) and (B8) up to first order in δφj around φ (0) j and neglecting terms ∝ δφjp y2 k leads to an explicit expression for δφj which can be substituted back to obtain ∂p y j = Lj(φ
steady states are characterized by f =0. Four qualitatively different stationary solutions are possible: i) twospot pattern T 1 with A 1 =0 and A 2 =0, ii) two-spot pattern T 2 with A 1 =0 and A 2 =0, iii) four-spot pattern F with A 1 =0 and A 2 =0, and iv) the trivial solution with A 1 =0 and A 2 =0. Here we are only interested in physical solutions A i ≥ 0, and therefore the state space is
Appendix C: Calculation of Model ParametersThe model paramaters can be calculated from the physical parameters via equation (B10). Their values, especially the signs, depend on the specific choice of the locked phases. The system's physical behavior observed in the full numerical simulations suggest destabilizing linear terms and stabilizing nonlinear terms in the PC model. Chosing phases satisfying this condition, characteristic valuesp y j,c =1 µm −1 and tc=1 ps, and anisotropy effects δ1=0.2 meV and δ2=0 meV, leads to the following model parameters for the switching simulation presented in section II: α1=0.49, α2=0.43, β1=0.007, β2=0.01, θ1=0.006, θ2=0.005. This corresponds to the case of larger self-saturation. The anisotropy is sufficiently high for the initial pattern-formation and back-switching, and also no hysteresis can occur.
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[
"SUBMITTED TO IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 1 ECG Identification under Exercise and Rest Situations via Various Learning Methods",
"SUBMITTED TO IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 1 ECG Identification under Exercise and Rest Situations via Various Learning Methods"
]
| [
"Zihan Wang ",
"Yaoguang Li ",
"Senior Member, IEEEWei Cui "
]
| []
| []
| As the advancement of information security, human recognition as its core technology, has absorbed an increasing amount of attention in the past few years. A myriad of biometric features including fingerprint, face, iris, have been applied to security systems, which are occasionally considered vulnerable to forgery and spoofing attacks. Due to the difficulty of being fabricated, electrocardiogram (ECG) has attracted much attention. Though many works have shown the excellent human identification provided by ECG, most current ECG human identification (ECGID) researches only focus on rest situation. In this manuscript, we overcome the oversimplification of previous researches and evaluate the performance under both exercise and rest situations, especially the influence of exercise on ECGID. By applying various existing learning methods to our ECG dataset, we find that current methods which can well support the identification of individuals under rests, do not suffice to present satisfying ECGID performance under exercise situations, therefore exposing the deficiency of existing ECG identification methods. | null | [
"https://arxiv.org/pdf/1905.04442v1.pdf"
]
| 152,282,902 | 1905.04442 | ad9099dd1d335c8be6bb6b05aade4e898d83fee5 |
SUBMITTED TO IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 1 ECG Identification under Exercise and Rest Situations via Various Learning Methods
Zihan Wang
Yaoguang Li
Senior Member, IEEEWei Cui
SUBMITTED TO IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 1 ECG Identification under Exercise and Rest Situations via Various Learning Methods
As the advancement of information security, human recognition as its core technology, has absorbed an increasing amount of attention in the past few years. A myriad of biometric features including fingerprint, face, iris, have been applied to security systems, which are occasionally considered vulnerable to forgery and spoofing attacks. Due to the difficulty of being fabricated, electrocardiogram (ECG) has attracted much attention. Though many works have shown the excellent human identification provided by ECG, most current ECG human identification (ECGID) researches only focus on rest situation. In this manuscript, we overcome the oversimplification of previous researches and evaluate the performance under both exercise and rest situations, especially the influence of exercise on ECGID. By applying various existing learning methods to our ECG dataset, we find that current methods which can well support the identification of individuals under rests, do not suffice to present satisfying ECGID performance under exercise situations, therefore exposing the deficiency of existing ECG identification methods.
I. INTRODUCTION
With the increasing demand for information security and reliability of protective measures, human identification has become an important research topic in corresponding field. However, traditional identification technologies such as certificates and passwords are likely to be forgotten, copied and even stolen, which may cause harm to people's personal information, therefore no longer meet their needs for security. Due to the rising awareness of people and the rapid development of science and technology, biometric identification technology has emerged, which is based on the unique anatomical, physiological or behavioural characteristics hardly possible to be forged. So far, common biometric recognition methods [1]- [6] include face, fingerprint, voice, which are mature and have high recognition rates, but not perfect enough. For example, a FaceID can be corrupted by photograph or make-up; a fingerprint can be copied and recreated with latex; a voice can be recorded or imitated as well. In order to strengthen the reliability and security of biometric identification technology, on the one hand, some scholars mix up multiple biometric technologies to make it harder for the system to be cracked. On the other hand, new reliable biometric identification techniques are being increasingly developed such as ECGID which, as one of the hottest research direction, has been developing for almost 20 years. The location, size and structure of the heart are so various from person to person that each person's ECG signal is regarded unique [7], and the difference of which provides a theoretical basis for the identification of ECG signals. Compared with the traditional biometric signal, ECG signals are bioelectrical signals generated by living bodies, making it more difficult to fake [8]- [10].
Containing abundant information about individual identity, ECG waveform has four fundamental characteristics required for biometric identification [11]: (1) Universality: the heart of every living human generates ECG signals at all times;
(2) Uniqueness: the ECG differences between individuals are mainly affected by body shape, age, weight, emotion, gender, heart location, heart size, geometric shape, physiological characteristics, chest structure and sports status, etc., so the ecg signals generated by different people are also unique [7]; (3) Stability: the structure and size of the adult heart are basically fixed. ECG waveform remains stable unless the heart has lesions; (4) Measurability: ECG acquisition equipment with miniaturization, portability and high precision reduces the cost and time of ECG acquisition and thus makes the measurement more convenient. In recent years, ECG signal has become the focus of major research in the field of human identification technology due to its good non-replicability and uniqueness. The methods of ECGID can be divided into two categories: (1) Fiducial approach: as shown in Fig. 1, ECG signal is composed by three main waves the P, the QRS and T waves [12], [13], between which the peaks, slopes, boundaries and intervals are depicted by Fiducial features. The fiducial method utilizes these features for human identification; (2) Non-fiducial approach: these methods often treats ECG signals as a whole without considering the details of waveform, and process signals in the frequency domain in most cases.
Biel et al. extract 30 time and amplitude features and use correlation matrix analysis to select 21 features to form the eigenvector [14]. It was demonstrated that soft independent modelling of class analogy method is implemented for classification. Israel et al. locates the key points by finding the local extremum of the waveform in each heart beat and extract 15 features to classify subjects by linear discriminant analysis [15]. This experiment included 29 subjects, with the rate of identification reaching 100% and the rate of heart beat recognition reaching 81%. Saechia [12].
ECG biometric recognition method requiring no waveform detection and analyzed the auto-correlation of ECGs while applying DCT to achieve dimensionality reduction [18]. Wang et al. followed the AC-DCT method and achieved satisfying results [19]. Agrafioti et al. combined the auto-correlation and LDA to make the classification [20]. Although these literatures have obtained good results, there remains some important problems that cannot be overlooked, such as robustness of ECGID, including, (1) heart disease: one was enrolled in good heart health, but tested in heart disease; (2) emotion: one was enrolled in calm, but tested in emotional state; (3) exercise: one was enrolled in rest, but tested after exercise(we call it rest-ex ECGID). Some researchers [21], [22] have explored the first question with the MIT-BIH database [23] or the PTB database [24], but the latter two issues have rarely been studied. Therefore, we would like to discuss the third issue i.e. the robustness of rest-ex ECGID in this manuscript.
From the 1970s to the 1990s, some researchers [25]- [28] attempted to dig out the changes of ECG waveform between rest and exercise, showed that during exercise the amplitude of P wave increases and the amplitude of T wave decreases, while the QRS duration remains nearly constant. After exercise, during the first minute, the amplitude of P and T wave increase and the other features start to return to normal. Then, P and T wave start to return to normal. But they only got qualitative results, not quantitative ones. Afterwards, the literature on the effects of exercise on ECGID also appears, but few good results on rest-ex ECGID have been achieved since postexercise is a period of recovery where various features change abruptly. Kim et al. [29] discussed the the effects of exercise on QRS interval, RT interval and QT interval. Furthermore, they used inverse Fourier transform to normalize features and improved rest-ex ECGID performance. Poree et al. [30] used correlation coefficient for template matching to find out the effects of exercise, waveform length and the number of leads on ECGID. Piao et al. [31] collected ECG records of only 5 normal people in rest and movement states. They used discrete wavelet transform for feature extraction, Euclidean distance and nearest neighbour for classification, with an accuracy of 100%. Sung et al. [32] collected ECG data of 55 subjects before and after exercise for 5 minutes respectively. The first and second order differential of waveform were used to help feature point detection. LDA is for feature extraction and classification, with the accuracy of identification being 59.64% within 1 minute after exercise, and more than 90% beyond 1 minute. Nobunaga et al. [33] showed that after bandpass filtering between 10 Hz and 80 Hz, the ECG waveforms in rest and post-exercise would be highly similar, and the rest-ex ECGID rate achieved 99.7% on 10 subjects. Komeili et al. [34] extracted different types of features and used feature selection algorithms to select the most stable ones in exercise.
This manuscript is aimed at applying existing methods to analyse the performance of rest-ex ECGID with our ECG database [35]. Through multiple sets of experiments, we find that for current methods, the rest-rest ECGID accuracy can reach more than 95% while the ex-ex ECGID performance become a little worse; As for the rest-ex ECGID performance, most methods collapse to 10% or even worse, except for the KL feature selection method, which can reach almost 65%. Although compared with other methods the KL feature selection method performs much better, its result still remains unsatisfactory. The rest of the paper is organized as follows. Section II introduces the main methods of the existing ECGID algorithms. In Sec. III, the process and results of the experiment with our ECGID database are described in detail. Finally, Sec. IV is the conclusion of the whole manuscript.
II. METHOD
A. Time domain features
A classical QRS detection algorithm based upon digital analyses of slopes, amplitude, and width, has been proposed by Pan and Tompkins [36], where the ideal pass band intended to maximize the QRS energy is approximately 5-15 Hz. After filtering, the signal is differentiated to provide the QRS complex slope information with the following transfer function
H(Z) = 1 8 [2 + Z −1 − Z −3 − 2Z −4 ].(1)
Then the signal is squared point by point in order to make positive all of the points and emphasize higher frequencies, given by the relation below
X = Y 2 .(2)
In order to obtain waveform feature information, a movingwindows integrator is passed through by squared signals, which is calculated from
H(Z) = 1 N [Z N −1 + Z N −2 + · · · + Z N −(N −1) + 1],(3)
where N is the number of samples in a given width of the integration window. In the last step, two thresholds are designed to make sure no peak is omitted: the first identifies peaks of the signal and when no peak has been detected by the first one in a certain time interval, the algorithm will search back promptly for the lost peak using the second threshold below the first one. Once a new peak is identified, it will be classified as a noise peak or as a signal peak if exceeding the first thresholds (or the second threshold if we search back in time for a lost peak). A peak to be identified as a QRS complex must be first identified as a QRS in both the integration and filtered waveforms by investigating the signals and trying different values of the above thresholds.
B. Short-time Fourier transform
Short-time Fourier transform (STFT) [37], [38] is one of the most commonly used time-frequency analysis method, which represents the signal characteristics at a certain moment through a segment of the signal in the time window. In the short-time Fourier transform process, the length of the window determines the time resolution and frequency resolution of the spectrum. The longer the window is, the higher the frequency resolution is and the lower the time resolution is after the Fourier transform. On the contrary, the shorter the window is, the lower the frequency resolution will be, and the higher the time resolution is. That is to say, in the STFT, the time resolution and frequency resolution cannot be achieved at the same time, and the choice should be made according to specific requirements. Due to the nonstability of ECG signal, STFT can not only transform ECG signal into frequency domain to avoid complex and difficult fiducial feature detection, but also overcome the disadvantage that the original Fourier transform cannot process non-stationary signal. Short-time Fourier transform of the continuous time signal s(t) is
ST F T (t, ω) = ∞ −∞ s(τ )γ * (τ − t)e −jωτ dτ = s(τ ), γ(τ − t)e jωτ = s(τ ), γ t,ω (τ )(4)
where γ(t) is the window function and generally γ(t) is real even function.
C. Wavelet transform
Wavelet transform(WT) [39]- [41] is a transform analysis method, which inherits and develops the idea of localization of STFT. It overcomes the disadvantages of unchangable window size with frequency, and can provide a "time-frequency" window changing with frequency. It is an ideal tool for signal time-frequency analysis and processing. The main characteristic is that WT can fully highlight some aspects of the problem characteristics, and can analyse the localization in time (space) frequency. In addition, signal is multi-scale tessellated through the telescopic translation operations, and ultimately adapts to the requirement of time-frequency signal analysis automatically. In other words, WT can focus on the arbitrary signal details, solved the difficulties of Fourier transform. The mathematical expression of the wavelet transform is
W T (a, τ ) = 1 √ a ∞ −∞ f (t) * Ψ( t − τ a )dτ,(5)
where a is the scale, corresponding to the frequency and controlling the scaling of the wavelet function and τ is the amount of translation, corresponding to the time and controlling the translation of the wavelet function.
D. Autocorrelation
The ECG is a nonperiodic but highly repetitive signal. The motivation behind the employment of auto correlation based features is to detect the nonrandom patterns. Autocorrelation provides an automatic, shift invariant accumulation of similarity characteristics over multiple heartbeat cycles, which represents the unique characteristics of a given ECG [19]. It provides the most robust representation of the heartbeat characteristics of an subject. In addition, AC is used to blend into a sequence of sums of products samples that would otherwise need to be subjected to fiducial detection. The AC method involves four stages: (1) windowing, where the preprocessed ECG trace is segmented into nonoverlapping windows, with the only restriction that the window has to be longer than the average heartbeat length so that multiple heartbeats are included; (2) estimation of the normalized autocorrelation of each window; (3) feature dimension reductioon, like DCT, LDA PCA etc; and (4) classification.
The autocorrelation coefficients R xx [m] can be computed as follows
R xx [m] = N −|m|−1 i=0 x [i] x [i + m] R xx [0](6)
where , eliminates the bias factor so that either biased or unbiased autocorrelation estimation can be performed. The main contributors to the autocorrelated signal are the P wave, the QRS complex, and the T wave. However, even among the pulses of the same subject, large variations in amplitude present and this makes normalization a necessity. It should be noted that a window is allowed to blindly cut out the ECG record, even in the middle of a pulse. This alone releases the need for exact heartbeat localization.
Autocorrleation provides important information in distinguish subjects. However, the dimension of the autocorrelation coefficient is determined by the size of M, which is generally large(e.g., M=100, 200, 300). So, a dimension reduction technique should be used before classification.
E. Deep learning
One of the biggest difficulties in ECGID is feature extraction. What features represent a person's ecg characteristics best is a question that we have been looking for a solution to for a long time. But so far, there are no perfect features that fit all situations. The greatest advantage of deep learning is the ability to learn features automatically, without human intervention.
Deep learning neural networks [42] is one of the most recent advancements in the machine learning field [43]. The term refers to multi layer neural networks similar to the ones used in the past. The main difference though is that now it is possible to use more hidden layers than we previously did. In deep learning, because of advancements in hardware and algorithms, there are efficient ways to train these deep neural nets. GPU computing has enabled researchers to utilize more processing power in order to train deep neural nets, while they have also implemented new algorithms that have made the whole procedure more efficient.
Additionally, deep neural nets have been proven successful in image recognition through the use of a new kind of hidden layers, convolutional layers [44]. These practically enable an hierarchical approach to data input that can detect recurring patterns in different parts of the data inputs. They have been very efficient in image recognition tasks, as they can practically detect objects present in images, irrespectively of the position of the object on the image, or the portion of the image size it corresponds to. Although ECG signal is a one-dimensional signal, it can be transformed into a two-dimensional image by processing in the case of multilead, and then be classified by using the convolutional neural network. Furthermore, Recurrent Neural Network(RNN), invented specifically to classify the time series signals, is another powerful tool that you can use for ECGID [45].
F. Feature selection
Feature selection approaches can be divided into two categories: supervised approaches and unsupervised approaches [46]. Supervised approaches use known label information to help select features nevertheless unsupervised approaches focus on mining the internal structure of the data in the absence of label information. In ECGID field, we usually adopt supervised approaches because in the biometric recognition application class labels are provided.
Supervised feature selection methods roughly include filters, wrappers and embedded methods. Wrapper methods, like genetic algorithm [47] and particle swarm optimization, select a feature subset (among all possible feature subsets) that gives the best performance with a specific model. In other words, such algorithms bind to specific models so that the results may be overfitting. In addition, wrappers are computationally very intensive specially if the feature space is too large or the chosen model is complex. Filter methods are relatively fast and do not suffer from aforementioned limitations. Filter method considers each feature individually and designs a criterion to determine whether a feature is useful for classification. Embedded methods embed feature selection in classification.
III. EXPERIMENT A. Database
Our database is collected in Prof. Cui's laboratory at South China University of Technology [35]. The database includes pre and post exercise recordings for 45 subjects. There are 33 males and 12 females whose age is between 18 to 22. Each subject in this set performed a few basic structural workouts such as steady running and climbing the stairs. The length of recordings in rest condition and post-exercise condition are around 5 minutes and 150 seconds respectively. The heart rate in post-exercise condition is range from 90 to 150 compared with around 70 in rest condition. The ECG signal was captured in lead II and the sampling frequency is 300 Hz.
B. Preprocessing
The raw ECG signals were filtered using a fourth-order bandpass Butterworth filter with cut-off frequencies 0.5 Hz and 40 Hz [19]. Under 0.5 Hz the signal is corrupted by baseline wander, and over 40 Hz there is distortion due to muscle movement, power-line noise etc.
C. Experiment on QRS complex
As mentioned before, QRS complex is the most stable part between rest and post-exercise. We use Pan & Tompkins algorithm to locate QRS complex. The length of QRS complex varies between 15 and 35 samples, so we normalize the duration of QRS complex to the same periodic length according to the procedure reported in [48]. That is, one of the ECG segments y i =[y i (1), y i (2), · · · , y i (n * )] can be converted into a segment x i =[x i (1), x i (2), · · · , x i (n)] that holds the same signal morphology, but in different data length (i.e., n * = n) using the following equation,
x i (j) = y i (j * ) + (y i (j * + 1) − y i (j * ))(r j − j * ),(7)
where r j = (j − 1)(n * − 1)/(n − 1) + 1, and j * is the integral part of r j . Here , we set n = 30. Therefore, the various lengths of the QRS complex will be compressed or extended into a set of ECG segments with the same periodic length. These 30 samples are considered to be the main description of a heart beat, and thus are regarded as the features of the heart beat.
Regarding the classification method, we choose RBF kernel with c = 100 and γ = 1 for SVM after reducing the feature dimension with PCA. In this manuscript, we mainly perform four types of ECGID test: (1) the training set and test set are both from rest condition (rest ECGID); (2) the training set and test set are both from post-exercise condition (ex-ex ECGID). According to the mode of distribution, this type is divided into two subtypes, which are the first 70% as the training set and the last 70% as the training set; (3) the training set is from rest condition and the test set is from post-exercise condition (restex ECGID).
D. Experiment on ECG beat
Pan & Tompkins algorithm is also utilized to locate R peak. Then we search the midpoint of two R-peak samples, and the signal between two consecutive midpoints is defined as a ECG segment. Like the way we used in the experiment on QRS complex, we also use the procedure reported in Ref. [48] to normalize the length of ECG segments. We set n = 300 here. These 300 samples are regarded as the features of the heart beat. We adopted SVM and deeplearning for classification respectively. For deep learning, the network structure we adopt is as shown in Fig. 2. And the results of experiment on ECG beat are shown in Tables 2-3:
E. Experiment on PQRST piecewise correction
In this experiment, we divide a complete heart beat into four parts: PQ segment, QRS segment, ST segment and T segment. Changes in heart rate are not evenly distributed across the P, R, and T complexes, so we adopt piecewise correction on these segments as follows. The PQ segment can be obtained using the following Equation
P Q(i) = S[R(i) − 230ms + dt : R(i) − 90ms],(8)
where dt is a variable threshold with changes in heart rate, and it can be described as follows
dt = −
The QRS-segment can be obtained using Equation (10)
QRS(i) = S[R(i) − 90ms : R(i) + 100ms].(10)
The ST-segment can be obtained using Equation (11) ST (i) = S[R(i)+100ms : R(i)+100ms+0.08 * RR], (11) where the RR is current RR interval. As for the T-segment, it is described by Equation (12) T (i) = S[R(i)+100ms+0.08 * RR : R(i)+0.42 * RR]. (12) After wave correction, we use the method mentioned above to resample the PQ segment to 450 milliseconds length, the ST segment to 110 milliseconds length, and the T segment to 50 milliseconds length. The QRS segment keeps unchanged for it remains fairly constant. These segments are combined to recreate the entire heartbeat. The normal length of a heartbeat is 800 milliseconds. Finally, the amplitude of each new heartbeat is also normalized to a mean zero. Next, we perform two kinds of processing. One is to classify the new heart beat as a feature vector through PCA and SVM, directly; the other is to extract the wavelet coefficient with the new heart beat and then conduct identity classification. The experiment result is shown in Table 4:
F. Experiment on band pass filtering
Ref. [33] showed that the ECG exercise waveform is similar to the rest waveform above 10 Hz, so in this experiment, after R peak detecting and ECG heartbeat segmentation, we filter the original ECG signal, including rest and post-exercise, from 10 Hz to 40 Hz. Then the heart beats are resampled to 300 points length and a rbf-SVM is used for classification. The experiment result is as follows (see Table 5):
G. Experiment on Short-time Fourier transform
In this experiment, we use short-time Fourier transform with Hamming window of the length 16 with step size of 13 computed over a 1 second window centered at R peak. The sampling frequency is 300 Hz, so this gives a total of 572 features. A PCA is for dimension reduction and a rbf-SVM is for classification. The experiment result is as follows (see Table 6):
H. Experiment on Wavelet transform
After preprocessing, Continuous wavelet transform with 32 scales and Daubechies 5 as mother wavelet is computed on a 1 second window centered at R peak, which gives a total of 9600 features. A PCA is for dimension reduction and a rbf-SVM is for classification. The experiment result is as follows (see Table 7):
I. Experiment on Autocorrelation
Autocorrelation upto n lags is computed on windows of the length L seconds, where n and L are adjustable parameters. The experiment results are shown in Tables 8 & 9:
J. Experiment on feature selection
We randomly select half of the subjects (22 subjects) as an auxiliary data set and utilize the other half (23 subjects) for registration and testing. In other word, the first half is picked for feature selection while the second one for calculating accuracy of identification where registration is performed in the resting state and testing in the post-exercise condition. We perform a short-time Fourier transform using a Hamming window of length 16 seconds with a step size of 13 to, and compute continuous wavelet transform with 32 scales and Daubechies 5 as mother wavelet over a window of length 1 second centered at R peak. Autocorrelation up to 80 lags is calculated over 1 second windows. In order to have zero mean and unit variance, every feature has to be normalized by z-score in the feature set which is collected by 10252 features.
The feature selection is based on Kullback-Leibler (KL) divergence. Following Ref. [34], feature weights, w(l), which measure the importance of a feature to rest-ex ECGID, are defined as linear combination of two terms
w(l) = λw 1 (l) − (1 − λ)w 2 (l).(13)
The first term w 1 (l) related to the class separability is defined as follows
w 1 (l) = 1 N N i=1 d(f (X i (l)), f (χ(l))),(14)
where f (X i (l)) is probability density function(pdf) of l-th feature computed over all samples of i-th subject including both rest and post-exercise samples. f (χ(l)) is pdf of lth feature computed over all samples of the auxiliary data set while N is total number of its subjects. d(·) is the symmetric KL divergence which can be estimated under a normal distribution as follows
d(f 1 , f 2 ) = σ 2 1 + (µ 1 − µ 2 ) 2 2σ 2 2 + σ 2 2 + (µ 1 − µ 2 ) 2 2σ 2 1 − 1.(15)
The second term in equation (13) related to the sensitivity of a feature to exercise is defined below
w 2 (l) = 1 N N i=1
(d(f (X rest i (l)), f (X i (l))) +d(f (X post−ex i (l)), f (X i (l)))),
where f (X rest i (l))(f (X post−ex i (l))) is part of l-th feature of subject i in rest(post-exercise) condition. w 2 (l) is small when the distribution corresponding to rest and post-exercise condition overlap with each other, suggesting the feature is more robust against exercise. Features can be selected by The ECGID performance on KL feature selection is shown in Table 10, while the details of rest-ex ECGID performance is shown in Fig. 3. It is shown that compared with other methods, though the KL feature selection method can significantly improve the restex ECGID accuracy, it is only 61.4%, far from satisfying the expectation.
IV. CONCLUSION
In this manuscript, we focus on the effect of exercise on ECGID and apply existing methods to analyse the performance of rest-ex ECGID with our ECG database. Through multiple sets of experiments, we find that for current methods, the rest-rest ECGID accuracy can reach more than 95% while the ex-ex ECGID performs a little worse; As for the rest-ex ECGID performance, most methods collapse to 10% or even worse, except for the KL feature selection method which can reach almost 65%. Although compared with other methods the KL feature selection method performs much better, its result still remains unsatisfactory. The current rest-ex ECGID solutions are too simple, but more stable features may be obtained and sent to feature selection algorithm in the future by filtering out the part with violent changes due to exercise. Furthermore, other more advanced methods in the field of identity recognition can be introduced to improve the accuracy of rest-ex ECGID.
This work was supported by the National Natural Science Foundation of China under Grant 61873317, and in part by the Fundamental Research Funds for the Central Universities. Corresponding author: Wei Cui. The authors are with the School of Automation Science and Engineering, South China University of Technology, Guangzhou 510641, China (e-mail: [email protected]).
x [i] is the windowed ECG and x [i + m] is the timeshift version of the windowed ECG with a time lag of m = 0, 1, ..., M − 1, M N . The division with the maximum value, R xx [0]
Fig. 2 :
2Network structure of LSTM on ECGID
Fig. 3 :
3Rest-ex ECGID accuracy of KL feature selection comparing the weights with a threshold and further sorted in descending order according to their weights, after which the top n features are picked. λ is chosen to be 0.3 empirically.
et al. uses Fourier coefficients to examine the effectiveness of segmenting ECG heartbeat [16]. Shen et al. proposed an identification method based on one-lead ECG [17]. They obtained 7 time domain features from QRS complex and used template matching and decisionbased neural network to complete the verification, with 100% correct verification rate. Plataniotis et al. first introduced an arXiv:1905.04442v1 [eess.SP] 11 May 2019 Fig. 1: A real time ECG signal
Table 1
1ECGID performance on QRS complexTraining set
Test set
Training accu-
racy
Test accuracy
rest record
rest record
98%
95%
post-exercise(first
70%)
post-exercise(last
30%)
94.1%
82.9%
post-exercise(last
70%)
post-exercise(first
30%)
94.8%
70.2%
rest
post-exercise
97.9%
17.5%
Table 2
2ECGID performance on ECG beat with SVMTraining set
Test set
Training accu-
racy
Test accuracy
rest record
rest record
100%
96.2%
post-exercise(first
70%)
post-exercise(last
30%)
100%
85.4%
post-exercise(last
70%)
post-exercise(first
30%)
100%
72.6%
rest
post-exercise
99.6%
2.7%
Table 3
ECGID performance on ECG beat with LSTM
Training set
Test set
Training accu-
racy
Test accuracy
rest record
rest record
97.4%
95.6%
post-exercise(first
70%)
post-exercise(last
30%)
96%
82.3%
post-exercise(last
70%)
post-exercise(first
30%)
97.2%
77.6%
rest
post-exercise
98.2%
12%
Table 4
4ECGID performance on PQRST piecewise correctionTraining set
Test set
Training accu-
racy
Test accuracy
rest record
rest record
100%
98.2%
post-exercise(first
70%)
post-exercise(last
30%)
100%
88%
post-exercise(last
70%)
post-exercise(first
30%)
100%
63.3%
rest
post-exercise
99.9%
4.9%
Table 5
5ECGID performance on band pass filteringTraining set
Test set
Training accu-
racy
Test accuracy
rest
rest
100%
92.7%
post-exercise(first
70%)
post-exercise(last
30%)
100%
85.4%
post-exercise(last
70%)
post-exercise(first
30%)
100%
52.4%
rest
post-exercise
98.6%
3%
Table 6
6ECGID performance on STFTTraining set
Test set
Training accu-
racy
Test accuracy
rest
rest
99.6%
98.3%
post-exercise(first
70%)
post-exercise(last
30%)
97.1%
75.3%
post-exercise(last
70%)
post-exercise(first
30%)
98.3%
66.1%
rest
post-exercise
99.6%
12.9%
Table 7
7ECGID performance on wavelet transformTraining set
Test set
Training accu-
racy
Test accuracy
rest
rest
99.9%
99.7%
post-exercise(first
70%)
post-exercise(last
30%)
99.8%
83%
post-exercise(last
70%)
post-exercise(first
30%)
99.9%
77.9%
rest
post-exercise
99.9%
7.1%
Table 8
8ECGID performance on AutocorrelationTraining set
Test set
Training accu-
racy
Test accuracy
rest
rest
100%
93.8%
post-exercise(first
70%)
post-exercise(last
30%)
99.9%
74.3%
post-exercise(last
70%)
post-exercise(first
30%)
100%
60.1%
rest
post-exercise
90%
11.3%
Table 9
9ECGID performance on beat with AutocorrelationTraining set
Test set
Training accu-
racy
Test accuracy
rest
rest
100%
95.2%
post-exercise(first
70%)
post-exercise(last
30%)
99.2%
67.3%
post-exercise(last
70%)
post-exercise(first
30%)
99.6%
63.9%
rest
post-exercise
98.3%
1.3%
Table 10
10ECGID performance on KL feature selectionTraining set
Test set
Training accu-
racy
Test accuracy
rest
rest
98.2%
93.8%
post-exercise(first
70%)
post-exercise(last
30%)
95.7%
85.1%
post-exercise(last
70%)
post-exercise(first
30%)
96.4%
73.9%
rest
post-exercise
98.7%
61.4%
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There are 33 males and 12 females whose age is between 18 to 22. Each subject in this set performed a few basic structural workouts such as steady running and climbing the stairs. The ECG signals are recorded from the wrist and the length of recordings in rest condition and post-exercise condition are around 5 minutes and 150 seconds respectively. The heart rate in post-exercise condition is range from 90 to 150 compared with around 70 in rest condition. The ECG signal was captured in lead II and the sampling frequency is 300 Hz. The database is collected in Prof. Cui's laboratory at South China University of Technology. The database includes pre and post exercise recordings for 45 subjectsThe database is collected in Prof. Cui's laboratory at South China University of Technology. The database includes pre and post exercise recordings for 45 subjects. There are 33 males and 12 females whose age is between 18 to 22. Each subject in this set performed a few basic structural workouts such as steady running and climbing the stairs. The ECG signals are recorded from the wrist and the length of recordings in rest condition and post-exercise condition are around 5 minutes and 150 seconds respectively. The heart rate in post-exercise condition is range from 90 to 150 compared with around 70 in rest condition. The ECG signal was captured in lead II and the sampling frequency is 300 Hz. Available: https://github.com/feimadada/exercise-ECGID-database-
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| [
"https://github.com/feimadada/exercise-ECGID-database-"
]
|
[
"A Probabilistic Logic for Verifying Continuous-time Markov Chains",
"A Probabilistic Logic for Verifying Continuous-time Markov Chains"
]
| [
"Ji Guan \nInstitute of Software\nState Key Laboratory of Computer Science\nChinese Academy of Sciences\nBeijingChina\n",
"Nengkun Yu [email protected] \nCentre for Quantum Software and Information\nUniversity of Technology Sydney\nSydneyAustralia\n"
]
| [
"Institute of Software\nState Key Laboratory of Computer Science\nChinese Academy of Sciences\nBeijingChina",
"Centre for Quantum Software and Information\nUniversity of Technology Sydney\nSydneyAustralia"
]
| []
| A continuous-time Markov chain (CTMC) execution is a continuous class of probability distributions over states. This paper proposes a probabilistic linear-time temporal logic, namely continuous-time linear logic (CLL), to reason about the probability distribution execution of CTMCs. We define the syntax of CLL on the space of probability distributions. The syntax of CLL includes multiphase timed until formulas, and the semantics of CLL allows time reset to study relatively temporal properties. We derive a corresponding model-checking algorithm for CLL formulas. The correctness of the model-checking algorithm depends on Schanuel's conjecture, a central open problem in transcendental number theory. Furthermore, we provide a running example of CTMCs to illustrate our method. | 10.1007/978-3-030-99527-0_1 | [
"https://arxiv.org/pdf/2004.08059v3.pdf"
]
| 234,486,946 | 2004.08059 | 5e0fb28f0554c4796a1e0ad45659ce92cccc1431 |
A Probabilistic Logic for Verifying Continuous-time Markov Chains
14 Apr 2022
Ji Guan
Institute of Software
State Key Laboratory of Computer Science
Chinese Academy of Sciences
BeijingChina
Nengkun Yu [email protected]
Centre for Quantum Software and Information
University of Technology Sydney
SydneyAustralia
A Probabilistic Logic for Verifying Continuous-time Markov Chains
14 Apr 2022
A continuous-time Markov chain (CTMC) execution is a continuous class of probability distributions over states. This paper proposes a probabilistic linear-time temporal logic, namely continuous-time linear logic (CLL), to reason about the probability distribution execution of CTMCs. We define the syntax of CLL on the space of probability distributions. The syntax of CLL includes multiphase timed until formulas, and the semantics of CLL allows time reset to study relatively temporal properties. We derive a corresponding model-checking algorithm for CLL formulas. The correctness of the model-checking algorithm depends on Schanuel's conjecture, a central open problem in transcendental number theory. Furthermore, we provide a running example of CTMCs to illustrate our method.
Introduction
As a popular model of probabilistic continuous-time systems, continuous-time Markov chains (CTMCs) have been extensively studied since Kolmogorov [1]. In the recent 20 years, probabilistic continuous-time model checking receives much attention. Adopting probabilistic computational tree logic (PCTL) [2] to this context with extra multiphase timed until formulas Φ 1 U T1 Φ 2 · · · U TK Φ K+1 , for state formula Φ and time interval T , Aziz et al. proposed continuous stochastic logic (CSL) to specify the branching-time properties of CTMCs and the modelchecking problem for CSL is decidable [3]. After that, efficient model-checking algorithms were developed by transient analysis of CTMCs using uniformization [4] and stratification [5] for a restricted version (path formulas are restricted to single until formulas Φ 1 U I Φ 2 ) and a full version of CSL, respectively. These algorithms have been practically implemented in model checkers PRISM [6], MRMC [7] and STORM [8]. Further details can be found in an excellent survey [9].
There are also different ways to specify the linear-time properties of CTMCs. Timed automata were first used to achieve this task [10,11,12,13,14], and then metric temporal logic (MTL) [15] was also considered in this context. Subsequently, the probability of "the system being in state s 0 within five-time units after having continuously remained in state s 1 " can be computed. However, some statements cannot be specified and verified because of the lack of a probabilistic linear-time temporal logic, for instance "the system being in state s 0 with high probability (≥ 0.9) within five-time units after having continuously remained in state s 1 with low probability (≤ 0.1)". Furthermore, this probabilistic property cannot be expressed by CSL because CSL cannot express properties that are defined across several state transitions of the same time length in the execution of a CTMC.
In this paper, targeting to express the mentioned probabilistic linear-time properties, we introduce continuous-time linear logic (CLL). In particular, we adopt the viewpoint used in [16] by regarding CTMCs as transformers of probability distributions over states. CLL studies the properties of the probability distribution execution generated by a given initial probability distribution over time. By the fundamental difference between the views of state executions and probability distribution executions of CTMCs, CLL and CSL are incomparable and complementary, as the relation between probabilistic linear-time temporal logic (PLTL) and PCTL in model checking discrete-time Markov chains [16,Section 3.3].
The atomic propositions of CLL are explained on the space of probability distributions over states of CTMCs. We apply the method of symbolic dynamics to the probability distributions of CTMCs. To be specific, we symbolize the probability value space where each symbol s, I asserts µ(s) ∈ I, i.e., the probability of state s in distribution µ falls in interval I. For example, s 0 , [0.9, 1] means the system is in state s 0 with a probability in 0.9 to 1. The symbolization idea of distributions has been considered in [16]: choosing a disjoint cover of [0, 1]:
I = {[0, p 1 ), [p 1 , p 2 ), ..., [p n , 1]}.
Here, we remove this restriction and enrich the expressiveness of I . A crucial fact about this symbolization is that the set S × I is finite. Consequently, the (probability distribution) execution path generated by an initial probability distribution µ induces a sequence of symbols in S × I over time. Therefore, the dynamics of CTMCs can be studied in terms of a (real-time) language over the alphabet S × I , which is the set of atomic propositions of CLL.
Different from non-probabilistic linear-time temporal logics -linear-time temporal logic (LTL) and MTL, CLL has two types of formulas: state formulas and path formulas. The state formulas are constructed using propositional connectives. The path formulas are obtained by propositional connectives and a temporal modal operator timed until U T for a bounded time interval T , as in MTL and CSL. The standard next-step temporal operator in LTL is meaningless in continuous-time systems since the time domain (real numbers) is uncountable. As a result, CLL can express the above mentioned probabilistic property "the system is at state s 0 with high probability (≥ 0.9) within 5 time units after having continuously remained at state s 1 with low probability (≤ 0.1)" in a path formula:
ϕ = s 1 , [0, 0.1] U [0,5] s 0 , [0.9, 1] .
In this single until formula, there is a time instant 0 ≤ t ≤ 5 at which state s 1 with low probability transits to state s 0 with high probability. Then we illustrate this on the following timeline.
s1,[0,0.1] ↓ t ≤ 5 ↓ 0 ↑ s 0 , [0.9, 1]
Furthermore, CLL allows multiphase timed until formulas. The semantics of the formulas focuses on relative time intervals, i.e., time can be reset as in timed automata [17,18], while those of CSL [3] are for absolute time intervals. Subsequently, CLL can express not only relatively but also absolutely temporal properties of CTMCs.
We illustrate the significant difference between relatively temporal properties and absolutely temporal properties of CTMCs. For instance, "before probability distributions transition ϕ happening in 3 to 7 time units, the system always stays at state s 0 with a high probability (≥ 0.9)" can be formalized in path formulae ϕ ′ = s 0 , [0.9, 1] U [3,7] ( s 1 , [0, 0.1] U [0,5] s 0 , [0.9, 1] ).
As we can see, there are two time instants, namely t 1 and t 2 , happening distribution transitions. Time is reset to 0 after the first distribution transition happens and thus t 2 is relative to t 1 . More clearly, we depict this on the following timeline.
↓ (t 2 + t 1 ) ≤ 12 =3 ↑ 0s ↓ t 1 ≤ 7 ↑ s 0 , [0.9, 1]
An absolute version is "probability distribution transition ϕ happens and the system always stays at state s 0 with a high probability (≥ 0.9) in 3 to 7 time units"
ϕ ′′ = [3,7] s 0 , [0.9, 1] ∧ s 1 , [0, 0.1] U [0,5] s 0 , [0.9, 1] ).
We can get a clear timeline representation by simply adding [3,7] s 0 , [0.9, 1] to that of ϕ. Assume that t < 3,
s1,[0,0.1] ↓ t < 3 ↓ 0 ↑ s 0 , [0.9, 1] ↓ 3 ↓ 7 s0,[0.9,1]
Time reset enriches the expressiveness of CLL but introduces more difficulties to model checking CLL than CSL. We cross this by translating relative time to the absolute one. As a result, we develop an algorithm to model check CTMCs against CLL formulas. More precisely, we reduce the model-checking problem to a reachability problem of absolute time intervals. The reachability problem corresponds to the real root isolation problem of real polynomial-exponential functions (PEFs) over the field of algebraic numbers, an extensively studied question in recent symbolic and algebraic computation community (e.g. [19,20,21]). By developing a state-of-the-art real root isolation algorithm, we resolve the latter problem under the assumption of the validity of Schanuel's conjecture, a central open question in transcendental number theory [22]. This conjecture has also been the footstone of the correctness of many recent model-checking algorithms, including the decidability of continuous-time Markov decision processes [23], the synthesizing inductive invariants for continuous linear dynamical systems [24], the termination analysis for probabilistic programs with delays [25], and reachability analysis for dynamical systems [20].
In summary, the main contributions of this paper are as follows.
-Introducing a probabilistic logic, namely continuous-time linear logic (CLL), for reasoning about CTMCs; -Developing a state-of-the-art real root isolation algorithm for PEFs over the field of algebraic numbers for checking atomic propositions of CLL; -Proving that model checking CTMCs against CLL formulas is decidable subject to Schanuel's conjecture.
Organization of this paper. In the next section, we give the mathematical preliminaries used in this paper. In Section 3, we recall the view of CTMCs as distribution transformers. After that, the symbolic dynamics of CTMCs are introduced by symbolizing distributions over states of CTMCs in Section 4. In the subsequent section, we present our continuous-time probabilistic temporal logic CLL. In Section 6, we develop an algorithm to solve the CLL model checking problem. A case study and related works are shown in Sections 7 and 8, respectively. We summarize our results and point out future research directions in the final section.
Preliminaries
For the convenience of the readers, we review basic definitions and notations of number theory, particularly Schanuel's conjecture.
Throughout this paper, we write C, R, Q and A for the fields of all complex, real, rational and algebraic numbers, respectively. In addition, Z denotes the set of all integer numbers. For F ∈ {C, R, Q, Z, A}, we use F[t] and F n×m to denote the set of polynomials in t with coefficients in F and n-by-m matrices with every entry in F, respectively. Furthermore, for F ∈ {R, Q, Z}, we use F + to denote the set of positive elements (including 0) of F.
A bounded (time) interval T is a subset of R + , which may be open, half-open or closed with one of the following forms:
[t 1 , t 2 ], [t 1 , t 2 ), (t 1 , t 2 ], (t 1 , t 2 ),
where t 1 , t 2 ∈ R + and t 2 ≥ t 1 (t 1 = t 2 is only allowed in the case of [t 1 , t 2 ]). Here, t 1 and t 2 are called the left and right endpoints of T , respectively. Conveniently, we use inf T and sup T to denote t 1 and t 2 , respectively. In this paper, we only consider bounded intervals.
For reasoning about the temporal properties, we further define the addition and subtraction of (time) intervals. The expression T + t or t + T , for t ∈ R + , denotes the interval {t + t ′ : t ′ ∈ T }. Similarly, T − t stands for the interval
{−t + t ′ : t ′ ∈ T } if t ≤ inf T .
Furthermore, for two intervals T 1 and T 2 ,
T 1 + T 2 = {t ∈ (t ′ + T 2 ) : t ′ ∈ T 1 } = {t 1 + t 2 : t 1 ∈ T 1 and t 2 ∈ T 2 }.
Two intervals T 1 and T 2 are disjoint if their intersection is an empty set, i.e., [3,4] are disjoint. It is obvious that all calculations of time intervals in the above are easy to be computed.
An algebraic number is a complex number that is a root of a non-zero polynomial in one variable with rational coefficients (or equivalent to integer coefficients, by eliminating denominators). An algebraic number α is represented by (P, (a, b), ε) where P is the minimal polynomial of α, a, b ∈ Q and a + bi is an approximation of α such that |α − (a + bi)| < ε and α is the only root of P in the open ball B(a + bi, ε). The minimal polynomial of α is the polynomial with the smallest degree in Q[t] such that α is a root of the polynomial and the coefficient of the highest-degree term is 1. Any root of f (t) ∈ A[t] is algebraic. Moreover, given the representations of a, b ∈ A, the representations of a ± b, a b and a · b can be computed in polynomial time, so does the equality checking [26].
Furthermore, a complex number is called transcendental if it is not an algebraic number. In general, it is challenging to verify relationships between transcendental numbers [27]. On the other hand, one can use the Lindemann-Weierstrass theorem to compare some transcendental numbers. The transcendence of e and π are direct corollaries of this theorem.
Theorem 1 (Lindemann-Weierstrass theorem). Let η 1 , · · · , η n be pairwise distinct algebraic complex numbers. Then k λ k e η k = 0 for non-zero algebraic numbers λ 1 , · · · , λ n .
The following concepts are introduced to study the general relation between transcendental numbers. Definition 1 (Algebraic independence). A set of complex numbers S = {a 1 , · · · , a n } is algebraically independent over Q if the elements of S do not satisfy any nontrivial (non-constant) polynomial equation with coefficients in Q.
By the above definition, for any transcendental number u, {u} is algebraically independent over Q, while {a} for any algebraic number a ∈ A is not. Thus, a set of complex numbers that is algebraically independent over Q must consist of transcendental numbers. {π, e π √ n } is also algebraically independent over Q for any positive integer n [28]. Checking the algebraic independence is challenging. For example, it is still widely open whether {e, π} is algebraically independent over Q.
Definition 2 (Extension field). Given two fields E ⊆ F , F is an extension field of E, denoted by F/E, if the operations of E are those of F restricted to E.
For example, under the usual notions of addition and multiplication, the field of complex numbers is an extension field of real numbers.
Definition 3 (Transcendence degree). Let L be an extension field of Q, the transcendence degree of L over Q is defined as the largest cardinality of an algebraically independent subset of L over Q.
For instance, let Q(e)/Q = {a + be | a, b ∈ Q} and Q( √ 2)/Q = {a + b √ 2 | a, b ∈ Q} be two extension fields of Q.
Then the transcendence degree of them are 1 and 0, respectively, by noting that e is a transcendental number and √ 2 is an algebraic number. Now, Schanuel's conjecture is ready to be presented.
Conjecture 1 (Schanuel's conjecture). Given any complex numbers z 1 , · · · , z n that are linearly independent over Q, the extension field Q(z 1 , ..., z n , e z1 , ..., e zn ) has transcendence degree of at least n over Q.
Stephen Schanuel proposed this conjecture during a course given by Serge Lang at Columbia in the 1960s [22]. Schanuel's conjecture concerns the transcendence degree of certain field extensions of the rational numbers. The conjecture, if proven, would generalize the most well-known results in transcendental number theory significantly [29,30]. For example, the algebraical independence of {e, π} would simply follow by setting z 1 = 1 and z 2 = πi, and using Euler's identity e πi + 1 = 0.
Continuous-time Markov Chains as Distributions Transformers
We begin with the definition of continuous-time Markov chains (CTMCs). A CTMC is a Markovian (memoryless) stochastic process that takes values on a finite state set S (|S| = d < ∞) and evolves in continuous-time t ∈ R + . Formally,
Definition 4. A CTMC is a pair M = (S, Q), where S (|S| = d)
is a finite state set and Q ∈ Q d×d is a transition rate matrix.
A transition rate matrix Q is a matrix whose off-diagonal entries {Q i,j } i =j are nonnegative rational numbers, representing the transition rate from state i to state j, while the diagonal entries {Q j,j } are constrained to be − j =i Q i,j for all 1 ≤ j ≤ d. Consequently, the column summations of Q are all zero. The evolution of a CTMC can be regarded as a distribution transformer. Given initial distribution µ ∈ Q d×1 ∈ D(S), the distribution at time t ∈ R + is:
µ t = e Qt µ,
where D(S) is denoted as the set of all probability distributions over S. We call D(S) the probability distribution space of CTMCs. An execution path of CTMCs is a continuous function indexed by initial distribution µ ∈ D(S):
σ µ : R + → D(S), σ µ (t) = e Qt µ.(1)
Example 1. We recall the illustrating example of CTMC M = (S, Q) in [3, Figure 1] as the running example in our work. In particular, M is a 5-dimensional CTMC with initial distribution µ, where S = {s 0 , s 1 , s 2 , s 3 , s 4 } and
Q = −3 0 0 0 0 1 0 0 0 0 2 0 −7 0 0 0 0 3 0 0 0 0 4 0 0 µ = 0.1 0.2 0.3 0.4 0 .
Symbolic Dynamics of CTMCs
In this section, we introduce symbolic dynamics to characterize the properties of the probability distribution space of CTMCs. First, we fix a finite set of intervals I = {I k ⊆ [0, 1]} k∈K , where the endpoints of each I k are rational numbers. With the states S = {s 0 , s 1 , · · · , s d−1 }, we define the symbolization of distributions as a function:
S : D(S) → 2 S×I S(µ) = { s, I ∈ S × I : µ(s) ∈ I},(2)
where × denotes the Cartesian product, and 2 S×I is the power set of S × I . s, I ∈ S(µ) asserts that the probability of state s in distribution µ is in the interval I. The symbolization of distributions is a generalization of the discretization of distributions with I k ∩ I m = ∅ for all k = m which was studied in [16]. This generalization increases the expressiveness of our continuous linear-time logic introduced in the next section. Now, we can represent any given probability distribution by finite symbols from S × I . For example, suppose
I = {[0, 0.1], (0.1, 0.9), [0.9, 1], [1, 1], [0.4, 0.4]},(3)
and then the initial distribution µ in Example 1 is symbolized as
S(µ) = { s 0 , [0, 0.1] , s 1 , (0.1, 0.9) , s 2 , (0.1, 0.9) , s 3 , (0.1, 0.9) , s 3 , [0.4, 0.4] , s 4 , [0, 0.1] }.(4)
As we can see from the above example, the symbolization of distributions on states considers the exact probabilities (singleton intervals) of the states and the range of their possibilities. Next, we introduce the symbolization to CTMCs,
Definition 5. A symbolized CTMC is a tuple SM = (S, Q, I ), where M = (S, Q) is a CTMC and I is a finite set of intervals in [0, 1].
As we can see, the set of intervals is picked depending on CTMCs. Then, we extend this symbolization to the path σ µ :
S • σ µ : R + → 2 S×I .(5)
Definition 6. Given a symbolized CTMC SM = (S, Q, I ), S•σ µ is a symbolic execution path of M = (S, Q).
Given a symbolized CTMC SM = (S, Q, I ), the path σ µ of CTMC M = (S, Q) over real numbers R + generated by probability distribution µ induces a symbolic execution path S • σ µ over finite symbols S × I . Subsequently, the dynamics of CTMCs can be studied in terms of a language over S × I . In other words, we can study the temporal properties of CTMCs in the context of symbolized CTMCs.
Continuous Linear-time Logic
In this section, we introduce continuous linear-time logic (CLL), a probabilistic linear-time temporal logic, to specify the temporal properties of a symbolized CTMC SM = (S, Q, I ). CLL has two types of formulas: state formulas and path formulas. The state formulas are constructed using propositional connectives. The path formulas are obtained by propositional connectives and a temporal modal operator timed until U T for a bounded time interval T , as in MTL and CSL. Furthermore, multiphase timed until formulas Φ 0 U T1 Φ 1 U T2 Φ 2 . . . U Tn Φ n are allowed to enrich the expressiveness of CLL. More importantly, time reset is involved in these multiphase formulas. Thus absolutely and relatively temporal properties of CTMCs can be studied.
Definition 7. The state formulas of CLL are described according to the following syntax:
Φ := true | a ∈ AP | ¬Φ | Φ 1 ∧ Φ 2
where AP denotes S × I as the set of atomic propositions. The path formulas of CLL are constructed by the following syntax:
ϕ := true | Φ 0 U T1 Φ 1 U T2 Φ 2 . . . U Tn Φ n | ¬ϕ | ϕ 1 ∧ ϕ 2
where n ∈ Z + is a positive integer, for all 0 ≤ k ≤ n, Φ k is a state formula, and T k 's are time intervals with the endpoints in Q + , i.e., each T k is one of the following forms:
(a, b), [a, b], (a, b], [a, b) ∀a, b ∈ Q + .
The semantics of CLL state formulas is defined on the set D(S) of probability distributions over S with the symbolized function S in Eq.(2) of Section 4.
(1) µ |= true for all probability distributions µ ∈ D(S);
(2) µ |= a iff a ∈ S(µ); (3) µ |= ¬Φ iff it is not the case that µ |= Φ (or written µ |= Φ ); (4) µ |= Φ 1 ∧ Φ 2 iff µ |= Φ 1 and µ |= Φ 2 .
The semantics of CLL path formulas is defined on execution paths {σ µ } µ∈D(S) of CTMC M = (S, Q).
(1) σ µ |= true for all probability distributions µ ∈ D(S);
(2) σ µ |= Φ 0 U T1 Φ 1 U T2 Φ 2 . . . U Tn Φ n iff there is a time instant t ∈ T 1 such that σ µt |= Φ 1 U T2 Φ 2 . . . U Tn Φ n , and for any t ′ ∈ T 1 ∩ [0, t), µ t ′ |= Φ 0 , where σ µt |= Φ iff µ t |= Φ, and µ t is the distribution of the chain at time instant t, i.e., µ t = e Qt µ ∀t ∈ R + ; (3) σ µ |= ¬ϕ iff it is not the case that σ µ |= ϕ (written σ µ |= ϕ ); (4) σ µ |= ϕ 1 ∧ ϕ 2 iff σ µ |= ϕ 1 and σ µ |= ϕ 2 .
Not surprisingly, other Boolean connectives are derived in the standard way, i.e., false = ¬true,
Φ 1 ∨ Φ 2 = ¬(¬Φ 1 ∧ ¬Φ 2 ) and Φ 1 → Φ 2 = ¬Φ 1 ∨ Φ 2 ,
and the path formula ϕ follows the same way. Furthermore, we generalize temporal operators ♦ ("eventually") and ("always") of discrete-time systems into their timed variant ♦ T and T , respectively, in the following:
♦ T Φ = trueU T Φ T Φ = ¬♦ T ¬Φ.
For n = 1 in multiphase timed until formulas, the until operator U T1 is a timed variant of the until operator of LTL; the path formula Φ 0 U T1 Φ 1 asserts that Φ 1 is satisfied at some time instant in the interval T 1 and that at all preceding time instants in T 1 , Φ 0 holds. For example,
ϕ = s 1 , [0, 0.1] U [0,5] s 0 , [0.9, 1] ,
as mentioned in introduction section.
For general n, the CLL path formula Φ 0 U T1 Φ 1 U T2 Φ 2 . . . U Tn Φ n is explained over the induction on n. We first mention that U T is right-associative, e.g.,
Φ 0 U T1 Φ 1 U T2 Φ 2 stands for Φ 0 U T1 (Φ 1 U T2 Φ 2 )
. This makes time reset, i.e., T 1 and T 2 do not have to be disjoint, and the starting time point of T 2 is based on some time instant in T 1 . Recall the multiphase timed until formula in introduction section and this formula expresses a relative time property:
ϕ ′ = s 0 , [0.9, 1] U [3,7] ( s 1 , [0, 0.1] U [0,5] s 0 , [0.9, 1] ),
which is different to the following CLL path formula representing an absolutely temporal property of CTMCs: ϕ ′′ = [3,7] As an example, we clarify the semantics of CLL by comparing the above two path formulas in general forms:
Φ 0 U T1 Φ 1 U T2 Φ 2 and Φ 0 U T1 Φ 1 ∧ Φ 1 U T2 Φ 2 . (1) σ µ |= Φ 0 U T1 Φ 1 U T2 Φ 2 asserts that there are time instants t 1 ∈ T 1 , t 2 ∈ T 2 such that µ t1+t2 |= Φ 2 and for any t ′ 1 ∈ T 1 ∩ [0, t 1 ) and t ′ 2 ∈ T 2 ∩ [0, t 2 ), µ t ′ 1 |= Φ 0 and µ t1+t ′ 2 |= Φ 1 , where µ t = e Qt µ ∀t ∈ R + .
This is more clear in the following timeline. , where the number of U [2,3] is 100, asserts that the probability of state s is beyond 0.7 in every time instant 2 to 3, and this happens at least 100 times.
Φ0 Φ1 =inf T2 ↓ Φ 2 =inf T1 ↑ time 0 ↑ t 1 ≤ sup T 1 ↑ (t 1 + t 2 ) ≤ sup(T 1 + T 2 ) (2) σ µ |= Φ 0 U T1 Φ 1 ∧ Φ 1 U T2 Φ 2 asserts that there are time instants t 1 ∈ T 1 , t 2 ∈ T 2 such that µ t1 |= Φ 1 and µ t2 |= Φ 2 , and for any t ′ 1 ∈ T 1 ∩ [0, t 1 ) and t ′ 2 ∈ T 2 ∩ [0, t 2 ), µ t ′ 1 |= Φ 0 and µ t ′ 2 |= Φ 1 , where µ t = e
Next, we can classify members of I as representing "low" and "high" probabilities. For example, if I contains 3 intervals {[0, 0.1], (0.1, 0.9), [0.9, 1]}, we can declare the first interval as "low" and the last interval as "high". In this case [10,1000) ( s 0 , [0, 0.1] → s 1 , [0.9, 1] ) says that, in time interval [10,1000), whenever the probability of state s 0 is low, the probability of state s 1 will be high.
CLL Model Checking
In this section, we provide an algorithm to model check CTMCs against CLL formulas, i.e., the following CLL model-checking problem -Problem 1 is decidable.
Problem 1 (CLL Model-checking Problem). Given a symbolized CTMC SM = (S, Q, I ) with an initial distribution µ and a CLL path formula ϕ on AP = S × I , the goal is to decide whether σ µ |= ϕ, where σ µ (t) = e Qt µ is an execution path defined in Eq.(1).
In particular, we show that Theorem 2. Under the condition that Schanuel's conjecture holds, the CLL model-checking problem in Problem 1 is decidable.
In the following, we prove the above theorem from checking basic formulas -atomic propositions to the most complex one -nontrivial multiphase timed until formulas. For readability, we put the proofs of all results in Appendix A.
We start with the simplest case of atomic proposition s, I . By the semantics of CLL, µ t |= s, I if and only if µ t = e Qt µ(s) ∈ I. To check this, we first observe that the execution path e Qt µ of CTMCs is a system of polynomial exponential functions (PEFs).
Definition 8. A function f : R → R is a polynomial-exponential function (PEF) if f has the following form:
f (t) = K k=0 f k (t)e λ k t(6)
where for all 0
≤ k ≤ K < ∞, f k (t) ∈ F 1 [t]
, f k (t) = 0, λ k ∈ F 2 and F 1 , F 2 are fields. Without loss of generality, we assume that λ k 's are distinct.
Generally, for a PEF f (t) with the range in complex numbers C, g(t) = f (t) + f * (t) is a PEF with the range in real numbers R, where f * (t) is the complex conjugate of f (t). The factor t is omitted whenever convenient, i.e., f = f (t). t is called a root of a function f if f (t) = 0. PEFs often appear in transcendental number theory as auxiliary functions in the proofs involving the exponential function [31]. By the above lemma, for a given t in some bounded time interval T (to be specific in the latter discussion), e Qt µ(s) ∈ I is determined by the algebraic structure of PEF g(t) = e Qt µ(s) in T . That is all maximum intervals T max ⊆ T such that g(t) ∈ I for all t ∈ T max , where interval T max = ∅ is called maximum for g(t) ∈ I if no sub-intervals T ′ T max such that the property holds, i.e., g(t) ∈ I for all t ∈ T ′ . Then e Qt µ(s) ∈ I if and only if t ∈ T max for some maximum interval T max . So, we aim to compute the set T of all maximum intervals. By the continuity of PEF g(t), this can be done by identifying a real root isolation of the following PEF f (t) in T : f (t) = (g(t) − inf I)(g(t) − sup I).
A (real) root isolation of function f (t) in interval T is a set of mutually disjoint intervals, denoted by Iso(f ) T = {(a j , b j ) ⊆ T } for a j , b j ∈ Q such that for any j, there is one and only one root of f (t) in (a j , b j ); -for any root t * of f (t), t * ∈ (a j , b j ) for some j.
Furthermore, if f has no any root in T , then Iso(f ) T = ∅.
Although there are infinite kinds of real root isolations of f (t) in T , the number of isolation intervals equals to the number of distinct roots of f (t) in T .
Finding real root isolations of PEFs is a long-standing problem and can be at least backtracked to Ritt's paper [32] in 1929. Some following results were obtained since the last century (e.g. [33,34]). This problem is essential in the reachability analysis of dynamical systems, one active field of symbolic and algebraic computation. In the case of F 1 = Q and F 2 = N + in [19], an algorithm named ISOL was proposed to isolate all real roots of f (t). Later, this algorithm has been extended to the case of F 1 = Q and F 2 = R [20]. A variant of the problem has also been studied in [21]. The correctness of these algorithms is based on Schanuel's conjecture. Other works are using Schanuel's conjecture to do the root isolation of other functions, such as exp-log functions [35] and tame elementary functions [36].
By Lemma 1, we pursue this problem in the context of CTMCs. The distinct feature of solving real root isolations of PEFs in our paper is to deal with complex numbers C, more specifically algebraic numbers A, i.e., F 1 = F 2 = A. At the same time, to the best of our knowledge, all the previous works can only handle the case over R. Here, we develop a state-of-the-art real root isolation algorithm for PEFs over algebraic numbers. Thus from now on, we always assume that PEFs are over A, i.e., F 1 = F 2 = A in Eq. (6). In this case, it is worth noting that whether a PEF has a root in a given interval, T ⊆ R + is decidable subject to Schanuel's Conjecture if T is bounded [37], which falls in the situation we consider in this paper.
Theorem 3 ([37]
). Under the condition that Schanuel's conjecture holds, there is an algorithm to check whether a PEF f (t) has a root in interval T , i.e., whether Iso(f ) T = ∅.
In this paper, we extend the above checking Iso(f ) T = ∅ to computing Iso(f ) T of PEF f (t).
Theorem 4. Under the condition that Schanuel's conjecture holds, there is an algorithm to find real root isolation Iso(f ) T for any PEF f (t) and interval T . Furthermore, the number of real roots is finite, i.e., |Iso(f ) T | < ∞.
We can compute the set T of all maximum intervals with the above theorem to check atomic propositions. Furthermore, we can compare the values of any real roots of PEFs, which is important in model checking general multiphase timed until formulas at the end of this section.
Lemma 2. Let f 1 (t) and f 2 (t) be two PEFs with the domains in T 1 and T 2 , and t 1 ∈ T 1 and t 2 ∈ T 2 are roots of them, respectively. Under the condition that Schanuel's conjecture holds, there is an efficient way to check whether or not t 1 − t 2 < g for any given rational number g ∈ Q.
For model checking general state formula Φ, we can also use real root isolation of some PEF to obtain the set of all maximum intervals T max such that µ t |= Φ for all t ∈ T max . The reason is that Φ admits conjunctive normal form consisting of atomic propositions. See the proof of the following lemma in Appendix A.
Lemma 3. Under the condition that Schanuel's conjecture holds, given a time interval T , the set T of all maximum intervals in T satisfying µ t |= Φ can be computed, where Φ is a state formula of CLL. Furthermore, the number of all intervals in T is finite; the left and right endpoints of each interval in T are roots of PEFs.
At last, we characterize the multiphase timed until formulas by the reachability analysis of time intervals (instants).
Lemma 4. σ µ |= Φ 0 U T1 Φ 1 U T2 Φ 2 · · · U Tn Φ n if and only if there exist time in- tervals {I k ⊆ R + } n k=0 with I 0 = [0, 0] such that -The satisfaction of intervals: for all 1 ≤ k ≤ n, µ t |= Φ k−1 for all t ∈ I k ,
and µ t * |= Φ n , where t * = sup I n and µ t = e Qt µ ∀t ∈ R + ; -The order of intervals: for all 1 ≤ k ≤ n, I k ⊆ I k−1 + T k and inf I k = sup I k−1 + inf T k .
By the above lemma, the problem of checking multiphase timed until formulas is reduced to verify the existence of a sequence of time intervals. Now we can show the proof of Theorem 2.
Proof. Recall that the nontrivial step is to model check multiphase timed until
formula Φ 0 U T1 Φ 1 U T2 Φ 2 · · · U Tn Φ n , where {T j } n j=1
is a set of bounded rational intervals in R + , and for 0 ≤ k ≤ n + 1, Φ k is a state formula.
By Lemma 4, for model checking the above formula, we only need to check the existence of time intervals {I k } n k=0 illustrated in the lemma. The following procedure can construct such a set of intervals if it exists:
-(1) Let I 0 = {I 0 = [0, 0]} ; -(2)
For each 1 ≤ k ≤ n, obtaining the set I k in [0, k j=1 sup T j ] of all maximum intervals such that µ t |= Φ k−1 for all t ∈ I of I ∈ I , where µ t = e Qt µ; this can be done by Lemma 3. Noting that I k can be the empty set, i.e., I k = ∅; -(3) Let k from 1 to n. First, updating I k :
I k = {I ∩ (I ′ + T k ) : I ∈ I k and I ′ ∈ I k−1 }.
The above updates can be finished by Lemma 2. If I k = ∅, then the formula is not satisfied; -(4) Updating I n : for each I ∈ I n , we replace I with [s − ε, s) for some constant ε > 0 if there is an s ∈ I with s − ε ∈ I such that µ s |= Φ n where µ s = e Qs µ; Otherwise, remove this element from I n . Again, this can be done by Lemma 3. If I n = ∅, then the formula is not satisfied; -(5) Finally, let k from n − 1 to 1, updating I k :
I k = {[s − inf T k , s − inf T k ] : [s − ε, s) ∈ I k+1 ]}.
Thus after the above procedure, we have non-empty sets {I k } n k=0 with the following properties.
for each 1 ≤ k ≤ n, µ t |= Φ k−1 for all t ∈ I k and I k ∈ I k , and µ t * |= Φ n , where t * = sup I n ; -for each 1 ≤ k ≤ n, I ∈ I k , there exists at least one I ′ ∈ I k−1 such that I ⊆ sup I ′ + T k and inf I = sup I ′ + inf T k .
Therefore, we can get a set of intervals {I k } n k=0 satisfying the two conditions in Lemma 4 if it exists. On the other hand, it is easy to check that all such {I k } n k=0 must be in {I k } n k=0 , i.e., for each k, I k ⊆ I for some I ∈ I k . This ensures the correctness of the above procedure.
By the above constructive analysis, we give an algorithm for model checking CTMCs against CLL formulas. Focusing on the decidability problem, we do not provide the pseudocode of the algorithm. Alternatively, we implement a numerical experiment to illustrate the checking procedure in the next section.
Numerical Implementation
In this section, we implement a case study of checking CTMCs against CLL formulas. Here, we consider a symbolized CTMC SM = (S, Q, I ), where M = (S, Q) is the CTMC in Example 1 and finite set I is the one considered in Eq.(3). We check the properties of M given by the following two CLL path formulas mentioned in the introduction for different initial distributions. Then, we consider an initial distribution µ as the same as the one in Example 1.
Then we have that the value of e Qt µ is as follows:
e −3t 0 0 0 0 − 1 3 (e −3t − 1) 1 0 0 0 1 2 (e −3t − e −7t ) 0 e −7t 0 0 3 14 e −7t − 1 2 e −3t + 2 7 0 − 3 7 e −7t + 3 7 1 0 2 7 e −7t − 2 3 e −3t + 8 21 0 − 4 7 e −7t + 4 7 0 1 0.1 0.2 0.3 0.4 0 = 1 10 e −3t − 1 30 e −3t + 7 30 1 20 e −3t + 1 4 e −7t − 1 20 e −3t − 3 28 e −7t + 39 70 − 1 15 e −3t − 1 7 e −7t + 22 105 .
As we only consider states s 0 and s 1 in formulas ϕ and ϕ ′ , we focus on the following PEFs: f 0 (t) = 1 10 e −3t and f 1 (t) = − 1 30 e −3t + 7 30 . Next, we initialize the model checking procedures introduced in the proof of Theorem 2. First, we compute the set T of all maximum intervals T ⊆ [0 , 5] such that e Qt µ |= s 0 , [0.9, 1] for t ∈ T , i.e., f 0 (t) ∈ [0.9, 1] for t ∈ T . We obtain T = ∅ by the real root isolation algorithm mentioned in Theorem 4, and this indicates that σ µ |= ϕ where σ µ (t) = e Qt µ is the path induced by µ and defined in Eq.(1).
To check whether σ µ |= ϕ ′ , we compute the set T of all maximum intervals T ⊆ [0, 12] such that e Qt µ |= s 0 , [0.9, 1] for t ∈ T , i.e., f 0 (t) ∈ [0.9, 1] for t ∈ T . Again, we obtain T = ∅ by the real root isolation algorithm in Theorem 4. Therefore, σ µ |= ϕ ′ .
In the following, we consider a different initial distribution µ 1 as follows:
e Qt µ 1 = e Qt 0.9 0 0.1 0 0 = 9 10 e −3t − 3 10 (e −3t − 1) 9 20 e −3t − 7 20 e −7t − 9 20 e −3t + 3 20 e −7t + 3 10 − 3 5 e −3t + 1 5 e −7t + 2 5 .
The key PEFs are: g 0 (t) = 9 10 e −3t and g 1 (t) = − 3 10 (e −3t − 1). Again, we initialize the model checking procedures introduced in the proof of Theorem 2. We first compute the set T of all maximum intervals T ⊆ [0, 5] such that e Qt µ 1 |= s 1 , [0, 0.1] for t ∈ T , i.e., g 1 (t) ∈ [0, 0.1] for t ∈ T . This can be done by finding a real root isolation of the following PEF: g 0 1 (t) = − 3 10 (e −3t − 1) − 1 10 . By implementing the real root isolation algorithm in Theorem 4, we have Iso(g 0 1 ) [0,5] = {(0.13, 0.14)} and then T = {[0, t * ]} for t * ∈ (0.13, 0.14).
Following the same way, we compute T for e Qt µ 1 |= s 0 , [0.9, 1] . Then we complete the model checking procedures in the proof of Theorem 2, and we conclude: σ µ1 |= ϕ. By repeating these, the result of the second formula ϕ ′ is σ µ1 |= ϕ ′ .
Related Works
Agrawal et al. [16] introduced probabilistic linear-time temporal logic (PLTL) to reason about discrete-time Markov chains in the context of distribution transformers as we did for CTMCs in this paper. Interestingly, the Skolem Problem can be reduced to the model checking problem for the logic PLTL [38]. The Skolem Problem asks whether a given linear recurrence sequence has a zero term and plays a vital role in the reachability analysis of linear dynamical systems. Unfortunately, the decidability of the problem remains open [39]. Recently, the Continuous Skolem Problem has been proposed with good behavior (the problem is decidable) and forms a fundamental decision problem concerning reachability in continuous-time linear dynamical systems [37]. Not surprisingly, the Continuous Skolem Problem can be reduced to model-checking CLL. The primary step of verifying CLL formulas is to find a real root isolation of a PEF in a given interval. Chonev, Ouaknine and Worrell reformulated the Continuous Skolem Problem in terms of whether a PEF has a root in a given interval, which is decidable subject to Schanuel's conjecture [37]. An algorithm for finding root isolation can also answer the problem of checking the existence of the roots of a PEF. However, the reverse does not work in general. Therefore, the decidability of the Continuous Skolem Problem cannot be applied to establish that of our CLL model checking.
Remark 1. By adopting the method in this paper, we established the decidability of model checking quantum CTMCs against signal temporal logic [40]. Again, we need Schanuel's conjecture to guarantee the correctness. A Lindblad's master equation governs a quantum CTMC and a more general real-time probabilistic Markov model than a CTMC, i.e., a CTMC is an instance of quantum CTMCs. We converted the evolution of Lindblad's master equation into a distribution transformer that preserves the laws of quantum mechanics. We reduced the model-checking problem of quantum CTMCs to the real root isolation problem, which we considered in this paper, and thus our method could be applied to it.
Conclusion
This paper revisited the study of temporal properties of finite-state CTMCs by symbolizing the probability value space [0, 1] into a finite set of intervals. To specify relatively and absolutely temporal properties, we propose a probabilistic logic for CTMCs, namely continuous linear-time logic (CLL). We have considered the model checking problem in this setting. Our main result is that a state-of-theart real root isolation algorithm over the field of algebraic numbers was proposed to establish the decidability of the model checking problem under the condition that Schanuel's conjecture holds. This paper aims to show decidability in as simple a fashion as possible without paying much attention to complexity issues. Faster algorithms on our current constructions would significantly improve from a practical standpoint.
A Appendices: Proofs
A.1 Proof of Lemma 1
We need to use the Jordan decomposition to prove Lemma 1.
Definition 9.
A Jordan block is a square matrix of the following form.
λ 1 0 · · · 0 0 λ 1 · · · 0 . . . . . . . . . 1 0 0 0 · · · λ . A square matrix J is in Jordan norm form if J = J 1 J 2 . . . J n ,
where J k is a Jordan block for each 1 ≤ k ≤ n.
Because A is algebraic closed, we know that Proposition 1 ( [41]). Any matrix A ∈ Q n×n is algebraically similar to a matrix in Jordan normal form over the algebraic number field A. Namely, there exists some invertible P ∈ A n×n and J ∈ A n×n in Jordan form such that A = P −1 JP , where A n×n is the set of all n-by-n matrices with every entry being algebraic numbers.
Now we can give the proof of Lemma 1.
Proof. As the elements of µ are rational, we only need to prove that any entry of e Qt can be expressed as a finite sum of k f k (t)e η k t for η k ∈ A and f k (t) ∈ A[t].
By Proposition 1, we have that there is a P ∈ A n×n such that Q = P −1 (⊕ k J k )P such that
J k = λ k 1 0 · · · 0 0 λ k 1 · · · 0 . . . 0 0 0 · · · λ k ,
where λ k is an eigenvalue of Q and Q ∈ Q d×d , so λ k is algebraic. Furthermore, J k ∈ A n k ×n k , where n k is the dimension of J k . Therefore, e Qt = P −1 e ⊕ k J k t P = P −1 (⊕ k e J k t )P . We complete the proof by proving that for each k, any entry of e J k t can be expressed as a finite sum of k f k (t)e η k t for η k ∈ A and f k (t) ∈ A[t]. Computing e J k t , we obtain that
e J k t = e λ k t te λ k t t 2 e λ k t /2! · · ·
t n k e λ k t /n k ! 0 e λ k t te λ k t · · · t n k −1 e λ k t /(n k − 1)! . . .
0 0 0 · · · e λ k t .
A.2 Proof of Lemma 2
Before proving, we need the following fact observed in [3] according to the Lindemann-Weierstrass theorem.
Observation 1 Given a real number r ∈ R of the form k µ k e η k where µ k ′ s and η k ′ s are algebraic complex numbers, and the η k ′ s are pairwise distinct, there is an effective procedure to compare the values of r and c for any c ∈ Q.
According to Lindemann-Weierstrass theorem, we know that r = c if and only if r = c = 0 or c = −µ k for some k with η k = 0. Otherwise, we can compute a good approximation of r. For each k, e η k can be approximated with an error, the norm of the difference between r and the approximation, of less than ε (for any ε < 1) by taking the first ⌈3|η k | 2 /ε⌉ + 1 terms of the Maclaurin expansion for e η k . This leads to an approximation of r within k |µ k |ε. Since the individual terms in the Maclaurin expansion are algebraic functions of the η k 's, it follows that the approximations are algebraic. Then we can check if r > c by the comparision between the approximations and c. See [3] for more details.
Proof. First, by Theorem 4, isolating the real roots of f 1 (t) and f 2 (t), we have t 1 ∈ (a 1 , b 1 ) and t 2 ∈ (a 2 , b 2 ) for a 1 , a 2 , b 1 , b 2 ∈ Q. Then we first check if t 1 −t 2 = g. Note that t 1 − t 2 = g if and only if f 2 1 (t) + f 2 2 (t + g) = 0 has a root in (a 1 − g, b 1 − g) ∩ (a 2 , b 2 ). f 2 1 (t) + f 2 2 (t + g) is still a PEF, then we can check whether or not there is a root of it in (a 1 , b 1 ) ∪ (a 2 , b 2 ) by Theorem 3.
If t 1 − t 2 = g, we answer whether or not t 1 − (t 2 + g) < 0 by narrowing (a 1 , b 1 ) and (a 2 , b 2 ) and maintaining the roots of f 1 (t) and f 2 (t) in the intervals. This can be done as we can arbitrarily narrow the interval (a 1 , b 1 ) by comparing the signs (> 0, < 0 or = 0) of f ( a1+b1 2 ) and f (a 1 ), and the same way works for narrowing (a 2 , b 2 ). The sign of f (a) for any a ∈ Q can be obtained by Observation 1 and Theorem 1. Moreover, there is a gap between t 1 and t 2 + g, and we can compare t 1 and (t 2 + g) by continuously narrowing (a 1 , b 1 ) and (a 2 , b 2 ), and comparing a 1 and b 2 (a 2 and b 1 ).
A.3 Proof of Lemma 3
Proof. Recall that state formulas of CLL are given by the following grammar:
Φ := true | a ∈ AP | ¬Φ | Φ 1 ∧ Φ 2 .
By the semantics of CLL state formulas, Φ k is a formula of propositional logics. So Φ admits conjunctive normal form (CNF) [42,Appendix A.3], i.e.,
Φ = ∧ j∈J ∨ l∈Lj lit j,l ,
where lit j,l is a literal of a ∈ AP or ¬a, and J k and L k are both finite sets. Furthermore, we observe that ¬a is (semantically equivalent to) either an atomic proposition or a disjunction of two atomic propositions. To prove this, let a = k, T for some interval T . We deal with the case of T = [c 1 , c 2 ] for 0 < c 1 < c 2 < 1, and the other situations of T can be done by the similar way. In this case, for any distribution µ ∈ D(S), µ |= ¬a if and only if µ |= a 1 ∨ a 2 , i.e. ¬a = a 1 ∨ a 2 , where a 1 = k, [0, c 1 ) and a 2 = k, (c 2 , 1] . Therefore,
Φ = ∧ j∈J ′ ∨ l∈L ′ j a j,l for some finite sets J ′ , L ′ j for j ∈ J ′ , {a j,l ∈ AP } j∈J ′ ,l∈L ′ j . From µ t |= ∧ j∈J ′ ∨ l∈L ′
j a j,l , by the semantic of CLL, we have that for all j ∈ J ′ , µ t |= a j,l for some l ∈ L ′ j . As J ′ and all L ′ j are finite sets, we can check one by one whether or not µ t |= a j,l . Let T j,l be the set of all the maximum intervals such that for each T ∈ T j,l , µ t |= a j,l for all t ∈ T . Then
T = ∩ j∈J ′ T j ,
where T j = {T 1 ∪T 2 : T 1 ∈ T j,l1 and T 2 ∈ T j,l2 for all distinct l 1 , l 2 ∈ L ′ j } for all j ∈ J ′ ,
and T i ∩ T j = {T 1 ∩ T 2 : T 1 ∈ T i and T 2 ∈ T j } for all i, j ∈ J ′ .
The left problem is to handle T 1 ∩ T 2 and T 1 ∪ T 2 , and by the continuity of PEFs, the left and right endpoints of I 1 and I 2 are all the roots of PEFs. It is equivalent to compare the values of two real roots of two (different) PEFs. This can be done by Lemma 2.
A.4 Proof of Lemma 4
Proof. We first prove the sufficient direction. Let
ϕ 1 = Φ 1 U T2 Φ 2 · · · U Tn Φ n .
Then the above formula is Φ 0 U T1 ϕ 1 . By the semantic of CLL, we have that there is a time t 1 ∈ T 1 such that σ µt 1 |= ϕ 1 , and for any t ′ 1 ∈ [0, t) ∩ T 1 , µ t ′ 1 |= Φ 0 . Then let ϕ 2 = Φ 2 U T3 Φ 3 · · · U Tn Φ n and we get ϕ 1 = Φ 1 U T2 ϕ 2 . In the similar way, we have that there is a time t 2 ∈ T 2 such that σ µt 1 +t 2 |= ϕ 2 , and for any t
′ 2 ∈ [0, t 2 ) ∩ T 2 , µ t1+t ′ 2 |= Φ 1 .
Iteratively, we get a set of time instants {t k } n k=0 with t 0 = 0. For all 1 ≤ k ≤ n, let
I k = k−1 j=0 t j + [0, t k ) ∩ T k .
Then it is easy to check that {I k ∈ R + } k=0 with I 0 = [0, 0] are the desired intervals satisfying the above two conditions.
Considering the necessary direction, by the above proof, we only need to identify {t k } n k=1 throughout intervals {I k } n k=0 . Let t k = sup I k − sup I k−1 for all 1 ≤ k ≤ n.
A.5 Proof of Theorem 4
Before presenting the proof of Theorem 4, we recall one useful technique from previous works -factoring PEFs.
Factoring PEFs An essential step of finding a real root isolation of a PEF is factoring it into a set of PEFs, such that the resulting PEFs have no multiple roots except for zero. In the following, we introduce a technique to implement this. Before that, we recall some concepts.
Given a root t * of a function f (t), i.e., f (t * ) = 0, the multiplicity of t * is the maximum number m such that (t − t * ) m is a factor of f (t), i.e., there exists a function g(t) such that f (t) = g(t)(t − t * ) m . In particular, if m = 1, then we call that t * is a single root ; otherwise t * is a multiple root.
Recall a PEF f : R → R has the following form:
f (t) = K k=0 f k (t)e λ k t(8)
where for all 0 ≤ k ≤ K < ∞, f k (t) ∈ A[t], λ k ∈ A. Without loss of generality, we assume λ k 's are distinct, and use Exp(f ) to denote the set of exponent of f (t), i.e., Exp(f ) = {λ k } K k=0 . Let {a j } n j=1 be a linearly independent basis over Q of the exponents Exp(f ) = {λ k } K k=0 appearing in f (t) such that f (t) is a multivariate polynomial with respect to t, e a1t , . . . , e ant , denoted by f (t, e a1t , . . . , e ant ).
This basis can be obtained by computing a simple extension (e.g. [43], [44, Algorithm 1]) Q(λ)/Q = Q(λ 0 , . . . , λ K )/Q. Next, we replace t by the indeterminate y 0 , and each exponential term e ait by y i for 1 ≤ i ≤ n. Then, we have a multivariate polynomial representation f (y 0 , . . . , y n ) of f (t) in the y ′ s. By the factorization of polynomials, we have f (y 0 , . . . , y n ) = f m1 1 (y 0 , . . . , y n ) · · · f ms s (y 0 , . . . , y n ) for m 1 , . . . , m s ∈ Z + whose greatest square free part is denoted bŷ f (y 0 , . . . , y n ) = f 1 (y 0 , . . . , y n ) · · · f 1 (y 0 , . . . , y n ).
Via switching y ′ s back by e ait and t, we have a PEF f (t) =f (t, e a1t , . . . , e ant ).
It is worth noting that the set of roots off (t) is the same to the one of f (t). In the latter, we will provef (t) only has single roots. At last, a corollary of Schanuel's conjecture is also needed to prove Theorem 4.
Corollary 1. [20, Corollary 3] Let a 1 , . . . , a n be algebraic numbers that are linearly independent over Q. Under the condition that Schanuel's conjecture holds, the transcendence degree of the extension field Q(t, e a1t , . . . , e ant ) is at least n if t = 0.
Main Proof In this subsection, we prove Theorem 4 with the help of the exclusion algorithm in [44, Algorithm 2] which can compute a real root isolation Iso(f ) T for a given simple PEF f (t) and interval T . Here, a PEF f (t) is simple if all roots of f (t) are single.
In particular, if f (t) has only single roots in T (there is no t ∈ T such that f (t) = f ′ (t) = 0), then we can get a real root isolation Iso(f ) T by [44,Algorithm 2]. Otherwise, f (t) has at least one multiple root, and thus the exclusion algorithm in [44] cannot be used to compute a real root isolation of f (t) directly. This can be dealt with by employing factoring PEFs in the last section and the following fact.
Lemma 5. Let f (t) = f (t, e a1t , . . . , e ant ) be a PEF with respect to t, and thus a polynomial with respect to t, e a1t , . . . , e ant , where {a i } n i=1 is linearly independent over Q. If the multivariate polynomial representation f (y 0 , . . . , y n ) of f (t) is square free, then, assuming that Schanuel's conjecture holds, f (t) has no multiple real root except 0.
Proof. Since f (y 0 , . . . , y n ) is square free, we may write f (y 0 , . . . , y n ) = f 1 (y 0 , . . . , y n ) · · · f 1 (y 0 , . . . , y n ) where for any 1 ≤ i, j ≤ n, i = j, f i (y 0 , . . . , y n ) is irreducible, and f i (y 0 , . . . , y n ) and f j (y 0 , . . . , y n ) are coprime, i.e., they do not share a polynomial g(y 0 , . . . , y n ) as a factor.
By contradiction, we first prove that f i (t, e a1t , . . . , e ant ) and f j (t, e a1t , . . . , e ant ) have no nonzero common real root with respect to t. Assume that t 0 = 0 is a common real root of f i (t, e a1t , . . . , e ant ) and f j (t, e a1t , . . . , e ant ). By Corollary 1, we have that the transcendence degree of Q(t 0 , e a1t0 , . . . , e ant0 ) is at
[0, 1] into a finite set of intervals I = {I k ⊆ [0, 1]} m k=1 . A probability distribution µ over its set of states S = {s 0 , s 2 , . . . , s d−1 } is then represented symbolically as a set of symbols S(µ) = { s, I ∈ S × I : µ(s) ∈ I}
T 1 ∩ T 2 = ∅. Let us see some concrete examples:
s 0 ,
0[0.9, 1] ∧ s 1 , [0, 0.1] U [0,5] s 0 , [0.9, 1] ).
Qt µ ∀t ∈ R + . Before solving the model-checking problem of CTMCs against CLL formulas in the next section, we shall first discuss what can be specified in our logic CLL. Given a CTMC (S, Q), CLL path formula ♦ [0,1000] s, [1, 1] expresses a liveness property that state s ∈ S is eventually reached with probability one before time instant 1000. In terms of safety properties, formula [100,1000] s, [0, 0] represents that state s ∈ S is never reached (reached with probability zero) between time instants 100 and 1000. Furthermore, setting the intervals nontrivial (neither [0, 0] or [1, 1]), liveness and safety properties can be asserted with probabilities, such as ♦ [0,1000] s, [0.5, 1] and [100,1000] s, [0, 0.5] . For multiphase timed until formula s, [0.7, 1] U [2,3] s, [0.7, 1] . . . U [2,3] s, [0.7, 1]
Lemma 1 .
1Given a CTMC M = (S, Q) with S = {s 0 , . . . , s d−1 }, Q ∈ Q d×d , and an initial distribution µ ∈ Q d×1 , for any 0 ≤ i ≤ d − 1 , e Qt µ(s i ), the i-th entry of e Qt µ, can be expressed as a PEF f : R + → [0, 1] as in Eq.(6) with F 1 = F 2 = A.
ϕ
= s 1 , [0, 0.1] U [0,5] s 0 , [0.9, 1] . ϕ ′ = s 0 , [0.9, 1] U [3,7] s 1 , [0, 0.1] U [0,5] s 0 , [0.9, 1] . By Jordan decomposition, we have Q = SJS −
AcknowledgmentsWe want to thank Professor Joost-Pieter Katoen for his invaluable feedback and for pointing out the references[12,13,23]. This work is supported by the National Keyleast n. Thus there must exist n elements in {t 0 , e a1t0 , . . . , e ant0 } that are algebraically independent. Without loss of generality, let {t 0 , e a1t0 , . . . , e an−1t0 } be the n elements that are algebraically independent. Considering the multivariate polynomial representation, let g(y 0 , . . . , y n ) be the resultant[45,Page 26]of f i (y 0 , . . . , y n ) and f j (y 0 , . . . , y n ) with respect to y n . Then (t 0 , e a1t0 , . . . , e an−1t0 ) is a root of g(y 0 , y 1 , . . . , y n−1 ), which is a nontrivial polynomial as f i (t, e a1t , . . . , e ant ) and f j (t, e a1t , . . . , e ant ) are coprime. Therefore, (t 0 , e a1t0 , . . . , e ant0 ) is a root of some nontrivial polynomial. This contradicts that {t 0 , e a1t0 , . . . , e ant0 } are algebraically independent.Next, we prove that f i (t, e a1t , . . . , e ant ) has no multiple real root. Supposewhere h 0 (t), . . . , h n (t) are nontrivial polynomials and b jk ∈ N for 1 ≤ j ≤ s and 1 ≤ k ≤ n. Then we haveMoreover, considering the corresponding multivariate polynomial representation, f i (y 0 , y 1 , . . . , y n ) = h 0 (y 0 ) + s k=1 h k (y 0 )(y 1 ) b k1 · · · (y n ) b kn f ′ i (y 0 , y 1 , . . . , y n )From the degree of h 0 (y 0 ) in the above two polynomials, it is evident to see that f i (y 0 , y 1 , . . . , y n ) is not a factor of f ′ i (y 0 , y 1 , . . . , y n ). Then, f i (y 0 , y 1 , . . . , y n ) and f ′ i (y 0 , y 1 , . . . , y n ) are coprime, since f i (y 0 , y 1 , . . . , y n ) is irreducible. For the same reason as above, f i (t, e a1t , . . . , e ant ) and f ′ i (t, e a1t , . . . , e ant ) have no common real roots. Therefore, f i (t, e a1t , . . . , e ant ) has no multiple real root.By the above lemma, for finding a real root isolation Iso(f ) T of a PEF f (t) in some time interval T , we first implement the method of factoring PEFs in the last section to get a simple PEFf (t), which shares the same roots with f (t). Then we can implement the exclusion algorithm in[44]to achieve this goal. Finally, we complete the proof of Theorem 4 by noting that the number of obtained real root isolations by the algorithm is finite (see the correctness analysis of [44, Algorithm 2]), i.e., |Iso(f ) T | < ∞.
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| []
|
[
"Magnetoelectric Jones Dichroism in Atoms",
"Magnetoelectric Jones Dichroism in Atoms"
]
| [
"D Budker ",
"J E Stalnaker ",
"\nDepartment of Physics\nNuclear Science Division\nUniversity of California at Berkeley\n94720-7300BerkeleyCalifornia\n",
"\nLawrence Berkeley National Laboratory\n94720BerkeleyCalifornia\n"
]
| [
"Department of Physics\nNuclear Science Division\nUniversity of California at Berkeley\n94720-7300BerkeleyCalifornia",
"Lawrence Berkeley National Laboratory\n94720BerkeleyCalifornia"
]
| []
| The authors suggest that atomic experiments measuring the interference between magnetic-dipole and electric-field-induced electric-dipole transition amplitudes provide a valuable system to study magnetoelectric Jones effects. PACS numbers: 42.25.Lc,32.60.+i A recent letter [1] reported the first observation of magnetoelectric Jones birefringence in liquids (see also Ref.[2]). This observation helped to clarify some of the longstanding theoretical confusion surrounding Jones birefringence and the associated Jones dichroism (collectively known as Jones effects)[3,4]. The interest in further understanding these effects has led to the investigation of other experimental systems which may exhibit Jones effects. These include the possibility of observing the effects through beam divergence in uniaxial crystals[5]and possible observation in the quantum vacuum[6]. In this letter we point out that Jones dichroism can be studied in atomic systems under much less severe experimental requirements. In addition, these atomic systems are more amenable to theoretical analysis than the relatively complicated condensed-matter systems that have been studied to date. The simplicity of these systems may help to expand the understanding of the manifestation of Jones effects in general. We also point out that our recent experiment [7] measuring interference between magneticdipole and electric-field-induced electric-dipole transition amplitudes in atomic ytterbium constitutes a measurement of Jones dichroism in a simple atomic system. | 10.1103/physrevlett.91.263901 | [
"https://arxiv.org/pdf/physics/0302096v3.pdf"
]
| 8,823,968 | physics/0302096 | 2a7070404b6da253b3012ed8c168823102a93717 |
Magnetoelectric Jones Dichroism in Atoms
26 Sep 2003
D Budker
J E Stalnaker
Department of Physics
Nuclear Science Division
University of California at Berkeley
94720-7300BerkeleyCalifornia
Lawrence Berkeley National Laboratory
94720BerkeleyCalifornia
Magnetoelectric Jones Dichroism in Atoms
26 Sep 2003(Dated: November 2, 2018)arXiv:physics/0302096v3 [physics.optics]PACS numbers: 4225Lc,3260+i
The authors suggest that atomic experiments measuring the interference between magnetic-dipole and electric-field-induced electric-dipole transition amplitudes provide a valuable system to study magnetoelectric Jones effects. PACS numbers: 42.25.Lc,32.60.+i A recent letter [1] reported the first observation of magnetoelectric Jones birefringence in liquids (see also Ref.[2]). This observation helped to clarify some of the longstanding theoretical confusion surrounding Jones birefringence and the associated Jones dichroism (collectively known as Jones effects)[3,4]. The interest in further understanding these effects has led to the investigation of other experimental systems which may exhibit Jones effects. These include the possibility of observing the effects through beam divergence in uniaxial crystals[5]and possible observation in the quantum vacuum[6]. In this letter we point out that Jones dichroism can be studied in atomic systems under much less severe experimental requirements. In addition, these atomic systems are more amenable to theoretical analysis than the relatively complicated condensed-matter systems that have been studied to date. The simplicity of these systems may help to expand the understanding of the manifestation of Jones effects in general. We also point out that our recent experiment [7] measuring interference between magneticdipole and electric-field-induced electric-dipole transition amplitudes in atomic ytterbium constitutes a measurement of Jones dichroism in a simple atomic system.
The development of the Jones matrix calculus for describing the propagation of light led to the prediction of two distinct types of linear birefringence and dichroism [8]. The two types of effects differ in the orientation of the birefringent and dichroic axes. The Jones formalism revealed that certain uniaxial media may exhibit birefringence and dichroism along axes which are at ±45 • relative to the axis of anisotropy. Birefringent and dichroic effects of this type are called Jones effects. They are distinct from the familiar birefringence and dichroism, which have axes parallel and perpendicular to the axis of anisotropy.
There has been theoretical discussion concerning the requirements for media that may exhibit Jones effects and what transition moments must be accounted for in order to describe it [3,4]. In Ref. [4], it was shown that Jones effects may be induced in isotropic media by the application of parallel electric and magnetic fields. If the direction of light propagation is perpendicular to the electric and magnetic fields, the Jones effects are described by
∆n J ≡ n +45 • − n −45 • ,(1)
where n ±45 • is the complex index of refraction for light polarized at ±45 • relative to the electric and magnetic fields. The real and imaginary parts of ∆n J describe Jones birefringence and dichroism, respectively. On a microscopic level, Jones dichroism may manifest itself as a difference between the rates with which atoms of the medium are transferred to the excited state in the presence of light polarized at ±45 • relative to the electric and magnetic fields given by
∆Γ J ≡ Γ +45 • − Γ −45 • .(2)
Jones effects generally occur in materials which exhibit the more familiar birefringence and dichroism. In addition, Jones effects are predicted to be significantly smaller than the usual birefringence and dichroism in most media. Consequently, magnetoelectric Jones birefringence has been observed only recently in molecular liquids under extreme experimental conditions [1,2].
To our knowledge, the observation of Jones dichroism has not been reported as such. Here we point out that Stark-interference experiments [9] which utilize parallel electric and magnetic fields provide a simple atomic system which exhibits Jones dichroism. The experiment [7] measuring a highly forbidden magnetic-dipole transition amplitude in atomic ytterbium using this technique constitutes such a system and its results can be interpreted as an observation of magnetoelectric Jones dichroism.
In the experiment [7], we studied a highly forbidden transition between states of the same parity. In the absence of external fields and neglecting paritynonconserving effects, the transition occurs only through a small magnetic-dipole amplitude (≈ 10 −4 µ B , where µ B is the Bohr magneton). By applying a static electric field, an electric-dipole amplitude is induced through mixing of opposite-parity states. An atomic beam of ytterbium was excited with resonant laser light propagating perpendicularly to parallel electric and magnetic fields. The excitation light was polarized at an angle θ relative to the external fields. For the transition studied in our experiment (between a ground state with total angular momentum equal to zero and an excited state with total angular momentum equal to one), the electric field, E, results in a Stark-induced electric-dipole transition amplitude to the M ′ J magnetic sublevel of the excited state given by [9]
A(E1 St ) = i β (E × ε) −M ′ J ,(3)
where ε is the electric-field amplitude of the laser light,
(E × ε) −M ′ J is the −M ′
J component of the vector in the spherical basis, and β is the vector transition polarizability [10]. The magnetic-dipole transition amplitude is given by
A(M 1) = µ(k × ε) −M ′ J ,(4)
wherek is the direction of propagation of the excitation light,k × ε is the magnetic-field amplitude of the light, and µ is the magnetic-dipole matrix element between the ground state and any of the M ′ J magnetic sublevels of the excited state. The transition rate is therefore
Γ ∝ M ′ J |A(E1 St ) + A(M 1)| 2 ∝ M ′ J |A(E1 St )| 2 + 2Re A(E1 St )A(M 1) * + |A(M 1)| 2 .(5)
As is discussed in Ref. [7], the interference term in Eq. (5) is of opposite sign for the M ′ J = +1 and M ′ J = −1 magnetic sublevels. It is therefore necessary to apply a magnetic field to resolve the different sublevels in order to observe the effect of this term. The signal due to the interference term is proportional to the rotational invariant
[(E × ε) × (k × ε)] ·B,(6)
which is also true in a more general case where E and B are not necessarily collinear. We define the z axis to be along the direction of the magnetic field, B = Bẑ, and define the x axis so that the electric field lies in the x-z plane, E = E xx + E zẑ . We assume that the light propagation is perpendicular to both fields, k = kŷ and that the light is linearly polarized at an angle θ relative to the magnetic field,ε = sinθx + cosθẑ (Fig. 1). The |A(E1 St )| 2 and |A(M 1)| 2 terms in Eq. (5) are independent of the sign of the angle of polarization while the interference term is odd with θ. Using expression (6) it is easily shown that the difference in transition rates for ±θ results in a Jones dichroism given by
Γ +θ − Γ −θ ∝ E z sinθ cosθ.(7)
The factor E z in Eqn. (7) shows that which is the predicted dependence of the magnetoelectric Jones effects on E and B [4]. It is interesting to note that the transition rate depends on the magnitude of the magnetic field only for values of the magnetic field which do not fully resolve the magnetic sublevels. This is analogous to the change in magnetic-field dependence of the resonant Faraday rotation (see for example Ref. [11]).
∆Γ J ∝ E ·B,(8)
Due to the weakness of the forbidden transition studied, we determined the transition rate by observing fluorescence in a decay branch of the excited state rather than detecting absorption. In our experiment the observed Jones dichroism is significantly smaller (≈ 5×10 −3 at the electric fields used in the experiment) than the normal dichroism, which is dominated by the Stark-induced amplitude. As can be seen from Eq.(3), only the component of the light electric field that is perpendicular to E contributes to the transition rate. Thus, the dominant fluorescence signal is proportional to sin 2 θ and to |E| 2 . These dependences were verified experimentally. The interference term responsible for Jones dichroism was isolated from the dominant signal by comparing the fluorescence spectra for opposite electric fields. In the data analysis we normalized the interference term to the dominant signal resulting in an asymmetry given by
Γ(E + ) − Γ(E − ) Γ(E + ) + Γ(E − ) = 2M 1 βE cos(θ) sin(θ) M J .(9)
The dependence of the asymmetry on the electric field, magnetic field, and polarization angle was verified experimentally (see Ref. [7] for figure showing the interference term versus the magnitude of the electric field). Figure 2 shows the experimental fractional asymmetry [Eq. (9)] plotted versus the polarization angle. We have normalized the signal to the magnitude of the electric field in order to combine data taken a different electric fields and leave only the polarization angle dependence. Also shown is the expected angular dependence of the asymmetry. Most of the data was taken with light polarized at θ = ±45 • relative to the electric field since the interference term is maximal at these values [see Eq. (7)].
The difference in the sign of the asymmetry for θ = ±45 • clearly indicates a nonzero value of ∆Γ J , verifying the key signature of Jones dichroism. We note that although the Jones dichroism was significantly smaller than the usual dichroism in our experiment, it is possible to significantly increase its size by using an allowed magnetic-dipole transition. In fact, it possible to have both the Jones dichroism and the regular dichroism of the same order as the overall absorption, which can be substantial in the case of an allowed magnetic-dipole transition.
Finally, we point out that atomic systems may be of use in measuring other types of magnetoelectric effects which are currently being studied in more complicated systems, such as more common forms of magnetoelectric linear birefringence [12] and magnetoelectric directional anisotropy [13]. In fact, expression (6) shows that this system exhibits both of these effects. It is interesting to note that a polarization-dependent directional anisotropy is present for both parallel and perpendicular electric and magnetic fields. For the case of perpendicular electric and magnetic fields a component of the directional anisotropy is present even when averaged over the polarization angle.
FIG. 1: Orientation of external fields.
FIG. 2 :
2Experimental results showing the dependence of the fractional transition-rate asymmetry, normalized by the magnitude of the electric field, on the angle of light polarization. Solid line shows the expected dependence. Data was taken in the work of Ref.[7] and experimental details are contained therein.
The authors acknowledge their co-authors in Ref. [
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The notation for the electric-field-induced transition amplitude and the magnetic-dipole transition amplitude used here differs from that of Ref. [7] in the sign of the MJ spherecial component. We adopt the present notation as it is more consistent with the standard convention of having a transition rate which is proportional to d · ε = q (−1) q dq ε−q. where d is the transition dipole moment and ε is the electric-field amplitude of the light. This change in notation does not change the results of [7The notation for the electric-field-induced transition am- plitude and the magnetic-dipole transition amplitude used here differs from that of Ref. [7] in the sign of the MJ spherecial component. We adopt the present nota- tion as it is more consistent with the standard conven- tion of having a transition rate which is proportional to d · ε = q (−1) q dq ε−q, where d is the transition dipole moment and ε is the electric-field amplitude of the light. This change in notation does not change the results of [7].
. D Budker, W Gawlik, D F Kimball, S M Rochester, V V Yashchuk, A Weis, Rev. Mod. Phys. 741153D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, Rev. Mod. Phys. 74, 1153 (2002).
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| []
|
[
"Electrical and Thermal Transport in Inhomogeneous Luttinger Liquids",
"Electrical and Thermal Transport in Inhomogeneous Luttinger Liquids"
]
| [
"Wade Degottardi \nMaterials Science Division\nArgonne National Laboratory\n60439ArgonneIllinoisUSA\n",
"K A Matveev \nMaterials Science Division\nArgonne National Laboratory\n60439ArgonneIllinoisUSA\n"
]
| [
"Materials Science Division\nArgonne National Laboratory\n60439ArgonneIllinoisUSA",
"Materials Science Division\nArgonne National Laboratory\n60439ArgonneIllinoisUSA"
]
| []
| We study the transport properties of long quantum wires by generalizing the Luttinger liquid approach to allow for the finite lifetime of the bosonic excitations. Our theory accounts for longrange disorder and strong electron interactions, both of which are common features of experiments with quantum wires. We obtain the electrical and thermal resistances and thermoelectric properties of such quantum wires. We cast our results in terms of the thermal conductivity and bulk viscosity of the electron liquid and give the temperature scale above which the transport can be described by classical hydrodynamics. PACS numbers: 71.10.Pm, 73.23.-bIntroduction.-The quantization of conductance exhibited by quantum wires in multiples of G 0 = 2e 2 /h highlights the crucial role played by the confinement of the electronic wave function in directions transverse to the wire [1-4]. Quantum wires potentially represent a window into the rich array of non-Fermi liquid phenomena predicted for one-dimensional electron systems[4,5]. The most direct means of probing such wires experimentally is through their transport properties. In addition to reduced dimensionality, aspects of these system which are crucial to understanding these properties include strong electron-electron interactions as well as slowly varying disorder potentials which inevitably arise in the fabrication of such devices. While fabrication techniques are now able to limit sources of short range disorder, such as impurities in heterostructure realizations, long range disorder remains a feature of these systems as it arises from the process of modulation doping used to populate the two-dimensional electron gas from which quantum wires are patterned[6]. | 10.1103/physrevlett.114.236405 | [
"https://arxiv.org/pdf/1412.0693v1.pdf"
]
| 29,794,995 | 1412.0693 | cdaf85c04d444a0ad6f26bdb62780bb7bd68830a |
Electrical and Thermal Transport in Inhomogeneous Luttinger Liquids
Wade Degottardi
Materials Science Division
Argonne National Laboratory
60439ArgonneIllinoisUSA
K A Matveev
Materials Science Division
Argonne National Laboratory
60439ArgonneIllinoisUSA
Electrical and Thermal Transport in Inhomogeneous Luttinger Liquids
(Dated: December 3, 2014)
We study the transport properties of long quantum wires by generalizing the Luttinger liquid approach to allow for the finite lifetime of the bosonic excitations. Our theory accounts for longrange disorder and strong electron interactions, both of which are common features of experiments with quantum wires. We obtain the electrical and thermal resistances and thermoelectric properties of such quantum wires. We cast our results in terms of the thermal conductivity and bulk viscosity of the electron liquid and give the temperature scale above which the transport can be described by classical hydrodynamics. PACS numbers: 71.10.Pm, 73.23.-bIntroduction.-The quantization of conductance exhibited by quantum wires in multiples of G 0 = 2e 2 /h highlights the crucial role played by the confinement of the electronic wave function in directions transverse to the wire [1-4]. Quantum wires potentially represent a window into the rich array of non-Fermi liquid phenomena predicted for one-dimensional electron systems[4,5]. The most direct means of probing such wires experimentally is through their transport properties. In addition to reduced dimensionality, aspects of these system which are crucial to understanding these properties include strong electron-electron interactions as well as slowly varying disorder potentials which inevitably arise in the fabrication of such devices. While fabrication techniques are now able to limit sources of short range disorder, such as impurities in heterostructure realizations, long range disorder remains a feature of these systems as it arises from the process of modulation doping used to populate the two-dimensional electron gas from which quantum wires are patterned[6].
We study the transport properties of long quantum wires by generalizing the Luttinger liquid approach to allow for the finite lifetime of the bosonic excitations. Our theory accounts for longrange disorder and strong electron interactions, both of which are common features of experiments with quantum wires. We obtain the electrical and thermal resistances and thermoelectric properties of such quantum wires. We cast our results in terms of the thermal conductivity and bulk viscosity of the electron liquid and give the temperature scale above which the transport can be described by classical hydrodynamics. Introduction.-The quantization of conductance exhibited by quantum wires in multiples of G 0 = 2e 2 /h highlights the crucial role played by the confinement of the electronic wave function in directions transverse to the wire [1][2][3][4]. Quantum wires potentially represent a window into the rich array of non-Fermi liquid phenomena predicted for one-dimensional electron systems [4,5]. The most direct means of probing such wires experimentally is through their transport properties. In addition to reduced dimensionality, aspects of these system which are crucial to understanding these properties include strong electron-electron interactions as well as slowly varying disorder potentials which inevitably arise in the fabrication of such devices. While fabrication techniques are now able to limit sources of short range disorder, such as impurities in heterostructure realizations, long range disorder remains a feature of these systems as it arises from the process of modulation doping used to populate the two-dimensional electron gas from which quantum wires are patterned [6].
At sufficiently high temperatures, the transport properties of electron liquids can be described by classical hydrodynamics [7]. This approach offers an ostensibly classical description of systems in cases for which the characteristic length scale associated with violations of momentum conservation is much longer than the electronelectron mean-free path. In one dimension, the conductance can be expressed in terms of the thermal conductivity κ and the bulk viscosity ζ of the electron liquid [7].
Ultimately, a complete theoretical description of these wires requires a quantum mechanical treatment. The case of weakly interacting electrons in a wire with long range disorder has been studied [8]. However, the applicability of this theory to experiments is limited since electron-electron interactions in wires are typically strong. A commonly employed theoretical framework for studying interacting electrons in one dimension is the Luttinger liquid (LL) formalism, a powerful nonperturbative approach which can account for electron interactions of arbitrary strength. In this framework, the excitations of the electron liquid are described by non-interacting bosons [4,5]. The effects of short range disorder on the conductance properties of a LL are well understood [9][10][11], however the effects of long range inhomogeneities have not been explored.
In this work, we study an inhomogeneous LL and calculate its transport properties. Standard LL theory, in which the bosonic excitations are infinitely long-lived, predicts that the conductance of quantum wire remains G 0 even for strongly interacting electrons [12]. Accounting for the scattering of the bosonic excitations, we find that the conductance is suppressed below G 0 . In addition to the electrical resistance, we obtain expressions for the thermal resistance, the Peltier coefficient, and the thermopower.
Our theory holds insofar as LL theory is applicable. In particular, the temperature T must be lower than the bandwidth D, which is typically on the order of the Fermi energy of the electron liquid. We identify a temperature scale T * such that for T * T D, our result for the resistance of a quantum wire with weak disorder reduces to that of the hydrodynamical theory [7]. An interesting aspect of our results is that although they differ from those of classical hydrodynamics at low temperatures, T T * , they can still be expressed in terms of the thermal conductivity κ and bulk viscosity ζ of the electron liquid.
The system of interest is shown in Fig. 1. and simplicity of presentation, we focus on the case of a spinless electron liquid, and briefly discuss the generalization to spinful and multi-channel LLs as well. We consider an inhomogeneous LL in which the electron density n(x) has spatial variations of characteristic length d n −1 . On length scales much smaller than d, segments of the wire can be treated as uniform LLs.
Uniform Luttinger liquid.-In the LL approximation, particle-hole excitations of the electron system are described by non-interacting bosons. The number of rightand left-moving electrons (denoted by N R/L ) is held fixed. The Hamiltonian and the momentum of the LL are given by
H = p ε p b † p b p + π 2L v N (N − N 0 ) 2 + v J J 2 , (1) P = p pb † p b p + p F J,(2)
where N = N R + N L , J = N R − N L , and b † p creates a bosonic excitation of momentum p and energy ε p . The Fermi momentum is proportional to the electron density, i.e. p F = π n. The quantity N 0 represents a fiducial number of electrons in the system. The velocities v N and v J are renormalized from the Fermi velocity by the interactions [5].
Standard LL theory holds that the bosons have an acoustic spectrum ε p = v|p|, where v depends on the interaction strength. While this description is appropriate for a liquid at rest, the spectrum is altered by bulk motion of the fluid. For a fluid moving with velocity v d , Galilean invariance gives that the spectrum in the lab frame is ε p = v|p| + v d p. This is consistent with the fact that in the lab frame, the right-and left-moving excitations have velocities ±v + v d , respectively.
Equilibration processes.-In a uniform LL, there are two types of relaxation processes which occur with widely separated timescales. Fast collisions among the bosons, which conserve their momentum, lead to a partially equilibrated distribution function
N p = 1 e (εp−up)/T − 1 ,(3)
where the parameter u fixes the value of the total momentum of the bosons. Physically, u is the velocity of the gas of the bosonic excitations. The rate at which bosons scatter, denoted here by τ −1 0 , is expected to scale as a power of temperature [13][14][15][16].
Equation (3) does not describe a fully equilibrated liquid. At full equilibrium, the velocity of the bosons u equals the velocity of the electronic fluid v d [17]. For u = v d , the velocity u can relax to v d by processes in which momentum is exchanged between the bosonic excitations and the zero mode J [see Eq. (2)] and this relaxation is described bẏ
u = − u − v d τ .(4)
The mechanism of relaxation of involves the backscattering of an electron from one Fermi point to the other and a transfer of 2p F of momentum to the bosons as dictated by the conservation of the total momentum P [Eq. (2)]. This process requires that a hole pass through the bottom of the band and thus the rate has an Arrhenius activated form τ −1 ∝ e −D/T , where the activation energy is on the order of the bandwidth of the system [17]. A detailed microscopic calculation of the quantity τ is given in [18].
The following analysis applies in the regime T D for which τ τ 0 . Our calculation of the transport properties of the LL involves the evaluation of the temperature gradient by tracking the momentum of the gas of excitations. We work in the linear response regime and thus retain any terms linear in u or v d . The momentum density of bosons with occupation numbers N p given by Eq. (3) is
ρ P = ∞ −∞ dp h pN p = πT 2 3 v 3 (u − v d ) .(5)
The momentum current of the gas of excitations is
j P = ∞ −∞ dp h v p pN p = πT 2 6 v ,(6)
where v p = sgn(p)v + v d . We will also make use of the energy density and current which are given by ρ E = j P and j E = T s 0 nu, expressions valid to linear order in u and v d . Here, s 0 is the entropy per particle of a fluid in full equilibrium, s 0 = πT /3 nv. Taking a time derivative of Eq. (5) yieldṡ ρ (κ)
P = − πT 2 3 v 3 u − v d τ = − s 0 n κ j E − T s 0 I e ,(7)
where we have used the expression (4) foru and the fact thatv d = 0. The quantity κ = πT vτ /3 has been introduced; we will see below that κ is the thermal conductivity of the electron liquid. Because relaxation is due to the backscattering of electrons, conservation of the total momentum (2) requires thaṫ
n R = −ρ (κ) P /2p F ,(8)
whereṅ R is the change in the number of right-movers per unit length, i.e.Ṅ R = dxṅ R . Inhomogeneous LL.-We now consider the effect of inhomogeneities of the LL characterized by the spatial variations of the electron density n(x). We assume that the scale of the inhomogeneities is large, i.e. d vτ 0 . In this limit, the boson distribution function in the wire is described by Eq. (3) with spatially varying u and T . Whilė ρ (κ) P involves a redistribution of the momentum between the bosons and the zero modes, inhomogeneities lead to a loss of the total momentum of the liquid.
Due to the fact that the velocity v is controlled by the electron density, the spatial dependence of n(x) leads to spatial variations in the velocity. Energy conservation dictates that as the velocity v(x) of a ballistically propagating boson changes, so does its momentum. In order to calculate the resultant contribution toρ P , we consider a semiclassical description of the bosons in which their energy ε p (x) is a function of momentum and position. Hamilton's equations of motion give,ṗ = −∂ x ε p = −∂ x v|p| − (∂ x v d )p. This gives rise to a change of the momentum density of the gas of excitationṡ
ρ (0) P = dp hṗ N p = − πT 2 6 v 2 ∂ x v,(9)
with no terms linear in u or v d appearing.
In addition to changing the momentum of ballistically propagating bosons, inhomogeneities give rise to scattering processes which also result in a contribution tȯ ρ P . In these dissipative processes, right-moving bosons scatter off inhomogeneities and become left-movers [19]. The reverse process also occurs, though not at the same rate. Given the linearity of the bosonic spectrum, u in Eq. (3) can be interpreted as giving rise to a difference in the effective temperatures of the right-(R) and left-(L) moving bosons, i.e. T R/L = T / [1 − u/(±v + v d )]. As a result of these scattering events, energy flows from the warmer subsystem (R) to the cooler one (L) as rightmoving bosons are converted to left-movers. For u = 0 (T R = T L ), the two branches are in equilibrium. This implies that for u v, the rate at which momentum is lost is proportional to u. Since these processes are driven by inhomogeneities, the scattering amplitude must be proportional to d −1 ∼ ∂ x n/n, i.e. the inverse length scale characterizing the disorder, while the corresponding rate is proportional to the square of this quantity. Therefore, we obtainρ (ζ)
P = −ζ ∂ x n n 2 u,(10)
where we have introduced a parameter ζ with units of momentum. That ζ is indeed the bulk viscosity will be demonstrated in our discussion of the transport properties. In contrast to Eq. (7) which describes the exchange of momentum between the bosons and the zero modes, Eq. (10) represents a net loss of the momentum of the full electron liquid as bosons scatter off inhomogeneities. The arguments leading to Eq. (10) are quite general and this result holds for multi-channel Luttinger liquids. For the particular case of a fully equilibrated, single channel LL,ρ (ζ) P was evaluated in Ref. [19]. The result is consistent with Eq. (10) provided
ζ = T v 4 ∂ n n v 2 .(11)
The bulk viscosity captures the response of a fluid to changes in its density and thus it is quite natural that Eq. (11) involves a derivative with respect to n.
Equation (10) describes damping which brings the gas of excitations to rest in the lab frame, i.e. u → 0. These processes counteract those described by Eq. (7) which drive u → v d . As is true for the case of weakly interacting electrons [8], the establishment of the velocity u(x) results from the competition between the effects of electron backscattering and the scattering of excitations by spatial inhomogeneities.
Transport properties.-The derivation of the transport coefficients requires that we relate ∆T and V to the energy and electric currents, j E and I (see Fig. 1). The temperature gradient can be obtained by tracking the momentum of the gas of excitations. In the steady-state regime, ∂ t ρ P = 0, and thus the gradient of momentum current obeys
∂ x j P =ρ (0) P +ρ (κ) P +ρ (ζ) P .(12)
In thermal equilibrium, u = v d = 0 and Eqs. (7) and (10) require thatρ (κ) P andρ (ζ) P vanish. Thus, we have ∂ x j P =ρ (0) P . Indeed, Eqs. (6) and (9) satisfy this relation for ∂ x T = 0, as is necessarily the case for a system in thermal equilibrium.
For a system out of equilibrium, Eqs. (7) and (10) show thatρ (κ) P andρ (ζ) P contribute terms linear in j E = T s 0 nu and I = env d to the right-hand side of Eq. (12). However, j P has no linear correction in these quantities. The only way that Eq. (12) can be satisified is for the temperature to acquire a spatial gradient. Substituting j P [Eq. (6)] into Eq. (12) reveals that the temperature gradient of the LL is
∂ x T = − j E κ + T s 0 I κe − ζ T s 2 0 ∂ x 1 n 2 j E .(13)
For I = 0, Eq. (13) establishes that κ, as defined after Eq. (7), is the thermal conductivity of a uniform electron liquid. The last term in Eq. (13) shows that spatial variations in n(x) give a correction to the thermal resistivity of the system. We now demonstrate that the quantity ζ is the bulk viscosity of the electron liquid. In order to make this identification, we consider an arbitrary point x 0 along the wire and adjust the currents such that j E /T s 0 (x 0 ) = I/e. This relation ensures that u(x 0 ) = v d (x 0 ) and thus the electron liquid is fully equilibrated at this point. In order to maintain steady-state flow, a force must be applied to counteract the damping force (10). Denoting this force (per electron) by f (ζ) and considering the small segment of the wire between the points x 0 ± ∆x/2 containing ∆N electrons, we have f (ζ) ∆N = −ρ
dW dx = ζ(∂ x v d ) 2 .(14)
This equation represents the dissipation of a fluid due to its bulk viscosity [20], thus confirming the physical meaning of ζ. We now consider the effects of contacts on the transport properties of the wire. The leads control the distribution function of the excitations entering the wire. For example, the temperature of the left lead coincides with the temperature of the right-moving electrons at x = 0. The latter is not given by T (0), but rather by T (0)/(1 − u/v). Therefore, there is a mismatch between the temperature of the leads and the temperature of the bosons at the end of the wire. This effect is fully accounted for by the contact thermal resistance R
T j E + L 0 dx j E κ − T s 0 I κe + ζ T s 2 0 ∂ x 1 n 2 j E ,(15)
where R (0) T = 6 /πT [21]. The electric current through a single conductance channel (appropriate for spinless electrons) is given by I = e 2 V /h. Accounting for electron backscattering, we have I = e 2 V /h + eṄ R , which may be derived by considering electron current conservation [17]. Solving for V and substituting Eqs. (7) and (8)
intoṄ R = ṅ R dx, we obtain V = R (0) I − 1 e L 0 s 0 κ j E − T s 0 I e dx,(16)
where R (0) = h/e 2 . Equations (15) and (16) relate {∆T, V } to the currents {j E , I}. The currents are independent of position and can be factored out of the integrals. The various transport coefficients will now be expressed as integrals of the density n(x), the entropy per particle s 0 (x), κ(x), and ζ(x) over the length of the system.
The thermal resistance is defined at zero current and is given by R T = ∆T /j E . Equation (15) with I = 0 gives
R T = R (0) T + dx κ + 1 T ζ s 2 0 ∂ x 1 n 2 dx.(17)
The Peltier coefficient is the ratio of energy current to electric current, i.e. Π = j E /I, when the left and right leads are at the same temperature. Setting ∆T = 0 in Eq. (15) and solving for the ratio of currents yields
Π = T eR T s 0 κ dx.(18)
Since Π ∝ u/v d , this coefficient is directly informed by the competition between the processes described by Eqs. (7) and (10). The thermopower is defined to be S = −V /∆T with I = 0 and is straightforwardly obtained by dividing Eq. (16) by Eq. (15). We find that S = Π/T , in accordance with the Onsager relations [22].
Of central interest in experimental realizations of quantum wires is the resistance, R = V /I, defined for ∆T = 0. By substituting j E = ΠI into Eq. (16), we obtain
R = R (0) + T e 2 s 2 0 κ dx − 1 R T s 0 κ dx 2 .(19)
Expressions (17), (18), and (19) give a complete description of the transport properties of an inhomogeneous LL at temperatures below D and represent our primary result. We now establish the conditions for which Eq. (19) agrees with the corresponding result of hydrodynamics. For long wires, Eq. (19) reduces to Eq. (4) of Ref. [7] provided
ζvτ v T d 2 1,(20)
and the variations of the density are small along the wire, i.e. ∆n n. The condition (20) is controlled by the competition between the long characteristic length scale of the inhomogeneities d and the exponentially long timescale τ ∼ e D/T associated with electron backscattering. The breakdown of the hydrodynamic theory occurs, to logarithmic accuracy, at a temperature T * D 2 ln (nd)
.
Thus, in the temperature range T * T D, the transport theory of Ref. [7] based on classical hydrodynamics applies to a quantum degenerate system.
The temperature dependence of the resistance (19) represents an important prediction of our theory. Since R − R (0) ∝ τ −1 ∝ e −D/T , the resistance increases monotonically with temperature. The opposite behavior is exhibited by corrections to the resistance arising from short range disorder [9][10][11]. So far, resistances increasing with temperature have been seen in conductance measurements of quantum point contacts [23,24], systems in which the effects of impurities are easier to eliminate. Observation of such behavior in long wires would represent a confirmation of our theory.
Discussion.-In this work, we have calculated the transport properties of a spinless quantum wire subject to long range disorder with arbitrary electron-electron interactions. This work is readily generalized to spinful and other multi-channel electron liquids by taking appropriate values of R (0) , R
T , s 0 , κ, and ζ. For sufficiently high temperatures, the degenerate electron liquid can be described within the hydrodynamic theory of Ref. [7]. From the vantage point of this more microscopic theory, we find that tacit to the hydrodynamic result is the requirement that the spatial variations in the density are small.
PACS numbers: 71.10.Pm, 73.23.-b
FIG. 1 :
1Sketch of an inhomogeneous quantum wire in contact with Fermi liquid leads with temperatures T and T + ∆T and electrochemical potentialsμ andμ + eV . The arrows indicate the flow of energy and electric currents, denoted by jE and I, respectively. The shading in the wire indicates the non-uniform electron density, n(x). The length scale d characterizes typical variations of the electron density. arXiv:1412.0693v1 [cond-mat.str-el] 1 Dec 2014
P
∆x. The power dissipated is then given by ∆W = f (ζ) v d ∆N = −v dρ (ζ) P ∆x. Using Eq. (10) with u = v d and the continuity equation ∂ x (nv d ) = 0, we find that
E due to the contacts and − ∂ x T dx [using Eq.
Acknowledgements.-We are grateful to A. V. Andreev for discussions. The work was supported by the U.S. Department of Energy, Office of Science, Materials Sciences and Engineering Division.
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[
"arXiv:gr-qc/9812069v1 18 Dec 1998 G 1 Cosmologies with Gravitational and Scalar Waves Typeset using REVT E X 1",
"arXiv:gr-qc/9812069v1 18 Dec 1998 G 1 Cosmologies with Gravitational and Scalar Waves Typeset using REVT E X 1"
]
| [
"Ruth Lazkoz \nSchool of Mathematical Sciences\nAstronomy Unit\nQueen Mary & Westfield College\nE1 4NSLondon\n\nU.K. and Fisika Teorikoaren Saila Euskal Herriko Unibertsitatea\n644 P.K48080BilbaoSpain\n"
]
| [
"School of Mathematical Sciences\nAstronomy Unit\nQueen Mary & Westfield College\nE1 4NSLondon",
"U.K. and Fisika Teorikoaren Saila Euskal Herriko Unibertsitatea\n644 P.K48080BilbaoSpain"
]
| []
| I present here a new algorithm to generate families of inhomogeneous massless scalar field cosmologies. New spacetimes, having a single isometry, are generated by breaking the homogeneity of massless scalar field G 2 models along one direction. As an illustration of the technique I construct cosmological models which in their late time limit represent perturbations in the form of gravitational and scalar waves propagating on a non-static inhomogeneous background. Several features of the obtained metrics are discussed, such as their early and late time limits, structure of singularities and physical interpretation. | 10.1103/physrevd.60.104008 | [
"https://arxiv.org/pdf/gr-qc/9812069v1.pdf"
]
| 119,010,827 | gr-qc/9812069 | 589bd1616612c91af34b67fbca84b7a43933f071 |
arXiv:gr-qc/9812069v1 18 Dec 1998 G 1 Cosmologies with Gravitational and Scalar Waves Typeset using REVT E X 1
Ruth Lazkoz
School of Mathematical Sciences
Astronomy Unit
Queen Mary & Westfield College
E1 4NSLondon
U.K. and Fisika Teorikoaren Saila Euskal Herriko Unibertsitatea
644 P.K48080BilbaoSpain
arXiv:gr-qc/9812069v1 18 Dec 1998 G 1 Cosmologies with Gravitational and Scalar Waves Typeset using REVT E X 1
(October 7, 2018)PACS numbers: 04.20.Jb, 98.80.Hw, 04.30-w
I present here a new algorithm to generate families of inhomogeneous massless scalar field cosmologies. New spacetimes, having a single isometry, are generated by breaking the homogeneity of massless scalar field G 2 models along one direction. As an illustration of the technique I construct cosmological models which in their late time limit represent perturbations in the form of gravitational and scalar waves propagating on a non-static inhomogeneous background. Several features of the obtained metrics are discussed, such as their early and late time limits, structure of singularities and physical interpretation.
I. INTRODUCTION
The high degree of isotropy observed in the Universe on large scales today is usually combined with the Copernican principle to justify the assumption of homogeneity on the same scales. However, there is no known reason to assume that the Universe was isotropic nor homogeneous at very early epochs. The puzzling question of how the Universe might have evolved from an initially irregular state into the current isotropic and apparently homogeneous state lacks a complete answer at present. To date, several regularization mechanisms have been put forward, such as Misner's chaotic cosmological program [1,2], the standard inflationary scenario [3][4][5][6] and, more recently, the alternative pre-Big Bang inflationary scenario [7]. However, none of these is completely satisfactory, and in general one cannot know for certain which range of initial conditions could have allowed the Universe evolve into its present form. Such scenarios can only provide one with partial indications of what initial conditions would have led a generic universe into the one observed at present. One way to maximize the amount of information obtainable from any cosmological program, such as those mentioned above, relies on studying the evolution of models with as many degrees of freedom as possible. This idea has motivated the special attention paid to inhomogeneous cosmological models in the last decades (see Krasinski [8] for a review).
In general, attempts to obtain new inhomogeneous metrics involve some symmetry assumption, so that the full implications of the non-linearity of the theory are compromised to some extent. In addition, non-vacuum spacetimes are required to have a physically meaningful matter content, so that they portray realistic situations. Motivated by the possibility of the existence of non-trivial massless scalar fields in the early universe, I will concern myself here with cosmological solutions to the Einstein equations induced by such matter sources. In particular, I present a new algorithm to generate families of massless scalar field G 1 cosmologies, i.e. time-dependent spacetimes with a single isometry. These new sets of metrics will be generated starting from generalized vacuum Einstein-Rosen spacetimes, which admit an Abelian group of isometries G 2 acting transitively on spacelike surfaces. In the last decades there has been intensive study of G 2 vacuum and matter filled cosmological models and several major reviews on the subject have been written [9,10,8].
Given the large number of known G 2 cosmologies and the various available techniques to generate further ones, algorithms transforming such spacetimes into G 1 metrics represent powerful tools for generating new inhomogeneous solutions. The new algorithm, which I shall present, displays the nice property of reducing the symmetry while keeping the type of matter source unaltered. Nonetheless, the input and output solutions may not admit the same physical interpretation or even share some of the same relevant features. For this reason, if one wishes to grasp the physical meaning of every new solution, an independent analysis of it will have to be carried out.
Here in I construct and analyze G 1 models representing the propagation of gravitational and matter waves on a non-spatially flat background. The study of primordial inhomogeneities in the form of waves is an active area of research. This is motivated by the fact that wave-like primordial perturbations originating from vacuum fluctuations during inflation may be responsible for structure formation. Unlike other types of inhomogeneities formed in the early universe, they would have remained nearly unaltered up to the present, and therefore allowing the possibility of their detection.
The exact inhomogeneous G 1 spacetimes seeded by gravitational and matter waves studied here represent a generalization of more symmetric configurations considered by Charach and Malin [11]; in those models the background hosting the waves was homogeneous. In the context of colliding plane waves, G 2 diagonal spacetimes with waves of scalar and gravitational nature have also been considered. Space-times such as those studied by Wu [12] or any solution generated by the methods of Barrow [13] and Wainwright et al. [14] could be taken as starting points to construct new G 1 metrics modeling interactions of waves on a curved background.
Furthermore, interest in this generation procedure is not restricted to the relativistic framework; solutions to Einstein's equations with a massless scalar field (in what follows m.s.f.) may be used to generate solutions to alternative theories of gravity, such as Brans-Dicke theory or string theory in its low energy limit. In the latter case, one could even take those spacetimes to generate new solutions with other massless modes in the characteristic spectrum of the theory.
The plot of the paper is as follows: First I introduce the G 1 massless scalar field solution generating algorithm itself. Then, I construct new inhomogeneous metrics starting from a infinite dimensional family of solutions which in the WKB limit admit an interpretation in terms of waves. It will be shown that at early times these solutions behave like a Belinskii-Khalatnikov generalized model [15] with homogeneity broken along one spatial direction.
The structure of spacelike singularities of the new solutions in the early time limit will be analyzed as well, and the special features due to the presence of the matter source will be indicated. Next I will consider the solutions' high frequency limit and show that they can be thought of in the same physical terms as their G 2 counterparts. In particular the new cosmological models represent a spacetime with a spatially inhomogeneous background filled with a non-static inhomogeneous scalar field and a null fluid of "gravitons" and "scalar particles." As time grows the null fluid's contribution to the energy-momentum tensor grows faster than that associated with the homogeneous part of the scalar field generating the background geometry. Thus, the model evolves into an inhomogeneous generalization of the cosmological model of Doroshkevich, Zeldovich and Novikov (DZN) [16]. Finally, the main conclusions are outlined.
II. SOLUTION GENERATION ALGORITHM
Basically, the new generating technique is a prescription to break homogeneity along one direction in G 2 m.s.f. cosmologies, ending up with new spacetimes possessing a single isometry but having the same type of matter content. Remarkably, the pioneering investigations on matter filled universes of Einstein-Rosen type, carried out respectively by Tabensky and Taub [17] and Liang [18] considered models with non interacting scalar fields, even though at the time there was no clear physical motivation.
According to a conjecture due to Belinskii, Lifshitz and Khalatnikov (BLK) [19][20][21][22][23][24], G 2 metrics seem to be specially relevant for the description of the early universe, as such solutions give the leading approximation to a general solution near the singularity at t = 0.
Their claim has very recently found the support of numerical results [25]. In particular BLK considered approximate Einstein-Rosen solutions and performed an analysis of their local behavior in the early and late time regime. An interesting result reached in the course of those investigations, which is specially relevant for the present paper, was the prediction of a high frequency gravitational wave regime in the late epochs of the Universe.
Before going any further it is convenient to explain how G 2 spacetimes induced by a m.s.f. can be generated starting from vacuum solutions to Einstein's equations with the same symmetry. For the sake of simplicity, the discussion will be restricted here to a particular case of a well-known general procedure [26,11,27,13,14,28,29]. The generic diagonal line-element with G 2 symmetry will be taken as a starting point:
ds 2 v = e fv(t,z) (−dt 2 + dz 2 ) + G v (t, z) e pv(t,z) dx 2 + e −pv(t,z) dy 2 ,(1)
where the subscript v stands for vacuum. A new solution g µν of the Einstein equations with a massless scalar field ϕ as a source and line-element
ds 2 = e f (t,z) (−dt 2 + dz 2 ) + G(t, z) e p(t,z) dx 2 + e −p(t,z) dy 2 ,(2)
can be obtained by the following transformations:
G = G v ,(3a)p = B p v + C log G v ,(3b)f = f v + E p v + F log G v ,(3c)ϕ = A p v + D log G v ; (3d)
provided that the constants A, B, C, D, E and F are subject to the constraints:
B C + 2 A D = E ,(4a)C 2 + 2 D 2 = 2 F ,(4b)B 2 + 2 A 2 = 1 . (4c)
The conditions (4) arise by demanding the following are satisfied:
R µν = ϕ ,µ ϕ ,ν ,(5a)g µν ∇ µ ∇ ν ϕ = 0 . (5b)
In principle a large number of new m.s.f. G 2 cosmologies can be obtained by simply applying the procedure sketched above to any of the representatives of the populated family of vacuum Einstein-Rosen spacetimes. However, generating m.s.f. solutions with a lower degree of symmetry is a more cumbersome task. At this point I would like to draw attention to a method given by Feinstein et al. [30] which allows one to generate families of solutions with a two-dimensional degree of inhomogeneity and a self-interaction term for the massless field of the form
V = V 0 (λ) e −λ ϕ .(6)
With these procedure new metrics are obtained by introducing an x-dependent conformal factor on the input G 2 metric, and where in general the potential term only vanishes for |λ| = 6. This difficulty in canceling the self-interaction term for the scalar field can be traced back to the high degree of symmetry of the x-dependence in the models considered in ref. [30].
In order to find a prescription for introducing an additional degree of symmetry in m.s.f. G 2 metrics without switching on a potential in the process, I have considered the possibility of having a more general x-dependence in the metric. In particular, I have sought metrics of the form:
ds 2 = Ω(x) e f (t,z) (−dt 2 + dz 2 ) + G(t, z) e p(t,z) dx 2 + Ξ(x) e −p(t,z) dy 2 ,(7)
and made the following ansatz for the scalar field:
φ(t, z, x) = ϕ(t, z) + Λ(x) .(8)
Note that the massless scalar field case included in the solutions of Feinstein et al. is also a particular case of the models here. Since the requirement is that no potential should arise in the transformations, the equations that must hold are:
R µν =φ ,µφ,ν ,(9a)g µν∇ µ∇νφ = 0 . (9b)
Explicitly, equation (9a) is equivalent to the set of equations:
R 00 = R 00 + e f −p Ξ ,x Ω ,x + 2 Ξ Ω ,xx 4 G Ξ =φ 2 ,t ,(10a)R 11 = R 11 − e f −p Ξ ,x Ω ,x + 2 Ξ Ω ,xx 4 G Ξ =φ 2 ,z ,(10b)R 01 = R 01 =φ ,tφ,z ,(10c)R 22 = R 22 + Ξ ,x 2 − 2 Ξ Ξ ,xx 4 Ξ 2 + Ω ,x 2 − 2 Ω Ω ,xx 2 Ω 2 =φ 2 ,x ,(10d)R 12 = G ,z Ω ,x 2 G Ω + Ξ ,x p ,z 2 Ξ =φ ,zφ,x ,(10e)R 02 = G ,t Ω ,x 2 G Ω + Ξ ,x p ,t 2 Ξ =φ ,tφ,x ,(10f)R 33 = R 33 + e −2 p Ξ ,x 2 − 2 Ξ Ξ ,xx 4 Ξ − e −2 p Ξ ,x Ω ,x 2 Ω = 0 ; (10g)
whereas equation (9b) in explicit form reads
g µν ∇ µ ∇ ν ϕ Ω + e −p (Ω Ξ 1/2φ ,x ) ,x Ξ 1/2 = 0 .(11)
Inspection of the equations (10) indicates that the three x-dependent metric functions must have the form:
Ω(x) = x k ,(12a)Ξ(x) = x n ,(12b)Λ(x) = m log |x| ;(12c)
subject to the following constraints on the parameters k, n and m:
k (2 k + n − 2) = 0 , (13a) n (2 k + n − 2) = 0 , (13b) k + n C = 2 D m , (13c) n B = 2 A m ,(13d)2 m 2 = 4 k − 3 k 2 .(13e)
In this case, the Klein-Gordon equation (9b) reduces to
m (2 k + n − 2) = 0 ,(14)
which is automatically satisfied provided that (13a,13b) and (13e) hold. Note that the parameter k must be non-negative and not larger than 4/3. The x-dependent term of the scalar field will be maximum for k = 2/3. Consistency of the solutions is reflected by the fact that for the following choice:
m = n = k = 0 .(15)
the set of equations (13,14) is satisfied for any value of A, B, C and D, and the input G 2 m.s.f is recovered. On the one hand, for m = 0 and k = 1, if k and C are taken as free parameters, one can parameterize the solution's constants in the form:
n = 2 − 2 k , (16a) m = sign (m) √ 4 k − 3 k 2 √ 2 , (16b) A = √ 2 sign (A) 1 − k 2 − k ,(16c)B = sign (A m) 1 − k 2 − k √ 4 k − 3 k 2 1 − k , (16d) D = sign (m) k + (2 − 2 k) C √ 8 k − 6 k 2 ,(16e)E = sign (A m) 2 − 2 k 2 − k C (2 − k) 2 + (2 − 2 k) k √ 4 k − 3 k 2 ,(16f)F = C 2 2 + (k + (2 − 2 k) C) 2 8 k − 6 k 2 .(16g)
Four different subcases can be distinguished, depending on the choice of the sign of A and m. On the other hand, in the particular case m = 0, k = 1, the constants take the values:
√ 2|m| = √ 2|D| = |B| (17a) A = 0 (17b) C 2 + 1 = 2 F (17c) E = sign (B) (17d)
In general the metricg µν admits only one Killing and one homothetic vector, namely:
ξ kv = ∂ ∂y (18a) ξ hv = y ∂ ∂y ,(18b)
where by definition:
L ξ kv g µν = 0, (19a) L ξ hv g µν = 2g µν ; (19b)
It is known that a matter source and the geometry it induces need not share the same symmetry properties unless it is a perfect fluid [31,32]. This property is usually referred to as the "inheritance problem". Inspection of the metrics obtained here show that for k = 4/3 the x-dependent term of the scalar field is absent, even though the metric depends on that coordinate in a non-trivial way. This is equivalent to:
L ξmc T µν = 0 (20a) L ξmc g µν = 0; (20b)
where ξ mc = ∂ ∂x and the vector ξ mc is a so-called matter collineation [10,33,32].
Bearing in mind the similarity between the new algorithm and the one of [30], one might wonder whether geometries like (7) can be also seeded by a scalar field with an exponential potential. Such a situation may be shown to be only possible if n = k, which is nothing but the case already found by Feinstein et al. In order to prove this, let us consider the case where the generic geometry (7), under the constraint (12), is induced by an exponential
potentialṼ (φ) = V 0 (λ) e −λφ .
In this case the field equations are:
R µν =φ ,µφ,ν +g µν V 0 e −λφ ,(21a)∇ µ∇ µφ = − ∂Ṽ (φ) ∂φ . (21b)
It is only necessary to look at the equations forR 00 andR 33 to realize the following constraint must hold:
k (2 − 2 k − n) = n (2 − 2 k − n) = 4 V 0 = 0.(22)
Compatibility of the latter set of equations in the case of a non vanishing potential requires n = k, or in other words, that the x-dependence of the metric is given by a global conformal factor as in the case studied by [30].
III. COSMOLOGIES WITH GRAVITATIONAL AND SCALAR WAVES
After having outlined our method to generate uniparametric families of G 1 cosmologies, I shall now construct the counterparts of a family of G 2 cosmologies found by Charach and
Malin [11]. That set of metrics, which can be thought of as inhomogeneous sinusoidal perturbations of the well-known Bianchi I spacetimes, represent the propagation of gravitational and scalar waves on an anisotropically expanding flat background. Simpler models of that sort, not including scalar degrees of freedom, were studied earlier on by Berger [25].
In this paper I am only considering Charach and Malin's solutions in the asymptotic limits t ∼ 0 and j t ≫ 1 for any value of j. It is convenient, however, to give here the full expressions of the metric functions of the vacuum solution from which they were derived, namely:
G v = t ,(23a)p v = β log t + ∞ j=1 cos[j(z − z n )]α j J 0 (j t) ,(23b)f v = β 2 − 1 2 + β ∞ j=1 α j cos[α j (z − z j )]J 0 (j t) + t 2 4 ∞ j=1 j 2 {[α j J 0 (j t)] 2 + [α j J 1 (j t)] 2 } − t 2 ∞ j=1 jα 2 j cos 2 [α j (z − z j )]J 0 (j t)J 1 (j t) + t 2 ∞ l=1 ∞ j=1 l j l 2 − j 2 × {sin[l(z − z l )] sin[j(z − z j )] [lα l α j J 1 (l t)J 0 (j t) − jα l α j J 0 (l t)J 1 (j t)] + cos[l(z − z l )] cos[j(z − z j )] [jα l α j J 1 (l t)J 0 (j t) − lα l α j J 0 (l t)J 1 (j t)]} .(23c)
Charach and Malin's solutions were obtained by breaking the spatial homogeneity of Belinskii-Khalatnikov homogeneous solution [15], which also has a m.s.f. as a seed. The gravitational and scalar degrees of freedom of those spacetimes satisfy linear wave equations, with the form of cylindrically symmetric waves propagating on Minkowski spacetime, for which the spatial and temporal coordinates have been interchanged. Since the solutions are of the standing wave form they are fully compatible with the S 1 ⊗ S 1 ⊗ S 1 topology (threetorus). However, the G 1 counterparts of those spacetimes cannot have the same topology,
x cannot be a cyclic coordinate in this case due to the presence of the term proportional to log |x| in the scalar fieldφ.
A. Early time behavior and singularities
From the analysis of the new G 1 solution's early time behavior it can be determined whether spacelike singularities arise at the time origin t = 0. A look at expressions (23) shows that in the case under discussion, the periodic inhomogeneities can be neglected in the very first stages of that spacetime's evolution. Our metrics will be then G 2 inhomogeneous generalizations of the cosmological models of Doroshkevich-Zeldovich-Novikov (DZN) [16], which have three commuting Killing vectors. In this limit the metric and scalar field read
g µν ∼ diag −x k ǫ 1 (t), x k ǫ 1 (t), ǫ 2 (t), x n ǫ 3 (t) (24a) ϕ = ϕ 0 log t + m log |x| (24b) where ǫ 1 = t f 0 ,(25a)ǫ 2 = t 1+p 0 ,(25b)ǫ 3 = t 1−p 0 ,(25c)
and
p 0 = Bβ + C ,(26a)f 0 = β 2 − 1 2 + Eβ + F ,(26b)ϕ 0 = Aβ + D .(26c)
In addition, the following relationship holds:
f 0 = p 2 0 − 1 2 + ϕ 2 0 .(27)
Following Charach and Malin, the metric can be rewritten using a synchronous set of coordinates in which the new time coordinate τ is defined dτ = ǫ 1 (t)dt ; (28) this way the metric can be recognized as a simple inhomogeneous generalization of a Belinskii-Khalatnikov [15] solution.
In broad terms, the presence of a spacelike singularity at early times will be reflected in the behavior of the curvature invariants R, R µν R µν and R µνσδ R µνσδ . Due to the inhomogeneous character of the metric it is possible in principle to have a conspiracy between the parameters, so that on certain hypersurfaces some those invariants are identically null and therefore do not reveal the presence of a singularity in the spacetime. Since the Kretchmann scalar K = R µνσδ R µνσδ is however non-null and positive everywhere I will use this in the search for singularities. In the case here it explicitly reads:
K = 3 f 2 0 8 t 6 f 0 +4 x 6 k (1 + p 2 0 ) 2 + 2 (f 0 + 1) 2 + 3 16 t 2 f 0 +4 x 2 k (1 − p 0 ) 2 (1 + f + p) 2 +(1 + p) 2 (1 + f − p) 2 .(29)
Since (27) holds it can be seen that K is singular at t = 0 for any value of the three free parameters of the solution, indicating thus the generic presence of a spacelike singularity at the time origin.
It is also possible to study the singularity structure of the solutions in a more refined way, in particular by studying the expansion along each spatial axis. In general, the behavior will strongly depend on the values of the parameters k, C and β. It can be shown analytically that for large enough β there will necessarily be contraction along the z axis, moreover if k > 1 that will be the case regardless the value of β and C. Another fact that can be easily checked is the impossibility of having simultaneous contraction along axes x and y. Three main types of singular behavior can be distinguished:
a) Point-like singularities (Quasi-Friedmann behavior)
All three spatial directions shrink as the initial time t = 0 is approached; or explicitly lim t→0 ǫ i = 0 ∀i. Depending on how many directions have the same expansion rate the behavior will be completely anisotropic, axially symmetric or isotropic.
b) Finite lines
This type of singular behavior occurs when in the vicinity of t = 0 one of the spatial directions neither expands nor contracts with time; in other words, it is said that the direction i is a finite line if lim t→0 ǫ i = 1. The subcases can be classified according to which the direction behaves in that way. In general, there will be a single finite line, though it is possible to have particular cases in which a second finite line exists. However, it is impossible to have such behavior along all three directions.
B. WKB limit
With regard to the physical interpretation of the new solutions constructed here, it is their high-frequency limit which turns out to be most interesting. This limit, also called the WKB regime, corresponds to the regime in which the time elapsed since the beginning of the Universe is much larger than the period of any perturbation mode. By taking j t ≫ 1, for every value of j, in the normal mode expansions of the scalar field and metric functions, Charach and Malin were able to show that the relativistic solutions taken here as input, represent scalar and gravitational waves propagating on an spatially flat background. In this limit, such universes are causally connected because the particle horizon is larger than the wavelength of any of the modes of the independent degrees of freedom, namely the transverse part of the gravitational field p and the scalar fieldφ.
In this vein, it will be proved here in two different ways that the G 1 counterparts to Charach and Malin's cosmological models can also be thought of in terms of waves propagating on a non-static background. Thus, the physical interpretation is not spoiled in the process of homogeneity breaking. A direct consequence of the additional degree of inhomogeneity present in the G 1 solutions, as compared to their more symmetric counterparts, is that the background on which the waves live is not spatially flat. Cosmological backgrounds perturbed by waves have also been considered in a series of paper by Centrella and Matzner [34][35][36] who studied collisions of plane gravitational waves in these settings.
Let us consider now the late time expressions for the metric functions and the scalar field of the G 1 spacetimes obtained by applying the new technique to the cosmological models given by expressions (23):
p ∼ p 0 log t + Bp(t, z) , (30a) f ∼ f 0 log t + t 2π ∞ j=1 jα 2 j , (30b) ϕ ∼ ϕ 0 log t + Ap + m log |x| , (30c) p = ∞ j=1 2 jπt α j cos(j t − π 4 ) cos[j(z − z j )] .(30d)
Since in this regimep ≪ 1, the metricg µν can be split into a backgroundη µν plus a perturbation metrich µν , that is:
g µν ∼η µν +h µν ,(31a)η µν = diag −x k t f 0 e f 1 t , x k t f 0 e f 1 t , t 1+p 0 , x n t 1−p 0 ,(31b)h µν = diag 0, 0, t 1+p 0 Bp, −x n t 1−p 0 Bp .(31c)
Here, in addition to (26), I have made the following definition 1 :
f 1 = ∞ j=1 jα 2 j 2 π (B 2 + 2 A 2 ) .(32)
A peculiarity regarding the perturbations on the scalar and gravitational degrees of freedom is that for 0 < k < 1 they will be on phase whereas for 1 < k < 4/3 they will be phase-shifted by π. Nonetheless, it will be seen later that whatever the value of k the scalar and gravitational perturbations contribute constructively to the energy momentum tensor.
A proof of the assertion made above regarding the non spatial flatness of the spacetimes described by the background metricη µν , is provided by the expression of the spatial curvature of the t = constant three dimensional hypersurfaces:
(3,η) R = m 2 2 x 2 t p 0 +2 .(33)
Let us now proceed to analyze theh µν tensor and thereby show that it represents the gauge-invariant perturbations of a background spacetimeη µν ; or in other words, that it satisfies the wave equation∇ γ∇ γhµν = 0, or equivalently, following [37], that:
h γ γ = 0 ,(34)∇ γh γλ = 0 ,(35)
with γ, λ = 2, 3, and where the D'Alambertian and the covariant derivative must be calculated using the corresponding background metricη µν .
Since it is straightforward to see that the trace-free condition (34) is satisfied by construction, the problem reduces to satisfying (35), which is equivalent to requiring:
1 Though the factor B 2 + 2 A 2 equates to unity in the simple case I am dealing with, it has been deliberately introduced in the definition of f 1 ; so that the trail of the separate contributions to the energy momentum tensor of the graviton and scalar field pair can be followed. Had I considered the general case of the procedure to generate a G 2 massless scalar field solution, then
f 1 = ∞ j=1 jα 2 j /(2 π) . h 22,x − 2Γ 2 22h 22 = 0 , (36a) h 33,x − 2Γ 3 32h 33 = 0 . (36b)
On the one hand, expression (36a) is identically null because neitherη 22 norh 22 are
x-dependent. On the other hand, it can be seen that (36b) is also satisfied by just having in mind that:h
33 = Bη 33 ,(37a)Γ 3 32 = log η 33 ,x .(37b)
Once it has been proved that the wave equation∇ γ∇ γhµν = 0 holds one can properly refer to the tensorh µν as describing metric perturbations in the form of gravitational waves.
In order to give additional arguments in favor of the interpretation of the solution in terms of waves propagating on a non flat background, I shall follow Charach and Malin [11] and analyze the energy-momentum tensor of the background metricη µν . It will be shown that the stress-energy tensor is naturally manifested in terms of two components. One of these corresponds to a null fluid, supporting thus the interpretation suggested above; while the other term corresponds to an inhomogeneous massless scalar field with no z-dependence.
In particular,
(η)T ν µ = (1)T ν µ + (2)T ν µ ,(38)
where (1)T ν ν = 0 and (2)T ν ν = 0. Explicitly
(1)T 0 0 = −t −(1+p 0 ) m 2 2 x 2 − t −(2+f 0 ) 1 + 2 f 0 − p 0 2 4 e f 1 t x k ,(39a)(1)T 1 1 = −t −(1+p 0 ) m 2 2 x 2 + t −(2+f 0 ) 1 + 2 f 0 − p 0 2 4 e f 1 t x k ,(39b)
(1)T 0
2 = t −(1+f 0 ) 2 p 0 (k − 1) − k 2 e f 1 t x 1+k ,(39c)
(1)T 2
2 = t −(1+p 0 ) m 2 2 x 2 + t −(2+f 0 ) 1 + 2 f 0 − p 0 2 4 e f 1 t x k ,(39d)
(1)T 3
3 = −t −(1+p 0 ) m 2 2 x 2 + t −(2+f 0 ) 1 + 2 f 0 − p 0 2 4 e f 1 t x k ,(39e)
(2)T 0
0 = −t −(1+f 0 ) f 1 2 e f 1 t x k ,(39f)
(2)T 1
1 = t −(1+f 0 ) f 1 2 e f 1 t x k .(39g)
I will now proceed to give an interpretation for the (1)T ν µ term. The Klein-Gordon equation for a scalar fieldψ, calculated using the metricη µν , takes the form
(xψ ,x ) ,x x 1−k − (tψ ,t ) ,t e f 1 t t f 0 −p 0 = 0,(40)
a solution of which isψ
= ϕ 0 log t + m log |x|.(41)
The energy momentum tensor for the fieldψ propagating on the spacetimeη µν yields (1)T ν µ exactly. That being so, one can conclude that the exact solution obtained after applying the generating technique to (23) asymptotically evolves into a solution with a single degree of inhomogeneity; that is, the sinusoidal inhomogeneities along the z-axis vanish with time.
In other words, (1)T ν µ corresponds to the energy-momentum tensor of the exact solution one would obtain by switching off the periodic inhomogeneities along z.
On the other hand, the traceless term (2)T ν µ can be shown to account for waves. It can also be separated into two parts, namely:
(2)T µ ν = (GW )T µ ν + (SW )T µ ν ,(42a)(GW )T µ ν = ∞ j=−∞ B 2 α 2 j 4 |j| κ µj κ νj ,(42b)(SW )T µ ν = ∞ j=−∞ A 2 α 2 j 2 |j| κ µj κ νj .(42c)
The null vector κ µ is defined by
κ µj = 1 √ π t (|j|, j, 0, 0).(43)
It is clear that (2)T µν corresponds to a null fluid describing a collisionless flow of "gravitons" and "scalar particles". It is easy to see that in the case under discussion gravitational waves will be absent only if the metric has no x-dependence.
I has been shown that the new solutions represent, in their WKB limit, waves propagating on an inhomogeneous non-static spacetime. In our case the only difference with respect to the case studied by Charach and Malin is that the background is not spatially flat.
As far as the further evolution of the model is concerned, the presence of the additional degree of inhomogeneity in the model plays a crucial role. The null fluid's contribution to the energy-momentum tensor dominates the one due to the homogeneous part of the scalar field, and in the t ≫ 1 limit the energy-momentum tensor has a term accounting for the waves, plus another corresponding to the x-dependent term in the scalar field. The scalar curvature of the background in this limit will be given by:
(η) R ∼ m 2 x 2 t 1+p 0 ,(44)
so that this universe can either become singular at t = ∞ or flat, depending on the sign of 1 + p 0 . The possibility of having a curvature spacelike singularity at late times is entirely due to the inhomogeneous character of the background.
Another proof of the interpretation of the solutions in terms of a null fluid propagating on the backgroundη µν can be given. Let us calculate the energy-momentum tensor that corresponds to the scalar fieldφ in the late time limit and just retain terms up to the order t −1 . Under this restriction the only non-null terms of the energy momentum tensor areT 00 andT 11 , which are given by:
T 00 = −T 11 = A 2 π t ∞ l=1 ∞ j=1 α l α j lj sin[l(z − z l )] sin[j(z − z j )] cos[l t − π 4 ] cos[j t − π 4 ] − cos[l(z − z l )] cos[j(z − z j )] sin[l t − π 4 ] sin[j t − π 4 ] + O(t −2 ).(45)
AveragingT 00 over a region 0 ≤ t ≤ 2 π, 0 ≤ z ≤ 2 π on the (t, z)-plane one obtains:
T 00 = − T 11 = 1 4 π 2 2 π 0 2 π 0 T 00 dzdt = 1 2 π t ∞ j=1 j A 2 .(46)
In the same approximation it can be seen that
T = T 2 2 = T 3 3 = O(t −2 ).(47)
So, essentially it has been found that
T µν = (SW )T µν .(48)
Summarizing, both methods have yielded the result that for t ≫ 1, and up to the order t −1 , the energy-momentum of the m.s.f. can be reduced to the null fluid form.
The WKB regime of the solutions under discussion admits a reformulation in terms of the density of particles contributing to the modes of two fields. I shall strictly follow Charach's approach here [38], which consists of performing a quasi-classical treatment based on the geometrical optics energy-momentum tensor. A family of Lorentz local frames is introduced so that the density of particles in each normal mode can be defined through:
λ (a) = |η νν | δ (a)
ν , (no summation over ν)
whereη νν represents the components of the inhomogeneous generalization of the DZN metric.
A set of observers corresponding to this tetrad are characterized by the 4-velocity:
u ν ≡ λ ν (0) = η 00 , 0, 0, 0 .
Let us consider now equations (42b,42c), which give the WKB stress-energy tensors of the cosmological model with gravitational and scalar wave perturbations, namely:
(GW )T µ ν j = B 2 α 2 j 4 |j| κ µ j κ ν j (51a) (SW )T µ ν j = A 2 α 2 j 2 |j| κ µ j κ ν j .(51b)
The density of scalar and gravitational particles in the n-th mode is given by
ρ S j =T (0)SW (0)j hκ (0) µj = A 2 α 2 j 2 t √ x k e f (52a) ρ G j =T (0)GW (0)j hκ (0) µj = B 2 α 2 j 4 t √ x k e f ,(52b)
where κ µν is a null vector with dimensions of length. Besides, since the description here is based on units c = G = 1, the Planck constant has dimensions of (length) 2 , where h ∼ 10 −66 cm 2 . The density does not depend on the direction along which the particles propagate, that is:
ρ S j = ρ S −j (53a) ρ G j = ρ G −j .(53b)
In the G 1 models considered here, the volume of the spatial sections t = constant is not finite; for that reason the total number of particles in each mode of the two degrees of freedom has no upper bound.
In light of this reformulation it was suggested that one should regard the evolution of G 2 models filled with waves as describing a process of transforming the initial inhomogeneities along z into quanta of various fields. Clearly, this interpretation's validity is extendible to models with just one isometry, such as those constructed in this paper.
IV. CONCLUSIONS
Before I finish, I will summarize the main results. I have presented the first method to generate uniparametric families of general relativistic cosmologies having a two-dimensional inhomogeneity and a m.s.f. as a source. In the context of either General Relativity or alternative theories of gravity, one can generate a large number of new inhomogeneous cosmologies using this algorithm, where moreover the only input needed is any of the many known vacuum relativistic cosmologies with two commuting Killing vectors.
It has also been shown that this technique allows one to construct families of cosmologies which represent waves propagating on a spatially curved cosmological background. The spacelike singularity structure of these solutions has been studied, and several peculiarities due to the matter content have been elucidated.
It is important to note here that the generation technique does not restrict the character of the gradient of metric function G(t, z) of the input metric. That function determines the local behavior of the spacetime, and its gradient can be globally timelike, spacelike, null orvary from point to point. Although in what follows I am focusing on a case with a timelike character of the gradient of G(t, z), a window is left open for the study of other physically appealing cases. Moreover, even though our generating prescription has been used to break homogeneity of an input m.s.f. solution with a single degree of inhomogeneity, it is also possible to construct an equivalent algorithm that would transform certain static spacetimes into nonstatic ones. One should start with a m.s.f. model with two commuting Killing vectors, one of them being timelike, and then generalize it by introducing time dependent factors in the metric and scalar field as I have done here.
FIG. 1 .
1An infinite line along the i direction exists when lim t→0 ǫ i = ∞. Again, three cases can be seen to occur, depending on which is the axis displaying that feature. For some particular values of the parameters the maximum allowed number of two infinite lines can be reached.Since the singular behavior of the metrics depends on three parameters it becomes rather complicated to represent graphically the structure of singularities in the general case. For that reason I will restrict myself to cases in which the value of one of the parameters is fixed,namely C. In the pictures here two different sets of lines can be distinguished. On the one hand, the black lines in each plot correspond to the curves along which a spatial direction becomes a finite line. In the region delimited by the black continuous line and the axes, a infinite line type of singularity arises along direction z. Here one can see how for C = 0.5 it is possible to have simultaneously the same behavior along directions y and z. For k > 1 in the region above the black dashed line there is contraction along direction y, this behavior gets reversed for k < 1. Similarly, for β values less than those along the dashed-dotted line contraction takes place, and the contrary happens for k > 1. The points where two black lines intersect correspond to having two finite lines. On the other hand, the grey lines represent the curves along which two spatial directions display the same expansion rate. The fact that at a given point the three grey lines intersect reflects the possibility of having isotropic expansion, and for the two C values chosen, that point lies in the k > 1 region. Structure of singularities of the inhomogeneous generalization of Belinskii-Khalatnikov's model for C = 0.5 (l.h.s. figure) and C = −0.5 (r.h.s. figure) with sign (A m). On the one hand, the black lines indicate the value of β as a function of k for which a given direction behaves as a finite line. The black continuous, dashed and dashed-dotted lines correspond to a finite line along z, y or x directions respectively. On the other hand, the grey lines indicate the value of β as a function of k for which two spatial directions have the same expansion rate. The grey continuous, dashed and dashed-dotted lines correspond respectively to ǫ 2 = ǫ 3 , ǫ 1 = ǫ 3 and ǫ 1 = ǫ 2 .
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| []
|
[
"Dynamic coupling design for nonlinear output agreement and time-varying flow control",
"Dynamic coupling design for nonlinear output agreement and time-varying flow control"
]
| [
"Mathias Bürger [email protected] \nInstitute for Systems Theory and Automatic Control\nUniversity of Stuttgart\nPfaffenwaldring 970550StuttgartGermany\n",
"Claudio De Persis [email protected] \nITM, Faculty of Mathematics and Natural Sciences\nUniversity of Groningen\nNijenborgh 49747 AGGroningenthe Netherlands\n"
]
| [
"Institute for Systems Theory and Automatic Control\nUniversity of Stuttgart\nPfaffenwaldring 970550StuttgartGermany",
"ITM, Faculty of Mathematics and Natural Sciences\nUniversity of Groningen\nNijenborgh 49747 AGGroningenthe Netherlands"
]
| []
| This paper studies the problem of output agreement in networks of nonlinear dynamical systems under time-varying disturbances, using dynamic diffusive couplings. Necessary conditions are derived for general networks of nonlinear systems, and these conditions are explicitly interpreted as conditions relating the node dynamics and the network topology. For the class of incrementally passive systems, necessary and sufficient conditions for output agreement are derived. The approach proposed in the paper lends itself to solve flow control problems in distribution networks. As a first case study, the internal model approach is used for designing a controller that achieves an optimal routing and inventory balancing in a dynamic transportation network with storage and time-varying supply and demand. It is in particular shown that the time-varying optimal routing problem can be solved by applying an internal model controller to the dual variables of a certain convex network optimization problem. As a second case study, we show that droop-controllers in microgrids have also an interpretation as internal model controllers. | 10.1016/j.automatica.2014.10.081 | [
"https://arxiv.org/pdf/1311.7562v1.pdf"
]
| 9,212,206 | 1311.7562 | 095c110ea1e59d2b86cb69cd6d5c65722e4652f0 |
Dynamic coupling design for nonlinear output agreement and time-varying flow control
December 2, 2013
Mathias Bürger [email protected]
Institute for Systems Theory and Automatic Control
University of Stuttgart
Pfaffenwaldring 970550StuttgartGermany
Claudio De Persis [email protected]
ITM, Faculty of Mathematics and Natural Sciences
University of Groningen
Nijenborgh 49747 AGGroningenthe Netherlands
Dynamic coupling design for nonlinear output agreement and time-varying flow control
December 2, 2013
This paper studies the problem of output agreement in networks of nonlinear dynamical systems under time-varying disturbances, using dynamic diffusive couplings. Necessary conditions are derived for general networks of nonlinear systems, and these conditions are explicitly interpreted as conditions relating the node dynamics and the network topology. For the class of incrementally passive systems, necessary and sufficient conditions for output agreement are derived. The approach proposed in the paper lends itself to solve flow control problems in distribution networks. As a first case study, the internal model approach is used for designing a controller that achieves an optimal routing and inventory balancing in a dynamic transportation network with storage and time-varying supply and demand. It is in particular shown that the time-varying optimal routing problem can be solved by applying an internal model controller to the dual variables of a certain convex network optimization problem. As a second case study, we show that droop-controllers in microgrids have also an interpretation as internal model controllers.
Introduction
Output agreement has evolved as one of the most important control objectives in cooperative control. It appears in various contexts, ranging from distributed optimization ( [TBA86]), formation control [OSFM07] up to oscillator synchronization ( [SS07]). Over the last years, it has become evident that the internal model principle takes a central role in output agreement problems, see e.g. [WSA11], [BAW11], [PJ12], [De 13]. The present paper studies output agreement in networks of heterogeneous nonlinear dynamical systems affected by external disturbances. Conditions on the dynamic couplings (or equivalently design principles for controllers placed on the edges of the network) are derived, that ensure output agreement. We follow here the trail opened in [PM08] for centralized output regulation and provide necessary and sufficient conditions for the solution of the output agreement problem for the class of incrementally passive systems. We propose an approach that is inherently different from other internal model approaches such as [WSA11] (see [WWA13] and [IMC13] for an extension to nonlinear system), where systems without external disturbances are considered. The conceptual idea of [WSA11] can be summarized as follows. Each node is augmented with a local controller that contains the model of a reference system, identical for all nodes. The local controllers are designed such that the node dynamics asymptotically track the reference system. The local ("virtual") copies of the reference system are then synchronized with static diffusive couplings. The approach considered in the present paper is inherently different. Most obviously, the objective of this paper is the design of dynamic couplings, rather than the design of local controllers. Furthermore, as external signals are assumed to affect the node dynamics, the assumptions of [WSA11] do not hold (e.g., the controllability of the complete node dynamics is not given) and therefore the approach of [WSA11] is not applicable. Incrementally passive systems and disturbance rejection are also dealt with in [PJ12]. However, the framework we propose here, inspired by [PM08], is completely different and leads to a family of new distinct results that have not been considered in [PJ12]. Therefore, our results complement the existing approaches and add a new perspective to internal model control for output agreement.
The contributions of this paper are as follows. We consider networks of nonlinear systems, interacting according to an undirected network topology. The design objective is to design controllers placed on the edges of the network that achieve output agreement. We present and discuss necessary conditions for the feasibility of the problem. For the class of linear systems, we provide an interpretation of these conditions, relating the node dynamics and the network topology, that explain the important role of passivity in output agreement problems. Following this, sufficient conditions for output agreement in networks of incrementally passive systems are provided. We prove that the output agreement problem is feasible if one can find an incrementally passive internal model controller. A relevant class of nonlinear systems is presented, for which the proposed internal model controller design is always possible. To clarify the relation to the existing literature, two special situations are discussed, where either output agreement can be reached with static diffusive couplings or where the disturbances are constant. Following the general theoretic discussion, the internal model control design approach is shown to be relevant for different applications. First, the problem of optimal routing control in distribution systems with time-varying demand is considered, as they appear, e.g., in supply chains ( [AGT11]) or data networks ( [MS83]). Following the internal model control design procedure, routing controllers are designed that achieve a balancing of the inventory levels and an optimal routing of the flow. Second, it is shown that droop-controllers in microgrids, as, e.g., studied in [SPDB13], turn out to be designed exactly in accordance to the internal model control approach. In view of this, the internal model control approach provides the theoretical framework for the analysis and design of networked systems.
The remainder of the paper is organized as follows. The problem formulation and necessary conditions for output agreement are presented in Section 2. Sufficient conditions for output agreement in networks of incrementally passive systems are discussed in Section 3. A constructive procedure for the design of such controllers for a class of nonlinear systems in presented in Section 4. In Section 5, the relation to known methods in the literature is formally discussed. The time-varying optimal distribution problem is presented in Section 6 and the interpretation of droop-controllers a internal model controllers is provided in Section 7.
Notation: The set of (positive) real numbers is denoted by R (R ≥ ). Given two matrixes A and B, the Kronecker product is denoted by A ⊗ B. The Moore-Penrose inverse (or pseudo-inverse) of a non-invertible matrix A is denoted by A † . The rangespace and null-space of a matrix B are denoted by R(B) and N (B), respectively. A graph G = (V, E) is an object consisting of a finite set of nodes, |V | = n, and edges, |E| = m. The incidence matrix B ∈ R n×m of the graph G with arbitrary orientation, is a {0, ±1} matrix with [B] ik having value '+1' if node i is the initial node of edge k, '-1' if it is the terminal node, and '0' otherwise.
Problem formulation and necessary conditions
We consider a network of dynamical systems defined on a connected, undirected graph G = (V, E). Each node represents a nonlinear systeṁ
x i = f i (x i , u i , w i ) y i = h i (x i , w i ), i = 1, 2, . . . , n,(1)
where x i ∈ R r i is the state, and u i , y i ∈ R p are the input and output, respectively. Each system (1) is driven by the time-varying signal w i ∈ R q i , representing, e.g., a disturbance or reference. We assume that the exogenous signals w i are generated by systems of the formẇ
i = s i (w i ), w i (0) ∈ W i ,(2)
where W i is a set whose properties are specified below.
Assumption 1 The vector field s i (w i ) satisfies for all w i , w i the inequality
(w i − w i ) T (s i (w i ) − s i (w i )) ≤ 0.(3)
This is going to be a standing assumption in this paper. As an example, consider the linear function with skew-symmetric matrix s i (w i ) = S i w i , S T i + S i = 0. We stack together the signals w i , for i = 1, 2, . . . , n, and obtain the vector w ∈ R q , which satisfies the equationẇ = s(w). In what follows, whenever we refer to the solutions ofẇ = s(w), we assume that the initial condition is chosen in a compact set W = W 1 × . . . × W n . The set W is assumed to be forward invariant for the systemẇ = s(w).
Similarly, let x, u, and y be the stacked vectors of x i , u i , and y i , respectively. Using this notation, the totality of all systems is given bẏ
w = s(w) x = f (x, u, w) y = h(x, w) (4)
with state space W × X and X a compact subset of R r 1 × . . . × R rn . The control objective is to reach output agreement of all nodes in the network, independent of the exact representation of the time-varying external signals. Therefore, between any pair of neighboring nodes, i.e., on any edge of G, a dynamic controller will be placed, taking the formξ
k = F k (ξ k , v k ) λ k = H k (ξ k , v k ), k = 1, 2, . . . , m,(5)
with state ξ k ∈ R ν k and input v k ∈ R p . When stacked together, the controllers (5) give raise to the overall controllerξ
= F (ξ, v) λ = H(ξ, v),(6)
where ξ ∈ Ξ, a compact subset of R ν 1 × . . . × R νm . Throughout the paper the following interconnection structure between the plants, placed on the nodes of G, and the controllers, placed on the edges of G, is considered. A controller (5), associated with edge k connecting nodes i, j, has access to the relative outputs y i −y j . In vector notation, the relative outputs of the systems are
z = (B ⊗ I p ) T y.(7)
The controllers are then driven by the systems via the interconnection condition
v = −z,(8)
where v are the stacked inputs of the controllers. Additionally, the output of the controllers influence the incident systems via the interconnection 1
u = (B ⊗ I p )λ.(9)
Due to this interconnection structure the dynamics on the network can be represented as a closed-loop dynamics as illustrated in Figure 1. We are now ready to formally introduce the output agreement problem.
Definition 1 (Output Agreement Problem) The output agreement problem is solvable for the process (4) under the interconnection relations (7), (8), (9), if there exists controllers (6), such that every solution (w(t), x(t), ξ(t)) originating from W × X × Ξ is bounded and satisfies lim t→∞ (B T ⊗ I p ) y(t) = 0. Figure 1: Structure of the internal model control scheme.
ẋ i = f i (x i , u i , w i ) y i = h i (x i , w i ) i = 1, 2, . . . , ṅ w i = s i (w i ) B ⊗ I p (B ⊗ I p ) T − ξ k = F k (ξ k , v k ) λ k = H k (ξ k , v k ), k = 1, 2, . . . , m, y(t) z(t) v(t) λ(t) u(t)
Necessary Conditions
To start the discussion, we first investigate the necessary conditions for the output agreement problem to be solvable. To this purpose, we strengthen the requirement on the convergence of the regulation error to the origin, requiring that lim t→∞ (B T ⊗ I p ) y(t) = 0 uniformly in the initial condition ( [IB08]). The closed-loop system (4), (6), (7), (8), (9) can be written asẇ = s(w)
x = f (x, (B ⊗ I p )H(ξ), w) ξ = F (ξ, −(B ⊗ I p ) T h(x, w)).
(10) Definition 2 (ω-limit set) The ω-limit set Ω(W ×X ×Ξ) is the set of points (w, x, ξ) for which there exists a sequence of pairs (t k , (w k , x k , ξ k )) with t k → +∞ and (w k , x k , ξ k ) ∈ W × X × Ξ such that ϕ(t k , (w k , x k , ξ k )) → (w, x, ξ) as k → +∞, where ϕ(·, ·) is the flow of (10).
If the output agreement problem is solvable, then the ω-limit set Ω(W × X × Ξ) is nonempty, compact, invariant and uniformly attracts W × X × Ξ under the flow of (10). Furthermore, the ω-limit set must satisfy
Ω(W × X × Ξ) ⊆ (w, x, ξ) ∈ W × X × Ξ : (B ⊗ I p ) T h(x, w) = 0 .
This set is the graph of a map defined on the whole W and is invariant for the closed-loop system. By the invariance, for any solution w of the exosystem originating from W, there exists (x w , u w , ξ w ) such thatẋ
w = f (x w , u w , w) 0 = (B ⊗ I p ) T h(x w , w)(11)andξ w = F (ξ w , 0) u w = (B ⊗ I p )H(ξ w , 0).(12)
Proposition 1 If the output agreement problem is solvable, then, for every w solution tȯ w = s(w) originating in W, there must exist solutions (x w , u w , ξ w ) such that the equations (11), (12) are satisfied.
In a controller-independent form, the constraints (11) and (12) require that there exists (x w , u w ) satisfyingẋ
w = f (x w , u w , w), y w = h(x w , w) u w ∈ R(B ⊗ I p ), y w ∈ N (B T ⊗ I p ),(13)
where u w ∈ R(B ⊗ I p ) denotes that at every time t the vector u w (t) is contained in the respective vector space. Let in the following u w be a solution to (13), and λ w p be a trajectory satisfying u w = (B ⊗ I q )λ w p . The trajectory λ w p is uniquely defined if and only if the graph G has no cycles. Otherwise, the matrix B has a nontrivial nullspace, see [GR01]. In the most general form, the existence of a feedforward controller is equivalent to the constraint that there exists an integer d and maps τ :
W → R d , φ : R d → R d and ψ : R d → R mp satisfying ∂τ ∂w s(w) = φ(τ (w)) λ w p + λ w 0 = ψ(τ (w)), λ w 0 ∈ N B ⊗ I p .(14)
Note that there might be an infinite number of possible controllers that can generate the desired steady state input u w . If the constraint (14) holds, the systeṁ
η = φ(η), η ∈ R d λ = ψ(η)(15)
has the property that if η 0 = τ (w(0)), then the solution η(t) to (15) starting from η 0 is such that (B ⊗ I p )λ(t) = u w (t) for all t ≥ 0. We denote by η w such a solution to (15) such that (B ⊗ I p )ψ(η w (t)) = u w (t) for all t ≥ 0. We then let λ w (t) := ψ(η w (t)). Here λ w (t) is one of the infinite many realizations of the map λ w p (t) + λ w 0 (t), with λ w 0 ∈ N B ⊗ I p . To design a controller that decomposes into controllers on the edges of G, we introduce a vector η k ∈ R d for each edge k = 1, . . . , m, and denote with ψ k the entries of the vector valued function ψ corresponding to the edge k. Each edge is now assigned a controller of the formη
k = φ(η k ), λ k = ψ k (η k ), k = 1, 2, . . . , m.(16)
With the stacked vector η = [η T 1 , . . . , η T m ] T , and vector valued functionsφ(η) = [φ(η 1 ), . . . , φ(η m )] T , ψ(η) = [ψ 1 (η 1 ), . . . , ψ m (η m )] T , the overall controller (6) iṡ
η =φ(η) λ =ψ(η).(17)
If the initial condition is chosen as η 0 = I m ⊗ τ (w(0)) then the solution η(t) to (15) starting from η 0 is such that λ(t) = λ w (t) for all t ≥ 0.
Discussion: The Regulator Equations
The necessary conditions (11) and (12) are a weaker form of the regulator equations of [IB90]. If the systems (1) are such that for each given exogeneous input w(t) there exists a unique steady state response, and the ω-limit set can be expressed as Ω(W × X × Ξ) = {(w, x, ξ) : x = π(w), ξ = π c (w)}, then x w = π(w) and the regulator equations (11) express the existence of an invariant manifold where the "regulation error" (B T ⊗ I p )y is identically zero provided that the control input u w is applied. Furthermore, (12) express the existence of a controller able to provide u w . In this case, (11), (12) take the familiar expressions, see e.g. [IB90]:
∂π ∂w s(w) = f (π(w), (B ⊗ I p )H(π c (w)), w) 0 = (B ⊗ I p ) T h(π(w), w)(18)
and ∂π c ∂w
s(w) = F (π c (w), 0)(19)
However, there is a substantial structural difference between the output agreement problem considered here and output regulation problems, that can be best seen for linear dynamical systems. Suppose each system (1) is of the forṁ
x i = A i x i + G i u i + P i w i y i = C i x i ,(20)
with a linear exosystemẇ i = S i w i . Let in the followingĀ = block.diag(A 1 , . . . , A n ),Ḡ = block.diag(G 1 , . . . , G n ),P = block.diag(P 1 , . . . , P n ) andC = block.diag(C 1 , . . . , C n ). The exosystems are stacked into the dynamics w =Sw, withS = block.diag(S 1 , . . . , S n ).
The classical result of [Fra76] states that one can take x w = Πw and λ w = Γw such that the regulator equations (18) take the form of Sylvester equations
ΠS =ĀΠ +Ḡ(B ⊗ I p )Γ +P (B T ⊗ I p )CΠ = 0.(21)
Under controllability and observability assumptions, feasibility of (21) is necessary and sufficient for output regulation of linear systems. We will see next, that due to the networked structure of the considered problems the assumptions fail to hold, although the output agreement problem is solvable (as we show in the next sections). First note that the regulator equations (21) have a solution if and only if rank
Ā − sI rḠ (B ⊗ I p ) (B ⊗ I p ) TC 0 np×np = #rows,(22)
for all s ∈ σ(S), where r = n i=1 r i and σ(S) is the spectrum ofS. The condition states that no pole of the stacked exosystem is a transmission zero of the system from input λ to output z = (B ⊗ I p ) T y. To focus the discussion on the impact of the constraints resulting from the network, we impose the following assumption:
Assumption 2 For each system i ∈ {1, . . . , n} rank A i − sI r i G i C i 0 p×p = r i + p, ∀ s ∈ σ(S).
The important observation is that the rank condition can be violated due to the networked structure of the problem. We summarize this in the result below.
Proposition 2 Suppose Assumption 2 holds. The rank condition (22) is violated if either of the following holds:
1. G contains a cycle; 2. R H (s)(B ⊗ I p ) ∩ N (B T ⊗ I p ) = {0} for some s ∈ σ(S), whereH(s) = C(sI −Ā) −1Ḡ .
Moroever, the conditions are necessary provided that for all s ∈ σ(S), s ∈ σ(Ā).
The proof is presented in the appendix. The first conditions shows that the regulator equations (21) have no solution if graph contains cycles or if the transfer functions of the dynamical systems "rotate" R(B ⊗ I p ) in such a way that it intersects nontrivially its orthogonal space N (B T ⊗ I p ). The previous result gives an intuition about a class of systems for which the output agreement problem is feasible.
Corollary 1 Assume Assumption 2 holds and G contains no cycles. Suppose furthermore that all eigenvalues ofS have zero real part. Then the equations (21) are feasible ifH(s) is strictly positive real. 2
The proof is presented in the appendix. The result suggests that passivity takes an outstanding role in the output agreement problem.
Output agreement under time-varying disturbances
In this section we highlight sufficient conditions that lead to a solution of the problem for a special class of systems, namely incrementally passive systems. Our approach follows the line of [PM08], where the following notion of a regular storage function was introduced.
Definition 3 ([PM08]) A storage function V (t, x, x ) is called regular if for any se- quence (t k , x k , x k ), k = 1, 2, . . ., such that x k is bounded, t k tends to infinity, and |x k | → ∞, it holds that V (t k , x k , x k ) → ∞, as k → ∞.
The dissipativity characterization of incremental passivity provided in [PM08] is as follows.
Definition 4 The system (1) is said to be incrementally passive if there exists a C 1 regular storage function V i : R ≥0 × R r i × R r i → R ≥0
such that for any two inputs u i , u i and any two solutions x i ,x i , corresponding to these inputs, the respective outputs y i , y i satisfy
∂V i ∂t + ∂V i ∂x i f i (x i , u i , w i ) + ∂V i ∂x i f i (x i , u i , w i ) ≤ (y i − y i ) T (u i − u i ).(23)
Example 1 Linear systems of the form (20) that are passive from the input u i to the output y i are also incrementally passive,
with V i = 1 2 (x i −x i ) T Q i (x i −x i ) and Q i = Q T i > 0 the matrix such that A T i Q i + Q i A i ≤ 0 and Q i G i = C T i .
Example 2 Nonlinear systems of the forṁ
x i = f i (x i ) + G i u i + P i w i y i = C i x i (24) with f i (x i ) = ∇F i (x i ), F i (x i ) twice continuously differentiable and concave, and G i = C T i are incrementally passive. In fact, by concavity of F i (x i ), (x i − x i ) T (f i (x i ) − f i (x i )) ≤ 0, and V i = 1 2 (x i − x i ) T (x i − x i ) is the incremental storage function.
For the sake of brevity, we will in the following sometimes writeV i for the directional
derivative ∂V i ∂t + ∂V i ∂x i f i (x i , u i , w i )+ ∂V i ∂x i f i (x i , u i , w i ).
Incremental passivity can also be defined for static nonlinear systems. A static system y i = h i (u i , t) is said to be incrementally passive if it satisfies the monotonicity condition
(h i (u i , t) − h i (u i , t)) T (u i − u i ) ≥ 0,(25)
for all input pairs u i , u i and all times t ≥ 0.
In the previous section, it was shown that the controllers at the edge have to take the form (16). Now, they must be completed by considering additional control inputs that guarantee the achievement of the steady state. While we require the internal model to be identical for all edges, i.e., φ(η k ), the augmented systems might be different. Then, the controllers (16) modify aṡ
η k = φ k (η k , v k ) λ k = ψ k (η k ), k = 1, 2, . . . , m,(26)
where all controllers reduce to the common internal model if no external forcing is applied, i.e., φ k (η k , 0) = φ(η k ). The controller is then said to have the internal model property.
The following is the main standing assumption that the controllers must satisfy.
Assumption 3 For each k = 1, 2, . . . , m, there exists regular storage functions
W k (η k , η k ), with W k : R q k × R q k → R + such that ∂W k ∂η k φ k (η k , v k ) + ∂W k ∂η k φ k (η k , v k ) ≤ (λ k − λ k ) T (v k − v k ).(27)
It is in general difficult to design the incrementally passive controllers above. An important example when the design is possible is when the feedforward control input is linear, that is (14) is satisfied with τ = Id, φ = s and ψ is a linear function of its argument. In this case, we let φ k (η k , 0) = s(η k ), ψ k (η k ) = M k η k and define
φ k (η k , v k ) = s(η k ) + M T k v k .(28)
Then, by definition of s as the gradient of a concave function, the storage function
W k (η k , η k ) = 1 2 (η k − η k ) T (η k − η k ) satisfies ∂W k ∂η k φ k (η k , v k ) + ∂W k ∂η k φ k (η k , v k ) = (η k − η k ) T (s(w k ) − s(w k ) ) + (η k − η k ) T M T k (v k − v k ) ≤ (ψ k (η k ) − ψ k (η k ))(v k − v k ),(29)
that is (27). We state below the main result of the section that, while extending to networked systems the results of [PM08], provides a solution to the output agreement problem in the presence of time-varying disturbances.
Theorem 1 Consider the network G with dynamics on the nodes (4). Suppose all exosystems satisfy (3), the regulator equations (11) hold, and all node dynamics are incrementally passive. Consider the controllerṡ
η =φ(η, v) λ =ψ(η) + ν (30)
whereφ andψ are the stacked functions of φ k (η k , v k ) and ψ k (η k ), and ν is an additional input to be designed. Suppose the controllers have the internal model property and satisfy Assumption 3. Then, the controller (30) with the interconnection structure
u = (B ⊗ I p )λ, v = −(B T ⊗ I p )y.(31)
and ν := v = −(B T ⊗ I p )y solves the output agreement problem, that is every solution starting from W × X × Ξ is bounded and
lim t→+∞ (B ⊗ I p ) T y(t) = 0.
Proof: By the incremental passivity property of the x subsystem in (4) and (11), it is true that
∂V ∂t + ∂V ∂x f (x, u, w) + ∂V ∂x w f (x w , u w , w) ≤ (y − y w ) T (u − u w ), where V = i V i .
Similarly by Assumption 3, the system (30) satisfies
∂W ∂ηφ (η, v) + ∂W ∂η wφ (η w ) ≤ (λ − λ w ) T v − (ν − ν w )v, with W = k W k andφ(η w ) = I m ⊗ φ(η w ). Bearing in mind the interconnection constraints u = (B ⊗ I p )λ, u w = (B ⊗ I p )λ w , and v = −(B T ⊗ I p )y, and letting U ((x, x w ), (η, η w )) = V (x, x w ) + W (η, η w ) we obtaiṅ U ((x, x w ), (η, η w )) :=V (x, x w ) +Ẇ (η, η w ) ≤ (y − y w ) T (u − u w ) + (λ − λ w ) T v − (ν − ν w ) T v = (y − y w ) T (B ⊗ I p )(λ − λ w ) − (λ − λ w ) T (B T ⊗ I p )y + (ν − ν w ) T (B T ⊗ I p )y.
By definition of output agreeement, (B ⊗ I p ) T y w = 0 and the previous equality becomeṡ
U ((x, x w ), (η, η w )) ≤ ν T (B T ⊗ I p )y = −||(B T ⊗ I p )y|| 2 = −z T z,(32)
by definition of ν = −z and ν w = 0. Since U is non-negative and non-increasing, then U (t) is bounded. As x w , η w are bounded 3 and U is regular, then x, η are bounded as well.
Hence the solutions exist for all t. Integrating the latter inequality we obtain
+∞ 0 z T (s)z(s)ds ≤ U (0). By Barbalat's lemma, if one proves that d dt z T (t)z(t) is bounded then one can conclude that z T (t)z(t) → 0. Now, z(t) = (B T ⊗ I p )y = (B T ⊗ I p )h(x, w) is bounded because x, w are bounded.
If h is continuously differentiable andẋ,ẇ are bounded, thenż is bounded and one can infer that d dt z T (t)z(t) is bounded. By assumption, w is the solution ofẇ = s(w) starting from a forward invariant compact set. Hence, both w andẇ are bounded. On the other hand,ẋ satisfieṡ
x = f (x, (B ⊗ I p )ψ(η) − z, w)
which proves that it is bounded because x, η, z were proven to be bounded, while w is bounded by assumption. Therefore,ẋ,ẇ are bounded and this implies that d dt z T (t)z(t) is bounded. Then by Barbalat's Lemma we have lim t→+∞ z(t) = 0 as claimed.
The result still holds true, if any of the dynamical systems on the nodes or on the edges, is replaced by a static incrementally passive systems. As a matter of fact, denoting byĪ ⊆ {1, 2, . . . , n} the subset of indices corresponding to dynamic incrementally passive systems, it is enough to replace the Lyapunov function V with i∈Ī V i . Then, exploiting (25), one can still prove that (32) holds. This proves that the states x i , i ∈Ī, η are bounded. Notice that the outputs of the static nonlinearities y i = h i (u i , t) are bounded provided that h i (u i , ·) is a bounded function for every u i ∈ R p . In fact, boundedness of the controller state η implies boundedness of u i since u i = k b ik λ k (η k ). Furthermore, the assumptions on the interconnections can be weakened, if stronger assumptions on the node dynamics are imposed.
Corollary 2 Let all assumptions of Theorem 1 hold, but assume furthermore that all node dynamics are output strictly incrementally passive, that is, there exists a C 1 regular storage function V i , and a positive definite function ρ i : R p → R, such that for any two inputs u i , u i and corresponding outputs
y i , y i V ≤ −ρ i (y i − y i ) + (y i − y i ) T (u i − u i ).(33)
Then the output agreement problem is feasible with the interconnection (31) and ν = 0.
Proof: Consider the storage function used in proof of Theorem 1, i.e., U ((x, x w ), (η, η w )) = V (x, x w ) + W (η, η w ). After repeating the steps of the proof of Theorem 1, but using the output strict passivity property and setting ν = 0, (32) is now replaced bẏ
U ((x, x w ), (η, η w )) ≤ − n i=1 ρ i (y i − y w i ),
where y w i = y w j for all i = j. Now, we can proceed as in the proof of Theorem 1.
Output agreement for a class of nonlinear systems
We propose now a fairly large class of nonlinear systems for which the sufficient conditions of Theorem 1 are satisfied. Consider the systems introduced in Example 2, namelẏ
w i = s i (w i ) x i = f 0 (x i ) + Gu i + P i w i y i = Cx i i = 1, 2, . . . , n(34)
where, compared with (24), we have chosen the systems to have the same dynamics, i.e. f i (x i ) = f 0 (x i ) for all i = 1, 2, . . . , n, and we have set
G i = C T i = G = C T .
Assuming that the dynamics of the systems are the same facilitate the design of incrementally passive distributed controllers, as we see in the proof below.
Proposition 3 Consider systems (34), where f 0 = ∇F and F is a twice continuously differentiable and concave map, G = C T and full column rank matrix, and the maps s i satisfy (3). Moreover, assume that R(P i ) ⊆ R(G) for i = 1, 2, . . . , n. Given the vector of disturbances w = (w 1 , . . . , w n ), assume there exists a bounded solution x * to the systeṁ
x = f 0 (x) + n i=1 P i w i n .(35)
Then:
1. there exists a bounded solution x w , u w to the regulator equations (11);
2. there exists controllers at the edges of the forṁ
η k = s(η k ) + H k v k λ k = H T k η k + ν k , k = 1, 2, . . . , m(36)
such that output agreement problem is solved for the systems (34), interconnected with the controllers (36) via the conditions (31).
Proof: Take any solution x w * to (35). By definitioṅ
x w * = f 0 (x w * ) + n i=1 P i w i n .(37)
Observe that such a solution x w * is necessarily bounded. As a matter of fact, in view of the assumptions on f 0 , the incremental dissipation inequality (23) hold and the incremental
storage function V (x w * , x * ) = 1 2 (x w * − x * ) T (x w * − x * ) satisfiesV ≤ 0 (in system (35) inputs are absent). Hence V (x w
* , x * ) is bounded and by regularity of V and boundedness of x * , x w * is bounded. Define now
Gu w i = − P i w i − n i=1 P i w i n .(38)
Observe that since n i=1 Gu w i = 0 by construction, and G is full-column rank, then u w ∈ R(B ⊗ I p ), i.e., the requirement imposed by the interconnection condition (31) is fulfilled. An explicit expression for u w can be given. Let
(I n ⊗ G)u w = (( 1 n 1 T n n − I n ) ⊗ I r )P w =: (Y ⊗ I r )P w
where r is the dimension of the state space of each system. Hence (38) can be rewritten as
Gu w i = n j=1 Y ij P j w j , with Y ij = [Y ] ij .
There exists a solution u w i to the latter equation if and only if GG
† b i = b i with b i = n j=1 Y ij P j w j ,
where G † is the Moore-Penrose pseudo inverse. Recalling that R(P j ) ⊆ R(G), we can assume the existence of matrices Γ j such that (37), the latter becomeṡ
P j w j = GΓ j w j . As a result b i = n j=1 Y ij P j w j = n j=1 Y ij GΓ j w j = G n j=1 Y ij Γ j w j that is b i ∈ R(G), and GG † b i = GG † G n j=1 Y ij Γ j w j = G n j=1 Y ij Γ j w j = n j=1 Y ij GΓ j w j = n j=1 Y ij P j w j = b i . Then the unique solution to (38) is u w i = G † N j=1 Y ij P j w j . Replacing (38) intox w * = f 0 (x w * ) + Gu w i + P i w i .
The latter holds true for all i = 1, 2, . . . , n thus showing that
(x w , u w ) = ((1 n ⊗ I r )x w * , ((u w ) T , . . . , (u w n ) T ) T )
solves the regulator equations. Bearing in mind that (B ⊗ I p )λ w = u w , we have
λ w = −(B † ⊗ I p ) I − 1 T n 1n⊗Ip n P w = − B † I − 1 T n 1n n ⊗ I p P w,
that is λ w = Hw. Using the embedding (14) with τ = Id, φ = s and ψ(η) = Hη, and an analogous decomposition as in (16), the internal model controller takes the forṁ
η k = s(η k ) λ k = H T k η k , k = 1, 2, . . . , m.
The addition of the control term H k v k
η k = s(η k ) + H k v k λ k = H T k η k , k = 1, 2, .
. . , m.
renders the system incrementally passive, in view of the incrementally passive nature of the map s(·). Recall (see Example 2) that the condition on F that defines the dynamics of the systems according to the identity f 0 = ∇F guarantees incremental passivity of systems (34). Hence, we are under the conditions of Theorem 1 and one concludes that the controllersη
k = s(η k ) − H k z k λ k = H T k η k − z k , k = 1, 2, . . . , m.(39)
with z = (B T ⊗ I p )y guarantee that the output agreement problem is solved.
The controllers, designed as in (39), can be stacked together to the dynamicṡ
η =s(η) −Hv λ =Hη − ν,(40)
where η = [η T 1 , . . . , η T m ] T ∈ R mr ,s(η) = [s(η 1 ) T , . . . , s(η m ) T ] T , andH = block.diag{H 1 , . . . , H m }. Bearing in mind the controllers (40), one conclusion that follows immediately from the proof of the result is that steady state solution of the controllers η w can be taken as η w = 1 n ⊗ w. That is, one possible steady state solution of the output agreement problem is that each controller dynamics reproduces exactly the disturbance signal. This observation can be used to redesign the controllers. In particular, additional communication between the different (distributed) controllers can be used to improve the convergence of the controllers.
Adding communication between controllers
Consider an additional communication network G comm , having one node for each controller, and one edge if the controllers can exchange data. 4 For simplicity, we assume that G comm is an undirected connected graph. The Laplacian matrix of the communication graph is denoted by L comm ∈ R m×m . As we shall see below, the additional communication term allows us to add a diffusive coupling between the various controllers that explicitly enforces the convergence of all the controllers states η k to the same signal. This in turn guarantees that the stacked vectorHη = block.diag{H T 1 , . . . , H T m }(η T 1 . . . η T m ) T converges to Hη * , for some η * . We recall that under the conditions that the convergence to the solution of the output agreement problem is uniform in the initial conditions, such a signal η * must satisfy (B ⊗ I p )Hη * = u w . If in addition the graph is acyclic and the systeṁ η = s(η), λ = Hη is incrementally observable 5 , then necessarily, η * = w, i.e. the internal model controllers asymptotically synchronize to the disturbance w.
By revisiting now the proof of Theorem 1, one can directly see that the assumption of incremental passivity of the controllers, i.e., Assumption 3, is stricter than necessary. In particular, one can require the incremental passivity property (27) not to hold with respect to any two trajectories, but only with respect to the real and the steady state trajectory, i.e., with η k = η w k , v k = 0, λ k = λ w . Thus, one can replace Assumption 3 with the following weaker assumption.
Assumption 3a Let η w = τ (w) and λ w = ψ(τ (w)) be a solution to (14), and let v w = 0. For each k = 1, 2, . . . , m, there exists regular storage functions W k (η k , η w k ), with
W k : R q k × R q k → R + such that ∂W k ∂η k φ k (η k , v k ) + ∂W k ∂η w k φ k (η w k , 0) ≤ (λ k − λ w k ) T v k .
It can be readily seen that the proof of Theorem 1 remains valid if Assumption 3 is replaced by Assumption 3a. In particular, the following result holds.
Proposition 4 Let all assumptions of Proposition 3 hold and let L comm ∈ R m×m be the Laplacian matrix of communication graph. Then the distributed controller with communication of the formη
=s(η) − (L comm ⊗ I r )η −Hv λ =H T η (41)
interconnected with the node dynamics (36) according to (31), solves the output agreement problem Furthermore, lim t→∞ ||η k (t) − η j (t)|| = 0 for all k = j.
Proof: Under the assumptions of Proposition 3 it holds that η w = w. Consequently (L contr ⊗ I r )η w = 0. The controller (41) satisfies Assumption 3a, since the directional
derivative of W = 1 2 (η −η w ) T (η −η w ) iṡ W ≤ −(η − η w ) T (L comm ⊗ I r )(η − η w ) + (λ − λ w ) T v.
Mirroring the proof of Theorem 1, the derivative of the storage function U ((x, x w ), (η, η w )) satisfiesU
≤ − B T ⊗ I p )y 2 − η T (L comm ⊗ I r )η.
Thus, with the same arguments as in the proof of Theorem 1, convergence can be concluded. Additionally, this proves that lim t→∞ η k (t) − η j (t) = 0 for all k = j.
Relation to known results
Next, we compare our results to known results.
Static Couplings
The papers [SS07], [SAS10] study synchronization of cocoercive (or semi-passive([PN01])) systems that are free from disturbances with purely static output feedback that uses relative measurements. The result extend to systems that have a "shortage" of incremental passivity, called relaxed cocoercive systems. 6 For identical relaxed cocoercive systems it was shown in [SAS10] (see also [DdR11]) that static couplings suffice to ensure synchronization, provided that the network features a sufficiently strong coupling.
Here, heterogeneous incrementally passive systems affected by disturbances are considered. The heterogeneity of the systems and the time varying external disturbances cause the need for dynamic couplings. However, if all systems already share a common internal model, static couplings are also sufficient in our approach.
Proposition 5 (Static Coupling) Consider the system (4) and suppose all node dynamics are incrementally passive. If there exists a solution to the regulator equations (11) with u w (t) = 0 7 , then, the static controller λ = ν with the interconnection u = (B ⊗ I p )λ, and ν = −(B T ⊗ I p )y solves the output agreement problem.
Proof: By the incremental passivity property of the subsystems it is true thaṫ
V ≤ (y − y w ) T (u − u w ), where V = n i=1 V i . Now, since u w = 0 and y w ∈ N ((B ⊗ I p ) T ) the coupling u = −Ly = −(B ⊗ I p )(B ⊗ I p ) T y = −(B ⊗ I p )(B ⊗ I p ) T (y − y w ) giveṡ V ≤ (y − y w ) T L(y − y w ).
Convergence and boundedness can now be shown as in the proof of Theorem 1.
Please note that the input to the systems computes in this case as
u = −(L ⊗ I p )y,
where L = BB T is the Laplacian matrix of the (undirected) graph. Thus, for homogeneous systems, our controller design method reduces to the well-known Laplacian coupling, as studied, e.g., in [SS07], [SAS10], [DdR11]. However, it should be remarked that, while incrementally passive systems strictly include the class of cocoercive systems, our results do not appear to be trivially extendable to the class of relaxed cocoercive systems. Moreover, while the results of [SS07], [SAS10] apply to networked systems over a balanced, directed graph (possibly even time-varying), our results are given for static undirected graphs.
Static Disturbances
The output agreement problem with constant disturbances deserves particular attention. Control of passive system with constant disturbances is studied, e.g., in [JOGC07] or [HAP11], where the notion of equilibrium independent passivity is introduced. Equilibrium independent passivity is closely related to incremental passivity, i.e., it is defined by a condition similar to (23) assuming that one of the trajectories (e.g., x ) is an equilibrium trajectory. Optimality properties and a network theoretic interpretation of networks of equilibrium independent passive systems are discussed in [BZA13a], [BZA13b]. The stability of passive networks with static disturbance signals has also been discussed in [vW12]. We derive here slightly more general 8 controllers (or dynamic couplings) as [BZA13a], [BZA13b], using the internal model control approach.
Proposition 6 Consider the network G with dynamics on the nodes (4). Suppose w i is some constant signal, i.e., s i (w i ) = 0, the regulator equations (11) hold and (4) are incrementally passive. Then, any controller of the forṁ
η k = v k λ k = ψ k (η k ) + ν k , k ∈ {1, . . . , m}(42)
with ψ k (·) satisfying the strong monotonicity condition
(ψ k (η) − ψ k (η )) T (η − η ) ≥ c η − η 2 , ∀η, η(43)
for some positive constant c, and interconnection constraints (31) solves the output agreement problem.
Note that the controller (42) is not necessarily incrementally passive, i.e., Assumption 3 is not met. However, we will show next that the controllers satisfy the weaker Assumption 3a.
Proof: Let x w and u w be solutions to the regulator equations (11). By the structure of (11) follows immediately that v w = −(B ⊗ I p ) T h(x w , w) = 0. Since the disturbance is static, i.e.,ẇ = 0, the conditions (14) are solved with φ(·) = 0 and τ (w) such that
λ w p + λ w 0 = ψ(τ (w))(44)
for some λ w p satisfying u w = (B ⊗ I p )λ w p and some λ w 0 ∈ N (B ⊗ I p ) and constant. Thus, there is not a unique solution to (14), but rather for any λ w 0 ∈ N (B ⊗ I p ) there exists exactly one τ (w) solving (14) (the existence of more than one solution τ (w) would contradict the strong monotonicity condition (43)). Select now η w = τ (w) as a solution to (44) an arbitrary λ w 0 ∈ N (B ⊗ I p ), and let λ w = λ w p + λ w 0 . One can construct now for each controller a storage function W k that satisfies Assumption 3a, i.e., that shows passivity with respect to the constant signals λ w k and v w k = 0. Let in the following Ψ k : R q → R be a twice continuously differentiable function such that ∇Ψ k (η k ) = ψ k (η k ). Since, by assumption, ψ k satisfy the monotonicity condition (43) all Ψ k are strongly convex. Consider now the following storage function ( [JOGC07], [BZA13a]):
W k (η k , η w k ) = Ψ k (η k ) − Ψ k (η w k ) − ∇Ψ T k (η w k )(η k − η w k ).(45)
Since Ψ k is convex and, by the global under-estimator property of the gradient, we have
Ψ k (η k ) ≥ Ψ k (η w k ) + ∇Ψ T k (η w k )(η k − η w k ) for each η k , η w k .
Since Ψ k is strongly convex, then it is in particular strictly convex and the previous inequality holds if and only if η k = η w k . Then W k is regular ( [JOGC07]). Hence, W k is a positive regular storage function. Furthermore,
∂W k ∂η k φ k (η k , v k ) = (ψ k (η k ) − ψ k (η w k )) T v k = (λ k − λ w k ) T v k .
In the case of constant disturbances Assumption 3a is always fulfilled by controllers of the form (42). Mimicking now the proof of Theorem 1, using the storage function
U ((x, x w ), (η, η w )) = V (x, x w )+W (η, η w ), with W (η, η w ) = m k=1 W k (η k , η w k ), one obtainṡ U ≤ − (B ⊗ I p ) T y 2 .
With the same arguments as used in the proof of Theorem 1, it follows that the controller (42) solves the output agreement problem in the case of static disturbances, that is lim t→∞ (B ⊗ I p ) T y(t) = 0.
Time-varying Optimal Distribution Control
We use now the output agreement theory for the design of (optimal) distribution control laws in distribution networks with storage.
Consider an inventory system with n inventories and m transportation lines, and let B be the incidence matrix of the transportation network. The dynamics of the inventory system is given asẋ
= Bλ + P w,(46)
where x ∈ R n represents the storage level, λ ∈ R m the flow along one line, and P w an external in-/outflow of the inventories, i.e., the supply or demand. This basic model is studied, e.g., in [BBP06], [vW12] or in a discrete-time form in [BB12], [DBO + 13].
We assume here that the exact realization of the supply/demand is unknown, while it is known that it is generated by the dynamicṡ
w = s(w).(47)
The distribution and balancing problem is to design controllers on the edges of the network, using only measurements of the storage levels of the incident inventories and regulating the flows λ k such that instantaneously all possible supply/demand is satisfied and all inventory levels evolve synchronously. The balancing problem has been recently studied in [DBO + 13] using predictive control. By choosing u = Bλ, the problem can be readily formulated as an output agreement problem with time varying disturbance. The regulator equations (13) for the distribution problem arė
x w (t) = u w (t) + P w(t),
u w (t) ∈ R(B), x w (t) ∈ N (B T ).(48)
The solution to the regulator equation (13) is
x w (t) = 1 n x w 0 + t 0 1 T n P w(s) n ds(49)
where x w 0 belongs to the projection of ω(W × X × Ξ) onto R r 1 +...+rn . To see (49), note that u w (t) ∈ R(B) ⇔ 1 T n u w (t) = 0. Let now x w (t) = 1 n x w * (t), for some x w * (t) ∈ R. Then, multiplying (48) from the left with the all ones vector gives nẋ w = 1 T P w(t), leading to the desired expression. The following observation is now a direct consequence.
Proposition 7
The output agreement problem is feasible only if the accumulated imbalancew(t) = t 0 1 T n P w(s) n ds is bounded for all t ≥ 0.
Otherwise the inventory levels (i.e., x w ) will grow unbounded. The corresponding input is naturally given as u w (t) = −∆ n P w(t), with ∆ n = (I n − 1 n 1 n 1 T n ), namely the projection of the supply/demand vector to the space orthogonal to span{1 n }. Next, we verify that the necessary conditions for the output agreement problem are satisfied by showing feasibility of (14). Note that the controller output must satisfy Bλ w = −(I n − 1 n 1 n 1 T n )P w(t).
Proposition 8 If the network contains a spanning tree then the condition (14) is feasible.
Proof: Let T ⊆ G be a spanning tree. Assume without loss of generality that the edges are labeled in such a way that the flow vector can be written as λ w = [λ wT T , λ wT T ] T , where λ w T are the flows on the edges in T and λ w T are the flows in all other edges. Similarly, the incidence matrix can be represented as B = [B T , BT ]. A feasible flow solution λ w p can now be chosen as λ w T = 0 and λ w
T = −(B T T B T ) −1 B T T P w(t).
Note that λ w T routes exactly the balanced component of the supply/demand through the network, since λ w
T = −(B T T B T ) −1 B T T (I n − 1 n 1 n 1 T n )P w(t) since B T T 1 n = 0 n . Define now H = −(B T T B T ) −1 B T T P 0 ,
and note that λ w p = Hw. Thus, τ = Id, φ(·) = s(·) and ψ being the linear function defined by H, solve (14).
After augmenting the controller with external outputs, a possible routing controller isη
= s(η) + H T v λ = Hη + ν.(50)
Note that if s(·) satisfies the standing assumption (3), then this controller is incrementally passive.
Optimal Distribution Control
We enlarge our control objective and aim to design a feedback controller that achieves an optimal routing. That is, we want to regulate the flows such that they minimize the quadratic cost function
P(λ) = 1 2 λ T Qλ,(51)
with Q = diag(q 1 , . . . , q m ) and q k > 0. We exploit therefore that the internal model controller achieving balancing is not unique.
In particular, we redesign the controller (50) in such a way that it routes the balanced component of the flow through the network in such a way that at each time instant the cost (51) is minimized. That is, asymptotically the routing should be such that at each time instant t the following static optimization problem is solved
min λ P(λ), s.t. 0 = Bλ + ∆ n P w,(52)
where w = w(t) is the supply at the respective time. Let now ζ ∈ R n be the multiplier for the equality constraint. The Lagrangian function of (52) is L(λ, ζ) = 1 2 λ T Qλ + ζ T (Bλ + ∆ n P w).
One can express the optimality conditions in terms of the dual solution as
Qλ + B T ζ = 0, Bλ + ∆ n P w = 0, from which B(Q −1 B T ζ)+∆ n P w = 0, with the optimal routing being λ = Q −1 B T ζ. Thus the optimal routing/supply pairs are defined as the set Γ = {(λ, w) : Qλ ∈ R(B T ), Bλ + ∆ n P w = 0}.
We formalize the optimal distribution problem as follows:
Definition 5 The time-varying optimal distribution problem is solvable for the system (46), if there exists a controller (6) such that any solution originating from W × X × Ξ satisfies (i) lim t→∞ B T x(t) = 0 and (ii) lim t→∞ dist Γ (λ(t), w(t)) = 0.
To solve the problem, we proceed in this way. Instead of designing the controller directly for the flows, we design the controllers for the multipliers. We take τ = Id and φ(·) = s(·) and design a controller of the formη
= s(η) + H v v ζ = H ζ η,
where H ζ and H v are suitable input and output matrices to be designed next. The routing will then be defined as λ(t) = Q −1 B T ζ(t). For designing H ζ , note that, provided that v = 0 and the initial condition η(0) is properly chosen, the system above generates the solution η w (t) = w(t). Then H ζ must be design in such a way that ζ w (t) = H ζ η w (t) satisfies the optimality condition
BQ −1 B T H ζ η w (t) + ∆ n P w(t) = 0.
The matrix L Q = BQ −1 B T is a weighted Laplacian matrix. As L Q has one eigenvalue at zero, with the corresponding eigenvector 1, it is not invertible. However, since η w (t) = w(t), one possible solution is
H ζ = −L † Q P,(53)
where L † Q is the Moore-Penrose-inverse of L Q , see e.g., [GX04]. From the properties of L † Q follows that BQ −1 B T H ζ η w +∆ n P w = −BQ −1 B T L † Q P η w +∆ n P w = −∆ n P η w +∆ n P w = 0 as desired. Now, as the controller should be incrementally passive with input v and output λ, we can design it in the form (50) taking
H = Q −1 B T H ζ .(54)
Then, to have incremental passivity, we simply choose H v = H T . This choice of the input and output matrix for the controller (50) ensures that the optimal distribution problem is solved.
Proposition 9 Consider the inventory system (46) with the supply generated by the linear dynamicsẇ = s(w), satisfying (3). Consider the controlleṙ
η = s(η) − H T z λ = Hη − z.
with the interconnection condition z = B T x. Then, every solution of the closed-loop system is bounded and (i) lim t→+∞ B T x = 0, and (ii) lim t→+∞ dist Γ (λ(t), w(t)) = 0, that is the time-varying optimal distribution problem is solvable.
Proof: First note that the optimal routing λ w (t) = Q −1 B T ζ w (t) satisfies the identity Bλ w (t) + P w(t) = −BQ −1 B T L † Q P η w (t) + P w(t) = −∆ n P η w (t) + P w(t) = 1 1 T P w(t) n .
Since 1 1 T P w(t) n =ẋ w (see (49)), the optimal routing is such thatẋ w (t) = Bλ w (t) + P w(t). Now, consider the storage function U (x − x w , η, η w ) = 1 2 x − x w 2 + 1 2 η − η w 2 along the solutions of the autonomous system d
dt (x − x w ) = B(λ − λ w ) = −BB T (x − x w ) + BHη − BHη w ,η = s(η) − H T B T (x − x w ),η w = s(η w ). It satisfieṡ U = − (x − x w ) T BB T x + (x − x w ) T BH(η − η w ) + (η − η w ) T (s(η) − s(η w )) − (η − η w ) T H T B T x ≤ − B T x 2 ,
due to the incremental passivity of the exosystem, i.e., (η − η w ) T (s(η) − s(η w )) ≤ 0. Since U is positive semidefinite and η w is bounded (again by the incremental passivity property of the exosystem), we have that x − x w , η, η w are all bounded. Then, by LaSalle's invariance principle, the trajectories converge to the largest invariant set such that B T (x − x w ) = B T x = 0. Thus, there exists x * such that on this set x − x w = x * 1 and the dynamics evolves aṡ
x * 1 = BHη − BHη w ,η = s(η),η w = s(η w ).(55)
After multiplying by 1 n 1 T from the left, it followsẋ * = 0, proving that x must approach x w modulo a constant. This proves the claim (i) of the statement. To prove claim (ii), note that insertingẋ * = 0 into (55) and bearing in mind that η w = w gives the necessary condition that in the set where B T (x − x w ) = 0 it must hold that
0 = BHη − BHη w = BHη − BHw = −BQ −1 B T L † Q P (η − w) = −∆ n P (η − w).
Hence, ∆ n P η = ∆ n P w. The flow on the invariant set is λ = Q −1 B T L † Q P η, while the optimal flow is λ w = Q −1 B T L † Q P w. Together with the previous condition, this implies that there is a vector ν ∈ N (B) such that λ = λ w + ν. We will show next that ν must be identical to zero. Note that ν = λ − λ w , and must therefore satisfy
ν = Q −1 B T L † Q P (η − w).
Multiplying the previous equation from the left by ν T Q leads to ν T Qν = 0 since ν T B T = 0. As Q is by assumption positive definite, the only solution is ν = 0. This proves that in the set where B T x = 0 it must hold that λ = λ w , completing the proof. If additionally communication between the controllers is allowed, the controller can be augmented with a consensus term of the form (41). (b) Flow λ 1 (t) (solid) and optimal flow on the corresponding edge (dashed).
Simulation Example
We illustrate the performance of the controller on a design example. Consider a network with four inventories and five transportation lines as illustrated in Figure 2. The supply/demand at each inventory is generated by the linear dynamicṡ (41), where H is chosen to satisfy (54), and G comm is chosen such that two controllers communicate if they are incident to the same inventory. The simulation results for the inventory levels are shown in Figure 3a. Note that the supply/demand is not balanced, but the accumulated imbalance is bounded. The controller achieves a balancing of the inventory levels. As an example, the flow λ 1 (t) is shown in Figure 3b. The flow approaches fairly quickly the time-varying optimal flow. The simulations illustrate that the controller achieves both objectives, the balancing of the inventory levels and the optimal routing of the flow through the network.
Power Systems Droop-Control as Internal Model Control
In [SPDB13] a dynamic oscillatory model of microgrids with frequency-droop controllers is investigated. We provide next an interpretation of the results of [SPDB13] in the context of internal model control. The model of [SPDB13] for the frequency-droop controller is
D iθi = P * i − P e,i , i ∈ {1, . . . , n},(56)
where D i is the inverse of the controller gain, P * i is the inverters nominal power, and P e,i is the active electric power. The active electric power is given by
P e,i = n j=1 α ij sin(θ i − θ j ),(57)
where α ij are constants depending on the node voltages and the line admittance. The coefficients are symmetric α ij = α ji and only non-zero if the two nodes i and j are connected by a line. We refer to [SPDB13] for a detailed discussion of the model. As in [SPDB13], we restrict the discussion in the following to acyclic networks. In the proposed model, the dynamics in the nodes (56) represents the controllers, while the couplings between the nodes, i.e., (57), are physical laws. Although the situation is reversed to the basic setup of this paper, we can still interpret the droop-controller as an internal model controller. Consider the node dynamics (1) as (56), with node state x i = θ i , constant external signal w i = P * i , satisfyingẇ i = 0, input u i = −P e,i and output y i =θ i , i.e.,
D iẋi = P * i + u i , y i =ẋ i .(58)
By defining the inputs and outputs in this way, the node dynamics is output strictly incrementally passive since for any to inputs u i , u i and the two corresponding outputs y i , y i , it holds that
(y i − y i )(u i − u i ) = (y i − y i )(D i y i − P * i − D i y i + P * i ) = D i y i − y i 2 .
From the interpretation of the node dynamics (1) as (56), one notices that u = −P e = −BA sin(B T x), where sin(z) = [sin(z 1 ), . . . , sin(z m )] T , A = diag{a 1 , . . . , a m }, a k = α ij = α ji , and k is the label of the edge connecting nodes i, j. One can then interpret the latter equation as the first one of the interconnection conditions (31), provided that
λ = A sin(B T x). Set now η = B T x. Theṅ η = −B T y λ = A sin(η).(59)
This can be understood as the stacked controllers (26), where, for all k, φ k (η k , v k ) = v k , v = B T y, ψ k (η k ) = a k sin(η k ).
Hence, rewriting the model (56)-(57) in this way leads directly to an interpretation as an internal model control loop of the form (42), where the feed-through term can be omitted, i.e., ν = 0, since the node dynamics is output strictly incrementally passive (see Corollary 2). We can now restate the result of [SPDB13] in the context of internal model control.
Proposition 10 Consider the droop-controller dynamics in the form (58) and (59) and let the underlying network G be acyclic. Then 1. the regulator equations (11) are solved byẋ w
i = n i=1 P * i n i=1 D i =: y w and u w i = D i n i=1 P * i n i=1 D i − P *
i for all i = 1, 2, . . . , n;
the embedding condition (14) is feasible if and only if
A −1 (B T B) −1 B T u w ∞ < 1;
3. if the necessary conditions (11) and (14) hold, the solutions to the closed loop dynamics (58) and (59) with interconnection u = Bλ and that originate sufficiently close to x w and η w :
= sin −1 (A −1 (B T B) −1 B T u w ) satisfy lim t→∞ y i − y w → 0.
The proof follows completely along the lines of the internal model control approach (except the local nature of the stability result) and exploits in particular the results for static disturbances of Section 5.2. For completeness, we provide the proof in the appendix.
Conclusions
The paper has investigated output agreement problems in the presence of time-varying disturbances and has discussed the role of dynamic internal-model-based controllers to tackle these problems. We focus on the case in which only relative measurements are available to the controllers and the control applied to the systems must lie in the range of the incidence matrix. This scenario in fact is very important in distribution networks and is motivated by the physics of the network (Kirchhoff's law). We have examined two of these distribution networks, namely an inventory system and a microgrid, and we have interpreted a load balancing controller and the frequency-droop controller within the proposed framework. Furthermore, in the case of the inventory system, we have shown controllers that achieve an optimal routing. The proposed methodology lends itself to several possible extensions. The use of dynamic controllers could be exploited not only to tackle the presence of exogenous disturbances but also to deal with synchronization problems of heterogenous systems for which a static diffusive coupling does not suffice. In many other distribution networks, and similarly to the inventory system, the constraints imposed by the network induces a non-unique solution to the output agreement problem. It is then meaningful to design controllers that lead to a solution with optimal features. Our approach naturally lends itself to providing such solutions. Other aspects that could be studied are the presence of uncertainties in the exosystems, larger classes of disturbance signals, robustness to other sources of uncertainties in the dynamical systems. In the current implementation, our internal model controllers depend on all the exosystems generating the disturbance, that could be unfeasible in practice and should be relaxed. Moreover, the potentials of our approach in the context of the two case studies have not been fully explored yet. Phenomena to be studied are for instance the presence of constraints on the input and state variables. For the case of power systems, other classes of controllers could be considered, dealing for instance with the presence of time-varying exogenous inputs.
Proof of Corollary 1
Proof: Under the given assumptions, the equations (21) are feasible if and only if Condition 2 in Proposition 2 does not hold. Suppose, by contradiction, that H(s) is strictly positive real and Condition 2 in Proposition 2 holds. The condition can equivalently be expressed as follows: there exist vectors v ∈ R np and 0 = β ∈ R p such that H(s)(B ⊗ I p )v = 1 n ⊗ β, ∀s ∈ σ(S).
Multiplying the previous condition from the left by v T (B ⊗ I p ) T leads to v T (B ⊗ I p ) TH (s)(B ⊗ I p )v = 0, ∀s ∈ σ(S).
Since all s ∈ σ(S) have zero real part, is equivalent toṽ T H (jω) +H(−jω) ṽ = 0 for allṽ = (B ⊗ I p )v and for some ω ∈ R. This is a contradiction since H(s) being strictly positive real implies that H (jω) +H(−jω) is positive definite for all ω ∈ R. This proves the statement.
Proof of Proposition 10
Proof: The first statement follows directly after summing all equations (58) and noting that there must be a scalar valued function y * ∈ R such that y i =ẋ w i = y * for all i ∈ {1, . . . , n}.
To prove the second statement, we choose φ(τ (w)) = 0 sinceẇ = 0. Now, note that u w = Bλ w , and since the network is acyclic, λ w is uniquely defined as λ w = (B T B) −1 B T u w . Since for an acyclic network N (B) = {0}, the second condition in (14) becomes (B T B) −1 B T u w = ψ(τ (w)), where we can take ψ(τ (w)) = A sin(τ (w)). Thus, we have τ (w) = sin −1 (A −1 (B T B) −1 B T u w ), which exists if and only if A −1 (B T B) −1 B T u w ∞ < 1. To prove local stability we use the standard storage function (45). Note that it can be defined with Ψ k (η k ) = a k (1 − cos(η k )). If the conditions (14) hold, choose η w = τ (w). Stability follows now with the storage function U ((x, x w ), (η, η w ) = n i=1 V i (x i , x w i ) + m k=1 W k (η k , η w k ), with V i (x i , x w i ) = 0 and W k defined as (45). Note that U is positive semidefinite in a neighborhood around x w and η w and such that η, η w ∈ (− π 2 , π 2 ) m , and satisfiesU
≤ − n i=1 D i y i (t) − y w i (t) 2 = − n i=1 D −1 i m k=1 b ik a k (sin(η k (t)) − sin(η w k (t))) 2 ,(62)
due to output strict incremental passivity of (58). Note that the latter inequality involves only the variables η, η w . Hence, it shows that the trajectories of the closed-loop systeṁ η = −B T y = −B T D −1 (P * + BAsin(η)) are bounded and converge to the set of points where BAsin(η) = BAsin(η w ) (i.e., to the set of points where sin(η) = sin(η w ), since the graph has no cycles and A is a diagonal matrix) or, equivalently, to the set of points where y i = y w =ẋ w * for all i. Thus, any trajectory originating sufficiently close to x w and η w satisfies lim t→∞ y i − y w → 0.
Figure 2 :
2Inventory network with four inventories and five transportation lines. levels x i (t), i ∈ {1, 2, 3, 4}, under time-varying supply/demand.
1 = 0.1, s 2 = 0.7, s 3 = −0.4 and s 4 = −0.2 and initial conditions w i (0) = [1, 1] T . The flow cost function is of the form (51) with Q = diag{1, 2, 3, 4, 5}. The controller is implemented in the distributed form with communication
The interconnection structure (7), (9) naturally represents a canonical structure for distributed control laws. This structure is often considered in the context of passivity-based cooperative control, see e.g.,[Arc07],[BAW11],[vdSM13],[DJ12],[BZA13a].
We refer to[Kha02, Def. 6.4] for the definition of a strictly positive real transfer function.
By definition, (w, x w , η w ) belongs to the ω-limit set, which is compact. Hence, x w , η w are bounded.
for instance, two controllers can exchange data if their corresponding edges are incident to the same node in the original graph G.5 The systemη = s(η), λ = Hη is incrementally observable if any two solutions η, η toη = s(η) which yield the same output necessarily coincide, i.e. η = η .
Relaxed cocoercive systems are related to QUAD systems[DdR11]. Any QUAD system, augmented with an additive input on each state and the output being the full state vector, is also relaxed cocoercive.
This holds in particular if all systems (and exosystems) are identical. More generally, it asserts that all systems incorporate the same internal model.8 In fact, differently from[vW12],[BZA13a],[BZA13b], we do not assume output strict incremental passivity but only incremental passivity.
A AppendixProof of Proposition 2 Proof: (Necessity) If the rank condition in (22) is violated then there exists a nonzero vector (x 0 , λ 0 ) such thatThis equality can be made explicit asAs s ∈ σ(Ā), then necessarily λ 0 = 0. The graph G may or may not contain a cycle. If it does, then statement (1) of the thesis holds. If not, then (B ⊗ I p )λ 0 = 0 (recall that λ 0 = 0). Since s is not an eigenvalue of any A i , then from (61)The latter shows that R(H(s)(B ⊗ I p )) ∩ N ((B ⊗ I p ) T ) = {0}, that is statement (2) of the thesis.
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| []
|
[
"HandVoxNet: Deep Voxel-Based Network for 3D Hand Shape and Pose Estimation from a Single Depth Map",
"HandVoxNet: Deep Voxel-Based Network for 3D Hand Shape and Pose Estimation from a Single Depth Map"
]
| [
"Jameel Malik ",
"Ibrahim Abdelaziz ",
"Ahmed Elhayek ",
"Soshi Shimada ",
"Sk Aziz Ali ",
"Vladislav Golyanik ",
"Christian Theobalt ",
"Didier Stricker ",
"T U Kaiserslautern ",
"Dfki Kaiserslautern ",
"Nust Pakistan ",
"Saudi Upm ",
"Arabia ",
"Mpii Saarland "
]
| []
| []
| 3D hand shape and pose estimation from a single depth map is a new and challenging computer vision problem with many applications. The state-of-the-art methods directly regress 3D hand meshes from 2D depth images via 2D convolutional neural networks, which leads to artefacts in the estimations due to perspective distortions in the images.In contrast, we propose a novel architecture with 3D convolutions trained in a weakly-supervised manner. The input to our method is a 3D voxelized depth map, and we rely on two hand shape representations. The first one is the 3D voxelized grid of the shape which is accurate but does not preserve the mesh topology and the number of mesh vertices. The second representation is the 3D hand surface which is less accurate but does not suffer from the limitations of the first representation. We combine the advantages of these two representations by registering the hand surface to the voxelized hand shape. In the extensive experiments, the proposed approach improves over the state of the art by 47.8% on the SynHand5M dataset. Moreover, our augmentation policy for voxelized depth maps further enhances the accuracy of 3D hand pose estimation on real data. Our method produces visually more reasonable and realistic hand shapes on NYU and BigHand2.2M datasets compared to the existing approaches. | 10.1109/cvpr42600.2020.00714 | [
"https://arxiv.org/pdf/2004.01588v1.pdf"
]
| 214,795,024 | 2004.01588 | ae335c92ee482f4d9db8321424e5746451679104 |
HandVoxNet: Deep Voxel-Based Network for 3D Hand Shape and Pose Estimation from a Single Depth Map
Jameel Malik
Ibrahim Abdelaziz
Ahmed Elhayek
Soshi Shimada
Sk Aziz Ali
Vladislav Golyanik
Christian Theobalt
Didier Stricker
T U Kaiserslautern
Dfki Kaiserslautern
Nust Pakistan
Saudi Upm
Arabia
Mpii Saarland
HandVoxNet: Deep Voxel-Based Network for 3D Hand Shape and Pose Estimation from a Single Depth Map
3D hand shape and pose estimation from a single depth map is a new and challenging computer vision problem with many applications. The state-of-the-art methods directly regress 3D hand meshes from 2D depth images via 2D convolutional neural networks, which leads to artefacts in the estimations due to perspective distortions in the images.In contrast, we propose a novel architecture with 3D convolutions trained in a weakly-supervised manner. The input to our method is a 3D voxelized depth map, and we rely on two hand shape representations. The first one is the 3D voxelized grid of the shape which is accurate but does not preserve the mesh topology and the number of mesh vertices. The second representation is the 3D hand surface which is less accurate but does not suffer from the limitations of the first representation. We combine the advantages of these two representations by registering the hand surface to the voxelized hand shape. In the extensive experiments, the proposed approach improves over the state of the art by 47.8% on the SynHand5M dataset. Moreover, our augmentation policy for voxelized depth maps further enhances the accuracy of 3D hand pose estimation on real data. Our method produces visually more reasonable and realistic hand shapes on NYU and BigHand2.2M datasets compared to the existing approaches.
Introduction
The problem of deep learning-based 3D hand pose estimation has been extensively studied in the past few years [33], and recent works achieve high accuracy on public benchmarks [32,18,22]. Simultaneous estimation of 3D hand pose and shape from a single depth map is a newly emerging computer vision problem. It is more challenging than the pose estimation because annotating real images for shape is laborious and cumbersome. Other salient challenges include varying hand shapes, occlusions, high number of degrees of freedom (DOF) and self-similarity. A 3D voxelized depth map and accurately regressed heatmaps of 3D joints (left block) are used to estimate two hand shape representations (middle block). To combine the advantages of these representations, we accurately register the shape surface to the voxelized shape (right block). Our architecture with 3D convolutions establishes a one-to-one mapping between voxelized depth map, voxelized hand shape and heatmaps of 3D joints.
The dense 3D hand mesh is a richer representation which is more useful than the sparse 3D joints, and it finds many applications in computer vision and graphics [27,23,17].
With the recent progress in deep learning, a few works [35,7,17,13,14] have proposed algorithms for simultaneous hand pose and shape estimation. Malik et al. [14] developed a 2D CNN-based approach that estimates shapes directly from 2D depth maps. The recovered shapes suffer from artifacts due to the limited representation capacity of their hand model [7,17]. The same problem can occur even by embedding a realistic statistical hand model (i.e., MANO) [23] inside a deep network [7,35]. In contrast to these model-based approaches [35,14], Ge et al. [7] proposed a more accurate direct regression-based approach using a monocular RGB image. Recently, Malik et al. [17] developed another direct regression-based approach from a single depth image. All of the approaches mentioned above treat and process depth maps with 2D CNNs, even though depth maps are intrinsically a 3D data. Training a 2D CNN to estimate 3D hand pose or shape given 2D representation of a depth map is highly non-linear and results in perspective distortions in the estimated outputs [18]. V2V-PoseNet [18] is the first work that uses 3D voxelized grid of depth map to estimate 3D joints heatmaps and, thus, avoids perspective distortions. However, extending this work for shape estimation by directly regressing 3D heatmaps of mesh vertices is not feasible in practice.
In this work, we propose the first 3D CNN architecture which simultaneously estimates 3D shape and 3D pose given a voxelized depth map (see Fig. 1) To this end, we introduce novel architectures based on 3D convolutions which estimate two different representations of hand shape (Secs. [3][4][5]. The first representation is the hand shape on a voxelized grid. It is estimated from a new voxel-to-voxel network which establishes a one-to-one mapping between the voxelized depth map and the voxelized shape. However, the estimated voxelized shape does not preserve the hand mesh topology and the number of vertices. For this reason, we also estimate hand surface (the second representation) with our voxel-to-surface network. Since this network does not establish a one-to-one mapping, the accuracy of the estimated hand surface is low but the hand topology is preserved. To combine the advantages of both representations, we propose registration methods to fit the hand surface to voxelized shape. Since real hand shape annotations are not available, we employ two 3D CNN-based synthesizers which act as sources of weak supervision by generating voxelized depth maps from our shape representations (see Fig. 2). To increase the robustness and accuracy of the hand pose estimation, we perform 3D data augmentation on the voxelized depth maps (Sec. 4.2).
We conduct ablation studies and perform extensive evaluations of the proposed method on real and synthetic datasets. Our approach improves the accuracy of hand shape estimation by 47.8% on SynHand5M dataset [14] and outperforms the state of the art. Our method produces visually more reasonable and plausible hand shapes of NYU and BigHand2.2M datasets compared to the state-of-the-art approaches (Sec. 6). To summarise, our contributions are:
1. The first voxel-based hand shape and pose estimation approach with the following novel components:
(i) Voxel-to-voxel 3D CNN-based network.
(ii) Voxel-to-surface 3D CNN-based network.
(iii) 3D CNN-based voxelized depth map synthesizers.
(iv) Hand shape registration components.
2.
A new 3D data augmentation policy on voxelized grids of depth maps.
Related Work
We now discuss the existing methods for deep hand pose and shape estimation. Moreover, we briefly report the most related works for depth-based hand pose estimation.
Deep Hand Pose and Shape Estimation. Malik et al. [14] proposed the first deep neural network for hand pose and shape estimation from a single depth image. To this end, they developed a model-based hand pose and shape layer which is embedded inside their deep network. Their approach suffers from artifacts due to the difficulty in optimizing complex hand shape parameters inside the network. Ge et al. [7] developed a direct regression-based algorithm for hand pose and shape estimation from a single RGB image. They highlight that the representation capacity of the statistical deformable hand model (i.e., MANO [23]) could be limited due to the small amount of training data and the linear bases utilized for the shape recovery. Zhang et al. [35] introduced a similar MANO model based approach using a monocular RGB image. Recently, Malik et al. [17] proposed a structured weakly-supervised deep learning-based approach using a single depth image. All of the above-mentioned methods use 2D CNNs and treat the depth maps as 2D data. Consequently, the deep network is likely to produce perspective distortions in the shape and pose estimations [18]. In contrast, we propose the first 3D convolutions based architecture which establishes a one-toone mapping between the voxelized depth map and the voxelized hand shape. This one-to-one mapping allows to more accurately reconstruct the hand shapes.
Hand Pose Estimation from Depth. In general, deep learning-based hand pose estimation methods can be classified into two categories. The first one encompasses the discriminative methods which directly estimate hand joint locations using CNNs [4,3,32,21,22,18,10,6,12]. The second category is hybrid methods which explicitly incorporate hand structure inside deep networks [16,30,8,19,15]. The disriminative methods achieve higher accuracy compared to the hybrid methods. The voxel-to-voxel approach [18] is powerful and highly effective because it uses 3D convolutions to learn a one-to-one mapping between the 3D voxelized depth map and 3D heatmaps of hand joints. Notably, the voxelized representation of depth maps is best suited for 3D data augmentation to improve the robustness and accuracy of the estimations. A few methods perform data augmentation on depth maps [19,28] or voxelized depth maps [18]. In this work, we integrate the voxel-to-voxel approach with our pipeline and, additionally, perform new 3D data augmentation on voxelized depth maps. Our 3D data augmentation policy helps to achieve a noticeable improvement in the 3D pose estimation accuracy on real datasets.
Method Overview
Given a single input depth image, our goal is to estimate N 3D hand joint locations J ∈ R 3×N (i.e., 3D pose) and K = 1193 3D vertex locations V ∈ R 3×K (i.e., 3D shape). Overview of our approach for 3D hand shape and pose recovery from a 3D voxelized depth map. V2V-PoseNet estimates 3D joints heatmaps (i.e., pose). Hand shape is obtained in two phases. First, V2V-ShapeNet and V2S-Net estimate the voxelized shape and shape surface, respectively. Thereby, V2V-SynNet and S2V-SynNet synthesize the voxelized depth acting as sources of weak-supervision. They are excluded during testing. In the second phase, shape registration accurately fits the shape surface to the voxelized shape. Fig. 2 shows an overview of the proposed approach. The input depth image is converted into a voxelized grid (i.e., V D ) of size 88 × 88 × 88, by using intrinsic camera parameters and a fixed cube size. For hand pose estimation, V D is provided as an input to the voxel-to-voxel pose regression network (i.e., V2V-PoseNet) that directly estimates 3D joint heatmaps {H j } N j=1 . Each 3D joint heatmap is represented as 44 × 44 × 44 voxelized grid. We resize V D to 44 × 44 × 44 voxel grid size (i.e., V D ) and concatenate it with the estimated H j , to provide as an input to our shape estimation network. We call this concatenated input as I S .
The voxelized hand shape (i.e., 64 × 64 × 64 grid size) is directly regressed via 3D CNN-based voxel-to-voxel shape regression network (i.e., V2V-ShapeNet), by using I S as an input. Notably, V2V-ShapeNet establishes a one-to-one mapping between the voxelized depth map and the voxelized shape. Therefore, it produces accurate voxelized shape representation but does not preserve the topology of hand mesh and the number of mesh vertices. To regress hand surface, I S is fed to the 3D CNN-based voxel-tosurface regression network (i.e., V2S-Net). Since the mapping between I S and hand surface is not one-to-one, it is therefore less accurate. Voxel-to-voxel and surface-tovoxel synthesizers (i.e., V2V-SynNet and S2V-SynNet) are connected after V2V-ShapeNet and V2S-Net, respectively. These synthesizers reconstruct V D and act as sources of weak supervision during training. They are excluded during testing. To combine the advantages of the two shape representations, we register the estimated hand surface to the estimated voxelized hand shape. We employ 3D CNN-based DispVoxNet [25] for synthetic data, and non-rigid gravitational approach (NRGA) [1] for real data.
The Proposed HandVoxNet Approach
In this section, we explain our proposed HandVoxNet approach by highlighting the function and effectiveness of each of its components. We develop an effective solution that produces reasonable hand shapes via 3D CNN-based deep networks. To this end, our approach fully exploits accurately estimated heatmaps of 3D joints as a strong pose prior, as well as voxelized depth maps. Given that collecting accurate real hand shape ground truth is hard and laborious, we develop a weakly-supervised network for real hand shape estimation by learning from accurately labeled synthetic data. Moreover, our 3D data augmentation on voxelized depth maps allows to further improve the accuracy and robustness of 3D hand pose estimation.
3D Hand Shape Estimation
As aforementioned, estimating 3D hand shape from a 2D depth map by using 2D CNN is a highly non-linear mapping. It compels the network to perform perspectivedistortion-invariant estimation which causes difficulty in learning the shapes. To address this limitation, we develop a full voxel-based deep network that effectively utilizes the estimated 3D pose and voxelized depth map to produce reasonable 3D hand shapes. Our proposed approach for 3D shape estimation comprises of two main phases. In the first phase, we estimate the shape surface and the voxelized hand shape. In the second phase, we register the estimated shape surface to the estimated voxelized hand shape by employing a 3D CNN-based registration for synthetic data and NRGAbased fitting process for real data. Voxelized Shape Estimation. Our idea is to estimate 3D hand shape in the voxelized form via 3D CNN-based net-work. It allows the network to estimate the shape in such a way that minimizes the chances for perspective distortion. Inspired by the approach proposed in the recent work [17], we consider sparse 3D joints as the latent representation of dense 3D shape. However, in this work, we combine 3D pose with the depth map which helps to represent the shape of hand more accurately. Furthermore, here we use more accurate and useful representations of 3D pose and 2D depth image which are 3D joints heatmaps and a voxelized depth map, respectively. The V2V-ShapeNet module is shown in Fig. 2. It can be considered as the 3D shape decoder:
V S ∼ Dec(H j ⊕ V D ) = p(V S |I S )(1)
where p(V S |I S ) is the decoded distribution. The decoder learns to reconstruct the voxelized hand shapeV S as close as possible to the ground truth voxelized hand shape V S . The V2V-ShapeNet is a 3D CNN-based architecture that directly estimates the probability of each voxel in the voxelized shape indicating whether it is the background (i.e., 0) or the shape voxel (i.e., 1). The per-voxel binary cross entropy loss L V S for voxelized shape reconstruction reads:
L V S = −(V S log(V S ) + (1 − V S ) log(1 −V S )) (2)
where V S andV S are the ground truth and the estimated voxelized hand shapes, respectively. The architecture of V2V-ShapeNet is provided in the supplement. Since the annotations for real hand shapes are not available, weak supervision is therefore essential in order to effectively learn real hand shapes. For this reason, we propose a 3D CNN-based V2V-SynNet (see Fig. 2) which acts as a source of weak supervision during training. This module is removed during testing. V2V-SynNet synthesizes the voxelized depth map from the estimated voxelized shape representation. The per-voxel binary cross entropy loss L v VD for voxelized depth map reconstruction is given by:
L v VD = −(V D log(V D ) + (1 − V D ) log(1 − log(V D )) (3)
where V D andV D are the ground truth and the reconstructed voxelized depth maps, respectively. The architecture of V2V-SynNet is provided in the supplement. Shape Surface Estimation. The hand poses of the shape surfaces and voxelized shapes need to be similar for an improved shape registration. To facilitate the registration, we employ V2S-Net deep network which directly regresses V. Based on the similar concept of hand shape decoding (as mentioned before), I S is provided as an input to this network while the decoded output is the reconstructed hand mesh (see Fig. 2). The hand shape surface reconstruction loss L V T is given by the standard Euclidean loss as:
L V T = 1 2 V T − V T 2 ,(4)
where V T andV T are the respective ground truth and reconstructed hand shape surfaces. As explained before, in the case of missing real hand shape ground truth, the weak supervision on mesh vertices is provided by S2V-SynNet. In this case, the input to the S2V-SynNet isV T which is in 3D coordinates form. The loss function L s VD for the S2V-SynNet is similar to Eq. (3). Further details on S2V-SynNet and V2S-Net can be found in the supplement. CNN-based Shape Registration. Thanks to fully connected (FC) layers, V2S-Net is able to estimate hand shapes while preserving the order and number of points. Losing local spatial information is also known as a drawback of FC layers. In contrast to FC layers, a lot of works show fully convolutional networks (FCN) perform well in geometry regression tasks [24,9,18,31]. However, estimating the voxelized hand shape by 3D convolutional layer results in an inconsistent number of points and loses point order. Hence, the ideal architecture is a network which estimates the hand shape without losing local spatial information while preserving the topology of the hand shape. To achieve this, we register the estimated shape by V2S-Net to the probabilistic shape representation estimated by FCN (V2V-ShapeNet) using DispVoxNets pipeline [25].
The original DispVoxNets pipeline is comprised of two stages, i.e., global displacement estimation and refinement stage. The refinement stage is used to remove roughness on the point set surface. In contrast to the original approach, we replace the refinement stage with Laplacian smoothing [29]. This is possible because we assume the mesh topology is already known, and it is preserved by our pipeline.
In the DispVoxNet pipeline, the hand surface shapeV T is first converted into a voxelized gridV T (i.e., 64 × 64 × 64 voxelized grid size). DispVoxNet estimates per-voxel displacements of the dimension 64 3 × 3 between the referencê V S and voxelized hand surfaceV T 1 . The displacement loss L Disp is given by:
L Disp. = 1 Q 3 d − D vn (V S ,V T ) 2 ,(5)
where Q and d are the voxel grid size and the ground truth displacement, respectively. Since it is difficult to obtain d between the voxelized shapeV S and hand surfaceV T , the displacements are first computed between V T andV T , and are discretized to obtain d. For more details of ground truth voxelized grid computation, please refer to [25].
NRGA-Based Shape Registration. In our voxel-based 3D hand shape and pose estimation pipeline (Fig. 2), the Dis-pVoxNet [25] component requires shape annotations in its source-to-target displacement field learning phase. These annotations are available only for the synthetic dataset which leaves a domain gap on the performance of Dis-pVoxNet when tested on real dataset. To bridge this gap, we apply NRGA [1] to improveV T by registering it withV S . NRGA is selected for this deformable alignment task over other methods [20,2], as it supports local topology preservation of input hand surface and is robust at noise handling. Although NRGA is a point cloud alignment method, it provides the option to relax the deformation magnitude in the neighbouring regions of the hand mesh vertices. The original NRGA estimates a rigid transformation for every vertex v ∈V T and diffuses the transformations in a subspace formed by a set of neighbourhood vertices of v. It builds a k-d tree on the template (V T in our case) and neighbourhood vertices are selected as the k-nearest neighbours (typically 0.1% − 0.2% of the total points in the template). We modify NRGA for the surface-to-voxel operation, i.e., instead of k-nearest neighbours, we use connected vertices in a 4-ring forV T . See more details in our supplementary material.
Data Augmentation in 3D
Our method for hand shape estimation relies on the accuracy of the estimated 3D pose. Therefore, the hand pose estimation method has to be accurate and robust. Training data augmentation helps to improve the performance of a deep network [19]. Existing methods for hand pose estimation [19,28] use data augmentation in 2D. This is mainly because these methods treat depth maps as 2D data. The representation of the depth map in voxelized form makes it convenient to perform data augmentation in all three dimensions. In this paper, we propose a new 3D data augmentation policy which improves the accuracy and robustness of hand pose estimation (see Sec. 6.3).
During V2V-PoseNet training, we apply simultaneous rotations in all three axes (x,y,z) to each 3D coordinate (i, j, k) of V D and H j by using Euler transformations:
[î,ĵ,k] T = [Rot x (θ x )] × [Rot y (θ y )] × [Rot z (θ z )][i, j, k] T ,(6)
The Network Training
V D is generated by projecting the raw depth image pixels into 3D space. Hand region points are then extracted by using a cube of size 300 that is centered on hand palm center position. 3D point coordinates of the hand region are discretized in the range [1,88]. Finally, to obtain V D , the voxel
Methods 3D V Err. (mm) V2S-Net (w/o H j ) 8.78 V2S-Net (w/o V D ) 3.54 V2S-Net (with H j ⊕ V D ) 3.36 Methods 3D S Err. V2V-ShapeNet (w/o H j ) 0.007 V2V-ShapeNet (w/o V D ) 0.016 V2V-ShapeNet (with H j ⊕ V D )
0.005 Table 1: Ablation study on inputs (i.e., Hj and V D ) to V2S-Net and V2V-ShapeNet. We observe that combining both inputs is useful for these two networks.
value is set to 1 for the 3D point coordinate of hand region, and 0 otherwise. Following [18], H j are generated as 3D Gaussians. Similar to the generating of V D , V S is obtained by voxelizing the hand mesh. V T is created by normalizing the mesh vertices in the range [−1, +1]. We perform this normalization by subtracting the vertices from the palm center and then dividing them by half of the cube size.
We train V2V-PoseNet [18] on NYU [28], BigHand2.2M [34] and SynHand5M [14] datasets separately with the 3D data augmentation technique mentioned in Sec. 4.2. For SynHand5M dataset, we train V2S-Net and V2V-ShapeNet (including the synthesizers S2V-SynNet and V2V-SynNet) separately using RMSProp as an optimization method with a batch size of 8 and a learning rate LR = 2.5 × 10 −4 . After training the pose and shape networks, we put these networks together in the pipeline (see Fig. 2) and fine-tune them in an end-to-end manner with synthetic, as well as combined real and synthetic data. The total loss L T read as follows:
L T = L H + 1L V S + 1L V T + L v VD + L s VD (7)
where L H is heatmaps loss [18] and 1 represents an indicator function layer. This layer forwards the estimations to the loss layer only for synthetic data using a flag value, which is 1 for synthetic and 0 for real data. It disables the gradients flow during the backward pass in the case of real data. For fine-tunings, we use RMSProp with a batch size of 6 and a learning rate 2.5 × 10 −5 . DispVoxNet is trained only on SynHand5M dataset due to the availability of the ground truth geometry. During the training, Adam optimizer [11] with a learning rate of 3.0 × 10 −4 was employed. The training continues until the convergence of L Disp with batch size 12. All models are trained until convergence on a desktop workstation equipped with Nvidia Titan X GPU.
Experiments
We perform qualitative and quantitative evaluations of our complete pipeline including ablation studies on the fully labeled SynHand5M [14] dataset. We qualitatively evaluate
Methods
3D V Err. (mm) DeepHPS [14] 11.8 WHSP-Net [17] 5.12 ours (w/o synthesizers) 2.92 ours (with synthesizers) 2.67 Table 2: Comparison with the state of the arts on SynHand5M [14]. Our full method, with V2V-SynNet and S2V-SynNet synthesizers, outperforms the WHSP-Net approach [17] by 47.85%.
real hand shape recovery on NYU [28] and BigHand2.2M [34] datasets. Furthermore, we study the impact of our 3D data augmentation on V2V-PoseNet [18].
Datasets and Evaluation Metrics
Although there are many depth-based hand pose datasets [34,5,26], only a few of them (i.e., BigHand2.2M [34], NYU [28], SynHand5M [14]) provide adequate training data and annotations which resemble the joint locations of a real hand. NYU real benchmark offers joint annotations for 72757 and 8252 RGBD images of the training (T N ) and test sets, respectively. Their hand model contains 42 DOF which makes it possible to combine this dataset with the recent benchmarks (e.g., BigHand2.2M). BigHand2.2M is a million-scale real benchmark. For pose estimation, it provides accurate joint annotations for 956k training (T B ) depth images acquired from 10 subjects. Their hand model contains 21 joint locations which resembles real hand skeleton. The size of the BigHand2.2M's test set is 296k. The annotation of hand palm center is not given in the BigHand2.2M dataset. Hence, we obtain the hand palm center position by taking the average of the metacarpal joints and the wrist joint positions. SynHand5M dataset contains fully annotated 5 million depth images for both the 3D hand pose and shape. The sizes of its training (T S ) and test sets are 4.5M and 500k, respectively. The joint annotations of Big-Hand2.2M are fully compatible with SynHand5M. Table 3: Runtime: (first four rows) forward-pass of deep networks on GPU. " * " shows that Laplacian smoothing runs on CPU.
We use three evaluation metrics: (i) the average 3D joint location error over all test frames (3D J Err.); (ii) mean vertex location error over all test frames (3D V Err.); and (iii) mean voxelized shape error (i.e., per-voxel binary cross entropy) over all test data (3D S Err.).
Evaluation of Hand Shape Estimation
In this subsection, we evaluate our method on Syn-Hand5M, NYU and BigHand2.2M benchmarks. Synthetic Hand Shape Reconstruction. We train our complete pipeline on the fully labeled SynHand5M dataset by following the training methodology explained in Sec. 5. We conduct two ablation studies to show the effectiveness of our design choice. First is the regression of V T and V S by using input V D (i.e., without H j ) and the synthesizers. Similar experiments are repeated by using H j (i.e., without V D ) and I S (i.e., with I S ⊕ V D ) as separate inputs to V2V-ShapeNet and V2S-Net. The results are summarized in Table 1 and clearly show the benefit of concatenating voxelized depth map with 3D heatmaps. The second ablation study is to observe the impact of V2V-SynNet and S2V-SynNet, given I S as an input to the complete shape estimation network. We train V2S-Net and V2V-ShapeNet with and without using their respective synthesizers (see Fig. 2). The quantitative results and comparisons with the state-of-the-art methods on SynHand5M test set are summarized in Table 2. Our method with synthesizers improves on ours without synthesizers, and achieves 47.8% improvement in the accuracy compared to the recent WHSP-Net [17]. Several synthesized samples of voxelized depth maps are shown in the supplement. The qualitative results of shape representations and poses are shown in Fig. 3. Dis-pVoxNet fits the estimated hand surface to the estimated voxelized hand shape, thereby improving the hand surface reconstruction accuracy by 20.5% (i.e., from 3.36mm to 2.67mm). Notably, the accuracy of our hand surface estimation is higher compared to WHSP-Net (cf . Tables 1 and 2), which clearly shows the effectiveness of employing 3D CNN based network for mesh vertex regression. Real Hand Shape Reconstruction. To estimate plausible real hand shape representations, the synthesizers are essential (see Fig. 2). For NYU hand surface and voxelized shape recovery, we combine the training sets of NYU and Syn-Hand5M (i.e., T NS = T N + T S ) by selecting closely matching 22 common joint positions in both datasets. However, note that the common joint positions are still not exactly similar in both the datasets. V2S-Net and V2V-ShapeNet recover plausible hand shape representations while NRGAbased method performs a successful registration (as shown in Fig. 4(a), (b) and (c)). It is observed that the voxelized shape is more accurately estimated than the hand surface. Thereby, the alignment further refines the hand surface. Using the similar training strategy, we combine BigHand2.2M and SynHand5M datasets and shuffle them (i.e., T BS = T B + T S ). Samples of the estimated hand shape representations for BigHand2.2M are shown in Fig. 5. Methods 3D J Err. (mm) DeepHPS [14] 6.30 WHSP-Net [17] 4.32 V2V-PoseNet [18] 3.81 our HandVoxNet (full method) 3.75 Table 4: 3D hand pose estimation results on SynHand5M [14] dataset. We compare the accuracy of our full method (i.e., Hand-VoxNet) with state-of-the-art methods.
We qualitatively compare our reconstructed hand shapes of NYU dataset with the state of the art. For better illustration of the shape reconstruction accuracy, we show the 2D overlay of hand mesh onto the corresponding depth image (as shown in Fig. 4-(d) and (e)). Model-based DeepHPS [14] suffers from artifacts, the regression-based WHSP-Net approach [17] produces perspective distortions and incorrect sizes of shapes. In contrast, HandVoxNet recovers visually more plausible hand shapes (Fig. 4-(c)). Table 3 Table 5: 3D hand pose estimation results on NYU [28] and BigHand2.2M [34] datasets using our 3D data augmentation.
Ours WHSP-Net DeepHPS Figure 7: Samples of NYU [28] depth images with 2D overlay of the estimated 3D hand pose. Our method produces more accurate results compared to WHSP-Net [17] and DeepHPS [14] methods.
missing information in the depth map (see Fig. 6).
Evaluation of Hand Pose Estimation
In our approach, the accuracy of the estimated hand shape is dependent on the accuracy of estimated 3D pose (see Sec. 4). Therefore, the hand pose estimation needs to be robust and accurate. Therefore, we perform a new 3D data augmentation on voxelized depth maps which further improves the accuracy of 3D hand pose estimation on real datasets. Notably, our focus is to develop an effective approach for simultaneous hand pose and shape estimation. However, for completeness, we show our results and comparisons of hand pose estimation with SynHand5M, NYU and BigHand2.2M datasets. SynHand5M dataset: We do not perform training data augmentation on SynHand5M because this dataset originally contains large viewpoint variations [14]. We train our full method and V2V-PoseNet [18] on SynHand5M dataset. The quantitative results on the test set are presented in Table 4. We observe that the backpropagation from the shape regression pipeline is effective and improves the accuracy of the estimated 3D pose. We achieve 13.19% improvement in the accuracy compared to WHSP-Net approach [17]. NYU and BigHand2.2M datasets: V2V-PoseNet [18] is a powerful pose estimation method that exploits the 3D data representations of hand pose and depth map. Thanks to our 3D data augmentation strategy (see Sec. 4.2), we improve Figure 8: We study the impact of our 3D data augmentation on the pose estimation accuracy of V2V-PoseNet [18] on NYU [28] dataset. The graph shows mean errors for individual hand joints. the accuracy by 5.42% and 6.83% compared to the original V2V-PoseNet models on NYU and BigHand2.2M datasets, respectively (see Table 5). Fig. 8 shows the average errors on individual hand joints. We observe a noticeable improvement in the accuracy of the finger tips. The qualitative results and comparisons with the state-of-the-art methods for hand pose estimation are shown in Fig. 7.
Conclusion and Future Work
We develop the first voxel-based pipeline for 3D hand shape and pose recovery from a single depth map, which establishes an effective inter-link between hand pose and shape estimations using 3D convolutions. This inter-link boosts the accuracy of both estimates, which is demonstrated experimentally. We employ 3D voxelized depth map and accurately estimated 3D heatmaps of joints as inputs to reconstruct two hand shape representations, i.e., 3D voxelized shape and 3D shape surface. To combine the advantages of both shape representations, we employ registration methods, i.e., DispVoxNet and NRGA, which accurately fit the shape surface to the voxelized shape.
The experimental evaluation further shows that our 3D data augmentation policy on voxelized grids enhances the accuracy of 3D hand pose estimation on real data. Hand-VoxNet produces visually more accurate hand shapes of real images compared to the previous methods. All these results indicate that the one-to-one mapping between voxelized depth map, voxelized shape and 3D heatmaps of joints is essential for an accurate hand shape and pose recovery.
In future work, generating a realistic synthetic dataset can further enhance the hand shape reconstruction from real images. The runtimes of the used registration methods can be improved by the parallelization on GPUs.
Figure 1 :
1Hand shape and pose estimation with HandVoxNet.
Figure 2 :
2Figure 2: Overview of our approach for 3D hand shape and pose recovery from a 3D voxelized depth map. V2V-PoseNet estimates 3D joints heatmaps (i.e., pose). Hand shape is obtained in two phases. First, V2V-ShapeNet and V2S-Net estimate the voxelized shape and shape surface, respectively. Thereby, V2V-SynNet and S2V-SynNet synthesize the voxelized depth acting as sources of weak-supervision. They are excluded during testing. In the second phase, shape registration accurately fits the shape surface to the voxelized shape.
where (î,ĵ,k) is the transformed voxel coordinate. Rot x (θ x ), Rot y (θ y ) and Rot z (θ z ) are 3 × 3 rotation matrices around x, y and z axes. The values for θ x , θ y and θ z are selected randomly in the ranges [−40 • , +40 • ], [−40 • , +40 • ] and [−120 • , +120 • ], respectively. In addition to rotations in 3D, following [18], we perform scaling and translation in the respective ranges [+0.8, +1.2] and [−8, +8].
Figure 3 :
3Qualitative results on SynHand5M[14] dataset. Estimated hand pose overlay (1 st col), voxelized shape (2 nd col), hand surface (3 rd col), final shape (4 th col), and the overlays of hand surface and final shapes with ground truth (gray color) are illustrated.
Figure 4 :Figure 5 :
45Shape reconstruction of NYU[28] dataset: (a), (b) and (c) show the 2D overlays and 3D visualizations of estimated voxelized hand shape, shape surface, and the final shape after registration, respectively. (d) and (e) show the corresponding results of hand shapes from DeepHPS[14] and WHSP-Net[17] methods. Our approach produces visually more accurate hand shapes than the existing approaches. Shape reconstruction of BigHand2.2M[34] dataset: (a) the 2D pose overlay; (b), (c) recovered voxelized shape and shape surface, respectively; (d) the overlays of shape surface and registered shape; (e) the final hand shape.
( a )
aDepth (b) 3D Pose (c) DispVoxNet[25] (d) NRGA[1]
Figure 6 :
6Failure case: our method is unable to produce plausible shapes in cases of severe occlusion and missing depth information.
provides the runtimes of different components of our pipeline. Failure Cases. Our approach fails to estimate plausible hand shapes in cases of severe occlusion of hand parts andDataset
Method
3D J Err. (mm)
NYU
V2V-PoseNet [18]
9.22
V2V-PoseNet (our 3D augm.)
8.72
BigHand2.2M
V2V-PoseNet [18]
9.95
V2V-PoseNet (our 3D augm.)
9.27
a larger grid size results in higher accuracy of DispVoxNet, and we hence set it to the maximum which our hardware supports.
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| []
|
[
"A Tighter Relation Between Hereditary Discrepancy and Determinant Lower Bound",
"A Tighter Relation Between Hereditary Discrepancy and Determinant Lower Bound"
]
| [
"Haotian Jiang ",
"Victor Reis "
]
| []
| []
| In seminal work, Lovász, Spencer, and Vesztergombi [European J. Combin., 1986] proved a lower bound for the hereditary discrepancy of a matrix A ∈ R m×n in terms of the maximum | det(B)| 1/k over all k × k submatrices B of A. We show algorithmically that this determinant lower bound can be off by at most a factor of O( log(m) · log(n)), improving over the previous bound of O(log(mn)· log(n)) given by Matoušek [Proc. of the AMS, 2013]. Our result immediately implies herdisc(F 1 ∪ F 2 ) ≤ O( log(m) · log(n)) · max(herdisc(F 1 ), herdisc(F 2 )), for any two set systems F 1 , F 2 over [n] satisfying |F 1 ∪ F 2 | = m. Our bounds are tight up to constants when m = O(poly(n)) due to a construction of Pálvölgyi [Discrete Comput. Geom., 2010] or the counterexample to Beck's three permutation conjecture | 10.1137/1.9781611977066.24 | [
"https://arxiv.org/pdf/2108.07945v2.pdf"
]
| 237,194,927 | 2108.07945 | 60c624a2ba9f4c0875c4b12dea4ad0f0466a5fca |
A Tighter Relation Between Hereditary Discrepancy and Determinant Lower Bound
1 Nov 2021 November 3, 2021
Haotian Jiang
Victor Reis
A Tighter Relation Between Hereditary Discrepancy and Determinant Lower Bound
1 Nov 2021 November 3, 2021* University of Washington, Seattle, USA. [email protected]. † University of Washington, Seattle, USA. [email protected].
In seminal work, Lovász, Spencer, and Vesztergombi [European J. Combin., 1986] proved a lower bound for the hereditary discrepancy of a matrix A ∈ R m×n in terms of the maximum | det(B)| 1/k over all k × k submatrices B of A. We show algorithmically that this determinant lower bound can be off by at most a factor of O( log(m) · log(n)), improving over the previous bound of O(log(mn)· log(n)) given by Matoušek [Proc. of the AMS, 2013]. Our result immediately implies herdisc(F 1 ∪ F 2 ) ≤ O( log(m) · log(n)) · max(herdisc(F 1 ), herdisc(F 2 )), for any two set systems F 1 , F 2 over [n] satisfying |F 1 ∪ F 2 | = m. Our bounds are tight up to constants when m = O(poly(n)) due to a construction of Pálvölgyi [Discrete Comput. Geom., 2010] or the counterexample to Beck's three permutation conjecture
Introduction
Given a matrix A ∈ R m×n , the discrepancy of A is disc(A) := min x∈{−1,+1} n Ax ∞ . The hereditary discrepancy of A is defined as herdisc(A) := max S⊆[n] disc(A S ), where A S denotes the restriction of the matrix A to columns in S. For a set system F , disc(F ) and herdisc(F ) are defined to be disc(A F ) and herdisc(A F ), where A F is the incidence matrix of F .
In seminal work, Lovász, Spencer, and Vesztergombi [LSV86] introduced a powerful tool, known as the determinant lower bound, for bounding hereditary discrepancy: where A S,T denotes the restriction of A to rows in S and columns in T . In particular, they showed that herdisc(A) ≥ 1 2 detLB(A) for any matrix A. A reverse relation was established by Matousěk [Mat13], who showed that herdisc(A) ≤ O(log(mn) log(n)) · detLB(A). However, Matousěk's bound does not match the largest known gap of Θ(log(n)) between herdisc(A) and detLB(A), given by a construction of Pálvölgyi [Pál10] or the counter-example to Beck's three permutation conjecture [NNN12].
Our main result is the following improvement over Matousěk's bound in [Mat13].
Theorem 1.1. Given a matrix A ∈ R m×n , one can efficiently find x ∈ {+1, −1} n such that Ax ∞ ≤ O( log(m) · log(n) · detLB(A)).
Restricting to an arbitrary subset of the columns of A, one immediately obtains the following:
Corollary 1.2. For any matrix A ∈ R m×n , herdisc(A) ≤ O( log(m) · log(n) · detLB(A)).
In light of the examples in [Pál10,NNN12] where herdisc(A) ≥ Ω(log n) · detLB(A), Theorem 1.1 is tight up to constants whenever m = poly(n). For the case where m ≫ poly(n), one cannot hope to improve the log(m) dependence on m in Theorem 1.1. In particular, the set system F = 2 [n] has herdisc(F ) = n, detLB(F ) = √ n and therefore
herdisc(F ) ≥ log(m) · detLB(F ).
It remains an open problem, however, whether one can improve the √ log n factor in the later regime.
Hereditary discrepancy of union of set systems. A question of V. Sós (see [LSV86]) asks whether herdisc(F 1 ∪ F 2 ) can be estimated in terms of herdisc(F 1 ) and herdisc(F 2 ), for any set systems F 1 and F 2 over [n]. This is, however, not possible without any dependence on m = |F 1 ∪ F 2 | or n, as first shown by an example of Hoffman (Proposition 4.11 in [Mat09]). This can also be seen from the examples in [Pál10,NNN12]. In [KMV05], it was shown that herdisc(F 1 ∪ F 2 ) ≤ O(log(n)) · herdisc(F 1 ) when F 2 contains a single set. For more general set systems, Matousěk [Mat13] proved that herdisc(F ) ≤ O( √ t log(mn) log(n)) · max i∈[t] (herdisc(F i )), where F = F 1 ∪ · · · ∪ F t and m = |F |.
Theorem 1.1 together with Lemma 4 in [Mat13] immediately imply the following improvement of this result, whose proof is the same as in [Mat13]. For t = 2 and m = poly(n), this bound is tight up to constants. Theorem 1.3. Let F be a system of m sets on [n] such that F = F 1 ∪ F 2 ∪ · · · ∪ F t . Then,
herdisc(F ) ≤ O t log(m) log(n) · max i∈[t]
(herdisc(F i )).
Approximating hereditary discrepancy. It was shown in [CNN11] that disc(A) cannot be approximated in polynomial time for an arbitrary matrix A ∈ {0, 1} m×n . The more robust notion of hereditary discrepancy, however, can be approximated within a polylog factor. The best-known result in this direction is a O(log(min(m, n)) · log(m))-approximation to hereditary discrepancy via the γ 2 -norm [MNT14]. When m = poly(n), this approximation factor is O(log 3/2 (n)).
Our result in Theorem 1.1 suggests a potential approach of approximating hereditary discrepancy by approximating the determinant lower bound. There has been a recent line of work in approximating the maximum k × k subdeterminant for a given matrix A. Overview of proof of Theorem 1.1. We follow the approaches in [Ban10] and [Mat13].
The key notion to prove Theorem 1.1 is that of hereditary partial vector discrepancy, which is defined as follows. Given a matrix A ∈ R m×n with entries a ij for i ∈ [m] and j ∈ [n], we consider the following SDP for a subset S ⊆ [n] and a parameter λ ≥ 0:
j∈S a ij v j 2 2 ≤ λ 2 ∀i ∈ [m], n j=1 v j 2 2 ≥ |S|/2, v j 2 2 ≤ 1 ∀j ∈ S, v j 2 2 = 0 ∀j ∈ [n] \ S.
SDP(A, S, λ)
Define the partial vector discrepancy of A, denoted as pvdisc(A), to be the smallest value of λ such that SDP(A, [n], λ) is feasible, and hereditary partial vector discrepancy herpvdisc(A) to be the smallest λ such that SDP(A, S, λ) is feasible for any subset S ⊆ [n].
Using the above definition, we show in Lemma 2.1 of Section 2.1 that the above SDP can be rounded efficiently to obtain a coloring with discrepancy at most O( log(m) log(n) · herpvdisc(A)). We then prove in Lemma 2.3 of Section 2.2 that herpvdisc(A) ≤ O(detLB(A)), from which Theorem 1.1 immediately follows. We conjecture that herpvdisc(A) is the same as detLB(A) up to constants (Conjecture 2.4).
Notations and preliminaries. Given a matrix A ∈ R m×n , its rows will be denoted by a 1 , . . . , a m ∈ R n . Define A S,T to be the matrix with rows restricted to some subset S ⊆ [m] and columns restricted to some T ⊆ [n], and A S := A [m],S . Theorem 1.4 (Freedman's Inequality, Theorem 1.6 in [Fre75]). Consider a real-valued martingale sequence {X t } t≥0 such that X 0 = 0, and E[X t+1 |F t ] = 0 for all t, where {F t } t≥0 is the filtration defined by the martingale. Assume that the sequence is uniformly bounded, i.e., |X t | ≤ M almost surely for all t. Now define the predictable quadratic variation process of the martingale to be W t = t j=1 E[X 2 j |F j−1 ] for all t ≥ 1. Then for all ℓ ≥ 0 and σ 2 > 0 and any stopping time τ , we have P τ j=0 X j ≥ ℓ ∧ W τ ≤ σ 2 for some stopping time τ ≤ 2 exp − ℓ 2 /2 σ 2 + Mℓ/3 .
2 Proof of Theorem 1.1
The Algorithm
The main result of this subsection is the following lemma.
Lemma 2.1. Given a matrix A ∈ R m×n , there exists a randomized algorithm that w.h.p. constructs a coloring x ∈ {+1, −1} n such that Ax ∞ ≤ O( log(m) log(n) · herpvdisc(A)). This implies that herdisc(A) ≤ O( log(m) log(n) · herpvdisc(A)).
The algorithm in Lemma 2.1 is given in Algorithm 1. This algorithm is a variant of the random walk in [Ban10], using the SDP for hereditary partial vector discrepancy.
Since Lemma 2.1 is invariant under rescaling of the matrix A, we may assume without loss of generality that that max i,j |a i,j | = 1. Given a coloring x ∈ [−1, 1] n , we say an element i ∈ [n] is alive if |x(i)| < 1 − 1/n. The following lemma from [Ban10] states that the number of alive elements halves after O(1/s 2 ) steps.
Lemma 2.2 (Lemma 4.1 of [Ban10]). Let y ∈ [−1, +1] n be an arbitrary fractional coloring with at most k alive variables. Let z be the fractional coloring obtained by running algorithm 1 with x ′ 0 = y for T ′ = 16/s 2 steps. Then the probability that z has at least k/2 alive variables is at most 1/4.
Proof of Lemma 2.1. We first argue that after T = 400 log(n)/s 2 steps, no element is alive with high probability. Divide the time horizon into epochs of size 16/s 2 . For each epoch,
Algorithm 1 HerpvdiscRounding(A)
1: λ ← herpvdisc(A) ⊲ The value of λ can be approximated with a binary search 2: x 0 ← 0 ∈ R n , S 0 ← [n], s ← 1/m 2 n 2 , T ← 200 log(n)/s 2 3: for t = 1, 2, · · · , T do 4: v 1 , · · · , v n ← SDP(A, S t−1 , λ)
5:
Sample r ∈ {−1, +1} n uniformly at random 6: Lemma 2.2 states that regardless of the past, the number of alive elements decreases by at least half with probability at least 3/4. It follows that no element is alive with high probability after 25 log(n) epochs. Note that when no element is alive for the coloring x T , one can round it to a full coloring without changing the discrepancy of each set by more than 1.
for i ∈ [n] do x t (i) ← x t−1 (i) + s · r, v i 7:
Next we prove that with high probability, the discrepancy of each row of A is at most O( log(m) log(n)) · λ. We consider any j ∈ [m], and denote disc t (j) = a j , x t the discrepancy of row j at the end of time step t ∈ [T ]. Note that E[disc t (j) − disc t−1 (j)|disc t−1 (j)] = 0 and E[(disc t (j) − disc t−1 (j)) 2 |disc t−1 (j)] ≤ λ 2 s 2 . It follows from Freedman's inequality (Theorem 1.4) that P |disc T (j)| ≥ 10 log(m) log(n) · λ ≤ 1/m 2 .
So by the union bound, the discrepancy of the obtained coloring is at most O( log(m) log(n)· herpvdisc(A)) with high probability. This completes the proof of Lemma 2.1.
Bounding Partial Vector Discrepancy
In this subsection, we prove the following lemma which upper bounds partial vector discrepancy in terms of the determinant lower bound. The proof can be seen as a simplification of Lemma 8 in [Mat13], which gives a corresponding upper bound for vector discrepancy that is weaker by a factor of √ log n due to a bucketing argument that is not needed here.
Lemma 2.3. For any A ∈ R m×n , we have herpvdisc(A) ≤ O(detLB(A)).
Proof. Recall that pvdisc(A) 2 is the optimal value of the SDP given by
min t n j=1 a ij v j 2 2 ≤ t ∀i ∈ [m] n j=1 v j 2 2 ≥ n/2 v j 2 2 ≤ 1 ∀j ∈ [n].
By denoting X ij := v i , v j , we may rewrite this SDP as follows:
min t a i a ⊤ i , X ≤ t ∀i ∈ [m] I n , X ≥ n/2 e j e ⊤ j , X ≤ 1 ∀j ∈ [n] X 0,
where e j denotes the vector with 1 on the j-th coordinate and 0 elsewhere. The dual formulation of the above SDP is given by the following:
max nγ − n j=1 z j m i=1 w i a i a ⊤ i + n j=1 z j e j e ⊤ j 2γ · I n m i=1 w i = 1 w, z ≥ 0.
Denote λ := pvdisc(A). By Slater's condition, there exists a feasible dual solution (w, z, γ) such that w, z ≥ 0 and nγ − n j=1 z j = λ 2 . Indeed, the dual has a feasible interior point (for example, w i = 1/m, z j = 1 and γ = 0) and is bounded, since we may rewrite the first constraint as
m i=1 w i a i a ⊤ i n j=1 (2γ − z j ) · e j e ⊤ j ,(1)
which implies
nγ − n j=1 z j ≤ nγ − 1 2 n j=1 z j ≤ 1 2 tr m i=1 a i a ⊤ i .
Letà be the matrix obtained from A by multiplying the i-th row by √ w i and J ⊆ [n] be the set of columns for which z j < 3 2 γ. Note that |J| ≥ 1 3 n, for otherwise n j=1 z j > 2 3 n · 3 2 γ = nγ. Since for each j ∈ J we have 2γ − z j ≥ 1 2 γ, for any vector x ∈ R J it follows by (1):
x ⊤ A ⊤ J A J x ≥ 1 2 γ · x 2 2 ≥ λ 2 2n · x 2 2 .
This implies that all eigenvalues of A ⊤ J A J are at least λ 2 /2n, so that det( A ⊤ J A J ) ≥ (λ 2 /2n) |J| . In the other direction, the Cauchy-Binet formula also gives
det( A ⊤ J A J ) = I⊆[m] |I|=|J| det( A I,J ) 2 = I⊆[m] |I|=|J| det(A I,J ) 2 i∈I w i ≤ detLB(A) 2|J| · I⊆[m] |I|=|J| i∈I w i ≤ detLB(A) 2|J| · 1 |J|! m i=1 w i |J| ,
where the last inequality follows as each term i∈I w i appears |J|! times in
m i=1 w i |J| . Since m i=1 w i = 1, we conclude detLB(A) 2|J| · 1 |J|! ≥ det( A ⊤ J A J ) ≥ (λ 2 /2n) |J| ,
from which detLB(A) ≥ Ω(λ · |J|/n) = Ω(λ) = Ω(pvdisc(A)). Applying this result to all subsets S ⊆ [n] of the columns of A proves the lemma.
We conjecture that the above Lemma 2.3 is tight up to constants.
Conjecture 2.4. For any matrix A ∈ R m×n , we have detLB(A) = Θ(herpvdisc(A)).
|
det(A S,T )| 1/k ,
For k = min(m, n), Nikolov[Nik15] gave a 2 O(k) -approximation; for general values of k, Anari and Vuong[AV20] showed a k O(k) -approximation algorithm. If these results can be strengthened to a 2 O(k) -approximation algorithm for general values of k, then together with Theorem 1.1, one would obtain the first O(log(n))-approximation algorithm for hereditary discrepancy when m = poly(n).
:
Round x T to a vector x ∈ {−1, +1} n 16: Return x
AcknowledgmentsWe thank the anonymous reviewers of SOSA 2022 for insightful comments. We also thank Aleksandar Nikolov, Nikhil Bansal and Mehtaab Sawhney for helpful discussions.
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. Schloss Dagstuhl-Leibniz-Zentrum für Informatik. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2020.
Constructive algorithms for discrepancy minimization. Nikhil Bansal, IEEE 51st Annual Symposium on Foundations of Computer Science. IEEENikhil Bansal. Constructive algorithms for discrepancy minimization. In 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, pages 3-10. IEEE, 2010.
Tight hardness results for minimizing discrepancy. Moses Charikar, Alantha Newman, Aleksandar Nikolov, Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms. the twenty-second annual ACM-SIAM symposium on Discrete AlgorithmsSIAMMoses Charikar, Alantha Newman, and Aleksandar Nikolov. Tight hardness re- sults for minimizing discrepancy. In Proceedings of the twenty-second annual ACM- SIAM symposium on Discrete Algorithms, pages 1607-1614. SIAM, 2011.
On tail probabilities for martingales. the Annals of Probability. A David, Freedman, 3David A Freedman. On tail probabilities for martingales. the Annals of Probability, 3(1):100-118, 1975.
Discrepancy after adding a single set. Jiří Jeong Han Kim, Matoušek, H Van, Vu, Combinatorica. 254499Jeong Han Kim, Jiří Matoušek, and Van H Vu. Discrepancy after adding a single set. Combinatorica, 25(4):499, 2005.
Discrepancy of setsystems and matrices. László Lovász, Joel Spencer, Katalin Vesztergombi, European Journal of Combinatorics. 72László Lovász, Joel Spencer, and Katalin Vesztergombi. Discrepancy of set- systems and matrices. European Journal of Combinatorics, 7(2):151-160, 1986.
Geometric discrepancy: An illustrated guide. Jiri Matousek, Springer Science & Business Media18Jiri Matousek. Geometric discrepancy: An illustrated guide, volume 18. Springer Science & Business Media, 2009.
The determinant bound for discrepancy is almost tight. Jiří Matoušek, Proceedings of the. theAmerican Mathematical Society141Jiří Matoušek. The determinant bound for discrepancy is almost tight. Proceedings of the American Mathematical Society, 141(2):451-460, 2013.
Factorization norms and hereditary discrepancy. Jiří Matoušek, Aleksandar Nikolov, Kunal Talwar, arXiv:1408.1376arXiv preprintJiří Matoušek, Aleksandar Nikolov, and Kunal Talwar. Factorization norms and hereditary discrepancy. arXiv preprint arXiv:1408.1376, 2014.
Randomized rounding for the largest simplex problem. Aleksandar Nikolov, Proceedings of the forty-seventh annual ACM symposium on Theory of computing. the forty-seventh annual ACM symposium on Theory of computingAleksandar Nikolov. Randomized rounding for the largest simplex problem. In Proceedings of the forty-seventh annual ACM symposium on Theory of computing, pages 861-870, 2015.
Beck's three permutations conjecture: A counterexample and some consequences. Alantha Newman, Ofer Neiman, Aleksandar Nikolov, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science. IEEEAlantha Newman, Ofer Neiman, and Aleksandar Nikolov. Beck's three permuta- tions conjecture: A counterexample and some consequences. In 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science, pages 253-262. IEEE, 2012.
Indecomposable coverings with concave polygons. Dömötör Pálvölgyi, Discrete & Computational Geometry. 443Dömötör Pálvölgyi. Indecomposable coverings with concave polygons. Discrete & Computational Geometry, 44(3):577-588, 2010.
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"Deep Incomplete Multi-View Multiple Clusterings",
"Deep Incomplete Multi-View Multiple Clusterings"
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"Shaowei Wei \nJoint SDU-NTU Centre for Artificial Intelligence Research\nShandong University\nJinanChina\n\nCollege of Computer and Information Sciences\nSouthwest University\nChongqingChina\n",
"Jun Wang \nJoint SDU-NTU Centre for Artificial Intelligence Research\nShandong University\nJinanChina\n\nCollege of Computer and Information Sciences\nSouthwest University\nChongqingChina\n\nSchool of Software\nShandong University\nJinanChina\n",
"Guoxian Yu [email protected] \nJoint SDU-NTU Centre for Artificial Intelligence Research\nShandong University\nJinanChina\n\nCollege of Computer and Information Sciences\nSouthwest University\nChongqingChina\n\nSchool of Software\nShandong University\nJinanChina\n\nCEMSE\nKing Abdullah University of Science and Technology\nThuwalSA\n",
"Carlotta Domeniconi [email protected] \nDepartment of Computer Science\nGeorge Mason University\nVAUSA\n",
"Xiangliang Zhang [email protected] \nCEMSE\nKing Abdullah University of Science and Technology\nThuwalSA\n"
]
| [
"Joint SDU-NTU Centre for Artificial Intelligence Research\nShandong University\nJinanChina",
"College of Computer and Information Sciences\nSouthwest University\nChongqingChina",
"Joint SDU-NTU Centre for Artificial Intelligence Research\nShandong University\nJinanChina",
"College of Computer and Information Sciences\nSouthwest University\nChongqingChina",
"School of Software\nShandong University\nJinanChina",
"Joint SDU-NTU Centre for Artificial Intelligence Research\nShandong University\nJinanChina",
"College of Computer and Information Sciences\nSouthwest University\nChongqingChina",
"School of Software\nShandong University\nJinanChina",
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"Department of Computer Science\nGeorge Mason University\nVAUSA",
"CEMSE\nKing Abdullah University of Science and Technology\nThuwalSA"
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| []
| Multi-view clustering aims at exploiting information from multiple heterogeneous views to promote clustering. Most previous works search for only one optimal clustering based on the predefined clustering criterion, but devising such a criterion that captures what users need is difficult. Due to the multiplicity of multi-view data, we can have meaningful alternative clusterings. In addition, the incomplete multi-view data problem is ubiquitous in real world but has not been studied for multiple clusterings. To address these issues, we introduce a deep incomplete multi-view multiple clusterings (DiMVMC) framework, which achieves the completion of data view and multiple shared representations simultaneously by optimizing multiple groups of decoder deep networks. In addition, it minimizes a redundancy term to simultaneously control the diversity among these representations and among parameters of different networks. Next, it generates an individual clustering from each of these shared representations. Experiments on benchmark datasets confirm that DiMVMC outperforms the state-of-the-art competitors in generating multiple clusterings with high diversity and quality. | 10.1109/icdm50108.2020.00074 | [
"https://arxiv.org/pdf/2010.02024v1.pdf"
]
| 222,133,117 | 2010.02024 | 695a24d54ddf563da0481c6d25e23c204fec6ccc |
Deep Incomplete Multi-View Multiple Clusterings
Shaowei Wei
Joint SDU-NTU Centre for Artificial Intelligence Research
Shandong University
JinanChina
College of Computer and Information Sciences
Southwest University
ChongqingChina
Jun Wang
Joint SDU-NTU Centre for Artificial Intelligence Research
Shandong University
JinanChina
College of Computer and Information Sciences
Southwest University
ChongqingChina
School of Software
Shandong University
JinanChina
Guoxian Yu [email protected]
Joint SDU-NTU Centre for Artificial Intelligence Research
Shandong University
JinanChina
College of Computer and Information Sciences
Southwest University
ChongqingChina
School of Software
Shandong University
JinanChina
CEMSE
King Abdullah University of Science and Technology
ThuwalSA
Carlotta Domeniconi [email protected]
Department of Computer Science
George Mason University
VAUSA
Xiangliang Zhang [email protected]
CEMSE
King Abdullah University of Science and Technology
ThuwalSA
Deep Incomplete Multi-View Multiple Clusterings
Index Terms-Multiple ClusteringsMulti-view ClusteringMissing Data ViewsQuality and Diversity
Multi-view clustering aims at exploiting information from multiple heterogeneous views to promote clustering. Most previous works search for only one optimal clustering based on the predefined clustering criterion, but devising such a criterion that captures what users need is difficult. Due to the multiplicity of multi-view data, we can have meaningful alternative clusterings. In addition, the incomplete multi-view data problem is ubiquitous in real world but has not been studied for multiple clusterings. To address these issues, we introduce a deep incomplete multi-view multiple clusterings (DiMVMC) framework, which achieves the completion of data view and multiple shared representations simultaneously by optimizing multiple groups of decoder deep networks. In addition, it minimizes a redundancy term to simultaneously control the diversity among these representations and among parameters of different networks. Next, it generates an individual clustering from each of these shared representations. Experiments on benchmark datasets confirm that DiMVMC outperforms the state-of-the-art competitors in generating multiple clusterings with high diversity and quality.
I. INTRODUCTION
With the wide-application of Internet of Things, many collected data are naturally represented with multiple feature views. For instance, an image can be encoded by its color, texture, shape and spatial descriptors. These feature views embody the consistent and complementary information of the same image, which spur extensive research of learning on multi-view data [1], [2]. Fusing these feature views can not only form a comprehensive description of the data, but also benefit the learning tasks on them, such as classification [3], clustering [4] and metric learning [5]. This work focuses on multi-view clustering (MVC), which aims to excavate complementary and consensus information across multiple views to identify the essential grouping structure with no requirement of labels from these data.
Various attempts have been made to find essential grouping structures of multi-view data. Some algorithms force the clustering results of different views being consistent with each other via correlation maximization [6], co-regularization [7], This paper was accepted by icdm 2020. * Corresponding author: [email protected] (Jun Wang). This work is supported by NSFC (No. 62072380, 61872300 and 62031003). fusing multiple similarity matrices of individual views [8], or exploring common and diverse information among views [9]. Other approaches assume the clusters of a clustering are embedded in different subspaces and try to explore these subspaces and find clusters therein [10], [11]. More recently, deep learning techniques also have been proposed to extract the high-order correlation and dependency between multi-view data for effective clustering [12], [13]. Besides, some other efforts study how to work on multi-view data with missing views, i.e., some objects are not available for all the views [14], [15].
These mentioned MVC algorithms account for the multiplicity of multi-view data, but focus on generating a single clustering result only. In practice, such multiplicity can also support to group the data in multiple different but meaningful clusterings [16], [17]. For example, a bunch of facial images represented with heterogeneous views can be separately grouped from the perspective of identity, sex and of emotions. All these groupings are different but yet meaningful. These reasonable groupings of the data are potentially useful for some purposes, regardless of whether or not it is optimal according to a specific clustering criterion [18]. Alike traditional clustering that focuses on the quality, multiple clusterings additionally pursue the diversity among alternative clusterings. However, it is a knotty task to balance the diversity and quality of these clusterings [19]. Previous works tried to obtain multiple clusterings in independent (or orthogonal) subspaces [20]- [22], by eliminating redundancy between the clusterings that are generated successively [23], [24], by executing clustering assignment again for the generated base clusterings [18], or by simultaneously gaining multiple clusterings and controlling the redundancy. However, they were designed for single-view data only.
A few efforts have been made toward exploring multiple clusterings on multi-view data. Multi-view multiple clusterings (MVMC) [17] mines the individual and shared information of multi-view data by utilizing self-representation learning [11], and then decomposes the combinations of the individuality feature matrices and commonality feature matrix by semi-nonnegative matrix factorization [25] to obtain multiple clusterings. DMClusts [26] is another multi-view multiple clusterings algorithm based on deep matrix factorization. It decomposes the multi-view data matrices layer-by-layer to obtain multiple common subspaces and generate corresponding clusterings therein. These two efforts still ideally assume all the data views are complete. However, this assumption is often violated in practice for some inevitable reasons [3], [14], such as the temporary failure of sensors or the human caused errors. As a result, the collected multi-view data are often incomplete. A simple strategy is to remove the samples with missing feature views, but this strategy obviously may throw away too much information, especially when with a high missing rate. Incomplete multi-view clustering (IMC) solutions have been proposed to address this practical issue. Some of them resort to matrix factorization to extract the shared subspace [14], [27], fill the missing information [28], or use Generate Adversarial Network (GAN) [29] to replenish the missing data [15], [30]. However, none of existing IMC methods can generate multiple clusterings with both high quality and diversity.
To address the drawbacks mentioned above, we propose a deep incomplete multi-view multiple clusterings framework (DiMVMC, as illustrated in Figure 1). DiMVMC adopts M decoder networks to generate M clusterings from M representational subspaces and to complete the missing features of instances. The input for the m-th decoder network is the m-th shared subspace H m which is randomly initialized at first, while the output is the reconstruction of multi-view data. By alternatively optimizing the decoder deep networks and {H m } M m=1 , DiMVMC can achieve the completeness and M individually shared subspaces {H m } M m=1 simultaneously. Moreover, these decoder networks are not isolated, but additionally controlled by a redundancy term based on Hilbert Schmidt Independence Criterion (HSIC) [31], which further enforces the diversity among subspaces and thus reduces the redundancy between clusterings. The main contributions of our work are summarized as follows:
(i) We study how to generate multiple clusterings on multiview data with missing samples, which is an important and practical topic, but more challenging and mostly overlooked by previous solutions. To our knowledge, DiMVMC is the first deep approach to generate multiple clusterings. (ii) DiMVMC discards the ad-hoc encoder part of autoencoder and works in a unsupervised way, as such DiMVMC has a lower network complexity and can more flexibly deal with data incompleteness in different views. In addition, it uses a redundancy quantification term to reduce the overlap among decoder networks for producing less overlapped representational subspaces, and finally generates diverse clusterings in these subspaces. (iii) DiMVMC can find multiple clusterings with higher quality and diversity than the state-of-the-art competitors [17], [21], [24], [26], [32], and it is robust to missing data in a wide range.
II. RELATED WORKS
Our work has close connections with two lines of related works, incomplete multi-view clustering and multiple cluster-ings.
A. Incomplete Multi-view Clustering
Various multi-view clustering solutions have been introduced, most of which focus on extracting the consistent/complementary information from different views to induce a consolidated clustering [6], [33], [34], while others additionally mine the individual information to achieve a more robust clustering [11]. These methods all build on the assumption that all the data views are complete. While in practice, it is more often that some samples are absent in some views. To handle such more challenging incomplete multi-view clustering (IMC), Li et al. [14] presented the first solution (named PVC) based on NMF (Nonnegative Matrix Factorization) [35], which learned common representations for complete instances and private latent representations for incomplete instances with the same basis matrices. Next, PVC used the common and private representations to seek a clustering. Zhao et al. [36] further integrated PVC and manifold learning to learn the global structure of multi-view data. Nevertheless, these NMF-based methods can only deal with two-view data, limiting their application scope. Weighted NMF-based approaches [37], [38] were also proposed to deal with more than two views by filling the missing data and assigning them with lower weights. Wen et al. [39] added an error matrix to compensate the missing data, and combined the original incomplete data matrix with the error matrix to form a completed data matrix for clustering. All these solutions in essence build on NMF, which performs shallow projection that cannot well mine the complex relationships between low-level features of multi-view data.
To mine nonlinear structures and complex correlations among multi-view data, Wang et al. [30] proposed the consistent GAN for the two-view IMC problem, which used one view to generate the missing data of the other view, and then performed clustering on the generated complete data. Xu et al. [15] sought the common latent space of multi-view data and performed missing data inference via combining GAN with autoencoder. Ngiam et al. [40] extracted shared representations by training a two-view deep autoencoder to best reconstruct the two-view inputs. Zhang et al. [41] combined auto-encoder with Bayesian framework to fully exploit partial multi-view data to produce a structured representation for classification. These shallow/deep multi-view clustering solutions still focus on producing a single clustering. Given the multiplicity of multi-view data, it is more desirable to find different clustering results from the same data and each clustering gives an independent grouping of the data.
B. Multiple Clusterings
Multiple clusterings focus on how to generate different clusterings with both high quality and diversity from the same data [19]. It is less well studied than single/multi-view clustering and ensemble clustering [42], [43], due to its requirement on generating multiple groups of results, and the difficulties on guaranteeing the good quality and diversity at the same time. Bae et al. [23] proposed a multiple clusterings solution based on hierarchical clustering (COALA). The main idea of COALA is that instances with higher intra-class similarity still gather in one cluster, while those with lower intra-class similarity are considered to be placed into different clusters for another clustering. Jain et al. presented Dec-kmeans [32], which obtained diverse clusterings simultaneously by finding multiple groups of mutually orthogonal cluster centroids. Unlike COALA and Dec-kmeans that directly control the diversity between clustering results, other solutions control the diversity between clustering subspaces and then generate different clusterings in these subspaces. Cui et al. [20] greedily found orthogonal projection matrices to get different feature representations of the original data and then found clusterings in these orthogonal subspaces. Mautz et al. [21] also tried to explore multiple mutually orthogonal subspaces, along with the optimization of k-means objective function, to find non-redundant clusterings. However, the orthogonal constraint is too strict to generate more than two clusterings. Wang et al. [22] generated multiple independent subspaces with semantic interpretation via independent subspace analysis, and then performed kernel-based clustering in these subspaces to explore diverse clusterings. Yang and Zhang [24] explicitly defined a regularization term to quantify and minimize the redundancy between the already generated clusterings and the to-be-generated one, and then plugged this regularization into the matrix factorization based clustering [25] to find another clustering. Wang et al. [44] and Yao et al. [45] directly minimized the redundancy between all the to-be-generated clusterings to simultaneously find alternative clusterings. Besides, Caruana et al. [18] firstly generated a number of useful high-quality clusterings, and then grouped these clusterings at the meta-level, and thus allowed the user to select a few high-quality and non-redundant clusterings for examination. However, these multiple clusterings methods are still restricted to single-view data.
Given the multiplicity of multi-view data, it is desirable but more difficult to generate multiple clusterings from the same multi-view data. Two approaches have been proposed for attacking this challenging task. MVMC [17] first extends multi-view self-representation learning [11] to explore the individuality information encoding matrices and the commonality information matrix shared across views, and then combines each individuality similarity matrix and the commonality similarity to generate a distinct clustering by matrix factorization. However, given the cubic time complexity of the self-representation learning, MVMC can hardly be applicable on datasets with a large number of samples. To alleviate this drawback, DMClusts extends the deep matrix factorization [13], [46] to collaboratively factorize the multi-view data matrices into multiple representational subspaces layer-bylayer, and seeks a different clustering of high quality per layer. In addition, it introduces a new balanced redundancy quantification term to guarantee the diversity among these clusterings, and thus reduces the overlap between the produced clusterings.
The above-mentioned single/multi-view multiple clusterings solutions assume all instances are complete across views, and project data into linear and shallow subspaces. Therefore, they cannot capture the complex correlations between views and nonlinear clusters in subspaces when data are incomplete. To address these issues, we introduce DiMVMC to mine multiple clusterings from multi-view data with missing instances. DiMVMC can capture the complex correlations among views and complete data by multiple decoder networks, and thus generate multiple nonlinear clusterings with quality and diversity.
III. THE PROPOSED METHOD
It was empirically demonstrated that different data views are complementary to each other, and they carry distinct information for generating diverse clusterings with quality [17], [26]. However, multi-view multiple clusterings is still challenging due to the difficulty in modeling the unknown and complex correlation among different views. Moreover, data with missing views and the required diversity between clusterings further upgrade the difficulty to address the incomplete multi-view multiple clusterings problem. Autoencoder is typically used to reconstruct the data with missing/noisy features [15], [41], [47]. The encoder takes input the incomplete data and learns a compact representation, from which the decoder recovers the missing values. To avoid the ad-hoc design of encoder for the incomplete cases in different views, we skip the encoder and take the shared subspace representation H m as the input for the m-th decoder network, from which the observed data are reconstructed and the missing data are completed, as shown in Fig. 1. In addition, we quantify and minimize the redundancy among these subspaces for generating diverse clusterings therein. The following subsections elaborate on the above procedure.
A. Generating Multiple Representation Subspaces
Suppose a multi-view dataset with V views has N instances. We use x v n ∈ R dv (v = 1, · · · , V ) to denote the feature vector for the v-th view of the n-th instance, where d v is the feature dimension of the v-th view. An indicator matrix Λ ∈ {0, 1} V ×N for all instances is defined as:
Λvn = 1 if the n-th instance has the v-th view 0 otherwise (1)
where each column of Λ is the status (present/absent) of instances for corresponding views. The relation 1 ≤ V v=1 Λ vn ≤ V holds, such that each instance is present in at least one view.
The aim of incomplete multi-view multiple clusterings is to integrate all the incomplete views to generate multiple clusterings. Inspired by cross partial multi-view networks for classification and by multi-view subspace learning [41], [48], we project instances with arbitrary view-missing patterns into the shared representational subspaces in a flexible way, where the subspaces include the information for observed views. That is, each view can be reconstructed by the obtained shared representation. Based on the reconstruction point of view [49], we use the joint distribution to concrete the above idea as follows:
P (Si|hi) = V v=1 P (x v i |hi)(2)
where h i ∈ R d is the multi-view shared representation of the ith instance, and
S i = {x v i } V v=1 , By maximizing P (S i |h i ), the common subspaces {h i } N i=1
can be obtained. However, just alike typical subspace learning methods [50], [51], (2) also optimizes one subspace and a single clustering result therein. Because of the multiplexes, multi-view data has a mix of diverse distributions. Therefore, multiple different subspaces and clusterings can co-exist. To gain multiple (M ) clusterings, we extend (2) as:
P (Si|h m i ) = V v=1 P (x v i |h m i )(3)
where h m i is the shared representation of the i-th instance in the m-th shared subspace. Based on different views in S i , we model the likelihood with respect to h m i given the observation x v i as:
P (x v i |h m i ) ∝ e −∆(x v i ,f v m (h m i ,Θ v m ))(4)
where ∆ represents the reconstruction loss. Here, we adopt the l 2 -norm for this reconstruction part. f v m is the mapping function from the common subspace H m to the v-th view and Θ v m are decoder network parameters of f v m .
Without loss of generality, suppose the data are independent and identically distributed, we can induce the log-likelihood function as follows:
L({H m } M m=1 , {Θ v m } M,V m=1,v=1 ) = M m=1 N i=1 lnP (Si|h m i ) (5)
Since maximizing the likelihood is equivalent to minimizing the loss ∆, by considering the missing case, we can obtain the following objective function for the decoder network:
min {H m } M m=1 ,{Θ v m } M,V m=1,v=1 M m=1 V v=1 N i=1 Λvi∆(x v i , f v m (h m i , Θ v m )) (6)
Optimizing the above equation can generate M shared representations {H m } M m=1 , each of which will be used to generate a clustering result. Unlike traditional autoencoder based solutions [15], [41], DiMVMC skips the encoder networks, but takes the shared subspace representation H m as the input for the m-th decoder to complete multi-view data, as done by f v m in (6). As such, DiMVMC does not need to specifically consider diverse missing cases of multi-view data, while still makes full use of observed data.
B. Reducing Redundancy between Subspaces
By minimizing (6), we can generate multiple common subspaces from incomplete multi-view data. For multiple clusterings, besides the quality of different clusterings, the diversity between clusterings is also important [19]. The diversity is usually approximately obtained by minimizing the redundancy between these subspaces. Orthogonality is the most common approach that forces two subspaces being orthogonal with each other. Orthogonality based methods may still generate multiple clusterings with high redundancy, since these orthogonal subspaces can still produce clusters with the same structure [26]. Furthermore, orthogonality does not specify which properties of the reference clustering should or should not be retained. Kullback Leibler (KL) divergence was also adopted to find diverse clusterings [52], but KL divergence is not symmetric and not applicable for high-dimensional data, due to its high time and space complexity.
Hilbert-Schmidt Independence Criterion (HSIC) [31] measures the squared norm of the cross-covariance operator over H m and H m in the Hilbert kernel space to estimate the dependency. It is empirically given by:
HSIC(H m , H m ) = 1 (N − 1) 2 tr(K m AK m A)(7)
where K m and K m ∈ R N ×N are Gram matrices, defined as an inner product between vectors in a specific kernel space.
A ij = δ ij − 1/N , δ ij = 1 if i = j, δ ij = 0 otherwise.
In this paper, we adopt the inner product kernel to specify K m = (H m ) T H m . A lower HSIC value means two subspaces are less correlated. This empirical estimation is simpler than any other kernel dependence test, and requires no user-defined regularisation. In addition, it has a solid theoretical foundation, a fast learning rate with guaranteed exponential convergence, and the capability in measuring both linear and nonlinear dependence between variables. For these merits, we adopt HSIC to quantify the overlap between generated subspaces {H m } M m=1 .
C. Unified Model
By integrating (6) with (7), we define the loss function of DiMVMC as:
J1(Θ v m , H m ) = Φ M m=1 V v=1 N i=1 Λvi∆(x v i , f v m (h m i , Θ v m ))+ λ M m=1,m =m HSIC(H m , H m ) (8) where Φ = 1 N 2 d 2 ave (d ave is the average of d v , v = 1, 2, · · · , V )
is the normalization factor, λ is the hyperparameter to balance the sought of M subspaces and diversity between them. DiMVMC can generate multiple common subspaces {H m } M m=1 and complete missing data simultaneously via minimizing (8). Since the optimal solution cannot be analytically given, we employ an optimization strategy that alternatively updates Θ v m or H m in an iterative way, while fixing the others. More specifically, Θ v m and H m are randomly initialized at first. The detailed optimization process is given in Algorithm 1. Once the optimization is done, k-means is implemented on each obtained subspace H m , and thus M clusterings with quality and diversity can be accordingly generated.
In the subspace clustering, it is desired that the subspace representation is sparse but captures the group-level semantic information. There are different ways to bring in sparsity. For example, we can add a drop-out layer for the deep learning approach. Here, to make an intuitive implementation, we add a sparsity-induced regularization into the above loss function and define a so-called Sparse DiMVMC:
J2(Θ v m , H m ) = Φ M m=1 V v=1 N i=1 Λvi∆(x v i , f v m (h m i , Θ v m ))+ λ M m=1,m =m HSIC(H m , H m ) + α M m=1 ||H m ||1(9)
where || · || 1 is the l 1 norm of the matrix. When α = 0, (9) goes back to the plain DiMVMC.
IV. EXPERIMENTAL RESULTS AND ANALYSIS A. Experimental Setup
In this section, we evaluate the performance of our proposed DiMVMC on four benchmark multi-view datasets, as described in Table I. Caltech20 [53] is a subset of Caltech-101 of 6 categories, which contains 2,386 instances and 20 clusters. We utilize 254D CENHIST vector, 512D GIST vector and 928D LBP vector as three views. Handwritten [53] is comprised of 2,000 data points from 0 to 9 digit classes, with 200 data points for each class. There are six public features available. We utilize 240D pixel averages feature, 216D profile correlations feature and 76D LBP feature as three views. Reuters [54] is a textual data set consisting of 111,740 documents in five different languages (English, French, German, Spanish and Italian) of 6 classes. We randomly sample 12000 documents from this collection in a balanced manner. We further do dimensionality reduction on the 12000 samples following the methodology of [55] and represent each view by 256 numeric features. Mirflickr includes 25,000 samples collected from Flicker with an image view and a textual view. Here, we remove textual tags that appear less than 20 times, and then remove samples without textual tags or semantic labels, finally we get 16,738 samples for experiments [17]. We compare DiMVMC against with six representative and recent multiple clusterings algorithms, including Dec-kmeans [32], Nr-kmeans [21], OSC [20], MNMF [24], MVMC [17] and DMClusts [26]. Dec-kmeans is a representative multiple clusterings solution based on orthogonalizing the clustering centroids. Nr-kmeans [21], OSC [20] and MNMF [24] attempt different techniques to seek subspaces and multiple clusterings therein. For this reason, they have close connections with our approach and are used for experimental comparison. MVMC [17] and DMClusts [26] are the only two multi-view multiple clusterings algorithms at present.
None of these compared multiple clusterings algorithms can directly handle missing data, we fill the missing features (instances) with average values for each view at first, and then apply these solutions. For those single-view algorithms, following the solution in [17], [26], we concatenate the feature vectors of multi-view data and then apply them on the concatenated vectors to generate different clusterings. Note, we do not take the deep/incomplete multi-view clustering solutions for experiments, since they can only output a single clustering. In the following experiments, the input parameters of the comparing methods are fixed (or optimized) as the authors suggested in their papers or generously shared codes.
DiMVMC selected the input parameters from the following ranges: λ ∈ {10 −3 , 10 −2 , · · · , 10 3 }, d ∈ {64, 128}, and M = 2. The alternate clusterings are generated by applying k-means algorithm on each shared subspace H m . The number of clusters for alternate clusterings was fixed to K of each dataset, as listed in Table I. In this paper, DiMVMC simply adopts two layers' network structure for each mapping f v m to reconstruct the v-th view from the m-th shared subspace H m . The demo code of DiMVMC is available at http://mlda.swu.edu.cn/codes.php?name=DiMVMC.
Following the evaluation protocol used by the baseline methods [17], [24], [44], we measure the quality of multiple clusterings via the average SC (Silhouette Coefficient) or DI (Dunn Index), and the diversity via the NMI (Normalized Mutual Information) or JC (Jaccard Coefficient). SC and DI quantify the compactness and separation of clusters within a clustering, while NMI and JC quantify the similarity of clusters of two clusterings C 1 and C 2 . We want to remark that unlike traditional clustering problem, a lower value of NMI and JC means the two alternative cluterings are less overlapped, so a smaller value of them is more preferred.
B. Discovering Multiple Clusterings
For the first experiment, we assume the four multi-view datasets are complete without any missing data. We report the average results of ten independent runs and standard deviations of each method on generating two alternative clusterings in Table II.
From Table II, we has the following important observations: (i) Multi-view vs. Single-view: DiMVMC, DMClusts and MVMC can be directly applied on multi-view data, and their generated two clusterings often have a lower redundancy than those generated by other compared methods. That is because these methods lack a redundancy control term or their redundancy strategies are difficult to be optimized. Our following experiment will further analyze the importance of redundancy control. DiMVMC frequently obtains a better quality than compared methods that can only work on the concatenated single view, which suggests that concatenating the feature vectors overrides the intrinsic nature of multi-view data, which helps to generate multiple clusterings with quality. These comparisons prove the effectiveness of our proposed approach in fusing multiple data views to generate multiple clusterings with diversity and quality. (ii) Shallow methods vs. DiMVMC: To our knowledge, DiMVMC is the first deep approach to generate multiple clusterings, and it often performs better on quality metrics (SC and DI), owing to the high-level expression ability of decoder networks. Even though, DiMVMC sporadically has a lower value on SC than some of compared methods. That is due to the widely-recognized dilemma of obtaining alternative clusterings with both high diversity and quality. DiMVMC has a larger diversity. That is explainable, since it can explore diverse nonlinear representation subspaces by decoder networks, while these shallow methods can only obtain lowlevel feature subspaces. Therefore, DiMVMC has a better tradeoff between quality and diversity than these compared methods. Although DiMVMC, DMClusts and MVMC can generate diverse clusterings from the same multi-view data, DiMVMC manifests a better performance than the latter two. That is because DiMVMC can mine the complex correlations between views and features via decoder networks, whereas these compared methods cannot.
In summary, even with complete data across views, DiMVMC outperforms compared methods across different multi-view datasets in terms of quality and diversity.
C. Impact of Missing Data
To study the performance of DiMVMC with missing data views, we define the missing rate as τ = 1 − v,n Λvn V ×N . The instances are randomly selected as missing ones, and the missing views are randomly erased by guaranteeing at least one of them is observed. In this paper, the missing rate is varied from 0 to 50% with an interval as 10%. Figure 2 shows the impact of missing rate of data on the clustering performance of DiMVMC and of compared methods. With the increase of missing rate, the performance of multiple clusterings methods does degrade. Nr-kmeans [21] and OSC [20] are always in a high position in terms of SC at the beginning. This indicates that they can obtain multiple clusterings of high quality under a small rate of missing instances. However, their SC values drop faster than others with the further increase of missing rate, since they do not take into account the missing instances/features. Furthermore, their diversity (1-NMI) is also at a low-level, suggesting their orthogonal subspaces still have a relatively high redundancy.
The performance curves of multi-view methods (MVMC, DMClusts and DiMVMC) drop more slowly than the singleview methods as the increase of missing rate. That is because the correlation between views helps to reduce the impact of missing data, and concatenating features cannot well capture this complementary information. In addition, although the SC curve of DiMVMC is not always in a relatively high level, it always holds better diversity than compared methods. In addition, it holds more stable quality (SC) and diversity (1-NMI) curves than compared methods. This observation again echoes the dilemma of balancing the quality and diversity of multiple clusterings.
Finally, DiMVMC can generate clusterings of diversity controlled by the HSIC term. It is the first multiple clusterings algorithm that considers the missing data views, it can reconstruct the incomplete multi-view dataset to complete the missing data views. As such, DiMVMC is more robust to missing data. By contrast, the simple data complement strategies used by compared methods are not so robust. As a result, the performance of the compared methods is not as stable as DiMVMC is. Figure 2 proves that DiMVMC has a better tradeoff between the quality and diversity of multiple clusterings, and is more competent in dealing with incomplete data than compared methods. That can be attributed to the adopted decoder networks and diversity control term, which can more well capture the correlations among different views and handle diverse missing patterns, and enforce the diversity among subspaces.
D. Parameter Analysis
Parameter λ balances the generation of multiple subspaces and the diversity control among these subspaces. We study the impact of λ by griding it from 10 −3 to 10 3 and plot the variation of quality (SC) and diversity (1-NMI, the larger the better) of DiMVMC on Caltech20 dataset in Figure 3a. We see that: (i) diversity (1-NMI) steadily increases at first and then gradually increases; (ii) the quality (SC) gradually decreases and then keeps relatively stable as λ further increases. Overall, SC keeps relatively stable as λ varies, but always below the starting point (λ = 0, no diversity control). This pattern is explainable, since the promotion of diversity between clusterings is often associated with the scarification of quality. We can conclude that λ indeed helps to boost the diversity between clusterings.
We vary M (number of alternative clusterings) from 2 to 6 on Caltech20 dataset to explore the variation of average quality (SC) and diversity (1-NMI) of multiple clusterings generated by DiMVMC. In Figure 3b, with the increase of M , the average quality (SC) fluctuates in a small range while the diversity (1-NMI) decreases slowly. Overall, DiMVMC can obtain M ≥ 2 alternative clusterings of quality and diversity.
Based on the base DiMVMC, we lead an l 1 norm for each common subspace H m , extend DiMVMC to a sparse DiMVMC. We apply DiMVMC and sparse DiMVMC (with α = 5 × 10 −2 ) on Caltech20 dataset, and report the results in Figure 4a. Although the SC values of these two models are both around 0.007, sparse DiMVMC has an average NMI (the lower the better) as 0.022, which is nearly 12% lower than DiMVMC (average NMI 0.025) . This comparison shows that sparsity helps to generate less correlated subspaces (fewer redundant features), and to better control the diversity of alternative clusterings.
E. Convergence and Complexity Analysis
From Figure 4b, we can find that DiMVMC and Sparse DiMVMC often converge within 15 epochs, while DiMVMC(λ = 0) without diversity control converges in 30 epochs. This trend not only proves the efficiency of our proposed alternative optimization strategy, but also shows that our diversity control term and the added l 1 norm do not increase the complexity.
The memory complexity of DiMVMC can be analyzed by two parts. For simplicity, suppose l is the number of layers, d is the dimension for any batch optimization technique. Table III gives the run-time of all compared methods. The compared methods are run on a linux server 1 . All methods are implemented by Matlab2014a, except Nr-kmeans, DMClusts and DiMVMC are implemented by Python. We observe that the three fastest methods are OSC, Dec-kmeans and DM-Clusts, respectively. OSC and Dec-kmeans do not consider the correlations between views and work on the concatenated views, so they run faster than others. DMClusts employs the efficient semi-NMF [25] to decompose multi-view data layer by layer and generates multiple clustering simultaneously. Although MNMF also builds on efficient semi-NMF, it is constrained by the reference clustering when seeking the other clustering, and has a larger run-time than DMClusts. Nrkmeans needs to update the clustering center many times, so it has a longer run-time than MNMF. MVMC involves with time demanding self-representation learning and the factorization of multiple representational matrices, so it has a longer runtime than others. Our DiMVMC has the largest run-time, since it has to capture the complex correlations between views and generate multiple clusterings with nonlinear clusters via optimizing multiple decoder networks. However, DiMVMC 1 Configuration: Intel Xeon8163, 1TB RAM with NVIDIA Tesla K80. almost always generate multiple clusterings with better quality and diversity than these compared methods.
V. CONCLUSIONS
In this paper, we introduced the DiMVMC model to explore alternative clusterings from the ubiquitous incomplete multiview data. DiMVMC can complete the missing data via a group of decoder networks, and seek multiple shared but diverse subspaces (clusterings therein) by further reducing the overlaps between subspaces. Experimental results on benchmark datasets confirm the effectiveness of DiMVMC. We will explore deep alternative clusterings by merging prior knowledge of different perspectives.
{H m } M m=1 , and d ave is the average dimension of V views. DiMVMC takes O(N M d + N M V d ave ) to save the data elements, and O(N M V ld ave ) to store the network parameters {Θ v m } M,V m=1,v=1 . So the memory complexity of DiMVMC for generating M clusterings on V views is O(N M (d+V d ave +V ld ave )). Since most multi-view data are typical sparse, the actual space complexity is much smaller. The time complexity of DiMVMC can be also analyzed by two sub-problems. DiMVMC takes O(N ld 2 d ave ) to update Θ v m , and O(N V ld 2 d ave +M N 2 d) to update H m . So the time complexity of DiMVMC for generating M clusterings on V views is O(tM N d(V ldd ave + N )), where t is the number of iterations. On the other hand, the time complexity of MVMC [17] is O(tM N 2 V (d + K)), and that of DMClusts [26] is O(tM d(M N d ave + M N d + M dd ave + V N d ave )). Thus, the time complexity of DiMVMC is quadratic to N , due to the use of HSIC term. MVMC is quadratic to N and DMClusts is linear to N . Note, both the memory complexity and time complexity of DiMVMC can drop an order of magnitude via Quality (SC) and Diversity (1-NMI) vs. M .
Fig. 3 :
3Parameter analysis of λ and M . Convergence trend of DiMVMC and its variants.
Fig. 4 :
4Sparse DiMVMC vs. DiMVMC .
Fig. 1: Main schema of DiMVMC to generate alternative clusterings from multi-view data with missing instances (gray blocks on the right). With the assumption that each view is conditionally independent given the shared multi-view representation P (S|H m ) (S stores the observed data views with missing instances), DiMVMC initializes a group of shared subspaces {H m } M m=1 , and then reconstructs the observed missing data views from H m via Decoderm. After the reconstruction, M representational subspaces {H m } M m=1 are simultaneously produced, and the features of missing instances are completed. It further uses the HSIC (Hilbert-Schmidt Independence Criterion) to reduce the overlap between these subspaces and thus to generate diverse clusterings.H 1
H 2
H M
…
…
HSIC
Decoder 1
Decoder 2
Decoder M
Instances
views
Generated data
views from H 1
Original multi-view data
with missing instances
Generated data
views from H 2
Generated data
views from H M
Reconstruction
Algorithm 1 DiMVMC: Deep incomplete Multi-view Multiple Clusterings Input: Incomplete multi-view dataset X , scalar parameters λ, number of subspaces M , learning rate η.Output: Networks parameters {Θ v
m } M,V
m=1,v=1 , M subspaces
{H m } M
m=1 and M alternative clusterings {C m } M
m=1 in these
subspaces.
1: Random initialization for {H m } M
m=1 and {Θ v
m } M,V
m=1,v=1
2: while not converged do
3:
for v = 1 : V do
4:
for m = 1 : M do
5:
Θ v
m ← Θ v
m − η∂J /∂Θ v
m ;
6:
end for
7:
end for
8:
for m = 1 : M do
9:
H m ← H m − η∂J /∂H m ;
10:
end for
11: end while
12: Grouping all instances via k-means in the representational
subspaces {H m } M
m=1 .
TABLE I :
IStatistics of used multi-view datasets. N , V and K are the numbers of instances, views and clusters. {dv} V v=1 are the feature dimensions of V views.Datasets
N , V
K
dv
Caltech20
2386, 3
20
[254, 512, 928]
Handwritten
2000, 3
10
[240, 216, 76]
Reuters
12000, 5
6
[256, 256, 256, 256, 256]
Mirflickr
16738, 2
24
[150, 500]
TABLE II :
IIQuality and Diversity of various compared methods on generating multiple clusterings from 'complete' multi-view datasets. ↑(↓) indicates the preferred direction for the corresponding evaluation metric. •/• indicates whether our DiMVMC is statistically (according to pairwise t-test at 95% significance level) superior/inferior to the other method.Dec-kmeans
Nr-kmeans
OSC
MNMF
MVMC
DMClusts
DiMVMC
Caltech20
SC↑
-0.107±0.002•
0.053±0.003• 0.190±0.001• -0.109±0.003•
-0.097±0.001• -0.033±0.002•
0.006±0.000
DI↑
0.032±0.000•
0.042±0.000• 0.045±0.002•
0.026±0.000•
0.009±0.000•
0.115±0.003•
0.265±0.006
NMI↓
0.053±0.002•
0.465±0.011• 0.667±0.007•
0.070±0.001•
0.025±0.003•
0.065±0.002•
0.024±0.001
JC↓
0.046±0.000•
0.176±0.008• 0.297±0.004•
0.045±0.000•
0.026±0.001•
0.050±0.001•
0.025±0.001
Handwritten
SC↑
0.043±0.001•
0.126±0.004• 0.371±0.003•
0.020±0.001•
-0.024±0.000•
0.020±0.001•
0.007±0.000
DI↑
0.056±0.000•
0.068±0.001• 0.074±0.002•
0.031±0.002•
0.004±0.000•
0.173±0.006•
0.604±0.009
NMI↓
0.057±0.001•
0.395±0.010• 0.756±0.003•
0.093±0.001•
0.006±0.001
0.061±0.005•
0.006±0.000
JC↓
0.065±0.001•
0.207±0.011• 0.637±0.020•
0.078±0.002•
0.091±0.001•
0.095±0.003•
0.052±0.001
Reuters
SC↑
-0.033±0.000•
-0.012±0.000• 0.013±0.001• -0.002±0.000•
-0.004±0.000•
0.015±0.001•
0.011±0.000
DI↑
0.047±0.002•
0.055±0.003• 0.068±0.002•
0.019±0.001•
0.013±0.000•
0.030±0.001•
0.434±0.000
NMI↓
0.231±0.003•
0.301±0.005• 0.236±0.011•
0.001±0.000
0.001±0.000
0.007±0.001•
0.001 ±0.000
JC↓
0.290±0.005•
0.284±0.012• 0.339±0.009•
0.094±0.000•
0.093±0.000•
0.114±0.003•
0.091 ±0.000
Mirflickr
SC↑
-0.004±0.000•
0.001±0.000• 0.017±0.000• -0.058±0.000•
-0.038±0.000•
0.336±0.008•
0.006±0.000
DI↑
0.061±0.002•
0.035±0.001• 0.059±0.002•
0.053±0.001•
0.173±0.005•
0.076±0.001•
0.536±0.013
NMI↓
0.427±0.012•
0.584±0.005• 0.575±0.011•
0.014±0.000•
0.005±0.000
0.043±0.001•
0.005±0.000
JC↓
0.878±0.022•
0.363±0.007• 0.368±0.011•
0.023±0.000•
0.022±0.000•
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| []
|
[
"Weighting gates in circuit complexity and holography",
"Weighting gates in circuit complexity and holography"
]
| [
"I Akal \nII. Institute for Theoretical Physics\nUniversity of Hamburg\nD-22761HamburgGermany\n"
]
| [
"II. Institute for Theoretical Physics\nUniversity of Hamburg\nD-22761HamburgGermany"
]
| []
| Motivated by recent studies of quantum computational complexity in quantum field theory and holography, we discuss how weighting certain classes of gates building up a quantum circuit more heavily than others does affect the complexity. Utilizing Nielsen's geometric approach to circuit complexity, we investigate the effects for a regulated field theory for which the optimal circuit is a representation of GL(N, R). More precisely, we work out how a uniformly chosen weighting factor acting on the entangling gates affects the complexity and, particularly, its divergent behavior. We show that assigning a higher cost to the entangling gates increases the complexity. Employing the penalized and the unpenalized complexities for the F κ=2 cost, we further find an interesting relation between the latter and the one based on the unpenalized F κ=1 cost. In addition, we exhibit how imposing such penalties modifies the leading order UV divergence in the complexity. We show that appropriately tuning the gate weighting eliminates the additional logarithmic factor, thus, resulting in a simple power law scaling. We also compare the circuit complexity with holographic predictions, specifically, based on the complexity=action conjecture, and relate the weighting factor to certain bulk quantities. Finally, we comment on certain expectations concerning the role of gate penalties in defining complexity in field theory and also speculate on possible implications for holography. * [email protected] arXiv:1903.06156v3 [hep-th] | 10.1093/ptep/ptab098 | [
"https://arxiv.org/pdf/1903.06156v3.pdf"
]
| 119,196,152 | 1903.06156 | 33d2ff124fb49599352b5181e2a8c9b06a474299 |
Weighting gates in circuit complexity and holography
21 Jul 2021
I Akal
II. Institute for Theoretical Physics
University of Hamburg
D-22761HamburgGermany
Weighting gates in circuit complexity and holography
21 Jul 2021(Dated: July 22, 2021)CONTENTS
Motivated by recent studies of quantum computational complexity in quantum field theory and holography, we discuss how weighting certain classes of gates building up a quantum circuit more heavily than others does affect the complexity. Utilizing Nielsen's geometric approach to circuit complexity, we investigate the effects for a regulated field theory for which the optimal circuit is a representation of GL(N, R). More precisely, we work out how a uniformly chosen weighting factor acting on the entangling gates affects the complexity and, particularly, its divergent behavior. We show that assigning a higher cost to the entangling gates increases the complexity. Employing the penalized and the unpenalized complexities for the F κ=2 cost, we further find an interesting relation between the latter and the one based on the unpenalized F κ=1 cost. In addition, we exhibit how imposing such penalties modifies the leading order UV divergence in the complexity. We show that appropriately tuning the gate weighting eliminates the additional logarithmic factor, thus, resulting in a simple power law scaling. We also compare the circuit complexity with holographic predictions, specifically, based on the complexity=action conjecture, and relate the weighting factor to certain bulk quantities. Finally, we comment on certain expectations concerning the role of gate penalties in defining complexity in field theory and also speculate on possible implications for holography. * [email protected] arXiv:1903.06156v3 [hep-th]
I. INTRODUCTION
The anti de-Sitter (AdS)/conformal field theory (CFT) correspondence [1] has substantially improved our understanding of strongly coupled systems and black hole (BH) micro states. As an explicit realization of the holographic principle [2,3], it basically dictates how quantum gravitational physics can nonperturbatively be formulated within the language of a pure quantum field theory (QFT). However, despite the enormous progress since its first proposal, many aspects of the duality still remain deeply mysterious.
In recent years, new concepts from quantum information and quantum computation have further helped to advance our understanding of the mechanisms behind the AdS/CFT correspondence [4][5][6]. The found relationships may be traced back to the much celebrated Bekenstein-Hawking entropy formula which relates an information theoretic quantity to a geometric characteristic of a BH; the area of the event horizon. Motivated by this striking prediction, one could already expect that a more fundamental information theoretic notion is related to other geometric properties of BHs.
In fact, it has been shown that entanglement properties of the boundary CFT are directly related to certain geometric quantities in the bulk [7][8][9]. These findings have provided surprising evidence that quantum entanglement plays a profound role in the emergence of spacetime in a gravitational theory. Since then, understanding the relation between quantum entanglement and the emergence of semiclassical geometry is being actively worked on, see e.g. [10][11][12].
In the context of BHs, such an information/geometry duality turns out to be even more astonishing. Namely, it has been shown that while the holographic entanglement entropy approaches a constant value with time during BH thermalization, certain bulk quantities such as the size of the Einstein-Rosen bridge (ERB) for the eternal AdS BH keeps increasing [13]. In view of that conundrum, it has been proposed that the boundary quantity which continues evolving after thermal equilibrium is quantum computational complexity [14].
Two separate conjectures have been proposed in the holographic context. The first one, the complexity=volume (CV) proposal, states that complexity is proportional to the volume of a maximal codimension one surface in the bulk which extends to the boundary [15][16][17].
A second, more precise one, is known as the complexity=action (CA) proposal [18,19]. It identifies the complexity of the boundary CFT state with the gravitational action evaluated on a specific bulk region known as the Wheeler-DeWitt (WDW) patch. The WDW patch is the domain of dependence of a bulk Cauchy surface attached to a specific time slice. Also note that the notion of complexity has recently been related to the spacetime volume of the WDW patch [20]. A sketch of the relevant geometric objects for the CA and CV proposals is depicted in figure 1. In both cases the corresponding quantity evolves with time even after thermal equilibrium sets in. Thus, allowing to probe the interior region of BHs, the mentioned proposals add two new classes of interesting gravitational observables with connections to quantum information and computation. Many aspects of the proposals and the related observables have recently been investigated, see e.g. . Even though both the CV and CA conjectures are interesting, much has to be done to advance our insights into the deep connection between quantum information and the structure of spacetime. For establishing the geometric dual of complexity, a precise definition of this quantity in strongly coupled theories or even in more general QFTs is for sure necessary.
The concept of compuational complexity is rooted in the field of theoretical computer science. Generally, an implemented algorithm for mapping an input (reference) quantum state |ψ R for a number of qubits to an output (target) quantum state |ψ T is determined by some function which is a unitary operation U , i.e. |ψ T = U |ψ R . In a quantum circuit model, such a unitary operation (or circuit) is constructed from elementary gates selected from a fixed set of universal gates, see figure 2. Accordingly, the circuit complexity can be defined as the minimal number of elementary gates required to construct the circuit. To define the circuit complexity for states in the boundary field theory [30,62], the corresponding task would then be identifying the minimum number of elementary gates required to prepare a desired target state by starting from a simpler reference state. For a construction built by a certain number of discrete elementary gates, the complexity of the target state explicitly depends on a specific reference, a choice of the gate set and an error tolerance.
Preliminary progress towards quantifying circuit complexity in field theory has recently been made. For instance, the formulation in [63] is based on earlier studies in quantum information showing that finding the optimal circuit is equivalent to finding the minimal geodesic in the space of unitaries [64][65][66]. This approach has also been generalized to fermionic [67,68] as well as interacting theories [69]; for other recent studies see e.g. [70][71][72][73][74][75][76][77][78][79]. Another field theory proposal is based on the Fubini-Study metric approach [80] recently applied in e.g. [75,81]. Even though both proposals have originally been made for free theories, it is remarkable that they give rise to similar leading order UV divergences as obtained in holographic computations. In addition, we also note that, motivated by the tensor network representation of the partition function, a notion of complexity has been proposed from the path integral point of view [82][83][84] which has given rise to interesting insights [85].
In the present paper, we focus on the notion of circuit complexity in QFT. More precisely, we investigate how weighting certain classes of gates building up a quantum circuit more heavily than others does affect the optimal circuit depth (i.e. complexity 1 ). Introducing such gate penalties may be motivated from different perspectives. Being already important concepts in quantum information and computation, such gate penalties may have interesting implications for complexity related aspects in QFT itself, but as well as for better understanding certain characteristics of quantum condensed matter systems. Moreover, it may shed light on recent observations resulting from holographic approaches to complexity. To be more precise, one may, for instance, ask for the consequence when complexity, if taken to be a physical attribute of a QFT, incorporates the notion of locality. In fact, if it is so, gates which entangle far separated points should require much higher costs in the geometric 1 We would like to note that the terms optimal circuit depth and (circuit) complexity are meant to mean the same. We use them interchangeably. distance in the space of unitaries than those which operate as entanglers of less separated points. In the simple case, namely, for a pair of coupled harmonic oscillators, it has been shown that introducing such penalties has a drastic effect on the complexity [63]. To be noted, there, the optimal circuit acts as a representation of GL(2, R).
} { } { FIG. 2.
A quantum circuit constructing the target state |ψ T from the initial reference state |ψ R .
The unitary U = g in g i n−1 . . . g i 2 g i 1 is a sequence of elementary gates.
Here, we work out how this change is reflected when similar penalties are introduced in the case of a field theory regulated by a lattice. In such a situation, the corresponding circuit becomes extended in form of a representation of GL(N, R). We should note that instead of using distance dependent weighting factors, we work with entangling gates which are uniformly weighted. Of course, this does not entirely correspond to the situation described above, i.e. concerning the notion of locality. However, doing so generalizes previous findings for states in field theory. This will enable us to make interesting comparisons with previous findings. Beyond that, it has recently been argued that holographic complexity might be nonlocal, means that the gate set in the CFT contains bilocal gates acting at arbitraryly distant points [44]. This additionally motivates us to penalize only one specific class of gates without implementing any distance dependencies.
Another motivation for considering the setup detailed above goes back to the found similarities between holographic complexity and circuit complexity in field theory. Particularly, we would like to understand how introducing gate penalties, even if only acting uniformly, will affect the leading order UV divergence. Such studies may be helpful for further comparisons with predictions from the gravity side. Closely related, introducing gate penalties may also be of particular importance for tensor network descriptions such as MERA [86] or cMERA [87,88]. It has been argued that these constructions provide a representation of a time slice of the AdS space [89,90].
The paper is structured as follows: in section II, we start with a short description of the geometric approach to circuit complexity proposed by Nielsen and collaborators [64][65][66]. In section III, we first introduce the setup under consideration and some basic notations. We then comment on the general procedure and summarize some of the earlier results. We would like to note that even though much of the content in this part has already been presented in [63], we recap the relevant findings which are needed for later computations. We also discuss some additional aspects which are important at the later stage. In section IV, we go through previous findings concerning the effect of implementing gate penalties in the case of two coupled oscillators. Afterwards, we comment on some further details and extensions related to other cost functions. Section V is the main part of this paper. We generalize the computations in penalized geometry for the regulated field theory and work out the differences to the unpenalized case. We then compare our results with holographic complexity computations, particularly, by referring to the findings based on the CA proposal. We also briefly comment on certain expectations concerning the implementation of distance dependent gates and on earlier findings which may share certain similarities with the presented results. In section VI, we finalize with a brief conclusion. In the remaining part, we mostly stick to the notations in [63] to allow for a direct comparison with the unpenalized case.
II. CIRCUIT COMPLEXITY
Let us begin by introducing the main idea of gemeotrizing circuit complexityà la Nielsen and collaborators. 2 The basic task is finding the optimal quantum circuit which implements a specific unitary operation U . This can be approached as a control problem of finding the right Hamiltonian which constructs the desired unitary or circuit. For doing so, one specifies a space of unitaries where the interesting paths satisfy the boundary conditions U (t = 0) = 1 and U (t = 1) ≡ U . 3 In addition, one also defines the corresponding cost function, call it F . Afterwards, by minimizing the following functional
D(U (t)) = 1 0 dt F U (t),U (t)(1)
one obtains the optimal path (or circuit depth) which yields the optimal quantum circuit.
Here, the mentioned cost F (U, V ) is a local function of U in the space of unitaries and V is a vector in the tangent space at the point U . It is argued that F must be (I) continuous, (II) positive, (III) homogeneous and (IV) should satisfy the triangle inequality. In the case when (I) is replaced with the criteria of smoothness, D(U ) defines a length in a Finsler manifold which corresponds to a class of differential manifolds with a quasimetric and the measure as defined above. Thus, a Finsler geometry generalizes the notion of a Riemannian manifold. According to this instruction, the problem of finding the optimal circuit becomes the problem of finding the geodesics in the resulting geometry. What remains to be done is fixing the form of the cost F . Once F is chosen appropriately, the complexity is simply defined as the length of the optimal path in the corresponding geometry.
III. SETUP
In the following, we specify the system we would like to focus on. We consider a free d dimensional scalar field theory described by the Hamiltonian H given by
H = 1 2 d d−1 x π 2 (x) + ∇φ 2 (x) + m 2 φ 2 (x) .(2)
We regulate this theory by putting it on a square lattice with spacing δ which describes an infinite number of coupled harmonic oscillators. The corresponding Hamiltonian then takes the form
H = 1 2 n P 2 ( n) M + M ω 2 X 2 ( n) + M Ω 2 j (X( n) − X( n −x j )) 2(3)
where the introduced definitions in H above read as follows
X( n) = δ d/2 φ( n), P ( n) = δ −d/2 p( n), M = 1/δ, ω = m, Ω = 1/δ.(4)
Let us first proceed with the simplest case, i.e. a system of two coupled oscillators.
A. Pair of oscillators
Position space
The system of two coupled harmonic oscillators is described by the following Hamiltonian
H = 1 2 p 2 1 + p 2 2 + ω 2 (x 2 1 + x 2 2 ) + Ω 2 (x 1 − x 2 ) 2(5)
which is written in terms of the physical variables where x j denote the spatial positions and M j = 1 is set for simplicity. The eigenstates and eigenenergies of (5) can be solved by rewriting the problem in terms of two decoupled oscillators via changing from the original position basis to the normal mode basis expressed in terms of the following normal mode coordinates and frequenciesx
± := 1 √ 2 (x 1 ± x 2 ),ω 2 + := ω 2 , ω 2 − := ω 2 + 2Ω 2 .(6)
The normalized 4 ground state wave function can be written as the product of the ground state wave functions for the two individual oscillators. In terms of the physical position coordinates, the resulting wave function reads
ψ 0 (x 1 , x 2 ) = (ω 2 − β 2 ) 1/4 √ π e − ω 2 (x 2 1 +x 2 2 )−βx 1 x 2(7)
where
ω =ω + +ω − 2 , β =ω + −ω − 2 .(8)
Following the same motivation as described in [63], we choose a factorized Gaussian reference state where both oscillators are disentangled, i.e.
ψ R (x 1 , x 2 ) = √ ω 0 √ π e − ω 0 2 (x 2 1 +x 2 2 ) .(9)
Here, ω 0 denotes some arbitrary frequency characterizing the reference state. Its explicit form will be discussed later. In order to construct U such that
|ψ 0 ≡ |ψ T = U |ψ R ,(10)
we need to fix the set of appropriate gates. A set of elementary gates which implement a unitary transformation as prescribed in (10) can be found in the literature. Without going into the details, here we will only mention that for our purpose the relevant one are the entangling and scaling gates, i.e.
Q ab = e i xap b ∀ a = b(entangling)
, 4 The normalization is chosen such that d 2 x |ψ 0 | 2 = 1.
Q aa = e /2 e i xapa (scaling),(11)
noting that a, b ∈ {1, 2}. The parameter appearing in (11) is chosen to be infinitesimal 5 in order ensure only small changes when the gates act on the wave function.
Next, we may express the circuit U in path ordered form,
U (s) = ← P exp s 0 ds Y I (s)O I .(12)
This representation will be used instead of a discrete gate representation as considered in the original formulation. The operators O I precisely correspond to the previous gates from (11) with Q ab = exp ( O ab ) and O ab = (ix a p b + δ ab /2) where ab ≡ I ∈ {11, 12, 21, 22}.
Accordingly, the path ordered 6 exponential (12) parametrizes the product of gates where the function Y I decides whether the gate of type I is switched on or off. Note that the differential ds behaves analogous to the infinitesimal parameter . By proceeding in this way, any circuit follows a particular trajectory determined by Y I through the space of unitaries such that ψ T = U (s = 1)ψ R and U (s = 0) = 1. In the following, we choose the cost F to be set by the following two norm
F = F 2 ≡ I (Y I ) 2 .(13)
For this choice, note that the optimal path is a geodesic in usual Riemannian geometry. So the functional for the circuit depth from (1) can be written as
D(U ) = 1 0 ds G IJ Y I (s)Y J (s)(14)
where we assume that the metric 7 is fixed by setting G IJ = δ IJ . Now, one can in principle express the function Y I which specifies the velocity vector tangent to the optimal trajectory in the space of unitaries in terms of the operators O I . However, it turns out to be more convenient to work with unitary matrices instead of operators. Accordingly, one just needs to reformulate the problem in matrix representation. In case of Gaussian states, as it is the case for the present setup, one may think of the space of states as the space of positive
quadratic forms, i.e. ψ ∼ exp − 1 2 x a A ab x b where A R = ω 0 1, A T = ω β β ω .(15)
5 Note that due to 1 we only need to find the circuit which minimizes the coefficient of the leading 1/ term in the complexity. 6 The operators at smallers act prior to those at largers. 7 In general, the metric G IJ allows to weight particular gates in the circuit. This will be discussed in the sections below.
It can be shown that the corresponding gate matrices take the form
Q I = exp ( M I )(16)
where M ab,cd = δ ac δ bd denotes a 2 × 2 matrix. For the present setup, the explicit generators
M I are M 11 = 1 0 0 0 , M 22 = 0 0 0 1 , M 12 = 0 1 0 0 , M 21 = 0 0 1 0 .(17)
In terms of the gates from (17), it becomes clear that the circuits form a representation of GL(2, R). Using the generator matrices in (17), the path ordered exponential in (12) can be replaced by
U (s) = ← P s 0 ds Y I (s)M I(18)
where
Y I (s) = tr(∂ s U (s)U −1 (s)M T I ).(19)
The expression in (19) drastically simplifies the form of the velocity vector Y I which determines the optimal circuit depth. However, in order use the expression from (19), it is required to construct an explicit parameterization of the mentioned transformations. After having done this, the main task reduces to finding the minimal 8 geodesic in the resulting metric on GL(2, R) which connects the matrices A T and A R according to A T = U (s = 1)A R U T (s = 1).
A general parameterization can be obtained by constructing U according to the decompo-
sition GL(2, R) = R × SL(2, R)
where certain coordinates, for instance, labeled by τ, θ and ρ, are chosen on the subgroup SL(2, R). A fourth coordinate, call it y, will be responsible for the remaining R fibre. In this way, inserting U ≡ U (τ, θ, ρ, y) into
ds 2 = G IJ tr(dU U −1 M T I ) tr(dU U −1 M T J )(20)
yields a right-invariant metric. To obtain the geodesic, one first identifies the corresponding Killing vectors. Afterwards, these can be used to find all conserved charges which simplify 8 Note that by definition the complexity is determined by the shortest geodesic which yields the desired transformation. In general, one may find a family of geodesics with various lengths.
solving the underlying geodesic equations. By proceeding as described, it has been found that the shortest geodesic is given by [63] y = y 1 s, ρ = ρ 1 s, θ = τ = 0 (21) where
y 1 = 1 2 log ω 2 − β 2 ω 0 , ρ 1 = 1 2 cosh −1 ω ω 2 − β 2 .(22)
Substituting this straight line geodesic-since the metric is flat-into the general expression for U ≡ U (τ, θ, ρ, y) ∈ GL(2, R) leads to the following optimal (straight line) circuit
U 0 (s) = exp y 1 −ρ 1 −ρ 1 y 1 s , 0 ≤ s ≤ 1.(23)
Here, we have expressed U 0 from (23) in exponential form which is more convenient for later purpose. For instance, using (23), we can immediately identify the corresponding velocity vector components, i.e. Y I 's. Having determined the latter, we just need to insert them into (14) which finally yields the complexity
C 2 ≡ D 2 (U 0 ) = 2y 2 1 + 2ρ 2 1(24)
for the F 2 cost from (13) as we have indicated by the subscript.
Normal mode subspace
Of course, one can express the complexity C 2 from (24) in terms of the physical parameters. For this, we just need to use the relations in (22). However, the final result would look rather complicated. Instead, in terms of the normal mode frequencies introduced in (6), which can be used to express (22) as
y 1 = 1 4 log ω +ω− ω 2 0 , ρ 1 = 1 4 log ω − ω + ,(25)
the complexity in (24) takes the form
C 2 = D 2 (U 0 ) = 1 2 log 2 ω + ω 0 + log 2 ω − ω 0 .(26)
This simple expression invites to investigate the optimal circuit in terms of the normal mode coordinatesx ± . Such an observation can also be anticipated from the ground state wave function which is a factorized Gaussian state in the normal mode basis, i.e.
ψ 0 (x + ,x − ) = (ω +ω− ) 1/4 √ π e − 1 2 (ω +x 2 + +ω −x 2 − ) .(27)
To work out the explicit form of the optimal circuit U 0 in normal mode basis, we first need to identify the required transformation matrix which yields the change [x + ,
x − ] T = R 2 [x 1 , x 2 ] T .
It turns out that such a transformation can be realized via the following orthogonal rotation matrix
R 2 = 1 √ 2 1 1 1 −1 with RR T = R T R = 1,(28)whereà T = ω + 0 0ω − ,à R = ω 0 1(29)
follow due toà = R 2 AR T 2 . Then, once the matrix R 2 is known, the optimal straight line circuit U 0 can be transformed to operate in the normal mode space via the transformatioñ
U 0 (s) = R 2 U 0 (s)R T 2 .
In contrast to U 0 , the latter transformation results in a considerable simplification of the normal mode circuit 9Ũ 0 with the following diagonal form
U 0 (s) = exp 1 2 log ω + ω 0 0 0 1 2 log ω − ω 0 s .(30)
It can be seen thatŨ 0 does not possess any off-diagonal entries. In other words, in normal mode subspace, there is no entanglement introduced. This is in line with the mentioned 9 Of course, the simple circuitŨ 0 can be obtained if the correct basis of generators in normal mode subspace are employed. Introducing the corresponding set of indices in that subspace, i.e.Ĩ ∈ {++, +−, −+, −−}, the generatorsMĨ are formally equal to MĨ , but act in a different space. Using the relatioñ
MĨ = R 2 MĨ R T 2 ⇒ MĨ = R T 2MĨ R 2
theMĨ generators can be transformed to act on states expressed in position basis. Then, transforming
(30) according to R T 2Ũ0 (s)R 2 , whereŨ 0 (s) = exp ỸĨ (s)MĨ ,
leads to the optimal circuit from (23), i.e.
U 0 (s) = exp YĨ (s)MĨ .
factorized Gaussian shape for the statesψ R andψ T . The factorization turns out to be a substantial simplification when generalizing the results for the regulated field theory. Now, if the analogous computation is made by using the generatorsMĨ and the velocity vector YĨ operating in the normal mode subspace, it follows that the complexity remains as in (24), i.e. the complexity C 2 for the F 2 cost is independent of the basis. This feature can be comprehended if one uses the transformation matrix which transforms the generators M J to MĨ. Let us note that the former one, i.e. M J = R T 2M J R 2 , act in the physical basis to scale/entangle the coordinates x 1,2 , whereasMĨ = R 2 MĨR T 2 scale/entangle the coordinates x +,− in the normal mode basis. The corresponding matrix is orthogonal and can be constructed according to the prescription [63]
RĨ J = R ka ⊗ R lb , k, l ∈ {+, −}, a, b ∈ {1, 2}(31)
such that MĨ = RĨ J M J . Utilizing the matrix RĨ J , the mentioned basis independence is nothing but the consequence of the equality
(20)| G IJ =δ IJ = δĨJ tr(dU U −1 M T I ) tr(dU U −1 M T J ).(32)
Very similar to the discussion above, one can show that such a basis independence also holds for the F κ=2 cost set by
F κ=2 = I G I |Y I | 2(33)
where we will use G I = 1 for the moment. In this case, the complexity in both bases turns out to be
C κ=2 = 1 4 log 2 ω + ω 0 + log 2 ω − ω 0(34)
i.e. C κ=2 = C 2 2 . However, this basis independence does generally not occur for costs of the form
F κ = I G I |Y I | κ ∀ κ = 2.(35)
We should add that the cost F κ=2 is of particular relevance for our studies, since it leads to the same leading UV divergence as found in holographic complexity computations. More details regarding this issue are discussed in section III C 2.
B. Lattice
One dimensional
The previous results can be generalized for the regulated field theory by a line lattice.
The corresponding Hamiltonian is of the form
H = 1 2 N −1 j=0 p 2 j + ω 2 x 2 j + Ω 2 (x j − x j+1 ) 2(36)
which satisfies the periodic boundary condition x j+N = x j . It describes a discretized field placed on a circle with length
L = N δ.(37)
Motivated by the simplifications in normal mode basis, we rewrite H in this representation,
i.e.
H = 1 2 N −1 k=0 |p k | 2 +ω 2 k |x k | 2 .(38)
The normal mode coordinates can be deduced from a discrete Fourier transform,
x k = 1 √ N N −1 j=0 exp 2πik N j x j ,(39)
where k ∈ {0, . . . , N − 1} andx † k =x N −k . For N = 2, we find the coordinates
x 0 = 1 √ 2 (x 0 + x 1 ),x 1 = 1 √ 2 (x 0 − x 1 )(40)
which of course correspond to the one introduced in (6) if we identifyx 0,1 ↔x +,− and
x 0,1 ↔ x 1,2 .
The normal mode frequencies are expressed in terms of the physical frequencies ω and Ω,ω
2 k = ω 2 + 4Ω 2 sin 2 πk N .(41)
Note that, in contrast to the system of two coupled harmonic oscillators, cf. equation (6),
here we have a factor 4 in front of the second term. This goes back to the periodic boundary condition. This can be also seen when we set N = 2 in (36) and demand x 2 ≡ x 0 such that the term being proportional to Ω 2 becomes doubled, see also section V C 1.
In normal mode basis, the factorized Gaussian shape of both the reference state and target ground state is preserved. Therefore, the previous studies for N = 2 can be generalized to the case of N oscillators. In total, one gets N 2 generators which give rise to N ×N matrices. This extends the initial group GL(2, R) to GL(N, R). For the F 2 cost from (13), an appropriate parameterization spanned with N 2 coordinates of a general element U ∈ GL(N, R) is needed.
Accordingly, the optimal circuit will then be determined by the optimal geodesic in this extended geometry characterized by the corresponding right-invariant metric ds 2 .
Following the usual procedure can be very challenging due to the large number of coordinates parameterizing the extended metric. Instead, one may expect that, similar to the previously discussed simplified diagonal form of the optimal circuit in normal mode basis, a simplification of such type also applies for the present problem. Hence, the most optimal circuit would simply amplify the Gaussian width for each of the normal mode coordinates.
In particular, the extended optimal circuit would not introduce any entanglement between the normal modes. Indeed, it has been shown that the optimal circuit takes the form of the generalized straight line circuitŨ
0 (s) = exp M 0 s (42) whereM 0 = diag 1 2 log ω 0 ω 0 , . . . , 1 2 log ω N −1 ω 0 .(43)
For the details showing thatŨ 0 from (42) indeed corresponds to the optimal circuit, we refer the interested reader to [63]. Bringing all together, the complexity for the one dimensional regulated field theory becomes
C 2 = 1 2 N −1 k=0 log 2 ω k ω 0 . (44) 2. d dimensional
The complexity for the one dimensional lattice can be easily extended to the (d − 1) dimensional lattice consisting of N d−1 oscillators. The final expression is of the from
C 2 = 1 2 N −1 {k j }=0 log 2 ω k ω 0 ,(45)
where k j denote the momentum vector components. The corresponding expression for the normal mode frequencies readsω
2 k = ω 2 + 4Ω 2 d−1 j=1 sin 2 πk j N .(46)
C. Comparison with holographic complexity
For the comparison with holographic complexity, it is advantageous to express the complexity from (45) in terms of the field theory parameters. Then, the normal mode frequencies
introduced in (46) becomeω 2 k = m 2 + 2 δ 2 d−1 j=1 sin 2 πk j N(47)
according to the definitions in (4). An estimation for the complexity (45) in terms of the volume of the system, V , is also very useful. This can be accomplished if we first write V = L d−1 by assuming an equisided lattice. Afterwards, applying the relation (37), the total number of oscillators can be expressed as
N d−1 = V δ d−1 .(48)
By using the relation in (48), it can be shown that the complexity scales as
C 2 ∼ N d−1 2 2 log ω k ω 0 .(49)
UV divergence in QFT
The leading order contribution to complexity in QFT is determined by the UV modes.
We can take δ to be the UV cutoff where δm 1. In this UV approximation, we may estimate the normal mode frequencies in (47) by assuming
ω k ∼ 1/δ.(50)
Using the approximation above, as well as the expressions in (48) and (49), the leading order contribution to the complexity depending on the UV cutoff scales as
C 2 ∼ V δ d−1 .(51)
Remember that for the two coupled oscillators we have seen that C κ=2 = C 2 2 . Taking this relation into account, we may immediately get the leading order UV divergence on the lattice for the F κ=2 cost which scales as
C κ=2 ∼ V δ d−1 .(52)
UV divergence in holography
Studies of the UV divergence in holographic complexity based on the CV and CA proposals have shown that the leading contribution scales as [28]
C V,A ∼ V δ d−1 .(53)
This behavior is similar to the scaling for the F κ=2 cost from (52). Apart from that, it has been argued that the following depth function for the circuit complexitỹ
D κ = 1 0 ds Ĩ |YĨ(s)| κ(54)
with κ ≥ 1 would give rise to the same UV divergence as in (53). On the field theory side we simply get the mentioned scaling,
C κ ∼ V δ d−1 log 1 δω 0 κ ,(55)
however, with an additional logarithmic factor which will be discussed in detail below. The reason for the same prefactor for all κ lies in the fact that for the depth function in normal mode basis, i.e.D κ , the previously discussed straight line circuit still corresponds to the optimal circuit [63]. This basis independence for all κ including the F 2 cost is the reason why the optimal geodesic remains the same. In the original position basis, i.e. D κ , this is generally not the case. The only exceptions are the F κ=2 and F 2 cost functions.
For further interesting similarities in the divergence structure, we can compare with the leading order contribution in the CA proposal,
C A ∼ V δ d−1 log L AdS Λδ(56)
where L AdS is the curvature scale of the AdS bulk spacetime, Λ is an arbitrary dimensionless coefficient fixing the null normals on the WDW patch boundary and δ is the short distance cutoff scale in the boundary CFT [28]. As suggested in [63], for eliminating L AdS from the expression-since C A is expected to be defined in the boundary CFT which therefore should not depend on any bulk AdS scale-one may fix the dimensionless coefficient as Λ = ω 0 L AdS .
Of course, ω 0 is still an arbitrary frequency. However, with the latter replacement, the complexity simply reduces to
C A ∼ V δ d−1 log 1 δω 0 .(57)
It is remarkable that this expression is similar to the QFT expression in (55) However, recalling our motivation in the introduction, we may consider a simpler setup via penalizing the gates uniformly. In the following, we first briefly discuss these effects for the system of two coupled harmonic oscillators subjected to the F 2 cost as already explored in [63]. Afterwards, we extend our discussion for F κ=2 and work out the differences. The results constitute the basis for the field theory computations later in section V.
A. F 2 cost
Introducing weighting factors for the entangling gates simply means modifying the offdiagonal directions in the circuit. To do so, we may write the tensor G IJ in (14) as
G IJ = diag[1, α 2 , α 2 , 1](58)
with α 2 > 1. Of course, for α = 1 one would arrive at the original situation in which no extra cost is assigned to the entangling gates. With the extended penalty tensor from (58), the corresponding metric would be modified due to the appearance of additional mixing terms. The final expression may principally be treated as usual. Namely, after identifying the corresponding Killing vectors, one would need to find all conserved charges in order to solve the geodesic equations. However, writing down the full metric, it turns out that this task is too complicated to be performed. This problem has been avoided by assuming α y 1 , ρ 1 1 and neglecting some components of the metric to simplify the full expression.
In this way, it is possible to construct a kind of segmented path which is not a geodesic, but is capable to come very close to the optimal geodesic [63]. The segmented circuit constructed by exploiting the described approximate path takes the form
U p (s) U pa (s), 0 ≤ s ≤ 1 2 U p b (s), 1 2 ≤ s ≤ 1 .(59)
The circuits for the two segments are determined by
U pa (s) = e −ρ 1 2s 0 0 e ρ 1 2s (60)
and
U p b (s) = e y 1 (2s−1) cos(φ)e −ρ 1 − sin(φ)e ρ 1 sin(φ)e −ρ 1 cos(φ)e ρ 1 (61)
where we have defined φ := π 4 (2s − 1). Instead of continuing with these expressions, we want to rewrite them in exponential form. This can be easily done by utilizing the relation (19) which yields the desired exponential forms
U pa (s) = exp −ρ 1 0 0 ρ 1 2s (62)
and
U p b (s) = exp y 1 π 4 π 4 y 1 s 2 .(63)
The expressions above turn out to be much simpler and more suitable for later use. Now, using (62) and (63) in combination with (18), we can easily find the velocity vector components. Note that the generators are the one which are given in (17). Once the components of Y I (s) are found, we can use the splitting
C = D(U p ) D(U pa ) + D(U p b )(64)
to deduce the complexity. The circuit depths on the right-hand side of (64) can be computed according to the integral form introduced in (14). Note that the latter is associated with the F 2 cost.
We should add that the given segmented construction can also be used for the F κ=2 cost.
Differently, for a general κ ≥ 1, it does not yield a circuit depth which comes close to the optimal one. One might argue that changing the basis and computingD κ instead of D κ may solve this problem. However, it has been discussed that, once the geometry is penalized, the depth function will not be invariant under a basis change [63]. Let us continue with the complexity for the F 2 cost. The final result, which can be obtained by plugging the mentioned segmented path into the normalization constant, 10 takes the form
D p 2 (U p ) √ 2ρ 1 + 2y 2 1 + 2α 2 π 4 2 √ 2ρ 1 + π 2 √ 2 α + O 1 α 2 .(65)
Here, the superscript p indicates that we work in penalized geometry. Now, if we go back and compare this result with the unpenalized straight line circuit depth in (24), we find that
D p 2 (U p ) D 2 (U 0 )(66)
10 The normalization constant gives the length of the geodesic. It corresponds to the complexity if the geodesic is the minimal one.
For Here, s is the scaled affine parameter. The components of x x x are the mentioned parameterization variables in section III A 1 for the subgroup SL(2, R) and the R fibre.
which follows due to the mentioned assumption α ρ 1 , y 1 1. Being in agreement with the naive expectation, the complexity increases when a higher cost is assigned to the entangling gates.
B. F κ=2 cost
Once the exponential forms in (62) and (63) are known, finding the complexity in penalized geometry for the F κ=2 cost becomes straightforward. Similar to the previous case, we first fix
G I = [1, α 2 , α 2 , 1] T .(67)
Then, by using the depth integral
D κ=2 = 1 0 ds I G I |Y I (s)| 2(68)
and the splitting in (64), we obtain
D p κ=2 (U p ) 2ρ 2 1 + 2y 2 1 + 2α 2 π 4 2 .(69)
As before, we can compare the result (69) with the complexity in unpenalized geometry, i.e.
D κ=2 (U 0 ) = D 2 2 (U 0 ). The difference between both complexities becomes
D p κ=2 (U p ) − D κ=2 (U 0 ) 2α 2 π 4 2 1(70)
showing again that assigning a higher cost to the entangling gates increases the complexity.
V. PENALIZED GEOMETRY II: FIELD THEORY
In this section, we study the effect of introducing weighting factors for the entangling gates in the regulated field theory placed on the lattice. As we have stressed above, for the two coupled oscillators, a direct attempt via analytically finding the minimal geodesic in full penalized geometry is already very challenging. However, certain assumptions allowed to simplify the problem under consideration. Even though the exact geodesic could not be analytically derived for the simplified case as well, an approximate segmented path has been constructed which comes very close to the optimal geodesic. This has been explicitly illustrated by comparing the corresponding complexities [63].
For the generalization on the lattice, the question is, how to find the the optimal geodesic in the extended geometry where a higher cost is attributed to the entangling gates? For instance, one may argue that by changing to the normal mode basis one could consider a perturbation around the previous straight line path, since this particular solution yields the optimal geodesic in unpenalized geometry for all F κ costs where κ ≥ 1. In general, this strategy will not simplify the problem either, since introducing a penalty tensor in one basis does not automatically lead to the same penalty tensor when working in a different basis.
However, for certain choices such as the F 2 and F κ=2 costs, as we will see, the straight line circuit may indeed be used in normal mode basis. Let us bring to mind that for these the complexity is basis independent if no extra cost is assigned to the entangling gates, see section III.
In the following, we use the results already discussed in the unpenalized case for deriving the complexity without solving the geodesic equations explicitly. For clarifying the underlying procedure, we start again with the simple case of two coupled oscillators. Please note, that all the following steps are described for the F 2 cost. But we know from earlier discussions that the findings can also be applied to the F κ=2 cost which will be our main choice later.
First, we begin by assuming that U 0 is the optimal circuit which implements the following
operation |ψ T = U 0 |ψ R(71)
in unpenalized geometry. The corresponding complexity shall be denoted by C(ψ T , ψ R |U 0 ).
In addition, we assume that the analogous optimal circuit U * in penalized geometry (i.e.
when the entangling gates are weighted as above) is known as well, which implements
|ψ T = U * |ψ R .(72)
For the latter, the complexity shall be denoted by C(ψ T , ψ R |U * ). As previously discussed, we know that both complexities now satisfy the following inequality
C(ψ T , ψ R |U * ) > C(ψ T , ψ R |U 0 ).(73)
Next, let us assume that we construct an artificial target state ψ * T which satisfies the implementation
|ψ * T = U 0 |ψ R(74)
in unpenalized geometry, but yields a different complexity C(ψ T , ψ R |U * ). Notice that the target state |ψ T depends on the physical parameters ω and δ by construction. Now, if we keep the frequency ω unchanged and demand (74), it is possible to fulfill such an implementation if we introduce a modified parameter δ * which is assigned to the mentioned state |ψ * T .
Then, we may replace the original cutoff scale δ in the optimal circuit U 0 by δ * . This will simply allow deriving the complexity in the penalized geometry by using the optimal geodesic in the unpenalized geometry. The procedure works for the states under consideration, since both are taken to be positive quadratic forms. As we will show soon, such a replacement will regulate the desired change in complexity.
It is important to note that this replacement is rather technical. The cutoff scale in the theory will still be determined by δ.
A. Modified parameter
In the following, we explicitly illustrate the described strategy from above for the system of two coupled oscillators subjected to the F κ=2 cost. For reasons which will become clear below, we cannot use the expression in (69) for the optimal circuit depth in the penalized geometry which we again denote by D p κ=2 . Recall that the result in (69) is based on the segmented path approach which is only valid when the weighting factor introduced in (58) satisfies α y 1 , ρ 1 1. A more general expression where the range of α is not restricted from below has been worked out for the F 2 cost [63]. According to our previous findings, we may also use this expression in the present case by simply squaring it. Proceeding in this way yields the desired circuit depth which takes the form
D p κ=2 = 2 α 2 tan −1 √ α 2 − 1 + ρ 1 2 + 2y 2 1 .(75)
Of course, for α → 1 it reduces to the straight line circuit depth D(U 0 ) which equals to the square of (24).
Having specified this, what remains to be done is finding the solution δ * which solves the equation D κ=2 = D p κ=2 . Note that the right-hand side of the latter equation has to be written in terms of the functions y 1 (δ, ω, ω 0 ) and ρ 1 (δ, ω) where we recall from (25) that
y 1 (δ, ω, ω 0 ) = 1 4 log ω 2 ω 2 0 1 + 2 δ 2 ω 2 , ρ 1 (δ, ω) = 1 4 log 1 + 2 δ 2 ω 2 .(76)
Contrary to that, the functions y 1 and ρ 1 on the left-hand side have to be written in terms of δ * . Finally, using the square of the straight line circuit depth in (24), the explicit equation we need to solve takes the form
2y 2 1 (δ * , ω, ω 0 ) + 2ρ 2 1 (δ * , ω) = 2y 2 1 (δ, ω, ω 0 ) + 2 α 2 tan −1 √ α 2 − 1 + ρ 1 (δ, ω) 2 .(77)
This equation can be analytically solved. An appropriate treatment yields the following
result δ * = √ 2 ω 2 0 e χ(α,ω,δ)+log 2 2+δ 2 ω 2 δ 2 ω 2 0 − ω 2 −1/2(78)
where we have defined
χ(α, ω, δ) := 4 2 α tan −1 √ α 2 − 1 α 2 tan −1 √ α 2 − 1 + 2ρ 1 (ω, δ)(79)
due to practical reasons. The expression in (78) is our key result which will be important in the remainder of this section. Of course, once we take the unpenalized limit, the only α dependent part in (78) vanishes, i.e. χ → 0, and we end up with the original cutoff scale δ,
lim α→1 δ * = δ.(80)
B. Small penalty
To highlight the effect of assigning a higher cost to the entangling gates, it is advantageous to use the perturbative expansion of δ * for small penalties. This can be achieved for moderate α. More precisely, the prefactor in front of 2ρ 1 in (79) which only depends on the weighting factor has to be sufficiently small, i.e. χ 1. This can be controlled with a weighting factor that is α = 1, but close enough to one. In this case, we may perturb δ * around χ = 0 which yields
δ * δ 1 − Rχ + O(χ 2 )(81)
where we have defined
R ≡ e √ K (ω 0 δ) 2 2 3 √ K , K ≡ log 2 2 + δ 2 ω 2 δ 2 ω 2 0 .(82)
Suppose that we fix the reference frequency ω 0 ∼ e −σ /δ at some scale ∼ 1/δ. The overall numerical parameter σ ensures that ω 0 > 1/δ. Furthermore, we would like to work in the UV approximation described in section III C 1. Using the corresponding assumptions, we may estimate K ∼ − log 2 (δω 0 ). Observe that the dimensionless factor R in (82) is positive in this regime. Also, χ is taken to be sufficiently small as described above. Therefore, we find that the expression in the brackets from (81) decreases as soon as the weighting factor α becomes different from one, i.e. δ * < δ. This change increases the complexity due to
y 1 (δ * , . . .) > y 1 (δ, . . .), ρ 1 (δ * , . . .) > ρ 1 (δ, . . .),(83)
see (76). This trend is for sure expected according to the nonperturbative expression from (75).
Up to a numerical factor N multiplied with R from (82), which is not relevant for our discussion later (thus we set N ≡ 1), we may also write an expansion of the following form
δ * δ 1 − Rα + O(α 2 ) , R := √ KR(84)
whereα := √ α − 1. Writing δ * δ(. . .) as in (84) will allow to make observations more comprehensible in certain cases.
C. On the lattice
1. N = 2, unpenalized
For the moment, let us consider a periodic lattice with N = 2. The related Hamiltonian describing this system takes the following form
H = 1 2 p 2 0 + p 2 1 + ω 2 (x 2 0 + x 2 1 ) + 2 δ 2 (x 0 − x 1 ) 2(85)
where we have imposed the boundary condition x 2 ≡ x 0 . Next, we write the Hamiltonian from (85) in terms ofδ
:= δ/ √ 2(86)
which just gives rise to the same Hamiltonian as introduced in (5). Recall that according to our previous discussion in section V A, the optimal circuit depth D κ=2 from (34) satisfies the relation
2y 2 1 (δ, ω, ω 0 ) + 2ρ 2 1 (δ, ω) = 1 4 1 k=0 log 2 ω k ω 0(87)
where the corresponding normal mode frequencies are given as
ω 2 0 = ω 2 ,ω 2 1 = ω 2 + 2/δ 2 .(88)
Now, based on the discussion in section III B 2, it is straightforward to generalize these results for the d dimensional case. For doing so, we may write
V = (N δ) d−1 ,(89)
although we should note that N has been fixed above. Then, working in the UV approximation, i.e.ω k → 1/δ, we end up with the following leading order contribution to the
complexity D κ=2 ∼ V δ d−1 log 2 1 δω 0 .(90)
Of course, the expression in (90) has been derived in unpenalized geometry.
N = 2, penalized
The previous findings can be easily extended to the penalized case when α > 1. We have seen that the periodic lattice with N = 2 reduces to the system of two coupled oscillators if δ from (86) replaces the original cutoff scale δ. Thus, we may use the previous straight line solution and insert √ 2δ * = δ * (δ → √ 2δ) into the corresponding places. In the mentioned UV approximation, this would simply mean thatω k → 1/δ * . Note that according to (80) we get lim α→1δ * = δ. The complexity in the penalized geometry then takes the form
D p κ=2 ∼ V δ d−1 log 2 1 δ * ω 0 .(91)
As in the preceding discussion, this expression has been derived by fixing N = 2. Even though (91) provides useful information on how (uniformly) weighting a certain class of gates (here, these are the entangling gates) would influence the complexity in the regulated field theory, we still need to extend the results for an arbitrary number of oscillators N . In the subsequent part, it will be argued that the relation from (91) also applies in the general case for any N .
As in the unpenalized case, let us close this part with expanding the α dependent parameterδ * for small penalties which yields
δ * δ 1 −Ȓχ + O(χ 2 )(92)
where we have definedȒ
:= e √K (ω 0 δ) 2 2 2 K ,K := log 2 1 + δ 2 ω 2 δ 2 ω 2 0 .(93)
Analogous to the expression in (84), we may also write an approximation of the form
δ * δ 1 −Ȓα + O(α 2 ) ,Ȓ := KȒ .(94)
The leading order contribution to the complexity in the smallα limit then becomes
D p κ=2 ∼ V δ d−1 log 2 1 δω 0 (1 −Ȓα) .(95)
Arbitrary N , penalized
We begin by noticing that the complexity in the unpenalized case is basis independent for the F 2 cost, i.e.
ds 2 = δ IJ dY I dY J * = δĨJ dYĨ dYJ * .(96)
In general, introducing a penalty tensor, i.e. δ IJ → G IJ , makes the metric depending on where the entangling gates are uniformly penalized with the weighting factor α 2 , it can be shown that
GĨJ = RĨ J G IJ R T IJ(97)
yields a matrix which takes the form
(1 + α 2 )1 + G off,ĨJ ,(98)
see e.g. [63]. The matrix G off,ĨJ is symmetric and only consists of off-diagonal entries of the form 1 − α 2 which do not distinguish between the different classes of gates. We should add that RĨ J denotes the generalized matrix with respect to the generators of GL(N, R), see section III A 2. So, the expression in (98) shows that in normal mode basis there is no extra cost attributed to the entangling gates relative to the scaling gates. Moreover, the dominant contribution to the metric GĨJ is determined by δ IJ multiplied with the prefactor 1 + α 2 .
Note that, except the difference due to the appearance of the prefactor, this is similar as in the unpenalized geometry where G IJ = δ IJ .
In the unpenalized case both the ground state ψ T and the reference state ψ R turn out to be factorized Gaussian states in normal mode basis, i.e. Similarly, we may also expect that the optimal circuit in the penalized geometry is a diagonal matrix. This goes back to observation that the metric in normal mode basis has the same shape as in the unpenalized case, i.e. GĨJ ∼ δĨJ . Thus, we may again use the straight line solution. The additional change for α > 1 can be incorporated by making the replacement δ →δ * as illustrated in section V C 2. Everything so far has been discussed for the F 2 cost. From earlier studies we know that the latter aspects also apply for the F κ=2 cost. Proceeding with the latter, the dominant contribution to the complexity for the regulated field theory is then given by (91). The corresponding smallα expansion of (91) has been introduced in (95).
ψ R (x k ) = ω 0 π N/4 exp − 1 2x †Ã Rx , ψ T (x k ) = N −1 k=0 ω k π 1/4 exp − 1 2x †Ã Tx ,(99)
D. Unpenalized vs penalized
Finally, once the complexity for the regulated field in penalized geometry is given, we can compute the difference ∆D := D p κ=2 − D κ=2 . From previous findings we expect the latter to be positive semidefinite. The corresponding complexities for both geometries can be found in (91) and (55). Using the smallα expansion from (95) for the former penalized one in (91), we find that
∆D V δ d−1 log 1 1 −Ȓα + 2 log 1 δω 0 log 1 1 −Ȓα .(100)
Of course, in the unpenalized limit, the expression in (100) vanishes, i.e. limα →0 ∆D = 0.
As soon as the gates are penalized, means when α > 1 orα > 0, respectively, the difference in the optimal depth starts to grow, similar to the complexity increase in the two coupled oscillator case from section V B. It should be noted that the sign of the approximate ∆D from (100) seems to depend on O(δω 0 ). This basically goes back to the smallα expansion used in (100) for which we need to distinguish between the different cases. For δω 0 1, we obviously find ∆D 0 as we would expect. In case of δω 0 1, we get a negative second term in the first line, but the leading positive term will ensure that ∆D is still positive.
However, ifȒα in the argument of the leading logarithm becomes too small, the logarithmic factor in the second line of course decreases either so that ∆D → 0. For simplifying the underlying expressions, let us assume 11 δω 0 1 for the remainder of this discussion.
According to our assumptions, we may in addition drop the leading term in the first line of (100) as well as the factor 2 in front of the second term which is assumed to be dominating. By doing so, we may further estimate
∆D V δ d−1 log 1 1 −Ȓα log 1 δω 0 .(101)
If we now carefully look at the right-hand side of (101), we notice that this expression is proportional to the complexity one would obtain for the F κ=1 cost. Of course, the absolute value bars in D κ=1 can be neglected due to δω 0 1, see also discussion below. Specifically, defining Θ := − log(1 −Ȓα) where 0 < Θ 1, the relation from above can equivalently be written as
D p κ=2 ΘD κ=1 + D κ=2 .(102)
Again, we would end up with the unpenalized complexity if Θ = 0. Recall that the F κ=1 cost yields the same leading order UV divergence as found via the holographic complexity proposals [63]. Even more, this particular cost-which gives rise to the so-called Manhattan metric-leads to the same logarithmic factor predicted on basis of the CA proposal, see section III C 2.
Let us here note that an interesting relation between the F 2 and F κ=1 costs has recently been found by exploring Nielsen's geometric approach to circuit complexity for two dimensional CFTs. 12 It has been shown that the difference between distinct metrics on the Virasoro circuits turns out to be vanishing in the holograhic large central charge limit [91].
Coming back to the present case, we can also write D κ=2 = ΦD κ=1 by simply defining Φ := − log(δω 0 ). This is possible, since by assuming δω 0 1 we can write | log( 1 δω 0 )| = Φ > 0 as noted before. Following these steps and replacing D κ=2 in (102) appropriately then yields
D p κ=2 (Θ + Φ)D κ=1 .(103)
This relation between complexities for two different costs is interesting. It somewhat resembles the situation described above in which a relation between two different costs has been found as well. Referring to the approximate expressions in (102) and (103), we may therefore ask whether weighting certain classes of gates-in particular the entangling one, since these also play an important role in holography related aspects-for a specific cost can compensate the geometry change in the space of unitaries for a different cost.
Of course, our previous estimations rely on certain simplifications. More detailed studies concerning such aspects may be important, particularly, if we presume that F κ=1 is the most holographic like cost which might be intrinsic to the CV and CA proposals. For that reason,
we are therefore confronted with the question whether weighting gates in a certain way are necessary for a consistent definition of complexity in QFT. Moreover, what would be the corresponding holographic interpretation?
On the other hand, let us bring to mind that notions such as reference, geometry or gate set are not explicitly set in the holographic CV and CA proposals. Instead, they appear to be implicit in the duality. Therefore, identifying possible similarities between QFT and holography along these lines, might serve valuable information and shed light on the role of gate weighting. 12 The corresponding submanifolds are associated to the underlying symmetry groups.
E. Divergence
Resorting to the UV approximation, we have seen that there appears an additional logarithmic factor in the complexity, see expression (55). In the limit δ → 0, this is the contribution which diverges much faster than the prefactor V δ 1−d , for more details see also [63]. In order to eliminate the additional divergence, one may, for instance, assume that the reference frequency ω 0 is fixed at some UV scale with ω 0 ∼ e −σ /δ. Again, the numerical factor σ would ensure that ω 0 > ω k . We should note that these estimations are of course very rough and are only meant to provide some hint about the approximate scaling behavior.
A more general approach is presented in [63] for the unpenalized κ = 1 measure. Nevertheless, as a first attempt, we would like to continue in the following with the more simple estimations. 13 Inserting the explicit choice for ω 0 from above into (55) yields
C UV,κ ∼ σ κ V δ d−1 ,(104)
so that the additional divergence is eliminated. As we can see, the leading order UV divergence becomes a power law. On the other hand, if we suppose to work in the IR approximation by making the following simple assumptioñ
ω k ∼ m,(105)
then the IR contribution to the complexity takes the form
C IR,κ ∼ − log κ (mδ)(106)
after having assumed mδ 1. From the approximation above we get that the cancellation of the additional factor engenders that the IR contributions to the complexity depend on the UV cutoff scale δ.
To avoid this feature, one may assume ω 0 1/δ and identify ω 0 with some IR frequency [63]. However, in this case, the additional logarithmic factor will again contribute to the leading divergence in the limit δ → 0.
As we can see, there are certain ambiguities in the discussions above. An important point is that the scaling of the complexity clearly depends on the explicit choice for the reference 13 Note that, at least, referring to the κ = 1 measure, the simple estimation here still leads to the same scaling behavior as present in the more general result derived by computing the sum over all normal mode frequenciesω k , cf. [63].
frequency ω 0 . The dependence on any reference state parameter is obviously unsatisfying from the holographic point of view, since in the AdS/CFT dictionary this choice is not explicit (this is not so for the target state which in contrast has an holographic interpretation, see e.g. [61]) and we do not exactly know how to get access to it at all.
Let us return to the UV approximation in the penalized geometry for the F κ=2 cost from (91). Assigning a higher cost to the entangling gates introduces an additional degree of freedom which is the weighting factor α. Contrary to the frequency ω 0 , whose explicit value plays a substantial role in the unpenalized case, the factor α does not characterize the reference state. It specifies the metric in the manifold of unitaries. Hence, instead of fixing the reference frequency and thus the reference state in a certain way, we may simply eliminate the additional logarithmic contribution by tuning α correspondingly.
To derive the explicit form of α, one may solve a similar equation as before, i.e.
δ * ω 0 = e −ν(107)
where ν is again taken to be some overall numerical factor. The leading contribution to the complexity would then read 14
C UV,κ=2 ∼ ν 2 V δ d−1 ,(108)
as in (104), i.e. without the additional logarithmic divergence dominating in the limit δ → 0.
Thus, avoiding any specific choice for ω 0 , a simple power law for the leading divergence multiplied by an arbitrary factor can be realized via choosing α to be the solution of (107).
In the mentioned UV approach with m 1/δ, it can be shown that the weighting factor is determined by
α tan −1 ( √ α 2 − 1) log 2 4 e −2ν + m 2 ω 2 0 − log 2 1 δ 2 ω 2 0 4 log 1 δ 2 m 2 .(109)
Of course, in the unpenalized case, i.e. α = 1, in other words, if we suppose ω 0 to be the only tunable parameter, we simply get ν = σ according to (109).
Similarly, we may focus on the IR contribution in the penalized case. In the latter case we may derive an analogous expression as in (106), where we need to take into account 14 As in the unpenalized case, although this rough estimation is in the first instance sufficient to deduce the characteristic leading order scaling, as well as the subleading IR contribution, a more detailed investigation will be interesting.
the modified parameterδ * . Apart from the IR contribution, according to the generalization derived for the unpenalized F κ=1 cost by summing over the normal mode frequencies [63], we may also expect some subleading UV contributions to the complexity which would roughly scale as
C UV-sub,κ ∼ (m * δ) 2(110)
in the smallα limit. Let us emphasize that, just formally, m * ≡ m 1 −Ȓα + O(α 2 ) may be seen as some effective mass which reduces to m when the entangling gates are not penalized, i.e. if we take the limitα → 0.
F. Comparison with C A
We can also compare the result in (91) where an extra logarithmic factor arises due to the asymptotic joint contributions in the WDW action [28]. Recall that in the unpenalized case, the QFT prediction (55) follows if the dimensionless coefficient Λ is set equal to ω 0 L AdS . This choice may be motivated by the fact that C A should be independent of the AdS curvature scale L AdS when the complexity is defined in the CFT.
In the penalized case, the comparison between the logarithmic factors becomes even more interesting. As before, we may identify the CFT cutoff scale in (56) with the scale parameter δ appearing in (91). Then, taking, for instance, the smallα expansion from (95), we may relate both contributions to each other. The cancellation of the bulk curvature scale L AdS may be realized if we set
Λ = (1 −Ȓα)ω 0 L AdS .(111)
Inserting Λ from (111) into the general expression (56) then yields
C A ∼ V δ d−1 log 1 δω 0 (1 −Ȓα) .(112)
In contrast to (57), we end up with an additional contribution in the denominator which is proportional toȒα and thus depends on the weighting factor α. Of course, the expression (112) still depends on the unspecified reference frequency ω 0 . At least when gate penalties are not implemented, we have encountered that this turns out to be a general feature of the notion of circuit complexity in QFT. Recall that in the unpenalized situation, eliminating the additional logarithmic contribution required fixing ω 0 at an appropriate scale.
However, introducing gate penalties, we have shown that tuning the weighting factor appropriately can eliminate the logarithmic contribution as well. Now, the logarithmic contribution in C A from (112) can be eliminated in the same way. In order to do so, we may for instance demand (107) to be fulfilled. In the present perturbative approximation (with respect toα), we would need to solve
δω 0 (1 −Ȓα) = e −ν(113)
to find the required α which eliminates the additional logarithmic factor. The latter would then yield the leading order contribution
C A ∼ ν V δ d−1 .(114)
As mentioned, the dimensionless coefficient Λ is related to the null normals on the boundary of the WDW patch in the bulk. To eliminate the curvature scale of the bulk spacetime L AdS , we have set Λ as in (111). Referring to such a relation between Λ and α (note that α 2 = α−1), we may ask how weighting a certain class of gates would be translated via holography. Of course, the present studies are clearly too premature to speculate towards this direction, but the found relations from the present and the previous sections may motivate further investigations.
Note that our findings have been obtained for a regulated free field. It is clear that this is far from dealing with strongly coupled theories with a large number of degrees of freedom with existent holographic duals. However, choosing the cost appropriately, the resulting similarities with holographic complexity proposals for the leading order UV divergence may provide useful insights. Namely, as noted above, such an accordance may suggest which cost function is intrinsic to the holographic complexity conjectures. Indeed, this was actually the main reason behind utilizing the F κ costs.
Despite the mentioned concordance, it will actually be necessary to extend the notion of complexity for strongly coupled theories with holographic duals, or at least, for some suitably excited states going beyond the free case. As mentioned, in the context of CFTs, interesting progress has been done in [91], also see [92]. Note also the recent studies in [82][83][84] introducing a notion of complexity which might shed light on this issue as well. Beyond these, circuit complexity has recently been studied for the TFD state in a free scalar field theory [74], also see [93] for similar considerations. The TFD state formed by entangling two copies of a CFT is especially important in holography, since, as already pointed out below figure 1, it is dual to an eternal BH in AdS [94].
Thus, it would be interesting to investigate the role of gate penalties in more holographic like setups in order to work out their possible holographic interpretation. On the other hand, as stressed before, such systems would particularly be interesting for making direct comparisons with recent studies arguing that holographic complexity might be nonlocal [44].
G. Distance dependent penalties
To find out what type of conditions are generically needed to be fulfilled by a field theory so that it has a bulk dual is important for better understanding the basic mechanisms behind holography. In previous studies, interesting conditions have been worked out by analyzing the corresponding boundary CFT, see e.g. [95,96].
More recently, it has been discussed that investigting the complexity growth for certain type of gauge theories may also shed light on such necessity conditions [97]. The motivation here is the conjectured criterion for the existence of a dual BH, namely, an upper bound in complexity with the scaling behavior C max ∼ e S where the exponent S denotes the corresponding entropy. Indeed, it has been shown that for both the classical as well as quantum cases, the complexity first grows and saturates afterwards at the maximum C max being in accordance with the second law of complexity [24,30]. More precisely, it has been found that the speed of growth increases when the time evolution is more nonlocal. It has been found that only the maximally nonlocal gauge theories satisfy the mentioned scaling behavior from above and thus possess the possibility of a gravity dual. Accordingly, the more nonlocal the interaction is the higher is the possibility for the existence of an holographic dual of the underlying theory.
On the other hand, as already mentioned, it has been discussed that complexity holographically defined via the CV and CA proposals may be nonlocal as well, but in a slightly different manner. To be more specific, it has been argued that any gate set for a CFT defining holographic complexity should necessarily contain bilocal gates acting at arbitraryly separated points [44]. Viewing the both conjectures as possible entries in the holographic dictionary, the presence of such bilocal gates constructing the corresponding field theory state may be taken as another necessity condition, similar to the one described above.
In the present paper, we use such kind of bilocal gates as well. However, we implement gate penalties for the entangling gates where those are uniformly weighted, thus, distinguishing between the different gate classes. It turns out that as long as we insert such distance independent penalties, we obtain a similar divergence structure in the complexity as seen in holographic computations, see (91). However, following the discussion in section V, this kind of agreement might not appear if certain gates are penalized according to the distance of points they act on. Even more, one may for instance think of entirely preventing such gates from participating in the construction of the circuit. In either cases, the characteristic power law scaling of the leading order UV divergence C UV,κ ∼ V δ 1−d , but of course the additional logarithmic factor as well, might be modified or even be rescinded completely.
Introducing locality in the way as described above may therefore lead to drastic differences between complexity in field theory and the holographic proposals. Taking into account that certain costs may be intrinsic to the holographic proposals due to the found similarities, such expectations may indeed favor the existence of bilocal gates in the gate set for holographic states as argued in [44]. Apart from that, the existence of bilocal gates in recent studies of circuit complexity for interacting [69] and time dependent nonlocal states [75] may explain the found behavior which is in line with the holographic predictions. It remains interesting to find out whether and how implementing distance dependent gate penalties would affect such agreements.
VI. CONCLUSION
We have investigated the effects of weighting certain classes of quantum gates constructing a quantum circuit for states in QFT. Our studies are based on the geometric approach to circuit complexity proposed by Nielsen and collaborators. Introducing such gate penalties may be motivated from various perspectives. For instance, they may incorporate the notion of locality in complexity when the latter is taken to be a physical attribute of a QFT. On the other hand, they may have important implications in the holographic dictionary. Closely related, also an application for tensor networks which are believed to provide a representation of a time slice of the AdS bulk space may motivate considering a penalized geometry in the space of unitaries.
In the present work, we have not introduced distance dependent weighting factors. Instead, we have worked with entangling gates which have been uniformly weighted. Of course, this does not introduce the notion of locality as described above. Nevertheless, an implementation of such kind has enabled us obtaining interesting insights which may motivate further investigations, in particular, by examining similar ideas for strongly coupled theories with holographic duals. Our results have generalized earlier findings dealing with the case of a pair of two coupled harmonic oscillators. More specifically, we have made the extension to the case of a regulated free field theory placed on the lattice for which the optimal circuit acts in form of a representation of GL(N, R).
Comparing both the penalized and the unpenalized results with each other, we have seen that assigning a higher cost to the entangling gates gives rise to a substantial increase in complexity. Furthermore, using the complexities for the F κ=2 cost function, we have found an interesting relation between these and the complexity for the unpenalized F κ=1 cost which is known to give rise to the Manhattan metric. From earlier studies we know that F κ=1 behaves as the most holographic like cost, since it yields the same leading order UV divergence as obtained via the holographic complexity proposals, including the additional logarithmic factor as appearing in the CA prediction. Referring to earlier results, the found relations have led us to speculate whether the change in the manifold of unitaries determined by a specific cost function may be compensated by introducing a penalty tensor for a different cost.
In addition, we have exhibited how the gate weighting modifies the leading order UV divergence in the complexity. In contrast to the unpenalized geometry, penalizing the entangling gates has introduced an additional degree of freedom, the latter corresponding to the mentioned weighting factor. We have shown that appropriately tuning this factor can eliminate the mentioned logarithmic contribution. Importantly, this procedure turned out to be independent from the choice for the reference frequency. Recall that in the penalized case, the latter is the only tunable control parameter. The dependence on any state parameter of this type is not explicit in the holographic proposals. Hence, it is of particular interest that the divergence can be modified by tweaking the weighting factor which specifies the metric in the space of unitaries instead of characterizing the reference state. In earlier studies it has been discussed that the geometric structure in the mentioned space may en-code important features of the holographic proposals. The main reason for such ideas lies in the found similarities in the divergence structure for some specific costs such as the F κ one.
Indeed, these similarities may suggest which cost is intrinsic to the holographic complexity conjectures. Hence, referring to our findings, the insertion of gate penalties in the space of unitaries makes such connections become even more interesting.
We have also compared our field theory results with the mentioned holographic proposals where we have given particular consideration to the predictions based on the CA conjecture.
FIG. 1 .
1The extended Penrose diagrams illustrate the relevant geometric objects for the CV (left) and CA (right) proposals for the two sided eternal AdS BH which is dual to the thermofield double state (TFD) formed by entangling two copies of a CFT. In the left diagram the blue solid curve represents the maximal bulk hypersurface extending to the AdS boundary, asymptoting to the indicated t L and t R time slices on which the CFT state is defined. It connects the latter through the ERB. The shaded blue region in the right diagram represents the WDW patch where the CFT state is again evaluated on the mentioned time slices on the left and right boundaries.
which also depends on some unspecified frequency ω 0 of the reference state. The found coincidence may therefore suggest an interesting relation between the notions of complexity defined in holography and QFT. IV. PENALIZED GEOMETRY I: COUPLED OSCILLATORS So far, both the scaling and entangling gates have been treated equally, i.e. each of them received the same cost when constructing the optimal circuit. As pointed out in the introduction, technically, for implementing the notion of locality one would need to introduce some weighting factors which vary the strength of the acting gates according to the separation between the corresponding points. Such a situation is sketched in figure 3. } { } { FIG. 3. A quantum circuit constructed by weighted gates. The color scaling for the gates shall indicate the difference in the gate weighting.
F 2
2, the normalization constant is simply given by the integral D(U ) = 1 0 ds g ijẋ i (s)ẋ j (s) where x x x(s) = [τ (s), ρ(s), θ(s), y(s)].
the choice of the basis. That means, choosing a specific metric G IJ in the original position basis would result in a different metric GĨJ in the normal mode basis. For the present setup,
withà R = ω 0 1,à T = diag[ω 0 , . . . ,ω N −1 ] andx = R N x where R N denotes the unitary matrix generalizing R 2 introduced in section III A 2. Such a factorization simplifies the optimal circuitŨ 0 = R N U 0 R † N in form of a diagonal matrix which amplifies each of the diagonal entries in the quadratic forms. As a consequence, the resulting geometry of the normal mode subspace remains flat, since the generators MĨ = RĨ J M J which construct the optimal circuit commute with one another. This property explains why the straight line solution yields the optimal geodesic for the regulated field on the lattice.
with the prediction (56) based on the CA proposal
By comparing both predictions with each other, we have related the weighting factor to quantities which are connected to the WDW patch in the bulk AdS spacetime. In view of these findings, we have finally commented on certain speculative expectations concerning the role of gate penalties in defining complexity for states in QFT and their possible implications in the holographic context. VII. ACKNOWLEDGEMENTS I would like to thank Arpan Bhattacharyya, Pawel Caputa and Tadashi Takayanagi for interesting conversations and helpful comments on this work. I also thank Mario Flory and Joan Simon for earlier conversations and Tadashi Takayanagi for the generous hospitality at the Yukawa Institute for Theoretical Physics at Kyoto University.
For more details we refer the interested reader to[63].3 The notation 1 stands for the identity matrix.
For instance, in cMERA, it is usually assumed that O(δω 0 ) ∼ 1.
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| []
|
[
"No-splitting property and boundaries of random groups",
"No-splitting property and boundaries of random groups",
"No-splitting property and boundaries of random groups",
"No-splitting property and boundaries of random groups"
]
| [
"François Dahmani [email protected]:[email protected] \nInstitut de Mathématiques de Toulouse\nUMR 5219\nUniversité de Toulouse et CNRS\n118 route de Narbonne31062, cedex 9ToulouseFrance\n",
"Vincent Guirardel \nInstitut de Mathématiques de Toulouse\nUMR 5219\nUniversité de Toulouse et CNRS\n118 route de Narbonne31062, cedex 9ToulouseFrance\n",
"Piotr Przytycki [email protected] \nInstitute of Mathematics, Polish Academy of Sciences\nIRMA, CNRS(UMR 7501)\nŚniadeckich 8, 00-956 WarsawPoland\n",
"François Dahmani [email protected]:[email protected] \nInstitut de Mathématiques de Toulouse\nUMR 5219\nUniversité de Toulouse et CNRS\n118 route de Narbonne31062, cedex 9ToulouseFrance\n",
"Vincent Guirardel \nInstitut de Mathématiques de Toulouse\nUMR 5219\nUniversité de Toulouse et CNRS\n118 route de Narbonne31062, cedex 9ToulouseFrance\n",
"Piotr Przytycki [email protected] \nInstitute of Mathematics, Polish Academy of Sciences\nIRMA, CNRS(UMR 7501)\nŚniadeckich 8, 00-956 WarsawPoland\n"
]
| [
"Institut de Mathématiques de Toulouse\nUMR 5219\nUniversité de Toulouse et CNRS\n118 route de Narbonne31062, cedex 9ToulouseFrance",
"Institut de Mathématiques de Toulouse\nUMR 5219\nUniversité de Toulouse et CNRS\n118 route de Narbonne31062, cedex 9ToulouseFrance",
"Institute of Mathematics, Polish Academy of Sciences\nIRMA, CNRS(UMR 7501)\nŚniadeckich 8, 00-956 WarsawPoland",
"Institut de Mathématiques de Toulouse\nUMR 5219\nUniversité de Toulouse et CNRS\n118 route de Narbonne31062, cedex 9ToulouseFrance",
"Institut de Mathématiques de Toulouse\nUMR 5219\nUniversité de Toulouse et CNRS\n118 route de Narbonne31062, cedex 9ToulouseFrance",
"Institute of Mathematics, Polish Academy of Sciences\nIRMA, CNRS(UMR 7501)\nŚniadeckich 8, 00-956 WarsawPoland"
]
| []
| We prove that random groups in the Gromov density model, at any density, satisfy property (FA), i.e. they do not act non-trivially on trees. This implies that their Gromov boundaries, defined at density less than 1 2 , are Menger curves.MSC: 20F65; 20F67; 20E08 | null | [
"https://arxiv.org/pdf/0904.3854v1.pdf"
]
| 15,116,750 | 0904.3854 | d77b5bd2e73ea9a1b659afe1e37d1d6158316490 |
No-splitting property and boundaries of random groups
24 Apr 2009
François Dahmani [email protected]:[email protected]
Institut de Mathématiques de Toulouse
UMR 5219
Université de Toulouse et CNRS
118 route de Narbonne31062, cedex 9ToulouseFrance
Vincent Guirardel
Institut de Mathématiques de Toulouse
UMR 5219
Université de Toulouse et CNRS
118 route de Narbonne31062, cedex 9ToulouseFrance
Piotr Przytycki [email protected]
Institute of Mathematics, Polish Academy of Sciences
IRMA, CNRS(UMR 7501)
Śniadeckich 8, 00-956 WarsawPoland
No-splitting property and boundaries of random groups
24 Apr 2009random groupproperty (FA)action on treessplittingword- hyperbolic groupGromov boundaryMenger curve
We prove that random groups in the Gromov density model, at any density, satisfy property (FA), i.e. they do not act non-trivially on trees. This implies that their Gromov boundaries, defined at density less than 1 2 , are Menger curves.MSC: 20F65; 20F67; 20E08
Introduction
The density model for random groups was introduced by Gromov. We adopt the following language from a survey by Ollivier.
Definition 1.1 ([Gro93, Section 9.B], [Oll05,Definition 7]). Let F n be the free group on n ≥ 2 generators s 1 , . . . , s n . For any integer L let R L ⊂ F n be the set of reduced words of length L in these generators.
Let 0 < d < 1. A random set of relators at density d, at length L is a ⌊(2n − 1) dL ⌋tuple of elements of R L , randomly picked among all elements of R L .
A random group at density d, at length L is the group G presented by S|R , where S = {s 1 , . . . , s n } and R is a random set of relators at density d, at length L.
Let I ⊂ N + . We say that a property of R, or of G, occurs with Ioverwhelming probability (shortly, w.I-o.p.) at density d if its probability of occurrence tends to 1 as L → ∞, for L ∈ I and fixed d. We omit writing "I-" if I = N + .
Note that the relators in R L need not be cyclically reduced. The case of another model is discussed in Section 4.
Gromov proved the following.
Theorem 1.2 ([Gro93, Section 9.B], [Oll04, Theorem 1]). A random group is with overwhelming probability (i) trivial or Z/2Z at density greater than 1 2 , (ii) word-hyperbolic, with aspherical presentation complex, at density less than 1 2 . Consequently (see e.g. [Oll05,Section I.3.b]) with overwhelming probability at density less than 1 2 a random group is torsion free, of cohomological dimension 2, and its Euler characteristic is positive.
We address the following question. At density less than 1 2 , what is the boundary at infinity of a random group G?
Since G is 2-dimensional, its boundary has topological dimension 1 (by [BM91,Corollary 1.4]). The list of possibilities for the boundary is therefore limited in view of the following. Theorem 1.3 ([KK00, Theorem 4]). Let G be a hyperbolic group which does not split over a finite or virtually cyclic subgroup, and suppose ∂ ∞ G is 1dimensional. Then one of the following holds:
(1) ∂ ∞ G is the Menger curve;
(2) ∂ ∞ G is the Sierpiński carpet;
(3) ∂ ∞ G is S 1 and G maps onto a Schwartz triangle group with finite kernel.
Moreover, Kapovich and Kleiner prove (see [KK00,Theorem 5(5)]) that ∂ ∞ G is a Sierpiński carpet only if (G; H 1 , . . . , H k ) is a 3-dimensional Poincaré duality pair, where H i are stabilizers of peripheral circles of the Sierpiński carpet. But since the groups H i are virtually Fuchsian [KK00, Theorem 5(2)], this implies that the Euler characteristic of G is negative. Hence w.o.p. this is not the case for a random group G at density less than 1 2 . Case (3) is also excluded, since G is w.o.p. torsion free.
In fact, at density d < 1 24 , it is known that w.o.p. the boundary of a random group is the Menger curve. Namely, w.o.p. at density d < 1 24 , a random group satisfies C ′ ( 1 12 ) small cancellation condition (see [Gro93,Section 9.B]). Champetier's theorem [Cha95,Theorem 4.18] states that this condition, together with the property that each word of length 12 is contained as a subword in one of the relators (this holds w.o.p. for random groups at any density), implies that the boundary is the Menger curve. But C ′ ( 1 12 ) small cancellation condition fails w.o.p. for a random group at density d > 1 24 (see [Gro93, Section 9.B]).
Nevertheless, from Żuk's theorem [Żuk03,Theorem 4] it follows (see [Oll05,Section I.3.g]) that a random group G at density greater than 1 3 satisfies w.o.p. Kazhdan's property (T). In particular w.o.p. G does not split and by Theorem 1.3 its boundary is the Menger curve.
We prove that this is the case at any density.
Theorem 1.4. Let 0 < d < 1 2 .
Then with overwhelming probability, the boundary of a random group at density d is the Menger curve.
Theorem 1.4 is a consequence of Theorem 1.3 together with the discussion after it, and the following, which is the main theorem of the article. Theorem 1.5. Let 0 < d < 1. Then with overwhelming probability, a random group at density d satisfies property (FA).
Recall [Ser77, Section I.6.1] that a group G satisfies property (FA) if each action of G on a simplicial tree has a global fixed point. When G is finitely generated, it satisfies property (FA) if and only if it does not admit an epimorphism onto Z and does not split as a free product with amalgamation (see [Ser77, Chapter I, Theorem 15]).
Here are additional corollaries to Theorem 1.5. Corollary 1.6. Let G be a random group at any density 0 < d < 1. Then with overwhelming probability, we have the following.
(1) Out(G) is finite.
(2) For any torsion free hyperbolic group Γ, Hom(G; Γ) is finite up to conjugacy.
Assertion (2) is equivalent to the fact that a random system of equations at density d has w.o.p. only finitely many conjugacy classes of solutions in any torsion-free hyperbolic group.
Assertion (1) is a well known (by experts) application of Bestvina-Paulin argument [Bes88,Pau91] and Rips theory [BF95,GLP94]. Assertion (2) is stronger. It follows from Sela's theory [Sel09] and the fact that property (FA) is inherited by quotients. More precisely, if Γ is hyperbolic, and Hom(G; Γ) is infinite modulo conjugacy, then Bestvina-Paulin argument provides an action of G on an R-tree T . This action factors through a group L (so called Γ-limit, possibly not finitely presented), such that L T is so-called superstable (see [Sel09, Lemma 1.3]). By [Sel97,Gui08], L has a non-trivial splitting, in contradiction with property (FA). This argument extends to the case where Γ is a toral relative hyperbolic group [Gro08], or where Γ has torsion.
We end the exposition with the following. Question 1.7. Is it true that, at any density, with overwhelming probability all finite index subgroups of a random group satisfy property (FA)?
If d > 1 3 then this question has positive answer, since Kazhdan's property (T) implies property (FA) for all finite index subgroups. But already for d < 1 5 , with overwhelming probability a random group does not have property (T) (see [OW05,Corollary 7.5]). Hence the answer to Question 1.7 cannot be only based on property (T). If we fix the index of the subgroups considered, Question 1.7 might have a positive answer justifiable in the spirit of our article. But we expect that the answer to Question 1.7 in general is much harder.
Our strategy of proof of Theorem 1.5 is the following. In the first part we describe a condition which guarantees property (FA). This part is inspired by an argument of Pride [Pri83], who gives examples of finitely presented groups of cohomological dimension 2 with property (FA). More precisely, we prove that certain finite collection of sets of words (the languages of what we call 1 3 -large basic automata) has the property that if we have at least one relator from each of those sets in the presentation of a group, then this group satisfies (FA). If we compute densities of those sets, they turn out to converge to 1, if the number of generators in the presentation converges to ∞. Hence this argument suffices to prove Corollary 2.8, which says that at each density d, if the number of generators is sufficiently large w.r.t. d, then a random group with overwhelming probability satisfies (FA).
In the second part we show that, to some extent, random groups with small number of generators have finite index subgroups which are quotients of random groups with large number of generators. This part of the proof is similar to the argument that random groups in the Gromov density model are quotients of the groups in the triangular model (see [Oll05,Section I.3.g]). We may then use the fact that property (FA) is inherited by quotients and by supergroups of finite index. This proves that with overwhelming probability random groups at any density and with any number of generators satisfy (FA) (Theorem 1.5).
If we require in Definition 1.1 that the relators are cyclically reduced, Theorem 1.5 is still valid, although the proof requires small changes. We decided to work mainly in the model in which we allow cyclically non-reduced words, since the proof in this setting is slightly simpler and easier to follow. However, we provide also the proof for the other model.
The article is organized as follows. In Section 2 we prove Proposition 2.6, which provides sufficient conditions for property (FA) and yields Corollary 2.8, which is a special case of Theorem 1.5. In Section 3 we use Proposition 2.6 to prove Theorem 1.5 in full generality. In Section 4 we give a proof of Theorem 1.5 in the model allowing only cyclically reduced relators.
The third author would like to thank Jacek Świątkowski for the introduction into the subject, and the people at the Institut de Mathématiques de Toulouse, where this work was carried out, for great atmosphere and hospitality.
Random groups with large number of generators
In this section we find conditions which guarantee property (FA) (Proposition 2.6).
We use the following language. An alphabet S is a finite set. Let S −1 denote the set of the formal inverses to the elements in S. Abbreviate S ± = S ∪ S −1 . Elements of S ± are called letters. A word over the alphabet S is a sequence of letters.
We fix, for the entire section, an alphabet S and we denote n = |S|. Below we define a restricted version of a classical notion of an automaton whose set of states is {∅} ∪ S ± .
Definition 2.1. A basic automaton (shortly a b-automaton) over an alphabet
S with transition data {σ s } is a pair (S, {σ s }), where {σ s } s∈{∅}∪S ± is a family of subsets of S ± .
The language of a b-automaton with transition data {σ s } is the set of all (nonempty) words over S beginning with a letter in σ ∅ and such that for any two consecutive letters ss ′ we have that s ′ ∈ σ s .
We say that a b-automaton is λ-large, for some λ ∈ (0, 1), if σ ∅ = ∅ and for each s ∈ S ± we have |σ s | ≥ λ2n.
Remark 2.2.
(i) There are exactly 2 2n(2n+1) b-automata (over the fixed alphabet S with n = |S|).
(ii) If a b-automaton is λ-large, then its language contains at least ⌈λ2n⌉ L−1 words of length L and at least (⌈λ2n⌉ − 1) L−1 reduced words of length L.
These estimates are useful in view of the following discussion. Definition 2.3. Let I ⊂ N + and let L be a set of reduced words over an alphabet S, containing for all but finitely many L ∈ I at least ck L words of length L, where c > 0, k > 1. Then we say that the I-growth rate of L is at least k.
Note that the value k is related to the classical notion of density d L of the set L by the relation k = (2n − 1) d L . A well known fact in random groups asserts that a random set of relators at density d intersects a fixed set of words of density greater than 1 − d. In the language of the growth rate this fact amounts to the following.
Lemma 2.4 ([Gro93, Section 9.A]). Let L be a set of reduced words over the alphabet S, of the I-growth rate at least k > (2n − 1) 1−d . Then with I-overwhelming probability, a random set of relators at density d intersects L.
We obtain the following corollary. Note that for fixed d and λ, its hypothesis is satisfied for sufficiently large n.
Corollary 2.5. If ⌈λ2n⌉ − 1 > 0 and (2n − 1) d ≥ 2
λ , then with overwhelming probability a random set of relators at density d intersects the languages of all λ-large b-automata over the alphabet S.
Proof. By Remark 2.2(ii) the N + -growth rate of the set of reduced words in the language L of a λ-large b-automaton is at least k = ⌈λ2n⌉ − 1. Since
2(⌈λ2n⌉ − 1) = (⌈λ2n⌉ − 2) + ⌈λ2n⌉ ≥ ⌈λ2n⌉ > λ(2n − 1),
we get that k > λ 2 (2n − 1) which is by hypothesis at least (2n − 1) 1−d . Hence by Lemma 2.4 a random set of relators at density d intersects L with overwhelming probability. By Remark 2.2(i), the number of b-automata over the fixed alphabet S depends only on n (and not on L), and we get the same conclusion for all languages simultaneously. Now we present the main result of this section.
Proposition 2.6. Let G be a group with presentation S|R such that R intersects the languages of all 1 3 -large b-automata over the alphabet S. Then G satisfies (FA).
Remark 2.7. The fact that the value 1 3 of the largeness constant in the hypothesis of Proposition 2.6 is greater than 1 4 will be surprisingly crucial in the proof of Theorem 1.5. In Section 4, we will need a modified version of Proposition 2.6, where the parameter 1 3 gets closer to 1 4 . Before we give the proof of Proposition 2.6, let us deduce the following consequence, which is a weak version of Theorem 1.5.
Corollary 2.8. Let 0 < d < 1 and let n satisfy (2n − 1) d ≥ 6. Then with overwhelming probability a random group with n generators at density d satisfies property (FA).
Proof. Let λ = 1 3 . Since n ≥ 2, we have ⌈λ2n⌉ − 1 ≥ ⌈ 4 3 ⌉ − 1 > 0. Moreover, (2n − 1) d ≥ 6 = 2
λ . By Corollary 2.5, with overwhelming probability R intersects the languages of all 1 3 -large b-automata over the alphabet S. Hence by Proposition 2.6 we have that w.o.p. G satisfies (FA).
The proof of Proposition 2.6 relies on the following lemmas.
Note that it is well known that random groups have trivial abelianization, hence they do not admit an epimorphism onto Z. However, it is convenient for us to include the proof of the following version of this assertion. Lemma 2.9. If G = S|R admits an epimorphism onto Z, then there is a 1 2 -large b-automaton over S with language disjoint with R.
Let us adopt the convention that if s ∈ S ± (resp. if w is a word over the alphabet S), then by s (resp. w) we denote the corresponding element in the group G.
Proof. If there is an epimorphism ψ : G → Z, we consider the following sets. Let
S + = {s ∈ S ± such that ψ(s) > 0}, S 0 = {s ∈ S ± such that ψ(s) = 0}.
Note that |S + ∪ S 0 | ≥ n and S + is nonempty. Consider the b-automaton A over the alphabet S with transition data σ ∅ = S + and σ s = S + ∪ S 0 , for s = ∅. Let w = s i 1 . . . s i L be a word in the language of A. Then ψ(w) = ψ(s i 1 ) + . . . + ψ(s i L ) > 0. In particular w = 0, hence w / ∈ R. On the other hand, the b-automaton A is 1 2 -large, as required. Before stating the next lemmas, we need the following discussion of free products with amalgamation. If G splits as A * C B, then we say that an element g ∈ G is written in a reduced form g = a 1 b 1 a 2 b 2 . . . a k b k w.r.t. this splitting, if a i ∈ A, b i ∈ B, and none of the terms a i , b i belong to C, with the exceptions that a 1 , b k are allowed to be trivial in G, and that if g ∈ C, we allow k = 1, a 1 = g, b 1 = e. The length of g ∈ G w.r.t. the splitting A * C B is the number of the terms a i , b i appearing in the reduced form of g. The length is well defined, i.e. it does not depend on the reduced form we choose (see [Ser77, Section I.1.2]). In particular, if g has a reduced form with at least 2 terms, then it is a nontrivial element of G.
If G = S|R splits as A * C B, we denote by A, B, C, D the sets of letters s ∈ S whose corresponding s lie, respectively, in A \ C, B \ C, C, and outside A ∪ B. (In particular, for s ∈ D we have that the length of s is at least 2.) We denote by α, β, γ, δ the cardinalities of these sets. We abbreviate A ± = A ∪ A −1 ⊂ S ± and similarly for B, C, D. Proof. For each s ′ ∈ S we consider the α+γ n -large b-automaton A s ′ over the alphabet S with transition data σ ∅ = {s ′ } and σ s = A ± ∪ C ± for s = ∅. We claim that at least for one s ′ ∈ S, the language of A s ′ is disjoint with R. Otherwise, for every s ′ ∈ S there is a relator r s ′ ∈ R contained in the language of A s ′ . Since r s ′ = 1, we obtain s ′ ∈ A. If this holds for every s ′ ∈ S, we obtain G ⊂ A, contradiction. The second construction is analogous.
Lemma 2.11. Assume that G = S|R splits as A * C B and the splitting is chosen so that the sum of the lengths of all generators s ∈ G, for s ∈ S, w.r.t this splitting is minimal. Then there is a 1 2n min{δ + β, δ + α}-large b-automaton with language disjoint with R.
We illustrate the idea of the proof by means of the following example. Assume that for all s ∈ S we have that s ∈ D and that the first term of the reduced form of s lies in A \ C, and its last term lies lies in B \ C. (One can check that in this case the minimality hypothesis is satisfied.) Consider the b-automaton with all σ s equal S ⊂ S ± . This b-automaton is 1 2 -large (its language consists of all "positive" words). Any word w = s i 1 . . . s i L in the language of this b-automaton has the following property. If we concatenate reduced forms of all s i l , we obtain w in a reduced form (there are no cancellations). Thus w has large length and cannot be trivial. Hence w / ∈ R. Thus we have constructed a 1 2 -large b-automaton, whose language does not intersect R.
Before we give the proof of Lemma 2.11, we need the following reformulation of the minimality assumption.
Sublemma 2.12. Assume that G = S|R splits as A * C B and the splitting is chosen so that the sum of the lengths of all generators s ∈ G, for s ∈ S, w.r.t this splitting is minimal. Then for each a ∈ A\C we have the following. There are at most β + δ letters s ∈ D ± with the property that the reduced form of the corresponding s ∈ G begins with a term a 1 ∈ A \ C, such that a −1 a 1 ∈ C. Similarly, for each b ∈ B \ C we have that there are at most α + δ letters s ∈ D ± with the property that the reduced form of the corresponding s ∈ G begins with a term b 1 ∈ B \ C, such that b −1 b 1 ∈ C.
Note that the first term of the reduced form (which in the notation of the definition of the reduced form is a 1 , or b 1 if a 1 = e) is determined uniquely modulo multiplying by an element from C on the right (see [Ser77, Section I.1.2]). We will write shortly modulo C instead of "modulo multiplying by an element from C on the right".
Proof. We prove the first assertion (the proof of the second one is analogous). Denote by F = F (a) the set of all letters in D ± , whose reduced forms begin with the term a modulo C. Let φ = |F |. Informally, φ is the number of "generators' extremities" whose reduced form cancels with a −1 or a depending on whether the "extremity" is the "beginning" or the "ending" of the generator.
Let us compute, how do lengths of generators change under conjugating by a (this is equivalent to computing the lengths of the generators with respect to the splitting obtained by conjugating the splitting A * C B by a). For s ∈ A ∪ C, the length of a −1 sa equals 1 which is the length of s. For s ∈ B, the length of a −1 sa equals 3, hence increases by 2 in comparison with the length of s, which is 1. For s ∈ D we study separately both "extremities" of s, which means that we study the first terms of reduced forms of s for all letters s ∈ D ± . Exactly φ of these first terms equal a modulo C. The other ones are either in A \ (C ∪ aC) or in B \ C, and we can denote their numbers, respectively, by φ ′ and 2δ−φ−φ ′ . This means that conjugating by a increases the sum of the lengths of the generators in D by −φ + (2δ − φ − φ ′ ).
To summarize, conjugating the splitting A * C B by a gives us a new splitting in which the sum of lengths of generators increases by at most 2β − φ + (2δ − φ − φ ′ ). By minimality hypothesis on the splitting A * C B, we get that this number is non-negative. Since φ ′ ≥ 0, this gives φ ≤ β + δ, as required.
Proof of Lemma 2.11. We define the following b-automaton A over the alphabet S.
Let σ ∅ = A ± ∪ B ± ∪ D ± . For s ∈ A ± let σ s be the union of B ± and those letters s ′ ∈ D ± for which the reduced form of s ′ does not begin with s −1 modulo C. Similarly, for s ∈ B ± let σ s be the union of A ± and those letters s ′ ∈ D ± for which the reduced form of s ′ does not begin with s −1 modulo C. Now suppose that s ∈ D ± , and that s ends, in the reduced form, with a term a k ∈ A \ C. Then let σ s be the union of B ± and the set of letters s ′ ∈ D ± for which the reduced form of s ′ does not begin with a −1 k modulo C. Analogously, if s ∈ D ± , and s ends, in the reduced form, with a term b k ∈ B \ C, then let σ s be the union of A ± and the set of letters s ′ ∈ D ± for which the reduced form of s ′ does not begin with b −1 k modulo C. Finally, for s ∈ C ± let σ s = S ± .
Step 1. The b-automaton A is 1 2n min{δ + β, δ + α}-large. If s ∈ A ± , then, by Sublemma 2.12, we have in D ± at least 2δ − (β + δ) elements of σ s . Adding elements of B ± , we get altogether at least δ + β elements of σ s . Analogously, if s ∈ B ± , there are at least δ + α elements in σ s . The computation is similar for s ∈ D ± .
Cases where s ∈ C ± or s = ∅ are obvious.
Step 2. The language of A is disjoint with R.
It is enough to prove that for any word w in the language of A, the element w ∈ G is nontrivial. This follows from the following stronger assertion.
Claim. For any word w of length k in the language of A, the length of w w.r.t. the splitting A * C B is at least k and the following holds. If we denote by s the last letter of w, we have that the last term of the reduced form of w equals the last term of the reduced form of s modulo multiplying by an element from C on the left.
We prove the claim by induction on k. For k = 1 this is obvious. Assume we have already proved the claim for k = l − 1 ≥ 1. Now let w be a word of length l in the language of A ending with the pair s ′ s. Then the word w ′ obtained from w by removing s from the end also lies in the language of A and we can apply to it the induction hypothesis. We get that w ′ has length at least l − 1 and its reduced form ends with the last term, say b ∈ B \ C, of the reduced form of s ′ .
If s ∈ D ± , then the length of s is at least 2 and by definition of σ s ′ the first term t of the reduced form of s does not equal b −1 modulo multiplying by an element from C on the right. Hence when we concatenate the reduced forms of w ′ and s and, if t ∈ B \ C, when we substitute the pair bt with a single term in B \ C, we obtain w in a reduced form of length at least l and whose last term equals the last term of the reduced form of s, as required.
By definition of σ s ′ , the only other possibility for s is that it lies in A ± , i.e. that s ∈ A \ C. Thus adjoining s at the end of the reduced form of w ′ gives a reduced form of w, which is of length at least l and whose last term equals the last (and only) term of the reduced form of s. This proves the claim for k = l, and ends the induction proof.
This ends the proof of Lemma 2.11.
We now collect all pieces of information.
Proof of Proposition 2.6. Since G is finitely generated, we need to prove that G does not admit an epimorphism onto Z and does not split as a free product with amalgamation. By Lemma 2.9, since 1 2 > 1 3 , we have that G does not admit an epimorphism onto Z. It remains to prove that G does not split as a free product with amalgamation. We prove this by contradiction.
Assume that G splits as A * C B and the splitting is chosen so that the sum of the lengths of all generators s ∈ G, for s ∈ S, w.r.t this splitting is minimal. By Lemma 2.11 we have that 1 2n min{δ + β, δ + α} < 1 3 . Assume, w.l.o.g., that β ≤ α. Then δ + β < 2n 3 . By Lemma 2.10 we have that α+γ n < 1 3 , i.e. that α + γ < n 3 . Adding up, we obtain δ + β + α + γ < n, contradiction.
Increasing the number of generators
In this section we demonstrate how to pass from a model where random groups have small number of generators to a model with large number of generators, where we can apply Proposition 2.6.
Recall that in our random model we denote the set of generators by S with n = |S|. The density of the random set of relators is denoted by d. For our argument we need to fix some natural number B which we will later require to be sufficiently large so that B √ 12 < (2n − 1) d (this estimate will be used only once at the end of the proof).
Let S denote the set of reduced words of length B over the alphabet S. The involution on S mapping each word to its inverse does not have fixed points. Thus we can partition S intoŜ andŜ −1 . We denoteŜ ± =Ŝ ∪Ŝ −1 (instead of S). Letn be the cardinality ofŜ, which equals n(2n − 1) B−1 .
Recall that L denotes the length of the random relators. Our proof is significantly simpler, if we consider only those L that are divisible by B. We will always distinguish this case and we recommend the reader to focus on this case during the first reading of the article. For 0 ≤ P < B let I P ⊂ N + denote the set of those L that can be written as L = BL + P .
Definition 3.1. Let r be a word of length L ∈ I 0 over the alphabet S. Divide the word r intoL blocks of length B. This determines a new wordr of lengthL over the alphabetŜ, which we call the word associated to r.
Definition 3.2. Given a set R of relators over S of equal length L ∈ I P , we define the associated groupĜ in the following way.
If P = 0, then we consider the setR of relators associated to relators in R. We defineĜ to be the group Ŝ |R .
If 1 ≤ P < B, then there is no natural way to associate relators over S to relators over S. We resolve this in the following way. Suppose that r 1 , r 2 ∈ R are two relators of length L over S, such that r 1 = q 1 v −1 and r 2 = vq 2 , for some word v over S of length P . We then obtain a (possibly nonreduced) word q 1 q 2 over S, of length 2BL, with the property that q 1 q 2 = 1 in G = S|R . To this word we can associate, as before, a relator overŜ, of length 2L, which we denote byr(r 1 , r 2 ). We denote byR the set of all r(r 1 , r 2 ) as above and we defineĜ = Ŝ |R . Proof. Indeed, we have a natural epimorphismĜ → H, where H is the subgroup of G generated by the words of length B over S. IfĜ satisfies (FA), then its quotient H also satisfies (FA). Moreover, since H ⊂ G is of finite (in fact at most 2n) index, we have by [Ser77, Section I.6.3.4] that G also satisfies (FA).
The idea behind the remaining part of the proof is the following. Assume that L = BL. Then each relator of lengthL over the alphabetŜ is associated to a relator of length L over the alphabet S. Consider the case of a model where we allow non-reduced relators. Then one can check that the density of a set R of relators over S equals the density of the setR of associated relators overŜ. This means thatĜ is a random group at density d in a model, where we allow non-reduced relators, with a large numbern = |Ŝ| of generators. Hence in this context, Corollary 2.8 implies, in view of Lemma 3.3, Theorem 1.5. However, we have decided not to work in this model, since it is less standard. For some reference on it, see [Oll04].
In our setting, we have to resolve the problem that some reduced words overŜ might be associated to non-reduced words over S. The key is the following.
Lemma 3.4. Let A be a λ-large b-automaton overŜ. Denote by L A the set of reduced words over the alphabet S, whose associated words lie in the language of A. Assume that λ ′ = λ − 1 2n > 0. Then the I 0 -growth rate of L A is at least
B √ λ ′ (2n − 1).
The outline of the proof is the following. We construct a b-automaton A red overŜ, whose language consists of elements of the language of A, which are associated to reduced words over S. In other words, the language of A red consists of words associated to elements of L A . Then we estimate from below the growth rate of the language of A red , hence the growth rate of L A , in terms of n, B, and λ.
Proof. Denote the transition data of A by {σŝ}. For anyŝ ∈Ŝ ± , let ρŝ ⊂Ŝ ± be the set ofŝ ′ such that the first letter ofŝ ′ interpreted as a word over the alphabet S is the inverse of the last letter ofŝ as a word over S. Observe that |ρŝ| = 1 2n 2n. Let A red be the b-automaton over the alphabetŜ with transition data σ red ∅ = σ ∅ and σ red s = σŝ \ ρŝ, forŝ ∈Ŝ ± . We have that |σ red s | ≥ |σŝ| − |ρŝ| ≥ λ2n − 1 2n 2n = λ ′ 2n. Hence A red is λ ′ -large. By Remark 2.2(ii), its language contains at least (⌈λ ′ 2n⌉)L −1 words of lengthL.
Observe that the language of A red has the following two properties. First, it is contained in the language of A. Second, for any word w of lengthL in the language of A red , if we substitute each letterŝ ∈ w with the corresponding word over the alphabet S, we obtain a reduced word of length BL over S.
This implies that the number of reduced words of length L = BL over the alphabet S, whose associated words lie in the language of A is bounded from below by
(⌈λ ′ 2n⌉)L −1 ≥ (λ ′ 2n) L B −1 ≥ c B √ λ ′ 2n L ,
for some c > 0. In other words, the I 0 -growth rate of L A is at least k = B √ λ ′ 2n. By definition ofn we have that
k > B λ ′ (2n − 1) B = B √ λ ′ (2n − 1).
This ends the proof of Lemma 3.4.
We obtain some corollaries for the case where P = 0. We recommend the reader only interested in the case of L divisible by B to skip them and proceed directly to the proof of Theorem 1.5.
Assume that 1 ≤ P < B. We keep the setting from Lemma 3.4. Let P P A (the "prefix set") denote the set of reduced words w over the alphabet S, whose length L lies in I P , such that the length BL = L − P prefix of w lies in L A .
Corollary 3.5. The I P -growth rate of P P A is at least k > B √ λ ′ (2n − 1).
Proof. This follows from the fact that any word of length BL in L A can be extended to a word of length L in P P A , and from Lemma 3.4. Consider the set of reduced words of length B over the alphabet S, which begin with s −1 , for some s ∈ S ± . View this set as a subset ρ s ofŜ ± . For anŷ s ∈Ŝ ± let Aŝ ,s be the b-automaton over the alphabetŜ with transition data equal to the transition data of A with the exception that we substitute σ ∅ with σŝ \ ρ s . This set is nonempty since |σŝ| ≥ λ2n and |ρ s | = 1 2n 2n. Hence Aŝ ,s is λ-large.
Let v be a reduced word of length P over the alphabet S ending with a letter s. Let Sŝ ,v A (the "suffix set") denote the set of words w over the alphabet S, whose length L lies in I P , such that the length P prefix of w equals v, and the length BL = L − P suffix of w lies in L Aŝ ,s . Since all words in the language of Aŝ ,s start with a letter outside ρ s , we have that all words in L Aŝ ,s start with a letter different from s −1 , and consequently all words in Sŝ ,v A are reduced. By Lemma 3.4 applied to Aŝ ,s we obtain immediately the following.
Corollary 3.6. The I P -growth rate of Sŝ ,v A is at least k > B √ λ ′ (2n − 1).
We are now ready for the following.
Proof of Theorem 1.5. Let R denote a random set of relators over S and G = S|R . We choose B sufficiently large so that B √ 12 < (2n − 1) d . Let G = Ŝ |R be the associated group. We want to verify, with overwhelming probability, the hypothesis of Proposition 2.6 forĜ, that the setR intersects the languages of all 1 3 -large b-automata over the alphabetŜ. By Remark 2.2(i) it is enough to prove this for a single 1 3 -large b-automaton A. As above, we denote by L A the set of reduced words over the alphabet S, whose associated words lie in the language of A. By Lemma 3.4 the I 0 -growth rate
of L A is at least k > B √ λ ′ (2n − 1), where (since n ≥ 2) we have λ ′ = 1 3 − 1 2n ≥ 1 3 − 1 4 = 1 12 .
(This is the point to which we refer in Remark 2.7.)
By the choice of B, the I 0 -growth rate of L A is at least 2n−1 B √ 12 > (2n − 1) 1−d . Thus we can apply Lemma 2.4 and we get that with I 0 -overwhelming probability there is a relator r ∈ R ∩ L A , hence there is a relatorr ∈R in the language of A, as required.
We have thus proved that the hypothesis of Proposition 2.6 for the group G is satisfied with I 0 -overwhelming probability. In that case, by Proposition 2.6,Ĝ satisfies property (FA). By Lemma 3.3 this implies that G satisfies (FA). This ends the proof of Theorem 1.5 under the assumption that we consider only L ∈ I 0 .
We now focus on the remaining case where L ∈ I P with P = 0. Since the number of the sets Sŝ ,v A (defined before Corollary 3.6) is finite and independent of L, we get by Corollaries 3.5 and 3.6, by the choice of B, and by Lemma 2.4, that with I P -overwhelming probability a random set of relators R contains an element in P P A and elements in Sŝ ,v A , for allŝ, v. In that case let r 1 ∈ R ∩ P P A . Denote by v −1 the word consisting of last P letters of r 1 and byŝ the letter inŜ ± associated to the length B block appearing before v −1 in r 1 . Let r 2 ∈ R ∩ Sŝ ,v A . Then the relatorr(r 1 , r 2 ) belongs to bothR and the language of the b-automaton A.
Hence, with I P -overwhelming probability, the hypothesis of Proposition 2.6 is satisfied and we can conclude thatĜ satisfies property (FA). By Lemma 3.3 this implies property (FA) for G and ends the proof of Theorem 1.5 in the case L ∈ I P for P = 0.
Cyclically reduced relators model
In this section we explain what changes need to be introduced in the proof of Theorem 1.5 in the case where we require random relators to be cyclically reduced.
Theorem 4.1. Let 0 < d < 1. Then in the model in which we allow only cyclically reduced relators, with overwhelming probability a random group at density d satisfies property (FA).
The problem is that the words in L A (see the proof of Theorem 1.5) might be not cyclically reduced. Moreover, we do not have a guarantee that for a given word of length B(L − 1) in L A , we can extend this word to any word of length BL in L A , which is cyclically reduced. This spoils the counting in Lemma 3.4.
To overcome this, we need to consider slightly wider class of automata than we have used so far, with richer languages. Shortly, we allow a different transition rule for the last letter, a rule that allows almost half of the letters to be put on the end. Definition 4.2. An enhanced basic automaton (shortly an e-automaton) over an alphabet S is a b-automaton over S together with a final transition data {τ s }, which is a family of sets τ s ⊂ S ± for all s ∈ S ± .
The language of an e-automaton is the set of all (nonempty) words in S beginning with a letter in σ ∅ and such that for any two consecutive letters ss ′ we have that s ′ ∈ σ s , if s ′ is not the last letter, and s ′ ∈ τ s , if s ′ is the last letter.
We say that an e-automaton is (λ, ε)-large, for some λ ∈ (0, 1), ε ∈ (0, 1 2 ), if its underlying b-automaton is λ-large, and for each s ∈ S ± we have |τ s | > ( 1 2 −ε)2n (the reason the latter condition is expressed in this way will become clear in Sublemma 4.5).
Remark 4.3. If λ > 1 2 − ε, then a λ-large b-automaton can be promoted to a (λ, ε)-large e-automaton with the same language by just putting τ s = σ s , for all s ∈ S ± .
First we show that using this notion we can save the counting argument from Lemma 3.4. For an e-automaton A e over the alphabetŜ, we denote by L cyc A e the set of cyclically reduced words over S, whose associated words lie in the language of A e . We have the following analogue (and consequence) of Lemma 3.4.
Lemma 4.4.
There is a constant ε 0 ∈ (0, 1 2 ) such that for any ε ∈ (0, ε 0 ] we have the following. Let B ≥ 3. Let A e be a (λ, ε)-large e-automaton with λ ′ = λ − 1 2n > 0. Then the I 0 -growth rate of L cyc A e is at least B √ λ ′ (2n − 1).
Proof. Let A be the underlying automaton of A e . By Lemma 3.4, the I 0growth rate of L A is at least B √ λ ′ (2n − 1). Hence, to prove Lemma 4.4 it suffices to show that any length B(L − 1) word in L A can be extended to a length BL word in L cyc A e . We need to compute the following. Sublemma 4.5. For any alphabet S with n = |S| ≥ 2, there is a constant ε 0 ∈ (0, 1 2 ) such that for any B ≥ 3 and any letters s, s ′ ∈ S ± , we have the following. Let 2n be the number of all reduced words of length B in S and let n ′ be the number of reduced words of length B over S which begin with s or end with s ′ . Thenn ′ ≤ ( 1 2 − ε 0 )2n. Actually, we can take the same ε 0 = 1 18 for any n. Proof. The number of reduced words which begin with s and the number of reduced words which end with s ′ equal 1 2n 2n, son ′ ≤ 1 n 2n. Hence if n ≥ 3, one can take ε 0 = 1 2 − 1 n . If n = 2, we need to estimate the number of words which simultaneously begin with s and end with s ′ . We have at least (2n − 1) B−3 (2n − 2) =n (2n−2) n(2n−1) 2 such words. Hencê
n ′ 2n ≤ 1 n − (2n − 2) 2n(2n − 1) 2 = 1 2 − 1 18 ,
Proof.
We take the b-automaton A described in the proof of Lemma 2.11, and promote it to an e-automaton A e by putting τ s = σ s ∪ A ± ∪ B ± ∪ C ± .
Step 1. A e is ( 1 2n min{δ + β, δ + α}, ε)-large. By step 1 in the proof of Lemma 2.11, we just need to estimate |τ s |. By Sublemma 2.12 we have that |σ s ∩ D ± | ≥ δ − β or |σ s ∩ D ± | ≥ δ − α. Since |τ s | = |σ s ∩ D ± | + |A ± ∪ B ± ∪ C ± |, we have that |τ s | ≥ δ − max{α, β} + 2(α + β + γ) ≥ α + β + γ + δ = n > 1 2 − ε 2n, as required.
Step 2. The language of A e is disjoint with the set of relators in R of length at least 3.
Otherwise, a word r ∈ R which is in the language of A e either lies in the language of the underlying b-automaton A (which is not possible by step 2 in the proof of Lemma 2.11) or is a concatenation of a word in the language of A and a letter in A ± ∪ B ± ∪ C ± . In the latter case, r is of the form ws, where w has length at least 2 (by the claim in the proof of Lemma 2.11) and (s) −1 has length 1. Contradiction. Now we are ready for the following.
Proof of Proposition 4.8. We argue as in the proof of Proposition 2.6. Again we need to prove that G does not admit an epimorphism onto Z and does not split as a free product with amalgamation.
By Lemma 2.9, in view of Remark 4.3 (note that 1 2 > 1 2 − ε), and since 1 2 > 1 4 + ε 2 we have that G does not admit an epimorphism onto Z. It remains to prove that G does not split as a free product with amalgamation. We prove this by contradiction.
Assume that G splits as A * C B and the splitting is chosen so that the sum of the lengths of all generators s ∈ G, for s ∈ S, w.r.t this splitting is minimal.
By Lemma 4.9 we obtain that 1 2n min{δ + β, δ + α} < 1 4 + ε 2 . Assume, w.l.o.g., that β ≤ α. Then δ+β n < 1 2 + ε, hence α + γ n = 1 − δ + β n > 1 2 − ε.
(We learned Pride's argument from an article by Delzant and Papasoglu [DP08, Theorem 4.1].)
Lemma 2 . 10 .
210If G = S|R splits as A * C B, then there are α+γ n -large and β+γ n -large b-automata with languages disjoint with R.
Lemma 3. 3 .
3IfĜ satisfies property (FA), then so does G = S|R .
Partially supported by ANR grant ANR-06-JCJC-0099-01. 2 Partially supported by MNiSW grant N201 012 32/0718, the Foundation for Polish Science, and ANR grant ZR58.
and we can take ε 0 = 1 18 . We return to the proof of Lemma 4.4. Let w be a length B(L − 1) word in L A , and letŵ be the word associated to it. Denote the last letter of w byŝ. Let s, s ′ denote the first and the last letter of w. The number of reduced words of length B over the alphabet S which start with (s ′ ) −1 or end with s −1 is at most ( 1 2 − ε)2n by Sublemma 4.5. Hence, by definition of largeness, there is a letter in τŝ (associated to some word v over S) without this property. Thus the word wv lies in L cyc A e . This ends the proof of Lemma 4.4. Now for 1 ≤ P < B let P P A be defined as in Section 3 and let (P P ) cyc A ⊂ P P A be the subset of cyclically reduced words. By Lemma 3.4 we obviously have the following.Corollary 4.6. The I P -growth rate of (P P ) cycA e be the subset of cyclically reduced words. The following analogue of Corollary 3.6 can be obtained from Sublemma 4.5 in the same way as Lemma 4.4.Now we prove the following version of Proposition 2.6. Let ε = min{ε 0 , 1 6 }, where ε 0 ∈ (0, 1 2 ) is the constant from Lemma 4.4 and Sublemma 4.5. Proposition 4.8. Let G be a group with presentation S|R such that the set of relators in R of length at least 3 intersects the languages of all ( 1 4 + ε 2 , ε)large e-automata over the alphabet S. Then G satisfies (FA).Before we give the proof, we need the following analogue of Lemma 2.11. Lemma 4.9. Assume that G splits as A * C B and the splitting is chosen so that the sum of the lengths of all generators s ∈ G, for s ∈ S, w.r.t this splitting is minimal. Then there is a ( 1 2n min{δ + β, δ + α}, ε)-large eautomaton with language disjoint with the set of relators in R of length at least 3.Proof of Theorem 4.1. Choose B such that B 2 ε < (2n − 1) d , where ε is the constant from Proposition 4.8. By Lemma 4.4 applied to λ = 1 4 + ε 2 , for any ( 1 4 + ε 2 , ε)-large e-automaton A, the I 0 -growth rate of L cyc A e is at leastwe have that this I 0 -growth rate satisfies k > (2n − 1) 1−d . By Lemma 2.4 (which is valid also in this model), w.I 0 -o.p. the setR, of relators associated to a random set R of cyclically reduced relators, intersects the languages of all ( 1 4 + ε 2 )-large eautomata over the alphabetŜ. Then, by Proposition 4.8, the associated groupĜ = Ŝ |R satisfies (FA) and thus, by Lemma 3.3, we have that G = S|R satisfies (FA).It remains to consider the case of L ∈ I P , for P = 0. This case follows from Corollaries 4.6 and 4.7. The argument is analogous to the one in the proof of Theorem 1.5, where we use Corollaries 3.5 and 3.6, and we omit it. This ends the proof of Theorem 4.1.
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[
"Acyclic polynomials of graphs",
"Acyclic polynomials of graphs"
]
| [
"Caroline Barton [email protected] \nDepartment of Mathematics and Statistics\nDepartment of Mathematics and Statistics Memorial\nDalhousie University Halifax\nUniversity of Newfoundland St. John's\nB3H 4R2, A1C 5S7Nova Scotia Canada, Newfoundland Canada\n",
"Jason I Brown [email protected] \nDepartment of Mathematics and Statistics\nDepartment of Mathematics and Statistics Memorial\nDalhousie University Halifax\nUniversity of Newfoundland St. John's\nB3H 4R2, A1C 5S7Nova Scotia Canada, Newfoundland Canada\n",
"David A Pike [email protected] \nDepartment of Mathematics and Statistics\nDepartment of Mathematics and Statistics Memorial\nDalhousie University Halifax\nUniversity of Newfoundland St. John's\nB3H 4R2, A1C 5S7Nova Scotia Canada, Newfoundland Canada\n"
]
| [
"Department of Mathematics and Statistics\nDepartment of Mathematics and Statistics Memorial\nDalhousie University Halifax\nUniversity of Newfoundland St. John's\nB3H 4R2, A1C 5S7Nova Scotia Canada, Newfoundland Canada",
"Department of Mathematics and Statistics\nDepartment of Mathematics and Statistics Memorial\nDalhousie University Halifax\nUniversity of Newfoundland St. John's\nB3H 4R2, A1C 5S7Nova Scotia Canada, Newfoundland Canada",
"Department of Mathematics and Statistics\nDepartment of Mathematics and Statistics Memorial\nDalhousie University Halifax\nUniversity of Newfoundland St. John's\nB3H 4R2, A1C 5S7Nova Scotia Canada, Newfoundland Canada"
]
| []
| For each nonnegative integer i, let a i be the number of i-subsets of V (G) that induce an acyclic subgraph of a given graph G. We define A(G, x) = i 0 a i x i (the generating function for a i ) to be the acyclic polynomial for G. After presenting some properties of these polynomials, we investigate the nature and location of their roots. | null | [
"https://arxiv.org/pdf/2011.01735v3.pdf"
]
| 226,237,210 | 2011.01735 | 92f0046fa0026fb8c433d395774841b587a8195d |
Acyclic polynomials of graphs
Jan 2022
Caroline Barton [email protected]
Department of Mathematics and Statistics
Department of Mathematics and Statistics Memorial
Dalhousie University Halifax
University of Newfoundland St. John's
B3H 4R2, A1C 5S7Nova Scotia Canada, Newfoundland Canada
Jason I Brown [email protected]
Department of Mathematics and Statistics
Department of Mathematics and Statistics Memorial
Dalhousie University Halifax
University of Newfoundland St. John's
B3H 4R2, A1C 5S7Nova Scotia Canada, Newfoundland Canada
David A Pike [email protected]
Department of Mathematics and Statistics
Department of Mathematics and Statistics Memorial
Dalhousie University Halifax
University of Newfoundland St. John's
B3H 4R2, A1C 5S7Nova Scotia Canada, Newfoundland Canada
Acyclic polynomials of graphs
Jan 2022acyclic graphsdecyclinggraph polynomialsacyclic polynomial AMS subject classifications: 05C3105C3805C30
For each nonnegative integer i, let a i be the number of i-subsets of V (G) that induce an acyclic subgraph of a given graph G. We define A(G, x) = i 0 a i x i (the generating function for a i ) to be the acyclic polynomial for G. After presenting some properties of these polynomials, we investigate the nature and location of their roots.
Introduction
A variety of graph polynomials have been studied, both for applied and theoretical considerations. Perhaps the best known family, chromatic polynomials, counts the number of proper colourings of graphs, and was introduced in the study of the Four Colour Problem, but has morphed over the years into an important field in its own right (see, for example, [28]). Reliability polynomials are a well-studied model of network robustness to probabilistic failures, and have attracted interest for both applied and pure perspectives (see [22]). Other graph polynomials have arisen as generating functions for subsets of vertex sets or edge sets of a graph, especially those having certain properties. Such polynomials allow for a finer investigation of the graph property in question, allowing an encoding of the minimum or minimum cardinality of such sets and the totality of the number of such sets. For example, independence polynomials are generating functions for independent sets of a graph, while domination polynomials enumerate dominating sets. For all these graph polynomials, work has varied from calculation and optimality to analytic properties and roots.
Given a (finite, undirected) graph G, we define the acyclic polynomial, A(G, x), of G to be the generating function for the number of acyclic subsets of V (G) (i.e., vertex subsets that induce acyclic subgraphs of G). Specifically,
A(G, x) = i 0 a i x i
where a i = |{S ⊆ V (G) : |S| = i and G[S] is acyclic}| is the number of acyclic vertex sets of cardinality i. 1 We remark that the collection A(G) of acyclic vertex subsets (that is, subsets of V (G) that induce an acyclic subgraph) of a graph G forms a (simplicial) complex, that is, it is closed under containment, and the acyclic polynomial of G is what is known as the face polynomial, or simply f -polynomial, of the complex (the (combinatorial) dimension of the complex is the maximum size of any set in the complex, and we shall say that G has acyclic dimension d if the acyclic complex of G, A(G), has dimension d).
In the remainder of this first section of the paper we discuss several aspects acyclic polynomials such as how they relate to decycling of graphs, how they do (or do not) encode certain graph invariants, the computational complexity of determining and/or evaluating them, and showing that they do not arise from evaluations of Tutte polynomials.
In Section 2 we focus on acyclic roots, namely the roots of acyclic polynomials. For acyclic polynomials of degree 3 we show that the roots all lie in the left half of the complex plane, and that arbitrarily large modulus is possible. For acyclic polynomials more generally (i.e., of any degree) we characterize those graphs for which the roots are all real as well as which rational numbers are acyclic roots. We also consider the maximum and minimum growth of the moduli of the roots, and we show that roots can exist in the right half-plane. In Section 3 we conclude with several open problems.
Acyclic Polynomials and Decycling
Any subset S of the vertex set V (G) of a graph G such that G − S (the subgraph of G that is induced by the vertices of V (G) \ S) is a forest (i.e., acyclic) is known as a decycling set or feedback vertex set for the graph G. For an introduction to the topic of decycling of graphs, see [10,12]. Determining whether an arbitrary graph G has a decycling set of a given cardinality is an NP-complete problem [39]. Nevertheless, calculating the size ∇(G) of a smallest decycling set for a graph G is a problem of practical interest as it has several natural applications, such as that of avoiding short-circuits and other forms of feedback in electrical networks [31]. For certain classes of graphs this problem is tractable, such as for complete graphs, complete bipartite graphs, cubic graphs [44,56], Cartesian products of cycles [50], and generalized Petersen graphs [32]. For instance, ∇(K n ) = n − 2 whenever n 2, ∇(K m,n ) = min{m, n} − 1 whenever min{m, n} 2, and ∇(C m ✷C n ) = ⌈ mn+2 3 ⌉ whenever m, n ∈ {3, 5, 6, 7, . . .}.
The complement V (G) \ S of a decycling set S in a graph G induces an acyclic subgraph of G. Let Υ(G) = |V (G)| − ∇(G) denote the acyclic dimension of G and observe that Υ(G) is also the degree of the acyclic polynomial A(G, x). Hence determining A(G, x) enables the decycling number ∇(G) to be found. This observation, and the potential that acyclic polynomials might address some problems involving decycling of graphs, provided our initial motivation for studying acyclic polynomials.
We note that the acyclic dimension of graphs has itself been an active area of research. At a conference in 1977, Albertson and Berman posed the still-open conjecture that Υ(G) |V (G)|/2 for any simple planar graph G [2]; a result of Borodin implies that Υ(G) 2|V (G)|/5 for every planar graph G [14]. For graphs in general (not necessarily planar) in 1987, Alon, Kahn and Seymour established a lower bound of Υ(G) v∈V (G) min{1, 2 1+deg(v) } and hence Υ(G) 2|V (G)|/(1 + ∆(G)), where ∆(G) denotes the maximum degree of G [4]. More recently it has been shown that Υ(G) (8|V (G)|−2|E(G)|− 2)/9 for every connected graph G [52] and also that Υ(G) 6|V (G)|/(2∆(G) + ω(G) + 2) where ω(G) denotes the order (i.e., the number of vertices) of a maximum clique in G [40].
Acyclic Polynomials and Graph Invariants
While determining the degree Υ(G) of A(G, x) is generally intractable, certainly some individual coefficients of the acyclic polynomial A(G, x) = i 0 a i x i of a given graph G can be easily be determined. Let σ i = σ i (G) denote the number of i-cycles in a graph G of order n and observe:
• Clearly a i n i for all i. Moreover, if a non-acyclic graph G has girth g then a i = n i for each i ∈ {0, 1, . . . , g − 1} and a g = n g − σ g < n g , whereas if G is acyclic then a i = n i for each i n. In particular, a 0 = 1, a 1 = n and a 2 = n 2 .
• As a i is the number of sets (or faces) of cardinality i in the complex A(G) and the complex clearly has dimension Υ(G) = n − ∇(G), we have a i > 0 for each i ∈ {0, 1, . . . , Υ(G)} and a i = 0 for each i > Υ(G).
• There are well known inequalities, known as Sperner bounds (see, for example, [53]), for the cardinalities of faces of each size in a complex, and these imply that
a i n − i + 1 i a i−1 .
We summarize these observations as follows: While the acyclic polynomial encodes the order, girth and decycling number of a graph, it does not encode some other basic invariants:
• The acyclic polynomial does not encode the number of edges. For example, any two acyclic graphs of order n share the same acyclic polynomial (1 + x) n , but they can have a different number of edges if n 2. Even for connected graphs that contain cycles, there are examples. Let n be an odd integer, let G 1 denote the graph obtained by adding a single pendant vertex to one of the vertices of K n−1 , and let G 2 denote the graph obtained by removing the n−1 2 edges of a maximum matching from the complete graph K n . Then
A(G 1 , x) = A(G 2 , x) = 1 + nx + n 2 x 2 + n−1 2 x 3 .
However, for all n 5, |E(G 1 )| = |E(G 2 )| and so as n varies we obtain an infinite family of pairs of distinct connected graphs that share the same acyclic polynomial but have different numbers of edges.
• The acyclic polynomial does not encode whether a graph is bipartite. To demonstrate this, it suffices to find two graphs, one bipartite and the other not bipartite, that share the same acyclic polynomial. Two such graphs are illustrated in Figure 1. These two graphs both have 1 + 7x + 21x 2 + 35x 3 + 32x 4 + 12x 5 as their acyclic polynomial.
• Whereas the acyclic polynomial does encode the girth of a graph, it can be observed from the graphs shown in Figure 1 that the circumference of a graph is not encoded (note that the graph on the right is Hamiltonian, but the one on the left is not Hamiltonian).
Theorem 1.2. The acyclic polynomial does not encode the number of edges, the bipartiteness or the circumference of a graph. ✷ Figure 1: A bipartite graph and a non-bipartite graph with identical acyclic polynomials
The Complexity of Calculating Acyclic Polynomials
For some families of graphs, the entire acyclic polynomial can be determined fairly easily. For instance, for complete graphs we have A(K n , x) = 1 + nx + n 2 x 2 and for cycles we have A(C n , x) = n−1 i=0 n i x i = (1 + x) n − x n . As a more interesting example, cographs are those graphs that can be built recursively from a single vertex via disjoint union and join operations. Equivalently, cographs can be characterized as those that do not contain any path P 4 of order 4 as an induced subgraph [24]. Given two graphs G and H, we denote their disjoint union as G ∪ H and their join as G + H (the join of two graphs G and H is formed from their disjoint union by adding in all edges between a vertex of G and a vertex of H).
It is elementary that
A(G ∪ H, x) = A(G, x) · A(H, x) (1.1)
as the union of acyclic vertex sets from disjoint graphs G and H is acyclic in G ∪ H. The effect of the join operation is more subtle, and it involves one other graph polynomial. For any graph G, let I(G, x) be the independence polynomial for G, that is, the generating function of the independent sets of G (a set of vertices is independent if it contains no edge). Independence polynomials have been well studied (see, for example, [34,42]). Their behaviour under disjoint union and join is quite straightforward:
I(G ∪ H, x) = (I(G, x)) · (I(H, x)) (1.2)
and
I(G + H, x) = I(G, x) + I(H, x) − 1. (1.3)
We now address how to calculate the acyclic polynomial of the join of two graphs from the acyclic and independence polynomials of each of the two graphs being joined. Theorem 1.3. For any graphs G and H, of orders n G and n H respectively,
A(G + H, x) = A(G, x) + A(H, x) + n G xI(H, x) + n H xI(G, x) + n 2 − 2n G n H − n G 2 − n H 2 x 2 − nx − 1,
where n = n G + n H . (iv) |S H | = 1 and S G consists of an independent set in G.
By using the notation [x t ] f (x) to denote the coefficient of the x t term of the polynomial f (x), then
[x i ] A(G + H, x) = [x i ] A(H, x) + |V (G)|[x i−1 ] I(H, x) + [x i ] A(G, x) + |V (H)|[x i−1 ] I(G, x)
when i 3. By summing over all i 3, it follows that
A(G + H, x) − n 2 x 2 − nx − 1 = A(H, x) − n H 2 x 2 − n H x − 1 + n G x (I(H, x) − n H x − 1) + A(G, x) − n G 2 x 2 − n G x − 1 + n H x (I(G, x) − n G x − 1) . Therefore A(G + H, x) = A(G, x) + A(H, x) + n G xI(H, x) + n H xI(G, x) + n 2 − 2n G n H − n G 2 − n H 2 x 2 − nx − 1.
✷ It was observed by one of our referees that the acyclic polynomial of a graph can be expressed in monadic second-order logic (MSOL). Specifically, note that
A(G, x) = A⊆V (G) : (V (G),E(G),A)|=ϕ(A) x |A|
where ϕ(A) is an MSOL expression indicating that A is an acyclic set in G. Equivalently, ϕ(A) indicates that the subgraph of G induced by A has no K 3 -minor. Section 1.3 of [26] shows how to formulate a logical expression to indicate that a graph has no H-minor, where H is any fixed simple loopless graph. As a consequence of being able to express A(G, x) in monadic second-order logic, it follows by a theorem due to Courcelle, Makowsky and Rotics [27] that the evaluation of A(G, x) for graphs of bounded clique-width is fixed parameter tractable. In the case of cographs (which have clique-width at most 2) we are able to do much better. Theorem 1.4. If G is a cograph, then A(G, x) can be calculated in linear time.
Proof. Recall that cographs are graphs that can be constructed through a collection of disjoint union and join operations. Recognizing that a graph is a cograph and determining the sequence of operations that comprise its construction can be accomplished in linear time [25]. For a cograph of order n, this sequence consists of n − 1 operations. For each disjoint union operation, the acyclic and independence polynomials of the graph arising from the operation can be calculated from the polynomials of the two ingredient graphs by using Equations (1.1) and (1.2). For each join operation, the acyclic and independence polynomials of the graph arising from the operation can be calculated by using Equation (1.3) and Theorem 1.3. For either operation these calculations take constant time. ✷
Since it is, in general, NP-hard to determine the decycling number ∇(G) of a graph G, it is likewise intractable to determine the acyclic polynomial A(G, x) for a general graph. However, we can ask whether the task of evaluating the acyclic polynomial for certain choices of x (without actually determining the polynomial itself) can be performed efficiently. This type of question has been investigated for other graph polynomials; for the chromatic polynomial see [33,45], and for the independence polynomial see [19]. For us to proceed, observe that Theorem 1.3 enables the following interesting and important connection between acyclic and independence polynomials for graphs to be derived.
Corollary 1.5. For any graph G, A(G + K 1 , x) = A(G, x) + xI(G, x).
In particular note that an oracle for determining (or evaluating) acyclic polynomials therefore enables the calculation (or evaluation) of independence polynomials. It now follows that the only acyclic polynomial evaluation that is tractable is for x = 0, for which A(G, 0) = 1 for every graph G. Theorem 1.6. Evaluating the acyclic polynomial for an arbitrary graph G and nonzero x is intractable.
Proof. It is known that evaluating I(G, x) is #P-hard for each x ∈ C \ {0} [19]. It therefore follows from Corollary 1.5 that evaluating the acyclic polynomial for an arbitrary graph G and nonzero x is also intractable. ✷ It follows immediately that determining A(G, 1), the total number of induced forests of a graph G, is #P-hard.
The Acyclic Polynomial is not an Evaluation of the Tutte Polynomial
If one concerns oneself with acyclic edge sets rather than vertex sets, then the corresponding complex is in fact a well known matroid, the graphic matroid of graph G, and its f -polynomial is a simple evaluation of G's Tutte polynomial. The acyclic polynomials we propose here do not arise as evaluations of Tutte polynomials, as can be seen by the following argument (which was provided to us by an anonymous referee). The most general edge elimination invariant (the ξ polynomial) was introduced in [7, 8] as a generalization of the Tutte and matching polynomials. In the definition below, −e, /e and †e denote the edge deletion, contraction and extraction of e from G (the extraction is the graph formed from G by removing the endpoints of e and all incident edges), and G ∪ H is the disjoint union of G and H. Definition 1.7. Let F be a graph parameter with values in a ring R. F is an EE-invariant if there exist α, β, γ ∈ R such that (ii) Every EE-invariant is a substitution instance of ξ(G; x, y, z) multiplied by some factor s(G) which only depends on the number of vertices, edges and connected components of G.
F (G) = F (G −e ) + αF (G /e ) + βF (G †e ),
(iii) Both the matching polynomial and the Tutte polynomial are EE-invariants given by
T (G; x, y) = (x − 1) −c(E(G)) (y − 1) −|V (G)| ξ(G; (x − 1)(y − 1), y − 1, 0), and M(G; w 1 , w 2 ) = ξ(G; w 1 , 0, w 2 ).
Theorem 1.9. The acyclic polynomial A(G, x) is not an EE-invariant, and hence not an evaluation (i.e., substitution instance) of the Tutte polynomial (or the matching polynomial).
Proof. Assume, to reach a contradiction, that the acyclic polynomial A(G, x) is an EEinvariant for some α, β, γ. For the path P n of order n with e an edge adjacent to a leaf, we have that
(P n ) −e = K 1 ∪ P n−1 , (P n ) /e = P n−1 , (P n ) †e = P n−2 ,
while for the cycle C n of order n with e any edge of the cycle,
(C n ) −e = P n , (C n ) /e = C n−1 , (C n ) †e = P n−2 . From A(P n , x) = (1 + x) n = (1 + x)(1 + x) n−1 + α(1 + x) n−1 + β(1 + x) n−2
we find that α = β = 0. However, then
A(C n , x) = (1 + x) n − x n = (1 + x) n + α((1 + x) n−1 − x n−1 ) + β(1 + x) n−2 ,
from which it follows that x n = 0, a contradiction. Thus A(G, x) is not an EE-invariant, and hence not an evaluation of the Tutte polynomial (or the matching polynomial).
Acyclic Roots
The roots of many graph polynomials have received considerable attention:
• For chromatic polynomials, the chromatic number of a graph G is simply the least positive integer that is not a root of its polynomial, and the infamous Four Colour Theorem can be stated as: 4 is never a root of a chromatic polynomial of a planar graph. There are many, many results on chromatic roots, that is, the roots of chromatic polynomials (see, for example, [28,), including that chromatic roots are dense in the whole complex plane, the closure of the real chromatic roots is [ 32 27 , ∞), and that the chromatic roots of a graph with maximum degree ∆ are within the disk {z ∈ C : |z| 8∆}.
• Given a graph G where vertices are always operational but each edge is independently operational with probability p, the all-terminal reliability (polynomial) of G is the probability that all vertices can communicate, that is, the operational edges contain a spanning tree of the graph. The roots of all-terminal reliability polynomials were studied first in [17], where it was conjectured that they all lay in the unit disk centered at z = 1 (the real roots were shown to be in the disk, and the closure of the roots contained the disk). While the conjecture remained open for approximately 10 years, it was finally shown [51] to be false, but only by the slimmest of margins (the furthest a root is known to be away from z = 1 is approximately 1.14 [20]). It is still unknown whether roots of all-terminal reliability are unbounded.
• The independence polynomial I(G, x) of a graph G is the generating polynomial i k x k for the number of independent sets i k of each cardinality k. There are many interesting results about the roots of independence polynomials (cf. [42]), including Chudnovsky and Seymour's beautiful paper [21] showing that the roots of independence polynomials of claw-free graphs (i.e., graphs that do not contain an induced star K 1,3 ) are real (and hence the coefficients are unimodal, that is, nondecreasing, then nonincreasing).
More generally, the roots of polynomials are interesting for a number of reasons. The nature and location of roots can also show relationships among the coefficients, and the regions that (or do not) contain roots can show interesting structure. For example, if a polynomial f with positive coefficients has all real roots, then the sequence of coefficients of f is unimodal, and extension shows the same is true provided all of the roots of f lie in the sector {z ∈ C : 2π/3 argz 2π/3} (see [16]). Michelen and Sahasrabudhe [49] prove that if a polynomial h is the probability generating function of a random variable X ∈ {0, 1, . . . , n} with sufficiently large standard deviation, and the polynomial has no roots "close" to 1, then X is approximately normally distributed. As yet another example, Barvinok [9] built deterministic quasi-polynomial-time approximation algorithms for approximating polynomial functions using zero-free regions of the complex plane. All of the above have been applied to various problems on graph polynomials. (We remark that Marden's book [48] on the geometry of polynomials is an excellent reference on classical results that we rely on for the rest of this section. See also [6,35,36].) As noted in [47], the location of the roots of graph polynomials can be indicative of various properties of the underlying graph. Here we investigate acyclic roots, their location and moduli, and also the properties of real roots. Our main results are:
(i) If G is a graph of order n 4 and A(G, x) is of degree 3, then A(G, x) has one real root and two nonreal roots s and s ′ (Theorem 2.6). Furthermore, both s and s ′ both tend to zero as n goes to infinity (Theorem 2.10).
(ii) The roots of an acyclic polynomial of degree 3 are all in the left half of the complex plane (Theorem 2.8).
(iii) There exist graphs with degree 3 acyclic polynomials having real roots of arbitrarily large modulus (Theorem 2.13).
(iv) The roots of A(G, x) (of any degree) are all real if and only if G is a forest (Theorem 2.16).
(v) There are graphs G of arbitrarily large order n which have a real acyclic root in [a(n), b(n)] and no acyclic root of modulus larger than |a(n)|, where a(n) = − n 2 2 − 5n 2 +17 and b(n) = − n 2 2 − 5n 2 + 18 (Theorem 2.19). (vi) There are arbitrarily large graphs with acyclic roots which have a positive real part (Theorem 2.24).
We begin our investigation of acyclic roots by looking at the roots of acyclic polynomials of small degree.
Roots of Acyclic Polynomials of Degree 3
The only graph whose acyclic polynomial is of degree 1 is K 1 , with polynomial 1 + x, with a root at x = −1. Acyclic polynomials of degree 2 are precisely those of K 2 (i.e., the complement of the complete graph of order 2) and complete graphs of order n 2, and have the form 1 + nx + n 2 x 2 . The acyclic polynomials of degree 2 have roots
−1 n − 1 ± √ n 2 − 2n n(n − 1) i.
The modulus of each such root is 2 n 2 −n , which is decreasing, bounded by 1 (and tends to 0 as n → ∞).
The roots so far are rather uninteresting, but the situation changes dramatically when we move to acyclic dimension 3 (see Figure 2). This subsection is devoted to such an investigation. We begin by characterizing when a graph has acyclic dimension 3.
Theorem 2.1. Let G be a graph of order n. Then A(G, x) has degree three if and only if one of the following holds: 2. G is connected and G is the disjoint union of at least two stars, at least one of which has an edge.
Proof. We begin by showing the reverse direction. First, note that the degree of the acyclic polynomial of a graph is the sum of the degrees of the acyclic polynomials of its components.
Consider the cases where G is disconnected. If G is K 3 , then it is clear that A(G, x) = (x + 1) 3 , so A(G, x) has degree three. If G = K 1 ∪ K n−1 with n 3, then, because A(K 1 , x)
has degree one and A(K n−1 , x) has degree two, A(G, x) has degree three. Now suppose G is connected and G is the disjoint union of at least two stars, at least one of which has an edge. Then G has two vertices that are not adjacent and G has at least three vertices. Since any set of three vertices, two of which are not adjacent, induces an acyclic subgraph of G then the degree of A(G, x) is at least three. It remains to show that any subset S ⊆ V (G) with four vertices contains a cycle. We will show this by considering cases based on how many vertices in S belong to the same component of G.
Suppose S contains at least three vertices from the same component of G. Then two of these vertices, call them u and v, must be leaves and are therefore not adjacent in G. Furthermore, both u and v are joined in G to the same vertex and to no other vertex. Thus, S must contain a vertex that is independent of both u and v in G (i.e., another leaf from the same component or a vertex from another component of G) as shown in Figure 3. Since S contains three vertices that are independent in G, S contains three vertices that form a 3-cycle G. If S does not contain three vertices from one component in G, but does contain exactly two vertices, u and v, from one component, then S must contain two other vertices to which neither u nor v are joined in G. Thus, these four vertices will contain a 4-cycle in G, as shown in Figure 4. Lastly, if S contains no more than one vertex from any given component of G, then S contains four vertices that are all independent in G. These four vertices form K 4 in G, which is clearly not acyclic. Thus, S is not an acyclic subset of G.
It follows that any subset of V (G) with at least four vertices must contain a cycle. Therefore, A(G, x) has degree three.
To show the forward direction, assume G has acyclic dimension at least three (so G has order n 3). We divide the proof into two cases based on the connectivity of G.
For the first case, suppose G is disconnected. Recall that the acyclic dimension of G equals the degree of A(G, x), and since the acyclic dimension of G is the sum of its components' acyclic dimensions, G cannot have more than three components. If G has three components, then the acyclic polynomial of each component has degree one. That is, G = K 3 . Otherwise, G has two components, with one component G 1 having an acyclic polynomial of degree one and the other G 2 degree two. It follows from our previous work that G 1 = K 1 and G 2 is complete of order at least 2, that is,
G 2 = K n−1 , so G = K 1 ∪ K n−1 .
Now for the second case, suppose G is connected. Notice that G cannot contain an induced P 4 for otherwise G would have acyclic dimension at least four. Hence G is a cograph. Since G is connected on n 3 vertices, G must be the join of k 2 smaller cographs, F 1 , . . . , F k . Thus G is the disjoint union of H 1 = F 1 , . . . , H k = F k , and without loss of generality we may assume that H 1 , . . . , H k are connected. Also notice that H 1 , . . . , H k must be cographs since P 4 is self-complementary. Suppose -to reach contradiction -that some H i is not a star.
If H i has a universal vertex u, then two vertices, v, w, both joined to u, must share an edge, as shown in Figure 5, as H i is not a star. So, u, v, w are three independent vertices in G. Thus taking u, v, w with any other vertex in G gives an acyclic subset with four vertices (G must contain another vertex as k 2). It follows that A(G, x) has degree greater or equal four, contradicting our assumption that A(G, x) has degree three.
If, on the other hand, H i does not have a universal vertex then H i contains at least four vertices. Let u be one of these vertices. Then u must be adjacent in H i to another vertex, v. A third vertex, w, must also be adjacent in H i to one of these two vertices, say v. Note that w cannot be adjacent to both u and v, as otherwise u, v, w will be an independent set of three vertices in G, which will form an acyclic subset of size four with any other vertex in u x y w v Figure 5: If H i has a universal vertex u, but is not a star, then two vertices not including u, for example v and w, must be joined.
G. Without loss of generality, w is joined to v (but not u).
Since v is not a universal vertex, we know that there exists another vertex x in H i that is not joined to v in H i . If x is joined to either u or w in G, then it is easy to see that {u, v, w, x} is an acyclic set in G (as the subgraph of G induced by {u, v, w, x} properly contains a P 4 as a subgraph), and we have a contradiction. Thus x is joined to both u and w (but not v). However, in this case {u, v, w, x} is again an acyclic set in G since the subgraph induced by {u, v, w, x} is K 2 ∪ K 2 (see Figure 6a). Hence, x is not adjacent to u, v, or w.
Consider a shortest path P in H i from x to the set {u, v, w}. Let y be the last vertex on this path that is not in {u, v, w}. By the same argument as for x, y is not joined to u or w in H i , so y is joined to v in H i . In particular, y = x, so there is a previous vertex y ′ on path P joined to y (y ′ could be x). However, then in H i , {u, v, y, y ′ } properly contains a P 4 as a subgraph, as shown in Figure 6b, and hence is acyclic in G, yielding a contradiction. Thus H i must be a star. It follows that G is the disjoint union of stars. Furthermore, one of these stars must have an edge, as otherwise, G will be the complete graph whose acyclic polynomial has degree two. ✷
As it is now straightforward to recognize graphs whose acyclic polynomials have degree three, we also wish to be able to find the acyclic polynomial for any one of these graphs. We already know a 0 , a 1 , and a 2 , so this task is equivalent to finding an expression for a 3 .
Theorem 2.2. Let G be a connected graph with order n and suppose A(G, x) has degree three so that the complement of G is the disjoint union of k stars H 1 , H 2 , . . . , H k . Let e i be the number of edges in H i . Then a 3 is given by
n − 3 2 (n − k) − 1 2 k i=1 e 2 i .
Proof. Suppose S is an acyclic subset of G with three vertices. Then S must contain two vertices that are not joined. This corresponds to two vertices that are joined in G. Since G is the disjoint union of k stars, G has n − k edges, i.e., k i=1 e i = n − k. Thus, there are n − k ways to choose two vertices joined by an edge in G.
Given
(n − 2)(n − k) − k i=1 e i (e i − 1) 2
ways to form an acyclic subset with three vertices. We then obtain
a 3 = (n − 2)(n − k) − k i=1 e 2 i − e i 2 = (n − 2)(n − k) − 1 2 k i=1 e 2 i + n − k 2 = n − 3 2 (n − k) − 1 2 k i=1 e 2 i ✷ Of course a 3 n 3
where n is the order of the graph. However, there is a much better upper bound on a 3 for graphs with acyclic dimension 3. To find such a bound, we first make the following observation:
k i=1 e i = n − k, then n − 3 2 (n − k) − 1 2 k i=1 e 2 i
is maximized when e 1 = e 2 = · · · = e k = n−k k .
Proof. Let u = (e 1 , . . . , e k ) and let v be the k-dimensional vector (1, . . . , 1). The well-known Cauchy-Schwartz inequality states that ||u||||v|| u · v with equality if and only if u = av for some scalar a. So,
k i=1 e 2 i = ||u|| 2 (u · v) 2 ||v|| 2 = ( k i=1 e i ) 2 k .
It follows that k i=1 e 2 i is minimized when u is a scalar multiple of v, that is, when each e i = n−k k . The result follows. ✷
This result allows us to find an upper bound on a 3 for graphs with acyclic dimension 3 that is an order of magnitude smaller than n 3 . Theorem 2.4. The leading coefficient a 3 of an acyclic polynomial for a graph of order n with acyclic dimension three is at most
f (n) = n 2 − n 2 √ 2n − 2 − n √ 2n − 2 2 − n 2 + n √ 2n − 2 Proof. Let n be fixed. First, if G is disconnected, then either (i) G = K 3 with a 3 = 1 3/2 = f (3)
, or (ii) G is the disjoint union of a single vertex and K n−1 , in which case
a 3 = n−1 2 = (n−1)(n−2) 2 .
In the second case, a 3 < f (n) is equivalent to
(n 2 + 2n − 2) √ 2n − 2 − 4n 2 + 4n 2 √ 2n − 2 > 0.
Clearly the denominator is positive, so we need only check that the numerator is positive as well. For n > 9, √ 2n − 2 > 4, and the numerator is greater than (n 2 + 2n − 2) · 4 − 4n 2 + 4n, which is positive. For 3 n 8, a simple calculation will verify that the numerator is again positive. In all cases, we find that (n 2 +2n−2)
√ 2n−2−4n 2 +4n 2 √ 2n−2
is positive so a 3 < f (n). We now assume that G is connected. From Lemma 2.3, we know that for a given k, a 3 is maximized when each e i = n−k k . Thus,
a 3 n − 3 2 (n − k) − 1 2 k i=1 n − k k 2 = n − 3 2 (n − k) − (n − k) 2 2k = k(1 − n) − n 2 2k + n 2 − 1 2 n. Let B(x) = x(1 − n) − n 2 2x + n 2 − 1 2 n for 1 x n − 1. At the maximum value of B, 0 = dB dx = 1 − n + n 2 2x 2 .
So, x = n √ 2n−2 is a critical point of B(x). The second derivative of B is
d 2 B dx 2 = − n 2 x 3 < 0.
Thus, x = n √ 2n−2 is in fact the absolute maximum of B. Therefore,
a 3 B n √ 2n − 2 = n 2 − n 2 √ 2n − 2 − n √ 2n − 2 2 − n 2 + n √ 2n − 2 ✷
(We remark that with more work, one can show that this upper bound is never achieved.) We shall need as well a lower bound for a 3 , and in this case the bound is tight.
Theorem 2.5. Suppose A(G, x) = a 3 x 3 + n(n−1) 2 x 2 + nx + 1 is an acyclic polynomial with degree three. Then a 3 n − 2, with equality if G = K n − e, the complete graph of order n 3 minus an edge.
Proof. Let G be an arbitrary graph of order n 3 with acyclic dimension three. Consider as well H = K n − e. The complement of H is the disjoint union of K 2 and n − 2 independent vertices. So, following from Theorem 2.1, A(H, x) has degree three. From Theorem 2.2, the coefficient of x 3 in A(H, x) is n − 2. Now clearly G is a spanning subgraph of H. Removing an edge does not change an acyclic subset to a cyclic subset, so any acyclic subset of H is also an acyclic subset of G. Thus, G has at least as many acyclic subsets has H. Therefore, a 3 n − 2. ✷
We are now ready to explore the roots of acyclic polynomials of degree three. We know at least one root of A(G, x) lies on the real axis and, for n > 3, there is only one such root. Theorem 2.6. If G has n > 3 vertices and has acyclic dimension 3, then A(G, x) has one real acyclic root and two nonreal acyclic roots.
Proof. For a general cubic, ax 3 + bx 2 + cx + d, the discriminant is defined to be ∆ = 18abcd − 4b 3 d + b 2 c 2 − 4ac 3 − 27a 2 d 2 . If ∆ > 0, then the cubic has three distinct real roots, and if ∆ < 0, the cubic has one real root and two nonreal roots (see, for example, [37]). We will show that the discriminant of A(G, x) is negative, from which the result follows.
Let
A(G, x) = ax 3 + n(n − 1) 2 x 2 + nx + 1 (2.1)
with fixed n > 3. Then the discriminant of A(G, x) can be calculated to be
∆(a) = 18a n(n − 1) 2 n − 4 n(n − 1) 2 3 + n(n − 1) 2 2 n 2 − 4an 3 − 27a 2 = − n 6 4 + n 5 − 5n 4 4 + 5a + 1 2 n 3 − 9an 2 − 27a 2 .
To show that ∆(a) is negative, observe that its derivative with respect to a, ∆ ′ (a) = 5n 3 − 9n 2 − 54a, is negative when a > 5 54 n 3 − 1 6 n 2 and positive when a < 5 54 n 3 − 1 6 n 2 . Thus ∆ is increasing to the left of 5 54 n 3 − 1 6 n 2 and decreasing to the right. Furthermore, at a = 5 54 n 3 − 1 6 n 2 , ∆ = − n 6 4 + n 5 − 5n 4 4 + 5 5 54
n 3 − 1 6 n 2 + 1 2 n 3 − 9 5 54 n 3 − 1 6 n 2 n 2 − 27 5 54 n 3 − 1 6 n 2 2 = − 1 54 n 6 + 1 6 n 5 − 1 2 n 4 + 1 2 n 3 .
For n 9, ∆ − 1 6 n 5 + 1 6 n 5 − 1 2 n 4 + 1 2 n 3 < 0. A quick check verifies that ∆ is negative for n = 4, . . . , 8 as well. Since a = 5 54 n 3 − 1 6 n 2 is a maximum, it follows that ∆(a) is negative for all a and for all n 4. As the discriminant of A(G, x) is negative, we conclude that A(G, x) has one real root and two nonreal roots. ✷ Figure 2 seems to suggest that all of the acyclic roots (real or otherwise) are in the left half-plane. Of course the real acyclic root of any acyclic polynomial is negative (as the polynomial has positive coefficients and hence is positive on the positive real axis), but what about the location of the two nonreal roots for acyclic polynomials of graphs of acyclic dimension three? A polynomial is said to be stable if all its roots lie in the left half-plane. To prove the stability of acyclic polynomials of degree three, so we will use the Hermite-Biehler Theorem. To state this theorem, we first define a few terms. A polynomial is real if each of its coefficients is real. Given a real polynomial f (x) = b i x i , then even and odd polynomials f e and f o , respectively, are given by f e (x) = b 2i x i and f o (x) = b 2i+1 x i (so that f (x) = f e (x 2 ) + xf o (x 2 )). A real polynomial is standard if and only if it is identically 0 or has positive leading coefficient. Finally, if α 1 α 2 · · · α k and β 1 β 2 · · · β ℓ are reals, then the sequence (α 1 , α 2 , . . . , α k ) interlaces the sequence (β 1 , β 2 , . . . , β ℓ ) if either 1. k = ℓ and α 1 β 1 α 2 β 2 · · · α k β ℓ , or 2. k = ℓ + 1 and α 1 β 1 α 2 β 2 · · · β ℓ α k .
The Hermite-Biehler Theorem (see, for example, [55]) states necessary and sufficient conditions for a real polynomial to be stable: Proof. First, if G is disconnected then either (i) G = K 3 , A(G, x) = (1 + x) 3 and the only root is −1, or (ii) G = K 1 ∪ K n−1 , A(G, x) = (1 + x)(1 + (n − 1)x + (n−1)(n−2) 2 x 2 ) and the roots have real part either −1 and or −1/(n − 2). Thus in either case, the roots are all in the left half-plane, so A(G, x) is stable. Now we assume that G is connected. Here, we will apply the Hermite-Biehler Theorem. Since A(G, x) has degree three, for some a 3 1,
A(G, x) = a 3 x 3 + n(n − 1) 2 x 2 + nx + 1 = f e (x 2 ) + xf o (x 2 )
where f e = n(n−1) 2
x + 1 and f o = a 3 x + n are the even and odd parts of f , respectively. Both of these functions are standard, as is f .
Let r e and r o be the roots of f e and f o respectively. Then r e = − 2 n(n−1) and r o = − n a 3 . Notice that both r e and r o are real and negative. Furthermore, by Theorem 2.4,
n(n − 1) 2 − a 3 n n(n − 1) 2 − n 2 − n 2 √ 2n−2 − n √ 2n−2 2 − n 2 + n √ 2n−2 n = n 2 2 − n 2 − n + n √ 2n − 2 + √ 2n − 2 2 + 1 2 − 1 √ 2n − 2 . Since n 3, it is clear that n √ 2n−2 > 1 √ 2n−2 . Hence, n(n − 1) 2 − a 3 n > n 2 2 − 3n 2 + √ 2n − 2 2 + 1 2 > n 2 2 − 3n 2 0.
So r o < r e and thus the roots of f o interlace the roots of f e . Therefore, by the Hermite-Biehler Theorem for stability, A(G, x) is stable.
✷ What more can we say about the location of the acyclic roots of graphs of acyclic dimension three? From Figure 2, we see that, in addition to being in the left half-plane, there are real acyclic roots far to the left, but the nonreal roots seem to be close to the origin. We shall make both of these observations more precise.
We begin with an observation about the roots of certain cubics that we can apply to acyclic polynomials. Lemma 2.9. Let a 3 , a ′ 3 , a 2 , a 1 , and a 0 be positive. Suppose
f (x) = a 3 x 3 + a 2 x 2 + a 1 x + a 0 g(x) = a ′ 3 x 3 + a 2 x 2 + a 1 x + a 0
have unique real roots r and r ′ respectively. If a 3 > a ′ 3 , then r > r ′ .
Proof. Note that r and r ′ must be negative, and
f (x) = a 3 x 3 + a 2 x 2 + a 1 x + a 0 = a ′ 3 x 3 + a 2 x 2 + a 1 x + a 0 + a 3 x 3 − a ′ 3 x 3 = g(x) + (a 3 − a ′ 3 )x 3
Since r is the only real root of f (x) and a 3 , . . . , a 0 are all positive, r is the unique place where f (x) changes from negative to positive. Similarly, r ′ is the unique place where g(x) changes from negative to positive.
Thus, f (r ′ ) = g(r ′ ) + (a 3 − a ′ 3 )(r ′ ) 3 = (a 3 − a ′ 3 )(r ′ ) 3 < 0, since a 3 > a ′
3 . Therefore, r > r ′ . ✷
We will use Lemma 2.9 to explain the behaviour of the nonreal roots of acyclic polynomials with degree three as the order of the graph increases.
A(G, x) = a 3 x 3 + n(n − 1) 2 x 2 + nx + 1,
where A(G, x) is an acyclic polynomial with a 3 1. Then, as n increases, s and its conjugate s go to zero.
Proof. We can assume that G is connected (as otherwise G is K 1 ∪K n−1 and its two nonreal acyclic roots are easily seen to head to 0). Suppose r is the real root of A(G, x) and let r ′ be the real root of
f (x) = n 2 − n 2 √ 2n − 2 − n √ 2n − 2 2 − n 2 + n √ 2n − 2 x 3 + n(n − 1) 2 x 2 + nx + 1.
As f (x) is of the same form as Equation (2.1) and n > 3, by the same reasoning in Theorem 2.6, r ′ is unique. From Theorem 2.4, we know
a 3 n 2 − n 2 √ 2n − 2 − n √ 2n − 2 2 − n 2 + n √ 2n − 2 .
Thus, from Lemma 2.9 it follows that r < r ′ . Furthermore, since r ′ is unique, f (x) will be negative for
x < r ′ . At x = − 1 2 , f − 1 2 = n 2 − n 2 √ 2n − 2 − n √ 2n − 2 2 − n 2 + n √ 2n − 2 − 1 2 3 + n(n − 1) 2 − 1 2 2 + n − 1 2 + 1 = n 2 8 √ 2n − 2 + n √ 2n − 2 16 − 9n 16 − n 8 √ 2n − 2 + 1
For n 42, it is straightforward to verify that
f − 1 2 42n 8 √ 2n − 2 + n √ 82 16 − 9n 16 − n 8 √ 2n − 2 + 1 > 0
Hence for n 42, r < r ′ < − 1 2 . Now notice that A(G, x) can be factored as follows:
A(G, x) = a 3 x 3 + n(n − 1) 2a 3 x 2 + n a 3 x + 1 a 3 = a 3 x 3 + (−r − s − s)x 2 + (rs + rs + ss)x − rss .
So, −rss = 1 a 3 . From Theorem 2.5, a 3 n − 2, and thus |s| = We can provide, depending on n, an annulus that contains the acyclic roots of graphs with acyclic dimension three. To do so, we will use the well known Eneström-Kakeya Theorem [30,38].
Theorem 2.11 (The Eneström-Kakeya Theorem). Let f (x) = a n x n + · · · + a 1 x 1 + a 0 x 0 be a real polynomial with degree greater than one. If r is a root of f (x), then min a 0 a 1 , a 1 a 2 , . . . , a n−1 a n < |r| < max a 0 a 1 , a 1 a 2 , . . . , a n−1 a n .
Lemma 2.12. Let G be a graph with n vertices. If r is a root of A(G, x), then 1 n < |r|. Moreover, if A(G, x) has degree three, then |r| < n(n−1) 2(n−2) .
Proof. As mentioned in the previous section, the Sperner bounds for complexes imply that n . From the Eneström-Kakeya Theorem we concude that |r| > 1 n . Now suppose A(G, x) = 1 + nx + n(n−1) 2 x 2 + a 3 x 3 has degree three. From Theorem 2.5, a 3 n − 2, and so a 2 n−2 a 2 a 3 . Also, n 3, and thus
a 2 n − 2 = n(n − 1) 2(n − 2) > 2n n(n − 1) = a 1 a 2 > 1 n = a 0 a 1 .
By the Eneström-Kakeya Theorem, we conclude that |r| < n(n−1) 2(n−2) . ✷
There are graphs whose acyclic polynomials have degree three and their real roots are within unit modulus. In particular, if G is the graph of order 2n 2 (n 2) whose complement is the disjoint union of n stars with 2n vertices, then A(G, x) = (4n 4 − 4n 3 − n 2 + n)x 3 + n 2 (2n 2 − 1)x 2 + 2n 2 x + 1 and max a 2 a 3 , a 1 a 2 , a 0 a 1 < 1. So via the Eneström-Kakeya Theorem, all three roots lie in the unit disk.
However, and more interestingly, we can show that there are graphs with acyclic dimension three and with real acyclic roots of large modulus. Theorem 2.13. There exist graphs with acyclic dimension three that have real acyclic roots of arbitrarily large modulus.
Proof. Let n 4 and let G be the graph K n minus an edge. Then as shown in Theorem 2.5, A(G, x) = (n − 2)x 3 + n(n−1) 2 x 2 + nx + 1. Let r be the unique real root of A(G, x). Note that at x = − n 2 ,
A G, − n 2 = − (n − 2)n 3 8 + n 3 (n − 1) 8 − n 2 2 + 1 = n 3 8 − n 2 2 + 1.
For n 4, A G, − n 2 1 > 0. It follows that r < − n 2 , since A G, − n 2 > 0 and all the coefficients of A(G, x) are positive. Therefore, as n → ∞, the root r → −∞.
We remark that A G, − n 2 − 1 = − n 3 8 − n 2 2 + n 2 + 3 < 0 for n 4, and so the real root r will lie in (− n 2 − 1, − n 2 ). ✷
Real acyclic roots
For graphs of acyclic dimension at most three, we have seen that an acyclic polynomial with all real roots is rare -it only happens for those graphs that are acyclic (so A(G, x) = (1 + x) n and the n roots are all −1). Does this behaviour extend past acyclic dimension three? The related question of when a particular graph polynomial has all real roots appears difficult. For example, an important result on independence polynomials, due to Chudnovsky and Seymour [21], states that the independence polynomials of claw-free graphs have all real roots, and yet this is far from a characterization of such graphs. There is much known on real-rooted chromatic polynomials [28] as well, and there are many examples of such families (such as the chromatic polynomials of chordal graphs). Surprisingly, we can precisely characterize graphs that have all real acyclic roots. To do so, we will need a theorem that provides a necessary and sufficient condition for a real polynomial to have all real roots. We begin with a definition.
The Sturm sequence of a real polynomial f is the sequence f 0 = f, f 1 = f ′ , f 2 , . . . , f k where, for i 2, f i is the negative of the remainder when f i−2 is divided by f i−1 , and f k is the last nonzero term in the sequence of polynomials of strictly decreasing degrees. Sturm proved the following result (see [35,48]). The Sturm sequence (f 0 , f 1 , . . . , f k ) of f has gaps in degree if the degree of one term is at least 2 lower than the preceding one; the Sturm sequence has a negative leading coefficient if one of the terms does. A consequence of Sturm's Theorem that is also due to Sturm (again, see [35,48]) will be very useful to us (we use the formulation stated in [18]). Corollary 2.15. Let f be a real polynomial whose degree and leading coefficient are both positive. Then f has all real roots if and only if its Sturm sequence has no gaps in degree and no negative leading coefficients.
We are ready to characterize graphs with all real acyclic roots.
Theorem 2.16. G has all real acyclic roots if and only if G is a forest.
Proof. We observe first that if G is a forest of order n, then A(G, x) = (1 + x) n , which clearly has all real roots (namely −1 with multiplicity n). We now assume that G is not a forest, so that it has a cycle. Let g be the order of a smallest cycle in G, so g 3. Suppose that A(G, x) has degree d = n − ∇(G) < n. Then
A(G, x) = 1 + n 1 x + n 2 x 2 + · · · + n g − 1 x g−1 + n g − α x g + · · · + a d x d
where α is a positive integer. It is clear that A(G, x) has all real roots if and only if
f (x) = x n A G, 1 x = x n + n 1 x n−1 + n 2 x n−2 + · · · + n g − 1 x n−(g−1) + n g − α x n−g + · · · + a d x n−d
has all real roots. We now consider the first few terms of the Sturm sequence of f and show that f = f 0 does not have all real roots. Since f 1 = f ′ , then
f 1 = nx n−1 + (n − 1) n 1 x n−2 + (n − 2) n 2 x n−3 + · · · + (n − (g − 1)) n g − 1 x n−g + (n − g) n g − α x n−g−1 + · · · + (n − d)a d x n−d−1 . Now f 0 = x n + 1 n f 1 + α n − 2g + 1 n x n−g + · · · , so f 2 = − f 0 − x n + 1 n f 1 .
As n − d 1, the coefficient of x n−d−1 in f 2 is clearly nonzero (since x n−d is the smallest power of x in f 0 that has a nonzero coefficient), so it follows that f 2 is a nonzero polynomial of degree at most n − g n − 3. This clearly implies that the Sturm sequence of f will have a gap in degree, and hence a nonreal root. We conclude that the acyclic polynomial of G has a nonreal root as well. ✷ A classical theorem of Newton states that if all of the roots of a polynomial with positive real coefficients are themselves real, then the polynomial's coefficients are log-concave and therefore also unimodal (see, for example, [23]). Hence it follows from Theorem 2.16 that if G is a forest then the coefficients of A(G, x) are unimodal, although this observation is also readily apparent from the fact that if G is a forest of order n then A(G, x) = (1 + x) n .
Instead of focussing on real roots, we could instead ask which rational numbers arise as acyclic roots. The answer is rather easy, in that the set of all such roots is {− 1 n : n 1}. The Rational Root Theorem shows that every root of an acyclic polynomial is of the form − 1 k for some positive integer k as acyclic polynomials have constant term 1 and have positive integer coefficients. On the other hand, note that A(K n,n , x) = 2nx (1 + x) n − nx − 1 + n 2 x 2 + 2(1 + x) n − 1 and hence A(K n,n , − 1 n ) = 0.
Acyclic roots of large modulus
The real acyclic roots of large modulus that we discovered in Section 2.1 had modulus Ω(n), but from calculations it seems that this is far from the true magnitude for graphs in general. To find (real) acyclic roots of larger modulus, we will examine the complement of the disjoint union of the star graph S n−4 of order n − 4 and the cycle C 4 . We denote this graph as J n = S n−4 ∪ C 4 , which we observe is the same as S n−4 + C 4 ; see Figure 7 for an example of such a graph. This family of graphs may seem to be an arbitrary choice of a family to examine, but for graphs with order between 5 and 8, the acyclic polynomial of S n−4 ∪ C 4 has the left-most real root, and the root of largest modulus. Question 2.17. Let n 5. Is it the case that the acyclic polynomial of the graph J n has a real root that is less than the real roots of all other acyclic polynomials of graphs of order n? Moreover, does J n have the acyclic root of largest modulus for graphs of order n?
We will show that the maximum modulus of the acyclic roots of such graphs is, in fact, quadratic in n. We begin by first finding the acyclic polynomial of these graphs. Lemma 2.18. Let n 5. The acyclic polynomial of the graph J n is
x 4 + n 2 + 5n − 34 2 x 3 + n 2 − n 2 x 2 + nx + 1
Proof. Note that J n = S n−4 ∪ C 4 = S n−4 + C 4 = (K 1 ∪ K n−5 ) + 2K 2 . It is easy to verify that:
A(K 1 ∪ K n−5 , x) = (1 + x) 1 + (n − 5)x + n − 5 2 x 2 A(2K 2 , x) = (1 + x) 4 I(K 1 ∪ K n−5 , x) = (1 + x)(1 + (n − 5)x) I(2K 2 , x) = (1 + 2x) 2 .
Applying Theorem 1.3, we derive that
A (K 1 ∪ K n−5 ) + 2K 2 , x = A(K 1 ∪ K n−5 , x) + A(2K 2 , x) + (n − 4)xI(2K 2 , x) + 4xI(K 1 ∪ K n−5 , x) + n 2 − 8(n − 4) − n − 4 2 − 4 2 x 2 − nx − 1 = x 4 + n 2 + 5n − 34 2 x 3 + n 2 − n 2 x 2 + nx + 1.
✷
Though we cannot show that the acyclic polynomial of J n has the minimum real root for arbitrary n, we can place bounds on the real root that appears to be the minimum.
Theorem 2.19. Let n 6. The acyclic polynomial of the graph J n has a real root between −n 2 2 − 5n 2 + 17 and −n 2 2 − 5n 2 + 18, and has no root of modulus larger than n 2 2 + 5n 2 − 17.
Proof It follows from the Intermediate Value Theorem that for n 6, the acyclic polynomial A(J n , x) has a real root r such that −n 2 2 − 5n 2 + 17 < r < −n 2 2 − 5n 2 + 18. That there is no root of modulus greater than n 2 2 + 5n 2 − 17 follows from the Eneström-Kakeya Theorem. ✷
Acyclic polynomials with roots in the right half-plane
Throughout this exposition, all the roots of acyclic polynomials that we have encountered lie in the left half-plane. This leads us to question whether there exist acyclic polynomials that have roots with a positive real component. Instead of directly finding an acyclic polynomial with a root in the right half-plane, we present a family of graphs with limits of roots in the right half-plane.
First, we make the following definition:
A (G[K k ], x) = A(G, kx) + I G, kx + k 2 x 2 − I(G, kx),
where I(G, x) is the independence polynomial of G.
Proof. Let H = K k and let V G and V H be the vertex sets of G and H respectively so that the vertex set of
G[H] is V G[H] = V G × V H . Suppose U ⊆ V G[H] and define support(U) = {v i : (v i , w) ∈ U for some w}. Suppose (v 1 , w 1 ), (v 2 , w 2 ) ∈ U for some v 1 , v 2 , w 1 , w 2 .
If v 1 , v 2 are adjacent in G, then (v 1 , w 1 ), (v 2 , w 2 ) must be adjacent in G [H]. Thus, if U is acyclic, support(U) must be acyclic as well. Partition the acyclic subsets of G into two sets:
1. The independent subsets of G (all of which are necessarily acyclic) 2. The acyclic subsets of G that are not independent.
Suppose support(U) belongs to the first set. Then, since no two vertices of support(U) are adjacent, two vertices (v 1 , w 1 ), (v 2 , w 2 ) ∈ U are adjacent if and only if v 1 = v 2 and w 1 is adjacent w 2 in H. Thus, since H is a complete graph, U is acyclic if and only if, for every v i ∈ support(U) there exist no more than two vertices in U of the form (v i , w) for some w. Hence, the contribution of these subsets to the acyclic polynomial of G[H] is I G, kx + k 2 x 2 . If instead support(U) is in the second set, then support(U) must contain two vertices that are adjacent in G. Because G is multipartite, this means that support(U) does not contain any isolated vertices. Thus, if there exist two vertices (v 1 , w 1 ), (v 1 , w ′ 1 ) ∈ U for some w 1 = w ′ 1 then there must exist a third vertex (v 2 , w 2 ) ∈ U such that v 2 is adjacent v 1 in G. These three vertices must then all be joined in G[H], forming a cycle. Hence, if U is acyclic, then for each v i ∈ support(U) there are no two vertices in U each of the form (v i , w) for some w.
On the other hand, if U is such that for each v i ∈ support(U) there are no two vertices in U each of the form (v i , w) for some w, then two vertices (v 1 , w 1 ), (v 2 , w 2 ) ∈ U are adjacent if and only if v 1 is adjacent v 2 in G. Since support(U) is acyclic, this means that U must also be acyclic. That is, U is acyclic if and only if for each v i ∈ support(U) there are no two vertices in U each of the form (v i , w) for some w. Hence, the contribution of the subsets of the second type to A (G[H], x) is A(G, kx) − I(G, kx).
Therefore, A (G[H], x) = I G, kx + k 2 x 2 + A(G, kx) − I(G, kx). ✷
We can now apply this formula to the lexicographic product of a star graph with a complete graph. Such are the graphs that will give us acyclic roots in the right half-plane (and indeed, we prove something stronger about the location of the limits of the roots).
First we present the Beraha-Kahane-Weiss Theorem which will be a useful tool [13]. For a family of (complex) polynomials {f n (x) : n ∈ N}, we say that z ∈ C is a limit of roots of {f n (x) : n ∈ N} if there is a sequence {z n : n ∈ N} such that f n (z n ) = 0 and z n → z as n → ∞.
f N (x) = α 1 (x)λ N 1 (x) + · · · + α k (x)λ N k (x)
where the α i 's and the λ i 's are polynomials in x such that no α i is the zero function and for no i = j does λ i = ωλ j where |ω| = 1. Then z is a limit of roots of f N if and only if either 1. for some ℓ 2, |λ i 1 (z)| = |λ i 2 (z)| = · · · = |λ i ℓ (z)| > |λ j (z)| for all j = i 1 , i 2 , . . . , i ℓ , or 2. for some i, α i (z) = 0 and for each j = i, |λ i (z)| > |λ j (z)|.
For the following result, we remind the reader that a limaçon is a plane curve of the form r = a cos θ ± b or r = a sin θ ± b. Theorem 2.23. Suppose G = K 1,n−1 [K 2 ] is the lexicographic product of a star graph K 1,n−1 and the complete graph K 2 . Then as n → ∞, the roots of A(G, x) approach a circle of radius 1 2 centered at − 1 2 and the truncated limaçon given by r = √ 2 − 2 cos θ with π 4 < θ < 7π 4 in the complex plane.
Proof. We have the following equations:
A(K 1,n−1 , x) = (1 + x) n I(K 1,n−1 , x) = (1 + x) n−1 + x.
Thus, from Lemma 2.21,
A (K 1,n−1 [K 2 ], x) = A(K 1,n−1 , 2x) + I(K 1,n−1 , 2x + x 2 ) − I(K 1,n−1 , 2x) = (1 + 2x) n + (1 + 2x + x 2 ) n−1 + 2x + x 2 − (1 + 2x) n−1 − 2x = 2x(1 + 2x) n−1 + (1 + x) 2 n−1 + x 2 = 2x(1 + 2x) N + (1 + x) 2 N + x 2 where N = n − 1. Let α 1 (x) = 2x λ 1 (x) = 1 + 2x α 2 (x) = 1 λ 2 (x) = (1 + x) 2 α 3 (x) = x 2 λ 3 (x) = 1 so that A (K 1,N [K 2 ], x) = α 1 (x)λ N 1 (x) + α 2 (x)λ N 2 (x) + α 3 (x)λ N 3 (x)
. From the Beraha-Kahane-Weiss Theorem, z ∈ C is a limit of roots of A (K 1,N [K 2 ], x) if and only if one of the following hold:
1. (a) |λ 1 (z)| = |λ 2 (z)| = |λ 3 (z)|, i.e., |λ 1 (z)| = |λ 2 (z)| = 1 (b) |λ 1 (z)| = |λ 3 (z)| > |λ 2 (z)|, i.e., |λ 1 (z)| = 1 > |λ 2 (z)| (c) |λ 2 (z)| = |λ 3 (z)| > |λ 1 (z)|,
i.e., |λ 2 (z)| = 1 > |λ 1 (z)|, or (d) |λ 1 (z)| = |λ 2 (z)| > |λ 3 (z)|, i.e., |λ 1 (z)| = |λ 2 (z)| > 1 or 2. (a) |λ 1 (z)| > max{|λ 2 (z)|, |λ 3 (z)|}, i.e., |λ 1 (z)| > max{|λ 2 (z)|, 1}, and α 1 (z) = 0 (b) |λ 2 (z)| > max{|λ 1 (z)|, |λ 3 (z)|}, i.e., |λ 2 (z)| > max{|λ 1 (z)|, 1}, and α 2 (z) = 0, or (c) |λ 3 (z)| > max{|λ 1 (z)|, |λ 2 (z)|}, i.e., 1 > max{|λ 1 (z)|, |λ 2 (z)|}, and α 3 (z) = 0.
It is easy to check that all of cases 2a, 2b and 2c lead to contradictions, so if z is a limit of roots, it must satisfy one of the conditions in case 1.
Case 1a is satisfied if and only if 1 = |1 + 2z| and 1 = |1 + z| 2 , i.e., when z lies on both the circle of radius 1 2 centered at − 1 2 and on the circle of radius 1 centered at −1. These two circles only intersect at the origin, as shown in Figure 8. Thus, z = 0 is the only limit of roots fulfilling case 1a. Next, z satisfies case 1b if and only if 1 = |1 + 2z| and 1 > |1 + z|. That is, when z lies on the circle of radius 1 2 centered at − 1 2 and z lies in the open circle of radius 1 centered at −1. As shown in Figure 8, the circle of radius 1 2 centered at − 1 2 is entirely contained within the circle of radius 1 centered at −1, except for the point at the origin. However, z = 0 is also a limit of roots from case 1a. So, z is a limit of roots satisfying case 1a or case 1b if and only if z lies on the circle of radius 1 2 centered at − 1 2 . If z satisfies case 1c, then 1 = |1+z| 2 and 1 > |1+2z|. This means that z lies on the circle of radius 1 centered at −1 and in the open circle of radius 1 2 centered at − 1 2 . However, the circle of radius 1 2 centered at − 1 2 is entirely contained within the circle of radius 1 centered at −1, so this is a contradiction. Hence, there is no z that satisfies case 1c.
Finally, z satisfies case 1d if and only if |1 + 2z| = |1 + z| 2 and |1 + 2z| > 1. Since |1 + 2z| > 1, z must lie outside the circle of radius 1 2 centered at − 1 2 . Furthermore, |1 + 2z| = |1 + z| 2 if and only if |1 + 2z| 2 = |1 + z| 4 . Letting z = a + bi with a, b ∈ R, we have, |1 + 2z| 2 = (1 + 2a) 2 + (2b) 2 = 1 + 4a + 4a 2 + 4b 2 and |1 + z| 4 = (1 + a) 2 + b 2 2 = 1 + 6a 2 + a 4 + b 4 + 4a + 2b 2 + 4a 3 + 4ab 2 + 2a 2 b 2 .
So, 1 + 4a + 6a 2 + 4b 2 = 1 + 4a 2 + a 4 + b 4 + 4a + 2b 2 + 4a 3 + 4ab 2 + 2a 2 b 2 .
Discussion and Open Problems
The results of the previous section point to two types of open problems -those that talk about the coefficients of acyclic polynomials, and those that talk about the location and nature of the acyclic roots -and the problems are not unconnected. In Section 2.1 we characterized those graphs having acyclic polynomials of degree 3, and as well we studied the coefficients and roots of their acyclic polynomials. For higher degree we can ask: Question 3.1. For fixed acyclic dimension Υ(G) 4, is there a characterization of graphs G with acyclic polynomials of degree Υ(G)?
Unimodality
Recall that a real polynomial f (x) = c 0 + c 1 x + c 2 x 2 + · · · is said to be unimodal if there exists an integer t such that c 0 c 1 · · · c t−1 c t and c t c t+1 c t+2 · · · . Clearly A(G, x) is unimodal in cases when G is a tree or a complete graph.
Graphic polynomials (that is, generating functions for the acyclic edge sets) were recently proved to always be unimodal [5,15,41], thereby settling what had been an outstanding conjecture for matroids in general for many years. Domination polynomials of graphs have also been conjectured to be unimodal (see [3] as well as [11] for some recent progress).
For the acyclic polynomial, it is not true that A(G, x) is unimodal for every graph G. When m 20, the acyclic polynomial A(K m + K 6 , x) is not unimodal because a 3 < a 2 and a 3 < a 4 . For example, observe that A(K 20 + K 6 , x) = 1 + 26x + 325x 2 + 320x 3 + 415x 4 + 306x 5 + 121x 6 + 20x 7 . As another example of a class of graphs with acyclic polynomials that are not unimodal, let H be complement of the join of a disjoint 4-cycles, and let G =
H + K b . So A(G, x) = 1 + (4a + b)x + 4a+b 2 x 2 + b 3 + 4 b 2 a + 4ab + 4a + 4a(4a − 4) x 3 + b 4 + 4 b 3 a + a x 4 + b i=5 b i + 4 b i−1 a x i .
Observe that when a 6 and b = 8, A(G, x) is not unimodal (because a 4 is less than a 3 and a 5 ). We ask: Inspired by questions about unimodality of independence polynomials of trees [1] and of bipartite graphs [43], we ask: Question 3.3. If a graph G is bipartite then is A(G, x) unimodal?
As pointed out by one of the referees, the entire setting for acyclic polynomials can be embedded in a much broader setting. Suppose that H is a hereditary class of graphs, that is, one closed under induced subgraphs (and isomorphism). For a graph G we can analogously define a generating function P H (G, x) = x |S| , where the sum is taken over all vertex subsets S of V (G) that induce a subgraph in H (when G consists of a forest, P H (G, x) = A(G, x)). A number of the results we have presented for acyclic polynomials can be carried over -in particular, that of Theorem 2.16, when a graph has all real acyclic roots (see [46]). In this light, it would also be interesting to find a hereditary family H for which P H (G, x) always has unimodal coefficient sequences, but may have nonreal roots.
Location and nature of acyclic roots
A well known theorem due to Newton (see, for example, [23]) states that if a polynomial with positive coefficients has all of its roots real (and on the negative real axis), then its coefficient sequence is unimodal. This result highlights the fact that the nature and location of the roots can inform the unimodality of the coefficient sequence of a real polynomial. In Section 2.4 we showed that for large enough n, the graph K 1,n−1 [K 2 ] has an acyclic root in the right half-plane. This is not the only class of graphs with this property. When m 18, we find that A(K m + K 6 , x) = 1 + (m + 6)x + m+6 Question 3.4. If a graph G has an acyclic polynomial that is not unimodal, then does it have acyclic roots in the right half-plane?
Continuing on the topic of acyclic roots, we pose the following problems: Question 3.5. What can be said about the nature and location of the roots of acyclic polynomials of degree 4 and higher? Question 3. 6. Are there open sets in C that are free of acyclic roots? What is the closure of the acyclic roots? Is the closure of the real acyclic roots equal to (−∞, 0]? Question 3.7. What is the maximum modulus of an acyclic root of a graph of order n?
Proof. Suppose S ⊆ V (G + H) induces an acyclic subgraph of G + H. Let S G = S ∩ V (G) and S H = S ∩ V (H). Necessarily min{|S G |, |S H |} 1 for otherwise the subgraph of G + H induced by S is not acyclic. Hence either (i) |S G | = 0 and S H induces an acyclic subgraph of H, or (ii) |S G | = 1 and S H consists of an independent set in H, or (iii) |S H | = 0 and S G induces an acyclic subgraph of G, or
where e ∈ E(G), with the additional conditions that F (∅) = 1,F (K 1 ) = γ and F (G ∪ H) = F (G) · F (H). )−cov(B) y |A|+|B|+c(A∪B)−cov(B) z cov(B) ,where the summation is over all subsets A and B of E(G) such that the vertex subsets V (A) and V (B) coveredby A and B, respectively, are disjoint, c(A)is the number of connected components in (V (G), A), and cov(B) is the number of connected components of (V (B), B).
Theorem 1. 8 .
8[7] Let G be a graph. Then
(i) ξ(G; x, y, z) is an EE-invariant.
Figure 2 :
2The acyclic roots of all graphs of order 12 with acyclic dimension 3.
Figure 3 :
3Example of G. If vertices u and v are part of S, then a or b must also be part of S, forming a 3-cycle in G.
Figure 4 :
4If u and v are from the same component in G, but a and b do not belong to this star, then these four vertices form a 4-cycle in G. The dotted lines represent edges that may or may not exist and the solid red lines show the edges that form a 4-cycle in G.
If x is joined to u and w in H i , then {u, v, w, x} is an acyclic subset of G. The black lines show the edges in H i (G) and the red shows the edge in H i (G).
) A shortest path P from a vertex x that is not joined to v must include a (noninduced) P 4 , shown in blue.
Figure 6 :
6If H i is acyclic, then H i must have a universal vertex.
the vertices of an edge of G, there are n − 2 ways to choose a third vertex to form an acyclic subset of G. However, if this third vertex belongs to the same component as the other two in G, then we have over-counted this subset once. There are e i 2 = e i (e i −1) 2 ways to choose such a 3-subset in H i . So, if G is the disjoint union of stars, then there are
Lemma 2. 3 .
3For fixed n and k, if e 1 , . . . , e k are nonnegative reals with
Theorem 2. 7 (
7The Hermite-Biehler Theorem for Stability). Define a standard polynomial to be a real polynomial with a positive leading coefficient (or the zero polynomial). Supposef (x) is standard. Write f (x) as f (x) = f e (x 2 ) + xf o (x 2 ). Then f (x)is stable if and only if the following hold• f e and f o are standard • both f e and f o have all real, nonpositive roots • the roots of f o interlace the roots of f e . Theorem 2.8. If A(G, x) is an acyclic polynomial with degree three, then A(G, x) is stable.
Theorem 2 . 10 .
210Suppose n 4 and s, s are the nonreal roots of
→ 0 .
0Hence, s and s → 0. ✷
. The term i n−i+1 is at a minimum when i = 1, so a
Theorem 2. 14 (
14Sturm's Theorem). Let f be a polynomial with real coefficients and let (f 0 , f 1 , . . . , f k ) be its Sturm Sequence. Let a < b be two real numbers that are not roots of f . Then the number of distinct roots of f in (a, b) is V (a) − V (b) where V (c) is the number of changes in sign in (f 0 (c), f 1 (c), . . . , f k (c)).
Figure 7 :
7The graph J 10 is the join of C 4 (in red) and S 6 (in blue). The dotted lines show the join.
.
By Lemma 2.18, A(J n , x) = x 4 + n 2 n = 6 and n = 7, we can verify that A(J n , m) is positive as well. Hence, for n 6, A(J n , x) is positive when x = −n 2 2 − 5n 2 + 17. On the other hand, when M = A(J n , M) is decreasing when n 8. At n = 8, A(J n , M) = −7207. So, A(J n , M) 0 when n 8. Again, we can also verify that when n = 6 and n = 7, A(J n , M) is negative.
Theorem 2 .
222 (The Beraha-Kahane-Weiss Theorem). Suppose
Figure 8 :
8The circle of radius 1 centered at −1 (in red) and the circle of radius 1 2 centered at − 1 2 (in blue).
Figure 10 :
10The curve given by r = √ 2 − 2 cos θ with π 4 < θ < 7π 4 .
Figure 11 :
11The limits of roots of A (K 1,n−1 [K 2 ], x).
Question 3 . 2 .
32Which classes of graphs have unimodal acyclic polynomials? And which ones do not?
Theorem 1.1. If G and H have the same acyclic polynomial, then they have the same order, girth and decycling number. ✷
Definition 2.20. Let G and H be two arbitrary graphs. Then G[H] is the graph formed by replacing each vertex of G by a copy of graph H, and inserting all edges between two copies of H if and only if the corresponding vertices of G were adjacent (we say that we have substituted H in for each vertex of G; G[H] is often called the lexicographic product of G and H).Lemma 2.21. Let G be an arbitrary complete multipartite graph. ThenThe following lemma allows us to compute the acyclic polynomial for G[K k ] where G is
a complete multipartite graph.
Note that the term "acyclic polynomial" already exists within the scientific literature. Historically, it referred to a polynomial that is now more commonly described as the "matching polynomial" and which is closely related to the generating function for the number of matchings of size i within a graph. See[29, page 263] for more details. Given that the term "acyclic polynomial" has fallen out of use from its original context, we now put the term to new use by defining it as the generating function for the number of acyclic induced subgraphs of order i that are within a given graph.
. G is disconnected and G = K 3 or G = K 1 ∪ K n−1 with n 3.
x 2 + (15m + 20)x 3 + (20m + 15)x 4 + (15m + 6)x 5 + (6m + 1)x 6 + mx 7 also has roots in the right half-plane. These examples lead us to ask:
AcknowledgementsAuthors J.I. Brown and D.A. Pike acknowledge research support from NSERC Discovery Grants RGPIN-2018-05227 and RGPIN-2016-04456, respectively. The authors would also like to thank the anonymous referees for their insightful comments. We especially thank one of the referees for pointing out that acyclic polynomials can be expressed within monadic second-order logic, for providing us with the content for Section 1.4, and also for noting that Theorem 2.16 can be extended to hereditary families of graphs.Rearranging this expression, we get 2a 2 + 2b 2 = 4a 2 + a 4 + b 4 + 4a 3 + 4ab 2 + 2a 2 b 2 = (a 2 + b 2 + 2a)2.This equation describes a limaçon. Thus, z is a limit of roots satisfying case 1d if and only if z lies on the limaçon in the complex plane whose equation in Cartesian form isand outside the circle of radius 1 2 centered at − 1 2 (seeFigure 9). In polar form, the equation of this limaçon is r = √ 2 − 2 cos θ. So, z is a limit of roots satisfying case 1d if and only if z lies on the curve given by r = √ 2 − 2 cos θ with π 4 < θ < 7π 4 , shown inFigure 10.Since z is a limit of roots if and only if z satisfies one of cases 1a, 1b, 1c, or 1d, z is a limit of roots if and only if z lies on the circle of radius 1 2 centered at − 1 2 (as in cases 1a and 1b) or on the curve given by r = √ 2 − 2 cos θ with π 4 θ 7π 4 (as in cases 1a and 1d). These limits of roots are shown inFigure 11. ✷ With this strong result, we can prove that we are guaranteed that there exist acyclic polynomials with roots in the right half-plane. Proof. From Theorem 2.23, we know that for large enough n, there exists a root of A (K 1,n−1 [K 2 ], x) arbitrarily close to the truncated limaçon given by r = √ 2 − 2 cos θ with π 4 < θ < 7π 4 . Take θ = π 3 . Then r = √ 2 − 2 cos π 3 = √ 2 − 1. So √ 2 − 1, π 3 is a point on this curve. In Cartesian coordinates, this means x = r cos θ = > 0, there must exist positive roots of A (K 1,n−1 [K 2 ], x) for some n. ✷
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| []
|
[
"Electron Tunneling Spectroscopy of the anisotropic Kitaev Quantum Spin Liquid Sandwiched with Superconductors",
"Electron Tunneling Spectroscopy of the anisotropic Kitaev Quantum Spin Liquid Sandwiched with Superconductors",
"Electron Tunneling Spectroscopy of the anisotropic Kitaev Quantum Spin Liquid Sandwiched with Superconductors",
"Electron Tunneling Spectroscopy of the anisotropic Kitaev Quantum Spin Liquid Sandwiched with Superconductors"
]
| [
"Shi-Qing Jia ",
"Liang-Jian Zou ",
"Ya-Min Quan ",
"Hai-Qing Lin ",
"\nInstitute of Solid State Physics, HFIPS\nand Science Island Branch of Graduate School\nKey Laboratory of Materials Physics\nChinese Academy of Sciences\n230031HefeiChina\n",
"\nInstitute of Solid State Physics, HFIPS\nKey Laboratory of Materials Physics\nUniversity of Science and Technology of China\n230026HefeiChina\n",
"\nBeijing Computational Science Research Center\nDepartment of Physics\nChinese Academy of Sciences\n230031, 100193Hefei, BeijingChina, China\n",
"\nBeijing Normal University\n100875BeijingChina\n",
"Shi-Qing Jia ",
"Liang-Jian Zou ",
"Ya-Min Quan ",
"Hai-Qing Lin ",
"\nInstitute of Solid State Physics, HFIPS\nand Science Island Branch of Graduate School\nKey Laboratory of Materials Physics\nChinese Academy of Sciences\n230031HefeiChina\n",
"\nInstitute of Solid State Physics, HFIPS\nKey Laboratory of Materials Physics\nUniversity of Science and Technology of China\n230026HefeiChina\n",
"\nBeijing Computational Science Research Center\nDepartment of Physics\nChinese Academy of Sciences\n230031, 100193Hefei, BeijingChina, China\n",
"\nBeijing Normal University\n100875BeijingChina\n"
]
| [
"Institute of Solid State Physics, HFIPS\nand Science Island Branch of Graduate School\nKey Laboratory of Materials Physics\nChinese Academy of Sciences\n230031HefeiChina",
"Institute of Solid State Physics, HFIPS\nKey Laboratory of Materials Physics\nUniversity of Science and Technology of China\n230026HefeiChina",
"Beijing Computational Science Research Center\nDepartment of Physics\nChinese Academy of Sciences\n230031, 100193Hefei, BeijingChina, China",
"Beijing Normal University\n100875BeijingChina",
"Institute of Solid State Physics, HFIPS\nand Science Island Branch of Graduate School\nKey Laboratory of Materials Physics\nChinese Academy of Sciences\n230031HefeiChina",
"Institute of Solid State Physics, HFIPS\nKey Laboratory of Materials Physics\nUniversity of Science and Technology of China\n230026HefeiChina",
"Beijing Computational Science Research Center\nDepartment of Physics\nChinese Academy of Sciences\n230031, 100193Hefei, BeijingChina, China",
"Beijing Normal University\n100875BeijingChina"
]
| []
| We present the electron tunneling transport and spectroscopic characters of a superconducting Josephson junction with a barrier of single anisotropic Kitaev quantum spin liquid (QSL) layer. We find that the dynamical spin correlation features are well reflected in the direct-current differential conductance dI c /dV of the single-particle tunneling, including the unique spin gap and dressed itinerant Majorana dispersive band, in addition to an energy shift 2∆ of two-lead superconducting gaps. From the spectral characters, we identify different topological quantum phases of the anisotropic Kitaev QSL. We also present the zero-voltage Josephson current I s which displays residual features of the anisotropic Kitaev QSL. These results pave a new way to measure the dynamical spinon or Majorana fermion spectroscopy of the Kitaev and other spin liquid materials. | null | [
"https://arxiv.org/pdf/2103.01003v3.pdf"
]
| 247,319,056 | 2103.01003 | de2558b65b5eb964b4c77d60f0478058c75026c6 |
Electron Tunneling Spectroscopy of the anisotropic Kitaev Quantum Spin Liquid Sandwiched with Superconductors
9 Mar 2022 (Dated: March 10, 2022)
Shi-Qing Jia
Liang-Jian Zou
Ya-Min Quan
Hai-Qing Lin
Institute of Solid State Physics, HFIPS
and Science Island Branch of Graduate School
Key Laboratory of Materials Physics
Chinese Academy of Sciences
230031HefeiChina
Institute of Solid State Physics, HFIPS
Key Laboratory of Materials Physics
University of Science and Technology of China
230026HefeiChina
Beijing Computational Science Research Center
Department of Physics
Chinese Academy of Sciences
230031, 100193Hefei, BeijingChina, China
Beijing Normal University
100875BeijingChina
Electron Tunneling Spectroscopy of the anisotropic Kitaev Quantum Spin Liquid Sandwiched with Superconductors
9 Mar 2022 (Dated: March 10, 2022)arXiv:2103.01003v3 [cond-mat.str-el]numbers: 7510Kt7510Jm7450+r
We present the electron tunneling transport and spectroscopic characters of a superconducting Josephson junction with a barrier of single anisotropic Kitaev quantum spin liquid (QSL) layer. We find that the dynamical spin correlation features are well reflected in the direct-current differential conductance dI c /dV of the single-particle tunneling, including the unique spin gap and dressed itinerant Majorana dispersive band, in addition to an energy shift 2∆ of two-lead superconducting gaps. From the spectral characters, we identify different topological quantum phases of the anisotropic Kitaev QSL. We also present the zero-voltage Josephson current I s which displays residual features of the anisotropic Kitaev QSL. These results pave a new way to measure the dynamical spinon or Majorana fermion spectroscopy of the Kitaev and other spin liquid materials.
INTRODUCTION
The quantum spin liquid (QSL) phase, which consists of various spin singlet pairings in the spin structure without breaking any constituent symmetries of their underlying lattice, has attracted great attention [1,2]. Enormous efforts have been made to understand the essence of the QSLs, and earlier studies focused on the geometrically and magnetic frustrated interaction [3,4]. However, the essence and unique characters of the QSL states remain great debates [5,6]. More than a decade ago Kitaev proposed an exactly solvable model on the twodimension (2D) honeycomb lattice [7], which shows that the interaction frustration drives a ground state of gapless or gapped Z 2 QSL with fractionalized excitations [8]. The QSL state with gapped excitations has the Abelian anyons [9], the one with gapless excitations may have the non-Abelian anyon excitations [10]. Due to topological protection and large degeneracy of these anyons, the Majorana fermion excitations and its braiding group in the gapless QSL state were expected to be applicable for the quantum computing storage and quantum computation [8,11]. However, how to excite and detect the dynamics of these Majorana fermion modes in Kitaev systems remains unknown.
On the other hand, the Josephson tunneling junctions, * [email protected] † [email protected] which are constructed of two superconducting (SC) leads separated by an insulating or metallic barrier, provide a well probe to measure the quasi-particle information of the central region through the quantum tunneling transport [12,13]. A great deal of central materials, such as insulators [14], normal metals [15], quantum dots [16][17][18], ferromagnets [19][20][21] and antiferromagnets [22,23] have been studied. In order to explore the exotic spin correlations and fractional excitations of the Majorana fermions through the transports of single electrons and Cooper pairs, especially the inelastic spin scattering process [24][25][26], it is worth constructing novel SC-Kitaev layer-SC tunneling junctions to reveal its current dynamics associated with exotic spin excitations in Kitaev layer. In realistic candidate materials for the Kitaev layer, the spin interactions are usually anisotropic [27][28][29][30], thus we employ anisotropic Kitaev layer in the designed SC Josephson junctions.
In this paper, we utilize the current and conductance features of the SC-anisotropic Kitaev layer-SC tunneling junctions to characterize the dynamical spin correlations of the central-zone Kitaev materials. We adopt the non-equilibrium Green's function [17] and the fewparticle response method [31,32] to obtain the formulae of the single-particle and Josephson tunneling currents. We find that the dynamical spin susceptibility explicitly displays in the direct current (DC) single-particle differential conductance spectrum dI c /dV , and from its spectral features, we could confirm the different topological quantum phases of the anisotropic Kitaev QSL. One ex-pects that the SC-anisotropic Kitaev QSL-SC mesoscopic hybrid systems with weak links may open a fruitful research field, not only because of the abundant fundamental features from the interplay between Kitaev physics and SC, but also of the potential application for design and development of new quantum devices.
II.
MODEL AND TUNNELING OF THE SC-KITAEV QSL-SC JUNCTION A. The SC-Kitaev QSL-SC junction and tunneling process
We construct a Kitaev Josephson junction, where a single-layer Kitaev insulator is the barrier, sandwiched by two leads consisting of two conventional s-wave superconductors. Here the SC leads may be Nb, or Pb metals, or their alloys NbTi and Nb 3 Sn, etc., and the central Kitaev layer may be α-RuCl 3 or Na 2 IrO 3 single layer, which are the candidate materials of the Kitaev QSL [29]. Such a SC-Kitaev QSL-SC Josephson junction is shown in Fig. 1. Since the Kitaev material is a kind of transition-metal Mott insulator with strong electronic correlation, the tunneling of conduction electrons between left and right SC leads is scattered by the local spins in the central region, as shown in Fig. 2. The scattering strength is s − d-type exchange coupling J. For this set-up, the tunneling current consists of normal single-particle one and Josephson one. We can describe the normal single-particle tunneling process as follows: firstly, the electrons at the bottom of SC gap in the right lead enter the Kitaev layer, and occupy the high energy levels to form the virtual double occupied states. The propagation of the electrons would be modulated by the dynamical spin susceptibility of the Kitaev QSL in the spin-conserving channel, as well as in the spin-flipping process with spin fluctuations. Finally, the electrons leave the Kitaev layer with constant or opposite spins and go to the top of the SC gap in the left SC lead. Moreover, the tunneling process of the SC Cooper pairs can be addressed as follows: the Cooper pair in the right lead firstly tunnels into the central Kitaev region, splitting as the quasi-electron and quasi-hole with opposite spins. Afterwards, the quasi-electron and quasi-hole would go through the similar virtual transitions as the single particles with the modulation of the Kitaev QSL. Once tunneling out of the central Kitaev region, the separated quasi-electrons and quasi-holes would recombine to SC Cooper pairs. These tunneling processes of single particles and Cooper pairs could be qualitatively described by the sketched diagram shown in Fig. 2. FIG. 2. (Color online) Sketched diagram of the single-particle (red) and Cooper pair (green) tunneling processes in the superconductor-Kitaev QSL-superconductor Josephson junction. The left and right sides are the bare density-of-states (DOS) distributions ρ(E) of the two SC leads, and the center is the Kitaev QSL layer. The circles indicate the s − d exchange processes of single particle and a Cooper pair with local spin, respectively.
B. Model Hamiltonian and Formulae
The total Hamiltonian of the SC-Kitaev QSL-SC tunneling junction shown in Fig. 1 and Fig. 2 consists of three parts as follows: the left and right SC electrodes H Lead,n (n = L, R), the single-layer Kitaev material in the central scattering region H cen , and the s−d exchange interaction part between the SC leads and central material H T . So H = n=L,R H Lead,n + H cen + H T , and H Lead,n = kσ ǫ 0 nkσ a † nkσ a nkσ + k ∆ n a n,−k↓ a nk↑ + h.c. ,
H cen = −K X ij Xσ x iσ x j − K Y ij Yσ y iσ y j − K Z ij Zσ z iσ z j ,(1)H T = − i 1 2J i (t) σ z i a † Li↑ a Ri↑ − a † Li↓ a Ri↓ +σ + i a † Li↓ a Ri↑ +σ − i a † Li↑ a Ri↓ + h.c. ,
where a † nkσ and c † iσ are the creation operators of electrons in the SC leads and Kitaev layer, respectively, and a † niσ is the Fourier transform of a † nkσ on the ith site of the 2D interface between the SC leads and Kitaev layer. σ
x(y,z) i = σσ ′ c † iσ σ x(y,z) σσ ′
c iσ ′ are the twice spin components,σ ± i =σ x i ± iσ y i , and σ
x(y,z) σσ ′ are the Pauli matrices. Let the two SC leads be the s − wave superconductors and their order parameters∆ n = ∆ n e −iφn with magnitudes ∆ n and phases φ n . ǫ 0 nkσ is the single-electron energy. K X , K Y and K Z are the spin coupling constants along the X, Y and Z bonds in the central Kitaev layer, and they satisfy the conditions K X = K Y > 0 and K X +K Y +K Z = 3K for the anisotropic Kitaev model. J i is the s−d exchange matrix element between the electrons in the SC leads and the local spins in Kitaev layer. In the presence of external electric potential V n (t)(n = L, R), the exchange parameter becomes voltage dependence of
J i (t) = J i exp[i(φ L −φ R )−(i/ ) t 0 e(V L (t 1 )−V R (t 1 ))dt 1 ]
through a unitary transformation, leaving only the perturbation term H T explicitly depends on time [16].
The tunneling current from the left SC lead to the central region reads,
I(t) = −e dN L (t) dt = ie [N L (t), H(t)] (2) = − e Re iJ i (t)i σ z i a † Li↑ a Ri↑ − a † Li↓ a Ri↓ +σ + i a † Li↓ a Ri↑ +σ − i a † Li↑ a Ri↓ .
It actually contains two parts: the normal single-particle tunneling current and SC Josephson current, and both of them stem from the inelastic scattering with the spinconserving (m = zz) and spin-flipping (m = xx, yy) processes,
I(t) = − 2e Re ij,m t −∞ dt 1 J i J j (3) e ieV (t−t 1 ) g r m,LR,ij (t, t 1 ) G < m,ji (t 1 , t) +g < m,LR,ij (t, t 1 ) G a m,ji (t 1 , t) + e ieV (t+t 1 ) e iφ g ′r m,LR,ij (t, t 1 ) G < m,ji (t 1 , t) +g ′< m,LR,ij (t, t 1 ) G a m,ji (t 1 , t) .
Throughout this paper we only consider the DC voltage V = V L − V R and φ = φ L − φ R is the phase difference between the left and right SC leads. Define G(g) r,a,< m,ji (t 1 , t) with superscripts r , a , and < as the dressed (bare) retarded, advanced, and lesser Green's functions of spin correlation in the central region, respectively.g r,a,< m,LR,ij (t, t 1 ) andg ′r,a,< m,LR,ij (t, t 1 ) are bare normal and anomalous Green's functions of electron-hole modes and Cooper pairs between left and right SC leads, respectively. For example, the advanced Green's functions can be written as follows:
g r m,ji (t 1 , t) = −iθ(t 1 − t) [0.5σ α j (t 1 ), 0.5σ α i (t)] , (4) g r m,LR,ij (t, t 1 ) = −iθ(t − t 1 ) [σ α σσ ′ a † Liσ a Riσ ′ (t),σ α σσ ′ a † Rjσ a Ljσ ′ (t 1 )] , g ′r m,LR,ij (t, t 1 ) = −iθ(t − t 1 ) [σ α σσ ′ a † Liσ a Riσ ′ (t),σ α σσ ′ a † Ljσ a Rjσ ′ (t 1 )] , where m = αα, α = x, y, z.
The details are shown in Sec. A of Supplementary Materials [33].
With zero bias voltage, we have only DC Josephson current I s generated by the tunneling of Cooper electron pairs through the Kitaev QSL. Moreover, at V = 0, we are much interested at the DC current I c and its conductance dI c /dV of the normal single-particle tunneling. Thus, the DC single-particle and Josephson current terms in the first-order approximation can be obtained as follows:
I c = 4e ij,m dǫ 2π J i J j Im g r m,LR,ij (eV − ǫ) Im g r m,ij (ǫ) [n (ǫ) − n (ǫ − eV )] ,(5)I s = 4e ij,m dǫ 2π J i J j Im g ′r m,LR,ij (ǫ) g r m,ji (ǫ) n (ǫ) sin φ, respectively, where n(ǫ) = 1/[exp(ǫ/k B T ) − 1]
is the Bose-Einstein distribution function. As seen in Eq. (5), I c obviously depends on the dynamical spin susceptibility S m ij (ǫ) = −2 Im[g r m,ij (ǫ)] of the Kitaev QSL, the spectral weight of electron-hole modes C m LR,ij (ǫ) = −2 Im[g r m,LR,ij (ǫ)] between the two SC leads and the occupation difference between spins and electron-hole modes. Similarly, I s is weighted by the hybridization spectrum of spins and Cooper pairs A m hy,ij (ǫ) = 2 Im[g ′r m,LR,ij (ǫ)g r m,ji (ǫ)] and the Bose-Einstein occupation n(ǫ). In these inelastic scattering processes, the electron-hole modes or Cooper pairs with charge between left and right SC leads transfer energy to the central spin system [25].
Actually, further analysis reveals that both the normal and anomalous Green's functions of the two leads have the same zz, xx and yy components because of the timereversal symmetry. We haveg r m,LR,ij (ǫ)=g r 0,LR,ij (ǫ) and g ′r m,LR,ij (ǫ) =g ′r 0,LR,ij (ǫ) for m = xx, yy, and zz, respectively. At the same time, the unique feature of QSL leads to that g r m,ji (ǫ) is a short-range spin correlation in real space and only the on-site and nearest-neighbour (NN) ones are nonzero, which is explained later in Sec. II C. So the currents have two part contributions from the on-site and NN X, Y , Z bonds. Therefore, we can simplify the tunneling currents I c and I s at zero temperature as
I c = 8e N J 2 m eV 0 dǫ 2π
Im g r 0,LR,AA (eV − ǫ) Im g r m,AA (ǫ) + AB Im g r 0,LR,BA (eV − ǫ) Im g r m,BA (ǫ)
,
I s = 8e N J 2 m ∞ 0 dǫ 2π sin φ Im g ′r 0,LR,AA (ǫ) g r m,AA (ǫ) + AB Im g ′r 0,LR,AB (ǫ) g r m,BA (ǫ) .(6)
Here the indexes of the sublattices, AA and AB, stand for the on-site and NN configurations, and J i = J for each site i. N is the number of unit cell of honeycomb lattice. Then, the normal and anomalous two-body Green's functions can be evaluated through the frequency summations over the combinations of left-and right-lead single-body Green's functions, in the 4 × 4 Nambu representation (a nk↑ a † n,−k↓ a nk↓ a † n,−k↑ ). The details can be seen in Sec. B of Supplementary Materials [33]. We thus obtain that
g r 0,LR,AA(BA) (ǫ) = s 2 2 d 2 k 4π 2 d 2 p 4π 2 e i(k+p)·R AA(BA) 1 ǫ − E Rp − E Lk + i0 + − 1 ǫ + E Rp + E Lk + i0 + (7a) g ′r 0,LR,AA(AB) (ǫ) = − s 2 2 d 2 k 4π 2 d 2 p 4π 2 e i(k+p)·R AA(AB) (7b) ∆ L ∆ R E Lk E Rp 1 ǫ − E Rp − E Lk + i0 + − 1 ǫ + E Rp + E Lk + i0 +
Here E nk(p) = ǫ 2 nk(p) + ∆ 2 n , and the parabolic energy dispersions ǫ Lk = 2 k 2 /2m * −E F , ǫ Rp = 2 p 2 /2m * −E F . m * is the effective mass of electron, E F is the Fermi energy level, and set = 1. s is the area of unit cell of SC-Kitaev layer-SC interface in the SC leads. R AA = 0 and R AB = X, Y, Z for the on-site and NN ones, respectively.
Assuming that k F = 1/a s and E F = 20K, where k F and a s are the Fermi wave vector and lattice constant of two SC leads. Since the exchanged momenta between the SC leads and Kitaev layer are constrained by 0 ≤ |q| ≤ 2k F , the product q · X(Y, Z) (q = k + p) can be taken to zero for simplicity [24,25] in the Green's functions with the NN contribution. This is suitable for the "bad metal" like Nb or Pb with the small Fermi wave vectors. Hence, in the leads, we haveg r 0,LR,AB(BA) (ǫ) ≈g r 0,LR,AA(BB) (ǫ) andg ′r 0,LR,AB(BA) (ǫ) ≈g ′r 0,LR,AA(BB) (ǫ). Then the imaginary part of the normal retarded Green's function of the two SC leads can be further simplified as follows,
Im g r 0,LR,AA(BA) (ǫ) = −2πρ L ρ R (8a) ǫ ∆L dE E E 2 − ∆ 2 L (ǫ − E) (ǫ − E) 2 − ∆ 2 R , ǫ ≥ E + ∆ R −ǫ ∆L dE E E 2 − ∆ 2 L (ǫ + E) (ǫ + E) 2 − ∆ 2 R , ǫ ≤ −E − ∆ R ,
as well as the imaginary and real parts of the anomalous retarded Green's function
Im g ′r 0,LR,AA(AB) (ǫ) = π 2 ρ L ρ R (8b) ǫ ∆L dE ∆ L E 2 − ∆ 2 L ∆ R (ǫ − E) 2 − ∆ 2 R , ǫ ≥ E + ∆ R −ǫ ∆L dE ∆ L E 2 − ∆ 2 L −∆ R (ǫ + E) 2 − ∆ 2 R , ǫ ≤ −E − ∆ R , Re g ′r 0,LR,AA(AB) (ǫ) = ∞ −∞ dω 2π (−2) Im g ′r 0,LR,AA(AB) (ω) ǫ − ω .
Here we calculate the real part of Green's function by the Kramers-Kronig transformation. The normal density of states (DOS) in the 2D interface ρ L(R) = m * a 2 s /2π 2 . More details can be seen in Sec. B of Supplementary Materials [33].
Therefore, we can obtain the DC single-particle differential conductance dI c /dV and the derivative of the DC Josephson current I s with respect to ∆, dI s /d∆, as
dI c dV = 2e 2 N J 2 m eV 0 dǫ 2π d[C 0 LR,AA (eV − ǫ)] dV S m (ǫ) , dI s d∆ = 4e N J 2 m ∞ 0 dǫ 2π d[A m hy (ǫ)] d∆ sin φ.(9)
Here the total dynamical spin susceptibility, the total hybridization spectrum of spins and Cooper pairs, and the equally weighted spectrum of electron-hole modes are defined as
S m (ǫ) = −2 Im[g r m (ǫ)], A m hy (ǫ) = 2 Im{g ′r 0,LR,AA (ǫ)g r m (ǫ)}, C 0 LR,AA (ǫ) = −2 Im[g r 0,LR,AA (ǫ)],(10)
respectively, where the total Green's function of spin correlation g r m (ǫ) = g r m,AA (ǫ) + Σ AB g r m,BA (ǫ). Once obtaining the Green's functionsg r 0,LR,AA (ǫ),g ′r 0,LR,AA (ǫ) and g r m (ǫ), we could get the DC single-particle current and its differential conductance numerically, as well as the zero-voltage Josephson current at zero temperature.
C. Dynamics of the Kitaev model
Next, we need the total Green's function of spin correlation of anisotropic Kitaev QSL, g r m (ǫ), whose imaginary part corresponds to the dynamical spin suscepti-bility, S m (ǫ). We would evaluate the S m (ǫ) by employing the few-particle-response method and g r m (ǫ) via the Kramers-Kronig transformation.
The Kitaev model H cen in Eq.
(1) can be exactly solved by introducing four Majorana fermions b α i (α = x, y, z) and c i per site for the local spins, i.e.σ α i = ic i b α i . Define the bond operatorsû α ij = ib α i b α j on the NN bond ij Λ (Λ = X, Y, Z), respectively. Their eigenvalues are u α ij = ±1, and they commute with H cen and with each other. So the Kitaev model can be expressed in terms of the different sets of {u α ij } and the Majorana fermions [8,32],
H cen = i Λ, ij Λ K Λ u α ij c i c j ,(11)
where the product of all bond operators around a plaquette, W p = i,j∈p u α ij .
= ±1, can define the flux sectors. The eigenstates of this model are Z 2 gauge fluxes threading the plaquettes and Majorana fermions (or spinons) propagating between sites in this Z 2 gauge field [32]. And their wave vectors |Φ are the direct product of bond (gauge flux) and Majorana-matter-fermion degrees of freedoms, |Φ = |F ⊗ |M . The ground state is within the zero-flux sector with W p = 1 (u α ij =1) for all plaquettes.
Through the diagonalization of the zero-flux Hamiltonian matrix in the momentum space, the ground-state spinon energy dispersion can be expressed as
E k = 2 K X e ik·X + K Y e ik·Y + K Z e ik·Z(12)
The ground-state parametric phase diagram is obtained [8], as shown in Fig. 3(a). From Eq. (12), one can find a van Hove singularity at E V 1 = 2|K Z | corresponding to the energy contour line PMP' in the first Brillouin region; and another van Hove singularity at E V 2 = 2|K X + K Y − K Z | when |K Z | < 1.5, or a spinon gap ∆ S = 2|K Z − K X − K Y | when 1.5 < |K Z | < 3.0, associated with the M' point. There is also a energy maximum E max = 2|K X + K Y + K Z | = 6 at Γ point. In probing into the dynamical features and evolution of the QSL ground states in the anisotropic Kitaev model, we take the range of the Kitaev couplings K Z along the line marked by red, blue, and green lines with arrows, labelling the gapped, gapless and another gapless QSL, in this phase diagram. The quantum phases in these three regions display distinct different quantum features [30]. The time-dependent dynamical spin susceptibility of the ground state, S αα ij (t) = 0.25 Φ 0 |σ α i (t)σ α j (0)|Φ 0 (|Φ 0 = |F 0 ⊗ |M 0 ) [32], can be derived as follows:
S αα ij (t) = −0.25i M 0 e iH0t c i e −i(H0+V ij Λ )t c j M 0 iδ ij +û α ij δ ij ,Λ ,(13)
where V ij Λ = −2iK Λ c i c j , i ∈ A, j ∈ B, and α = x, y, z corresponds to Λ = X, Y, Z one-to-one. We can find that only the on-site (δ ij ) and NN (δ ij ,Λ ) ones of the dynamical spin correlation are non-zero, and S αα ij only has the α = z(x, y) component in the NN X(Y, Z)-bond.
S αα ij has the Lehmann representation by inserting the identity 1 = λ |λ λ| of the two-flux sector with a flipping bond u α ij = −1. The main contributions are from the zero-, one-and two-particle of |λ , which occupy the 98% of the total [32]. We thus can obtain the dynamical spectrums in the frequency ω-space as
S αα AA (ω) = π 2 λ M 0 | c A |λ λ| c A |M 0 δ[ω − E F λ − E 0 ], S αα BA (ω) = π 2 i λ M 0 | c B |λ λ| c A |M 0 δ[ω − E F λ − E 0 ].(14)
Here E 0 is the ground-state energy of the zero-flux sector, and E F λ is the energy eigenvalue of the [32]state |λ of two-flux sector, while the lowest-energy is E F 0 with the state |M z(x,y) F . |λ and E F λ are obtained through the diagonalization of the two-flux Hamiltonian matrix in the real space. Further we can calculate the overlaps M 0 |M z(x,y) F 2 and "vison" gap ∆ z(x,y) F = E F 0 − E 0 due to the gauge-flux excitation, as shown in Fig. 3(b)(c), consistent with Knolle's results [32].
From the dynamical phase diagrams in Fig. 3(b)(c), we can see that the lowest-energy states of the zero-flux sector H 0 and two-flux sector H 0 + V z(x,y) , |M 0 and |M z(x,y) F conserve the parity owing to the spatial inversion symmetry. Along the line in Fig. 3(a), |M 0 and |M z F have the same parity when 1.24 < |K Z | < 3.0 and the opposite parity when 0 < |K Z | < 1.24; |M 0 and |M x(y) F have the same parity all the way. In the case with the same parity, |λ must contain the odd number of excitations, mainly the single-particle contribution. This dynamical spin susceptibilities could be evaluated by Eq. (12). In the opposite case, |λ must contain the even number of excitations, mainly the zeroand two-particle contributions. Actually the Lehmann representation is modified by inserting the identity 1 = λ c A(B) |λ λ|c A(B) of two-flux sector H 0 + V x + V y with two flipping bonds u x ij , u y ij = −1. Its lowest-energy state |M x,y F have the same parity with |M 0 [32], as shown in Fig. 3(b). We also plot the lowest-energy state |M y,z F for H 0 + V y + V z shown in Fig. 3(c). It has the opposite parity with |M 0 all the way. Therefore, we can explicitly express Eq. (11) for the zero-and two-particle contributions,
S αα AA (ω) = π 2 λ M 0 |λ λ|M 0 δ ω − E F λ − E 0 ,(15)S αα BA (ω) = π 2 i λ M 0 | c B c A |λ λ|M 0 δ ω − E F λ − E 0 .
Then the dynamical spin correlation S m (ǫ) = S m AA (ǫ) + S m BA (ǫ) (m = αα,α = x, y, z). More details are shown in Sec. C of Supplementary Materials [33]. Hence, combining the dynamical and parametric phase diagrams, we choose four representative points K Z = 1.8, 1.4, 1.0 and 0.6, respectively, among the phase transition points about the parity relationship and spinon gap, K Z = 1.24 and 1.5. Substituting the Eq. (8), (14) and (15) into Eq. (6), (9) and (10), we can obtain the tunneling current I c,s and differential conductances dI c /dV and dI s /d∆. Throughout this paper the SC order parameters∆ L and∆ R in the left and right leads have the same modulus ∆ L = ∆ R = ∆, but different phase φ L(R) . In this paper, all of the energies are measured in terms of the Kitaev coupling K, which can be taken as K = 1.
III. RESULTS AND DISCUSSION
A. Dynamical spin correlations of the anisotropic Kitaev model At first, we plot the dynamical spin susceptibilities of the anisotropic Kitaev model, including the components S αα (E)(α = x, y, z) and their total S tot , as functions of energy E [24-26, 31, 32], as shown in Fig. 4
(a)-(d).
Here S xx = S yy because K X = K Y . From this, we can see that the anisotropic components of dynamical spin susceptibilities, S zz and S xx(yy) , and their total S tot reveal remarkable different features in these four quantum phases.
When K Z = 1.8 with a gapped QSL, the parities between |M z F and |M 0 are opposite. As shown in Fig. 4(a), in S xx(yy) , we can see the total QSL gap ∆ t ≈ 1.2. It actually contains the spinon gap ∆ S = 2|K Z − K X − K Y | = 1.2 and vison gap ∆ x F ≈ 0.0. There is a dip at E ≈ 3.6 owing to the van Hove singularity of spinon spectrum at 2K Z and an energy shift of ∆ x F . And an upper edge emerges at about 6.0 which equals ∆ x F + 2|K X + K Y + K Z |. So these three feature points in S xx(yy) correspond to the ones of spinon dispersion at E V (∆ S or E V 1 , E V 2 and E max ) one-to-one, and move towards ∆
x(y) F + E V .
There is a new peak at about 2.5 caused by the interacting vison and spinon. However, in S zz , we can observe the total gap ∆ ′ t ≈ 2.4, which stems from the ∆ z F ≈ 0.0 and the new spinon gap ∆ ′ S = 2∆ S . A peak appears at E ≈ 7.2 resulted from the van Hove singularity, and the upper edge emerges at E ≈ 12.0. The three feature points at ∆ z F + 2E V in S zz are from the virtual transitions to the eigenstates of two-flux sector with two flipping bonds. Moreover, we can see a distinct sharp peak at ∆ z F , stemed from the virtual transitions to the lowest-energy state, |M x,y F . There is also a new peak around 5.0 due to the interaction of vison and spinon. Note that S xx(yy) is an order of magnitude bigger than S zz . As for the sum of S zz , S xx and S yy , S tot can exhibit the complete information of vison, spinon and their interaction, except some feature points because of the resolution of S zz .
When K Z = 1.4 shown in Fig. 4(b), the ground-state is gapless QSL and |M z F and |M 0 have the opposite parity. Hence, the three feature points are displayed on S xx(yy) and S zz in the similar way as K Z = 1.8, except the van Hove singularity instead of the spinon gap. In S xx , we can observe two dips at E ≈ 0.5 and 2.9 corresponding to the van Hove singularities, and a upper edge at about 6.1 with ∆ x F ≈ 0.11. These three feature points emerge at ∆ x(y) F + E V . There are two new interaction peaks at about 0.4 and 1.5. In S zz , there is a dip and an inflection point associated with the van Hove singularities at E ≈ 1.0 and 5.8, and a boundary at about 12.2 with ∆ z F ≈ 0.17. So these feature points are shown at ∆ z F +2E V . A remarkable sharp peak appears at ∆ z F , and a new interaction peak emerges at about 0.8. Since S zz has the same order in magnitude to S xx,yy , the total one S tot could reveal the full dynamical features of Kitaev QSL well.
When K Z = 1.0 and 0.6, as shown in Fig. 4(c)(d), |M z(x,y) F and |M 0 have the same parity. At K Z = 1.0, the ground-state of the isotropic Kitaev model is a C 6 gapless QSL, and S xx(yy) and S zz components are equal, ∆ x(y) F = ∆ z F ≈ 0.26. From S zz we can find that there is only one dip related to the two-in-one van Hove singularity at E ≈ 2.26, and a upper edge at about 6.26. There is also a new interaction peak at about 0.5. At K Z = 0.6, two dips resulted from the van Hove singularities, a upper edge and a new sharp peak are shown at E ≈ 1.45, 3.85, 6.25 and 0.3 in S xx(yy) with ∆ x(y) F ≈ 0.25. There is a dip and a inflection point, a upper boundary and a new peak at E ≈ 1.32, 3.72, 6.12 and 0.8 in S zz with ∆ z F ≈ 0.12. Therefore, the feature points emerge at ∆ x(y) F + E V and ∆ z F + E V in S xx(yy) and S zz , respectively, because of the virtual transitions to the eigenstates of two-flux sector with only a flipping X(Y )-or Z-bond. The total spin correlations S tot also have the entire characters of Kitaev QSL when K Z = 1.0 and 0.6.
In a word, the dynamical spin susceptibility components S zz and S xx (= S yy ) reveal different vison gaps ∆ z F and ∆
x(y) F , respectively. Every component can only reveal the partial features of the Majorana fermion (spinon) dispersions influenced by the gauge fluxes, including the two van Hove singularities (or a spinon gap and a van Hove singularity) and the energy upper edge, and in different ways. Therefore, the total S tot can exhibit the complete information of Kitaev QSL well. There are some new peaks between these feature points, which stem from the interaction between the vison and spinon excitations.
B. DC Josephson current with zero voltage
In the absence of the bias voltage, only the DC Josephson current with the tunneling of the Cooper pairs is presented in the SC-Kitaev QSL-SC junction. The SC gap ∆ dependences of the derivative of the DC Josephson current I s with respect to ∆, G tot = dI s /d∆, and its components G z(x,y) (G x = G y ) have been described in Fig. 5(a)-(d) for different Kitaev couplings K Z = 1.8, 1.4, 1.0 and 0.6, respectively. Here the phase difference φ = 3π/2 and we define the dimensionless constant g 0 = 4πρ L ρ R J 2 .
As seen in Fig. 5(a)-(d), when K Z = 1.8, we can see a peak at ∆ ≈ 0.6, and an inflection point at about 1.25. The former corresponds to the total QSL gap ∆ t ≈ 1.2, which originates from the resonant tunneling when 2∆ = ∆ t , while the latter stems from the interaction between the vison and spinon excitations near 2∆ ≈ 2.5. Similarly, at K Z = 1.4, a distinct peak, corresponding to the total QSL gap, emerges around 2∆ ≈ 0.17. Another peak at about 0.9 stemming from the response to the interaction peak appears near 2∆ ≈ 1.8. When K Z = 1.0 and 0.6, we can only observe the peaks at about 0.25 and 0.4, which are due to the response to the interaction peaks around 2∆ ≈ 0.5 and 0.8, respectively. Thus, dI s /d∆ curves mainly provide the information about the interaction of gauge fluxes and Majorana fermion mode, as well as the total QSL gap. These peaks in dI s /d∆ could be seen from the dynamical spin susceptibilities in Fig. 4, however, only partially provide full information of the Kitaev QSLs.
To understand this reason, we plot the energy E dependences of the total dynamical hybridization spectral functions between local spins of the Kitaev layer and Cooper pairs of the two SC leads, A hy = α A αα hy (α = x, y, z), in Fig. 6(a)-(d). The SC gaps are set as ∆=1, 3, and 5, respectively. From Fig. 6 we can see the whole dynamical spin correlation characters clearly and A hy > 0 before E = 2∆. When E > 2∆, these spin correlation features appear with a reversal sign, i.e. A hy < 0 in the same magnitude. Thus, the DC Josephson current at zero bias, as the frequency integration of the hybridization spectrum, is partially cancelled; hence, it only keeps partial information of Kitaev QSL. This arises from the fact that in the inelastic tunneling, the quasi-electrons and quasiholes of the SC Cooper pairs contribute the positive and negative parts of the A hy , respectively. Therefore, the total response to the dynamical spin correlation spectrum is cancelled out due to the spin-singlet Cooper pairs. C. DC conductance of the normal single-particle tunneling Further, in the presence of a DC bias voltage in the SC-Kitaev QSL-SC junction, one could reveal more characters of the Kitaev QSL. The bias potential eV dependences of the DC single-particle differential conductance G tot =dI c /dV , as well as its zz(xx, yy) components G z(x,y) , have been described in Fig. 7(a)-(d) for different Kitaev couplings K Z = 1.8, 1.4, 1.0 and 0.6, respectively. Here we define the conductance constant G 0 = g 0 e 2 /h, and G tot = G x + G y + G z with G x = G y .
From Fig. 7(a)-(d), the single-particle DC differential conductance spectrums of the SC junction, G tot and G z(x,y) , show the distinct different characters in the four quantum phases. To clearly see the dynamical behaviors of G tot in present anisotropic Kitaev layer, we first describe the bias voltage dependence of G z(x,y) . When K Z = 1.8, as seen in Fig. 7(a), contrast with the dynamical spin susceptibility in Fig. 4(a), the threshold of the conductance G x(y) is modulated up to about 3.2, i.e. ∆ t + 2∆. This arises from the fact that the electrons at the bottom of the SC gap in the right lead need a high enough bias potential eV = ∆ + ∆ + ∆ t to overcome the right and left SC gaps and total QSL gap of the central layer along the X-(Y -) bond, and finally reach the empty state on the top of the SC gap in the left lead. When eV > 2∆ + ∆ t , with the open of the channel of the Majorana bond state, the conductance G x(y) starts to rise rapidly and goes up to a sharp peak at about 4.5. This peak corresponds to the interaction peak of the dynamical spin correlation around 2.5 shown in Fig. 4(a), and results from the dynamical creation of the Majorana fermions (or spinon) interacting with the NN two gauge fluxes in the virtual transition. Soon afterwards, a re- markable dip have been observed at about 5.6, which is associated with the dip of dynamical spectrum around 3.6 and due to the van Hove singularities of the DOS of the Majorana dispersive band. Finally, the singleparticle conductance approaches to a constant after the upper edge at about 8 due to the one of Majorana dispersive band around 6. Hence, the features of the singleparticle DC differential conductance spectrums G x(y) in Fig. 7(a)correspond to those of the dynamical spin susceptibility one-to-one in Fig. 4(a).
Meanwhile, we can see a remarkable sharp peak at eV ≈ 2 in G z related to the one in the dynamical spectrum near ∆ F z ≈ 0.0, which originates from the δfunction contribution of the virtual transition between the ground state |M 0 and the excited state |M x,y F . When eV > ∆ F z , no obvious characters in G z is observed since S zz is an order of magnitude smaller than S xx(yy) . Summing the three components gives rise to the total conductance G tot , which contains the complete characters of the spinon spectrums, vison excitation and their interaction of Kitaev QSL. Thus, compared to the normal-metal junction situation [24,25], the present differential conductance spectrums G tot have a more intuitive and sensitive response to the characters of dynamical spin correlation components of Kitaev QSL, S tot .
When K Z = 1.4, 1.0 and 0.6, similar to K Z = 1.8, the single-particle DC differential conductance spectrums G z(x,y) can reflect the features of dynamical spin susceptibility components S zz(xx,yy) well, except some feature points due to the resolution in numerical integration. Fortunately, in the present situations with K Z = 1.4, 1.0 and 0.6, the z-component of dynamical spin correlations S zz are the same order in magnitude to S xx,yy , so G z can resolve the complete features of S zz . Hence, from the single-particle tunneling spectrums, we could get insight into the features of the dynamical spin susceptibilities of Kitaev QSL.
IV. CONCLUSION
In our present theory, we point out two possible improvements to the present results. On the one hand, with the condition of q · X(Y, Z) ≈ 0, we obtain the features of the total dynamical spin susceptibility S tot . When q · X(Y, Z) = 0, the individual contribution of each component of the NN spin correlation S αα BA to the tunneling currents would be slightly different from the result above. Our further study reveals that in this situation the correction to Eq. (8) only quantitatively alters the tunneling current, nevertheless, it is qualitatively consistent with the above conclusion. On the other hand, although the zero-voltage Josephson current fails to measure the full information of the Kitaev QSL in the elastic scattering process, we expect that the AC Josephson currents with DC bias voltage can reveal more features of dynamical spin correlation, which goes beyond the scope of this paper. As a summary, in investigating the electron tunneling transport and its spectroscopic features in an SCanisotropic Kitaev QSL-SC Josephson junction with the weak link, we assume that the inelastic scattering tunneling of the single particle and Cooper pair is realized by the s − d exchange interaction. As expected, the DC differential conductance dI c /dV of the normal singleparticle tunneling succeeds in exhibiting the dynamical spin susceptibility characters of the anisotropic Kitaev QSL, including the unique spin gaps even in gapless QSL, the sharp or broad peaks, the small dips and the upper edge of the itinerant Majorana fermion dynamics, except an energy shift of two-SC-lead gap 2∆. The different topological quantum phases of anisotropic Kitaev QSL can be distinguished by the tunneling spectral features well. Unusually, the zero-voltage DC Josephson currents I s only have some residual information of Kitaev QSL, which stems from the spin singlet of Cooper pairs. Our results may pave a new way to measure the Majoranafermion dynamical correlation features of the anisotropic Kitaev and other spin liquid materials. We expect that our theoretical results could be confirmed by future experiments and be applied in the SC junction devices.
FIG. 1 .
1(Color online) Schematic superconductor-Kitaev QSL-superconductor tunneling junction. The left (right) side is the SC lead with gap ∆L (∆R), phase φL (φR) and electric potential VL(t) (VR(t)). The central region is a single-layer Kitaev material in the ab plane.
FIG. 3 .
3(Color online) (a) The variation range of Kitaev coupling strengths in the parametric phase diagram of the Kitaev model with the conditions KX = KY and KX + KY + KZ = 3.0, marked by red, blue and green arrows. Six points are marked with black dots, KZ = 1.8, 1.5, 1.4, 1.24, 1.0, and 0.6. Kitaev coupling KZ dependences of (b) the overlaps M z F |M0 2 , M x,y F |M0 2 and vison gap ∆ z F , and (c) the overlaps M x F |M0 2 , M y,z F |M0 2 and vison gap ∆ x F in the variation range of (a).
FIG. 4 .FIG. 5 .
45(Color online) Energy E dependences of the dynamical spin susceptibilities of the anisotropic Kitaev QSL, including the total S tot and its three components S αα (q = 0, E)(α = x, y, z) for different Kitaev coupling KZ = 1.8 (a), 1.4 (b), 1.0 (c) and 0.6 (d), respectively, in units of the energy K. Here S xx = S yy (Color online) Derivative of the DC Josephson tunneling current I s with respect to the SC gap, dI s /d∆, including the components G z(x,y) (Gx = Gy) and the total Gtot, as functions of the SC gap ∆ for different Kitaev coupling KZ = 1.8 (a), 1.4 (b), 1.0 (c) and 0.6 (d), respectively. Here φ = 3π/2
FIG. 6 .FIG. 7 .
67(Color online) Energy E dependences of the total hybridization spectral functions between spins of the central Kitaev QSL layer and Cooper pairs of the two SC leads, A hy , with ∆=1, 3 and 5 for different Kitaev coupling KZ = 1.(Color online) DC differential conductances of the single-particle tunneling dI c /dV , including the components G z(x,y) (Gx = Gy) and the total Gtot, as functions of the bias potential eV for different Kitaev coupling KZ = 1.8 (a), 1.4 (b), 1.0 (c) and 0.6 (d), respectively.
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See supplementary materials to this publication. See supplementary materials to this publication.
| []
|
[
"The Quantum-Statistical Condensate of One-Dimensional Anyons",
"The Quantum-Statistical Condensate of One-Dimensional Anyons",
"The Quantum-Statistical Condensate of One-Dimensional Anyons",
"The Quantum-Statistical Condensate of One-Dimensional Anyons"
]
| [
"Thore Posske \nInstitut für Theoretische Physik\nUniversität Hamburg\nJungiusstraße 920355HamburgGermany\n",
"Björn Trauzettel \nInstitute for Theoretical Physics and Astrophysics\nUniversity of Würzburg\n97074WürzburgGermany\n",
"Michael Thorwart \nInstitut für Theoretische Physik\nUniversität Hamburg\nJungiusstraße 920355HamburgGermany\n",
"Thore Posske \nInstitut für Theoretische Physik\nUniversität Hamburg\nJungiusstraße 920355HamburgGermany\n",
"Björn Trauzettel \nInstitute for Theoretical Physics and Astrophysics\nUniversity of Würzburg\n97074WürzburgGermany\n",
"Michael Thorwart \nInstitut für Theoretische Physik\nUniversität Hamburg\nJungiusstraße 920355HamburgGermany\n"
]
| [
"Institut für Theoretische Physik\nUniversität Hamburg\nJungiusstraße 920355HamburgGermany",
"Institute for Theoretical Physics and Astrophysics\nUniversity of Würzburg\n97074WürzburgGermany",
"Institut für Theoretische Physik\nUniversität Hamburg\nJungiusstraße 920355HamburgGermany",
"Institut für Theoretische Physik\nUniversität Hamburg\nJungiusstraße 920355HamburgGermany",
"Institute for Theoretical Physics and Astrophysics\nUniversity of Würzburg\n97074WürzburgGermany",
"Institut für Theoretische Physik\nUniversität Hamburg\nJungiusstraße 920355HamburgGermany"
]
| []
| We develop an exact many-body formalism for one-dimensional anyons, the hybrid particles between bosons and fermions. Besides providing characteristic observables, we reveal the quantumstatistical condensate. This genuine many-body condensate is created purely by quantum-statistical attraction. It is potentially more stable than a Bose-Einstein condensate and carries a rich structure of degenerate internal excitations. | 10.1103/physrevb.96.195422 | [
"https://arxiv.org/pdf/1609.06314v2.pdf"
]
| 54,819,040 | 1609.06314 | e5617b1ae08dec24ff4222720316658e47618852 |
The Quantum-Statistical Condensate of One-Dimensional Anyons
20 Sep 2016
Thore Posske
Institut für Theoretische Physik
Universität Hamburg
Jungiusstraße 920355HamburgGermany
Björn Trauzettel
Institute for Theoretical Physics and Astrophysics
University of Würzburg
97074WürzburgGermany
Michael Thorwart
Institut für Theoretische Physik
Universität Hamburg
Jungiusstraße 920355HamburgGermany
The Quantum-Statistical Condensate of One-Dimensional Anyons
20 Sep 2016numbers: 0365Ge7110Pm7343Lp7420Mn0367Lx
We develop an exact many-body formalism for one-dimensional anyons, the hybrid particles between bosons and fermions. Besides providing characteristic observables, we reveal the quantumstatistical condensate. This genuine many-body condensate is created purely by quantum-statistical attraction. It is potentially more stable than a Bose-Einstein condensate and carries a rich structure of degenerate internal excitations.
Introduction -Modern physics has made it possible to construct nanoscopic systems in which electronic excitations are effectively confined to a lower-dimensional world. An unexpected consequence of a reduced spatial dimension is the occurrence of particles that neither obey Fermi nor Bose statistics: anyons [1][2][3][4][5][6]. While anyons in two dimensions have been theoretically extensively studied [7][8][9] and indicated to exist in several experimental systems [10][11][12][13][14][15][16], they have been comparably neglected in one dimension. This may be because the spatial exchange of two-dimensional anyons results in fascinating physics [17], while in one dimension, anyons only collide, remaining their order. However, exchangeability is installable considering ringlike systems or T-structures [18,19]. Sparked by this idea, the interest in lowerthan-two-dimensional anyons has recently risen, especially in conjunction with the possible detection of Majorana bound states in quantum wires [20][21][22][23][24]. Those are expected to be non-abelian anyons with potential application to topological quantum computing [19,25].
There exist different theories for one-dimensional particles of intermediate quantum statistics that are all referred to as anyons [5,20,[26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41]; see [42] for a brief summary. Within this work, we employ the concept of anyons introduced in the seminal work by Leinaas and Myrheim [1]. Their approach advantageously bases only on one fundamental idea: to set up the proper classical theory of indiscernible particles and then quantize it. In two dimensions, this results in "standard" anyons, imaginable as bosons with an attached flux acquiring an Aharonov-Bohm phase when physically exchanged [6,9]. Applied to one dimension, Leinaas' and Myrheim's idea leads to a description of either intrinsically one-dimensional anyons or quasi-one-dimensional anyons that are created from two-dimensional ones due to confinement by an external potential [33]. Concretely, we might imagine a fractional quantum Hall insulator [11], where anyonic bulk excitations are confined to one dimension by an electric potential. Despite its fundamentality, Leinaas' and Myrheim's approach to one dimensional anyons has rarely been applied [8,33] and has remained, to the best of our knowledge, in its elementary state until now.
In this manuscript, we develop the quantum manybody formalism for one-dimensional anyonsà la Leinaas and Myrheim. We construct the space of admissible wave functions, develop the second quantization formalism and calculate observables [20,21,[43][44][45] for confined anyons: the energy spectrum, momentum density, and finite size density oscillations. Our results are readily applicable to quasi-particle excitations in quasi one-dimensional systems, like interacting cold atom/ion chains and edge liquids of topological insulators, that potentially carry anyonic excitations [20,21,41,46,47]. While developing the formalism, we take particular care to include complex momenta, which leads to peculiar statistical bound states. Although the two-particle precursor of these states has already been discovered at the origins of anyons [1], they have been rather neglected in subsequent works [8,33]. Their full importance unfolds in the exact many-body solutions, where they generalize to the quantum-statistical condensate, a remarkably stable quantum phase exhibiting a complex structure of degenerate internal excitations. Our work shows that onedimensional anyons offer rich and elegant physics even in the absence of exchangeability.
Model -Let us concisely recapitulate and slightly extend the theory of Leinaas and Myrheim [1]. Consider n classical, indiscernible particles in a region M of real space. The spatial configurations of a system of discernible particles would be described by tuples of positions x = (x 1 , . . . , x n ), where x j lies in M . Because the particles are indiscernible, however, using tuples is prodigal: for n = 2, (x 1 , x 2 ) and (x 2 , x 1 ) label the same configuration. Instead, we employ the sets {x 1 , . . . , x n } of n distinct positions. The family of all these sets is called configuration space C and inherits various properties by local equivalence to M n . For n indiscernible particles on a line, the real variables x 1 < · · · < x n parametrize C, where x 1 denotes the leftmost and x n the rightmost particle. To obtain the quantum mechanical theory, space and momentum variables get promoted to the usual operators acting on the wave functions Ψ : C → C. Additionally, the Hamiltonian H must be hermitian. For concreteness,
∂ 2 xj ,(1)
where m denotes the mass of the particles. Electromagnetic potentials and particle interactions can be added without changing the formalism. Hermiticity is granted by fulfilling the Robin boundary conditions
∂ xj+1 − ∂ xj Ψ(x) | xj→xj+1 = η Ψ(x) | xj →xj+1 ,(2)
for each j between 1 and n − 1. These boundary conditions classify the scattering behavior of indiscernible particles. As such, it is not surprising that they depend on the relative momentum i (∂ xj −∂ xj+1 ) of adjacent anyons [48]. The statistical parameter η is a real momentum that characterizes the anyonic species interpolating between bosons and fermions. Coherently, Eq.
(2) reduces to the Neumann and Dirichlet boundary conditions of bosons (at η = 0) and fermions (η = ±∞) [49].
Construction of the wave functions -We next construct all wave functions that fulfill Eq. (2) by deriving a proper basis. To this end, we write a general wave function as an integral over momentum space by Ψ (x) = k∈C n d n k α (k) e ikx . Complex momenta are explicitly included. These are needed to describe anyonic bound states that may potentially form for a negative statistical parameter. To understand why anyons can bind, note that a scattering event of two anyons with a negative statistical parameter results in a negative time shift of the corresponding matter wave [33]. Thus, anyons effectively scatter backwards in time. They then have the chance to scatter again and form a bound state by infinite repetition [50]. In momentum space, the boundary conditions translate to
α (k) = e −iφη(kj+1−kj ) α (σ j k) if k j+1 − k j = iη, (3) α (k) = 0 if k j+1 − k j = iη. (4)
Here, σ j denotes the elementary permutation which permutes the j th and (j + 1) th element of a tuple and
φ η (k j+1 − k j ) = 2 arctan [η/(k j+1 − k j )](5)
is the statistical phase. By iteration, these conditions connect coefficients of relatively permuted momenta α(k) = e iφ P η (k) α(P k). Here, if P = σ j1 . . . σ jr is a general permutation written with an r as small as possible, then
φ P η (k) = r i=1 φ η (σ j1 . . . σ ji k) ji − (σ j1 . . . σ ji k) ji+1 . The basis functions are therefore of the form Ψ k ∝ P ∈Sn e iφ P η (k) e i(P k)x .
Divergent elements of this set are excluded by only permitting k that are the concatenation [51] of disjoint tuples which are rigorously protected against permutation by Eq. (4). We call these tuples clusters motivated by their physical interpretation, which is discussed in a moment. Technically, a cluster is a tuple of complex momenta µ = µ 1 , . . . , µ nµ ordered by imaginary part, where the sum of all momenta is real and the difference of adjacent momenta µ j+1 − µ j is either zero or −iη. Thereby, a cluster defines an integer partition ν of n µ by counting the numbers of equal momenta starting from the left. Examples of clusters and their integer partitions are sketched in Fig. 1. We call a cluster irreducible if its elements can not be reordered to form two (nonempty) clusters.
Physically, irreducible clusters with more than one element represent composite anyons whose constituents are localized within a characteristic length scale of 1/η from each other and move collectively. They should be conceived as individual particle themselves. Each possible cluster structure, as defined by the integer partition, represents a different composite anyon species. We elaborate on this below. For positive η, irreducible clusters only consist of single particles, which describes free, unbound, anyons. In order to uniquely label a basis function, we introduce the cluster ordering O. To apply O to a tuple D of irreducible clusters, first merge clusters that are not disjoint to larger clusters by taking their union and reordering appropriately. Then, sort the resulting clusters by real part and lexicographically by their integer partitions. Finally, concatenate the clusters in this order to obtain the tuple O (D).
In conclusion, the basis wave functions describe composite and free anyons in momentum space. Given an ordered tuple O (D) of irreducible clusters, the correspond-ing basis function obtains the form
Ψ (k=O(D))) (x) = N k P ∈Sn e iφ P η (k) e i(P k)x ,(6)
where N k is the normalization [52] and φ P η (k) plays the role of a generalized Slater determinant.
Second quantization -Given the basis wave functions of Eq. (6), second quantization amounts to defining creation operators to construct all basis states from a vacuum state [53]. For an irreducible cluster µ, we define its creation operator by
a † µ Ψ O(D) = n µ + 1 e iΦ µ η (D) Ψ O({µ}∪D)(7)
and linear continuation to all states. Here, n µ is the number of irreducible clusters µ in D. The phase Φ µ η (M ) = μ<µ ϕμ ,µ η , is composed of the cluster-cluster exchange phases ϕμ ,µ
η = nµ i=1 nμ i=j φ η (μ j − µ i ).
Employing the latter, the algebra of the cluster creation operators is
a † µ1 a † µ2 = e iϕ µ 1 ,µ 2 η a † µ2 a † µ1 .(8)
To be concrete, we consider the case of unbound anyons, described by clusters with exactly one element. Here,
a † p a † q =e iφη(p−q) a † q a † p a p a † q =e −iφη(p−q) a † q a p + δ(p − q),(9)
where the annihilation operator a p is the hermitian conjugate of a † p . The real space algebra for
Ψ † (x) = ∞ −∞ dp e ipx √ 2π a † p is readily obtained [42] to be Ψ † (x), Ψ † (y) = ∞ 0 dz 2e − z |η| |η| Ψ † (y − z)Ψ † (x + z),(10)
which yields a smeared anyonic Pauli principle for x = y. Finally, it is well known that the concept of statistics in one dimension is fuzzy as there exist several ways to transform between different statistics [20,54]. Likewise here, there is a generalized Jordan-Wigner transformation from anyons to bosons, which we derive in [42]. Systems of finite size -When anyons are confined to the length L, one would expect the Dirichlet boundary conditions Ψ (0, x 2 , . . . , x n ) = Ψ (x 1 , . . . , x n−1 , L) = 0 to quantize the allowed momenta, similar to the particle in a box problem. In fact, the conditions translate to
α (−k 1 , . . . , k n ) = − α (k) , α (k 1 , . . . , −k n ) = − e 2iknL α (k) .(11)
These constraints are only consistent with Eqs.
(3) and (4) if the system of transcendental equations
Lk j + 1≤(i =j)≤n [φ η (k i − k j ) − φ η (k i + k j )] /2 = πz j (12)
is fulfilled for j between 1 and n. Here, the z j are positive integers. The momenta that solve Eq. (12) are discrete and readily numerically obtainable. Application -Equipped with the developed formalism, we now consider observables of experimental interest. First, we calculate the spectrum of two confined anyons numerically by solving Eq. (12) as depicted in Fig. 2a. The anyonic spectra interpolate between the familiar bosonic and fermionic particle-in-a-box spectra for positive η. For instance, in units of 2 π 2 /(2mL 2 ), the bosonic level at energy 2 continuously evolves to the fermionic level at energy 5. At negative η, the anyonic levels form two-anyon bound states with an energy proportional to −η 2 in the infinite-size limit L → ∞. Energetically higher anyonic bound states correspond to kinetic excitations of the composite anyon in analogy to the behavior of a single particle in a box. Some anyonic levels refuse to form bound states and instead converge to fermionic energies as η → −∞. These levels ensure that the finite-size spectrum coherently converges to the infinite-size spectrum. Energy spectra could be a viable observable in systems with few anyons, like, potentially, interacting cold-atom chains [20,21,41], and detectable by spectroscopic techniques. Turning to systems containing many anyons, as possibly the case in solid state systems, unavoidable level broadening renders an accurate measurement of the discrete spectrum unfeasible; yet the momentum distribution could uncover the character of the anyons [20]. We depict the momentum density n k at zero temperature in Fig. 2b. This function gives the number of anyons with momentum between k 1 and k 2 by k2 k1 dk n k . For bosons and fermions, it is proportional to the Bose-Einstein and Fermi-Dirac distribution, respectively. Anyons with a positive statistical parameter transform these distributions into each other, still remaining a sharply defined chemical potential reflected by a discontinuity in n k . If the spectral properties of a system are inaccessible, the statistics is still inferable via local properties, e.g., the finite size density fluctuations [21,45]. While bosons condense to the middle of the system, fermions distribute equally spaced (by Pauli repulsion), resulting in oscillations of the particle density. Figure 2c depicts the scenario for four anyons in the ground state. Unbound anyons suppress the fermionic peaks and broaden the bosonic one, which is characteristic to intermediate statistics [21,45].
The quantum-statistical condensate -For η < 0, the anyonic ground state is a maximally bound cluster of the form µ j = iη[j − (n − 1)/2] as depicted in Fig. 1d. We call this state the quantum-statistical condensate. Being an irreducible cluster it can be conceived as a single composite anyon. Therefore, its local density is the same as for the Bose-Einstein condensate, as depicted in Fig. 2c. In fact, the Bose-Einstein condensate can be interpreted as the limit of the quantum-statistical condensate as η → 0 − . Besides this, both condensates dif- fer profoundly: bosons condense into their single-particle ground state, but anyons into an inseparable many-body ground state. Let us derive further characteristics of the quantum-statistical condensate. First, we obtain its ground state energy
ǫ GS = − 2 24m
η 2 (n − 1)n(n + 1) (13) by Eq. (1). The proportionality to n 3 reveals an exceptional stability of the condensate. Let us, for a moment, regard charged anyons exhibiting Hubbard repulsion, which is proportional to n 2 . Then, providing a sufficiently large number of anyons, the negative statistical energy outperforms the positive one created by charge repulsion. The quantum-statistical condensate is hence stable against the introduction of charge. We conjecture that this property could lead to remarkable anyon superconductivity [55][56][57]. Next, we consider the excitations of the statistical condensate, i.e., the anyonic clusters themselves. A cluster behaves as an individual anyon, the energy of which separates into a kinetic and an internal part. Additionally, by Eq. (8), clusters acquire different statistical phases than their constituents. For instance, clusters of two anyons behave like anyons with the statistical phase 2φ η + φ 2η [42]. In the vocabulary of topological field theories for two-dimensional anyons [9,17,58], the formation of clusters is linked to anyon fusion [59]. Interestingly, the composition process is not unique for clusters containing more than two anyons. This property corresponds to the existence of different fusion channels, leading to non-abelian anyons by the condensation of abelian ones [58,60]. Non-abelian anyons should be reflected by a systematic degeneracy of the spectrum. To see this degeneracy, note that inversing and negating a cluster leaves its energy unchanged. We call this operation cluster conjugation. Examples for conjugate clusters are −iη 3 (−1, −1, 2) and −iη 3 (−2, 1, 1), depicted in Fig. 1c. Given the aforementioned results, the anyons treated in this manuscript neither behave like the more familiar Ising anyons (or Majorana zero modes [24]) nor like Fibonacci anyons [61] but generally in a more complex way.
Conclusions -On the basis of the general assumptions of [1], we derive an exact quantum many-body formalism for one-dimensional anyons including the exact wave functions, the second quantization, and the momentum discretizing equations for anyons in a box. Employing this formalism, we numerically calculate characteristic observables, namely, the energy spectrum, the momentum statistics, and the finite-size density fluctuations. For a negative statistical parameter, anyons attract each other with a force purely induced by their quantumstatistics and form the quantum-statistical condensate. This genuine many-body phase possesses a rich structure of degenerate internal excitations and is more stable than the Bose-Einstein condensate. In particular, the statistical condensate is stable for charged anyons in the presence of Coulomb repulsion. Because of these properties, we deem the statistical condensate a promising candidate for exhibiting anyonic superconductivity and carrying non-abelian anyons. Our work shows that one-dimensional anyons exhibit original and interesting physics even in the absence of exchangeability.
We would like to acknowledge interesting discussions with Pablo Burset, François Crépin, Daniel Hetterich, Axel Pelster, and Nils Rosehr. BT thanks the DFG for financial support through the SFB 1170 ("ToCoTronics").
[1] J. Leinaas
SUPPLEMENTAL MATERIAL
With this supplemental material, we support the main manuscript by giving an overview about intermediate statistics in one dimension, elaborate on the interpretation of anyon clusters as individual anyons, provide the real space algebra for anyonic creation and annihilation operators, and derive the generalized Jordan-Wigner transformation to convert from an anyonic second quantized algebra to a bosonic one.
Notions of intermediate statistics in one spatial dimension
There exists a variety of formalisms describing particles of intermediate statistics in one dimension, which are expected to be applicable to different physical situations. Albeit they differ in their phenomenology, these particles are all occasionally called anyons. For clarity, we shortly discuss the most broadly known theories that are applicable to one spatial dimension.
If the occupation number of a single particle quantum state is restricted to maximally assume a given integer, the particles can be described as parafermions, which are closely related to Potts and clock models [27,28] and Gentile statistics [26]. Such particles are, amongst others, expected to exist as magnetic excitations [29,30]. Another kind of intermediate statistics considers the representations of the local current algebra (the commutation relations between the particle density and the particle currents in all spatial dimensions) [5,31] or the quantization of the algebra of allowed observables of indistinguishable particles. The latter has been applied to superconducting vortices [32] and two-dimensional anyons effectively confined to one dimension by a strong magnetic field [33]. Yet another notion of anyons in one dimension can be derived from Haldane's generalization of the Pauli principle [34], which is, for instance, applicable to spinon excitations in spin chains. In this approach, the single-particle Hilbert space dimension depends on the total number of particles in the system. Finally, the name anyons is used in one dimension to describe lowenergy quasi-particle excitations of interacting fermionic systems [35] linked to the Calogero-Sutherland model [36,40], the Haldane-Shastry chain, and the fractional excitations in Tomonaga-Luttinger liquids [20,[37][38][39]. It is known that these particles (considering each channel separately in the case of a Tomonaga-Luttinger liquid) break time reversal symmetry on the fundamental level of their quantum brackets, reflected by an asymmetric momentum distribution [20,41]. S1. Geometrical interpretation of the statistical phase of two clusters, each consisting of two anyons. The radius of the circle denotes the relative momentum between clusters. The statistical angle is obtained by adding up three summands. Two of these summands are the normal statistical angle φη, the third summand is the statistical angle of the doubled statistical parameter φ2η. The statistical parameter η appears in this picture as the length of the drawn tangential segments.
φη(p) η η φ2η(p) 2η 2η φη(p) (K2 − K1) η η FIG.
Interpretation of anyon clusters as individual anyons
We want to show how the exchange phase of clusters can be interpreted as the statistical phase of a composite species of anyons reaching further than the interpretation supported by Eq. (8). To this end, we consider two clusters of anyons µ 1 = (K 1 + iη/2, K 1 − iη/2) and µ 2 = (K 2 + iη/2, K 2 − iη/2), the cluster structures of which are depicted in Fig. 1b. We introduce the center of mass coordinates X 1 = (x 1 + x 2 ) /2 and X 2 = (x 3 + x 4 ) /2, as well as the separation coordinates Z 1 = (x 2 − x 1 ) /2 and Z 2 = (x 4 − x 3 ) /2. Under the assumption that the two clusters are sufficiently far away from each other, i.e., X 2 − X 1 → ∞ and Z 1 , Z 2 finite, we obtain Ψ (µ1,µ2) (X 1 , X 2 , Z 1 , Z 2 ) ∝ e 2i(K1X1+K2X2) + e iϕ µ 1 ,µ 2 η e 2i(K2X1+K1X2) e η(Z1+Z2) .
(S1) This wave function obtains the form of a wave function of two composite anyons with an altered statistical phase of ϕ µ1,µ2 η , especially if we recall that Z 1 and Z 2 are of the order of 1/η. This can be physically interpreted as the fusion of anyons to clusters which themselves behave as a composite anyon species. Interestingly, the new statistical phase is ϕ µ1,µ2 η = 2φ η (K 2 − K 1 ) + φ 2η (K 2 − K 1 ),
where φ η is the statistical phase defined in Eq. (5) of the main manuscript. This has an appealing geometric interpretation, which we depict in Fig. S1.
FIG. 2 .
2Observables for confined anyons. The anyonic properties for 0 < η < ∞ continuously interpolate between bosons (η = 0) and fermions (η = ∞). The statistical condensate forms at η < 0. (a) Discrete energy spectrum of two anyons. To depict the full range of η, we plot against φη(1/L). The two-anyon bound state and its excitations emerge for negative η. (b) Momentum density at zero temperature in the limit of infinitely many particles (numerical calculations for n = 512 anyons, where the curves have well converged to the limit n → ∞). (c) Finite size oscillations of the particle density ρ for four anyons.
Real space operator algebra for single anyonsGiven the momentum space operator algebra of Eq.(9), we can ask about the algebra of the real space operators Ψ † (x) = ∞ −∞ dp e ipx √ 2π a † p . We obtainwhere {. . . , . . . } is the anticommutator.Here, lim η→0 ∞ 0 1/η e −z/η f (z) = f (0) yields the bosonic commutation algebra, while the fermionic anticommutation relations for η → ∞ are trivially contained. If we set x = y, we obtain a generalized, smeared Pauli principle in real space represented byGeneralized Jordan Wigner transformation: anyons to bosonsIt is possible to describe the anyonic creation and annihilation operators in terms of bosonic ones, as inspired by the Jordan-Wigner transformation of Ref. 20. Ultimately, this reflects the fact that the Fock space of anyons is isomorphic to the one of bosons (if η = ±∞). To this end, consider the bosonic operators b with the algebra b k , b † l = δ(k − l) and b k , b † l = 0 with k, l ∈ R. For η = ±∞, we define the generalized Jordan-Wigner transformationãCalculating the algebra ofã, we findã jãk =ã kãj e iφη(k−j) andã jã † k =ã † kã j e −iφη(j−k) +δ(j −k), which is exactly the anyonic algebra described in Eq.(9). Interestingly, we haveã † kã k = b † k b k , which results in the same free Hamiltonian using either the bosonic or the anyonic description.Hence, Hamiltonians that are diagonal in momentum space acquire the same form when represented either in anyonic or bosonic degrees of freedom. In order to distinguish between anyons or bosons on a line, it is therefore important to regard scattering processes of anyons, or non-momentum-conserving interactions, or single-particle observables that are not diagonal in momentum space. For finite systems, the difference between anyons and bosons is more obvious by the momentum quantization described in the main manuscript.
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In two dimensions, the constraints to the wave functions are independent of momentum because the exchange of anyons is generally conducted without scattering. In two dimensions, the constraints to the wave functions are independent of momentum because the exchange of anyons is generally conducted without scattering.
There also is a more down-to-earth explanation: the boundary conditions, Eq. (2), induce a local phase shift of the wave function, which is related to an effective local two-particle interaction. similar to [62There also is a more down-to-earth explanation: the boundary conditions, Eq. (2), induce a local phase shift of the wave function, which is related to an effective local two-particle interaction, similar to [62].
The concatenation of tuples means to write their elements in a row to form a combined tuple, i.e., k = µ 1 , . . . , µ m = µ 1 1 , µ 1 2 , . . . , µ m nm −1 , µ m nm. if µ 1 , . . . , µ m are tuples of lengths n1, . . . , nm and k is their concatenationThe concatenation of tuples means to write their ele- ments in a row to form a combined tuple, i.e., k = µ 1 , . . . , µ m = µ 1 1 , µ 1 2 , . . . , µ m nm −1 , µ m nm , if µ 1 , . . . , µ m are tuples of lengths n1, . . . , nm and k is their concate- nation.
For real vectors k, we have N k = 1/ (2π n n!). For real vectors k, we have N k = 1/ (2π n n!).
Technically, we define the vacuum state as Ψ {} = 1. Technically, we define the vacuum state as Ψ {} = 1.
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| []
|
[
"Microscopic approach to large-amplitude deformation dynamics with local QRPA inertial masses",
"Microscopic approach to large-amplitude deformation dynamics with local QRPA inertial masses"
]
| [
"Koichi Sato [email protected] \nRIKEN Nishina Center\n351-0198WakoJapan\n",
"Nobuo Hinohara \nRIKEN Nishina Center\n351-0198WakoJapan\n",
"Takashi Nakatsukasa \nRIKEN Nishina Center\n351-0198WakoJapan\n",
"Masayuki Matsuo \nDepartment of Physics\nNiigata University\n950-2181NiigataJapan\n",
"Kenichi Matsuyanagi \nRIKEN Nishina Center\n351-0198WakoJapan\n\nYukawa Institute for Theoretical Physics\nKyoto University\n606-8502KyotoJapan\n"
]
| [
"RIKEN Nishina Center\n351-0198WakoJapan",
"RIKEN Nishina Center\n351-0198WakoJapan",
"RIKEN Nishina Center\n351-0198WakoJapan",
"Department of Physics\nNiigata University\n950-2181NiigataJapan",
"RIKEN Nishina Center\n351-0198WakoJapan",
"Yukawa Institute for Theoretical Physics\nKyoto University\n606-8502KyotoJapan"
]
| []
| We have developed a new method for determining microscopically the fivedimensional quadrupole collective Hamiltonian, on the basis of the adiabatic self-consistent collective coordinate method. This method consists of the constrained Hartree-Fock-Bogoliubov (HFB) equation and the local QRPA (LQRPA) equations, which are an extension of the usual QRPA (quasiparticle random phase approximation) to non-HFB-equilibrium points, on top of the CHFB states. One of the advantages of our method is that the inertial functions calculated with this method contain the contributions of the time-odd components of the mean field, which are ignored in the widely-used cranking formula. We illustrate usefulness of our method by applying to oblate-prolate shape coexistence in 72 Kr and shape phase transition in neutron-rich Cr isotopes around N = 40. | 10.1088/1742-6596/381/1/012103 | [
"https://arxiv.org/pdf/1111.0359v1.pdf"
]
| 118,425,710 | 1111.0359 | fb5475f1de3fd3ecd5421fe1401b3dc36feeb846 |
Microscopic approach to large-amplitude deformation dynamics with local QRPA inertial masses
2 Nov 2011
Koichi Sato [email protected]
RIKEN Nishina Center
351-0198WakoJapan
Nobuo Hinohara
RIKEN Nishina Center
351-0198WakoJapan
Takashi Nakatsukasa
RIKEN Nishina Center
351-0198WakoJapan
Masayuki Matsuo
Department of Physics
Niigata University
950-2181NiigataJapan
Kenichi Matsuyanagi
RIKEN Nishina Center
351-0198WakoJapan
Yukawa Institute for Theoretical Physics
Kyoto University
606-8502KyotoJapan
Microscopic approach to large-amplitude deformation dynamics with local QRPA inertial masses
2 Nov 2011
We have developed a new method for determining microscopically the fivedimensional quadrupole collective Hamiltonian, on the basis of the adiabatic self-consistent collective coordinate method. This method consists of the constrained Hartree-Fock-Bogoliubov (HFB) equation and the local QRPA (LQRPA) equations, which are an extension of the usual QRPA (quasiparticle random phase approximation) to non-HFB-equilibrium points, on top of the CHFB states. One of the advantages of our method is that the inertial functions calculated with this method contain the contributions of the time-odd components of the mean field, which are ignored in the widely-used cranking formula. We illustrate usefulness of our method by applying to oblate-prolate shape coexistence in 72 Kr and shape phase transition in neutron-rich Cr isotopes around N = 40.
Introduction
The five-dimensional (5D) quadrupole collective Hamiltonian is a powerful tool to describe large-amplitude deformation dynamics including triaxial shape degree of freedom as well as axially symmetric deformation. The 5D quadrupole collective Hamiltonian is characterized by seven functions: the collective potential, three vibrational and three rotational inertial functions (collective masses). To calculate the inertial functions, the Inglis-Belyaev (IB) cranking formula [1,2] has been widely used. However, it is well known that IB cranking masses do not contain the contributions from the time-odd components of the mean field and underestimate the collective masses.
In this presentation, we introduce a new method of determining microscopically the collective potential and the inertial functions which can overcome the shortcoming of the IB cranking formula. This method is derived from the adiabatic self-consistent collective coordinate (ASCC) method [3,4], which is based on the adiabatic time-dependent Hartree-Fock-Bogoliubov (TDHFB) theory and, with which one can extract the collective subspace (collective manifold) from the large-dimensional TDHFB configuration space. The method we employ in this study is the version of the ASCC method simplified by assuming the one-to-one correspondence between the collective manifold and the quadrupole deformation parameter space (β, γ). The main concept of this method is the local normal modes on top of the constrained Hartree-Fock-Bogoliubov (CHFB) states at every point of the (β, γ) plane. In this method, we first solve the CHFB equations imposing the constraints on the proton and neutron numbers and deformation parameters (β, γ). Then, we solve local QRPA (quasiparticle random-phase approximation) equations, which are an extension of the usual QRPA to the non-HFB-equilibrium points. Hereafter, we call the method "the constrained HFB plus local QRPA (CHFB+LQRPA) method."
We shall show some results of the applications of the CHFB+LQRPA method to the oblateprolate shape coexistence in 72 Kr and to the shape phase transition in neutron-rich Cr isotopes around N = 40.
Theoretical framework
Here, we briefly explain the theoretical framework of the CHFB+LQRPA method (see Ref. [5] for details). The 5D quadrupole collective Hamiltonian is written in terms of the magnitude β, the degree of triaxiality γ of quadrupole deformation, three Euler angles, and their time derivatives as
H coll = T vib + T rot + V (β, γ),(1)T vib = 1 2 D ββ (β, γ)β 2 + D βγ (β, γ)βγ + 1 2 D γγ (β, γ)γ 2 ,(2)T rot = 1 2 3 k=1 J k (β, γ)ω 2 k ,(3)
where T vib , T rot and V represent the vibrational, rotational and collective potential energies, respectively. We determine the three vibrational inertial masses, three rotational moments of inertia, and collective potential in the collective Hamiltonian, by solving the CHFB+LQRPA equations.
In the CHFB+LQRPA method, we first solve the CHFB equation to determine the collective potential V (β, γ):
δ φ(β, γ)|Ĥ CHFB (β, γ) |φ(β, γ) = 0,(4)whereĤ CHFB =Ĥ − τ λ (τ )Ñ (τ ) − m µ (m)D 2m
. Then, we solve the LQRPA equations on top of the CHFB states obtained above,
δ φ(β, γ)| [Ĥ CHFB (β, γ),Q i (β, γ)] − 1 iP i (β, γ) |φ(β, γ) = 0,(5)δ φ(β, γ)| [Ĥ CHFB (β, γ), 1 iP i (β, γ)] − C i (β, γ)Q i (β, γ) |φ(β, γ) = 0, (i = 1, 2).(6)
The vibrational inertial functions are calculated by transforming two LQRPA modes to the (β, γ) degrees of freedom. We also solve the LQRPA equations for rotation to determine the moments of inertia. In this work, we adopt a version of the pairing-plus-quadrupole (P+Q) interaction including the quadrupole pairing interaction as well as the monopole pairing interaction and take two major harmonic oscillator shells as a model space.
Finally, we solve the collective Schrödinger equation for the 5D quadrupole collective Hamiltonian quantized according to Pauli's prescription to obtain the excitation energies and collective wave functions, with which the electric quadrupole transitions and moments are calculated.
3. Application to oblate-prolate shape coexistence in 72 Kr In this section, we briefly show the numerical results of the application of the CHFB+LQRPA method to oblate-prolate shape coexistence in 72 Kr. The effective charges are adjusted to [7]. The detailed results of this calculation are shown in Ref. [6] together with the results for 74,76 Kr. Figure 1 shows the collective potential V (β, γ), the vibrational inertial function D ββ (β, γ), and the rotational moment of inertia J 1 (β, γ) for 72 Kr. The collective potential has two local minima: the oblate minimum is lower than the prolate one. The spherical shape is a local maximum. One can see that D ββ indicates strong β − γ dependence and that the moment of inertia J 1 deviates strongly from the irrotational moment of inertia. In Ref. [6], we have seen the collective wave function is localized on the oblate (prolate) side in the (β, γ) plane in the ground (excited) band and that the localization of the wave function develops with increasing angular momentum. The development of the localization is strongly related to the β − γ dependence of the moments of inertia seen in Fig. 1(c).
In Fig. 2, the excitation energies and B(E2) values for 72 Kr are shown together with the experimental data. We show the result obtained using the IB cranking masses for comparison. The excitation energies obtained with the LQRPA masses are lower than those obtained with the IB cranking masses and agree better with the experimental data except for the 2 + 1 state. In particular, the observed 0 + 2 excitation energy which is close to the 2 + 1 excitation energy is well reproduced with the LQRPA masses. One can see from the E2 transition strengths in Fig. 2 that the shape-coexistence-like character becomes stronger with increasing angular momentum: the interband transitions between the initial and final states having equal angular momentum become weaker and weaker, which reflects the development of the localization of the vibrational wave function mentioned above.
Application to shape transition in neutron-rich Cr isotopes
In this section, we shall show the results of the application to the CHFB+LQRPA method to shape phase transition in neutron-rich Cr isotopes around N = 40, where the development of deformation has been suggested by recent experimental data. More details will be discussed in our forthcoming paper [11].
In this calculation, we have determined the interaction parameters as follows: the parameters for 62 Cr are determined such that the calculated monopole pairing gaps and deformation at the HFB equilibrium point as well as the pairing gaps at the spherical shape reproduce those obtained with the Skyrme-HFB calculation with the SkM* functional using the HFBTHO code [12]. For the other nuclei 58,60,64,66 Cr, we assume the simple mass number dependence according to Baranger and Kumar [13]. We take two major harmonic oscillator shells with N = 3, 4 and N = 2, 3 for neutrons and protons, respectively. The single particle energies are determined from those obtained with the constrained Skyrme-HFB calculation at spherical shape. We scale them according to the effective mass of the SkM* functional m * /m = 0.79. For the calculation of the E2 transitions and moments, we have used a standard value of the effective charges (e (n) eff , e (p) eff ) = (0.5, 1.5) . We show the collective potentials for 58−64 Cr in Fig. 3. The location of the absolute minimum is indicated by the triangle. In 58 Cr, the absolute minimum is located at a nearly spherical shape. Although the minimum shifts to larger β in 60 Cr, the potential is extremely soft in the β direction. In 62 Cr, a more definite local minimum appears and the minimum gets still deeper in 64 Cr. The potential energy surfaces seem to indicate a shape transition from a spherical to a prolately deformed shape along the isotopic chain toward N = 40.
We show in Fig. 4 the calculated excitation energies of the 2 + 1 and 4 + 1 states, their ratios R 4/2 = E(4 + 1 )/E(2 + 1 ), B(E2; 2 + 1 → 0 + 1 ), and the spectroscopic quadrupole moments of the 2 + 1 states Q(2 + 1 ) calculated for 58−66 Cr in comparison with the available experimental data. Both of the experimental excitation energies and R 4/2 are reproduced well. The decrease in the excitation energies and the increase in R 4/2 reflect the development of deformation with increasing the neutron number. The E2 transition and moments also indicate the enhancement of collectivity: the magnitude of B(E2) and Q(2 + 1 ) increase as the neutron number increases and they take a maximum at N = 40.
Concluding Remarks
We have proposed the CHFB+LQRPA method for determining the inertial functions in the 5D quadrupole collective Hamiltonian, with which one can take into account the contributions from the time-odd components of the mean field to the inertial functions. We applied this method to the oblate-prolate shape coexistence in the low-lying states of 72 Kr and shape transition in neutron-rich Cr isotopes around N = 40. The calculated results are in good agreement with the available experimental data.
The CHFB+LQRPA method is based on the ASCC method and it can be used in conjunction with any interaction in principle. Nevertheless, in this study, we have employed a rather simple interaction, the P+Q model including the quadrupole pairing interaction, for simplicity. The implementation with modern energy density functionals, such as Skyrme energy functionals and density-dependent pairing interaction is an issue for future under progress. Figure 4. Excitation energies of the 2 + 1 and 4 + 1 states, their ratios R 4/2 , B(E2; 2 + 1 → 0 + 1 ), and the spectroscopic quadrupole moments of the 2 + 1 states in comparison with the available experimental data [14,15,16,17]. The B(E2) values are shown in Weisskopf units. work is supported by KAKENHI (Nos. 21340073 and 20105003).
Figure 1 .
1Collective potential energy surfaces V (β, γ) in units of MeV, LQRPA vibrational inertial mass D ββ (β, γ) and LQRPA rotational moment of inertia J 1 (β, γ) in unit of MeV −1 for 72 Kr. = (0.658, 1.658) such that the calculated result of B(E2; 2 + 1 → 0 + 1 ) reproduces the experimental value for 74 Kr
Figure 2 .
2Excitation spectra and B(E2) values calculated for 72 Kr by means of the CHFB+LQRPA method (denoted LQRPA) and experimental data[8,9,10]. For comparison, results calculated using the IB cranking masses (denoted Inglis-Belyaev) are also shown. Only B(E2)'s larger than 1 Weisskopf unit are shown in units of e 2 fm 4 .
(Note that Q(2 + 1 )'s are negative indicating prolate shapes.) To sum up, our result suggests the development of prolate deformation from N = 34 to N = 40 and the largest collectivity at N = 40.
Figure 3 .
3Collective potential energy surfaces V (β, γ) for 58−64 Cr in units of MeV. The triangle denotes the position of the absolute minimum.
AcknowledgmentsOne of the authors (N. H.) is supported by the Special Postdoctoral Researcher Program of RIKEN. The numerical calculations were carried out on SR16000 at Yukawa Institute for Theoretical Physics in Kyoto University and RIKEN Cluster of Clusters (RICC) facility. This
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[
"IDEAL GROWTH IN AMALGAMATED POWERS OF NILPOTENT RINGS AND ZETA FUNCTIONS OF QUIVER REPRESENTATIONS",
"IDEAL GROWTH IN AMALGAMATED POWERS OF NILPOTENT RINGS AND ZETA FUNCTIONS OF QUIVER REPRESENTATIONS"
]
| [
"Tomer Bauer ",
"Michael M Schein "
]
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| []
| Let L be a nilpotent algebra of class two over a compact discrete valuation ring A of characteristic zero or of sufficiently large positive characteristic. Let q be the residue cardinality of A. The ideal zeta function of L is a Dirichlet series enumerating finite-index ideals of L. We prove that there is a rational function in q, q m , q −s , and q −ms giving the ideal zeta function of the amalgamation of m copies of L over the derived subring, for every m ≥ 1, up to an explicit factor. More generally, we prove this for the zeta functions of nilpotent quiver representations of class two defined by Lee and Voll, and in particular for Dirichlet series counting graded submodules of a graded A-module. If the algebra L, or the quiver representation, is defined over Z, then we obtain a uniform rationality result. | null | [
"https://arxiv.org/pdf/2202.10865v1.pdf"
]
| 247,025,725 | 2202.10865 | 51f5ccd4621f8795eb26c7be3e47c0decb98fea5 |
IDEAL GROWTH IN AMALGAMATED POWERS OF NILPOTENT RINGS AND ZETA FUNCTIONS OF QUIVER REPRESENTATIONS
22 Feb 2022
Tomer Bauer
Michael M Schein
IDEAL GROWTH IN AMALGAMATED POWERS OF NILPOTENT RINGS AND ZETA FUNCTIONS OF QUIVER REPRESENTATIONS
22 Feb 2022
Let L be a nilpotent algebra of class two over a compact discrete valuation ring A of characteristic zero or of sufficiently large positive characteristic. Let q be the residue cardinality of A. The ideal zeta function of L is a Dirichlet series enumerating finite-index ideals of L. We prove that there is a rational function in q, q m , q −s , and q −ms giving the ideal zeta function of the amalgamation of m copies of L over the derived subring, for every m ≥ 1, up to an explicit factor. More generally, we prove this for the zeta functions of nilpotent quiver representations of class two defined by Lee and Voll, and in particular for Dirichlet series counting graded submodules of a graded A-module. If the algebra L, or the quiver representation, is defined over Z, then we obtain a uniform rationality result.
Let m ∈ N. We define the (direct) amalgamated m-th power L * m to be the amalgamated product of m copies of L over their derived sublattices. More precisely, given x ∈ L and 1 ≤ i ≤ m set e i (x) ∈ L m to be the m-tuple with x in the i-th component and 0 elsewhere. Then
L * m = L m / e i (x) − e j (x) : x ∈ [L, L], 1 ≤ i, j ≤ m .
A motivation of this note is to study the behavior of ideal zeta functions with respect to amalgamated powers.
In this paper A will denote a discrete valuation ring that is compact or, equivalently, complete with finite residue field. Let q be the residue cardinality. For the abelian A-lattice A n with [x, y] = 0 for all x, y ∈ A n , it is known (cf. [13,Proposition 1.1]) that (1) ζ ⊳A A n (s) =
n−1 i=0 (1 − q i−s ) −1 .
An A-Lie lattice is an A-lattice in which the multiplication is anti-symmetric and satisfies the Jacobi identity. Consider now the Heisenberg Z-Lie lattice H = x, y, z Z with product determined by [x, y] = z and z ∈ Z(H). It was shown by Grunewald, Segal, and Smith [13,Proposition 8.4] that (2) ζ ⊳A (H⊗ Z A) * m (s) = ζ ⊳A A 2m (s)
1 1 − q 2m−(2m+1)s .
The ideal zeta functions ζ ⊳A (H r ⊗ Z A) * m (s), for r ∈ N, were computed by the first author [1]; while they are considerably more complicated than the right-hand side of (2), the dependence on m is of the same type. See [4, §5.4] for further variations. It is thus natural to ask whether the form of the dependence of ζ ⊳A (H⊗ Z A) * m (s) on m exhibited in (2) is a general phenomenon. We prove that it is. Theorem 1.1. Let n, d ∈ N. There exists M n,d ∈ N such that the following holds:
Let A be a compact discrete valuation ring either of characteristic zero or of characteristic at least M n,d . Let q be the residue cardinality of A. Let L be a nilpotent A-lattice of class at most two such that L/[L, L] is a free A-module of rank n and [L, L] is a free A-module of rank d. There exists a rational function W L ∈ Q(X, Y 0 , Y 1 , Y 2 ) such that ζ ⊳A L * m (s) = ζ ⊳A A mn (s)W L (q, q m , q −s , q −ms ) for all m ∈ N and all s ∈ C with Re s ≫ 0. The denominator of W L is a product of factors of the form (1 − X b Y a 0 0 Y a 1 1 Y a 2 2 ) with a 0 , a 1 , a 2 ∈ N ∪ {0} and b ∈ Z.
For any nilpotent A-lattice L, the ideal zeta function ζ ⊳A L (s) may be expressed as a rational function in q −s ; see [13,Theorem 3.5] for the first general theorem of this form. Theorem 1.1 may be viewed as a refinement of this result that makes precise the dependence of ζ ⊳A L * m (s) on m; see Remark 1.4 below. Since only one residue cardinality q is considered in Theorem 1.1, the numerator of W L is not uniquely defined. However, if the lattice L is defined over Z, and hence may naturally be interpreted by extension of scalars as a lattice over any compact discrete valuation ring A, then we prove a version of Theorem 1.1 that is uniform over all such A of sufficiently large residue characteristic. Theorem 1.2. Let L be a nilpotent Z-lattice of class at most two, and suppose that L/[L, L] is a free Z-module of rank n. Then there exist r, g ∈ N, rational functions W 1 , . . . , W r ∈ Q(X, Y 0 , Y 1 , Y 2 ) with denominators of the same form as in Theorem 1.1, and a collection of g-ary formulae ψ 1 , . . . , ψ r in the language of rings, such that the following holds for all compact discrete valuation rings A of sufficiently large residue characteristic and all m ∈ N: For all s ∈ C with Re s ≫ 0 we have
ζ ⊳A (L⊗ Z A) * m (s) = ζ ⊳A A mn (s) r i=1 m i (k)W i (q, q m , q −s , q −ms ),
where k is the residue field of A, with q = |k| and m i (k) = |{ξ ∈ k g : k |= ψ i (ξ)}|. Remark 1.3. If L = H, then (2) fits into the framework of Theorem 1.2 as follows:
ζ ⊳A (H⊗ Z A) * m (s) = ζ ⊳A A 2m (s) m(k) W (q, q m , q −s , q −ms ), where m(k) = |{ξ ∈ k : k |= ψ(ξ)}| for ψ = {ξ = 0} and W (X, Y 0 , Y 1 , Y 2 ) = 1 1−Y 2 0 Y 1 Y 2 2 .
Remark 1.4. A consequence of Theorems 1.1 and 1.2 is that taking amalgamated powers does not increase the complexity of ideal zeta functions, apart from the explicit factor ζ ⊳A A mn (s). Indeed, let L be a nilpotent Z-lattice of class at most two as in Theorem 1.2. Then, for all compact discrete valuation rings A of sufficiently large residue characteristic, the ideal zeta function ζ ⊳A L * m (s) has the form
ζ ⊳A L * m (s) = ζ ⊳A A mn (s) M i=1 κ i q a i0 m+a i1 −(b i0 m+b i1 )s N i=1 (1 − q c i0 m+c i1 −(d i0 m+d i1 )s ) for suitable coefficients κ i ∈ Q and a ij , b ij , c ij , d ij ∈ Z that are independent of m.
Thus, apart from the explicit factor ζ ⊳A A mn (s), the only dependence of ζ ⊳A L * m (s) on m is that the exponents of monomials appearing in the rational function are linear in m.
The hypothesis of nilpotency class at most two in our results is truly necessary. The ideal zeta functions ζ ⊳Zp M 3 ⊗ Z Zp (s) and ζ ⊳Zp (M 3 ⊗ Z Zp) * 2 (s) are computed in Theorems 2.26 and 2.30 of [12]; here M 3 is a certain nilpotent Z-Lie lattice of class three and M * 2 3 is the amalgamation of two copies of M 3 over the center. The latter function is much more complex than the former, indicating that no analogue of Theorem 1.2 applies.
The generality of Theorems 1.1 and 1.2 stands in stark contrast to the difficulty of computing ideal zeta functions explicitly. No explicit ideal zeta functions are known for nilpotent A-lattices of class five or greater. Even in class two, the known examples either have small rank, such as those listed in [13, §8] and [12, §2], or have a particular combinatorial structure rendering them tractable, such as those considered in [10,21,22,26,4]. The rational functions describing ideal zeta functions of base extensions to compact discrete valuation rings of nilpotent Z-lattices of class two generically satisfy functional equations upon inversion of the variables [24, Theorem C].
1.2. Counting normal subgroups. Theorem 1.2 may be interpreted in the context of counting normal subgroups of finitely generated nilpotent groups. Indeed, let G be a finitely generated torsion-free nilpotent group. Grunewald, Segal, and Smith [13] introduced the function
(3) ζ ⊳ G (s) = H G [G : H] −s ,
where the sum runs over all normal subgroups of G of finite index. They proved the existence of an Euler decomposition ζ ⊳ G (s) = p ζ ⊳ G,p (s), where the product runs over all primes and ζ ⊳ G,p (s) is defined as in (3) but counts only the normal subgroups of p-power index. If G is nilpotent of class two, then L(G) = G/Z(G) × Z(G) is a nilpotent Z-Lie lattice of class two, and there is a natural correspondence between normal subgroups of G and ideals of L(G) that preserves inclusion and index. Thus ζ ⊳ G (s) = ζ ⊳,Z L(G) (s) and ζ ⊳ G,p (s) = ζ ⊳Zp L(G)⊗ Z Zp (s) for all primes p. Any class-two nilpotent Z-Lie lattice L is isomorphic to L(G) for a suitable group G, and it is easy to see that the operation L * m corresponds to amalgamating m copies of G over the derived subgroup [G, G]. . . , ψ r and rational functions W 1 , . . . , W r ∈ Q(X, Y 1 , Y 2 ) such that, for any number field K of degree m and any prime p such that L ⊗ Z Z p is rigid in the sense of [3, Definition 3.8], we have
(4) ζ ∧Zp (L⊗ Z Zp)⊗ Z O K (s) = p|p r i=1 m i (k p ) W i (q p , q m p , q −s p ) ,
where the product runs over places p of K dividing p, and q p is the cardinality of the residue field k p = O K /p. Here m i (k p ) = |{ξ ∈ k e p : k p |= ψ i (ξ)}|. Interpreting (4) as a rational function in q p and q −s p , this means that the exponents of q p depend linearly on m, while all other coefficients and exponents are independent of m.
Although [3,Theorem 1.3(2)] involves a fine Euler decomposition, whereas Theorem 1.2 includes the factor ζ ⊳A A mn (s), the two theorems and their proofs share a similar spirit. In both cases, the relevant zeta functions are interpreted as p-adic integrals whose domains of integration are definable sets and integrands are powers (possibly depending on m) of valuations of definable functions that do not depend on m. The theorems thus become rationality and uniformity results for such integrals and are proved using established arguments from the theory of p-adic integration.
By contrast, pro-isomorphic zeta functions do not behave well under amalgamated products. For instance, the functions ζ ∧A (H⊗ Z A) * m (s) are computed in [11, Example 1.4(2)] for m ∈ {2, 3} and [3, Theorem 5.10] in general; their complexity appears to grow factorially with m.
Quiver representations.
Our main results hold in greater generality than presented above. We take the point of view of zeta functions of nilpotent quiver representations over rings, which were introduced by Lee and Voll [15]. Let Q = (V, E) be a finite quiver, namely a directed graph, possibly containing loops and multiple edges. Here V and E are the sets of vertices and edges of Q, respectively. Given an edge e ∈ E, we denote its tail by t(e) and its head by h(e). The definitions below are tailored to the specific cases considered in this paper; we refer the reader to [15] for further details and a more general perspective on their zeta functions. In particular, the A-representations of Q called admissible below are the homogeneous nilpotent A-representations of class at most two that are free of finite rank in the terminology of [15].
We denote N 0 = N ∪ {0} and [n] = {1, 2, . . . , n} for all n ∈ N. For the rest of the introduction, A is an arbitrary ring unless otherwise stated.
Definition 1.5. Let A be a ring, and let Q be a quiver.
(1) An A-representation of Q is a pair (L, f ), where L = {L v } v∈V is a family of A-modules and f = {f e : L t(e) → L h(e) } e∈E is a family of A-module homomor- phisms. We often write L for (L, f ). (2) An A-representation (L, f ) is called admissible if for every v ∈ V there is a decomposition L v = L v,1 ⊕ L v,2 such that L v,1 and L v,2 are free A-modules of finite rank, and if 0 ⊕ L t(e),2 ⊆ ker f e and im f e ⊆ 0 ⊕ L h(e),2 for every e ∈ E. (3) Given an admissible A-representation L, set n v,i = rk A L v,i for v ∈ V and i ∈ [2]. The rank vector of L is n(L) = (n v,1 , n v,2 ) v∈V ∈ (N 2 ) V . (4) A (finite-index) A-subrepresentation of L is a family Λ = {Λ v ≤ L v } v∈V of A-submodules such that [L v : Λ v ] < ∞ for every v ∈ V and f e (Λ t(e) ) ⊆ Λ h(e) for every e ∈ E. Note that (Λ, {(f e ) | Λ t(e) } e∈E ) is an A-representation of Q. (5) The zeta function of L is the Dirichlet series ζ L (s) = Λ≤L v∈V [L v : Λ v ] −sv .
Here the sum runs over all finite-index subrepresentations of L, there is a complex variable s v for every v ∈ V , and s = (s v ) v∈V . . The previous definition is compatible with the notion of amalgamated powers of Alattices via the dictionary of Remark 1.6. We can now state our main results in terms of zeta functions of quiver representations. Theorem 1.8. Let Q = (V, E) be a quiver, and let n ∈ (N 2 ) V . There exists M n ∈ N such that the following holds:
Let A be a compact discrete valuation ring either of characteristic zero or of characteristic at least M n , and let q be its residue cardinality. Let L be any admissible A-representation of Q with rank vector n(L) = n. There exists a rational function
W L ∈ Q(X, Y 0 , Y 1 , Y 2 ), where Y 1 = (Y 1,v ) v∈V and Y 2 = (Y 2,v ) v∈V , such that ζ L * m (s) = v∈V ζ ⊳A A mn v,1 (s v ) W L (q, q m , (q −sv , q −msv ) v∈V ) for all m ∈ N and s ∈ C V with Re s v ≫ 0 for all v ∈ V . The denominator of W L is a product of factors of the form (1 − X b Y a 0 0 v∈V (Y a 1,v 1,v Y a 2,v 2,v )) with a 0 , a 1,v , a 2,v ∈ N 0 and b ∈ Z.
Given a Z-representation L of Q, we may naturally extend scalars to any ring A to obtain an A-representation L ⊗ Z A of Q; cf. [15, §1.1]. Zeta functions of base extensions to compact discrete valuation rings of admissible quiver Z-representations generically satisfy functional equations [15,Theorem 1.7], generalizing the result for ideal zeta functions mentioned above. By Remark 1.6 the following claim generalizes Theorem 1.2, just as Theorem 1.8 generalizes Theorem 1.1. Theorem 1.9. Let L be an admissible Z-representation of a quiver Q. There exist r, g ∈ N, rational functions W 1 , . . . , W r ∈ Q(X, Y 0 , Y 1 , Y 2 ), and a collection of g-ary formulae ψ 1 , . . . , ψ r in the language of rings, such that the following holds for all compact discrete valuation rings A of sufficiently large residue characteristic and for all m ∈ N:
for all s ∈ C V such that Re s v ≫ 0 for all v ∈ V we have ζ (L⊗ Z A) * m (s) = v∈V ζ ⊳A A mn v,1 (s v ) r i=1 m i (k)W i (q, q m , (q −sv , q −msv ) v∈V ),
where k is the residue field of A, with q = |k| and m i (k) = |{ξ ∈ k g : k |= ψ i (ξ)}|.
1.5. Structure of the paper. In Section 2 below we obtain three auxiliary results on which the proofs of our main theorems rely. Proposition 2.1 states that certain p-adic integrals, over a fixed local field, are expressed by rational functions. Proposition 2.2 is a uniformity result for p-adic integrals of a particular type, whereas the crucial combinatorial Lemma 2.3 counts A-submodules Λ of a direct product L m such that the sum of the projections of Λ to the components of L m is a fixed submodule of L. In Section 3 we carry out the program outlined in Section 1.3 above by expressing the zeta functions of amalgamated powers of admissible quiver representations as p-adic integrals and applying the results of Section 2 to prove Theorems 1.8 and 1.9.
Preliminaries
2.1.
A rationality result. Let Loc denote the collection of pairs (F, π F ), where F is a non-Archimedean local field with normalized additive valuation v F and π F ∈ F is a uniformizer; we will generally omit π F in the notation. Write O F for the valuation ring of F ∈ Loc, and let k F = O F /(π F ) be the residue field and q F = |k F | its cardinality. The multiplicative valuation on F is given by
|x| F = q −v F (x) F
for x ∈ F . For every n ∈ N, we denote by µ F the Haar measure on F n normalized so that µ F (O n F ) = 1. Let Loc ≫ ⊂ Loc denote the collection of F ∈ Loc with char k F ≥ M for a suitable M . Let Loc 0 ⊂ Loc denote the collection of local fields of characteristic zero.
Let F ∈ Loc 0 . A set Y ⊂ F ℓ is semi-algebraic [9, Definition 1.2] if it may be constructed from sets of the form {x ∈ F ℓ : ∃z ∈ F, f (x) = z n }, where n ≥ 2 and f ∈ F [x 1 , . . . , x ℓ ], by taking finitely many unions, intersections, and complements. By Theorem 1.3 and Lemma 2.1 of [9], a set Y ⊂ F ℓ is semi-algebraic if and only if Y is definable in Macintyre's language [16]. A function is called semi-algebraic if its graph is a semi-algebraic set [9, Remark 1.5].
The following claim generalizes Denef's classical rationality theorem [8,Theorem 7.4]. If s = (s 1 , . . . , s n ) ∈ C n and α = (α 1 , . . . , α n ) ∈ R n , we say that Re s > α if Re s i > α i for all i ∈ [n].
Proposition 2.1. Let F ∈ Loc 0 , let Y ⊂ F ℓ be a semi-algebraic set, and consider semi-algebraic functions f 0 , f 1 , . . . , f n : Y → F . If there exists α ∈ (R >0 ) n such that the function
I(s 1 , . . . , s n ) = Y |f 0 (y)| F |f 1 (y)| s 1 F · · · |f n (y)| sn F dµ F
is defined for all s = (s 1 , . . . , s n ) ∈ C n such that Re s > α, then I(s 1 , . . . , s n ) is a rational function in the variables q −s 1 F , . . . , q −sn F . The denominator is a product of factors of the form (1 − q −a 0 −a 1 s 1 −···−ansn F ) with (a 1 , . . . , a n ) ∈ N n 0 and a 0 ∈ Z such that (a 0 , . . . , a n ) = (0, . . . , 0).
Proof. We imitate the proof of [7,Corollary 15], which is our claim in the case n = 1.
For every t ∈ N set O (t) F = {π v F (1 + π t F α) : v ∈ N 0 , α ∈ O F } ⊂ O F .
By [7,Theorem 7] there is a finite partition of Y such that each part B either has measure zero (this corresponds to the map g being a composition of bijections that include maps of the type f 0 , in the notation of [7, Theorem 8]), or there exists a bijective semi-algebraic
function ϕ : (O (t) F ) ℓ → B such that, for every 0 ≤ j ≤ n, we have v F (f j (ϕ(x))) = v F β j ℓ i=1 x µ ij i
for some β j ∈ F and (µ 1j , . . . , µ ℓj ) ∈ Z ℓ . Moreover, the valuation of the Jacobian determinant of ϕ has the same form. Thus, possibly adjusting the integers µ i0 to absorb the Jacobian, we decompose I(s 1 , . . . , s n ) into a sum of finitely many integrals of the form
(O (t) F ) ℓ β 0 ℓ i=1 x µ i0 i F n j=1 β j ℓ i=1 x µ ij i s j F dµ F .
Since the integrand decomposes into a product of factors depending on a single-variable
x i , we may assume without loss of generality that ℓ = 1. Then we obtain the integral
O (t) F |β 0 x µ 0 | F n j=1 |β j x µ j | s j F dµ F (x) = ∞ v=0 µ F (π v F (1 + π t F O F ))|β 0 | F n j=1 |β j | s j F q −v(µ 0 +µ 1 s 1 +···µnsn) F = q −t F |β 0 | F n j=1 |β j | s j F 1 − q −1−µ 0 − n j=1 µ j s j F = q −t−v F (β 0 )− n j=1 v F (β j )s j F 1 − q −1−µ 0 − n j=1 µ j s j F .
Here the second equality follows by our assumptions regarding the convergence of I(s 1 , . . . , s n ). Note that all terms of the series above are positive, so we must have µ j ≥ 0 for all j ∈ [n]. Moreover, if µ j = 0 for all j ∈ [n], then µ 0 + 1 > 0. This completes the proof.
2.2.
A uniformity result. The rationality result of Proposition 2.1 may be upgraded to a claim providing uniformity as F varies over all local fields of sufficiently large residue characteristic.
The Denef-Pas language L DP [18, Definition 2.3] has three sorts: the valued field sort VF and the residue field sort RF are endowed with the language of rings, and the valued group sort VG, which we will simply call Z, is endowed with the Presburger language of ordered abelian groups. Moreover, L DP has two function symbols: v : VF \ {0} → VG and ac : VF → RF, interpreted as a valuation map and an angular component map, respectively. Any formula ϕ in L DP with c 1 free VF-variables, c 2 free RF-variables, and c 3 free Z-variables yields a subset ϕ(F ) ⊆ F c 1 × k c 2 F × Z c 3 for any F ∈ Loc. A collection X = (X F ) F ∈Loc≫ with X F = ϕ(F ) is called an L DP -definable set. A collection of functions f = (f F : X F → Y F ) F ∈Loc≫ is called an L DP -definable function if the associated collection of graphs is an L DP -definable set.
Proposition 2.2. Let Y = (Y F ) F be an L DP -definable subset of VF c , let n ∈ N, and let Φ 0 , Φ 1 , . . . , Φ n : Y → Z be L DP -definable functions. Suppose that there exists α ∈ R n such that for all F ∈ Loc ≫ the integral Y F q Φ 0 (y)+Φ 1 (y)s 1 +···+Φn(y)sn F dµ F converges for all s = (s 1 , . . . , s n ) ∈ C n such that Re s > α.
Then there exist r, g ∈ N, a collection of g-ary formulae ψ 1 (ξ), . . . , ψ r (ξ) in the language of rings, and rational functions W 1 , ..., W r ∈ Q(X, Y 1 , . . . , Y n ) such that the following holds for all F ∈ Loc ≫ and all s ∈ C n satisfying Re s > α:
Y F q Φ 0 (y)+Φ 1 (y)s 1 +···+Φn(y)sn F dµ F = r i=1 m i (k F ) · W i (q F , q −s 1 F , . . . , q −sn F ),
where m i (k F ) denotes the cardinality of the set {ξ ∈ k g F : k F |= ψ i (ξ)} for all i ∈ [r]. Moreover, for each i ∈ [r] the denominator of W i is a product of factors of the form (1−X a 0 Y a 1 1 · · · Y an n ) for (a 1 , . . . , a n ) ∈ N n 0 and a 0 ∈ Z such that (a 0 , . . . , a n ) = (0, . . . , 0). . Moreover, it generalizes in an obvious way to arbitrary n ∈ N.
2.3.
A combinatorial lemma. The following observation is key to the method of this paper. Let A be a compact discrete valuation ring.
Λ ≤ Ω such that m i=1 ϕ i (π i (Λ)) = H. Then Λ∈S H [Ω : Λ] −s = ζ ⊳A A mn (s) ζ ⊳A A n (ms) [A n : H] −ms .
Proof. Fix a uniformizer π ∈ A. There exists an A-basis (β 1 , . . . , β n ) of A n and integers ν 1 , . . . , ν n ∈ N 0 such that H is the A-linear span of {π ν 1 β 1 , . . . , π νn β n }. Set β
(i) j = ϕ −1 i (β j ) for every i ∈ [m] and j ∈ [n]. Then {β (i) j : (i, j) ∈ [m] × [n]} is an A-basis of Ω. Consider the A-module endomorphism ψ : Ω → Ω determined by ψ(β (i) j ) = π ν j β (i) j . Let ψ be the endomorphism of A n given by ψ(β j ) = π ν j β j . For any finite-index sublattice Λ ≤ Ω, it is clear that m i=1 ϕ i (π i (ψ(Λ))) = ψ m i=1 ϕ i (π i (Λ)) .
In particular, since ψ(A n ) = H, the map ψ induces a map S A n → S H . This map is injective since ψ is. Moreover, it is easy to see that if Λ ≤ H, then there exists an A-sublattice Λ ′ ≤ Ω satisfying Λ = ψ(Λ ′ ). Thus ψ induces a bijection between S A n and S H . Moreover, where the sums run over finite-index sublattices. The claim is immediate from (5) and (6).
Ideal growth in nilpotent Lie lattices
Fix a quiver Q = (V, E). Throughout this section, A is a compact discrete valuation ring with residue field of cardinality q.
′ = (H ′ v ), we write H ≤ H ′ if H v ≤ H ′ v is a finite-index A-submodule for all v. For H ≤ L 1 and v ∈ V , set f (H) v = e∈E h(e)=v f e (H t(e) ) ≤ L v,2 .
This defines a family f (H) ≤ L 2 . The following claim, for which admissibility is essential, is the analogue for quiver representations of [13, Lemma 6.1].
Lemma 3.1. Let L be an admissible A-representation of Q. Then ζ L (s) = H≤L 1 M ≤L 2 f (H)≤M v∈V [L v,1 : H v ] −sv [L v,2 : M v ] n v,1 −sv .
Proof. For every v ∈ V , let b v,1 and b v,2 be A-bases of L v,1 and L v,2 , respectively. Concatenate them to obtain an A-basis of L v . For every finite-index A-submodule Λ v ≤ L v , there exists a unique matrix Mat(Λ v ) in Hermite normal form such that Λ v is spanned by the rows of Mat(Λ v ), viewed as elements of L v with respect to the chosen basis. Consider Mat(Λ v ) as a block matrix The next proposition allows us to rewrite the zeta function ζ L * m (s) as a product of an explicit factor and an infinite sum in which only the summands depend on m, but the set parametrizing them is independent of m.
Mat(Λ v ) = Mat(H v ) C v 0 Mat(M v ) , where H v ≤ L v,1 and M v ≤ L v,2 are finite-index A-submodules and C v is an n v,1 × n v,2 matrix with entries in A. It is clear that Λ = (Λ v ) v∈V is an A-subrepresentation of L if and only if f (H) ≤ M . Clearly [L v : Λ v ] = [L v,1 : H v ][L v,2 : M v ],ζ L * m (s) = v∈V ζ ⊳A A mn v,1 (s v ) ζ ⊳A A n v,1 (ms v ) H≤L 1 M ≤L 2 f (H)≤M v∈V [L v,1 : H v ] −msv [L v,2 : M v ] mn v,1 −sv .
Proof. For v ∈ V and i ∈ [m], let π v,i : L m v,1 → L v,1 be the projection onto the i-th component and let ϕ v,i : π v,i (L m v,1 ) ∼ → L v,1 be the natural identification, so that for each v we are in the setup of Lemma 2.3 with Ω = (L * m ) v = L m v,1 . By Lemma 3.1 we have
(7) ζ L * m (s) = Λ≤L m 1 M ≤L 2 f * m (Λ)≤M v∈V [L m v,1 : Λ v ] −sv [L v,2 : M v ] mn v,1 −sv . Given Λ ≤ L m 1 , set S(Λ) v = m i=1 ϕ v,i (π v,i (Λ v )) for every v ∈ V . Observe that f * m (Λ) v = e∈E h(e)=v m i=1 f e,i (Λ t(e) ) = e∈E h(e)=v f e m i=1 ϕ t(e),i (π t(e),i (Λ t(e) )) = f (S(Λ)) v .
Hence the right-hand side of (7) may be expressed as
H≤L 1 M ≤L 2 f (H)≤M Λ≤L m 1 S(Λ)=H v∈V [L m v,1 : Λ v ] −sv [L v,2 : M v ] mn v,1 −sv .
Our claim follows from Lemma 2.3, since S(Λ) = H if and only if Λ v ∈ S Hv for all v ∈ V in the notation of that lemma.
3.2.
Restatement in terms of p-adic integrals. Let F be the fraction field of A, and fix a uniformizer π ∈ A. Then (F, π) ∈ Loc and A = O F [23, §II. [4][5]. Let (L, f ) be an admissible A-representation of Q, and consider the infinite sum (8)
H≤L 1 M ≤L 2 f (H)≤M v∈V [L v,1 : H v ] −msv [L v,2 : M v ] mn v,1 −sv ,
which by Proposition 3.2 is equal to ζ L * m (s) up to an explicit factor. Our aim is to express it as a p-adic integral to which the results of Section 2 are applicable. This type of argument, for sums running over subgroups, was introduced in [13, §2]. Consider the additive algebraic group T r of upper triangular r × r matrices, defined over Z. For any F ∈ Loc we identify T r (F ) with F ( r+1 2 ) in the natural way and let µ F be the additively invariant measure on T r (F ) normalized so that µ F (T r (O F )) = 1. For M ≤ U as above, let T (M) ⊂ T r (A) be the set of all matrices C ∈ T r (A) such that the rows of C, interpreted as elements of U with respect to the A-basis (b 1 , . . . , b r ), constitute a good basis of M. The following claim is analogous to [13,Lemma 2.5].
µ F (T (M)) = (1 − q −1 ) r q − r i=1 iλ i , where λ i , for all i ∈ [r], is determined by q λ i = [U i : (M ∩ U i ) + U i+1 ].
Moreover, for any C = (c ij ) ∈ T (M) the following holds:
[U : M] = q λ 1 +···+λr = r i=1 |c ii | −1 F .
Proof. We construct T (M) inductively from the bottom row up. For any i ∈ [r] the i-th row of a matrix C ∈ T (M) necessarily has a leading term of additive valuation v F (c ii ) = λ i . This implies the second part of our claim, since [U : M] = | det C| −1 F for any C ∈ T (M). If the lower rows of C have been constructed already, then the i-th row is well-defined modulo addition of A-multiples of the lower rows. Thus the set of possible matrix entries c ii has measure (1 − q −1 )q −λ i in F , whereas the set of possible (r − i)-tuples (c i,i+1 , . . . , c ir ) has measure q −(λ i+1 +···+λr) in F r−i . It is clear that the conditions of Definition 3.5 can be expressed by an L DP -formula ψ((C v,1 , C v,2 ) v ) that depends only on the quiver Q, the rank vector n(L), and the structure constants a e ij ∈ A. Observe that
For every v ∈ V , fix A-bases b v = (b v 1 , . . . , b v n v,1 ) of L v,1 and β v = (β v 1 , . . . , β v n v,2 ) of L v,(C v,1 , C v,2 ) v ∈ v∈V (T n v,1 (F ) × T n v,2 (F )) = v∈V F ( n v,1 +1 2 )+( n v,2+1(9) Y L = H≤L 1 M ≤L 2 f (H)≤M v∈V T (H v ) × T (M v ),
where the union is disjoint. Thus the expression (8) may be rewritten as (10)
H≤L 1 M ≤L 2 f (H)≤M v∈V T (Hv )×T (Mv) [L v,1 : H v ] −msv [L v,2 : M v ] mn v,1 −sv µ F (T (H v )) · µ F (T (M v )) dµ F .
Lemma 3.6. Let L be an admissible A-representation of Q. Then
ζ L * m (s) = v∈V ζ ⊳A A mn v,1 (s v ) ζ ⊳A A n v,1 (ms v ) (1 − q −1 ) −(n v,1 +n v,2 ) × Y L v∈V n v,1 i=1 |c v,1 ii | msv −i F n v,2 j=1 |c v,2 jj | sv−mn v,1 −j F dµ F .
Proof. The equality follows from Proposition 3.2 and the observation (9), after substituting the claims of Lemma 3.4 into (10).
Y F = Y L⊗ Z O F = (C v,1 , C v,2 ) v ∈ v∈V F ( n v,1 +1 2 )+( n v,2 +1 2 ) : F |= ψ((C v,1 , C v,2 ) v )
as in Definition 3.5. The collection Y = (Y F ) F ∈Loc is then an L DP -definable set. We treat m as a formal complex variable. Applying Proposition 2.2 to the statement of Lemma 3.6, where we take n = 2|V | + 1 and the variables s 1 , . . . , s n to be −m in addition to s v and ms v for all v ∈ V , we find for all F ∈ Loc ≫ that
(11) ζ (L⊗ Z O F ) * m (s) = v∈V ζ ⊳O F O mn v,1 F (s v ) ζ ⊳O F O n v,1 F (ms v ) · q n v,1 +n v,2 F (q F − 1) n v,1 +n v,2 · r i=1 m i (k F ) · W i (q F , q m F , (q −sv F , q −msv F ) v∈V )
for rational functions W 1 , . . . , W r ∈ Q(X, Y 0 , (Y 1,v , Y 2,v ) v∈V ) and for g-ary formulae ψ 1 (ξ), . . . , ψ r (ξ), where m i (k F ) = |{ξ ∈ k g F : k F |= ψ i (ξ)}|. Here q F is the cardinality of the residue field k F of F . Set
W i (X, Y 0 , Y 1 , Y 2 ) = W i (X, Y 0 , Y 1 , Y 2 ) v∈V −X 1 − X n v,1 +n v,2 n v,1 −1 j=0 (1 − X j Y 2,v ) .
Recall that any complete discrete valuation ring with finite residue field is the valuation ring of a local field. We deduce from (1) and (11) that these rational functions, and the formulae ψ 1 (ξ), . . . , ψ r (ξ), satisfy the statement of Theorem 1.9.
3.4. Proof of Theorem 1.8. First let A be a compact discrete valuation ring of characteristic zero, and let F ∈ Loc 0 be its fraction field. Let L be an admissible Arepresentation of Q. The domain of integration Y L of Definition 3.5 is semi-algebraic by [9, Lemma 2.1]. Hence Theorem 1.8 follows by applying Proposition 2.1 to the statement of Lemma 3.6 and proceeding analogously to the end of the proof of Theorem 1.9.
In order to treat equal-characteristic local fields of sufficiently large characteristic, we need a bit more technology. Fix n ∈ (N 2 ) V , and let L n be the expansion of the Denef-Pas language L DP by e∈E n t(e),1 n h(e),2 constant symbols a e ij , for e ∈ E and (i, j) ∈ [n t(e),1 ]×[n h(e),2 ], of the valuation ring. These will be interpreted as the structure constants of a representation of the quiver Q. Thus, a model will be a triple (F, π F , L), where (F, π F ) ∈ Loc, while L is an 2 ) as in Definition 3.5, and the collection {Y L } (F,π F ,L)∈LocRep is an L n -definable set. Consider the L n -formula ϕ (C v,1 , C v,2 , D v,1 , D v,2 , t v,1 , t v,2 ) v∈V , u , where u ∈ Z and
t v,i ∈ Z, while C v,i , D v,i ∈ F ( n v,i +1 2 )
for v ∈ V and i ∈ [2], expressing the following: there exist H ≤ L 1 and M ≤ L 2 such that:
• [L v,1 : H v ] = q t v,1 F for all v ∈ V ; • [L v,2 : M v ] = q t v,2 F for all v ∈ V ; • v∈V [L v,2 , M v ] n v,1 = q u F ; • (C v,1 , C v,2 ) v and (D v,1 , D v,2 ) v are contained in v∈V (T (H v ) × T (M v )) ⊂ Y L .
More precisely, ϕ may be taken to be the conjunction of the conditions:
• (C v,1 , C v,2 ) v , (D v,1 , D v,2 ) v ∈ Y L ; • n v,ℓ i=1 v F (c v,ℓ ii ) = n v,ℓ i=1 v F (d v,ℓ ii ) = t v,ℓ for all v ∈ V and ℓ ∈ [2]; • u = v∈V n v,1 t v,2 ;
• For every v ∈ V and ℓ ∈ [2] there exists B v,ℓ ∈ T n v,ℓ (O F ) with unit diagonal entries such that C v,ℓ = B v,ℓ D v,ℓ ; this condition is obviously L DP -definable.
The formula ϕ affords an L n -definable equivalence relation on Y L . Set t i = (t v,i ) v∈V for i ∈ [2], and let δ ϕ,L,t 1 ,t 2 ,u be the number of equivalence classes. This is the number of pairs ( W L (q m , (q −sv , q −msv ) v∈V ) by Proposition 3.2, from which our claim follows.
Amalgamated powers. Let A be a commutative unital ring. An A-lattice is a free A-module L of finite rank equipped with an A-bilinear multiplication [ , ] : L × L → L. Here we consider nilpotent A-lattices of class at most two, namely those for which [L, L] ⊆ Z(L) = {x ∈ L : ∀y ∈ L, [x, y] = [y, x] = 0}. A (two-sided) A-ideal of L is an A-submodule I ≤ L such that [x, y] ∈ I and [y, x] ∈ I for all x ∈ I and y ∈ L. We consider the ideal zeta function ζ ⊳A L (s) = I≤L [L : I] −s , where the sum runs over A-ideals of finite index.
1. 3 .
3Comparison with related zeta functions. The "preservation of complexity" under amalgamated powers observed in Remark 1.4 contrasts with another natural operation on Lie lattices, namely base extension: given an A-Lie lattice L and an extension B/A of rings such that B is free of finite rank as an A-module, we may consider the restriction of scalars of L ⊗ A B as an A-Lie lattice and seek to studyζ ⊳A L⊗ A B (s). For instance, if B = A r , then L ⊗ A B = Lr is the direct product of r copies of L. It appears that the complexity of ζ ⊳A L⊗ A B (s) grows rapidly with rk A B. For the class of Lie lattices considered in [4, Theorem 4.21], for instance, it is possible to express ζ ⊳A L⊗ A B (s) as a sum of explicit combinatorially defined functions, where the number of summands grows super-exponentially in rk A B. Nearly the opposite behavior occurs if we consider the pro-isomorphic zeta functions ζ ∧A L (s) = M ≃L [L : M ] −s , where the sum runs over all A-Lie sublattices M ≤ L that are isomorphic to L; these are necessarily of finite index. Let L be a nilpotent Z-Lie algebra, of arbitrary class, such that Z(L) ≤ [L, L]. By [3, Theorem 1.3(2)] there exist formulae ψ 1 , .
Remark 1. 6 .
6Let L be a nilpotent A-lattice of class at most two such that [L, L] and L/[L, L] are free A-modules of finite rank. Let (b 1 , . . . , b n+d ) be an A-basis of L such that [L, L] = b n+1 , . . . , b n+d A . Set L 1 = b 1 , . . . , b n A and L 2 = [L, L]. Consider the quiver consisting of a single vertex v 0 and 2n loops. It has an admissible A-representation (L, f ), where L v 0 = L = L 1 ⊕ L 2 , and the 2n maps f e : L → L are the left and right multiplications by b 1 , . . . , b n . It is clear that ζ ⊳A L (s) = ζ L (s). Further important examples of quiver representation zeta functions are Dirichlet series enumerating graded ideals of a graded A-lattice and submodules of an A-lattice that are invariant under the action of a collection of endomorphisms [19, 20, 25]. Indeed, quiver representation zeta functions are equivalent to the class of Dirichlet series counting graded submodules invariant under a collection of endomorphisms [15, §1.3.3]. Let Q = (V, E) be a quiver and m ∈ N. Set Q * m = (V, E × [m]), where t(e, i) = t(e) and h(e, i) = h(e) for every i ∈ [m]. Informally, the vertex set of Q * m is the same as that of Q, but every edge of Q is replaced by m edges with the same tail and head. Definition 1.7. Let L be an admissible A-representation of Q and m ∈ N. The amalgamated m-th power of L is the following admissible A-representation (L * m , f * m ) of Q * m . For every v ∈ V , set L * m v = (L v,1 ) m ⊕ L v,2 . For every e ∈ E and i ∈ [m], the map f e,i : L * m t(e) → L * m h(e) is equal to f e on the i-th component of L m t(e),1 and zero on the remaining components of L * m t(e)
Lemma 2. 3 .
3Let m, n ∈ N, and let Ω be a free A-module of rank mn. Suppose that we are given a decomposition Ω = Ω 1 ⊕· · ·⊕Ω m and an A-module isomorphism ϕ i : Ω i → A n for every i ∈ [m]. Let π i : Ω → Ω i be the projection onto the i-th component. For every finite-index A-sublattice H ≤ A n , let S H be the set of finite-index sublattices
j [Ω : Λ] = q m n j=1 ν j [Ω : Λ] = [A n : H] m [Ω : Λ] for any lattice Λ ≤ Ω.
3. 1 .
1Rewriting the ideal zeta function. Let (L, f ) be an admissible A-representation of the quiver Q. For i ∈ [2], define L i to be the family (L v,i ) v∈V . Given two families H = (H v ) and H
and, given H v and M v , there are [L v,2 : M v ] n v,1 possible choices of C v admitted by the Hermite normal form. The claim follows.
Proposition 3 . 2 .
32Let L be an admissible A-representation of Q, and let m ∈ N. Then
Definition 3. 3 .
3Let U be a free A-module of rank r with a fixed basis (b 1 , . . . , b r ). SetU i = b i , . . . , b r A for every i ∈ [r + 1]. In particular, U r+1 = 0. If M ≤ U is a finite-index A-submodule,we say that an A-basis (c 1 , . . . , c r ) of M is a good basis if c i , . . . , c r A = M ∩ U i for every i ∈ [r].
Lemma 3 . 4 .
34Let U be a free A-module of rank r, and let M ≤ U be an A-submodule of finite index. Then T (M) is an open subset of T r (A) and
.
2 . Having chosen these bases, given families H ≤ L 1 and M ≤ L 2 we can define good bases of H v and M v for every v ∈ V . For every edge e ∈ E and (i, j) ∈ [n t(e),1 ] × [n h(e),2 ] let a e ij ∈ A be the structure constants satisfying f e (b Definition 3.5. Let Y L be the set of tuples
the A-linear spans of the rows of C v,1 = (c v,1 ij ) and C v,2 = (c v,2 ij ), with respect to the bases b v and β v , are finite-index A-submodules H v ≤ L v,1 and M v ≤ L v,2 , respectively, satisfying f (H) ≤ M . Equivalently, (C v,1 , C v,2 ) v ∈ Y L if and only if the following conditions are satisfied: • v F (c v,ℓ ij ) ≥ 0 for all v ∈ V and ℓ ∈ {1, 2} and 1 ≤ i ≤ j ≤ n v,ℓ . • det C v,ℓ = 0 for all v ∈ V and ℓ ∈ {1, 2}. • For all e ∈ E and (i, j) ∈ [n t(e),1 ] × [n h(e),2 ] there exists d e ij ∈ F such that v F (d e ij ) ≥ 0 and the following equality holds for all (i, k) ∈ [n t(e),1 ] × [n h(e),2 ]:
3. 3 .
3Proof of Theorem 1.9. Let L be an admissible Z-representation of Q. For every v ∈ V , fix Z-bases b v of L v,1 and β v of L v,2 . Recall the structure constants a e ij ∈ Z for e ∈ E and (i, j) ∈ [n t(e),1 ] × [n h(e),2 ]. For every F ∈ Loc, the O F -representation L ⊗ Z O F is determined by the constants a e ij , interpreted as elements of O F . Set
admissible O F -representation with O F -bases b v of L v,1 and β v of L v,2 for every v ∈ V and arrows determined by f e (b e ∈ E and i ∈ [n t(e),1 ]. Let LocRep be the set of such triples. With these data we can define Y L ⊂ v∈V F (
H, M ), where H ≤ L 1 and M ≤ L 2 and f (H) ≤ M , such that [L v,1 :H v ] = q t v,1 F and [L v,2 : M v ] = q t v,2 F for all v ∈ V .By [17, Theorem 4.1.1], whose hypotheses hold by [17, Example 2.3.7], there exists M n ∈ N such that the Poincaré series(t 1 ,t 2 ,u)∈N 2|V |+1 0 δ ϕ,L,t 1 ,t 2 ,u T u 0 T is a rational function W L ∈ Q(T 0 , (T 1,v , T 2,v ) v∈V ) withdenominator of the requisite form provided that the residue characteristic of F is at least M n . Alternatively, Theorem 6.1 and Corollary 6.8 of [14] may be used in place of [17, Theorem 4.1.1], with the observation that, by the Ax-Kochen-Ershov principle, the proof of [14, Corollary 6.8] applies to all local fields of sufficiently large positive characteristic. Hence ζ L * m (s) = v∈V ζ ⊳A A mn v,1 (s v ) ζ ⊳A A n v,1 (ms v )
t v,1 1,v T t v,2 2,v
Acknowledgements. We are very grateful to Itay Glazer, Immanuel Halupczok, Seungjai Lee, and Christopher Voll for helpful correspondence and comments on earlier versions of this work.
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]
| []
| Coherent configurations (CCs) are highly regular colorings of the set of ordered pairs of a "vertex set"; each color represents a "constituent digraph." CCs arise in the study of permutation groups, combinatorial structures such as partially balanced designs, and the analysis of algorithms; their history goes back to Schur in the 1930s. A CC is primitive (PCC) if all its constituent digraphs are connected.We address the problem of classifying PCCs with large automorphism groups. This project was started in Babai's 1981 paper in which he showed that only the trivial PCC admits more than exp( O(n 1/2 )) automorphisms. (Here, n is the number of vertices and the O hides polylogarithmic factors.)In the present paper we classify all PCCs with more than exp( O(n 1/3 )) automorphisms, making the first progress on Babai's conjectured classification of all PCCs with more than exp(n ǫ ) automorphisms.A corollary to Babai's 1981 result solved a then 100-year-old problem on primitive but not doubly transitive permutation groups, giving an exp( O(n 1/2 )) bound on their order. In a similar vein, our result implies an exp( O(n 1/3 )) upper bound on the order of such groups, with known exceptions. This improvement of Babai's result was previously known only through the Classification of Finite Simple Groups(Cameron, 1981), while our proof, like Babai's, is elementary and almost purely combinatorial.Our analysis relies on a new combinatorial structure theory we develop for PCCs. In particular, we demonstrate the presence of "asymptotically uniform clique geometries" on PCCs in a certain range of the parameters. * An extended abstract of this paper appeared in the Proceedings of the 47th ACM Symposium on Theory of Computing (STOC'15) under the title Faster canonical forms for primitive coherent configurations.† | null | [
"https://arxiv.org/pdf/1510.02195v2.pdf"
]
| 2,829,031 | 1510.02195 | 91a9030085a73d06f689288bc05dc140d1f5e09e |
Structure and automorphisms of primitive coherent configurations *
25 Aug 2016
Xiaorui Sun [email protected].
Columbia University
University of Chicago
John Wilmes
Columbia University
University of Chicago
Structure and automorphisms of primitive coherent configurations *
25 Aug 20161
Coherent configurations (CCs) are highly regular colorings of the set of ordered pairs of a "vertex set"; each color represents a "constituent digraph." CCs arise in the study of permutation groups, combinatorial structures such as partially balanced designs, and the analysis of algorithms; their history goes back to Schur in the 1930s. A CC is primitive (PCC) if all its constituent digraphs are connected.We address the problem of classifying PCCs with large automorphism groups. This project was started in Babai's 1981 paper in which he showed that only the trivial PCC admits more than exp( O(n 1/2 )) automorphisms. (Here, n is the number of vertices and the O hides polylogarithmic factors.)In the present paper we classify all PCCs with more than exp( O(n 1/3 )) automorphisms, making the first progress on Babai's conjectured classification of all PCCs with more than exp(n ǫ ) automorphisms.A corollary to Babai's 1981 result solved a then 100-year-old problem on primitive but not doubly transitive permutation groups, giving an exp( O(n 1/2 )) bound on their order. In a similar vein, our result implies an exp( O(n 1/3 )) upper bound on the order of such groups, with known exceptions. This improvement of Babai's result was previously known only through the Classification of Finite Simple Groups(Cameron, 1981), while our proof, like Babai's, is elementary and almost purely combinatorial.Our analysis relies on a new combinatorial structure theory we develop for PCCs. In particular, we demonstrate the presence of "asymptotically uniform clique geometries" on PCCs in a certain range of the parameters. * An extended abstract of this paper appeared in the Proceedings of the 47th ACM Symposium on Theory of Computing (STOC'15) under the title Faster canonical forms for primitive coherent configurations.†
Introduction
Let V be a finite set; we call the elements of V "vertices." A configuration of rank r is a coloring c : V × V → {0, . . . , r − 1} such that (i) c(u, u) = c(v, w) for any v = w, and (ii) for all i < r there is i * < r such that c(u, v) = i iff c(v, u) = i * . The configuration is coherent (CC) if (iii) for all i, j, k < r there is a structure constant p i jk such that if c(u, v) = i, there are exactly p i jk vertices w such that c(u, w) = j and c(w, v) = k. The diagonal colors c(u, u) are the vertex colors, and the off-diagonal colors are the edge colors. A CC is homogeneous (HCC) if (iv) there is only one vertex color. We denote by R i the set of ordered pairs (u, v) of color c(u, v) = i. The directed graph X i = (V, R i ) is the colori constituent digraph. An HCC is primitive (PCC) if each constituent digraph is strongly connected. An association scheme is an HCC for which i = i * for all colors i (so the constituent graphs X i are viewed as undirected).
The term "coherent configuration" was coined by Donald Higman in 1969 [17], but the essential objects are older. In the case corresponding to a permutation group, CCs already effectively appeared in Schur's 1933 paper [24]. This grouptheoretic perspective on CCs was developed further by Wielandt [28].
CCs appeared for the first time from a combinatorial perspective in a 1952 paper by Bose and Shimamoto [10]. They, along with many of the subsequent authors, consider the case of an association scheme, which is essential for understanding partially balanced incomplete block designs, of interest to statisticians and to combinatorial design theorists. The generalization of an association scheme to an HCC was considered by Nair in 1964 [21]. The algebra associated with a CC, which already appeared in Schur's paper, was rediscovered in 1959 in the context of association schemes by Bose and Mesner [9].
Weisfeiler and Leman [27] and Higman [17] independently defined CCs in their full generality, including the associated algebra (called "cellular algebras" by Weisfeiler and Leman), in the late 1960s. For Higman, CCs were a generalization of permutation groups, whereas Weisfeiler and Leman were motivated by the algorithmic Graph Isomorphism problem. In the intervening years, CCs, and association schemes in particular, have become basic objects of study in algebraic combinatorics [8,11,7,29]. CCs also continue to play a role in the study of permutation groups [16,20]. Recent algorithmic applications of the CC concept include the Graph Isomorphism problem [3] and the complexity of matrix multiplication [15].
PCCs are in a sense the "indivisible objects" among CCs and are therefore of particular interest.
In this paper we classify the PCCs with the largest automorphism groups, up to the threshold stated in the following theorem. (See Defintion 1.3 and Theo-rem 1.4 for a more detailed statement, and see Section 1.6 for an explanation of the asymptotic notation used throughout, including O, O, Θ, Ω, ∼, and o.) Theorem 1.1. If X is a PCC not belonging to any of three exceptional families, then | Aut(X)| ≤ exp(O(n 1/3 log 7/3 n)).
Primitive permutation groups of large order were classified by Cameron [12]. We refer to the orbital, or Schurian configurations of these groups as "Cameron schemes" (see Sections 1.1 and 1.2). For every ǫ > 0, for every n there is only a bounded number of primitive groups of order greater than exp(n ǫ ) (the bound depends on ǫ but not on n); we refer to this stratification as the "Cameron hierarchy." Theorem 1.1 represents progress on the following conjectured classification of PCCs with large automorphism groups. Conjecture 1.2 (Babai). For every ε > 0, there is some N ε such that if X is a PCC on n ≥ N ε vertices and | Aut(X)| ≥ exp(n ε ), then X is a Cameron scheme. In particular, Aut(X) is a known primitive group.
This conjecture would be a far-reaching combinatorial generalization of Cameron's classification of large primitive permutation groups. In particular, while Cameron's result is only known through the Classification of Finite Simple Groups (CFSG), Conjecture 1.2 would imply (at least for orders greater than exp(n ǫ )) a CFSG-free proof of Cameron's result, giving a different kind of insight into the structure of large primitive permutation groups.
Babai [1] established Conjecture 1.2 for all ε > 1/2 (the "first level of the Cameron hierarchy"). As a corollary, he solved a then 100-year-old problem on primitive but not doubly transitive permutation groups, giving an nearly tight, exp( O(n 1/2 )) bound on their order. The tight bound was subsequently found by Cameron, using CFSG; our result implies a CFSG-free proof of the same tight bound. Moreover, our Theorem 1.1 confirms the conjecture to all ε > 1/3, the first improvement since Babai's paper. An elementary proof of an exp( O(n 1/3 )) upper bound on the order of primitive permutation groups, with known exceptions (the "second level of the Cameron hierarchy") follows.
For the proof of Theorem 1.1, we find new combinatorial structure in PCCs, including "clique geometries" in certain parameter ranges (Theorem 2.4). An overview of our structural results for PCCs is given in Section 2.
Our motivation is thus twofold. First, we develop a structure theory for PCCs, the most general objects in a hierarchy of much-studied highly regular combinatorial structures. Second, as a corollary to our main result, we obtain a CFSG-free proof for the second level of the Cameron hierarchy of large primitive permutation groups.
Additional motivation for our work comes from the algorithmic Graph Isomorphism problem. We explain this connection in Section 1.5.
Exceptional coherent configurations
We now give a precise statement of our main combinatorial results.
Given a graph X = (V, E), we associate with X the configuration X(X) = (V ; ∆, E, E) where E denotes the set of edges of the complement of X. (We omit E if E = ∅ and omit E if E = ∅.) So graphs can be viewed as configurations of rank ≤ 3.
Given an (undirected) graph H, the line-graph L(H) has as vertices the edges of H, with two vertices adjacent in L(H) if the corresponding edges are incident in H. The triangular graph T (m) is the line-graph of the complete graph K m (so n = m 2 ). The lattice graph L 2 (m) is the line-graph of the complete bipartite graph K m,m (on equal parts) (so n = m 2 ). The configurations X(T (m)) and X(L 2 (m)) are coherent, and in fact primitive for m > 2.
Definition 1.3. A PCC is exceptional if it is of the form X(X)
, where X is isomorphic to the complete graph K n , the triangular graph T (m), or the lattice graph L 2 (m), or the complement of such a graph.
We note that the exceptional PCCs have n Ω( √ n) automorphisms. Indeed, the exceptional PCCs are exactly the "orbital configurations" of large primitive permutation groups, as explained below.
Our main result is that all the non-exceptional PCCs have far fewer automorphisms. Theorem 1.4. If X is a non-exceptional PCC, then | Aut(X)| ≤ exp(O(n 1/3 log 7/3 n)). The exceptional PCCs correspond naturally to the largest primitive permutation groups. Given a permutation group G ≤ Sym(V ), we define the orbital configuration X(G) on vertex set V with the R i given by the orbitals of G, i.e., the orbits of the induced action on V × V . CCs of this form were first considered by Schur [24], and are commonly called Schurian. Note that G ≤ Aut(X(G)).
The Schurian CC X(G) is homogeneous if and only if G is transitive, and primitive if and only if G is a primitive permutation group. If G is doubly transitive, then X(G) = X(K n ). We also have X(S (2) m ) = X(T (m)) and X(S m ≀ S 2 ) = X(L 2 (m)).
Primitive permutation groups
Following the completion of the Classification of Finite Simple Groups (CFSG), one of the tasks has been to obtain elementary proofs of results currently known only through CFSG. One such result is Cameron's classification of all primitive permutation groups of large order, obtained by combining CFSG with the O'Nan-Scott theorem [12]. Cameron's threshold for the order is n O(log log n) , but we state Maróti's refinement of his classification of permutation groups of order greater than n c log n [18]. Theorem 1.5 (Cameron, Maróti). If G is a primitive permutation group of degree n > 24, then one of the following holds:
(a) there are positive integers d, k, and m such that (A
(k) m ) d ≤ G ≤ S (k) m ≀ S d ; (b) |G| ≤ n 1+log 2 n .
We call the primitive groups G of Theorem 1.5 (a) Cameron groups. Given a Cameron group G with parameters d and k bounded, we obtain a PCC X(G) with exponentially large automorphism group H ≥ G, in particular, of order |H| ≥ exp(n 1/(kd) ). We call the PCCs X(G) Cameron schemes when G is a Cameron group.
Hence, Conjecture 1.2 states that Cameron's classification of primitive permutation groups transfers to the combinatorial setting of PCCs. Furthermore, the conjecture entails Cameron's theorem, above the threshold |G| ≥ exp(n ε ) (see [5]). Hence, confirmation of Conjecture 1.2 would yield a CFSG-free proof of Cameron's classification (above this threshold).
It seems unlikely that combinatorial methods will match Cameron's n O(log log n) threshold for classification of primitive permutation groups. An n O(log n) threshold (as in stated Theorem 1.5) via elementary techniques might be possible, since above this threshold the socle of a primitive permutation group is a direct product of alternating groups, whereas below this threshold, simple groups of Lie type may appear in the socle.
However, until the present paper, the only CFSG-free classification of the large primitive permutation groups was given by Babai in a pair of papers in 1981 and 1982 [1,2]. Babai proved that |G| ≤ exp(O(n 1/2 log 2 n)) for primitive groups G other than A n and S n [1]. A corollary of our work gives the first CFSG-free improvement to Babai's bound, by proving that |G| ≤ exp(O(n 1/3 log 7/3 n)) for primitive permutation groups G, other than groups belonging to three exceptional families.
In the following corollary to Theorem 1.1, S
m and A (2) m denote the actions of S m and A m , respectively, on the m 2 pairs, and G ≀ H denotes the wreath product of the permutation groups G ≤ S n by H ≤ S m in the product action on a domain of size n m . Corollary 1.6. Let Γ be a primitive permutation group of degree n. Then either |Γ| ≤ exp(O(n 1/3 log 7/3 n)), or Γ is one of the following groups:
(a) S n or A n ;
(b) S (2) m or A (2) m , where n = m 2 ; (c) a subgroup of S m ≀ S 2 containing (A m ) 2 , where n = m 2 .
The slightly stronger bound |Γ| ≤ exp(O(n 1/3 log n)) follows from CFSG [12]. By contrast, our proof is elementary.
For given n = m 2 , there are exactly three primitive groups in the third category of Corollary 1.6. We note that the groups of categories 1-3 of the corollary have order exp(Ω(n 1/2 log n)). Corollary 1.6 follows from Theorem 1.4 by classifying the large primitive groups G for which X(G) is an exceptional PCC, as in the following proposition.
Proposition 1.7.
There is a constant c such that the following holds. Let G ≤ S n be primitive, and suppose |G| ≥ n c log n .
1. If X(G) = X(K n ), then G belongs to category (a) of Corollary 1.6.
2. If X(G) = X(T (m)), then G belongs to category (b) of Corollary 1.6.
3. If X(G) = X(L 2 (m)), then G belongs to category (c) of Corollary 1.6. Proposition 1.7 as stated requires CFSG, but an elementary proof is available under the weaker bound of |G| ≥ exp(c log 3 n) using [23]. For the proof and a more general classification, we refer the reader to [5].
Individualization and refinement
We now introduce the individualization/refinement heuristic. We shall use individualization/refinement to find bases of automorphism groups of configurations.
A base for a group G acting on a set V is a subset S ⊆ V such that the pointwise stablizer G (S) of S in G is trivial. If S is a base, then |G| ≤ |V | |S| .
Let Iso(X, Y) denote the set of isomorphisms from X to Y, and Aut(X) = Iso(X, X).
Individualization means the assignment of individual colors to some vertices; then the irregularity so created propagates via some canonical color refinement process. For a class C of configurations (not necessarily coherent), an assignment X → X ′ is a color refinement if X, X ′ ∈ C have the same set of vertices and the coloring of X ′ is a refinement of the coloring of X. Such an assignment is canonical if for all X, Y ∈ C, we have Iso(X, Y) = Iso(X ′ , Y ′ ). In particular, Aut(X) = Aut(X ′ ).
Repeated application of the refinement process leads to the stable refinement after at most n − 1 rounds.
If after individualizing the elements of a set S ⊆ V , all vertices get different colors in the resulting stable refinement, we say that S completely splits X (with respect to the given canonical refinement process). If S completely splits X, then S is a base for Aut(X). Hence, to prove Theorem 1.4, it suffices to show that some set of O(n 1/3 log 4/3 n) vertices completely splits X after canonical color refinement.
For our purposes, the simple "naive vertex refinement" will suffice as our color refinement procedure. Under naive vertex refinement, the edge-colors do not change, only the vertex-colors are refined. The refined color of vertex u of the configuration X encodes the following information: the current color of u and the number of vertices v of color i such that c(u, v) = j, for every pair (i, j), where i is a vertex-color and j is an edge-color.
We now state our main technical result, from which Theorem 1.4 immediately follows. Theorem 1.8 (Main). Let X be a non-exceptional PCC. Then there exists a set of O(n 1/3 log 4/3 n) vertices that completely splits X under naive refinement.
This improves the main result of [1], which stated that if X is a PCC other than X(K n ), then there is a set of O(n 1/2 log n) vertices which completely splits X under naive refinement.
Naive vertex refinement is the only color refinement used in the present paper. However, we remark that coherent configurations were first studied by Weisfeiler and Leman in the context of their stronger canonical color refinement [27,26].
Given a configuration X, the Weisfeiler-Leman (WL) canonical refinement process [27,26] produces a CC X ′ on the same vertex set with Aut(X) = Aut(X ′ ), by refining the coloring until it is coherent. More precisely, in every round of the refinement process, the color c(u, v) of the pair u, v ∈ V is replaced with a color c ′ (u, v) which encodes c(u, v) along with, for every pair j, k of original colors, the number of vertices w such that c(u, w) = i and c(w, v) = k. This refinement is iterated until the coloring stabilizes, i.e., the rank no longer increases in subsequent rounds of refinement. The stable configurations under WL refinement are exactly the coherent configurations.
Relation to strongly regular graphs
An undirected graph X = (V, E) is called strongly regular (SRG) with parameters (n, k, λ, µ) if X has n vertices, every vertex has degree k, each pair of adjacent vertices has λ common neighbors, and each pair of non-adjacent vertices has µ common neighbors.
We note that a graph X is a SRG if and only if the configuration X(X) is coherent. If a SRG X is nontrivial, i.e., it is connected and coconnected, then X(X) is a PCC.
All of our exceptional PCCs are in fact SRGs. Our classification of PCCs, Theorem 1.4, was established in the special case of SRGs by Spielman in 1996 [25], on whose results we build. In fact, Chen, Sun, and Teng have now established a stronger bound for SRGs: a non-exceptional SRG has at most exp( O(n 9/37 )) automorphisms [14].
The results of Spielman and Chen, Sun, and Teng both rely on Neumaier's structure theory [22] of SRGs to separate the exceptional SRGs with many automorphisms from those to which I/R can be effectively applied. However, no generalization of Neumaier's results to PCCs has been known. We provide a weak generalization, sufficient for our purposes, in Section 2.
Graph Isomorphism
The "Graph Isomorphism (GI) problem" is the computational problem to decide whether or not a pair of given graphs are isomorphic. This problem is of great interest to complexity theory since it is one of a very small number of natural problems in NP of intermediate complexity status (unlikely to be NP-complete but not known to be solvable in polynomial time).
In recent major development, Babai [3] announced a quasipolynomial-time (exp(O((log n) c ))) algorithm.
Babai's algorithm reduces the problem to the isomorphism problem of PCC's and then uses his (rather involved) "split-or-Johnson" procedure for further reduction.
Babai conjectures that a considerably simpler algorithm might succeed; unless the PCC is a Cameron scheme, individualization of a small number of vertices may completely split the vertex set. This is a more explicit version of Conjecture 1.2.
Our result proves that this is indeed the case if "small number" means O(n 1/3 ), improving Babai's O(n 1/2 ). We hope that further refinement of our structure theory will yield further progress in this direction.
Asymptotic notation
To interpret asymptotic inequalities involving the parameters of a PCC, we think of the PCC as belonging to an infinite family in which the asymptotic inequalities hold.
For functions f, g : N → R >0 , we write f (n) = O(g(n)) if there is some constant C such that f (n) ≤ Cg(n), and we write f (n) = Ω(g(n)) if g(n) = O(f (n)). We write f (n) = Θ(g(n)) if f (n) = O(g(n)) and f (n) = Ω(g(n)).
We use the notation f (n) = O(g(n)) when there is some constant c such that f (n) = O(g(n)(log n) c ). We write f (n) = o(g(n)) if for every ε > 0, there is some N ε such that for n ≥ N ε , we have f (n) < εg(n). We write f (n) = ω(g(n)) if g(n) = o(f (n)). We use the notation f (n) ∼ g(n) for asymptotic equality, i.e., lim n→∞ f (n)/g(n) = 1. The asymptotic inequality f (n) g(n) means g(n) ∼ max{f (n), g(n)}.
Structure theory of primitive coherent configurations
To prove Theorem 1.8, we need to develop a structure theory of PCCs. The overview in this section highlights the main components of that theory.
Throughout the paper, X will denote a PCC of rank r on vertex set V with structure constants p i jk for 0 ≤ i, j, k ≤ r −1. We assume throughout that r > 2, since the case r = 2 is the trivial case of X(K n ), listed as one of our exceptional PCCs. We also assume without loss of generality that color 0 corresponds to the diagonal, i.e.,
R 0 = {(u, u) : u ∈ V }.
For any color i in a PCC, we write n i = n i * = p 0 ii * = p 0 i * i , the out-degree of each vertex in X i .
We say that color i is dominant if n i ≥ n/2. Colors i with n i < n/2 are nondominant. We call a pair of distinct vertices dominant (nondominant) when its color is dominant (nondominant, resp.). We say color i is symmetric if i * = i. Note that when color i is dominant, it is symmetric, since n i * = n i ≥ n/2.
Our analysis will divide into two cases, depending on whether or not there is a dominant color. In fact, many of the results of this section will assume that there is an overwhelmingly dominant color i satisfying n i ≥ n−O(n 2/3 ). The reduction to this case is accomplished via Lemma 3.1 of the next section. The main structural result used in its proof is Lemma 2.1 below, which gives a lower bound on the growth of "spheres" in a PCC.
For a color i and vertex u, we denote by X i (u) the set of vertices v such that c(u, v) = i. We denote by dist i (u, v) the directed distance from u to v in the colori constituent digraph X i , and we write dist i (j) = dist i (u, v) for any vertices u, v with c(u, v) = j. (This latter quantity is well-defined by the coherence of X.) The δ-sphere X
(δ) i (u) in X i centered at u is the set of vertices v with dist i (u, v) = δ.
Lemma 2.1 (Growth of spheres). Let X be a PCC, let i, j ≥ 1 be nondiagonal colors, let δ = dist i (j), and u ∈ V . Then for any integer 1 ≤ α ≤ δ − 2, we have
|X (α+1) i (u)||X (δ−α) i (u)| ≥ n i n j .
We note that Lemma 2.1 is straightforward when X i is distance-regular. Indeed, a significant portion of the difficulty of the lemma was in finding the correct generalization.
Overview of proof of Lemma 2.1. The bipartite subgraphs of X i induced on pairs of the form (X j (u), X k (u)), where j, k are colors and u is a vertex, are biregular by the coherence of X i . We exploit this biregularity to count shortest paths in X i between a carefully chosen subset of X (δ−α) i (u) and X j (u), for an arbitrary vertex u.
The details of the proof are given in Section 4.
In the rest of the paper, we assume without loss of generality that n 1 = max i n i . We write ρ = i≥2 n i = n − n 1 − 1. For the rest of the section, color 1 will in fact be dominant. In fact, every theorem in the rest of this section will state the assumption that ρ = o(n 2/3 ). Lemma 2.2 below demonstrates some of the power of this supposition. Lemma 2.2. Let ε > 0 and let X be a PCC with ρ < (1 − ε)n 2/3 ). Then, for n sufficiently large, dist i (1) = 2 for every nondominant color i. Consequently,
n i ≥ √ n − 1 for i = 0.
Overview of proof of Lemma 2.2. We will prove that if dist i (1) ≥ 3 for some color i, then ρ n 2/3 . Without loss of generality, we assume n 1 ∼ n, since otherwise we are already done. Fix an arbitrary vertex u and consider the bipartite graph B between X
(δ−1) i (u)
and X 1 (u), with an edge from x ∈ X (δ−1) i (u) to y ∈ X 1 (u) when c(x, y) = i. By the coherence of X, the bipartite graph is regular on X 1 (u); call its degree γ. An obstacle to our analysis is that the graph need not be biregular. Nevertheless, we estimate the maximum degree β of a vertex in X
(δ−1) i (u) in B.
We first note that n 1 γ ≤ βρ.
Let w be a vertex satisfying c(u, w) = i. We pass to the subgraph B ′ induced on (X
(δ−2) i (w), X (δ−1) i (w))
, and observe that the degree of vertices in X
(δ−1) i (u) ∩ X (δ−2) i (w) is preserved, while the degree of vertices in X 1 (u) ∩ X (δ−1) i (w) does not increase.
Let v be a vertex of degree β in B ′ , and let j = c(w, v). We finally consider the bipartite graph B ′′ on (X j (w), X w ), where X w is the set of vertices x ∈ X (δ−1) i (w) with at most γ in-neighbors in X i lying in the set X j (w). In
particular, X 1 (u) ∩ X (δ−1) i (w) ⊆ X w . This graph B ′′ is now regular (of degree ≥ β) on X j (w). Since X w ⊆ X (δ−1) i (w)
, we have |X w | ≤ ρ, which eventually gives the bound β ≤ γρ 2 /n 1 . Combining this with our earlier estimate βρ ≥ n 1 γ proves the lemma.
The details of the proof are given in Section 6.
Notation. Let G(X) be the graph on V formed by the nondominant pairs. So G(X) is regular of valency ρ, and every pair of distinct nonadjacent vertices in G(X) has exactly µ common neighbors, where µ = i,j≥2 p 1 ij . The graph G(X) is not generally SR, since pairs of adjacent vertices in G(X) of different colors in X will in general have different numbers of common neighbors. However, intuition from SRGs will prove valuable in understanding G(X).
We write N (u) for the set of neighbors of u in the graph G(X). For i nondominant, we define λ i = |X i (u) ∩ N (v)|, where c(u, v) = i. So, the parameters λ i are loosely analogous to the parameter λ of a SRG.
A clique C in an undirected graph G is a set of pairwise adjacent vertices; its order |C| is the number of vertices in the set.
Definition 2.3.
A clique geometry on a graph G is a collection G of maximal cliques such that every pair of adjacent vertices in G belongs to a unique clique in G. A clique geometry of a PCC X is a clique geometry on G(X). The clique geometry G is asymptotically uniform (for an infinite family of PCCs) if for every C ∈ G, u ∈ C, and nondominant color i, we have either |C ∩ X i (u)| ∼ λ i or |C ∩ X i (u)| = 0 (as n → ∞).
We have the following sufficient condition for the existence of clique geometries in PCCs.
Theorem 2.4. Let X be a PCC satisfying ρ = o(n 2/3 ), and fix a constant ε > 0. If λ i ≥ εn 1/2 for every nondominant color i, then for n sufficiently large, there is a clique geometry G on X. Moreover, G is asymptotically uniform. Theorem 2.4 provides a powerful dichotomy for PCCs: either there is an upper bound on some parameter λ i , or there is a clique geometry. Adapting a philosophy expressed in [4], we note that bounds on λ i are useful because they limit the correlation between the i-neighborhoods of two random vertices. Similar bounds on the parameter λ of a SRG were used in [4].
On the other hand, Theorem 2.4 guarantees that if all parameters λ i are sufficiently large, the PCC has an asymptotically uniform clique geometry. This is our weak analogue of Neumaier's geometric structure. Clique geometries offer their own dichotomy. Geometries with at most two cliques at a vertex can classified; this includes the exceptional PCCs (Theorem 2.5 below). A far more rigid structure emerges when there are at least three cliques at every vertex. In this case, we exploit the ubiquitous 3-claws (induced K 1,3 subgraphs) in G(X) in order to construct a set which completely splits X (Lemma 3.4 (b)).
Overview of proof of Theorem 2.4. The existence of a weaker clique structure follows from a result of Metsch [19]. (See Lemma 7.1 below and the comments in the paragraph preceding it.) Specifically, under the hypotheses of Theorem 2.4, for every nondominant color i and vertex u, there is a partition of X i (u) into cliques of order ∼ λ i in G(X). We call such a collection of cliques a local clique partition (referring to the color-i neighborhood of any fixed vertex).
The challenge is to piece together these local clique partitions into a clique geometry. An obstacle is that Metsch's cliques are cliques of G(X), not X i ; that is, the edges of the cliques partitioning X i (u) have nondominant colors but not in general color i. In particular, for two vertices u, v ∈ V with c(u, v) = i, the clique containing v in the partition of X i (u) may not correspond to any of the cliques in the partition of X i (v).
We first generalize these local structures. An I-local clique partition is a partition of i∈I X i (u) into cliques of order ∼ i∈I λ i . We study the maximal sets I for which such I-local clique partitions exist, and eventually prove that these maximal sets I partition the set of nondominant colors, and the corresponding cliques are maximal in G(X).
Finally, we prove a symmetry condition: given a nondominant pair of vertices u, v ∈ V , the maximal local clique at u containing v is equal to the maximal local clique at v containing u. This symmetry ensures the cliques form a clique geometry, and this clique geometry is asymptotically uniform by construction.
The details of the proof are given in Section 7.
The case that X has a clique geometry with some vertex belonging to at most two cliques includes the exceptional CCs corresponding to T (m) and L 2 (m). We give the following classification.
Theorem 2.5. Let X be a PCC such that ρ = o(n 2/3 ).
Suppose that X has an asymptotically uniform clique geometry G and a vertex u ∈ V belonging to at most two cliques of G. Then for n sufficiently large, one of the following is true:
(a) X has rank three and is isomorphic to X(T (m)) or X(L 2 (m));
(b) X has rank four, X has a non-symmetric non-dominant color i, and G(X) is isomorphic to T (m) for m = n i + 2.
Overview of proof of Theorem 2.5. We first use the coherence of X to show that every vertex u ∈ V belongs to exactly two cliques of G, and these cliques have order ∼ ρ/2. By counting vertex-clique incidences, we then obtain the estimate ρ 2 √ 2n. On the other hand, by Lemma 2.2, every nondominant color i satisfies n i √ n. Hence, there are at most 2 nondominant colors.
Since every vertex belongs to exactly two cliques, the graph G(X) is the linegraph of a graph. If there is only one nondominant color, then G(X) is strongly regular, and therefore, for n sufficiently large, G(X) is isomorphic to T (m) or L 2 (m). On the other hand, if there are two nondominant colors, by counting paths of length 2 we show that G(X) must again be isomorphic to T (m). By studying the edge-colors at the intersection of the cliques containing two distinct vertices and exploiting the coherence of X, we finally eliminate the case that the two nondominant colors are symmetric.
The details of the proof are given in Section 8.
Overview of analysis of I/R
We now give a high-level overview of how we apply our structure theory of PCCs to prove Theorem 1.8.
Most of the results highlighted in Section 2 assumed that ρ = o(n 2/3 ). Hence, the first step is to reduce to this case, which we accomplish via the following lemma.
Lemma 3.1. Let X be a PCC. If ρ ≥ n 2/3 (log n) −1/3 , then there is a set of size O(n 1/3 (log n) 4/3 ) which completely splits X.
We remark that in the case that the rank r of X is bounded, our Lemma 3.1 follows from a theorem of Babai [1,Theorem 2.4]. Following Babai [1], we analyze the distinguishing number.
Definition 3.2. Let u, v ∈ V . We say w ∈ V distinguishes u and v if c(w, u) = c(w, v)
. We write D(u, v) for the set of vertices w distinguishing u and v, and
D(i) = |D(u, v)| where c(u, v) = i. We call D(i) the distinguishing number of i. Hence, D(i) = j =k p i jk * . If w ∈ D(u, v)
, then after individualizing w and refining, u and v get different colors.
Babai observed that in order to completely split a PCC X, it suffices to individualize some set of
O(n log n/D min ) vertices, where D min = min i =0 {D(i)} [1, Lemma 5.4]. Thus, to prove Lemma 3.1, we show that if ρ ≥ n 2/3 (log n) −1/3 then for every color i = 0, we have D(i) = Ω(n 2/3 (log n) −1/3 ).
The following bound on the number of large colors in a PCC becomes powerful when D(i) is small.
Lemma 3.3. Let X be a PCC. For any nondiagonal color i, the number of colors
j such that n j > n i /2 is at most O((log n + n/ρ)D(i)/n i ).
Overview of proof of Lemma 3.3. Let I α be the set of colors i such that D(i) ≤ α, and J β the set of colors j such that n j ≤ β. For a set I of colors, let n I = i∈I n i be the total degree of the colors in I.
First, we prove that ⌊α/(3D(i))⌋n i ≤ n Iα , a lower bound on the total degree of colors with distinguishing number ≤ α. Next, we prove a lemma that allows us to transfer estimates for the total degree of colors with small distinguishing number into estimates for the total degree of colors with low degree. Specifically, we prove that n J β ≤ 2α, where β = n Iα/2 . Together, these two results allow us to transfer estimates on total degree between the sets I α and J β , as α and β increase.
The details of the proof are given in Section 5.
Overview of proof of Lemma 3.1. Fix a color i ≥ 1. We wish to give a lower bound on D(i). Babai observed that for any color j ≥ 1, we have D(i) ≥ D(j)/ dist i (j) [1, Proposition 6.4 and Theorem 6.1]). Hence, we wish to give an upper bound on dist i (j) for some color j with D(j) large. We analyze two cases: n 2/3 (log n) −1/3 ≤ ρ < n/3 and ρ ≥ n/3. In the former case, when n 2/3 (log n) −1/3 ≤ ρ < n/3, we first observe that D(1) = Ω(ρ). Hence, the problem is reduced to bounding the quantity dist i (1) for every color i. Our bound in Lemma 2.1 on the size of spheres suffices for this task since n 1 is large.
In the case that ρ ≥ n/3, Babai observed that the color j maximizing D(j) satisfies D(j) = Ω(n). We partition the colors of X according to their distinguishing number, by first partitioning the positive integers less than D(j) into cells of length 3D(i). (Specifically, we partition the colors of X so that the α-th cell contains the colors k satisfying 3D(i)α ≤ D(k) < 3D(i)(α + 1), and there are O(D(j)/D(i)) cells.) Each cell of this partition P is nonempty. In fact, we show that the sum of the degrees of the colors in each cell is at least n i .
On the other hand, Lemma 3.3 says that there are few colors k satisfying n k > n i /2, and we show that the total degree of the colors k with n k ≤ n i /2 is also small. Since each cell of the partition has degrees summing to at least n i , these together give an upper bound on the number of cells, and hence a lower bound on D(i).
The details of the proof are given in Section 5.
We have now reduced to the case that ρ = o(n 2/3 ). Our analysis of this case is inspired by Spielman's analysis of SRGs [25].
Lemma 3.4.
There exists a constant ε > 0 such that the following holds. Let X be a PCC with ρ = o(n 2/3 ). If X satisfies either of the following conditions, then there is a set of O(n 1/4 (log n) 1/2 ) vertices which completely splits X.
(a) There is a nondominant color i such that λ i < εn 1/2 .
(b) For every nondominant color i, we have λ i ≥ εn 1/2 . Furthermore, X has an asymptotically uniform clique geometry C such that every vertex belongs to at least three cliques of C.
Overview of proof of Lemma 3.4. We will show that if we individualize a random set of O(n 1/4 (log n) 1/2 ) vertices, then with positive probability, every pair of distinct vertices gets different colors in the stable refinement. Let u, v ∈ C, and fix two colors i and j. Generalizing a pattern studied by Spielman, we say a triple (w, x, y) is good for u and v if c(u, x) = c(u, y) = c(x, y) = 1, c(u, w) = i, and c(w, x) = c(w, y) = j, but there exists no vertex z such that c(v, z) = i and c(z, x) = c(z, y) = j. (See Figure 3). To ensure that u and v get different colors in the stable refinement, it suffices to individualize two vertices x, y ∈ V for which there exists a vertex w such that (w, x, y) is good for u and v. We show that if there are many good triples for u and v, then individualizing a random set of O(n 1/4 (log n) 1/2 ) vertices is overwhelming likely to result in the individualization of such a pair x, y ∈ V .
Condition (a) of the lemma is analogous to the asymptotic consequences of Neumaier's claw bound used by Spielman [25] (cf. [6, Section 2.2]), except that the bound on λ i does not imply a similar bound on λ i * . We show that a relatively weak bound on λ i * already suffices for Spielman's argument to essentially go through.
However, if even this weaker assumption fails, then we turn to our local clique structure for the analysis (as described in the overview of Theorem 2.4).
When condition (b) holds, we cannot argue along Spielman's lines, and instead analyze the structural properties of our clique geometries to estimate the number of good triples.
The details of the proof are given in Section 9.
By Theorem 2.4, either the hypotheses of of Lemma 3.4 are satisfied, or X has an asymptotically uniform clique geometry C, and some vertex belongs to at most two cliques of C. Theorem 2.5 gives a characterization PCCs X with the latter property: X is one of the exceptional PCCs, or X has rank four with a nonsymmetric non-dominant color i and G(X) is isomorphic to T (m) for m = n i + 2.
We handle this final case via the following lemma, proved in Section 8. We conclude this overview by observing that Theorem 1.8 follows from the above results.
Proof of Theorem 1.8. Let X be a PCC. Suppose first that ρ ≥ n 2/3 (log n) −1/3 . Then by Lemma 3.1, there is a set of size O(n 1/3 (log n) 4/3 ) which completely splits X.
Otherwise, ρ < n 2/3 (log n) −1/3 = o(n 2/3 ). By Theorem 2.4, either the hypotheses of Lemma 3.4 are satisfied, or the hypotheses of Theorem 2.5 are satisfied. In the former case, some set of O(n 1/4 (log n) 1/2 ) vertices completely splits X. In the latter case, either X is exceptional, or, by Lemma 3.5, some set of O(log n) vertices completely splits X.
Growth of spheres
In this section, we will prove Lemma 2.1, our estimate of the size of spheres in constituent digraphs.
We start from a few basic observations.
|N (A ′ )|/|A ′ | ≥ |B|/|A| where N (A ′ ) is the set of neighbors of vertices in A ′ , i.e., N (A ′ ) = {y ∈ B : ∃x ∈ A ′ , {x, y} ∈ E}. Proof. Let A ′ ⊆ A. By the pigeonhole principle, there is some i such that |A ′ ∩ A i |/|A i | ≥ |A ′ |/|A|. Let α be the degree of a vertex in A i and let β be the number of neighbors in A i of a vertex in B. We have α|A i | = β|B|, and β|N (A ′ ∩ A i )| ≥ α|A ′ ∩ A i |. Hence, |N (A ′ )| ≥ |N (A ′ ∩ A i )| ≥ |A ′ ∩ A i |α β = |A ′ ∩ A i ||B| |A i | ≥ |B||A ′ | |A| .
Suppose A, B ⊆ V are disjoint set of vertices. We denote by (A, B, i) the bipartite graph between A and B such that there is an edge from
x ∈ A to y ∈ B if c(x, y) = i. For I ⊆ [r −1]
a set of nondiagonal colors, we denote by (A, B, I) the bipartite graph between A and B such that there is an edge from
x ∈ A to y ∈ B if c(x, y) ∈ I. Fact 4.2. For any vertex u, colors 0 ≤ j, k ≤ r − 1 with j = k, and set I ⊆ [r − 1] of nondiagonal colors, the bipartite graph (X j (u), X k (u), I) is biregular. Proof. The degree of every vertex in X j (u) is i∈I p j ik * . And the degree of every vertex in X k (u) is i∈I p k ji . Recall our notation X (δ) i (u) for the δ-sphere centered at u in the color-i con- stituent digraph, i.e., the set of vertices v such that dist i (u, v) = δ.
For the remainder of Section 4, we fix a PCC X, a color 1 ≤ i ≤ r − 1, and a vertex u. For a color 1 ≤ j ≤ r − 1 and an integer 1 ≤ α ≤ dist i (j), we denote by S (j) α the set of vertices v ∈ X (α) i (u) such that there is a vertex w ∈ X j (u) and a shortest path in X i from u to w passing through v, i.e.,
S (j) α = {v ∈ X (α) i (u) : ∃w ∈ X j (u) s.t. dist i (u, v) + dist i (v, w) = dist i (u, w)}.
Note that these sets S (j) α are nonempty by the primitivity of X, and in particular, if Figure 1 for a graphical explanation of the notation.
α = dist i (j), then S (j) α = X j (u). For v ∈ V and an integer dist i (u, v) < α ≤ dist i (j), we denote by S (j) α (v) ⊆ S (j) α the set of vertices x ∈ S (j) α such that there is a shortest path in X i from u to x passing through v, i.e. S (j) α (v) = S (j) α ∩ X (α−dist i (u,v)) i (v) = {x ∈ S (j) α : dist i (u, v) + dist i (v, x) = dist i (u, x)}. SeeCorollary 4.3. Let 1 ≤ j ≤ r − 1 be a color such that δ = dist i (j) ≥ 3. Let 1 ≤ α ≤ δ − 2 be an integer, and let v ∈ S (j) α . Then |S (j) δ (v)| |S (j) α+1 (v)| ≥ n j |S (j) α+1 | . u w v . . . . . . X (1) i (u) . . . . . . X (α) i (u) S (j) α X (α+1) i (u) S (j) α+1 (v) S (j) α+1 X j (u) Figure 1: S (j) α and S (j) α+1 (v). Proof. Consider the bipartite graph (S (j) α+1 , X j (u), I) with I = {k : 1 ≤ k ≤ r − 1 and dist i (k) = dist i (j) − α − 1}.
There is an edge from x ∈ S (j) α+1 to y ∈ X j (u) if there is a shortest path from u to y passing through x.
By the coherence of X, if X ℓ (u) ∩ S (j) α+1 is nonempty for some color ℓ, then
X ℓ (u) ⊆ S (j) α+1 . Hence, S (j)
α+1 is partitioned into sets of the form X ℓ (u) with dist i (ℓ) = α + 1. For such colors ℓ, by Fact 4.2, (X ℓ (u), X j (u), I) is biregular, and by the definition of S (j) α+1 , then (X ℓ (u), X j (u), I) is not an empty graph. Therefore, the result follows by applying Proposition 4.
1 with A = S (j) α+1 , B = X j (u), A ′ = S (j) α+1 (v) ⊆ S (j) α+1 , and (hence) N (A ′ ) = S (j) δ (v). Fact 4.4. Let 1 ≤ j ≤ r − 1 be a color such that δ = dist i (j) ≥ 3, and w be a vertex in X j (u). Let 1 ≤ α ≤ δ − 2, and let v be a vertex in S (j) α . If dist i (v, w) = δ − α, then {x : x ∈ X i (v) and dist i (x, w) = δ − α − 1} ⊆ S (j) α+1 (v). Proof. For any x ∈ X i (v), we have dist i (u, x) ≤ α+1. If dist i (x, w) = δ −α−1, then x ∈ X (α+1) i (u), because otherwise dist(u, w) < δ. Then x is in S (j) α+1 (v),
since there is a shortest from u to w passing through x.
Proposition 4.5. Let 1 ≤ j ≤ r − 1 be a color such that δ = dist i (j) ≥ 3. Let 1 ≤ α ≤ δ − 2, and let v ∈ S (j) α . Then |X δ−α i (u)| ≥ n i |S (j) δ (v)| |S (j) α+1 (v)| . Proof. Let k be a color satisfying dist i (k) = δ −α and X k (v)∩S (j) δ (v) = ∅. Let w be a vertex in X k (v)∩S (j) δ (v). Consider the bipartite graph B = (X i (v), X k (v), I), where I = {ℓ : dist i (ℓ) = δ − α − 1}.
By Fact 4.2, B is biregular, and by Fact 4.4 the degree of w in B is at most |S
(j) α+1 (v)|. Denote by d k the degree of a vertex x ∈ X i (v) in B, so n k |S (j) α+1 (v)| ≥ n i d k . Hence, summing over all colors k such that X k (v) ∩ S (j) δ (v) = ∅, we have |X (δ−α) i (v)| ≥ k n k ≥ k n i d k |S (j) α+1 (v)| ≥ n i |S (j) δ (v)| |S (j) α+1 (v)| .
Finally, by the coherence of X, we have |X
(δ−α) i (u)| = |X (δ−α) i (v)|.
We now complete the proof of Lemma 2.1.
Proof of Lemma 2.1. Combining Corollary 4.3 and Proposition 4.5, for any
1 ≤ α ≤ δ − 2 we have |X (δ−α) i (u)| ≥ n i n k |S (k) α+1 | and so since S (k) α+1 ⊆ X (α+1) i
(u) by definition, we have the desired inequality.
Distinguishing number
In this section, we will prove Lemma 3.1, which will allow us to assume that our PCCs X satisfy ρ = o(n 2/3 ).
We recall that the distinguishing number D(i) of a color i is the number of vertices w such that c(w, u) = c(w, v), where u and v are any fixed pair of vertices such that c(u, v) = i. Hence, D(i) = k =j p i jk * . If D(i) is large for every color i > 0, then for every pair of distinct vertices u, v ∈ V , a random individualized vertex w gives different colors to u and v in the stable refinement with good probability. This idea is formalized in the following lemma due to Babai [1]. We give the following lower bound on ζ when ρ is sufficiently large.
Lemma 5.2. Let X be a PCC and suppose that ρ ≥ n 2/3 (log n) −1/3 . Then D(i) = Ω(n 2/3 (log n) −1/3 ) for all 1 ≤ i ≤ r − 1.
Lemma 3.1 follows immediately from Lemmas 5.1 and 5.2. We will prove Lemma 5.2 by separately addressing the cases ρ ≥ n/3 and ρ < n/3. The case ρ < n/3 will rely on our estimate for the size of spheres in constituent digraphs, Lemma 2.1. For the case ρ ≥ n/3, we will rely on Lemma 3.3, which bounds the number of large colors when D(i) is small for some color i ≥ 1. We prove Lemma 3.3 in the following subsection.
We first recall the following observation of Babai [1, Proposition 6.3].
Proposition 5.3 (Babai). Let X be a PCC. Then
1 n − 1 r−1 j=1 D(j)n j ≥ ρ + 2.
The following corollary is then immediate. The following facts about the parameters of a coherent configuration are standard. . Let X be a CC. Then for all colors i, j, k, the following relations hold:
(i) n i = n i * (ii) p i jk = p i * k * j * (iii) n i p i jk = n j p j ik * (iv) r−1 j=0 p i jk = r−1 j=0 p i kj = n k
Bound on the number of large colors
We now prove Lemma 3.3, using the following preliminary results.
Lemma 5.6. Let X be a PCC, let I be a nonempty set of nondiagonal colors, let n I = i∈I n i , and let J be the set of colors j such that n j ≤ n I /2. Then j∈J n j ≤ 2 max{D(i) : i ∈ I}.
Proof. For any color i, by Proposition 5.5, we have
D(i) = r−1 j=0 k =j p i jk * = r−1 j=0 k =j n j p j ik n i = 1 n i r−1 j=0 n j k =j p j ik = 1 n i r−1 j=0 n j (n i − p j ij ).
Therefore,
n I max{D(i) : i ∈ I} ≥ i∈I n i D(i) ≥ i∈I j∈J n j (n i − p j ij ) ≥ j∈J n j i∈I (n i − p j ij ) ≥ j∈J n j (n I − n j ) ≥ n I 2 j∈J n j .
Lemma 5.7. Let X be a PCC, and suppose p i jk > 0 for some i, j, k. Then
D(j) − D(k) ≤ D(i) ≤ D(j) + D(k).
Proof. Fix vertices u, v, w ∈ V with c(u, w) = i, c(u, v) = j, and c(v, w) = k.
(These vertices exist since p i jk > 0.) For any vertex x such that c(x, u) = c(x, w),
we have c(x, u) = c(x, v) or c(x, v) = c(x, w). Therefore, D(j) + D(k) ≥ D(i).
For the other inequality, if p i jk > 0 then p j ik * > 0 by Proposition 5.5, and D(k * ) = D(k) by the definition of distinguishing number. So we have D(i) + D(k) = D(i) + D(k * ) ≥ D(j), using the previous paragraph for the latter inequality.
Lemma 5.8. Let X be a PCC. Then for any nondiagonal color i and number 0 ≤ η ≤ ρ − D(i), there is a color j such that η < D(j) ≤ η + D(i).
Proof. By Corollary 5.4, there is a color k with D(k) > ρ. Now consider a shortest path u 0 , . . . , u ℓ in X i with c(u 0 , u ℓ ) = k. (By the primitivity of X, the digraph X i is strongly connected, and such a path exists.) Let δ j = D(c(u 0 , u j )) for 1 ≤ j ≤ k. By Lemma 5.7, we have |δ j − δ j+1 | ≤ D(i). Hence, one of the numbers δ j falls in the interval (η, η + D(i)] for any 0 ≤ η ≤ ρ − D(i).
We denote by I α the set of colors i with D(i) ≤ α. Lemma 5.9. Let X be a PCC with ρ > 0. Let i be a nondiagonal color and let 0 ≤ η ≤ ρ − 2D(i). Then
n i ≤ j∈I η+3D(i) \Iη n j .
Proof. By Lemma 5.8, the set I η+2D(i) \ I η+D(i) is nonempty. Let k ∈ I η+2D(i) \ I η+D(i) . We have r−1 j=0 p k ij = n i by Proposition 5.5. On the other hand, if p k ij > 0 for some j, then D(j) − D(i) ≤ D(k) ≤ D(j) + D(i) by Lemma 5.7, and so j ∈ I η+3D(i) \ I η . Hence,
n i = r−1 j=0 p k ij = j∈I η+3D(i) \Iη p k ij ≤ j∈I η+3D(i) \Iη n j .
Lemma 5.10. Let X be a PCC with ρ > 0, let i be a nondiagonal color, and let
0 ≤ η ≤ ρ. Then η 3D(i) n i ≤ j∈Iη n j .
Proof. If η < 3D(i), the left-hand side is 0, so assume η ≥ 3D(i). For any integer 1 ≤ α ≤ ⌊η/(3D(i))⌋, let S α = I 3D(i)α \I 3D(i)(α−1) . Then
⌊η/(3D(i))⌋ α=1 S α ⊆ I η
By the disjointness of the sets S α and Lemma 5.9, we have
j∈Iη n j ≥ ⌊η/(3D(i))⌋ α=1 j∈Sα n j ≥ η 3D(i) n i .
Finally, we are able to prove Lemma 3.3.
Proof of Lemma 3.3. Fix an integer 0 ≤ α ≤ ⌊log 2 (ρ/(3D(i)))⌋. For any number β, let J β denote the set of colors j such that n j ≤ β. We start by estimating |J 2 α n i \ J 2 α−1 n i |, i.e., the number of colors j with 2 α−1 n i < n j ≤ 2 α n i . By Lemma 5.10, we have
j∈I 2 α (3D(i)) n j ≥ 2 α n i .
Therefore, applying Lemma 5.6 with I = I 2 α (3D(i)) and J = J 2 α n i , we have
j∈J 2 α n i n j ≤ 2 max{D(i) : i ∈ I 2 α ·3D(i) } ≤ 2 α+1 (3D(i)),
with the second inequality coming from the definition of I 2 α (3D(i)) . It follows that the number of colors j such that j ∈ J 2 α n i \ J 2 α−1 n i is at most 2 α+1 (3D(i))/(2 α−1 n i ) = 12D(i)/n i . Overall, the number of colors j satisfying (1/2)n i < n j ≤ 2 ⌊log 2 (ρ/3D(i))⌋ n i is at most 12(log 2 n + 1)D(i)/n i . Furthermore, the number of colors j satisfying
n j > 2 ⌊log 2 (ρ/3D(i))⌋ n i ≥ ρn i 6D(i)
is at most (6D(i)/(ρn i ))n, since r−1 j=0 n j = n. Hence, the number of colors j such that n j > n i /2 is at most O((log n + n/ρ)D(i)/n i ).
Estimates of the distinguishing number
We now prove Lemma 5.2, our lower bound for D(i).
First, we recall the following two observations made by Babai [1, Proposition 6.4 and Theorem 6.11].
Proposition 5.11 (Babai). Let X be a PCC. For colors 0 ≤ i, j ≤ r − 1, we have
D(j) ≤ dist i (j)D(i).
Proposition 5.12 (Babai). Let X be a PCC. For any color 1 ≤ i ≤ r − 1, we have n i D(i) ≥ n − 1.
We prove the following two estimates of the distinguish number Lemma 5.13. Let X be a PCC. Fix nondiagonal colors i, j ≥ 1 and a vertex u ∈ V . Let δ = dist i (j), and γ = δ−1 α=2 |X
(α) i (u)|. If δ ≥ 3, then D(i) = Ω D(j) √ nn j γ 2/3 . Proof. By Lemma 2.1, for any 1 ≤ α ≤ δ − 2 we have |X (α+1) i (u)||X (δ−α) i (u)| ≥ n i n j
and in particular,
max{|X (α+1) i (u)|, |X (δ−α) i (u)|} ≥ √ n i n j .
Hence,
γ = δ−1 α=2 |X (α) i (u)| = Ω(δ √ n i n j ) = Ω δ √ nn j D(i) ,(1)
where the last inequality comes from Proposition 5.12. Now by Proposition 5.11 and Eq. (1), we have
D(i) ≥ D(j) δ = Ω D(j) √ nn j γ D(i) ,
from which the desired inequality immediately follows.
Lemma 5.14. Let X be a PCC with ρ = Ω(n). Then every nondiagonal color i with n i ≤ ρ satisfies
D(i) = Ω ρn i log n .
Proof. Fix a nondiagonal color i with n i ≤ ρ, and suppose D(i) < ρ/6 (otherwise the lemma holds trivially). Let J β denote the set of colors j such that n j ≤ β. Applying Lemma 5.6 with the set I = {i}, we have
j∈J n i /2 n j ≤ 2D(i).(2)
On the other hand, by Lemma 5.9, for every integer η with
0 ≤ η ≤ ρ/2 − 3D(i), n i ≤ j∈I η+3D(i) \Iη n j .
Thus, for every such η, at least one of following two conditions hold:
(a) there exists a color j ∈ I η+3D(i) \ I η satisfying n j > n i /2;
(b) j∈I η+3D(i) \Iη: n j ≤n i /2 n j ≥ n i .
There are at least ⌊ρ/(6D(i))⌋ disjoint sets of the form I η+3D(i) \ I η with We recall that when color 1 is dominant, it is symmetric. In this case, we recall our notation µ = |N (x) ∩ N (y)|, where x, y ∈ V are any pair of vertices with c(x, y) = 1 and N (x) is the nondominant neighborhood of x. Hence, µ = i,j>1 p 1 ij . Lemma 5.15. Let X be a PCC with n 1 ≥ n/2. Then µ ≤ ρ 2 /n 1 .
0 ≤ η ≤ ρ/2 − 3D(i). By
Proof. Fix a vertex u. There are at most ρ 2 paths of length two from u along edges of nondominant color, and exactly n 1 vertices v such that c(u, v) = 1. For any such vertex y, there are exactly µ paths of length two from u to v along edges of nondominant color. Hence, µ ≤ ρ 2 /n 1 .
Proof of Lemma 5.2. First, suppose n 2/3 (log n) −1/3 ≤ ρ < n/3. We have n 1 = n − ρ − 1 > 2n/3 − 1. Consider two vertices u, v ∈ V with c(u, v) = 1. Note that for any vertex w ∈ N (v) \ N (u), we have c(w, u) = 1 and c(w, v) > 1. Hence, by Lemma 5.15 and the definition of D(1),
D(1) ≥ ρ − µ ≥ ρ − ρ 2 n 1 ≥ 1 2 − o(1) ρ = Ω(n 2/3 (log n) −1/3 ).
Fix a color i = 1. If dist i (1) = 2, then by Proposition 5.11,
D(i) ≥ D(1) 2 ≥ Ω(n 2/3 (log n) 1/3 ).
Otherwise, if dist i (1) ≥ 3, by applying Lemma 5.13 with j = 1, we have
D(i) = Ω D(1) √ nn 1 n − n 1 2/3 = Ω ρn ρ − 1 2/3
= Ω(n 2/3 ). Now suppose ρ ≥ n/3. By Lemma 5.14 and Proposition 5.12, for every color i with n i ≤ ρ, we have (D(i)) 3/2 = Ω ρn i D(i) log n = Ω ρn log n , and hence D(i) = Ω(n 2/3 (log n) −1/3 ). If n 1 ≤ ρ, then n i ≤ ρ for all i, and we are done. Otherwise, if n 1 > ρ, we have only to verify that D(1) = Ω(n 2/3 (log n) −1/3 ). Consider two vertices u, w with dist 1 (u, w) = 2. (Since we assume the rank is at least 3, we can always find such u, w by the primitivity of X.) Let i = c(u, w). Then i > 1 and so n i ≤ ρ. Since D(i) = Ω(n 2/3 (log n) −1/3 ) for every color 1 < i ≤ r − 1, and dist 1 (i) = 2, we have D(1) = Ω(n 2/3 (log n) −1/3 ) by Proposition 5.11.
Diameter of constituent graphs
We now prove Lemma 2.2, which states that dist i (1) = 2 for any nondominant color i, assuming that inequality ρ = o(n 2/3 ).
We start from a few basic observations. Observation 6.1. Let X be a PCC. For any nondominant color i, we have
n i ≥ n 1 /ρ. Proof. Fix a vertex u ∈ V . Since X i is connected (X is primitive), for any v ∈ X 1 (x), there is a shortest path in X i from u to v, hence there exists a vertex w ∈ N (u) such that c(w, v) = i, so |N (x)|n i ≥ |X 1 (x)|.
Lemma 6.2. Let X be a PCC with a nondominant color i, let δ = dist 1 (i), and suppose δ ≥ 3. Any vertices u, w with c(u, w) = i satisfy the following two properties: Proof. In fact, B is regular on X 1 (u) by Fact 4.2, so every vertex in X 1 (u) has degree γ in B.
(i) If v ∈ X (δ−1) i (u) ∩ X (δ−2) i (w), then X i (v) ∩ X 1 (u) ⊆ X i (v) ∩ X (δ−1) i (w); (ii) If z ∈ X 1 (u) ∩ X (δ−1) i (w), then X i * (z) ∩ X (δ−2) i (w) ⊆ X i * (z) ∩ X (δ−1) i (u). Proof. If v ∈ X (δ−1) i (u) ∩ X (δ−2) i (w) then dist i (u, v) = δ − 1 and dist i (w, v) = δ − 2. So for any vertex x ∈ X i (v), we have dist i (w, x) ≤ δ − 1. If x ∈ X 1 (u), then dist i (w, x) = δ − 1, since otherwise dist i (u, x) < δ. Similarly, z ∈ X 1 (u) ∩ X (δ−1) i (w) means dist i (u, z) = δ and dist i (w, z) = δ − 1. So for any y satisfying dist i (w, y) = δ − 2, we have dist i (u, y) ≤ δ − 1. If z ∈ X i (y), then dist i (u, y) = δ − 1, since otherwise dist i (u, z) < δ.
Let v ∈ X (δ−1) i (u) achieve degree β in B, and let w ∈ X i (u) be such that
dist i (w, v) = δ − 2. Let B ′ be the subgraph of B given by (S 1 , S 2 , i), where S 1 = X (δ−1) i (u) ∩ X (δ−2) i (w) and S 2 = X 1 (u) ∩ X (δ−1) i (w)
. Note that v ∈ S 1 . By Lemma 6.2 (i), every neighbor of v in B is also a neighbor of v in B ′ , and in particular, the degree of v in B ′ is again β.
Let j = c(w, v), and let H = (X j (w), S 3 , i), where
S 3 = z ∈ X (δ−1) i (w) : |X i * (z) ∩ X j (w)| ≤ γ .
Recall that v ∈ X j (w), so v is also a vertex of H. We claim that every neighbor of
v in B ′ is also a neighbor of v in H, so the degree of v in B ′ is again ≥ β. Indeed, let z ∈ X i (v) ∩ S 2 . Then z ∈ X (δ−1) i (w), and furthermore |X i * (z) ∩ X j (w)| ≤ |X i * (z) ∩ X (δ−2) i (w)| ≤ |X i * (z) ∩ X (δ−1) i (u)| = γ.
So, z ∈ S 3 , and every neighbor of v in B ′ is also a vertex of H as claimed. Now by Fact 4.2, H is regular on X j (w) with degree ≥ β. Hence,
βn j ≤ |E(H)| ≤ γ|S 3 | ≤ γ|X (δ−1) i (w)| ≤ γρ.
The lemma follows by Observation 6.1.
Proof of Lemma 2.2.
We prove that if dist i (1) ≥ 3 for some color i, then ρ n 2/3 . Without loss of generality, we assume n 1 ∼ n, since otherwise we are already done.
Let i be such that dist i (1) ≥ 3, and write δ = dist i (1). Fix a vertex u and let B, γ, and β be as in Lemma 6.3, so β ≤ γρ 2 /n 1 . By Lemma 6.3, B is regular on X 1 (u). Let γ, β be defined as Lemma 6.3. By Lemma 6.3, we have β ≤ γρ 2 /n 1 . Therefore, by counting the number of edges in B
ρ 3 γ n 1 ≥ β|X (δ−1) i (x)| ≥ |E(B)| = n 1 γ.
The lemma is then immediate since n 1 ∼ n.
Clique geometries
In this section, we prove Theorem 2.4, giving sufficient conditions for the existence of an asymptotically uniform clique geometry in a PCC. We use the word "geometry" in Definition 2.3 because the cliques resemble lines in a geometry: two distinct cliques intersect in at most one vertex. Indeed, a regular graph G has a clique geometry G with cliques of uniform order only if it is the point-graph of a geometric 1-design with lines corresponding to cliques of G.
Theorem 2.4 builds on earlier work of Metsch [19] on the existence of similar clique structures in "sub-amply regular graphs" (cf. [6]) via the following lemma. The lemma can be derived from [19, Theorem 1.2], but see [6, Lemma 4] for a self-contained proof.
Lemma 7.1. Let H be a graph on k vertices which is regular of degree λ and such that any pair of nonadjacent vertices have at most µ common neighbors. Suppose that kµ = o(λ 2 ). Then there is a partition of V (H) into maximal cliques of order ∼ λ, and all other maximal cliques of H have order o(λ).
Metsch's result, applied to the graphs induced by G(X) on sets of the form X i (u), gives collections of cliques which locally resemble asymptotically uniform clique geometries. These collections satisfy the following definition for a set I = {i} containing a single color. Definition 7.2. Let I be a set of nondominant colors. An I-local clique partition at a vertex u is a collection P of subsets of X I (u) satisfying the following properties:
(i) P is a partition of X I (u) into maximal cliques in the subgraph of G(X)
induced on X I (u);
(ii) for every C ∈ P u and i ∈ I, we have |C ∩ X i (u)| ∼ λ i .
We say X has I-local clique partitions if there is an I-local clique partition at every vertex u ∈ V .
To prove Theorem 2.4, we will stitch local clique partitions together into geometric clique structures.
Note that from the definition, if P is an I-local clique partition (at some vertex) and i ∈ I, then |P| ∼ n i /λ i .
Corollary 7.3. Let X be a PCC and let i be a nondominant color such that
n i µ = o(λ 2 i )
. Then X has {i}-local clique partitions.
Proof. Fix a vertex u, and apply Lemma 7.1 to the graph H induced by G(X) on X i (u). The Lemma gives a collection of cliques satisfying Definition 7.2.
The following simple observation is essential for the proofs of this section.
Observation 7.4. Let X be a PCC, let C be a clique in G(X), and suppose u ∈ V \ C is such that |N (u) ∩ C| > µ. Then C ⊆ N (u).
Proof. Suppose there exists a vertex
v ∈ C \ N (u), so c(u, v) = 1. Then |N (u) ∩ N (v)| = µ by the definition of µ in a PCC. But |N (u) ∩ N (v)| ≥ |N (u) ∩ C ∩ N (v)| = |N (u) ∩ (C\{v})| = |N (u) ∩ C| > µ, a contradiction.
Under modest assumptions, if local clique partitions exist, they are unique.
Lemma 7.5. Let X be a PCC, let i be a nondominant color such that n i µ = o(λ 2 i ), and let I be a set of nondominant colors such that i ∈ I. Suppose X has I-local clique partitions. Then for every vertex u ∈ V , there is a unique I-local clique partition P at u.
Proof. Let u ∈ V and let P be an I-local clique partition at u. Let C and C ′ be two distinct maximal cliques in the subgraph of G(X) induced on X I (u). We show that |C ∩ C ′ | < µ. Suppose for the contradiction that |C ∩ C ′ | ≥ µ. For a vertex v ∈ C \ C ′ , we have |N (v) ∩ (C ′ ∪ {u})| > µ, and so C ′ ⊆ N (v) by Observation 7.4. But since v / ∈ C ′ , this contradicts the maximality of C ′ . So in fact |C ∩ C ′ | < µ. Now let C / ∈ P be a maximal clique in the subgraph of G(X) induced on X I (u). Since P is an I-local clique partition, it follows that
|C| = C ′ ∈P |C ′ ∩ C| < µ|P| ∼ n i µ/λ i = o(λ i ).
Then C does not belong to an I-local clique partition, since it fails to satisfy Definition 7.2 (ii).
Local cliques and symmetry
Suppose X has I-local clique partitions, and c(u, v) ∈ I for some u, v ∈ V . We remark that in general, the clique containing v in the I-local clique partition at u will not be in any way related to any clique in the I-local clique partition at v. In particular, we need not have c(v, u) ∈ I. However, even when c(v, u) ∈ I as well, there is no guarantee that the clique at u containing v will have any particular relation to the clique at v containing u. This lack of symmetry is a fundamental obstacle that we must overcome to prove Theorem 2.4. Lemma 7.7 below is the main result of this subsection. It gives sufficient conditions on the parameters of a PCC for finding the desired symmetry in local clique partitions satisfying the following additional condition. Definition 7.6. Let I be a set of nondominant colors, let u ∈ V , and let P be an I-local clique partition at u. We say P is strong if for every C ∈ P, the clique C ∪ {u} is maximal in G(X). We say X has strong I-local clique partitions if there is a strong I-local clique partition at every vertex u ∈ V .
We introduce additional notation. Suppose I is a set of nondominant colors, and i ∈ I satisfies n i µ = o(λ i ) 2 . If X has I-local clique partitions, then for every u, v ∈ V with c(u, v) ∈ I, we denote by K I (u, v) the set C ∪ {u}, where C is the clique in the partition of X I (u) containing v (noting that by Lemma 7.5, this clique is uniquely determined). Lemma 7.7. Let X be a PCC with ρ = o(n 2/3 ), let i be a nondominant color, and let I and J be sets of nondominant colors such that i ∈ I, i * ∈ J, and X has strong I-local and J-local clique partitions. Suppose λ i λ i * = Ω(n). Then for
every u, v ∈ V with c(u, v) = i, we have K I (u, v) = K J (v, u).
We first prove two easy preliminary statements. Lemma 7.9. Let X be a PCC and let I and J be sets of nondominant colors such that X has strong I-local and J-local clique partitions. Suppose that for some vertices u, v, x, y ∈ V we have |K
I (u, v) ∩ K J (x, y)| > µ. Then K I (u, v) = K J (x, y). Proof. Suppose there exists a vertex z ∈ K J (x, y) \ K I (u, v). We have |N (z) ∩ K I (u, v)| ≥ |K J (x, y) ∩ K I (u, v)| > µ. Then K I (u, v) ⊆ N (z) by Observa- tion 7.4, contradicting the maximality of K I (u, v). Thus, K J (x, y) ⊆ K I (u, v). Similarly, K I (u, v) ⊆ K J (x, y).
Proof of Lemma 7.7. Without loss of generality, assume λ i ≤ λ i * .
Suppose for contradiction that there exists a vertex u ∈ V such that for every v ∈ X i (u), we have
K I (u, v) = K J (v, u). Then |K I (u, v) ∩ K J (v, u)| ≤ µ by Lemma 7.9. Fix v ∈ X i (u), so for every w ∈ K I (u, v) ∩ X i (u), we have |K J (w, u) ∩ K I (u, v)| ≤ µ. Hence, there exists some sequence w 1 , . . . , w ℓ of ℓ = ⌈λ i /(2µ)⌉ vertices w α ∈ K I (u, v)∩X i (u) such that K J (w α , u) = K J (w β , u) for α = β. But by Lemma 7.9, for α = β we have |K J (w α , u) ∩ K J (w β , u)| ≤ µ.
Hence, for any 1 ≤ α ≤ ℓ we have
K J (w α , u) \ β =α K J (w β , u) λ i * − µλ i /(2µ) ≥ λ i * /2. But K J (w α , u) ⊆ N (u), so |N (u)| ≥ ℓ α=1 K J (w α , u) λ i λ i * 4µ = ω(ρ)
by Proposition 7.8. This contradicts the definition of ρ.
Hence, for any vertex u, there is some
v ∈ X i (u) such that K I (u, v) = K J (v, u). Then, in particular, |X i * (v) ∩ X I (u)| λ i * by the definition of a J-local clique partition. By the coherence of X, for every v ∈ X i (u), we have |X i * (v) ∩ X I (u)| λ i * .
Recall that X I (u) is partitioned into ∼ n i /λ i maximal cliques, and for each of these cliques C other than K I (u, v), we have |N (v)∩C| ≤ µ. Hence,
|X i * (v) ∩ K I (u, v)| λ i * − O µn i λ i = λ i * − o n λ i ∼ λ i *
by Proposition 7.8. Since the J-local clique partition at v partitions X i * (v) into ∼ n i /λ i * cliques, at least one of these intersects K I (u, v) in at least ∼ λ 2 i * /n i = ω(µ) vertices. In other words, there is some
x ∈ X i * (v) such that |K J (v, x) ∩ K I (u, v)| = ω(µ). But then K J (v, x) = K I (u, v) by Lemma 7.9. In particu- lar, u ∈ K J (v, x), so K J (v, x) = K J (v, u). Hence, K J (v, u) = K J (v, x) = K I (u, v), as desired.
Existence of strong local clique partitions
Our next step in proving Theorem 2.4 is showing the existence of strong local clique partitions. We accomplish this via the following lemma. Lemma 7.10. Let X be a PCC such that ρ = o(n 2/3 ), and let i be a nondominant color such that n i µ = o(λ 2 i ). Suppose that for every color j with n j < n i , we have λ j = Ω( √ n). Then for n sufficiently large, there is a set I of nondominant colors with i ∈ I such that X has strong I-local clique partitions.
We will prove Lemma 7.10 via a sequence of lemmas which gradually improve our guarantees about the number of edges between cliques of the I-local clique partition at a vertex u and the various neighborhoods X j (u) for j / ∈ I. Lemma 7.11. Let X be a PCC, and let i and j be nondominant colors. Then for any 0 < ε < 1 and any u, v ∈ V with c(u, v) = j, we have
|X i (u) ∩ N (v)| ≤ max λ i + 1 1 − ε , n i µ εn j Proof. Fix u, v ∈ V with c(u, v) = j and let α = |X i (u) ∩ N (v)|.
We count the number of triples (x, y, z) of vertices such that x, y ∈ X i (u) ∩ N (z), with c(u, z) = j and c(x, y) = 1. There are at most n 2 i pairs x, y ∈ X i (u), and if c(x, y) = 1 then there are at most µ vertices z such that x, y ∈ N (z). Hence, the number of such triples is at most n 2 i µ. On the other hand, by the coherence of X, for every z with c(u, z) = j, we have at least α(α − λ i − 1) pairs x, y ∈ X i (u) ∩ N (z) with c(x, y) = 1. Hence, there are at least n j α(α − λ i − 1) total such triples. Thus,
n j α(α − λ i − 1) ≤ n 2 i µ.
Hence, if α ≤ (λ i +1)/(1−ε), then we are done. Otherwise, α > (λ i +1)/(1−ε), and then λ i + 1 < (1 − ε)α. So, we have
n 2 i µ ≥ n j α(α − λ i − 1) > εn j α 2 ,
and then α < n i µ/(εn j ).
Lemma 7.12. Let X be a PCC, and let i be a nondominant color such that n i µ = o(λ 2 i ). Let I be a set of nondominant colors with i ∈ I such that X has I-local clique partitions. Let j be a nondominant color such that n i µ/n j < ( √ 3/2)λ i . Let u ∈ V , let P u be the I-local clique partition at u, and let v ∈ X j (u). Suppose some clique C ∈ P u is such that c(u, v) = j and |N (v) ∩ C| ≥ µ. Then for every vertex x, y ∈ V with c(x, y) = j, letting P x be the I-local clique partition at x, the following statements hold:
(i) there is a unique clique C ∈ P x such that C ⊆ N (y); (ii) |N (y) ∩ X i (x)| ∼ λ i .
Proof. Letting C = C ∪ {u}, we have | C ∩ N (v)| ≥ µ + 1 > µ. Therefore, by Observation 7.4, we have C ⊆ N (v). In particular, |N (v)∩X i (u)| λ i , and so by the coherence of X, |N (y) ∩ X i (x)| λ i for every pair x, y ∈ V with c(x, y) = j.
Now fix x ∈ V , and let P x be the I-local clique partition at x. By the definition of an I-local clique partition, we have |P x | ∼ n i /λ i . For every y ∈ X j (x), by assumption we have
|N (y) ∩ X i (x)| λ i = ω(µn i /λ i ).(3)
Then it follows from the pigeonhole principle that for n sufficiently large, there is some clique C ∈ P x such that |N (y) ∩ C| > µ, and then C ⊆ N (y) by Observation 7.4. Now suppose for contradiction that there is some clique C ′ ∈ P x with C ′ = C, such that C ′ ⊆ N (y).
|N (y) ∩ X i (x)| ≥ |C ∪ C ′ | 2λ i ∼ 2(λ i + 1)(4)
(with the last relation holding since λ i = ω( √ n i µ) = ω(1).) However, by Lemma 7.11 with ε = 1/3, we have
|X i (x) ∩ N (y)| ≤ max 3 2 (λ i + 1), n i 3µ n j = 3 2 (λ i + 1),
with the last equality holding by assumption. This contradicts Eq. (4), so we conclude that C is the unique clique in P x satisfying C ⊆ N (y). In particular, by Observation 7.4, we have |N (y) ∩ C ′ | ≤ µ for every C ′ ∈ P x with C ′ = C. Finally, we estimate |N (y) ∩ X i (x)| by
|N (y) ∩ X i (x) ∩ C| + C ′ =C |N (y) ∩ X i (x) ∩ C ′ | λ i + µn i /λ i ∼ λ i , which, combined with Eq. (3), gives |N (y) ∩ X i (x)| ∼ λ i .
Lemma 7.13. Let X be a PCC, and let i be a nondominant color such that n i µ = o(λ 2 i ). There exists a set I of nondominant colors with i ∈ I such that X has I-local clique partitions and the following statement holds. Suppose j is a nondominant color such that n i µ/n j = o(λ i ), let u ∈ V , and let P be the I-local clique partition at u. Then for any C ∈ P and any vertex v ∈ X j (u) \ C, we have |N (v) ∩ C| < µ.
Proof. By Corollary 7.3, X has {i}-local clique partitions. Let I be a maximal subset of of the nondominant colors such that i ∈ I and X has I-local clique partitions. We claim that I has the desired property.
Indeed, suppose there exists some color j / ∈ I satisfying n i µ/n j = o(λ i ), some vertices u, v with c(u, v) = j, and some C ∈ P with |N (v) ∩ C| ≥ µ, where P is the I-local clique partition at u. By Lemma 7.12, for n sufficiently large, for every vertex u, v ∈ V with c(u, v) = j, and I-local clique partition P at u, there is a unique clique C ∈ P such that C ⊆ N (v), and furthermore
|N (v) ∩ X i (u)| ∼ λ i .(5)
Now fix u ∈ V and let P be the I-local clique partition at u. Let P ′ be the collection of sets C ′ of the form
C ′ = C ∪ {v ∈ X j (u) : C ⊆ N (v)}
for every C ∈ P. Let J = I ∪ {j}. We claim that P ′ satisfies Definition 7.2, so X has local clique partitions on J. This contradicts the maximality of I, and the lemma then follows.
First we verify Definition 7.2 (i). By the second paragraph of this proof, P ′ partitions X J (u). Furthermore, the sets C ∈ P ′ are cliques in G(X), since for any C ∈ P ′ and any distinct v, w ∈ C ∩ X j (u), we have
|N (v) ∩ N (w)| ≥ |C ∩ X I (u)| λ i = ω(µ)
and so c(v, w) is nondominant by the definition of µ. Furthermore, the cliques C ∈ P ′ are maximal in the subgraph of G(X) induced on X J (u), since they are maximal when restricted to X I (u) and each vertex v ∈ X j (u) extends a unique clique in the restriction of P ′ to X I (u).
We now verify Definition 7.2 (ii). By the pigeonhole principle, there is some
C ∈ P ′ with |C ∩ X j (u)| n j |P ′ | = n j |P| ∼ λ i n j n i .
But since C is a clique in G(X), we have |C ∩ X j (u)| ≤ λ j + 1. So, from the defining property of j,
λ j + 1 λ i n j n i = ω( √ µn j )
Since n j and µ are positive integers, we have in particular λ j = ω(1), and thus
λ j λ i n j n i = ω( √ µn j ).(6)
Hence, n j µ = o(λ 2 j ), and so by Corollary 7.3, X has {j}-local clique partitions.
Let C ′ ⊆ X j (u) be a maximal clique in G(X) of order ∼ λ j . By Eq. (5) there are ∼ λ j λ i nondominant edges between C ′ and X i (u), so some x ∈ X i (u) satisfies
|N (x) ∩ C ′ | λ j λ i /n i = ω(λ j µ/n j ) = ω(µ).
(The last equality uses Eq. (6).) Furthermore, by Eq. (6), we have
n j (n i /λ i )λ j = o( n i /µλ j ),
where the last inequality comes from the assumption that √ n i µ = o(λ i ). So by applying Lemma 7.12 with {j} in place of I, it follows that for every x ∈ X i (u),
we have |N (x) ∩ X j (u)| ∼ λ j .
We count the nondominant edges between X i (u) and X j (u) in two ways: there are ∼ λ j such edges at each of the n i vertices in X i (u), and (by Eq. (5)) there are ∼ λ i such edges at each of the n j vertices in X j (u). Hence, n i λ j ∼ n j λ i . Now, using Eq. (6), µ|P ′ | ∼ µn i /λ i ∼ µn j /λ j = o(λ j ). By the maximality of the cliques C ∈ P ′ in the subgraph of G(X) induced on X J (u), for every distinct
C, C ′ ∈ P ′ and v ∈ C, we have |N (v) ∩ C ′ | ≤ µ. Therefore, for v ∈ C ∩ X j (u), we have λ j − |X j (u) ∩ C| = |X j (u) ∩ N (v)| − |X j (u) ∩ C| ≤ |(N (v) ∩ X j (u)) \ C| ≤ µ|P ′ | = o(λ j ),
so that |X j (u) ∩ C| ∼ λ j , as desired. Now P ′ satisfies Definition 7.2, giving the desired contradiction.
Proof of Lemma 7.10. Suppose for contradiction that no set I of nondominant colors with i ∈ I is such that X has strong I-local clique partitions. Without loss of generality, we may assume that n i is minimal for this property, i.e., for every nondominant color j with n j < n i , there is a set J of nondominant colors with j ∈ J such that X has strong I-local clique partitions. Let I be the set of nondominant colors containing i guaranteed by Lemma 7.13. Let u ∈ V be such that some clique C in the I-local clique partition at u is not maximal in G(X). In particular, let v ∈ V \ C be such that C ⊆ N (v), and let j = c(u, v). Then j is a nondominant color, and j / ∈ I. Furthermore, by the defining property of I (the guarantee of Lemma 7.13), it is not the case that n i µ/n j = o(λ i ). In particular we may take n j < n i , since otherwise, if n j ≥ n i , then n i µ/n j ≤ √ n i µ = o(λ i ) by assumption. Now since n j < n i , also λ j = Ω( √ n) by assumption. Furthermore, by the minimality of n i , there is a set J of nondominant colors with j ∈ J such that X has strong J-local clique partitions on J. In particular, i / ∈ J. By the definition of I-local clique partitions,
|N (v) ∩ X i (u)| ≥ |N (v) ∩ X i (u) ∩ C| λ i .
Now let D be the clique containing v in the J-local clique partition at u. By the coherence of X, for every x ∈ X j (u) ∩ D, we have |N (x) ∩ X i (u)| λ i . Hence, there are λ j λ i nondominant edges between X j (u) ∩ D and X i (u). So, by the pigeonhole principle, some vertex y ∈ X i (u) satisfies
|N (y) ∩ D ∩ X j (u)| λ i λ j n i = ω µ n i λ j = ω µn n i = ω(µ).
(The second inequality uses the assumption that √ n i µ = o(λ i ). The last inequality uses Proposition 7.8.) But then D \ {y} ⊆ N (y) by Observation 7.4. Then y ∈ D by the definition of a strong local clique partition, and so i ∈ J, a contradiction.
We conclude that in fact X has strong I-local clique partitions.
We finally complete the proof of Theorem 2.4.
Proof of Theorem 2.4. By Lemma 7.10, for every nondominant color i there is a set I such that X has strong local clique partitions on I. We claim that these sets I partition the collection of nondominant colors. Indeed, suppose that there are two sets I and J of nondominant colors such that i ∈ I ∩J and X has strong I-local and J-local clique partitions. Let u, v ∈ V be such that c(u, v) = i. By the uniqueness of the induced {i}-local clique partition at u (Lemma 7.5), we have , v), and I = J. In particular, for every nondominant color i, there exists a unique set I of nondominant colors such that X has strong I-local clique partitions. We simplify our notation and write K(u, v) = K I (u, v) whenever c(u, v) ∈ I and X has strong I-local clique partitions. By Lemma 7.7, we have K(u, v) = K(v, u) for all u, v ∈ V with c(u, v) nondominant. Let G be the collection of cliques of the form K(u, v) for c(u, v) nondominant. Then G is an asymptotically uniform clique geometry.
|K I (u, v) ∩ K J (u, v)| λ i = ω(µ), so K I (u, v) = K J (u
Consequences of local clique partitions for the parameters λ i
We conclude this section by analyzing some consequences for the parameters λ i of our results on strong local clique partitions. Lemma 7.14. Let X be a PCC with ρ = o(n 2/3 ). For every nondominant color i, we have λ i < n i − 1.
Proof. Suppose for contradiction that λ i = n i − 1 for some nondominant color i. For every nondominant color j, by Proposition 7.8, we have n i µ/n j = o(n i ) = o(λ i ). Furthermore, n i µ = o(λ 2 i ). Let I be the set of nondominant colors with i ∈ I guaranteed by Lemma 7.13. In particular, X has I-local clique partitions. In fact, since λ i = n i − 1, for every vertex u and every clique C in the I-local clique partition at u, we have C ∩ X i (u) ∼ n i . Hence, there is only one clique in the I-local clique partition at u, and so X I (u) is a clique in G(X). For every vertex u, let K I (u) = X I (u) ∪ {u}. Then for every vertex v / ∈ X I (u), we have |N (v) ∩ K I (u)| ≤ µ. In particular, K I (u) is a maximal clique in G(X), and X has strong I-local clique partitions.
Let U, v ∈ V with v ∈ X I (u), let j = c(u, v) ∈ I, and suppose |K I (v) ∩ K I (u)| > µ. Then K I (v) = K I (u) by Lemma 7.9. Hence, by the coherence of X, for any w, x ∈ V with x ∈ X j (w), K I (w) = K I (x). By applying this fact iteratively, we find that for any two vertices y, z ∈ V such that there exists a path from y to z in X j , we have z ∈ K I (y), contradicting the primitivity of X. We conclude that |K I (v) ∩ K I (u)| ≤ µ if c(u, v) ∈ I. Hence, if we fix a vertex u and count pairs of vertices (v, w) ∈ X i (u) × X I (u) with c(w, v) = i, we have
n i j∈I p i ji ≤ n I µ,
where n I = i∈I n i . In particular, for any vertex u and v ∈ X i (u), we have
|X i * (v) ∩ X I (u)| ≤ µn I /n i .
Fix a vertex v ∈ V . For some integer ℓ, we fix distinct vertices u 1 , . . . , u ℓ in X i * (v) such that for all 1 ≤ α, β ≤ ℓ, we have u α / ∈ X I (u β ). Since |X i * (v) ∩ X I (u α )| ≤ µn I /n i , we may take ℓ = ⌊n i /(2µ)⌋. As µ = o(n i ) by Proposition 7.8, we therefore have ℓ = Ω(n i /µ). By Lemma 7.9, for α = β, we have |X I (u α ) ∩ X I (u β )| ≤ µ. Hence, for any 1 ≤ α ≤ ℓ, we have
X I (u α ) \ β =α X I (u β ) n I − n i 2µ µ ≥ n I 2 . But c(u α , v) = i, so v ∈ K I (u α ), and so X I (u α ) \ {v} ⊆ K I (u α ) ⊆ {v} ⊆ N (v). Then |N (v)| ≥ ℓ α=1 X I (u α , v) \ {v} n I ℓ 2 = Ω( n 2 i µ ) = ω(ρ)
by Proposition 7.8. But this contradicts the definition of ρ. We conclude that λ i < n i − 1.
Lemma 7.15. Let X be a PCC. Suppose for some nondominant color i we have
λ i < n i − 1. Then λ i ≤ (1/2)(n i + µ).
Proof. Fix a vertex u, and suppose λ i < n i − 1. Then there exist vertices v, w ∈ Proof. Fix a nondominant color i and a vertex u, and let m i be the number of cliques C ∈ C such that u ∈ C and X i (u) ∩ C = ∅. So n i /m i ∼ λ i . But by Corollary 7.16, we have λ i n i /2, so m i ≥ 2.
X i (u) such that c(v, w) is dominant. Then |N (v) ∩ N (w)| = µ. Therefore, 2λ i − µ ≤ |(N (u) ∪ N (v)) ∩ X i (u)| ≤ n i .
Clique geometries in exceptional PCCs
In this section we will classify PCCs X having a clique geometry C and a vertex belonging to at most two cliques of C. In particular, we prove Theorem 2.5. We will assume the hypotheses of Theorem 2.5. So, X will be a PCC such that ρ = o(n 2/3 ), with an asymptotically uniform clique geometry C and a vertex u ∈ V belonging to at most two cliques of C.
Proof.
Recall that by the definition of a clique geometry, for every vertex x ∈ V , every nondominant color i, and every clique C in the geometry containing x, we have |C∩X i (x)| λ i . Thus, by Corollary 7.16, every vertex belongs to at least two cliques. In particular, u belongs to exactly two cliques of C, and (by Corollary 7. 16) it follows that λ i ∼ n i /2 for every nondominant color i. Hence, by the definition of a clique geometry, for every vertex x and every nondominant color i, there are exactly two cliques C ∈ C such that x ∈ C and X i (x) ∩ C = ∅.
Let i and j be nondominant colors, and let v ∈ X j (u), and let C ∈ C be the clique containing u and v. Since
|X i (u) ∩ C| ∼ λ i ∼ n i /2, we have |N (v) ∩ X i (u)| n i /2.(7)
Now suppose for contradiction that some x ∈ V belongs to at least three cliques of C. Then there is some C ∈ C and nondominant color i such that x ∈ C but X i (x) ∩ C = ∅. Let j be a nondominant color such that X j (x) ∩ C = ∅, and let y ∈ X j (x)∩C. By the coherence of X and Eq. (7), we have |N (y)∩X i (x)| n i /2.
But since there are exactly two cliques C ′ ∈ C such that x ∈ C ′ and X i (x) ∩ C ′ = ∅, then one of these cliques C ′ is such that |N (y) ∩ X i (x) ∩ C ′ | n i /4. By Proposition 7.8, n i /4 = ω(µ) for n sufficiently large. But then C ′ ⊆ N (y), and y / ∈ C ′ , contradicting the maximality of C ′ . So every vertex x ∈ V belongs to exactly two cliques of C, and for each clique C ∈ C containing x and each nondominant color i, we have |X i (x) ∩ C| ∼ n i /2. It follows that |C| ∼ ρ/2 for each C ∈ C. Proof. Counting the number of vertex-clique incidences in G(X), we have 2n ∼ |C|(ρ/2 + 1) ∼ |C|ρ/2 by Lemma 8.1. On the other hand, every pair of distinct cliques C, C ′ ∈ C intersects in at most one vertex in G(X) by Property 2 of Definition 2.3, and so |C| 2 /2 n. It follows that ρ √ 8n. On the other hand, by Lemma 2.2, we have n i √ n for every i > 1. Since ρ = i>1 n i , for n sufficiently large there are at most two nondominant colors.
v = w in C 1 , we have |N (v) ∩ (C 2 \ {w})| ≤ 1.
Proof. We first note that by Lemma 8.1, there are indeed exactly two cliques containing w. Note that v / ∈ C 2 , since otherwise there are two cliques in C containing both w and v. Suppose v has two distinct neighbors x, y in C 2 \ {w}, so x, y / ∈ C 1 for the same reason. Let C 3 ∈ C \ {C 1 } be the unique clique containing v other than C 1 . We have x, y ∈ C 3 , but then |C 2 ∩ C 3 | ≥ 2, a contradiction. So v has at most one neighbor in C 2 \ {w}.
The following result is folklore, although we could not find an explicit statement in the literature. A short elementary proof can be found inside the proof of [13,Lemma 4.13].
Lemma 8.4. Let G be a connected and co-connected strongly regular graph. If G is the line-graph of a graph, then G is isomorphic to T (m), L 2 (m), or C 5 .
Proof of Theorem 2.5. Let H be the graph with vertex set C, and an edge {C, C ′ } whenever |C ∩ C ′ | = 0. Then G(X) is isomorphic to the line-graph L(H).
By Lemma 8.2, X has rank at most four. By assumption (see Section 3), X has rank at least three.
Consider first the case that X has rank three. The nondiagonal colors i, j of a rank three PCC X satisfy either i * = i and j * = j, in which case X is a strongly regular graph, or i * = j, in which case X is a "strongly regular tournament," and ρ = (n − 1)/2. We have assumed ρ = o(n 2/3 ), so X is a strongly regualr graph. But G(X) is the line-graph of the graph H, so by Lemma 8.4, for n > 5, X is isomorphic to either X(T (m)) or X(L 2 (m)).
Suppose now that X has rank four, and let I = {2, 3} be the nondominant colors. Fix u ∈ V , and let C 1 , C 2 ∈ C be the cliques containing u by Lemma 8.1. By Corollary 7.16 and Lemma 8.3, for any i, j ∈ I, not necessarily distinct, there exist v ∈ C 1 and w ∈ C 2 with c(v, w) = 1, c(v, u) = i, and c(u, w) = j. Therefore, p 1 ij ≥ 1, and so µ i,j>1 p 1 ij ≥ 4. Now let x ∈ V be such that c(u, x) = 1, and let D 1 , D 2 ∈ C be the cliques containing x. For any α, β ∈ {1, 2}, we have |C α ∩ D β | ≤ 1, and so µ ≤ 4. Hence, µ = 4, and |C α ∩ D β | = 1 for every α, β ∈ {1, 2}. Therefore, for any pair of distinct cliques C, C ′ ∈ C we have |C ∩ C ′ | = 1, and so H is isomorphic to K m , where m = |C|.
In particular, every clique C ∈ C has order m − 1, and so n 2 + n 3 = 2(m − 2). Now we prove 2 * = 3 and 3 * = 2. Suppose for contradiction that colors 2 and c(w, z) = 3, which contradicts the fact that p 1 23 = p 1 33 = 1 for c(z, y) = 1. We conclude that 2 * = 3 and 3 * = 2.
Finally, we prove that individualizing O(log n) vertices suffices to completely split the PCCs of situation (b) of Theorem 2.5.
Proof of Lemma 3.5. By Theorem 2.5, we may assume that X is a rank four PCC with a non-symmetric nondominant color i, and G(X) is isomorphic to T (m) for m = n i + 2. (The other nondominant color is i * .) In particular, every clique in C has order n i + 1. We show that there is a set of size O(log n) which completely splits X.
Note that p i ii * = p i ii = p i i * i by Proposition 5.5. For any edge {u, v} in T (m), there are exactly m − 2 = n i vertices w adjacent to both u and v. Hence, considering all the possible of colorings of these edges in X, we have
n i = p i ii + p i ii * + p i i * i + p i i * i * = 3p i ii + p i i * i * .
Therefore, p i ii + p i i * i * ≥ n i /3, and
p i ii * + p i i * i ≤ 2n i /3.(8)
Fix an arbitrary clique C ∈ C and any pair of distinct vertices u, v ∈ C. in C \ {u, v}, at most 2n i /3 of these have c(w, u) = c(w, v), by Eq. (8). So, including u and v themselves, there are at least n i /3 − 1 + 2 = n i /3 + 1 vertices w ∈ C such that c(w, u) = c(w, v). Thus, if we individualize a random vertex w ∈ C, then Pr[c(w, u) = c(w, v)] > 1/3. If this event occurs, then u and v get different colors in the stable refinement. Hence, if we individualize each vertex of C independently at random with probability 6 ln(n 2 i )/n i , then u and v get the same color in the stable refinement with probability ≤ 1/n 4 i . The union bound then gives a positive probability to every pair of vertices getting a different color, so there is a set A of size O(log n i ) such that after individualizing each vertex in A and refining to the stable coloring, every vertex in C has a uniqe color. We repeat this process for another clique C ′ , giving every vertex in C ′ a unique color at the cost of another O(log n i ) individualizations.
On the other hand, every other clique C ′′ ∈ C intersects C ∪ C ′ in two uniquely determined vertices, since G(X) is isomorphic to T (m). So, if u ∈ C ′′ and v / ∈ C ′′ , then u and v get different colors in the stable refinement. Since every vertex lies in two uniquely determined cliques by Lemma 8.1, it follows that every vertex gets a different color in the stable refinement.
Good triples
In this section, we finally prove Lemma 3.4. For given nondominant colors i and j, we will be interested in quadruples of vertices (u, w, x, y) with the following property:
Property Q(i, j): c(x, y) = c(u, x) = c(u, y) = 1, c(u, w) = i, and c(w, x) = c(w, y) = j (See Figure 3) Definition 9.1 (Good triple of vertices). For fixed nondominant colors i, j and vertices u, v, we say a triple of vertices (w, x, y) is good for u and v if (u, w, x, y) has Property Q(i, j), but there is no vertex z such that (v, z, x, y) has Property Q(i, j).
We observe that if (w, x, y) is good for vertices u and v, and both x and y are individualized, then u and v receive different colors after two refinement steps.
In the case of SRGs, there is only one choice of nondominant color, and Property Q(i, j) and Definition 9.1 can be simplified: a triple (w, x, y) is good for u and v if w, x, y, u induces a K 1,3 , but there is no vertez z such that z, x, y, v induces a K 1,3 . Careful counting of induced K 1,3 subgraphs formed a major part of Spielman's proof of Theorem 1.4 in the special case of SRGs [25]. Spielman's ideas inspired parts of this section. In particular, the proof of the following lemma directly generalizes Lemmas 14 and 15 of [25]. Lemma 9.2. Let X be a PCC with ρ = o(n 2/3 ). Suppose that for every distinct u, v ∈ V there are nondominant colors i and j such that there are α = Ω(n i n 2 j ) good triples (w, x, y) of vertices for (u, v). Then there is a set of O(n 1/4 (log n) 1/2 ) vertices that completely splits X.
Proof. Let S be a random set of vertices given by including each vertex in V independently with probability p. Fix distinct u, v ∈ V . We estimate the probability that there is a good triple (w, x, y) for u and v such that x, y ∈ S.
Let T denote the set of good triples (w, x, y) of vertices for (u, v). Observe that any vertex w ∈ X i (u) appears in at most n 2 j good triples (w, x, y) in T . On the other hand, if w ∈ X i (u) is a random vertex, and X is the number of pairs x, y such that (w, x, y) ∈ T , then E[X] ≥ α/n i . Therefore, we have
n 2 j Pr[X ≥ α/(2n i )] + (1 − Pr[X ≥ α/(2n i )])α/(2n i ) ≥ E[X] ≥ α/n i ,
and so, since α = Ω(n i n 2 j ) and α < n i n 2 j by definition,
Pr[X ≥ α/(2n i )] ≥ 1 2n 2 j n i /α − 1 = Ω(1).
Let U be the set of vertices w ∈ X i (u) appearing in at least α/(2n i ) triples (w, x, y) in T , so |U | = Ω(n i ). Now let W ⊆ U be a random set given by including each vertex w ∈ U independently with probability n/(3n i n j ). Fix a vertex w ∈ W and a triple (w, x, y) ∈ T . Note that there are at most p 1 ij n i n j /n vertices w ′ ∈ U such that c(w ′ , x) = j. Therefore, by the union bound, the probability that there is some
w ′ = w with w ′ ∈ X j * (x) ∩ W is ≤ 1/3.
Similarly, the probability that there is some w ′ ∈ X j * (y) ∩ W with w ′ = w is at most 1/3. Hence, the probability that
X j * (x) ∩ W = X j * (y) ∩ W = {w}(9)
is at least 1/3. Now for any w ∈ W , let T w denote the set of pairs x, y ∈ V such that (w, x, y) ∈ T and Eq. (9) holds. We have E[|T w |] ≥ α/(6n i ) = Ω(n 2 j ). But in any case, |T w | ≤ n 2 j . Therefore, for any w ∈ W , we have |T w | = Ω(n 2 j ) with probability Ω(1). Let W ′ ⊆ W be the set of vertices w with |T w | = Ω(n 2 j ). Since E[|W ′ |] = Ω(|W |), we have |W ′ | = Ω(|W |) with probability Ω(1). Furthermore, |W | = Ω(n/n j ) with high probability by the Chernoff bound.
Thus, there exists a set W ⊆ X i (x) with a subset W ′ ⊆ W of size Ω(n/n j ) such that |T w | = Ω(n 2 j ) for every w ∈ W ′ . Now fix a w ∈ W ′ . The probability that there are at least two vertices in X j (w) ∩ S is at least
1 − (1 − p) n j − pn j (1 − p) n j −1 > 1 − e −pn j − pn j e −pn j = Ω(p 2 n 2 j )
if pn j < 1, using the Taylor expansion of the exponential function. Since |T w | = Ω(n 2 j ), the probability that there is a pair (x, y) ∈ T w with x, y ∈ S is Ω(p 2 n 2 j ). Therefore, the probability that there is no w ∈ W ′ with a pair (x, y) ∈ T w such that x, y ∈ S is at most
(1 − Ω(p 2 n 2 j )) |W ′ | ≤ (1 − Ω(p 2 n 2 j )) εn/n j ,
for some constant 0 < ε < 1. For p = β log n/(nn j ) with a sufficiently large constant β, this probability is at most 1/(2n 2 ). Since n j √ n for all j by Lemma 2.2, we may take p = β log n/n 3/2 with a sufficiently large constant β. Then, for any pair u, v ∈ V of distinct vertices, the probability no good triple (w, x, y) for u and v has x, y ∈ S is at most 1/(2n 2 ). By the union bound, the probability that there is some pair u, v ∈ V of distinct vertices such that no triple (w, x, y) has the desired property is at most 1/2. Therefore, after individualizing every vertex in S, every vertex in V gets a unique color with probability at least 1/2. By the Chernoff bound, we may furthermore assume that |S| = O(n 1/4 (log n) 1/2 ).
We will prove that the hypotheses of Lemma 9.2 hold separately for the case that λ k is small for some nondominant color k and the case that X has an asymptotically uniform clique geometry. Specifically, in Section 9.1 we prove the following lemma. Lemma 9.3. There is an absolute constant ε > 0 such that the following holds. Let X be a PCC with ρ = o(n 2/3 ) and a nondominant color k such that λ k < εn 1/2 . Then there are two nondominant colors i and j such that for every pair of distinct vertices u, v ∈ V , there are Ω(n i n 2 j ) good triples of vertices for u and v with respect to the colors i and j.
Then, in Section 9.2, we prove the following lemma. Lemma 9.4. Let X be a PCC with ρ = o(n 2/3 ) and a asymptotically uniform clique geometry C such that every vertex u ∈ V belongs to at least three cliques in C. Suppose that n i µ = o(λ 2 i ) for every nondominant color i. Then there are nondominant colors i and j such that for every pair of distinct vertices u, v ∈ V , there are Ω(n i n 2 j ) good triples of vertices for u and v with respect to the colors i and j.
Lemma 3.4 follows from Lemmas 9.2, 9.3, and Lemma 9.4 (noting for the latter that λ i = Ω(n 1/2 ) implies n i µ = o(λ 2 i ) by Proposition 7.8). Before proving Lemmas 9.3 and 9.4, we prove two smaller lemmas that will be useful for both. Lemma 9.5. Let X be a PCC with ρ = o(n 2/3 ). Let i be a nondominant color, and let u, v ∈ V be distinct vertices. We have |X i (u) \ N (v)| n i /2.
Proof. Let j = c(u, v), and let ε = 2µ/n j , so ε = o(1) by Proposition 7.8. By Corollary 7.16, we have λ i n i /2. Therefore, by Lemma 7.11, we have |X i (u) ∩ N (v)| ≤ max λ i + 1 1 − ε , n i µ εn j n i /2. Lemma 9.6. Let X be a PCC with ρ = o(n 2/3 ). Let i and j be nondominant colors and let u and w be vertices with c(u, w) = i. Suppose that |X j (w) ∩ N (u)| n j /3. Then there are (1/9)n 2 j pairs of vertices (x, y) with c(x, y) = 1 such that (u, w, x, y) has Property Q(i, j).
Proof. By Corollary 7.16, we have λ j n j /2. Thus, for every vertex x ∈ X j (w)\ N (u), there are at least n j − |X j (w) ∩ N (u)| − λ j n j /6 vertices y ∈ X j (w) \ N (u) such that (u, w, x, y) has Property Q(i, j). Since |X j (w) ∩ N (u)| n j /3, the number of pairs (x, y) with c(x, y) = 1 such that (u, w, x, y) has Property Q(i, j) is (2n j /3)(n j /6) = (1/9)n 2 j .
Good triples when some parameter λ k is small
We prove Lemma 9.3 in two parts, via the following two lemmas.
Lemma 9.7. There is an absolute constant ε > 0 such that the following holds. Let X be a PCC with ρ = o(n 2/3 ) and a nondominant color k such that λ k < εn 1/2 and λ k * n k /3. Then there are two nondominant colors i and j such that for every pair of distinct vertices u, v ∈ V , there are Ω(n i n 2 j ) good triples of vertices for u and v with respect to the colors i and j. Lemma 9.8. Let τ be an arbitrary fixed positive integer. Let X be a PCC with ρ = o(n 2/3 ) and a nondominant color k such that λ k < n 1/2 /(τ + 1) and λ k * n k /τ . Then for every pair of distinct vertices u, v ∈ V , there are Ω(n 3 k ) good triples of vertices for u and v with respect to the colors i = k * and j = k * .
We observe that Lemma 9.3 follows from these two.
Proof of Lemma 9.3. Let ε ′ be the absolute constant given by Lemma 9.7, and let ε = min{ε ′ , 1/4}. Let X be a PCC with ρ = o(n 2/3 ) and a nondominant color k such that λ k < εn 1/2 . If λ k * n k /3, then Lemma 9.8 gives the desired result. Otherwise, the result follows from Lemma 9.7.
We now turn our attention to proving Lemmas 9.7. Lemma 9.9. Let 0 < ε < 1 be a constant, and X be a PCC with ρ = o(n 2/3 ). Let i and k be nondominant colors, and let w and v be vertices such that c(w, v) is dominant. Suppose n i ≤ n ℓ for all ℓ, and λ k < εn 1/2 . Then there are ε 2 n 2 k triples (z, x, y) of vertices such that x, y ∈ X k * (w), c(x, y) = 1 and (v, z, x, y) has Property Q(i, k * ).
Proof. First we observe that (p 1 k * k ) 2 n i (n 2 k /n) 2 n i ≤ n 2 k (ρ 3 /n 2 ) = o(n 2 k ). For every color ℓ, there are exactly p 1 ℓi * vertices z such that c(v, z) = i and c(w, z) = ℓ. For every such vertex z, there are at most (p ℓ k * k ) 2 pairs x, y ∈ X k * (w) with c(x, y) = 1 such that (v, z, x, y) has Property Q(i, k * ). Thus, by Proposition 5.5, the total number of such triples is pairs x, y ∈ X j (w) with c(x, y) = 1 such that (v, z, x, y) ∈ T , for a total of at most O(n j µ(ρn i )/n) = o(n 2 j ) triples (z, x, y) ∈ T with c(z, w) = 1.
r−1 ℓ=1 p 1 ℓi * (p ℓ k * k ) 2 p 1 1i * (p 1 k * k ) 2 +
Proof of Lemma 9.8. For every nondominant color ℓ, by Proposition 7.8, we have n k µ/n ℓ = o(n k ) = o(λ k * ). Similarly, n k µ = o(λ 2 k * ). Hence, by Lemma 7.13 and Definition 7.6, there is a set I of nondominant colors with k * ∈ I such that X has strong I-local clique partitions. Since λ k < n 1/2 /(τ + 1) n k /(τ + 1) by Lemma 2.2, and since λ k * n k /τ , the by the definition of an I-local clique partition, k / ∈ I. Hence, by the definition of a strong I-local clique partition and Observation 7.4, for a vertex x and a vertex y ∈ X k (x), |N (y) ∩ X k * (x)| ≤ τ µ = o(n k ).
On the other hand, by Corollary 7.16, we have λ k * n k /2, and hence λ k ≤ n 1/2 /3 n k /3 by Lemma 2.2.
Fix u, v ∈ V . By Lemma 9.5, we have |X k * (u) \ N (v)| n k * /2. Let w ∈ X k * (u) \ N (v). We have c(w, u) = k and so |X k * (w) ∩ N (u)| = o(n k ). By Lemma 9.6, there are Ω(n 2 k ) pairs of non-adjacent vertices (x, y) such that (u, w, x, y) has Property Q(k * , k * ). But by Lemma 9.10, there are o(n 2 k ) triples (x, y, z) of vertices such that x, y ∈ X k * (w), c(x, y) = 1 and (v, z, x, y) has Property Q(k * , k * ). So there are Ω(n 2 k ) pairs (x, y) of vertices such that (w, x, y) is good for u and v with respect to colors i = k * and j = k * . Since we have Ω(n k ) choices for vertex w, there are in total Ω(n 3 k ) good triples, as desired.
Good triples with clique geometries
We now prove Lemma 9.4. Lemma 9.11. Let X be a PCC with ρ = o(n 2/3 ) and an asymptotically uniform clique geometry C such that every vertex u ∈ V belongs to at least three cliques in C. Suppose that n i µ = o(λ 2 i ) for every nondominant color i. Then, for any nondominant color i, there is a nondominant color j such that for every u, w with w ∈ X i (u), we have |X j (w) ∩ N (u)| n j /3.
Proof.
Let C ∈ C be the unique clique such that u, w ∈ C.
If λ i * n i /3, we let j = i * . Then by the maximality of the cliques in C partitioning X j (w), we have |(X j (w) ∩ N (u)) \ C| ≤ µn j /λ j = o(λ j ) by Observation 7.4. So |X j (w) ∩ N (u)| n j /3.
Otherwise, by Corollary 7.17, λ i * ∼ n i /2, and so there is at most one clique C ′ ∈ C with C ′ = C such that |X i * (w) ∩ C ′ | = 0. Therefore, there is a clique C ′′ such that |X i * (w) ∩ C ′′ | = 0. Let j be a nondominant color such that |X j (w) ∩ Lemma 9.12, for all but o(n 2 j ) of these pairs (x, y), the triple (w, x, y) is good for u and v. Since there are n i /2 such vertices w, we have a total of Ω(n i n 2 j ) triples (w, x, y) that are good for u and v.
Conclusion
We have proved that except for the readily identified exceptions of complete, triangular, and lattice graphs, a PCC is completely split after individualizing O(n 1/3 ) vertices and applying naive color refinement. Hence, with only those three classes of exceptions, PCCs have at most exp( O(n 1/3 )) automorphisms. As a corollary, we have given a CFSG-free classifcation of the primitive permutation groups of sufficiently large degree n and order not less than exp( O(n 1/3 )).
As we remarked in the introduction, Theorem 1.4 is tight up to polylogarithmic factors in the exponent, as evidenced by the Johnson and Hamming schemes. However, further progress may be possible for Babai's conjectured classification of PCCs with large automorphism groups, Conjecture 1.2.
The PCCs with large automorphism groups appearing in Conjecture 1.2 are all in fact association schemes, i.e., they satisfy i * = i for every color i. Intuitively, the presence of asymmetric colors (oriented constituent graphs) should reduce the number of automorphisms. On the other hand, the possibility of asymmetric colors greatly complicates our analysis. For example, situation (2) of Theorem 2.5 and Lemma 3.5 could be eliminated, and the proof of Lemma 2.2 would become straightforward, for association schemes. Hence, a reduction to the case of association schemes would be desirable.
Question 1.
Is it the case that every sufficiently large PCC with at least exp(n ε ) automorphisms is an association scheme?
The best that is known in this direction is the result of the present paper: if X is a PCC that is not an association scheme, then | Aut(X)| ≤ exp( O(n 1/3 )).
We comment on the bottlenecks for the current analysis. Below the threshold ρ = o(n 1/3 ), we in fact have the improved bound | Aut(X)| ≤ exp( O(n 1/4 )) when X is nonexceptional, by Lemma 3.4. This region of the parameters is therefore not a bottleneck for improving the current analysis. On the other hand, Conjecture 1.2 suggests that nonexceptional PCCs X with ρ = o(n 1/3 ) should satisfy | Aut(X)| ≤ exp(O(n o(1) )).
When ρ = Θ(n 2/3 ), the Johnson scheme J(m, 3) and H(3, m) emerge as additional exceptions, with automorphism groups of order exp(Θ(n 1/3 log n)). The bottleneck for the current analysis is above this threshold. In this region of the parameters, we analyze the distinguishing number D(i) of the edge-colors i. When
We remark that the bound of Theorem 1.4 is tight, up to polylogarithmic factors in the exponent. Indeed, the Johnson scheme J(m, 3) and the Hamming scheme H(3, m) both have exp(Θ(n 1/3 log n)) automorphisms. (The Johnson scheme J(m, 3) has vertices the 3-subsets of a domain of size m and c(A, B) = |A \ B|, and the Hamming scheme H(3, m) has vertices the words of length 3 from an alphabet of size m with color c given by the Hamming distance.)
Lemma 3. 5 .
5Let X be a PCC satisfyingTheorem 2.5 (b). Then some set of size O(log n) completely splits X.
Proposition 4. 1 .
1Let G = (A, B, E) be a bipartite graph, and let A 1 ∪· · ·∪A m be a partition of A such that the subgraph induced on (A i , B) is biregular of positive valency for each 1 ≤ i ≤ m. Then for any A ′ ⊆ A, we have
Lemma 5. 1 (
1Babai[1, Lemma 5.4]). Let X be a PCC and let ζ = min{D(i) : 1 ≤ i ≤ r − 1}. Then there is a set of size O(n log n/ζ) which completely splits X.
Corollary 5. 4 .
4Let X be a PCC. There exists a nondiagonal color i with D(i) > ρ.
Lemma 3.3, at most O((log n + n/ρ)D(i)/n i ) = O((log n)D(i)/n i ) of these satisfy (a). By Eq. (2), at most 2D(i)/n i satisfy (b). Hence, ⌊ρ/(6D(i))⌋ = O((log n)D(i)/n i ), giving the desired inequality.
Lemma 6. 3 .
3Let X be a PCC with a nondominant color i, let δ = dist 1 (i), and suppose δ ≥ 3. Fix a vertex u ∈ V and let B be the bipartite graph (X (δ−1) i (u), X 1 (u), i). Let γ denote the minimum degree in B of a vertex in X 1 (u), and let β denote the maximum degree in B of a vertex in X (δ−1) i (u). Then β ≤ γρ 2 /n 1 .
Proposition 7. 8 .
8Suppose ρ = o(n 2/3 ). Then µ = o(n 1/3 ) and µρ = o(n). Furthermore, for every nondominant color i, we have µ = o(n i ). Proof. By Lemma 5.15, µ ≤ ρ 2 /n 1 = o(n 1/3 ), and then µρ = o(n). The last inequality follows by Lemma 2.2.
Corollary 7. 16 .
16Suppose X is a PCC with ρ = o(n 2/3 ). Then for every nondominant color i, we have λ i n i /2. Proof. For every nondominant color i we have λ i < n i − 1 by Lemma 7.14. Then by Lemma 7.15 and Proposition 7.8, we have λ i ≤ (1/2)(n i + µ) ∼ n i /2.Corollary 7.17. Let X be a PCC with ρ = o(n 2/3 ) with an asymptotically uniform clique geometry C. Then for every nondominant color i there is an integer m i ≥ 2 such that λ i ∼ n i /m i .
Lemma 8. 1 .
1Under the hypotheses of Theorem 2.5, for n sufficiently large, every vertex x ∈ V belongs to exactly two cliques of C, each of order ∼ ρ/2.
Lemma 8. 2 .
2Under the hypotheses of Theorem 2.5, for n sufficiently large, X has rank at most four.
Lemma 8. 3 .
3Under the hypotheses of Theorem 2.5, let w be a vertex, and C 1 , C 2 ∈ C be the two cliques containing w. Then for any
Figure 2 :
23 are symmetric. Fix two vertices u and v with c(u, v) = 1. (See Figure 2.) ThenN (u) ∩ N (v) = {w,x, y, z} for some vertices w, x, y, z ∈ V , and there are four distinct cliques C 1 , C 2 , C 3 , C 4 ∈ C such that every vertex in A = {u, v, w, x, y, z} lies in the intersection of two of these cliques. Without loss of generality, assume c(w, x) and c(y, z) are dominant, and all other distinct pairs in A except (u, v) have nondominant color. Since for any i, j ∈ I we have p 1 ij = 1, then without loss of generality, by considering the paths of length two from u to v in G(X), we have c(u, w) = c(u, x) = 2, c(u, y) = c(u, z) = 3, c(v, w) = c(v, y) = 2, and c(v, x) = c(v, z) = 3. Now c(w, u) = c(w, v) = 2, and so c(w, y) = c(w, z) = 3 since p 1 ij = 1 for all i, j ∈ I and c(w, x) = 1. But now c(u, z) = c(v, z) Two non-adjacent vertices u, v and their common neighbors w, x, y, z. The dashed line represents color 1. The red line represents color 2. The blue line represent color 3.
Figure 3 :
3(By possibly exchanging u and v, we have c(u, v) = i.) Of the n i − 1 (u, w, x, y) has Property Q(i, j). The dotted line represents the dominant color.
AcknowledgementsThe authors are grateful to László Babai for sparking our interest in the problem addressed in this paper, providing insight into primitive coherent configurations and primitive groups, uncovering a faulty application of previous results in an early version of the paper, and giving invaluable assistance in framing the results.We finally complete the proof of Lemma 9.7.Proof ofLemma 9.7. Let ε = 1/18. Let i be a nondominant color minimizing n i . By Lemma 9.5, we have |X i (u) \ N (v)| n i /2 for any pair of distinct vertices u, v ∈ V .Let color j = k * . Fix two distinct vertices u and v. Let w ∈ X i (u) \ N (v). Since λ j n j /3, we have |X j (w) ∩ N (u)| n j /3 by Lemma 7.11 (with ε = 9µ/n i = o(1)). By Lemma 9.6, there are (1/9)n 2 j pairs of vertices (x, y) with c(x, y) = 1 such that (u, w, x, y) has Property Q(i, j). Furthermore, by Lemma 9.9, for all but (1/18)n 2 j of these pairs (x, y), the triple (w, x, y) is good for u and v. Since there are n i /2 such vertices w, we have a total of (1/36)n i n 2 j triples (w, x, y) that are good for u and v.Now we prove Lemma 9.8. Lemma 9.10. Let X be a PCC with ρ = o(n 2/3 ) and strong I-local clique partitions for some set I of nondominant colors. Let j ∈ I be a color such that λ j = Ω(n j ). Let w and v be vertices such that c(w, v) = 1. Then for any nondominant color i with n i ≤ n j , there are o(n 2 j ) triples (z, x, y) of vertices such that x, y ∈ X j (w), c(x, y) = 1 and (v, z, x, y) has Property Q(i, j).Proof. Fix a nondominant color i, and let T be the set of triples (z, x, y) such that x, y ∈ X j (w), c(x, y) = 1 and (v, z, x, y) has Property Q(i, j).If c(z, w) = 1, then |X j (z) ∩ X j (w)| = p 1 jj * , and so there are at most (p 1 jj * ) 2 pairs x, y ∈ X j (w) with c(x, y) = 1 such that (z, x, y) ∈ T . Then since c(v, z) = i whenever (z, x, y) ∈ T , the total number of triples (z, x, y) ∈ T such that C(z, w) = 1 is at mostwhere the first inequality follows from Proposition 5.5 (iii) and (iv), and the relationLet C denote the collection of cliques partitioning X I (w). If some clique in C contains z, let C be that clique; otherwise, let C = ∅. Since C partitions X j (w) into ∼ n j /λ j = O(1) cliques for each w ∈ V , and |X j (z) ∩ C ′ | ≤ µ for every clique C ′ ∈ C with C = C ′ , we therefore have |(X j (w) ∩ X j (z)) \ C| µn j /λ j = O(µ). But then there are at mostAgain, by the maximality of the cliques in C partitioning X j (w) and Observation 7.4, we haveas desired. Furthermore, this inequality does not depend on the choice of w by the coherence of X. Lemma 9.12. Let X be a PCC with ρ = o(n 2/3 ) and an asymptotically uniform clique geometry C, let w and v be vertices such that c(w, v) is dominant, and let j be a nondominant color such that µ = o(min{λ j , λ 4 j /n 2 j }). Then for any nondominant color i, there are o(n 2 j ) triples (z, x, y) of vertices such that x, y ∈ X j (w), c(x, y) = 1 and (v, z, x, y) has Property Q(i, j).Proof. Fix a nondominant color i, and let T be the set of triples (z, x, y) such that x, y ∈ X j (w), c(x, y) = 1 and (v, z, x, y) has Property Q(i, j).If c(z, w) = 1, then |X j (z) ∩ X j (w)| = p 1 jj * , and so there are at most (p 1 jj * ) 2 pairs x, y ∈ X j (w) such that (z, x, y) ∈ T , for a total of at mosttriples (z, x, y) ∈ T with c(z, w) = 1.Since c(w, v) = 1, there are ≤ µ vertices z ∈ X i (v) ∩ N (w). Suppose z ∈ X i (v) ∩ N (w). Let C be the clique in C containing both z and w. Note that |C ∩ X j (w)| λ j . For any C w , C z ∈ C with w ∈ C w and z ∈ C z such that C w = C and C z = C, we have |C w ∩ C z | ≤ 1. Since C partitions X j (u) into ∼ n j /λ j cliques for each u ∈ V , we therefore have |(X j (w) ∩ X j (z)) \ C| (n j /λ j ) 2 . But then there are at most |X j (w) ∩ X j (z)| · |(X j (w) ∩ X j (z)) \ C| (λ j + (n j /λ j ) 2 )(n j /λ j ) 2 n 2 j /λ j + (n j /λ j ) 4 = o(n 2 j /µ) pairs x, y ∈ X j (w) with c(x, y) = 1 such that (v, z, x, y) ∈ T , for a total of at most o(n 2 j ) triples (z, x, y) ∈ T with c(z, w) = 1.Proof ofLemma 9.4. Let u and v be two distinct vertices. By Lemma 9.5, there is a nondominant color i such that |X i (u) \ N (v)| n i /2. By Lemma 9.11, there is a nondominant color j such that for every w ∈ X i (u), we have |X j (w) ∩ N (u)| n j /3. Let w ∈ X i (u) \ N (v). By Lemma 9.6, there are Ω(n 2 j ) pairs of vertices (x, y) with c(x, y) = 1 such that (u, w, x, y) has Property Q(i, j). Furthermore, since µ is a positive integer and µ = o(λ 2 j /n j ), we have µ = o(min{λ j , λ 4 j /n 2 j }). By Again, the best known bound is | Aut(X)| ≤ exp( O(n 1/3 )), from the present paper, although the conjecture has been confirmed for PCCs of bounded rank[1].
Then there is a set of O((n/ρ) log n) vertices which completely splits X under naive refinement. Conjecture 10.1. Let X be a PCC. particular, Aut(X) ≤ expConjecture 10.1. Let X be a PCC. Then there is a set of O((n/ρ) log n) ver- tices which completely splits X under naive refinement. In particular, Aut(X) ≤ exp((n/ρ) log 2 n).
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| []
|
[
"DUST AND THE TYPE II-PLATEAU SUPERNOVA 2004DJ",
"DUST AND THE TYPE II-PLATEAU SUPERNOVA 2004DJ"
]
| [
"W P S Meikle ",
"R Kotak ",
"D Farrah ",
"S Mattila ",
"S D Van Dyk ",
"A C Andersen ",
"R Fesen ",
"A V Filippenko ",
"R J Foley ",
"C Fransson ",
"C L Gerardy ",
"P A Höflich ",
"P Lundqvist ",
"M Pozzo ",
"J Sollerman ",
"J C Wheeler "
]
| []
| []
| We present mid-infrared (MIR) spectroscopy of a Type II-plateau supernova, SN 2004dj, obtained with the Spitzer Space Telescope, spanning 106-1393 d after explosion. MIR photometry plus optical/near-IR observations are also reported. An early-time MIR excess is attributed to emission from non-silicate dust formed within a cool dense shell (CDS). Most of the CDS dust condensed between 50 d and 165 d, reaching a mass of 0.3 × 10 −5 M ⊙ . Throughout the observations much of the longer wavelength (> 10 µm) part of the continuum is explained as an IR echo from interstellar dust. The MIR excess strengthened at later times. We show that this was due to thermal emission from warm, non-silicate dust formed in the ejecta. Using optical/near-IR line-profiles and the MIR continua, we show that the dust was distributed as a disk whose radius appeared to be shrinking slowly. The disk radius may correspond to a grain destruction zone caused by a reverse shock which also heated the dust. The dust-disk lay nearly face-on, had high opacities in the optical/near-IR regions, but remained optically thin in the MIR over much of the period studied. Assuming a uniform dust density, the ejecta dust mass by 996 d was (0.5 ± 0.1) × 10 −4 M ⊙ , and exceeded 10 −4 M ⊙ by 1393 d. For a dust density rising toward the center the limit is higher. Nevertheless, this study suggests that the amount of freshly-synthesized dust in the SN 2004dj ejecta is consistent with that found from previous studies, and adds further weight to the claim that such events could not have been major contributors to the cosmic dust budget. | 10.1088/0004-637x/732/2/109 | [
"https://arxiv.org/pdf/1103.2885v2.pdf"
]
| 119,118,294 | 1103.2885 | d0994f6fa76397fda7940419112729efc46a7220 |
DUST AND THE TYPE II-PLATEAU SUPERNOVA 2004DJ
13 Apr 2011
W P S Meikle
R Kotak
D Farrah
S Mattila
S D Van Dyk
A C Andersen
R Fesen
A V Filippenko
R J Foley
C Fransson
C L Gerardy
P A Höflich
P Lundqvist
M Pozzo
J Sollerman
J C Wheeler
DUST AND THE TYPE II-PLATEAU SUPERNOVA 2004DJ
13 Apr 2011To be submitted to Astrophysical Journal To be submitted to Astrophysical JournalPreprint typeset using L A T E X style emulateapj v. 11/10/09Subject headings: circumstellar matter -dust, extinction -supernovae: general -supernovae: individual (SN 2004dj)
We present mid-infrared (MIR) spectroscopy of a Type II-plateau supernova, SN 2004dj, obtained with the Spitzer Space Telescope, spanning 106-1393 d after explosion. MIR photometry plus optical/near-IR observations are also reported. An early-time MIR excess is attributed to emission from non-silicate dust formed within a cool dense shell (CDS). Most of the CDS dust condensed between 50 d and 165 d, reaching a mass of 0.3 × 10 −5 M ⊙ . Throughout the observations much of the longer wavelength (> 10 µm) part of the continuum is explained as an IR echo from interstellar dust. The MIR excess strengthened at later times. We show that this was due to thermal emission from warm, non-silicate dust formed in the ejecta. Using optical/near-IR line-profiles and the MIR continua, we show that the dust was distributed as a disk whose radius appeared to be shrinking slowly. The disk radius may correspond to a grain destruction zone caused by a reverse shock which also heated the dust. The dust-disk lay nearly face-on, had high opacities in the optical/near-IR regions, but remained optically thin in the MIR over much of the period studied. Assuming a uniform dust density, the ejecta dust mass by 996 d was (0.5 ± 0.1) × 10 −4 M ⊙ , and exceeded 10 −4 M ⊙ by 1393 d. For a dust density rising toward the center the limit is higher. Nevertheless, this study suggests that the amount of freshly-synthesized dust in the SN 2004dj ejecta is consistent with that found from previous studies, and adds further weight to the claim that such events could not have been major contributors to the cosmic dust budget.
INTRODUCTION
Massive stars explode via core collapse and ejection of their surrounding layers (e.g. Arnett et al. 1989, and references therein). The extent to which core-collapse supernovae (CCSNe) are, or have been, a major source of dust in the Universe is of great interest. Of particular concern is the evidence of enormous amounts of dust ( 10 8 M ⊙ ) in galaxies at high redshifts (z 5). This comes from a variety of observations such as sub-mm and near-infrared (NIR) studies of the most distant quasars (Bertoldi et al. 2003;Maiolino et al. 2004), obscuration by dust of quasars in damped Ly-α systems (Pei et al. 1991), and measurements of metal abundances in these systems (Pettini et al. 1997). Until recently, the scenario of dust from AGB stars tended to be rejected since it was thought that their progenitors would not yet have evolved off the main sequence. However, Valiante et al. (2009) and Dwek & Cherchneff (2011) have argued that, under certain circumstances, asymptotic giant branch (AGB) stars may make some contribution to the dust budget at high redshifts. Both studies nevertheless cannot rule out a supernova contribution. In this paper, we examine the supernova option through observations of a nearby core-collapse event.
CCSNe arising from short-lived Population III stars might seem to be a viable alternative. It is estimated that each supernova (SN) must produce 0.1-1 M ⊙ of dust to account for the high-redshift observations (Dwek et al. 2007;Meikle et al. 2007). Such masses have been predicted in models of dust formation in CCSNe (Todini & Ferrara 2001;Nozawa et al. 2003), although more recent calculations by Cherchneff & Dwek (2010) revise such estimates downward by a factor of ∼5. Perhaps even more problematic is the fact that actual dust-mass measurements in CCSNe and SN remnants yield values not exceeding, respectively, 10 −3 M ⊙ and 10 −2 M ⊙ , although only a handful of such measurements exist. (For a summary of this topic see, for example, Kotak et al. 2009, §1.) The Spitzer Space Telescope (hereafter, Spitzer; Werner et al. 2004) provided an excellent opportunity for us to test the ubiquity of dust condensation in a larger number of CCSNe.
Newly-condensed dust in CCSNe can be detected by its attenuating effects on optical/NIR light and/or via thermal emission from the grains in the ejecta. Prior to Spitzer, the only evidence of dust condensation in typical CCSNe was in the Type II-plateau (IIP) SN 1988H (Turatto et al. 1993) and SN 1999em (Elmhamdi et al. 2003). However the light curve data used to type SN 1988H was sparse. In the case of SN 1999em, Elmhamdi et al. used optical line suppression to infer a dust mass lower limit of about 10 −4 M ⊙ . With the launch of Spitzer, we were at last provided with a facility for high-sensitivity spectroscopy and imaging of nearby CCSNe over the mid-infrared (MIR) range, covering the likely peak of the dust thermal emission spectrum. This can provide a superior measure of the total flux, temperature, and possibly dust emissivity than can be achieved at shorter wavelengths. Moreover, the longer-wavelength coverage of Spitzer allow us to detect cooler grains and see more deeply into dust clumps than was previously possible for typical nearby CCSNe. In this paper, we present our late-time Spitzer observations of the Type IIP SN 2004dj. We use these observations to study the dust production in this supernova.
The paper is arranged as follows. In §1.1 we summarize and discuss previous observations of SN 2004dj. In §2 we present MIR (Spitzer) photometric and spectroscopic observations of SN 2004dj, extending to more than 3 years after the explosion. This MIR coverage is one of the most extensive ever achieved for a SN IIP. We also present late-time optical and NIR photometry and spectra of SN 2004dj. In §3 we analyze these data. Corrections are derived in §3.1 for the effects of the line-of-sight cluster S96, and in §3.2 the mass of ejected 56 Ni is determined. We compare the data with blackbodies in §3.3, in order to assess the likely number and nature of the contributing sources. In §3.4 the origins of the IR radiation are examined in detail. The work is then summarized in §4.
1.1. SN 2004dj SN 2004dj was discovered in the nearby spiral galaxy NGC 2403 on 2004 July 31 by Nakano et al. (2004) and was classified as a normal Type IIP SN by Patat et al. (2004). It was the nearest such event in over three decades; the host galaxy lies within the M81 group. In Kotak et al. (2005) we adopted a distance to NGC 2403 (and the SN) of 3.13 ± 0.15 Mpc (statistical errors only), this being the Cepheid-derived, zero metallicity value reported by Freedman et al. (2001) using the Udalski et al. (1999) period-luminosity slopes.
We continue to use this distance in the work presented here. Vinkó et al. (2006) have estimated the distance to SN 2004dj using a combination of the Freedman et al. value plus their own expanding photosphere method (EPM) and standard candle estimates. This yields an average distance of 3.47 ± 0.29 Mpc, implying that the SN luminosity could be ∼20% larger than the values used herein.
The progenitor of SN 2004dj was almost certainly a member of the compact star cluster Sandage 96 (S96) (Sandage 1984;Bond et al. 2004;Filippenko et al. 2004;Maíz-Apellániz et al. 2004;Wang et al. 2005;Vinkó et al. 2006). The cluster age is variously estimated to be 14 ± 2 Myr (Maíz-Apellániz et al. 2004), ∼20 Myr (Wang et al. 2005), and 10-16 Myr (Vinkó et al. 2009). The main-sequence mass of the progenitor is estimated at ∼15 M ⊙ (Maíz- Apellániz et al. 2004;Kotak et al. 2005), ∼12 M ⊙ (Wang et al. 2005), and 12-20 M ⊙ (Vinkó et al. 2009). Maíz-Apellániz et al. (2004) and Kotak et al. (2005) favor a red supergiant (RSG) progenitor. Guenther & Klose (2004) used echelle spectroscopy of Na I D absorption lines in NGC 2403 along the line of sight to SN 2004dj to infer a heliocentric velocity of +164.8 ± 0.1 km s −1 . This is somewhat larger than the heliocentric velocity of the nuclear region of NGC 2403 of ∼130 km s −1 given in the SIMBAD and NED databases, but this is not surprising given the likely dispersion of velocities within the host galaxy. Indeed, Vinkó et al. (2006) point out that H I mapping of NGC 2403 (Fraternali et al. 2001) suggests that the true radial velocity of the SN 2004dj barycenter is about +221 km s −1 . We adopt this value here.
There is no firm consensus about the value of the reddening to SN 2004dj. Stellar population fitting for S96 yields total (Galactic + host) E(B − V ) values of 0.17 ± 0.02 mag (Maíz-Apellániz et al. 2004), 0.35 ± 0.05 mag (Wang et al. 2005), and 0.1 ± 0.05 mag (Vinkó et al. 2009).
Direct color comparisons of SN 2004dj with other CCSNe yield E(B − V ) values of ∼0 mag (Zhang et al. 2006), 0.06 mag (Chugai 2006), and 0.07 ± 0.1 mag (Vinkó et al. 2006). Perhaps most significantly, the resolved Na I D observations of Guenther & Klose (2004) yield a host-only E(B − V ) value of just 0.026 ± 0.002 mag. The smaller values of E(B − V ) obtained using direct measurements toward SN 2004dj suggests that the SN actually lies near the front of S96. Based on the extinction maps of Schlegel et al. (1998), Chugai et al. (2005) find a Galactic reddening of E(B − V ) = 0.062 mag while Zhang et al. (2006) report E(B − V ) = 0.04 mag (the same value as obtainable from NED). We therefore adopt a Galactic E(B − V ) = 0.05 ± 0.01 mag. If we add the Guenther & Klose (2004) host value to the Galactic value, we obtain E(B − V ) = 0.076 ± 0.01 mag, or A V = 0.24 ± 0.03 mag for a Cardelli et al. (1989) extinction law with R V = 3.1. Given the range of published values, in the present work we used a total extinction A V = 0.31 mag, the same as that preferred by Vinkó et al. (2009). Adoption of even the largest published value of E(B − V ) would increase our shortest MIR wavelength (3.6 µm) flux by just a few percent.
Estimates of the explosion date of SN 2004dj vary by several weeks. On the basis of an early-time spectrum, Patat et al. (2004) placed the explosion at about 2004 July 14. This is consistent with the date obtained by Beswick et al. (2005), who used the radio L peak versus rise-time relation (Weiler et al. 2002) to yield an explosion date between 2004 July 11 and July 31. On the other hand, based on the EPM method, Vinkó et al. (2006) derive an explosion date as early as 2004 June 30. On the assumptions that the light curve of SN 1999gi was typical of SNe IIP and that the SN 2004dj plateau was of similar length, Chugai et al. (2005) obtained an even earlier explosion date: 2004 June 13. Likewise, assuming similar evolution between SN 2004dj and SN 1999em, Zhang et al. (2006) find an explosion date of 2004 June 11. Nevertheless, partly as a compromise with the later Patat et al. (2004) value, Chugai (2006) subsequently adopted 2004 June 28 as the date of the explosion. Chugai et al. (2007) also used this explosion date. Given the weight of evidence for a later explosion date, we reject that preferred by Zhang et al. (2006). In Kotak et al. (2005) we adopted an explosion date of 2004 July 10, or MJD = 53196.0. We use the same explosion date here but recognize that there is an uncertainty of about ±7 d. All epochs will be with respect to MJD = 53196.0 (i.e., t = 0 d).
Optical light curves of SN 2004dj are presented by Korcáková et al. (2005); Chugai et al. (2005); Leonard et al. (2006); Vinkó et al. (2006); Zhang et al. (2006); Vinkó et al. (2009). The V R light curves fell by 10% and 90% of the total decline from the plateau to the start of the radioactive tail at, respectively, epochs 70 ± 5 d and 96 ± 3 d. At the start of the radioactive tail the SN luminosity was about 15% of the plateau value. During the early nebular phase (up to ∼300 d) the V band declined at about 1.1 mag (100 d) −1 (Vinkó et al. 2006), which is typical for a SN IIP. On the basis of the light curves, the mass of ejected 56 Ni has been estimated at 0.02-0.03 M ⊙ (Chugai et al. 2005;Kotak et al. 2005;Vinkó et al. 2006;Zhang et al. 2006). Notwithstanding, in the present work we argue that these authors have overestimated the 56 Ni mass and that the true mass is more like 0.01 M ⊙ .
At the end of the plateau phase, SN 2004dj exhibited a remarkable and rapid change in some of its prominent optical lines, especially Hα. At 89 d the Hα profile still had a typical P Cygni morphology with a symmetric peak blueshifted by only about −270 km s −1 after correction for the heliocentric velocity of the SN. Yet, by just ∼10 d later the profile had developed a strong asymmetric profile with a peak at −1610 km s −1 (Chugai et al. 2005). As time went by during the first year, this blueshift gradually decreased and the asymmetry became less pronounced. Chugai (2006) interpreted this unusual behavior as being due to the gradual emergence of an asymmetric, bipolar jet whose more massive component is travelling towards the observer. They propose that the lines are driven by the radioactive decay of spherical fragments of 56 Co cocooned in metals and helium, lying within the core. Using spectropolarimetry Leonard et al. (2006) also found evidence for departure from spherical symmetry. The polarization was observed to increase dramatically at the end of the plateau phase, implying the presence of significant asphericity in the inner regions of the ejecta.
Early-time evidence of a significant circumstellar medium (CSM) around the progenitor of SN 2004dj has also been reported. Radio emission was detected by Stockdale et al. (2004) at 23 d, by Beswick et al. (2005) between 26 d and 145 d, and by Chandra & Ray (2004) at 33 d and 43 d. The SN was also detected at X-ray wavelengths at 30 d. The X-ray luminosity was about three times that of the Type IIP SN 1999em and nearly fifteen times that of the Type IIP SN 1999gi at similar epochs. Beswick et al. (2005) point out that both types of emission arise from a shocked CSM. Chugai et al. (2007) have used the presence of a high-velocity absorption component in the Hα line during the late photospheric phase to deduce the presence of a cool dense shell (CDS), with a mass of 3.2 × 10 −4 M ⊙ , produced by interaction of the ejecta with the pre-existing CSM. To this evidence for a CDS we add our observation of an early-time IR echo in SN 2004dj (see §3.4.2).
In Kotak et al. (2005) we presented MIR photometric and spectroscopic observations of SN 2004dj at epochs 97-137 d after explosion. Simultaneous modelling of the fundamental (1-0) and first overtone (2-0) of CO was carried out. The results favor a 15 M ⊙ RSG progenitor and indicate post-explosion CO formation in the range 2000-4000 km s −1 . Kotak et al. (2005) also noted an underlying NIR continuum. A possible origin in CSM dust was suggested, but RSGs in S96 were favored as the more likely cause. Nevertheless, in the present work ( §3.2.2) we find that the bulk of the early-time NIR and MIR continuum is most plausibly explained as an IR echo from CDS dust. The presence of 1.7 × 10 −4 M ⊙ of Ni + in the ejecta was also deduced by Kotak et al. (2005).
In summary, while the early-time optical light curves and spectra of SN 2004dj are typical of a Type IIP event, its early radio, MIR, and X-ray behavior point to an exceptionally strong ejecta/CSM interaction. Moreover its earlier nebular-phase spectra imply an atypically asymmetric core. SN 2004dj is only "typical" in some respects.
OBSERVATIONS
Mid-Infrared Photometry
Imaging at 3.6, 4.5, 5.8, and 8.0 µm was obtained with the Infrared Array Camera (IRAC) (Fazio et al. 2004), at 16 and 22 µm with the Peak-up Array (PUI) of the Infrared Spectrograph (IRS) (Houck et al. 2004), and at 24 µm with the Multiband Imaging Photometer for Spitzer (MIPS) (Rieke et al. 2004). Imaging observations spanned epochs 89.1 to 1393.3 d, plus four observations, at 3.6 µm only, covering 1953.9-2143.5 d. Aperture photometry was performed on the images using the Starlink package GAIA (Draper et al. 2002). A circular aperture of radius ∼ 3. ′′ 7 was used for the photometry. The background flux was measured and subtracted by using a concentric sky annulus having inner and outer radii of 1.5 and 2.2 times the aperture radius, respectively. These parameters were chosen as a compromise between maximizing the sampled fraction of source flux and minimizing the effects of the bright, complex background. The aperture radius corresponds to ∼55 pc at the distance of SN 2004dj. The aperture was centered according to the SN WCS coordinates. Aperture corrections were derived from the IRAC and MIPS point-response function frames available from the Spitzer Science Center, and ranged from ×1.16 at 3.6 µm to ×2.79 at 24 µm. A 2σ clipped mean sky estimator was used, and the statistical error was estimated from the variance within the sky annuli. Fluxing errors due to uncertainties in the aperture corrections are about ±5%.
The MIR photometry is presented in Table 1. The Spitzer programs from which the imaging data were taken are listed in the caption. The pre-explosion MIR flux of S96 has not been measured, so the tabulated values are uncorrected for S96. An estimate of the S96 contribution is given at the bottom of the Table. The effect of emission from S96 is discussed in §3.1. A temporally varying point source at the SN position is clearly visible in all bands. Figure 1 shows a sequence of images at 8.0 µm at 257, 621, and 996 d. The SN is clearly brighter at 621 d (about ×2 relative to 257 d). This is due to the epoch being close to the peak of the thermal emission from the dust (see §3.4.4.2). It can also be seen that the SN lies in a region of relatively bright, complex background emission. The MIR photometry is displayed as light curves and spectral energy distributions (SEDs) in Fig. 2 and Fig. 3, respectively.
In Fig. 2, a rapid initial decline is seen at wavelengths 3.6-8.0 µm. With the exception of 4.5 µm, all of the light curves (with sufficient temporal coverage) exhibit secondary maxima, with the peak emission occurring at epochs of ∼450 d at 3.6 µm to ∼850 d at 24 µm. As will be discussed later, these second maxima constitute strong evidence of dust formation. The absence of a delayed peak at 4.5 µm is due to the earlier appearance and dominance of CO fundamental emission in this band. After ∼1000 d, the 24 µm light curve starts to climb again. The evolution of the 24 µm flux is complex, as it is a combination of the detailed behaviors of emission from the ejecta dust and from an interstellar (IS) IR echo (see §3.4.3.1). In Fig. 3 The jittered on-source exposures were median-combined to form sky frames. In each band the sky-subtracted frames were then aligned and median-combined.
Aperture photometry was performed on the reduced images using the Starlink package GAIA (Draper et al. 2002) with the same aperture and sky annuli as for the MIR photometry. The aperture was centered by centroiding on the sources. The magnitudes at J, H, and K s were obtained by comparison with four field stars lying within ∼ 100 ′′ of SN 2004dj. The field-star magnitudes were acquired by measurement of 2MASS images (Skrutskie et al. 1997). For the single Z-band measurement, the magnitude was obtained by comparison with the four field stars with their JHK s SEDs extrapolated to the Z band. The resulting SN photometric measurements are listed in Table 2 and plotted in Fig. 4. Errors shown include uncertainties in the magnitudes of the four 2MASS comparison field stars. Pre-explosion JHK s fluxes of S96 were measured from the 2MASS survey (see Table 2) and used to correct the JHK s light curves. Also shown for comparison is the 3.6 µm light curve from the present work. In H and K s the slopes flatten after ∼300 d accompanied by a rise at 3.6 µm. This is suggestive of radiation from warm, newly-forming dust.
Optical photometry was taken from Vinkó et al. (2006,2009). The optical light curves are displayed in Fig. 4 and have been corrected for emission from the S96 cluster using the BV RI magnitudes given by Vinkó et al. (2006). In addition, the V and R points of Zhang et al. (2006) around the end of the plateau were added to enhance the detail of this phase. Also shown for comparison (labelled "Rad") is the temporal evolution of the radioactive energy deposition for SN 1987A as specified by Li et al. (1993) (0 to 1200 days) and Timmes et al. (1996) (500 to 3500 days) with the addition of the early-time contribution of 56 Ni decay assuming complete absorption. The radioactive isotopes include 56 Ni, 56 Co, 57 Co, 60 Co, 22 Na, and 44 Ti. (In subsequent use of the Li et al. and Timmes et al. deposition specifications, our addition of the early-time contribution of 56 Ni decay is assumed.) In the optical light curves, for about 35 d (115-150 d) after the end of the fall from the plateau the decline rate matches the radioactive deposition quite closely, indicating that this was powering the emission during this phase. The optical decline rates then flatten during ∼150-250 d, indicating the emergence of an additional power source (see below). After about 250 d, the optical light curves exhibit a steepening (possibly also present in the J band) which increases up to the final observations.
Mid-Infrared Spectroscopy
1 IRAF is written and supported by the IRAF programming group at the National Optical Astronomy Observatories (NOAO) in Tucson, Arizona, which are operated by the Association of Universities for Research in Astronomy (AURA), Inc., under cooperative agreement with the National Science Foundation (NSF).
Low-resolution (R ≈ 60-127) MIR spectroscopy between 5.2 and 14.5 µm was acquired at nine epochs between 106.3 d and 1393.3 d with the IRS in lowresolution mode. Long-Low (LL; 14-38 µm) observations were also attempted. Unfortunately, the LL observations were unusable. The LL slit lies at 90 • to the SL slit. This meant that, given the scheduling constraints, the LL slit always lay across the host galaxy, resulting in heavy contamination. The MIR spectroscopic observations were drawn from the MISC programs plus one epoch at 1207 d from the SEEDS program. The log of spectroscopic observations is given in Table 3.
The data were processed through the Spitzer Science Center's pipeline software, which performs standard tasks such as ramp fitting and dark-current subtraction, and produces Basic Calibrated Data (BCD) frames. Starting with these data, we produced reduced spectra using both the SPICE and SMART v6.4 software packages. We first cleaned individual frames of rogue and otherwise "bad" pixels using the IRSCLEAN task. The first and last five pixels, corresponding to regions of reduced sensitivity on the detector, were then removed. The individual frames at each nod position were median-combined with equal weighting on each resolution element. Sky background was removed from each combined frame by subtracting the combined frame for the same order taken with the other nod position. We also experimented with background removal by subtracting the adjacent order. In general, nod-nod subtraction was preferred, as the background sampled in this way is expected to most closely represent the background underlying the SN. Any residual background was removed by fitting low-order polynomials to regions immediately adjacent to the SN position.
One-dimensional spectra were then extracted using the optimal extraction tool within the SPICE software package, with default parameters. We found that in all cases the source was point-like, with a full width at half-maximum intensity (FWHM) that was never wider than the point-spread function (PSF). This procedure results in separate spectra for each nod and for each order. The spectra for each nod were inspected; features present in only one nod were treated as artifacts and removed. The two nod positions were subsequently combined. The nod-combined spectra were then merged to give the final spectrum for each epoch. Overall, we obtained excellent continuum matches between different orders.
Despite our careful reduction procedure, the fluxes of the IRS spectra and the IRAC photometry were not completely consistent. This was due to (a) differences in the fixed sizes of the spectrograph aperture slits and the circular apertures used for the image photometry, and (b) the fact that the spectra were generally taken some days before or after the imaging data, during which time the SN flux changed. We therefore recalibrated the IRS spectra against contemporaneous photometry in the 8 µm band obtained by interpolation of the light curve. This band was chosen since it was completely spanned by the short-low (SL) spectrum. For each epoch, the IRAC 8 µm transmission function was multiplied by the MIR spectra and by a model spectrum of Vega 2 . The resulting MIR spectra for the SN and for Vega were integrated over wavelength. The total SN spectral flux in the 8 µm band was then obtained from the ratio of the two measurements using a zero (Vega) magnitude of 64.1 Jy (IRAC Data Handbook, Table 5.1). These were then compared with the 8 µm photometry to derive scaling factors by which the spectra were multiplied. The spectra are plotted in Fig. 5, together with contemporary photometric data.
The MIR spectra comprise both continua and emission features. Up to at least 281 d, strong emission from the CO fundamental was present in the IRS spectra and IRAC photometry. This had disappeared by 500 d. Fig. 5). Apart from the CO region during 106-281 d, the 5-14 µm region was dominated by continuum emission. Moreover, simple extrapolation below and above the spectral coverage to the limits of the photometric coverage suggests that the continuum dominated over at least 3.6-24 µm.
Optical and Near-Infrared Spectroscopy
We acquired optical spectra using ISIS on the WHT, La Palma, and DEIMOS (Faber et al. 2003) on the 10 m Keck 2 telescope, Hawaii. The 895 d Keck spectrum has already been presented by Vinkó et al. (2009). We also made use of earlier post-plateau optical spectra obtained by Vinkó et al. (2006) at 89 d and 128 d, Leonard et al. (2006) at 95 d, and Chugai et al. (2005) at 100 d. NIR spectroscopy of SN 2004dj was obtained using LIRIS on the WHT and with the OSU-MDM IR Imager/Spectrograph on the 2.4 m Hiltner Telescope of the MDM Observatory, Arizona. The data were reduced using standard procedures in Figaro (Shortridge 2002) and IRAF. The observing log for the optical and NIR spectra is given in Li et al. (1993) and Timmes et al. (1996) for SN 1987A, but scaled down to an initial 0.0095 M ⊙ of 56 Ni (see §3.2). The evolution of the luminosities is plotted in Fig. 8. This indicates that from just after the plateau phase to ∼460 d, the Hα and Paβ luminosities declined at a rate roughly comparable to that of the radioactive deposition. Table 6 lists profile parameters expressed as velocities for the more isolated lines over a range of epochs, shifted to the center-of-mass rest frame of SN 2004dj. Also listed (Col. 7) for 461-925 d are the maximum blue-wing velocities derived from profile model matches (see §3.4.4.1). Preliminary inspection indicated that for lines within a given element (e.g., Hα and Paβ), the velocities exhibited similar values and evolution. Therefore, in order to improve the temporal coverage and sampling, Hα and Paβ were grouped together, as were [Fe II] 7155Å and [Fe II] 12570Å. The evolution of the line velocities (shifted to the center-of-mass rest frame of the SN) is plotted in Fig. 9, and in more detail in Fig. 10. These plots reveal a complex velocity evolution.
Up to 138 d, of the three elements considered, only hydrogen lines could be reliably identified. At 89 d (corresponding to about half-way down the plateau-edge), the half width at half-maximum (HWHM) velocity was 1820 ± 60 km s −1 , although the red and blue wings extended to much higher values. In addition, the peak emission exhibited a blueshift of −450 ± 50 km s −1 . Then, as already described in §1, a strong asymmetry rapidly developed, this being attributed to the emergence of an asymmetric, bipolar core (Chugai 2006). Rather than being entirely due to the bulk motion of the ejecta however, some of the width of the Hα and Paβ lines may also have been produced by scattering from thermal electrons as in the cases of SN 1998S (Chugai 2001) and SN 2006gy (Smith et al. 2010, but this effect is unlikely to have a significant influence at later epochs. By the time of the next observation at 283 d, the asymmetry had diminished, with the blueshift of the peak now only −730 ± 50 km s −1 . By this time the lines of [O I] and [Fe II] had emerged, also with asymmetric blueshifted profiles. As the SN continued to evolve, the line widths narrowed and by 461 d we see the first signs of a sharp suppression of the red wing. By the time of the next season's observations (895 d, 925 d), this suppression is very pronounced in all three species. This phenomenon suggests dust formation resulting in the obscuration of the far side of the ejecta and will be examined in more detail in §3.4.4.1. (Table 5) as well as being much wider (Table 6). Therefore an underlying, weaker, redshifted component could have been present at 467 d or earlier, but was swamped by the main component of the line. The 895 d spectrum has a blueshift in the main peak in Hα of −140±10 km s −1 with the minor peak showing a redshift of about +160 ± 10 km s −1 . In [O I] 6300Å the corresponding velocities are, respectively, −210 ± 10 km s −1 and +170 ± 10 km s −1 . Thus, the main (blueshifted) peak and the weaker (redshifted) peak lie roughly symmetrically about the local zero velocity. This suggests that a minor fraction of the line flux originates in an emission zone centered on the SN and having the geometry of an expanding ring, jet, or cone. The [Fe II] 7155Å line also shows a secondary peak but at a much larger redshift of +480 ± 10 km s −1 . There may actually be a peak also at around +170 km s −1 , similar to those seen in Hα and [O I] 6300Å, but which is swamped by the stronger peak at the larger redshift. The profiles are analyzed in §3.4.4.1.
We do not consider the MIR line-profile kinematics due to the much lower resolution (R FWHM ≈ 100). The formation and evolution of the MIR lines will be analyzed in a future paper.
ANALYSIS
The evolution of the MIR spectral continuum indicates IR emission from dust playing a major role in the postplateau flux distribution of SN 2004dj. We now make use of the observations described above to explore the origin, location, distribution, energy source(s), and nature of these grains. This will be done by comparison of a variety of simple models with the observations. 3.1. Correction of the Supernova Flux for S96. The position of SN 2004dj coincides with that of the compact star cluster S96 (Sandage 1984), and it seems likely that the progenitor was a member (Wang et al. 2005). Regardless of whether or not this is the case, it is still important to correct for the contribution of the cluster to the photometry and spectra, especially at the later epochs, before we embark on modelling the observed SED. To do this, in the optical region we made use of the pre-explosion optical photometry compiled in table 4 of Vinkó et al. (2009). NIR photometry of S96 was obtained from 2MASS (see Table 2). Unfortunately there are no pre-explosion MIR images of S96 and so its contribution to the flux had to be assessed indirectly.
We found that longward of ∼0.4 µm the optical/NIR photometric points could be fairly represented by a combination of two blackbodies, reddened according to the Cardelli et al. (1989) law with E(B − V ) = 0.1 mag and R V = 3.1. This is illustrated in Fig. 11. In this representation, the fluxes longward of ∼1 µm are dominated by a component with a temperature of 3500 K. At shorter wavelengths, the hotter component (50,000 K) becomes increasingly important. This hot blackbody is not intended as an explanation for the shortwave radiation, but rather it simply serves as a means of representing and extrapolating the optical SED. The contribution of S96 to the MIR photometric points was then obtained by extrapolation of the cooler blackbody. (We did not make use of the Vinkó et al. (2009) models to correct for S96 as it was unclear how they should be extrapolated into the MIR region.) Some support for the effectiveness of our estimation method, at least for the shorter wavelengths, comes from four serendipitous 3.6 µm images of the SN 2004dj field spanning 1954-2143 d (Table 1), obtained in Spitzer program 61002 (PI W. Freedman). These show that, by this period, the light curve at this wavelength had levelled out at a mean value of 0.32 ± 0.02 mJy which in good agreement with our estimate for S96 of 0.28 ± 0.05 mJy (see Fig. 2).
It is possible that the above procedure could underestimate a contribution from cooler material but it is unlikely that S96 would be the source of such emission. Maíz-Apellániz et al. (2004) estimate a cluster age of 13.6 Myr and point out that by this age its parent molecular cloud should have been dispersed by stellar winds and SN explosions. Wang et al. (2005) find an age of ∼20 Myr while Vinkó et al. (2009) obtain ∼ 10 − 16 Myr. This suggests that the flux contribution from S96 to longer MIR wavelengths would be small. On the other hand a significant SN-driven IR echo from the general IS dust of the host galaxy is quite likely (see §3.4.3.1).
We conclude that the 3500 K blackbody extrapolation provides a reasonable estimate of the MIR flux contributions from S96. The inferred S96 fluxes in the 3.6 − 24 µm range are shown in Table 1. The values are insensitive to the extinction over the ranges of E(B − V ) (0.06 − 0.35 mag) suggested in the literature. It can be seen that the contribution of S96 at 3.6 µm is significant as early as ∼250 d, and dominates by ∼1000 d. As we move to longer wavelengths, the effect of S96 declines, becoming negligible for wavelengths longward of ∼10 µm even at the latest epochs. In the optical-NIR-MIR continuum modelling (see §3.4.4.2), we use the S96 blackbody representation to correct for its contribution to the SED.
Mass of 56 Ni in the Ejecta
It is important to establish the mass of 56 Ni in the ejecta of SN 2004dj since this will allow us to test for the presence of energy sources, other than radioactive decay, which might be responsible for the SN luminosity. In Fig. 12 we show the bolometric light curves (BLCs) of Vinkó et al. (2006) (open circles) and Zhang et al. (2006) (open triangles). The phases of these have been shifted to our adopted explosion date of MJD=53196.0. This date is nearly one month later than that of Zhang et al. reducing their derived 56 Ni mass by about 25%. The explosion date of Vinkó et al. is about 10 d earlier than ours but it is not clear if this would significantly affect the 56 Ni mass they derived. In addition to the phase shifts, the BLCs of these authors have been scaled downward to our adopted distance of 3.13 Mpc. This has the effect of reducing the 56 Ni masses of Vinkó et al. and Zhang et al. by 19% and 10% respectively. We scaled the Vinkó et al. (2006) BLC by a further ×1.1 in order to allow approximately for the higher total extinction (A V = 0.31 mag) adopted by Vinkó et al. (2009) and the present work. We scaled the BLC of Zhang et al. (2006) by a further factor of 0.68 to force agreement with our adjusted version of the Vinkó et al. BLC. The need for this was due to the much larger extinction, A V = 1.02 mag, adopted by Zhang et al., compared with the A V = 0.31 mag adopted in the present work. It was found that these adjustments brought our 89 d, 106 d and 129 d blackbody total luminosities (see §3.3 and Table 7, col. 11.) into fair coincidence with the other two BLCs (see Fig. 12). Vinkó et al. (2006) constructed their BLC by integrating observed fluxes in the BV RI bands and then extrapolating linearly from the B and I fluxes assuming zero flux at 3400Å and 23,000Å. Their BLC extended to 307 d. Zhang et al. (2006) simply integrated the observed fluxes in 12 narrow bands between 4000Å and 10,000Å. Their BLC extended to 154 d, with five additional points to 180 d obtained by interpolation within a reduced number of bands. Thus, neither of these BLCs included unobserved excess flux beyond about 1 µm. However, for 89-129 d the unobserved MIR flux made up no more than 10% of the total luminosity (see §3.4.2). Moreover, the optical/NIR region was dominated by continuum emission at this time. Consequently, the hot+warm continuum luminosities obtained from the present work via blackbody matching (see §3.3 and Table 7 col. 12.) and plotted in Fig. 12 (solid squares) agree well with the two adjusted BLCs. (We exclude the cold component because, as argued in §3.4.3.1, it is due to an IS IR echo which was predominantly powered by the peak luminosity of the SN prior to the earliest epoch of observation.) By 251/281 d our hot+warm continuum luminosities make up only ∼60% of the adjusted Vinkó et al. BLC. This is due to the relatively strong contribution of line emission to the total luminosity during this time. Line emission luminosity was not included in our blackbody matches (see §3.3).
In Fig. 12, the SN BLCs are compared with the radioactive deposition power in SN 1987A, as specified by Li et al. (1993) & Timmes et al. (1996. These radioactive decay light curves are scaled to, respectively, 0.0095 M ⊙ (solid line) and 0.016 M ⊙ (dashed line) of 56 Ni. Also shown (red) is the total radiocative luminosity in the case of 0.0095 M ⊙ . We also show (dotted lines) the actual UV-augmented BLCs of SN 1987A (Pun et al. 1995) derived from observations at ESO and CTIO, scaled to an initial 56 Ni mass of 0.0095 M ⊙ . It can be seen that the 0.0095 M ⊙ case provides a good match to the SN 2004dj BLC during 115−150 d, just after the end of the plateau phase. After 150 d, unlike SN 1987A, the SN 2004dj BLC begins to exceed the luminosity of the 0.0095 M ⊙ case with the discrepancy growing steadily with time. Indeed, even the total radioactive luminosity of the scaled SN 1987A is exceeded by the SN 2004dj light curve, implying that an additional source of energy has appeared.
Our 0.016 M ⊙ deposition plot corresponds approximately to the 0.02 M ⊙ case of Vinkó et al. (2006) in their Fig. 18. We agree that this case provides a fair match to the BLC during ∼ 260 − 310 d. Nevertheless, viewed within the context of the whole BLC, it can be seen that this "match" is actually due to an inflection section during the growth of the BLC excess relative to the true radioactive deposition. Adoption of the 0.016 M ⊙ case would imply an unexplained BLC deficit during ∼ 95 − 250 d. Given the phase of the event and the unexceptional progenitor mass it is difficult to see how such a discrepancy would come about. Consequently, we argue that only during the 115 − 150 d phase was the BLC of SN 2004dj actually dominated by radioactive decay. Beyond this period, and as deduced also in §2.2 and §2.3, an additional luminosity source emerged. We therefore reject the 56 Ni mass deduced by Vinkó et al. (2006).
We also reject larger 56 Ni masses reported by other authors. Kotak et al. (2005) used the V band exponential tail method of Hamuy (2003) to derive a 56 Ni mass of ∼ 0.022 M ⊙ . This was based on the V magnitude at 100 d which the subsequently more complete database shows was not quite yet on the radioactive tail (see Fig. 4), thus leading to an overestimate of the 56 Ni mass. This method was also one of those used by Zhang et al. (2006) who obtained 0.025 ± 0.010 M ⊙ of 56 Ni. As already indicated, their larger value was due mostly to their much earlier explosion epoch and much larger extinction, neither of which we view as likely. Chugai et al. (2005) obtained 0.020 ± 0.006 M ⊙ of 56 Ni based on comparison of the V magnitude at 200 d with that of SN 1987A. The difficulty here is that by this epoch (as also in the Vinkó et al. (2006) case) an additional power source had appeared in SN 2004dj, biasing the derived 56 Ni mass to higher values. In addition, Chugai et al. (2005) used an exceptionally early explosion date, pushing their result even higher. Finally we note that both Chugai et al. (2005) and Zhang et al. (2006) also used the V -light curve "steepness" method of Elmhamdi et al. (2003), which is insensitive to distance and extinction uncertainties. Chugai et al. obtained 0.013 ± 0.004 M ⊙ , consistent with our result. Zhang et al. applied the same method to a number of wavebands, including V , but obtained a larger 0.020 ± 0.002 M ⊙ . Their steepness parameter at just V yields about 0.019 M ⊙ suggesting that their use of multiple bands is not the cause of the apparent disagreement with Chugai et al. However, the difference between the Chugai et al. and Zhang et al. determinations is only at the level of ∼ 1.5σ significance.
We conclude that, taking into account the uncertainties in fluxing, adopted distance, extinction and explosion epoch (see §1.1) the mass of 56 Ni ejected by SN 2004dj was 0.0095 ± 0.002 M ⊙ . We adopt this value for the rest of the paper.
Comparison of Observed Continua with Blackbody
Radiation Here we begin to consider the location and energy source of the SN continuum, especially longward of 2 µm where thermal emission from dust would appear. To take an initially neutral standpoint on the interpretation, we compared optical, NIR and MIR spectra and photometry with blackbody continua. This provides us with the minimum radii of the emitting surfaces. The epochs were selected primarily as those for which MIR spectra were available, although the earliest such epoch, 106 d, was already during the nebular era. The earliest MIR photometry was acquired at 89 d when the SN light curve was only about half way down the fall from the plateau to the nebular level. Given the potential interest of this epoch we began our model comparisons at this epoch despite the lack of an MIR spectrum. In addition, to compensate for the large gap between 859 d and 1207 d we also considered the 996 d SED based on MIR photometry only. Optical photometry was taken from Vinkó et al. Zhang et al. (2006). Details about the sources of the other data are given in Tables 1-4. Apart from 89 d and 996 d, all the optical and NIR data plus the MIR photometry were flux-scaled by interpolation of the light curves to the epochs of contemporaneous MIR spectra.
The contribution of S96, represented by a 3500 K blackbody of radius 1.5 × 10 14 cm (see §3.1), was first subtracted from all the data. To model the resulting 0.4 − 24 µm SN continuum it was found to be necessary to use three blackbodies ("hot": 5300 − 10000 K, "warm": 320 − 1750 K and "cold": ∼200 K). These were reddened and then matched visually to the continua. The hot blackbody was first added and adjusted to match the optical continuum. While the hot blackbody provides some information about the energy budget of the shorter wavelength part of the spectrum, the main reason for its inclusion in this study was to allow correction for its effect on the net continuum in the NIR where, up to about 500 d, it is comparable in strength to the warm component. The warm blackbody was then added and adjusted to match the ∼ 5 − 10 µm continuum plus the long wavelength end (2 − 2.4 µm) of the NIR continuum. It was found that, starting at the earliest epoch, as the SN evolved the hot+warm blackbody flux longward of 10 µm increasingly fell below that of the observations. Therefore a cold blackbody was added and adjusted to provide the final match. For epochs where photometry but no spectra were available we used the temporally nearest spectral matches to indicate the likely position of the underlying continuum. The expansion velocities of the blackbody surfaces, v hot , v warm and v cold , and temperatures, T hot , T warm and T cold , for these matches are tabulated in Table 7. The warm blackbody radii, R warm are also listed. The model matches are displayed in Figs. 13 (89 − 500 d) and 14 (652 − 1393 d). In Table 7 we also show the luminosities, L hot , L warm and L cold , of the three blackbody components together with the sum of the hot and warm components (L total ). We exclude the cold component because, as will be argued in §3.4.3.1, it is due to an IS IR echo which was predominantly powered by the peak luminosity of the SN prior to the earliest epoch of observation. In the final column is listed the radioactive deposition power corresponding to the ejection of 0.0095 M ⊙ of 56 Ni, scaled from the SN 1987A case specified by Li et al. (1993) & Timmes et al. (1996.
We stress that the blackbody matches were to the underlying spectral continua where this could be reasonably judged, and not to the photometric points which also contained flux from line emission. Thus, the models sometimes lie below the average level of the spectra. This is particularly so for later epochs at wavelengths shortward of 2 µm where the spectra are dominated by broad, blended emission lines. This tends to mask most of the underlying thermal continuum, leading to a possible overestimation of the hot continuum. Nevertheless, the blackbody luminosities tend to underestimate the total luminosity as they do not allow for the total line emission. This is particularly the case around 200−500 d when the relative contribution of nebular line emission to the total luminosity is at a maximum. There is less of a problem before this era when the hot continuum is relatively strong, or afterwards when the warm/cold continuum increasingly dominates. Also, by 652 d, the relative weakening of the hot continuum means that it has a negligible effect at wavelengths longward of 2 µm. In the 2 − 14 µm region the true continuum level is easier to judge. By 251 d and later, the total 2 − 14 µm flux exceeds that of the continuum model by no more than 25%.
The hot continuum declined monotonically and dominated the total SN continuum luminosity up to about a year post-explosion. It presumably arose from hot, optically-thick ejecta gas. At 89 d most of the hot continuum was probably still driven by the shock-heated photosphere, with the remainder being due to radioactive decay (see Table 7 and Fig. 12). After 500 d the hot continuum became relatively weak and of low S/N. In addition, the NIR spectral dataset ended on 554 d. For subsequent epochs the strength of the hot component was estimated by extrapolation. At 500 d the hot blackbody match was achieved with a temperature of 10,000 K. Between 251 d and 500 d the blackbody velocity declined exponentially with an e-folding time of ∼130 d and so, for epochs after 500 d the hot blackbody temperature was fixed at 10,000 K and the velocity obtained by extrapolation of the earlier exponential behavior (see Table 7). While this is likely to be increasingly inaccurate with time, it is unlikely to be a serious source of error in determining the warm continuum; for example, by 500 d the hot blackbody contributed barely 1% of the flux at 3.6 µm. Consequently, and in order to illustrate the MIR behavior in more detail, the optical/NIR region is not shown in the later plots (Fig. 14).
The warm component luminosity declined monotonically to 281 d but then, unlike the hot component, increased by a factor of 3.5 by 500 d. By that time the hot+warm SN continuum luminosity exceeded that of the radioactive energy input by a factor of 4 and this excess continued to increase with time. The cause of the growing excess was the warm component luminosity. We also note that R warm more than doubled between 281 d and 500 d, but remained roughly constant thereafter. In fact, as we show later ( §3.4.4.1, §3.4.4.2), after 500 d slow shrinkage in the size of the warm emission region occurred.
The cold component is primarily defined by the 24 µm point, and can be fairly reproduced using a range of temperatures (150−300 K) and velocities. Its luminosity remained roughly constant throughout the observations. In Table 7 we show the case with the temperature fixed at 200 K.
It is interesting that, even as early as 89 d, warm and cold blackbodies had to be included to achieve a fair match to the observed fluxes in the NIR-MIR region. This will be discussed in §3.4.2 and 3.4.3.1. By 106 d, the nebular phase was just beginning and by 129 d the hot+warm luminosity was driven predominantly by radioactive decay. By 251 d and 281 d, the radioactive deposition exceeded the sum of the hot+warm blackbody luminosity by about 15%. This excess probably went into powering the line emission not included in the blackbody matches. As already indicated, the nebular line emission was particularly strong at this time. Indeed it was responsible for a ∼ 50% excess of the total bolometric luminosity relative to the radioactive deposition ( Fig. 12), indicating the emergence of an additional energy source, probably the reverse shock ( §3.4.4.2). The appearance of an additional power source has already been indicated by the flattening of the optical light curves ( §2.2). Indeed, after ∼150 d the true BLC exhibited a steadily growing excess relative to the radioactive deposition (see Fig. 12).
The hot and warm blackbody velocities never exceeded 1750 km s −1 , indicating continuum emission consistent with an origin in the ejecta or ejecta/CSM interface. The cold component exhibits velocities between 4500 km s −1 and 8500 km s −1 during the earlier phase pointing to an origin more likely to be outside the ejecta, specifically an IR echo from pre-existing dust.
We conclude that the IR continuum comprised at least two components. The temperatures and temporal variation of these components point to thermal emission from dust whose energy source is ultimately the supernova. The surge in the luminosity of the warm component by 500 d suggests the emergence of an additional source of radiation -i.e., that the warm component was driven by different energy sources at, respectively, early and late times. It also raises the possibility that distinct dust populations were responsible for the early and late-time warm components.
Origin of the Infrared Radiation
We now explore the origin of the IR continuum radiation from SN 2004dj, especially the warm component. To do this we have constructed a model continuum comprising hot gas, warm local dust, and cold IS dust. We also made use of spectral line red-wing suppression in the study of the warm dust. The continuum model was adjusted to provide visual matches to the observations.
The Hot Component
As in §3.3, the hot continuum component, presumably due to optically-thick ejecta gas, was represented using a hot blackbody having a temperature of 5300-10000 K. This blackbody radius and temperature was adjusted to obtain a match to the optical-NIR continuum. The warm component model (see §3.4.2 and §3.4.4.2) was then added and adjusted to match the ∼5-10 µm continua plus the long-wavelength end (2-2.4 µm) of the NIR continuum. As with the pure blackbody matches it was found that, at all epochs, the warm component model flux longward of 10 µm increasingly fell below that of the observations. This excess is attributed to an IS IR echo (see §3.4.3.1).
The Warm Component: Early Phase IR Excess due to a CDS
As pointed out above, a striking result from the hot-blackbody matches is that an NIR-MIR excess (relative to the hot component flux) was present as early as 89 d post-explosion. (For brevity we henceforth refer to the NIR-MIR continuum excess as the "IR excess".) The MIR spectra from 106 d onward show that this was primarily due to continuum emission. We have no reason to suspect that the IR excess in the 89 d SED was not also due to continuum emission. Indeed, the CO peak at 4.5 µm is noticeably suppressed at this epoch, compared with 106 d (Fig. 3). The obvious interpretation of the IR excess is that it arose from warm dust heated by the supernova. The very early appearance of this emission, when the H-recombination front had not yet reached the He-metal core, argues against an origin in newly-formed ejecta dust.
In addition, SN-ejecta dust-formation models (e.g. Todini & Ferrara 2001;Nozawa et al. 2003) suggest that dust formation in a CCSN is unlikely to occur until after one year post-explosion. A second possibility of direct shock-heating of pre-existing circumstellar dust is also ruled out. The SN UV flash would evaporate dust out to (0.5 − 1.0) × 10 17 cm (0.016 − 0.032 pc) (see §3.4.3.1). To reach this distance by 89 d would require a shock velocity of at least 65,000 km s −1 . Yet the velocities of the Hα trough (Korcáková et al. 2005;Vinkó et al. 2006) as well as of the extreme blue-profile edge (our measurements of the spectra of Korcáková et al. (2005); Vinkó et al. (2006)) during the period 25 − 90 d suggests that the bulk of the ejecta never exceeded velocities much more than ∼15,000 km s −1 . Indeed Chugai et al. (2007) adopt a velocity of 13,000 km s −1 as the boundary velocity in their treatment of SN 2004dj. A third possibility, CSM dust heating by X-rays, is also implausible. The luminosity of the warm continuum component at 89 d was 1.3 × 10 40 erg s −1 (Table 7) but the X-ray luminosity at 30 d was only about 1% of this . In the case of dust in a CDS, collisional heating is ruled out as the energy available in the CDS is about a factor of 10 5 less than that required to account for the observed IR excess. The heat capacity of the grains is also insufficient to account for required energy.
The most likely explanation for the early-phase IR excess is an IR echo of the SN early-time luminosity from circumstellar dust. There are two possible scenarios here. In the first of these, the IR echo is from pre-existing dust in the CSM. Such early IR echoes have been suspected before in other CCSNe. Wooden et al. (1993) found an IR excess in SN 1987A as early as 260 d and possibly also at just 60 d. They hypothesised that the origin of the excess was warm, SN-heated dust in the CSM. Wooden (1997) reiterates that the cause was CSM dust "echoing the light curve". Fassia et al. (2000) reported a strong K − L ′ excess in the emission from the Type IIn SN 1998S at 130 d. They attributed this to pre-existing CSM dust heated either by the SN luminosity (a conventional IR echo) or by X-rays from the CSM-shock interaction. Pozzo et al. (2004) argued in favor of the former scenario.
In the second IR echo scenario, as the fast moving ejecta collides with the CSM, a CDS forms between the forward and reverse shocks. As noted in §1, Chugai et al. (2007) have used the Hα spectrum to deduce the existence of a CDS in SN 2004dj. Within the CDS conditions can allow new dust to form (Pozzo et al. 2004). CDS dust has been invoked as the origin of the early-time IR excess of SN 2006jc (Smith et al. 2008;Di Carlo et al. 2008;Mattila et al. 2008). Mattila et al. (2008) showed that the IR emission was probably an IR echo from the CDS dust. IR excesses in CCSNe at later times have also been attributed to emission from newly-condensed CDS dust viz. in the Type IIn SN 1998S (Pozzo et al. 2004) and the Type IIP SN 2007od (Andrews et al. 2010).
We explored the possibility that the early IR excess was due to an IR echo of the SN luminosity from spherical distributions of either pre-existing circumstellar dust or newly-formed CDS dust. Details of the model are given in Meikle et al. (2006). It fully allows for the effects of light-travel time across the dust distribution. Versions of this model have also been used in Meikle et al. (2007); Mattila et al. (2008); Botticella et al. (2009);Kotak et al. (2009). The model assumes a spherically symmetric cloud of grains centered on the SN, with a concentric dust-free cavity at the center. The SN is treated as a point source. For simplicity, a single grain radius, a, is adopted. For ease of computation, we assumed that the grain material was amorphous carbon where, for wavelengths longer than 2πa, the grain absorptivity/emissivity can be well-approximated as being proportional to λ −1.15 (Rouleau & Martin 1991). For shorter wavelengths, an absorptivity/emissivity of unity was used. The material density is 1.85 g cm −3 (Rouleau & Martin 1991). Free parameters are the grain size, grain number density, radial density law and extent. The input luminosity is a parametrized description up to 550 d of the BLC shown in Fig. 12 viz.:
L bol = L 0 exp(−t/τ ), where L 0 = 57.0, τ = 1000.0 d for t ≤ 0.2 d, L 0 = 1.70, τ = 23.4 d for 0.2 < t ≤ 23.0 d, L 0 = 0.66, τ = 141.9 d for 23.0 < t ≤ 76.4 d, L 0 = 50.7, τ = 15.9 d for 76.4 < t ≤ 112.0 d. L 0 = 0.079, τ = 174.8 d for 112.0 < t ≤ 550.0 d L 0 = 0 for t > 550.0 d L 0 is in units of 10 42 erg s −1 .
The brief but highly luminous first term represents the energy in the UV flash. The second term, which covers the pre-discovery phase, was estimated by using the BLC of SN 1999em (Elmhamdi et al. 2003), adjusted so that the epoch and luminosity of the beginning of its plateau phase coincided approximately with the earliest observed point on the SN 2004dj BLC.
For the third, fourth and fifth terms, the adjusted BLCs of Vinkó et al. (2006) and Zhang et al. (2006) were used (see §3.2). As already pointed out, these BLCs did not include excess flux beyond about 1 µm, nor any flux beyond 2.3 µm; virtually all of the warm and cold components were excluded. Thus, use of the above parametrized description avoids "doublecounting" of a putative CSM/CDS IR echo. Indeed the BLC description slightly underestimates the true SN luminosity input as it excludes all IR emission longward of 2.3 µm, as well as all UV emission shortward of 0.34 µm. Consequently the input luminosity was scaled by a factor of about 1.1 to allow for unobserved UV and IR fluxes.
For the case of pre-existing CSM dust the model was adjusted to reproduce the IR-excess SEDs for the three earliest epochs (89, 196 and 129 d). The outer limit of the circumstellar dust was initially set at 10 times that of the cavity radius, and a r −2 (steady wind) density profile was assumed. However, it was impossible to reproduce the quite rapid temporal decline of the IR excess without raising the temperature of the hottest grains to above their evaporation temperature. While a better match to the SED shape and evolution was obtained by setting the density profile to steeper than r −4 , the best match was achieved with a discrete, thin shell. We therefore adopted this configuration, setting the shell thickness at ×0.1 the cavity radius.
We investigated a range of cavity radii. For preexisting CSM dust the minimum size of the concentric dust-free cavity is fixed by the extent to which the dust was evaporated by the initial UV flash from the supernova. In this scenario, while the bolometric light curve (BLC) dominates the heating of the surviving dust, the size of the dust-free cavity is determined by the luminosity peak of the UV flash, with r evap ∝ L 0.5 peak (Dwek 1983). For a Type IIP SN the flash luminosity is estimated to peak at about 10 45 erg s −1 (Klein & Chevalier 1978;Tominaga et al. 2009) although it has never been observed directly. A similar peak luminosity is estimated for the Type IIpec SN 1987A (Ensman & Burrows 1992). Dwek (1983) provides an approximate estimate of the flash-evaporated cavity size for a Type II supernova. More recently, in a detailed study Fischera et al. (2002) determined that for SN 1987A the UV flash would have totally evaporated graphite CSM dust out to a radius of (0.5 − 0.9) × 10 17 cm (0.016 − 0.029 pc). We therefore adopt the Fischera et al. estimates and apply them to the case of SN 2004dj. In any case, matches to the data are fairly insensitive to the cavity radii. Fair matches to the early-time IR-excess SEDs were obtainable with cavity radii (0.5 − 1.3) × 10 17 cm (0.016 − 0.042 pc) and corresponding grain radii of 0.1 − 0.04 µm. For cavity radii exceeding ∼0.05 pc the model continuum slopes were inconsistent with the observations.
As an example of a pre-existing CSM dust model match we have: cavity radius: 0.7 × 10 17 cm (0.023 pc), shell thickness: 0.7 × 10 16 cm (0.0023 pc), grain size: 0.07 µm, grain number density: 6.0 × 10 −9 cm −3 , total grain mass: 0.38 × 10 −5 M ⊙ .
The optical depth through the shell in the optical band is 0.006, which is easily encompassed within the observed total extinction of A V = 0.3 mag. For a dust/gas mass ratio of 0.005 (see §3.4.3.1), the dust mass corresponds to a total shell mass of 0.76 × 10 −3 M ⊙ . For the adopted shell thickness and a typical RSG wind velocity of 20 km s −1 , this mass would be produced by a mass loss event about 1100 years ago, lasting for 110 years, with a mass loss rate rate of 7 × 10 −6 M ⊙ yr −1 . However, while this mass loss and loss rate are plausible for RSGs, such discrete events are not thought to occur in this type of star. This prompts us to seek a more natural explanation for the thin dust shell. An obvious candidate is the CDS inferred by Chugai et al. (2007).
For the CDS scenario the size of the cavity is essentially the radius of the CDS. From Chugai et al. (2007), the CDS radius is given by (1) where t (in days) is with respect to our explosion epoch. Thus, at the earliest of our epochs, 89 d, the CDS radius was just 7.7 × 10 15 cm, or about one tenth of the pre-existing CSM dust cavity. This is only 3 light days implying that light travel time effects are small. Nevertheless, for convenience and ease of comparison with the pre-existing dust case, we applied the IR echo model to the CDS case. Although the Chugai et al. study terminates at just 99 d, we assume that the CDS radius continues to increase as described in equation (1) until at least 500 d, by which time its contribution to the IR excess is small.
r CDS = 5.2 × 10 15 × ((t + 11.5)/64.0) 6/7 cm
We compared the CDS case of the IR echo model with the observations, with the dust lying in a thin shell of radius r CDS as given above. We found that setting the dust mass at a constant value produced a poor match to the observations. With a match at 106 d, the model at 89 d yielded a continuum which matched the observed IR excess at 8 µm but exceeded the IR excess by nearly ×2 at 3.6 µm. On the other hand, at 129 days, while the model continuum slope was similar to that of the observed IR excess, it significantly underproduced it with the deficit being as large as ∼35% at 8 µm. We propose that this problem is due to the unjustified assumption of a fixed CDS dust mass. CDS dust could not form until the supernova flux at the CDS had faded sufficiently for proto-dust material to cool below the condensation temperature. This occurred at about 50 d, assuming amorphous carbon dust. Therefore, following Mattila et al. (2008) we allowed the CDS dust mass to grow as
M d = M 0 (1 − exp(−(t − t 0 )/t d ))
where t is time, t 0 is the time at which dust condensation began (set at 50 d), t d is the characteristic grain growth timescale and M 0 is the asymptotically-approached final mass. (We note that Mattila et al. also deduced an epoch of 50 d for the start of the CDS dust condensation in SN 2006jc.) No attempt was made to simulate the growth of individual grains which were assumed to appear instantaneously at their final size. Owing to the light travel time differences across the CDS, the grain condensation is seen to commence during the epochs (t 0 − (r CDS /c)) to (t 0 + (r CDS /c)) days. Yet, even as late as 500 d, (r CDS /c) was only about 12 light days and so this effect was ignored.
In Fig. 15, we compare the CDS model light curves and spectra with, respectively, the observed MIR excess fluxes at 3.6 µm and 8.0 µm (LH panel), and with the 89, 106 and 129 d MIR excess SEDs (RH panel). The free parameters for the CDS case were the dust mass scaler, the grain radius and the grain growth timescale. Satisfactory reproduction of the IR excess was achieved for all five epochs spanning 89-281 d with t d = 50 d, a grain radius of 0.2 µm and a final dust mass of M 0 = 0.33 × 10 −5 M ⊙ , similar to the dust mass derived in the pre-existing CSM case above. The CDS mass is 3.2 × 10 −4 M ⊙ (Chugai et al. 2007) indicating a plausible final dust/gas mass ratio of 0.01. The hottest dust ranged from 1330 K at 89 days declining to about 640 K at 281 days. The optical depth in the UV/optical range was 0.077 at 89 d falling to 0.011 by 500 di.e., consistent with the observed total extinction of A V = 0.3 mag. By 500 d the MIR excess exceeds that of the model by a significant factor (especially at longer wavelengths) (Fig. 15) implying the appearance of an additional energy source. The CDS contribution to the SN continua for 89-500 d is plotted in Fig. 18. The luminosity contribution of the CDS IR echo, L CDS , is listed in Table 9, Col. 3. For epochs after 500 d the CDS component was negligible and so was ignored.
We conclude that the early-time IR excess was probably due primarily to an IR echo from newly-formed dust lying within the CDS. The rapid decline of the IR excess flux, especially between 89 d and 106 d, is due largely to the ongoing fall from the BLC plateau, tempered by the growth of CDS dust during this period. Owing to the small size of the CDS, light travel time effects are small. In contrast, in the pre-existing CSM dust scenario, the rapid decline of the input luminosity is tempered by the larger size of the dusty shell which produces significant light travel time effects, smoothing the observed IR excess light curve over longer timescales. Nevertheless, given the natural explanation by the CDS model of the required thin shell dust distribution, in the completion of the analysis below we use the CDS scenario.
The Cold Component
An Interstellar Echo
The cold component is defined primarily by the 24 µm data and can be fairly reproduced using a range of temperatures (150 − 300 K) and velocities. As described above ( §3.3), the luminosity of the cold component remained roughly constant throughout the observations. In addition, the cold blackbody velocities were high, arguing against an origin in ejecta dust. An obvious alternative is an IS IR echo. The possibility of detecting the reflection of SN optical light from IS grains was first suggested by van den Bergh (1965). In the IR a potentially much more important phenomenon is the absorption and re-radiation by the grains of the SN BLC energy (the "IR echo"). The possibility of detecting an IS IR echo from a SN was first proposed by Bode & Evans (1980). 3 The occurrence of cold dust IS IR echoes should be relatively common for SNe occurring in dusty, late-type galaxies. The SEDs of such echoes tend to peak in the 20 − 100 µm region, allowing echo detection in nearby galaxies by Spitzer during its cold mission. Spitzer-based evidence of this phenomenon in the Milky Way Galaxy have been presented for the Cassiopeia A SN (Krause et al. 2005;Kim et al. 2008;Dwek & Arendt 2008). Meikle et al. (2007) showed that an IS IR echo provided a natural explanation for the strength and decline of the 24 µm flux between 670 − 681 d and 1264 d in SN 2003gd in the SA(s)c galaxy NGC 628 (M74). Kotak et al. (2009) showed that the cold component of the SN 2004et SED was most likely due to an IS IR echo in the SAB(rs)cd host galaxy NGC 6946. The host galaxy of SN 2004dj, NGC 2403, is of type SAB(s)cd and so there is a good likelihood of a similar IS IR echo occurring. Therefore, we included an IS IR echo component in our modelling of the SED.
Our IS IR echo model is the same as that used for the early-time CDS IR echo ( §3.4.2). Only the dust distribution and grain radii are different. We ignore the CDS dust emission derived in §3.4.2 since this is already invoked in the CDS echo model and included in the continuum modelling up to 500 d. We recognise that a more extended, lower density CSM may also have existed. However, we found that the addition of yet another model component was unnecessary to provide plausible matches to the continua and so for simplicity the possible effects of an extended CSM were ignored.
As described in §3.4.2, pre-existing dust surrounding the SN would have been evaporated by the SN flash out to a distance of about 0.025 pc. At sufficiently late epochs, enlargement of this cavity can be produced by the forward shock. Assuming that the shock velocity is comparable to the highest ejecta velocity viz. ∼15,000 km s −1 (see §3.4.2), and that there was no deceleration, the edge of the UV-flash-determined cavity would be reached after about 600 d. After this time, the SN shock would evaporate the dust and enlarge the dust-free cavity. Therefore, for epochs earlier than 600 d we fixed the inner limit of the IS dust at 0.025 pc. For later epochs the inner limit of the IS dust increases from 0.028 pc (34 light days) at 662 d to 0.058 pc (70 light days) by 1393 d. It might be objected that this ignores the possibility of shock deceleration. However, the IR echo contribution which the shock-evaporated dust would otherwise have made, for 600 > t > 1393 d, to the cold component is negligible since the cavity radius was never more than 70 light days i.e. during this late period the SN peak luminosity would have long since passed by the dust near the cavity.
The outer limit of the IS dust is much less certain. Bendo et al. (2010) have used Spitzer imaging of NGC 2403 at 70 µm and 160 µm to map the dust column density via its thermal emission. They also used the far-IR images in conjunction with H I observations to map the gas/dust ratio. The objection to using such measurements in the present work is that they only provide total densities through the disk (or to the mid-plane assuming symmetry), and not directly to SN 2004dj. While it was argued above that SN 2004dj actually lies towards the front of S96, we are still faced with the uncertainty of the depth of S96 in the galaxy plane. Our approach, therefore, is to assume a spherically symmetric dust distribution centered on the SN, with an outer limit of 100 pc. While not appropriate in principle for the outer limits of the dust in the galactic disk, for the early era being considered spherical symmetry provides a good approximation; at this epoch, the echo ellipsoid is extremely elongated and so the region of the spherical model outer surface intercepted by the ellipsoid is small. The plane of NGC 2403 appears to be tilted at an inclination of about 55 • roughly doubling the face-on column density, and so the adopted outer limit is equivalent to a ∼60 pc scale height for the IS dust with the host galaxy face-on.
The free parameters were (i) the grain size, which influenced the dust temperatures, and (ii) the grain number density, which determined the luminosity. These parameters were adjusted in conjunction with the warm dust models (see §3.4.2 and §3.4.4.2) to provide a match to the longwave excess. It was found that satisfactory matches at all epochs were obtained with a grain radius of 0.1 µm and a dust number density of 2.6 × 10 −13 cm −3 (i.e., an IS gas number density of 0.24 cm −3 and a dust/gas mass ratio of 0.005 - Bendo et al. 2010.) This is comparable to the typical value of 5 × 10 −13 cm −3 for the Milky Way (Allen 1973) and 7.0 × 10 −13 cm −3 obtained by Kotak et al. (2009) for NGC 6946. In the wavelength region (λ > 15 µm) where the cold component makes a significant contribution (> 20%) to the total flux, the IS IR echo model Thus, we have a ∼ ×8 discrepancy in column density between Bendo et al. and the present work. Part of this discrepancy may be that the adopted dust outer limit is too small. The derived IS dust density is fairly insensitive to the extent of the outer limit. For example, with a 75% increase in the outer limit the model match to the longwave continuum is retained with a reduction of just ∼10% in the IS dust density. Thus, at least some of the discrepancy could be removed by simply increasing the dust outer limit. An explanation for the remaining discrepancy is that the SN and S96 actually lie significantly above the mid-plane of NGC 2403. This is confirmed as follows. The optical depth to UV/optical photons yielded by the model is about 0.026. We can use this to estimate independently the host extinction to SN 2004dj. The optical depth translates to an absorption-only extinction of 0.028 magnitudes. Assuming an albedo of about 0.4 (Draine 2003), we obtain a total host extinction of 0.071 mag. This is reasonably consistent with the host-only value of A V = 0.081 mag (R = 3.1) obtained via the Na I D observations of Guenther & Klose (2004). A similar result is obtainable from our study of SN 2004et (Kotak et al. 2009). For SN 2004et, the IS IR echo model yields an absorption-only extinction of 0.081 mag implying a total host extinction of about A V = 0.20 mag. High resolution spectra of Na I D lines to SN 2004et gave a total (host plus Galaxy) E(B − V ) = 0.41 mag (Zwitter et al. 2004), or A V = 1.27 mag (R V = 3.1). Subtracting the estimated Galactic contribution of A V = 1.06 mag (Schlegel et al. 1998;Misra et al. 2007) yields a host-only value of A V = 0.21 mag, in excellent agreement with the IS IR echo-based value.
One possible objection to the IS IR echo interpretation of the cold component of SN 2004dj is that the steady component of the flux around 24 µm could be due to cool IS dust in S96, insufficiently corrected for by the 3500 K blackbody extrapolation ( §3.1). We regard this as unlikely. As already indicated, Maíz-Apellániz et al. (2004) argue that the parent molecular cloud of S96 should have been dispersed by stellar winds and SN explosions. In addition we have found that (a) a single set of IS IR echo parameters provided a fair match throughout the 89 − 1393 d covered and (b) the derived dust column density is comparable to that obtained from IS Na I D spectroscopy.
A second possible objection to the IS IR echo interpretation is that the cold component is actually due to free-free emission. This is discussed and dismissed in §3.4.3.2.
We conclude that the cold component of SN 2004dj was due to an IS IR echo. As originally suggested by Bode & Evans (1980), the study of IS IR echoes from SNe can provide an independent method of measuring the IS extinction in galaxies.
Free-Free and Free-Bound Radiation
In their MIR study of SN 1987A, Wooden et al. (1993 suggest that during the 60 − 415 d period, a significant proportion of the MIR flux longward of ∼20 µm was due to free-free (ff) emission, with ff and free-bound (fb) emission also contributing at shorter wavelengths. For SN 2004dj we have looked at the possible contribution of fb-ff radiation. It is particularly important to consider ff emission as this rises in flux towards longer wavelengths, thus potentially reducing or even dismissing an IS IR echo contribution. For consistency in this analysis fb emission must also be included. For the wavelengths covered this is strongest in the optical/NIR region. Free-bound radiation has a strong "sawtooth" structure and the extent to which this structure is undetectable in the optical/IR continuum places a limit on its strength. For a particular temperature this, in turn, constrains the strength of the ff emission (
F f f /F f b ∝ √ T /ν 3 ).
The fb-ff radiation was assumed to arise in a hydrogen envelope centered on the supernova. Kozma & Fransson (1998) modelled the time-dependent behavior of the temperature and ionization of SN 1987A, including the hydrogen envelope, for epochs after 200 d. Therefore, we made use of their study to model the ff and fb emission from SN 2004dj at epoch 251 d and later. We adopted a hydrogen envelope of a few solar masses and used their density profile:
ρ = 9.1×10 −16 (t/(500 d)) −3 (v/(2000 km s −1 ) −2 g cm −3 .
(2) By 251 d, the highest velocity of observable hydrogen in SN 2004dj was no more than ∼ 3100 km s −1 (Table 8). By 925 d the maximum H velocity observed in SN 2004dj was no more than 1400 km s −1 . There were no optical or NIR spectra covering the final three MIR epochs. We therefore assumed that the maximum hydrogen velocity at these late epochs remained at 1400 km s −1 . The hydrogen inner limit is less certain; a value of 1000 km s −1 was adopted for all epochs. In any event, the fb-ff luminosity from 500 d onwards was completely negligible for any plausible hydrogen velocity limits (see below).
We estimated the fb-ff emission from the hydrogen within the velocity limits. This was done using the escape probability formalism (Osterbrock 1989), although in practice this was unnecessary as the matches to the data always showed that the hydrogen was very optically thin. The mass of the fb-ff-emitting hydrogen fell from ∼4 M ⊙ at 251 d to 0.75 M ⊙ at 996, 1207 and 1393 d. Using modest extrapolation of figure 7 in Kozma & Fransson (1998) to the velocities observed in SN 2004dj, we deduced that the hydrogen at the observed velocity limits stayed at a temperature of T ∼5000 K up to 859 d and then declined to ∼2000 K by 1393 d. The fractional ionization, χ e , was obtained from figure 9 of Kozma & Fransson by interpolation. This indicated χ e ∼ 0.02 at 251 d, falling to χ e ∼ 0.0005 by 1393 d. At each epoch, the T and χ e values were assumed to apply to all the hydrogen within the velocity limits. The Kozma & Fransson study did not extend to epochs as early as the earliest MIR observations (89, 106, 129 d) of SN 2004dj. We therefore adopted 6000 K as a plausible temperature for these epochs and allowed the ionization to take the largest value consistent with the overall match to the observed continuum. Plausible values were obtained, viz: χ e = 0.003 at 89 d rising to χ e = 0.005 at 129 d. We used Gaunt factors tabulated by Hummer (1988). For the fb radiation we used the continuum recombination coefficients of Ercolano & Storey (2006).
The hydrogen maximum velocity, temperature and free electron density for each epoch are listed in Table 8. The fb-ff luminosities are listed in Table 9, col. 4. It can be seen that by 500 d the fb-ff contribution to the total luminosity is small, becoming increasingly negligible at later epochs The fb-ff components for 89-281 d are plotted in Fig.18. At 500 d the fb-ff flux is too weak to appear on the 500 d plot which has been scaled to allow easy comparison with the earlier epochs. A more comprehensive estimation of the fb-ff flux, taking into account the temperature and ionization gradients, is beyond the scope of this paper.
As noted in §3.4.3.1, in the wavelength region (λ > 15 µm) where the cold component makes a significant contribution (> 20%) to the total flux, the IS IR echo maintained a near-constant luminosity (15-150 µm) of ∼ 2.0 × 10 38 erg s −1 between 89 d and 1393 d. Moreover, in setting the IS IR model to reproduce the longwave excess at the latest epochs, when the fb-ff flux was negligible, it was found that a satisfactory match to the longwave (λ > 15 µm) excess was automatically achieved for all epochs. Consequently, the fb-ff flux could make, at most, only a minor contribution even at the earliest epochs. We conclude that any free-free emission was too weak to account for the cold component at any epoch.
The Warm Component
Late-Time Optical and NIR Line Profiles
In §2.4 we described the evolution of optical and NIR line profiles in a number of species. In particular, we noted the development of a sharp suppression in the red wing, suggesting dust formation causing obscuration of the far side of the ejecta. Could this dust also be responsible for the warm component at later epochs? In this and the next sections we explore this possibility. Here we examine the distributions of dust that might give rise to the late-time optical/NIR line profiles. In The period 461 − 925 d was studied since this was mostly overlapped by the 500 − 1393 d during which the blackbody analysis and more detailed studies of the MIR emission ( §3.4.4.2) suggest that substantial quantities of dust were present in the ejecta. We note ( Table 6) that the Hα line widths towards the end of the bright plateau phase greatly exceeded those observed on 461 d and later, implying that there was a negligible contribution of any light echo of the early phase profiles to the late-time profiles.
We did also examine the blue asymmetry of the Hα and [Fe II] 7155Å profiles at a much earlier epoch (283 d) but were unable to achieve a satisfactory match using the model adopted for later epochs. In any case, as already demonstrated in §3.4.2, we were able to account for the 251/81 d IR excess as thermal emission from dust formed in the CDS. Moreover, Chugai et al. (2005) and Chugai (2006) successfully explained the Hα profile up to ∼300 d by invoking the emergence of an intrinsic asymmetric, bipolar core (i.e. no dust involved). We conclude that there is no evidence for the formation of new ejecta dust earlier than 461 d.
The line-profile red wings exhibit increasingly abrupt declines after ∼1 year (Fig. 10) suggesting the condensation of attenuating dust. In contrast, the extended blue wings exhibit little sign of developing suppression.
This behavior points to the attenuating dust lying at low line-of-sight velocities, such as a face-on disk-like distribution centered on the SN center of mass. Nevertheless, in our initial considerations of possible dust configurations we included the case of a spherically symmetric dust sphere. Our line-profile model comprises a homologously expanding sphere of gas responsible for most of the observed line flux, with the emission being attenuated by a dust zone lying concentrically with the gas sphere. The dust zone is also assumed to participate in the expansion. The gas emissivity is assumed to have a power-law dependence with radius.
Several arrangements of dust were examined. These were (i) a uniform opaque sphere (Fig. 16, upper panels), (ii) a face-on opaque ring, and (iii) a thin face-on disk (Fig. 16, lower panels) whose opacity was uniform or varied radially as an r −δ power law along the disk plane but was always uniform normal to the disk plane. (For the disks, "opacity" refers to the value normal to the disk plane.) For the radially-varying version of (iii), a power law index of δ = 1.9 was chosen as typical of the power law indices determined for the gas emission profiles. Configurations (i)-(iii) were initially tested against the 895 d optical profiles since these were (a) of the highest available resolution, and (b) corresponded to a time by which the SN spectral continuum was overwhelmingly dominated by thermal emission from dust. Configurations (i)-(ii) could not reproduce the observed line profiles. This is illustrated in Fig. 16 (upper panels) where we show two examples of the line profile produced by an opaque, concentric dust sphere, compared with the observed [O I] 6300Å profile at 895 d. Only configuration (iii) -the dust disk -was able to reproduce the observed profiles. Within this configuration we examined cases where the dust distribution (a) extended beyond the gas limit, (b) ceased abruptly at a given velocity within the gas sphere, or (c) extended uniformly to a given velocity, v duni , and then declined as a power law to the edge of the gas sphere. Case (a) failed to reproduce the observed line profiles. Cases (b) and (c) are discussed below.
As described in §2.4, by 895 d and 925 d, a "secondary", weaker, redshifted peak had also appeared in the Hα and [O I] 6300Å profiles, suggesting that a fraction of the line flux originated in an emission zone centered on the SN and having the geometry of an expanding ring, jet or cone. The second peak had a redshift of only +160 to +170 km s −1 . Given the preference for a thin face-on disk of attenuating dust to account for the asymmetry of the line profiles, a natural explanation for the low velocity of the second peak is that its source was actually at a velocity comparable to that of the gas sphere, but moving in a way which was roughly coplanar with the dust disk. Consequently the low inclination angle led to the low observed redshift in the second peak. The simplest geometry which accounts for the second peak is a ring-like emission zone whose plane lay at a small angle to the "face-on" plane. A cone or jet geometry is more problematic. If we insisted on maintaining axial symmetry then a cone/jet would have to be normal to the disk but with a mysteriously low velocity. Alternatively, a cone/jet lying near the disk plane, while possibly allowing a high intrinsic velocity, would have the unattractive feature of breaking the axial symmetry. Therefore, to account for the second peak, we added a thin, near-coplanar ring of emitting gas to the line-profile model (cf. Gerardy et al. 2000;Fransson et al. 2005, for the case of SN 1998S). The ring was assumed to participate in the overall homologous expansion of the gas.
We set the intrinsic ring velocity equal to the fastest moving gas observed viz. hydrogen at 2400 km s −1 as indicated by the Hα blue wing on 895 and 925 d, although higher velocities could have been used. Consequently the ring emission is unattenuated by the dust. For ease of computation, we retained a face-on disk (i = 0 • ). Tilting the disk to become exactly coplanar with the ring would have had only a small effect on the line profile.
In matching the model to the observed profiles the free parameters for the gas are the maximum (outer) velocity of the gas sphere, v gmax , the gas emissivity scaling factor and power-law index β, the inclination, i r , of the ring component assuming an intrinsic expansion velocity of 2400 km s −1 , and the ring component emissivity assumed uniform. In addition, the overall wavelength positions of the observed profiles were allowed to vary by small amounts (see below). For ease of comparison, the dust-disk dimensions were also expressed as velocities within the homologous expansion. Thus, for the case (b) dust disk the free parameters are the maximum radial velocity of the dust disk, v dmax , the maximum velocity, v dth , perpendicular to the disk plane and the magnitude of the dust radial density power-law index, δ. Multiplication of v dmax and 2v dth by the epoch yields, respectively, the disk radius and thickness at that time. For case (c) v dmax is replaced with v duni . We adopted amorphous carbon as the grain material (see §3.4.4.2) and a power law grain-size distribution, index m = 3.5 (Mathis et al. 1977), with a (min) = 0.005 µm and a (max) = 0.05 µm. The optical depth normally through the disk was then calculated. For example, for case (b) the optical depth for emission from gas lying behind the disk at wavelength λ, epoch t and radial velocity v d , τ λ (t, v d ), is given by:
τ λ (t, v d ) = 4 3 πρκ λ k 1 4 − m [a 4−m (max) − a 4−m (min) ]v −δ d 2v dth t −2
(3) where k is proportional to the grain number density, at a fiducial epoch, to the radial velocity and to the grain radius, ρ is the grain material density and κ λ is the mass absorption coefficient of the grain material at wavelength λ. For emission from gas lying within the disk the same formula is used but replacing 2v dth with the velocity equivalent to the path from the emission point to the near side of the disk. The disk thickness was always < 20% of the disk diameter and so edge-effects at the disk's outer limit were assumed to be negligible.
The gas emission profiles (i.e., those for the attenuated sphere and unattenuated ring) were individually convolved with the appropriate slit PSF. The PSF FWHM values were 6.5, 4.2, 6.5, 1.25, and 1.75Å for 461, 467, 554, 895, and 925 d, respectively. The two components were then summed to form the final model profile which was compared with the observed profile. The profile model (case (b) disk) was adjusted to match the observed profiles as follows. The parameters v gmax , β and the gas sphere emissivity scaling factor were adjusted to provide a match to the blue wing of the profile. Then, for a given value of δ (0 or 1.9), v dth (equivalent to the disk half-thickness) together with the grain number density scaler, k, were adjusted to reproduce the sharp-decline section of the profile. The velocity v dmax was varied to reproduce the suppressed red wing. The gas ring emissivity and inclination were then adjusted to reproduce the secondary peak. Some iteration of all the model parameters was necessary to reach the final profile match. For the profile model (case (c) disk) v dmax was replaced with v duni , and δ was allowed to vary to arbitrarily large steepness.
The offset of the sharp-decline section of the profiles from zero velocity was only about −100 km s −1 after correction for the +221 km s −1 of the heliocentric velocity. Consequently, the match to the sharp-decline section of the profile (i.e., determination of the disk half-thickness) could be significantly affected by errors in individual wavelength measurements or intrinsic variations in the distributions of different gas species. Forcing a match in each case by letting the wavelength position of each observed profile to vary by small amounts, we were able to estimate the disk half-thickness and its uncertainty. For a heliocentric velocity of +221 km s −1 we obtained a disk half-thickness equivalent to v dth = +112 ± 18 km s −1 . There was no significant variation in v dth during the period covered by the profile study. Note that a change in the heliocentric velocity estimate would yield the same change in the disk half-thickness velocity.
As indicated above, we first carried out model matches to the 895 d profiles. It was found that the best case (b) matches were obtained when the disk was highly opaque (say, τ > 5) at all radial locations. For case (c), comparably good matches were achieved by setting the uniform zone at a high opacity (τ > 5) and δ at values steeper than ∼ 10 -i.e., the dust density beyond the uniform zone had to be extremely steep. Less steep declines tended to suppress the visibility of the central minimum. We conclude that the dust disk was highly opaque in the optical region but terminated abruptly at an approximately fixed radius (see §3.4.4.2). Given that this condition was indicated by both cases (b) and (c), we abandoned the more complicated case (c) model and completed the analysis for all the 461-925 d spectra using only case (b) -i.e., a dust density which was uniform or declined radially as r −1.9 and which terminated abruptly at v dmax .
The models for 895 d were initially adjusted to determine the minimum opacity -that is, the minimum dust mass that was needed to provide a satisfactory match to the observed profile. The dust masses were obtained from the following:
M d = 8 3 π 2 ρκ λ k 1 4 − m [a 4−m (max) − a 4−m (min) ] 1 2 − δ v 2−δ dmax 2v dth .(4)
At 895 d, with a uniform disk (δ = 0), the dust extended to v dmax = 590 ± 25 km s −1 i.e. r dmax = (4.6 ± 0.2) × 10 15 cm. The disk half-thickness was 19% of this. Simultaneous matches to all three profiles (Hα, [O I] 6300Å, [Fe II] 7155Å) required a minimum τ of 13 at 6300Å, corresponding to a minimum dust mass of 0.27 × 10 −4 M ⊙ . Note that the high optical depth was demanded by the very sharp decline, unresolved even at the 1.25Å resolution of 895 d. Other parameters are: r gmax = 7.65 × 10 15 cm, β = 1.9, (ring flux)/(total flux) = 0.07 and ring tilt = 6 • .
In using the dust-disk model to reproduce the contemporary MIR continua ( §3.4.4.2) it was found that the opacities and dust masses for all lines and epochs had to be somewhat higher than the minimum values required to match the observed optical and NIR line profiles. For example, for [O I] 6300Å at 895 d and a uniform disk it was necessary to set τ ≥ 18 ± 3, corresponding to a minimum dust mass of (0.38 ± 0.06) × 10 −4 M ⊙ . In the rest of this subsection, therefore, we present and consider results for profile matching incorporating dust-disk masses obtained by interpolating to the profile epochs the values obtained from the MIR continuum modelling. A sketch of the case (b) [O I] 6300Å uniform (δ = 0) disk model at 895 d, is shown in Fig. 16 (lower LH panel) together with an illustration (lower RH panel) of the model match (solid red line) to the observed profile (blue). Also shown are the intrinsic line-profile contributions from the attenuated gas sphere (green) and unattenuated gas ring (cyan) components, as well as the final profile (dotted red line) which would result if the attenuating dust were removed. In Fig. 17 This suggests an additional, faster moving component of iron, and may be indicative of 56 Ni asymmetry or "bullets" in the initial explosion (e.g. Burrows et al. 1995). At the earliest two epochs (461 d and 467 d) the model matches are also poorer. Specifically, if the model was adjusted to match the full suppression of the extreme red wing, then it also over-attenuated the less redshifted portion of the red wing; in other words, the steep red decline in the observed profile is generally less pronounced at these earliest epochs. Alternatively, matching to the less redshifted portion of the red wing meant that the full suppression of the extreme red wing was not reproduced. These points suggest that, during the 461-467 d period, dust formation was less complete and was not yet fully opaque over the whole disk. We therefore adjusted the models to reproduce the suppression of the extreme red wing, but recognise that the derived dust mass lower limits for 461-467 d may well be overestimated.
Between 554 d and 925 d the minimum dust mass increased from 0.25 × 10 −4 M ⊙ to 0.4 × 10 −4 M ⊙ . During this time the radius of the dust disk appeared to actually decrease slightly from 5.3 × 10 15 cm to 4.6 × 10 15 cm. We explored the possibility that this was due to a declining optical depth as the disk expanded -i.e., that while the dust disk continued to expand, the optically-thick/thin boundary declined in radius. We found that the requirement of a steep, sharp cutoff at the disk limit (see above) ruled out this explanation implying that the decrease in the dust radius was real. This issue will be considered further in §3.4.4.2.
The above analysis was repeated with δ = 1.9. Similar disk radii were obtained, but the minimum dust masses were ∼ ×20 larger than for δ = 0. This is as one would expect. To maintain the line-profile matches with δ = 1.9 required the maintenance of a high optical depth at the disk edge in the optical/NIR wavelength range. If we demand that the optical depths be the same at r dmax for both δ = 0 and δ > 0 then it may be shown that
M d (δ)/M d (0) = 2/(2 − δ), where M d (δ)
is the dust mass for a given value of δ and M d (0) is the mass for δ = 0. Thus, for δ = 1.9, M d (δ)/M d (0) = 20. We note that this assumes that the power law continues to the center, which is unlikely to be the case. Indeed for δ = 2 or more, the mass would be infinite. A more plausible behavior would be that the density law flattens toward the center. For example, if we assume that within 0.05r dmax the dust density became uniform, with δ = 1.9 outside 0.05r dmax , then the total minimum dust mass would only be about a factor of 3 larger than for the totally uniform disk case -that is, about 10 −4 M ⊙ fallen to less than 1000 km s −1 , perhaps suggesting some stratification by element. At 554 d and earlier epochs, the ring flux was negligible. At the later epochs, the ring inclination is 6 • (i.e., close to face-on). The fractional contribution of the ring to the total observed flux never exceeded 20%. Indeed, were it not for the dust disk, it is unlikely that the relatively weak ring emission would have been detected since it would have been swamped by the emission from the gas sphere (see Fig. 16). We therefore regard the ring component as an interesting but minor effect.
We conclude from the line-profile analysis that, in the 461-925 d period, the mass of the dust disk and its radial extent can be small enough to be consistent with an origin as newly-condensed ejecta dust. In §3.4.4.2 we shall show that the same dust disk can account for the MIR continua.
Later Phase IR Excess and Ejecta Dust Condensation
We have argued that, up to at least 281 d, the warm component could be fully accounted for in terms of an IR echo from a CDS. By 500 d the warm component luminosity exceeded the CDS contribution by a factor of ∼ 20. This "late IR excess", together with the steady shift of the peak emission of the SED to longer wavelengths in the period ∼ 250 − 1200 d (Figs. 3, 5) suggests the appearance of an additional population of warm, but cooling dust. What is the location, distribution and heating mechanism of this dust? Such IR emission can arise from (i) heating, by a number of possible mechanisms (see below), of new dust formed either in the ejecta, or in a late surge of dust growth within the CDS or (ii) heating of pre-existing circumstellar dust by the early-time SN luminosity (a conventional IR echo), or possibly the forward shock travelling into undisturbed CSM beyond the CDS. We can dismiss immediately hypothesis (ii):
(a) An IR echo does not naturally produce the long delay (at least ∼ 300 d) before the commencement of the warm component excess flux rise. To account for this within an IR echo scenario, we would have to invoke an ad hoc asymmetry in the CSM.
(b) IR-echo or forward-shock heating would not account for the observed late-time red wing suppression in the spectral line profiles discussed in §3.4.4.1. In contrast, red-wing attenuation can easily be produced by dust formation in the ejecta. New CDS dust could conceivably also have produced such an effect provided that the dust did not lie completely outside the optical/IR emission zone.
(c) Pre-existing CSM dust, whether heated by the SN luminosity or a forward shock, would not produce steepening of the optical-NIR decline rate simultaneously with the appearance of the late IR excess. In contrast, new dust in the ejecta or CDS can be indicated by optical-NIR steepening. In Fig. 4, after ∼470 d the V RIJ light curves show weak evidence of steepening, providing a minor additional argument against a CSM IR echo origin for the later warm component, although the errors on the latest points are large. We also note that an alternative explanation for such steepening could be a faster-than-expected decrease in the γ-ray absorption relative to that invoked by Li et al. (1993) for the radioactive energy deposition in SN 1987A (Fig. 4).
The above points, especially (a) and (b), leaves us with hypothesis (i) viz. that the late IR excess is due to emission from new dust formed in either the ejecta or CDS. The later-era appearance of the red wing suppression coincides roughly with the emergence of the late IR excess, suggesting that the same dust could have been responsible for both the optical/NIR attenuation effects and the rise of the MIR emission. That this dust formed in the ejecta rather than the CDS is indicated by the fact that the extent of the attenuating dust derived from late-time line-profile analysis was comparable to that of the MIR-emitting dust as derived from blackbody analysis; for example, at 859 d the blackbody radius is 3.3 × 10 15 cm compared with a profile-derived radius of (4.4 ± 0.2) × 10 15 cm at 895 d. Moreover, the line-profile analysis demonstrates that the dust was of high opacity in the optical/NIR, and distributed as a near face-on disk lying concentrically with the SN center of mass.
Attenuation by a CDS could not have produced the late-time line profile behavior.
In view of the above points, we shall now give detailed consideration to the scenario where the warm component emission originated in newly-formed ejecta dust. Such dust could be heated by a number of mechanisms including radioactive decay, ionization freeze-out effect, embedded pulsar or reverse-shock radiation. In the case of SN 2004dj we can rule out radioactivity as the principal heating mechanism of any putative new ejecta dust. Inspection of Table 7 shows that, as early as 500 d, the warm component flux exceeded the total deposited radioactive luminosity by a factor of 2 growing to a factor of ∼ 450 by 1393 d. Indeed, by the latest epoch, the warm component luminosity was still as high as ∼ 4 × 10 38 erg s −1 . We also rule out the freeze-out effect as the main energy source (Clayton et al. 1992;Fransson & Kozma 1993) since, at ∼ 1400 d, its contribution would only be ∼ 5 × 10 36 ergs.
Power input from an embedded pulsar, via a pulsar wind nebula, is another possibility Chevalier & Fransson 1992 was covered by these spectra. We co-added the spectra at the two epochs and searched for the stronger [O III] 5007Å component. A broad, low S/N emission feature was detected with a dereddened luminosity of (0.8 ± 0.2) × 10 36 erg s −1 . This is about half the luminosity predicted by Chevalier & Fransson. A serious difficulty is that the redshift-corrected position of the feature lies about +400 km s −1 from where it would be if due to an [O III] 5007Å line subject to the same dust attenuation deduced in other lines ( §3.4.4.1). Even without dust attenuation, the feature would still be +200 km s −1 too far to the red. This, together with the low S/N, leads us to conclude that there is no persuasive evidence for the presence of pulsar-driven high-ionization features in the latest spectra of SN 2004dj. This leaves us with reverse-shock heating of the ejecta following the ejecta-CSM collision as the most promising mechanism for the late-time energy source
To test further the hypothesis that the same dust was responsible for the late-time line-profile red-wing suppression and the MIR emission we modelled the IR emission over a range of late-time epochs using the same dust-disk configuration as was employed in the line-profile analysis. The emission was derived following a similar procedure to that used in the isothermal dust model by Meikle et al. (2007) and Kotak et al. (2009). The resulting flux was then added to the other continuum components and the net model compared with the observed continua.
We considered the thermal radiation from a warm, isothermal, face-on disk of dust located symmetrically about the SN center of mass. We shall refer to this as the IDDM (isothermal dust-disk model). The disk radius is r dmax and, as in §3.4.4.1, the dust number density declines as r −δ d where δ is set as 0 or 1.9. As mentioned above the disk thickness was < 20% of the disk diameter and so we judged that a thin-disk treatment would provide an adequate means of estimating the flux from the disk (i.e., edge effects are ignored). The observed IDDM flux, dF λ (t, r d ), at wavelength λ and time t from an elemental ring lying between radii r d and r d + dr d is:
dF λ (t, r d ) = 2πr d dr d D −2 B λ (t)(1 − exp(−τ λ (t, r d ))) (5)
where D is the distance of the SN, B λ (t) is the Planck function, and τ λ (t, r d ) is the optical depth perpendicularly through the disk. The total flux is then found by integrating from r d = 0 to r d = r dmax . The dust mass is obtained from equation (4). Amorphous carbon grains were assumed (silicate grains are discussed below). As explained before, during the 89 − 281 d period ejecta dust formation was unlikely and in any case the early IR excess was explainable as emission from CDS dust. Therefore the IDDM was introduced at 500 d and used at all subsequent epochs. By 652 d the CDS IR emission was negligible and so was not included in the modelling of this or subsequent phases. We first describe the results with δ = 0 in the IDDM. The matches together with the observations are shown in Figs. 18 and 19. The overall model parameters are listed in Table 8.
The three MIR observation epochs 500 d, 652 d and 859 d lay within the timespan of the line-profile analysis. We therefore imposed the constraint that the disk radius had to be consistent with those values derived from the line profiles -that is, r dmax ≈ 5 × 10 15 cm and r dth = v dth × t where v dth = +112 km s −1 , with k (and therefore M d ) being constrained by the demand that the disk have a high optical depth in the optical/NIR region ( §3.4.4.1). In practice, matching the IDDM to the MIR continuum demanded optical depths which were higher than the minimum values obtained from the line-profile analysis. The disk parameters for the specific MIR epochs were set by linear interpolation of the line-profile-derived values for r dmax and r dth . The k parameter could take values at or above those set in the line-profile analysis. Only T d was a completely free IDDM parameter. By 500 d the MIR continuum up to ∼20 µm was overwhelmingly due to the warm dust disk, with the CDS component yielding no more than ∼8% of the total flux at any wavelength. For λ 20 µm, the IS IR echo dominated. Also from 500 d onwards the hot continuum and fb-ff contributions to the MIR were negligible. In other words, from 500 d onwards, only the dusty disk and IS IR echo made significant contributions to the total MIR continuum. Consequently, and in order to show the MIR behavior in more detail, the optical/NIR region is not shown in the plots for 652-1393 d (Fig. 19).
In spite of the line-profile constraints on r dmax and r dth , for epochs 500 d, 652 d, and 859 d we were nevertheless able to obtain fair matches to the observed continuum (Figs. 18, 19). It was found that the matches required the dust to be optically thick (τ > 1) in the optical/NIR region, but optically thin in the MIR region. A blackbody spectrum matched to the shorter MIR wavelengths overproduced the flux at longer wavelengths. This is illustrated in Fig. 18 (500 d) and Fig. 19 as dotted cyan lines. This restriction, together with the fixed values for r dmax and r dth allowed us to obtain specific values, not limits, for T d , τ and M d . At 500 d the ejecta dust mass was (0.22 ± 0.02) × 10 −4 M ⊙ , with T d = 650 ± 15 K. The optical depths through the disk were ∼10 in the V band and 0.45 ± 0.05 at 10 µm. By 859 d the dust mass had increased to (0.33 ± 0.05) × 10 −4 M ⊙ , while T d fell to 570 ± 15 K. A particular value of this 500-859 d study is that it demonstrates that the same dust-disk parameters can account for the line profiles and the MIR continuum. This adds considerable weight to our contention that the two disks are one and the same. Moreover specific values, not limits, for the dust masses and temperatures were determined for the 500-859 d period.
As the SN aged between 500 d and 996 d it was found that the IDDM steadily approached the blackbody case. Moreover, there were no line profiles available after 925 d. Linear extrapolation to 996 d of the line-profilederived values for r dmax gave r dmax ∼ (4 ± 1) × 10 15 cm. The model match yielded dust with a high optical depth in the optical/NIR but with τ ∼ 1 in the MIR. The dust mass was M d ≈ 0.5×10 −4 M ⊙ and T d = 520±30 K.
By epochs 1207 d and 1393 d we found that the MIR continuum was best reproduced by allowing a continuing increase in the IDDM optical depth such that the disk was optically thick at all observed wavelengths (see Table 8). Indeed, given the uncertainties in the observed fluxes, we cannot rule out a totally opaque disk at all wavelengths covered. Consequently, the IDDM could provide only dust mass lower limits of 1.0 × 10 −4 M ⊙ at 1207 d and 1.5 × 10 −4 M ⊙ at 1393 d. The 1207 d and 1393 d plots shown in Fig. 19 are with r dmax and T d set at the limiting values (Table 8).
In Table 9 we show the luminosities of the IDDM (δ = 0) components compared with the radioactive deposition power of 0.0095 M ⊙ of 56 Ni, including 56 Ni decay. The IS IR echo component is excluded since it is powered primarily by the SN peak luminosity. The luminosity of the thin disk, L IDDM , is approximated by:
L IDDM ≈ 2πR 2 d πB ν (1 − exp(−2τ ν )).(6)
As already indicated by the BLC analysis ( §3.2) the post-30 d evolution of SN 2004dj falls into three phases. At 89 d, (about half-way down the plateau-edge) the luminosity is still dominated by the shock-ionized ejecta, with radioactive decay contributing a small proportion of the total. There is then a short period just after the end of the plateau when radioactive decay deposition dominated the luminosity. This is supported by the fact that at 129 d, the radioactive luminosity, L rad , is highly similar to the total continuum model luminosity, L total . At the nebular phases of 251 d and 281 d, the small excess in L rad , relative to L total , presumably went into powering the line emission which is not included in the model. By 500 d, in spite of the exclusion of much of the line luminosity from the model, L total is ×2.5L rad , rising to ×200L rad by 1393 d. As argued above, the most plausible additional power source available at this stage is a reverse shock.
Model matches were also carried out with δ = 1.9 in the IDDM component. Similar dust parameters were found to those obtained for δ = 0. The only significant difference was that the dust masses were ×3 larger, assuming a uniform density distribution within the inner 0.05r dmax (cf. §3.4.4.1). These larger values are due to the growing proportion of dust mass concentrated in the optically-thick region of the disk.
We also considered a dusty disk of warm silicate grains, with the radii and thickness determined by the line-profile analysis as before. The demand that the disk should be of optical depth in the optical/NIR up to its edge meant that the replacement of amorphous carbon dust with silicate dust had no effect on these dimensions. The problem with silicate dust continuum matching is the absence of the 8 − 14 µm silicate feature in the observed continua during the period when the dust was optically thin in the MIR, up to 996 d. Attempts to suppress the feature in the IDDM by increasing the dust mass and hence the optical depth yielded a continuum that was too bright. Only for 1207 d and 1393 d was silicate dust able to reproduce the observed continuum. This is not surprising since, by these two epochs, the warm ejecta dust was close to being opaque over the MIR range observed. Between 500 d and 996 d the proportion of silicate grains by mass could have been no more than 20%, and was usually significantly less than this. Indeed, the data were always consistent with there being no silicate grains at all. We also note the absence of the SiO feature at 7.5 − 9.3 µm. SiO formation is a necessary step on the way to silicate grains (Todini & Ferrara 2001;Nozawa et al. 2003).
The points made in the previous paragraph argue against a substantial amount of silicate dust in the ejecta of SN 2004dj. The absence of silicate grains is consistent with the presence of strong CO fundamental and first overtone emission in the period ∼100 d to 300 − 500 d. A high C/O ratio in the ejecta could result in most of the oxygen being absorbed as CO leaving behind an excess of carbon to provide carbon grains, but little oxygen to provide silicate grains. However, in the SN environment, the net grain population can be affected by factors in addition to the C/O ratio. These include molecule destruction by high-energy electrons, charge transfer reactions and ejecta density (Liu & Dalgarno 1996;Nozawa et al. 2003;Deneault et al. 2006). Nevertheless, we conclude that the dust grains in SN 2004dj were predominantly composed of non-silicate material.
The success in reproducing the MIR continua using the same dust disk as was invoked to explain contemporary line profiles tends to support the curious result from §3.4.4.1 that, rather than expanding, the radius of the dust disk actually shrunk by a small amount. Indeed, if we include the disk radii for 1207 d and 1393 d derived from the IDDM matches (Table 8, col. 7) we find a shrinkage of 27% since 500 d. In SN 2004et (Kotak et al. 2009) it was found that the dust radius remained roughly constant. The explanation offered was that the dust was contained within an optically-thin cloud of optically-thick, pressure-confined clumps. But the availability of late-time optical/NIR spectra for SN 2004dj and the profile modelling presented here shows that a clumping explanation for the non-expanding dust-disk radius would not work. We note that, during the 500-859 d period in SN 2004dj, the product of the blue-wing, half-maximum (BHM) velocities and epoch for the line profiles, R BHM , yields a roughly constant value of ∼ 4 × 10 15 cm (see Table 6) -similar to that obtained for the dust-disk radius. This coincidence may imply that the extent of both the dust and the bulk of the ejecta gas emission was physically constrained within a radius of ∼ 5 × 10 15 cm.
We suggest that the apparent shrinkage of the dustdisk radius may have been due to the presence and inward motion of the reverse shock such that the shock position defined the disk radius. Dust formation could have continued within the disk with the dust taking part in the overall ejecta outflow, but as the ejecta passed through the reverse shock it would have been destroyed. There would have been a net increase in dust mass with time, as observed, if the dust growth rate within the disk exceeded the destruction rate. A possible difficulty with this scenario is that the dust mass appeared to continue to grow right up to the final epoch at 1393 d. This is rather later than dust formation studies suggest. Todini & Ferrara (2001) find that all grain condensation would be complete by 800 days. Moreover, both Todini & Ferrara (2001) and Nozawa et al. (2003) find carbon dust condensation is complete in not much more than one year. An alternative explanation, therefore, might be that the dust mass in SN 2004dj did not increase after ∼400 d, but rather that the density gradient of the outer region of the disk was actually steeper than r −2 . Thus, as the ejecta expanded, the dust density at a fixed location in the outer region would have grown, yielding an apparently higher total dust mass -i.e., more dust emerged from the optically-thick inner regions.
Radiation from the reverse shock could have been primarily responsible for heating the dust. As more and more ejecta passed through, the shock and its radiation would have weakened causing the dust to cool, as observed. The reverse shock may also have been responsible for the approximate constancy of the radius of the line emitting gas sphere. Further examination of the reverse-shocked dust-disk hypothesis is beyond the scope of this paper. We note that in a recent observational study of Cassiopeia A Delaney et al. (2010) deduce a flattened ejecta distribution or "thick disk" containing all the ejecta structures. They also deduce the occurrence of a roughly spherical reverse shock.
Intrinsic axial asymmetry in the form of a bipolar jet biased towards the observer has been invoked by Chugai et al. (2005) to account for the Hα profile in SN 2004dj up to about 1 year. We note that the jet makes an angle of only ∼ 15 • to the normal to the dust disk plane derived from our line-profile analysis, perhaps indicating a physical connection. However, line-profile asymmetry in the earlier (pre-∼1 year) nebular spectra of other CCSNe also tend to be blue-biased. This implies that, in general, an intrinsic bias towards the observer cannot be the explanation for such line blueshifts (cf. Milisavljevic et al. 2010). At later (post-∼1 year) nebular epochs, line blueshifts are also often seen (Lucy et al. 1989;Spyromilio et al. 1990;Turatto et al. 1993;Fesen et al. 1999;Gerardy et al. 2000;Leonard et al. 2000;Fassia et al. 2002;Elmhamdi et al. 2003;Pozzo et al. 2004) and these are usually attributed to attenuation by newly-formed dust in the ejecta or CDS. Our dust-disk model invokes this scenario. This has the advantage of explaining the red-wing suppression without invoking an intrinsic observer-biased axial asymmetry in the SN. Moreover, our model uses the same dust disk to simultaneously account for the line-profile attenuation and the MIR emission. We consider it unlikely that the Chugai et al. model can provide a superior alternative explanation for the line profiles presented and analysed in §3.4.4.1. Further examination of the relationship between our model and that of Chugai et al. is beyond the scope of this paper.
We conclude that dust formation in the ejecta of SN 2004dj had commenced, and may even have been completed, by 500 d with a near-face-on, disk-like distribution. This dust was responsible both for the late-time line-profile red-wing suppression and the bulk of the MIR luminosity up to ∼ 20µm. The main source of dust heating was probably reverse-shock radiation. Assuming δ = 0, the dust mass was at least 10 −4 M ⊙ . For δ = 1.9, the lower limit rises to about 3 × 10 −4 M ⊙ . The value of δ is poorly constrained. We reject silicates as the grain material.
CONCLUSIONS
We have presented optical, NIR, and MIR observations of the Type IIP SN 2004dj. The combination of wavelength and temporal coverage achieved makes this SN one of the most closely studied of such events. In the present work we have analyzed the SN continuum over a period spanning 89-1393 d, augmented by a line-profile analysis over 461-925 d. Our conclusions are as follows.
(1) A mass of 0.0095 ± 0.002 M ⊙ of 56 Ni was ejected from SN 2004dj, which is less than that reported by other authors. The period during which the radioactive tail dominated the bolometric light curve lasted for an unusually short period of only ∼35 d. Subsequently, a different energy source dominated; we suggest reverseshock heating.
(2) At early times the optical/NIR ("hot") part of the continuum provided most of the SN luminosity. This emission is attributed to hot, optically-thick ejecta gas. At later epochs the optical/NIR luminosity was increasingly due to nebular emission, and formed only a minor, ultimately negligible, proportion of the BLC.
(3) At both early and late times, the long-wave portion of the MIR continuum (the "cold" component) was primarily due to an IS IR echo. Free-free radiation made only a minor contribution. As originally suggested by Bode & Evans (1980), the analysis of SN IR echoes may provide a useful way of studying IS dust in nearby galaxies.
Subsequent to the submission of this paper, Szalai et al. (2011) reported their findings on SN 2004dj, making use of some of the data presented in this work. They dismiss the IS IR echo, and argue against pre-existing dust since the estimated extinction of SN 2004dj was lower than that of other SNe. We do not concur. We have shown that an extinction of only Av=0.02 is all that is required to account for an adequate IR echo. Szalai et al. (2011) also argue that the early UV/X-ray flash would create a dust-free cavity of up to 10 17 cm and that OB-stars would have expelled the ISM/dust from the cluster. In fact most of the IS IR echo longward of ∼ 10 µm comes from dust lying at considerably more than 1 parsec. We therefore strongly favour a scenario whereby the long-wave component was primarily due to an IS IR echo.
(4) The early-time NIR/MIR ("warm") component was probably due to thermal emission from non-silicate dust formed in the CDS. The CDS dust growth began at about 50 d, reached 90% of maximum by 165 d, and approached a maximum of 0.33 × 10 −5 M ⊙ ; the dust mass produced in this way was small. Heating of the dust by the contemporary optical-NIR BLC completely accounts for the strength and evolution of the early-time warm component. Szalai et al. (2011) did not present contemporary optical or NIR data and consequently did not identify the early warm component.
(5) The late-time warm component was dominated by the luminosity of newly-formed, non-silicate dust in the ejecta. The dust growth commenced at some time between 281 d and 461 d. The same dust was responsible for the late-time red-wing attenuation of optical and NIR spectral line profiles. The dust was distributed as a nearly face-on disk, within a spherical cloud of emitting gas. During 500-996 d the disk was effectively opaque in the optical/NIR region, but was optically thin at longer wavelengths. The dust mass appeared to grow during this period, attaining (0.5 ± 0.1) × 10 −4 M ⊙ by 996 d for a uniform density disk, or a few times more than this for an r −1.9 gradient outside 0.05r dmax and uniform within. However, it may be that the dust mass "growth" was really due to the emergence of previously formed (i.e., pre-500 d) dust from optically-thick regions of a disk with an even steeper density gradient. For the latest two epochs (1207 d and 1393 d) the dust was optically thick at all wavelengths and only lower limits could be obtained for a given gradient -for example, > 10 −4 M ⊙ for a flat gradient and a factor of 3 higher limit for an r −1.9 gradient outside 0.05r dmax and uniform within. This is broadly in agreement with Szalai et al. (2011). For a smooth distribution, they found a dust mass of ∼ 10 −5 − 10 −4 M ⊙ though, with clumping, up to ∼ 10 −3 M ⊙ would also be possible.
(6) Rather than expanding, the dust-disk radius appeared to slowly shrink. This may have been due to the dust extent being confined by the reverse shock, which also heated the grains radiatively.
(7) While the latest epochs provide only lower limits to the mass of dust produced by SN 2004dj, these limits are at least a factor of 100 below the 0.1 M ⊙ of grains per SN required to account for the dust observed at high redshifts. Moreover, measurements as late as 996 d yield actual dust masses of only ∼ 10 −4 M ⊙ . While not completely ruling out the possibility that typical CCSN ejecta are major contributors to cosmic dust production, this study does suggest that, at least for SN 2004dj, the dust-mass production was small. (2) 0.28(5) 0.21(4) 0.14(3) 0.08(2) 0.024 (7) 0.013(4) 0.011 (3) Note. - * We assume an explosion date of 2004 Jul. 10.0 (MJD=53196.0). * * All measurements were made in a ∼ 3. ′′ 7 radius circular aperture, with annular sky measured between ×1.5 and ×2.2 the aperture radius. The aperture was centered on the WCS co-ordinates of the SN. Statistical uncertainties in the last one or two significant figures are shown in brackets. The fluxes have not been corrected for reddening, nor for the contribution of the S96 cluster. The last line shows the estimated contribution to the total flux by S96 (see §3.1). The exposure times in Column 4 are per band. It can be seen that some same-wavelength observation epochs have negligible temporal spacing as well as differing exposure times (viz. 840/3 d, 859 d, 987 d, 996 d, 1201 d, 1227/31/32, 1244/54, 1367/72, 1374/5). This was due to the impact of Spitzer scheduling constraints on the MISC and SEEDS programs. . * * All measurements were made in a 3.7" radius circular aperture, with annular sky measured between ×1.5 and ×2.2 the aperture radius.
The aperture was centered on the centroid of the SN. The error on the last one or two figures is given in parentheses. Errors shown include uncertainties in the magnitudes of the four 2MASS comparison field stars. The fluxes have not been corrected for reddening, nor for the contribution of the S96 cluster (see §3.1). The last line shows magnitudes of S96 measured from 2MASS images (Skrutskie et al. 1997). 1 Estimated by extrapolation of JHKs SEDs of calibration field stars to Z band. (Vinkó et al. 2006) Lick 3m: Shane 3m telescope at Lick Observatory (Leonard et al. 2006) LAT Scorpio: Scorpio on the 6m large azimuthal telescope (Chugai et al. 2005;Chugai 2006 Vinkó et al. (2006). In the final column is shown L rad , the radioactive deposition power corresponding to the ejection of 0.0095 M ⊙ of 56 Ni, scaled from the SN 1987A case specified by Li et al. (1993) & Timmes et al. (1996. (6000) 100 ------106 275 6000 3420 (6000) 86 ------129 180 6000 3420 (6000) 56 ------251 50 6000 3080 5000 36 ------281 50 5300 2980 5000 24 ------500 9.0 6000 2470 5000 1.8 5. ). An example of uncertainties in the optical depths is shown in col. 10 for 10 µm. For the hot blackbody at 652 d and later, the temperature was fixed at 10,000 K and the velocity obtained by extrapolation, and so these parameters are shown in brackets. For the fb-ff modelling, the hydrogen temperature and electron density (cols. 5 & 6) for 251-1393 d were derived from the observed hydrogen velocities for SN 2004dj (col. 4) in conjunction with the late-time SN 1987A study of Kozma & Fransson (1998) (see §3.4.3.2). For 89-129 d, the hydrogen temperature is shown in brackets as it is a rough estimate (see §3.4.3.2). Cols. 9-11 give the optical depths perpendicularly through the dust disk at the wavelengths indicated. The matches also included a CDS IR echo at epochs 89 − 500 d and an IS IR echo at all epochs (see §3.4.2 and §3.4.3.1). - * L IDDM is the luminosity of the warm, dusty disk model. For the hot blackbody (col. 2) at 652 d and later, the temperature was fixed at 10,000 K and the velocity obtained by extrapolation, and so the luminosities derived from these parameters are shown in brackets. The post-652 d contribution of the hot blackbody to the total luminosity is negligible. In col. 7, L rad is the radioactive deposition corresponding to the ejection of 0.0095 M ⊙ of 56 Ni, scaled from the SN 1987A case described by Li et al. (1993) & Timmes et al. (1996. No IS IR echo luminosities are shown since these were predominantly powered by the peak luminosity of the SN prior to the earliest epoch of observation. In the wavelength region (λ > 15 µm) where the cold component makes a significant contribution (> 20%) to the total flux, the IS IR echo model described in the text maintained a near-constant luminosity (15-150 µm) of ∼ 2.0 × 10 38 erg s −1 between 89 d and 1393 d. The supernova is clearly brighter on 621 d. This is due to the epoch being close to the peak of the thermal emission from the ejecta dust at this wavelength (see Fig. 2). The fields are about 2 arcmin across (∼ 1.8 kpc at the distance of the supernova). North is ∼ 30 • clockwise from the upward vertical (Table 1). They show little significant change during this period, with a mean flux of 0.32 ± 0.02 mJy. This is consistent with there having been no decline since 1372 d. The MIR flux of S96 alone has not been measured and so the light curves are uncorrected for S96. An estimate of the S96 contribution is given in Table 1 and is discussed in §3.1. Vinkó et al. (2006Vinkó et al. ( , 2009) (solid dots). In addition the V and R points of Zhang et al. (2006) (open circles) around the end of the plateau have been added to enhance the detail of this phase. For all data sets the epoch has been adjusted to our explosion epoch of MJD=53196.0. The NIR data are from the present work. Also shown is the 3.6 µm light curve from the present work. The vertical dashed line is at the epoch of the earliest 3.6 µm point and indicates the corresponding phase in the other light curves. S96 fluxes at BV RI and at JHKs have been subtracted from the data. Also shown for comparison (labelled "Rad.") is the temporal evolution of the radioactive energy deposition as specified by Li et al. (1993) & Timmes et al. (1996 for SN 1987A with the addition of the early-time contribution of 56 Ni assuming complete absorption. Table 3). Also shown are the fluxes obtained through aperture photometry of the IRAC, PUI and MIPS images, interpolated to the epochs of the spectra. These photometric data are represented by round or square points respectively on alternate epochs. All plots except the latest have been displaced vertically for clarity, with the zero flux levels indicated by the horizontal dotted lines on the left. The stronger features are identified, with the fiducials redshifted to the SN rest frame. The wavelengths are labelled in microns. Vinkó et al. (2006) and the 895 d spectrum has already been presented in Vinkó et al. (2009). The spectra have not been dereddened. In addition, the S96 flux has not been subtracted and this accounts for the relatively large continuum fluxes at epochs 895 d and 925 d. No correction has been applied for the heliocentric velocity of the SN. All plots except the latest have been displaced vertically for clarity, with the zero flux levels indicated by the horizontal dotted lines on the left. Note that the zero flux shown at 23 mJy corresponds to the 895 d spectrum. Also, the spectra have been flux-scaled by the amounts shown in brackets. Locations of spectral lines of interest are identified, with the fiducials redshifted to the SN rest frame. The wavelengths are labelled in microns. No correction has been applied for the heliocentric velocity of the SN. All plots except the latest have been displaced vertically for clarity, with the zero flux levels indicated by the horizontal dotted lines on the left. Also, the spectra have been flux-scaled by the amounts shown in brackets. The stronger features are identified, with the fiducials redshifted to the SN rest frame. The wavelengths are labelled in microns. Li et al. (1993) & Timmes et al. (1996. The errors in the fluxes and luminosities are primarily due to uncertainties in the absolute fluxing and the levels of the underlying continua. Not included in the luminosity errors are systematic uncertainties in the distance and extinction (see §1.1). Vinkó et al. (2006), while the 95 d and 100 d spectra are from, respectively, Leonard et al. (2006) and Chugai et al. (2005). All other spectra are from the present work. The horizontal axes are in terms of equivalent velocity with respect to the SN center of mass, which has a velocity of +221 km s −1 relative to the Earth. The (Vinkó et al. 2009). The latter data were acquired at 800 d. In the model, the fluxes longward of ∼1 µm are dominated by the cold component, with a temperature of 3500 K. The temperature of the hotter component, 50,000 K, is not intended to have a particular physical meaning, but simply serves as a means of representing and extrapolating the optical SED. Table 7, col. 11.). The phases of the Vinkó et al. and Zhang et al. BLCs have been shifted to an explosion date of MJD=53196.0, the luminosities scaled to 3.13 Mpc, and the extinction to A V = 0.31 mag. These BLCs are compared with the radioactive deposition power for SN 1987A as specified by Li et al. (1993) & Timmes et al. (1996. Two scaled cases are shown viz. 0.0095 M ⊙ (solid line) and 0.016 M ⊙ (dashed line) of 56 Ni. We also show the total radioactive luminosity (red) for the 0.0095 M ⊙ case. In addition (dotted line) are the actual UV-augmented bolometric light curves of SN 1987A (Pun et al. 1995) scaled to the case of an initial 56 Ni mass of 0.0095 M ⊙ . The divergence between the two SN 1987A datasets at late times is ascribed to differences in the IR flux measurements between the two observatories at, respectively, CTIO (lower curve) and ESO (upper curve). The cyan line shows the total model spectrum. A hot component was also included in the matches but had a negligible effect on the warm-cold matches to the MIR continua. Consequently, and in order to show the MIR behavior in more detail, the optical/NIR region is not shown. Estimates of the hot component contributions are given in Table 7. The model profiles are derived from a homologously-expanding, emitting gas sphere attenuated by an embedded, opaque dust distribution. A minor additional contribution to the model is provided by a thin, expanding concentric ring of emitting gas oriented with the normal to the ring plane at 6 • to the line-of-sight. The gas-dust configurations are illustrated in the LH panels which show sections through the SN system in the plane defined by the thin ring axis and the line-of-sight. The gas sphere is shaded green and the attenuating dust shaded magenta. The thin ring, viewed edge-on in the figure, is represented by two large cyan dots. The region of line emission occluded by the dust is shown by the black shading. The expansion velocities are indicated by the axes. The final model profiles are obtained by combination of the attenuated sphere and unattenuated ring spectra. The resulting spectrum is then smoothed to the spectral resolution. The RH panels show the resulting model line profiles (solid red), plotted in velocity space, compared with the observed spectrum (blue). Also shown (dotted red) is the line profile that would result in the absence of dust attenuation. In this illustration we compare the model profiles with the observed 895 d [O I] 6300Å line. In this case, the gas sphere emissivity declines as r −1.9 .
5.
The upper panels illustrate the case where the dust is distributed as an opaque, concentric sphere. Two sizes of sphere are considered. Profiles a and b (upper RH panel) correspond, respectively, to the large and small dust spheres (upper LH panel). It can be seen that neither reproduce the observed line profile. The lower panels illustrate the case where the dust is composed of amorphous carbon and is distributed as a uniform density, nearly face-on concentric disk with τ = 18 perpendicularly through the disk plane. This optical depth is the value required for the model to simultaneously match the line profile and the contemporaneous MIR continuum (see Fig. 19). While the disk is shown as being coplanar with the ring, for ease of computation the disk is taken to be exactly face-on. Shifting the disk tilt to match that of the ring would have had only a small effect on the model profile. No correction is made for minor edge effects at the disk's outer limit (radius). Also shown (lower RH panel) are the intrinsic line profiles from the attenuated gas sphere (green solid line) and the thin ring (cyan solid line). The thin ring emission is not affected by the dust. For clarity, both of these intrinsic profile plots have been scaled downward in flux by the same amount relative to the total profile. It can be seen that the final dust-disk model profile provides an excellent match to the data. . The velocities of the observed spectra have been shifted by -221 km s −1 to match the center-of-mass rest frame of the SN. The latest two epochs also include a small contribution from a ring of emission. Also, the velocity scale of these two epochs has been expanded to provide more detail. The matches incorporate dust disk masses obtained by interpolating to the profile epochs the values obtained from the MIR continuum modelling. Between 554 d and 925 d the dust mass increased from 0.25 × 10 −4 M ⊙ to 0.4 × 10 −4 M ⊙ . The radius of the dust disk decreased slightly from 5.3 × 10 15 cm to 4.6 × 10 15 cm. The disk half-thickness expanded from 0.45 × 10 15 cm to 0.90 × 10 15 cm. For each profile the optical depth had a value in the range 3 < τ < 19 (i.e. the disk was always of high optical depth). , free-bound (dashed blue), free-free (dashed red), CDS IR echo (green), IS IR echo (solid red) and total flux (solid cyan). The observed spectra and contemporary photometry (dots) of SN 2004dj are plotted in black. 500 d: As for earlier epochs but with the addition of IR emission from a warm disk of dust (magenta). The dotted cyan line represents the model continuum when the IDDM is replaced with a blackbody adjusted to match the shortwave region of the MIR continuum. At this epoch the ff and fb fluxes are too weak to appear on the plot. -Model continua at epochs 652 − 1393 d, comprising IR flux from a warm disk of dust (magenta), IS IR echo (red) and total flux (solid cyan). The dotted cyan line represents the model continuum when the IDDM is replaced with a blackbody adjusted to match the shortwave region of the MIR continuum. By 1393 d the blackbody has fully merged with the IDDM; i.e., the model is effectively opaque over the whole observed wavelength range. The observed spectra and contemporary photometry (dots) of SN 2004dj are plotted in black. A hot blackbody plus free-bound and free-free continua were also included in the matches but these had a negligible effect on the MIR continua. Consequently, and in order to show the MIR behavior in more detail, the optical/NIR region is not shown. Estimates of the hot and fb-ff component contributions are given in Tables 8 and 9.
By 895 d and 925 d a second, weaker peak redshifted by ∼170 km s −1 had also appeared in the Hα and [O I] 6300Å profiles (Fig. 10). (The redshifted peak can also be seen in the weaker [O I] 6364Å component lying at ∼+3000 km s −1 in the [O I] 6300Å rest-frame plots.) The previous observations of these lines were at 467 d when the line luminosities were about a factor of 20 greater
(15-150 µm) maintained a near-constant luminosity of ∼ 2.0 × 10 38 erg s −1 between 89 d and 1393 d. The Bendo et al. dust column density map of NGC 2403 indicates 0.05 M ⊙ pc −2 at the position of SN 2004dj. From the above dust number density from the echo model and integrated over 100 pc we obtain just 0.003 M ⊙ pc −2 . The absorption opacities used by Bendo et al. are consistent, to within a factor of ∼ 2, with the opacity law used in the present work.
§3.4.4.2 we shall then test the hypothesis that the same dust distributions are responsible for the warm component. We modelled the line profiles of Hα, Paβ, [O I] 6300Å, [Fe II] 7155Å and [Fe II] 12567Å.
We thank J. Vinkó for providing us with digitized versions of his optical spectra. The work presented here is based on observations made with the Spitzer Space Telescope, the W. M. Keck Observatory, the William Herschel Telescope (WHT), and the 2.4 m Hiltner telescope of the MDM Observatory. The Spitzer Space Telescope is operated by the Jet Propulsion Laboratory, California Institute of Technology, under a contract with NASA. The W. M. Keck Observatory is operated as a scientific partnership among the California Institute of Technology, the University of California, and NASA; it was made possible by the generous financial support of the W. M. Keck Foundation. We wish to extend special gratitude to those of Hawaiian ancestry on whose sacred mountain we are privileged to be guests. The WHT is operated on the island of La Palma by the Isaac Newton Group in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias. Financial support for this research was provided by NASA through an award issued by JPL/Caltech (specifically grant number 1322321 in the case of A.V.F.). A.V.F. gratefully acknowledges additional support from NSF grant AST-0908886 and the TABASGO Foundation. P.A.H. was supported by NSF grants AST-1008962 and 0708855. S.M. acknowledges support from the Academy of Finland (project 8120503). J.S. is a Royal Swedish Academy of Sciences Research Fellow supported by a grant from the Knut and Alice Wallenberg Foundation. J.C.W gratefully acknowledges support from NSF grant AST-0707769. The Dark Cosmology Centre is funded by the Danish National Research Foundation.
a PID. 00226 Van Dyk (MISC); b PID. 00159 Kennicutt et al.(SINGS); c PID. 20256 Meikle et al.(MISC); d PID. 20321 Zaritsky; e PID. 30292 Meikle et al.(MISC); f PID. 30494 Sugerman et al.(SEEDS); g PID. 40010 Meixner et al.(SEEDS); h PID. 40619 Kotak et al.(MISC); i PID. 61002 Freedman et al. † Target ∼50% off edge of field. Total flux estimated by extrapolation. † † Contamination to north-east.
Note. -Line fluxes, F are in units of 10 −15 erg cm −2 s −1 while dereddened line luminosities, L, are in units of 10 38 erg s −1 . Figures in brackets give the error on the last one or more significant figures. The errors in the fluxes and luminosities are primarily due to uncertainties in the absolute fluxing and the levels of the underlying continua. Not included in the luminosity errors are systematic uncertainties in the distance and extinction (see §1.1). * Spectra from
R BHM is the product of v BHM and the elapsed time (epoch). * : Vinkó et al. (2006), † : Leonard et al. (2006), * * Chugai et al. (2005) (N IR): Indicates NIR lines Paβ and [Fe II] 12570Å. In col. 3, 895 d and 925 d, the velocities of the stronger, blueshifted peak and the weaker, redshifted peak are given (see §2.4 and §3.4.4.1). a : Based on blueshifted component only. b : Blueshifted component insufficiently resolved. Typical uncertainties in the period 89-554 days, in km s −1 , are: ±50 (BHM), ±50 (peak), ±100 (RHM), ±60 (HWHM). At 895 d and 925 d the uncertainties are smaller by a factor of ∼ 4. The last column, "Model", gives the maximum velocity derived from profile (blue wing) matches (δ = 0) -see §3.4.4.1.
Fig. 1 .
1-Sequence of images at 8 µm at 257 d (LHS), 621 d (middle) and 996 d (RHS). SN 2004dj is the point source at the center of each image.
Fig. 2 .
2-MIR light curves of SN 2004dj. They are uncorrected for reddening. For clarity, the plots have been shifted vertically by the amounts shown in brackets (mJy). Off to the right of the plot there are four additional 3.6 µm points spanning 1954 d to 2143 d
Fig. 3 .
3-Evolution of the MIR spectral energy distribution of SN 2004dj. The SEDs are uncorrected for reddening. For clarity, the individual SEDs have been scaled by the amounts shown in brackets. As inFig. 2, the fluxes from S96 have not been subtracted. The large peak at 4.5 µm present at 114 d and 257 d is due to the dominance of CO fundamental emission in this band during this period.
Fig. 4 .
4-Optical and NIR light curves of SN 2004dj. For clarity, they have been displaced vertically by the values shown in brackets. The optical light curves are from
Fig. 5 .
5-Evolution of the MIR spectra of SN 2004dj. The spectra have not been dereddened and the S96 flux has not been been subtracted. No correction has been applied for the heliocentric velocity of the SN. The 1207 d spectrum has a small exposure time and so is of relatively low S/N compared with those of 859 d and 1393 d (see
Fig. 6 .
6-Evolution of the optical spectra of SN 2004dj. The 128 d spectrum is from
Fig. 7 .
7-Evolution of the NIR spectra of SN 2004dj. The spectra have not been dereddened and the S96 flux has not been subtracted.
Fig. 8 .
8-Evolution of spectral line luminosities of SN 2004dj. The luminosities have been corrected for extinction. Also shown is the deposition power of 0.0095 M ⊙ of 56Ni, scaled from the SN 1987A case described by
Fig. 9 .
9-Evolution of spectral line velocities of SN 2004dj (shifted to the center-of-mass rest frame of the SN). The plots are as follows: triangles=red wings, half-width half-maximum, round dots=peak intensity, squares=blue wings, half-width half-maximum. On 895 d and 925 d, the red wing half-width half-maximum refers to the blueshifted component only. Typical uncertainties in the period 89-554 days, in km s −1 , are: ±100 (RHM), ±50 (peak), ±50 (BHM). At 895 d and 925 d the uncertainties are smaller by a factor of ∼ 4.
Fig. 10 .
10-Evolution of the spectral line profiles of SN 2004dj. The 89 d and 128 d spectra are from
LH panel shows Hα and Paβ profiles. The Paβ profiles are labelled "IR". The middle panel shows [O I] 6300Å ([O I] 6364Å can also be seen lying at ∼+3000 km s −1 in the [O I] 6300Å rest-frame plots). The RH panel shows [Fe II] 7155Å and [Fe II] 12567Å profiles. The [Fe II] 12567Å profiles are labelled "IR". All the profiles have been flux scaled and shifted vertically for clarity.
Fig. 11 .
11-Two-blackbody representation of the S96 SED. The solid and open dots represent the pre-explosion and post-explosion fluxes respectively
Fig. 12 .
12-BLCs ofVinkó et al. (2006) (open circles),Zhang et al. (2006) (open triangles) and the hot+warm luminosities obtained in the present work via blackbody continuum matching (solid squares) (see §3.3 and
Fig. 13 .
13-Comparison of blackbody continua with the optical-NIR-MIR photometric fluxes and spectra of SN 2004dj at six epochs spanning 89 − 500 d. The solid dots show contemporary photometry. The blue, green and red lines show, respectively, the hot, warm and cold blackbodies. The cyan line shows the total model spectrum.
Fig. 14 .
14-Comparison of blackbody continua with the MIR photometric fluxes and spectra of SN 2004dj at five epochs spanning 652 − 1393 d. The solid dots show contemporary photometry. The green and red lines show, respectively, the warm and cold blackbodies.
Fig. 15 .
15-CDS IR echo model compared with the 3.6 µm and 8.0 µm MIR excess light curves of SN 2004dj (LH panel), and the 89, 106 and 129 d MIR excess SEDs (RH panel). The grain growth began at 50 d with a growth timescale also of 50 d. Amorphous carbon grains were adopted of radius 0.2 µm. The final dust mass is 0.33 × 10 −5 M ⊙ .
Fig. 16 .
16-Illustration of the line-profile models.
Fig. 17 .
17-Individual dust-disk model matches (red) to observed line profiles (blue) of SN 2004dj for the case where δ = 0 (uniform dust density)
Fig. 18 .
18-89-281 d: Model continua comprising hot blackbody (solid blue)
Fig. 19 .
19Fig. 19.-Model continua at epochs 652 − 1393 d, comprising IR flux from a warm disk of dust (magenta), IS IR echo (red) and total flux (solid cyan). The dotted cyan line represents the model continuum when the IDDM is replaced with a blackbody adjusted to match the shortwave region of the MIR continuum. By 1393 d the blackbody has fully merged with the IDDM; i.e., the model is effectively opaque over the whole observed wavelength range. The observed spectra and contemporary photometry (dots) of SN 2004dj are plotted in black. A hot blackbody plus free-bound and free-free continua were also included in the matches but these had a negligible effect on the MIR continua. Consequently, and in order to show the MIR behavior in more detail, the optical/NIR region is not shown. Estimates of the hot and fb-ff component contributions are given in Tables 8 and 9.
Optical and Near-Infrared Photometry NIR imaging of SN 2004dj was obtained using LIRIS (Long-slit Intermediate Resolution Infrared Spectrograph) on the 4.2 m William Herschel Telescope (WHT), La Palma, and at an effectively single epoch (spanning two days) with the OSU-MDM IR Imager/Spectrograph on the 2.4 m Hiltner Telescope of the MDM Observatory, Arizona. The wavebands are Z (1.033 µm), J(1.250 µm), H (1.635 µm), and K s (2.15 µm). The LIRIS data were reduced using standard IRAF routines. 1we see a steady reddening of the MIR
SED with time. It is argued below ( §3.4.4.2) that this
effect also constitutes strong evidence of dust formation
and cooling in the SN ejecta. The large peak at 4.5 µm
at 114 d and 257 d is due to the aforementioned domi-
nance of CO fundamental emission in this band. As in
Table 1, neither Fig. 2 nor Fig. 3 have been corrected for
S96.
2.2.
Strong lines of H I, [Ni I], [Ni II], [Co II], and [Ne II] were also present during the first year, but by 500 d only [Ni II] 6.64 µm, [Ni I] 7.51 µm, and [Ne II] 12.81 µm were still relatively strong (see
Table 4 .
4The post-100 d spectra are plotted in Figs. 6 and 7. The earlier post-plateau spectra still exhibited pronounced P Cygni features in Hα, He I 5876Å + Na I D, He I 10830Å, He I 20581Å, and O I 7771Å + K I 7665/99Å. By 461/467 d the absorption components had largely vanished, with a broad-line emission spectrum now being observed. A few lines persisted to as late as the final optical spectroscopy epoch at 925 d. We examined in detail the evolution of the more isolated of these lines, specifically Hα, Paβ, [O I] 6300Å, [Fe II] 7155Å, and [Fe II] 12567Å. Table 5 lists the line luminosities (dereddened) versus epoch, together with the radioactive deposition power specified by
The R. Kurucz Stellar Atmospheres Atlas, 1993, ftp://ftp.stsci.edu/cdbs/grid/k93models/standards. contrast, from their earliest observation at ∼300 d, the [O I] and [Fe II] lines decline significantly more slowly than the radioactive rate. Moreover, by 895 d the summed luminosity of just the Hα, [O I] 6300Å, and [Fe II] 7155Å lines exceeds that of the radioactive input by ∼40%, rising to over 60% by 925 d. Thus, as with the optical light curves, we deduce the appearance of an additional source of energy, possibly earlier than 300 d.2
In
we show all the individual matches to the line profiles for 461-925 d. In general, good matches were achieved for epochs 554 d, 895 d, and 925 d. Poorer matches were obtained at 461 d and 467 d. On 895 d and 925 d the model did not reproduce the extended red wing seen in the [Fe II] 7155Å profile.
The model gas sphere expansion velocity remained at around 2400 km s −1 in hydrogen at all epochs. Similar velocities were obtained in [Fe II] 12567Å at 461 d and 554 d, and in [O I] 6300Å and [Fe II] 7155Å at 467 d. By 895 d and 925 d the expansion velocities in [O I] 6300Å and [Fe II] 7155Å had
). Chevalier & Fransson predicted that a distinctive feature of such a source would be the presence of certain high ionization lines. The earliest epoch they studied was at 1500 d. The nearest of our optical/NIR spectra to this epoch are those at 895 d and 925 d. Of the high ionization lines for which Chevalier & Fransson make luminosity predictions, only [O III] 4959/5007Å
TABLE 1
1Mid-IR Photometry of SN 2004dj.Flux (mJy) * *
TABLE 2
2Near-IR Photometry of SN 2004dj Note. -The 2005 May 11,12 images are from the TIFKAM IR camera on the 2.4m Hiltner Telescope of the MDM Observatory. All the other images are from the LIRIS IR imager/spectrograph on the 4.2m William Herschel Telescope of the Observatorio del Roque de los Muchachos, La Palma, * We assume an explosion date of 2004 Jul. 10.0 (MJD=53196.0)Date
MJD
Epoch *
texp
Magnitudes * *
(UT)
(d)
(s)
Z
J
H
Ks
2004 Nov 25 53334.0
138.0
J(70) H(135)
-
13.56(4)
13.24(4)
-
2005 Mar 17 53446.0
250.0
Z(60) J(60) H(60) K(50) 15.4(1) 1
14.35(4)
14.15(6)
14.13(5)
2005 May 11 53501.2
305.2
J(300) H(225) Ks(180)
-
14.88(1)
14.54(2)
14.42(4)
2005 May 12 53502.1
306.1
J(300) H(225)
-
14.88(1)
14.52(2)
-
2006 Jan 14
53749.9
553.9
J(675) H(1025) Ks(1215)
-
15.87(5)
15.19(5)
14.75(5)
Sandage 96
-
15.93(11) 15.70(12) 15.15(14)
TABLE 3
3Mid-IR Spectroscopy Log of SN 2004dj.Date
MJD
Epoch * texp Program
(UT)
(d)
(s)
2004 Oct 24
TABLE 4
4Optical and Near-Infrared Spectroscopy Log of SN 2004dj. Note. - * We adopt an explosion date of 2004 Jul. 10.0 (MJD=53196.0). Column 5 details: DDO 1.88m: 1.88m telescope of the David Dunlap ObservatoryDate
MJD
Epoch * Spectral range Telescope
(UT)
(d)
(µm)
2004 Oct 05
53284.9
88.4
0.420-0.766
DDO 1.88m
2004 Oct 13
53291
95
0.480-0.910
Lick 3m
2004 Oct 18
53296
100
0.370-0.750
LAT Scorpio
2004 Nov 15 53324.21
128.2
0.402-0.839
DDO 1.88m
2004 Nov 25 53334.06
138.1
0.884-2.406
WHT 4.2m
2005 Apr 19 53479.86
283.9
0.477-0.944
WHT 4.2m
2005 May 12 53502.15
306.1
0.827-2.400
Hiltner 2.4m
2005 Oct 14
53657.08
461.1
0.889-2.453
WHT 4.2m
2005 Oct 20
53663.21
467.2
0.417-0.928
WHT 4.2m
2006 Jan 15
53750.03
554.0
0.890-2.343
WHT 4.2m
2006 Dec 22
54091.5
895.5
0.458-0.723
Keck 10m
2006 Jan 21
54121.5
925.5
0.444-0.890
Keck 10m
) WHT 4.2m: 4.2m William Herschel Telescope of the Observatorio del Roque de los Muchachos. Slit width = 1". Hiltner 2.4m: 2.4m Hiltner Telescope of the MDM Observatory. Keck 10m: 10m telescope of the W. M. Keck Observatory.
TABLE 5
5Optical and Near-Infrared Line Fluxes, F , and Luminosities, L.Hα
Paβ
[O I] 6300Å
[Fe II] 7155Å
[Fe II] 12570Å
Epoch (d)
F
L
F
L
F
L
F
L
F
L
L rad
89
5300 * (260)
63(3)
-
-
-
-
-
-
-
-
586.9
128
2120 * (100)
25(1)
-
-
-
-
-
-
-
-
417.2
138
-
-
380(20)
4.5(2)
-
-
-
-
-
-
381.4
283
1535(80)
18.1(9)
-
-
250(10)
3.0(2)
82(14)
1.0(2)
-
-
101.7
306
-
-
137(7)
1.62(8)
-
-
-
-
35(10)
0.4(1)
81.5
461
-
39(4)
0.46(5)
-
-
-
-
20(2)
0.24(2)
16.4
467
230(10)
2.75(15)
-
-
134(7)
1.58(8)
41(7)
0.49(8)
-
-
15.35
554
-
-
9.7(8)
0.11(1)
-
-
-
-
12(3)
0.14(3)
6.0
895
10.7(7)
0.13(1)
-
-
7.6(9)
0.09(1)
3.0(2)
0.035(3)
-
-
0.18
925
9.5(7)
0.12(1)
-
-
6.9(3)
0.082(4) 2.52(8) 0.031(1)
-
-
0.14
TABLE 6
6Optical and Near-Infrared Line Profile Parameters Note. -All the velocities are with respect to the center-of-mass rest frame of SN 2004dj. Velocity subscripts: BHM: Blue wing, half-maximum RHM: Red wing, half-maximum. HWHM: Half-width, half-maximum.Epoch
Velocity (km s −1 )
R BHM
Model
(d)
v BHM
v peak
v RHM v HWHM (10 15 cm) (km s −1 )
Hα and Paβ
89 *
-2010
-450
1630
1820
1.5
-
95 †
-2120
-270
2210
2160
1.7
-
100 * *
-2550
-1620
2000
2270
2.2
-
128 *
-2670
-1770
-700
980
2.9
-
138(N IR) -2190
-1780
-310
940
2.6
-
283
-1710
-730
990
1350
4.2
-
306(N IR) -1840
-350
910
1370
4.9
-
461(N IR) -1170
-180
300
730
4.7
-2400
467
-1240
-160
550
890
5.0
-2400
554(N IR)
-800
15
500
650
3.8
-2400
895
-310
-140, 160
-10 a
150 a
2.4
-2400
925
-430
-80, 210
-b
-
3.4
-2400
[O I] 6300Å
283
-1760
-140
-
-
4.3
-
467
-1040
-300
400
720
4.5
-2250
895
-440
-210, 170
-90 a
170 a
3.4
-990
925
-440
-170, 210
20 a
230 a
3.5
-990
[Fe II] 7155Å & 12570Å
283
-1600
-830
370
980
3.9
-
306(N IR) -1670
-510
400
1030
4.4
-
461(N IR) -1610
-600
160
880
6.4
-3000
467
-1130
-330
510
820
4.6
-2250
554(N IR)
-940
-230
260
600
4.5
-2250
895
-400
-210, 490 -120 a
140 a
3.1
-780
925
-410
-170, 480
-30 a
190 a
3.3
-900
TABLE 7
7Parameters for Triple-Blackbody Matches to SN 2004dj ContinuaEpoch
v hot
T hot
vwarm
Rwarm
Twarm
v cold
T cold
L *
hot
L *
warm
L *
cold
L †
total
L rad
(d)
(km s −1 )
(K)
(km s −1 ) (10 15 cm)
(K)
(km s −1 )
(K)
(10 38
(10 38
(10 38
(10 38
(10 38 )
erg s −1 )
erg s −1 ) erg s −1 ) erg s −1 ) erg s −1
89
450
7000
Note. - * r dmax and T d are, respectively, the radius and temperature of the warm, dusty disk model. Uncertainties in the last one or two figures are shown in brackets. Likewise for the estimated disk dust mass, M dust (last col.2(5)
650(15)
9.6
0.45(5)
0.16
0.22(2)
652
(1.0)
(10000)
2090
5000
0.6
4.9(5)
610(15)
15.3
0.73(10)
0.26
0.32(5)
859
(0.16)
(10000)
1590
5000
0.25
4.6(5)
570(15)
17.4
0.82(10)
0.29
0.33(5)
996
(0.05)
(10000)
1400
2500
0.14
4(1)
520(30)
36.7
1.7(5)
0.62
0.5(1)
1207
(0.01)
(10000)
1400
2500
0.05
3.8(6)
460(30)
>80
>4
>1.5
>1.0
1393
(0.0025) (10000)
1400
2000
0.024
3.8(6)
430(20)
>120
>6
>2
>1.5
TABLE 9 Hot
9Blackbody, fb+ff and Warm (δ = 0) Dust-disk Luminosities compared with Radioactive Input. Fixed disk radiusEpoch
L hot
L CDS
L ff+fb
L *
IDDM
L total
L rad
(d)
(10 38
(10 38
(10 38
(10 38
(10 38
(10 38 )
erg s −1 )
erg s −1 ) erg s −1 ) erg s −1 ) erg s −1 ) erg s −1
89
2050
128
8.0
-
2186
592
106
586
55.7
10.0
-
652
507
129
372
26.8
7.7
-
407
411
251
109
6.6
3.3
-
119
137
281
82.9
4.8
4.2
-
91.9
103
500
14.0
0.6
0.20
12.4
27.2
10.7
652
(2.3)
-
0.03
9.9
12.2
2.1
859
(0.10)
-
0.005
6.9
7.0
0.26
996
(0.032)
-
0.0025
3.9
3.9
0.077
1207
(7.7 × 10 −4 )
-
0.0006
2.3
2.3
0.019
1393
(6.5 × 10 −5 )
-
0.0002
1.8
1.8
0.009
Note.
Wright (1980) also considered an IR echo from a SN but only in the more restricted case of an explosion within a molecular cloud.
Note. - * The continuum luminosities tend to underestimate the total SN luminosity as they do not include line emission. This is particularly the case around 200 − 500 d when the relative contribution of nebular line emission is at a maximum. There is less of a problem before this era when the hot continuum dominates, or afterwards when the warm/cold continuum increasingly dominates. † : L total excludes L cold . This is because the cold component is due to an IS IR echo (see §3.4.3.1) which was predominantly powered by the peak luminosity of the SN prior to the earliest epoch of observation. * * : Hot blackbody velocities and temperatures estimated by fixing the temperature at the 500 d value and extrapolating the earlier velocity evolution. In col. 13, L rad is the radioactive deposition corresponding to the ejection of 0.0095 M ⊙ of 56 Ni, scaled from the SN 1987A case specified byLi et al. (1993) & Timmes et al. (1996. The 89 d and 996 d warm and cold parameters are based on photometry only.
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Spectral ranges at each epoch were Short-Low (SL) first and second orders: 7.4 − 14.5 and 5.2 − 8.7 µm, respectively. texp gives the exposure time per order, in seconds. * We assume an explosion date of. 10.0 (MJD=53196.0)Note. -Spectral ranges at each epoch were Short-Low (SL) first and second orders: 7.4 − 14.5 and 5.2 − 8.7 µm, respectively. texp gives the exposure time per order, in seconds. * We assume an explosion date of 2004 Jul. 10.0 (MJD=53196.0).
| []
|
[
"ON INTERLACING OF ZEROS OF CERTAIN FAMILY OF MODULAR FORMS",
"ON INTERLACING OF ZEROS OF CERTAIN FAMILY OF MODULAR FORMS"
]
| [
"Ekata Saha ",
"N Saradha "
]
| []
| []
| Let k = 12m(k) + s ≥ 12 for s ∈ {0, 4, 6, 8, 10, 14}, be an even integer and f be a normalised modular form of weight k with real Fourier coefficients, written asUnder suitable conditions on aj (rectifying an earlier result of Getz), we show that all the zeros of f , in the standard fundamental domain for the action of SL(2, Z) on the upper half plane, lies on the arc A := e iθ : π 2 ≤ θ ≤ 2π 3 . Further, extending a result of Nozaki, we show that for certain family {f k } k of normalised modular forms, the zeros of f k and f k+12 interlace on A • := e iθ : π 2 < θ < 2π 3 . | 10.1016/j.jnt.2017.07.013 | [
"https://arxiv.org/pdf/1702.07296v1.pdf"
]
| 119,136,441 | 1702.07296 | 49b3f167d315baf5e7d8cb0c7d2cf5e79da2e3aa |
ON INTERLACING OF ZEROS OF CERTAIN FAMILY OF MODULAR FORMS
Ekata Saha
N Saradha
ON INTERLACING OF ZEROS OF CERTAIN FAMILY OF MODULAR FORMS
Let k = 12m(k) + s ≥ 12 for s ∈ {0, 4, 6, 8, 10, 14}, be an even integer and f be a normalised modular form of weight k with real Fourier coefficients, written asUnder suitable conditions on aj (rectifying an earlier result of Getz), we show that all the zeros of f , in the standard fundamental domain for the action of SL(2, Z) on the upper half plane, lies on the arc A := e iθ : π 2 ≤ θ ≤ 2π 3 . Further, extending a result of Nozaki, we show that for certain family {f k } k of normalised modular forms, the zeros of f k and f k+12 interlace on A • := e iθ : π 2 < θ < 2π 3 .
Introduction
Let H denote the complex upper half plane. Then the full modular group SL(2, Z) acts on H by the transformation law ≤ (z) ≤ 0 ∪ |z| > 1, 0 < (z) < 1 2 .
Throughout this article we take k ≥ 4 to be an even integer. For z ∈ H, the Eisenstein series of weight k for the full modular group SL(2, Z) is defined by the following absolutely convergent series,
E k (z) := 1 2 c,d∈Z (c,d)=1 1 (cz + d) k .
Rankin and Swinnerton-Dyer [8] proved that for k ≥ 4, all the zeros of the Eisenstein series E k lie in the arc
A := |z| = 1, − 1 2 ≤ (z) ≤ 0 = e iθ : π 2 ≤ θ ≤ 2π 3 .
In 2004, extending the arguments of Rankin and Swinnerton-Dyer, Getz [4] gave a criterion for a normalised modular form of weight k for SL (2, Z), written as f = E k + m(k) j=1 a j E k−12j ∆ j , to have all its zeros on the arc A, in terms of a j 's. However, there is a rectifiable error in his proof. While estimating H(θ) in [4, p. 2225, eq. (2.5)], he used an upper bound for R k−12j , 1 ≤ j ≤ m(k) from [4, p. 2224, eq. (2.3)] which is valid if k − 12j ≥ 12. But k − 12m(k) is always less than 12, unless it is 14. We present below a corrected version of his theorem.
Let us define Theorem 1. Let k ≥ 12 and f be a normalised modular form of weight k, written as
(1) δ t = −2 if t = 0,f = E k + m(k) j=1 a j E k−12j ∆ j ,
with a j ∈ R for 1 ≤ j ≤ m(k). Let := sup z∈A |∆(z)|. Suppose that Then f has m(k) zeros (other than possible zeros at i, ρ := e 2πi/3 ) in the fundamental domain F and they all lie on the arc A.
Remark 1. Note that δ 12 above is smaller than the δ of [4,Theorem 1]. This better value is due to a more accurate estimation of a finite sum using computation. See §2, Lemma 1. Getz [4] computed ∼ 0.004809 . . ..
Apart from this kind of normalised modular forms there are other examples of families of modular forms, which have been shown to have their zeros on the arc A (see [9,1,2]). Now for the zeros of these families of modular forms, one interesting question is to ask about their possible interlacing property.
Definition 1. Let α < β. Suppose that f, g are two complex valued functions with simple zeros in the open interval (α, β). Let α < t 1 < · · · < t m < β and α < t * 1 < · · · < t * m+1 < β be the zeros of f and g, respectively. We say that zeros of f and g interlace in
(α, β) if t * j < t j < t * j+1 for 1 ≤ j ≤ m.
For example, the zeros of cos(nθ) and cos((n + 1)θ) interlace in (0, π) for an integer n ≥ 1. Rankin and Swinnerton-Dyer [8] proved that the Eisenstein series E k has m(k) simple zeros in the open arc A • := e iθ : π 2 < θ < 2π 3 . For this, they considered the function
(3) F k (θ) := e ikθ/2 E k (e iθ ).
This is a real valued function for θ real and it has m(k) zeros in the interval (π/2, 2π/3) and so does E k (e iθ ) in A • . Note that m(k + 12) = m(k) + 1. Hence one may look for the interlacing property for the zeros of F k (θ) and F k+12 (θ) for θ ∈ (π/2, 2π/3). In this instance, we say that the zeros of E k (e iθ ) and E k+12 (e iθ ) interlace in A • . This was predicted by Gekeler [3] and proved by Nozaki [7]. For the zeros of certain families of weakly holomorphic modular forms considered by Asai, Kaneko and Ninomiya [1], their interlacing property was established by Jermann [6]. Similar properties for the zeros of the weakly holomorphic modular forms, studied by Duke and Jenkins [2], were proved by Jenkins and Pratt [5]. Here we establish the interlacing property of the zeros of certain family of normalised modular forms that were considered in Theorem 1.
Theorem 2. For each k ≥ 12, let (a (k) j ) 1≤j≤m(k) be real numbers such that (4) (3 + δ 12 ) m(k)−1 j=1 |a (k) j | j + (3 + δ s )|a (k) m(k) | m(k) ≤ 20 1 2 k/2 ,
where δ s is as in (1). Then for the family of normalised modular forms (f k ) k for SL(2, Z) defined by
f k := E k + m(k) j=1 a (k) j E k−12j ∆ j ,
the zeros of f k in the fundamental domain F lie on the arc A. Further, the zeros of f k and those of f k+12 interlace in A • for each k ≥ 12.
Remark 2. By (4), we see that (2) is satisfied. Hence by Theorem 1, all the zeros of (f k ) k lie on the arc A, thus giving the first assertion of Theorem 2.
Remark 3. Nozaki's result is a special case of Theorem 2 when a (k)
j = 0 for all 1 ≤ j ≤ m(k).
For proving the interlacing of the zeros of the Eisenstein series E k (z), Nozaki showed that F k (θ) as defined in (3), is very well approximated by 2 cos(kθ/2) for θ ∈ (π/2, 2π/3). This is an important step in his method. We are able to show that
(5) G k (θ) := e ikθ/2 f k (e iθ )
is also well approximated by 2 cos(kθ/2) for θ ∈ (π/2, 2π/3). See §3.1 for details. Both these functions have m(k) zeros in (π/2, 2π/3). If α is a zero of cos(kθ/2) for θ ∈ (π/2, 2π/3), then there is a neighbourhood of α, say (α − , α + ), containing exactly one zero α * of G k (θ). It can be easily seen that, the zeros of cos(kθ/2) and cos((k + 12)θ/2) interlace in (π/2, 2π/3) (see §3.2). Thus there exist successive zeros β, γ of cos((k + 12)θ/2) with β < α < γ.
Again, there exist intervals of the form (β − δ, β + δ) and (γ − µ, γ + µ), each containing exactly one zero of cos((k + 12)θ/2), say β * , γ * respectively. Thus if
β + δ < α − < α + < γ − µ,
then we obtain that
β * < α * < γ * .
This argument is used to show that the zeros of G k (θ) and G k+12 (θ) interlace in (π/2, 23π/36) (see §3.2, 4.1). This method does not work as we approach 2π/3. For proving the interlacing property in the remaining interval we consider the interval (19π/32, 2π/3), which overlaps with (π/2, 23π/36).
Here the method depends on analysing different cases according to the increasing or decreasing property of the cosine function at their respective zeros (see §4.2).
A lemma and Proof of Theorem 1
We begin this section with the following lemma. This will be used in the proof of both the Theorems 1 and 2.
Lemma 1. Let k ≥ 4 and F k (θ) := e ikθ/2 E k (e iθ ) for θ ∈ [π/2, 2π/3]. Then F k (θ) = 2 cos(kθ/2) + 1 2 cos(θ/2) k + 1 2i sin(θ/2) k + P k (θ), where |P k (θ)| < 0.759 if k = 4, 0.179 if k = 6, 0.059 if k = 8, 0.019 if k = 10, 0.359( 1 2 ) k/2 if k ≥ 12. Remark 4. From Lemma 1 we obtain that sup θ∈[π/2,2π/3] |F k (θ)| = sup z∈A |E k (z)| < 4.009 if k = 4, 3.304 if k = 6, 3.122 if k = 8, 3.051 if k = 10, 3 + 1.359( 1 2 ) k/2 if k ≥ 12.
In particular, we shall use the following bounds for the proof of Theorem 1. For k ≥ 4,
sup θ∈[π/2,2π/3] |F k (θ)| ≤ 3 + 1 2i sin(θ/2) k + |P k (θ)| < 3 + δ k , where δ k is as in (1). 2.1. Proof of Lemma 1. For N ≥ 1, let us define σ N (θ) := 1 2 c,d∈Z c 2 +d 2 =N (c,d)=1 1 (ce iθ/2 + de −iθ/2 ) k .
In the above definition, whenever empty sum appears, it is assumed to be 0. Now
F k (θ) = 1 2 c,d∈Z (c,d)=1 1 (ce iθ/2 + de −iθ/2 ) k ,
for θ ∈ [π/2, 2π/3]. Since the series defining the Eisenstein series of weight k ≥ 4 is absolutely convergent, we can write
F k (θ) = N ≥1 σ N (θ).
One can easily see that
σ 1 (θ) = 2 cos(kθ/2) and σ 2 (θ) = 1 2 cos(θ/2) k + 1 2i sin(θ/2) k . Hence (6) F k (θ) = 2 cos(kθ/2) + 1 2 cos(θ/2) k + 1 2i sin(θ/2) k + P k (θ), where P k (θ) := N ≥5 σ N (θ).
Now we split the above sum and write
(7) P k (θ) = 5≤N ≤A σ N (θ) + N ≥B σ N (θ),
for some integer A ≥ 5 and B, the least integer larger than A which can be written as sum of squares of two co-prime integers. Note that for any two real numbers c, d one has |cd| ≤ c 2 +d 2 2 . Since −1 ≤ 2 cos θ ≤ 0 for θ ∈ [π/2, 2π/3], we obtain
(8) |ce iθ/2 + de −iθ/2 | 2 = c 2 + d 2 + 2cd cos θ ≥ c 2 + d 2 − |cd| ≥ c 2 + d 2 2 .
Thus we get
5≤N ≤A σ N (θ) ≤ 1 2 5≤N ≤A c,d∈Z c 2 +d 2 =N (c,d)=1 1 (c 2 + d 2 − |cd|) k/2 .
For any natural number N , there are at most 2(2N 1/2 +1) pairs (c, d) of integers such that c 2 +d 2 = N . Hence for the second sum in (7), using (8)
we get N ≥B σ N (θ) ≤ N ≥B 2 N k/2 (2N 1/2 + 1) ≤ 2 + 1 √ B 2 k/2 N ≥B N (1−k)/2 ≤ 2 + 1 √ B 2 k/2 ∞ B−1 x (1−k)/2 dx = 2 + 1 √ B 2 (k+2)/2 k − 3 1 B − 1 (k−3)/2 . So we have (9) |P k (θ)| ≤ 1 2 5≤N ≤A c,d∈Z c 2 +d 2 =N (c,d)=1 1 (c 2 + d 2 − |cd|) k/2 + 2 + 1 √ B 2 (k+2)/2 k − 3 1 B − 1 (k−3)/2 .
Let us denote the right hand side of (9) by p k . Using a C++ programming we obtain optimal values of p k as given in Table 1. The values of p k is best possible up to second decimal place. Our choice of A is also indicated in the table. Table 1 The computation of the values of p k , when k ≥ 6 has taken only a few seconds, whereas for p 4 , the programme ran for about two hours. Our proof is now complete.
2.2.
Proof of Theorem 1. It is easy to see that F k (θ) is real valued for θ ∈ [π/2, 2π/3]. Further,
∆(e iθ )e 6iθ = F 3 4 (θ) − F 2 6 (θ) 1728 ∈ R for θ ∈ [π/2, 2π/3]. Now, G(θ) := e ikθ/2 f (e iθ ) = e ikθ/2 E k (e iθ ) + m(k) j=1 a j e i(k−12j)θ/2 E k−12j (e iθ )∆(e iθ ) j e 6ijθ = F k (θ) + m(k) j=1 a j F k−12j (θ)∆(e iθ ) j e 6ijθ .
Since a j ∈ R for 1 ≤ j ≤ m(k), we see that G(θ) is real valued for θ ∈ [π/2, 2π/3]. Also by Lemma 1 we have,
G(θ) = 2 cos(kθ/2) + 1 2 cos(θ/2) k + 1 2i sin(θ/2) k + P k (θ) + m(k) j=1 a j F k−12j (θ)∆(e iθ ) j e 6ijθ .
We show that (10) |G(θ) − 2 cos(kθ/2)| < 2.
By Remark 4,
|G(θ) − 2 cos(kθ/2)| < 1 + δ 12 + m(k) j=1 |a j ||F k−12j (θ)| j ≤ 1 + δ 12 + (3 + δ 12 ) m(k)−1 j=1 |a j | j + (3 + δ s )|a m(k) | m(k) .
Thus, |G(θ) − 2 cos(kθ/2)| < 2 by our hypothesis. Now we argue as in [8]. By (10), we get that between two consecutive extremum points of cos(kθ/2), there is a zero of G(θ). Now the extremum points of cos(kθ/2) for θ ∈ [π/2, 2π/3] are given by 2πn
k , where k 4 ≤ n ≤ k 3 i.e. 3m(k) + s 4 ≤ n ≤ 4m(k) + s 3 .
So we have m(k) + 1 such n's. Therefore G(θ) has m(k) zeros on (π/2, 2π/3). This, together with the valence formula, completes the proof.
3. Properties of G k (θ) 3.1. Approximation of G k (θ) by 2 cos(kθ/2). We write G k (θ) = e ikθ/2 f k (e iθ ) as G k (θ) = 2 cos(kθ/2) + S k (θ), where by Lemma 1,
S k (θ) = 1 2 cos(θ/2) k + 1 2i sin(θ/2) k + P k (θ) + m(k) j=1 a (k) j F k−12j (θ)∆(e iθ ) j e 6ijθ , with |P k (θ)| < 0.359 1 2 k/2 . Let (11) Q k (θ) := 1 2i sin(θ/2) k + P k (θ) + m(k) j=1 a (k) j F k−12j (θ)∆(e iθ ) j e 6ijθ .
Then by Lemma 1, Remark 4 and our hypothesis we get Further, we argue as in the proof of [7,Lemma 4.4] to obtain that for θ ∈ [19π/32, α m(k) ], the function g(θ) := (2 cos(θ/2)) −k − (2 cos(θ/2)) −(k+12)
(12) |Q k (θ)| ≤ 1 2i sin(θ/2) k + |P k (θ)| + m(k) j=1 |a (k) j ||F k−12j (θ)| j < 21.
is minimised at θ = 19π/32. Hence for k ≥ 24
S k (θ) − S k+12 (θ) = g(θ) + Q k (θ) − Q k+12 (θ) >
S k (θ) > S k+12 (θ).
3.2.
Zeros of G k (θ) and G k+12 (θ). Let α 1 < · · · < α m(k) and β 1 < · · · < β m(k)+1 denote the zeros of cos(kθ/2) and cos((k + 12)θ/2) in (π/2, 2π/3), respectively. Then
(15) α j = ( 1 2 + 2j−1 k )π if k ≡ 0 mod 4, ( 1 2 + 2j k )π if k ≡ 2 mod 4, for 1 ≤ j ≤ m(k) and (16) β j = ( 1 2 + 2j−1 k+12 )π if k ≡ 0 mod 4, ( 1 2 + 2j
k+12 )π if k ≡ 2 mod 4, for 1 ≤ j ≤ m(k) + 1. We observe the following properties: for 1 ≤ j ≤ m(k),
(17) β j < α j < β j+1 (18) min 1≤j≤m(k) {α j − β j , β j+1 − α j } ≥ 12π k(k + 12) and (19) α j+1 − α j = 2π k for 1 ≤ j ≤ m(k) − 1.
Hence we have,
(20) cos k 2 α j − 6π k(k + 12) cos k 2 α j + 6π k(k + 12) < 0 and (21) cos k 2 α j − π 3k cos k 2 α j + π 3k < 0.
Remark 5. From (15) and (16) it follows that k ≥ 24, whenever α j + 6π k(k+12) > 23π 36 or α j ≥ 19π 32 + π 3k or β j + 6π (k+12)(k+24) > 23π 36 .
We now associate the zeros of G k (θ) with the zeros of cos(kθ/2).
Lemma 2. Let k ≥ 12 and α * 1 < · · · < α * m(k) be the zeros of G k (θ) in (π/2, 2π/3). Then
α * j ∈ α j − 6π k(k + 12) , α j + 6π k(k + 12)
for all j such that α j + 6π k(k+12) ≤ 23π 36 and
α * j ∈ α j − π 3k
, α j + π 3k for all j such that α j ≥ 19π/32 + π/3k. In particular, α * j ∈ α j − π 3k , α j + π 3k for 1 ≤ j ≤ m(k).
Proof of Lemma 2. Observe that
2 cos k 2 α j − 6π k(k + 12) = 2 cos k 2 α j + 6π k(k + 12) = 2 sin 3π k + 12 .
We first claim that 2 sin 3π k+12 > |S k (θ)| for θ ∈ (π/2, 23π/36]. Note that 2 sin 3π k + 12
> 2 3π k + 12 − 1 6 3π k + 12 3 > 18.365 k + 12 .
Hence it is enough to show that
18.365 k + 12 − |S k (θ)| = 18.365 k + 12 − 1 2 cos(θ/2) k − |Q k (θ)| > 18.365 k + 12 − 1 2 cos(θ/2) k − 21.359 2 k/2 > 0.
Now this is true if 2 k/2 18.365(2 cos(θ/2)) k − (k + 12) − 21.359(k + 12) 2 k/2 (2 cos(θ/2)) k > 0,
In particular if 10.355(2 cos(θ/2)) k − (k + 12) > 0.
The quantity k k+12 10.355 is maximum for k = 12, and therefore k k+12 10.355 < 1.073. Note that for θ ∈ (π/2, 23π/36], 2 cos(θ/2) > 1.074. This proves our first claim. Since
2 cos k 2 α j ± 6π k(k + 12) > |S k (θ)|
for θ ∈ (π/2, 23π/36], we get that
G k α j ± 6π k(k + 12) = 2 cos k 2 α j ± 6π k(k + 12) + S k α j ± 6π k(k + 12)
have same sign as 2 cos k 2 α j ± 6π k(k+12)
. Then from (20) we get that
G k α j − 6π k(k + 12) G k α j + 6π k(k + 12) < 0.
This implies that there exists a zero of G k (θ) in the interval α j − 6π k(k+12) , α j + 6π k(k+12) for all j such that α j + 6π k(k+12) ≤ 23π 36 .
It is easy to see from (15) that α m(k) ≤ 2π 3 − π k . Hence we get α m(k) + π 3k ≤ 2π 3 − 2π 3k . Since cos(kα j /2) = 0, we have 2 cos k 2 α j − π 3k = 2 cos k 2 α j + π 3k = 1.
As G k (θ) = 2 cos(kθ/2) + S k (θ), by (13) we get that the sign of G k α j ± π 3k is same as that of 2 cos k 2 α j ± π 3k , whenever α j ≥ 19π 32 + π 3k . Hence it follows from (21) that
G k α j − π 3k G k α j + π 3k < 0.
This implies that there exists a zero of G k (θ) in the interval α j − π 3k , α j + π 3k for all j such that α j ≥ 19π 32 + π 3k . From Remark 5 we deduce that a zero α j of cos(kθ/2) in (π/2, 2π/3) satisfies α j + 6π k(k+12) ≤ 23π 36 or α j ≥ 19π 32 + π 3k . Moreover,
α j − 6π k(k + 12) , α j + 6π k(k + 12) ⊂ α j − π 3k , α j + π 3k .
By (19) all the intervals of the form α j − π 3k , α j + π 3k are disjoint for 1 ≤ j ≤ m(k). We have shown above that each of them contains at least one zero of G k (θ). We also know that G k (θ) has exactly m(k) zeros in (π/2, 2π/3). Since the zeros of cos(kθ/2) and G k (θ) have been labelled as per the increasing order of magnitude, the assertion of the lemma follows.
Proof of Theorem 2
We closely follow the arguments of Nozaki [7]. However at several places our arguments are simpler. We know that a zero α j of cos(kθ/2) in (π/2, 2π/3) satisfies α j + 6π k(k+12) ≤ 23π 36 or α j ≥ 19π 32 + π 3k . Hence we prove Theorem 2 for these two cases.
4.1. Case I. Let α j ∈ π 2 , 23π 36 − 6π k(k+12) . Then we prove the following:
(i) β * j < α * j . (ii) α * j < β * j+1 if β j+1 + 6π (k+12)(k+24) ≤ 23π 36 .
4.1.1. Proof of (i). Since α j + 6π k(k+12) ≤ 23π 36 , we have β j + 6π (k+12)(k+24) ≤ 23π 36 . Applying Lemma 2 for G k (θ) and G k+12 (θ), we get α * j ∈ α j − 6π k(k + 12)
, α j + 6π k(k + 12) and β * j ∈ β j − 6π (k + 12)(k + 24)
, β j + 6π (k + 12)(k + 24) .
Therefore by (18), we have
β * j < β j + 6π (k + 12)(k + 24) < β j + 6π k(k + 12) ≤ α j − 6π k(k + 12) < α * j .
4.1.2.
Proof of (ii). When β j+1 + 6π (k+12)(k+24) ≤ 23π 36 , by Lemma 2 β * j+1 ∈ β j+1 − 6π (k + 12)(k + 24)
, β j+1 + 6π (k + 12)(k + 24) .
Therefore by (18), we have
α * j < α j + 6π k(k + 12) ≤ β j+1 − 6π k(k + 12) < β j+1 − 6π (k + 12)(k + 24) < β * j+1 .
4.2.
Case II. Let α j ∈ 19π 32 + π 3k , 2π 3 . Then we prove the following:
(iii) β * j < α * j if β j ≥ 19π 32 + π 3(k+12) . (iv) α * j < β * j+1
. For an integer n,
cos(nπ/2) = 1 if n ≡ 0 mod 4, −1 if n ≡ 2 mod 4, 0 otherwise.
Further cos(nθ) is decreasing at π/2 if n ≡ 1 mod 4 and increasing at π/2 if n ≡ 3 mod 4. Thus if α denotes the first zero of cos(nθ) in (π/2, ∞), then cos(nθ) is decreasing at α if n ≡ 0, 3 mod 4 and increasing at α if n ≡ 1, 2 mod 4. Hence if β denotes the first zero of cos((n + 6)θ) in (π/2, ∞), then cos((n + 6)θ) is increasing at β if n ≡ 0, 3 mod 4 and decreasing at β if n ≡ 1, 2 mod 4. Therefore, for all 1 ≤ j ≤ m(k), if cos(kθ/2) is increasing (resp. decreasing) at α j , then cos((k + 12)θ/2) is decreasing (resp. increasing) at β j . We consider two subcases.
(a) The function cos(kθ/2) is increasing at α j . (b) The function cos(kθ/2) is decreasing at α j .
Note that by Remark 5, we have k ≥ 24. Now for k ≥ 24 and θ ∈ [19π/32, 2π/3 − 2π/3k], S k (θ) > 0 (see (13)). Hence α * j ≶ α j according as cos(kθ/2) is increasing or decreasing at α j .
4.2.1.
Proof of (iii) and (iv) when (a) occurs. It follows by our earlier analysis that cos((k + 12)θ/2) is increasing at β j+1 . Also β j+1 > α j > 19π 32 + π 3(k+12) . Hence (22) α * j < α j and β * j+1 < β j+1 .
Further
β j < β * j if β j ≥ 19π 32 + π 3(k + 12)
.
The Case (a) is described pictorially below.
In the above one and other picture below, the respective arrows indicate the direction in which β * j , α * j and β * j+1 lie. We first prove (iii). In fact, we show that the intersection point of the two cosine curves between β j and α j separates β * j and α * j . The function 2 cos((k + 12)θ/2) − 2 cos(kθ/2) = −4 sin((k + 6)θ/2) sin(3θ), has a zero between β j and α j , say γ j . For 1 ≤ j ≤ m(k),
(23) γ j = ( 1 2 + 2j−1 k+6 )π if k ≡ 0 mod 4, ( 1 2 + 2j k+6 )π if k ≡ 2 mod 4.
We prove that β * j ∈ (β j , γ j ) and α * j ∈ (γ j , α j ). By (13), G k (α j ) > 0 and G k+12 (β j ) > 0. Thus, we are led to show that G k (γ j ), G k+12 (γ j ) < 0. As in the proof of [7,Lemma 4.2], it follows that |2 cos(kγ j /2)| > S k (θ) and |2 cos((k + 12)γ j /2)| > S k+12 (θ), for any j such that γ j ≥ 19π 32 and θ ∈ [19π/32, 2π/3). Since cos(kθ/2) is increasing at α j and cos(kα j /2) = 0, we get cos(kγ j /2) < 0. Hence G k (γ j ) < 0. Similarly one obtains that G k+12 (γ j ) < 0. This completes the proof of (iii) when (a) occurs.
Next we show (iv) i.e. α * j < β * j+1 . Now if α j ≤ β j+1 − π 3(k+12) , then by (22) and Lemma 2,
α * j < α j ≤ β j+1 − π 3(k + 12) < β * j+1 ,
which proves (iv). Hence we need to consider only β j+1 − π 3(k + 12) < α j and also α * j , β * j+1 ∈ β j+1 − π 3(k + 12)
, α j .
From (14) we have, S k (θ) > S k+12 (θ) for θ ∈ [β j+1 − π/3(k + 12), α j ]. Further, the function cos(kθ/2) − cos((k + 12)θ/2) takes positive value in the interval (γ j , γ j+1 ), where γ j 's are as in (23). It is easy to check that [β j+1 − π/3(k + 12), α j ] ⊂ (γ j , γ j+1 ). Therefore, we obtain that
G k (θ) − G k+12 (θ) = 2 cos(kθ/2) − 2 cos((k + 12)θ/2) + S k (θ) − S k+12 (θ) > 0
for θ ∈ [β j+1 − π/3(k + 12), α j ]. Taking θ = α * j , we find that G k+12 (α * j ) < 0. On the other hand, G k+12 (β j+1 ) > 0. Hence α * j < β * j+1 . This completes the proof of Case (a).
4.2.2.
Proof of (iii) and (iv) when (b) occurs. In this case cos((k + 12)θ/2) is decreasing at β j+1 . Hence (24) α * j < α j and β * j+1 < β j+1 .
Further
(25) β j < β * j if β j ≥ 19π 32 + π 3(k + 12) . r(θ) is minimised at α j + π/3k for θ ∈ [β j+1 , α j + π/3k]. Note that 2 cos k 2 α j + π 3k = −1. Now α j + π 3k = β j+1 + 72j−5k−24 3k(k+12) π if k ≡ 0 mod 4, β j+1 + 72j−5k+12 3k(k+12) π if k ≡ 2 mod 4. This is a contradiction since k ≥ 24.
4.3.2.
Proof of (vi). As in (v), we can deduce that α j + 6π k(k+12) ≤ 23π 36 . Hence β * j < α * j by §4.1.
F
SL(2, Z). The standard fundamental domain for this action of SL(2, Z) on H is the following subset of H, :
if t ≥ 12.
j | j + (3 + δ s )|a m(k) | m(k) ≤ 1 − δ 12 .
.
Thus G k (θ) = 2 cos(kθ/2) + S k (θ) = 2 cos(kθ/2) + 1 2 cos(θ/2)k + Q k (θ)with |Q k (θ)| < 21.359 1 2 k/2 . By following the proof of [7, Lemma 4.1], we observe that (13) 0 < S k (θ) < 1, whenever k ≥ 24 and θ ∈ [19π/32, 2π/3 − 2π/3k].
Thus for k ≥ 24 and θ ∈ [19π/32, α m(k) ],g
19π
32
−
21.359
2 k/2 −
21.359
2 (k+12)/2
> 2 cos
19π
64
−k
1 − 2 cos
19π
64
−12
−
24.919
2 k/2
>
0.877
(1.192) k −
24.919
(1.414) k > 0.
From (26) and the condition j ≤ (k − s)/12, we get that if k ≡ 0 mod 4,The equation of the tangent line of the sine curve at π/6 is Proof of (v). We show that α j ≥ 19π 32 + π 3k . Hence by §4.2, α * j < β * j+1 . Suppose that α j < 19π 32 + π 3k . Then using (15) we get that 576j − 384 if k ≡ 0 mod 4, 576j − 96 if k ≡ 2 mod 4.Hence
2 cos
k + 12
2
α j +
π
3k
=
2 sin 72j−5k−24
6k
π
if k ≡ 0 mod 4,
2 sin 72j−5k+12
6k
π
if k ≡ 2 mod 4.
0 <
72j − 5k − 24
6k
≤
k − 6s − 24
6k
≤
1
6
−
4
k
and if k ≡ 2 mod 4,
0 <
72j − 5k + 12
6k
≤
k − 6s + 12
6k
≤
1
6
−
4
k
.
Therefore,
2 cos
k + 12
2
α j +
π
3k
≤ 2 sin
π
6
−
4π
k
.
y =
1
2
+
4.3.1. (28)
27k >
Since β j+1 +
6π
(k+12)(k+24) > 23π
36 , it follows from Remark 5 and (16) that k ≥ 24 and
(29)
0 <
(k + 24)(72j − 5k − 24) + 216 if k ≡ 0 mod 4,
(k + 24)(72j − 5k + 12) + 216 if k ≡ 2 mod 4
respectively. Using (28) in (29) we get that
0 <
(k + 24)(−13k + 192) + 1728 if k ≡ 0 mod 4,
(k + 24)(−13k) + 1728
if k ≡ 2 mod 4.
Acknowledgement:We would like to thank Biswajyoti Saha and Jhansi Bhavani V. for helping us with the computations in Lemma 1.From (24) and (25) we easily getwhich proves (iii). Thus we proceed to prove (iv) i.e. α * j < β * j+1 . If α j + π 3k ≤ β j+1 , then by Lemma 2 and (24), we haveHence we can assume thatNote that β j+1 < α j + π 3k implies Here the aim is to show thatIf (27) holds, then G k (β * j+1 ) < 0. Since G k (α j ) > 0, we get α j < α * j < β * j+1 as required. Thus it remains to prove (27). Note thatLet r(θ) := 2 cos((k +12)θ/2)−2 cos(kθ/2). We first find a lower bound for r(θ) in [β j+1 , α j +π/3k]. For this we show that r(θ) is decreasing in this interval. Note that r(θ) = −4 sin((k +6)θ/2) sin(3θ). Now − sin(3θ) is decreasing from 1 to 0 in (π/2, 2π/3). Also r(θ) > 0 for θ ∈ (γ j , γ j+1 ). Thus sin((k + 6)θ/2) > 0 for θ ∈ (γ j , γ j+1 ). As γ j and γ j+1 are two consecutive zeros of sin((k + 6)θ/2), we therefore obtain that sin((k + 6)θ/2) is decreasing in γ j +γ j+1 2 , γ j+1 . From (15), (16) and (23), we deduce that γ j + γ j+1 2 < β j+1 and α j + π 3k < γ j+1 .Hence sin((k + 6)θ/2) is decreasing in [β j+1 , α j + π/3k]. As both the functions sin((k + 6)θ/2) and − sin(3θ) are positive and decreasing, we get that r(θ) is decreasing in [β j+1 , α j + π/3k]. ThereforeHence 2 sin π 6 − 4πk . Next let t(θ) := (2 cos(θ/2)) −(k+12) − (2 cos(θ/2)) −k . Then t (θ) = sin(θ/2)(2 cos(θ/2)) −(k+1) (k + 12)(2 cos(θ/2)) −12 − k .So t(θ) is minimised when (2 cos(θ/2)) −12 = k k+12 . We thus get,This completes the proof of Case (b).4.3.Remaining cases. By Cases I and II, in order to complete the proof of Theorem 2, we need to prove the following two cases.(v) α * j < β * j+1 when α j ∈ π 2 , 23π 36 − 6π k(k+12) and β j+1 + 6π (k+12)(k+24) > 23π 36 . (vi) β * j < α * j when α j ∈ 19π 32 + π 3k , 2π 3 and β j < 19π 32 + π 3(k+12) .
Zeros of certain modular functions and an application. T Asai, M Kaneko, H Ninomiya, Comment. Math. Univ. St. Paul. 461T. Asai, M. Kaneko and H. Ninomiya, Zeros of certain modular functions and an application, Comment. Math. Univ. St. Paul. 46 (1997), no. 1, 93-101.
On the zeros and coefficients of certain weakly holomorphic modular forms. W Duke, P Jenkins, Special Issue: In honor of Jean-Pierre Serre, Part. 44Pure Appl. Math. Q.W. Duke and P. Jenkins, On the zeros and coefficients of certain weakly holomorphic modular forms, Pure Appl. Math. Q. 4 (2008), no. 4, Special Issue: In honor of Jean-Pierre Serre, Part 1, 1327-1340.
Some observations on the arithmetic of Eisenstein series for the modular group SL(2, Z). E.-U Gekeler, Arch. Math. (Basel). 77E.-U. Gekeler, Some observations on the arithmetic of Eisenstein series for the modular group SL(2, Z), Arch. Math. (Basel) 77 (2001), 5-21.
A generalization of a theorem of Rankin and Swinnerton-Dyer on zeros of modular forms. J Getz, Proc. Amer. Math. Soc. 1328J. Getz, A generalization of a theorem of Rankin and Swinnerton-Dyer on zeros of modular forms, Proc. Amer. Math. Soc. 132 (2004), no. 8, 2221-2231.
Interlacing of zeros of weakly holomorphic modular forms. P Jenkins, K Pratt, Proc. Amer. Math. Soc. Ser. B. 1P. Jenkins and K. Pratt, Interlacing of zeros of weakly holomorphic modular forms, Proc. Amer. Math. Soc. Ser. B 1 (2014), 63-77.
Interlacing property of the zeros of jn(τ ). J Jermann, Proc. Amer. Math. Soc. 14010J. Jermann, Interlacing property of the zeros of jn(τ ), Proc. Amer. Math. Soc. 140 (2012), no. 10, 3385-3396.
A separation property of the zeros of Eisenstein series for. H Nozaki, Bull. Lond. Math. Soc. 402H. Nozaki, A separation property of the zeros of Eisenstein series for SL(2, Z), Bull. Lond. Math. Soc. 40 (2008), no. 1, 26-36.
On the zeros of Eisenstein series. F K C Rankin, H P F Swinnerton-Dyer, Bull. London Math. Soc. 2F. K. C. Rankin and H. P. F. Swinnerton-Dyer, On the zeros of Eisenstein series, Bull. London Math. Soc. 2 (1970), 169-170.
The zeros of certain Poincaré series. R A Rankin, Compositio Math. 463R. A. Rankin, The zeros of certain Poincaré series, Compositio Math. 46 (1982), no. 3, 255-272.
. Ekata Saha, N Saradha, Homi Bhabha Road. 400005School of Mathematics, Tata Institute of Fundamental ResearchIndia E-mail address: [email protected] E-mail address: [email protected] Saha and N. Saradha, School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Navy Nagar, Mumbai, 400 005, India E-mail address: [email protected] E-mail address: [email protected]
| []
|
[
"Sparse approximate solution of fitting surface to scattered points by MLASSO model",
"Sparse approximate solution of fitting surface to scattered points by MLASSO model"
]
| [
"Y-X Hao ",
"C-J Li ",
"R-H Wang "
]
| []
| [
"SCIENCE CHINA Mathematics journal sample. Sci China Math"
]
| The goal of this paper is to achieve a computational model and corresponding efficient algorithm for obtaining a sparse representation of the fitting surface to the given scattered data. The basic idea of the model is to utilize the principal shift invariant (PSI) space and the l 1 norm minimization. In order to obtain different sparsity of the approximation solution, the problem is represented as a multilevel LASSO (MLASSO) model with different regularization parameters. The MLASSO model can be solved efficiently by the alternating direction method of multipliers. Numerical experiments indicate that compared to the AGLASSO model and the basic MBA algorithm in[35], the MLASSO model can provide an acceptable compromise between the minimization of the data mismatch term and the sparsity of the solution. Moreover, the solution by the MLASSO model can reflect the regions of the underlying surface where high gradients occur. | 10.1007/s11425-016-9087-y | [
"https://arxiv.org/pdf/1704.07058v2.pdf"
]
| 119,136,589 | 1704.07058 | 0cc5eac26a75ac7153823edf4c787b24151c1ef2 |
Sparse approximate solution of fitting surface to scattered points by MLASSO model
Apr 2017. 2010. 2013
Y-X Hao
C-J Li
R-H Wang
Sparse approximate solution of fitting surface to scattered points by MLASSO model
SCIENCE CHINA Mathematics journal sample. Sci China Math
651756Apr 2017. 2010. 201310.1007/s11425-000-0000-0Received ; acceptedSparse solutionPrinciple shift invariant spaceL 1 norm minimizationAlternating direction method of multipliersMLASSO model
The goal of this paper is to achieve a computational model and corresponding efficient algorithm for obtaining a sparse representation of the fitting surface to the given scattered data. The basic idea of the model is to utilize the principal shift invariant (PSI) space and the l 1 norm minimization. In order to obtain different sparsity of the approximation solution, the problem is represented as a multilevel LASSO (MLASSO) model with different regularization parameters. The MLASSO model can be solved efficiently by the alternating direction method of multipliers. Numerical experiments indicate that compared to the AGLASSO model and the basic MBA algorithm in[35], the MLASSO model can provide an acceptable compromise between the minimization of the data mismatch term and the sparsity of the solution. Moreover, the solution by the MLASSO model can reflect the regions of the underlying surface where high gradients occur.
Introduction
Sparse representation of a function via a linear combination of a small number of functions has recently received a lot of attention in several mathematical fields such as approximation theory [13,33,41,42], compressed sensing, signal and image processing [7][8][9][10] etc. The problem can be described as follows. Consider a linearly dependent set of n functions {ϕ i } n i=1 and a function f represented as f = n i=1 X i ϕ i . Since the set of functions is not linearly independent, this representation is not unique. The problem is then to find the sparsest solution, i.e., the coefficient vector X = (X 1 , X 2 , . . . , X n ) has as many zero components as possible (referred to minimizing the l 0 norm of the vector X). This optimization problem is NP-hard, since the l 0 norm is nonconvex and discontinuous. Hence, much attention has been paid to solutions minimizing X 1 = n i=1 |X i | instead. In this paper, we consider the problem of reconstructing a surface from scattered data using a sparse representation. The scattered data fitting problem arises in many applications, such as signal processing, computer graphics and neural networks [32]. In a typical scattered data reconstruction problem, we are given a set of scattered points Ξ = {x 1 , x 2 , . . . , x N } ⊆ R 2 and associated noisy function values f = f | Ξ + n = {f 1 , f 2 , . . . , f N }, where n is the error vector. Then we seek a function g which fits the given data {(x i , f i )} N i=1 well. There are a lot of existing methods and algorithms in the literature. Various methods can be found in a survey on scattered data interpolation [38]. For approximation methods, Bsplines have a solid mathematical foundation and have been used in many literatures, such as [34,51] etc. Wavelet frames have also been used to reconstruct implicit surfaces from unorganized point sets in R 3 [16]. In order to control the local and global fitting error simultaneously, adaptive methods are presented in [11,44]. The adaptivity is achieved by a portion of the data with a patch, testing the fit for satisfaction within a given tolerance, and subdividing the patch if the tolerance is not met [23]. In addition, several approximation methods employ a multilevel structure to approximate data efficiently. In particular, a multilevel scheme based on B-splines is proposed in [35] to approximate scattered data. These methods run on the approximation space S = J j=1 S j , where S j ⊆ S j+1 are principal shift invariant (PSI) spaces generated by a single compactly supported function ϕ. The multilevel approximation procedure is as follows: for each level j, the point set (Ξ, △g j−1 ) is approximated by a function g j ∈ S j obtained by the least square method, where △g j−1 | Ξ =f − (g 1 + · · · + g j−1 )| Ξ . The procedure is terminated until certain conditions are satisfied. Then the final approximation surface is g = g 1 + g 2 + · · · + g J , g j ∈ S j , j = 1, . . . , J.
However, the methods above do not produce the sparse representation of the surface.
In this paper, we present an efficient method to obtain a sparse representation of the fitting surface to the given scattered points. We still choose the space S defined above as the approximation space. But instead of using the multilevel scheme, we put all the basis functions of S j , 1 j J together as a frame of S. Denote the basis functions of S j as {ϕ j i } nj i=1 , 1 j J, then S = span{ϕ j i } nj ,J i,j=1 . We then try to find the fitting surface g ∈ S as:
g = J j=1 g j , g j = nj i=1 X j i ϕ j i .
Since the functions {ϕ j i } nj,J i,j=1 are linearly dependent, the representation of g as above is not unique and we will seek a relatively sparse one. The choice of the space S makes a sparse representation of g exist and the function ϕ j i can be constructed in a multilevel way. We use the similar approach as those used in compressed sensing, i.e., to use the l 1 norm of the coefficient vector as the regularization term. Thus, the problem can be represented by the following minimization
min g∈S N i=1 (g(x i ) − f i ) 2 + J j=1 λ j X j 1 , (1.1) λ j X j 1 ,(1.2)
where A j is the observation matrix defined by
A j (i, k) = ϕ j k (x i ), i = 1, 2, .
.., N, k = 1, 2, ..., n j , j = 1, 2, ..., J.
For the case λ 1 = · · · = λ J , the model reduces to the model presented in [30]. Moreover, the l 1 related minimization resulting from our proposed model can be efficiently solved using the alternating direction method of multipliers (ADMM) [24,54].
This framework combines the ideas developed in compressed sensing with well-known concepts arising in adaptive and multilevel finite element methods. The solution of the l 1 minimization problem and the multilevel basis functions are used to control the grid refinement and adaptivity. Since we aim at finding a sparse representation of the surface, we discard those coefficients which are smaller than a certain threshold. Then only large coefficients of the solution are left which indicate important contributions of the underlying surface. Moreover, these large coefficients belong to the parts of the surface that have large fluctuations. It seems a little similar for our method and the approach in [15], both using the PSI space. However, the method in [15] was used to approximate functions expressed as a infinite sum of wavelet decomposition by a finite sum, while we deal with scattered data fitting problem by representing the fitting function as a finite sum directly with certain accuracy and more sparse coefficients. The behavior of our method is demonstrated via four examples: a discontinuous function, a non-smooth function, a smooth function and the Franke test function. In addition, we compare the numerical results with the AGLASSO model and the basic MBA algorithm in [35] followed by the same thresholding.
The rest of the paper is organized as follows. In the next section, we recall the main ingredients of the PSI space which will be used here. Moreover, we will propose the sparsity based regularization model for scattered data fitting. In Section 3, the ADMM algorithm will be applied to solve the minimization problem resulted from the proposed model. Numerical experiments are also performed to illustrate the algorithm. Section 4 is the conclusion.
2 Sparse solution of PSI approach to scattered data approximation For a given set of scattered points {x i } N i=1 ⊆ Ω ⊆ R 2 and the corresponding noisy data {f i } N i=1 , our task is to reconstruct a fitting surface with a sparse representation.
PSI space and l 1 regularization
Let Ω ⊆ R 2 be a bounded domain of interest where all data lie in and let ϕ be a carefully chosen, compactly supported function (e.g. uniform B-spline, box spline, radial basis functions). Denote
Λ = {k ∈ Z 2 : supp(ϕ(·/h − k)) ∩ Ω = ∅}, S h (ϕ, Ω) = {f |f = k∈Λ c k ϕ(·/h − k)},
where h > 0 is a scaling parameter that controls the refinement of the space. Denote S j = S h/2 j−1 (ϕ, Ω), S 1 ⊆ · · · ⊆ S J , we then look for a function g ∈ S J which fits closely the given data. Then g is composed of a sequence of functions as g = g 1 + g 2 + · · · + g J ,
where g i ∈ S i , i = 1, 2, ..., J.
Here we choose a proper PSI space generated by B-spline as the approximation space S since it enjoys desirable properties for data fitting. It has a simple structure and provides good approximation to smooth functions, which leads to simple and accurate algorithms. Moreover, it can be associated to a wavelet or frame system and hence one can solve the fitting problem by making use of the advantages that a wavelet (frame) system can offer [30]. These advantages include sparse approximation of functions in the wavelet (frame) domain, multilevel structure of basis functions, adaptivity to the data, norm equivalence, etc.
Recall that a function ϕ is said to satisfy the Strang-Fix conditions of order m if ϕ(0) = 0, D αφ (2πj) = 0, ∀j ∈ Z 2 \ 0, |α| < m.
Denote W m 2 = {f ∈ L 2 (R 2 ) : f W m 2 = √ 2π (1 + | · | m )f L2(R 2 ) < +∞},f − s L 2 (R 2 ) = O(h m ).
Particularly, the B-spline B m of order m satisfies the Strang-Fix conditions of order 2 for all m 2 [16]. For more detailed discussions on PSI space, see [14]. Obviously, the union set of the basis functions of S j is not linearly independent. Thus the representation of g is not unique and we want to determine a relatively sparse one, i.e., a representation with as many vanishing coefficients as possible. Every function g j ∈ S j can be written as
g j = k∈Ij X j k ϕ(2 j−1 x/h − k), where I j = {k ∈ Z 2 , supp(ϕ(2 j−1 · /h − k)) ∩ Ω = ∅}.
Let X j and f denote the column vector {X j k } k∈Ij and {f i } 1 i N respectively, then the problem can be formulated as follows.
min Xj ,1 j J J j=1 A j X j − f 2 2 + J j=1 λ j X j 1 , (2.1)
where A j is the observation matrix defined by
A j (i, k) = ϕ(2 j−1 x i /h − k), k ∈ I j , i = 1, 2, ..., N, j = 1, 2, ..., J.
Obviously, the model (2.1) balances the fitting accuracy and the l 1 norm. In order to achieve a sparse representation, small coefficients are neglected. That is, after obtaining the solution {X j } J j=1 of the model (2.1), we discard the small elements of X j , 1 j J. Then the final solution only has large values left which indicate important contributions (fluctuations) of the real surface. Furthermore, comparing with the multilevel approximation approach given in [35], our method has the advantages of simplicity. Another important distinction is that it can be interpretable as a sparse strategy for reconstructing scattered data.
The MLASSO model
The model (2.1) is related to the LASSO model in some extent. Recall that the mathematical model of LASSO is: min
X AX − f 2 2 + µ X 1 . X j 2 ,
where µ > 0 is a regularization parameter. The GLASSO model was proposed to perform variable selection on groups of variables for linear regression models. It has many applications in areas such as computer vision, data mining, etc. Meier et al. in [39] extended the GLASSO to logistic regression. The GLASSO does not, however, yield sparsity within a group. Moreover, GLASSO suffers from estimation inefficiency and selection inconsistency. To remedy these problems, the adaptive GLASSO method (AGLASSO) is proposed in [49] as:
min Xj ,1 j J J j=1 A j X j − f 2 2 + J j=1 µ j X j 2 .
Obviously, the model (2.1) reduces to the LASSO model when µ = λ 1 = · · · = λ J . Moreover, the model (2.1) acts like the LASSO at the multilevel. Therefore, we denote the model (2.1) as the multilevel LASSO model (MLASSO). Compared with the AGLASSO model, the MLASSO model considers an additional penalty on the l 1 norm instead of l 2 norm of the regression coefficient vector, and it produces as election of variables with sparsity among different levels. It is known that when l 2 norm regularization term is applied to the data set, the resulting surface tends to be smooth without sharp discontinuities but have undesirable oscillations near the discontinuities [30]. Recently, several surface reconstruction approaches have been proposed to preserve surface discontinuity by replacing l 2 regularization using more sophisticated regularization, e.g., the Huber approximation of l 2 norm of function derivatives in [47], the local kernel regularization in [25] and the non-local means regularization in [16]. In our method, we use the l 1 regularization instead, in order to obtain the relatively sparse solution. The regularization term · 1 in (2.1) penalizes the roughness of the solution.
The parameter selection
The determination of the proper value of {λ j } in the MLASSO model is an important problem and depends on the variance of the noise n, the properties of {A j } and · 1 . An appropriate choice of the regularization parameters is of vital importance for the quality of the resulting estimate and has been the subject of extensive research [3]. In recent years, there has been a growing interest in sophisticated regularization techniques which use multiple constraints as a mean of improving the quality of the solution [3]. Among them, only a few papers discussed the choice of multiple regularization parameters. However, most of them discuss the case that the regularization term is the l 2 norm. For example, a multi-parameter generalization of heuristic L-curve has been proposed in [3], a knowledge of noise (covariance) structure is required for a choice of parameter in [1,2,12], some reduction to a single parameter choice is suggested in [6]. At the same time, the discrepancy principle, which is widely used and known as the first parameter choice strategy proposed in the regularization theory [40], has been discussed in a multi-parameter context in [37]. Of course, there might be many different methods to choose the regularization parameters satisfying certain principle. In fact, the choice of the regularization parameters in the regularization modes, such as the LASSO model and the standard Tikhonov regularization model, have not yet solved. No choice is available for all the models. Basically, different models need different methods to decide the parameters, and even the same one may have different methods to choose, such as [27,28,49,50,55].
For the MLASSO model, instead of discussing similar methods as listed above, we propose two simple criteria for the choice of the parameters {λ j } J j=1 according to the sparsity and the support of the basis functions of the different levels. On one hand, since the length of X j is less than that of X j+1 , so if one wants to obtain sparser solution, the parameters of the last several levels should be larger. In particular, choosing the same value for all the parameters, i.e., λ 1 = · · · = λ J , obtains the global sparsity in the whole space S. On the other hand, the support of the basis functions of the j-th level is larger than that of the (j + 1)-th level, so if one wants smaller support, the parameters of the first several levels should be larger. Hence, the more appropriate choice of the parameters is to make the parameters of the first and last several levels greater than the middle several levels. We can only give such a qualitative guideline, since quantitative guidance for regularization parameters is rarely available. In this way, the solution has smaller support and sparser representation with good approximation accuracy. Numerical results also confirm this selection method as shown in the Tables 1-8 3 Numerical algorithm
Algorithm
In this section, we use the ADMM algorithm to solve the minimization model (2.1) for experimental evaluation. It turns out that ADMM is equivalent to or closely related ro many famous algorithms, such as the Douglas-Rachford splitting method in PDE literature [18,19,36], the Bregman iterative algorithms for l 1 problems in signal processing [26] and many others. In particular, we refer to [21,45,46,52] for the relationship between ADMM and the split Bregman iteration scheme which is very influential in the area of image processing. Convergence analysis of the ADMM was given in [5,20,29].
In order to solve the MLASSO model (2.1) and guarantee the convergence, we denote
A = (A 1 , A 2 , . . . , A J ), X = (X T 1 , X T 2 , . . . , X T J ) T , λ = diag(λ 1 , λ 2 , . . . , λ J ) is a diagonal matrix with λ j , j = 1, ..., J as the main diagonal. Then we can rewrite (2.1) as min X AX − f 2 2 + λX 1 . (3.1)
By introducing an auxiliary variable d = λX, we convert the unconstrained minimization problem (3.1) into a constrained one: min
X,d=λX AX − f 2 2 + d 1 . (3.2)
In this way, the MLASSO model (2.1) is turned into a classical l 1 minimization problem. The augmented Lagrangian function of problem (3.2) is
L(X, d, b) = AX − f 2 2 + d 1 + β 2 λX − d 2 2 + < b, λX − d >,
where β > 0 is a parameter of the algorithm. Then by applying the ADMM method, given the initialization {X 0 , d 0 , b 0 } and the parameters {λ j } J j=1 , it results in the following optimization algorithm:
X k+1 = arg min X AX − f 2 2 + β 2 λX − d k + b k β 2 2 , d k+1 = arg min d d 1 + β 2 λX k+1 − d + b k β 2 2 , b k+1 = b k + (λX k+1 − d k+1 ). (3.3)
First of all, the above algorithm is convergent, since it is just the classical ADMM for two block of variables. Secondly, the system can be simply solved. The first step of each iteration in (3.3) is
(2A T A + βλ T λ)X = 2A T f + βλ T (d k − b k β ).
This linear system is positive definite and therefore it can be solved by the conjugate gradient method (CG). When the matrix is ill-posed, i.e., its condition number is huge, the convergence rate of the CG will be very slow. Under this case, the preconditioned CG [4,22,43] can be used instead to reduce the condition number of the coefficient matrix and improve the convergence speed. The second subproblem has a simple analytical solution based on soft-thresholding operator [17], that is
d k+1 = T 1 β (λX k+1 + b k β ),
where T θ is the soft-thresholding operator defined by
T θ : x = [x 1 , x 2 , . . . , x M ] → T θ (x) = [t θ (x 1 ), t θ (x 2 ), . . . , t θ (x M )],
where t θ (ξ) = sgn(ξ)max{0, |ξ| − θ}.
The complete description of the algorithm for solving the model (2.1) is provided as Algorithm 1 as follows:
Algorithm 1. (Adapted ADMM for solving the MLASSO model (2.1))
Step 1) Set J and the initial values {X 0 , d 0 , b 0 }, choose appropriate sets of parameters {λ j } J j=1 , β and two thresholds σ, ǫ;
Step 2) For k = 0, 1, . . ., perform the iteration (3.3) until convergence;
Step 3) AssumeX is the solution obtained from Step 2), if |X(i)| σ, i.e., the absolute value of the i-th element ofX is less than σ, setX(i) = 0;
Step 4) The final solution X areX after the treatment of Step 3).
In our numerical experiments, the initializations are X 0 = d 0 = b 0 = 0, β = 1 and the stopping criteria is
d k − λX k 2 ǫ.
Numerical experiments
Given a test function f ( ).
Then apply different methods to obtain the approximation function g. The difference between f and g is measured by computing the normalized RMS (root mean square) error [35]. That is,
RM S = M1,N1 i,j=1 (g(x i ,ỹ j ) − f (x i ,ỹ j )) 2 M 1 N 1 , wherex i = −1 + 2i M1−1 , i = 0, 1, . . . , M 1 − 1,ỹ j = −1 + 2j N1−1 , j = 0, 1, . . . , N 1 − 1, M 1 = N 1 = 50. Moreover, denote Error = N i=1 (g(x i , y i ) − f (x i , y i )) 2 N
as the fitting error of the given scattered points
{(x i , y i )} N i=1 ⊆ Ω.
To demonstrate the accuracy of the proposed algorithm, we perform experiments with four functions: a discontinuous function f 1 , a non-smooth function f 2 , a smooth function f 3 and the Franke test function f 4 as follows.
f 1 (x, y) = x 2 y x 2 +y 2 , x 2 + y 2 1, x + y, x 2 + y 2 > 1. f 2 (x, y) = xy √
x 2 +y 2 , x 2 + y 2 1, xy, x 2 + y 2 > 1.
f 3 (x, y) = 1.25 + cos(5.4y) 6 + 6(3x − 1) 2 . (m) 900 randomly scattered data points for f 4 .
f 4 (x, y) = 0.75 exp[− (9x − 2) 2 + (9y − 2) 2 4 ] + 0.75 exp[− (9x + 1) 2 49 − (9y + 1) 2 10 ] + 0.5 exp[− (9x − 7) 2 + (9y − 3) 2 4 ] − 0.2 exp[−(9x − 4) 2 − (9y − 7) 2 ].✲ ✲ ✁ ✂ ✄ ✁ ✁ ✂ ✄ ✲ ✲ ✁ ✂ ✄ ✁ ✁ ✂ ✄ ✁ ✁ ✂ ✄ ✂ ✄ ✷ (n)
The method in [35]. (l 0 (X 1 ), l 0 (X 2 ), l 0 (X 3 )) Error RMS Iterations Time(sec) [ (l 0 (X 1 ), l 0 (X 2 ), l 0 (X 3 ), l 0 (X 4 )) Error RMS Iterations Time(sec) [ Fig. 4 The distribution of the support for f 2 with J = 3. Table 3. The l 0 norm and the approximation errors for different parameters shown in Fig.4 for f 2 with J = 3.
(l 0 (X 1 ), l 0 (X 2 ), l 0 (X 3 )) Error RMS Iterations Time(sec) [ (l 0 (X 1 ), l 0 (X 2 ), l 0 (X 3 ), l 0 (X 4 )) Error RMS Iterations Time(sec) [ Fig. 6 The distribution of the support for f 3 with J = 3. Table 5. The l 0 norm and the approximation errors for different parameters shown in Fig.6 for f 3 with J = 3.
(l 0 (X 1 ), l 0 (X 2 ), l 0 (X 3 )) Error RMS Iterations Time(sec) [35] (24, 63, 193) 3.0285e-3 5.8147e-3 3 0.0617 Table 6. The l 0 norm and the approximation errors for different parameters shown in Fig.7 for f 3 with J = 4.
(l 0 (X 1 ), l 0 (X 2 ), l 0 (X 3 ), l 0 (X 4 )) Error RMS Iterations Time(sec) [ Table 7. The l 0 norm and the approximation errors for different parameters shown in Fig.8 for f 4 with J = 3.
(l 0 (X 1 ), l 0 (X 2 ), l 0 (X 3 )) Error RMS Iterations Time(sec) [ Table 8. The l 0 norm and the approximation errors for different parameters shown in Fig.9 for f 4 with J = 4. (l 0 (X 1 ), l 0 (X 2 ), l 0 (X 3 ), l 0 (X 4 )) Error RMS Iterations Time(sec) [ For the numerical implementation in this paper, we employ the 2D tensor product quadratic B-spline as the function ϕ. We show that such a simple system with the Algorithm 1 can be used to effectively reconstruct surface of sparse representation from a scattered data set. The choices of the threshold σ, ǫ can be chosen according to the accuracy and sparsity. We consider the four functions with σ = 10 −3 , ǫ = 10 −4 , N = 900 and the 900 scattered points are chosen randomly. Moreover, we compare Algorithm 1 with the basic MBA algorithm presented in [35] under the termination condition |△g j | O(10 −3 ) and the AGLASSO model, both applying the same thresholding σ = 10 −3 as Algorithm 1 to their results.
We experiment the above three methods for J = 3 and J = 4 starting from level 1 with 5 × 5 biquadratic B-spline functions respectively. Fig.1 shows the scattered data points and the corresponding approximations of the four functions with Algorithm 1, the method in [35] and the AGLASSO model respectively. Fig.2-9 illustrate the distribution of the support of the B-spline functions with nonzero coefficients for J = 3 and J = 4 respectively. The red, green, blue and black rectangles denote the support of the B-splines corresponding to X 1 , X 2 , X 3 and X 4 respectively. Moreover, Tables 1-8 give the approximation accuracy, the iterations, the running time and the l 0 norm of the solution with different parameters for f 1 − f 4 , where the length of X 1 , X 2 , X 3 and X 4 is 25, 64, 196 and 625 respectively. In addition, all our calculations are done in Matlab on a laptop with Inter Core i7 (2.90GHZ) CPU and 8.0G RAM. Discussion 1. The numerical results demonstrate that Algorithm 1 and the method in [35] have almost the same approximation errors, while Algorithm 1 obtains the sparse solution. Compared with the AGLASSO model, Algorithm 1 provides the sparser solutions with less error, though more iterations and more time.
Through the first two steps of Algorithm 1, we can obtain an approximation solution of the MLASSO model with no sparsity and the solution has some big components and some small ones which reflect different importance and contribution. Then by step three, we throw out small components of the computed solution which means we keep only the important ones with great contribution to the solution. Therefore, by choosing appropriate regularization parameters, the final solution can indicate the important parts we are interested in and identify the important features within the selected levels simultaneously of the exact surface since they are all determined by the big components. Experiment results verify this conclusion.
Taken together, Algorithm 1 can reconstruct the test functions reasonably with a sparse representation within a few levels by choosing some appropriate regularization parameters.
Conclusion
This paper presents an approach for scattered data fitting using the PSI space and the l 1 regularization. It is concluded into the MLASSO model which allows us to balance the accuracy and the sparsity of the fitting surface. The model can be solved using Algorithm 1 with the ADMM algorithm. Numerical examples demonstrate that compared to the basic MBA algorithm in [35] and the AGLASSO model, the MLASSO model provides an efficient, sparse, flexible and reasonable solution. Moreover, the distribution of the basis functions of the sparse solution can identify the regions of the underlying surface where large fluctuations occur.
. Moreover, in contrast with the methods listed above, this choice is easier and cheaper in the sense of computational cost.Based on the above, the proposed method offers some interesting advantages: 1) Related with the LASSO model, the MLASSO model has a strong statistical background; 2) Compared to the LASSO model, the parameters in the MLASSO model can be chosen differently, thus one can attain high flexibility for the approximation accuracy and the sparsity of the solution; 3) Compared to the AGLASSO model, the MLASSO model can provide the sparsity of the solution by choosing different parameters and the distribution of the solution can reflect the large fluctuations of the underlying surface; 4) Utilizing the ADMM algorithm, the MLASSO model can be efficiently solved.
x, y) with Ω = [−1, 1] × [−1, 1], we first sample data points with certain noises from it, i.e., {(x i , y i , f (x i , y i )+ε i )}. The error vector n = (ε i ) i , whose entries consist of the pseudorandom values drawn from the standard uniform distribution on the open interval (
randomly scattered data points for f 1 .
randomly scattered data points for f 2 .
randomly scattered data points for f 3 .
Fig. 1 .
1The scattered data points and the corresponding approximation surfaces: (a)-(d) for f 1 , (e)-(h) for f 2 , (i)-(l) for f 3 , (m)-(p) for f 4 .
) J = 3 for AGLASSO: λ 1 = 0.02, λ 2 = 0.01, λ 3 = 0.03.
Fig. 2
2The distribution of the support for f 1 with J = 3.Table 1. The l 0 norm and the approximation errors for different parameters shown inFig.2for f 1 with J = 3.
) J = 4 for AGLASSO: λ 1 = λ 4 = 0.02, λ 2 = 0.001, λ 3 = 0.01.
Fig. 3
3The distribution of the support for f 1 with J = 4 .Table 2. The l 0 norm and the approximation errors for different parameters shown inFig.3for f 1 with J = 4.
) J = 3 for AGLASSO: λ 1 = 0.03, λ 2 = 0.01, λ 3 = 0.02.
) J = 4 for MLASSO: λ 1 = λ 2 = λ 3 = λ 4 = 0.001.
) J = 4 for AGLASSO: λ 1 = 0.05, λ 2 = 0.01, λ 3 = 0.02, λ 4 = 0.05.
Fig. 5
5The distribution of the support for f 2 with J = 4 .
Table 4 .
4The l 0 norm and the approximation errors for different parameters shown in Fig.5 for f 2 with J = 4.
) J = 3 for MLASSO: λ 1 = 0.03, λ 2 = 0.02, λ 3 = 0.05.
) J = 3 for AGLASSO: λ 1 = 0.03, λ 2 = 0.02, λ 3 = 0.05.
Fig. 6
6) J = 4 for AGLASSO: λ 1 = 0.05, λ 2 = 0.01, λ 3 = 0.02, λ 4 = 0.05.
Fig. 7
7The distribution of the support for f 3 with J = 4.
) J = 3 for AGLASSO: λ 1 = 0.03, λ 2 = 0.02, λ 3 = 0.05.
Fig. 8
8The distribution of the support for f 4 with J = 3.
) J = 4 for MLASSO: λ 1 = λ 2 = λ 3 = λ 4 = 0.001.
) J = 4 for MLASSO: λ 1 = λ 2 = λ 3 = λ 4 = 0.01.
) J = 4 for MLASSO: λ 1 = 0.03, λ 2 = 0.02, λ 3 = 0.02, λ 4 = 0.04.
) J = 4 for AGLASSO: λ 1 = 0.03, λ 2 = 0.02, λ 3 = 0.02, λ 4 = 0.04.
) J = 4 for AGLASSO: λ 1 = 0.05, λ 2 = 0.03, λ 3 = 0.02, λ 4 = 0.04.
Fig. 9
9The distribution of the support for f 4 with J = 4.
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| []
|
[
"Ultraviolet Number Counts of Galaxies from Swift UV/Optical Telescope Deep Imaging of the Chandra Deep Field South",
"Ultraviolet Number Counts of Galaxies from Swift UV/Optical Telescope Deep Imaging of the Chandra Deep Field South"
]
| [
"E A Hoversten ",
"C Gronwall ",
"D E Vanden Berk ",
"T S Koch ",
"A A Breeveld ",
"P A Curran ",
"D A Hinshaw ",
"F E Marshall ",
"P W A Roming ",
"M H Siegel ",
"M Still "
]
| []
| []
| Deep Swift UV/Optical Telescope (UVOT) imaging of the Chandra Deep Field South is used to measure galaxy number counts in three near ultraviolet (NUV) filters (uvw2: 1928Å, uvm2: 2246Å, uvw1: 2600Å) and the u band (3645 A). UVOT observations cover the break in the slope of the NUV number counts with greater precision than the number counts by the Hubble Space Telescope (HST) Space Telescope Imaging Spectrograph (STIS) and the Galaxy Evolution Explorer (GALEX ), spanning a range from 21 m AB 25. Number counts models confirm earlier investigations in favoring models with an evolving galaxy luminosity function. | 10.1088/0004-637x/705/2/1462 | [
"https://arxiv.org/pdf/0910.0033v1.pdf"
]
| 14,270,642 | 0910.0033 | 3048106b8ccf31bd4ec3d25f6b2b651df6867a04 |
Ultraviolet Number Counts of Galaxies from Swift UV/Optical Telescope Deep Imaging of the Chandra Deep Field South
30 Sep 2009
E A Hoversten
C Gronwall
D E Vanden Berk
T S Koch
A A Breeveld
P A Curran
D A Hinshaw
F E Marshall
P W A Roming
M H Siegel
M Still
Ultraviolet Number Counts of Galaxies from Swift UV/Optical Telescope Deep Imaging of the Chandra Deep Field South
30 Sep 2009Subject headings: galaxies: evolution -galaxies: UV properties -galaxies: number density -Chandra Deep Field South
Deep Swift UV/Optical Telescope (UVOT) imaging of the Chandra Deep Field South is used to measure galaxy number counts in three near ultraviolet (NUV) filters (uvw2: 1928Å, uvm2: 2246Å, uvw1: 2600Å) and the u band (3645 A). UVOT observations cover the break in the slope of the NUV number counts with greater precision than the number counts by the Hubble Space Telescope (HST) Space Telescope Imaging Spectrograph (STIS) and the Galaxy Evolution Explorer (GALEX ), spanning a range from 21 m AB 25. Number counts models confirm earlier investigations in favoring models with an evolving galaxy luminosity function.
Introduction
Galaxy number counts as a function of magnitude provide direct constraints on galaxy evolution in both luminosity and number density. Number counts in the UV, in particular, can help trace the star formation history of the universe. Until recently obtaining faint galaxy number counts in the UV has been difficult due to the small areas surveyed (Gardner, Brown, & Ferguson 2000;Deharveng et al. 1994;Iglesias-Páramo et al. 2004;Sasseen et al. 2002;Teplitz et al. 2006). While the Galaxy Evolution Explorer (GALEX) has allowed for the measurement of UV galaxy number counts over a wide field of view (Xu et al. 2005, ∼ 20 deg 2 ), the confusion limit of GALEX restricts the magnitude range covered to 14 to 23.8 m AB . The deepest UV number counts from HST range from m AB = 23 to 29 over an extremely small field of view of ∼ 1.3 square arcminutes.
Here, we present galaxy number counts obtained in 3 near UV filters (1928Å, 2246Å, 2600Å) as well as in the u band (3645Å) obtained using the Swift UV/Optical Telescope (UVOT; Roming et al. 2005). Deep exposures were taken of a 289 square arcminute field of view overlapping the Chandra Deep Field South (CDF-S; Giacconi et al. 2002) allowing for the measurement of number counts from m AB = 21 to 26. UVOT data covers the break in the slope of the NUV number counts with greater precision than the existing GALEX and HST number counts. We use the UVOT number counts to explore the evolution of star-forming galaxies out to z ∼ 1.
Data & Analysis
The CDF-S was observed with UVOT, one of three telescopes onboard the Swift spacecraft (Gehrels et al. 2004), the primary mission of which is to study gamma-ray bursts (GRB). The UVOT is a 30 cm telescope with f -ratio 12.7 (Roming et al. 2005). It has two grisms and seven broadband filters. The central wavelengths and widths of the uvw2, uvm2, uvw1, and u filters used in this paper can be found in Table 1. For a more detailed discussion of the filters, as well as plots of the responses, see (Poole et al. 2008).
Observations of the CDF-S were made between July 7, 2007 andDecember 29, 2007. CDF-S images are unbinned with a pixel scale of 0.5 arcseconds. UVOT data processing is described in the UVOT Software Guide 1 . The data were processed with a version of the UVOT pipeline in which exposure maps are aspect corrected. This feature is not currently available for data currently in the archive but will appear in future versions of the pipeline. Image files and exposure maps were summed using UVOTIMSUM from the publicly available UVOT FTOOLS (HEAsoft 6.6.1) 2 . This involves two flux conserving interpolations of the images, the first to convert from the raw frame to sky coordinates, and the second when summing the images. Bad pixels are known and a correction is applied. UVOT, as is the case with all microchannel plate intensified CCDs, is insensitive to cosmic rays. The maximum exposure time in each filter is given in Table 1.
Because Swift is optimized for fast slewing to accomplish its GRB mission, the pointing 1 http://heasarc.gsfc.nasa.gov/docs/swift/analysis 2 http://heasarc.gsfc.nasa.gov/docs/software/lheasoft/ accuracy is of order 1 to 2 arcminutes. In addition the requirement that the solar panels face towards the Sun causes the field of view to rotate over the course of the year. As a result the exposure times vary significantly across the summed images. Exposure maps are nearly uniform in the center but become complicated on the edges. Table 1 gives the area covered where the exposure time is at least 98% of the maximum exposure time in each filter. The 98% value was chosen to maximize the area used in this study while simultaneously maintaining a magnitude limited sample.
The area in each filter covered by the 98% exposure time criterion is shown in Figure 1. For comparison the area covered by the CDF-S, Hubble Ultra Deep Field (Beckwith et al. 2006), and Great Observatories Origins Deep Survey (Giavalisco et al. 2004) is shown by the labeled contours. A false color image of the central region of the CDF-S using the uvw2, uvm2, and uvw1 images is shown in Figure 2.
Photometry was performed using SExtractor (Bertin & Arnouts 1996), a publicly available code designed for the identification and measurement of astrophysical sources in large scale galaxy survey data. A full listing of the SExtractor parameters used is provided in the online version of Table 2. The background map, which measures the local background due to the sky and other sources, was generated internally by SExtractor. To improve the detectability of faint extended sources the filtering option was used with a Gaussian filter. The filter size was selected to match the full width half maximum (FWHM) of the point spread function (PSF) as recommended in the SExtractor manual. The PSF was measured from the CDF-S image for each filter using one star. There was only one star which was bright enough and isolated enough to accurately measure the outer regions of the PSF. The PSFs used were 3.30" in uvw2, 2.87" in uvm2, 2.86" in uvw1, and 2.67" in u. Magnitudes were calculated from MAG AUTO which is designed to be the best measure of the total magnitudes of galaxies. SExtractor was used to process count rate images created by dividing the summed images by the exposure map. The resulting output was converted to flux using the values given by Poole et al. (2008) for stellar spectra. The fluxes were then converted to AB magnitudes (Oke 1974). The number of sources detected in each band ranges from 888 to 1260 and is given in Table 1 along with the area covered in each image.
The UVOT detector is a microchannel plate intensified CCD which operates in photon counting mode. As such it is subject to coincidence loss which occurs when two or more photons arrive at a the same location on the detector within a single CCD readout interval of 11 ms (Fordham, Moorhead, & Galbraith 2000). When this happens only one photon will be counted, which systematically undercounts the true number of photons. The coincidence loss correction is at the 1% level for m AB ∼ 19 in the UVOT filters we use in this paper. For the magnitude ranges considered in our number counts the coincidence loss is insignificant and no attempts are made to correct for it.
By design the CDF-S is on a line of sight with very low Galactic extinction. In addition the area covered by the UVOT observation is around 130 square arcminutes, depending on the filter, so variations in extinction across the field are small. According to the dust maps of Schlegel, Finkbeiner, & Davis (1998) the range of Galactic extinction in our field is 0.020 ≤ A V ≤ 0.030. Our photometry is corrected for Galactic extinction based on the position of the source assuming the Milky Way dust curve of Pei (1992). The extinction correction is largest in the uvm2 filter as it is centered on the 2175Å dust feature which is pronounced in the Milky Way. The extinction correction ranges from 0.053 ≤ A uvm2 ≤ 0.086 across the field in uvm2 which demonstrates that the extinction correction is not a significant source of error in any of the filters.
Bias Corrections
The raw number counts suffer from several biases which need to be quantified. Completeness addresses the inability to detect an object either due to confusion with other sources or limitations in the photometry. Eddington bias (Eddington 1913) occurs because magnitude errors will preferentially scatter objects into brighter magnitude bins because there are generally more objects at fainter magnitudes. There is also the potential for false detections of objects due to noise.
These first two problems can be addressed simultaneously with a Monte Carlo simulation, following the procedure set out in Smail et al. (1995). For each of the four images, synthetic galaxies were added and the analysis repeated. Synthetic galaxies were placed at random locations on the image. The magnitudes of the synthetic galaxies were between 21 and 27 in uvw2, uvm2 and uvw1 and between 20 and 25.5 in u and the relative numbers by magnitude follow the observed distribution from the original SExtractor photometry in the relevant filter. The synthetic galaxies are given exponential profiles with semi-major axes and ellipticities that match the observed distribution as a function of magnitude. Individual photon arrivals are modeled using Poisson statistics and following the galaxy profile. The resulting image is then convolved with the UVOT PSF for the final image.
For each filter a single synthetic galaxy was added to the real image and the photometry process described in §2 was redone. The resulting photometry catalog was checked to determine if the synthetic galaxy was detected and at what magnitude. This was repeated 50,000 times for each filter to build up statistics on the completeness. The number counts were corrected by dividing by the fraction of synthetic galaxies detected in the relevant mag-nitude bin. These values are tabulated in Table 3. Following Smail et al. (1995) the number counts are truncated where the completeness correction exceeds 50%. The Poisson error bars on the number counts are also divided by the completeness correction to take into account uncertainties introduced by the completeness correction.
Correcting for false detections was also done using the methods of Smail et al. (1995). Using the exposure time and background count rate calculated from the background map output by SExtractor noise frames were simulated for each filter. The photometry methods described in §2 were repeated for each frame and the number of false detections recorded as a function of magnitude. For each filter 100 noise frames were analyzed. The number of spurious sources, N spur , is shown as a function of magnitude for each filter in Table 3. Out of all the simulated frames only one spurious source was detected. Given our deep exposures the completeness correction truncates our number counts well before background noise becomes an issue.
Galaxy number counts can be overestimated due to contamination by Galactic stars and quasars. The fraction of objects in the field that are quasars is estimated by position matching the four UVOT photometry catalogs with the Extended Chandra Deep Field South X-ray point source catalog (Lehmer et al. 2005). Objects with X-ray detections are assumed to be quasars. The number of such sources in each band is 11, 11, 14, and 21 for the uvw2, uvm2, uvw1, and u bands respectively. This represents 1.2, 1.0, 1.1, and 2.3% of the total sample. These sources have been removed from the number counts. The number of AGN per magnitude bin, N AGN , is tabulated in Table 3.
The problem of stellar contamination is greatly reduced by the fact that the line of sight towards the CDF-S is out of the plane of the Milky Way. The CDF-S field was explicitly chosen to be particularly sparse. As a result the field is a statistical outlier, and the stellar contamination in this field will be unusually low. In addition, the fraction of stars with significant UV flux is low, particularly when the field points toward the Galactic halo where the stellar population is very old. This is another reason the stellar contamination in the three NUV filters should be low.
The contamination due to stars is estimated by position matching the UVOT photometry catalogs with objects in the field with stellar classifications in the COMBO-17 survey (Wolf et al. 2004). The COMBO-17 survey includes photometry in 17 passbands for a 30 × 30 arcminute field surrounding the CDF-S. It also contains photometric redshifts and classifications of objects in the survey. The UVOT positions were compared with the objects classified as stars or white dwarfs in COMBO-17. This yields 24, 15, and 40 stars in the uvw2, uvm2, and uvw1 NUV filters which corresponds to 2.7, 1.4, and 3.2% of the total sample. The number of stars per magnitude bin, N star is shown in Table 3. Not all NUV number counts are corrected for stellar contamination (e.g. Gardner et al. 2000). Given the numbers provided in Table 3 the number counts can easily be recalculated without the stellar contamination correction.
However it is different in the u band where more stars have significant fluxes. Position matching yielded 48 stars in the u band which is 5.1% of the total sample. As in the NUV counts the stellar contamination has been corrected for and the details are in Table 3. Capak et al. (2004) provide both raw number counts and the number counts corrected for stellar contamination in the U band from observations around the Hubble Deep Field North (HDFN). The u and U filters are comparable, and the HDFN is similar to the CDF-S in being one of the darkest areas of the sky pointed out of the Galactic disk with low Galactic extinction. The level of stellar contamination in Capak et al. (2004) ranges from 66% at u = 20 to 6% at u = 25. At the bright end of this scale the values are comparable, but at the faint end they are roughly twice as high as in the CDF-S. One possible explanation for this discrepancy is that the Capak et al. (2004) sample covers ∼ 720 arcsec 2 compared to 137 arcsec 2 for this sample. Over this larger area one would expect the number of stars to be closer to the average number expected for that line of sight to the halo, while in our relatively smaller area the number of stars can remain a statistical outlier.
Cosmic variance is another potential source of bias which arises due to local inhomogeneities in the Universe. Galaxies are known to cluster on many different length scales. As a result the density of galaxies will differ along different lines of sight. The smaller the area covered by a survey the more the results will be biased by cosmic variance. A publicly available code from Trenti & Stiavelli (2008) was used to estimate the errors due to cosmic variance in our number counts. This code is based in part on N-body simulations of galaxy structure formation. It uses the area of the survey, mean redshift, range of redshifts observed and the number of objects detected to calculate the error due to cosmic variance. The mean redshift and redshift range of each of our luminosity bins was estimated from the model number counts described in §4. The results show that the uncertainty due to cosmic variance are of the same order as the Poisson errors for all of the filters and magnitude bins used here. We therefore multiply our Poisson errors by a factor of √ 2 to take into account the effects of cosmic variance.
The resulting corrected number counts are shown in Figures 3 (uvw2), 4 (uvm2), 5 (uvw1), and 6 (u). The number counts are also given in Table 3. In Figures 3, 4, and 5 the number counts are plotted along side the NUV number counts from GALEX (Xu et al. 2005) and STIS (Gardner et al. 2000). A color conversion has been applied to shift the GALEX NUV filter and STIS F25QTZ filter in the NUV channel by generating synthetic magnitudes from a catalog of spectral synthesis models with a range of ages and star formation histories and estimating the typical color offset. The GALEX and STIS NUV filters have very similar bandpasses which typically differ by less than 0.01 magnitudes. The uvm2 filter has the tightest relationship with the NUV filters with a color correction of 0.013. The spread is larger in the uvw2 and uvw1 filters, but is still only of order 0.05. In Figure 6 the UVOT u band number counts are compared to the U band counts of Capak et al. (2004) and Eliche-Moral et al. (2006) and the u band measurements of Metcalfe et al. (2001) and Yasuda et al. (2001). Color corrections to the UVOT u band were determined in the same fashion and are equal to 0.81 in U and 0.06 in u.
Models
Simple models of number counts were constructed for both non-evolving and evolving luminosity functions in the UVOT filters. For each model the luminosity function is summed over redshift. This summation includes two corrections. The first is a filter correction to convert the GALEX NUV filter to the UVOT uvw2, uvm2, and uvw1 filters and the U band to the UVOT u filter. The second is a K-correction to convert the observed UVOT filter to the rest-frame UVOT filter. Both of these corrections are a function of redshift.
These corrections were calculated using a model galaxy spectrum generated with the publicly available PÉGASE spectral synthesis code (Fioc & Rocca-Volmerange 1997). For the uvw2, uvm2, and uvw1 filters a starburst galaxy model was used with a constant star formation rate, Solar metallicity, and standard Salpeter IMF at an age of 800 Myr. This was chosen to match the model number counts in Xu et al. (2005) which used the SB4 starburst template of Kinney et al. (1996) because it most closely matched the ratio of the local FUV to NUV luminosity densities described by Wyder et al. (2005). The PÉGASE model is very nearly the SB1 template of Kinney et al. (1996), however we model a range of internal extinctions. An Ω M = 0.3, Ω Λ = 0.7, H 0 = 70 km s −1 Mpc −1 cosmology is used throughout.
The u band models were calculated assuming the cosmic spectrum of Baldry et al. (2002) in addition to the starburst spectrum . The cosmic spectrum is a luminosity weighted average spectrum of galaxies with z 0.1 which makes it a good choice for a template representative of all galaxies. The empirical cosmic spectrum does not extend far enough into the blue to be useful for modeling the UVOT u band let alone passing it through the filters at increasing redshifts. To extend the spectrum into the ultraviolet a template spectrum was created in PÉGASE from the best fitting parameters given by Baldry et al. (2002). The model number counts were corrected for a range of models of internal extinction. Models were calculated for 0 ≤ A V ≤ 2 for the Milky Way, LMC, and SMC dust models of Pei (1992) and the starburst dust model of Calzetti, Kinney, & Storchi-Bergmann (1994).
Galaxy luminosity functions have traditionally been fit empirically using Schechter functions (Schechter 1976
where φ(M)dM is the number of galaxies with absolute magnitude between M and M + dM per Mpc 3 . Three free parameters are fit using an empirical luminosity function; α is the slope at the faint end of the luminosity function, M * is the luminosity where the luminosity function turns over, and φ * is the density normalization.
For the non-evolving models the local galaxy luminosity function was used at all redshifts. The GALEX NUV galaxy luminosity function of Wyder et al. (2005) was used for the uvw2, uvm2, and uvw1 models. In the u band the models are based on the local U band luminosity function from Ilbert et al. (2005). In the evolving models the Schechter function parameters α, φ * , and M * vary with redshift. In the uvw2, uvm2, and uvw1 bands the evolution of the Schechter function parameters is based on their evolution at 1500Å as found by Arnouts et al. (2005), normalized to match the Wyder et al. (2005) NUV parameters for the local universe. For the u band the evolution of the Schechter function parameters comes from Ilbert et al. (2005). In neither the non-evolving nor evolving models does the dust extinction change as a function of redshift, nor does the underlying galaxy template evolve. A model with that level of complexity would be beyond the scope of this paper.
The model number counts are also corrected for the Lyman forest and continuum using the methods described by Madau (1995). With the exception of Hubble Deep Field U band number counts (Metcalfe et al. 2001;Volonteri et al. 2000) NUV and U model number counts are generally not corrected for Lyman absorption. Our modeling reveals that this is justified. In the bands considered in this paper this affects the models by a few percent at 29th magnitude and much less at brighter magnitudes. Although the models described here are plotted mainly for context the Lyman absorption corrections are included.
Example models are plotted with the number counts in Figures 3, 4, 5, and 6. Figures 3 and 4 show that in the uvw2 and uvm2 filters the number counts are in excellent agreement with the NUV results from GALEX (Xu et al. 2005) and HST (Gardner et al. 2000). Furthermore, Figures 3 and 4 demonstrate the unique contribution of UVOT. The UVOT number counts have a significant overlap with GALEX, however they continue ∼ 1.5 magnitudes deeper with error bars comparable to those of GALEX. In this magnitude range they overlap with the HST number counts, but are much less uncertain due to the wider field of view of UVOT as compared to STIS. While UVOT is not able to go as deep as HST, it provides more precise number counts in the magnitude range where there is a knee in the slope of the number counts.
Results
Figures 3 and 4 also show some of the models discussed in §4. The models shown are for the star forming galaxy template with Calzetti et al. (1994) dust models and A V = 1.0. The solid line is a model with a non-evolving galaxy luminosity function and the dashed line is an evolving model following the evolution of the Schechter function parameters described by Arnouts et al. (2005). The underlying models are the same in the two figures, but have been calculated for the different filters. In both cases the non-evolving luminosity function model under-predicts the number counts given the galaxy template and extinction assumptions. However the evolving luminosity function model is simultaneously in good agreement with the uvw2 and uvm2 number counts. This is an independent confirmation that the evolution in the luminosity function parameters found by Arnouts et al. (2005) are reasonable. Figure 5 shows that the uvw1 number counts are significantly higher than the GALEX NUV counts. This can be explained by the fact that the uvw1 filter has a tail in the red with significant sensitivity between 3000 and 4000Å. This extends redward of the limits of the GALEX and STIS NUV filters. At this point bright elliptical galaxies can be detected in spite of the fact that they do not produce an appreciable flux in the NUV. Beyond the extreme case of ellipticals, post-starburst galaxies with substantial populations of A type stars and even to a lesser extent regular spiral galaxies will also be over represented in the uvw1 number counts compared to the NUV due to light being detected in the red wing of the uvw1 filter.
The black models in Figure 5 are the same as those in Figures 3 and 4. The evolving luminosity function model is still better than the no evolution model, and is fairly representative of the GALEX and HST number counts. However it does not agree with the uvw1 number counts as well as in the uvw2 and uvm2. This is due to the fact that the starburst galaxy template is too blue to take into account the red objects which may be detected by the red end of the uvw1 filter. The red models assume the same evolutionary parameters as the black models but uses the redder cosmic spectrum of Baldry et al. (2002) as the galaxy template. The models using the cosmic spectrum template are below their respective star forming template counterparts. Thus the cosmic spectrum model has the opposite problem in that it undercounts galaxies experiencing strong star formation. This shows that the simple modeling used here is less successful for describing the uvw1 number counts, but also suggests that the uvw1 filter could be useful in constraining the relative numbers of different galaxy types over time. Figure 6 shows that the u band number counts are generally in good agreement with other observations. On the faint end of the number counts the UVOT observations are in excellent agreement with the U band counts of Capak et al. (2004) and Eliche-Moral et al. (2006) and the u band counts of Metcalfe et al. (2001). At around magnitude 22 to 23 the UVOT number counts appear about 50% higher. One explanation for this is that the Yasuda et al. (2001) u number counts are also higher than the other observations on the faint end. Modeling galaxy colors shows that the SDSS u is a much better proxy for UVOT u than Johnson U. The higher number counts may be due to additional blue sensitivity. Figure 6 also reveals that in the u band UVOT does not have the unique advantage it has in the NUV filters as it covers the same magnitude range as the ground based observations and does not go as deep. However it provides an independent check on the ground based results. Figure 6 also shows u band model number counts for both the starburst (black) and cosmic spectrum (red) templates, and both non-evolving luminosity functions (solid) and those which evolve with the parameters of Ilbert et al. (2005). In the u band the evolving luminosity function models with the starburst and cosmic spectrum templates bracket the observed number counts, but then turn over at u ∼ 25 faster than the observed counts.
In summary, the UVOT is uniquely positioned to cover the knee in the galaxy number counts compared to GALEX and HST in the NUV. Due to its smaller PSF it can go deeper than the GALEX confusion limit, and it's larger field of view provides better statistics on the bright end of the STIS number counts. The simple model number counts used here strongly point to an evolving galaxy luminosity function in agreement with earlier studies. More detailed models are needed to explain the number counts in the uvw1 and u filters, but are beyond the scope of this paper. However the measurements provided by this paper in the magnitude range where the number counts turn over will enable a more precise differentiation between models. In addition, the three NUV filters of UVOT are narrower than the single NUV filter of STIS and GALEX so more color information is provided which is potentially useful for more involved modeling. Future plans include measurements of the UV galaxy luminosity function as a function of redshift.
We acknowledge support from NASA Astrophysics Data Analysis grant, #NNX09AC87G. This work is sponsored at PSU by NASA contract NAS5-00136 and at MSSL by funding from the Science and Technology Facilities Council (STFC).
Facilities: Swift (UVOT) (Giacconi et al. 2002), Hubble Ultra Deep Field (Beckwith et al. 2006) and Great Observatories Origins Deep Survey (Giavalisco et al. 2004) are denoted by thick contours which are labeled. (Xu et al. 2005, green diamonds), and STIS NUV number counts Gardner et al. (2000, blue X's) are also plotted with a conversion to the uvw2 filter as described in the text. Model number counts are also plotted for a starburst template galaxy and Calzetti et al. (1994) dust model with A V = 1 and galaxy luminosity function parameters from Wyder et al. (2005). Models for a non-evolving (solid line) and an evolving (dashed line) galaxy luminosity function following Arnouts et al. (2005) are shown. Yasuda et al. (2001) (cyan plus signs) are also plotted with a conversion to the UVOT u filter as described in the text. Yasuda et al. (2001) and Metcalfe et al. (2001) do not tabulate their errors. Model number counts are also plotted for a starburst template galaxy and Calzetti et al. (1994) dust model with A V = 1, and galaxy luminosity function parameters from Ilbert et al. (2005) are shown in black for non-evolving (solid line) and an evolving (dashed line) galaxy luminosity functions. Model number counts assuming the cosmic spectrum of Baldry et al. (2002) as a template are shown in red for both non-evolving (solid line) and evolving (dashed line) luminosity functions.
10 0.4(M * −M ) α+1 exp −10 0.4(M * −M ) dM
Fig. 1 .
1-Field of view for UVOT CDF-S observations. Contours indicate the area covered with at least 98% of the maximum exposure time as described in the text and tabulated inTable 1. The contours are uvw2 (thin solid line), uvm2 (dotted line), uvw1 (dashed line), and u (dot-dashed line). For reference the extent of the Chandra Deep Field South
Fig. 2 .
2-Synthetic color image of a portion of the UVOT CDF-S deep field. This image includes uvw2 (blue), uvm2 (green), and uvw1 (red). The u band is not included in the image. For reference the green bar is 1 arcminute long.
Fig. 3 .
3-UV number counts in the uvw2 filter (red triangles). GALEX NUV number counts
Fig. 4 .
4-UV number counts in the uvm2 filter (red triangles). The rest of the description followsFigure 3.
Fig. 5 .
5-UV number counts in the uvw1 filter (red triangles). In addition to the description from Figure 3, model number counts assuming the cosmic spectrum of Baldry et al. (2002) as a template are shown in red for both non-evolving (solid line) and evolving (dashed line) luminosity functions.
Fig. 6 .
6-Galaxy number counts in the UVOT u filter (red triangles). U number counts from Capak et al. (2004) (green diamonds), Eliche-Moral et al. (2006) (blue X's), and Metcalfe et al. (2001) (black circles), as well as SDSS u number counts from
)
Table 1 .
1Swift UVOT observations of the CDF-SFilter
Central Wavelength (Å)
FWHM (Å)
Exposure (s)
Area (arcmin 2 ) a # Sources
uvw2
1928
657
144763
132.7
888
uvm2
2246
498
136286
112.0
1061
uvw1
2600
693
158334
143.2
1260
u
3465
785
124787
136.6
931
a Area used for number counts where the exposure time is greater than or equal to 98% of the maximum
exposure time at the center of the image.
Table 2 .
2SExtractor parameters for CDF-S photometryNote.-Table 2appears in its entirety in the online version of the Astrophysical Journal. A portion is provided here for guidance regarding its form and content.Parameter Name
Parameter Value
ANALYSIS THRESH
5.0
Table 3 .
3Swift UVOT galaxy number counts in the CDF-Sm AB
Filter
Counts
Nraw
Nstars
N AGN
Nspur
Completeness
Area
(deg −2 mag −1 )
(arcmin 2 )
21.375
uvw2
566 ± 358
7
2
0
0.00
0.957
132.708
21.625
uvw2
673 ± 388
6
0
0
0.00
0.967
132.708
21.875
uvw2
458 ± 324
4
0
0
0.00
0.946
132.708
22.125
uvw2
1494 ± 586
14
1
0
0.00
0.944
132.708
22.375
uvw2
1731 ± 632
17
2
0
0.00
0.940
132.708
22.625
uvw2
2656 ± 783
24
1
0
0.00
0.940
132.708
22.875
uvw2
2401 ± 741
21
0
0
0.00
0.949
132.708
23.125
uvw2
3811 ± 938
36
3
0
0.00
0.939
132.708
23.375
uvw2
6018 ± 1191
56
2
3
0.00
0.919
132.708
23.625
uvw2
6011 ± 1190
52
0
1
0.00
0.921
132.708
23.875
uvw2
8655 ± 1432
76
2
1
0.00
0.915
132.708
24.125
uvw2
10108 ± 1569
89
4
2
0.00
0.891
132.708
24.375
uvw2
14694 ± 1921
117
0
0
0.00
0.864
132.708
24.625
uvw2
14980 ± 1967
122
5
1
0.00
0.840
132.708
24.875
uvw2
17648 ± 2223
129
1
2
0.00
0.775
132.708
25.125
uvw2
19683 ± 2584
118
1
1
0.00
0.639
132.708
21.375
uvm2
687 ± 435
5
0
0
0.00
0.943
110.982
21.625
uvm2
411 ± 335
3
0
0
0.00
0.946
110.982
21.875
uvm2
953 ± 509
7
0
0
0.00
0.952
110.982
22.125
uvm2
1221 ± 575
12
3
0
0.00
0.956
110.982
22.375
uvm2
1866 ± 705
14
0
0
0.00
0.973
110.982
22.625
uvm2
2293 ± 786
17
0
0
0.00
0.962
110.982
22.875
uvm2
3259 ± 940
26
2
0
0.00
0.955
110.982
23.125
uvm2
4510 ± 1110
35
1
1
0.00
0.949
110.982
23.375
uvm2
6757 ± 1365
49
0
0
0.00
0.941
110.982
23.625
uvm2
7894 ± 1478
58
1
0
0.00
0.937
110.982
23.875
uvm2
11535 ± 1801
82
0
0
0.00
0.922
110.982
24.125
uvm2
12975 ± 1913
94
2
0
0.00
0.920
110.982
24.375
uvm2
18846 ± 2337
130
0
0
0.00
0.895
110.982
24.625
uvm2
18397 ± 2336
125
0
1
0.00
0.875
110.982
24.875
uvm2
20142 ± 2537
131
2
3
0.00
0.812
110.982
25.125
uvm2
25642 ± 3022
147
1
2
0.00
0.729
110.982
25.375
uvm2
28135 ± 3647
126
3
4
0.00
0.549
110.982
20.875
uvw1
301 ± 246
3
0
0
0.00
1.000
143.192
21.125
uvw1
545 ± 345
6
1
0
0.00
0.921
143.192
21.375
uvw1
430 ± 304
6
2
0
0.00
0.934
143.192
21.625
uvw1
1060 ± 474
13
3
0
0.00
0.948
143.192
21.875
uvw1
1073 ± 479
13
3
0
0.00
0.937
143.192
22.125
uvw1
1885 ± 628
19
0
1
0.00
0.960
143.192
22.375
uvw1
1590 ± 580
18
2
1
0.00
0.948
143.192
22.625
uvw1
4935 ± 1029
50
4
0
0.00
0.937
143.192
22.875
uvw1
5968 ± 1138
56
1
0
0.00
0.927
143.192
23.125
uvw1
8594 ± 1376
81
2
1
0.00
0.913
143.192
23.375
uvw1
9689 ± 1460
90
2
0
0.00
0.913
143.192
23.625
uvw1
10364 ± 1519
98
3
2
0.00
0.902
143.192
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| []
|
[
"Tuning of metal-insulator transition of two-dimensional electrons at parylene/SrTiO 3 interface by electric field",
"Tuning of metal-insulator transition of two-dimensional electrons at parylene/SrTiO 3 interface by electric field"
]
| [
"H Nakamura \nDepartment of Advanced Materials\nUniversity of Tokyo\n277-8561KashiwaJapan\n\nCorrelated Electron Research Center (CERC)\nNational Institute of Advanced Industrial Science and Technology (AIST)\nAIST Tsukuba Central 4\n305-8562TsukubaJapan\n",
"H Tomita \nDepartment of Advanced Materials\nUniversity of Tokyo\n277-8561KashiwaJapan\n\nCorrelated Electron Research Center (CERC)\nNational Institute of Advanced Industrial Science and Technology (AIST)\nAIST Tsukuba Central 4\n305-8562TsukubaJapan\n",
"H Akimoto \nAdvanced Science Institute\nRIKEN (The Institute of Physical and Chemical Research)\n351-0198WakoJapan\n",
"R Matsumura \nDepartment of Advanced Materials\nUniversity of Tokyo\n277-8561KashiwaJapan\n",
"I H Inoue \nCorrelated Electron Research Center (CERC)\nNational Institute of Advanced Industrial Science and Technology (AIST)\nAIST Tsukuba Central 4\n305-8562TsukubaJapan\n",
"T Hasegawa \nCorrelated Electron Research Center (CERC)\nNational Institute of Advanced Industrial Science and Technology (AIST)\nAIST Tsukuba Central 4\n305-8562TsukubaJapan\n",
"K Kono \nAdvanced Science Institute\nRIKEN (The Institute of Physical and Chemical Research)\n351-0198WakoJapan\n",
"Y Tokura \nCorrelated Electron Research Center (CERC)\nNational Institute of Advanced Industrial Science and Technology (AIST)\nAIST Tsukuba Central 4\n305-8562TsukubaJapan\n\nDepartment of Applied Physics\nUniversity of Tokyo\n113-8656TokyoJapan\n",
"H Takagi \nDepartment of Advanced Materials\nUniversity of Tokyo\n277-8561KashiwaJapan\n\nCorrelated Electron Research Center (CERC)\nNational Institute of Advanced Industrial Science and Technology (AIST)\nAIST Tsukuba Central 4\n305-8562TsukubaJapan\n\nAdvanced Science Institute\nRIKEN (The Institute of Physical and Chemical Research)\n351-0198WakoJapan\n"
]
| [
"Department of Advanced Materials\nUniversity of Tokyo\n277-8561KashiwaJapan",
"Correlated Electron Research Center (CERC)\nNational Institute of Advanced Industrial Science and Technology (AIST)\nAIST Tsukuba Central 4\n305-8562TsukubaJapan",
"Department of Advanced Materials\nUniversity of Tokyo\n277-8561KashiwaJapan",
"Correlated Electron Research Center (CERC)\nNational Institute of Advanced Industrial Science and Technology (AIST)\nAIST Tsukuba Central 4\n305-8562TsukubaJapan",
"Advanced Science Institute\nRIKEN (The Institute of Physical and Chemical Research)\n351-0198WakoJapan",
"Department of Advanced Materials\nUniversity of Tokyo\n277-8561KashiwaJapan",
"Correlated Electron Research Center (CERC)\nNational Institute of Advanced Industrial Science and Technology (AIST)\nAIST Tsukuba Central 4\n305-8562TsukubaJapan",
"Correlated Electron Research Center (CERC)\nNational Institute of Advanced Industrial Science and Technology (AIST)\nAIST Tsukuba Central 4\n305-8562TsukubaJapan",
"Advanced Science Institute\nRIKEN (The Institute of Physical and Chemical Research)\n351-0198WakoJapan",
"Correlated Electron Research Center (CERC)\nNational Institute of Advanced Industrial Science and Technology (AIST)\nAIST Tsukuba Central 4\n305-8562TsukubaJapan",
"Department of Applied Physics\nUniversity of Tokyo\n113-8656TokyoJapan",
"Department of Advanced Materials\nUniversity of Tokyo\n277-8561KashiwaJapan",
"Correlated Electron Research Center (CERC)\nNational Institute of Advanced Industrial Science and Technology (AIST)\nAIST Tsukuba Central 4\n305-8562TsukubaJapan",
"Advanced Science Institute\nRIKEN (The Institute of Physical and Chemical Research)\n351-0198WakoJapan"
]
| []
| Electrostatic carrier doping using a field-effect-transistor structure is an intriguing approach to explore electronic phases by critical control of carrier concentration[1,2]. We demonstrate the reversible control of the insulator-metal transition (IMT) in a two dimensional (2D) electron gas at the interface of insulating SrTiO3 single crystals. Superconductivity was observed in a limited number of devices doped far beyond the IMT, which may imply the presence of 2D metal-superconductor transition. This realization of a two-dimensional metallic state on the most widely-used perovskite oxide is the best manifestation of the potential of oxide electronics. | 10.1143/jpsj.78.083713 | [
"https://arxiv.org/pdf/0809.4774v1.pdf"
]
| 119,113,298 | 0809.4774 | 0087c78efa9bc3c9415e45304d3c2c39513177ae |
Tuning of metal-insulator transition of two-dimensional electrons at parylene/SrTiO 3 interface by electric field
27 Sep 2008 (Dated: September 27, 2008)
H Nakamura
Department of Advanced Materials
University of Tokyo
277-8561KashiwaJapan
Correlated Electron Research Center (CERC)
National Institute of Advanced Industrial Science and Technology (AIST)
AIST Tsukuba Central 4
305-8562TsukubaJapan
H Tomita
Department of Advanced Materials
University of Tokyo
277-8561KashiwaJapan
Correlated Electron Research Center (CERC)
National Institute of Advanced Industrial Science and Technology (AIST)
AIST Tsukuba Central 4
305-8562TsukubaJapan
H Akimoto
Advanced Science Institute
RIKEN (The Institute of Physical and Chemical Research)
351-0198WakoJapan
R Matsumura
Department of Advanced Materials
University of Tokyo
277-8561KashiwaJapan
I H Inoue
Correlated Electron Research Center (CERC)
National Institute of Advanced Industrial Science and Technology (AIST)
AIST Tsukuba Central 4
305-8562TsukubaJapan
T Hasegawa
Correlated Electron Research Center (CERC)
National Institute of Advanced Industrial Science and Technology (AIST)
AIST Tsukuba Central 4
305-8562TsukubaJapan
K Kono
Advanced Science Institute
RIKEN (The Institute of Physical and Chemical Research)
351-0198WakoJapan
Y Tokura
Correlated Electron Research Center (CERC)
National Institute of Advanced Industrial Science and Technology (AIST)
AIST Tsukuba Central 4
305-8562TsukubaJapan
Department of Applied Physics
University of Tokyo
113-8656TokyoJapan
H Takagi
Department of Advanced Materials
University of Tokyo
277-8561KashiwaJapan
Correlated Electron Research Center (CERC)
National Institute of Advanced Industrial Science and Technology (AIST)
AIST Tsukuba Central 4
305-8562TsukubaJapan
Advanced Science Institute
RIKEN (The Institute of Physical and Chemical Research)
351-0198WakoJapan
Tuning of metal-insulator transition of two-dimensional electrons at parylene/SrTiO 3 interface by electric field
27 Sep 2008 (Dated: September 27, 2008)
Electrostatic carrier doping using a field-effect-transistor structure is an intriguing approach to explore electronic phases by critical control of carrier concentration[1,2]. We demonstrate the reversible control of the insulator-metal transition (IMT) in a two dimensional (2D) electron gas at the interface of insulating SrTiO3 single crystals. Superconductivity was observed in a limited number of devices doped far beyond the IMT, which may imply the presence of 2D metal-superconductor transition. This realization of a two-dimensional metallic state on the most widely-used perovskite oxide is the best manifestation of the potential of oxide electronics.
The perovskite SrTiO 3 is sometimes called the "silicon of oxide electronics" because it is commonly used as a substrate for epitaxial growth of a variety of oxide films [3]. Not only it is an insulator with a band gap of 3.2 eV but also a quantum paraelectric with an extremely large dielectric constant of more than 10 6 at low temperatures [4]. N-type conduction has been realized by introducing a small number of oxygen vacancies or by cation substitutions. An insulator to metal transition occurs at a rather low carrier concentration of n ∼10 18 cm −3 [5,6] for a doped oxide system, which owes largely to the high mobility exceeding 10 4 cm 2 /Vs. Superconductivity emerges in a limited concentration range from n = 7 × 10 18 to 5 × 10 20 cm −3 and the transition temperature T c shows a bell shaped dependence on n with an optimum T c ∼ 0.35 K at n ∼ 10 20 cm −3 [7]. The carrier concentration required to achieve superconductivity is two orders of magnitude smaller than that for high-T c cuprates which is a great advantage to realize electrostatic control of insulator-metal and superconductor transitions.
Such a unique doping properties of SrTiO 3 has triggered attempts to dope charged carriers electrostatically by using a field effect transistor (FET) structure. However, so far, attempts to realize metallic state have been unsuccessful [8]. This is due to several reasons, notably carrier trapping at the gate insulator/SrTiO 3 interface, Schottky barrier formation between SrTiO 3 and source and drain electrodes, and insufficient breakdown voltage of the gate insulators. An alternative approach has shown that two dimensional electron gas can be created at SrTiO 3 /LaAlO 3 hetero-interface by a charge-transfer due to a polar discontinuity [9,10,11,12,13,14] or possible oxygen defects [15]. This electronically tailored interface was found to be superconducting as in the bulk material [14].
Recently the performance of SrTiO 3 FET has been drastically improved [16,17] by overcoming the contacts and interface problems. These new devices showed metallic behavior down to 7 K [17]. In this Letter, we extend our study to lower temperatures below 1 K, and show that the two-dimensional insulator-metal transition is indeed achieved at the interface between parylene and SrTiO 3 reversibly while sweeping the gate voltage. A superconducting transition to zero-resistance state was observed at 0.13 K only in a sample where a large enough number of carriers were successfully introduced beyond the critical carrier number for the insulatormetal transition (IMT). The observation of two dimensional insulator-metal-superconductor transition represents major progress not only in the attempts for future oxide electronics but also in oxide physics.
A top gate FET was fabricated consisting of Au/parelne/SrTiO 3 layers. Al electrodes of the Hall bar geometry (Fig.1a) were evaporated on an atomically flat surface of SrTiO 3 (100) before depositing the gate insulator parylene. The use of Al electrode is the key to suppressing Schottky barrier formation. Details of the device fabrication are described elsewhere [17]. We have measured six samples (Sample A-F) with different parylene thickness (0.53-0.63 µm), corresponding to capacitance C i = 4.4 − 5.3 nF/cm 2 allowing a wide-ranging threshold gate voltage (V th = 97-170 V). Resistivity measurements above 2 K were carried out by dc four-probe method with temperature control by the Quantum Design Physical Property Measurement System, while those below 1 K were carried out by ac four-probe method (frequency 45 Hz) in a dilution refrigerator. Figure 1a shows the temperature dependence of sheet resistance for various gate voltages (or electron densities). As the gate voltage is increased, the temperature dependence changes from insulating to metallic. At high enough gate voltages, the resistance eventually becomes metallic all the way down to a low tempera- ture. The sheet carrier density (n 2D ) here is estimated as C i (V − V th )/e, where C i is the capacitance per unit area. The sheet carrier density (n 2D ) at the insulator to metal transition is approximately 1×10 12 cm −2 , which corresponds to a Fermi energy of about 1 meV assuming that the system is a two-dimensional (2D) free-electron gas. We also note that the sheet resistance (R 2D ) in this regime is close to the two dimensional quantum sheet resistance h/e 2 =25.8 kΩ. In the insulating state, R 2D shows logarithmic temperature dependence at lower temperatures as shown in Fig. 1b, and the slope is of the order of the quantum conductance. At first glance these observations suggest two-dimensional weak-localization [18,19]. However, we show that the conduction in this weakly insulating regime is highly likely to be percolative in nature and not described by the weak localization scenario.
The above behavior suggests the emergence of 2D metal at the parylene/SrTiO 3 interface. This is in contrast to the scaling theory of localization, which predicts that a 2D system should become insulating at zero temperature [20]. Recent experiments on very high mobility 2D electron gases formed in Si and GaAs hetero-structure, however, demonstrated the existence of insulator-metal transitions even in 2D and have been attracting a considerable attention [21]. To explore the ground state of electrostatically induced metallic inter-face between parylene and non-doped SrTiO 3 single crystal, we extend the transport measurements down to 15 mK using a dilution refrigerator for the samples with R 2D well below the quantum resistance. Figure 1c shows a typical result. The metallic behavior (dR/dT > 0) is observed down to ∼15 mK. As shown in Fig. 1a, when the gate voltage is reduced, R 2D shows an insulating (dR/dT < 0) behavior. We emphasize here that the two ground states can be tuned reversibly by the electrostatic control through the parylene gate insulator.
Compared to conventional semiconductors like Si or GaAs, SrTiO 3 has a large dielectric constant of the order of 10 6 at low temperatures [4] which weakens the confining potential at the interface. Therefore, it should be clarified whether or not the electronic state formed at parylene/SrTiO 3 interface is two-dimensional. We have checked this experimentally by measuring the anisotropy of transverse magnetoresistance ∆R/R 0 . Here, ∆R is the resistance change in a magnetic field and R 0 is the zero-field resistance. Figure 2 shows the magnetoresistance taken under B perpendicular to the plane, and parallel to the plane. The large positive magnetoresistance, indicative of a high mobility, is observed only for the field perpendicular to the plane. No appreciable magnetoresistance is seen for the field parallel to the plane, which implies that the thickness of the conducting layer is much thinner than the cyclotron radius of electrons, r c = m * vF eB =h kF eB , (m * ; the electron effective mass [25], v F ; Fermi velocity, k F ; Fermi wave number,h = h/2π, where h is Planck's constant), which we estimate r c ∼15 nm at B = 6 T. Numerical calculation based on the Schrödinger-Poisson equation with a triangular potential approximation estimates the conducting layer depth to be approximately 10 nm in metal/insulator/SrTiO 3 structure [22] and support the presence of two-dimensional electron system. It should also be noted that this estimation of conducting layer depth may become even smaller when the rapid decrease of dielectric constant of SrTiO 3 at large electric field is taken into account [23,24]. From these results, we conclude that the metal-insulator transition observed in the parylene/SrTiO 3 -FET is two-dimensional in nature. Figure 3a shows the sheet resistance R 2D as a function of the sheet carrier density n 2D at 4.2 K. A notable feature of the electrostatically-induced IMT is the sharp and almost discontinuous decrease of the resistance with increasing the sheet carrier density. With further increasing the carrier density, a "kink" in R 2D -n 2D is clearly recognized, and above the kink, much more moderate decrease is observed, where metallic behavior of resistivity is seen down to the lowest temperature. The comparison of two kinds of mobilities deduced from the magnetoresistance and the field-effect, respectively, suggests phase separation into metallic and insulating domains in the transition region before reaching the kink, implying that the insulator to metal transition is very likely discontinuous. Figure 3b shows the evolution of carrier mobility near the transition, where Magnetoresistance ∆R/R0 for two different orientations of magnetic field B at 4.2 K: perpendicular and parallel fields to the SrTiO3/parylene interface. Positive magnetoresistance for the perpendicular field is caused by the cyclotron motions of electrons and its magnitude is related to the mobility µ of the electron gas (∆R/R0 ∼ (µB) 2 ). Little effect for the parallel field indicates that the conducting layer is much thinner compared to the cyclotron radius of electrons.
two independent mobilities are compared; the field-effect mobility µ FE = (1/e)(dσ 2D /dn 2D ), where σ 2D = 1/R 2D , n 2D = C i V G /e, and the magnetoresistance mobility µ MR estimated by the magnitude of the quadratic part of magnetoresistance ∆R/R 0 = (µ MR B) 2 [26]. While the two mobilities agree very well in the larger gate voltage region above the kink, they are distinct from each other in the transition region (Fig. 3b). The µ MR is as large as 2000 cm 2 /Vs and surprisingly independent of carrier density above and below the kink. In contrast, µ FE shows a rapid decrease with decreasing voltage around the transition region. This contrasting behavior between the two mobilities may be understood naturally in terms of phase separation. The conduction should be dominated by the metallic domains. The magnetoresistance is then almost identical to those of metallic domains independent of the gate voltage. On the other hand, the value of µ FE reflects the channel resistance. Narrowing of the metallic domain during the separation enhances the channel resistance and hence µ FE decreases rapidly. Thus, the transition between metallic and insulating state tuned by a gate voltage is very likely a first order transition accompanied by a phase separation. The origin of seemingly weak localization behavior in the transition region shown in Fig. 1 is not clear. We suspect that narrow percolation path in the mixed phase might be responsible for the apparent logarithmic temperature dependence.
The discontinuous transition presented here is reminiscent of the "steplike" change in conductance observed in SrTiO 3 /LaAlO 3 hetero-structures as a function of LaAlO 3 thickness [11], where the electric field produced by polar discontinuity is believed to play a role. In both cases, the sheet resistance of the conducting side of the "discontinuous change" is of order 10 kOhm, suggesting there exists a universal minimum metallic conductivity of around ∼ 10 −4 S. Although the mechanism of first-orderlike metal-insulator transition is currently unclear, the observation of similar behavior in different types of device structure seems to indicate that the origin of discontinuous metalization may be intrinsic at least to SrTiO 3 .
In only a few samples, we were able to apply a large enough gate voltage without breaking the gate insulator (parylene) to go far beyond the discontinuous IMT. With successful accumulation of an enough number of carriers, we observed zero-resistance state associated with superconductivity. As shown in Fig. 4, at 128 mK the resistance decreases gradually with increasing gate voltage, namely the carrier concentration, and eventually becomes zero within our resolution above n 2D ∼ 2 × 10 12 cm −2 . With decreasing gate voltage, then, the resistance goes back to a finite value reproducibly. With increasing temperature from 128 mK at n 2D ∼ 2.2 × 10 12 cm −2 , the resistive transition to a normal state is observed around 130-160 mK as seen in the inset of Fig. 4. The application of magnetic field of 0.03 T perpendicular to the plane was found to suppress the zero resistance state.
These observations demonstrate that superconductivity is marginally achieved around n 2D ∼ 2 × 10 12 cm −2 . This sheet carrier density corresponds to n 3D ∼ 2 × 10 18 cm −3 assuming the thickness of 2D electron layer to be 10 nm. Though the reason is not obvious, this is similar to the onset n 3D ∼ 7 × 10 18 cm −3 reported for bulk, given the uncertainty in 2D electron layer thickness. Note that we did not see any trace of superconductivity in the metallic sample shown in Fig. 1c, down to 15 mK at n 2D ∼ 1 × 10 12 cm −2 . These suggest that the 2D insulator-metal transition occurs first and that, to achieve 2D metal-superconductor transition, extra charge carrier must be introduced. We, however, cannot rule out completely superconductivity at an extremely low temperature in the sample near IMT. It should be noted that by applying a large gate field for a long period (typically V G > 150 V for one week), another superconductor-like transition without a zero resistance state appears around 0.35 K which does not disappear even after reducing the gate field. This is highly likely to be the superconductivity associated with the creation and/or migration of oxygen defects at the interface.
In conclusion, we have demonstrated that the parylene/SrTiO 3 interface shows a transition from an insulator into two-dimensional metal by the electrostatic carrier doping as observed in high mobility 2D electron gas formed in Si and GaAs hetero-structures. We believe that this represents a breakthrough in oxide electronics. One of the unique features of the parylene/SrTiO 3 interface is a zero-resistance state emerging at 130 mK. The emergent superconductivity is thought-provoking not only because of the possible 2D character but also because it develops in a electron gas with a small Fermi energy of a few meV-usually trivial interactions like spinorbit coupling [27] can be comparable with Fermi energy in this system and may give rise to an exotic superconductivity. Therefore, what we have demonstrated in this study is not simply a hectic chase of oxide electronics after the conventional semiconductor but also a prologue to new paradigm of transition metal oxide physics.
We thank R. Perry for critical reading of the manuscript. This work was supported in part by MEXT, Grant-in Aid for Scientific Research (S)(19104008).
FIG. 1 :
1Electrostatic control of metal-insulator transition of non-doped SrTiO3 single crystal. (a) Temperature dependence of the sheet resistance (R2D) tuned by the gate voltage near the metal-insulator transition. The parylene gate insulator thickness (dins) was 0.6 µm. Sheet carrier density (n2D) in units of 10 11 cm −2 is shown. n2D is derived from the gate voltage VG, capacitance Ci of parylene, and threshold gate voltage V th (n2D = Ci(VG−V th )/e). The inset shows an photograph of sample with Al electrodes, parylene, and Au gate electrode. Channel area is 400 µm×200 µm. (b) Sheet conductance in units of e 2 /h plotted against logT . (c) Sheet resistance versus temperature at T < 1 K in the case of R2D << h/e 2 . Metallic behavior (dR/dT > 0) is seen down to the base temperature (15 mK).
FIG. 2 :
2Anisotropic magnetoresistance.
FIG. 3 :
3Electrostatic tuning of the sheet resistance and the mobilities. (a) Sheet resistance R2D plotted against the sheet carrier density n2D at 4 K. The position of a "kink" is shown by a thick arrow. (b) Field-effect mobility: µFE = (1/e)(dσ2D/dn2D), and magnetoresistance mobility µMR: ∆R/R0 = (µMRB) 2 . Two mobilities show significant difference near the threshold of gate voltage (early stage after the accumulation of mobile charge), but coincide at large n2D.
FIG. 4: Continuous and reversible electrostatic tuning of superconductivity at the surface of undoped SrTiO3 single crystal. Reversible and continuous control between the zeroresistance and the finite resistance was achieved by sweeping the gate voltage at T = 128 mK. The inset shows temperature dependence of the sheet resistance at a constant sheet carrier density at B =0 T (continuous line) and at B =0.03 T (dashed line). Zero-resistance is seen below 130 mK for B =0 T.Sample-E
V th = 96 V
d ins = 0.56 µm
n 2D = 2.2×10 12 cm -2
1.6
1.8
2
2.2
0
100
200
300
400
500
n 2D (×10
12 cm
-2 )
R
2D (Ω)
T = 128 mK
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. H Uwe, R Yoshizaki, T Sakudo, A Izumi, T Uzumaki, Jpn. J. Appl. Phys. 335Suppl.24-2H. Uwe, R. Yoshizaki, T. Sakudo, A. Izumi, and T. Uzu- maki, Jpn. J. Appl. Phys. Suppl.24-2, 335 (1985).
Solid-State Electron. T R Jervis, E F Johnson, 13181T. R. Jervis and E. F. Johnson, Solid-State Electron. 13, 181 (1970).
Spin-orbit coupling effects in twodimensional electron and hole systems. R Winkler, SpringerBerlinR. Winkler, Spin-orbit coupling effects in two- dimensional electron and hole systems., (Springer, Berlin, 2003).
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[
"Spin rotation and oscillation of high energy particles in storage ring",
"Spin rotation and oscillation of high energy particles in storage ring"
]
| [
"V G Baryshevsky \nResearch Institute of Nuclear Problems\nByelorussian State University\n220050MinskBelarus\n"
]
| [
"Research Institute of Nuclear Problems\nByelorussian State University\n220050MinskBelarus"
]
| []
| The phenomenon of light polarization plane rotation (e.g., the Faraday effect, the natural rotation of a light polarization plane) as well as light birefringence (e.g., in matter placed in an electric field due to the Kerr effect) are the well known optical coherent phenomena. For the first glance they distinguish photons from other particles (nucleons, electrons, etc.) for which these effects for a long time have been considered nonexistent.In [1-8] a wide range of phenom-ena similar to the effects of light polarization plane rotation and birefringence was shown to exist for particles other than photons. In particular it has been shown that as particles (neutrons, protons, neutrinos, etc.) pass through matter with polarized nuclei, the particle spin undergoes a rotation in an effective pseudomagnetic field of the matter induced by both strong and weak interactions. This effect is kinematically analogous to the phenomenon of light polarization plane rotation due to the Faraday effect. Experimentally, the effect of neutron spin precession in polarized target has been studied for neuterons[4][5][6].As it was ascertained in [5-8] the analogue of birefringence phenomenon exists for particles, too.As a matter of fact, the Faraday effect and the ef-fect of birefringence are caused by the dependence of the coherent photon-medium interaction energy on the photon spin state. This property unites the quasioptical phenomena discovered in [1-8] for interaction of spin-particles in matter with nuclei with the phenomena existing in light optics. However, attention should be drawn to the fact that whereas the photon spin is equal to unity, the particle (atom, nucleus) spin may take on different values. For particles of spin S = 1/2 there exists only one effect -that of spin rotation, i.e. a kinematic analog of the effect of light polarization plane rotation. An effect similar to birefringence exists for spin S ≥ 1 particles. It is very interesting to mention that phenomena of rotation and oscillations of particle spin (birefringence effect) exist for particles with spin S ≥ 1 in a medium with unpolarized scatterer spins, too. The fact that these effects are described by spin-dependent part of scattering amplitude allows to use them for the measurement of this amplitude at different energies of colliding particles.0.1Spin rotation of high-energy particles in polarized targets.As a result of numerous studies (see, for example, [12]), a close connection between the coherent elastic scattering amplitude f (0) and the refraction index of a medium N has been established: 1 | 10.1007/s10582-002-0101-5 | [
"https://export.arxiv.org/pdf/hep-ph/0109099v1.pdf"
]
| 14,115,956 | hep-ph/0109099 | 33fdc28b1efd2d97d4ee5bc39e6cd45d75a49fc2 |
Spin rotation and oscillation of high energy particles in storage ring
arXiv:hep-ph/0109099v1 12 Sep 2001
V G Baryshevsky
Research Institute of Nuclear Problems
Byelorussian State University
220050MinskBelarus
Spin rotation and oscillation of high energy particles in storage ring
arXiv:hep-ph/0109099v1 12 Sep 2001
The phenomenon of light polarization plane rotation (e.g., the Faraday effect, the natural rotation of a light polarization plane) as well as light birefringence (e.g., in matter placed in an electric field due to the Kerr effect) are the well known optical coherent phenomena. For the first glance they distinguish photons from other particles (nucleons, electrons, etc.) for which these effects for a long time have been considered nonexistent.In [1-8] a wide range of phenom-ena similar to the effects of light polarization plane rotation and birefringence was shown to exist for particles other than photons. In particular it has been shown that as particles (neutrons, protons, neutrinos, etc.) pass through matter with polarized nuclei, the particle spin undergoes a rotation in an effective pseudomagnetic field of the matter induced by both strong and weak interactions. This effect is kinematically analogous to the phenomenon of light polarization plane rotation due to the Faraday effect. Experimentally, the effect of neutron spin precession in polarized target has been studied for neuterons[4][5][6].As it was ascertained in [5-8] the analogue of birefringence phenomenon exists for particles, too.As a matter of fact, the Faraday effect and the ef-fect of birefringence are caused by the dependence of the coherent photon-medium interaction energy on the photon spin state. This property unites the quasioptical phenomena discovered in [1-8] for interaction of spin-particles in matter with nuclei with the phenomena existing in light optics. However, attention should be drawn to the fact that whereas the photon spin is equal to unity, the particle (atom, nucleus) spin may take on different values. For particles of spin S = 1/2 there exists only one effect -that of spin rotation, i.e. a kinematic analog of the effect of light polarization plane rotation. An effect similar to birefringence exists for spin S ≥ 1 particles. It is very interesting to mention that phenomena of rotation and oscillations of particle spin (birefringence effect) exist for particles with spin S ≥ 1 in a medium with unpolarized scatterer spins, too. The fact that these effects are described by spin-dependent part of scattering amplitude allows to use them for the measurement of this amplitude at different energies of colliding particles.0.1Spin rotation of high-energy particles in polarized targets.As a result of numerous studies (see, for example, [12]), a close connection between the coherent elastic scattering amplitude f (0) and the refraction index of a medium N has been established: 1
The phenomenon of light polarization plane rotation (e.g., the Faraday effect, the natural rotation of a light polarization plane) as well as light birefringence (e.g., in matter placed in an electric field due to the Kerr effect) are the well known optical coherent phenomena. For the first glance they distinguish photons from other particles (nucleons, electrons, etc.) for which these effects for a long time have been considered nonexistent.
In [1][2][3][4][5][6][7][8] a wide range of phenom-ena similar to the effects of light polarization plane rotation and birefringence was shown to exist for particles other than photons. In particular it has been shown that as particles (neutrons, protons, neutrinos, etc.) pass through matter with polarized nuclei, the particle spin undergoes a rotation in an effective pseudomagnetic field of the matter induced by both strong and weak interactions. This effect is kinematically analogous to the phenomenon of light polarization plane rotation due to the Faraday effect. Experimentally, the effect of neutron spin precession in polarized target has been studied for neuterons [4][5][6].
As it was ascertained in [5][6][7][8] the analogue of birefringence phenomenon exists for particles, too.
As a matter of fact, the Faraday effect and the ef-fect of birefringence are caused by the dependence of the coherent photon-medium interaction energy on the photon spin state. This property unites the quasioptical phenomena discovered in [1][2][3][4][5][6][7][8] for interaction of spin-particles in matter with nuclei with the phenomena existing in light optics. However, attention should be drawn to the fact that whereas the photon spin is equal to unity, the particle (atom, nucleus) spin may take on different values. For particles of spin S = 1/2 there exists only one effect -that of spin rotation, i.e. a kinematic analog of the effect of light polarization plane rotation. An effect similar to birefringence exists for spin S ≥ 1 particles. It is very interesting to mention that phenomena of rotation and oscillations of particle spin (birefringence effect) exist for particles with spin S ≥ 1 in a medium with unpolarized scatterer spins, too. The fact that these effects are described by spin-dependent part of scattering amplitude allows to use them for the measurement of this amplitude at different energies of colliding particles.
0.1
Spin rotation of high-energy particles in polarized targets.
As a result of numerous studies (see, for example, [12]), a close connection between the coherent elastic scattering amplitude f (0) and the refraction index of a medium N has been established:
N = 1 + 2πρ k 2 f (0) (1)
where ρ is the number of particles per cm 3 , k is the wave number of a particle incident on a target. In 1964 it was shown [1] that while slow neutrons are propagating through the target with polarized nuclei a new effect of nucleon spin precession occured. It is stipulated by the fact that in a polarized target the neutrons are characterized by two refraction indices (N ↑↑ for neutrons with the spin parallel to the target polarization vector and N ↑↓ for neutrons with the opposite spin orientation , N ↑↑ = N ↑↓ ). According to the [2], in the target with polarized nuclei there is a nuclear pseudomagnetic field and the interaction of an incident neutron with this field results in neutron spin rotation. The results obtained in [1], initiated experiments which proved the existence of this effect [9][10][11].
Thus, let us consider the amplitude of elastic coherent zero-angle scattering of nucleon by polarized nucleon (nucleous).
The general form of this amplitude with allowance for strong electromagnetic and weak interactions is given in [2]. Below we shall consider more concretely the effect of a relativistic nucleon spin rotation in the target with polarized protons (nuclei with spin 1/2), caused by strong interaction. In this case, the explicit structure of the elastic scattering amplitude of a particle with spin 1/2 by a particle with spin 1/2 (see, for example, [13,14]) proceeds from the following simple discussions. In our case, the elastic scattering amplitude at zero angle depends on the spin operators 1 2 σ, 1 2 σ 1 of an incident particle and that of a target , and also on the momentum of the incident particle , that is, on − → n = − → k k . Operators − → σ , − → σ 1 can be contained in the expression for the amplitude only in the first degree, as higher degrees of − → σ reduce either to a number or to − → σ . The combinations − → σ , − → σ 1 and − → n must be such that the scattering amplitudes are a scalar and invariant in space and time reflections. These conditions defenetely determine its general form:
F = A + A 1 ( − → σ · − → σ 1 ) + A 2 ( − → σ · − → n ) ( − → σ 1 · − → n ) .(2)
By averaging the amplitude F with the help of a spin matrix of the density of scatters ρ s the elastic coherent scattering amplitude may be written as:
f = Sp ρ s F = A + A 1 ( − → σ · − → p ) + A 2 ( − → σ · − → n ) ( − → n · − → p )(3)
where − → p = Sp ρ s − → σ 1 is the polarization vector of a scatterer in a target. Amplitude f can be expressed as
f = A + − → σ · − → g(4)
where
− → g = A 1 − → p + A 2 − → n ( − → n · − → p ).
To simplify further reasoning let us consider the situation when vector − → n is either parallel to − → p ( − → n − → p ) or perpendicular to − → p ( − → n ⊥ − → p ).
In this case one has that g ( − → n − → p ) = (A 1 + A 2 ) − → p and g ( − → n ⊥ − → p ) = A 1 − → p . Thus in these cases vector − → g is directed along − → p . Selecting quantization axes parallel to − → p , one can see that scattering amplitude f ↑↑ = A+g of nucleon with spin parallel to − → p is not equal to scattering amplitude f ↑↓ = A − g of nucleon with spin antiparallel to − → p . Hence, the corresponding refractive indices are not equal to each other (i.e. N ↑↑ = N ↑↓ ).
Considering a wave passes through a layer of polarized medium with finite thickness one can express refractive index of nuclon with spin parallel to − → p as follows:
N ↑↑ = 1 + 2πρ k 2 f ↑↑ = 1 + 2πρ k 2 (A + g)(5)
and for nucleon with opposite polarization
N ↑↓ = 1 + 2πρ k 2 f ↑↓ = 1 + 2πρ k 2 (A − g)(6)
then the difference
∆N = N ↑↑ − N ↑↓ = 2πρ k 2 (f ↑↑ − f ↑↓ ) = 4πρ k 2 g(7)
is determined by the difference in correspondent coherent scattering amplitudes and differs from zero only in polarized medium.
Suppose that nucleon passes through polarized medium and their polarizations are oriented at certain angle to the vector − → p . This state of nucleon can be described as superposition of two states with polarizations directed along and opposite to the vector − → p . Initial wave function of nucleon can be expressed as:
ψ ( − → r ) = e i − → k − → r χ n , χ n = c 1 c 2 (8) or ψ ( − → r ) = c 1 e i − → k − → r 1 0 + c 2 e i − → k − → r 0 1 .(9)
Suppose quantization axes z coincides with vector − → p and particle falls onto the target orthogonally to its surface. As the state 1 0 possesses refration index N ↑↑ and the state 0 1 is characterized by N ↑↓ , then the wave function of nucleon in polarized medium changes with penetration depth as follows:
ψ ( − → r ) = c 1 ψ ↑↑ ( − → r ) c 2 ψ ↑↓ ( − → r ) = c 1 e ikN ↑↑ l 1 0 + c 2 e ikN ↑↓ l 0 1 , .(10)
l is the pass length of nucleon in target. Using (10) one can find nucleon polarization vector p n = ψ| σ|ψ (11) and as a result
p nx = 2Rec * 1 c 2 ψ * + ψ − ψ|ψ −1 , p ny = 2Imc * 1 c 2 ψ * + ψ − ψ|ψ −1 ,(12)p nz = |c 1 ψ + | 2 − |c 2 ψ − | 2 ψ|ψ −1 .
Suppose that nucleon spin in vacuum is perpendicular to the polarization vector of nuclei. Direction of nucleon spin in vacuum define as axes x. In this case c 1 = c 2 = 1/ √ 2. Using (12) we obtain
p nx = cos [kRe (N ↑↑ − N ↑↓ ) l] e −kIm(N ↑↑ −N ↑↓ )l ψ|ψ −1 , p ny = − sin [kRe (N ↑↑ − N ↑↓ ) l] e −kIm(N ↑↑ −N ↑↓ )l ψ|ψ −1 ,(13)p nz = 1 2 e −2kImN ↑↑ l − e −2kImN ↑↓ l ψ|ψ −1 = e −2kImN ↑↑ l − e −2kImN ↑↓ l e −2kImN ↑↑ l + e −2kImN ↑↓ l . ψ|ψ = 1 2 e −2kImN ↑↑ l + e −2kImN ↑↓ l . It should be reminded that Im f (0) = k 4π σ tot ,
where σ tot is the total crosssection of scatterring of nucleon by nucleon (nulei, atoms).
According to (13) when nucleon penetrate deep into target, its polarisation vector rotates around nuclei polarization vector at the angle
ϑ = kRe (N ↑↑ − N ↑↓ ) l = 2πρ k Re (f ↑↑ − f ↑↓ ) l (14)
This rotation is similar to the spin rotation appearing in magnetic field. Then we can conclude that polarized nuclear target acts on spin likewise the area occupied with nuclear pseudomagnetic field [1].
It is important to emphasize that in experiments with gas target scattering amplitude f (0), being contained in expression for N , is the elastic coherent amplitude of zero-angle scattering of a nucleon by an atom (of hydrogen, deuterium and so on). This fact should be taken into consideration at particular analysis, because atom electrons can contribute to the rotation angle.
Let us consider proton beam passing through polarized gas hydrogen target. Let polarized atoms of hydrogen in magnetic field are described by the magnetic quantum number M = 1. It means that in hydrogen atom spins of proton and electron are parallel. Polarized atoms of hydrogen possess magnetic moment directed along spin, which produces magnetic field B H ∼ ρ µ H , where ρ is the number of atoms of hydrogen in 1 cm 3 , µ H is the magnetic moment of hydrogen atom. As the magnetic moment of proton is much less than that of electron, then magnetic field B H of polarized hydrogen atoms is mainly produced by their electrons.
Hence the angle of rotation of proton spin can be expressed as a sum of two additives ϑ B and ϑ S
ϑ = ϑ B + ϑ S ,
where ϑ B is the angle of rotation caused by magnetic field and ϑ S is the angle of rotation appearing due to strong interactions i.e. nuclear pseudomagnetic field.
Thus, considering rotation phenomena as a tool for experimental investigation of zero-angle nucleon-nucleon scattering amplitude one come to neccessity to extract addition caused by magnetic field of the target
− → B = − → H + −→ B H , here − →
H is the external magnetic field. This can be done:
1. by calculation using equations of Bargman-Michel-Telegdi (BMT) [14] as magnetic moments of proton and electron (hydrogen atom) are known with high accuracy or 2. by experimental separating of additions caused by magnetic and nuclear pseudomagnetic fields. This possibility is due to the fact that magnetic field induced by polarized magnetic moment depends on the shape of the target because of long-distance action of electromagnetic interaction. Whereas the nuclear pseudomagnetic field (being short-range action) does not depend on the shape of the target.
Moreover, from the analysis of BMT-equations follows that if proton velocity is directed along − → B then angle of rotation of proton spin orthogonal to − → B for nonrelativistic protons is determined by magnetic moment. If proton velocity is orthogonal to − → B than spin rotation in magnetic field is determined by anomalous magnetic moment, since due to cyclotron movement of proton in magnetic field the addition to the angle of rotation, caused by Dirac magnetic moment (equal to the nuclear magneton), yields to the conservation of angle between proton spin and momentum (if the anomalous magnetic moment ∆µ is equal to zero). Hence the observed deviation of proton spin in magnetic field − → B is conditioned by ∆µ.
Let us evaluate the effect magnitude for particular setup [15]. According to [15] target thickness is n = ρl = 10 14 atoms/cm 2 , a revolution frequency of proton beam ν ∼ 10 6 s −1 .
From (14) we can obtain for rotation angle caused by nuclear pseudomagnetic field :
ϑ S = 2πρl k Re f N N ↑↑ (0) − f N N ↑↓ (0) · νT = 4π nνT k Reg = = 4πnνT 2MĒ h 2 Reg = 4πhnνRegT √ 2M E ∼ = 1 rad
T is the observation time. (for example, T = 10 hours = 3.6 · 10 4 s, g ∼ = 2 · 10 −13 cm) It is interesting to note that we can obtain simple estimation, showing relation between angle of proton spin rotation caused by magnetic field − → B e , produced by polarized magnetic moment of electrons (atoms), and that in nuclear pseudomagnetic field. It should be reminded that B e = η4πρµ e , where η depends on the target shape (for example, for sphere η = 2 3 ) [16]. Suppose polarization vector − → p is orthogonal to the particle momentum. In this case the addition to the rotation angle caused by B e is expressed
ϑ Be = k∆µ p B e l E = η k∆µ p 4πρµ e l E ,
where ∆µ p is the anomalous magnetic moment of proton , l is the length of gas target , E is the particle energy.
Then we obtain ϑ Be ϑ s = η ∆µ p µ e gh 2 2M ∼ = 0.9 η r 0 g , where r 0 = e 2 mc 2 is the electromagnetic radius of electron. And for g = 2 · 10 −13 and r 0 = 2.8 · 10 −13 ratio ϑ Be ϑ s ∼ = η.
Deuteron spin rotation and oscillations in a nonpolarized target
According to the above, the refractive index of neutral and charged particles of spin S can be written as
N = 1 + 2πρ k 2 f (0)(15)
where f (0) is the amplitude of particle zero-angle elastic coherent scattering by a scattering center, which is an operator acting in the particle spin space, f (0) = Sp ρ J F (0) ; ρ J is the spin density matrix of the scatterer; F (0) stands for the forward scattering operator amplitude acting in the spin space of the particle and the scatterer of spin − → J . If the wave function of the particle entering a target is ψ 0 , then that after travelling a distance z in the target is written as
ψ (z) = exp i k N z ψ 0 (16)
The explicit form of the amplitude f (0) for particles with arbitrary spin S has been obtained in [6]. According to these articles even for an unpolarized target is a function of the incident particle spin operator and can be written as
f (0) = d + d 1 S 2 z + d 2 S 4 z + ... + d s S 2s z(17)
The quantization axis z is directed along n = − → k k . Consider a specific case of strong interactions invariant under space and time reflections. For this reason, the terms containing odd powers of S are neglected. Correspondingly, the refractive index
N = 1 + 2πρ k 2 d + d 1 S 2 z + d 2 S 4 z + ... + d s S 2s z(18)
From eq. (18) one can draw an important conclusion about the refractive index being dependent on the spin orientation with respect to the pulse direction. Let m denote a magnetic quantum number, then for a particle in a state that is an eigenstate of the spin projection operator onto the z axis, S z , the refractive index is written as
N (m) = 1 + 2πρ k 2 d + d 1 m 2 + d 2 m 4 + ... + d s m 2s(19)
According to eq. (19), the particle states with quantum numbers m and −m have the same refractive indices. For a spin-1 particle (for example, a J/ψ particle, a deuteron) and a spin-3 2 particle (i.g. an Ω − hyperon).
N (m) = 1 + 2πρ k 2 d + d 1 m 2
As seen,
Re N (±1) = Re N (0) , Im N (±1) = Im N (0) , Re N ± 3 2 = Re N ± 1 2 Im N ± 3 2 = Im N ± 1 2
Since we have obtained the explicit spin structure of the refractive index and the wave function (16) is known, in every specific case we can find all spin properties of a beam in the target at depth z.
Let us come to consideration of deuteron passing through medium (particle with spin = 1). The wave function can be represented as a superposition of basis spin wave functions χ m , which are eigenfunctions of the operators S 2 and S z , S z χ m = mχ m :
ψ = m=±1,0 a m χ m .
Let us look for the mean value S = ψ S ψ / ψ|ψ of the spin operator in state ψ.
Suppose the particle enter the target at z = 0. Wave function of the particle insid the medium at the depth z can be expressed as:
Ψ = a 1 a 0 a −1 = a e iδ1 e ikN1z b e iδ0 e ikN0z c e iδ−1 e ikN−1z = a e iδ1 e ikN1z b e iδ0 e ikN0z c e iδ−1 e ikN1z it should be mentioned that N 1 = N −1
Let us choose coordinate system in which plane (xz) coincides with that formed by vector − → S (< S>= <ψ| S|ψ> |ψ| 2 ) before entering the target and deuteron momentum. In this case δ 1 − δ 0 = δ −1 − δ 0 = 0 and components of vector at z = 0 S x = 0, S y = 0.
As a result we obtain:
< S x >= √ 2e − 1 2 ρ(σ0+σ1)z b(a + c) cos[ 2πρ k Red 1 z]/|ψ| 2 , < S y >= − √ 2e − 1 2 ρ(σ0+σ1)z b(a − c) sin[ 2πρ k Red 1 z]/|ψ| 2 ,(20)< S z >= e ρσ1z (a 2 − c 2 )/|ψ| 2 ,
Particle with spin 1 also possesses tensor polarization i.e. tensor of quadrupo-larizationQ ij = 3/2(Ŝ iŜj +Ŝ jŜi − 4/3δ ij ) for it we can obtain
< Q xx >= − a 2 + c 2 1 2 e −ρσ1 z + b 2 e −ρσ0 z + 3e −ρσ1 z ac cos [δ 1 − δ −1 ] /|ψ| 2 < Q yy >= − a 2 + c 2 1 2 e −ρσ1 z + b 2 e −ρσ0 z − 3e −ρσ1 z ac cos [δ 1 − δ −1 ] /|ψ| 2 < Q zz >= a 2 + c 2 1 2 e −ρσ1 z − 2b 2 e −ρσ0 z /|ψ| 2 ,(21)< Q xy >= −3e −ρσ1 z ac sin [δ 1 − δ −1 ] /|ψ| 2 , < Q xz >= 3 √ 2 e − 1 2 ρ(σ0+σ1)z b(a − c) cos[ 2πρ k Red 1 z]/|ψ| 2 , < Q yz >= − 3 √ 2 e − 1 2 ρ(σ0+σ1)z b(a + c) sin[ 2πρ k Red 1 z]/|ψ| 2 ,
where |ψ| 2 = 2(a 2 + c 2 ) e −ρσ1 z + b 2 e −ρσ0 z and z is the length of particle path inside a medium.
According to (20,21) rotation appears if angle between polarization vector and momentum of particle differs from π 2 . At this for acute angle between polarization vector and momentum the sign of rotation is opposite than that for obtuse angles.
If spin is orthogonal to momentum then (a = c) particle spin (tensor of quadrupolarization) oscillate (do not rotate)
< S x >= √ 2e − 1 2 ρ(σ0+σ1)z 2 cos[ 2πρ k Red 1 z]/|ψ| 2 , < S y >= 0, < S z >= 0,
And tensor of quadrupolarization:
< Q xx >= 2 2 e −ρσ1 z + 2 e −ρσ0 z }/|ψ| 2 , < Q yy >= −4 2 e −ρσ1 z /2 + 2 e −ρσ0 z /|ψ| 2 , < Q zz >= 2 2 e −ρσ1 z − 2 2 e −ρσ0 z /|ψ| 2 ,(22)
< Q xy >= 0,
< Q xz >= 0, < Q yz >= − 3 √ 2 e (σ0+σ1)z 2ab sin[ 2πρ k Red 1 z] /|ψ| 2 ,
where |ψ| 2 = 2 2 e −ρσ1 z + 2 e −ρσ0 z and z is the length of particle path inside a medium.
Let us evaluate phase of oscillation
ϕ = 2πρ z k Red 1
for COSY. In this case total phase storing during experiment Φ = ϕνT . For particular conditions ρ z = n = 10 14 cm −2 , Red 1 ∼ 10 −13 (for scattering of deuteron by hydrogen), ν ∼ 10 6 , T ∼ 70 hours = 2.52 · 10 5 s and Φ can be estimated as Φ ∼ 1 rad.
As you can see the effect magnitude is large enough to be observable at COSY. It is very important that in considered case (deuterons passing through nonpolarized medium) there are no magnetic fields at the target area.
Experimental study of f (0) at different particle energies is an important tool for investigation of particles interaction properties. Phenomena of spin rotation and oscillation, described above, give the basis for methods of investigation of amplitude f (0) spin-dependendent part, examination of dispesion relations for it and testing of P-,T-violations at particle interactions for different energies. Of the essence, interactions at final state make the measurement of T-odd contributions difficult at experimental studies of T-violation by scattering of particles by each other. This problem is absent for T-odd particle spin rotation due to refraction, because it is determined by the elastic coherent zero-angle scattering amplitude. In this case initial and final states of target coincides and the above difficulty does not occur. As a result, methods based on the measurement of spin rotation angle (and f (0) measurement, as a sequence) can provide the most tight restrictions to the possible value of T-odd interactions.
One more promising aria of experimental investigations can be mentined, where phenomena, caused by nuclear optics of polarized medium, can appear efficient. Storage rings become now the more and more important investigation tool. Life-time of particle beam in storage ring can reach several hours and even days. During this time particles circulate in storage ring with frequency of several MHz and even small spin-dependent interactions of beams with each other can significantly influence on polarization state of beams.
We considered the rest target above. But in storage rings moving bunch is usually used as a target. No problems appear at study of spin rotation effect in this case -you should consider the effect in the rest frame of one of the beams and then reduce the result to the laboratory frame.
As an example let us consider cross-collision of two bunches of polarized particles. Suppose particles of the first beam have mass m 1 , energy E 1 and Lorentz factor γ 1 , whereas particles of the second beam are characterized by mass m 2 , energy E 2 and Lorentz factor γ 2 . Choose the frame where the second beam rest. In this frame the energy of the particles of the first beam E ′ 1 = E 1 γ 2 = m 1 c 2 γ 1 γ 2 . Refractive index is expressed in conventional form:
N = 1 + δ N + 2πρ ′ 2 k ′2 1 f (E ′ 1 , 0)
where δ N is the contribution to refractive index caused by refraction of particles in electric and magnetic fields of the bunch, ρ ′ 2 = γ −1 2 ρ 2 is the density of bunch 2 in its rest frame and ρ 2 is the density of the second bunch in laboratory frame, k ′ 1 = k 1 γ 2 . is the wave number of particles of the first bunch in the rest frame of bunch 2 and k 1 is the wave number of these particles in laboratory frame. Contribution to the particle spin rotation angle caused by f (E ′ 1 , 0) is expressed:
ϑ ′ = 2πρ ′ 2 k ′ 1 (Re f ↑↑ (E ′ 1 , 0) − Re f ↑↓ (E ′ 1 , 0)) L,
where L = γ 2 l is the length of the bunch 2 in its rest frame, l is the length of this bunch in laboratory frame. Lorentz factor of particle 1 in rest frame of particle 2 is γ = γ 1 γ 2 and this factor can be educed in scattering amplitude: f (E ′ 1 , 0) = γ 1 γ 2 f ′ (E ′ 1 , 0). As a result:
ϑ ′ = 2πρ 2 λ 1c Re f ′ ↑↑ (E ′ 1 , 0) − Re f ′ ↑↓ (E ′ 1 , 0) l,
where λ 1c = h/m 1 c.
Angle of rotation is invariant. Then in laboratory frame it will be the same. Evidently, particles 2 also experience refraction on the bunch 1. As a result mutual rotation of polarizatuion vectors of both bunches occurs. Taking into consideration the fact that in storage ring particle passes through the target (bunch) many times one can see that total angle of spin rotation, caused by refraction, many times rises. The above makes measurement of zero-angle scattering amplitude in wide particle energy range very promising.
Very interesting possibility appears when one of the beams (e.g. beam 2) has very low energy. It can be beam of ultracold atoms. Then index of refraction of beam 2 by beam 1 is high and we can apply atom interferometry methods for scattering amplitude measurement.
It should be mentioned that unique possuibility to measure spin-independent zero-angle scattering amplitude at high energies also appears.
It is important to note that photon beam formed by laser wave can be used as one of the beams. In this case effect of spin rotation is determined by elastic coherent amplitude of zero-angle scattering of particle by photon. According to the above analysis both electromagnetic and P-,T-odd weak interactions contribute to the refration index of particle in the area occupied by photons [17]. As a result area occupied by photons can be described as optically anisotropic medium.
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| []
|
[
"Robust and rate-optimal Gibbs posterior inference on the boundary of a noisy image",
"Robust and rate-optimal Gibbs posterior inference on the boundary of a noisy image"
]
| [
"Nicholas Syring ",
"Ryan Martin "
]
| []
| []
| Detection of an image boundary when the pixel intensities are measured with noise is an important problem in image segmentation. From a statistical point of view, the challenge is that likelihood-based methods require modeling the pixel intensities inside and outside the image boundary, even though these are typically of no practical interest. Since misspecification of the pixel intensity models can negatively affect inference on the image boundary, it would be desirable to avoid this modeling step altogether. Towards this, we develop a robust Gibbs approach that constructs a posterior distribution for the image boundary directly, without modeling the pixel intensities. We prove that the Gibbs posterior concentrates asymptotically at the minimax optimal rate, adaptive to the boundary smoothness. Monte Carlo computation of the Gibbs posterior is straightforward, and simulation results show that the corresponding inference is more accurate than that based on existing Bayesian methodology. | 10.1214/19-aos1856 | [
"https://arxiv.org/pdf/1606.08400v4.pdf"
]
| 88,514,791 | 1606.08400 | 9cd06281f7fcc4c8a9bbbe1af5a92fe8f2df8f47 |
Robust and rate-optimal Gibbs posterior inference on the boundary of a noisy image
June 4, 2018
Nicholas Syring
Ryan Martin
Robust and rate-optimal Gibbs posterior inference on the boundary of a noisy image
June 4, 2018and phrases: Adaptationboundary detectionlikelihood-free infer- encemodel misspecificationposterior concentration rate
Detection of an image boundary when the pixel intensities are measured with noise is an important problem in image segmentation. From a statistical point of view, the challenge is that likelihood-based methods require modeling the pixel intensities inside and outside the image boundary, even though these are typically of no practical interest. Since misspecification of the pixel intensity models can negatively affect inference on the image boundary, it would be desirable to avoid this modeling step altogether. Towards this, we develop a robust Gibbs approach that constructs a posterior distribution for the image boundary directly, without modeling the pixel intensities. We prove that the Gibbs posterior concentrates asymptotically at the minimax optimal rate, adaptive to the boundary smoothness. Monte Carlo computation of the Gibbs posterior is straightforward, and simulation results show that the corresponding inference is more accurate than that based on existing Bayesian methodology.
Introduction
In image analysis, the boundary or edge of the image is one of the most important features of the image, and extraction of this boundary is a critical step. An image consists of pixel locations and intensity values at each pixel, and the boundary can be thought of as a curve separating pixels of higher intensity from those of lower intensity. Applications of boundary detection are wide-ranging, e.g., Martin et al. (2004) use boundary detection to identify important features in pictures of natural settings, Li et al. (2010) identifies boundaries in medical images, and in Yuan et al. (2016) boundary detection helps classify the type and severity of wear on machines. For images with noiseless intensity, boundary detection has received considerable attention in the applied mathematics and computer science literature; see, e.g., Ziou and Tabbone (1998), Maini andAggarwal (2009), Li et al. (2010), and Anam et al. (2013). However, these approaches suffer from a number of difficulties. First, they can produce an estimate of the image boundary, but do not quantify estimation uncertainty. Second, these methods use a two-stage approach where the image is first smoothed to filter out noise and then a boundary is estimated based on a calculated intensity gradient. This two-stage approach makes theoretical analysis of the method challenging, and no convergence results are presently known to the authors. Third, in our examples, these methods perform poorly on noisy data, and we suspect one reason for this is that the intensity gradient is less informative for the boundary when we observe noisy data. In the statistics literature, Gu et al. (2014) take a Bayesian approach to boundary detection and emphasize borrowing information to recover boundaries of multiple, similar objects in an image. Boundary detection using wombling is also a popular approach; see Liang et al. (2009), with applications to geography (Lu and Carlin 2004), public health (Ma and Carlin 2007), and ecology (C. et al. 2010). However, these techniques are used with areal or spatially aggregated data and are not suitable for the pixel data encountered in image analysis.
In Section 2, we give a detailed description of the image boundary problem, following the setup in Li and Ghosal (2017). They take a fully Bayesian approach, modeling the probability distributions of the pixel intensities both inside and outside the image. This approach is challenging because it often introduces nuisance parameters in addition to the image boundary. Therefore, in Section 3, we propose to use a so-called Gibbs model, where a suitable loss function is used to connect the data to the image boundary directly, rather than a probability model (e.g., Bissiri et al. 2016;Catoni 1997;Jiang and Tanner 2008;Syring and Martin 2017;Zhang 2006).
We investigate the asymptotic convergence properties of the Gibbs posterior, in Section 4, and we show that, if the boundary is α-Hölder smooth, then the Gibbs posterior concentrates around the true boundary at the rate {(log n)/n} α/(α+1) , which is minimax optimal (Mammen and Tsybakov 1995) up to the logarithmic factor, relative to neighborhoods of the true boundary measured by the Lebesgue measure of a symmetric difference. Moreover, as a consequence of appropriately mixing over the number of knots in the prior, this rate is adaptive to the unknown smoothness α. To our knowledge, this is the first Gibbs posterior convergence rate result for an infinite-dimensional parameter, so the proof techniques used herein may be of general interest. Further, since the Gibbs posterior concentrates at the optimal rate without requiring a model for the pixel intensities, we claim that the inference on the image boundary is robust.
Computation of the Gibbs posterior is relatively straightforward and, in Section 5, we present a reversible jump Markov chain Monte Carlo method; R code to implement to the proposed Gibbs posterior inference is available at https://github.com/nasyring/ GibbsImage. A comparison of inference based on the proposed Gibbs posterior to that based on the fully Bayes approach in Li and Ghosal (2017) is shown in Section 6. For smooth boundaries, the two methods perform similarly, providing very accurate estimation. However, the Gibbs posterior is easier to compute, thanks to there being no nuisance parameters, and is notably more accurate than the Bayes approach when the model is misspecified or when the boundary is not everywhere smooth. The technical details of the proofs are deferred to Appendix A.
Problem formulation
Let Ω ⊂ R 2 be a bounded region that represents the frame of the image; typically, Ω will be a square, say, [− 1 2 , 1 2 ] 2 , but, generally, we assume only that Ω is scaled to have unit Lebesgue measure. Data consists of pairs (X i , Y i ), i = 1, . . . , n, where X i is a pixel location in Ω and Y i is an intensity measurement at that pixel. The range of Y i is contextdependent, and we consider both binary and real-valued cases below. The model asserts that there is a region Γ ⊂ Ω such that the intensity distribution is different depending on whether the pixel is inside or outside Γ. We consider the following model for the joint distribution P Γ of pixel location and intensity, (X, Y ):
X ∼ g(x), Y | (X = x) ∼ f Γ (y) 1(x ∈ Γ) + f Γ c (y) 1(x ∈ Γ c ),(1)
where g is a density on Ω, f Γ and f Γ c are densities on the intensity space, F Γ and F Γ c are their respective distribution functions, and 1(·) denotes an indicator function. That is, given the pixel location X = x, the distribution of the pixel intensity Y depends only on whether x is in Γ or Γ c . Of course, more complex models are possible, e.g., where the pixel intensity distribution depends on the specific pixel location, but we will not consider such models here. We assume that there is a true, star-shaped set of pixels, denoted by Γ , with a known reference point in its interior. That is, any point in Γ can be connected to the reference point by a line segment fully contained in Γ . The observations {(X i , Y i ) : i = 1, . . . , n} are iid samples from P Γ , and the goal is to make inference on Γ or, equivalently, its boundary γ = ∂Γ . The density g for the pixel locations is of no interest and is taken to be known. The question is how to handle the two conditional distributions, f Γ and f Γ c . Li and Ghosal (2017) take a fully Bayesian approach, modeling both f Γ and f Γ c . By specifying these parametric models, they are obligated to introduce priors and carry out posterior computations for the corresponding parameters. Besides the efforts needed to specify models and priors and to carry out posterior computations, there is also a concern that the pixel intensity models might be misspecified, potentially biasing the inference on Γ .
Since the forms of f Γ and f Γ c , as well as any associated parameters, are irrelevant to the boundary detection problem, it is natural to ask if inference can be carried out robustly, without modeling the pixel intensities.
We answer this question affirmatively, developing a Gibbs model for Γ in Section 3. In the present context, suppose we have a loss function Γ (x, y) that measures how well an observed pixel location-intensity pair (x, y) agrees with a particular region Γ. The defining characteristic of Γ is that Γ should be the unique minimizer of Γ → R(Γ), where R(Γ) = P Γ Γ is the risk, i.e., the expected loss. A main contribution here, in Section 3.1, is specification of a loss function that meets this criterion. A necessary condition to construct such a loss function is that the distribution functions F Γ and F Γ c are stochastically ordered. Imagine a gray-scale image; then, stochastic ordering means this image is lighter, on average, inside the boundary than outside the boundary, or viceversa. In the specific context of binary images the pixel densities f Γ and f Γ c are simply numbers between 0 and 1, and the stochastic ordering assumption means that, without loss of generality, f Γ > f Γ c , while for continuous images, again without loss of generality, F Γ (y) < F Γ c (y) for all y ∈ R. If we define the empirical risk,
R n (Γ) = 1 n n i=1 Γ (X i , Y i ),(2)
then, given a prior distribution Π for Γ, the Gibbs posterior distribution for Γ, denoted by Π n , has the formula
Π n (B) = B e −nRn(Γ) Π(dΓ) e −nRn(Γ) Π(dΓ) ,(3)
where B is a generic Π-measurable set of Γ's. Of course, (3) only makes sense if the denominator is finite; in Section 3, our risk functions R n are non-negative so this integrability condition holds automatically. Proper scaling of the loss in the Gibbs model is important (e.g., Bissiri et al. 2016;Syring and Martin 2016), and here we will provide some context-specific scaling; see Sections 4 and 5.2. Our choice of prior Π for Γ is discussed in Section 3.2. Together, the loss function Γ and the prior for Γ define the Gibbs model, no further modeling is required.
3 Gibbs model for the image boundary
Loss function
To start, we consider inference on the image boundary when the pixel intensity is binary, i.e., Y i ∈ {−1, +1}. In this case, the densities, f Γ and f Γ c , in (1) must be Bernoulli, so the likelihood is known. Even though the parametric form of the conditional distributions is known, the Gibbs approach only requires prior specification and posterior computation related to Γ, whereas the Bayes approach must also deal with the nuisance parameters in these Bernoulli conditional distributions. The binary case is relatively simple and will provide insights into how to formulate a Gibbs model in the more challenging continuous intensity problem.
A reasonable choice for the loss function Γ is the following weighted misclassification error loss, depending on a parameter h > 0:
Γ (x, y) = Γ (x, y | h) = h 1(y = +1, x ∈ Γ c ) + 1(y = −1, x ∈ Γ).(4)
Note that putting h = 1 in (4) gives the usual misclassification error loss. In order for the Gibbs model to work the risk, or expected loss, must be minimized at the true Γ for some h. Picking h to ensure this property holds necessitates making a connection between the probability model in (1) and the loss in (4). The condition in (5) below is just the connection needed. With a slight abuse of notation, let f Γ and f Γ c denote the conditional probabilities for the event Y = +1, given X ∈ Γ and X ∈ Γ c , respectively. Recall our stochastic ordering assumption implies f Γ > f Γ c .
Proposition 1. Using the notation in the previous paragraph, if h is such that
f Γ > 1 1 + h > f Γ c ,(5)
then the risk function R(Γ) = P Γ Γ is minimized at Γ .
Either one-but not both-of the above inequalities can be made inclusive and the result still holds. The condition in (5) deserves additional explanation. For example, if we know f Γ ≥ 1 2 > f Γ c , then we take h = 1, which means that in (4) we penalize both intensities of 1 outside Γ and intensities of −1 inside Γ by a loss of 1. If, however, we know the overall image brightness is higher so that f Γ ≥ 4 5 > f Γ c then we take h = 1/4 in (4) and penalize bright pixels outside Γ by less than dull pixels inside Γ. To see why this loss balancing is so crucial, suppose the second case above holds so that f Γ = 4/5 and f Γ c = 3/4, but we take h = 1 anyway. Then, in (4), 1(y = +1, x ∈ Γ c ) is very often equal to 1 while 1(y = −1, x ∈ Γ) is very often 0. We will likely minimize the expected loss then by incorrectly taking Γ to be all of Ω so that the first term in the loss vanishes. Knowing a working h corresponds to having some prior information about f Γ and f Γ c , but we can also use the data to estimate a good value of h and we describe this data-driven strategy in Section 5.2.
Next, for the continuous case, we assume that the pixel intensity takes value in R. The proposed strategy is to modify the misclassification error (4) by working with a suitably discretized pixel intensity measurement, reminiscent of threshold modeling. In particular, consider the following version of the misclassification error, depending on parameters (c, k, z), with c, k > 0:
Γ (x, y) = Γ (x, y | c, k, z) = k 1(y > z, x ∈ Γ c ) + c 1(y ≤ z, x ∈ Γ).(6)
Again, we claim that, for suitable (c, k, z), the risk function is minimized at Γ . Let F Γ and F Γ c denote the distribution functions corresponding to the densities f Γ and f Γ c in (1), respectively. Recall our stochastic ordering assumption implies
F Γ (z) < F Γ c (z). Proposition 2. If (c, k, z) in (6) satisfies F Γ (z) < k k + c < F Γ c (z),(7)
then the risk function R(Γ) = P Γ Γ is minimized at Γ .
Again, either one-but not both-of the above inequalities in (7) can be made inclusive and the result still holds. The parameters k and c in (6) determine the scale of the loss as mentioned in Section 1, while z determines an intensity cutoff. According to the loss in (6), if a given pixel is located at x ∈ Γ c , and with intensity y larger than cutoff z, it will incur a loss of k > 0. This implies that the true image Γ can be identified by working with a suitable version of the loss (6). A similar condition to (7), see Assumption 1 in Section 4, says what scaling is needed in order for the Gibbs posterior to concentrate at the optimal rate. Although the conditions on the scaling all involve the unknown distribution P Γ , a good choice of (c, k, z) can be made based on the data alone, without prior information, and we discuss this strategy in Section 5.2.
Prior specification
We specify a prior distribution for the boundary of the region Γ by first expressing the pixel locations x in terms of polar coordinates (θ, r), an angle and radius, where θ ∈ [0, 2π] and r > 0. The specific reference point and angle in Ω used to define polar coordinates are essentially arbitrary, subject to the requirement that any point in Γ can be connected to the reference point by a line segment contained in Γ . Li and Ghosal (2017) tested the influence of the reference point in simulations and found it to have little influence on the results. Using polar coordinates the boundary of Γ can be determined by the parametric curve (θ, γ(θ)). We proceed to model this curve γ.
Whether one is taking a Bayes or Gibbs approach, a natural strategy to model the image boundary is to express γ as a linear combination of suitable basis functions, i.e., γ(θ) =γ D,β (θ) = D j=1 β j B j,D (θ). Li and Ghosal (2017) use the eigenfunctions of the squared exponential periodic kernel as their basis functions. Here we consider a model based on free knot b-splines, where the basis functions are defined recursively as
B i,1 (θ) = 1(θ ∈ [t i , t i+1 ]) B i,k (θ) = θ − t i t i+k−1 − t i B i,k−1 (θ) + t i+k − θ t i+k − t i+1 B i+1,k−1 (θ), where θ ∈ [0, 2π], t −2 , t −1 , t 0 and t D+1 , t D+2 , t D+3
are outer knots, t 1 , ..., t D are inner knots, and β = (β 1 , ..., β D ) ∈ (R + ) D is the vector of coefficients. Note that we restrict the coefficient vector β to be positive because the function values γ(θ) measure the radius of a curve from the origin. In the simulations in Section 6, the coefficients β 2 , ..., β D are free parameters, while β 1 is calculated deterministically to force the boundary to be closed, i.e. γ(0) = γ(2π), and we require t 1 = 0 and t D = 2π; all other inner knots are free. Our model based on the b-spline representation seem to perform as well as the eigenfunctions used in Li and Ghosal (2017) for smooth boundaries, but a bit better for boundaries with corners; see Section 6. Therefore, the boundary curve is γ is parametrized by an integer D and a D-vector β. We introduce a prior Π on (D, β) hierarchically as follows: D has a Poisson distribution with rate µ D and, given D, the coordinates β 1 , . . . , β D of β are iid exponential with rate µ β . These choices satisfy the technical conditions on Π detailed in Section 4. In our numerical experiments in Section 6, we take µ D = 12 and µ β = 10.
Gibbs posterior convergence
The Gibbs model depends on two inputs, namely, the prior and the loss function. In order to ensure that the Gibbs posterior enjoys desirable asymptotic properties, some conditions on both of these inputs are required. The first assumption listed below concerns the loss; the second concerns the true image boundary γ = ∂Γ ; and the third concerns the prior. Here we will focus on the continuous intensity case, since the only difference between this and the binary case is that the latter provides the discretization for us.
Assumption 1. Loss function parameters (c, k, z) in (6) satisfy F Γ (z) < e k − 1 e c+k − 1 and F Γ c (z) > e k − 1 e k − e −c .(8)
Compared to the condition (7) that was enough to allow the loss function to identify the true Γ , condition (8) in Assumption 1 is only slightly stronger. This can be seen from the following inequality:
e k − 1 e k − e −c > k k + c > e k − 1 e c+k − 1.
However, if (c, k) are small, then the three quantities in the above display are all approximately equal, so Assumption 1 is not much stronger than what is needed to identify Γ . Again, these conditions on (c, k, z) can be understood as providing a meaningful scale to the loss function. Intuitively, the scale of the loss between observations receiving no loss versus some loss, expressed by parameters k and c, should be related to the level of information in the data. When F Γ (z) and F Γ c (z) are far apart, the data can more easily distinguish between F Γ and F Γ c , so we are free to assign larger losses than when F Γ (z) and F Γ c (z) are close and the data are relatively less informative. The ability of a statistical method to make inference on the image boundary will depend on how smooth the true boundary is. Li and Ghosal (2017) interpret γ as a function from the unit circle to the positive real line, and they formulate a Hölder smoothness condition for this function. Following the prior specification described in Section 3.2, we treat the boundary γ as a function from the interval [0, 2π] to the positive reals, and formulate the smoothness condition on this arguably simpler version of the function. Since the reparametrization of the unit circle in terms of polar coordinates is smooth, it is easy to check that the Hölder smoothness condition (9) below is equivalent to that in Li and Ghosal (2017).
Assumption 2. The true boundary function γ : [0, 2π] → R + is α-Hölder smooth, i.e., there exists a constant L = L γ > 0 such that
|(γ ) ([α]) (θ) − (γ ) ([α]) (θ )| ≤ L|θ − θ | α−[α] , ∀ θ, θ ∈ [0, 2π],(9)
where (γ ) (k) denotes the k th derivative of γ and [α] denotes the largest integer less than or equal to α. Following the description of Γ in the introduction, we also assume that the reference point is strictly interior to Γ meaning that it is contained in an open set itself wholly contained in Γ so that γ is uniformly bounded away from zero. Moreover, the density g for X, as in (1), is uniformly bounded above by g := sup x∈Ω g(x) and below by g := inf x∈Ω g(x) ∈ (0, 1) on Ω.
General results are available on the error in approximating an α-Hölder smooth function by b-splines of the form specified in Section 3.2. Indeed, Theorem 6.10 in Schumaker (2007) implies that if γ satisfies (9), then
∀ d > 0, ∃ β d ∈ (R + ) d such that γ −γ d,β d ∞ d −α .(10)
Since γ (θ) > 0, we can consider all coefficients to be positive; i.e. β d ∈ (R + ) d and see Lemma 1(b) in Shen and Ghosal (2015). The next assumption about the prior makes use of the approximation property in (10).
Assumption 3. Let β d , for d > 0, be as in (10). Then there exists C, m > 0 such that the prior Π for (D, β) satisfies, for all d > 1,
log Π(D > d) −d log d, log Π(D = d) −d log d, log Π( β − β d 1 ≤ kd −α | D = d) −d log{1/(kd −α )}, log Π(β ∈ [−m, m] d | D = d) log d − Cm.
The first two conditions in Assumption 3 ensure that the prior on D is sufficiently spread out while the second two conditions ensure that there is sufficient prior support near β's that approximate γ well. Assumption 3 is also needed in Li and Ghosal (2017) for convergence of the Bayesian posterior at the optimal rate. However, the Bayes model also requires assumptions about the priors on the nuisance parameters, e.g., Assumption C in Li and Ghosal (2017), which are not necessary in our approach here.
In what follows, let A B denote the symmetric difference of sets in Ω and λ(A B) its Lebesgue measure.
Theorem 1. With a slight abuse of notation, let Π denote the prior for Γ, induced by that on (D, β), and Π n the corresponding Gibbs posterior (3). Under Assumptions 1-3, there exists a constant M > 0 such that
P Γ Π n ({Γ : λ(Γ Γ) > M ε n }) → 0 as n → ∞,
where ε n = {(log n)/n} α/(α+1) and α is the smoothness coefficient in Assumption 2.
Theorem 1 says that, as the sample size increases, the Gibbs posterior places its mass on a shrinking neighborhood of the true boundary γ . The rate, given by the size of the neighborhood, is optimal according to Mammen and Tsybakov (1995), up to a logarithmic factor, and adaptive since the prior does not depend on the unknown smoothness α.
Computation
Sampling algorithm
We use reversible jump Markov chain Monte Carlo, as in Green (1995), to sample from the Gibbs posterior. These methods have been used successfully in Bayesian free-knot spline regression problems; see, e.g., Denison et al. (1998) and DiMatteo et al. (2001). Although the sampling procedure is more complicated when allowing the number and locations of knots to be random versus using fixed knots, the resulting spline functions can do a better job fitting curves with low smoothness.
To summarize the algorithm, we start with the prior distribution Π for (D, β) as discussed in Section 3.2. Next, we need to initialize values of D, the knot locations {t −2 , ..., t D+3 }, and the values of β 2 , ..., β D . The value of β 1 is then calculated numerically to force closure. We choose D = 12 with t −2 = −2, t −1 = −1, t 0 = −0.5, t 13 = 2π + 0.5, t 14 = 2π + 1, t 15 = 2π + 2 and t 1 , ..., t 12 evenly spaced in [0, 2π]. We set inner knots t 0 = 0 and t D = 2π while the other inner knot locations remain free to change in birth, death, and relocation moves; we also set β 2 = β 3 = ... = β 12 = 0.1. Then the following three steps constitutes a single iteration of the reversible jump Markov chain Monte Carlo algorithm to be repeated until the desired number of samples are obtained:
1. Use Metropolis-within-Gibbs steps to update the elements of the β vector, again solving for β 1 to force closure at the end. In our examples we use normal proposals centered at the current value of the element of the β vector, and with standard deviation 0.10.
2. Randomly choose to attempt either a birth, death, or relocation move to add a new inner knot, delete an existing inner knot, or move an inner knot.
Attempt the jump move proposed in
Step 2. The β vector must be appropriately modified when adding or deleting a knot, and again we must solve for β 1 . Details on the calculation of acceptance probabilities for each move can be found in Denison et al. (1998) and DiMatteo et al. (2001).
R code to implement this Gibbs posterior sampling scheme, along with the empirical loss scaling method described in Section 5.2, is available at https://github.com/nasyring/ GibbsImage.
Loss scaling
It is not clear how to select (c, k, z) to satisfy Assumption 1 without knowledge of F Γ and F Γ c . However, it is fairly straightforward to select values of (c, k, z) based on the data which are likely to meet the required condition. First, we need a notion of optimal (c, k, z) values. If we knew F Γ and F Γ c , then we would select z to maximize F Γ c (z) − F Γ (z) because this choice of z gives us the point at which F Γ and F Γ c are most easily distinguished. Then, we would choose (c, k) to be the largest values such that (8) holds. Intuitively, we want large values of (c, k) so that the loss function in (6) is more sensitive to departures from γ . Since we do not know F Γ and F Γ c , we estimate F Γ (z) and F Γ c (z) from the data. In order to do this, we need a rough estimate of γ to define the regions Γ and Γ c . Our approach is to model γ with a b-spline, as before, and estimate γ several times by minimizing (6) using several different values of the loss scaling parameters (c, k, z). Specifically, set a grid of z values z 1 , z 2 , ..., z g , and for each z j , find (c, k) = (c j , k j ) that satisfy k j k j + c j = |{i : y i ≤ z j }| n .
Next, estimate γ by minimizing (6) using (c j , k j , z j ). The estimate of γ provides estimates of the regions Γ and Γ c which we use to calculate the sample proportionŝ
F Γ := |{i : y i ≤ z j , x i ∈Γ }| |{i : x i ∈Γ }| andF Γ c := |{i : y i ≤ z j , x i ∈Γ c }| |{i : x i ∈Γ c }| .
Then, the approximately optimal value is z = arg max z jF Γ c (z j )−F Γ (z j ). Finally, choose the approximately optimal values of (c, k) to satisfy (8) replacing F Γ (z) and F Γ c (z) by their estimatesF Γ (z) andF Γ c (z). Based on the simulations in Section 6, this method produces values of (c, k, z) very close to their optimal values. Importantly, the estimated (c, k) are more likely to be smaller than their optimal values than larger, which makes our estimates more likely to satisfy (8). This is a consequence of the stochastic ordering of F Γ and F Γ c . Unless the classifier we obtain by minimizing (6) is perfectly accurate, we will tend to mix together samples from F Γ and F Γ c in our estimates. If we estimate F Γ (z) with some observations from F Γ and some from F Γ c , we will tend to overestimate F Γ (z), and vice versa we will tend to underestimate F Γ c (z). These errors will cause (c, k) to be underestimated, and therefore more likely to satisfy (8).
Numerical examples
We tested our Gibbs model on data from both binary and continuous images following much the same setup as in Li and Ghosal (2017). The pixel locations in Ω = [− 1 2 , 1 2 ] 2 are sampled by starting with a fixed m × m grid in Ω and making a small random uniform perturbation at each grid point. Several different pixel intensity distributions are considered. We consider two types of shapes for Γ : an ellipse with center (0.1, 0.1), rotated at an angle of 60 degrees, with major axis length 0.35 and minor axis length 0.25; and a centered equilateral triangle of height 0.5. The ellipse boundary will test the sensitivity of the model to boundaries which are off-center while the triangle tests the model's ability to identify non-smooth boundaries.
We consider four binary and four continuous intensity images and compare with the Bayesian method of Li and Ghosal (2017) B3. Ellipse image, m = 500, and F Γ and F Γ c are Bernoulli with parameters 0.25 and 0.2, respectively.
B4. Same as B3 but with triangle image.
C1. Ellipse image, m = 100, and F Γ and F Γ c are N (4, 1.5 2 ) and N (1, 1), respectively.
C2. Same as C1 but with triangle image.
C3. Ellipse image, m = 100, and F Γ and F Γ c are 0.2 N (2, 10) + 0.8 N (0, 1), a normal mixture, and N (0, 5), respectively.
C4. Ellipse image, m = 100, and F Γ and F Γ c are t distributions with 3 degrees of freedom and non-centrality parameters 1 and 0, respectively.
For binary images, the likelihood must be Bernoulli, so the Bayesian model is correctly specified in cases B1-B4. For the continuous examples in C1-C4, we assume a Gaussian likelihood for the Bayesian model. Then, cases C1 and C2 will show whether or not the Gibbs model can compete with the Bayesian model when the model is correctly specified, while cases C3 and C4 will demonstrate the superiority of the Gibbs model over the Bayesian model under misspecification. Again, the Gibbs model has the added advantage of not having to specify priors for or sample values of the mean and variance associated with the normal conditional distributions.
We replicated each example scenario 100 times for both the Gibbs and Bayesian models, each time producing a posterior sample of size 4000 after a burn in of 1000 samples. We recorded the errors-Lebesgue measure of the symmetric difference-for each run along with the estimated loss function parameters for the Gibbs models for continuous images. The results are summarized in Tables 1-2. We see that the Gibbs model is competitive with the fully Bayesian model in Examples B1-B4, C1, and C2, when the likelihood is correctly specified. When the likelihood is misspecified, there is a chance that the Bayesian model will fail, as in Examples C3 and C4. However, the Gibbs model does not depend on a likelihood, only the stochastic ordering of F Γ and F Γ c , and it continues to perform well in these non-Gaussian examples. From Table 2, we see that the empirical method described in Section 5.2 is able to select parameters for the loss function in (6) close to the optimal values and meeting Assumption 1. Figures 1 and 2 show the results of the Bayesian and Gibbs models for one simulation run in each of Examples B1-B2 and C1-C4, respectively. The 95% credible regions, as in Li and Ghosal (2017), are highlighted in gray around the posterior means. That is, let u i = sup θ {|γ i (θ) −γ(θ)|/s(θ)}, where γ i (θ) is the i th posterior boundary sample,γ(θ) is the pointwise posterior mean and s(θ) the pointwise standard deviation of the γ(θ) samples. If τ is the 95 th percentile of the u i 's, then a 95% uniform credible band is given byγ(θ) ± τ s(θ). The results of cases B2 and C2 suggest that free-knot b-splines may do a better job of approximating non-smooth boundaries than the kernel basis functions used by Li and Ghosal (2017). In particular, the RJ-MCMC sampling method with its relocation moves allowed knots to move towards the corners of the triangle, thereby improving estimation of the boundary over b-splines with fixed knots.
A Proofs
A.1 Proof of Proposition 1
By the definition of the loss function in (4), for a fixed h and for any Γ ⊂ Ω, we have Then the expectation of the loss difference above is
Γ (X, Y ) − Γ (X, Y ) = h 1(Y = +1, X ∈ Γ c ) − h 1(Y = +1, X ∈ Γ c ) + 1(Y = −1, X ∈ Γ) − 1(Y = −1, X ∈ Γ ) = h 1(Y = +1, X ∈ Γ \ Γ) − 1(Y = −1, X ∈ Γ \ Γ) + 1(Y = −1, X ∈ Γ \ Γ ) − h 1(Y = +1, X ∈ Γ \ Γ ).P (X ∈ Γ \ Γ) (hf Γ + f Γ − 1) + P (X ∈ Γ \ Γ ) (1 − f Γ c − hf Γ c ),
where the probability statement is with respect to the density g of X. By Assumption 2, the density g is bounded away from zero on Ω, so the expectation of the loss difference is zero if and only if Γ = Γ . The expectation can also be lower bounded by
P (X ∈ Γ Γ ) min{hf Γ + f Γ − 1, 1 − f Γ c − hf Γ c }.
Given the condition (5) in Proposition 1, both terms in the minimum are positive. Therefore, R(Γ) ≥ R(Γ ) with equality if and only if Γ = Γ , proving the claim.
A.2 Proof of Proposition 2
The proof here is very similar to that of Proposition 1. By the definition of the loss function in (6), for any fixed (c, k, z) and for any Γ ⊂ Ω, we get Then, the expectation of the loss difference above is given by
Γ (X, Y ) − Γ (X, Y ) = k 1(Y ≥ z, X ∈ Γ c ) − k 1(Y ≥ z, X ∈ Γ c ) + c 1(Y < z, X ∈ Γ) − c 1(Y < z, X ∈ Γ ) = k 1(Y ≥ z, X ∈ Γ \ Γ) − c 1(Y < z, X ∈ Γ \ Γ) + c 1(Y < z, X ∈ Γ \ Γ ) − k 1(Y ≥ z, X ∈ Γ \ Γ ).P (X ∈ Γ \ Γ) {k − kF Γ (z) − cF Γ (z)} + P (X ∈ Γ \ Γ ) {cF Γ c (z) − k + kF Γ c (z)},
where the probability statement is with respect to the density g of X. As in the proof of Proposition 1, this quantity is zero if and only if Γ = Γ . It can also be lower bounded by
P (X ∈ Γ Γ ) min k − kF Γ (z) − cF Γ (z), cF Γ c (z) − k + kF Γ c (z) .
Given the condition (7) in Proposition 2, both terms in the minimum are positive. Therefore, R(Γ) ≥ R(Γ ) with equality if and only if Γ = Γ , proving the claim.
A.3 Preliminary results
Towards proving Theorem 1, we need several lemmas. The first draws a connection between the distance between defined by the Lebesgue measure of the symmetric difference and the sup-norm between the boundary functions.
Lemma 1. Suppose Γ , with boundary γ = ∂Γ , satisfies Assumption 2, in particular, γ := inf θ∈[0,2π] γ (θ) > 0. Take any Γ ⊂ Ω, with γ = ∂Γ, such that λ(Γ Γ ) > δ for some fixed δ > 0, and any Γ ⊂ Ω such thatγ = ∂ Γ satisfies γ − γ ∞ < ωδ, where ω ∈ (0, 1). Then Proof. The first step is to connect the symmetric difference-based distance to the L 1 distance between boundary functions. A simple conversion to polar coordinates gives
λ( Γ Γ ) > 4δ γ 1 diam(Ω) − πω , where diam(Ω) = sup x,x ∈Ω x − x is the diameter of Ω. So, if ω < {π diam(Ω)} −1 ,λ(Γ Γ ) = Γ Γ dλ = 2π 0 γ(θ)∨γ (θ) γ(θ)∧γ (θ) r dr dθ = 1 2 2π 0 {γ(θ) ∧ γ(θ )} 2 − {γ(θ) ∨ γ(θ )} 2 dθ = 1 2 2π 0 |γ(θ) − γ (θ)| |γ(θ) + γ (θ)| dθ.
If we let γ = inf θ γ (θ), then it is easy to verify that
γ ≤ |γ(θ) + γ (θ)| ≤ diam(Ω), ∀ θ ∈ [0, 2π].
Therefore,
1 2 γ γ − γ 1 ≤ λ(Γ Γ ) ≤ 1 2 diam(Ω) γ − γ 1 .(11)
Next, if λ(Γ Γ ) > δ, which is positive by Assumption 2, then it follows from the rightmost inequality in (11) that diam(Ω) γ − γ 1 > 2δ and, by the triangle inequality,
diam(Ω){ γ −γ 1 + γ − γ 1 } > 2δ.
We have γ −γ 1 ≤ 2π γ −γ ∞ which, by assumption, is less than 2πωδ. Consequently,
diam(Ω){2πωδ + γ − γ 1 } > 2δ
and, hence,
γ − γ 1 > 2δ diam(Ω) − 2πωδ.
By the left-most inequality in (11), we get
λ( Γ Γ ) > 4δ γ diam(Ω) − 4πωδ γ = 4δ γ 1 diam(Ω) − πω ,
which is the desired bound. It follows immediately that the lower bound is a positive multiple of δ if ω < {π diam(Ω)} −1 .
The next lemma shows that we can control the expectation of the integrand in the Gibbs posterior under the condition (8) on the tuning parameters (c, k, z) in the loss function definition.
Lemma 2. If (8) holds, then P Γ e −( Γ − Γ ) < 1 − ρλ(Γ Γ) for a constant ρ ∈ (0, 1).
Proof. From the proof of Proposition 2, we have
Γ (x, y) − Γ (x, y) = k 1(y ≥ z, x ∈ Γ \ Γ) − c 1(y < z, x ∈ Γ \ Γ) + c 1(y < z, x ∈ Γ \ Γ ) − k (y ≥ z, x ∈ Γ \ Γ ).
The key observation is that, if x ∈ Γ Γ , then the loss difference is 0 and, therefore, the exponential of the loss difference is 1. Taking expectation with respect P Γ , we get
P Γ e −( Γ − Γ ) = P g (X ∈ Γ Γ) + {e −k (1 − F Γ (z)) + e c F Γ (z)}P g (X ∈ Γ \ Γ) + {e −c F Γ c (z) + e k − e k F Γ c (z)}P g (X ∈ Γ \ Γ ).
From (8), we have
κ := max{e −k (1 − F Γ (z)) + e c F Γ (z), e −c F Γ c (z) + e k − e k F Γ c (z)} < 1, so that P Γ e −( Γ − Γ ) ≤ 1 − P g (X ∈ Γ Γ ) + κP g (X ∈ Γ Γ ) = 1 − (1 − κ)P g (X ∈ Γ Γ ).
Then the claim follows, with ρ = (1 − κ)g < 1, since P g (X ∈ Γ Γ ) ≥ gλ(Γ Γ ).
The next lemma yields a necessary lower bound on the denominator of the Gibbs posterior distribution. The neighborhoods G n are simpler than those in Lemma 1 of Shen and Wasserman (2001), because the variance of our loss difference is automatically of the same order as its expectation, but the proof is otherwise very similar to theirs, so we omit the details.
Lemma 3. Let t n be a sequence of positive numbers such that nt n → 0 and set G n = {Γ : R(Γ) − R(Γ ) ≤ Ct n } for some C > 0. Then e −[Rn(Γ)−Rn(Γ )] dΠ Π(G n ) exp(−2nt n ) with P Γ -probability converging to 1 as n → ∞.
Our last lemma is a summary of various results derived in Li and Ghosal (2017) towards proving their Theorem 3.3. This helps us to fill in the details for the lower bound in Lemma 3 and to identify a sieve-a sequence of subsets of the parameter space-with high prior probability but relatively low complexity.
Lemma 4. Let ε n be as in Theorem 1 and let D n = ( n log n ) 1 α+1 . Then, γ −γ Dn,β ∞ ≤ Cε n for some C > 0, β = β Dn ∈ (R + ) Dn from (10).
Define the neighborhood
B n = {(β, d) : β ∈ R d , d = D n , γ −γ d,β ∞ ≤ Cε n }.
Then Π(B n ) exp(−anε n ) for some a > 0 depending on C.
2. Define the sieve Σ n = {γ : γ =γ d,β , β ∈ R d , d ≤ D n , β ∞ ≤ n/K 0 }. Then Π(Σ c n ) exp(−Knε n ) for some K, K 0 > 0.
3. The bracketing number of Σ n satisfies log N (ε n , Σ n , · ∞ ) nε n .
A.4 Proof of Theorem 1
Define the set A n = {Γ : λ(Γ Γ) > M ε n }.
For the sieve Σ n in Lemma 4, we have Π n (A n ) ≤ Π n (Σ c n ) + Π n (A n ∩ Σ n ). We want to show that both terms in the upper bound vanish, in L 1 (P Γ ). It helps to start with a lower bound on I n = e −n{Rn(Γ)−Rn(Γ )} Π(dΓ), the denominator in both of the terms discussed above. First, write where G n is defined in Lemma 3 with t n = ε n and C > 0 to be determined. From Proposition 2 and Lemma 1 R(Γ) − R(Γ ) ≤ P g (X ∈ Γ Γ ) min{k − kF Γ (z) − cF Γ c (z), cF Γ c (z) + kF Γ c (z)} ≤ 1 2 V g diam(Ω) γ − γ 1
where V = V c,k,z is the min{· · · } term in the above display. Let B ∞ (γ ; r) denote the set of regions Γ with boundary functions γ = ∂Γ that satisfy γ − γ ∞ ≤ r. If Γ ∈ B ∞ (γ ; C 0 ε n ), then we have γ − γ 1 ≤ 2πgC 0 ε n and, therefore, R(Γ) − R(Γ ) ≤ Cε n , where C = C 0 πV g 2 diam(Ω). From Lemma 3, we have I n Π(G n )e −2Cεn , with P Γ -probability converging to 1. Since G n ⊇ B ∞ (γ ; C 0 ε n ), it follows from Lemma 4, part 1, I n Π{B ∞ (γ ; C 0 ε n )}e −2Cεn e −C 1 nεn , with P Γ probability converging to one, and where C 1 > 0 is a constant depending on C 0 and C. Now we are ready to bound Π n (Σ c n ). Write this quantity as Π n (Σ c n ) = N n (Σ c n ) I n = 1 I n Σ c n e −n{Rn(Γ)−Rn(Γ } Π(dΓ).
It will suffice to bound the expectation of N n (Σ c n ). By Tonelli's theorem, independence, and Lemma 2, we have ≤ Π(Σ c n ).
By Lemma 4, part 2, we have that Π(Σ c n ) ≤ e −Knεn . Next, we bound Π n (A n ∩ Σ n ). Again, it will suffice to bound the expectation of N n (A n ∩ Σ n ). Choose a covering A n ∩ Σ n by sup-norm balls B j = B ∞ (γ j ; ωM n ε n ), j = 1, . . . , J n , with centers γ j = ∂Γ j in A n and radii ωM n ε n , where ω < {π diam(Ω)} −1 is as in Lemma 1. Also, from Lemma 4, part 3, we have that J n is bounded by e K 1 nεn for some constant K 1 > 0. For this covering, we immediately get P Γ N n (A n ∩ Σ n ) ≤ Jn j=1 P Γ N n (B j ).
For each j, using Tonelli, independence, and Lemma 2 again, we get
P Γ N n (B j ) = B j {P Γ e −( Γ − Γ ) } n Π(dΓ) ≤ B j e −nρλ(Γ Γ ) Π(dΓ).
By Lemma 1, for Γ in B j , since the center γ j is in A n , it follows that λ(Γ Γ ) is lower bounded by ηM ε n , where η = η(Γ ) is given by
η = 4 γ 1 diam(Ω) − πω > 0.
Therefore, from the bound on J n , P Γ N n (A n ∩ Σ n ) ≤ Jn j=1 P Γ N n (B j ) ≤ e −nηM εn J n ≤ e −(ηM −K 1 )nεn .
Finally, we have Π n (A n ) ≤ Π n (A n ∩ Σ n ) + Π n (Σ c n ) = N n (A n ∩ Σ n ) I n + N n (Σ c n ) I n ≤ N n (A n ∩ Σ n ) e −C 1 nεn 1(I n > e −C 1 nεn ) + N n (Σ c n ) I n 1(I n ≤ e −C 1 nεn ) ≤ N n (A n ∩ Σ n ) e −C 1 nεn + 1(I n ≤ e −C 1 nεn ).
Taking P Γ -expectation and plugging in the bounds derived above, we get P Γ Π n (A n ) ≤ e −(ηM −K 1 −C 1 )nεn .
If M > (K 1 + C 1 )/η, then the upper bound vanishes, completing the proof.
as implemented in the BayesBD package (Syring and Li 2017) available on CRAN . B1. Ellipse image, m = 100, and F Γ and F Γ c are Bernoulli with parameters 0.5 and 0.2, respectively.B2. Same as B1 but with triangle image.
Figure 1 :
1From top, Examples B1-B2. In each row, the observed image is on the left, the Bayesian posterior mean estimator(Li and Ghosal 2017) is in the middle, and the Gibbs posterior mean estimator is on the right. Solid lines show the true image boundary, dashed lines are the estimates, and gray regions are 95% credible bands.
Figure 2 :
2Same asFigure 1, but for Examples C1-C4.
then the lower bound is a positive multiple of δ.
n{Rn(Γ)−Rn(Γ )} Π(dΓ)
P
Γ e −( Γ − Γ ) } n Π(dΓ) ρλ(Γ Γ )} n Π(dΓ)
Table 1 :
1Average errors (and standard deviations) for each example.Model
B1
B2
B3
B4
C1
C2
C3
C4
Bayes
0.00
(0.00)
0.02
(0.00)
0.01
(0.00)
0.02
(0.01)
0.03
(0.03)
0.04
(0.03)
0.11
(0.06)
0.10
(0.05)
Gibbs
0.01
(0.00)
0.01
(0.00)
0.02
(0.01)
0.02
(0.01)
0.01
(0.00)
0.01
(0.00)
0.01
(0.01)
0.01
(0.01)
Table 2 :
2Average (and optimal) values of the parameters (c, k, z) in (6).
Parameter
C1
C2
C3
C4
c
1.45
(1.86)
1.47
(1.86)
0.80
(1.27)
0.71
(0.80)
k
2.30
(2.36)
2.29
(2.36)
0.26
(0.34)
0.71
(0.75)
z
2.43
(2.40)
2.39
(2.40)
-1.83
(-1.76)
0.46
(0.46)
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"Institut für Theoretische Physik\nGoethe-Universität Frankfurt\nMax-von-Laue-Straße 160438Frankfurt am MainGermany",
"Institut für Theoretische Physik\nGoethe-Universität Frankfurt\nMax-von-Laue-Straße 160438Frankfurt am MainGermany",
"Institut für Theoretische Physik\nGoethe-Universität Frankfurt\nMax-von-Laue-Straße 160438Frankfurt am MainGermany"
]
| []
| Recently, KFe2As2 was shown to exhibit a structural phase transition from a tetragonal to a collapsed tetragonal phase under applied pressure of about 15 GPa. Surprisingly, the collapsed tetragonal phase hosts a superconducting state with Tc ∼ 12 K, while the tetragonal phase is a Tc ≤ 3.4 K superconductor. We show that the key difference between the previously known non-superconducting collapsed tetragonal phase in AFe2As2 (A= Ba, Ca, Eu, Sr) and the superconducting collapsed tetragonal phase in KFe2As2 is the qualitatively distinct electronic structure. While the collapsed phase in the former compounds features only electron pockets at the Brillouin zone boundary and no hole pockets are present in the Brillouin zone center, the collapsed phase in KFe2As2 has almost nested electron and hole pockets. Within a random phase approximation spin fluctuation approach we calculate the superconducting order parameter in the collapsed tetragonal phase. We propose that a Lifshitz transition associated with the structural collapse changes the pairing symmetry from d-wave (tetragonal) to s± (collapsed tetragonal). Our DFT+DMFT calculations show that effects of correlations on the electronic structure of the collapsed tetragonal phase are minimal. Finally, we argue that our results are compatible with a change of sign of the Hall coefficient with pressure as observed experimentally. * [email protected] 1 M. Rotter, M. Tegel, and D. Johrendt, Superconductivity at | 10.1103/physrevb.91.140503 | [
"https://arxiv.org/pdf/1501.03068v3.pdf"
]
| 55,943,397 | 1501.03068 | c45e8c2ac6281036b0a855f7b53b2dc0cf420877 |
Origin of the superconducting state in the collapsed tetragonal phase of KFe 2 As 2
Daniel Guterding
Institut für Theoretische Physik
Goethe-Universität Frankfurt
Max-von-Laue-Straße 160438Frankfurt am MainGermany
Steffen Backes
Institut für Theoretische Physik
Goethe-Universität Frankfurt
Max-von-Laue-Straße 160438Frankfurt am MainGermany
Harald O Jeschke
Institut für Theoretische Physik
Goethe-Universität Frankfurt
Max-von-Laue-Straße 160438Frankfurt am MainGermany
Roser Valentí
Institut für Theoretische Physik
Goethe-Universität Frankfurt
Max-von-Laue-Straße 160438Frankfurt am MainGermany
Origin of the superconducting state in the collapsed tetragonal phase of KFe 2 As 2
numbers: 7115Mb7118+y7420Pq7470Xa
Recently, KFe2As2 was shown to exhibit a structural phase transition from a tetragonal to a collapsed tetragonal phase under applied pressure of about 15 GPa. Surprisingly, the collapsed tetragonal phase hosts a superconducting state with Tc ∼ 12 K, while the tetragonal phase is a Tc ≤ 3.4 K superconductor. We show that the key difference between the previously known non-superconducting collapsed tetragonal phase in AFe2As2 (A= Ba, Ca, Eu, Sr) and the superconducting collapsed tetragonal phase in KFe2As2 is the qualitatively distinct electronic structure. While the collapsed phase in the former compounds features only electron pockets at the Brillouin zone boundary and no hole pockets are present in the Brillouin zone center, the collapsed phase in KFe2As2 has almost nested electron and hole pockets. Within a random phase approximation spin fluctuation approach we calculate the superconducting order parameter in the collapsed tetragonal phase. We propose that a Lifshitz transition associated with the structural collapse changes the pairing symmetry from d-wave (tetragonal) to s± (collapsed tetragonal). Our DFT+DMFT calculations show that effects of correlations on the electronic structure of the collapsed tetragonal phase are minimal. Finally, we argue that our results are compatible with a change of sign of the Hall coefficient with pressure as observed experimentally. * [email protected] 1 M. Rotter, M. Tegel, and D. Johrendt, Superconductivity at
The family of AFe 2 As 2 (A= Ba, Ca, Eu, K, Sr) superconductors, also called 122 materials, has been intensively investigated in the past due to their richness in structural, magnetic and superconducting phases upon doping or application of pressure 1-6 . One phase whose properties have been recently scrutinized at length is the collapsed tetragonal (CT) phase present in BaFe 2 As 2 , CaFe 2 As 2 , EuFe 2 As 2 , and SrFe 2 As 2 under pressure and in CaFe 2 P 2 7-14 . The structural collapse of this phase has been shown to be assisted by the formation of As 4p z -As 4p z bonds between adjacent Fe-As layers giving rise to a bonding-antibonding splitting of the As p z bands 15 . It has been argued that this phase does not support superconductivity due to the absence of hole cylinders at the Brillouin zone center and the corresponding suppression of spin fluctuations 10,16,17 . However, recently Ying et al. 18 investigated the hole-doped end member of Ba 1−x K x Fe 2 As 2 , KFe 2 As 2 , under high pressure and observed a boost of the superconducting critical temperature T c up to 12 K precisely when the system undergoes a structural phase transition to a CT phase at a pressure P c ∼ 15 GPa. These authors attributed this behavior to possible correlation effects. Moreover, measurements of the Hall coefficient showed a change from positive to negative sign upon pressure, indicating that the effective nature of charge carriers changes from holes to electrons with increasing pressure. Similar experiments are also reported in Ref. 19.
KFe 2 As 2 has a few distinct features: at ambient pressure, the system shows superconductivity at T c = 3.4 K and follows a V-shaped pressure dependence of T c for moderate pressures with a local minimum at a pressure of 1.55 GPa 20 . The origin of such behavior and the nature of the superconducting pairing symmetry are still under debate [21][22][23][24][25][26][27] . However, it has been es-tablished by a few experimental and theoretical investigations based on angle-resolved photoemission spectroscopy, de Haas-van Alphen measurements, and density functional theory combined with dynamical mean field theory (DFT+DMFT) calculations that correlation effects crucially influence the behavior of this system at FIG. 1. (Color online) Crystal structure, schematic Fermi surface (dashed lines) and schematic superconducting gap function (background color) of KFe2As2 in the one-Fe Brillouin zone before and after the volume collapse. The Lifshitz transition associated with the formation of As 4pz-As 4pz bonds in the CT phase changes the superconducting pairing symmetry from dxy to s±. arXiv:1501.03068v3 [cond-mat.supr-con] 8 Apr 2015 P = 0 GPa [28][29][30][31][32][33][34][35] . Application of pressure should nevertheless reduce the relative importance of correlations with respect to the bandwidth increase. In fact, recent DFT+DMFT studies on CaFe 2 As 2 in the high-pressure CT phase show that the topology of the Fermi surface is basically unaffected by correlations 36,37 . One could argue though, that at ambient pressure CaFe 2 As 2 is less correlated than KFe 2 As 2 and therefore, in KFe 2 As 2 correlation effects may be still significant at finite pressure.
In order to resolve these questions, we performed density functional theory (DFT) as well as DFT+DMFT calculations for KFe 2 As 2 in the CT phase. Our results show that the origin of superconductivity in the collapsed tetragonal phase in KFe 2 As 2 lies in the qualitative changes in the electronic structure (Lifshitz transition) experienced under compression to a collapsed tetragonal phase and correlations play only a minor role. Whereas in the tetragonal phase at P = 0 GPa KFe 2 As 2 features predominantly only hole pockets at the Brillouin zone center, at P ∼ 15 GPa in the CT phase significant electron pockets emerge at the Brillouin zone boundary, which together with the hole pockets at the Brillouin zone center favor a superconducting state with s ± symmetry, as we show in our calculations of the superconducting gap function using the random phase approximation (RPA) spin fluctuation approach. Moreover, our results in the tetragonal phase of KFe 2 As 2 at P = 10 GPa suggest a change of pairing symmetry from d xy (tetragonal) to s ± upon entering the collapsed phase (see Fig. 1). This scenario is distinct from the physics of the CT phase in CaFe 2 As 2 , where the hole pockets at the Brillouin zone center are absent. For comparison, we will present the susceptibility of collapsed tetragonal CaFe 2 As 2 , which is representative for the collapsed phase of AFe 2 As 2 (A= Ba, Ca, Eu, Sr). Our findings also suggest an explanation for the change of sign in the Hall coefficient upon entering the CT phase in KFe 2 As 2 .
Density functional theory calculations were carried out using the all-electron full-potential local orbital (FPLO) 38 code. For the exchange-correlation functional we use the generalized gradient approximation (GGA) by Perdew, Burke, and Ernzerhof 39 . All calculations were converged on 20 × 20 × 20 k-point grids.
The structural parameters for the CT phase of KFe 2 As 2 were taken from Ref. 18. We used the data points at P ≈ 21 GPa, deep in the CT phase, where a = 3.854Å and c = 9.6Å. The fractional arsenic zposition (z As = 0.36795) was determined ab-initio via structural relaxation using the FPLO code. We also performed calculations for the crystal structure of Ref. 19, where a preliminary experimental value for the arsenic z-position was given. The electronic structure is very similar to the one reported here. For the CT phase of CaFe 2 As 2 we used experimental lattice parameters from Ref. 40 (T = 40 K, P ≈ 21 GPa) and determined the fractional arsenic z-position (z As = 0.37045) using FPLO. All Fe 3d orbitals are defined in a coordinate system rotated by 45 • around the z-axis with respect to the conventional I 4/mmm unit cell. The electronic bandstructure in the collapsed tetragonal phase of CaFe 2 As 2 and KFe 2 As 2 is shown in Fig. 2. These results already reveal a striking difference between the CT phases of CaFe 2 As 2 and KFe 2 As 2 : while the former does not feature hole bands crossing the Fermi level at Γ and only one band crossing the Fermi level at M (π, π, 0), the latter does feature hole-pockets at both Γ and M in the one-Fe equivalent Brillouin zone. The reason for this difference in electronic structure is that KFe 2 As 2 is strongly hole-doped compared to CaFe 2 As 2 .
In Fig. 3 we show the Fermi surface in the one-Fe equivalent Brillouin zone at k z = 0. In both cases, the Fermi surface is dominated by Fe 3d xz/yz character. The hole cylinders in KFe 2 As 2 span the entire k z direction of the Brillouin zone, while only a small three-dimensional hole-pocket is present in CaFe 2 As 2 (see Ref. 41 ). For KFe 2 As 2 , the hole-pockets at M (π, π, 0) and the electron pockets at X (π, 0, 0) are clearly nested, while no nesting is observed for CaFe 2 As 2 . It is important to note here, that the folding vector in the 122 family of iron-based superconductors is (π, π, π), so that the hole-pockets at M (π, π, 0) will be located at Z (0, 0, π) after unfolding the bands to the effective one-Fe picture 42 .
After qualitatively identifying the difference between the CT phases of CaFe 2 As 2 and KFe 2 As 2 , we calculate the non-interacting static susceptibility to verify that the better nesting of KFe 2 As 2 generates stronger spin fluctuations. For that we constructed 16-band tight-binding models from the DFT results using projective Wannier functions as implemented in FPLO 43 . We keep the Fe 3d and As 4p states, which corresponds to an energy win- dow from −7 eV to +6 eV. Subsequently, we unfold the 16-band model using our recently developed glide reflection unfolding technique 42 , which produces an effective eight-band model of the three-dimensional one-Fe Brillouin zone.
-π 0 π -π 0 π (a) k y k x CaFe 2 As 2 (CT) -π 0 π -π 0 π (b) k y k x KFe 2 As 2 (CT) d z 2 d x 2 -y 2 d xy d xz d yz
We analyse these eight-band models using the 3D version of random phase approximation (RPA) spin fluctuation theory 44
with a Hamiltonian H = H 0 + H int ,
where H 0 is the eight-band tight-binding Hamiltonian derived from the ab-initio calculations, while H int is the Hubbard-Hund interaction. The arsenic states are kept in the entire calculation, but interactions are considered only between Fe 3d states. Further information is given in Ref. 41.
The non-interacting static susceptibility in orbitalspace is defined by Eq. (1), where matrix elements a s µ ( k) resulting from the diagonalization of the initial Hamiltonian H 0 connect orbital and band-space denoted by indices s and µ respectively. The E µ are the eigenvalues of H 0 and f (E) is the Fermi function.
χ pq st ( q) = − 1 N k,µ,ν a s µ ( k)a p * µ ( k)a q ν ( k + q)a t * ν ( k + q) × f (Eν ( k+ q))−f (Eµ( k)) Eν ( k+ q)−Eµ( k)
(1) The observable static susceptibility 41 is defined as the sum over all elements χ bb aa of the full tensor χ( q) =
1 2 a,b
χ bb aa ( q). The effective interaction in the singlet pairing channel is constructed from the static susceptibility tensor χ pq st which measures strength and wave-vector dependence of spin fluctuations, via the multiorbital RPA procedure. Both the original and effective interaction are discussed, e.g. in Ref. 45. We have shown previously that our implementation is capable of capturing effects of fine variations of shape and orbital character of the Fermi surface 46 . At first glance, the observable static susceptibility displayed in Fig. 4 is comparable for CaFe 2 As 2 and KFe 2 As 2 . A key difference is however revealed upon investigation of the largest elements, i.e. the diagonal entries χ aa aa . These show that in CaFe 2 As 2 the susceptibility has broad plateaus, while in KFe 2 As 2 the susceptibility has a strong peak at X (π, 0, 0) in the one-Fe Brillouin zone, which corresponds to the usual s ± pairing scenario that relies on electron-hole nesting. In CaFe 2 As 2 the pairing interaction is highly frustrated because there is no clear peak in favor of one pairing channel.
KFe 2 As 2 (CT) 0.1 0.2 0.3 0.4 Γ X M Γ (c) χ(q) [1/eV] Γ X M Γ (d) d z 2 d x 2 −y 2 d xy d xz d yz
We have also performed spin-polarized calculations for KFe 2 As 2 at P ≈ 21 GPa in order to confirm the antiferromagnetic instability we find in the linear response calculations. Out of ferromagnetic, Néel and stripe antiferromagnetic order only the stripe antiferromagnet is stable with small moments of 0.07µ B on Fe, in agreement with our calculations for the susceptibility.
The leading superconducting gap function of KFe 2 As 2 in the CT phase is shown in Fig. 5. As expected from our susceptibility calculations, the pairing symmetry is s-wave with a sign-change between electron and holepockets. While the superconducting gap is nodeless in the k z = 0 plane, the k z = π plane does show nodes where the orbital character changes from Fe 3d xz/yz to Fe 3d xy . Note that this k z = π structure of the superconducting gap is exactly the same as in the well studied LaFeAsO compound 44 , which shows that the CT phase of KFe 2 As 2 closely resembles usual iron-based superconductors although it is much more three-dimensional than, e.g. in LaFeAsO.
We have also calculated the superconducting gap function for KFe 2 As 2 at P = 10 GPa in the tetragonal phase and find d xy as the leading pairing symmetry 41 based on rigid band shifts 22,24 is also present in our calculation, but as a sub-leading solution. Our results strongly suggest that the Lifshitz transition, which occurs upon entering the collapsed tetragonal phase, changes the symmetry of the superconducting gap function from d-wave (tetragonal) to s-wave (CT) (see Fig. 1). The possible simultaneous change of pairing symmetry, density of states and T c potentially opens up different routes to understanding their quantitative connection. In order to estimate the strength of local electronic correlations in collapsed tetragonal KFe 2 As 2 , we performed fully charge self-consistent DFT+DMFT calculations. We used the same method as described in Ref. 35. The DFT calculation was performed by the WIEN2k 47 implementation of the full-potential linear augmented plane wave (FLAPW) method in the local density approximation (LDA) with 726 k-points in the irreducible Brillouin zone. We checked that the results of FPLO and WIEN2k agree on the DFT level. The Bloch wave functions are projected to the localized Fe 3d orbitals as described in Refs. 48 . Table I displays the orbital-resolved massrenormalizations m * /m LDA for KFe 2 As 2 in the collapsed tetragonal phase. The obtained values show that local electronic correlations in the CT phases of KFe 2 As 2 and CaFe 2 As 2 36,37 are comparable. As in CaFe 2 As 2 , the effects of local electronic correlations on the Fermi surface are negligible (see Ref. 41). The higher T c of the collapsed phase in absence of strong correlations raises the question how important strong correlations are in general for iron-based superconductivity. This issue demands further investigation.
Finally, the change of dominant charge carriers from hole to electron-like states measured in the Hallcoefficient under pressure 18 is naturally explained from our calculated Fermi surfaces. While KFe 2 As 2 is known to show only hole-pockets at zero pressure, the CT phase features also large electron pockets. On a small fraction of these electron pockets, the dominating orbital character is Fe 3d xy (Fig. 3). It was shown in Ref. 53 that quasiparticle lifetimes on the Fermi surface can be very anisotropic and long-lived states are favored where marginal orbital characters appear. As Fe 3d xy character is only present on the electron pockets, these states contribute significantly to transport and are responsible for the negative sign of the Hall coefficient.
In summary, we have shown that the electronic structure of the collapsed tetragonal phase of KFe 2 As 2 qualitatively differs from that of other known collapsed materials. Upon entering the CT phase, the Fermi surface of KFe 2 As 2 undergoes a Lifshitz transition with electron pockets appearing at the Brillouin zone boundary, which are nested with the hole pockets at the Brillouin zone center. Thus, the spin fluctuations in collapsed tetragonal KFe 2 As 2 resemble those of other iron-based superconductors in non-collapsed phases and the superconducting gap function assumes the well-known s ± symmetry. This is in contrast to other known materials in the CT phase, like CaFe 2 As 2 , where hole pockets at the Brillouin zone center are absent and no superconductivity is favored. Based on our LDA+DMFT calculations, the CT phase of KFe 2 As 2 is significantly less correlated than the tetragonal phase, and mass enhancements are comparable to the CT phase of CaFe 2 As 2 . Finally, we suggest that the change of dominant charge carriers from hole to electron-like can be explained from anisotropic quasiparticle lifetimes.
Origin of the superconducting state in the collapsed tetragonal phase of KFe 2 As 2 :
Supplemental Information For calculating the susceptibility we used 30 × 30 × 10 k-point grids and an inverse temperature of β = 40 eV −1 . To calculate the pairing interaction, the susceptibility is needed at k-vectors that do in general not lie on a grid. Those susceptibility values are obtained using trilinear interpolation of the gridded data.
The pairing interaction is constructed using ∼ 800 points on the three-dimensional Fermi surface. Consequently, the solution of the gap equation is available only on those points scattered in three-dimensional space.
In order to obtain a graphical representation of the gap function on two-dimensional cuts through the Brillouin zone, we used multiscale radial basis function interpolation as implemented in ALGLIB (http://www.alglib.net).
The Hubbard-Hund interaction H int includes the onsite intra (inter) orbital Coulomb interaction U (U ), the Hund's rule coupling J and the pair hopping energy J . We assume spin rotation-invariant interaction parameters U = 2.4 eV, U = U/2, and J = J = U/4. Because of the large bandwidth in the collapsed tetragonal phase, these comparatively large values are necessary to bring the system close to the RPA instability. Note however, that the symmetry of the superconducting gap in this system does not change, even if significantly reduced parameter values are considered.
The parameter values used in the RPA method are renormalized with respect to those in the DFT+DMFT method, because the electronic self-energy is neglected in the usual RPA scheme 1,2 .
II. THREE DIMENSIONAL FERMI SURFACE
We extracted the three-dimensional Fermi surface of collapsed tetragonal CaFe 2 As 2 and KFe 2 As 2 from FPLO using 50 × 50 × 25 k-point grids in the two-Fe equivalent Brillouin zone (Fig. 1). For CaFe 2 As 2 only electron pockets in the Brillouin zone corners are observed. The three-dimensional hole pocket that arises from the bands at M (π, π, 0) is so small that it is not detected here with the given k-resolution. In KFe 2 As 2 one of the hole cylinders is highly dispersive, while the central hole cylinder and the electron pockets located in the corners are nested throughout the entire Brillouin zone.
III. DFT BANDSTRUCTURE AND FERMI SURFACE OF THE NON-COLLAPSED PHASE UNDER PRESSURE
To verify that electron pockets in KFe 2 As 2 do not grow continously with applied pressure, but rather appear as a result of the structural collapse, we have also prepared a crystal structure at P ∼ 10 GPa, using the data from Ref.
3. The lattice parameters were chosen as a = 3.663Å and c = 13.0Å. The fractional arsenic z-position (z As = 0.35750) was again determined ab-initio via structural relaxation using the FPLO code.
No qualitative changes in the bandstructure and Fermi surface (Fig. 2) are observed compared to the ambient pressure structure 4 . Most importantly, electron pockets which are nested with the hole pockets in the Brillouin zone center are not present, even at high pressures slightly below the transition from the tetragonal to the collapsed tetragonal phase. We conclude that the appearance of electron pockets is directly linked to the structural collapse.
IV. BANDSTRUCTURE, FERMI SURFACE AND SUSCEPTIBILITY AT LOW PRESSURES
In the main paper we used crystal structures at P = 21 GPa for both materials of interest to ensure that results are comparable. Here we show the electronic bandstructure, Fermi surface and static susceptibility of CaFe 2 As 2 at low pressure (P = 0.53 GPa), right after the structural collapse. We used the experimental structural parameters from Ref. 5, which are the same as in our previous GGA+DMFT study 6 . The differences compared to the high pressure results in the main text are purely quantitative.
The electronic bandstructure and Fermi surface are shown in Fig. 3. The static susceptibility is shown in Fig. 4. The overall enhancement of the susceptibility compared to the figures in the main text is a consequence of the smaller bandwidth at low pressure. The particular enhancement of the peak between X and M in the d xz di- agonal element of the susceptibility has its origin in the hole band crossing the Fermi level at the M point, which in turn contributes a small three-dimensional pocket.
V. DETAILS OF THE LDA+DMFT CALCULATION
For our DMFT calculations we used a temperature of 290 K, i.e. an inverse temperature β = 40 eV −1 . The interaction parameters are defined in terms of Slater integrals 7 with an on-site Coulomb interaction U = 4 eV and Hund's rule coupling J = 0.8 eV. As the double counting correction we used the fully localized limit 8,9 (FLL) scheme. Before the continuation of the imaginaryfrequency self-energy to the real axis by stochastic analytic continuation 10 , the calculation was converged with 2 × 10 7 Monte-Carlo sweeps for the solution of the impurity model.
From the LDA+DMFT calculation we obtain the spectral function on real frequencies from the analytically continued self-energy as
A vv (k, ω) = − 1 π 1 (ω + µ)δ vv − vv (k) − Σ vv (k, ω) ,(1)
where µ is the chemical potential, vv (k) = v (k)δ vv are the eigenenergies of the LDA Hamiltonian and Σ vv (k, ω) is the impurity self-energy upfolded to Bloch space by projectors P mv (k)(see Ref. 11 and 12)
Σ vv (k, ω) = m P † mv (k) (Σ mm (ω) − Σ DC ) P mv (k),(2)
with the double-counting correction term Σ DC taken in the fully-localized limit (FLL) 8,9 . The diagonal entries of this object are then integrated over all bands v and k-points k to obtain the total density of states within LDA+DMFT shown in Fig. 5 (a). Additional projection onto the Fe 3d orbitals (denoted by index m) by
A mm (k, ω) = − 1 π vv P mv (k) 1 (ω + µ)δ vv − vv (k) − Σ vv (k, ω) P † m v (k) ,(3)
is then used in the same fashion to obtain the orbital resolved density of states shown in Fig. 5 (b)-(e). The k-resolved spectral function shown in Fig. 6 was obtained in the same manner, but only integrating over all bands v in order to keep the k-dependence. The orbital resolved Fermi surfaces shown in Fig. 7 and Fig. 8 were obtained by evaluating A mm (ω) at the Fermi energy (ω = ω F ) on a 200 × 200 k-grid in the two-Fe Brillouin zone.
We observe moderate effects of renormalization and broadening in the density of states (see Fig. 5) with no significant change at the Fermi level. This is confirmed by the k-resolved spectral function (see Fig. 6). Using the FLL double-counting correction, we do not observe any topological changes in the Fermi surface, which closely resembles the DFT result (see Fig. 7 and Fig. 8). However, we observe a closing of the pockets at Γ when the nominal double-counting correction 13 is considered. This indicates that the LDA+DMFT results slightly depend on the double-counting. Note that these changes do not affect our conclusions regarding the superconducting pairing symmetry since only a small fraction of the Brillouin zone around k z = 0 is affected. As the relevant electron-hole nesting takes place around k z = π, our superconducting pairing calculations based on the DFT bandstructure should be robust with respect to the effects of correlations.
Note that the Fermi surface plots ( Fig. 7 and Fig. 8) have to be rotated by 45 • in the plane in order to compare with the Fermi surface plots in the original paper. In 122 compounds the Z-point of the two-Fe Brillouin zone corresponds to the M-point of the one-Fe Brillouin zone.
FIG. 2 .
2(Color online) Electronic bandstructure of the collapsed tetragonal phase in (a) CaFe2As2 and (b) KFe2As2. The path is chosen in the one-Fe equivalent Brillouin zone. The colors indicate the weights of Fe 3d states.
FIG. 3 .
3(Color online) Fermi surface of the collapsed tetragonal phase in (a) CaFe2As2 and (b) KFe2As2 at kz = 0. The full plot spans the one-Fe equivalent Brillouin zone, while the area enclosed by the grey lines is the two-Fe equivalent Brillouin zone. The colors indicate the weights of Fe 3d states.
FIG. 4 .
4(Color online) Summed static susceptibility (top) and its diagonal components χ aa aa (bottom) in the eight-band tight-binding model for [(a) and (c)] CaFe2As2 and [(b) and (d)] KFe2As2 in the one-Fe Brillouin zone. The colors identify the Fe 3d states.
FIG. 5 .
5. The dominant d x 2 −y 2 -solution obtained in model calculations (Color online) Leading superconducting gap function (s±) of the eight-band model in the one-Fe Brillouin zone of KFe2As2 in the CT phase at (a) kz = 0 and (b) kz = π.
and 49 .
49The energy window for projection was chosen from −7 to +13 eV, with the lower boundary lying in a gap in the density of states. For the solution of the DMFT impurity problem the continuoustime quantum Monte Carlo method in the hybridization expansion 50 as implemented in the ALPS 51,52 project was employed (see Ref.41 for more details). The massrenormalizations are directly calculated from the analytically continued real part of the impurity self-energy Σ
FIG. 1 .
1Three-dimensional Fermi surface of (a) collapsed tetragonal CaFe2As2 and (b) collapsed tetragonal KFe2As2 (both at P = 21 GPa) shown in the two-Fe equivalent Brillouin zone. The Γ-point is located in the center of the displayed volume.
FIG. 2 .
2(a) Electronic bandstructure in the one-Fe equivalent Brillouin zone and (b) Fermi surface of the P ∼ 10 GPa non-collapsed tetragonal (TET) phase of KFe2As2 at kz = 0. The full plot of the Fermi surface spans the one-Fe equivalent Brillouin zone, while the area enclosed by the grey lines is the two-Fe equivalent Brillouin zone. The colors indicate the weights of Fe 3d states.
FIG. 3 .
3(a) Electronic bandstructure in the one-Fe equivalent Brillouin zone and (b) Fermi surface of the P = 0.53 GPa collapsed tetragonal (CT) phase of CaFe2As2 at kz = 0. The full plot of the Fermi surface spans the one-Fe equivalent Brillouin zone, while the area enclosed by the grey lines is the two-Fe equivalent Brillouin zone. The colors indicate the weights of Fe 3d states.
FIG. 4 .
4Summed static susceptibility (a) and its diagonal components χ aa aa (b) in the eight-band tight-binding model for CaFe2As2 (CT, P = 0.53 GPa) in the one-Fe Brillouin zone. The colors identify the Fe 3d states.
FIG. 5 .
5The k-integrated spectral function (DOS) of KFe2As2 at P = 21 GPa as obtained within DFT compared to LDA+DMFT. The upper panel (a) shows the total DOS, with the DFT result (black line) and the renormalized LDA+DMFT result (red line). The lower four panels show the partial DOS of the Fe 3d orbitals: (b) 3d z 2 , (c) 3d x 2 −y 2 , (d) 3dxy and (e) 3d xz/yz orbital.
FIG. 6 .
6The k-resolved spectral function of KFe2As2 at P = 21 GPa as obtained within LDA+DMFT. The red lines show the DFT bandstructure for comparison, rescaled by the average mass enhancement of 1.37. The labels on the x-axis correspond to high-symmetry points in the two-Fe Brillouin zone.
PACS numbers: 71.15.Mb, 71.18.+y, 74.20.Pq, 74.70.Xa
TABLE I .
IMass renormalizations m * /mLDA of the Fe 3d or-
bitals in the collapsed tetragonal phase of KFe2As2 calculated
with the LDA+DMFT method.
d z 2
d x 2 −y 2
dxy
d xz/yz
1.318
1.309
1.319
1.445
Daniel Guterding, * Steffen Backes, Harald O. Jeschke, and Roser ValentíInstitut für Theoretische Physik, Goethe-Universität Frankfurt,
Max-von-Laue-Straße 1, 60438 Frankfurt am Main, Germany
I. DETAILS OF THE SUSCEPTIBILITY AND
PAIRING CALCULATION
ACKNOWLEDGMENTSWe thank the Deutsche Forschungsgemeinschaft for financial support through Grant No. SPP 1458.
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The kz = 0 Fermi surface of KFe2As2 at P = 21 GPa as obtained within LDA+DMFT in the two-Fe Brillouin zone. Panel (a) shows the total spectral function, while panels (b)-(f) show the orbital resolved spectral function for the Fe 3d orbitals. FIG. 7. The kz = 0 Fermi surface of KFe2As2 at P = 21 GPa as obtained within LDA+DMFT in the two-Fe Brillouin zone. Panel (a) shows the total spectral function, while panels (b)-(f) show the orbital resolved spectral function for the Fe 3d orbitals.
The kz = π Fermi surface of KFe2As2 at P = 21 GPa as obtained within LDA+DMFT in the two-Fe Brillouin zone. Panel (a) shows the total spectral function, while panels (b)-(f) show the orbital resolved spectral function for the Fe 3d orbitals. FIG. 8. The kz = π Fermi surface of KFe2As2 at P = 21 GPa as obtained within LDA+DMFT in the two-Fe Brillouin zone. Panel (a) shows the total spectral function, while panels (b)-(f) show the orbital resolved spectral function for the Fe 3d orbitals.
| []
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[
"ON SERRIN'S OVERDETERMINED PROBLEM IN SPACE FORMS",
"ON SERRIN'S OVERDETERMINED PROBLEM IN SPACE FORMS"
]
| [
"Giulio Ciraolo ",
"Luigi Vezzoni "
]
| []
| []
| We consider an overdetermined Serrin's type problem in space forms and we generalize Weinberger's proof in [17] by introducing a suitable P-function.1991 Mathematics Subject Classification. Primary 35R01, 35N25, 35B50; Secondary: 53C24, 58J05. | 10.1007/s00229-018-1079-z | [
"https://arxiv.org/pdf/1702.05277v1.pdf"
]
| 119,278,151 | 1702.05277 | 790af64a7c892f957897d7ef41ab9dd59290fea0 |
ON SERRIN'S OVERDETERMINED PROBLEM IN SPACE FORMS
17 Feb 2017
Giulio Ciraolo
Luigi Vezzoni
ON SERRIN'S OVERDETERMINED PROBLEM IN SPACE FORMS
17 Feb 2017
We consider an overdetermined Serrin's type problem in space forms and we generalize Weinberger's proof in [17] by introducing a suitable P-function.1991 Mathematics Subject Classification. Primary 35R01, 35N25, 35B50; Secondary: 53C24, 58J05.
Introduction
In the seminal paper [15] Serrin proved that if there exists a solution to (1.1) ∆v + f (v) = 0 in a bounded domain Ω ⊂ R n such that v = 0 and v ν = const on ∂Ω ,
then Ω must be a ball and v radially symmetric. The proof in [15] makes use of the method of moving planes and actually applies to more generally uniformly elliptic operators (see [15]).
In [17] Weinberger considerably simplified the proof of Serrin's result in the case ∆v = −1 by considering what is nowadays called P-function and using some integral identities. The approach of Weinberger, as well as the use of a P-function, inspired several works in the context of elliptic partial differential equations (see e.g. [1,3,4,5,6,10,14,16] and references therein).
In the present paper we investigate such symmetry results from a broader perspective of the ambient space. We consider overdetermined problems in space forms by assuming the ambient space to be a complete simply-connected Riemannian manifold with constant sectional curvature K. Up to homoteties we may assume K = 0, −1, 1; the case K = 0 corresponds to the case of the Euclidean space, K = −1 is the Hyperbolic space and K = 1 is the unitary sphere with the round metric.
In space forms, Serrin's symmetry result was proved in [7] and [8] by adapting the proof of Serrin [15], i.e. by using the method of moving planes. The aim of the present paper is to prove Serrin's result in space forms by using an approach analogous to the one of Weinberger by using a suitable P-function associated to the equation ∆v + nKv = −1. As in the Euclidean case, our approach is suitable only for the equation that we are considering, and does not fit with more general equations of the form (1.1).
Let (M, g) be a Riemannian manifold isometric to one of the following three models: the Eucliden space R n , the Hyperbolic space H n , the hemisphere S n + . The three models can be described as the warped product space M = I × S n−1 equipped with the rotationally symmetric metric g = dr 2 + h 2 g S n−1 , where g S n−1 is the round metric on the (n − 1)-dimensional sphere S n−1 and -I = [0, ∞) and h(r) = r in the Euclidean case (K = 0); -I = [0, ∞) and h(r) = sinh(r) in the hyperbolic case (K = −1); -I = [0, π/2) and h(r) = sin(r) in the spherical case (K = 1). Our main result is the following.
Theorem 1.1. Let Ω ⊂ M be a bounded connected domain with boundary ∂Ω of class C 1 . Let v be the solution to (1.2) ∆v + nKv = −1 in Ω , v = 0 on ∂Ω ,
and assume that
(1.3) |∇v| = c on ∂Ω ,
for some postive constant c.
Then Ω is a geodesic ball B R and v depends only on the distance from the center of B R .
In theorem 1.1 we may assume, up to isometries, that B R is centered at the origin. In this case v is given by
v(r) = H(R) − H(r) nḣ(R) ,
with H =´r 0 h(s)ds. Indeed, since the Laplacian of a radial radial function u = u(r) is given by ∆u =ü + (n − 1)ḣh −1u , a straightforward computation yields that v solves (1.2). Furthermore, by computing the first derivative of v, we deduce that c and R are related by
c = h(R) nḣ(R) .
We notice that for K = 0, (1.2) reduces to the classical model problem ∆v = −1 in the Euclidean space. The extra term nKv is the one needed to obtain that the Hessian of the solution in the radial case is proportional to the metric. Moreover, this allows us to consider the P -function
(1.4) P (v) = |∇v| 2 + 2 n v + Kv 2 ,
which is subharmonic when v solves (1.2). Equation (1.2) arises from the study of constant mean curvature hypersurfaces in space forms. Indeed, it is known from Reilly's paper [12] that a possible approach to prove Alexandrov soap bubble theorem in the Euclidean space is by considering the torsion potential, i.e. the solution to ∆v = −1, and apply Reilly's identity. In space forms this approach was generalized by Qui and Xia in [11] by replacing equation ∆v = −1 with ∆v + nKv = −1.
We also mention that Alexandrov's soap bubble theorem in the Euclidean space can be proved via Serrin's overdetermined problem for the equation ∆v = −1 (see [12][remark at p. 468]). Hence, theorem 1.1 can be used to give an alternative proof to Alexandrov theorem in space forms by using the generalization of Reilly's identity in [11].
In the next section we write ∇ 2 to denote the Hessian of a function and, for X, Y vector fields, we write X · Y instead of g(X, Y ).
Proof of the result
We first prove that the P -function (1.4) is subharmonic.
Moreover, ∆P (v) = 0 if and only if
(2.1) ∇ 2 v = − 1 n + Kv g .
Proof. From the Bochner-Weitzenböck formula in space forms 1 2 ∆|∇v| 2 = |∇ 2 v| 2 + D(∆v) · ∇v + (n − 1)K∇v · ∇v and from Cauchy-Schwartz inequality we obtain that 1 2 ∆|∇v| 2 ≥ 1 n (∆v) 2 + D(∆v) · ∇v + (n − 1)K∇v · ∇v .
From (1.2) we find that
1 2 ∆|∇v| 2 ≥ 1 n (∆v)(−1 − nKv) + D(−1 − nKv) · ∇v + (n − 1)K∇v · ∇v = − 1 n ∆v − Kv∆v − K∇v · ∇v = − 1 n ∆v − K 2 ∆v 2 ,
where in the last inequality we have used that div (v 2 /2) = v∆v + |∇v| 2 . Hence ∆P (v) ≥ 0. From the argument above, we readily see that ∆P (v) = 0 if and only if
n|∇ 2 v| 2 = (∆v) 2 ,
which implies that ∇ 2 v is a multiple of the metric g. Since v satisfies (1.2) then we obtain (2.1).
Lemma 2.1 will be used in the following form.
Corollary 2.2.
Let v be the solution to (1.2) and assume that (2.1) holds. Then either
(2.2) P (v) = c 2 inΩ or (2.3) c 2ˆΩḣ > 1 + 2 n ˆΩḣ v − KˆΩ hvv r .
Proof. From lemma 2.1 we have that ∆P (v) ≥ 0. Since P (v) = c 2 on ∂Ω, by the strong maximum principle either P (v) = c 2 inΩ or P (v) < c 2 in Ω. If we assume that P (v) < c 2 in Ω then, beingḣ > 0,
c 2ˆΩḣ >ˆΩḣ|∇v| 2 + 2 nˆΩḣ v + KˆΩḣv 2
and since (2.4) div (ḣv∇v) =ḣ|∇v| 2 +ḣv∆v +ḧvv r and v = 0 on ∂Ω, beingḧ = −Kh, we obtain that c 2ˆΩḣ > −ˆΩḣv∆v −ˆΩḧvv r + 2 nˆΩḣ v + KˆΩḣv 2 = (n + 1)KˆΩḣv 2 + 1 + 2 n ˆΩḣ v + KˆΩ hvv r , which completes the proof.
As we will see, (2.3) will give a contradiction which follows from Pohožaev inequality.
Lemma 2.3.
Let v be the solution to (1.2) and assume that v satisfies (2.1). Then
(2.5) c 2ˆΩḣ = 1 + 2 n ˆΩḣ v − KˆΩ hvv r .
Proof. We first consider a generic sufficiently smooth function v (not necessarily a solution to (1.2)). We consider the Pohožaev identity in space forms (see e.g. [2])
(2.6) div |∇v| 2 2 X − hv r ∇v = n − 2 2ḣ |∇v| 2 − hv r ∆v ,
where X is the radial vector field X = h∂ r .
Since
X · ∇v = div (vX) − nḣv ,
we have that
1 n div |∇v| 2 X − 2(X · ∇v)∇v − n − 2 n div (ḣv∇v) −ḣv∆v + KvX · ∇v + 2 n (div (vX) − nḣv)∆v = 0 ,
which we write as
1 n div |∇v| 2 X − 2(X · ∇v)∇v − n − 2 n div (ḣv∇v) −ḣv(∆v + nKv) + nKḣv 2 + KvX · ∇v + 2 n (div (vX) − nḣv)(∆v + nKv) − 2(div (vX) − nḣv)Kv = 0 ,
i.e.
(2.7) 1 n div |∇v| 2 X − 2(X · ∇v)∇v − n − 2 n div (ḣv∇v) − n + 2 nḣ v(∆v + nKv) + 2 n (div (vX))(∆v + nKv)
+ (n + 2)Kḣv 2 − n − 2 n KvX · ∇v − 2Kvdiv (vX) = 0 .
Now we assume that v is a solution to (1.2) satisfying (2.1), and we integrate (2.7)
(2.8) − c 2 nˆ∂ Ω X · ν + n + 2 nˆΩḣ v + (n + 2)KˆΩḣv 2 − n − 2 n KˆΩ vX · ∇v − 2KˆΩ vdiv (vX) = 0 , i.e. − c 2 nˆ∂ Ω X · ν + n + 2 nˆΩḣ v − (n − 2)KˆΩḣv 2 + 2 n − 3 KˆΩ vX · ∇v = 0 ,
and since div X = nḣ we obtain
c 2ˆΩḣ = n + 2 nˆΩḣ v − (n − 2)KˆΩḣv 2 + 2 n − 3 KˆΩ vX · ∇v .
From (2.4) we obtain (2.5).
The following result is known but, beside the Euclidean case (see [13][Lemma 3]), we didn't find a proof in letterature and we provide a detailed proof for reader's convenience.
Lemma 2.4.
Let Ω be a bounded connected domain in M and assume that there exists a function v :Ω → R, with v ∈ C 1 (Ω) ∩ C 2 (Ω), such that
(2.9) ∇ 2 v = (− 1 n − Kv)g in Ω , v = 0 on ∂Ω .
Then Ω is a geodesic ball B R and v depends only on the center of B R .
Proof. We first notice that v > 0 in Ω. This follows from the standard maximum principles when K = 0, −1 (see e.g. [16][theorem 2.6]). Now we consider the case K = 1. We recall that the first eigenvalue of the Dirichlet Laplacian on the hemisphere is n and the corresponding eigenfunction φ is strictly positive. By writing v = wφ we see that w satisfies ∆w + 2 ∇φ φ · ∇w < 0 in Ω w = 0 on ∂Ω , which implies that w > 0 in Ω again by [16][Theorem 2.6]. Hence v > 0 in Ω.
Since v > 0 it achieves the maximum at a point p ∈ Ω, with v(p) = a > 0. Let γ : I → M be a unit speed maximal geodesic satisfying γ(0) = p and let f (s) = v(γ(s)). This implies that v has the same expression along any geodesic starting from p, and hence v depends only on the distance from p, which completes the proof.
Proof of Theorem 1.1. Corollary 2.2 and Lemma 2.3 imply that P (v) = c 2 and from Lemma 2.1 we find that v satisfies (2.1). Lemma 2.4 gives that Ω is a geodesic ball B R and v depends only on the distance from the center of B R .
Lemma 2. 1 .
1Let v be a solution to ∆v + nKv = −1 and let P be givenby (1.4). Then∆P (v) ≥ 0 in Ω .
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. Giulio Ciraolo, Dipartimento Di Matematica E Informatica, Via Archirafi. 34Università di Palermo90123 Palermo, Italy E-mail address: [email protected] Ciraolo, Dipartimento di Matematica e Informatica, Università di Palermo, Via Archirafi 34, 90123 Palermo, Italy E-mail address: [email protected]
Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy E-mail address: luigi. Luigi Vezzoni, G Dipartimento Di Matematica, Peano, [email protected] Vezzoni, Dipartimento di Matematica G. Peano, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy E-mail address: [email protected]
| []
|
[
"UNSUPERVISED VEHICLE RECOGNITION USING INCREMENTAL RESEEDING OF ACOUSTIC SIGNATURES",
"UNSUPERVISED VEHICLE RECOGNITION USING INCREMENTAL RESEEDING OF ACOUSTIC SIGNATURES"
]
| [
"Justin Sunu [email protected] \nInstitute of Mathematical Sciences\nDepartment of Mathematical Sciences\nClaremont Graduate University Claremont\nClaremont McKenna College Claremont\n91711, 91711CA, CA\n",
"Allon G Percus [email protected] \nInstitute of Mathematical Sciences\nDepartment of Mathematical Sciences\nClaremont Graduate University Claremont\nClaremont McKenna College Claremont\n91711, 91711CA, CA\n",
"Blake Hunter [email protected] \nInstitute of Mathematical Sciences\nDepartment of Mathematical Sciences\nClaremont Graduate University Claremont\nClaremont McKenna College Claremont\n91711, 91711CA, CA\n"
]
| [
"Institute of Mathematical Sciences\nDepartment of Mathematical Sciences\nClaremont Graduate University Claremont\nClaremont McKenna College Claremont\n91711, 91711CA, CA",
"Institute of Mathematical Sciences\nDepartment of Mathematical Sciences\nClaremont Graduate University Claremont\nClaremont McKenna College Claremont\n91711, 91711CA, CA",
"Institute of Mathematical Sciences\nDepartment of Mathematical Sciences\nClaremont Graduate University Claremont\nClaremont McKenna College Claremont\n91711, 91711CA, CA"
]
| []
| Vehicle recognition and classification have broad applications, ranging from traffic flow management to military target identification. We demonstrate an unsupervised method for automated identification of moving vehicles from roadside audio sensors. Using a short-time Fourier transform to decompose audio signals, we treat the frequency signature in each time window as an individual data point. We then use a spectral embedding for dimensionality reduction. Based on the leading eigenvectors, we relate the performance of an incremental reseeding algorithm to that of spectral clustering. We find that incremental reseeding accurately identifies individual vehicles using their acoustic signatures. | 10.1007/978-3-030-01851-1_15 | [
"https://arxiv.org/pdf/1802.06287v1.pdf"
]
| 3,344,935 | 1802.06287 | b0695fd775295136c65b6a15ec657c71b904d21c |
UNSUPERVISED VEHICLE RECOGNITION USING INCREMENTAL RESEEDING OF ACOUSTIC SIGNATURES
Justin Sunu [email protected]
Institute of Mathematical Sciences
Department of Mathematical Sciences
Claremont Graduate University Claremont
Claremont McKenna College Claremont
91711, 91711CA, CA
Allon G Percus [email protected]
Institute of Mathematical Sciences
Department of Mathematical Sciences
Claremont Graduate University Claremont
Claremont McKenna College Claremont
91711, 91711CA, CA
Blake Hunter [email protected]
Institute of Mathematical Sciences
Department of Mathematical Sciences
Claremont Graduate University Claremont
Claremont McKenna College Claremont
91711, 91711CA, CA
UNSUPERVISED VEHICLE RECOGNITION USING INCREMENTAL RESEEDING OF ACOUSTIC SIGNATURES
Index Terms-Spectral ClusteringMachine learningvehicle audio
Vehicle recognition and classification have broad applications, ranging from traffic flow management to military target identification. We demonstrate an unsupervised method for automated identification of moving vehicles from roadside audio sensors. Using a short-time Fourier transform to decompose audio signals, we treat the frequency signature in each time window as an individual data point. We then use a spectral embedding for dimensionality reduction. Based on the leading eigenvectors, we relate the performance of an incremental reseeding algorithm to that of spectral clustering. We find that incremental reseeding accurately identifies individual vehicles using their acoustic signatures.
INTRODUCTION
Recognizing and distinguishing moving vehicles based on their audio signals are problems of broad interest. Applications range from traffic analysis and urban planning to military vehicle recognition. Audio data sets are small compared to video data, and multiple audio sensors can be placed easily and inexpensively. However, challenges arise due to equipment as well as to the underlying physics. Microphone sensitivity can result in disruption from wind and ambient noise. The Doppler shift can make a vehicle's acoustic signature differ according to its position.
In order to interpret acoustic signatures, one must extract information contained within the raw audio data. A natural feature extraction method is the short-time Fourier transform (STFT), with time windows chosen large enough that they carry sufficient frequency information but small enough that they can localize vehicle events. STFT has been used previously for classifying cars vs. motorcycles with principle component analysis [1], for characterizing -neighborhoods in vehicle frequency signatures [2], for vehicle classification based on power spectral density [3], and for estimating the fundamental frequency in engine sounds [4]. Other related feature extraction approaches have included the wavelet transform [3,5] and the one-third-octave filter bands [6].
Our study uses a spectral embedding approach to identify different individual vehicles. Representing each time window as an individual data point, we define a similarity measure between two points based on the cosine distance between their sets of Fourier coefficients, and then cluster according to the symmetric normalized graph Laplacian [7]. We relate the eigenvectors of the Laplacian to a recently proposed clustering method, incremental reseeding (INCRES) [8], that iteratively propagates cluster labels across a graph. We compare the performance of INCRES with spectral clustering on the vehicle audio data. We find that both are promising unsupervised methods for vehicle identification, with INCRES correctly clustering 91.7% of the data points in a sequence of passages of three different vehicles.
ALGORITHMS
Our clustering algorithms are based on the use of spectral embedding for dimensionality reduction. Consider a signal of length n, with feature vector x i ∈ R m associated with data point i ∈ {1, . . . , n}. A spectral embedding represents data as vertices on a weighted graph, with edge weights S ij expressing a similarity measure between data points i and j. The graph is encoded using the symmetric normalized graph Laplacian matrix [7]
L s = I − D −1/2 SD −1/2 where D is a diagonal matrix with D ii = j S ij .
We use this embedding for vehicle identification with two related clustering methods: spectral clustering, and a recently developed incremental reseeding approach [8].
Spectral clustering
The eigenvectors of L s associated with the leading nontrivial eigenvalues λ 2 , . . . λ k form a (k − 1)-dimensional approximation to x i . The approximation is justified when the spectral gap |λ k+1 − λ k | is large, which occurs when the data naturally form k clusters [7]. Spectral clustering uses k-means to cluster this R k−1 projection of the data.
Incremental Reseeding (INCRES) Algorithm
The INCRES algorithm [8] is a diffusive method that propagates cluster labels across the graph specified by L s . The approach (Algorithm 1) is incremental: it plants cluster seeds among nodes, grows clusters from these seeds, and then reseeds among the grown clusters. PLANT (Initialize random walk matrix) 5: GROW (Propagate the random walk matrix) 6: HARVEST (Finalize the clusters) 7: end for
Since the random walk process is governed by the graph Laplacian, INCRES is closely connected with spectral clustering. Eigenvectors of L s are organized hierarchically: the second eigenvector separates data into two clusters at the coarsest resolution, the third eigenvector identifies a third cluster at a finer resolution, and so on. An application of INCRES with parameter k propagates k labels through the graph, resulting in k clusters governed by the spectral properties of L s .
To illustrate the relation between the two algorithms, consider a similarity matrix S ij given by a block matrix form with added "salt and pepper" noise. This is shown in Figure 1, with lighter colors representing greater similarities. Figure 2 shows the second and third eigenvectors of L s , along with the results of INCRES for k = 2 and k = 3. The binary clustering results of both methods split the data into the same larger classes, while the third eigenvector and INCRES at k = 3 find the same subdivision of one of these classes. In less straightforward clustering examples, the reseeding process can allow INCRES to learn partitions that are not apparent to spectral clustering. Furthermore, the formulation of INCRES allows it to be applied even in cases of larger datasets where eigenpairs cannot be easily computed.
DATA AND FEATURE EXTRACTION
Our audio data consists of recordings, provided by the US Navy's Naval Air Systems Command [9], of different vehicles moving multiple times through a parking lot at approximately 15mph. The original dataset consists of MP4 videos taken from a roadside camera; we extract the dual channel audio signal, and average the channels together into a single channel. The audio signal has a sampling rate of 48,000 frames per second. Video information is used only to ascertain ground truth (vehicle identification) for training data. Each extracted audio signal is a sequence of a vehicle approaching from a distance, becoming audible after 5 or 6 seconds, passing the microphone after 10 seconds, and then leaving. An example of the raw audio signal is shown in Figure 3. We form a composite sequence, shown in Figure 4, from multiple passages of three different vehicles (a white truck, black truck, and jeep), cropping the two seconds where the vehicle is closest to the camera. The goal is to test the clustering algorithm's ability to differentiate the vehicles.
We preprocess the data by grouping audio frames into larger windows. With windows of 1/8 of a second, or 6000 frames, we find both a sufficient number of windows and sufficient information per window. While there is no clear standard in the literature, this window size is comparable to those used in other studies [1]. Discontinuities between successive windows can in some cases be reduced by applying a weighted window filter such as a Hamming filter, or by allowing overlap between windows [1]. However, in our study we found no conclusive benefit from either of these, and therefore used standard box windows with no overlap.
Relevant features are extracted from the raw audio signal using the short-time Fourier transform (STFT). The Fourier decomposition contains 6000 symmetric coefficients, leaving 3000 usable coefficients. Figure 5 shows the first 1000 Fourier coefficients for a time window representing a sedan passing, and a time window representing a truck passing, both in similar positions. Note that a clear frequency signature is apparent for each vehicle, with much of the signal concentrated within the first 200 coefficients.
Each time window in the audio signal is taken as an independent data point to be clustered: we define the feature vector x i ∈ R m as the set of m Fourier coefficients associated with that window. Since many of these coefficients are relatively insignificant, we consider the cosine distance measure between data points Fig. 6. Spectrum of L s for vehicle data. Largest gap is after third eigenvalue.
d ij = 1 − x i · x j x i x j .
We then construct an M -nearest neighbor graph, where the edge {i, j} is present if j is among the M closest neighbors of i or vice-versa, for a fixed value of M . Following standard methods [7], the similarity S ij is taken to be a Gaussian function of distance,
S ij = e −d 2 ij /σ 2 i ,
where σ i is defined adaptively [10] as the distance to vertex i's M th neighbor.
RESULTS
Our composite vehicle dataset contained 18 seconds of raw audio, resulting in n = 144 data points each representing 1/8-second time windows. We used only the first m = 1500 Fourier coefficients. We set M = 15 for the M -nearest neighbor graph, so that neighborhoods contain the 16 data points used in the 2-second clips of a single vehicle passage. Figure 6 shows the eigenvalues of the Laplacian for the vehicle data. The largest gap follows the third eigenvalue, consistent with three clusters representing the three vehicles actually present in the data. We therefore set k = 3 for both spectral clustering and INCRES. Figure 7 shows the second and third eigenvectors of L s , along with typical results of INCRES for k = 2 and k = 3 (INCRES is stochastic, but results vary little from run to run). As in our earlier synthetic example, the second eigenvector and k = 2 INCRES result provide comparable binary separations of the data. Thresholding the eigenvector just above zero would place all of the vehicle 1 data in one cluster, and most a) Eigenvectors b) INCRES results Note that unlike in the straightforward synthetic data problem, the third eigenvector is not by itself sufficient to separate the three clusters. Figure 8 shows the results of kmeans clustering, with k = 3, on the third eigenvector alone. While all vehicle 1 data points are clustered together, a significant fraction of vehicle 2 and 3 data points are incorrectly placed in that cluster as well. Figure 9 shows results of the more conventional spectral clustering method, using k-means on the R 2 projection of the data given by the 2nd and 3rd eigenvectors. The inclusion of the 2nd eigenvector is sufficient to cluster the vast majority of vehicle 2 and 3 data points correctly.
Tables 1 and 2 interpret the spectral clustering results of Figure 9 and the INCRES k = 3 results of Figure 7b as classifications. Both methods classify all of vehicle 1 correctly. but INCRES performs noticeably better than spectral clustering on vehicle 2, and they perform comparably on vehicle 3. Overall purity scores are 87.5% for spectral clustering, and 91.7% for INCRES, with misclassifications again occurring primarily at the beginning or end of a vehicle passage.
CONCLUSIONS
We have presented a method to identify moving vehicles from audio recordings, by clustering their frequency signatures with an incremental reseeding method (INCRES) [8]. We decompose the audio signal with a short-time Fourier transform (STFT), and treat each 1/8-second time window as an individual data point. We then apply a spectral embedding and consider the symmetric normalized graph Laplacian. We find that spectral clustering, which uses the leading eigenvectors of the Laplacian, correctly clusters 87.5% of the data points. INCRES, which directly uses the Laplacian to construct a random walk on the graph, correctly clusters 91.7% of the data points. Almost all incorrectly clustered points lie at the very beginning or very end of a vehicle passage, when the vehicle is furthest from the recording device. The vast majority of data points result in correct vehicle recognition. We observe that there is a close relation between the kth eigenvector and the INCRES output for k clusters. This suggests that clustering results might be improved by simultaneously taking the INCRES output for 2 through k clusters, and then using k-means on this R k−1 projection of the data just as spectral clustering does on the 2nd through kth eigenvectors. While doing so does not noticeably change our INCRES k = 3 results, the difference could be significant for larger values of k. This could be tested, using a dataset with a larger number of vehicles.
Finally, we note that, since time windows are treated as independent data points, our approach ignores most temporal information. Explicitly taking advantage of the time-series nature of our data in the clustering algorithm could improve results, by clustering data points according not only to their own frequency signatures but also to those of preceding or subsequent time windows. Furthermore, while the STFT is a standard method for processing audio signals, it suffers from two drawbacks: the use of time windows imposes a specific time scale for resolving the signal that may not always be the appropriate one, and vehicle sounds may contain too many distinct frequencies for the Fourier decomposition to yield easily learned signatures. These difficulties may best be addressed by using multiscale techniques such as wavelet decompositions that have been proposed for vehicle detection and classification [3,5], as well as more recently developed sparse decomposition methods that learn a set of basis functions from the data [11,12,13,14].
Input Similarity Matrix S, number of clusters k, number of iterations s 2: Initialize Random partitioning of data 3: for i = 1 to s do 4:
Fig. 1 .
1Synthetic similarity matrix with salt and pepper noise. White represents a similarity value of 1
Fig. 2 .
22nd and 3rd eigenvectors of L s as well as INCRES results, for synthetic similarity matrix. Note identical separation into two and three clusters.
Fig. 3 .
3Raw audio signal for vehicle passage.
Fig. 4 .
4Raw audio signal for composite data. Images show the three different vehicles, as seen in accompanying video (not used for analysis).
Fig. 5 .
5First 1000 Fourier coefficients for a car and a truck, after applying a moving mean of size 5.
Fig. 7 .
72nd and 3rd eigenvectors of L s as well as INCRES results, for vehicle data.
Fig. 8 .
8k-means on third eigenvector of L s for vehicle data.
Fig. 9 .
9k-means on second and third eigenvectors of L s (standard spectral clustering) for vehicle data. of the vehicle 2 and 3 data in the other cluster (the exceptions are primarily data points at the beginning and end of a vehicle passage, where the signal is weakest). The third eigenvector mostly distinguishes vehicle 2 (negative values) and vehicle 3 (positive values). The k = 3 INCRES result recognizes the three vehicles very accurately, and is discussed below.
Table 1 .
1Vehicle clustering results using spectral clustering.True
Obtained
cluster
Vehicle 1
(w. truck)
Vehicle 2
(b. truck)
Vehicle 3
(jeep)
Vehicle 1 (white truck)
64
0
0
Vehicle 2 (black truck)
5
24
3
Vehicle 3 (jeep)
8
2
38
Table 2 .
2Vehicle clustering results using INCRES with k = 3.True
Obtained
cluster
Vehicle 1
(w. truck)
Vehicle 2
(b. truck)
Vehicle 3
(jeep)
Vehicle 1 (white truck)
64
0
0
Vehicle 2 (black truck)
1
29
2
Vehicle 3 (jeep)
6
3
39
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Synchrosqueezed wavelet transforms: An empirical mode decomposition-like tool. Ingrid Daubechies, Jianfeng Lu, Hau-Tieng Wu, Applied and Computational Harmonic Analysis. 30Ingrid Daubechies, Jianfeng Lu, and Hau-Tieng Wu, "Synchrosqueezed wavelet transforms: An empirical mode decomposition-like tool," Applied and Computa- tional Harmonic Analysis, vol. 30, no. 2, pp. 243 -261, 2011.
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| []
|
[
"Towards a New Proof of Anderson Localization",
"Towards a New Proof of Anderson Localization"
]
| [
"Robert Brandenberger \nDepartment of Physics\nMcGill University\n3600 Rue UniversitéH3A 2T8MontréalQuébecCanada\n",
"Walter Craig [email protected] \nDepartment of Mathematics and Statistics\nMcMaster University\nL8S 4K1HamiltonOntarioCanada\n"
]
| [
"Department of Physics\nMcGill University\n3600 Rue UniversitéH3A 2T8MontréalQuébecCanada",
"Department of Mathematics and Statistics\nMcMaster University\nL8S 4K1HamiltonOntarioCanada"
]
| []
| The wave function of a non-relativistic particle in a periodic potential admits oscillatory solutions, the Bloch waves. In the presence of a random noise contribution to the potential the wave function is localized. We outline a new proof of this Anderson localization phenomenon in one spatial dimension, extending the classical result to the case of a periodic background potential. The proof makes use of techniques previously developed to study the effects of noise on reheating in inflationary cosmology, employing methods of random matrix theory. | 10.1140/epjc/s10052-012-1881-9 | [
"https://arxiv.org/pdf/0805.4217v1.pdf"
]
| 33,129,759 | 0805.4217 | eb45f6f57e035cdc974fb85d28c5fafa7810cd25 |
Towards a New Proof of Anderson Localization
27 May 2008
Robert Brandenberger
Department of Physics
McGill University
3600 Rue UniversitéH3A 2T8MontréalQuébecCanada
Walter Craig [email protected]
Department of Mathematics and Statistics
McMaster University
L8S 4K1HamiltonOntarioCanada
Towards a New Proof of Anderson Localization
27 May 2008
The wave function of a non-relativistic particle in a periodic potential admits oscillatory solutions, the Bloch waves. In the presence of a random noise contribution to the potential the wave function is localized. We outline a new proof of this Anderson localization phenomenon in one spatial dimension, extending the classical result to the case of a periodic background potential. The proof makes use of techniques previously developed to study the effects of noise on reheating in inflationary cosmology, employing methods of random matrix theory.
I. INTRODUCTION
A classic problem in non-relativistic quantum mechanics is the propagation of a particle, e.g. an electron, in a periodic potential, set up e.g. by a lattice of ions. It is known that the time-independent Schrödinger equation for this problem admits modulated periodic solutions, the so-called Bloch waves [1]. In the real world, however, we expect that the potential will not be perfectly periodic due to the presence of various types of disorder, e.g. thermal noise. As was first shown by Anderson [2], if the noise is modeled as a random contribution to the potential, then for sufficiently large disorder the wave functions become localized. In one spatial dimension, a stronger result holds: it can be shown [3] that for any amount of disorder, the wave functions become exponentially localized, i.e. instead of being oscillatory, the solutions now are exponentially decaying in space about the location of the center of the wave packet. This result was extended to two spatial dimensions in [4]. Anderson localization is reviewed in [5,6,7]. A mathematically rigorous proof of one dimensional Anderson localization for constant background potential is given in [8], and there is an extensive mathematical literature on the subject.
As is well known, in one spatial dimension the timeindependent Schrödinger equation can be mapped into a classical time-dependent wave equation by a simple variable transformation in which the wave function Ψ becomes a classical field χ and the spatial variable x is transformed to time t. After this mapping, the Schrödinger equation for the wave function Ψ in a potential which is the sum of a periodic term V p (x) and a random noise term V R (x) becomes the relativistic wave equation (in momentum space) for a scalar field with a mass which contains one contribution to the mass which is periodically varying in time, and a second contribution which is random in time.
The inflationary scenario [9] (see also [10] for a recent review emphasizing both successes and problems of inflation), the current paradigm of early universe cosmology, is based on the dynamics of classical scalar fields coupled as matter source to Einstein's theory of General Relativity. According to the inflationary universe scenario, there is a period in the very early universe in which space expands almost exponentially. This is obtained by making use of a scalar field ϕ whose energy functional is dominated by a potential energy density contribution which is almost constant in time. Regular matter can be modeled as a second scalar field χ.
During the period of inflation, an exponentially increasing fraction of the energy density of the universe is stored in the field ϕ. Hence, to make contact with what is observed today, a phase at the end of the period of inflation when the energy transfers from ϕ to regular matter is crucial. This phase is called the "reheating stage", and works in the following way. The period of inflation terminates once ϕ begins to oscillate about the minimum of its potential. Due to a resonant coupling between ϕ and the matter field χ, energy can be transferred from ϕ to χ. The theory of reheating is obtained by studying the evolution of χ in the presence of an oscillating ϕ field.
The equation of motion for χ is the Klein-Gordon equation with a correction term due to the coupling with ϕ. Treating χ at the linearized level, each Fourier mode of χ evolved independently. Neglecting the effects of the expansion of the universe [27], and for the specific coupling between the two fields which we specify below, the equation of motion for such a Fourier mode of χ is that of a harmonic oscillator whose mass has two time-dependent contributions, one periodic in time (due to the oscillatory dynamics of ϕ during the reheating period, the other being an aperiodic random noise term (due e.g. to quantum fluctuations in ϕ).
In the absence of noise, the equation of motion is the Mathieu equation and falls into the class of equations first studied by Hill, Floquet and Legendre [11]. As a function of the value of the bare mass (the term in the mass independent of time), there are stability bands (for which the solutions are oscillatory) and instability bands (for which the solutions are characterized by overall exponential growth or decay. The coefficient describing the exponential growth is called the "Lyapunov exponent". Applied to the case of cosmology, then in the absence of noise there are bands of Fourier modes for which the mass lies in the instability band and for which there is resonant increase in the amplitude of χ, which in terms of physics corresponds to resonant production of particles [12].
Not too long ago [13], the effects of a particular type of random noise on the resonant production of particles during inflationary reheating was studied. It was shown that the Lyapunov exponent µ k (q) which describes the exponential growth of each Fourier mode k of the scalar field in the presence of the noise q is strictly larger than the corresponding Lyapunov exponent µ k (0) in the absence of noise [28]. In particular, this implies that in the presence of noise, every Fourier mode grows exponentially. The stability bands which are present in the absence of noise disappear.
In this Letter, we review the above analysis, translate the results into the language of the time-independent Schrödinger equation, and in this way immediately obtain a new proof of Anderson localization which extends to the case of a periodic background potential.
II. REVIEW OF RESULTS ON THE EFFECTS OF NOISE ON REHEATING IN INFLATIONARY COSMOLOGY
We begin by reviewing the results of [13]. We consider the simplest model which describes both the inflationary phase of the very early Universe and the period of "reheating" which terminates the phase of inflation and during which the energy of the Universe is transferred to regular matter. This model contains two scalar matter fields, ϕ and χ. The first, ϕ, has a large potential energy which is changing only very slowly in time, dominates the total energy density of the Universe and thus gives rise to inflation [9]. The second scalar field, χ, represents regular matter. It is in its vacuum state initially and gets excited by a coupling with ϕ during the final stages of inflation when ϕ(t) oscillates about the minimum of its potential [29]. It is of great interest in cosmology to study the dynamics of matter production at the end of inflation.
We will assume that the interaction Lagrangian density takes the form [30]
L int = 1 2 gϕχ 2 ,(1)
and that the Lagrangian for χ alone is that of a free scalar field with mass m χ . We will also consider ϕ to be spatially homogeneous. In the absence of quantum fluctuations, this is a reasonable assumption since during the period of inflation, spatial fluctuations are red-shifted exponentially. In this case, the equation of motion for χ can be solved independently for each Fourier mode. In the absence of expansion of the Universe, the resulting equation is [31]χ
k + ω 2 k + gϕ(t) χ k = 0 ,(2)
where ω 2 k = k 2 + m 2 χ . At the end of the period of inflation, the scalar field ϕ is oscillating about its ground state value, which we without loss of generality can take to be ϕ = 0. The frequency ω of the oscillation is set by the mass of ϕ. In this case, the equation (2) has the form of a Mathieu equation. As is well known (see e.g. [11,20,21]), there are bands of values of ω k for which χ k grows exponentially. The exponential growth is governed by a Lyapunov exponent µ(k), which can be extracted from the time evolution of χ k (t) as follows:
µ k = lim t→∞ 1 t log|χ k (t)| .(3)
The time scale of exponential growth is in many concrete inflationary models much shorter than the typical expansion time of the Universe, thus justifying the neglect of the expansion. Thus [12], the parametric instability leads to a very efficient energy transfer [32] However, in cosmology we expect fluctuations of thermal or quantum nature to be super-imposed on the homogeneous oscillation of ϕ. In fact, we believe that quantum vacuum fluctuations of ϕ in inflationary cosmology are the seeds for the inhomogeneities in the galaxy distribution and the anisotropies in the temperature of the cosmic microwave background observed today. Thus, it becomes important to study the sensitivity of the parametric resonance instability of (2) in the presence of an oscillating ϕ(t) to the presence of random noise in ϕ(t). In [13], this problem was studied for homogeneous, aperiodic noise, modeled as the addition of a stochastic contribution to the time dependent mass. More precisely, we studied the equation
χ k + ω 2 k + p(ωt) + q(t) χ k = 0 ,(4)
where p is a periodic function with period 2π and q(t) is an aperiodic, random noise contribution [33] To derive rigorous inequalities on the magnitudes of the Lyapunov exponent with and without noise, it is necessary to make certain assumptions on the noise q(t). We consider noise drawn from some sample space Ω = C(R) with a translationally invariant probability measure dP (κ), where κ labels an element in Ω, and assume:
• The noise is ergodic, i.e. the ensemble average of a function of the noise equals the time average for almost all realizations of the noise.
• The noise is uncorrelated in time on scales larger than T , the period of the oscillatory contribution to the mass in (4).
• Restricting the noise to the time interval 0 ≤ t < T , the samples q(t; κ) : 0 ≤ t < T within the support of the probability measure fill a neighborhood of the origin in Ω.
Given these assumptions, we were able to use a theorem of random matrix theory (applied to the transfer matrix corresponding to the differential equation (4)) to prove that for almost all realizations of the noise
µ(q) > µ(0)(5)
for all values of the momentum k. In particular, this implies that the stability bands of the Mathieu equation disappear once noise of the type considered here is added to the system. The noise in fact strengthens the instability. Note that the result (5) was proven non-perturbatively in [13] for values of k in the stability bands, the case of interest to us here. We conjecture that the result is true in general, and have been able to show this at least perturbatively [14].
III. APPLICATION TO ANDERSON LOCALIZATION
Let us start from Eq. (4) and transform variables. We replace the time coordinate t by a spatial coordinate x (space being infinite, i.e. R), and substitute the field variable χ by the variable ψ representing a wave function. With these substitutions, Eq. (4) becomes the time-independent Schrödinger equation for the wave function ψ in one spatial dimension for a system with a potential which is the superposition of a periodic piece V p (x) (coming from p(ωt) in (4) and a random noise piece V R (x) coming from q(t) in (4):
Hψ = Eψ(6)
with H = − 1 2m
∂ 2 ∂x 2 + V p (x) + V R (x) .(7)
The energy eigenvalue E is given by the momentum k of (4). We use units in whichh = 1.
In the absence of noise, the spectrum of the Schrödinger equation (6) consists of bands of continuous spectrum, for which bounded quasi-periodic (modulated periodic) solutions exist, alternating with instability gaps (resonance intervals) in which the solution behavior is exponential. In both regions the solutions, the so-called Bloch waves [1], exhibit Floquet behavior, namely they are of the form ψ(x) = exp(±(µ E + iα E )x)p(x), where p(x) is periodic of the same period as the potential V p . The Floquet exponent m E = (µ E + iα E ) has real part the Lyapunov exponent and imaginary part the rotation number of the solution ψ(x). The Lyapunov exponent vanishes in the stable bands, while the rotation number takes (a fixed multiple of) integer values in the gaps. In the classical field theory problem these bands of spectrum correspond precisely to the stability bands of the Mathieu equation (4) (for q(t) = 0).
Let us now turn on noise satisfying the conditions listed in the previous section and look for solutions of (6). Due to the translation invariance of the problem, we can take x = 0 to be the point at which ψ takes on its maximum. In the language of the quantum mechanics problem, the Lyapunov exponent can be extracted as follows:
µ E = − lim x→∞ 1 x log|ψ(x)| .(8)
Since the wave function in quantum mechanics must be normalized, no exponentially growing solutions are allowed. An eigenstate corresponds to a wave function which does not grow either for x → ∞ or x → −∞.
Thus, a positive real Lyapunov exponent corresponds to an exponentially decaying wave function, whereas an imaginary Floquet exponent with absolute value 1 corresponds to Bloch waves. A standard mathematical argument [8,24,25,26] shows that the eigenstates are dense and the spectrum is pure point. If we consider an energy eigenvalue E which lies in a conduction band (i.e. an energy band for which Bloch wave solutions exist) in the absence of noise, then our result (5) implies that as soon as a random noise term is added to the potential in (7), the Lyapunov exponent becomes positive, and that thus the corresponding wave function are exponentially decaying. This means that the addition of any noise satisfying the conditions listed in the previous section exponentially localizes the wave function. The localization length is inversely proportional to the Lyapunov exponent.
IV. DISCUSSION
We have shown that the time-independent Schrödinger equation for the non-relativistic Hamiltonian (7) does not admit Bloch wave solutions for any amplitude of the noise. Given a wave function centered at some point x (which we can without loss of generality take to be x = 0), we have shown that the wave function decays exponentially away from x = 0. Thus, we have given a new proof of the classic result [3] of exponential localization of states in one spatial dimension which extends the known result to the case of a periodic background potential.
Our methods are based on translating the timeindependent quantum mechanical problem to a classical dynamical systems problem, and thus are only applicable to the case of one spatial dimension and cannot be used to study Anderson localization in a higher number of spatial dimensions.
The dynamical systems problem turns out to be a problem concerning the effects of noise on the resonant production of particles during the reheating phase of inflationary cosmology. In mathematical language, our problem concerns adding aperiodic noise to the mass term in the Mathieu equation. We have shown that the Lyapunov exponent increases for any amplitude of the noise. Thus, the solutions exhibit exponential behavior both forwards and backwards in time (starting from the initial time t = 0 corresponding to x = 0 for the quantum problem). Normalizability of the wave function in the quantum problem implies that the eigenstates are precisely those solutions which decay away from the source point.
AcknowledgmentsWe are grateful to Profs. A. Maia and V. Zanchin for collaborating with us on the earlier work on the effects of noise in inflationary cosmology. One of us (RB) wishes to thank Prof. T. D. Lee for asking key questions during a seminar at Columbia University. We would like to acknowledge the hospitality of the Fields Institute for Research in Mathematical Sciences during its 2003-2004 Thematic Program on Partial Differential Equations which brought us together to study this problem. This research is partly supported by NSERC Discovery Grants to R.B. and W.C., and by the Canada Research Chair program.
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This is self-consistent if the period of energy transfer turns out to be short compared to the time scale of the expansion of space. This is self-consistent if the period of energy transfer turns out to be short compared to the time scale of the expansion of space.
this was only shown for values of k in the stability bands. Perturbatively, the statement also holds for values of k in the instability bands. Non-Perturbatively, What is relevant for this paper is the result for values of k in the stability bandsNon-perturbatively, this was only shown for values of k in the stability bands. Perturbatively, the statement also holds for values of k in the instability bands [14]. What is relevant for this paper is the result for values of k in the stability bands.
This toy model was already used in the first studies of reheating in inflationary cosmology. 15This toy model was already used in the first studies of reheating in inflationary cosmology [15, 16].
The quantity g is a constant which has dimensions of mass. It is the coupling constantThe quantity g is a constant which has dimensions of mass. It is the coupling constant.
The expansion of the Universe can be taken into account. 17, 18, 19] without affecting the result that there are exponential instabilitiesThe expansion of the Universe can be taken into account [17, 18, 19] without affecting the result that there are exponential instabilities.
At some point, the back-reaction of the energy in the χ field on the dynamics of space-time will become important. and this could terminate the energy transfer [22At some point, the back-reaction of the energy in the χ field on the dynamics of space-time will become impor- tant, and this could terminate the energy transfer [22].
The mathematically much more complicated problem of inhomogeneous noise, in which case the inhomogeneity in the noise couples the different Fourier modes of χ, thus leading to a problem in the field of partial differential equations. was studied in [23The mathematically much more complicated problem of inhomogeneous noise, in which case the inhomogeneity in the noise couples the different Fourier modes of χ, thus leading to a problem in the field of partial differential equations, was studied in [23].
| []
|
[
"Space-Efficient Merging of Succinct de Bruijn Graphs",
"Space-Efficient Merging of Succinct de Bruijn Graphs"
]
| [
"Lavinia Egidi [email protected] ",
"Giovanni Manzini [email protected] \nUniversity of Eastern Piedmont\nAlessandriaItaly\n\nIIT CNR\nPisa Italy\n",
"\nDepartment of Computing and Mathematics\nUniversity of São Paulo\nBrazil\n"
]
| [
"University of Eastern Piedmont\nAlessandriaItaly",
"IIT CNR\nPisa Italy",
"Department of Computing and Mathematics\nUniversity of São Paulo\nBrazil"
]
| []
| We propose a new algorithm for merging succinct representations of de Bruijn graphs introduced in [Bowe et al. WABI 2012]. Our algorithm is based on the lightweight BWT merging approach by Holt and McMillan [Bionformatics 2014, ACM-BCB 2014]. Our algorithm has the same asymptotic cost of the state of the art tool for the same problem presented by Muggli et al. [bioRxiv 2017, 2019], but it is more space efficient. A novel feature of our algorithm not found in the existing tools is that it can compute the Variable Order succinct representation of the union graph starting from the plain representation of the input graphs. | 10.1007/978-3-030-32686-9_24 | [
"https://arxiv.org/pdf/1902.02889v3.pdf"
]
| 59,842,916 | 1902.02889 | 9147b1cae995aceee2c014633fa17c4c0ac17e59 |
Space-Efficient Merging of Succinct de Bruijn Graphs
Lavinia Egidi [email protected]
Giovanni Manzini [email protected]
University of Eastern Piedmont
AlessandriaItaly
IIT CNR
Pisa Italy
Department of Computing and Mathematics
University of São Paulo
Brazil
Space-Efficient Merging of Succinct de Bruijn Graphs
de Bruijn graphs · succinct data structures · merging · variable-order · colored graphs
We propose a new algorithm for merging succinct representations of de Bruijn graphs introduced in [Bowe et al. WABI 2012]. Our algorithm is based on the lightweight BWT merging approach by Holt and McMillan [Bionformatics 2014, ACM-BCB 2014]. Our algorithm has the same asymptotic cost of the state of the art tool for the same problem presented by Muggli et al. [bioRxiv 2017, 2019], but it is more space efficient. A novel feature of our algorithm not found in the existing tools is that it can compute the Variable Order succinct representation of the union graph starting from the plain representation of the input graphs.
Introduction
The de Bruijn graph for a collection of strings is a key data structure in genome assembly [18]. After the seminal work of Bowe et al. [5], many succinct representations of this data structure have been proposed in the literature [2,3,4,17] offering more and more functionalities still using a fraction of the space required to store the input collection uncompressed. In this paper we consider the problem of merging two existing succinct representations of de Bruijn graphs built for different collections. Since the de Bruijn graph is a lossy representation and from it we cannot recover the original collection, the alternative to merging is storing a copy of each collection to be used for building new de Bruijn graphs from scratch.
Recently, Muggli et al. [16,15] have proposed a merging algorithm for colored de Bruijn graphs and have shown the effectiveness of the merging approach for the construction of de Bruijn graphs for very large datasets. The algorithm in [15] is based on an MSD Radix Sort procedure of the graph nodes and its running time is O(m max(k, c)), where m is the total number of edges, k the order of the de Bruijn graph and c the total number of colors.
In this paper we present a new merging algorithm based on a mixed LSD/MSD Radix Sort which is inspired by the lightweight BWT merging algorithm introduced by Holt and McMillan [10,11] and later improved in [7,8]. Our algorithm has the same time complexity of the one in [16,15] but it is much more space efficient (we defer the detailed comparison to the end of Section 4). In addition, our algorithm can compute, with no additional cost, the LCS (Longest Common Suffix) between the node labels, thus making it possible to construct succinct Variable Order de Bruijn graph representations [4], a feature not shared by any other merging algorithm.
Our algorithm works by accessing sequentially the input data and some auxiliary arrays, so it is suitable for execution in external memory. Combining our merging algorithm with recent results on external memory de Bruijn graph construction [6], we provide a space efficient external memory procedure for building succinct representations of de Bruijn graphs for very large collections.
Notation
Given the alphabet Σ = {1, 2, . . . , σ} and a collection of strings C = s 1 , . . . , s d over Σ, we prepend to each string s i k copies of a symbol $ / ∈ Σ which is lexicographically smaller than any other symbol. The order-k de Bruijn graph G(V, E) for the collection C is a directed edge-labeled graph containing a node v for every unique k-mer appearing in one of the strings of C. For each node v we denote by
− → v = v[1, k] its associated k-mer, where v[1] . . . v[k] are symbols. The graph G contains an edge (u, v), with label v[k], iff one of the strings in C contains a (k+1) -mer with prefix − → u and suffix − → v . The edge (u, v) therefore represents the (k + 1)-mer u[1, k]v[k]
. Note that each node has at most |Σ| outgoing edges and all edges incoming to node v have label v[k].
BOSS succinct representation. In 2012, Bowe et al. [5] introduced a succinct representation for the de Bruijn graphs, usually referred to as BOSS representation, for the authors initials. The authors showed how to represent the graph in small space supporting fast navigation operations. The BOSS representation of the graph G(V, E) is defined by considering the set of nodes v 1 , v 2 , . . . v n sorted according to the colexicographic order of their associated k-mer. Hence, if [1] denotes the string − → v reversed, the nodes are ordered so that
← − v = v[k] . . . v← − v 1 ≺ ← − v 2 ≺ · · · ≺ ← − v n(1)
By construction the first node is ← − v 1 = $ k and all ← − v i are distinct. For each node v i , i = 1, . . . , n, we define W i as the sorted sequence of symbols on the edges leaving from node v i ; if v i has out-degree zero we set W i = $. Finally, we define The length m of the arrays W , W − , and last is equal to the number of edges plus the number of nodes with out-degree 0. In addition, the number of 1's in last is equal to the number of nodes n, and the number of 1's in W − is equal to the number of nodes with positive in-degree, which is n − 1 since v 1 = $ k is the only node with in-degree 0. Note that there is a natural one-to-one correspondence, called LF for historical reasons, between the indices i such that
$ $ $ $ $ $ A C A T C A $ GA $ T A C A C GA C T A C T A C C T C C T C $ $ G T C G $ $ T A C T T G C $ C C T T A T A G A $ A C 0 1 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 last Nodes W W −W [i] < W [j] =⇒ LF (i) < LF (j), (W [i] = W [j]) ∧ (i < j) =⇒ LF (i) < LF (j).(2)
In [5] it is shown that, enriching the arrays W , W − , and last with the data structures from [9,19] supporting constant time rank and select operations, we can efficiently navigate the graph G. The authors defined the following basic queries: outgoing(v i , c) returns the node v j reached from v i by an edge with label c, or −1 if no such node exists; incoming(v i ) returns the nodes v j with an edge from v j with v i ; and lastchar(v i ) returns the last symbol of − → v i .
Colored BOSS. The colored de Bruijn graph [12] is an extension of the de Bruijn graphs for a multiset of individual graphs, where each edge is associated with a set of "colors" that indicates which graphs contain that edge. The BOSS representation for a set of graphs G = {G 1 , . . . , G t } contains the union of all individual graphs. In its simplest representation, the colors of all edges W [i] are stored in a two-dimensional binary array C, such that C[i, j] = 1 iff the i-th edge is present in graph G j . There are different compression alternatives for the color matrix C that support fast operations [2,14,17]. Recently, Alipanah et al. [1] presented a different approach to reduce the size of C by recoloring.
Variable-order BOSS. The order k (dimension) of a de Bruijn graph is an important parameter for genome assembling algorithms. The graph can be very small and uninformative when k is small, whereas it can become too large with unrelated parts when k is large. To add flexibility to the BOSS representation, Boucher et al. [4] suggest to enrich the BOSS representation of an order-k de Bruijn graph with the length of the longest common suffix (LCS) between the k-mers of consecutive nodes v 1 , v 2 , . . . , v n sorted according to (1). These lengths are stored in a wavelet tree using O(n log k) additional bits. The authors show that this enriched representation supports navigation on all de Bruijn graphs of order k ≤ k and that it is even possible to vary the order k of the graph on the fly during the navigation up to the maximum value k.
The LCS between − → v i and − − → v i+1 is equivalent to the length of the longest common prefix (LCP) between their reverses ← − v i and ← − − v i+1 . The LCP (or LCS) between the nodes v 1 , v 2 , · · · , v n can be computed during the k-mer sorting phase. In the following we denote by VO-BOSS the variable order succinct de Bruijn graph consisting of the BOSS representations enriched with the LCS/LCP information.
Merging plain BOSS representations
Suppose we are given the BOSS representations of two de Bruijn graphs W 0 , W − 0 , last 0 and W 1 , W − 1 , last 1 obtained respectively from the collections of strings C 0 and C 1 . In this section we show how to compute the BOSS representation for the union collection C 01 = C 0 ∪ C 1 . The procedure does not change in the general case when we are merging an arbitrary number of graphs. Let G 0 and G 1 denote respectively the (uncompressed) de Bruijn graphs for C 0 and C 1 , and let v 1 , . . . , v n0 and w 1 , . . . , w n1 their respective set of nodes sorted in colexicographic order. Hence, with the notation of the previous section we have
← − v 1 ≺ · · · ≺ ← − v n0 and ← − w 1 ≺ · · · ≺ ← − − w n1(3)
We observe that the k-mers in the collection C 01 are simply the union of the k-mers in C 0 and C 1 . To build the de Bruijn graph for C 01 we need therefore to: 1) merge the nodes in G 0 and G 1 according to the colexicographic order of their associated k-mers, 2) recognize when two nodes in G 0 and G 1 refer to the same k-mer, and 3) properly merge and update the bitvectors W − 0 , last 0 and W − 1 , last 1 .
Phase 1: Merging k-mers
The main technical difficulty is that in the BOSS representation the k-mers associated to each node − → v = v[1, k] are not directly available. Our algorithm will reconstruct them using the symbols associated to the graph edges; to this end the algorithm will consider only the edges such that the corresponding entries in W − 0 or W − 1 are equal to 1. Following these edges, first we recover the last symbol of each k-mer, following them a second time we could recover the last two symbols of each k-mer and so on. However, to save space we do not explicitly maintain the k-mers; instead, using the ideas from [10,11] we compute a bitvector Z (k) representing how the k-mers in G 0 and G 1 should be merged according to the colexicographic order. In addition, we maintain just enough information, to recognize when two k-mers are identical, so that their incoming and outgoing edges can be merged.
To this end, our algorithm executes k −1 iterations of the code shown in Fig. 3. For h = 2, 3, . . . , k, during iteration h, we compute a bitvector Z (h) [1, n 0 + n 1 ] containing n 0 0's and n 1 1's such that Z (h) satisfies the following property Property 2. For i = 1, . . . , n 0 and j = 1, . . . n 1 the i-th 0 precedes the j-th 1 in
Z (h) if and only if ← − v i [1, h] ← − w j [1, h].
Property 2 states that if we merge the nodes from G 0 and G 1 according to the bitvector Z (h) the corresponding k-mers will be sorted according to the lexicographic order restricted to the first h symbols of each reversed k-mer. As a consequence, Z (k) will provide us the colexicographic order of all the nodes in G 0 and G 1 . To prove that Property 2 holds, we first define Z (1) and show that it satisfies the property, then we prove that for h = 2, . . . , k the code in Fig. 3 computes Z (h) that still satisfies Property 2.
For c ∈ Σ let 0 (c) and 1 (c) denote respectively the number of nodes in G 0 and G 1 whose associated k-mers end with symbol c. These values can be computed with a single scan of W 0 (resp. W 1 ) considering only the symbols W 0
[i] (resp. W 1 [i]) such that W − 0 [i] = 1 (resp. W − 1 [i] = 1). By construction, it is n 0 = 1 + c∈Σ 0 (c), n 1 = 1 + c∈Σ 1 (c)
where the two 1's account for the nodes v 1 and w 1 whose associated k-mer is $ k . We define
Z (1) = 01 0 0(1) 1 1(1) 0 0(2) 1 1(2) · · · 0 0(σ) 1 1(σ)(4)
the first pair 01 in Z (1) accounts for v 1 and w 1 ; for each c ∈ Σ group 0 0 (c) 1 1(c) accounts for the nodes ending with symbol c. Note that, apart from the first two symbols, Z (1) can be logically partitioned into σ subarrays one for each alphabet symbol. For c ∈ Σ let
start(c) = 3 + i<c ( 0 (i) + 1 (i))
then the subarray corresponding to c starts at position start(c) and has size 0 (c) + 1 (c). As a consequence of (3), the i-th 0 (resp. j-th 1) belongs to the subarray associated to symbol
c iff ← − v i [1] = c (resp. ← − w j [1] = c)
. To see that Z (1) satisfies Property 2, observe that the i-th 0 precedes j-th 1 iff the i-th 0 belongs to a subarray corresponding to a symbol not larger than the symbol corresponding to the subarray containing the j-th 1; this implies
← − v i [1, 1] ← − w j [1, 1].
The bitvectors Z (h) computed by the algorithm in Fig. 3 can be logically divided into the same subarrays we defined for Z (1) . In the algorithm we use an array F to keep track of the next available position of each subarray. Because of how the array F is initialized and updated, we see that every time we read a symbol c at line 14 the corresponding bit b = Z (h−1) [k], which gives us the graph containing c, is written in the portion of Z (h) corresponding to c (line 16). The only exception are the first two entries of Z (h) which are written at line 6 which corresponds to the nodes v 1 and w 1 . We treat these nodes differently since they are the only ones with in-degree zero. For all other nodes, we implicitly use the one-to-one correspondence (2) between entries W [i] with W − [i] = 1 and nodes v j with positive in-degree.
Note that lines 8-10 and 17-22 of the algorithm are related to the computation of the B array that is used in the following section and do not influences the computation of Z (h) or Property 2. The following Lemma, proven in the Appendix, proves the correctness of the algorithm in Fig. 3.
Lemma 1.
For h = 2, . . . , k, the array Z (h) computed by the algorithm in Fig. 3 satisfies Property 2.
Phase 2: Recognizing identical k-mers
Once we have determined, via the bitvector Z (h) [1, n 0 + n 1 ], the colexicographic order of the k-mers, we need to determine when two k-mers are identical since in this case we have to merge their outgoing and incoming edges. Note that two identical k-mers will be consecutive in the colexicographic order and they will necessarily belong one to G 0 and the other to G 1 .
b ← Z (h−1) [p] Read bit b from Z (h−1) 12:
repeat
Current node is from graph G b 13: if W − b [i b ] = 1 then 14: c ← W b [i b ] GetZ (h) [1, 1 ], Z (h) [ 1 + 1, 2 ], . . . , Z (h) [ b(h) + 1, n 0 + n 1 ](5)
such that each block corresponds to a set of k-mers which are prefixed by the same length-h substring. Note that during iterations h = 2, 3, . . . , k the k-mers within an h-block will be rearranged, and sorted according to longer and longer prefixes, but they will stay within the same block.
In the algorithm of Fig. 3, in addition to Z (h) , we maintain an integer array B[1, n 0 + n 1 ], such that at the end of iteration h it is B[i] = 0 if and only if a block of Z (h) starts at position i. Initially, for h = 1, since we have one block per symbol, we set B = 10 10 0(1)+ 1(1)−1 10 0(2)+ 1(2)−1 · · · 10 0(σ)+ 1(σ)−1 . The above lemma shows that using array B we can establish when two k-mers are equal and consequently the associated graph nodes should be merged.
Phase 3: Building BOSS representation for the union graph
We now show how to compute the succinct representation of the union graph G 0 ∪ G 1 , consisting of the arrays W 01 , W − 01 , last 01 , given the succinct representations of G 0 and G 1 and the arrays Z (k) and B.
The arrays W 01 , W − 01 , last 01 are initially empty and we fill them in a single sequential pass. For q = 1, . . . , n 0 + n 1 we consider the values Z (k) [q] and B[q]. If B[q] = 0 then the k-mer associated to Z (k) [q − 1], say ← − v i is identical to the k-mer associated to Z (k) [q], say ← − w j . In this case we recover from W 0 and W 1 the labels of the edges outgoing from v i and w j , we compute their union and write them to W 01 (we assume the edges are in the lexicographic order), writing at the same time the representation of the out-degree of the new node to last 01 . If instead B[q] = 0, then the k-mer associated to Z (k) [q − 1] is unique and we copy the information of its outgoing edges and out-degree directly to W 01 and last 01 .
When
Implementation details and analysis
Let n = n 1 + n 0 denote the sum of number of nodes in G 0 and G 1 , and let m = |W 0 | + |W 1 | denote the sum of the number of edges. The k-mer merging algorithm as described executes in O(m) time a first pass over the arrays W 0 , W − 0 , and W 1 , W − 1 to compute the values 0 (c) + 1 (c) for c ∈ Σ and initialize the arrays start [1, σ] and Z (1) We now analyze the space usage of the algorithm. Since space is a major issue to give a more accurate estimate we indicate the alphabet size with σ, even if in the known applications it is σ = 4 or σ = 5 depending on whether we represent explicitly the $ symbol. In addition to the input and the output, our algorithm uses 2n bits for two instances of the Z (·) array (for the current Z (h) and for the previous Z (h−1) ), plus n log k bits for the B array. Note however, that during iteration h we only need to check whether B
[i] = h − 1; while if h is odd the correspondence is 2 → h − 1, 1 → h.
The reason for this apparently involved scheme, first introduced in [6], is that during phase h, an entry in B 2 can be modified either before or after we have read it at Line 9. Using this technique, the working space of the algorithm, i.e., the space in addition to the input and the output, is 4n bits plus O(σ) words of RAM for the arrays start, F , and Block id.
Note that during Phase 1, at each iteration h = 2, . . . , k, the arrays W 0 , W − 0 , last 0 , W 1 , W − 1 , last 1 , Z (h−1) , and B 2 are read sequentially from beginning to end. At the same time, the arrays Z (h) and B 2 are written sequentially but into σ different partitions whose starting positions are the values in start which are the same for each iteration. Thus, if we split Z (·) and B 2 into σ different files, all accesses are sequential and Phase 1 is suitable for execution in external memory using only O(σ) words of RAM. Since during Phases 2 and 3 all input and output arrays are accessed sequentially in linear time, we can summarize our analysis as follows. Comparison with the state of the art. The de Bruijn graph merging algorithm by Muggli et al. [16,15] is similar to ours in that it has a planning phase consisting of the colexicographic sorting of the (k + 1)-mers associated to the edges of G 0 and G 1 . To this end, the algorithm uses a standard MSD radix sort. However only the most significant symbol of each (k + 1)-mer is readily available in W 0 and W 1 . Thus, during each iteration the algorithm computes also the next symbol of each (k + 1)-mer that will be used as a sorting key in the next iteration.
The overall space for such symbols is 2m log σ bits, since for each edge we need the symbol for the current and next iteration. During the sorting the algorithm uses up to 2(n + m) bits to maintain the set of intervals consisting in edges whose associated reversed (k + 1)-mer have a common prefix; these intervals correspond to the blocks we implicitly maintain in the array B 2 using 2n bits.
Summing up, the algorithm by Muggli et al. takes O(mk) time, and uses 2(m log σ + m + n) bits plus O(σ) words of space. Our algorithm has the same time complexity but uses less space: assuming σ = 4 the version in Theorem 1 uses less than half the space (4n bits vs. 6m + 2n bits).
Note also that our algorithm is based on radix sorting, however, it uses a mixed LSD/MSD strategy. The algorithm by Muggli et al. follows the traditional MSD radix sort strategy; hence it establishes, for example, that ACG ≺ ACT when it compares the third 'digits' and finds that G < T . In our algorithm we also find that ACG ≺ ACT during the third iteration, but this is established without comparing directly G and T , which are not explicitly available. Instead, during the second iteration the algorithm finds that CG ≺ CT and during the third iteration it uses this fact to infer that ACG ≺ ACT : this is indeed a remarkable sorting trick first introduced in [11] and adapted here to de Bruijn graphs.
Merging colored and VO-BOSS representations
Our algorithm can be easily generalized to merge colored and VO (variable-order) BOSS representations. Note that the algorithm by Muggli et al. can also merge colored BOSS representations, but it cannot be easily adapted to merge, or compute, VO representations.
Given the colored BOSS representation of two de Bruijn graphs G 0 and G 1 , the corresponding color matrices C 0 and C 1 have size m 0 ×c 0 and m 1 ×c 1 , respectively. We initially create a new color matrix C 01 of size (m 1 + m 2 ) × (c 0 + c 1 ) with all entries empty. During the merging of the union graph (Phase 3), for q = 1, . . . , n, we write the colors of the edges associated to Z (h) [q] to the corresponding line in C 01 possibly merging the colors when we find nodes with identical k-mers in O(c 01 ) time, with c 01 = c 0 + c 1 . To make sure that colors id from C 0 and C 1 do not intersect in the new graph we just need to add the constant c 0 (the number of distinct colors in G 0 ) to any color id coming from the matrix C 1 . We now show that we can compute the variable order VO-BOSS representation of the union of two de Bruijn graphs G 0 and G 1 given their plain, eg. non variable order, BOSS representations. For the VO-BOSS representation we need the LCS array for the nodes in the union graph W 01 , W − 01 , last 01 . Notice that after merging the k-mers of G 0 and G 1 with the algorithm in Fig. 3 (Phase 1) the values in B[1, n] already provide the LCP information between the reverse labels of all consecutive nodes (Lemma 2). When building the union graph (Phase 3), for q = 1, . . . , n, the LCS between two consecutive nodes, say v i and w j , is equal to the LCP of their reverses ← − v i and ← − w j , which is given by B[q] − 1 whenever B[q] > 0 (if B[q] = 0 then ← − v i = ← − w j and nodes v i and v j should be merged). Note that in this case we cannot use the compact (2-bit) representation of B suggested in Section 4, however we can re-use the space of B[1, n] to store the LCS array so in this case the space for B [1, n] is not part of the working space. 6 Space-efficient construction of succinct de Bruijn graphs Using our merging algorithm it is straightforward to design a complete spaceefficient algorithm to construct succinct de Bruijn graphs.
Assume we are given a string collection C = s 1 , . . . , s d of total length N , and the desired order k, and the amount of available RAM M . First, we split C into smaller subcollections r i = s j , . . . , s j , such that we can compute the BWT and LCP array of each subcollection in linear time in RAM using M bytes, using e.g. the suffix sorting algorithm gSACA-K [13]. For each subcollection we then compute, and write to disk, the BOSS representation of its de Bruijn graph using the algorithm described in [6,Section 5.3]. Since these are linear algorithms the overall cost of this phase is O(N ) time and O(N ) sequential IOs.
Finally, we merge all de Bruijn graphs into a single BOSS representation of the union graph with the external memory variant of the merging algorithm (Theorem 1). Since the number of subcollections is O(N/M ), a total of log(N/M ) merging rounds will suffice to get the BOSS representation of the union graph. Note that our construction algorithm can be easily extended to generate the colored/variable order variants of the de Bruijn graph. For the colored variant it suffices to use gSACA-K to generate also the document array and then use the colored merging variant. For the variable order representation, it suffices to store the LCP/LCS values during the very last merging phase.
Fig. 2 .
2BOSS representation of the graph inFig. 1. The colored lines connect each label in W to its destination node; edges of the same color have the same label. Note that edges of the same color do not cross because of Property 1.
W − [i] = 1 and the the set {2, . . . , n}: in this correspondence LF (i) = j iff v j is the destination node of the edge associated to W [i]. See example in Figs. 1 and 2.
Property 1 .
1The LF map is order preserving in the following sense: if W − [i] = W − [j] = 1 then
Fig. 3 .
3Main procedure for merging succinct de Bruijn graphs.Following Property 2, and a technique introduced in [7], we identify the i-th 0 in Z (h) with ← − v i and the j-th 1 in Z (h) with ← − w j . Property 2 is equivalent to state that we can logically partition Z (h) into b(h) + 1 h-blocks
During iteration h, new block boundaries are established as follows. At line 9 we identify each existing block with its starting position. Then, at lines 17-22, if the entry Z (h) [q] has the form cα, while Z (h) [q − 1] has the form cβ, with α and β belonging to different blocks, then we know that q is the starting position of an h-block. Note that we write h to B[q] only if no other value has been previously written there. This ensures that B[q] is the smallest position in which the strings corresponding to Z (h) [q − 1] and Z (h) [q] differ, or equivalently, B[q] − 1 is the LCP between the strings corresponding to Z (h) [q − 1] and Z (h) [q]. The above observations are summarized in the following Lemma, which is a generalization to de Bruijn graphs of an analogous result for BWT merging established in Corollary 4 in [7].Lemma 2. After iteration k of the merging algorithm for q = 2, . . . , n 0 + n 1 if B[q] = 0 then B[q] − 1 is the LCP between the reverse k-mers corresponding to Z (k) [q − 1] and Z (k) [q], while if B[q] = 0 their LCP is equal to k, hence such k-mers are equal.
[1, n] (Phase 1). Then, the algorithm executes k − 1 iterations of the code in Fig. 3 each iteration taking O(m) time. Finally, still in O(m) time the algorithm computes the succinct representation of the union graph (Phases 2 and 3). The overall running time is therefore O(m k).
[i] is equal to 0, h, or some value within 0 and h. Similarly, for the computation of W − 01 we only need to distinguish between the cases where B[i] is equal to 0, k or some value 0 < B[i] < k. Therefore, we can save space replacing B[1, n] with an array B 2 [1, n] containing two bits per entry representing the four possible states {0 , 1 , 2 , 3 }. During iteration h, the values in B 2 are used instead of the ones in B as follows: An entry B 2 [i] = 0 corresponds to B[i] = 0, an entry B 2 [i] = 3 corresponds to an entry 0 < B[i] < h − 1. In addition, if h is even, an entry B 2 [i] = 2 corresponds to B[i] = h and an entry B 2 [i] = 1 corresponds to B
Theorem 1 .
1The merging of two succinct representations of de Bruijn graphs can be done in O(m k) time and 4n bits plus O(σ) words of working space. The algorithm can be executed in external memory using O(σ) words of RAM and O(m k) sequential IOs.
Theorem 2 .
2The merging of two succinct representations of colored de Bruijn graphs takes O(m max(k, c 01 )) time and 4n bits plus O(σ) words of working space, where c 01 = c 0 + c 1 .
Theorem 3 .
3The merging of two succinct representations of variable order de Bruijn graphs takes O(mk) time and 2n bits plus O(σ) words of working space.
Theorem 4 .
4Given a collection of strings collection C = s 1 , . . . , s d of total length N , the construction of the succinct order k de Bruijn graph takes O(N k log(N/M )) time using O(M ) words of RAM.
1. W [1, m] as the concatenation W 1 W 2 · · · W n ; 2. W − [1, m] as the bitvector such that W − [i] = 1 iff W [i] corresponds to the label of the edge (u, v) such that ← − u has the smallest rank among the nodes that have an edge going to node v; 3. last[1, m] as the bitvector such that last[i] = 1 iff i = m or the outgoing edges corresponding to W [i] and W [i + 1] have different source nodes.Fig. 1. de Bruijn graph for C = {TACACT, TACTCA, GACTCA}.$$$
$$T
$TA
$$G
$GA
ACA
CAC
TCG
TAC
ACT
CTC
GAC
TCA
$
$
T
A
T
C
C
A
G
T
T
G
A
C
A
C
we write the symbol W 01 [i] we simultaneously write the bit W − 01 [i] according to the following strategy. If the symbol c = W 01 [i] is the first occurrence of c after a value B[q], with 0 < B[q] < k, then we set W − 01 [i] = 1, otherwise we set W − 01 [i] = 0. The rationale is that if no values B[q]with 0 < B[q] < k occur between two nodes, then the associated (reversed) k-mers have a common LCP of length k − 1 and therefore if they both have an outgoing edge labelled with c they reach the same node and only the first one should have W − 01 [i] = 1.
AcknowledgmentsFunding. L.E. was partially supported by the University of Eastern Piedmont project Behavioural Types for Dependability Analysis with Bayesian Networks.A Proof of Lemma 1To prove the "if" part, let 1 ≤ f < g ≤ n 0 + n 1 denote two indexes such that Z (h) [f ] is the i-th 0 and Z (h) [g] is the j-th 1 in Z (h) for some 1 ≤ i ≤ n 0 and 1 ≤ j ≤ n 1 . We need to show that, since otherwise during iteration h the j-th 1 would have been written in a subarray of Z (h) preceding the one where the i-th 0 is written.In this case during iteration h the i-th 0 and the j-th 1 are both written to the subarray of Z (h) associated to symbol c. Let f , g denote respectively the value of the main loop variable p in the procedure ofFig. 3when the entriesBy construction there is an edge labeled c from v i to v i and from w j to w j henceh] for some i ≥ 1 and j ≥ 1. We need to prove that in Z (h) the i-th 0 precedes the j-th 1.Let i and j be such that[2, h]. By induction hypothesis, in Z (h−1) the i -th 0 precedes the j -th 1.During phase h, the i-th 0 in Z (h) is written to position f when processing the i -th 0 of Z (h−1) , and the j-th 1 in Z (h) is written to position g when processing the j -th 1 of Z (h−1) . Since in Z (h−1) the i -th 0 precedes the j -th 1 and since f and g both belong to the subarray of Z (h) corresponding to the symbol c, their relative order does not change and the i-th 0 precedes the j-th 1 as claimed.
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| []
|
[
"Safety Verification and Controller Synthesis for Systems with Input Constraints",
"Safety Verification and Controller Synthesis for Systems with Input Constraints"
]
| [
"Han Wang ",
"Kostas Margellos ",
"Antonis Papachristodoulou "
]
| []
| []
| In this paper we consider the safety verification and safe controller synthesis problems for nonlinear control systems. The Control Barrier Certificates (CBC) approach is proposed as an extension to the Barrier certificates approach. Our approach can be used to characterize the control invariance of a given set in terms of safety of a general nonlinear control system subject to input constraints. From the point of view of controller design, the proposed method provides an approach to synthesize a safe control law that guarantees that the trajectories of the system starting from a given initial set do not enter an unsafe set. Unlike the related control Barrier functions approach, our formulation only considers the vector field within the tangent cone of the zero level set defined by the certificates, and is shown to be less conservative by means of numerical evidence. For polynomial systems with semi-algebraic initial and safe sets, CBCs and safe control laws can be synthesized using sum-of-squares decomposition and semi-definite programming. Examples demonstrate our method. | 10.48550/arxiv.2204.09386 | [
"https://arxiv.org/pdf/2204.09386v2.pdf"
]
| 248,266,526 | 2204.09386 | bebb5927b10a9fab4a00f9ce4c7ac6e2ebed41ce |
Safety Verification and Controller Synthesis for Systems with Input Constraints
Han Wang
Kostas Margellos
Antonis Papachristodoulou
Safety Verification and Controller Synthesis for Systems with Input Constraints
arXiv:2204.09386v2 [math.OC] 21 Apr 2022
In this paper we consider the safety verification and safe controller synthesis problems for nonlinear control systems. The Control Barrier Certificates (CBC) approach is proposed as an extension to the Barrier certificates approach. Our approach can be used to characterize the control invariance of a given set in terms of safety of a general nonlinear control system subject to input constraints. From the point of view of controller design, the proposed method provides an approach to synthesize a safe control law that guarantees that the trajectories of the system starting from a given initial set do not enter an unsafe set. Unlike the related control Barrier functions approach, our formulation only considers the vector field within the tangent cone of the zero level set defined by the certificates, and is shown to be less conservative by means of numerical evidence. For polynomial systems with semi-algebraic initial and safe sets, CBCs and safe control laws can be synthesized using sum-of-squares decomposition and semi-definite programming. Examples demonstrate our method.
I. INTRODUCTION
Safety-critical systems are commonly used in modern autonomous applications, such as unmanned aerial vehicles, autonomous driving and surgical robotics [1]. Their safetycritical nature requires the behaviour of these systems to remain within a given safe set for an infinite time horizon. For a model of these systems, such a property is straightforwardly related to reachability analysis and reach-avoid games [2], [3], i.e. finding an initial set so that trajectories reach a target set without entering an unsafe region. However, verify safety for general nonlinear systems using these methods is hard due to the computational difficulty of solving the underlying Hamilton Jacobi Isaacs PDE, especially when control actuation constraints are considered.
To overcome this issue, the connection between forward invariance and safety was established in [4]. Forward invariance is a system-set property which guarantees that the trajectories entering a set cannot escape it [5]. By finding an invariant subset of a safe region, the system is ensured to be safe. To identify a candidate invariant set, the Barrier certificates approach which takes the invariant set as the certificate's sub-zero level set, was proposed in [6], [7]. Although the properties of this framework have been demonstrated for autonomous systems with and without stochasticity, there is no systematic formulation for the case where control inputs are present. To address this issue, the control Barrier functions (CBF) approach was proposed in [8]. The Control Barrier functions are a class of functions that are negative in the unsafe regions, and can be used to verify the safety property. Unlike Lyapunov-like Barrier certificates, control Barrier functions are less restrictive by introducing an additional relaxation term in the constraint. Forward invariance is proved by satisfying the constraints and utilizing the comparison lemma [9]. The approach can be easily combined with the control Lyapunov functions approach [10] under a unified quadratic programming framework that compromises safety and controller performance [8]. It was also shown to be applicable and promising in many applications such as adaptive cruise control [11], bipedal robots [12], multi-robot collision avoidance [13] and others.
Later on, extensive methods to improve the feasibility when input limits are taken into account were proposed, such as adaptive CBF [14], [15], higher relative degree CBF [16], backup CBF [17], singular CBF [18]. These methods aim at addressing the cases where the CBF based QP is infeasible. Many times a CBF is assumed to be constructed directly from a physical property such as kinodynamics of the vehicle. How to synthesize a CBF efficiently is still an open question, and has attracted significant attention in recent years.
Direct numerical synthesis approaches by sum-of-squares programming [19], [11], machine learning [20], and deep learning [21] have been proposed. All these methods, either via convex optimisation, or learning techniques, consider the synthesis of a CBF with a relaxation term included in the synthesis procedure. From the standpoint of control invariant sets, it is guaranteed that there exists a class-K relaxation term to bound the safety variation, but imposing such a term at every point inside the set during the control synthesis introduces conservativeness. Abandoned this term during the synthesis process has been considered in [22], and using Positivstellensatz, a weaker condition on invariance is imposed for systems without input limits.
In this work we revisit the Barrier certificates approach, and extend it for nonlinear control systems with actuation constraints. Our formulation is a direct interpretation of control invariance and safety guarantee, thus alleviating conservativeness. The existence of a CBC is proved to be sufficient to guarantee safety, hence the approach can be used for safety verification. For systems with polynomial dynamics and semi-algebraic safe and initial sets, we use sumof-squares programming and the generalised S-procedure to synthesize a CBC, as well as a Lipschitz continuous safe control law which fulfills the actuation constraints.
The remainder of this paper is organized as follows.
The notion of control Barrier certificates is introduced in Section II. The computation methods with sum-of-squares programming and the S-procedure is presented in Section III. Several simulation results on synthesizing CBCs and safe controllers are shown in Section IV. Section V concludes the paper.
II. CONTROL BARRIER CERTIFICATES
In this section, we consider the controller synthesis problem under the promise of safety for nonlinear systems. Existing work on Barrier certificates synthesis either limits the analysis to noisy autonomous systems, or tries to design a control law in an online quadratic programming framework. There is no work focusing on combining Barrier certificates construction with controller design, which only requires safety on the boundary of the invariant set. Here, we extend the results of Barrier certificates to control Barrier certificates for safety verification and safe controller design. We also compare our results with the CBF approach.
Notation: R represents the space of real numbers, and R n denotes the n−dimensional real space. For a set S, IntS, ∂S,S are the interior, boundary and complementary set, respectively. A 0 means matrix A is positive semi-definite. Σ[x] and R[x] denote the set of sum-of-squares polynomials and polynomials in x with real coefficients.
A. Control Barrier Certificates Formulation
We start the formulation for a continuous-time nonlinear system for generality. The system is described by an ordinary differential equation:ẋ
= f (x, u),(1)
where x(t) ∈ R n denotes the state vector, and u(t) ∈ U ⊆ R m is the control input, where U is a bounded set denoting actuation limits and f (·, ·) is a locally Lipschitz continuous vector field. We assume that the solution to (1) is unique. The flow ψ(x, t, u) denotes the solution of (1) at time t from initial condition x under control u. The definitions of a reachable set, forward invariance and safety can be extended to the control system setting.
Definition 1 (Control Reachable Set). Consider a vector field f (·, ·), a set X ⊆ R n and time horizon T ∈ R. Then the control reachable set of X with respect to vector field f (·, ·), control law u and time horizon T is R T f,u (X) := {ψ(x, t, u)|x ∈ X, t ∈ T, u ∈ U}.
We note here that although the vector field f (·, ·) in (1) already includes the control input u, we still denote the input explicitly in the subscript to distinguish this from the reachable set R T f (X) for the autonomous systemẋ = f (x). Definition 2 (Control Invariant Set). A set X is said to be control invariant with respect to vector field f (·, ·) if there exits u, such that R ∞ f,u (X) ⊆ X. If u = 0, we call the set X positive invariant. The difference between positive invariance and control invariance is obvious: the control effort allows guaranteeing that the flow stays in the set. Hence, the safety of the control system not only depends on the vector field and the predefined safe set S, but also on the control admissible set U.
Definition 3 (Safety). Given system (1), an initial set I and a safe set S, we say that the system is safe if there exits u ∈ U such that R ∞ f,u (I) ∩S = ∅. The definition of safety of a controlled system is similar to that of an autonomous system. To incorporate safety for the nonlinear control system (1), we aim at finding controller u and a control invariant set W , which includes I in its interior and is a subset of the safe set S. In particular, W and u fulfill: We now define the notion of Control Barrier Certificates (CBC) for finding a feasible candidate control invariant set, and a controller according to condition (2). Proof. Equation (3a) indicates that for any x ∈S, we have x ∈B, thus B ⊆ S, which shows that condition (2c) holds. Similarly, Equation (3b) demonstrates that I ∈ B. Therefore, we only need to prove that R ∞ f,u (B) ∈ B to show conditions (2a) -(2b). We recall that under control input u, the vector field f (x, u) is locally Lipchitz continuous, and the solution is unique. This indicates that the flow ψ(x, t, u) is continuous over t. Besides, the fact that B(x) is a C 1 function guarantees that trajectories starting from Int(B) toB will cross ∂B. Thus, any bounded input u ∈ U at x ∈ Int(B) = {x|B(x) > 0} shows the positivity of B(ψ(x, t, u)) when t → 0. Regarding the boundary, for
R ∞ f,u (W ) ⊆ W, (2a) R ∞ f,u (I) ⊆ W, (2b) W ⊆ S.(2c
Definition 4 (Control Barrier Certificates). Let a continuous time control system denoted byẋ
= f (x, u), with initial set I ⊆ R n , safe set S ⊆ R n , and input constraints U ⊆ R m . A C 1 function B : R n → R is called a Control Barrier Certificate (CBC) if B(x) < 0, ∀x ∈S, (3a) B(x) ≥ 0, ∀x ∈ I, (3b) sup u∈U ∂B(x) ∂x f (x, u) > 0, ∀x ∈ ∂B. (3c) Let K CBC (x) := u| ∂B(x) ∂x f (x, u) > 0 ∩ U, if B(x) = 0 U, otherwise(4)any x ∈ ∂B,Ḃ(x) = ∂B(x) ∂x dx dt = ∂B(x)
∂x f (x, u) > 0 from the definition of CBC. Thus, the vector field f (x, u) ∈ Tang B (x) for x ∈ ∂B and u ∈ U. According to the subtangenality Theorem [23], the set B is control invariant with vector field f (x, u), which directly indicates that B is control invariant. According to Lemma 1, B is a feasible candidate control invariant set verifying the safety of the control system (1).
Theorem 1 shows that the existence of a control Barrier certificate B(x) ensures safety for safety-critical systems. Meanwhile, the control admissible set (4) certifies the selection of control effort. For problems that the control Barrier certificate B(x) can be easily constructed and verified through physical properties, one can formulate a quadratic program to synthesize the safe controller u at x.
min u ||u − u * (x)|| s.t. u ∈ K CBC (x),(5)
where u * (x) is a nominal control input designed from other tools, e.g. PID, MPC. We note here the formulation (5) is different from that of CBF based QP. Here, in the interior of the control invariant set B, the solution of (5) is the direct projection from u * (x) on the control admissible set U.
For the scenario where the control Barrier certificate is unknown, the problem is to synthesise the control Barrier certificates together with the safe controller design. To begin with, B(x) is parameterized by
B(x) = p 1 Λ 1 (x) + . . . + p k Λ k (x),(6)
where p := {p 1 , . . . , p k } denotes a series of parameters which will be decision variables in an optimisation problem, and Λ 1 (x), . . . , Λ k (x) are a class of function basis. The new optimisation problem for constructing the CBC and controller is
find u(x), p s.t. (6), (3); u(x) ∈ K CBC (x).(7)
Compared to quadratic programming (5) with known control Barrier certificates, (7) is computationally intractable since it involves solving an infinitely constrained optimisation problem. We will show how to address this difficulty by sum-of-squares programming in Section III.
III. COMPUTATION METHOD
In this section we show how to construct the control Barrier certificates and the safe control law for polynomial systems with semi-algebraic safe and initial sets. The nonlinear control affine system is represented bẏ
x = f (x) + g(x)u,(8)
where f (x) and g(x) are locally smooth polynomial functions and u ∈ U := {u|Au + b ≥ 0}. Even for such a simplified system model, solving the parametric optimisation problem (6) -(7) involves solving an infinite set of non-negative inequalities and hence is computationally intractable. However, for systems with polynomial functions f (x), g(x) and semi-algebraic sets I, S, a tractable method for tackling the infinite inequalities is sum-of-squares (SOS) programming, which is a convex relaxation method based on the sum-of-squares decomposition of multivariate polynomials and semidefinite programming.
A SOS program is a convex optimisation problem of the following form:
min p k j=1 w j p j s.t. h 0 (x) + k j=1 p j h j (x) ∈ Σ[x],(9)
where the decision variables p 1 , . . . , p k are real parameters, and w 1 , . . . , w k are predefined weight constants. Also,
[h 0 (x), . . . , h k (x)] is a polynomial basis in x. A multivariate polynomial s.t. h 0 (x) + k j=1 p j h j (x) with x ∈ R n is a SOS polynomial if there exists k polynomials f 1 (x) . . . f k (x) such that f (x) = k i=1 f 2 i (x). Then it directly follows that a SOS f (x) is non-negative for any x ∈ R n . A SOS program can be transformed into a semi-definite program with f (x) = Z ⊤ QZ(x), where Q 0 and Z(x) is a monomial vector.
A. SOS for CBC Synthesis
To interpret the constraints (3) into SOS constraints, we assume that the resulting control Barrier certificate is a polynomial function parameterized by real coefficients p 1 , . . . , p m in the following way
B(x) = p 0 + m j=1 p j b j (x),(10)
where b j (x)s are polynomial or monomial function bases, and p 0 is a positive real scalar. Similarly, the control input is parameterized by real scalar coefficients k 1 , . . . , k l , and a real vector coefficient
k 0 ∈ R m with u(x) = k 0 + l j=1 k j v j (x),(11)
where v j (x)s are polynomial or monomial vector bases. We note here there the reason why we use the constant term k 0 is different from that of p 0 . From the view of control, k 0 introduces a feedforward term, which in some cases is important for safety, for example at some singular points where l j=1 k j v j (x) = 0.
σ safe ∈ Σ[x], σ init ∈ Σ[x], λ 1 ∈ R[x], λ 2 ∈ R[x] h , polynomials B(x) ∈ R[x], u(x) ∈ R[x], predefined small positive real scalars ǫ 1 > 0, ǫ 2 > 0, such that − B(x) + σ safe s(x) − ǫ 1 ∈ Σ[x], (12a) B(x) − σ init w(x) ∈ Σ[x], (12b) ∂B(x) ∂x (f (x) + g(x)u(x)) + λ 1 B(x) − ǫ 2 ∈ Σ[x], (12c) − λ 2 B(x) + Au(x) + b ∈ Σ[x] h ,(12d)
then B(x) fulfills the conditions (3) and B = {x|B(x) ≥ 0} is a control invariant set with respect to vector field f (x) + g(x)u(x).
Proof. Condition (12a) indicates that for any x, −B(x) + σ safe s(x) − ǫ 1 ≥ 0, thus for any x, −B(x) + σ safe s(x) > 0. Therefore, for any x ∈S, we directly have that σ safe s(x) ≤ 0, and further B(x) < 0, i.e., (3a) holds. Similarly (12b) can be shown to satisfy (3b) following the same arguments. Based on the S-procedure, condition (12c) implies condition (3c), because when
B(x) = 0, ∂B(x) ∂x (f (x) + g(x)u(x)) − ǫ 2 ≥ 0, and thus B(x) ∂x (f (x) + g(x)u(x)) > 0. Condition (12d) implies that Au(x) + b is elementary-wise nonnegative for x ∈ ∂B.
The small positive real scalars ǫ 1 , ǫ 2 ensure strict inequality for (3a) and (3c).
We note that in Theorem 2 we only require a polynomial multiplier λ, but not a SOS one since the condition
∂B(x) ∂x (f (x) + g(x)u(x)
) ≥ 0 is only imposed on the boundary B(x) = 0. Condition (12c) introduces products of decision variables, i.e. λB(x), which results in bilinearity. However, there is no guaranteed solver for nonconvex, or specifically bilinear constrained SOS programs. Here, like existing work of using SOS to synthesize Barrier certificates, we use an iterative procedure for control Barrier certificate synthesis and safe control law design. Different from the iterative algorithm for Barrier certificate synthesis, our problem involves an additional polynomial variable u in the SOS program. Thus, an additional round for controller synthesis is required in our algorithm. 1) Initialization: We first fix the degree of polynomials B(x), σ safe , σ init , λ 1 , λ 2 and u(x). The polynomial/monomial scalar/vector bases b j (x)s and v j (x)s in (10) and (11) have degree upper bounded by the aforementioned degrees of B(x). ǫ 1 and ǫ 2 are chosen to be small real numbers. Unlike the iterative procedure proposed in [11] which initializes the control law by a scaled LQR controller, we find the initialized feasible control input u 0 (x) by solving a feasibility SOS program.
find k 1 , . . . , k l , σ cont
s.t. A(k 0 + l j=1 k j v j (x)) + b · σ cont ∈ Σ[x] h .(13)
We note here that there is no assumed control Barrier certificate B(x) at this stage of finding the initial feasible control input u 0 (x). Therefore, u 0 (x) can not be restricted to the domain of ∂B as that in (12d). Other than directly interpreting A(k 0 + l j=1 k j v j ) + b ∈ Σ[x] h , we add an additional positive multiplier σ cont which satisfies σ cont − ǫ 3 ∈ Σ[x], ǫ 3 > 0 to avoid introducing constant terms in the SOS constraints, as well as improving feasibility. The resulting initial controller u 0 (x) is derived by the parameters k 1 , . . . , k l and the scaled term σ cont from the solution of (13)
u 0 (x) = 1 σ cont · (k 0 + l j=1 k j v j (x)).(14)
The feasibility of such an initialized controller is guaranteed by the following proposition. (13). Because of the positivity of the multiplier σ cont , we
Proposition 1. The initialized control input u 0 (x) in (14) satisfies Au 0 (x) + b ≥ 0. Proof. We have A(k 0 + l j=1 k j v j ) + b · σ cont ≥ 0 from the SOS constraints A(k 0 + l j=1 k j v j ) + b · σ cont ∈ Σ[x] h indirectly have A( 1 σcont · (k 0 + l j=1 k j v j )) + b ≥ 0, which indicates u 0 (x) ∈ U.
Given initial input u 0 (x), the corresponding scaled multiplier σ cont , the initial control Barrier certificate B 0 (x) can be found by solving an initial feasibility SOS program as
find p 0 , . . . , p m , σ safe , σ init s.t. − B(x) + σ safe s(x) − ǫ 1 ∈ Σ[x], B(x) − σ init w(x) ∈ Σ[x], σ cont · ∂B(x) ∂x (f (x) + g(x)u 0 (x)) − ǫ 2 ∈ Σ[x], B(x) from (10).(15)
The boundary condition (12c) is strengthened to be
∂B(x) ∂x (f (x) + g(x)u(x)) − ǫ 2 ∈ Σ[x]
for convexity and simplicity of computing. This condition is also referred to be the weak Barrier certificate in [7]. σ cont · ∂B(x) ∂x (f (x) + g(x)u 0 (x)) − ǫ 2 is guaranteed to be a polynomial, since σ cont · u 0 (x) is a polynomial.
After obtaining a feasible initial control input u 0 (x) and control Barrier certificate B 0 (x), the problem of control Barrier certificates synthesis can be regarded as a Barrier certificates synthesis problem with vector field f (x) + g(x)u 0 (x). The multipliers λ 0 1 , λ 0 2 are fixed to be 0 or 1 in initialization for simplicity. The initial control Barrier certificate B 0 (x) is used to enlarge the size of the control invariant set incrementally. The following steps of the algorithm iteratively solve the SOS program to address the bisecting terms λ 1 B(x) and ∂B(x) ∂x (f (x) + g(x)u(x)) in (12c). 2) Update the control input u k (x): At iteration k, given a control Barrier certificate from (15) (when k = 1) or (17) (when k ≥ 2), the controller synthesis is constrained to (12d). Fixing B(x) = B k−1 (x), a convex programming synthesis procedure for u k (x) is
find k 0 , . . . , k l , λ 1 , λ 2 s.t. − λ 2 B k−1 (x) + A(k 0 + l j=1 k j v j ) + b ∈ Σ[x] h , ∂B k−1 (x) ∂x (f (x) + g(x)u(x)) + λ 1 B k−1 (x) − ǫ 2 ∈ Σ[x],(16)
and we have that u k (x) = (k 0 + l j=1 k j v j ). Here we use λ 1 other than λ k−1 1 since B(x) has been substituted by B k (x), thus there is no bilinear term anymore. By limiting the domain of the controller to ∂B, there is no need to have additional multiplier σ cont as that has been used in initial controller design for feasibility.
3) Synthesize the control Barrier certificate B k (x): After obtaining a feasible control input u k−1 (x), the synthesis of a control Barrier certificate B k (x) relies on fixed multipliers λ k−1 1 , λ k−1 2 to bypass the bilinear terms. Searching for B k (x) and the remaining multipliers follows the following SOS program
find p 0 , . . . , p m , σ safe , σ init , σ enl s.t. − B(x) + σ safe s(x) − ǫ 1 ∈ Σ[x], B(x) − σ init w(x) ∈ Σ[x], ∂B(x) ∂x (f (x) + g(x)u k (x)) + λ k−1 1 B(x) − ǫ 2 ∈ Σ[x], − λ k−1 2 B(x) + Au k (x) + b ∈ Σ[x] h , B(x) − σ enl B k−1 (x) ∈ Σ[x], B(x) in (10),(17)
where σ enl ∈ Σ[x]. Here the control law u k−1 (x) is substituted for the variable u, and the multipliers λ 1 λ 2 are substituted by λ k−1 1 and λ k−1 2 , respectively. We introduce additional constraints B(x)−σ enl B k−1 (x) ∈ Σ[x] to enlarge the volume of the control invariant set B k by enforcing B k−1 ⊆ B k . A similar technique is also used in [24]. 4) Update the multipliers: The multiplier λ k 1 updates rely on a fixed control Barrier certificate B k (x) and input u k (x). Clearly, there is no bilinearity in the control input update procedure (16). The multipliers λ k 1 and λ k 2 are obtained by directly solving it. There is no need to fix B(x) and re-solve the programming problem.
Remark. For the case where (13)
IV. SIMULATION RESULTS AND DISCUSSION
In this section we show numerical simulation results on synthesizing control Barrier certificates and safe controllers under different system settings. The SOS toolbox SOS-TOOLS [25] [26] is used with version v401 for parsing the SOS programs, while SeDuMi is used for solving the resulting semidefintie program [27]. We also give a comparison between the CBC proposed in this paper and CBF mainly from the view point of synthesis.
A. Nonlinear Control Affine Systems
We first consider a general second order polynomial nonlinear control affine system. This system is defined by
ẋ 1 x 2 = x 2 x 1 + 1 3 x 3 1 + x 2 + x 2 1 + x 2 + 1 x 2 2 + x 1 + 1 u 1 u 2 ,(18)
where the control input is box constrained, i.e. u 1 ∈ [−1.5, 1.5], u 2 ∈ [−1.5, 1.5]. The safe set is defined by a disc S = {x|x 2 1 + x 2 2 − 3 ≤ 0}, and initial set defined by
I = {x|(x 1 − 0.4) 2 + (x 2 − 0.4) 2 − 0.16 ≤ 0}.
We leverage the control Barrier certificates synthesis procedures (12) to find a polynomial CBC B 1 (x), and compare the results with the CBF synthesis procedure proposed in [11]. To synthesize a candidate CBF B 2 (x), an alternative constraint for (12c) is introduced
∂B(x) ∂x (f (x) + g(x)u(x)) − σ cbf B(x) + αB(x) − ǫ 2 ∈ Σ[x],(19)
where the class-K function is selected to be αB(x) with α > 0, and σ cbf ∈ Σ[x] is a SOS multiplier. Here we restrict the definition domain for CBF to be B 2 . ǫ 2 is set to be the same with that in (12c). Instead of the feasibility SOS program used for CBC, we set an objective function α which is maximized for CBF as in [11]. Figure 1 shows the control invariant sets defined by CBF and CBC. The red and light blue disc represent the safe and initial sets, respectively. The interior of the deep blue curve is the invariant set B 2 defined by CBF, and the interior of the black curve is the invariant set B 1 defined by CBC. It can be seen from the figure that B 1 is "larger" than B 2 . Actually we have B 2 ⊂ B 1 , which is proved by there exists a SOS multiplier σ, such that B 1 (x) − σB 2 (x) ∈ Σ[x]. The reason is that, we trivially have σ cbf + α ∈ R[x]. A larger search area enables us to find a larger control invariant set. On the other hand, the additional term λ 1 B 1 (x) can be regarded as an adapted relaxation term compared with a fixed class-K function used in CBF approach. By using a zeroth order base for the polynomial multiplier λ 1 and expanding the definition domain of CBF to the whole real space, our formulation is equivalent to CBF. Higher order basis selections hereby reduce conservativeness. Figure 2 shows the value of relaxation coefficient λ 1 and α. The multiplier λ 1 includes the following monomial basis:
[x 2 1 , x 1 x 2 , x 2 2 , x 1 , x 2 , 1].
It can be seen that λ 1 varies in the control invariant set, which therefore endows the formulation flexibility. An interesting property here is that α cannot be too large, this is because for x ∈B 1 , αB 2 (x) < 0. In addition, with a non-empty safe set S ⊂ R n , we directly have
B 1 (x) / ∈ Σ[x], and αB 1 (x) / ∈ Σ[x].
B. LTI Systems
Consider a second order linear model
ẋ 1 x 2 = 2 1 3 1 x 1 x 2 + u 1 u 2 ,(20)
where u 1 ∈ [−2.5, 2.5], u 2 ∈ [−2.5, 2.5]. The system is unstable since the eigenvalues of the state matrix 2 1 3 1 are 3.3 and −0.3, whereas it is locally stabilizable. The safe set is defined by a disc S = {x|x 2 1 + x 2 2 − 3 ≤ 0}. The trajectories of the system start from the following initial set
I = {x|(x 1 − 0.4) 2 + (x 2 − 0.4) 2 − 0.16 ≤ 0}.
Clearly, all trajectories starting from the initial set tend to infinity, since the system is unstable. Safety is therefore violated with a closed safe region set.
Using a second degree basis [1, x 1 , x 2 , x 1 x 2 , x 2 1 , x 2 2 ], a feasible candidate CBC is given by B 1 (x) = −7.635x 2 1 − 3.439x 1 x 2 − 3.4024x 2 2 + 0.5x 1 − 0.4x 2 + 7.402. The corresponding control inputs lying inside [−2.5, 2.5] when x ∈ ∂B 1 are u 1 (x) = −2.32x 1 − 1.11x 2 + 0.022, u 2 (x) = −2.12x 1 − 1.27x 2 − 0.046. Obviously u 1 (x) and u 2 (x) are admissible only within some local regions.
More specifically, within B 1 . We can show the boundary
condition ∂B1(x) ∂x (f (x) + g(x)u(x)) + λ 1 B 1 (x) − ǫ 2 ≥ 0 holds by exploiting the SOS decomposition ∂B1(x) ∂x (f (x) + g(x)u(x)) + λ 1 B 1 (x) − ǫ 2 = Z(x) ⊤ QZ(x), where Z(x) = [1, x 1 , x 2 , x 1 x 2 , x 2
1 , x 2 2 ] ⊤ and Q 0. Figure 4(a) shows the zero level set of the quadratic CBC B 1 (x). With controller u 1 (x) and u 2 (x), vector field in (20) guarantees safety with avoiding the unsafe set. For this case, the system admits an ellipsoidal control invariant set. The level sets of u 1 (x) and u 2 (x) are shown in Figure 4(b)-4(c). It can be seen that u(x) ∈ U for any x ∈ B 1 .
C. Comparison with Control Barrier Functions
We end this section by a brief comparison between CBF and CBC.
From the point of view of set invariance, the zero-super level set of both CBC and CBF are control invariant. CBC, which is a direct interpretation of control invariance to ensure safety, takes initial conditions into consideration as wellwithout initial conditions, the CBC formulation is equivalent to CBF. Although the definition of CBF involves the existence of a class-K function, this, however is a straightforward property that holds for both CBC and CBF.
From the aspect of controller design, the CBF-QP approach relies on a given safe control invariant set, which is free for our approach (7). For the case where the control invariant set is constructed a priori, although the CBF approach endows Lipschitz continuity for the resulting controller, it also introduces unnecessary conservativeness sinceḂ 2 (x) is bounded by a fixed additional relaxation term. Although there are existing works propose to tune the relaxation coefficient α online [14], additional computational complexity and necessary cost trade-off are also introduced. Our approach (5), on the other hand, is less restricted with an adapted relaxation coefficient λ 1 . For systems with mode switching such as power systems, formulation (5) ensures safety. For continuous controller synthesis, we can also formulate a QP with using λ 1 B 1 (x) as a relaxation term
min u∈U ||u − u * (x)|| s.t. ∂B 1 (x) ∂x (f (x) + g(x)u) + λ 1 B 1 (x) ≥ 0,(21)
we recall here λ 1 is a polynomial of x, the argument is dropped for simplicity.
V. CONCLUSION
In this paper we investigate the problem of safety verification and controller design for safety critical systems. Our approach depends on the evaluation of a control invariant set which encloses the initial set whereas avoiding the unsafe set. We prove that the existence of a control invariant set inside the safe region is sufficient for safety of nonlinear control systems. The formulation only imposes boundary conditions, thus alleviating conservatism. For polynomial systems with semi-algebraic initial and safe sets, we propose an iterative procedure with using SOS program to synthesize the CBC with encoding general affine control limits. We also show (c) Level set of u 2 Fig. 3. The interior of the red disc represents the safe set, the interior of the blue disc represents the initial set from which the trajectories start. The black closed curve encircling the initial set is the control invariant set, defined by the super-zero level set of B 1 (x). The arrows in the figure represent the vector field. The colorful lines are the trajectories starting from ∂B 1 . (c) Level set of u 2 Fig. 4. The safe and initial set are defined to be the same as in Figure 3. Safety is ensured with the polynomial control law.
that CBC has less conservativeness compared with CBF from numerical simulations. In the future we aim at extending the formulation to discrete time systems.
denote the admissible set of control inputs for a CBC B(x). Let B := {x|B(x) ≥ 0} denote the zero-super level set of B(x). We then have the following result on safety.
Theorem 1 .
1Consider (1), a safe set S and an initial set I. If there exists a CBC B(x) that satisfies conditions (3), then for any state x and any u ∈ K CBC (x), the safety of system (1) is guaranteed.
Theorem 2 .
2Consider a polynomial nonlinear system (8), semi-algebraic safe set S = {x|s(x) ≥ 0}, initial set I = {x|w(x) ≥ 0}, and control admissible set U := {u|Au+b ≥ 0}, where A ∈ R h×m , and b ∈ R h . If there exit multipliers
or (15) is infeasible, there are two options for ensuring feasibility: (i) Increase the degree of the polynomial bases v 1 , . . . , v l , b 1 , . . . , b m ; (ii) Re-solve the problem (13) with an alternative objective function for a different initialization.
Fig. 1 .
1Control invariant sets defined by CBC or CBF
Fig. 2 .
2Relaxation coefficients λ(x) for CBC, and α for CBF The control invariant set B 1 obtained by CBC design and values of the safe controllers are shown inFigure 3. The vector field, which is represented by the arrows inFigure 3(a) point inside B 1 on ∂B 1 . The value of the polynomial control law u ( x) is within [−1.5, 1.5] in both coordinates.
authors are with the Department of Engineering Science, University of Oxford, Oxford, United Kingdom. E-mails: {han.wang, kostas.margellos, antonis}@eng.ox.ac.uk
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| Non-local carrier injection/detection schemes lie at the very foundation of information manipulation in integrated systems. This paradigm consists in controlling with an external signal the channel where charge carriers flow between a "source" and a well separated "drain". The next generation electronics may operate on the spin of carriers instead of their charge [1, 2] and germanium appears as the best hosting material to develop such a platform for its compatibility with mainstream silicon technology and the long electron spin lifetime at room temperature[3]. Moreover, the energy proximity between the direct and indirect bandgaps [4] allows for optical spin injection and detection within the telecommunication window[5]. In this letter, we demonstrate injection of pure spin currents (i.e. with no associated transport of electric charges) in germanium, combined with non-local spin detection blocks at room temperature.Spin injection is performed either electrically through a magnetic tunnel junction (MTJ) or optically, exploiting the ability of lithographed nanostructures to manipulate the distribution of circularly-polarized light in the semiconductor.Pure spin current detection is achieved using either a MTJ or the inverse spin-Hall effect (ISHE) across a platinum stripe. These results broaden the palette of tools available for the realization of opto-spintronic devices.Spintronics aims at exploiting the spin degree of freedom to manipulate information, while in conventional electronics, information is associated with the charge of carriers[6]. In this regard, n-type germanium appears as the best hosting material for spin transport and manipulation. The electron spin lifetime can reach several nanoseconds at room temperature [3] and the compatibility with mainstream silicon technology allows exploiting the spinrelated properties of low dimensional SiGe-heterostructures[7]. Electrical spin injection and detection has been explored in Ge films or nanowires using either non-local measurements in lateral or vertical spin valves[8][9][10][11]or the Hanle effect in three-terminal devices[12][13][14][15][16][17][18][19].So far, the non-local lateral geometry is the most interesting one for the development of spintronics since the spin can be manipulated in the Ge channel between the spin injector and detector. However, experimental measurements have been limited in temperature to 225 K [8] and the only demonstration at room temperature used an indirect method based on the combination of spin pumping and inverse spin Hall effect (ISHE) [20].Another important feature of bulk Ge is related to the energy proximity between its di- | 10.1063/1.5003244 | [
"https://arxiv.org/pdf/1612.09136v1.pdf"
]
| 118,973,928 | 1612.09136 | f8fc00f9066685a540d37152d081ee9f58346e4b |
Non-local opto-electrical spin injection and detection in germanium at room temperature
29 Dec 2016
F Rortais
Institut Nanosciences et Cryogénie
Univ. Grenoble Alpes
Spintec
CEA
CNRS
F-38000GrenobleFrance
C Zucchetti
Dipartimento di Fisica
LNESS
Politecnico di Milano
20133MilanoItaly
L Ghirardini
Dipartimento di Fisica
LNESS
Politecnico di Milano
20133MilanoItaly
A Ferrari
Institut Nanosciences et Cryogénie
Univ. Grenoble Alpes
Spintec
CEA
CNRS
F-38000GrenobleFrance
C Vergnaud
Institut Nanosciences et Cryogénie
Univ. Grenoble Alpes
Spintec
CEA
CNRS
F-38000GrenobleFrance
J Widiez
Univ. Grenoble Alpes
F-38000GrenobleFrance
CEA
LETI
MINATEC Campus
F-38054GrenobleFrance
A Marty
Institut Nanosciences et Cryogénie
Univ. Grenoble Alpes
Spintec
CEA
CNRS
F-38000GrenobleFrance
J.-P Attané
Institut Nanosciences et Cryogénie
Univ. Grenoble Alpes
Spintec
CEA
CNRS
F-38000GrenobleFrance
H Jaffrès
Unité Mixte de Physique
CNRS
Univ. Paris-Sud
Univ. Paris-Saclay
91767PalaiseauThalesFrance
J.-M George
Unité Mixte de Physique
CNRS
Univ. Paris-Sud
Univ. Paris-Saclay
91767PalaiseauThalesFrance
M Celebrano
Dipartimento di Fisica
LNESS
Politecnico di Milano
20133MilanoItaly
G Isella
Dipartimento di Fisica
LNESS
Politecnico di Milano
20133MilanoItaly
F Ciccacci
Dipartimento di Fisica
LNESS
Politecnico di Milano
20133MilanoItaly
M Finazzi
Dipartimento di Fisica
LNESS
Politecnico di Milano
20133MilanoItaly
F Bottegoni
Dipartimento di Fisica
LNESS
Politecnico di Milano
20133MilanoItaly
M Jamet
Institut Nanosciences et Cryogénie
Univ. Grenoble Alpes
Spintec
CEA
CNRS
F-38000GrenobleFrance
Non-local opto-electrical spin injection and detection in germanium at room temperature
29 Dec 2016
Non-local carrier injection/detection schemes lie at the very foundation of information manipulation in integrated systems. This paradigm consists in controlling with an external signal the channel where charge carriers flow between a "source" and a well separated "drain". The next generation electronics may operate on the spin of carriers instead of their charge [1, 2] and germanium appears as the best hosting material to develop such a platform for its compatibility with mainstream silicon technology and the long electron spin lifetime at room temperature[3]. Moreover, the energy proximity between the direct and indirect bandgaps [4] allows for optical spin injection and detection within the telecommunication window[5]. In this letter, we demonstrate injection of pure spin currents (i.e. with no associated transport of electric charges) in germanium, combined with non-local spin detection blocks at room temperature.Spin injection is performed either electrically through a magnetic tunnel junction (MTJ) or optically, exploiting the ability of lithographed nanostructures to manipulate the distribution of circularly-polarized light in the semiconductor.Pure spin current detection is achieved using either a MTJ or the inverse spin-Hall effect (ISHE) across a platinum stripe. These results broaden the palette of tools available for the realization of opto-spintronic devices.Spintronics aims at exploiting the spin degree of freedom to manipulate information, while in conventional electronics, information is associated with the charge of carriers[6]. In this regard, n-type germanium appears as the best hosting material for spin transport and manipulation. The electron spin lifetime can reach several nanoseconds at room temperature [3] and the compatibility with mainstream silicon technology allows exploiting the spinrelated properties of low dimensional SiGe-heterostructures[7]. Electrical spin injection and detection has been explored in Ge films or nanowires using either non-local measurements in lateral or vertical spin valves[8][9][10][11]or the Hanle effect in three-terminal devices[12][13][14][15][16][17][18][19].So far, the non-local lateral geometry is the most interesting one for the development of spintronics since the spin can be manipulated in the Ge channel between the spin injector and detector. However, experimental measurements have been limited in temperature to 225 K [8] and the only demonstration at room temperature used an indirect method based on the combination of spin pumping and inverse spin Hall effect (ISHE) [20].Another important feature of bulk Ge is related to the energy proximity between its di-
rect and indirect bandgaps [4]. This "quasi-direct" band structure was previously used to investigate spin dynamics in bulk Ge [21,22] and SiGe nanostructures [23]. It also gives the opportunity to develop opto-spintronic devices where photons can generate and detect spin currents. Here, we implement novel non-local spin injection/detection building blocks in germanium at room temperature, adding new functionalities to the common architectures available for spintronic devices. We demonstrate the lateral spin transport in n-type germanium-on-insulator (GeOI, n=2×10 19 cm −3 ) and a lightly n-doped bulk Ge sample (n=1.7×10 16 cm −3 ), exploiting electrical and optical spin generation respectively. The nonlocal spin detection is achieved using a MTJ or the ISHE in a Pt bar. By this, we can accurately extract the spin diffusion length l sf in the GeOI sample and the Ge substrate at room temperature, obtaining l sf =650±30 nm and l sf =12.4±0.4 µm, respectively. We also show direct optical mapping of spin diffusion in Ge and, by combining optical spin orientation and the ISHE in Pt, we build a non-local spin injection/detection scheme without the use of any ferromagnetic metal.
For electrical spin injection and detection, we use lateral spin valves (LSVs) fabricated on GeOI. The Ge layer is 1 µm-thick with uniform n-type heavy doping (n=2×10 19 cm −3 ) to favor electrical conduction and reduce the width of the Schottky barrier (see Methods for details). The SiO 2 buried oxide layer (BOX) is also 1µm-thick. Electrical spin injection and detection are achieved using a MgO-based MTJ to avoid the impedance mismatch issue [24]. Moreover, in order to reduce the density of localized states at the MgO/Ge interface [15,25] Fig. 1a.
The sample is then processed into LSVs made of two magnetic tunnnel junctions and two ohmic contacts as schematically shown in Fig. 1b and detailed in Methods. An example of LSV is shown in Fig. 1c where the gap, defined as the distance between the ferromagnetic electrodes edges, is 0.5 µm. The soft and hard magnetic electrodes have been processed along the [100] crystal axis of Fe and their dimensions are 1×20 µm 2 and 0.5×20 µm 2 respectively. The MTJs I(V ) curves are almost linear and their resistance-area product is 430 Ωµm 2 . Electrical measurements are performed in 3 different geometries as depicted in Fig. 2 using a 2 µm gap between the two MTJs. The applied DC electrical current is 10 mA. In Fig. 2a and 2b, the magnetic field is applied at 45 • from the Fe electrode axis i.e. along the [110] crystal direction which corresponds to the hard axis. This geometry allows to enhance and better separate the switching fields of the ferromagnetic electrodes in order to stabilize the antiparallel state.
In Fig. 2a, the DC electrical current is applied between the two ferromagnetic electrodes and the voltage is measured on the same contacts. This configuration is called the local or GMR (for giant magnetoresistance) configuration. The resistance is minimum when the Fe electrode magnetizations are parallel (R ↑↑ ) and maximum when they are antiparallel (R ↑↓ ).
We find ∆R GM R = R ↑↓ − R ↑↑ ≈45 mΩ.
The configuration of Fig. 2b is the non-local (NL) geometry: the current is applied between one pair of ferromagnetic-ohmic contacts and the voltage measured on the other pair. By this, only a pure spin current flows between the two ferromagnets without any charge current avoiding the contribution from spurious tunneling magnetoresistance effects to the detected signal [26,27]. The measured magnetoresistance signal ∆R N L = R ↑↓ − R ↑↑ ≈-25 mΩ is approximately half the GMR signal in amplitude and is independent on the applied bias voltage (not shown). It can be written as [28]: ∆R N L = − P I P D σA l sf exp − L l sf . P I (resp. P D ) is the spin polarization of the tunnel current at the injection (resp. detection) electrode, σ and A are the conductivity and cross-sectional area of the Ge channel respectively. L is the distance between the two ferromagnetic electrodes and l sf the spin diffusion length in Ge.
In the last geometry of Fig. 2c, the magnetic field is applied perpendicular to the Ge film and we measure the Hanle effect. In this configuration, injected spins experience the Larmor precession and the spin signal ∆R Hanle N L decays following roughly a Lorentzian curve.
Starting from the parallel state, the spin signal writes [28]:
∆R Hanle N L (B ⊥ ) = P I P D D σA ∞ 0 P (t)cos(ω L t)exp(− t τ sf )dt(1)
where D is the electron diffusion coefficient we determined independently using double
Hall crosses at room temperature: D=23.5 cm 2 s −1 .
P (t) = [1/ √ 4πDt]exp[−L 2 /(4Dt)],
ω L = gµ B B ⊥ / is the Larmor angular frequency with g=1.6 the g-factor of electrons in the Ge conduction band [29], µ B the Bohr magneton and the reduced Planck's constant.
τ sf is the electron spin lifetime in Ge. By fitting the Hanle curve in Fig. 2c, we find:
P I P D =0.
28±0.05 and l sf =650±30 nm corresponding to a spin lifetime τ sf =180±20 ps.
The spin diffusion length is in very good agreement with that obtained by Dushenko et al.
[20] who found l sf =660±200 nm at room temperature in a Ge film with comparable n-type doping by non-local spin pumping. This value is also in good agreement with that obtained by three-terminal measurements at room temperature: 1300 nm for n=10 18 cm −3 [15] and 530 nm for n=1.0×10 19 cm −3 [17].
This agreement between non-local and three-terminal Hanle measurements can be attributed to the fact that interface states are only weakly confined at room temperature and have little influence on the spin injection mechanism [15]. Finally, the GMR and NL magnetoresistance signals in Fig. 2 both exhibit an overall Lorentzian shape around µ 0 H=0 T. This is due to the unavoidable oblique Hanle effect since the magnetic field is applied at 45 • from the injected spin direction. This interpretation is supported by the fact that the full width at half maximum (FWHM) of the oblique Hanle curves in Fig. 2a and 2b is exactly √ 2 times the FWHM of the Hanle curve of Fig. 2c where the applied field is perpendicular to the injected spins.
As a further step, we implement an optical spin injection block, combined with non-local lateral spin detection using a MTJ or a Pt stripe to detect the ISHE. In this case, a net electron spin polarization is generated by optical spin orientation [30]. In this process, the absorption of circularly-polarized light generates spin polarized electron-hole pairs at the Γ point of the Brillouin zone. The spin polarization of photogenerated electrons in the conduction band is P = (n ↑ − n ↓ ) / (n ↑ + n ↓ ), being n ↑(↓) the up-(down-) spin densities referred to the quantization axis given by the direction of light propagation in the material. Photogenerated holes are rapidly depolarized due to their very short spin lifetime [14]. If the incident photon energy is tuned to the direct Ge bandgap, an electron spin polarization P = 50%
can be achieved [30]. Right after the photogeneration, spin-oriented electrons thermalize from the Γ to the L-valleys within ≈ 300 fs, maintaining most of their spin polarization [4].
Here, we use a diffraction-limited confocal setup shown in Fig. 3a. The Ge(001) substrate is 450-µm-thick and lightly As-doped (n=1.7×10 16 cm −3 ). At normal incidence, only an outof-plane spin polarization is generated, preventing any electrical spin detection in Ge with in-plane MTJs or the ISHE in Pt stripes. We circumvent this limitation by patterning Pt or Pt/MgO nanostructured stripes on the Ge substrate [31]. When the sample is illuminated with circularly-polarized light, the metallic pattern introduces a modulation of the amplitude and phase of the incoming electromagnetic wavefront. As shown in Fig. 3b and 3c, the absorption of the scattered light in the Ge substrate results in the generation of complementary opposite electron spin populations inside Ge in correspondence to the edges of Pt or Pt/MgO nanostructures, with a net spin polarization lying in the plane of the device [31].
The resulting spin accumulation at each edge of a nanostructure creates a pure spin current detected non-locally by an in-plane magnetized MTJ or by ISHE in a Pt bar. As an example, the spin detection with a MTJ is detailed in Fig. 3d. The two non-local devices are depicted in Fig. 4a and 4b Fig. 5a and 5b. The voltage signals are recorded at the same time as the optical images at room temperature. They are reported in Fig. 5c for the MTJ (incident power W = 60 µW) and in Fig. 5d for the Pt ISHE pad (incident power W = 1.8 mW) respectively. In Fig. 5e and 5f, the average across the Pt stripes of the MTJ and ISHE voltage signals are reported as a function of the distance to the detector for the MTJ and ISHE devices respectively. We clearly see an alternating signal that decays when the illuminating beam is moved away from the detector as a consequence of the finite spin diffusion length in Ge. As shown in Fig. 5e and 5f, the voltage ∆V normalized to the light power W can be easily fitted using the following expression: ∆V /W ∝ sin(ωx) × exp(−x/l sf ).
ω=2π/L where L=2 µm is the pattern periodicity and l sf is the spin diffusion length. x and l sf are in µm and x=0 corresponds to the position of the detector. By using such simple expression, we assume a one-dimensional spin diffusion model which is a rough approximation considering the three-dimensional geometry of our system. However, the very good agreement between the fitting curve and the experimental data suggests that the spin diffusion mostly takes place along x which is probably due to the partial spin absorption by , we obtain a spin lifetime τ sf ≈12 ns which is longer than the value predicted by Li et al. [3]. However, the presence of Pt stripes might modify the atomic and electronic structure at the Pt/Ge interface where the spin transport takes place. Finally, the combination of optical spin orientation with in-plane polarization and ISHE in a Pt bar defines an original non-local spin injection/detection scheme without the use of any ferromagnetic metal which represents a new paradigm in the field of semiconductor spintronics.
In summary, we have demonstrated spin transport in Ge at room temperature using nonlocal electrical detection. Two different systems have been investigated: a heavily n-doped GeOI for electrical spin injection using a magnetic tunnel junction and a high quality lightly doped Ge substrate for optical spin orientation. In GeOI, we used a second MTJ for electrical spin detection in a lateral spin valve geometry whereas we used both a MTJ and the inverse spin Hall effect in a Pt stripe to detect the spin signal in the Ge substrate. In the latter case, we are only sensitive to in-plane spin polarization which is achieved by optical spin orientation at the edge of Pt nanostructures grown on Ge. In GeOI, we find a spin diffusion length l sf =650±30 nm in perfect agreement with previous estimations at room temperature. In the Ge substrate, we find: l sf =12.4±0.4 µm and only 8±2 µm when the Pt stripes partly absorb the spin current. Finally, we conclude that optical spin orientation in a confocal microscope with in-plane spin polarization is a very powerful tool to directly image the spin diffusion in Ge and more generally in any direct bandgap semiconductors.
METHODS
Electrical spin injection and detection
GeOI was used to well-define the conduction channel and it was fabricated using the Smart Cut T M process from Ge epitaxially grown on Si at low temperature (400 • C) [33].
By using short duration thermal cycling under H 2 atmosphere, the threading dislocation density was reduced down to 10 7 cm −2 . However, a residual tensile strain of +0.148 % (as determined by grazing incidence x-ray diffraction) built up during the cooling-down to room temperature after the thermal cycling due to the difference of thermal expansion coefficients between Ge and Si. The Ge layer is protected against oxidation by a 10 nm-thick SiO 2 film which is removed using hydrofluoric acid before the introduction into the molecular beam RA of the tunnel junction is much higher than the spin resistance of Ge r Ge =ρ × l sf , ρ and l sf being the Ge resistivity and the spin diffusion length. In our case, RA is two orders of magnitude larger than r Ge which justifies the use of Eq. 1.
Optical spin orientation and electrical spin detection
The bulk Ge sample was first cleaned in acetone and isopropyl alcohol into an ultrasonic bath for 5 min and then rinsed into deionized water before being loaded into the MBE Orange (resp. black) is for increasing (resp. decreasing) magnetic field. All the curves have been shifted vertically so that the high field magnetoresistance ∆R is zero.
µm 2 and separated by 1 µm (see Methods). For the MTJ device, the MgO layer separating Pt from the Ge substrate is meant to avoid spin absorption in the Pt layer whereas, for the detection block based on the Pt bar, spin absorption by the Pt layer is necessary to detect the ISHE signal and Pt is in direct contact with the Ge surface. Scanning electron microscopy images of the final devices are shown in Fig. 4c for the MTJ device and Fig. 4d for the ISHE device. The measurement details are given in the Methods. The optical images of the nanostructures are shown in
epitaxy (MBE) chamber. The native Ge oxide top layer was then thermally removed by annealing under ultrahigh vacuum. After this cleaning procedure, the reflection high-energy electron diffraction (RHEED) pattern exhibited a well-defined and high-quality (2×1) surface reconstruction as the one shown in Fig. 1a. The nanofabrication of LSVs required 5 successive electron beam lithography levels and the key steps are: (i) the ion beam etching of the ferromagnetic electrodes using metallic hard masks, (ii) the growth of ohmic contacts made of Au(250nm)/Ti(10nm) by electron beam evaporation and lift-off technique and (iii) the deposition of a 100 nm-thick SiO 2 passivation layer by ion beam deposition (IBD) to insulate the bonding pads from the Ge channel. The GMR and NL measurements could be repeated on 5 different lateral spin valves on the same chip with different gaps. However, we could not obtain a proper gap dependence of the spin signal to extract the spin diffusion length. For the quantitative analysis, Eq. 1 is only valid when the resistance-area product
FIG. 1 .FIG. 2 .
12Lateral spin valve fabricated on GeOI. (a) RHEED patterns recorded along the [110] and [100] crystal axes of Ge at different stages of the epitaxial growth of the magnetic tunnel junction on Ge(100). (b) Sketch of the lateral spin valve used for non-local electrical spin injection and detection in n-Ge. The electrical current is applied between the hard magnetic layer and one ohmic contact in electron spin injection conditions. (c) Scanning electron microscopy image of the lateral spin valve. The magnetic field is applied at 45 • from the injected spin direction. Room temperature magnetoresistance measurements. Three different measurement geometries are used: (a) local or GMR, (b) non-local and (c) Hanle. The applied DC current is 10 mA. The magnetic field is applied in-plane at 45 • from the Fe electrode axis in (a) and (b) and perpendicular to the film plane in (c). The horizontal arrows indicate the field sweep directions.
FIG. 3 .FIG. 4 .
34Setup and principle of optical spin orientation and electrical spin detection. (a) Sketch of the confocal microscope setup used for optical spin orientation and electrical detection.(b) and (c) Right and left circularly polarized light impinging the Pt or Pt/MgO nanostructures and resulting in-plane spin polarization of electrons with opposite chirality at the two edges. (d) Voltage recorded between the MTJ and one ohmic contact while sweeping the laser spot along the x axis. The white arrow indicates the magnetization direction of the MTJ. V σ+ (x) (resp. V σ− (x)) is for right (resp. left) circularly polarized light. ∆V (x) = V σ+ (x) − V σ− (x) is the voltage recorded on the lock-in amplifier. Here, the distance between the MTJ and the Pt stripe is assumed to be much shorter than the spin diffusion length in germanium. Lateral devices for optical spin generation and non-local electrical spin detection. (a) and (b) Sketch of the MTJ and ISHE devices respectively. (c) and (d) Corresponding scanning electron microscopy images. For the MTJ device, the electrical contacts are taken between the MTJ and one ohmic contact made of Au(250nm)/Ti(10nm) directly grown on Ge. For the ISHE device, the contacts are directly taken on the Pt stripe.
Figure 1
1Figure 1
Figure 2
2Figure 2
Figure 3
3Figure 3
Figure 4
4Figure 4
Figure 5
5Figure 5
, we have grown the magnetic tunnel junction Pd(5nm)/Fe(15nm)/MgO(2.5nm) by epitaxy on Ge(100). The overall epitaxial relationship is Fe[100]||MgO[110]||Ge[100] as illustrated by the RHEED patterns along the [110] and [100] crystal axes of Ge in
the Pt bars. We find: l sf =12.4±0.4 µm for the MTJ device and l sf =8±2 µm for the ISHE device. The difference between these values is related to the different sample geometry: in the MTJ device, a MgO layer separates Pt bars and Ge, whereas in the other one, Pt is directly in contact with Ge. Platinum acts as a spin sink: the presence of Pt bars between the generation and detection points reduces the number of spins reaching the detector. In the MTJ device, MgO prevents this mechanism and the exponential decay can be mostly related to depolarization in the semiconductor. On the contrary, for ISHE detection, spin absorption in the Pt pads cannot be neglected and this reduces the effective spin diffusion length as well as the detected electrical signal. To confirm this picture we fabricated a test ISHE device with MgO separating Pt and Ge, thus lowering the Pt spin-absorption,i.e. the ISHE detector efficiency. As expected, this dramatically suppressed the detected
signal. Using an upper bound of 100 cm 2 s −1 for the diffusion coefficient of electrons in the
substrate[32]
chamber. The native Ge oxide top layer was then thermally removed by annealing under ul-The sample is illuminated with a continuous wave (CW) laser diode working at a wavelength of 1550 nm (hν=0.8 eV) to be resonant with the direct bandgap of Ge. The light is circularly polarized using the combination of a polarizer (POL) rotated at 45 • with respect to the neutral lines of a photoelastic modulator (PEM). The numerical aperture (NA) of the objective is 0.7 giving a full-width at half maximum beam size w ≈ 1.5µm. The circular polarization is modulated at 50 kHz which allows for the synchronous detection of the electrical signal ∆V using a lockin amplifier. Finally, the optical image is recorded using a near infrared (NIR) InGaAs detector after the light crosses a beamsplitter (BS).It is worth noticing that the detection pad for MTJ and ISHE measurements has been de-signed in order to avoid spurious electrical effects related to the electron diffusion in Pt. At variance from Ref. 31, where the ISHE detection was performed with a continuous thin Pt film, with a bar-shaped ISHE pad we do not detect any signal related to a component of the spin-polarization parallel to the Pt stripe edge. This indicates that the electromagnetic field modulation, operated by the Pt scatterers, generates only two complementary in-plane components of the spin polarization, perpendicular to the stripe edges.trahigh vacuum to obtain a well-defined (2×1) surface reconstruction. For the MTJ device,
we first deposited a 8 nm-thick MgO layer at 310 • C followed by 10 minutes of annealing
at 650 • C, 15 nm of Pt were then deposited on top at room temperature. Eight stripes of
Pt/MgO with dimensions 1×2 µm 2 and separated by 1 µm were then patterned by electron
beam lithography and ion beam etching. Finally, the MTJ Pt(5nm)/Fe(15nm)/MgO(3.5nm)
was grown at room temperature by electron beam evaporation and patterned using electron
beam lithography and the lift-off process. For the ISHE device, starting from the same Ge
surface, only a 15 nm-thick Pt layer was grown on Ge at room temperature. The 8 Pt stripes
and the Pt stripe with dimensions 3×1 µm 2 for the ISHE detection were then patterned by
electron beam lithography and ion beam etching. For both the MTJ and ISHE devices, after
passivating the surface with a 100 nm-thick SiO 2 layer, a Au(250nm)/Ti(10nm) stack was
deposited to contact the MTJ and ISHE detector.
ACKNOWLEDGEMENTSThe authors acknowledge the financial support from the French National Research Agency through the ANR project SiGeSPIN #ANR-13-BS10-0002. Partial funding is acknowledged to the CARIPLO project SEARCH-IV (grant 2013-0623). Dr. Edith Bellet-Amalric is also acknowledged for the x-ray diffraction analysis of GeOI.COMPETING FINANCIAL INTERESTSThe authors declare no competing financial interests.
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Experimental evidence of optical spin generation and non-local electrical spin detection in Ge at room temperature. (a) and (b) Optical images recorded on the MTJ and ISHE devices respectively using the confocal microscope. (c) and (d) Voltage signals simultaneously recorded using a lockin amplifier at the PEM modulation frequency. (e) and (f) Voltage intensity profiles along x for y=0. x=0 is the detector position. Dots are experimental data and the solid lines correspond to the fits. The signals have been normalized to the laser power and expressed in V /W . As illustrated by the red dotted vertical line. the signal is zero at the center of the Pt stripe, positive (resp. negative) at the left (resp. right) edgeFIG. 5. Experimental evidence of optical spin generation and non-local electrical spin detection in Ge at room temperature. (a) and (b) Optical images recorded on the MTJ and ISHE devices respectively using the confocal microscope. (c) and (d) Voltage signals simultaneously recorded using a lockin amplifier at the PEM modulation frequency. (e) and (f) Voltage intensity profiles along x for y=0. x=0 is the detector position. Dots are experimental data and the solid lines correspond to the fits. The signals have been normalized to the laser power and expressed in V /W . As illustrated by the red dotted vertical line, the signal is zero at the center of the Pt stripe, positive (resp. negative) at the left (resp. right) edge.
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[
"Exploiting Workload Cycles for Orchestration of Virtual Machine Live Migrations in Clouds",
"Exploiting Workload Cycles for Orchestration of Virtual Machine Live Migrations in Clouds"
]
| [
"Artur Baruchi \nUniversity of Sao Paulo\nBrazil\n",
"Edson T Midorikawa \nUniversity of Sao Paulo\nBrazil\n",
"Liria M Sato \nUniversity of Sao Paulo\nBrazil\n",
"Marco A S Netto \nIBM Research\nBrazil\n"
]
| [
"University of Sao Paulo\nBrazil",
"University of Sao Paulo\nBrazil",
"University of Sao Paulo\nBrazil",
"IBM Research\nBrazil"
]
| []
| Virtual machine live migration in cloud environments aims at reducing energy costs and increasing resource utilization. However, its potential has not been fully explored because of simultaneous migrations that may cause user application performance degradation and network congestion. Research efforts on live migration orchestration policies still mostly rely on system level metrics. This work introduces an Application-aware Live Migration Architecture (ALMA) that selects suitable moments for migrations using application characterization data. This characterization consists in recognizing resource usage cycles via Fast Fourier Transform. From our experiments, live migration times were reduced by up to 74% for benchmarks and by up to 67% for real applications, when compared to migration policies with no application workload analysis. Network data transfer during the live migration was reduced by up to 62%. | null | [
"https://arxiv.org/pdf/1607.07846v1.pdf"
]
| 1,256,002 | 1607.07846 | 88643b6206bfdd1762ea692bedda0ac899e79660 |
Exploiting Workload Cycles for Orchestration of Virtual Machine Live Migrations in Clouds
Artur Baruchi
University of Sao Paulo
Brazil
Edson T Midorikawa
University of Sao Paulo
Brazil
Liria M Sato
University of Sao Paulo
Brazil
Marco A S Netto
IBM Research
Brazil
Exploiting Workload Cycles for Orchestration of Virtual Machine Live Migrations in Clouds
Live MigrationCloud ComputingVirtual MachineWorkload Cycle RecognitionServer ConsolidationFast Fourier Transform
Virtual machine live migration in cloud environments aims at reducing energy costs and increasing resource utilization. However, its potential has not been fully explored because of simultaneous migrations that may cause user application performance degradation and network congestion. Research efforts on live migration orchestration policies still mostly rely on system level metrics. This work introduces an Application-aware Live Migration Architecture (ALMA) that selects suitable moments for migrations using application characterization data. This characterization consists in recognizing resource usage cycles via Fast Fourier Transform. From our experiments, live migration times were reduced by up to 74% for benchmarks and by up to 67% for real applications, when compared to migration policies with no application workload analysis. Network data transfer during the live migration was reduced by up to 62%.
Introduction
Through multiplexing techniques, virtualization allows for different operating systems and workloads to co-exist on the same hardware, without users' perception and interference among themselves. This feature is the core of cloud computing, which is a concept that dates back to the 1960s [1].
Live migration also allows a Virtual Machine (VM) to be moved across physical hosts with minimal interruption. This feature brings several benefits to cloud providers, including policy creation for physical host maintenance and energy consumption reduction via server consolidation [2]. Another benefit is to distribute VM load among physical hosts to meet circumstantial computing demand [3].
Despite the resource usage optimization brought by server consolidation or load balancing policies, they still generate VM performance degradation [4]. This problem comes mainly from live migration algorithms, whose performance is sensitive to VM memory usage. These algorithms can generate large data traffic to migrate VMs across hosts, especially when several VMs are moved simultaneously.
To address this problem, our previous work [5] introduced an Applicationaware Live Migration Architecture (ALMA) which determines, before hand, suitable moments to move VMs among physical hosts according to VM workloads. Our hypothesis is that, we can reduce live migration side effects, such as data traffic, by choosing the right moment to trigger the migration process. This hypothesis comes from two observations. The first is live migration algorithms are sensitive to VM memory usage and the second is several industry and scientific workloads follow a resource usage cyclic pattern. Examples of these cycle scenarios are: (i) a Web service with higher access during some periods of day or due to application characteristics that can present long periods of processor usage after I/O access and (ii) parallel applications with processes exchanging synchronization messages.
This work presents a detailed description of ALMA algorithms and the characterization process. Moreover, a new set of experiments to evaluate the effectiveness of using application characterization data to trigger migrations and a scalability analysis of our solution is presented. Therefore, we extended our work with the following contributions:
• Characterization and workload cycle recognition using Fast Fourier Transform with benchmarks and real scientific applications ( § 4);
• Live migration orchestration based on workload cycle recognition ( § 5);
• Evaluation of the architecture that implements the migration orchestration on a private cloud environment, including scalability analysis of the architecture with data from up to 1,000 VMs ( § 6).
Motivation and Problem Description
The overhead of live migrations comes mainly from algorithms, like precopy [6] and post-copy [7] and, at first, it is not the scope of server consolidation policies to deal with migration algorithm issues. As a result, server consolidation is barely used inside the cloud provider data centers [8].
The proposed architecture explores live migration algorithm characteristics and mediates between the consolidation policies and live migration algorithms, reducing the impact created by multiple concurrent migrations. Our architecture is based on the observation that several workloads have cyclic behaviors. Two examples of real workloads with cyclic behavior are presented in Figure 1, which illustrates the resource usage of a production data base from a telco company during a week and the resource usage of a pool of VMs of a big magazine publisher. We can see a cyclic behavior in both graphs and the best periods to perform live migrations is during the valleys of the graph, which consist of low resource usage. postponed, its workload would be in a suitable moment, avoiding network congestion-which is the second scenario.
The main goal of the cyclic analysis is to identify and extract the workload's execution pattern and postpone live migrations with potential to harm the network. In Figure 2b, all live migrations were postponed to a suitable moment to reduce live migration time and network congestion. With our architecture, algorithms for server consolidation and live migration (pre-copy and post-copy) are not modified. Our solution intercepts all pending migrations, and orchestrates them according to the workload cycles. In practice, there should be modification only in the consolidation strategy APIs for concurrent live migrations.
The problems tackled in this paper are therefore: (i) how to detect application resource consumption cycles and (ii) how to use this information to know suitable moments for moving VMs across physical hosts in order to minimize network traffic and migration times.
Background
In this section we discuss concepts to understand how our architecture can reduce live migration overhead. These concepts cover pre-copy live migration algorithm and the basics of server consolidation.
Live Migration
The encapsulation of the operating system execution environment offered by virtualization is fundamental to live migrations [9]. This feature allows for: (1) at any time, a VM to be frozen, (2) all necessary information from restarting the execution to be stored in a file and (3) using this information to restart a VM on any physical host from the breakpoint.
Live migration algorithms can be classified according to the moment in which VM's memory content is copied to its replica on the destination host. The main migration algorithms are: pre-copy and post-copy. The pre-copy algorithm copies VM memory before it starts its execution in the destination host, whereas the post-copy copies VM memory after it starts its execution.
Research studies use four metrics to compare these algorithms-two are related to performance evaluation and two refer to VM workload overhead caused by live migration:
• Live migration total time: time interval between the start of the migration process and the VM execution beginning in destination host;
• Downtime: time interval in which VM is not running nor available to the user;
• Execution time: execution time of an application, including a possible migration time;
• Throughput: amount of data processed in a time interval with and without live migrations.
Pre-Copy
As pre-copy moves VMs only when their memory is already copied, this algorithm is more robust and widely used in commercial Virtual Machine Monitor (VMM) compared to post-copy-we used this algorithm in this paper. The memory is copied between hosts in several iterations and can be split into five stages [10]:
1. Resource reservation: it checks whether destination host has available resources to the VM; 2. Iterative copy: the VM's memory is entirely copied to destination host in the first iteration. In next iterations only the memory changed in last iteration is copied (dirty pages);
3. Stop and copy: the VM is suspended in source host and the last copy is performed; 4. Shutdown: the VM is stopped in source host and all resources allocated to the VM are released; 5. Activation: the VM is activated in destination host.
One of the main problems of this algorithm is the possibility of unlimited cycles of memory copy. To avoid this issue, some VMMs impose conditions to stop the copy iterations. Taking Xen VMM [11] as example, the stop conditions are: (i) less than 50 pages marked as dirty since the last iteration; (ii) maximum of 29 iterations; and (iii) amount of data transferred greater than three times of memory assigned to the VM.
Furthermore, this algorithm is sensitive to VM dirty page rate and network throughput. Some studies, such as the one from Strunk [12], formalized the dirty page rate and network throughput dependencies. The author defines an upper and lower limit of the pre-copy algorithm that are presented in Inequalities 1 and 2 containing limits to migration and downtime time:
V mem B ≤ T mig ≤ (M + 1) * V mem B (1) 0 ≤ T down ≤ (M + 1) * V mem B(2)
Where: V mem : amount of memory assigned to VM to be moved; B: network throughput available; T mig : live migration duration; T down : downtime duration; M: number of times allowed to copy the entire memory in iteration phase.
The lower limits of Inequalities 1 and 2 refer to an idle VM. In this situation, the migration duration is limited only by the network throughput between the two hosts and the downtime duration due to the low dirty page rate. However, the upper limit is related to a high dirty page rate, leading to the occurrence of several copy iterations. The worst case happens when the dirty page rate is higher than network throughput.
There are other factors that influence live migration duration, as observed by Xu et al. [13], such as the number of concurrent migrations. However, the dominant factor to live migration performance is the dirty page rate.
Server Consolidation
Server consolidation policies consist in choosing VMs, according to a given criterion, and concentrating them into a few physical hosts. Server consolidation is fundamental to help cloud providers reduce energy consumption. The main problem of consolidation policies is to find an optimal combination of VM placement in physical hosts. Another dilemma is how to avoid concurrent migrations to accomplish an objective [14].
The most common policies of consolidation are based on heuristics [15] or linear programming [14], where the former is more explored by researchers due to scalability issues. Consolidation based on heuristics are more flexible and the final solution (usually suboptimal) is obtained faster. Implementations based on linear programming are more efficient when dealing with several restrictions, such as Service Level Agreements (SLAs) and maximum number of concurrent live migrations.
Workload Characterization and Cycle Recognition
Workload characterization is the strategy used to collect information about what resources and their consumption by applications under analysis. An application can use several computing resources at the same time, but it is likely that a specific resource is being used more than others. This behavior can be static, meaning that an application can use a given resource more from the beginning to the end or dynamic, when during the application execution, it can use different resources and at various utilization levels. In this work, the workload characterization is defined in time unit to perceive fluctuations in resource usage during the application run time. We characterize the workload every fifteen seconds.
Server consolidation relies on workload characterization to avoid the placement of VMs competing for the same resources in the same physical hosts. There are challenges in cloud workload characterization due to the features of this paradigm, such as dynamic resource usage [16] and multi-tenancy. These factors can produce ambiguous signals to the workload classifier and generate wrong results.
Another common challenge in workload characterization is the data interpretation and gathering from VMs. Virtual Machine Monitors (VMMs) add a second level of indirection in order to isolate VMs located in the same host. This extra level, known as Semantic Gap [17], imposes several challenges to interpret and gather performance metrics.
The majority of traditional workload characterization strategies are computationally expensive and prohibitive to be implemented in a cloud data center running thousands of VMs simultaneously. Recent research findings specialized in VM characterization are difficult to implement in the cloud, because such strategies are mostly based on specific VMM metrics [18] or in VMM's source code modification [19].
A load index is a metric that aims at quantifying the system load in a given moment or during a time interval. In this work we used load indexes related to processor and memory resources, which are enough to identify other types of load, such as I/O.
Naive Bayes Classifier
The Naive Bayes (NB) classifier is based on Bayes's theorem, which is broadly used in the probability field. The Naive term is due to the assumption the events' probability are independent of one another.
The aim of Bayesian classifiers is to estimate the most probable class of a set of characteristics using probabilities known a priori, which are computed used a training data set. The Bayes's theorem requires at least three termsone conditional probability and two unconditional probabilities-to compute a third conditional probability [20].
The quantitative results of the NB classifier is one of its main features. When submitting data to the classification, the NB classifier returns the most likely class of that data and their probability, allowing the implementation of optimization strategies.
Cyclic Analysis
Once the workload characterization has been completed as LM (live migration) or NLM (no live migration), cyclic patterns can be extracted, if any, from the characterization data collected over time. Given that possible classes are only LM or NLM, workload cycles can be identified and decomposed.
The extraction process is made by Fast Fourier Transform (FFT) [21]. FFT has O(n log n) complexity where n is the number of samples used to compute the cycle. FFT allows to convert time (or space) in frequency.
In this work we define a cycle as a recurrent pattern of a workload, which can be composed of several moments, suitable or non-suitable to live migration. Additionally, in a same cycle, there can be several compositions, as presented in Figure is in 0, the cycle's shape will be like A, if t 0 is in 10 or 30, the cycle's shape will be like A' and A", respectively. A cycle can be simple or complex. A simple cycle is composed of up to three interleaved intervals, that is, there is a single occurrence of one type of workload (LM or NLM). In Figure 3 the cycle is simple, even in shape A", where we observe three interleaved intervals (LM, NLM and LM). In Figure 4 is presented a complex cycle example, which contains two intervals of NLM and LM. FFT can identify both types of cycles.
ALMA -Application Aware Live Migration Architecture
Cyclic analysis, based on workload characterization, can be used to define suitable moments to trigger live migrations. To this end, we propose an architecture, called Application-aware Live Migration Architecture (ALMA) [5], which intermediates all live migration requests from massive migration strategies and the VM monitor.
Architecture Overview
Efforts to solve macro problems in a data center, such as energy consumption, computational waste and live migration algorithm optimizations do not address problems related to pre-copy and post-copy algorithm's limitations. Our architecture avoids live migration drawbacks by choosing suitable moments to trigger this operation, which benefits strategies broadly used to solve macro problems. Figure 5 illustrates ALMA and the other two most common architectures for live migration. Figure 5a presents an architecture with no live migration control-once a VM needs to be moved across hosts, the architecture does so without any concern about network traffic and other ongoing live migrations [22]. The architecture presented in Figure 5b implements a control over the live migrations. However, this control comes from the VM monitor and it orchestrates ongoing live migrations. The orchestration in this architecture considers only one or two metrics, such as available network bandwidth.
Our architecture, presented in Figure 5c, has a different approach. When the live migration plan is created, the architecture receives all live migration submissions and orchestrates the migrations. The main component of ALMA is the Live Migration Control Module (LMCM), which is responsible for migration scheduling based on the VM workload. It can postpone, run immediately or cancel a live migration. The postponement can occur when the VM workload is under a not suitable moment to trigger live migration and then it has to wait for a better moment. However, if the workload is suitable to be moved, ALMA triggers the migration immediately. If the workload is almost at the end and the live migration cost is higher than keeping the VM running on the current physical server, ALMA can cancel the live migration request. LMCM accepts parameters from cloud service provider to impose limits, such as the maximum time a VM can wait to be migrated. This could avoid long waiting time due to long cycle periods. On the customer side, there are some parameters that could be implemented, for example the expected time to finish a given workload and with this information ALMA could avoid migrations that harm customer-defined thresholds.
Algorithms
ALMA implementation is based on two algorithms. The first one finds and extracts cyclic pattern from a workload, while the second computes the waiting time for a suitable moment for live migration. These algorithms are complementary to one another, since the output from one is used by the other and could be implemented together. However, for a better understanding of the strategy, they will be described separately.
The first algorithm uses the workload classification collected for a given time interval, which is sorted chronologically. Each array position has a VM characterization for the period of the data collected. Moments are represented by the array index and the Fast Fourier Transform computes the cycle size (line 2) based on this array. Thereafter, all analyses occur in the interval of the array that represents the cycle size.
Algorithm 1 Cycle decomposition in two arrays.
Require: An array C with VM workload classification data for a given time interval. The array should be chronologically ordered.
1: function Decomposition(C) 2: CycleSize ← F F T (C)
Fast Fourier Transform. The part of the array representing an entire cycle is then split into two smaller arrays (line 5 to 13). One array stores only suitable moments for live migration (ArrayLM) and the other stores moments not favorable for live migration (ArrayNLM).
The second algorithm aims to find the instant in which the workload is inside a cycle. We use two known variables: (1) the cycle length time, which is computed in the first algorithm and (2) the workload execution time, which is the elapsed time of the workload execution. The relative time (M relative ) can be computed as the module between elapse time of the workload (M current ) and the cycle length time (CycleSize, line 2 of Algorithm 2).
After the computation of M relative , the next step is to find in which array it is placed (ArrayLM or ArrayNLM). If the relative moment is placed in ArrayNLM (line 3), we need to find out when the workload will be in suitable moment to be migrated (ArrayLM). In order to compute the remaining time of this period, we look for the first instant longer than the relative moment in ArrayLM (NextLM, in line 4). The difference between both instants (NextLM and M relative ) is the remaining time (RemainT ime) to the workload be feasible to be migrated. N extLM ← f indGreater(M relative , ArrayLM )
5:
RemainT ime ← N extLM − M relative 6: else 7:
RemainT ime ← 0 8: return RemainT ime
Evaluation
Our previous results showed substantial reduction in network data traffic and live migration time when using workload cycle recognition for live migration [5,23]. Here we present a more detailed evaluation with more VMs, additional applications, and new insights and discussions.
Metrics evaluated in this work are:
• Total migration time (secs): time between the start of a migration submission and the moment the VM is released from the source host;
• Downtime duration (secs): time in which the migrated VM is unreachable from network. Data is collected using ICMP;
• Network data transfer (MB): amount of data transferred in the network during the live migration;
• Cycle accuracy identification: used to show moments where migration actually occurred-not when they are requested.
Host_A Host_B Host_C Host_D Host_E We organized our experiments in three parts: (i) workload characterization and cycle recognition analysis; (ii) orchestration analysis with benchmarks and real scientific applications; and (iii) scalability tests to handle data from hundreds of VMs.
Experiment Setup
We setup a private cloud with five physical hosts, one Network Attached Storage (NAS) and ten VMs equally distributed among physical hosts. We used VMs with three computational resource configurations (Table 1). VMs were initially placed in four physical hosts and during the workload execution, they were consolidated into two physical hosts ( Figure 6). The consolidation moments were randomly chosen to explore various points in time, having preferences for points where all machines were running workloads in order to stress the consolidation policies.
Benchmark / Application
Description Experiment
TeraSort (MapReduce)
Sort algorithm which uses MapReduce paradigm [24]. Scientific Application Evaluation
BRAMS
Brazilian atmospheric model used for weather forecast [25].
Scientific Application Evaluation
OpenModeller Scientific application used to mode specimen distribution [26].
Scientific Application and Characterization Evaluation
SPEC CPU 2k
It is a broadly used benchmark to compare computational systems. It has several subprograms which stress the processor [27].
Benchmark and Characterization Evaluation BT Part of NASA Parallel Benchmark. This program has a memory footprint of 650 MB with high rate of dirty page [28].
Benchmark Evaluation
IOZONE
Benchmark with high usage of I/O subsystem. To avoid cache usage effect, files are larger than available memory [29]. Benchmark Evaluation sleep 1 Linux command used to delay processing for a given time. Benchmark Evaluation LAME LAME is an MP3 codifier used as benchmark [30].
We used input file of 2.3 GB. Characterization Evaluation
Benchmarks and applications (Table 2) were used to evaluate the characterization strategy and ALMA. OpenModeller, LAME, and SPEC were used with different VM configurations for workload characterization. The ALMA evaluation consists of two experimental scenarios in order to create a controlled scenario. In first scenario we created artificial cycles running benchmarks with a specific behavior in a given order. The evaluation contains the SPEC benchmark as CPU intensive workload, BT as Memory intensive workload, IOZone as I/O intensive workload and, sleep command to simulate IDLE periods. The artificial cycles and VMs used are described in Table 3. The second scenario contains BRAMS, OpenModeller, and TeraSort where the former two are scientific applications and the latter represents a typical data-intensive cloud workload, all running simultaneously in different VMs.
Workload Classification and Cycle Recognition
The evaluation of the NB classifier was based on two benchmarks and one scientific application. Benchmarks behavior is more constant during the execution and, due to this characteristic, we can verify the NB classifier precision. On the other hand, the scientific application presents several oscillations during its execution and enables the analysis of the classifier sensibility to peaks of usage and workload changes.
A new subclass of four VM configurations, summarized in Table 4, was used for this experiment. For each configuration, benchmarks (SPEC and LAME) and an application (OpenModeller) were run ten times and during the test load indexes were collected. VMs were installed in a hardware consisting of a 2.66 MHz Intel Core 2 Quad processor, 2 GB of memory, and a 5400 RPM hard disk with nominal throughput of 3 GB/second. As software configuration, the VMM used was Xen 4.1.3 and OpenSuse 12.1 running Kernel 3.1.10 as host OS. All VM images use CentOS 5.9 running Kernel 2.6.18.
All results are summarized in Table 5, which also presents resource usage average, the standard deviation between parenthesis and Naive Bayes classification in last column. Since classification is done during the benchmark and application execution, it is expected to observe some classification oscillation, which we then captured as primary and secondary workload.
In the SPEC characterization, running in configurations C1 and C2, with only one processor available, NB classified as CPU intensive workload, with memory or I/O fluctuations during the benchmark execution. Operations of I/O occurred when SPEC wrote statistics in control files, such as FLOPS and elapsed time. Memory classification was also expected, since the MCF has a high memory usage profile.
For the LAME benchmark, the workload profile is CPU and I/O intensive usage. While an input file is being processed, LAME creates the MP3 file, resulting in high I/O operations, with simultaneous read and write operations. Figure 7 presents the characterization over time for configuration C3 running the workload LAME. The top of the graph contains the VM resource usage (CPU, memory, and I/O per second) and the bottom represents how NB characterized the workload at a given moment.
I/O+MEM I/O LAME C1 98 (±15) 7 (±1) CPU+I/O I/O+CPU C2 98 (±15) 5 (±1) CPU+I/O I/O C3 50 (±11) 7 (±1) I/O I/O C4 51 (±11) 5 (±1) I/O I/O OpenModeller C1 100 (±5) 15 (±1) CPU+I/O I/O C2 99 (±11) 8 (±1) CPU+I/O I/O C3 51 (
OpenModeller has a high CPU usage, with some memory access and I/O operations during the application initialization, when it reads the input file and during the finalization, when the benchmark writes the output file. Characterization for C1 and C2 configuration was CPU intensive, which is expected in configurations with only one processor available. The processor usage is more evident for all benchmarks/application for configurations C1 and C2 due to the availability of one processor. When adding a second processor, NB identifies other workload profiles.
The NB's asymptotic complexity is linear which, as previously discussed, is necessary for any characterization strategy in cloud computing. Considering the discretization steps and the probability computation, the complexity is Θ(n + k), where k is the number of indexes to be discretized and n is the number of classes to be evaluated.
By using the NB characterization we identify when a VM can be moved across physical hosts. Using NB characterization, we can identify the primary workload and, instead of usual classification as CPU, MEM, I/O or IDLE, it is classified as suitable to LM or non-suitable to LM (NLM).
Orchestration Analysis
This experiment evaluates the orchestration considering suitable moments to trigger live migration. A live migration representation graph illustrates the workload behavior over time and the moments when live consolidation were submitted and the instant when live migration actually occurred. Figure 8 presents the migration diagram for the four VMs running benchmarks. Line in blue is the workload behavior over time, where valleys are periods not suitable to trigger live migration (NLM) and peaks are periods where the workloads are suitable to live migration (LM). The workload is executed at the same time across all VMs. Dashed lines in red represent consolidation instants. Lines in black are instants where ALMA actually triggered live migration.
Benchmark Experiments
In order to compare the consolidation strategy under control of our architecture and without any surveillance we run two sets of experiments. In the first set, VMs were actually consolidated in instants represented in dashed red lines and we left the workload run to the end. During the second set, our architecture was in place and according to the workload and the cycle analy- figure). Ideally, when ALMA is in place, live migration (lines in black) should be triggered during the peaks. In this experiment, our architecture was able to migrate VMs at suitable workload moments, thus reducing data transferred and live migration time ( Table 6). The reduction in live migration time was up to 74% (vm02_A) and data traffic reduction was up to 21%, representing a reduction of about 2.3 GB. For the downtime metric, with 95% of confidence, it is not possible to infer any improvements when using ALMA or not.
Application Experiments
We used two scientific applications, BRAMS and OpenModeller running in vm02_C and vm03_A, respectively and a typical cloud workload, represented by Hadoop cluster, running in vm01_B, vm02_C and vm01_C. The vm01_B VM was not moved from the physical host because it already was in one of the physical hosts in which the workload was consolidated. That is the reason why metrics of this VM were suppressed from presented results. Figure 9 shows long suitable periods to live migration, such as the one of the vm01_C. There are also workloads with long periods not suitable to live migration, like vm03_A's workload, and complex cycles as observed in vm02_C. However, even in this scenario, ALMA was able to identify and successfully postpone the live migration to a suitable moment. The ALMA accuracy leads to the results presented in Table 7. Reduction in live migration time (up to 67%) and amount of data transferred in network (up to 62%) were significant. This result is due to the Hadoop cluster behavior, which exchanges larges amount of data across the cluster nodes. This application, particularly, was benefited by our architecture, as observed in Table 7.
The application behavior is not known a priori (as opposed to the benchmarks evaluation) and it is sensitive to the initial setup, such as command line parameters and the input data. In both experiments, downtime did not show improvements or deterioration. Statistically, with 95% of confidence, it is not possible to determine if using ALMA or not can achieve performance improvements.
The explanation for this is in the TCP behavior and how the hypervisor contacts the network devices that a given IP address is hosted by a given physical host. Once a VM is moved across physical hosts, the hypervisor needs to update the ARP table and, to this end, it sends an ICMP packet to the network gateway. This process is not part of the migration algorithm; it is an independent process. Moreover, downtime is sensitive to TCP. Our results corroborate observations from Kikuchi and Matsumoto [10]: when a packet does not arrive to the destiny, sender will try again when retransmission time out (RTO) ends. The RTO is computed using round time trip (RTT) which is, initially, equal to three seconds. Every time a retransmission is needed, RTO value is doubled, increasing downtime.
Scalability
Scalability is a factor to be considered when dealing with cloud computing environments. Among the main features of such environment we can cite resource elasticity, which enables the user to increase or decrease the amount of available computational resources in response to a given demand [16]. Also, users could add or remove VMs to their environment. These features make cloud computing environments sensitive to solutions with low scalability characteristic.
In this section we present an analysis of scalability of our solution. Despite the fact that previous experiments were run in a real private cloud, performing a scalability test would require a large testbed, which we do not have access to. Therefore, we used traces of the previous executed benchmark experiments to measure the overhead caused by ALMA, specifically the Live Migration Control Module (LMCM) that performs the classification and cycle analysis. The baseline used to infer the overhead is the Linux kernel compilation with no other processes competing for resources. Next, traces were submitted to the LMCM and the amount of extra time to perform the same compilation was considered as the overhead. For each VM a new process was created inside LMCM and we started the analysis with five VMs and gradually increased to up to thousand VMs. Figure 10 shows the results from ten-run experiments which highlight the fact that the overhead has a linear tendency (line in red) that increases proportionally with the increase of VMs. The overhead has increased, in average, 0.21% for every five VMs added. According to this result, and in this configuration testbed, LMCM saturation would be achieved with 1,800 VMs running and submitting the live migration request at the same time. We considered the saturation when the overhead caused by LMCM reached 100% when compiling the linux kernel.
Data Gathering Overhead in Virtual Machines
The overhead imposed by index data gathering in VMs needs to be considered. As mentioned earlier, data indexes were collected by SNMP version 2. For each SNMP request, a script is executed, which returns a given value according to the index requested. The overhead evaluation conducted in this section is similar to the evaluation that has been conducted to infer the LMCM overhead. We changed the VM configuration during the experiment (processor and memory) and our main objective is to observe and quantify the overhead caused by index data gathering according to the amount of available computational resource.
The first experiment set was conducted using one VM with one processor and memory increased from 256 MB to 1,080 MB. Results are presented in Figure 11 and indicate the overhead is about 0.75% and 0.5% for VMs with one and two processors, respectively. The overhead is constant with small fluctuation when varying the memory size. However, it is more sensitive to the available processors.
Related Work
The main research efforts on live migration overhead for cloud environments are related to VM workload consolidation techniques [31,32,13]. The motivation comes from the substantial cost reduction for service providers that can be achieved using optimized consolidation techniques. Due to the different aspects of existing efforts in this area, we split them into two categories. The first one is on live migration overhead as a factor to trigger live migration. The efforts in second category present strategies to control the live migration to avoid network congestion caused by massive migrations.
Live Migration with Overhead Constraints
As discussed earlier, consolidation deals with the selection and transfer of VMs to a common physical host in order to reduce power consumption of the source physical hosts. Since this is an NP-hard problem, several solutions based on heuristics are available in literature. Resource reservation was also considered as a mechanism to optimize migrations [32]. However, most of solutions do not consider live migration overhead in infrastructure and VMs collocated.
Xu et al. [13] present a strategy that takes into account the computational cost of live migration in physical hosts (source and target) that are involved in the migration process and in collocated VMs. This strategy, called iAware, aims to avoid service level agreements violation, preventing that a given live migration harms other VMs. Authors also present a strategy to model the impact of live migration in collocated VMs, which is based on available physical resources and the amount of interruptions generated by VMs. Similar to our work, iAware can be embedded into existing consolidation or load balance strategies. Furthermore, both strategies aim to reduce performance degradation caused by live migration.
Despite the features in common, iAware does not consider the VM workload as it only relies on resource usage of the physical host. Live migration itself has a computational cost, therefore, postponing it to more suitable moment according to the VM workload can reduce overhead in physical host.
Verma et al. [33] present CosMig, which is a model for live migrations, including time estimation to perform them. CosMig is based on processor and memory usage parameters and determines the live migration impact of a VM. Verma et al. also identified that: (1) an effective live migration model must take into account application behavior, (2) only live migration does not improve application performance; other factors can promote performance improvement such as target host computing power and VM memory fragmentation. The main similarity of CosMig and our work is the evaluation of live migration in VM workload. Despite the fact that metrics used to model live migration impact are different, both studies present models to infer live migration impact in workload. A fundamental difference of is how information about live migration impact is used by the proposed strategy. In CosMig, the question asked is related to "if " live migration of a given VM will lead to performance gains or not. On the other hand, ALMA asks "when" a live migration can be performed in order to avoid infrastructure damage and, consequently, in application.
Finally, Stage and Setzer [34] introduce a live migration scheduling strategy that classifies migrations according to the current workload and identifies the minimal network resources to perform a migration. According to the authors, a migration of a single VM can consume significant network bandwidth during a long period (about 500 Mb/s for ten seconds to migrate a VM run-ning a web server). The architecture presented by Stage and Setzer has similarities with our work. Like in ALMA, there is a live migration scheduler management, which decides when a VM can be migrated. However, their architecture only observes network parameters (available bandwidth and a live migration time constraint). Also, there is a workload classifier based on the following attributes: (i) predictable: workload is considered predictable when its behavior has a reliable prognosis for a given period; (ii) tendentious: refers to fluctuations of a tendency; and (iii) cyclic: indicates how often a pattern occurs in a given workload. The main difference from our work is that, in Stage and Setzer work, live migration will take place according to the network bandwidth consumption. From the estimative based on workload type and live migration duration threshold, which can be defined by user or service provider, the architecture schedules live migrations in order to meet live migration duration threshold. In addition, the characterization of Stage and Setzer aims to group VMs with similar workload and performs the live migration of these groups. In ALMA, the workload characterization is the main criterion to define the suitable moment to trigger live migrations.
Live Migration Control Strategies
Beloglazov and Buyya [35] propose a dynamic strategy for VM consolidation that considers suitable moments to perform live migrations. Their goal is to minimize power consumption and maximize quality of service (QoS) delivered by service provider which, according to the authors, composes the trade-off between energy and performance. Their strategy identifies physical hosts overloaded and live migrations intervals are defined in order to keep QoS. In our work, ALMA can postpone a live migration according to the VM workload and in Beloglazov and Buyya study, live migrations can be postponed according to the physical host workload.
Ye et al. [36] present a framework, called VC-Migration, which controls live migrations in a cluster composed of VMs. The VC-Migration has strategies previously configured which decides how many VMs (granularity) will be considered for migration in a given moment. The decision is based on current computational resource usage of physical hosts. The strategies defined by the framework are:
• Concurrent migration: this strategy performs the live migration of several VMs, simultaneously, running in the same cluster;
• Mutual migration: strategy which is applied when physical hosts involved in live migration process have VMs moved between each other;
• Homogeneous migration in multi-cluster: strategy applied when several virtual clusters, with the same number of VMs, are being migrated;
• Heterogeneous migration in multi-cluster: same strategy as homogeneous migration, but virtual clusters have different sizes.
The framework chooses the best strategy according to the number of VMs being migrated and network bandwidth consumption. Authors argue that application interdependence, which is common in a cluster environment, reduces the infrastructure impact for network and applications.
Concluding Remarks
Live migration algorithms are known to be sensitive to memory usage. However, during an application execution these algorithms can present periods of high memory usage or high processor usage. These periods can float according to the day of the week, period of the year, or even with application input. Therefore, the challenge is to identify workloads with cyclic pattern and, once the cycle is identified, how to postpone live migrations to reduce their overhead.
We proposed and evaluated a migration strategy and architecture using a private cloud running benchmarks and real applications. The architecture was able to reduce up to 74% and 62% in live migration time and data traffic, respectively. The scalability analysis showed a host with 6 GB of memory is capable of handling data of up to 1,800 VMs. Based on evaluation results, our main finding is that using workload cycle recognition it is possible to choose suitable moments to trigger live migration, thus leading to a significant reduction in migration time and data traffic, confirming our hypothesis.
Figure 1 :
1Real workloads with cyclic behaviors.
Figure 2
2shows two scenarios for the timing of triggering migrations: one in which the consolidation triggers live migration just after the definition of the new physical hosts (without cyclic analysis), and the other scenario, where VM live migrations are orchestrated according to the their workload (with cyclic analysis). In the first scenario, live migrations produce more network traffic, since two (VM01 and VM03) out of three VMs are migrated during a not suitable moment. Moreover, if VM03 live migration were
Figure 2 :
2Live migration orchestration using cyclic analysis: valleys represent moments where live migrations have potential to congest the network and peaks represent suitable moments for live migration.
Figure 3 :
33. The cycle shape depends on the initial instant t 0 . If t 0 Shapes for the same cycle.
Figure 4 :
4A complex cycle.
Figure 5 :
5Architectures for virtual machine live migration.
ArrayN LM [N LM Count] ← i 11: N LM Count ← N LM Count + 1 12: return ArrayN LM, ArrayLM
Algorithm 2
2Identification of live migration moment. Require: Cycle size and current moment. 1: function Postpone(CycleSize, M current ) 2: M relative ← M current % CycleSize 3: if f ind(M relative , ArrayN LM ) then 4:
Figure 6 :
6Testbed topology.
Figure 7 :
7Characterization over time in configuration C3 running LAME.
Figure 8 :
8Cycle accuracy identification diagram for benchmarks.
Figure 9 :
9Cycle accuracy identification diagram for applications.
Figure 10 :
10Overhead caused by LMCM with data from up to thousand VMs.
Figure 11 :
11Overhead in virtual machine.
Table 1 :
1Virtual Machine configurations used in testbed.Configuration VCPUS
Memory
(MB)
Virtual Machine
(hostname)
Small
1
768
vm02_A vm03_A
vm01_B vm02_B
Medium
2
1024
vm01_A vm01_C
vm01_D vm02_D
Large
2
2048
vm03_B vm02_C
Table 2 :
2Benchmarks used in testbeds.
Table 3 :
3Artificial cycles used to evaluate ALMA.Virtual Machine
Artificial Cycles
vm03_A
I/O CPU CPU I/O CPU
CPU I/O CPU CPU
vm02_C
MEM IDLE CPU MEM IDLE CPU
MEM IDLE CPU MEM IDLE CPU
vm02_A
MEM CPU CPU MEM CPU CPU
MEM CPU CPU MEM CPU CPU
vm01_C
MEM IDLE CPU MEM IDLE CPU
Table 4 :
4Virtual Machine configurations.Configuration
ID
Processor
(VCPUs)
Memory
(GB)
C1
1
1
C2
2
C3
2
1
C4
2
Table 5 :
5Naive Bayes classification summary.Average Resource Usage
Naive Bayes
Characterization
Benchmark/Application Conf. ID
CPU
(%)
MEM
(%)
CPU (Prim/Sec) MEM (Prim/Sec)
SPEC
C1
96 (±18) 17 (±5)
CPU+I/O
CPU+MEM
C2
96 (±18) 9 (±2)
CPU+I/O
MEM
C3
49 (±12) 17 (±5)
IO
I/O+MEM
C4
49 (±11) 10 (±3)
Table 6 :
6Results with four VMs running benchmarks.Metric
Virtual
Machine
Traditional
Consolidation
ALMA
Reduction
(%)
Downtime
(sec)
vm03_A
20.06
20.44
-1.87
vm02_C
18.63
17.75
4.70
vm02_A
20.75
23.69
-14.16
vm01_C
19.25
18.94
1.62
Live Migration
Time
(sec)
vm03_A
28.81
12.00
58.35
vm02_C
87.56
42.31
51.68
vm02_A
43.81
11.13
74.61
vm01_C
54.31
26.81
50.63
Data
Traffic
(MB)
11,557.50
9,159.60
21.56
sis, it triggered or postponed live migration to a better instant (represented
by the black lines in the
Table 7 :
7Results with four VMs running real applications.Metric
Virtual
Machine
Traditional
Consolidation
ALMA
Reduction
(%)
Downtime
(sec)
vm03_A (OpenModeller)
21.80
23.00
-5.50
vm02_C (BRAMS)
22.60
20.73
8.26
vm01_C (Hadoop)
19.07
22.33
-17.13
vm02_A (Hadoop)
12.67
17.20
-35.79
Live Migration
Time
(sec)
vm03_A (OpenModeller)
31.27
12.73
59.28
vm02_C (BRAMS)
12.93
10.60
18.04
vm01_C (Hadoop)
39.20
18.67
52.38
vm02_A (Hadoop)
38.20
12.40
67.54
Data
Traffic
(MB)
14,566.47
5,504.98
62.21
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| []
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[
"Polarization singularities on high index nanoparticles",
"Polarization singularities on high index nanoparticles"
]
| [
"Aitzol Garcia-Etxarri [email protected] \nDonostia International Physics Center (DIPC)\nDonostia -San Sebastian 20018Spain\n"
]
| [
"Donostia International Physics Center (DIPC)\nDonostia -San Sebastian 20018Spain"
]
| []
| In this article, we study the emergence of polarization singularities in the scattered fields of optical resonators excited by linearly polarized plane waves. First, we prove analytically that combinations of isotropic electric and magnetic dipoles can sustain L surfaces, and C lines that propagate from the near-field to the far field. Moreover, based on these analytical results, we derive anomalous scattering Kerker conditions trough singular optics arguments. Secondly, through exact full-field calculations, we demonstrate that high refractive index spherical resonators present such topologically protected features. Furthermore, we calculate the polarization structure of light around the generated C lines, unveiling a Möbius strip structure in the main axis of the polarization ellipse. These results prove that high-index nanoparticles are excellent candidates for the generation and control of polarization singularities and that they may lead to new platforms for the experimental study of the topology of light fields around optical antennas. | 10.1021/acsphotonics.7b00002 | [
"https://arxiv.org/pdf/1601.04365v1.pdf"
]
| 118,628,528 | 1601.04365 | 9d9b17a2ffbaf7af040b31e631176446f6d374a8 |
Polarization singularities on high index nanoparticles
17 Jan 2016
Aitzol Garcia-Etxarri [email protected]
Donostia International Physics Center (DIPC)
Donostia -San Sebastian 20018Spain
Polarization singularities on high index nanoparticles
17 Jan 2016* To whom correspondence should be addressed 1Singular opticsPolarization singularitiesHigh refractive index nanoparticlesKerker conditions
In this article, we study the emergence of polarization singularities in the scattered fields of optical resonators excited by linearly polarized plane waves. First, we prove analytically that combinations of isotropic electric and magnetic dipoles can sustain L surfaces, and C lines that propagate from the near-field to the far field. Moreover, based on these analytical results, we derive anomalous scattering Kerker conditions trough singular optics arguments. Secondly, through exact full-field calculations, we demonstrate that high refractive index spherical resonators present such topologically protected features. Furthermore, we calculate the polarization structure of light around the generated C lines, unveiling a Möbius strip structure in the main axis of the polarization ellipse. These results prove that high-index nanoparticles are excellent candidates for the generation and control of polarization singularities and that they may lead to new platforms for the experimental study of the topology of light fields around optical antennas.
Mathematical singularities may be defined as points where the value of a function in not defined or is not well behaved. In physics, singularities arise whenever a mathematical formalism fails at describing some particular physical phenomenon. Often, these singularities drop hints on interesting phenomena happening in a particular physical process. Singularities in wave physics have been studied since 1830 in different kinds of waves (for a good overview, see 1 ). In wave optics, two general types of singularities have been identified and studied. 2 In certain situations, the polarization of light is uniform throughout space and thus, light can be mathematically described as a position dependent complex scalar wave multiplied by a constant polarization vector. In these situations, singularities arise whenever the amplitude of the wave is zero. In such cases, even though the amplitude of the wave si well defined, its phase cannot be unambiguously determined. These singularities receive the name of Phase singularities. Around these singularities, the phase of the scalar field varies gradually from 0 to 2πq, where q is an integer (positive or negative) named the topological charge of the singularity. These phase singularities, also known as nodal lines or vortices, have been widely studied in the recent past due to their ability to carry orbital angular momentum in their vicinity. 3,4 Nevertheless, in most situations, light presents a spatially varying polarization structure and the vectorial nature of electromagnetic fields cannot be disregarded. In these situations, optical vortices occur very rearly since all of the three field components need to be exactly zero in order to have a real singularity. Over a frequency cycle, the real part of the complex vector draws an ellipse (the polarization ellipse) and, as we proceed to detail, singularities arise as Polarization Singularities either when light is circularly polarized or linearly polarized. Following the formalism introduced by M. R. Dennis and M. V. Berry, 2,5 the vectors defining the major (α) and minor (β ) axes of the polarization ellipse and its surface normal (N) (see Fig.1) can be expressed as: Green arrows represent the cartesian coordinates, α and β the major and minor axes of the polarization ellipse and N its surface normal.
α = 1 | √ E · E| ℜ E * √ E · E (1) β = 1 | √ E · E| ℑ E * √ E · E(2)N = ℑ (E * × E)(3)
where E = E xx + E yŷ + E zẑ is the complex electric field vector, ℜ(·) and ℑ(·) represent the real and imaginary parts of a complex function and the polarization scalar
ϕ = E · E = E x E x + E y E y + E z E z .
If light is linearly polarized, the polarization ellipse collapses to a line and it is not possible to define a unique vector which is normal to the polarization ellipse. In those situations, |N| = 0. In three dimensions, the spatial positions where this condition is fulfilled arrange in surfaces, named L surfaces.
On the other hand, when light is circularly polarized, the major and minor axes of polarization cannot be unambiguously determined, and both α and β vectors become identically zero. Since ϕ is a complex scalar function, its vanishing requires both its real and the imaginary parts to be zero.
Thus, in 3D space, the points of circular polarization form lines (C lines). On the other hand, as in scalar optical vortices, the phase of ϕ around any zero spans all the possible values from 0 to 2πl, l being the topological charge of the singularity. Around any small closed curve surrounding a C line, the major and minor axes of the polarization ellipse generate a multi twist ribbon with l half twists. 6,7 Thus, as it was recently experimentally demonstrated, 8 for l = 1, both α and β form a Möbius strip around any small closed path arund a C line.
Since the seminal contributions by Nye 9-11 polarization singularities in optical fields have been studied extensively for their intrinsic theoretical interest. C lines and L surfaces have been identified in contexts as disparate as the skylight, 12 speckle fields, 13,14 tightly focused beams, 15,16 crystal optics, 17 photonic crystals 18,19 and plasmonic systems. 20,21 Nevertheless, experimental applications of such topological features remained elusive until very recently. In the recent past, polarization singularities have proved to be useful in quantum information applications. 22,23 In this paper, we explore the possibility of creating such polarization singularities in optical antennas illuminated by linearly polarized light. We prove that systems presenting a simultaneous isotropic electric and a magnetic polarizability (α e and α m respectively) are among the simplest nanostructures capable of sustaining C lines and L surfaces. Moreover, we prove the anomalous scattering Kerker conditions based on singular optics arguments alone. We verify our ideas by performing full field simulations on the optical response of a high index dielectric nanosphere iluminated by a linearly polarized planewave. We track the C lines generated by a Si nanoparticle over all the radiation regions and unveil the Möbius strip structure of the main axis of the polarization ellipse around them.
Results and Discussion
Dielectric nanoantennas made of materials with a high index of refraction have been intensely studied in the recent past due to their ability to sustain lossless electric and magnetic resonances in the visible and IR parts of the spectrum. 24 On the one hand, their lossless character has proved crucial in the fabrication of metamaterials and metasurfaces not limited by ohmic losses. [24][25][26] On the other hand, the combination of electric and magnetic dipolar responses allows for an additional degree of freedom in the manipulation of light that has proved instrumental in surface enhanced chiral spectroscopy, 27 and in shaping the radiation properties of optical antennas. 28,29 In this ar- For isolated electric and magnetic dipoles (a and b) iluminated by linearly polarized light, all the scattered fields are linearly polarized. On the contrary, for a system radiating as a simultaneous set of electric and magnetic dipoles (c), interference makes the scattered field to be generally elliptically polarized.
ticle, we focus in the polarization aspect of the fields scattered by such resonators illuminated by linearly polarized light.
Let us start by considering the far field radiation characteristics of an electric dipole. Excited by linearly polarized light (E inc = E 0x , H inc = H 0ŷ ), the induced electric dipolar moment can be expressed as p = ε 0 α e E inc = E 0 ε 0 α ex , ε 0 being the vacuum permittivity and α e the electric dipolar polarizability The far field scattering of such dipolar moment is given by:
E ED scat = k 2 ε 0 G e p = k 2 ε 0 [(n × p) × n] e ikr 4πr(4)
where G e is the electric Green's tensor in the far field approximation and k is the wavenumber of light in vacuum. r = |r − r 0 | is the distance between the position of the dipole (r 0 ) and the observation point r, and n = n xx + n yŷ + n zẑ is the unit vector in the direction of r − r 0 . Figure 2a plots the far field electric field amplitude (|E ED scat |) with superimposed electric field polarization vectors.
As expected, the fields scattered by a linearly polarized electric dipole are linearly polarized in all spatial directions.
The situation is similar for an isotropic magnetic dipole polarizability α m excited by linearly polarized light. In such a system, the induced magnetic dipolar moment can be expressed as m = α m H inc = H 0 α mŷ , and the scattered fields in the far field can be calculated from:
E MD scat = iZ 0 k 2 G m m = −Z 0 k 2 (n × m) e ikr 4πr(5)
where Z 0 in the impedance of free space and G m is the magnetic Green's tensor in the far field approximation. Figure 2b presents the far field electric field amplitude (|E MD scat |) with superimposed electric field polarization vectors. In this case, the electric field vectors circulate around theŷ polarized magnetic dipole, but the polarization is also linear in the entire solid sphere.
Nevertheless, this situation changes when considering the scattering coming from a particle with simultaneous isotropic electric and magnetic polarizabilities excited by linearly polarized light. In this case, the scattered fields are a superposition of the fields radiated by both the electric and magnetic dipole:
E ED+MD scat = k 2 ε 0 G e p + iZ 0 k 2 G m m.(6)
E ED scat and E MD scat are not necessarily in phase and they are not parallel. Thus, the polarization state in these systems (Figure 2c) becomes spatially inhomogeneous. Since the scattered fields become generally elliptically polarized, it is logical to think that these kind of systems may sustain polarization singularities.
We will first search for L surfaces. Calculating |N| by substituting Eq.6 into Eq.3, one obtains:
|N| = ℑ(α e α * m )n x n y .(7)
Thus, both thexz plane and theŷz plane are L surfaces (|N| = 0) in the radiation region.
The analytical determination of C lines is more intricate. Deriving the polarization scalar ϕ from Eq.6 yields:
ϕ = α 2 m n 2 x + α 2 e + (α 2 e + α 2 m )n 2 z + 2α e α m n z .
Consequently, light will be circularly polarized if ℜ α 2 m n 2 x + α 2 e n 2 y + (α 2 e + α 2 m )n 2 z + 2α e α m n z = 0 (9) ℑ α 2 m n 2 x + α 2 e n 2 y + (α 2 e + α 2 m )n 2 z + 2α e α m n z = 0
.
Since n is a unit vector, a third condition also holds,
n 2 x + n 2 y + n 2 z = 1(11)
Note that none of these equations do depend on r. Thus, for some particular α e and α m , n x , n y and n z are the only unknowns in this set of 3 equations and their solutions specify the directions on which the C lines propagate on the far field. It is interesting to note that geometrically, Eq.9
and Eq.10 describe two hyperbolic functions, and that Eq. 11 describes a unitary sphere. So,
for particular values of the complex electric and magnetic polarizability, the direction of the C lines will be determined by the intersection of these three surfaces. An analytic solution to this equations does exist, but unfortunately, it is too cumbersome to provide any intuitive interpretation.
Nevertheless some interesting conclusions can be extracted under some particular assumptions.
If α e = α m , it can be easily proved that equations 9-11 determine that a single C line exists in the direction of n = −ẑ. Nonetheless, according to Eq.7, theẑ axis belongs to a L surface. So, according to this formalism, light should be both linearly and circularly polarized in the back-scattering trajectory. This is only feasible if no fields are radiated in this direction. This phenomenon is the well known Kerker condition for zero back-scattering, 28-31 but, to the best of our knowledge, it has never been proved before based on singular optics arguments. For α e = −α m , the second Kerker condition for zero forward-scattering can also be easily derived following the same procedure.
To verify the existence of C lines in a realistic system, we consider the optical response of a 150 nm silicon sphere illuminated by linearly polarized light. Figure 3a shows the geometrically Figure 3b plots the surfaces defined by equations 9, 10 and 11 for λ = 1110nm. Four intersection points can be identified (blue dots).
These points indicate the directions on which C lines generated by the silicon nanoparticle will propagate through the far field.
To verify our analytical predictions, we solve exactly Maxwell's equations using the Boundary Element Method 32-34 for such a system. In particular, we calculate ϕ at a distance 20 µm away from the Si nanoantenna. Figure 3c and Figure 3d plots the phase of ϕ and its absolute value.
It easy to see that, at the predicted positions, |ϕ| presents exact zeros while the phase becomes
Conclusions
In conclusion, we have analytically and numerically proved that high index nanoparticles support polarization singularities that propagate from the near-field to the far-field. In particular, C lines arise from the interference of the fields scattered by electric and magnetic isotropic dipolar excitations. The study of polarization singularities is a rapidly growing field of research, on one hand due to the intrinsic theoretical interest of the topology of light fields and on the other hand because of the upcoming applications on quantum information systems. This theoretical contribution proves that combinations of electric and magnetic dipoles supported by high index nanoparticles create such topologically protected features and may facilitate new experimental platforms to study polarization singularities.
Methods
Identification and tracking of C lines around optical antennas
In order to identify and track the C lines emerging from an optical antenna, we first calculate the vectorial electric fields on a meshed surface covering the nanoparticle at a short distance form its periphery (E 0 (r)). In particular, to compute the results presented in Fig. 4, we choose to use the Metallic Nanoparticle Boundary Element Method MATLAB toolbox. 34 Starting from E 0 (r), the polarization scalar can be readily computed as: ϕ 0 (r) = u(r) + iv(r) = E 0 (r) · E 0 (r) = E x (r)E x (r) + E y (r)E y (r) + E z (r)E z (r).
The objective is to find the exact zeros of ϕ on the surface surrounding the nanoparticle. In order to do so, we first find the local minima of |ϕ| on the calculated mesh, and then run a minimization routine (fminsearch.m in Matlab) on |ϕ(r)| using the identified minima as seeds for the minimization routine. After verifying that these minima are actual zeros of ϕ(r), we take them as the starting points of the C lines (r 0 ). The direction of the C line can be calculated though the following expression: 2 d(r 0 ) = 1 2 ∇ϕ * (r 0 ) × ∇ϕ(r 0 ) = ∇u(r 0 ) × ∇v(r 0 )
Having calculated the origin of the C line and its direction, it is easy to calculate the seed of the next point on the C line as r s 1 = r 0 + ld(r 0 ),
with l = l 0 d(r 0 ) ·r |d(r 0 ) ·r| (15) being the target distance to the next point on the line, l 0 , corrected for the fact that d may point towards the nanostructure. Minimizing ϕ around this new position, it is easy to calculate the exact position of the next point on the C line. One can compute the entire trajectory of the C line by following this procedure recursively.
Figure 1 :
1Polarization ellipse traced by an arbitrary electric field as a function of time (t).
Figure 2 :
2Electric far-field amplitude and polarization distribution of: (a) an electric dipole, blue arrow, (b) a magnetic dipole, red arrow, and (c) a combination of an electric and a magnetic dipole.
Figure 3 :b 1 .
31(a) Extinction cross section of a Si nanosphere (150 nm radius) illuminated by linearly polarized light. Blue line represents the geometrically normalized total extinction cross section. The yellow and red lines are the magnetic and electric dipolar contributions to the total extinction cross section. (b) Geometrical solution to equations 9 -11 for λ = 1110 nm. Eq.9 (red surface) and Eq.10 (green surface) describe two hyperbolic functions, while Eq. 11 (yellow surface) describes the unitary sphere. (c) Phase and (d) amplitude of ϕ as a function of position, for the 150 nm Si sphere illuminated by a 1110 nm linearly polarized plane wave, calculated in the far-field, 20 µm away from the origin. Blue and red arrows indicate the direction of the induced electric and magnetic dipoles respectively. As expected, zeros in (d) correspond to phase singularities in (c), indicating the presence of polarization singularities in those directions. . normalized extinction cross section of this system (blue line) calculated using Mie theory. Yellow and red lines plot the contribution of the dipolar electric and magnetic terms in the Mie series to the total extinction spectrum. As expected, the lowest energy peak at 1110 nm corresponds to the first order b term in the Mie series (b 1 ) . This resonance is of magnetic dipolar character and itcan be associated to a magnetic polarizability through 24 α m = i On the contrary, the peak at 860 nm is the lowest order electric resonance and its dipolar polarizability can be inferred from the a 1 term in the Mie series through α e
singular. Along these lines, the scattered fields are exactly circularly polarized, their handedness being determined by the sign of n · N which is positive for right handed fields and negative left handed fields [refDennis thesis].Finally, we calculate the polarization structure on a 3D volume encompassing all the radiation regions around the 150 nm Si nanoparticle excited by a 1100 nm linearly polarized plane wave. We track the C lines from the near-field to the far-field by the procedure described in the Methods section.Figure 4, compiles these results. The blue sphere represents the Si nanoparticle while the red and green lines emerging from the sphere are the left and right handed C lines respectively. These calculations reveal that even though in the far field the C lines evolve radially as straight lines, in the near-field they curve in a non trivial manner. Moreover, according to Freunds's predictions,6,7 taking a closed loop around any point on a C line, any of the axes of the polarization ellipse should form a multi-twist strip, 8 with the number of twists being the equal to the topological charge of the C line. In this particular case, since |l| = 1 2 for all of the C lines, a Möbius strip should be formed around any loop around any of the C lines. We verify this polarization structure through the calculation of the major axis of the polarization ellipse on several circles encompassing the C lines along their trajectory. The inset in figure 4, is a zoom of one of these calculations. For the purpose of displaying the Möbius strip clearer, one half of the major axis of the polarization ellipse is depicted as a green arrow and the other half in blue. The single twist in the ribbon is evident and consistent all along every C line.
Figure 4 :
4C lines around the Si nanoparticle for λ = 1110nm in all the radiation regions. The silicon nanoparticle is represented by the blue sphere. Green and red lines are the left and right handed C lines respectively. The main axis of the polarization ellipse forms a single twist Möbius strip in any curve enclosing a C line. The inset is a zoom of this polarization structure around one particular C line.
AcknowledgementWe thank J. J Saenz for insightful discussions and XX for the careful reading of the manuscript. A.G.-E. received funding from the Fellows Gipuzkoa fellowship of the Gipuzkoako Foru Aldundia through FEDER "Una Manera de hacer Europa"Supporting Information AvailableThis material is available free of charge via the Internet at http://pubs.acs.org/.
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| []
|
[
"Anomalous local distortion in BCC refractory high-entropy alloys",
"Anomalous local distortion in BCC refractory high-entropy alloys"
]
| [
"Yang Tong \nDivision of Materials Science and Technology\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA\n",
"Shijun Zhao \nDivision of Materials Science and Technology\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA\n",
"Hongbin Bei \nDivision of Materials Science and Technology\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA\n",
"Takeshi Egami \nDivision of Materials Science and Technology\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA\n\nDepartment of Materials Science and Engineering\nThe University of Tennessee\n37996KnoxvilleTNUSA\n\nDepartment of Physics and Astronomy\nThe University of Tennessee\n37996KnoxvilleTNUSA\n",
"Yanwen Zhang \nDivision of Materials Science and Technology\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA\n\nDepartment of Materials Science and Engineering\nThe University of Tennessee\n37996KnoxvilleTNUSA\n",
"Fuxiang Zhang \nDivision of Materials Science and Technology\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA\n"
]
| [
"Division of Materials Science and Technology\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA",
"Division of Materials Science and Technology\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA",
"Division of Materials Science and Technology\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA",
"Division of Materials Science and Technology\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA",
"Department of Materials Science and Engineering\nThe University of Tennessee\n37996KnoxvilleTNUSA",
"Department of Physics and Astronomy\nThe University of Tennessee\n37996KnoxvilleTNUSA",
"Division of Materials Science and Technology\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA",
"Department of Materials Science and Engineering\nThe University of Tennessee\n37996KnoxvilleTNUSA",
"Division of Materials Science and Technology\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA"
]
| []
| Whereas exceptional mechanical and radiation performances have been found in the emergent medium-and high-entropy alloys (MEAs and HEAs), the importance of their complex atomic environment, reflecting diversity in atomic size and chemistry, to defect transport has been largely unexplored at the atomic level. Here we adopt a local structure approach based on the atomic pair distribution function measurements in combination with density functional theory calculations to investigate a series of body-centered cubic (BCC) MEAs and HEAs. Our results demonstrate that all alloys exhibit local lattice distortions (LLD) to some extent, but an anomalous LLD, merging of the first and second atomic shells, occurs only in the Zr-and/or Hf-containing MEAs and HEAs. In addition, through the ab-initio simulations we show that charge transfer among the elements profoundly reduce the size mismatch effect. The observed competitive coexistence between LLD and charge transfer not only demonstrates the importance of the electronic effects on the local environments in MEAs and HEAs, but also provides new perspectives to in-depth understanding of the complicated defect transport in these alloys. | null | [
"https://arxiv.org/pdf/1902.09279v1.pdf"
]
| 119,484,184 | 1902.09279 | e80a95ae327208f2aff6ffdd7264ac8bdc3f27ba |
Anomalous local distortion in BCC refractory high-entropy alloys
Yang Tong
Division of Materials Science and Technology
Oak Ridge National Laboratory
37831Oak RidgeTNUSA
Shijun Zhao
Division of Materials Science and Technology
Oak Ridge National Laboratory
37831Oak RidgeTNUSA
Hongbin Bei
Division of Materials Science and Technology
Oak Ridge National Laboratory
37831Oak RidgeTNUSA
Takeshi Egami
Division of Materials Science and Technology
Oak Ridge National Laboratory
37831Oak RidgeTNUSA
Department of Materials Science and Engineering
The University of Tennessee
37996KnoxvilleTNUSA
Department of Physics and Astronomy
The University of Tennessee
37996KnoxvilleTNUSA
Yanwen Zhang
Division of Materials Science and Technology
Oak Ridge National Laboratory
37831Oak RidgeTNUSA
Department of Materials Science and Engineering
The University of Tennessee
37996KnoxvilleTNUSA
Fuxiang Zhang
Division of Materials Science and Technology
Oak Ridge National Laboratory
37831Oak RidgeTNUSA
Anomalous local distortion in BCC refractory high-entropy alloys
This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).
Whereas exceptional mechanical and radiation performances have been found in the emergent medium-and high-entropy alloys (MEAs and HEAs), the importance of their complex atomic environment, reflecting diversity in atomic size and chemistry, to defect transport has been largely unexplored at the atomic level. Here we adopt a local structure approach based on the atomic pair distribution function measurements in combination with density functional theory calculations to investigate a series of body-centered cubic (BCC) MEAs and HEAs. Our results demonstrate that all alloys exhibit local lattice distortions (LLD) to some extent, but an anomalous LLD, merging of the first and second atomic shells, occurs only in the Zr-and/or Hf-containing MEAs and HEAs. In addition, through the ab-initio simulations we show that charge transfer among the elements profoundly reduce the size mismatch effect. The observed competitive coexistence between LLD and charge transfer not only demonstrates the importance of the electronic effects on the local environments in MEAs and HEAs, but also provides new perspectives to in-depth understanding of the complicated defect transport in these alloys.
Introduction
High-entropy alloys (HEAs), or concentrated solid solution alloys (CSAs), are a new type of structural materials, with chemical complexity achieved by mixing of multiple elements (at least four) at equiatomic or near-equiatomic concentrations to form a single solid solution phase 1,2 . This new alloying strategy based on high configurational entropy greatly expands the composition space 3,4 , offering opportunities for discovering alloys with new properties for advanced applications. CSAs based on refractory elements are of particular interest from both theoretical and practical points of view. With the yield strength at 1473 K far surpassing Ni-based superalloys 4,5 , refractory CSAs (RCSAs) are potential hightemperature structural materials to meet the aggressive demand on raising operation temperature in next-generation jet engine and nuclear reactors, and re-entry vehicles.
Recent experimental investigations on oxygen doping effect in a model TiZrHfNb RCSA have shown not only strength enhancement but also substantial ductility improvement 6 , in contrast to the known interstitial-induced embrittlement in conventional refractory alloys.
Moreover, some RCSAs exhibit abnormal behaviors, e.g., anomalous phonon broadening 7 , decomposition from one BCC phase to two BCC solid-solution phases [8][9][10][11] . All these phenomena hint at the existence of unexplored novel properties in RCSAs.
But deeper knowledge needs to be created for these chemically complex RCSAs to assist the alloy design without wandering in the endless compositional space. For this purpose, we focus on the local structure of RCSAs. Previously, strongly-distorted local structure was observed for the ZrNbHf refractory medium-entropy alloy 12 . In order to elucidate the physical origin of the strongly-distorted local structure in the RCSA, we examined the local structure of fifteen BCC equiatomic RCSAs with different combinations of refractory elements, by integrating pair distribution function measurement and density functional theory (DFT) calculations. The observed deviations of local structure from average structure demonstrate the existence of local lattice distortions (LLD) with varying extent in these alloys. The discovered charge transfer competes against the LLD to stabilize these metastable RCSAs. Our findings are expected to help design advanced structural materials towards excellent strength and radiation resistance by retarding defect transport.
Methods
Sample preparation. Elemental Ti, V, Zr, Nb, Mo, Hf, Ta, W and Re (>99% pure) were carefully weighted and mixed into fifteen equiatomic concentrated solid solution alloys by arc melting. The arc-melted buttons were flipped and re-melted at least five times to improve homogeneity. Then, fine powders were ground from the as-cast buttons for synchrotron X-ray characterization.
Materials characterization. The diffraction and total scattering measurement were carried out at 28-ID-2 beamline of the NSLS II with an X-ray energy of 67 keV (λ = 0.185 Å) and 11-ID-B beamline of the APS with an X-ray energy of 58.66 keV (λ = 0.2114 Å). A twodimensional stationary detector with 200200 m 2 pixel size was used to collect data.
Calibration was performed using either CeO2 or Ni NIST powder standard. Fit2D software 24 was used to correct for a beam polarization and a dark current. X-ray diffraction profiles were analyzed with the Rietveld refinement using the program GSAS 25 .
PDFgetX2 26 was used to obtain real-space PDF by a Fourier transformation of the measured reciprocal-space structure function, ( ), in a range of 25 Å -1 ,
G( ) = 2 ∫ [ ( ) − 1] sin( )
where is the magnitude of scattering vector 27 . By using PDFGUI software 28 the measured PDFs were refined with different structure models.
Density functional theory calculations. DFT calculations were performed using the Vienna ab initio simulation package (VASP) 29 . The exchange and correlation interactions were described by a gradient corrected functional in the Perdew-Burke-Ernzerhof (PBE) form 30 . Standard projector-augmented-wave (PAW) pseudopotentials distributed with VASP were used to treat electron-ion interactions 31 . The energy cutoff for the plane-wave basis set was set to be 300 eV. A -centered 2×2×2 k-points was used to sample the Brillouin zone. The energy and force convergence criterion were set to be 10 -4 eV and 0.01 eV/Å. The chemical disorder of alloys was modeled by special quasi-random structures (SQS) 32 that were constructed using a simulated annealing Monte-Carlo algorithm. The supercells built based on the SQS contain 250 atoms. The optimal volume of the supercell is determined by calculating the energy-volume curve and then fixed during atomic relaxation.
Results
Average structure by X-ray diffraction
We determined the average structure of the RCSAs by synchrotron X-ray diffraction with a two-dimensional (2-D) detector. Two representative 2-D diffraction images are shown in
Local structure by the PDF
We further conducted the pair distribution function (PDF) study to examine the local structure of the RCSAs. The PDF is obtained from a Fourier transformation of the total scattering data, i.e. Bragg peaks and diffuse scattering induced by local distortion, and describes the local structure of the RCSAs in terms of interatomic distance, as presented in PDFs, in which the first and second PDF shells overlap to form a broad peak with a hump on its right shoulder (Fig. 2b) accompanied by a rapid damping of structure features with increasing r. The overlap of the first two PDF peaks in these ZH-RCSAs is a clear feature of a strong LLD because for phonon vibrations the first PDF peak is sharper due to an effect of correlated motion among neighboring atoms 16 . The LLD is also evidenced by the shift of the first PDF peak. As compared in Fig. 2b, the value of the first PDF peak position (P 1st ) is larger than the value expected from the average structure (P avg = √3 2 * , where a is the lattice constant determined by the Bragg peaks), indicating the presence of internal strain in the first atomic shell of these ZH-RCSAs. This local lattice strain can be quantified as 1st = (P 1st − P avg )/P avg , and the value of 1st was determined for all 15 RCSAs (see Table 1). Five of these RCSAs have 1st higher than the value reported so far for the facecentered cubic FeCoNiCrPd HEA (0.79%) 17,18 .
The distortion in the second atomic shell of these ZH-RCSAs is not obvious from Fig. 2b. However, an anomalous shift of the second PDF peak exists in the ZH-RCSAs. To demonstrate this point, we fitted the undistorted BCC random-alloy structure to the measured PDFs and two representative fitting cases are shown in Fig. 3. Note that in this fitting process, the lattice constant was fixed as the value determined from the diffraction pattern. The difference curves in Fig. 3a, c reveal that the undistorted BCC model can be fitted to the average structure in the high r range but not the local structure in the low r region. However, this local structure deviation only happens within the first atomic shell for the VNbTaTa RCSA, whereas the LLD in the TiZrNbHf RCSA involves atoms in both the first and second atomic shells. After examining the fitting for all fifteen RCSAs, we found a general trend that the LLD in the X-RCSAs is localized within the first atomic shell while the LLD in the ZH-RCSAs extends up to the second atomic shell. In contrast to the expansion of the first PDF peaks, the second PDF peaks in these ZH-RCSAs surprisingly shift to the low r region to reduce interatomic distances. Clearly the first and second atomic shells tend to merge in these ZH-RCSAs to form a combined atomic shell involving fourteen neighbors.
Distorted BCC crystal structure by DFT simulation
To elucidate the microscopic origin of the local distortions in the measured PDF through the DFT calculations, we firstly built supercell models with 250 atoms based on special quasi-random structure method, and then performed volume and ionic relaxations to find atomic positions for the lowest energy state (see Methods). Two representative relaxed supercells for the ZH-RCSAs and the X-RCSAs, respectively, are shown in Fig. 4a, c. The DFT results clearly demonstrate the major differences between ZH-RCSAs and X-RCSAs in the local structure. It can be seen that atoms in the TiZrNbHf RCSA displace much more severely from their perfect lattice sites when compared with the VNbTaTi RCSA. We then fitted the distorted structure models to the measured PDFs (Fig. 4b, d). Compared with the fitting based on undistorted structure (Fig. 3), the distorted structure model much more successfully reproduces both the average structure in the high-r region and the local structure in the low-r range, leading to a significant improvement on the goodness of the fit, Rw. In the fitting process, the lattice constant parameter was refined, and its value perfectly agrees with the one determined from the average structure.
To further understand the anomalous shift in the PDF peaks of the ZH-RCSAs, we examined the variations in the atomic pair distances particularly in the first and second atomic shells after the relaxation processes. We show the distributions of atomic pair distances for the 1NN and 2NN atoms (defined from the unrelaxed supercells) (Fig. 4e, f, g). The distributions of the 1NN and 2NN atomic distances are broad enough to overlap with each other, resulting in an undistinguished gap between the first and second atomic shells in the relaxed structure. The small atoms, Ti, Nb, Ta and V in these RCSAs, however, has the widest spreading of interatomic distances, as revealed from the distance distribution of the partial atomic pairs. In contrast, the distribution of the interatomic distances is relatively narrow for large Zr and Hf atoms, but these atoms tend to escape from the first and second atomic shells and meanwhile prefer staying in between the first and second atomic shells, causing the shift in the first and second PDF peak positions. The physical origin of these anomalous shifts is more complex, as illustrated later.
Charge transfer effect
The energy associated with LLD is exceptionally large 19 , which destabilizes these ZH-RCSAs especially at intermediate temperatures where entropy contribution to the phase stability is decreased. For instance, the separation of large Zr and Hf atoms from other small ones was found in the TiZrNbHfTa RCSA through a decomposition of the single BCC phase to two NbTa-and ZrHf-rich BCC ones [8][9][10][11] . However, DFT calculations revealed that the Gibbs free energy of the TiZrNbHfTa and TiZrNbHf RCSAs rather decreases after relaxing atomic positions due to a considerable devaluation of enthalpy 19 . Therefore, besides the entropy effect, other factor related to chemistry competes with the substantial energy increase induced by LLD. In the fusion of different metals to form substitutional alloy, the difference in the Fermi levels among pure metals drives the displacement of electrons from one element to the other 20,21 . This charge transfer effect (CTE) not only changes the energy balance but also alters the atomic radius of each constituent element.
To demonstrate this atomic size variation, we calculated the charge difference of the d (eg and t2g) orbitals for each atom in the supercells between the unrelaxed and relaxed states.
Meanwhile, the relation of the charge difference in both eg and t2g orbitals with Wigner-Seitz (WS) cell volume is examined for some typical RCSAs (Fig. 5). Among all studied In general a parameter based on the fixed metallic radii is widely used to estimate the size mismatch in HEAs,
= √∑ (1 − / ∑ =1 ) 2 =1
, where N is the total number of the constituent elements, , and , denote the atomic fraction and atomic radius of the ith or jth element, respectively 23 . The metallic radii of Zr and Hf elements are 9.4%-21.6% larger than the Ti, V, Nb and Ta elements 14 , and the δ parameter estimates the size mismatch in the TiZrNbHfTa, TiZrNbHf, and TiVZrNb RSCAs to be 7.4%, 7.3% and 10.1%, respectively. Note that the δ value calculated from the WS radii of pure metals in Table 2 is equivalent to the one from the metallic radii since the ratio of WS radius to metallic radius for the BCC lattice is constant. However, after considering the CTE, the value of δ becomes to 1.2%, 1.4% and 1.8% for the TiZrNbHfTa, TiZrNbHf, and TiVZrNb RCSAs, respectively, which are remarkably smaller (~80% drop) than the estimations based on the metallic radii or WS radii of pure refractory metals. Therefore, we conclude that CTE can adjust the atomic radius of the constituents in the BCC RCSAs to profoundly relax their size mismatch and stabilize the solid solution.
Discussion and conclusion
The DFT calculation showed that the atomic mismatch in RCSAs considered here is greatly reduced because of the charge transfer from the early TEs and 4d elements which are large to later TEs and 3d elements which are smaller. Therefore one would expect reduced local distortion in all RCSAs. Then why the local environment in ZH-RCSAs is more distorted than in X-RCSAs? The answer is that the atomic size mismatch is reduced only in average, but the actual interatomic distances depends on chemistry. As shown in Fig. 5 In summary, through pair distribution function technique, we investigated the local structure variation in fifteen loosely-packed BCC CSAs, obtained by mixing multiple refractory elements with different size and chemistry at the same crystallographic sites. We found these RCSAs have distorted local structure, characterized by the deviation of their local structure from average structure. However, the length-scales of this deviation are different between the ZH-RCSAs and the X-RCSAs and closely related to the large size and Fermi level differences between Zr/Hf atoms and other constituent atoms. For the X-RCSAs, the LLD only occurs within the first atomic shell, whereas the LLD in the ZH-RCSAs is larger as a consequence of more extensive charge transfer, involving atoms in both the first and second atomic shells. In particular, we revealed that whereas charge transfer greatly reduces the atomic size mismatch in average and contributes to the stability of the solid solution, it broadens the distribution of local interatomic distances. As a final remark, the obtained insights on not only local lattice distortion but also charge transfer effect in the BCC refractory concentrated solid solution alloys are fundamentally important for defect (dislocation, radiation-induced defects) transport physics and therefore have significant practical implications as they offer a quantitative guidance to design strong and radiation-tolerant structural materials. Note: Lattice constant and thermal factor, Uiso, are obtained from the refinement of average structure. Local lattice distortion parameter, , is from the refinement of local structure.
Fig. 1a ,
1ab. The homogeneous diffraction intensity distribution of each diffraction ring in the 2-D diffraction images demonstrates the sample is powder without texture. However, strong intensity decay in the Bragg peaks at high diffraction angles was observed especially in the TiZrNbHfTa, TiZrNbHf and TiVZrNb RCSAs when compared to the seven-element VNbTaTiMoWRe RCSA, indicating large local atomic displacements in these three RCSAs.The 2-D diffraction images were further integrated over azimuthal angles to obtain the one-dimensional (1-D) powder diffraction profiles, as shown inFig. 1c. The diffraction profiles of all fifteen RCSAs can be indexed to a pure BCC phase. Structural information including lattice constants and Debye Waller factor (DWF, Uiso) were extracted from these diffraction patterns based on the Rietveld method 13 , as listed in
Fig. 2a .
2aThe RCSAs without Zr and/or Hf elements (hereafter denoted as X-RCSAs) have well-separated first and second PDF peaks corresponding to the first and second atomic shells. For the ZH-RCSAs, however, an unusual feature can be readily identified in their
RCSAs, the charge transfer in the t2g orbitals follows a linear correlation with the WS volumes whereas charge transfer is not found in the eg orbitals. The distinct feature of charge transfer in different d orbitals is related to the directional nature of the eg and t2g orbitals in BCC structure that the t2g orbital points to the 1 st nearest neighbor, whereas the eg orbital points to the 2 nd neighbors 22 . The results, presented in Fig. 5, reveal that in the RCSAs CTE tends to mitigate the size mismatch among their constituents by transferring electrons from the early transition elements (TEs) which are large to the later TEs which are smaller. We further calculated the average WS radius, rws, for each element in some exemplary RCSAs and compared them with the WS radius of pure metals in Table 2. The WS radii of these pure refractory metals are calculated from their lattice constant according to 3/4 ws 3 = 3 / , where a and N are lattice constant and number of atoms in a unit cell, respectively. We found that in these RCSAs the largest atoms, Zr and Hf, reduce their rws by 4-6%, the rws of the smallest V atom can increase it by 4-6.7%, and the rws change in the medium-sized Ti, Nb and Ta atoms can be zero, positive or negative depending on the difference between their radii and the average radius of all elements in one RCSA, which are consistent with the findings in Fig. 5. The trend of transferring charge from large earlyTEs to small ones qualitatively agrees with the electronegativity difference between two elements. For instance, the large Zr and Hf atoms are more electronegative than other small atoms.
considerably smaller and the B-B distances are considerably larger than the average, even though the average size mismatch is small, because charge transfer is much weaker for the A-A pairs and B-B pairs whereas it is strong for the A-B pairs. Thus charge transfer increases the dispersion in the interatomic distances increases. In particular Zr and Hf are large, and have high Fermi levels. Thus the interatomic distances around Zr and Hf are widely distributed as shown in Fig. 5, resulting in the merger of the 1 st and 2 nd neighbor shells.
Table 1 .
1The TiZrNbHfTa, TiZrNbHf and TiVZrNb RCSAs have larger lattice constants than other RCSA without Zr and/or Hf since a large space is needed to accommodate the large Zr and/or Hf atoms 14 . The DWFs of the VNbTaTiMoWRe, VNbTaMoWRe and NbTaMoWRe RCSAs are comparable with the DWFs of their constituent metals, whereas the rest of RCSAs have DWFs one order of magnitude larger than the pure metal cases for which the phonon thermal factor is dominant 15 . Note that the TiZrNbHfTa, TiZrNbHf and TiVZrNb RCSAs have extraordinarily large DWFs when compared to the rest alloys. These abnormally-large DWFs indicate that the existence of large LLD in these three Zr-and/or Hf-containing RCSAs (hereafter denoted as ZH-RCSAs). But a quantitative analysis remains difficult from the average structure analysis because the DWF contains contributions from both dynamic and static atomic displacements.
Table 1 .
1Structure and LLD parameters.Composition
Lattice constant
(Å)
Uiso (Å 2 )
(%)
TiZrNbHfTa
3.4088
0.0291
0.91
TiZrNbHf
3.4359
0.0292
2.39
TiVZrNb
3.3181
0.0302
2.00
TiVNbMo
3.1846
0.0144
0.01
VNbTaTiMoWRe
3.1678
0.0083
0.08
VNbTaMoWRe
3.1653
0.0077
0.23
NbTaMoWRe
3.1901
0.0064
0.05
VNbTaTiMoW
3.1911
0.0106
0.21
VNbTaTiRe
3.1865
0.0136
0.02
VNbTaTiMo
3.2119
0.0148
0.52
VNbTaRe
3.1750
0.0110
0.22
VNbTaW
3.2043
0.0126
0.32
VNbTaMo
3.1998
0.0126
0.65
VNbTaTi
3.2319
0.0150
0.82
VNbTa
3.2291
0.0146
0.99
Table 2 .
2WS radii of each element in BCC refractory metals and RCSAs.Composition WS radius (Å)Note: WS radius of pure metals is calculated from their lattice constants.Ti
V
Zr
Nb
Mo
Hf
Ta
Ti
1.63
__
__
__
__
__
__
V
__
1.49
__
__
__
__
__
Zr
__
__
1.78
__
__
__
__
Nb
__
__
__
1.63
__
__
__
Mo
__
__
__
__
1.55
__
__
Hf
__
__
__
__
__
1.78
__
Ta
__
__
__
__
__
__
1.63
TiZrNbHfTa
1.67
__
1.70
1.66 __
1.70
1.66
TiZrNbHf
1.67
__
1.71
1.66 __
1.71
__
TiVZrNb
1.63
1.59
1.67
1.62 __
__
__
TiVNbTa
1.60
1.57
__
1.60
__
__
1.60
TiVNbMo
1.57
1.55
__
1.58
1.56
__
__
AcknowledgementsThis work was supported as part of the Energy Dissipation to Defect Evolution (EDDE),
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| []
|
[
"Tracers of the ionization fraction in dense and translucent gas: I. Automated exploitation of massive astrochemical model grids",
"Tracers of the ionization fraction in dense and translucent gas: I. Automated exploitation of massive astrochemical model grids"
]
| [
"Emeric Bron \nLERMA\nObservatoire de Paris\nPSL Research University\nCNRS\nSorbonne Universités\n92190MeudonFrance\n",
"Evelyne Roueff \nLERMA\nObservatoire de Paris\nPSL Research University\nCNRS\nSorbonne Universités\n92190MeudonFrance\n",
"Maryvonne Gerin \nLERMA\nObservatoire de Paris\nPSL Research University\nCNRS\nSorbonne Universités\n75014ParisFrance\n",
"Jérôme Pety \nLERMA\nObservatoire de Paris\nPSL Research University\nCNRS\nSorbonne Universités\n75014ParisFrance\n\nIRAM\n300 rue de la Piscine38406Saint Martin d'HèresFrance\n",
"Pierre Gratier \nLaboratoire d'Astrophysique de Bordeaux\nUniv. Bordeaux\nCNRS\nAllee Geoffroy Saint-Hilaire\nB18N, 33615PessacFrance\n",
"Franck Le Petit \nLERMA\nObservatoire de Paris\nPSL Research University\nCNRS\nSorbonne Universités\n92190MeudonFrance\n",
"Viviana Guzman \nInstituto de Astrofísica\nPontificia Universidad Católica de Chile\nAv. Vicuña Mackenna 48607820436Macul, SantiagoChile\n",
"Jan H Orkisz \nDepartment of Space, Earth and Environment\nChalmers University of Technology\n412 93GothenburgSweden\n",
"Victor De Souza Magalhaes \nIRAM\n300 rue de la Piscine38406Saint Martin d'HèresFrance\n",
"Mathilde Gaudel \nLERMA\nObservatoire de Paris\nPSL Research University\nCNRS\nSorbonne Universités\n75014ParisFrance\n",
"Maxime Vono \nIRIT/INP-ENSEEIHT\nUniversity of Toulouse\nCNRS\n2 rue Charles CamichelBP 712231071Toulouse cedex 7France\n",
"Sébastien Bardeau \nIRAM\n300 rue de la Piscine38406Saint Martin d'HèresFrance\n",
"Pierre Chainais \nUMR 9189 -CRIStAL\nUniv. Lille\nCNRS\nCentrale Lille\n59651Villeneuve d'AscqFrance\n",
"Javier R Goicoechea \nInstituto de Física Fundamental (CSIC)\nCalle Serrano 12128006MadridSpain\n",
"Annie Hughes \nInstitut de Recherche en Astrophysique et Planétologie (IRAP)\nUniversité Paul Sabatier\nToulouse cedex 4France\n",
"Jouni Kainulainen \nDepartment of Space, Earth and Environment\nChalmers University of Technology\n412 93GothenburgSweden\n",
"David Languignon \nLERMA\nObservatoire de Paris\nPSL Research University\nCNRS\nSorbonne Universités\n92190MeudonFrance\n",
"Jacques Le Bourlot \nLERMA\nObservatoire de Paris\nPSL Research University\nCNRS\nSorbonne Universités\n92190MeudonFrance\n",
"François Levrier \nLaboratoire de Physique de l'Ecole normale supérieure\nENS\nUniversité PSL\nCNRS\nSorbonne Université\nUniversité Paris-Diderot\nSorbonne Paris Cité\nParisFrance\n",
"Harvey Liszt \nNational Radio Astronomy Observatory\n520 Edgemont Road22903CharlottesvilleVAUSA\n",
"Karin Öberg \nHarvard-Smithsonian Center for Astrophysics\n60 Garden Street02138CambridgeMAUSA\n",
"Nicolas Peretto \nSchool of Physics and Astronomy\nCardiff University\nQueen's buildingsCF24 3AACardiffUK\n",
"Antoine Roueff \nInstitut Fresnel\nAix Marseille Univ\nCNRS\nCentrale Marseille\nMarseille\n",
"Albrecht Sievers \nIRAM\n300 rue de la Piscine38406Saint Martin d'HèresFrance\n"
]
| [
"LERMA\nObservatoire de Paris\nPSL Research University\nCNRS\nSorbonne Universités\n92190MeudonFrance",
"LERMA\nObservatoire de Paris\nPSL Research University\nCNRS\nSorbonne Universités\n92190MeudonFrance",
"LERMA\nObservatoire de Paris\nPSL Research University\nCNRS\nSorbonne Universités\n75014ParisFrance",
"LERMA\nObservatoire de Paris\nPSL Research University\nCNRS\nSorbonne Universités\n75014ParisFrance",
"IRAM\n300 rue de la Piscine38406Saint Martin d'HèresFrance",
"Laboratoire d'Astrophysique de Bordeaux\nUniv. Bordeaux\nCNRS\nAllee Geoffroy Saint-Hilaire\nB18N, 33615PessacFrance",
"LERMA\nObservatoire de Paris\nPSL Research University\nCNRS\nSorbonne Universités\n92190MeudonFrance",
"Instituto de Astrofísica\nPontificia Universidad Católica de Chile\nAv. Vicuña Mackenna 48607820436Macul, SantiagoChile",
"Department of Space, Earth and Environment\nChalmers University of Technology\n412 93GothenburgSweden",
"IRAM\n300 rue de la Piscine38406Saint Martin d'HèresFrance",
"LERMA\nObservatoire de Paris\nPSL Research University\nCNRS\nSorbonne Universités\n75014ParisFrance",
"IRIT/INP-ENSEEIHT\nUniversity of Toulouse\nCNRS\n2 rue Charles CamichelBP 712231071Toulouse cedex 7France",
"IRAM\n300 rue de la Piscine38406Saint Martin d'HèresFrance",
"UMR 9189 -CRIStAL\nUniv. Lille\nCNRS\nCentrale Lille\n59651Villeneuve d'AscqFrance",
"Instituto de Física Fundamental (CSIC)\nCalle Serrano 12128006MadridSpain",
"Institut de Recherche en Astrophysique et Planétologie (IRAP)\nUniversité Paul Sabatier\nToulouse cedex 4France",
"Department of Space, Earth and Environment\nChalmers University of Technology\n412 93GothenburgSweden",
"LERMA\nObservatoire de Paris\nPSL Research University\nCNRS\nSorbonne Universités\n92190MeudonFrance",
"LERMA\nObservatoire de Paris\nPSL Research University\nCNRS\nSorbonne Universités\n92190MeudonFrance",
"Laboratoire de Physique de l'Ecole normale supérieure\nENS\nUniversité PSL\nCNRS\nSorbonne Université\nUniversité Paris-Diderot\nSorbonne Paris Cité\nParisFrance",
"National Radio Astronomy Observatory\n520 Edgemont Road22903CharlottesvilleVAUSA",
"Harvard-Smithsonian Center for Astrophysics\n60 Garden Street02138CambridgeMAUSA",
"School of Physics and Astronomy\nCardiff University\nQueen's buildingsCF24 3AACardiffUK",
"Institut Fresnel\nAix Marseille Univ\nCNRS\nCentrale Marseille\nMarseille",
"IRAM\n300 rue de la Piscine38406Saint Martin d'HèresFrance"
]
| []
| Context. The ionization fraction in the neutral interstellar medium (ISM) plays a key role in the physics and chemistry of the ISM, from controlling the coupling of the gas to the magnetic field to allowing fast ion-neutral reactions that drive interstellar chemistry. Most estimations of the ionization fraction have relied on deuterated species such as DCO + , whose detection is limited to dense cores representing an extremely small fraction of the volume of the giant molecular clouds (GMC) they are part of. As large field-of-view hyperspectral maps become available, new tracers may be found. The growth of observational datasets is paralleled by the growth of massive modeling datasets, and new methods need to be devised to exploit the wealth of information they contain. Aims. We search for the best observable tracers of the ionization fraction based on a grid of astrochemical models, with the broader aim of finding a general automated method applicable to the search of tracers of any unobservable quantity based on grids of models. Methods. We build grids of models that sample randomly a large space of physical conditions (unobservable quantities such as gas density, temperature, elemental abundances, etc.) and compute the corresponding observables (line intensities, column densities) and the ionization fraction. We estimate the predictive power of each potential tracer by training a Random Forest model to predict the ionization fraction from that tracer, based on these model grids.Results. In both translucent medium and cold dense medium conditions, several observable tracers with very good predictive power for the ionization fraction are found. Many tracers in cold dense medium conditions are found to be better and more widely applicable than the traditional DCO + /HCO + ratio. We also provide simpler analytical fits for estimating the ionization fraction from the best tracers, and for estimating the associated uncertainties. We discuss the limitations of the present study and select a few recommended tracers in both types of conditions. Conclusions. The method presented here is very general and can be applied to the measurement of any other quantity of interest (cosmic ray flux, elemental abundances, etc.) from any type of model (PDR models, time-dependent chemical models, etc.). | 10.1051/0004-6361/202038040 | [
"https://arxiv.org/pdf/2007.13593v1.pdf"
]
| 220,793,509 | 2007.13593 | fd45bda9029da8d9380c1a0664cb11bc9288ebca |
Tracers of the ionization fraction in dense and translucent gas: I. Automated exploitation of massive astrochemical model grids
July 28, 2020
Emeric Bron
LERMA
Observatoire de Paris
PSL Research University
CNRS
Sorbonne Universités
92190MeudonFrance
Evelyne Roueff
LERMA
Observatoire de Paris
PSL Research University
CNRS
Sorbonne Universités
92190MeudonFrance
Maryvonne Gerin
LERMA
Observatoire de Paris
PSL Research University
CNRS
Sorbonne Universités
75014ParisFrance
Jérôme Pety
LERMA
Observatoire de Paris
PSL Research University
CNRS
Sorbonne Universités
75014ParisFrance
IRAM
300 rue de la Piscine38406Saint Martin d'HèresFrance
Pierre Gratier
Laboratoire d'Astrophysique de Bordeaux
Univ. Bordeaux
CNRS
Allee Geoffroy Saint-Hilaire
B18N, 33615PessacFrance
Franck Le Petit
LERMA
Observatoire de Paris
PSL Research University
CNRS
Sorbonne Universités
92190MeudonFrance
Viviana Guzman
Instituto de Astrofísica
Pontificia Universidad Católica de Chile
Av. Vicuña Mackenna 48607820436Macul, SantiagoChile
Jan H Orkisz
Department of Space, Earth and Environment
Chalmers University of Technology
412 93GothenburgSweden
Victor De Souza Magalhaes
IRAM
300 rue de la Piscine38406Saint Martin d'HèresFrance
Mathilde Gaudel
LERMA
Observatoire de Paris
PSL Research University
CNRS
Sorbonne Universités
75014ParisFrance
Maxime Vono
IRIT/INP-ENSEEIHT
University of Toulouse
CNRS
2 rue Charles CamichelBP 712231071Toulouse cedex 7France
Sébastien Bardeau
IRAM
300 rue de la Piscine38406Saint Martin d'HèresFrance
Pierre Chainais
UMR 9189 -CRIStAL
Univ. Lille
CNRS
Centrale Lille
59651Villeneuve d'AscqFrance
Javier R Goicoechea
Instituto de Física Fundamental (CSIC)
Calle Serrano 12128006MadridSpain
Annie Hughes
Institut de Recherche en Astrophysique et Planétologie (IRAP)
Université Paul Sabatier
Toulouse cedex 4France
Jouni Kainulainen
Department of Space, Earth and Environment
Chalmers University of Technology
412 93GothenburgSweden
David Languignon
LERMA
Observatoire de Paris
PSL Research University
CNRS
Sorbonne Universités
92190MeudonFrance
Jacques Le Bourlot
LERMA
Observatoire de Paris
PSL Research University
CNRS
Sorbonne Universités
92190MeudonFrance
François Levrier
Laboratoire de Physique de l'Ecole normale supérieure
ENS
Université PSL
CNRS
Sorbonne Université
Université Paris-Diderot
Sorbonne Paris Cité
ParisFrance
Harvey Liszt
National Radio Astronomy Observatory
520 Edgemont Road22903CharlottesvilleVAUSA
Karin Öberg
Harvard-Smithsonian Center for Astrophysics
60 Garden Street02138CambridgeMAUSA
Nicolas Peretto
School of Physics and Astronomy
Cardiff University
Queen's buildingsCF24 3AACardiffUK
Antoine Roueff
Institut Fresnel
Aix Marseille Univ
CNRS
Centrale Marseille
Marseille
Albrecht Sievers
IRAM
300 rue de la Piscine38406Saint Martin d'HèresFrance
Tracers of the ionization fraction in dense and translucent gas: I. Automated exploitation of massive astrochemical model grids
July 28, 2020Received 27 March 2020 / Accepted 6 July 2020Astronomy & Astrophysics manuscript no. paper_RF_ionizAstrochemistryISM: moleculesISM: cloudsISM: lines and bandsMethods: statisticalMethods: numerical
Context. The ionization fraction in the neutral interstellar medium (ISM) plays a key role in the physics and chemistry of the ISM, from controlling the coupling of the gas to the magnetic field to allowing fast ion-neutral reactions that drive interstellar chemistry. Most estimations of the ionization fraction have relied on deuterated species such as DCO + , whose detection is limited to dense cores representing an extremely small fraction of the volume of the giant molecular clouds (GMC) they are part of. As large field-of-view hyperspectral maps become available, new tracers may be found. The growth of observational datasets is paralleled by the growth of massive modeling datasets, and new methods need to be devised to exploit the wealth of information they contain. Aims. We search for the best observable tracers of the ionization fraction based on a grid of astrochemical models, with the broader aim of finding a general automated method applicable to the search of tracers of any unobservable quantity based on grids of models. Methods. We build grids of models that sample randomly a large space of physical conditions (unobservable quantities such as gas density, temperature, elemental abundances, etc.) and compute the corresponding observables (line intensities, column densities) and the ionization fraction. We estimate the predictive power of each potential tracer by training a Random Forest model to predict the ionization fraction from that tracer, based on these model grids.Results. In both translucent medium and cold dense medium conditions, several observable tracers with very good predictive power for the ionization fraction are found. Many tracers in cold dense medium conditions are found to be better and more widely applicable than the traditional DCO + /HCO + ratio. We also provide simpler analytical fits for estimating the ionization fraction from the best tracers, and for estimating the associated uncertainties. We discuss the limitations of the present study and select a few recommended tracers in both types of conditions. Conclusions. The method presented here is very general and can be applied to the measurement of any other quantity of interest (cosmic ray flux, elemental abundances, etc.) from any type of model (PDR models, time-dependent chemical models, etc.).
Introduction
The so-called neutral component of the interstellar medium, despite being shielded from EUV (13.6 to 124 eV) stellar photons able to ionize hydrogen, retains a small ionization fraction (x(e − ) = n(e − )/n H ). The ionization mechanism depends on the type of region: FUV (6 to 13.6 eV) photons ionizing C and S Send offprint requests to: E. Bron, e-mail: [email protected] in the low A V surface layer of clouds, or cosmic rays, X rays, shocks, etc., in the densest parts. As a result, ionization fractions range between ∼ 10 −4 in low A V cloud surfaces and down to ∼ 10 −9 in dense cores (e.g. Goicoechea et al. 2009;Draine 2011).
This ionization fraction controls several key aspects of neutral interstellar clouds. It determines the degree of coupling of the gas to the magnetic field: the neutrals, accounting for most of the mass of the fluid, are only indirectly sensitive to the presence of a magnetic field through their friction with the ions that remain coupled to the field, a process called ion-neutral friction. This coupling can provide a significant magnetic support against gravitational collapse of dense cores despite the low ionization fractions values found there, between 10 −9 and 10 −7 (Mestel & Spitzer 1956;Mouschovias 1976;Basu & Mouschovias 1994). The ionization fraction also controls the onset of the magnetorotational instability (Balbus & Hawley 1991), the main mechanism of angular momentum transport in accretion disks. Moreover, the gas phase chemistry in dense molecular clouds is to a large extent driven by fast ion-neutral reactions (Herbst & Klemperer 1973;Oppenheimer & Dalgarno 1974). The build-up of chemical complexity thus depends on the ionization fraction of the medium. Finally, some common molecular tracers with high dipole moments, such as HCN and HCO + , have high inelastic collision cross sections with electrons, and their excitation can be significantly affected by electron collisions for ionization fractions 10 −5 (Black & van Dishoeck 1991;Liszt 2012;Liszt & Pety 2016;Goldsmith & Kauffmann 2017). This makes the interpretation of their emission (e.g. to estimate gas density) sensitive to our knowledge on the local ionization fraction .
Direct observational estimation of the ionization fraction in neutral clouds is difficult, except in very specific regions (e.g. Goicoechea et al. 2009;Cuadrado et al. 2019, at the dissociation front in a photodissociation region). Direct estimation of the total charge accounted for by observable molecular ions in molecular clouds only yields a loose lower limit (e.g. Miettinen et al. 2011). Indirect methods based on tracers that are chemically sensitive to the ionization fraction have thus been commonly used. These methods have mostly involved measuring the deuterium fractionation through abundance ratios involving simple molecular ions like DCO + /HCO + (Guelin et al. 1977(Guelin et al. , 1982Dalgarno & Lepp 1984;Caselli et al. 1998) or N 2 D + /N 2 H + which is less affected by depletion (Caselli 2002). The idea is that the deuterium enrichment (defined as H 2 D + /H + 3 ), initiated by the exchange reaction H + 3 + HD H 2 D + + H 2 (1) at low temperature, is limited by electronic dissociative recombination of H 2 D + , and that the resulting ratio is transmitted (with a known prefactor) to the deuteration fraction of other molecules such as HCO + . Using such tracers, the ionization fraction is deduced either by using approximate analytical formulae representing simplified networks Miettinen et al. 2011;Caselli 2002), or by adjusting an astrochemical model including a full chemical network to the observations, using stationary chemical models (Williams et al. 1998;Bergin et al. 1999;Caselli et al. 2002;Fuente et al. 2016), time-dependent ones (Maret & Bergin 2007;Shingledecker et al. 2016), or PDR models (Goicoechea et al. 2009). Despite the variety of determination methods, using different deuterated molecules, only very few works have proposed using other tracers than deuterated species. For instance, Flower et al. (2007) proposed the C 6 H − /C 6 H ratio as a tracer of the ionization fraction, and Fossé et al. (2001) have investigated the relationship between the cyclic-to-linear ratio of C 3 H 2 and the ionization fraction. Deuteration-based approaches however suffer from several limitations due to the fact that they depend on other physical or chemical parameters that need to be determined independently. The initial deuteration reaction (Eq. 1) is sensitive not only to the gas temperature but also to the essentially unmeasurable orthoto-para ratio of H 2 (Pagani et al. 1992(Pagani et al. , 2011Shingledecker et al. 2016). Indeed, the endothermicity of the reaction in the backward direction (192K) is very close to the J=1 to J=0 energy difference of H2 (170.5K). Then, even a small fraction of o-H2 (J=1) contributes to H2D+ destruction (with a reduced endothermicity of 20K) and restricts the deuteration process. In addition, ratios such as DCO + /HCO + are linked to H 2 D + /H + 3 through reactions with neutral species like CO. The estimated deuterium fraction is therefore sensitive to the depletion factors of carbon, oxygen and nitrogen that are not easy to evaluate (Caselli 2002). Moreover, deuterated tracers such as DCO + are typically only detectable in cold dense cores, representing only a tiny fraction of the observable area of a giant molecular cloud (GMC). Deuteration-based approaches are thus inadequate for an unbiased characterization of the conditions in GMCs as a whole.
Despite the common use of advanced chemical models computing the abundances of hundreds of species, the observed tracers to which these models are compared to estimate x(e − ) (deuterated molecules such as DCO + ) are still those initially proposed based on analytical reasoning using simplified chemical networks. The wealth of data produced in large chemical model grids remains largely unexploited. Their exploration of wide parameter spaces might reveal less intuitive but more efficient tracers. Based on this approach, we propose here a general and largely automatic method to identify the best observational predictors of the ionization fraction, when other important parameters such as the gas density, temperature or H 2 ortho-topara ratio are unknown. We apply this method to propose new predictors of the ionization fraction as a function of the molecular cloud conditions. We use simple stationary chemical models with a complete up-to-date chemical network (Roueff et al. 2015), and use molecular ratios (column density ratios or integrated line intensity ratios) as observable tracers from which we seek to predict the ionization fraction. We base our investigation on the observed range of physical conditions and detected tracers in the IRAM-30m Large Program ORION-B (Outstanding Radio-Imaging of OrioN B, co-PIs: J. Pety and M. Gerin) 1 . In this program, we imaged 5 square degrees towards the southern part of the Orion B giant molecular cloud over most of the 3 mm atmospheric window Gratier et al. 2017;Orkisz et al. 2017;Bron et al. 2018;Orkisz et al. 2019;Roueff et al. 2020;Gratier et al. subm.).
In the context of dense cores, the ionization fraction is linked with the cosmic ray ionization rate (CRIR) and both are often studied from the same molecular ratios (although direct tracers of the cosmic ray ionization rate can also be used in more diffuse medium, in particular H + 3 , e.g. Indriolo & McCall 2012;Le Petit et al. 2016). In our context of the Orion B giant molecular cloud, where UV illumination controls the ionization fraction in large parts of the cloud, we focus here on the question of estimating the ionization fraction only, independently of the source of ionization. The task of tracing the cosmic ray ionization rate (which has also been attempted using astrochemical model grids, e.g. Barger & Garrod 2020) will be considered in a future application of the method presented here.
In this first article, we present a generic method to find the best tracers of an unobservable physical parameter and apply it to the search of new tracers of the ionization fraction among the species that are detectable in the ORION-B dataset. The observational application of the tracers found here to study the ionization fraction in the Orion B molecular cloud will be presented in a second paper (Guzman et al. in prep.).
In Sect. 2, we describe our general statistical method for determining the best predictors of a given unobservable parameter based on the results of a model grid. In Sect. 3, we present the models used in this study for the search of ionization fraction tracers. We then present the ranking of observable predictors in Sect. 4. For ease of application of our results, we provide in Sect. 5 analytical fit formulae to deduce the ionization fraction from each of the proposed best predictors. We finally discuss our results in Sect. 6 and present our conclusions in Sect. 7.
Method
Both observable line intensities (or column densities) on the one hand, and the ionization fraction x(e − ) on the other hand, depend on multiple, unobservable physical parameters (e.g. gas density, elemental abundances, cosmic ray flux, ...). Our goal is to find reliable relationships between observable quantities and ionization fraction, despite lacking estimations of these hidden physical parameters. To do this, we first run model grids covering the whole possible parameter space. We then use a flexible regression method to fit x(e − ) as a function of one of the potential observational tracers through the whole grid of chemical models. This means that we treat the effects of the variations of the hidden parameters as sources of noise on the prediction of the ionization fraction. Finally, we use a quantitative measurement of the fit quality as an estimate of the predictive power of each potential tracer. These estimates are used to rank the tracers and highlight the most powerful predictors of the ionization fraction. The fitted models for the best tracers will provide ready-to-use tools to be applied to observations.
Predicting the ionization fraction from one ratio of line intensities (or column densities) with Random Forests
Any a priori information could easily be included in the method by sampling the parameters of the model grid according to a specific prior distribution. However, we wish to minimize the amount of a priori information injected in the method and to avoid making assumptions on the shape of the distributions of physical parameters (e.g. gas density, elemental abundances, cosmic ray flux, ...). We thus build a model grid that samples uniformly the possible range of values (see Sect. 3). Our model grids provide us with a dataset comprising ionization fraction values and corresponding values of observable quantities. We will consider line intensity ratios or column density ratios as our observable quantities in this paper. The hidden physical parameter values introduce a non deterministic aspect to the relationship between x(e − ) and the observables: models might have identical values of an observable but different x(e − ) if the underlying physical parameters are different. Learning to predict x(e − ) from a given observable is then a regression problem, with the uncertainty introduced by the hidden physical parameters playing the role of noise. Determining the best tracers of x(e − ) is thus equivalent to finding observables for which the relationship to x(e − ) is least affected by this noise (i.e. by the hidden physical conditions). This means finding the observables for which the most accurate regression model can be found.
For this regression problem, we choose to use Random Forests (Breiman 2001) because their flexibility makes it possible to fit general non-linear shapes, while their simplicity provides reasonable computational costs. This makes the method presented here very general and applicable to finding tracers of other physical parameters without any assumption on the shape of the relationship between the tracers and the target parameter.
We will use RF for Random Forest in the rest of the paper. RF regression models are based on the concept of regression trees (Breiman et al. 1984), where a succession of binary decisions are made based on the input variables (e.g. x 3 < 2 or ≥ 2) and constant values are predicted in each of the subsets of the partition that the decision tree defines. While such decision trees are easily interpretable, they require large tree depths to be flexible but are prone to overfitting if this depth is too large. RF tackle this overfitting problem by using the simple idea that multiple overfitted regression models will, when averaged, give a better prediction as long as the errors they individually make are uncorrelated between models. In a RF, the individual trees are made as independent as possible by introducing randomness in two aspects: 1) the building of each tree only considers a random subset of the input variables, and 2) each tree is given a bootstrapped sample (i.e. drawn by random sampling with replacement from the original dataset, Breiman 1996) instead of the original full sample. This way, the datasets seen by the different trees are independent and each tree only sees a subset of the dataset (bootstrapped datasets typically contain only 63% of the points of the original dataset as repetition is allowed). This provides a very flexible regression model, which retains some of the interpretability of decision trees. RF have thus quickly become a standard method in Machine Learning (see e.g. Hastie et al. 2001). In addition, they allow to estimate the generalization error of the fit (i.e., the error made when predicting data not seen during training): as each tree has only seen a random bootstrapped sample from the data, it is possible to estimate for each datapoint a partial prediction using only the trees that have not seen this datapoint during training. As the sample seen by a given tree is called a bag, these partial predictions are called outof-bag predictions (OOB). Gratier et al. (subm.) also use RF in the context of the interstellar medium and introduce the method in detail.
We thus train RF regression models for each observable (using only one observable at a time) and estimate the accuracy of the regression models. This accuracy is taken as an estimate of the predictive power of the observable quantity considered, for the purpose of predicting x(e − ). The different observables can then be ranked according to this predictive power estimate. The accuracy of the regression model is estimated with the OOB R 2 R 2 = 1 − SS res SS tot with the sums of squares
SS res = i y pred i − y true i 2 and SS tot = i y true − y true i 2 ,
where the index i runs across data points (individual model results), y true i is the true value of ionization fraction (computed by the chemical model), y pred i is the OOB prediction value from the RF, and y true is the average of the (true) ionization fraction over the model grid. This coefficient R 2 gives the fraction of the total ionization fraction variance (across the full model grid) that the RF model can explain from the given observable predictor alone (i.e. it measures the fractional decrease from the initial variance of x(e − ) to the variance of the residuals).
It is thus <1, with 1 representing perfect prediction (zero residual variance). Note that it can take a negative value when the model performs worse than predicting a constant value set at the average x(e − ) of the dataset. A value of 0 indicates a performance equivalent to this constant prediction of the average. This R 2 value is used for the ranking of tracers. For information, we Table 1. Range of physical parameters explored for each of our two classes of medium: gas density n H , gas temperature T gas , incident FUV radiation field intensity G 0 , line-of-sight visual extinction A V , cosmicray ionization rate ζ, H 2 ortho-to-para ratio OPR H 2 , depletion factor and sulfur gas-phase elemental abundance [S]. translucent medium cold dense medium n H [cm −3 ] 3 × 10 2 − 3 × 10 3 10 3 − 10 6 T gas [K] 15 − 100 7 − 20
G 0 1 − 1000 1 A V 2 − 6 5 − 20 ζ [s −1 ] 10 −17 − 10 −15 10 −17 − 10 −16 OPR H 2 0.1 − 3 10 −4 − 10 −1 depletion factor 1 1 − 10 [S]
1.86 × 10 −8 − 1.86 × 10 −5 1.86 × 10 −8 − 1.86 × 10 −5 also provide below the root mean square error
RMSE = 1 N i y true i − y pred i 2 ,
where N is the number of chemical models in our grid. The RMSE is completely univocally related to the R 2 value, but is more interpretable in terms of the amplitude of typical errors. We also provide the maximum absolute error
max. abs. err. = max i y true i − y pred i .
This quantity, estimating the maximum error made by our regression model, is not guaranteed to converge when increasing the size of the dataset. It should thus not be interpreted further than being the largest error we observed in our limited-size sample. The RF model depends on a few internal parameters (number of trees, maximum depth of trees, etc...). Their values can affect the quality of the model and its tendency to overfit. We used a number of trees in the forest N trees = 400 and a maximum tree depth d max = 4. The procedure used to select these values is described in Appendix A. Our tests show that this optimization scheme is not critical for our purpose: while the choice of parameter values does affect the quality of the best fit RF model, it does not change significantly the ranking of the predictors that we deduce from it.
The pipeline tool implementing this procedure is available at [link to be added before publication].
Chemical models
We use here the chemical code presented in Roueff et al. (2015) to study isotopic fractionation of deuterium, carbon and nitrogen compounds. Single zone models with fixed density, temperature, visual extinction, radiation field, cosmic ray ionization rate, ortho-to-para H 2 ratio and depletion factors are computed at steady state.
We consider for the present study a chemical network including deuterium, and isotopic carbon and oxygen species where the deuterium, carbon and oxygen fractionation reactions have been introduced following the recent determinations of exothermicities by Mladenović & Roueff (2017). We introduce in particular D 13 CO + . Apart from these specific fractionation reactions, the chemistry of isotopically substituted species is built automatically from the chemical network of the major components. The chemical reactions involving one single carbon-containing reactant and one single carbon-containing product are duplicated with the same reaction rate coefficient. Simple statistical assumptions are introduced when two carbon containing molecules are implied in the reaction. Consider for example the case of the reaction CX + CY → CX + CY , taking place with a reaction rate coefficient k. The reactions introduced for the isotopically substituted species are the following:
13 CX + CY → 13 CX + CY with k/2 13 CX + CY → CX + 13 CY with k/2 CX + 13 CY → CX + 13 CY with k/2 CX + 13 CY → 13 CX + CY with k/2 13 CX + 13 CY → 13 CX + 13 CY with k Such a procedure leads to an ensemble of 310 species linked through 8711 chemical reactions.
These models allow us to compute observable column density ratios for the hundreds of species included. Although commonly derived by observers, column densities are not the primary observable quantities, and we thus also compute line intensity ratios. To do so, we post-process the results of our chemical models using a non-LTE excitation and radiative transfer model (RADEX, van der Tak et al. 2007) assuming a typical linewidth of 1 km/s (observed linewidths in the Orion B are typically of a few km/s ). Results based on column density ratios and based on line intensity ratios will be presented separately in the following sections.
It is unlikely that a single tracer will provide a good estimate of the ionization fraction x(e − ) in all possible physical conditions. Either the tracer will lose its relationship with x(e − ) in some conditions, or it might be too weak to be observable in other conditions. We thus decided to divide the range of possible conditions into subregions corresponding to the different types of environments found in GMCs Bron et al. 2018). We focus on two kinds of environments : the translucent medium and the cold and dense gas, and we derive separate rankings of tracers for these two environments. The range of physical conditions explored for each of these environments are chosen based on our previous studies of Orion B and listed in Table 1.
In both grids, the gas density and gas temperatures were varied covering the typical ranges for translucent medium (3×10 2 − 3 × 10 3 cm −3 , 15-100 K) and cold and dense medium (10 3 − 10 6 cm −3 , 7-20 K). In the translucent model grid, external FUV photons still play an important role in the chemistry and in controlling the ionization fraction. This FUV illumination is controlled through an external FUV field strength G 0 (see e.g. Hollenbach & Tielens 1999, p. 177) and an extinction A V representing the amount of shielding between the FUV source and the gas under consideration (also used as the depth of the slab when computing line intensities). We take into account self-shielding of H 2 by using the approximate expression of Draine & Bertoldi (1996) and introduce also the shielding of CO by H 2 from Heays et al. (2017). We consider lower extinctions and higher G 0 values in translucent medium (A V in the range 2-6, G 0 of the external field in the range 1 -1000) than in cold dense medium (A V in the range 5-20, external G 0 set to 1). We explore average to moderately strong FUV illumination values in the low density translucent grid. Regions with both high density high FUV illumination correspond to dense photodissociation regions (PDR), in which strong chemical and physical stratification on small spatial scale is critical. These regions would thus require the use of complete PDR models, such as the Meudon PDR Code (Le Petit et al. 2006). We thus did not explore this type of conditions in the present study.
Given the uncertainties about the cosmic ray ionization rate in molecular clouds (Lepp 1992;McCall et al. 2003;Indriolo et al. 2007), we consider the range of value 10 −17 − 10 −15 s −1 . In the cold gas conditions, in order to account for the reduced cosmic ray fluxes (Padovani et al. 2009), we limit this range to 10 −17 − 10 −16 s −1 . As sulfur can be an important contributor of electrons in neutral gas but has a highly uncertain gas-phase elemental abundance (Agúndez & Wakelam 2013;Goicoechea et al. 2006), we explore in both grids values of [S], the relative sulfur abundance with respect to H, in the range 1.86 × 10 −8 − 1.86 × 10 −5 .
We also explore ranges of H 2 ortho-para ratio (OPR H 2 ) which impact significantly two reactions: i.e. H 2 D + + o-H 2 → H + 3 + HD (Pagani et al. 1992) where the energy endothermicity is reduced to 61.5 K (compared to 232K with p-H 2 ) and N + + o-H 2 → NH + + H which is slightly endothermic (∼ 44 K) whereas N + + p-H 2 → NH + + H is more strongly endothermic (∼ 170 K), as first emphasized by Le Bourlot (1991). For this reaction, we follow the prescription of Dislaire et al. (2012) which is derived from experimental results. We use higher values of the OPR in the warmer translucent medium (0.1−3) than in cold dense gas (10 −4 −10 −1 ). Finally, cold dense cores offer conditions where molecules can freeze out on dust grains, depleting the gas phase abundances of elements such as C and O. In our cold dense medium grid, we thus in addition explore depletion factors going from 1 (no depletion) to 10 (C elemental abundance 10 times lower than the reference values) with a constant C/O elemental ratio value of 0.6 (the elemental abundance of carbon is taken to be [C]= 1.32 × 10 −4 when there is no depletion). Other parameters that might have an impact (although second order compared to the parameters considered here), such as variations of the metal elemental abundances or PAH abundance, were not considered in this study. The gas-phase elemental abundances for metals (relative to H) are taken to be [Fe]= 1.5 × 10 −8 , [Cl]= 1.8 × 10 −7 , [Si]= 8.2 × 10 −7 , [F]= 1.8 × 10 −8 , [Ar]= 3.29 × 10 −6 .
For each medium type, a set of 5000 models was run, sampling randomly and uniformly within the chosen parameter space. The adequacy of this number of models for our purpose is ascertained later when estimating the uncertainties on our results.
Given the variation by orders of magnitude both in the parameter values and the computed observables, we choose to work with the logarithm of all quantities. We sampled uniformly on the logarithm of each parameter within the ranges indicated above. The method described in Sect. 2 is applied on the logarithm of all quantities (i.e., training RF models using the logarithm of column density ratios or line intensity ratios to predict the logarithm of x(e − )). Note that representative error values such as the RMSE on logarithms are equivalent to representative error factors on the actual quantity. These corresponding error factors are given in parenthesis in the result tables of the following sections.
To get rid of possible instrumental, calibration and other source geometry effects, we choose to work only on ratios of observable quantities (either column density ratios or line intensity ratios). In the following, we use the term tracers for ratios of observable quantities.
Among all the species computed in our chemical model, we made a selection of species that are detected in the radio observa- Table 2. We list here the shorthand names, full quantum number designation, and frequency of the molecular lines we considered.
Short name Full quantum numbers
Frequency (GHz) 13 CO (1-0) J = 1 → J = 0 110.201354
C 18 O (1-0) J = 1 → J = 0 109.782173 HCO + (1-0) J = 1 → J = 0 89.188396 HCN (1-0) J = 1 → J = 0 88.631602 HNC (1-0) J = 1 → J = 0 90.663568 CN (1-0) N = 1, J = 3/2, F = 5/2 → N = 0, J = 1/2, F = 3/2 113.490970 C 2 H (1-0) N = 1, J = 1/2, F = 0 → N = 0, J = 1/2, F = 1 87.40716 CS (2-1) J = 2 → J = 1 97.980953 SO (3-2) J = 3, N = 2 → J = 2, N = 1 99.299870 HCS + (2-1) J = 2 → J = 1 85.347890 CF + (1-0) J = 1 → J = 0 102.587533 H 2 CS (3-2) J = 3, K p = 0, K o = 3 → J = 2, K p = 0, K o = 3 103.040452 DCO + (1-0) J = 1 → J = 0 72.039354 N 2 H + (1-0) J = 1, F1 = 2, F = 3 → J = 0, F1 = 1, F = 2 93.173764
tions of the ORION-B project and potentially linked to the ionization fraction. Our search for the best tracers is made among the ratios of these selected species. For the translucent medium condition, we selected 13 CO, C 18 O, HCO + , HCN, HNC, CN, C 2 H, CS, SO, H 2 CS, HCS + , CF + . The search for a best ratio was thus done among 66 possible column density ratios (and 66 line intensity ratios). For dense cold medium conditions, we considered the same selection with the addition of DCO + and N 2 H + . We thus had 91 possible column density ratios (and 91 line intensity ratios) in this case. For line intensities, the exact transition and frequency considered for each species are listed in Table 2. In the RADEX computations of line intensities, we account for collisional excitation with electrons (using the ionization fraction computed by the chemical model) for species for which collisional data with electrons are available in RADEX (HCO + , HCN and C 2 H). We note that for CN, excitation by electrons was not included as in the current version of RADEX, the collisional data that includes the hyperfine structure of CN does not include collisions with electrons. We chose to privilege the fact of accounting for the hyperfine structure here.
Tracers rankings
We applied the method described in Sect. 2 to the two chemical model grids (translucent medium conditions and cold dense medium conditions) presented in Sect. 3 in order to obtain a ranking of the selected potential tracers according to their usefulness for predicting the ionization fraction. Figure 1 presents the predictive power (estimated as the R 2 of a RF fit) of each tracer for the best 20 tracers. The left panel shows the result when taking column density ratios as observable quantities, and the right panel the results when considering line intensity ratios. We see that the ranking is similar in both cases, suggesting that excitation and radiative transfer only have a moderate effect on the relationship between these tracers and the ionization fraction. A more complete ranking (covering all tracers having R 2 > 0.5) is given in ( 1 0 the results tables of this Section and of Sect. 5 are placed in Appendix B) 2 .
Translucent medium
In both cases, about ten different ratios are found to be each able to explain more than 80% of the ionization fraction variance (R 2 > 0.8). We emphasize that this means that an accurate prediction of the ionization fraction is possible from each of these tracers despite not knowing the values of the 7 physi-contours in shades of blue indicating iso-PDF contours encompassing 25%, 50%, and 75% of the distribution) and the prediction of the fitted RF model (solid red line), which is found in this case to explain 95.7% of the ionization fraction variance in our grid. The remaining scatter around the relationship represents the effect of ignoring all other parameters (gas density, temperature, UV field, H 2 OPR,...). The best ranked line intensity ratio (C 2 H (1-0) / HCN (1-0)) is similarly shown on the right panel of Fig. 2 with the corresponding fitted RF model (solid red line).
In translucent gas, the ionization is still dominated by the effect of external FUV photons ionizing carbon (and to a lesser extent sulfur and chlorine), and is slowly decreasing as the total extinction increases. In the conditions covered by our translucent grid, we find x(e − ) ranging from 2×10 −4 to 2×10 −7 . C 2 H, which we find in several of the best ratios, is known to be enhanced in FUV illuminated environments (Pety et al. 2005;Cuadrado et al. 2015;Guzmán et al. 2015;Gratier et al. 2017;Pety et al. 2017): as explained in Beuther et al. (2008), C 2 H traces the amount of carbon not locked into CO, and is thus sensitive to the FUV flux through CO photodissociation and the presence of C and C + at a significant abundance level. In our translucent medium model grid, we indeed find C + to be the main charge carrier and thus to very strongly correlate with x(e − ). H + and H + 3 may also contribute to the electronic fraction in environments where the cosmic ionization reaches values above 10 −16 s −1 (Le Petit et al. 2016). However, C + , an open shell ion, is chemically reactive with various molecules, except H 2 3 , and is at the origin of a complex chemistry with insertion of carbon atoms. C + itself is not straightforwardly detectable as its fine structure transition at 158 µm requires spaceborne or airborne observations. But we may expect that molecules involving C + in the initial chemical steps allow to probe the electronic fraction. Our finding of ratios involving C 2 H (relative to e.g. HCN, HNC or CN) as good proxies of the electronic fraction is a natural consequence of the relevance of C + as one of the main positive charge carriers. The initial step of C 2 H formation involves indeed the C + + CH → C + 2 + H reaction, followed by subsequent reactions with H 2 up to C 2 H + 2 , which recombines to form C 2 H. Molecules such as HCN, HNC or CO and its isotopologues, on the other hand, are saturated stable molecules which scale with column density. As a result, ratios such as C 2 H/HCN, whose transitions are easily detectable, offer a convenient diagnostic tool of the electronic fraction in translucent medium. The electronic fraction is then, as shown in Fig. 2 (left panel), an increasing function of the C 2 H/HCN ratio. CF + is another proxy of C + , as described in Neufeld et al. (2006); Guzmán et al. (2012), and ratios involving this ion are also found here to be good tracers of the ionization fraction. However, this ion is relatively scarce since it involves fluorine, which has a low relative abundance to H 2 and has only been detected in PDR environments so far (detectability issues are investigated in Sect. 6.2).
In order to estimate the reliability of our results and determine if our 5000-model grid is sufficient to explore the chosen parameter space for our purpose, we compute errorbars on the predictive power estimate (the R 2 of the RF fit). To do so, we use 10-fold cross validation: the model grid is randomly split into 10 parts, and for each of these parts, a RF model is trained on the other 9 parts and tested on the remaining part which it has not seen during training. From these 10 estimates of the R 2 (all made on samples unseen during training), a reliable estimate of the predictive power on unseen data is made, as well as an estimate of the standard error on this predictive power. The corresponding error bars are also shown in Fig. 1. For most of the points, the errorbars are smaller than the marker, and the inset on the left panel presents a zoom on the first five ratios, showing the magnitude of the errorbars. This shows that the uncertainty induced by the finite size of our model grid in negligibly small and that our 5000-model grid is sufficient for our purpose. However, this conclusion should be taken with some caution as it has been shown that there exists no unbiased estimator of the variance of a cross-validation estimate (Bengio & Grandvalet 2004) and that a naive estimation of this variance tends to underestimate it by a factor of up to 4 (Varoquaux 2018). Figure 3 presents the ranking of the best tracers in cold dense conditions (cf. Table 1), for both column density ratios (left panel) and line intensity ratios (right panel). Error bars computed by the cross validation procedure described above are also shown, confirming that the size of our model grid is sufficient for estimating the quality of the fits based on each tracer. We see that the R 2 values of the best tracers are slightly lower than in the translucent medium case, indicating slightly stronger degeneracies with unknown parameters in this case (note that depletion was varied in this cold dense medium, in addition to the parameters varied in the translucent medium grid). However, we still find several tracers explaining more than 80% of the variance in x(e − ). The best column density ratio is here found to be CN/N 2 H + . The cold dense environments are essentially ionized through cosmic rays and secondary UV photons induced by cosmic rays. Electrons are primarily produced by cosmic ray ionization of H 2 and destroyed in the efficient dissociative recombination reactions of the various molecular ions. He + ions, produced by cosmic ray ionization of He, are also particularly efficient in ionizing the molecular reservoirs, CO, HCN, N 2 , H 2 O. This contributes to forming atomic ions, in addition to the H + 3 molecular ion resulting from H 2 ionization and the other stable molecular ions resulting from proton transfer of H + 3 with stable molecules giving ions such as H 2 D + , HCO + , H 3 O + , or N 2 H + . The ionization carriers are then shared amongst several different species, going from the simple atomic ions that do not react with H 2 (i.e. C + , S + , H + ) and closed shell molecular ions such as H + 3 , HCO + , H 3 O + , and N 2 H + . Molecular ions are principally destroyed by dissociative recombination reactions whereas atomic ions rather react with the present neutral molecules since radiative recombination is not efficient. One can thus expect that molecular ions are inversely proportional to the electron abundances, as seen with the CN/N 2 H + ratio which is found to increase monotonically with x(e − ) (cf. Fig. 4, left panel).
Cold dense medium
As in the translucent case, we find slightly lower R 2 values for the best line intensity ratios than for the best column density ratio. However, we find a few ratios that have better scores as intensity ratios than as column density ratios. In particular, while the 13 CO/HCO + column density ratio is found to be a poor predictor of the ionization fraction, the 13 CO (1-0) / HCO + (1-0) line intensity ratio appears as one of the best tracers. Contrary to the previous cases, this is entirely an excitation effect. The abundance ratio of 13 CO/HCO + is found to be mostly uncorrelated with x(e − ) and mostly constant in the cold dense medium model grid (ratio of about 10 3 with a typical scatter of a factor of 2-3). However, HCO + and 13 CO have strongly different critical densities: ∼ 2 × 10 5 cm −3 for HCO + (in cold dense medium condi- tions, x(e − ) is too low for electron collisions to play a significant role) in comparison to ∼ 2 × 10 3 cm −3 for 13 CO. In the range of densities considered in the cold dense medium grid (10 3 − 10 6 cm −3 ), 13 CO excitation is thus mostly at local thermodynamic equilibrium (LTE) and its emissivity per molecule is thus constant with gas density, while HCO + is transitioning from the sub-thermally excited regime to the LTE regime, and its emissivity per molecule thus increases with density. The 13 CO (1-0) / HCO + (1-0) ratio thus decreases with gas density. On the other hand, we find the gas density to be very strongly anti-correlated with x(e − ) in these conditions (with a typical scatter of a factor of ∼ 3), as cosmic ray ionization is the dominant source of ionization here and the recombination rates per ion scale with the gas density. These two effects combine to give a 13 CO (1-0) / HCO + (1-0) ratio that increases with x(e − ) with a relatively tight correlation (cf. Fig. C.4, top right panel). Fig. 4 shows the relation between x(e − ) and the best column density ratio, CN/N 2 H (left panel), and the best line intensity ratio, CF + (1-0) / DCO + (1-0) (right panel). We see a larger scatter than in Fig. 2, but a clear relationship is still found.
CF + (1 0) / DC O + (1 0) 13 CO (1 0) / HC O + (1 0) CN (1 0) / N 2H + (1 0) C 2H (1 0) / N 2H + (1 0) HC O + (1 0) / CF + (1 0) C 2H (1 0) / HC N (1 0) 13 CO (1 0) / DC O + (1 0) C 2H (1 0) / HN C (1 0) C 18 O (1 0) / DC O + (1 0) C 2H (1 0) / DC O + (1 0) CF + (1 0) / N 2H + (1 0) C 2H (1 0) / HC O + (1 0) CN (1 0) / DC O + (1 0) C 18 O (1 0) / HC O + (1 0) HC O + (1 0) / CN (1 0) HC N (1 0) / CN (1 0) HN C (1 0) / CN (1 0) SO (3 2) / C 2H (1 0) SO (3 2) / CN (1 0) CS (2 1) / C 2H (1 0
We note that the classical DCO + /HCO + ratio does not appear among the best tracers found here for cold dense medium conditions. This point is discussed in Sect. 6.1.
Analytical fit formulas
If possible, we recommend using the RF models described in the previous section when attempting to estimate x(e − ) from one of the best tracers listed above. However, the provided datafiles are dependent on a specific implementation of Random Forests (the scikit-learn module for Python, Pedregosa et al. 2011). For a simpler and more persistent solution (independent of any external software), we provide in this section simple analytical fit formulae for the best tracers found in Sect. 4. While the RF models are flexible enough to make the method described in Sect. 2 generally applicable to any model grid and any physical quantity we want to find tracers of, the analytical fits provided here use formulas that have been specifically chosen for the application presented here (finding predictors of the ionization fraction from our chemical model grid). There is no guarantee that these same formulae would perform adequately to find analytical fits with other model grids and/or another quantity to predict.
Prediction formulae
We use simple polynomial formulae (working as before on the logarithm of both the observable ratios and the ionization fraction) to fit the non-linear relationships between the best tracers found in Sect. 4 and x(e − ). This is applied to all tracers which where found to have R 2 > 0.5 in the previous RF analysis.
In the cold dense medium conditions, we thus use a simple polynomial of order 5:
f dense (x) = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + a 4 x 4 + a 5 x 5 (2)
where x is the log 10 of the column density ratio or line intensity ratio from which we want to predict log 10 (x(e − )), f dense (x) is our fitting function to log 10 (x(e − )) in cold dense gas conditions 4 , and the parameters a 0 to a 5 are our fit parameters. A fit is made for each of the tracers that had R 2 > 0.5 (more than half of the variance explained) in the rankings of Section 4. The corresponding fit coefficient values for each column density ratio are given in Table B.7 and the coefficient values for line intensity ratios are listed in Table B.8. In the translucent medium conditions, x(e − ) naturally reaches a plateau at the fractional abundance of carbon (1.32 × 10 −4 in our undepleted models). We thus use a modified formula combining a polynomial of order 5 and a saturation:
f translucent (x) = f max − log e 1 + e −(a 0 +a 1 x+a 2 x 2 +a 3 x 3 +a 4 x 4 +a 5 x 5 ) (3)
The fit parameters here are f max and a 0 to a 5 . As in the cold dense medium case, f translucent (x) is our fitting function to log 10 (x(e − )) in translucent gas conditions. The corresponding fit coefficient values for each column density ratio are listed in Table B.5 and the coefficient values for line intensity ratios are listed in Table B.6.
The order of the polynomial in these functions is chosen so that further increasing it yields only marginal increase in R 2 (estimated by cross-validation to avoid overfitting). The R 2 values found for the best tracers in each case are close to the values initially found with the RF models, indicating that our analytical fits are not significantly worse than the RF models, at least for the high-R 2 tracers. As an example, Figures 2 and 4 also show the analytical fit (solid red line) in comparison to the RF model (solid black line). For a few of the lower R 2 tracers, the analytical fits perform significantly worse as can be seen on the tables of Appendix B by comparing the R 2 values of the RF models with the R 2 values of the analytical fits.
Uncertainty formulae
Finally, a key point is to estimate uncertainties on our prediction of the ionization fraction. We separate here two sources of uncertainty.
Our best analytical fit is determined on a finite sample of models, so that the fit coefficients are only estimates of the theoretical best fit coefficients. Estimating again these fit coefficients from a different sample of models (drawn from the same distribution) would result in slightly different values, and these uncertainties on the fit coefficients in turn imply an uncertainty on the ionization fraction value predicted by the fit formula at any given value of the observable quantity (intensity ratio or column density ratio). In order to estimate this uncertainty on the predicted value, we proceed by bootstrapping: we repeat the fitting procedure on 100 bootstrapped samples (drawn from the original model grid) and report the standard deviation of the value of the fit function as its uncertainty.
The left panel of Fig. 5 shows the corresponding uncertainty (showing the 3σ level in dashed curves around the main prediction curve) in the case of the best column density ratio in cold dense gas conditions (CN/N 2 H + ). We name this uncertainty the fit coefficients uncertainty to distinguish it from the second form of uncertainty below. We define the validity domain of our fit as the range of values of the observable ratio where our best fit is sufficiently constrained by our finite grid of models for this fit coefficient uncertainty to be negligible. In practice, we define it as the range of ratios where the above-defined uncertainty remains lower than 2% of the predicted value. In the following, we will thus assume this uncertainty to be negligible inside of this validity domain and focus on the second form of uncertainties. The limits of the corresponding validity range are also shown on the left panel of Fig. 5 as blue vertical lines. Due to the tendency of high order polynomial fits to diverge quickly outside of the domain of the fitted dataset, the analytical fit formulae should not be used outside of the validity range defined here.
The second form of uncertainties comes from the unobservable parameters (density, temperature,...), which induce a scatter in the relationship between any of the line intensity ratios or column density ratios and the ionization fraction. Inside of the validity domain of the fit, this scatter in the residuals is much larger than the fit coefficients uncertainty, as can be seen on the left panel of Fig. 5), and is thus the dominant source of uncertainties. When applying the fit formula to real observations, in the ideal case where the chemical model used in this paper would be a perfect model of reality, we would expect the value predicted by the fit formula to commit a mean squared error equal to the variance of this scatter of the residuals, and this error represents our lack of information on the underlying physical conditions. As can be seen on the previous figures however, this residual scatter varies as a function of the observable predictor (the vertical scatter is smaller in some regions of the plot than in others). This residual variance as a function of the predictor can be estimated by a moving average method (shown in the right panel of Fig. 5, we used a window of 0.1 dex). In order to provide a simpler way of estimating this residual variance function, we fitted the simple analytical function
g(x) = b 0 + b 1 x + b 2 x 2 + b 3 x 3 + b 4 x 4 + b 5 x 5(4)
to the squared residuals, thus providing a fit to the local variance. Note that this function provides a fit to the variance on the prediction of log 10 (x(e − )), and, as previously, x is the log 10 of the observable ratio. The absolute value in this function was chosen because the residual variance is by definition a positive quantity. The best fit coefficients to this residual variance function are given in Tables B.5, B.6, B.7 and B.8 in Appendix B. An example of the corresponding residual standard deviation function is shown in the right panel of Fig. 5, where we compare its movingaverage estimate (dotted red curves around the main prediction curve) to its squared polynomial fit (dashed black curve), showing in both cases the 3σ level, for the best column density ratio in the cold dense medium case (CN/N 2 H).
The resulting fits are also presented for the best ratio in the different cases in Fig. 2 and 4, showing the RF fit, the analytical fit and the scatter fit. Similar figures for each of the 6 best tracers for both the translucent medium and the cold dense medium conditions, and for both column density ratios and line intensities ratios, are presented in Appendix C, .
Discussions
Traditional ionization tracers
One of the most commonly used ionization fraction tracers is DCO + /HCO + (Guelin et al. 1977(Guelin et al. , 1982Dalgarno & Lepp 1984;Caselli et al. 1998). It is used mainly in cold dense cores, where the temperature is low enough to allow sufficient deuterium enrichment, and the column density is large enough to make DCO + detectable. However, our results show that even in the cold dense gas regime, and despite including DCO + in our list of potential tracers, the DCO + /HCO + column density ratio is not ranked among the best tracers of the ionization fraction. In fact, it is ranked as the 38 th best tracer in dense cold gas conditions, with a R 2 of 0.57 only (cf. Table B.7). Note that this ratio is often determined using observations of H 13 CO + as H 12 CO + can be optically thick in high-column-density lines of sight. We leave this aspect aside in this discussion by showing that the column density ratio DCO + /HCO + itself (however it might be determined Chemical model results Models with OPR < 2.5 × 10 3 RF model Fig. 6. DCO + over HCO + column density ratio in our grid of dense cold gas models (blue points and contours), shown as a scatter plot, with the central crowded regions replaced by PDF isocontours containing 25%, 50%, and 75% of the points. Our best fit Random Forest model is shown as a black line. The red dashed contours shows the distributions of the models with OPR H 2 < 2.5 × 10 −3 . observationally) suffers from several limitations as a tracer of the ionization fraction.
The relationship between the DCO + /HCO + abundance ratio and the ionization fraction found in our (cold and dense) model grid is shown in Fig. 6. The blue distribution shows the results of the model grid and the black line our best RF model, with a R 2 of 0.57. We see that two main problems limit the usability of DCO + /HCO + .
First, a large scatter of the ionization values (by up to 3 orders of magnitude) is present at all values of the ratio, despite having a significant fraction of the distribution tightly located around a clear relationship (the outermost blue contour encloses 75% of the distribution). As a result, the best fit RF model tries to make a compromise between the tightly located part of the distribution, and the scattered points at lower ionization fraction values. We found most of this scatter to be related to variations in the ortho-to-para ratio of H 2 (OPR H 2 ), an unobservable parameter whose value remains difficult to estimate in observations of dense cold cores. For instance, selecting only the models having OPR H 2 < 2.5 × 10 −3 , we see in Fig. 6 (red dashed contour) that we retain only the unscattered part of the distribution. Thus the difficulty of obtaining reliable estimates of the OPR of H 2 in dense cores limits the use of DCO + /HCO + as a tracer of the ionization fraction.
Second, even when selecting the low OPR H 2 models, we see that the relationship presents a very steep slope at high ratio values (low ionization fraction values). As a result, for DCO + /HCO + ratios above 10 −1.6 , a range of ionization fractions of more than two orders of magnitude is possible. The ratio would then only be usable for lower ratios, equivalent to ionization fractions larger than 10 −6.5 . Even if the model grid presented no scatter at all, a steep slope implies that small observational uncertainties on the ratio will induce large uncertainties on the predicted ionization fraction. Thus, relationships with steep slopes are of limited use.
These different effects combine to make the DCO + /HCO + ratio a poor predictor of the ionization fraction, compared to the best ranked tracers found by our method.
Detectability constraints
So far, detectability constraints have been ignored. Predictive power has been tested from noiseless values of column density or line intensity ratios. However, the different lines considered here have widely different brightnesses and thus differ in terms of detectability with current instruments. We explore here the effect of various noise levels on the predictive power of the different line ratios. We will consider two noise setups correspond- ing to two observation scenarios: the case of one constant noise level for all lines, corresponding to the typical case of a line survey where faint lines are detected with a lower signal-to-noise ratio than bright ones, and the case of a fixed signal-to-noise ratio (SNR) for all lines, corresponding to observations being designed to reach a set signal-to-noise ratio for a few desired lines. These two cases correspond to the two opposite extremes of possible observation scenarios and will give us a general overview of the possible impact of noise on the performance of the tracers. In both cases, synthetic noise is added to the line integrated intensity values and we consider the SNR on the integrated intensity and not the peak intensity. Note that all noise and SNR values quoted are for individual lines, not for line ratios.
In both cases, in order to measure how the predictive power (measured as the R 2 value) is affected by noise, we perform a modified cross-validation. The model grid is randomly split in ten parts. For each of these tenths and for each line ratio :
1. A RF model is trained from the other nine parts of the model grids, without any noise added. 2. The trained RF is tested on the tenth under consideration, with added noise (either with a constant noise variance σ 2 noise for all models and all lines, or with a constant SNR for each model and line). Since the RF models take the log 10 of the intensity ratio as input and since the addition of noise can produce negative values, we only apply the RF model when both line integrated intensity values are above 1 σ. Otherwise, we do not use the RF model and simply take the average ionization fraction value of the grid as our prediction.
We finally take the average R 2 value obtained over the ten noisy tests as the predictive power of the tracer under noisy conditions. We thus avoid estimating R 2 from datapoints that have been seen during training, and we estimate the predictive power of a model trained on noiseless data when applied to noisy data (as would be the case when applying the results of this article to real observations).
The results for the translucent medium grid, for a few possible line ratios, are presented in Fig. 7 and 8. Figure 7 presents the scenario of a constant noise level σ noise for all lines and all models, and shows how the R 2 of the prediction from different Table 3. Impact of adding noise to the line intensities on the predictive power R 2 (for translucent medium), measured by the noise level σ 1/2 (in a constant noise level situation) and the signal-to-noise ratio SNR 1/2 (in a constant SNR situation) for which R 2 reaches one half of its value in the absence of noise R 2 (noiseless) .
Line intensity ratio R 2 (noiseless) (1-0), and the four tracers found to be the least sensitive to noise. We define the tracers least sensitive to noise as those having the highest σ 1/2 among tracers with R noiseless ≥ 0.7, thus giving a compromise between a good fit quality (high R noiseless ) and a slow decrease with increasing noise level (high σ 1/2 ). The four best ratios found according to this definition and shown on the figure are 13 CO (1-0) / C 18 O (1-0), C 18 O (1-0) / CF + (1-0), 13 CO (1-0) / CF + (1-0), and HCN (1-0) / CF + (1-0). We see that the best three tracers, all including C 2 H, have their predictive power decreasing sharply at a relatively low noise level (their R 2 decreases by 50% at σ noise ∼ 5 × 10 −3 K km s −1 ). This is due to the relatively low brightness of the C 2 H line in translucent conditions (median integrated intensity of ∼ 7 × 10 −3 K km s −1 in our translucent medium grid). In comparison, some line ratios built from brighter lines, despite a lower 9. Evolution of the R 2 performance of the Random Forest predictors (for cold dense gas) when adding noise of constant variance σ 2 noise to the line intensities, as a function of the noise level σ noise , for the 3 best tracers for cold dense medium (first three curves) and for the four tracers least affected by the noise (next four curves). R 2 on noiseless data, are found to perform better in the presence of noise : the R 2 for the 13 CO (1-0) / C 18 O (1-0) ratio decreases by 50% at σ noise ∼ 4 × 10 −2 K km s −1 (C 18 O (1-0) has a median integrated intensity of ∼ 3 × 10 −1 K km s −1 in this grid), the C 18 O (1-0) / CF + (1-0) ratio has its R 2 decreased by 50% at σ noise ∼ 10 −2 K km s −1 (CF + (1-0) has a median integrated intensity of ∼ 1.2 × 10 −2 K km s −1 in this grid). We note, however, that ratios built from CF + (1-0) actually only perform marginaly better than the best ratios involving C 2 H at high noise levels (see Fig. 7) due to the low brightness of CF + (1-0). For a more exhaustive comparison of the line ratios, Tab. 3 gives for each line ratio the noise level σ 1/2 at which the R 2 is decreased by half. Figure 8 similarly shows the results for the scenario of a fixed SNR for all lines and all models, still in the case of translucent medium conditions. The variations of the R 2 of the prediction with the SNR are shown for the best seven tracers (according to the noiseless ranking of Table B.2). In this scenario, we find as expected that the predictive power drops at a SNR of order unity. The only exception is the 13 CO (1-0) / C 18 O (1-0) ratio, which decreases slightly earlier. This is due to this ratio spanning a relatively small range of values in our grid of models (approximately one order of magnitude) while the other line ratios span ranges of four to six orders of magnitude. As the ionization fraction spans a range of values of 2.5 orders of magnitude in the grid, this implies that the relationship between the ionization fraction and the 13 CO (1-0) / C 18 O (1-0) ratio has a much steeper slope than for the other line ratios. A steep slope then implies that small errors in the line ratios result in large errors in the predicted ionization fraction. As a result, the 13 CO (1-0) / C 18 O (1-0) ratio requires significantly higher SNRs than the other line ratios. Similarly to σ 1/2 , we define SNR 1/2 as the SNR value at which R 2 is half of its value on noiseless data. The SNR 1/2 values for all line ratios are also given in Table 3.
σ 1/2 SNR 1/2 K.km/s C 2 H (1 − 0) / HCN (1 − 0) 0.
The results for dense and cold medium conditions are similarly shown in Fig. 9 and 10 and Table 4. In the case of a constant noise level, we find that the R 2 drop generally occurs at higher noise levels than in the translucent medium case, as expected because the lines are brighter in the cold dense medium due to higher column densities. Figure 9 shows the decrease of R 2 with σ noise for the best three dense gas tracers (according to the noiseless ranking of Tab. B.2), and for the four ratios least sensitive two noise (according to the same definition as previously). We note that 13 CO (1-0) / HCO + (1-0), the second best ratio on noiseless data (therefore already shown on the figure), has also the highest σ 1/2 value among ratios with R noiseless ≥ 0.7, so that we show the next four ratios least sensitive to noise on the figure. These four ratios are found to be C Table 4. When considering a constant SNR scenario for the dense cold medium case, we again find that the R 2 drop occurs at a SNR value of order unity, as shown in Fig. 10. The SNR 1/2 values for all line ratios are listed in Table 4.
When interpreting the results of this study, one must keep in mind that the decrease in R 2 in our two scenarios (constant σ noise or constant SNR) does not come from the same effect. In the constant σ noise scenario, at a given σ noise level one part of the grid has undetected or very low SNR values for the ratio under consideration, while the other part has high SNR. The R 2 decrease is indicative of the growing fraction of the parameter space with undetected/low SNR ratios. As a result, even when finding a low overall R 2 , there might remain a fraction of the parameter space were the predictor remains very good (usually, high column density, high volume density, high temperature,...), which we did not characterize here. In the constant SNR scenario on the other hand, the SNR is by design constant over all models, independent of the physical parameters. The decrease in overall R 2 is then more representative of the decrease in predictive power at any point in the parameter space. As a result, a ratio with a low σ 1/2 value might still be usable in real observations with a higher noise level but would be restricted to high brightness regions of GMCs (in the corresponding lines), while a ratio cannot be used at all in observations with SNR significantly lower than its SNR 1/2 value.
Chemical model reliability
Independently of the statistical method that we present in this article, the results obtained rely on the chosen chemical model Table 4. Impact of adding noise to the line intensities on the predictive power R 2 (for cold dense medium), measured by the noise level σ 1/2 (in a constant noise level situation) and the signal-to-noise ratio SNR 1/2 for which the R 2 reaches one half of its value in the absence of noise R 2 (noiseless) .
Line intensity ratio R 2 (noiseless) and its limitations. Previous works on ionization fraction tracers have mostly used stationary-state results of single-zone chemical models (although some works have used time-dependent models, e.g., Maret & Bergin 2007;Shingledecker et al. 2016). As these previous studies have been mostly limited to deuterationbased tracers, the focus of the present article has been on highlighting the non-deuteration-based tracers that can be found for the ionization fraction from similar chemical models. We discuss here the impacts of our model's limitations on our results. Our single-zone model cannot include a detailed treatment of UV radiative transfer through the cloud. While most photodissociation rates can be simply estimated based on an assumed optical depth of dust protecting each model from UV photons (parameter A V in our models), species such as H 2 , CO and its isotopologues can be protected from photodissociation by self-or mutual-shielding. While self-shielding of H 2 and mutual shielding of CO by H 2 are included in our model using approximations (Draine & Bertoldi 1996;Heays et al. 2017), mutual shielding of 13 CO and C 18 O by H 2 and 12 CO are not included. As a result, in our translucent medium models where photodissociation by external UV photons still plays an important role, we expect the abundances of the rarer CO isotopologues to be less reliable than the other species. Note that the observations of 13 CO and C 18 O in our ORION-B dataset indeed present specificities (systematic excitation temperature differences with 12 CO, Bron et al. 2018;Roueff et al. 2020) that remain unexplained even by more complex 1D PDR models.
σ 1/2 SNR 1/2 K.km/s CF + (1 − 0) / DCO + (1 − 0) 0.
The only explicit surface reaction in our chemical model is H 2 formation, however we account for the freeze-out of CO through our depletion parameter. The list of species that we consider as possible tracers has been restricted to species that are not strongly affected by surface chemistry beyond the depletion effect. Extension to more complex molecules would require a chemical model including a more complete treatment of surface chemistry.
Another source of uncertainties comes from the experimental or theoretical estimates of the reaction rate coefficients used in our chemical network. While we did not directly perform a sensitivity analysis of the reaction rate coefficients, our model grids include temperature variations which subsequently impact the reactions rate coefficients through their temperature dependence. This is especially true for the important dissociative recombination reactions which display significative temperature dependences. Temperature is then considered as an unobserved parameter when searching for good tracers of x(e − ). The discovery of strong relationships between some of the ratios and x(e − ), despite temperature variations in the grid, thus indicates that these relationships are to some extent robust to the reaction rates. A careful analysis of the magnitude and correlations of model uncertainties resulting from reaction rate uncertainties in the chemical network would deserve a separate study.
Time-dependent effects are expected to be more important in cold dense medium conditions than in translucent medium conditions as photochemistry has shorter timescales in the latter case. Time-dependent effects that result from the time evolution of some physical parameter (e.g. density and temperature during the contraction of a core) while the chemistry follows in a quasi-stationary way are in part accounted for in our models by exploring a large range of the various physical parameters that can be subject to variations (see Table 1). In addition, the progressive freeze out of CO on dust grains is accounted for by considering a range of depletion factors for carbon and oxygen. Similarly, the slow evolution of OPR H 2 in cold gas can keep parts of the chemistry (deuterium chemistry, nitrogen chemistry) in a time-dependent evolution that depends mainly on the evolution of OPR H 2 . This is also in part accounted for by exploring a large range of OPR H 2 values in our models. The tracer-finding method presented in this article will be applied to time-dependent chemical models in a future study.
The final and most important limitation of our model is that it does not include a spatial dimension : gas at a single value of density, temperature, etc. is assumed to be exposed to a given radiation field and protected by a given column density. As a result the possibility that different emission lines originate in separate layers of gas on the line of sight is completely neglected. Variations of the physical conditions along the line of sight can indeed have an important impact on the observables (e.g. Levrier et al. 2012). This limitation could be important both in the translucent medium where physical and chemical gradients are present due to the progressive extinction of the external UV field, and in dense cores with density and temperature gradients. As a result some caution must be exercised when choosing the line ratios to consider, ratios involving two species expected to emit in completely different regions should be avoided. For instance, the C 2 H (1-0) / N 2 H + (1-0) that is found as the fourth best line ratio in dense cold medium should be avoided : C 2 H is known to be a tracer of UV illumination and thus more likely to be emitted at the external surface of a given clump, while N 2 H + is abundant in the inner regions of the core where CO is already significantly depleted. An application of our method to 1D PDR models to better account for this effect in translucent medium conditions will be carried out in a future work.
Parameter PDF in the model grids
In the model grids used in this study, we sampled uniformly (in logarithm) for the values of the unobservable physical conditions (gas density, temperature, UV field, etc.) in an hypercube defined by lower and upper bounds for each of the parameters. The results of our ranking method will depend on this assumed PDF (probability density function) for the physical conditions in the ISM. We made here the choice of making the minimal assumption: knowing only reasonable lower and upper bounds on each of the parameters, the uniform distribution is the maximum entropy PDF (i.e. the PDF that best represents our assumed state of knowledge). As a perspective, if more a priori knowledge is available, then more accurate assumptions for the PDF (in particular for the correlations between the different physical parameters) could reveal additional tracers.
We note that this assumption of a uniform PDF over a maximum support (in the sense that any more accurate PDF would have almost all of its weight enclosed in this support) makes it likely that any tracer found to have a very good relationship with the ionization fraction would keep a strong relationship for more accurate PDF choices (if the relationship is strong over the full hypercube, it should with high likelihood stay strong on subregions of this hypercube). In this sense, we expect the tracers found here to remain reliable, but a more accurate PDF choice might strongly increase the performance of some tracers found here to perform poorly and thus reveal additional tracers of the ionization fraction. This argument remains however qualitative as pathological cases of PDF might be constructed that would radically change the rankings of the tracers.
As a result, the precise rankings presented in this article could slightly change, but we expect the good tracers highlighted by these ranking to be reliable for more realistic PDFs of the physical conditions in GMCs.
Final recommandation
Based on the limitations discussed above (detectability and model reliability), we recommend the use of the following integrated line intensity ratios to trace the ionization fraction. This list is of course not exhaustive, and other ratios can give satisfactory predictions (see Tables 3 and 4) if the species listed above are not available. In translucent gas conditions, this recommendation is based on the following points. After eliminating rarer CO isotopologues based on our discussion of mutual-shielding effects on selective photodissociation in low/moderate A V regions (cf Sect. 6.3), and eliminating ratios involving sulfur species that were found to require unreasonably low noise levels of 10 −4 − 10 −5 K km s −1 , the three remaining best line intensity ratios are C 2 H (1-0) / HCN (1-0) and C 2 H (1-0) / HNC (1-0) and C 2 H (1-0) / CN (1-0). If noise sensitivity is critical, the tracer found to have the best predictive power at high noise levels is found in Sect. 6.2 to be HCN (1-0) / CF + (1-0) but is negligibly better than the three previously mentioned at high noise levels.
In cold dense gas conditions, the three best ratios are CF + (1-0) / DCO + (1-0), 13 CO (1-0) / HCO + ,and CN (1-0) / N 2 H + (1-0). If noise sensitivity is critical, we found in Sect. 6.2 that the ratios with the best predictive power at high noise levels are 13 CO (1-0) / HCO + and C 18 O (1-0) / HCO + (1-0).
Conclusions
We have presented a general statistical method to find the best observable tracers of an unobservable parameter based on a grid of models spanning the range of possible values for all the unknown underlying physical parameters (e.g., gas density, temperature, depletion, etc.). Our method estimates the predictive power of each potential observable tracer by training a flexible, non-linear regression model (a Random Forest model, making no assumption on the non-linear shape of the relationship to be found) on the task of predicting the target quantity from each of the potential tracers. The fit quality on test data, measured as the R 2 coefficient by cross-validation and out-of-bag estimation, is used to rank the potential tracers by order of predictive power.
In the context of our recent studies of the Orion B GMC Gratier et al. 2017;Orkisz et al. 2017;Bron et al. 2018;Orkisz et al. 2019), we have applied this method to the important astrophysical question of tracing the ionization fraction in the neutral ISM, with the goal of being able to probe its variations across a whole GMC, from its translucent enveloppe to its dense cores. We considered grids of single-zone, stationary state astrochemical models exploring wide ranges of values in gas density, temperature, external UV field, A V on the line of sight, cosmic ray ionization rate, ortho-to-para ratio of H 2 , depletion factor, and sulfur elemental abundance. For a finer exploration of the possible conditions, we considered two grids corresponding to translucent medium conditions and cold dense medium conditions respectively, based on the different types of environments found in the Orion B GMC Bron et al. 2018).
We considered successively column density ratios and line intensity ratios as potential tracers, focusing on species observable in the band at 100 GHz of our observations of Orion B. We find that in both cases and in both types of physical conditions, multiple ratios allow accurate predictions of the ionization fraction, with R 2 > 0.8 (and up to 0.96). We investigated the impact of the noise level on the predictive capability of the different ratios After accounting for detectability and model reliability, we recommend : In order to simplify the use of these predictors, we constructed ad hoc analytical fits (using polynomials or saturated polynomials) of the relationship of each observable tracer to the ionization fraction. Contrary to the Random Forest models, the choice of the analytical form of these fits is specific to the types of relationships observed in this specific application (different analytical forms might be necessary for other applications). We also provide analytical formulae to estimate the uncertainty on any measurement of the ionization fraction from these tracers. These tracers will be used to study the ionization fraction in the Orion B molecular cloud in a second paper (Guzman et al. in prep.). The method presented here is very general and could be easily applied to finding tracers of other related (cosmic ray ionization rate, absolute electron abundance) or unrelated (gas density, temperature, OPR H 2 ,...) unobservable quantities. This method can also be extended simply to simultaneously use pairs (or more) of line ratios (by training RF models on the possible combinations of line ratios), which would likely further increase the quality of the prediction.
-
Appendix A: Random Forest parameter optimization
The Random Forest contains a few tuning parameters for which a value needs to be chosen before training. In particular, we will consider only the two most important here : the number of trees in the forest N trees , and the maximum depth allowed for each tree, d max . Another usual parameter is not considered here: the number of predictors in the random subset considered when choosing along which axis to make a split. Indeed, we will only train RF models with a single predictor at a time, so that this number is necessarily 1.
Since optimizing the choice of these parameters on the full model grids (that will then be used for training) and for each potential tracer will lead both to an increased risk of overfitting and a heavy computational time cost, we opt for a very limited optimization, where a single set of parameter values is used for all tracers (i.e. for all RF models trained on a single ratio) and for all model grids. The selection of these parameter values is done with a simplified procedure:
1. For each model grid, RF models are first trained on each ratio using default parameter values (default sklearn values, N trees = 100 and d max = ∞), and the OOB R 2 value is calculated for each ratio. 2. For the two ratios having the best and worst R 2 in the previous step, RF models are trained for a grid of parameter values exploring N trees = 50 − 800 and d max = 1 − 12, and the OOB R 2 is again computed for RF models with each possible combination of parameter values (the OOB value is used to limit the risks of overfitting). 3. As the R 2 maps obtained show very flat minima, we found a common set of parameter values assuring a R 2 within 0.01 of the best value in all cases. The parameter values found are N trees = 400, and d max = 4.
In order to illustrate this procedure, Fig. A.1 shows for instance the resulting R 2 maps as a function of N trees and d max for the preliminary best (top) and worst (bottom) tracers in the translucent model grid when using line integrated intensity ratios. The red contours delimit the region where R 2 is within 0.01 of its maximum value. We see that the variations with N trees are limited to a decrease at low values. Any large enough value could thus be chosen but larger values will induce higher computation time cost, so that a minimal acceptable value had to be chosen. The variations with d max show a clear maximum (although rather flat). The R 2 value decreases for decreasing values of d max below the optimum as the RF model then becomes not flexible enough to capture the relationship between the predictor and the target variable. Above the optimum, R 2 decreases for increasing values of d max as overfitting starts to arise. The RF model becomes too flexible and starts to learn noise artefacts (we recall that the "noise" here is induced by the random sampling of the unobservable physical parameters of the chemical model).
Appendix B: Tables for tracers ranking, performance, and fit coefficients
Appendix B.1: Tracers ranking and performance
We provide here the ranking and performance of single ratio RF models, obtained following the method described in Sect. 2.
Appendix B.1.1: Translucent medium Tables B.1 and B.2 present the ranking we obtain for translucent medium conditions, respectively for column density ratios and integrated line intensity ratios. See Sect. 2 for a description of the method used, and Sect. 4 for a discussion of these results. For each ratio, we list the performance of the corresponding RF model measured through the (cross-validated) R 2 , the equivalent root mean square error on log 10 (x(e − )) and the corresponding error factor on x(e − ), the maximum absolute error on log 10 (x(e − )) and the corresponding error factor on x(e − ). We also list for comparison the R 2 value obtained with the analytical fit described in Sect. 5.
Appendix B.1.2: Cold dense medium Tables B.3 and B.4 present the ranking we obtain for cold dense medium conditions, respectively for column density ratios and integrated line intensity ratios. Ranking of column density ratios according to their usefulness to predict the ionization fraction in translucent medium conditions (measured through the R 2 of a fitted Random Forest model). Additional error measures of the Random Forest model (root mean square error and maximum absolute errors) are also given. As these errors concern the logarithm of the ionization fraction, we also provide the equivalent error factors on the ionization fraction. For comparison, the R 2 obtained with the analytical fit described in Sect. 5 is also listed in the last column. 3) for predicting log 10 (x(e − )), the quality of this fit estimated as the (cross-validated) R 2 , the root mean square error on log 10 (x(e − )) and corresponding error factor on x(e − ), the maximum absolute error on log 10 (x(e − )) and corresponding error factor on x(e − ), the fit coefficients (corresponding to the fit formula given in Eq. 4) for estimating the uncertainty on the prediction, and the limits of the validity range of the fit (given as log 10 of the ratio values.
Appendix B.2.2: Cold dense medium
Tables B.7 and B.8 list the fit coefficients for cold dense medium conditions, respectively for column density ratios and integrated line intensity ratios. For each ratio, ranked according the results of Sect 4, we list the fit coefficients (corresponding to the fit formula given in Eq. 2) for predicting log 10 (x(e − )), the quality of this fit estimated as the (cross-validated) R 2 , the root mean square error on log 10 (x(e − )) and corresponding error factor on x(e − ), the maximum absolute error on log 10 (x(e − )) and corresponding error factor on x(e − ), the fit coefficients (corresponding to the fit formula given in Eq. 4) for estimating the uncertainty on the prediction, and the limits of the validity range of the fit (given as log 10 of the ratio values. Ranking of line intensity ratios according to their usefulness to predict the ionization fraction in translucent medium conditions (measured through the R 2 of a fitted Random Forest model). Additional error measures of the Random Forest model (root mean square error and maximum absolute errors) are also given. As these errors concern the logarithm of the ionization fraction, we also provide the equivalent error factors on the ionization fraction. For comparison, the R 2 obtained with the analytical fit described in Sect. 5 is also listed in the last column.
Line intensity ratio Random Forest Model Analytical fit R 2 Root mean square error Maximum absolute error R 2 dex (equ. factor) dex (equ. factor) C 2 H (1 − 0) / HCN (1 − 0)
Appendix C: Analytical model visualization
We present in this section figures of the RF models and analytical fits for the best six tracers in each case.
Appendix C.1: Translucent medium Ranking of column density ratios according to their usefulness to predict the ionization fraction in dense cold medium conditions (measured through the R 2 of a fitted Random Forest model). Additional error measures of the Random Forest model (root mean square error and maximum absolute errors) are also given. As these errors concern the logarithm of the ionization fraction, we also provide the equivalent error factors on the ionization fraction. For comparison, the R 2 obtained with the analytical fit described in Sect. 5 is also listed in the last column. Table B.4. Ranking of line intensity ratios according to their usefulness to predict the ionization fraction in dense cold medium conditions (measured through the R 2 of a fitted Random Forest model). Additional error measures of the Random Forest model (root mean square error and maximum absolute errors) are also given. As these errors concern the logarithm of the ionization fraction, we also provide the equivalent error factors on the ionization fraction. For comparison, the R 2 obtained with the analytical fit described in Sect. 5 is also listed in the last column.
Line intensity ratio Random Forest Model Analytical fit R 2 Root mean square error Maximum absolute error R 2 dex (equ. factor) dex (equ. factor) CF A&A proofs: manuscript no. paper_RF_ioniz Table B.5. Fit coefficients (for our main fit and scatter fit) and fit quality for column density ratios in translucent medium conditions. We list the fit coefficients for predicting log 10 (x(e − ) (according to the fit formula given in Eq. 3), the quality of this fit estimated as the (cross-validated) R 2 , root mean square error (RMSE) on log 10 (x(e − ) and corresponding error factor on x(e − ), maximum absolute error factor on log 10 (x(e − )) and corresponding error factor on x(e − ), the fit coefficients for estimating the uncertainty on the prediction (according to the fit formula given in Eq. 4), and the validity range of the fit (given as log 10 of the ratio values).
Column density ratio Main fit coefficients Table B.6. Fit coefficients (for our main fit and scatter fit) and fit quality for line intensity ratios in translucent medium conditions. We list the fit coefficients for predicting log 10 (x(e − ) (according to the fit formula given in Eq. 3), the quality of this fit estimated as the (cross-validated) R 2 , root mean square error (RMSE) on log 10 (x(e − ) and corresponding error factor on x(e − ), maximum absolute error factor on log 10 (x(e − )) and corresponding error factor on x(e − ), the fit coefficients for estimating the uncertainty on the prediction (according to the fit formula given in Eq. 4), and the validity range of the fit (given as log 10 of the ratio values).
Line intensity ratio
Main fit coefficients Table B.7. Fit coefficients (for our main fit and scatter fit) and fit quality for column density ratios in cold dense medium conditions. We list the fit coefficients for predicting log 10 (x(e − ) (according to the fit formula given in Eq. 2), the quality of this fit estimated as the (cross-validated) R 2 , root mean square error (RMSE) on log 10 (x(e − ) and corresponding error factor on x(e − ), maximum absolute error factor on log 10 (x(e − )) and corresponding error factor on x(e − ), the fit coefficients for estimating the uncertainty on the prediction (according to the fit formula given in Eq. 4), and the validity range of the fit (given as log 10 of the ratio values).
Column density ratio
Main fit coefficients Table B.8. Fit coefficients (for our main fit and scatter fit) and fit quality for line intensity ratios in cold dense medium conditions. We list the fit coefficients for predicting log 10 (x(e − ) (according to the fit formula given in Eq. 2), the quality of this fit estimated as the (cross-validated) R 2 , root mean square error (RMSE) on log 10 (x(e − ) and corresponding error factor on x(e − ), maximum absolute error factor on log 10 (x(e − )) and corresponding error factor on x(e − ), the fit coefficients for estimating the uncertainty on the prediction (according to the fit formula given in Eq. 4), and the validity range of the fit (given as log 10 of the ratio values).
Line intensity ratio
Main fit coefficients Ionization fraction versus column density ratio for the best six ratios found in Sect.4 for cold dense medium conditions. The chemical model grid is shown as a scatter plot, with the central crowded regions replaced by PDF isocontours containing 25%, 50%, and 75% of the points. Superimposed are the RF model (red line), the analytical fit (solid black line), and the analytical fit of the 1σ uncertainty (dashed black lines). The quality estimates of the two models are indicated on the figure.
Fig. 1 .
1Ranking of column density ratios (left) or line intensity ratios (right) of observable tracers by order of the predictive power for predicting the ionization fraction (measured by the R 2 coefficient), in the case of translucent medium conditions (showing only the first 20). Errorbars of the R 2 estimates are computed by cross-validation (see text for explanations). The inset in the left panel shows a zoom on the first five ratios in order to make the magnitude of the errorbars visible.
Fig. 2 .
2Ionization fraction versus the best column density ratio, C 2 H/HCN (left panel), and the best line intensity ratio, C 2 H (1-0) / HCN (1-0) (right panel) for tracing the ionization fraction in translucent medium conditions. The model grid is shown as a scatter plot, with the central crowded regions replaced by PDF isocontours containing 25%, 50%, and 75% of the points. Superimposed are the RF model (red line), the analytical fit (solid black line, presented in Sect. 5), the analytical fit of the 1σ uncertainty (dashed black lines, presented in Sect. 5), and the bounds of the validity range of the analytical fit (vertical lines, presented in Sect. 5). The quality estimates of the two models are indicated on the figure.
Fig. 3 .
3Ranking of column density ratios (left) or line intensity ratios (right) of observable tracers by order of the predictive power for predicting the ionization fraction (measured by the R 2 coefficient), in the case of dense cold medium conditions (showing only the first 20). Errorbars of the R 2 estimates are computed by cross-validation (see text for explanations).
dex (fact. 1.69) max. abs. error = 1.44 dex (fact. 27.85) RF model : R 2 = 0.919 RMSE = 0.23 dex (fact. 1.70) max. abs. error = 1.49 dex (fact. 31.
Fig. 4 .
4Same as Fig. 2 for cold dense medium conditions.
Fig. 5 .
5Illustration of the two sources of uncertainties for the best column density ratio for tracing the ionization fraction in dense cold medium, CN/N 2 H + . The left panel shows the analytical fit (solid black line), the standard deviation around this curve corresponding to the uncertainties on the fit coefficients (thin dashed red lines representing the 3σ level), and the bounds of the validity range defined in the text (vertical blue lines). The right panel shows the analytical fit (solid black line) and two estimates of the standard deviation corresponding to the residual scatter of the data points around the curve: the red dotted line presents a moving-average estimate of the local standard deviation, and the dashed black lines shows our analytical fit of the residual standard deviation. On both panels, the chemical model grid is shown as a scatter plot, with the central crowded regions replaced by PDF isocontours containing 25%, 50%, and 75% of the points.
Fig. 7 .
7Evolution of the R 2 performance of the Random Forest predictors (for translucent gas) when adding noise of constant variance σ 2 noise to all line intensities, as a function of the noise level σ noise , for the 3 best tracers for translucent medium (
Fig. 8 .
8Evolution of the R 2 performance of the Random Forest predictors (for translucent gas) when adding noise of constant signal-to-noise ratio to the line intensities, as a function of the SNR, for the 7 best tracers for translucent medium.
66 × 10 −5 1.57 CS (2 − 1) / CF + (1 − 0) 0.50 8.93 × 10 −4 1.57 line ratios decreases when the noise level is increased. We show only the three best line ratios (according to the noiseless ranking of Tab. B.2): C 2 H (1-0) / HCN (1-0), C 2 H (1-0) / 13 CO (1-0), and C 2 H (1-0) / C 18 O
Fig. 9. Evolution of the R 2 performance of the Random Forest predictors (for cold dense gas) when adding noise of constant variance σ 2 noise to the line intensities, as a function of the noise level σ noise , for the 3 best tracers for cold dense medium (first three curves) and for the four tracers least affected by the noise (next four curves).
Fig. 10 .
10Evolution of the R 2 performance of the Random Forest predictors (for cold dense gas) when adding noise of constant signal-to-noise ratio to the line intensities, as a function of the SNR, for the 7 best tracers for cold dense medium.
18 O (1-0) / HCO + (1-0), HCO + (1-0) / CN (1-0), SO (3-2) / CN (1-0), and HCN (1-0) / CN (1-0). The σ 1/2 values for all line ratios are listed in
perform as well as the previously listed ratios.-In cold dense gas conditions, we recommend the use of CF +
/ CN (1-0), for cold dense medium conditions, CF + (1-0) / DCO + (1-0), 13 CO (1-0) / HCO + (1-0) or CN (1-0) / N 2 H + (1-0) at low enough noise level, or 13 CO (1-0) / HCO + (1-0) or C 18 O (1-0) / HCO + (1-0) if sensitivity is an issue.
Fig
. A.1. R 2 (OOB value) as a function of the tuning parameters of the Random Forest (N trees and d max ), for the preliminary best (top panel) and worst (bottom panel) ratios found though a preliminary estimate using defaults values of the parameters, for line integrated intensity ratios and translucent medium conditions. The red contours show the region where the R 2 value is within 0.01 of the maximum.
Figure C. 1
1presents scatter plots of the ionization fraction versus each of the best six column density ratios in translucent medium conditions. Superimposed are the RF model (red line), the analytical fit (solid black line), the uncertainty fit (dashed lines), and the validity range (blue vertical lines).Figure C.2 similarly shows scatter plots of the ionization fraction versus each of the best six integrated line intensity ratios in translucent medium conditions. Appendix C.2: Cold Dense medium Figure C.3 and C.4 similarly present the scatter plots (for column density ratios and integrated line intensity ratios respectively) for the best six tracers in cold dense medium conditions.
Fig. C. 3 .
3Fig. C.3. Ionization fraction versus column density ratio for the best six ratios found in Sect.4 for cold dense medium conditions. The chemical model grid is shown as a scatter plot, with the central crowded regions replaced by PDF isocontours containing 25%, 50%, and 75% of the points. Superimposed are the RF model (red line), the analytical fit (solid black line), and the analytical fit of the 1σ uncertainty (dashed black lines). The quality estimates of the two models are indicated on the figure.
Table B .
B1 (due to their sizes,
A&A proofs: manuscript no. paper_RF_ioniz
C 2H
/ HC N
C 2H
/ 13 CO
C 18 O / CF +
HC N / CF +
C 2H
/ C 18 O
HC N / CN
13 CO / CF +
SO / HC S +
C 2H
/ HN C
HC O + / CF +
C 2H
/ CN
13 CO / C 18 O
C 2H
/ HC O +
SO / C 2H
HN C / CF +
H 2CS
/ C 2H
SO / CF +
CN / CF +
CS / SO
C 18 O / CN
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
R 2
of RF model
Column density ratios ranking for translucent medium
C 2H
/ HC N
C 2H
/ 13 CO
C 18 O / CF +
HC N / CF +
C 2H
/ C 18 O
0.90
0.92
0.94
0.96
C 2H
Table B .1.
B
Table B .2.
B
Table B .3.
B
RMSE = 0.23 dex (fact. 1.69) max. abs. error = 1.44 dex (fact. 27.85) RF model : R 2 = 0.919 RMSE = 0.23 dex (fact. 1.70) max. abs. error = 1.49 dex (fact. 31.07)Analytical model :
R 2 = 0.920
Chemical model results
RF model
Analytical model
1.0
0.5
0.0
0.5
1.0
1.5
2.0
log 10( HNC
CN )
8.5
8.0
7.5
7.0
6.5
6.0
5.5
5.0
log
10 (Ionisation fraction)
Analytical model :
R 2 = 0.886
RMSE = 0.27 dex (fact. 1.88)
max. abs. error = 1.14 dex (fact. 13.87)
RF model :
R 2 = 0.886
RMSE = 0.27 dex (fact. 1.88)
max. abs. error = 1.19 dex (fact. 15.48)
Chemical model results
RF model
Analytical model
Analytical model : R 2 = 0.830 RMSE = 0.33 dex (fact. 2.16) max. abs. error = 1.77 dex (fact. 58.35) RF model : R 2 = 0.883 RMSE = 0.28 dex (fact. 1.90) max. abs. error = 1.43 dex (fact. 26.85)Chemical model results
RF model
Analytical model
1
0
1
2
3
4
5
6
log 10( CN
DCO + )
8.5
8.0
7.5
7.0
6.5
6.0
5.5
5.0
log
10 (Ionisation fraction)
Analytical model :
R 2 = 0.877
RMSE = 0.28 dex (fact. 1.92)
max. abs. error = 1.17 dex (fact. 14.69)
RF model :
R 2 = 0.877
RMSE = 0.28 dex (fact. 1.93)
max. abs. error = 1.14 dex (fact. 13.93)
Chemical model results
RF model
Analytical model
0.5
0.0
0.5
1.0
1.5
2.0
2.5
log 10( HCN
CN )
8.5
8.0
7.5
7.0
6.5
6.0
5.5
5.0
log
10 (Ionisation fraction)
Analytical model :
R 2 = 0.876
RMSE = 0.29 dex (fact. 1.93)
max. abs. error = 1.28 dex (fact. 19.02)
RF model :
R 2 = 0.876
RMSE = 0.29 dex (fact. 1.93)
max. abs. error = 1.27 dex (fact. 18.41)
Chemical model results
RF model
Analytical model
3
2
1
0
1
2
log 10( HCO +
CN )
8.5
8.0
7.5
7.0
6.5
6.0
5.5
5.0
log
10 (Ionisation fraction)
Analytical model :
R 2 = 0.823
RMSE = 0.34 dex (fact. 2.20)
max. abs. error = 1.61 dex (fact. 41.13)
RF model :
R 2 = 0.823
RMSE = 0.34 dex (fact. 2.20)
max. abs. error = 1.61 dex (fact. 40.40)
Chemical model results
RF model
Analytical model
Informations and data related to the ORION-B program can be found at http://www.iram.fr/~pety/ORION-B/ Article number, page 2 of 29 Emeric Bron et al.: Tracers of the ionization fraction in molecular clouds: I.
Article number, page 4 of 29 Emeric Bron et al.: Tracers of the ionization fraction in molecular clouds: I.
Datafiles containing the training RF models and tables of the rankings presented in this section are available at [the link will be added in the published version]. cal and chemical parameters that have been varied in our model grid (cf.Table 1). The R 2 values are slightly lower when using line intensity ratios rather than column densities ratios, indicating that excitation and radiative transfer effects tend to increase the degeneracy between the ionization fraction and other unknown parameters, but this effect remains moderate. To illustrate the performance of the tracers found with this ranking, we show on the left panel ofFig. 2the ionization fraction versus the best ranked column density ratio (C 2 H/HCN) in our grid of models for translucent medium conditions (blue symbols, with Article number, page 6 of 29 EmericBron et al.: Tracers of the ionization fraction in molecular clouds: I.
Except in strong PDR environments, where a small fraction of vibrationally excited H 2 may overcome the endothermicity barrier. Strong PDR environments are not considered here.
Article number, page 8 of 29 Emeric Bron et al.: Tracers of the ionization fraction in molecular clouds: I.
We use the notation log 10 for the logarithm in base 10, and log e for the natural logarithm.Article number, page 9 of 29 A&A proofs: manuscript no. paper_RF_ioniz
Article number, page 12 of 29 Emeric Bron et al.: Tracers of the ionization fraction in molecular clouds: I.
Acknowledgements. We thank the anonymous referee for his comments that helped improve this article. We thank the CIAS for their hospitality during the many workshops devoted to the ORION-B project. This work was supported in part by the Programme National "Physique et Chimie du Milieu Interstellaire" (PCMI) of CNRS/INSU with INC/INP, co-funded by CEA and CNES. This project has received financial support from the CNRS through the MITI interdisciplinary programs. The authors also acknowledge funding by Paris Observatory through the AF Astrochimie program. JRG thanks Spanish MICI for funding support under grant AYA2017-85111-P.A&A proofs: manuscript no. paper_RF_ioniz . Ionization fraction versus integrated line intensity ratio for the best six ratios found in Sect.4 for cold dense medium conditions. The chemical model grid is shown as a scatter plot, with the central crowded regions replaced by PDF isocontours containing 25%, 50%, and 75% of the points. Superimposed are the RF model (red line), the analytical fit (solid black line), and the analytical fit of the 1σ uncertainty (dashed black lines). The quality estimates of the two models are indicated on the figure.
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| []
|
[
"Coulomb Blockade in an Open Quantum Dot",
"Coulomb Blockade in an Open Quantum Dot"
]
| [
"S Amasha \nDepartment of Physics\nStanford University\n94305StanfordCaliforniaUSA\n",
"I G Rau \nDepartment of Applied Physics\nStanford University\n94305StanfordCaliforniaUSA\n",
"M Grobis \nDepartment of Physics\nStanford University\n94305StanfordCaliforniaUSA\n",
"R M Potok \nDepartment of Physics\nStanford University\n94305StanfordCaliforniaUSA\n\nDepartment of Physics\nHarvard University\n02138CambridgeMassachusettsUSA\n",
"H Shtrikman \nDepartment of Condensed Matter Physics\nWeizmann Institute of Science\n96100RehovotIsrael\n",
"D Goldhaber-Gordon \nDepartment of Physics\nStanford University\n94305StanfordCaliforniaUSA\n"
]
| [
"Department of Physics\nStanford University\n94305StanfordCaliforniaUSA",
"Department of Applied Physics\nStanford University\n94305StanfordCaliforniaUSA",
"Department of Physics\nStanford University\n94305StanfordCaliforniaUSA",
"Department of Physics\nStanford University\n94305StanfordCaliforniaUSA",
"Department of Physics\nHarvard University\n02138CambridgeMassachusettsUSA",
"Department of Condensed Matter Physics\nWeizmann Institute of Science\n96100RehovotIsrael",
"Department of Physics\nStanford University\n94305StanfordCaliforniaUSA"
]
| []
| We report the observation of Coulomb blockade in a quantum dot contacted by two quantum point contacts each with a single fully-transmitting mode, a system previously thought to be well described without invoking Coulomb interactions. At temperatures below 50 mK we observe a periodic oscillation in the conductance of the dot with gate voltage that corresponds to a residual quantization of charge. From the temperature and magnetic field dependence, we infer the oscillations are Mesoscopic Coulomb Blockade, a type of Coulomb blockade caused by electron interference in an otherwise open system. | 10.1103/physrevlett.107.216804 | [
"https://arxiv.org/pdf/1009.5348v1.pdf"
]
| 16,231,215 | 1009.5348 | 14d58a63088e40d0db2d5537dbbd07d59281c450 |
Coulomb Blockade in an Open Quantum Dot
27 Sep 2010
S Amasha
Department of Physics
Stanford University
94305StanfordCaliforniaUSA
I G Rau
Department of Applied Physics
Stanford University
94305StanfordCaliforniaUSA
M Grobis
Department of Physics
Stanford University
94305StanfordCaliforniaUSA
R M Potok
Department of Physics
Stanford University
94305StanfordCaliforniaUSA
Department of Physics
Harvard University
02138CambridgeMassachusettsUSA
H Shtrikman
Department of Condensed Matter Physics
Weizmann Institute of Science
96100RehovotIsrael
D Goldhaber-Gordon
Department of Physics
Stanford University
94305StanfordCaliforniaUSA
Coulomb Blockade in an Open Quantum Dot
27 Sep 2010
We report the observation of Coulomb blockade in a quantum dot contacted by two quantum point contacts each with a single fully-transmitting mode, a system previously thought to be well described without invoking Coulomb interactions. At temperatures below 50 mK we observe a periodic oscillation in the conductance of the dot with gate voltage that corresponds to a residual quantization of charge. From the temperature and magnetic field dependence, we infer the oscillations are Mesoscopic Coulomb Blockade, a type of Coulomb blockade caused by electron interference in an otherwise open system.
Mesoscopic systems are conventionally divided into two classes. In closed systems electrons are localized and Coulomb interaction effects determine the transport properties, while in open systems the Coulomb interaction can be neglected at low energies. The class of a system is thought to depend on the contacts between the mesoscopic region and the surrounding electrons. If the contacts contain a large number of poorly transmitting channels, such as in metallic nanostructures, then the crossover from closed to open is smooth and occurs when the total conductance of the contacts is on the order of e 2 /h [1,2]. If the contacts each have one mode, such as can happen in semiconductor nanostructures, then the transition from closed to open is sharp: Coulomb blockade occurs when the mode in each contact is partially transmitting, and in the absence of phase coherence Coulomb blockade disappears when the mode in either contact becomes fully transmitting [3].
This transition from the closed to the open regime has been demonstrated with laterally gated quantum dots [4][5][6]. Such a dot is contacted via one-dimensional channels called quantum point contacts (QPCs). A QPC is tunable, and its conductance G QP C is directly related to the transmission of its modes: G QP C = 2e 2 /h corresponds to a single fully transmitting spin-degenerate mode, while G QP C ≪ 2 e 2 /h corresponds to the tunneling regime. A dot is typically contacted by two QPCs. As the conductance of either QPC increases, the energy to add an additional electron to the dot (the charging energy U ) is reduced [7,8] and capacitance measurements show that the Coulomb oscillations decrease in amplitude, disappearing entirely at G QP C = 2 e 2 /h [9,10].
Phase coherence complicates the transition from closed to open. In a one-leaded dot (a dot where the conductance of one QPC is adjustable, while the other is kept ≪ 2 e 2 /h) Coulomb blockade features are observed even if the adjustable QPC is set at 2 e 2 /h [11]. Coherent electron paths in the dot interfere at this QPC and can reduce its transmission, trapping electrons on the dot. This leads to a type of Coulomb blockade called Mesoscopic Coulomb Blockade (MCB) [12]. Electron interference, and hence MCB, is strongest at zero magnetic field because a closed path that begins and ends at the same QPC interferes constructively with its time-reversed pair, an effect called Weak Localization (WL). In contrast to the one-leaded case, a coherent dot where both QPCs have many fully transmitting modes is predicted not to have MCB [13,14]: there are now multiple escape paths and it is unlikely that interference will reduce the transmission of all paths simultaneously. This result is expected to extend to the case where each QPC has just one fully transmitting spin-degenerate mode and MCB should be absent. To date transport measurements have confirmed that this system is open: while there are hints of MCB [6,15], most experimental results [16,17] have been well understood using Random Matrix theory that neglects explicit Coulomb interactions [18].
In this Letter, we report measurements of MCB in a dot where each QPC has a fully transmitting mode. We reach an electron temperature of 13 mK, lower than previously attained in such systems, and this allows us to observe a periodic oscillation in the conductance of the dot as a function of gate voltage. Finite bias and capacitance measurements demonstrate that this oscillation corresponds to a residual quantization of charge, with a renormalized charging energy. We find that the amplitude of the oscillation depends sensitively on both temperature and magnetic field; in particular, the field scale on which the oscillation decreases is that on which time-reversal symmetry is broken. This demonstrates the oscillation is MCB, and reveals how phase coherence leads to the emergence of Coulomb interactions at low temperatures in a system previously thought to be open.
We measure a quantum dot fabricated from an epitaxially grown AlGaAs/GaAs heterostructure with a twodimensional electron gas (2DEG) located at an interface 68 nm below the surface. The 2DEG has a density of 2 × 10 11 cm −2 and a mobility of 2 × 10 6 cm 2 /Vs. ure 1(a) shows an electron micrograph of the metallic gates we pattern on the surface. Negative voltages are applied to the gates to form a large dot of area ≈ 3 µm 2 that we study, as well as an adjacent small dot that is used as a charge sensor [19,20] in the capacitance measurements. The gates bw1, n, and bw2 define the QPCs of the dot, while the gates c1, c2, and bp are used to change the shape of the dot [16,21]. Gates c1 and c2 have a small effect on the conductance on of the QPCs, and this effect is compensated by adjusting the gates bw1 and bw2 to maintain the conductance through the QPCs. For all measurements, the gates sn1 and sn2 are kept sufficiently negative that there is no measurable conductance through the channel between them. We measure the conductance using standard lock-in techniques [22]. Figure 1(b) shows the zero-bias conductance of the large dot G dot as a function of the voltages on the gates bw1 and bw2 (V bw1 and V bw2 , respectively) that control the two QPCs. These data are taken at T = 540 mK to suppress Universal Conductance Fluctuations (UCFs) [16] and at B > 5 mT to avoid WL, and in this regime G dot is just the series conductance of the two QPCs. In particular, there is a plateau at G dot = 1 e 2 /h, corresponding to the 2 e 2 /h plateaus in the conductance of both QPCs [22]. Figure 1(c) shows data taken at 13 mK at different values of (V bw1 , V bw2 ); the gate voltage settings are indicated by the vertical lines in Fig. 1(b). When the QPCs are in the tunneling regime, clear Coulomb blockade peaks are observed (bottom trace in Fig. 1(c)), we observe a residual oscillation in the conductance with the same period as the Coulomb blockade peaks. Figure 2(a) shows this oscillation at B = 0 with voltage settings that correspond to the middle of the dot's 1 e 2 /h plateau. The large variations in conductance on the scale of tens of mV in gate voltage are caused by the UCFs, while the rapid periodic oscillation is superimposed on top. The fact that the Coulomb blockade peaks and the oscillation have the same periodicity suggests that they have the same cause: quantization of the charge in the dot.
This hypothesis is further supported by measurements of differential conductance dI/dV ds as a function of the bias voltage V ds . Figure 2(b) shows dI/dV ds vs both bias and the voltage on gate n (V n ), which controls the conductance of both QPCs. For V n < ∼ −370 mV the QPCs are not fully transmitting and we see clear Coulomb diamonds associated with charge quantization. These diamonds correspond to a charging energy of 110 µeV. As V n is made less negative, the conductance of the QPCs increases. This causes U to be renormalized [7,8] and as a consequence the vertical size of the diamonds shrinks. At V n ≈ −315mV the QPCs are fully trans- mitting, and in this regime we see that the oscillations correspond to Coulomb diamond features, with a renormalized U * ≈ 16µeV (see [22]). These diamonds are superimposed on larger UCFs which form a Fabry-Perot pattern in gate voltage and bias [23]. The diamonds associated with the oscillations are shown in more detail in Fig. 2(c), where the gates are set to the middle of the dot's 1 e 2 /h plateau and the voltage on gate c2 is varied. The presence of Coulomb diamonds support the hypothesis that the oscillations are connected to a residual quantization of charge.
To confirm this hypothesis, we directly observe this residual charge quantization with capacitive measurements using the adjacent charge sensor. Making the voltage on a dot gate less negative increases the charge on the dot. The electric fields from both the gate and the additional charge change the conductance of the charge sensor by ∆G CS . We convert ∆G CS into an effective voltage change V eff , which if applied to the gate sp would produce the same ∆G CS (Fig. 3(a)). When the conductances of both QPCs are less than 2e 2 /h we have welldefined Coulomb blockade in G dot as shown in Fig. 3(b) (black trace, right axis). A simultaneous measurement of the charge sensor (blue trace, left axis) shows that V eff initially increases as V n is made less negative because of the capacitance between the gate and the charge sensor. However at the value of V n where an electron is added to the dot, there is a sharp decrease in V eff . To highlight the correspondence between the decrease in V eff and the peaks in G dot , we take the derivative D = dV eff /dV n . These data are shown in Fig. 3(c) over a range in V n that goes from the tunneling to the open regime. For V n < −375 mV the Coulomb blockade peaks are well defined and correspond to large dips in D. As V n is increased, the dip size decreases but the dips remain aligned to peaks in G dot . Figure 3(d) shows measurements in the range of V n where the QPC conductances are at 2 e 2 /h. We see a periodic variation in D, with the dips corresponding to peaks in G dot . This measurement confirms that the conductance oscillation corresponds to a residual quantization of charge on the dot.
We quantitatively analyze these data to estimate the magnitude of the residual quantization. D is determined by the capacitances of the dot d and the charge sensor CS [9]:
D = R n + R d (C n,d − e dN d /dV n )/C * d,tot .
Here N d is the number of electrons on the dot and R n = C n,CS /C sp,CS where C sp,CS and C n,CS are the capacitances of gates sp and n to the charge sensor respectively. Also R d = C d,CS /C sp,CS where C d,CS is the capacitance between the dot and the charge sensor. C n,d is the capacitance of gate n to the dot and C * d,tot is the re-normalized total capacitance of the dot with U * = e 2 /C * d,tot . For V n < −385 mV the lineshapes are well described by theoretical predictions for dN d /dV n [24,25]. The solid red lines in Fig. 3(c) shows the results of simultaneously fitting the G dot and D data to the theory of Schoeller and Schön, using values of R n , C n,d , and C * d,tot estimated from other measurements (see [22] for details). This fit gives R d = 0.93 (we estimate an error of ±0.21, see [22]), and we use this value to analyze the data in other gate voltage regions. For −370 < V n < −340 mV the solid line in Fig. 3(c) shows a fit to Matveev's prediction for a one-leaded dot without phase coherence, with the adjustable QPC near 2e 2 /h [3,9]. In the limit of a perfectly transmitting contact, this theory predicts there should not be a periodic variation in the charge sensing signal, so we fit the data in Fig. 3(d) to a model for MCB in a one-leaded dot [12]: e dN d /dV n = C n,d (1 + (A/e) cos(2πC n,d V n /e)) where A gives the residual charge quantization. We find that A/e = 0.27 +0. 21 −0.08 , indicating that a significant amount of charge is still quantized.
Theoretical results imply that the oscillation depends on phase coherence in the dot. In a two-leaded dot without phase coherence, G dot should not oscillate when the QPCs are at 2 e 2 /h. Even if the QPCs have a small reflection coefficient r 2 (defined by G QP C = 2e 2 /h (1−r 2 )) the lowest order effect is to decrease the average dot conductance, whereas any oscillations are order r 4 or higher [26]. The fact that phase coherence is important suggests that the oscillation is MCB. If the conductance oscillation is MCB, then it should be sensitive to an applied magnetic field which disrupts the constructive interference between time-reversed paths that causes WL. Figure 4(a) shows G dot as a function of V c2 at several magnetic fields, and the size of the oscillation quickly decreases with increasing field. To quantitatively analyze these data, we follow Cronenwett et al. [11], Fourier transforming the data and integrating the power spectral density around the frequency of the oscillation to find the power P MCB . The results are shown as the solid line in Fig. 4(b). The dotted line in Fig. 4(b) shows G dot averaged over an ensemble of dot shapes obtained by changing the voltages on gates c1 and c2 [21,27]. The dip around B = 0 is caused by WL, and the width of the dip is the magnetic field scale necessary to break time-reversal symmetry. The fact that the amplitude of the oscillation decreases over the same field scale is strong evidence that the oscillation is MCB. For B > 5 mT P MCB is small but non-zero because while the oscillations are weaker and less frequent, they are still present at some gate voltages and magnetic fields, eg. the top two traces in Fig. 1(c).
MCB should also depend sensitively on temperature: the dephasing time, which describes the time scale on which electrons in the dot lose phase coherence, decreases with increasing temperature and interference effects become weaker [21,27]. Figure 4(c) shows measurements of G dot at different temperatures, and Fig. 4(d) show the results of extracting P MCB from data at B = 0 (filled circles) and B = 30 mT (open squares). The amplitude of the oscillation decreases quickly with increasing temperature (the saturation at P MCB = 2 × 10 −5 e 4 /h 2 is from the noise floor).
In conclusion, we observe an oscillation in the conductance of an open dot that we identify as MCB, a type of Coulomb blockade that depends on electron interference. Previously, a dot with four total modes (one spin degenerate mode in each QPC) was thought to be well described by the theory for the many mode limit, which predicts that MCB should be absent. Our results demonstrate that the understanding of this system, and more generally two terminal mesoscopic systems with several transmitting modes and long coherence times at low temperatures, is incomplete and that theoretical calculations are necessary to explain the interplay of coherence and Coulomb interactions.
We are grateful to P. W. Brouwer
Supplementary Information for Coulomb Blockade in an Open
Quantum Dot
QUANTUM DOT AND CHARGE SENSOR CONDUCTANCE MEASUREMENTS
We measure the large quantum dot by placing a small oscillating voltage on top of the dc bias voltage and measuring the resulting current with a DL Instruments Model 1211 current pre-amplifier and a Princeton Applied Research 124A lock-in amplifier (the circuit for the large dot is sketched in Fig. 1(a) in the main text). For the measurements of the large dot we use an oscillation frequency of 17 Hz and an excitation voltage V exc from 1 to 5 µVrms. We use a separate but identically constructed circuit to measure the charge sensor. For the charge sensor measurements we use a frequency of 97 Hz and an excitation voltage of 5 µVrms. Measurements of the individual QPCs show clear conductance plateaus quantized at integer multiples of 2 e 2 /h, demonstrating that the QPCs do not have spurious resonances that could cause the observed oscillation in the dot conductance. Figure S1(a) shows the conductance of QPC 1, formed by the gates bw1 and n, with no voltage applied to any other dot gates. These data show plateaus at G = 2 e 2 /h and 4 e 2 /h. Figure S1(b) shows similar data for QPC 2, with voltages applied only to the gates bw2 and n. The clear plateaus in these data show there is no evidence of spurious resonances. The figure also shows the results of measuring the QPCs at 600 mK. The slope of the increase in conductance between the plateaus at low and high temperatures is approximately equal, indicating that the dominant energy scale for the opening of a new mode in the QPC is greater than ≈ 50 µeV.
QPC CONDUCTANCE PLATEAUS
For the measurements in the main text, the device was cooled to 4 K with a positive bias of ≈ +200 mV applied to all gates. The positive bias "pre-depletes" the gates, preventing us from characterizing the individual QPCs even if all other gates are set to 0 V. The measurements in Fig. S1(a) and (b) have been performed after all the dot measurements reported in the paper, and following a partial thermal cycle to a temperature on the order of or greater than 100 K to reduce the effects of the positive bias voltage applied when the dot was initially cooled down to 4 K. The measurements in Fig. S1(a) and (b) were taken with a small excitation voltage and little averaging, which cause the noise on the measurement.
ESTIMATION OF THE REFLECTION COEFFICIENT
Through careful analysis of the temperature dependence of the average dot conductance at finite magnetic field we have determined that the reflection coefficients of the QPCs are on the order of 2% or less. The analysis is discussed below.
For the work reported in this paper we have carefully tuned both QPCs to their 2 e 2 /h plateaus so that there is one fully transmitting spin-degenerate mode in each QPC. However, even at the optimal QPC settings there may still be a small reflection coefficient in the QPCs. This reflection coefficient affects the dot conductance, and this can be observed in the conductance at finite magnetic field where weak localization is absent. To extract the conductance at finite field, we analyze measurements of the ensemble averaged dot conductance as a function of magnetic field like that shown by the dotted line in Fig. 4(b) of the main text. In this data we see the weak localization dip at B= 0, and following Huibers et al. [27] we fit this dip to a Lorentzian:
< G dot (B) >=< G dot > B =0 − A 1 + (2B/B c ) 2
where < G dot > B =0 is the average conductance at finite field, and A and B c are the size and width of the weak localization dip, respectively. The temperature dependence of < G > B =0 depends on the reflection coefficient of the QPCs.
To characterize the temperature dependence, we find the difference
δ < G dot >=< G dot > B =0 (T )− < G dot > B =0 (T 0 )
where T 0 = 435 mK and this quantity is plotted in Fig. S2. We see there is a very small temperature dependence of the dot conductance. There is no explicit theoretical prediction for the temperature dependence of a coherent quantum dot with one fully transmitting spindegenerate mode in each QPC (this is the N = 4 case, where N is the total number of transmitting channels in both QPCs). However, for a coherent dot with N ≫ 1 Brouwer et al. [13] have calculated the temperature dependence and find that it is the same as that for an incoherent dot [14] with N ≫ 1 and is given by:
δ < G dot >= e 2 h −r 2 2 ln T 0 T (S1)
In this equation the reflection coefficients r 2 of the two QPCs are assumed to be equal and are defined by G QP C = 2 e 2 /h (1 − r 2 ). The solid red line in Fig. S2 shows the results of fitting the data to this equation, and it appears the N ≫ 1 theoretical prediction gives a decent fit for N = 4. From the fit we obtain r 2 ≈ 0.02. Furusaki and Matveev [26] have calculated the temperature dependence for an incoherent dot with N = 4 and found
δ < G dot >= e 2 h −4r 2 Γ( 3 4 ) Γ( 1 4 ) γU k B π (T −1/2 − T −1/2 0 ) (S2)
In this equation γ = exp(C) where C = 0.5772 . . .. The dashed blue line shows a fit to this equation and we obtain r 2 ≈ 0.007. We note that the theoretical prediction for a coherent dot fits the data better than the prediction for the incoherent dot, indicating the importance of phase coherence to understanding the dot properties. These fits allow us to conclude that the reflection coefficients of the QPCs are small (on the order of 2% or less) and hence effects that are higher order in r 2 should be suppressed.
DETERMINING THE RENORMALIZED CHARGING ENERGY
In this section, we describe how we analyze the data in figure 2(b) in the main text to extract the renormalized charging energy U * ≈ 16 µeV near V n = −315 mV. This value is used to determine C * d,tot = e 2 /U * which is an input for the fit in figure 3(d) of the main text.
In figure 2(b) of the main text the diamonds are on top of a background conductance caused by Fabry-Perot interference of electrons in the big dot, making determination of the charging energy more difficult. The procedure for subtracting this background is demonstrated in Fig. S3. Figure S3(a) shows the data from figure 2(b) in the main text. To isolate the background Fabry-Perot pattern, we smooth the data by averaging in gate voltage over the period of the Coulomb diamonds, ∆V n ≈ 3 mV. The result of this averaging is shown in Fig. S3(b). We then subtract these averaged data from the raw data in Fig. S3(a) to isolate the oscillation. The result is shown in Fig. S3(c), with dashed white lines as guides to the eye. From the Coulomb diamonds, we find a renormalized charging energy U * ≈ 16 µeV at V n = −315 mV.
CHARGE SENSING FITS
In this section we describe how we fit the charge sensing data in the range V n < −385 mV in figure 3 of the main text to determine the capacitance ratio R d ≈ 0.93 ± 0.21. This ratio is used as an input for the fits in the other two V n ranges discussed in the main text.
To fit the charge sensing data in figure 3 in the main text we use a model similar to that in Berman et al. [9] illustrated in Fig. S4(a). In this diagram, C sp,CS and C n,CS are the capacitances of gates sp and n to the charge sensor, respectively. C n,d is the capacitance of gate n to the large dot, C d,CS is the capacitance between the dot and the charge sensor, and C * d,tot is the re-normalized total capacitance of the large dot, related to the re-normalized charging energy by U * = e 2 /C * d,tot . Based on this model for two quantum dots [28] (the large dot and the charge sensor), we can derive the dependence of D = dV eff /dV n on the capacitances, giving rise to the equation
D = R n + R d C n,d − e dN d /dV n C * d,tot(S3)
given in the main text, with R n = C n,CS /C sp,CS and R d = C d,CS /C sp,CS . Figure S4(b) shows a simultaneous measurement of the charge sensing signal and conductance for V n < −385 mV (magnification of the data in figure 3c of the main text). In this region, a calculation of the individual QPC conductances based on measurements of the dot conductance show that one of the dot QPCs is partially transmitting, while the other is in the tunneling regime. For this configuration, theoretical work by Schoeller and Schön [24] and Grabert [25] predict dN d /dV n . We fit the data in Fig. S4(b) to these theoretical predictions using equation S3. For these fits, we input C n,d ≈ 58 aF estimated from the spacing of the Coulomb Blockade peaks and U * ≈ 115 µeV estimated from the Coulomb diamonds in figure 2(a) of the main text. For the Schoeller and Schön fits we input g = G QP C1 + G QP C2 ≈ 0.8 e 2 /h estimated from the calculation of the QPC conductances (the fits do not depend sensitively on this estimate). To estimate R n , we use the fact that in the region −330 mV < V n < −300 mV when both QPCs are open, dN d /dV n is close to its classical value C n,d /e, with small perturbations from the residual charge quantization. So the average charge sensing signal in this region is < D >≈ R n , and from the data we estimate R n ≈ 0.008. This value is close to what we expect: from measurements of the charge sensor response to voltage changes on individual gates, we measure C n,CS = 0.08 aF and C sp,CS = 12 aF, giving R n = 0.007.
Using these values as inputs we fit the data in Fig. S4(b). The solid red line shows a simultaneous fit to both the charge sensing data and the transport data from the theory of Schoeller and Schön, a fit also shown in figure 3(c) of the main text. The solid black line shows a fit of the same theory to only the charge sensing data. Finally, the solid blue line shows a fit to the theoretical prediction of Grabert. The theoretical predictions agree well with the data, and from these and other fits, we extract R d ≈ 0.93±0.21. A value of R d near 1 is reasonable, because the large dot essentially "gates" the charge sensor, and so we expect it to have a capacitance comparable to one of the charge sensor's gates. Using C sp,CS = 12 aF estimated from measurements of the charge sensor, we obtain C d,CS ≈ 11 aF.
The fits return temperatures T ≈ 54 ± 20 mK, which are significantly higher than our electron temperature of 13 mK. The high temperatures extracted from the fits are caused by the Coulomb blockade peaks being broadened by the back-action of the charge sensor on the large quantum dot [29]. When an electron tunnels on and off the charge sensor, it "gates" the large dot and acts like a fluctuating gate voltage that affects the energy of the large dot. The fluctuations of the large dot energy have energy scale (C d,CS /C * d,tot )(e 2 /C s,tot ) = (C d,CS /C s,tot )U * ≈ 8 µeV, where U * = 115 µeV in this gate voltage range, and C s,tot ≈ 150 aF is the total capacitance of the small dot. These fluctuations broaden the Coulomb Blockade peaks, and this broadening appears as an increased temperature in our fits: the energy scale of 8 µeV converts (via Boltzmann's constant) to a temperature of 93 mK, which is on the order of the broadening that we obtain in the fits. We note that as we increase the conductance of both QPCs in the large dot, the renormalized charging energy U * decreases and the effect of the back-action on the large dot grows smaller. When both QPCs are fully transmitting, U * ≈ 16 µeV and the energy scale of the back-action is ≈ 1 µeV, which is on the order of our temperature of 13 mK.
1. (color online) (a) Electron micrograph of a device nominally identical to the measured device. (b) Conductance of the large dot (G dot ) at T = 540 mK and B = 25 mT. The vertical lines mark the gate voltages at which the cuts in (c) are taken. (c) G dot as a function of V bw2 for different settings of the QPCs at T = 13 mK and B = 25 mT. The bottom trace is taken at V bw1 = −477 mV while the top trace is taken at V bw1 = −390 mV.
2. (color online) (a) G dot measured at T = 13 mK (charge sensor not active) and B = 0 with the gate voltage settings corresponding to the middle of the dot's 1 e 2 /h plateau. (b) dI/dV ds as a function of V ds and Vn at 13 mK (with the charge sensor active) and B = 0. The QPCs each have a fully transmitting mode at Vn = −315mV. (c) dI/dV ds as a function of the voltage on the gate c2, when the conductance through each QPC is kept at 2 e 2 /h. The data are taken at T = 13 mK (charge sensor not active) and B = 0. The dashed white lines are guides to the eye.
Fig. 1
1(c)). However, when both QPCs are set to 2 e 2 /h (top two traces in in
FIG. 3 .
3(color online) (a) Small dot Coulomb blockade peak used for charge sensing. We convert the change in the conductance ∆GCS of the charge sensor into an effective voltage change V eff . (b) Simultaneous measurement of charge sensing signal V eff (left axis) and transport (right axis) in the large dot, when the conductances of both QPCs are less than 2 e 2 /h. (c) Simultaneous measurement of dV eff /dVn (blue dots, left axis) and transport (black dots, right axis). The solid line shows fits discussed in the text. (d) Charge sensing data (left axis) and transport (right axis) at the values of Vn for which the QPCs are open to 2 e 2 /h conductance. The solid red line is a fit described in the text.
FIG. 4 .
4(color online) (a) Conductance as a function of Vc2 at several different magnetic fields at 13mK. (b) The solid line (left axis) shows PMCB obtained by Fourier transforming data like those in (a). The dotted line shows the ensembleaveraged conductance of the dot as a function of magnetic field. All data are taken at 13mK. (c) Conductance as a function of Vc2 at B = 0 and several different temperatures. (d) PMCB averaged over data taken over a wider range of Vc2 than in (c) at several different values of Vc1.
FIG. S1. (a) Conductance measurements of QPC 1 (a) and QPC 2 (b) at 13 mK and 600 mK. The 600 mK plateaus have been shifted horizontally for clarity.
FIG. S2 .
S2Temperature dependence of the change in the ensemble averaged dot conductance at finite magnetic field. The solid and dashed lines are fits discussed in the text.
FIG. S3. (a) Conductance as a function of Vn at 13 mK (with charge sensor active) and B = 0. The QPCs each have a fully transmitting mode at Vn = −315 mV. (b) Result of averaging (a) over ∆Vn ≈ 3 mV, which is one period of the Coulomb diamonds. The averaging leaves only the background Fabry-Perot resonance. (c) Result of subtracting the background in (b) from the data in (a). These subtracted data allow us to identify Coulomb diamonds (dashed white lines are guides to the eye) and to extract a renormalized charging energy U * .
FIG. S4. (a) Schematic of the large quantum dot and charge sensor, showing the capacitances relevant to determining the charge sensing signal as described by equation 1. (b) Simultaneous measurement of transport and charge sensing. The solid lines are fits discussed in the text.
, K. A. Matveev, J. von Delft, Y. Oreg, and I. L. Aleiner for discussions. This work was supported by the NSF under DMR-0906062 and CAREER grant No. DMR-0349354, as well as by the U.S.-Israel BSF grants No. 2008149 and No. 2004278. D.G.-G. thanks the Sloan and Packard Foundations for financial support, and acknowledges a Research Corporation Research Innovation grant.
CA 95135 ‡ Present address: Solyndra. Gst Hitachi, San Jose, Fremont, CA 94538address: Hitachi GST, San Jose, CA 95135 ‡ Present address: Solyndra, Fremont, CA 94538
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| []
|
[
"INTEGRATED SYMBOLIC CONTROL DESIGN FOR NONLINEAR SYSTEMS WITH INFINITE STATES SPECIFICATIONS",
"INTEGRATED SYMBOLIC CONTROL DESIGN FOR NONLINEAR SYSTEMS WITH INFINITE STATES SPECIFICATIONS"
]
| [
"Giordano Pola ",
"Alessandro Borri ",
"Maria D Di Benedetto "
]
| []
| []
| Discrete abstractions of continuous and hybrid systems have recently been the topic of great interest from both the control systems and the computer science communities, because they provide a sound mathematical framework for analysing and controlling embedded systems. In this paper we give a further contribution to this research line, by addressing the problem of symbolic control design of nonlinear systems with infinite states specifications, modelled by differential equations. We first derive the symbolic controller solving the control design problem, given in terms of discrete abstractions of the plant and the specification systems. We then present an algorithm which integrates the construction of the discrete abstractions with the design of the symbolic controller. Space and time complexity analysis of the proposed algorithm is performed and a comparison with traditional approaches currently available in the literature for symbolic control design, is discussed. Some examples are included, which show the interest and applicability of our results. | null | [
"https://arxiv.org/pdf/1006.2853v1.pdf"
]
| 50,662,065 | 1006.2853 | 01940ea3a9d1b74ad8d50b40730e8e7f039b65df |
INTEGRATED SYMBOLIC CONTROL DESIGN FOR NONLINEAR SYSTEMS WITH INFINITE STATES SPECIFICATIONS
Giordano Pola
Alessandro Borri
Maria D Di Benedetto
INTEGRATED SYMBOLIC CONTROL DESIGN FOR NONLINEAR SYSTEMS WITH INFINITE STATES SPECIFICATIONS
Discrete abstractions of continuous and hybrid systems have recently been the topic of great interest from both the control systems and the computer science communities, because they provide a sound mathematical framework for analysing and controlling embedded systems. In this paper we give a further contribution to this research line, by addressing the problem of symbolic control design of nonlinear systems with infinite states specifications, modelled by differential equations. We first derive the symbolic controller solving the control design problem, given in terms of discrete abstractions of the plant and the specification systems. We then present an algorithm which integrates the construction of the discrete abstractions with the design of the symbolic controller. Space and time complexity analysis of the proposed algorithm is performed and a comparison with traditional approaches currently available in the literature for symbolic control design, is discussed. Some examples are included, which show the interest and applicability of our results.
Introduction
Discrete abstractions of continuous and hybrid systems have been the topic of intensive study in the last twenty years from both the control systems and the computer science communities [EFP06]. While physical world processes are often described by differential equations, digital controllers and software and hardware at the implementation layer, are usually modelled through discrete/symbolic processes. This mathematical models heterogeneity has posed during the years interesting and challenging theoretical problems that are needed to be addressed, in order to ensure the formal correctness of control algorithms. One approach to deal with this heterogeneity is to construct symbolic models that are equivalent to the continuous process, so that the mathematical model of the process, of the controller, and of the software and hardware at the implementation layer, are of the same nature. Several classes of dynamical and control systems admitting symbolic models, were identified during the years. We recall timed automata [AD94], rectangular hybrid automata [HKPV98], and o-minimal hybrid systems [LPS00] in the class of hybrid automata. Control systems were considered further. Early results in this regard are reported in the work of [CW98], [MRO02], [FJL02] and [BMP02]. Recent results include the work of [TP06], which showed existence of symbolic models for controllable discretetime linear systems, and the work of [HCS06,BH06] for piecewise-affine and multi-affine systems. Many of the aforementioned work are based on the notion of bisimulation equivalence, introduced by Milner and Park [Mil89,Par81] in the context of concurrent processes, as a formal equivalence notion to relate continuous and hybrid processes to purely discrete/symbolic models. A new insight in the construction of symbolic models has been recently placed through the notion of approximate bisimulation introduced by Girard and Pappas in [GP07]. Based on the above notion, some classes of incrementally stable [Ang02] control systems were recently shown to admit symbolic models: discrete-time linear control systems [Gir07], nonlinear control systems with and without disturbances [PGT08,PT09], nonlinear time-delay systems [PPDT10] and switched nonlinear systems [GPT10]. Recent results in the work of [ZPT10] have also shown the existence of symbolic models for unstable nonlinear control systems, satisfying the so-called incremental forward completeness property. The use of symbolic models in the control design of continuous and hybrid systems has been investigated in the work of [TP06,YB09,Tab08], among many others. The work in [TP06] considers discrete-time linear control systems, the work in [YB09] considers piecewise-affine systems while the work in [Tab08] considers stabilizable nonlinear control systems. In this paper we give a further contribution to this research line and in This work has been partially supported by the Center of Excellence for Research DEWS, University of L'Aquila, Italy. particular, in the direction of [Tab08]. We consider symbolic control design of nonlinear control systems where specifications are characterized by an infinite number of states and modelled through differential equations: Given a plant nonlinear control system and a specification nonlinear (autonomous) system, we investigate conditions for the existence of a symbolic controller that implements the behaviour of the specification, with a precision that can be rendered as small as desired. In other words, we look for a symbolic controller so that the interconnection between the plant and the controller satisfies or conforms [CGP99] the specification with an arbitrarily small precision. The symbolic controller is furthermore requested to be non-blocking in order to prevent the occurrence of deadlocks in the interaction between the plant and the symbolic controller. This control design problem can be seen as an approximated version of similarity games, as discussed in [Tab09]. Similar problems have been studied in the literature (in a non-approximating settings) in the context of supervisory control [CL99], symbolic control design for piecewise-affine systems enforcing temporal logic specifications [YB09], among many others. The control design problem that we consider in this paper has been solved by following the so-called correctby-design approach, see e.g. [TP06,Tab08,YB09]. We first construct the symbolic models of the plant and the specification by making use of (some variations of) the results established in [PGT08]. We then solve the control design problem at the symbolic layer, to finally come back at the continuous layer, by providing appropriate approximating bounds in the quantization errors which guarantee the solution to the control design problem under study. The solution of the control design problem at the symbolic layer is shown to be the maximal non-blocking part of the (exact) parallel composition [CL99] of the symbolic models associated with the plant and the specification. By following the correct-by-design approach, the design of the symbolic controller solving the problem at hand, requires a first computation of the plant and the specification symbolic models, then a construction of the (exact) parallel composition of the symbolic systems obtained and finally a computation of the maximal non-blocking part of the composed system. While being formally correct from the theoretical point of view, this approach is in general rather demanding from the computational point of view, because of the large size of the symbolic models needed to be constructed, in order to synthesize the symbolic controller solving the design problem. This drawback is common with other approaches currently available in the literature on symbolic control design of continuous and hybrid systems, see e.g. [TP06,YB09,Tab08] and motivated some researchers to propose solutions to cope with complexity. For example, the work in [TiI09] proposes nonuniform state quantizations in the construction of the symbolic models of the to-be-controlled plant system. In this paper we propose an alternative solution to the one studied in [TiI09]. Inspired by on-the-fly verification and control of timed or untimed transition systems (see e.g. [CVWY92,TA99]), we approach the design of symbolic controllers by advocating an "integration" philosophy: instead of computing separately the symbolic models of the plant and of the specification to then design the controller at the symbolic layer, we integrate each step of the procedure in only one algorithm. Space and time complexity analysis of the proposed algorithm is performed and a comparison with traditional approaches currently available in the literature, is discussed. Some examples are included which show the interest and applicability of our results. For the sake of completeness, a detailed list of the employed notation is included in the Appendix (Section 8).
Preliminary Definitions
2.1. Control Systems. The class of control systems that we consider in this paper is formalized in the following definition.
Definition 2.1. A control system is a quintuple:
(2.1) Σ = (X, X 0 , U, U, f ),
where:
• X ⊆ R n is the state space;
• X 0 ⊆ X is the set of initial states;
• U ⊆ R m is the input space;
• U is a subset of the set of all locally essentially bounded functions of time from intervals of the form ]a, b[⊆ R to U with a < 0, b > 0; • f : R n × U → R n is a continuous map satisfying the following Lipschitz assumption: for every compact set K ⊂ R n , there exists a constant κ ∈ R + such that
f (x, u) − f (y, u) ≤ κ x − y ,
for all x, y ∈ K and all u ∈ U .
A curve ξ :]a, b[→ R n is said to be a trajectory of Σ if there exists u ∈ U satisfying: . We also write ξ xu (τ ) to denote the point reached at time τ under the input u from initial condition x; this point is uniquely determined, since the assumptions on f ensure existence and uniqueness of trajectories [Son98]. A control system Σ is said to be forward complete if every trajectory is defined on an interval of the form ]a, ∞[. Sufficient and necessary conditions for a system to be forward complete can be found in [AS99]. The above formulation of control systems can be also used to model autonomous nonlinear systems, i.e. systems with no control inputs. With a slight abuse of notation we denote an autonomous system Σ by means of the tuple (X, X 0 , f ).
(2.2)ξ(t) = f (ξ(t), u(t)),
2.2. Systems. We will use systems to describe both control systems as well as their symbolic models. For a detailed exposition of the notion of systems and of their properties we refer to [Tab09].
Definition 2.2. [Tab09] A system S is a sextuple:
S = (X, X 0 , U, -, Y, H),
consisting of:
• a set of states X;
• a set of initial states X 0 ⊆ X;
• a set of inputs U ;
• a transition relation
-⊆ X × U × X; • an output set Y ; • an output function H : X → Y . A transition (x, u, x ) ∈ -of system S is denoted by x u -x .
System S is said to be:
• countable, if X and U are countable sets;
• symbolic, if X and U are finite sets;
• metric, if the output set Y is equipped with a metric d : Y × Y → R + 0 ; • deterministic, if
for any x ∈ X and u ∈ U there exists at most one x ∈ X such that (x, u, x ) ∈ -; • non-blocking, if for any x ∈ X there exists (x, u, x ) ∈ -; • accessible, if for any x ∈ X there exists a finite number of transitions
x 0 u1 -x 1 u2 -. . . u N -x
starting from an initial state x 0 in X 0 and ending up in x.
We now introduce some notions which will be employed in the further developments. We start by introducing the notion of sub-system which formalizes the idea of extracting from the original system a subset of states, inputs and transitions. Definition 2.3. Given two systems S 1 = (X 1 , X 0,1 , U 1 , 1 -, Y 1 , H 1 ) and S 2 = (X 2 , X 0,2 , U 2 , 2 -, Y 2 , H 2 ), system S 1 is a sub-system of S 2 , denoted S 1
S 2 , if X 1 ⊆ X 2 , X 0,1 ⊆ X 0,2 , U 1 ⊆ U 2 , 1 -⊆ 2 -, Y 1 ⊆ Y 2 and H 1 (x) = H 2 (x) for any x ∈ X 1 .
The following notion formalizes the idea of extracting the maximal non-blocking sub-system from a system, where maximality is given with respect to the notion of sub-system, which naturally induces a preorder on the class of systems.
Definition 2.4. Given a system S = (X, X 0 , U, -, Y, H) the non-blocking part of S is a system N b(S) so that:
(i) N b(S) is a non-blocking system; (ii) N b(S) is a sub-system of S; (iii) S N b(S)
, for any non-blocking S S.
We finally introduce the notion of accessible part [CL99] which formalizes the idea of extracting the maximal accessible sub-system from a system.
Definition 2.5. Given a system S = (X, X 0 , U, -, Y, H) the accessible part of S is a system Ac(S) so that:
(i) Ac(S) is an accessible system; (ii) Ac(S) is a sub-system of S; (iii) S Ac(S), for any accessible S S.
In this paper we consider simulation and bisimulation relations [Mil89,Par81] that are useful when analyzing or designing controllers for deterministic systems [Tab09]. Bisimulation relations are standard mechanisms to relate the properties of systems. Intuitively, a bisimulation relation between a pair of systems S 1 and S 2 is a relation between the corresponding state sets explaining how a state trajectory s 1 of S 1 can be transformed into a state trajectory s 2 of S 2 and vice versa. While typical bisimulation relations require that s 1 and s 2 are observationally indistinguishable, that is H 1 (s 1 ) = H 2 (s 2 ), we shall relax this by requiring H 1 (s 1 ) to simply be close to H 2 (s 2 ) where closeness is measured with respect to the metric on the output set. A simulation relation is a one-sided version of a bisimulation relation. The following notions have been introduced in [GP07] and in a slightly different formulation in [Tab08].
Definition 2.6. Let S 1 = (X 1 , X 0,1 , U 1 , 1 -, Y 1 , H 1 ) and S 2 = (X 2 , X 0,2 , U 2 , 2 -, Y 2 , H 2 ) be metric systems with the same output sets Y 1 = Y 2 and metric d, and consider a precision ε ∈ R + 0 . A relation R ⊆ X 1 × X 2 , is said to be an ε-approximate simulation relation from S 1 to S 2 , if the following conditions are satisfied:
(i) for every x 1 ∈ X 0,1 , there exists x 2 ∈ X 0,2 with (x 1 , x 2 ) ∈ R; (ii) for every (x 1 , x 2 ) ∈ R we have d(H 1 (x 1 ), H 2 (x 2 )) ≤ ε; (iii) for every (x 1 , x 2 ) ∈ R we have that:
x 1 u1 1 -x 1 in S 1 implies the existence of x 2 u2 2 -x 2 in S 2 satisfying (x 1 , x 2 ) ∈ R.
System S 1 is ε-approximately simulated by S 2 or S 2 ε-approximately simulates S 1 , denoted by S 1 ε S 2 , if there exists an ε-approximate simulation relation from S 1 to S 2 . When ε = 0, system S 1 is said to be 0-simulated by S 2 or S 2 is said to 0-simulate S 1 .
By symmetrizing the notion of approximate simulation we obtain the notion of approximate bisimulation, which is reported hereafter. 1 -, Y 1 , H 1 ) and S 2 = (X 2 , X 0,2 , U 2 , 2 -, Y 2 , H 2 ) be metric systems with the same output sets Y 1 = Y 2 and metric d, and consider a precision ε ∈ R + 0 . A relation R ⊆ X 1 × X 2 , is said to be an ε-approximate bisimulation relation between S 1 and S 2 , if the following conditions are satisfied:
(i) R is an ε-approximate simulation relation from S 1 to S 2 ; (ii) R −1 is an ε-approximate simulation relation from S 2 to S 1 .
System S 1 is ε-approximately bisimilar to S 2 , denoted by S 1 ∼ =ε S 2 , if there exists an ε-approximate bisimulation relation R between S 1 and S 2 . When ε = 0, system S 1 is said to be 0-bisimilar or exactly bisimilar to S 2 .
We now introduce the notion of approximate composition of systems which is employed in the further developments to formalize the interconnection between a nonlinear control system representing the plant, and a symbolic system representing the symbolic controller.
Definition 2.8. [Tab08] Given two metric systems S 1 = (X 1 , X 0,1 , U 1 , 1 -, Y 1 , H 1 ) and S 2 = (X 2 , X 0,2 , U 2 , 2 -, Y 2 , H 2 ), with the same output sets Y 1 = Y 2 and metric d and a precision ε ∈ R + 0 , the ε-approximate composition of S 1 and S 2 is the system:
S 1 ε S 2 := (X, X 0 , U, -, Y, H), where: • X = {(x 1 , x 2 ) ∈ X 1 × X 2 : d(H 1 (x 1 ), H 2 (x 2 )) ≤ ε}; • X 0 = X ∩ (X 0,1 × X 0,2 ); • U = U 1 × U 2 ; • (x 1 , x 2 ) (u1,u2) -(x 1 , x 2 ) if x 1 u1 1 -x 1 and x 2 u2 2 -x 2 ; • Y = Y 1 ; • H : X 1 × X 2 → Y is given by H(x 1 , x 2 ) := H 1 (x 1 ), for any (x 1 , x 2 ) ∈ X.
The above notion of composition is asymmetric. This is because it models the interaction of systems S 1 and S 2 which play different roles in the composition. As it will be clarified in the next section, we interpret system S 1 as the plant system, i.e. the to-be-controlled process, and system S 2 as the controller.
Problem Statement
In this paper we address the problem of symbolic control design for nonlinear systems with infinite states specifications modelled by differential equations. In order to formally define the control design problem under consideration, we first need to provide a formal notion of symbolic controllers. Given a control system Σ = (X, X 0 , U, U, f ) and a sampling time parameter τ ∈ R + , we associate the following system to Σ:
(3.1) S τ (Σ) := (X, X 0 , U τ , τ -, Y, H),
where:
• U τ = {u ∈ U| the domain of u is [0, τ ]}; • x u τ -x if there exists a trajectory ξ : [0, τ ] → X of Σ satisfying ξ xu (τ ) = x ; • Y = X; • H = 1 X .
System S τ (Σ) is metric when we regard Y = X as being equipped with the metric d(p, q) = p − q . The above system can be thought of as the time discretization of the control system Σ. Definition 3.1. Given the control system Σ, a sampling time τ ∈ R + , a state quantization θ ∈ R + and an input quantization µ ∈ R + , a symbolic controller for Σ is formalized by means of the system:
C := (X c , X c,0 , U c , c -, Y c , H c ),
where 1 :
• X c = [X] 2θ ; • X c,0 ⊆ X c ; • U c = {u ∈ U τ | the co-domain of u is [U ] 2µ }; • c -⊆ X c × U c × X c ; • Y c = X c ; • H c = 1 Xc .
We denote by C τ,θ,µ (Σ) the class of symbolic controllers with sampling time τ , state quantization θ and input quantization µ, associated with Σ. The θ-approximate composition between the time discretization S τ (Σ) of a control system Σ and a symbolic controller C ∈ C τ,θ,µ (Σ) formalizes classical static state feedback control schemes with digital controllers, studied in the literature, see e.g. [FPW98], as illustrated in Figure 1: The state signal ξ x0u (t) at time t ∈ R + is firstly sampled with sampling time τ ∈ R + , then quantized through an Analog-to-Digital (A/D) converter with precision θ ∈ R + which associates to a state ξ x0u (τ ), the unique state x ∈ X c for which 2 ξ x0u (τ ) ∈ B [θ[ (x); the obtained digital/symbolic signal is then plugged as input to the digital/symbolic controller C which outputs a symbolic signal taking values in [U ] 2µ . Such symbolic signal is then plugged into a Zero order Holder (ZoH) with sampling time parameter τ which outputs in turn, a piecewise-constant signal u that is finally plugged as digital/symbolic control input to the control system Σ.
We are now ready to formally state the symbolic control design problem that we consider in this paper. Consider a plant nonlinear control system:
(3.2) P = (X p , X p,0 , U p , U p , f p ),
and a specification nonlinear autonomous system:
Q = (X q , X q,0 , g q ).
For the sake of homogeneity in the notation of the plant P and the specification Q we rephrase the above tuple by means of:
(3.3) Q = (X q , X q,0 , U q , U q , f q ),
where U q = {u q } with u q = 0, U q = {u q } with u q = 0, the signal 0 being the identically null function, and
f q (x, u) = g q (x) + u for any (x, u) ∈ X q × U q .
Problem 3.2. Given a plant nonlinear control system P as in (3.2), a specification nonlinear autonomous system Q as in (3.3) and a desired precision ε ∈ R + , find quantization parameters τ, θ, µ ∈ R + and a symbolic controller C ∈ C τ,θ,µ (P ) such that:
(i) (S τ (P ) θ C) ε S τ (Q); (ii) S τ (P ) θ C is non-blocking.
The above control design problem asks for a symbolic controller C that implements the behaviour of the specification Q, up to a precision ε that can be chosen as small as desired. In other words, in Problem 3.2 we look for a symbolic controller C so that the approximate composition between the plant P and the controller C satisfies or conforms [CGP99] the specification Q with an arbitrarily small precision. The symbolic controller is furthermore requested to be non-blocking in order to prevent occurrence of deadlocks in the interaction between the plant and the symbolic controller. This control design problem can be seen as an approximated version of similarity games, as discussed in [Tab09]. Similar problems have been studied in the literature (in a non-approximating settings) in the context of supervisory control [CL99], symbolic control design for piecewise-affine systems enforcing temporal logic specifications [YB09], among many other work.
Symbolic Control Design with Infinite States Specifications
In this section we provide the solution to Problem 3.2. Inspired by the so-called correct-by design approach, see e.g. [TP06, Tab08, YB09], we first construct the symbolic systems associated with the plant P and the specification Q in Section 4.1, we then solve the control design problem at the symbolic layer in Section 4.2 to finally come back at the continuous layer in Section 4.3 by providing the bounds in the approximation scheme that we propose, which guarantee the solution to Problem 3.2.
4.1. From the Continuous Layer to the Symbolic Layer. In this section we present some results based on the work of [PGT08] for constructing symbolic systems associated with the plant P and the specification Q. We start by recalling from [Ang02], the notion of incremental input-to-state stability for nonlinear control systems.
Definition 4.1. A control system Σ is incrementally input-to-state stable (δ-ISS) if it is forward complete and there exist a KL function β and a K ∞ function γ such that for any t ∈ R + 0 , any x, x ∈ R n , and any u, u ∈ U the following condition is satisfied:
(4.1) ξ xu (t) − ξ x u (t) ≤ β ( x − x , t) + γ ( u − u ∞ ) .
A characterization of the above incremental stability notion in terms of dissipation inequalities can be found in [Ang02]. Given a δ-ISS nonlinear control system Σ of the form (2.1), a sampling time τ ∈ R + , a state quantization η ∈ R + and an input quantization µ ∈ R + consider the following system:
(4.2) S τ,η,µ (Σ) := (X τ,η,µ , X 0,τ,η,µ , U τ,η,µ , τ,η,µ -, Y τ,η,µ , H τ,η,µ ),
where:
• X τ,η,µ = [X] 2η ; • X 0,τ,η,µ = X τ,η,µ ∩ X 0 ; • U τ,η,µ = [U ] 2µ ; • x u τ,η,µ -y if ξ xu (τ ) ∈ B [η[ (y) ∩ X; • Y τ,η,µ = X; • H τ,η,µ = ı : X τ,η,µ → Y τ,η,µ .
It is readily seen from the definition of X τ,η,µ and U τ,η,µ that system S τ,η,µ (Σ) is countable and becomes symbolic when the state space X and the input space U are bounded sets. System S τ,η,µ (Σ) is basically equivalent to the symbolic model proposed in [PGT08]. The main difference is that, while the symbolic model in [PGT08] is not guaranteed to be deterministic, system S τ,η,µ (Σ) is so, as formally stated in the following result:
Proposition 4.2. System S τ,η,µ (Σ) is deterministic.
Proof. The existence and uniqueness of a trajectory from an initial condition x ∈ X τ,η,µ with input u ∈ U τ,η,µ guarantees that ξ xu (τ ) is uniquely determined. Since the collection of sets
{B [η[ (y) ∩ X} y∈Xτ,η,µ is a partition of X, there exists at most one state y ∈ X τ,η,µ such that ξ xu (τ ) ∈ B [η[ (y) ∩ X.
We stress that determinism in the symbolic system S τ,η,µ (Σ) is an important property because algorithmic synthesis of symbolic systems simplifies when systems are deterministic [Tab09]. We can now give the following result that establishes sufficient conditions for the existence and construction of symbolic systems for nonlinear control systems.
Theorem 4.3. Consider a δ-ISS nonlinear control system Σ = (X, X 0 , U, U, f ) and a desired precision θ ∈ R + . For any sampling time τ ∈ R + , state quantization η ∈ R + and input quantization µ ∈ R + satisfying the following inequality:
(4.3) β(θ, τ ) + γ(µ) + η ≤ θ, systems S τ,η,µ (Σ) and S τ (Σ) are θ-approximately bisimilar.
Proof. The proof of the above result can be given along the lines of Theorem 5.1 in [PGT08]. We include it here for the sake of completeness. Consider the relation
R ⊆ X × X τ,η,µ defined by (x, y) ∈ R if and only if x ∈ B [θ[ (y) ∩ X.
We start by showing that condition (i) of Definition 2.6 holds. Consider an initial condition x 0 ∈ X 0 . By definition of the set X 0,τ,η,µ there exists y 0 ∈ X 0,τ,η,µ so that (x 0 , y 0 ) ∈ R. Condition (ii) in Definition 2.6 is satisfied by the definition of R. Let us now show that condition (iii) in Definition 2.6 holds.
Consider any (x, y) ∈ R. Consider any u 1 ∈ U τ and the transition x u1 τ -w in S τ (Σ). There exists u 2 ∈ U τ,η,µ such that:
(4.4) u 2 − u 1 ∞ ≤ µ. Set z = ξ yu2 (τ ). Since X = v∈Xτ,η,µ B [η[ (v) ∩ X, there exists v ∈ X τ,η,µ such that: (4.5) z ∈ B [η[ (v),
and therefore y u2 τ,η,µ -v in S τ,η,µ (Σ). Since Σ is δ-ISS, by the definition of R and by condition (4.4), the following chain of inequalities holds:
w − z ≤ β( x − y , τ ) + γ( u 1 − u 2 ∞ ) ≤ β(θ, τ ) + γ(µ),
which implies:
(4.6) w ∈ B β(θ,τ )+γ(µ) (z).
By combining the inclusions in (4.5) and (4.6), it is readily seen that
w ∈ B [β(θ,τ )+γ(µ)+η[ (v). By the inequality in (4.3), B [β(θ,τ )+γ(µ)+η[ (v) ⊆ B [θ[ (v)
, which implies (w, v) ∈ R and hence, condition (iii) in Definition 2.6 holds. Thus, condition (i) in Definition 2.7 is satisfied. By using similar arguments it is possible to show condition (ii) of Definition 2.7.
The above result is conceptually equivalent to Theorem 5.1 in [PGT08]. The main difference is that while Theorem 4.3 relates nonlinear systems to deterministic symbolic systems, Theorem 5.1 in [PGT08] relates nonlinear systems to symbolic models which are in general nondeterministic. The above result is now employed to define symbolic systems for the plant and the specification. Consider a plant system P as defined in (3.2) and a specification system Q as defined in (3.3). Suppose that P and Q are δ-ISS and choose a precision θ p ∈ R + and a precision θ q ∈ R + , required in the construction of the symbolic systems for P and Q, respectively. Let β p and γ p be a KL function and a K ∞ function guaranteeing the δ-ISS stability property for P and β q be a KL function guaranteeing the δ-ISS stability property for Q. Find quantization parameters τ, η, µ ∈ R + such that:
β p (θ p , τ ) + γ p (µ) + η ≤ θ p , β q (θ q , τ ) + η ≤ θ q , (4.7)
It is readily seen that parameters τ, η, µ ∈ R + satisfying the above inequalities always exist. By Theorem 4.3, S τ,η,µ (P ) is θ p -approximately bisimilar to S τ (P ) and S τ,η,0 (Q) is θ q -approximately bisimilar to S τ (Q). For the sake of notational simplicity in the further developments we refer to the systems S τ,η,µ (P ) and S τ,η,0 (Q), by means of S p and S q , respectively. 4.2. Control Design at the Symbolic Layer. Problem 3.2 translates to the following problem at the symbolic layer:
Problem 4.4. Given system S p and system S q , find a symbolic controller C ∈ C τ,θ,µ (P ) such that:
(i) (S p 0 C) 0 S q ; (ii) S p 0 C is non-blocking.
We start by introducing a technical lemma that will be used in the sequel.
S i = (X i , X 0,i , U i , i -, Y, H i ), i = 1, 2, 3.
The following properties hold:
(i) [GP07] For all ε 1 ∈ R + 0 , if S 1 ε1 S 2 then S 1 ε2 S 2 , for all ε 2 ≥ ε 1 ; (ii) [GP07] For all ε 1 , ε 2 ∈ R + 0 , if S 1 ε1 S 2 and S 2 ε2 S 3 , then S 1 ε1+ε2 S 3 ; (iii) For all ε ∈ R + 0 , S 1 ε S 2 ε S 2 .
Proof of (iii). Denote S 1 ε S 2 by the tuple (X, X 0 , U, -, Y, H) and define: R ={((x 1 , x 2 ), x) ∈ X × X 2 : x 2 = x}.
We start by showing that condition (i) in Definition 2.6 holds. Consider any initial condition (x 0,1 , x 0,2 ) ∈ X 0 . Since x 0,2 ∈ X 2 , by choosing x 0 = x 0,2 we have that ((x 0,1 , x 0,2 ), x 0 ) ∈ R. We now show that also condition (ii) in Definition 2.6 holds. Consider any ((x 1 , x 2 ), x) ∈ R. Since x 2 = x, then H 2 (x 2 ) = H 2 (x), hence by Definition 2.8 of approximate composition d(H(x 1 , x 2 ), H 2 (x)) = d(H 1 (x 1 ), H 2 (x 2 )) ≤ ε. We conclude by showing that condition (iii) in Definition 2.6 holds. Consider any ((x 1 , x 2 ), x) ∈ R and any transition
(x 1 , x 2 ) (u1,u2) -(x 1 , x 2 ) in S 1 ε S 2 . Choose the transition x u2 2 -x in S 2 so that x = x 2 .
By definition of the systems involved such transition exists. This implies that ((x 1 , x 2 ), x ) ∈ R, which concludes the proof.
We are now ready to provide the solution to Problem 4.4. Define:
(4.8) C * = S p 0 S q .
Theorem 4.6. N b(C * ) solves Problem 4.4.
Proof. We start by proving condition (i) of Problem 4.4. By Lemma 4.5 (iii), we obtain:
(4.9)
S p 0 N b(C * ) 0 N b(C * ).
By the definition of N b(C * ) it is readily seen that:
(4.10)
N b(C * ) 0 C * .
By the definition of C * = S p 0 S q and Lemma 4.5 (iii), one gets:
(4.11) C * 0 S q . By combining conditions in (4.9), (4.10), (4.11) and by Lemma 4.5 (ii) we obtain:
S p 0 N b(C * ) 0 S q .
Hence, condition (i) of Problem 4.4 is proved. We now prove condition (ii) of Problem 4.4. Consider any state (p 1 , p 2 , q) of S p 0 N b(C * ). Since N b(C * ) is non-blocking there exists a state (p + 2 , q + ) of N b(C * ) so that (p 2 , q) u -(p + 2 , q + ) is a transition of N b(C * ) for some input u = (u 2 , u 3 ). Since N b(C * ) is a sub-system of C * = S p 0 S q , then by choosing p + 1 = p + 2 and u 1 = u 2 , the transition p 1 u1 -p + 1 is a transition of S p . Since by construction p + 1 = p + 2 then (p + 1 , p + 2 , q + ) is a state of S p 0 N b(C * ) and therefore (p 1 , p 2 , q) (u1,u) -(p + 1 , p + 2 , q + ) is a transition of S p 0 N b(C * ), which concludes the proof.
We conclude this section by showing that the controller N b(C * ) is the maximal system solving Problem 4.4 in the sense of the preorder naturally induced by the notion of 0-simulation relations.
Theorem 4.7. For any system C solving Problem 4.4
(S p 0 C) 0 (S p 0 N b(C * )).
Proof. Denote by S 1 p and S 2 p copies of S p that are connected to C and N b(C * ), respectively; denote by X pc and X pc * the state spaces of S 1 p 0 C and S 2 p 0 N b(C * ) and by X 0 pc and X 0 pc * the corresponding sets of initial states. Moreover let C * = S c p 0 S c q , where S c p and S c q are the copies of S p and S q in the controller and define:
R = {((p 1 , c), (p 2 , p 3 , q)) ∈ X pc × X pc * : ((p 1 , c), q) ∈ R 1 ∧ p 1 = p 2 },
where R 1 is a 0-simulation relation from S 1 p 0 C to S q . We start by showing that condition (i) in Definition 2.6 holds. Consider any initial condition (p 0 1 , c 0 ) ∈ X 0 pc . Since (S p 0 C) 0 S q there exists q 0 ∈ S q s.t. ((p 0 1 , c 0 ), q 0 ) ∈ R 1 . By choosing p 0 2 = p 0 3 = p 0 1 , we have (p 0 2 , p 0 3 , q 0 ) ∈ X 0 pc * and hence, ((p 0 1 , c 0 ), (p 0 2 , p 0 3 , q 0 )) ∈ R. We now show that also condition (ii) in Definition 2.6 holds. Since H p (p 1 ) = H p (p 2 ), we can conclude d(H pc (p 1 , c), H ppq (p 2 , p 3 , q)) = d(H p (p 1 ), H p (p 2 )) = 0. We conclude by showing that condition (iii) in Definition 2.6 holds. Consider any ((p 1 , c), (p 2 , p 3 , q)) ∈ R and any transition (p 1 , c) (up,uc) -(p + 1 , c + ) in S 1 p 0 C. Since ((p 1 , c), q) ∈ R 1 , there exists a transition q v q + in S q so that ((p + 1 , c + ), q + ) ∈ R 1 . Hence H pc (p + 1 , c + ) = H p (p + 1 ) = H q (q + ) and q + = p + 1 . Now, since p 1 = p 2 = p 3 = q, we consider the transitions p 2 up p + 2 in S 2 p , p 3 up p + 3 in S c p and q v q + in S c q with p + 3 = p + 2 = p + 1 = q + . Notice that such transitions exist. Hence (p + 2 , p + c , q + ) is a state of S 2 p 0 N b(C * ) and the transition (p 2 , p 3 , q)
(up,up,v) -(p + 2 , p + 3 , q + ) is in S 2 p 0 N b(C * ), which implies ((p + 1 , c + ), (p + 2 , p + 3 , q + )) ∈ R.
This result is important because it shows that the controller N b(C * ) implements the maximal non-blocking behaviour of the specification symbolic system S q , which can be implemented by the plant symbolic system S p .
From the Symbolic Layer to the Continuous Layer.
We now have all the ingredients to present one of the main results of this paper which shows that there exists an appropriate choice of quantization parameters so that the symbolic controller N b(C * ) with C * defined in (4.8) solves Problem 3.2.
Theorem 4.8. Consider the plant system P as in (3.2), the specification system Q as in (3.3) and a precision ε ∈ R + . Suppose that P and Q are δ-ISS and choose parameters θ p , θ q ∈ R + so that:
(4.12) θ p + θ q ≤ ε.
Furthermore choose parameters τ, η, µ ∈ R + satisfying the inequalities in (4.7). Then the symbolic controller N b(C * ) ∈ C τ,θ,µ (P ) with θ = θ p and C * defined in (4.8) with S p = S τ,η,µ (P ) and S q = S τ,η,0 (Q), solves Problem 3.2.
Proof. We start by proving condition (i) of Problem 3.2. By Lemma 4.5 (iii), we obtain:
(4.13) S τ (P ) θp N b(C * ) θp N b(C * ).
By the definition of N b(C * ) it is readily seen that:
(4.14)
N b(C * ) 0 C * .
By the definition of C * = S p 0 S q and Lemma 4.5 (iii), one gets:
(4.15) C * 0 S q . Since S q is θ q -approximately bisimilar to S τ (Q) then:
(4.16) S q θq S τ (Q).
By combining conditions in (4.13), (4.14), (4.15), (4.16) and by Lemma 4.5 (ii) we obtain:
S τ (P ) θp N b(C * ) θp+θq S τ (Q).
Since by (4.12), θ p + θ q ≤ ε, by Lemma 4.5 (i), condition (i) of Problem 3.2 is proved. We now prove condition (ii) of Problem 3.2. Consider any state (p 1 , p 2 , q) of S τ (P ) θp N b(C * ). Since N b(C * ) is non-blocking there exists a state (p + 2 , q + ) of N b(C * ) so that (p 2 , q) u -(p + 2 , q + ) is a transition of N b(C * ) for some input u = (u 2 , u 3 ). Since S τ (P ) and S p are θ p -approximately bisimilar, for the transition p 2 u2 -p + 2 in S p there exists a transition p 1 u1 -p + 1 in S τ (P ) so that d(H p (p + 1 ), H p (p + 2 )) ≤ θ p . This implies that (p + 1 , p + 2 , q + ) is a state of S τ (P ) θp N b(C * ) and therefore that (p 1 , p 2 , q) (u1,u) -(p + 1 , p + 2 , q + ) is a transition of S τ (P ) θp N b(C * ), which concludes the proof.
Integrated Symbolic Control Design
The construction of the symbolic controller N b(S p 0 S q ) solving Problem 3.2 relies upon the basic-steps procedure illustrated in Algorithm 1.
Construct system S p , θ p -approximately bisimilar to S τ (P ); 1 Construct system S q , θ q -approximately bisimilar to S τ (Q); 2 Construct the composition S p 0 S q ; 3 Compute the non-blocking part N b(S p 0 S q ) of S p 0 S q .
4
Algorithm 1: Construction of N b(S p 0 S q ).
The procedure in Algorithm 1 is common with other approaches currently available in the literature for symbolic control design of continuous and hybrid systems, see e.g. [TP06,YB09,Tab08]. Software implementation of Algorithm 1 requires that:
• State space X p and set of input values U p of P are bounded; • State space X q of Q is bounded.
The above assumptions, while being reasonable in many realistic engineering control problems, are also needed to store the transitions of systems S p and S q in a computer machine, whose memory resources are limited by their nature. In this section, we suppose that the plant P and the specification Q satisfy the above assumptions. The procedure illustrated in Algorithm 1 is not efficient from the space and time complexity point of view 3 because:
• It considers the whole state spaces of the plant P and the specification Q. A more efficient algorithm would consider only the intersection of the accessible parts of P and Q. • For any source state x and target state y it includes all transitions (x, u, y) with any control input u by which state x reaches state y. A more efficient algorithm would consider for any source state x and target state y only one control input u and hence, only one transition. • It first construct the symbolic models S p and S q , then the composed system S p 0 S q to finally eliminate blocking states from S p 0 S q . A more efficient algorithm would eliminate blocking states as soon as they show up.
Inspired from the research line in the context of on-the-fly verification and control of timed or untimed transition systems (see e.g. [CVWY92,TA99]), we now present an algorithm which integrates each step of the four sub-algorithms in Algorithm 1 in only one algorithm.
The proposed procedure is composed of Algorithm 2 and Algorithm 3. Algorithm 2 is the main one while Algorithm 3 introduces Function NonBlock, which is recursively used in Algorithm 2. The outcome of Algorithm 2 is the symbolic controller C * * which will be shown in the further results to solve Problem 3.2. Given a set T ⊆ X × U × Y , the set X source (T ) ⊆ X denotes the projection of T onto X, i.e.
X source (T ) = {x ∈ X : ∃y ∈ Y ∧ ∃u ∈ U s.t. (x, u, y) ∈ T }.
Given a vector x ∈ R n and a precision η ∈ R + , the symbol [x] 2η denotes the unique vector in [R n ] 2η such that
x ∈ B [η[ ([x] 2η
). Algorithm 2 proceeds as follows. The set of states X 0 of C * * is initialized to be [X p,0 ∩ X q,0 ] 2η in line 2.8 and the set of states to be processed, denoted by X target , is initialized to the set of initial states in line 2.9. The set T of transitions and the set Bad of blocking states of C * * are initialized to be the empty-sets (lines 2.10, 2.11). At each basic step, Algorithm 2 processes a (non processed) state in line 2.13, by computing the state y = [ξ q x (τ )] 2η (line 2.14). If the state y is non-blocking (line 2.15), the algorithm looks for a control input u ∈ [U ] 2µ such that the plant P meets the specification Q, i.e. z = y (line 2.20). If such a control input u exists, then boolean variable F lag is updated to 1 (line 2.21), the transition (x, u, y) is added to the set of transitions T (line 2.25), and the state y is added to the set of the to-be-processed states (line 2.26). If either state y is blocking or no inputs are found for the plant P to meet the specification Q, then state x is declared blocking, and Function NonBlock(T, x, Bad) in Algorithm 3 is invoked (line 2.30), in order to remove all blocking states originating from x. Algorithm 2 proceeds with further basic steps, until there are no more states to be processed. When Algorithm 2 terminates, it returns in line 2.34 the symbolic controller C * * . Function NonBlock(T, x, Bad) extracts the non-blocking part of T . The set Badx includes the states to be processed and is initialized to contain the only state x (line 3.3). At each basic step, for any y ∈ Badx, Function NonBlock removes from the set T any transition (z, u, y) ending up in y (line 3.7), it adds z to the set Badx of states to be processed (line 3.8) and adds y to the set Bad of blocking states (lines 3.11, 3.12). Function NonBlock terminates when there are no more states to be processed and returns in line 2.14 the updated sets of transitions T of and blocking states Bad. Termination of Algorithm 2 is discussed in the following result:
Theorem 5.1. Algorithm 2 terminates in a finite number of steps.
Proof. Algorithm 2 terminates when there are no more states x in X target to be processed. For each state x, either line 2.25 or line 2.30 is executed (depending on the value of the boolean variable F lag); this ensures by line 2.13 that state x cannot be processed again in future iterations. Furthermore, the set X target is nondecreasing (see line 2.26) and always contained in the finite set [X p ] 2η ∩ [X q ] 2η . Hence, provided that Algorithm 3 terminates in finite time, the result follows. Regarding termination of Algorithm 3, in the worst case the set Bad ends up in coinciding with the accessible states of S p and S q (line 3.12) and the set Badx ends up in being empty (line 3.11). Hence from line 3.4, finite termination of Algorithm 3 is guaranteed.
Formal correctness of Algorithm 2 is guaranteed by the following result.
Theorem 5.2. Controllers N b(C * ) and C * * are exactly bisimilar.
Input:
1 Plant: P = (X p , X p,0 , U p , U p , f p ); 2 Specification: Q = (X q , X q,0 , U q , U q , f q ); 3 Precision: ε ∈ R + ; 4 Parameters: θ p , θ q ∈ R + satisfying (4.12); 5 Parameters: τ, η, µ ∈ R + satisfying (4.7); 6 Init: Proof. (Sketch.) For any state (x p , x q ) of the accessible part Ac(N b(C * )) of N b(C * ) there exists a state x c of C * * so that x p = x q = x c (see lines 2.14, 2.19, 2.20 and 2.25 in Algorithm 2). Consider the relation defined by ((x p , x q ), x c ) ∈ R if and only if x p = x c . It is readily seen that R is a 0-bisimulation relation between N b(C * ) and C * * .
7 X 0 := [X p,0 ∩ X q,0 ] 2η ; 8 X target = X 0 ; 9 T := ∅; 10 Bad := ∅; 11 foreach x ∈ [X p ∩ X q ] 2η do 12 if x ∈ X target \(X source (T ) ∪ Bad)
By the above result the controller N b(C * ) solves Problem 3.2 if and only if the controller C * * solves Problem 3.2. Hence, it shows that Algorithm 2 is correct. While the controllers N b(C * ) and C * * are exactly bisimilar, the number of states of C * * is in general, smaller than the one of N b(C * ). In fact the controller N b(C * ) may contain spurious states, e.g. states which are not accessible from a quantized initial condition in S p and a quantized initial condition in S q , since in general Ac(N b(C * )) is a (strict) sub-system of N b(C * ). On the other hand, a straightforward inspection of Algorithm 2 reveals that: Hence, the aforementioned spurious states of N b(C * ) are not included in C * * . The above remarks suggest the following result:
Theorem 5.4. C * * is the minimal 0-bisimilar system of N b(C * ).
Proof. The proof can be given by using standard arguments on bisimulation theory [CGP99]. Briefly, since by Proposition 5.3 Ac(C * * ) = C * * and since the output function H τ,η,µ of C * * is the natural inclusion from X source (T ) to X, the maximal 0-bisimulation relation R * between C * * and itself is the identity relation, i.e. R * = {(x 1 , x 2 ) ∈ X source (T ) × X source (T ) : x 1 = x 2 }. Since R * is the identity relation, the quotient of C * * induced by R * , coincides with C * * . Finally, since by Theorem 5.2 systems C * * and N b(C * ) are 0-bisimilar, the result follows.
The above result is important because it shows that the controller C * * is the system with the smallest number of states which is equivalent by bisimulation to the solution N b(C * ) of Problem 3.2.
Space and Time Complexity Analysis
In this section we provide a formal comparison in terms of space and time complexity analysis, between the procedure illustrated in Algorithm 1 and Algorithm 2.
Proposition 6.1. Space complexity of Algorithm 1 is O(max{card([X p ] 2η ) · card([U p ] 2µ ), card([X q ] 2η )}).
Proof. Since by Proposition 4.2 system S p is deterministic, the number of transitions of S p amounts to card([X p ] 2η ) · card([U p ] 2µ ). For the same reason, the number of transitions of S q is given by card([X q ] 2η ). By definition of exact composition (see Definition 2.8 with ε = 0), the number of transitions in S p 0 S q amounts in the worst case to (card([X p ] 2η ) ∩ card([X q ] 2η )) · card([U p ] 2µ ). By definition of the N b operator, the number of transitions in N b(C * ) is less than or equal to the one of S p 0 S q . Hence, by comparing the above worst case bounds, the result follows.
Proposition 6.2. Space complexity of Algorithm 2 is O(card([X p ] 2η ∩ [X q ] 2η )).
Proof. By lines 2.14, 2.15, 2.20, and 2.25 in Algorithm 2, the triple (x, u, y) is added to the set T of transitions of C * * , if (x, u, y) is a transition of S p and (x, y) is a transition of S q . Hence, the result follows from determinism of systems S p and S q , which is guaranteed by Proposition 4.2.
By comparing Propositions 6.1 and 6.2, it is readily seen that space complexity of Algorithm 2 is smaller than or equal to space complexity of Algorithm 1. In particular, when the plant system P and the specification system Q coincide, implying [X p ] 2η = [X q ] 2η and card([U p ] 2µ ) = 1, the space complexity of the procedure in Algorithm 1 and of Algorithm 2 coincides, resulting in O(card([X p ] 2η )) = O(card([X q ] 2η )). This is indeed consistent with the integration philosophy that we advocated in Algorithm 2. Algorithm 2 becomes more and more efficient from the space complexity point of view as much as the behaviours of the plant and of the specification differ. When P and Q coincide there is no gain in terms of space complexity, in the use of Algorithm 2. We now proceed with a further step by providing a comparison in terms of time complexity analysis.
Proposition 6.3. Time complexity of Algorithm 1 is O(card([X q ] 2η ) · card([X p ] 2η ) · card([U p ] 2µ )).
Proof. The number of steps needed in the construction of S p and S q amounts to card([X p ] 2η )·card([U p ] 2µ ) and card([X q ] 2η ), respectively. Since as shown in Proposition 6.1 the number of transitions in S p and S q is given respectively by card([X p ] 2η ) · card([U p ] 2µ ) and card([X q ] 2η ), the number of steps needed in the construction of
S p 0 S q is given by card([X q ] 2η ) · card([X p ] 2η ) · card([U p ] 2µ )
. Regarding the computation of the non-blocking part N b(S p 0 S q ), in the worst case for any state of S p 0 S q , i.e. for any state in [X q ∩ X p ] 2η , all transitions in S p 0 S q are needed to be processed in order to find blocking states. Since the number of transitions in
S p 0 S q is card([X q ∩ X p ] 2η ) · card([U p ] 2µ )
, the overall number of steps needed in the computation of
N b(S p 0 S q ) is given by card([X q ∩ X p ] 2η ) 2 · card([U p ] 2µ )
. By comparing the above worst case bounds, the result follows.
Proposition 6.4. Time complexity of Algorithm 2 is
O(max{card([X q ∩ X p ] 2η ) · card([U p ] 2µ ), card([X q ∩ X p ] 2η ) 2 }).
Proof. By exploring Algorithm 2, it is easy to see that the number of steps needed in the computation of C * * is upper bounded by:
(6.1) N1 i=0 (N 2 + N 3 ),
where N 1 = card([X p ∩ X q ] 2η ), N 2 is an upper bound to the number of steps needed in the execution of lines 2.13/27 in Algorithm 2, and N 3 is an upper bound to the number of steps needed in the execution of lines 2.28/30 in Algorithm 2. Quantity in (6.1) can be rewritten as the sum of the term N1 i=0 N 2 and the term N1 i=0 N 3 , the first of which is upper bounded by card([X p ∩ X q ] 2η ) · card([U p ] 2µ ). Regarding the term N1 i=0 N 3 , whenever Algorithm 2 executes line 2.30, i.e. (T, Bad) :=NonBlock(T, x, Bad), states x involved are different. Indeed suppose by contradiction that at step i state x is processed in line 2.30 and at step j state x is processed in line 2.30 with i < j and x = x . When at step i Algorithm 3 is invoked, state x is added to the set Bad (see lines 3.3, 3.4 and 3.12). Since at the end of step i state x ∈ Bad, in the further steps and in particular at step j, state x will be no longer processed (see line 2.13). Since x = x, then at step j state x cannot be processed in line 2.30. Hence a contradiction holds. Since any time Algorithm 3 is invoked it processes different states, the overall time complexity due to the term N1 i=0 N 3 is upper bounded by the time complexity needed in computing the non-blocking part of S p 0 S q which, from Proposition 6.3 amounts to card([X q ∩ X p ] 2η ) 2 · card([U p ] 2µ ). By comparing the above worst case bounds, the result follows.
By comparing Propositions 6.3 and 6.4, it is readily seen that time complexity of Algorithm 2 is smaller than or equal to time complexity of Algorithm 1. In particular, when the plant system P and the specification system Q coincide, implying [X p ] 2η = [X q ] 2η and card([U p ] 2µ ) = 1, the time complexity of the procedure in Algorithm 1 and of Algorithm 2 coincides, resulting in O(card([X p ] 2η ) 2 ) = O(card([X q ] 2η ) 2 ).
Examples
In this section we present some examples of application of the results illustrated in the previous sections. In particular, we consider in Section 7.1 symbolic control design problem for a nonlinear control system and in Section 7.2 symbolic control design for linear control systems. The results shown hereafter are based on computations performed on an Intel Core 2 Duo T5500 1.66GHz laptop with 4 GB RAM.
7.1. Nonlinear Control Systems. Consider the following plant nonlinear control system: P :
ẋ 1 = −2x 1 + x 2 3 − u x 2 = 2x 1 − 7e x2 + 7 x 3 = −3x 3 + 3
4 u 2 , and an infinite states specification, expressed by the following differential equation:
Q : ẋ 1 = −3x 1 + x 3 3 x 2 = x 1 − 5 sin x 2 x 3 = −x 2 2 − 4x 3 .
We suppose for simplicity that the plant and the specification systems share the same state space, chosen as:
X p = X q = [−1, 1[×[−1, 1[×[−1, 1[,
the same set of initial states, chosen as:
X 0 p = X 0 q = [−1, 0[×[−1, 0[×[−1
, 0[, and that the plant input space is: U = [−1, 1]. By using the δ-ISS Lyapunov characterization in [Ang02] it is possible to show the plant system P is δ-ISS with functions: β p (r, s) := √ 2 e −1.21 s r, γ(r) := √ 14.88 r, r, s ∈ R + 0 . Analogously the specification system Q can be shown to be δ-ISS with function: β q (r, s) := √ 2 e −s r, r, s ∈ R + 0 . For a precision ε = 0.2, we can choose the following quantization parameters for the plant and the specification systems:
θ p = 0.13, θ q = 0.07, η = 1/30, τ = 1, µ = 0.001. The above choice of quantization parameters guarantees that the inequalities in (4.7) and (4.12) are fulfilled. By running Algorithm 2 the integrated symbolic controller C * * has been designed. Given the large size of the controller obtained (3152 states) we do not report in the paper further details on it. Figures 2 shows the evolution of the plant system P when interconnected with the symbolic controller C * * and the evolution of the specification system Q, with initial condition x 0 = (−1, −1, −1 + 4 η). It is readily seen from the plots that for the initial condition x 0 the specification is fulfilled, up to the precision ε = 0.2 chosen in this example.
We conclude this section by discussing a comparison between the "integrated" approach formulated in Algorithm 2 and the "non-integrated" approach described in Algorithm 1. Experimental results associated with the computation of C * * and of N b(C * ) are reported in Tables 1 and 2. In particular, Table 1 shows details in the computation of the controller N b(C * ) performed by running Algorithm 1. Table 2 reports a comparison between the computation of the controllers C * * and N b(C * ). The computation time needed in the construction of the controllers is expressed in seconds and the maximal memory occupation is given in terms of the maximal number of data needed in the construction of the controllers. In particular, the maximal memory occupation in the construction of N b(C * ) is expressed as the sum of the number of transitions of S p , the Table 1. Details on the computation of N b(C * ).
N b (C * ) C * * Ratio States 21894 3152 0.14 Transitions 1265217 3152 2.5 · 10 −3 Max memory occupation 93347397 10400 1.11 · 10 −4 Time 147487 11144 0.08 Table 2. Comparison in the computation of N b(C * ) and C * * .
number of transitions of S q and the number of transitions of S p S q , while the maximal memory occupation in the construction of C * * is given as the sum of the number of transitions in C * * and the number of states in Bad. For both controllers N b(C * ) and C * * each transition is weighted as three data and each state as one datum. The experimental results shown in Table 2 can be summarized, as follows:
• The number of states of C * * is 14% times the number of states of N b(C * ); • The number of transitions of C * * is 0.25% times the number of transitions of N b(C * ); • The maximal memory occupation of C * * is 0.011% times the maximal memory occupation of N b(C * ); • The time needed in the computation of C * * is 8% times the time of computation of N b(C * ). Table 3 shows the experimental results. In particular lines 1.1, 1.2 and 1.3 show respectively, dynamical matrices A p and B p of the plant P and dynamical matrices A q of the specification Q. It is readily seen that the above parameters satisfy the inequalities in (4.7) and (4.12). Experimental results associated with the computation of the controller N b(C * ) are reported in lines 2.1/2.10. In particular, line 2.10 shows the time of computation needed in the construction of N b(C * ) and line 2.9 shows the maximal memory occupation in the construction of N b(C * ). Experimental results associated with the computation of the controller C * * are reported in lines 3.1/3.5. In particular line 3.5 shows the time of computation needed in the construction of C * * and line 3.4 shows the maximal memory occupation in the construction of C * * . Table 4 summarizes the results shown in Table 3:
• Line 4.1: Gain in terms of number of states. The minimum gain of the integrated procedure versus the non-integrated procedure is obtained in Example # 5, resulting in 100% (meaning that in this example there is no gain in the integrated procedure) and, the maximum gain is obtained in Example # 7, resulting in 53%. • Line 4.2: Gain in terms of number of transitions. The minimum gain of the integrated procedure versus the non-integrated procedure is obtained in Example # 3, resulting in 5% and, the maximum gain is obtained in Example # 7, resulting in 2%. • Line 4.3: Gain in terms of maximal memory occupation. The minimum gain of the integrated procedure versus the non-integrated procedure is obtained in Examples # 2 and 4, resulting in 0.017% and, the maximum gain is obtained in Example # 8, resulting in 0.007%. • Line 4.4: Gain in terms of time of computation. The minimum gain of the integrated procedure versus the non-integrated procedure is obtained in Example # 3, resulting in 28% and, the maximum gain is obtained in Example # 7, resulting in 9%.
Discussion
In this paper we addressed the problem of symbolic control design of nonlinear systems with infinite states specifications, modelled by differential equations. After having provided an explicit solution to the symbolic control design problem, we presented Algorithm 2 which integrates the design of the symbolic controller with the construction of the symbolic systems of the plant and of the specification. Although the focus of the present paper is on infinite states specifications, it can be shown that the results here presented can be easily adapted to consider finite states specifications which include language specifications, formalized through automata theory [CL99]. This is important because, as shown in the work of [TP06,Tab08,BH06], automata theory provides a novel class of specifications which were traditionally not addressed before, in the control design of continuous (nonlinear) systems. Future work will focus on more efficient techniques at the software layer which can further reduce space and time complexity in the implementation of Algorithm 2. Useful insights in Example # 1 Example # 2 Example # 3 Example # 4 4.1 States(C * * )/States(N b(C * )) 0.59 0.54 0.58 0.55 4.2 Transitions(C * * )/Transitions(N b(C * )) 0.04 0.04 0.05 0.04 4.3 Max(C * * )/Max(N b(C * )) 1.5 · 10 −4 1.7 · 10 −4 1.5 · 10 −4 1.7 · 10 −4 4.4 Time(C * * )/Time(N b(C * )) 0.17 0.25 0.28 0.19 Example # 5 Example # 6 Example # 7 Example # 8 4.1 States(C * * )/States(N b(C * )) 1.00 0.84 0.53 0.81 4.2 Transitions(C * * )/Transitions(N b(C * )) 0.04 0.03 0.02 0.03 4.3 Max(C * * )/Max(N b(C * )) 0.9 · 10 −4 0.8 · 10 −4 0.9 · 10 −4 0.7 · 10 −4 4.4 Time(C * * )/Time(N b(C * )) 0.15 0.12 0.09 0.10 Table 4. Comparison between the computation of N b(C * ) and of C * * . this direction can be found in the tool Pessoa [Pes09] which employes binary decision diagrams [Pac] as data structures to encode symbolic systems.
Figure 1 .
1Approximate composition of the plant and specification systems.
Lemma 4. 5 .
5Consider three metric systems
34 Algorithm 2 :
342: C * * = (X source (T ), X 0 ∩ X source (T ), [U p ] 2µ , T, Y τ,η,µ , H τ,η,µ ) Integrated Symbolic Control Design.
. 3 .
3z ∈ X source (T ) do 5 if ∃u ∈ [U ] 2µsuch that (z, u, y) ∈ T then 6 T := T \{(z, Ac(C * * ) = C * * .
Figure 2 .
2Trajectories of the controlled plant and the specification systems with initial condition (−1,
For simplicity we consider in the eight examples the same state space of the plant and the specification, chosen as: X p = X q = [set of initial states of P and Q, chosen as: parameters in the construction of the symbolic systems S p and S q are the same in all the examples and chosen as: ε = 0.1, τ = 0.5, µ = 0.001, η = 0.01, θ p = 0.05, θ q = 0.05.
for almost all t ∈ ]a, b[. Although we have defined trajectories over open domains, we shall refer to trajectories ξ :[0, τ ] → R n defined on closed domains [0, τ ], τ ∈ R + with the understanding of the existence of a trajectory ξ :]a, b[→ R n such that ξ = ξ | [0,τ ]
7.2. Linear Control Systems. In this section we consider eight examples randomly chosen in the class of linear systems, characterized by different properties regarding controllability and eigenvalues of dynamical matrices. We consider controllable, versus noncontrollable plant systems (Examples no. 1, 2, 3, 4 vs. 5, 6, 7, 8), plant dynamical matrices A p with real, versus complex eigenvalues (Examples no. 3, 4, 5, 6, 7, 8 vs. 1, 2), specification dynamical matrices A q with real, versus complex eigenvalues (Examples no. 2, 3, 7 vs. 1, 4, 5, 6, 8).
Department of Electrical and Information Engineering, Center of Excellence DEWS, University of L'Aquila, Poggio di Roio, 67040 L'Aquila, Italy E-mail address: {giordano.pola,alessandro.borri,mariadomenica.dibenedetto}@univaq.it1
The sets [X] 2θ and [U ] 2µ , are lattices embedded in the sets R n and U , with precisions θ and µ respectively, as formally defined in the Appendix.2 The set B [θ[ (x) is defined in the Appendix.
This qualitative claim will be substantiated in terms of complexity analysis in the next section.
Acknowledgement. The first author would like to thank Paulo Tabuada for having inspired the idea of integration of control algorithms with the construction of the symbolic systems of the plant and the specification.Example # 1Example # 2 Example # 3 Example # 4 1. DataAAppendix: NotationThe identity map on a set A is denoted by 1 A . Given two sets A and B, if A is a subset of B we denote by 1 A : A → B or simply by ı the natural inclusion map taking any a ∈ A to ı (a) = a ∈ B. Given a function f :is a preorder if it is reflexive, transitive but not symmetric. The symbols N, Z, R, R + and R + 0 denote the set of natural, integer, real, positive real, and nonnegative real numbers, respectively. Given a vector x ∈ R n , we denote by x i the i-th element of x and by x the infinity norm of x, we recall that x = max{|x 1 |, |x 2 |, ..., |x n |}, where |x i | denotes the absolute value of x i . Given a measurable function f : R + 0 → R n , the (essential) supremum of f is denoted by f ∞ ; we recall that f ∞ = (ess) sup { f (t) , t ≥ 0}; f is essentially bounded if f ∞ < ∞. Given x ∈ R n and ε ∈ R + , the symbol B ε (x) denotes the set {x ∈ R n : x ≤ ε} and the symbol B [ε[ (x) denotes the setIt is readily seen that if x ∈ B ε (y) and y ∈ B [θ[ (z) then x ∈ B [ε+θ[ (z). For any A ⊆ R n and µ ∈ R + , define [A] µ = {a ∈ A | a i = k i µ, k i ∈ Z, i = 1, 2, ..., n}. The set [A] µ will be used as an approximation of the set A with precision µ/2. For a given time τ ∈ R + , define f τ so that f τ (t) = f (t), for any t ∈ [0, τ [, and f (t) = 0 elsewhere; f is said to be locally essentially bounded if for any τ ∈ R + , f τ is essentially bounded. A continuous function γ : R + 0 → R + 0 , is said to belong to class K if it is strictly increasing and γ(0) = 0; γ is said to belong to class K ∞ if γ ∈ K and γ(r) → ∞ as r → ∞. A continuous function β : R + 0 × R + 0 → R + 0 is said to belong to class KL if, for each fixed s, the map β(r, s) belongs to class K ∞ with respect to r and, for each fixed r, the map β(r, s) is decreasing with respect to s and β(r, s) → 0 as s → ∞.
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"INVARIANCE OF QUANTUM RINGS UNDER ORDINARY FLOPS: III"
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"Yuan-Pin Lee \nAND CHIN-LUNG WANG\n\n",
"Hui-Wen Lin \nAND CHIN-LUNG WANG\n\n",
"Feng Qu \nAND CHIN-LUNG WANG\n\n"
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| The paper is a sequel to[16,17], as part of our project to study a case of Crepant Transformation Conjecture: K-equivalence Conjecture for ordinary flops. In this paper we prove the invariance of quantum rings for general ordinary flops, whose local models are certain nonsplit toric bundles over arbitrary smooth base. An essential ingredient in the proof is a quantum splitting principle which reduces a statement in Gromov-Witten theory on non-split bundles to the case of split bundles. | 10.4310/cjm.2016.v4.n3.a2 | [
"https://arxiv.org/pdf/1401.7097v2.pdf"
]
| 119,314,774 | 1401.7097 | 57ab1e737cd02bd111edcc51a593185aed063c43 |
INVARIANCE OF QUANTUM RINGS UNDER ORDINARY FLOPS: III
31 Mar 2014
Yuan-Pin Lee
AND CHIN-LUNG WANG
Hui-Wen Lin
AND CHIN-LUNG WANG
Feng Qu
AND CHIN-LUNG WANG
INVARIANCE OF QUANTUM RINGS UNDER ORDINARY FLOPS: III
31 Mar 2014arXiv:1401.7097v2 [math.AG]
The paper is a sequel to[16,17], as part of our project to study a case of Crepant Transformation Conjecture: K-equivalence Conjecture for ordinary flops. In this paper we prove the invariance of quantum rings for general ordinary flops, whose local models are certain nonsplit toric bundles over arbitrary smooth base. An essential ingredient in the proof is a quantum splitting principle which reduces a statement in Gromov-Witten theory on non-split bundles to the case of split bundles.
The paper is Part III of our ongoing efforts to establish the K-equivalence conjecture [32,33] for general ordinary flops, a special but pivotal case of the Crepant Transformation Conjecture. (See Section 1 for definition of ordinary flops.) In Part I and Part II [16,17], we solve the genus zero case for which the local models are split projective bundles over smooth bases. In this paper, we extend the result to general ordinary flops whose local models are non-split projective bundles over arbitrary smooth bases. As far as we know, this is the first general result for non-split bundles (which can not be transformed via deformations etc. into split bundles) in Gromov-Witten theory. The previous results are mostly based on localization of fiberwise C * action for which bundles must be direct sums of line bundles. We believe that the techniques developed here can be applied to other problems in Gromov-Witten theory. For example, they will form the backbone of the quantum splitting principle [22]. Meanwhile, the full procedure on analytic continuations of quantum cohomolgy rings initiated in [15] and completed here, together with the inductive structure on stratified flops established in [9], will form the foundation to attack the K-equivalence conjecture. 0.1. Main results. Recall that given an ordinary flop
f : X X ′
there is a canonically induced isomorphism of Chow motives by the graph closureΓ f . See [15,16] for definitions and results. In particular, it induces an isomorphism of Chow groups and Cohomology groups
F = [Γ f ] * : H(X) → H(X ′ ),
with the Poincaré pairing preserved. To extend the correspondence to the context of Gromov-Witten theory, the quantum variables still need to be identified in order for the comparison to work. To that end, we set
F (q β ) = q F (β) .
Our main result is the following theorem.
Theorem 0.1.1. For a general ordinary flop X X ′ , F induces an isomorphism of big quantum rings of X and X ′ after an analytic continuation over the Novikov variables corresponding to the extremal rays. Furthermore, the same results hold for relative primary invariants and (relative) ancestors.
Some explanation is in order. Firstly, the cohomological ring structures are not preserved under F [15]. Since the quantum ring is a deformation of the classical ring, it might seem impossible to have isomorphic quantum rings without isomorphic classical ones. Secondly, for an effective curve class β in X, F (β) is in general not an effective curve in X ′ , and in that case q F (β) is not in the Novikov ring of X ′ . These two problems are solved simultaneously with analytic continuation. More precisely, the comparison of Gromov-Witten theory is only valid between generating functions summing over contributions from the extremal rays. See Section 1.2 for definition of the term generating functions. Theorem 0.1.1 implies that these generating functions on X are analytic in q ℓ , the Novikov variable corresponding to the extremal ray ℓ. Similarly, the corresponding generating functions on X ′ are analytic in ℓ ′ . In this sense F identifies the corresponding generating functions on X and on X ′ as analytic functions in q ℓ (while remaining formal in other Novikov variables).
We remark that analytic continuations can also be formulated on the complexified Kähler moduli as done in [15,13,16]. We note also that Theorem 0.1.1 does not hold for descendants. Examples are given in [15]. 0.2. Outline of the strategy. The strategy of the proof involves steps of reduction. In Part I [16], a degeneration argument together with various reconstruction results reduce the proof to the corresponding statements on the local models. We note that a local model of an ordinary flop is constructed from a triple (S, F, F ′ ) consisting of two vector bundles F and F ′ of equal rank over a smooth projective variety S. Indeed, the f exceptional loci Z ⊂ X and Z ′ ⊂ X ′ are projective bundles ψ : Z = P S (F) → S,ψ ′ : Z ′ = P S (F ′ ) → S, and the local models are toric (double projective) bundles over S:
X = P Z (O Z (−1) ⊗ψ * F ′ ⊕ O), X ′ = P Z ′ (O Z ′ (−1) ⊗ψ ′ * F ⊕ O).
The flop f : X X ′ is the blowup of X along Z followed by contracting the exceptional divisor along the other ruling. The local model of ordinary flops can be viewed as a functor over the triple (S, F, F ′ ). (See Section 1.1.)
In the next step, we modify the triple by blowing up the base S with the aim of simplifying the structure of the bundles F and F ′ . Starting with (S 0 , F 0 , F ′ 0 ) = (S, F, F ′ ), we construct a sequence of triples (S i , F i , F ′ i ) i≥0 , such that S i+1 is obtained by blowing up S i along some smooth subvariety Z i , and F i+1 and F ′ i+1 are the pullback of F i and F ′ i from S i to S i+1 respectively. We will show (i) F -invariance for (S i , F i , F ′ i ) can be reduced to the F -invariance for the triple in the next stage (S i+1 , F i+1 , F ′ i+1 ). (ii) After a finite number of blowups, we obtain a triple (S n , F n , F ′ n ) such that F n and F ′ n can be deformed to a direct sum of line bundles. That is, we end up with an ordinary flop of splitting type, for which the F -invariance is proved in Part II via a quantum Leary-Hirsch theorem for split toric bundles [17]. Theorem 0.1.1 then follows from (i).
Recall that the Quantum Leray-Hirsch says that for X → S a split (iterated) projective bundle, the Dubrovin connection on X can be constructed from a (carefully chosen) lifting of the Dubrovin connection on S and the Picard-Fuchs system associated to the fiber. When f : X X ′ is an ordinary flop, the naturality of the construction in [17] allows us to perform analytic continuations along the fiberwise Novikov variables by way of the Picard-Fuchs systems on X and X ′ . They turn out to coincide after analytic continuations.
The splitting principle in (ii) via blow-ups is in fact a simple consequence of Hironaka's theorem on resolution of indeterminacies (cf. Section 1.8). Thus the major efforts made in this paper is indeed to prove (i). To relate the F -invariance for (S i , F i , F ′ i ) to that for (S i+1 , F i+1 , F ′ i+1 ), we consider the deformation to the normal cone for T i ֒→ S i . This is the family
Bl T i ×{0} (S i × A 1 ) → A 1
with smooth fiber S i over A 1 − {0} and singular fiber
S i+1 ∪ E i P i ,
where E i is the exceptional divisor in S i+1 = Bl T i S i , and P i is the exceptional divisor in Bl T i ×{0} (S i × A 1 ). By construction, P i and E i are themselves projective bundles over T i :
P i = P T i (N T i /S i ⊕ O), E i = P T i (N T i /S i ).
The bundlesF i+1 andF ′ i+1 on P i and E i are pulled backs from the induced bundles on T i , which are compatible with the restrictions of F i+1 and F ′ i+1 on S i+1 . In order to relate the F -invariance for (S i , F i , F ′ i ) with that for (S i+1 , F i+1 , F ′ i+1 ) and (P i ,F i+1 ,F ′ i+1 ) (as well as for (E i ,F i+1 ,F ′ i+1 )), the degeneration formula [25,24] is used, and the corresponding statements must be generalized to relative Gromov-Witten invariants of smooth pairs.
To avoid cumbersome notations, we will omit the bundles in the notation when there is no danger of confusion. By the degeneration formula, Finvariance for S i (absolute invariants) is implied by those for (S i+1 , E i ) and (P i , E i ) (relative invariants). We then show that (iii) F -invariance for relative invariants on (S i+1 , E i ) follows from the Finvariance for absolute invariants on S i+1 and E i . Similarly F -invariance for (P i , E i ) follows from that for P i and E i . This is the most intricate part of the reduction. Before we discuss it, we explain why (iii) allows us to run an inductive proof on the dimension of the base S. Clearly dim E i = dim S i − 1 and F -invariance for E i follows by induction. However, dim P i = dim S i does not drop. A key observation is that P i = P T i (N T i /S i ⊕ O) is constructed from T i and dim T i < dim S i . If N T i /S i splits, then a variant of the quantum Leray-Hirsch still applies. Namely analytic continuations along the base T i (F -invariance) implies the F -invariance for P i . In fact, the fiberwise Novikov variables in P i → T i need no analytic continuations at all. However, the bundle N T i /S i may not be deformable to split bundles. This suggests that we should have taken into account the splitting procedure of those relevant normal bundles during the induction process. It is thus plausible to expect that some kind of refined induction procedure may lead to F -invariance for P i . Such a delicate induction is indeed possible, as in Section 8. Hence F -invariance for S i+1 implies that for S i under the validity of (iii). Note that in the precess we have to go back and forth between absolute and relative invariants without involving descendants in order to keep the F -invariance in all inductive steps.
It remains to prove (iii). Similar ideas were already used by Maulik and Pandharipande in [28], where it was shown that the relative descendants of a smooth pair (S, D) (D a smooth divisor in S) are determined by the absolute descendants of S and D. This is accomplished by considering the trivial deformations to the normal cone
Bl D×{0} (S × A 1 ) → A 1 with smooth fiber S over A 1 \ {0} and singular fiber S ∪ D P, where P = P D (N D/S ⊕ O)
is a projectivized split bundle of rank two. In our case the degeneration formula (which we recall in Section 1.6) decompose absolute invariants into convolution of relative invariants:
α X(S) β = ∑ η=(Γ 1 ,Γ 2 ),I C η α 1 | µ, e I •(X(S),X(D)) Γ 1 µ, e I | α 2 •(X(P),X(D)) Γ 2 .
It gives rise to a system of linear equations where the relative invariants for the P 1 bundle X(P) → X(D) are treated as coefficients of the system. To handle these coefficients, fiberwise localization on the P 1 bundle is used and that inevitably introduces the descendants and, as noted above, breaks F -invariance in the induction.
While we employ some ideas from [28], the arguments there have to be considerably modified to be useful for our purpose. In particular, we substitute ancestors for descendants in key steps of degeneration analysis. Furthermore, the localization was replaced by more complex degeneration argument and the strong virtual pushforward property developed by Cristina Manolache in [27]. These form the basis of inversion of degeneration arguments, and of the proof of (iii).
Remark 0.2.1. Indeed, the vital role played by ancestors in the study of Crepant Transformation Conjecture and the K-equivalence Conjectures had been first advocated and studied in [13,14] where the invariance of higher genus Gromov-Witten theory was established for simple ordinary flops.
Remark 0.2.2. Even though the theory of algebraic cobordism [19,23] is not explicitly utilized, it inspires many ideas in this paper. See Section 8.3 for some comments. The complexity of the arguments could have been reduced and results greatly generalized had the ideas inspired by algebraic cobordism been applicable. 0.3. Outline of the contents. The sections of this paper are arranged in its logical order towards the proof of Theorem 0.1.1. The order of presentation is different from that described in the strategy outlined above. Thus we would like to briefly describe the contents in order for the readers to quickly locate necessary details for each step.
In Section 1, we give some basic definitions and recall some facts on ordinary flops and Gromov-Witten theory.In particular we define, in Section 1.3, the notion of F -invariance for a projective smooth variety S, or rather a triple (S, F, F ′ ) with F and F ′ two vector bundles on S of equal rank. We also recall, in Proposition 1.1.1, that the proof of Theorem 0.1.1 can be reduced to that of "local models", a notion defined in Section 1.1.
The degeneration formula and the product formula for relative invariants are briefly described in Sections 1.6 and 1.7. As explained above, we our goal is to show a quantum splitting principle. In Section 1.8, we explain how splitting a bundle is done by birational modification and deformation. The entire paper can be considered as a verification that this simple proceedure can be carried out while keeping F -invariance in all steps.
Sections 2 to 4 are technical results needed for our proof, and can be omitted in the first reading by assuming the results. In Section 2, we recall and prove some general results about the fiber integrals. Section 3 is devoted to some structural results relating invariants of a P 1 bundle with those of its base, without invoking fiberwise localization. In particular, we prove a strong virtual pushforward property between various relative moduli spaces with P 1 bundles considered as rigid or non-rigid targets. The deformation invariance of F -invariance is established in Section 4.
In Sections 5 and 6, F -invariance for ancestor invariants of P 1 bundle is proved, assuming F -invariance for the base. These results can be viewed as the ancestor version of the corresponding quantum Leray-Hirsch theorem in genus zero. A splitting principle for vector bundles under birational modification (statement (ii) in Section 0.2) is described in Section 1.8. It is already used there to determine the fiber class type II invariants in terms of the simple flop case and the classical cohomology rings on the base. The non-fiber class type II invariants as well as the rubber invariants are then handled inductively by the strong virtual pushforward property and related results established in Section 3. A technically important result is the inversion of degeneration argument for ancestors of P 1 bundles in Section 6 (Proposition 6.5.4).
Section 7 (Proposition 7.4.1 and Theorem 7.4.2) proves the statement (iii) mentioned in Section 0.2. Namely F -invariance for S and D implies Finvariance for the smooth pair (S, D), assuming the F -invariance of the related P 1 bundles proved in Section 6.
The last section (Section 8) concludes the proof by reducing to the case of split bundles, whose proof was done in Part II [17]. The proof consists of a refined induction procedure on ordinary flops constructed out of the triple (S, F, F ′ ) such that S is of the form
S = P T (N 1 ) × T · · · × T P T (N k )
with {N i } a finite collection of vector bundles over T and F and F ′ are pullbacks from T. We proceed by induction on the dimensional of the base T.
The topological recursion relation (TRR) for the ancestors in genus zero relative Gromov-Witten theory, whose precise formula is not used in the main text, is discussed in the appendix.
Remark 0.3.1. By the comparison results proved in [1] and [5], the five different models of genus zero relative invariants are equivalent. Although we have employed primarily the logarithmic approach to relative Gromov-Witten invariants [2,10], results concerning the genus zero relative Gromov-Witten invariants can also be deduced using, for instance, the orbifold approach of Cadman.
In this paper, we freely switch between the usual notations of the relative Gromov-Witten and the log Gromov-Witten perspectives. It is certainly possible to formulate everything in one perspective only. However, the available literature is scarce in this area and it is often easier to make connection with the existing literature by discussing one thing in the relative notation (e.g., the degeneration formula) and another in the log notation (e.g., the perfect obstruction theory).
NOTATIONS AND BASIC FACTS
In this section, we review notations and facts on ordinary flops from Part I and II [16,17], including the notion of F -invariance on generating functions of GW invariants and the degeneration formula. We also introduce the partial ordering on weighted partitions for relative invariants.
1.1. Ordinary flops.
1.1.1. Local models of ordinary flops. Given a triple (S, F, F ′ ), where S is a smooth projective variety over C and F, F ′ two vector bundles of rank r + 1, we build the local model of ordinary flop f : X X ′ as follows:
• Z = Z(S, F, F ′ ) = P S (F). There is a natural projection ψ S : Z → S. • X = X(S, F, F ′ ) = P Z (ψ * S F ′ ⊗ O Z (−1) ⊕ O) with projection p S : X → Z.
• Denote by i S : Z → X the inclusion which identifies Z as the zero section ofψ *
S F ′ ⊗ O Z (−1). • Using the projective bundle structure p S , we define ξ = c 1 (O X (1)). Similarly, h = c 1 (O Z (1)) forψ S .
h is understood either as a class in Z or its pullback to X. • (Leray-Hirsch) H * (X) is generated by h, ξ and (pull-back of) H * (S). ξ can be represented by the infinity divisor disjoint from Z.
• Z ′ = Z ′ (S, F, F ′ ),ψ ′ S , X ′ = X ′ (S, F, F ′ ), p ′ S , i ′ S
, h ′ and ξ ′ are defined as above by switching the roles of F and F ′ . For example,
X ′ (S, F, F ′ ) := X(S, F ′ , F). • Let Y = Y(S, F, F ′ ) be the common blow-up Y = Bl Z (X) = Bl Z ′ (X ′ ) with blowdown maps φ S : Y → X and φ ′ S : Y → X ′ .
• The local model for the ordinary flop associated to (S, F, F ′ ) is the birational map
f S : φ ′ S • φ −1 S : X Y → X ′ .
• The correspondence
F S = [Γ f S ] * = φ ′ S * • φ * S : H * (X) → H * (X ′ ) is an isomorphism preserving the Poincaré pairing.
If the triple (S, F, F ′ ) is clear from the context, we will use the shorthand notation Z(S), X(S). The letter Z, Y will be used locally to stand for other objects, but their meaning will be made clear in the context. When S is a point Spec C, we will use X(C) to denote
X(Spec C, O ⊕(r+1) , O ⊕(r+1)
).
General ordinary flops.
An ordinary flop f : X X ′ is a birational map which is locally isomorphic to the local models in the neighborhood of the exceptional loci. We summarize these in the following commutative diagram
Ẽ~⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ❇ ❇ ❇ ❇ ! ! ❇ ❇ ❇ j / / Y } } ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ! ! ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ Zψ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ i / / X ψ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ Z ′ ⑤ ⑤ ⑤ 5 ψ ′~⑤ ⑤ ⑤ i ′ / / X ′ ψ ′ } } ④ ④ ④ ④ ④ ④ ④ ④ ④ S j ′ / / X
Note that ψ and ψ ′ are flopping contractions which contract Z and Z ′ to S and S ′ inX, which is singular. (We will not useX except its existence, which guarantees the projectivity of X ′ from that of X andX.) There are two main differences between a general ordinary flop and its local model. Obviously, the latter allows simpler topological and geometric description. More importantly for us, from the functorial perspective, the local models have an additional smooth morphism from X to Z, and both X → Z, X → S are fiber bundles with fibers toric manifolds. However, these toric bundles do not come from toric construction. Thus, the localization technique does not apply. We will henceforth assume X and X ′ are the local models.
Generating functions of Gromov-Witten invariants.
1.2.1. Curve classes. We first discuss the notation of curve classes in our generating functions.
Given X(S) → Z(S) → S, denote by γ the curve class of the fiber of X(S) → Z(S), and ℓ the curve class of the fiber of Z(S) → S. When S is a point, N 1 (X(S)) ≃ Zγ ⊕ Zℓ. If a curve class is effective, its γ coefficient is non negative. Similarly we have γ ′ and ℓ ′ for
X ′ (S) → Z ′ (S) → S.
Given a curve class β S ∈ H 2 (S, Z) and integer d, we say β ∈ H 2 (X, Z) belongs to (β S , d) if β ξ = d, and β projects to β S under X(S) → S. If β belongs to (β S , d), then any other element belonging to (β S , d) is of the form β + kℓ for some integer k. Straightforward modification of the above definition applies to X ′ . (Notice that d was called d 2 in Part I, II [16,17]. ) We will refer to (β S , d) as curve classes (modulo the extremal ray). (β S , d) = (0, 0) are the extremal classes. Other classes are called non-extremal.
It is proved in Part I [16,Lemma 3.5], that β belongs to (β S , d) if and only if F (β) belongs to (β S , d).
Gromov-Witten invariants.
We consider only genus zero GW invariants.
Let M 0,n (X, β) be the moduli stack of stable maps from n-pointed, genus zero pre stable curves to X with curve class β ∈ H 2 (X).
We have evaluation maps (1.2.1) ev j : M 0,n (X, β) → X for each 1 ≤ j ≤ n. When n ≥ 3, there is a stabilization map st : M 0,n (X, β) → M 0,n .
The notation
n ∏ j=1τ k j (α j ) X β = [M 0,n (X,β)] vir ∏ j ev * j (α j ) st * ( ∏ j ψ k j j )
stands for the ancestor invariant, where k j 's are nonnegative integers, α j ∈ H * (X), and ψ j ∈ H 2 (M 0,n ) the cotangent class for the marked point labeled by j.
We also define the cycle
(1.2.2) n ∏ j=1τ k j (α j ) X β := [M 0,n (X, β)] vir ∩ ∏ j ev * j (α j ) ∩ st * ( ∏ j ψ k j j ). Note that st * n ∏ j=1τ k j (α j ) X β
is a nonzero cycle on M 0,n only if
n − 3 ≥ ∑ k j .
Similar notations are used for relative invariants. Let B be a smooth divisor in X, µ a partition of β B, and M 0,n+m (X, B; β, µ) the moduli stack of relative stable maps to (X, B) with indicated discrete contact data. Here m = l(µ) is the number of contact points in B.
When n + m ≥ 3, by abusing notation, we also use st to denote the stabilization map
(1.2.3) st : M 0,n+m (X, B; β, µ) → M 0,n+m .
In addition to the evaluation maps (1.2.1) for the internal marked points, we have the evaluation maps for the relative marked point
(1.2.4) ev B i =: ev i : M 0,n+m (X, B; β, µ) → B, i = n + 1, . . . , n + m.
We will abused the notation and dropped the superscript B when there is no confusion.
For α j ∈ H * (X), δ i ∈ H * (B), non-negative integers {k j } and {l i }, we use n ∏ j=1τ k j (α j ) | m ∏ i=1τ l i (δ i ) (X,B) β to represent the relative invariant [M 0,n+m (X,B;β,µ)] vir ∏ j ev * j (α j ) ∏ i ev * i (δ i ) st * ( ∏ j ψ k j j ∏ i ψ l i i ), and (1.2.5) n ∏ j=1τ k j (α j ) | m ∏ i=1τ l i (δ i ) (X,B) β,µ the cycle [M 0,n+m (X, B; β, µ)] vir ∩ ∏ j ev * j (α j ) ∏ i ev * i (δ i ) ∩ st * ( ∏ j ψ k j j ∏ i ψ l i i ).
The following shorthand notation for relative invariants will also be used:
µ, γ | α (X,B) β , α | µ (X,B) β , µ | α (X,B) β , µ | α | ν (X,B) β .
The first three are all type I invariants and the last one is referred as a type II invariant, which is used when B = B 0 ∐ B ∞ is a disjoint union of two divisors. Here α denote insertions for non-relative marked points, γ insertions for relative marked points, and µ a weighted partition defined in Section 1.5, which combines the relative profile µ and relative insertions. The symbols µ, ν or λ denote partitions, and µ = (µ, γ) etc. weighted partitions.
Generating functions.
A superscript is used to indicate the target and a subscript for the curve class modulo the extremal ray. Denote by α
X(S) (β S ,d) the generating function α X(S) (β S ,d) = ∑ β∈(β S , d) α X(S) β q β ,
where α
X(S) β
is the genus zero GW invariant of X(S) with class β and insertions α. Similar generating functions are used for relative invariants. Given a smooth divisor D ֒→ S, not necessarily connected,
α | ν (X(S), X(D)) (β S , d) = ∑ β∈(β S , d) α | ν (X(S), X(D)) β q β .
By abuse of the language, we sometimes refer to a generating function as invariants when confusion is unlikely to arise.
1.3. Definition of F -invariance. Given a triple (S, F, F ′ ), we say that Finvariance holds for (S, F, F ′ ) with curve classes (β S , d) (modulo extremal ray) if for any ancestor insertions ω,
(1.3.1) F ω X(S) (β S , d) = F (ω) X ′ (S) (β S , d)
after analytic continuation. If ω is a primary insertion, we require the number of insertions ||ω|| ≥ 3. When the context is clear, we sometimes omit the bundles and say "F -invariance holds for S". We explicate the meaning of analytic continuation. As explained earlier, both sides of (1.3.1) are generating functions which, a priori, are formal power series in q ℓ and q ℓ ′ respectively. When we say the equality holds in (1.3.1) after analytic continuation, we mean that the LHS is an analytic function in q ℓ , the RHS is an analytic function in q ℓ ′ , and they are equal as analytic functions after the identification
F q ℓ = q F ℓ = (q ℓ ′ ) −1 .
The last equality follows from F (ℓ) = −ℓ ′ , which was shown in [15].
We also note that the condition ||ω|| ≥ 3 is necessary for primary insertions. (Ancestors by definition satisfy the condition.) Counterexamples of F -invariance were given in [15] for ||ω|| < 3.
Given a smooth divisor D ֒→ S, we say F -invariance holds for (S, D) (bundles omitted) if whenever ||ω|| + l( ν) ≥ 3,
F ω | ν (X(S), X(D)) (β S , d) = F (ω) | F ( ν) (X ′ (S), X ′ (D)) (β S , d)
.
Note that by the product rule, F -invariance for disconnected domain curve also holds when there is no contracted component and each component contains at least 3 insertions.
F -invariance for non-rigid targets, also termed rubber targets [28], can be formulated similarly. The definition of relative invariants for rubbers can be found in [11,Section 2.4]. Remark 1.3.1. In fact, for non-extremal (β S , d), the condition ||ω|| ≥ 3 is not necessary. For primary invariants, the divisor equation allows us to create extra insertions, upholding F -invariance for absolute and relative invariants for few insertions. More precisely, for β S = 0, pick an ample divisor H on S.
Then
ω X(S) (β S , d) = 1 β S H φ * H, ω X(S) (β S , d) .
Since H is a pullback class,
β φ * H = β S H.
The key point is that the intersection number of H and any β ∈ (β S , d) is constant and the generating function remains unchanged. By ampleness of H on S, β S H = 0. This way, the F -invariance of generating functions with fewer insertions is implied by that with arbitrary number of insertions. If d = 0, replace H by ξ. Since F (ξ) = ξ ′ , the same argument works.
1.4. Some geometric properties of the local models.
1.4.1. Functoriality. We view Z, X, Z ′ , X ′ as functors over the category of smooth projective schemes (over S). Given a map g : T → S, we have the pullback triple (T, g * F, g * F ′ ). Any relevant diagram constructed from (T, g * F, g * F ′ ) can be obtained by base change from T → S. For instance, we have
X(T) p T / / g X Z(T)ψ T / / g Z T g X(S) p S / / Z(S)ψ S / / S.
Later, for simplicity, we will denote by | X(S) the pull back under X(S) → S. Due to the motivic property of F , it is compatible with proper pushforward and pullback. Denote F S the correspondence F modeled on S. Given a map g : T → S, we have
F S • g X * = g X ′ * • F T , F T • g * X = g * X ′ • F S . 1.4.2
. F in terms of a basis and its dual basis. Let
{T i } be a basis of H * (S), {T ∨ i } its dual basis, then {T i h j ξ k | 0 ≤ j ≤ r, 0 ≤ k ≤ r + 1} is a basis of H * (X(S)), with dual basis {T ∨ i H r−j Θ r+1−k }. See [16, Lemma 1.4, Lemma 2.1].
We do not need the precise formulas for H i or Θ j in this paper. We only remark that H i 's (resp. Θ j 's) are the Chern classes of certain quotient bundles of rank r over S (resp. rank r + 1 over Z). In particular they are defined only for 0 ≤ i ≤ r (resp. 0 ≤ j ≤ r + 1). For notational convenience, we define H i = 0 = Θ j for i, j outside the above range. Similar remarks apply to H ′ i and Θ ′ j on the X ′ side as well.
Proposition 1.4.1 ([16]). F S is H * (S)-linear and for j ≤ r or k = 0, F (h j ξ k ) = (ξ ′ − h ′ ) j ξ ′ k .
Since F preserves the Poincaré pairing, this implies
Corollary 1.4.2. There are constants C(i, j, k) such that F (H j Θ k ) = j+k ∑ m=0 C(j, k, m)H ′ m Θ ′ j+k−m .
Proof. Let pt be the point class in H * (S). Then
F (pt.h j ξ k ) = pt.(ξ ′ − h ′ ) j ξ ′ k = j ∑ m=0 C j m (−1) m pt.h ′ j−m ξ ′ m+k .
The Chern polynomial equations allow us to rewrite the sum as
r ∑ m=0 C(j, k, m) pt.h ′ m ξ ′ j+k−m , where C(j, k, m) = 0 if j + k − m < 0 or j + k − m > r + 1.
The result follows by Poincaré duality.
F -homogeneous basis. Let
G = GL r+1 × GL r+1 ,
then the pair of vector bundles (F, F ′ ) determines a map
S → BG = BGL r+1 × BGL r+1 ,
where BG is the classifying stack of G bundles. We have a cartesian diagram
X(S) / / [X(C)/G] S / / BG. Let {T i } be a basis of H * (S). Let {V m } be classes on [X(C)/G] such that the pullbacks of V m 's to X(C) via X(C) → [X(C)/G] form a basis of H * (X(C)). Then {T i V m }
Definition 1.4.3. A F -homogeneous basis for X(S, F, F ′ ) is a basis of its cohomology of the form {T
i V m }, where T i are homogeneous classes on S, V m are homogeneous classes on [X(C)/G]. We further require (1) {T i } is a basis of H * (S). (2) V m is a linear combination of {h j ξ k } {0≤j≤r, 0≤k≤r+1}
, and the pullbacks of V m 's to X(C) form a basis of H * (X(C)). In this case, we
say {T i V m } is compatible with {T i }. As X ′ (S, F, F ′ ) = X(S, F ′ , F). A F -homogeneous basis for X ′ (S, F, F ′ ) is by definition a F -homogeneous basis for X(S, F ′ , F). Lemma 1.4.4. If {T i V m } is a F -homogeneous basis for X(S), then {T i F (V m )} is a F -homogeneous basis for X ′ (S).
Proof. This follows from Proposition 1.4.1.
The significance of requiring V m to be linear combinations of {h j ξ k } is stated in the following lemma.
Lemma 1.4.5. If {T i V m } is a F -homogeneous basis for X(S), then its dual basis is of the form {T ∨ i W m }. Here {T ∨ i } is the dual basis of {T i }, W m is a linear combination of {H r−j Θ r+1−k }, and when pulled back to X(C), {W m } is the dual basis of {V m }.
Proof. This follows from Corollary 1.4.2.
By abuse of notation, we will write the dual basis of {T i V m } as
{T ∨ i V ∨ m }, although V ∨ m is not the dual of V m in [X(C)/G].
The abstraction to F -homogeneous basis serves to streamline arguments involving the choice of a basis. In practice, we only need the particular Fhomogeneous basis {T i h j ξ k } for X(S) and {T i F (h j ξ k )} for X ′ (S).
Weighted partitions and partial orderings.
1.5.1. Weighted partitions. As above, let (S, D) be a smooth pair. We will order relative invariants with respect to a chosen F -homogeneous basis for X(D).
Let {δ a V m } be a F -homogeneous basis of H * (X(D)) compatible with {δ a }. Consider generating functions of the form
(1.5.1) α | {(µ i , δ a i V m i )} (X(S), X(D)) (β S , d) ,
where α denotes a sequence of primary insertions with length ||α||. Following [28], we call the contact data
µ := {(µ 1 , δ a 1 V m 1 ), · · · , (µ l(µ) , δ a l(µ) V m l(µ) )} a (cohomology) weighted partition.
Here l(µ) is the length of the relative profile µ, which is called the number of contact points ρ in [15,16,17]. We emphasis that the cohomology classes in a weighted partition are always chosen from a F -homogeneous basis. Some notations will be used throughout the paper. For a weighted par-
tition µ = {(µ i , δ a i V m i )}, define (1.5.2) k( µ) := max{0, ∑ {i | µ i >1} µ i − Id( µ)}, where Id( µ) := #{i | (µ i , deg δ a i , deg V m i ) = (1, 0, 0)}. k( µ)
is the number of extra divisorial insertions to ensure the existence of M 0,n+m in (1.2.3) and hence that of the corresponding ancestors. It will be used in Sections 6 and 7. See, for example, Proposition 6.5.4.
We use deg D ( µ) to denote the total degree of the relative insertions
deg D ( µ) := ∑ deg δ a i .
1.5.2. Partial orderings. This is a variant of the partial ordering in [28].
Given µ, the set {(µ i , deg δ a i , deg V m i ) ∈ N × Z 2 ≥0 | i = 1, .
. . , l( µ)} can be rearranged in the decreasing lexicographical order. For two such sets associated to µ and µ ′ with the same l( µ) = l( µ ′ ), we say that
{(µ i , deg δ a i , deg V m i )} > l {(µ ′ i , deg δ a ′ i , deg V m ′ i )} if,
after placing them in the decreasing lexicographical order, the first triple for which they differ is greater for
{(µ i , deg δ a i , deg V m i )}.
Now we define a partial ordering ≻ on the weighted partition µ by lexicographic order on the triple
(deg D ( µ), l(µ), {(µ i , deg δ a i , deg V m i )}), where (i) For deg D ( µ)
, smaller number corresponds to higher order in ≻, (ii) For l(µ), larger number corresponds to higher order in ≻,
(iii) For {(µ i , deg δ a i , deg V m i )},
lower order in > l corresponds to higher order in ≻. Based on it, generating functions of the above form (1.5.1) can be partially ordered lexicographically by the triple
((β S , d), ||α||, µ) : (1) Curve class (β S , d): (β S , d) > (β ′ S , d ′ ) if β S − β ′ S > 0, or β S = β ′ S and d > d ′ .
(2) Number of internal insertions ||α||: more insertions corresponds to higher order.
(3) µ = {(µ i , δ a i V m i )} are ordered by ≻ defined above.
A partial ordering on a set satisfies the DCC if any descending chain has only finite length. DCC is an essential condition for the induction process. The partial ordering ≻ on the set of all weighted partitions does not satisfy DCC. However, the partial ordering on the set of generating functions does, as the following lemma shows. Lemma 1.5.1. The partial ordering on the generating functions of (primary) relative GW invariants (1.5.1) associated to a fixed triple (S, F, F ′ ) satisfies the DCC.
Proof. Given ((β S , d), ||α||, µ), the virtual dimension of the moduli space of relative stable maps is given by
(β S ,d) c 1 (X(S)) + ||α|| + l( µ) − l( µ) ∑ i=1 µ i .
The first term depends only on (β S , d) since ℓ c 1 (X(S)) = 0. In order to get non-trivial invariants it must agree with deg D µ may increase in a descending chain. However, since
||α|| ∑ i=1 deg α i + deg D µ + deg V m i . For a fixed β • S , the set {(β • S , d ′ ) | (β • S , d ′ ) < (β • S , d • )} is clearly finite. Furthermore, the set {β ′ S | β ′ S < β S } isl( µ) − l( µ) ∑ i=1 µ i ≤ 0,
the virtual dimension count gives an upper bound for it and then deg D µ stabilizes. Then the number of contact points l( µ) stabilizes as well. It is clear that the remaining choices for µ form a finite set.
1.6. Degeneration formula. We will apply the degeneration formula to a family obtained from deformation to the normal cone.
1.6.1. Deformation to the normal cone. Given a smooth pair (S, Z) with Z is a smooth closed subvariety of S, we introduce the following notations. Let
N = N Z/S be the normal bundle of Z in S,S = Bl Z S be the blow up of S along Z, E = P Z (N) the exceptional divisor inS, and P = P Z (N ⊕ O). Let p : P → Z → S, φ :S → S
be the natural morphisms. The deformation to the normal cone for the pair (S, Z) is simply
W = W(S, Z) = Bl Z×{0} (S × A 1 ).
Then W t = S when t = 0 and W t=0 is obtained by gluingS and P along E.
It is easy to see that X(W(S, Z)) → A 1 and W(X(S), X(Z)) → A 1 are isomorphic.
1.6.2. Degeneration formula. Applying the degeneration formula to the family X(W) → A 1 , we get a degeneration of absolute invariants:
(1.6.1) α X(S) (β S , d) = ∑ I ∑ η=(Γ 1 ,Γ 2 ) C η × φ X * ( α 1 | µ, e I •(X(S),X(E)) Γ 1 ) · p X * ( µ, e I | α 2 •(X(P),X(E)) Γ 2
).
Here η = (Γ 1 , Γ 2 ) is a splitting of the discrete data. C η is a constant determined by η. Γ i specifies curve classes modulo extremal rays, non-relative and relative marked points, and contact orders of relative marked points encoded in a partition µ for each component. I is an index set of l(µ) elements, e I ∈ H * (X(E)) ⊕l(µ) , e I its dual with respect to some basis of H * (X(E)). α represents primitive insertions, α i are the corresponding insertions specified by Γ i . As we are dealing with generating functions, we identify variables using φ X and p X induced from φ :S → S and p :
P → S. For instance φ X * (q β ) = q φ X * (β) .
Assume the total curve classes of Γ 1 is (βS, dS) and that of Γ 2 is (β P , d P ). The constrains on curve classes are
β S = φ * (βS) + p * (β P ), βS E = β P E, d = dS + d P .
Further analyzing these constraints on each connected component of Γ i , we see that if (β S , d) is non-extremal, then any component specified by Γ 1 or Γ 2 has non-extremal curve classes. We note that (1.6.1) has a natural extension which allows the left hand side to be a relative invariant. It comes in different variants, but all "obvious" extensions of (1.6.1). The readers who are unfamiliar with them can consult [28] for example. 1.7. A product formula for the relative invariants. We recall a product formula for relative invariants proved in [21].
Let X and Y be nonsingular projective varieties , and D a a smooth divisor in Y. We further assume
H 1 (Y) = 0, so a curve class of X × Y is of the form (β X , β Y ) where β X (resp. β Y ) is a curve class of X (resp. Y).
The product formula is best formulated in terms of the Gromov-Witten correspondence. Let Γ X = (g, n + ρ, β X ). The Gromov-Witten correspondence For the relative Gromov-Witten correspondence for the smooth pair (Y, D), there is a similarly defined Gromov-Witten correspondence
R Γ X : H * (X) ⊗(n+ρ) → H * (M g,n+ρ ), is defined by R Γ X (α) := PD st * ev * X (α) ∩ [M Γ X (X)] vir ,R Γ (Y,D) : H * (Y) ⊗n ⊗ H * (D) ⊗ρ → H * (M g,n+ρ ) defined as R Γ (Y,D) (α ′ ; δ) := PD st * ev * Y (α)ev * D (δ) ∩ [M Γ Y (Y, D)] vir ,
where, by a slight abuse of notation, Γ (Y,D) encodes all discrete data of the relative moduli.
Theorem 1.7.1 (The product formula for X × (Y, D) [21, Corollary 3.1]). Let Γ X×(Y,D) = (g, n, (β X , β Y ), ρ, µ)
be the relative data for the product X × (Y, D). We have (ii) The above product formula can be reformulated in terms of more refined GW invariants, with additional insertions from arbitrary cycles in M g,n . For example, for any class γ ∈ H * (M g,n ),
R Γ X×(Y,D) ((α 1 ⊗ α ′ 1 ) ⊗ ... ⊗ ((α n ⊗ α ′ n )); (α n+1 ⊗ δ 1 ) ⊗ ... ⊗ ((α n+ρ ⊗ δ ρ )) =R Γ X (α 1 ⊗ ... ⊗ α n+ρ )R Γ (Y,D) (α ′ 1 ⊗ ... ⊗ α ′ n ; δ 1 ⊗ ... ⊗ δ ρ ), where α i ∈ H * (X), α ′ i ∈ H * (Y) and δ j ∈ H * (D).M g,n st * (γ)R Γ X (α) = γ, st * (ev * X (α) ∩ [M Γ X (X)] vir ) ,
where · is the pairing between cohomology and homology of M g,n . Since the Poincaré duality holds for H * (M g,n ) Q , the above integral with arbitrary γ gives R Γ X (α). In genus zero, however, H * (M 0,n ) is generated by ψ-classes and the refined GW invariants above are simply ancestors. Similar discussion applies to the relative invariants.
1.8. Splitting bundles. Our main task is to reduce the proof of F -invariance of (S, F, F ′ ) from the non-split bundles to split bundles. Here we explain how the bundles can be split "classically". Lemma 1.8.1. Given a rank r + 1 vector bundle F → S, there exists a sequence of blowing-ups on smooth centers such that the pullback of F, denoted π * (F), admits a filtration of subbundles
0 = F 0 ⊂ F 1 ⊂ . . . ⊂ F r+1 = π * (F), satisfying rank(F i+1 /F i ) = 1 for all i.
Proof. Consider the complete flag bundle over S p : Fl(F) → S.
By local triviality, p admits a rational section s. Resolving the rational map S Fl(T) by a sequence of blowing-ups along smooth centers, one gets π :S → S such that π * Fl(F) admits a section.
We say that π * F admits complete flags when the conclusion of Lemma 1.8.1 holds.
Lemma 1.8.2. Let F → S be a vector bundle admitting complete flags. Then it can be deformed to F
1 → S such that F 1 is a split bundle.
Proof. The non-triviality of complete flags is governed by multiple extension classes. There is a deformation of the bundles, with the base S fixed, sending all extension classes to zero.
FIBER INTEGRALS
In this section, we assemble some results about fiber integrals. By fiber integrals we mean, after [28], the GW invariants of G-principal bundles or their associated fiber bundles p : E → B with the curve class β such that p * (β) = 0. This should not be confused with the similarly named fiber curve class (modulo the extremal rays) in Section 5.2.1.
We first recast results from [28, Section 1.2] in the form needed for our purpose. This also allows us to deduce Proposition 2.1.6. All results in this section are for fiber integrals only.
All the schemes we consider are smooth. Let G be a group scheme over C, and BG = [Spec C/G] the classifying stack of G bundles. Given a smooth map f :
B → BG and a G-equivariant smooth pair (F, D F ), where D F is a G-divisor in F, define the fiber bundle pairs (E, D E ) over B as the fiber product (E, D E ) g / / p ([F/G], [D F /G]) B f / / BG.
Now we switch to the notation of log geometry (see Remark 0.3.1 for justification). Let E † (resp. F † ) be the log scheme with the divisorial log structure determined by D E (resp. D F ). We have a cartesian diagram of log stacks
E † g / / [F † /G] π B f / / BG,
where B and BG are equipped with the trivial log structure. In particular f is strict.
Given a G-invariant curve class β in F, we have a cartesian diagram between log stacks
(2.1.1) M 0,n (E † /B, β)ḡ / / M 0,n ([F † /G]/BG, β) π B f / / BG. Here M 0,n (E † /B, β) is[M 0,n (E † /B, β)] vir =ḡ * [M 0,n ([F † /G]/BG, β)] vir .
Proof. The cartesian diagram
M 0,n (E † /B, β) / / M 0,n ([F † /G]/BG, β) B × M 0,n / / BG × M 0,n induces another M 0,n (E † /B, β) / / M 0,n ([F † /G]/BG, β) B × T or M 0,n / / BG × T or M 0,n ,
where M 0,n is the moduli stack of genus zero, n-pointed prestable curves and T or M is the classifying stack of fine and saturated log schemes over M [30,Remark 5.26].
Since
Ω E † /B ≃ g * Ω π ,M 0,n (F † , β) / / M 0,n ([F † /G]/BG, β) Spec C / / BG we see Corollary 2.1.4. M 0,n ([F † /G]/BG, β) = [M 0,n (F † , β)/G], and [M 0,n ([F † /G]/BG, β)] vir corresponds to the G-equivariant virtual class on [M 0,n (F † , β)/G].
We note that M 0,n (E † /B, β) can be identified with M 0,n (E † , β) as moduli stacks by forgetting the map to the base B, and their virtual classes are the same since we are considering curves of genus zero (cf. [28, Equation (3)]).
We spell out the consequence of Proposition 2.1.3 in numerical form.
i ∈ A * (B), α r ∈ A * ([F/G]), θ s ∈ A * ([D F /G]), we have ∏ 1≤r≤nτ k r (p * (δ r ) ∪ g * (α r )) | ∏ 1≤s≤mτ k s+n (p * (δ s+n ) ∪ g * (θ s )) (E,D E ) β = B ∏ 1≤i≤n+m δ i ∪ f * ∏ 1≤r≤nτ k r α r | ∏ 1≤s≤mτ k s+n θ s ([F/G],[D F /G]) β , (2.1.2) where ∏ 1≤r≤nτ k r α r | ∏ 1≤s≤mτ k s+n θ s ([F/G],[D F /G]) =π * ∏ 1≤r≤nτ k r α r | ∏ 1≤s≤mτ k s+n θ s is an equivariant GW invariant of (F, D F ) with the cycle · | · defined in (1.2.5) andπ defined in (2.1.1). In particular, the LHS of (2.1.2) is nonzero only if ∑ deg δ i ≤ dim B.
We will not make explicit use of this corollary. The interested readers may consult [28] for details. Proposition 2.1.6. Given a birational map x : B ′ → B between smooth projective varieties, let E ′ † be the fiber product in the following diagram
E ′ † y / / E † B ′ x / / B, and M 0,n (E ′ † , β)ȳ / / M 0,n (E † , β)
be the induced map. Then for any fiber curve β, we havē
y * ([M 0,n (E ′ † , β)] vir = [M 0,n (E † , β)] vir .
Proof. Consider the cartesian diagram
M 0,n (E ′ † /B ′ , β) / / M 0,n (E † /B, β) B ′ × T or M 0,n / / B × T or M 0,n induced from M 0,n (E ′ † /B ′ , β) / / M 0,n (E † /B, β) B ′ × M 0,n / / B × M 0,n .
Since
y * Ω E † /B ≃ Ω E ′ † /B ,
INVARIANTS OF A P 1 BUNDLE IN TERMS OF THOSE OF THE BASE
We discuss the relation of the genus zero relative and rubber invariants of a P 1 bundle with those of its base. The main tools used here are C. Manolache's virtual pullback and pushforward [26,27].
In [26,Section 5.4], the absolute invariants of a P 1 bundle and its base are related by the strong virtual pushforward property as in [27]. We adapt the arguments there and establish similar results for relative and rubber invariants. For rubber invariants, the rubber calculus in [28] is also used.
3.1. Terminologies and notations. Let X be a smooth projective variety and L a line bundle on X. Let Y = P X (L ⊕ O), which has a natural projection π : Y → X. π has two sections Y 0 , Y ∞ . Denote by
i 0 : Y 0 → Y and i ∞ : Y ∞ → Y the inclusions.
Recall some terminologies used in [28]. Relative invariants coming from (Y, Y 0 ) and (Y, Y ∞ ) are called type I; those from (Y, Y 0 , Y ∞ ) are called type II.
A variant of type II relative invariants are the invariants of the non-rigid targets, called rubber targets, whose relative invariants are called rubber invariants. See [11,Section 2.4] and [28, Section 1.5] for precise definitions and references. The rubbers naturally occur in the expanded targets of the usual relative maps.
A cohomology class of the form
i 0 * (ω) or i ∞ * (ω) is called distinguished. Note that [Y 0 ] · π * α = i 0 * (α), [Y ∞ ] · π * α = i ∞ * (α)
. In this section, we will use ω to denote a non-distinguished class, i.e. ω ∈ π * H * (X) ⊂ H * (Y).
Relative invariants with rigid targets.
Log notations. We have
A 1 (Y) = i 0 * A 1 (X) ⊕ Z[P 1 ],
where [P 1 ] is the class for the fiber of π. For an effective curve class β of Y, it is determined by θ = π * (β) and β Y ∞ by
β = i 0 * (θ) + β Y ∞ [P 1 ].
We use (Y, Y 0 , Y ∞ ) to denote the log scheme whose underlying scheme is Y equipped with the divisorial log structure determined by the divisor Y 0 ⊔ Y ∞ . Locally it is the product of U (a Zariski open subset of Y) with the trial log structure and the log scheme (P 1 , {0}, {∞}). Similarly we have a log scheme (Y, Y ∞ ). They are log smooth and integral.
Let M 0,n ((Y, Y 0 , Y ∞ ), β; µ, ν) be the log stack of stable log maps from genus zero , n-pointed log curves to (Y, Y 0 , Y ∞ ). Here β is the curve class, µ a partition of d 0 = β Y 0 and ν a partition of d ∞ = β Y ∞ . This is equivalent to specifying the contact orders of the marked points with Y 0 and Y ∞ (see [2, Section 3.2] and [10]). As µ and ν encode the log structure we are considering on Y and ν determines d ∞ , we will use the notation M 0,n (Y; µ, ν) when θ = π * (β) is clear from the context.
For relative invariants of (Y, Y ∞ ) with class
β = i 0 * (θ) + d ∞ [P 1 ]
and a partition ν of d ∞ , we have the log stack
M 0,n ((Y, Y ∞ ), β; ν)
or M 0,n (Y; ν) for short.
A virtual dimension count.
View X as a log scheme with the trivial log structure. The projections
(Y, Y 0 , Y ∞ ) → X and (Y, Y ∞ ) → X
are log maps. When θ is nonzero or n ≥ 3, we have induced maps between log stacks:
p X : M 0,n (Y; µ, ν) → M 0,n (X, θ), q X : M 0,n (Y; ν) → M 0,n (X, θ). (3.2.1)
The following lemma follows from virtual dimension count.
(1) dim [M g,n (Y; µ, ν)] vir = dim [M g,n (X, θ)] vir + 1 − g. (2) dim [M g,n (Y; ν)] vir = dim [M g,n (X, θ)] vir + 1 − g + β Y 0 .
When g = 0, the lemma suggests we might prove strong virtual pushforward properties for p X and, when 1 + β Y 0 ≥ 0, for q X .
3.2.3.
Compatibility of obstruction theories. Let X and X ′ be log smooth projective varieties. Consider a strict map i : X → X ′ ; assume the underlying map of i is either a closed immersion or induces an injective map on the Chow group A 1 as in [26,Section 5].
The map i induces a map between log stacks
i : M 0,n (X) → M 0,n (X ′ ),
where M 0,n (X) (resp. M 0,n (X ′ )) is the log stack of stable log maps to X (resp. X ′ ) from genus zero, n-pointed log curves . (We do not specify curve class or contact orders for ease of notation.) By our assumption on i, there is a commutative diagram
M 0,n (X)¯i / / ρ ' ' P P P P P P P M 0,n (X ′ ) ρ ′ v v ♥ ♥ ♥ ♥ ♥ ♥ ♥
M 0,n and the horizontal arrow is strict. This induces a commutative diagram between stacks M 0,n (X)¯i
/ / ( ( ◗ ◗ ◗ ◗ ◗ ◗ ◗ M 0,n (X ′ ) v v ♠ ♠ ♠ ♠ ♠ ♠ ♠ T or M 0,n . Define E ∨ i := Rπ * f * (L ∨ i ),
where π, f are maps from the universal curve C over M 0,n (X) in the fol-
lowing diagram C f / / π X M g,n (X).
Then it is straightforward to check we have compatible obstruction theories
(3.2.2)ī * E ′ / / E / / E¯i i * L ρ ′ / / L ρ / / L¯i ,
where E → L ρ ′ and E ′ → L ρ are the perfect obstruction theories for ρ and ρ ′ respectively.
The bottom row of (3.2.2) can be identified with the transitivity triangle of Olsson's log cotangent complexes, while the top row is related to the transitivity triangle on Xī where N(µ, ν) is a rational number determined by µ and ν.
* Ω X ′ → Ω X → L i .(p X ) * [M 0,n (Y; µ, ν)] vir = 0 in A * (M 0,n (X, θ)) and (p X ) * ([M 0,n (Y; µ, ν)] vir ∩ ev * 1 [Y 0 ]) = N(µ, ν)[M 0,n (X, θ)] vir ,(2) Assume β Y 0 ≥ 0, then (q X ) * [M 0,n (Y; ν)] vir = 0.
Proof. We will prove the strong virtual pushforward property [27, Definition 4.1] for p X and q X , which consists of mainly checking certain compatibility of perfect obstruction theories. Then the above equations follow from the virtual dimension counts in Lemma 3.2.1.
For (1)
M 0,n (Y; µ, ν)¯j / / p X M 0,n (P(O(−1, 1) ⊕ O); µ, ν) p M 0,n (X, θ)¯i / / M 0,n (P |M| × P |L⊗M| , ( θ M, θ L ⊗ M)).
Here we use p for p P |M| ×P |L⊗M| andī,j to denote the horizontal maps. Asī is strict, the underlying diagram between stacks is also cartesian.
Recall we have defined obstruction theory E¯i (resp. E¯j) forī (resp.j) in (3.2.2). They fit in the following diagram Note that by Lemma 3.2.1, p satisfies the strong virtual pushforward property since P |M| × P |L⊗M| is homogeneous. Then we can transfer this property to p X usingī ! . To determine the number N, consider a point l :
p * X E¯i ≈ / / E¯j p * X L¯i / / L¯j .P 1 → P |M| × P |L⊗M| in M 0,n (P |M| × P |L⊗M| , ( π * β M, π * β L ⊗ M)).
As
A 1 (P 1 ) → A 1 (P |M| × P |L⊗M| ) is injective, we have a cartesian diagram M 0,n (P(O(d 0 − d ∞ ) ⊕ O); µ, ν) / / p P 1 M 0,n (P(O(−1, 1) ⊕ O); µ, ν) p M 0,n (P 1 , 1)¯l / / M 0,n (P |M| × P |L⊗M| , ( θ M, θ L ⊗ M)). Note that θ L = d 0 − d ∞ . Let P = P(O(d 0 − d ∞ ) ⊕ O).
The number N is determined by N(µ, ν)[M 0,n (P 1 , 1)] vir . This completes the proof of (1). The proof of (2) is entirely similar and is omitted. Consider the fiberwise C * action on Y and the trivial action on X. Under these actions, π : Y → X is C * equivariant, and the induced map p X and
(p P 1 ) * [M 0,n (P; µ, ν)] vir ∩ ev * 1 [P 0 ] =q X are C * equivariant. Assume β satisfies β Y 0 ≥ 0, β Y ∞ ≥ 0.
For type II invariants, assume 2g − 2 + n > 0, then the pushforward of the equivariant virtual class under p X lies in t −1 A * (M g,n (X, γ))[[t For type I, when g = 0, n ≥ 3, the pushforward of the equivariant virtual class under q X lies in t −1 A * (M 0,n (X, γ))[[t −1 ]] by dimension count and therefore vanishes.
Special two-pointed fiber integrals.
When θ is zero, β is a fiber class for π and d 0 = d ∞ > 0 for type II invariants. In particular,
n ≥ l(µ) + l(ν) ≥ 1 + 1 = 2.
If n = 2, we see l(µ) = l(ν) = 1 and there are no non-relative marked points. Let d : . In particular, it is smooth with virtual class equal to its fundamental class. Consequently, for α 1 , α 2 ∈ H * (X),
= d 0 = d ∞ .[M 0,2 (Y;(d),(d))] vir ev * 1 (α 1 ) ∪ ev * 2 (α 2 ) = 1 d X α 1 ∪ α 2 .
Proof. We show that first that the source curve has no contracted component. Assume there are v contracted components and h non contracted components, then there are v + h − 1 nodes. Consider the number of special points (nodes or marked points) on each component. On a contracted component, there are at least 3 of them. There are at least 2 special points on a non contracted component, which are points mapped into Y 0 and Y ∞ . As each node is counted twice, we have
2(v + h − 1) + 2 ≥ 3v + 2h.
Thus v = 0. This implies in fact the source curve must be smooth as this is a fiber integral with 2 totally ramified relative points. It is then easy to see a stable log map C → Y is determined by its underlying map. The moduli space being unobstructed follows from
H 1 (C, O C ) = 0.
Since β is a fiber class for π, the last integral can be evaluated by first integrating over the fiber of e : M 0,2 (Y; (d), (d)) = r √ L/X → X which has degree 1/d. The last statement then follows.
Rubber invariants.
Let M Γ (Y, β; µ, ν) •∼ be the moduli stack of relative maps to rubber targets. Here Γ specifies the discrete data for the genus zero source curve, including the number of components, the curve class of each component, and the distribution of marked points among these components.
We call a component unstable if its curve class is a fiber class for π, and there are only two relative marked points with no other marked points. Otherwise the component is stable. Unstable component might appear for disconnected rubber invariants.
We treat the rubber invariants in two steps.
Lemma 3.3.1. Suppose there is no unstable component in Γ. In this case
π : Y → X induces r X : M Γ (Y, β; µ, ν) •∼ → M Γ (X, θ) • . Furthermore, under the same assumption, we have dim[M Γ (Y, β; µ, ν) •∼ ] vir = dim[M Γ (X, θ) • ] vir + c(Γ) − 1,
where c(Γ) is the number of components (of domain curve) in Γ.
Proof. The first statement is an easy consequence of the definitions of the moduli and the second follows from a straightforward dimensional count.
Proposition 3.3.2. When Γ does not contain an unstable component, we have
(r X ) * ([M Γ (Y, β; µ, ν) •∼ ] vir = R(µ, ν)[M Γ (X, θ) • ] vir , if c(Γ) = 1. 0, otherwise.
where R(µ, ν) is a rational number determined by µ, ν.
Proof. The proof is similar to the one given in Proposition 3.2.2. In general, insertions ω for genus zero rubber invariants
µ | ω | ν •Y∼ Γ
are necessarily non-distinguished. Γ might contain unstable components. We will show that it can be calculated as a product of contribution from the unstable components and the contribution from the stable component.
The unstable contribution will be treated using Lemma 3.2.4 and the stable contribution will be converted to type II invariant with rigid Y. We decompose Γ into stable and unstable parts
Γ = Γ s ⊔ Γ u ,
where Γ s and Γ u ) consist of stable and unstable components respectively. We then divide weighted partitions and insertions accordingly:
µ = µ u ⊔ µ s , ω = ω u ⊔ ω s , ν = ν u ⊔ ν s . Proposition 3.3.4. (3.3.1) µ | ω | ν •Y∼ Γ = µ s | ω s | ν s •Y∼ Γ s · µ u | ω u | ν u •Y Γ u , where ω is a non-distinguished insertion. Furthermore, µ s | ω s | ν s •Y∼ Γ s is determined
by invariants on X by Proposition 3.3.2, and µ
u | ω u | ν u •Y Γ u is determined by Lemma 3.2
.4. In particular, a genus zero rubber invariant is nonzero only if there is exactly one stable component in Γ.
Proof. Note that the stable part Γ s is non-empty by the stability condition. If there is no internal marked point in Γ s , the rubber calculus may be used to create a point as in [28, Section 1.5]. Then we apply the rigidification and replace the rubber invariants by type II invariants (with rigid target Y), If the extra marked point created above (by rubber calculus) goes to the unstable part, the contribution from that configuration is zero. Therefore, we may assume the extra point goes to the stable part, which is not empty by the stability requirement. Once it is rigidified, the product formula applies and we conclude that the contribution splits into the stable contribution and the unstable contribution of the rigid target. Reverse the rigidification process in the stable contribution, we obtain (3.3.1).
For the contribution from the stable components, Proposition 3.2.2 (1) implies that the stable contribution can be determined by the invariants on X since ω is non-distinguished (pull-backed from X). It vanishes unless the number of connected components c(Γ) is 1.
The curve classes for unstable components are necessarily fiber curve classes. After the rigidification, they can be treated by Lemma 3.2.4 for the same reason that ω is a non-distinguished class.
DEFORMATION INVARIANCE OF F -INVARIANCE PROPERTY
In this section we prove that F -invariance is stable under deformation of vector bundles. This is an easy consequence of the deformation invariance for GW invariants. Proposition 4.1.5. Let E, E ′ be two vector bundles of rank r + 1 over S × T, where S, T are smooth projective varieties. If for some t 0 ∈ T, F -invariance holds for (S, E t 0 , E ′ t 0 ), then F invariance holds for (S, E t , E ′ t ) any t ∈ T. Here E t , E ′ t are the restrictions of E, E ′ to S × {t} for a closed point t ∈ T. Proof. We abuse notation and use i t to denote both the inclusions
S × {t} ֒→ S × T and X(S, E t , E ′ t ) ֒→ X(S × T)
. Because T is irreducible, it is connected as a complex manifold. Therefore
i t * : H 2 (S) → H 2 (S × T) is independent of t.
Since
S × {t} i t / / S × T pr S / / S
is the identity map, the induced map
H 2 (S × {t}) → H 2 (S × T)
is injective. It follows from the commutative diagram
X(S, E t , E ′ t ) X(S × T) S × {t} / / S × T that H 2 (X(S, E t , E ′ t )) → H 2 (X(S × T)) is injective. Given β ∈ H 2 (X(S × T)), consider the cartesian diagram M 0,n (X(S, E t , E ′ t ), β t ) / / M 0,n (X(S × T), β) {t} / / T.
Here β t ∈ H 2 (X(S, E t , E ′ t )), if exists, is the class satisfying i t * (β t ) = β. By the compatibility of virtual classes, we have
∏ i pr * S (α i )Q i X(S×T) β = ∏ i α i Q i X(S,E t ,E ′ t ) β t · [T]
as top dimensional cycles in H * (T). Here α i are classes on S, Q i are classes on [X(C)/G]. Now we see the generating function
∏ i α i Q i X(S,E t ,E ′ t ) (β S , d) is determined by ∏ i pr * S (α i )Q i X(S×T) ((i t ) * (β S ), d) = ∏ i α i Q i X(S,E t ,E ′ t ) (β S , d) · [T] .
Note that the LHS is independent of t.
Similarly, using the fact that F commutes with pullback, we see that
∏ i F (pr * S (α i )Q i ) X ′ (S×T) ((i t ) * (β S ), d) = ∏ i F (α i Q i ) X ′ (S,E t ,E ′ t ) (β S , d) · [T].
As the classes ((i t ) * (β S ), d)) is independent of T, we see that F -invariance for (S, E t , E ′ t ) is independent of t.
F -INVARIANCE FOR P 1 BUNDLES: ABSOLUTE, TYPE II AND RUBBERS
In this and the next sections, we show that F -invariance for (D, F, F ′ ) implies F -invariance for (P, π * F, π * F ′ ) for π : P = P D (N ⊕ O) → D a split P 1 bundle, in the absolute and relative settings. In this section, we treat the absolute and type II invariants, including the rubber invariants.
Absolute invariants of split projective bundles.
For absolute invariants, a variant (and easier version) of the quantum Leray-Hirsch theorem proved in Part II [17] leads to a stronger result for ordinary flops with base being a split projective bundle.
Theorem 5.1.1. Let V = m i=1 L i → T
such that L i 's are line bundles and π : P = P T (V) → T be its associated split projective bundle. Then, F -invariance for generating function of absolute invariants on (T, F, F ′ ) implies that on (P, π * F, π * F ′ ).
Proof. This follows from the techniques in [16,17]. Notice that
X(P) = P X(T) (V) → X(T),
is a split P m−1 bundle with V pulled-back from the base T. The I-function of X(P) is a hypergeometric modification of the J-function of X(T), as explained in [17, §2]. Symbolically, we can write
I X(P) = I X(P)/X(T) * J X(T)
where the hypergeometric factor I X(P)/X(T) is determined by the Chern classes of the line bundles L i 's. Since these bundles are pulled-backs from T, the F -invariance holds by assumption.
Using the techniques in [17, §2 and 3], in particular the F -invariance of the Birkhoff factorization procedure, we conclude that F -invariance for T implies that for P.
Applying the theorem to T = D, V = N ⊕ O, we obtain the desired result.
Corollary 5.1.2. F -invariance for generating function of absolute invariants on
(D, F, F ′ ) implies that on (P, π * F, π * F ′ ).
We now proceed to establish similar results for relative and rubber invariants.
Type II.
Fiber classes.
We define fiber curve classes (modulo extremal rays) to be curve classes (β P , d) such that β P is a fiber class for P → D. Equivalently, (β P , d) is a fiber curve class in this sense if any β ∈ (β P , d) is a fiber class for the fiber bundle X(P) → D in the sense of section 2.
Theorem 5.2.1. Fiber class type II invariants of (P, P 0 ,
P ∞ ) are F -invariant. Proof. Let G := GL r+1 × GL r+1 , T := 2r+2 ∏ 1 G m ,
where G m = GL 1 (C). Fiber class invariants for (P, P 0 , P ∞ ) are fiber integrals. On the X(P) side, they are fiber integrals for the bundle X(P) → D, which fits in a cartesian diagram
(X(P), X(P 0 ), X(P ∞ )) / / ([P 1 /G m ], [{0}/G m ], [{∞}/G m ]) × [X(C)/G] D / / BG m × BG.
By Proposition 2.1.6, we only need to prove invariance after passing to D ′ via a birational map D ′ → D. Then by Lemmas 1.8.1, 1.8.2, and Proposition 4.1.5, we can assume the fiber bundle below has a smaller structure group:
(X(P ′ ), X(P ′ 0 ), X(P ′ ∞ )) / / ([P 1 /G m ], [{0}/G m ], [{∞}/G m ]) × [X(C)/T] D ′ / / BG m × BT.
Here P ′ is the pullback of P via D ′ → D. Now fiber class type II invariants of X(P ′ ) are determined by (i) the classical intersection product on D ′ , which are the same on both sides of the flop, and (ii) the equivariant invariants of P 1 , {0}, {∞} × X(C), which by the product formula [21] are determined by the equivariant invariants of (P 1 , {0}, {∞}) and X(C). From here we see fiber class Finvariance for (P, P 0 , P ∞ ) follows from that for T-equivariant invariants of (Spec C, O ⊕r+1 , O ⊕r+1 ), which can be deduced from the split case for
Non-fiber classes.
Theorem 5.2.2. F -invariance for D implies F -invariance for (P, P 0 , P ∞ ) with curve classes (β P , d) such that β P is not a fiber class for P → D.
Proof. We prove it by induction on the number of distinguished insertions (defined in Section 3.1). If the number of distinguished insertions is less than 2, we apply Proposition 3.2.2 (1) to conclude the proof.
For type II invariants of (X(P), X(P 0 ), X(P ∞ )) with l ≥ 2 distinguished insertions, we use the degeneration formula. As
(5.2.1) [X(P 0 )] − [X(P ∞ )] = π * c 1 (N| X(D) ), (pullback of N under X(D) → D)
, modulo type II invariants with l − 1 distinguished insertions, we can assume one of the distinguished insertions is of the form i 0 * (α) and the other ones are of the form i ∞ * (α i ).
Consider the family
W(X(P), X(P ∞ )) = Bl X(P ∞ )×{0} (X(P) × A 1 ) with divisors X(P 0 ) × A 1 and X(P ∞ ) × A 1 , the strict transformation of X(P ∞ ) × A 1 under the blowing-up W → X(P) × A 1 .
The degeneration formula for this family implies
µ | ω · i 0 * (α) l−1 ∏ i=1 i ∞ * (α i ) | ν (X(P), X(P 0 ), X(P ∞ )) (β P ,d) = ∑ I ∑ η C η µ | ω 1 · i 0 * (α) | λ, e I Γ 1 · p X * λ, e I | ω 2 · l−1 ∏ i=1 i ∞ * (α i ) | ν Γ 2 .
Note that the RHS is determined by type II generating functions with at most l − 1 distinguished insertions. This relation is clearly compatible with F . The theorem now follows by induction. In this section, we prove F -invariance for type I relative invariants of the P 1 bundle P, assuming invariance of the base D.
Rubber invariants.
As the strong virtual pushforward property for type I invariants is conditional, the arguments in this section is not as straightforward as those for type II and rubber in the last section. The starting point is that, up to type II invariants, distinguished insertions in an invariant can be removed using the degeneration formula (Lemma 6.5.1).
Consider a type I invariant of (P, P ∞ ) without distinguished insertions of classes (β P , d). If β P P 0 ≥ 0, then the invariant is zero by the strong virtual push forward property (Proposition 3.2.2 (2)). If β P P 0 < 0, using an inversion of degeneration argument, we show in Proposition 6.5.4 that the invariant is determined by an absolute invariant, lower order type I invariants, type II and rubber invariants. As absolute, type II and rubber invariants are F -invariant, F -invariance for type I invariants follows.
The proof of Proposition 6.5.4 requires some preparation. The argument involves applying the degeneration formula to an ancestor invariant. We then need to understand certain fiber integral and relative invariants with ancestor insertions. A number of lemmas in this section are about nonvanishing fiber integrals, which ensures certain inversion of degeneration arguments. Ancestor insertions in a relative invariant can be removed using TRR for ancestors (Appendix A). We need to keep track of the orders of type I invariants obtained from removing ancestors. To do so, we introduce in Subsection 6.2 a relation between type I invariants called dominance and the order of those relative invariants are characterized in Proposition6.2.6. 6.1. Nonvanishing conditions. In this subsection, we discuss some nonvanishing conditions on X(C), or X(Spec C, O ⊕r+1 , O ⊕r+1 ). These conditions give "selection rules" of the cohomology classes in the highest order terms arising from the degeneration arguments. See for example the end of the proof of Proposition prop:inversion' and Section ss:dom.
We use Q and R to represent elements in a chosen (homogeneous) basis of X(C); deg Q and deg R their degrees with respect to the real grading of H * (X(C)). A superscript ∨ will be used to represent elements in the dual basis. In particular, when Q = R,
X(C) Q ∨ ∪ R = 0.
We use notations form Subsection 1.2.2, especially (1.2.2) and (1.2.5) on certain cycles in absolute and relative moduli stacks. Proof. By calculating the dimension of the cycle Q ∨ , R, 1, · · · , 1
X(C) nℓ , we see that deg Q ∨ + deg R = 2r + 1.
We can write Q ∨ and R uniquely as polynomials in h and ξ of the form
∑ i≤r,j≤r+1 a ij h i ξ j .
Here a ij are complex numbers. We will write Q ∨ (h, ξ) or R(h, ξ) when we view them as polynomials in h and ξ. When n = 0, a stable map to X(C) factors through Z(C). As ξ| Z(C) = 0,
Q ∨ (h, ξ), R(h, ξ), 1, · · · , 1 X(C) nℓ = Q ∨ (h, 0), R(h, 0), 1 · · · 1) X(C) nℓ , but then deg Q ∨ (h, 0) + deg R(h, 0) ≤ r + r < 2r + 1.
When n = 0,
st * Q ∨ , R, 1, · · · , 1 m X(C) nℓ = X(C) Q ∨ ∪ R [M 0,m+2 ].
It is nonzero if and only if Q = R.
(c, Q ∨ ) | tR (P 1 , {0})×X(C) c[P 1 ]+nℓ = 1 c (c, Q ∨ ) | t, tR (P 1 , {0})×X(C) c[P 1 ]+nℓ .
By the product formula [21], {0}) is zero unless c = 1, which follows from calculating the virtual dimension. For d = 1, (1, 1) | t, t (P 1 , {0}) = 1.
(c, Q ∨ ) | t, tR (P 1 , {0})×X(C) c[P 1 ]+nℓ = (c, 1) | t, t (P 1 , {0}) Q ∨ , 1, R X(C) nℓ . (c, 1) | t, t (P 1 ,
If Q ∨ , 1, R X(C) nℓ is nonzero, then by Lemma 6.1.1, n = 0 and Q = R.
Dominance relation for weighted partitions.
In this subsection, the base D is general. Let {δ i } be a basis of H * (D), we fix a F -homogeneous basis compatible with {δ i }. We will use Q and R with possible subscripts to represent cohomology classes in H * ([X(C)/G]) that appear in this Fhomogeneous basis. Notations for weighted partitions are introduced in Section 1.5.
6.2.1. Dominance relation. We start with the definition of splitting. They will be used in the definition of dominance relation (Definition 6.2.3).
Definition 6.2.1. We say that we can split (δ a 1 ,
Q) into {(δ b m , R m )}, if D δ ∨ a 1 · ∏ m δ b m = 0,
and for some n ≥ 0,
st * Q ∨ · ∏ m R m X(C) nℓ is a nonzero top dimensional cycle on M 0,1+|{m}| . In particular, when (δ a 1 , Q) splits into {(δ b m , R m )}, we have deg δ a 1 = ∑ deg δ b m , deg Q = ∑ deg R m .
Remark 6.2.2. In the definition of splitting, classes Q and R m are classes on [X(C)/G]. We abuse notation to use the same symbols for corresponding classes on X(C).
Definition 6.2.3 (Dominance relation). Let
µ = {(µ i , δ a i Q i )} i∈I and ν = {(ν j , δ b j R j )} j∈J
be two weighted partitions such that
deg D µ = deg D ν.
We say µ dominates ν if there exists a partition of J
J = ⊔ i∈I J i ,
such that for every i, one of the following conditions is satisfied.
• J i = ∅ and µ i − 1 > ∑ j∈J i (ν j − 1), • J i = ∅, µ i − 1 = ∑ j∈J i (ν j − 1), and {(δ b j , R j )} j∈J i can be obtained from (δ a i , Q i ) by splittings. • J i = ∅ and µ i − 1 > 0, • J i = ∅, µ i = 1 and δ a i = 1, Q i = 1.
The following lemma is an easy consequence of the definition of dominance.
Lemma 6.2.4.
(1) If λ dominates µ and µ dominates ν, then λ dominates ν. (2) If µ i dominates ν i , then ⊔ i µ i dominates ⊔ i ν i . Lemma 6.2.5. If µ dominates ν and ∑ µ i = ∑ ν j , then either µ is of lower order than ν, or it can be identified with ν.
Proof. Assume µ is not of lower order than ν. Let
µ = {(µ i , δ a i Q i )}, ν = {(ν j , δ b j R j )}.
By the definition of dominance,
µ i − 1 ≥ ∑ j∈J j (ν j − 1),
where the right hand side is understood to be zero if J i = ∅. Summing up all i, we have
∑ µ i − l(µ) ≥ ∑ ν j − l(ν), or l(µ) ≤ l(ν).
Since µ is not of lower order than ν, we should l(µ) = l(ν), or equivalently
µ i − 1 = ∑ j∈J j (ν j − 1). Assume 1 ∈ J 1 and {(ν j , deg δ b j , deg R j )} is arranged lexicographically. We shall compare (ν 1 , deg δ b 1 , deg R 1 ) with (µ 1 , deg δ a 1 , deg Q 1 ).
From the following equations
µ 1 − 1 = ∑ j∈J 1 (ν j − 1), deg δ a 1 = ∑ j∈J 1 deg δ b j , deg Q 1 = ∑ j∈J 1 deg R j ,
we conclude that either (µ 1 , deg δ a 1 , deg Q 1 ) is of lower order or
(µ 1 , deg δ a 1 , deg Q 1 ) = (ν 1 , deg δ b 1 , deg R 1 ).
In the latter case
(ν j , deg δ b j , deg R j ) = (1, 0, 0) for all j ∈ J 1 , j = 1. When (µ 1 , deg δ a 1 , deg Q 1 ) = (ν 1 , deg δ b 1 , deg R 1 ),
δ a 1 can be split into δ b 1 , 1, · · · , 1 and Q 1 can be split into R 1 , 1, · · · , 1. It is easy to see δ a 1 = δ b 1 from the definition of a splitting. By Lemma 6.1.1, we see Q 1 = R 1 . Now we find the highest order term of ν in µ. The same argument will match the other terms of ν with terms in µ.
Generating functions with relative ancestor insertions.
Here we consider the only type of generating functions whose relative insertions contain ancestors. It naturally occur in our induction process. Let D ⊂ S be a smooth divisor as before. Consider a special type of relative weighted partition with ancestors
(6.2.1) γ = {(µ i , δ a i Q i )} ⊔ {(1,τ ν j −1 (δ b j R j )}.
That is, the relative insertions with ancestors always have contact order (multiplicity) 1. We refer to γ as specified by two (primary) weighted par-
titions µ = {(µ i , δ a i Q i )} and ν = {(ν j , δ b j R j )}. Define deg D (γ) = deg D ( µ) + deg D ( ν).
The relative generating functions
ω | γ (X(S),X(D)) (β S , d)
with γ of the above form will be useful in our induction process. In particular, the multiplicity 1 condition comes from Lemma 6.1.2.
The following proposition shows that the relative generating functions of the above type can be determined inductively by generating functions whose relative insertions contains no ancestors. can be determined by the following three types of generating functions:
(i) primary relative invariants that are of strictly lower order than
{(β S , d), ||ω||, deg D (γ)},
(ii) rubber invariants, and (iii) primary relative invariants of order
{(β S , d), ||ω||, deg D (γ)}, whose weighted partition dominates µ ⊔ ν.
Proof. If there is no ancestor involved, the statement is trivial by (iii). Otherwise we apply the topological recursion relation for ancestors and perform induction on the number of ancestor insertions. If we apply TRR once to ω | γ (X(S),X(D)) (β S , d) and lowerτ ν j −1 in its relative insertion toτ ν j −2 , we get a positive linear combination of terms either of the form (see Appendix A)
ω A · α | γ A (X(S),X(D)) (β A , d A ) · ω B · α ∨ | γ B (X(S),X(D)) (β B , d B ) where ||ω A || + ||ω B || = ||ω||, (β A , d A ) + (β B , d B ) = (β S , d),
or of the form
ω Ξ | γ Ξ •(X(S),X(D)) Ξ · p X * γ Ξ ′ | ω Ξ ′ | γ •X(P)∼ Ξ ′ ,
where p X is the composition of maps
X(P) → X(D) ֒→ X(S).
For a term of the form
ω A · α | γ A (X(S),X(D)) (β A , d A ) · ω B · α ∨ | γ B (X(S),X(D)) (β B , d B ) , if (β A , d A ) equals (β S , d)
then there is no relative insertion in γ B . As either factor contains at least 3 marked point, ||ω B || + 1 ≥ 3. Then
||ω A || + 1 = ||ω|| − ||ω B || + 1 < ||ω||.
Thus either factor gives rise to terms of lower order than {(β S , d), ||ω||}. For a term of the form
ω Ξ | γ Ξ •(X(S),X(D)) Ξ · p X * γ Ξ ′ | ω Ξ ′ | γ •X(P)∼ Ξ ′ , the factor ω Ξ | γ Ξ
•(X(S),X(D)) Ξ will produce primary relative invariants of order {(β S , d), ||ω||} only when Ξ specifies a connected curve with classes (β S , d) and ω Ξ = ω. Then it is of the form ω | γ ′ (X(S),X(D)) (β S , d) and the curve
classes for γ Ξ ′ | ω Ξ ′ | γ •(X(P))∼ Ξ ′ are fiber classes. If γ Ξ ′ | ω Ξ ′ | γ •X(P)∼ Ξ ′
is nonzero, by Proposition 3.3.4 we see that Ξ ′ contains a stable component with at least 3 marked points, and all other possible components are unstable. For such a configuration, the ancestor insertionτ ν j −2 belongs to the stable component, and on any unstable component the contact order and insertion for γ and γ ′ should match . This implies γ ′ is specified by some partitions µ ′ and ν ′ , and it has at least one less ancestor insertions than γ. Therefore we can assume the proposition holds for ω | γ ′ (X(S),X(D)) (β S , d)
.
As γ Ξ ′ | ω Ξ ′ | γ •X(P)∼ Ξ ′ is a non-zero fiber integral, we have deg D (γ ′ ) ≥ deg D (γ) = deg D ( µ ⊔ ν). If µ ′ ⊔ ν ′ dominates µ ⊔ ν when deg D (γ ′ ) = deg D (γ)
, then by the transitivity of dominance, the proposition is proved for
ω | γ (X(S),X(D)) (β S , d)
.
The fact that µ ′ ⊔ ν ′ dominates µ ⊔ ν follows from a simple dimensional count that the stable rubber component should have enough points to guarantee the non-vanishing of ancestors invariants.
A dominance lemma.
The following lemma will be used in the proof of Proposition 6.5.4.
Lemma 6.2.7. If the generating function
D δ ∨ a ∏ j δ b j (c, Q ∨ ) | t k ∏ jτ ν j−1 (tR j ) (P 1 , {0})×X(C) (c[P 1 ], 0) is nonzero, then (c, δ a Q) dominates {(ν j , δ b j R j )}.
Proof. The generating function
(c, Q ∨ ) | t k ∏τνj−1 (tR j ) (P 1 , {0})×X(C) (c[P 1 ],0) is the following sum ∑ n≥0 (c, Q ∨ ) | t k ∏τνj−1 (tR j ) (c, nℓ) q (c,nℓ) ,
where (c, nℓ) represents the curve class c[P 1 ] + nℓ on P 1 × X(C).
By the product formula [21],
(c, Q ∨ ) | t k ∏τνj−1 (tR j ) (c,nℓ)
is the intersection on M 0,1+k+|{j}| of the cycles
st * (c, 1) | t k ∏τνj−1 (t) (P 1 , {0}) c[P 1 ],(c) and st * Q ∨ , 1 k ∏ R j X(C) nℓ .
(In case there is only one internal insertion, since there is no ancestor, we use the divisor equation to create a divisor insertion t, and then the following argument goes through.) The cycle from (P 1 , {0}) has dimension c − 1 − ∑(ν j − 1), which is nonzero only if
c − 1 ≥ (ν j − 1). And if c − 1 = ∑ (ν j − 1),
the cycle is zero dimensional, and then the cycle from X(C) is top dimensional. From here we see (c, δ a Q) dominates {(ν j , δ b j R j )}.
6.3. Fiber class. Theorem 6.3.1. Fiber class type I invariants of (P, P 0 ) and (P, P ∞ ) are F -invariant.
Proof. The proof is entirely similar to the proof of Theorem 5.2.1 and is omitted.
Positivity of certain relative two-pointed invariants on (P 1 , {∞}).
Here we establish some positivity lemmas of certain relative integrals on (P 1 , {∞}). These integrals naturally occur in the degeneration process as"coefficients" of the highest order terms. We need the non-vanishing in order for the inversion of degeneration to work. (See, e.g., the proof of Proposition 6.5.4.)
∏ ev i ×st / / ∏ c+1 i=1 P 1 × M 0, c+2 ,
where ev i 's are the c + 1 evaluation maps determined by the internal marked points, and st is the the map stabilizing the source curve. We have
( ∏ ev i × st) * [M 0,c+2 (P 1 , {∞}; (c, 1))] vir = (c, 1) | t cτ c−1 (t) c+1 ∏ i=1 P 1 × M 0, c+2 .
As M 0,c+2 (P 1 , {∞}; (c, 1)) is smooth of dimension 2c, (c, 1) | t cτ c−1 (t) = #( ∏ ev i × st) −1 p 1 , · · · , p c+1 , (P 1 , x 1 , · · · , x c+2 ) for generic p 1 , · · · , p c+1 , (P 1 , x 1 , · · · , x c+2 ) . It is not hard to see this equals one. (Consider f :
P 1 → P 1 such that f −1 (∞) = cx c+2 , f (x i ) = p i . Assume that x c+2 = ∞,(c, 1) | t c ′τ c−1 (t) c[P 1 ] > 0 when c ′ > c.
Proof. Note that t is ample on P 1 and ψ classes are ample on M 0,n (by the stability condition). The divisor equation then implies
(c, 1) | t c ′τ c−1 (t) c[P 1 ] ≥ c[P 1 ] t · (c, 1) | t c ′ −1τ c−1 (t) c[P 1 ] .
The lemma follows by induction and Lemma 6.4.1.
6.5. Non-fiber class.
Reduction to non-distinguished insertions.
Consider Type I invariants of (X(P), X(P 0 )) with l ≥ 1 distinguished insertions. Lemma 6.5.1. As before, ω stands for a non-distinguished insertion. We have
ν | ω · l ∏ i=1 i 0 * (α i ) (X(P),X(P 0 )) (β P , d) = ∑ I ∑ η C η ν | ω 1 · l ∏ i=1 i 0 * (α i ) | µ, e I Γ 1 · p X * ( µ, e I | ω 2 Γ 2 ).
Proof. This follows from the degeneration formula applied to the A 1 family
W X(P), X(P ∞ ) , X(P 0 ) × A 1
of pairs, with special fiber (X(P), X(P 0 )) ∪ X(P ∞ ) X(P).
Corollary 6.5.2. The F -invariance of generating functions of type I without distinguished insertions and of type II implies the F -invariance of type I in general.
Proof. Using (5.2.1), we can assume all of the distinguished insertions are of the form i 0 * (α). By Lemma lem:rmdist, those generating functions are determined by type II invariants and type I invariants without distinguished insertions.
We will therefore assume that ω contains no disntinguished insertions in the remaining of this section. 6.5.2. The case β P P 0 ≥ 0. Consider a type I invariants of (X(P), X(P ∞ )):
ω | ν (X(P),X(P ∞ )) (β P , d) , where ν = {(ν j , δ b j R j )},
and δ b j R j are taken from a F -homogeneous basis of H * (X(D)). Proposition 6.5.3. If β P P 0 ≥ 0 and ω has no distinguished insertions, then
ω | ν (X(P),X(P ∞ )) (β P , d) = 0.
Proof. This follows directly from Proposition 3.2.2 (2). 6.5.3. The case β P P 0 < 0. For a weighted partition ν,
k( ν) = max{0, ∑ {i | ν i >1} ν i − Id( ν)}
is the number defined in (1.5.2). In the following, we apply divisorial insertion [X(P ∞ )] to increase the number of internal marked point by k( ν), in order to ensure the existence of the corresponding ancestors
ω[X(P ∞ )] k(ν)τ ν−1 (i ∞ * (·)) X(P) (β P , d)
. Proposition 6.5.4. Assume β P P 0 < 0.
(1) If ν is not empty, then there exists a positive number C( ν) such that
C( ν) ω | ν (X(P),X(P ∞ )) (β P , d) − ω · [X(P ∞ )] k( ν) · ∏ jτ ν j −1 (i ∞ * (δ b j R j )) X(P) (β P , d)
is generated by generating functions of relative and rubber invariants on X(P) of class at most (β P , d), and those of of (X(P), X(P ∞ )) involving class (β P , d) whose orders are lower than ω | ν (β P , d) .
(2) If ν is empty, then ω | ν (X(P),X(P ∞ )) (β P , d) − ω X(P) (β P , d)
is generated by generating functions of relative invariants on X(P) with curve classes lower than (β P , d).
Here we say a formal power series f is generated by { f m } if it belongs to the subalgebra of formal power series generated by { f m }.
Proof. We prove the first part, and the second part can be proved similarly.
Consider the family W(X(P), X(P ∞ )) → A 1 and a generating function of a general fiber
ω · [X(P ∞ )] k · ∏ jτ ν j−1 (i ∞ * (δ b j R j )) X(P) (β P , d)
.
We can lift the insertions [X(P ∞ )] and i ∞ * (δ b j R j ) to (X(P), X(P 0 )) in the singular fiber. The degeneration formula allows us to express this function in terms of the ancestor relative invariants of (X(P), X(P ∞ )) and (X(P), X(P 0 )).
By choosing the splitting properly, the resulting expression is as follows:
∑ (γ 1 ,γ 2 ) ∑ η C η ω 1 | µ, γ 1 •(X(P),X(P ∞ )) Γ 1 × π X * µ, γ 2 | ω 2 · [X(P ∞ )] k · ∏ jτ ν j −1 ([X(P ∞ )] · δ b j R j ))
•(X(P),X(P 0 ))
Γ 2 .
Here π X : X(P) → X(P) is induced from π : P → D ≃ P ∞ ֒→ P, and γ 1 , γ 2 are used to denote insertions without specifying their forms due to the behavior of ancestors. Denote the curve classes of Γ i by (β i , d i ), then β 2 < β P since β P P 0 < 0. We know that β 1 ≤ β P , and when β 1 = β P , β 2 is a fiber class for P → D.
When Γ 1 is connected, Γ 2 is a disjoint union of l(µ) rational curves. Denote by p j the marked point corresponding to the insertion i ∞ * (δ b j R j ), and by C i the curve with relative condition µ i . If Γ 2 specifies that some C i has only two marked points and one of them is p j with ν j > 1, then the ancestor τ ν j −1 appears in the relative insertion of (X(P), X(P ∞ )), and in this case we say Γ 2 is unstable (otherwise it is stable).
We divide η = (Γ 1 , Γ 2 ) into three types:
(1) (β 1 , d 1 ) < (β P , d), or (β 1 , d 1 ) = (β P , d) and Γ 1 is not connected.
(2) (β 1 , d 1 ) = (β P , d), Γ 1 is connected, and Γ 2 is stable.
(3) (β 1 , d 1 ) = (β P , d), Γ 1 is connected, and Γ 2 is unstable.
It is easy to see that type (1) terms are generated by (connected) ancestor relative invariants of (X(P), X(P 0 )) and (X(P), X(P ∞ )) with curve classes less than (β P , d). If we apply TRR to remove ancestor insertions then we get primitive relative rubber invariants of X(P) with curve classes less than (β P , d).
For type (2) and type (3) terms, the factor with discrete data Γ 2 is generated by relative and rubber invariants of X(P) with class less than (β P , d).
We will show that for type (2) and (3) terms, the Γ 1 factor might produce the term
ω | ν (β P , d)
after removing ancestors using TRR. And for its 'coefficients', a priori functions generated by lower order invariants, is always a positive number.
A type (2) term can be written as
ω 1 | µ, e I (X(P),X(P ∞ )) (β P , d) × π X * µ, e I | ω 2 · [X(P ∞ )] k · ∏ jτ ν j −1 ([X(P ∞ )] · δ b j R j ))
•(X(P),X(P 0 )) Γ 2
.
If e I = {δ a i Q i } 1≤i≤l(µ) , its dual e I is given by {δ ∨ a i Q ∨ i }. Recall we abuse the notation to use Q ∨ i to denote the class on [X(C)/G] which is dual to Q i in X(C).
We will show that the order of
ω 1 | µ, e I (X(P),X(P ∞ )) (β P , d)
is no greater than
ω | ν (X(P),X(P ∞ )) (β P , d)
.
We can assume ω 1 = ω or ω 2 is empty, for otherwise ω 1 | µ, e I (X(P),X(P ∞ ))
(β P , d)
is of lower order. The factor
µ, e I | [X(P ∞ )] k · ∏ jτ ν j−1 ([X(P ∞ )] · δ b j R j ))
•(X(P),X(P 0 )) Γ 2 is determined by fiber integrals for the the following bundle:
(X(P), X(P 0 )) / / ([P 1 /G m ], [{0}/G m ]) × [X(C)/G] D / / BG m × BG, where G = GL r+1 × GL r+1 .
The contribution from C i is the generating function
µ i , δ ∨ a i Q ∨ i | [X(P ∞ )] k i · ∏ p j ∈C iτ ν j−1 ([X(P ∞ )] · δ b j R j )) (X(P),X(P 0 )) ,
where k i of those k marked points with X(P ∞ )-insertion are distributed to C i . If this fiber integral is nonzero, then
deg δ ∨ a i + ∑ p j ∈C i deg δ b j ≤ dim D.
By summing up all i we get
∑ i deg δ a i = ∑ i (dim D − deg δ ∨ a i ) ≥ ∑ j deg δ b j .
This is the same as deg D ( µ) ≥ deg D ( ν). A potential highest order term should satisfy
deg D ( µ) = deg D ( ν),
and then
µ i , δ ∨ a i Q ∨ i | [X(P ∞ )] k i · ∏ p j ∈C iτ ν j−1 ([X(P ∞ )] · δ b j R j ) (X(P),X(P 0 )) simplifies to D δ ∨ a i ∏ p j ∈C i δ b j (µ i , Q ∨ i ) | t k i ∏ p j ∈C iτ ν j−1 (tR j ) (P 1 , {0})×X(C)
. By Lemma 6.2.7, we see that
(µ i , δ a i Q i ) dominates {(ν j , δ b j R j )} {p j ∈C i } .
This implies that µ dominates ν. As
∑ µ i = ∑ ν j = β S D,
by Lemma 6.2.5, µ should be identified with ν.
In fact, if (ν j , deg δ b j , deg R j )) = (1, 0, 0), then p j must be distributed on C i for which (u i , δ a i Q i ) = (ν j , δ b j R j ). When (ν j , deg δ b j , deg R j ) = (1, 0, 0), p j can be distributed freely. By further taking into account of the non vanishing ofτ ν j −1 , we see that if k( ν) > 0, there is a unique configuration up to
Aut({(ν j , b j , R j }), whose 'coefficient' is determined by ∏ ν j >1 (ν j , 1) | t ν jτ ν j−1 (t) .
If k( ν) = 0, we have Id( ν) − ∑ ν j >1 ν j extra t insertions which can be distributed freely among C i 's. These are positive numbers by the lemmas in 6.4. Now we turn to type (3) terms. The Γ 1 factor of a type (3) term is of the form ω 1 | µ, γ 1 (X(P),X(P ∞ )) (β P , d)
, and the Γ 2 factor is of the form
µ, γ 2 | ω 2 · [X(P ∞ )] k · ∏ jτ * ([X(P ∞ )] · δ b j R j )) •(X(P),X(P 0 )) Γ 2 .
If C i has only two marked points with a non relative marked point p j with ν j > 1, then its contribution to the Γ 2 factor is
µ i , δ ∨ a i Q ∨ i | [X(P ∞ )] · δ b j R j ) (X(P),X(P ∞ )) .
As a fiber integral, it is nonzero only if
deg δ a i ≥ deg δ b j .
Together with the estimate for fiber integrals appearing in type (2) terms above, we see that
deg D µ ≥ deg D ν.
When deg D µ = deg D ν, we have deg δ a i = deg δ b j , and
µ i , δ ∨ a i Q ∨ i | [X(P ∞ )] · δ b j R j ) (X(P),X(P ∞ )) simplifies to D δ ∨ a i δ b j (µ i , Q ∨ i ) | tR j (P 1 , {0})×X(C) .
So δ a i = δ b j , and then by Lemma 6.1.2 it is nonzero if and only if
µ i = 1, Q i = R j .
Therefore the ancestor relative insertions in ω | µ, γ (X(P),X(P ∞ )) (β P , d)
are of the form (1,τ ν j −1 (δ b j R j )). Applying Proposition 6.2.6 and necessary dominance results for type (2) terms established above, we conclude the highest order term is ω | ν (X(P),X(P ∞ )) (β P , d)
. Its coefficients are again positive numbers. 6.5.4. Conclusion of type I. Finally we are ready to prove the F -invariance for type I relative generating functions on P, assuming the F -invariance on D.
Theorem 6.5.5. F -invariance for D implies F -invariance for (P, P 0 ) and (P, P ∞ ).
Proof. For fiber curve classes this is proved in Theorem 6.3.1, so we consider the case with non-fiber curve classes.
Assuming F -invariance for invariants without distinguished insertions, it is easy to prove invariance inductively on the number of distinguished insertions using Lemma 6.5.1.
For invariants without distinguished insertions, Proposition 6.5.4 allows us to perform induction using the partial ordering defined in Section 1.5.2, expressing a type I invariant in terms of an absolute invariant, type II invariants, rubber invariants and type I invariants of lower order. As Invariance for absolute invariants, type II and rubber invariants are shown, inductively the theorem is proved. Proof. Consider deformation to the normal cone for Z ֒→ S. The degeneration formula shows that
α X(S) (β S , d) = ∑ I ∑ η C η φ X * α 1 | µ, e I •(X(S),X(E)) Γ 1 · p X * µ, e I | α 2
•(X(P),X(E)) Γ 2 and F (α)
X ′ (S) (β S , d) = ∑ I ∑ η C η φ X ′ * F (α 1 ) | µ, F (e I ) •(X ′ (S), X ′ (E)) Γ 1 · p X ′ * µ, F (e I ) | F (α 2 ) •(X ′ (P), X ′ (E)) Γ 2 .
As φ X (ℓ) = p X (ℓ) = ℓ and φ X ′ (ℓ ′ ) = p X ′ (ℓ ′ ) = ℓ ′ , the analytic continuations for (S, E) and (P, E) are compatible with that for S. 7.2. Primaries imply ancestors. This is achieved by the topological recursion relations (TRR) for ancestors, which express the ψ-classes on genus zero moduli spaces of curves in terms of the boundary classes. Therefore, we have the following Lemma 7.2.1. F -invariance for all absolute (resp. relative) primary invariants with curve classes less than or equal to (β, d) implies invariance for absolute (resp. relative) ancestors with curve classes less than or equal to (β, d).
The explicit form of TRR is not important. In the absolute case, TRR is well known. We include some discussions on TRR for relative invariants in the appendix. Proof. Consider deformation to the normal cone for X(D) → X(S). By the degeneration formula we have
ω X(S) (0, 0) = ω (X(S), X(D)) (0, 0) + ω (X(P), X(D)) (0, 0) . By Proposition 3.2.2, ω (X(P), X(D)) (0, 0) = 0.
Therefor F -invariance for (S, D) with extreme curve classes follows from that for S which was proved in Part I [16].
Absolute implies relative. Recall D is a smooth divisor in S and
P = P(N ⊕ O)
is a P 1 bundle over D. It has two sections P 0 and P ∞ = P(N) = D. We assume F -invariance for P 1 bundles proved in previous sections.
Let ν = {(ν j , δ b j R j )} be a weighted partition and i X : X(D) → X(S) be the inclusion. Again k( ν) is the number defined in (1.5.2).
Proposition 7.4.1. Assume (β S , d) is non-extremal.
(1) If ν is non empty, then there exists a positive constant C( ν) such that
C( ν) ω | ν (X(S), X(D)) (β S , d) − ω · [X(D)] k( ν) · ∏ jτ ν j−1 (i X * (δ b j R j )) X(S) (β S , d)
is generated by generating functions on (X(S), X(D)) of lower order, and relative and rubber invariants on X(P).
(2) If ν is empty, then ω X(S) (β S , d) − ω | ν (X(S), X(D)) (β S , d)
is generated by generating functions on (X(S), X(D)) of lower order, and those of relative invariants on X(P).
Proof. For (1), consider the family W(X(S), X(D)) → A 1 and a generating function of a general fiber
ω · [X(D)] k( ν) · ∏ jτ ν j−1 (i X * (δ b j R j )) X(S) (β S , d)
.
We can lift the insertions X(D) and i X * (δ b j R j ) to X(P) in the singular fiber. The degeneration formula allows us to express this function in terms of the ancestor relative invariants of (X(S), X(D)) and (X(P), X(D)). For (2), apply the degeneration formula to ω
X(S) (β S , d) .
Analyzing the invariants involved in the degeneration formula by the arguments in the proof of Proposition 6.5.4, the proposition is proved.
F ω | ν (X(S), X(D)) (β S , d) = F (ω) | F ( ν) (X ′ (S), X ′ (D)) (β S , d) ,
where we define
F ( ν) = ν j , F (δ b j R j ) .
Proof. This can be proved inductively using Proposition 7.4.1. Using the argument for Proposition 7.4.1, we can show that
F (ω) · [X ′ (D)] k( ν) · ∏ jτ ν j−1 i X ′ * F (δ b j R j ) X ′ (S) (β S , d)
has a highest order term
F (ω) | F ( ν) (X ′ (S), X ′ (D)) (β S , d)
.
The lower order terms are F -invariant by induction. The theorem now follows from combining Theorems 5.3.1, 6.3.1 and 6.5.5. To prove F -invariance for (S, F, F ′ ), we reduce the general case to the case when F and F ′ admit complete flags, then by deformation invariance of F -invariance, to the split case.
Motivation of the refined induction.
Before we go to the actual proof, we would like to motivate the (refined) induction procedure by looking at some starting cases.
Let (S, F, F ′ ) be the triple defining the local model of an ordinary flop. If dim S = 0 this is the simple flop case and the F -invariance was proved in [15]. Indeed, if dim S = 1 then F and F ′ admit complete flags and the F -invariance is reduced to the split case proved in [17].
If dim S = 2, F and F ′ may not admit complete flags. We need to perform a sequence of blow-ups S i+1 → S i (S 0 = S) to achieve this property. Fortunately each blow-up has only points as its center and the normal bundles are all trivial. In particular the easier quantum Leray-Hirsch (Theorem 5.1.1) and the "F -invariance for P 1 bundles" established in the previous sections all apply and the F -invariance is again reduced to S i for i large where the pullbacks of F and F ′ admits complete flags and the proof is done.
Essentially the same argument applies to the case dim S = 3: Let T ⊂ S be the blow-up center. During the applications of degeneration formulas and deformation to the normal cone, the essential objects to take care are the normal bundles
N = N T/S and N T×{0}/(S×A 1 ) = N ⊕ O.
If dim T ≤ 1, then N is either trivial or deformable to split bundles. If dim T = 2, then N is already a line bundle. In all cases the easier quantum Leray-Hirsch and F -invariance for P 1 bundles apply and the proof is done without the need of any refinement to the blow-ups.
The situation changes when dim S = 4 and S 1 → S is the blow-up along T with dim T = 2. In this case N = N T/S may not admit complete flags anymore. The space P T (N) is a 3-fold whose F -invariance can be assumed by induction. However, P = P T (N ⊕ O) is also of the same dimension as S, with N ⊕ O being non-deformable to split bundles. In particular we are unable to deduce F -invariance for P from that for T (which is known by induction since dim T < dim S) via Theorem 5.1.1. Thus, an additional sequence of blow-ups on the surface T is indispensable in order for the proof to proceed.
Indeed, for each step of blow-up S i+1 → S i along some T i ⊂ S i , the normal bundle N i = N T i /S i has to be treated similarly. It is therefore more economic to use a refined induction presented in the following subsection.
8.2.
Proof of the main theorem by reduction to split bundles. Given a triple (T, G, G ′ ) and a finite set of vector bundles N 1 , N 2 , · · · , N k over T, we will use the notation (T, G, G ′ ; {N i })
to represent a triple (S, F, F ′ ), where S = P T (N 1 ) × T · · · × T P T (N k ) → T is the fiber product and F, F ′ are the pullbacks of G, G ′ from T to S. We introduce this notation to streamline the induction argument in Theorem 8.2.3. Proof. This is an immediate consequence of Lemma 1.8.1.
Let Y be a smooth subvariety of T and T = Bl Y T → T the blowing-up of T along Y. Denote byG,G ′ ,Ñ i the pullbacks of G, G ′ , N i toT respectively.
Lemma 8.2.2. F -invariance for
(T,G,G ′ ; {Ñ i }), (Y, G| Y , G ′ | Y ; {N i | Y } ∪ {N Y/T }), and (Y, G| Y , G ′ | Y ; {N i | Y } ∪ {N Y/T ⊕ O})
implies F -invariance for (T, G, G ′ ; {N i }) with non-extremal (β S , d).
Proof. The lemma follows from Proposition 7.1.1 and Theorem 7.4.2:
We have a natural projection
S = ∏ T P T (N i ) → T.
Let Z be the fiber product S × T Y, which is ∏ Y P(N i | Y ). Note that the normal bundle of Z in S is the pullback from Y if its normal bundle in T. Let S = Bl Z S → S be the blow-up of S along Z with exceptional divisor E. Recall Proposition 7.1.1 says that invariance for S follows from invariance for (S, E) and (P, E). By Theorem 7.4.2, invariance of (S, E) (resp. (P, E)) follows from those forS and E (resp.P and E). So invariance ofS, E and P implies invariance for S, this is exactly what the lemma claims. Proof. If (β S , d) = (0, 0), F -invariance was proved in Part I [16].
For non-extremal generating functions with curve classes (β S , d), we will prove F -invariance for all (T, G, G ′ ; {N i }) by induction on the dimension of T. We simply call this statement as "invariance for T".
Note that by Proposition 4.1.5, we can assume F -invariance holds for any triple with bundles admitting complete flags. When dim T ≤ 1, G, G ′ and N i 's all admit complete flags, so F -invariance holds for (T, G, G ′ ) and then it holds for (S, F, F ′ ) (i.e. (T, G, G ′ ; {N i })) by Theorem 5.1.1.
When dim T ≥ 2, for any Y a smooth subvariety of T, we may assume that invariance is proved for Y. Then by Lemma 8.2.2, invariance for T follows from that forT. Then by Lemma 8.2.1, after a finite number of blowing-ups, we are left to proving invariance for a triple with bundles admitting complete flags, which is guaranteed by our hypothesis.
Proof of Theorem 0.1.1. Since F -invariance of split bundles was proved in Part II [17], Theorem 8.2.3 implies Theorem 0.1.1.
Comments on algebraic cobordism of bundles on varieties.
A "dream proof" of Theorem 0.1.1 would be to apply the idea of the algebraic cobordism in [23,19]. Theorem 1 in [19] implies that, up to double point degeneration, any list of vector bundles is equivalent to a Q combination of split vector bundles on products of projective spaces, whose genus zero theory can be easily computed as toric varieties. Furthermore, as the quantum cohomology of a toric variety is semisimple, the higher genus theory can be deduced from genus zero theory by Givental's quantization formalism.
The major obstacle of this approach is the lack of "inversion of degeneration" for a general double point degeneration. Section 7.1 says that the Finvariance is preserved under a "forward degeneration". In Section 7.4 we established the inversion of degeneration for deformation to normal cones, which we will call "backward DNC" for convenience. Now, let us define a "directed" (non-reflexive) double point degeneration relation which allows forward degenerations and backward DNC.
Question 8.3.1.
Can results similar to [23,19] (in particular Theorem 1 in [19]) be obtained by "directed double point degeneration" above?
If so, the story will be much simpler. At this moment, we have no idea whether this is possible. Nonetheless, the forward degenerations and backward DNC can lead us a bit further than we have employed in this paper. For example, if dim S = 1, i.e. a curve, one sees immediately that only forward double point degenerations, which includes deformations, are needed to reduce the proof of F -invariance of (S, F, F ′ ) to to (P 1 , F, F ′ ), with F and F ′ being of the form ⊕O(k i ) and ⊕O(k ′ i ). If we assume results in Sections 2-5 can be generalized to higher genera, the above strategy will also apply to any genus. Therefore, we just need to establish the F -covariance for the absolute invariants on (P 1 , F, F ′ ) in all genera. However, since the local models built upon (P 1 , F, F ′ ) are toric, F -invariance of higher genera follows from that of genus zero by the strategy in [13].
For general S, the reduction in genus zero can also go further than we have used in Section 8.2. Tracing the arguments in Sections 3.1 and 2.5 in [19], one sees that the triple (S, F, F ′ ) can be reduced to the following two special types by deformations to the normal cones only:
(i) F, F ′ are direct sum of globally generated (bpf) line bundles; (ii) S = (P 1 ) ×n , and F = F ′ = O(l 1 , . . . , l r+1 ) ⊕(r+1) .
APPENDIX A. TRR FOR RELATIVE ANCESTORS
For Γ = (0, [n + m], β, µ), let K Γ (X, D) be Kim's moduli stack of relative stable maps from genus 0, n + m marked curve to (X, D), with curve class β, relative profile µ; n is the number of internal (or non-relative) marked points, and m = l(µ). When n + m ≥ 3, consider ρ Γ : K Γ (X, D) → M 0,n+m , which maps a relative stable map to the stabilization of the source curve.
For a partition of [n + m] into a disjoint union A ⊔ B, denote by D A|B : M 0,A∪{• A } × M 0,B∪{• B } → M 0,n+m the map gluing • A and • B . By abusing notation, we will also use D A|B to represent the corresponding divisor of M n+m . We will use α, δ to represent cohomology classes in H * (X), H * (D) respectively, adding subscripts to number different classes, and superscript ∨ to represent a dual class with respect to some chosen basis. We use the notation for the virtual cycles
n ∏ j=1 α j | m ∏ i=1 δ i (X,D) Γ := [K Γ (X, D)] vir ∩ n ∏ j=1 ev * j (α j ) m ∏ i=1 ev * i (δ i ).
For cycles on a moduli stack to a rubber target, we will use the same notation except adding a superscript ∼. We might add a superscript • to emphasis that the source curve is possibly disconnected. For even classes {α j } 1≤j≤n , {δ i } 1≤i≤m . We have )'s such that β A + β B = β, and µ is the disjoint union of µ A and µ B . The summation ∑ α runs over a basis of H * (X). η = (Ξ, Ξ ′ ) is a splitting of Γ into two modular graphs with necessary compatibility conditions. See [3,Definition 4.8.1,5.1.1]. Ω is the set of all possible splittings. M is a labeling of the set of roots for Ξ and Ξ ′ . (i.e. the set of marked points mapped into the divisor D or P ∞ ). N (reps. N ′ ) is a labeling of the set of legs of the modular graph Ξ (reps. Ξ ′ ) (i.e. the set of the remaining marked points). m(η) is the product of the contact orders of the roots in M. By the compatibility of η and Γ, there is a root • Ξ in Ξ corresponding to a root • Ξ ′ in Ξ ′ such that if we glue all the roots but • Ξ and • Ξ ′ , we get a disconnect curve in M 0,A∪{• A } × M 0,B∪{• B } . c(η) is the contact order for the root • Ξ or • Ξ ′ . ρ (Ξ,Ξ ′ ) is the map
ρ Γ * n ∏ j=1 α j | m ∏ i=1 δ i (X,D) Γ ∩ D A|B = ∑ (Γ A ,Γ B ) ∑ α (D A|B ) * ρ Γ A * ∏ j∈A α j · α | ∏ i∈A δ i (X,D) Γ A × ρ Γ B * ∏ j∈B α j · α ∨ | ∏K Ξ (X, D) × K Ξ ′ (P, P 0 , P ∞ ) ∼ → ∏ v∈V(Ξ) M g(v),n(v) × ∏ v∈V(Ξ ′ ) M g(v),n(v) → M 0,n+m .
The first arrow is mapped to the source curve; the second arrow is the gluing of all the roots.
Using these it is straightforward to deduce a TRR using
ψ i = ∑ i∈A, j,k∈B D A|B ∈ A * (M 0,n+m )
Here i, j, k are three indexes in [n + m]. These are the ancestors we consider in the paper.
of a P 1 bundle in terms of those of the base 22 4. Deformation invariance of F -invariance property 29 5. F -invariance for P 1 bundles: absolute, type II and rubbers 31 6. F -invariance for P 1 bundles: type I invariants 34 7. F -invariance between relative and absolute invariants 47 8. Conclusion of the proof of Theorem 0.
Proposition 1.1.1 ([16, Proposition 3.3]). To prove Theorem 0.1.1, it is enough to prove the corresponding statements for the local models.
form a basis of H * (X(S)) by Leray-Hirsch theorem. Here T i , V m are viewed as classes on X(S) via X(S) → S and X(S) → [X(C)/G] respectively. Note that the classes "h, ξ, H j and Θ k " are indeed defined on [X(C)/G], and their pullbacks are the classes h, ξ, H j and Θ k in H * (X(S)).
also finite. Thus in any descending chain, (β S , d) is stabilized in finite steps. Similarly the number of insertions ||α|| stabilizes in finite steps.
where PD stands for the Poincaré duality and st and ev are defined in (1.2.3) and (1.2.1).
Remark 1.7.2. (i) In this paper, only the special case with g = 0 and (Y, D) = (P 1 , {pt}) is used.
the perfect obstruction theories for the vertical arrows are compatible, and the bottom arrow is projective of degree 1. Thereforē y * ([M 0,n (E ′ † , β)] vir = [M 0,n (E † , β)] vir by [26, Theorem 4.1 (3)].
/
, we embed X into a homogeneous variety. Choose two line bundles M and L on X such that both M and L ⊗ M are very ample. These line bundles induce an embedding i : X → P |M| × P |L⊗M| such that L is the pullback of O(−1, 1) on P |M| × P |L⊗M| . Then we have / P |M| × P |L⊗M| , which induces a cartesian diagram between log stacks
(cf. [ 26 ,
26Propsition 5.4 (ii)].) As P |M| × P |L⊗M| is homogeneous, E¯i is perfect in [−1, 0]. This implies there exists a virtual pullbackī ! such that i ! [M 0,n (P |M| × P |L⊗M| , ( θ M, θ L ⊗ M))] vir = [M 0,n (X, θ)] vir andī ! [M 0,n (P(O(−1, 1) ⊕ O); µ, ν)] vir = [M 0,n (Y; µ, ν)] vir .
Remark 3.2.3. Using localization, one can show similar vanishing results.
−1 ]] by dimension count. Since pushforward of equivariant class should be an equivariant class, i.e. in A * (M g,n (X, γ))[[t]], it must vanish.
Lemma 3 .2. 4 .
34When the partitions µ = (d), ν = (d) are totally ramified and β a fiber class, M 0,2 (Y; (d), (d)) is isomorphic to the root stack d √ L/X ([4, Appendix B.1])
Remark 3.3. 3 .
3One may use a variant of Bumsig Kim's log moduli stack, adapted to rubber targets in[29], to check compatibilities of perfect obstruction theories in Proposition 3.3.2. The log moduli stack of Kim is the saturation of the log moduli stack of Jun Li. See[10, Section 6].
Chow group A * (BT) is determined by {A * (∏ P n )} n≥1 .
Q ∨ , R, 1, · · · , 1 m X(C) nℓ is a nonzero top dimensional cycle on M 0,m+2 if and only if n = 0 and Q = R.
Lemma 6 .1. 2 .
62For any c > 0, n ≥ 0 integers, consider the two-pointed relative invariant on (P 1 , {0}) × X(C) (c, Q ∨ ) | tR (P 1 , {0})×X(C) c[P 1 ]+nℓ with curve class d[P 1 ] + nℓ, where [P 1 ] the fundamental class of P 1 , and t = c 1 (O P 1 (1)). Then this invariant is nonzero if and only if c = 1, n = 0 and Q = R.Proof. Using the divisor equation, we have
Proposition 6 .2. 6 .
66Let ω | γ (X(S),X(D)) (β S , d) be the relative generating functions, with γ specified by (6.2.1).This type of relative invariants ω | γ (X(S),X(D)) (β S , d)
Lemma 6 .4. 1 .
61For relative invariants of (P 1 , {∞}), we have(c, 1) | t cτ c−1 (t) c[P 1 ] = 1.Proof. Let M 0, c+2 (P 1 , {∞}; (c, 1)) be Kim's moduli stack of log maps of degree c with one fully ramified marked point. Consider the map M 0,c+2 (P 1 , {∞}; (c, 1))
then such a map looks like f : [x, y] → c ∑ i=0 a i x i y c−i , y c . The constraints f (x i ) = p i determine the coefficients {a i } uniquely.) Lemma 6.4.2.
7. F -INVARIANCE BETWEEN RELATIVE AND ABSOLUTE INVARIANTS 7.1. Relative implies absolute. We use the notations introduced in Section 1.6.1. Proposition 7.1.1. F -invariance for the pair (S, E) and (P, E) implies the Finvariance for S with non-extremal (β S , d).
7. 3 .
3Relative invariants associated to extremal rays. In this subsection we show the F -invariance for (S, D) with extremal curve classes (β S , d) = (0, 0). Proposition 7.3.1. The relative generating functions of (S, D) with (β S , d) = (0, 0) are F -invariant.
Theorem 7.4.2. Let (S, D) be a smooth pair. If F -invariance holds for S and D, then it holds for (S, D) with non-extremal curve classes:
8. CONCLUSION OF THE PROOF OF THEOREM 0.1.1
Lemma 8.2. 1 .
1(T, G, G ′ ; {N i }) becomes a triple with bundles admitting complete flags after a sequence of blowing-ups.
Theorem 8.2.3. F -invariance holds for all (T, G, G ′ ; {N i }) provided it holds for any triple with split vector bundles.
Γ
A = (0, A ∪ {• A }, β A , µ A ) and Γ B = (0, B ∪ {• B }, β B , µ B ),then • A and • B denote internal marked points. The summation ∑ (Γ A ,Γ B ) are over those (Γ A , Γ B
the log stack of stable log maps to E † over the category of log schemes over B with the prescribed discrete invariants. It can also be viewed as the log stack of stable maps to the family E † → B over the category of log schemes. M 0,n ([F † /G]/BG, β) is defined similarly.Proposition 2.1.3.
the perfect obstruction theories for the vertical arrows are compatible. By [26, Theorem 4.1 (3)], virtual pullbacks commute with flat pullbacks, the proposition is proved.When B is a point, from the diagram
Corollary 2.1.5 (cf. [28, Equation (4))]). Let A * be the operational Chow ring. For any non-negative integers k i , and classes δ
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E-mail address: [email protected]. Y.-P Lee: Department Of, Mathematics, University, Salt Utah, U S A Lake City, ; H.-W. Lin: Department Of Mathematics And Taida Institute Of Mathemati-Cal Sciences ( Tims), National Taiwan University, TAIWAN E-mail address: [email protected] F. QU: DEPARTMENT OF MATHEMATICSCENTER FOR ADVANCED STUDIES IN THEORETICAL SCIENCES (CASTS), AND TAIDA INSTITUTE OF MATHEMATICAL SCIENCES (TIMS). UNIVERSITY OF UTAH, SALT LAKE CITY, UTAHNATIONAL TAIWAN UNIVERSITY84112-0090, U.S.A. E-mail address: [email protected] C.-L. WANG: DEPARTMENT OF MATHEMATICS. TAIPEI 10617, TAIWAN E-mail address: [email protected]. LEE: DEPARTMENT OF MATHEMATICS, UNIVERSITY OF UTAH, SALT LAKE CITY, UTAH 84112-0090, U.S.A. E-mail address: [email protected] H.-W. LIN: DEPARTMENT OF MATHEMATICS AND TAIDA INSTITUTE OF MATHEMATI- CAL SCIENCES (TIMS), NATIONAL TAIWAN UNIVERSITY, TAIPEI 10617, TAIWAN E-mail address: [email protected] F. QU: DEPARTMENT OF MATHEMATICS, UNIVERSITY OF UTAH, SALT LAKE CITY, UTAH 84112-0090, U.S.A. E-mail address: [email protected] C.-L. WANG: DEPARTMENT OF MATHEMATICS, CENTER FOR ADVANCED STUDIES IN THEORETICAL SCIENCES (CASTS), AND TAIDA INSTITUTE OF MATHEMATICAL SCIENCES (TIMS), NATIONAL TAIWAN UNIVERSITY, TAIPEI 10617, TAIWAN E-mail address: [email protected]
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[
"Discovery of Drifting High-frequency QPOs in Global Simulations of Magnetic Boundary Layers",
"Discovery of Drifting High-frequency QPOs in Global Simulations of Magnetic Boundary Layers"
]
| [
"M M Romanova \nDept. of Astronomy\nCornell University\n14853IthacaNY\n",
"A K Kulkarni \nDept. of Astronomy\nCornell University\n14853IthacaNY\n"
]
| [
"Dept. of Astronomy\nCornell University\n14853IthacaNY",
"Dept. of Astronomy\nCornell University\n14853IthacaNY"
]
| [
"Mon. Not. R. Astron. Soc"
]
| We report on the numerical discovery of quasi-periodic oscillations (QPOs) associated with accretion through a non-axisymmetric magnetic boundary layer in the unstable regime, when two ordered equatorial streams form and rotate synchronously at approximately the angular velocity of the inner disk The streams hit the star's surface producing hot spots. Rotation of the spots leads to high-frequency QPOs. We performed a number of simulation runs for different magnetospheric sizes from small to tiny, and observed a definite correlation between the inner disk radius and the QPO frequency: the frequency is higher when the magnetosphere is smaller. In the stable regime a small magnetosphere forms and accretion through the usual funnel streams is observed, and the frequency of the star is expected to dominate the lightcurve. We performed exploratory investigations of the case in which the magnetosphere becomes negligibly small and the disk interacts with the star through an equatorial belt. We also performed investigation of somewhat larger magnetospheres where one or two ordered tongues may dominate over other chaotic tongues. In application to millisecond pulsars we obtain QPO frequencies in the range of 350 Hz to 990 Hz for one spot. The frequency associated with rotation of one spot may dominate if spots are not identical or antipodal. If the spots are similar and antipodal then the frequencies are twice as high. We show that variation of the accretion rate leads to drift of the QPO peak. | 10.1111/j.1365-2966.2009.15175.x | [
"https://arxiv.org/pdf/0812.0384v4.pdf"
]
| 15,606,059 | 0812.0384 | f8308dc632fce5337d2a6a68550e1dde624f45d4 |
Discovery of Drifting High-frequency QPOs in Global Simulations of Magnetic Boundary Layers
2 June 2009 2 June 2009
M M Romanova
Dept. of Astronomy
Cornell University
14853IthacaNY
A K Kulkarni
Dept. of Astronomy
Cornell University
14853IthacaNY
Discovery of Drifting High-frequency QPOs in Global Simulations of Magnetic Boundary Layers
Mon. Not. R. Astron. Soc
00000002 June 2009 2 June 2009Printed (MN L A T E X style file v2.2)accretion, dipole -plasmas -magnetic fields -stars
We report on the numerical discovery of quasi-periodic oscillations (QPOs) associated with accretion through a non-axisymmetric magnetic boundary layer in the unstable regime, when two ordered equatorial streams form and rotate synchronously at approximately the angular velocity of the inner disk The streams hit the star's surface producing hot spots. Rotation of the spots leads to high-frequency QPOs. We performed a number of simulation runs for different magnetospheric sizes from small to tiny, and observed a definite correlation between the inner disk radius and the QPO frequency: the frequency is higher when the magnetosphere is smaller. In the stable regime a small magnetosphere forms and accretion through the usual funnel streams is observed, and the frequency of the star is expected to dominate the lightcurve. We performed exploratory investigations of the case in which the magnetosphere becomes negligibly small and the disk interacts with the star through an equatorial belt. We also performed investigation of somewhat larger magnetospheres where one or two ordered tongues may dominate over other chaotic tongues. In application to millisecond pulsars we obtain QPO frequencies in the range of 350 Hz to 990 Hz for one spot. The frequency associated with rotation of one spot may dominate if spots are not identical or antipodal. If the spots are similar and antipodal then the frequencies are twice as high. We show that variation of the accretion rate leads to drift of the QPO peak.
INTRODUCTION
In disk-accreting magnetized stars, disk-magnetosphere interaction and channeling of matter to the magnetic poles may determine the majority of observational features of such stars (e.g., Ghosh & Lamb 1979;Königl 1991). These include young solar-type stars (e.g., Bouvier et al. 2007) and very old stars such as magnetized white dwarfs (Warner 1995) or accreting millisecond pulsars (van der Klis 2000). In many cases no clear period is observed, and high-frequency quasiperiodic oscillations (QPOs) with a frequency corresponding to a fraction of the Keplerian frequency at the stellar surface are seen instead. It is often the case that a second, lower frequency peak appears and drifts in concert with the main frequency, forming two-peak QPOs. Such high-frequency oscillations are observed in both accreting millisecond pulsars (e.g., and in dwarf novae, which are a sub-class of cataclysmic variables E-mail: [email protected] † E-mail: [email protected] Warner & Woudt 2006). It has been suggested that the highfrequency QPOs may be associated with accretion to a star with a very weak magnetic field, when some sort of weakly magnetized equatorial belt forms (Paczyński 1978;. Miller et al. (1998) and Lamb & Miller (2001) suggested that the high-frequency QPOs in accreting millisecond pulsars may result from rotation of clumps in the inner disk around a weakly magnetized star. They suggested that some of the disk matter may penetrate through the magnetosphere until stopped, e.g., by the radiation pressure. In this paper we investigate accretion to stars with small magnetospheres, where the energy density of the disk matter dominates up to very small distances from the star, or even up to the stellar surface, while magnetospheres are small compared to the radius of the star. We call the region around the star with a small magnetosphere the Magnetic Boundary Layer (MBL), and consider different cases. We notice that as for large magnetospheres, matter may accrete either in the stable (e.g., Romanova et al. 2003, 2004, hereafter RUKL03 andRUKL04) or unstable regime (Kulkarni & Romanova 2008a,b, hereafter KR08a,b; , hereafter RKL08).
The most striking result is the fact that in the unstable regime, an ordered structure of two tongues forms, and rotates with the angular velocity of the inner disk, showing clear highfrequency QPO peaks associated with this rotation 1 . Unstable accretion with a preferred number of tongues as a possible origin for QPOs has been discussed earlier by Li & Narayan (2004). In the present research we have shown that an even more ordered situation is possible with small magnetospheres, where two ordered tongues rotate with the frequency of the inner disk. Variation of the accretion rate leads to drifting of this QPO frequency. Correlation between the frequency and accretion rate has been observed in a number of accreting millisecond pulsars (e.g., van der Klis 2000). Below we describe this and other results in detail. In §2 we discuss a possible classification scheme for magnetic boundary layers. In §3 we describe our model and reference parameters. In §4 we consider MBLs during stable accretion. In §5 we describe MBLs and formation of QPOs in the unstable regime. We present some discussion and conclusions in §6.
CLASSIFICATION OF BOUNDARY LAYERS
In accreting magnetized stars the disk is stopped by the stellar magnetosphere at the magnetospheric, or truncation, radius (e.g., Elsner & Lamb 1977):
rm = kA(GM * ) −1/7Ṁ −2/7 µ 4/7 ,
where M * and µ are the mass and magnetic moment of the star,Ṁ is the accretion rate through the disk, and the coefficient kA is of the order of unity (e.g., Long et al. 2005;Bessolaz et al. 2008). The magnetospheric radius may be small if the magnetic moment of the star µ is small, or if the accretion rate is high. At a given magnetic moment of the star, variation of the accretion rate will lead to expansion or compression of the magnetosphere, and so at very high accretion rates the magnetosphere may be completely buried by accreting matter (e.g., Cumming et al. 2001;Lovelace et al. 2005), while at very low accretion rates it strongly expands, and different types of accretion are expected at differentṀ (e.g., ). An important parameter of the problem is the size of the magnetosphere in units of the stellar radius, rm/R * . Depending on this parameter we can define three types of boundary layers:
(1). Hydrodynamic BL. If the ratio rm/R * << 1, then the magnetic field does not influence the dynamics of matter flow in the vicinity of the star, and a purely hydrodynamic boundary layer is expected where the magnetic field can be neglected. In this case the disk matter comes to the surface of the star and interacts with the star in the equatorial zone (e.g., Lynden-Bell & Pringle 1974;Pringle 1981;Popham & Narayan 1995). Such interaction leads to release of a significant amount of energy at the surface of the star, about a half of the gravitational energy,LBL ≈ 0.5GM * Ṁ /R * . Interaction through such a hydrodynamic boundary layer is expected in many non-magnetic white dwarfs and neutron stars and in cases where accretion rate is so high that the magnetic field is buried by the accreting matter. It has been noted that the boundary layer matter will not interact with the star as a thin 1 See animations at http://wwww.astro.cornell.edu/∼romanova/qpo.htm layer, but will spread to higher latitudes along the surface of the star due to matter pressure (Ferland et al. 1982). This effect has been calculated by Inogamov & Sunyaev (1999) and Piro & Bildsten (2004) and has recently been observed in axisymmetric hydrodynamic simulations by Fisker & Balsara (2005). Much less work has been done for analysis of the magnetic boundary layers (MBL).
(2). Magnetic Boundary Layer (MBL) With Magnetospheric Gap. If the magnetosphere is only slightly larger than the radius of the star, rm/R * > 1, then it stops the disk at a small distance from the star. The simulations described in this paper show that in this case, accretion may be either in the stable regime producing funnel streams, or in the unstable regime where most of the matter accretes through instabilities (KR08a; RKL08). We find that an unusual type of MBL forms in the unstable regime with two ordered streams rotating in the equatorial plane at approximately the angular velocity of the disk. These streams may be responsible for high-frequency QPOs.
(3). MBL Without a Magnetospheric Gap. If the magnetospheric radius is comparable to the stellar radius, rm/R * ≈ 1, then the magnetosphere cannot stop the disk, and the disk interacts with the stellar surface. In this case the magnetic field may be somewhat enhanced in the accretion disk due to wrapping of the magnetic field lines by the disk matter. Preliminary simulations have shown that interesting phenomena may happen at the disk-magnetosphere boundary, such as interaction through a Kelvin-Helmholtz-type instability.
Global 3D simulations of accretion to stars with large magnetospheres (a few stellar radii in size) have been performed earlier (e.g., RUKL03,04; KR05). It was recently found that stars may be either in the stable or unstable regime of accretion (KR08a,b; RKL08). In this paper we show results of global 3D simulations of accretion to star with very small magnetospheres, in both stable and unstable regimes, and some results for direct interaction between the disk and the star.
MODEL
We solve the 3D MHD equations presented, e.g., in RUKL03, using the "cubed sphere" second order Godunov-type code described in Koldoba et al. (2002). The equations are written in a coordinate system rotating with a star. The energy equation is solved for the entropy, and the equation of state is that for an ideal gas. Viscosity is incorporated into the code with a viscosity coefficient proportional to α (Shakura & Sunyaev 1973), where α < 1. The action of the viscosity is restricted to regions of high density (the disk). Viscosity helps bring matter towards the star from the disk.
Dimensionalization and Reference Values
The MHD equations are solved in dimensionless form so that the results can be readily applied to different types of stars. We have the freedom to choose three parameters from which the reference scales are derived, and we choose those to be the star's mass M , radius R * and magnetic moment µ * (with corresponding magnetic field B * = µ * /R 3 * ). We take the reference mass M0 to be the mass M of the star. The reference radius is about three times the radius of the star, R0 = R * /0.35 ≈ 3 × R * . In our past research (RUKL03; RUKL04) this radius approximately corresponded to the truncation (or magnetospheric) radius. In the present research the truncation radius is much smaller, but we keep this unit for consistency with our prior work. The reference velocity is v0 = (GM/R0) 1/2 . The reference time-scale t0 = R0/v0, and the reference angular velocity ω0 = 1/t0. We measure time in units of P0 = 2πt0 (which is the Keplerian rotation period at r = R0). In the plots we use the dimensionless time T = t/P0.
The reference magnetic moment µ0 = µ * /e µ, where e µ is a dimensionless parameter which determines the dimensionless size of the magnetosphere. The reference magnetic field is B0 = µ0/R 3 0 . The reference density is ρ0 = B 2 0 /v 2 0 . The reference accretion rate isṀ0 = ρ0v0R 2 0 . Taking into account relationships for ρ0, B0, v0 we obtainṀ0 = µ 2 * /(e µ 2 v0R 4 0 ). The dimensional accretion rate isṀ = ḟ MṀ0, where ḟ M is the dimensionless accretion rate onto the surface of the star which is obtained from the simulations. Substituting parameters of the star to the reference values, we obtain the dimensional accretion rate:
M = ḟ M e µ 2 ! µ 2 * (GM ) 1/2 (R * /0.35) 7/2 .(1)
One can see that at fixed parameters of the star (M, R * , µ * ), the accretion rate depends on the ratio ḟ M /e µ 2 . Further analysis shows that parameter e µ 2 varies more strongly than ḟ M and determines the variation of the accretion rate. In the paper we vary parameter e µ which leads to variation of the dimensional accretion rate. Table 1 shows examples of reference variables for different stars. We solve the MHD equations using the normalized variables e ρ = ρ/ρ0, e v = v/v0, e B = B/B0, etc. Most of the plots show the normalized variables (with the tildes implicit). To obtain dimensional values one needs to multiply values from the plots by the corresponding reference values from Table 1.
Initial and Boundary Conditions
Here we briefly summarize the initial and boundary conditions, described in detail in our earlier papers (e.g. in RUKL03). The simulation region consists of the accretion disk, the corona and the star. The star has a dipole magnetic field which is frozen into its surface. The disk is cold and dense. The corona is 100 times hotter and 100 times less dense (at the fiducial point at the inner edge of the disk). Initially we rotate the disk with Keplerian velocity and the star with the angular velocity of the inner disk, so as to avoid discontinuity of the magnetic field lines at the disk-star boundary. We also rotate the corona above the disk with the Keplerian velocity of the disk so as to avoid discontinuity at the disk-corona boundary. This is necessary since the discontinuity may lead to artificially strong forces on the disk and strong magnetic braking. In addition, we derive such a distribution of the density and pressure in the disk and corona, that the sum of the gravitational, pressure and centrifugal forces is equal to zero everywhere in the simulation region . This initial condition ensures a very smooth gradual start-up, where the disk matter accretes slowly inwards due to viscous stresses. On the other hand, this condition dictates a density distribution that does not correspond to that in an equilibrium viscous disk (e.g. Shakura & Sunyaev 1973). That is why our disk starts reconstructing itself on the viscous time-scale. Fortunately, matter from the inner parts of the disk flows inward for a long time, and the matter flux increases with α, though dependence is not linear (as one would expect from the theoretical prediction for viscous disks). In addition, interaction of the disk with the external magnetosphere leads to some magnetic braking, leading to an enhancement of the accretion rate which may be smaller or larger than the effect of the viscosity and also to oscillations of the matter flux. That is why we cannot use the dimensionless accretion rate ḟ M as a parameter of the problem. Instead, we choose e µ as the main parameter which is responsible for the size of the magnetosphere. Equation (1) shows that the dimensional accretion rate onto the star is regulated by the combination ḟ M /e µ 2 . In this paper we consider small magnetospheres with e µ ∼ 0.08 − 0.5, which is about 10 times smaller than in our past research (RUKL03; RUKL04; KR05).
The size of the simulation region is Rmax = 15R0 ≈ 45R * . We initially place the inner radius of the disk at a distance r d from the star which is either 1.2R0 ≈ 3.6R * (in stable cases) or 1.8R0 ≈ 4.8R * (in the unstable cases). Simulations show that the result does not depend on r d , because in both cases the disk matter moves inward and is stopped by the magnetosphere at small distances from the star. Then we have the freedom to spin the star up or down gradually. In this paper we choose a slowly rotating star with angular velocity ω * = (GM/rcor) 1/2 corresponding to the corotation radius rcor = 2R0. In dimensionless units, f ω * = 0.354, and in application to neutron stars, P * = 6.2ms.
The boundary conditions are similar to those used in our other papers (e.g., RUKL03). On the star (the inner boundary) the magnetic field is frozen into the surface of the star (the normal component of the field, Bn, is fixed), though all three magnetic field components may vary. For all other variables A, free conditions are prescribed: ∂A/∂n = 0. The total velocity vector is adjusted to be parallel to the total magnetic field to enhance the frozen-in condition. At the external boundary, r = Rout, again free boundary conditions are prescribed for all variables. In some cases we fix the density on the external boundary so that matter of the disk stays in the region during the viscous reconstruction mentioned above. We do not supply matter to the disk from the external boundary, because the amount of matter in the inner regions of the disk is sufficient to support accretion over the duration of our simulations.
The grid. Simulations of accretion to stars with small magnetospheres require a finer grid compared to the larger magnetosphere cases (e.g. RUKL03). We use a grid resolution of Nr × N 2 = 100 × 41 2 (in each of the 6 blocks of the cubed sphere) for most of our cases, and Nr × N 2 = 130 × 61 2 for the smallest magnetospheres. The simulations were done using 60-180 processors each on the NASA HPC clusters.
ACCRETION IN THE STABLE REGIME
Below we describe the results of simulations of stable accretion to stars with small magnetospheres e µ = 0.08 − 0.2.
CTTSs
White dwarfs Neutron stars
M (M ) 0.8 1 1.4 R * 2R 5000 km 10 km B * (G)
10 3 10 6 3 × 10 8 R 0 (cm) 4 × 10 11 1.4 × 10 9 2.9 × 10 6 v 0 (cm s −1 )
1.6 × 10 7 3 × 10 8 8.1 × 10 9 ω 0 (s −1 ) 4 × 10 −5 0.21 2.8 × 10 3 ν 0 0.55 day −1 3.2 × 10 −2 Hz 4.5 × 10 2 Hz P 0 1.8 days 29 s 2.2 ms Table 1. Sample reference values of the dynamical quantities for different types of stars. We choose the mass, radius and magnetic field of the star, and define the other variables in terms of these three quantities. Note that the frequency ν 0 is the Keplerian frequency at radius R 0 . Simulations show that the boundary between the stable and unstable regimes depends on a number of factors, such as the accretion rate (determined by the viscosity parameter α), the rotation period of the star P * , and the misalignment angle Θ of the dipole (KR08a,b). If a star with a small misalignment angle, Θ < 5 • , rotates slowly (the disk rotates much faster than the star), then the accretion has a tendency to be unstable irrespective of α (KR08a,b). We have chosen slowly rotating stars with dimensionless angular velocity widetildeω * = 0.354 (corresponding to a corotation radius of rcor = 2). We take slowly rotating stars in order to model the situation in which the star initially has a large magnetosphere and is in the rotational equilibrium state, but later the accretion rate increases and the magnetosphere becomes small. To stabilize accretion we chose Θ = 15 • . We also used α = 0.04. In the description of the results we measure time in units of the Keplerian rotation period at r = 1. We consider two main cases, one with e µ = 0.2 which has a small magnetosphere, and the other with e µ = 0.08 which has a tiny magnetosphere. Fig. 1 shows an example of accretion to a star with e µ = 0.2. A small magnetosphere forms, in which the magnetic field of the dipole dominates. The position of the inner disk edge is determined by the balance between the magnetic and kinetic matter pressure. Matter is lifted above the magnetosphere forming two funnel streams, and accretes to the surface of the star forming two hot spots. The hot spots show the distribution of the total energy flux on the surface of the star (see RUKL04 and KR05 for details).
The hot spots have a preferred position, although in case of small misalignment angles the funnel streams are dragged by the rapidly rotating disk and may rotate faster than the star. Such spot rotation has been observed in earlier simulations (RUKL03; Romanova et al. 2006). In the opposite situation when the star spins fast, both funnels and spots may rotate slower than the star (e.g. Romanova et al. 2002). Faster or slower rotation of spots may lead to a QPO feature with frequency higher or lower than the stellar frequency (Romanova et al. 2006). It has been suggested that at small misalignment angles, the spots may wander around the magnetic poles, possibly causing intermittency in some millisecond pulsars Lamb et al. (2008a,b) (see also Casella et al. 2008)). On the other hand, if the misalignment angle of the dipole is not very small, then such faster or slower spot motions become less significant. The spots acquire a preferred place on the surface of the star and corotate with the star (see also RUKL03; RUKL04; KR05). In simulations with Θ = 15 • we observe a mixture of spots: on the one hand, the disk drags the funnel around the weak magnetosphere in spite of the relatively high misalignment angle, and frame to frame analysis shows that the spots often move faster than the star. On the other hand, the hot spots spend a somewhat longer time in the µ − Ω plane compared with other positions, due to which we observe a definite peak associated with star's spin in the Fourier spectrum. Longer simulation runs are probably required to see the QPO peak. Fig. 2 shows an example of accretion to a star with a tiny magnetosphere, e µ = 0.08. It is amazing that in this case too, the magnetosphere channels the accreting matter, forming tiny funnel streams hitting the surface of the star. Density waves form in the equatorial plane of the disk. Sometimes density fluctuations in the accreting matter push the disk to the stellar surface. However, later the magnetosphere is restored again. The hot spots have a preferred position but they often rotate faster than the star as a result of the difference in angular velocities between the disk and the star. In this case again, a QPO at the stellar frequency is expected, and also a QPO corresponding to the frequency at the inner edge of the disk and beats between these two frequencies. Longer simulation runs are required to extract these frequencies.
The main result of this section is that even small and tiny magnetospheres disrupt the disk and channel matter to the star, forming hot spots. It is also important to note that the high-frequency QPO associated with rotation of the inner edge of the disk is expected if the disk rotates faster (slower) than the star. Signs of faster rotation of the spot are clearly observed. Drifting of the high-frequency QPO is expected if the inner edge of the disk varies as a result of variation of the accretion rate. Future (longer) simulations will help obtain different frequencies.
ACCRETION IN THE UNSTABLE REGIME: A NEW TYPE OF BOUNDARY LAYER AND QPOS
Formation of two symmetric streams and spots
Now we take a star with a small misalignment angle, Θ = 5 • , and investigate accretion in the unstable regime. First we choose a small magnetosphere with magnetic moment e µ = 0.2. We take the viscosity coefficient α = 0.1 in all runs below. A star rotates slowly with angular velocity f ω * = 0.354 (corresponding to rcor = 2). We observe that the instability starts, and matter penetrates through the equatorial region of the magnetosphere through a number of tongues. Later, however, two symmetric ordered tongues form and rotate synchronously with angular velocity approximately equal to the angular velocity in the inner region of the accretion disk, that is, much faster than the star. These tongues reach the surface of the star and produce two hot spots at the star's equator which move faster than the star. The tongues deposit a significant amount of energy to the surface of the star. Part of this is the gravitational energy associated with acceleration of matter towards the star. We take this energy into account while plotting hot spots. In addition, there is energy released due to friction between the surface of the slowly rotating star and the foot-points of the rapidly rotating tongue. Fig. 3 shows snapshots of rotation of the tongues, slices in the equatorial plane and hot spots (energy flux) on the star's surface. The slices show that the magnetosphere has a strongly modified shape, particularly in those places where the tongues reach the surface of the star. They push the magnetosphere to the surface of the star and when the tongue moves, the magnetosphere re-emerges, while the tongue pushes another part of the magnetosphere to the surface of the star. Two strong hot spots form close to equatorial region. Sometimes they are not symmetric because the tongues push the magnetosphere more to one side than another. Some matter accretes to the poles through weak funnel streams. It is interesting that the polar spots also rotate with the angular velocity of the tongues. Thus, in the case of a small magnetosphere we observe a new phenomenon -a modified boundary layer, where the funnel streams and hot spots move with the angular velocity of the disk -much faster than the star. This is a new type of boundary layer where matter of the disk interacts with the star through unusual symmetric tongues. Rotation of the hot spots along the surface of the star is expected to give strong QPO peaks at twice the frequency of the inner disk.
To check this phenomenon, we performed a set of simulation runs at a variety of stellar magnetic moments: e µ = 0.16, e µ = 0.15, e µ = 0.12, e µ = 0.1, keeping all the other parameters fixed. We observed that similar symmetric streams form and rotate around the star. However, at smaller e µ the disk stops closer to the star. In the case of e µ = 0.1, only a tiny magnetosphere forms with tiny streams (see Fig. 4). The bottom panels of the Fig. 4 show the density distribution in the hot spots instead of the emitted flux, because it is expected that energy would be released mainly due to friction between the tongues and the surface of the star, which we do not calculate in this paper. In spite of the fact that the tongues are very small, there is still energy associated with gravitational acceleration of matter in the tongues and the corresponding heating of the stellar surface. The hot spots associated with gravitational acceleration are located in the regions of high-est density and occupy a much smaller area than the spots shown in Fig. 4. At the end of the e µ = 0.1 simulation run the accretion rate increased, the disk reached the surface of the star and started interacting with the star's surface through an equatorial belt (see §4.3).
We also performed exploratory simulations of accretion to stars with larger magnetospheres, e µ = 0.5 and observed that in many cases accretion through two or one ordered tongues dominates. We varied the α parameter (which regulates the accretion rate) and rotation rate of the star and obtained tongues and hot spots with different levels of order. In some cases two identical, diametrically opposite streams dominate accretion during the whole simulation run. In other cases only one stream dominates while the other is weaker, and therefore only one hot spot will determine the frequency observed in the light curve. In yet another type of case, multiple tongues are observed (like in cases of large magnetospheres, e.g. KR08a; KR08b), but one or two tongues are stronger than others, and can give rise to a QPO peak. Below we include the e µ = 0.5 case as an example of accretion to slightly larger magnetospheres. This case links the very small magnetospheres considered in this paper with the much larger ones considered in KR08b, where QPOs of similar origin are observed at some sets of parameters.
Frequency analysis
We performed frequency analysis for all the above cases. First we perform what we call spot-omega analysis, which we find to be the most informative for analysis of our simulations. Namely, we plot the equatorial distribution of the emitted energy flux at different moments of time (see Fig. 5, left column). We obtain diagonal lines which show that the spots have a definite order in their rotation along the star's equator. The slopes of these lines are proportional to the angular velocity of the spots. We obtained a number of almost parallel lines which reflect multiple rotations of a single spot with approximately the same angular velocity. We performed this spotomega analysis for all cases. Fig. 5 shows the spot-omega diagrams for the same time interval. One can see that at smaller e µ, the lines are steeper because the inner disk is closer to the star and the rotation of the streams/spots is faster. Near the lines we show the rotation frequencies of the corresponding spots in units of the stellar rotation frequency. Note that in the case of the smallest magnetosphere (e µ = 0.1), the streams become very small and the spots associated with the energy flux due to gravitational acceleration become weak. In this case we plot the distribution of the density along the equator. Closer to the end of this simulation run, the disk comes to the surface of the star, and the streams (and ordered lines) disappear (see bottom left panel of Fig. 5). In the case of the largest magnetosphere, e µ = 0.5, the spot-omega diagram shows a little less order because the magnetosphere is stronger and the rotation of the tongues is less synchronized.
Next we performed wavelet analysis of the light-curves from hot spots. In the cases with e µ = 0.5, 0.2, 0.15 and 0.12, the light-curve is calculated with the suggestion that all the gravitational and thermal energy of the matter falling onto the star is converted into thermal energy, which is radiated isotropically. The light from the surface of the star is integrated in the direction of the observer, whose line of sight makes an angle of ı = 90 • with the rotational axis of the star (see RUKL04 and KR05 for details). In the case with e µ = 0.1, it is expected that energy is released mainly due to friction between the rapidly moving streams and the star. Calculation of the radiation from such friction is beyond of the scope of this paper. As a first approximation we suggest that the emitted flux is proportional to the matter density at the star's surface. Fig. 5 shows wavelet spectra for all these cases. These plots exclude early times, when the streams have yet to reach the star's surface. Since the wavelet transform uses a window of width ∆t centered at time t to calculate the time-dependent frequency spectrum of the lightcurve from time t1 to time t2, it is necessary to have t1 + ∆t/2 t t2 − ∆t/2. This results in a portion of the wavelet plot being cut off. The width ∆t is inversely related to the frequency, so lower frequencies are more strongly affected by this restriction. One can see that in each case several frequencies are observed. Comparisons of the wavelet frequencies with the spot-omega frequencies (left panels) show that one of the frequencies is approximately The slope of the lines is proportional to the angular velocity of the hot spots. Right two columns: Wavelet and Fourier spectra of the light curves from the hot spots obtained at an observer inclination angle of i = 90 • (i.e., when the observer is in the equatorial plane). The arrows show the frequency corresponding to the presence of two rotating spots. In the bottom row the lines show the distribution of the density, and not the flux, in the spots, and the density varies between 3.8 (red) and 0.01 (blue). In this case the lightcurve for which the wavelet and Fourier spectra are shown is calculated assuming that the emission of the star is proportional to the density. twice as high as the frequency of the single spots shown in the spot-omega diagram, and hence corresponds to the motion of two spots along the surface of the star.
We also performed Fourier analysis of the light-curves. For this analysis we chose only the time intervals during which steady rotation of the spots was observed. In the case with e µ = 0.1, we also excluded the late times after the spots disappear and the boundary layer forms. The Fourier spectra show similar peaks as the wavelet. Between all these plots, the peak corresponding to rotation of two spots is most important. Thus we get an approximate agreement between the high-frequency QPOs obtained in all three methods of the frequency analysis.
There are also lower frequency peaks observed in both the wavelet and Fourier plots. These frequencies may either reflect the shape of the spots, or be beat frequencies (see also Smith et al. 1995;Psaltis et al. 1998). We do not see the star's rotation frequency itself. However, it is possible that our simulations are not long enough for the wavelet analysis to capture possible QPOs associated with the slowly rotating star. On the other hand, the spot-omega diagrams do not show signs of stellar rotation, so it is possible that the corresponding QPO is very weak. Note that spiral density waves often form in the disk (see Fig. 4, middle panels). The density waves tend to be stationary in the coordinate system rotating with a star, that is, they are generated by the inclination of the dipole. The rotating tongues disrupt the inner part of this pattern, but the external pattern approximately corotates with the star (see Fig. 4). The figure also shows that sample magnetic field lines starting out equidistantly from the star's surface also accumulate in the density waves. Note that these density waves are similar in shape to those suggested by Miller et al. (1998), except that here the streams are guided by magnetic fields instead of being produced by radiation drag.
Interaction through the boundary layer
Closer to the end of the simulation run with the smallest magnetosphere (e µ = 0.1), the accretion rate increased and the disk matter started interacting with the surface of the star along a belt-like path (see Fig. 6). Later, the belt became wider, and an unusual pattern formed, connected with some instability. This is probably the Kelvin-Helmholtz instability which develops between the slowly rotating stellar surface and the much more rapidly rotating disk, which is expected to rotate at a frequency ν disk ≈ 8ν * at the surface of the star.
We should note that in all cases, when the modified tongue or a boundary layer reaches the surface of the star, it spreads to higher latitudes as it was predicted by Inogamov & Sunyaev (1999) (see also Piro & Bildsten 2004;Fisker & Balsara 2005). Fig. 7 shows such spreading and also a closer view of the unstable boundary layer on the star's surface. During such close contact between the disk and the star, energy is expected to be released mainly from friction between the disk matter and the stellar surface, like in the usual hydrodynamic boundary layer. Compared to all the cases in the previous sections, no energy is associated with the matter falling onto the star's surface. Such boundary layers require special investigation. Note that even in the case with no magnetosphere, a weak magnetic field threads the disk, and magnetic field lines are wrapped in the inner parts of the disk.
At later times, when the disk comes closer to the star, the magnetic field lines trapped in the equatorial region are pushed closer to the star or buried. Note that at the same time the field lines above and below the disk are not buried (see also Fig. 7). Possibly the process of field burial by the thin disk through the boundary later should be reconsidered taking into account the fact that a significant amount of magnetic flux may inflate into the corona and stay there.
Example: Application to Accreting Millisecond Pulsars
Here we show a sample application of our simulations to accreting millisecond pulsars. We take a neutron star with mass M * = 1.4M , radius R * = 10 km and surface magnetic field B * = 3 × 10 8 G. The corresponding reference values are in Table 1. We convert the dimensionless results obtained in §4.1 and §4.2 into dimensional values. The dimensionless stellar angular velocity is f ω * = 0.354. The corresponding dimensional angular velocity is ω * = f ω * ω0 = 1002 s −1 , dimensional period is P * = 2π/ω * = 6.3 ms, and dimensional frequency is ν * = 159 Hz. The time T used in the figures has the reference value P0 = 2.2 ms which is the Keplerian rotation period at r = 1 (3R * = 30 km). The spot-omega diagrams in Fig. 5 (left panels) show the frequencies associated with the rotation of one spot for different values of e µ. In the case of the larger magnetosphere (e µ = 0.2) the frequency is ν1spot ≈ 3.4ν * = 541 Hz. Since there are two antipodal hot spots, we expect the observer to see twice that frequency, ν2spots ≈ 1082 Hz. In the case with e µ = 0.15, the disk is closer to the star and the frequencies are ν1spot ≈ 4.5ν * ≈ 715 Hz and ν2spots ≈ 1430 Hz. For e µ = 0.12 the frequencies are ν1spot ≈ 5.3ν * = 843 Hz and ν2spots ≈ 1686 Hz. And for e µ = 0.1 the frequencies are ν1spot ≈ 6.2ν * = 986 Hz and ν2spots ≈ 1972 Hz. The frequencies are quite high because the disk is close to the surface of the star. In simulations with a larger magnetosphere, e µ = 0.5, we obtain lower frequencies, ν1spot ≈ 2.2ν * = 350 Hz and ν2spots ≈ 700 Hz.
In application to neutron stars the formulae (1) can be re-written as:
M = 2.2 × 10 −9 ḟ M e µ 2 ! B * 3 × 10 8 G ! 2 M * 2.8 × 10 33 g ! − 1 2 R * 10 6 cm ! 5 2Ṁ yr .(2)
One can see that the accretion rate depends on the dimensionless parameter ḟ M /e µ 2 . Table 2 shows that the ratio ḟ M /e µ 2 decreases systematically with e µ, and so do the accretion rate and the frequencies ν1spot and ν2spots. So, the considered above cases with different e µ correspond to different accretion rates 1.9 × 10 −9 6.6 × 10 −9 9.0 × 10 −9 9.7 × 10 −9 1.2 × 10 −8 1. where, depending on accretion rate, the frequency drifts between ν2spots ≈ 700 Hz when the accretion rate is lower and the magnetosphere is larger, and ν2spots ≈ 1972 Hz when the accretion rate is higher and the magnetosphere is very small. The ratio between these two frequencies is 2.8 and can be smaller or possibly larger depending on the variation of the accretion rate, and is close to the observed drifts of the main high-frequency QPO in, e.g., millisecond pulsars (van der Klis 2000) and dwarf novae (Warner & Woudt 2006).
Only if the two hotspots are antipodal and identical, do we expect ν1spot to be absent from the power spectrum. This is true for the small magnetosphere (e µ ≤ 0.2) cases. In the large magnetosphere (e µ = 0.5) case, we see that the spots are not identical; one spot is often much larger than the other, and hence the frequency ν1spot may dominate. If the magnetic field of the star is not an ideal dipole field (that is, if it is a slightly misplaced dipole, or a more complex field (e.g., Long et al. 2007Long et al. , 2008 then the symmetry breaks and one spot will be always larger than the others, and the lower frequency ν1spot will dominate. Then we expected the frequency frift between 350 Hz and 990 Hz.
Simulations of stars with larger magnetosphere sizes, e µ 0.5, usually show much more chaotic behavior in the unstable regime (KR08a,b; RKL08). In spite of stochastic accretion in the unstable regime, the spots on the surface of the star show a component which rotates with the angular velocity of the inner disk (KR08b). This component analyzed through rotation of spots (like in the left column of Fig. 5) shows clear inclination of lines associated with motion of different temporary spots with angular velocity approximately equal to the angular velocity of the inner disk (KR08a). This frequency component is weaker than the stochastic components associated with unstable accretion. However in longer simulation runs this QPO peak associated with the inner disk frequency may by amplified with time and may become significant, because the high-amplitude stochastic components are relatively incoherent and their frequencies constantly drift.
Although we consider accretion to a very slowly rotating c 0000 RAS, MNRAS 000, 000-000 star, similar MBLs and QPOs are expected for more rapidly rotating stars.
Disk oscillations
Note that some peaks observed in the wavelets and Fourier spectra may be associated with the disk-star interaction. An accretion disk can have different modes of oscillation. These include bending oscillations of the inner disk driven by the star's rotating misaligned dipole field (e.g., Lai & Zhang 2008) and radial oscillations of the inner disk (e.g., Lovelace et al. 2009;Erkut et al. 2008). The bending oscillations lead to the formation of an m = 1 mode spiral wave rotating with the frequency of the star. In our simulations we do see traces of bending waves. The radial oscillations can arise from a linear Rossby wave instability (RWI) where the angular frequency of the oscillations (with mode number m = 1) is less than the peak in the angular velocity in the inner disk, Ωmax Lovelace et al. 2009). This mode is radially trapped in a narrow region inside the radius at which Ωmax occurs. The beat between this mode and the stellar rotation frequency may lead to the coupled twin peak QPOs observed in some millisecond pulsars (e.g., van der Klis 2006).
Here we derive from our simulations the radius at which the disk angular velocity matches that of the spot. Fig. 8 shows this analysis for cases with different e µ. First, we plot the radial dependence of the angular velocity in the disk at sample times (see the top panels of the Fig. 8). Then we take the angular velocity of the spot from the spot-omega diagram (Fig. 5, left panels) and find the point where angular velocity of the spot equals the angular velocity in the disk (see red dots). Next, we plot the radial dependence of angular momentum (Fig. 8 middle panels) and density (bottom panels) at same moment of time. The circular line in the plots is the line on which the angular velocity equals that of the spot. One can see that in all cases the rotational velocity of the spot corresponds to the inner edge of the disk. It corresponds to a distance only slightly larger than the Alfvén surface (red line), where the kinetic plasma parameter β1 = (p + ρv 2 )/(B 2 /8π) = 1. Note that the magnetosphere is strongly non-axisymmetric, and therefore the Ω distribution in the disk varies with time.
Different oscillation modes in the disk may influence the position or brightness variation of the spots on the surface of the star. We do observe some frequencies in the power spectra of the light-curves, but at present we are not sure that the observed frequencies reflect disk oscillations. Future (longer) simulation runs may help reveal such a possibility. On the other hand, disk oscillations can be investigated directly from the simulations. We plan to do this in the future. Fig. 5 shows that there are several peaks observed in the wavelet and Fourier spectra. We know the origin of only one of them, associated with rotation of the unstable tongues. Other peaks may be associated with disk oscillations or with beat frequencies between the disk and the star. However, at present our runs are not long enough to establish the origin of these peaks.
CONCLUSIONS
We considered accretion to slowly rotating stars with small and tiny magnetospheres in the stable and unstable regimes. We conclude that:
(1) In the stable regime, a small magnetospheric cavity and tiny funnel streams form, producing hot spots on the star's surface which tend to be in a preferred position determined by the dipole inclination (we had example runs for Θ = 15 • ). In this case the frequency of the star is expected to dominate. However, the rapidly rotating disk has a tendency to "drag" the funnel stream to faster rotation, due to which parts of the hot spots often rotate much faster than the star. At lower Θ this effect becomes even more significant, and the spots may rotate faster than the star for a long time, leading to QPOs (see also RUKL04), or slower than the star leading to matter accreting through a trailing funnel, (e.g., ) producing a lower-frequency QPO.
(2) In the unstable regime we observed that matter accretes through two ordered streams which rotate with the angular velocity of the inner disk, that is, much faster than the star. They hit the surface of the star forming two antipodal hot spots. Rotation of these spots along the surface of the star leads to high-frequency QPOs. Such persistent streams/spots have been observed at a variety of parameters and are seen to be quite long-lived. Coherent rotation of this kind has been observed for small magnetospheres. For large magnetospheres we find that the spots are much more chaotic in the sense that they form at different parts of the star (RKL08; KR08a). In intermediate cases we observed that one or two ordered streams may form, which are less ordered than in the cases of small magnetospheres, but still may give QPO peaks (KR08b). On the other hand, the accretion may be stochastic, but one or two streams may be stronger than the others, and the hot spots associated with these streams may lead to QPOs.
(3) Correlation is observed between the size of the magnetosphere and the QPO frequency: the frequency is higher at smaller magnetospheric sizes. We expect that secular variation of the accretion rate will lead to drifting of the QPO frequency. Correlation between the frequency and the accretion rate has been observed in a number of accreting millisecond pulsars (e.g., van der Klis 2000).
(4) We were able to model a wide range of QPO frequencies. In application to millisecond pulsars the frequency associated with rotation of one spot varies between ν1spot = 350 Hz and 990 Hz. In cases of small magnetospheres, the two spots are very similar in brightness, and the expected frequencies are twice as high. Only if the two spots are antipodal and identical, do we expect ν1spot to be absent from the power spectrum. This is true for the small magnetosphere (e µ ≤ 0.2) cases. In the large magnetosphere (e µ=0.5) case, we expect to see ν1spot.
The most striking result of this paper is the discovery of a new regime of unstable accretion which shows clear highfrequency QPO peaks. Angular velocity distribution. The background shows the angular velocity, while the foreground shows the kinetic plasma parameter β 1 = 1 (red thick line) and the circle shows where the angular velocity equals that of the spot (black dash-dot line). Bottom panels: Same, but for the density distribution.
NASA grant NNX08AH25G and by NSF grants AST-0607135 and AST-0807129. We are thankful to NASA for using NASA High Performance Facilities.
Figure 1 .
1Stable accretion to a star with a small magnetosphere, where the magnetic moment of the star is e µ = 0.2 and is misaligned relative to the rotational axis Ω at Θ = 15 • . Left panel: Density distribution and sample magnetic field lines in the xz (µ − Ω) plane. Right panel: The same, but in the xy− (equatorial) plane. The dimensionless density varies between 0.8 (red) and 0.006 (blue).
Figure 2 .
2Stable accretion to a star with a very small (tiny) magnetosphere (e µ = 0.08) with Θ = 15 • (α = 0.04). Top two panels: Density distribution and sample magnetic field lines in the xz (µ − Ω) plane and the xy− (equatorial) plane in a coordinate system rotating with the star. The dimensionless density varies between 1.4 (red) and 0.006 (blue). Bottom panel: Distribution of the dimensionless energy flux of matter onto the surface of the star (pole-on), varying between 0.4 (red) and 0.0009 (blue). The time T is measured in units of ∆T = 1/16P 0 , where P 0 is the Keplerian period at r = 1.
Figure 3 .
3Snapshots of MBL accretion through the synchronized instability tongues in the case of a relatively large magnetosphere (e µ = 0.2 in dimensionless units). The time interval shown is a small portion of the simulation; the tongue pattern is steady for the entire duration of the simulation. Top panel: A surface of constant density (0.4 in dimensionless units) and sample magnetic field lines. Middle panel: Density distribution and sample magnetic field lines in the xy-plane. The density varies from 6.7 (red) to 0.01 (blue). Bottom panel: Energy flux onto the star's surface, ranging from 3.4 (red) to 0.008 (blue). The figures are shown in a coordinate system rotating with the star. The time T is measured in periods of Keplerian rotation at r = 1.
Figure 4 .
4Top panel: A surface of constant density (0.47 in dimensionless units) and sample magnetic field lines in the case of a small magnetosphere, e µ = 0.1. Middle panel: xy-slice showing the density distribution and sample magnetic field lines. Bottom panel: density distribution on the surface of the star, ranging from 3.8 (red) to 0.01 (blue). The figures are shown in a coordinate system rotating with the star. The time T is measured in periods of Keplerian rotation at r = 1.
Figure 5 .
5Frequency analysis for runs with different magnetospheric sizes. Left column: Spot-omega analysis showing the emitted energy flux distribution in the equatorial plane at different times. The flux varies from 3.4 (red) to 0.008 (blue). The lines reflect the motion of individual spots on the stellar surface.
Figure 7 .
7Enlarged view of two snapshots fromFig. 6, showing a surface of constant density (0.43 in dimensionless units) and sample magnetic field lines. The top and bottom panels show different projections. One can see that matter is lifted along the surface of the star to larger latitudes. In the case when the disk comes to the surface of the star, a new instability appears at the disk-star boundary, which is probably of the Kelvin-Helmholtz type.
Figure 6 .
6Same plot as inFig. 4but closer to the end of the run when the disk approaches the star. The surface shown in the top panels has a density of 0.43. In the bottom panels the density varies between 2.6 (red) and 0.007 (blue). The figures are shown in a coordinate system rotating with the star. The time T is measured in periods of Keplerian rotation at r = 1.
Figure 8 .
8Top panels: Radial dependence of the disk angular velocity for the moments of time corresponding to the lower panels (thick blue line) and for other moments of time (black line). The red spot marks the point where the angular velocity matches that of the spots. Middle panel:
c 0000 RAS arXiv:0812.0384v4 [astro-ph] 2 Jun 2009
c 0000 RAS, MNRAS 000, 000-000
AcknowledgmentsThe authors thank R.V.E. Lovelace for discussions and the referee M.A. Alpar for comments and suggestions which helped improve the paper. The authors were supported in part by
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| []
|
[
"Laplacian-level density functionals for the exchange-correlation energy of low-dimensional nanostructures",
"Laplacian-level density functionals for the exchange-correlation energy of low-dimensional nanostructures"
]
| [
"S Pittalis \nDepartment of Physics and Astronomy\nUniversity of Missouri\n65211ColumbiaMissouriUSA\n",
"E Räsänen \nNanoscience Center\nDepartment of Physics\nUniversity of Jyväskylä\nFI-40014JyväskyläFinland\n"
]
| [
"Department of Physics and Astronomy\nUniversity of Missouri\n65211ColumbiaMissouriUSA",
"Nanoscience Center\nDepartment of Physics\nUniversity of Jyväskylä\nFI-40014JyväskyläFinland"
]
| []
| In modeling low-dimensional electronic nanostructures, the evaluation of the electron-electron interaction is a challenging task. Here we present an accurate and practical density-functional approach to the two-dimensional many-electron problem. In particular, we show that spin-density functionals in the class of meta-generalized-gradient approximations can be greatly simplified by reducing the explicit dependence on the Kohn-Sham orbitals to the dependence on the electron spin density and its spatial derivatives. Tests on various quantum-dot systems show that the overall accuracy is well preserved, if not even improved, by the modifications. PACS numbers: 71.15.Mb, 31.15.E-, 73.21.La | 10.1103/physrevb.82.165123 | [
"https://arxiv.org/pdf/1009.4292v1.pdf"
]
| 118,518,970 | 1009.4292 | a399baa3a26215ff7b26b4bab8c590253fbd45b5 |
Laplacian-level density functionals for the exchange-correlation energy of low-dimensional nanostructures
22 Sep 2010 (Dated: September 23, 2010)
S Pittalis
Department of Physics and Astronomy
University of Missouri
65211ColumbiaMissouriUSA
E Räsänen
Nanoscience Center
Department of Physics
University of Jyväskylä
FI-40014JyväskyläFinland
Laplacian-level density functionals for the exchange-correlation energy of low-dimensional nanostructures
22 Sep 2010 (Dated: September 23, 2010)numbers: 7115Mb3115E-7321La
In modeling low-dimensional electronic nanostructures, the evaluation of the electron-electron interaction is a challenging task. Here we present an accurate and practical density-functional approach to the two-dimensional many-electron problem. In particular, we show that spin-density functionals in the class of meta-generalized-gradient approximations can be greatly simplified by reducing the explicit dependence on the Kohn-Sham orbitals to the dependence on the electron spin density and its spatial derivatives. Tests on various quantum-dot systems show that the overall accuracy is well preserved, if not even improved, by the modifications. PACS numbers: 71.15.Mb, 31.15.E-, 73.21.La
I. INTRODUCTION
With the present technology the electron gas can be confined in various ways to create nanoscale devices of lower dimension. The field of two-dimensional (2D) physics has grown rapidly alongside the development of electronic devices such as quantum Hall bars and point contacts, and semiconductor quantum dots. When modeling these systems, the finite extent in the growth direction (say z) can often be neglected, so that the system is well described by a 2D Hamiltonian in the effectivemass approximation. 1 In this respect, the building block is the 2D electron gas (2DEG), whose properties are well known in the literature. 2 Density-functional theory (DFT) and its extensions have become the method of choice to describe the electronic properties of three-dimensional (3D) systems such as atoms, molecules and solids. 3,4 Despite the fact that the many 3D density functionals developed for the exchange-correlation (xc) energy and potential fail in the quasi-2D limit, 5-8 the derivation of explicitly 2D xc functionals has started only very recently. [9][10][11][12][13][14][15][16][17] In finite 2D systems, most of these functionals overperform the 2D local spin-density approximation (LSDA), which is a combination of the analytic exchange energy of the 2DEG 18 and the corresponding correlation energy having a parametrized form. 19,20 The encouraging results obtained with the new functionals indicate that DFT in 2D has entered in a more mature phase.
Among the newly proposed functionals, our focus is on those approximations that have as ingredients the electron density and its spatial derivatives, the kinetic energy density and the paramagnetic current. [9][10][11][12][13][14][15] In other words, the expressions of these functionals are current-dependent meta-generalized-gradient approximations (meta-GGAs), and therefore they explicitly depend on the Kohn-Sham (KS) orbitals. As it is valid for 3D systems, also for 2D systems the meta-GGAs are very accurate. However, the price for their accuracy is the more involved numerical implementation (if applied self-consistently) as well as the (case-dependent) numerical burden in the applications. In order to simplify both the mentioned tasks, we explore here to which extent and how the explicit dependence on the KS orbitals may be reduced to the dependence on the electron density and its spatial derivatives, yet possibly maintaining a satisfactory level of accuracy. In other words, we examine the path from implicit to explicit density functionals in the class of meta-GGAs. A similar study has been already carried out for 3D systems, introducing some Laplacianlevel meta-GGA functionals. 21 With the present work, we explore this possibility for low-dimensional systems. We find that the performance after the simplifications is well preserved, and in some cases even improved.
II. REVIEW AND MODIFICATIONS OF THE FUNCTIONALS
In the following, we review the main ingredients of recently derived functionals and suggest how their expressions may be significantly simplified in a consistent fashion. Ideally, the ultimate goal is to obtain the best performance with the least (numerical) effort. In particular, we consider (i) an exchange-energy functional obtained through the modeling of the exchange-hole (xhole) functions, 9 (ii) an exchange-energy functional obtained from the one-body-density-matrix 11 (1BSDM), and (iii) correlation-energy functionals obtained through the modeling of the correlation hole. 12,13 Details of the derivation of the functionals can be found the in the mentioned literature.
A. Exchange energy from the exchange-hole functions
The exchange energy E x can be expressed through the x-hole functions as 3,4
h σ x (r 1 , r 2 ) E x = 1 2 σ d 2 rρ σ (r) ∞ 0 ds 2π 0 dφ s h σ x (r, r+s), (1) with h σ x (r 1 , r 2 ) = − | Nσ k=1 ψ * k,σ (r 1 )ψ k,σ (r 2 )| 2 ρ σ (r 1 ) ,(2)
where ψ kσ (r) are the KS (spin) orbitals. From Eq. (1), it is apparent that an approximation of the x-hole also provides an approximation for the exchange energies. Equation (1) also suggests that the details of the angular dependence of the x-hole are energetically negligible: all we need is the cylindrical average of the x-hole, i.e.h σ x (r; s). As the basis of our model, 9 we have chosen
h σ x (r; s) = 1 2π 2π 0 dφ s h σ x (r; r + s) ≈ − a π exp −a(r) b(r) + s 2 × I 0 2a(r) b(r)s ,(3)
where I 0 (x) is the zeroth-order modified Bessel function of the first kind. This model provides the correct sign of the x-hole and the correct normalization d 2 s h σ x (r, s) = −1. The non-negative functions a(r) and b(r) are introduced to reproduce the short-range behavior of the x-hole. It is important to note that the curvature of the x-hole in 2D is given by 9
C σ x (r) = 1 4 ∇ 2 ρ σ (r) − 2τ σ (r) + 1 2 (∇ρ σ (r)) 2 ρ σ (r) + 2 j 2 p,σ (r) ρ σ (r) ,(4)
where τ σ is (twice) the spin-dependent kinetic-energy density, and j p,σ is the spin-dependent paramagnetic current density.
In Ref. 9, we have applied the above scheme in two ways. In the first instance, we have employed it in its full spirit by numerically determining a and b at each point in space. Then, in a fully self-consistent application, the scheme should be implemented within the optimizedeffective-potential (OEP) method. [22][23][24][25][26] This is necessary because τ σ and j p,σ (in non-current-spin-density functional calculations 27,28 ) make the overall scheme explicitly dependent on the KS orbitals. As a consequence, the corresponding numerical task would be highly nontrivial.
In the second instance, we have analyzed the 2DEG limit, for which
C σ x (r) → C σ h,x (r) = −πρ 2 σ (r) .(5)
In this way, a local density functional has been recovered, which was seen to improve over the exchange energies obtained within the standard LSDA when applied to fewelectron quantum dots. Next we proceed from the review of the functional to its modification. As discussed above, the task is to remove the explicit reference to the KS orbitals, yet retaining some degree of flexibility in dealing with inhomogeneous systems. This may be achieved by introducing the following modification: 15,29,30
τ σ (r) = Nσ k=1 ψ kσ (r) →τ σ (r) = 2πρ 2 σ (r)+ 1 3 ∇ 2 ρ σ (r)+ j 2 p,σ (r) ρ σ (r) .
(6) Correspondingly, we obtain
C σ x (r) →C σ x (r) = −πρ 2 σ (r) + 1 12 ∇ 2 ρ σ (r) + 1 8 (∇ρ σ (r)) 2 ρ σ (r) .(7)
It is worth mentioning that due to the last (currentdependent) term on the right hand side of Eq. (6), the modified x-hole curvature manifestly preserves its gauge invariance. As described in Ref. 9, the functions a and b are determined from a = πρ σ exp (y),
and b = y πρ σ exp (−y)(8)
where y = ab satisfies
(y − 1) exp(y) =C σ x πρ 2 σ = −1 + 1 12π ∇ 2 ρ σ ρ 2 σ + 1 8π (∇ρ) 2 ρ 3 σ .
(10) If a solution does not exist, we set 9 y ≡ 0. This corresponds to the 2DEG limit mentioned just above. From Eq. (10), it is apparent that the second and third terms are relevant when the curvature and the gradient of the spin-density are non-negligible.
Going back to the first modification τ σ →τ σ , we refer to the new (angular-averaged) x-hole function as h σ x (r, s). The corresponding x-hole potentials denoted as U σ x,model (r), read as follows 9
U σ x,model (r) = 2π ∞ 0 dsh σ x (r, s)(11)
from which, the exchange energy is obtained as
E model x = 1 2 σ d 2 rρ σ (r)Ũ σ x (r) .(12)
The last two quantities calculated without the modification of τ σ are denoted below simply without the tilde symbols, i.e., U σ x,model and E σ x,model . Finally, we point out that the above modifications offer a straightforward way to calculate the corresponding KS exchange potential as a a functional derivative, v σ x = δẼ x /(δρ σ ). The properties and performance of these potentials will be assessed elsewhere.
B. Exchange energy from the one-body-spin-density matrix
Another way to express the exchange energy is to make use of the 1BSDM γ σ (r 1 , r 2 ), 3,4 so that
E x = − 1 2 σ=↑,↓ d 2 r ∞ 0 ds × 2π 0 dφ s γ σ r + s 2 , r − s 2 2 (13) with γ σ (r 1 , r 2 ) = Nσ k=1 ψ k,σ (r 1 )ψ * k,σ (r 2 ) ,(14)
where ψ kσ (r) are the KS (spin) orbitals. Clearly, an approximation for the 1BSDM implies an approximation for the exchange energy. Also, it is apparent that the angular dependence of the 1BSDM is energetically of minor importance. Therefore, as a basis of our approximation we have considered the following expression 11
1 2π ∞ 0 dφ s γ σ r + s 2 , r − s 2 2 ≈ ρ 2 σ (r)e − s 2 βσ (r) × 1 + s β σ (r) 2 A σ (N σ ) ,(15)
where β(r) is chosen to reproduce the exact-short behavior of the 1BSDM
β −1 σ (r) = 1 2 τ σ (r) ρ σ (r) − 1 8 ∇ 2 ρ σ (r) ρ σ (r) − 1 2 j p,σ (r) ρ σ (r) 2(16)
and A σ (N σ ) is obtained through the normalization of the particle number for each spin channel. In Ref. 11 we have employed this scheme leading to accurate results for various quantum-dot systems. In addition, we have observed that this approach and the one of Sec. II A coincide in the 2DEG limit.
Here we suggest to simplify the present functional by making use of Eq. (6) in Eq. (16). This yields
β −1 σ (r) = πρ σ (r) + 1 24 ∇ 2 ρ σ (r) ρ σ (r) .(17)
We emphasize that no gradients of the spin-density appear in this expression. The resulting expression is manifestly gauge invariant. The relevance of the gauge invariance of the expression in Eq. (16) has been already verified in Ref. 11. The final expression reads as
E 1BSDM x = − π 2 σ=↑,↓ d 2 r √ π + 3 4Ã σ × ρ 2 σ (r)β 1/2 σ (r) ,(18)
whereà σ is determined through the normalization as
N σ = π d 2 r 1 + 2Ã σ ρ 2 σ (r)β σ (r) .(19)
Equation (18)
U σ x,1BSDM (r) = −π √ π + 3 4Ã σ ρ σ (r)β 1/2 σ (r) . (20)
Similarly to the previous section, the exchange-hole potentials and exchange energies calculated without the modification of τ σ are denoted below simply without the tilde symbols, i.e., U σ x,1BSDM and E σ x,1BSDM .
C. Correlation energy from the correlation-hole functions
High predictive power in the application of DFT requires the accurate treatment of the electronic correlation in both inhomogeneous systems and in the limit of the homogeneous electron gas. We have achieved this goal in 2D by generalizing our previous approximation 12 to a parameter-free form, 13 which reproduces the correlation energy of the 2DEG while preserving the ability to deal with inhomogeneous systems (quantum dots).
The correlation energy is expressed in terms of the cylindrical average of the (coupling-constant dependent) correlation-hole (c-hole) functionsh σσ ′ c,λ (r, s) as follows: 3,4
E σσ ′ c = π dr ρ σ (r) ∞ 0 ds 1 0 dλh σσ ′ c,λ (r, s) .(21)
It is obvious that an approximation forh σσ c,λ (r, s) implies an approximation for E σσ ′ c . In modeling these quantities, we have proposed a form 12
h σσ c,λ (r, s) ≈ 2λs 2 3 (s − z σσ (r)) D σ (r) 1 + 2 3 λz σσ (r) exp − 9πs 2 16z 2 σσ (r) (22) h σσ c,λ (r, s) ≈ 2λρσ(r) s − z σσ (r) 1 + 2λz σσ (r) exp − πs 2 4z 2 σσ (r)(23)
for the same-and opposite-spin cases, σσ ′ = σσ and σσ ′ = σσ, respectively. Here
D σ (r) := 1 2 τ σ − 1 4 (∇ρ σ ) 2 ρ σ − j 2 p,σ ρ σ(24)
and
z σσ (r) := 2c σσ |U σ x (r)| −1 ,(25)z σσ (r) := c σσ |U σ x (r)| −1 + |Uσ x (r)| −1 .(26)
Equations (25) and (26) are proportionality relations that may be enforced locally in space [see Eqs. (27) and (28) below]. It is apparent that z σσ ′ (r) set the characteristic sizes of the c-hole functions in terms of the sizes of the x-hole functions. The idea behind this assumption is the following: the smaller the x-hole around each electron is, the more tightly the electrons are screened. Therefore, they are expected to be correlated much less. The above modeling provides 12 : (i) zero correlation energy for one-particle systems (as the exact one); (ii) exact short-range behavior of the λ-dependent c-hole functions; (iii) a "reasonable" decay in the limit s → ∞; (iv) exact normalization of the λ-dependent c-hole functions. Furthermore, (v) c σσ ′ can be defined in such a way that the total correlation energy of the 2DEG is exactly reproduced. 13 As a result, when the (average) density has a realistic range, 0 < r s = 1/ √ πρ < 20, we can use the following (approximate) parameterizations:
with δ = 0.663 25 and ξ = 0.123 96. When using the present correlation functional, the coefficients c σσ ′ [r s ](r) must be calculated at each point in space by making use of the density, that is, r s (r) = 1/ πρ(r).
Here we propose a few simplifications along the lines of the previous sections. First, we may replace U σ x with an approximate expression obtained in Sec. II A. Secondly, we apply Eq. (6) to Eq. (24) leading to
D σ (r) →D σ (r) = πρ 2 σ (r) + 1 6 ∇ 2 ρ σ (r) − 1 8
(∇ρ σ (r)) 2 ρ σ (r) .
(29) Now, conditions (i) and (ii) given above are no longer satisfied. The latter modification clearly affects the samespin c-hole functions. Applying both of the described simplifications -which is naturally required in order to make the functional orbital-free -the correlation energy can be expressed in terms of the (spin-dependent) c-hole potentials, U σσ ′ c (r), as follows:
E σσ ′ c = 1 2 dr ρ σ (r)Ũ σσ ′ c (r) ,(30)withŨ σσ c (r) = 16 81π (8 − 3π)D σ (r)z 2 σσ (r) × 2z σσ (r) − 3 ln 2 3z σσ (r) + 1 ,(31)
andŨ σσ c (r) = (2 − π)ρσ(r)
× [2z σσ (r) − ln (2z σσ (r) + 1)] ,(32)
wherez σσ ′ (r) are obtained by replacing U σ x withŨ σ x in Eqs. (25) and (26).
III. TESTING THE MODIFICATIONS
Next we test the modifications for different 2D quantum-dot systems. As a standard test set we consider parabolic (harmonic) dots consisting of N electrons confined in an external potential v ext (r) = ω 2 r 2 /2. First we use the octopus code 31 to solve the KS problem self-consistently using the exact-exchange (EXX) functional within the Krieger-Li-Iafrate (KLI) approximation. 32 Then the resulting KS orbitals -and for the modified functionals solely the electron density and its gradients -are used to compute the energy expressions and their ingredients introduced in the previous sections. The obtained exchange energies can be directly compared with the EXX-KLI results E EXX
x . In addition to this test set, we also consider a large quantum dot where we compare with the LSDA, as well as a rectangular quantum slab. In the case of correlation, we exploit the numerically exact configuration-interaction data 33
A. Exchange energies
First we test the ingredients entering in the expressions for the exchange energy in Secs. II A and II B. Figure 1(a) shows the original and modified kineticenergy densities [as defined in Eq. (6)] of a spin-polarized three-electron parabolic quantum dot with ω = 1/4. The characteristic step in τ σ at the shell of the quantum dot at r ∼ 3 is significantly smoother inτ σ . However, this difference is partly washed away in the local curvature of the x-hole shown in Fig. 1(b).
In the exchange-hole potential shown in Fig. 2(a)computed with the functional described in Sec. II Bthe difference is reduced further, so that the results are almost identical. Moreover, they agree very well with the EXX-KLI result corresponding to the Slater potential (dotted line). Naturally, this similarity leads to precise exchange energies as explicitly shown below.
In Fig. 2(b) we show the exchange-hole potentials calculated with the functional described in Sec. II B. Again, the results with and without the modification of the functional are similar, although the relative difference is larger than in the previous case [ Fig. 2(a)]. Both potentials, however, deviate rather strongly from the EXX-KLI result. This tendency is already present in the original functional, which has been tailored mainly to produce accurate exchange energies, 11 which is indeed the case as shown below. Table I shows the exchange energies calculated for several quantum dots. The set includes four cases with orbital currents (rows 1, 3, 4, and 7), in two cases arising from an external magnetic field perpendicular to the 2D plane (rows 3 and 7). 34 We consider modifications for both E model Table I). For the 1BSDM approximation the modification even improves the performance. The LSDA is giving clearly the worst accuracy of the tested functionals.
In addition to the test set of Table I that covers only few-electron quantum dots, we now consider two rather different cases. First we focus on a large 48-electron parabolic quantum dot with ω = 0.3373 at a magnetic field of B = 3.05 T. This partially spin-polarized (total spin S = 3) ground state has a compact "spin droplet" on the second-lowest Landau level, and its existence has been confirmed in recent spin-blockade experiments. 35,36 Here we have performed a LSDA calculation and use that density as an input in the functionals. Figure 3 shows the kinetic-energy densities and exchange-hole potentials calculated with the functionals obtained from the modeling of the exchange hole (see Sec. II A). The orig- In lack of a reliable EXX reference result for a system of this size it is not possible to judge whether the exchange energy from the model(s) or from the LSDA is more accurate. However, knowing that the LSDA typically underestimates the (absolute value of) E x , our results for E model x andẼ model x deviate from the LSDA in the correct direction. Most importantly, the exchange-hole potentials from the model are more accurate, especially in the asymptotic region. 9 Our second example of a quantum-dot system that differs from those in Table I is a 16-electron rectangular hard-wall quantum slab with size 2 √ 2π × √ 2π ), so that the last row shows the mean absolute error in percentage. corresponding to ∼ 90 nm × 45 nm in SI units. 34 In Fig. 4 we compare the exchange-hole potentials given by the 1BSDM functionals (see Sec. II B) to the EXX-KLI (Slater) potential computed. The overall shapes are very similar, but as expected, the EXX-KLI potential is considerably smoother. Interestingly, however, the modified potential is qualitatively closer to the EXX-KLI result than the original one. Regarding the exchange energies both functionals perform similarly: E 1BSDM x = 13.13, E 1BSDM x = 13.15, and E EXX x = 12.7. Thus, the modification in τ σ is well justified also when considering a 2D system with a hard-wall geometry. Figure 5 shows the spin-pair components of the correlation-hole potentials for a spin-unpolarized (total spin S = 0) six-electron parabolic quantum dot (ω = 1/4) calculated with the model of Ref. 13 in comparison with its modifications introduced in Sec. II C. The modification induces clear devitations from the original potential, especially for the same-spin component affected by modifications in D σ . For example, the bump at r ∼ 4 is due to the change of sign inD σ in that regime. In contrast, the opposite-spin component is independent of D σ [see Eq. (32)], so that the modified functionals are almost the same; here the choice ofŨ σ
N B(T ) E model xẼ model x E 1BSDM xẼ 1BSDM x E LSDA x E EXX x 2 0 -0.
B. Correlation energies
x,model instead of U σ
x,EXX has a negligible effect (solid and dashed lines overlap).
In lack of an exact reference result we cannot assess the quality of the correlation-hole potential(s). Hence, in the following we will focus on the correlation energies for which reference results can be obtained as described in the beginning of Sec. III.
In Table II we test the effect of the modifications for the accuracy of the correlation-energy functional. As discussed at the end of Sec. II C, we consider two approxi- mations, where we either do not approximate U x in the same framework but use the EXX result, or then we apply the modification also to U x . Interestingly, the best result -apart from the original functional which is very close in accuracy -is given byẼ c (Ũ σ x ) for both spin-polarized and unpolarized cases. This finding may demonstrate the compatibility between the corresponding exchangeand correlation-energy functionals. Nevertheless, all the functionals introduced here are superior to the LSDA, whose error is an order of magnitude larger. 12,13 Finally, in Fig. 6 we plot the relative errors of the correlation-energy functionals as a function of N (cf. Fig. 2 in Ref. 13). It is interesting to note that, at least for this set of systems, the modified functionals show consistent behavior as a function of the number of electrons. Thus, it may be expected that the good performance continues further to larger N . Unfortunately, a throughout testing of this is beyond the capability of numerically exact methods to provide accurate reference data.
IV. CONCLUSIONS AND OUTLOOK
In this work we have explored the possibility to modify meta-generalized-gradient approximations (meta-GGAs) for the exchange and correlation energies of twodimensional systems to Laplacian-level meta-GGA ones. We have analyzed the effects of the according modifications on various systems. Although the differences in the kinetic-energy densities can be considerable, the functionals considered in this work preserve well the quality of the exchange-and correlation-hole potentials, and in particular the corresponding energies. Overall, we find that the performance is well preserved, if not even improved, by the modifications.
The simplified meta-GGAs provide significant prac- (color online) Spin-pair components of the correlation-hole potential for a spin-unpolarized six-electron parabolic (ω = 1/4) quantum dot calculated using the functional obtained from the correlation-hole modeling 13 (solid line), its modification described in Sec. II C with U σ x,EXX (obtained within the KLI approximation) as an input (dashed line), and withŨ σ x,model as an input (dotted line). II C), its modification with U σ x,EXX (in the KLI approximation) as an input, the modified form withŨ σ x as an input, and the local-spin-density approximation. 37 The last column shows the numerically exact reference result. The last row shows the mean absolute error in percentage. ticality and numerical efficiency in the application of the functionals. Therefore, a self-consistent and multipurpose implementation of the present toolbox of functionals is now within reach, enabling the investigation of (quasi-)two dimensional electronic nanostructures of experimental and technological relevance.
c σσ [r s ] = α log(r s ) + β r γ s (27) with α = −0.1415 1, β = 1.226 1, γ = 0.144 99, and c σσ [r s ] = δ r ξ s
FIG
x and E 1BSDMx , respectively. Overall, the modifications preserve the excellent performance of the . 1: (color online) (a) Comparison of the original (solid line) and modified (dashed line) kinetic-energy density of a spin-polarized three-electron parabolic (ω = 1/4) quantum dot. (b) Local curvature of the exchange hole calculated from the original (solid line) and modified (dashed line) kineticenergy density. functionals very well (see the last row of
FIG
. 2: (color online) Exchange-hole potentials of a spinpolarized three-electron parabolic (ω = 1/4)quantum dot. (a) Result of the functionals in Ref. 9 without (solid line) and with (dashed line) the modification (see Sec. II A). (b) The same as (a) for the functional in Ref. 11 (see Sec. II B). The dotted line corresponds to the EXX-KLI result. inal and modified τ σ for both spin-up and spin-down electrons are very similar, and the resulting exchangehole potentials are practically the same. The exchange energies are E model x = −22.18 andẼ model x = −22.25 (difference of 0.3%). In comparison, the LSDA yields E LSDA x = −21.11.
FIG
. 3: (color online) (a) Original (solid lines) and modified (dashed lines) kinetic-energy densities for spin-up and spin-down electrons in a 48-electron parabolic (ω = 0.3373) quantum dot at B = 3.05 T. The dotted lines show the spin densities. (b) Resulting spin-up and spin-down exchange-hole potentials using the functionals obtained from the modeling of the exchange hole (see Sec. II A).
FIG
. 4: (color online) Exchange-hole potentials of a 16electron rectangular quantum dot calculated with the original (a) and modified (b) functionals obtained from the one-body spin-density matrix (see Sec. II B) in comparision with the exact-exchange (Slater) potential (c).
FIG. 5: (color online) Spin-pair components of the correlation-hole potential for a spin-unpolarized six-electron parabolic (ω = 1/4) quantum dot calculated using the functional obtained from the correlation-hole modeling 13 (solid line), its modification described in Sec. II C with U σ x,EXX (obtained within the KLI approximation) as an input (dashed line), and withŨ σ x,model as an input (dotted line).
FIG. 6 :
6(color online) Error in the correlation energy of spin-polarized parabolic (ω = 1/4) quantum dots with N electrons obtained using different approximations. The circles correspond to the functional of Ref.13 and the squares show the performance of the modified functional introduced in Sec. II C, when the exact exchange-energy potential has been used in the expression. The triangles correspond to the case when also the exchange-energy potential has been used in a (similar) modified form.
together with Eq.(19) provide another density functional for the exchange energy. Finally, the x-hole potential has a form
TABLE I :
IExchange energies for fully spin-polarized parabolic
quantum dots calculated using the functional of Ref. 9
(E model
x
), its modification described in Sec. 2 (Ẽ model
x
), the
functional in of Ref. 11 (E 1BSDM
x
), its modification (Ẽ 1BSDM
x
)
described in Sec. II B, and the local spin-density approxima-
tion (E LSDA
x
. They are compared with the EXX-KLI result
(E EXX
x
TABLE II :
IICorrelation energies for spin-polarized (S = N/2) and unpolarized (S = 0) parabolic quantum dots calculated using the functional in Ref. 13 (see Sec.
AcknowledgmentsThis work has been supported by DOE grant DE-FG02-05ER46203 (S.P.) and by the Academy of Finland (E.R.).
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we use the effective-mass approximation with the typical GaAs parameters: m * = 0.067 m0 and ε = 12.4 ε0. Hence, the energies, lengths, and magneticfield strengths scale as E * h = (m * /m0)/(ε/ε0) 2 E h ≈ 12 meV, a * 0 = (ε/ε0)/(m * /m0)a0 ≈ 10 nm, and B * 0 = (m * /m0) 2 /(ε/ε0) 2 B0 ≈ 6.9 T, respectively. When transforming the (effective) atomic units given in the paper to SI units. In the paper the magnetic fields are always given in Tesla for clarityWhen transforming the (effective) atomic units given in the paper to SI units, we use the effective-mass approximation with the typical GaAs parameters: m * = 0.067 m0 and ε = 12.4 ε0. Hence, the energies, lengths, and magnetic- field strengths scale as E * h = (m * /m0)/(ε/ε0) 2 E h ≈ 12 meV, a * 0 = (ε/ε0)/(m * /m0)a0 ≈ 10 nm, and B * 0 = (m * /m0) 2 /(ε/ε0) 2 B0 ≈ 6.9 T, respectively. In the paper the magnetic fields are always given in Tesla for clarity.
. M C Rogge, E Räsänen, R J Haug, Phys. Rev. Lett. 10546802M. C. Rogge, E. Räsänen, and R. J. Haug, Phys. Rev. Lett. 105, 046802 (2010).
. E Räsänen, H Saarikoski, A Harju, M Ciorga, A S Sachrajda, Phys. Rev. B. 7741302E. Räsänen, H. Saarikoski, A. Harju, M. Ciorga, and A. S. Sachrajda, Phys. Rev. B 77, 041302(R) (2008).
Note that here the LSDA correlation energies have been calculated from the self-consistent EXX-KLI densities, whereas in Refs. 12 and 13 the LSDA correlation energies have been obtained self-consistently. The differences between the results are small. Note that here the LSDA correlation energies have been calculated from the self-consistent EXX-KLI densities, whereas in Refs. 12 and 13 the LSDA correlation ener- gies have been obtained self-consistently. The differences between the results are small.
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"On the Brannan's conjecture",
"On the Brannan's conjecture"
]
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"Róbert Szász "
]
| []
| []
| We will prove the Brannan conjecture for particular values of the parameter. The basic tool of the study is an integral representation published in a recent work[3]. 1 | 10.1007/s00009-019-1469-9 | [
"https://arxiv.org/pdf/1710.09153v1.pdf"
]
| 119,118,076 | 1710.09153 | 0277c32c7ef60986619eef13337e1c08a69a27b1 |
On the Brannan's conjecture
25 Oct 2017
Róbert Szász
On the Brannan's conjecture
25 Oct 2017
We will prove the Brannan conjecture for particular values of the parameter. The basic tool of the study is an integral representation published in a recent work[3]. 1
Introduction
We consider the following Mac-Laurin development
(1 + xz) α (1 − z) β = ∞ n=0 A n (α, β, x)z n ,(1)
where α > 0, β > 0, x = e iθ , θ ∈ [−π, π], and z ∈ U. It is easily seen, that the radius of convergence of the series (1) is equal to 1. In [5] the author conjectured, that if α > 0, β > 0 and |x| = 1, then
|A 2n−1 (α, β, x)| ≤ A 2n−1 (α, β, 1),
where n is a natural number. Partial results regarding this question were already proved in [1], [2], [5], [8].
The case β = 1, α ∈ (0, 1) is still open. Regarding this case partial results were obtained in [3], [4], [6], [7]. We will prove some partial results regarding the case β = 1, and α ∈ (0, 1). In order to do this we will use an integral representation proved in [3], and the fact that the conjecture was proved for 2n − 1 ≤ 51 in [6]. 1
Preliminaries
We inroduce the notation:
A n (α, 1, x) = A n (α, x).
It is easily seen that
A 2n−1 (α, x) = 2n−1 k=0 (−α) k (−x) k k! = 1 + α 1! x − α(1 − α) 2! x 2 + α(1 − α)(2 − α) 3! x 3 (2) − α(1 − α)(2 − α)(3 − α) 4! x 4 + . . . + α(1 − α)(2 − α) . . . (2n − 2 − α) (2n − 1)! x 2n−1 .
We denote B(t, θ) = cos θ + t + t 2n−1 cos 2nθ + t 2n cos(2n − 1)θ 1 + t 2 + 2t cos θ , C(t, θ) = sin θ + t 2n−1 sin 2nθ + t 2n sin(2n − 1)θ 1 + t 2 + 2t cos θ .
We need the following lemmas in our study.
where F ′ (t) = −t −1−α (1 − t) α−1 , and F (1) = 0.
Proof. We use two equalities in order to prove the assertion of the lemma, the first one is the following:
A n (α, x) = 1 + 1 Γ(α)Γ(−α) 1 0 n k=1 (−tx) k k t −α−1 (1 − t) α−1 dt,(3)
which has been deduced in [3], while the second one is the well-known equality
B(p, q) = 1 0 t p−1 (1 − t) q−1 dt = Γ(p)Γ(q) Γ(p+q) .
Replacing p = α and q = 1 − α, we get
1 0 t α−1 (1 − t) −α dt = Γ(α)Γ(1 − α) Γ(1) = −αΓ(α)Γ(−α) = 1 0 t −α (1 − t) α−1 dt.
This equality and (3) imply that
Γ(α)Γ(−α)A n (α, x) = 1 α 1 0 tF ′ (t)dt − 1 0 n k=1 (−tx) k k F ′ (t)dt, or equivalently −Γ(α)Γ(−α)A n (α, x) = 1 0 F ′ (t) −t α + n k=1 (−tx) k k dt.
Now integrating by parts and using that lim t→0
F (t) t α + n k=1 (−tx) k k = 0 and lim t→1 F (t) t α + n k=1 (−tx) k k = 0, we infer that −Γ(α)Γ(−α)A n (α, x) = 1 0 F (t) 1 α + n k=1 (−t) k−1 x k dt.
We get the desired equality replacing n by 2n − 1.
Remark 1. It is easily seen that the condition α ∈ (0, 1) implies the existence of the integrals in the previous lemma and its proof.
We denote
Φ(θ) = −Γ(α)Γ(−α)A 2n−1 (α, e iθ ) = 1 0 F (t) 1 α + cos θ + t + t 2n−1 cos 2nθ + t 2n cos(2n − 1)θ 1 + t 2 + 2t cos θ(4)
+i sin θ + t 2n−1 sin 2nθ + t 2n sin(2n − 1)θ 1 + t 2 + 2t cos θ dt.
Since |Φ(θ)| 2 ∈ R it follows that
|Φ(θ)| 2 = Φ(θ)Φ(θ) = 1 0 F (t) 1 α + cos θ + t + t 2n−1 cos 2nθ + t 2n cos(2n − 1)θ 1 + t 2 + 2t cos θ +i sin θ + t 2n−1 sin 2nθ + t 2n sin(2n − 1)θ 1 + t 2 + 2t cos θ dt 1 0 F (v) 1 α + cos θ + v + v 2n−1 cos 2nθ + v 2n cos(2n − 1)θ 1 + v 2 + 2v cos θ (5) −i sin θ + v 2n−1 sin 2nθ + v 2n sin(2n − 1)θ 1 + v 2 + 2v cos θ dv = 1 0 1 0 F (t)F (v) 1 α + cos θ + t + t 2n−1 cos 2nθ + t 2n cos(2n − 1)θ 1 + t 2 + 2t cos θ 1 α + cos θ + v + v 2n−1 cos 2nθ + v 2n cos(2n − 1)θ 1 + v 2 + 2v cos θ + sin θ + t 2n−1 sin 2nθ + t 2n sin(2n − 1)θ 1 + t 2 + 2t cos θ sin θ + v 2n−1 sin 2nθ + v 2n sin(2n − 1)θ 1 + v 2 + 2v cos θ . Lemma 2.
(a) Let f, g : [0, 1] → R be two continuous function. If there is a point t * ∈ (0, 1) such that f is decreasing on (t * , 1), and the equation
g(t) = 0 has a unique root t 0 ∈ [t * , 1), such that g(t) ≤ 0, t ∈ [t 0 , 1], g(t) ≥ 0, t ∈ [0, t 0 ], and f (v) ≥ f (t 0 ) for v ∈ [0, t * ], then we have 1 0 f (t)g(t)dt ≥ f (t 0 ) 1 0 g(t)dt.
(b) Let f, g : [0, 1] → R two continuous functions. If f is a decreasing function, and if there is a point t 0 ∈ (0, 1) such that g(t) ≥ 0, t ∈ (0, t 0 ) and
g(t) ≤ 0, t ∈ (t 0 , 1), then 1 0 f (t)g(t)dt ≥ f (t 0 ) 1 0 g(t)dt.
The statement (b) is a particular case of (a). (c) A well-known result is the following statement. (Chebyshev's inequality) If f and g are monotonic functions with different monotony, then
1 0 f (t)g(t)dt ≤ 1 0 f (t)dt 1 0 g(t)dt
and in case of the same monotony we have
1 0 f (t)g(t)dt ≥ 1 0 f (t)dt 1 0 g(t)dt. Proof. We have 1 0 f (t)g(t)dt = t 0 0 f (t)g(t)dt + 1 t 0 f (t)g(t)dt ≥ t 0 0 f (t)g(t 0 )dt + 1 t 0 f (t)g(t 0 )dt = 1 0 f (t)g(t 0 )dt. Lemma 3. If θ ∈ [0, π 2 ]
, and n ≥ 27, then
(a) 1 0 F (t) B(t, 0) − B(t, θ) dt ≥ 1 0 F (t) 1 2 (1 − cos θ) + t 2n−1 (1 − cos 2nθ) + t 2n (1 − cos(2n − 1)θ) (1 + t)(1 + t 2 + 2t cos θ) dt, (6) (b) 1 0 F (t) 1 + B(t, θ) dt ≥ 1 0 F (t) (1 + t)(1 + cos θ) + t 2n−1 (1 + cos 2nθ) + t 2n (1 + cos(2n − 1)θ) 1 + t 2 + 2t cos θ dt. (7)
Proof. According to Lemma 2 (c), we have
1 0 t 2n 1 + t dt ≤ 1 0 t 2n dt 1 0 1 1 + t dt = ln 2 2n + 1 . (8) We use assertion (b) of Lemma 2 putting f (t) = F (t) 1+t 2 +2t cos θ and g(t) = 1 2 −t−2t 2n 1+t . If θ ∈ 0, π 2 , then the mapping f : [0, 1] → [0, ∞) is strictly de- creasing and we get 1 0 F (t) 1 2 − t − 2t 2n (1 + t)(1 + t 2 + 2t cos θ) dt ≥ F (t n ) 1 + t 2 n + 2t n cos θ 1 0 1 2 − t 1 + t −2 t 2n 1 + t dt ≥ F (t n ) 1 + t 2 n + 2t n cos θ 3 2 ln 2 − 1 − 2 ln 2 2n + 1 > 0,(9)
where t n denotes the unique root of the equation 1 2 − t − 2t 2n = 0, in the interval t ∈ (0, 1). The following equality holds:
1 0 F (t) B(t, 0) − B(t, θ) dt = 1 0 F (t) 1 2 (1 − cos θ) (1 + t)(1 + t 2 + 2t cos θ) dt + 1 0 F (t) (t 2n + t 2n−1 )(1 − cos 2nθ) (1 + t)(1 + t 2 + 2t cos θ) dt + 1 0 F (t) (t 2n + t 2n+1 )(1 − cos(2n − 1)θ) (1 + t)(1 + t 2 + 2t cos θ) dt + 1 0 F (t) ( 1 2 − t − 2t 2n )(1 − cos θ) (1 + t)(1 + t 2 + 2t cos θ) dt. (10)
The equality (10) and the inequality (9) imply that
1 0 F (t) B(t, 0) − B(t, θ) dt ≥ 1 0 F (t) 1 2 (1 − cos θ) (1 + t)(1 + t 2 + 2t cos θ) dt + 1 0 F (t) t 2n−1 (1 − cos 2nθ) (1 + t)(1 + t 2 + 2t cos θ) dt + 1 0 F (t) t 2n (1 − cos(2n − 1)θ) (1 + t)(1 + t 2 + 2t cos θ) dt. (11) We use assertion (b) of Lemma 2 putting f (t) = F (t) 1+t 2 +2t cos θ and g(t) = t 2 − 2t 2n − t 2n−1 . If θ ∈ [0, π 2 ]
, then f is strictly decreasing and we get
1 0 F (t) t 2 − t 2n − t 2n−1 1 + t 2 + 2t cos θ dt ≥ F (t n ) 1 + t 2 n + 2t n cos θ 1 0 (t 2 − t 2n − t 2n−1 )dt = F (t n ) 1 + t 2 n + 2t n cos θ 1 3 − 1 2n + 1 − 1 2n > 0. (12)
In order to finish the proof of the second inequality, we take notice of the fact that in case θ ∈ [0, π 2 ] each member of the following sum is positive:
1 0 F (t) 1 + B(t, θ) dt = 1 0 F (t) (1 + t)(1 + cos θ) 1 + t 2 + 2t cos θ dt + 1 0 F (t) t 2n−1 (1 + cos 2nθ) (1 + t)(1 + t 2 + 2t cos θ) dt + 1 0 F (t) t 2n (1 + cos(2n − 1)θ) 1 + t 2 + 2t cos θ dt + 1 0 F (t) t 2 − t 2n − t 2n−1 1 + t 2 + 2t cos θ dt + 1 0 F (t) t cos θ 1 + t 2 + 2t cos θ dt.
Thus we get
1 0 F (t) 1 + B(t, θ) dt ≥ 1 0 F (t) (1 + t)(1 + cos θ) (1 + t)(1 + t 2 + 2t cos θ) dt + 1 0 F (t) t 2n−1 (1 + cos 2nθ) (1 + t)(1 + t 2 + 2t cos θ) dt + 1 0 F (t) t 2n (1 + cos(2n − 1)θ) (1 + t)(1 + t 2 + 2t cos θ) dt,
and the proof is done.
Lemma 4. If θ ∈ π 2 , 2π 3 , then 5 2 + t + cos θ 1 + t 2 + 2t cos θ ≥ 50 23 (1 + t)(1 + cos θ) 1 + t 2 + 2t cos θ , (∀) t ∈ [0, 1].(13)
Proof. The inequality (13) is equivalent to
f (t) = 5 2 t 2 + t 65 23 cos θ − 27 23 + 15 46 − 27 23 cos θ ≥ 0, (∀) t ∈ [0, 1]. (14)
The function f has a minimum point at t * = 65 115 cos θ − 27 115 ∈ (0, 1), for every θ ∈ π 2 , 2π 3 . Thus we get where g(x) = − 1 5 + 5 13 x − 1 10 x 2 . and consequently (13) holds.
f (t) ≥ f (t * ) =
Lemma 5. If θ ∈ π 2 , 2π 3 , and n ≥ 27, then
1 0 F (t) B(t, 0) − B(t, θ) dt ≥ 1 0 F (t) 27 50 (1 − cos θ) + t 2n−1 (1 − cos 2nθ) + t 2n (1 − cos(2n − 1)θ) (1 + t)(1 + t 2 + 2t cos θ) dt (15) 1 0 F (t) 2 + B(t, 0) + B(t, θ) dt ≥ 1 0 F (t)F (t) 27 50 − t − 2t 2n (1 + t)(1 + t 2 + 2t cos θ) dt ≥ F (t n ) (1 + t n )(1 + t 2 n + 2t n cos θ) 1 0 27 50 − t −2t 2n dt > F (t n ) (1 + t n )(1 + t 2 n + 2t n cos θ) 27 50 − 1 2 − 2 2n + 1 > 0 (17)
where t n denotes the unique root of the equation 27 50 − t − 2t 2n = 0, in the interval (0, 1).
1 0 F (t) B(t, 0) − B(t, θ) dt = 1 0 F (t) 23 50 (1 − cos θ) (1 + t)(1 + t 2 + 2t cos θ) dt + 1 0 F (t) (t 2n + t 2n−1 )(1 − cos 2nθ) (1 + t)(1 + t 2 + 2t cos θ) dt + 1 0 F (t) (t 2n + t 2n+1 )(1 − cos(2n − 1)θ) (1 + t)(1 + t 2 + 2t cos θ) dt + 1 0 F (t) 27 50 − t − 2t 2n (1 − cos θ) (1 + t)(1 + t 2 + 2t cos θ) dtv(18)
The equality (18) and the inequality (17) imply (15). In order to prove the second inequality, we remark that
1 0 F (t) 2 + B(t, 0) + B(t, θ) dt = 1 0 F (t) 5 2 + 1 + t 2n−1 1 + t − 1 2 + B(t, θ) dt = 1 0 F (t) 5 2 + 1 + 2t 2n−1 − t 2(1 + t) + t + cos θ + t 2n cos(2n − 1)θ + t 2n−1 cos(2nθ) 1 + t 2 + 2t cos θ dt (19) = 1 0 F (t) 5 2 +
t + cos θ 1 + t 2 + 2t cos θ + 1 + 2t 2n−1 − t 2(1 + t) + t 2n cos(2n − 1)θ + t 2n−1 cos(2nθ) 1 + t 2 + 2t cos θ dt = 1 0 F (t) 5 2 + t + cos θ 1 + t 2 + 2t cos θ + 2t 2n (1 + cos(2n − 1)θ) + 2t 2n−1 (1 + cos(2nθ)) 1 + t 2 + 2t cos θ dt
+ 1 0 F (t) 2t 2n−1 1 + t dt + 1 0 F (t) 1 − t 2(1 + t) − t 2n (2 + cos(2n − 1)θ) + t 2n−1 (2 + cos(2nθ)) 1 + t 2 + 2t cos θ dt
We put g 1 (t) = 1 − 2t + 2t 2 − t 3 − 24t 2n−1 g 2 (t) = 2t 3 − 8t 2n−1 , and f (t) = F (t) 2(1+t 3 ) in the assertion (b) of Lemma 2, and we get
1 0 F (t) 1 − t 2(1 + t) − t 2n (2 + cos(2n − 1)θ) + t 2n−1 (2 + cos(2nθ)) 1 + t 2 + 2t cos θ dt ≥ 1 0 F (t) 1 − t 2(1 + t) − 3t 2n + 3t 2n−1 1 + t 2 + 2t cos θ dt ≥ 1 0 F (t) 1 − t 2(1 + t) (20) − 6t 2n−1 1 + t 2 − t dt ≥ 1 0 F (t) 2(1 + t 3 ) 1 − 2t + 2t 2 − t 3 − 12t 2n−1 (1 + t) dt ≥ 1 0 F (t) 2(1 + t 3 ) 1 − 2t + 2t 2 − t 3 − 24t 2n−1 dt F (t * ) 2(1 + (t * ) 3 ) 1 0 1 − 2t + 2t 2 − t 3 − 24t 2n−1 dt = F (t * ) 2(1 + (t * ) 3 ) 5 12 − 12 n > 0, θ ∈ π 2 , 2π 3 .
Finally equality (19), Lemma 4 and inequality (20) imply (16), and the proof is done.
The Main Result
Theorem 1. If n is a natural number, n ≥ 52 and α ∈ (0, 1) then the following inequality holds
|A 2n−1 (α, e iθ )| ≤ |A 2n−1 (α, 1)|, for all θ ∈ [− π 2 , π 2 ].(21)
Proof. According to (4) and (5) the inequality (21) is equivalent to
|Φ(0)| 2 ≥ |Φ(θ)| 2 , θ ∈ [−π, π] ⇔ |Φ(0)| 2 − |Φ(θ)| 2 ≥ 0, θ ∈ [− π 2 , π 2 ]. (22)
We denote B(t, θ) = cos θ + t + t 2n−1 cos 2nθ + t 2n cos(2n − 1)θ 1 + t 2 + 2t cos θ , C(t, θ) = sin θ + t 2n−1 sin 2nθ + t 2n sin(2n − 1)θ 1 + t 2 + 2t cos θ .
The equality (5) implies that
Φ 2 (0) − |Φ(θ)| 2 = 1 0 1 0 F (t)F (v) 1 α + B(t, 0) 1 α + B(v, 0) (23) − 1 α + B(t, θ) 1 α + B(v, θ) − C(t, θ)C(v, θ) dtdv = 1 0 1 0 F (t)F (v) 1 α B(t, 0) − B(t, θ) + 1 α B(v, 0) − B(v, θ) +B(t, 0)B(v, 0) − B(t, θ)B(v, θ) − C(t, θ)C(v, θ) dtdv.
It is easily seen that Φ 2 (0) − |Φ(θ)| 2 is an even function with respect to θ, consequently we have to prove (22) only for θ ∈ [0, π 2 ]. Lemma 3, (a) implies that in case θ ∈ [0, π 2 ], the following inequality holds
1 0 F (t) B(t, 0) − B(t, θ) dt ≥ 0. Thus we infer that Φ 2 (0) − |Φ(θ)| 2 = 1 0 1 0 F (t)F (v) B(t, 0) − B(t, θ) 1 + B(v, θ) + B(v, 0) − B(v, θ) 1 + B(t, 0) dtdv − 1 0 1 0 F (t)F (v)C(t, θ)C(v, θ)dtdv + 1 0 1 0 F (t)F (v) B(t, 0) − B(t, θ) B(v, 0) − B(v, θ) dtdv = 1 0 1 0 F (t)F (v) B(t, 0) − B(t, θ) 1 + B(v, θ) + B(v, 0) − B(v, θ) 1 + B(t, 0) dtdv − 1 0 1 0 F (t)F (v)C(t, θ)C(v, θ)dtdv + 1 0 F (t) B(t, 0) − B(t, θ) dt 1 0 F (v) B(v, 0) − B(v, θ) dv ≥ 1 0 1 0 F (t)F (v) B(t, 0) − B(t, θ) 1 + B(v, θ) + B(v, 0) − B(v, θ) 1 + B(t, 0) dtdv − 1 0 1 0 F (t)F (v)C(t, θ)C(v, θ)dtdv
This inequality is equivalent to
Φ 2 (0) − |Φ(θ)| 2 ≥ 1 0 F (t) B(t, 0) − B(t, θ) dt 1 0 F (v) 1 + B(v, θ) dv + 1 0 F (t) 1 + B(t, θ) dt 1 0 F (v) B(v, 0) − B(v, θ) dv (24) − 1 0 1 0 F (t)F (v)C(t, θ)C(v, θ)dtdv
The inequality between the arithmetic and geometric means leads to
1 0 F (t) B(t, 0) − B(t, θ) dt 1 0 F (v) 1 + B(v, θ) dv + 1 0 F (t) 1 + B(t, θ) dt 1 0 F (v) B(v, 0) − B(v, θ) dv (25) − 1 0 1 0 F (t)F (v)C(t, θ)C(v, θ)dtdv ≥ 2 1 0 F (t) B(t, 0) − B(t, θ) dt 1 0 F (v) 1 + B(v, θ) dv 1 0 F (t) 1 + B(t, θ) dt 1 0 F (v) B(v, 0) − B(v, θ) dv 1 2 − 1 0 1 0 F (t)F (v)C(t, θ)C(v, θ)dtdv = 2 1 0 F (t) 1 + B(t, θ) dt 1 0 F (t) B(t, 0) − B(t, θ) dt − 1 0 F (t)C(t, θ)dt 2
The Cauchy-Schwarz inequality for integrals implies
F (t) 1 + B(t, θ) B(t, 0) − B(t, θ) 1 2 dt 2 (27) − 1 0 F (t)C(t, θ)dt 2 ≥ 2 1 0 F (t) 1 2 (1 − cos θ) + t 2n−1 (1 − cos 2nθ) + t 2n (1 − cos(2n − 1)θ) (1 + t)(1 + t 2 + 2t cos θ)
(1 + t)(1 + cos θ) + t 2n−1 (1 + cos 2nθ) + t 2n (1 + cos(2n − 1)θ) 1 + t 2 + 2t cos θ dt
2 − 1 0 F (t)C(t, θ)dt 2 .
Putting a 2 1 = 1 2 (1−cos θ) (1+v)(1+v 2 +2v cos θ) , b 2 1 = (1+t)(1+cos θ) 1+v 2 +2v cos θ and so an, in the inequality
a 2 1 + a 2 2 + a 2 3 b 2 1 + b 2 2 + b 2 3 ≥ |a 1 b 1 | + |a 2 b 2 | + |a 3 b 3 |, we get 2 1 0 F (t) 1 2 (1 − cos θ) + t 2n−1 (1 − cos 2nθ) + t 2n (1 − cos(2n − 1)θ) (1 + t)(1 + t 2 + 2t cos θ) (1 + t)(1 + cos θ) + t 2n−1 (1 + cos 2nθ) + t 2n (1 + cos(2n − 1)θ) 1 + t 2 + 2t cos θ dt 2 (28) − 1 0 F (t)C(t, θ)dt 2 ≥ 2 1 0 F (t) 1 2 (1 + t)| sin θ| + t 2n−1 | sin 2nθ| + t 2n | sin(2n − 1)θ| (1 + t 2 + 2t cos θ) √ 1 + t dt 2 − 1 0 F (t)C(t, θ)dt 2 .
On the other hand we have
2 1 0 F (t) 1 2 (1 + t)| sin θ| + t 2n−1 | sin 2nθ| + t 2n | sin(2n − 1)θ| (1 + t 2 + 2t cos θ) √ 1 + t dt 2 (29) − 1 0 F (t)C(t, θ)dt 2 ≥ 1 0 F (t) (1 + t)| sin θ| + √ 2t 2n−1 | sin 2nθ| + √ 2t 2n | sin(2n − 1)θ| (1 + t 2 + 2t cos θ) √ 1 + t dt 2 − 1 0 F (t)C(t, θ)dt 2 ≥ 1 0 F (t) | sin θ| + t 2n−1 | sin 2nθ| + t 2n | sin(2n − 1)θ| 1 + t 2 + 2t cos θ dt 2 − 1 0 F (t)
sin θ + t 2n−1 sin 2nθ + t 2n sin(2n − 1)θ 1 + t 2 + 2t cos θ dt
2 ≥ 0, θ ∈ [0, π 2 ],
Finally the inequalities (24), (25), (26), (27), (28) and (29) imply that
Φ 2 (0) − |Φ(θ)| 2 ≥ θ ∈ [0, π 2 ],
and consequently inequalities (22) and (21) hold in case θ ∈ [− π 2 , π 2 ].
Theorem 2. If n is a natural number, n ≥ 51 and α ∈ (0, 1), then the following inequality holds
|A 2n−1 (α, e iθ )| ≤ |A 2n−1 (α, 1)|, for all θ ∈ [− 2π 3 , − π 2 ] ∪ [ π 2 , 2π 3 ](30)
Proof. Equality (23) can be rewritten as follows
2 Φ 2 (0) − |Φ(θ)| 2 = 1 0 F (t) B(t, 0) − B(t, θ) dt 1 0 F (v) 2 α + B(v, 0) + B(v, θ) dv (31) + 1 0 F (v) B(v, 0) − B(v, θ) dv 1 0 F (t) 2 α + B(t, 0) + B(t, θ) dt −2 1 0 1 0 F (t)F (v)C(t, θ)C(v, θ)dtdv.
Since 2 Φ 2 (0) − |Φ(θ)| 2 defines an even function in order to prove (30) it is enough to show that
Φ 2 (0) − |Φ(θ)| 2 ≥ 0, θ ∈ π 2 , 2π 3 .(32)
The inequality between the arithmetic and geometric means and the condition α ∈ (0, 1] imply (1 + t)(1 + cos θ) + 2t 2n−1 (1 + cos 2nθ) + 2t 2n (1 + cos(2n − 1)θ) 1 + t 2 + 2t cos θ dt − 1 0 F (t)C(t, θ)dt 2 We apply twice the Cauchy-Schwarz inequality jest like in the proof of the previous theorem and we get that in order to prove (34) it is enough to show that Φ 2 (0) − |Φ(θ)| 2 ≥ 1 0 F (t) √ 1 + t| sin θ| + √ 2t 2n−1 | sin 2nθ| + √ 2t 2n | sin(2n − 1)θ| (1 + t 2 + 2t cos θ) √ 1 + t dt 2 − 1 0 F (t) sin θ + t 2n−1 sin 2nθ + t 2n sin(2n − 1)θ 1 + t 2 + 2t cos θ dt 2 This inequality is equivalent to Φ 2 (0) − |Φ(θ)| 2 ≥ 1 0 F (t) | sin θ| 1 + t 2 + 2t cos θ dt + 1 0 F (t) √ 2t 2n−1 | sin 2nθ| (1 + t 2 + 2t cos θ) √ 1 + t dt (35)
2 Φ 2 (0) − |Φ(θ)| 2 ≥ 2 1 0 F (t) B(t, 0) − B(t, θ) dt 1 0 F (v) 2 + B(v, 0) (33) +B(v, θ) dv 1 0 F (v) B(v, 0) − B(v, θ) dv+ 1 0 F (t) √ 2t 2n | sin(2n − 1)θ| (1 + t 2 + 2t cos θ) √ 1 + t dt 2 − 1 0 F (t)
sin θ 1 + t 2 + 2t cos θ dt + 1 0 F (t) t 2n−1 sin 2nθ (1 + t 2 + 2t cos θ) dt
+ 1 0 F (t) t 2n sin(2n − 1)θ (1 + t 2 + 2t cos θ) √ 1 + t dt 2 ≥ 0, θ ∈ π 2 , 2π 3 .
and the proof is done.
Theorem 3. If n is a natural number, n ≥ 27, and α ∈ [ 1 3 , 1), then the following inequality holds
Lemma 1 .
1For α ∈ (0, 1) the following equality holds:Φ(θ) = −Γ(α)Γ(−α)A 2n−1 (α, e iθ )
+ B(t, θ) + iC(t, θ) dt,
t)(1 + cos θ) + 2t 2n−1 (1 + cos 2nθ) + 2t 2n (1 + cos(2n − 1)θ)1 + t 2 + 2t cos θ dt (16) Proof. We use assertion (b) of Lemma 2, putting f (t) = F (t) (1+t)(1+t 2 +2t cos θ) and g(t) = 27 50 − t − 2t 2n . If θ ∈ π 2 ,2π 3 the mapping f : [0, 1] → [0, ∞) is strictly decreasing and we get 1 0
F
(t) B(t, 0) − B(t, θ) dt
1 − cos θ) + t 2n−1 (1 − cos 2nθ) + t 2n (1 − cos(2n − 1)θ) (1 + t)(1 + t 2 + 2t cos θ)
Proof. We will use the Taylor formula with an integral remainder. Let f : (−a, a) → R be a 2n times derivabile function, such that f (2n) is continous. If x ∈ (−a, a), thenLet f be the function defined by f : (−1, 1) → R, f (x) = (1+x) α , α ∈ (0, 1) and we getSince the mapping f : U → R, f (z) = (1 + z) α is well defined(we take the principal branch of the multi valued function) it follows that the equalityholds for every z ∈ U. The mapping f : U → R, f (z) = (1 + z) α is radially continuous and so we infer thatTaking the absolute value, this equality implies thatOn the other hand we haveThus in order to prove (36) we have to show that the following inequality holdsWe denote x = − cos θ, and the inequality (37) will be equivalent toand this inequality can be rewritten in the following form:It is easily seen thatis decreasing with respect to n and x, and< 0, for all α ∈ (0, 1), and x ∈ [ 1 2 , 1]. Thus in order to prove (38) it is enough to prove the inequalityin case α = 1, that is).This inequality holds and the proof is done.Concluding RemarcsThe following two corollaries are the proof of the Brannan conjecture in two different particular cases. Theorem 1, Theorem 2 and the result of[6]imply the following corollary.Corollary 1. If x ∈ C with | arg x| ≤ 2π 3 , and |x| = 1, then the inequalityholds for every α ∈ (0, 1).Theorem 1, Theorem 2, Theorem 3 and the result of[6]imply the following corollary. This corollary is the solution of the Brannan conjecture in case α ∈ [ 1 2 , 1). Corollary 2. The inequalityholds for every α ∈ [ 1 3 , 1), and x ∈ C, with |x| = 1. Conjecture 1. Numerical resuls suggest that the inequalityholds for every α ∈ (0, 1 3 ). Remark 2. If the previous conjecture holds, then the conjecture of Brannan holds in case β = 1 and every α ∈ (0, 1).
On an inequality connected with the coefficient conjecture for functions of bounded boundary rotation. D Aharonov, S Friedland, 14. MR0322155 (48Ann. Acad. Sci. Fenn. Ser. A I. 524519D. Aharonov and S. Friedland, On an inequality connected with the coefficient conjecture for functions of bounded boundary rotation, Ann. Acad. Sci. Fenn. Ser. A I 524 (1972), 14. MR0322155 (48, 519)
Brannans coefficient conjecture for certain power series, Open problems and conjectures in complex analysis. R W Barnard, Computational Methods and Function Theory. SpringerLecture notes in Math. MR1071758 (91j:12001R. W. Barnard, Brannans coefficient conjecture for certain power se- ries, Open problems and conjectures in complex analysis, Computational Methods and Function Theory (Valparaso, 1989), 1-26. Lecture notes in Math. 1435, Springer, Berlin, 1990. MR1071758 (91j:12001)
Solynin, Brannan's conjecture and trigonometric sums. R W Barnard, C Udaya, Alexander Jayatilake, Yu, 2117-2128 S 0002- 9939Proc. Amer. Math. Soc. 1435R. W. Barnard, Udaya C. Jayatilake, and Alexander Yu. Solynin, Brannan's conjecture and trigonometric sums, Proc. Amer. Math. Soc. Volume 143, Number 5, May 2015, Pages 2117-2128 S 0002- 9939(2015)12398-2
On a coefficient conjecture of Brannan. Roger W Barnard, Kent Pearce, William Wheeler, Complex Variables Theory Appl. 331-4, 5161. MR1624894 (98m:30021Roger W. Barnard, Kent Pearce, and William Wheeler, On a coefficient conjecture of Brannan, Complex Variables Theory Appl. 33 (1997), no. 1-4, 5161. MR1624894 (98m:30021)
On coefficient problems for certain power series. D A Brannan, Proceedings of the Symposium on Complex Analysis (Univ. Kent, Canterbury, 1973). the Symposium on Complex Analysis (Univ. Kent, Canterbury, 1973)LondonCambridge Univ. Press537No. 12. MR0412411 (54D. A. Brannan, On coefficient problems for certain power series, Pro- ceedings of the Symposium on Complex Analysis (Univ. Kent, Canter- bury, 1973), Cambridge Univ. Press, London, 1974, pp. 1727. London Math. Soc. Lecture Note Ser., No. 12. MR0412411 (54 ,537)
Brannans conjecture for initial coefficients. C Udaya, Jayatilake, Complex Var. Elliptic Equ. 585Udaya C. Jayatilake, Brannans conjecture for initial coefficients. Com- plex Var. Elliptic Equ. 58 (2013), no. 5, 685-694.
On a coefficient conjecture of Brannan. John G Milcetich, DOI10.1016/0022-247X(89)90125-X.MR996975J. Math. Anal. Appl. 139290d:30006John G. Milcetich, On a coefficient conjecture of Brannan, J. Math. Anal. Appl. 139 (1989), no. 2, 515-522, DOI 10.1016/0022- 247X(89)90125-X. MR996975 (90d:30006)
On Brannans coefficient conjecture and applications. Stephan Ruscheweyh, Luis Salinas, DOI10.1017/S0017089507003400.MR2337865Glasg. Math. J. 49130048Stephan Ruscheweyh and Luis Salinas, On Brannans coefficient con- jecture and applications, Glasg. Math. J. 49 (2007), no. 1, 45-52, DOI 10.1017/S0017089507003400. MR2337865 (2008f:30048)
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[
"INVERTIBILITY THRESHOLD FOR H ∞ TRACE ALGEBRAS, AND EFFECTIVE MATRIX INVERSIONS",
"INVERTIBILITY THRESHOLD FOR H ∞ TRACE ALGEBRAS, AND EFFECTIVE MATRIX INVERSIONS"
]
| [
"Nikolai Nikolski ",
"Vasily Vasyunin "
]
| []
| []
| For a given δ, 0 < δ < 1, a Blaschke sequence σ = {λ j } is constructed such that every function f , f ∈ H ∞ , having δ < δ f = inf λ∈σ |f (λ)| ≤ f ∞ ≤ 1 is invertible in the trace algebra H ∞ |σ (with a norm estimate of the inverse depending on δ f only), but there exists f with δ = δ f ≤ f ∞ ≤ 1, which does not. As an application, a counterexample to a stronger form of the Bourgain-Tzafriri restricted invertibility conjecture for bounded operators is exhibited, where an "orthogonal (or unconditional) basis" is replaced by a "summation block orthogonal basis". | null | [
"https://arxiv.org/pdf/1010.6090v1.pdf"
]
| 119,128,553 | 1010.6090 | 0b75cdf735d1b69511aec2cc7d735dee6afb4c24 |
INVERTIBILITY THRESHOLD FOR H ∞ TRACE ALGEBRAS, AND EFFECTIVE MATRIX INVERSIONS
28 Oct 2010
Nikolai Nikolski
Vasily Vasyunin
INVERTIBILITY THRESHOLD FOR H ∞ TRACE ALGEBRAS, AND EFFECTIVE MATRIX INVERSIONS
28 Oct 2010Dedicated to the memory of M. S. Birman, from whom both of us were learned a lot (and not only mathematics)
For a given δ, 0 < δ < 1, a Blaschke sequence σ = {λ j } is constructed such that every function f , f ∈ H ∞ , having δ < δ f = inf λ∈σ |f (λ)| ≤ f ∞ ≤ 1 is invertible in the trace algebra H ∞ |σ (with a norm estimate of the inverse depending on δ f only), but there exists f with δ = δ f ≤ f ∞ ≤ 1, which does not. As an application, a counterexample to a stronger form of the Bourgain-Tzafriri restricted invertibility conjecture for bounded operators is exhibited, where an "orthogonal (or unconditional) basis" is replaced by a "summation block orthogonal basis".
Introduction
The paper deals with a numerical control of inverses (condition numbers) for functions T = f (A) of large matrices in terms of the lower spectral parameter δ = δ(T ) = min |λ j (T )| Precisely, our problem is the following. Given a sequence σ = {λ j } in the unit disk D = {z ∈ C : |z| < 1} of the complex plain, we consider all normalized matrices A, A ≤ 1 (or Hilbert space operators) such that σ(A) ⊂ σ (counting multiplicities) and look for a numerical function c(δ) = c(δ, σ) bounding the inverses
T −1 ≤ c(δ)
for all T = f (A) having δ ≤ |λ j (T )| ≤ T ≤ 1, where λ j (T ) mean eigenvalues of T = f (A). The best possible upper bound c(δ) is called Date: September 12, 2010. Key words and phrases. Effective inversions H ∞ trace algebra, invisible spectrum, critical constant, interpolation Blaschke product, Bourgain-Tzafriri restricted invertibility conjecture.
V. Vasyunin's research was supported in part by RFBR (grant 08-01-00723). N. Nikolski's research was partially supported by the French ANR Projects DYNOP and FRAB. c 1 (δ) = c 1 (δ, σ),
c 1 (δ, σ) = sup T −1 : T = f (A), δ ≤ |λ j (T )| ≤ T ≤ 1, σ(A) ⊂ σ, A ≤ 1 .
Here f can be a polynomial (if A is a finite matrix) or an H ∞ function (if A is a Hilbert space contraction). Recall that H ∞ = f : f holomorphic on D and f ∞ = sup z∈D |f (z)| < ∞ .
Since δ → c 1 (δ, σ), 0 < δ < 1, is a decreasing function, we can define a critical constant (or, an invertibility threshold) δ 1 = δ 1 (σ), 0 ≤ δ 1 ≤ 1, by the following properties 0 < δ < δ 1 =⇒ c 1 (δ) = ∞ ,
δ 1 < δ ≤ 1 =⇒ c 1 (δ) < ∞ .
The number δ 1 can be considered as a threshold of bounded invertibility or as a threshold for an operator algebra to be inverse closed : operators T from our collection with a "scattered " spectral data (i. e., inf j |λ j (T )| < δ 1 , T = 1) are, in general, not invertible, whereas those with "flat " spectral data δ 1 < δ ≤ |λ j (T )| ≤ T ≤ 1 are invertible.
The principal result of this paper is a construction of a Blaschke sequence σ with a given in advance value of the critical constant δ 1 (σ) = δ 1 , 0 ≤ δ 1 ≤ 1 (Section 2 below).
The case, where δ 1 = 0, was considered in [GMN]; moreover, the paper quoted contains necessary and sufficient conditions for δ 1 (σ) = 0, which reduces to the so-called Weak (Carleson) Embedding Property (WEP). See the statement of the result at the end of this Introduction.
It is worth mentioning that, strictly speaking, the properties of a function algebra A on a set σ to be inverse closed (i. e., the property f ∈ A, inf z∈σ |f (z)| > 0 =⇒ 1/f ∈ A) does not imply that δ 1 (σ, A) = 0 (this fact was already mentioned in [GMN]; the constants c 1 (σ, A) and δ 1 (σ, A) are defined for an algebra A in a similar way). Indeed, for an arbitrary Blaschke sequence σ = {λ j }, the trace algebra A = C a (D)|σ of the disk algebra C a (D) = H ∞ ∩ C(D), is always inverse closed, whereas c 1 (δ, C a (D)|σ) = c 1 (δ, H ∞ |σ) for every δ, 0 < δ < 1, and hence δ 1 (σ, C a (D)|σ) = δ 1 (σ, H ∞ |σ), but the algebra H ∞ |σ can be not inverse closed (i. e. possibly δ 1 (σ, C a (D)|σ) > 0). These properties are shown in [GMN].
The constant c 1 (δ, σ) has a meaning of "the best estimate for the worst case" when bounding inverse matrices in terms of the lower spectral parameter δ. Moreover, we can describe it in two more ways, at least in the case of the "simple spectrum" (the points λ j of the sequence σ are pairwise different). First, it is the optimal upper bound for inverses in the trace algebra
H ∞ |σ = {a : σ → C : ∃f ∈ H ∞ such that a = f |σ} endowed with the trace norm a = inf{ f ∞ : a = f |σ}.
Examples of such algebras with a given threshold δ 1 of the bounded invertibility (Section 2 below) are, probably, of interest for the H ∞ interpolation theory.
Secondly, in the definition of c 1 (δ, σ), we can restrict ourselves to a just one ("the worst") contraction A and the algebra generated by H ∞ functions of it. This is the so-called model contraction M B , which can be defined as follows. Given a Blaschke product
B = B σ B σ = j≥1 b λ j , where b λ = λ−z 1−λz · |λ| λ , λ ∈ D, and σ = {λ j }, j (1 − |λ j |) < ∞ (the Blaschke condition), we set M * B f = f − f (0) z , f ∈ K B ,
where K B = H 2 ⊖ BH 2 (the orthogonal complement of BH 2 in H 2 ) and H 2 stands for the standard Hardy space of the disk,
H 2 = f = k≥0 a k z k : k≥0 |a k | 2 = f 2 2 < ∞ .
It is well known (and easy to verify, see [Nik1], [Nik2]) that
M B K B ⊂ K B , M B = 1,
and σ(M B ) = clos{λ j : j = 1, 2, . . . }.
Moreover, M −1 B = 1/B(0). It is also known that for every matrix A with A ≤ 1 and σ(A) ⊂ σ, one has f (A) ≤ f (M B ) for every function f . This entails that the question on the invertibility and the norm control of inverses can be reduced to functions f (M B ) of the model operator only, and inverse f (
M B ) −1 , if it exists, is again a H ∞ -function of M B .
The above discussion easily implies the following.
(1) If the set {λ ∈ σ : δ ≤ |λ| ≤ δ ′ } is infinite for some 0 < δ ≤ δ ′ < 1, then c 1 (δ, σ) = ∞.
(2) If σ is a sequence tending to the unit circle (i. e., {λ ∈ σ : |λ| ≤ δ} is finite for every δ < 1) and λ∈σ (1 − |λ|) = ∞, then c 1 (δ, σ) = ∞ for every δ, 0 < δ < 1.
These properties show that, in fact, the Blaschke condition is necessary in order our questions (to find or to estimate c 1 (δ, σ) and δ 1 (σ)) to be nontrivial. In what follows we always assume this property (if the converse does not stated explicitly). Now, we can give the following expression for c 1 (δ, σ).
Lemma 1. Let σ be a Blaschke subset of the unit disk D. Then
c 1 (δ, σ) = c 1 (δ, H ∞ |σ) = c 1 (δ, H ∞ /BH ∞ ) for every δ, 0 < δ < 1, where B = B σ and c 1 (δ, H ∞ /BH ∞ ) =: sup 1 f H ∞ /BH ∞ : f ∞ ≤ 1, δ ≤ f (λ) for λ ∈ σ = sup inf g ∞ : gf + hB = 1 : δ ≤ f (λ) ≤ f ∞ ≤ 1 for λ ∈ σ and h H ∞ /BH ∞ means inf{ g ∞ : g(λ) = h(λ) for λ ∈ σ}.
Proof. For every matrix A, A ≤ 1, and f ∈ H ∞ , the von Neumann inequality entails
f (A) ≤ f ∞ . Since B(A) = 0, B = B σ , for A having σ(A) ⊂ σ, we get f (A) ≤ inf g∈H ∞ f + Bg ∞ = f H ∞ /BH ∞ . This implies f (A) −1 = h(A) and f (A) −1 ≤ h H ∞ for every solution h of the equation f h + Bk = 1, and therefore f (A) −1 ≤ 1 f H ∞ /BH ∞ . Thus, c 1 (δ, σ) ≤ c 1 (δ, H ∞ /BH ∞ ).
On the other hand, there exists an "extreme operator" (matrix) for which the above calculus inequality becomes an identity. Indeed, if A = M B , the "model operator" mentioned above, then h(M B ) = h H ∞ /BH ∞ for every h ∈ H ∞ (Sarason's commutant lifting theorem, see for example, [Nik1] or [Nik2]). Hence, c 1 (δ, σ) ≥ c 1 (δ, H ∞ /BH ∞ ).
Finally, we quote the principal result from [GMN] Theorem 2. ( [GMN]) Let σ = {λ j } be a Blaschke sequence in the disk D. The following are equivalent.
(1) δ 1 (H ∞ |σ) = 0.
(2) The following Weak Embedding Property holds: for every ε > 0 there exists C such that
j≥1 (1 − |λ j | 2 )(1 − |z| 2 ) |1 −λ j z| 2 ≤ C for every z ∈ D \ λ∈σ ζ : |b λ (ζ)| < ε . (3) For every ε > 0 there exists η such that |B(z)| ≤ η implies inf λ∈σ |b λ (z)| ≤ ε; here B is the corresponding Blaschke product B = λ∈σ b λ .
Moreover, if η(ε) = max{η} over all η admitted in (3), then
1 η(δ) ≤ c 1 (δ, H ∞ |σ) ≤ a η(δ/3) 2 log 1 η(δ/3) for every δ, 0 < δ < 1; a > 0 is a numerical constant.
The paper is organized as follows. Section 2 contains our principal result: for a given δ, 0 < δ < 1, there exists a Blaschke product B = B σ such that δ 1 (σ, H ∞ |σ) = δ. We also exhibit an upper estimate for c 1 (δ, H ∞ |σ) for δ 1 < δ ≤ 1. Since the problem (and our result) on the invertibility threshold is conformally invariant, we will change the variable and work (in Section 2) in the upper half-plane C + = {z ∈ C : Im(z) > 0} instead of the unit disk D.
In Section 3, we use the above result in order to give a counterexample to a stronger form of the so-called (Bourgain-Tzafriri) restricted invertibility conjecture. The conjecture claims (see [CCLV] and comments in Section 3): for every unconditional normalized basic sequence {x j } j∈J in a Hilbert space H and for every bounded operator T : H → H having inf j∈J T x j > 0 there exists a partition J = r i=1 J i such that all restrictions T |H J i , i = 1, . . . , r, are left invertible; here H J ′ = span{x j : j ∈ J ′ } for every J ′ ⊂ J. The conjecture is still open (June 2010). A stronger form (which is disproved in Section 3) claims the same property but for all summation basic sequences {x j }.
2.
Algebras H ∞ |σ with a given constant δ 1
We start with some geometrical considerations. In this section the symbol b λ always means the Blaschke factor with the zero λ in the upper half-plane, i. e.,
b λ (z) = z − λ z − λ · |1 + λ 2 | 1 + λ 2 .
Lemma 3. The rectangle
z : √ 1 + ε 2 √ 1 + ε 2 + √ 2 ε ≤ Im z Im λ ≤ √ 1 + ε 2 √ 1 + ε 2 − √ 2 ε , | Re(z − λ)| Im λ ≤ √ 2 ε √ 1 − ε 2 is inscribed into the circle {z : |b λ (z)| ≤ ε}. Proof. Put a = Re(z − λ) Im λ and b = Im z Im λ , then |b λ (z)| 2 = z − λ z −λ 2 = a 2 + (b − 1) 2 a 2 + (b + 1) 2 .
We have to check that the vertices of the rectangle are on the mentioned circle, i. e., we need to check that the equality
a 2 + (b − 1) 2 a 2 + (b + 1) 2 = ε 2 holds if a = ± √ 2 ε √ 1 − ε 2 , b = √ 1 + ε 2 √ 1 + ε 2 ± √ 2 ε .
We shall verify the required identity in the form (a 2 + b 2 + 1)(1 − ε 2 ) = 2b(1 + ε 2 ):
(a 2 + b 2 + 1)(1 − ε 2 ) = 2ε 2 1 − ε 2 + 1 + ε 2 ( √ 1 + ε 2 ± √ 2 ε) 2 + 1 (1 − ε 2 ) = 2ε 2 + (1 + ε 2 ) √ 1 + ε 2 ∓ √ 2 ε √ 1 + ε 2 ± √ 2 ε + 1 − ε 2 = (1 + ε 2 ) 2 √ 1 + ε 2 √ 1 + ε 2 ± √ 2 ε = 2b(1 + ε 2 ).
Now, we are using Frostman shifts of an inner function Θ:
Θ c def = Θ + c 1 +cΘ ,
which is known to be a Blaschke product for almost all values c, |c| < 1. In some cases it is easy to check that this is a Blaschke product for all c = 0. For example, this is the case for Θ = e iaz , a > 0. Indeed, the inner function Θ c is analytic in a neighborhood of any real point, therefore it could have a singular factor with a mass at infinity only. But there is no such factor because lim y→+∞ Θ c (iy) = c = 0. For more details, see, for example, [Gar] or [Nik1].
Lemma 4. Let z k be zeroes of the Blaschke product
B α,γ = e πiγz + e −πα 1 + e π(iγz−α) = ∞ k=−∞ b z k ,
i. e., z k = (2k + 1 + iα)/γ, z k ∈ Z. Then the strip
S α,γ = z : α √ 1 + α 2 √ 1 + α 2 + 1 ≤ γ Im z ≤ α √ 1 + α 2 √ 1 + α 2 − 1 is in the set ∞ k=−∞ {z : |b z k (z)| < ε}, if ε > 1/ √ 1 + 2α 2 .
Proof. Apply Lemma 3 with λ = z k and ε = 1/ √ 1 + 2α 2 . Then the sides of the rectangle are
√ 1 + ε 2 √ 1 + ε 2 ± √ 2 ε = √ 1 + α 2 √ 1 + α 2 ± 1 and √ 2 ε √ 1 − ε 2 = 1 α ,
i. e., the rectangle from Lemma 3 is
z : √ 1 + α 2 √ 1 + α 2 + 1 ≤ γ α Im z ≤ √ 1 + α 2 √ 1 + α 2 − 1 , | Re(z − z k )| ≤ 1 γ .
It is clear that the union of these rectangles gives just the required strip.
Remark 1. Let us note that the set
S α,γ \ ∞ k=−∞ z : |b z k (z)| < 1 √ 1 + 2α 2
consists of a discrete set of points
1 γ 2m + α √ 1 + α 2 √ 1 + α 2 ± 1 i
on the upper and lower boundaries of the strip S α,γ and the distance from any such point to the set of zeroes {z k } is equal to 1/ √ 1 + 2α 2 .
Lemma 5. The Blaschke product B = ∞ n=0 B α,β n ρ converges for all α, β, ρ such that α > 0, 0 < β < 1, ρ > 0. If
β = √ 1 + α 2 − 1 √ 1 + α 2 + 1 , 6 - q q q q q q q q q q q q q q q z k = 2k+1+iα ρ q q q q q q q q q q q z k = 2k+1+iα ρβ q q q q q q q z k = 2k+1+iα ρβ 2 q q q q q z k = 2k+1+iα ρβ 3 "! # "! # Figure 1. then the half-plane Π α,ρ = z : Im z ≥ α √ 1 + α 2 ρ( √ 1 + α 2 + 1) is in the set λ∈σ(B) {z : |b λ (z)| < ε} for ε > 1/ √ 1 + 2α 2 .
( Fig. 1 illustrates zeroes of B and four circles |b λ (z)| = 1 √ 1+2α 2 for zeroes λ = ±1+iα ρ and λ = ±1+iα βρ ) Proof. The following estimate implies convergence of B:
1 − B α,β n ρ (i) =1 − e −πβ n ρ + e −πα 1 + e −π(α+β n ρ) = (1 − e −πβ n ρ )(1 − e −πα ) 1 + e −π(α+β n ρ) ≤ (1 − e −πα )πβ n ρ.
It remains to note that for β = ( √ 1 + α 2 − 1)/( √ 1 + α 2 + 1) and γ n = β n ρ the upper boundary of the strip from Lemma 4 for γ = γ n−1 coincides with the lower boundary of the strip for γ = γ n . Therefore the union of these strips gives just the required half-plane.
Remark 2. The set of points on the imaginary axis
v n = 1 ρβ n α √ 1 + α 2 √ 1 + α 2 + 1 i
(the points of intersection of four corresponding circles as on Fig.1
) is included into Π α,ρ \ ∞ λ∈σ(B) z : |b λ (z)| < 1 √ 1 + 2α 2 .
Every point v n has four nearest zeroes of B, namely, (iα±1)/(ρβ n ) and (iα ± 1)/(ρβ n−1 ) with the pseudohyperbolic distance just 1/ √ 1 + 2α 2 from each of them.
Recall that pseudohyperbolic distance between two points z, w ∈ C + is defined by
|b w (z)| = z − w z −w
and between two points z, w ∈ D:
|b w (z)| = z − w 1 −wz .
Lemma 6. For the Blaschke product B = ∞ n=0 B α,β n ρ a lower estimate
|B(x + iy)| ≥ exp − (1 + e −πα ) πρy (e −πρy − e −πα ) (1 − β)
is true in the strip 0 < y < α ρ . In the complementary half-plane y > α ρ we have the following upper estimate
|B(x + iy)| ≤ exp − log(cosh πα) · log ρy α log 1 β .
Proof. For the product B α,γ we have
|B α,γ (x + iy)| 2 = e −2πγy + e −2πα + 2e −π(γy+α) cos πγx 1 + e −2π(γy+α) + 2e −π(γy+α) cos πγx
and therefore
e −πγy − e −πα 1 − e −π(γy+α) ≤ |B α,γ (x + iy)| ≤ e −πγy + e −πα 1 + e −π(γy+α) .
Now, we deduce an estimate from below assuming 0 < y < α ρ :
log 1 |B(x + iy)| = ∞ n=0 log 1 |B α,β n ρ (x + iy)| ≤ ∞ n=0 log 1 − e −π(β n ρy+α) e −πβ n ρy − e −πα ≤ ∞ n=0 1 − e −π(β n ρy+α) e −πβ n ρy − e −πα − 1 = ∞ n=0 (1 + e −πα ) 1 − e −πβ n ρy e −πβ n ρy − e −πα ≤ 1 + e −πα ∞ n=0 πβ n ρy e −πρy − e −πα = (1 + e −πα ) πρy (e −πρy − e −πα ) (1 − β) ,
as it was claimed. To estimate |B(x + iy)| from above we replace B by a finite product 0≤n≤N B α,β n ρ , where
N def = log ρy α log 1 β .
The number of such indices n is [N] + 1 > N. Since for these n we have β n ρy ≥ α , for each factor we get an estimate
|B α,β n ρ (x + iy)| ≤ e −πβ n ρy + e −πα 1 + e −π(β n ρy+α) ≤ 2e −πα 1 + e −2πα = 1 cosh πα .
Therefore for the whole product we have
|B(x + iy)| ≤ (cosh πα) −N .
From now on, we fix
β = √ 1 + α 2 − 1 √ 1 + α 2 + 1 ,
and consider B = ∞ n=0 B α,β n ρ corresponding to this β. Theorem 7.
δ 1 (H ∞ /BH ∞ ) = 1 √ 1 + 2α 2 .
Moreover, there exists an absolute constant c and another constant C = C(δ 1 ) such that
c 1 (δ) ≤ max c (δ − δ 1 ) 2 log 1 δ − δ 1 , C
for every δ, δ 1 < δ ≤ 1.
The proof of the Theorem is contained in two following lemmata, where δ 1 means simply the number 1 √ 1+2α 2 . After proving these lemmata we can conclude that δ 1 = δ 1 (H ∞ /BH ∞ ).
Lemma 8. Let δ > δ 1 . Then
c 1 (δ) ≤ max c (δ − δ 1 ) 2 log 1 δ − δ 1 , C
for some an absolute constant c and another constant C = C(δ 1 ).
Proof. First we check that the function |f (z)|+|B(z)| can be separated from zero by some constant η depending on α and δ only. By Lemma 6 in the strip
0 < yρ ≤ α √ 1 + α 2 √ 1 + α 2 + 1 we have the estimate |B(x + iy)| ≥ exp − α √ 1 + α 2 (e πα + 1) 2 e πα √ 1+α 2 +1 + 1 . Now we check that |f (z)| is separated from zero in the half-plane yρ ≥ α √ 1 + α 2 √ 1 + α 2 + 1 .
Fix any ε, δ > ε > δ 1 . By Lemma 5
z : Im z ≥ α √ 1 + α 2 ρ √ 1 + α 2 + 1 ⊂ λ∈σ(B) z : |b λ (z)| < ε ,
and therefore it is enough to verify that f is separated from zero on each disk {z : |b λ (z)| < ε}, λ ∈ σ(B), uniformly with respect to λ. By the Schwarz lemma we have
f (z) − f (λ) 1 − f (λ)f (z) ≤ |b λ (z)|,
i. e., for all point z of the disk {z : |b λ (z)| < ε} we have
f (z) − f (λ) 1 − f (λ)f (z) < ε.
Rewriting the inequality a + b 1 +āb ≤ |a| + |b| 1 + |a| |b| (which is, in fact, the triangle inequality for the hyperbolic metric for the points a, b, and 0) in the form
|b| ≥ |a| − a+b 1+āb 1 − |a| a+b 1+āb with a = f (λ) and b = −f (z) we get |f (z)| ≥ δ − ε 1 − δε > δ − δ 1 1 − δδ 1 .
Therefore, in the whole half-plane we have
|f (z)| + |B(z)| ≥ η , where η = min exp − α √ 1 + α 2 (e πα + 1) 2 e πα √ 1+α 2 +1 + 1 , δ − δ 1 1 − δδ 1 .
( 2.1) Finally, by the Carleson corona theorem (see, e.g. [Nik2]), we know that there exists a solution h of the Bezout equation f h + Bg = 1 with a norm estimate
h ∞ ≤ c η 2 log 1 η ,
which means that
c 1 (δ) ≤ c η 2 log 1 η .
Recall that δ 1 = 1 √ 1+2α 2 . If the first term in (2.1) is less than the second one, we have
η = η(δ 1 ) = exp − α √ 1 + α 2 (e πα + 1) 2 e πα √ 1+α 2 +1 + 1 ,
and we can put
C(δ 1 ) = c η 2 (δ 1 ) log 1 η(δ 1 )
.
If the second term is smaller, we have
c 1 (δ) ≤ c (δ − δ 1 ) 2 log 1 δ − δ 1 .
Lemma 9. Let δ ≤ δ 1 . Then c 1 (δ) = +∞.
Proof. Consider a sequence of points v n from Remark 2 and put f n = b vn . As it was mentioned in Remark 2,
|b vn (λ)| ≥ 1 √ 1 + 2α 2 = δ 1 ≥ δ ∀λ ∈ σ(B) .
We would like to estimate from below the H ∞ norm of a solution g n of the Bezout equation g n f n + Bh n = 1. Since
g n ∞ = 1 − Bh n ∞ = h n −B ∞ ≥ h ∞ − 1 and h n ∞ ≥ |h n (v n )| = 1 |B(v n )| ,
by the estimate of Lemma 6 we obtain
g n ∞ → ∞ ,
what yields c 1 (δ) = +∞.
Remark 3. Taking an arbitrary δ, δ < δ 1 , and using the above construction, it is easy to construct a function f with the properties f ∞ ≤ 1, |f (λ)| ≥ δ for every λ ∈ σ, which is not invertible in H ∞ /BH ∞ , so that there is no bounded solution g, h to the Bezout equation gf + Bh = 1. Indeed, it is sufficient to take for f a product of the factors b vn with sufficiently rare subsequence of zeroes v n to ensure the condition |f (λ)| ≥ δ. However, for the Blaschke product B from Theorem 7, we do not know whether there exists such a function in the case δ = δ 1 . In order to guarantee this property, i. e., to have a noninvertible element f of the algebra H ∞ /BH ∞ with δ 1 ≤ f (λ) ≤ f ∞ ≤ 1 (λ ∈ σ(B)), we need a Blaschke product B with more sophisticated zero set, which will be exhibited in the following theorem.
Theorem 10. For an arbitrary fixed number δ 1 from (0, 1) there exists a Blaschke product B such that
1) c 1 (δ, H ∞ /BH ∞ ) < ∞ for every δ, δ 1 < δ ≤ 1; 2) there exists a function f satisfying δ 1 ≤ |f (λ)| ≤ f ∞ ≤ 1 for λ ∈ σ(B), but 1 f / ∈ H ∞ /BH ∞ .
Proof.
Step 1. We start with an arbitrary bounded increasing sequence of positive number α n with α = lim α n , δ 1 def = 1 √ 1+2α 2 . Our Blaschke product B will be of the form
B(z) = ∞ n=1 mn−1 m=0 B αn,β m n ρn (z) ,
where β n = 1 + α 2 n − 1 1 + α 2 n + 1 and ρ n+1 = ρ n β mn n 1 + α 2 n + 1 α n 1 + α 2 n · α n+1 1 + α 2 n+1 1 + α 2 n+1 + 1 .
The initial value ρ 1 = ρ can be taken arbitrarily. The claimed noninvertible function f will be the following Blaschke product
f (z) = ∞ n=0 b vn (z) , where v n = 1 ρ n+1 · α n+1 1 + α 2 n+1 1 + α 2 n+1 + 1 i = 1 ρ n β mn−1 n · α n 1 + α 2 n 1 + α 2 n − 1 i ,
i. e., we put the root v n on the common boundary of the last strip defined by α n and the first strip defined by α n+1 . So, the only parameters, which are in our disposition, are the numbers m n of strips of equal hyperbolic width or, in other words, the distances between the neighbor roots v n . We subordinate these distance to the following condition
v l − v k v l + v k ≥ δ 1 + 2α 2 l+1 2 −k .
(2.2)
If we take any zero λ of the Blaschke product B with Im v n−1 < Im λ < Im v n , then
|f (λ)| = ∞ k=0 |b v k (λ)| = n−2 k=0 |b v k (λ)| · |b v n−1 (λ)b vn (λ)| · ∞ k=n+1 |b v k (λ)| ≥ n−2 k=0 |b v k (v n−1 )| · |b v n−1 (λ)b vn (λ)| · ∞ k=n+1 |b v k (v n )| ≥ n−2 k=0 δ 1 + 2α 2 n 2 −k · |b v n−1 (λ)b vn (λ)| · ∞ k=n+1 δ 1 + 2α 2 n+1 2 −k ≥ δ 1 + 2α 2 n 1−3·2 −n · |b v n−1 (λ)b vn (λ)| .
Thus, would we guarantee for any root λ in the strip between v n−1 and v n the estimate
|b v n−1 (λ)b vn (λ)| ≥ δ 1 + 2α 2 n 3·2 −n 1 + 2α 2 n ,
(2.3)
we will immediately obtain the required estimate for f : |f (λ)| ≥ δ.
Step 2. We shall construct the roots v n by induction. Assume that all v k for k < n are already fixed and we need to choose v n . First of all we have to take v n far enough from the preceding roots in order to satisfy (2.2) for k = n and all l < n as well as for l = n and all k < n.
Note that we need to check condition (2.3) only for the roots λ of B with positive real part and the nearest to the imaginary axis, i. e., for λ = (1 + iα n )/ρ n β m n , because the hyperbolic distance between all other λ with positive real part and any v k is strictly larger, but the consideration for λ with negative real part can be omitted due to the symmetry. Now, we would like to reduce the problem to the case of two roots of B only, the nearest roots to one of the zeroes of f , either v n−1 or v n , i. e., for m = 0 and m = m n − 1.
For the root λ = (1 + iα n )/ρ n , we have |b v n−1 (λ)| = 1 1 + 2α 2 n , and hence (2.3) turns into
|b vn (λ)| ≥ δ 1 + 2α 2 n 3·2 −n .
(2.4)
For the root λ = (1 + iα n )/ρ n β mn−1 n we have |b vn (λ)| = 1 1 + 2α 2 n , therefore (2.3) turns into
|b v n−1 (λ)| ≥ δ 1 + 2α 2 n 3·2 −n .
(2.5)
In fact, both (2.4) and (2.5) follow from (2.2), however we do not want to enter into these additional estimations and simply add (2.4)-(2.5) to the list of requirements for the inductive choice of m n . Now, we check that (2.4)-(2.5) are fulfilled, as well as (2.2), for m n sufficiently large. Let us consider the behavior of the function φ a (t),
φ a (t) def = |b ia (1 + iα n )t | 2 = (a − α n t) 2 + t 2 (a + α n t) 2 + t 2 . Since φ ′ a (t) φ a (t) = 4α n a[t 2 (1 + α 2 n ) − a 2 ] [t 2 (1 + α 2 n ) + a 2 ] 2 − 4α 2
n a 2 t 2 , the function φ a monotonously decreases from 1 to its minimal value, when t changes from 0 to a/ 1 + α 2 n , and then increases tending again to 1 as t → ∞. If we consider the product of two such functions φ a (t)φ b (t) with sufficiently large b/a, then it is clear that this product has two local minima: first of them tends to a/ 1 + α 2 n monotonously decreasing as b → ∞, and the second one tends to b/ 1 + α 2 n monotonously increasing as a → 0. We apply these arguments to our requirement (2.3). To this end, we set a = |v n−1 | = α n 1 + α 2 n ρ n ( 1 + α 2 n + 1)
, b = |v n | = α n 1 + α 2 n ρ n β mn n ( 1 + α 2 n + 1)
, and λ = 1 + iα n ρ n β m n , and will compare the values of our function at the points t = t m = 1/ρ n β m n , m = 0, . . . , m n − 1, in order to guarantee that the minimal value is attained either for m = 0 or for m = m n − 1, where by our assumption either (2.4) or (2.5) is fulfilled. To this aim, we note that t 0 = 1 ρ n > α n ρ n ( 1 + α 2 n + 1) = a 1 + α 2 n , and therefore, for m n sufficiently large, the point of the minimum is less then t 0 and the function φ a (t)φ b (t) is increasing at t 0 . Symmetrically, t mn−1 = 1 ρ n β mn−1 n < α n ρ n β mn−1 n ( 1 + α 2 n − 1) = b 1 + α 2 n , and therefore, for m n sufficiently large, the point of the minimum is bigger then t mn−1 and the function φ a (t)φ b (t) is decreasing at t mn−1 . It follows that conditions (2.3) is fulfilled for m n large enough.
Thus, we have proved that conditions (2.2) and (2.3) are fulfilled if the sequence m n increases fast enough. Therefore, the construction of the Blaschke product B and a function f such that f ∞ = 1, |f (λ)| ≥ δ 1 for all λ, λ ∈ σ(B), is completed. The function f represents a noninvertible element of H ∞ /BH ∞ . Indeed, if we assume that f is invertible in H ∞ /BH ∞ , i. e., there exist two H ∞ -functions g and h such that f g + Bh = 1, then we come to a contradiction, because
lim n→∞ f (v n )g(v n ) + B(v n )h(v n ) = 0 .
So, we have finished the proof of the second statement of the Theorem.
Step 3. In order to complete the proof, we need to check that any other function f satisfying conditions f ∞ = 1, |f (λ)| ≥ δ for all λ, λ ∈ σ(B), and for arbitrary δ, δ > δ 1 , represents an invertible element of H ∞ /BH ∞ .
Fix such a δ and such a function f . We need to check that inf Im z>0
(|f (z)| + |B(z)|) > 0. (2.6)
As in the proof of Theorem 7, we take an arbitrary ε, δ 1 < ε < δ, and split the upper half-plane in two parts:
Π ε def = ∪ λ∈σ(B) {z : |b λ (z)| ≤ ε}
and its complement Π c ε . We will check that f is bounded away from zero on Π ε and B does it on Π c ε . If |b λ (z)| ≤ ε, then by Schwarz' lemma
f (z) − f (λ) 1 − f (λ)f (z) ≤ |b λ (z)| ≤ ε ,
and again, as in the proof of Lemma 8, using the triangle inequality in the form
|b| ≥ |a| − a+b 1+āb 1 − |a| a+b 1+āb
with a = f (λ) and b = −f (z) we get
|f (z)| ≥ δ − ε 1 − δε for arbitrary z from Π ε .
On the complement Π c ε , we estimate |B(z)| splitting the product B into two subproducts B = B ′ B ′′ . Namely, we fix a number N so that δ 1 + 2α 2 n > 1 for n ≥ N and put
B ′ = N −1 n=1 mn−1 m=0 B αn,β m n ρn , B ′′ = ∞ n=N mn−1 m=0 B αn,β m n ρn .
Note that the first product is an interpolating Blaschke product. Indeed, all B α,γ are interpolating, because due to the relation
B b λ (λ) = 2i · Im λ · dB dz (λ) we have B k (z k ) = πα sinh πα ,
where B k = B α,γ /b z k , z k = (2k + 1 + iα)/γ. Therefore zeroes of B ′ form a finite union of interpolating sets. Since they are uniformly separated, the whole product B ′ is interpolating as well. Using a generalized form of the Carleson condition (see, e.g., [Nik1]- [Nik2])
|B ′ (z)| ≥ c inf λ∈σ(B ′ ) |b λ (z)| , we get |B ′ (z)| ≥ cε in Π c ε .
As to the second product B ′′ , we can use the estimate of Lemma 6 for ρ = ρ N . The estimate of Lemma 6 was obtained for the strips of equal hyperbolic width, but in our situation the width of the strips decreases, because α n is increasing. This means that the hyperbolic distance from any point below the first strip to the corresponding zero of B αn,γn,m is strictly bigger than that distance in the case when all α n are equal to α. Therefore, below the first strip, each factor |B αn,γn,m | is strictly larger than in the equidistant case. The whole half-plane Im z > Im v N −1 = 1 ρ N · α N 1 + α 2 N 1 + α 2 N + 1 is in the set Π ε by Lemma 5, therefore B ′′ is separated from zero in the set Π c ε , whence the whole B is separated from zero on Π c ε . So, condition (2.6) is fulfilled what means the invertibility of f in the algebra H ∞ /BH ∞ , i. e., c(δ) < ∞ for any δ, δ > 1/ √ 1 + 2α 2 .
A version of the restricted invertibility conjecture
3.1. Bourgain-Tzafriri's restricted invertibility theorem. The following statement is known as Bourgain-Tzafriri's restricted invertibility theorem. where H I = span{e j : j ∈ I}.
See also [SS] for a generalization and a simpler proof of a matrix version of Bourgain-Tzafriri's Theorem.
The following conjecture often is quoted as Bourgain-Tzafriri's restricted invertibility conjecture (RIC) (it seems although that these authors never actually stated this as a conjecture). It is known that the famous Kadison-Singer conjecture on pure states on C * -algebras (see [KS], [CT]) implies RIC: it is proved in [CCLV] that if Kadison-Singer problem has a positive solution then the RIC has as well. For more details about these conjectures we refer to the papers mentioned above, as well as to a WEB page [ARCC]. It is easy to see that the RIC is equivalent to require the same quality partitions for every bounded T and every unconditional basis in H (in place of orthogonal ones). Knowing no much progress in this conjecture during the last 20 years, we can try to approach the truth treating first some stronger conjectures. Namely, we can replace here an "unconditional basis" by a "Schauder basis", and even by a "summation basis". We denote the corresponding conjectures by B-RIC and SB-RIC, respectively.
Precisely, a summation basis relative to a (triangular) matrix V = {v nj } of scalars v nj is a sequence {e j } j∈N in H such that for every x ∈ H there exists a unique sequence of scalars {a j } j∈N satisfying x = (V ) j≥1 a j e j , which means the following:
• v nj = 0 for j > n; • x = lim n→∞ n j=1 v nj a j e j = x (norm convergence). Clearly, SB-RIC =⇒ B-RIC =⇒ RIC. Here, we present a counterexample to the SB-RIC.
Counterexample.
Theorem 12. Given δ, 0 < δ < 1, there exists a sequence {e j } j∈N in a Hilbert space H satisfying the following properties.
(1) {e j } j∈N is a summation basis (relative to a triangular matrix ).
(2) {e j } j∈N is block orthogonal : there exists an increasing sequence of integers n s such that H [ns,n s+1 ) ⊥ H [nt,n t+1 ) for every s = t,
where H [ns,n s+1 ) = span{e j : n s ≤ j < n s+1 }.
(3) There exists a bounded operator A : H → H satisfying A ≤ 1,
Ae j = λ j (A)e j , δ ≤ |λ j (A)| = Ae j e j ≤ A ≤ 1 (j ∈ N),
and such that for every finite partition r s=1 I s = N there is a restriction A|H Is (1 ≤ s ≤ r), which is NOT left invertible.
(4) Every bounded operator T : H → H satisfying T e j = λ j (T )e j (j ∈ N) and 1 ≥ T ≥ inf j |λ j (T )| > δ is invertible.
Proof. We use our main construction from Theorem 10 replacing the upper half-plane by the unit disk. Namely, given δ (δ 1 in the Theorem), 0 < δ < 1, there exists a Blaschke sequence σ = {z j } of distinct points in D such that a) for every f ∈ H ∞ with δ < inf j |f (z j )| and f ∞ ≤ 1, we have
1 f ∈ H ∞ |σ; b) there is an H ∞ function g such that δ ≤ |g(z j )| ≤ g ∞ ≤ 1 but 1 g ∈ H ∞ |σ. Now,f ∈ H ∞ , δ < inf j |f (z j )| and f ∞ ≤ 1.
Hence, 1 f ∈ H ∞ |σ, which means that T is invertible (and proves point (4)).
Secondly, in order to fix statements (1)-(3), we restate item b) above in terms of the same model operator. Namely, for an operator
T = g(M B ) * with a function g from b) we have T ≤ 1, T x j = λ j (T )x j , δ ≤ |λ j (T )| = |g(z j )| ≤ T ≤ 1 (j ∈ N), and inf{ T x : x ∈ K B , x = 1} = 0.
Notice that if we would like to restrict ourselves to properties (1) and (3) only, we simply set A = T = f (M B ) * . Property (1) follows from the fact that the sequence {x j } j∈N corresponding to a Blaschke sequence {z j } j∈N is a summation basis, [Nik1],p. 194. In order to check (3), suppose that there exists a partition r s=1 σ s = N such that all restrictions T |H σs are left invertible:
0 < inf{ T x : x ∈ H σs , x = 1}
for every s, 1 ≤ s ≤ r. We lead this to contradiction as follows. Let B s be the Blaschke product whose zero sequence is σ s . Since the restriction T |H σs = g(M Bs ) * is, in fact, invertible, there exist functions f s , h s ∈ H ∞ such that gf s + B s h s = 1. Hence, B · r s=1 h s = r s=1 (1 − gf s ) = 1 − gF , where F ∈ H ∞ . This shows that the operator T = g(M B ) * is invertible, what contradicts the construction of T . Therefore, a counterexample satisfying properties (1), (3), and (4) of the Theorem is constructed.
In order to satisfy property (2), we modify the previous construction in the following way. Let N ∈ N and T N = T |H N be the restriction of T to H N = span{x j : 1 ≤ j ≤ N} . Then
T N ≤ 1, T N x j = λ j (T N )x j , δ ≤ |λ j (T N )| ≤ T ≤ 1 (1 ≤ j ≤ N) , and lim N →∞ inf{ T N x : x ∈ H N , x = 1} = 0 . Now, we set A = N ≥1 ⊕T N ,
which is defined coordinate-wise on an (l 2 ) orthogonal sum
H = N ≥1 ⊕H N .
In particular, this means that the point spectrum of A is {λ j (T )} j≥1 but each eigenvalues is repeated infinitely many times. Next, we denote {e j } j∈N the sequence of eigenvectors of A ordered naturally: if f k = (δ N,k ) N ≥1 ∈ l 2 , then {e j } j∈N = (x 1 f 1 , x 1 f 2 , x 2 f 2 , . . . , x 1 f N , x 2 f N , . . . , x N f N , x 1 f N +1 , . . . ) , or, more formally,
e j = x m f N ,
where N = [ 2j + 1 2 ], m = j − N(N − 1) 2 .
Show that {e j } j∈N satisfies properties (1) and (2), and A fulfils all requirements of (3).
Indeed, properties (1) and (2) for {e j } easily follow from the property (1) for {x j } and a block orthogonal nature of {e j }.
In order to prove (3), suppose the contrary, i. e., that there exists a finite partition r s=1 I s = N such that all restrictions A|H Is (1 ≤ s ≤ r) are left invertible. Taking an intersection of r s=1 I s = N with the N-th group of indices corresponding to the eigenfunctions {x m f N } 1≤m≤N , we obtain a partition r s=1 I s,N = I N of the set I N = {1, 2, . . . , N}, where index m runs. Reasoning by induction, assume we have an infinite subsequence {N i } of N such that for a given N all partitions r s=1 (I s,N i I N ) = I N , i ≥ 1, are the same. Since there is only a finite number of partitions of I N +1 , we can choose an infinite subsequence of {N i }, say {N ′ l }, such that all partitions r s=1 (I s,N ′ l I N +1 ) = I N +1 , l ≥ 1, are the same. Applying a diagonal process to this table of sequences, we obtain a growing sequence of integers {M i } i≥1 such that all partitions r s=1 (I s,M i I N ) = I N , i ≥ 1, are the same, for all N = 1, 2, . . . . This means that we have a partition r s=1 σ s = N, I s,M i I N = σ s I N . Next, we observe that, for every s, 1 ≤ s ≤ r, 0 < δ := inf{ Ax : x ∈ H Is , x = 1}
≤ inf{ T N f : f ∈ H I s,M i ∩I N , f = 1} for every N ≥ 1. Taking N → ∞, we get 0 < δ ≤ inf{ T f : f ∈ H σs , f = 1} for every s, 1 ≤ s ≤ r.
But, as we saw above, this is impossible.
Theorem 11 .
11([BTz]) Whatever are a bounded operator T on a Hilbert space H and an orthogonal basis {e j } j∈N satisfying inf j T e j e j > 0, there exists a subset I ⊂ N of positive upper density 0 < dens(I) def = lim sup n→∞ |I ∩ {1, 2, . . . , n}| n such that the restriction T |H I is left invertible:inf{ T x : x ∈ H I , x = 1} > 0 ,
Restricted Invertibility Conjecture (RIC). Conjecture. For every bounded operator T on a Hilbert space H and every orthogonal basis {e j } j∈N satisfying inf j T e j e j > 0, there exists a finite partition r s=1 I s = N such that all restrictions T |H Is are left invertible.
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. / V A University Bordeaux1, Steklov, Math. Inst. [email protected] Bordeaux1/V.A. Steklov. Math. Inst., St. Petersburg E-mail address: [email protected]
Petersburg Branch of the V. St, A. Steklov. Math. Inst., Fontanka. 27Russia E-mail address: [email protected]. Petersburg Branch of the V.A. Steklov. Math. Inst., Fontanka 27, St. Petersburg, 191023, Russia E-mail address: [email protected]
| []
|
[
"Angle Optimization of Graphs Embedded in the Plane",
"Angle Optimization of Graphs Embedded in the Plane"
]
| [
"Sergey Bereg ",
"Timothy Rozario "
]
| []
| []
| In this paper we study problems of drawing graphs in the plane using edge length constraints and angle optimization. Specifically we consider the problem of maximizing the minimum angle, the MMA problem. We solve the MMA problem using a spring-embedding approach where two forces are applied to the vertices of the graph: a force optimizing edge lengths and a force optimizing angles. We solve analytically the problem of computing an optimal displacement of a graph vertex optimizing the angles between edges incident to it if the degree of the vertex is at most three. We also apply a numerical approach for computing the forces applied to vertices of higher degree. We implemented our algorithm in Java and present drawings of some graphs. | null | [
"https://arxiv.org/pdf/1211.4927v2.pdf"
]
| 13,265,308 | 1211.4927 | 600401136f703a659e5da609e3152e9d9a3cc8f5 |
Angle Optimization of Graphs Embedded in the Plane
Sergey Bereg
Timothy Rozario
Angle Optimization of Graphs Embedded in the Plane
In this paper we study problems of drawing graphs in the plane using edge length constraints and angle optimization. Specifically we consider the problem of maximizing the minimum angle, the MMA problem. We solve the MMA problem using a spring-embedding approach where two forces are applied to the vertices of the graph: a force optimizing edge lengths and a force optimizing angles. We solve analytically the problem of computing an optimal displacement of a graph vertex optimizing the angles between edges incident to it if the degree of the vertex is at most three. We also apply a numerical approach for computing the forces applied to vertices of higher degree. We implemented our algorithm in Java and present drawings of some graphs.
Introduction
Angular resolution is one of the aesthetic criteria measuring the quality of graph drawings in terms of human comprehension. The angular resolution of a straight-line drawing in the plane is the minimum angle between any two incident edges. The study of graph drawing with angular resolution started by Formann et al. [12] in 1990. They introduced the angular resolution of a graph as the supremum angular resolution over all straight-line drawings of the graph. The problem of computing the angular resolution of a graph is NP-hard (even the problem of deciding if R = π/2 for graphs with vertex degrees at most four is NP-hard).
The main focus in the early investigation [12,18] was on bounding the angular resolution of a graph in terms of the maximum vertex degree d. The obvious upper bound is R(d) ≤ 2π/d. A lower bound R(d) ≥ Ω(1/d) has been proved [12] for many graphs including planar graphs and complete graphs. A lower bound R(d) ≥ Ω(1/d 2 ) holds for all graphs [12]. If we insist on planar straight-line drawings, then R(d) ≥ α d for some constant 0 < α < 1 [18].
There was also a study of the optimization problem where the angular resolution of a given graph is maximized. Matousek et al. [19] considered an angle-optimal placement of point in polygon. In this problem, the task is to find a point p in the kernel of a star-shaped polygon P such that after connecting p to all the vertices of P by straight edges, the minimum angle between two adjacent edges is maximized. The kernel of P is defined as the locus of all points inside P that see all the edges and vertices of P (it is not empty since P is star-shaped). They showed that it is an LP-type problem of combinatorial dimension 3. Amenta et al. [2] studied various problems of optimal point placement for mesh smoothing (using different mesh quality measures). A related result is the polynomial-time algorithm for computing a Steiner point in a star-shaped polygon, minimizing the maximum angle. A parallel algorithm for mesh smoothing is presented in [13].
Carlson and Eppstein [6] considered tree drawings such that all faces form convex polygons (the infinite faces are created by extending the edges incident on leaves to the infinity). They showed that the optimal angular resolution can be computed in linear time and even the lengths of the tree edges may be chosen arbitrarily. In the recent paper, Eppstein and Wortman [11] considered graph drawing in the plane with faces drawn as centrally symmetric convex polygons. They found a polynomial time algorithm for computing a drawing maximizing the angular resolution.
Another direction in graph drawing with the angle resolution is Lombardi drawing where the graph edges are represented as circular arcs instead of straight-line segments [7,8,9]. Relaxing the condition of straight-line segments allows to achieve perfect angular resolution where the edges are equiangularly spaced around each vertex. The classes of graphs admitting Lombardi drawings are presented in [9] (and the algorithms for finding these drawings). Duncan et al. [8] found that unrooted trees can drawn with perfect angular resolution and polynomial area.
In this paper we study the problem of maximizing the angular resolution using the force-directed approach. This idea is not new and some algorithms for optimizing angular resolution using the force-directed or spring-embedding approach [3,7]. The common feature of these approaches is that the forces are directed from one vertex to another vertex. Our approach is different in the sense that we want to optimize the dislocation of a vertex. It can be viewed as a restricted version of the angle-optimal placement of point in polygon where a graph is embedded in the plane with straight-line edges and we want to find a new position of a given vertex by moving it at distance at most r > 0 (for a given r) and optimizing the incident angles. We call it Max-Min Angle Problem (MMA). This perturbation problem is interesting in its own right. From combinatorial point of view, it is an LP-type problem with the same combinatorial dimension as the angle-optimal placement of point in polygon [19] since only one new constraint is added. However, the situation is quite different from the algebraic point of view.
The angle-optimal placement of point in polygon is related to the famous Fermat problem (which appears as a special case when the vertex degree is three). In general it is known as the Fermat-Weber problem: given n points in a Euclidean space R d , find a point minimizing the sum of the distances to n given points. This point is called the geometric median. We assume d = 2. If n ≤ 4 then the geometric median can be computed exactly. However, it cannot be computed exactly if n ≥ 5, in general (i.e. for some instances) [4].
The MMA problem is harder algebraically than the angle-optimal placement of point in polygon since the optimal point can be at distance r from the initial point. In fact, we show that even for a vertex of degree three, the solution involves polynomials of degree 6 in some (difficult) cases. The polynomials of degree at least five cannot be solved exactly in general. The main result of this paper is that the solution of the MMA problem for a vertex of degree three can be computed exactly (we show that it can be expressed using a polynomial of degree four only).
Another reason why we introduce and study the MMA problem is that the parameter r allows to control the strength of the angular resolution force applied to vertices. This works well if we use more than one force. For example, we applied it to the spring embedding with length constraints.
The paper is organized as follows. In Section 2 we briefly describe force-directed graph drawing and introduce Max-Min-Angle problem. In Section 3 we recall the classical Fermat problem that appears as the special case of the MAA problem. In Section 4 we provide a solution of the MMA problem for vertex degree two. Optimal solution for degree three vertices is provided in Section 5. The algorithm and its performance is discussed in Section 6. Finally we conclude in Section 7.
Spring Embedder and Problem Statement
Force-directed graph drawing is a popular technique and there is a growing literature on force-directed drawing algorithms, see the recent survey by Kobourov [17]. Eades [10] introduced a mechanical model for graph drawing. To achieve aesthetically pleasing layouts and capture the edge length constraints he applied attracting/repelling force between two vertices if the distance between them is less/greater than the desired length. He found that the Hookes Law (linear) springs are too strong when the vertices are far apart but the logarithmic force solves this problem. As initial embedding of the graph the algorithm places its vertices at random locations. The algorithm stops after a sufficient number of iterations. If a state of equilibrium is reached i.e. all forces are zero, then the graph embedding reaches the desired positioning in the plane and remains static. An example of such drawing is shown in Figure 1.
Fruchterman and Reingold [14] added a condition of "even vertex distribution" which is modeled by attractive forces between adjacent vertices and repulsive forces between all pairs of vertices. This increases the number of forces (|V | 2 repulsive forces for a graph G = (V, E)) and slows down the algorithm.
The algorithms by Eades [10] and Fruchterman and Reingold [14] are just two examples of force-directed graph drawing. There are many spring embedders nowadays [17]. We consider a new force-directed approach for angular resolution where the vertex displacement is optimized locally.
Problem Statement
We consider one step of the spring embedder where the vertices of a graph G are embedded in the plane and they are allowed to move slightly. The spring embedder may take into account several forces, for example the forces aiming to achieve the desired lengths of the edges of G. We introduce a force that aims to optimize, for all vertices v, the angles between edges incident to v and embedded in the plane. Ideally, we want the edges to spread evenly around v. This may not be possible due to other constraints of the drawing (edge lengths, for example). Our goal is to maximize, for every v, the smallest angle between two edges incident to v. For this, we formulate our problem. Max-Min-Angle (MMA) Problem. Let G be a graph embedded in the plane. Let P be the position of a vertex v of G and let A 1 , A 2 , . . . , A k be the positions of the adjacent vertices of P in G. Let r > 0 be a radius. Find a point P * (the best position to move P within distance r) such that |P P * | ≤ r and the smallest angle ∠A i P * A j is maximized.
Let Z be the circle centered at P and radius r. We will solve the MMA problem in Sections 4 and 5 by considering degree k of vertex v.
Fermat problem
When the degree of v is three, the MMA problem is related to the classical Fermat problem: Given triangle ABC in the plane, find a point F such that the total distance from the three vertices of the triangle to F is the minimum possible. The solution of the Fermat problem (called Fermat point or Torricelli point) depends on triangle ABC. Case 1: All angles of triangle ABC are smaller than 120 • . Construct three new regular triangles ABC , AB C, and A BC out of the three sides of triangle ABC. Then the point F is the intersection of three lines AA , BB , and CC , see Figure 2(a). In this case the Fermat point F is coincident with the first isogonic center of the triangle [22], where the angles subtended by the sides of the triangle are all equal, i.e. ∠AF B, ∠BF C, ∠CF A = 120 • .
A B C F (a) (b) C B A A B C = F
Case 2:
There exists an angle in triangle ABC greater than or equal to 120 • . Only one angle of the triangle is greater than or equal to 120 • , say ∠ACB ≥ 120 • , see Figure 2(b) for example. Then the Fermat point is coincident with C.
The MMA problem
We solve the MMA problem depending on the degree of vertex v.
Vertex of Degree 2
Suppose that the degree of v is two. Let A and B be the positions of its adjacent vertices. The task is to find a point P * with |P P * | ≤ r maximizing angle θ = min(∠AP B, ∠BP A) (the angles are in the counterclockwise order). Suppose that segment AB intersects circle Z. Obviously, θ = 180 • is achieved by any point P * in the intersection of AB and Z. The solution is unique if AB ∩ Z is a single point (for example, AB may be tangent to Z).
Z B P A P * Z B P A P * Q (a) (b) Figure 3: The case where AB ∩ Z = ∅. (a) P * = Q where Q is defined by Equation(1). (b) Point Q lies not in AB ∩ Z.
Suppose that AB ∩Z is a segment of positive length, see Figure 3(a). There are infinitely many positions of P * (all in AB ∩Z achieving θ = π). If P is not on segment AB then any location of P * will change distances to A and B. We would like to preserve the ratio |AP |/|BP | and compute point Q in segment AB such that |AQ|/|BQ| = |AP |/|BP |. This implies
|AQ| |AB| = |AP | |AP | + |BP | .(1)
Then the coordinates of point Q can be computed using
Q x = A x + |AP | |AP | + |BP | (B x − A x ),(2)Q y = A y + |AP | |AP | + |BP | (B y − A y ).(3)
If point Q lies in circle Z then P * = Q as shown in Figure 3(a). Otherwise we select the endpoint of segment AB ∩ Z that is closer to Q, see Figure 3(b) for an example.
We assume now that AB and Z do not intersect.
Proposition 1 If segment AB and circle Z do not intersect then P * is on the boundary of Z. Furthermore the circle passing through A, B, and P * is tangent to Z.
Proof: Suppose P * = (x, y) where x and y are unknown. Consider circular arc AP * B as shown in Figure 4(a). The angles ∠AP B are equal for all points P from arc AP * B (since the inscribed angle of a chord equals half of the central angle ,see Figure 8(b)). Therefore we can assume that P * is on the boundary of Z. Furthermore arc AP * B cannot intersect the boundary of Z twice since any point P from the arc of Z cut off by arc AP * B makes an angle larger than P * makes, i.e. ∠AP B > ∠AP * B as shown in Figure 4. Thus, arc AP * B must be tangent to Z as shown in Figure 5(a). (4) and (5).
Let O be the center of the circle passing through A, B, and P * . It suffices to find point O since point P * is the intersection point of the circle Z and the line passing through points O and P . The first equation is (P * x − P x ) 2 + (P * y − P y ) 2 = r 2 and the second equation is linear
P * x P * y 1 P x P y 1 O x O y 1 = 0,
where the subscripts denote the coordinates.
Using coordinate transformations we can assume without loss of generality that A(0, −1) and B(0, 1), see Figure 5(b). Then |OA| = |OB| implies that O(x, 0) where x is unknown. Let r = |OA| be the radius of the circle centered at O. Then |OP | = r + r and |OA| = r . They can be written as
(x − P x ) 2 + P 2 y = (r + r ) 2 x 2 + 1 = r 2 (4)(5)
Subtracting Equation (4) from Equation (5) , we obtain
2P x x = P 2 x + P 2 y − 2r r − r 2 − 1.(6)
By plugging x from this equation into Equation of (5) we obtain a quadratic equation in the variable r . There are two roots of the quadratic equation and they correspond to two circles shown in Figure 6. Therefore we take the smallest root of the quadratic equation.
Therefore we proved the folowing claim. (4) and (5).
Proposition 2
The maximum angle problem can be solved using a quadratic equation.
Vertex of degree 3
In this Section we consider the case where the degree of v is three. Let A, B and C be the positions of its adjacent vertices in G. We consider the first case of the Fermat point of triangle ABC where all angles of ABC are less than 120 • .
Proposition 3 If all angles of ABC are less than 120 • and the Fermat point F of ABC lies within circle Z, then P * = F .
Proof: Without loss of generality we assume that A, B, C are in counterclockwise order around P * , see Figure 7 (a). Suppose to the contrary that P * is a point inside Z different from F as shown in Figure 7 (b). The Fermat point lies in one of triangles AP * B, BP * C, or AP * C. Suppose that it belongs to triangle AP * C. Then ∠AP * C < ∠AF C = 120 • and P * is not the optimal point. Therefore P * = F , see Figure 7 (c). Note that in the case of Proposition 3 all angles around P * are equal 120 • . In the rest of this section we consider cases where not all angles ∠AP * B, ∠BP * C, and ∠AP * C are equal. If the smallest angle of ∠AP * B, ∠BP * C, and ∠AP * C is unique, say it is ∠AP * B, then it can be computed by solving the maximum angle problem for A, B using the method from the previous section. It remains to consider the case of two smallest angles.
Two smallest angles
In this section we assume that there are exactly two smallest angles from {∠AP * B, ∠AP * C, ∠BP * C}. First, we show that P * must be on the boundary of circle Z.
Lemma 4
If the degree of v is 3 and there are two smallest angles around P * then P * lies on the boundary of Z.
Proof: We assume that (i) A, B, C are in counterclockwise order around P * , and (ii) ∠AP * B = ∠BP * C < ∠CP * A. Suppose to the contrary that P * lies inside Z as shown in Figure 8. Draw two circles, one passing through points A, B, P * and the second passing through points B, C, P * . Since these circles intersect by two points B and P * , then their interiors intersect by a lune. Since P * is inside circle Z, then the intersection of the lune and the interior of Z is a non-empty set I, see the shaded area in Figure 8(a). For any point P in I, ∠AP B > ∠AP * B and ∠BP C > ∠BP * C (this can be seen, for example, by the fact that the inscribed angle of a chord equals half of the central angle, see Figure 8(b)). We choose P from I close enough to P * such that ∠AP B, ∠BP C < ∠CP A. Then the smallest angle around P is greater than ∠AP * B. Therefore P * is not the solution of the MMA problem. Contradiction. The main result in this section is the following Theorem 5 If the degree of vertex v is three and there are two smallest angles around P * then P * can be computed by solving a polynomial equations of degree at most four.
Proof: First we show that it is possible that there are two smallest angles. An example where two smallest angles around P * are equal is shown in Figure 9 where points P, P * and B are collinear and points A and C are symmetric about the line P B. Therefore angles ∠AP 2 B < ∠AP * B < ∠AP 1 B and P * is the solution of the MMA problem. We now prove the theorem. Suppose that ∠AP * B = ∠BP * C < ∠CP * A. We need to find coordinates of P * (x, y). By the law of cosines we write the equation cos(∠AP * B) = cos(∠BP * C) as
|BP * | 2 + |AP * | 2 − |AB| 2 2 · |BP * | · |AP * | = |BP * | 2 + |CP * | 2 − |BC| 2 2 · |BP * | · |CP * |(7)|CP * | · (|BP * | 2 + |AP * | 2 − |AB| 2 ) = |AP * | · (|BP * | 2 + |CP * | 2 − |BC| 2 )(8)
Without loss of generality we can assume that P = (0, 0) and A y = C y . By Lemma 4 we assume |P P * | = r. Then x 2 + y 2 = r 2 and
|CP * | 2 = r 2 + C 2 x + C 2 y − 2C x x − 2C y y = a 1 x + a 2 y + a 3 (9) 1 2 (|BP * | 2 + |AP * | 2 − |AB| 2 ) = r 2 + A x B x + A y B y − (A x + B x )
x − (A y + B y )y = a 4 x + a 5 y + a 6 (10)
|AP * | 2 = r 2 + A 2 x + A 2 y − 2A x x − 2A y y = b 1 x + b 2 y + b 3 (11) 1 2 (|BP * | 2 + |P * C| 2 − |BC| 2 ) = r 2 + B x C x + B y C y − (B x + C x )x − (B y + C y )y = b 4 x + b 5 y + b 6 ,(12)
A B C P P * P 1 P 2 r Figure 9: Points P 1 and P 2 are the points on circle Z maximizing angles ∠AP 1 B and ∠BP 2 C, respectively. Point P * is different from P 1 and P 2 and angles ∠AP * B = ∠BP * C.
where
a 1 = −2C x , a 2 = −2C y , a 3 = r 2 + C x 2 + C y 2 , a 4 = −(A x + B x ), a 5 = −(A y + B y ), a 6 = B x A x + B y A y + r 2 , b 1 = −2Ax, b 2 = −2A y , b 3 = r 2 + A x 2 + Ay 2 , b 4 = −(B x + C x ), b 5 = −(By + Cy), b 6 = B x C x + B y C y + r 2 .
Take the square of both sides of Equation (8) (a 1 x + a 2 y + a 3 )(a 4 x + a 5 y + a 6 )
2 = (b 1 x + b 2 y + b 3 )(b 4 x + b 5 y + b 6 ) 2(13)
It can be written as c 1 x 3 + c 2 y 3 + c 3 x 2 + c 4 y 2 + c 5 x 2 y + c 6 xy 2 + c 7 x + c 8 y + c 9 xy + c 10 = 0
where c 1 = a 1 a 2
4 − b 1 b 2 4 , c 2 = a 2 a 2 5 − b 2 b 2 5 , c 3 = (2a 1 a 4 a 6 + a 3 a 2 4 − (2b 1 b 4 b 6 + b 3 b 2 4 )), c 4 = (2a 2 a 5 a 6 + a 3 a 2 5 − (2b 2 b 5 b 6 + b 3 b 2 5 )), c 5 = (2a 1 a 4 a 5 + a 2 a 2 4 − (2b 1 b 4 b 5 + b 2 b 2 4 )), c 6 = (2a 2 a 4 a 5 + a 1 a 2 5 − (2b 2 b 4 b 5 + b 1 b 2 5 )
), c 7 = (2a 3 a 4 a 6 + a 1 a 2 6 − (2b 3 b 4 b 6 + b 1 b 2 6 )), c 8 = (2a 3 a 5 a 6 + a 2 a 2 6 − (2b 3 b 5 b 6 + b 2 b 2 6 )), c 9 = 2(a 1 a 5 a 6 + a 2 a 4 a 6 + a 3 a 4 a 5 − (b 1 b 5 b 6 + b 2 b 4 b 6 + b 3 b 4 b 5 )), c 10 = a 3 a 2 6 . Using y 2 = r 2 − x 2 we reduce powers of y in Equation (14) c
1 x 3 + y(c 2 r 2 + c 8 ) + c 3 x 2 + c 4 − c 4 x 2 + x 2 y(c 5 − c 2 ) + c 6 x − c 6 x 3 + c 7 x + c 8 y + c 9 xy + c 10 = 0 (15) d 1 x 3 + d 2 x 2 + d 3 x + d 4 + (d 5 x 2 + d 6 x + d 7 )y = 0(16)
where
d 1 = c 1 − c 6 , d 2 = c 3 − c 4 , d 3 = c 6 r 2 + c 7 , d 4 = c 4 r 2 + c 10 , d 5 = c 5 − c 2 , d 6 = c 9 , d 7 = c 2 r 2 + c 8 . d 1 x 3 + d 2 x 2 + d 3 x + d 4 2 − ((d 5 x 2 + d 6 x + d 7 ) 2 · (r 2 − x 2 )) = 0(17)
Simplifying Equation (17) ), e 6 = 2d 3 d 4 − (2d 6 d 7 r 2 ), e 7 = d 2 4 − d 2 7 r 2 . By Lemma 6 the sextic equation has a quadratic factor. Dividing by it, we obtain a polynomial of degree four (Equation (19), (20), or (21)) and the theorem follows.
Lemma 6
The polynomial equation (18) has factor x 2 + A 2 y − r 2 .
Proof: We consider 3 cases. Case I: A y = r. Then e 5 = e 6 = e 7 = 0 and the polynomial (18) has factor x 2 . The polynomial is reduced to the polynomial of degree 4
e 1 x 4 + e 2 x 3 + e 3 x 2 + e 4 x = 0.
Case II: A 2 y − r 2 > 0. We scale the coordinates such that A 2 y − r 2 = 1. Then e 6 = e 4 − e 2 , e 7 = e 3 − e 1 and the polynomial (18) has factor x 2 + 1. The polynomial is reduced to the polynomial of degree 4 e 1 x 4 + e 2 x 3 + (e 3 − e 1 )x 2 + (e 4 − e 2 )x + e 5 − (e 3 − e 1 ) = 0.
Case III: A 2 y − r 2 < 0. We scale the coordinates such that A 2 y − r 2 = −1. Then e 6 = e 4 + e 2 , e 7 = e 3 + e 1 and the polynomial (18) has factor x 2 − 1. The polynomial is reduced to the polynomial of degree 4 e 1 x 4 + e 2 x 3 + (e 3 + e 1 )x 2 + (e 4 + e 2 )x + e 5 + (e 3 + e 1 ) = 0.
The lemma follows.
Algorithm
In this Section we discuss how to modify the spring embedder in order to take into account the angles between embedded edges. Let G = (V, E) be an input graph. At the initial step the spring embedder randomly places the vertices of G in the plane. Then, it iterates a simultaneous motion of the vertex positions based on one or several forces (springs). We describe a new force using angle optimization. For every vertex v with a position P in the plane, the algorithm computes a new position P * and uses vector P P * as a force applied to vertex v. Angle Optimization Algorithm Input: Graph G = (V, E) embedded in the plane and radius r. Output: New embedding of graph G where each vertex is translated within distance r.
For each vertex v ∈ V do the following steps.
1. Let P be the position of v in the plane. We compute P * as follows. Every time P * is assigned, we proceed to the next vertex v.
2. If the degree of v is at most one then set P * = P .
3. Suppose that the degree of v is equal to two. Let A and B be the positions of vertices adjacent to v.
(a) Suppose AB ∩Z = ∅. Compute point Q using Equations (2) and (3). If |P Q| ≤ r then P * = Z; otherwise assign P * to the endpoint of segment AB ∩ Q that is closer to Q.
(b) If AB ∩ Z = ∅ then set compute P * using Proposition 2.
4. Suppose that the degree of v is equal to three. Let A, B, and C be the positions of vertices adjacent to v. Compute Fermat point F of triangle ABC. If |P F | ≤ r then set P * = F . Otherwise, for each segment ab ∈ {AB, AC, BC}, compute point P ab maximizing angle ∠aP ab b (Section 4.1). If angle ∠aP ab b is the smallest angle from {∠AP AB B, ∠AP AC C, ∠BP BC C} then set P * = P ab . Otherwise compute P * as the best solution using two smallest angles for every two segments from {AB, AC, BC} (Section 5.1).
5.
For the remaining vertices of degree at least four, apply the following grid approach. Pick a grid stepsize δ, for example δ = r/3. Consider a grid with the origin at P . For every grid point Q with |P Q| ≤ r, compute the smallest angle α Q if P is moved to Q. Find Q maximizing α Q . Assign P * = Q.
We implemented this algorithm 1 and run it on several graphs. First, we tested the algorithm on graph T 10 from [21] since it contains vertices of degree two. The program draws T 10 with angles optimized, see Figure 10 We also tested the algorithm on the phylogenetic networks for Algae example [20]. The drawing of the Algae network by the spring embedder [1] is shown in Figure 1. It can be compared with our drawings in Figure 11. In all drawings (in Figures 1 and Figure 11) the edge length constraints are satisfied but the angle resolution in drawings in Figure 11 is significantly larger. The drawing shown in Figure 11 (a) uses the weighted version of the graph and the drawing in Figure 11 Finally, we run our program on the well-known graphs: the Petersen graph [16], the Heawood graph [15], and the Herschel graph [5], see Figure 12. The Petersen graph is drawn with two crossings (the Petersen graph is in fact the smallest 2-crossing cubic graph), see Figure 12(a). The Heawood graph has crossing number three (it is actually the smallest 3-crossing cubic graph) and our program found a drawing with exactly three crossings, see Figure 12(b).
Conclusion
We proposed a novel approach to the problem of optimizing the angular resolution of a drawing. It has been applied to the spring embedder and the results with good angular resolution were obtained. It is known that the running time of the spring embedder can increase with the size of the graph. Therefore it is important to perform better the intermediate steps. The optimal solution of the MMA problem provides an opportunity to decrease the number of iterations of the spring embedder.
The main result of this paper states that a vertex of degree at most three can be displaced optimally by solving a polynomial equation of degree at most four (which is interesting since the straightforward approach leads to a polynomial of degree six). A special case of MMA problem for degree three verices is the classical Fermat problem and the Fermat point is the solution for the special case. An interesting question for future research is to find the algebraic complexity of MMA problem for higher vertex degrees.
Figure 1 :
1A drawing of Algae graph[20] produced by the spring embedder [1].
Figure 2 :
2The Fermat point F in (a) Case 1 and (b) Case 2. (a) Three angles ∠AF B, ∠BF C, and ∠CF A are equal. (b) The Fermat point is at C.
Figure 4 :Figure 5 :
45Arc AP * B. Wrong location of P * since ∠AP B > ∠AP * B. The MMA problem can be formulated now as the following problem dealing with only one angle. Maximum Angle Problem. Let A, B, and P be points in the plane and a number r > 0 such that |AP |, |BP | > r. Compute a point P * with |P P * | = r maximizing angle ∠AP * B(a) The maximum angle problem. (b) The circles centered at O and O are the two solutions of the system of equations
Figure 6 :
6An example of the maximum angle problem. The circles centered at O and O are the two solutions of the system of equations
Figure 7 :
7(a) Setting of Proposition 3. (b) Contradiction when P * = F . (c) Points P * and F coincide.
Figure 8 :
8(a) For all points P in the shaded area ∠AP B > ∠AP * B and ∠BP C > ∠BP * C. (b) The inscribed angle ∠AP * B of chord AB equals half of the central angle ∠AOB.
we get a sextic equation e 1 x 6 + e 2 x 5 + e 3 x 4 + e 4 x 3 + e 5 x 2 + e 6 x + e 7 = 0 (18) where e 1 = d 2 1 + d 2 5 , e 2 = 2(d 1 d 2 + d 5 d 6 ), e 3 = (d 2 2 + 2d 1 d 3 − (d 2 5 r 2 − (d 2 6 + 2d 5 d 7 ))), e 4 = (2(d 1 d 4 + d 2 d 3 ) − (2(d 5 d 6 r 2 − d 6 d 7 ))), e 5 = ((d 2 3 + 2d 2 d 4 ) − ((d 2 6 + 2d 5 d 7 )r 2 − d 2 7 )
(b). It can be compared with the drawing of the original embedder [1], see Figure 10 (a).
Figure 10 :
10Graph T 10 from [21]. (a) Drawing by Spring Embedder [1]. (b) Drawing by our program with angle optimization.
Figure 11 :
11(b) uses the graph with intermediate points on the edges. Phylogenetic network for the Algae example [20]. (a) Weighted graph. (b) Graph with intermediate points on the edges.
Figure 12 :
12(a) The Petersen graph. (b) The Heawood graph. (c) The Herschel graph.
Demo is available at http://www.utdallas.edu/˜sxb027100/soft/AngleOpt/.
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| []
|
[
"A generally covariant measurement scheme for quantum field theory in curved spacetimes",
"A generally covariant measurement scheme for quantum field theory in curved spacetimes"
]
| [
"Christopher J Fewster "
]
| []
| []
| We propose and develop a measurement scheme for quantum field theory (QFT) in curved spacetimes, in which the QFT of interest, the "system", is dynamically coupled to another, the "probe", in a compact spacetime region. Measurements of observables in the probe system then serve as proxy measurements of observables in the system, under a correspondence which depends also on a preparation state of the probe theory. All our constructions are local and covariant, and the conditions may be stated abstractly in the framework of algebraic quantum field theory (AQFT). The induced system observables corresponding to probe observables may be localized in the causal hull of the coupling region and are typically less sharp than the probe observable, but more sharp than the actual measurement on the coupled theory. A formula is given for the post-selected system state, conditioned on measurement outcomes, which is closely related to the notion of an instrument as introduced by Davies and Lewis. This formula has the important property that individual measurements form consistent composites, provided that their coupling regions can be causally ordered and a certain causal factorisation property holds for the dynamics; the composite is independent of the causal order chosen if more than one exists. The general framework is amenable to calculation, as is shown in a specific example. This contribution reports on joint work with R. Verch, arXiv:1810.06512.IntroductionA typical first course on quantum mechanics will include some straightforward rules describing measurements. It might be said, for example, that an ideal measurement of a quantum mechanical observable returns one of the eigenvalues of the operator representing the observable, and the system is to be found in the corresponding CJ Fewster | 10.1007/978-3-030-38941-3_11 | [
"https://arxiv.org/pdf/1904.06944v1.pdf"
]
| 119,283,637 | 1904.06944 | 8a230a14323f8906a5f07281bcd3bce4534c3174 |
A generally covariant measurement scheme for quantum field theory in curved spacetimes
15 Apr 2019
Christopher J Fewster
A generally covariant measurement scheme for quantum field theory in curved spacetimes
15 Apr 2019arXiv:1904.06944v1 [gr-qc]
We propose and develop a measurement scheme for quantum field theory (QFT) in curved spacetimes, in which the QFT of interest, the "system", is dynamically coupled to another, the "probe", in a compact spacetime region. Measurements of observables in the probe system then serve as proxy measurements of observables in the system, under a correspondence which depends also on a preparation state of the probe theory. All our constructions are local and covariant, and the conditions may be stated abstractly in the framework of algebraic quantum field theory (AQFT). The induced system observables corresponding to probe observables may be localized in the causal hull of the coupling region and are typically less sharp than the probe observable, but more sharp than the actual measurement on the coupled theory. A formula is given for the post-selected system state, conditioned on measurement outcomes, which is closely related to the notion of an instrument as introduced by Davies and Lewis. This formula has the important property that individual measurements form consistent composites, provided that their coupling regions can be causally ordered and a certain causal factorisation property holds for the dynamics; the composite is independent of the causal order chosen if more than one exists. The general framework is amenable to calculation, as is shown in a specific example. This contribution reports on joint work with R. Verch, arXiv:1810.06512.IntroductionA typical first course on quantum mechanics will include some straightforward rules describing measurements. It might be said, for example, that an ideal measurement of a quantum mechanical observable returns one of the eigenvalues of the operator representing the observable, and the system is to be found in the corresponding CJ Fewster
eigenstate immediately afterwards, having changed instantaneously. Later, one learns that essentially every part of this rule is a considerable oversimplification, either technically or conceptually. Not least, the idea of an instantaneous transition is obviously problematic in relativistic theories. Quantum measurement theory (QMT) takes as its task the problem of refining rules of this type and putting them into an operational context. See [8] for a recent comprehensive account. One strand of this work is the description of measurement schemes that describe part of the process by which physical quantities may be measured. A system of interest is prepared, and then coupled to a probe, itself a quantum mechanical system, which is later measured. The probe result is then interpreted as a measurement of the system. The combination of the system, probe, coupling, and the probe observable measured, is called a measurement scheme for the system observable indirectly measured in this way.
This contribution describes joint work with Rainer Verch [16], which aims to adapt these ideas to describe measurements in quantum field theory in possibly curved spacetimes, using the methods of algebraic quantum field theory (AQFT) [17,1]. In so doing, we bridge a surprising gap in the literature: research on AQFT, despite its focus on local observables and local operations, has largely avoided the question of how such observables may be measured or operations performed; on the other hand, QMT has typically concentrated on quantum mechanical problems and avoided questions relating to spacetime localisation. Naturally, there are exceptions, and the works of Hellwig and Kraus [18,20,19] and more recent works [22,21] may be mentioned as discussions in which questions of measurement in AQFT have been addressed. The reader will find much food for thought in the paper of Peres and Terno [23]. None of these, however, take quite the line that we describe here and, in particular, none of them consider the curved spacetime context. Again, this is a surprise, because of the prominence of the Unruh effect [25,9] in QFT in curved spacetimes (QFT in CST). (See [7] for a recent discussion of some operational and interpretative aspects of the Unruh effect.) As mentioned, a full presentation of this work is given in [16]; our aim here is to present the outlines of the results obtained, while remaining reasonably self-contained. The main general questions considered are:
• what is the correspondence between probe observables and system observables? • in particular, what can be said concerning the spacetime localisation of the latter? • what is the appropriate description of state change following measurement, consistent with covariance? • how can one combine multiple observations in different spacetime regions, and the consequent changes of state, in a consistent way?
We emphasise that the framework is on the one hand sufficiently broad to stand as a general description of measurement schemes in AQFT, while on the other it is sufficiently concrete to permit calculation in particular models; we will report the results of such calculations later on. To start, however, we begin by describing the nature of the system, probe and the coupling between them.
System, probe, and coupling
In QMT the system and probe are usually quantum mechanical systems, with individual Hamiltonians H S and H P on respective Hilbert spaces H S and H P . The two can be treated as a single system with uncoupled Hamiltonian
H U = H S ⊗ 1 + 1 ⊗ H P(1)
on the combined Hilbert space H = H S ⊗ H P . A coupling can be introduced and removed by modifying the Hamiltonian to
H C (t) = H U + H int (t)(2)
with H int (t) = 0 for sufficiently early and sufficiently late times t. Alternatively one might simply specify a unitary time evolution U(t) on the combined Hilbert space, with U(t) = 1 for early and late t. The goal is then to understand how measurements of observables on H P can be interpreted as measurements of observables on H S . In this section we will introduce analogous structures for two quantum field theories with a coupling confined to a compact region of spacetime. There are two problems. First, we wish to maintain covariance and therefore avoid introducing preferred time coordinates. This will be dealt with by adapting ideas from AQFT, particularly in the locally covariant description developed for curved spacetimes [6,14]. Second, one might wonder about the physical status of coupling together quantum fields. After all, the interactions of nature are not ours to change -what, then, is the relevance, of a formalism based on such modifications? The answer to this criticism is that the couplings represent a proxy for an experimental design that engineers interactions to occur in the apparatus and tries to screen out extraneous influences. For example, electromagnetism and QCD are coupled within the standard model. But if we probe the structure of a nucleus by directing a laser at it, we produce interactions in a particular spacetime region, and it becomes reasonable to neglect the interactions between electromagnetism and QCD elsewhere in spacetime. It is also true that by choosing interaction energies we may exploit the running of coupling constants to modify the strength of various interactions.
We now explain the formalism in more detail, which requires a short discussion of AQFT in curved spacetimes. A recent introduction to AQFT in flat spacetimes can be found in [12]; for an exposition of the locally covariant approach to QFT in CST, introduced in [6], see [15].
Recall that a time-oriented Lorentzian manifold spacetime M is globally hyperbolic if it contains no closed causal curves and each of its compact sets K has a compact causal hull J + (K) ∩ J − (K), where J +/− (S) represent the causal future/past of a set S (where needed, we will denote J(S) = J + (S) ∪ J − (S)). Equivalently, M contains a Cauchy surface -a subset met exactly once by every inextendible timelike curve. A subset of M will be called causally convex if it is equal to its causal hull, which means that it contains every causal curve between any pair of its points. If O is any open causally convex subset of a globally hyperbolic spacetime M, then O, equipped with the induced metric and causal structure, is a globally hyperbolic spacetime in its own right, to be denoted M | O .
We shall be interested in QFTs that can be formulated on a class of globally hyperbolic spacetimes that is closed under the formation of subspacetimes in the manner just described. To each such spacetime M, we assume that such the QFT is Comparing with assumptions standard in flat spacetime (see, e.g., [12]) one notes that Poincaré covariance has been replaced by the compatibility condition. What we have described is a cut-down version of locally covariant QFT [6,15], the full version of which would describe the theory as a functor from the category of globally hyperbolic spacetimes to a category of unital * -algebras. Einstein causality may be built in by suitable monoidal [5] or operadic [3] refinements. However we will not need this level of generality here. On a point of terminology, if A ∈ A(M; O) we will say that A is localisable in O, or that O is a localisation region for A; due to A1 and A4 there will be many possible localisation regions for a given A. Finally, recall that a state of a theory is a positive normalised linear map ω : A(M) → C, and the expectation value of observable A ∈ A(M) in state ω is ω(A). Let two theories A and B be given, each of which obeys the above axioms, with compatibility maps α M;N and β M;N respectively. We will regard A as describing the system and B the probe. Then we obtain a further theory U, by defining
U(M) = A(M) ⊗ B(M),(7)
for each spacetime M, and using υ M;N = α M;N ⊗ β M;N : U(N ) → U(M) as the compatibility map when N is an open causally convex subset of M. This is the analogue of the tensor product of the system and probe in quantum mechanics; as ever in AQFT, the primary focus is on algebras of observables rather than on Hilbert spaces. We wish to describe a coupling between the system and probe in an abstract way, without needing a specific Lagrangian description of the theories involved, and also without introducing privileged coordinates. Let K be a compact subset of spacetime M. Then the regions M ± = M \ J ∓ (K) are causally convex and open and constitute covariantly defined 'in' (−) and 'out' (+) regions, so that the 'out' region has no intersection with the causal past of K and the 'in' region has no intersection with the causal future of K.
Definition 1 A theory C, with compatibility maps γ M;N , is a coupling of A and B within K if, there is a family of isomorphisms
χ L : U(L) → C(L)(8)
labelled by the open causally convex subsets L of M \ (J + (K) ∩ J − (K)), with the following compatibility property: for any two such subsets L, L ′ with L ′ ⊂ L, one has a commuting diagram
U(L ′ ) U(L) C(L ′ ) C(L) υ L;L ′ χ L ′ χ L γ L;L ′ .(9)
In short, this definition asserts that the theories U and C coincide outside the causal hull of K, capturing the idea that the coupling is only switched on in this region. There is a close link to the discussion of equivalence between theories in local covariant QFT [6,14,15]. Many properties of the coupling can be described in terms of a scattering map defined as follows. For brevity, we denote the compatibility maps α M;M ± associated with the regions M ± by α ± , and the isomorphisms χ M ± by χ ± (by construction, M ± are indeed open causally convex subsets not intersecting the causal hull of K). By the timeslice property, the monomorphisms α ± , β ± , υ ± , γ ± , are isomorphisms, so the same applies to the compositions
κ ± = γ ± • χ ± : U(M ± ) → C(M),(10)
and also to the retarded (+) and advanced (−) response maps
τ ± = κ ± • (υ ± ) −1 : U(M) → C(M),(11)
and finally to the scattering morphism
Θ = (τ − ) −1 • τ + : U(M) → U(M),(12)
which maps algebra elements from late to early times. The scattering morphism has important properties [16, Prop. 1]: in particular, it is unchanged if the notional coupling region K is expanded, but keeping C the same; it also acts trivially on any subalgebra A(M; L) labelled by a subset L ⊂ K ⊥ .
Induced system observables
The measurement scheme may be described loosely as follows. At early times, the system and probe are separately prepared in states ω and σ respectively. They are then coupled; finally, at late times, a probe observable B ∈ B(M) is measured. This description is somewhat imprecise because the actual world of the experiment is described by the coupled theory C, rather than the separate theories A and B, or their uncoupled combination U. A little more precisely, what is meant is that, at early times, one prepares a state of C(M) that has no correlations between system and probe degrees of freedom, and at late times an observation is made that only tests degrees of freedom in the probe. The problem of translating statements about the coupled theory into the language of the uncoupled theory is solved by the response maps. Specifically, the state of C(M) corresponding at early times to the product state ω ⊗ σ of U(M) is given by
ω σ = (τ − ) −1 * (ω ⊗ σ)(13)
where the star denotes the adjoint map, i.e., ω σ (C) = (ω ⊗ σ)(τ − C). Likewise, the observable B ∈ B(M), may be identified with 1 ⊗ B ∈ U(M) and hence corresponds to the observable
B = τ + (1 ⊗ B)(14)
of C(M), with the identification being made at late times. The expected measurement outcome from the experiment is the expectation value for the observable B in the state ω σ , which may be written
ω σ ( B) = (ω ⊗ σ)(Θ(1 ⊗ B)).(15)
Notice that we use the advanced response to identify the states of the uncoupled and coupled systems at early times, and the retarded response to identify observables at late times. This reflects the fundamental time-asymmetry of the measuring process, which may be sloganized as prepare early and measure late.
The goal is to interpret ω σ ( B) as the expectation value of a system observable in state ω. To this end, we introduce the map η σ :
A(M) ⊗ B(M) → A(M) defined by η σ (A ⊗ B) = σ(B)A,(16)
extended by linearity, which is a completely positive map closely related to a conditional expectation. Then the map ε σ :
B(M) → A(M) defined by ε σ (B) = (η σ • Θ)(1 ⊗ B)(17)
clearly satisfies
ω(ε σ (B)) = ω((η σ • Θ)(1 ⊗ B)) = (ω ⊗ σ)(Θ(1 ⊗ B)) = ω σ ( B),(18)
which provides an interpretation of the measurement in terms of the induced system observable ε σ (B). In other words, we have a measurement scheme for ε σ (B). The following is proved as [16, Thm 2]
Theorem 1 For each probe preparation state σ, A = ε σ (B) is the unique ω- independent solution to ω(A) = ω σ ( B), provided that A(M)
is separated by its states. In general, the map ε σ is a completely positive linear map with the properties
ε σ (1) = 1, ε σ (B * ) = ε σ (B) * , ε σ (B) * ε σ (B) ≤ ε σ (B * B) .(19)
For fixed B, the map σ → ε σ (B) is weak- * continuous.
An immediate consequence is that each self-adjoint B = B * ∈ B(M) is mapped to a self-adjoint element ε σ (B) of A(M). It is important to note that ε σ is generally neither an injection, nor an algebra homomorphism. Indeed, the inequality in (19) implies
Var( B; ω σ ) ≥ Var(ε σ (B); ω),(20)
that is, the variance of measurement results in the actual experiment of measuring B in state ω σ is at least the variance of the hypothetical equivalent measurement of ε σ (B) in state ω. The failure of ε σ to be a homomorphism is intimately connected to the existence of experimental noise, i.e., fluctuations of the probe. As another illustration, if B is a projection, then ε σ (B) is in general not a projection, but rather an effect; that is, an algebra element A so that both A and 1 − A are positive, i.e., 0 ≤ A ≤ 1. In general, an effect A models an unsharp zero-one measurement, with
Prob(A = 1 | ω) = ω(A), Prob(A = 0 | ω) = ω(1 − A)(21)
in state ω. Thus, sharp (projective) measurements of the probe are to be interpreted as unsharp measurements of the system. Combining our two observations, it may be said that ε σ (B) is typically less sharp than B, but more sharp than the actual experimental observation.
An important question concerns the localisation of the induced system observables, which turns on the locality properties of the scattering morphism. Let L be an open causally convex subset of K ⊥ , so Θ acts trivially on U(M; L). This has the following consequences. First, if B ∈ B(M; L), we have 1 ⊗ B ∈ U(M; L) and hence the corresponding induced system observable is
ε σ (B) = η σ (Θ(1 ⊗ B)) = η σ (1 ⊗ B) = σ(B)1.(22)
As one would hope, no information concerning the system can be obtained by measuring a probe observable at spacelike separation from the coupling region; our framework does not allow for nonlocal signalling. Second, suppose that A ∈ A(M; L) be any system observable localised in L. Then for an arbitrary probe observable B ∈ B(M) one has
[ε σ (B), A] = [η σ (Θ(1 ⊗ B)), A] = η σ ([Θ(1 ⊗ B), A ⊗ 1)]) (23) = η σ (Θ[1 ⊗ B, A ⊗ 1]) = 0,
as Θ leaves A ⊗ 1 invariant. By the Haag property, we infer that ε σ (B) may be localised in any connected open causally convex set containing the coupling region K (and hence containing its causal hull). We have proved:
Theorem 2 ([16, Thm 3]) For each probe observable B ∈ B(M), the induced system observable ε σ (B) may be localised in any connected open causally convex set containing K. If B may be localised in
K ⊥ then ε σ (B) = σ(B)1.
The failure of ε σ to be a homomorphism is natural from another perspective. Suppose two incompatible (noncommuting) system observables A i (i = 1, 2) are localisable in the causal hull of K and are in fact induced by probe observables B i , A i = ε σ (B i ). If ε σ were a homomorphism, we would conclude that the B i are incompatible, which (by Einstein causality) would prohibit the possibility of localising them in spacelike separated regions. Turning this around, probe measurements in spacelike separated regions that correspond to the A 1 and A 2 provide an unsharp joint measurement of incompatible system observables.
Instruments and change of state
Suppose an effect B of the probe is measured, resulting in the value 1 (we also say that the effect has been observed). How should the system state ω be updated in consequence? One way of addressing this question is to require that the updated state ω ′ on A(M) should have the property that
Prob(A; ω ′ ) = Prob σ (A | B; ω) = Prob σ (A&B; ω) Prob σ (B; ω)(24)
for all system effects A measured at late times. Here, the middle quantity is the classical conditional probability, in the original state, that the effect A is observed, given that B is also observed, with the subscript indicating the likely dependence on the probe preparation state σ. On the right-hand side, A&B is the joint effect corresponding to the operator A ⊗ B ∈ U(M). By the reasoning used in the previous section, we have (25) and so our requirement on ω ′ is
Prob σ (A&B; ω) = ω(η σ Θ(A ⊗ B)), Prob σ (B; ω) = ω(η σ Θ(1 ⊗ B))ω ′ (A) = (I σ (B)(ω))(A) (I σ (B)(ω))(1) ,(26)
where
(I σ (B)(ω))(A) := (ω ⊗ σ)(Θ(A ⊗ B)) = (Θ * (ω ⊗ σ))(A ⊗ B)(27)
and the forest of parentheses indicates that I σ (B) is to be regarded as a map on the dual space of A(M). Assuming that the algebra separates the states, equation (26) implies that
ω ′ := I σ (B)(ω) I σ (B)(ω)(1)(28)
is the post-selected system state after selective measurement of the probe.1 The term 'post-selection' is used in various ways in the literature; to be clear, the meaning intended here is given by (24), i.e., the post-selected state is the one in which the probability of observing a system effect A is equal to the conditional probability, in the original state, of observing A given that the probe effect is observed.
The pre-instrument I σ (B) defined by (27) is a positive map on the dual space A(M) * ; that is, it maps positive linear functionals to positive linear functionals. Davies and Lewis [10] introduced the term instrument to describe a measure on the σ-algebra of measurement outcomes valued in positive maps A(M) * ; our preinstruments are related to instruments as follows: given any measure E valued in the effects of A(M), then X → I σ (E(X)) is a (completely positive) instrument in the Davies-Lewis sense.
The discussion above includes non-selective probe measurement as the special case B = 1, because the absence of any filtering on the measurement outcome is equivalent to employing a probe effect that is observed with probability 1 in all states. Therefore the updated state resulting from the non-selective measurement is
ω ′ ns (A) = I σ (1)(ω)(A) = (Θ * (ω ⊗ σ))(A ⊗ 1),(29)
which is the partial trace of the state Θ * (ω ⊗ σ) over the probe. By the locality properties of Θ, it follows immediately that ω ′ ns (A) = ω(A) if A ∈ A(M; K ⊥ ). This shows that a non-selective measurement cannot influence the results of measurements in causally disjoint regions. The general situation for selective probe measurements is the following [16,Thm 4]. (ε σ (B)).
Theorem 3 Consider a measurement of a probe effect B in which the effect is observed. For each
This includes our non-selective result because ε σ (1) = 1 is uncorrelated with every system observable in every state. In general, updating the state by post-selection changes expectation values at spacelike separation from the measurement region (and, one expects, also in its past and future). Theorem 3 shows that the explanation is simply to do with correlation and inference. To give a non-quantum, non-relativistic example: if two envelopes, one red and the other blue, are sent through the post to two addresses, then the recipient of the red envelope may infer that the envelope at the other address is blue.
It may be clear that our account leans towards, though is not predicated upon, the view that the state is not a physical entity,2 and away from attempts to extend ideas like the 'instantaneous collapse of the wavefunction' to QFT (cf. [19]). Particularly if one adopts a more subjective view on the nature of the state, there is a significant question of consistency to be addressed. Experiments conducted by multiple students in mutually spacelike separated laboratories might comprise a single experiment controlled by their supervisor. How can the updated states obtained by the students be reconciled, covariantly, with the overall update made by the supervisor?
Suppose that two probes are coupled to the system in compact interaction regions K 1 and K 2 , so that no point of K 2 lies to the past of any point in K 1 . At least some observers regard K 2 as occurring later than K 1 , though it is not excluded that some might regard K 2 as earlier than K 1 (which happens if K 1 and K 2 are causally disjoint). To each probe system B i there is a coupled system C i and a scattering morphism
Θ 1 = Θ 1 ⊗ 3 id B 2 (M) ,Θ 2 = Θ 2 ⊗ 2 id B 1 (M) ,(30)
where the subscript on the tensor product indicates the slot into which the identity is inserted in each case. Alternatively, the two probes may be described as a single probe with algebra B 1 (M) ⊗ B 2 (M), coupling region K 1 ∪ K 2 and a combined scattering morphismΘ on A(M) ⊗ B 1 (M) ⊗ B 2 (M). A natural assumption is that the causal factorisation formulaΘ
=Θ 1 •Θ 2(31)
holds -this can be checked in examples but is expected on the basis e.g., of Bogoliubov's factorisation formula in perturbative QFT [24,11,4]. Note that the mapΘ 2 appears to the right because our scattering morphisms map from the future to the past. One of the main results of [16] is the following consistency result.
Theorem 4 ([16, Thm 5])
Consider two probes as described above, with K 2 ∩ J − (K 1 ) = ∅. For all probe preparations σ i of B i (M) and all probe observables B i ∈ B i (M), the following identity for the pre-instruments holds:
I σ 2 (B 2 ) • I σ 1 (B 1 ) = I σ 1 ⊗σ 2 (B 1 ⊗ B 2 ).(32)
If, in fact, K 1 and K 2 are causally disjoint, we have
I σ 2 (B 2 ) • I σ 1 (B 1 ) = I σ 1 ⊗σ 2 (B 1 ⊗ B 2 ) = I σ 1 (B 1 ) • I σ 2 (B 2 ).(33)
The reader is referred to [16] for the proof, which is a direct calculation. Suppose that ω 1 (B 1 ) 0, so there is a nonzero probability of observing B 1 , and that ω ′ 1 (B 2 ) 0, where ω ′ 1 is the updated state conditioned on the observation of B 1 . Then another direct calculation shows that the post-selected state ω ′′ 12 conditioned on the observation of B 2 in state ω ′ 1 coincides with the post-selected state ω ′ 12 conditioned on the combined effect B 1 ⊗ B 2 being observed in state ω. In other words, the updating may be made in stages, and in any order if the B i have causally disjoint localisation. No additional assumptions are needed beyond those we have mentioned; no geometric boundaries across which state reduction occurs are needed. The cornerstone is locality of the interactions in the coupled theories, expressed through (31).
A specific model
The general formalism described above is amenable to practical calculation [16]. For instance, consider a simple situation in which both the system and the probe are quantized real scalar fields of possibly different mass, with classical action
S 0 = 1 2 ∫ M dvol (∇ a Φ)(∇ a Φ) − m 2 Φ Φ 2 + (∇ b Ψ)(∇ b Ψ) − m 2 Ψ Ψ 2(34)
for the uncoupled combination. Here, Φ will be the system field and Ψ the probe field, with respective masses m Φ and m Ψ . A simple interaction term that couples them together in a local region is
S int = − ∫ M dvol ρΦΨ,(35)
where ρ is a real, smooth function compactly supported in compact region K. It is convenient to write the field equations in a matrix form
T Φ Ψ := P R R Q Φ Ψ = 0,(36)
where P = + m 2 Φ , Q = + m 2 Ψ are the free Klein-Gordon operators for the two fields and RΦ = ρΦ. Writing the advanced (−) and retarded (+) Green operators for P, Q and T by E ± P and so forth, the Green functions of T may be established via a Born series
E ± T = ∞ r=0 (−1) r E ± P 0 0 E ± Q 0 R R 0 E ± P 0 0 E ± Q r ,(37)
which converges at least for sufficiently weak coupling ρ [13]. Thus, the (un)coupled classical field equation is Green hyperbolic [2]. The (un)coupled theory may be quantized by standard means; evidently the uncoupled theory has as its algebra the algebraic tensor product of the algebras for the free scalar fields of masses m Φ and m Ψ respectively. The generators of these algebras will be denoted Φ( f ) and Ψ( f ), labelled by test functions on M. It is of particular relevance to find the scattering morphism acting on elements 1 ⊗ Ψ(h) in the uncoupled algebra. In the case where h is supported in the out region M + , there is a simple formula
Θ(1 ⊗ Ψ(h)) = Φ( f − ) ⊗ 1 + 1 ⊗ Ψ(h − ),(38)
where
f − h − = 0 h − 0 R R 0 E − T 0 h (h ∈ C ∞ 0 (M + )).(39)
As Θ is a homomorphism, we immediately obtain the identity
Θ(1 ⊗ e iΨ(h) ) = e iΦ(f − ) ⊗ e iΨ(h − )(40)
between formal power series in h ∈ C ∞ 0 (M + ), from which the induced system observables corresponding to probe observables Ψ(h) n may be computed. Indeed, the definition (17) allows us to determine the generating function ε σ (e iΨ(h) ) = η σ (Θ(1 ⊗ e iΨ(h) )) = σ(e iΨ(h − ) )e iΦ(f − )
for any probe preparation state σ, with f − and h − as before. Specific induced observables may be obtained by differentiation. For example,
ε σ (Ψ(h)) = Φ( f − ) + σ(Ψ(h − ))1 (42) ε σ (Ψ(h) 2 ) = Φ( f − ) 2 + σ(Ψ(h − ) 2 )1(43)
and so on. A point of interest here is that none of the computations requires a Hilbert space representation -everything takes place at the level of the algebras. A number of general features become apparent in this model. We give two examples. First, recall that the actual experiment takes place on the coupled system and that probe observable Ψ(h) is identified at late times with observable Ψ(h) of the coupled system, while the system state is identified with a state ω σ . A simple calculation with the generating functions gives ω σ (e i Ψ(h) ) = σ(e iΨ(h − ) )ω(e iΦ(f − ) ).
(44)
Given sufficient regularity, λ → ω(e iλΦ(f − ) ) and λ → σ(e iλΨ(h − ) ) are the characteristic functions of probability measures for measurement outcomes of Φ( f − ) in state ω and Ψ(h − ) in state σ. Using (44) we see that the measurement outcomes of Ψ(h) in state ω σ -the outcomes of the actual experiment performed -are distributed according to the convolution of these measures, with characteristic function λ → ω σ (e iλ Ψ(h) ). This illustrates the general fact that the actual measurement is less sharp than the hypothetical measurement of the induced observable, due to fluctuations in the probe system. In particular, one has
Var( Ψ(h); ω σ ) = Var(Φ( f − ); ω) + σ(Ψ(h − ) 2 )(45)
for the variance of the measured observable, assuming for simplicity that σ has vanishing one-point function.
Second, the localisation of the induced observables can be determined. Taking the probe observable Ψ(h) as our example (with h ∈ C ∞ 0 (M + )), the induced observable has localisation determined by the support of f − , contained within the intersection J − (supp h) ∩ supp ρ. Unsurprisingly, the localisation of the probe observable must intersect the causal future of the coupling in order to constitute a nontrivial system measurement.
The general formalism allows one to assert that the induced observable may be localised roughly within the causal hull of J − (supp h) ∩ supp ρ. It is very tempting to try to ascribe it the localisation supp f − , particularly if ρ were localised in a thin tube near a timelike curve, for the spatial dimensions of the causal hull are proportional (with constant equal to the speed of light) to the temporal duration of the coupling and it might be convenient to replace this with a much smaller spacetime volume. A simple argument can be given to disabuse the reader of such temptations. Consider two induced observables, ε σ (Ψ(h)) and ε σ (Ψ(h ′ )). Their commutator may be computed as
[ε σ (Ψ(h)), ε σ (Ψ(h ′ ))] = [Φ( f − ), Φ( f ′− )] = iE P ( f − , f ′− )1,(46)
where
E P ( f 1 , f 2 ) = ∫ M dvol M f 1 E − P f 2 − E + P f 2 .(47)
Crucially, the right-hand side of (46) depends on the geometry throughout the region
S = (J + (supp f − ) ∩ J − (supp f ′− )) ∪ (J − (supp f − ) ∩ J + (supp f ′− )).(48)
Typically, it will be possible to find h ′ so that the supports of f ′− and f − are equal, whereupon S is the causal hull of supp f − . Consequently, there are questions concerning the induced observable ε σ (Ψ(h)), e.g., it compatibility or otherwise with another observable, that cannot be answered without knowledge of the geometry of the causal hull of supp f − . It seems unhelpful or misleading, therefore, to ascribe any smaller localisation to ε σ (Ψ(h)). For example, if the coupling is supported precisely on a timelike curve segment γ : [0, τ] → M, then the localisation region must include J + (γ(0)) ∩ J − (γ(τ)). For a uniformly accelerated trajectory of infinite proper duration, as in the Unruh-deWitt detector analysis, the localisation region becomes an entire wedge region of infinite spatial extent.
Conclusion
The framework set out here, and in full detail in [16], provides a fully covariant measurement scheme for general QFTs in curved spacetimes, which brings together quantum measurement theory and algebraic QFT. It is not tied to particular models, and formulates its assumptions in abstract terms; on the other hand, it allows for practical computations in specific models. It describes both the correspondence between observables of the probe and induced system observables, and also the updating of states by post-selection, with non-selective measurement as a special case. Several more topics are discussed in [16] but not here, including the role of gauge invariance and symmetries, and a perturbative treatment of the specific model studied in 5. It was also shown -for our model -how the product on the probe algebra could be deformed to make the mapping ε σ into a homomorphism, providing a sense in which the system observables are partially represented within the probe algebra.
Above all, we wish to emphasise three points in particular. First, the localisation properties of the induced observables were discussed; they may all be localised within any connected open causally convex neighbourhood of the coupling region K. Such neighbourhoods necessarily contain the causal hull of K and we have argued in the context of our model that no smaller localisation region can be expected. Second, the post-selected states satisfy an important consistency condition that allows multiple measurements to be combined into overarching measurements whenever they are subject to a causal order (and independently of the choice of order where relevant). Finally, incorporating the central insight of AQFT, our approach puts the principle of locality at its centre.
described on M by means of a unital * -algebra A(M) and subalgebras A(M; O) sharing the unit with A(M) and labelled by open causally convex (but not necessarily precompact) subsets O of M. The assumptions needed here are A1 Isotony Whenever O 1 ⊂ O 2 ⊂ M, the corresponding local algebras are nested, A(M; O 1 ) ⊂ A(M; O 2 ). (3) A2 Einstein causality If O 1 and O 2 are causally disjoint subsets of M then A(M; O i ) commute element-wise, [A(M; O 1 ), A(M; O 2 )] = 0. (4) A3 Compatibility If N is an open causally convex subset of M, then there is an injective unit-preserving * -homomorphism (to be described henceforth as a monomorphism) α M;N : A(N ) → A(M) whose image is precisely A(M; N), where N = M | N . We will refer to α M;N as a compatibility map. Whenever M 3 ⊂ M 2 ⊂ M 1 the compatibility maps obey the relation α M 1 ;M 2 • α M 2 ;M 3 = α M 1 ;M 3 . (5) A4 Timeslice property (existence of dynamics) If N is an open causally convex subset of M and contains at least one Cauchy surface of M, then α M;N is an isomorphism; equivalently, A(M; N) = A(M). In conjunction with compatibility, we may deduce that whenever O 1 ⊂ O 2 ⊂ M and O 1 contains a Cauchy surface of O 2 , then A(M; O 1 ) = A(M; O 2 ). (6) A5 Haag property Let K be a compact subset of M. Suppose that A ∈ A(M) commutes with every element of A(M; N) for every region N contained in the causal complement K ⊥ = M \ J(K) of K. Then A ∈ A(M; L) whenever L is a connected open causally convex subset containing K. This is a weakened form of Haag duality [17].
A ∈ A(M; K ⊥ ), the expectation value of A is unchanged in the post-selected state ω ′ if and only if A is uncorrelated with ε σ (B) in the original system state ω, i.e., ω(Aε σ (B)) = ω(A)ω
Θ i on the uncoupled algebra U i (M) = A(M) ⊗ B i (M). On the three-fold tensor product A(M) ⊗ B 1 (M) ⊗ B 2 (M), these scattering maps may be represented bŷ
Actually, one must check that ω ′ is indeed a state; see[16].
'[A] wavefunction is not a physical object. It is only a tool for computing the probabilities of objective macroscopic events' -Peres and Terno in[23].
Acknowledgements I thank Rainer Verch for useful comments on the text.References
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"S-AMP for Non-linear Observation Models",
"S-AMP for Non-linear Observation Models"
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| [
"Burak Ç Akmak \nDepartment of Electronic Systems\nDTU Compute Technical\nAalborg University\n9220AalborgDenmark\n",
"Ole Winther \nDepartment of Electronic Systems\nUniversity of Denmark\n2800LyngbyDenmark\n",
"Bernard H Fleury [email protected] \nAalborg University\n9220AalborgDenmark\n"
]
| [
"Department of Electronic Systems\nDTU Compute Technical\nAalborg University\n9220AalborgDenmark",
"Department of Electronic Systems\nUniversity of Denmark\n2800LyngbyDenmark",
"Aalborg University\n9220AalborgDenmark"
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| Recently we extended Approximate message passing (AMP) algorithm to be able to handle general invariant matrix ensembles. In this contribution we extend our S-AMP approach to non-linear observation models. We obtain generalized AMP (GAMP) algorithm as the special case when the measurement matrix has zero-mean iid Gaussian entries. Our derivation is based upon 1) deriving expectation propagation (EP) like algorithms from the stationary-points equations of the Gibbs free energy under first-and second-moment constraints and 2) applying additive free convolution in free probability theory to get low-complexity updates for the second moment quantities. | 10.1109/isit.2015.7282968 | [
"https://arxiv.org/pdf/1501.06216v1.pdf"
]
| 9,482,867 | 1501.06216 | 661d6926950fb12aa31cac79d2dbbb521665b2eb |
S-AMP for Non-linear Observation Models
25 Jan 2015
Burak Ç Akmak
Department of Electronic Systems
DTU Compute Technical
Aalborg University
9220AalborgDenmark
Ole Winther
Department of Electronic Systems
University of Denmark
2800LyngbyDenmark
Bernard H Fleury [email protected]
Aalborg University
9220AalborgDenmark
S-AMP for Non-linear Observation Models
25 Jan 2015arXiv:1501.06216v1 [cs.IT]Index Terms-Approximate Message PassingVariational In- ferenceExpectation PropagationFree Probability
Recently we extended Approximate message passing (AMP) algorithm to be able to handle general invariant matrix ensembles. In this contribution we extend our S-AMP approach to non-linear observation models. We obtain generalized AMP (GAMP) algorithm as the special case when the measurement matrix has zero-mean iid Gaussian entries. Our derivation is based upon 1) deriving expectation propagation (EP) like algorithms from the stationary-points equations of the Gibbs free energy under first-and second-moment constraints and 2) applying additive free convolution in free probability theory to get low-complexity updates for the second moment quantities.
I. INTRODUCTION
Approximate message passing techniques, e.g. [1]- [3], have recently received significant attention by the signal processing community. Essentially, these methods are based on taking the large system limit of loopy belief propagation where the central limit theorem can be applied when the underlying measurement matrix has independent and zero-mean entries.
Variational inference techniques are well-established in the field of information theory e.g. [4], [5] and machine learning e.g. [6], [7]. For example, it is well-known that exact inference can be formulated as the solution to a minimization problem of the Gibbs free energy of the underlying probabilistic model under certain marginalization consistency constraints [4]. We have recently shown in [8] that for the zero-mean independent identically distributed (iid) measurement matrix, approximate message passing (AMP) algorithm [1] can also be obtained from the stationary-points equations of the Gibbs energy under first-and second-moment consistency constraints. Furthermore, AMP can be extended to general invariant 1 matrix ensembles by means of the asymptotic spectrum of the measurement matrix. We call this approach S-AMP (where S comes from the fact that the derivation uses the S-transform).
AMP is an estimation algorithm for the linear observation models. However many interesting cases occur in practice where the observation model is non-linear, e.g non-linear form of compressed sensing, Gaussian processes for classification. In this article we extend S-AMP approach [8] to general observation models. Specifically we address the sum-product generalized AMP (GAMP for short) algorithm [3].
The derivation of GAMP is based on certain approximations (mainly Gaussian and quadratic approximations) of loopy belief propagation. If the measurement matrix is large and has zero mean and iid entries, GAMP provides excellent performance, e.g. [3], [9]. Furthermore, for general matrix ensembles it can show quite reasonable accuracy [10]. However the algorithm itself and its derivation are not well-understood.
To better understand GAMP, in [11] the authors characterize its fixed points. Specifically, they show that GAMP can be obtained from the stationary-point equations of some implicit approximations of naive mean-field approximation [11]. These implicit approximations only provide limited insight. Furthermore, the naive mean-field interpretation is misleading, because the fixed points of AMP-type algorithms are typically known as the TAP-like equations, i.e. they include a correction term to naive mean-field solution. In fact GAMP can also be obtained from the stationary-points equations of the Bethe free energy (BFE) of the underlying loopy graph under first-and second-moment constraints. However, this approach also limits our understanding, because the BFE formulation of a loopy graph is suitable for sparsely connected systems.
In this work we focus on the BFE formulation of a tree graph, i.e. an exact Gibbs free energy formulation. We note that our approach coincides with the expectation prorogation (EP) [12]- [14] since the fixed points of EP are the stationary points of BFE of the underlying probabilistic graph under a set of moment consistency constraints [6].
Notations: The entries of the N × K matrix X are denoted by either X nk or [X] nk , n ∈ N {n : 1 ≤ n ≤ N } and k ∈ K {k : 1 ≤ k ≤ K}. Without loss of generality we assume that K and N are disjoint. (·) † denotes the transposition. We denote by ℜz and ℑz the real and imaginary part of z ∈ C, respectively. The entries of a vector u ∈ R T ×1 are indicated by either u t or [u] t , t ∈ [1, T ]. Furthermore u T t=1 u t /T . Moreover, diag(u) is a diagonal matrix with the elements of vector u on the main diagonal. For a square matrix X, diag(X) is a column vector containing the diagonal elements of X. Furthermore Diag(X) diag(diag(X)). The Gaussian probability density function (pdf) with mean µ and the covariance Σ is denoted by N (·; µ, Σ). Throughout the paper when referring to "in the large system limit" we imply that N, K tends to infinity with the ratio α N/K fixed. All large system limits are assumed to hold in the almost sure sense, unless explicitly stated.
II. SYSTEM MODEL AND REVIEW OF GAMP Consider the estimation of a random vector x ∈ R K×1 which is linearly transformed by A ∈ R N ×K as z Ax, then passed through a noisy channel whose output is given by y ∈ R N ×1 . We assume that the conditional pdf of the channel factorizes according to
p(y|z) = n∈N p(y n |z n ).(1)
Furthermore for the Bayesian setting we assign a prior pdf for x that is assumed to be factorized as
p(x) = k∈K p(x k ).(2)
A. GAMP summarized We summarize GAMP here for the sake of streamlining and making the connection to the derivation of S-AMP. We separate the GAMP iteration rules [3] into two parts: (i) GAMP-1st order that initializesx t , τ t x and m t from tabula rasa at t ≤ 0 and proceeds iteratively as
κ t z = Ax t − ( Λ t z ) −1 m t−1 (3) z t = µ z (κ t z ; Λ t z ) (4) τ t z = σ z (κ t z ; Λ t z ) (5) m t = Λ t z (ẑ t − κ t z ) (6) κ t x = ( Λ t x ) −1 A † m t +x t (7) x t+1 = µ x (κ t x ; Λ t x ) (8) τ t+1 x = σ x (κ t x ; Λ t x ).(9)
(ii) GAMP-2nd order are the update rules for Λ
t z and Λ t x : Λ t z = (diag((A • A)τ t x )) −1 (10) τ t m = Λ t z (1 − Λ t z τ t z ) (11) Λ t x = diag((A • A) † τ t m ).(12)
In these expressions 1 is the all-ones vector of appropriate dimension and µ x and σ x are scalar functions. Specifically, if Λ is a K × K diagonal matrix and κ is a K × 1 vector; then for k ∈ K, [µ x (κ;
Λ )] k and [σ x (κ; Λ )] k are respectively the mean and the variance taken over the pdf
q k (x k ) ∝ p k (x k ) exp − Λ kk 2 (x k − κ k ) 2 .(13)
Similarly, µ z and σ z are scalar functions such that if Λ is a N × N diagonal matrix and κ is a N × 1 vector, for n ∈ N , [µ z (κ;
Λ )] n and [σ z (κ; Λ )] n are respectively the mean and the variance taken over the pdf
q n (z n ) ∝ p(y n |z n ) exp − Λ nn 2 (z n − κ n ) 2 .(14)
If the entries of A are iid with zero mean and variance 1/N , the iteration steps for the GAMP-2nd order simplify as
Λ t z = α τ t x I, Λ t x = τ t m I,(15)
where I is the identity matrix of appropriate dimension. We note that if in addition p(y|z) = N (y; z, σ 2 I), GAMP yields AMP, see e.g. [2, Appendix C].
III. GIBBS FREE ENERGY WITH MOMENT CONSTRAINTS
For the sake of notational compactness, consider s = (x, z). Furthermore we introduce the set V K ∪ N and assume that K and N are disjoint. Moreover we define
f A (s) δ(z − Ax) (16) f v (s v ) p v (x v ) v ∈ K p(y v |z v ) v ∈ N .(17)
With these definitions, the posterior pdf of s reads
p(s|y, A) = 1 Z f A (s) v∈V f v (s v ).(18)
with Z denoting a normalization constant. The factor graph representing (18) is a tree. Thus the BFE of (18) is equal to its Gibbs free energy [4]:
G({b v , b A , b v }) − v∈V b v (s v ) logb v (s v )ds v − b A (s) log f A (s) b A (s) ds − v∈V b v (s v ) log f v (s v ) b v (s v ) ds v .(19)
Here b A and b v , v ∈ V, denote the beliefs of the factors in (18), whileb v , v ∈ V, denote the beliefs of the unknown variables in (18). Without loss of generality we assume that the expressions (19) are strictly continuous; so that the Gibbs free energy is welldefined. Indeed this is what we will end up with in the analysis. If we define a Lagrangian for (19) that accounts for certain marginalization consistency constrains, then at its stationary point, the beliefb v (s v ) is equal to p(s v |y, A) for all v ∈ V [4]. Instead, following the arguments of [6], we define the Lagrangian on the basis of a set of moment consistency constraints as
f A (s)/b A (s) and f v (s v )/b v (s v ) inL({b v , b A , b v }) G({b v , b A ,b v }) + Z − v∈V ν † v φ(s v ) b A (s) −b v (s v ) ds − v∈Vν † v φ(s v ) b v (s v ) −b v (s v ) ds v .(20)
Here we consider constraints on the mean and variance, i.e.
φ(s v ) = (s v , s 2 v ), v ∈ V.
For convenience we write the Lagrangian multipliers as
ν v γ v , − Λ vv 2 ,ν v ρ v , − Λ vv 2 , v ∈ V.
The term Z accounts for the normalization constraints:
Z −β A 1 − b A (s)ds − v∈Vβ v 1 − b v (x v )ds v − β v 1 − b v (s v )ds v
where β A , β v ,β v are the associated Lagrange multipliers.
We formulate the estimate of s v , v ∈ V, aŝ
s v s vb ⋆ v (s v )ds v ,(21)
whereb ⋆ v (s v ) representsb v (s v ) at a stationary point of (20).
A. The Stationary Points of the Lagrangian
For notational convenience we introduce first the (K +N )× (K +N ) diagonal matrices Λ and Λ as well as the (K +N )×1 vectors γ and ρ whose entries are respectively Λ vv , Λ vv , γ v and ρ v , v ∈ V. In connection with variables x and z we write
Λ = Λ x 0 0 Λ z , γ = (γ x , γ z ) (22) Λ = Λ x 0 0 Λ z , ρ = (ρ x , ρ z ).(23)
The dimensions of Λ x and Λ x are K × K; vectors γ x and ρ x have dimension K × 1.
Following the arguments of [6], we have the stationary points of the Lagrangian (20)
in the form b ⋆ v (s v ) = 1 Z v exp((ν v +ν v ) † φ(s v )), v ∈ V (24) b ⋆ v (s v ) = 1 Z v f v (s v ) exp(ν † v φ(s v )), v ∈ V (25) b ⋆ A (s) = 1 Z A f A (s) exp − 1 2 s † Λs + s † γ(26)
where Z A , Z v ,Z v are the associated normalization constants. Let us first consider the marginalization of the belief b ⋆ A (s) with respect to z:
b ⋆ A (x) = b ⋆ A (x, z)dz = N (x;x, Σ x )(27)
where
Σ x (Λ x + A † Λ z A) −1 ,x Σ x (γ x + A † γ z ). (28)
Here we note that Σ x is positive definite since b ⋆ A (x) is a well-defined pdf. Second, let us consider the marginalization over x, which basically follows from the linear transformation property of a Gaussian random vector:
b ⋆ A (z) = e − 1 2 z † Λzz+z † γ z Z A δ(z − Ax)e − 1 2 x † Λxx+x † γ x dx = δ(z − Ax)N (x;x, Σ)dx = N (z;ẑ, Σ z ) (29)
whereẑ Ax and Σ z AΣ x A † . At this stage it is convenient to define
κ (κ x , κ z ) = ( Λ −1 x ρ x , Λ −1 z ρ z )(30)
with κ x ∈ R K . In this way we can write the belief in (25) as
b ⋆ v (s v ) ∝ f v (s v ) exp − Λ vv 2 (s v − κ v ) 2 .(31)
Thereby (31) has a form identical to (13) and (14) for v ∈ N and for v ∈ K, respectively. Then let us define
µ(κ; Λ ) (µ x (κ x ; Λ x ), µ z (κ z ; Λ z )) (32) σ(κ; Λ ) (σ x (κ x ; Λ x ), σ z (κ z ; Λ z )) .(33)
The entries [µ(κ; Λ )] v and [σ(κ; Λ )] v are respectively the mean and variance of the belief (25). Moreover we introduce
Σ Σ x 0 0 Σ z ,ŝ = (x,ẑ).(34)
With these definitions, the identities resulting from the moment consistency constraints are given bŷ
s = Diag(Σ)(γ + ρ),ŝ = µ(κ; Λ ) (35) Diag(Σ) = (Λ + Λ ) −1 , diag(Σ) = σ(κ; Λ ).(36)
B. The TAP-like Equations and GAMP-1st Order
By using the fixed-point identities presented in Section III-A, one can introduce numerous fixed-point algorithms.
In this work we restrict our attention to TAP-like algorithms, e.g. [3], [12], [14]. To that end we start with the definitions in (28) and write
γ x = −A † γ z + (Λ x + A † Λ z A)x.(37)
Furthermore by making use of the identities in (35) and (36) we have
ρ x = A † γ z − (Λ x + A † Λ z A)x + (Λ x + Λ x )x (38) = A † (γ z − Λ z Ax) + Λ xx (39) = A † m + Λ xx with m (γ z − Λ z Ax).(40)
Moreover, by the definition of m we also point out that
m = (Λ z + Λ z )ẑ − ρ z − Λ z Ax (41) = Λ zẑ − ρ z = Λ z (ẑ − κ z ).(42)
Thereby we exactly obtain the fixed points of GAMP-1st order, i.e. (3)- (9). Now let us keep the iterations step of GAMP-1st order but define the update rule for Λ t x and Λ t z on the basis of the fixed point identities in (36). For example:
Λ t z = (diag(τ t−1 z )) −1 − Λ t−1 z (43) Σ t x = (Λ t−1 x − A † Λ t z A) −1 .(44)Λ t z = Diag AΣ t x A † −1 − Λ t z (45) Λ t x = diag(τ t x ) −1 − Λ t−1 x (46) Λ t x = Diag Σ t x −1 − Λ t x .(47)
In this way we obtain a new fixed point algorithm whose fixed points are the stationary point of Lagrangian (20). However from the complexity point of view these updates are problematic due to the matrix inversion in (44). In the sequel we will address how to bypass (44) as K, N are large.
C. The Large-System Simplifications
To circumvent the complexity problem (44), we utilize the so-called additive free convolution in free probability theory [15]. The reduction that we obtain in this way can be also obtained by means of the self-averaging ansatz in [14,Section 3.1].
In order to make use of additive free convolution we need to restrict our consideration to the invariant matrix ensembles: ASSUMPTION 1 Consider the singular value decomposition A = U DV where U N ×N and V K×K are orthogonal matrices and D is a N × K non-negative diagonal matrix. We distinguish between the invariance assumption on A from right and from left: a) A is invariant from right, i.e. V is
Haar distributed; b) A is invariant from left, i.e. U is Haar distributed.
It indeed makes sense to distinguish between the invariance from right and the invariance from left. For example, once we consider the classical linear observation model such as p(y|z) = N (y; z, σ 2 I), then Λ z = I/σ 2 . In this case we do not need to consider Assumption 1-b).
Second, we make the following technical assumption on the limiting spectrum of the respective matrices: ASSUMPTION 2 As N, K → ∞ with the ratio α = N/K fixed let the spectra of Λ x , Λ z and A † A converge almost surely to some limiting spectra whose supports are compact.
Due to lack of an explicit definition of the "Lagrangian" matrix Λ, Assumption 2 is rather implicit. Nevertheless it can be considered in the same vein as the so-called thermodynamic limit in statistical physics: all microscopic variables converge to deterministic values in the thermodynamic limit [16].
For example, under Assumption 1-a) and Assumption 2, it turns out that Λ x and J z A † Λ z A are asymptotically free [17] and from [15,Lemma 3.3.4] we have that 2
R K Λx+Jz (ω) ≃ R K Λx (ω) + R K Jz (ω), ℑω < 0.(48)
Here for a T × T symmetric matrix X R T X denotes the Rtransform of the spectrum of X (see Appendix A) and ≃ stands for the large system approximation that turns to an almost surely equality in the large system limit. Furthermore we introduce
R T X (r) lim ω→r ℜR T X (ω), ℑr = 0(49)
whenever the limit exists. It turns out that by solely invoking "additive free convolution", e.g. (48), we can easily solve the complexity issue of the fixed point identities for Λ x and Λ z which do not require matrix inversion. First we consider the simplification for Λ x . To than end let us first define the auxiliary variable 2 In fact we can define the R-transform on negative real line. However in the exposition it requires an implicit assumption that Λ is being positive-definite.
q 1 K tr{(Λ x + J z ) −1 } = 1 K k∈K 1 [Λ x ] kk + [ Λ x ] kk .(50)
Then by invoking (48) we easily obtain that (see Appendix B)
q ≃ 1 K k∈K 1 [Λ x ] kk + R K J z (−q) .(51)
Thereby, we conclude that
[ Λ x ] kk ≃ R K J z (−q), k ∈ K.(52)
The average of (52) over the random matrix A agrees with [14,Eq. (50)]. Note that the simplification in (52) is still implicit due to the definition of q in (51). Subsequently we present an explicit complexity simplification for [ Λ x ] kk . First we note that (52) states that we can replace all the elements [ Λ x ] kk , k ∈ K by a single scalar quantity, say Λ x . This allows us to write q ≃ σ x (κ x , Λ x I) with κ x = Λ −1
x A † m +x. Then, from (52) we write an explicit fixed point identity for
Λ x as Λ x = R K J z (− σ x (κ x ; Λ x I) ).(53) Note that we omit to mention the invariance property in[8]. It is however crucial for the derivation.
As a second part we address similar complexity simplification for [ Λ z ] nn for n ∈ N . To that end let us introduce an auxiliary N × 1 vectorτ m whose entries are defined aswhere (55) follows directly from Woodbury's matrix inversion lemma. Furthermore by making use of (36) for (54) we can write the following fixed-point identityThus, we can invoke identical arguments on the additive free convolution approximation above for [ Λ z ] nn as well. Specifically, under Assumption 1-b) and Assumption 2, for a large N, K we haveThe complexity simplification (58) is still implicit due the definition ofτ m . To present an explicit form of it consider first (56) and (57) such that we can writeOn the other hand, (58) implies that we can replace all the elements [ Λ z ] nn , n ∈ N by a single scalar quantity, say Λ z . Now for convenience let us define N × 1 vector τ m whose entries are given byThen following (58) we introduce an explicit fixed-point identity for Λ z asSo far we have shown in (53) and (62) how to bypass the need for matrix inversion to "update" Λ x and Λ z , respectively. However this treatment require solving a highly non-trivial random matrix problem i.e. deriving the closed form solution for R K J z and R N J x . This is usually, though not always, not possible. On the other hand deriving the solution of e.g R K Jz in the limiting case, denote R J z , is rather simpler. Due to the uniform convergence property of the R-transform[15,Lemma 3.3.4], this approach would allow us to accurately predict, for example R K J z , for large N, K. This is what we show in the next subsection for the zero mean iid Gaussian matrix ensemble.Example: The zero-mean and iid case, i.e. GAMP: In this section we provide the explicit solutions for Λ x and Λ z when the entries of A are assumed to be iid Gaussian with zero mean and variance 1/N .From the well-known Marchenko-Pastur theorem we obtain that (see Appendix C)Then we obtain the following expression for Λ x and Λ z asFrom these equations one can conclude thatThus we recover the fixed point of the GAMP-2nd order updates for the zero-mean iid matrix ensemble as in(15).IV. CONCLUSIONFor the given zero-mean iid Gaussian matrix ensemble, the fixed points of GAMP "asymptotically" coincide with the stationary points of Gibbs free energy under first-and secondmoment constraints. It turns out that the only critical issue for GAMP is the update rules for "variance" parameters Λ x and Λ z . These parameters play a central role. Specifically a crude update rule for a given measurement matrix ensemble would completely spoil the optimality of the algorithm. If for general invariant matrix ensembles, Λ x and Λ z can be updated based on the R-transform formulation in (53) and (62); the algorithm "asymptotically" fulfills the stationary points identities of Gibbs free energy formulation. Once the closed form expressions of (53) and (62) are obtained, the resulting algorithm includes solely O(N ) operations. But the computation of the solutions to these identities is not trivial. Nevertheless it is sometimes doable, e.g. the random row orthogonal matrix ensembles. Furthermore once either the prior or the likelihood is expressed in terms of a Gaussian function, the R-transform formulation becomes rather trivial. In general updating Λ x and Λ z requires a matrix inversion at each iteration, e.g. see (43)-(47). An alternative, but suboptimal, method would be the Swept-AMP algorithm[10]that is based on the GAMP methodology and includes O(N 2 ) operations.APPENDIX A PRELIMINARIESLet P X a probability distribution on real line. We denote the Stieltjest transform of P X aswhere ℑG X (s) > 0[18]. The R-transform of P X is defined as[15]with G −1 X denoting the inverse of G X . Equivalently,Here we draw the attention of the reader that ℑR X (ω) < 0, for ℑω < 0 unless P X is a Dirac distribution. This fact follows from the following property of the Stieltjest transform [18, Proposition 2.2]: for ℑs > 0, ℑ{ 1 GX (s) + s} ≤ 0 where the equality holds if, and only if, P X is a Dirac distribution.Then we have the following identityConsider an T × T symmetric matrix X. Let L be the set containing the eigenvalues of X. The spectrum of X is denoted byWe denote the Stieltjest transform and the R-transform of P T X by G T X and R T X , respectively. Furthermore if for T → ∞, X has a limiting spectrum almost surely it is denoted by P X . Moreover, the Stieltjest transform and the R-transform of P X are denoted by G X and R X , respectively.APPENDIX B PROOF OF (51)Note that from definition in (50) we haveBy invoking Remark 1 and (48) (under the Assumption 1-a) and Assumption 2), successively we can writeHere, without loss of generality, we can define q ǫ q + ǫ. On the other hand, from the definition of the R-transform in (70) we haveHence we can writeThis completes the proof.APPENDIX C PROOF OF (63) & (64)Let us first consider J z = A † Λ z A. Note that we do not assume that A and Λ z are independent. On the other hand, Assumption 2 results in that A † A and Λ z are asymptotically free of each others. In this way we can find R Jz by means of the so-called multiplicative free convolution[19]. However this requires the reader to be familiar with the S-transform in free probability. In fact, by invoking standard random matrix results we can bypass the need for using the S-transform. Specifically, from the well-known Marchenko-Pastur theorem, we can writeThe result (79) is proven under the assumptions that the entries of A are iid (not necessarily Gaussian) with zero mean and A is independent of Λ z[20]. Due to the asymptotic freeness, this result holds when the entries are restricted to Gaussian but without restriction that A and Λ z are independent. Now by letting s = G −1 J z (−ω) in (79) and from the definition of the R-transform in (69) we haveFurthermore following the identical arguments for J x we findDue to[15,Lemma 3.3.4], the right hand side of the expressions in (63) and (64) converge uniformly to (80) and (81), respectively. This completes the proof.
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| []
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[
"Deciphering the nonlocal entanglement entropy of fracton topological orders",
"Deciphering the nonlocal entanglement entropy of fracton topological orders"
]
| [
"Bowen Shi \nDepartment of Physics\nThe Ohio State University\n43210ColumbusOHUSA\n",
"Yuan-Ming Lu \nDepartment of Physics\nThe Ohio State University\n43210ColumbusOHUSA\n"
]
| [
"Department of Physics\nThe Ohio State University\n43210ColumbusOHUSA",
"Department of Physics\nThe Ohio State University\n43210ColumbusOHUSA"
]
| []
| The ground states of topological orders condense extended objects and support topological excitations. This nontrivial property leads to nonzero topological entanglement entropy Stopo for conventional topological orders. Fracton topological order is an exotic class of models which is beyond the description of TQFT. With some assumptions about the condensates and the topological excitations, we derive a lower bound of the nonlocal entanglement entropy S nonlocal (a generalization of Stopo). The lower bound applies to Abelian stabilizer models including conventional topological orders as well as type I and type II fracton models, and it could be used to distinguish them. For fracton models, the lower bound shows that S nonlocal could obtain geometry-dependent values, and S nonlocal is extensive for certain choices of subsystems, including some choices which always give zero for TQFT. The stability of the lower bound under local perturbations is discussed.PACS numbers:arXiv:1705.09300v2 [cond-mat.str-el] | 10.1103/physrevb.97.144106 | null | 118,865,040 | 1705.09300 | aba3ca06fc23f7ef4a5e40ff340cc69fae2d817e |
Deciphering the nonlocal entanglement entropy of fracton topological orders
Bowen Shi
Department of Physics
The Ohio State University
43210ColumbusOHUSA
Yuan-Ming Lu
Department of Physics
The Ohio State University
43210ColumbusOHUSA
Deciphering the nonlocal entanglement entropy of fracton topological orders
(Dated: April 17, 2018)PACS numbers:
The ground states of topological orders condense extended objects and support topological excitations. This nontrivial property leads to nonzero topological entanglement entropy Stopo for conventional topological orders. Fracton topological order is an exotic class of models which is beyond the description of TQFT. With some assumptions about the condensates and the topological excitations, we derive a lower bound of the nonlocal entanglement entropy S nonlocal (a generalization of Stopo). The lower bound applies to Abelian stabilizer models including conventional topological orders as well as type I and type II fracton models, and it could be used to distinguish them. For fracton models, the lower bound shows that S nonlocal could obtain geometry-dependent values, and S nonlocal is extensive for certain choices of subsystems, including some choices which always give zero for TQFT. The stability of the lower bound under local perturbations is discussed.PACS numbers:arXiv:1705.09300v2 [cond-mat.str-el]
I. INTRODUCTION
Topological order [1] is a gapped quantum phase of matter beyond the description of the Landau-Ginzburg theory of symmetry breaking. Many of the early examples of topological orders (which we will refer to as conventional topological orders) share the following properties: robust ground state degeneracy which depends on the topology of the manifold [1], the ground states are locally indistinguishable [1][2][3], the existence of integer dimensional condensates and logical operators that can be topologically deformed [4], nontrivial braiding statistics of anyons or other topological excitations (or topologically charged excitations) e.g., excitations which could not be created alone by local operators [5,6], they are effectively described by topological quantum field theory (TQFT) at low temperatures [1], and they can be used to do faulttolerant quantum information processing [7]. And it is well known that, in 2D, a suitable linear combination of entanglement entropy with local contributions canceled is a topological invariance called the topological entanglement entropy [8,9]. Topological entanglement entropy is a property of the ground state wave function and it has been used to identify quantum spin liquid phases [10]. It also contains information about the ground state degeneracy [11] and the forms of low-energy excitations [12]. Generalizations of topological entanglement entropy into 3D bulk [13] and boundary [12] are studied.
On the other hand, there are recently discussed 3D exotic topological ordered models [5,[14][15][16][17][18][19][20][21][22][23][24] that do not fit very well into the pictures above. These models have recently been classified into fracton topological orders [18]. While fracton models have locally indistinguishable ground states when placed on nontrivial manifolds and the ground state degeneracy is robust under local perturbations [2,3], the ground state degeneracy depends on the system size (geometry) rather than merely the topology of the manifold [5,6,16]. While fracton models possess topological excitations [5,6], these topological excitations are constrained to move in lower dimensional submanifolds rather than the whole system [17,18]. The condensates and logical operators can be fractal dimensional [16,18] instead of integer dimensional. These models are beyond the description of TQFT.
There are type I and type II fracton topological orders. The type I fracton models include the Chamon-Bravyi-Leemhuis-Terhal (CBLT) model [5,14], the Majorana cubic model [17], and the X-cube model [18], etc.; they have integer dimensional condensates and logical operators. The type II fracton models include Haah's code [15] and many of the fractal spin liquid models [16] (see Sec.III D). Type II fracton models possess fractal condensates and logical operators and the excitations are fully immobile [18].
For the ground state entanglement properties of fracton models, a relation to the ground state degeneracy is implied in [11] and the entanglement renormalization group transformation of Haah's code is studied in [25]. In this work, we construct a direct analogy of topological entanglement entropy by doing linear combinations of entanglement entropies of different subsystems in such a way that the local contributions (from each boundary or corner of the subsystems) are canceled, and we call the linear combination S nonlocal , the nonlocal entanglement entropy. While S nonlocal is topologically invariant for conventional topological orders, it is geometry-dependent for fracton models.
Explicitly, we choose a conditional mutual information form used by Kim and Brown [12] and define the nonlocal entanglement entropy S nonlocal ≡ (S BC +S CD −S C − S D )| ρ = I(A : C|B)| ρ . Where ρ = |ψ ψ| with |ψ being a ground state. The whole system is the union of subsystems A, B, C, and D with A and C separated by distance l ξ, the correlation length. One can check that the local contributions from the boundaries are canceled, and this is why we use the name "nonlocal entanglement entropy." This construction can be used in any dimensions and an example in 2D is shown in Fig.1. With several assumptions about the condensates and topological excitations, a lower bound of S nonlocal is derived.
When applied to known conventional Abelian topological orders e.g. the 2D toric code model and the 3D toric code model [26], the lower bound is topologically invariant and it is identical to the exact result, i.e. the lower bound is saturated. When applied to fracton models [5,[14][15][16][17][18], the lower bound depends on the sizes and relative locations of the subsystems. For fracton models, there exist choices of subsystems for which the lower bound is nonzero and extensive. It is possible to have S nonlocal > 0 for subsystem choices, that are expected to have S nonlocal = 0 if TQFT holds. This method observes an intimate relation between S nonlocal and topological excitations created in D by a unitary operator U stretched out in CD which could be "deformed" into a unitary operator U def in AD, and since deformable U is intimately related to condensate operator W (for more details of U , U def and condensate operator W see Sec.II D), this method observes an intimate relation between S nonlocal and ground state condensates as well. This method allows a lower bound of S nonlocal to be obtained without calculating the entanglement entropy of any individual subsystem. Furthermore, it provides us with a unified viewpoint to understand the topology-dependent S nonlocal in the conventional topological orders and the geometry-dependent S nonlocal in fracton topological orders. Also discussed is the stability of the lower bound under local perturbations.
Two additional papers appeared after our work which also study the entanglement entropy of fracton phases, using explicit computation [27] and tensor network [28].
For non-Abelian models, some of our assumptions breakdown, and our original method does not apply. Nevertheless, a variant of our lower bound is applicable to non-Abelian models [29].
The structure of the paper is as follows: In Sec.II we provide a derivation of the lower bound from some assumptions about topological excitations and condensates; In Sec.III we apply our lower bound to several exactly solved Abelian stabilizer models of 2D, 3D conventional topological orders and type I, type II fracton topological orders. In Sec.IV we discuss the stability of the lower bound under local perturbations. Sec.V is discussion and outlook.
II. THE LOWER BOUND
A. A few notations and definitions
We will consider a infinite system without boundaries. The system is divided into subsystems A, B, C, D (nonoverlapping regions in real space, the union of which is the whole system). Each subsystem has a size large compared to the correlation length ξ, and the subsystems A and C are separated by a distance much larger than the correlation length. One example is shown in Fig.1, and similar constructions can apply to any dimensions. For all the examples in this paper, we have chosen B, C, D to be local subsystems while A is not, but there exist other possible choices, say A, B, C local. A local subsystem is a subsystem which can be contained in a ball-shaped subsystem of finite radius. ∂A, ∂B are the boundaries of the subsystems. We useĀ to denote the complement of A.
We will use ρ, σ to represent density matrices. In this paper we always use ρ for the ground state density matrix, ρ = |ψ ψ|, and |ψ is the ground state. We use ρ ABC , σ BC when we want to specify the subsystems. The entanglement entropy is defined in terms of the (reduced) density matrix as usual S = −tr[σ ln σ]. We use S ABC | ρ and S ABC | σ to distinguish the entanglement entropy on region ABC with different density matrices ρ, σ. Define conditional mutual information
I(A : C|B) ≡ S AB + S BC − S B − S ABC
and we use I(A : C|B)| ρ when we want to specify a density matrix. It is known that the conditional mutual information is always nonnegative I(A : C|B) ≥ 0. We say σ ABC is conditionally independent if I(A : C|B)| σ = 0.
For unitary operators U and U which create excitations in D when acting on the ground state |ψ , we say U ∼ U or U is similar to U if the states U |ψ and U |ψ have identical reduced density matrices on ABC, i.e. tr D [U ρU † ] = tr D [U ρU † ]. Otherwise, we say U and U are distinct.
B. Prepare for the lower bound
If there is a density matrix σ which is related to the ground state density matrix ρ by ρ AB = σ AB and ρ BC = σ BC , then we have: For a pure state, the entanglement entropy of a subsystem equals the entanglement entropy of its complement, e.g. S Ω = SΩ for any subsystem Ω. Therefore:
I(A : C|B)| ρ ≥ S ABC | σ − S ABC | ρ(1)I(A : C|B)| ρ = (S BC + S CD − S B − S D )| ρ .
Observe that the local contributions of the entanglement entropy get canceled due to the fact that A and C are separated. Let us define the nonlocal entanglement entropy (of the ground state)
S nonlocal ≡ (S BC + S CD − S B − S D )| ρ .(2)
The nonlocal entanglement entropy is just another way to write down the conditional mutual information, S nonlocal = I(A : C|B)| ρ , and therefore S nonlocal ≥ 0. The form in Eq.
(2) has the advantage that it involves only local systems B, C, D. When the system is placed on a torus or other nontrivial manifolds instead of a infinite manifold, the system may have several locally indistinguishable ground states, this form of S nonlocal in terms of local subsystems is more convenient, and even if ρ is a mixed state density matrix of different locally indistinguishable ground states, S nonlocal still has the same value.
C. The key idea about the lower bound
The discussion above suggests a way to obtain a lower bound of S nonlocal . For any σ satisfying σ AB = ρ AB and σ BC = ρ BC ,
S nonlocal ≥ S ABC | σ − S ABC | ρ .(3)
The density matrix σ does not have to be a density matrix of a pure state. If we could find a σ satisfying the above requirement and S ABC | σ > S ABC | ρ , a nonzero lower bound is obtained, and then the existence of nonzero nonlocal entanglement entropy is established. Now, let us assume that we could find a set of σ I with I = 1, . . . , N such that σ I AB = ρ AB and σ I BC = ρ BC . Then we can do superpositions and define σ ≡ N I=1 p I σ I with {p I } being a probability distribution, i.e. p I ∈ [0, 1] and N I=1 p I = 1. The σ is a new density matrix which satisfies σ AB = ρ AB and σ BC = ρ BC . So we have a whole parameter space of σ to try.
If lucky, we may even be able to find a σ * ABC which is conditionally independent (satisfying I(A : C|B)| σ * = 0) and we have an exact result
S nonlocal = S ABC | σ * − S ABC | ρ .(4)
Or, if we find the lower bound is saturated, we know the σ we used to obtain the lower bound is conditionally independent. We note that, in the quantum case (unlike the classical case), it is not always possible to find a conditionally independent σ * such that σ * AB = ρ AB and σ * BC = ρ BC for ρ being a general density matrix [30]. Therefore, the existence of such σ * in some system might be interesting by itself. On the other hand, the conditional independent state σ * is known to exist for models satisfying simple conditions (I)(II) in [31].
D. Calculate the lower bound for Abelian models employing assumptions
We make a few assumptions about the condensates and operators creating topological excitations in order to develop a way to find σ I and calculate a lower bound of S nonlocal . These assumptions are applicable to Abelian models with commuting projector Hamiltonians (with each term acting on a few sites localized in real space), including conventional models e.g. the toric code model in various dimensions [26], quantum double models with any Abelian finite group, and fracton models [5,[14][15][16][17][18] of type I and type II which we will be focusing on in this work. In the context of fracton models, Abelian means no protected degeneracy associated with excitations.
Our way of employing the operators is inspired by a method by Kim and Brown [12], where an interesting connection between conditional mutual information and deformable operator U is obtained. While the subsystems we choose have only an unimportant difference from [12], our result is different. The result in [12] shows that if S nonlocal = 0, there will be no topological excitations, and therefore, S nonlocal > 0 is needed for the existence of topological excitations. The result is very general since very small amount of assumptions was used. Nevertheless, the result was not powerful as a lower bound for S nonlocal . In this work, on the other hand, we use more detailed properties of topological excitations and condensates to obtain a powerful lower bound of S nonlocal . Our method shows that the key to have S nonlocal > 0 in these models is the nonlocal nature of the ground state condensates and the operators creating topological excitations. S nonlocal can be extensive in the subsystem size and it is not necessarily topologically invariant. Whether S nonlocal is topologically invariant or not depends on whether the operators can be deformed topologically.
Before rigorously stating the assumptions and deriving the lower bound, here are a few words about the physical picture. The ground states of topological orders condense extended objects. If a unitary operator W (which has an extended support) acting on the ground state |ψ gives you W |ψ = e iϕ |ψ , we call the operator W a condensate operator with eigenvalue e iϕ . Whenever confusion can be avoided, we may call W a condensate for short. Let us further assume W to be a tensor product of operators acting on each site. Then, a suitably defined "truncation" of a condensate operator W onto a subsystem Ω gives you a new operator U . U |ψ is an excited state with topologically excitations located around ∂Ω ∩ W , and U can be deformed in the sense that you could choose a "truncation" of W † ontō Ω and call it U def , which creates the same topological FIG. 2: Condensate operators from different topological orders W , W , W (in orange color), and the truncations of corresponding operators give us deformable operators U , U , U (in blue color) which create topological excitations (in red color) when acting on the ground state. The operators W , U are from the 2D toric code model; the operators W , U are from the 3D toric code model; the operators W , U are from a fractal spin liquid model (note that W is not a precise depiction). The support of W is a closed loop; the support of W is a closed membrane, i.e. the 2D boundary of the box; the support of W has fractal structure. The picture only shows part of W explicitly, the rest of W is embedded in the dotted 2D surfaces with dashed 1D edges, and it may contain fractal parts or 1D parts. excitations and satisfies U |ψ = U def |ψ . One the other hand, if we have unitary operators U and U def satisfying U |ψ = U def |ψ , then [U def ] † U |ψ = |ψ , and therefore [U def ] † U is a condensate operator with eigenvalue +1. Intuitively, a condensate operator W can be "truncated" into a deformable operator U which creates topological excitations, and a pair of deformable operators U and U def can be "glued together" into a condensate operator W . Therefore U and W are closely related. Some examples of condensate operator W and deformable operator U , are shown in Fig.2. Once we have condensate operators W i and deformable operators U j , we use U j to create topological excitations in D which result in some states σ I which is identical to the ground state ρ on AB and BC. If the excitations created in D are topological excitations, they can not be created by an operator supported on D, and σ I ABC and ρ ABC will have some difference. The difference is detected by a change of the eigenvalue of condensate operator W i supported on ABC. Then, we use σ I to obtain a lower bound of S nonlocal .
The following are our assumptions U-1, U-2, U-3; W-1, W-2:
Assumption U-1: There exists a set of unitary operators {U i } supported on CD and a set of unitary operators {U def i } supported on AD, with i = 1, · · · , M . See Fig.3 for an example. When acting on the ground state |ψ , U i can be "deformed" into U def i , i.e.:
U i |ψ = U def i |ψ .(5)
Assumption U-2: For any subsystem Ω, the unitary operator U i can always be written as a direct product of unitary operators U i Ω and U iΩ which act on the subsystem Ω,Ω respectively:
U i = U i Ω ⊗ U iΩ .(6)
Assumption U-3: There are integers n i such that U ni i = 1, and when multiple U i act on a ground state |ψ , we have the following:
M i=1 U ki i |ψ = e iδ({ki}) M i=1 [U def i ] ki |ψ ,(7)
where integer 0 ≤ k i ≤ n i − 1, and we allow possible phase factors e iδ({ki}) . Assumption W-1: There exists a set of unitary operators {W i } supported on subsystem ABC such that
W i |ψ = |ψ i = 1, . . . , M.(8)
Assumption W-2: The following relation between W i and U j holds:
W i U j = U j W i e iθij with θ ij = 2π n i δ ij ,(9)
where δ ij is the Kronecker delta. are shown in red color. The color setting of the operators and excitations will be used throughout the paper. We do not assume these operators to be integer dimensional and the construction applies to models in different dimensions.
Comments about the assumptions: 1) U-1 implies that when U i is acting on the ground state, it can create excitations only in D, but not in ABC.
2) We do not assume U i , W i to be string operators and not even assume U i , W i to be integer dimensional operators. In fact, we will apply this method to fractal operators later.
3) In U-1 we assumed U can be deformed without changing the excitations. Nevertheless, we do not assume U can be deformed into all topologically equivalent configurations, and we do not assume U can be deformed continuously. As we will see below, in fracton models, deformations exist in weird form, U may not be topologically deformed (i.e. deformed continuously into any topologically equivalent configuration), and U can sometimes be deformed in discontinuous ways into topologically inequivalent configurations. 4) U-2 is not true when a local perturbation is added. We will address the stability of the lower bound under local perturbations separately in Sec.IV. 5) For the non-Abelian case, the entanglement entropy of an excited state could depend on the quantum dimension of the anyon [8,29], and U-2 does not apply. On the other hand, the idea in Sec.II C still holds, a saturated lower bound for non-Abelian models is recently discussed in [29]. 6) For systems with boundaries, one may choose D being region attached to boundaries, as is done in [12]. An alternative way is to identify D with a boundary region ∂Ω; in this case, U-1 should be understood as: U i being an operator supported on C and attached to ∂C ∩ ∂Ω, and U def i being an operator supported on A and attached to ∂A ∩ ∂Ω.
7)
According to W-2, |ψ and U i |ψ are eigenstates of W i with different eigenvalues, where |ψ is the ground state. Since W i is supported on ABC, this implies that U i is distinct from the identity operator. Similarly, U i and U j are distinct for i = j. We will refer to this change of eigenvalue of W i as a detection, e.g. U i is detected by W i . The requirement θ ij = 2π ni δ ij is not crucial, and it can be replaced by other numbers as long as the operator set {W i } can detect the difference among the set of operators {U i }.
8) For a relatively simple class of models, which has the Hamiltonian
H = − k h k , [h i , h j ] = 0 and h 2 k = 1, there is an obvious class of operators that satisfy Eq.(8) in W-1, namely W i = k∈Ei h k ,
where E i is a subset of the stabilizer generators. For the ground state |ψ , we have h i |ψ = |ψ , it follows that W i |ψ = |ψ . As U i could not flip stabilizers in ABC, E i must contain some h j in D. It turns out that this simple observation applies to all the Abelian stabilizer models we will use as examples in Sec.III. However, we do not provide a general procedure to find the subset E i for fracton models. On the other hand, our method works for models not in this simple class also, such as quantum double models [7,32] with Abelian finite groups.
Define the set of states
|{k i }; ψ ≡ M i=1 U ki i |ψ with 0 ≤ k i ≤ n i − 1 with k i being integers. Define σ({k i }) ≡ |{k i }; ψ {k i }; ψ|. Note the total number of σ({k i }) is M i=1 n i . Relabel σ({k i }) using a new index I = 1, · · · , N with N = M i=1 n i and call them σ I . One immediately varifies that 1) U-1, U-3 ⇒ σ I AB = ρ AB and σ I BC = ρ BC ; 2) W-1, W-2 ⇒ σ I ABC · σ J ABC = 0 for I = J; 3) U-2 ⇒ σ I ABC = V I ρ ABC V † I and S ABC | σ I = S ABC | ρ .
Where V I is some unitary operator acting on subsystem ABC and recall that ρ is the ground state density matrix.
Let
σ = N I=1 p I σ I with probability distribution {p I }, one derives that S ABC | σ − S ABC | ρ = − N I=1 p I ln p I ≤ ln N "=" if and only if p I = 1 N for all I. From Eq.(3) we find S nonlocal | ρ ≥ ln N = M i=1 ln n i .(10)
E. When is the lower bound topologically invariant?
It is instructive to think of the conditions under which our lower bound of S nonlocal is topologically invariant.
Consider a chosen set of subsystems A, B, C, D and the operator sets {U i }, {U def i } and {W i }. Let us do "topological deformations" of the subsystems and the operator sets. Here, by "topological deformation" of the operator sets we mean that we can topologically deform the support of each operator to get new operator sets which preserve the algebra in Eq. (5,6,7,8,9). Note that, these deformations generally change the positions of the excitations, which should be contrasted with the type of deformation in U-1, in which the positions of the excitations never change. When these conditions are satisfied, the lower bound for the two topologically equivalent choices of subsystems are the same. If such conditions are satisfied for each pair of topologically equivalent choices of subsystems, then our lower bound will be topologically invariant.
As is shown in the examples below in Sec.III, S nonlocal can be either topologically invariant or not, and it is instructive to think of how the conditions above are violated in fracton models [5,[14][15][16][17][18] in which S nonlocal depends on the geometry of subsystems.
III. APPLICATIONS
In this section, our lower bound is applied to several stabilizer models of Abelian phases: the 2D and 3D conventional topological orders and type I, type II fracton phases.
A. The 2D Toric Code Model
For a 2D topological order, choose the subsystems A, B, C, D of the same topology as is shown in Fig.1. From the well-known results [8,9], one derives S nonlocal = 2γ where −γ is the topological entanglement entropy. For the 2D toric code model [7]: On a square lattice with a qubit on each link, the Hamiltonian is
H = − s A s − p B p ,
where A s is a product of X r of a "star" or vertex, B p is a product of Z r of a plaquette.
A s = r∈s X r B p = r∈p Z r ,
where X r , Z r are Pauli operators acting on the qubit on link r. The ground state of toric code model condenses two types of closed string operators, and the corresponding open string operators (which could be regarded as truncations of closed string operators) create topological excitations at the endpoints.
We find the following unitary operators U 1 , U 2 , W 1 , W 2 as is shown in Fig.4. U 1 and W 2 are products of X r ; U 2 and W 1 are products of Z r . Also notice the feature that the closed string operators W 1 (W 2 ) can be written as a product of stabilizers B p (A s ) on a 2D disk region surrounded by the corresponding closed strings.
U-1, U-2, U-3, W-1, W-2 can be checked. The operators satisfy:
U 2 1 = U 2 2 = W 2 1 = W 2 2 = 1 W i U j = U j W i e iπδij i, j = 1, 2.
Therefore M = 2, n 1 = n 2 = 2 and N = n 1 n 2 = 4. Using the result in Eq.(10), one derives S nonlocal ≥ 2 ln 2. By comparing with the known result γ = ln 2, S nonlocal = 2γ = 2 ln 2, we find that our lower bound is saturated. A by-product of a saturated lower bound is an explicit construction of a conditionally independent σ * . In the toric code case:
σ * = 1 4 (ρ + U 1 ρU † 1 + U 2 ρU † 2 + U 1 U 2 ρU † 2 U † 1 )(11)
and σ * satisfies:
1) σ * AB = ρ AB , σ * BC = ρ BC ; 2) I(A : C|B)| σ * = 0.
Where ρ = |ψ ψ| is the ground state density matrix.
The observation in Sec.II E explains why the lower bound is topologically invariant in the toric code model: the operators U i and W i can be topologically deformed together with the subsystems A, B, C and D, without changing the algebra in Eq. (5,6,7,8,9).
This method can be applied to other 2D Abelian topological orders, e.g., quantum double models with Abelian finite groups, and the lower bounds are saturated. For a variant of the method for non-Abelian models, see [29].
B. The 3D Toric Code model
The 3D toric code model [33] is defined on a cubic lattice, with one qubit on each link. The Hamiltonian is of exactly the same form as the one of the 2D toric code model:
H = − s A s − p B p ; A s = r∈s X r ; B p = r∈p Z r .
Here a star s includes the 6 links around a vertex, and a plaquette p is a square consistent of 4 links.
Subsystem types for 3D models
The 3D toric code model is the first 3D model we discuss, and it is a good place to introduce subsystem types for 3D models which will be discussed for all 3D models. We focus on the following three topologically distinct subsystem types, i.e. the type-α,β,γ shown in Fig.6, although other choices are possible. We will use the notation S
(α) nonlocal , S (β) nonlocal , S (γ)
nonlocal to distinguish the nonlocal entanglement entropy for the three topological types.
FIG. 6: Three topologically distinct choices of subsystems, e.g. type-α, type-β and type-γ. We make subsystem B transparent in order to see C and D more clearly. It is understood that B ∪ CD is the box with blue edges and A is the complement of the box. We also apply this convention to pictures below.
Type-α has D, which consists of two disconnected boxes, while CD is connected, and it can be used to detect open string-like U i , which is attached to the two boxes of D. Type-β has D of the topology of a solid torus (and therefore D is not simply connected), while CD is simply connected. It can be used to detect open membrane-like U i supported on CD which create excitations in D as noncontractible loops. Type-γ has C and CD of the same topology, e.g. the topology of a solid torus, and B is simply connected.
Type-α and type-β have already been implied in paper by Kim and Brown [12], in which, similar subsystems types are used to study different types of boundaries of 3D models.
For type-γ, S (γ) nonlocal = 0 for models satisfying the assumptions in [13], e.g. the entanglement entropy of a general subsystem Ω (which has large size compared to correlation length) can be decomposed into local plus topological parts:
S Ω = S Ω,local + S Ω,topological ⇒ S For type-β, we find M = 1, n 1 = 2 where the operator U 1 is an open membrane operator which creates a looplike topological excitation at the edge of the membrane (the loop could not continuously shrink within D into a point), and W 1 is a closed string operator. Therefore N = 2 and S (β) nonlocal ≥ ln 2. For type-γ, operators supported on CD, which create excitations in D, could always be deformed into D. This is because, for 3D toric code, the operators that create topological excitations can be topologically deformed keeping the excitations fixed. Therefore we obtain a lower bound S (γ) nonlocal ≥ 0. Comparing with the known entanglement properties [12,13] of the 3D toric code model, our lower bounds are identical to the exact results:
C. The X-Cube Model
The X-cube model is a 3D exactly solved stabilizer model, and it is an example of type I fracton phase [18]. The model is defined on a cubic lattice with one qubit on each link. The Hamiltonian is
H = − c A c − s (B (xy) s + B (yz) s + B (zx) s ),(12)
where A c is a product of Z r on a cube (which includes 12 links), and B are products of X r of 4 links around a vertex which are parallel to the xy-plane, yz-plane, zx-plane respectively.
Here we focus on type-α and consider the geometry dependence of S nonlocal , see Fig.7. We find the following lower bounds: The types of U i that contribute to the S (α) nonlocal in Fig.7a are illustrated in Fig.8a, each U i stretches out in directions parallel to the yz-plane and creates a pair of "dimension-2 anyons." The translations of the operators U i in Fig.8a in (1, 0, 0) Translations can produce distinct operators, this indicates a breakdown of topological deformation: in the X-cube model, U i is deformable but not topologically deformable. Another nice example of the breakdown of topological deformation is shown in Fig.8c, in which U and its translations U T1 , U T2 , and U T1+T2 are distinct; nevertheless, by thinking of the condensate in Fig.8b,
one can show U ∼ U T1 · U T2 · U T1+T2 .
The result S Fig.7c comes from the fact that the types of U i discussed above could not connect the two boxes of D separated by a displacement vector d = (d x , d y , d z ) with |d x |, |d y |, |d z | > l D and the inability to find U i gives M = 0.
These results for S (α) nonlocal may also be calculated using the method in [34] and an independent estimation agrees with our lower bounds up to O(1) contributions.
The lower bounds of S (α) nonlocal for all the cases in Fig.7 depend on the length scale l D and the displacement vector d but are not sensitive to other details. This is due to the fact that we have chosen "big enough" A, B, C, so that they do not block any U i . If we consider another extreme, say the subsystem C has a very narrow neck, then S (α) nonlocal will be sensitive to the geometry of the neck which determines how many U i could pass through.
One may also apply the same idea to subsystems of type-β and type-γ and find extensive values of S
D. Fractal spin liquids
Fractal spin liquids [16] is a generalization of Haah's code [15]. A common feature of fractal spin liquid models is the existence of fractal condensates. Fractal structures have discrete scale symmetries, and this results in a more complicated dependence of the ground state degeneracy on the system size [6,16] compared to the type I fracton models.
Some of the fractal models possess "hybrid" condensates W i having both 1D parts and fractal parts, and the truncations of the condensates give you U i which can be either a string-like operator or a fractal operator. Note that, they do not fit into the definition of type I due to the existence of fractal operators. On the other hand, they do not fit into type II because the excitations created by the string-like operator are mobile excitations. Some fractal models have only fractal condensates, and no string-like U i exists. These models are type II fracton models.
Discussed in the following are ways to detect stringlike U i and fractal U i using condensates. Then, S nonlocal is shown to be extensive for certain choices of subsystem geometry.
The Sierpinski Prism Model
As an example, we consider the model (d) in Yoshida's paper [16]. Let us call this model the Sierpinski prism model, named after the shape of the condensate in Fig.9a, which looks like a prism with three legs decorated with Sierpinski triangles. This model lives on a 3D cubic lattice with two qubits (A and B) on each site. The Hamiltonian can be written as
H = − i,j,k h Z (i,j,k) − i,j,k h X (i,j,k) ,(13)
where i, j, k are integers labeling the sites on cubic lattice, and the Hamiltonian involves all the translations of the A condensate that can be thought of as a "deformed version" of Fig.9a, and now the upper surface is a Sierpinski triangle perpendicular to (1,-1,1) direction and it is a product of both Z A r and Z B r , its lower surface is the same as that in Fig.9a, i.e. a product of Z B r . (c) A condensate different from the former two, its upper and lower surfaces parallel to the xy-plane are products of X A r and the strings parallel to z-axis are products of X B r .
operator h Z (0,0,0) and h X (0,0,0) . Explicitly, in terms of Pauli operators acting on each A and B qubit on different sites, we have
h Z (0,0,0) = Z A r=(0,0,0) Z A r=(0,1,0) Z A r=(1,1,0) ×Z B r=(0,0,0) Z B r=(0,0,1) h X (0,0,0) = X A r=(0,0,0) X A r=(0,0,−1) ×X B r=(0,0,0) X B r=(0,−1,0) X B r=(−1,−1,0) .
It is easy to check that all terms in the Hamiltonian commute and [h Z (i,j,k) ] 2 = [h X (i,j,k) ] 2 = 1. Using Yoshida's notation:
h Z (0,0,0) = Z 1 + y + xy 1 + z ; h X (0,0,0) = X 1 +z 1 +ȳ +xȳ .
Wherex ≡ x −1 ,ȳ ≡ y −1 andz ≡ z −1 . In terms of polynomials f (x), g(x) with coefficients over F 2 , i.e. the coefficients can take 0 or 1:
h Z (0,0,0) = Z 1 + f (x)y 1 + g(x)z ; h X (0,0,0) = X 1 +ḡ(x)z 1 +f (x)ȳ .
Here f (x) = 1 + x and g(x) = 1 for the Sierpinski prism model. The polynomials with F 2 coefficients indicate the locations and the numbers of Pauli Z or X operators in the product; the upper row is for A qubits and the lower row is for B qubits.
Choosing other polynomials f (x), g(x) or changing F 2 into F p (p > 2 prime number) will generally give you other fractal models.
The Sierpinski prism model possesses hybrid condensates which consist of 1D parts and fractal parts, see Fig.9. The condensates in Fig.9a and Fig.9b can be constructed as a product of h Z (i,j,k) and the condensate in Fig.9c can be constructed as a product of h X (i,j,k) . As is suggested by the discrete scaling symmetry of fractal structure and the continuous scaling symmetry of a 1D line: the upper and lower surfaces can be separated by an arbitrary distance in z-direction (without changing the size of upper/lower surfaces), and under a rescaling l → 2 m l the condensates look similar. Under other rescaling factors, the condensates look different but they could be constructed using a product of condensates that looks similar to the ones in Fig.9. While this model does not have any logical qubits under periodic boundary conditions on an L x × L y × L z lattice, i.e. x Lx = y Ly = z Lz = 1, it does have logical qubits under some "twisted" boundary conditions (say x Lx = 1, y Ly = x, z Lz = 1 with L x = 2 m + 1, L y = 2 m , ∀ L z and integer m), or under open boundary conditions.
Despite the fact that the Sierpinski prism model is one of the simplest fractal models, it nicely illustrates all the important ingredients needed in order to understand how our method works in fractal models. To be specific, it illustrates the following three types of detections: 1) The detection of a string-like U using a fractal W .
2) The detection of a fractal U using a string-like W .
3) The detection of a fractal U using a fractal W . As is shown in Fig.9, we have condensates with stringlike parts and fractal parts. String-like U i can be obtained from a truncation of the condensates. In Fig.10a, translations of a string-like U , e.g. U T1 and U T2 (with T 1 = (−2 m , 0, 0), T 2 = (0, 2 m , 0) and an integer m) are distinct from U , but U ∼ U T1 · U T2 . It is different from what happens in conventional topological orders but similar to what happens in type I fracton models, see Fig.8.
The distinctness of a string-like U and its translations indicates a lower bound of S nonlocal extensive in the subsystem size, and indeed we can use the fractal part of W i to detect different string-like U i , see Fig.10b, and get an extensive lower bound.
The detection of fractal U by fractal W
Another way to detect fractal U is to use a fractal part of a condensate W , see Fig.11b, and it is the only way to detect U for those fractal models without string operators i.e., when the condensates contain only a fractal structure without string-like parts (i.e., type II). Therefore, it is important to understand this case.
The key features, which can be observed in Fig.11b are the following.
1) The fractal condensates are supported on 2D surfaces with "holes" of different (discrete) length scales, in other words it is less than 2D.
2) Fractal U and fractal (part of) W lie in distinct intersecting surfaces, and it is possible to make the operators intersecting at a point. Certain translation of U has a non-overlapping support with W and therefore it commutes with W . This implies that translations of U can give you distinct operators.
After some thought, one finds it is possible to get an extensive lower bound of S 2) When rescaling the subsystem sizes according to the discrete scale symmetries (for the Sierpinski prism model, it is L → 2 m L), the change of S nonlocal could be investigated using entanglement renormalization group transformation [25]. For a rescaling by a factor which is not in the discrete scaling group, S nonlocal may change in more complicated way.
3) It is possible to have a fractal model with a unique ground state on a T 3 , in which case, there still exists nonzero S nonlocal . This indicates that S nonlocal is in some sense more universal than the ground state degeneracy. 4) For models with only fractal condensates (i.e. type II), although it is possible to find S (α) nonlocal > 0 for some choices of A, B, C, D, we get a lower bound 0 when, for example, the two boxes of D of length l D × l D × l D are separated by a displacement vector d satisfying | d| > λl D with some constant λ depending on the model. It might be a general exact result that S (α) nonlocal = 0 when | d| > λl D no matter how you choose A, B, C but we do not have a proof. The case for Haah's code is a conjecture by Kim [43]. If the conjectured results are true, this may be used to make a clear distinction between type I and type II fracton models.
IV. PERTURBATIONS
The stability of quantities under local perturbations is an extremely important topic. If some property of an exactly solved model is totally changed when a tiny local perturbation is added, this property could never be observed in real systems.
The ground state degeneracy of topological orders is robust (stable) to arbitrary local perturbations. It is known that the toric code model is stable under arbitrary local perturbations [7]. The stability of ground state degeneracy is proved [2,3] for a very general class of models which satisfy assumptions TQO-1 and TQO-2. In the proof, Osborne's modification [35] of the quasi-adiabatic continuation [36] is employed. This proof is applicable to both conventional and fracton topological orders.
We would like to understand the stability of S nonlocal under local perturbations. It turns out that the stability of S nonlocal is a trickier problem compared to the stability of the ground state degeneracy. The corresponding problem for the conventional topological orders, e.g. the stability of topological entanglement entropy is not solved completely without additional assumptions. It is known from Bravyi's counterexample (see [37] for a published reference) that the arguments provided in the original works [8,9] about the invariance of the topological entanglement entropy under perturbation are not complete. Kim obtained a bound of the change of topological entanglement entropy with a 1st order perturbation [38] assuming the conditionally independence of certain subsystems.
Here we study the stability of our lower bound of S nonlocal under a finite depth quantum circuit for simplicity, since it is known from the viewpoint of quasiadiabatic evolution [36] that local perturbations for gapped systems can be approximated by finite depth quantum circuit [39,40].
Assumption S: For subsystems A, B, C as is shown in Fig.12. δQ is a unitary operator which has support intersecting with A and C. B is separated from δQ by a distance d. σ is a density matrix of a state with correlation length ξ and replica correlation length [37] ξ α and σ ≡ δQσδQ † . And σ is a density matrix such that
S AB − S A | σ S AB − S A | σ ,(14)
where " " means there is a correction that is negligible when d is large compared to ξ and ξ α .
The assumption S should be understood as an assumption about the density matrix σ. It is trivial to check that " " can be replaced by "=" when σ AB = σ A ⊗ σ B , and it may seem intuitive that the difference between the left-hand side and the right-hand side of Eq. (14) should decay as e −d/ξ . Nevertheless, the original suggestion [8] that S is true for ξ d is violated in Bravyi's counterexample. It is observed in [37] that this is due to the fact that the replica correlation length ξ α is infinity for the cluster state in Bravyi's counterexample. When the cluster state is deformed, ξ α becomes finite. For generic local perturbations without symmetry requirement, it is fine-tuned to have ξ α = +∞ but ξ α can be arbitrarily large compared to ξ. Judging from a recent conjecture [37] , ξ α d may be the condition required for S to be true. In the following, we discuss the stability of our lower bound of S nonlocal under a depth-R quantum circuit Q which creates a perturbed ground stateρ satisfying S. We take R ∼ ξ, and assume ξ and ξ α much smaller than the length scales of the subsystems. This analysis does not cover all possible local perturbations (especially those with ξ α → +∞), but we believe it covers a large class of interesting local perturbations.
Let Q be the depth-R quantum circuit (QQ † = 1) which is responsible for the local perturbation. In other words, we assume the following objects in the perturbated model are related to the corresponding objects in the unperturbed model by 1) The new Hamiltonian:H = QHQ † ;
2) The new (dressed) operators:
Ū i = QU i Q † ,Ū def i = QŪ def i Q † , andW i = QW i Q † ;
3) The new ground state: |ψ = Q|ψ ; 4) The new density matrices:σ I = Qσ I Q † andρ = QρQ † . Fig.3, the width of the support of dressed operators can increase at most by 2R due to the depth-R quantum circuit.
The dressed operators typically have a "fatter" support than the corresponding operators in the unperturbed stabilizer model, see Fig.(13) for an illustration. It is possible that the support of someŪ i will overlap with AB, the support of someŪ def i will overlap with BC and the support of some W i will overlap with D. For those operators, we need to throw them away, and supply with other operators if possible. We label the remaining operators usingī,ī = 1, · · ·M , whereM ≤ M , and we call the remaining density matricesσĪ , withĪ = 1, · · · ,N , wherē N = m i=1nī . With U-1, U-2, U-3, W-1, W-2 satisfied for the unperturbed system, we supply with S in order to complete a result about the stability of the lower bound. With these assumptions, one could verify the following results: 1 ) U-1, U-3 ⇒σĪ AB =ρ AB andσĪ BC =ρ BC ; 2 ) W-1, W-2 ⇒σĪ ABC ·σJ ABC = 0 forĪ =J; 3 ) U-1, U-2, U-3, S ⇒ S ABC |σĪ S ABC |ρ.
The derivation of 1 ) and 2 ) are parallel to what is done in Sec.II D. The derivation of 3 ) follows. There exists a unitary operator δQ supported on a region within a distance R around ∂C ∩ ∂D, See Fig.14, such that ρ = δQρ δQ † ,σĪ = δQσ Ī δQ † andσ Ī ABC =VĪρ ABCV † I . VĪ is some unitary operator supported on ABC. It is always possible to find such δQ andVĪ given that U-2 is satisfied for the unperturbed case. Therefore, S ABC |σ Ī = S ABC |ρ . Then, by applying U-1, U-3, and S we find that U-1, U-3 ⇒ S BC |σ Ī = S BC |ρ S BC |σĪ = S BC |ρ ; S ⇒ (S ABC − S BC )|ρ (S ABC − S BC )|ρ (S ABC − S BC )|σĪ (S ABC − S BC )|σ Ī .
After simple algebra one arrives at the result 3 ) i.e. S ABC |σĪ S ABC |ρ.
FIG. 14: The support of the operator δQ (in purple color) is within a distance R around ∂C ∩ ∂D.
Assuming the error caused by " " could be neglected, one could apply the same method as Sec.II D to arrive at a lower bound
S nonlocal |ρ ≥ lnN =M ī =1 lnnī.(15)
For models with topologically deformable U i operators, like the 2D, 3D toric code models, our lower bound is invariant under perturbation. Because we can always move the excitations deep inside D, such that the distance from any excitation to the boundary d e ξ ∼ R. For large subsystems, we would have M = M and we do not lose any U i , U def i and W i . For models with U i not topologically deformable, e.g. the X-cube model and fractal spin liquids. There is usually some excitation that could not be moved deep inside subsystem D. Therefore, after adding perturbations, we typically lose a few U i and U def i , such thatM < M and N < N . But for large subsystems which possess extensive S nonlocal before perturbation is added, this modification is small comparing to the leading contribution.
To summarize, for local perturbations satisfying assumption S, and subsystem sizes much larger than ξ and ξ α we expect:
S nonlocal (perturbed) S nonlocal (unperturbed) − µξ.
(16) For conventional topological orders µ = 0, and for fracton topological orders µ > 0 being a number depends on the model and subsystem geometry.
V. DISCUSSION AND OUTLOOK
In this paper, we have obtained a lower bound of the nonlocal entanglement entropy S nonlocal from assumptions about the topological excitations and the ground state condensates of Abelian topological orders and applied our method to several examples. For conventional topological orders, e.g. the 2D toric code model and the 3D toric code model, our lower bounds are saturated and topologically invariant. Whenever the lower bound is saturated, we get an explicit construction of a conditionally independent density matrix σ * . For fracton topological orders [5,[14][15][16][17][18], e.g. the X-cube model and the Sierpinski prism model, our lower bound depends on the geometry of the subsystems and S nonlocal is extensive for certain subsystem choices.
This method observes an intimate relation between S nonlocal and the topological excitations and the ground state condensates, and it obtains a lower bound of S nonlocal without calculating the entanglement entropy of any subsystem. A nonzero lower bound of S nonlocal is a result of the nonlocal nature of topological excitations, i.e. the fact that topological excitations could not be created alone by local operators. This nonlocal nature of topological excitations does not guarantee the operators which create the topological excitations to be topologically deformable and S nonlocal is not necessarily a topological invariance. Geometry-dependent S nonlocal is what appears in fracton models. It is beyond an established paradigm, i.e., the topological entanglement entropy, and should be treated as its generalization. The stability of the lower bound is discussed for local perturbations satisfying assumption S, which should cover a large class of interesting local perturbations.
The different behaviors of S nonlocal may be used to distinguish fracton topological orders from conventional topological orders. Together with other methods being developed so far [11,25], our result provides a better understanding of the entanglement properties of fractal models. Furthermore, the lower bound suggests (but not prove) different behaviors of S nonlocal between type I and type II fracton models. These different behaviors may be proven or disproven by later works.
Some of the assumptions in our method do not apply to non-Abelian models, a variant of our lower bound of S nonlocal for non-Abelian models is presented in [29]. Also, it might be interesting to investigate possible implications of our method on relations among topological order, topological entanglement entropy and quantum black holes [41,42].
FIG. 1 :
1A system is divided into subsystems A, B, C and D. Geometrically and topologically distinct choices will be used throughout the paper. They share the following features: ∂A∩∂C = 0 and other pairs of subsystems have shared boundaries.
and the "=" happens if and only if I(A : C|B)| σ = 0. For a proof, observe that I(A : C|B)| ρ and I(A : C|B)| σ has only a single different term, and that I(A : C|B)| σ ≥ 0.
FIG. 3 :
3An illustration of the support of different operators: Wi (in orange color) is supported on ABC; Ui (in blue color) is supported on CD; U def i (in blue color) is supported on AD. The topological excitations created by Ui or U def i
FIG. 4 :
4Deformable operators U1, U2 supported on open strings and condensate operators W1, W2 supported on closed strings. U1, U2 create topological excitations around their endpoints. In other words, U1 flips two plaquettes and U2 flips two stars.
The ground state of the 3D toric code model condenses one type of closed string and one type of closed membrane. There is one type of open string operator that creates point-like topological excitations at the endpoints and one type of open membrane operator which creates loop-like topological excitations at the edge of the membrane, see Fig.5.
FIG. 5 :
5Topological excitations in the 3D toric code model and their detection using condensates.
a model with S (γ)nonlocal > 0 is a model beyond the description of[13]. Fractal models do have Code model has saturated lower bounds for each subsystem typeLet us go back to the 3D toric code model. For type-α, we find M = 1, n 1 = 2 where the operator U 1 is an open string operator which creates a point-like topological excitation in each box of D and W 1 is a closed membrane operator. Therefore N = 2 and S
≥ (2l D + O(1)) ln 2; Fig.7b, S (α) nonlocal ≥ (4l D + O(1)) ln 2; Fig.7c, S (α) nonlocal ≥ 0.Where O(1) denotes order one contributions which dependent on the detailed shapes of the subsystems, which is not crucial for our discussion.
FIG. 7 :
7Type-α subsystem choices: The two boxes of D are of the same size lD × lD × lD, and they are separated by a displacement vector d = (dx, dy, dz). ∂D ∩ ∂A contains two 2D pieces parallel to the xz-plane. (a) dx = 0, |dy|, |dz| > lD. (b) dx = dz = 0, |dy| > lD. (c) |dx|, |dy|, |dz| > lD.
direction give you distinct operators (while translations in (0, 1, 0) or (0, 0, 1) directions do not give you distinct operators). This gives M = 2l D + O(1). FIG. 8: About U and W in the X-cube model: (a) The two types of U and their translations in (1, 0, 0) direction contribute to S nonlocal for the configuration in Fig.7a. (b) A product of Ac gives you a W that is a product of Zr on the 1D edges of a cuboid. (c) U is a product of Zr on a line parallel to z-axis and it creates point-like excitations in D. UT 1 , UT 2 , and UT 1 +T 2 are translations of U by vectors T1 = (−a, 0, 0), T2 = (0, b, 0), and T1+ T2 = (−a, b, 0), and a and b are positive integers in unit of lattice spacing.
≥ (4l D + O(1)) ln 2 for Fig.7b can be understood by thinking of contributions from operators parallel to the yz-plane and the xy-plane, which gives M = 4l D + O(1). The result S (α) nonlocal ≥ 0 in
for certain choices of subsystems.
FIG. 9 :
9Examples of condensates in the Sierpinski prism model: (a) The fractal structures (here are Sierpinski triangles) contained in the upper and lower surfaces parallel to the xy-plane are products of Z B r . The three strings parallel to the z-axis are products of Z A r . (b)
2 .
2The detection of sting-like U by fractal W FIG. 10: A string-like U which can be thought of as a truncation of the condensate in Fig.9c. It creates point-like excitations at its endpoints. (a) Translating U by vectors T1, T2, which naturally appear in the fractal structure, to get UT 1 and UT 2 , where T1 = (−2 m , 0, 0) and T2 = (0, 2 m , 0) with an integer m. (b) The detection of U using the fractal part of a condensate of the same type as Fig.9a.
3 .
3The detection of fractal U by string-like W FIG. 11: The fractal U comes from a truncation of the condensate in Fig.9c, and it creates point-like excitations at the three vertices of the Sierpinski triangle. (a) Detecting U using the string-like part of W of the type in Fig.9a. (b) Detecting U using the fractal part of a W similar to the upper surface of Fig.9b. The fractal structure of U and W lie in different planes and the dashed line is the intersection of the two planes. Shown in yellow color is a choice of subsystem D for type-β or type-γ.Very similarly, fractal U can be detected by string-like parts of W , seeFig.11a. It is clear that a translation of a fractal U will be a distinct operator if it anticommutes with a different W . Therefore, by suitably choosing the geometrical shapes of subsystems A, B, C, D, it is possible to get an extensive lower bound of S
by suitably choosing the geometrical shapes of the subsystems A, B, C, D. A choice of D for type-β or type-γ is shown in yellow color inFig.11b. may consider other subsystem types, for example: type-δ with D consists of three disconnected boxes, and suitably chosen A, B, C. When putting the three boxes on the positions of the three excitations of a
FIG. 12 :
12Subsystems A, B, C and the unitary operator δQ (in purple color) which appears in assumption S. δQ is separated from B by a distance d.
FIG. 13 :
13Dressed operatorsWi andŪi. Compare with the operators Wi and Ui shown in
a discussion about the entanglement in non-Abelian phases. This work is supported by the startup funds at OSU and the National Science Foundation under Grant No. NSF DMR-1653769. BS,YMLa discussion about the entanglement in non-Abelian phases. This work is supported by the startup funds at OSU and the National Science Foundation under Grant No. NSF DMR-1653769 (BS,YML).
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| []
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[
"Monte Carlo methods for linear and non-linear Poisson-Boltzmann equation *",
"Monte Carlo methods for linear and non-linear Poisson-Boltzmann equation *"
]
| [
"Mireille Bossy ",
"Nicolas Champagnat ",
"Hélène Leman ",
"§ Sylvain ",
"Maire ¶ Laurent ",
"Violeau Mariette Yvinec "
]
| []
| []
| The electrostatic potential in the neighborhood of a biomolecule can be computed thanks to the non-linear divergence-form elliptic Poisson-Boltzmann PDE. Dedicated Monte-Carlo methods have been developed to solve its linearized version (see e.g.[7],[27]). These algorithms combine walk on spheres techniques and appropriate replacements at the boundary of the molecule. In the first part of this article we compare recent replacement methods for this linearized equation on real size biomolecules, that also require efficient computational geometry algorithms. We compare our results with the deterministic solver APBS. In the second part, we prove a new probabilistic interpretation of the nonlinear Poisson-Boltzmann PDE. A Monte Carlo algorithm is also derived and tested on a simple test case. * This work was carried out and financed within the framework of the WalkOnMars project of the CEMRACS 2013. † INRIA Sophia Antipolis -Méditerranée, TOSCA project-team, 2004 route des Lucioles, | 10.1051/proc/201448020 | [
"https://arxiv.org/pdf/1411.2304v3.pdf"
]
| 119,699,872 | 1411.2304 | f0bf0020c00fddc2cb93ba09164b9af7f2a58e36 |
Monte Carlo methods for linear and non-linear Poisson-Boltzmann equation *
November 13, 2014
Mireille Bossy
Nicolas Champagnat
Hélène Leman
§ Sylvain
Maire ¶ Laurent
Violeau Mariette Yvinec
Monte Carlo methods for linear and non-linear Poisson-Boltzmann equation *
November 13, 2014
The electrostatic potential in the neighborhood of a biomolecule can be computed thanks to the non-linear divergence-form elliptic Poisson-Boltzmann PDE. Dedicated Monte-Carlo methods have been developed to solve its linearized version (see e.g.[7],[27]). These algorithms combine walk on spheres techniques and appropriate replacements at the boundary of the molecule. In the first part of this article we compare recent replacement methods for this linearized equation on real size biomolecules, that also require efficient computational geometry algorithms. We compare our results with the deterministic solver APBS. In the second part, we prove a new probabilistic interpretation of the nonlinear Poisson-Boltzmann PDE. A Monte Carlo algorithm is also derived and tested on a simple test case. * This work was carried out and financed within the framework of the WalkOnMars project of the CEMRACS 2013. † INRIA Sophia Antipolis -Méditerranée, TOSCA project-team, 2004 route des Lucioles,
Introduction
The goal of this paper is to study Monte Carlo methods to solve both linear and nonlinear versions of the Poisson-Boltzmann PDE. In the linear case, we implement these algorithms on real size biomolecules, the geometrical complexity being handled thanks to efficient computational geometry algorithms.
The Poisson-Boltzmann equation describes the electrostatic potential of a biomolecular system in a ionic solution with permittivity ε, assuming a mean-field distribution of solvent ions-an assumption known as the implicit solvent approximation [2]. In this work, we are interested in the case of a neutral ionic solution with two kinds of ions with opposite charge, where the Poisson-Boltzmann equation takes the form − ∇ · (ε(x)∇u(x)) + κ 2 (x) sinh(u(x)) = f (x), ∀x ∈ R 3 ,
(1.1) where ε(x) = ε in > 0 if x ∈ Ω in , ε out > 0 if x ∈ Ω out , κ(x) = 0 if x ∈ Ω in , κ := √ ε out κ out > 0 if x ∈ Ω out . (1.2)
Here, Ω out represents the domain of the ionic solvent, and Ω in = Ω c out represents the interior of the molecule, defined as the union of N spheres of centers c 1 , c 2 , . . . , c N and radii r 1 , r 2 , . . . , r N respectively, representing the atoms of the molecule, that is
Ω in = N i=1 S(c i , r i ),
where S(c, r) = {x ∈ R 3 : |x − c| < r}. We denote by Γ the boundary of the bounded domain Ω in . More precisely, Γ can be either the van der Waals surface when the r i are the radii of the atoms, or the Solvent Accessible Surface (SAS), which is the set of spheres with radii r i = ρ i + r s , where r s is the radius of solvent molecules and ρ i are the radii of the atoms. The positive numbers ε in and ε out are the relative permittivity of each medium, and u(x) represents the dimensionless potential at x ∈ R 3 , defined as u(x) = e c (k B T ) −1 Φ(x) where Φ is the potential and the constants e c , k B and T are respectively the charge of an electron, Boltzmann's constant and the absolute temperature. Some differences may be found in the normalizing constants depending on the bibliographic references. We focus here on the choice of the normalization, the values of each constants and the derivation of Poisson-Boltzmann's equation given in [18], also used in the Poisson-Boltzmann solver APBS [3].
The source term f is given as a sum of Dirac measures:
f (dx) = N i=1 e 2 c k B T ε 0 z i δ ci (dx),(1.3)
where ε 0 represents the absolute permittivity of vacuum, and z i is the relative charge of the i-th atom of the molecule (relative meaning its actual charge divided by e c ). Note that, even though ε and κ are discontinuous and f is a measure, a proper notion of solution can be found in [10]. Note also that κ is sometimes considered discontinuous at the ion accessible surface (obtained as the SAS surface with r s replaced by the radius of ions in the solvent) and ε at the van der Waals or the SAS surface. In this work we present our methods for a single discontinuity surface for both κ and ε (either Van-der-Walls or SAS), but the methods can easily treat the double discontinuity surfaces model. Finally,κ −1 is the Debye length in the ionic solution (see e.g. [18]), (1.4) where N A is the Avogadro constant, and I = c z 2 is the ionic strength of the solvent, where c is the concentration of one of the ion species in the ionic solution and z its relative charge (we recall that the two ion species have the same concentration and opposite charges). The values of all the physical constants used in the simulations are reported in Appendix A.
κ −2 = 2N A e 2 c I ε 0 k B T ,
In structural biology, the Poisson-Boltzmann equation is often used in its linearized form:
− ∇(ε(x)∇u(x)) + κ 2 (x)u(x) = f (x), ∀x ∈ R 3 (1. 5) which gives a good approximation of the electrostatic potential around uncharged molecules. We refer to Folgari et al. [13] for a survey on applications in structural biology. Monte Carlo (MC) algorithms for the linearized Poisson-Boltzmann equation were proposed first by Mascagni and Simonov in [26,27] and improved or extended in Mascagni et al. [31,32,12,23], Bossy et al. [7]. These MC algorithms involve double randomization techniques [27], walk on spheres techniques [30] to simulate Brownian motion exit times and positions in Ω out and Ω in , and asymmetric jump methods from the boundary Γ to take into account the discontinuity of ε in the divergence form of the PDE (1.5). Recently, Lejay and Maire [22] and Maire and Nguyen [24] have proposed new replacement methods from the boundary Γ which improve the order of convergence of the original algorithm. Section 2 is devoted to MC algorithms for the linearized Poisson-Boltzmann equation. After recalling their general forms, we compare these MC algorithms on biomolecule geometries, i.e. for domains Ω in defined from measured biomolecular data. The key issue to deal with real-size biomolecule geometries consists in finding efficient algorithms to locate the closest atom from any position in R 3 . In this work, we use the efficient power diagram construction, search and exploration tools developed in the CGAL library [1] to solve this specific problem.
Section 3 is devoted to the study of probabilistic interpretations and Monte Carlo methods for the nonlinear Poisson-Boltzmann equation. We show how branching versions of diffusion processes may be used to deal with the nonlinear version of the PDE (Sections 3.1.2 and 3.1.3). We derive the corresponding Monte Carlo algorithm in Sections 3.2.1 and 3.2.2. We test the numerical method on simple molecule test-cases with one or two atoms (Section 3.3).
Linear case
This section deals with the linearized Poisson-Boltzmann equation (1.5). The Monte-Carlo algorithm used to solve this equation was first proposed by Mascagni and Simonov in [27], and the probabilistic interpretation of the PDE and improved replacement algorithms are given in Bossy et al [7]. This last reference also presents numerical tests on cases where the molecule has one or two atoms.
In this work, the simulation code that we implement can deal with molecules having an arbitrary number of atoms. We use the PDB format files of biomolecules which can be found for example on the RCSB Protein Data Bank, and convert it into PQR files containing the positions, radii and charges of all the atoms of the biomolecule using PDB2PQR [11]. This gives all the parameters of the Poisson-Boltzmann equation, except the Debye length κ −1 out , which is computed from (1.4) (see the Appendix A for the explicit values).
When designing a Monte-Carlo method for the linearized Poisson-Boltzmann equation, the main difficulty is to deal correctly with the discontinuous coefficient in the divergence form operator. The key point is the equivalent formulation of the equation (1.5) as two subproblems
−ε in ∆u(x) = f (x) for x ∈ Ω in u(x) = h(x) for x ∈ Γ, (2.1) −ε out ∆u(x) +κ 2 u(x) = 0 for x ∈ Ω out u(x) = h(x) for x ∈ Γ, (2.2)
with a transmission condition (see e.g. [20]), which holds true in general for smooth Γ (see [7]): The general principle of the Monte Carlo algorithms used here is given by the following extended Feynman-Kac formula for the solution u of (1.5) (see [7]) ∀x ∈ R d \ {c 1 , . . . , c N },
h = u| Γ , u is continuous on R 3 , and ε in ∇ in u(x) · n(x) = ε out ∇ out u(x) · n(x), ∀x ∈ Γ,(2.u(x) = E x +∞ k=1 u 0 (X τ k ) − u 0 (X τ k ) exp − τ k 0 ε out κ 2 (X s )ds (2.4)
where for all h > 0, we denote Ω h in := {x ∈ Ω in , d(x, Γ) ≥ h} and define τ 0 = 0 and ∀k ≥ 1,
τ k = inf{t ≥ τ k−1 , X t ∈ Ω h in } τ k = inf{t ≥ τ k , X t ∈ Γ}.
The function
u 0 (x) := e 2 c k B T ε 0 N i=1 z i 4πε in |x − c i | (2.5)
is solution to −ε in ∆u 0 = f in R 3 , and (X t , t ≥ 0) is the weak solution to the stochastic differential equation with weighted local time at the boundary Γ
X t = x + t 0 2ε(X θ )dB θ + ε out − ε in 2ε out t 0 n(X θ )dL 0 θ (Y ) Y t is the signed distance of X t to Γ (positive in Ω in ), L 0 (Y ) is the local time at 0 of the semimartingale Y.
The formula (2.4) suggests to use a Monte Carlo approximation of u based on M independent simulations of (X t , t ≥ 0). The MC algorithm computes the solution u at specific points (as needed for the computation of the solvation free energy [2]) and of the same PDEs with different parameters [31]. Moreover, the MC method takes advantage of the geometry of the problem (the molecule is a union of spheres):
• away from Γ, since the paths of X are scaled Brownian paths, we can use (centered and uncentered) walk on spheres techniques [30] as fast numerical scheme.
• close to Γ, since (2.4) only involves the position of X t and the amount of time spent in Ω out by X between τ k and τ k for all k ≥ 1, we approximate X by a process that jumps away from Γ when it hits Γ.
The main steps of the Monte-Carlo algorithm
Let us describe the steps of the simulation of X for the Monte Carlo algorithms.
Outside the molecule: the walk on spheres (WOS) algorithm
Recall that in Ω out , u is a solution to − 1 2 ∆u+λu = 0 with λ := κ 2 out /2. Therefore,
u(x) = E x [u| Γ (B τ )e −λτ ]
, where under P x , (B t ) t≥0 is a Brownian motion started at x and τ is its first hitting time of Γ.
It is well-known [30,27,7,22] that the WOS algorithm simulates exactly successive positions of B in Ω out until it reaches -a small neighborhood of-Γ, taking into account the exponential term in the probabilistic interpretation by means of a constant rate of killing λ.
Starting at a point y 0 := x ∈ Ω out , we find the largest sphere S(y 0 , r 0 ) included in Ω out and centered at y 0 . In that sphere, the killed Brownian motion either dies before reaching the boundary of S(y 0 , r 0 ) with probability r 0 √ 2λ/ sinh(r 0 √ 2λ), or it reaches the sphere boundary at a point y 1 uniformly distributed on S(y 0 , r 0 ). We then start again the same procedure from y 1 , and we obtain thus a sequence (y k ) k≥0 , possibly killed. Except in very specific situations, this sequence will a.s. never hit Γ with a finite number of steps. Hence it is classical to introduce a small parameter > 0 and to stop the algorithm either at the first killing or at the first step where y k is in the -neighborhood of Γ and project y k on Γ to obtain an approximation of the exit point. More formally, this algorithm can be written as follows.
WOS algorithm in the domain Ω out . Set k = 0. Given y 0 ∈ Ω out , λ ≥ 0, and ε > 0 1. Let S(y k , r k ) be the largest open sphere included in Ω out centered at y k . 2. Kill the particle with probability 1 − r k √ 2λ/ sinh(r k √ 2λ), and goto END if killed. 3. Sample y k+1 according to the uniform distribution on ∂S(y k , r k ). 4. IF d(y k+1 , ∂Ω out ) ≤ ε, THEN set exit position(y 0 ) as the closest point of ∂Ω out from y k+1 and goto END. ELSE, set k = k + 1 and return to Step (1). END.
At least for smooth Γ, it is known [30] that this algorithm stops a.s. in finite time, after a mean number of steps of order O(| log( )|). Moreover when u is continuous on Γ,
E[u(exit position(y 0 ))1 {exit before killing} ] = E y0 [u(B τ )e −λτ ] + O( ).
(2.6)
CGAL Library: search for the closest atom
Given a position y ∈ Ω out , the first step of the WOS algorithm requires to construct the biggest open sphere S = S(y, r) in Ω out with center y. In other words, it requires to find the nearest atom of the molecule from y, i.e. the atom with index argmin 1≤i≤N ( y − c i − r i ), where · is the Euclidean norm in R 3 . The simplest way to achieve such a search consists in doing a "brute force", naive search among the N atoms to find the closest, with a computational cost of order N . However, for such a minimization problems, it is generally possible to construct search algorithms with computational cost of order log N , if one can afford to build an appropriate search tree in a precomputation step. Since N could be large for biomolecules (several thousands) and since our algorithm requires to search for the closest atom a very large number of times (of order log times for each independent simulation of the Monte-Carlo method), it is clearly interesting for us to use this second method.
The C++ library CGAL (Computational Geometry Algorithms Library) [1] proposes geometric algorithms allowing to solve this problem. The idea is to construct first the power diagram associated to the set of spheres (S(c i , r i )) 1≤i≤N , i.e. the partition of the space into polygonal cells, each of which are associated to an index j ∈ {1, . . . , N }, such that the power distance x − c i 2 − r 2 i , 1 ≤ i ≤ N , of points x of the cell, is minimal for i = j. Given a point y ∈ R 3 , the library can then easily compute, with complexity O(|log( )|), the index of the cell containing y. Since it minimizes the power distance, this index is not necessarily the one that minimizes the Euclidean distance, so we need to check that none of the neighboring cells are closer. CGAL also provides tools to explore the neighboring cells in the power diagram, which we used. This last step is needed since otherwise one could use the WOS algorithm on a too large sphere and obtain points that would belong to Ω in . We ran several tests to find the proportion of such events. Without the local search step, this proportion is roughly 8%, whereas with the local search step, it drops down to roughly 2% 1 . We use this local search in all the numerical tests presented here.
For comparison, we implement three methods to find the nearest atom from a point:
• The brute localization method is the naive but exact method, of complexity O(N ).
• The power diagram method uses the CGAL tools described above, of complexity O(log N ).
• The power diagram with hint: in the WOS algorithm, the closest atom in the previous step was already computed. Therefore, it can be used as a hint to find the atom with minimal power distance among the neighboring atoms of the previous one. CGAL proposes options enabling to start the search from specific points. This method is also of complexity O(log N ), but hopefully with a smaller constant multiplying log N .
The computational costs of those three methods are reported in Figure 1(a). Note that other ideas to improve the computational speed of this localization step were recently developed in [23].
Inside the molecule: uncentered walk on spheres (UWOS) algorithm
If the current position of the particle belongs to Ω in , we use the function u 0 in (2.5) as the unique bounded solution of the PDE (2.1) in the domain R 3 . Thus u − u 0 is harmonic in Ω in , and using the notation of Section 2.2.1, for all x ∈ Ω in ,
(u − u 0 )(x) = E [(u| Γ − u 0 )(B τ )] .
(2.7)
Again, one can use a walk on spheres algorithm to compute this expectation. This can be done by taking advantage of the union of spheres geometry of the molecule. In this case, it is convenient to use an uncentered walk on spheres method: at each step we use the atom sphere to which the simulated path's position belongs, instead of drawing a virtual sphere centered around the current position. The exit position from the sphere is not uniformly distributed on the sphere, but can be explicitly computed and exactly simulated [27]. The following algorithm allows the exact simulation of the exit position of a Brownian motion from Ω in .
UWOS algorithm in the domain Ω in . Set k = 0. Given y 0 ∈ Ω in , 1. Choose i ∈ {1, . . . , n} such that y k ∈ S(c i , r i ). 2. Simulate y k+1 = (r i , θ, ϕ) where θ is uniform on [0, 2π] and ϕ is independent of θ with cumulative distribution function F ri,|y k −ci| , in the spherical coordinates centered at c i such that y k = (|y k − c i |, 0, 0). 3. IF y k+1 ∈ ∂Ω in , THEN set exit(y 0 ) = y k+1 and goto END.
ELSE, set k = k + 1 and return to Step (1). END.
The cumulative distribution function F R,r is explicitly invertible and is given by
F R,r (α) := R 2 − r 2 2Rr R R − r − R √ R 2 − 2Rr cos α + r 2 .
On the boundary of the molecule: the jump method
The last ingredient of the MC algorithm is the discretization procedure to apply when the process X hits the boundary Γ. We use approximations of the process X that jumps immediately after hitting Γ either in Ω in at a distance h from Γ (as in the probabilistic representation of the solution u in (2.4)), or in Ω out at a distance αh from Γ, for some constant α > 0. More formally, we associate to each x ∈ Γ a random variable p(x) a.s. belonging to (R 3 \ Γ) ∪ {∂}, distributing the new position in R 3 \ Γ of the process after its jump, or killing the process when it belongs to the cemetery point ∂.
The following algorithm computes a score along the trajectory of an approximation of (X t ) t≥0 . In view of the probabilistic representation of u (2.4), the Monte-Carlo average of this score approximates u(x 0 ).
Given x 0 ∈ {c 1 , . . . , c N }, set k = 0 and score = u 0 (x 0 ) if x 0 ∈ Ω in or score = 0 otherwise. 1. IF x k ∈ Ω in ,
(a) THEN use the UWOS algorithm to simulate exit position(x k ) and set score = score− u 0 (exit position(x k )), (b) ELSE use the WOS algorithm with λ =κ 2 /2ε out to simulate exit position(x k ).
IF the particle has been killed, THEN return score and goto END.
2. Set x k+1 equal to an independent copy of p(exit position(x k )).
3. IF x k+1 ∈ Ω in , THEN set score = score + u 0 (x k+1 ). 4. Set k = k + 1 and return to Step (1). END.
In this work, we consider three different jump methods, i.e. three different families of r.v. (p(x)) x∈Γ . All are based on a finite difference approximation of the transmission condition (2.3). Note that other types of jump methods have been studied in the literature, among which "jump on spheres" techniques [32] and neutron transport approximations [7]. For the first two methods, we follow the terminology of [7].
Symmetric normal jump (SNJ):
This method is the one proposed by Mascagni and Simonov in their seminal paper [27]. It can be justified by a first-order expansion in (2.3): for all x ∈ Γ,
u(x) = ε out ε in + ε out u(x + hn(x)) + ε in ε in + ε out u(x − h n(x)) + remainder,
where the remainder is of order O(h 2 ) provided that the solution u to the Poisson-Boltzmann equation has uniformly bounded second-order derivatives in Ω out and Ω in . This holds true at least if Γ is a C ∞ manifold [7, Thm. 2.17]. The expansion can be written as an expectation involving a Bernoulli r.v. B with parameter εout εin+εout as
u(x) = E[u(x + (2B − 1)hn(x))] + O(h 2 ). (2.8)
This suggests the following choice for the r.v. p(x): fix h > 0, then for all x ∈ Γ,
p SNJ (x) = x + h n(x) with probability ε out ε in + ε out x − h n(x) with probability ε in ε in + ε out .
(2.9)
Note that the full simulation algorithm with the SNJ jump method (called the SNJ algorithm) can also be obtained by successive iterations of the formulas (2.7), (2.6) and (2.8), as explained in [27]). This suggests that an error of order h 2 accumulates at each time the discretized process hits Γ. Since this number of hitting times is of order 1/h, this suggests a global error of order h. Taking into account the additional error in the WOS algorithm, one can actually prove for smooth Γ that the error between u(x) and the expectation of the score of the SNJ algorithm is of
order O(h + /h) when h, ε → 0 [7, Thm. 4.7].
Asymmetric normal jump (ANJ): This method, proposed in [7], can also be deduced from the transmission condition (2.3), by introducing different finite difference steps to approximate the interior and exterior gradients. We fix h > 0 and introduce a fixed parameter α > 0. Then, for all x ∈ Γ, we set
p ANJ (x) = x + αh n(x) with probability ε out ε out + αε in x − h n(x)
with probability αε in ε out + αε in .
(2.10)
The error of the ANJ algorithm obtained with this jump method is also of order O(h + /h) [7, Thm. 4.7], but, if α > 1, the process is moved further away from Γ when it jumps in Ω out . Since the process is killed with a larger probability when it starts in Ω out further away from Γ, assuming α > 1 makes the computational cost of a simulation score smaller than for the SNJ algorithm. Of course, a compromise must be found with the increased bias for increased α > 1, which is analysed in [7].
Totally Asymmetric Jump (TAJ): This jump method from Γ is a new proposition in the context of the Poisson-Boltzmann equation. It was originally proposed in a two-dimensional context by Lejay and Maire [22] and for more general equations and boundary conditions by Maire and Nguyen in [24]. These methods apply to linear divergence form equations with damping. It is based on a higher order expansion of the transmission conditions of the PDE on Γ and extensively use the linearity. It does not apply in the nonlinear case, and will not be used in Section 3. When the process hits the boundary, we replace it as follows:
• with probability αε in αε in + ε out + 1 2κ 2 α 2 h 2 , the process moves toward Ω in at one of the following four points with uniform probability:
x − hn(x) + √ 2hm(x), x − hn(x) + √ 2hq(x), x − hn(x) − √ 2hm(x) or x − hn(x) − √ 2hq(x) ,
where m(x) and q(x) are any two orthonormal vectors in the tangent plane of Γ at the point x,
• with probability ε out αε in + ε out + 1 2κ 2 α 2 h 2 , the process moves toward Ω out at one of the four points with uniform probability:
x + αhn(x) + √ 2αhm(x), x + αhn(x) + √ 2αhq(x), x + αhn(x) − √ 2αhm(x) or x + αhn(x) − √ 2αhq(x) ,
• with probability 1 2κ 2 α 2 h 2 αε in + ε out + 1 2κ 2 α 2 h 2 , the process is killed.
As stated in Theorem 1 below, the TAJ method is of order 2, whereas SNJ and ANJ are first order methods.
Theorem 1. Assume that Γ is a C ∞ compact manifold in R 3 .
Then, for all x ∈ {c 1 , . . . , c N }, the expectationū h,α, (x) of the score of the TAJ algorithm with parameters h, α and , started from
x 0 = x, satisfies |ū(x) − u(x)| ≤ C h 2 + h ,
for a constant C depending only on α and the finite constant sup y ∈Γ, y ≤R (|u(y)|+ ∇u(y)
+ ∇ 2 u(y) + ∇ 3 u(y) ), for R large enough such that Γ ⊂ B(0, R), where B(0, R) = {z ∈ R 3 : z < R}.
While the proof is based on similar computations in [24], for the sake of completeness we give a detailed proof in the context of Poisson Boltzmann equation. Note that the above TAJ replacement formulas are more convenient than the formulas derived in [24], as they do not require to impose some constraints on h.
Proof. In the case where Γ is C ∞ , it has been proved in [7,Thm. 2.17] that the solution u to the Poisson-Boltzmann equation satisfies that u |Γ is C ∞ . Hence, the solutions of the two subproblems (2.1) and (2.2) admit derivatives of any order which are continuous up to Γ, i.e. they belong to C ∞ (Ω in ) and C ∞ (Ω out ), respectively (see [15]). In particular, ∇ k u is bounded on B(0, R) for all k ≥ 0.
Hence the following Taylor expansions are valid for all x ∈ Γ and y ∈ B(0, R) \ Γ:
u(y) = u(x) + ∇ in u(x) · (y − x) + 1 2 (y − x) ∇ 2 in u(x)(y − x) + O( y − x 3 ), if y ∈ Ω in , (2.11) u(y) = u(x) + ∇ out u(x) · (y − x) + 1 2 (y − x) ∇ 2 out u(x)(y − x) + O( y − x 3 ), if y ∈ Ω out , (2.12)
where the O( y−x 3 ) are bounded by y−x 3 times a constant depending only on sup z ∈Γ, z ≤R ∇ 3 u(z) , and where the notation ∇ in and ∇ out are extended to higher-order derivatives in an obvious way. Fix x ∈ Γ, η ∈ R and γ > 0. Without loss of generality, we can assume that x = 0, n(x) = (1, 0, 0), q(x) = (0, 1, 0) and m(x) = (0, 0, 1). We define
E η,γ u = u(η, γη, 0) + u(η, −γη, 0) + u(η, 0, γη) + u(η, 0, −γη) 4 .
Applying (2.11), we obtain
E −h,γ = u(0) − h∇ in u(0) · n(0) + 1 4 2h 2 ∂ 2 in u(0) ∂x 2 + γ 2 h 2 ∂ 2 in u(0) ∂y 2 + ∂ 2 in u(0) ∂z 2 + O(h 3 ),
and applying (2.12),
E αh,αγ = u(0) + αh∇ out u(0) · n(0) + 1 4 2α 2 h 2 ∂ 2 out u(0) ∂x 2 + γ 2 α 2 h 2 ∂ 2 out u(0) ∂y 2 + ∂ 2 out u(0) ∂z 2 + O(h 3 ).
Now, the relations ∆u(x) = 0 in Ω in in the neighborhood of Γ and ∆u(x) = κ 2 out u(x) in Ω out can be extended by continuity to Γ, so that
∂ 2 in u(0) ∂y 2 + ∂ 2 in u(0) ∂z 2 = − ∂ 2 in u(0) ∂x 2 , and ∂ 2 out u(0) ∂y 2 + ∂ 2 out u(0) ∂z 2 = − ∂ 2 out u(0) ∂x 2 + κ 2 out u(0).
This entails
E −h,γ = u(0) − h∇ in u(0) · n(0) + 2 − γ 2 4 h 2 ∂ 2 in u(0) ∂x 2 + O(h 3 )
and
E αh,αγ = 1 + κ 2 out α 2 γ 2 h 2 4 u(0) + αh∇ out u(0) · n(0) + 2 − γ 2 4 α 2 h 2 ∂ 2 out u(0) ∂x 2 + O(h 3 ).
Hence, choosing γ = √ 2, we obtain
αε in αε in + ε out (1 + κ 2 out α 2 h 2 /2) E −h, √ 2 + ε out αε in + ε out (1 + κ 2 out α 2 h 2 /2) E αh, √ 2α = u(0) + O(h 3 ), (2.13)
where the gradient terms canceled because of (
Numerical experiments
Parallel version of the algorithm
It is usually very simple to implement a parallel version of a Monte-Carlo algorithm. This is the case for our algorithm. The only delicate issue for reliable and statistically sound calculations is the parallel generation of pseudo random numbers. In our MPI 2 parallel implementation of the code, we used the SPRNG 4.4 library [25].
Comparison of the 'locate nearest atom' methods
As described in Subsection 2.2.2, we implemented three different methods to approximate the closest atom from a given particle position in R 3 . In Figure 1(a), we present the CPU time for each method as a function of the size of the molecule. We use 5 molecules of different sizes: the molecule composed of N = 103 atoms described in the next subsection, and the molecules 2KAM, 4HHF, 1KDM, 1HHO, 1HFO and 4K4Y of RCSB Protein Data Bank, with sizes ranging from N = 416 to N = 29420 atoms. We start each simulation from a location close to the alpha-carbon atom of the first residue of the molecule. We use the SNJ algorithm with h = 0.1 and = 10 −4 (the shape of the curves is roughly independent on the jump method). We run 10 5 independent simulations for each molecule on a laptop computer. Of course, the computational time is very dependent of the shape of the molecule and of the initial position of the algorithm, so that the CPU time does not necessarily increase with N , as observed in Figure 1(a) for the largest molecules.
However, as expected, the power diagram with hints method is the fastest, at least for large molecules. The power diagram method is slightly slower, but the difference is not very significant. The brute localization method is up to 10 times slower than the power diagram with hints method for large molecules, but is actually faster for molecules of sizes smaller than a thousand atoms.
Comparison with APBS
As a validation of the algorithm, we compared our Monte-Carlo estimate of u(x) with the value computed by a deterministic solver of Poisson-Boltzmann equation. We choose APBS solver [4], which uses adaptive finite element methods and algebraic multilevel methods.
This numerical experiment is done on a small peptide composed of 6 residues (GLU-TRP-GLY-PRO-TRP-VAL) and N = 103 atoms. To produce the plots in Figure 1(b), we have calculated u at 30 different points of the space located on a line close to the alpha-carbon of the first residue (GLU). Our Monte-Carlo method was run with the SNJ jump method, h = 0.1, = 10 −4 and 4 × 10 4 independent simulations for each initial points to compute the Monte-Carlo average. The agreement between the two curves is quite good, although there are some differences, which might be due either to the Monte-Carlo error, or to the discretization in APBS.
The TAJ methods' convergence order in the single atom case
The goal of this experiment is to check that the expected error of the TAJ method converges faster than ANJ methods, as suggested by Theorem 1. We used the simplest molecule, composed of a single atom, which is the only practical case where Γ is C ∞ in Poisson-Boltzmann equation. In [7], some comparisons were done on the SNJ and ANJ convergences in such a case. We consider an atom with radius 1Å and charge 1. We compute the approximation of (u − u 0 )(x 0 ), where x 0 is the center of the atom, using SNJ, ANJ (α = 3 and α = 10) and TAJ (α = 1, α = 3 and α = 10) methods. The Monte-Carlo average is computed from 10 7 independent simulations for each method and each values of h, ranging from 0.003 to 0.9 and we took = 10 −5 . The results are compared with the exact value of (u − u 0 )(x 0 ) for which an exact expression is known [7]. The results are shown in Figure 2 and Figure 4(a). As expected, we observe a faster convergence for the TAJ methods, although the second order is not very clear because of the Monte-Carlo error. The first order of convergence for the SNJ and ANJ methods can be observed much more clearly.
It seems that for a given h, the error is smaller for a smaller parameter α both for ANJ and TAJ methods, but this needs to be compared with the CPU time of the simulations. The performance plot of Figure 4(a) shows the expected error of the Monte-Carlo algorithm as a function of CPU time. It reveals that, for a given CPU time of computation and choosing an appropriate value of h, the expected error of the algorithm is comparable for the three different values of α, and is slightly smaller for α = 10 for ANJ algorithms. This is consistent with the tests realized in [7]. This plot also confirms the better efficiency of the TAJ methods.
Comparison between the different jump methods on a biomolecule
The goal of this experiment is to compare the SNJ, ANJ and TAJ methods on a small molecule, but with realistic geometry. We use the molecule composed of 103 atoms described in Section 2.3.3. We use the jump methods SNJ, ANJ (α = 3 and α = 10) and TAJ (α = 1, α = 3 and α = 10). The simulations are done with = 10 −6 and with different values for h, ranging from 0.9 to 0.003, and we take the same number of Monte-Carlo simulations (10 6 ) for each run, large enough to be able to detect and compare the convergence of the expectation of the score computed by our algorithms. We also compared the result with the value computed with the APBS method, but it differs from the Monte-Carlo reference value of roughly 7%, which is too large to detect the rate of convergence for h small. We suspect that this difference is due to the finite element discretization used in APBS. The value computed by APBS is shown in Figure 3(a).
All 6 methods show a good convergence. The higher order of convergence of the TAJ method cannot be detected because of the statistical error of the Monte Carlo method. However, we observe a smaller error for larger values of h for the TAJ method than for the ANJ ones. Of course, the TAJ methods might not converge with order 2 as in Theorem 1, because Γ is not smooth. This indeed also causes some difficulties in the implementation of the method, since it might occur, even for small h, that a jump from Γ to the direction of Ω out actually gives a new position inside the molecule. This is particularly true for the TAJ algorithm. Our tests show that the result of the algorithm is quite sensitive to the method used in this particular situation, and can produce differences up to 5%. When this situation occurs, we chose here to push the particle outside of the molecule, at a distance h of Γ, close to its position before the jump. This choice shows good agreement between the values given by the ANJ and TAJ algorithms.
As in the case of a single atom, we compare the performance of the 6 methods in Figure 4(b). This plot does not show as clear conclusions as for the case of a single atom, but it confirms that TAJ has a better performance at small CPU times. a jump and scoring procedure when the particle hits Γ (SNJ and ANJ jump methods), and a killing rate outside of the molecule. In the nonlinear case, our method also involves a possible reproduction of a particle when it dies. This modifies the scoring procedure which is now based on the product of the scores of the daughter particles.
Similarly to the linear case, the Poisson-Boltzmann equation (1.1) can be reformulated as two PDEs in Ω in and Ω out , and the transmission condition (2.3). The PDE in Ω in is still (2.1), but in Ω out the PDE (2.2) is replaced by
− 1 2 ∆u(x) + λ 0 sinh u(x) = 0, for x ∈ Ω out u(x) = h(x) for x ∈ Γ,(3.
Quasi-linear elliptic PDEs and Branching Brownian motion
Branching Brownian motion
Let us construct a branching Brownian motion (denoted BBM below) in Ω out by defining the dates of birth (θ) and death (σ) of each particle and the positions of the particles between their birth and death times. We use the classical Ulam-Harris-Neveu labelling for individuals, i.e. each individual is labeled by an element of H = n≥0 N n . The ancestor is denoted by ∅ (= N 0 by convention), and the j-th child of an individual u ∈ H is denoted by uj, where uj stands for the concatenation of the vector u and the number j. Some of the particles of H never appear in the population, so we need to introduce a cemetery point ∂ to code for those individuals. More precisely, when v ∈ H satisfies θ v = σ v = ∂, it means that the individual v never lived in the BBM.
We now introduce the following independent stochastic objects:
• let (B v t , t ≥ 0) v∈H be i.i.d. Brownian motions;
• let (E v ) v∈H i.i.d. exponential r.v. of parameter λ;
• let (K v ) v∈H i.i.d. r.v. in N with distribution (p k ) k≥0 ,
the parameters λ and p k , k ≥ 0, will be chosen later. Fix x ∈ Ω out . We construct the BBM started from x (i.e. the r.v. θ v , σ v in [0, +∞) ∪ {∂} and X v t ∈ Ω out for t ∈ [θ v , σ v ) for all v ∈ H) as follows:
1. The initial particle has position x ∈ Ω out ; its label is ∅,
θ ∅ = 0, σ ∅ = inf{t ≥ 0 : x + B ∅ t ∈ Γ} ∧ E ∅ and X ∅ t = x + B ∅ t for all t ≤ σ ∅ . 2. Suppose that the r.v. θ v (birth time of individual v) and σ v (death time of individual v) and X v t , t ∈ [θ v , σ v ] are constructed (a) If σ v = ∂ and X v σv ∈ Γ, then for all 1 ≤ i ≤ K v , θ vi = σ v , and for all i > K v , θ vi = σ vi = ∂, for all 1 ≤ i ≤ K v , we also define σ Γ vi = inf{t ≥ θ vi : X v σv + B vi t − B vi θvi ∈ Γ}, σ vi = (θ vi + E vi ) ∧ σ Γ vi , and X vi t = X v σv + B vi t − B vi θvi for all t ∈ [θ vi , σ vi ]. (b) If σ v = ∂ and X v σv ∈ Γ, then θ vi = σ vi = ∂ for all i ≥ 1. (c) If σ v = ∂, then θ vi = σ vi = ∂ for all i ≥ 1.
We denote by P x the law of the BBM when the first particle has initial position x, and E x the corresponding expectation.
Note that, in the case where Ω out = R 3 , we obtain the standard branching Brownian motion (cf. e.g. [28]), in which the number of particles is a continuous-time branching process (Z t , t ≥ 0) with branching rate λ and offspring distribution (p k ) k≥0 . For general Ω out , the BBM process can be coupled with the standard one such that the number of particles alive at time t is smaller than Z t for all t ≥ 0.
In particular, in the case where k≥0 kp k ≤ 1, the branching process Z is sub-critical or critical and hence all the particles of the BBM process either die or reach Γ after an almost surely finite time.
A general probabilistic interpretation of quasi-linear elliptic PDEs in terms of BBM
The general link between branching Markov processes and non-linear parabolic PDEs is known since [33,19]. The particular case of the binary branching Brownian motion (with p 2 = 1 and p k = 0 for all k = 2) has been particularly studied because of its link with the Fisher-Kolmogorov-Petrovskii-Piskunov PDE [28,8]. This approach has been also used to give a probabilistic interpretation of the Fourier transform of Navier-Stokes equation [21]. In contrast, the link between branching diffusions and elliptic PDEs like Poisson-Boltzmann equation has not been exploited a lot in the literature. Surprisingly, the probabilistic interpretations of parabolic or elliptic non-linear PDEs seem not to have been used a lot for numerical purpose either (see e.g [34,29,16,17]). The general result we present here about the link between non-linear elliptic PDEs and the BBM, and our method of proof, are original (as far as we know).
Let g : Z + → R and h : Γ → R be bounded functions and assume that the power series k≥0 p k g(k)x k has infinite radius of convergence. We consider the following quasi-linear elliptic PDE:
− 1 2 ∆u(x) + λu(x) − λ k≥0 p k g(k)u(x) k = 0, x ∈ Ω out ,(3.2)
with boundary condition u(x) = h(x) for all x ∈ Γ.
Theorem 2. Consider the BBM of the previous subsection and assume that the PDE (3.2) has a C 2 solution u, bounded and with bounded first-order derivatives. For all n ≥ 0, let τ n be the first time where the BBM had n alive particles in the whole elapsed time. Then, for all n ≥ 0 and x ∈ Ω out ,
E(Y τn ) = u(x), (3.3) where Y t = v∈H g(K v )1 {σv≤t, X v σv ∈Γ} + h(X v σv )1 {σv≤t, X v σv ∈Γ} + u(X v t )1 {θv≤t<σv} = v∈H g(K v )1 {σv≤t, X v σv ∈Γ} + u(X v σv∧t )1 {X v σv ∈Γ or θv≤t<σv}
and Y ∞ = lim t→∞ Y t is well-defined on the event {τ n = ∞}. Assume further that k≥0 kp k < 1. Then, P(τ n < ∞) → 0 when n → ∞ and Y ∞ is well-defined a.s. Then, in the case where E(|Y ∞ |) < ∞, we have, for all x ∈ Ω out ,
u(x) = E x v∈H:σv =∂ 1 {X v σv ∈Γ} g(K v ) + 1 {X v σv ∈Γ} h(X v σv ) .
(3.4)
The last formula extends the classical Feynman-Kac formula to a class of non-linear elliptic PDEs. Note that the existence of a C 2 solution to (3.2) holds for example if Γ and h are C ∞ [15]. In the case of Poisson-Boltzmann PDE, a PDE of the form (3.2) is coupled with a PDE in Ω in . In this case, on can also prove that u is C 2 on Ω out if Γ is C ∞ [9].
A simple way to get the heuristics behind the probabilistic interpretation is the following. If we assume that the r.h.s. of (3.4), denoted below byû(x), is well-defined and smooth, then we can use the following standard technique for branching processes: distinguish between the different events that may occur between time 0 and h, apply the Markov property, and let h converge to 0. More precisely, let us denote by Z the r.v. in the expectation in the r.h.s. of (3.4). First, we have of courseû(x) = h(x) for all x ∈ Γ. Next, for x ∈ Ω out , we can writê
u(x) = E x 1 {σ ∅ ≤h} g(K ∅ )û(X ∅ σ ∅ ) K ∅ + E x 1 {σ ∅ >h}û (X ∅ h ) . Now, since x ∈ Γ, the first hitting time τ Γ of Γ by (x + B ∅ t , t ≥ 0) satisfies P(τ Γ ≤ h) = o(h)
at least for smooth Γ (it actually decreases exponentially in 1/h, since τ Γ is larger than the exit time of a Brownian motion from a fixed ball, the distribution of which is known [6]). Therefore, except on an event of probability o(h), σ ∅ = E ∅ on the event {σ ∅ ≤ h}, and
E x 1 {σ ∅ ≤h} g(K ∅ )û(X ∅ σ ∅ ) K ∅ = E x 1 {E ∅ ≤h} g(K ∅ )û(X ∅ σ ∅ ) K ∅ + o(h) = (1 − e −λh ) k≥0 p k g(k)E[û(x + B ∅ E (h) ) k ] + o(h),
where E (h) is an exponential r.v. with parameter λ, conditioned to be smaller than h, independent of B ∅ . Note that, in the last equation, we implicitly extended the functionû as a bounded function on R 3 . Under the assumption thatû is bounded and continuous, and the power series k p k g(k)x k has an infinite radius of convergence, it is elementary to prove that lim h→0 k≥0
p k g(k)E[û(x + B E ) k ] = k≥0 p k g(k)û(x) k .
Similarly,
E x 1 {σ ∅ >h}û (X ∅ h ) = E x 1 {E ∅ >h}û (x + B ∅ h ) + o(h).
Since E ∅ is independent of (B ∅ t ) andû has been assumed to be twice continuously differentiable, Itô's formula gives
E x 1 {σ ∅ >h}û (X ∅ h ) = e −λh û(x) + 1 2 h 0 E[∆û(x + B ∅ s )]ds + o(h).
Again, if ∆û is bounded and continuous, one has
lim h→0 1 h h 0 E[∆û(x + B ∅ s )]ds = ∆û(x).
Combining all the previous estimates, we obtain
u(x) = λh k≥0 p k g(k)û(x) k + (1 − λh) û(x) + h ∆û(x) 2 + o(h),
which entails (3.2) in the limit h → 0.
The last argument, although intuitive, requires a priori regularity for the functionû(x). In addition, this method does not allow to obtain results in cases where the expectation in the right-hand side of (3.4) is not finite. This is why we prefer to give a new proof extending the classical method of proof of Feynman-Kac formula [14] to the case of branching diffusions. The idea is to compute the semimartingale decomposition of the process (Y t , t ≥ 0).
Proof of Theorem 2. It is more convenient to use an equivalent construction of the BBM: let us consider the following independent stochastic objects:
• let (B v t , t ≥ 0) v∈H be i.i.d. Brownian motions; • let (N v (dt, dk)) v∈H be i.i.d. Poisson point measures on [0, ∞)×Z + with intensity measure q(dt, dk) = λ i≥0 p i δ i (dk)dt,
where δ i is the Dirac measure at the point i.
The following construction of the BBM amounts to define E v as the time before the first atom of N v after θ v , and K v as the second coordinate of this atom. The state of the BBM will be represented as the counting measure
ν t = v∈H 1 {θv≤t<σv} δ (X v t ,v) ,
constructed as follows: for all bounded function f : R 3 × H → R, twice continuously differentiable w.r.t. the first variable with uniformly bounded derivatives, the measure ν t satisfies
ν t , f = f (x, ∅) + t 0 v∈H ν s− , ∇f e v dB v s + 1 2 t 0 v∈H ν s− , ∆f e v ds + t 0 N v∈H k j=1 ν s− , f ( · , vj)e v − ν s− , f e v N v (ds, dk),
where ν, f stands for R 3 ×H f (x, v)ν(dx, dv) and e v (x, w) = 1 {x∈Ωout} 1 {w=v} , so that ν s , e v = 1 {θv≤t<σv} . These equations characterize the measure ν t for all t ≥ 0, and they simply rephrase the algorithmic construction of the BBM given above.
Introducing the compensated Poisson point measuresÑ v (ds, dk) = N v (ds, dk) − q(ds, dk), the last formula immediately gives the semimartingale decomposition of functionals of ν t of the form ν t , f :
ν t , f = f (x)+ 1 2 t 0 v∈H ν s− , ∆f e v ds+λ t 0 N v∈H k≥0 p k k j=1 ν s− , f ( · , vj)e v − ν s− , f e v ds+M f t ,
where M f t is the local martingale
M f t = t 0 v∈H ν s− , ∇f e v dB v s + t 0 v∈H k j=1 ν s− , f ( · , vj)e v − ν s− , f e v Ñ v (ds, dk).
Of course, the semimartingale decomposition of other functionals of (ν t , t ∈ [0, T ]) can be obtained in a similar way. Given x ∈ Ω out , under P x , we obtain for the process Y t
Y t = u(x) + t 0 Y s− v∈H ν s− , ∇u u e v dB v s + 1 2 t 0 Y s− v∈H ν s− , ∆u u e v ds + t 0 N Y s− v∈H ν s− , u k−1 g(k)e v − ν s− , e v N v (ds, dk) = u(x) + t 0 Y s− v∈H ν s− , e v u 1 2 ∆u + λ k≥0 p k g(k)u k−1 − λu ds + M t = u(x) + M t ,
where
M t = t 0 Y s− v∈H ν s− , ∇u u e v dB v s + t 0 N Y s− v∈H ν s− , u k−1 g(k)e v − ν s− , e v Ñ v (ds, dk)
is a local martingale. Note that in the previous computation, we made the convention that, for any
t ≥ 0, v ∈ H and x s.t. ν t ({(x, v)}) = 1, Y t /u(x) = w∈H, w =v g(K w )1 {σw≤t, X w σw ∈Γ} + u(X w σw∧t )1 {X w σw ∈Γ or θw≤t<σw} .
Since u and ∇u are bounded functions, we have of course that (M t∧τn , t ≥ 0) are martingales for all n ≥ 0, and so
E(Y t∧τn ) = u(x)
for all t ≥ 0 and n ≥ 1. Since (Y t∧τn , t ≥ 0) is uniformly bounded, we obtain (3.3).
In the case where k≥0 kp k < 1, the number of particles in the BBM is smaller than the number of particles in a sub-critical continuous-time branching process defined as Z t = µ t , 1 , where
µ t = δ ∅ + t 0 N v∈H k j=1 δ vj − δ v µ s− ({v})N v (ds, dk).
Then of course P(τ n < ∞) → 0 when n → 0, Y ∞ is a.s. well-defined, and (3.4) is clear if E|Y ∞ | < ∞.
A probabilistic interpretation of the nonlinear Poisson-Boltzmann equation
The last theorem gives a probabilistic interpretation of PDEs of the form of Poisson-Boltzmann equation (outside the molecule), which suggests to use again a Monte-Carlo method to estimate u(x). The difficulty is to find the good probabilistic interpretation allowing to use (3.4) rather than (3.3) which involves the unknown function. One possible choice of the function g and the probability distribution (p k ) k≥0 to recover the PDE (3.1) is as follows:
λ = λ 0 ; g(k) = −1 for all k ≥ 1, g(0) = 0; p 2k+1 = 1 (2k + 1)!
and p 2k = 0 for all k ≥ 1, p 1 = 0;
p 0 = 1 − k≥1 p 2k+1 = 2 − sinh 1 > 0.
(3.5)
With these parameters, we have k≥0 kp k = cosh 1 − 1 < 1, so the BBM goes extinct after an a.s. finite time.
In this case, Theorem 2 gives the following probabilistic interpretation for the Poisson-Boltzmann PDE.
Corollary 3. Let u be the solution of the PDE (3.1): provided that the expectation is well-defined, for all x ∈ Ω out ,
u(x) = E x (−1) #{v:X v σv ∈Γ} v:σv =∂ 1 {X v σv ∈Γ} 1 {Kv≥1} + 1 {X v σv ∈Γ} u(X v σv ) , (3.6)
where the BBM (X v t , t ≥ 0, v ∈ H) has parameters (3.5).
In view of the last formula, it is clear that the expectation in the right-hand side is finite if u is bounded by 1 on Γ. When u takes larger values on Γ, the product involved in (3.6) can be bounded by some exponential moment of the total number of particles which hit Γ in the BBM, but this bound is not finite in general. Therefore, we cannot ensure in general that the expectation is well-defined with so simple estimates. Note that the random variable inside the expectation in (3.6) is zero when one of the particles in the BBM dies before hitting Γ and gives birth to no children, so one can expect better bounds on the expectation, but such bounds seem very delicate to obtain. We will not study this question here.
Remark 4. In this work, we concentrate on the characterization of the solution of Poisson-Boltzmann equation as the expectation of a random variable Z defined from a BBM as in (3.6). We do not provide a complete study of the variance of Z, which requires a deeper analysis. One can give an idea of the possible difficulties with the previous tools as follows: because Z is defined as a product, it can be seen from Theorem 2, using the parameter values (3.5), that v(x) = E x (Z 2 ) should be (at least formally) solution to the PDE
− 1 2 ∆v(x) + 3λv(x) − λ sinh v(x) = 0, v |Γ = u 2 |Γ .
(3.7)
This elliptic PDE has a non-linear term which is concave where the classical theory would require it to be convex (as in Poisson-Boltzmann equation). In particular, this prevents to use classical convexity arguments to characterize the solution of this PDE as the solution of a variational problem (see for example [9] for the application of these arguments for Poisson-Boltzmann PDE). This problem will be crucial if the solution takes large values, typically when the boundary condition is too large, because then one cannot expect a good approximation of v by the linearization of (3.7), and the classical arguments cannot ensure that a solution to this PDE exists.
The main steps of the Monte-Carlo algorithm
Outside the molecule: branching walk on spheres (BWOS) algorithm
The walk on spheres (WOS) algorithm of Section 2.2.1 can be extended in order to deal with possible branching. The only difference is that the precise location of death of the particle must be simulated to obtain the initial position of its daughters. Consider y ∈ Ω out and a radius R such that B(y, R) ⊂ Ω out . Consider also a Brownian motion (B t , t ≥ 0) such that B 0 = y and let τ R be the first hitting time by B t of the sphere ∂B(y, R). A WOS algorithm sampling the system of branching Brownian particles requires to simulate the position of B τ R ∧E , where E is an independent exponential random variable of parameter λ. Because of the spherical symmetry of the Brownian path, the only relevant information is the p.d.f. of R τ R ∧E , where R t = |B t − y| is a Bessel process of dimension 3. This p.d.f. can be computed from the formula [6, Ch. 5,
Formula 1.1.6] P 0 sup 0<s<E R s < R, R E ∈ dr = 2λr sinh (R − r) √ 2λ sinh R √ 2λ , ∀0 ≤ r ≤ R,
where, under P 0 , B is a standard Brownian motion started from 0. From this, we obtain for all 0 ≤ r ≤ R
P 0 sup 0<s<E R s < R, R E ≤ r = 1 − r √ 2λ cosh (R − r) √ 2λ + sinh (R − r) √ 2λ sinh R √ 2λ .
Note that, by taking r = R, we recover the death probability given in the WOS algorithm of Section 2.2.1, as expected. Hence, we obtain the explicit cumulative distribution function of R E conditionally on
{E < τ R } F (r) = P R E ≤ r | sup 0<s<R R s < R = sinh[r √ 2λ] − r √ 2λ cosh[(R − r) √ 2λ] − sinh[(R − r) √ 2λ] sinh[R √ 2λ] − R √ 2λ .
(3.8) One can sample from this distribution by several ways, among which we tested acceptance-rejection methods with several proposition distributions (among which the Beta(2, 2) distribution), and Newton's algorithm to invert the cumulative distribution function. It appears that Newton's algorithm gives a precise sampling and is much faster than acceptance-rejection algorithms, so we use this method in our code. More precisely, we will denote by split(y, R) the random variable uniformly distributed on the sphere of center y and radius r obtained by applying Newton's method with 4 iterations to approximate F −1 (U ), where U is a uniform r.v. on [0, 1] and F is given by (3.8).
The BWOS algorithm started at x ∈ Ω out returns a random position which belongs to Γ ∪ Ω out . This random variable is an approximation of B τ ∧E , where B is a standard Brownian motion started from x, τ its first hitting time of Γ, and E an independent exponential r.v. of parameter λ.
BWOS Algorithm
Given x ∈ Ω out , λ ≥ 0, and ε > 0 1. Use the WOS algorithm. 2. IF the particule is not alive, THEN (a) Denote y the last known position of the particle before the death, given by the WOS algorithm, (b) let S(y, r) be the largest open sphere included in D centered at y, (c) return an independent copy of split(y).
ELSE return exit(x) simulated by WOS algorithm. END.
We denote by split or exit(x) the position returned by this algorithm.
Branching Algorithm
The idea of our algorithm is to use the double randomization technique, as in the linear case and as in [27], that consists to use the approximations (2.7), (2.8) and (3.6) recursively to estimate the unknown function u appearing on the right-hand side of each of these formulas. Therefore, the method consists when the initial particle starts in Ω in to use the UWOS algorithm to simulate its first hitting point of Γ, or when the initial particle starts in Ω out the WOS algorithm, then use a jump method as in Section 2.2.4 to move the particle away from Γ, and continue to use inductively the UWOS and WOS algorithm depending whether the particle entered inside the molecule or exited outside the molecule. This part is exactly the same as in the linear case, and stops when the particle is killed outside the molecule. This time of killing is actually a branching time, since a random number of new particles, with distribution (p k ) k≥0 , appears at the death position of the initial particle, and each new particle continues to evolve independently as the first one.
As in the linear case, we can use the SNJ and ANJ jump methods to move a particle after it reaches Γ. Therefore, we set p(x) = p SNJ (x) (2.9) or p ANJ (x) (2.10) depending on the method chosen. Because of the non-linearity, as explained in Section 2.2.4, one cannot expect a better precision for the TAJ method, which is therefore not used here.
One of the difficulties of the algorithm consists in the computation of the global score of the algorithm. Before the first particle is killed, the score is computed exactly as in the linear case. Then, if this particle has daughters, because of (3.6), we need to add to the score of the first particle minus the product of the scores of its daughters (the minus comes from the term (−1) #{v:X v σv ∈Γ} in the r.h.s. of (3.6)). These scores might be 0 if the daughter dies before hitting Γ and gives birth to no children, or might be different from zero if the particle hits Γ. Indeed, in this case, the particle first jumps according to (2.8) and then scores some value if it jumps inside the molecule, according to (2.7). But its score might also involve the scores of its own daughters if it is non-zero. Therefore, the convenient way to formulate our algorithm is by a recursive procedure.
Let M be a r.v. distributed according to the offspring distribution of the BBM:
M = 0
with probability 2 − sinh 1 2k + 1 with probability 1 (2k + 1)! , for all k ≥ 1.
(3.9) Given x 0 ∈ {c 1 , . . . , c N }, our branching algorithm can be described recursively as follows.
BA(x 0 ) algorithm.
Set k = 0 and score = u 0 (x 0 ) if x 0 ∈ Ω in or score = 0 otherwise 1. IF x k ∈ Ω in (a) THEN
i. Use the UWOS algorithm to simulate exit(x k ) ii. Set score = score − u 0 (exit(x k )) iii. Set x k+1 equal to an independent copy of p(exit(x k )) (b) ELSE i. Use the BWOS algorithm with λ =κ 2 /2ε out to simulate split or exit(x k ) ii. IF split or exit(x k ) ∈ Ω out iii. THEN A. Sample m as an independent copy of M . B. Let score 1 , . . . , score m be the scores returned by m independent runs of the BA(split or exit(x k )) algorithm. C. Return score − m j=1 score j if m = 0, or score if m = 0. D. Goto END. iv. ELSE set x k+1 equal to an independent copy of p(split or exit(x k )).
IF x k+1
∈ Ω in , THEN set score = score + u 0 (x k+1 ). 3. Set k = k + 1 and return to Step (1).
END.
As in the linear case, the approximation of u(x 0 ) is obtained by computing the Monte-Carlo average of the score returned by the BA(x 0 ) algorithm.
The convergence of our algorithm could be analyzed following exactly the same lines as Thm. 4.7 of [7], giving an error of the same order as the SNJ and ANJ algorithms of Section 2.2.4, provided one could ensure that all the expectations involved in this algorithm are finite. Contrary to the linear case, because of the products involved in Substep (C) above, this is not obvious at all, even if the function u is bounded by 1, so we leave aside this question for a future work.
Numerical experiments
Implementation
As in the linear case, the localization of the closest atom from the particle is done using CGAL library, and the parallelization of the code is done using MPI and SPRGN libraries.
Note that, since the number of children of all particles are i.i.d. and independent of their positions of death, the set of particles simulated in our algorithm and their genealogical relations are the same as those of a Galton-Watson process with reproduction distribution (p k ) k≥0 , the distribution of the r.v. M of (3.9).
Hence, for the practical implementation of the algorithm, we started by sampling first a Galton-Watson tree with offspring distribution (p k ) k≥0 conditioned to be smaller than 10 (the probability to have 11 children or more is smaller than 5 · 10 −8 ). Then, we simulate the trajectory of each particle as above and we follow the genealogical structure of the tree previously sampled when a particle is killed.
To avoid the use of recursive functions, which can be time consuming, the simulation of the trajectories of each particle and the scores obtained along each of these trajectories is done first by exploring the Galton-Watson tree forward in time (from the root to the leaves), whereas the computation of the score is done in the end by exploring the tree backward in time (from the leaves to the root).
Since after a branching the score is evaluated as a product of scores of daughters, it is important to detect fast when one of these scores is zero, which happens for example when one of the daughters dies without children before hitting Γ. Hence, when exploring the Galton-Watson tree forward in time, at each birth event, it is more advantageous in terms of computational time to deal first with daughters particles that have no children.
The fact that the genealogical tree of our particles is sampled before the simulation of the particles' motion allows us to easily implement a stratification Monte-Carlo algorithm to reduce the variance of the method. Each stratum is a given Galton-Watson genealogical tree. Its probability of occurrence is easy to compute. We restricted in our simulation to trees of height 2 or less, because the probability that the Galton-Watson tree has depth 3 or more is less than 1.3 · 10 −4 . We run 500 trajectories for each stratum to estimate the variance within each stratum and allocate the number of runs in a given stratum proportionally to the probability of the stratum times the empirical standard deviation within the stratum.
The single atom case
To evaluate the convergence and the efficiency of the previous algorithm, we made some simulations with a monoatomic molecule. Indeed an approximation of the exact solution of the non-linear Poisson-Boltzmann equation (1.1) can be easily computed in this case. Assuming the unique atom is centered at 0, the solution u(x) to the Poisson-Boltzmann equation can be written as v( x ) for some function v because of the spherical symmetry of the problem. Now, u − u 0 is harmonic in B(0, r), with a constant Dirichlet boundary condition thanks again to the spherical symmetry of the problem. with boundary conditions ε out v (r) = ε in v 0 (r) and v(x) → 0 when x → +∞. We approximate this function as the solution of the same differential equation on [r, R] for a large R with u(R) = 0. The numerical results presented here are obtained using the branching algorithm with jump methods SNJ and ANJ (α = 3 and 10), taking the same number of Monte-Carlo simulations (1 · 10 5 ) for each run, with a value of h varying from 0.003 to 0.9, and with = 10 −5 . The reference values are computed with h = 0.001 and 1 · 10 6 Monte-Carlo simulations. We take a radius r = 1Å for the atom centered at zero, and we compute the value of (u − u 0 )(0) using our method and compare it to the value v(r) − v 0 (r) computed above.
Hence u − u 0 is constant in B(0, r), so that v − v 0 is constant in [0, r], where v 0 ( x ) = u 0 (x) for all x ∈ R 3 .
We have tested different values of the charge z, and it appears that the variance of the algorithm is very sensitive to this parameter. As explained above, this can be understood from formula (3.6) when |v(r)| > 1, since then the value computed is similar to an exponential moment of the number of individuals in a sub-critical Galton-Watson tree. In practice, above some threshold on the parameter z, the convergence of the algorithm is much slower. A finer numerical analysis reveals that this is due to very unprobable Galton-Watson trees, with several large numbers of offsprings, which contribute for an important part to the empirical variance. Because of the large value of the constant in front of the Dirac masses in (1.3), the threshold above which the variance starts to increase drastically is slightly larger than z = 1. This is an important issue of our algorithm, which needs some new ideas to be solved (see the perspectives of Section 4). However, for small enough values of z, the algorithm behaves very well. We present the numerical results for a much smaller value z = 0.2 in Figure 5, which show a good performance of the algorithm. Figure 3.10 confirms our conjecture that the error is of the order of h. The performance plots in Figure 6 indicate that the value of α has a negligible influence on the error of the algorithm for a given CPU time. Note also that the confidence intervals shown in Figure 5(a) are very small, despite the relatively small number of Monte-Carlo simulation we used (3 · 10 4 ). The value z = 1 (which corresponds to a monoatomic ion of valence ±1) also has a satisfactory behavior, shown in Fig 7, although the convergence is not as good as for z = 0.2, as appears in Figure 7(a) for h = 0.003 and in Figure 7(b). We also observe a very unstable behavior of the ANJ method with α = 3 for large values of h. Still, for all values of h between 0.003 and 0.1, the relative error of the algorithm is less than 1%. Note that the same simulation, run without stratified Monte-Carlo, gives a very large variance of the result. According to our tests, the stratification method allowed to increase the threshold of variance explosion from roughly z = 0.3 to more than z = 1.
If one takes a larger value for z, the variance starts increasing drastically and the method no longer converges for h small. A finer analysis of the results of the algorithm shows that the variance is extremely large in a few very unlikely strata, corresponding to genealogical trees with many children at each generation. Some other runs show a very small empirical variance, because the very unlikely trajectories with a very large score did not occur by chance.
The case with two atoms close with opposite charges
It is generally accepted (cf. e.g. [2]) that for uncharged molecules, the approximation of the non-linear Poisson-Boltzmann equation by the linear one is not too bad, meaning that the potential u does not take too large values in Ω out , and in particular close to Γ. Since the explosion of the variance observed in the last test case for large values of z seems to be related to the fact that u takes too large values on Γ, this suggests to look at uncharged molecules. The simplest uncharged molecules with non-trivial electrostatic potential u are composed of N = 2 atoms with opposite charges. We focus here on this situation, assuming that the two atoms have same radius r 1 = r 2 = 1Å and opposite charges z 1 = 1 and z 2 = −1. We denote by a the distance between the centers of the two atoms (inÅ).
As in the first case, assuming a too large value of a increases the values of u on Γ and hence produce variance explosion of the method. We run similar tests as in the previous example, with 10 5 Monte-Carlo simulations for several values of h between 0.003 and 0.9. We take = 10 −5 and we run our branching algorithm to compute u at a point x 0 on the line linking the centers of the two atoms, at a distance 1.5Å from the closest center. We used the SNJ jump method, and the ANJ jump method for α = 3 and α = 10.
The reference value involved in the error estimate cannot be computed as above because the spherical symmetry is broken. We could use APBS to solve the nonlinear Poisson-Boltzmann PDE, but our tests in the linear case show that the adaptive finite element method of this solver can produce small errors which could prevent us from observing the convergence of our methods. This is why we use a reference value computed with the ANJ jump method (α = 10) with a large number of Monte-Carlo simulations (10 6 ), h = 0.001 and = 10 −6 .
We tested several values of a. The results are very similar to those obtained in the case of a single atom. The results for a small value a = 0.2Å are shown in Figure 8. As in the case of a single atom for z = 0.2, the algorithm behaves nicely. We observe as before small confidence intervals and a convergence of the error to 0 at a speed of the order of h. The performance plot of Figure 9 shows similar performances for the three values of α, with a slight advantage for the largest value of α, which gives the better error for a given CPU time and an appropriate value of h.
The value a = 0.5Å also has a relatively good behavior, shown in Figure 10, although the convergence is not as good as for a = 0.2Å, similarly as for z = 1 in the case of a single atom. In particular, the convergence for small h is not as clear as for a = 0.2Å, but, again, for h between 0.003 and 0.1, the relative error of the method is smaller than 2%.
As in the case of a single atom, higher values of a lead to larger values of the variance and the error of the algorithm and the convergence fails.
Conclusion and perspectives
Our numerical experiments on the linear case show that the TAJ jump method can be used without significant increase in computational time, and with a slightly improved expected error. Therefore, it allows to take a larger value of h for a given error threshold, hence actually reducing the computational time. This is a new argument which, together with those developed in [23], allows to expect that optimized walk-on-spheres Monte-Carlo solvers for the linear Poisson-Boltzmann equation can be made competitive in terms of computational time with respect to classical deterministic methods.
Our preliminary tests to solve the nonlinear Poisson-Boltzmann PDE using branching particle systems show that our method has roughly equivalent performances than the walk-on-spheres solver in the linear case (with SNJ and ANJ methods). However, in some situations, typically when the electrostatic potential u is large on Γ, the variance of the method might explode. This issue requires to develop adequate variance reduction techniques, to be discussed in future work. We have already tested a stratification technique, which is quite efficient in reducing the variance of the method and make it converge in cases where the unstratified algorithm shows variance explosion. However, this method fails again for too large values of u on Γ.
First, a deeper theoretical analysis of the variance of the algorithm is needed. In particular, a study of the influence of the parameters and the boundary conditions (see Remark 4) on the variance can give insights on adequate variance reduction techniques or on appropriate values of the parameters λ, g(k), p k for the stratified Monte-Carlo method. The parameter values (3.5) proposed in Section 3.1.3 are one possible choice, but other possibilities might be considered and tuned in order to optimize the variance.
Additional variance reduction techniques have to be explored. For example, one could try to reduce the variance of the score within each stratum. To undertake this, we can analyze the probabilistic interpretation (3.6). Let us denote by X the r.v. inside the expectation in the r.h.s. of this equation. X might be 0 if one of the particles dies without children before leaving Ω out , or might be a product of values of u on Γ if all the particles hit Γ before dying. If u takes large values on Γ, this product might be very large, and hence the variance of X is very large. One could reduce this variance by reducing the probability that X = 0. This could be done using importance sampling techniques, for example by adding to the Brownian motion of each particle a drift towards the center of the molecule.
We can also study pruning techniques in the spirit of [5], where pruning of genealogical trees of branching particle systems is used to study the probabilistic interpretation of the Fourier transform of Navier-Stokes equation.
the solutions with APBS and WOS algorithm (b) Comparison with APBS. The blue curve is produced with our code, and the red curve with APBS.
Figure 1 :
1Comparison of the localization methods and adequation with a deterministic method.
a) Monte-Carlo average as a function of h (Log scale).
Error (Log scale) as a function of h (Log scale).
Figure 2 :
2Convergence and error of the linear with jump methods SNJ, ANJ (α = 3 and α = 10) and TAJ (α = 1, α = 3 and α = 10) for the single atom case.
a) Monte-Carlo average as a function of h (Log scale).
Error (Log scale) as a function of h (Log scale).
Figure 3 :
3Convergence and error of the linear with jump methods SNJ, ANJ (α = 3 and α = 10) and TAJ (α = 1, α = 3 and α = 10) for a molecule composed of 103 atoms.
Figure 3
3shows the approximation of u(x 0 ) by Monte-Carlo average, and the associated error relative to a reference value computed using ANJ (α = 3) algorithm, h = 0.001, = 10 −7 and (15 · 10 6 ) runs for the Monte Carlo average. The point x 0 is chosen out to the molecule, but close (at 1Å) to the Trp amino acid.
error (Log scale) as a function of CPU time (Log scale), on a signe atom.
error (Log scale) as a function of CPU time (Log scale), on a molecule of 103 atoms.
Figure 4 :
4Performance plots of the 6 linear with jump methods SNJ, ANJ and TAJ for a single atom (a), and a molecule composed of 103 atoms (b).
Subsection 3.1 is devoted to the description of the branching Brownian motion and its link with elliptic non-linear PDEs. Our algorithms are then described in Subsection 3.2, and numerical experiments are described in Subsection 3.3.
Using a spherical coordinates change of variables in spherical coordinates, the transmission problem (2.1), (2.3) and (3.1) can be written as v (x) + 2 x v (x) = κ 2 out sinh v(x) = 0, for all x ∈ (r, +∞), (3.10)
a) Monte-Carlo average of the 3 jump methods as a function of h (Log scale).
Error (Log scale) of the 3 jump methods w.r.t. the reference value given by (3.10) as a function of h (Log scale).
Figure 5 :
5Convergence and error of the branching algorithm with jump methods SNJ and ANJ (α = 3 and α = 10) in the case of a single atom with charge z = 0.2.
Figure 6 :
6Performance of the branching algorithm with 3 jump methods for a single atom with z = 0.2: error (Log scale) as a function of CPU time (Log scale).
a) Monte-Carlo average of the 3 jump methods as a function of h (Log scale).
Error (Log scale) of the 3 jump methods w.r.t. the reference value given by (3.10) as a function of h (Log scale).
Figure 7 :
7Convergence and error of the branching algorithm with jump methods SNJ and ANJ (α = 3 and α = 10) in the case of a single atom with charge z = 1.
a) Monte-Carlo average for the 3 jump methods as a function of h (Log scale).
Error (Log scale) of the 3 jump methods as a function of h (Log scale).
Figure 8 :
8Convergence and error of the branching algorithm with jump methods SNJ and ANJ (α = 3 and α = 10) in the case of two atoms with distance a = 0.2.
Figure 9 :
9Performance of the branching algorithm with 3 jump methods for two atoms with a = 0.2Å: error (Log scale) as a function of CPU time (Log scale).
a) Monte-Carlo average of the 3 jump methods as a function of h (Log scale).
Error (Log scale) of the 3 jump methods as a function of h (Log scale).
Figure 10 :
10Convergence and error of the branching algorithm with jump methods SNJ and ANJ (α = 3 and α = 10) in the case of two atoms with distance a = 0.5.
[ 3 ]
3N. A. Baker, D. Sept, M. J. Holst, and J. A. McCammon. The adaptive multilevel finite element solution of the Poisson-Boltzmann equation on massively parallel computers. IBM J. Res. & Dev., 45(3/4):427-437, 2001.
and n(x) is the normal vector to Γ at x ∈ Γ pointing towards Ω out .3)
where
∇ in u(x) :=
lim
y∈Ωin, y→x
∇u(y),
∇ out u(x) :=
lim
y∈Ωout, y→x
∇ϕ(y), ∀x ∈ Γ
2.1 A general probabilistic interpretation for (2.1)-(2.2) -(2.3)
2.3) and where the O(h 3 ) is bounded by h 3 times a constant depending only on sup z ∈Γ, z ≤R ∇ 3 u(z) . The TAJ jump method corresponds exactly to the probabilistic interpretation of this formula.Theorem 1 then follows from (2.13) exactly as Theorem 4.7 of [7] follows from Equation (4.19)
of [7].
symbol value Boltzmann constant k B 1, 3806488 × 10 −23 m 2 kg s −2 K −1Charge of an electron
e c
1, 602176565 × 10 −19 C
Temperature
T
298 K
Vacuum permittivity
ε 0
8, 854187817 × 10 −12 F m −1
Avogadro constant
N A
6, 02214129 × 10 23 mol −1
Molecule relative permittivity
ε in
2
(dimensionless)
Solvent relative permittivity
ε out
80
(dimensionless)
Solvent ion concentration
c
1 mol L −1
Solvent ion relative charge
z
±1
(dimensionless)
Inverse Debye length (1.4)κ
2.9132Å
−1
Table 1 :
1The physical constants used in the simulations in the the International System of Units.
These proportions of course depend on the biomolecule and on the starting point for the simulation of X; the values given here were obtained from a molecule with 103 atoms described in section 2.3.
Message Passing Interface (MPI) is a standardized and portable message-passing system available on a wide variety of parallel computers.
Non-linear caseThe method we present here extends the previous probabilistic interpretation to the non-linear Poisson-Boltzmann equation (1.1), by making use of the link between PDEs with non-linearity of order 0 (in u) and branching diffusions. Our algorithm is based on the simulation of a family of branching particles in R 3 . As in the linear case, our method involves walk on spheres algorithms inside and outside the molecule,
Acknowledgments.The authors thank Sélim Kraria (from DREAM, INRIA Sophia Antipolis Méditerranée) and Pierre Navarro (from IRMA, Université de Strasbourg) for their precious help regarding the software development aspect.This work was granted access to the HPC resources of Aix-Marseille Université financed by the project Equip@Meso (ANR-10-EQPX-29-01) of the program "Investissements d'Avenir" supervised by the Agence Nationale pour la Recherche.Appendix A Values of constantsThe following table gives the values of the constants involved in the Poisson-Boltzmann PDEs we consider in this work.
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| []
|
[
"Locating Charging Stations: Connected, Capacitated and Prize-Collecting",
"Locating Charging Stations: Connected, Capacitated and Prize-Collecting"
]
| [
"Rajni Dabas \nDepartment of Computer Science\nDepartment of Computer Science\nUniversity of Delhi\n[Delhi-110007]India\n",
"Neelima Gupta \nUniversity of Delhi\n[Delhi-110007]India\n"
]
| [
"Department of Computer Science\nDepartment of Computer Science\nUniversity of Delhi\n[Delhi-110007]India",
"University of Delhi\n[Delhi-110007]India"
]
| []
| In this paper, we study locating charging station problem as facility location problem and its variants (k-Median, k-Facility location and k-center). We study the connectivity and the capacity constraints in these problem.Capacity and connectivity constraints have been studied in the literature separately for all these problems. We give first constant factor approximations when both the constraints are present. Extending/modifying the techniques used for connected variants of the problem to include capacities or for capacitated variants of problem to include connectivity is a tedious and challenging task. In this paper, we combine the two constraints by reducing the problem to underlying well studied problems, solving them as black box and combine the obtained solutions. We also, combine the two constraints in the prize-collection set up.In the prize-collecting set up, the problems are not even studied when one of the constraint is present. We present constant factor approximation for them as well.2012 ACM Subject Classification CCS → Theory of computation → Design and analysis of algorithms → Approximation algorithms analysis → Facility location and clustering | null | [
"https://arxiv.org/pdf/2202.06262v1.pdf"
]
| 246,823,689 | 2202.06262 | 4223e1994dcbe344c4c021ac19b6618a95c55cae |
Locating Charging Stations: Connected, Capacitated and Prize-Collecting
Rajni Dabas
Department of Computer Science
Department of Computer Science
University of Delhi
[Delhi-110007]India
Neelima Gupta
University of Delhi
[Delhi-110007]India
Locating Charging Stations: Connected, Capacitated and Prize-Collecting
10.4230/LIPIcsFunding Rajni Dabas: Supported by a UGC-JRFand phrases Facility LocationConnected Facility LocationCapacitated Facility LocationPrize Collecting Facility LocationPenaltiesLower Bounds
In this paper, we study locating charging station problem as facility location problem and its variants (k-Median, k-Facility location and k-center). We study the connectivity and the capacity constraints in these problem.Capacity and connectivity constraints have been studied in the literature separately for all these problems. We give first constant factor approximations when both the constraints are present. Extending/modifying the techniques used for connected variants of the problem to include capacities or for capacitated variants of problem to include connectivity is a tedious and challenging task. In this paper, we combine the two constraints by reducing the problem to underlying well studied problems, solving them as black box and combine the obtained solutions. We also, combine the two constraints in the prize-collection set up.In the prize-collecting set up, the problems are not even studied when one of the constraint is present. We present constant factor approximation for them as well.2012 ACM Subject Classification CCS → Theory of computation → Design and analysis of algorithms → Approximation algorithms analysis → Facility location and clustering
Introduction
To address the increasing environmental health issues, countries around the globe are planning to phase out combustion engine vehicles. Automobile manufacturers, like Nissan Leaf, Tesla, Mahindra and Tata Motors, are switching to produce electric vehicles. One of the major challenge posed by this shift is to strategically identify the locations to set up the charging stations. Due to the short range of electric vehicles, the existing refueling station model is not sufficient. Governments across the world are trying to address the issue, but, high costs associated with equipment and installation of charging stations at public spaces are currently obstructing the build-out of such a network. So, what we need is a cost effective solution for locating the charging stations. The problem of locating charging stations can be formulated as a facility location problem. Facility Location Problem (FL) is a well known and well studied problem in operations research and computer science [53,18,37,9,30,40,17,48,50,14,6,58,43]. In (uncapacitated) FL, we are given a set of facilities and a set of clients. Every facility has an associated facility opening cost. Serving a client from a facility incurs service cost. We assume that the service cost is a metric. The goal is to open a subset of facilities so as to minimise the sum of facility opening costs (facility cost) and the total service cost of serving all the clients from the opened facilities. In case of charging stations, the facilities are charging stations and the consumers are clients. Another closely related problem is k-Median problem (kM). In The constraints, capacities and connectivity, have been studied separately for all the basic problems (FL, kM, kFL and kC). Extending/modifying the techniques used for connected variants of the problem to include capacities or for capacitated variants of problem to include connectivity is a tedious and challenging task. In this paper, we combine the two constraints using reduction to underlying well studied problems. Instead of extending and modifying the solutions to the underlying problems, we use them as black box and combine the obtained solutions. We also, combine the two constraints in the prize-collection set up. Figures 1 and 2 give a broad idea of our plan. Figure 1 depicts the reduction of Connected -Capacitated variant of the problems to two problems, each with only one constraint, either Connectivity or Capacities. Figure 2 depicts the reduction of Connected -Capacitated -Prize-Collecting variant of the problems to two problems: Connected -Prize Collecting and Capacitated -Prize Collecting variants of the problem. Figure 2 also depicts the reduction of Connected/Capacitated -Prize Collecting variant to Connected/Capacitated variant of FL. The last reduction does not work in presence of the cardinality constraint as depicted in (c) i.e., we are not able to reduce ConPkFL/ConPkM to ConkFL/ConkM. However, we are able to solve ConCPkFL/ConCPkM by reducing them to ConPFL instead, as shown in Figure 3. In particular, we present the following results: Theorem 1. Given an α-factor approximation for for ConFL and a β-factor approximation for CFL/CkM/CkFL with γ-factor violation in capacity/cardinality, a (α + 2β)-factor approximation for ConCFL/ConCkM/ConCkFL preserving the violations in capacities/cardinality can be obtained in polynomial time.
Theorem 2.
Given an α-factor approximation for CkC and a β-factor approximation for ConkC, a (2α + β)-factor approximation for ConCkC can be obtained in polynomial time.
Theorem 3.
Given an α-factor approximation for ConPFL and a β-factor approximation for CPFL/CPkM/CPkFL with γ-factor violation in capacities, a (α + 2β)-approximation for ConCPFL/ConCPkM/ConCPkFL preserving the violations in capacities/cardinality can be obtained in polynomial time.
Theorem 4.
Given an α-factor approximation for ConFL (using LP optimal to lower bound the cost of the optimal), a 2α-factor approximation for ConPFL can be obtained in polynomial time.
Theorem 5.
Given an β-factor approximation for CFL, an β-factor approximation for CPFL can be obtained in polynomial time.
To the best of our knowledge, our results are the first approximations for ConCkFL, ConCkM, ConCkC, ConCPFL, ConCPkM, ConCPkFL and ConPFL. The only known result for ConCFL in the literature is by Friggstad et al. [25]. They gave a constant factor approximation for lower and (uniform) upper bounded ConFL violating both the upper and the lower bounds. We cannot get rid of the violations in the upper bounds even when there are no lower bounds using their technique as they use LP rounding and the LP is known to have an unbounded integrality gap. We give the first true constant factor approximation for the problem. The only result known for CPFL is by Gupta and Gupta [33] using local search technique. They give (5.83 + ) factor for the case of uniform capacities and (8.532 + ) factor for non-uniform capacities. Our result is interesting as it is much simpler and uses the underlying problem CFL as a black box without increasing the cost. We also improve the factor to 5. The factor will improve, if the factor for the underlying problem (CFL) improves in future.
Tables 1 and 2 summarise the results we obtain by plugging in the best results known for the underlying problems. Approximation guarantees will improve if better solutions are obtained for the underlying problems.
Problem
Sub
O(1/ 2 ) (2 + )u 21.32 O(1/ 2 ) (2 + )u (U)ConCPkFL ConPFL [This Paper] (U)CPkFL [22] O(1/ 2 ) (2 + )u 21.32 O(1/ 2 ) (2 + )u ConPFL ConFL [31] - 21.32 Nil 10.66 - (U/NU)CPFL - (NU)CFL [7] 5
Nil -5 Nil Table 2 Results corresponding to Theorems 3, 4 and 5 based on current best approximation for the underlying problem. (U-Uniform, NU-Non-Uniform)
Our Techniques
The Combining Technique: In ConCX/ConCPX problems (where X is FL, kM, kFL or kC as applicable), we reduce our problem to two sub-problems in a way that the connectivity constraint move to one sub problem (ConFL/ConPFL) and the capacity constraints to another (CX/CPX). We use the openings and the assignments of the solution of CPX and connect them using the connectivity of the solution of ConFL via clients. ConPFL: We reduce an instance of ConPFL to an instance of ConFL using LP rounding and thresholding techniques. A client paying penalty to an extent of at least half in LP optimal is removed by paying full penalty. Assignment of the remaining clients is raised so that they are served to full extent. The openings are raised accordingly. Note that the reduction does not work in presence of cardinality constraints as increasing the opening of facilities can violate the cardinality constraint by a factor of 2. CPFL: We reduce an instance of CPFL to an instance of CFL by creating a dummy facility, with capacity 1, collocated with every client. The facility opening cost of a dummy facility is equal to the penalty cost of the respective collocated client.
Previous and Related Work
Capaciated variants of the problem (CFL, CkM, CkFL and CkC): Shmoys et al. [53] gave the first constant factor(7) approximation when the capacities are uniform, with a capacity blow-up of 7/2, by rounding the solution to standard LP. Grover et al. [28] reduced the capacity violation to (1 + ) thereby showing that the capacity violation can be reduced to arbitrarily close to 0 by rounding a solution to standard LP. An et al. [5] gave the first true constant factor(288) approximation for the problem (non-uniform), by strengthening the standard LP. Local search has been successful in dealing with capacities, several results [40,19,3,51,49,57] have been successfully obtained using local search with the current best being 5-factor for non-uniform [7] and 3-factor for uniform capacities [2]. No true approximation is known for CkM, till date. Constant factor approximations [13,15,41,20,10,1,28] are known, that violate capacities or cardinality by a factor of 2 or more. Strengthened LPs and dependent rounding techniques have been used to give constant factor approximation with small violations in capacities/cardinality. For uniform capacities Byrka et al. [12] and for non-uniform capacities Demirci et al. [23] gave the approximations with (1 + ) violation in capacities whereas Li [44,45] gave the approximations using at most (1 + )k facilities for uniform as well as non-uniform capacities. For uniform CkFL, a constant factor approximation, with (2 + ) violation in capacities, was given by Byrka et al. [10] using dependent rounding. This was followed by two constant factor approximations by Grover et al. [28], one with (1 + ) violation in capacities and 2-factor violation in cardinality and the other with (2 + ) factor violation in capacities only. For (uniform) CkC, a 10-factor approximation was given by Bar-Ilan et al. [8] which was improved to 6 by Khuller and Sussman [39]. For non-uniform capacities, Cygan et al. [21] gave a large constant factor approximation, improved to 9 by An et al. [4] which is also also the current best.
Connected Variants of the problem (ConFL, ConkM, ConkFL and ConkC):
ConFL was first introduced by Gupta et al. [31] where they gave a 10.66-factor approximation for the problem using LP-rounding technique. Gupta et al. [32] described a random facility sampling algorithm for the problem giving 9.01-factor approximation. Primal and dual technique was first used by Swamy and Kumar [54] to improve the factor to 8.55 which was then improved to 6.55 by Jung et al. [38]. Eisenbrand et al. [24] used random sampling to improve the factor to 4. The factor was improved to 3.19 using similar techniques by Grandoni and Rothvob [27] which is also the current best for the problem. Eisenbrand et al. [24] extends their algorithm to ConkFL giving 6.98-factor approximation for ConkM and ConkFL . For ConkC, Ge et al. [26] gave a 3-factor approximation when k is fixed and a 6-factor approximation for arbitrary k. Liang et al. [47] also gave a simpler 6-factor approximation for the problem. Prize-Collecting FL (PFL): For PFL, a 3-factor approximation using primal dual techniques was given by Charikar et al. [16] which was subsequently improved to 2 by Jain et al. [35] using dual-fitting and greedy approach. Wang et al. [55] also gave a 2-factor approximation using a combination of primal-dual and greedy technique. Later Xu and Xu [56] gave a 2 + 2/e using LP rounding. The factor was improved to 1.8526 by the same authors in [29] using a combination of primal-dual schema and local search. For linear penalties Li et al. [46] gave a 1.5148-factor using LP-rounding. For CPFL , Dabas and Gupta [22] gave an O(1/ ) approximation with (1+ ) factor violation in capacities using LP rounding techniques and reduction to CFL. The only true approximation is due to Gupta and Gupta [33]. They uses local search to obtain a 5.83-factor and 8.532factor approximation for uniform and non-uniform variants respectively. For CPkFL , Dabas and Gupta [22] gave an O(1/ 2 )-approximation algorithm with (2 + )-factor violation in capacities.
ConCkFL, ConCkM, ConCkC, ConCPFL, ConCPkM, ConCPkFL and ConPFL:
To the best of our knowledge, no result is known for these problems in the literature.
Organisation of the Paper
In Section 2, we present our combining technique to obtain constant factor approximations for ConCFL , ConCkM , ConCkFL and ConCkC. In Section 3, we present our results for ConCPFL , ConCPkM and ConCPkFL. In Sections 4 and 5, we give the results for ConPFL and CPFL respectively. Finally, we conclude with future work in Section 6.
Connected and Capacitated variants of the problem
Let V be a set of locations and G be a complete graph on V . Let c : V × V → R + be a metric cost function. Let F ⊆ V and C ⊆ V be the set of facilities and clients respectively. Each facility i ∈ F has an associated opening cost f i and serving a client j from facility i incurs a cost c(i, j). In facility location problem (FL), we wish to open F ⊆ F of facilities and compute an assignment function φ : C → F . Our goal is to minimize the total cost of opening the facilities in F and serving the clients in C.
ConFL:
In this variant, we wish that the set F of the opened facilities is connected by a steiner tree. We call the total cost of the steiner tree, that is the sum of the costs on edges of the steiner tree, as connection cost. Our goal now is to minimise the total cost of opening the facilities, serving the clients and the connection cost. CFL: In this variant, a facility i has a bound u i on the maximum number of clients it can serve i.e., |φ −1 (i)| ≤ u i for all i ∈ F .
In this section, we present a constant factor approximation for ConCFL which is a common generalisation of ConFL and CFL, that is, we have both connectivity and capacity constraints in facility location problem. The result is obtained by combining the results of ConFL and CFL. Let I concf l be an instance of ConCFL. We first make two instances I conf l and I cf l ConFL and CFL by dropping the capacity and the connectivity constraints respectively.
Note that, an optimal solution O concf l of ConCFL forms a feasible solution for instances I conf l and I cf l . Hence the cost of the optimal solutions O conf l and O cf l of I conf l and I cf l respectively are bounded. Next we solve these instance using approximation algorithms for ConFL and CFL to obtain two approximate solutions AS conf l and AS cf l respectively. Finally, we combine AS conf l and AS cf l (presented in subsection 2.1) to obtain our final approximate solution AS concf l . For a solution S to an instance I, let Cost I (S) denote the cost of S, we will drop I wherever it will clear from the context for brevity of notations. Refer Algorithm 1 for details of the algorithm.
Algorithm 1 Algorithm for ConCFL
Input : I concf l (F, C, f, d, c, u) 1 Create an instance I conf l (F, C, f, d, c) of ConFL using I concf l , by dropping the capacities on facilities. Since any solution to instance I concf l is feasible for I conf l as well, we have Cost(O conf l ) ≤ Cost(O concf l ). 2 Create an instance I cf l (F, C, f, c, u) of CFL using instance I concf l (F, C, f, d, c, u) of CFL , by dropping the connection cost and the connectivity constraint. Since any solution to I concf l is feasible for I cf l as well, we have Cost(O cf l ) ≤ Cost(O concf l ). 3 Obtain a α-approximate solution AS conf l to instance I conf l using approximation algorithm for ConFL given by Grandoni and Rothvob [27]. 4 Obtain a β-approximate solution AS cf l to instance I cf l using approximation algorithm for CFL given by Bansal et al. [7]. 5 Combine solutions AS conf l and AS cf l to obtain a solution AS concf l = (F , φ) to I concf l such that Cost(AS concf l ) ≤ αCost(AS conf l ) + 2βCost(AS cf l ).
Combining AS conf l and AS cf l
In this section, we combine AS conf l and AS cf l to obtain an approximate solution AS concf l for ConCFL. Refer Figure 4. For every facility i opened in AS cf l , we open i in AS concf l and assign j to i if it was assigned to i in AS cf l . The cost of opening i and assigning j to i is bounded by cost of AS cf l . The capacities are respected because they are respected in AS cf l .We connect the facilities opened in AS cf l to facilities opened in AS conf l which are already connected via a steiner tree. We bound this cost (additonal)as follows: let i be a facility in F cf l . Let j be a client served by i in AS cf l and by i in AS conf l (wlog, we assume that such a client exists for otherwise i can be closed). The cost of connecting i to i is bounded by c(i, i ) ≤ c(i, j) + c(j, i ) (by triangle inequality (refer Figure 5)). Summing over all i ∈ F cf l , the cost is bounded by S conf l + S cf l where S conf l and S cf l denote the service cost of AS conf l and AS cf l respectively.
The overall cost is bounded by Cost(AS conf l ) + 2Cost(AS cf l ) which is ≤ αCost(O conf l ) + 2βCost(O cf l ). Using Rothvob and Grandoni [27] for AS conf l we have α = 3.19. For non-uniform capacities, using Bansal et al. [7] for AS cf l , we have β = 5, which gives us a 13.19-factor approximation. For uniform capacities, we get 9.19 factor using 3-factor approximation of Aggarwal et al. [2] for AS cf l .
Connected, Capacitated and Prize-Collecting varaints of the problem
Prize-collecting facility location problem is a generalisation of FL where every client j has an associated penalty cost p j . We wish to open F ⊆ F of facilities, select C p ⊆ C of clients to pay penalty and compute an assignment function φ : (C \ C p ) → F for the remaining clients.
Our goal is to minimize the total cost of opening the facilities in F , paying penalty of clients in C p and serving clients in C \ C p .
In this section, we present a constant factor approximation for ConCPFL which is a common generalisation of ConPFL and CPFL, that is, we have both the connectivity as well as the capacity constraints in prize-collecting facility location problem. The result is obtained by combining the solutions of ConPFL and CPFL (obtained in Section 4 and 5 respectively). The idea is similar to the one presented in Section 2. Let I concpf l be an instance of ConCPFL. We first make two instances I conpf l and I cpf l of ConPFL and CPFL by dropping the capacity and the connectivity constraints respectively. Note that, an optimal solution O concpf l of ConCPFL forms feasible solution for instances I conpf l and I cpf l . Hence the cost of the optimal solutions O conpf l and O cpf l of I conpf l and I cpf l respectively are bounded. Next we solve these instance using approximation algorithms for ConPFL and CPFL to obtain two approximate solutions AS conpf l and AS cpf l respectively. Finally, we combine AS conpf l and AS cpf l to obtain our final approximate solution AS concpf l . Refer Figure 6. As in Section 2 we would like to open all the facilities that were opened in AS cpf l , connect them using the solution to AS conpf l and serve the clients that were served in AS cpf l paying penalty for the rest. However, we do not know how to bound the cost of connecting the facilities in AS cpf l to the facilities opened in AS conpf l in this case: some clients served in AS cpf l may be paying penalty in AS conpf l . As a result, it is possible that all the clients served by a facility opened in AS cpf l are paying penalty in AS conpf l ; we do not know how to bound the cost of connecting such a facility to the facilities in AS concpf l (for example, facility i in Figure 6). Thus, we decide to pay penalty for clients that were paying penalty in any one of the solutions. The penalty costs of these clients are paid by AS cpf l and AS conpf l . Next, we look at the remaining clients, open the facilities each of which is serving at least one such client in AS cpf l , assign clients and connect the opened facilities to the facilities in AS conpf l in the same manner as in Section 2. Capacities are respected as they are respected in the underlying problem. The bound on facility cost, service cost and connection cost is obtained in a similar manner as done in section 2.
Using 21.32-factor approximation for ConPFL and 5-factor approximation for CPFL (presented in sections 4 and 5 resp.), we get a 31.32-factor approximation for ConCPFL. The result holds for both uniform and non-uniform capacities.
Introducing cardinality: The same algorithm extends to ConCPkM/ConCPkFL if we take an instance of ConPFL on one side and CPkM/CPkFL on the other side. The violation in capacities/cardinality is preserved from underlying problem. Using 21.32-factor approximation (presented in section 4) for ConPFL and O(1/ ) with (2 + ) violation in capacities by Dabas et al. [22] for CPkM/CPkFL, we get a constant factor approximations for (uniform) ConCPkM/ConCPkFL with (2 + ) factor violation in capacities. Our algorithm will extend to ConCPkC if, in future, we have solutions for the underlying problems viz ConPkC and CPkC.
Solving ConPFL via reduction to ConFL
In this section, we present a constant factor approximation for ConPFL using LP rounding techniques and reduction to ConFL. Let us guess a vertex v which is opened as a facility in the optimal solution. We can run our algorithm for all possible choices of v and choose the best among them. For a non-trivial set S ⊆ F, let δ(S) represents the cut defined by S and F \ S. ConPFL can be formulated as the following integer program:
Constraints 8 holds trivially if
w i = 1 as x ij ≤ 1. Otherwise, for x * ij > 0, w i ≥ (w * i /x * ij )x ij ≥ x ij as w * i /x * ij ≥ 1 4.
For y e = 1, constraints 9 holds because i∈S x ij ≤ 1. Otherwise,
e∈δ(S) y e = 2 e∈δ(S) y * 2 ≥ 2 i∈S x * ij ≥ i∈S x * ij i∈F x * ij = i∈S x ij
where the last inequality follows because i∈F x * ij ≥ 1/2 for all j ∈ C \ C .
Cost Bound: Next we will bound the cost of our feasible solution ρ . Note that, (i)
x ij = x * ij / i∈F x * ij ≤ 2x * ij , (ii) w i ≤ 2w * i and (iii) y e ≤ 2y * e . Therefore, Cost(w , x , y ) = i∈F w i y i + i∈F j∈C\C c(i, j)x ij + e∈F ×F d e y e ≤ 2( i∈F w * i y i + i∈F j∈C\C c(i, j)x * ij + e∈F ×F d e y * e ).
Next, we solve the instance I conf l using any algorithm that uses LP optimal as a lower bound on the optimal cost. In particular, we use approximation algorithm by Gupta et al. [31] as black-box to obtain an approximate solution AS conf l . The solution AS conf l along with clients in C p forms an approximate solution AS conpf l for ConPFL. The final cost bound is as follows:
Cost(AS conpf l ) = Cost(AS conf l ) + j∈Cp p j z j ≤ βCost(O conf l ) + 2 j∈Cp p j z * j ≤ 2β( i∈F w * i y i + i∈F j∈C\Cp c(i, j)x * ij + e∈F ×F d e y * e ) + 2 j∈Cp p j z * j = 2βLP opt .
Using Gupta et al. [31], we have α = 10.66 which gives us 21.32-factor approximation.
Solving CPFL via reduction to CFL
In this section, we present a 5-factor approximation for CPFL by reducing it to CFL. To create an instance I cf l of CFL from an instance I cpf l of CPFL, we create a set of dummy facilities D as follows: for every client we create a collocated dummy facility having facility opening cost equal to the penalty cost of the client and 1 unit of capacity. That is, ∀j ∈ C, add a facility i j to D such that c(j, i j ) = 0, f ij = p j and u ij = 1. Open a facility i if it is opened in O cf l and assign client j to it if it was assigned to i in O cf l where O cf l is an optimal solution for I cf l . For a client j for which O cf l pays the penalty, we open the facility i j collocated with j and assign j to it. Clearly the cost of the solution is bounded by the cost of the optimal (O cpf l ). We obtain an approximate solution AS cf l to I cf l using a β-approximation to CFL. Next, we create an approximate solution for I cpf l from AS cf l : open a true facility i if it is opened in AS cf l and assign client j to it if it was assigned to i in AS cf l ; if a dummy facility i j collocated with a client j is opened in AS cf l we pay penalty for j in our solution. Since u ij = 1 wlog we may assume that if i j is opened in AS cf l then j is assigned to i j (for if this is not the case and j is assigned to a facility i and j is assigned to i j , we can obtain another solution by assigning j to i j and j to i without increasing the cost). Clearly, the cost of the solution so obtained is bounded by the cost of AS cf l . Using result of Bansal et al. [7] to obtain AS cf l , we have a 5-factor approximation for the problem.
Conclusion and Future Work
In this paper, we presented constant factor approximations for connected, capacitated and prize-collecting facility location problem and variants. The approximations were obtained using the solutions of the underlying problems as black-box. We successfully reduced ConPFL and CPFL to ConFL and CFL respectively. Obtaining similar reductions when we have cardinality constraints is an interesting open problem.
Though we feel that reducing ConPkC and CPkC to ConkC and CkC respectively would be challenging, the techniques of ConkC [47] and CkC [4] should be extendable (modifiable) to accommodate penalties. Then our technique in section 3 can be used to give an approximation for ConCPkC as depicted in Figure 2.
This paper used reductions and combining techniques to introduce notoriously hard capacity constraints to connected-and prize-collecting variants of some important classical problems. It will interesting to see similar results for other constraints, for example, the outlier constraint where you are allowed to leave some specified number of clients unserved. Note that our technique for prize-collecting variants gives similar results for outlier version with 2-factor violation in outliers; the challenge would be to get rid of this violation.
Figure 1
1Combining connectivity and capacities. Using ConFL instead of ConkM and ConkFL as black-box in (b) and (c) will also work.
Figure 2
2Combining connectivity, capacities and penalties. Connected Capacitated k-Center (ConCkC) Connected Prize Collecting Facility Location (ConPFL) Capacitated Prize Collecting Facility Location (CPFL) Connected Capacitated Prize Collecting Facility Location (ConCPFL) Connected Capacitated Prize Collecting k-Median (ConCPkM) Connected Capacitated Prize Collecting k-Facility Location (ConCPkFL)
Figure 3
3Combining connectivity, capacities and penalties. Using ConPFL instead of ConPkM and ConPkFL as black-box.
Figure 2
2Figure 2 (b) and (c) i.e., we are not able to reduce ConPkFL/ConPkM to ConkFL/ConkM. However, we are able to solve ConCPkFL/ConCPkM by reducing them to ConPFL instead, as shown in Figure 3. In particular, we present the following results:
Connected and Capacitated FL (ConCFL): A constant factor approximation for (uniform) ConCFL with violation in capacities follows as a special case from approximation algorithm of Friggstad et al. [25] on Connected Lower and Upper Bounded Facility Location problem. Capacitated and Prize-Collecting variants of the problem (CPFL and CPkFL):
Figure 4
4(a) Represents solution AS conf l and AS cf l with respective opened facilities(circles) and assignment of clients(squares). (b) Construction of Solution AS concf l . The filled circles represent the opened facilities, the thick lines represent the assignment of clients and the dashed lines represents the connection of opened facilities to the steiner tree.
Figure 5 c
5(i, i ) ≤ c(i, j) + c(j, i )
Figure 6
6(a) Represents solution AS conpf l and AS cpf l with respective opened facilities(circles) and assignment of clients(squares). The clients that pay penalty in one of the solutions are represented by light grey dashed boundary squares. (b) Construction of Solution AS concpf l . The filled circles represent the opened facilities, the thick lines represent the assignment of clients and the dashed lines represents the connection of opened facilities to the steiner tree.
Results corresponding to Theorems 1 and 2 based on current best approximation for the underlying problem. (U-Uniform, NU-Non-Uniform)-Problem 1
Sub-Problem 2
Factor
Violation
Factor
Factor
Violation
(U)ConCFL
ConFL [27]
(U)CFL [2]
9.19
Nil
3.19
3
Nil
(NU)ConCFL
ConFL [27]
(NU)CFL [7]
13.19
Nil
3.19
5
Nil
(U)ConCkM
ConFL [27]
(U)CkM [12]
O(1/ )
(1 + )u
3.19
O(1/ )
(1 + )u
(NU)ConCkM
ConFL [27]
(NU)CkM [23]
O(1/ )
(1 + )ui
3.19
O(1/ )
(1 + )u
(U)ConCkM
ConFL [27]
(NU)CkM [44]
O(1/ )
(1 + )k
3.19
O(1/ )
(1 + )k
(NU)ConCkM
ConFL [27]
(NU)CkM [45]
O(1/ )
(1 + )k
3.19
O(1/ )
(1 + )k
(U)ConCkFL
ConFL [27]
(U)CkFL [10]
O(1/ 2 )
(2 + )u
3.19
O(1/ 2 )
(2 + )u
(U)ConCkC
ConkC [26]
(U)CkC [39]
18
Nil
6
6
Nil
(NU)ConCkC
ConkC [26]
(NU)CkC [4]
24
Nil
6
9
Nil
Table 1 Problem
Sub-Problem 1
Sub-Problem 2
Factor
Violation
Factor
Factor
Violation
(U/NU)ConCPFL ConPFL [This Paper] (U/NU)CPFL [This Paper]
31.32
Nil
21.32
5
Nil
(U)ConCPkM
ConPFL [This Paper]
(U)CPkM [22]
Introducing cardinality: The same algorithm extends to ConCkM/ConCkFL/ConCkC if we take an instance of connected kM/kFL/kC on one side and capacitated kM/kFL/kC on the other side. Note that the violations in capacities/cardinality are preserved from the underlying solutions. Plugging in the current best results of ConkM/ConkFL/ConkC and CkM/CkFL/CkC, we get the results stated inTable 1.
Minimise Cost(w, x, y, z) = i∈F w i y i + i∈F j∈C c(i, j)x ij + e∈F ×F d e y e + j∈C p j z j subject to i∈F x ij + z j = 1 ∀j ∈ C (1)x ij ≤ w i ∀i ∈ F, ∀j ∈ C (2)where variable w i denotes whether facility i is open or not, x ij indicates if client j is served by facility i or not, y e denoted whether edge e belongs to steiner tree or not and z j denotes if client j pay penalty or not. Constraints 1 ensure that every client is either served or pays penalty. Constraints 2 ensure that a client is assigned to an open facility only. Constraint 3 ensures that the guessed facility v is opened. Constraints 4 ensure that if a facility is opened then it is connected to v. LP-Relaxation of the problem is obtained by allowing the variables w i , x ij , y e , z j ∈ [0, 1]. Let ρ * =< w * , x * , y * , z * > denote the optimal solution of the LP and LP opt denote the cost of ρ * .Identifying the Clients that will Pay Penalty (C )We first identify a set C p of clients that pay penalty in our solution AS conpf l via thresholding technique. For every client j, if j was paying penalty to an extent of at least 1/2, we add j to C p . Formally, ∀j such that z * j ≥ 1/2, add j to C p , set z j = 1 and x ij = 0 for every i. Note that, this can be done within 2 factor of penalty cost, that is, j∈Cp p j z j ≤ 2 j∈Cp p j z * j .An instance I conf l of ConFL :To handle the remaining clients, we create an instance I conf l of ConFL wherein the client set is reduced with C \ C p . An LP for I conf l (LP2) can be formulated as follows: Minimise Cost(w, x, y) = i∈F w i y i + i∈F j∈C\C c(i, j)x ij + e∈F ×F d e y e subject to i∈F x ij = 1 ∀j ∈ C \ C (6)x ij ≤ w i ∀i ∈ F, ∀j ∈ C \ C (7)The following lemma shows existence of a feasible fractional solution to I conf l such that the cost is bounded within a constant factor of LP optimal (LP opt ).Lemma 6. There exist a fractional feasible solution ρ =< w , x , y > such that cost is bounded byProof. Recall that, every client j ∈ C \ C is served to an extent of at least 1/2. For all i ∈ F : x * ij > 0, we raise the assignment of j on i proportionately so that j is fully served. Variables w and y are raised accordingly to satisfy constraints (7) and constraints(9)respectively. Formally, ∀j ∈ C \ C , ∀i ∈ F and ∀e ∈ F × F, set (i)and (iii) y e = max{1, 2y * e }. Next, we will see that < w , x , y > is a feasible solution to LP of ConFL. 1. Constraints 6 are satisfied because for all j ∈ C \C , i∈F x ij = i∈F (x * ij / i∈F x * ij ) = 1.Constraint 7 is satisfied because
Approximation algorithms for hard capacitated k-facility location problems. Karen Aardal, Pieter L Van Den, Dion Berg, Shanfei Gijswijt, Li, EJOR. 2422Karen Aardal, Pieter L. van den Berg, Dion Gijswijt, and Shanfei Li. Approximation algorithms for hard capacitated k-facility location problems. EJOR, 242(2):358-368, 2015.
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| []
|
[
"Chern-Simons Gravity in Four Dimensions",
"Chern-Simons Gravity in Four Dimensions"
]
| [
"Ivan Morales \nDepartamento de Física\nUniversidade Federal de Viçosa (UFV) Viçosa\nMGBrazil\n",
"Bruno Neves \nDepartamento de Física\nUniversidade Federal de Viçosa (UFV) Viçosa\nMGBrazil\n",
"Zui Oporto \nDepartamento de Física\nUniversidade Federal de Viçosa (UFV) Viçosa\nMGBrazil\n",
"Olivier Piguet [email protected] \nDepartamento de Física\nUniversidade Federal de Viçosa (UFV) Viçosa\nMGBrazil\n"
]
| [
"Departamento de Física\nUniversidade Federal de Viçosa (UFV) Viçosa\nMGBrazil",
"Departamento de Física\nUniversidade Federal de Viçosa (UFV) Viçosa\nMGBrazil",
"Departamento de Física\nUniversidade Federal de Viçosa (UFV) Viçosa\nMGBrazil",
"Departamento de Física\nUniversidade Federal de Viçosa (UFV) Viçosa\nMGBrazil"
]
| []
| Five dimensional Chern-Simons theory with (anti-)de Sitter SO(1,5) or SO(2,4) gauge invariance presents an alternative to General Relativity with cosmological constant. We consider the zero-modes of its Kaluza-Klein compactification to four dimensions. Solutions with vanishing torsion are obtained in the cases of a spherically symmetric 3-space and of a homogeneous and isotropic 3-space, which reproduce the Schwarzshild-de Sitter and ΛCDM cosmological solutions of General Relativity. We also check that vanishing torsion is a stable feature of the solutions.1 valid in the quantum microscopic realm, and the centenary General Relativity (GR) valid in the classical macroscopic realm, from GPS monitoring in the planetary scale up to the cosmic scale, the evolution of the Universe from the Big Bang up to an unforeseeable future. No observation neither experiment have shown any falsification of both theories, up to now 1 .An important problem, however, is theoretical: the contradiction of GR being classical and SM being quantum. Two cures may be conceived. The more radical one may be the construction of a new framework, beyond "quantum" and "classical", in which GR and SM would stay as approximations of a unique theory, each being valid in its respective domain. String theory represents an effort in this direction.Another, more obvious and (only apparently) straightforward cure is the direct quantization of GR, along the canonical lines of Loop Quantum Gravity [2], for instance. The latter is based on a first order formulation of GR, which has two local symmetries: the invariances under the space-time diffeomorphisms and the local Lorentz transformations. In Dirac's canonical formalism [3], a constraint is associated to each local invariance, which has to be solved at the quantum level. Unfortunately, one of these constraints, namely the one associated with the time diffeormophism invariance -called the hamiltonian or scalar constraint -has resisted to any tentative of solving it, up to now -although important progresses have been made[4,5,6].It happens that the de Sitter or anti-de Sitter ((A)dS) gravitation theory in 5D space-time defined by a Chern-Simons theory with the gauge groups SO(n,6 − n) for n = 1 or 2 [7] shows the remarkable property of its time-diffeomorphism constraint being a consequence of its gauge invariance and its invariance under the space diffeomorphisms[8]. It follows that the scalar constraint is then an automatic consequence of the other ones. This yields a first motivation for studying this particular theory of gravity.A second motivation is given by the fact that the presence of a cosmological constant, hence of a fundamental scale at the classical level, happens as a necessary feature of this theory, as we shall verify, in contrast with usual GR where its presence or not is the result of an arbitrary choice.The Chern-Simons (A)dS theory is a special case of the extensions of Einstein theory known as Lovelock theories [9] which, despite of containing higher powers of the curvature, obey second order field equations. There exists a vast literature 2 on Lovelock theories, beginning with the historical papers[10,11,12]. The paper[13]already gives explicit solutions of the Schwarzschild, Reissner-Nordström and Kerr 1 See however [1] for an experimental result hinting to a possible problem with the Standard Model.2 Only a few references are given here. A rather complete list may be found in the book [7], which offers an up-to-date review on Lovelock and Chern-Simons theories of gravitation. | 10.1140/epjc/s10052-017-4653-8 | [
"https://arxiv.org/pdf/1701.03642v1.pdf"
]
| 119,234,726 | 1701.03642 | 1264d65301734fa65a8fe47ed9a9fcafca710e2e |
Chern-Simons Gravity in Four Dimensions
13 Jan 2017 January 2017
Ivan Morales
Departamento de Física
Universidade Federal de Viçosa (UFV) Viçosa
MGBrazil
Bruno Neves
Departamento de Física
Universidade Federal de Viçosa (UFV) Viçosa
MGBrazil
Zui Oporto
Departamento de Física
Universidade Federal de Viçosa (UFV) Viçosa
MGBrazil
Olivier Piguet [email protected]
Departamento de Física
Universidade Federal de Viçosa (UFV) Viçosa
MGBrazil
Chern-Simons Gravity in Four Dimensions
13 Jan 2017 January 2017PACS numbers: 04.20.Cv, 04.50.CdTopological gravityGeneral RelativityCosmologyHigher dimen- sions
Five dimensional Chern-Simons theory with (anti-)de Sitter SO(1,5) or SO(2,4) gauge invariance presents an alternative to General Relativity with cosmological constant. We consider the zero-modes of its Kaluza-Klein compactification to four dimensions. Solutions with vanishing torsion are obtained in the cases of a spherically symmetric 3-space and of a homogeneous and isotropic 3-space, which reproduce the Schwarzshild-de Sitter and ΛCDM cosmological solutions of General Relativity. We also check that vanishing torsion is a stable feature of the solutions.1 valid in the quantum microscopic realm, and the centenary General Relativity (GR) valid in the classical macroscopic realm, from GPS monitoring in the planetary scale up to the cosmic scale, the evolution of the Universe from the Big Bang up to an unforeseeable future. No observation neither experiment have shown any falsification of both theories, up to now 1 .An important problem, however, is theoretical: the contradiction of GR being classical and SM being quantum. Two cures may be conceived. The more radical one may be the construction of a new framework, beyond "quantum" and "classical", in which GR and SM would stay as approximations of a unique theory, each being valid in its respective domain. String theory represents an effort in this direction.Another, more obvious and (only apparently) straightforward cure is the direct quantization of GR, along the canonical lines of Loop Quantum Gravity [2], for instance. The latter is based on a first order formulation of GR, which has two local symmetries: the invariances under the space-time diffeomorphisms and the local Lorentz transformations. In Dirac's canonical formalism [3], a constraint is associated to each local invariance, which has to be solved at the quantum level. Unfortunately, one of these constraints, namely the one associated with the time diffeormophism invariance -called the hamiltonian or scalar constraint -has resisted to any tentative of solving it, up to now -although important progresses have been made[4,5,6].It happens that the de Sitter or anti-de Sitter ((A)dS) gravitation theory in 5D space-time defined by a Chern-Simons theory with the gauge groups SO(n,6 − n) for n = 1 or 2 [7] shows the remarkable property of its time-diffeomorphism constraint being a consequence of its gauge invariance and its invariance under the space diffeomorphisms[8]. It follows that the scalar constraint is then an automatic consequence of the other ones. This yields a first motivation for studying this particular theory of gravity.A second motivation is given by the fact that the presence of a cosmological constant, hence of a fundamental scale at the classical level, happens as a necessary feature of this theory, as we shall verify, in contrast with usual GR where its presence or not is the result of an arbitrary choice.The Chern-Simons (A)dS theory is a special case of the extensions of Einstein theory known as Lovelock theories [9] which, despite of containing higher powers of the curvature, obey second order field equations. There exists a vast literature 2 on Lovelock theories, beginning with the historical papers[10,11,12]. The paper[13]already gives explicit solutions of the Schwarzschild, Reissner-Nordström and Kerr 1 See however [1] for an experimental result hinting to a possible problem with the Standard Model.2 Only a few references are given here. A rather complete list may be found in the book [7], which offers an up-to-date review on Lovelock and Chern-Simons theories of gravitation.
Introduction
Our present understanding of the fundamental processes in Nature is dominated by two extremely efficient theories: the already half a century old Standard Model (SM) type in higher dimension Einstein theory with cosmological constant. More recent works may be divided into general Lovelock models [14,15,16,17,18], Chern-Simons model based on (A)dS gauge invariance [19], and Chern-Simons models based on larger gauge groups, see in particular [20]- [23]. It is worth noticing the work of [22], where the choice of the gauge group extension leads to a theory which reduces to 5D Einstein theory with cosmological constant in the case of a vanishing torsion. We may also mention genuinely 4-dimensional models together with the search for physically reliable solutions of them, such as the Chamseddine model [24,25] obtained from the 5D (A)dS Chern-Simons by dimensional reduction and truncation of some fields, or the model of [30] obtained by adding to the Einstein-Hilbert action the coupling of a scalar field with the 4D Euler density.
The aim of the present work is an investigation of the classical properties of the 4D theory obtained from the 5D (A)dS Chern-Simons theory by a Kaluza-Klein compactification, find solutions of the field equations with spherical symmetry and solutions of the cosmological type, and comparisons of these solutions with the results of usual GR. This is intended to be a preliminary step to any attempt of quantization, the latter deserving future care.
(A)dS theory and its reduction to 4 dimensions are reviewed in Section 2, solutions with spherical symmetry and cosmological solutions are showed in Sections 3 and 4. Conclusions are presented in Section 5. Appendices present details omitted in the main text.
(A)dS Chern-Simons theory for 5D and 4D gravity 2.1 (A)dS Chern-Simons theory as a 5D gravitation theory
Apart of some considerations from the authors, the content of this subsection is not new. A good review may be found in the book [7] together with references to the original literature 3 .
Chern-Simons theories are defined in odd-dimensional space-times, we shall concentrate to the 5-dimensional case. We first define the gauge group as the pseudoorthogonal group SO (1,5) or SO (2,4), the de Sitter or anti-de Sitter group in 5 dimensions, generically denoted by (A)dS. These are the matrix groups leaving invariant the quadratic forms η (A)dS = diag (−1, 1, 1, 1, s) , s = 1 for de Sitter , s = −1 for anti-de Sitter . 3 Notations and conventions are given in Appendix A A convenient basis of the Lie algebra of (A)dS is given by 10 where η AB := diag(−1, 1, 1, 1, 1) is the D = 5 Minkowski metric.
We then define the (A)dS connexion 1-form, expanded in this basis as
A(x) = 1 2 ω AB (x)M AB + 1 l e A (x)P A , (2.2)
where l is a parameter of dimension of a length. ω AB will play the role of the 5D Lorentz connection form and e A of the "5-bein" form in the corresponding gravitation theory. We may already note that the presence of the parameter l, which will be related to the cosmological constant (see Eq. (2.14)), is necessary in order to match the dimension of the 5-bein form e A , which is that of a length, to that of the dimensionless Lorentz connection form ω AB .
The (A)dS gauge transformations of the connection read, in infinitesimal form, as
δA = dǫ − [A, ǫ] ,
where the infinitesimal parameter ǫ expands as
ǫ(x) = 1 2 ǫ AB (x)M AB + 1 l β A (x)P A .
From this follow the transformations rules of the fields ω and e:
δω AB = dǫ AB + ω A C ǫ CB + ω B C ǫ AC − s l 2 e A β B − e B β A , δe A = e C ǫ C A + dβ A + ω A C β C . (2.3)
Desiring to construct a background independent theory, we assume a dimension 5 manifold M 5 without an a priori metric. Then the unique (A)dS gauge invariant action -up to boundary terms -which may constructed with the given connection is the Chern-Simons action for the group (A)dS, which in our notation reads
S CS = 1 8κ M 5 ε ABCDE e A ∧ R BC ∧ R DE − 2s 3l 2 e A ∧ e B ∧ e C ∧ R DE + 1 5l 4 e A ∧ e B ∧ e C ∧ e D ∧ e E ,(2.4)
where κ is a dimensionless 4 coupling constant and
R A B = dω A B + ω A C ∧ ω C B (2.5) 4
In our units c = 1.
is the Riemann curvature 2-form associated to the Lorentz connection ω. We may add to the action a part S matter describing matter and its interactions with the geometric fields ω AB and e A , which leads to a total action S = S CS + S matter . The resulting field equations read
δS δe A = 1 8κ ε ABCDE F BC ∧ F DE + T A = 0 , δS δω AB = 1 2κ ε ABCDE T C ∧ F DE + S AB = 0 ,(2.6)
where
T A = De A = de A + ω A B ∧ e B (2.7)
is the torsion 2-form,
F AB = R AB − s l 2 e A ∧ e B (2.8)
is the (A)dS curvature, and
T A := δS matter δe A , S AB := δS matter δω AB (2.9)
are the energy-momentum 4-form, related to the energy-momentum components T A B in the 5-bein frame, by 10) and the spin 4-form S AB .
T A = 1 4! ε BCDEF T B A e C ∧ e D ∧ e E ∧ e F ,(2.
A generalized continuity equation for energy, momentum and spin results from the field equations (2.6), the zero-torsion condition De A = 0 and the Bianchi identity DR AB = 0:
DT A + s l 2 S AB ∧ e B = 0 ,(2.11)
which reduces to the energy-momentum continuity equation in the case of spinless matter:
DT A = 0 . (2.12)
We observe that the sum of the second and third term of the action (2.4) is proportional to the 5D Einstein-Palatini action with cosmological constant, which is equivalent to the more familiar 5D Einstein-Hilbert action in the metric formulation:
S EH = 1 16πG (5D) M 5 d 5 x √ −g(R − 2Λ) ,
with G (5D) the 5D gravitation constant, Λ the cosmological constant, R the Ricci scalar and g = −det(e A α ) 2 the determinant of the 5D metric
g αβ = η AB e A α e B β .(2.13)
This allows us to express the parameters κ and l in terms of the 5D physical parameters G (5D) and Λ as 3s
l 2 = Λ , κ = − 8π 3 Λ G (5D) .
(2.14)
The coefficient of the first term in the action (2.4) -the so-called Gauss-Bonnet term -is of course fixed by (A)dS gauge invariance in terms of the two parameters of the theory. This is a special case of the more general Lanczos-Lovelock or Lovelock-Cartan theory [26,7]
A trivial solution
In the vacuum defined by the absence of matter, a special class of solutions of the field equations (2.6) is that of the solutions of the stronger equations
F AB = 0 ,
with the (A)dS curvature 2-forms F AB given by (2.8). In fact the solution is unique up to an arbitrary torsion as it is readily seen by inspection of the second of the field equations (2.6). This is a solution of constant curvature and corresponds to an empty de Sitter or anti-de Sitter 5D space-time with a 5-bein form
e A = 1 − Λ 3 r 2 dt + 1 1 − Λ 3 r 2
dr + r (dθ + sin θ dφ + sin θ sin φ dψ) , (2.15) or its Lorentz transforms, leading to the metric ds 2 = − 1 − Λ 3 r 2 dt 2 + dr 2 1 − Λ 3 r 2 + r 2 dθ 2 + sin 2 θ dφ 2 + sin 2 θ sin 2 φ dψ 2 , (2.16) in spherical 4-space coordinates t, r, θ, φ, ψ. This metric has the symmetry O(4) of 4-space rotations.
Compactification to 4 dimensions
In order to connect the theory with 4-dimensional physics, we choose to implement a Kaluza-Klein type of compactification [27], considering the 4th spatial dimension to be compact. In other words, we consider a 5D space-time with the topology of M 5 = M 4 × S 1 , the first factor being a 4-dimensional manifold and S 1 the circle representing the compactified dimension 5 . Any space-time function admits a Fourier expansion in the S 1 coordinate χ, its coefficients -the Kaluza-Klein modes -being functions in M 4 . In the applications presented in Sections 3 and 4 only the zero mode is considered, which amounts to consider all functions as constant in χ.
Note that the zero (A)dS curvature solution (2.15,2.16) is not a solution of the compactified theory.
Solutions with zero torsion 2.3.1 On the number of degrees of freedom
The number of local physical degrees of freedom of the theory is best calculated by means of a canonical analysis. It is known [8] that the present theory has 75 constraints of first and second class. The number n 2 of the latter is equal, in the weak sense 6 , to the rank r of the matrix formed by the Poisson brackets of the constraints. The number of first class constraints -which generate the gauge transformationsis thus equal to n 1 = 75 − n 2 . Moreover, the number of generalized coordinates 7 is equal to 60. Thus the number n d.o.f. of physical degrees of freedom, at each point of 4-space, is given by
n d.o.f. = 1 2 (2 × 60 − 2n 1 − n 2 ) = r 2 − 15 .
The authors of [8] have shown that the result for the rank r, hence for n d.o.f. , depends on the region of phase space where the state of the system lies. They have computed it in the "generic" case, i.e., the case where the rank r is maximal, corresponding to the situation with the minimal set of local invariances, namely that of the fifteen (A)dS gauge invariances and the four 4-space diffeomorphism invariances. This results in r = 56, i.e., in 13 physical degrees of freedom.
The case with zero torsion is non-generic in the sense given above. We have checked by numerical tests that the rank r is then at most equal to 40, which shoes that n d.o.f. ≤ 5 in the case of a zero torsion.
On the stability of solutions with zero torsion
The second of the field equations (2.6) is identically solved by assuming zero torsion. We would like to know in which extent solutions with zero torsion are stable under small perturbations. More precisely, considering a field configuration with a torsion of order ǫ, we will look for conditions ensuring its vanishing as a consequence of the equations.
The second of Eqs. (2.6), written in 5-bein components as
1 8 ε ABCDE ε XY ZT U T C XY F DE ZT = 0 ,
can be rewritten in the form Let us write the infinitesimal torsion as T A BC = ǫ t A BC . We note that the connection ω (B.2) constructed from the 5-bein and the torsion is linear in the torsion components, thus in ǫ, hence F is a polynomial in ǫ, and so is the matrix M. This implies that its inverse M −1 exists and is analytic in ǫ in a neighborhood
T i M i j = 0 , (2.17) with M i j = 1 8 ε ABCDE ε XY ZT U F DE ZT ,(2.of ǫ = 0, if the matrix M i j(0) = M j i ǫ=0 (2.19)
is regular. It then follows, under the latter assumption, that the torsion vanishes.
We can summarize this result as the following This criterion is important in view of the difference between the number of physical degrees of freedom for states with zero torsion and this number for generic states 8 , as discussed in Subsection 2.3.1. Indeed, if the state of the system lies in the sub-phase space of zero torsion states, the fulfillment of the condition of the criterion guaranties that the state will evolve staying in that subspace.
Solutions with 3D rotational symmetry
The most general metric and torsion tensor components compatible with the rotational symmetry of 3-space are calculated in Appendix C, with the metric given by (C.1) and the torsion by (C.5) in a system of coordinates t, r, θ, φ, χ. All component fields depend on t, r, χ. But we shall restrict ourselves here to look for stationary solutions, neglecting also the higher Kaluza-Klein modes. Thus only a dependence on the radial coordinate r is left. In this situation the metric takes the simpler form (C.4) with only one non-diagonal term, thanks to some suitable coordinate transformations, as explained in Appendix C.
Through the definition (2.13), this metric leads to the 5-bein e A = e A α dx α , up to local Lorentz transformations e ′A = Λ A B e B , with
(e A α ) = n(r) 0 0 0 c(r) 0 a(r) 0 0 0) 0 0 r 0 0 0 0 0 r sin θ 0 0 0 0 0 b(r) ,(3.1)
and the relations
g tt (r) = −n 2 (r) , g rr (r) = a 2 (r) , g tχ (r) = −n(r)c(r) , g χχ (r) = b 2 (r) − c 2 (r) .
Beyond the spherical symmetry of 3-space, the stationarity and the restriction to the zero KK mode, we still make the following hypotheses:
(i) The torsion (2.7) is zero: T A = 0, hence the second of the field equations (2.6) is trivially satisfied;
(ii) We look for static solutions, hence g tχ (r) = 0, and c(r) = 0 in (3.1).
(iii) We restrict the discussion to the de Sitter case, i.e., with a positive cosmological constant: s = 1, which corresponds to the present data [28].
Consistently with the symmetry requirements and the hypotheses above, the tensor T A B appearing in the definition (2.10) of the energy-momentum 4-form reads 9
T A B = diag (−ρ(r),p(r),p(r),p(r),λ(r)) ,(3.2)
We also assume that the spin current 4-form S AB in (2.9) is vanishing. In the present setting, the continuity equation (2.11) takes then the form
p ′ (r) + p(r) n ′ (r) n(r) − b ′ (r) b(r) − λ(r) b ′ (r) b(r) + ρ(r) b ′ (r) b(r) = 0 . (3.3)
We shall consider the case of an empty physical 3-space, which means zero energy density and pressure, i.e., ρ(r) = p(r) = 0, keeping only the "compact dimension pressure" λ(r) = 0 (We shall see that the solution of interest indeed has a nonvanishing λ). The continuity equation thus implies the 5-bein component b(r) to be a constant:
b(r) = R = constant . (3.4)
The parameter R, which has the dimension of a length, defines the compactification scale.
With all of this, the field equations (first of (2.6)) reduce to the three independent equations
1 − 3r 2 l 2 a(r) 2 n(r) − 2rn ′ (r) − n(r) = 0 , −2ra ′ (r) + a(r) 3 3r 2 l 2 − 1 + a(r) = 0 , κl 2 r 2 a(r) 5 n(r)λ(r) − (l 2 a(r) + (r 2 − l 2 )a(r) 3 ) n ′′ (r) + (3l 2 + (r 2 − l 2 )a(r) 2 ) a ′ (r)n ′ (r) + 2ra(r) 2 n(r)a ′ (r) −2ra(r) 3 n ′ (r) − a(r) 3 + 3r 2 l 2 − 1 a(r) 5 n(r) = 0 . (3.5)
Note that these equations do not depend on the compactification scale R. The second equation solves for a(r), and then the first one yields n(r):
n(r) = 1 − 2µ r − r 2 l 2 , a(r) = 1/n(r) , (3.6)
after a time coordinate re-scaling is made. The Schwarzschild mass µ is an integration constant as in GR. The third equation yields λ(r) in terms of the functions a(r) and n(r), with the final result:
λ(r) = 6µ 2 κ 1 r 6 . (3.7)
The final 5-bein and metric thus read
(e A α ) = 1 − 2µ r − r 2 l 2 0 0 0 0 0 1 − 2µ r − r 2 l 2 −1 0 0 0 0 0 r 0 0 0 0 0 r sin θ 0 0 0 0 0 R , ds 2 = −(1 − 2µ r − r 2 l 2 )dt 2 + dr 2 1 − 2µ r − r 2 l 2 + r 2 (dθ 2 + sin 2 θ dφ 2 ) + R 2 dχ 2 . (3.8)
This result is just the generalization of the Schwarzschild solution in a space-time which is asymptotically de Sitter, with cosmologial constant Λ = 3/l 2 . One remembers that we have described the "vacuum" as described by an energy-momentum tensor (3.2) with one possibly non-zero component: the "compact dimension pressure" λ(r). Our result is that this "pressure" is indeed non-vanishing, singular at the origin and decaying as the inverse of the sixth power of the radial coordinate as shown in Eq. (3.7).
We must emphasize that this result follows uniquely from the hypotheses we have made.
Finally, we have checked the condition of stability of the zero-torsion solutions of the model according to the criterion proved in Subsection 2.3.2: a computation of the matrix M i j(0) (2.19) using the 5-bein (3.1) (with the non-diagonal component c(r) = 0) indeed shows that its rank takes the maximum value, 50, hence it is regular. We have also computed its determinant for the case of the solution (3.8):
Det (M i j(0) ) = − 4608µ 6 (2r 3 + l 2 µ) 3 l 88 r 27 ,
which is clearly not vanishing as long as the mass µ is not equal to zero.
Cosmological solutions
We turn now to the search for cosmological solutions, again the case of a positive cosmological constant Λ, e.g., taking the parameter s equal to 1.
This search is based on the hypothesis of isotropy and homogeneity of the physical 3-space. The space-time coordinates are taken as t, r, θ, φ, χ as in Section 3, r, θ, φ being spherical coordinates for the 3-space and χ ∈ (0, 2π) the compact subspace S 1 coordinate. We shall only consider here the zero modes of Kaluza-Klein, i.e., all functions will only depend on the time coordinate t.
The most general metric satisfying our symmetry requirements, up to general coordinate transformations, is given by Eq. (D.1) of Appendix D. In the present case of χ-independence, we can perform another time coordinate transformation in order to eliminate the factor in front of dt 2 , which yields the metric
ds 2 = −dt 2 + a 2 (t) 1 − kr 2 dr 2 + r 2 dθ 2 + sin 2 θ dφ 2 + b 2 (t)dχ 2 ,(4.1)
which is of the FLRW type in what concerns the 4D sub-space-times at constant χ. We shall restrict on the case of a flat 3-space, i.e., k = 0. This metric can then be obtained, using (2.13), from the 5-bein
e A α = diag (1, a(t), a(t)r, a(t)r sin θ, b(t)) ,(4.2)
up to a 5D Lorentz transformation. We shall not assume from the beginning a null torsion T A (2.7). Due to the isotropy and homogeneity conditions, the torsion depends on five independent functionsf (t), h(t),h(t), u(t) andũ(t), as shown in Eq. (D.2) of Appendix D. The equations resulting from the field equations (2.6) are also displayed in this appendix.
Let us now show that two components of the torsion, namely u andũ, can be set to zero by a partial gauge fixing condition. The two gauge invariances which are fixed in this way are the ones generated by P 0 and P 4 , i.e., the transformations (2.3) for the parameters β 0 (t) and β 4 (t). The torsion components which transform non-trivially are T 0 tχ and T 4 tχ :
δT 0 tχ = F 04 tχ β 4 , δT 4 tχ = −F 04 tχ β 0 .
These transformations are non-trivial as a consequence of the non-vanishing of the F -curvature component occurring here:
F 04 tχ = −b + ∂ t (ḃũ) + b l 2 ,
as it can be read out from (D.5). It follows therefore that the gauge fixing conditions
u = 0 ,ũ = 0 ,(4.3)
are permissible.
We shall describe matter with the perfect fluid energy-momentum tensor (D.7) with zero pressure, p = 0, a non-vanishing energy density and a possibly nonvanishing "compact dimension pressure":
T A B = diag − ρ(t) 2πb(t) , 0, 0, 0, λ(t) 2πb(t)
.
(4.4)
The first entry here is the energy densityρ(t) in 4-space, written in terms of the effective 3-space energy density ρ(t). We have correspondingly redefined λ for the sake of homogeneity in the notation (see (D.8) yields ρ(t) = 0, which contradicts the hypothesis of a non-vanishing energy density. We thus conclude thatf (t) = 0, which finally means a vanishing torsion 10 :
T A = 0 .
In order to solve now for the remaining field equations, we make the simplifying hypothesis that the compactification scale is constant:
b(t) = R .
(4.6)
At this stage, the field equations reduce to the systeṁ
a(t) 2 a(t) 2 − Λ 3 = 8πG 3 ρ(t) a(t) a(t) +ȧ (t) 2 2a(t) 2 − Λ 2 = 0 , λ(t) = −3a 2 (t)ä(t) − 3a(t)ȧ 2 (t) + 9 Λȧ 2 (t)ä(t) + Λa 3 (t) 16π 2 GR a 3 (t) , (4.7)
where we have expressed the parameters κ and l in terms of the Newton constant G and the cosmological constant Λ as
κ = − 16π 2 3 GRΛ , l = 3 Λ . (4.8)
We recognize in the first two equations the Friedmann equations for dust. The third equation gives the "compact dimension pressure" λ.
With the Big Bang boundary conditions a(0)=0 , the solution of the system reads
a(t) = C sinh 2 3 √ 3Λ 2 t , ρ(t) = Λ 8πG C a(t) 3 , λ(t) = − Λ 32π 2 GR C a(t) 6 . (4.9)
where C is an integration constant. The first line of course reproduces the ΛCDM solution for dust matter, whereas the second line shows a decreasing of λ as the sixth inverse power of the scale parameter a.
As it should, the solution obeys the continuity equation (D.18), which now readṡ ρ + 3ρȧ a = 0 .
The continuity equation (D.19) is trivially satisfied.
We recall that we have made the assumption of a constant scale parameter b for the compact dimension. This assumption is not necessary, but it is interesting to note, as it can be easily checked, that solving the equation in which we insert the ΛCDM expression of (4.9) for the 3-space scale parameter a(t), implies the constancy of b.
We have also explicitly checked the validity of the condition for stability according to the criterion of Subsection 2.3.2: the matrix M i j(0) (2.19) calculated using the 5-bein (4.2) has its maximum rank, 50, hence it is regular. We have also computed its determinant for the case of the solution (4. which is generically not vanishing as a function of t.
Conclusions
After recalling basic facts on the five-dimensional Chern-Simons gravity with the five dimensional (anti)-de Sitter ((A)dS) gauge group, we have studied some important aspects of this theory in comparison with the results of General Relativity with cosmological constant.
First of all, the cosmological constant is here a necessary ingredient due to the (A)dS algebraic structure, although it remains a free parameter. It cannot be set to zero.
We have shown that, for a spherically symmetrical 3-space, the "vacuum" Schwarzschild-de Sitter solution (3.8) follows uniquely from the hypotheses of a zerotorsion, stationary and static geometry. However, the existence of this solution implies the presence of a non-vanishing "compact dimension pressure" λ(r) as given by (3.7), a fact of non-easy interpretation, in particular due to the expected smallness of the compactification scale.
For the other physically interesting case of a cosmological model based on an homogeneous and isotropic 3-space, where we have restricted ourselves to the observationally favored flatness of 3-space, we have shown that the equations for the Friedmann scale parameter a(t) and the energy density ρ(t) are identical to the well-known Friedmann equations of General Relativity under the hypothesis that the compact scale parameter b(t) be a constant. Conversely, only this constancy is compatible with the Friedmann equations. There is also a non-vanishing "compact dimension pressure", decreasing in time as the sixth inverse power of the scale pa-rameter a(t). We have also seen that the vanishing of the torsion follows from the full (A)dS gauge invariance and of the field equations.
An important aspect of this work is the establishment of a criterion guarantying the stability of the zero-torsion solutions if a certain condition based on zero-torsion geometrical quantities is fulfilled. We have also checked that this condition is indeed met in the two situations considered in this paper.
Summarizing all these considerations, we can conclude that the two families of solutions investigated here coincide with the corresponding solutions of General Relativity in presence of a (positive) cosmological constant. However we recall that we have only examined the Kaluza-Klein zero modes of the theory. Possible deviations from the results of Einstein General Relativity could follow from the consideration of higher modes. Also, solutions with torsion would be interesting in view of its possible physical effects.
• A hat on a symbol means a 5D quantity, like e.g.,ρ(t, x 1 , x 2 , x 3 , x 4 ) for the energy density in 4-space.
B Construction of the spin connection
We recall here how the spin connection ω can be constructed from the 5-bein e and the torsion T [30]. First, given the 5-bein, one constructs the torsion-free connection ω, solution of the zero torsion equation de A +ω A B ∧ e B = 0. The result [31] is
ω A Bµ = 1 2 ξ C AB + ξ B C A − ξ AB C e C µ ,
with
ξ AB C = e µ A e ν B (∂ µ e C ν − ∂ ν e C µ )
. One then defines the contorsion 1-form C A B by the equation
T A = C A B ∧ e B , which solves in C AB = − 1 2 T AB C + T C AB − T B C A e C , (B.1)
where T A BC = e A µ T µ νρ e ν B e ρ C are the torsion components in the 5-bein basis 11 . From this we get the full connection form as
ω A B =ω A B + C A B , (B.2)
obeying the full torsion equation T A = de A + ω A B ∧ e B = 0.
C Metric and torsion for 3-space spherical symmetry
In this Appendix, we derive the metric g µν and torsion tensors T ρ µν in the case of a 3-space with spherical symmetry around the origin r = 0. Accordingly, observables such as the metric and the torsion components in the coordinate basis must satisfy Killing equations, which are the vanishing of the Lie derivatives of the fields along the vectors ξ which generate the symmetries.
The set o Killing vectors ξ are the generators J i (i = 1, 2, 3) of SO(3), which generate the spatial rotations. In the coordinate system t, r, θ, φ, χ, where r, θ, φ are spherical coordinates for 3-space, and χ the compact subspace coordinate, these vectors read
J 1 = − sin φ∂ θ − cot θ cos φ∂ φ , J 2 = cos φ∂ θ − cot θ sin φ∂ φ , J 3 = ∂ φ ,
and obey the commutation rules
[J i , J j ] = ε ijk J k ,
The Killing equations for the metric and the torsion read, for ξ = J 1 , J 2 , J 3 ,
£ ξ g µν = ξ ρ ∂ ρ g µν + g ρµ ∂ ν ξ ρ + g νρ ∂ µ ξ ρ = 0 , £ ξ T δ µν = ξ ρ ∂ ρ T δ µν − T ρ µν ∂ ν ξ δ + T δ ρν ∂ µ ξ ρ + T δ µρ ∂ ν ξ ρ = 0 .
This yields, for the metric: ds 2 = g tt (t, r, χ)dt 2 + 2g tr (t, r, χ)dtdr + 2g tχ (t, r, χ)dtdχ + 2g rχ (t, r, χ)drdχ + g rr (t, r, χ)dr 2 + g χχ (t, r, χ)dχ 2 + g θθ (t, r, χ) dθ 2 + sin 2 (θ)dφ 2 .
(C.1) If we perform a change of radial coordinate r to r ′ = (g θθ (t, r, χ)) 1/2 , and after that drop the primes, the line element becomes ds 2 = g tt (t, r, χ)dt 2 + 2g tr (t, r, χ)dtdr + 2g tχ (t, r, χ)dtdχ + 2g rχ (t, r, χ)drdχ +g rr (t, r, χ)dr 2 + g χχ (t, r, χ)dχ 2 + r 2 dθ 2 + sin 2 (θ)dφ 2 .
(C.2)
We shall consider the stationary case, i.e., where the components of the metric are independent of the coordinate t. For this case we can consider the differential g χχ (r, χ)dχ + g rχ (r, χ)dr, and from the theory of partial differential equations we know that we can multiply it by an integrating factor I 1 = I 1 (r, χ) which makes it an exact differential. Using this result to define a new coordinate χ ′ by requiring dχ ′ = I 1 (r, χ)(g χχ (r, χ)dχ + g rχ (r, χ)dr), substituting this in the latter expression of the line element and again dropping the prime, the line element simplifies to 12 , ds 2 = g tt (r, χ)dt 2 + 2g tr (r, χ)dtdr + 2g tχ (r, χ)dtdχ +g rr (r, χ)dr 2 + g χχ (r, χ)dχ 2 + r 2 dθ 2 + sin 2 (θ)dφ 2 .
(C.3)
Now we go to the special case where the components of the metric depend only on the radial variable r, which amounts to restrict to the Kaluza-Klein zero modes.
We can now consider the differential form g tt (r)dt + g tr (r)dr and multiply it by an integral factor I 2 (t, r) that permits to write it as a perfect differential, dt ′ = I 2 (t, r)(g tt (r)dt + g tr (r)dr). Substituting in the line element and dropping the prime we finally get ds 2 = g tt (r)dt 2 + 2g tχ (r)dtdχ + g rr (r)dr 2 + g χχ (r)dχ 2 + r 2 dθ 2 + sin 2 (θ)dφ 2 . (C.4)
For the torsion, the Killing equations leave the following non-vanishing components:
T t tr = h 1 (t, r, χ), T t tχ = q 1 (t, r, χ), T t rχ = q 2 (t, r, χ), T r tr = h 2 (t, r, χ), T r tχ = q 3 (t, r, χ), T r rχ = q 4 (t, r, χ), T χ tr = h 5 (t, r, χ), T χ tχ = q 5 (t, r, χ), T χ rχ = q 6 (t, r, χ), T t θφ = sin(θ)f 1 (t, r, χ, T r θφ = sin(θ)f 2 (t, r, χ), T χ θφ = sin(θ)f 5 (t, r, χ),
T θ tθ = h 3 (t, r, χ) = T φ tφ , T θ tφ = sin(θ)f 3 (t, r, χ), T φ tθ = − f 3 (t, r, χ) sin(θ) , T θ rθ = h 4 (t, r, χ) = T φ rφ , T θ rφ = sin(θ)f 4 (t, r, χ), T φ rθ = − f 4 (t, r, χ) sin(θ) T θ θχ = h 6 (t, r, χ) = T φ φχ , T θ φχ = sin(θ)f 6 (t, r, χ), T φ θχ = − f 6 (t, r, χ) sin(θ) .
(C.5)
A derivation of the connection ω(t, r, χ) from the general metric (C.2) and the torsion (C.5) following the lines of Appendix B, hence of the curvature forms and the field equations, may be found in [33]. In the present work we shall restrict to solutions which are independent of t (stationary) and independent of χ (Kaluza-Klein zero modes). The metric (C.4) will be used.
D Equations in the case of an isotropic and homogeneous 3-space
We give here the derivation of the general set of field equations with full dependence on the compact dimension coordinate χ, in the case of a 5D space-time with an isotropic and homogeneous 3D subspace. All fields are functions of the time coordinate t and the compact coordinate χ. 3-space coordinates are spherical: r, θ, φ.
D.1 Metric, 5-bein, torsion and curvature
The cosmological principle demands that the 3D spatial section of space-time be isotropic and homogeneous. Therefore the fields involved in the model must be compatible with this assumption. Isotropy of space-time means that the same observational evidence is available by looking in any direction in the universe, i.e., all the geometric properties of the space remain invariant after a rotation. Homogeneity means that at any random point the universe looks exactly the same. These two assumptions are translated in Killing equations, which are the vanishing of the Lie derivatives of the fields along the vectors ξ which generate the symmetries.
The set o Killing vectors ξ are the generators J i (i = 1, 2, 3) of SO (3), which generate the spatial rotations, and the generators of spatial translations P i , satisfying the commutation rules
[J i , J j ] = ε ijk J k [J i , P j ] = ε ijk P k [P i , P j ] = −kε ijk J k ,
where k is the 3-space curvature parameter: k = 0, 1, −1 for plane, closed or open 3-space, respectively. In our coordinate system, these vectors read
J 1 = − sin φ∂ θ − cot θ cos φ∂ φ , J 2 = cos φ∂ θ − cot θ sin φ∂ φ , J 3 = ∂ φ ,
and
P 1 = √ 1 − kr 2 sin θ cos φ ∂ r + cos θ cos φ r ∂ θ − sin φ r sin θ ∂ φ , P 2 = √ 1 − kr 2 sin θ sin φ ∂ r + cos θ sin φ r ∂ θ + cos φ r sin θ ∂ φ , P 3 = √ 1 − kr 2 cos θ ∂ r − sin θ r ∂ θ .
The Killing conditions must hold for (A)dS gauge invariant tensors. We are interested here in these conditions for the metric tensor g αβ = η AB e A α e B β and the torsion tensor T γ αβ = e γ A T A αβ :
£ ξ g µν = ξ γ ∂ γ g µν + g γµ ∂ ν ξ γ + g νγ ∂ µ ξ γ = 0 , £ ξ T δ µν = ξ γ ∂ γ T δ µν − T γ µν ∂ ν ξ δ + T δ γν ∂ µ ξ γ + T δ µγ ∂ ν ξ γ = 0 ,
with ξ = J 1 , J 2 , J 3 , P 1 , P 2 , P 3 . The Killing conditions for the metric yield the line element ds 2 = g αβ dx α dx β = g tt (t, χ)dt 2 + g χχ (t, χ)dχ 2 + α(t, χ) dr 2 1 − kr 2 + r 2 dθ 2 + r 2 sin 2 θ dφ 2 +2g tχ (t, χ)dtdχ .
In the same way as we did in Appendix C, we can eliminate the cross term in dt dχ through a change of the time coordinate defined by [32] dt ′ = I(t, χ) (g tt (t, χ) + g tχ (t, χ)) ,
where I(t, χ) is an integrating factor turning the right-hand side into an exact differential. Dropping the prime and redefining the coefficients we write the resulting line element as ds 2 = −n 2 (t, χ)dt 2 + b 2 (t, χ)dχ 2 + a 2 (t, χ) dr 2 1 − kr 2 + r 2 dθ 2 + r 2 sin 2 θ dφ 2 .
(D.1)
The non-vanishing components of the torsion left by the Killing conditions are:
T t tχ = u(t, χ), T r tr = T θ tθ = T φ tφ = −h(t, χ)
T χ tχ =ũ(t, χ), T r rχ = T θ θχ = T φ φχ =h(t, χ)
T r θφ = r 2 √ 1 − kr 2 sin θf (t, χ), T θ rφ = − sin θf (t, χ)
√ 1 − kr 2 , T φ rθ =f (t, χ) sin θ √ 1 − kr 2 .
(D.
2)
The 5-bein e A α corresponding to the metric(D.1) may be written in a diagonal form by fixing the 10 local invariances generated by the Lorentz generators M AB (see Eqs.
(2.1)). The result is
e A α = n(t, χ) 0 0 0 0 0 a(t, χ) √ 1 − kr 2 0 0 0 0 0 a(t, χ) r 0 0 0 0 0 a(t, χ)r sin θ 0 0 0 0 0 b(t, χ) , (D.
3)
The 5-bein form e A = e A α dx α read e 1 = a(t, χ) √ 1 − kr 2 dr , e 2 = a(t, χ)rdθ , e 3 = a(t, χ)r sin θdϕ, e 0 = n(t, χ)dt , e 4 = b(t, χ)dχ .
To find the connection compatible with the 5-bein (D.
3)] and the torsion T A (see (D.2)), i.e., a connection ω AB such that (2.7) holds, is a lengthy but well-known procedure (see, e.g., [30]), summarized in Appendix B. The result reads, in the 5-bein basis, with Q = −∂ χ (nQ) − ∂ t (bQ) , A = 1 a n ∂ t (aU) − QŨ ,
A 1 = 1 a b ∂ χ (aU) +QŨ , B = − 1 a n ∂ t (aŨ) −QU , B 1 = 1 a n ∂ t (aŨ) − QU , K = k a 2 + U 2 −Ũ 2 − f 2a 2 .
We will also need the torsion components T A BC = e A α e β B e γ C T α βγ in the 5-bein basis:
T 0 04 = u b , T 4 04 =ũ n , T i 0i = h n , T i i4 =h b , T i jk = ε i jkf 2a
, (i, j, · · · , = 1, 2, 3) .
As a -2-form, the torsion reads
D.2 Field equations
Matter will be assumed to consist in a spinless perfect fluid described by the energymomentum 4-form
T A = 1 4! ε BCDEF T B A e C ∧ e D ∧ e E ∧ e F ,
M AB = −M BA and 5 "translation" generators P A , where A, B, etc., are Lorentz indices taking the values 0, · · · , 4. These generators obey the commutation rules [M AB , M CD ] = η BD M AC + η AC M BD − η AD M BC − η BC M AD , [M AB , P C ] = η AC P B − η BC P A , [P A , P B ] = −s M AB , (2.1)
18) the index i standing for (C, [XY ]) and j for (U, [AB]). If the 50 × 50 matrix M is inversible, then (2.17) implies the vanishing of the torsion.
Stability criterion: A sufficient condition for the stability of the solutions at zero torsion under possible fluctuations of the torsion is that the matrix (2.18) restricted to zero torsion, M (0) , be regular.
Appendix A). With this form of the energymomentum tensor and the assumptions made at the beginning of this section, the continuity equation (D.19) is trivially satisfied, whereas (D.to the usual continuity equation for dust in the case of a constant compactification scale b. Let us now solve the field equations (D.8 -D.17). A key observation is that none of both expressions (K − s l 2 ) and (B − s l 2 ) can vanish, since we assume a non-zero energy density. (Remember the gauge conditions (4.3), and that all derivatives in χ vanish since we only consider the zero KK modes.) Then, Eqs. (D.17), (D.13) and (D.16), taken in that order, implỹ h(t) = 0 , h(t) = 0 ,f (t) = constant , respectively. Now, solving (D.15) leads to two possibilities:f vanishing or not. Let us first show that the latter case leads to a contradiction. (D.15) withf = 0 implies the equationb/b − 1/l 2 = 0, which solves in b(t) = b 0 exp(±t/l). Then Eq. (D.12) reads (ä − 1/l 2 )(ȧ − 1/l) which solves in a(t) = a 0 exp(±t/l). Inserting this into Eq.
ω 04 =FF
04Qe 0 −Qe 4 , ω 0i = Ue i , ω i4 =Ũe i , (H + h) ,Ũ = 1 b (H +h)) , H = ∂ t a a ,H = ∂ χ a a .(D.4) From the connection and the 5-bein we can calculate the Riemann curvature R AB (2.5) and the (A)dS curvatureF AB = R ABs l 2 e A ∧ e B 0i = A − s l 2 e 0 e i − A 1 e i e 4 + Uf 2a ε i jk e j e k , F i4 = B − s l 2 e i e 4 + B 1 e 0 e i +Ũf 2a ε i jk e j e k , ij
∧ e i +h b e i ∧ e 4
4+f 2a e j ε i jk ∧ e k .
The coordinates of M 5 are denoted by x α (α = 0, · · · , 4) and those of M 4 by x µ (µ = 0, · · · , 3). The coordinate of S 1 is denoted by χ, with 0 ≤ χ < 2π.
A "weak equality" is an equality valid up to the constraints.7 The generalized coordinates are the space components of the connection and 5-bein fields: ω AB a and e A a , with a = 1, · · · , 4.
We thank Jorge Zanelli for pointing out this problem to us.
The hats onρ, etc., mean energy density, etc. in 4-space.
The attentive reader may -correctly -find that the gauge fixing conditions (4.3) are not necessary in order to achieve the result in the casef = 0: Their are in fact consequences, together with h =h = 0, of the field equations (D.13-D.17). But the result above forf = 0 indeed does need these gauge fixing conditions.
In the 5-bein basis:T A = 1 2 T A BC e B ∧ e C .
This well known argument may be found in the textbook[32].
The hats onρ, etc., mean energy density, etc in 4-space.
AcknowledgmentsUse has been made of the differential geometry computation program "matrixEDC for Mathematica"[29]for various calculations.ThisAppendices A Notations and conventions• Units are such that c = 1.• Indices α, β, · · · = 0, · · · , 4, also called t, r, θ, φ, χ, are 5D space-time coordinates.• Indices. A, B, · · · = 0, · · · , 4 are 5-bein frame indices.• Indices A, · · · , are raised or lowered with the Minkowski metric (η AB ) = diag(−1, 1, 1, 1, 1).• Indices α, · · · may be exchanged with indices A, · · · using the 5-bein e A α or its inverse e α A .and the spin 4-form S AB = 0.With the expressions above for the curvature and torsion components and for the matter content, we can now write the explicit form of the field equations (2.6):D.3 Continuity equationsFor the spinless perfect fluid considered in the latter Subsection, the continuity equation (2.12), consequence of the field equations, takes the form of a system of two equations: ∂ tρ + 3 (ρ +p) ∂ t a a + ρ +λ ∂ t b b + 3ph −λũ = 0 , (D.18) ∂ χλ + 3 λ −p ∂ χ a a + ρ +λ ∂ χ n n − 3ph +ρu = 0 . (D.19)
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| []
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[
"Reliable Restricted Process Theory",
"Reliable Restricted Process Theory"
]
| [
"Fatemeh Ghassemi [email protected] \nUniversity of Tehran\nTehranIran\n",
"Wan Fokkink [email protected] \nVrije Universiteit Amsterdam\nAmsterdamThe Netherlands\n"
]
| [
"University of Tehran\nTehranIran",
"Vrije Universiteit Amsterdam\nAmsterdamThe Netherlands"
]
| [
"Fundamenta Informaticae XX"
]
| Malfunctions of a mobile ad hoc network (MANET) protocol caused by a conceptual mistake in the protocol design, rather than unreliable communication, can often be detected only by considering communication among the nodes in the network to be reliable. In Restricted Broadcast Process Theory, which was developed for the specification and verification of MANET protocols, the communication operator is lossy. Replacing unreliable with reliable communication invalidates existing results for this process theory. We examine the effects of this adaptation on the semantics of the framework with regard to the non-blocking property of communication in MANETs, the notion of behavioral equivalence relation and its axiomatization. We illustrate the applicability of our framework through a simple routing protocol. To prove its correctness, we introduce a novel proof process, based on a precongruence relation.Keywords: Mobile ad hoc network, restricted broadcast, process algebra, behavioral congruence, refinement. F. Ghassemi, W. Fokkink / Reliable Restricted Process Theory write {B A, C, B D, E} instead of {B A, B C, B D, B E}.The set Loc is extended with the unknown address ? to represent the address of a node which is still not known or concealed from an external observer. For instance, the leader address of a node can be initialized to this value. Furthermore, to define the semantics of communicating nodes in terms of restrictions over the topology in a compositional way, the semantics of receive actions can be defined through an unknown sender, which will be replaced by a known address when the receive actions are composed with the corresponding send action at a specific node (see Section 4).A network constraint C is said to be well-formed if ∀ℓ, ℓ ′ ∈ Loc (ℓ ℓ ′ ∈ C ∨ ℓ ℓ ′ ∈ C). Let C v (Loc) denote the set of well-formed network constraints that can be defined over the network addresses in Loc. We define an ordering on network constraints. We say that C 1denotes the substitution of d 1 for d 2 in d; this can be extended to process terms. For instance, {B A} {? A} and {B A, B C} {B A}. Each well-formed network constraint C represents the set of network topologies that satisfy the (dis)connectivity pairs in C, i.e., Γ(} extracts all one-hop (dis)connectivity information from γ. So the empty network constraint {} still denotes all possible topologies over Loc. The negation ¬ C of network constraint C is obtained by negating all its (dis)connectivity pairs. Clearly, if C is well-formed then so is ¬ C.Constrained labeled transition systems (CLTSs) provide a semantic model for the operational behavior of MANETs. Let Msg denote a set of messages communicated over a network and ranged over by m. Let Act be the network send and receive actions with signatures nsnd : Msg × Loc and nrcv : Msg, respectively. The send action nsnd (m, ℓ) denotes that the message m is transmitted from a node with the address ℓ, while the receive action nrcv (m) denotes that the message m is ready to be received. Let Act τ = Act ∪ {τ }, ranged over by η.Definition 2.1. A CLTS is a tuple S, Λ, →, s 0 , with S a set of states, Λ ⊆ C v (Loc) × Act τ , → ⊆ S × Λ × S a transition relation, and s 0 ∈ S the initial state. A transition (s, (C, η), s ′ ) ∈ → is denoted by sGenerally speaking, the transition s (C,η) − −−− → s ′ expresses that a MANET protocol in state s with an underlying topology γ ∈ Γ(C) can perform action η to evolve to state s ′ .The semantics of broadcast communication is defined to be reliable if and if only the nodes that are connected to the sender, as defined by its corresponding network constraint, receive the message. We remark that the status of the links from the receivers to the sender or between two arbitrary receivers are not of importance and hence, they are abstracted away. Therefore, by constructing such network constraints through the semantic rules, reliable communication is brought into our framework.Syntax of RRBPTLet A denotes a countably infinite set of process names which are used as recursion variables in recursive specifications. Besides network send and receive actions, i.e., nsnd (m, ℓ) and nrcv (m), we assume protocol send and receive actions, denoted by snd , rcv : Msg, i.e., parametrized by messages. Furthermore, let IAct be a set of internal actions. The syntax of RRBPT is given by the following grammar: | 10.3233/fi-2019-1775 | [
"https://arxiv.org/pdf/1705.02600v1.pdf"
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| 32,938,916 | 1705.02600 | 139e0c885aa39c15e5778e902cba9b788a2d8908 |
Reliable Restricted Process Theory
May 2017. 2017
Fatemeh Ghassemi [email protected]
University of Tehran
TehranIran
Wan Fokkink [email protected]
Vrije Universiteit Amsterdam
AmsterdamThe Netherlands
Reliable Restricted Process Theory
Fundamenta Informaticae XX
May 2017. 201710.3233/FI-2016-00001 IOS PressMobile ad hoc networkrestricted broadcastprocess algebrabehavioral congru- encerefinement
Malfunctions of a mobile ad hoc network (MANET) protocol caused by a conceptual mistake in the protocol design, rather than unreliable communication, can often be detected only by considering communication among the nodes in the network to be reliable. In Restricted Broadcast Process Theory, which was developed for the specification and verification of MANET protocols, the communication operator is lossy. Replacing unreliable with reliable communication invalidates existing results for this process theory. We examine the effects of this adaptation on the semantics of the framework with regard to the non-blocking property of communication in MANETs, the notion of behavioral equivalence relation and its axiomatization. We illustrate the applicability of our framework through a simple routing protocol. To prove its correctness, we introduce a novel proof process, based on a precongruence relation.Keywords: Mobile ad hoc network, restricted broadcast, process algebra, behavioral congruence, refinement. F. Ghassemi, W. Fokkink / Reliable Restricted Process Theory write {B A, C, B D, E} instead of {B A, B C, B D, B E}.The set Loc is extended with the unknown address ? to represent the address of a node which is still not known or concealed from an external observer. For instance, the leader address of a node can be initialized to this value. Furthermore, to define the semantics of communicating nodes in terms of restrictions over the topology in a compositional way, the semantics of receive actions can be defined through an unknown sender, which will be replaced by a known address when the receive actions are composed with the corresponding send action at a specific node (see Section 4).A network constraint C is said to be well-formed if ∀ℓ, ℓ ′ ∈ Loc (ℓ ℓ ′ ∈ C ∨ ℓ ℓ ′ ∈ C). Let C v (Loc) denote the set of well-formed network constraints that can be defined over the network addresses in Loc. We define an ordering on network constraints. We say that C 1denotes the substitution of d 1 for d 2 in d; this can be extended to process terms. For instance, {B A} {? A} and {B A, B C} {B A}. Each well-formed network constraint C represents the set of network topologies that satisfy the (dis)connectivity pairs in C, i.e., Γ(} extracts all one-hop (dis)connectivity information from γ. So the empty network constraint {} still denotes all possible topologies over Loc. The negation ¬ C of network constraint C is obtained by negating all its (dis)connectivity pairs. Clearly, if C is well-formed then so is ¬ C.Constrained labeled transition systems (CLTSs) provide a semantic model for the operational behavior of MANETs. Let Msg denote a set of messages communicated over a network and ranged over by m. Let Act be the network send and receive actions with signatures nsnd : Msg × Loc and nrcv : Msg, respectively. The send action nsnd (m, ℓ) denotes that the message m is transmitted from a node with the address ℓ, while the receive action nrcv (m) denotes that the message m is ready to be received. Let Act τ = Act ∪ {τ }, ranged over by η.Definition 2.1. A CLTS is a tuple S, Λ, →, s 0 , with S a set of states, Λ ⊆ C v (Loc) × Act τ , → ⊆ S × Λ × S a transition relation, and s 0 ∈ S the initial state. A transition (s, (C, η), s ′ ) ∈ → is denoted by sGenerally speaking, the transition s (C,η) − −−− → s ′ expresses that a MANET protocol in state s with an underlying topology γ ∈ Γ(C) can perform action η to evolve to state s ′ .The semantics of broadcast communication is defined to be reliable if and if only the nodes that are connected to the sender, as defined by its corresponding network constraint, receive the message. We remark that the status of the links from the receivers to the sender or between two arbitrary receivers are not of importance and hence, they are abstracted away. Therefore, by constructing such network constraints through the semantic rules, reliable communication is brought into our framework.Syntax of RRBPTLet A denotes a countably infinite set of process names which are used as recursion variables in recursive specifications. Besides network send and receive actions, i.e., nsnd (m, ℓ) and nrcv (m), we assume protocol send and receive actions, denoted by snd , rcv : Msg, i.e., parametrized by messages. Furthermore, let IAct be a set of internal actions. The syntax of RRBPT is given by the following grammar:
Introduction
The applicability of wireless communication is growing rapidly in areas like home networks and satellite transmissions, due to their broadcasting nature. Mobile ad hoc networks (MANETs) consist of several portable hosts with no pre-existing infrastructure, such as routers in wired networks or access points in managed (infrastructure) wireless networks. The design of MANET protocols is complicated, because due to mobility of nodes the topology of communication links is dynamic. Important MANET protocols such as the Ad hoc On Demand Distance Vector (AODV) routing protocol [1] contained flaws in their original design and have been revised accordingly. Formal methods can be applied in the early phases of the protocol development to analyze and capture conceptual errors before their implementation. For instance, some errors in the design of AODV were found in [2,3,4,5] using formal techniques.
There are numerous applications of existing formal frameworks such as SPIN [6,7,2] and UP-PAAL [7,8,9,10,11,12] for the analysis of MANET protocols. Lack of support for compositional modeling and arbitrary topology changes motivates developing a new approach, tailored to the domain of MANETs, with a primitive for local broadcast and supporting the verification of MANET protocols against changes of the underlying topology. The tailored formal modeling framework should provide some form of wireless communication which varies at the different layers of the Open Systems Interconnection (OSI) model: physical, data link, network, transport, session, presentation, and application. For instance, the data link layer is responsible for transferring data across the physical link and handling conflicts due to simultaneous accesses to the shared media. In contrast, communication at the network layer provides point-to-point communication between two nodes that are not directly connected through appropriate routing of messages by using the communication service of the data link layer. Most frameworks for the formal analysis of MANET protocols, such as [13,14,15,16,17,18,19,20,21,5], focus on protocols above the data link layer; hence they support the core services of this layer, which means that local broadcast is the primitive means of communication. Wireless communication at this layer is non-blocking, i.e., the sender broadcasts irrespective of the readiness of its receivers, and is asynchronous, i.e., received packets are buffered at the receiver. The data link layer of a node processes the packet if it is an intended destination. While a node is busy processing a message, it can still receive messages, buffer them and process them later. However, if two different nodes broadcast simultaneously with a common node in their range, the latter node cannot receive both messages and drops one of them, which is called the hidden node problem. We say that wireless communication is reliable if the intended receivers successfully receive the packet. In other words, message delivery is guaranteed to all connected neighbors.
Although lossy communication is an integral part of MANETs, mimicking it faithfully in a formal framework can hamper the formal analysis of MANET protocols. To obtain a deeper understanding of a malfunctioning of such a protocol due to a conceptual mistakes in its design rather than unreliable communication, it may be helpful to consider communication reliable, meaning that the possibility of the hidden node problem is omitted from the framework. Therefore we introduced the process algebra Reliable Restricted Broadcast Process Theory (RRBPT) in [22], to perform model checking of MANET protocols in a setting where communication is reliable. It is a variant of Restricted Broadcast Process Theory (RBPT) that we introduced previously in [23] for the modeling and analysis of protocols above the data link layer. The underlying semantic model of RBPT, a so-called constrained labeled transition system (CLTS), implicitly considers mobility of nodes with the novel notion of a network constraint, which abstractly defines a set of topologies: those satisfying the given connectivity constraints. The transitions of a CLTS are annotated with appropriate network constraints to restrict the behavior to MANETs with a topology of the specified ones. RBPT was extended with a set of auxiliary operators to reason about MANETs by equational reasoning, so-called Computed Network Process Theory (CNT) [24]. We provided a sound and complete axiomatization for CNT terms with finite-state behaviors, modulo so-called rooted branching computed network bisimilarity. This axiomatization enables linearization of processes at the syntactic level to take advantage of symbolic verification [25,26], especially when the network is composed of similar nodes [27,28].
Somewhat surprisingly, all these results do not carry over in a straightforward fashion from RBPT to RRBPT. To put the model checking approach presented in [22] on a firm basis, the current paper develops the formal foundations for RRBPT and modifies the core of CNT. In a lossy setting, the nonblocking property of local broadcast communication is an immediate consequence of the rule Par and its counterpart for the parallel composition:
t 1 a − → t ′ 1 t 1 t 2 a − → t ′ 1 t 2
, which expresses that if a node is not ready to participate in a communication, then we can assume that either it was disconnected from the sender or it was connected but has lost the message. However, in the reliable setting, to guarantee the non-blocking property, nodes should always be input-enabled. RRBPT provides a sensing operator which allows to change the control flow of a process depending on the status of node connectivity with other nodes. The input-enabledness feature is ensured through the RRBPT operational rules, where the main difference between RRBPT and RBPT is: in RRBPT, nodes lose a communication only when they are disconnected and are always input-enabled. We recap challenges of bringing inputenabledness feature in the semantics of RRBPT in the presence of the sensing operator. Furthermore, the behavioral equivalence relation of CNT setting is not a congruence with respect to parallel composition anymore. To support the desired distinguishing power, we provide a new bisimulation relation which guarantees the congruence property for MANETs. RRBPT can be extended in the same way as RBPT with computed network terms and the auxiliary operators left merge ( ) and communication merge (|) to provide a sound and complete axiomatization for the parallel composition. However, the input-enabledness feature and the new sensing operator require new auxiliary operators to assist their axiomatization. To this aim, we discuss the appropriate axioms of RRBPT. We utilize our axioms to analyze the correctness of protocols at the syntactic level. To this aim, we facilitate the specification of the protocol behaviors preconditioned to multihop constraints and then introduce a new notion of refinement among protocol implementations and their specifications. We demonstrate the applicability of our framework by analyzing and proving the correctness of a simple routing protocol inspired by the AODV protocol. This paper is organized as follows. Sections 2 and 4 introduce our semantic model and explain how it is helpful in giving semantics to reliable communication. Section 3 introduces the syntax of RRBPT. Sections 5 and 6 provide the appropriate notion of behavioral equivalence and axioms in the reliable setting, respectively. We demonstrate the applicability of our new framework by analyzing a simple routing protocol in Section 7. We review and compare the related process algebraic frameworks in depth in Section 8 before concluding the paper.
Constrained Labeled Transition Systems
Let Loc denote a set of network addresses, ranged over by ℓ. Viewing a network topology as a directed graph, it can be defined as γ : Loc → IP (Loc), where γ(A) expresses the set of nodes that are directly connected to A, and hence, can receive message from A. A network constraint C is a set of connectivity pairs : Loc × Loc and disconnectivity pairs : Loc × Loc. In this setting, nonexistence of (dis)connectivity information between two addresses implies lack of information about this link (which can e.g. be helpful when the link has no effect on the evolution of the network). For instance, B A denotes that A is connected to B directly and consequently A can receive data sent by B as before, while B A denotes that A is not connected to B directly and consequently cannot receive any message from B. The direction of an arrow shows the direction of information flow. We
The deadlock process is modeled by 0. The process α.t performs action α and then behaves as process t, where α is either an internal action or a protocol send/receive action snd (m)/rcv (m). Internal actions are useful in modeling the interactions of a process with other applications running on the same node. Protocol send/receive actions specify the interaction of a process with its data-link layer protocols: these protocols are responsible for transferring messages reliably throughout the network. These actions are turned into their corresponding network ones via the semantics (see Section 4). The process t 1 + t 1 behaves non-deterministically as t 1 or t 2 . The simplest form of a MANET is a node, represented by the network deployment operator [[t]] ℓ , denoting process t deployed on a node with the known network address ℓ = ? (where ? denotes the unknown address). A MANET can be composed by putting MANETs in parallel using ; the nodes communicate with each other by reliable restricted broadcast. A process name is specified by a recursive equation A def = t where A ∈ A is a name. MANET protocols may behave based on the (non-)existence of a link. A neighbor discovery service can be implemented at the data link layer, by periodically sending hello messages and acknowledging such messages received from a neighbor. The sensing operator sense(ℓ ′ , t 1 , t 2 ) examines the status of the link from the node, say with address ℓ, that the sensing is executed on to the node with the address ℓ ′ ; in case of its existence it behaves as t 1 , and otherwise as t 2 . For instance, the term [[sense(ℓ ′ , t 1 , t 2 )]] ℓ examines the existence of the link ℓ ℓ ′ , and then behaves accordingly. As a running example, P def = sense(B, snd (data B ).P, 0) denotes a process that recursively broadcasts a data message data B as long as it is connected to B; and Q def = rcv (data B ).deliver .Q a process that recursively receives a data message data and then the internal action deliver upon successful receipt of data. The network process [[P ]] A [[Q]] B specifies an ad hoc network composed of two nodes with the network addresses A and B deploying processes P and Q, respectively.
The hide operator (νℓ)t conceals the address ℓ in the process t, by renaming this address to ? in network send/receive actions. For each message m ∈ Msg, the abstraction operator τ m (t) renames network send/receive actions over messages of type m to τ , and the encapsulation operator ∂ m (t) forbids receiving messages of type m. Let τ {m 1 ,...,mn} (t) and ∂ {m 1 ,...,mn} (t) denote τ m 1 (. . . (τ mn (t)) . . .) and ∂ m 1 (. . . (∂ mn (t)) . . .).
For example, τ Msg (∂ Msg ([[P ]] A [[Q]] B )
) specifies an isolated MANET that cannot receive any message from the environment, while its communications (i.e., send actions) are abstracted away.
Terms should be grammatically well-defined, meaning that processes deployed at a network address are only defined by action prefix, choice, sense and process names. Furthermore, the application of action prefix, choice, sense and process names is restricted to the deployment operator.
Semantics of RRBPT
The operational rules in Table 1 induce a CLTS with transitions of the form t β − → t ′ , where β ∈ C v (Loc) × Act τ where Act = {NAct ∪ IAct}, NAct denotes the set of network send and receive actions, and IAct the set of internal actions ranged over by i. Assume that α denotes actions of the form {rcv (m), snd (m) | m ∈ Msg}. In these rules, t
(C, nrcv (m)) − −−−−−−−−− → denotes that there exists no t ′ such that t (C ′ , nrcv (m)) − −−−−−−−−− → t ′ and C ′ C.
The symmetric counterparts of the rules Choice, Bro, and Par hold, but have been omitted for the brevity.
Rule Prefix assigns an empty network constraint to each prefixed action, which may be accumulated by further constraints through application of rules Rcv 1 or Sen 1,2 . The rule Int indicates that a
t 1 (C,α) − −−− → t ′ 1 sense(ℓ, t 1 , t 2 ) ({ℓ ?}∪C,α) −−−−−−−−−−−−→ t ′ 1 : Sen 1 α.t ({},α) −−−−→ t : Prefix t 2 (C,α) − −−− → t ′ 2 sense(ℓ, t 1 , t 2 ) ({ℓ ?}∪C,α) −−−−−−−−−−−−→ t ′ 2
:
Sen 2 t β − → t ′ (νℓ)t β[?/ℓ] − −−− → (νℓ)t ′ :
Hid
t (C, snd(m)) − −−−−−−−− → t ′ [[t]] ℓ (C[ℓ/?], nsnd (m,ℓ)) − −−−−−−−−−−−−−− → [[t ′ ]] ℓ : Snd t (C, rcv (m)) −−−−−−−−→ t ′ [[t]] ℓ (C[ℓ/?]∪{? ℓ}, nrcv (m)) − −−−−−−−−−−−−−−−−−−−− → [[t ′ ]] ℓ : Rcv 1 [[t]] ℓ (C, nrcv (m)) − −−−−−−−−− → ∄C ′ ([[t]] ℓ (C ′ , nrcv (m) − −−−−−−−−− → ∧ C C ′ ) [[t]] ℓ (C, nrcv (m)) −−−−−−−−−→ [[t]] ℓ : Rcv 2 t 1 β − → t ′ 1 t 1 + t 2 β − → t ′ 1 : Choice t 1 (C 1 , nsnd(m,ℓ)) − −−−−−−−−−−− → t ′ 1 t 2i (C 2 , nrcv (m)) − −−−−−−−−− → t ′ 2 t 1 t 2 (C 1 ∪ C 2 [ℓ/?], nsnd(m,ℓ)) −−−−−−−−−−−−−−−−−−−→ t ′ 1 t ′ 2 : Bro t β − → t ′ A β − → t ′ : Inv , A def = t t 1 (C 1 ,nrcv (m)) − −−−−−−−−− → t ′ 1 t 2 (C 2 ,nrcv (m)) − −−−−−−−−− → t ′ 2 t 1 t 2 (C 1 ∪C 2 ,nrcv (m)) −−−−−−−−−−−−−→ t ′ 1 t ′ 2 : Recv t (C,η) − −−− → t ′ t (C ′ ,η) − −−− → t ′ : Exe, C ′ C t (C, i) − −−− → t ′ [[t]] ℓ (C, i) − −−− → [[t ′ ]] ℓ : Int t 1 (C,η) − −−− → t ′ 1 η ∈ IAct ∪ {τ } t 1 t 2 (C,η) − −−− → t ′ 1 t 2 : Par t (C,η) − −−− → t ′ η = nrcv (m) ∂ m (t) (C,η) − −−− → ∂ m (t ′ ) : Encap t (C,η) − −−− → t ′ τ m (t) (C,τm(η)) − −−−−−− → τ m (t ′ )
: Abs node progresses when the deployed process on the node performs an internal action. Interaction between the process t and its data-link layer is specified by the rules Snd and Rcv 1,2 : when t broadcasts a message, it is delivered to the nodes in its transmission range, disregarding their readiness. Rcv 1 specifies that a process t with an enabled receive action can perform it successfully if it has a link to a sender (not currently known). If a node does not have any enabled receive action nrcv (m) for the network constraint C, then receiving the message has no effect on the node behavior, as explained by Rcv 2 . This rule also implicitly implies that an enabled receive action cannot be performed when the node is disconnected from the sender (not currently known). Consequently, this rule makes nodes input-enabled, meaning that a node not ready to receive a message will drop it. Rule Rcv 2 adds a network receive action (C, nrcv (m)) to the behavior of a network node, specified by [[t]] ℓ , if it has no transition (C ′ , nrcv (m)) such that C ′ C. Furthermore, this rule ensures that a most general C is selected, and hence, the receive action nrcv (m) is defined for all possible network constraints (when combined with rule Exe). Therefore, [[P ]] A has a ({}, nrcv (data B ))-transition by application of this rule.
Rules Sen 1,2 explain the behavior of the sense operator. In case there is a link to the node with the address ℓ from the node that is running the sense operator, and currently its address is unknown, then it behaves like t 1 ; in case this link is not present, it behaves like t 2 . Therefore, the link status is combined with the network constraint C generated by its first or second term argument, as given by Sen 1,2 respectively. For instance, by Prefix and Sen 1 , P only generates a ({?
B}, snd (data B ))transition.
In rules Snd and Rcv 1 , the network constraint C may have the unknown address due to sensing operators, which is replaced by the address of the deployment operator, i.e., C[ℓ/?]. Therefore, by applying Snd to the only transition of P ,
[[P ]] A generates a ({A B}, nsnd (data B ))-transition.
Rule Recv synchronizes the receive actions of processes t 1 and t 2 on message m, while combining together their (dis)connectivity information in network constraints C 1 and C 2 . Rule Bro specifies how a communication occurs between a receiving and a sending process. This rule combines the network constraints, while the unknown location (in the network constraint of the receiving process) is replaced by the concrete address of the sender. In Bro and Recv it is required that the union of network constraints on the transition in the conclusion be well-formed.
The rule Par prevents evolution of sub-networks on network actions, in contrast to lossy settings, and enforces all nodes to specify their localities with respect to the sender before evolving the whole network via Recv or Bro rules. It only allows a process to evolve by performing an internal or silent action. Exe explains that a behavior that is possible for a network constraint, is also possible for a more restrictive network constraint.
For instance, the MANET
[[P ]] A [[Q]] B
can generate the ({B A}, nsnd (data B , A)) transition induced by the deduction tree below, where y ≡ deliver .Q:
:Prefix P ({}, snd (dataB )) − −−−−−−−−−−−− → P :Sen 1 P ({? B}, snd (dataB )) − −−−−−−−−−−−−−−−−− → P :Snd [[P ]] A ({A B}, nsnd (dataB ,A)) − −−−−−−−−−−−−−−−−−−−−− → [[P ]] A :Prefix Q ({}, rcv(data B )) −−−−−−−−−−−−→ y :Rcv 1 [[Q]] B ({? B}, nrcv (dataB )) − −−−−−−−−−−−−−−−−−− → [[y]] B :Bro [[P ]] A [[Q]] B ({B A}, nsnd (data B ,A)) − −−−−−−−−−−−−−−−−−−−−− → [[P ]] A [[y]] B
Rule Hid replaces every occurrence of ℓ in the network constraint and action of β by ?, and hence hides activities of a node with address ℓ from external observers. According to Abs, the abstraction operator τ m converts all network send and receive actions with a message of type m to τ and leaves other actions unaffected, as defined by the function τ m (η). The encapsulation operator ∂ m disallows all network receive actions on messages of type m, as specified by Encap.
The semantics of RRBPT was first introduced in [22] with the aim of defining CLTSs with negative connectivity pairs to illustrate their benefit for model checking MANET protocols. In this research, we modify its semantics to properly define the behavior of MANETs in the reliable setting. To this end, two groups of rules have been modified substantially: those of receive actions and the sensing operator. More specifically, the operational semantics of receive action in [22] explicitly specifies the locality of the receiver node with respect to the sender (that could be connected, disconnected, or unknown) through three semantic rules. Furthermore, the semantics of the sensing operator in [22] makes
[[P ]] A move by ({B A, ? A}, nrcv (data B )) and ({B A, ? A}, nrcv (data B )) to [[0]] A while
here it has a self-loop with the label of ({B A}, nrcv (data B )). In other words, the chance of sending data B is lost after dropping a received message of data B . Such a drawback is resolved by the newly introduced rule Rcv 2 and removing two previous rules of the sensing operator.
Rooted Branching Reliable Computed Network Bisimilarity
Terms of the lossy framework RBPT are considered modulo rooted branching computed network bisimilarity [24]. This equivalence relation is defined using the following notations:
• ⇒ denotes the reflexive and transitive closure of unobservable actions:
-t ⇒ t; -if t (C,τ ) − −−− → t ′ for some arbitrary network constraint C and t ′ ⇒ t ′′ , then t ⇒ t ′′ . • t (C,η) − −−−− → t ′ iff t (C,η) − −−− → t ′ or t (C[ℓ/?],η[ℓ/?]) − −−−−−−−−−− → t ′ and η is of the form nsnd (m, ?) for some m.
Intuitively t ⇒ t ′ expresses that after a number of communications, t can behave like t ′ . Furthermore, an action like ({? B}, nsnd (req (?), ?)) can be matched to an action like ({A B}, nsnd (req (A), A)), which is its − counterpart.
Definition 5.1. A binary relation R on RBPT terms is a branching computed network simulation if
t 1 Rt 2 and t 1 (C,η) − −−− → t ′ 1 implies that either: • η is of the form nrcv (m) or τ , and t ′ 1 Rt 2 ; or • there are t ′ 2 and t ′′ 2 such that t 2 ⇒ t ′′ 2 (C,η) − −−−− → t ′ 2 , where t 1 Rt ′′ 2 and t ′ 1 Rt ′ 2 .
R is a branching computed network bisimulation if R and R −1 are branching computed network simulations. Two terms t 1 and t 2 are branching computed network bisimilar, denoted by t 1 ≃ b t 2 , if t 1 Rt 2 for some branching computed network bisimulation relation R.
This definition distinguishes process terms according to their abilities to broadcast messages, and therefore, MANET protocols that can only receive are treated as deadlock as they cannot send any observable message.
Definition 5.2. Two terms t 1 and t are rooted branching computed network bisimilar, written t 1 ≃ rb t 2 , if:
• t 1 (C,η) − −−− → t ′ 1 implies there is a t ′ 2 such that t 2 (C,η) − −−−− → t ′ 2 and t ′ 1 ≃ b t ′ 2 ; • t 2 (C,η) − −−− → t ′ 2 implies there is a t ′ 1 such that t 1 (C,η) − −−−− → t ′ 1 and t ′ 1 ≃ b t ′ 2 .
Rooted branching computed network bisimilarity does not constitute a congruence with respect to the RRBPT operators. We still want that a receiving MANET (after its first action) be equivalent to deadlock. In this setting, still
[[0]] A ≃ b [[rcv (m).0]] A , but [[0]] A [[snd (m).0]] B ≃ b [[rcv (m).0]] A [[snd (m).0]] B
, since by application of Rcv 1,2 , Snd, and Bro:
[[rcv (m).0]] A [[snd (m).0]] B ({B A},nsnd(m,B)) − −−−−−−−−−−−−−−−− → [[rcv (m).0]] A [[0]] B [[rcv (m).0]] A [[snd (m).0]] B ({B A},nsnd (m,B)) − −−−−−−−−−−−−−−−− → [[0]] A [[0]] B
while by application of Rcv 2 , Snd, Bro:
[[0]] A [[snd (m).0]] B ({},nsnd (m,B)) −−−−−−−−−−−− → [[0]] A [[0]] BC 1 , . . . , C n such that ∀i, j ≤ n (i = j ⇒ Γ(C i ) ∩ Γ(C j ) = ∅) ∧ n k=1 Γ(C k ) = Γ(C).
Definition 5.3. A binary relation R on RRBPT terms is a branching reliable computed network sim-
ulation if t 1 R t 2 and t 1 (C,η) − −−− → t ′ 1 imply that either: • η is a τ action, and t ′ 1 R t 2 ; or • there are s ′′ 1 , . . . , s ′′ k and s ′ 1 , . . . , s ′ k for some k > 0 such that ∀i ≤ k(t 2 ⇒ s ′′ i (C i ,η) −−−−−→ s ′ i , with t 1 R s ′′ i and t ′ 1 R s ′ i ), and C 1 , . . . , C k constitute a partitioning of C .
R is a branching reliable computed network bisimulation if R and R −1 are branching reliable computed network simulations. Two terms t 1 and t 2 are branching reliable computed network bisimilar, denoted by t 1 ≃ br t 2 , if t 1 R t 2 for some branching reliable computed network bisimulation relation R.
Trivially (t 1 ≃ b t 2 ) ⇒ (t 1 ≃ br t 2 ).
Theorem 5.4. Branching reliable computed network bisimilarity is an equivalence.
See Section A for the proof of this theorem.
Definition 5.5. Two terms t 1 and t are rooted branching reliable computed network bisimilar, written t 1 ≃ rbr t 2 , if:
• t 1 (C,η) − −−− → t ′ 1 implies there is a t ′ 2 such that t 2 (C,η) − −−−− → t ′ 2 and t ′ 1 ≃ br t ′ 2 ; • t 2 (C,η) − −−− → t ′ 2 implies there is a t ′ 1 such that t 1 (C,η) − −−−− → t ′ 1 and t ′ 1 ≃ br t ′ 2 .
Corollary 5.6. Rooted branching reliable computed network bisimilarity is an equivalence.
Corollary 5.6 is an immediate consequence of Theorem 5.4 and Definition 5.5.
Theorem 5.7. Rooted branching reliable computed network bisimilarity is a congruence for RRBPT operators.
See Section B for the proof.
Axiomatization for RRBPT
To provide a sound and complete axiomatization for closed RRBPT terms with respect to rooted branching reliable computed network bisimilarity, the framework should be extended with the computed network terms, i.e., (C, η).t which expresses that action η is possible for topologies belonging to C, in the same way as [24]. This prefix operator is helpful to transform protocol send/receive actions into their corresponding network ones. Furthermore, it borrows the operators left merge ( ) and communication merge from the process algebra ACP [29] to axiomatize parallel composition. Note that the interleaving semantics for parallel composition is only valid for internal and unobservable actions (see rule Par ). To axiomatize the behavior of nodes while being input-enabled, we also exploit two novel auxiliary operators.
RRBPT is extended with new operators and called Reliable Computed Network Process Theory (RCNT). Its syntax contains:
t ::= 0 | β.t | t + t | A , A def = t | t | t | t t | t t | recA · t sense(ℓ, t, t) | (νℓ)t | τ m (t) | ∂ m (t) | ℓ : t : t | C ⊲ t | [[t]] ℓ
The prefix operator in β.t again denotes a process which performs β and then behaves as t. The action β can now be of two types: either an internal action or a send/receive action snd (m)/rcv (m), denoted by α, or actions of the form (C, nrcv (m)), (C, nsnd (m, ℓ)) and (C, τ ), denoted by (C, η), where the first two actions are called the network receive and send actions, respectively. The new operator ℓ : t 1 : t 2 , so-called local deployment, defines the behavior of process t 2 deployed at the network address ℓ while it only considers the input-enabledness feature with regard to the behavior of t 2 . In cases that it should drop a message (i.e., processing the message has not been defined by t 2 ), it behaves as t 1 . This operator is helpful to axiomatize the behavior of the deployment operator in the reliable setting. To axiomatize the behavior of the sense operator, the framework is extended with the topology restriction operator C ⊲ t which restricts the behavior of t by taking restrictions of C into account.
Due to the input-enabledness feature of nodes, their behavior is recursive: upon receiving a message for which no receive action has been defined, a node drops the message. To this aim, we exploit the recursion operator recA · t, which specifies the solution of the process name A, defined by the
equation A def = t. The process term t A is a solution of the equation A def = t if the replacement of A by t A on both sides of the equation results in equal terms, i.e. t A ≃ rb t[t A /A].
As we are interested in equations with exactly one solution, we define a guardedness criterion for network names, in the same way as [24]. A free occurrence of a network name A in t is called guarded if this occurrence is in the scope of an action prefix operator (not (C, τ ) prefix) and not in the scope of an abstraction operator [30]; in other words, there is a subterm
(C, η).t ′ in t such that η = τ , and A occurs in t ′ . A is (un)guarded in t if (not) every free occurrence of A in t is guarded. A RCNT term t is guarded if for every subterm recA · t ′ , A is guarded in t ′ .
This guardedness criterion ensures that any guarded recursive term has a unique solution.
A term is grammatically well-defined if its processes deployed at a network address through either a network or local deployment operator, are only defined by action prefix, choice, sense, and process names.
The operational semantic rules of the new operators are given in Table 2 while the counterpart of
Sync 2 holds. In these rules, t rcv (m) − −−−− → denotes that there exists no t ′ such that t (C ′ , rcv (m)) − −−−−−−−− → t ′ for some network constraint C ′ .
The behavior of the local deployment operator is almost similar to the deployment operator. Its rules Inter ′ 1 and Inter ′ 2 are the same as Snd and Rcv 1 , respectively. However, it substitutes Inter ′ 3 for Rcv 2 by which it only adds transitions containing the disconnectivity pair ? ℓ for those possible receive actions of t 2 (generated by Rcv 1 ). Rules Sen 3,4 make the behavior of sense(ℓ ′ , t 1 , t 2 ) input-enabled toward receive actions that are possible by t 1 but not t 2 and vice versa. The constraints of the topology restriction operator C ⊲ t is added to the behaviors of t as explained by the rule TR.
The main differences of extended RCNT with CNT are that its deployed nodes are input-enabled and its communication primitive is reliable. We use the notation m∈M t to define t[m 1 /m] + . . .
+ t[m k /m], where M = {m 1 , . . . , m k }. Furthermore, if (b, t 1 , t 2 )
behaves as t 1 if the condition b holds and otherwise as t 2 .
The axioms regarding the choice, deployment, left and communication merge, and parallel operators are given in Table 3. The axioms Ch 1−4 , Br , LM 2,3 and S 1−4 are standard (cf. [31]). The axiom Ch 5 denotes that a network send action whose sender address is unknown can be removed if its counterpart action exists. The axiom Ch 6 explains that a more liberal network constraint allows more behavior. Axioms Dep 0−7 , LM ′ 1,2 , and TRes 1−5 are new in comparison with the lossy setting of [24]. The axiom (C, η).t 1 t 2 = (C, η).(t 1 t 2 ) has been replaced by LM ′ 1,2 which only allow internal or unobservable actions of the left operand to be performed.
To axiomatize the behavior of a node considering the input-enabledness feature, we need to find the messages that it cannot currently respond to and then add a summand which receives those message without processing them. To this aim, axiom Dep 0 expresses the behavior of [[t]] ℓ as a recursive specification which drops messages that it does not handle with the help of the auxiliary function Message(t, S), and the behavior of t with the help of the local deployment operator ℓ : Q : t. The function Message(t, S) returns the set of messages that can be currently processed by t and is defined
t 2 (C, snd(m)) − −−−−−−−− → t ′ 2 ℓ : t 1 : t 2 (C[ℓ/?], nsnd(m,ℓ)) − −−−−−−−−−−−−−− → [[t ′ 2 ]] ℓ : Inter ′ 1 t[recA · t/A] (C,η) − −−− → t ′ recA · t (C,η) − −−− → t ′ : Rec t 2 (C, rcv (m)) −−−−−−−−→ t ′ 2 ℓ : t 1 : t 2 (C[ℓ/?]∪{? ℓ}, nrcv (m)) − −−−−−−−−−−−−−−−−−−−− → [[t ′ 2 ]] ℓ : Inter ′ 2 t 2 (C, rcv (m)) −−−−−−−−→ t ′ 2 ℓ : t 1 : t 2 (C[ℓ/?]∪{? ℓ}, nrcv (m)) − −−−−−−−−−−−−−−−−−−−− → t 1 : Inter ′ 3 t (C ′ ,η) − −−− → t ′ C ⊲ t (C ′ ∪C,η) − −−−−−− → t ′ : TR t 1 rcv (m) − −−−− → t 2 (C, rcv (m)) −−−−−−−−→ t ′ 2 ℓ : t 3 : sense(ℓ ′ , t 1 , t 2 ) ({? ℓ ′ }∪C, nrcv (m)) − −−−−−−−−−−−−−−−−− → t 3 : Sen 3 t 1 (C, rcv (m)) −−−−−−−−→ t ′ 1 t 2 rcv (m) − −−−− → ℓ : t 3 : sense(ℓ ′ , t 1 , t 2 ) ({? ℓ ′ }∪C, nrcv (m)) − −−−−−−−−−−−−−−−−− → t 3
:
Sen 4 t 1 β − → t ′ 1 t 1 t 2 β − → t ′ 1 t 2 : LExe t 1 (C 1 ,nrcv (m)) − −−−−−−−−− → t ′ 1 t 2 (C 2 ,nrcv (m)) − −−−−−−−−− → t ′ 2 t 1 | t 2 (C 1 ∪C 2 ,nrcv (m)) −−−−−−−−−−−−−→ t ′ 1 t ′ 2 : Sync 1 t 1 (C 1 ,nsnd(m,ℓ)) −−−−−−−−−−−→ t ′ 1 t 2 (C 2 ,nrcv (m)) − −−−−−−−−− → t ′ 2 t 1 | t 2 (C 1 ∪C 2 [ℓ/?],nsnd(m,ℓ)) − −−−−−−−−−−−−−−−−−− → t ′ 1 t ′ 2
: Sync 2 using structural induction:
Message(0, S) = ∅ Message(i.t, S) = ∅, i ∈ IAct Message(snd (m).t, S) = ∅ Message(rcv (m).t, S) = {m} Message(t 1 + t 2 , S) = Message(t 1 , S) ∪ Message(t 2 , S) Message(sense(ℓ, t 1 , t 2 ), S) = Message(t 1 , S) ∪ Message(t 2 , S) Message(A, S) = Message(t, S ∪ {A}), A ∈ S, A def = t Message(A, S) = ∅, A ∈ S
where S keeps track of process names whose right-hand definitions have been examined. We remark that Dep 0 extends the deployment behavior of the lossy setting with the input enabledness feature with the help of operator ℓ : Q : t. The axioms Dep 1−7 specify the behavior of the operator ℓ : t 1 : t 2 . Axiom Dep 1 defines the interaction between the network and data link layers. The protocol send action (at the network layer) is transformed into its network version (at the data link layer). Axiom Dep 2 indicates that when ℓ is connected to a sender (which is unknown yet), the receive action is successful and its behavior proceeds as [[t]] ℓ . Otherwise, the receive action is unsuccessful and its behavior is defined by t ′ . Axioms Dep 3,4,5 express the effect of the local deployment on choice, deadlock, and process names, respectively while axioms Dep 6,7 define its effect on the prefixed internal actions and sense operator, respectively. The behavior of the topology restriction operator is defined by the axioms TRes 1−5 in Table 3. Axiom TRes 1 considers the restrictions of C 1 by integrating its restrictions with C 2 in the computed network term (C 2 , η).t if C 1 ∪ C 2 is well-formed. Axiom TRes 2 defines that topology restriction can be distributed over the choice operator. Axiom TRes 3 expresses that the topology restriction operator can be moved inside and outside of a recursion operator. Axioms TRes 4,5 explain that the topology restriction operator has no effect on a process name and deadlock, respectively. Table 3. Axioms for the choice, deployment, left and communication merge, and parallel operators. The sets M 1 and M 2 denote Message(t 2 , ∅) \ Message(t 1 , ∅) and Message(t 1 , ∅) \ Message(t 2 , ∅) respectively. nsnd(m, ?)).t + (C, nsnd (m, ?)) .t = (C, nsnd (m, ?)) .t Table 4. Axiomatization of hiding, abstraction and encapsulation operators. nsnd(m, ℓ)).t) = (C, nsnd(m, ℓ)).∂ m (t) nsnd(m, ℓ)).t) = if ((m = m), (C, τ ).τ m (t), (C, nsnd(m, ℓ)).τ m (t))
Ch 1 0 + t = t Ch 2 t 1 + t 2 = t 2 + t 1 Ch 3 t 1 + (t 2 + t 3 ) = (t 1 + t 2 ) + t 3 Ch 4 t + t = t Ch 5 (C,Ch 6 (C 1 , η).t + (C 2 , η).t = (C 1 , η).t, C 2 C 1 Dep 0 [[t]] ℓ = recQ · m ′ ∈Message(t,∅) ({}, nrcv(m ′ )).Q + ℓ : Q : t Dep 2 ℓ : t ′ : rcv (m).t = ({? ℓ}, nrcv(m)).t ′ + ({? ℓ}, nrcv(m)).[[t]] ℓ Dep 7 ℓ : t 3 : sense(ℓ ′ , t 1 , t 2 ) = m ′ ∈M1 ({ℓ ℓ ′ }, nrcv(m ′ )).t 3 + m ′ ∈M2 ({ℓ ℓ ′ }, nrcv(m ′ )).t 3 + {ℓ ℓ ′ } ⊲ ℓ : t 3 : t 1 + {ℓ ℓ ′ } ⊲ ℓ : t 3 : t 2 Dep 1 ℓ : t ′ : snd (m).t = ({}, nsnd(m, ℓ)).[[t]] ℓ Dep 6 ℓ : t ′ : i.t = ({}, i).[[t]] ℓ Dep 3 ℓ : t 3 : t 1 + t 2 = ℓ : t 3 : t 1 + ℓ : t 3 : t 2 Dep 4 ℓ : t : 0 = 0 Dep 5 ℓ : t ′ : A = ℓ : t ′ : t, A def = t TRes 1 C 1 ⊲ (C 2 , η).t = (C 1 ∪ C 2 , η).t, if C 1 ∪ C 2 ∈ C v (Loc) TRes 2 C ⊲ (t 1 + t 2 ) = (C ⊲ t 1 ) + (C ⊲ t 2 ) TRes 3 C ⊲ recA · t = recA · (C ⊲ t) TRes 4 C ⊲ A = A TRes 5 C ⊲ 0 = 0 Br t 1 t 2 = t 1 t 2 + t 2 t 1 + t 1 | t 2 S 1 t 1 | t 2 = t 2 | t 1 LM ′ 1 (C, η).t 1 t 2 = 0, η ∈ IAct ∪ {τ } S 2 (t 1 + t 2 ) | t 3 = t 1 | t 3 + t 2 | t 3 LM 2 (t 1 + t 2 ) t 3 = t 1 t 3 + t 2 t 3 S 3 0 | t = 0 LM 3 0 t = 0 S 4 (C, η).t 1 | t 2 = 0, η ∈ IAct ∪ {τ } LM ′ 2 (C, η).t 1 t 2 = (C, η).(t 1 t 2 ), η ∈ IAct ∪ {τ } Sync 1 (C 1 , nsnd(m 1 , ℓ)).t 1 | (C 2 , nrcv(m 2 )).t 2 = if ((m 1 = m 2 ), (C 1 ∪ C 2 [ℓ/?], nsnd(m 1 , ℓ)).t 1 t 2 , 0) Sync 2 (C 1 , nrcv(m 1 )).t 1 | (C 2 , nrcv(m 2 )).t 2 = if ((m 1 = m 2 ), (C 1 ∪ C 2 , nrcv (m 1 )).t 1 t 2 , 0) Sync 3 (C 1 , nsnd(m 1 , ℓ 1 )).t 1 | (C 2 , nsnd(m 2 , ℓ 2 )).t 2 = 0Res 1 (νℓ)(t 1 + t 2 ) = (νℓ)t 1 + (νℓ)t 2 Res 3 (νℓ)0 = 0 Res 2 (νℓ)(C, η).t = (C[?/ℓ], η[?/ℓ]).(νℓ)t Ecp 1 ∂ m ((C,Ecp 2 ∂ m ((C, nrcv(m)).t) = if ((m = m), (C, nrcv(m)).∂ m (t), 0) Abs 1 τ m ((C, nrcv (m)).t) = if ((m = m), (C, τ ).τ m (t), (C, nrcv (m)).τ m (t)) Abs 2 τ m ((C,Abs 3 τ m (t 1 + t 2 ) = τ m (t 1 ) + τ m (t 2 ) Ecp 3 ∂ m (t 1 + t 2 ) = ∂ m (t 1 ) + ∂ m (t 2 ) Abs 4 τ m (0) = 0 Ecp 4 ∂ m (0) = 0 T 1 (C ′ , η).((C 1 , η).t + (C 2 , η).t + t ′ ) = (C ′ , η).((C, η).t + t ′ ) iff ∃ℓ, ℓ ′ ∈ Loc, ∃C ∈ C v (Loc) · (C 1 = C ∪ {ℓ ℓ ′ } ∧ C 2 = C ∪ {ℓ ℓ ′ } T 2 (C, η).((C ′ , τ ).(t 1 + t 2 ) + t 2 ) = (C, η).(t 1 + t 2 )
The axioms of hiding and encapsulation are given in Table 4. Axiom T 1 accumulates the network constraints that constitute a partitioning while T 2 removes a τ action which preserves the behavior of a network after some topology changes. The remaining axioms in this table are similar to the lossy setting.
Axioms for process names are given in Table 5. which explains that in case A is connected to B, each sending of data B is followed by the internal action deliver It is not hard to see that the axioms of Table 3, Table 4 and Table 5 provide a sound axiomatization of RCNT. This can be checked by verifying soundness for each axiom individually.
Theorem 6.1. The axiomatization is sound, i.e. for all closed RCNT terms t 1 and t 2 , if t 1 = t 2 then t 1 ≃ rb t 2 .
Our axiomatization is also ground-complete for terms with a finite-state CLTS, but not for infinitestate CLTSs. For example, recW · ({}, nsnd (req (A), A)).W lx:Loc ({? B}, nrcv (req (lx))).W produces an infinite-state CLTS, since at each recursive call a new parallel operator is generated. Its equality to recH · ({}, nsnd (req (A), A)).H cannot be proved by our axiomatization. Table 5. Axioms for process names.
recA · t = t{recA · t/A} Unfold t 1 = t 2 {t 1 /A} ⇒ t 1 = recA · t 2 , if A is guarded in t 2 Fold recA · (A + t) = recA · t Ung recA · ((C, τ ).((C ′ , τ ).t ′ + t) + s) = WUng 1 recA · ((C, τ ).(t ′ + t) + s), if A is unguarded in t ′ recA · ((C, τ ).(A + t) + s) = recA · ((C, τ ).(t + s) + s) WUng 2 τ m (recA · t) = recA · τ m (t), if A is serial in t Hid
Theorem 6.2. The axiomatization is ground-complete, i.e., for all closed finite-state reliable computed network terms t 1 and t 2 , t 1 ≃ rb t 2 implies t 1 = t 2 .
See sections C and D for the proofs of theorems 6.1 and 6.2, respectively.
Case Study
In MANETs, nodes communicate through others via a multi-hop communication. Hence, nodes act as routers to make the communication possible among not directly connected nodes. We illustrate the applicability of our axioms in the analysis of MANET protocols through a simple routing protocol inspired by the AODV protocol.
Protocol Specification
The protocol consists of three processes P , M , and Q, each specifying the behavior of a node as the source (that finds a route to a specific destination), middle node (that relays messages from the source to the destination), and destination. The description of these process are given in Figure 1. Process P , deployed at the address A, uses the neighbor discovery service of the data link layer to examine if it has a direct link to the destination with the address B. If it is connected, then it sends its data directly by broadcasting the message data B ; otherwise, it initiates the route discovery procedure by sending the message req , then behaving as P 1 . This process waits until it receives a reply from a middle name with the address C or B. In the former case, it behaves as P 2 which indicates that A sends it data through C as long as C is connected to A. In the latter case, it behaves as P which indicates that A sends it data as long as B is directly connected to A.
Process M relays req messages to find a route to B and then behaves as M 1 . This process waits until it receives a reply. To model waits with a timeout, it non-deterministically sends a request again. Upon receiving a reply from C it behaves as M 2 , indicating that it relays data messages of A as long as it has a link to C. Finally, process Q sends a reply upon receiving a request message and receives data messages.
To simplify the route maintenance procedure of AODV, the middle node takes advantage of the sensing operator when it behaves as M 2 . Whenever it finds out that it has no link to C, it sends an error message to its upstream node, i.e., A, to inform it that its route to B through C is not valid. Afterwards, they both execute the route discovery procedure by sending a request message.
The network with the three nodes of a source, middle, and destination is specified by
N ≡ τ Msg (∂ Msg ([[P ]] A [[M ]] C [[Q]] B )).
Analyzing (νA)(νB)(νC)N , whose network addresses have been abstracted away, reveals that it is rooted branching bisimilar to recX · τ.deliver .X + τ.0. Thus, possibly a deadlock occurs where data is not delivered to B. Such behavior may be the result of a conceptual mistake in the protocol design or lossy communication between A and B. However, the latter one does not exist in our reliable setting. We propose a technique in Section 7.2 to discover only those faulty behaviors that are due to an incorrect protocol design. (rep B , B)
Next, we simplify
∂ Msg ([[P 1 ]] A [[snd (req ).M 1 ]] C [[Q]] B ) as ∂ Msg ([[P 1 ]] A [[snd (req ).M 1 ]] C [[Q]] B ) = (2) ({A B}, nsnd (req , A)).∂ Msg ([[P 1 ]] A [[snd (req ).M 1 ]] C [[snd (rep B ).Q]] B )+ ({A B}, nsnd (req , A)).∂ Msg ([[P 1 ]] A [[snd (req ).M 1 ]] C [[Q]] B )+ ({C B}, nsnd (req , C)).∂ Msg ([[P 1 ]] A [[M 1 ]] C [[snd (rep B ).Q]] B )+ ({C B}, nsnd (req , C)).∂ Msg ([[P 1 ]] A [[M 1 ]] C [[Q]] B ).∂ Msg ([[P ]] A [[snd (rep C ).M 2 ]] C [[Q]] B ) = ({ }, nsnd (rep C , C)).∂ Msg ([[P ]] A [[M 2 ]] C [[Q]] B )+ ({A B}, nsnd (data B , A)).∂ Msg ([[P ]] A [[snd (rep C ).M 2 ]] C [[deliver .Q]] B )+ ({A B}, nsnd (req , A)).∂ Msg ([[P 1 ]] A [[snd (rep C ).M 2 ]] C [[Q]] B ). By extending ∂ Msg ([[P ]] A [[M 2 ]] C [[Q]] B ), we have: ∂ Msg ([[P ]] A [[M 2 ]] C [[Q]] B ) = ({A B}, nsnd (data B , A)).∂ Msg ([[P ]] A [[M 2 ]] C [[deliver .Q]] B )+ ({A B}, nsnd (req , A)).∂ Msg ([[P 1 ]] A [[M 2 ]] C [[Q]] B ). Finally extending ∂ Msg ([[P 1 ]] A [[M 2 ]] C [[Q]] B ) results: ∂ Msg ([[P 1 ]] A [[M 2 ]] C [[Q]] B ) = ({A B}, nsnd (req , A)).∂ Msg ([[P 1 ]] A [[M 2 ]] C [[snd (rep B ).Q]] B )+ ({A B}, nsnd (req , A)).∂ Msg ([[P 1 ]] A [[M 2 ]] C [[Q]] B )+ ({C B}, nsnd (error , C)).∂ Msg ([[P 1 ]] A [[snd (req ).M 1 ]] C [[Q]] B ).
The following scenario, found by above equations, is valid for a topology in which A has only a multi-hop link to B via C, but B has a direct link to A:
∂ Msg ([[P ]] A [[M ]] C [[Q]] B ) ({A B,A C},nsnd (req ,A)) − −−−−−−−−−−−−−−−−−−−−−−− → ∂ Msg ([[P 1 ]] A [[snd (req ).M 1 ]] C [[Q]] B ) ({C B},nsnd (req,C)) −−−−−−−−−−−−−−−−−−→ ∂ Msg ([[P 1 ]] A [[M 1 ]] C [[snd (rep B ).Q]] B ) ({B A,C},nsnd (rep B ,B)) − −−−−−−−−−−−−−−−−−−−−− → ∂ Msg ([[P ]] A [[snd (rep C ).M 2 ]] C [[Q]] B ) ({ },nsnd (rep C ,C)) − −−−−−−−−−−−−−− → ∂ Msg ([[P ]] A [[M 2 ]] C [[Q]] B ) ({A B},nsnd(req ,A)) − −−−−−−−−−−−−−−−−− → ∂ Msg ([[P 1 ]] A [[M 2 ]] C [[Q]] B ) ({A B},nsnd(req ,A)) − −−−−−−−−−−−−−−−−− → ∂ Msg ([[P 1 ]] A [[M 2 ]] C [[Q]] B ) . . .
The reason is found in the specification of M 2 which does not handle request messages, and hence, for such a topology no data will be received by B although there is a path form A to B and from B to A. Therefore, we revise M 2 as:
M 2 def
= sense(B, rcv (data C ).snd (data B ).M 2 + rcv (req ).snd (rep C ).M 2 , snd (error).snd (req ).M 1 )
The path above also exists in the lossy setting, but with all disconnectivity pairs removed from the network constraints. However, an exhaustive and therefore expensive inspection of this path is needed to determine that it is due to a design error. The first transition carries the label ({A B, A C}, nsnd (req , A)) in the reliable setting, meaning that B is not ready to receive, and the label ({A C}, nsnd (req , A)) in the lossy setting. The latter label indicates that either B was not ready to receive or it was not connected to A. So in the lossy setting one has to examine the origin state to find out if B had an enabled receive action or not. The concept of not being ready to receive is treated in the same way as a lossy communication. Since only the former may be due to a conceptual design in the protocol, finding design errors is not straightforward in the lossy setting. In general the lossy setting will produce a large number of possible error traces that all need to be examined exhaustively, while the reliable setting will produce no spurious error traces.
Protocol Analysis
The properties of wireless protocols, specially MANETs, tends to be weaker in comparison with wired protocols. For instance, the simple property of packet delivery from node A to B is specified as "if there is a path from A to B for a long enough period of time, any packet sent by A, will be received by B" [21]. The topology-dependent behavior of communication, and consequently the need for multihop communication between nodes, make their properties preconditioned by the existence of some paths among nodes.
To investigate the topology-dependent properties of MANETs by equational reasoning, it is necessary to enrich our process theory RCNT to specify behaviors constrained by multi-hop constraints. To this aim, we extend the action prefix operator of RCNT with actions that are paired with multi-hop constraints, first introduced in [22] and here extended by negative multi-hop connectivity pairs. Viewing a network topology as a directed graph, a multi-hop constraint is represented as a set of multi-hop (dis)connectivity pairs : Loc × Loc and : Loc × Loc. For instance, A C denotes there exists a multi-hop connection from A to C, and consequently C can indirectly receive data from A. Let M(Loc) denote the set of multi-hop constraints that can be defined over network addresses in Loc, ranged over by M. Term (M, ι).t, where ι ∈ IAct ∪ {τ }, denotes that the action ι is possible if the underlying topology satisfies the multi-hop network constraint M. Formally, a topology like γ satisfies the multi-hop network constraint M, denoted by γ |= M iff for each ℓ ℓ ′ in M, there is a multi-hop connection from ℓ to ℓ ′ in γ, and for each ℓ ℓ ′ in M, there is no multi-hop connection from ℓ to ℓ ′ in γ. To define a well-formed RCNT term, the rule which restricts the application of the new prefixed-actions to sequential processes, is added to the previous ones. Furthermore, a term cannot have two summands such that one is prefixed by an action of the form (C, η) and the other by an action of the form (M, ι). So terms with an action of the form (M, ι) only contain action prefix (with multi-hop constraints), choice and recursion operators.
To reason about the correctness of a MANET protocol, its behavior can be abstractly specified by observable internal actions with the required conditions on the underlying topology, i.e., ι-actions with multi-hop constraints. Intuitively, each communication of a protocol implementation triggers an internal action. Such communications are abstracted away by τ -transitions. Therefore, we define a novel preorder relation to examine if a protocol refines its specification. To this aim, a sequence of τ -transitions is allowed to precede an action that is matched to an action of the specification, as long as the accumulated network constraints of the τ -transitions satisfy the multi-hop network constraint of the matched action. Hence our preorder relation is parametrized by a network constraint to reflect such accumulated network constraints.
To provide such a relation, we use the notation C = ⇒ which is the reflexive and transitive closure of τ -relations while their network constraints are accumulated:
• t { } = = ⇒ t; • if t (C,τ ) − −−− → t ′ for some arbitrary network constraint C and t ′ C ′ = ⇒ t ′′ , then t C ′ ∪C = == ⇒ t ′′ , where C ′ ∪ C is well-formed.
Furthermore, the network constraint C satisfies the multi-hop constraint C, denoted by C |= M iff ∃ γ ∈ Γ(C) (γ |= M). We remark that a network constraint like {A B} may satisfy both {A B} and {A B}, but {A B} only satisfies {A B}.
Definition 7.1. A binary relation R C on RCNT terms is a refinement relation if t R C s implies:
• if t (C ′ ,η) − −−− → t ′ , where C ∪ C ′ ∈ C v (Loc), then -η = τ and t ′ R C∪C ′ s with C ∪ C ′ |= M, or -there is an s ′ such that s (C,η) − −−− → s ′ , and t ′ R C∪C ′ s ′ , and C ∪ C ′ |= M, or -η = ι for some ι ∈ IAct ∪ {τ } and there is an s ′ such that s (M,ι) −−−−→ s ′ with t ′ R C∪C ′ s ′ ; • if s (M,ι) −−−−→ s ′ , then there are t ′′ and t ′ such that t C ′ = ⇒ t ′′ (C ′′ ,ι) − −−− → t ′ with t ′′ R C∪C ′ s and t ′ R C∪C ′ ∪C ′′ s ′ ; • if s (C,η) − −−− → s ′ , then there is a t ′ such that t (C ′ ,η) − −−− → t ′ with t ′ R C∪C ′ s ′ .
The protocol t refines the specification s, denoted by t ⊑ s, if t R { } s for some refinement relation R { } . Thus, we use Equation 1 to show that:
τ Msg (∂ Msg ([[P ]] A [[M ]] C [[Q]] B )) ⊑ S ⇔ {A B} ⊲ τ Msg (∂ Msg ([[P ]] A [[M ]] C [[deliver .Q]] B )) ⊑ S ∧ {A B, A C} ⊲ τ Msg (∂ Msg ([[P 1 ]] A [[snd (req ).M 1 ]] C [[Q]] B )) ⊑ S ∧ (3) {A B, A C} ⊲ τ Msg (∂ Msg ([[P 1 ]] A [[M ]] C [[Q]] B )) ⊑ 0
To prove the refinement relation 3, we use the Equation 2 to show that
{A B, A C} ⊲ τ Msg (∂ Msg (P 1 ]] A [[snd (req ).M 1 ]] C [[Q]] B )) ⊑ S ⇔ {A B, A C, C B} ⊲ τ Msg (∂ Msg ([[P 1 ]] A [[M 1 ]] C [[snd (rep B ).Q]] B )) ⊑ S ∧ {A B, A C, C B} ⊲ τ Msg (∂ Msg ([[P 1 ]] A [[M 1 ]] C [[Q]] B )) ⊑ 0(4)
This proof process stops when we reach to the predicate C ⊲ t ⊑ (M, ι).s to prove for which either we have previously examined C ′ ⊲ t ⊑ (M, ι).s where C C ′ , or it holds trivially. For instance, the refinement relation (4)
Related Work
Related calculi to ours are CBS# [13], CWS [32], CMAN [14,15], CMN [16] and its timed version [33], bKlaim [17], ω-calculus [18], SCWN [19], CSDT [20], AWN [21] and its timed extension [34], and the broadcast psi-calculi [35]. These approaches have already been compared in [24] with regard to modeling issues, such as topology and mobility, as well as behavioral congruence relations, in particular observables and distinguishing power. As all these approaches, except [32], focus on protocols above the data like layer, we investigate their capabilities to faithfully support the properties of wireless communication at this layer, i.e., being non-blocking and asynchronous. Furthermore, we compare our behavioral equivalence relation to those with a reliable setting. All these approaches, except [5], provide an algebraic framework. Among them only [17] is asynchronous, centered around the tuple space paradigm; broadcast messages are output into the tuple spaces of neighboring nodes to the sending node.
The non-blocking property is a consequence of either nodes being input-enabled or the communication primitives being lossy. In the former case, the asynchronous property is achieved through abstract data specifications [36] in line with the approach from [37,38], in which the sum operator plays a pivotal role. Each process is then parametrized by a variable of the queue type with a summand which receives all possible messages (if the queue is empty). Among these approaches, CMN, CMAN, ω-calculus, SCWN, and the broadcast psi-calculi are lossy.
To make a process input-enabled while communication is synchronous, three approaches are followed. In the first approach, followed by AWN, the semantics is equipped with a rule similar to our Rcv 2 with a negative premise which expresses that if a node is not ready to receive, the message is simply ignored [21]. Due to our implicit modeling of topology, the negative premise of our rule is more complicated to characterize the unreadiness of nodes regarding the underlying topology. In the second approach, followed by CDST, counterparts for the rules Bro and Recv are defined with negative premises to cover cases when a process cannot participate in the communication message [20]. The third approach, provided by CSB#, eliminates negative premises, to remain within the de Simone format of structural operational semantics [21], in favor of actions which discard messages [13]. Therefore, the semantics is augmented by rules that trigger the ignore actions for any sending node, receiving nodes for disconnected locations, and deadlock. Furthermore, the rules Bro and Recv are modified to cover cases when a process ignores a message.
Among the reliable settings, only CDST provides a behavioral equivalence relation, based on the notion of observational congruence: the receive and send actions are observable while transitions changing the underlying topology are treated as unobservable. However, due to implicit modeling of topology and mobility, our behavioral equivalence relation has been parametrized with network constraints while it considers the branching structure of MANETs.
Conclusion
We introduced the reliable framework RRBPT, suitable to specify and verify MANETs, with the aim to catch errors in design decisions. We discussed the required changes at the semantic model by extending the network constraints with negative connectivity links. Furthermore, we revised the equivalence relation of the lossy setting to preserve required behavior in the reliable framework. Then we demonstrated which axioms should be added to /removed from the reliable setting. We provided an analysis approach at the syntactic level, exploiting a precongruence relation and our axiomatization. We applied our analysis approach to a simple routing protocol to prove that it correctly finds routes among connected nodes.
A. Branching Reliable Computed Network Bisimilarity is an Equivalence
To prove that branching reliable computed network bisimilarity is an equivalence, we exploit semibranching reliable computed network bisimilarity, following [41].
Definition A.1. A binary relation R on computed network terms is a semi-branching reliable computed network simulation, if t 1 Rt 2 implies whenever t 1
(C,η) − −−− → t ′ 1 : • either η = τ and there is a t ′ 2 such that t 2 ⇒ t ′ 2 with t 1 Rt ′ 2 and t ′ 1 Rt ′ 2 ; or
• there are s ′′ 1 , . . . , s ′′ k and s ′ 1 , . . . , s ′ k for some k > 0 such that ∀i ≤ k (t 2 ⇒ s ′′
i (C i ,η) −−−−−→ s ′ i , with t 1 Rs ′′ i and t ′ 1
Rs ′ i ), and C 1 , . . . , C k constitute a partitioning of C .
R is a semi-branching reliable computed network bisimulation if R and R −1 are semi-branching reliable computed network simulations. Computed networks t 1 and t 2 are semi-branching reliable computed network bisimilar if t 1 Rt 2 , for some semi-branching reliable computed network bisimulation relation R.
Lemma A.2. Let t 1 and t 2 be computed network terms, and R a semi-branching reliable computed network bisimulation such that t 1 Rt 2 .
• If t 1 ⇒ t ′ 1 then ∃t ′ 2 · t 2 ⇒ t ′ 2 ∧ t ′ 1 Rt ′ 2 • If t 2 ⇒ t ′ 2 then ∃t ′ 1 · t 1 ⇒ t ′ 1 ∧ t ′ 1 Rt ′ 2
Proof:
We only give the proof of the first property. The second property can be proved in a similar fashion. The proof is by induction on the number of ⇒ steps from t 1 to t ′ 1 :
• Base: Assume that the number of steps equals zero. Then t 1 and t ′ 1 must be equal. Since t 1 Rt 2 and t 2 ⇒ t 2 , the property is satisfied.
• Induction step: Assume t 1 ⇒ t ′ 1 in n steps, for some n ≥ 1. Then there is t ′′ 1 such that t 1 ⇒ t ′′ 1 in n − 1 ⇒ steps, and t ′′ 1 (C,τ ) − −−− → t ′ 1 . By the induction hypothesis, there is a t ′′ 2 such that t 2 ⇒ t ′′ 2 and t ′′ 1 Rt ′′ 2 . Since t ′′
(C,τ )
− −−− → t ′ 1 and R is a semi-branching reliable computed network bisimulation, there are two cases to consider:
-there is a t ′ 2 such that t ′′ 2 ⇒ t ′ 2 , t ′′ 1 Rt ′ 2 , and t ′ 1 Rt ′ 2 . So t 2 ⇒ t ′ 2 such that t ′ 1 Rt ′ 2 .
or there are s ′′′ 1 , . . . , s ′′′ k and s ′ 1 , . . . , s ′ k for some
k > 0 such that ∀i ≤ k (t ′′ 2 ⇒ s ′′′ i (C i ,τ ) − −−− → s ′ i , with t ′′ 1
Rs ′′′ i and t ′ 1 Rs ′ i ), and C 1 , . . . , C k constitute a partitioning of C. By definition,
s ′′′ i (C i ,τ ) − −−− → s ′ i yields s ′′′ i ⇒ s ′ i . Consequently for any arbitrary i ≤ k, t 2 ⇒ s ′ i such that t ′ 1 Rs ′ i . ⊓ ⊔ Proposition A.3.
The relation composition of two semi-branching reliable computed network bisimulations is again a semi-branching reliable computed network bisimulation.
Proof:
Let R 1 and R 2 be semi-branching reliable computed network bisimulations with t 1 R 1 t 2 and t 2 R 2 t 3 .
Let t 1
(C,η) − −−− → t ′ 1 . It must be shown that • either η = τ and there is a t ′ 3 such that t 3 ⇒ t ′ 3 with t 1 R 1 • R 2 t ′ 3 and t ′ 1 R 1 • R 2 t ′ 3 ; or • ∃s ′ 1 , . . . , s ′ k , s ′′ 1 , . . . , s ′′ k ∀i ≤ k (t 3 ⇒ s ′′ i (C i ,η) −−−−−→ s ′ i ∧ t 1 R 1 • R 2 s ′′ i ∧ t ′ 1 R 1 • R 2 s ′ i ), where C 1 , .
. . , C k constitute a partitioning of C .
Since t 1 R 1 t 2 , two cases can be considered:
• η = τ and there is a t ′ 2 such that t 2 ⇒ t ′ 2 with t 1 R 1 t ′ 2 and t ′ 1 R 1 t ′ 2 . Lemma A.2 yields that there is a t ′ 3 that t 3 ⇒ t ′ 3 with t ′ 2 R 2 t ′ 3 . It immediately follows that t 1 R 1 • R 2 t ′ 3 and t ′ 1 R 1 • R 2 t ′ 3 .
• there exist s * * 1 , . . . s * * j , s * 1 . . . s * j for some j > 0 such that ∀i ≤ j (t 2 ⇒ s * *
i (C i ,η) −−−−−→ s * i , t 1 R 1 s * * i , t ′ 1 R 1 s * i )
, and C 1 , . . . , C j is a partitioning of C . Since t 2 R 2 t 3 and t 2 ⇒ s * * i , Lemma A.2 yields that there are s ′′′ 1 , . . . , s ′′′ j such that ∀i ≤ j (t 3 ⇒ s ′′′ i ∧ s * * i R 2 s ′′′ i ). Two cases can be distinguished:
-either η = τ and for some i ≤ j, s * * i (C i ,τ ) − −−− → s * i implies there is an s ′′ i such that s ′′′ i ⇒ s ′′ i with s * * i R 2 s ′′ i and s * i R 2 s ′′ i . It follows immediately that there is an s ′′ i such that t 3 ⇒ s ′′ i with t 1 R 1 • R 2 s ′′ i and t ′ 1 R 1 • R 2 s ′′ i ; or -for all i ≤ j, s * * i (C i ,η) −−−−−→ s * i implies there are s ′′ i 1 , . . . , s ′′ i k i and s ′ i 1 , . . . , s ′ i k i for some k i > 0 such that ∀o ≤ k i (s ′′ i ⇒ s ′′ io (C io ,η) − −−−−− → s ′ io , s * * i R 2 s ′′ io , s * i R 2 s ′ io ), and C i 1 , . . . , C i k i is a partitioning of C i . Since t 3 ⇒ s ′′ i , we have ∀i ≤ j, ∀o ≤ k i (t 3 ⇒ s ′′ io (C io ,η) − −−−−− → s ′ io with t 1 R 1 • R 2 s ′′ io , t ′ 1 R 1 • R 2 s ′ io ), and { C io | i ≤ j, o < k i } is a partitioning of C . ⊓ ⊔
Corollary A.4. Semi-branching reliable computed network bisimilarity is an equivalence relation.
It is not hard to see that the union of semi-branching reliable computed network bisimulations is again a semi-branching reliable computed network bisimulation.
with (s * * * 1z j , s * * 2 io ) ∈ R ∧ (s * * 2 io , s ′′′ 1 i ) ∈ R −1 ∧ (s ′′′ 1 i , t 2 ) ∈ R ⇒ (s * * * 1z j , t 2 ) ∈ R • R −1 • R (s ′ 1z j , s * 2 io ) ∈ R ∧ (s * 2 io , s ′′ 1 i ) ∈ R −1 ∧ (s ′′ 1 i , t ′′ 1 ) ∈ R ⇒ (s ′ 1z j , t ′′ 2 ) ∈ R • R −1 • R, where z = ( i−1 l=1 k l ) + o), and { C io j | i ≤ k, o ≤ k i , j ≤ k ′ z } is a partitioning of C .
By Proposition A.3, R • R −1 • R is a semi-branching reliable computed network bisimulation.
Since R is the largest semi-branching reliable computed network bisimulation, and clearly R ⊆
R • R −1 • R, we have R = R • R −1 • R.
So R ′ is a semi-branching reliable computed network bisimulation. Since R is the largest semibranching reliable computed network bisimulation, R ′ = R.
We will now prove that R is a branching reliable computed network bisimulation. Let t 1 Rt 2 , and
t 1 (C,η) − −−− → t ′ 1 .
We only consider the case when η = τ , because for other cases, the transfer condition of Definition 5.3 and Definition A.1 are the same. Two cases can be distinguished:
1. There is a t ′ 2 such that t 2 ⇒ t ′ 2 with t 1 Rt ′ 2 and t ′ 1 Rt ′ 2 : we proved above that t ′ 1 Rt 2 . This agrees with the first case of Definition 5.3.
2. There are s ′′ 1 , . . . , s ′′ k and s ′ 1 , . . . , s ′ k for some
k > 0 such that ∀i ≤ k (t 2 ⇒ s ′′ i (C i ,τ ) −−−−−→ s ′ i with t 1
Rs ′′ i and t ′ 1 Rs ′ i ) and C 1 , . . . , C k constitute a partitioning of C . This agrees with the second case of Definition 5.1.
Consequently R is a branching reliable computed network bisimulation.
⊓ ⊔
Since any branching reliable computed network bisimulation is a semi-branching reliable computed network bisimulation, this yields the following corollary.
Corollary A.6. Two computed network terms are related by a branching reliable computed network bisimulation if and only if they are related by a semi-branching reliable computed network bisimulation.
Corollary A.7. Branching reliable computed network bisimilarity is an equivalence relation.
Corollary A.8. Rooted branching reliable computed network bisimilarity is an equivalence relation.
Proof:
It is easy to show that rooted branching reliable computed network bisimilarity is reflexive and symmetric. To conclude the proof, we show that rooted branching reliable computed network bisimilarity is transitive. Let t 1 ≃ rbr t 2 and t 2 ≃ rbr t 3 . Since t 1 ≃ rbr t 2 , if t 1
(C,η) − −−− → t ′ 1 , then there is t ′ 2 such that t 2 (C,η) − −−−− → t ′ 2 and t ′ 1 ≃ br t ′ 2 . Since t 2 ≃ rbr t 3 , there is a t ′ 3 such that t 3 (C,η) − −−−− → t ′ 3 and t ′ 2 ≃ br t ′ 3
. Equivalence of branching reliable computed network bisimilarity yields t 3
(C,η) − −−−− → t ′ 3 with t ′ 1 ≃ br t ′ 3 .
The same argumentation holds when t 3
(C,η) − −−− → t ′ 3 .
Consequently the transfer conditions of Definition 5.5 holds and t 1 ≃ rbr t 3 .
⊓ ⊔ B. Rooted Branching Reliable Computed Network Bisimilarity is a Congruence Theorem B.1. Rooted branching reliable computed network bisimilarity is a congruence for terms with respect to RCNT operators.
Proof:
We need to prove the following cases:
1. [[t 1 ]] ℓ ≃ rbr [[t 2 ]] ℓ implies [[α.t 1 ]] ℓ ≃ rbr [[α.t 2 ]] ℓ ; 2. [[t 1 ]] ℓ ≃ rbr [[t 2 ]] ℓ and [[t ′ 1 ]] ℓ ≃ rbr [[t ′ 2 ]] ℓ implies [[t 1 + t ′ 1 ]] ℓ ≃ rbr [[t 2 + t ′ 2 ]] ℓ ; 3. [[t 1 ]] ℓ ≃ rbr [[t 2 ]] ℓ and [[t ′ 1 ]] ℓ ≃ rbr [[t ′ 2 ]] ℓ implies [[sense(ℓ ′ , t 1 , t ′ 1 )]] ℓ ≃ rbr [[sense(ℓ ′ , t 2 , t ′ 2 )]] ℓ ; 4. [[t 1 ]] ℓ ≃ rbr [[t 2 ]
] ℓ implies ℓ : t : t 1 ≃ rbr ℓ : t : t 2 for any arbitrary term t;
5. t 1 ≃ rbr t 2 implies (C, η).t 1 ≃ rbr (C, η).t 2 ; 6. t 1 ≃ rbr t 2 and t ′ 1 ≃ rbr t ′ 2 implies t 1 + t ′ 1 ≃ rbr t 2 + t ′ 2 ; 7. t 1 ≃ rbr t 2 implies (νℓ)t 1 ≃ rbr (νℓ)t 2 ; 8. t 1 ≃ rbr t 2 and t ′ 1 ≃ rbr t ′ 2 implies t 1 t ′ 1 ≃ rbr t 2 t ′ 2 ;
9. t 1 ≃ rbr t 2 and t ′ 1 ≃ rbr t ′ 2 implies t 1 t ′ 1 ≃ rbr t 2 t ′ 2 ;
10. t 1 ≃ rbr t 2 and t ′ 1 ≃ rbr t ′ 2 implies t 1 | t ′ 1 ≃ rbr t 2 | t ′ 2 ;
11. t 1 ≃ rbr t 2 implies ∂ M (t 1 ) ≃ rbr ∂ M (t 2 );
12. t 1 ≃ rbr t 2 implies τ M (t 1 ) ≃ rbr τ M (t 2 );
13. t 1 ≃ rbr t 2 implies C ⊲ t 1 ≃ rbr C ⊲ t 2 .
Clearly, if t 1 ≃ rbr t 2 then t 1 ≃ br t 2 is witnessed by the following branching reliable computed network bisimulation relation:
R ′ = {R | t 1 (C,η) − −−− → t ′ 1 ⇒ ∃t ′ 2 · t 2 (C,η) − −−−− → t ′ 2 ∧ t ′ 1 ≃ br t ′ 2 is witnessed by R} ∪ {R | t 2 (C,η) − −−− → t ′ 2 ⇒ ∃t ′ 1 · t 1 (C,η) − −−−− → t ′ 1 ∧ t ′ 1 ≃ br t ′ 2 is witnessed by R} ∪ {(t 1 , t 2 )}.
We prove the cases 1, 2, 4, 7, 10, 11, and 13 since the proof of the cases 3 and 6 are similar to the case 2, the case 5 is similar to the case 1, the cases 8 and 9 are similar to the case 10, and the case 12 is similar to the case 11. and (C, η).t + t ′ are branching reliable computed network bisimilar, witnessed by the relation R constructed as follows: R = {((C 1 , η).t + (C 2 , η).t + t ′ , (C, η).t + t ′ ), (t, t) | t ∈ RCNT } The pair ((C 1 , η).t + (C 2 , η).t + t ′ , (C, η).t + t ′ ) satisfies the transfer conditions of Definition 5.3. Because every initial transition that (C 1 , η).t + (C 2 , η).t + t ′ can perform owing to t ′ , (C, η).t + t ′ can perform too. If (C 1 , η).t + (C 2 , η).t + t ′ can perform a (C 1 , η) or (C 2 , η)-transition, (C, η).t + t ′ can also perform it by application of Exe. Vice versa, if (C, η).t + t ′ can perform a (C, η)-transition, then as C 1 and C 2 form a partitioning of C, (C 1 , η).t + (C 2 , η).t + t ′ can perform a corresponding (C 1 , η)or (C 2 , η)-transition.
D. Completeness of RCNT axiomatization
To define RCNT terms with a finite-state behavior, we borrow the syntactical restriction of [24] on recursive terms recA · t, following the approach of [30]. We consider so-called finite-state Reliable Computed Network Theory (RCNT f ), obtained by restricting recursive terms recA · t to those that of which the bound network names do not occur in the scope of parallel, communication merge, left merge, hide, encapsulation and abstraction operators in t.
We follow the corresponding proof of [24] to prove Theorem 6.2 by performing the following steps:
E. Proofs of Section 7.2
We first prove Theorem 7.2 which indicates that the refinement relation is a preorder relation and has the precongruence property, and then we discuss the proof of Proposition 7.3.
E.1. Proof of Theorem 7.2
We first show that the refinement relation is a preorder relation and then discuss its precongruence property. To prove that refinement is a preorder, we must show that it is reflexive and transitive. As it is trivial that Definition 7.1 is reflexive, we focus on its transitivity property.
Regrading the well-formedness conditions imposed on RCNT terms, the transitivity property of our refinement relation, i.e., t 1 ⊑ t 2 and t 2 ⊑ s implies that t 1 ⊑ s, can be only proved when t 1 and t 2 have no prefixed-actions with a multi-hop network constraint. For such terms, Definition 7.1 enforces they mimic the behavior of each other by the first and third conditions. In other words, for reliable computed network terms with no prefixed-actions with multi-hop network constraints, refinement and strong bisimulation of [51] coincide.
Lemma E.1. (Transitive property) t 1 ⊑ t 2 and t 2 ⊑ s implies that t 1 ⊑ s.
Proof:
Assume sets of refinement relations R 1 C and R 2 C witnessing t 1 ⊑ t 2 and t 2 ⊑ s, respectively. We construct a set of refinement relations
R ′ C = {(t ′ 1 , s ′ ) | (t ′ 2 , s ′ ) ∈ R 2 C ∧ t ′ 1 R 1 C t 2 }
for any wellformed network constraint C. We show that t ′ 1 R ′ C s ′ satisfies the transfer conditions of Definition 7.1.
Assume t ′ 1 (C ′ ,η) − −−− → t ′′ 1 . By assumption t ′ 1 R 1 C t ′ 2 implies that t ′ 2 (C ′ ,η) − −−− → t ′′ 2 such that t ′′ 1 R 1 C∪C ′ t ′′ 2 .
By the assumption t ′ 2 R 2 C s ′ , there are three cases to consider:
• η = τ and t ′′ 2 R 2 C∪C ′ s ′ with C ∪ C ′ |= M. Thus by construction, t ′′ 1 R ′ C∪C ′ s ′ .
• There is an s ′′ such that s ′ (C,η) − −−− → s ′′ , and t ′′ 2 R 2 C∪C ′ s ′′ , and C ∪ C ′ |= M. Thus by construction, t ′′ 1 R ′ C∪C ′ s ′ .
• η = ι for some ι ∈ IAct ∪ {τ } and there is an s ′′ such that s ′ (M,ι) −−−−→ s ′′ with t ′′ 2 R 2 C∪C ′ s ′′ . Thus by construction, t ′′ 1 R ′ C∪C ′ s ′′ .
Assume s ′ (C ′ ,η) − −−− → s ′′ . Hence t ′ 2 R 2 C∪C ′ s ′ implies that there is a t ′′ 2 such that t ′ 2 (C ′ ,η) − −−− → t ′′ 2 with t ′′ 2 R 2 C∪C ′ s ′′ . By assumption t ′ 1 R 1 C t ′ 2 implies that t ′ 1 (C ′ ,η) − −−− → t ′′ 1 such that t ′′ 1 R 1 C∪C ′ t ′′ 2 ,
which cannot be matched to any transition of [[rcv (m).0]] A [[snd (m).0]] B according to the second condition of Definition 5.1. However, we observe that the ({}, nsnd (m, B))-transition can be matched to the transition sets of actions ({B A}, nsnd (m, B)) and ({B A}, nsnd (m, B)), as the network constraints {B A} and {B A} provide a partitioning of {} while the resulting states of their corresponding transitions are equivalent. Thus, we revise our Definition 5.1 by generalizing its second condition. Intuitively, two MANETs are equivalent if they have the same observable behaviors for all possible underlying topologies. In the lossy setting, the observable behaviors exclude receive actions, as the node [[rcv (a).snd (a).0]] A can be distinguished from [[rcv (a).0]] A due to its capability to send a after its receipt. However, the capability of receiving messages implicitly defines a restriction on the underlying topology. For instance, the sending action snd (a) in [[rcv (a).snd (a).0]] A is only possible if the node in question was previously connected to a sender and successfully received a. Thus to distinguish [[rcv (a).snd (a).0]] A from [[snd (a).0]] A , receive actions are included in the observables in the reliable setting. Furthermore, as dropping a message may have the same effect as its processing (as explained above), a transition cannot be matched in the same way as in Definition 5.1 and it may be matched to multiple transitions. A partitioning of a network constraint C consists of network constraints
For instance, the behavior of the MANET [[P ]] A , where P def = sense(B, snd (data B ).P, 0), Msg = {data B }, is simplified as:[[P ]] A = Dep 0,5 recQ · ({}, nrcv (data B )).Q + A : Q : sense(B, snd (data B ).P, 0) = Dep 7 recQ · ({}, nrcv (data B )).Q + {A B} ⊲ A : Q : snd (data B ).P + {A B} ⊲ A : Q : 0 = Dep 1,4 recQ · ({}, nrcv (data B )).Q + {A B} ⊲ ({}, nsnd (data B , A)).Q + {B A} ⊲ 0 = TRes 1,5 recQ · ({}, nrcv (data B )).Q + ({A B},nsnd (data B , A)).QThe behavior of [[Q]] B , where Q def = rcv (data B ).deliver .Q, is equated to: [[Q]] B = Dep 0,5 recQ · A : Q : rcv (data B ).deliver .Q = Dep 2 recQ · ({? B}, nrcv (data B )).[[deliver .Q]] A + ({? B}, nrcv (data B )).Q
Unfold and Fold express existence and uniqueness of a solution for the equation A def = t, which correspond to Milner's standard axioms, and the Recursive Definition Principle (RDP) and Recursive Specification Principle (RSP) in ACP. Unfold states that each recursive operator has a solution (whether it is guarded or not), while Fold states that each guarded recursive operator has at most one solution. The behavior of τ Msg (∂ Msg ([[P ]] A [[Q]] B )) by using the axioms of Table 5 is expressed by: τ Msg (∂ Msg ([[P ]] B [[Q]] B )) = recQ · ({A B}, τ ).({}, deliver ).Q + ({A B}, τ ).0
=
sense(B, snd (data B ).P, snd (req ).P 1 )P 1 def = [rcv (rep C ).P 2 + rcv (rep B ).P + snd (req ).P 1 ] P 2 def = sense(C, rcv (error).P + snd (data C ).P 2 , snd (req ).P 1 ) M def = rcv (req ).snd (req ).M 1 M 1 def = rcv (rep B ).snd (rep C ).M 2 + snd (req ).M 1 M 2 def = sense(B, rcv (data C ).snd (data B ).M 2 , snd (error).snd (req ).M 1 ) Q def = rcv (req ).snd (rep B ).Q + rcv (data B ).deliver.Q
Figure 1 .
1The specification of processes P , M , and Q as a part of our simple routing protocol.
The network ∂ Msg ([[P ]] A [[M ]] C [[Q]] B ) can be simplified as: ∂ Msg ([[P ]] A [[M ]] C [[Q]] B ) = (1) ({A B}, nsnd (data B , A)).∂ Msg ([[P ]] A [[M ]] C [[deliver .Q]] B )+ ({A B, A C}, nsnd (req , A)).∂ Msg ([[P 1 ]] A [[snd (req ).M 1 ]] C [[Q]] B )+ ({A B, A C}, nsnd (req , A)).∂ Msg ([[P 1 ]] A [[M ]] C [[Q]] B ).
Now, we continue by extending∂ Msg ([[P 1 ]] A [[M 1 ]] C [[snd (rep B ).Q]] B ): ∂ Msg ([[P 1 ]] A [[M 1 ]] C [[snd (rep B ).Q]] B ) = ({ }, nsnd (req , A)).∂ Msg ([[P 1 ]] A [[M 1 ]] C [[snd (rep B ).Q]] B )+ ({ }, nsnd (req , C)).∂ Msg ([[P 1 ]] A [[M 1 ]] C [[snd (rep B ).Q]] B )+ ({B A, C}, nsnd (rep B , B)).∂ Msg ([[P ]] A [[snd (rep C ).M 2 ]] C [[Q]] B )+ ({B A, B C}, nsnd
Theorem 7 . 2 .
72Refinement is a preorder relation and has the precongruence property.See Section E for its proof. To analyze the correctness of our simple routing protocol, we investigate if it has the packet delivery property. To this end, we verify whetherτ Msg (∂ Msg ([[P ]] A [[M ]] C [[Q]] B )) refines S, where S def = ({A B,B A}, deliver ).S + ({A B}, τ ).0 + ({A B, B A}, τ ).0. To this aim, we match all the resulting terms of τ -transitions to S as long as their accumulated network constraints satisfy {A B, B A}. If a τ -transition violates {A B, B A} but satisfies {A B}, then it will be matched to the transition ({A B}, τ ). Otherwise, it will be matched to the transition ({A B, B A}, τ ). Therefore, we exploit the provided equations together with the precongruence property of our refinement for the choice operator and the rules of Proposition 7.3.
Proposition 7 . 3 .
73Suppose ι ∈ IAct. The following rules holds(C, τ ).t ⊑ (M, ι).s ⇔ C ⊲ t ⊑ (M, ι).s ∧ C |= M (C, ι).t ⊑ (M, ι).s ⇔ C ⊲ t ⊑ s
trivially holds as it can be proved with the help of our axiomatization, especially the rules Fold and TRes 1,2 , that {A B, A C, C B} ⊲ τ Msg (∂ Msg ([[P 1 ]] A [[M 1 ]] C [[Q]] B ))is the answer to the equation Q def = ({A B, A C, C B}, τ ).Q, and trivially recQ · ({A B, A C, C B}, τ ).Q ⊑ 0. So, it can be easily proved that τ Msg (∂ Msg ([[P ]] A [[M ]] C [[Q]] B )) ⊑ S .
Case 1 .
1The first transitions of [[α.t 1 ]] ℓ and [[α.t 2 ]] ℓ are the same with application of the rule Snd (if α is a send action), Rcv 1 (if α is a receive action), or Rcv 2 (for receiving (C, nrcv (m))which are not derivable from Rcv 1 ), and by assumption [[t 1 ]] ℓ ≃ rbr [[t 1 ]] ℓ implies [[t 1 ]] ℓ ≃ br [[t 1 ]] ℓ . Thus the transfer conditions of Definition 5.5 hold. above, ∂ M (t 2 )
First
we show that (C, τ ).t ⊑ (M, ι).s ⇒ C ⊲ t ⊑ (M, ι).s ∧ C |= M. The only transition
Table 1 .
1Semantics of RRBPT operators.
Table 2 .
2Semantics of the new operators of RCNT
).∂ Msg ([[P ]] A [[M 1 ]] C [[Q]] B )+ ({B A, B C}, nsnd (rep B , B)).∂ Msg ([[P 1 ]] A [[snd (rep C ).M 2 ]] C [[Q]] B ). simplifying the term ∂ Msg ([[P ]] A [[snd (rep C ).M 2 ]] C [[Q]] B ), which indicates that A and C have found a direct route to B, we reach ∂ Msg ([[P ]] A [[M 2 ]] C [[Q]] B ):By
and consequently t ′′ 1 R ′ C∪C ′ s ′′ . Assume s (M,ι) −−−−→ s ′ . Therefore t ′ 2 R 2 C∪C ′ s ′ implies that there are t ′′′ 2 and t ′′ 2 such that t ′ C∪C ′ s ′ and t ′′ 2 R 2 C∪C ′ ∪C ′′ s ′′ . As every transition of t ′ 2 is mimicked by t ′ 1 , there are t ′′′ 1 and t ′′ 1 such that t ′ − −−− → t ′′ 1 with t ′′′ 1 R 1 C∪C ′ t ′′′ 2 and t ′′ 1 R 1 C∪C ′ ∪C ′′ t ′′ 2 . Concluding, there are t ′′′ 1 and t ′′ 1 such that t ′ − −−− → t ′′ 1 with t ′′′ 1 R ′ C∪C ′ s ′ and t ′′ 1 R ′ C∪C ′ ∪C ′′ s ′′ . ⊓ ⊔ Theorem E.2.Refinement is a precongruence for terms with respect to the RCNT operators.The above cases result from the congruence property of strong bisimilarity. ⊓ ⊔ The proof of Theorem 7.2 is an immediate result of Lemma E.1 and Theorem E.2. E.2. Proof of Proposition 7.32
C ′
= ⇒
t ′′′
2
(C ′′ ,ι)
− −−− → t ′′
2 with t ′′′
2 R 2
1
C ′
= ⇒ t ′′′
1
(C ′′ ,ι)
1
C ′
= ⇒ t ′′′
1
(C ′′ ,ι)
t ⇒ s ′′′ i t (C 1 i ∪C[ℓ/?],nsnd(m,ℓ)) − −−−−−−−−−−−−−−−−−− → s ′′ i t ′ , and (t ′ 1 t, s ′′′ i t), (t ′′ 1 t ′ , s ′′ i t ′ ) ∈ R ′ ) and C 1 1 ∪ C[ℓ/?], . . . , C 1 k ∪ C[ℓ/?] constitute a partitioning of C 1 ∪ C[ℓ/?].
Proposition A.5. The largest semi-branching reliable computed network bisimulation is a branching reliable computed network bisimulation.Proof:Suppose R is the largest semi-branching reliable computed network bisimulation for some given CLTS. Let t 1 Rt 2 , t 2 ⇒ t ′ 2 , t 1 Rt ′ 2 and t ′ 1 Rt ′ 2 . We show that R ′ = R ∪ {(t ′ 1 , t 2 )} is a semi-branching reliable computed network bisimulation.• either η = τ and there is a t ′′ 2 such that t ′ 2 ⇒ t ′′ 2 with t ′ 1 Rt ′′ 2 and t ′′ 1 Rt ′′ 2 . Finally t 2 ⇒ t ′ 2 results t ′ 1 R ′ t ′′ 2 and t ′′ 1 R ′ t ′′ 2 ; or• there are s ′′′ 1 , . . . , s ′′′ k and s ′′ 1 , . . . , s ′′ k for some k > 0 such thatIf t 2 (C,η)− −−− → t ′′ 2 , then it follows from (t 1 , t 2 ) ∈ R that• either η = τ , and there is a t ′′ 1 such that t 1 ⇒ t ′′ 1 with t ′′ 1 Rt 2 and t ′′ 1 Rt ′′ 2 . Furthermore, (t 1 , t ′ 2 ) ∈ R, t 1 ⇒ t ′′ 1 , and Lemma A.2 imply there is a t ′′′ 2 such that t ′ 2 ⇒ t ′′′ 2 with (t ′′ 1 , t ′′′ 2 ) ∈ R. Similarly (t ′ 1 , t ′ 2 ) ∈ R, t ′ 2 ⇒ t ′′′ 2 , and Lemma A.2 imply there is a t ′′′ 1 such that t ′ 1 ⇒ t ′′′ 1 with (t ′′′ 1 , t ′′′ 2 ) ∈ R. From (t ′′′ 1 , t ′′′ 2 ) ∈ R, (t ′′′ 2 , t ′′ 1 ) ∈ R −1 , and (t ′′ 1 , t 2 ) ∈ R, we conclude (t ′′′ 1 , t 2 ) ∈ R • R −1 • R. And from (t ′′′ 1 , t ′′′ 2 ) ∈ R, (t ′′′ 2 , t ′′ 1 ) ∈ R −1 , and (t ′′ 1 , t ′′ 2 ) ∈ R, we conclude (t ′′′ 1 , t ′′ 2 ) ∈ R • R −1 • R. • or there are s ′′′ 1 1 , . . . , s ′′′ 1 k and s ′′ 1 1 , . . . , s ′′ 1 k for some k > 0 such that ∀i ≤ k (t 1 ⇒ s ′′′Choice, or is implied by application of Rcv 2 , i.e.,ThusFor the latter case by Choice, there exists noWe remark that transitions derived by application of Rcv 2 are those that cannot be derived from Rcv 1 . The greatest value of the network constraints of such transitions either have the disconnectivity pair in the form of ? ℓ or have no connectivity pair in the form of ? ℓ. This implies that such transitions can not be mimicked by application of Rcv 1 (since it will add constraints of the form ? ℓ) . Therefore,−−−−→ t * , then three cases can be distinguished:• It owes to t 1• It owes to t 1This case is proved with the same argumentation as the previous case.• If t 1 and t 2 are of the form sense(ℓ ′ , t * 1 , t * * 1 ) and sense(ℓ ′ , t * 2 , t * * 2 ) respectively, and the transition owes to either Sen 3 or Sen 4 . Assume it was derived by Sen 3 , as the other case can be proved with the same argumentation.by application of Sen 2 and Rcv 1 ,′ (as the only way to generate the pair ℓ ′ ℓ is through the sense operator) and ℓ : t : t 2Case 7. We prove that if t 1 ≃ br t 2 then (νℓ)t 1 ≃ br (νℓ)t 2 . Let t 1 ≃ br t 2 be witnessed by the branching reliable computed network bisimulation relation R. We defineWe prove that R ′ is a branching reliable computed network bisimulation relation. Supposethere are two cases; in the first case η is a τ action and (t ′′, and C 1 . . . , C k is a partitioning of C . By application of Hid,There are two cases to consider:Owing to the fact that a subset of CTo this aim we examine the root condition in Definition 5.5. Suppose (νℓ)t 1With the same argument as above, (νℓ)t 2Case 10. From the three remaining cases, we focus on the most challenging case, which is the communication merge operator |, as the other operators are proved in a similar way. First we prove that if t 1 ≃ br t 2 , then t 1 t ≃ br t 2 t. Let t 1 ≃ br t 2 be witnessed by the branching reliable computed network bisimulation relation R. We defineNow suppose that the send action was performed by t, and t ′ 1 participated in the communication. That is, t ′, and C 1 1 , . . . , C 1 k is a partitioning of C 1 . Therefore, ∀i ≤ k (t ′• The case where η is a receive action is proved in a similar way to the previous case.• Suppose η is a τ action. Assume it originates from t 1 by application of Par . Thus t ′t ′ by application of Par is straightforward.• The case when η is an internal action is easy to prove (similar to the second case of the previous case).Likewise we can prove that t 1 ≃ rbr t 2 implies t t 1 ≃ rbr t t 2 .Now let t 1 ≃ rbr t 2 . To prove t 1 | t ≃ rbr t 2 | t, we examine the root condition from Definition 5.5.There are two cases to consider:• This send action was performed by t 1 at node ℓ, and t participated in the communication. That is,• The send action was performed by t at node ℓ, and t 1 participated in the communication. ThatFinally, the case where t 1 | t (C * ,nrcv (m)) − −−−−−−−−− → t * can be easily dealt with. This receive action was performed by both t 1 and t.Concluding, t 1 | t ≃ rbr t 2 | t. Likewise it can be argued that t | t 1 ≃ rbr t | t 2 . Case 11. We prove that if t 1 ≃ br t 2 , then ∂ M (t 1 ) ≃ br ∂ M (t 2 ). Let t 1 ≃ br t 2 be witnessed by the branching reliable computed network bisimulation relation R. We defineWe prove that R ′ is a branching reliable computed network bisimulation relation.) ∈ R, two cases can be considered: either η is a τ action and (t ′′ 1 , t ′ 2 ) ∈ R, or there are s ′′′ 1 , . . . , s ′′′ k and s ′′ 1 , . . . , s ′′ k for some k > 0 such thatand C 1 , . . . , C k is a partitioning of C . In the former case, (∂ M (t ′′ 1 ), ∂ M (t ′ 2 )) ∈ R ′ . In the latter case, by application of Par and Encap,. Likewise we can prove that t 1 ≃ rbr t 2 implies ∂ M (t 1 ) ≃ rbr ∂ M (t 2 ). To this aim we examine the root condition in Definition 5.5. Suppose ∂ M (t 1 ). With the same argument asTherefore, by application of TR, C⊲t 2C. Soundness of RCNT axiomatizationAs two rooted branching computed network bisimilar terms are also rooted branching reliable computed network bisimilar, the soundness of axioms which are in common with the lossy setting are established[24]. Thus, to prove Theorem 6.1, it suffices to prove the soundness of each new axiom in comparison with the lossy setting, i.e., Dep 0−7 , TRes 1−5 , LM ′ 1,2 , and T 1 , modulo rooted branching reliable computed network bisimilarity.We focus on the soundness of Dep 0 and T 1 , as the soundness of the remaining axioms can be argued in a similar fashion. To prove Dep 0 , we show that both sides of the axiom satisfy the transfer conditions of Definition 5.5. Three cases can be distinguished. In following cases, for the sake of brevity, we write X for recQ · m ′ ∈Message(t,∅) ({}, nrcv (m ′ )).Q + ℓ : Q : t:ℓ ∈ C ′′ due to application of Rcv 1 . Then by application of Inter ′ 3 , ℓ : X : t We focus on the soundness of T 1 . The only transition that the terms (C ′ , η).((C 1 , η).t + (C 2 , η).t + t ′ )and (C ′ , η).((C, η).t + t ′ ) in T 1 can do is− −−− → and the resulting terms (C 1 , η).t + (C 2 , η).t + t ′ Proof: Assume that t 1 ⊑ s 1 and t 2 ⊑ s 2 . We first show that t 1 + t 2 ⊑ s 1 + s 2 . There are sets of refinement relations R 1 C and R 2 C witnessing t 1 ⊑ s 1 and t 2 ⊑ s 2 , respectively. We construct a set of refinement{ } satisfies the transfer conditions of Definition 7.1.AssumeThree cases can be considered:• There is an s ′ 1 such that s 1 −−−−→ s ′ 1 and by construction t ′ 1 R C∪C ′ s ′ 1 .• η = ι for some ι ∈ IAct ∪ {τ } and there is an s ′ 1 such that s 1Thus by the sos rule Choice, there is an s ′ 1 such that s 1 + s 2The same discussion holds if t 1 + t 2 −−−−→ s ′ 1 . By assumption t 1 R 1 C s 1 implies there are t ′′ 1 and t ′ 1 such that t 1 − −−− → s ′ 1 . By assumption t 1 R 1 C s 1 implies there is a t ′ 1 such that t 1Hence, there is a t ′ 1 such that t 1 + t 2 (C ′ ,η) − −−− → t ′ 1 with t ′ 1 R C∪C ′ s ′ 1 . The above discussions together yield t 1 + t 2 ⊑ s 1 + s 2 . If s 1 and s 2 have no prefixed-action with a multi-hop network constraint, then we must show the following cases:(C, τ ).t can make is (C, τ ).t(C,τ )− −−− → t. As ι = τ , according to Definition 7.1, t R C (M, ι).s. We construct R ′ { } = R C and show that it induces C ⊲ t ⊑ (M, ι).s. This is trivial as any transition C ⊲ t
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first we show that each RCNT f term can be turned into a normal form consisting of only 0, (C, η).t ′ , t ′ + t ′′ and recA · t ′ , where A is guarded in t ′. first we show that each RCNT f term can be turned into a normal form consisting of only 0, (C, η).t ′ , t ′ + t ′′ and recA · t ′ , where A is guarded in t ′ ;
next we define recursive network specifications and prove that each guarded recursive network specification has a unique solution. 2. next we define recursive network specifications and prove that each guarded recursive network specification has a unique solution;
finally we show that our axiomatization is ground-complete for normal forms, by showing that equivalent normal forms are solutions for the same guarded recursive network specification. 3. finally we show that our axiomatization is ground-complete for normal forms, by showing that equivalent normal forms are solutions for the same guarded recursive network specification.
Completeness of our axiomatization for all RCNT f terms results from the steps 1 and 3. We only discuss the first step, as others are exactly the same as in the lossy setting. Completeness of our axiomatization for all RCNT f terms results from the steps 1 and 3. We only discuss the first step, as others are exactly the same as in the lossy setting.
We prove this by structural induction over the syntax of terms t (possibly open). The base cases of induction for t ≡ 0 or t ≡ A are trivial because they are in normal form already. Proposition D.1. Each closed term t of RCNT f whose network names do not occur in the scope of one of the operators. |, (νℓ), τ M or ∂ M for some ℓ ∈ Loc and M ⊆ Msg. The inductive cases of the induction are the following ones: • if t ≡ [[0]] ℓ , then by application of Dep 0,4 and Ch 1 we have t = recQ· m ′ ∈Msg ({}, nrcv (m ′ )).Q, which is in normal formProposition D.1. Each closed term t of RCNT f whose network names do not occur in the scope of one of the operators , , |, (νℓ), τ M or ∂ M for some ℓ ∈ Loc and M ⊆ Msg, can be turned into a normal form. We prove this by structural induction over the syntax of terms t (possibly open). The base cases of induction for t ≡ 0 or t ≡ A are trivial because they are in normal form already. The inductive cases of the induction are the following ones: • if t ≡ [[0]] ℓ , then by application of Dep 0,4 and Ch 1 we have t = recQ· m ′ ∈Msg ({}, nrcv (m ′ )).Q, which is in normal form.
. • , α.t ′. t ′ + t ′′. ℓ or. sense(ℓ ′ , t ′ , t ′′ ). ℓ or [[A]] ℓ , then t can be turned into a normal form by application of axioms Dep 0−5,6,7 and induction over [[t ′ ]] ℓ and [[t ′′ ]] ℓ• if t ≡ [[α.t ′ ]] ℓ or t ≡ [[t ′ + t ′′ ]] ℓ or [[sense(ℓ ′ , t ′ , t ′′ )]] ℓ or [[A]] ℓ , then t can be turned into a normal form by application of axioms Dep 0−5,6,7 and induction over [[t ′ ]] ℓ and [[t ′′ ]] ℓ .
• if t ≡ (C, η).t ′ or t ≡ t ′ + t ′′ , then t can be turned into normal form by induction over t ′ and t ′′. • if t ≡ (C, η).t ′ or t ≡ t ′ + t ′′ , then t can be turned into normal form by induction over t ′ and t ′′ .
. C , t 1 ⊑ (C, η).t 2(C, η).t 1 ⊑ (C, η).t 2 ;
. ∂ M ; ⊑ ∂ M, ∂ M (t 1 ) ⊑ ∂ M (t 2 );
| []
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[
"On Dimensional Analysis, Redundancy in set of fundamental quantities and Proposal of a New Set",
"On Dimensional Analysis, Redundancy in set of fundamental quantities and Proposal of a New Set"
]
| [
"Mohd Abubakr [email protected] \nMicrosoft R & D\nHyderabad\n"
]
| [
"Microsoft R & D\nHyderabad"
]
| []
| Inclusion of redundant fundamental quantities in SI system has resulted in lot of ambiguities and confusion in modern theories. The incompatibilities between the existing theories can possibly be due to incorrect assumption of fundamental quantities and existence of newer fundamental quantities. This paper is an attempt to eliminate the redundancy in the SI system using random mathematical mappings done through a computer program on SI system to New System. Each mathematical mapping is studied manually. Out of 1000 random mathematical mappings generated using a Computer Program, a mapping with "three fundamental quantities" is found to describe a non-redundant and intuitive set of Fundamental quantities. This paper also proposes the possible existence of a new fundamental quantity. A significant attempt is made to understand the advantages of new system over SI system. The relations between the set {Mass, length, time, current, and temperature} and new set of three fundamental quantities are calculated. The paper also describes the intuitive reasons favoring the new fundamental set of quantities. | null | [
"https://arxiv.org/pdf/0710.3483v1.pdf"
]
| 117,537,739 | 0710.3483 | 7103ff95ed47c9baa3d4953f62f935ead94f57d1 |
On Dimensional Analysis, Redundancy in set of fundamental quantities and Proposal of a New Set
Mohd Abubakr [email protected]
Microsoft R & D
Hyderabad
On Dimensional Analysis, Redundancy in set of fundamental quantities and Proposal of a New Set
Inclusion of redundant fundamental quantities in SI system has resulted in lot of ambiguities and confusion in modern theories. The incompatibilities between the existing theories can possibly be due to incorrect assumption of fundamental quantities and existence of newer fundamental quantities. This paper is an attempt to eliminate the redundancy in the SI system using random mathematical mappings done through a computer program on SI system to New System. Each mathematical mapping is studied manually. Out of 1000 random mathematical mappings generated using a Computer Program, a mapping with "three fundamental quantities" is found to describe a non-redundant and intuitive set of Fundamental quantities. This paper also proposes the possible existence of a new fundamental quantity. A significant attempt is made to understand the advantages of new system over SI system. The relations between the set {Mass, length, time, current, and temperature} and new set of three fundamental quantities are calculated. The paper also describes the intuitive reasons favoring the new fundamental set of quantities.
INTRODUCTION
The idea of fundamental quantities of physics is nothing new to the scientific literature. Time and again there have been modifications in the notion of the primary constituents of the matter and Universe. As per the ancient Greek Philosophers, matter constituted of one single element. Later, it was modified to four fundamental elements i.e. fire, earth, water and air. Ancient Indians had theories with five fundamental elements i.e. Agni (fire), Pritvi (earth), jala (water), vayu (air) and akasha (Ether). Though scientifically this idea has been abandoned since a long time, nevertheless it was the stepping-stone about understanding the fundamental constituents of the Universe. Analyzing the reasons of attributing a certain physical quantity as fundamental reveals that such attribution is primarily dependent on the science of the existing era. If the researchers were unable to probe further into a quantity or were unable to split it into more fundamental gradients, they attributed such quantities as 'fundamental'.
The fundamental quantities that are currently being accepted by the scientific community are mass, time, length, current, temperature, luminous intensity and amount of substance also referred as SI units [9,10]. Amount of substance and luminous intensity are mere numbers and are continued as fundamental quantities only due to historic reasons. Ignoring luminous intensity and amount of substance from the further discussion, the present fundamental set consists of five constituents, namely mass, time, length, current and temperature. In order to reduce the numeric complexity of fundamental constants, most of the scientific literature is expressed in Natural Units such as Planck's units [11]. However, even the so-called Natural Units can dimensionally expressed in SI Units.
As our modern theories are built on the assumptions of fundamental quantities, any conflict in the choosing the fundamental quantities could result into chaos. If the fundamental quantities turn out to be redundant or incorrect, then the very foundation of any modern theory will fall trembling. With the multiple theories existing in modern day physics with conflicting results, it is noteworthy to doubt the correctness of existing set of fundamental quantities. In case, the existing set of fundamental quantities is wrong, then there are different possibilities such as 1. Redundancy in Fundamental quantities 2. Existence of new fundamental quantities 3. Redundancy in fundamental quantities and Existence of new fundamental quantities.
This paper proceeds with an assumption that "there exists an inherent redundancy in the existing fundamental quantities and existence of new fundamental quantities". If the assumption is wrong, then there should not exist a new set that can satisfy the existing scientific literature. The simplest ways to prove the correctness of the assumption is discover a new fundamental quantity and experimentally locate a redundancy in fundamental quantities. However, the method adopted to prove the assumption in this paper is rather different.
HUNT FOR FUNDAMENTAL SET
The assumption of existence of inherent redundancy and existence of new fundamental quantities can be interpreted as following, "there exists a new fundamental set with less than five fundamental quantities and Mass, Time, Length, current and temperature can be dimensionally expressed in terms of new fundamental quantities". In order to achieve this task a computer program is written that maps the existing five fundamental quantities into a new smaller set of fundamental quantities. The new fundamental set is purely mathematical. The algorithm used in the computer program has no intelligence and it merely generates different mappings. The only advantage of this computer program is it generates hundreds of mappings within a fraction of a second. The algorithm was run to produce more than 1000 different mappings.
Once the mappings are generated, each mapping was manually analyzed. While analyzing the mapping of fundamental quantities into new fundamental quantities, it is sensible to take the existing knowledge base (existing literature) into account. The new prediction can only be justifiable if it satisfies the all constraints of the existing knowledge base and also opens horizons for future research. If the assumption about the 'existing redundancy in the fundamental quantities and existence of new fundamental quantities' is incorrect, then there exists no new set that satisfies the existing knowledge base.
ALGORITHM
A program was written to map the five base quantities {M, L, T, K and A} into new three base quantities {X, Y and Z}. A simple three-step algorithm followed by the program is given below. Let B and N represent the set of base quantities {M, L, T, K and A} and {X, Y and Z} respectively. The algorithm is as follows
Step 1: Assign the dimensional formula X a Y b Z c to each base quantity of set B, where a, b and c are three random variables chosen differently for each base quantity.
Step 2:
Substitute the new dimensional formula of set B in the dimensional formula of the derived quantities such as force, energy etc.
Step 3:
Save all the dimensional formulae and different values of {a, b, c}.
The saved results in the Step 3 in the algorithm are analyzed manually. Note that the algorithm purely relies on the probability of the random combination of the N assigned to B. The above algorithm was implemented for 1000 random combinations of N for B. The next section of the paper puts light on the process that was carried out to implement the above algorithm.
PROGRAM OVERVIEW
A computer program was written to generate 3 random values {a, b and c} from the set R {-6.00, -5.50, -5.00, -4.50, -4.00, -3.50, -3.00, -2.50, -2.00, -1.50, -1.00, -0.50, 0.00, 0.50, 1.00, 1.50, 2.00, 2.50, 3.00, 3.50, 4.00, 4.50, 5.00, 5.50, 6.00}. Set R consists of total 25 values. Once the 3 random values are obtained, the new dimensional formula X a Y b Z c for a particular quantity from the Set B is created. Once all the quantities in the Set B are assigned a new dimensional formula, the corresponding values of {M, L, T, K and A} are substituted in the dimensional formulae of derived quantities such as velocity, force, energy, charge, permittivity, permeability, specific heat etc. In all, the program contains a database of 40 derived quantities. The resultant set of dimensional formulae of the derived quantities is taken out in a printed sheet and analysis is done manually. Presence of ambiguities in the dimensional formulae reflects that random combination of Set N assigned to Set B is redundant and wrong. Below is the example of random combination that resulted in the ambiguities.
EXAMPLE FOR A RANDOM COMBINATION
One of the random combinations generated by the program is given as below
M = X 1 Y 0 Z 0 for {a = 1, b = 0 and c = 0) L = X 0 Y 1 Z 0 for {a = 0, b = 1 and c = 0) T = X 0 Y 0 Z 1 for {a = 0, b = 0 and c = 1) K = X 1 Y 1 Z 0 for {a = 1, b = 1 and c = 0) A = X 0 Y 1 Z 1 for {a = 0, b = 1 and c = 1)
The new dimensional formulae obtained for {M, L, T, K and A} are substituted in the dimensional formulae of derived quantities. Consider the dimensional formulae of derived quantities obtained for the above substitutions,
Velocity = [L 1 T -1 ] = [X 0 Y 1 Z -1 ] Acceleration = [L 1 T -2 ] = [X 0 Y 1 Z -2 ] Force = [M 1 L 1 T -2 ] = [X 1 Y 1 Z -2 ] Energy = [M 1 L 2 T -2 ] = [X 1 Y 2 Z -2 ] Universal Gravitational Constant = [M -1 L 3 T -2 ] = [X -1 Y 3 Z -2 ] Charge = [AT] = [X 0 Y 1 Z 2 ] Voltage = [M 1 L 2 T -3 A -1 ] = [X 1 Y 1 Z -4 ] Magnetic Pole Strength = [A 1 L 1 ] = [X 0 Y 2 Z 1 ] Magnetic Moment = [A 1 L 2 ] = [X 0 Y 3 Z 1 ] Intensity of magnetization = [A 1 L -1 ] = [X 0 Y 0 Z 1 ] Specific Heat = [L 2 T -2 K -1 ] = [X -1 Y 1 Z -2 ] Thermal Capacity = [M 1 L 2 T -1 K -1 ] = [X 0 Y 1 Z -1 ] Permeability = [M 1 L 1 T -2 A -2 ] = [X 1 Y -1 Z -4 ] Permittivity = [M -1 L -3 T 4 A 2 ] = [X -1 Y -1 Z 6 ] Impedance = [M 1 L 2 T -3 I -2 ] = [X 1 Y 0 Z -5 ]
It can be seen that derived quantities have retained their uniqueness even after the 5 base quantities were transformed into 3 base quantities. Had their been no redundancy in the Set B {M, L, T, K and A}, the resultant set of formulae of derived quantities in the above example should have conflicted with each other. One of the challenging tasks here is to make the dimensional formulae for derived quantities more perceptive to practical observations. Hence the more number of random combinations were further probed to find the combinations that will have additional advantages over others. The next section of the paper gives the detailed analysis of a new dimensional system that is found to be most intuitively represents the practical observations out of 1000 mappings that were manually analyzed.
NEW DIMENSIONAL SET
Out of 1000 random combinations that have been manually tested, the following combination was found to be very close to the threshold of being more perceptive towards practical observations. However, there is a possibility of existence of more advantageous combinations cannot be ruled out. Since L = L = X 0 Y 1 Z 0 , we can interpret that Y = L. Similarly, T = X 0 Y 0 Z 1 can be interpreted as Z = T. Hence the new system is introducing only one unknown quantity "X". To make the new dimensional set sound more realistic, lets call this new base quantity as "Chakr" (Wheel) and represent it with letter "C". Hence the The construction set is distinct, non-redundant and satisfies all the norms of existing literature. The next section throws light on the intuitive reasons favoring the construction set.
INTUITIVE REASONS FAVORING THE CONSTRUCTION SET
Here are few highlights of the construction set 1. One of the first remarkable changes brought by the Construction set is the change in the dimensional formula of Mass. There has been no reported discovery of any particle that has non-zero mass and has volume equal to zero. It can be argued that mass occupies volume. The answer to the above thought example is "no" since Alice doesn't recognize something called as "mass". The 3. It is known that Vacuum has impedance, equal to 376.7 Ω. This is widely used in communication system as "free space impedance" [18]. Let us revisit the thought example that we considered previously, "Now Alice decides to measure the free space impedance, will Alice includes "Mass" in the dimensional formula of impedance?" The answer is no, since Alice doesn't recognize something called as "mass"
Again, the answer is no, since Alice doesn't recognize something called as 'mass'. If vacuum has impedance, then it is obvious that it should not have 'mass' in dimensional formula. The new dimensional system developed here rightly depicts this scenario. The new dimensional formula of "Impedance" is [L -5 T 2 ].
4. One of the significant results obtained through this new dimensional system is that fact that the dimensional formula of "Energy" is equivalent to Temperature. It is known in literature that at increase in temperature corresponds to increase in rate of movements of atom in the system. This is also attributed as increase in the thermal energy of the system. In the fields of plasma physics and quantum electrodynamics, temperature is calculated in terms of electron volts and then converted into Kelvin by multiplying with suitable constant. From the above cases, it can be seen that Temperature is nothing but a form of energy and hence the dimensional formula of Temperature and Energy are equivalent. Rightly, the gas constant, which represents a mere statistical number, is a dimensionless quantity.
5. Considering the fact that "Length" and "Time" have retained their position as fundamental quantities and quantities such as "Mass", "Temperature" and "Current" are derived in terms of "Length", "Time" and "Chakr". The proposal of existence of unknown fundamental quantity "Chakr" is the most noteworthy concept of this new dimensional system. Potentially the concept of "Chakr" solves the trivial problem existing in the current physics. According to the standard model, fundamental particles obtain their masses upon interacting with the scalar background Higg's field, which happens to be in non zero ground state [13][14][15][16][17]. However there is an ambiguity about the existence of the Higg's field. The new dimensional formula of "mass" entirely changes the point of view regarding "mass". As per the new dimensional system, mass is a mere derived quantity.
6. Law of Conservation of Fundamental quantities of Construction Set: Any fundamental quantity of the Construction set can neither be created nor destroyed. This is a new insightful aspect of Construction Set unlike the older set of {M, L, T, A and K}. In the older set, we had the possibility such as "Mass can be totally destroyed to obtain energy and this energy can be converted into electrical energy or heat energy". The simplest of the argument that can be made here is that "if a particular quantity is fundamental, it can neither be created or destroyed, it exists for eternal".
7. In the older dimensional system, the presence of mass in the dimensional formula of Energy has resulted in constraints such as zero rest mass. It is to say that any physical entity that has energy and non-existent mass is attributed as "zero rest mass". The new dimensional system resolves this issue, as mass and energy are not directly related dimensionally.
8. As per the new dimensional system, the dimensional formula of derived quantities is more logical compared with previous one. Gravitational and Electrical forces are approximated with inverse square law. This effect is reflected in the dimensional formula of Force, which has [L 2 ] in it. 9. In the quantum field theories, charge is treated as fundamental quantity whereas the dimensional system treats electric current as a fundamental quantity. It is startling to note that "charge" has "time" in its dimensional formula even though existence of charge is independent of time. As per the construction set, the dimensional formula of charge is independent of time and the dimensional formula of current is dependent on time.
10. With the ongoing conflict between quantum mechanics and relativity, the Construction set offers a new paradigm to unify both the theories. The construction set is perfectly compatible with the existing literature plus it offers a whole new dimensional quantity that can be defined as per the needs of unification of the two big theories [1][2][3][4][5][6][7][8].
CONCLUSIONS
Incorrect assumption of fundamental quantities can negatively affect the success of the theories. The incompatibility of the modern day theories is a possible example of incorrect choice of fundamental quantities. In this paper, a new set of fundamental quantities is described by removing the redundancy in the SI fundamental units. The New set, referred in the paper as "Construction Set" proposes a new fundamental quantity called as "Chakr" [C]. The Construction set includes three fundamental quantities namely, Length, Time and Chakr. All the other derived quantities are dimensionally described in terms of Length, Time and Chakr. The basic mapping between the set {Length, Time and Chakr} and {Mass, Length, Time, Current and Temperature} is evaluated after analyzing 1000 mappings generated by the Computer Program. The paper also describes the intuitive reasons supporting the new "Construction Set" as ideal set of fundamental quantities. The scope of future work includes studying the properties of Chakr and description of modern theories in terms of Construction set.
Heat ML 2 T -2 CL -1 T 36
Coeff. Of thermal expansion K -1 C -1 LT -1 37
Specific heat L 2 T -2 K -1 C -1 L 3 T -3 38
Thermal capacity ML 2 T -1 K -1 T 39
Gas constant ML 2 T -2 K -1 Dimensionless 40
Botlzmann constant and entropy ML 2 T -2 K -1 Dimensionless 41
Latent heat L 2 T -2 L 2 T -2 42
Coeff. Of thermal conductivity MLT -3 K -1 L -1 T -1 43
Stefan's constant MT -3 K -4 C -3 LT -4 44
Magnetic pole strength AL C 1/2 L 3 T -1 45
Magnetic moment AL 2 C 1/2 L 4 T -1 46
Flux density MT -2 A -1 C 1/2 L -5 T 2 47
Magnetic flux ML 2 T -2 A -1 C 1/2 L -3 T 2 48
Intensity of magnetic field, Intensity of magnetization AL -1 C 1/2 LT -1 49
Permittivity M -1 L -3 T 4 A 2 L 4 T -1 50
Permeability MLT -2 A -2 L -6 T 3 51
Magnetic susceptibility Nil Dimensionless 52
Electric potential, E.M.F ML 2 T -3 A -1 C 1/2 L -3 T 53
Electric capacity M -1 L -2 T 4 A 2 L 5 T -1 54
Intensity of electric field MLT -3 A -1 C 1/2 L -4 55
Electric resistance ML 2 T -3 A -2 L -5 T 2 56 Specific resistance ML 3 T -3 A -2 L -4 T 2 57
Conductance M -1 L -2 T 3 A 2 L 5 T -2 58
Self inductance, mutual inductance ML 2 T -2 A -2 L -5 T 3 59
Rydberg constant, wave number L -1 L -1 60 Compressibility M -1 LT 2 C -1 L 4 T -1
Fig 1 .
1Schematic View of Mapping done between Construction set and Reduced SI set
new dimensional system consists of Length [L], Time [T] and Chakr [C], lets call this new set as "Construction Set". values of {M, L, T, K and A} in the dimensional formulae of the derived quantities, Appendix-I for other derived quantities]
new dimensional system accurately depicts this scenario unlike the older version with only length and time in their dimensional formulae. The new dimensional formulae of permittivity and permeability are [L 4 T -1 ] and [L 6 T 3 ].
Appendix -ISerial NumberQuantitySI dimensions Construction Set 1 Length L L 2 Time T T 3 Chakr -C 4EnergyVelocity LT -1 LT -1 8AccelerationForce MLT -2 Cl -2 T 10Mass M CL -3 T 3 11Mass density ML -3 CL -6 T 3 12ElectricPower Velocity gradient T -1 T -1 34Universal gravitational constant M -1 L 3 T -2 C -1 L 6 T -5
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Antennas -For All Applications, Tata Mc-Graw-hill. John D Kraus, Ronalad J Marhefka, John D. Kraus, Ronalad J.Marhefka, Antennas -For All Applications, Tata Mc-Graw-hill, 2003.
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[
"Quantum coadjoint orbits of GL(n) and generalized Verma modules",
"Quantum coadjoint orbits of GL(n) and generalized Verma modules"
]
| [
"J Donin \nDepartment of Mathematics\nBar Ilan University\n52900Ramat GanIsrael\n",
"A I Mudrov \nDepartment of Mathematics\nBar Ilan University\n52900Ramat GanIsrael\n"
]
| [
"Department of Mathematics\nBar Ilan University\n52900Ramat GanIsrael",
"Department of Mathematics\nBar Ilan University\n52900Ramat GanIsrael"
]
| []
| In our previous paper, we constructed an explicit GL(n)-equivariant quantization of the Kirillov-Kostant-Souriau bracket on a semisimple coadjoint orbit. In the present paper, we realize that quantization as a subalgebra of endomorphisms of a generalized Verma module. As a corollary, we obtain an explicit description of the annihilators of generalized Verma modules over U gl(n) . As an application, we construct real forms of the quantum orbits and classify finite dimensional representations. We compute the non-commutative Connes index for basic homogenous vector bundles over the quantum orbits. | 10.1023/b:math.0000035038.98122.93 | [
"https://arxiv.org/pdf/math/0212318v1.pdf"
]
| 18,638,339 | math/0212318 | cd9cc003141dc385fe7071000d45f01fefdf3efe |
Quantum coadjoint orbits of GL(n) and generalized Verma modules
23 Dec 2002
J Donin
Department of Mathematics
Bar Ilan University
52900Ramat GanIsrael
A I Mudrov
Department of Mathematics
Bar Ilan University
52900Ramat GanIsrael
Quantum coadjoint orbits of GL(n) and generalized Verma modules
23 Dec 2002Kirillov-Kostant-Souriau bracketequivariant quantizationgeneralized Verma modules AMS classification codes: 53D5553D0522E47
In our previous paper, we constructed an explicit GL(n)-equivariant quantization of the Kirillov-Kostant-Souriau bracket on a semisimple coadjoint orbit. In the present paper, we realize that quantization as a subalgebra of endomorphisms of a generalized Verma module. As a corollary, we obtain an explicit description of the annihilators of generalized Verma modules over U gl(n) . As an application, we construct real forms of the quantum orbits and classify finite dimensional representations. We compute the non-commutative Connes index for basic homogenous vector bundles over the quantum orbits.
to an orbit is called the Kirillov- Kostant-Souriau (KKS) bracket. Importance of this bracket is accounted for the fact that every G-homogeneous symplectic manifold is locally isomorphic to a G-coadjoint orbit via the moment map. Construction of G-equivariant quantization of the KKS bracket is a classic problem of the deformation quantization theory, [BFlFrLSt]. There are various approaches to quantization of Poisson manifolds. One of them, the * -product, presents a deformed multiplication in A(M) ⊗ C C[[t]] (the tensor product is completed in the t-adic topology) as a formal t-series with coefficients being bi-differential operators. Famous Fedosov's construction guarantees existence of an equivariant * -product on a symplectic manifold with a G-invariant connection. However, Fedosov's quantization is not given by an explicit formula for a particular manifold. Another approach to quantization is to describe the algebra A t (M) in terms of generators and relations. While associativity holds by the very construction and G-equivariance can be easily guaranteed, the principal difficulty is to ensure flatness of an algebra built in such a way.
There is a universal approach to equivariant quantization of semisimple coadjoint orbits of complex reductive Lie groups which is based on generalized Verma modules, [DGS]. By a generalized Verma module V p,λ we mean the left U(g)-module U(g) ⊗ U (p) C λ , where p ⊂ g is a parabolic subalgebra and C λ is a one-dimensional representation of U(p). It is generated by a p-character λ, which can be identified with a certain element of g * . The approach of [DGS] presents a quantized orbit as a subalgebra of endomorphisms of a generalized Verma module.
Consider the Lie algebra g t over C[t] which coincides with g[t] as a C[t]-module and whose Lie bracket [ . , . ] t extends from the bracket of g: [x, y] t = t [x, y] for all x, y ∈ g. The universal enveloping algebra U(g t ) is a G-equivariant quantization of the polynomial algebra on g * .
There is a family of isomorphisms U(g t ) → U(g) at t = 0 (this extends to a natural embedding of C[t]-algebras) such that the image of the composite map U(g t ) → U(g) → End(V p,λ/t ) gives an equivariant quantization of the orbit passing through λ.
The approach of [DGS] also presents quantized orbits as quotients of U(g t ). This fact was used in [DM2] for explicit description of quantized semisimple GL(n)-orbits in terms of generators and relations. There was built a C[t]-algebra A m,µ,t giving quantization of the polynomial algebra on the orbit of matrices with eigenvalues µ = (µ 1 , . . . , µ k ) of multiplicities m = (m 1 , . . . , m k ). The algebra A m,µ,t is a quotient of U g t by a certain ideal whose generators are given explicitly. This ideal can be realized (not uniquely) as the annihilator of a generalized Verma module V p,λ/t , where p and λ depend on m, µ, and t. In the present paper, we find this dependence explicitly.
Our first result is the following. We relate the two approaches to equivariant quantization of semisimple orbits: the quantization via generalized Verma modules and the one in terms of generators and relations.
As our second result, we describe the annihilator of a generalized Verma module V p,λ for generic λ as an ideal in U(g) by presenting its set of generators.
As an application, we classify all finite dimensional representations of the algebras A m,µ,t for fixed m, µ, and t. We show that they all are factored through generalized Verma modules. For a finite dimensional representation to exist, the numbers (µ i − µ j )/t should satisfy certain integral conditions. When µ is fixed, the dimension of representations grows as the deformation parameter t tends to zero. This process yields "fuzzyfication" for semisimple orbits of GL(n). Let us note that a reversed approach from fuzzy geometry to * -product quantization was used for quantization of grassmanian spaces in [DolJ]. For applications of fuzzy geometry to theoretical physics see e.g. [ARSch].
As another application, we construct real forms A m,µ,t that are compatible with the standard real forms of gl(n).
Finally, we show that special rational functions defining the central character of a quantum orbit, [DM2], give the Connes index for the basic quantum homogeneous vector bundles on the orbit. The paper is organized as follows. In Sections 2 and 3 we recall respectively the quantization via generalized Verma modules and the quantization in terms of generators and relations.
The correspondence between these two approaches is established in Section 4. Subsection 4.6 is devoted to finite-dimensional representations of the algebras A m,µ,t . In Section 5 we construct real forms on A m,µ,t . In Subsection 6.2, we compute the non-commutative Connes index for the case of two-parameter quantization of [DM2]. In Appendix we deduce an explicit polynomial expression for the central characters of quantum orbits.
2 Quantization via generalized Verma modules.
2.1. Consider a complex reductive Lie algebra g and let G be the corresponding connected Lie group. The dual space g * is a Poisson G-manifold endowed with the standard Poisson-Lie structure induced by the Lie bracket of g. This bracket can be restricted to every orbit of G in g * making it a homogeneous symplectic manifold. That symplectic structure is called Kirillov-Kostant-Souriau bracket. An element from g * is called semisimple if it is the image of a semisimple element under the G-equivariant isomorphism g → g * via a non-degenerate adinvariant inner product on g. An orbit is called semisimple if it passes through a semisimple element.
Semisimple coadjoint orbits are closed affine varieties in g * . By the (polynomial) function algebra on a subvariety M ⊂ g * we mean the algebra A(M) consisting of restrictions of polynomial functions on g * . In the present paper a quantization of M is a flat 1 C[t]-algebra
A t (M) together with an isomorphism κ : A t (M)/tA t (M) → A(M).
The quantization is called G-equivariant if there is a G-action on A t (M) by algebra automorphisms, an extension of the natural action on A(M). Such an action gives rise to a Hopf algebra action of U(g),
x ⊲ (ab) = (x ⊲ a)b + a(x ⊲ b), x ∈ g, a, b ∈ A t (M),
and vice versa, so we also call a G-equivariant quantization U(g)-equivariant.
2.2.
Let h be a Cartan subalgebra in g and g = n − ⊕ h ⊕ n + a triangular decomposition relative to h. Let p be a parabolic subalgebra in g containing h and n + and let l be its Levi factor. The projection g → h along the triangular decomposition gives rise to an embedding h * → g * . Denote by c the center of l; then c * is identified with the annihilator of [l, l] in h * via decomposition h = [l, l] ∩ h ⊕ c. Any λ ∈ c * defines a one-dimensional representation C λ of p ⊃ l, which extends to a representation of the universal enveloping algebra U(p). By definition, the generalized Verma module V p,λ is the left U(g)-module U(g) ⊗ U (p) C λ . When p coincides with the Borel subalgebra b = h ⊕ n + , V b,λ is the ordinary Verma module, i.e. the maximal object in the category of U(g)-modules with the highest weight λ.
Denote by g t the Lie algebra over C[t] which coincides with g[t] as a C[t]-module and equipped with the Lie bracket [x, y] t = t[x, y] for all x, y ∈ g. The universal enveloping
algebra U(g t ) is a G-equivariant quantization of g * . If v is a subspace in g, we put v t = v[t] ⊂ g[t]
. Given a parabolic subalgebra p t ⊂ g t and its character λ (one dimensional representation on C[t]) we denote by V pt,λ the U(g t )-module U(g t ) ⊗ U (pt) C[t] and call it a generalized Verma module over U(g t ).
Introduce the subset c * reg of elements in c * ⊂ h * whose stabilizer in g is l ⊃ c. We call such elements regular. The set c * reg is a dense cone in c * . Let us fix an element λ ∈ c * reg .
Theorem 2.1 ( [DGS]). The image A pt,λ of U(g t ) in End(V pt,λ ) is a G-equivariant quantization of the KKS bracket on the semisimple orbit O λ ⊂ g * passing through λ.
1 By flatness of a C[t]-module we mean flatness of its t-adic completion over C [[t]].
Let J pt,λ ⊂ U(g t ) denote the annihilator of the module V pt,λ . Theorem 2.1 says that the G-equivariant quantization of a semisimple orbit O λ can be realized as a quotient of the algebra U(g t ) by the ideal J pt,λ .
3 Explicit quantization for the GL(n)-case 3.1. From now on g stands for the Lie algebra gl(n) of the complex general linear group G = GL(n). Let us specialize the definitions of the previous subsection to this case. The
Cartan subalgebra h is chosen to be the subalgebra of diagonal matrices; the nilpotent subalgebras n ± consist of respectively upper-and lower-triangular matrices. Using the trace pairing, we identify the dual space g * with End(C n ) and h * with the subspace of diagonal matrices in End (C n The Levi subalgebra l ∈ g consists of block diagonal matrices, l = ⊕ k i=1 g i , where k is the number of blocks and g i = gl(m i ). We denote by m(l) the k-tuple (m 1 , . . . , m k ) ∈ {n : k} + . Clearly the parabolic subalgebras in p containing upper-triangular matrices are in one-to-one correspondence with elements of ∪ n k=1 {n : k} + . Consider the decomposition C n = C m 1 ⊕ . . . ⊕ C m k and denote by p i : C n → C m i the corresponding projectors. The center of p is the linear space c = ⊕ k i=1 Cp i . Under the identification g * ≃ g, the dual space c * coincides with c, and the linear isomorphism between C k and c * ⊂ h * is given by the map
µ → k i=1 µ i p i . The image of the subspace C k reg ⊂ C k is the set of regular elements c * reg ⊂ c * .
3.2. The algebra U(g t ) is generated by the set {E i j } n i,j=1 ⊂ g t of elements satisfying the relations
E i j E m l − E m l E i j = t(δ m j E i l − δ i l E m j ), i, j, l, m = 1, . . . , k.(1)
They can be arranged into a matrix, E = ||E i j || n i,j=1 . The matrix E can be considered as
an element E = n i,j=1 E i j ⊗ e i j ∈ U(g t ) ⊗ C End(C n ), where {e i j } n i,j=1
is the standard base in End (C n ). It is invariant with respect to the diagonal action of U(g) on U(g t ) ⊗ C End(C n ).
We fix a multiplication in End(C n ) by setting it on the base by the formula e i j e l m = δ l j e i m , where δ l j is the Kronecker symbol. One can consider polynomials in E as an element of the algebra U(g t ) ⊗ C End(C n ). Explicitly, the ℓ-th power E ℓ is a matrix with the entries
(E ℓ ) i j = n α 1 ,...,α ℓ−1 =1 E α 1 j E α 2 α 1 . . . E i α ℓ−1 .(2)
We will also consider the matrix algebra End • (C n ) whose multiplication is opposite to that of End(C n ). The matrix E may be thought of as an element from U(g t )
⊗ C End • (C n ), then its ℓ-th power E •ℓ is explicitly (E •ℓ ) i j = n α 1 ,...,α ℓ−1 =1 E i α 1 E α 1 α 2 . . . E α ℓ−1 j .
(3)
In [DM2], a special series ϑ ℓ (m, µ, q −2 , t) ∞ ℓ=0
of polynomials in µ ∈ C k and t, q −2 ∈ C was introduced in connection with a two-parameter quantization on semisimple orbits. The notationm stands for the vector (m 1 , . . . ,m k ), where the hat denotes the q-integers,m =
1−q −2m 1−q −2 , m ∈ N. Here we will use the restriction ϑ ℓ (m, µ, t) = ϑ ℓ (m, q −2 , µ, t)| q=1 . An explicit polynomial expression for ϑ ℓ (m, µ, t) is presented in Appendix, formula (28). Definition 3.1. Let m = (m 1 , . . . , m k ) ∈ {n : k} + and µ = (µ 1 , . . . , µ k ) ∈ C k . 1. The C[t]-algebra A m,µ,t is a quotient of U(g t ) by the ideal specified by the relations (E − µ 1 ) . . . (E − µ k ) = 0,(4)Tr E ℓ = ϑ ℓ (m, µ, t), ℓ ∈ 1, . . . , k − 1.(5)
2. The C[t]-algebra A • m,µ,t is a quotient of U(g t ) by the ideal specified by the relations
(E − µ 1 ) • . . . • (E − µ k ) = 0,(6)Tr E •ℓ = ϑ ℓ (m, µ, −t), ℓ ∈ 1, . . . , k − 1.(7)
The algebras A m,µ,t and A • m,µ,t are related by a certain transformation of parameters whose exact form will be presented in Subsection 5.2, formula (19). We will use that relation in Subsection 5.4 concerning real forms of quantum orbits.
Theorem 3.2 ( [DM2]). Given µ ∈ C k reg and m ∈ {n : k} + the algebra A m,µ,t (A • m,µ,t ) is a G-equivariant quantization of the semisimple orbit of matrices with eigenvalues µ of multiplicities m.
This theorem was proven in [DM2] for the algebra A m,µ,t . For the algebra A • m,µ,t the proof is analogous. Besides, the family A • m,µ,t can be obtain from A m,−µ,t via the automorphism of the C-algebra U(g t ) extending the map
E i j → −E i j , t → −t.
3.3. Let S k be the symmetric group of permutations of a k-element set. It acts on the family of algebras A m,µ,t through the action on the k-tuples m and µ. Denote by l the element
(1, . . . , 1) ∈ C k . Proposition 3.3. The C[t]-algebras A m,µ,t and A m ′ ,µ ′ ,t are isomorphic if and only if there
is an element τ ∈ S k and a complex number b such that m ′ = τ (m) and µ ′ = τ (µ) + bl. The same holds for the family A • m,µ,t as well.
Proof. The proof is straightforward in one direction. Indeed, equations (4) and (5) are
symmetric with respect to permutations of the pairs (m i , µ i ). The correspondence E i j → E i j − bδ i j extends to an isomorphism A m,µ,t → A m,µ+bl,t ,
as it is seen from relations (1), (4), and (5).
Conversely, suppose A m,µ,t and A m ′ ,µ ′ ,t are isomorphic as C[t]-algebras. They are quantizations of orbits characterized by eigenvalues of constituent matrices and their multiplicities.
Those orbits are isomorphic as symplectic manifolds if and only if there is a permutation τ and a complex number b such that m ′ = τ (m) and µ ′ = τ (µ) + bl.
Remark 3.4. Given a C[t]-module V we will often treat it as a family of C-modules. Special-
ization of V at a point t = t 0 is the C-module V ⊗ C[t] C corresponding to the C-homomorphism C[t] → C[t]/(t − t 0 ) ≃ C.
3.4 (Cayley-Hamilton identity). The elements Tr E ℓ , ℓ = 1, . . . , n, generate the center Z(g t ) of the algebra U(g t ). There is another set of generators of Z(g t ), namely, the coefficients, {c i } n i=1 , of the "characteristic" polynomial equation (8) is a non-commutative analog of the Cayley-Hamilton identity in the classical polynomial algebra on matrices. Its existence immediately follows from representation theory arguments and the fact that U(g t ) is an equivariant quantization on g * , see [DM2].
P(E) = E n − c 1 E n−1 + . . . + (−1) n c n = 0 (8) identically held in U(g t ) ⊗ C End(C n ). Equation
Let us define two characters of the center Z(g t ) that will appear in what follows.
1. The assingment Tr E ℓ → ϑ ℓ (m, µ, t), ℓ ∈ N, defines a central character, [DM2], which we denote by χ m,µ .
2. Given a weight λ ∈ h * t we denote by χ λ the central character whose kernel lies in the annihilator of the Verma module V bt,λ . It follows that if λ is a character of a parabolic subalgebra p t ⊃ b t , then ker χ λ lies in J pt,λ , the annihilator of the generalized Verma module V pt,λ . One has zv 0 = χ λ (z)v 0 , where z ∈ Z(g t ) and v 0 ∈ V pt,λ is the highest weight vector.
Let χ be a character of the center Z(g t ). Consider its specialization at t = 0, which is a C-algebra homomorphism Z(g t ) → C. We denote by R(χ) the set of roots of the polynomial
x n − χ(c 1 )x n−1 + . . . + (−1) n χ(c n )(9)
with complex coefficients χ(c i ).
Proposition 3.5. Given m ∈ {n : k} + and µ ∈ C k one has
R(χ m,µ ) = µ 1 , µ 1 − t, . . . , µ 1 − (m 1 − 1)t; . . . ; µ k , µ k − t, . . . , µ k − (m k − 1)t . (10)
Proof. This follows from the construction of the functions ϑ ℓ (m, µ, t), [DM2].
Remark 3.6. Relations (5) specify an ideal in U(g t ) which is generated by the kernel of the character χ m,µ . Remark that for k = n matrix polynomial (4) is obtained from (8) by the substitution c i → χ m,µ (c i ), as follows from (10). Thus equation (4) becomes superfluous, being a consequence of (5) and (8). This situation corresponds to an orbit of maximal rank, which is determined solely by a central character, [Kost].
4 Generalized Verma modules and quantum orbits 4.1. Theorem 2.1 describes quantization of the semisimple orbit O λ as a quotient of the algebra U(g t ) by the ideal J pt,λ . Assuming λ to be a formal function, λ = λ(µ, t), such that
λ(µ, 0) = µ ∈ c * reg , we obtain a family, A pt,λ(µ,t) , of non-isomorphic quantizations of O µ . The vector µ is an element of C k under the identification c * ∼ C k . On the other hand, the algebra A m,µ,t , where m = m(l), is a quantization of the same orbit O µ and it is a quotient of U(g t ), too. The question is what dependence λ(µ, t) ensures an isomorphism A pt,λ(µ,t) ≃ A m,µ,t .
Lemma 4.1 ( [DM2]). Let p be a parabolic subalgebra with the center c ≃ C k . For any λ ∈ c * reg there exists a polynomial in one variable,
p(x) = x k − σ 1 x k−1 + . . . + (−1) k σ k (11) with coefficients σ ℓ ∈ C[[t]] such that the entries of the matrix p(E) = ||p(E) i j || lie in J pt,λ .
For λ ∈ c * reg , the polynomial p has only simple roots. Rephrasing this lemma, the entries of the matrix p(E) = ||p(E) i j || annihilate the Verma module V pt,λ . Thus the coefficients σ i of the polynomial p depend on λ and t. At the same time they are symmetric polynomials of its roots {µ}. Our goal is to find the relation between µ, λ, and t. The present section is devoted to a proof of the following theorem.
Theorem 4.2. Let p ⊂ g be a parabolic subalgebra with the Levi factor l and the center c ≃ C k . Put m = m(l) ∈ {n : k} + and take a regular element λ ∈ c * reg ≃ C k reg as a p-character. Then
1. the algebra A m,µ,t is isomorphic to A pt,λ over C[t] with µ i = λ i − i−1 α=1 m α t, i = 1, . . . , k. (12) 2. the algebra A • m,ν,t is isomorphic to A pt,λ over C[t] with ν i = λ i + k α=i+1 m α t, i = 1, . . . , k.(13)
The rest of the section is devoted to the proof of Theorem 4.2. We prove only statement 1; statement 2 is verified in the same manner.
4.2.
Consider the coefficients σ ℓ of p(x) in (11) as the elementary symmetric polynomials in µ ∈ C k , σ ℓ (µ) = 1≤i 1 <...<i ℓ ≤m µ i 1 . . . µ i ℓ , ℓ = 1, . . . , k. First of all, to prove Theorem 4.2, we must show that the matrix entries p(E) i j annihilate the generalized Verma module V pt,λ provided condition (12) holds. Let us check the following elementary lemma.
Lemma 4.3. Let W ⊂ U(g t ) be a submodule with respect to the adjoint representation. The left ideal J W = U(g t )W coincides with the right ideal W U(g t ), which is therefore a two-sided ideal. Then J W ⊂ J pt,λ if and only if W annihilates the highest weight vector of V pt,λ .
Proof. Let ∆ and γ denote the comultiplication and antipode of the Hopf algebra U(g t ). In the standard Sweedler notation with implicit summation, ∆(x) = x (1) ⊗ x (2) . Let x ∈ U(g t ) and w ∈ W . Then wx = x (1) γ x (2) wx (3) ∈ U(g t )W ; this proves that U(g t )W ⊂ W U(g t ).
The reversed inclusion is checked similarly. This proves the first assertion of the lemma.
If J W ⊂ J pt,λ , than W annihilates the highest weight vector v 0 ∈ V pt,λ . Conversely, suppose W v 0 = 0. An element v ∈ V pt,λ can be represented as xv 0 , where x ∈ U(g t ). Hence
W v = (W x)v 0 ⊂ J W v 0 = 0 and therefore J W ⊂ J pt,λ .
We will use Lemma 4.3 in the situation when the module W is spanned by the entries of the polynomial p(E), that is, W = W p = Span p(E) i j .
4.3. Consider the map σ : C n → C n , λ i → σ i (λ), where σ i are the elementary symmetric polynomials in λ ∈ C n . The differential of this map is a matrix
D(λ) = ||D ij (λ)|| n i,j=1 , where D ij (λ) = ∂σ i ∂λ j (λ). Lemma 4.4. The determinant det D(λ) is equal to 1≤i<j≤n (λ i − λ j ).
Proof. Multiply the top row of D(λ), which is ||D 1i (λ)|| = (1, . . . , 1), by D(λ) i1 and subtract it from the i-th row, i = 2, . . . , n. This kills the entries of the first column except for the upper one. The determinant is preserved and it equals to its minor det||D ij (λ)|| n i,j=2 . It is easy to see, that the j-th column of the matrix ||D ij (λ)|| n i,j=2 turns zero at λ 1 − λ j and, when divided by λ 1 − λ j , it forms the (j − 1)-th column of the (n − 1) × (n − 1) matrix D(λ ′ ), λ ′ = (λ 2 , . . . , λ n ). It remains to apply induction on n.
4.4.
The following lemma is a key step in the proof of Theorem 4.2.
Lemma 4.5. Let p ∈ g be a parabolic subalgebra with the center c ≃ C k . There is a vector a(p) ∈ C k such that for every λ ∈ C k ≃ c * the ideal J pt,λ contains the module W p for the
polynomial p(x) = (x − µ 1 ) . . . (x − µ k ) with roots {µ i } = {λ i − a i (p)t}.
Proof. First of all, let us prove this lemma assuming that p is the Borel subalgebra b = h + n + , i.e. for k = n. Denote by p the polynomial (9) obtained from the characteristic polynomial P by substitution c i → χ λ (c i ). Its entries annihilate the module V pt,λ because p(E) i j v 0 = P(E) i j v 0 = 0. So we have W p ⊂ J pt,λ . We will prove the statement for p = b if we show that the roots {µ} of the polynomial p has the form stated in the lemma. It suffices to consider only λ ∈ C n reg . The coefficients χ λ (c i ) are homogeneous polynomials in (λ, t) of degree i. On the other hand, they are elementary symmetric polynomials σ i (µ).
The map C n → C n , µ → σ(µ), is locally invertible at regular µ ∈ C n reg . Hence every point in C n reg × {0} has a neighborhood U ⊂ C n reg × C and an analytic function ψ : U → C n such that µ = ψ(λ, t), (λ, t) ∈ U, are roots of the polynomial p. We want to show that ψ(λ, t) = λ − at for some a = a(b) ∈ C n .
Observe that dilatation E i j → 1 c E i j with c = 0 extends to a C-algebra isomorphism A pt,λ → A pct,cλ . Similarly, the dilatation E i j → 1 c E i j extends to an isomorphism of A m,µ,t → A m,cµ,ct . This implies ψ(cλ, ct) = cψ(λ, t). Consider the t-expansion of the function ψ:
ψ(λ, t) = λ + j>0 t j ψ (j) (λ),(14)
where ψ (j) : C n → C n are homogeneous functions of degree −j + 1. Substituting µ = ψ(λ, t)
to σ i (µ) and comparing coefficients before t j , we find
n l=1 D il (λ)ψ (j)
l + (terms depending on ψ (l) with l < j) ∈ C n [λ].
Using induction on j and Lemma 4.4, we see that ψ Therefore ψ (j) ∈ C n [λ] and, taking into account their homogeneity degree −j + 1, all they are zero except for the term ψ (1) = −a(b). Thus we have proven the statement for p = b.
As a corollary, we obtained R(χ λ ) = {λ − a(b)t}.
Let us consider the situation of a general parabolic subalgebra p ⊃ b. Take λ ∈ c * reg and denoteλ its image under the canonical embedding in c * ⊂ h * . By Lemma 4.1, there is a polynomial p of degree k whose entries lie in J pt,λ . Let µ be its roots. The central character χ m,µ coincides with χλ, therefore {µ} ⊂ {λ − a(b)t} by Proposition 3.5. This proves the lemma when λ ∈ c * reg . But the condition W p v 0 = 0 determines the coefficients of p as rational functions of (λ, t). We have already proven that they are in fact polynomials, being the elementary symmetric polynomials in the roots. Therefore the polynomial p with the property W p ⊂ J pt,λ does exist for all λ. Its roots {µ} are contained in {λ − a(b)t}, so µ is related to λ as stated in the lemma.
Remark 4.6. The module W p is unique for regular λ and sufficiently small t. Indeed, the condition W p v 0 = 0 gives rise to a system of equations on the polynomial p. It goes over to the system p(λ i ) = 0, i = 1 . . . k, at t = 0. This uniquely determines p up to a factor at λ ∈ c * reg and t = 0, hence p is unique for λ ∈ c * reg and small t = 0.
To finish the proof of Theorem 4.2, it remains to determine the vector a(p) ∈ C k .
(Proof of Theorem 4.2).
Let p be the parabolic subalgebra with the center c ≃ C k and put m = m(l) ∈ {n : k}. In this subsection we show that the j-th coordinate of the vector a(p) ∈ C k from Lemma 4.5 is equal to j−1 i=1 m i . This will prove Theorem 4.2. Let us first consider the case k = 2. Suppose λ ∈ c * reg . Only two equations are independent in the system p(E) i i v 0 = 0, i = 1, . . . , n, say for i = 1 and i = m 1 + 1:
λ 2 1 − (µ 1 + µ 2 )λ 1 + µ 1 µ 2 = 0, λ 2 2 − (µ 1 + µ 2 + tm 1 )λ 2 + µ 1 µ 2 + tm 1 λ 1 = 0.
This system has a solution (λ 1 , λ 2 ) = (µ 1 , µ 2 + tm 1 ) satisfying the condition λ = µ at t = 0.
Thus we have proven Theorem 4.2 for k = 2.
We deduce the case k = n from the studied case k = 2. Recall that given an element
λ ∈ h * the set R(χλ) is equal to λ i − a i (b)t n i=1
. Puttingλ i = λ 1 for i = 1, . . . , m 1 and λ i = λ 2 for i = m 1 + 1, . . . , n, we obtain a central character χ m,µ associated with the algebra A m,µ,t , for m ∈ {n : 2} + , and µ ∈ C 2 . We have already shown that µ 1 = λ 1 and µ 2 = λ 2 −tm 1 . Thus we have, by Proposition 3.5,
R(χ m,µ ) = λ 1 − (i − 1)t} m 1 i=1 ∪ {λ 2 − (m 1 + i − 1)t m 2 i=1 .
On the other hand,
R(χλ) = λ 1 − a i (b)t m 1 i=1 ∪ λ 2 − a m 1 +i (b)t m 2 i=1 .
The sets R(χ m,µ ) and R(χλ) coincide since χ m,µ = χλ. Let us put t = 1 and take λ 1 and λ 2 to be positive and negative real numbers, respectively. We can assume them big enough in their absolute values so as to ensure coincidence of the subsets λ 1 − (i − 1)
m 1 i=1 ⊂ R(χ m,µ ) and λ 1 − a i (b) m 1 i=1 ⊂ R(χλ).
Using this argument, we can apply induction on m 1 and prove the case k = n of Theorem 4.2. This gives a i (b) = (i − 1), i = 1, . . . , n.
It remains to consider the case 2 < k < n. Let us set t = 1 and assume that the coordinates of λ ∈ c * reg ≃ C k are real and form a strictly decreasing sequence, λ j > λ j+1 . The vector λ ∈ C k corresponds to the vectorλ ∈ h * ≃ C n via the canonical embedding C k ≃ c * → h * ≃ C n ; the coordinates ofλ are (λ 1 , . . . , λ 1 ; . . . ; λ k , . . . , λ k ), where each λ i is taken m i times. The set R(χλ) is a union ∪ k j=1 R j of non-intersecting subsets
R j = λ j − ( j−1 l=1 m l + i − 1) m j i=1 .(15)
We have min R j > max R j+1 . On the other hand, we know from Proposition 3.5 and Lemma
that R(χλ) is a union the of subsets
R ′ j = µ j − (i − 1)} m j i=1 = {λ j − (a j (p) + i − 1) m j i=1 .(16)
We can chose such λ j that R ′ j do not intersect and, moreover, min R ′ j > max R ′ j+1 (for that to be true, it is enough to assume λ j − λ j+1 > a j (p) − a j+1 (p) + m j − 1). Taking into account # R ′ j = m j = #R j we conclude that R ′ j = R j for all j. Comparing (15) and (16), we find a(p) and thus prove Theorem 4.2.
Remark 4.7. As follows from Lemma 4.5, the identity map g t → g t extends to an epimorphism A m,µ,t → U(g t )/J pt,λ of C-algebras if the pair (p t , λ) is related to the pair (m, µ) as in (12). For regular λ this map is an isomorphism of C[t]-algebras. It is known, [J], that for almost all λ ∈ c * the annihilator of V p,λ is generated by a copy of adjoint representation and the kernel of a central character. Theorem 4.2 gives an explicit description of the annihilator.
4.6 (Representations of quantum orbits). This subsection is devoted to finite dimensional representations of the C-algebras A m,µ,t . Our principal tool is the established correspondence between the family A m,µ,t and generalized Verma modules. Note that this correspondence is not one-to-one. The algebra A m,µ,t can be realized as the algebra A pt,λ for some p t and λ in different ways. This freedom comes out from permutations (m, µ) → τ (m), τ (µ) , τ ∈ S k , leaving A m,µ,t invariant but changing the generalized Verma modules. The situation is exactly the same as in the case p = b when the Weyl group transitively acts on the set of the ordinary Verma modules with isomorphic annihilators.
Any representation of A m,µ,t is at the same time a representation of U(g t ). Therefore to describe representations of A m,µ,t it is necessary and sufficient to determine those representation of U(g t ) that are factored through the ideal specifying A m,µ,t . We do it for finite dimensional representations.
τ ∈ S k such that (µ ′ i − µ ′ i+1 )/t − m ′ i , where µ ′ = τ (µ)
and m ′ = τ (m), are non-negative integers for i = 1, . . . , k − 1. If exists, such a representation is unique and it is factored through a generalized Verma module.
Proof. As was already mentioned in the proof of Lemma 4.5, the correspondence E i j → 1 c E i j , where c = 0, extends to an isomorphism between the algebras A m,µ,t and A m,cµ,ct . Therefore we can assume t = 1 in the proof.
Suppose there is such element τ ∈ S k as stated in the proposition. Take the parabolic subalgebra p with the Levi factor l such that m(l) = m ′ . Consider the U(g)-module V p,λ , where λ is related to µ ′ by formula (12), where one should set t = 1. By assumption, the numbers λ i = µ ′ i + i−1 α=1 m ′ α define a dominant integral weight of sl(n) ⊂ g, so the module V p,λ is projected onto a finite dimensional U(g)-module W with the highest weight λ. The homomorphism of modules induces a representation of A m,µ,1 ≃ A m ′ ,µ ′ ,1 on W .
Conversely, suppose W is a finite dimensional module over A m,µ,1 . Then it is a module over U(g) as well. There is a cyclic U(g)-submodule in W with the highest weight λ. We can think that this submodule coincides with W . The highest weight gives rise to a central character, χ λ . On the other hand, one has χ λ = χ m,µ , and the set R(χ λ ) = λ j − (j − 1) n j=1 contains {µ i } k i=1 as a subset, by Proposition 3.5. Further, W is a finite dimensional U(g)module with the highest weight λ, therefore λ j −λ j+1 are non-negative integers. The elements of R(χ λ ) are ordered by their real components; then the numbers λ j − (j − 1)t ∈ R(χ λ ), j = 1, . . . , n, form a strictly decreasing sequence. The subset {µ} ⊂ R(χ λ ) is ordered by inclusion, hence there is a permutation τ such that µ ′ i > µ ′ i+1 , µ ′ = τ (µ). The permutation τ satisfies the condition of the theorem. Uniqueness of the module W follows from the ordering on {µ ′ }.
Let us show that the module W is a quotient of a generalized Verma module associated with A m,µ,1 . The elements λ j − (j − 1) belonging to the interval µ ′ i , µ ′ i+1 ⊂ R(χ λ ) form an arithmetic progression with the initial term µ ′ i and decrement 1, as follows from (10). Thus λ j is stable within this interval. Therefore λ defines a character of the parabolic subalgebra p with Levi factor l such that m(l) = m ′ . Let ρ denote the homomorphism U(g) → End(W ).
As dim W < ∞, for all x ∈ l one has ρ(x)w 0 = λ(x)w 0 , where w 0 ∈ W is the highest weight vector. Therefore the subspace Cw 0 ⊂ W forms a one dimensional U(p)-module C λ . The map U(g) ⊗ C C → W , x ⊗ 1 → ρ(x)w 0 , is a homomorphism of left U(g)-modules, which is obviously factored through the generalized Verma module U(g) ⊗ U (p) C λ .
Real forms of quantum orbits
As an application of the established relation between the algebras A m,µ,t (A • m,µ,t ) and the generalized Verma modules we construct real forms of quantum orbits.
5.
1. An anti-linear endomorphism of a complex vector space V is an additive map f : V → V satisfying f (βa) =βf (a) for any a ∈ V and β ∈ C; the bar stands for the complex conjugation. When V is a C[t]-module, we assume f (t) = t.
Definition 5.1. A * -structure on an associative algebra A over C or C[t] is an anti-linear involution * : A → A that is an anti-automorphism with respect to the multiplication, (ab) * = b * a * .
Let H be a Hopf algebra over C or C[t] with the multiplication m, comultiplication ∆, and antipode γ, [ChPr].
Definition 5.2. A real form on H is an anti-linear involution such that
θ • m = m • (θ ⊗ θ), τ • ∆ • θ = (θ ⊗ θ) • ∆,(17)
where τ is the flip on H ⊗ H.
When H is the universal enveloping algebra U(g) of a complex Lie algebra g, the involution restricts to g. The set of θ-fixed points in U(g) is the universal enveloping algebra U(g R ) over R of the Lie algebra g R , a real form of g.
Proposition 5.3. Let θ be a real form of a Hopf algebra H with invertible antipode γ. Then
γ • θ • γ = θ.
Proof. Consider the map θ•γ •θ : H → H. It satisfies the axioms of antipode for the opposite comultiplication and therefore equals to γ −1 , due to uniqueness of the antipode.
This proposition implies that the composition of θ with any odd power of γ makes H an involitive algebra. We are going to define involutions on H-module algebras that will be compatible with real forms on H in the sense of the following definition.
Definition 5.4. Let θ be a real form of a Hopf algebra H. A real form of an H-module
algebra A is a * -structure on A such that (x ⊲ a) * = θ(x) ⊲ a * , x ∈ H, a ∈ A.(18)
Example 5.5. Let H be a Hopf algebra equipped with a real form θ. Consider H as a left module algebra over itself with respect to the adjoint action x ⊲ y = x (1) yγ x (2) . Define a * -structure on H by the involution γ • θ. Then, * is a real form of the self-adjoint module algebra H. Indeed, one has
(x ⊲ y) * = (x (1) yγ(x (2) )) * = (γ • θ • γ)(x (2) )y * (γ • θ)(x (1) ) = θ(x (2) )y * (γ • θ)(x (1) ),
which is equal to θ(x) ⊲ y * .
5.2.
Before proceeding with real forms as applied to G-equivariant quantization, we study a relation between the two algebras A m,µ,t and A • m,µ,t from Definition 3.1.
Proposition 5.6. The identity map g t → g t induces an isomorphism of algebras A m,µ,t and A • m,ν,t over C, provided
ν i = µ i + (n − m i )t, i = 1, . . . , k.(19)
Proof. Both algebras are quotients of U(g t ) by the ideals generated by certain copies of the adjoint and trivial submodules. Denote them by W µ and W • ν correspondingly. It is sufficient to show that W µ and W • ν are related by the transformation of parameters (19). With m given, both modules are determined by their adjoint components, generated by the matrix coefficients of polynomials in E. Every monomial (2) is expressed as a linear combination of monomials (3) with coefficients being polynomial functions in t. A polynomial (4) can be represented as a polynomial in the sense of (3), and the coefficients of the latter will be polynomials in µ and t. Therefore it suffices to prove the proposition only for generic µ.
Take the parabolic subalgebra p ⊃ l such that m(l) = m. Assuming µ regular and t small, there is a generalized Verma module such that W µ ⊂ J pt,λ . Further, there is a unique, as emphasized in Remark 4.6, module W • ν such that J pt,λ ⊃ W • ν . Hence W µ = W • ν , and the parameters µ, λ, and ν are related by transformations (12) and (13). This proves the statement for regular µ and small t and therefore for all µ and t.
5.3.
It is known, [Kn], that all real forms of the Lie algebra gl(n, C) are isomorphic to gl(n, R), u ⋆ (2m) when n = 2m is even, and u(r, s) with r + s = n, r ≥ s ≥ 0. When s = 0, u(r, s) turns into the compact real form u(n). The corresponding involutions are defined on
generators {E i j } k i,j=1 as θ gl(n,R) (E) = E, θ u(r,s) (E) = −J (r,s) E ′ J −1 (r,s) , θ u ⋆ (2m) (E) = J m EJ −1 m , where J (r,s) = r i=1 e i i − n i=r+1 e i i , J m = m i=1 (e i i+m − e i+m i )
, and the prime stands for the matrix transposition.
5.4.
Take µ ∈ C k , m ∈ {n : k} and denote by S m k the stabilizer of m in the symmetric group S k .
Proposition 5.7. The map E → J (r,s) E ′ J −1 (r,s) extends to a θ u(r,s) -compatible * -real form of the algebra A m,µ,t , providedμ = τ (µ) for some τ ∈ S m k .
Proof. Consider U(g) as a module algebra over itself with respect to the adjoint representation. Define a θ u(r,s) -compatible * -real form on U(g) as in Example 5.5. It is naturally extended to a θ u(r,s) -compatible * -real form on U(g)[t] considered as a U(g)-module algebra.
This real form is restricted to the U(g)-equivariant embedding U(g t ) → U(g) [t]. On the gen- r,s) . Let us prove that it induces an isomorphism (A m,µ,t ) * ≃ A m,μ,t . Setting J = J r,s we have J 2 = 1 and J ′ = J; hence for
erators {E i j } ⊂ g t , it is defined by the map E → J (r,s) E ′ J −1 (m > 0 (E ℓ ) i j = α 1 ,...,α ℓ−1 E α 1 j E α 2 α 1 . . . E i α ℓ−1 * → α 0 ,...,α ℓ J α ℓ i E α ℓ−1 α ℓ . . . E α 1 α 2 E i α 1 J j α 0 .
In the concise matrix form this reads (E ℓ ) * = (JE ℓ J) ′ . If p(x) is a polynomial in one variable andp(x) is obtained from p(x) by the complex conjugation of its coefficients, then p(E) * → Jp(E)J ′ . Relations (5) go over to Tr E ℓ = ϑ ℓ (m,μ, t), ℓ ∈ N. Therefore (A m,µ,t ) * ≃ A m,μ,t = A m,µ,t ifμ = τ (µ) for some τ ∈ S m k .
Proposition 5.8. Suppose −μ = τ (µ − tm) + tnl ∈ C k for some τ ∈ S m k . Then 1. the map E → E extends to a θ gl(n,R) -compatible * -real form of A m,µ,t
2. the map E → −J m EJ −1 m extends to a θ u ⋆ (2m) -compatible * -real form of A m,µ,t Proof. Let J denote either the unit matrix or J m = m i=1 (e i i+m − e i+m i
). We will consider the two cases simultaneously. Define a * -real form on the algebra U(g t ) in the same way as in the proof of Proposition 5.7. It is compatible with the corresponding real form of the Hopf algebra U(g). On the generators {E i j } ⊂ g t , it is defined by the map E → −JEJ −1 . We have
(E ℓ ) i j = α 1 ,...,α ℓ−1 E α 1 j E α 2 α 1 . . . E i α ℓ−1 * → α 0 ,...,α ℓ (−1) ℓ J i α ℓ E α ℓ α ℓ−1 . . . E α 2 α 1 E α 1 α 0 J −1 α 0 j .
Thus every matrix monomial E ℓ is transformed into the matrix (−1) ℓ JE •ℓ J −1 ; therefore (A m,µ,t ) * ≃ A • m,−μ,t , with respect to either sl(n, R)-or u ⋆ (2m)-involutions. Now the rest of the proof follows from (19).
6 Non-commutative Connes index 6.1. The purpose of the section is to give an interpretation of certain rational functions arising in our theory of quantum orbits, [DM2]. We show that those functions give the non-commutative Connes index, [C], of basic projecitve modules over quantum orbits. We will consider the two-parameter U q (g)-equivariant quantization of [DM2] (see also [DM1]) including the G-equivariant quantization as the limit case q → 1. A two-parameter quantum orbit is a quotient of the so called modified reflection equation algebra, [KSkl,IP,DM3], L q,t , which itself is a U q (g)-equivariant quantization of the polynomial algebra on g * . The algebra L q,t is generated by the elements {L i j } n i,j=1 subject to certain quadratic-linear relations turning to commutation relations (1), where E i j = lim q→1 L i j . There is a U q (g)-equivariant generalization, Tr q , of the trace operation such that Tr q L ℓ , ℓ ∈ N, belong to the center of L q,t . For details the reader is referred to [DM2] and references therein.
The functions ϑ ℓ (m, µ, t) in the right-hand side of (5) are the specialization of certain functions ϑ ℓ (m, µ, q −2 , t) introduced in [DM2] (see also Appendix). Here,m = (m 1 , . . . ,m k ) andm = 1−q −2m 1−q −2 for m ∈ N. Also, we introduced in [DM2] rational functions C j (m, µ, q −2 , t) satisfying relation (30).
Theorem 6.1 ( [DM2]). For any m ∈ {n : k} and µ ∈ C k reg , the quotient Am ,µ,q,t of the algebra L q,t by the relations
(L − µ 1 ) . . . (L − µ k ) = 0,(20)Tr q (L ℓ ) = ϑ ℓ (m, q −2 , µ, t), ℓ = 1, . . . , k − 1,(21)
is a U q (g)-equivariant quantization on the orbit of semisimple matrices with eigenvalues µ of multiplicities m. In the two-parameter quantization setting an idempotent π from A q,t (M) ⊗ C End(V ) defines the projective A q,t -module π A q,t (M)⊗ C V , which we consider as a quantized vector bundle over M. Let us built the idempotents for the quantized basic vector bundles over the orbits.
Proposition 6.2. Let L = ||L i j || k i,j=1 be the matrix of generators of the algebra Am ,µ,q,t and suppose µ ∈ C k reg . The elements π j (L) = i=1,...,k i =j L − µ i µ j − µ i ∈ Am ,µ,q,t ⊗ C End(C n ), j = 1, . . . , k,
are invariant projectors. One has Tr q π j (L) = C j (m, µ, q −2 , t), j = 1, . . . , k, see (30).
Proof. The matrix L ∈ Am ,µ,q,t ⊗ C End(C n ) satisfies (20). This implies that the elements (22) are orthogonal idempotents, and L = k j=1 µ j π j (L). Therefore, every positive integer power of the matrix L has the decomposition over the basis π j (L), j = 1, . . . , k:
L ℓ = k j=1 µ ℓ j π j (L), ℓ = N.(23)
Taking trace on the both sides of this equation and using condition (21) extended for all ℓ ∈ N (see [DM2]) and representation (30), we get k j=1 µ ℓ j Tr q π j (L) = k j=1 µ ℓ j C j (m, µ, q −2 , t) for all ℓ = N. Since the numbers {µ i } are pairwise distinct, this proves the statement.
The quantities Tr q π j (L) are central elements of Am ,µ,q,t . Therefore we can take the function C j (m, µ, q −2 , t) as the "universal" Connes index of the family of quantum vector bundles π j Am ,µ,q,t ⊗ C C n .
Note that the problem of U q (g)-equivariant quantization of vector bundles on orbits as one-and two-sided projective modules over quantized function algebras was considered in [D]. An interesting problem is to construct the quantum bundles explicitly, in terms of projectors. This problem is solved for non-commutative sphere S 2 q in [GLS].
If one puts, by definition,θ 0 (ν, µ, ω) = ν ω , where ν ω is determined by the equation (1−ων ω ) = k i=1 (1 − ων i ), then representation (26) is valid for ℓ = 0 as well. Suppose ν has a polynomial dependence in ω and lim ω→0 ν(ω) = m. Recall that l ∈ C k denotes the vector (1, . . . , 1).
Proposition 6.3. The function k j=1 µ ℓ jC j ν(ω), µ + t ω , ω is a polynomial in all its arguments. Its specialization ω = 0 is a polynomial ϑ ℓ (m, µ, t) =
Proof. It seen from (27) that the functionsC j ν(ω), µ + t ω , ω are regular at ω = 0; this proves the first assertion. Consider the specialization at ω = 0:
C j (m, µ, t) =C j ν(ω), µ + t ω , ω | ω=0 .(29)
We assume µ i = 0 for all i and put m/µ = (m 1 /µ 1 , . . . , m k /µ k ). Now the statement follows from formulas (24) and (26) if one observes that C j (m, µ, t) =C j m/µ, µ, t from (27).
Specialization at ν =m, ω = 1 − q −2 yields polynomials ϑ ℓ (m, µ, q −2 , t) participating in the two-parameter U q (g)-equivariant quantization of orbits, [DM2]:
ϑ ℓ (m, µ, q −2 , t) = k j=1 µ ℓ j C j (m, µ, q −2 , t),
where C j (m, µ, q −2 , t) =C j m, µ + t ω , ω | ω=1−q −2 . The coefficients C j (m, µ, q −2 , t) were shown above to give the non-commutative Connes index for basic homogeneous vector bundles over the two-parameter quantum orbits.
By a quantization of a manifold M we mean a flat C[[t]]-algebra A t (M) with an isomorphism A t (M)/tA t (M) → A(M), where A(M) is the function algebra on M. A quantization of a G-space M is called equivariant if G acts on A t (M) by algebra automorphisms and that action extends the original action of G on A(M).
); the coadjoint action of the group G on g * becomes the similarity transformation under this identification. Let us define the set {n : k} ⊂ Z k of k-tuples m = (m 1 , . . . , m k ) such that 0 ≤ m i and m 1 + . . . + m k = n; the subset {n : k} + ⊂ {n : k} consists of m with all m i positive. Let C k reg denote the subspace in C k of µ = (µ 1 , . . . , µ k ) with pairwise distinct µ i .
functions in λ maybe having poles only atλ i = λ l , i = l. But ψ (j) are bounded at λ i = λ l since µ = ψ(λ, t)are roots of the polynomial p with the coefficients χ λ (c i ) being regular functions in (λ, t).
Proposition 4 . 8 .
48The quantum orbit A m,µ,t has a finite dimensional representation if and only if there is an element
6. 2 .
2In the classical geometry, a vector bundle over a manifold M may be given by anidempotent π of the algebra A(M) ⊗ C End(V ), where V is a finite dimensional vector space. It forms a projective A(M)-module π A(M) ⊗ C V . If M is a G-manifold and V a G-module, a G-equivariant bundle corresponds to an invariant idempotent. For every semisimple coadjoint orbit O µ in End(C n ) of rank k −1 there are k projector-valued functions π i : O µ → End(C n ). At a point A ∈ O µ , they commute with A and map the linear space C n onto the A-eigenspaces. These basic vector bundles generate the Grothendieck ring of equivariant vector bundles over O µ .
AppendixIn this subsection, we study the functions ϑ ℓ entering the right-hand side of (5). They were introduced in[DM2]as specialization at q = 1 of certain functions participating in the two-parameter quantization of orbits. Assuming ν, µ ∈ C k and ω ∈ C consider the functionsThey satisfy the recurrent relatioñwhere µ ′ = (µ 1 , . . . , µ k−1 ) and ν ′ = (ν 1 , . . . , ν k−1 ). Using this relation, it is easy to prove by induction on k thatθwhereC j (ν, µ, ω) = ν j + ν j k−1 ℓ=1 ω ℓ 1≤i 1 <...<i ℓ ≤k i 1 ,...,i ℓ =j
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Covariant differential complexes on quantum linear groups. A Isaev, P Pyatov, J. Phys. A. 28A. Isaev, P. Pyatov, Covariant differential complexes on quantum linear groups, J. Phys. A, 28 (1995) 2227-2246.
B Fedosov, Deformation Quantization and Index Theory, Mathematical Topics 9. BerlinAcademie VerlagB. Fedosov, Deformation Quantization and Index Theory, Mathematical Topics 9, Academie Verlag, Berlin, 1996.
Saponov q-Index on braided spheres. D Gurevich, R Leclercq, P , math.QA/0207268D. Gurevich, R. Leclercq, P. Saponov q-Index on braided spheres, math.QA/0207268.
A W Knapp, Lie Groups Beyond an Introduction. Birkhäuser140A. W. Knapp, Lie Groups Beyond an Introduction, Progress in Mathematics, 140, Birkhäuser, 1996.
A Criterion for an Ideal to Be Induced. A Joseph, J. Algebra. 110A. Joseph, A Criterion for an Ideal to Be Induced, J. Algebra 110 (1987) 480-497.
Lie group representation on a polynomial ring. B Kostant, Amer. J. Math. 85B. Kostant, Lie group representation on a polynomial ring, Amer. J. Math. 85 (1963), 327-404.
Algebraic structure related to the reflection equation. P P Kulish, E K Sklyanin, J. Phys. A. 25P. P. Kulish, E. K. Sklyanin, Algebraic structure related to the reflection equation, J. Phys. A 25 (1992) 5963-2389.
| []
|
[
"A REMARK ON \"CONNECTIONS AND HIGGS FIELDS ON A PRINCIPAL BUNDLE\"",
"A REMARK ON \"CONNECTIONS AND HIGGS FIELDS ON A PRINCIPAL BUNDLE\""
]
| [
"\nINDRANIL BISWAS AND CARLOS FLORENTINO\n\n"
]
| [
"INDRANIL BISWAS AND CARLOS FLORENTINO\n"
]
| []
| We show that a unipotent vector bundle on a non-Kähler compact complex manifold does not admit a flat holomorphic connection in general. We also construct examples of topologically trivial stable vector bundle on compact Gauduchon manifold that does not admit any unitary flat connection.2000 Mathematics Subject Classification. 32L05, 53C07. | 10.1007/s10455-011-9257-1 | [
"https://arxiv.org/pdf/1102.4216v1.pdf"
]
| 121,941,680 | 1102.4216 | c60e65172a90890cbf7a0584c93aae256c729667 |
A REMARK ON "CONNECTIONS AND HIGGS FIELDS ON A PRINCIPAL BUNDLE"
21 Feb 2011
INDRANIL BISWAS AND CARLOS FLORENTINO
A REMARK ON "CONNECTIONS AND HIGGS FIELDS ON A PRINCIPAL BUNDLE"
21 Feb 2011and phrases Unipotent bundleconnectionCalabi-Eckmann manifold
We show that a unipotent vector bundle on a non-Kähler compact complex manifold does not admit a flat holomorphic connection in general. We also construct examples of topologically trivial stable vector bundle on compact Gauduchon manifold that does not admit any unitary flat connection.2000 Mathematics Subject Classification. 32L05, 53C07.
Let M be a compact connected complex manifold. A holomorphic vector bundle E −→ M is called unipotent if there is a filtration of holomorphic subbundles
(1) 0 = E 0 ⊂ E 1 ⊂ · · · ⊂ E ℓ−1 ⊂ E ℓ = E
such that for every i ∈ [1 , ℓ], the quotient E i /E i−1 is a holomorphically trivial vector bundle. If M is Kähler, and E is unipotent as above, then there is a flat holomorphic connection on E that preserves each subbundle in (1) [BG,p. 21, Corollary 1.2] (take the Higgs field in [BG,Corollary 1.2] to be zero); see [FL] for related results.
It is natural to ask whether the same result holds for unipotent bundles on compact complex manifolds. We will construct an example showing that it does not hold.
Fix integers m , n ≥ 1, and fix τ ∈ C with Im τ > 0. This gives the elliptic curve T := C/(Z ⊕ τ · Z). Let M be the corresponding Calabi-Eckmann manifold [CE]. We recall that M is diffeomorphic to S 2m+1 × S 2n+1 , and it is the total space of a holomorphic principal T -bundle over CP m ×CP n . The complex manifold M does not admit any Kähler metric because H 2 (M, Z) = 0.
Let O M be the sheaf of holomorphic functions on M. A computation of Borel shows that H 1 (M, O M ) = C [Bo,p. 216,Theorem 9.5] (see also [Ho,p. 232]). Let
(2) 0 −→ O M ι −→ E p −→ O M −→ 0
be a nontrivial extension given by some nonzero element of H 1 (M, O M ).
Since the extension in (2) is nontrivial, it can be shown that the holomorphic vector bundle E is nontrivial. Indeed, if E is holomorphically trivial, then take any holomorphic section
s : O M −→ E
that is not a multiple of the section given by ι in (2). Since the composition p • s is nonzero, it must be an automorphism of O M . Hence s generates a line subbundle of I. BISWAS AND C. FLORENTINO E which splits the short exact sequence in (2). But this exact sequence does not split. Therefore, we conclude that E is not holomorphically trivial.
Consequently, E is a nontrivial unipotent vector bundle. But E does not admit any flat holomorphic connection because M being simply connected, any holomorphic vector bundle on M admitting a flat holomorphic connection is holomorphically trivial, while we know that E is not holomorphically trivial.
Consider the short exact sequence of sheaves on M
0 −→ 2π √ −1 · Z −→ O M exp −→ O * M −→ 0 ,
where O * M is the multiplicative sheaf of nowhere vanishing holomorphic functions. Since Any holomorphic line bundle on M is stable with respect to any Gauduchon metric on M. Consequently, M has topologically trivial stable vector bundles that do not admit any unitary flat holomorphic connection. (Compare this with [Bi].)
We recall that the Hitchin-Kobayashi correspondence implies that any stable vector bundle E with c 1 (E) = 0 = c 2 (E) on a compact Kähler manifold admits a unique unitary flat holomorphic connection (here c i are rational Chern classes).
Z) = 0 for i = 1 , 2, from the long exact sequence of cohomologies for this short exact sequence it follows thatPic(M) := H 1 (M, O * M ) = H 1 (M, O M ) = C .Since Pic(M) is connected, all holomorphic line bundles on M are topologically trivial.Therefore, any nontrivial holomorphic line bundle on M is topologically trivial but does not admit any unitary flat holomorphic connection (because M being simply connected there are no nontrivial flat holomorphic connection on M).
Acknowledgements. The first author wishes to thank Instituto Superior Técnico, where the work was carried out, for its hospitality. The visit to IST was funded by the FCT project PTDC/MAT/099275/2008.
I Biswas, T L Gómez, Connections and Higgs fields on a principal bundle. 33I. Biswas and T. L. Gómez, Connections and Higgs fields on a principal bundle, Ann. Glob. Anal. Geom. 33 (2008), 19-46.
I Biswas, Stable Higgs bundles on compact Gauduchon manifolds. 349I. Biswas, Stable Higgs bundles on compact Gauduchon manifolds, Comp. Ren. Acad. Sci. Paris 349 (2011), 71-74.
A spectral sequence for complex analytic bundles, Appendix Two. A Borel, Topological methods in algebraic geometry, by F. Hirzebruch. BerlinSpringer-VerlagA. Borel, A spectral sequence for complex analytic bundles, Appendix Two, in: Topological methods in algebraic geometry, by F. Hirzebruch, Springer-Verlag, Berlin, 1995.
A class of compact, complex manifolds which are not algebraic. E Calabi, B Eckmann, Ann. of Math. 58E. Calabi and B. Eckmann, A class of compact, complex manifolds which are not algebraic, Ann. of Math. 58 (1953), 494-500.
Unipotent Schottky bundles on Riemann surfaces and complex tori. C Florentino, T Ludsteck, preprintC. Florentino and T. Ludsteck, Unipotent Schottky bundles on Riemann surfaces and complex tori, preprint (2011).
Remarks on torus principal bundles. T Höfer, Jour. Math. Kyoto Univ. 33Homi Bhabha RoadTata Institute of Fundamental ResearchSchool of MathematicsT. Höfer, Remarks on torus principal bundles, Jour. Math. Kyoto Univ. 33 (1993), 227-259. School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India
| []
|
[
"INFORMATION-DIRECTED EXPLORATION FOR DEEP REINFORCEMENT LEARNING",
"INFORMATION-DIRECTED EXPLORATION FOR DEEP REINFORCEMENT LEARNING"
]
| [
"Nikolay Nikolov [email protected] \nImperial College London\nETH Zurich\n\n",
"Johannes Kirschner \nImperial College London\nETH Zurich\n\n",
"Felix Berkenkamp \nImperial College London\nETH Zurich\n\n",
"Andreas Krause [email protected] \nImperial College London\nETH Zurich\n\n",
"Eth Zurich \nImperial College London\nETH Zurich\n\n"
]
| [
"Imperial College London\nETH Zurich\n",
"Imperial College London\nETH Zurich\n",
"Imperial College London\nETH Zurich\n",
"Imperial College London\nETH Zurich\n",
"Imperial College London\nETH Zurich\n"
]
| []
| Efficient exploration remains a major challenge for reinforcement learning. One reason is that the variability of the returns often depends on the current state and action, and is therefore heteroscedastic. Classical exploration strategies such as upper confidence bound algorithms and Thompson sampling fail to appropriately account for heteroscedasticity, even in the bandit setting. Motivated by recent findings that address this issue in bandits, we propose to use Information-Directed Sampling (IDS) for exploration in reinforcement learning. As our main contribution, we build on recent advances in distributional reinforcement learning and propose a novel, tractable approximation of IDS for deep Q-learning. The resulting exploration strategy explicitly accounts for both parametric uncertainty and heteroscedastic observation noise. We evaluate our method on Atari games and demonstrate a significant improvement over alternative approaches. | null | [
"https://arxiv.org/pdf/1812.07544v2.pdf"
]
| 56,177,829 | 1812.07544 | 7ae3c62481338c5888de5d0abe5dc88884b24035 |
INFORMATION-DIRECTED EXPLORATION FOR DEEP REINFORCEMENT LEARNING
Nikolay Nikolov [email protected]
Imperial College London
ETH Zurich
Johannes Kirschner
Imperial College London
ETH Zurich
Felix Berkenkamp
Imperial College London
ETH Zurich
Andreas Krause [email protected]
Imperial College London
ETH Zurich
Eth Zurich
Imperial College London
ETH Zurich
INFORMATION-DIRECTED EXPLORATION FOR DEEP REINFORCEMENT LEARNING
Published as a conference paper at ICLR 2019
Efficient exploration remains a major challenge for reinforcement learning. One reason is that the variability of the returns often depends on the current state and action, and is therefore heteroscedastic. Classical exploration strategies such as upper confidence bound algorithms and Thompson sampling fail to appropriately account for heteroscedasticity, even in the bandit setting. Motivated by recent findings that address this issue in bandits, we propose to use Information-Directed Sampling (IDS) for exploration in reinforcement learning. As our main contribution, we build on recent advances in distributional reinforcement learning and propose a novel, tractable approximation of IDS for deep Q-learning. The resulting exploration strategy explicitly accounts for both parametric uncertainty and heteroscedastic observation noise. We evaluate our method on Atari games and demonstrate a significant improvement over alternative approaches.
INTRODUCTION
In Reinforcement Learning (RL), an agent seeks to maximize the cumulative rewards obtained from interactions with an unknown environment. Given only knowledge based on previously observed trajectories, the agent faces the exploration-exploitation dilemma: Should the agent take actions that maximize rewards based on its current knowledge or instead investigate poorly understood states and actions to potentially improve future performance. Thus, in order to find the optimal policy the agent needs to use an appropriate exploration strategy.
Popular exploration strategies, such as -greedy (Sutton & Barto, 1998), rely on random perturbations of the agent's policy, which leads to undirected exploration. The theoretical RL literature offers a variety of statistically-efficient methods that are based on a measure of uncertainty in the agent's model. Examples include upper confidence bound (UCB) (Auer et al., 2002) and Thompson sampling (TS) (Thompson, 1933). In recent years, these have been extended to practical exploration algorithms for large state-spaces and shown to improve performance (Osband et al., 2016a;O'Donoghue et al., 2018;Fortunato et al., 2018). However, these methods assume that the observation noise distribution is independent of the evaluation point, while in practice heteroscedastic observation noise is omnipresent in RL. This means that the noise depends on the evaluation point, rather than being identically distributed (homoscedastic). For instance, the return distribution typically depends on a sequence of interactions and, potentially, on hidden states or inherently heteroscedastic reward observations. Kirschner & Krause (2018) recently demonstrated that, even in the simpler bandit setting, classical approaches such as UCB and TS fail to efficiently account for heteroscedastic noise.
In this work, we propose to use Information-Directed Sampling (IDS) (Russo & Van Roy, 2014;Kirschner & Krause, 2018) for efficient exploration in RL. The IDS framework can be used to design exploration-exploitation strategies that balance the estimated instantaneous regret and the expected information gain. Importantly, through the choice of an appropriate information-gain function, IDS is able to account for parametric uncertainty and heteroscedastic observation noise during exploration.
As our main contribution, we propose a novel, tractable RL algorithm based on the IDS principle. We combine recent advances in distributional RL (Bellemare et al., 2017;Dabney et al., 2018b) and approximate parameter uncertainty methods in order to develop both homoscedastic and heteroscedastic variants of an agent that is similar to DQN (Mnih et al., 2015), but uses informationdirected exploration. Our evaluation on Atari 2600 games shows the importance of accounting for heteroscedastic noise and indicates that at our approach can substantially outperform alternative state-of-the-art algorithms that focus on modeling either only epistemic or only aleatoric uncertainty. To the best of our knowledge, we are the first to develop a tractable IDS algorithm for RL in large state spaces.
RELATED WORK
Exploration algorithms are well understood in bandits and have inspired successful extensions to RL (Bubeck & Cesa-Bianchi, 2012;Lattimore & Szepesvári, 2018). Many strategies rely on the "optimism in the face of uncertainty" (Lai & Robbins, 1985) principle. These algorithms act greedily w.r.t. an augmented reward function that incorporates an exploration bonus. One prominent example is the upper confidence bound (UCB) algorithm (Auer et al., 2002), which uses a bonus based on confidence intervals. A related strategy is Thompson sampling (TS) (Thompson, 1933), which samples actions according to their posterior probability of being optimal in a Bayesian model. This approach often provides better empirical results than optimistic strategies (Chapelle & Li, 2011).
In order to extend TS to RL, one needs to maintain a distribution over Markov Decision Processes (MDPs), which is difficult in general. Similar to TS, Osband et al. (2016b) propose randomized linear value functions to maintain a Bayesian posterior distribution over value functions. Bootstrapped DQN (Osband et al., 2016a) extends this idea to deep neural networks by using an ensemble of Qfunctions. To explore, Bootstrapped DQN randomly samples a Q-function from the ensemble and acts greedily w.r.t. the sample. Fortunato et al. (2018) and Plappert et al. (2018) investigate a similar idea and propose to adaptively perturb the parameter-space, which can also be thought of as tracking an approximate parameter posterior. O'Donoghue et al. (2018) propose TS in combination with an uncertainty Bellman equation, which propagates agent's uncertainty in the Q-values over multiple time steps. Additionally, propose to use the Q-ensemble of Bootstrapped DQN to obtain approximate confidence intervals for a UCB policy. There are also multiple other ways to approximate parametric posterior in neural networks, including Neural Bayesian Linear Regression (Snoek et al., 2015;Azizzadenesheli et al., 2018), Variational Inference (Blundell et al., 2015), Monte Carlo methods (Neal, 1995;Mandt et al., 2016;Welling & Teh, 2011), and Bayesian Dropout (Gal & Ghahramani, 2016). For an empirical comparison of these, we refer the reader to Riquelme et al. (2018).
A shortcoming of all approaches mentioned above is that, while they consider parametric uncertainty, they do not account for heteroscedastic noise during exploration. In contrast, distributional RL algorithms, such as Categorical DQN (C51) (Bellemare et al., 2017) and Quantile Regression DQN (QR-DQN) (Dabney et al., 2018b), approximate the distribution over the Q-values directly. However, both methods do not take advantage of the return distribution for exploration and use -greedy exploration. Implicit Quantile Networks (IQN) (Dabney et al., 2018a) instead use a risksensitive policy based on a return distribution learned via quantile regression and outperform both C51 and QR-DQN on Atari-57. Similarly, Moerland et al. (2018) and Dilokthanakul & Shanahan (2018) act optimistically w.r.t. the return distribution in deterministic MDPs. However, these approaches to not consider parametric uncertainty.
Return and parametric uncertainty have previously been combined for exploration by Tang & Agrawal (2018) and Moerland et al. (2017). Both methods account for parametric uncertainty by sampling parameters that define a distribution over Q-values. The former then act greedily with respect to the expectation of this distribution, while the latter additionally samples a return for each action and then acts greedily with respect to it. However, like Thompson sampling, these approaches do not appropriately exploit the heteroscedastic nature of the return. In particular, noisier actions are more likely to be chosen, which can slow down learning.
Our method is based on Information-Directed Sampling (IDS), which can explicitly account for parametric uncertainty and heteroscedasticity in the return distribution. IDS has been primarily studied in the bandit setting (Russo & Van Roy, 2014;Kirschner & Krause, 2018). Zanette & Sarkar (2017) extend it to finite MDPs, but their approach remains impractical for large state spaces, since it requires to find the optimal policies for a set of MDPs at the beginning of each episode.
BACKGROUND
We model the agent-environment interaction with a MDP (S, A, R, P, γ), where S and A are the state and action spaces, R(s, a) is the stochastic reward function, P (s |s, a) is the probability of transitioning from state s to state s after taking action a, and γ ∈ [0, 1) is the discount factor. A policy π(·|s) ∈ P(A) maps a state s ∈ S to a distribution over actions. For a fixed policy π, the discounted return of action a in state s is a random variable Z π (s, a) = ∞ t=0 γ t R(s t , a t ), with initial state s = s 0 and action a = a 0 and transition probabilities s t ∼ P (·|s t−1 , a t−1 ), a t ∼ π(·|s t ). The return distribution Z statisfied the Bellman equation,
Z π (s, a) D = R(s, a) + γZ π (s , a ),(1)
where D = denotes distributional equality. If we take the expectation of (1), the usual Bellman equation (Bellman, 1957) for the Q-function, Q π (s, a) = E[Z π (s, a)], follows as
Q π (s, a) = E [R(s, a)] + γE P,π [Q π (s , a )] .(2)
The objective is to find an optimal policy π * that maximizes the expected total discounted return E[Z π (s, a)] = Q π (s, a) for all s ∈ S, a ∈ A.
UNCERTAINTY IN REINFORCEMENT LEARNING
To find such an optimal policy, the majority of RL algorithms use a point estimate of the Q-function, Q(s, a). However, such methods can be inefficient, because they can be overconfident about the performance of suboptimal actions if the optimal ones have not been evaluated before. A natural solution for more efficient exploration is to use uncertainty information. In this context, there are two source of uncertainty. Parametric (epistemic) uncertainty is a result of ambiguity over the class of models that explain that data seen so far, while intrinsic (aleatoric) uncertainty is caused by stochasticity in the environment or policy, and is captured by the distribution over returns (Moerland et al., 2017). Osband et al. (2016a) estimate parametric uncertainty with a Bootstrapped DQN. They maintain an ensemble of K Q-functions, {Q k } K k=1 , which is represented by a multi-headed deep neural network. To train the network, the standard bootstrap method (Efron, 1979;Hastie et al., 2001) constructs K different datasets by sampling with replacement from the global data pool. Instead, Osband et al. (2016a) trains all network heads on the exact same data and diversifies the Q-ensemble via two other mechanisms. First, each head Q k (s, a; θ) is trained on its own independent target head Q k (s, a; θ − ), which is periodically updated (Mnih et al., 2015). Further, each head is randomly initialized, which, combined with the nonlinear parameterization and the independently targets, provides sufficient diversification.
Intrinsic uncertainty is captured by the return distribution Z π . While Q-learning (Watkins, 1989) aims to estimate the expected discounted return Q π (s, a) = E[Z π (s, a)], distributional RL approximates the random return Z π (s, a) directly. As in standard Q-learning (Watkins, 1989), one can define a distributional Bellman optimality operator based on (1),
T Z(s, a) D := R(s, a) + γZ(s , arg max a ∈A E[Z(s , a )]).(3)
To estimate the distribution of Z, we use the approach of C51 (Bellemare et al., 2017) in the following. It parameterizes the return as a categorical distribution over a set of equidistant atoms in a fixed interval [V min , V max ]. The atom probabilities are parameterized by a softmax distribution over the outputs of a parametric model. Since the parameterization Z θ and the Bellman update T Z θ have disjoint supports, the algorithm requires an additional step Φ that projects the shifted support
of T Z θ onto [V min , V max ].
Then it minimizes the Kullback-Leibler divergence D KL (ΦT Z θ ||Z θ ).
HETEROSCEDASTICITY IN REINFORCEMENT LEARNING
In RL, heteroscedasticity means that the variance of the return distribution Z depends on the state and action. This can occur in a number of ways. The variance Var(R|s, a) of the reward function itself may depend on s or a. Even with deterministic or homoscedastic rewards, in stochastic environments the variance of the observed return is a function of the stochasticity in the transitions over a sequence of steps. Furthermore, Partially Observable MDPs (Monahan, 1982) are also heteroscedastic due to the possibility of different states aliasing to the same observation.
Interestingly, heteroscedasticity also occurs in value-based RL regardless of the environment. This is due to Bellman targets being generated based on an evolving policy π. To demonstrate this, consider a standard observation model used in supervised learning
y t = f (x t ) + t (x t ),
with true function f and Gaussian noise t (x t ). In Temporal Difference (TD) algorithms (Sutton & Barto, 1998), given a sample transition (s t , a t , r t , s t+1 ), the learning target is generated as y t = r t + γQ π (s t+1 , a ), for some action a . Similarly to the observation model above, we can describe TD-targets for learning Q * being generated as
y t = f (s t , a t ) + π t (s t , a t ), with f and π t given by f (s t , a t ) = Q * (s t , a t ) = E[R(s t , a t )] + γE s ∼p(s |st,at) [max a Q * (s , a )] π t (s t , a t ) = r t + γQ π (s t+1 , a ) − f (s t , a t ) = (r t − E[R(s t , a t )]) + γ Q π (s t+1 , a ) − E s ∼p(s |st,at) [max a Q * (s , a )](4)
The last term clearly shows the dependence of the noise function π t (s, a) on the policy π, used to generate the Bellman target. Note additionally that heteroscedastic targets are not limited to TDlearning methods, but also occur in TD(λ) and Monte-Carlo learning (Sutton & Barto, 1998), no matter if the environment is stochastic or not.
INFORMATION-DIRECTED SAMPLING
Information-Directed Sampling (IDS) is a bandit algorithm, which was first introduced in the Bayesian setting by Russo & Van Roy (2014), and later adapted to the frequentist framework by Kirschner & Krause (2018). Here, we concentrate on the latter formulation in order to avoid keeping track of a posterior distribution over the environment, which itself is a difficult problem in RL. The bandit problem is equivalent to a single state MDP with stochastic reward function R(a, s) = R(a) and optimal action a * = arg max a∈A E[R(a)]. We define the (expected) regret ∆(a) := E [R(a * ) − R(a)], which is the loss in reward for choosing an suboptimal action a. Note, however, that we cannot directly compute ∆(a), since it depends on R and the unknown optimal action a * . Instead, IDS uses a conservative regret estimate∆ t (a) = max a ∈A u t (a ) − l t (a), where [l t (a), u t (a)] is a confidence interval which contains the true expected reward E[R(a)] with high probability.
In addition, assume for now that we are given an information gain function I t (a). Then, at any time step t, the IDS policy is defined by
a IDS t ∈ arg min a∈A∆ t (a) 2 I t (a) .(5)
Technically, this is known as deterministic IDS which, for simplicity, we refer to as IDS throughout this work. Intuitively, IDS chooses actions with small regret-information ratioΨ t (a) :=∆ t(a) 2 It(a) to balance between incurring regret and acquiring new information at each step. Kirschner & Krause (2018) introduce several information-gain functions and derive a high-probability bound on the cumulative regret,
T t=1 ∆ t (a IDS t ) ≤ O( √ T γ T ).
Here, γ T is an upper bound on the total information gain T t=1 I t (a t ), which has a sublinear dependence in T for different function classes and the specific information-gain function we use in the following (Srinivas et al., 2010). The overall regret bound for IDS matches the best bound known for the widely used UCB policy for linear and kernelized reward functions.
One particular choice of the information gain function, that works well empirically and we focus on in the following, is I t (a) = log 1 + σ t (a) 2 /ρ(a) 2 (Kirschner & Krause, 2018). Here σ t (a) 2 is the variance in the parametric estimate of E[R(a)] and ρ(a) 2 = Var[R(a)] is the variance of the observed reward. In particular, the information gain I t (a) is small for actions with little uncertainty in the true expected reward or with reward that is subject to high observation noise. Importantly, note that ρ(a) 2 may explicitly depend on the selected action a, which allows the policy to account for heteroscedastic noise.
We demonstrate the advantage of such a strategy in the Gaussian Process setting (Murphy, 2012). In particular, for an arbitrary set of actions a 1 , . . . , a N , we model the distribution of R(a 1 ), . . . , R(a N ) by a multivariate Gaussian, with covariance Cov[R(a i ), R(a j )] = κ(x i , x j ), where κ is a positive definite kernel. In our toy example, the goal is to maximize R(x) under heteroscedastic observation noise with variance ρ(x) 2 ( Figure 1). As UCB and TS do not consider observation noise in the acquisition function, they may sample at points where ρ(x) 2 is large. Instead, by exploiting kernel correlation, IDS is able to shrink the uncertainty in the high-noise region with fewer samples, by selecting a nearby point with potentially higher regret but small noise.
INFORMATION-DIRECTED SAMPLING FOR REINFORCEMENT LEARNING
In this section, we use the IDS strategy from the previous section in the context of deep RL. In order to do so, we have to define a tractable notion of regret ∆ t and information gain I t .
ESTIMATING REGRET AND INFORMATION GAIN
In the context of RL, it is natural to extend the definition of instantaneous regret of action a in state s using the Q-function
∆ π t (s, a) := E P max a Q π (s, a ) − Q π t (s, a)|F t−1 ,(6)
where F t = {s 1 , a 1 , r 1 , . . . s t , a t , r t } is the history of observations at time t. The regret definition in eq. (6) captures the loss in return when selecting action a in state s rather than the optimal action. This is similar to the notion of the advantage function. Since ∆ π t (s, a) depends on the true Qfunction Q π , which is not available in practice and can only be estimated based on finite data, the IDS framework instead uses a conservative estimate.
To do so, we must characterize the parametric uncertainty in the Q-function. Since we use neural networks as function approximators, we can obtain approximate confidence bounds using a Bootstrapped DQN (Osband et al., 2016a). In particular, given an ensemble of K action-value functions, we compute the empirical mean and variance of the estimated Q-values,
µ(s, a) = 1 K K k=1 Q k (s, a), σ(s, a) 2 = 1 K K k=1 (Q k (s, a) − µ(s, a)) 2 .(7)
Based on the mean and variance estimate in the Q-values, we can define a surrogate for the regret using confidence intervals,
∆ π t (s, a) = max a ∈A (µ t (s, a ) + λ t σ t (s, a )) − (µ t (s, a) − λ t σ t (s, a)) .(8)
where λ t is a scaling hyperparameter. The first term corresponds to the maximum plausible value that the Q-function could take at a given state, while the right term lower-bounds the Q-value given the chosen action. As a result, eq. (8) provides a conservative estimate of the regret in eq. (6).
Algorithm 1 Deterministic Information-Directed Q-learning
Input: λ, action-value function Q with K outputs {Q k } K k=1 , action-value distribution Z for episode i = 1 : M do Get initial state s 0
for step t = 0 : T do µ(s t , a) = 1 K K k=1 Q k (s t , a) σ(s t , a) 2 = 1 K K k=1 [Q k (s t , a) − µ(s t , a)] 2 ∆(s t , a) = max a ∈A [µ(s t , a ) + λσ(s t , a )] − [µ(s t , a) − λσ(s t , a)]
ρ(s t , a) 2 = Var (Z(s t , a)) / 1 + 1 |A| a ∈A Var (Z(s t , a )) I(s t , a) = log 1 + σ(st,a) 2 ρ(st,a) 2 + 2 Compute regret-information ratio:Ψ(s t , a) =∆ (st,a) 2
I(st,a)
Execute action a t = arg min a∈AΨ (s t , a), observe r t and state s t+1 end for end for Given the regret surrogate, the only missing component to use the IDS strategy in eq. (5) is to compute the information gain function I t . In particular, we use I t (a) = log 1 + σ t (a) 2 /ρ(a) 2 based on the discussion in (Kirschner & Krause, 2018). In addition to the previously defined predictive parameteric variance estimates for the regret, it depends on the variance of the noise distribution, ρ. While in the bandit setting we track one-step rewards, in RL we focus on learning from returns from complete trajectories. Therefore, instantaneous reward observation noise variance ρ(a) 2 in the bandit setting transfers to the variance of the return distribution Var (Z(s, a)) in RL. We point out that the scale of Var (Z(s, a)) can substantially vary depending on the stochasticity of the policy and the environment, as well as the reward scaling. This directly affects the scale of the information gain and the degree to which the agent chooses to explore. Since the weighting between regret and information gain in the IDS ratio is implicit, for stable performance across a range of environments, we propose computing the information gain I(s, a) = log 1 + σ(s,a) 2 ρ(s,a) 2 + 2 using the normalized variance ρ(s, a) 2 = Var (Z(s, a))
1 + 1 |A| a ∈A Var (Z(s, a )) ,(9)
where 1 , 2 are small constants that prevent division by 0. This normalization step brings the mean of all variances to 1, while keeping their values positive. Importantly, it preserves the signal needed for noise-sensitive exploration and allows the agent to account for numerical differences across environments and favor the same amount of risk. We also experimentally found this version to give better results compared to the unnormalized variance ρ(s, a) 2 = Var (Z(s, a)).
INFORMATION-DIRECTED REINFORCEMENT LEARNING
Using the estimates for regret and information gain, we provide the complete control algorithm in Algorithm 1. At each step, we compute the parametric uncertainty over Q(s, a) as well as the distribution over returns Z(s, a). We then follow the steps from Section 4.1 to compute the regret and the information gain of each action, and select the one that minimizes the regret-information ratioΨ(s, a).
To estimate parametric uncertainty, we use the exact same training procedure and architecture as Bootstrapped DQN (Osband et al., 2016a): we split the DQN architecture (Mnih et al., 2015) into K bootstrap heads after the convolutional layers. Each head Q k (s, a; θ) is trained against its own target head Q k (s, a; θ − ) and all heads are trained on the exact same data. We use Double DQN targets (van Hasselt et al., 2016) and normalize gradients propagated by each head by 1/K.
To estimate Z(s, a), it makes sense to share some of the weights θ from the Bootstrapped DQN. We propose to use the output of the last convolutional layer φ(s) as input to a separate head that estimates Z(s, a). The output of this head is the only one used for computing ρ(s, a) 2 and is also not included in the bootstrap estimate. For instance, this head can be trained using C51 or QR-
) = i p i (z i − E[Z(s, a)]) 2 ,
where z i denotes the atoms of the distribution support, p i , their corresponding probabilities, and E[Z(s, a)] = i p i z i . To isolate the effect of noise-sensitive exploration from the advantages of distributional training, we do not propagate distributional loss gradients in the convolutional layers and use the representation φ(s) learned only from the bootstrap branch. This is not a limitation of our approach and both (or either) bootstrap and distributional gradients can be propagated through the convolutional layers. Importantly, our method can account for deep exploration, since both the parametric uncertainty σ(s, a) 2 and the intrinsic uncertainty ρ(s, a) 2 estimates in the information gain are extended beyond a single time step and propagate information over sequences of states. We note the difference with intrinsic motivation methods, which augment the reward function by adding an exploration bonus to the step reward (Houthooft et al., 2016;Stadie et al., 2015;Schmidhuber, 2010;Bellemare et al., 2016;Tang et al., 2017). While the bonus is sometimes based on an information-gain measure, the estimated optimal policy is often affected by the augmentation of the rewards.
EXPERIMENTS
We now provide experimental results on 55 of the Atari 2600 games from the Arcade Learning Environment (ALE) (Bellemare et al., 2013), simulated via the OpenAI gym interface (Brockman et al., 2016). We exclude Defender and Surround from the standard Atari-57 selection, since they are not available in OpenAI gym. Our method builds on the standard DQN architecture and we expect it to benefit from recent improvements such as Dueling DQN (Wang et al., 2016) and prioritized replay . However, in order to separately study the effect of changing the exploration strategy, we compare our method without these additions. Our code can be found at https:// github.com/nikonikolov/rltf/tree/ids-drl.
We evaluate two versions of our method: a homoscedastic one, called DQN-IDS, for which we do not estimate Z(s, a) and set ρ(s, a) 2 to a constant, and a heteroscedastic one, C51-IDS, for which we estimate Z(s, a) using C51 as previously described. DQN-IDS uses the exact same network architecture as Bootstrapped DQN. For C51-IDS, we add the fully-connected part of the C51 network (Bellemare et al., 2017) on top of the last convolutional layer of the DQN-IDS architecture, but we do not propagate distributional loss gradients into the convolutional layers. We use a target network to compute Bellman updates, with double DQN targets only for the bootstrap heads, but not for the distributional update. Weights are updated using the Adam optimizer (Kingma & Ba, 2015). We evaluate the performance of our method using a mean greedy policy that is computed on the bootstrap heads arg max
a∈A 1 K K k=1 Q k (s, a).(10)
Due to computational limitations, we did not perform an extensive hyperparameter search. Our final algorithm uses λ = 0.1, ρ(s, a) 2 = 1.0 (for DQN-IDS) and target update frequency of 40000 agent steps, based on a parameter search over λ ∈ {0.1, 1.0}, ρ 2 ∈ {0.5, 1.0}, and target update in {10000, 40000}. For C51-IDS, we put a heuristically chosen lower bound of 0.25 on ρ(s, a) 2 to prevent the agent from fixating on "noiseless" actions. This bound is introduced primarily for numerical reasons, since, even in the bandit setting, the strategy may degenerate as the noise variance of a single action goes to zero. We also ran separate experiments without this lower bound and while the per-game scores slightly differ, the overall change in mean human-normalized score was only 23%. We also use the suggested hyperparameters from C51 and Bootstrapped DQN, and set learning rate α = 0.00005, ADAM = 0.01/32, number of heads K = 10, number of atoms N = 51. The rest of our training procedure is identical to that of Mnih et al. (2015), with the difference that we do not use -greedy exploration. All episodes begin with up to 30 random no-ops (Mnih et al., 2015) and the horizon is capped at 108K frames (van Hasselt et al., 2016). Complete details are provided in Appendix A.
To provide comparable results with existing work we report evaluation results under the best agent protocol. Every 1M training frames, learning is frozen, the agent is evaluated for 500K frames and performance is computed as the average episode return from this latest evaluation run. Table 1 shows the mean and median human-normalized scores (van Hasselt et al., 2016) of the best agent performance after 200M training frames. Additionally, we illustrate the distributions learned by C51 and C51-IDS in Figure 3.
We first point out the results of DQN-IDS and Bootstrapped DQN. While both methods use the same architecture and similar optimization procedures, DQN-IDS outperforms Bootstrapped DQN by around 200%. This suggests that simply changing the exploration strategy from TS to IDS (along with the type of optimizer), even without accounting for heteroscedastic noise, can substantially improve performance. Furthermore, DQN-IDS slightly outperforms C51, even though C51 has the benefits of distributional learning.
We also see that C51-IDS outperforms C51 and QR-DQN and achieves slightly better results than IQN. Importantly, the fact that C51-IDS substantially outperforms DQN-IDS, highlights the significance of accounting for heteroscedastic noise. We also experimented with a QRDQN-IDS version, which uses QR-DQN instead of C51 to estimate Z(s, a) and noticed that our method can benefit from better approximation of the return distribution. While we expect the performance over IQN to be higher, we do not include QRDQN-IDS scores since we were unable to reproduce the reported QR-DQN results on some games. We also note that, unlike C51-IDS, IQN is specifically tuned for risk sensitivity. One way to get a risk-sensitive IDS policy is by tuning for β in the additive IDS formulationΨ(s, a) =∆(s, a) 2 − βI(s, a), proposed by Russo & Van Roy (2014). We verified on several games that C51-IDS scores can be improved by using this additive formulation and we believe such gains can be extended to the rest of the games.
CONCLUSION
We extended the idea of frequentist Information-Directed Sampling to a practical RL exploration algorithm that can account for heteroscedastic noise. To the best of our knowledge, we are the first to propose a tractable IDS algorithm for RL in large state spaces. Our method suggests a new way to use the return distribution in combination with parametric uncertainty for efficient deep exploration and demonstrates substantial gains on Atari games. We also identified several sources of heteroscedasticity in RL and demonstrated the importance of accounting for heteroscedastic noise for efficient exploration. Additionally, our evaluation results demonstrated that similarly to the bandit setting, IDS has the potential to outperform alternative strategies such as TS in RL.
There remain promising directions for future work. Our preliminary results show that similar improvements can be observed when IDS is combined with continuous control RL methods such as the Deep Deterministic Policy Gradient (DDPG) (Lillicrap et al., 2016). Developing a computationally efficient approximation of the randomized IDS version, which minimizes the regret-information ratio over the set of stochastic policies, is another idea to investigate. Additionally, as indicated by Russo & Van Roy (2014), IDS should be seen as a design principle rather than a specific algorithm, and thus alternative information gain functions are an important direction for future research.
A HYPERPARAMETERS
score = agent − random human − random × 100(11)
where agent, human and random represent the per-game raw scores. Plots obtained by sampling a random batch of 32 states from the replay buffer every 50000 steps and computing the estimates for ρ 2 (s, a) based on eq. (9). A histogram over the resulting values is then computed and displayed as a distribution (by interpolation). From top to bottom, the lines on each plot correspond to standard deviation boundaries of a normal distribution [max, µ + 1.5σ, µ + σ, µ + 0.5σ, µ, µ − 0.5σ, µ − σ, µ − 1.5σ, min]. The x-axis indicates number of training frames. Dabney et al. (2018b) and Dabney et al. (2018a). C51-IDS averaged over 3 seeds.
Figure 1 :
1Gaussian Process setting. R: the true function, ρ 2 : true observation noise variance, blue: confidence region with µ indicating the mean, blue dots: sampled evaluation points. (a): prior, (b), (c), (d): UCB, TS, IDS posteriors respectively after 20 samples.
Figure 2 :
2Training curves for DQN-IDS and C51-IDS averaged over 3 seeds. Shaded areas correspond to min and max returns.
Figure 3 :
3The return distributions learned by C51-IDS and C51.
Table 1 :
1Mean and median of best scores computed across the Atari 2600 games fromTable 3and 4 in the appendix, measured as human-normalized percentages(Nair et al., 2015). QR-DQN and IQN scores obtained fromTable 1inDabney et al. (2018a), by removing the scores of Defender and Surround. DQN-IDS and C51-IDS averaged over 3 seeds.Mean
Median
DQN
232%
79%
DDQN
313%
118%
Dueling
379%
151%
NoisyNet-DQN
389%
123%
Prior.
444%
124%
Bootstrapped DQN 553%
139%
Prior. Dueling
608%
172%
NoisyNet-Dueling
651%
172%
DQN-IDS
757%
187%
C51
721%
178%
QR-DQN
888%
193%
IQN
1048% 218%
C51-IDS
1058% 253%
DQN, with variance Var (Z(s, a)
Table 2 :
2ALE hyperparameters Agent step at which learning starts. Random policy beforehand number of bins 51 Number of bins for Categorical DQN (C51) [V MIN , V MAX ] [-10, 10] C51 distribution range number of quantiles 200 Number of quantiles for QR-DQN target network update frequency 40000 Number of agent steps between consecutive target updates evaluation length 125K Number of agent steps each evaluation window lasts for. Equivalent to 500K frames evaluation frequency 250K The number of steps the agent takes in training mode between two evaluation runs. Equivalent to 1M frames eval episode length 27K Number of maximum agent steps during an evaluation episode. Equivalent to 108K frames max no-ops 30 Maximum number no-op actions before the episode starts B SUPPLEMENTAL RESULTS Human-normalized scores are computed as (van Hasselt et al., 2016),Hyperparameter
Value
Description
λ
0.1
Scale factor for computing regret surrogate
ρ 2
1.0
Observation noise variance for DQN-IDS
1 , 2
0.00001 Information-ratio constants; prevent division by 0
mini-batch size
32
Size of mini-batch samples for gradient descent step
replay buffer size
1M
The number of most recent observations stored in the replay
buffer
agent history length
4
The number of most recent frames concatenated as input to
the network
action repeat
4
Repeat each action selected by the agent this many times
γ
0.99
Discount factor
training frequency
4
The number of times an action is selected by the agent be-
tween successive gradient descent steps
K
10
Number of bootstrap heads
β 1
0.9
Adam optimizer parameter
β 2
0.99
Adam optimizer parameter
ADAM
0.01/32
Adam optimizer parameter
α
0.00005 learning rate
learning starts
50000
Table 3 :
3Raw evaluation scores. Episodes start with up to 30 no-op actions. Reference values fromWang et al. (2016) andOsband et al. (2016a). DQN-IDS averaged over 3 seeds. Bootstrap DQN scores for Berzerk, Phoenix, Pitfall!, Skiing, Solaris and Yars' Revenge obtained from our custom implementation.DQN
DDQN
Duel.
Bootstrap Prior.Duel. DQN-IDS
Alien
1,620.0
3,747.7
4,461.4
2,436.6
3,941.0
9,780.1
Amidar
978.0
1,793.3
2,354.5
1,272.5
2,296.8
2,457.0
Assault
4,280.4
5,393.2
4,621.0
8,047.1
11,477.0
9,446.7
Asterix
4,359.0
17,356.5
28,188.0
19,713.2
375,080.0
50,167.3
Asteroids
1,364.5
734.7
2,837.7
1,032.0
1,192.7
1,959.7
Atlantis
279,987.0 106,056.0 382,572.0 994,500.0 395,762.0
993,212.5
Bank Heist
455.0
1,030.6
1,611.9
1,208.0
1,503.1
1,226.1
Battle Zone
29,900.0
31,700.0
37,150.0
38,666.7
35,520.0
67,394.2
Beam Rider
8,627.5
13,772.8
12,164.0
23,429.8
30,276.5
30,426.6
Berzerk
585.6
1,225.4
1,472.6
1,077.9
3,409.0
4,816.2
Bowling
50.4
68.1
65.5
60.2
46.7
50.7
Boxing
88.0
91.6
99.4
93.2
98.9
99.9
Breakout
385.5
418.5
345.3
855.0
366.0
600.1
Centipede
4,657.7
5,409.4
7,561.4
4,553.5
7,687.5
5,860.2
Chopper Command
6,126.0
5,809.0
11,215.0
4,100.0
13,185.0
13,385.4
Crazy Climber
110,763.0 117,282.0 143,570.0 137,925.9 162,224.0
194,935.7
Demon Attack
12,149.4
58,044.2
60,813.3
82,610.0
72,878.6
130,687.2
Double Dunk
-6.6
-5.5
0.1
3.0
-12.5
1.2
Enduro
729.0
1,211.8
2,258.2
1,591.0
2,306.4
2,358.2
Fishing Derby
-4.9
15.5
46.4
26.0
41.3
45.2
Freeway
30.8
33.3
0.0
33.9
33.0
34.0
Frostbite
797.4
1,683.3
4,672.8
2,181.4
7,413.0
5,884.3
Gopher
8,777.4
14,840.8
15,718.4
17,438.4
104,368.2
47,826.2
Gravitar
473.0
412.0
588.0
286.1
238.0
771.0
H.E.R.O.
20,437.8
20,818.2
23,037.7
21,021.3
21,036.5
15,165.4
Ice Hockey
-1.9
-2.7
0.5
-1.3
-0.4
1.7
James Bond
768.5
1,358.0
1,312.5
1,663.5
812.0
1,782.2
Kangaroo
7,259.0
12,992.0
14,854.0
14,862.5
1,792.0
15,364.5
Krull
8,422.3
7,920.5
11,451.9
8,627.9
10,374.4
10,587.3
Kung-Fu Master
26,059.0
29,710.0
34,294.0
36,733.3
48,375.0
38,113.5
Montezuma's Revenge 0.0
0.0
0.0
100.0
0.0
0.0
Ms. Pac-Man
3,085.6
2,711.4
6,283.5
2,983.3
3,327.3
7,273.7
Name This Game
8,207.8
10,616.0
11,971.1
11,501.1
15,572.5
15,576.7
Phoenix
8,485.2
12,252.5
23,092.2
14,964.0
70,324.3
176,493.2
Pitfall!
-286.1
-29.9
0.0
0.0
0.0
0.0
Pong
19.5
20.9
21.0
20.9
20.9
21.0
Private Eye
146.7
129.7
103.0
1,812.5
206.0
201.1
Q*Bert
13,117.3
15,088.5
19,220.3
15,092.7
18,760.3
26,098.5
River Raid
7,377.6
14,884.5
21,162.6
12,845.0
20,607.6
27,648.3
Road Runner
39,544.0
44,127.0
69,524.0
51,500.0
62,151.0
59,546.2
Robotank
63.9
65.1
65.3
66.6
27.5
68.6
Seaquest
5,860.6
16,452.7
50,254.2
9,083.1
931.6
58,909.8
Skiing
-13,062.3
-9,021.8
-8,857.4
-9,413.2
-19,949.9
-7,415.3
Solaris
3,482.8
3,067.8
2,250.8
5,443.3
133.4
2,086.8
Space Invaders
1,692.3
2,525.5
6,427.3
2,893.0
15,311.5
35,422.1
Star Gunner
54,282.0
60,142.0
89,238.0
55,725.0
125,117.0
84,241.0
Tennis
12.2
-22.8
5.1
0.0
0.0
23.6
Time Pilot
4,870.0
8,339.0
11,666.0
9,079.4
7,553.0
13,464.8
Tutankham
68.1
218.4
211.4
214.8
245.9
265.5
Up and Down
9,989.9
22,972.2
44,939.6
26,231.0
33,879.1
85,903.5
Venture
163.0
98.0
497.0
212.5
48.0
389.1
Video Pinball
196,760.4 309,941.9 98,209.5
811,610.0 479,197.0
696,914.0
Wizard Of Wor
2,704.0
7,492.0
7,855.0
6,804.7
12,352.0
19,267.9
Yars' Revenge
18,098.9
11,712.6
49,622.1
17,782.3
69,618.1
25,279.5
Zaxxon
5,363.0
10,163.0
12,944.0
11,491.7
13,886.0
16,789.2
Table 4 :
4Raw evaluation scores. Episodes start with up to 30 no-op actions. Reference values (available for a single seed) for C51, QR-DQN and IQN taken from
ACKNOWLEDGMENTSWe thank Ian Osband and Will Dabney for providing details about the Atari evaluation protocol. This work was supported by SNSF grant 200020 159557, the Vector Institute and the Open Philanthropy Project AI Fellows Program.
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Name This Game 2. 292Name This Game 2,292.3
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[
"MuZero with Self-competition for Rate Control in VP9 Video Compression",
"MuZero with Self-competition for Rate Control in VP9 Video Compression"
]
| [
"Amol Mandhane ",
"Anton Zhernov ",
"Maribeth Rauh ",
"Chenjie Gu ",
"Miaosen Wang ",
"Flora Xue ",
"Wendy Shang ",
"Derek Pang ",
"Rene Claus ",
"Ching-Han Chiang ",
"Cheng Chen ",
"Jingning Han ",
"Angie Chen ",
"Daniel J Mankowitz ",
"Jackson Broshear ",
"Julian Schrittwieser ",
"Thomas Hubert ",
"Oriol Vinyals ",
"Timothy Mann "
]
| []
| []
| Video streaming usage has seen a significant rise as entertainment, education, and business increasingly rely on online video. Optimizing video compression has the potential to increase access and quality of content to users, and reduce energy use and costs overall. In this paper, we present an application of the MuZero algorithm to the challenge of video compression. Specifically, we target the problem of learning a rate control policy to select the quantization parameters (QP) in the encoding process of libvpx, an open source VP9 video compression library widely used by popular video-on-demand (VOD) services. We treat this as a sequential decision making problem to maximize the video quality with an episodic constraint imposed by the target bitrate. Notably, we introduce a novel self-competition based reward mechanism to solve constrained RL with variable constraint satisfaction difficulty, which is challenging for existing constrained RL methods. We demonstrate that the MuZero-based rate control achieves an average 6.28% reduction in size of the compressed videos for the same delivered video quality level (measured as PSNR BD-rate) compared to libvpx's two-pass VBR rate control policy, while having better constraint satisfaction behavior. * Equal contributions | null | [
"https://arxiv.org/pdf/2202.06626v1.pdf"
]
| 246,823,228 | 2202.06626 | 5a3fe4482c8d754b290cf90c0fc03362d02ee1a4 |
MuZero with Self-competition for Rate Control in VP9 Video Compression
Amol Mandhane
Anton Zhernov
Maribeth Rauh
Chenjie Gu
Miaosen Wang
Flora Xue
Wendy Shang
Derek Pang
Rene Claus
Ching-Han Chiang
Cheng Chen
Jingning Han
Angie Chen
Daniel J Mankowitz
Jackson Broshear
Julian Schrittwieser
Thomas Hubert
Oriol Vinyals
Timothy Mann
MuZero with Self-competition for Rate Control in VP9 Video Compression
Video streaming usage has seen a significant rise as entertainment, education, and business increasingly rely on online video. Optimizing video compression has the potential to increase access and quality of content to users, and reduce energy use and costs overall. In this paper, we present an application of the MuZero algorithm to the challenge of video compression. Specifically, we target the problem of learning a rate control policy to select the quantization parameters (QP) in the encoding process of libvpx, an open source VP9 video compression library widely used by popular video-on-demand (VOD) services. We treat this as a sequential decision making problem to maximize the video quality with an episodic constraint imposed by the target bitrate. Notably, we introduce a novel self-competition based reward mechanism to solve constrained RL with variable constraint satisfaction difficulty, which is challenging for existing constrained RL methods. We demonstrate that the MuZero-based rate control achieves an average 6.28% reduction in size of the compressed videos for the same delivered video quality level (measured as PSNR BD-rate) compared to libvpx's two-pass VBR rate control policy, while having better constraint satisfaction behavior. * Equal contributions
Introduction
In recent years, video traffic has dominated global internet traffic and is expected to grow further [Cisco, 2020]. Efficient video encoding algorithms are key to reducing bandwidth and storage costs, as well as improving user Quality of Experience (QoE) in scenarios such as video-on-demand (VOD) and live streaming.
In video compression, rate control is a critical component that determines the trade-off between rate (video size) and distortion (video quality) by assigning a quantization parameter (QP) for each frame in the video. A lower QP results in lower distortion (higher quality), but uses more bits. In this paper, we focus on the rate control problem in libvpx [WebM, 2010b], an open source VP9 codec library (b) Example of Rate-Distortion (RD) curve. Figure 1: Illustrative example of the rate control process, Rate-Distortion curve, and the BD-rate metric for encoding efficiency.
[ Mukherjee et al., 2013]. The objective in this setting is to select a QP for each of the video frames to maximize the quality of the encoded video subject to a bitrate constraint as seen in (1).
The rate control problem can be seen as a constrained planning problem. In particular, the QPs are chosen sequentially for each video frame as seen in Figure 1a, with the goal of optimizing for video quality under a bitrate constraint. Classical optimization algorithms such as mixed integer linear programming cannot be used for rate control as the codec operates as a black box without an explicit equation and the interactions between the video frames cannot easily be modeled by a linear objective.
To provide some intuition for the planning element, imagine a video that has little motion for the first half of the video (e.g., a person talking or a static scene), and then has a lot of motion for the latter half of the video (e.g., a sports highlight). In this case, using up more of the bitrate budget on the first half of the video will cause the second half of the video to have lower quality. Intuitively, the rate control algorithm needs to allocate more bits to complex/dynamic frames and less bits to simple/static frames. This requires the algorithm to understand the rate-distortion tradeoff for each frame and their relationship to the QP values. However, the inter-dependency of QP decisions and the rate-distortion metrics are often non-trivial, making it challenging to engineer optimal rate control algorithms that work for a diverse set of videos.
Reinforcement Learning (RL) [Sutton and Barto, 2018] is a sequential decision making framework that has had recent success in solving numerous planning problems (in many cases achieving superhuman performance) ranging from games [Mnih et al., 2013, Tessler et al., 2017, Vinyals et al., 2019, Silver et al., 2016 to chip design [Mirhoseini et al., 2020] and robotics [Levine et al., 2016]. RL algorithms are now solving problems at scale and are potentially well-suited for solving rate control.
In this work, we apply the MuZero RL algorithm [Schrittwieser et al., 2020] along with a novel selfcompetition based constrained RL method to rate control in VP9. Specifically, our main contributions are as follows:
• We propose a self-competition based reward mechanism that optimizes for the constrained rate control objective.
• We combine this self-competition based reward mechanism with the MuZero algorithm to create 'MuZero-RC', an RL based rate control agent.
• We evaluate this agent against libvpx's two-pass VBR rate control implementation [WebM, 2017] on 3062 video clips, each of which are 5-seconds in length, obtained from the YouTube UGC dataset [Wang et al., 2019] which has been created for evaluating video compression algorithms. We show that MuZero-RC achieves an average 6.28% reduction in bitrate (measured as PSNR BD-rate) compared to the libvpx baseline which is canonically used for VP9 encoding, while having better bitrate constraint satisfaction.
Related Work
Rate control in video codecs: Traditionally, rate control algorithms are based on empirical models using relevant features, such as mean absolute differences (MAD) and sum of absolute transformed differences (SATD) from past encoded frames. Common empirical models include quadratic models used in MPEG-4's VM8 [Tihao Chiang and Ya-Qin Zhang, 1996] and H.264/AVC [Ma et al., 2005, Minqiang Jiang and Nam Ling, 2005, Kwon et al., 2007, ρ-domain models that map between rate and the proportion of non-zero quantized coefficients [He and Mitra, 2002], and dynamic programming based models that exploit interframe dependency [Jiangtao Wen et al., 2000]. In modern codecs, such as HEVC and VVC, Li et al. [2014] introduced λ-domain model to control both rate and mode selections through rate-distortion optimization instead of relying on QP decisions. Despite recent advances in rate control models, the empirical rate control models require complex hand-designed heuristics to effectively adapt to different dynamic video sequences and satisfy different requirements for different applications.
ML for rate control: Machine learning enables more complex and nonlinear models to be formulated. Sanz-Rodriguez and Diaz-de-Maria [2011] use a radial basis function network to predict QP offset and ensure quality consistency. Gao et al. [2016] utilize a game theory method to allocate CTU-level bit allocation and optimize for SSIM in HEVC. More recently, Reinforcement learning approaches in rate control have also been proposed to HEVC , Ho et al., 2021, Chen et al., 2018. Mao et al. [2020] use an imitation learning approach on evolutionary search based policy [Salimans et al., 2017] with a feedback-based correction for rate control in VP9.
Unlike the previously proposed ML based approaches, our proposed solution can work with a wide range of target bitrates specified for longer video durations with multiple groups-of-pictures (GOPs) instead of specifying bitrates for each GOP, which is close to the practical setting. The solution does not rely on demonstrations from prior experience, which allows discovery of rate control strategies from scratch.
Background
VP9 Video Compression
In this section, we briefly describe the two-pass variable bitrate (VBR) mode, which is a popular libvpx-vp9 rate control mode for Videos-On Demand (VOD). In this work, we use the statistics collected in the first-pass and the second-pass encoding as the input features to a neural network model that predicts the QP for every frame.
First-pass encoding: The encoder computes statistics for every frame in the video, by dividing the frame into non-overlapping 16 × 16 blocks followed by per-block intra and inter-frame prediction (of each block) and calculation of the residual error. The statistics contain information such as average motion prediction error, average intra-frame prediction error, average motion vector magnitude, percentage of zero motion blocks, noise energy, etc. [WebM, 2010a].
Second-pass encoding: The encoder uses the first-pass statistics to decide key-frame locations and insert hidden alternate reference frames. The key-frames and alternate reference frames are used as references for encoding other frames, so their encoding quality affects other frames in the video as well. With those decisions made, the encoder starts to encode video frames sequentially as seen in Figure 1a. The rate controller regulates the trade-off between rate and distortion by specifying a QP to each frame in order to maximize the quality and reduce the bits as explained in section 4.1. The QP is an integer in range [0, 255] that can be monotonically mapped to a quantization step size which is used for quantizing the frequency transform of the prediction residue for entropy coding. Smaller quantization step sizes lead to smaller reconstruction error (measured by mean squared error), but also higher bits usage for the frame.
Constrained Markov Decision Process
A Markov Decision Process (MDP) [Sutton and Barto, 2018] M is defined by the tuple S, A, P, R, γ where S is the set of states; A is the set of available state-dependent actions; R : S × A → R is a bounded reward function; P : S × A × S → [0, 1] is the transition function, where P (s |s, a) is the probability of transitioning to state s from s when action a was taken; γ ∈ [0, 1] is the discount factor. A policy π : S × A → [0, 1] is a mapping from a state to a distribution over actions where π (a|s) denotes the probability of taking action a in state s. The goal is to obtain a policy which maximizes the expected sum of discounted rewards (J R π ).
max π J R π (s) := E a∼π,s∼P t γ t R(s t , a t )
A Constrained Markov Decision Process (CMDP) [Altman, 1999] extends the MDP formulation by introducing k constraint functions c k : S × A → R and corresponding thresholds β k . The goal is to obtain a policy which maximizes the expected sum of discounted rewards subject to the discounted constraints (J c k π ) being under the target thresholds β k .
max π J R π s.t. J c k π ≤ β k ∀k(2)
There are a number of approaches to solving this constrained RL objective [Altman, 1999, Tessler et al., 2018, Efroni et al., 2020, Achiam et al., 2017, Bohez et al., 2019, Chow et al., 2018, Paternain et al., 2019, Zhang et al., 2020. One approach involves solving the Lagrangian relaxation of the original problem as max π min λ≥0 J R π − λ(J c k π − β k ). Here, the typical approach is to alternate the optimization between the policy π and the Lagrangian parameter λ [Tessler et al., 2018, Calian et al., 2021. Another powerful approach is to fix the Lagrangian parameter and perform reward shaping [Achiam et al., 2017, Tessler et al., 2018.
MuZero
MuZero [Schrittwieser et al., 2020] is a Reinforcement Learning algorithm based on deep neural networks and Monte-Carlo tree search (MCTS) planning algorithm [Coulom, 2006] which has achieved state-of-the-art performance in various domains like Chess, Go, Shogi, and Atari. MuZero uses a learned model of the environment dynamics with MCTS to look ahead and plan from a given state in order to identify good actions. It uses a policy neural network as a prior for the MCTS, and a value network to truncate the MCTS. Unlike its predecessors, MuZero does not need to interact with the environment for planning; instead it uses a learned model of environment dynamics to look ahead.
Rate Control with MuZero
The following section is divided into three parts. First, we formulate the rate control problem as a CMDP and define the rate control CMDP objective that we wish to optimize. Next, we introduce the self-competition based reward mechanism to address this constrained objective that can be incorporated into any RL agent. Finally, we provide an overview of the architecture and training setup of our MuZero agent combined with the self-competition based reward for rate control (MuZero-RC).
Rate Control as a Constrained RL Problem
In this paper, we focus on the 'two-pass, variable bitrate (VBR)' mode in the libvpx implementation of VP9. However, our methods do not focus on the specifics of this mode and can be generalized to other settings. The objective of rate control in this mode is to maximize the quality of the encoded video while keeping the size under a user-specified target. Quality of the encoding is commonly measured as Peak Signal-to-Noise Ratio (PSNR) := 10 log 10 255 2 /MSE(original, encoded) , although any other quality measure such as SSIM [Wang et al., 2004] or VMAF [Li et al., 2016] can be used.
The rate control problem can be formulated as the Constrained MDP (CMDP) tuple S, A, P, R PSNR , γ = 1, c Bitrate , β Target . The state space S consists of the first-pass statistics and the frame information generated by the encoder (described in section 4.3). The action space A consists of an integer QP in the range [0,255] which is applied to the frame being encoded. The transition function P (s |s, a) transitions to a new state s ∈ S (next frame to be encoded) given that action a ∈ A (QP) was executed from state s (current frame being encoded) as seen in Figure 1a. The bounded reward function R PSNR is defined as the PSNR of the encoded video at the final timestep of the episode and 0 otherwise. The constraint function c Bitrate is the bitrate of the encoded video at the final timestep of the episode and 0 otherwise. β Target is the user-specified size target for the encoding. We consider the size target range of [256,768] kbps (kilobits-per-second) for 480p videos in this paper, although our methods can be extended to larger ranges. The rate control constrained objective to optimize is defined as:
max π(QP t |st,βtarget) J PSNR π s.t. J Bitrate π ≤ β Target(3)
As mentioned in Section 3.2, it is possible to solve the unconstrained Lagrangian relaxation of the CMDP objective by performing alternating optimization on the Lagrange parameter λ and the policy π. However, as mentioned in previous works (e.g., [Tessler et al., 2018]), multi-timescale stochastic approximation is very sensitive to learning rates and is therefore typically difficult to tune. Additionally, for learning a rate control policy, a large number of Lagrangian parameters would be needed to solve the constrained RL objective (per video and per target bitrate) as there is no single multiplier that is optimal across videos and bitrates. We validated this by combining MuZero with the Lagrangian relaxation method and observed that it was both difficult to tune during training and was unable to provide a reasonable solution in terms of quality and constraint satisfaction during evaluation. A more in-depth analysis can be found in the appendix A.4. As such, we propose a different approach which we refer to as self-competition which solves a slightly different form of the above CMDP objective and removes the need to use Lagrangian parameters.
Constrained RL via Self-Competition
In this section, we present the self-competition reward mechanism that enables solving the rate control constrained optimization problem defined in Equation 3. The high-level intuition of this reward mechanism is that the agent attempts to outperform its own historical performance on the constrained objective over the course of the training. We adapt MuZero to perform self-competition and refer to this agent as MuZero-Rate-Controller (MuZero-RC). Note that this mechanism differs from MuZero's self-play as self-play runs two-player games with the latest version of the agent, while self-competition agent is competing against its own historical performance in a single player setting.
To track the historical performance of the agent, we maintain exponential moving averages (EMA) of PSNR and overshoot (bitrate -target β) for each unique [video, target bitrate β] pair over the course of the training in a buffer. When the agent newly encodes a video with a target bitrate β, the encoder computes the PSNR and overshoot of the episode (P episode and O episode respectively). The agent looks up the historical PSNR and overshoot (P EMA and O EMA respectively) for that [video, target bitrate β] pair from the buffer. We first compare O episode with the O EMA , and then compare P episode with P EMA . The return for the episode is set to ±1 depending on whether the episode is better than the EMA-based historical performance according to equation 4. We also update the EMAs in the buffer for the [video, target bitrate β] pair with the P episode and O episode . Algorithm 1 provides a detailed pseudocode of this self-competition algorithm.
Return = sgn(P episode − P EMA ) Improve PSNR 1 OEMA≤0,Oepisode≤0 if constraints are satisfied − sgn( O episode + − O EMA + )
otherwise, ensure constraints are satisfied (4) where sgn is the sign function, 1 is the indicator function, and x + = max(x, 0).
To understand why this solution is suitable for variable difficulty constrained RL, we can interpret the EMA-based historical performance as a baseline for the agent to beat. As the agent learns to beat this baseline, the baseline itself improves via EMA updates. This leads the agent to improve its performance further to beat the newer baseline, which creates a cycle of performance improvement.
Since the performance is measured by comparing bitrate constraint satisfaction first, the agent learns to satisfy the constraint first, and then learns to improve the PSNR. Unlike Lagrangian relaxation methods, this mechanism doesn't contain any direct parameters which represent the difficulty of constraint satisfaction. This allows the agent to infer the difficulty of satisfying the constraints from the observations. This formulation can be related to applying Lagrangian relaxation methods on an augmented CMDP. The state of the CMDP can be augmented with EMAs, and the constrained objective becomes max π(QP t |st,PEMA,OEMA,βTarget)
sgn(P episode − P EMA )1 OEMA≤0,Oepisode≤0 s.t. sgn( O episode + − O EMA + ) ≤ 0
Using a Lagrangian relaxation method with λ = 1 on this objective results in the same return as equation 4. Reward mechanisms with historical performance based self-competition have been proposed in the past [Laterre et al., 2018, Schmidt et al., 2019, although they haven't been applied to constrained RL to the best of our knowledge.
Agent Architecture
Our training setup closely follows Schrittwieser et al. [2020]. We train this agent in the canonical asynchronous distributed actor-learner setting with experience replay [Lin, 1992, Horgan et al., 2018. We maintain a shared buffer across all actors to track agent's EMA-based historical performance for each [video, target bitrate] pair. Actor processes sample [video, target bitrate] pairs randomly from the training dataset, and generate QPs for encoding them using the latest network parameters and 200 simulations of MuZero's MCTS algorithm. The self-competition based reward for the episode is computed using equation 4 and the episode is added to the experience replay buffer. The learner process samples transitions from the experience replay buffer to train the networks, and sends the updated parameters to the actor processes at regular intervals.
Features: At every step in the encoding process, the agent receives an observation from the encoder containing first-pass statistics for the video, PSNR and bits for the frames in the video encoded so far, index and type of the next frame to be encoded, and the target bitrate for the encoding. Appendix A.1 lists the features and preprocessing methods in detail.
Network Architecture: Similar to MuZero, the agent has three subnetworks. The representation network maps the features to an embedding of the state using Transformer-XL [Dai et al., 2019] encoders and ResNet-V2 style blocks [He et al., 2016]. The dynamics network generates the embedding of the next state given the embedding of the previous state and an action using ResNet-V2 style blocks. Given an embedding, the prediction network computes a policy using a softmax over the QPs, the value of the state, and several auxiliary predictions of the metrics related to the encoding using independent layer-normalised [Ba et al., 2016] feedforward networks. Appendix A.1 lists these auxiliary predictions and describes the architecture of the subnetworks in detail.
Training: At every step in the training, the learner uniformly samples a batch of states, five subsequent actions, and the necessary labels from the experience replay buffer. The representation network generates the embedding of the state, and the dynamics network is unrolled five times to generate the subsequent embeddings. The prediction network outputs the predictions for policy, value, and auxiliary metrics for the current and the five subsequent states. The policy predictionπ t is trained to match the MCTS policy generated at acting time π t using cross-entropy loss. The value prediction v t is trained to match the self-competition reward v t for the episode using IQN loss [Dabney et al., 2018a], and the auxiliary predictionsŷ t are trained to match the corresponding metrics y t using a quantile regression loss [Dabney et al., 2018b]. We use equation 5 to combine the losses.
Loss = 1 6 5 t=0 LCE(πt,πt) Policy + 0.5 · L IQN (v t ,v t ) Value +0.1 · Auxiliary L QR (y t ,ŷ t ) + 10 −3 · θ 2 2 L2-Reg(5)
In our experiments, we found the auxiliary losses crucial for learning the dynamics of the encoding in order to perform the MCTS. The hyperparameters of the training are described in appendix A.2.
Augmented MuZero-RC
We also propose a small variant of the MuZero-RC agent. We augment the self-competition mechanism in equation 4 by replacing the PSNR term with 'PSNR − 0.005 × overshoot' in order to get the agent to reduce bitrate if it can't improve PSNR. We label this agent 'Augmented MuZero-RC' and report the results alongside MuZero-RC in section 5. The factor of 0.005 was selected as it is roughly the average slope of VP9's RD curve across the training dataset. In our experiments, we didn't find significant performance difference with small adjustments to this factor.
Experiments
In this section, we first describe the experiment setup. This includes an overview of the environment, the dataset of videos used for training and evaluation, and the evaluation metrics. We then describe the performance of MuZero-RC agents in this setup compared to the baselines.
Setup
Environment setup and baseline: We use the libvpx implementation of VP9 in our experiments. In particular, we implemented an RL environment based on the SimpleEncode API [WebM, 2019] in libvpx. This allows us to override the libvpx's QP decision with an external QP computed by our agent. We train and evaluate the learned rate control policy using the environment. In our experiments, we use libvpx's two-pass variable bitrate (VBR) mode [WebM, 2017] at encode speed 1 (i.e., --cpu-used=1), which is a commonly used setting for video-on-demand (VOD). We use libvpx's default rate control implementation as our baseline as it is canonically used in VP9. Metrics: To evaluate the coding efficiency and overall bitrate reduction, we measure the Bjontegaarddelta rate (BD-rate) [Bjontegaard, 2001] using libvpx's default VBR rate control policy as the reference. Given the bitrate v.s. PSNR curves of two policies, BD-rate computes the average bitrate difference for the same PSNR across the overlapped PSNR range, and therefore, measures the average bitrate reduction for encoding videos at the same quality. See figure 1b for an example. We report the mean PSNR BD-rate difference compared to libvpx on the evaluation set, as well as BD-rate difference with respect to perceptual quality measures SSIM and VMAF. To evaluate the bitrate constraint satisfaction, we report the fraction of evaluation cases on which the policies violate the bitrate constraint. Since a small amount of constraint violation is acceptable in practice, we also report the fraction of cases on which the policies violate the constraint by >5% of the target bitrate. We also note that since PSNR and bitrate are competing objectives, the optimal behavior is to use the target bitrate budget fully in order to maximize the PSNR. To evaluate this, we report the fraction of cases on which the policies achieve bitrate within 5% of the target. We report the histograms of BD-rate differences and bitrates in appendices A.6 and A.7.
Evaluation: We use the final network checkpoint generated by the training process for evaluation. Practically, the rate control step should not take a large amount of time. So, instead of performing MuZero-style MCTS, we select the QPs with the highest probability assigned by the policy output of the prediction network. We iterate over the evaluation dataset and encode each video with libvpx rate control and both the MuZero-RC agents at nine target bitrate values uniformly spaced between [256,768] kbps. We evaluate the constraint satisfaction behavior of the agent for all nine target bitrates, and use the RD-curve generated by these encodings for each video to compute the BD-rate difference.
Results
PSNR BD-Rate improvement: First, we evaluate the PSNR BD-rate of MuZero-RC agents compared to libvpx. Table 1 shows the difference of PSNR BD-rate achieved by MuZero-RC over the evaluation set videos compared to libvpx's rate control. It shows that MuZero-RC can achieve better PSNR v.s. bitrate tradeoff with 4.72% average bitrate savings for encoding videos at same quality as libvpx, and Augmented MuZero-RC increases it further to 6.28%. The agent reduces the BD-rate with respect to perceptual quality measures SSIM and VMAF as well. We present the histogram of BD-rate differences over the evaluation video set in appendix A.6. MuZero-RC -4.72% ± 0.32% -3.68% ± 0.33% -0.53% ± 0.21% Augmented MuZero-RC -6.28% ± 0.19% -5.11% ± 0.24% -1.88% ± 0.12%
Constraint satisfaction: Then, we evaluate the bitrate constraint satisfaction behaviour. Table 2 lists the constraint satisfaction metrics achieved by libvpx baseline policy and the MuZero-RC agents. The MuZero-RC agent violates the constraints on substantially fewer videos compared to libvpx. Even when the constraints are violated, large magnitude violations by MuZero-RC are less frequent compared to libvpx. Also, MuZero-RC agent encodes videos with bitrates within 5% margin around the target bitrate more frequently than libvpx. This indicates that MuZero-RC not only overshoots less, but it also tends to fully use the target bitrate budget more frequently compared to libvpx.
We also see that Augmented MuZero-RC satisfies constraints just as well as MuZero-RC, and its bitrate accuracy is comparable to libvpx. We present the histogram of overshoots over the evaluation video set for different target bitrates in appendix A.7. Figure 2 demonstrates a typical behavior of MuZero-RC regarding bitrate control. As shown in Figure 2a, the libvpx baseline overshoots the target bitrate by about 5%, while MuZero-RC is accurate in allocating the target bitrate (marked by black line). MuZero-RC avoids overshooting by using fewer bits (and thus sacrificing the PSNR) in early frames of the video as shown in Figure 2b. This is a good decision because the PSNR of the first part of the video is relatively high. In fact, if a policy sacrifices the PSNR in the last 40 frames, it would hurt the overall PSNR. This example highlights agent's ability to reason about frame complexity and bitrate-PSNR tradeoff, and plan accordingly. Planning performance: We investigate the encoding trajectories of videos and observe a few recurring patterns where MuZero-RC demonstrates better planning performance than libvpx. Figure 3 shows examples where MuZero-RC achieves better BD-rate. In Figure 3a, MuZero-RC sacrifices the PSNR of the first 60 frames in order to improve the PSNR of the rest of the frames. Because the latter part of the video is more complex (indicated by lower PSNR), improving the PSNR of the latter part of the video improves the overall PSNR. In Figure 3b, MuZero-RC boosts PSNR of a reference frame, leading to significant PSNR improvement for all the frames following that frame. In Figure 3c, the video has a scene change towards the very end. Because libvpx runs out of bitrate budget, it decides to encode the last few frames with low quality. In contrast, MuZero-RC correctly anticipates the scene change, and saves some bitrate budget from the first 140 frames so that it can obtain a reasonable PSNR for the last few frames. Figure 3d shows another pattern where MuZero-RC has overall less frame PSNR fluctuation. In particular, it avoids having extremely low PSNR on some frames, which in turn improves the overall PSNR.
Conclusions
In this paper, we have demonstrated that the MuZero reinforcement learning algorithm can be used for rate control in VP9. Our formulation of the self-competition based reward mechanism allows the agent to tackle the complex constrained optimization task and achieve better quality-bitrate tradeoff and better bitrate constraint satisfaction than libvpx's VBR rate control algorithm. The final agent results in 6.28% average reduction in bitrate (measured as PSNR BD-rate) on videos from the evaluation set, and can be readily deployed in libvpx via the SimpleEncode API.
Limitations: The self-competition based reward mechanism requires that every unique [video, target bitrate] pair be encoded a few times so that the historical performance converges and provides a reasonable baseline for reward computation. Because of this, the amount of data the actors need to generate increases linearly with the number of videos in the training dataset and the number of target bitrate samples. For very large training datasets, this method might not scale well. However, in future work, it may be possible to learn these baseline values based on observations using a neural network which can generalise to unseen videos in a large dataset.
Future Work: Our proposed methods are agnostic to the specifics of VP9/libvpx, and they can potentially be generalized not only to other coding formats and implementations, but also to other components within video encoders such as block partitioning and reference frame selection. Our method also opens the possibility of allowing codec developers and users to develop new rate control modes. For example, we can replace PSNR with other video quality metrics such as VMAF. We can also modify the reward to minimize bitrate given a minimum PSNR constraint -which is similar to the constrained quality (CQ) mode in libvpx, but reinforcement learning is likely to learn a policy that has more precise control of the PSNR.
A Appendix
A.1 Network Architecture
As required by the MuZero algorithm, the MuZero-RC agent has three subnetworks.
Representation Network: This network takes the features provided by the environment as the input and produces an embedding of the current state of the environment as the output. For any state, the environment generates following observations. 1. A sequence of first-pass statistics for all the show frames in the video. 2. A sequence of PSNR, number of used bits, and applied QPs for all the previously encoded frames in the video so far, along with indices of those frames. 3. The index and the type of the frame to be encoded next. The type can be one of five frame types from the SimpleEncode API. 4. The duration of the video. 5. Target bitrate for the encoding.
We use the same first pass statistics and features normalization methods used by Mao et al. [2020]. Additionally, we generate the fraction of the target bitrate used so far in the encoding using the bits used by previously encoded frames and video duration. We use this fraction as an additional scalar feature.
The representation network aligns the first two sequential features along the indices of the frames and concatenates the aligned sequences along the feature dimension. This concatenated sequence is then processed using a series of 4 Transformer-XL encoder blocks [Dai et al., 2019]. From this sequence, the entry at index of the frame to be encoded next is extracted. This entry is concatenated with the remaining scalar features and processed using two feedforward layers with intermediate layer normalization [Ba et al., 2016]. The network processes the output of these layers with a series of 4 layer normalized ResNet-V2 blocks [He et al., 2016]. The output of these blocks is the embedding of the state. We use an embedding of 512 units in all our experiments. All the layers use ReLU as the activation function. Figure 4 shows a diagram of this network. Dynamics Network: This network takes an embedding of the state and the QP to be applied in that state as the input. It produces an embedding of the next state reached after applying the QP as output. This network processes the QP using two feedforward layers with intermediate layer normalization to output a vector with the same dimension as the embedding of the previous state. It performs elementwise addition of this vector and the embedding of the previous state, and processes the result with a series of 4 layer normalized ResNet-V2 blocks. The output of these blocks is the embedding of the next state reached after applying the QP. All the layers use ReLU as the activation function. Figure 5 shows a diagram of this network. Prediction Network: This network takes the embedding of a state and produces the policy, value, and several auxiliary predictions as the output. For the policy prediction, the network processes the state embedding with two feedforward layers with 256 hidden units and layer normalization followed by a linear layer of 256 units representing the logits for each QP value. A softmax function is applied to these logits to produce the policy. For the value prediction, the network processes the state embedding with two feedforward layers with 256 hidden units and layer normalization followed by a linear layer of 64 units. The output of this layer is used as an embedding for the IQN layer which produces samples of the value prediction. We apply the tanh function to these samples to limit them in range (-1, 1) as the value in the self-competition based reward mechanism is limited to [-1, 1]. At training time, we draw 8 samples from the IQN layer to match the self-competition reward. At inference time, we use the expected value instead of sampling.
Padding
For each of the auxiliary predictions, the network processes the state embedding with two feedforward layers with 256 hidden units and layer normalization followed by a linear layer of 64 units. The output of this layer represents the uniformly spaced quantiles of the corresponding auxiliary prediction. In our experiments, we predict the following metrics as auxiliary predictions.
1. The PSNR of the last encoded frame (0 when no frames are encoded).
2. The log of the number of bits used by the last encoded frame (0 when no frames are encoded).
3. The expected PSNR of the video being encoded.
4. The expected bitrate of the video being encoded.
These auxiliary predictions help the agent understand the dynamics of the video encoding process, which we found to be crucial in our experiments. All the layers use ReLU as the activation function unless specified otherwise. Figure 6 shows a diagram of this network. All experiments in this paper are implemented using the JAX framework [Bradbury et al., 2018]. We use the Haiku library to implement the subnetworks .
A.2 Training
We train all the subnetworks jointly similar to Schrittwieser et al. [2020] in order to minimize the loss in equation 5. We use the SGD algorithm with momentum factor of 0.9 for optimization. The learning rate for the optimization reduces over the course of training as follows
lr(t) = lr init · decay t interval
where t is the training step, lr init = 0.05, decay is 0.1 and interval is 300,000. We use the Optax library [Hessel et al., 2020] for optimization. We train the agent for 1 million steps. 400 million frames are generated by the actor processes over the course of training. For experience replay, we keep a buffer of 50,000 latest episodes generated by the actor processes, and draw samples with batch size 512 from the replay buffer.
A.3 Computational Resources
All experiments in this paper are run using Google Cloud TPUs [Google, 2018]. The learner processes use two 3rd generation TPUs, and the actor processes use four 2nd generation TPUs for 15 hours. Additionally, we use 3000 CPU cores from a shared heterogeneous compute cluster for running video encoding with libvpx for actors.
A.4 Lagrangian-relaxation Method for Rate Control
To evaluate the hypothesis that Lagrangian-relaxation methods for constrained RL are not suitable for rate control, we train a MuZero agent with Lagrangian-relaxation based value function similar to Tessler et al. [2018]. We maintain all other hyperparameters and architecture same as MuZero-RC apart from the tanh activation on value network, which is removed in this agent as the value is no longer in [-1, 1]. We observe that this agent was difficult to tune, and we had to run multiple hyperparameter sweeps to get a stable instance. We evaluated this instance using the protocol described in section 5 and report the metrics in tables 3 and 4 respectively. We notice that the agent exhibits poor performance compared to libvpx in terms of the PSNR BD-rate reduction. We also note that while the agent satisfies the constraint for a reasonable number of test videos, it tends to use very little of the bitrate target budget as seen in the last column of table 4. Also, the number of videos on which agent overshoots by >5% of the target is significantly higher compared to the other policies.
A.5 Pseudocode for Self-competition Reward
Algorithm 1 illustrates the pseudocode for the self-competition reward mechanism. In our experiments, we set the parameters PSNR 0 = 30 dB and α = 0.9. Algorithm 1 Self-competition reward mechanism for MuZero-RC
B [video i , β i ] [0] ← (1 − α) · B [video i , β i ] [0] + α · PSNR Episode 15: B [video i , β i ] [1] ← (1 − α) · B [video i , β i ] [1] + α · overshoot
A.7 Histograms of bitrate constraint satisfaction
This section presents histograms of constraint satisfaction behaviors of libvpx, MuZero-RC, and Augmented MuZero-RC policies on 3062 videos from evaluation set. We encode the videos using each of these policies with nine uniformly spaced target bitrates from the [256,768] kbps. We report the histograms of overshoot (bitrate -target) of the encodings done by each of the policies on aggregate as well as sliced by each of the target bitrate value. Figure 10 shows the histogram of the overshoots for all the policies with all the target bitrates together. Figures 11-19
Quality s.t. Bitrate ≤ Target
Figure 2 :
2Encoding trajectories of video LyricVideo_480P-1484. The learned policy sacrifices PSNR in early frames in order to save bits to avoid bitrate overshooting in the end.
Figure 3 :
3Show frame PSNR trajectories of 4 video clips.
Figure 4 :
4MuZero-RC Representation Network
Figure 5 :
5MuZero-RC Dynamics Network
Figure 6 :
6MuZero-RC Prediction Network
Figure 7 :Figure 8 :Figure 9 :
789Histogram of PSNR BD-Rate difference of agent compared to libvpx on evaluation set. Histogram of SSIM BD-Rate difference of agent compared to libvpx on evaluation set. Histogram of VMAF BD-Rate difference of agent compared to libvpx on evaluation set.
Figure 10 :
10Histogram of overshoots of the agents on the evaluation set for all target bitrates.
Figure 12 :
12Histogram of overshoots of the agents on the evaluation set for 320 kbps target bitrate.
Figure 13 :Figure 14 :Figure 15 :Figure 16 :
13141516Histogram of overshoots of the agents on the evaluation set for 384 kbps target bitrate. Histogram of overshoots of the agents on the evaluation set for 448 kbps target bitrate. Histogram of overshoots of the agents on the evaluation set for 512 kbps target bitrate. Histogram of overshoots of the agents on the evaluation set for 576 kbps target bitrate.
Figure 17 :Figure 18 :
1718Histogram of overshoots of the agents on the evaluation set for 640 kbps target bitrate. Histogram of overshoots of the agents on the evaluation set for 704 kbps target bitrate.
Figure 19 :
19Histogram of overshoots of the agents on the evaluation set for 768 kbps target bitrate.
varying frame rates, and diverse content types. The target bitrate for training is uniformly sampled from {256, 384, 512, 640, 768} kbps. For evaluation, we use videos with resolution 480p and higher from the YouTube UGC dataset[Wang et al., 2019], designed for benchmarking video compression and quality assessment research available under CC-BY 3.0 license. We resize the videos to 480p, and chunk them into 5-second clips, resulting in 3062 clips.Dataset: We limit the focus of our experiments to 480p videos with target bitrate in [256, 768] kbps
range, although our methods can be generalised to other resolutions and bitrates. For training, we use
20,000 randomly selected 480p videos that have varying duration (3 to 7 seconds), varying aspect
ratios,
Table 1 :
1BD-rate difference of MuZero-RC agents compared to libvpx with respect to various quality measures. We report ±1 standard error interval computed over 5 random seeds. As seen in the table, both MuZero-RC and Augmented MuZero-RC lead to BD-rate reductions compared to the libvpx baseline with respect to PSNR, SSIM and VMAF metrics.Mean PSNR
Mean SSIM
Mean VMAF
Agent
BD-rate difference BD-rate difference BD-rate difference
Table 2 :
2Bitrate constraint satisfaction behavior over the evaluation set. We report ±1 standard error interval computed over 5 different random seeds for MuZero-RC agents. As libvpx rate control is a deterministic process, we do not report the standard error intervals for it. As seen in the table, both MuZero-RC and Augmented MuZero-RC overshoot the target bitrate for substantially fewer test cases compared to libvpx. MuZero-RC agent achieves higher bitrate accuracy compared to libvpx while Augmented MuZero-RC achieves a comparable accuracy. RC 16.10% ± 2.06% 2.25% ± 0.15% 70.12% ± 1.15%Fraction of videos with
overshoot > 5% bitrate within 5%
Agent
overshoot > 0
of target
of target
libvpx
64.00%
6.13%
71.34%
MuZero-RC
20.34% ± 2.16% 2.04% ± 0.20% 84.14% ± 0.68%
Augmented MuZero-0
20
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(a) Cumulative frame bits trajectories.
0
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Table 3 :
3BD-rate difference of Lagragnian-relaxation method compared to libvpx. Columns in this table are same as those in table 1.Mean PSNR
Mean SSIM
Mean VMAF
Agent
BD-rate difference BD-rate difference BD-rate difference
Lagrangian
256.37%
41.44%
195.12%
MuZero-RC
-4.72% ± 0.32%
-3.68% ± 0.33%
-0.53% ± 0.21%
Table 4 :
4Bitrate constraint satisfaction behavior of Lagrangian-method over the evaluation set.Columns in this table are same as those in table 2.
Fraction of videos with
overshoot > 5% bitrate within 5%
Agent
overshoot > 0
of target
of target
libvpx
64.00%
6.13%
71.34%
Lagrangian
25.83%
25.49%
0.68%
MuZero-RC 20.34% ± 2.16% 2.04% ± 0.20% 84.14% ± 0.68%
13: procedure UPDATE_BUFFER(B, video i , PSNR Episode , overshoot Episode , β i , α)1: Dataset D of videos with assigned target bitrates
2: Buffer B ← [[Video i , β i ] → (PSNR 0 , 0) ∀ videos]
3: α : Buffer update factor
4: repeat
Actor Process
5:
video i , target_bitrate β i ∼ D
Sample video and target bitrate from dataset
6:
PSNR Episode , Bitrate Episode ← encode_video(video i |β i , agent policy)
7:
Encode video using agent policy and compute PSNR and bitrate
8:
9:
UPDATE_BUFFER(B, video i , PSNR Episode , overshoot Episode , β i , α)
10:
Return ← SELF_COMPETITION_REWARD(B, video i , PSNR Episode , overshoot Episode , β i )
11: until end
12:
14:
18: procedure SELF_COMPETITION_REWARD(B, video i , PSNR Episode , overshoot Episode , β i )PSNR EMA , overshoot EMA ← B [video i , β i ] if overshoot Episode > 0 or overshoot EMA > 0then 22: if overshoot Episode ≤ overshoot EMA then if PSNR Episode ≥ PSNR EMA then end if 34: end procedure A.6 Histograms of BD-Rate differences compared to libvpxThis section presents the histograms of BD-rate difference of the behaviors of MuZero-RC and Augmented MuZero-RC on 3062 videos from evaluation set compared to libvpx. See figure 1b for a qualitative example of RD curves and BD-rate difference. The RD curves for all the policies are computed by encoding each video from the evaluation set with nine uniformly spaced target bitrates from the[256, 768] kbps. The figures 7, 8, and 9 show the histogram of BD-rate differences for PSNR, SSIM, and VMAF respectively for a single run of both agents compared to libvpx.Episode
16: end procedure
17:
19:
Equivalent to equation 4
20:
21:
23:
return 1
24:
else
25:
return −1
26:
end if
27:
else
28:
29:
return 1
30:
else
31:
return −1
32:
end if
33:
show the histograms of the overshoots for all the policies with target bitrate of 256, 320, 384, 448, 512, 576, 640, 704, and
768 kbps respectively.
40
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Mean = -32.9 Kbps
Median = 2.7 Kbps
Figure 11: Histogram of overshoots of the agents on the evaluation set for 256 kbps target bitrate.40
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Mean = -8.1 Kbps
Median = 2.6 Kbps
(a) libvpx
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Mean = -5.8 Kbps
Median = -1.0 Kbps
(b) MuZero-RC
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Mean = -4.6 Kbps
Median = -2.7 Kbps
(c) Augmented MuZero-RC
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Mean = -15.2 Kbps
Median = 2.5 Kbps
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Mean = -11.4 Kbps
Median = -1.2 Kbps
(b) MuZero-RC
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Mean = -14.0 Kbps
Median = -3.8 Kbps
AcknowledgmentsWe would like to thank Balu Adsumilli, Ross Wolf, Yaowu Xu, James Bankoski, Nishant Patil, Marisabel Hechtman, Dan A. Calian, Luis C. Cobo, Chris Gamble, and David Silver for the insightful discussions about this work and for their support throughout the project.
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| Long-term hourly time series representing the PV generation in European countries have been obtained and made available under open license. For every country, four different PV configurations, i.e. rooftop, optimum tilt, tracking, and delta, have been investigated. These are shown to have a strong influence in the hourly difference between electricity demand and PV generation. To obtain PV time series, irradiance from CFSR reanalysis dataset is converted into electricity generation and aggregated at country level. Prior to conversion, reanalysis irradiance is bias corrected using satellite-based SARAH dataset and a globally-applicable methodology. Moreover, a novel procedure is proposed to infer the orientation and inclination angles representative for PV panels based on the historical PV output throughout the days around summer and winter solstices. A key strength of the methodology is that it doesn't rely on historical PV output data. Consequently, it can be applied in places with no existing knowledge of PV performance. | 10.1002/pip.3126 | [
"https://arxiv.org/pdf/1807.10044v1.pdf"
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| 119,243,848 | 1807.10044 | 0273718ceef197a204a676e3ddbea050911b027b |
Using validated reanalysis data to investigate the impact of the PV system configurations at high penetration levels in European countries
Marta Victoria
Department of Engineering
Aarhus University
Inge Lehmanns Gade 108000Aarhus CDenmark
Gorm B Andresen
Department of Engineering
Aarhus University
Inge Lehmanns Gade 108000Aarhus CDenmark
Using validated reanalysis data to investigate the impact of the PV system configurations at high penetration levels in European countries
energy modelingreanalysistime seriesclearness indexduck curve
Long-term hourly time series representing the PV generation in European countries have been obtained and made available under open license. For every country, four different PV configurations, i.e. rooftop, optimum tilt, tracking, and delta, have been investigated. These are shown to have a strong influence in the hourly difference between electricity demand and PV generation. To obtain PV time series, irradiance from CFSR reanalysis dataset is converted into electricity generation and aggregated at country level. Prior to conversion, reanalysis irradiance is bias corrected using satellite-based SARAH dataset and a globally-applicable methodology. Moreover, a novel procedure is proposed to infer the orientation and inclination angles representative for PV panels based on the historical PV output throughout the days around summer and winter solstices. A key strength of the methodology is that it doesn't rely on historical PV output data. Consequently, it can be applied in places with no existing knowledge of PV performance.
Introduction
The cost decrease experienced by photovoltaic (PV) solar energy throughout the last decade has been so dramatic that the projected installed capacities have been persistently underestimated by almost every relevant actor, e.g., the International Energy Agency (IEA) [1] or Greenpeace [2]. In 2016 and 2017, PV was the technology with the highest installed capacity among the renewable energy sources, and its cumulative installed capacity world-wide reached 390 GW at the end of 2017 [3].
With very low capital and operational costs and using a resource widely available, PV is seen as one of the main key enabling technologies in the transition towards a low-carbon energy system in almost every country. In fact, energy modeling efforts aiming to attain low cost highly renewable penetration in Europe include a significant share of PV generation. This result is obtained both when modeling the power system [4,5] or when taking into consideration its coupling with other sectors [6,7]. PV also plays a major role when individual countries are modeled. Since the literature is vast in this case, the reader is referred to two papers in which a large number of countries are investigated with a consistent approach [8,9].
Since most of the energy models use PV generation time series as inputs and PV is expected to play a prominent role, we must ensure that its representation is as ac-curate as possible to reduce modeling uncertainties. In fact, the generation, bias correction and validation of PV time series at national scale lay within one of the main challenges for PV research in the near future identified by Kurtz et al. [10], namely "research enabling more effective integration of solar electricity into the grid at high penetration levels". Moreover, Schlachtberger et al. [11] showed that using different time series for renewable generation may significantly impact the energy model outcomes.
Hourly capacity factors are typically employed to deal with the inherent variability of renewables. For every hour, the capacity factor is calculated as the ratio between the delivered power and the cumulative installed capacity, i.e., the rated power of that technology. To obtain PV hourly capacity factors representative for a certain country two main strategies can be followed. First, historical data comprising generated electricity and installed capacity can be used to compute the hourly capacity factors. However, national Transmission System Operators (TSO) do not provide this information for every country. When they do, it would be, to some extent, based on modeling since monitoring a myriad of small PV installations is in practice impossible. An additional drawback of this approach is, of course, that it does not allow to obtain hourly capacity factors for those countries where there is no capacity installed at the moment. The second strategy consists in using irradiance and temperature time series, together with a model for the PV systems, to compute hourly capacity factors. The resulting time series could be later bias-corrected using historical data when available. In this case, using ac-Pfenninger and Staffell [17] used the SARAH satellite dataset as well as MERRA-2 (Modern-Era Retrospective Analysis for Research and Applications) reanalysis dataset to obtain hourly capacity factors for European countries and make their results available through the very convenient Renewables Ninja website [18]. To correct the bias of both modeled time series, Europe-wide multiplicative scaling factors are obtained by computing the average difference (modeled minus measured capacity factors) for all the individual sites whose data is available. The main limitation of this approach is the fact that the availability of historical data for individual sites is restricted to a few European countries and, consequently, the scaling factor applied to the whole Europe could be impact by the local climate of those countries. The authors also processed metadata for a large number of individual PV sites to estimate the configuration (orientation and tilt angle) of the installations. Moraes et al. [19] carried out a comparison on wind and PV time series from EMHIRES and Renewables Ninja in terms of correlation among hourly capacity factors, duration curves, annual full loads, weakly average, and seasonal ratios. Time series for 5 countries were compared for the period from year 2012 until 2014. Greater similarity between TSOs time series and EMHIRES was found for PV, however, the authors stress the significant differences among available time series and encourage future works validating and comparing them with actual renewable generation in additional years and countries. Lingfors et al. developed PV time series for Sweden and validated them against historical data [20]. They used a local reanalysis model (STRÅNG) to derive irradiance and discarded using the global reanalysis MERRA due to the lower accuracy attained. There are other previous works that use reanalysis datasets to estimate PV output time series but either they don't mention any validation procedure [21,22] or the validation is limited to one country, Czech Republic in [23] and Germany in [24].
When comparing to the historical PV output of individual sites, Pfenninger and Staffell [17] found that modeled series using satellite-based or reanalysis dataset as input have a systematic error of the same order but the former showed lower RMSE 1 . Satellite-based datasets are known to capture better the local atmospheric phenomena and, in particular, the clouds dynamics which directly influence the irradiance at ground level [13,25]. In addition, SARAH dataset provides values for direct and global irradiance at ground level avoiding the modeling of diffuse irradiance that is required for reanalysis data and, hence, reducing uncertainties. Nevertheless, using irradiance from global reanalysis datasets to generate PV time series has significant advantages. Firstly, since reanalysis datasets cover the entire globe, the methodology can be replicated to obtain hourly capacity factors in every country no matter whether ground measurements or satellite images are available or not. Secondly, reanalysis datasets usually expand several decades enabling the generation of PV capacity factors for long-time periods. Thirdly, reanalysis dataset can be used to validate and bias correct climate models enabling the assessment of climate change impacts on energy system as in [26]. Fourthly, reanalysis datasets can also be used to generate time series representing the wind or hydroelectricity production and, hence, a consistent set of renewable generation time series to feed-in a certain energy model can be obtained. Finally, reanalysis data is usually freely available making it suitable for scientific analysis and the replicability of results.
In summary, we know that the accuracy of reanalysis may not be sufficient for detailed studies on the performance of individual sites but we also know that there are significant benefits from using reanalysis to represent country-wise PV time series. Then, we can formulate the fundamental research questions of this paper as follows:
1. Can we use bias-corrected global reanalysis to obtain PV time series integrated over large-scale regions and attain a similar accuracy than when using satellitebased irradiance? 2. Can we develop a methodology to bias-correct reanalysis irradiance that is globally applicable?
The Renewable Energy Atlas (REatlas) from Aarhus University was introduced in [27] where it was used to obtain time series for onshore and offshore wind generation in Denmark that were validated against historical data. The REatlas uses as input the Climate Forecast System Reanalysis (CFSR) from the National Center for Environmental Prediction (NCEP) [28]. In this paper, we present a methodology to obtain PV hourly capacity factors at a national level based on irradiance from reanalysis data. The method comprises two major innovations: -In the first place, we introduce a procedure to biascorrect irradiance at a country level using 12-values, one per month, which can be derived from the best available source, either satellite datasets or ground measurements. The bias-corrected irradiance is used, together with a model of the PV system, to generate hourly capacity factors for this technology at a national scale. These are validated against historical data. This allows us to retain the previously stated advantages of using irradiance from reanalysis datasets while reducing the errors due to a poor representation of the local atmosphere.
-In the second place, we propose an indirect procedure to infer the configuration (orientation and inclination) of PV panels in a country based on two artificial clear-sky days. By selecting historical PV electricity generation in hours with clear-sky conditions within days close to the summer and winter solstices we produce these two artificial clear-sky days corresponding to the days in which the sun is highest and lowest above the horizon.
Finally, the bias-corrected irradiance is used to generate 38 years-long hourly PV time series for every country in Europe (EU-28 plus Serbia, Bosnia-Herzegovina, Norway, and Switzerland) and the modeled time series are validated using historical data for 2015 and 15 countries. The accuracy of the modeled time series is compared to that of EMHIRES [14] and Renewables Ninja [17] datasets. The 38 years-long time series are computed for every European country and 4 possible configurations for the PV systems: (a) rooftop installations, (b) optimum orientation and inclination, (c) 2-axis tracking and (d) delta configuration. The time series are open-licensed and can be retrieved from the Zenodo repository. As a proof of concept, these time series are used to investigate the evolution of the mismatch curve, i.e., the electricity demand minus the PV generation, in two representative days (winter and summer solstice) in every country.
The paper is organized as follows. Section 2 summarizes all the data used. Methods are described in Sections 3 and 4. First, Section 3 describes the determination of monthly correction factors for reanalysis data. Then, a general overview of REatlas is provided on Section 4 while annex A includes a detailed description of the model to convert irradiance into electricity generated by PV systems. The methodology to infer the orientation and tilt representative angles for every country based on two artificial clear-sky days is described in Section 5. Section 6 compares the modeled time series with historical values provided by national TSOs. The analysis of mismatch curves is carried out in Section 7. Finally, Section 8 gathers some conclusions. The Baseline Surface Radiation Network (BSRN) provides high temporal resolution measurements over long periods with high data quality. Global horizontal irradiance time series measured in the 9 ground stations located in Europe ( Figure 1) were downloaded and processed. The BSRN uses Secondary Standard ventilated pyranometers and the global measurement accuracy is estimated to be approximately 5 W/m 2 [29]. Besides the quality assurance protocols [30] implemented in the BSRN, we also applied the recommended "Extreme Rare Limits" and the comparison test to ensure the consistency of the global, direct and diffuse radiation measurements [31]. Since BSRN provides minute-resolved measurements, a 60 minutes-wide averaging window has been applied to calculate hourly values that can be compared to those included in the CFSR reanalysis dataset. Assuming a horizontal wind speed of 10 m/s, the averaging window represents the movement of an air parcel of 36 km over the ground station. This can be compared to the irradiance data in reanalysis dataset whose spatial resolution is 40x40km 2 . BSRN measurements are only used for the preliminary investigation on the capability of CFSR reanalysis dataset to capture local atmospheric effects carried out in Section 3.1 but this information is not used in the methodology to determine the monthly correction factors proposed in this paper.
Climate Forecast System Reanalysis
The Climate Forecast System Reanalysis (CFSR) is provided by the National Center for Environmental Prediction (NCEP) [28]. It comprises a 38 years-long global high-resolution dataset (hourly time resolution and spatial resolution of 0.3125 • x 0.3125 • which in Europe is roughly equivalent to 40 x 40 km 2 ). The irradiance data included in the CFSR dataset is used as input for the REatlas to generate PV capacity factors time series. Prior to the conversion, the monthly bias correction described in Section 3.2 is carried out. Figure 1 shows the irradiance from CFSR reanalysis dataset on the 21 st of June, 2015 at 12:00 UTC
Solar Surface Radiation -Heliostat (SARAH)
The Meteosat-based SARAH (Solar SurfAce RAdiation -Heliostat) satellite dataset provides hourly resolution and very high spatial resolution (0.05 • x0.05 • ) for 30 years (1986-2015) although a significant percentage of the values are missing for the initial years [17]. The SARAH irradiance dataset has been used to retrieve ground horizontal irradiance time series for every location corresponding to a point in the CFSR grid. Those time series are aggregated to generate irradiance time series at country level that are compared to those generated using CFSR reanalysis to determine monthly correction factors. The procedure is described in Section 3.2. The SARAH irradiance time series for different locations are downloaded using PV-GIS version 5 [16]. This is also ensures the replicability of the method. For additional countries to those included in this paper, the best available irradiance data can be used to correct CFSR reanalysis irradiance on a monthly basis. For instance, the National Surface Radiation Database (NSRDB) maintained by NREL [32] can be used to determine correction coefficients in North and South America.
PV cumulative installed capacity
The cumulative installed capacities for every European country in 2015 were obtained from the following sources: ENTSO-E, Eurostat, EurObservER [33], IRENA [3], BP [34]. The data from the first two sources were retrieved through the convenient compilation carried out by the Open Power System Data (OPSD) initiative [35]. Figure 2 shows the installed capacities according to various sources. The large discrepancy found is probably a combined effect of a fast changing scenario, where significant capacities relative to cumulative values are installed every year, and the difficulties associated to monitoring a myriad of new small installations. Since the discrepancy between different sources is significant, an averaged value was calculated and used to compute the historical hourly capacity factors in the next section. This is the same approach followed in [17]. Cumulative installed capacities are considered to be constant throughout 2015, since differences among the capacities reported by different sources are in some cases larger than the difference among two consecutive years reported by the same source.
PV generation time series and hourly capacity factors
Actual PV generation time series are reported by ENTSO-E or national TSOs for 2015 for the following countries: Austria, Belgium, Bulgaria, Czech Republic, Denmark, France, Germany, Greece, Italy, Lithuania, Netherlands, Portugal, Romania, Slovakia, Slovenia and Spain. The data is accessed through the OPSD file [36]. For the case of Spain, ENTSO-E data cannot be used since only the solar aggregated time series, including the generation from Concentrated Solar Power (CSP) and PV plants, is reported. Hence, hourly values for every day starting from May, 1 st , 2015 have been retrieved from [37] and assembled. The hourly capacity factors for every country in 2015 are computed by dividing the PV hourly electricity generation by the cumulative installed capacity, averaged values among those reported by different sources.
METHODS: Determination of country-wise monthly correction factors for reanalysis irradiance
3.1. Preliminary analysis: reanalysis capability to capture local atmospheric effects
The clearness index K t is defined as the ratio between the global irradiance at the ground G(0) and the extraterrestrial irradiance B 0 , that is, at the top of the atmosphere (equation A.2). K t is influenced by the thickness of the atmosphere, which in turns depends on the time, date, and location, as well as by its composition and cloud content. Furthermore, K t is usually employed to calculate the fraction of direct to global irradiance at ground level (equation A.12 and [38]). We follow the approach proposed in [39] to evaluate the capability of the CFSR reanalysis dataset to represent the local atmosphere filtering properties. In Figure 3, the probability density function (PDF) of K t for every hour throughout 2015 is depicted for the BSRN ground station located in Palaiseau, France. The figure shows the PDF obtained from the time series corresponding to irradiance at the CFSR grid point closest to the station together with the PDF obtained from ground measurements. When compared to experimental data, the CFSR PDF shows a higher probability for verylow and very-high clearness indices. The same result is consistently found for the 9 ground stations within the BSRN located in Europe. The associated figures are provided in the Supplementary Materials. The impact of the atmosphere in the CFSR dataset is more extreme than in reality. For clear-sky days, the modeled atmosphere is more transparent than in measurements overestimating global horizontal irradiance, while, on cloudy days, the modeled atmosphere is more absorbing/scattering than in reality underestimating global horizontal irradiance.
Monthly correction factors
The preliminary analysis demonstrated that the CFSR reanalysis dataset does not properly represent the light filtering effect of the local atmosphere. Since PV generation is directly proportional to the irradiance, any bias in irradiance must be corrected to avoid a significant error when computing PV time series. A classic approach followed to estimate the energy produced by a PV power plant consists in using monthly values for K t to characterize the solar climate at a particular location. By using one K t value per month, the dispersion is reduced and a better match with the real performance of the power plant in achieved [38]. Based on that experience, we propose the following procedure to bias correct irradiance values from CFSR. series G CF SR n (0, t) for every CFSR grid point n within a country are retrieved and aggregated to obtain a time series G CF SR s (0, t) representative for the country s.
G CF SR s (0, t) = G CF SR n (0, t)(1)
where n ∈ s. 2. Global horizontal irradiance time series for the same locations G SARAH n (0, t) are obtained from SARAH dataset and aggregated to obtain G SARAH s (0, t).
G SARAH s (0, t) = G SARAH n (0, t)(2)
where n ∈ s. Satellite-based SARAH dataset was selected since it is probably the most accurate dataset available [13]. Alternatively, for countries not included in SARAH dataset, another satellite dataset or measurements from a ground stations network can be used. 3. For every month m, the correction factor C m,s is estimated as that of the median day, that is
C m,s = median day G SARAH s (0, t)dt day G CF SR s (0, t)dt(3)
where day ∈ m. 4. For every country, C m,s are determined using irradiance data for years in the period 2005-2014 and the average is calculated. Then, C m,s are used to correct CFSR time series modeled for 2015.
G CF SR,corrected s (0, t) = C m,s G CF SR,uncorrected s (0, t) (4) where t ∈ m.
The correction factors C m,s represent the solar climate characteristic of a country and correct the irradiance from CFSR in the cases where the reanalysis is not capable of adequately capturing it. In general, irradiance values in the CFSR dataset are higher than those obtained from the SARAH dataset and, consequently, monthly correction factors are lower than 1. This was expected as other reanalysis datasets are known to underestimate the presence of clouds [25,13]. This is also in agreement with the Europe-wide scaling factor equal to 0.935 applied to correct the Renewables Ninja time series obtained using irradiance from the MERRA-2 reanalysis dataset [17]. C m,s show noticeable differences among countries. For instance, C m,s determined for Spain depicted in Figure 4 are very close to one for every month. Conversely, monthly correction factors for Denmark varies throughout the year and are higher for winter months. The large variability in C m,s winter values for Denmark is caused by the unstable and cloudy local climate during those months. C m,s values are summarized in Appendix B and figures for every European country are provided in the Supplementary Materials.
METHODS: Computation of PV hourly capacity factors at country level using REatlas
The REatlas from Aarhus University is used for converting irradiance and temperature time series into countrywise PV hourly capacity factors. The REatlas was introduced in [27] and a detailed description of the methodology and equations involved in the solar conversion can be found in Appendix A. The procedure can be summarized as follows. For every point in the CFSR grid, the irradiance time series is bias corrected as described in Section 3.2. The global irradiance at ground level is first decomposed into direct and diffuse irradiance (eq. A.12). Then, the direct, diffuse, and global irradiances on a tilted panel are calculated and aggregated (eq. A.13). The global irradiance at the entrance of the solar panel is converted into electricity using a simplified model for the PV system. The PV output is assumed to be proportional to the irradiance at its entrance. The impact of temperature on efficiency is assessed using the classic approach based on the Nominal Operating Cell Temperature (NOCT) (eq. A.23 and [38] ). The efficiency temperature coefficient of crystalline silicon flat panel is assumed since this is the most spread technology. Finally, the time series for every point in the CFSR grid data within a country are aggregate to obtain hourly capacity factors representative for every country. A uniform capacity distribution across every country is assumed. In practice, this implies assuming one PV panel is installed in every point in the CFSR grid. Although detailed databases including the location of wind turbines [40] and conventional power plants exists, this is not the case of PV plants. The uncertainty associated to the real position of thousands of small PV installations lead us to select the uniform distribution hypothesis.
METHODS: Determination of the configuration of PV panels based on artificial clear-sky days.
We assume that the configuration of PV panels in a country can be represented using two normal distributions, that is, the tilt angles are assumed to follow a Gaussian distribution centered in µ β with a standard deviation σ β while the orientation angles follow a Gaussian distribution centered in µ α = 0 • , on average PV panels are south oriented, and with a standard deviation σ α . In principle, if winter and summer solstices happen to be clear-sky days, the parameters σ β , µ α , and σ α could be inferred by comparing the modeled hourly capacity factors at national level throughout those days to those reported by the TSOs. Under the clear-sky assumption, differences between the PV generation in winter and summer solstices are directly related to the inclination and orientation of PV panels.
The main problem is, of course, that clear-sky conditions may not occur in winter and summer solstices. In fact, since we are aiming to uniform clear-sky conditions across the whole country, it is possible that there is not a single clear-sky day throughout the year. However, we can take advantage of the fact that the Sun path across the sky is almost constant for the days close to the solstice. We have generated two artificial clear-sky days by selecting, for every hour, the maximum capacity factor found for the days around the summer and winter solstice. For instance, Figure 5 depicts the artificial clear-sky summer solstice reconstructed for Denmark. We have selected a ±10 days window around the 21 s t of June and the 21 s t of December. Within those days and for the latitudes in Europe, the maximum daily irradiance varies within 5% of the value found in the solstice. A figure showing this variation for different countries is provided in the Supplementary Materials.
The theorical hourly capacity factor for winter and summer solstices is calculated as
CF s (t) = n,β,α ζCF n (t, β, α)P DF (µ β , σ β )P DF (µ α , σ α ) (5)
where the parameter ζ represents the decrease in the maximum capacity factor due to one of the following causes: (a) the time evolution of irradiance in different location is asynchronous due to different latitudes and longitudes in a country, (b) presence of clouds in some part of the country, (c) systems out of production due to repairing or maintenance, (d) systems under curtailment following TSO orders. The parameters ζ and µ β are inferred by minimizing the normalized Root Mean Square Error (RM SE n ) calculated by comparing the modeled capacity and the artificial clear-sky winter and summer solstices recreated from historical PV generation reported by TSO (and assuming µ α =0 • , σ β =20 • , and σ α =30 • ). Appendix C gathers the optimum parameters estimated for different countries. We did not find a direct correlation between µ β and the latitude representative for every country (Figure is provided in the Supplementary Materials). This indicates that the panel tilt angles are probably more influenced by the rooftop inclination than by the local latitude. This is in agreement with Pfenninger and Staffel [17] findings by processing metadata of individual sites. Consequently, for the system configuration named as rooftop in section 6, we assumed the same values for every country (ζ = 1, µ α =0 • , µ β =25 • , σ α =40 • , and σ β =15 • ).
RESULTS: Modeled vs historical time series
The country-wise capacity factors modeled using REatlas and bias-corrected reanalysis irradiance for 2015 are compared to historical values. Figure 6 depicts the QQ plots for Germany when the capacity factors are integrated using different time scales (year, month, day, and hour) as
RM SE p = s,p ( p CF mod s (t)dt − p CF hist s (t)dt) 2 n s · n t (6) M E p = s,p ( p CF mod s (t)dt − p CF hist s (t)dt) n s · n t(7)
where n s is the number of countries and n p the number of periods. Figure 9 also includes the errors calculated using time series for 2015 provided by Renewables Ninja (using either SARAH or MERRA-2 dataset as input) [17] and the EMHIRES time series (using SARAH dataset as input) [14]. Figures 7 and 8 depict, for Germany, the time evolution of PV throughout two representative weeks, modeled by different sources together with the historical data provided by the TSO. From Figures 6 and 9 it can be clearly observed that, for all datasets, the RM SE increases as we try to model capacity factors integrated over shorter time periods. This is a classic result when modeling the generation of PV plants: the uncertainty in the prediction of the energy generated by the plant in a certain month is far lower than the uncertainty when attempting to predict the generation in a particular day of that month [38]. It is also one of the main reasons that led us to propose monthly correction factors for the irradiance.
Although most of the energy models use hourly values, it is important to realize that uncertainty in the energy model outcomes may be influenced by a different timescale. For example, let's imagine that we try to optimize the capacity mix necessary to supply the electricity demand in a country using mainly renewables and that it is cost bene-ficial to provide a significant share of the electricity using PV. In that case, several works [5,6,41] have found that electric batteries must be installed to counterbalance the daily cycles of PV generation and that the necessary en-ergy and power capacity of those batteries are heavily impacted by the PV generation throughout winter weeks with low renewable generation. Then, in order to reduce the uncertainty of the model results, it becomes more important to properly simulate the daily PV generation throughout those weeks than to match exactly the hourly generation. Figure 9 shows how the irradiance-corrected REatlas time series adequately predict the country-wise performance of PV at different time scales. The RMSE calculated for the irradiance-corrected REatlas time series show, for all the different integration periods, similar RMSE to Renewables Ninja time series (either using MERRA-2 or SARAH dataset as input) and EMHIRES. It should be reminded here that the monthly correction factors for the irradiance time series have been calculated using irradiance from the previous years (2005-2014). However, the EMHIRES time series have been corrected using the PV generation reported from TSO in 2015, that is, the same historical data against which they are evaluated, so a very good match was expected. For the Renewables Ninja time series, Figure 9 confirms the lower RMSE when using SARAH dataset as input found for 2014 in [17]. Figure 9, one meaningful question arises: Why the SARAH-based Renewables Ninja time series shows a positive ME while the corrected REatlas time series, whose bias correction is based on SARAH database, doesn't? We discuss bellow several possible explanations. First, SARAH-based Renewables Ninja time series were bias corrected using a Europe-wide scaling factor of 1.094. On the one hand, this factor was determined to minimize the mean bias when comparing the modeled time series with historical data in more than 1000 individual sites across Europe. On the other hand, the M E for SARAH-based Renewables Ninja time series is computed by comparing the modeled time series with the historical data reported by TSOs. We agree with Pfenninger and Staffel on their affirmation that "the output reported by the TNOs is not necessarily more accurate than simulations" so the M E shown in Figure 9 may not be the best metrics to assess the time series. Nevertheless, this is the most accessible historical data at these moments. There are two alternative explanations to this discrepancy. First, REatlas uses a composed circumsolar and isotropic model to represent diffuse irradiance on solar panels (equation A.17) while Renewables Ninja uses a isotropic-sky model. The latter is know to underestimate the diffuse irradiance on PV panels tilted to the equator, that could have led to the positive bias-correction used in Renewables Ninja time series. Second, SARAH is probably the most accurate available irradiance dataset and it has been validated against a dense network of ground stations showing a extremely low mean error for the whole Europe [13]. However, it is know to slightly underestimate irradiance at high latitudes, overestimate it in the south and attain unbiased estimation in central Europe. Since the individual PV sites used in Renewables Ninja are not uniformly distributed across Europe, this could have influenced the calculation of the scaling factor.
The distribution of ramps, i.e., the temporal variability of the hourly capacity factor, is a usual metric to evaluate modeled time series for wind energy [27,42]. Extreme ramps in the PV generation time series might be critical for grid stability and require a fast response from backup technologies. Modeling the dynamics of ramps is easier for PV time series since they are mainly determined by the Sun daily cycle. Figure 10 depicts, for Germany, the QQ plots for the duration curve of the modeled and historical ramps. The hourly ramp ∆CF is defined as the difference between the capacity factor CF (h) and its value in a previous hour CF (h − 1):
∆CF = CF (h) − CF (h − 1)(8)
Equivalent figures for other countries are provided in the Supplementary Materials. In general, the ramp rates modeled using reanalysis irradiance are statistically consistent with the observed ones. Maximum ramps of ∆CF ≈ 0.25 are found for Germany, this implies that the power supplied by PV increases or decreases by 25% of the installed capacity in an hour. Nevertheless, it should be mentioned here that ramps are caused by two different phenomena. On the one hand, the Sun trajectory in the sky which can be accurately predicted. On the other hand, the country-integrated effect of clouds.
RESULTS: Impact of the PV system configuration on the mismatch curves
Future PV capacity factors at country level are difficult to predict. For wind energy, higher capacity factors are expected for future years as wind turbines increase their size and efficiency [27,42]. In particular, the deployment of offshore wind turbines will rise wind hourly capacity factors when aggregated at a country level. Conversely, PV capacity factors mainly depend on the system configuration (static vs. tracking, orientation and tilt angles). In fact, if the cost decrement tendency keeps as it is today, country-wise annual capacity factors will probably decrease. With very cheap PV panels, static configuration will be preferred, as the extra energy provided by the tracker will not pay off, PV will be integrated into building with non-optimum tilt and orientation angles and even delta configuration (a row of tilted panels where those on one side face east while those on the other face east) may result cost effective.
Using bias-corrected reanalysis irradiance and the methodology described in Section 4, we have generated 38 yearslong time series representing the PV hourly capacity factors in every country in Europe (EU-28 plus Serbia, Bosnia-Herzegovina, Norway, and Switzerland). A uniform capacity layout is used, that is, one PV panel is assumed to be installed in every point in the CFSR grid data, every 40x40 km 2 , and the configurations of the panels is one of the followings: The dataset comprising the time series for the four configurations is under open license and can be downloaded from the Zenodo repository. Then, to obtain the PV hourly capacity factors for a country, different assumptions on the share of the alternative configurations can be made and the weighted time series can be aggregated accordingly. For instance, Figure 11 shows the duration curves for Spain in 2016 assuming different configurations. They are plotted against duration curve obtained using historical data [37]. A very good agreement is found when the duration curves for optimum tilt and tracking are added assuming 70%/30% proportion which roughly represent the current configuration of PV plants in Spain. The produced time series have also been used to assess the impact of PV on the hourly operation of electricity system when large capacities are deployed. The mismatch curve, i.e., electricity demand minus PV generation, is usually employed to investigate this effect. The mistmatch curve is also known in the literature as the 'duck curve' [43]. Figure 12 depicts the evolution of the mismatch curve in Spain throughout the winter and summer solstices when a PV capacity equals to the yearly-averaged hourly load (av.h.l.) is installed. In the case of Spain, this corresponds to 28.4 GW. The red line represents the average value for the ±10 days around the solstice in 2015, while the red area includes the minimum and maximum values obtained within that time period. Equivalent curves for other countries in Europe are provided in the Supplementary Materials. The mismatch curves are shown for the four PV configurations previously described. The operation of power systems with high PV penetration requires that throughout the evening, backup technologies increase their production to counterbalance the drop in PV generation. Several strategies have been proposed to re-shape the curve and allow more PV on the grid such us changing operational practices in power system to enable more frequent plant cycling or demand shifting [43]. As Figure 12 shown, the configuration of PV directly impact the ramp rate and range of backup generation necessary throughout the evening and, consequently, selecting a proper mix of different configurations could ease the operation of the power system when large PV capacities are achieved.
Conclusions
The research questions that we raised in the introduction can now be answered affirmatively. Irradiance from CFSR reanalysis can be used to obtain PV time series integrated over large-scale regions with a reasonable accuracy. To that end, the bias in reanalysis irradiance must be corrected. We propose a methodology that doesn't require using historical PV output data and, consequently, can be universally applied. When compared to historical PV hourly capacity factors at national scale an error lower or similar to Renewables Ninja and EMHIRES time series, both relaying on satellite-based SARAH irradiance as input, was achieved.
Using global reanalysis irradiance to model PV generation at country level enables the production of consistent long-term time series dataset, including other renewables such as wind and hydroelectricity that can be used as inputs for energy models. The validated methodology was used to produce open-license long-term country-wise PV time series for European countries under four different assumptions for the PV systems configurations (rooftop, optimum tilt, tracking, and delta).
Acknowledgments
The authors are fully or partially funded by the RE-INVEST project, which is supported by the Innovation it can be described by means of two normal distributions, i.e., the panels tilt angles β are assumed to follow a Gaussian distribution with mean µ β and standard deviation σ β while the panels orientation angles α are assumed to follow a Gaussian distribution with mean µ α and standard deviation σ α . In the northern hemisphere, µ α can be assumed to be zero, that is, in average panels are south oriented.
Finally, the capacity factors time series for all the panel configurations and all the grid points CF n (t, β, α) are aggregated together to obtain a time series representative for the PV electricity generation in a certain country.
CF y,s (t) = n,β,α {C n } y,s CF n (t, β, α)P DF (µ β , σ β )P DF (µ α , σ α ) (A.1) where C n ∈ {C n } y,s
In this paper, a uniform capacity layout has been assumed for every country, i.e., one solar panel is installed in every CFSR grid point.
Appendix A.1. Irradiance on a tilted surface
This Section includes a description of the approach followed to: (a) determine the clearness index of the atmosphere for every hour; (b) decompose the global horizontal irradiance at ground level into direct and diffuse irradiance; and (c) estimate the direct, diffuse, and albedo irradiances on a tilted surface and their aggregate value.
Clearness Index
The extraterrestrial irradiance B 0 (t), i.e., the irradiance at the top of the atmosphere, travels through the atmosphere where it is absorbed, reflected and scattered. The clearness index K t measures the effect of the atmosphere and is influenced by time, date, and location, as well as by the atmosphere composition and cloud content.
K t = G(0, t) B 0 (t) (A.2)
The CFSR provides hourly values for G(0,t) at ground level. B 0 (t) can be analytically calculated using the following equation.
B 0 (t) = B 0 (t) sin γ s (A.3)
where B 0 is the solar constant B 0 =1367 W/m 2 . The eccentricity is calculated using equation A.11, and γ s is the solar altitude obtained from equation A.4.
For a certain location on the surface of the Earth with latitude φ and longitude ξ, the position of the Sun in the sky, at any moment, can be described by the solar altitude γ s (angle between the radio-vector of the Sun and the horizontal plane) and the solar azimuth ψ s (angle between the projection of the Sun radio-vector over the horizontal plane and the south direction). γ s and ψ s depend on the time and date trough equations A. 6 where LT is the hour of the day (expressed using Universal Coordinated Time, UCT), ξ is the longitude and ET is a correction for the different lengths of the days in a year due to the fact the Earth orbit is not circular but elliptical. The true solar time ω can also be expressed as an angle. In this case ω = 0 corresponds to the moment when the Sun is at the highest position, that is, ST = 12.
Decomposition of global irradiance into direct and diffuse irradiance
The global horizontal irradiance on a horizontal surface G(0), either measured or, as in this case, modeled, includes the direct horizontal irradiance B(0), that is, the irradiance that reaches the horizontal surface from the Sun without being scattered and the diffuse horizontal irradiance D(0), that is, the irradiance that reaches the horizontal surface after being scattered in the atmosphere. The diffuse fraction of the global irradiance F = D(0)/G(0) is estimated based on the clearness index K t as initially proposed by Liu and Jordan [44]. The piecewise function proposed by Reindlt [45], which is similar to the classic Page model [46], is used.
F = min(1, 1.02 − 0.254K t + 0.0123 sin γ s ), if 0 ≤ K t ≤ 0.3 min(0.97, max(0.1, 1.4 − 1.749K t + 0.177 sin γ s ), if 0.3 < K t ≤ 0.78 max(0.1, 0.486K t − 0.182 sin γ s ), if K t > 0.78 (A.12)
Irradiance on a tilted surface The irradiance G(β, α) on a surface with tilt angle β and orientation α is calculated by summing the direct B(β, α), diffuse D(β, α) and albedo irradiance R(β, α). The diffuse irradiance B(β, α) is calculated using the anisotropic model proposed by Hay and Davies [47], where D(β, α) is composed of a circumsolar component coming directly from the direction of the Sun D circumsolar (β, α), and an isotropic component coming from the entire celestial hemisphere D isotropic (β, α). The anisotropy index k 1 weights both components. k 1 is estimated by the ratio of the direct irradiance on the ground B(0) and at the top of the atmosphere B 0 (0).
k 1 = B(0) B 0 (0) (A.16) D(β, α) = D circumsolar (β, α) + D isotropic (β, α) (A.17)
where D circumsolar (β, α) = k 1 D(0) max(0, cos θ s ) sin γ s (A. 18) and the classic definition of isotropic diffuse irradiance D isotropic (β, α) has been modified to include the horizon brightening effect k horizon as proposed in [45].
D isotropic (β, α) = k horizon (1 − k 1 )D(0) 1 + cos β 2 (A.19) k horizon = 1 + k 1 sin 3 ( γ s 2 ) (A.20)
Finally, the albedo irradiance R(β, α) is calculated as
R(β, α) = ρG(0) 1 − cos β 2 (A.21)
where ρ is the reflectivity of the ground. ρ is determine for every grid point using the information provided by the CFSR database which includes downward shortwave radiation at surface, i.e. G(0), and upward shortwave at surface R(0).
ρ = R(0) G(0) (A.22)
Appendix A.2. Model of the PV panel The REatlas includes a simple model for the PV panel whose parameters can be obtained from commercial data sheet of PV panels. It is worth noticing that the conversion efficiency η ST C assumed for the PV panel under Standard Test Conditions (STC) does not have any influence on the hourly capacity factor time series. In fact, η ST C is only necessary if one wants the compute the area that must be covered with PV panels in order to install a certain capacity. Conversely to wind capacity factors, where the turbine power curve directly influences the time series, PV capacity factors are mainly influenced by the configuration of the panels (tilt and orientation angles if fixed or tracking strategy) but not by η ST C .
The temperature dependence of the efficiency has been included in the model using the efficiency thermal coefficient γ T .The efficiency is then computed as where T cell,ST C is the temperature of the cell under Standard Test Conditions. For silicon flat panels, γ T = -0.4%/ • C is assumed.
T cell is calculated using the Nominal Operating Cell Temperature (NOCT). NOCT conditions include G N OCT = 800 W/m 2 and T amb,N OCT =20 • C. From the ambient temperature at every grid point T amb , T cell can be computed applying the energy balance equation to the PV module. Finally, the model includes a system efficiency η system =0.9 which represents the aggregate effect of wiring losses, inverter efficiency, and voltage conversion efficiency. 0.9 is the average value found only for the inverter efficiency for all the individual sites processed in [17]. However, the inverter efficiency of new installation is expected to be better [48]. Hence, 0.9 is selected as a coefficient representing the aggregate efficiency of the system (wiring, inverter and voltage conversion). The possible curtailment due to PV generated power exceeding the rated power of the inverted is not included in the model. The hourly capacity factors can be then computed as the ratio of the energy produced at any hour and the generation under Standard Test Conditions (STC).
Figure 1 :
1Global Horizontal Irradiance from CFSR reanalysis dataset on the 21st of June, 2015 at 12:00 UTC. The location of every BSRN station corresponds to the lower-left corner of the 3-letter code labels shown in the map.
Radiation Network (BSRN)
Figure 2 :
2AT BE BG HR CY CZ DK FR GR HU LT LU MT NL PL PT RO SK SI ES SE CH Cumulative installed capacities for European countries in 2015 reported by different sources.
1 .Figure 3 :
13For every country, global horizontal irradiance time Probability Density Function (PDF) of the clearness index Kt calculated from measurements at the BSRN station located in Palaiseau, France and from the CFSR reanalysis dataset. The bin sizes are 0.02.
Figure 4 :
4Monthly correction factors Cm,s for Spain and Denmark. Figures for other European countries are included in the Supplementary Materials.
Figure 5 :
5Capacity factor throughout the artificial clear-sky summer solstice in Denmark obtained by selecting, for every hour, the maximum capacity factors for days in June close to the 21 st .
well as the QQ plot for the duration curve (sorted hourly capacity factors). Figures for other countries are provided in the Supplementary Materials. To enable a global assessment of the modeled time series,Figure 9depicts the RMSE and Mean Error (ME) calculated including every country s where historical data is available (see Section 2.3) and using different time periods p. The RM SE p and M E p are defined as follows:
Figure 6 :
6QQ plots comparing modeled and historical capacity factors for Germany integrated throughout four different time periods (from left to right: year, month, day, hour). The rightest column shows the QQ plot for the duration curve (sorted hourly capacity factors).
Figure 7 :
7Modeled time series vs historical data throughout a week in February for Germany.
Figure 8 :
8Modeled time series vs historical data throughout a week in June for Germany.
Figure 9
9also shows the Mean Error (M E) calculated for all the hours in 2015 and the 15 countries with available historical data. The uncorrected REatlas time series shows a positive bias of 0.011 that is reduced to 0.001 for the irradiance-corrected REatlas time series. For 2016, the bias is 0.013 and 0.003 for corrected/uncorrected REatlas time series. For 2017, the attained values are 0.016 and 0.006 respectively. Equivalent figures using data for 2016 and 2017 are provided in the Supplementary Materials. When observing
Figure 9 :
9(left) Root Mean Square Error (RM SE, eq. 6) and Mean Error (M E, eq. 7) calculated comparing simulated and historical time series for 2015 and 15 countries.
Figure 10 :
10QQ plot for the duration curve of the ramps ∆CF in Germany.
1 .
1Rooftop installation where tilt angles follow a Gaussian distribution with µ β = 25 • and σ β = 15 • and orientation angles follows a Gaussian distribution centered around 0 • (south orientation) with σ α = 40 • . These are the same distributions assumed in [17]. 2. South-orientation (α=0 • ) and optimum tilt angle for every panel. 3. 2-axis tracking.
4 .
4Delta configuration where tilt angle is 30 • , one panel is oriented to the east (α=90 • ) and the other is is oriented to the west (α=-90 • ).
Figure 11 :
11Duration curve (sorted hourly capacity factor) modeled for different PV system configurations vs. historical data provided by REE [37]. Data for Spain in 2016.
Figure 12 :
12Electricity demand minus PV generation for winter and summer solstice in Spain, assuming different configurations for the PV panels in the country. The installed PV capacity is assumed to be equal to the yearly-average hourly load(28.4 GW) Fund Denmark under grant number 6154-00022B. The responsibility for the contents lies solely with the authors.
] H.-J. Fell, C. Breyer, M. Metayer, The Projections for the Future and Quality in the Past of the World Energy Outlook for Solar PV and Other Renewable Energy Technologies, 31 st European Photovoltaic Solar Energy Conference and Exhibition
Figure A. 13 :
13Scheme of the methodology followed to calculate the electricity output of a PV panel based on the temperature and irradiance data from reanalysis.
and A.10 . sin γ s = sin δ sin φ + cos δ cos φ cos ω (A.4) cos ψ s = (sin γ s sin φ − sin δ) cos γ s cos φ [sign(φ)] (A.5) The declination δ is computed using the number of the day d n (counted from the first day of the year) δ = 23.45 • sin( 360(d n + 284) 365 ) (A.6) The true solar time ST , expressed in hours, can be calculated with the formula ST = LT + ET 60 − ξ 15 (A.7)
ω
= 15 • (ST − 12) (A.10) Finally, the eccentricity is calculated as = 1 + 0.033 cos( 360d n 365 ) (A.11)
G
(β, α) = B(β, α) + D(β, α) + R(β, α) (A.13) If B(0) is known, the direct irradiance B(β, α) can be calculated by straightforward geometrical considerationsB(β, α) = B(0) max(0, cos θ s ) sin γ s (A.14)where γ s is the solar altitude (see equation A.4) and θ s is the angle that forms the radio-vector of the Sun and the normal of the surface.cos θ s = sin δ sin φ cos β − [sign(φ)] sin δ cos φ sin β cos α + cos δ cos φ cos β cos ω + [sign(φ)] cos δ sin φ sin β cos α cos ω + cos δ sin α sin ω sin β (A.15)
η
= η ST C [1 − γ T (T cell − T cell,ST C )] (A.23)
CF n (t, β, α) = η system G n (t, β, α)η(T amb )G ST C η(T ST C ) = 0.9 G n (t, β, α)η(T amb ) 1000W/m 2 η(298K) (A.25)Appendix B. Monthly correction factors
Table B .
B1: Monthly correction factors Cm,s for European countries Appendix C. Gaussian distributions for tilt and orientation anglesTable C.2: Set of parameters to describe the inclination and orientation angles of PV panels in every country.Jan
Feb Mar
Apr May
Jun
Jul
Ago
Sep
Oct
Nov
Dic
AUT
0.78 0.82
0.84 0.87
0.86 0.85 0.88 0.91 0.89 0.89 0.81 0.79
BEL
1.04 0.90
0.94 0.94
0.91 0.93 0.94 0.94 0.98 1.02 1.07 1.09
BGR
0.85 0.78
0.87 0.89
0.90 0.94 0.96 0.99 0.96 0.95 0.89 0.88
BIH
0.88 0.83
0.85 0.86
0.84 0.89 0.91 0.94 0.91 0.92 0.90 0.87
CHE
0.62 0.67
0.78 0.86
0.84 0.87 0.88 0.90 0.89 0.81 0.73 0.63
CYP
0.95 0.92
0.94 0.94
0.96 0.98 0.98 0.98 0.98 0.96 0.97 0.97
CZE
0.84 0.80
0.84 0.89
0.86 0.88 0.88 0.89 0.93 0.92 0.87 0.90
DEU
0.97 0.87
0.91 0.90
0.88 0.90 0.90 0.91 0.95 0.97 1.00 1.00
DNK 0.90 0.84
0.91 0.93
0.91 0.91 0.93 0.92 1.00 1.01 1.08 0.95
ESP
1.00 0.99
0.97 0.97
0.96 0.97 0.98 0.98 0.98 0.98 1.02 1.00
EST
0.79 0.84
1.01 0.95
0.92 0.91 0.90 0.89 0.97 1.07 0.97 0.85
FIN
0.80 0.84
1.02 1.12
1.00 0.94 0.89 0.91 0.98 1.01 0.91 0.81
FRA
0.96 0.93
0.94 0.94
0.91 0.93 0.95 0.96 0.97 0.96 1.01 1.01
GBR
1.02 0.96
0.96 0.96
0.95 0.99 0.98 1.07 1.08 1.06 1.11 1.03
GRC
0.93 0.87
0.90 0.91
0.93 0.98 1.01 1.00 0.97 0.94 0.93 0.92
HRV
0.92 0.86
0.91 0.90
0.90 0.93 0.96 0.96 0.94 0.94 0.91 0.92
HUN
0.88 0.79
0.92 0.92
0.91 0.95 0.97 0.96 0.96 0.98 0.94 0.89
IRL
1.06 1.02
1.03 0.98
1.03 1.03 1.10 1.30 1.22 1.12 1.10 1.10
ITA
0.91 0.87
0.90 0.90
0.91 0.94 0.97 0.98 0.95 0.93 0.91 0.91
LTU
0.94 0.90
0.92 0.95
0.92 0.94 0.93 0.94 0.99 1.09 1.08 1.02
LUX
1.02 0.93
0.97 0.96
0.93 0.93 0.93 0.95 0.97 0.99 1.05 1.15
LVA
0.89 0.92
0.95 0.98
0.92 0.93 0.93 0.94 0.99 1.10 1.04 0.96
MLT
0.95 0.93
0.94 0.96
0.98 0.99 1.00 1.00 0.99 0.96 0.94 0.95
NLD
1.09 0.95
0.92 0.93
0.90 0.95 0.94 0.95 1.02 1.05 1.14 1.11
NOR
0.57 0.73
0.87 0.98
0.97 0.97 0.94 0.93 0.97 0.92 0.77 0.56
POL
0.96 0.87
0.89 0.93
0.91 0.93 0.93 0.92 0.98 1.00 1.03 1.04
PRT
1.05 1.01
0.97 0.96
0.96 0.96 0.97 0.97 0.98 0.98 1.01 1.03
ROU
0.80 0.76
0.87 0.89
0.91 0.91 0.95 0.95 0.96 0.95 0.89 0.80
SRB
0.83 0.74
0.86 0.87
0.88 0.91 0.94 0.95 0.92 0.93 0.90 0.84
SVK
0.90 0.86
0.90 0.90
0.86 0.88 0.90 0.90 0.95 0.94 0.94 0.92
SVN
0.92 0.89
0.93 0.89
0.90 0.92 0.94 0.94 0.93 0.95 0.96 0.89
SWE
0.80 0.81
0.92 1.01
0.96 0.93 0.91 0.90 0.95 0.96 0.91 0.82
country
µ β
σ β
µα
σα
ζ
AUT
40
20
0
30
1
BEL
10
20
0
30
0.8
BGR
40
20
0
30 0.75
CZE
25
20
0
30
0.7
DEU
20
20
0
30 0.65
DNK
20
20
0
30 0.95
ESP
10
20
0
30
1
FRA
20
20
0
30
0.8
GRC
40
20
0
30 0.85
ITA
35
20
0
30 0.65
LTU
10
20
0
30
1
NLD
40
20
0
30 0.95
PRT
25
20
0
30
1
ROU
35
20
0
30
0.7
SVK
10
20
0
30
1
SVN
10
20
0
30
1
Root Mean Square Error (RMSE) calculate on the hourly capacity factors.
In the CFSR terminology, G(0, t) is named downward shortwave radiation at surface.
Appendix A. Solar conversion in REatlasThis annex provides details on the implemented methodology to obtain hourly capacity factors for PV generation at a national scale. The global renewable energy atlas (REatlas) from Aarhus University[27]uses as input the Climate Forecast System Reanalysis (CFSR) dataset from the National Center for Environmental Prediction (NCEP)[28]. CFSR dataset comprises a 38 years-long global high-resolution dataset (hourly time resolution and approximately 40 x 40 km 2 space resolution). The procedure to obtain PV hourly capacity factors representative for a country comprises the following steps.1. At every point in the CFSR grid, the time series for temperature T amb (t) and global horizontal irradiance G(0, t) at ground level 2 are used to compute the irradiance at the entrance of the PV panel and its electricity output. The former is calculated as described in Subsection A.1 (either fixed tilted surface or tracking can be assumed) and the latter is computed using the model for the PV panel described in Section A.2. 2. For every country s and year y, a capacity layout {C n } y,s is built representing the cumulative PV capacity installed at every point n of the CFSR grid. In addition, the configuration of the PV panels in the country under study is also needed. For instance,
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| Department of Electrical Engineering (Affiliated member), Columbia University, USANeural network models can now recognise images, understand text, translate languages, and play many human games at human or superhuman levels. These systems are highly abstracted, but are inspired by biological brains and use only biologically plausible computations. In the coming years, neural networks are likely to become less reliant on learning from massive labelled datasets, and more robust and generalisable in their task performance. From their successes and failures, we can learn about the computational requirements of the different tasks at which brains excel. Deep learning also provides the tools for testing cognitive theories. In order to test a theory, we need to realise the proposed information-processing system at scale, so as to be able to assess its feasibility and emergent behaviours. Deep learning allows us to scale up from principles and circuit models to end-to-end trainable models capable of performing complex tasks. There are many levels at which cognitive neuroscientists can use deep learning in their work, from inspiring theories to serving as full computational models. Ongoing advances in deep learning bring us closer to understanding how cognition and perception may be implemented in the brainthe grand challenge at the core of cognitive neuroscience. | null | [
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| 67,877,113 | 1903.01458 | d74037476b85dc02eb74369b12e0a5848617ecbc |
Deep Learning for Cognitive Neuroscience
Katherine R Storrs
Department of Psychology
Department of Neuroscience
Abteilung Allgemeine Psychologie
Justus-Liebig University
GiessenGermany
Nikolaus Kriegeskorte
Department of Psychology
Department of Neuroscience
Abteilung Allgemeine Psychologie
Justus-Liebig University
GiessenGermany
Deep Learning for Cognitive Neuroscience
2. Mortimer B. Zuckerman Mind Brain Behavior Institute,
Department of Electrical Engineering (Affiliated member), Columbia University, USANeural network models can now recognise images, understand text, translate languages, and play many human games at human or superhuman levels. These systems are highly abstracted, but are inspired by biological brains and use only biologically plausible computations. In the coming years, neural networks are likely to become less reliant on learning from massive labelled datasets, and more robust and generalisable in their task performance. From their successes and failures, we can learn about the computational requirements of the different tasks at which brains excel. Deep learning also provides the tools for testing cognitive theories. In order to test a theory, we need to realise the proposed information-processing system at scale, so as to be able to assess its feasibility and emergent behaviours. Deep learning allows us to scale up from principles and circuit models to end-to-end trainable models capable of performing complex tasks. There are many levels at which cognitive neuroscientists can use deep learning in their work, from inspiring theories to serving as full computational models. Ongoing advances in deep learning bring us closer to understanding how cognition and perception may be implemented in the brainthe grand challenge at the core of cognitive neuroscience.
We can also compare each model's internal representations to human perceptual judgements. The similarity of stimuli or conditions in the model's internal activation patterns should predict perceived similarity to humans (Kubilius, Bracci & Op de Beeck, 2016). Stimuli that elicit identical responses within the model should appear identical to humans (Wallis et al., 2017). As stimuli are degraded or distorted, the model's performance should decline in a quantitatively similar way to human performance (contrary to this, Geirhos et al. (2018) found that CNNs were far less robust to image noise than humans). Models should be able to predict patterns of confusions and errors, ideally at the single-stimulus level. For example, in one study CNN classifications accurately predicted object-matching confusions in monkeys and humans at the object-category level, but failed to predict which specific images were confused (Rajalingham et al., 2018).
At the level of the internal representation, a model should go through the same sequences of representational transformations across space (brain regions) and time (sequence of processing). Comparing the internal representations between model and brain is complicated by the fact that we may not know the detailed spatial and temporal correspondence mapping between model activity patterns and brain activity patterns. Diedrichsen (2018) introduces the framework of representational models, which can be used to test neural network models, by comparing their internal representations with brain-activity measurements.
Briefly, encoding models predict each measured response channel as a linear combination of the units of the neural network (Kay et al., 2008;Dumoulin et al., 2008;Mitchell et al., 2008;Naselaris & Gallant, 2011). Representational similarity analysis (Kriegeskorte et al., 2008;Nili et al., 2014;Kriegeskorte & Diedrichsen, 2016) and pattern component modelling (Diedrichsen et al., 2011) use a stimulus-by-stimulus matrix of dissimilarities or similarities to characterise the representational space and compare model layers to brain regions. These approaches have complementary pros and cons. For example, encoding models lend themselves to analysing responses in each voxel separately and mapping these out over the cortex (Güçlü & van Gerven 2015;Eickenberg et al., 2017;Wen et al., 2017); representational similarity analysis and pattern component modeling obviate the need for fitting thousands of parameters of a linear encoding model, while still allowing the flexibility of estimating weights to model the relative prevalence of different features in a brain representation (e.g. Khaligh-Razavi & Kriegeskorte, 2014;Khaligh-Razavi, Henriksson, Kay & Kriegeskorte, 2017). Encoding models, representational similarity analysis, and pattern component modelling are best thought of as part of a toolbox of representational modelling techniques that can be combined as appropriate to the goals of a study (Diedrichsen & Kriegeskorte, 2016). Khaligh-Razavi & Kriegeskorte (2014) found that the later layers of a categorisation-trained CNN outperformed any of 27 shallow computer vision models in predicting the representation of natural object images in inferior temporal cortex. Categorisation-trained CNNs also best predict visual cortical activity whether measured via electrophysiology (Cadieu et al., 2014), fMRI (Güçlü & van Gerven 2015Eickenberg et al., 2017;Wen et al., 2017), MEG (Cichy et al., 2016;Cichy et al., 2017) or EEG (Greene & Hansen, 2018, and a DNN trained on speech and music best predicted auditory cortical activity (Kell et al., 2018).
Variants of this approach which focus on decoding or reconstructing stimuli from neural activity likewise find that CNNs provide excellent feature spaces for decoding activity in visual areas (Wen et al., 2017;Horiwaka & Kamitani, 2017).
Whatever method is used to evaluate a DNN model, it is crucial to include comparisons to control models, and/or to test multiple DNNs that instantiate alternative theories of computation or learning. Even trivial models of stimulus processing explain small but significant variance in sensory brain regions (e.g. the pixel-based control models in Khaligh-Razavi & Kriegeskorte, 2014), as do randomly-weighted untrained DNNs (e.g. Cichy et al., 2016).
Interpreting neural network models: doing cognitive neuroscience in silico
Once we have a DNN that can perform some task and explain brain and behavioural data (up to the limit determined by noise and intersubject variability), we have reached an important milestone. However, our job is not done. Because the model has been trained on the task and has many parameters, we may not understand its computational mechanism. We therefore need to analyse the representations and dynamics in the model. Like brains, DNNs can be studied at many different levels of resolution and abstraction. Unlike brains, they permit perfect access to the entire system. We can stop and restart a network, have it re-learn under different environments or task demands, gather data from it continuously without fatigue or damage, lesion and reinstate any combination of its components, use stimulus optimisation techniques to see what features it has learned, and even analytically prove some of its properties 1 .
Because of our unfettered access, DNNs are far more amenable to "electrophysiological" methods than real brains are. To analyse the feature preferences learned by individual units in a network, experimenters can present thousands or millions of stimuli and record each unit's activation (e.g. Yosinski et al., 2015), or complementarily, occlude different regions of an image stimulus to find which features are most crucial for each unit's activation (Zhou et al., 2014). Figure 2c shows how the activation of a single LSTM unit inside a recurrent text-prediction network trained on movie reviews changes as the passage the network is predicting unfolds (Radford, Jozefowicz & Sutskever, 2018). In this striking example, the visualised unit appears to have learned to represent whether the movie review is expressing a positive or negative sentiment.
DNNs are also amenable to novel interrogation techniques that could never be applied to a brain. Because neural network models are differentiable, it is possible to use gradient-descent optimisation to create stimuli that elicit specific patterns of network activation, including generating optimal stimuli for individual units (e.g. Yosinski et al., 2015). Several techniques exist for doing this, as beautifully summarised and illustrated by Olah (2017). Figure 2a shows a noise image which has been iteratively optimised to strongly activate each of four different layers in two category-trained CNNs using Google's "DeepDream" algorithm (Mordvintsev, Olah & Tyka, 2015). The flow of information can be traced through networks to reveal where or when in the stimulus evidence was drawn in order to support a certain output decision (again beautifully illustrated by Olah (2018)), or to which part of the stimulus a network with dynamic "attention" is currently devoting its resources (Figure 2b). Direct optimisation methods have revealed that the hierarchy of features learned by deep visual networks bears a striking similarity to that in the mammalian ventral stream (see e.g. Yamins & DiCarlo, 2016).
They also allow us to explore how feature preferences develop over the course of training, suggesting tantalising similarities with human perceptual learning (Wenliang & Seitz, 2018). Task The accessibility of neural network models raises new possibilities for efficient model search and falsification that could substantially change how we design experiments. Currently experimental conditions are usually chosen "by hand" to differentiate among hypotheses suggested by verbal theories. In the future, conditions could be optimised algorithmically to best differentiate between explicit computational models (e.g. Wang & Simoncelli, 2008).
Neural networks as models of diverse cognitive functions
Visual object and face recognition have received a great deal of attention from engineers and cognitive neuroscientists because DCNNs have been able to reach near-human-level performance using static stimuli and a single feedforward sweep of processing (He et al., 2015;Taigman et al., 2014). However, the deep learning revolution has touched almost all domains in cognition and perception, including our higher-level abilities such as language and reasoning.
Language in neural networks
Language-processing DNNs are widely used in automatic translation, for example of websites and online search results. Such high-level tasks tend to be achieved by more complex compositional network The backpropagation algorithm used to train networks can also be used to visualise features in trained networks. Here, a noise image has been iteratively optimised to increase how strongly it activates units in each of four layers in a 16-layer DCNN, which has been trained either to categorise objects and animals (top row) or identify faces (bottom row). Training has a striking impact on the features learned. (b) "Attention networks" have a spatial attention mask (a multiplicative weighting of the input) which they learn to allocate in ways that improve performance at their training task. Here, the white patches show which regions of this image were attended to at each time step by a recurrent network trained to generate text captions of images, outputting one word per time step (reproduced with permission from Xu et al., 2015). (c) Visualisation of the output activation of a single LSTM unit in a recurrent network trained to predict text one character at a time, for an example passage the network has never seen before. High activation is indicated by green, and low by red. Responses of this unit, when analysed across a large sample of reviews, predicted whether a review was positive or negative in sentiment (reproduced with permission from Radford, Jozefowicz & Sutskever, 2018). architectures than we have yet encountered. The state-of-the-art Google Neural Machine Translation system (Wu et al., 2016) consists of two eight-layer recurrent neural networks, one which encodes the text from its original language to a latent representation within the network, and another which takes the latent representation and decodes it into the target language, using "attentional" weights to change which parts of the encoded representation it processes at each time step. The entire dynamic process is learned through backpropagation by training on tens of millions of human-translated sentences, including the form of the latent representation and the allocation of attention during decoding (Wu et al., 2016). A similar approach has been used to generate verbal descriptions of images (Xu et al., 2016). There, an image-processing convolutional network is used to transform the image into a latent representation of high-level image features, and a recurrent network is trained to output a sequence of words based on those features, using spatial attention to modulate which parts of the image most guide word choice at each time step ( Figure 2b).
Although these networks are engineering solutions, it is striking how the composition of multiple pattern-recognition DNNs, coupled with attentional guidance (a psychologically inspired feature), can enable such uniquely human skills as language translation and image description. Cognitive neuroscientists are just beginning to bring such principles into neural network models of human cognition and language.
Devereux, Clarke and Tyler (2018) used a modular visual-semantic model to predict fMRI activation patterns as human observers named objects in images. In the model, images were first processed by a feedforward image CNN trained on object classification, and the penultimate-layer features from the CNN were then fed into a single-layer recurrent network, which was trained to encode the presence or absence of a large number of semantic properties of the pictured objects ("is a fruit", "grows on trees", etc.). Similarity of representations within the CNN better predicted similarity in human visual cortex, while similarity within the recurrent semantic network better predicted similarity in human perirhinal cortex, an area associated with semantic processing.
Reasoning in neural networks
Artificial general intelligence may not yet be in sight, but some deep networks display impressively flexible reasoning. Relation Networks (Santoro et al., 2017) achieved superhuman performance on the CLEVR benchmark, a notoriously difficult task involving reasoning about the relations among objects in images. Example questions include "How many objects have the same shape as the green object?" and "Are there any rubber things that have the same size as the yellow metallic cylinder?" (Santoro et al., 2017).
A feedforward image CNN extracts high-level visual features from the image, and a recurrent language network encodes the verbal question. The Relation Network then considers all possible pairs of "objects" in the image (operationalised as pairs of spatial locations), along with the question, and evaluates the likelihood of each possible response. Relation Networks achieved a 95% accuracy rate on the CLEVR task; human performance is around 93% (Santoro et al., 2017). The image, language, and relation-reasoning sub-networks form and end-to-end differentiable system and are simply trained together using backpropagation.
Compositional network architectures have proved an important engineering trick for achieving intelligent behaviour in DNNs. Recent work by Yang, Song, Newsome & Wang (2017) begins to unpick how task requirements might determine neural architectures. A recurrent network was trained simultaneously to perform twenty very simple tasks from human and animal cognitive research, such as speeded classification and delayed match-to-sample. The authors were then able to analyse which units were involved in which tasks. They found that the network had learned a compositional representation of the twenty tasks, in that clusters of units specialised in the components of the tasks (e.g. "remember the direction of the cue" or "respond in the direction opposite to the cue" ), and any given task was performed by a coalition of clusters, according to the composition of its sub-tasks.
Embodied neural networks
DNNs are not necessarily disembodied. For example, a recurrent network trained to simulate the muscle movements made by monkeys reaching towards targets in an experiment learned internal representations that predicted the firing patterns of motor neurons recorded while the monkeys were reaching (Sussilo, Churchland, Kaufman & Shenoy, 2015). Neural networks trained to navigate in a simulated environment developed units with similar tuning properties to place, grid, border, and head-direction cells found in rat entorhinal cortex (Cueva & Wei, 2018;Banino et al., 2018). Embodied models, whether with physical or virtual bodies, may be able to learn more efficiently than disembodied ones by using active learning methods to seek out informative training data (e.g. Haber, Mrowca, Fei-Fei & Yamins, 2018).
Current challenges for deep neural networks in cognitive neuroscience
Cognitive neuroscientists have much work to do. Any particular current instance of a deep neural network, presented in the machine learning literature for a particular practical application, will almost certainly not prove to be a fully satisfying model of how similar functions are performed by human brains. We must seek out ways to make cognition in DNNs more flexible and generalisable, learning more ecologically plausible, and turn the predictive power of neural network models into theoretical understanding. Happily, many of these goals align at least in part with those of industrial AI researchers.
Modelling more robust and flexible cognition and perception
Although DNNs are architecturally deep, they can seem conceptually shallow. Their intelligence is more fragile than that of humans, apparently failing to develop the causal and conceptual understanding that allows us to generalise performance across tasks and environments (Marcus, 2017;Lake, Ullman, Tenenbaum & Gershman, 2017). Visual object-recognition DNNs can be tricked into drastic misclassifications using minute image perturbations imperceptible to humans (Szegedy et al., 2013). Reading comprehension networks able to answer questions based on passages of text can be similarly tricked by small changes to the text which do not distract human readers (Jia & Liang, 2017). Standard DNNs are poor at estimating the confidence of their decisions, and may confidently return seemingly nonsense classifications when shown stimuli with different statistics than those in their training dataset (Nguyen, Yosinski & Clune, 2015;Gharamanhi, 2015). Reinforcement-learning networks can outcompete humans at playing arcade video games (Mnih et al., 2015), but appear to depend on features which humans would consider superficial or irrelevant -changing the shape or position of the player's paddle by just a few pixels can disastrously undermine the network's performance (Kansky et al., 2017).
It is worth bearing in mind that current DNNs are still very limited in computational scale compared to biological brains. A modern DNN might contain hundreds of thousands or millions of units, on the same order as the number of cortical neurons in a handful of fMRI voxels 2 . Importantly, the units in neural network models cannot be equated to biological neurons, which are more complex in structure and dynamics, and thus potentially more computationally powerful. As a result, DNNs likely fall short by a larger factor than counting units and neurons would suggest. The numbers of units and neurons simply cannot meaningfully be compared.
To compound their disadvantage in scale, DNNs so far have tended to live exclusively within a single modality and task. At best this makes them models of specific sensory processing areas, but even these areas in human brains have extensive interaction with other modalities, executive control, and action. The examples described in the section "Reasoning in neural networks" above hint at some of the possible solutions, involving compositional architectures (Santoro et al., 2017;Xu et al., 2015), as well as hybrid solutions which combine neural networks with symbolic reasoning models (e.g. Battaglia et al., 2016;Evans & Grefenstette, 2018).
Cognitive neuroscience is well placed to devise benchmark tasks that capture important hallmarks of human cognition. Many existing experimental paradigms may be ideal, for example "violation of expectation" tasks used to measure the development of intuitive physics concepts in children can be applied to assess models' reasoning about physical interactions (Piloto, 2018). Lake and Baroni (2017) and Marcus (2017) suggest a number of other tasks to evaluate the profundity and generality of machine cognition.
Formalising these as benchmarks would provide concrete goals for the modelling community and drive progress (Kriegeskorte & Douglas 2018), as object recognition benchmarks have done in computer vision (Russakovsky et al., 2015).
Building networks that learn in more ecologically and biologically plausible ways
Both supervised and unsupervised learning depend on the backpropagation algorithm, which is traditionally considered biologically implausible (Glaser, Benjamin, Farhoodi & Kording, 2018). Recent theoretical work suggests that deep learning with an algorithm similar to backpropagation might be biologically feasible (Lillicrap et al., 2016;Guerguiev, Lillicrap, and Richards, 2017;Kording & König 2001;Scellier & Bengio, 2016;Hinton and McClelland 1988). However, supervised training on millions of labelled stimuli is still an ecologically unrealistic requirement. Animal brains are capable learning far more data-efficiently, often without explicit supervision, and sometimes even from single examples.
Unsupervised, weakly-supervised and one-shot learning are current focuses of the machine learning community (Hassabis et al., 2017), and we can expect advances in these areas over the next few years.
Cognitive neuroscientists may be able to draw from brain theory and data for creative and biologically feasible solutions. One influential idea within neuroscience is that prediction may be fundamentally important to how brains learn and perceive (Hawkins & Blakeslee, 2007). Predictive coding theories, in particular, propose that, during perception, neural activity primarily encodes the differences between predicted and actual sensory information (Srinivasan, Laughlin & Dubs, 1982). Explicit computational models of predictive coding have previously been formulated at the neuronal-circuit level (Rao & Ballard, 1999), and have received some support from electrophysiological (Rao & Ballard, 1999) and fMRI data (Muckli et al., 2015).
In machine learning, prediction offers a rich unsupervised training signal requiring no additional reward or label information. When implemented in a modern deep learning framework, a predictive coding network was capable of predicting future frames in video of natural environments (Lotter, Kreiman & Cox, 2017;2018). Furthermore, the network spontaneously discovered, in its deeper layers, higher-level properties of the objects depicted in videos, such as facial identity and pose (Lotter, Kreiman & Cox, 2017), and its individual units reproduced certain temporal dynamics of primate visual neurons (Lotter, Kreiman & Cox, 2018).
"Curiosity-based" learning is another tantalising method motivated by animal learning, in which networks embodied in simulated environments actively seek out the most informative parts of those environments during learning (Haber, Mrowca, Fei-Fei & Yamins, 2018).
Illuminating the black box: from prediction to explanation
A criticism levelled at neural networks as cognitive models is that success at predicting brain and behavioural data, which DNNs have unarguably enjoyed (Khaligh-Razavi et al. 2014;Cadieu et al. 2014;Güçlü & van Gerven 2015;Cichy et al., 2016;Eickenberg et al., 2017;Kubilius et al., 2016;Wen et al., 2017), does not constitute success at explaining or understanding the brain or mind (Kay, 2017). If a model exhaustively predicted neural and behavioural data with generalisation to new instances of the task, we might be convinced that it is an accurate model of a certain cognitive function. But unless we can express more concisely how it performs that function, we are unlikely to be fully satisfied as scientists.
We have described some techniques of in silico electrophysiology and visualisation of internal representations for illuminating the black box above. An even more satisfying approach is to go from a neural network model to a concise mathematical description. Gonçalves and Welchmann (2017) trained a convolutional neural network to judge the relative depth of objects in scenes from pairs of stereo images, requiring it to solve the classic "correspondence problem" (which points in the two retinal images correspond to the same point on an external object?). By tracing the strength and sign of connection weights in the trained network, they discovered a surprising computational strategy: rather than using positive weights to find positive matches between points in the two images, the network primarily used negative weights to suppress incorrect matches. The authors distilled this strategy into a succinct mathematical image-filtering model.
The model displayed several unusual and previously unexplained phenomena in human depth perception, and substantially changes our understanding of the correspondency problem, which has stood as a computational puzzle for at least 100 years (Gonçalves & Welchmann, 2017). Deep neural network models do not replace intuitive explanations, verbal theories, and concise mathematical descriptions. Instead they make complex hypotheses testable and help us bridge the levels of description between verbally communicable theory and neural implementation.
Conclusion
Deep neural networks are currently at the cutting edge of artificial intelligence, and are the most powerful predictive models yet discovered for many aspects of behaviour ( Deep learning, therefore, is relevant to everyone interested in how perception and cognition arise from neural activity. There are many levels at which cognitive neuroscientists can use deep learning in their work, from considering the modelling literature in theoretical discussions, through testing models build by others, and on to building new models based on cognitive and neuroscientific theories. Conversely, the cognitive neuroscience community may help drive engineering progress, bringing DNNs that learn in more ecologically feasible ways, learn more profound conceptual representations, and display more robust and generalisable task performance. Machine learning of computations capable of real-world tasks in biologically plausible systems will play a major role in understanding how intelligent behaviour arises from brains.
Appendix: resources for getting started with deep learning Articles
There are many excellent review articles on cognitive neuroscientific applications of deep neural network models. For a recent overview of the complementary exchanges between neuroscience and artificial intelligence, see Hassabis et al. (2017). For a more extensive treatment of neural network architectures, principles, and their application as models of brain function, see Kriegeskorte (2015) and Kietzmann, Mc-Clure & Kriegeskorte (2017). For reviews of neural network models in sensory and systems neuroscience, see Yamins & DiCarlo (2016) and Glaser, Benjamin, Farhoodi & Kording (2018).
Tutorials, courses and books
For accessible introductions to machine learning and deep neural networks, the free online courses currently offered by Geoff Hinton (https://www.coursera.org/learn/neural-networks) and Andrew Ng (https://www.coursera.org/learn/machine-learning) are invaluable, as is the online book by Michael Nielsen (http://neuralnetworksanddeeplearning.com/). Deep Learning (Goodfellow, Bengio & Courville, 2016) is a comprehensive book written by some of the leaders in the field.
Software
Software frameworks to implement deep neural network models are rapidly evolving. At the time of writing, the leading frameworks are free and Python-based: Tensorflow (developed by Google), PyTorch (developed by Facebook) and Caffe (developed by Berkeley University). Keras and Sonnet are higher-level packages which work atop Tensorflow to provide readier access to many advanced features. MATLAB (MathWorks, Inc.) is less widely supported or used by machine learning teams in industry, but is worth mentioning due to its popularity in psychology labs. The Neural Network Toolbox in versions of MATLAB from 2017 onward allows all core deep learning operations such as defining and training feedforward and recurrent models on CPUs or GPUs, loading pre-trained models, importing models defined in other frameworks (currently Keras and Caffe are supported), and visualising units within trained networks.
-performing networks may help settle longstanding debates in cognitive neuroscience such as how sparse or distributed neural codes are likely to be (Agrawal, Girschick & Malik, 2014; Zhuang, Wang, Yamins & Hu, 2017; Morcos, Barrett, Rabinowitz & Botvinick, 2018). Network representations can also be summarised at higher levels of abstraction. Traditional functional localisation methods can identify units or layers that are preferentially selective for particular classes of stimuli, with initially counter-intuitive results such as the emergence of object-selective units in a visual network trained only to classify scenes (Zhou et al., 2014). Representational similarity analysis (RSA) can help show how a network transforms stimulus information across the layers (e.g. Khaligh-Razavi & Kriegeskorte, 2014) or time-steps (e.g. Cichy et al., 2016) of its processing.
Figure 2 :
2The relative transparency of deep neural networks. (a)
Bracci & Op de Beeck, 2016; Wallis et al., 2017), single-neuron activity (e.g. Cadieu et al., 2014), and large-scale cortical activity (Khaligh-Razavi & Kriegeskorte, 2014; Cichy et al., 2016; Cichy et al., 2017; Wen et al., 2017; Güçlü & van Gerven 2015; Eickenberg et al., 2017; Greene & Hansen, 2018; Kell et al., 2018; Horiwaka & Kamitani, 2017). With biologically plausible components and structures, they are the closest we have yet come to explicit end-to-end models of how perception and cognition might be performed in brains.
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"True Contextuality Beats Direct Influences in Human Decision Making",
"True Contextuality Beats Direct Influences in Human Decision Making"
]
| [
"Irina Basieva [email protected] \nCity University of London\nUK\n",
"Víctor H Cervantes [email protected] \nPurdue University\nUSA\n",
"Ehtibar N Dzhafarov [email protected] \nPurdue University\nUSA\n",
"Andrei Khrennikov [email protected] \nLinnaeus University\nSweden\n"
]
| [
"City University of London\nUK",
"Purdue University\nUSA",
"Purdue University\nUSA",
"Linnaeus University\nSweden"
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| []
| In quantum physics there are well-known situations when measurements of the same property in different contexts (under different conditions) have the same probability distribution, but cannot be represented by one and the same random variable. Such systems of random variables are called contextual. More generally, true contextuality is observed when different contexts force measurements of the same property (in psychology, responses to the same question) to be more dissimilar random variables than warranted by the difference of their distributions. The difference in distributions is itself a form of context-dependence, but of another nature: it is attributable to direct causal influences exerted by contexts upon the random variables. The Contextuality-by-Default (CbD) theory allows one to separate true contextuality from direct influences in the overall context-dependence. The CbD analysis of numerous previous attempts to demonstrate contextuality in human judgments shows that all context-dependence in them can be accounted for by direct influences, with no true contextuality present. However, contextual systems in human behavior can be found. In this paper we present a series of crowdsourcing experiments that exhibit true contextuality in simple decision making. The design of these experiments is an elaboration of one introduced in the "Snow Queen" experiment (Decision 5, 193-204, 2018), where contextuality was for the first time demonstrated unequivocally. | 10.1037/xge0000585 | [
"https://arxiv.org/pdf/1807.05684v3.pdf"
]
| 49,864,257 | 1807.05684 | e3c8cf81a6730947cabac140f2ca9970ea44e09a |
True Contextuality Beats Direct Influences in Human Decision Making
Irina Basieva [email protected]
City University of London
UK
Víctor H Cervantes [email protected]
Purdue University
USA
Ehtibar N Dzhafarov [email protected]
Purdue University
USA
Andrei Khrennikov [email protected]
Linnaeus University
Sweden
True Contextuality Beats Direct Influences in Human Decision Making
concept combinationscontext-dependencecontextualitydirect influences
In quantum physics there are well-known situations when measurements of the same property in different contexts (under different conditions) have the same probability distribution, but cannot be represented by one and the same random variable. Such systems of random variables are called contextual. More generally, true contextuality is observed when different contexts force measurements of the same property (in psychology, responses to the same question) to be more dissimilar random variables than warranted by the difference of their distributions. The difference in distributions is itself a form of context-dependence, but of another nature: it is attributable to direct causal influences exerted by contexts upon the random variables. The Contextuality-by-Default (CbD) theory allows one to separate true contextuality from direct influences in the overall context-dependence. The CbD analysis of numerous previous attempts to demonstrate contextuality in human judgments shows that all context-dependence in them can be accounted for by direct influences, with no true contextuality present. However, contextual systems in human behavior can be found. In this paper we present a series of crowdsourcing experiments that exhibit true contextuality in simple decision making. The design of these experiments is an elaboration of one introduced in the "Snow Queen" experiment (Decision 5, 193-204, 2018), where contextuality was for the first time demonstrated unequivocally.
Introduction
A response to a stimulus (say, a question) is generally a random variable that can take on different values (say, Yes or No) with certain probabilities. The identity of a random variable, in nontechnical terms, is what uniquely distinguishes this random variable from other random variables. 1 The distribution of this random variable (probabilities with which it takes on different values) is part of this identity, but clearly not the entire identity: think of a handful of fair coins -a set of distinct random variables with the same distribution. Other stimuli (e.g., other questions posed together or prior to a given one) may directly influence the identity of the response to the given stimulus by changing its distribution. In fact, this change in the distribution, mathematically, is how the "directness" of the influence is defined. True contextuality is such dependence of the identity of a response to a stimulus on other stimuli that cannot be wholly explained by such direct influences. We will elaborate this definition below.
Contextuality is at the very heart of quantum mechanics (see, e.g., Liang, Spekkens, & Wiseman, 2011), where it can be observed by eliminating (or at least greatly reducing) all direct influences by experimental design. (In quantum physics "response to a stimulus" has to be replaced with "measurement of a property," but this is in essence the same input-output relation.) This paper addresses a question that ever since the 1990's interested researchers in physics, computer science, and psychology, the question of whether true contextuality can be observed outside quantum mechanics, with special interest (largely for philosophical reasons we will not be discussing) in whether it is present in human behavior. Many previous behavioral experiments designed to answer this question (e.g., Aerts have been shown to result in systems of random variables that are noncontextual. This prompted Dzhafarov, Zhang, and Kujala (2015) to consider the possibility that human behavior may never exhibit true contextuality. It turns out, however, that contextual systems in human behavior can be found. In this paper we describe a series of experiments that, added to one previously conducted (Cervantes & Dzhafarov, 2018), demonstrate this unequivocally.
It should be emphasized at the outset that it would be incorrect to think of contextuality as being "surprising" and "strange" while noncontextuality is "trivial" and "expected." In the absence of constraints imposed by a general psychological theory, comparable to quantum mechanics, we have no justification for such judgements. One might argue in fact that it is most surprising that so many experiments in psychology are described by noncontextual systems of random variables. Nor would it be correct to assume that typical psychological models, even very simple ones, can only predict noncontextual systems: thus, in the concluding section of this paper we mention a simple model that, on the contrary, predicts only contextual systems (and has to be dismissed because of this). Contextuality analysis is not a predictive model of behavior, and both contextual and noncontextual systems are compatible with "ordinary" psychological models. In that, as we point out in Section 4, psychology is not different from quantum physics, where (non)contextuality of a system is established based on the laws of quantum physics but is not used to derive or revise them. What contextuality analysis elucidates is the nature and structure of random variables -arguably, the most basic and mandatory construct in the scientific analysis of empirical systems, whether in psychology or elsewhere. In a well-defined and mathematically rigorous sense, in a contextual system random variables form true "wholes" that cannot be reduced to sets of distinct random variables measuring or responding to specific elements of contexts while being also cross-influenced by other elements of contexts. This makes contextuality analysis inherently interesting, but we need much greater knowledge of which behavioral systems are contextual and which are not in order to determine what other properties of behavior these characteristics are related to. We will return to the role and meaning of contextuality after we introduce necessary definitions, theoretical results, and empirical evidence.
Direct influences and true contextuality
We introduce the basic notions related to contextuality analysis using a simple example -responses to three Yes/No questions asked two at a time. Most of the experiments reported below are of this kind. Let, e.g., the three questions be q 1 :
Do you like chocolate?
q 2 :
Are you afraid of pain?
q 3 :
Do you see your dentist regularly?
Let a very large group of people be divided into three subgroups: in the first subgroup each respondent is asked questions q 1 and q 2 ; in the second subgroup each respondent is asked questions q 2 and q 3 ; and in the third subgroup the questions are q 3 and q 1 . We call these pairwise arrangements of questions contexts, and we denote them c 1 , c 2 , c 3 , respectively. It does not matter for the example whether the questions are asked in a fixed order, randomized order, or (if in writing) simultaneously. A response to question q i asked in context c j is a random variable that we denote R j i : some of the people in the subgroup corresponding to context c j will answer question q i with Yes, others with No. Assuming the subgroups are so large that statistical issues can be ignored, by counting the numbers of responses we can get a good estimate of the probability distribution for our random variable:
R j i : Y es N o response p j i 1 − p j i probability .(1)
All in all we have six random variables in play, and they can be arranged in the form of the following content-context matrix :
R 1 1 R 1 2 c 1 R 2 2 R 2 3 c 2 R 3 1 R 3 3 c 3 q 1 q 2 q 3 system R 3 .
(
Now, the distributions of the responses to question q i should be expected to differ depending on the context in which it is asked. For instance, when q 1 (Do you like chocolate?) is asked in combination with q 2 (Are you afraid of pain?), the probability of R 1 1 ="Yes, I like chocolate" may be relatively high, because chocolate is usually liked, and the mentioning of pain in q 2 may make it sound especially comforting. However, when the same question q 1 is asked in context c 3 , in combination with mentioning a dentist, the probability of R 3 1 ="Yes, I like chocolate" may very well be lower. The same reasoning applies to the two other questions: the responses to each of them will generally be distributed differently depending on its context. This type of influence exerted by a context on the responses to questions within this context can be called direct influence. Indeed, the dependence of R 1 1 (responding to q 1 ) on q 2 (another question in the same context) is essentially of the same nature as the dependence of R 1 1 on q 1 : a response to q 1 is based on the information contained in q 1 and (even if to a lesser extent) on the information contained in q 2 . The other question in the same context can be viewed as part of the question to which a response is given.
Is all context-dependence of this direct influence variety? As it turns out, the answer is negative. Imagine, e.g., that all direct influences are eliminated by some procedural trick, and each question in each context is answered Yes with probability 1/2. This means, in particular, that R 1 1 and R 3 1 have one and the same distribution,
R 1 1 : Y es N o 1/2 1/2 , R 3 1 : Y es N o 1/2 1/2 ,(3)
and if one does not take into account their relations to R 1 2 (in context c 1 ) and to R 3 3 (in context c 3 ), one could consider R 1 1 and R 3 1 as if they were always equal to each other -essentially one and the same random variable. 2 And similarly for R 1 2 and R 2 2 , and for R 2 3 and R 3 3 . If one looks at each column of matrix (2) separately, ignoring the row-wise joint distributions, then one can write
R 1 1 = R 3 1 R 1 2 = R 2 2 R 2 3 = R 3 3 .(4)
Consider, however, the possibility that no respondent ever gives the same answer to both questions posed to her. Thus, if she answers Yes to q 1 in context c 1 (which can happen with probability 1/2), she always answers No to q 2 , and vice versa. Denoting Yes and No by +1 and −1, respectively, we have a chain of equalities
R 1 1 = −R 1 2 R 2 2 = −R 2 3 R 3 1 = −R 3 3 ,(5)
and it is clear that (4) and (5) cannot be satisfied together: combining them would lead to a numerical contradiction. We should conclude therefore that when the joint distributions within contexts are taken into account, R 1 1 and R 3 1 , or R 1 2 and R 2 2 , or R 2 3 and R 3 3 cannot be considered always equal to each other. In at least one of these pairs, the two random variables should be more different than it is warranted by their individual distributions (which are, in this example, identical). This is a situation in which we can say that the system exhibits true contextuality, the kind of context-dependence that is not reducible to direct influences (in this example, absent).
Empirical data, especially outside quantum physics, almost always involve some direct influences, but the logic of finding out whether they also involve true contextuality remains the same. Continuing to use matrix (2) as a demonstration tool, we first look at the columns of the matrix one by one, ignoring the contexts. For each pair of random variables in a column (responses to the same question), we find out how close to each other they could be made if they were jointly distributed. In other words, we find the maximal probabilities with which each of the equalities in (4) can be satisfied. Then we investigate whether all the variables in our system can be made jointly distributed while preserving these maximal probabilities. If the answer is negative, we conclude that the contexts force the random variables sharing a column to be more dissimilar than warranted by direct influences (differences in their individual distributions). We then call such a system contextual . Otherwise it is noncontextual . This is the gist of the approach to contextuality called Contextuality-by-Default (CbD), and we illustrate it in the next section by a detailed numerical example.
CbD forms the theoretical basis for the design and analysis of our experiments. For completeness, however, another approach to the notion of contextuality should be mentioned, one treating context-dependent probabilities as a generalization of conditional probabilities defined through Bayes's formula (Khrennikov, 2009). With some additional assumptions these contextual probabilities can be represented by quantum-theoretical formalisms -state vectors in complex Hilbert space and Hermitian operators or their generalizations. Applications of such approach to cognitive psychology can be found in Khrennikov (2010) and Busemeyer and Bruza (2012), among other monographs and papers. CbD, by contrast, is squarely within classical probability theory. Although contextuality in CbD can be called "quantum-like" due to the origins of the concept in quantum physics, CbD uses no quantum formalisms. 2 The "as if" here serves to circumvent the technicalities associated with the fact that, strictly speaking, we are dealing here not with R 1 1 and R 3 1 themselves but with their probabilistic copies (couplings) that are jointly distributed. See Dzhafarov and Kujala (2014a, 2017b) for details.
A numerical example and interpretation
The following numerical example illustrates how CbD works. Let there be just two dichotomous questions, q 1 and q 2 , answered in two contexts, c 1 and c 2 (e.g., in two different orders, as in Wang & Busemeyer, 2013). The content-context matrix here is
R 1 1 R 1 2 c 1 R 2 1 R 2 2 c 2 q 1 q 2 system R 2 .(6)
Assume that the joint distributions along the rows of the matrix are as shown:
c 1 R 1 2 = Y es R 1 2 = N o R 1 1 = Y es 1/2 0 1/2 R 1 1 = N o 0 1/2 1/2 1/2 1/2 c 2 R 2 1 = Y es R 2 1 = N o R 2 2 = Y es a 1/2 − a 1/2 R 2 2 = N o 3/4 − a a − 1/4 1/2 3/4 1/4 ,(7)
where a is some value between 1/4 and 1/2. Knowing these distributions means that, for any filling of the matrix (6) with values of the random variables R 1 1 , R 1 2 , R 2 1 , R 2 2 (Yes or No, for a total of 16 combinations), we know the row-wise probabilities: e.g.,
R 1 1 = Y es R 1 2 = Y es p 1 (Y es, Y es) = 1/2 R 2 1 = Y es R 2 2 = N o p 2 (Y es, N o) = 3/4 − a .(8)
We see from (7) that R 1 2 and R 2 2 (the responses to question q 2 ) are distributed identically. Because of this, if they were jointly distributed (see footnote 1), the maximal probability with which they could be equal to each other would be 1:
q 2 R 2 2 = Y es R 2 2 = N o R 1 2 = Y es 1/2 0 1/2 R 1 2 = N o 0 1/2 1/2 1/2 1/2 .(9)
The responses to question q 1 , however, are distributed differently, and in the imaginary matrix of their joint distribution,
q 1 R 2 1 = Y es R 2 1 = N o R 1 1 = Y es 1/2 0 1/2 R 1 1 = N o 1/4 1/4 1/2 3/4 1/4 ,(10)
the maximal possible probability of R 1 1 = R 2 1 = Y es is 1/2, and the maximal possible value of R 1 1 = R 2 1 = N o is 1/4. Therefore, if they were jointly distributed, the maximal probability with which R 1 1 = R 2 1 would be 3/4. Now, with these imaginary distributions, for any filling of the matrix (6) with Yes-No values of the random variables R 1 1 , R 1 2 , R 2 1 , R 2 2 , we also have the column-wise probabilities: e.g.,
R 1 1 = Y es R 1 2 = Y es R 2 1 = Y es R 2 2 = N o p 1 (Y es, Y es) = 1/2 p 2 (Y es, N o) = 0 .(11)
The problem we have to solve now is: are these column-wise probabilities compatible with the row-wise probabilities in (8)?
The compatibility means that, to any of the 16 filling of the matrix (6) with values of the random variables R 1 1 , R 1 2 , R 2 1 , R 2 2 , we can assign a probability, e.g., p R 1 1 = Y es, R 1 2 = Y es, R 2 1 = Y es, R 2 2 = N o , such that the row-wise sums of these probabilities agree with (8) and the column-wise sums agree with (11). This is a classical linear programming problem: for any given value of a it is guaranteed that either such an assignment of probabilities will be found (so that the system is noncontextual) or the determination will be made that such an assignment does not exist (the system is contextual). In our case, however, one need not resort to linear programing to see that no such assignment of probabilities is possible for any value of a other than 1/2. Indeed, we see from the c 1 -distribution in (7) and from (9) that, with probability 1,
R 1 1 = R 1 2 = R 2 2 .(12)
So, R 1 1 and R 2 2 are essentially the same random variable, say X. But, from (10), this X equals R 2 1 with probability 3/4, whereas from the c 2 -distribution in (7), this X equals R 2 1 with probability 2a − 1/4, which is not 3/4 if a = 1/2. The conclusion is that the joint distributions along the two rows of the content-context matrix (6) prevent the responses to the same questions in the two columns of the matrix to be as close to each other as they can be if the two columns are viewed separately. The system therefore is contextual for any a = 1/2.
Why is this interesting? In psychological terms, the interpretation of the question order effect seems straightforward: the first question reminds something or draws one's attention to something that is relevant to the second question. What is shown by the contextuality analysis of our hypothetical question-order system is that this interpretation is only sufficient for a = 1/2, being incomplete in all other cases. The responses to two questions posed in a particular order form a "whole" that cannot be reduced to an action of the first question upon the second response: the identity of the two random variables changes beyond the effect of this action on their distributions. We will return to this issue in the concluding section of the paper.
The reader should not forget that we are discussing a numerical example rather than experimental data. The large body of experimental data on the question-order effect collected by Wang and Busemeyer (2013) has been subjected to contextual analysis in Dzhafarov, Zhang, and , the result being that the responses to any of the many pairs of questions studied exhibit no contextuality. In fact, almost all question pairs are in a good agreement with the "QQ law" discovered by Wang and Busemeyer (2013),
Pr R 1 1 = R 1 2 = Pr R 2 1 = R 2 2 ,(13)
and, as shown in Dzhafarov, Zhang, and , this law implies no contextuality: this system of random variables is entirely describable in terms of each response being dependent on "its own" question, plus the second respond being also influenced by the first question. The idea of a "whole" being irreducible to interacting parts is not therefore an automatically applicable formula. To see if it is applicable at all, in psychology, one should look for empirical evidence elsewhere. Such evidence is presented below. . In such a system n questions and n contexts can be arranged as
Contextuality-by-Default
q 1 c1 q 2 c2 · · · cn−2 q n−1 cn−1 q n cn (14)
The number n is referred to as the rank of the system. The question-order system (6) considered in Section 1.2 is the smallest possible cyclic system, of rank 2,
q 1 c1 q 2 . c2(15)
The system (2) in Section 1.1 is a cyclic system of rank 3,
q 1 c1 q 2 c2 q 3 , c3(16)
and it is used in four of the six experiments reported below. The remaining two are analyzed as cyclic systems of rank 4,
q 1 c1 q 2 c2 q 3 cn−1 q 4 cn ,(17)
with the content-context matrix
R 1 1 R 1 2 c 1 R 2 2 R 2 3 c 2 R 3 3 R 3 4 c 3 R 4 1 R 4 4 c 4 q 1 q 2 q 3 q 4 system R 4 .(18)
To formulate the criterion of contextuality in cyclic systems, we encode the values of our random variables by +1 and −1. Then the products of the random variables in the same context, such as R 1 1 R 1 2 , are well-defined, and so are the
expected values E R 1 1 R 1 2 , E R 2 2 R 2 3 , etc. For instance, if the joint distribution of R 1 1 and R 1 2 (responses to questions q 1 and q 2 in context c 1 ) is c 1 R 1 2 = +1 R 1 2 = −1 R 1 1 = +1 a b a + b R 1 1 = −1 c d c + d a + c b + d ,(19)then R 1 1 R 1 2 has the distribution R 1 1 R 1 2 = +1 R 1 1 R 1 2 = −1 a + d b + c ,(20)
and the distribution of R 1 1 and R 1 2 is described by the expected values
E R 1 1 = (a + b) − (c + d) , E R 1 2 = (a + c) − (b + d) , E R 1 1 R 1 2 = (a + d) − (b + c) .(21)
We will also need a special function, s odd : given some real numbers x 1 , . . . , x n ,
s odd (x 1 , . . . , x n ) = max (±x 1 ± . . . ± x n ) ,(22)
where each ± is to be replaced with + or −, and the maximum is taken over all choices that contain an odd number of minus signs. Thus,
s odd (x, y) = max (−x + y, x − y) , s odd (x, y, z) = max (−x + y + z, x − y + z, x + y − z, −x − y − z) etc. ,(23)
The theorem proved by Kujala and Dzhafarov (2016) says that a cyclic system of rank n is contextual (exhibits true contextuality) if and only if
D = s odd E R 1 1 R 1 2 , E R 2 2 R 2 3 , . . . , E [R n n R n 1 ] − (n − 2) − ∆ > 0,(24)
where
∆ = E R 1 1 − E [R n 1 ] + E R 1 2 − E R 2 2 + . . . + E R n−1 n − E [R n n ] .(25)
The value of ∆ is a measure of direct influences, or of inconsistent connectedness. It shows how much, overall, the distributions of responses to one and the same question differ in different contexts. If ∆ = 0, the system is consistently connected: the response to a given question is not influenced by the other questions with which it co-occurs in the same context. 3 One can loosely interpret s odd as a measure of the "potential true contextuality": it shows how much, overall, the identities of the random variables responding to the same question differ in different contexts. The contextuality test for a cyclic system therefore can be viewed as a test of whether these differences exceed those due to direct influences alone. The failure of the previous attempts to find contextuality in behavioral data may be described by saying that the empirical situations chosen for investigation had too strong direct influences for the amount of potential true contextuality they contained. The idea of the "Snow Queen" experiment was to make the value of s odd as large as possible, increasing its chances of "beating" ∆, a quantity that cannot be controlled by experimental design. 4 The formal structure of the experiment was a cyclic system of rank 4, with q 1 and q 3 being two choices of characters from a story (Snow Queen, by H.C. Andersen), and q 2 and q 4 being two choices of attributes of these characters.
R 1 1 R 1 2 c 1 R 2 2 R 2 3 c 2 R 3 3 R 3 4 c 3 R 4 1 R 4 4 c 4 q 1 :
Gerda Troll q 2 : beautiful unattractive q 3 : Snow Queen old Finn woman q 4 :
kind evil system SQ 4 .(26)
For instance, in context c 3 , a respondent could choose either Snow Queen or old Finn woman, and also choose either "kind" or "evil." The instruction said the choices had to match the story line. The respondents knew, e.g., that Snow Queen is beautiful and evil, and that the old Finn woman is unattractive and kind. 5 It is easy to show that if all respondents followed the instruction correctly, s odd in this experiment had to have the maximal possible value of 4. The amount of direct influences measured by ∆ was considerable, but the left-hand side expression in (24) was well above zero, with very high statistical reliability (evaluated by 99.99% bootstrap confidence intervals). One possible criticism of the "Snow-Queen" experiment can be that the paired choices were too "asymmetric": choice of a character, such as Gerda, and choice of a characteristic, such as "beautiful," seem too different in nature. In the experiments reported below the paired choices were "on a par." Otherwise, the experiments followed the same logic, ensuring the highest possible value for s odd . This value equals n, the rank of the cyclic system. In quantum physics, the systems with this property (if, additionally, they are consistently connected, i.e., ∆ = 0), are called PR-boxes, after Popescu and Rohrlich (1994). In our experiments n was 3 or 4.
Method
Participants
We recruited 6192 participants on CrowdFlower (2018) between February 7 and 12, 2018. They agreed to participate in this study by accepting a standard consent from. The consent form and the interactive experimental procedure were provided via a Qualtrics survey hosted by City University London. The study was approved by City University London Research Ethics Committee, PSYETH (S/L) 17/18 09. (The number of participants was chosen so that we could construct reliable 99.99% bootstrap confidence intervals for each context in each experiment, as described below.)
Materials and procedure
Each respondent participated in all six experiments, in a random order. For each of the experiments, each participant was randomly and independently assigned to one of the conditions (contexts). In each context, a participant was introduced to a pair of choices to be made by a fictional Alice; each choice was between two alternatives. There were three contexts in Experiments 1-4, and four contexts in Experiments 5 and 6. Figure 1 shows the way the instruction and choices were presented to respondents in one context of Experiment 1.
Experiments 1 to 4
In experiments 1-4, in each context, the character Alice was faced with two choices out of a set of three dichotomous choices. The participant was asked to select a pair of responses that respected Alice's preferences as stated in the instructions (see Fig. 1). The system would not allow the respondent to make only one choice or two choices contradicting the instructions. The following depicts the situations presented, while table 1 summarizes the sets of dichotomous choices.
Experiment "Meals." Alice wishes to order a two-course meal. For each course she can choose a high-calorie option (indicated by H) or a low-calorie option (indicated by L). Alice does not want both courses to be high-calorie nor does she want both of them to be low-calorie.
Experiment "Clothes." Alice is dressing for work, and chooses two pieces of clothing. She does not want both of them to be plain, nor does she want both of them to be fancy.
Experiment "Presents." Alice wishes to buy two presents for her nephew's birthday. She can choose either a more expensive option (indicated by E) or a cheaper option (indicated by C). Alice does not want both presents to be expensive or both presents to be cheap.
Experiment "Exercises." Alice is doing two physical exercises. Alice does not want both exercises to be hard or both to be easy.
Experiments 5 and 6
In experiments 5 and 6, in each context, the character Alice was faced with two choices out of a set of four. In all other respects the procedure was similar to that in Experiments 1-4. The participant was asked to select a pair of responses that respected the character's preferences as stated in the instructions. The following depicts the situations presented, while table 2 summarizes the sets of dichotomous choices.
Experiment "Directions." Alice goes for a walk, and has to choose path directions at forks. Alice wants the two directions to be as similar as possible (i.e., the angle between them to be as small as possible).
Experiment "Colored figures." Alice is taking a drawing lesson, and is presented with two pairs consisting of a square and a circle (the pairs being labeled as "Section 1" and "Section 2"). Alice needs to choose one figure from each section, and she wants the two figures chosen to be of similar color. Table 2: Dichotomous choices in experiments 5 and 6. Each respondent was asked to make two choices (q 1 &q 2 or q 2 &q 3 or q 3 &q 4 or q 4 &q 1 ), randomly and independently assigned to this respondent in each experiment. q 1 q 2 q 3 q 4
Directions
West-East fork NorthWest-SouthEast fork North-South fork NorthEast-SouthWest fork ← or → or ↑ or ↓ or 6. Colored figures one of one of one of one of
For each choice q i , the response encoded by +1 is the one on the left: e.g., for q 1 in Experiment 5, the response ← was encoded by +1.
Results
In Experiments 1-4, irrespective of the specific content of the questions, there were three dichotomous choices, q 1 , q 2 , q 3 , offered to the respondents two at a time. Denoting, for each of the choices, one of the response options +1 and the other −1, the results have the following form:
c 1 R 1 2 = 1 R 1 2 = −1 R 1 1 = 1 0 p 1 p 1 R 1 1 = −1 1 − p 1 0 1 − p 1 1 − p 1 p 1 c 2 R 2 3 = 1 R 2 3 = −1 R 2 2 = 1 0 p 2 p 2 R 2 2 = −1 1 − p 2 0 1 − p 2 1 − p 2 p 2 c 3 R 3 1 = 1 R 3 1 = −1 R 3 3 = 1 0 p 3 p 3 R 3 3 = −1 1 − p 3 0 1 − p 3 1 − p 3 p 3(27)
In reference to the CbD criterion (24)- (25), it follows that in these experiments
s odd E R 1 1 R 1 2 , E R 2 2 R 2 3 , E R 3 3 R 3 1 = s odd (−1, −1, −1) = 3,(28)
so that D in (24) is
D = 2 − ∆,(29)
where Table 3 presents the observed values ofp 1 ,p 2 andp 3 for each context of each of Experiments 1-4, and the corresponding numbers of participants from which these probabilities were estimated. Table 4: Probability estimatesp 1 ,p 2 ,p 3 ,p 4 that determine the outcomes of Experiments 5 and 6 in accordance with (31), and the sizes N 1 , N 2 , N 3 , N 4 of the samples from which these estimates were computed. In Experiments 5 and 6 there were four dichotomous choices, q 1 , q 2 , q 3 , q 4 , and each respondent was offered two of them, forming one of four possible contexts. Denoting, again, for each of the choices, one of the response options +1 and another −1, the results have the following form:
∆ = E R 1 1 − E R 3 1 + E R 2 2 − E R 1 2 + E R 3 3 − E R 2 3 = 2 |p 1 + p 3 − 1| + 2 |p 2 + p 1 − 1| + 2 |p 3 + p 2 − 1| . ,(30)Experiment c 1 c 2 c 3 c 4 p 1 N 1p2 N 2p 3 N 3p4c 1 R 1 2 = 1 R 1 2 = −1 R 1 1 = 1 p 1 0 p 1 R 1 1 = −1 0 1 − p 1 1 − p 1 p 1 1 − p 1 c 2 R 2 3 = 1 R 2 3 = −1 R 2 2 = 1 p 2 0 p 2 R 2 2 = −1 0 1 − p 2 1 − p 2 p 2 1 − p 2 c 3 R 3 4 = 1 R 3 4 = −1 R 3 3 = 1 p 3 0 p 3 R 3 3 = −1 0 1 − p 3 1 − p 3 p 3 1 − p 3 c 4 R 4 1 = 1 R 4 1 = −1 R 4 4 = 1 0 p 4 p 4 R 4 4 = −1 1 − p 4 0 1 − p 4 1 − p 4 p 4(31)
In reference to the CbD criterion (24)- (25), it follows that in these experiments
s odd E R 1 1 R 1 2 , E R 2 2 R 2 3 , E R 3 3 R 3 4 , E R 4 4 R 4 1 = s odd (1, 1, 1, −1) = 4,(32)
whence, once again,
D = 2 − ∆,(33)
where Table 4 presents the observed values ofp 1 ,p 2 ,p 3 ,p 4 in Experiment 5 and 6, and the corresponding numbers of participants from which these probabilities were estimated. Table 5 shows the estimated values of D = 2 − ∆ in all our experiments. We see that contextuality is observed in Experiments 1-4 and 5. Experiment 6, however, shows no contextuality: the negative value in the bottom row indicates that direct influences here are all one needs to account for the results.
∆ = E R 1 1 − E R 4 1 + E R 1 2 − E R 2 2 + E R 2 3 − E R 3 3 + E R 3 4 − E R 4 4 = 2 |p 1 + p 4 − 1| + 2 |p 2 − p 1 | + 2 |p 3 − p 2 | + 2 |p 4 − p 3 | .(34)
We evaluate statistical reliability of these results in two ways. The first way is to compute an upper bound for the standard deviation ofD and use it to conservatively test the null-hypothesis D = 0 (the maximal noncontextual value) against D > 0 (contextuality). In Experiment 6 the alternative hypotheses changes to D < 0 (noncontextuality), with Table 5: Estimated values of D = 2 − ∆ in Experiments 1-4 (n = 3) and 5-6 (n = 4). Positive (negative) values of D indicate contextuality (resp., noncontextuality). Tables 3 and 4. Using the independent coupling of stochastically unrelatedp i 's, commonly adopted in statistics, each summand in (30) and (34) Table 6. If we assume applicability of the central limit theorem, given the very large sample sizes, the t-distribution-based p-values are essentially zero. If we make no assumptions, the maximally conservative p-values based on Chebyshev's inequality are still below the conventional significance levels.
In our second statistical analysis, we computed bootstrap distributions and constructed the 99.99% bootstrap confidence intervals for D from 500000 independent resamples for each context of each experiment (Davison & Hinkley, 1997). These are presented in Figure 2. As we see, the left endpoints of the confidence intervals for experiments 1-5 are well above zero. For experiment 6, the 99.99% bootstrap confidence interval (Fig. 2) has the right endpoint well below zero, indicating reliable lack of contextuality.
Discussion
Our results confirm beyond doubt the presence of true contextuality, separated from direct influences, in simple decision making. Compared to the "Snow Queen" experiment (Cervantes & Dzhafarov, 2018), where the paired choices belonged to different categories (choice of characters, such as "Gerda or Troll," was paired with the choice of characteristics, such as "kind or evil"), in our experiments the paired choices belonged to the same category (e.g., two levels of arm exercises were paired with two levels of leg exercises). The fact that our results are similar to those of the "Snow Queen" experiment shows that this difference is immaterial. What is material is the design that ensures a very large value of s odd in the contextuality criterion (24). In our experiments it was in fact the largest possible value, one equal to the rank of the cyclic system, n. This value in all but one of our experiments was sufficient to "beat" direct influences, measured by ∆ (in the sense that their difference exceeded n − 2). The one exception we got, with "Colored figures," is also valuable, as it shows that the presence of true contextuality in our experiment is an empirical finding rather than mathematical consequence of the design: even with s odd maximal in value, direct influences may very well exceed the value of s odd − (n − 2), making the the value of D in (24) negative.
As explained in Cervantes and Dzhafarov (2018), in much greater detail than in the present brief recap, it is important that the design we used was between-subjects, i.e. each respondent in each experiment was assigned to a single context only. The reason for this is that if a single respondent were asked to make pairs of choices in all three contexts (in or in all four contexts (in Experiments 5 and 6), it would have created an empirical joint distribution of all the random variables in the respective systems. This would contravene the logic of CbD, in which different contexts are mutually exclusive, and the random variables in different rows of content-context matrices are stochastically unrelated (have no joint distribution). One might question another aspect of our experimental design: the fact that the respondents were not allowed to contravene their instructions and make incorrect choices (e.g., choose two "high" options or two "low" options in Experiments 1-4). The main reason for this is that in a crowdsourcing experiment, with no additional information about the respondents, it is difficult to understand what could lead a person not to follow the simple instructions. Ideally, one would want to separate data due to deliberate non-compliance or disregard from "honest mistakes," and this is impossible. In fact, it is hard to fathom what an "honest mistake" in a situation as simple as ours might be. In the "Snow Queen" experiment (Cervantes and Dzhafarov, 2018), where the choices were, arguably, less simple than in the present experiments, incorrect responses were allowed, and their percentage was just over 8%. Their inclusion or exclusion did not make any difference for analysis and conclusions.
In the opening of the paper and at the end of Section 1.2 we alluded to the interpretation of true contextuality in terms of the "wholes" irreducible to interacting "parts." One must not mistake this interpretation for the old adage that "the whole is something besides the parts" (Aristotle) or, as reformulated by Kurt Koffka (1935), "the whole is something else than the sum of its parts" (p.176). These and similar statements are not only vague, they have also been rendered essentially meaningless by their indiscriminate application to all kinds of situations. In most of cases one has a justifiable suspicion that what is meant is that parts interact, or that someone can discern a pattern in them. This is probably always true when the parts are deterministic entities. In the case of random variables, however, there is a rigorous analytic meaning of saying that the whole is different from, and indeed greater than a system of parts with all their interactions. Random variables measuring or responding to one and the same "part" (property or stimulus) have different identities in different "wholes" (contexts), with the difference being greater than warranted by the mere distributional differences caused by their interactions with other elements of the "wholes." If this sounds too philosophical to be of importance in scientific practice, we have an example of quantum mechanics to counter this view.
Contextuality in quantum mechanics is not a predictive theory, and it is never used to derive any parts of quantummechanical theory. Rather the other way around, quantum-mechanical theory is used to determine a system's behavior, from which it is possible to establish if the system is contextual. Thus, in the most famous example of quantum contextuality, involving spins of entangled particles (Bell, 1964(Bell, , 1966, the correlations between spins are computed by standard quantum-theoretic formulas, and the results are used to establish that, for certain choices of axes along which the spins are measured, the system is contextual. The computations themselves make no use of contextuality, nor are they being amended in any way as a result of establishing contextuality or lack thereof. Nevertheless, the contextuality analysis of spins of entangled particles (Bell, 1964(Bell, , 1966Clauser, Horne, Shimony, & Holt, 1969;Fine 1982), mathematically related to a special case of our contextuality criterion (24), with n = 4 and ∆ = 0, is considered highly significant. A prominent experimental physicist, Alain Aspect, called it "one of the profound discoveries of the [20th] century" (Aspect, 1999), and teams of experimentalists have put much effort into verifying that the quantum-mechanical predictions used to derive it are correct (Handsteiner et al., 2017). The reason for this is, of course, that contextuality reveals something about one of the most fundamental aspects of quantum theory: the nature of random variables used to describe quantum phenomena. Thus, it is significant that typical systems of random variables describing classical mechanics happen to be noncontextual, while some quantum-mechanical systems are contextual. In time it has also become clear that, in addition to its foundational significance, quantum contextuality correlates with physical properties that can be used for practical purposes. Physicists and computer scientists at present are beginning to pose the question of "contextuality advantage" or "contextuality as a resource," which is the question of whether contextuality or noncontextuality of a system can be utilized for practical purposes. It is argued, e.g., that the degree of contextuality (a notion we have not discussed in this paper, see Dzhafarov Psychology shares the mandatory use of random variables with quantum physics: stochasticity of responses in most areas of psychology is inherent, it cannot be reduced by progressively greater control of stimuli and conditions. The status and role of contextuality therefore can be expected to be similar. The same as in quantum physics, contextuality analysis is not a predictive model competing with other models. Thus, in constructing a model to fit our data, contextuality analysis can help only in the trivial sense: as with any other property of the data, if contextuality or noncontextuality of them is established, a model is to be rejected if it fails to predict this property. As an example, one could attempt to fit our data by a model with responses being chosen from some "covertly" evoked initial responses actualized with the aid of some conflict resolution scheme. Assume that each question q has a probability h of being "covertly" answered +1 (standing here for one of the two options), and that in a context c = (q, q ) these covert responses occur independently, so that (+1, +1) occurs with probability hh , (+1, −1) with probability h (1 − h ) etc. If the combination of covert responses is allowed by the instructions (e.g., West and North-West in Experiment 5, or Red and Orange in Experiment 6), they turn into observed responses; if the combination is prohibited (say, West and South-East, or Red and Blue), the respondent randomly flips one of the two responses, say, with probability 1/2. Then the observed probability of choosing an allowed combination (+1, −1) is computed as h (1 − h ) + hh /2 + (1 − h) (1 − h ) /2. This model can be shown to predict that a system in our experiments is contextual, but it is incompatible with the noncontextuality in Experiment 6. This was only one example, however. Simple models that can predict both contextual and noncontextual outcomes in our experiments can be readily constructed, because all one has to predict are three probabilities (p 1 , p 2 , p 3 ) in (27) for Experiments 1-4, and four probabilities (p 1 , p 2 , p 3 , p 4 ) in (31) for Experiments 5-6. Consider, e.g., a model with eight triples (+1, +1, +1) , (+1, +1, −1) , . . . , (−1, −1, −1), mental states evoked with certain probabilities, with the following decision rule: if the context is (q i , q j ), i, j = 1, 2, 3 and the mental state contains r i (+1 or −1) and r j (+1 or −1) in the ith and jth positions, respectively, then respond (r i , r j ) if this response combination is allowable; if the combination is forbidden, choose one of the allowable combinations with probability 1/2. The model has 7 free parameters, and it can fit (p 1 , p 2 , p 3 ) in Experiments 1-4 precisely. For Experiments 5 and 6, the eight triples have to be replaced with 16 quadruples. We need not get into discussing such models here: it was not a purpose of our experiments to achieve a deeper understanding of how someone chooses to eat soup and beans over burger and salad. Rather our aim was to capitalize on the psychological transparency and modeling simplicity of such choices to firmly establish that "quantumlike" contextuality can be observed outside quantum physics, in human behavior. Recall that many previous attempts to demonstrate behavioral contextuality have failed, so our paper is only one of the first two steps (the other one being the "Snow Queen" experiment in Cervantes & Dzhafarov, 2018) on the path of identifying contextual systems in human behavior.
Thinking by analogy with the "contextuality advantage" mentioned above, can we, at this early stage of exploration, point out any properties of human behavior as correlating with or being indicated by contextuality? One obvious fact is that in our experiments contextuality is negatively related to the value of ∆, the amount of direct influences. Lack of direct influences means that the probability of choosing a particular option, say, burger, is the same irrespective of what context this option is included in (e.g., whether the plain skirt is chosen in the skirt-blouse combination or in the jacket-skirt one). The lack of direct influences would result in the maximal possible value of D = 2. This simplicity, however, is specific to our design, in which s odd function does not vary. For a more general class of systems of random variables, one cannot simply replace contextuality with a measure inversely related to the amount of direct influences (we even have examples when the two are synergistic rather than antagonistic). Another dimension of human behavior that can be related to contextuality can be called the degree of "similarity" or "unanimity" of decisions across pools of respondents, or across repeated responses by the same person when a within-subject design is possible (as in Dzhafarov, 2017a, b, and. Consider, e.g., one of our Experiments 1-4, and assume that the respondents agreed among themselves on what option to choose in each context. The system then would become deterministic and noncontextual, with D = −4 or D = 0, depending on the pattern of choices agreed upon. Small deviations from an agreed-on pattern would result in small deviations from the corresponding values of D. On the other extreme we have maximal diversity, when in each context the opposite options are chosen with equal probabilities. In this case the system would reach the maximal possible degree of contextuality. Again, it is not possible to simply replace contextuality with some measure of unanimity, such as variance: the maximal value of contextuality can also be achieved without maximal diversity of responses, and "deep noncontextuality," with D between −4 and 0, can be achieved with non-deterministic systems. With due caution, one can conjecture that the degree of (non)contextuality, for a given format of the content-context matrix, may reflect a combination of the two dimensions mentioned: (in)consistency of choices across contexts (reflecting the amount of direct influences) and unanimity/diversity of choices made in each context across a pool of respondents or repeated in a within-subject design (reflecting the amount of determinism/stochasticity). We will not know if this or other relations of contextuality to various aspects of behavior can be established until we broaden our knowledge of the degree of (non)contextuality to a much larger class of behavioral systems.
, 2014 ;
2014Aerts, Gabora, & Sozzo, 2013; Asano, Hashimoto, Khrennikov, Ohya, & Tanaka, 2014; Bruza, Kitto, Nelson, & McEvoy, 2009; Bruza, Kitto, Ramm, & Sitbon,
was developed (Dzhafarov, Cervantes, Kujala, 2017; Dzhafarov & Kujala 2014a, 2016, 2017a, 2017b; Kujala, Dzhafarov, & Larsson, 2015) as a generalization of the quantum-mechanical notion of contextuality (Abramsky & Brandenburger, 2011; Fine, 1982; Kochen & Specker, 1967; Kurzynski, Ramanathan, & Kaszlikowski, 2012). The latter only applies to consistently connected systems, those in which direct influences are absent, i.e., responses to the same stimulus (or measurements of the same property) in different contexts are distributed identically. In physics this requirement is known by such names as "no-signaling," "no-disturbance," etc.; in psychology it is known as marginal selectivity (Dzhafarov, 2003; Townsend & Schweickert, 1989). This requirement is never satisfied in behavioral experiments (Dzhafarov & Kujala, 2014b; Dzhafarov, Kujala, Cervantes, Zhang, & Jones, 2016; Dzhafarov, Zhang, & Kujala, 2015), and it is often violated in quantum physical experiments too (Adenier & Khrennikov, 2017; Kujala, Dzhafarov, & Larsson, 2015). The main difficulty faced by many previous attempts to reveal contextuality in human behavior was that they could not apply mathematical tests predicated on the assumption of consistent connectedness to systems in which this requirement does not hold. As mentioned in the introduction, a CbD-based analysis of these experiments(Dzhafarov & Kujala, 2014b; Dzhafarov, Kujala, Cervantes, Zhang, & Jones, 2016; Dzhafarov, Zhang, & showed that all context-dependence in them was attributable to direct influences. The first unequivocal evidence of the existence of contextual systems in human behavior was provided by Cervantes and Dzhafarov's (2018) "Snow Queen" experiment.The idea underlying the design of the "Snow Queen" experiment (and all the experiments reported below) is suggested by the criterion (necessary and sufficient condition) of contextuality when CbD is applied to cyclic systems with dichotomous random variables
Figure 1 :
1The appearance of the computer screen to the participant if assigned to context c 1 in experiment 1. The participant was required to choose an option for each question, in this case each menu section; the next experiment or the end of the survey would be reached by clicking the 'Next' arrow. If the participant had made both choices in accordance with the instructions, in this case having chosen Soup (H) with Beans (L) or Salad (L) with Burger (H), clicking the 'Next' arrow allowed the survey to continue; otherwise the participants were prompted to revise or complete their responses.
participant assigned to context c 4 was excluded from Experiment 6 because she or he did not complete the responses in accordance with the instructions.
has a variance bounded by 2 Nmin . The different summands are not independent, but the standard deviation of the sum cannot exceed the sum of their standard deviations. upper bounds for the standard deviation ofD. These values are reported in
Figure 2 :
2Histograms of the bootstrap values ofD = 2 −∆ for Experiments 1-6. The solid vertical line indicates the location of the observed sample value. The vertical dotted lines indicate the locations of the 99.99% bootstrap confidence intervals.
, Cervantes & Kujala, 2017; Kujala & Dzhafarov, 2016) is directly related to computational advantage of quantum computing over conventional one (Abramsky, Barbosa, & Mansfield, 2017; Frembs, Roberts, & Bartlett, 2018).
Table 1 :
1Dichotomous choices in experiments 1 to 4. Each respondent was asked to make two choices (q 1 &q 2 or q 2 &q 3 or q 3 &q 1 ), randomly and independently assigned to this respondent in each experiment.q 1
q 2
q 3
1. Meals
Starters:
Main course:
Dessert:
Soup (H)* or Salad (L)
Burger (H)* or Beans (L) Cake (H)* or Coffee (L)
2. Clothes
Skirt:
Blouse:
Jacket:
Plain* or Fancy
Plain* or Fancy
Plain* or Fancy
3. Presents
Book:
Soft toy (bear):
Construction set:
Big expensive book (E)* or Smaller book(C)
(E)* or (C)
(E)* or (C)
4. Exercises
Arms:
Back:
Legs:
Hard* or Easy
Hard* or Easy
Hard* or Easy
* Denotes the response encoded with +1
Table 3 :
3Probability estimatesp 1 ,p 2 ,p 3 that determine the outcomes of Experiments 1-4 in accordance with(27), and the sizes N 1 , N 2 , N 3 of the samples from which these estimates were computed.Experiment
c 1
c 2
c 3
p 1
N 1p2
N 2p3
N 3
1. Meals
0.349 2090 0.658 2052 0.653 2050
2. Clothes
0.639 1996 0.566 2086 0.435 2110
3. Presents
0.547 2081 0.387 2052 0.515 2059
4. Exercises 0.590 2058 0.306 2024 0.580 2110
Table 6 :
6Statistical significance of contextuality in Experiment 1-5 and of noncontextuality in Experiment 6.
Experiment: 1. Meals 2. Clothes 3. Presents 4. Exercises 5. Directions 6. Colored
D = 2 −∆
1.361
1.440
1.548
1.223
0.758
−0.984
N min
2050
1996
2052
2024
1504
1482
Upper bound for st. dev. ofD
0.094
0.095
0.094
0.095
0.146
0.147
Number of st. dev. from zero
> 14.5
> 15.1
> 16.5
> 12.9
> 5.1
> 6.6
t-distribution p-value < 10 −45
< 10 −48
< 10 −57
< 10 −36
< 10 −6
< 10 −10
Chebyshev p-value
< 0.005
< 0.005
< 0.004
< 0.006
< 0.038
< 0.023
D = 0 in the null hypothesis interpreted as the infimum of contextual values. We begin by observing that eachp i has
variance p(1−p)
Ni
≤ 1
4Ni ≤ 1
4Nmin , where N min is the smallest among N i for a given experiment, as shown in
In rigorous mathematical terms, a random variable is defined as a (measurable) function mapping a domain probability space into another (measurable) space. Its distribution is just one property of this function, the probability measure it induces on the codomain space.
The special case of (24) for ∆ = 0 was proved, by very different mathematical means, in Araújo, Quintino, Budroni, Cunha, & Cabello (2013).4 In physics the situation is different: one can eliminate or greatly reduce direct influences by, e.g., separating two entangled particles by a space-time interval that prevents transmission of a signal between them.5 This instruction is an analogue of the quantum-mechanical preparation, an empirical procedure preceding an experiment with the aim of creating a specific pattern of high correlations between measurements.
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| []
|
[
"Fundamental diagram of urban rail transit considering train-passenger interaction *",
"Fundamental diagram of urban rail transit considering train-passenger interaction *"
]
| [
"Toru Seo ",
"Kentaro Wada ",
"Daisuke Fukuda "
]
| []
| []
| Urban rail transit often operates with high service frequencies to serve heavy passenger demand during rush hours. Such operations can be delayed by two types of congestion: train congestion and passenger congestion, both of which interact with each other. This delay is problematic for many transit systems, since it can be amplified due to the interaction. However, there are no tractable models describing them; and it makes difficult to analyze management strategies of congested transit systems in general and tractable ways. To fill this gap, this article proposes simple yet physical and dynamic model of urban rail transit. First, a fundamental diagram of transit system (i.e., theoretical relation among train-flow, train-density, and passenger-flow) is analytically derived considering the aforementioned physical interaction. Then, a macroscopic model of transit system for dynamic transit assignment is developed based on the fundamental diagram. Finally, accuracy of the macroscopic model is investigated by comparing to microscopic simulation. The proposed models would be useful for mathematical analysis on management strategies of urban rail transit systems, in a similar way that the macroscopic fundamental diagram of urban traffic did. Keywords: public transport; rush hour; fundamental diagram; macroscopic fundamental diagram; dynamic transit assignment * This paper is an enhanced version of Seo et al. (2017a,b). Submitted to an international journal. | 10.1007/s11116-022-10281-0 | [
"https://arxiv.org/pdf/1708.02147v2.pdf"
]
| 237,635,495 | 1708.02147 | 551ae36cdd7b43ab1b2a07aac357262d7bda96d0 |
Fundamental diagram of urban rail transit considering train-passenger interaction *
September 27, 2021
Toru Seo
Kentaro Wada
Daisuke Fukuda
Fundamental diagram of urban rail transit considering train-passenger interaction *
September 27, 2021public transportrush hourfundamental diagrammacroscopic fundamental diagramdynamic transit assignment
Urban rail transit often operates with high service frequencies to serve heavy passenger demand during rush hours. Such operations can be delayed by two types of congestion: train congestion and passenger congestion, both of which interact with each other. This delay is problematic for many transit systems, since it can be amplified due to the interaction. However, there are no tractable models describing them; and it makes difficult to analyze management strategies of congested transit systems in general and tractable ways. To fill this gap, this article proposes simple yet physical and dynamic model of urban rail transit. First, a fundamental diagram of transit system (i.e., theoretical relation among train-flow, train-density, and passenger-flow) is analytically derived considering the aforementioned physical interaction. Then, a macroscopic model of transit system for dynamic transit assignment is developed based on the fundamental diagram. Finally, accuracy of the macroscopic model is investigated by comparing to microscopic simulation. The proposed models would be useful for mathematical analysis on management strategies of urban rail transit systems, in a similar way that the macroscopic fundamental diagram of urban traffic did. Keywords: public transport; rush hour; fundamental diagram; macroscopic fundamental diagram; dynamic transit assignment * This paper is an enhanced version of Seo et al. (2017a,b). Submitted to an international journal.
Introduction
Urban rail transit systems such as metro is handling significant transportation needs of metropolitan areas (Vuchic, 2005). Its most notable usage is the morning commute, in which heavy passenger demand is concentrated in a short time period. It is known that such transit systems often suffer from delays caused by congestion, even if no serious incidents or accidents occur (Kato et al., 2012;Tirachini et al., 2013;Kariyazaki et al., 2015). Therefore, appropriate management of transit systems is required; especially, travel demand management for mass transit systems has been gaining attention recently (Halvorsen et al., 2019;Huan et al., 2021).
One of the approaches to find management strategies of transit systems is theoretical analysis with simplifications, such as use of certain static models with constant travel time (de Cea and Fernández, 1993;Tabuchi, 1993;Kraus and Yoshida, 2002;Tian et al., 2007;Gonzales and Daganzo, 2012;Trozzi et al., 2013;de Palma et al., 2015a,b). In this approach, general policy implications can be obtained thanks to the simplicity and tractability of the analysis. However, they may not be sufficient to investigate dynamic operation and demand management strategies.
In congested transit systems, dynamical interaction among trains and passengers plays essential roles to determine the system's operational behavior, and the travel time can be dynamically and significantly changed due to this interaction. For instances, there are two types of congestion in transit systems:
• train-congestion: congestion involving consecutive trains using the same tracks,
• passenger-congestion: congestion of passengers who are boarding to a train, namely, bottleneck congestion at the doors of a train while it is stopped at a station (Lam et al., 1998;Wada et al., 2012;Kariyazaki et al., 2015), and these two types of congestion interact with each other and cause delay (Newell and Potts, 1964;Kusakabe et al., 2010;Wada et al., 2012;Kato et al., 2012;Tirachini et al., 2013;Kariyazaki et al., 2015;Cuniasse et al., 2015). The most typical phenomena involving the dynamic train-passenger interaction would be the "knock-on delay" (Carey and Kwieciński, 1994)-this is a train equivalent of the "bus bunching" (Newell and Potts, 1964;Daganzo, 2009). For example, assume that passenger-congestion happened temporally due to high demand. It would extend the dwelling time of a train at a station. Then, this extended dwell time would interrupt the operation of subsequent trains, and cause train-congestion on the track. It would deteriorate the passenger throughput, and thus the passenger-congestion at stations would intensify. This kind of dynamical phenomena cannot be captured by static models.
A consequence of such dynamical passenger-train interaction can be found in macroscopic states of transit systems. Fig. 1 shows observed 3-dimensional relations among states of transit systems, that is, train-flow (train/h), train-density (train/km), and passenger-flow (passenger/h). The visualization is based on the concepts of the fundamental diagram (Greenshields, 1935) of urban traffic. Although Fig. 1a and Fig. 1b show data from completely different transit systems, they have remarkable similarities. First, as the passenger-flow increases, the train-density increases; this could be a result of transit operators responding to increased passenger-demand. Second, as the passenger-flow increases, the average service speed of trains and train-flow decrease; this could be a result of the aforementioned congestion due to the interaction among trains and passengers. It would be preferable if we have a theoretical model of this phenomena, because it would be useful to obtain general principles on transit operations; however, to our knowledge, such a model does not exist in the literature.
This study derives a theoretical relation among the state variables of transit systems similar to Fig. 1 based on the microscopic operation principles. It is modeled as a fundamental diagram (FD), which is a well-known concept in vehicular traffic flow theory. The original FD describes relation between vehicular flow and density, and it can be used to describe dynamic evolution of traffic by combining with other principles in a tractable manner (Lighthill and Whitham, 1955;Richards, 1956;Mahmassani et al., 1984;Geroliminis and Daganzo, 2007). In fact, several recent studies have employed FDs of transit systems to describe train-congestion by modeling the relation between train-flow and Note that passenger-congestion differs from in-vehicle passenger-crowding (Kumagai et al., 2020), which results in discomfort due to standing and crowding, but is not necessarily cause any delay directly.
Several detailed operation models have been proposed to capture the detailed mechanism of the dynamics of interaction (see Vuchic, 2005;Koutsopoulos and Wang, 2007;Parbo et al., 2016;Cats et al., 2016;Li et al., 2017;Alonso et al., 2017;Cunha et al., 2021, and references therein), and these have been used to develop efficient operation schemes. However, these models are often based on microscopic simulation, and thus their purposes tend to be case-specific optimization and evaluation. It would be difficult to use them to derive the relation depicted in Fig. 1 or to obtain general policy implications for management strategies, as they are essentially complex and intractable. x-coordinates represent the train-flow (i.e., the number of trains per a kilometer), the y-coordinates represent the train-flow (i.e., the number of passing trains per an hour), the slope from the origin to an arbitrary point indicates the train-speed (i.e., the average speed of trains) of that point, and the color of each point represents the passenger-flow (i.e., average demand of passengers per an hour) of that point. "pax" in the figure is an abbreviation for "passenger". Data sources: Fukuda et al. (2019), Zhang and Wada (2019), Tokyu Railways, Massachusetts Bay Transportation Authority.
train-density (Cuniasse et al., 2015;Corman et al., 2019;de Rivera and Dick, 2021). The novel feature of this study is the incorporation of passengers in an analytical way. Furthermore, this study develops a dynamic transit assignment method based on the proposed FD. This study proposes tractable models of the dynamics of urban rail transit considering the physical interaction between train-congestion and passenger-congestion. In Section 2, a microscopic model of a rail transit system is introduced based on a passenger boarding model and a train cruising model. In Section 3, the operation performance of the microscopic model is analyzed. Specifically, a mathematically tractable relation among train-flow, train-density, and passenger-flow is derived-that is, a fundamental diagram (FD). The model can be also viewed as a variation of 3-dimensional macroscopic fundamental diagram (Mahmassani et al., 1984;Geroliminis and Daganzo, 2007) with an analytical derivation. This is the key contribution of this study. In Section 4, a macroscopic loading model of a transit system is developed based on the proposed FD. The model describes the aggregated behavior of trains and passengers in a urban-scale spatial domain based on the FD. In Section 5, the approximation accuracy and other properties of the proposed macroscopic model are investigated through a comparison with microscopic simulation. Section 6 concludes this article. Note that empirical validation of the proposed model based on actual data is out of scope of this study. Such validation is now being conducted by some of the authors and preliminary results that support the model have been obtained (Fukuda et al., 2019;Zhang and Wada, 2019).
Microscopic Model of Rail Transit System
This section introduces a microscopic model of rail transit system, from which we derive the FD in Section 3. It consists of two microscopic operation principles, namely, a passenger boarding model which describes the train's dwell behavior at a station for passenger boarding and a train cruising model which describes the cruising behavior on the railroad. This microscopic model has been proposed by Wada et al. (2012) to analyze train bunching.
Rail Transit Operation Principles
Consider a railway system on a single line track, where trains and stations are indexed by m and i, respectively. We assume that all trains stop at every station. Let t m,i be the arrival time of train m at station i. Then, a dynamical system that represents each train motion is given by
t m,i+1 = t m,i + b m,i + c m,i ∀m, i(1)
where b m,i is the passenger boarding time of train m at station i, and c m,i is the trip time of train m between stations i and i + 1, which are determined by the two operational submodels (see Figure 2). The passenger boarding time is modeled using a queuing model. That is, the flow-rate of passenger boarding is assumed to be constant, µ p ; and there is a buffer time (e.g., time required for door opening/closing), g b , for the dwell time. Then, the dwell time of a train at a station, b m,i , is represented as
b m,i = q p,i h m,i µ p + g b,i ,(2)
where q p,i is the (possibly time-dependent) passenger demand flow rate at station i, h m,i ≡ t m,i −t m−1,i is the time-headway, and thus q p,i h m,i is the number of waiting passengers at the station. This can be considered as a special case of Lam et al. (1998). All passengers waiting a train at a station are assumed to board the first train arrived.
In reality, there are passengers alighting a train, in addition to ones boarding. By carefully distinguishing the two types of passengers and replacing the terminology in the main text, the discussions in the main text are valid and the final results are not altered. For example, "the number of boarding passengers" can be replaced with "the sum of the number of boarding passengers and the number of alighting passengers". However, it will complicate the discussions; therefore, we ignore passengers alighting a train.
The cruising behavior of a train is modeled using the Newell's simplified car-following model (Newell, 2002). In this model, a train travels by maintaining the minimum safety clearance. Specifically, x m,i (t), position of a train m between stations i and i + 1 at time t, is described as
x m,i (t) = min {x m,i (t − τ ) + vτ, x m−1,i (t − τ ) − δ} ,(3)
where m − 1 indicates the preceding train of train m, τ is the physical minimum time-headway, v is the desired cruising speed that is determined by a (fixed-block or moving-block) signal control system, and δ is the minimum spacing.
We call the traffic is in free-flowing regime if the train travels between stations at the free-flow speed v f (the maximum speed of the track), i.e., the train motion is represented by the first term with v = v f in the minimum operation of Eq. (3). We call the traffic is in congested regime if the train is required to decrease its speed to maintain both the safety headway τ and distance δ. In this case, the second term of Eq. (3) is active. The speed profile in this regime may differ in different train operators (i.e., signal control systems) and drivers. We employ one of the simplest approximations of this speed profile, that is, the train travels between stations at a constant speed while maintaining the minimum safety clearance.
Validity of Assumptions
In this model, the dwell time of a train is determined by the number of boarding passengers, not by a pre-determined timetable. Although this seems like inconsistency between the proposed model and actual schedule-based train operations, this can be considered as a reasonable approximation of average operation pattern of schedule-based train operations. The reasons are as follows. First, in a congested urban areas, it is common that passenger boarding time is not negligible and occasionally delay transit operation, as reviewed in Section 1. Therefore, in order to maintain a scheduled operation based on a timetable, this timetable has to be determined considering the passenger demand (e.g., Niu and Zhou, 2013). Consequently, the dwelling time in such timetable can be considered as similar to the proposed passenger boarding model (2) where q p,i h m,i is interpreted as an average number of waiting passenger and g b is interpreted as a buffer time to deal with fluctuation of the demand.
Second, the passenger boarding model with a constant capacity is consistent with the modelling of ordinary pedestrian flows for a fixed-width bottleneck (Lam et al., 1998;Hoogendoorn and Daamen, 2005). Meanwhile, there is no stock capacity for passengers in the presented model; in other words, a train can transport infinite number of passengers. This is a limitation of the current model; however, unless the passenger demand is excessive level (e.g., where not all of the waiting passenger can board an arriving train), this limitation will not be problematic. The train cruising model (3) can be considered as a lower order but a reasonable approximation of the train movement in the sense that the fundamental operating principle is consistent with practical controls and existing studies (Carey and Kwieciński, 1994;Higgins and Kozan, 1998;Huisman et al., 2005). That is, each train has to maintain a headway and a spacing that are greater than the given minimum ones. The model directly corresponds to the "moving block control," (Dicembre and Ricci, Newell's simplified car-following model is a special case of the well-known road traffic flow model, the Lighthill-Whitham-Richards (LWR) model (Lighthill and Whitham, 1955;Richards, 1956;Newell, 1993). Although the LWR model is known as a "macroscopic" model based on continuum fluid approximation, Newell (2002) showed that it is equivalent to a microscopic car-following model proposed by his paper. 2011) but it can be also viewed as an approximation of traditional "fixed block control ." The main assumption here is the constant speed assumption in the congested regime. This approximates a train operation with low acceleration rates for ensuring the comfort of passengers and less energy loss. In addition, this assumption can describe the congested situation where the train stops between stations, on average, as we will show in the numerical experiment in Section 5.
Fundamental Diagram of Rail Transit System
In this section, we derive an FD of a rail transit system described by the microscopic model formulated in Section 2. The FD is defined as the relation among train-flow, train-density, and passenger-flow.
Steady State of Rail Transit System
We consider the steady state of the proposed microscopic model. The steady state is an idealized traffic state that does not change over time, and its traffic state variables (typically combination of flow, density, and speed) are characterized by special relation (called an FD) of the traffic flow (Daganzo, 1997). Let us consider a homogeneous rail transit system in which the stations and passenger demands are homogeneously distributed over the line, i.e., l i = l, q p,i = q p and other parameters (g b , v f , τ , and δ) are the same for all stations and trains. Then, the steady state is defined as a state that, for a given steady passenger demand q p , the time-headway between successive trains, H, is time-independent. Note that q p < µ p must be satisfied; otherwise, passenger boarding will never end.
Transit systems under different steady states are illustrated as time-space diagrams in Fig. 3. In each sub-figure, train m arrives at and departs from station i, then travels to station i + 1 at cruising speed v, and finally arrives at station i + 1. The main differences between each sub-figure are density of trains and, consequently, traffic regime. In Fig. 3a, the density is small so that the speed v is equal to the free-flow speed v f and h f is greater than zero; therefore the state is classified into the free-flowing regime. In Fig. 3b, the density is medium so that the speed is equal to v f and h f is equal to zero; therefore, the state is classified into the critical regime. In Fig. 3c, the density is large so that the speed is less than v f ; therefore, the state is classified into the congested regime.
Fundamental Diagram
In general, the followings are considered as the traffic state variables of a rail transit system:
• train-flow q, • train-density k, • train-mean-speedv, • passenger-flow q p , • passenger-density k p , • passenger-mean-speedv p .
Among these, there are three independent variables: for example, the combination of q, k, and q p . This is because of the identities q = kv and q p = k pvp , andv =v p .
The equivalent cellular automaton models of Newell's car-following model (Daganzo, 2006) may represent the fixed block control as in the existing studies (e.g., Li et al., 2005;Kariyazaki et al., 2015).
Note that the mean speedv differs from the cruising speed v; the former takes the dwelling time at a station and cruising between stations into account, whereas the latter only considers the cruising time. Now suppose that the relation among the independent variables of the traffic state under every steady state is expressed using a function Q as q = Q(k, q p ).
Time t Space x Train m Train m − 1 Station i Station i + 1 τ δ H q p H/µ p + g b v l h f (a) Free-flowing regime: v = v f , h f > 0. Time t Space x Train m Train m − 1 Station i Station i + 1 τ δ H q p H/µ p + g b v l (b) Critical regime: v = v f , h f = 0. Time t Space x Train m Train m − 1 Station i Station i + 1 τ δ H q p H/µ p + g b v l (c) Congested regime: v < v f , h f = 0.
(4)
The function Q is regarded as an FD of the rail transit system. In fact, by assuming that the rail transit operation principle follows Eqs. (2) and (3), the FD function is analytically derived as
Q(k, q p ) = lk − q p /µ p g b + l/v f , if k < k * (q p ), − lδ (l − δ)g b + τ l (k − k * (q p )) + q * (q p ), if k ≥ k * (q p ),(5)
with
q * (q p ) = 1 − q p /µ p g b + δ/v f + τ , (6) k * (q p ) = − (l − δ)/v f − τ (g b + δ/v f + τ )µ p l q p + g b + l/v f (g b + δ/v f + τ )l ,(7)
where q * (q p ) and k * (q p ) represent train-flow and train-density, respectively, at a critical state with passenger-flow q p . For the derivation, see Appendix A. Although the FD equations (5)-(7) look complicated, they represent a simple relation: a piecewise linear (i.e., triangular) relation between q and k under fixed q p . See Fig. 4 for a numerical example of the FD which we will explain later.
Discussions
The FD is interpreted as a function that describes transit operation performance (train-flow q, headway H = 1/q, and mean-speedv = q/k) under a given train supply (train-density k) and passenger demand (passenger-flow q p ) for the given technical parameters of the transit system (µ p , g b , v f , τ, δ, l). Therefore, it can be considered as a similar concept to the macroscopic fundamental diagram (MFD) (Geroliminis and Daganzo, 2007;Daganzo, 2007), which describes a road network throughput under a given number of vehicles and technical parameters of the road network.
Numerical Example
First of all, for ease of understanding, we show a numerical example of the FD in Fig. 4. The parameter values are presented in Table 1. In the figure, the horizontal axis represents train-density k, the vertical axis represents train-flow q, and the plot color represents passenger-flow q p . The slope of the straight line from a traffic state to the origin represents the mean speedv of the state. For example, the figure can be read as follows. Suppose that the passenger demand per station is q p = 16000 (pax/h). If the number of trains in the transit system is given by the train-density k = 0.3 (train/km), then the resulting train traffic has a train-flow of q 15 (train/h) and a mean speed ofv 50 (km/h). This is the traffic state in the free-flowing regime. There is a congested state corresponding to a free-flowing state: for the aforementioned state with (q, k,v) fastest mean speed under the given passenger demand. The triangular q-k relation mentioned before is clearly shown in the figure; the "left edge" of the triangle corresponds to the free-flowing regime, the "top vertex" corresponds to the critical regime, and the "right edge" corresponds to the congested regime. By comparing the theoretical FD ( Fig. 4) with the actual data ( Fig. 1), some similarities can be found. The two features found in the actual data (as the passenger-flow increases, the traindensity increases; and as the passenger-flow increases, the train-speed and train-flow decrease) can be interpreted that the actual data are from a part of free-flow regime of the theoretical FD. Furthermore, in the high train-density regime in the Tokyo data (Fig. 1a), we observed a slight drop of passenger-flow; this might be a congested regime of the theoretical FD. From these results, we can say that the theoretical model explains the actual data to some extent.
Detailed Features of Fundamental Diagram
The FD has the following theoretical features which are analytically derived from Eq. (5). They can easily be found in the numerical example in Fig. 4.
As mentioned, the traffic state of a transit system is categorized into three regimes (free-flowing, critical, and congested), as in the standard traffic flow theory. Therefore, there is a critical train-density k * (q p ) for any given q p . Train traffic is in the free-flowing regime if k < k * (q p ), in the critical regime if k = k * (q p ), or in the congested regime otherwise. The congested regime can be considered as inefficient compared with the free-flowing regime, because the congested regime takes more time to transport the same volume of passengers. The critical regime is the most efficient in the sense that its travel time (i.e., 1/v, 1/v p ) and in-vehicle crowding (i.e., number of passengers per train, q p /q) are minimum under a given passenger demand. However, the critical regime requires more trains (i.e., higher train-density) than the free-flowing regime; therefore, it may not be the most efficient if the operation cost is taken into account.
Even in the critical regime, the mean speedv is inversely proportional to passenger demand q p . This means that travel time increases as passenger demand increases. In addition, the size of the feasible area of (q, k) narrows as q p increases. Thus, the operational flexibility of the transit system declines as the passenger demand increases.
Flow and density of trains in the critical regime satisfy the following relations:
q * (q p ) = l (l − δ)/v f − τ k * (q p ) − 1 (l − δ)/v f − τ .(8)
(Here, we have assumed (l − δ)/v f − τ = 0.) Therefore, the critical regime is represented as a straight line whose slope (l/[(l − δ)/v f − τ ]) is either positive or negative in the q-k plane. This implies a qualitative difference between transit systems. Specifically, if the slope is positive, a transit operation with constant train-density would transition from the free-flowing regime to the congested regime as passenger demand increases (Fig. 4). On the contrary, if the slope is negative, such an operation would transition from free-flowing to congested as passenger demand decreases. This seems paradoxical, but it is actually reasonable because the operational efficiency can be degraded if the number of trains is excessive compared to passenger demand. The FD describes an transit system's performance under a steady state operation as mentioned. Under the presence of well-designed adaptive control strategies, such as schedule-based and headwaybased control (Daganzo, 2009;Wada et al., 2012), the steady state is likely to be realized. This is because the aim of such adaptive control is usually to eliminate bunching-in other words, such control makes the operation steady. Therefore, it can be expected that the FD could be useful to describe average performance of actual transit system, which is usually not steady due to heterogeneity among passenger demand and train supply. This issue is numerically validated in Section 5.
Last but not least, it is worth mentioning that all parameters in the proposed model have an explicit physical meaning. Therefore, the parameter calibration required to approximate an actual transit system is relatively easy.
Relation to the Macroscopic Fundamental Diagram
The proposed FD resembles the MFD (Geroliminis and Daganzo, 2007;Daganzo, 2007) and its extensions (e.g., Geroliminis et al., 2014;Chiabaut, 2015) as mentioned. They are similar in the following sense. First, they both consider dynamic traffic. Second, they both describe the relations among macroscopic traffic state variables in which the traffic is not necessarily steady or homogeneous at the local scale (i.e., they use area-wide aggregations based on Edie's definition; see Appendix B). Third, they both have unimodal relations, meaning that there are free-flowing and congested regimes, where the former has higher performance than the latter; in addition, there is a critical regime where the throughput is maximized. Therefore, it is expected that existing approaches for MFD applications, such as modeling, control and the optimization of transport systems (e.g., Daganzo, 2007;Geroliminis and Levinson, 2009;Geroliminis et al., 2013;Fosgerau, 2015), are also suitable for the proposed transit FD.
However, there are substantial differences between the proposed FD and the existing MFD-like concepts. In comparison with the original MFD (Geroliminis and Daganzo, 2007;Daganzo, 2007) and its railway variant (Cuniasse et al., 2015), the proposed FD has an additional dimension, that is, passenger-flow. In comparison with the three-dimensional MFD of Geroliminis et al. (2014), which describes the relations among total traffic flow, car density, and bus density in a multi-modal traffic network, the proposed FD explicitly models the physical interaction among the three variables. In comparison with the passenger MFD of Chiabaut (2015), which describes the relation between passenger flow and passenger density when passengers can choose to travel by car or bus, in the proposed FD, passenger demand can degrade the performance (i.e., speed) of the vehicles because of the inclusion of the boarding time.
Dynamic Model Based on Fundamental Diagram
Recall that the proposed FD describes the relationship among traffic variables under the steady state. It means that the behavior of a dynamical system in which demand and supply change over time is not described solely by the FD. This feature is the same as in the road traffic FD and MFDs. In this section, we formulate a model of urban rail transit operation where the demand (i.e., passenger-flow) and supply (i.e., train-density) change dynamically. In this proposed model, individual train and passenger trajectories are not explicitly described; therefore, the model is called macroscopic.
The proposed model is based on an exit-flow model (Merchant and Nemhauser, 1978;Carey and McCartney, 2004) in which the proposed FD is employed as the exit-flow function. Specifically, the transit system is considered as an input-output system, as illustrated in Fig. 5. The exit-flow modeling approach is often employed for area-wide traffic approximations and analysis using MFDs, such as optimal control to avoid congestion (Daganzo, 2007) and analyses of user equilibrium and social optimum in morning commute problems (Geroliminis and Levinson, 2009). The advantage of this approach is that it would be possible to conduct mathematically tractable analysis of dynamic, large-scale, and complex transportation systems, where the detailed traffic dynamics are difficult to model in a tractable manner-this is the case for transit operations.
Railway system internal average train-flow: Q(k(t) a p (t)) dynamics of internal average train-density:
dk(t) dt = a(t) − Q(k(t), a p (t))
Formulation
Let a(t) be the inflow of trains to the transit system, a p (t) be the inflow of passengers, d(t) be the outflow of trains from the transit system, and d p (t) be the outflow of passengers, on time t respectively. We set the initial time to be 0. Let A(t), A p (t), D(t), and D p (t) be the cumulative values of a(t), a p (t), d(t), and d p (t), respectively (e.g., A(t) = t 0 a(s)ds). Let T (t) be the travel time of a train that entered the system at time t, and let its initial value T (0) be given by the free-flow travel time under q = a(0) and q p = a p (0). To simplify the formulation, the trip length of the passengers is assumed to be equal to that of the trains. This means that T (·) is the travel time of both the trains and the passengers. These functions are interpreted as follows:
• a(t): trains' departure rate from their origin station at time t.
• a p (t): passengers' arrival rate at the platform of their origin station at time t.
• d(t): trains' arrival rate at their final destination station at time t.
• d p (t): passengers' arrival rate at their destination station at time t.
• T (t): travel time of a train and passengers from origin (departs at time t) to destination. Note that the arrival time at the destination is t + T (t).
Therefore, in reality, a(·) and a p (·) will be determined by the transit operation plan and passenger departure time choice, respectively. Then, d(·), d p (·), and T (·) are endogenously determined through the operational dynamics.
In accordance with exit-flow modeling, the train traffic is modeled as follows. First, the exit flow d(t) is assumed to be
d(t) = Q(k(t), a p (t))(9)
where the FD function Q(·) is considered to be an exit-flow function. This means that the dynamics of the transit system are modeled by taking the conservation of trains into account as follows:
L dk(t) dt = a(t) − Q(k(t), a p (t)),(10)
where L represents the length of the transit route. This exit-flow model has been employed in several studies to represent the macroscopic behavior of a transportation system (e.g., Merchant and Nemhauser, 1978;Carey and McCartney, 2004;Daganzo, 2007). Note that the average train-density k(t) is defined as
k(t) = A(t) − D(t) L ,(11)
which is consistent with Eq. (10). Based on above functions and equations, d(t) and D(t) are sequentially computed-in other words, the train traffic is computed using the initial and boundary conditions and the exit-flow model based on the FD. The passenger traffic is derived as follows. By the definition of the travel time of trains,
A(t) = D(t + T (t))(12)
holds. As A(t) and D(t) have already been obtained, the travel time T (t) such that Eq. (12) holds is computed. Then, D p (t) and d p (t) are computed from the definition of the travel time of passengers, which is also T (t):
A p (t) = D p (t + T (t)).(13)
This assumption is reasonable if the average trip length is shared by trains and passengers. If they are different, a modification such as Tp(t) = T (t)/λ, where λ is the ratio of average trip length of the passengers to that of the trains, would be possible.
If np is considered as the sum of the number of passengers who are boarding and alighting (as mentioned in note 3), we can simply define d(t) to be equal to Q(k(t), ap(t) + dp(t)). Such a model is also computable using a similar procedure.
Discussion
The proposed macroscopic model computes train out-flow d(t) and passenger out-flow d p (t) based on the FD function Q(·), the initial and boundary conditions a(t), a p (t), and T (0). The notable feature of the model is its high tractability and computational efficiency, as it is based on an exit-flow model. Therefore, we expect the proposed model to be useful for analyzing various management strategies for transit systems (e.g., dynamic pricing during the morning commute).
It is reasonable to expect that the proposed model can accurately approximate the macroscopic behavior of a transit operation with high-frequency operation (i.e., small time-headway) under moderate changes in demand and/or supply. This is because exit-flow models are reasonably approximate a dynamical system's behavior when the changes in inflow are moderate compared with the relaxation time of the system. In the next section, the quantitative accuracy of the model is validated through numerical experiments.
Validation of the Macroscopic Model
In this section, we validate the quantitative accuracy of the macroscopic model by comparing its results with that of the microscopic model (i.e., Eqs. (2) and (3)).
Simulation Setting
The parameter values of the transit operation are listed in Table 1 for both the microscopic and macroscopic models. The railroad is considered to be a one-way corridor. The stations are equally spaced at intervals of l, and there are a total of 10 stations. Trains enter the railroad with flow a(t); in the microscopic model, a discrete train enters the railroad from the upstream boundary station if A(t) (i.e., integer part of A(t)) is incremented. In the microscopic model, trains leave the railroad from the downstream boundary station without any restrictions, other than the passenger boarding and minimum headway clearance. Passengers arrive at each station with flow a p (t).
The functions a(t) and a p (t) are exogenously determined to mimic morning rush hours, with each having a peak at t = 2. The flow before the peak time increases monotonically, whereas the flow after the peak time decreases monotonically-in other words, the so-called S-shaped A(t) and A p (t) (c.f., Fig. 7) are considered. The parameters of these functions are the minimum train supply a min , the maximum train supply a max , the minimum passenger demand a min p , and the maximum passenger demand a max p . The functional forms are described in Appendix C. The simulation duration is set to 4 h for the baseline scenario in Section 5.2.1 and to 8 h for the sensitivity analysis in Section 5.2.2 (the reason will be explained later).
The microscopic model without any control is asymptotically unstable, as proven by Wada et al. (2012); this means that time-varying demand and supply always cause train bunching, making the experiment unrealistic and useless. Therefore, the headway-based control scheme proposed by Wada et al. (2012) is implemented in the microscopic model to prevent bunching and stabilize the operation. This scheme has two control measures: holding (i.e., extending the dwell time) and an increase of free-flow speed, similar to Daganzo (2009). The former is activated by a train if its following train is delayed, and is represented as an increase in g b in the microscopic model. The latter is activated by a train if it is delayed, and is represented as an increase in v f up to a maximum allowable speed v max . In this experiment, v max is set to 80 km/h and v f is 70 km/h. This control scheme can be considered realistic and reasonable, as similar operations are executed in practice. See Appendix D for further
Results
First, to examine how well the proposed model reproduces the behavior of the transit system under time-varying conditions, the results for the baseline scenario are presented in Section 5.2.1. Then, a sensitivity analysis of the demand/supply conditions is conduced and applicable ranges of the proposed model are investigated in Section 5.2.2.
Baseline scenario
The baseline scenario with parameter values a min = 10 (train/h), a max = 15 (train/h), a min p = 0.1µ p (pax/h), and a max p = 0.5µ p (pax/h) is investigated first. A solution of the microscopic model is shown in Fig. 6 as a time-space diagram. The colored curves represent the trajectories of each train traveling in the upward direction while stopping at every station. Around the peak time period (t = 2), train congestion occurs; namely, some of the trains stop occasionally between stations in order to maintain the safety interval. The congestion is caused by heavy passenger demand; therefore, the situation during rush hour is reproduced. The result given by the macroscopic model is shown in Fig. 7 as cumulative plots. Fig. 7a shows the cumulative curves for the trains, where the blue curve represents the inflow A and the red curve represents the outflow D. Fig. 7b shows those of passengers in the same manner. Congestion and delay are observed around the peak period (it is more remarkable in the passenger traffic). For example, during the peak time period, d p (t) is less than a p (t) and a p (t ), where t is time such that t = t + T (t ). This means that the throughput of the transit system is reduced by the heavy passenger demand. Consequently, T (t) is greater during peak hours than in off-peak periods such as T (0), meaning that delays occur due to the congestion.
The macroscopic and microscopic models are compared in terms of the cumulative number of trains in Fig. 8. In the figure, the solid curves denote the macroscopic model and the dots denote the microscopic model. It is clear that D in the macroscopic model follows that of the microscopic model fairly precisely. For example, the congestion and delay during the peak time period are captured very well. However, there is a slight bias: the macroscopic model gives a slightly shorter travel time. This is mainly due to the large-scale unsteady state (i.e., train bunching) generated in the microscopic model; the delay caused by such large-scale bunching cannot be recovered by the microscopic model under the implemented headway-based control scheme (for details, see Appendix D). It means that if the control is schedule-based, the bias could be reduced.
Sensitivity analysis of the demand/supply conditions
The accuracy of the macroscopic model regarding the dynamic patterns of demand/supply is now examined. This is worth investigating it quantitatively, because it is qualitatively clear that the exit-flow model is valid if the speed of demand/supply changes is "sufficiently" small as discussed in Section 4.2. Specifically, the sensitivity of the peak passenger demand a max p and train supply a max is evaluated by assigning various values to these parameters. The simulation duration is set to 8 h to take the residual delay after t = 4 (h) in some scenarios into account. The other parameters are the same as in the baseline scenario.
The results are summarized in Fig. 9. It shows the relative difference in total travel time (TTT) of trains between the microscopic and macroscopic models for various peak train supply a max and peak passenger demand a max p . The minimum train supply and passenger demand are set as a min = 10 (train/h) and a min p = 0.1µ p = 6000 (pax/h). The relative difference can be considered as an error index of the macroscopic model. The negative values indicate that TTT of the macroscopic model is smaller. According to the results in Fig. 9, the accuracy of the macroscopic model is high when the maximum passenger demand is not extremely large. This is an expected result, as the speed of demand change is slow in these cases. TTT given by the macroscopic model is almost always less than that of the microscopic model; this might be due to the aforementioned inconsistency between the steady state assumption of the macroscopic model and headway-based control of the microscopic model.
The relative error increases suddenly when the demand exceeds a certain value, around 20000-22000. This sudden change is a result of extraordinary large-scale train bunching in the microscopic model. This bunching often occurs in cases with excessive passenger demand, such as a max p > µ p /2. Such demand can be considered as unrealistically excessive, as the dwell time of a train at a station is longer than the cruising time between adjacent stations in such situations; this usually does not occur even in rush hours.
As for the sensitivity of the train supply a(·), there is a weak tendency for faster variations in supply to cause larger errors. This is also an expected result. In any case, the error is small.
From these results, we conclude that the proposed model is fairly accurate under ordinary passenger demand, although it is not able to reproduce extraordinary and unrealistic situations with excessive train bunching. This might be acceptable for representing transit systems during normal rush hours.
Conclusion
The main contribution of this paper is that it analytically derived a closed-form expression of FD of rail transit systems based on microscopic operation principles. The FD determines operation performance of rail transit systems (i.e., flow, headway, and mean speed) based on supply of trains and passenger demand. Furthermore, this paper proposed an efficient, macroscopic dynamic assignment method based on the FD, and numerically showed that the method is fairly accurate under realistic situations.
Specifically, the following three models of an urban rail transit system have been analyzed in this paper:
• Microscopic model: A model describing the trajectories of individual trains and passengers based on Newell's car-following model and passenger boarding model. This is represented in Eqs. (2) and (3), and is solved using simulations. • Fundamental diagram: An exact relationship among train-flow, train-density, and passenger-flow in the microscopic model under a steady state. This is represented in Eqs. (4)-(7). It is a closed-form equation. • Macroscopic model: A model describing train and passenger traffic using an exit-flow model whose exit-flow function is the FD. This is represented in Eqs. (9), (11), (12), and (13), and is solved using simple simulations.
The FD and macroscopic model are the original contributions of this study, whereas the microscopic model was proposed by Wada et al. (2012). The FD itself implies several insights on transit system, such as relation between mean speed of the system and passenger demand. In addition, according to the results of the numerical experiment, the macroscopic model can reproduce the behavior of the microscopic model accurately, except for cases with unrealistically excessive demands. Because of the simplicity, mathematical tractability, and good approximation accuracy of the proposed FD and macroscopic model in ordinary situations, it can be expected that they will contribute for obtaining general policy implications on management strategies of rail transit systems, such as pricing and control for morning commute problems.
Following future works are considerable. First, rigorous empirical validation on the existence of the FD is required. In fact, several preliminarily results on it have been reported (Fukuda et al., 2019;Zhang and Wada, 2019) as shown in Fig. 1. Second, as an application of the FD, analysis of operation and demand management for transit systems is important. For example, the morning commute problem (Zhang et al., 2021) has been analyzed, and its departure time choice equilibrium and optimal pricing have been derived.
Appendix A Derivation of FD
This appendix describes derivation of the FD expressed in Eqs. (5)-(7). Consider a looped rail transit system under steady state operation. Let L be the length of the railroad, S be the number of the stations, M be the number of trains, H be the time-headway of the operation, t b be the dwelling time of a train at a station, t c be the cruising time of a train between adjacent stations, and q p be the passenger demand flow rate per station. Note that the distance between adjacent stations l is L/S and the number of passengers boarding a train at each station is q p H.
The time-headway of the operation is derived as follows. The round trip time of a train in the looped railroad is S(t b + t c ), and M trains pass the station during that time. Then, the identities N H = S(t b + t c ) and
H = g b + t c M/S − q p /µ p (A.1)
hold. Moreover, by the definition of headway and Newell's car-following rule, the time-headway H must satisfy
H = t b + δ + vτ v + h f . (A.2)
This reduces to
H = g b + δ/v + τ 1 − q p /µ p + h f . (A.3)
The q-k relation in a free-flowing regime is derived as follows. As the train-flow is 1/H and train-density is M/L by definition, Eq. (A.1) is transformed to
q(k) = kl − q p /µ p g b + l/v f . (A.4)
The train-flow and train-density under a critical state, (q * , k * ), are derived as follows. By substituting v = v f and h f = 0 into Eq. (A.3) and using the identity q = kv, we obtain
q * = 1 − q p /µ p g b + δ/v f + τ , (A.5) k * = k 0 + (1 − q p /µ p )(g b + l/v f ) (g b + δ/v f + τ )l , (A.6)
where k 0 is the minimum train-density where the train-flow is zero, namely, k 0 = q p /(µ p l).
The q-k relation in a congested regime is derived as follows. First, the k-v relation in a congested regime is easily derived from the q-v relation (A.3) with h f = 0 and the identity q = kv:
k(v) = k 0 + (1 − q p /µ p )(g b + l/v) (g b + δ/v + τ )l . (A.7)
Now, consider dq/dk, which is identical to (dq/dv) · (dv/dk). This is derived as
dq dk = lδ (δ − l)g b − τ l , (A.8)
which is constant and negative; therefore, the q-k relation is linear in a congested regime. Then, recalling that the linear q-k curve passes the point (q * , k * ) with a slope of dq/dk, the q-k relation in a congested regime is derived as q(k) = lδ (δ − l)g b − τ l k + q 0 (A.9) with q 0 = q * − dq dk · k * .
Appendix B Consistency of the FD and Edie's generalized definition of traffic state
It is noteworthy that Eqs. (4) and (5) are consistent with Edie's generalized definition (Edie, 1963) of traffic states; because from this consistency we can confirm that the FD is consistent with the fundamental definition of traffic. For steady-state transit operation, Edie's traffic state is derived as This means that, in the event of a delay, the train tries to catch up by increasing its cruising speed up to the maximum allowable speed v max (which implies that the free-flow speed v f is a "buffered" maximum speed). Meanwhile, the proposed train operation model in this study does not have a schedule-it is a frequency-based operation. Therefore, in this study, the scheduled headway in the scheme (T m,i − T m−1,i ) is approximated by the planned frequency (1/a(t m (i))). Thus, we set α = 1 and substitute E m (i) with µ p t m (i) − t m−1 (i) − 1/a(t m (i)) .
(D.4)
The stationary state of the operational dynamics under the original scheme is basically identical to the steady state defined in Section 3.1. There may be small difference in the congested regime because of the operation scheme; however, this will not be problematic since heavily congested regime will not occur. In the case of α < 1, the scheme makes the train operation asymptotically stable, meaning that the operation schedule is robust to small disturbances. In the case of α = 1, the scheme prevents the propagation and amplification of delay, but does not recover the original schedule (the small 'shift' found in Fig. 8 is due to α = 1). Note that these control measures do not interrupt passenger boarding or violate the safety clearance between trains, meaning that most of the fundamental assumptions of the proposed FD are satisfied.
Figure 1 :
1Observed 3-dimensional relations among train-flow, train-density, and passenger-flow. The
Figure 2 :
2Illustration of the microscopic model of rail transit system.
Figure 3 :
3Time-space diagrams of rail transit system under steady states.
Figure 4 :
4Numerical example of the FD.
Figure 5 :
5travel time: T (t) train in-flow: a(t) its cumulative:A(t) passenger in-flow: a p (t) its cumulative: A p (t)train out-flow: Q(k(t), a p (t)) its cumulative: D(t) passenger out-flow: d p (t) (determined by the model) its cumulative: D p (t) Railway system as an input-output system.
Figure 6 :
6Result of the microscopic model in the baseline scenario.
Figure 7 :
7Result of the macroscopic model in the baseline scenario. details of the control scheme. Note that the boundary conditions are the trajectory of the first train x 0 (t) ∀t, the initial position of all the trains x m (0) ∀m (this is converted to the departure time of all the trains from the most upstream station x m (t 0 m ) ∀m where t 0 m denotes the departure time of train m), and the passenger demand to each station q p .
Figure 8 :
8Comparison between the macroscopic and microscopic models in the baseline scenario.
Figure 9 :
9Comparison between the microscopic and macroscopic models under different demand/supply conditions.
are constructed based on Eqs. (A.4), (A.5), (A.6), (A.9), and (A.10).
are derived by applying Edie's definition to the "minimum component of the time-space diagram" of the steady state, which is a parallelogram-shaped area in Fig. 3 whose vertexes are time-space points of (i) train m departs from station i, (ii) train m arrives at station i + 1, (iii) train m − 1 arrives at station i + 1, and (iv) train m − 1 departs from station i. One can easily confirm that Eqs. (B.1)-(B.3) satisfy the FD equation. In fact, the FD equation is also derived from Eqs. (B.1)-(B.3) and the constraint (A.3) induced by Newell's car-following model. min l/v f − l/v max , max{0, E m (i) − g b } .
(15 veh/h, 0.3 veh/km, 50 km/h), the corresponding congested state is (15 veh/h, 0.55 veh/km, 27 km/h). The critical state under q p = 16000 (pax/h) is (22 veh/h, 0.42 veh/km, 52 km/h). Notice that this state has the
Table 1 :
1Parameters of the numerical example.parameter value
u 70 km/h
τ 1/70 h
δ 1 km
µp 36000 pax/h
g b 10/3600 h
l 3 km
Appendix C S-shaped supply and demand functionsThe train supply and passenger demand in the experiments are given by the following functions:Both functions have a minimum value at t = 0 and t ≥ 4 and a minimum value at t = 2, and change linearly in between.Appendix D Adaptive control scheme in the microscopic modelThis appendix briefly explains the adaptive control scheme for preventing train bunching, proposed byWada et al. (2012). This scheme consists of two control measures: holding at a station and increasing the maximum speed during cruising. First, the scheme modifies the buffer time for dwelling (originally defined as g b in Eq.(2)) of train m at station i towhere ε m (i) ≡ t m (i) − T m,i represents the delay, t m (i) represents the time at which train m arrives at station i, T m,i represents the scheduled time (i.e., without delay) at which train m should arrive at station i, and α ∈ [0, 1] is a weighting parameter. This scheme represents a typical holding control strategy, similar to the bunching prevention method ofDaganzo (2009), which extends the dwelling time of a vehicle if the headway to the preceding vehicle is too small and vice versa. Second, the scheme modifies the free-flow cruising speed v f such that the interstation travel time is reduced by
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| []
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| [
"Euclid Collaboration ",
"Andrea Moneti [email protected] ",
"H J Mccracken ",
"M Shuntov ",
"O B Kauffmann ",
"P Capak ",
"I Davidzon ",
"O Ilbert ",
"C Scarlata ",
"S Toft ",
"J Weaver ",
"R Chary ",
"J Cuby ",
"A L Faisst ",
"D C Masters ",
"C Mcpartland ",
"B Mobasher ",
"D B Sanders ",
"R Scaramella ",
"D Stern ",
"I Szapudi ",
"H Teplitz ",
"L Zalesky ",
"A Amara ",
"N Auricchio ",
"C Bodendorf ",
"D "
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| []
| [
"J. Garcia-Bellido"
]
| We present a new infrared survey covering the three Euclid deep fields and four other Euclid calibration fields using Spitzer's Infrared Array Camera (IRAC). We have combined these new observations with all relevant IRAC archival data of these fields in order to produce the deepest possible mosaics of these regions. In total, these observations represent nearly 11 % of the total Spitzer mission time. The resulting mosaics cover a total of approximately 71.5 deg 2 in the 3.6 and 4.5 µm bands, and approximately 21.8 deg 2 in the 5.8 and 8 µm bands. They reach at least 24 AB magnitude (measured to 1σ, in a 2 . 5 aperture) in the 3.6 µm band and up to ∼ 5 mag deeper in the deepest regions. The astrometry is tied to the Gaia astrometric reference system, and the typical astrometric uncertainty for sources with 16 < [3.6] < 19 is 0 . 15. The photometric calibration is in excellent agreement with previous WISE measurements. We have extracted source number counts from the 3.6 µm band mosaics and they are in excellent agreement with previous measurements. Given that the Spitzer Space Telescope has now been decommissioned these mosaics are likely to be the definitive reduction of these IRAC data. This survey therefore represents an essential first step in assembling multi-wavelength data on the Euclid deep fields which are set to become some of the premier fields for extragalactic astronomy in the 2020s. | 10.1051/0004-6361/202142361 | [
"https://arxiv.org/pdf/2110.13928v1.pdf"
]
| 239,998,315 | 2110.13928 | 95b6ac8081984ccf14519f317d9a657ff4ccae80 |
Hudelot 1 , K. Jahnke 58 , S. Kermiche 59 , A. Kiessling 14 , M. Kilbinger 48 , T. Kitching 41 , R. Kohley 39 , M. Kümmel 54 , M. Kunz 60 , H. Kurki-Suonio 61 , S. Ligori 18 , P.B. Lilje 55 , I. Lloro 62 , E. Maiorano 16 , O. Mansutti 49 , O. Marggraf 63 , K. Markovic 14 , F. Marulli 16
October 28, 2021
Euclid Collaboration
Andrea Moneti [email protected]
H J Mccracken
M Shuntov
O B Kauffmann
P Capak
I Davidzon
O Ilbert
C Scarlata
S Toft
J Weaver
R Chary
J Cuby
A L Faisst
D C Masters
C Mcpartland
B Mobasher
D B Sanders
R Scaramella
D Stern
I Szapudi
H Teplitz
L Zalesky
A Amara
N Auricchio
C Bodendorf
D
Hudelot 1 , K. Jahnke 58 , S. Kermiche 59 , A. Kiessling 14 , M. Kilbinger 48 , T. Kitching 41 , R. Kohley 39 , M. Kümmel 54 , M. Kunz 60 , H. Kurki-Suonio 61 , S. Ligori 18 , P.B. Lilje 55 , I. Lloro 62 , E. Maiorano 16 , O. Mansutti 49 , O. Marggraf 63 , K. Markovic 14 , F. Marulli 16
J. Garcia-Bellido
57October 28, 2021Released on TBD / Accepted date TBDAstronomy & Astrophysics manuscript no. Cosmic_dawn_spitzer (Affiliations can be found after the references)cosmology: observations -cosmology: large scale structure of universe -cosmology: dark matter -galaxies: formation - galaxies: evolution -surveys
We present a new infrared survey covering the three Euclid deep fields and four other Euclid calibration fields using Spitzer's Infrared Array Camera (IRAC). We have combined these new observations with all relevant IRAC archival data of these fields in order to produce the deepest possible mosaics of these regions. In total, these observations represent nearly 11 % of the total Spitzer mission time. The resulting mosaics cover a total of approximately 71.5 deg 2 in the 3.6 and 4.5 µm bands, and approximately 21.8 deg 2 in the 5.8 and 8 µm bands. They reach at least 24 AB magnitude (measured to 1σ, in a 2 . 5 aperture) in the 3.6 µm band and up to ∼ 5 mag deeper in the deepest regions. The astrometry is tied to the Gaia astrometric reference system, and the typical astrometric uncertainty for sources with 16 < [3.6] < 19 is 0 . 15. The photometric calibration is in excellent agreement with previous WISE measurements. We have extracted source number counts from the 3.6 µm band mosaics and they are in excellent agreement with previous measurements. Given that the Spitzer Space Telescope has now been decommissioned these mosaics are likely to be the definitive reduction of these IRAC data. This survey therefore represents an essential first step in assembling multi-wavelength data on the Euclid deep fields which are set to become some of the premier fields for extragalactic astronomy in the 2020s.
Introduction
The Euclid mission will survey 15 000 deg 2 of the extragalactic sky to investigate the nature of dark energy and dark matter, and Article number, page 1 of 15 arXiv:2110.13928v1 [astro-ph.GA] 26 Oct 2021 to study the formation and evolution of galaxies (Laureijs et al. 2011). To this end Euclid will obtain high resolution and high signal-to-noise imaging of a billion galaxies in a broad optical filter to measure their shapes and in three near-infrared (NIR) filters to measure their colours. It will also obtain high signal-tonoise NIR spectroscopy of about thirty million of these galaxies to measure abundances and redshifts. Additionally, photometric redshifts will be determined by combining the Euclid data with optical photometry from external surveys.
To reach the required precision on cosmological parameters and satisfy the stringent mission requirements on completeness, spectroscopic purity and shape noise bias, Euclid must also obtain observations with 40 times longer exposure per pixel than the main survey over regions covering at least 40 deg 2 . To this end, three 'deep' fields have been selected by the Euclid Consortium. They are described in detail in Scaramella et al. (2021) and we give just a very brief description here. They are: (1) the Euclid Deep Field North (EDF-N), a roughly circular, 10 deg 2 region centred on the well-studied north Ecliptic pole, (2) the Euclid Deep Field Fornax (EDF-F), also roughly circular and of 10 deg 2 , centred on the Chandra Deep Field South and including the GOODS-S (Giavalisco et al. 2004) and the Hubble Ultra Deep Field (Beckwith et al. 2006), and (3) the Euclid Deep Field South (EDF-S), a pill-shaped area of 20 deg 2 with no previous dedicated observations. In addition to these three deep fields Euclid will observe several fields for the calibration of photometric redshifts (photo-z). These fields need to be observed to a level 5 times deeper than the main survey and they are centred on some of the best studied extragalactic survey fields that already have extensive spectroscopic data: (1) the COSMOS field (2) the Extended Groth Strip (EGS), (3) the Hubble Deep Field North (HDF, also GOODS-N), and (4) the XMM-Large Scale Structure Survey field, which includes the Subaru XMM Deep Survey field (SXDS), and VIMOS VLT Deep Survey (VVDS) 1 .
While Euclid will observe these fields primarily for calibration purposes, those observations will provide an unprecedented data set to study galaxies to faint magnitudes and high redshifts. The survey efficiency of Euclid in the NIR bands is orders of magnitudes greater than that of ground-based telescopes (e.g., VISTA). The Euclid deep fields alone will be 30 times larger and one magnitude deeper than the latest UltraVISTA data release covering the COSMOS field, and will reach a depth of 26 mag in the Y, J, H filters (5 σ). In addition, Euclid carries a wide-field near-infrared grism spectrograph, the Near Infrared Spectrometer and Photometer (NISP), covering the 0.92 < λ < 1.85 µm region, which will provide multiple spectra at numerous grism orientations for more than one million sources to a line flux limit similar to 3D-HST (Brammer et al. 2012) and over an area 200 times larger than the COSMOS field (depending on the scheduling of the blue grism observations). The observations of the deep fields will result in the most complete and deepest spectroscopic coverage produced by Euclid. Such a spectroscopic data set will be unique for the reconstruction of the galaxy environment at cosmic noon and for measuring the star formation rate from the Hα emission line intensity.
The deep and wide NIR data from Euclid are also ideal for detecting significant numbers of high-redshift (7 < z < 10) galaxies, as the Lyman α line is redshifted out of the optical into the NIR. However, in order to distinguish galaxy candidates from stars (primarily brown dwarfs), faint Balmer-break galaxies, and dusty star-forming galaxies at lower redshifts which all can have similar NIR magnitudes and colours, deep optical and mid-infrared (MIR) data are also needed (Bouwens et al. 2019;Bridge et al. 2019;Bowler et al. 2020).
The Cosmic Dawn Survey (Toft et al., in prep) aims to obtain uniform, multi-wavelength imaging of the Euclid deep and calibration fields to limits matching the Euclid data for characterisation of high-redshift galaxies. The optical data will be provided by the Hawaii-Two-0 Subaru telescope/Hyper-SuprimeCam (HSC) survey (McPartland et al., in prep.) for the EDF-N and EDF-F and likely by the Vera C. Rubin Observatory for EDF-S and EDF-F. For the COSMOS and SXDS fields optical data are provided by the Subaru HSC Strategic program (HSC-SSP Aihara et al. 2011).
In this paper, we present the Spitzer Space Telescope (Werner et al. 2004) component of the Cosmic Dawn Survey, consisting primarily of 3.6 and 4.5 µm observations of the three deep fields and parts of the calibration fields acquired with Spitzer's IRAC camera (Fazio et al. 2004). Two dedicated programs were submitted for this purpose: the Euclid/WFIRST Spitzer Legacy Survey (SLS, requesting 5 286 h, PI: Capak) covering the EDF-N and EDF-F fields, and the EDF-S survey (requesting 687 h, PI: Scarlata). These programmes were established based on the Euclid plans for the deep fields that were available at the time. All the fields had been observed, at least in part, before our new observations, and we processed our new data together with all relevant archival IRAC data, thus including data obtained during the cryogenic mission, i.e., data at 5.8 and 8.0 µm. In this way we strive to produce the deepest possible MIR images (mosaics) of these fields to date. A significant improvement in our processing is that our pipeline ties the astrometry to the Gaia reference system which, given its higher precision, will greatly facilitate cross-identification with other data, which will of course also have to be tied to Gaia.
In addition to being essential for the identification of highredshift galaxies, MIR data are crucial to reveal the stellar mass content of the high-redshift Universe (which is outside the scope of the Euclid core science). The Euclid data alone are not sufficient to characterise the stellar masses at z > 3.5, as the Balmer break is redshifted out of the reddest band of the NISP. Without MIR data, the interpretation of spectral energy distributions (SEDs) would rely on rest-frame ultraviolet emission which is strongly affected by dust attenuation and dominated by stellar light of new-born stars. Therefore, integrated quantities like the stellar mass would be highly unreliable (Bell & de Jong 2001). Moreover, photometric redshifts would be prone to catastrophic failures resulting from the mis-identification of the Lyman and Balmer breaks (e.g. Le Fèvre et al. 2015;Kauffmann et al. 2020). In summary, Spitzer/IRAC data are crucial for identifying the most distant objects (e.g. Bridge et al. 2019), for improving the accuracy of their photometric redshifts and for deriving their physical properties such as stellar masses, dust content, age, and star-formation rate from population synthesis models (e.g. Pérez-González et al. 2008;Caputi et al. 2015;Davidzon et al. 2017). The build-up of stellar mass, especially when confronted with the amount of matter residing in dark matter halos at high redshifts can be a highly discriminating test for galaxy formation models (Legrand et al. 2019). Extrapolation of recent work in the COSMOS field (Bowler et al. 2020) suggests that hundreds of the rarest, brightest z > 7 galaxies are expected to be discovered in the Euclid deep fields. These provide unique constraints on cosmic reionisation, as the brightest galaxies form in the highest density regions of the Universe which are expected to be the sites of the first generation of stars and galaxies, and thus of reionisation bubbles (Trac et al. 2008).
The layout of this paper is as follows: Sect. 2 describes the observations, Sect. 3 presents our data processing techniques, and Sect. 4 compares our results to previous ones.
Observations
All observations described here were made with IRAC. In brief, IRAC is a four-channel array camera on the Spitzer Space Telescope, observing simultaneously four fields slightly separated on the sky at 3.6, 4.5, 5.8, and 8.0 µm, known as channels 1-4, respectively. Spitzer science observations began in August 2003 but observations in channels 3 and 4 ceased once the on-board cryogen was exhausted (May 15, 2009). During the following 'warm mission' phase, channels 1 and 2 continued to operate until the end of operations in late January 2020, albeit with somewhat lower but still comparable performance. The earliest observations presented here are archival observations that were obtained in September 2003; the observations of the dedicated Capak program began in 2017 and the ones of the dedicated Scarlata program began in 2019. The dedicated observations continued until January 2020, shortly before the shutdown of the satellite. Figure 1 shows a histogram of the integration time accumulated in bins of 30 days over the observing period. These observations account for almost 1.5 million frames, a total integration time of 34 000 hr, all channels combined, and a total ontarget time, omitting overheads, of just over 15 600 hr, or nearly 1.8 yr, which is approximately 11 % of the Spitzer Space Telescope mission time.
For our dedicated observations of EDF-N, EDF-S and EDF-F we adopted a consistent observing strategy that comprises blocks of 3 × 3 maps with a step size of 310 , a large three-point dither pattern and four repeats per position. Each block covers a 15 .1 × 15 .1 region with a coverage of 3 × 4 × 100 s exposures per pixel. The block centres are offset between passes in order to ensure uniform coverage and enable self-calibration. Each block forms an AOR, or Astronomical Observation Request, in IRAC jargon. All other data included in our processing is archival data. It was obtained with a variety of observing strategies which we did not investigate in detail and which we do not attempt to summarise here. In Appendix C we list the Program IDs of all the observations processed; in bold the ones of our dedicated observations. The combination of the archival data with our own dedicated data produces a spatially variable depth in most fields; this is discussed further in Sect. 4.
A total of 292 IRAC observing programs are used in this work. Table 1 lists the ten largest programs in terms in terms of observing time together with the PI of the program, the field concerned and program's total integration time.
All observations are summarised in Table 2 which gives, for each field and channel, the number of frames (Data Collection Events or DCEs in IRAC terminology) used to produce the mosaics (note that this can be lower than the number of frames downloaded as some were discarded, see Sect. 3) together with the total observing time. For channels 1 and 2, on the left side of the table, the information is subdivided into the cryogenic part and the warm part of the mission.
Processing
Pre-processing and calibration
Processing begins with the Level 1 data products generated by the Spitzer Science Center via their 'Basic Calibrated Data' pipeline (Lowrance et al. 2016), which were downloaded from the NASA/IPAC Infrared Science Archive (IRSA 2 ). They have had all well-understood instrumental signatures removed, have been flux-calibrated in units of MJy sr −1 , and are delivered with an uncertainty image and a mask image; they are described in detail in the IRAC Instrument Handbook 3 . More precisely, we begin from the 'corrected basic calibration data' products, which have file extensions .cbcd for the image, .cbunc for the uncertainty, and .bimsk for the mask. The files are grouped by AORs, namely sets of a few to several hundred DCEs obtained sequentially. All frames are 256 × 256 pixels, the pixels are 1 . 2 wide, and the image file header contains the photometric solution and an initial astrometric solution.
The processing is done region by region. A first pass over the files is used to check the headers for completeness and to discard a few incomplete AORs, which accounts for most of the differences in the number of frames listed in Table 2 between channels 1 and 2, or 3 and 4. This is followed by the correction of the 'first frame' bias effect 4 . Next, the positions and magnitudes of WISE ( (Wright et al. 2010), (Mainzer et al. 2011)) and Gaia DR2 (Gaia Collaboration et al. 2018) sources falling within the field are downloaded. The Gaia sources are first 'projected' to their location at the time of the observations using the Gaia proper motions. Next they are identified on each IRAC frame, their observed fluxes and positions determined in each frame using the APEX software (the point-source extractor in MOPEX 5 ) in forced-photometry mode, and the positions are used to update the astrometric solution of each frame. There are typically 30-40 Gaia DR2 sources available for each frame. In channels 1 and 2 most of them are detected and used for the astrometric correction. In the longer-wavelength channels 3 and 4 only a few sources in total are detected and usable but that is still sufficient to determine an astrometric solution with negligible distortion as shown in Sect. 4.2.
An attempt was made to subtract bright stars in order to recover faint sources in their wings. For each AOR a model star built from the template PSFs described in the IRAC Instrument Handbook 6 (see Fig. 4.9 there) is scaled to the median of the fluxes of the star measured in that AOR, and is subtracted from each frame (of the AOR). Different templates are available for each filter and separately for the cryogenic and the warm missions. While this procedure worked quite well for moderately bright stars (which are of course the vast majority and which represent only a small loss in area), it introduced significant artefacts around the (few) very bright stars in the final mosaics. These artefacts included diffraction spikes corrected only out to a certain distance (out to where the template extends beyond the frame), other edge effects, and the subtraction of the core of bright galaxies. For these reasons the bright-star subtraction was not performed and the bright stars are left as they are.
Stacking and image combination
In the next step we compute a median image for all frames within an AOR which corrects for persistence in the detectors and also for any residual first-frame pattern that introduces structure in the background. In parallel, a background map is also created by iteratively clipping objects and masking them, and finally that background is subtracted from each frame of the AOR. The final processing steps consist of resampling the background-subtracted frames onto a common grid with a scale of 0 . 6 pix −1 , i.e., half the instrument pixel size, that covers all data in all channels and which is the same in all channels. We experimented with two MOPEX interpolation schemes to produce our final mosaics. We first tried the 'drizzling' (Fruchter & Hook 2002) scheme in which the final value of the output pixels is computed by considering the contribution of each input pixel in a smaller pixel grid in the output image. This procedure has excellent noise properties (it does not suffer from correlated pixels) when many input frames are available, but with few input frames it can produce artefacts in the output images. The second, simpler approach is to compute the value of each output pixel as a linear combination of the input pixel values. Although this procedure produces correlated noise, it works reliably for all the fields considered in this work which can have widely varying numbers of input images. Noise correlations can be estimated through simulations or by comparing sources in our drizzled and non-drizzled images. These comparisons show that the linear interpolation procedure leads to an underestimation of aperture magnitude errors by 30 − 40% while the magnitudes themselves are unaffected.
Next, we use MOPEX to produce an average-combined image while rejecting outliers and excluding masked regions. The stacking pipeline also produces the following ancillary characterisation maps: (1) an uncertainty map produced by stacking the input uncertainty maps using the same shifts as for the signal stack, (2) a coverage map giving the number of frames contributing to each pixel, and (3) an exposure time map giving the total exposure time per pixel. As the exposure times are not the same for all the observing programmes, these last two maps are not simply scaled versions of each other.
On the spatial variation of the PSF in the stacks
The observations described here were made at many different satellite position angles (PAs), and thus when the images are stacked they must be rotated back to North upwards. This has the effect of rotating the PSF, which is fixed in the satellite's reference frame. Since the PSF is not rotationally symmetric, due in particular to the diffraction spikes, the stacked image of a star will depend on when it was observed. As all parts of the stack Notes. Here 'Num' is the number of frames used, and 'Time' is the integration time, in hours, they contribute. The left part of the table is for channels 1 and 2, split between cryogenic and warm mission, the right part is for channels 3 & 4 which were used during the cryogenic mission only. Note that the EDF-S field was observed only during the warm mission.
were not observed at the same time (or at the same PA), the PSF varies spatially in the stack. The COSMOS field, which is near the Equator, was observable only at specific times and therefore with a very restricted range of PAs; the PSF in the COSMOS stacks is thus quite homogeneous. But in the EDF-N, which was in a continuous viewing zone, observations were obtained at many different PAs, yielding more complicated and more spatially variable PSF. This effect is very important for PSF-based photometry: the PSF at each position of the stack has to be reconstructed by stacking the nominal PSF at the PAs of the observations at that position, as did e.g., Labbe et al. (2015) for the GOODS-South and HUDF fields, and also Weaver et al. (submitted) for the production of the COSMOS2020 catalogue. The latter used the PRFmap code by Andreas Faisst, available at https://github.com/cosmic-dawn/prfmap for that purpose. While doing such photometry is beyond the scope of this paper, we nevertheless provide, for each stack, a table of the PAs of each frame used in the stack. For completeness those tables also contain the frame coordinates, the MJD of the observation, and the exposure time; see Appendix A for more details.
Products
As an example of the data quality, Fig. 2 shows a zoomed section of the EDF-F mosaic in the four channels near the region of maximum coverage. We do not provide here figures of the full mosaics as they would be physically too small to show anything informative other than the overall coverage.
Maps of the integration time per pixel for channels 1 and 3 of all the fields are presented in Appendix B. Since channel 2 is observed together with channel 1, and similarly for channels 4 and 3, the paired channels have very similar coverage, albeit slightly shifted in position. The 10 deg 2 circular area of EDF-N and EDF-F and the 20 deg 2 pill-shaped area of EDF-S are easily seen on those figures. Also, and with the exception of EDF-S, for which there are only observations done specifically for this programme and no archival data, the integration time per pixel, and consequently the depth reached, is far from uniform, with only a small part of the total area of each field having been observed for more than a few hours. In fact, the median integration time per pixel is larger than 1 hr for only two fields. Table 3 gives the median and maximum pixel integration time for each field and each channel. That variation of area covered as a function of exposure time for channels 1 and 3 and for all fields is shown graphically in Fig. 3 which presents a cumulative histogram of the area covered vs. exposure time. The intersection of the curve with the vertical axis thus gives the total area covered for that field and these areas are also listed in Table 4. EDF-S is the most uniformly observed field and it covers the largest area, but it is also the shallowest, with only 0.1 hr per pixel on average, and it is also the only field with no channel 3 and 4 data. EDF-F and EDF-N reach the target coverage of 10 deg 2 with about 1 hr of exposure time, with the latter showing deeper coverage over smaller zones. The other fields were covered by many observing programs with different objectives and which covered specific areas to different depths. The combination of these programs with our own yields a curve with many plateaus. Finally, there are a few small parts of the EDF-F and HDFN fields that have more than 100 hr of exposure time.
Final sensitivities
We estimate the sensitivities of the stacked images by measuring the flux in circular 2 . 5 diameter apertures randomly placed across each image after masking the regions with detected objects using the SExtractor (Bertin & Arnouts 1996) segmentation map. The sensitivity is then computed as the standard deviation of these fluxes (3σ clipped). This procedure is done in 200 × 200 pixel cells (4 arcmin 2 ). Figure 4 shows the cumulative area covered as a function of sensitivity for the channel 1 mosaics. Note the similarity between this figure and the top panel of Fig
Validation and quality control
As part of our validation process we compare photometry and astrometry of sources in our stacks with reference catalogues and also extract number counts that can be compared to previous works.
Catalogue extraction
We begin by extracting source catalogues from the channels 1 and 2 stacks of all fields using SExtractor. We adopt the usual approach of searching for objects that contain a minimum number of connected pixels above a specified noise threshold (in this case 2 σ) and measuring their aperture magnitudes. In the case of our moderately deep IRAC data, where many sources are blended due to the large IRAC PSF, this approach is known to miss faint sources. However, these faint sources are not required for our quality assessment purposes and a shallower catalogue is entirely sufficient. SExtractor estimates a global background on a grid with mesh size of 32 × 32 pixels (recall that pixels are 0 . 6 wide). This background is smoothed with a 5 × 5 pixel Gaussian kernel with FWHM = 1 . 5. For each source, the flux is measured within a circular aperture of 7 diameter and a local background is estimated within an annulus of width 32 pixels around the isophotal limits. The measured fluxes were converted from MJy/sr to AB magnitude using a zero-point of 21.58 (which accounts for a zero-magnitude flux of 3631 Jy and a pixel size of 0 . 6 7 ), and the latter were converted to total magnitude using the aperture corrections given in the IRAC Instrument Handbook for the warm mission (−0.1164 and −0.1158 for channel 1 and channel 2 respectively), which covers the vast majority of the data, while the correction for the cryogenic mission differs at only the 1-2% level 8 . A list of relevant SExtractor parameters used for the catalogue extraction can be found in Table 5.
Astrometric and photometric validation
Using the catalogues extracted above, we evaluate the astrometric accuracy of our stacked images. For each field we crossmatch sources with magnitude 16 < [3.6] < 19 within 1 of their counterparts in the Gaia DR2 catalogue. This magnitude range was adopted to ensure only bright, non-blended sources were chosen. We now present a detailed analysis for EDF-N but other fields are similar. Figure 5 shows the difference between reference and measured coordinates (for clarity, only one point in ten is shown). The heavy blue dashed line gives the size of one pixel in the stacked image (which is half the size of the instrument pixel). Similarly (again showing only one in ten points), Fig. 6 shows, for each coordinate, the difference between the reference and the measured value as a function of position along the other coordinate. The thick red dashed line shows a running median computed over a bin containing 20 points. The flatness of this line indicates that there is no significant spatial variation in astrometric precision. Considering all fields, we find that the 1 σ precision (measured as the RMS of the difference between positions in our catalogue and those in Gaia DR2) is 0 . 15. Furthermore, the median value is always < ∼ 0 . 1, with the exception of the sparselycovered HDF-N field where it is < ∼ 0 . 2.
These measurements demonstrate that the astrometric solutions have been correctly applied to the individual images and that the combined images are free of residuals on a scale much smaller than an individual mosaic pixel, which is more than sufficient to measure precise infrared and optical-infrared colours. Finally, we perform a simple check on the photometric calibration of our mosaics. As described previously, individual images are photometrically calibrated by the Spitzer Science Center (SSC). Following the validation procedures outlined by the SSC, we compare magnitudes of objects in our catalogues with
Magnitude number counts
We compute the differential number counts in channel 1 in each field using the corrected 7 aperture magnitudes. Since the IRAC PSF is too large to perform morphological source classification, we simply include all objects detected. These are shown in Fig. 8, where the red circles with uncertainties present our measurements and the lines show the number counts from the literature; the bottom-right panel shows the mean of all fields. We compare our number counts with those presented in Ashby et al. (2013) who also surveyed many of our fields and also with those computed using the new COSMOS2020 photometric catalogue (Weaver et al., submitted) which we use as a reference.
There is a general agreement in the number counts in all the fields with Ashby et al. (2013) and COSMOS2020 for 16 < [3.6] < 22. At brighter magnitudes the COSMOS2020 counts drop off as bright sources were not included. At fainter magnitudes, our aperture-based catalogues are confusion-limited and thus incomplete. Conversely, the COSMOS2020 catalogue, which uses a high-resolution prior for the detection and a profilefitting method for the measurement, is complete up to significantly fainter magnitudes.
EDF-N counts are slightly higher than the other fields at bright magnitudes. To investigate this difference we simulated a stellar catalogue of 1 deg 2 centred on EDF-N using TRILEGAL (Girardi et al. 2005) and compared counts from this simulated catalogue with our observations, shown in Fig. 9. At bright magnitudes, where stars are expected to outnumber galaxies, our counts are in reasonable agreement with TRILEGAL predictions, and in excellent agreement with the number counts extracted from the AllWISE (Wright et al. 2010) catalogue for this field. These comparisons indicate that the difference between EDF-N and other fields is largely due to the higher density of stellar sources in there, consistent with its lower Galactic latitude.
Summary
We have presented the Spitzer/IRAC mid-infrared component of the Cosmic Dawn Survey: an effort to complement the Euclid mission's observations of deep and calibration fields with deep longer-wavelength data to enable high redshift legacy science.
The survey consists of two major new programs covering the three Euclid deep fields (EDF-N, EDF-F and EDF-S) and a homogeneous reprocessing of all existing data in Euclid's four calibration fields (COSMOS, XMM, EGS and HDFN). We have processed new data together with all relevant archival data to produce mosaics of these fields covering a total of ∼ 71 deg 2 in IRAC channels 1 and 2. Furthermore, the new mosaics are tied to the Gaia astrometric reference system. The MIR data will be essential for a wide range of legacy science with Euclid, including improved star/galaxy separation, more accurate photometric redshifts, determination of stellar masses of galaxies, and the construction of complete galaxy samples at z > 2 with well understood selection effects.
We validated our final products by comparing catalogues extracted from channels 1 and 2 to external catalogues. In all fields, comparing with Gaia DR2, the residual astrometric uncertainty for sources with total magnitudes 16 < [3.6] < 19 is around 0 . 15 (1σ). Our photometric measurements are in excellent agreement with WISE photometry and our number counts are consistent with previous determinations.
The Cosmic Dawn Survey Spitzer survey presented here represents the first essential step in assembling the required multiwavelength coverage in the Euclid deep fields which are set to Fig. 9. Magnitude number counts in the EDF-N field compared to All-WISE and the predicted stellar number counts from TRILEGAL.
become some of the most important fields in extragalactic astronomy for the coming decade. Since the Spitzer mission has finished, and all available data in these fields have been pro-cessed with the latest reduction pipeline, the resulting mosaics will remain the deepest and widest MIR imaging survey for the foreseeable future. No existing or approved future observatories are capable of obtaining such data. While JWST is more sensitive and has higher spatial resolution at these wavelengths, its mapping speed is too slow to cover comparable degree-scale areas.
In the context of the Cosmic Dawn Survey, several programs are currently underway to add data at other wavelengths to the Euclid deep fields and calibration fields. In particular deep optical data in the EDF-N and EDF-F are currently being obtained with the Subaru's Hyper-Suprime-Cam instrument as part of the Hawaii-Two-0 program (McPartland et al., in prep). These fields are also being targeted with high spatial resolution millimeter observations as part of the planned Large-scale Structure Survey with the Toltech Camera 9 on the Large Millimeter Telescope (LMT Pope et al. 2019). A deep U-band survey is also underway with the CFHT (Zalesky et al., in prep). EDF-S is being covered with K-band observations from the VISTA telescope (Nonino et al., private communication), and planning is ongoing to obtain optical data with the Vera C. Rubin Observatory.
The Cosmic Dawn Survey Spitzer mosaics and associated products described here can be downloaded from the IRSA web site, Appendix A gives the details of the download site and the naming convention used. The community is encouraged to make use of them for their science.
Fig. 1 .
1Histogram of the exposure time of the data analysed here (including the few discarded observations) using bins of 30 days. Our dedicated observations began in November 2016 and comprise most of the data after that date. The red part of each bar accounts for observations in channels 3 and 4, the blue part for those in channels 1 and 2; the vertical dotted line at 2009.37 indicates the end of the cryogenic mission. No observations are made in channel 3 and 4 after the end of the cryogenic mission.
Fig. 3 .
3Cumulative area coverage as a function of exposure time for channels 1 and 3, for all fields. The figures for channels 2 and 4 are similar to the ones above, as explained in the text.
. 3 once the latter is rotated by 90 degrees. The solid line shows our total depth, summed over all our survey fields. Also shown in the figure are the published sensitivities of the surveys that are included in our data and analyses. Generally, our measured sensitivities are consistent with literature measurements for surveys of equivalent exposure time.
Fig. 4 .
4Sensitivity of the Spitzer/IRAC channel 1 data as a function of cumulative area coverage. The coloured lines illustrate 1 σ depths measured in empty 2 . 5 diameter apertures in each field. The grey solid line is the total area observed to a given depth summed over different surveys. The data points indicate point-source sensitivities at 1σ compiled inAshby et al. (2018) (note that some of these data are included in our stacks). The circles and squares represent surveys executed during cryogenic and warm missions, respectively.
Fig. 5 .
5The difference between the reference and the measured position, in arcseconds, of Gaia DR2 catalogue sources with 16 < [3.6] < 19 total magnitudes extracted from the EDF-N channel 1 mosaic. The blue dashed lines indicate the size of one mosaic pixel. The blue dotted lines go through the origin. The shaded regions are ellipses containing 68 % and 99 % of all sources respectively. For clarity, only one in ten sources is plotted.
Fig. 6 .Fig. 7 .
67The difference between the reference Gaia DR2 catalogue and the measured RA (top panel) and DEC (bottom panel) of sources in the EDF-N channel 1 mosaic with 16 < [3.6] < 19 total magnitudes as a function of the coordinate. The solid red line shows a running median computed in bins of 20 points, and the shaded areas indicate the regions containing 68 % and 99 % of all sources respectively. those in the WISE survey. Because of the difference between the WISE W1 and IRAC channel 1 filter profiles, we select objects with [3.6] − [4.5] ∼ 0.Figure 7shows the magnitude difference for the EDF-N field, and the agreement is excellent. Further comparisons with photometric measurements in previous COSMOS IRAC surveys can be found in the Appendix of Weaver et al. Photometric comparison with the WISE survey. The magnitude measured in 7 apertures for flat-spectrum objects ([3.6] − [4.5] ∼ 0) is compared with W1 magnitudes in the ALLWISE survey. The shaded area represents the 68% confidence interval.
FigFig. B. 2 .
2. B.1. Integration time maps for channel 1 Article number, Integration time maps for channel 3. A blank field is included for EDF-S which was not observed in that channel and in order to have the same structure as B.1. Article number, page 14 of 15
Table 1 .
1The ten largest programmes, by Program IDPID Principal Investigator Field
Time (hr) Reference
61041 Giovanni Fazio
XMM
847 SEDS; Ashby et al. (2013)
61040 Giovanni Fazio
HDFN
914 SEDS; Ashby et al. (2013)
14235 Claudia Scarlata
EDF-S
1086 this paper
169 Mark Dickinson
HDFN
1104 GOODS; Labbé et al. (2015)
10042 Peter Capak
XMM
2033 this paper
90042 Peter Capak
COSM
2167 this paper
13094 Ivo Labbe
COSM
2483 GOODS; Labbé et al. (2015)
11016 Karina Caputi
COSM
3021 SMUVS; Ashby et al. (2013)
13058 Peter Capak
EDF-F
3162 this paper
13153 Peter Capak
EDF-N
4625 this paper
Table 2 .
2Valid observationsField
Ch.
cryo
warm
total
Ch.
total
Num Time
Num Time
Num Time
Num Time
EDF-N
1
5 859
52 113 521 2 380 119 380 2 432
3
5 856
52
EDF-N
2
5 857
52 113 204 2 467 119 061 2 519
4
7 667
50
EDF-F
1 14 299 363 105 781 2 672 120 080 3 035
3 14 301 363
EDF-F
2 14 299 363 105 779 2 764 120 078 3 127
4 29 686 352
EDF-S
1
n/a
n/a 21 982 534 21 982 534
3
n/a
n/a
EDF-S
2
n/a
n/a 21 982 552 21 982 552
4
n/a
n/a
COSMOS 1
7 014 185 191 072 4 886 198 086 5 071
3
7 011 185
COSMOS 2
7 013 185 191 031 5 052 198 044 5 237
4 13 894 179
EGS
1
4 673 192 44 101 551 48 774 743
3
4 672 192
EGS
2
4 673 192 44 101 569 48 774 761
4 14 535 186
HDFN
1
6 253 298 36 485 930 42 738 1 228
3
6 252 298
HDFN
2
6 253 298 36 485 962 42 738 1 260
4 22 496 288
XMM
1 10 264 154 98 027 2 410 108 291 2 564
3 10 265 154
XMM
2 10 265 154 98 030 2 495 108 295 2 649
4 14 321 151
Table 3 .
3Median and maximum pixel integration time in hoursField
ch1
ch2
ch3
ch4
COSMOS 0.51 93.7 0.50 97.1 0.38 5.1 0.38 5.5
EDF-F
1.33 199.7 1.33 149.5 0.03 47.3 0.03 54.2
EDF-N
1.47 23.4 1.56 21.3 0.04 20.4 0.04 19.6
EDF-S
0.13
0.5 0.16
0.5
-
-
-
-
EGS
0.16 71.1 0.16 71.6 0.93 5.4 0.95 4.8
HDFN
0.16 236.2 0.16 224.4 0.13 91.2 0.13 95.2
XMM
0.31 65.9 0.33 67.1 0.04 2.0 0.04 2.0
Table 4 .
4Location and area, in deg 2 , covered in each fieldField
RA
Dec
ch1
ch2 ch3 ch4
EDF-N
17 h 58 m 66°36
11.74 11.54 0.61 0.62
EDF-F
3 h 32 m −28°12
10.52 11.05 7.78 7.77
EDF-S
4 h 5 m −48°30
23.60 23.14
-
-
COSMOS 10 h 0 m
2°12
5.37 5.46 2.72 2.72
EGS
14 h 19 m 52°42
1.76 1.80 0.97 0.98
HDFN
12 h 37 m 62°24
0.91 0.91 0.57 0.63
XMM
2 h 27 m −4°36
17.54 17.48 9.09 9.10
Table 5 .
5SExtractor parameters used for detection and photometry.Parameter name
Value
DETECT_MINAREA
5
DETECT_MAXAREA
1000000
THRESH_TYPE
RELATIVE
DETECT_THRESH
2
ANALYSIS_THRESH
2
FILTER_NAME
gauss_2.5_5x5.conv
DEBLEND_NTHRESH
32
DEBLEND_MINCONT
0.00001
BACK_SIZE
32
BACKPHOTO_THICK
32
BACK_FILTERSIZE
3
BACKPHOTO_TYPE
LOCAL
MAG_ZEROPOINT
21.58
PHOT_AUTOPARAMS
2.5,5.0
PIXEL_SCALE
0.60
Article number, page 8 of 15 Moneti et al.: The Cosmic Dawn SurveyFig. 8. Magnitude number counts in channel 1 (red circles) together with COSMOS2020 (long dashed lines) and Ashby et al. (2013, hereafter A13; short dotted green lines). The bottom right panel shows the mean of all fields compared to , and the legend there applies to all panels.10 2
10 4
101
103
105
EDF-N
EDF-F
10 2
10 4
101
103
105
N mag
−1 deg
−2
EDF-S
COSMOS
10 2
10 4
101
103
105
HDFN
EGS
15
20
25
10 2
10 4
101
103
105
XMM
15
20
25
Magnitude [3.6 µm]
All fields
A13 Total
COSMOS2020
All fields
14
16
18
20
22
24
Magnitude [3.6 µm]
10 1
10 2
10 3
10 4
10 5
N mag
−1 deg
−2
TRILEGAL
AllWISE
Cosmic Dawn
These are now known as the "Euclid Auxiliary Fields" in Euclid terminology
Article number, page 2 of 15 Moneti et al.: The Cosmic Dawn Survey
https://irsa.ipac.caltech.edu 3 https://irsa.ipac.caltech.edu/data/SPITZER/docs/ irac/iracinstrumenthandbook/home 4 https://irsa.ipac.caltech.edu/data/SPITZER/docs/ irac/iracinstrumenthandbook/26/ 5 https://irsa.ipac.caltech.edu/data/SPITZER/docs/ dataanalysistools/tools/mopex/ 6 https://irsa.ipac.caltech.edu/data/SPITZER/docs/ irac/iracinstrumenthandbook/19/ Article number, page 3 of 15 A&A proofs: manuscript no. Cosmic_dawn_spitzer
https://irsa.ipac.caltech.edu/data/SPITZER/docs/ irac/iracinstrumenthandbook/19/ 8 https://irsa.ipac.caltech.edu/data/SPITZER/docs/ irac/calibrationfiles/ap_corr_warm/
http://toltec.astro.umass.edu/ Article number, page 9 of 15 A&A proofs: manuscript no. Cosmic_dawn_spitzer
Article number, page 12 of 15
Acknowledgements. We thank the MOPEX support team for fixing issues that appeared when combining large numbers of files. This publication is based on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA, and has made use of the NASA/IPAC Infrared Science Archive, which is funded by the National Aeronautics and Space Administration and operated by the California Institute of Technology. This publication has also made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/ gaia/dpac/consortium). Funding for the Gaia Data Processing and Analysis Consortium (DPAC) has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. This publication makes use of data products from the Wide-field Infrared Survey Explorer (WISE), which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. H.J.McC. acknowledges support from the PNCG. This work used the CANDIDE computer system at the IAP supported by grants from the PNCG and the DIM-ACAV and maintained by S. Rouberol. S.T. and J.W. acknowledge support from the European Research Council (ERC) Consolidator Grant funding scheme (project ConTExt, grant No. 648179Appendix A: Delivered data productsThe new mosaics and associated products can be obtained from the IRSA website at https://irsa.ipac.caltech. edu/data/SPITZER/Cosmic_Dawn (ATTN: the products will become available once the paper is accepted; the URL may be updated at publication time). The file naming convention for the stacks is as follows:CDS_{field}_ch{N}_{type}_v24.fitswhere field is the field name, N is the channel number, and type is one of ima: for the flux image, cov: for the coverage in terms of number of frames used to build each pixel of the mosaic, tim: for the exposure time in sec of the pixel, and unc: for the uncertainty as determined from the standard deviation of the image pixels that contributed to the mosaic pixel.Also,Table A.1 gives the precise J2000 coordinates of the field tangent point in decimal degrees, the reference pixel corresponding to that tangent point, and size, in pixels, of the mosaics. These values are the same for all channels of a field and for all the ancillary images. The pixel scale is 0 . 60 per pixel for all mosaics.The tables with the observation date, coordinates, position angles, and exposure times of the input frames are provided in IPAC format and are gzipped to reduce their size. Their names are as follows:CDS_{field}_ch{N}_info_v24.tbl.gzThe first few lines of the The coordinates are in degrees of longitude and latitude (Equatorial, J2000) and the PAs are measured eastward of North.Appendix B: Coverage mapsFiguresB.1 and B.2show the full set of pixel exposure time maps for channels 1 and 3; channels, 2 and 4 are similar though slightly shifted in location. A square root scaling is applied in order to emphasise the differences at the low levels, and the same maximum is used for all fields in each channel. As EDF-S was not observed in channel 3, a blank field is placed there. Notes. Longitude and latitude are Equatorial and J2000, for the image tangent point. These values are valid for all four channels of each field and for their ancillary images.-1700 1910-1949 1951 1953-1961 1963-1983Appendix C: PID numbers
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| [
"https://github.com/cosmic-dawn/prfmap"
]
|
[
"Noncommutative geometry of phase space",
"Noncommutative geometry of phase space"
]
| [
"Maja Burić *[email protected]†[email protected] \nFaculty of Physics\nUniversity of Belgrade\nP.O. Box 368SR-11001Belgrade\n",
"John Madore \nLaboratoire de Physique Théorique\nUniversité de Paris-Sud\nBâtiment 211F-91405Orsay\n"
]
| [
"Faculty of Physics\nUniversity of Belgrade\nP.O. Box 368SR-11001Belgrade",
"Laboratoire de Physique Théorique\nUniversité de Paris-Sud\nBâtiment 211F-91405Orsay"
]
| []
| A version of noncommutative geometry is proposed which is based on phase-space rather than position space. The momenta encode the information contained in the algebra of forms by a map which is the noncommutative extension of the duality between the tangent bundle and the cotangent bundle. | 10.1007/s10714-011-1231-5 | [
"https://arxiv.org/pdf/1110.0592v2.pdf"
]
| 119,118,466 | 1110.0592 | e3dc731fd99557da3a848824e9469f3b71444d21 |
Noncommutative geometry of phase space
5 Oct 2011
Maja Burić *[email protected]†[email protected]
Faculty of Physics
University of Belgrade
P.O. Box 368SR-11001Belgrade
John Madore
Laboratoire de Physique Théorique
Université de Paris-Sud
Bâtiment 211F-91405Orsay
Noncommutative geometry of phase space
5 Oct 2011$Id: ncp.tex,v 1.76 2011/07/06 02:36:58 madore Exp $
A version of noncommutative geometry is proposed which is based on phase-space rather than position space. The momenta encode the information contained in the algebra of forms by a map which is the noncommutative extension of the duality between the tangent bundle and the cotangent bundle.
Introduction
It has been long conjectured [1] that quantum fluctuations of the gravitational field might soften the singularities which appear in almost all solutions to the gravitational field equations without source. We here assume that this role is to be played by the effect of noncommutativity. We assume that is that noncommutativity is an effective if but partial description of quantum fluctuations; it would play in this respect the same role that thermodynamics plays with respect to statistical physics. Noncommutative geometry to a large extent follows a path previously taken by quantum mechanics with one essential difference; there is no experimental motivation for it. We consider the subject of interest in so far that it can be considered as an extension of the theory of gravity which enables us to better understand the role of the gravitational field as a regulator for classical as well as quantum singularities.
Over the passed few years a noncommutative generalization of the Cartan frame formalism has been developed [3,4] and applied [5,6,7,8] with varying degrees of success to problems in gravitational physics, notably to the possibility of blowing up [9] the Big Bang. One distinguishing feature of this formalism is that the field equations are not derived from an action principle but rather from constraints imposed on the frame arising from Jacobi identities. These identities, we conjecture, fix the Ricci tensor. In particular then the Einstein tensor is determined and the value it takes can be interpreted as an effective source due to the existence of the noncommutative structure. In the quasi-classical approximation this can be more precisely interpreted as the energy of the Poisson structure. One of the motivations of this modification of the field equations is the further conjecture that curvature as is usually defined in differential geometry cannot be used without modification in the noncommutative generalization. It is hoped that once the correct expression is found then the Ricci contraction will yield an Einstein tensor which includes the Poisson energy.
The notation is drawn from previous publications [3]. A tilde is used to designate commutative geometry or a classical limit of a noncommutative one. We use the convention of distinguishing the frame components of a vector from the natural components by a choice in index: the Greek or Latin indices from the middle of the alphabet are coordinate and those from the beginning are frame indices. For example if J µν are the natural components of an antisymmetric tensor then the frame components will be written as J αβ . If J µν is a commutator this is quite consistent provided we remain at the quasi-classical approximation. We shall however apply the convention also when describing the nonperturbative solutions in Section 2 in which case it can be ambiguous.
Algebraic approach
Consider a smooth manifold M with a moving frameθ α . Let A be a noncommutative deformation of the algebra C(M) of smooth functions on M defined by a symplectic structure J and let θ α be a noncommutative deformation of the moving frame. As we shall see, the connection on A can be defined to satisfy both a left and right Leibniz rule, a condition which is intimately connected with the existence of a reality condition; the metric can be defined to be a bilinear map, a condition connected with locality. The classical limit of the geometry is thus naturally equipped with a linear connection and a metric as well as with a Poisson structure. Under the assumptions which we shall impose the Poisson structure is non-degenerate. We shall be more precise about these extensions below. We would like to be able to show that in the weak-field, quasi-classical approximation they imply that the metric defined by the frame cannot be arbitrary and that the Ricci tensor is fixed by Jacobi identities.
Let A be a noncommutative * -algebra generated by four hermitian elements x µ which satisfy the commutation relations:
[x µ , x ν ] = ikJ µν (x σ ). (1.1)
As a measure of noncommutativity, and to recall the many parallels with quantum mechanics, we use the symbolk, which designates the square of a real number whose value could lie somewhere between the Planck length and the proton radius m −1 P . The value becomes important when we consider perturbations. The J µν are of course restricted by Jacobi identities; we shall see below that there are two other natural requirements which also restrict them.
Let L be a macroscopic length scale. In the Schwarzschild geometry defined by a source of mass µ the gravitational field is weak if the parameter G N µL −1 is small. In de Sitter geometry with cosmological constant Λ the corresponding parameter is ΛL 2 . In the microscopic domain we have two length scales determined by respectively the square G N of the Planck length and byk. These two scales are not necessarily related, although both are of course much smaller than L 2 . It would be reasonable to identifyk with G N thus presumably identifying quantum gravity with noncommutativity; a weaker assumption, that noncommutativity gives just an effective description of quantum gravity, would correspond to a vague inequality G N ≃k.
One can also compare a classical gravitational field with noncommutativity. As noted, the gravitational field is weak if the dimensionless parameter ǫ GF = (G N µ) 2 L −2 is small; on the other hand, the space-time is almost commutative if the dimensionless parameter ǫ N C =kL −2 is small. If noncommutativity is not related to gravity then it makes sense to speak of ordinary gravity as the limitk → 0 with G N µ non vanishing. However one could assume that noncommutativity and gravity are directly related: in that case, both should vanish withk. We shall consider the first situation with
k ≪ (G N µ) 2 ≪ L 2 , ǫ = ǫ N C /ǫ GF ≪ 1,(1.2)
and expand in the parameter ǫ which is a measure of the relative dimension of a typical 'space-time cell' compared with a typical 'quantity of gravitational energy'.
Differential calculi
Assume that there is over A a differential calculus which is such [4] that the module of 1-forms is free as a left or right module and possesses a preferred basis θ α which commutes
[x µ , θ α ] = 0 (1.3)
with the algebra. Such a basis we call a frame. The space which one obtains in the commutative limit is therefore parallelizable, with a global moving frameθ α which is the commutative limit of θ α . We can write the differential
dx µ = e µ α θ α , e µ α = e α x µ . (1.4)
The algebra is defined by product (1.1) that is by the matrix of elements J µν ; the metric is defined we shall see below as in the commutative case, by the matrix of elements e µ α . Consistency requirements impose relations between these two matrices which in simple situations allow us to find a one-to-one correspondence between the structure of the algebra and the metric.
The input of which we shall make the most use is the Leibniz rule
ike α J µν = [e µ α , x ν ] − [e ν α , x µ ]. (1.5)
One can see here a differential equation for J µν in terms of e µ α . The momenta p α introduced as in quantum mechanics stand in duality to the position operators x µ by the relation
[p α , x µ ] = e α x µ = e µ α .
(1.6)
The right-hand side of this identity defines the gravitational field. The left-hand side must obey Jacobi identities. These identities yield relations between quantum mechanics in the given space-time and the noncommutative structure of the algebra.
The three aspects of reality then, the curvature of space-time, quantum mechanics and the noncommutative structure are intimately connected. We shall explore here an even more exotic possibility that the field equations of general relativity are encoded in the structure of the algebra so that the relation between general relativity and quantum mechanics can be understood by the relation which each of these theories has with noncommutative geometry.
In spite of the rather lengthy formalism the basic idea is simple. We start with a classical geometry described by a moving frameθ α and we quantize it by replacing the moving frame by a frame θ α , as we shall describe in some detail below. The easiest cases would include those frames which could be quantized without ordering problems. Letẽ α be the vector fields dual to the frameθ α ; we quantize them as in (1.6) by imposing the rulẽ
e α → e α = −1 ad p α . (1.7)
Finally, we must construct a noncommutative algebra consistent with the assumed differential calculus; this defines the image of the map
θ α −→θ α −→ J µν . (1.8)
More details of this map will be given in the Appendix. The algebra we identify with 'position space'.
To construct phase space we must add the momenta p α . In ordinary geometry there is but one way to do so; the derivations e α are outer and the associated momenta do not belong to the algebra generated by the position variables. That is, we add the extra elements which are necessary in order that the derivations be inner, as one does in ordinary quantum mechanics. In noncommutative geometry there are more possibilities; in particular, p α can belong to the initial algebra A. Also the new element is that the consistency relations in the algebra such as Jacobi identities
ik[p α , J µν ] = [x [µ , [p α , x ν] ]]
. (1.9) in principle restrict θ α and J µν .
If the space is flat, e µ α = δ µ α and the frame is the canonical flat frame then the right-hand side of (1.9) vanishes and it is possible to consistently choose the expression J µν to be equal to a constant. On the other hand, if the space is curved the right-hand side cannot vanish except of course in the limitk → 0. The map (1.8) is not single valued since any constant J has flat space as inverse image.
We must insure also that the differential is well defined. A necessary condition is that d[x µ , θ α ] = 0, from which it follows that the momenta p α must satisfy the consistency condition
2p γ p δ P γδ αβ − p γ F γ αβ − K αβ = 0. (1.10)
The P γδ αβ define the exterior product in the algebra of forms,
θ γ θ δ = P γδ αβ θ α ⊗ θ β . (1.11)
We write P αβ γδ in the form
P αβ γδ = 1 2 δ [α γ δ β] δ + iǫQ αβ γδ (1.12)
of a standard antisymmetrizer plus a peturbation. If further [10] we decompose Q αβ γδ as the sum of two terms
Q αβ γδ = Q αβ − γδ + Q αβ + γδ (1.13)
symmetric (antisymmetric) and antisymmetric (symmetric) with respect to the upper (lower) indices then the condition that P αβ γδ be a projector is satisfied to first order ink because of the property that
Q αβ γδ = P αβ ζη Q ζη γδ + Q αβ ζη P ζη γδ .
(1.14)
The compatibility condition
(P αβ ζη ) * P ηζ γδ = P βα γδ (1.15)
with the product is satisfied provided Q αβ γδ is real.
From (1.3) it follows that d[x µ , θ α ] = [dx µ , θ α ] + [x µ , dθ α ] = e µ β [θ β , θ α ] − 1 2 [x µ , C α βγ ]θ β θ γ = 0. (1.16)
We have here introduced the Ricci rotation coefficients C α βγ . We find then that multiplication of 1-forms must satisfy
[θ α , θ β ] = 1 2 θ β µ [x µ , C α γδ ]θ γ θ δ . (1.17)
Consistency requires then that
θ [β µ [x µ , C α] γδ ] = 0; (1.18)
because of the condition (1.3) consistency also requires that
θ (α µ [x µ , C β) γδ ] = Q αβ − γδ . (1.19)
We have in general four consistency relations which must be satisfied in order to obtain a noncommutative extension of a commutative manifold. They are the Leibniz rule (1.5), the Jacobi identity and the conditions (1.18) and (1.19) on the differential. The first two constraints are not completely independent of the differential calculus since one involves the momentum operators. The condition (1.19) follows in general from the expression [11]
C α βγ = F α βγ − 4iǫp δ Q αδ − βγ (1.20)
for the rotation coefficients. It follows also from general considerations that the rotation coefficients must satisfy the gauge condition
e α C α βγ = 0. (1.21)
We shall refer to all these conditions as the Jacobi constraints.
Phase space
The classical phase space associated to Minkowski space-time has eight dimensions. The associated algebra A can be considered as the algebra with four position generatorsx µ and four momentum generatorsp α subject to the sole relation that they commute. The quantized phase space is the same but with the Heisenberg commutation relations
[p α , x µ ] = δ µ α . (2.1)
Over this algebra there is a natural moving frame θ α = δ α µ dx µ the dual derivations of which are given by
e β = δ ν β ∂ ν = −1 ad p β . (2.2)
We shall set = 1 so that the momenta are normalized as we wish them to be, with dx µ (e β ) = δ µ β . The moving frame θ α is clearly a frame in the sense we have defined it. As in the commutative case, since the frame is exact the curvature vanishes: the associated geometry is flat. We have thus a map
Minkowski → Heisenberg (2.3)
the extension of which to a geometry with generic curvature we wish to construct. There exists the commutative limit as special case; the map takes each 4-geometry into a subalgebra of the algebra of sections of the form bundle defined by the moving frame. We shall extend this map to one which includes the corrections of first order in the noncommutativity parameterk. We shall see that the existence of the extension imposes integrability conditions on the classical limit of the map. These conditions we conjecture (guess) replace the field equations. An analogous situation exists in classical gravity when one attempts to find solutions as perturbation expansions in the gravitational coupling constant: the existence for example of the first-order corrections implies that the sources obey the conservation laws of the flat space-time.
Momenta and representations
To make transparent the dual roles played by the position and the momenta we introduce the index i = (λ, α) and write a point in phase space as y i = (x λ , p α ). We lower the index with the metric components g ij = (g µν , g αβ ). The classical Heisenberg commutation relations can be written then as
[y i , y j ] = J ij ,(2.4)
with
J ij = 0 δ µ β −δ ν α 0 . (2.5)
The matrix J contains all the information of the system. The even elements J + describe the algebra and the noncommutative differential calculus, the odd elements J − depend directly on the frame which in turn determines the limiting commutative (de Rham) differential calculus. The extension of the map (2.3) is equivalent then to a map
J + → J − . (2.6)
The diagonal elements J + consist of the six position commutators
[x µ , x ν ] = ikJ µν (2.7)
as well as of the 'dual' momentum commutators
[p α , p β ] = K αβ + F γ αβ p γ − 2iǫQ γδ αβ p γ p δ . (2.8)
which measure the curvature. There seems to be no obvious property which would characterize the map (2.6). As solution to a differential equation it is non-local. Even in the semi-classical limit it has no obvious characterization.
The unusual new feature of the noncommutative extension is the possibility that the momenta which are functions of coordinates p α (x σ ) and satisfy all the constraints exist. In the commutative limit this would correspond to a section of the frame bundle; locally at least there are many. We shall refer to the p α (x σ ) as a 'section' of the noncommutative frame bundle although we have not defined the latter. If represented by operators on some Hilbert space, the representation will be irreducible; the commutant will reduce to the identity. For example if the matrix J is constant and invertable then the elements π α defined by π α = p α − p 0α ,
p 0α = −(ikJ) −1 αµ x µ (2.9)
commute with all the position generators. They can then be set to zero; the momenta and the position operators generate then the same algebra. We shall assume this case to be generic. We assume that is there be a solution p 0α (x µ ) to the equations
[p 0α , x µ ] = e µ α .(2.10)
But if we have independent momenta p α then also [p α , x µ ] = e µ α .
(2.11)
We have then two solutions and the differences π α = p α − p 0α commute with the generators x µ .
It would be natural to set π α = 0 to obtain an irreducible representation of the position algebra; this would lead however to a manifestly singular commutative limit. We resolve this by requiring only that the 'phase algebra' T be irreducible. The projection of the tangent bundle onto the manifold is in the algebraic transcription an injection of A into T .
The p α then are decomposed as a sum of a section p 0α of this bundle and a remainder π α . The subalgebra A ′ of T generated by the π commutes with A. The condition we impose is
T = A ⊗ A ′ . (2.12)
This is certainly true in the commutative limit. The conditions we have placed on the manifold imply that the tangent bundle is trivial; the condition could be considered as the statement that this remains so in the non-commutative extension.
We can write the relation (2.9) as the definition of a 'covariant momentum'
π α = p α + Z α (x µ ), Z α = −p 0α (2.13)
This is somewhat analogous to gauge transformations; a covariant derivative is constructed to commute with them.
The correspondence
We shall here consider phase space with eight generators, the position generators x µ and the momenta p α defining the exterior derivations. The latter we suppose admits to first order a bracket of the form [p α , p β ] = C γ αβ p γ + K αβ .
(2.14)
This is a central extension of the classical relation satisfied by the derivations. In general the relation is given by (1.10). The center is non-trivial and is generated by the elements π α .
The rotation coefficients are directly related to the commutators of the momentum generators. We have seen that the former are given by the expression (1.20) and the latter by (1.10), which we write in the form
[p α , p β ] = 1 ik L αβ (2.15) with L αβ = K αβ + ikF γ αβ p γ − 2(ik) 2 µ 2 Q γδ αβ p γ p δ . (2.16) [p α , x µ ] = e µ α (2.17)
There is therefore a direct connection between the rotation coefficients and the commutators J µν . This relation can be derived directly without explicitly referring to the momenta.
It is easy to see that the Jacobi identity from which one derives in the semi-classical approximation a differential equation
[x ν ,e α J µν − J [µρ ∂ ρ e ν] α = 0. (2.21)
This condition can be expressed uniquely in terms of frame components using the sequence of identities
θ β µ θ γ ν (e α J µν − J [µρ ∂ ρ e ν] α ) = e α J βγ − J µν e α (θ β µ θ γ ν ) − J [βδ [p δ , [p α , x ν ]]θ γ ν = e α J βγ − J µν e α (θ β µ θ γ ν ) − J [βδ e α e ν δ θ γ] ν + J [βδ [x ν , [p δ , p α ]]θ γ] ν (2.22)
which follow immediately from the Leibniz rule. We find then the condition
e α J βγ + C [β αδ J δγ] =
Metrics and connections
We have defined a notion of antisymmetry by the array P αδ γβ . To define symmetry we introduce a flip σ which exchanges in a twisted way the two factors of a tensor product. In terms of the frame it can be written
σ(θ α ⊗ θ δ ) = S αδ γβ θ γ ⊗ θ β . (2.26)
If we require that the map be bilinear then the coefficients must be constant. The relation between P αδ γβ and S αδ γβ is the condition Some further relations are given in the Appendix.
π • (1 + σ) = 0,(2.
With the frame one can construct a metric just as one does in the commutative case. It is the bilinear map of the tensor product of the module of 1-forms by itself into the algebra. It associates therefore a function to each pair of vector fields. We consider this metric in the classical approximation. It is defined by the frame and a set of coefficients, necessarily in the center, by the expression
g αβ = g(θ α ⊗ θ β ). (2.30)
We choose the frame to be orthonormal in the commutative limit; we can write therefore
g αβ = η αβ − iǫh αβ . (2.31)
In the linear approximation, the condition of the reality of the metric becomes h αβ +h αβ = T βα γδ η γδ .
(2.32)
The covariant derivative is given by
Dξ = σ(ξ ⊗ θ) − θ ⊗ ξ. (2.33)
In particular
Dθ α = −ω α γ ⊗ θ γ = −(S αβ γδ − δ β γ δ α δ )p β θ γ ⊗ θ δ = −iǫT αβ γδ p β θ γ ⊗ θ δ ,(2.34)
so the connection-form coefficients are linear in the momenta
ω α γ = ω α βγ θ β = iǫp δ T αδ βγ θ β . (2.35)
On the left-hand side of the last equation is a quantity ω α γ which measures the variation of the metric; on the right-hand side is the array T αδ βγ which is directly related to the anticommutation rules for the 1-forms, and more importantly the momenta p δ which define the frame. Ask → 0 the right-hand side remains finite and ω α γ →ω α γ .
(2.36)
The connection is torsion-free if the components satisfy the constraint ω α ηδ P ηδ βγ = 1 2 C α βγ .
(2.37)
The connection is metric if ω α βγ g γδ + ω δ γη S αγ βζ g ζη = 0, (2.38) or linearized,
T (αγ δ β) = 0. (2.39)
The equation can be solved, so in the linear approximation every metric has associated to it a unique torsion-free metric connection.
The perturbation expansion
We must now examine the conditions under which the noncommutative frame can be considered as having the classical frame as limit. By classical we refer here to ordinary quantum mechanics. We have three commutators, position space, momentum space and the cross terms, given respectively by
[x µ , x ν ] = ikJ µν (3.1) [p α , p β ] = 1 ik L αβ (3.2) [p α , x µ ] = e µ α (3.3)
with L αβ given by Equation (2.16).
General quasi-classical limit
Let us discuss in more detail the limit of the frame geometry. Recall the definitions
f θ α = θ α f, θ α = θ α µ (x σ )dx µ ,(3.4)
the second being the inverse of (1.4). As functions of coordinates are given through the Taylor expansion, the commutator [x λ , f (x σ )] can be expressed in terms of the basic commutators J λµ . Neglecting the operator ordering that is in linear order ink we obtain
[x λ , f (x σ )] = ikJ λµ ∂ µ f = ikJ λα e α f. (3.5)
This is the quasi-classical approximation. We have denoted
J λµ = J αβ e λ α e µ β .(3.6)
In particular,
[x λ , dx µ ] = −e µ α [x λ , θ α ν ]dx ν = −ikJ λβ e µ α e β θ α ν dx ν = ikJ λβ e β e µ α θ α . (3.7)
Using the quasi-classical approximation we can obtain an equation for J. From
dJ λµ = dJ αβ e λ α e µ β + J [λβ e α e µ] β θ α (3.8) and (3.5) we have dJ αβ + J γ[α C β] γδ θ δ = 0. (3.9)
This equation has the integrability condition
d(J γ[α C β] γδ θ δ ) = 0, (3.10) that is d(J γ[α C β] γδ )θ δ + J γ[α C β] γδ dθ δ = 0. (3.11)
Using the known expression for dJ αβ and the Bianchi identities
ǫ αβγδ (e δ C ζ βγ + C ζ δη C η βγ ) = 0 (3.12)
one shows that the first term is identically zero. The condition reduces to
J γ[α C β] γδ dθ δ = 0. (3.13)
Conclusion
We have argued in favor of considering noncommutative geometry as a deformation of phase space rather than position space. From this point of view the algebra and the calculus are on the same footing and in fact one could avoid to a certain extent at least using the later since the 1-forms have been encoded in the momenta. Derivations can be almost identified with momenta but only if one neglects an additive constant in the latter. We have shown that in noncommutative geometry there is a preferred origin to the momenta, somewhat analogous to the preferred origin in the module of 1-forms. This is quite consistent with previous results to the effect that the commutators determine not only the structure of the algebra but also the metric of the associated geometry.
Appendices
Gauge dependence
We have found a map between the symplectic and the metric structures on a manifold. The definition is valid only in the semi-classical approximation and relies essentially on the existence of a frame. An interesting problem is the study of the variation of the map. For example one might inquire into the type of variations of the frame which leave the symplectic structure invariant and inversely. Both of these variations could be considered as 'gauge' transformations. We have succeeded in solving only partially this problem.
Consider two choices of commutator J αβ and J ′αβ with
J ′αβ = J αβ + δJ αβ (5.1)
Then the corresponding variation of the rotation coefficients δC γ 1αβ is given by a solution to the constraints
δe γ J αβ + e γ δJ αβ − δC [α γδ J β]δ − C [α γδ δJ β]δ = 0, (5.2)
ǫ αβγδ δJ αǫ e ǫ J βγ + ǫ αβγδ J αǫ δe ǫ J βγ + ǫ αβγδ J αǫ e ǫ δJ βγ = 0, In particular the rotation coefficients vary only with a change in the coefficients.
Rotation coefficients
One can show quite generally that the momenta p α have a bracket of the form
[p α , p β ] = K αβ + F γ αβ p γ − 2iǫQ γδ αβ p γ p δ . (5.7)
For the formalism to work we must be able to impose the gauge condition
E αβ = 0, E αβ = e γ C γ αβ . (5.8)
There is yet another stronger condition to be considered below.
Consider the equations [e α , e β ] = C γ αβ e γ (5.9)
To obtain them as the commutative limits of a set of noncommutative extensions we introduce the commutators [p α , p β ] = X αβ . (5.10) We can consider the left-hand side of (5.9)as a limit of the left-hand side of (5.10) For the right-hand sides to satisfy such a relation we require coefficients K αβ , F γ αβ , Q γδ αβ such that the matrix of quadratic polynomials
X αβ = K αβ + F γ αβ p γ − 2iǫQ γδ αβ p γ p δ (5.12)
have the property that lim
k→0 [X αβ , f ] = C γ αβ lim k→0 [p γ , f ]. (5.13)
and so we must find coefficients Q γδ αβ as well as momenta p α such that
C γ αβ lim k→0 [p γ , f ] = F γ αβ lim k→0 [p γ , f ] − 4 lim k→0 iǫQ γδ αβ p δ [p γ , f ]. (5.14)
Since f is arbitrary we must have therefore to lowest order ink
C γ αβ = F γ αβ − 4iǫQ γδ αβ p δ . (5.15)
This is formally and to first order the same as
C γ αβ = ∂ ∂p γ X αβ . (5.16)
If we take another derivative we obtain
∂ ∂p δ C γ αβ = ∂ 2 ∂p δ ∂p γ X αβ = −4iǫQ γδ αβ . (5.17)
From this follows the condition
∂ ∂p δ C γ αβ − ∂ ∂p γ C δ αβ = 0. (5.18)
We see then also that (5.19) and from this the integrability conditions e γ C γ αβ = 0 (5.20)
e δ C γ αβ = X ζδ ∂ ∂p δ C γ αβ = −4iǫX δζ Q γζ αβ
If the classical rotation coefficients do not satisfy this condition for some choice of pvariables they cannot be 'quantized'. We see here that the p-variables are somewhat analogous to the special variables of the classical Darboux theorem. In this case the transformation to the special coordinates is the Fourier transform x µ → p α .
The cocycle
We return now to the cocycle condition (5.31). Either F = dA for some 1-form A or there is no such 1-form. We know of no F which is not of the form F = dA but there is a case with F = dA D where A D has no regular commutative limit. We know that the Dirac operator θ = −p α θ α diverges in the commutative limit and that the limit of
A D = −ikθ (5.21)
is finite but not everywhere well defined. We notice then that the square of θ can be written as
θ 2 = 1 2 [p α , p β ]θ α θ β . (5.22)
From this it follows that
dθ + θ 2 = − 1 2 [p α , p β ]θ α θ β + 1 2 p γ C γ αβ θ α θ β = − 1 2 [p α , p β ]θ α θ β + 1 2 p γ F γ αβ θ α θ β − 2iǫp γ p δ Q γδ αβ = − 1 2 ([p α , p β ] − p γ F γ αβ + 4iǫp γ p δ Q γδ αβ )θ α θ β = −K.
(5.23)
where we have set K = 1 2 K αβ θ α θ β . We can conclude then that It follows that in the quasi-classical approximation, the linear curvature is a polynomial in the commutator J and its inverse and their derivatives.
dA D + A 2 D = ikK(
If we consider F as a Maxwell field strength then there is a source given by e α F αβ = F αγ C αβγ .
also from the condition (1.21) that the commutator must necessarily satisfy the constraint e α (J αη e η F βγ ) = 0.(5.35) This can also be written as(e α J αζ + J αη C ζ αη ) e ζ F βγ = 0. (5.36)If we equate the Expression (5.33) for the rotation coefficients with that in terms of the components of the frame we find after a few simple applications of the Leibniz rule that(dF ) αβγ = e µ [β e γ] F αµ. (5.37) The cocyle condition (5.31) is equivalent to the condition e µ [β e γ] F αµ. = 0. (5.38) An interesting particular solution is given by constants: F αµ. = F 0αµ. . αγ = J αη e η F αγ = e η J αη F αγ (5.41) and so the left-hand side vanishes if and only if e β J αβ = 0. (5.42)
27 )
27the antisymmetric part of a symmetrized tensor should vanish. The conditions satisfied by the flip are quite simple in the first approximation we are here considering. If we writeS αβ
γδ = δ β
γ δ α
δ + iǫT αβ
γδ ,
(2.28)
then we find that
Q αβ
+ γδ + T [αβ]
γδ = 0.
(2.29)
[δp α , p β ] + [p α , δp β ] = δC γ αβ p γ + C γ αβ δp γ .3)
[δe α , e β ] + [e α , δe β ] = δC γ
αβ e γ + C γ
αβ δe γ .
(5.4)
This can be simplified if we use the momenta. We conclude from (5.7) that a variation of
the momenta must satisfy the constraint
(5.5)
with
δC γ
αβ = δF γ
αβ − 4iǫδQ γδ
αβ p δ .
(5.6)
in the sense that [[p α , p β ], f ] = [[e α , e β ], f ]. (5.11)
5.24)Equations (2.23) can be written also in terms of the dual quantities It will be convenient to introduce the suggestive notationF αβ = (J −1 ) αβ .We can now rewrite the equations in terms of the inverse.From Equation (2.23) one can derive the identity e α F βγ + F αδ C δ for the derivative of the inverse if it exists. This can also be written as a 'cocycle condition' One can solve (5.30) for the rotation coefficients. One obtains C α βγ = J αη e η F βγ . (5.33)J *
αβ = 1
2 ǫ αβγδ J γδ
(5.25)
as
e α J *
βγ + C δ
α[β J *
γ]δ + C δ
αδ J *
βγ = 0.
(5.26)
C α
[αγ J *
β]δ J δγ = 0.
(5.27)
(5.28)
We could also have written
F αβ = |J| −1 J *
αβ ,
|J| 2 =
1
4
J *
αβ J αβ .
(5.29)
βγ = 0
(5.30)
dF = 0
(5.31)
if we introduce the 2-form
F = 1
2 F αβ θ α θ β .
(5.32)
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| []
|
[
"Curves on surfaces and surgeries",
"Curves on surfaces and surgeries"
]
| [
"Abdoul Karim Sane \nUMPA-ENS Lyon\n\n"
]
| [
"UMPA-ENS Lyon\n"
]
| []
| We study collections of curves in generic position on a closed surface whose complement consists of one disk only, up to orientationpreserving homeomorphism of the surface. We define a surgery operation on the set of such collections and prove that any two of them can be connected by a sequence of such surgeries. | 10.1016/j.ejc.2020.103281 | [
"https://arxiv.org/pdf/1902.06436v1.pdf"
]
| 119,152,454 | 1902.06436 | 8ce3cc236a82bfbd520582f2a25481e49dd8fe1b |
Curves on surfaces and surgeries
January 30th 2019.
Abdoul Karim Sane
UMPA-ENS Lyon
Curves on surfaces and surgeries
January 30th 2019.
We study collections of curves in generic position on a closed surface whose complement consists of one disk only, up to orientationpreserving homeomorphism of the surface. We define a surgery operation on the set of such collections and prove that any two of them can be connected by a sequence of such surgeries.
Introduction
Curves on surfaces is a rich domain in low dimensional topology and natural questions emerge from it. One of them is the counting problem under topological constraints. For instance it is well known that on a genus g surface, there are g 2 + 1 homeomorphism classes of simple closed curves. A collection of curves Γ on a genus g oriented closed surface Σ g is filling if its complement Σ g − Γ is a union of topological disks. Considering Γ as a graph embedded in Σ g whose vertices are the intersection points of Γ and whose edges are the arcs connecting intersection points, a filling collection is the same object as a map -an embedded graph whose complement consists of disks. If Γ is in generic position, all vertices have degree 4.
If Σ g − Γ consists of exactly one disk, we say that the collection is onefaced. In terms of maps one usually speaks of a unicellular map.
The counting problem for (unicellular) maps has been first considered by W. Tutte in the planar case [8,9,10,11], followed by Lehman-Walsh [12] and Harer-Zagier [5] in higher genus. They give a counting formula for the number of unicellular maps in a genus g surface and that formula has been reproved by direct bijective methods by G. Chapuy [2]. An exact formula for rooted unicellular maps (that is maps with one marked oriented edge) was provided by A. Goupil and G. Schaeffer [4]. For the number of quadrivalent maps on a genus g surface with one marked oriented edge, their formula greatly simplifies into the number (4g−2)! 2 2g−1 g! . Since marking one oriented edge multiplies the number of such maps by at most the number of oriented edges, 8g −4 in this case, the number of one-faced collections up to homeomorphism grows exponentially with the genus.
In a more topological context, T. Aougag and S. Shinnyih studied minimally intersecting pairs : those one-faced collections made of exactly two simple closed curves. They show that their number also grows exponentially with the genus. Recently we studied one-faced collections in order to find symmetric polytopes in R 4 which are not (dual) unit ball of intersection norms [7].
In this article, we endow the set of one-faced collections on a genus g surface Σ g (up to homeomorphism of Σ g ) with an additional graph structure.
Given a one-faced collection Γ and a simple arc λ connecting two edges x and y, we obtain a new collection Γ by "cutting-open" Γ along λ (see Figure 4). When the final collection is one-faced, we speak of a surgery and denote it by σ x,y (Γ). We then define the surgery graph K g as the graph whose vertices are homeomorphism classes of one-faced collections of curves on Σ g and whose edges are pairs of collections connected by surgery.
For example, if g = 1 there is only one one-faced collection, so K 1 consists of one isolated vertex. Using the Goupil-Schaeffer formula for g = 2, one checks that there are exactly six one-faced collections on Σ 2 . 1 By inductively trying all possible surgeries, one obtains those six one-faced collections.
One notices that K 2 is connected (see Figure 1). Our main result is :
Theorem 1. For every integer g the graph K g is connected.
Our proof is not straightforward as one may hope. We define a connected sum operation on one-faced collections that turns two one-faced collections on Σ g 1 and Σ g 2 into a new one-faced collection on Σ g 1 +g 2 . We then considers the surgery-sum graph K g as the union i≤g K i where one adds an edge between Γ 1 and Γ 2 if Γ 2 can be realized as a connected sum of Γ 1 with the unique one-faced collection on the torus. We then have : Theorem 2. For every g, the graph K g is connected.
It is obvious that Theorem 1 implies Theorem 2. However we were not able to find a direct proof of Theorem 1, that is why we first prove Theorem 2 and then use the main ingredient of the proof (Proposition 2) to prove Theorem 1.
1. The Goupil-Schaeffer Formula gives 45 marked collections, but every one-faced collection corresponds to 3, 6, or 12 marked collections depending on the number of symmetries of the collection. Now, denoting by D g the diameter of K g , we also prove the following :
Theorem 3. For every g, we have D g ≤ 3g 2 + 9g − 12.
Organization of the article : In Section 2, we recall some facts on onefaced collections. In Section 3, we define the surgery operation on a one-faced collection and the connected sum of two one-faced collections. In Section 4, we prove a couple of lemmas which are going to be useful for the proof of Theorem 1, Theorem 2 and Theorem 3 in section 5.
Combinatorial description of one-faced collections
In the whole paper, Σ g denotes an oriented closed surface of genus g. Let Γ be a filling collection on Σ g . One can consider Γ as a graph embedded in Σ g . Let V be the number of vertices of Γ, E the number of edges and F the number of faces. The Euler characteristic of Σ g is given by χ(Σ g ) = 2 − 2g = V − E + F . As Γ defines a regular 4-valent graph we have E = 2V , which implies V = 2g − 2 + F. If Γ is one-faced, we then have
V = 2g − 1, E = 4g − 2, F = 1.
From this one can see that there is only one one-faced collection on the torus. We call it Γ T (see Figure 2).
Gluing pattern for a one-faced collection : Let Γ be a one-faced collection on Σ g , then Σ g − Γ is a polygon with 8g − 4 edges, that we denote by P Γ . The polygon P Γ comes with a pairwise identification of its edges. Choosing an edge on P Γ as an origin, one can label the edges of P Γ from the origin in a clockwise manner ; thus obtaining a word W Γ on 8g − 4 letters. If two edges are identified, we label them with the same letter with a bar on the letter of the second edge. The word W Γ is a gluing pattern of (Σ g , Γ) and two gluing patterns of (Σ g , Γ) differ by a cyclic permutation and a relabeling. A letter of a gluing pattern associated to a one-faced collection corresponds to a side of an edge of Γ ; thus we can see a letter as an oriented edge.
Exemple 2.1. The word W = abāb is a gluing pattern for the one-faced collection Γ T on the torus. We have the following : Proposition 1. Let Γ 1 and Γ 2 be two one-faced collections on Σ g . Then Γ 1 and Γ 2 are topologically equivalent if and only if they have the same gluing patterns up to cyclic permutation an relabeling.
Proof. Let us assume that Γ 1 and Γ 2 have the same gluing patterns. Since Σ g − Γ 1 and Σ g − Γ 2 are disks, they are homeomorphic. The fact that Γ 1 and Γ 2 have the same gluing patterns implies that one can choose a homeomorphismφ
:
P Γ 1 −→ P Γ 2
such thatφ maps a couple of identified sides on P Γ 1 to a couple of identified sides on P Γ 2 . Therefore,φ factors to φ on the quotient (identification of sides) and φ(Γ 1 ) = Γ 2 .
Conversely, if φ(Γ 1 ) = Γ 2 then Γ 1 and Γ 2 have the same gluing pattern.
One-faced collections as permutations : Maps on surfaces can be described by a triple of permutations which satisfy some conditions (see [6] for the combinatorial definition of maps). We restrict that definition to the case of one-faced collection.
To a one-faced collection we associate H : the set of oriented edges, an involution α of H which maps an oriented edge to the same edge with opposite orientation and a permutation µ whose cycles are oriented edges emanating from vertices when we turn counter-clockwise around them. If W Γ is a gluing pattern of Γ, we can take H to be the set of letters of W Γ . The permutation γ is then the shift to the right and it corresponds to the unique face. The cycles of α and µ correspond to the edges and vertices of Γ, respectively. Moreover, if we fix an origin x ∈ H we get a natural order from γ :
x < γ(x) < ...... < γ 8g−3 (x).
Changing the origin, the order above changes cyclically. The cycles of µ are in one to one correspondence with the vertices of Γ. (see Figure 3).
If x and y are two oriented edges with x < y, they define two intervals in a gluing pattern for Γ :
Surgery and connected sum on one-faced collections
In this section, we define two topological operations on the set of onefaced collections.
Surgery on a one-faced collection : Let Γ be a one-faced collection on Σ g , x and y be two oriented edges of Γ (x and y correspond to two sides of P Γ ). Since Γ is one-faced, there is a unique homotopy class of simple arcs whose interiors are disjoint from Γ and with endpoints in x and y ; let us denote it by λ x,y . We obtain a new collection denoted by σ x,y (Γ) by "cuttingopen" Γ along λ x,y (see Figure 4). The collection σ x,y (Γ) is not necessarily one-faced. Otherwise, we say that {x, y} and {x,ȳ} are not intertwined. By abuse, we will just say that x and y are intertwined or not intertwined. Now, the following lemma gives a necessary and sufficient condition for the above operation to preserve the one-faced character.
Lemma 3.1. Let Γ be a one-faced collection, x and y be two oriented edges of Γ. Then σ x,y (Γ) is one-faced if and only if x and y are intertwined. In this case, we call the operation a surgery on Γ between x and y.
Moreover, if w 1 xw 2x w 3 yw 4ȳ is a gluing pattern for Γ then,
w 3 Xw 2X w 1 Y w 4Ȳ
is a gluing pattern for σ x,y (Γ).
Proof. Since the operation along λ x,y leads to a new collection Γ := σ x,y (Γ), all we have to do is to prove that Γ is one-faced. We use a cut and past argument similar to several proofs of the classification of surfaces (see Figure 5).
Assume first that x and y are intertwined. When we "cut-open" along λ x,y , the edges {x,x} and {y,ȳ} get replaced by new edges {X,X} and {Y,Ȳ }. When we cut along the two new edges (in the polygonal description) and glue along the old ones (see Figure 5), we obtain a polygon ; that is Γ is one-faced with gluing pattern On the other hand, if x and y are not intertwined, one constructs an essential curve disjoint from Γ , so Γ is not one-faced (see Figure 6).
w 3 Xw 2X w 1 Y w 4Ȳ . w 1 x w 2 x w 3 y w 4 y λ x,y / / w 1 x w 2 x w 3 y w 4 y w 1 x w 2 x w 3 y w 4 y / / w 3 X w 2 X w 1 Y w 4Ȳ
Remark 3.1. The word W σx,y in Lemma 3.1 is obtained by permuting w 1 and w 3 . It is also equivalent to permute w 2 and w 4 .
x y Figure 6 -The two arcs in red color define an essential closed curve on Σ g disjoint from Γ since it intersects Γ algebraically twice.
Remark 3.2. If x and y are two intertwined oriented edges, thenx and y are also intertwined. Moreover one has σ x,y (Γ) = σx ,ȳ (Γ). In fact, by
Lemma 3.1, if W Γ = w 1 xw 2x w 3 yw 4ȳ is a gluing pattern for Γ, W σx,ȳ(Γ) = w 1 Xw 4X w 3 Y w 2Ȳ
is a gluing pattern for σx ,ȳ (Γ), and it is equivalent to W σx,y(Γ) = w 3 Xw 2X w 1 Y w 4Ȳ up to cyclic permutation and relabeling. Remark 3.3. Given a one-faced collection, there are always intertwined pairs unless it is the one-faced collection in the torus. Indeed, if all pairs of Γ are not intertwined a gluing pattern for Γ is given by
W Γ = x 1 x 2 .....x 4g−2x1x2 ....x 4g−2 .
After identifying the sides of P Γ , all the vertices of P Γ get identified. Thus Γ has only one self-intersection point. It follows that g = 1 and that Γ is the only one-faced collection with one self-intersection point, namely Γ T .
Connected sum : Let Γ 1 and Γ 2 be two one-faced collections on two surfaces Σ 1 and Σ 2 , respectively. Let D 1 and D 2 be two open disks on Σ 1 and Σ 2 , disjoint from Γ 1 and Γ 2 , respectively. Let Σ g 1 #Σ g 2 be the connected sum along D 1 and D 2 . Then (Σ g 1 #Σ g 2 , Γ 1 ∪ Γ 2 ) is a genus g 1 + g 2 surface endowed with a collection Γ 1 ∪ Γ 2 . Since Γ 1 and Γ 2 are one-faced, the complement of Γ 1 ∪ Γ 2 in Σ g 1 #Σ g 2 is an annulus. Now, let x and y be two oriented edges of Γ 1 and Γ 2 respectively, and λ x,y a simple arc on Σ g 1 #Σ g 2 from x to y whose interior is disjoint from Γ 1 ∪ Γ 2 . The arc λ x,y joins the two boundary components of
Σ g 1 #Σ g 2 − Γ 1 ∪ Γ 2 . The- refore the graph Γ 1 ∪ Γ 2 ∪ λ x,y fills Σ g 1 #Σ g 2 with one disk in its complement.
Thus, the collection Γ := (Γ 1 ∪Γ 2 ∪λ x,y )/λ x,y -the quotient here means the contraction of λ x,y to a point-(see Figure 7) is a one-faced collection. We say that Γ is the connected sum of the marked collections (Γ 1 , x) and (Γ 2 , y).
Definition 3.2. Let Γ be a one-faced collection, x and y be two oriented edges of Γ. We say that x and y are symmetric if the gluing pattern for Γ starting at x is the same as the one starting at y up to relabeling. The following lemma states how we obtain a gluing pattern for (Γ 1 , x)#(Γ 2 , y) from gluing patterns for Γ 1 and Γ 2 .
Lemma 3.2. If W Γ 1 = xw 1x w 2 (respectively W Γ 2 = yw 1ȳ w 2 ) is a gluing pattern for Γ 1 (respectively Γ 2 ), then x 1 w 1x1 x 2 w 2x2 y 1 w 1ȳ 1 y 2 w 2ȳ 2
is a gluing pattern for (Γ 1 , x)#(Γ 2 , y).
Moreover, (Γ 1 , x)#(Γ 2 , y) and (Γ 1 , x)#(Γ 2 , y ) are topologically equivalent if y and y are symmetric.
Proof. The proof can be read on Figure 7. Lemma 3.2 implies that the connected sum of a one-faced collection Γ with Γ T depend only on the oriented edge we choose on Γ, since all oriented edges of Γ T are symmetric.
Surgery classification of one-faced collections
In this section, we study how surgeries connects one-faced collections.
Simplification of one-faced collections : Let Γ be a one-faced collection, (H, α, µ, γ) the permutations associated to Γ (Definition 2.1) and x an oriented edge of Γ. Then the oriented edges x and C(x) := γαγ(x) belong to the same curve β ∈ Γ ; x and C(x) are consecutive along β (see Figure 8). Moreover, the sequence (C n (x)) n is periodic and it travels through all edges of β. Definition 4.1. Let Γ be a one-faced collection and θ ∈ Γ a simple curve. The curve θ is 1-simple if θ intersects exactly one time Γ − θ.
Note that we have C(x) = x if and only if x is a side of a 1-simple curve θ ∈ Γ. We denote by S Γ the number of 1-simple closed curves in Γ.
x y = γ(x)ȳ = αγ(x) C(x) := γαγ(x)Γ into σ x,C(x) (Γ) is called a simplification.
The name is explained by :
Lemma 4.1. If Γ := σ x,C(x) (Γ) is a simplification, then S Γ = S Γ + 1.
In other words, a surgery on Γ between x and C(x) creates an additional 1-simple curve in Γ .
Proof. Suppose that x and C(x) are intertwined, a gluing pattern for Γ is given by :
W Γ = (tw 1 )x(yw 2t )x(w 3ȳ )C(x)w 4 C(x).
Therefore, by Lemma 3.1 a gluing pattern for Γ := σ x,C(x) (Γ) is given by :
W Γ = w 3ȳ Xyw 2tX tw 1 Zw 4Z . So, in Γ we have C(X) = γαγ(X) = γα(y) = γ(ȳ) = X.
It implies that X is the side of simple curve which intersects Γ only once. Now if θ 1 and θ 2 are two 1-simple curves of a one-faced collection, then θ 1 and θ 2 are disjoint ; otherwise θ 1 ∪θ 2 would be disjoint from Γ, that is absurd since Γ is connected. So the number S Γ of 1-simple curves on a one-faced collection Γ is bounded by the genus g of the underlying surface. Therefore a sequence of simplifications on a one-faced collection stabilizes at a collection on which no simplification can be applied anymore. Order around vertices of a non simplifiable collection : In this paragraph, we will show that vertices of non simplifiable one-faced collections are of certain types.
Let Γ be a one-faced collection and (H, α, µ, γ) the permutations associated to Γ ; H being the set of letters of a gluing pattern for Γ. If we fix an origin x 0 ∈ H, we then get an order on H :
x 0 < γ(x 0 ) < ... < γ 8g−3 (x 0 ).
Therefore, if v is a vertex of Γ defined by a cycle (txyz) of µ, we get a local order around v by comparing t, x, y and z. Since each letter corresponds to an oriented edge which leaves an angular sector of v (see Figure 3), the local order around v corresponds also to a local order on the four angular sectors around v when running around Γ with γ.
Definition 4.4.
Let v be a vertex defined by the oriented edges (t, x := µ(t), y := µ 2 (t), z := µ 3 (t)) with t = min{t, x, y, z} relatively to an order of edges on Γ. Then,
v is a vertex of Type 1 if t < x < y < z ; v is a vertex of Type 2 if t < z < y < x. Otherwise, the vector v is a vertex of Type 3.
Up to rotation and change of origin, we have the three cases depicted on Figure 9. Proof. All we have to do is to write the possible gluing patterns of Γ by figuring out the order of the edges around a vertex and look when consecutive edges are intertwined or not.
Case 1 : If v is a vertex of Type 1, then a gluing pattern for Γ is given by :
W Γ = w 1z tw 2t xw 3x yw 4ȳ z.
Therefore, one cheks thatx and C(x) = z are not intertwined ; so aret and C(t) = y. Hence, no simplification is possible around v.
Case 2 :
If v is a vertex of Type 2, the gluing pattern for Γ is
W Γ = w 1z tw 2ȳ zw 3x yw 4t x.
Then,x and C(x) = z are not intertwined ; so aret and C(t) = y. Again, no simplification is possible around v in this case.
Case 3 :
If v is a vertex of Type 3, then
W Γ = w 1z tw 2t xw 3ȳ zw 4x y.
Here,t and C(t) = y are intertwined and a simplification is possible. So Γ is non simplifiable if and only all is vertices are of Type 1 or Type 2.
Number of vertices of Type 1 and 2 in a non simplifiable one-faced collection : In [2], G. Chapuy has defined a notion which catches the topology of a unicellular map : trisection. We recall one of his results about trisection. Let G be an unicellular map and (H, α, µ, γ) the permutation associated to G. Let v be a degree d vertex of G defined by a cycle (x 1 x 2 ...x d ) of µ, with x 1 = min{x 1 , ..., x d } relatively to an order on H.
If x i > x i+1 we say that we have a down-step. Since x 1 = min{x 1 , ..., x d }, one has x d > x 1 ; the other down-steps around v are called non trivial. [2]). Let G be unicellular map on a genus g surface. Then G has exactly 2g trisections.
Applying the trisection lemma to one-faced collections, we get : Corollary 4.1. A non simplifiable one-faced collection on Σ g has g vertices of Type 2 and g − 1 vertices of Type 1.
Proof. A vertex of Type 2 (respectively a vertex of Type 1) has two trisections (respectively zero trisection) (see Figure 9). If N i is the number of vertices of Type i (i=1,2), by the trisection lemma we have 2N 2 = 2g, so
N 2 = g. Since V Γ = N 1 + N 2 = 2g − 1, it follows that N 1 = g − 1.
Repartition of vertices on a non simplifiable collection : Now, we show that using surgeries, we can re-order the vertices of a non simplifiable one-faced collection. If v 1 and v 2 are two vertices represented by the cycles (abcd) and (ef gh), respectively, they are adjacent if and only if there exist x ∈ {a, b, c, d} such thatx ∈ {e, f, g, h}.
We now show that some configurations of vertices "hide" simplifications ; that is from those configurations we can create new simplifications after a suitable surgery without killing the old ones.
Lemma 4.4. Let Γ be a non simplifiable one-faced collection. If Γ contains two vertices of Type 2 which are adjacent, then there is a sequence of surgeries Γ = Γ 0 −→ Γ 1 −→ ... −→ Γ n from Γ to Γ n such that Γ n is non simplifiable and S Γ < S Γn .
Proof. Let v 1 and v 2 be two adjacent vertices of Type 2 defined by the cycles (bfḡā) and (cdēb), respectively (see Figure 10).
Let us fix an oriented edge as an origin, so that a = min{a,c, d, e, f, g};
that is the first time we enter in the local configuration is by the oriented edge a. Then we have the following order :
a < b < c < g <ā < f <ḡ < e <b <f < d <ē <c <d.
Otherwise, it would contradict the fact that the two vertices are of Type 2. A gluing pattern for Γ is given by : The oriented edges b andḡ are intertwined, so we can define Γ := σ b,ḡ (Γ). By Lemma 3.1, a gluing pattern for Γ is :
W Γ = w 1 abcw 2 gāw 3 fḡw 4 ebf w 5 dēw 6cd .W Γ =āw 3 f Bcw 2 Gf w 5 dēw 6cd w 1 aḠw 4 eB.
The cycles (Ḡf Bā) and (Bcdē) define the two vertices of Γ in Figure 10 and the orders around these two vertices are : G <ā < B <f ;B < c <ē <d.
Therefore, the vertex (B, c,d,ē) is a vertex of Type 3 and it implies that Γ is simplifiable. Indeed the operation on Figure 10 does not touch any 1simple curve of Γ and each simplification increases strictly the number of 1-simple. Let Γ n be a non simplifiable collection obtained after finitely many simplifications on Γ ; so S Γ < S Γn .
Let v 1 and v 2 be two vertices of Type 1 and Type 2 defined by the cycles (cdef ) and (gābc), respectively, such that v 1 and v 2 are adjacent. The local configuration in this case is depicted on Figure 11 and we assume that a = min{a,b,d,ē,f ,ḡ}. Proof. Since v 2 is a vertex of Type 2, min{b,d,ē,f ,ḡ} is different fromb and f . Case 1 : If min{b,d,ē,f ,ḡ} =d, the fact that the vertices are of Type 1 and 2 implies that the local order is either a < b <d < e <ē < f <ḡ <ā <f <c < g <b < c < d, or
a < b <d < e <ḡ <ā <ē < f <f <c < g <b < c < d.
Sub-case 1 : If a < b <d < e <ē < f <ḡ <ā <f <c < g <b < c < d, then W Γ = w 1 abw 2d ew 3ē f w 4ḡā w 5fc gw 6b cd is a gluing pattern for Γ. The oriented edges a andē are intertwined and a gluing pattern for Γ := σ a,ē (Γ) is given by
W Γ = w 3 Abw 2d Ew 5fc gw 6b cdw 1Ē f w 4ḡĀ .
The cycles (bcgĀ) and (Efcd) define the two vertices in Figure 12. Moreover, b < g < c <Ā and E <c < d < f that is they are vertices of Type 3. Therefore, Γ is simplifiable and there is sequence of simplification from Γ to Γ n such that Γ n is non simplifiable and N Γn > N Γ = N Γ . The equality N Γ = N Γ holds since the surgery in this case does not touch a 1-simple curve.
Sub-case 2 : If a < b <d < e <ḡ <ā <ē < f <f <c < g <b < c < d, then a gluing pattern for Γ is W Γ = w 1 abw 2d ew 3ḡā w 4ē f w 5fc gw 6b cd. Here again, the oriented edges a and f are intertwined and a gluing pattern for Γ := σ a,f (Γ) is given by :
W Γ = w 4ē Abw 2d ew 3ḡĀc gw 6b cdw 1 F w 5F .
The two vertices in Figure 13 are defined by the cycles (bcgĀ) and (Acde). Moreover, b <Ā < g < c and A < e <c < d. It follows that the vertex (Acde) is a vertex of Type 3 and therefore, Γ is simplifiable. Case 2 : if min{b,d,ē,f ,ḡ} =ē, then the local order is given by : a < b <ē < f <ḡ <ā <f <c <ḡ <b < c < d <d < e, and a gluing pattern for Γ is given by :
W 4 = w 1 abw 2ē f w 3ḡā w 4fc gw 5b cdw 6d e.
In this case, a and d are intertwined. A gluing pattern for Γ := σ a,d (Γ) is given by :
W Γ = w 4fc gw 5b cAbw 2ē f w 3ḡĀ ew 1 Dw 6D .
The cycles (cAef ) and (gĀbc) represent the two vertices in Figure 14 and c < A < f < e. So, the vertex (cAef ) is a vertex of Type 3 and Γ is simplifiable. Figure 11) Let v 1 and v 2 be two adjacent vertices of Type 1 and 2, respectively. We say that we have a good order around v 1 and v 2 ifḡ = min{b,d,ē,f ,ḡ}. -Γ is non simplifiable, -no two vertices of Type 2 are adjacent, -the local orders around two adjacent vertices of Type 1 and 2 are good.
Lemma 4.6. Let Γ be a non simplifiable one-faced collection. Then there is a sequence of surgeries Γ 0 = Γ −→ Γ 1 ... −→ Γ n such that Γ n is an almost toral one-faced collection.
Proof. If Γ is not almost toral, by Lemma 4.4 and Lemma 4.5 there is a sequence of surgeries Γ 0 = Γ −→ Γ 1 −→ ... −→ Γ n such that Γ n is non simplifiable and S Γ < S Γn , i.e, we create new 1-simple curves after some suitable surgeries. Since the number of 1-simple curves is bounded by the genus, those operations stabilize to an almost toral one-faced collection. Now, we are going to improve the configuration of the vertices of an almost toral one-faced collection.
Lemma 4.7. Let Γ be an almost toral one-faced collection, v 1 and v 2 be two adjacent vertices of Type 1 and Type 2, respectively ; with a = min{a,b,d,ē,f ,ḡ} (see Figure 11). If x := min{b,d,ē,f } is adjacent to a vertex of Type 2, then there is a sequence of surgeries Γ 0 = Γ −→ Γ 1 −→ ... −→ Γ n such that Γ n is almost toral and N Γ < N Γn .
Proof. If x := min{b,d,ē,f } is adjacent to a vertex v := (xȳtū), a gluing pattern of Γ is given by (see Figure 15) : Figure 15 Since v is a vertex of Type 2, x <ū <t <ȳ. It implies thatt ∈ w 4 and u ∈ w 4 .
W Γ = w 1 aw 2ā w 3 uxw 4xȳ . ā a x x ū u t t yȳ
The oriented edges a and x are intertwined and
W Γ = w 3 uAw 2Āȳ w 1 Xw 4X
is a gluing pattern for Γ := σ a,x (Γ). The vertex v in Γ is defined by the cycle (Aȳtū) and one checks that A <ȳ <ū <t ; that is v is a vertex of Type 3 and Γ is simplifiable. Hence, there is a sequence of simplification which strictly increases the number of 1-simple curves.
Definition 4.9. Let Γ be a one-faced collection. We say that Γ is a toral one-faced collection if Γ is an almost toral one-faced collection and if every vertex of Type 1 is adjacent to at most two vertices of Type 2.
Lemma 4.8. Let Γ be an almost toral one-faced collection. Then there is a sequence of surgeries Γ 0 = Γ −→ Γ 1 −→ ... −→ Γ n such that Γ n is a toral one-faced collection.
Proof. Let v be a vertex of Γ of Type 1. If v is adjacent to 4 vertices of Type 2 or 3 vertices all of which are of Type 2, Lemma 4.7 implies that there is a sequence of surgeries Γ 0 = Γ −→ Γ 1 −→ ... −→ Γ n such that S Γ < S Γn . Since the number of 1-simple curves is bounded by the genus g, there is a sequence of surgeries Γ 0 = Γ −→ Γ 1 −→ ... −→ Γ n such that Γ n is almost toral and such that every vertex of Type 1 adjacent to three vertices of Type 2 is also adjacent to a fourth of Type 1. The local configuration around those vertices is depicted in Figure 16, withē = min{d,ē,f } . A gluing pattern for Γ n is given by :
W Γn = w 1 aw 2ā w 3 dw 4d .
The oriented edges a and d are intertwined and the surgery σ a,d (Γ n ) decreases the number of adjacent vertices to v (see Figure 16). Following this process, we get a toral one-faced collection after finitely many surgeries. Proof. We have to show that there is a vertex of Type 2 which is adjacent to exactly one vertex of Type 1.
Assume that every vertex of Type 2 is adjacent to at least two vertices of Type 1. Let N 1 and N 2 be the number of vertices of Type 1 and Type 2 respectively, and let N 1,2 be the number of pairs of vertices of Type 1 and Type 2 which are adjacent.
Since Γ is a toral one-faced collection, any vertex of Type 1 has at most two vertices of Type 2. It implies that,
N 1,2 < 2N 1 .
On the other part, we have assumed that every vertex of Type 2 is adjacent to at least two vertices of Type 1. Therefore,
2N 2 ≤ N 1,2 .
Combining the two inequalities above, we get N 2 ≤ N 1 which contradicts the fact that we have g − 1 vertices of Type 1 and g vertices of Type 2 in a non simplifiable one-faced collection.
So, there is a vertex v 0 of Type 2 which is adjacent to exactly one vertex v 1 of Type 1. As vertices of Type 2 are not adjacent, v 0 lies on a 1-simple curve. ( * )
Next, we show that v 1 can be transformed into a self-intersection point (if it is not the case) by a surgery. Assume that v 1 is not a self-intersection point. Then a gluing pattern for Γ is given by :
W Γ = w 1 xyw 2ȳ zw 3z tw 4tx ;
where v 1 is defined by the cycle (yztx) (Figure 17). The oriented edges y and z are intertwined and a gluing pattern for Γ := σ y,z (Γ) is given by :
W Γ = Y w 2Ȳ tw 4tx w 1 xZw 3Z
The vertex v 1 in Γ is defined by the cycle (Y txZ) and Y < t <x < Z ; that is v 1 is still a vertex of Type 1. Moreover, v 1 get transformed to a selfintersection point. So, the surgery on Γ between y and z has transformed v 1 to a self-intersection point of Type 1. ( * * )
Finally, ( * ) and ( * * ) implies that
Γ = (Γ , x)#(Γ T , x 0 );
with Γ a one-faced collection on Σ g−1 .
Proof of the main theorems
In this section, we prove Theorem 1, Theorem 2 and Theorem 3. We recall that the graph K g is the graph whose vertices are homeomorphism classes of one-faced collections on Σ g , and on which two vertices Γ 1 and Γ 2 are connected by an edge if there is a surgery which transforms Γ 1 into Γ 2 (if a surgery on Γ fix Γ, we do not put a loop). The graph K g := i≤g K i is the disjoint union of the graphs K i on which we add an edge between two one-faced collections Γ 1 and Γ 2 on Σ i and Σ i+1 respectively if Γ 2 is a connected sum of Γ 1 with the one-faced on the torus.
The following proposition is the main technical result : it easily implies Theorem 2. It also implies Theorem 1 with a bit of extra-work and its proof uses most lemmas of Section 4.
Proposition 2.
Let Γ be a one-faced collection on Σ g+1 . Then there is a finite sequence of surgeries Γ := Γ 0 −→ ... −→ Γ n and a one-faced collection Γ on Σ g with a marked edge x such that Γ n = (Γ , x)#Γ T .
Proof. Let Γ be a one-faced collection on Σ g , there is a sequence of surgeries Γ −→ ... −→ Γ 1 such that Γ 1 is non simplifiable. By Lemma 4.6, there is a sequence of surgeries Γ 1 −→ ... −→ Γ 2 such that Γ 2 is almost toral and by Lemma 4.7, there is a sequence of surgeries Γ 2 −→ ... −→ Γ 3 such that Γ 3 is toral. Lemma 4.9 implies that Γ 3 = (Γ , x)#Γ T with Γ a one-faced collection in Σ g−1 .
We can now prove Theorem 2, which states that for every g the graph K g is connected.
Proof of Theorem 2. Let Γ ∈ K g . By Proposition 2, there is path in K g from Γ to (Γ , x)#Γ T where Γ ∈ K g−1 . Thus, there is path in K g from Γ to Γ . By induction on g, we deduce a path from Γ to Γ T . So, K g is connected.
Now, we turn to the proof of Theorem 1. Let us start with some preliminaries.
Lemma 5.1. Let Γ be a one-faced collection on Σ g and x an oriented edge of Γ. Then there is a surgery from (Γ, x)#Γ T to (Γ,x)#Γ T .
Lemma 5.1 states that up to surgery the connected sum of Γ with Γ T depends only on the edge we choose on Γ but not on its orientation.
Proof. Let W Γ := w 1 xw 2x be a gluing pattern for Γ, Γ 1 := (Γ, x)#Γ T and Γ 2 := (Γ,x)#Γ T . We recall that W Γ T = abāb is a gluing pattern for Γ T . By Lemma 3.2,
W Γ 1 = x 1 w 1x1 x 2 w 2x2 a 1 bā 1 a 2bā2
and
W Γ 2 = x 1 w 2x1 x 2 w 1x2 a 1 bā 1 a 2bā2
are gluing patterns of Γ 1 and Γ 2 , respectively. Thus, in W Γ 1 , x 1 and x 2 are intertwined, Γ := σ x 1 ,x 2 (Γ 1 ) is one-faced, with gluing pattern
W Γ = x 1 w 2x1 x 2 w 1x2 a 1 bā 1 a 2bā2 .
We check that W Γ = W Γ 1 . So, σ x 1 ,x 2 (Γ 1 ) = Γ 2 . Definition 5.1. We call g-necklace the homeomorphism class of the onefaced collection on Σ g , denoted by N g , with g 1-simple curves and one spiraling curve η with g − 1 self intersection points (see Figure 18 for the 5necklace).
Remark 5.1. Intersection points between 1-simple curves and γ in N g are of Type 2 ; the others are Type 1 vertices. There are g − 1 vertices of Type 2 which are adjacent to exactly one vertex of Type 1 and one special vertex of Type 2 which is adjacent to two vertices of Type 1.
We can now prove Theorem 1, namely given Γ 1 and Γ 2 are two one-faced collections on a genus g surface Σ g there is a finite sequence of surgeries from Γ 1 to Γ 2 .
Proof of Theorem 1. We give a proof by induction on g. Assume K g is connected. Let Γ an one-faced collection on Σ g+1 . By Proposition 2, there exists a sequence of surgeries Γ = Γ 0 −→ ... −→ Γ n where Γ n is of the form (Γ , x)#Γ T .
Since we have assumed that K g is connected, then there is a sequence of surgeries Γ = Γ 0 −→ ... −→ Γ n = N g from Γ to N g (the g-necklace). This sequence lifts to a sequence Γ = (Γ , x)#Γ T −→ ... −→ (N g , x n )#Γ T of surgeries on Σ g+1 . Indeed if x is an oriented edge of Γ, then x brokes into two oriented edges x 1 and x 2 . The surgery σ x,y (Γ ) (respectively σ z,y (Γ )) lift to σ x 1 ,y (Γ) (respectively σ z,y (Γ)).
By Lemma 5.1, up to surgery the way we glue Γ T on N g depends only on the edges of N g but not on there sides. It follows that there are three situations depending whether : x n is the side of an edge connecting two vertices of Type 1, or one vertex of Type 1 and the special vertex of Type 2, x n is the side of an edge connecting one vertex of Type 1 and one vertex of Type 2 which is not the special one. x n lies on a 1-simple closed curve. The first situation leads to the (g + 1)-necklace N g+1 , and for the other two situations there is a path to the (g + 1)-necklace. We give the paths for the genus 5 case in Figure 19 ; the other cases inductively follow the same sequence of surgeries.
Since K 1 (a single vertex) is connected, by induction K g is also connected. Figure 19 -Sequence of surgeries to the necklace. The sequence on the first arrow goes from the case where Γ T is glued on a 1-simple curve to the case where Γ T is glued between one vertex of Type 1 and one vertex of Type 2. The second arrow leads to the (g + 1)-necklace. The red arcs are the arcs on which we apply surgeries.
/ / / / / / / / / /
Now we turn to the question of the diameter of K g that we denote by D g . We prove Theorem 4. For every g we have D g ≤ 3g 2 + 9g.
Proof. Let d g := max{d(Γ, N g )} be the maximal distance to the necklace. By Proposition 2, if Γ is a one-faced collection, there is sequence s n of surgeries from Γ to Γ n such that Γ n is toral. In this sequence, we have three kind of steps :
-making an apparent simplification on a vertex of Type 3 ; let m be their numbers, -making a hidden simplification, that is a simplification which follows a suitable surgery as in Lemma 4.4 ; let n be their numbers, -making a surgery which are not followed by simplification as in Figure 16 ; let k be their numbers. It follows that the length l(s n ) is equal to m + 2n + k ; with m + n ≤ g and k ≤ g − 1 (since the last step correspond to a surgery around vertices of Type 1).
The maximum is reached when every simplification follows a suitable surgery ; that is m = 0 and n = g. So we have l(s n ) ≤ 3g − 1.
Since Γ n = (Γ , x)#Γ T , it follows that Γ is at most at distance 3g−1+d g−1 of (N g−1 , y)#Γ T . So, d(Γ, N g ) ≤ 3g + 3 + d g−1 ,
for (N g−1 , y)#Γ T is at most at distance 4 of N g (see Figure 19). Hence d g ≤ 3g + 3 + d g−1 ;
and by induction on g D g ≤ 2d g ≤ 3g 2 + 9g − 12.
Question 1. The characterization of a surgery on a one-faced (Lemma 3.1) collection still holds for the general case, namely for unicellular maps. Therefore, one can wonder whether the surgery graph of unicellular maps is connected.
A surgery on a unicellular map Γ leaves the degree partition of Γ invariant. Therefore, the surgery graph for unicellular maps is for unicellular maps with the same degree partition.
Among one-faced collections, there is a big class of those made by only simple curves. We know that their number grows exponentially with the genus [1]. Question 2. Is the surgery graph of those one-faced collection made by simple curves connected ? Is the surgery graph of minimally intersecting pairs connected ?
In the first case, surgeries are allowed only between intertwined oriented edges belonging to the same curve or to two disjoint curves. For minimally intersecting filling pairs, surgery are allowed only between intertwined oriented edges on different sides of the same curve.
Using the Goupil-Schaeffer formula for the number of one-faced collections, one can show that D g is (asymptotically) at least linear on g.
Question 3. Is the diameter D g linear on g ? Is the family (K g ) an expander ?
Bibliography
Figure 1 -
1The graphs K 2 and K 1 . The dashed edge indicates the vertex obtained by the connected sum of two copies of the unique collection in K 1 .
Figure 2 -
2The one-faced collection Γ T on the torus.
Definition 2 . 1 .
21The triple (H, α, µ) is the combinatorial definition of a one-faced collection and γ := αµ describes the face of Γ.
[x, y] := {t, x < t < y}, [y, x] := {t, t < x} ∪ {t,
Figure 3 -
3A vertex, the four oriented edges emanating from it, and the four letters of the gluing pattern. The cycle (ijkl) is a cycle of µ.
Exemple 2 . 2 .
22The one-faced collection on the torus is given by the following permutations : µ = (abāb); α = (aā)(bb); γ = (abāb).
Figure 4 -
4Surgery between two oriented edges x and y along λ x,y . On the left-hand side, we have the arc λ x,y from x to y and on the right-hand side the cutting-open operation along λ x,y .Definition 3.1. Let Γ be a one-faced collection, x and y be to oriented edges of Γ. We say that {x, y} and {x,ȳ} are intertwined ifx andȳ are not both in [x, y] and not both in [y, x]. It means that Γ admits a gluing pattern of the form w 1 xw 2x w 3 yw 4ȳ .
Figure 5 -
5Cut and paste on the polygon P Γ .
Figure 7
Exemple 3 . 1 .
31On Γ T , any two oriented edges are symmetric.
Figure 8 -
8An oriented edge x and its consecutive C(x).
Definition 4 . 2 .
42Assume that x is an edge not lying on a 1-simple component of Γ, and that x and C(x) are intertwined. The operation transforming
Definition 4. 3 .
3A collection Γ is non simplifiable if one cannot do a simplification from it, i.e, x and C(x) are always non intertwined.
Figure 9 -
9Different types of vertices of a one-faced collection.
Lemma 4. 2 .
2A one-faced collection Γ is non simplifiable if and only if all of its vertices are of Type 1 or Type 2.
Definition 4.5 (G. Chapuy). A trisection is a down-step which is not a trivial one.
Lemma 4 . 3 (
43The trisection lemma ; G. Chapuy
Definition 4 . 6 .
46Let G be a graph. Two vertices are adjacent if they share an edge.
Figure 10 -
10Surgery which creates new simplifications. On the left figure, a < g < f < e < d <c is the order by which we pass through the eight sectors. On the figure on the right, we focus on the angular order around v 1 and v 2 . The order on the figure on the right comes from that of the figure on the left. At each time we leave the local configuration on the figure on the right, we come back on it in the same way like in the figure on the left.
Lemma 4 . 5 .
45If min{b,d,ē,f ,ḡ} =ḡ, then there is a sequence Γ = Γ 0 −→ Γ 1 −→ ... −→ Γ n from Γ to Γ n such that Γ n is non simplifiable and S Γ < S Γn .
Figure 11
Figure 12
Figure 13
Figure 14
Definition 4.8. A one-faced collection Γ is almost toral if :
Figure 16
Lemma 4 . 9 .
49Let Γ be a toral one-faced collection in Σ g+1 . Then Γ = (Γ , x)#Γ T where Γ is a one-faced collection in Σ g .
Figure 17 -
17Transforming a Type 1 vertex to a self-intersection point.
Figure 18 -
18The 5-necklace N 5 .
Acknowledgments : I am thankful to my supervisors P. Dehornoy and J.-C Sikorav for their support during this work. I am also grateful to Gregory Miermont for his indication to the works of G. Chapuy.
Minimally intersecting filling pairs on surfaces. T Aougab, S Huang, Algebr. Geom. Topol. 152T. Aougab and S. Huang , Minimally intersecting filling pairs on surfaces. Algebr. Geom. Topol., 15(2) :903-932, 2015.
A new combinatorial identity for unicellular maps, via a direct bijective approach. G Chapuy, Advances in Applied Mathematics. 474G. Chapuy, A new combinatorial identity for unicellular maps, via a direct bijective approach. Advances in Applied Mathematics, 47(4) :874- 893, 2011.
G Chapuy, V Féray, E Fusy, arXiv:1604.06688A simple model of trees for unicellular maps. G. Chapuy, V. Féray , E. Fusy, A simple model of trees for unicellular maps, arXiv:1604.06688.
Factoring n-cycles and counting maps of a given genus. A Goupil, G Schaeffer, European J. Combin. 197A. Goupil and G. Schaeffer, Factoring n-cycles and counting maps of a given genus, European J. Combin., 19(7) : 819-834, 1998.
The Euler characteristic of the moduli space of curves. J Harer, D Zagier, Invent. Math. 853J. Harer and D. Zagier, The Euler characteristic of the moduli space of curves, Invent. Math., 85(3) : 457-485, 1986.
S K Lando, A K Zvonkin, Graphs on surfaces and their applications. SpringerS. K. Lando and A. K. Zvonkin, Graphs on surfaces and their applica- tions. Springer, 2004.
A Sane, arXiv:1809.03190Intersection norm and one-faced collections. A. Sane, Intersection norm and one-faced collections, arXiv:1809.03190.
A census of Hamiltonian polygons. W T Tutte, Canad. J. Math. 14W. T. Tutte, A census of Hamiltonian polygons, Canad. J. Math., 14 :402- 417, 1962.
A census of planar triangulations. W T Tutte, Canad. J. Math. 14W. T. Tutte, A census of planar triangulations, Canad. J. Math., 14 :21- 38, 1962.
A census of slicing Canad. W T Tutte, J. Math. 14W. T. Tutte, A census of slicing Canad. J. Math., 14 :708-722, 1962.
A census planar graph. W T Tutte, Canad. J. Math. 15W. T. Tutte, A census planar graph, Canad. J. Math., 15 :249-271, 1963.
Counting rooted maps by genus. T R S Walsh, A B Lehman, Unité de Mathématiques Pures et Appliquées (UMPA). ENS-Lyon13T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. I. J. Combin. Theory Ser. B, 13 :192-218, 1972. Unité de Mathématiques Pures et Appliquées (UMPA), ENS-Lyon.
E-mail address : [email protected]. E-mail address : [email protected]
| []
|
[
"Merger of black hole and neutron star in general relativity: Tidal disruption, torus mass, and gravitational waves",
"Merger of black hole and neutron star in general relativity: Tidal disruption, torus mass, and gravitational waves"
]
| [
"Masaru Shibata \nGraduate School of Arts and Sciences\nDepartment of Physics\nUniversity of Tokyo\n153-8902KomabaMeguro, TokyoJapan\n",
"Keisuke Taniguchi \nUniversity of Illinois at Urbana-Champaign\n61801IllinoisUSA\n"
]
| [
"Graduate School of Arts and Sciences\nDepartment of Physics\nUniversity of Tokyo\n153-8902KomabaMeguro, TokyoJapan",
"University of Illinois at Urbana-Champaign\n61801IllinoisUSA"
]
| []
| We systematically perform the merger simulation of black hole-neutron star (BH-NS) binaries in full general relativity, focusing on the case that the NS is tidally disrupted. We prepare BH-NS binaries in a quasicircular orbit as the initial condition in which the BH is modeled by a nonspinning moving puncture. For modeling the NS, we adopt the Γ-law equation of state with Γ = 2 and the irrotational velocity field. We change the BH mass in the range MBH ≈ 3.3-4.6M⊙, while the rest mass of the NS is fixed to be M * = 1.4M⊙ (i.e., the NS mass MNS ≈ 1.3M⊙). The radius of the corresponding spherical NS is set in the range RNS ≈ 12-15 km (i.e., the compactness GMNS/RNSc 2 ≈ 0.13-0.16). We find for all the chosen initial conditions that the NS is tidally disrupted near the innermost stable circular orbit. For the model of RNS = 12 km, more than 97 % of the rest mass is quickly swallowed into the BH and the resultant torus mass surrounding the BH is less than 0.04M⊙. For the model of RNS ≈ 14.7 km, by contrast, the torus mass is about 0.16M⊙ for the BH mass ≈ 4M⊙. The thermal energy of the material in the torus increases by the shock heating occurred in the collision between the spiral arms, resulting in the temperature 10 10 -10 11 K. Our results indicate that the merger between a low-mass BH and its companion NS may form a central engine of short gamma-ray bursts (SGRBs) of the total energy of order 10 49 ergs if the compactness of the NS is fairly small < ∼ 0.145. However, for the canonical values MNS = 1.35M⊙ and RNS = 12 km, the merger results in small torus mass, and hence, it can be a candidate only for the low-energy SGRBs of total energy of order 10 48 ergs. We also present gravitational waveforms during the inspiral, tidal disruption of the NS, and subsequent evolution of the disrupted material. We find that the amplitude of gravitational waves quickly decreases after the onset of tidal disruption. Although the quasinormal mode is excited, its gravitational wave amplitude is much smaller than that of the late inspiral phase. This reflects in the fact that the spectrum amplitude sharply falls above a cut-off frequency which is determined by the tidal disruption process. We also find that the cut-off frequency is 1.25-1.4 times larger than the frequency predicted by the study for the sequence of the quasicircular orbits and this factor of the deviation depends on the compactness of the NS. | 10.1103/physrevd.77.084015 | [
"https://arxiv.org/pdf/0711.1410v1.pdf"
]
| 119,211,797 | 0711.1410 | f7a337dc1edcaf501ec646160061795a0d2ce53c |
Merger of black hole and neutron star in general relativity: Tidal disruption, torus mass, and gravitational waves
9 Nov 2007
Masaru Shibata
Graduate School of Arts and Sciences
Department of Physics
University of Tokyo
153-8902KomabaMeguro, TokyoJapan
Keisuke Taniguchi
University of Illinois at Urbana-Champaign
61801IllinoisUSA
Merger of black hole and neutron star in general relativity: Tidal disruption, torus mass, and gravitational waves
9 Nov 2007numbers: 0425Dm0430-w0440Dg
We systematically perform the merger simulation of black hole-neutron star (BH-NS) binaries in full general relativity, focusing on the case that the NS is tidally disrupted. We prepare BH-NS binaries in a quasicircular orbit as the initial condition in which the BH is modeled by a nonspinning moving puncture. For modeling the NS, we adopt the Γ-law equation of state with Γ = 2 and the irrotational velocity field. We change the BH mass in the range MBH ≈ 3.3-4.6M⊙, while the rest mass of the NS is fixed to be M * = 1.4M⊙ (i.e., the NS mass MNS ≈ 1.3M⊙). The radius of the corresponding spherical NS is set in the range RNS ≈ 12-15 km (i.e., the compactness GMNS/RNSc 2 ≈ 0.13-0.16). We find for all the chosen initial conditions that the NS is tidally disrupted near the innermost stable circular orbit. For the model of RNS = 12 km, more than 97 % of the rest mass is quickly swallowed into the BH and the resultant torus mass surrounding the BH is less than 0.04M⊙. For the model of RNS ≈ 14.7 km, by contrast, the torus mass is about 0.16M⊙ for the BH mass ≈ 4M⊙. The thermal energy of the material in the torus increases by the shock heating occurred in the collision between the spiral arms, resulting in the temperature 10 10 -10 11 K. Our results indicate that the merger between a low-mass BH and its companion NS may form a central engine of short gamma-ray bursts (SGRBs) of the total energy of order 10 49 ergs if the compactness of the NS is fairly small < ∼ 0.145. However, for the canonical values MNS = 1.35M⊙ and RNS = 12 km, the merger results in small torus mass, and hence, it can be a candidate only for the low-energy SGRBs of total energy of order 10 48 ergs. We also present gravitational waveforms during the inspiral, tidal disruption of the NS, and subsequent evolution of the disrupted material. We find that the amplitude of gravitational waves quickly decreases after the onset of tidal disruption. Although the quasinormal mode is excited, its gravitational wave amplitude is much smaller than that of the late inspiral phase. This reflects in the fact that the spectrum amplitude sharply falls above a cut-off frequency which is determined by the tidal disruption process. We also find that the cut-off frequency is 1.25-1.4 times larger than the frequency predicted by the study for the sequence of the quasicircular orbits and this factor of the deviation depends on the compactness of the NS.
I. INTRODUCTION
The merger of black hole (BH)-neutron star (NS) binaries is one of the most promising sources of kilo-meter size laserinterferometric gravitational wave detectors such as LIGO and VIRGO. Although such system has not been observed yet in our Galaxy in contrast to NS-NS binaries [1], statistical studies based on the stellar evolution synthesis suggest that the merger will happen 1-10% as frequently as the merger of the NS-NS binaries in the universe [2]. Because the BH mass should be more than twice as large as the canonical NS mass ∼ 1.35M ⊙ , the typical amplitude of gravitational waves from the BH-NS binaries will be larger than that from the NS-NS binaries even if the distance is larger. This indicates that the detection rate of the BH-NS binaries by the gravitational wave detectors may be more than 10% as high as that for the NS-NS binaries, suggesting that the detection of such system will be achieved in the near future.
The final fate of the BH-NS binaries is divided into two cases depending primarily on the BH mass: (1) when the BH mass is small enough, the NS will be tidally disrupted before it is swallowed by the BH; (2) when the BH mass is large enough, the NS will be swallowed into the BH without tidal disruption. The tidal disruption occurs when the tidal force of the BH is stronger than the self-gravity of the NS. Such condition is approximately written as
M BH R NS r 3 ≥ C 2 M NS R 2 NS ,(1)
where M NS is the NS mass, R NS is the circumferential radius of the NS, M BH is the BH mass, r is the orbital separation, and C is a nondimensional coefficient. Using the orbital angular velocity Ω, Eq. (1) may be written as
Ω 2 M BH M 0 ≥ C 2 GM NS R 3 NS ,(2)
where G is the gravitational constant and M 0 = M BH + M NS . Actually, the latest high-precision numerical study for the quasicircular states of the BH-NS binaries shows that the tidal disruption occurs if the following condition is satisfied [3,4]:
Ω ≥ C GM NS R 3 NS 1/2 1 + M NS M BH 1/2 ,(3)
where the value of C is ≈ 0.270 for Γ = 2 polytropic equation of state (EOS). We note that in [3,4] the NS is assumed to be irrotational and the BH spin is set to be zero. According to an approximate general relativistic study [5], the value of C depends weakly on the stiffness of the EOSs for 2 ≤ Γ ≤ 3, and hence, it would be close to 0.27 for the NS which likely has such stiff EOS. Substituting the typical values considered in this paper, the tidal disruption is expected to set in at GM 0 Ω c 3 = 0.0708
C 0.270 M NS 1.30M ⊙ 3/2 × R NS 13.0 km −3/2 q 1/3 −1 1 + q 4/3 3/2 ,(4)
where c is the speed of the light and q denotes the mass ratio q ≡ M NS /M BH . The corresponding frequency of gravitational waves is calculated from f ≡ Ω/π as According to the third post-Newtonian (PN) study, the innermost stable circular orbit (ISCO) for q ≈ 1/3 is located at GM 0 Ω/c 3 ≈ 0.11 [6]. The tidal effect reduces this value slightly to be GM 0 Ω/c 3 ∼ 0.08-0.09 [3,4]. Adopting this value, we expect that the tidal disruption of the NSs of mass 1.3M ⊙ occurs for q > ∼ 1/4 if R NS = 13 km, and for q > ∼ 1/3 if R NS = 11 km. This indicates that the tidal disruption occurs only for the binaries of low-mass BH and NS at an orbit very close to the ISCO.
The tidal disruption of NSs by a BH has been studied with great interest because of the following reasons: (i) Gravitational waves during the tidal disruption may bring the information about the NS radius because the orbital frequency at the onset of tidal disruption depends strongly on the compactness of the NS (GM NS /R NS c 2 ) [3,4,5]. The NS mass will be determined by the data analysis for observed gravitational waves in the inspiral phase [7]. If the NS radius could be determined from observed gravitational waves emitted during the tidal disruption, the resultant relation between the mass and the radius of the NSs may constrain the EOSs of the high density nuclear matter [8,9]. (ii) The tidally disrupted NSs may form a disk or torus of mass larger than 0.01M ⊙ around the BH if the tidal disruption occurs outside the ISCO. Systems consisting of a rotating BH surrounded by a massive, hot torus have been proposed as one of the likely sources for the central engine of gamma-ray bursts with a short duration [10] (hereafter SGRBs). Hence, the merger of a low-mass BH and its companion NS can be a candidate central engine. According to the observational results by the Swift and HETE-2 satellites [11], the total energy of the SGRBs is larger than ∼ 10 48 ergs, and typically 10 49 -10 50 ergs. The question is whether or not the mass and thermal energy of the torus are large enough for driving the SGRBs of such huge total energy.
The tidal disruption of NSs by a BH has been investigated in the Newtonian [12] and approximately general relativistic [13,14] simulations. However, the criterion of the tidal disruption and the evolution of the tidally disrupted NS material would depend strongly on general relativistic effects around the BH, and hence, a simulation in full general relativity is obviously required. In the previous paper [15], we presented our first numerical results for fully general relativistic simulation performed by our new code which had been improved from the code used for the NS-NS merger [16,17]: We handle an orbiting BH employing the moving puncture method, which has been recently developed (e.g., [18,19]). We prepared a quasicircular state composed of a nonspinning moving puncture BH and a corotating NS as the initial condition. We computed the BH-NS quasicircular state by employing our new formalism based on the moving puncture framework. From the merger simulation, we found that a torus with mass ∼ 0.1M ⊙ is an outcome for M NS = 1.3M ⊙ and R NS = 13-14 km, and for M BH = 3-4M ⊙ .
In the previous work, we assumed that the NSs are corotating around the center of mass of the system for simplicity. However, this is not very realistic velocity field for the NS in close compact binaries [20]. Recently, we have developed a new code for computing quasicircular states of a BH and an irrotational NS. For obtaining the irrotational velocity field of the NS, one has to solve the equation for the velocity potential [21]. We employ the same method as that for computing the NS-NS binaries (e.g., [22,23]), which has been used in our project for the NS-NS merger [16,17]. (For solving the basic equations for the initial data, we use the spectral method library LORENE developed by the Meudon relativity group [24].) In this paper, we perform the simulations employing these new quasicircular states as the initial conditions. In addition, we modify our simulation code for the Einstein evolution equation to the fourthorder scheme for more accurate computation. Changing the BH mass and NS radius, we systematically investigate the dependence of the torus mass formed after the merger on M BH and R NS . We also compute gravitational waveforms during the tidal disruption and their spectrum.
The paper is organized as follows. Section II summarizes the initial conditions chosen in this paper. Section III briefly describes the formulation and numerical methods for the simulation. Section IV presents the numerical results of the simulation for the merger of BH-NS binaries. Section V is devoted to a summary. In the following, the geometrical units of c = G = 1 are used.
II. INITIAL CONDITION
We compute the initial condition for the numerical simulation by employing a formulation in the moving punc- I: Parameters for the quasicircular states of BH-NS binaries. The mass parameter of the puncture, the BH mass, the rest mass of the NS, the mass, radius, and normalized mass of the NS in isolation, the ADM mass of the system, the total angular momentum in units of M 2 , the orbital period in units of M0 = MBH + MNS, and the compactness of the system defined by (M0Ω) 2/3 . The BH mass is computed from the area of the apparent horizon A as (A/16π) 1 ture framework which is described in our previous papers [15]. This formulation is slightly different from that in [3,4,25,26], although both formulations are based on the confmormal flatness formulation for the three-metric, and solve the equations of the maximal slicing condition, and Hamiltonian and momentum constraints. (See [27] for the simulations preformed employing the initial data presented in [3]). Although the basic equations adopted in this paper are the same as those described in [15], we slightly change the method for defining the quasicircular state as follows: In the previous paper, we determined the center of mass of the system imposing that the dipole moment of the conformal factor at spatial infinity is zero. In this paper, the center of mass is determined from the condition that the azimuthal component of the shift vector β ϕ at the location of the puncture is equal to −Ω where Ω is the orbital angular velocity. Namely, we impose the corotating gauge condition at the location of the puncture. The reason for this change is that with this new method, the curve of the angular momentum J as a function of the orbital angular velocity Ω along quasicircular sequences is closer to the curve derived from the third PN equation of motion [6]. Furthermore, the relation among the orbital angular velocity, ADM mass M , and angular momentum agrees with the results obtained by a different method [3,4] fairly well; for the mass ratio and NS radii adopted in this paper, the disagreement of the angular momentum for a given angular velocity is ∼ 1%.
As the velocity field of the NS, we assume the irrotational one because it is believed to be the realistic velocity field [20]. The density profile and velocity field are determined by solving the hydrostatic equation (the first integral of the Euler equation) and an elliptic-type equation for the velocity potential [21].
The NSs are modeled by a polytropic EOS
P = κρ Γ ,(6)
where P is the pressure, ρ the rest-mass density, κ the polytropic constant, and Γ the adiabatic index for which we set Γ = 2. κ is a free parameter and in the following we choose it so as to fix the rest mass of the NSs to be M * = 1.4M ⊙ because this value is close to the canonical value [1]. We note that the mass, radius, and time can be rescaled arbitrarily by changing κ: i.e., if we change the value of κ from κ 1 to κ 2 , these quantities are rescaled systematically by a factor of (κ 2 /κ 1 ) 1/2 for Γ = 2. By contrast, the nondimensional parameters such as M NS /R NS , M Ω, M κ −1/2 , and R NS κ −1/2 are invariant. In this paper, We present the results for the specific values of κ for the help to the general readers who are not familiar with the nondimensional units. Because we choose M * = 1.4M ⊙ , the NS mass is in the narrow range M NS ≈ 1.29-1.31M ⊙ irrespective of the NS radius R NS . The NS radius is chosen to be 12.0, 13.2, and 14.7 km. These values agree approximately with those predicted by the nuclear EOSs in which R NS ≈ 10-15 km for M NS = 1.3M ⊙ [28]. The value R NS = 12 km agrees approximately with that predicted by the stiff nuclear EOSs such as the Akmal-Pandharipande-Ravenhall EOS [29].
The BH mass is chosen in the range M BH ≈ 3.3-4.6M ⊙ . As we described in Sec. I, the tidal disruption of the NS occurs for such low-mass BH.
In the present work, we prepare the quasicircular states with M 0 Ω ≈ 0.04. With such initial condition, the BH-NS binary experiences about one and half orbits before the onset of merger. In order to reduce the spurious ellipticity and to take into account the nonzero radial velocity associated with the gravitational radiation reaction, we may need to perform a simulation for several orbits before the onset of merger, starting from a more distant orbit [30,31,32]. In the present paper, however, we focus primarily on the qualitative study of the tidal disruption events. For such purpose, the adopted initial conditions are acceptable. The simulation of the longterm evolution for the inspiral phase is left for the future.
In Table I, the parameters of the quasicircular states adopted in this paper are listed. Specifically, we prepare six models for clarifying the dependence of the tidal disruption events on the NS radius and mass ratio. The NSs for models A, B, and C and for models D and E, respectively, have approximately the same mass ratio q = M NS /M BH , although the NS radii are different each other. Comparing the numerical results among these models, the dependence of the properties of the tidal dis-ruption event on the NS radius is clarified. The NS radius for models A, D, and F and for models B and E are the same, although the mass ratios are different each other. Thus, comparison of the numerical results for these models clarifies the dependence of the properties of the tidal disruption event on the mass ratio.
III. NUMERICAL METHODS
The numerical code for the hydrodynamics is the same as that in [15] where we use a high-resolution central scheme with the third-order piece-wise parabolic interpolation and with a steep min-mod limiter in which the limiter parameter b is set to be 2.5 (see appendix A of [34]). We have already used this numerical code for the simulation of the NS-NS binary merger [17,33]. We adopt the Γ-law EOS in the simulation as
P = (Γ − 1)ρε,(7)
where ε is the specific internal energy and Γ = 2.
For solving the Einstein evolution equation, we use the original version of the BSSN formalism [35]. Namely, we evolve the conformal part of the three-metric φ ≡ (ln γ)/12, the trace part of the extrinsic curvature K, the conformal three-metricγ ij ≡ γ −1/3 γ ij , the tracefree extrinsic curvatureà ij ≡ γ −1/3 (K ij − Kγ ij /3), and a three auxiliary variable F i ≡ δ jk ∂ jγik . Here, γ ij is the three-metric, K ij the extrinsic curvature, γ ≡ det(γ ij ), and K ≡ K ij γ ij . In the previous paper [15], we solved the equation for γ −1/2 . Since then, we have learned that we do not have to evolve the inverse of the conformal factor even in the moving puncture framework because we use the cell-centered grid in which the puncture is never located on the grid points.
For the condition of the lapse function α and the shift vector β i , we adopt a dynamical gauge condition in the following form:
(∂ t − β i ∂ i ) ln α = −2K,(8)∂ t β i = 0.75γ ij (F j + ∆t∂ t F j ).(9)
Here, ∆t denotes the time step in the simulation and the second term in the right-hand side of Eq. (9) is introduced for the stabilization of the numerical computation. The numerical code for solving the Einstein equation is slightly modified from the previous version as follows: (i) All the spatial finite differencing including the advection term such as β i ∂ i φ are evaluated with the fourth-order finite differencing scheme. For the advection term, an upwind method is adopted as proposed, e.g., in [19]. (ii) For the time-integration, a third-order Runge-Kutta scheme is employed. With this scheme, the stable evolution is feasible. Furthermore, the numerical accuracy is signigicantly improved, and as a result, the convergent results can be obtained by a relatively wider grid spacing than by that adopted in the previous papers [15].
The location and properties of the BH such as the area and circumferential radii are determined using an apparent horizon finder, for which our method is described in [39].
For extracting gravitational waves from the geometrical variables, the gauge-invariant Moncrief variables R lm in the flat spacetime [36] has been computed in our series of papers (e.g., [16,17,37]). From R lm , the plus and cross modes of gravitational waves, h + and h × , are obtained by a simple algebraic calculation. The detailed equations are described in [16,37] to which the reader may refer. In this method, we split the metric in the wave zone into the flat background and linear perturbation. Then, the linear part is decomposed using the tensor spherical harmonics, and the gauge-invariant variables are constructed for each mode of the eigen values (l, m). The gauge-invariant variables of l ≥ 2 can be regarded as gravitational waves in the wave zone, and hence, we derive such modes in the numerical simulation.
In the present work, we also compute the outgoing part of the Newman-Penrose quantity (the so-called ψ 4 ; see, e.g., [19,30] for definition of ψ 4 ). By twice time integration of 2ψ 4 (and with an appropriate choice of the integration constants), one can compute the gravitational waveforms. We compare these gravitational waveforms with those computed from the gauge-invariant waveextraction method. It is found that the wave phases of both waveforms agree well throughout the inspiral phase to the tidal disruption phase. However, the amplitude does not agree well in the early inspiral state; the magnitude of the disagreement is ∼ 20-30%, although the disagreement is much smaller during the tidal disruption. The reason is that we extract gravitational waves near the outer boundary which is located in the local wave zone in the present work, and the amplitude is not the correct asymptotic amplitude. To obtain the asymptotic amplitude, it is necessary to perform the extrapolation using the data of different extraction radii. One point worthy to note is that the convergence of the amplitude of Dψ 4 with changing the extraction radius D is much faster than that for DR lm . This indicates that ψ 4 is a better tool than R lm for evaluating the asymptotic waveforms with smaller systematic error, in the simulation that the outer boundary is located in the local wave zone. Hence, we decide that in this paper, we evaluate the waveform, energy, angular momentum, and linear momentum fluxes analyzing ψ 4 .
We compute the modes of 2 ≤ l ≤ 4 for ψ 4 , and found that the mode of (l, |m|) = (2, 2) is dominant, but l = |m| = 3 and l = |m| = 4 modes also contribute to the energy and angular momentum dissipation by more than 1% for the merger of the chosen initial data.
We also estimate the kick velocity from the linear momentum flux of gravitational waves. The linear momentum flux dP i /dt is computed from the same method as that given in [19,38]. Specifically, the coupling terms between l = |m| = 2 and l = |m| = 3 modes and between l = |m| = 2 and (l, |m|) = (2, 1) modes primarily con- II: Parameters for the grid in the numerical simulation. The grid number for covering one positive direction N , the grid number for the inner uniform-grid domain N0, the parameters for the non-uniform grid domain (∆i, ξ), the grid spacing in the uniform domain in units of Mp, the grid number covered for the major diameter of the NS (LNS), and the ratio of the location of the outer boundary along each axis to the gravitational wavelength at t = 0.
Model N N0 ∆i ξ ∆x/Mp LNS/∆x L/λ A∆P i = dP i dt dt,(10)
the kick velocity is defined by ∆P i /M where M is the initial ADM mass of the system.
IV. NUMERICAL RESULTS
A. Choosing the grid points
In the simulation, the cell-centered Cartesian, (x, y, z), grid is adopted to avoid the situation that the location of the puncture (which always stays in the z = 0 plane) coincides with one of the grid points. The plane symmetry is assumed with respect to the equatorial plane. The computational domain of −L ≤ x ≤ L, −L ≤ y ≤ L, and 0 < z ≤ L is covered by the grid size of (2N, 2N, N ) for x-y-z where L and N are constants. Following [15,35], we adopt a nonuniform grid as follows; an inner domain of (2N 0 , 2N 0 , N 0 ) grid zone is covered with a uniform grid of the spacing ∆x and outside this inner domain, the grid spacing is increased according to the relation of ξ tanh[(i − N 0 )/∆i]∆x where i denotes the i-th grid point in each positive direction, and N 0 , ∆i, and ξ are constants. Then, the location of the i-th grid, x k (i), in each direction is
x k (i) = (i − 1/2)∆x 1 ≤ i ≤ N 0 (i − 1/2)∆x + ξ∆i∆x × log[cosh{(i − N 0 )/∆i}] i > N 0(11)and x k (−i) = −x k (i) i = 1, 2, · · · , N for x k = x and y.
For all the simulations, we chose N = 225 and ∆i = 30. For most of the simulation, we determined the grid spacing for the inner uniform domain according to the rule ∆x/M p = 1/15 where M p is the mass parameter of the puncture. We have found that with this choice of the grid size, a convergent result for the BH orbit is obtained. Actually, the simulations for the BH-BH binary in the moving puncture framework (e.g., [18,19,31]) show that this choice is appropriate for obtaining a convergent result in the fourth-order scheme. With increasing the mass ratio M p /M NS , however, the grid number of covering the NS becomes small for the fixed value of ∆x. We require that the major diameter of the NS, L NS , should be covered by > ∼ 50 grid points, because a result with small error is obtained with such setting (see Appendix A). By this reason, for model F, we choose ∆x/M p = 1/16.
In Table II, we list the parameters for the grid coordinates. λ denotes the wavelength of gravitational waves at t = 0 (λ = π/Ω). N 0 is chosen so as to put the NS in the inner uniform-grid domain initially. ξ is determined from the conditions (i) L/λ > ∼ 0.75 and (ii) the grid spacing near the outer boundaries is smaller than 1.05M . Note that λ decreases with the time because the orbital radius decreases due to the gravitational radiation reaction. Hence, L is smaller than λ at the onset of tidal disruption. The gravitational wavelength after the merger sets in is expected to be larger than 11M which is approximately equal to the wavelength of the quasinormal mode (QNM) of the BH excited in the final phase of the merger. Because the wavelength is covered by at least 10 grid points, we may expect that gravitational waves are extracted with a good accuracy near the outer boundary.
To see the dependence of the numerical results on the grid resolution, we performed test simulations for ∆x = M p /12 (models A2 and C2), and for ∆x = M p /13.5 (model A1). For models A, A1, and A2 and for models C and C2, the locations of the outer boundaries are approximately the same, respectively (see Table II). We found that the numerical results (see Appendix A) show an acceptable convergence for the case that the grid spacing satisfies the conditions (i) ∆x ≥ M p /12 and (ii) that the diameter of the NS is covered by more than ≈ 50 grid points. As we discuss in Appendix A, the errors for the estimate of the mass surrounding the BH and area of the apparent horizon at the end of the simulation are within ∼ 10%-30% (∼ 0.01-0.02M ⊙ ) and ∼ 0.3%-0.5%, respectively, for such setting. The total energy and angular momentum emitted by gravitational waves are also computed within ∼ 10%-20% error. In the previous papers [15], we performed simulations in the second-order accurate finite-differencing for solving the Einstein equation. In such case, the convergence of the numerical results with improving the grid resolution is not very fast. In the present work, we employ the fourth-order scheme, with which the convergence is achieved much faster. The solid contour curves are drawn for ρ = 2i × 10 14 g/cm 3 (i=1-4) and for 10 14−i g/cm 3 (i = 1 ∼ 5). The maximum density at t = 0 is ≈ 8.86 × 10 14 g/cm 3 . The blue, cyan, magenta, and green curves denote 10 14 , 10 13 , 10 12 , and 10 11 g/cm 3 , respectively. The vectors shows the velocity field (v x , v y ), and the scale is shown in the upper right-hand corner. The thick (red) circles are the apparent horizons. The 2nd and 3rd panels denote the states after one and one-and-half orbits, respectively. The last panel plots the density contours and velocity vectors in the x = 0 plane at the same time as that for the 8th panel.
Because any conservation scheme of hydrodynamics is unable to evolve a vacuum, we have to introduce an artificial atmosphere outside the NS. However, if the density of the atmosphere was too large, it might affect the orbital motions of the BH and NS and moreover we might overestimate the total amount of the rest mass of the formed torus surrounding a BH. Thus, we initially assign a small rest-mass density as follows:
ρ = ρ at r ≤ x(N 0 ) ρ at e 1−r/x(N0) r > x(N 0 ).(12)
Here, we choose ρ at = ρ max ×10 −8 where ρ max is the maximum rest-mass density of the NS. With such choice, the total amount of the rest mass of the atmosphere is about 10 −5 of the rest mass of the NS. Thus, the accretion of the atmosphere onto the NS and BH plays a negligible role for their orbital evolution in the present context. In the following, we discuss the rest mass of the torus surrounding a BH. As we show below, the rest mass is larger than 0.01M ⊙ which is much larger than the atmosphere mass. Hence, the atmosphere also plays a negligible role for determining the properties of the torus.
C. General process of tidal disruption
All the numerical simulations were performed from about one and half orbits before the onset of tidal disruption to the time in which the accretion time scale of the rest mass of material into the BH is much longer than the dynamical time scale. The duration of the simulations is 400-500M in units of the ADM mass. In this section, we first describe the general process of tidal disruption, subsequent evolution of the BH, and formation of the accretion disk (torus) showing the numerical results for model A. Figure 1 plots the evolution of the contour curves for ρ and the velocity vectors for the three velocity v i (= dx i /dt) in the equatorial plane for model A. The location of the apparent horizons is shown together. Due to gravitational radiation reaction, the orbital radius decreases and then the NS is elongated gradually (panels 2 and 3). The tidal disruption of the NS by the BH sets in at t ∼ 4.3 ms (at about one and half orbits; panel 3). Soon after this time, the outer part of the NS expands outward due to angular momentum transport by the hydrodynamic interaction. This material subsequently forms a torus around the BH. However, the tidal disruption sets in at an orbit very close to the ISCO and hence the material in the inner part is quickly swallowed into the BH (4rd and 5th panels). By the outward angular momentum transport, the material in the outer part of the NS forms a one-armed spiral arm (5th and 6th panels). The spiral arm then winds around the BH, and the material, which does not have angular momentum large enough to orbit the BH, shrinks and falls into the BH (7th panel). A relatively large fraction of the material falls in particular for t ≈ 9-11 ms (see also the upper panel of Fig. 5(a)). In this phase, the inner and outer parts of the spiral arm collide and thermal energy is generated due to the shock heating. The high-density part of the spiral arm then expands to form a compact torus with the maximum density ∼ 10 12 g/cm 3 (8th and 9th panels; see also Fig. 2). We stopped the simulation at t ≈ 12.5 ms, at which the rest mass of the material located outside the apparent horizon is ≈ 0.092M ⊙ .
We define the specific thermal energy generated by the shocks by
ε th ≡ ε − κ Γ − 1 ρ Γ ,(13)
where the second term in the right-hand side is the polytropic term, i.e., the specific internal energy in the absence of the shocks. ε th is zero in the adiabatic evolution (in the absence of the shocks), and hence, ε th may be regarded as the specific thermal energy generated by the shocks. Figure 2 (top panel of each) plots ε th along x and y axes at t = 7.117, 9.070, and 11.02 ms. The ratio of the thermal energy to the total internal energy ε th /ε and the rest-mass density are shown together. This figure shows that the generated thermal energy is much larger than the polytropic one for most region except for the high-density region of the spiral arms which do not experience the shock heating. However, during the evolution, the spiral arms winds around the BH and the shocked region increases. As a result, ε th /ε eventually becomes approximately equal to unity for all the regions. Figure 2 indicates that a hot torus with the density ∼ 10 11 -10 12 g/cm 3 is the final outcome. This is a favorable property for producing a large amount of neutrinos which may drive SGRBs (e.g., [40,41,42,43]). Although the EOS adopted in this paper is idealized one, we approximately estimate the temperature of the torus. Assuming that the specific thermal energy is composed of those of the gas, photons, and relativistic electrons (and positrons), we have [42,43]
ε th = 3kT 2m + 11a r T 4 ρ ,(14)
where k, a r , T , and m are the Boltzmann constant, radiation-density constant, temperature, and mass of the main component of nucleons. For simplicity, we assume that the gas is composed of the neutron and set m = 1.66 × 10 −24 g. Because the temperature is higher than ∼ 6 × 10 9 K (see below), we assume that the electrons and positrons are relativistic to describe Eq. (14). If the density is high, the neutrinos are optically thick and contribute to the thermal energy. Although this effect is important for ρ > ∼ 10 12 g/cm 3 [40,41,42,43], the density of the outcome is at most 10 12 g/cm 3 , and hence, we ignore it. Then, the radiation energy is smaller than the gas energy for ρ > ∼ 10 11 g/cm 3 and ε th /c 2 < ∼ 0.01. From Eq. (14), the temperature is approximately given by T ≈ 7.2 × 10 10 ε th 0.01 K.
Thus, the estimated temperature is between 10 10 and ∼ 10 11 K for the torus. (Note that ε ∼ 0.1 inside the apparent horizon, but we do not pay attention to the inside of the BH.) For clarification of the dependence of the density and the thermal energy on the radius of the tidally disrupted NSs, R NS , we show the same plots as Fig. 2 for models B and C in Fig. 3. We also plot the density contour curves and velocity vectors at t = 11.02 ms for these models in Fig. 4. Figure 3 shows that the value of ε th increases above 0.001 dominating the specific thermal energy irrespective of models. Figure 4 shows that irrespective of the models, the outcome after the tidal disruption is a compact torus surrounding the BH. Thus, the hot torus is the universal outcome irrespective of R NS as long as it is in the range between 12 and 15 km.
By contrast, the rest-mass density depends sensitively on the torus mass (see Table III and next section for the final values of the rest mass of the material located outside the apparent horizon which is approximately equal to the torus mass). For model B, the torus mass is much smaller than that for model A. This is simply because the NS radius is smaller and hence the tidal disruption sets in at an orbit very close to the ISCO. For model C, by contrast, the torus mass is much larger than those for models A and B because of the larger NS radius. The averaged rest-mass density has a clear correlation with the torus mass. For model B, the density is smaller than 10 12 g/cm 3 for most region of the torus, whereas for model C, the density is larger than 10 12 g/cm 3 for most region. For such high-density case, the neutrinos are likely to be optically thick [40], and the neutrinodominated accretion flow will be subsequently formed.
D. Torus mass
The upper panels of Fig. 5(a)-(c) plot the evolution of the fraction of the rest mass of the material located outside the apparent horizon M r>rAH (a) for models A, D, and F, (b) for models B and E, and (c) for models A-C. Here, M r>rAH is defined by
M r>rAH ≡ r>rAH ραu t √ γd 3 x,(16)
and u t is the time component of the four-velocity. We find that a large fraction of the material is swallowed into the BH soon after the onset of merger irrespective of the models (see also Table III). However, the final value of M r>rAH depends strongly on the NS radius; e.g., for R NS = 14.7 km and q ≈ 0.33 (model C), ≈ 12% of the material is located outside the BH at the end of the simulation, whereas for R NS = 12.0 km and q ≈ 0.33 (model B), only ≈ 2% of the material is located outside the BH at the end of the simulation (see Fig. 5(c)). The final value of M r>rAH depends also on the mass ratio q. Figure 5(a) illustrates that the final value of M r>rAH increases by ∼ 75% for the increase of the value of q from 0.28 to 0.39. Figure 5(b) also illustrates that the final value of M r>rAH increases by ∼ 50% for the small increase of q from 0.33 to 0.39. Figure 5(b) and (c) (in particular (c)) clarifies the dependence of the infall process of the tidally disrupted material into the BH on the NS radius. With the decrease of the NS radii, the accretion rate of the rest mass into the BH at the onset of tidal disruption increases steeply. Furthermore, the accretion time scale shortens; e.g., for model B, the duration of the quick accretion is shorter than 1 ms, whereas for model C, it is ∼ 1.5 ms.
E. Black hole mass and spin
The lower panels of Fig. 5(a)-(c) plot the evolution of the area of the apparent horizon in units of 16πM 2 . The area of the BHs is constant before the onset of tidal disruption. The value of the area fluctuates only by ∼ 0.03% in such phase. After the onset of tidal disruption, the area quickly increases as a result of the mass accretion, and finally, approaches an approximate constant. From the final values of the area of the apparent horizon together with the estimated BH mass at the final stage, M BH,f , the nondimensional spin parameter of the formed BHs, a, is approximately derived from
A ≡ A 16πM 2 BH,f = 1 + √ 1 − a 2 2 .(17)
To approximately estimate M BH,f , we use the following equation as in [15] M
BH,f ≡ M − M r>rAH − E GW ,(18)
where E GW is the total radiated energy by gravitational waves [52]. The results for E GW , M r>rAH , M BH,f , and are listed in Table III. The gravitational wave energy is primarily carried by the l = |m| = 2 modes. We found that l = |m| = 3 and l = |m| = 4 modes are subdominant The total radiated energy in units of the initial ADM mass M , the total radiated angular momentum in units of the initial angular momentum J, the kick velocity, the rest mass of the material located outside the apparent horizon, the estimated BH mass, the area of the apparent horizon in units of 16πM 2 BH,f , the ratio of the polar circumferential radius to the equatorial one of the apparent horizon (Cp/Ce), and the estimated spin parameters of the final state of the BH. a f1 , a f2 , and a f3 are computed from the area of the apparent horizon, the estimated angular momentum and mass of the BH, and Cp/Ce, respectively. All the values presented here are measured for the states obtained at the end of the simulations. We note that Cp/Ce varies with time by ≈ 0.05%. The energy, angular momentum, and linear momentum fluxes of gravitational waves are evaluated changing the location of the extraction, and we found that the energy and angular momentum fluxes converge within ∼ 1% error, whereas the error of the linear momentum flux is ∼ 10%. modes; for q ≈ 0.33 (0.39), the total emitted energies carried by l = |m| = 3 and l = |m| = 4 modes are ∼ 5-6% (4%) and ∼ 1-2% (1%) of that of l = |m| = 2 modes, respectively. The nondimensional spin parameter determined from A is listed in Table III as a f1 . We find that a f1 is between ∼ 0.5 and ∼ 0.6 for all the models, implying that rotating BHs of moderate magnitude of the spin are the final outcomes. For the larger mass ratio q, the spin parameter is larger for the same NS radius. The reason for this is that for the larger mass ratio, the initial total angular momentum of the system J is larger (see Table I). For the smaller NS radius, the spin parameter is larger for the same mass ratio. This is simply because the larger fraction of the material falls into the BH resulting in the spin up.
Model ∆E/M ∆J/J V kick (km) Mr>r AH (M⊙) M BH,f (M⊙)ÂAH Cp/Ce a f1 a
To check the validity of the above estimate, we evaluate the nondimensional spin parameter employing other two methods by which the BH spin may be approximately calculated. In the first method, the spin is evaluated directly from the angular momentum of the BH defined by
J BH,f ≡ J − J r>rAH − J GW ,(19)
where J GW is the total radiated angular momentum by gravitational waves and J r>rAH is the angular momentum of the material located outside the apparent horizon, which is defined by
J r>rAH ≡ r>rAH ραhu t u ϕ √ γd 3 x.(20)
Here, u ϕ is the ϕ component of the four-velocity, and h is the specific enthalpy defined by 1+ε+P/ρ. J BH,f exactly gives the angular momentum of the material in the axisymmetric and stationary spacetime. In the late phase of the merger, the spacetime relaxes to a quasistationary and nearly axisymmetric state. Thus, we may expect that J r>rAH will provide an approximate magnitude of the angular momentum of the torus.
From J BH,f and M BH,f , we define the nondimensional spin parameter by a f2 ≡ J BH,f /M 2 BH,f (see Table III). It is found that a f2 is systematically larger than a f1 but these two values, independently determined, still agree within ≈ 2.5% disagreement. This suggests that both quantities denote the BH spin within the error of ∼ 2.5% (∼ 0.015). We note that because the relation a f1 < a f2 holds systematically, this disagreement would primarily result from the systematic error associated with the approximate definitions of the spin. For example, we use the rest mass M r>rAH to estimate the disk mass in Eq. (18), ignoring the gravitational binding energy between the BH and the torus. Also, Eq. (20) may include an error because the torus is neither exactly axisymmetric nor stationary.
In the second method, we use the ratio of the polar circumferential length to the equatorial one of the BH, C p /C e . We compare this ratio to that of the Kerr BH, and estimate the spin parameter. The results are listed in Table III as a f3 . Again we find that this agrees with other two with a good accuracy (within ∼ 1.5% disagreement with a f1 ). We note that the axial ratio varies with time by ∼ 0.05%. This implies that the error for the estimation of a f3 is ∼ 0.5% for a f3 = 0.5-0.6. We also note that the relation between C p /C e and the spin parameter in the presence of the torus is slightly different from that of the Kerr BH [44]. However, the systematic error is negligible because the torus mass is only < ∼ 3% of the BH mass. From these results, we conclude that the spin parameter obtained by three different methods agree within ∼ 2.5% error. We note that all three methods can only approximately determine the spin, and the systematic error associated with their definition should be included. Actually, the systematic relation a f2 > a f1 > a f3 holds. Hence, the error of ∼ 2.5% results not only from the numerical error but also from the systematic error.
The spin parameter of the formed BHs is smaller than the initial value of J/M 2 of the system by 14-20%. The primary reason for this in the case that most of the NS matter falls into the BH, i.e., for R NS = 12 km, is that gravitational waves carry away the angular momentum by ∼ 13% of J (see Table II). By contrast, for the case that M r>rAH is large, the angular momentum of the torus occupies a large fraction of J; e.g., for model C, in which M r>rAH ≈ 0.16M ⊙ , J r>rAH is about 15% of J. In such case, the angular momentum of the torus is about twice larger than that carried by gravitational waves. For t ret < ∼ 4.5 ms, the inspiral waveforms are seen: The amplitude increases and characteristic wavelength decreases with time. The wavelength at the last inspiral orbit is ∼ 0.8 ms, indicating that the orbital period at the complete tidal disruption is ∼ 1.6 ms, i.e., the angular velocity is ∼ 0.10M −1 . This value coincides approximately with that of the ISCO, but does not agree with Ω calculated from Eq. (4) (see Sec. I for discussion). This indicates that the tidal disruption may set in approximately at the orbit predicted by the study of the quasicircular orbits but completes near the ISCO.
For 4.5 ms < ∼ t ret < ∼ 5.5 ms, the amplitude of gravitational waves decreases quickly. In this phase, the tidal disruption and resultant quick accretion of the material into the BH proceed (see Fig. 5). Thus, gravitational waves could be emitted both by the matter motion and by the quasinormal mode (QNM) oscillation of the BH. The characteristic oscillation period in this phase is ≈ 2.9 kHz. This value is slightly smaller than the value predicted from the perturbation study for the fundamental QNM of BH mass M BH,f ≈ 5.10M ⊙ and spin a = 0.55.
Here, the perturbation study predicts the frequency and damping time scale as [45] f qnm ≈ 3.23
M BH,f 10M ⊙ −1 [1 − 0.63(1 − a) 0.3 ] kHz, (23) t d ≈ 2(1 − a) −0.45 πf ,(24)
which gives f qnm ≈ 3.19 kHz. This indicates that gravitational waves are primarily emitted by the matter motion in this phase; the damping of the amplitude is not due to the QNM damping but to the fact that the compactness and degree of nonaxisymmetry of the matter distribution decrease by the tidal disruption. For t > ∼ 5.5 ms, the rapid accretion stops and the BH approaches a nearly stationary state (see Fig. 5(a)). Hence, for t ret > ∼ 5.5 ms, the ring-down waveforms of the QNM should be seen. Because gravitational waves are also emitted by the matter moving around the BH, the waveforms are modulated and the waveforms associated only by the QNM are not clearly seen. However, ψ 4 with 5.4 ms < ∼ t ret < ∼ 6 ms can be fitted by the damping waveforms of the fundamental QNM fairly well. Figure 6(b) plots the l = |m| = 2 mode of ψ 4 and a fitted waveform as
Ae −t/t d sin(2πf qnm t + δ),(25)
where A and δ are constants. For the fitting, we choose the BH mass and spin as 5.10M ⊙ and a = 0.55. In this case, f qnm = 3.19 kHz and t d = 0.286 ms, respectively [45]. This figure shows that gravitational waves are primarily characterized by the fundamental QNM, and the gravitational wave amplitude of the QNM is much smaller than that at the last inspiral orbit. The possible reason for this small amplitude is that the QNM is excited only incoherently by the infall of the tidally disrupted noncompact material into the BH. The characteristic feature of the gravitational waveforms emitted after the tidal disruption sets in is that the amplitude damps quickly even in the formation of the nonstationary torus of mass ∼ 0.1M ⊙ . This is due to the fact that the compactness and degree of nonaxisymmetry of the torus decrease in a very short time scale (∼ 2-3 ms). Moreover, the gravitational wave amplitude of the QNM is very small. This implies that the amplitude of the Fourier spectrum steeply decreases in the high-frequency region for f > ∼ 1/0.8 ms ≈ 1.2 kHz (see Sec. IV G; cf. Fig. 7).
Gravitational waveforms are qualitatively similar for other models. However, quantitative differences due to the difference of the NS radius are clearly seen. Figure 6 (c) plots gravitational waves for model C (solid curves) together with those for model A (dashed curves). For t ret < ∼ 4 ms, the inspiral waveforms for two models agree very well, because gravitational waves at such phase are primarily determined by the total mass and mass ratio. However, for t ret > ∼ 4 ms, the waveforms disagree because the amplitude for model C starts decreasing (compare the waveforms for 4.5 ms < ∼ t ret < ∼ 6 ms). This earlier decrease results from the onset of tidal disruption of the NS at a larger orbital separation due to the larger NS radius for model C (see Eq. 4). Soon after the onset of tidal disruption, the material spreads around the BH at a relatively large orbital separation. Because the compactness and degree of nonaxisymmetry decrease as the tidally disrupted material spreads, the amplitude for model C quickly damps. Furthermore, the fraction of the material coherently falling into the BH is not large. As a result, the amplitude of the QNM is negligible (compare ψ 4 of models A and C; because it is smaller than the amplitude of the wave modulation due to numerical error or matter motion, we cannot extract the QNM for model C).
By contrast, the QNM is clearly seen for the case that the NS radius is smaller. Figure 6(d) plots the l = |m| = 2 mode of ψ 4 for model B. In this case, the tidal disruption sets in at an orbit closer to the ISCO than that for models A and C. Consequently, a larger fraction of the material coherently falls into the BH, and relatively coherently excites the QNM. In the lower panel of Fig. 6(d), we again fit the waveforms by Eq. (25) with f qnm = 3.23 kHz and t d = 0.289 ms assuming that the BH mass and spin are 5.12M ⊙ and 0.57, respectively. Figure 6 (d) shows that the waveforms for t ret > ∼ 5 ms agree well with the hypothetical fitting formula. A distinguishable feature is that the amplitude of the QNM is by a factor of ∼ 2 larger than that for model A. However, this amplitude is still much smaller than the amplitude at the last inspiral orbit.
The kick velocity is also estimated from the total linear momentum carried by gravitational waves. We find that it is < ∼ 20 km/s for all the models (see Table III) [53]. This value is much smaller (by one order of magnitude) than that in the case of the BH-BH merger [38]. The reason for this is that the gravitational wave amplitude damps quickly during the merger due to the tidal disruption, and the amplitude associated with the QNM is very small. Actually, a PN study [46] illustrates that the kick velocity is ≈ 20 km/s for q = 1/3 if the kick only by gravitational waves from the inspiral motion is taken into account. This value agrees fairly with our results. In the BH-BH merger, gravitational waves associated with the QNM have a large amplitude and as a result, the linear momentum flux is significantly enhanced. If the NS escapes the tidal disruption during the merger (i.e., for the case that the BH mass is large enough or the NS radius is small enough), a large kick velocity of order 100 km/s might be induced.
G. Gravitational wave spectrum
To determine the characteristic frequency of gravitational waves emitted during the tidal disruption, we compute the Fourier spectrum of gravitational waves of l = |m| = 2 modes defined by
h(f ) ≡ D M 0 |h + (f )| 2 + |h × (f )| 2 2 ,(26)
where
h + (f ) = e 2πif t h + (t)dt,(27)
h × (f ) = e 2πif t h × (t)dt.
In the present simulations, we followed the inspiral phase only for about one and half orbits before the onset of tidal disruption. Thus, the Fourier spectrum for the low-frequency region is absent if we perform the Fourier transformation for the purely numerical data (see the dashed curve of Fig. 7(a)). To artificially compensate the Fourier spectrum for the low-frequency region, we combine the hypothetical waveforms computed from the third PN approximation for two structureless, inspiraling compact objects [6]. Specifically we determine the evolution of the orbital angular velocity Ω using the so-called Taylor T4 formula (e.g. [32] for a detailed review) by solving
dx dt = 64ηx 5 5M 0 1 − 743 +η 2 πx 7/2 ,(29)
where x = [M 0 Ω(t)] 2/3 is a function of time [54], η is the ratio of the reduced mass to the total mass M 0 , and γ E = 0.577 · · · is the Euler constant. Then, gravitational waveforms are determined from
h + (t) = 4ηM 0 x D A(x) cos[Φ(t) + δ],(30)h × (t) = 4ηM 0 x D A(x) sin[Φ(t) + δ],(31)
where A(x) is a nondimensional function of x for which A(x) → 1 for x → 0, δ is an arbitrary phase, and
Φ(t) = 2 Ω(t)dt.(32)
For A(x), we choose the 2.5 PN formula (e.g., [32]). The PN waveforms are calculated for M 0 Ω ≥ 0.005. In this method, it is not clear at which frequency we should combine the PN waveforms with the numerical waveforms. In the present paper, we match the two waveforms at M 0 Ω ≈ 0.04 (at f ≈ 517(M 0 /5M ⊙ ) −1 Hz). The Fourier spectrum for the late inspiral phase depends on this matching frequency, in particular, around the matching frequency ∼ 500-700 Hz. However, the spectrum around the frequency for f > ∼ f tidal , for which the tidal disruption proceeds, depends very weakly on the matching frequency. Thus, the present method is acceptable for studying the Fourier spectrum during the tidal disruption. Figure 7(a) plots h(f )f for model A. We also plot the spectra for the analytical third PN waveforms (dotted curve) and for the purely numerical waveforms with no matching (dashed curve). For πM 0 f ≪ 1, the spectra of the matched and PN waveforms behave [7] h(f )f = 5 24π
η 1/2 (πM 0 f ) −1/6 .(33)
We note that the effective amplitude defined by the average over the source direction and the direction of the binary orbital plane is
h eff ≡ 2 5 h(f )f M 0 D = 9.6 × 10 −23 h(f )f 0.1 M 0 5M ⊙ × D 100 Mpc −1 .(34)
Thus, the effective amplitude is ≈ 10 −22 at f ≈ 1 kHz for D = 100 Mpc and M 0 = 5M ⊙ . It is found that the spectrum for the purely numerical data agrees with the spectrum of the matched data for f > ∼ 800 Hz besides the spurious modulation for the spectrum of the purely numerical data which results from the incomplete data sets for the inspiral phase. Here, f = 800 Hz is slightly smaller than f tidal predicted from Eq. (5) (see also Table IV). This illustrates that the matching does not affect the global shape of the spectrum for f > ∼ f tidal . We still see the modulation and dip of the spectrum for the matched data in the frequency band between ∼ 500 and 800 Hz. The possible physical reason for the dip at f < ∼ 1 kHz is that the radial-approach velocity of the BH and NS steeply increases near the ISCO (f ∼ 1.2 kHz), and hence, the integration time for gravitational waves decreases. As a result, the effective amplitude at such high-frequency band slightly decreases. This feature is well-known for the NS-NS merger [47]. However, there are also the possible unphysical reasons as follows: (i) the orbit of the BH-NS has a nonzero eccentricity because the initial data is not exactly the circular orbit [48], and (ii) the matching of the numerical and PN data induces a systematic error. Thus, the spectrum for this band may not be very accurate. We do not pay attention to this low-frequency band in the following. The noteworthy feature of the spectrum is that the spectrum amplitude does not damp even for f tidal < f < ∼ 1.3f tidal ≡ f cut (see the solid circles in Fig. 7(b)-(d) for the approximate location of f cut ). The reason for this is that the NS orbiting near the ISCO has a radialapproach velocity of order ∼ 10% of the orbital velocity due to the radiation reaction of gravitational waves, and hence, the NS is not immediately tidally disrupted at the orbit of f = f tidal , although the tidal disruption of the NS likely sets in at f = f tidal . As a result, the inspiral orbit is maintained for a while even inside the predicted tidal disruption orbit and gravitational waves of a large amplitude are emitted for f > f tidal . Because the tidal disruption completes at f ≈ f cut > f tidal , it is not straightforward to determine f tidal from the spectrum of gravitational waves.
We note that our finding of f cut > f tidal is the unique feature for the BH-NS binary. If the companion of the BH is not as compact as the NS (e.g., for the BH-white dwarfs binary), f cut would be approximately equal to f tidal because the radial-approach velocity due to the gravitational radiation reaction is negligible and hence the tidal disruption would complete at the predicted tidal disruption orbit.
Because the amplitude of gravitational waves quickly decreases after the tidal disruption completes, the spectrum also sharply falls for the high-frequency band; i.e., above a cut-off frequency f cut , the spectrum amplitude steeply decreases. Figure 7(a) shows that for model A, the value of f cut is approximately 1.16 kHz (i.e., M 0 Ω ≈ 0.094). This confirms the finding in Sec. IV F about the quick decrease of the gravitational wave amplitude occurs near the ISCO in this case.
To see the dependence of the spectrum shape on the mass ratio and NS radius, we plot Fig. 7(b)-(d). Figure 7(b) and (c) compare the spectra for models A, D, and F (R NS = 13.2 km) and for models B and E (R NS = 12.0 km), for which the NS radii are identical each other, whereas the mass ratio q is different. It is found that the spectrum shape depends weakly on the mass ratio. The reason for this is that the gravitational wave frequency at which the tidal disruption occurs is determined primarily by the radius and mass of the NS. For models A, D, and F, f cut ∼ 1.1-1.2 kHz, and for models B and E, f cut ∼ 1.4 kHz (see Table IV). Obviously, for the smaller radii, the value of f cut is larger because the tidal disruption completes for the smaller orbital separation. We also calculate the ratio of f cut to f tidal (see Table IV). For models A, D, and F, the ratio is ∼ 1.3, whereas for models B and E, it is slightly larger, ∼ 1.4. The reason is that the compact NS has the larger radial-approach velocity at f = f tidal , and hence, the tidal disruption completes at a smaller orbital separation. Figure 7(d) compares the spectra for models A-C, for which the mass ratio q is approximately identical, whereas the NS radii are different. We find that the value of f cut depends strongly of the NS radius and is smaller for larger NS radii, indicating that the tidal disruption sets in at a larger orbital separation. The ratio f cut /f tidal is also smaller for larger NS radii (see Table IV). The reason for this is that for the larger NS, the tidal disruption sets in at a larger orbital separation, and hence, the radial-approach velocity is smaller. However, even for model C, the ratio is ≈ 1.25. This implies that gravitational waves of a high amplitude are emitted well inside the orbit of the onset of tidal disruption, even for a less compact NS of nearly upper-limit radius ∼ 15 km.
Finally, we note the following: Figure 7 and Equation (34) show that the effective amplitude at f = f cut ∼ 1 kHz is ≈ 10 −22 for the typical distance and total mass as D = 100 Mpc and M 0 = 5M ⊙ . Even for the optimistic direction of the source and its binary orbital plane, the effective amplitude is at most ≈ 2.5 × 10 −22 . The designed sensitivity of the advanced LIGO is ∼ 3 × 10 −22 [49]. This implies that it will not be possible to detect gravitational waves during the tidal disruption by the detectors of standard design. To detect gravitational waves at such high frequency, the detectors of special instrument (e.g., resonant-side band extraction [50]) which is sensitive to high-frequency gravitational waves is necessary.
V. SUMMARY
We have presented the numerical results of fully general relativistic simulation for the merger of BH-NS binaries, focusing on the case that the NS is tidally disrupted by a nonspinning low-mass BH of M BH ≈ 3.3-4.6M ⊙ . The Γ-law EOS with Γ = 2 and irrotational velocity field are employed for modeling the NSs. To see the dependence of the tidal disruption event on the NS radius, we choose it in a wide range as R NS = 12.0-1.47 km whereas the NS mass is fixed to be ≈ 1.3M ⊙ . The resulting mass ratio q ≡ M NS /M BH is in the range between ≈ 0.28 and 0.4.
As predicted by the study for the quasicircular states [3,4], for all the chosen models, the NS is tidally disrupted at the orbits close to the ISCO. The BH of the spin of 0.5-0.6 is formed and a large fraction of the material is quickly swallowed into the BH, whereas 2-12% of the material forms a hot and compact torus around the BH. The resultant mass and density of the torus depend strongly on the mass ratio q and NS radius R NS , in particular on R NS . For R NS = 12.0 km, the torus mass is at most 0.05M ⊙ even for the large value of the mass ratio q ≈ 0.39. For R NS = 14.7 km, by contrast, the torus mass is ≈ 0.16M ⊙ for q ≈ 0.33. Extrapolating the results in this paper, the torus mass would be smaller than 0.01M ⊙ for R NS = 11 km even for q ∼ 0.4. The stiff nuclear EOSs predict the radius of ≈ 11-12 km for M NS = 1.3-1.4M ⊙ [29]. This suggests that the torus mass is likely to be ≪ 0.1M ⊙ even for the BH mass 3-4M ⊙ . This point should be more rigorously clarified for the future simulation employing the realistic nuclear EOSs.
For the optimistic cases in which R NS > ∼ 13 km or of q > ∼ 0.4, the torus mass can be > ∼ 0.05M ⊙ . As we found from the value of the ε th , the resultant torus likely has high temperature as 10 10 -10 11 K. This suggests that such outcomes are promising candidates for driving SGRBs. According to the latest simulations for the hot and compact torus around the BH by Setiawan et al. [41], the total energy deposited by neutrino-antineutrino annihilation is ∼ 10 49 (M torus /0.01M ⊙ ) ergs for 0.01M ⊙ < ∼ M torus < ∼ 0.1M ⊙ and a = 0.6. Here, M torus is the initial torus mass for the simulation. Assuming that the conversion rate to the gamma-ray energy is 10% [51], the total energy of SGRBs is ∼ 10 48 (M torus /0.01M ⊙ ) ergs according to their numerical results. This indicates that if the NS radius is > ∼ 13 km with M BH = 3.3-4M ⊙ , the SGRB of the total energy ∼ 10 49 ergs may be driven. By contrast, for R NS < ∼ 12 km with M BH > ∼ 3.3M ⊙ , the total energy is likely to be at most several ×10 48 ergs; i.e., only the weak SGRBs can be explained. We note that the final spin parameter depends on the initial spin of the BH. In the merger of the rapidly rotating BH and NS, the resuling spin of the BH will be close to unity. For such BHs, the total deposited energy will be enhanced by the spin effects [43].
Gravitational waves are also computed. It is found that the amplitude of gravitational waves damps during the tidal disruption, and when the tidal disruption completes, the amplitude becomes much smaller than that at the last inspiral orbit. Although we find that the waveforms in the final phase of the tidal disruption are characterized by the QNM ringing, its amplitude is much smaller than that of gravitational waves at the last in-spiral phase, in particular, for the large value of R NS . The reason for this is that the NS is tidally disrupted before plunging into the BH, and hence, a significant excitation of the QNM by the coherently infalling material is not achieved in contrast to the BH-BH merger (e.g., [18,19,30,31]). For the case that the NS is compact with R NS = 12 km, the tidal disruption occurs at an orbit very close to the ISCO and a large amount of the material falls fairly coherently into the BH, resulting in a relatively high amplitude of the QNM. However, the amplitude is still much smaller than that at the last inspiral orbit.
The Fourier spectrum of gravitational waves is also analyzed. Because the amplitude of gravitational waves quickly damps during the tidal disruption, the spectrum amplitude steeply decreases above a cut-off frequency f cut . The noteworthy feature is that the cut-off frequency does not agree with the predicted frequency at which the tidal disruption sets in, f tidal . The reason for this is that the NS in the compact binaries has a radial-approach velocity of order ∼ 10% of the orbital velocity and is not immediately tidally disrupted at f = f tidal . We have found that f cut /f tidal is in the range between 1.25 and 1.4 in our results depending primarily on the NS radius R NS ; this value is larger for the smaller values of R NS . These results imply that it is not straightforward to determine f tidal from gravitational wave observation. If the dependence of f cut /f tidal on the NS radius and mass ratio is clarified in detail by the numerical simulation, f tidal may be inferred from f cut . If such a relation is found, f cut will be useful for determining the properties of NSs such as the radius and density profile. For this purpose, a detailed simulation taking into account the nuclear EOSs is necessary. We plan to perform such simulation in the next step. Figure 8 plots the evolution of the rest mass of the material located outside the apparent horizon M r>rAH and the area of the apparent horizon in units of 16πM 2 for models A1, A2, and C2. For comparison, the results for models A and C are shown together.
We find that the results for models A1 and A2 agree qualitatively well with those for model A. Because of the difference of the grid structure for covering the BH and NS and/or because of the larger numerical dissipation for the low-resolution runs, the time of the onset of tidal disruption t disr slightly disagrees among three models, but the difference is only ≈ 0.2-0.3 ms (∼ 10% of the orbital period at the last inspiral orbit). The value of M r>rAH (A AH /16πM 2 ) is systematically overestimated (underestimated) for the lower-resolution runs. However, the magnitude of the disagreement is not large. For model A2 (A1), the relative errors of M r>rAH and A AH to the values for model A is ∼ 7% (∼ 4%) and ∼ 0.3% (∼ 0.3%), respectively. Furthermore, the values systematically converge, although the order of the convergence is not exactly specified because the difference among three results is small. (Note that the order of the convergence in the presence (absence) of shocks should be the first (third) order for the hydro part whereas that for the gravitational field is fourth order, and thus, the order of the convergence is not simply determined). For determining the order of the convergence, it is necessary to perform simulations with a better accuracy. However, it is not feasible to do it in our present computational resources. Assuming that the second-order convergence holds for M r>rAH , the extrapolation gives the exact value of M r>rAH , and we find that the errors for M r>rAH at t − t disr = 7 ms is ∼ 0.019M ⊙ (∼ 22%), 0.024M ⊙ (∼ 30%), and 0.030M ⊙ (∼ 35%) for models A, A1, and A2, respectively. For A AH /16πM 2 , the relative errors at t − t disr = 7 ms are ∼ 0.3%, 0.5%, and 0.5% for models A, A1, and A2.
We find that the results for model C2 agree well with those for model C. In this case, the time of the onset of tidal disruption t disr agrees within ≈ 0.04 ms (∼ 2% of the orbital period at the last inspiral orbit). Furthermore, the value of M r>rAH and the area of the apparent horizon agree within ∼ 3% and ∼ 0.1% errors, respectively. Assuming the second-order convergence, the exact values of M r>rAH and A AH /16πM 2 at t − t disr = 8 ms are determined, and by comparison with these values, we find that the errors of these values for model C (C2) are 4% (6%) and 0.2% (0.4%), respectively. For model C2, the grid spacing for the inner computational domain is ∆x = M p /12, with which the major diameter of the NS is covered by the 50 grid points (see Table II). Thus, with such setting, the orbits of the BH and NS are computed with an accuracy enough at least for the qualitative study. Figure 9 plots h + and h × (a) for models A and A2 and (b) for models C and C2. This shows that the waveforms for the high and low-resolution runs agree qualitatively well. For the inspiral phase, the difference of the waveforms for the two different resolutions is very small. In particular, for models C and C2, the amplitude at a given time agrees each other within ∼ 3% error for t ret < ∼ 4.5 ms. During the tidal disruption phase, the amplitude is systematically smaller for the low-resolution runs. The possible reason for this is that the compactness of the NS is more quickly lost with the lower resolution, resulting in the less coherent excitation of gravitational waves. However, the difference of the amplitude is still at most ∼ 10% for model C2. For model A2, the error is larger in particular for 4.5 ms < ∼ t ret < ∼ 5 ms, although the waveforms agree qualitatively with those for model A. The primary source of the error is the phase difference caused by the fact that for the lower resolution, the tidal disruption sets in earlier. During the tidal disruption, the wave amplitude for model A2 is by a factor of ∼ 2 smaller than that for model A. This indicates that the simulation of the poor resolution significantly underestimates the amplitude during the tidal disruption phase. Because of the underestimation of the amplitude, the total energy and angular momentum emitted by gravitational waves are underestimated by ∼ 5% for model C2 and by ∼ 10% for model A2 in comparison with models C and A.
FIG. 1 :
1Snapshots of the density contour curves for ρ and the three velocity for v i (= dx i /dt) in the equatorial plane for model A.
FIG. 2 :
2Profiles of ε th , ε th /ε, and ρ along x-axis (solid curves) and y-axis (dashed curves) at t = 7.117 ms (left), 9.070 ms (middle), and 11.02 (right).FIG. 3: Profiles of ε th , ε th /ε, and ρ along x-axis (solid curves) and y-axis (dashed curves) at t = 11.02 for models B (left) and C (right).
FIG. 4 :
4The same asFig. 1but for models B and C at t = 11.02 ms. The upper two panels plot the contour plots and velocity vectors for x-y and y-z planes for model B. The lower two panels show the plots for model C.
Figure 5 (
5d) plots the values of M r>rAH at the end of the simulations as a function of R NS for given values of the mass ratio q. It is clearly seen that the fraction of the material located outside the BH steeply decreases with the decrease of the NS radius. Extrapolating the results for the smaller values of R NS , we expect that M r>rAH would become close to zero for R NS < ∼ 11 km both for q = 0.33 and 0.39. For such case, a very small value of the BH mass as M BH < 3M ⊙ would be required for the formation of torus of mass > ∼ 0.01M ⊙ .FIG. 5: The evolution of the rest mass of the material located outside the apparent horizon (upper panel of each figure) and the evolution of the area of the apparent horizon in units of 16πM 2 (lower panel of each figure) (a) for models A, D, and F (for RNS = 13.2 km), (b) for models B and E (for RNS = 12 km), and (c) for models A, B, and C (for q ≈ 0.33). t disr in the horizontal axis denotes the approximate time at which the tidal disruption sets in, and we determine it to be 4.5 ms for model A, 4.2 ms for mode B, 4.3 ms for mode C, 4.2 ms for model D, 3.9 ms for model E, and 5.6 ms for model F, respectively. (d) Mr>r AH at the end of the simulations as a function of RNS.
Figure 6 Figure 6
66(a) and (b) plot gravitational waveforms observed along the z-axis as a function of the retarded time for model A. The retarded time is approximately defined byt ret ≡ t − D − 2M ln(D/M ),(21)where D is the distance to the observer. The upper panel ofFig. 6(a) plots + and × modes of gravitational waves and the lower one the Newman-Penrose quantity. The gravitational waveforms are obtained from twice time integration of the Newman-Penrose quantity. Because the l = |m| = 2 mode is dominant in the observation along the z-axis, we determine the waveforms only from this mode.From the values of h + and h × , the amplitude of gravitational waves at a distance D is evaluated h gw ≈ 5.0 × 10 (a) implies that the maximum amplitude at a distance of D = 100 Mpc is ≈ 4 × 10 −22 for model A.
FIG. 6 :
6(a) upper panel: + and × modes of gravitational waveforms observed along the z-axis for model A (solid and dashed curves, respectively). tret denotes the retarded time [see Eq.(21)] and M0 is the total mass defined in the caption ofTable I. The amplitude at a distance of observer can be found from Eq.(22). (a) lower panel: The real and imaginary parts of ψ4 (solid and dashed curves). (b) The real (upper panel) and imaginary (lower panel) parts of ψ4 for model A together with the hypothetical curves of gravitational waves associated with the QNM (dotted curves) for 5 ms < ∼ tret < ∼ 6.5 ms. (c) h+ (upper panel) and Re(ψ4) (lower panel) for models A (dashed curve) and C (solid curve). (d) The real (solid curve) and imaginary (dashed curve) parts of ψ4 for model B (upper panel). The waveforms for 4.8 ms < ∼ tret < ∼ 6 ms are magnified in the lower panel together with the hypothetical curves of gravitational waves associated with the QNM (dotted curves). For all the plots, only the l = |m| = 2 modes are taken into account.
FIG. 7 :
7(a) Spectrum of gravitational waves h(f )f for model A (solid curve). The dashed and dotted curves are the spectra for the purely numerical (short-term) data and for the analytic third PN waveform. (b) the same as (a) but for models A, D, and F. The vertical dotted lines denote the expected frequencies at which the tidal disruption sets in for models D and F; f tidal ≈ 0.84 kHz for model F and 0.88 kHz for model D. (c) the same as (b) but for models D and E. The vertical dotted line denotes the expected frequency at which the tidal disruption sets in, f tidal ≈ 1.0 kHz. (d) the same as (b) but for models A-C. For (b)-(d), the solid circles denote the location of fcut.
FIG. 8 :
8The same as Fig. 5(c) but for models A, C, A1, A2, and C2. We choose the values of t disr = 4.30 ms, 4.17 ms, and 4.50 ms for A2, A1, and C2, respectively. For models A and C, see the caption of Fig. 5.
TABLE
/2 . M0Ω is ≈ 0.040-0.041 for all the models.Model Mp(M⊙) MBH(M⊙) M * (M⊙) MNS(M⊙) q RNS (km) M * /κ 1/2 M (M⊙) J/M 2 P0/M0 (M0Ω) 2/3
A
3.906
3.975
1.400
1.302
0.327
13.2
0.150
5.227 0.662 154
0.118
B
3.881
3.950
1.400
1.294
0.327
12.0
0.160
5.193 0.660 154
0.119
C
3.930
4.001
1.400
1.310
0.328
14.7
0.140
5.260 0.663 152
0.119
D
3.255
3.321
1.400
1.302
0.392
13.2
0.150
4.576 0.725 157
0.117
E
3.234
3.300
1.400
1.294
0.392
12.0
0.160
4.546 0.721 154
0.118
F
4.557
4.627
1.400
1.302
0.281
13.2
0.150
5.878 0.612 158
0.117
TABLE
TABLE III :
III
TABLE IV :
IVThe expected frequency of gravitational waves at the tidal disruption f tidal , the peak frequency of gravitational wave spectrum near the sharp cut-off in the high-frequency region fcut, and the ratio fcut/f tidal .Model f tidal (kHz) fcut (kHz) fcut/f tidal
A
0.856
1.16
1.36
B
0.997
1.41
1.41
C
0.736
0.92
1.25
D
0.877
1.14
1.30
E
1.021
1.40
1.37
F
0.840
1.09
1.30
AcknowledgmentsWe deeply thank members in the Meudon relativity group, in particular, Eric Gourgoulhon, for developing the free library LORENE, which is used for computation of the initial conditions. Numerical computations were performed on the FACOM-VPP5000 at CfCA at National Astronomical Observatory of Japan and on the NEC-SX8 at Yukawa Institute of Theoretical Physics of Kyoto University. This work was supported by a Monbukagakusho Grant (No. 19540263).APPENDIX A: COMPARISON WITH THE LOW-RESOLUTION RESULTSIn this appendix, we present the results for models A1, A2 and C2 for which the physical parameters of the initial conditions are the same as those for models A and C but the simulations were performed with the poorer grid resolutions (seeTable II). the BH and surrounding material. Thus, M BH,f likely overestimates the true BH mass slightly.[53] The studies for the kick velocity in the numerical relativity[38]has clarified that the numerical results of its magnitude depend sensitively on the initial condition.For the initial condition of small orbital separation, the given quasicircular orbit includes nonrealistic radial velocity and/or nonzero ellipticity. Due to such unrealistic elements, the estimated linear momentum flux includes an systematic error. The results of[38]show that the systematic error could be as large as the magnitude of the kick velocity. Because the simulation is started from a fairly close orbit in this work, the systematic error may be as large as the magnitude for the kick velocity.[54] In this subsection, x is different from the one of the Cartesian coordinates.
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| []
|
[
"Dependence of Hall Coefficient on Grain Size and Cosmic Ray Rate and Implication for Circumstellar Disk Formation",
"Dependence of Hall Coefficient on Grain Size and Cosmic Ray Rate and Implication for Circumstellar Disk Formation"
]
| [
"Shunta Koga \nDepartment of Earth and Planetary Sciences\nFaculty of Sciences\nKyushu University\n819-0395FukuokaJapan\n",
"Yusuke Tsukamoto \nDepartment of Earth and Space Science\nGraduate Schools of Science and Engineering\nKagoshima University\nKagoshimaJapan\n",
"Satoshi Okuzumi \nDepartment of Earth and Planetary Sciences\nTokyo Institute of Technology\nTokyoJapan\n",
"Masahiro N Machida \nDepartment of Earth and Planetary Sciences\nFaculty of Sciences\nKyushu University\n819-0395FukuokaJapan\n"
]
| [
"Department of Earth and Planetary Sciences\nFaculty of Sciences\nKyushu University\n819-0395FukuokaJapan",
"Department of Earth and Space Science\nGraduate Schools of Science and Engineering\nKagoshima University\nKagoshimaJapan",
"Department of Earth and Planetary Sciences\nTokyo Institute of Technology\nTokyoJapan",
"Department of Earth and Planetary Sciences\nFaculty of Sciences\nKyushu University\n819-0395FukuokaJapan"
]
| []
| The Hall effect plays a significant role in star formation because it induces rotation in the infalling envelope, which in turn affects the formation and evolution of the circumstellar disk. The importance of the Hall effect varies with the Hall coefficient, and this coefficient is determined by the fractional abundances of charged species. These abundance values are primarily based on the size and quantity of dust grains as well as the cosmic ray intensity, which respectively absorb and create charged species. Thus, the Hall coefficient varies with both the properties of dust grains and the cosmic ray rate (or ionization source). In this study, we explore the dependence of the Hall coefficient on the grain size and cosmic ray ionization rate using a simplified chemical network model. Following this, using an analytic model, we estimate the typical size of a circumstellar disk induced solely by the Hall effect. The results show that the disk grows during the main accretion phase to a size of ∼ 3 − 100 au, with the actual size depending on the parameters. These findings suggest that the Hall effect greatly affects circumstellar disk formation, especially in the case that the dust grains have a typical size of ∼ 0.025 µm − 0.075 µm. | 10.1093/mnras/sty3524 | [
"https://arxiv.org/pdf/1812.07131v2.pdf"
]
| 119,104,118 | 1812.07131 | aeff325bdb7f972f482d1efb89f9dda327a7827d |
Dependence of Hall Coefficient on Grain Size and Cosmic Ray Rate and Implication for Circumstellar Disk Formation
25 Dec 2018 27 December 2018
Shunta Koga
Department of Earth and Planetary Sciences
Faculty of Sciences
Kyushu University
819-0395FukuokaJapan
Yusuke Tsukamoto
Department of Earth and Space Science
Graduate Schools of Science and Engineering
Kagoshima University
KagoshimaJapan
Satoshi Okuzumi
Department of Earth and Planetary Sciences
Tokyo Institute of Technology
TokyoJapan
Masahiro N Machida
Department of Earth and Planetary Sciences
Faculty of Sciences
Kyushu University
819-0395FukuokaJapan
Dependence of Hall Coefficient on Grain Size and Cosmic Ray Rate and Implication for Circumstellar Disk Formation
25 Dec 2018 27 December 2018MNRAS 000, 000-000 (0000) Preprint 27 December 2018 Compiled using MNRAS L A T E X style file v3.0stars: formation -stars: magnetic field -ISM: clouds -cosmic rays-dust, extinction ⋆
The Hall effect plays a significant role in star formation because it induces rotation in the infalling envelope, which in turn affects the formation and evolution of the circumstellar disk. The importance of the Hall effect varies with the Hall coefficient, and this coefficient is determined by the fractional abundances of charged species. These abundance values are primarily based on the size and quantity of dust grains as well as the cosmic ray intensity, which respectively absorb and create charged species. Thus, the Hall coefficient varies with both the properties of dust grains and the cosmic ray rate (or ionization source). In this study, we explore the dependence of the Hall coefficient on the grain size and cosmic ray ionization rate using a simplified chemical network model. Following this, using an analytic model, we estimate the typical size of a circumstellar disk induced solely by the Hall effect. The results show that the disk grows during the main accretion phase to a size of ∼ 3 − 100 au, with the actual size depending on the parameters. These findings suggest that the Hall effect greatly affects circumstellar disk formation, especially in the case that the dust grains have a typical size of ∼ 0.025 µm − 0.075 µm.
INTRODUCTION
It is important to elucidate the process by which stars are formed, because stars are the fundamental constituents of the universe and because star formation is associated with the origin of planets. Observations indicate that the magnetic energy in prestellar clouds is comparable to the gravitational energy (Crutcher et al. 2010). This large quantity of magnetic energy (that is, a strong Lorentz force) affects the star formation process. For example, magnetic fields play a significant role in determining the angular momentum distribution in star-forming cores. In the star formation process, the magnetic fields suppress disk formation. Some researchers pointed out that no disk appears in the early star formation phase, because the disk angular momentum is excessively transferred by the magnetic effect (the socalled 'magnetic braking catastrophe' problem) (Mellon & Li 2008). There is another issue to the effect of magnetic fields during the star formation process (the so-called magnetic flux problem). The magnetic flux of a star-forming core is approximately five orders of magnitude greater than that of a protostar (e.g. Nakano 1984). Because magnetic flux is a conserved quantity, the magnetic field should dissipate during the course of the star formation.
The magnetic field is coupled with charged particles, and the amount of charged particles is considerably less in star-forming clouds due to their low temperature and high density.
However, during the early gas collapse phase, which is associated with a number density of n 10 10 cm −3 , the magnetic field is closely coupled to the neutral gas as a result of efficient momentum exchange between charged particles and neutral species. Conversely, as the cloud density increases, dust grains absorb charged particles and the momentum exchange becomes inefficient (Umebayashi & Nakano 1980). Therefore, the magnetic field cannot couple with neutral species and the magnetic field dissipates to a greater extent (Nakano et al. 2002). The magnetic dissipation helps disk formation, because the magnetic braking, which suppresses the formation and evolution of the disk, is alleviated by the dissipation. Thus, the magnetic field and its dissipation are related to major issues (magnetic braking catastrophe and magnetic flux problems) of the star formation process.
Both the degree of ionization (that is, the abundance of charged particles) and the extent of magnetic dissipation during star formation had been investigated in detail using one-zone calculations by Nakano and his collaborators for approximately three decades (Umebayashi & Nakano 1990). The methodology of this group is well established and is frequently applied to estimate the abundance of various chemicals as well as magnetic dissipation during the star formation process (e.g. Tsukamoto et al. 2017;Wurster et al. 2018b).
Beginning in the 1990s and subsequent to the introduction of the one-zone calculations, which basically estimate the chemical abundance of each charged species, multi-dimensional magnetohydrodynamics (MHD) simulations of collapsing star-forming clouds were reported.
In this process, calculations are performed until a protostar begins to form from the prestellar cloud core stage. One pioneering work by Tomisaka (1998Tomisaka ( , 2000Tomisaka ( , 2002 investigated cloud evolution up to the point of protostar formation using two-dimensional ideal MHD simulations. There are three non-ideal MHD effects: Ohmic dissipation, ambipolar diffusion and the Hall effect. Machida et al. (2006Machida et al. ( , 2007 first calculated protostar formation using non-ideal MHD simulations, although only Ohmic dissipation was considered. Duffin & Pudritz (2009) performed three dimensional non-ideal MHD simulations including only ambipolar diffusion and showed the early formation of outflow and disk. Subsequently, Tsukamoto et al. (2015a) and Tomida et al. (2015) included ambipolar diffusion in addition to Ohmic dissipation.
However, until recently, the Hall effect has been ignored in multi-dimensional core collapse simulations because it is believed to be not directly related to the magnetic flux problem. It should also be noted that, although both Ohmic dissipation and ambipolar diffusion substantially reduce the magnetic flux in the star-forming core, the Hall effect only changes the magnetic field direction to generate toroidal magnetic fields from global poloidal fields (Wardle & Ng 1999).
Recently, the Hall effect has been considered with regard to the formation and evolution of circumstellar disks (Krasnopolsky et al. 2011;Tsukamoto et al. 2015bTsukamoto et al. , 2017Wurster et al. 2016). Krasnopolsky et al. (2011) have reported that, without rotation, a toroidal magnetic field component that is decoupled from the neutral gases is generated by the Hall effect. In the case that coupling between the magnetic field (or charged particles) and neutral species is recovered, the toroidal field (or twisted magnetic field) imparts a rotation to the infalling gas as a result of magnetic tension. In summary, even when the initial prestellar cloud has no rotation, rotation can be induced by the Hall effect. It should be noted that when a rotation motion is induced by the Hall effect, the magnetic field produces an counter-rotation in a star forming core due to the angular momentum conservation law, in which the angular momentum of the anti-rotation is transferred to the outer envelope from the cloud centre by the Alfvén wave. In other words, both clockwise and anti-clockwise rotations coexist in the initially non-rotating cloud, as reported by past studies (Krasnopolsky et al. 2011;Tsukamoto et al. 2015b;Wurster et al. 2016;Tsukamoto et al. 2017;Marchand et al. 2018).
Thus, it is important to consider the Hall effect when examining the angular momentum problem and the formation and evolution of the circumstellar disk during the star formation process. Note that we should not ignore the Hall effect even when considering the dissipation of magnetic fields (i.e. the magnetic flux problem). Since the Hall effect changes the configuration of the magnetic field, it would affect the dissipation process of the magnetic field (Bai & Stone 2017). Wardle & Ng (1999) also pointed out that the Hall effect (or Hall conductivity) strongly depends on the chemical abundance, which in turn is determined by the properties of the dust in star the formation process and the ionization (that is, cosmic ray) rate. Thus, it is important to estimate the effect of the Hall conductivity (or Hall coefficient) in conjunction with various grain sizes and cosmic ray rates and, for this purpose, some previous studies have examined grain size ranges and various cosmic ray rates. Hirashita & Lin (2018) investigated the grain sizes in galaxy halos using the extent of cosmic extinction of distant background quasars, and found that the most likely size range is 0.01-0.3 µm. In addition, based on theoretical and observational studies, cosmic ray rates have been estimated to be 2 × 10 −16 s −1 (Padovani et al. 2009) and 3.4 × 10 −18 s −1 (Farquhar et al. 1994) in a diffuse core and in the Bok globule B335, respectively. Thus, it is necessary to consider a wide range of possible grain sizes and cosmic ray rates.
However, it is quite difficult to perform multi-dimensional simulations of cloud collapse including non-ideal MHD effects and incorporating different dust properties and cosmic ray rates because of the limited CPU resources. For example, the calculation time of the simulation including the Hall effect is about 10 times longer than that of the simulation not including the Hall effect, because the whistler waves, which are the right-handed waves induced by the Hall term, should be resolved and implicit methods can not be applied to the Hall effect (Tsukamoto et al. 2015b). In addition, calculations regarding the formation and evolution of circumstellar disks that resolve the protostar using multi-dimensional non-ideal MHD
simulations have yet to be realized. It is important to note that erroneous results tend to be produced when introducing a sink method to model the long-term evolution of the circumstellar disk (Machida et al. 2014). Also, performing calculations to resolve both a protostar and circumstellar disk having large resistivities (or small conductivities) requires exceedingly short time steps during simulations (Vaytet et al. 2018), and so such calculations cannot be executed. Multi-dimensional calculations up to the point of several years after protostar for-mation are possible without employing a sink (e.g., Tomida et al. 2015;Masson et al. 2016;Tsukamoto et al. 2017;Wurster et al. 2018a;Tsukamoto et al. 2018), but it is important to keep in mind that a sink can introduce numerical artifacts and these should be carefully considered, as discussed in Bate (1998), Machida et al. (2010), ) & Machida et al. (2014. In addition to the large computational cost and sink problems, there is a further serious problem in non-ideal MHD simulations including the Hall term. Marchand et al. (2018) pointed out that the angular momentum is not conserved after the first core formation in their Hall MHD simulations. Therefore, the disk formation simulation including the Hall term is not an easy task.
In the present study, we analytically estimated the influence of the Hall effect on the formation and evolution of a circumstellar disk. In particular, we focused on the dependence of the Hall coefficient on the grain size (distribution) and the cosmic ray rate. We initially calculated the fractional abundance of charged species by solving chemical networks in conjunction with the grain size (distribution) and cosmic ray strength as input parameters, and derived the Hall coefficient in a realistic parameter space. Then, using the Hall coefficient, we estimated the disk size. On this basis, we discuss the importance of the Hall effect with regard to the disk evolution.
The remainder of the present paper is organized as follows. In §2, we describe the method used to calculate the Hall coefficient, while in §3 we describe the chemical abundance and
Hall coefficient results obtained in association with various grain sizes (distributions) and cosmic ray strengths. In §4, we describe the disk model obtained by considering the Hall effect and the resulting disk sizes based on the parameters employed. In §5, we discuss the time scale for the Hall effect and the effect of the magnetic field strength. Finally, a summary is presented in §6.
CHEMICAL REACTIONS AND HALL COEFFICIENT
The Hall Effect and Hall Drift Velocity
The induction equation including the Hall term is written as
∂B ∂t = ∇ × (v × B) − ∇ × η H (∇ × B) ×B ,(1)
where v, B,B and η H are the fluid velocity, the magnetic field, the unit vector for the magnetic flux density and the Hall coefficient, respectively. Ohmic dissipation and ambipolar diffusion terms are not included in this section because this study focuses on the Hall effect.
We define the Hall drift velocity as
v Hall ≡ −η H ∇ × B |B| .(2)
Substituting equation (2) into equation (1), the induction equation can be rewritten as
∂B ∂t = ∇ × [(v + v Hall ) × B] .(3)
Equation (3) indicates that the magnetic field can be amplified by the Hall velocity even when the gas fluid has no velocity (i.e., v = 0). In addition, the Hall velocity is rewritten as
v Hall = −η H cJ 4π|B| ,(4)
and is also parallel to the current (v Hall J ). The equations (3) and (4) indicate that magnetic field drifts to the direction parallel to J with the velocity of v Hall . The drifted magnetic field (or toroidal field) induces gas rotation due to the magnetic tension force.
Assuming only uniform vertical magnetic fields in the initial state, the toroidal component of the magnetic field is continuously amplified by the Hall effect until v = −v Hall is realized. Thus, the Lorentz force changes the gas velocity unless the condition v + v Hall = 0 is fulfilled, and the generation of the toroidal field gradually decreases as the gas velocity, v, approaches −v Hall . Finally, further generation of the toroidal field is completely suppressed and the gas has a velocity of v = −v Hall , such that the right hand side of equation (3) becomes zero. The timescale for the gas velocity to reach v = −v Hall is discussed in §5.2.1.
The gas fluid should have a rotational velocity of v = −v Hall without the initial cloud rotation. That is, even in the absence of rotation of the initial cloud, a rotational motion will result from the Hall effect (Krasnopolsky et al. 2011;Tsukamoto 2016;Marchand et al. 2018).
The Hall Coefficient
Investigating the influence of the Hall effect requires calculation of the Hall coefficient, η H , as described in Section §2.1. The Hall coefficient is written as
η H = c 2 σ H 4πσ 2 ⊥ ,(5)
where c is the speed of light and σ ⊥ is described as
σ ⊥ = σ 2 p + σ 2 H ,(6)
in which the Hall σ H and Petersen σ P conductivities are defined as
σ H = − c B ν β 2 ν 1 + β 2 ν Q ν n ν ,(7)
and
σ P = c B ν β ν 1 + β 2 ν |Q ν |n ν .(8)
Here, B is the magnetic field strength, ν indicates the physical quantity of each chemical species (for details, see Section §2.4), and Q ν , n ν and β ν are the charge number, number density and Hall parameter for each charged species, respectively. The Hall parameter, β ν , is defined as
β ν = τ ν |Q ν |B m ν c ,(9)
where τ ν and m ν are the stopping time (see below) and mass of each species, respectively.
The number density, n ν , of each chemical species, ν, is calculated using the chemical reaction networks described in Section §2.4.
Based on the abundance of each species, ν, relative to that of atomic hydrogen (hereafter termed the fractional abundance), x ν (≡ n ν /n H ), the Hall and Pedersen conductivities can be rewritten as
σ H = − c n H B ν Q ν x ν β 2 ν 1 + β 2 ν ,(10)
and
σ P = c n H B ν |Q ν |x ν β ν 1 + β 2 ν ,(11)
where n H is the number density of hydrogen atoms and x ν is calculated according to the procedure described in Section §2.4.
To calculate the Hall coefficient η H or Hall parameter β ν , one needs to know the stopping time, τ ν , in equation (9). In a weakly ionized gas, the stopping times for electrons, ions and dust grains are determined by collisions with neutral gas particles. The stopping times for electrons and ions are given by
τ e = m e + m n σν en ρ n ,(12)
and
τ i = m i + m n σν in ρ n ,(13)
respectively. Here, m n , m e and m i are the masses of neutrals, electrons and ions, respectively, ρ n is the number density of the neutrals, and σν en and σν in are the momentum-transfer rate coefficients for electron-neutral and ion-neutral collisions. According to Draine et al. (1983), the scattering cross-sections between a neutral and an electron, σν en , and between a neutral and an ion, σν in are respectively described by
σν en = 1.0 × 10 −15 cm 2 8k B T πm e ,(14)
and σν in = 1.8 × 10 −9 cm 3 s −1 ,
where k B is the Boltzmann constant and T is the gas temperature. In equations (14) and (15), the suffix n indicates neutral atoms and molecules (H, H 2 , He). The stopping time for dust grains can be written as
τ dust = ρ d a ρ n [8k B T /(πm n )] 1/2 ,(16)
where a and ρ d are the radius and total density of the dust grains (i.e., the total mass of the dust per cm 3 ), respectively. Equation (16) is the stopping time based on the Epstein drag law. Using the process described in §2.4, we calculated x ν for the density range of 10 4 cm −3 < n H < 10 14 cm −3 , within which the sizes of dust grains are much smaller than the mean free path of the neutrals, meaning that equation (16) can be utilized. The grain size (distribution) a is discussed in the following section.
Grain Size Distribution
Dust grains play a significant role in determining the Hall coefficient, η Hall , because they greatly affect the chemical abundance that in turn governs η Hall . Thus, calculation of the Hall coefficient requires the dust grain density and size (distribution) to be ascertained.
Although the true size distribution of dust grains is uncertain, the sizes can be estimated as described in §1, and so the present work includes the grain size as a parameter. In this subsection, we describe the two types of grain size distributions utilized in this study: MRN (Mathis et al. 1977) and single-sized distributions. The MRN grain size distribution can be written as
dn d,tot (a) = Ca −q da,(17)
where n d,tot and a are the dust grain number density and radius, respectively, q is typically given a value of 3.5 (Mathis et al. 1977)and C is a normalization factor determined by the total dust grain mass. Herein, we define the dust-to-gas mass ratio f dg = ρ d /ρ g , where ρ g is the gas density. Although f dg = 0.01 is adopted in the present calculations as the fiducial value, the value was varied over the range of 0.005 ≤ f dg ≤ 0.0341 so as to investigate the dependence of the amount of dust grains on the Hall coefficient. In the case on an MRN distribution, the dust density can be calculated as
ρ d = 4 3 πρ s amax a min a 3 dn d,tot da da = 4 3 πρ s C amax a min a 3 a −q da = 4 3 πρ s C 1 4 − q a 4−q max − a 4−q min ,(18)
where ρ s , a max and a min are the density of a dust grain and the maximum and minimum grain radius, respectively. Using equation (18), the normalization factor C is derived as
C = 3(4 − q) f dg ρ g 4πρ s a 4−q max − a 4−q min .(19)
In this work, the dust grain density was defined as ρ s = 2 g cm −3 and the MRN dust distribution was discretized into ten logarithmically-spaced bins. The number of partitions was s max (having a value of 10 in the present case), where s (with values from 1 -10) indicates the bin number. We define the typical (or averaged) dust grain radius included in the s-th bin by dividing the size range (a min ≤ a < a max ) on the log scale and the typical radius in the s-th bin can be written as
a s = a min a max a min 2s−1 2smax .(20)
In addition, we calculated the typical (or averaged) dust grain cross-section σ s in the s-th bin as
σ s = a s+0.5 a s−0.5 πa 2 dn d,tot da da a s+0.5 a s−0.5 dn d,tot da da = a s+0.5 a s−0.5 πa 2 Ca −q da a s+0.5 a s−0.5 Ca −q da = π 1 − q 3 − q a 3−q s+0.5 − a 3−q s−0.5 a 1−q s+0.5 − a 1−q s−0.5 ,(21)
in order to define the collisional rate between dust grains and charged particles (see Appendix
Appendix §C) where a s+0.5 and a s−0.5 are the maximum and minimum dust grain radius values in the s-th bin, respectively. The number density of the dust grains in the s-th bin is also calculated as
n d,s = a s+0.5 a s−0.5 dn d,tot da da = C 1 − q a 1−q s+0.5 − a 1−q s−0.5 ,(22)
using n d,s as the initial abundance of dust grains in each bin.
In addition to the MRN distribution calculations, we also constructed single-sized dust grain models. To make the model easier to understand and analogous to the MRN distribution, we defined the dust radius for single-sized grains as a single as a single = a max = a min and set s max =1. Based on this treatment, the collision cross-section, σ s , dust number density, n d , and total dust density, ρ d , were, respectively, described as
σ s = πa single 2 ,(23)n d = 3ρ g 4πa single 3 ρ s f dg ,(24)
and
ρ d = f dg ρ g = 1.4f dg m n n H .(25)
The factor of 1.4 is derived from the assumption of the neutral gas component n He = 0.2 n H 2 in neutral gas. Substituting ρ d and a s into equation (16) allows the collisional timescale between neutrals and dust grains, τ ν,dust , to be calculated. We employed n d as the initial abundance of dust grains in the single-sized dust models while, in the MRN distribution model, we calculated τ ν,dust for each bin (that is, the s = 1 − 10 th bins).
In this study, we considered four MRN distribution models having different values of a min and a max along with six single-sized grain models. The minimum and maximum radii in each MRN dust grain model (s1a, s1b, s1c and s2) are provided in Table 1, along with the grain radii for the single-sized models (s3 -s8). The dust-to-gas ratio and total dust grains surface area for each model are presented in the sixth and seventh columns of the table, respectively. Table 2. Abundance of each neutral species relative to the number of hydrogen atoms (Sano et al. 2000). δ indicates the fraction of each element remaining in the gas phase. The fractional abundances of molecules, δm, and of metal ions, δM, in the interstellar gas phase are δm = 0.2 and δM = 0.02, respectively, while the abundance of oxygen molecules in the gas phase is δO 2 = 0.7 (Morton 1974).
Chemical Species Relative Abundances
H 2 xH 2 = 0.5 He xHe = 9.75 × 10 −2 CO xCO = 3.62 × 10 −4 δm O 2 xO 2 = 0.5 δO 2 (8.53 − 3.62) × 10 −4 δm O xO = (1 − δO 2 )(8.53 − 3.62) × 10 −4 δm Mg xMg = 7.97 × 10 −5 δM
Chemical Reactions and Networks
Calculation of the Hall coefficient also required determination of the fractional abundance of each charged species, x ν (see equations (10) and (11)). In this subsection, we describe the method by which the chemical abundance of each charged particle was determined, based on a series of papers by Umebayashi & Nakano (1990). In these calculations, the relative abundance of each neutral species (H 2 , He, CO, O 2 , O and Mg) relative to the number of hydrogen atoms was fixed as shown in Table 2. As described in §2.2, H 3 + , m + , Mg + , He + , C + and H + were considered as the charged species in addition to the charged dust grains (see below),
where m + represents molecular ions other than H + 3 . Because the most abundant charged molecular ion species is HCO + , we identified m + with HCO + . A total of 25 chemical reactions was considered.
The four types of charged particle reactions incorporated in the calculations were (1) j → i + e − (ionization of neutral particles),
(2) i + e − → j (recombination of positive ions and electrons),
(3) i + k → j + * (recombination of charged ions), and (4) G(q) + i/e − (ion or electron) → G(q ± 1) (absorption of charged particles onto dust grains following collisions).
Here, i indicates a charged species, j and k represent different neutral species, * is the product of a charged species and G(q) is dust having charge q. In the following discussions, for convenience, we assign a number to each charged particle (ion or electron), as shown in Table 3.
In the case of reaction (1), cosmic rays represent the ionization source. Here, the parameter ζ is applied, equal to the total ionization rate of a hydrogen molecule, and we set the reaction rate of H 2 and He as shown in Table 2 in Umebayashi & Nakano (1990). The cosmic ray rates employed in this study are provided in Table 4, with ζ = 10 −17 s −1 as the fiducial value.
The chemical reactions and the associated reaction rates in points (2) and (3), above, are taken from the UMIST database (McElroy et al. 2013) and are summarized in Table A1 in Appendix §A.
Since dust grains absorb charged particles, they play a significant role in determining the fractional abundances of such particles. In addition, because the dust grains themselves are charged, they also contribute to the Hall coefficient. In Appendix §B, we discuss reaction (4), which is based on collisions between dust grains and charged particles with subsequent absorption of the charged particles on the grain surfaces.
At this point, we explain the calculations used to derive the abundances of charged particles and dust grains. For the sake of convenience, these calculations are based on the abundance of charged particles and dust grains relative to the number of hydrogen atoms, and so the abundances of each charged species and dust grain is described as
x i = n i /n H and x d = n d /n H .
In this process, the maximum and minimum charges of the dust grains at each gas density are determined (for details, see Okuzumi 2009), and the time-derivative equation for each species is provided in Appendix §C. The fractional abundance of each charged particle and dust grain as a function of the gas density was calculated according to the procedure:
(a) beginning from the initial chemical abundance values in Table 2, the equations given in Appendix §C were implicitly solved for a fixed gas density and gas temperature (see §2.5) until an equilibrium state was realized, and the initial dust grains abundance values were pre-calculated for each dust distribution model (using equation (22) for MRN models and equation (24) for single-sized models), following which (b) the gas density was increased by 0.1 dex and step (a) was repeated.
Using this procedure, the fractional abundance of each species was derived over the range of 10 4 cm −3 < n H < 10 14 cm −3 . It should be noted that we confirmed that the elapsed time required to obtain an equilibrium state was much shorter than the freefall time scale t ff (=(3π/(32 G ρ)) 1/2 ) in the density range considered in this study. We compare the elapsed timescale and freefall timescale and show that equilibrium time is much shorter than freefall time in Appendix §D. Finally, using the abundance of each species and the equations described in §2.2, we derived the Hall coefficient, η H , for each gas density. Employing ten different dust grain models (Table 1) and five different cosmic ray rates (Table 4), we calculated the chemical abundance values and the Hall coefficients for 50 models in total.
Gas Temperature and Magnetic Field Strength
When solving the chemical network for a given gas density, it is necessary to assume the gas temperature. According to Larson (1969), Masunaga & Inutsuka (2000) and Tomida et al. (2013), the gas temperature can be obtained using the barotropic equation of state
T = 10 1 + γ n H 10 11 cm −3 γ−1 K,(26)
where n H and γ (=5/3) are the number density and specific heat ratio of hydrogen. Because the Hall coefficient depends on the magnetic field strength, B, as discussed in Section §2, we also need to assume the magnetic field strength. In the present work, the plasma beta, β, was employed as an index of the magnetic field, defined as
β ≡ P th P mag ,(27)
where the thermal P th and magnetic P mag pressures are respectively written as
P th = ρ g k B T µm H(28)
and
P mag = B 2 8π .(29)
Here, ρ g , k B , µ and m H are the fluid density, Boltzmann constant, mean molecular weight and proton mass, respectively. In this study, a constant mean molecular weight of µ = 2.4 was adopted because the work focused solely on the low-temperature region of the collapsing cloud (see Section §4.1). Based on equations (28) and (29), the magnetic field strength can be written as
B = 8πρ g k B T µm H β .(30)
The gas density is related to the number density according to the relationship ρ g = 1.4n H m H .
Thus, the gas temperature is a function of the (number) density (eq. 26) and the magnetic field strength is a function of both the number density, n H , and the plasma beta, β. Because the plasma beta is not uniquely determined, we referred to recent publications regarding non-ideal MHD simulations (e.g. Wurster et al. 2018b), and on this basis adopted β = 100
as the fiducial value. Note that the effect of varying β on the Hall coefficient is discussed in Section §5.3. (30) with β = 100 against the number density. These plots are in good agreement with recently published one-and multi-dimensional simulations (Tsukamoto et al. 2015a;Tomida et al. 2015).
RESULTS OF CHEMICAL REACTION CALCULATIONS
Chemical reactions were calculated according to the method outlined in §2. To begin with, we present the chemical abundance results obtained using different dust models in conjunction with ζ = 10 −18 s −1 (Fig. 2), 10 −17 s −1 (Fig. 3) and 10 −16 s −1 (Fig. 4). These data indicate that the chemical abundances are significantly affected by both the dust distribution and cosmic ray rate. As the grain size becomes smaller, the adsorption of charge onto the grains becomes more effective, such that the number of ions and electrons decreases, as shown in Fig. 3. In addition, we checked the relation between the number of grains and f dg comparing models s1a, s1b and s1c, and confirmed that the fractional abundances of grains increase as f dg decreases. Figs. 2 -4 also show that the fractional abundances of dust grains in single sized models s3 -s8 are roughly proportional to a −3 because the total mass of grains is fixed.
Models s1a and s7 in Fig. 3 give similar results because the average size in MRN model s1a is 0.035µm (as calculated using equation (20)), which is close to the grain size assumed in model s7 (0.025 µm). In contrast, models s3 (a single = 0.3µm) and s8 (a single = 0.01µm) provide noticeably different results, with the abundances of Mg + and e − at n H = 10 8 cm −3 from model s3 being two and four orders of magnitudes higher, respectively, than those from model s8. These large differences in the fractional abundances of charged species greatly affects the Hall coefficient. In addition, a comparison of Fig. 2, Fig. 3 and Fig. 4 shows that the cosmic ray ionization rate significantly affects the fractional abundances of charged particles. To a first approximation, the fractional abundances of charged particles scale linearly with the ionization rate.
APPLICATION: DISK SIZE INDUCED BY THE HALL EFFECT
As described in Section §1, the purpose of this study was to estimate the disk size considering solely the Hall effect. To correctly determine the disk size, non-ideal MHD simulations were required (e.g. Tsukamoto et al. 2015b). However, during the star formation process, In addition, the Hall coefficient, which is related to the angular momentum associated with transport in the infalling gas and disk evolution, is greatly affected by specific dust properties and the environmental effects of the ionization source (or the cosmic ray strength).
Thus, to determine the disk size, it is necessary to calculate the disk evolution over a span of at least ∼ 10 5 yr, while resolving the protostar using the grain size and cosmic ray strength as parameters. Since such calculations are difficult to execute even using the fastest supercomputers presently available, we analytically estimated the disk size by referring to the simulation results.
Because the Hall coefficient is determined by the abundance of charged particles, the charged particle properties and chemical networks were discussed in the previous sections.
Below, in order to simplify our assumptions, we estimate and discuss the influence of the Hall effect on the evolution and formation of the circumstellar disk. This discussion begins with an explanation of the process used to estimate the disk size induced by the Hall effect.
The (specific) angular momentum of the infalling gas resulting from the Hall effect must first be evaluated, as this determines the disk size. Note that the angular momentum is redistributed in the disk and a part of the angular momentum is also ejected by protostellar outflow, meaning that the estimates provided herein actually refer to the upper limit of the disk size. Even so, it is useful to assess the effects of both the dust properties and cosmic ray strength on the Hall coefficient and resultant disk size. Our approach to estimating the disk size is illustrated schematically in Fig. 7. To begin, we assume a non-rotating prestellar cloud. Although this assumption is not actually correct, it is a useful approach to estimating the angular momentum resulting solely from the Hall effect. Next, we calculate the chemical networks and derive the Hall coefficient as a function of cloud density, which can be converted to a length scale (or Jeans length). Subsequently, the angular momentum of the infalling gas is derived by defining r Hall and assuming that the infalling gas instantaneously acquires the angular momentum at r Hall . In this study, we use the term r Hall for the Hall radius or
Hall point and employ a value of r Hall = 300 au, which is similar to the pseudo disk size (Tsukamoto et al. 2017). A Jeans length of r Hall = 300 au corresponds to a number density of n Hall = 10 8 cm −3 .
Even without rotation, the magnetic field is twisted by the Hall effect such that a toroidal component is produced. Subsequently, after coupling between the neutral gas and magnetic field (or charged particles) is recovered, the magnetic tension force caused by the toroidal component of the magnetic field imparts a rotational motion. This is the reason why the non-rotating gas acquires an angular momentum as a result of the Hall effect.
Hall Velocity
We assume that the gas velocity passing through the Hall radius (r = r Hall ) converges to the Hall drift velocity. Thus, the azimuthal component of the gas velocity can be written as
v φ ∼ −v Hall,φ , = η H (∇ × B) φ |B z | .(31)
The same definition of the Hall drift velocity is also used in Tsukamoto (2016). The convergence of the gas rotation velocity is explained in Section §2.1, and the validity of the gas velocity assumption is demonstrated in Fig. 3 of Krasnopolsky et al. (2011), which confirms that the azimuthal gas velocity converges to the Hall drift velocity. The timescale of the v φ change is discussed in Section §5.2.1.
Using a cylindrical coordinate system, the Hall-induced rotational velocity can be written
as v φ = η H B z ∂B r,s ∂z − ∂B z ∂r ,(32)
where B r,s is the magnetic field strength in the radial direction at the surface of the pseudodisk. These calculations allow the approximation |∂B r,s /∂z| ∼ B r,s /H env , where H env is the scale height of the infalling envelope. In addition, the radial derivative term is assumed to be zero (i.e., ∂Br,s ∂z ≫ ∂Bz ∂r ; Lubow et al. 1994). Thus, equation (32) can in turn be approximated by
v φ = η H |B z | B r,s H env .(33)
In these calculations, B r,s is rewritten in terms of the mass-to-flux ratio. The magnetic flux threading the core is given by
Φ tot = 2πr 2 B r,s ,(34)
based on the monopole approximation employed in Contopoulos et al. (1998) and Braiding & Wardle (2012).
We also define the mass-to-flux ratio normalized by the critical value as
µ tot ≡ M tot Φ tot / M Φ crit ,(35)
where (M/Φ) crit = (0.53/3π)(5/G) 1/2 is the critical mass-to-flux ratio (Mouschovias & Spitzer 1976;Mac Low & Klessen 2004) and M tot is the total core mass, as discussed below (see equation (42)). Using these equations, equation (33) can be rewritten as
v φ = η H |B z | B r,s H env (M/Φ) tot µ tot (M/Φ) crit = η H |B z | 1 µ tot (M/Φ) crit M tot 2πr 2 H env .(36)
Here, µ tot = 1 is adopted based on the simulation results of .
The vertical direction of the magnetic field is given by equation (30), using B = B z .
According to Saigo & Hanawa (1998), the scale height of the envelope can be approximated by
H env = √ 2c s √ πGρ env ∼ c s √ Gρ env ,(37)
where ρ env = 3.8×10 −16 g cm −3 (n env = 10 8 cm −3 ) is adopted on the basis of Tsukamoto et al. (2017). The speed of sound is determined employing equation (26). Using this approach, the scale height of the envelope is estimated to be H env = 360 au, in good agreement with recently-reported simulation results (e.g., Tsukamoto et al. 2017).
Disk Model and Angular Momentum Estimated using the Hall Effect
Recent simulations of disk formation have indicated that the circumstellar disk in the early star formation phase is massive Tsukamoto et al. 2015a;Tomida et al. 2015;Vaytet et al. 2018). Thus, Toomre's Q-parameter, which is given by Q = c s Ω kep /(πGΣ disk ), is expected to be small. Using the Q-parameter, we can represent the disk surface density
as Σ disk = c s Ω K πGQ ,(38)
where the value of Q is assumed to be constant over the entire disk. According to Kusaka et al. (1970) and Chiang & Goldreich (1997), the disk temperature can be determined using the equation
T disk = 150 r 1 au − 3 7 K.(39)
Substituting the temperature, T disk , and Keplerian angular velocity, Ω k , the dimensional surface density of the disk can be described as
Σ disk = c sd Σ 0 Q 1 −1 M star 0.1M ⊙ 1 2 r 100au − 12 7 g cm −2 ,(40)
where M star is the protostellar mass, Σ 0 is a constant estimated to have a value of Σ 0 = 8.2 g cm −2 based on the above equations, and c sd is a control parameter used to adjust the disk surface density. Using equation (40), the disk mass is estimated as
M disk = r 0 Σ disk 2πrdr.(41)
In addition, we define the total mass (which includes the disk and protostellar masses) as
M tot = M star + M disk .(42)
The total mass, M tot , should equal the overall mass of the infalling gas passing through the Hall radius. Note that we ignore mass ejection via protostellar outflow in this study.
The infalling gas passing through the Hall radius, r Hall , is assumed to acquire an angular momentum via the Hall effect, and this momentum can be estimated as
J Hall ∼ M tot j Hall = M tot r Hall v φ ,(43)
where r = r Hall and equation (36) To determine the size of the circumstellar disk, the angular momentum induced by the Hall effect is assumed to produce a Keplerian disk around the protostar. Consequently, the infalling mass is distributed into the protostar and Keplerian disk, although only the Keplerian disk is assumed to have angular momentum. This angular momentum is given by
J Kepler = r 0 Σ disk (r)2πrj Kepler dr = r 0 Σ disk (r)2πr 2 v Kepler dr = r 0 Σ disk (r)2πr 2 GM star r dr.(44)
Therefore, using solely the total and protostellar masses, we can determine the size of the disk outer edge at which the Keplerian angular momentum corresponds to the angular momentum of the total gas that has already undergone infall. Hereafter, we define the size of disk's outer edge as r disk .
As described above, the disk surface density is determined based on the Q-parameter.
However, since disk fragmentation may occur when Q 1, it is difficult to retain a simple disk. Thus, we consider that c sd = 1 (i.e. Q = 1) gives the maximum surface density. In reality, because the disk surface density is expected to be less than c sd = 1, c sd is parameterized and the value adopted in this study is provided in Table 5. The protostellar mass, which is also employed as a parameter to determine the disk radius, is included in the same table.
Keplerian Disk Induced by the Hall effect
The disk size induced by the Hall effect was determined according to the procedure:
(i) the protostellar mass, M star , was assumed to determine the total mass, M tot (equation (42)),
(ii) the specific angular momentum, j Hall = r Hall v Hall,φ , was estimated using equation (36), (iii) the total angular momentum of the accreted material was calculated as J Hall = M tot j Hall , (iv) the total angular momentum of the Keplerian disk was estimated based on equation (44), assuming that the disk extends to a very large radius, and
(v) the disk size, r disk , at which the angular momentum of the Keplerian disk equals that of the accreted material (the rotation of which is determined by the Hall effect as in steps (i) -(ii)) was determined.
After varying the protostellar mass, steps (i)-(v) were repeated using the different models listed in Table 1 While varying the protostellar mass, M star , we derived the corresponding disk sizes according to the procedure described above. Fig. 9 plots the disk sizes obtained using the single-sized (left) and MRN (right) dust grain size distribution models as functions of the protostellar mass, based on using c sd = 1 and ζ = 10 −17 s −1 . These plots demonstrate that the size of the Keplerian disk varies considerably depending on the grain size (distribution), even if the protostellar mass is kept constant. As an example, at M ps = 0.7 M ⊙ , the disk size for a single = 0.025µm (model s7) is r disk = 80 au, while the size for 0.3µm (model s3)
is r disk = 3 au (left panel of Fig. 9), corresponding to a variation in the disk size by a factor of 27. This result indicates that both the Hall effect and dust properties can affect the formation and evolution of the circumstellar disk.
Parameter Effects
The effects of the various parameters on the size of the Keplerian disk induced by the Hall effect were examined by constructing a fiducial model with the surface density parameter c sd = 1, protostellar mass M star =0.1M ⊙ and a cosmic ray ionization rate of ζ =10 −17 s −1 . In this subsection, we discuss the results. Tables 1and 5, obtained using a protostellar mass of M star = 0.1 M ⊙ and an ionization rate of ζ =10 −17 s −1 .
As described in §4.2, we constructed the disk model on the basis of Q = 1, which corresponds to a marginally gravitationally unstable disk. Subsequently, we calculated the disk sizes, while adjusting the disk surface density using the parameter c sd . As can be seen from these data, the disk size increases as the disk surface density decreases. This occurs because a less massive disk requires a large radius to have the same angular momentum as a more massive disk. However, the disk size is not strongly correlated with the surface density or with c sd . As demonstrated by Fig. 10, the disk radius increases by approximately one order of magnitude when the disk surface density is reduced by about two orders of magnitude (c sd = 0.01). intensity is strong. However, an exception is also evident such that, in the case of the small dust grain models (models s1a, s1c and s8), a disk radius peak is seen in the vicinity of ζ ∼ 10 −17.5 − 10 −16.5 s −1 . These three models have greater proportions of small dust particles than the other seven models (as seen in Table1) along with a smaller proportion of electrons, and the peak is attributed to these differences. The calculations required to obtain the Hall coefficient are highly complex and so at present we cannot conclusively identify the cause of the peak. However, this phenomenon will be addressed in a forthcoming paper.
CAVEATS
As reported in §2, previous simulation studies have identified the possible importance of the Hall effect during the star formation process. However, long-term simulations (up to the formation of a mature circumstellar disk) cannot be performed because the whistler waves propagating to the star forming cloud must be resolved using a very short time step to properly estimate the Hall effect during simulations (e.g. Sano & Stone 2002). Thus, in this study, we analytically estimated the impact of the Hall effect on disk evolution in starforming clouds, and found that this effect significantly influences the star and disk formation processes depending on the dust properties and the strength of ionization sources. However, there are some caveats associated with this investigation of the circumstellar disk induced by the Hall effect. In this section, we discuss the validity of the assumptions used in this study.
Other non-ideal MHD effects
In this study, we ignored ambipolar diffusion and Ohmic dissipation in the induction equation. Only considering the ambipolar diffusion, Hennebelle et al. (2016) also analytically estimated the disk radius as
r ∼ 18 au η AD 0.1s 2 9 B z 0.1G − 4 9 M 0.1M ⊙ 1 3 ,(45)
where M is the total mass of disk and protostar and η AD 1 is the ambipolar diffusion coefficient. When the protostellar mass is M star = 0.1M solar , the disk radius derived from Hennebelle et al. (2016) is comparable to that derived from this study (see Fig.9), which implies that we cannot ignore both ambipolar diffusion and the Hall effect when considering the disk formation.
In addition to the Hall effect and ambipolar diffusion, Ohmic dissipation also affects the disk formation (e.g. ). Moreover, in analytical works, we ignored some mechanisms of angular momentum transfer such as protostellar outflow that significantly reduces the angular momentum of the disk. Thus, in order to fully investigate the disk evolution, we need to execute non-ideal MHD simulations including all the non-ideal MHD terms in future.
Assumptions and limitations of the disk model
In this subsection, we discuss the assumptions and limitations used in this study. Especially, we focus on the timescale for convergence of the Hall velocity, the initial rotation, and the size growth of grains.
Convergence timescale of the Hall velocity
In Section §4.1, we assumed that gas particles instantaneously receive the angular momentum (or the Hall velocity) at the moment at which they pass through the Hall radius (see Section §4.2). In the collapsing cloud, the Hall velocity results from the Lorenz force of the toroidal field component induced by the Hall effect. We assumed the toroidal field produced by the Hall effect at a specific, constant value. However, it is necessary to estimate the saturation timescale in order to confirm that the gas particles indeed obtain the Hall velocity estimated in Section §4.1.
The specific angular momentum of the gas fluid induced by the Hall effect is estimated from the magnetic torque exerted by the magnetic tension and is described as
j(t Hall ) = r × (∇ × B) × B 4πρ z t Hall , ∼ r 4πρ B r,s B z H t Hall ,(46)
where t Hall is the timescale until a constant (or time-independent) toroidal field is induced by the magnetic tension force or Hall effect. We use the same approximation as used in the derivation of the Hall velocity (see eqs. (32) and (33)). In these calculations, a Hall radius of r Hall = 300 au and a density of ρ = 10 −15 g cm −3 are employed (as per Section §4.1). Using B z (eq. 30), H (equation (37)) and B r,s (equation (34)) and their fiducial values, equation (46) can be written as
j(t Hall ) ∼ 4.0 × 10 11 t Hall 1 s cm 2 s −1 .(47)
In addition, using the saturated Hall velocity (equation (36)), we can write the specific angular momentum at r Hall = 300 au as j(t sat ) = rv φ ∼ 2.1 × 10 18 cm 2 s −1 .
When the steady state (or saturation) is reached, j(t Hall ) = j(t sat ) is established. Thus, the saturation timescale is estimated as t Hall ∼ 5.1 × 10 6 s.
The dynamic timescale of the collapsing cloud roughly corresponds to the freefall timescale, t ff . The t ff at n Hall = 10 8 cm −3 is given by
t ff = 3π 32Gρ = 1.4 × 10 11 s.(50)
Thus, the saturation timescale is much shorter than the freefall timescale t Hall ≪ t ff , suggesting that the (saturated) toroidal field is instantaneously generated under a given set of cloud parameters, and supplies the Hall velocity estimated in this study to the infalling gas particles.
Initial Rotation
In this study, we ignored the rotation of prestellar clouds for simplicity. Observations show that molecular cloud cores have non-negligible rotations or angular momenta (e.g. Goodman et al. 1993;Caselli et al. 2002). Thus, our assumption of non-rotating cores is not very correct.
However, we could estimate the angular momentum purely induced by the Hall effect, and found that the Hall effect significantly influences the disk formation. This indicates that the Hall effect cannot be ignored to investigate the disk formation, independent of the initial cloud rotation.
Grain growth during star formation process
We do not include the time-dependent grain growth. However, in reality, grains would grow by collisions between grains or sticking molecules onto grains. In addition, collisional destruction should be occurred in the star forming cloud. Chiaki et al. (2014) executed the simulations to estimate the dust growth under very lowmetallicity environments. They showed that the grain growth can change the star formation process under such environments. Our results showed that the grain sizes strongly affect the Hall effect and resulting disk size. Thus, we require simulations including the time-dependent change of grain sizes (or grain growth) in a future study.
The Effect of Magnetic Field Strength
A value of β=100 was used to estimate the magnetic field strength in conjunction with equation (30), which is consistent with recent numerical simulations (e.g., Wurster et al. 2018b). However, since only the very early phase of disk formation has been investigated by simulations, the magnetic field strength and plasma beta of the disk would be expected to vary over time.
In this subsection, we assess the effect of the magnetic field strength (or plasma beta) on the disk size. Based on §4.2, we again calculated the disk size, employing β values over the range of 1 ≤ β ≤ 500. Fig. 12 plots the disk size values against the plasma beta, as determined using the fiducial parameters a single = 0.1 µ m and ζ = 10 −17 s −1 . This figure confirms that the disk size decreases as β increases over the range of 1 β 50, but increases in the case of β values in the range of β 50. At n Hall = 10 8 cm −3 , the Hall coefficient, η H , becomes large as the magnetic field is strengthened (see Fig. 6 in Marchand et al. (2016)).
In the present analysis, when 1 β 50, variations in η H are primarily responsible for changes in β . In contrast, when 50 β 500, the decrease in the vertical component of the magnetic field, B z , determines the disk size (see equation (36)). However, the dependence of the magnetic field on η H is not straightforward (see equations (6) -(9)), and thus we cannot provide a simple explanation of the trend seen in Fig. 12. During the star formation process, although the fluctuations in the magnetic field strength are complex, our results suggest that it is important to consider the magnetic field strength when assessing the Hall effect.
It is evident that the disk size varies with the magnetic field strength, and that the magnetic field is accumulated and amplified in the disk because it is coupled with the infalling matter. In contrast, the magnetic field is dissipated by Ohmic dissipation and ambipolar diffusion in the disk, and the magneto-rotational instability plays a role in determining the magnetic field strength in the disk. These effects are beyond the scope of this study, and the determination of magnetic field strength in the disk would not be possible even using
SUMMARY
In this study, we analytically estimated the influence of the Hall effect on the formation and evolution of circumstellar disks. The chemical reactions in the collapsing cloud were initially determined to derive the Hall coefficient based on prior publications, but using a variety of dust grain models and ionization intensities. The results show that the Hall coefficient strongly depends both on the assumed dust model and the ionization intensity.
Subsequently, a simple disk model was employed to evaluate the size of the circumstellar disk solely on the basis of the Hall effect. Depending on the grain size and ionization intensity values selected, disk sizes of ∼ 3 − 100 au (∼ 2 − 20 au) were obtained for a protostar mass of 0.5 M ⊙ (0.1 M ⊙ ). It was expected that at n Hall = 10 8 cm −3 , which number density is used in the disk model, smaller dust grains tend to form a larger disk in the grain size range of 0.025µm a single 0.3µm, because small grains significantly reduce the fraction of charged particles and amplify the non-ideal MHD effects. Conversely, in the range of 0.01µm a single 0.025µm, the disk size decreases as the grain size becomes smaller, because charged dust grains are much more abundant than charged particles. As such, a peak in the disk size change data appears at a single = 0.025µm in the case of ζ = 10 −17 s −1 .
In addition, large disks tend to be associated with weak ionization intensities, as a result of coupling between charged species and neutral gas.
The Keplerian disks observed around Class 0 objects have a size of ∼ 10 − 100 au, which is comparable to the disk size induced by the Hall effect. Thus, the Hall effect evidently contributes to the disk evolution, depending on the star-forming environment. It is important to note that the simple disk model in the work reported herein was not comprehensive. In particular, the time required for the growth of dust grains was neglected. Nevertheless, this work demonstrates that the Hall coefficient is significantly dependent on the average grain size, and so investigations of disk evolution must take into account the growth of dust particles.
APPENDIX B: DUST REACTIONS
In this appendix, we address collisions between dust grains and charged particles and the absorption of charged particles onto the surfaces of dust grains. Three types of dust grains are considered: positively charged, negatively charged and neutral. The reactions of these grains with ions and electrons are also considered, such that six reactions are required, as described below. According to Draine & Sutin (1987), the reaction rates can be summarized as itemized below.
[1] The collisional rate between charged grains and ions (i = 1 -6), written as
k id (s, z) = 8k B T πm i σ s exp − z (1+|z | −1 /2 )τs 1 + 1 4τs+3z 1/2 2 (z > 0)
Here, z, m e , σ s and τ s indicate the charge on the dust particles, the electron mass, the collision cross-section and the reduced temperature, respectively. The reduced temperature, τ s , is described as
τ s = a s k B T e 2 ,(B4)
and the collision cross-section, σ s , is
σ s = πa s 2 ,(B5)
where the suffix s indicates the dust grains in the s-th bin (see Section §2.3).
i and a dust grain, and is given by Figure D1. Chemical abundances of charged species for different gas number density 10 4 , 10 8 , 10 11 , 10 14 cm −3 as a function of ζt (ζ = 10 −17 s −1 ). The freefall timescale is given by t ff = 3π 32Gρ .
δ i,d =
APPENDIX D: EQUILIBRIUM TIMESCALE VS. FREEFALL TIMESCALE
In this appendix, we compare the elapsed time required to obtain an equilibrium state with the freefall timescale at a given density. Figure D1 shows that chemical abundances of charged species considered in this study reach an equilibrium state within the freefall timescale.
In all density, the equilibrium is achieved before the freefall time. Thus we confirm that the elapsed time necessary for reaching an equilibrium state is much shorter than the freefall timescale over the density range of 10 4 cm −3 < n H < 10 14 cm −3 . The validity of the chemical equilibrium is also seen in Fig.13 of Marchand et al. (2016).
This paper has been typeset from a T E X/L A T E X file prepared by the author.
Figure 1 .
1Gas temperature (left) and magnetic field strength (right) as functions of number density.
Fig 1
1plots the gas temperatures (left) derived from equation(26)and the magnetic field strengths (right) derived from equation
Figure 2 .
2Chemical abundances of charged species, neutral species and charged dust grains (Z = ±1) for different dust models (s1a, s3, s4, s5, s7 and s8), obtained using a constant ionization rate of ζ = 10 −18 s −1 , as functions of the number density.
Figure 3 .
3The same data as inFig. 2but for ζ = 10 −17 s −1 using models (s1a, s3, s4, s5, s7 and s8).
Fig. 5 and
5Fig. 6show the Hall coefficients calculated for the different dust models using equations (5)-(16). These plots indicate that the absolute value of the Hall coefficient is modified by both the grain size and grain size distribution, as well as by the cosmic ray strength. Furthermore, as seen inFig. 5 and Fig. 6, the sign of the Hall coefficient can change depending on the grain size or grain size distribution. As an example, in the center panel of Fig. 5 (ζ = 10 −17 s −1 ), the Hall coefficient is positive over the entire range of 10 4 cm −3 n H 10 14 cm −3 for model s3, while it is positive only in the range of n H > 10 13 cm −3 for model s5. Thus, even a change in the grain size by a factor of four can cause a flip in the sign of the Hall coefficient at n H ∼ 10 13 cm −3 . Zhao et al. (2018) also showed that large-sized grains (or removal of small-sized grains; 0.1µm) noticeably decrease the Hall coefficient, which agrees well with our results (see the center panel of Fig. 5).
Figure 4 .
4The same data as inFig. 2but for ζ = 10 −16 s −1 using models (s1a, s3, s4, s5, s7 and s8).
Figure 5 .
5Hall coefficients for single size dust models at ζ = 10 −18 s −1 (left), 10 −17 s −1 (center) and 10 −16 s −1 (right). The results from models s3, s4, s5, s6, s7 and s8 are plotted in blue, red, green, yellow, orange and purple. For each line, the positive and negative values of η H are indicated by broken and solid lines, respectively.
Figure 6 .
6Hall coefficients for various MRN dust distributions. Models s1a, s1b, s1c, s2 and s4 are plotted in green, light green, dark green, gray and red, respectively. For each line, positive and negative values of η H are indicated by solid and broken lines, respectively. a small disk formed around a protostar gradually evolves. Thus, at minimum, the spatial scale of the protostar should be resolved(Vaytet et al. 2018) when examining the formation and evolution of the circumstellar disk. Despite this, the timescale becomes extremely short when resolving the protostar, and so it is impossible to simulate disk evolution and determine the disk size with adequate spatial resolution. As an example, a very recent simulation managed to calculate the disk evolution over a time span of several months following protostar formation (e.g.,Tomida et al. 2015;Masson et al. 2016;Tsukamoto et al. 2017). In contrast, observations have indicated that the main accretion phase, during which the disk evolves via mass accretion, lasts for ∼ 10 5 yr(Andre & Montmerle 1994;Enoch et al. 2008).
Figure 7 .
7Schematic of the envelope, pseudo-disk and disk.
are employed to obtain v φ . Thus, the Hall angular momentum, J Hall , can be written as J Hall ∼ M tot r Hall v Hall,φ .
Figure 8 .
8and 4, to determine r disk for each model. The resulting values are presented in Fig. 8 as filled magenta circles. As an example of the estimation of the disk induced by the Hall effect, Fig. 8 plots the angular momentum values for Keplerian and Hall disks (hereafter, we refer to disks formed by the Hall effect as Hall disks). These values were obtained using c sd = 1 and ζ = 10 −17 s −1 for different grain size models and assuming a protostellar mass of M star = 0.1 M ⊙ . Additionally, the specific angular momentum induced by the Hall effect, j Hall , was calculated for each model (see Section §4.2). For each line (that is, each grain size model) in this figure, the angular momentum of the Hall disk is less than that of the Keplerian disk outside the point denoted by the purple circle, meaning that a rotationally-supported disk will not form in such regions. Conversely, in the region (or at the radius) inside the purple circle, the angular momentum of the Hall disk is greater than that of the Keplerian disk and so a rotationally-supported disk (i.e., a Keplerian disk) can form. Because angular momentum can be transferred inside the disk, it is difficult to precisely determine the disk radius values in this study. Even so, a comparison of the angular momenta of the Hall disk and Keplerian disk allows a rough estimation of the disk size for The angular momentum values of the Hall disks for different grain size models versus the radius, as obtained using c sd = 1 and ζ = 10 −17 s −1 , with a protostellar mass of Mstar = 0.1 M ⊙ . The thick black solid line corresponds to the angular momentum of the Keplerian disk. The filled magenta circles indicate the intersections of the angular momenta of Hall and Keplerian disks, and correspond to the radii of the Hall disks. a given grain size and protostellar mass. The data in this figure indicate that the disk size is r disk = 2.13 au in the case of model s4 (a single = 0.1 µm) and 5.01 au in the case of model s8 (a single = 0.01 µm).
Figure 9 .Figure 10 .
910Disk sizes obtained using single-sized (left) and MRN (right) grain distributions as functions of the protostellar mass, as calculated with c sd = 1 and ζ = 10 −17 s −1 . Disk sizes obtained from the single-sized (left) and MRN (right) dust grain distributions versus the disk surface density control parameter c sd , obtained using a prestellar mass of Mstar = 0.1 M ⊙ and a cosmic ray ionization rate of ζ = 10 −17 s −1 .
Fig. 10
10plots the disk sizes as functions of c sd for each dust model listed in
Figure 11 .
11Disk sizes for the single-sized (left) and MRN (right) dust grain distributions as functions of the cosmic ray ionization rate, ζ, obtained using c sd = 1 and Mstar = 0.1 M ⊙ .
Fig. 11
11summarizes the effect of the cosmic ray ionization rate ζ on the disk size for different grain size models. Basically, the disk size decreases as ζ increases, because the nonideal effect (or the Hall effect) becomes ineffective in an environment in which the ionization
Figure 12 .
12Calculated disk size as function of plasma β.
Table 1 .
1Dust grain model parameters.Model
a min [µm] a single [µm] amax [µm]
smax
f dg
σtot/n H [cm 5 ]
s1a
0.005
0.25
10
0.01
2.83052 × 10 −21
s1b
0.005
0.25
10
0.005
1.41526 × 10 −21
s1c
0.005
0.25
10
0.02
5.66105 × 10 −21
s2
0.0181
0.9049
10
0.0341
2.66649 × 10 −21
s3
0.3
1
0.01
5.83531 × 10 −22
s4
0.1
1
0.01
1.74486 × 10 −21
s5
0.075
1
0.01
2.33644 × 10 −21
s6
0.05
1
0.01
3.5035 × 10 −21
s7
0.025
1
0.01
7.00006 × 10 −21
s8
0.01
1
0.01
1.74486 × 10 −20
Table 3 .
3Chemical species considered in this study and assigned numbers.Table 4. Cosmic ray rates employed in this study.species
assigned number i
e −
0
H +
3
1
m +
2
Mg +
3
He +
4
C +
5
H +
6
ζ [ s −1 ]
10 −18
10 −17.5
10 −17
10 −16.5
10 −16
Table 5 .
5Parameters determining the disk surface density and protostellar mass.c sd
1, 0.1, 0.01
Mstar [M ⊙ ]
0.03, 0.05, 0.07, 0.1, 0.3, 0.5, 0.7
state-of-art simulations. The results of this work imply that the magnetic field strength is improtant for investigating the angular momentum induced by the Hall effect. Thus, both the magnetic field strength and Hall effect would influence the formation and evolution of circumstellar disks.
MNRAS 000, 000-000 (0000)
The unit of η AD in equation(45)differs from the unit used in this study (cm 2 s −1 ). The difference in resistivities between them is c 2 /4π (for detailed definition, seeMarchand et al. 2016).MNRAS 000, 000-000(0000)
ACKNOWLEDGEMENTSWe thank the referee for his/her useful comments. The authors gratefully acknowledge helpful discussions with K. Tomida. This work was supported by JSPS KAKENHI grants (numbers 17H02869, JP17K05387, JP17H06360 and 17KK0096). This research used computational resources within the high-performance computing infrastructure (HPCI) systemAPPENDIX A: REACTION RATESAll the reactions used in this study, along with the associated reaction rates, are summarized inTable A1. The reaction between a charged particle i (that is, an ion) and an electron, e − , is written as k i , while k ij represents the reaction rate between charged species i and neutral species j, as inTable 3. Note that the HCO + reaction rate was used as the m + reaction rate, because HCO + is the most abundant molecular ion after H + 3 . In the case that a single charged species could undergo multiple reactions, these were distinguished using the symbols a − e. As an example, H + 3 can participate in five different reactions, with reaction rates k 12a , k 12b , k 12c , k 12d and k 12e . As described in Section §2.4, these reaction rates were acquired from the UMIST database(McElroy et al. 2013).APPENDIX C: REACTION EQUATIONSThe reaction equations used in this study were as follows. The term x i (i=0 -6) refers to the fractional abundance of each charged species (for each index, seeTable 3) and x d (s, z) refers to the fractional abundance of dust grains, where s is the bin number and z is the charge of dust grains. The reaction rates are summarized in Appendix §A and the cosmic ray rates, ζ, can be found in Section §2.4.dx 6 dt = 0.03ζ − n H (k 6 x 0 − k 46 x 4 + (k 62a + k 62b )x 2 + k 63 x 3 ) x 6 − δ 6,d .k id (s, z)x d (s, z)x i + k ed (s, z + 1)x d (s, z + 1)x 0 + k id (s, z − 1)x d (s, z − 1)x i ,In the above, δ i,d indicates the reaction following a collision between a charged particle
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"Abasy Atlas: a comprehensive inventory of systems, global network properties and systems-level elements across bacteria",
"Abasy Atlas: a comprehensive inventory of systems, global network properties and systems-level elements across bacteria"
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"Miguel A Ibarra-Arellano \nGroup of Regulatory Systems Biology\nEvolutionary Genomics Program\nCol. Chamilpa\nUniversidad Nacional Aut onoma De México\nAv. Universidad S/N\nMé xico62210Cuernavaca, Morelos\n\nCenter for Genomics Sciences\nCol. Chamilpa\nUndergraduate Program in Genomic Sciences\nUniversidad Nacional Aut onoma De México\nAv. Universidad S/N\nMé xico62210Cuernavaca, Morelos\n",
"Adri ",
"I Campos-Gonz Alez \nGroup of Regulatory Systems Biology\nEvolutionary Genomics Program\nCol. Chamilpa\nUniversidad Nacional Aut onoma De México\nAv. Universidad S/N\nMé xico62210Cuernavaca, Morelos\n\nCenter for Genomics Sciences\nCol. Chamilpa\nUndergraduate Program in Genomic Sciences\nUniversidad Nacional Aut onoma De México\nAv. Universidad S/N\nMé xico62210Cuernavaca, Morelos\n",
"Luis G Treviño-Quintanilla \nDepartamento De Tecnolog ıa Ambiental\nUniversidad Polité cnica Del Estado De Morelos\nBlvd. Cuauhn ahuac 566\n\nCol. Lomas Del Texcal\n62550Jiutepec, MorelosMéxico\n",
"Andreas Tauch \nDepartment of Molecular Genetics and Microbiology\nCollege of Medicine\nCentrum Für Biotechnologie (CeBiTec)\nUniversit€ at Bielefeld\nUniversit€ atsstraße 2733615BielefeldGermany\n\nUniversity of Florida\n2033 Mowry Road32610-0266GainesvilleFLUSA\n",
"Julio A Freyre-Gonz Alez \nGroup of Regulatory Systems Biology\nEvolutionary Genomics Program\nCol. Chamilpa\nUniversidad Nacional Aut onoma De México\nAv. Universidad S/N\nMé xico62210Cuernavaca, Morelos\n",
"Miguel A Ibarra-Arellano "
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"Center for Genomics Sciences\nCol. Chamilpa\nUndergraduate Program in Genomic Sciences\nUniversidad Nacional Aut onoma De México\nAv. Universidad S/N\nMé xico62210Cuernavaca, Morelos",
"Group of Regulatory Systems Biology\nEvolutionary Genomics Program\nCol. Chamilpa\nUniversidad Nacional Aut onoma De México\nAv. Universidad S/N\nMé xico62210Cuernavaca, Morelos",
"Center for Genomics Sciences\nCol. Chamilpa\nUndergraduate Program in Genomic Sciences\nUniversidad Nacional Aut onoma De México\nAv. Universidad S/N\nMé xico62210Cuernavaca, Morelos",
"Departamento De Tecnolog ıa Ambiental\nUniversidad Polité cnica Del Estado De Morelos\nBlvd. Cuauhn ahuac 566",
"Col. Lomas Del Texcal\n62550Jiutepec, MorelosMéxico",
"Department of Molecular Genetics and Microbiology\nCollege of Medicine\nCentrum Für Biotechnologie (CeBiTec)\nUniversit€ at Bielefeld\nUniversit€ atsstraße 2733615BielefeldGermany",
"University of Florida\n2033 Mowry Road32610-0266GainesvilleFLUSA",
"Group of Regulatory Systems Biology\nEvolutionary Genomics Program\nCol. Chamilpa\nUniversidad Nacional Aut onoma De México\nAv. Universidad S/N\nMé xico62210Cuernavaca, Morelos"
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| These authors contributed equally to this work. Citation details: Ibarra-Arellano,M.A, Campos-Gonz alez,A.I., Treviño-Quintanilla,L.G et al. Abasy Atlas: a comprehensive inventory of systems, global network properties and systems-level elements across bacteria.AbstractThe availability of databases electronically encoding curated regulatory networks and of high-throughput technologies and methods to discover regulatory interactions provides an invaluable source of data to understand the principles underpinning the organization and evolution of these networks responsible for cellular regulation. Nevertheless, data on these sources never goes beyond the regulon level despite the fact that regulatory networks are complex hierarchical-modular structures still challenging our understanding. This brings the necessity for an inventory of systems across a large range of organisms, a key step to rendering feasible comparative systems biology approaches. In this work, we take the first step towards a global understanding of the regulatory networks organization by making a cartography of the functional architectures of diverse bacteria. Abasy (Across-bacteria systems) Atlas provides a comprehensive inventory of annotated functional systems, global network properties and systems-level elements (global regulators, modular genes shaping functional systems, basal machinery genes and intermodular genes) predicted by the natural decomposition approach for reconstructed and meta-curated regulatory networks across a large range of bacteria, including V C The Author(s) | 10.1093/database/baw089 | null | 17,149,883 | 1605.04959 | 3b39feba20f6167b888d95f45be398a7846deee6 |
Abasy Atlas: a comprehensive inventory of systems, global network properties and systems-level elements across bacteria
2016. 2016
Miguel A Ibarra-Arellano
Group of Regulatory Systems Biology
Evolutionary Genomics Program
Col. Chamilpa
Universidad Nacional Aut onoma De México
Av. Universidad S/N
Mé xico62210Cuernavaca, Morelos
Center for Genomics Sciences
Col. Chamilpa
Undergraduate Program in Genomic Sciences
Universidad Nacional Aut onoma De México
Av. Universidad S/N
Mé xico62210Cuernavaca, Morelos
Adri
I Campos-Gonz Alez
Group of Regulatory Systems Biology
Evolutionary Genomics Program
Col. Chamilpa
Universidad Nacional Aut onoma De México
Av. Universidad S/N
Mé xico62210Cuernavaca, Morelos
Center for Genomics Sciences
Col. Chamilpa
Undergraduate Program in Genomic Sciences
Universidad Nacional Aut onoma De México
Av. Universidad S/N
Mé xico62210Cuernavaca, Morelos
Luis G Treviño-Quintanilla
Departamento De Tecnolog ıa Ambiental
Universidad Polité cnica Del Estado De Morelos
Blvd. Cuauhn ahuac 566
Col. Lomas Del Texcal
62550Jiutepec, MorelosMéxico
Andreas Tauch
Department of Molecular Genetics and Microbiology
College of Medicine
Centrum Für Biotechnologie (CeBiTec)
Universit€ at Bielefeld
Universit€ atsstraße 2733615BielefeldGermany
University of Florida
2033 Mowry Road32610-0266GainesvilleFLUSA
Julio A Freyre-Gonz Alez
Group of Regulatory Systems Biology
Evolutionary Genomics Program
Col. Chamilpa
Universidad Nacional Aut onoma De México
Av. Universidad S/N
Mé xico62210Cuernavaca, Morelos
Miguel A Ibarra-Arellano
Abasy Atlas: a comprehensive inventory of systems, global network properties and systems-level elements across bacteria
Database
20162016. 201610.1093/database/baw089Received 30 March 2016; Revised 27 April 2016; Accepted 5 May 2016Original article Present address: Page 1 of 16 This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. (page number not for citation purposes) Original article
These authors contributed equally to this work. Citation details: Ibarra-Arellano,M.A, Campos-Gonz alez,A.I., Treviño-Quintanilla,L.G et al. Abasy Atlas: a comprehensive inventory of systems, global network properties and systems-level elements across bacteria.AbstractThe availability of databases electronically encoding curated regulatory networks and of high-throughput technologies and methods to discover regulatory interactions provides an invaluable source of data to understand the principles underpinning the organization and evolution of these networks responsible for cellular regulation. Nevertheless, data on these sources never goes beyond the regulon level despite the fact that regulatory networks are complex hierarchical-modular structures still challenging our understanding. This brings the necessity for an inventory of systems across a large range of organisms, a key step to rendering feasible comparative systems biology approaches. In this work, we take the first step towards a global understanding of the regulatory networks organization by making a cartography of the functional architectures of diverse bacteria. Abasy (Across-bacteria systems) Atlas provides a comprehensive inventory of annotated functional systems, global network properties and systems-level elements (global regulators, modular genes shaping functional systems, basal machinery genes and intermodular genes) predicted by the natural decomposition approach for reconstructed and meta-curated regulatory networks across a large range of bacteria, including V C The Author(s)
pathogenically and biotechnologically relevant organisms. The meta-curation of regulatory datasets provides the most complete and reliable set of regulatory interactions currently available, which can even be projected into subsets by considering the force or weight of evidence supporting them or the systems that they belong to. Besides, Abasy Atlas provides data enabling large-scale comparative systems biology studies aimed at understanding the common principles and particular lifestyle adaptions of systems across bacteria. Abasy Atlas contains systems and system-level elements for 50 regulatory networks comprising 78 649 regulatory interactions covering 42 bacteria in nine taxa, containing 3708 regulons and 1776 systems. All this brings together a large corpus of data that will surely inspire studies to generate hypothesis regarding the principles governing the evolution and organization of systems and the functional architectures controlling them.
Database URL: http://abasy.ccg.unam.mx
Background
Bacterial regulatory networks (RNs) are responsible for sense stimuli and environmental cues and respond accordingly. In complex environments, they 'take composite decisions' to prioritize, for example, the transport and catabolism of carbon sources according to the metabolic preferences of each organism. To accomplish this, RNs composed by thousands of regulatory interactions, must follow well-defined organization principles governing their dynamics. In the last decades of the 20th century, the first levels of gene organization were unveiled as the operon and the regulon.
Currently, a few databases [RegulonDB (1), SubtiWiki (2), DBTBS (3), CoryneRegNet (4) and RegTransBase (5)] extract, by manual curation of literature, the molecular knowledge of gene regulation in different organisms, thus providing an invaluable source of data. Nevertheless, data within these databases never goes beyond the regulon level, whereas cumulated evidence has shown that RNs are complex hierarchical-modular networks (6-11) whose organizational and evolutionary principles are pivotal for determining the dynamics of the cell and still challenging our understanding.
The promising field of synthetic biology aims to apply engineering principles for designing and constructing biological systems and devices. To fulfill this aim, it is crucial to understand the set of organizational principles underpinning how cellular systems interconnect, work and evolve. In synthetic biology, and even in biotechnology, a deep understanding of the alternative regulatory patterns, which have evolved as adaptations to different environments, will inspire better synthetic designs, thus enhancing our ability to optimize a genetic regulatory circuit. To accomplish this, we need an inventory of systems and their properties across a large range of organisms, a key step to rendering feasible comparative systems biology approaches.
From a genomic perspective, identifying clusters of genes working together in a system to achieve a particular physiological function is important. It enables the use of a guilt-by-association strategy to propose hypotheses regarding the physiological function of genes whose annotation by homology analysis is not enough mainly because no homologs have been characterized, e.g. the annotation is 'hypothetical predicted/conserved protein'.
On the other hand, a recent study has shown that mathematical models, like the ones used in synthetic biology, are highly sensitive to structural variations in the system under study, and that failure to consider the relevance of this structural uncertainty gives rise to bias in the analysis and potentially misleading conclusions (12). Therefore, the ability to infer and delimit the members, inputs and outputs of a system, is key to construct reliable and unbiased models.
In this work, we take the first step towards a global understanding of the RNs organization by making a cartography of the functional architectures of diverse bacteria. Abasy (Across-bacteria systems) Atlas provides a comprehensive atlas of annotated functional systems (hereinafter also referred as modules), global network properties and systems-level elements predicted by the natural decomposition approach (NDA) (9,10) for reconstructed and metacurated RNs across a large range of bacteria, including pathogenically and biotechnologically relevant organisms. This biologically motivated mathematical approach uses the global structural properties of a given RN to derive its architecture and classify its genes into categories of systems-level elements (global regulators, modular genes, basal machinery genes and intermodular genes). The decomposition of diverse RNs into their systems and systems-level elements beyond the regulon allows unraveling the complexity of these networks and provides new insights into the organizational principles governing them.
This atlas also provides data enabling large-scale comparative systems biology studies aimed at understanding the common principles and particular lifestyle adaptions of systems across bacteria. The meta-curation of regulatory datasets provides the most complete and reliable set of regulatory interactions currently available, which can even be projected into subsets by considering the force or weight of evidence supporting them or the systems that they belong to.
The natural decomposition approach: a primer There are different levels of description in models of genetic networks (13). The NDA is a large-scale modeling approach characterizing the circuit wiring and its global architecture. It defines a mathematical-biological framework that provides criteria to identify the four classes of systems-level elements shaping RNs: global regulators, modular genes shaping functional systems, basal machinery genes and intermodular genes. Studies have shown that regulatory networks are highly plastic (14). Despite this plasticity, by applying the NDA our group has found that there are organizational principles conserved by convergent evolution in the RNs of phylogenetically distant bacteria (10). The high predictive power of the NDA has been proven in previous studies by applying it to the phylogenetically distant Escherichia coli K-12 (9) and Bacillus subtilis 168 (10) and by comparing it with other methods to identify modules (11).
The NDA defines objective criteria (e.g. the j-value to identify global regulators) to identify functional systems and systems-level elements in a RN, and rules to reveal its functional architecture by controlled decomposition (Figure 1). It is based on two biological premises (10): (i) a module is a set of genes cooperating to carry out a particular physiological function, thus conferring different phenotypic traits to the cell. (ii) Given the pleiotropic effect of global regulators, they must not belong to modules but rather coordinate them in response to general-interest environmental cues.
According to the NDA, every gene in a RN is predicted to belong to one out of four possible classes of systemslevel elements, which interrelate in a non-pyramidal, threelayer, hierarchy shaping the functional architecture (9,10) as follows ( Figure 2): (i) Global regulators are responsible for coordinating both the (ii) basal cell machinery, composed of strictly globally regulated genes and (ii) locally autonomous modules (shaped by modular genes), whereas (iv) intermodular genes integrate, at promoter level, physiologically disparate module responses eliciting a combinatorial processing of environmental cues.
Construction and content
Abasy Atlas has a three-tier, web-based and client-server architecture. PHP programs are responsible for querying the MySQL database, executing Python scripts and rendering HTML with CSS webpages containing the extracted information. Interactive network panels display gene neighborhood (i.e. the set of all the genes/complexes regulating or being regulated by a given gene, including the latter) and modules, providing alternative layouts and different visualization options. These network panels are rendered by software running on the web browser of the client using serialized data and were developed using JavaScript, Python and the state-of-the-art Cytoscape.js library (15).
Data extraction to create a compendium of regulatory network models across bacteria
To create a comprehensive compendium of bacterial RNs we first compiled regulatory interaction data for the bestcurated RNs available in organism-specific databases: E. coli K-12 MG1655 from RegulonDB (1). B. subtilis 168 from DBTBS (3) and SubtiWiki (2), and Corynebacterium glutamicum ATCC 13032 from CoryneRegNet (4). We then identified six additional regulatory datasets by carrying out a literature review: (i) A recent experimentally supported predicted RN for B. subtilis 168 (16). (ii) A ChIP-seq based reconstruction of the RN of Mycobacterium tuberculosis H37Rv (17) plus three studies also reporting experimentally supported and literature-curated RNs for M. tuberculosis H37Rv (18)(19)(20). (iii) A study reporting a literature-curated RN for Pseudomonas aeruginosa PAO1 (21). Finally, we expanded this compendium by retrieving the regulatory dataset for 477 bacteria available in RegTransBase, a database of regulatory interactions based on literature (5).
Regulatory interaction data was obtained in a diversity of formats: tab-delimited flatfiles, Excel spreadsheet (XLS), XML and HTML. Ad-hoc developed Python scripts parsed and converted regulatory interaction data into a unique intermediate tab-delimited format that we defined. RNs were modeled as directed graphs where nodes represent genes (or regulatory complexes, as we will discuss below) and directed edges stand for regulatory interactions. Regulatory data coming from different sources were reconciled through the process of gene symbols disambiguation and meta-curation as described below.
Sigma factors
Sigma factors play an active role in response to stimuli by differentially redirecting the transcriptional machinery towards specific sets of promoters. The competition for a limited resource, the RNA polymerase core, requires a strict regulation on the availability of alternative sigma factors and even housekeeping sigma factors, to control the delicate interplay among them properly (22). Therefore, sigma factors transcriptionally control and are controlled by the products of other genes, and as such, they are important players in the systems-level organization of RNs. To consider this alternative regulatory layer, we included sigma factors as activators in our RN models.
Regulatory complexes
Proteins encoded by different genes can form heteromeric complexes that exhibit regulatory activity. As available regulatory datasets are gene-based, genes encoding subunits of a regulatory complex may regulate the same set of genes, thus duplicating interactions in the RN model. To remove this redundancy, we first curated and compiled a list of heteromeric regulatory complexes for E. coli K-12 and B. subtilis 168, no data was found for other organisms. Next, for each set of subunits composing a complex, if the subunit-encoding genes are regulated by the same set of regulators and do not exhibit regulatory activity independently, we collapsed the genes encoding subunits into a single node representing the regulatory complex. Otherwise, we added regulatory complexes as nodes that are activated by their subunit-encoding genes. We kept outgoing regulations mediated by a subunit that exhibit regulatory activity independently, and we removed the remaining outgoing regulations from the subunit-encoding genes and added them to the complex according manual curation and removing duplicates.
Regulation mediated by small RNAs
Small RNAs have emerged as a novel and relevant regulatory element (23). Given their importance, we retrieved the sRNA-mediated regulatory interactions curated by RegulonDB team and created an additional expanded version of the E. coli K-12 RN model based on the all evidences version, the latter because all the evidences supporting sRNA-mediated interactions are weak. Other databases or datasets (e.g. DBTBS and RegTransBase) already include sRNA-mediated interactions, and even attenuators or other RNA-mediated regulatory interactions, and these were added to the atlas.
Classification of regulatory interactions according to their supporting evidences
Curation of RNs proceeds by identifying evidences that a certain regulator is able to affect the expression of a gene. These evidences have varying degrees of confidence: some evidences (e.g. footprinting) are strong enough to show that the regulator is able to bind to the upstream region of the regulated gene possibly altering its expression, whereas weak evidences just suggest a hypothetical DNA binding site (e.g. bioinformatics predictions) or a possible indirect effect (e.g. gene expression experiments). Weakly supported interactions that eventually turn out to be spurious Figure 2. Common functional architecture identified by the NDA in bacteria. The functional architecture unveiled by the NDA is a diamond-shaped, three-layer, hierarchy, exhibiting some feedback between processing and coordination layers, which is shaped by four classes of systems-level elements: global transcription factors, locally autonomous modules, basal machinery and intermodular genes.
act like noise affecting the conclusions drawn from RN analyses and mathematical models disregarding this confounding factor (12). We therefore created two versions of the E. coli K-12 and B. subtilis 168 (DBTBS) RN models: all evidences (containing both weakly and strongly supported interactions) and strong evidences (accounting for strongly supported evidences only).
RegulonDB team has proposed a classification scheme for the strength of single evidences into 'weak' and 'strong' (24). They define weak evidences as a 'single evidence with more ambiguous conclusions, where alternative explanations, indirect effects, or potential false positives are prevalent, as well as computational predictions; for instance, gel mobility shift assays with cell extracts or gene expression analysis'. On the other hand, strong evidence is defined as a 'single evidence with direct physical interaction or solid genetic evidence with a low probability for alternative explanations; for instance, footprinting with purified protein or site mutation'.
Using this classification scheme, we retrieved all evidences supporting each regulatory interaction for the E. coli K-12 RN model and computed its confidence level as weak or strong according the following rule: if a regulatory interaction is supported by at least one strong evidence it is classified as strong, otherwise it is weakly supported. For B. subtilis 168, we developed an analogous classification scheme by manual curation of the set of experiments supporting regulatory interactions in DBTBS, and we next computed the confidence level for each regulatory interaction as previously described.
An unambiguous and integrative compendium of gene names, locus tags and synonyms to cope with disagreement among regulatory datasets Disagreement in gene names is an important confounding factor in genomic analyses. For instance, when RNs are compared, or any other genomic data is mapped onto them, disagreement in gene names is responsible for missing or duplicated interactions or genes potentially causing bias in conclusions (25). Besides, the herein compiled regulatory datasets use different formats for naming genes: some only use gene names, others only locus tags, and some others a mixture of gene names (including synonyms and database IDs) and locus tags.
To cope with the problem of disagreement in gene names, we compiled a dictionary of gene names, locus tags and synonyms for each gene in the regulatory datasets. This compilation of identity information was carried out by integrating information from UniProtKB (26) and NCBI GenBank (27) for all organisms, and additionally RegulonDB for E. coli K-12, SubtiWiki for B. subtilis 168 and RegTransBase for organisms obtained from this source (Supplementary material, File 1). In this step, we also extracted accession IDs to provide cross-links to external databases.
Using this identity information, we developed an algorithm to map gene symbols onto unambiguous canonicalform gene names and to complete the identity information by finding missing locus tags and synonyms. We defined a canonical-form or canonical gene name as a UniProtproposed gene name (http://www.uniprot.org/help/gene_ name) for protein coding genes, or if gene codes for an RNA, an NCBI GenBank gene name, or for the cases of E. coli K-12 and B. subtilis 168, main names proposed by the corresponding curated database RegulonDB or SubtiWiki, respectively, or a locus tag if no names were available.
We represented locus tags and their corresponding gene names (including synonyms) as the two disjoint sets in a bipartite graph where an edge represents an equivalence relation between sets ( Figure 3). We then found that gene names exhibit degeneracy, i.e. some gene names map to different locus tags. Therefore, to remove this ambiguity, we first mapped all gene names onto locus tags by using canonical gene names as keys. Next, we transformed the remaining non-mapped gene names into locus tags by a constrained mapping using the previously constructed bipartite graph as follows (Figure 3): if a gene name maps onto more than one locus tag, the gene name is marked as taboo (forbidden) so that the algorithm does not consider it for future mappings and we conserved it as is for historical consistency, otherwise we mapped it onto that unique locus tag. Finally, we mapped all locus tags onto canonical names, when possible. As a result, we collapsed nodes representing genes that are synonyms into a single node and removed duplicated edges in RN models.
Meta-curation of regulatory network models
The development of high-throughput interaction discovery methodologies (e.g. ChIP-seq) and gene-expression-based methods for inference of RNs has produced novel regulatory datasets with high genomic coverage. Examples of this are the datasets for B. subtilis 168, M. tuberculosis H37Rv and P. aeruginosa PAO1 found in our literature review described above (Table 1).
We analyzed these datasets to quantify the overlap among them. We found that the first regulatory dataset reported for M. tuberculosis H37Rv (18) is completely included in the second regulatory dataset reported (19), so we discarded the former. Interestingly, we discovered that the remaining independent regulatory datasets share a poor amount of genes and interactions (Figure 4). The lack of redundancy between them offers an opportunity to obtain more complete RN models through meta-curation. Briefly, we translated the regulatory datasets into locus tags using the constrained mapping strategy described above. We then merged the datasets avoiding duplicating genes and interactions. Regulatory effects and evidences supporting interactions were inherited, if available, from the original datasets. Finally, the meta-curated datasets were processed by the pipeline for data integration as a normal dataset for their inclusion in Abasy Atlas.
These novel meta-curated datasets represent the most complete RNs currently reported for B. subtilis 168, M. tuberculosis H37Rv and P. aeruginosa PAO1, as determined by their genomic coverage: 75, 77 and 16%, respectively. Remarkably, these meta-curated regulatory networks, regardless of their coverage, exhibit properties that characterize hierarchical-modular RNs (see their global network properties in Abasy Atlas) (10,11). All this is suggestive of the good quality of these novel meta-curated RN models that is inherently dependent on the quality of the source datasets.
Datasets obtained from RegTransBase were filtered to reconstruct only the best RN models and to eliminate redundancy from Abasy Atlas as follows. After modeling and analyzing the 477 datasets obtained from RegTransBase, we retained 115 datasets and discarded the remaining 362 for being very small networks (<110 genes) mostly composed by disconnected, small (around five genes on average), tree-like structures (regulons). During the construction of Abasy Atlas, UniProt team identified a high level of proteome redundancy in unreviewed UniProtKB (TrEMBL) mainly due to the sequencing and submitting of different strains of the same species (http:// www.uniprot.org/help/proteome_redundancy). Since release 2015_04, UniProt discarded redundant entries. To remove redundancy from Abasy Atlas, we removed the RNs of organisms with redundant proteomes as identified by UniProt. This step affected only RNs obtained from RegTransBase. After these filtering, we retained 37 RNs from RegTransBase that were integrated in Abasy Atlas. Mapping from gene names into locus tags can follow any vertical link but diagonal links are taboo (red node) except if node degree is exactly one (e.g. yvbD). In this example, opuCB, dnaE, dnaG, dnaN, dnaA and dnaX are solved via canonical gene names; yvbD is found to be a synonym for opuCB and both are the same gene; whereas dnaH cannot be solved because it exhibits degeneracy (node degree greater than one, and no vertical link denoting it as a canonical gene name). Genomic coverage for datasets coming from RegTransBase range from 24% (Staphylococcus aureus USA300) to 2% (Pseudomonas aeruginosa PA7). Remarkably, across strains of the same species, genomic coverage extends through a large range. For example, E. coli K-12 MG1655 (strong evidences) has a genomic coverage of 42%, whereas E. coli K-12 DH10B has only 7%. This is explained by the fact that regulatory data curation is strain specific and mapped to a specific genome sequence. Coding sequences are highly conserved across strains of the same species, but the same is not true for regulatory sequences that are highly plastic (14). Therefore, regulatory data cannot be extrapolated to other strains even if they belong to the same species.
Extraction of functional information
To provide a functional context to RN model genes, we collected the annotated product functions, gene ontology (GO) terms and clusters of orthologous genes (COGs). We extracted product functions from NCBI GenBank, RegTransBase and RegulonDB; GO terms from UniProt-GOA (GO annotation) (28), and COGs from EMBL eggNOG (29) ( Figure 5 and Supplementary material, File 1).
Computation of predictions
The procedure to populate Abasy Atlas is mostly automated. We developed a Python script ( Figure 5) that is responsible for mapping gene symbols onto canonical gene names, collecting accession IDs for cross-linking and gene symbols for search in the Abasy Atlas, computing systems and systems-level elements by implementing the NDA (Figure 1), computing network global properties and annotating each system identified by the NDA.
Among the global properties computed for a RN model, we computed the degree distribution, out-degree distribution and clustering coefficient distribution. Commonly these distributions are fitted to a power-law by ordinary least squares, which is a method that is highly sensitive to outliers in data. Instead, we fitted these distributions using robust linear regression of log-log-transformed data with Huber's T for M-estimation to overcome the negative effect of outliers.
The pipeline annotated each NDA-predicted module by computing the functional enrichment of the cluster of genes. Functional enrichment analysis was carried out by computing P-values for each GO term in the cluster of genes using a hypergeometric distribution probability test. We considered only GO terms in the biological process namespace because they reflect physiological processes. We then corrected P-values for multiple tests into Q-values by controlling the false discovery rate (FDR) at level 0.05. We selected UniProt GO annotation as the functional classifications schema for annotating systems because it is actively and manually curated (28) and provides a shared vocabulary among organisms and databases (30).
A standard for modeling regulatory networks, interaction effects and evidences and NDA predictions
We represented our RN models using JSON (JavaScript Object Notation) open standard format (http://www.json. org/), which is a lightweight, language independent, widely used, data-interchange format supported by > 50 programming languages. JSON is easy for humans to read and write and for machines to parse and generate. These properties also simplified the serialization of RN models, allowing their processing and visualization in interactive network panels.
Database schema
A relational database implemented in MySQL models the metadata and predictions for all RN models, including the RN models in JSON format, and the identity and functional information throughout 14 tables (Supplementary material, File 2). Three tables contain manually curated information (reg_complexes, sources and source_urls). Five (organisms, genes, gene_ids, modules and computational_ annot) were generated by the pipeline (Figure 5), which is responsible for computing NDA predictions, RN model metadata containing global network properties and systems computational annotation for each RN model. Six tables store functional data extracted from external sources (gos, goa, cogs, cog_families, cog_classes and cog).
Utility and discussion
Biological diversity in Abasy Atlas
Abasy Atlas goes beyond current databases of regulatory information by inventorying systems, global network properties, and systems-level elements for 50 reconstructed and meta-curated RNs (78 649 regulatory interactions) covering 42 bacteria (including pathogenically and biotechnologically relevant organisms) in nine taxa, containing 3708 regulons and 1776 systems ( Figure 6). Given that nowadays, there are no models estimating the total number of . We also reconstructed the most complete RN model for P. aeruginosa PAO1 by meta-curation, which has a genomic coverage of 16%.
All the RN models in Abasy Atlas have genes in the four classes of systems-level elements predicted by the NDA, except for four organisms whose RN models (currently) do not have intermodular genes: Streptococcus pyogenes serotype M18 (strain MGAS8232), Streptococcus pneumoniae (strain Hungary19A-6), Streptococcus pyogenes serotype M12 (strain MGAS9429) and E. coli (strain SMS-3-5/ SECEC). Nevertheless, these are quite incomplete organisms having a genomic coverage <9%.
Overview of the Abasy Atlas interface
Abasy Atlas has a hierarchical structure enabling users to browse through the available RN models, their systems and systems-level elements. In every section of Abasy Atlas, a top menu bar displays five options and a search interface (Figure 7). The options are Homepage, Browse, Downloads, About and Contact. Homepage is a shortcut to the homepage. Browse enables the user to explore through the atlas. Download displays an interface to download data as flatfiles. About provides background information on the atlas and the NDA. Finally, Contact provides a form for feedback from users. The Browse option and the search interface provide the two main ways to start exploring data contained in Abasy Atlas. They will be discussed in the following sections along with graphical browsing of RN models. Abasy Atlas is cross-linked to various external databases and sites providing biological, genomic, and molecular details. When a user visits an external database or site this opens in a new browser tab, whereas hyperlinks redirecting to other sections in Abasy Atlas open in the same tab.
Browsing Abasy Atlas
The Browse option is the entry point for browsing the whole atlas. This redirects to a section listing all the RN models contained in the atlas. Each RN model lists genome size, RN genomic coverage, PubMed IDs of the data source, number and percentage of global regulators, modular genes, intermodular genes and basal machinery genes, number of systems (modules) and a link to a separated section listing the global properties of the RN model. From here, the user can list all the genes in the RN model, retrieve a list of all the genes in a given class, or list all the identified systems.
In the list of genes, genes are listed along with their product description and the predicted class of system-level element (the latter only applicable for all and modular genes). Each gene name is canonical (see 'Construction and content' section) and links to a gene details section showing identity (locus tag, UniProt ID, NCBI GeneID and synonyms) and functional (product function, GO terms, and COGs) information on the gene along with their indegree, out-degree and clustering coefficient and an interactive network panel displaying the gene and their graph neighborhood (see 'Interactive network panel' section).
Here the canonical gene name cross-links to specialized databases containing genomic and molecular details.
By following the modules link in the RN models listing, the users get a list of all the systems identified in that RN model (Figure 7). Data displayed in that list comprise the number of genes belonging to that system, the enriched GO terms and their q-value, and a module ID linking to another section displaying the system in an interactive network panel and listing the genes belonging to the system along with their product descriptions.
Search by gene
An interface to search Abasy Atlas is also available in the top menu bar of every section (Supplementary material, File 3). Users may search a gene in all the RN models (by default) or in a subset by selecting the proper option in a searchable dropdown list allowing multiple selections. Search using this interface is case-insensitive and the user may employ wildcards in the search string. Supported wildcards are '?' to match any single character and '*' to match any arbitrary number of characters including zero.
The search engine looks for the query in the set of canonical gene names, synonyms and locus tags returning a composite set of results grouped for each RN model and ordered by canonical gene name (Supplementary material, File 3). The results section lists all the matches found providing canonical gene names, NCBI GeneIDs, locus tags, UniProt IDs, synonyms and classes of systems-level elements. Gene names link to the details section described above. If the class of systems-level elements is a system, this links to the corresponding entry in the list of all the systems identified in that RN model.
Interactive network panel
Abasy Atlas displays gene graph neighborhoods (i.e. the set of all the genes/complexes regulating or being regulated by a given gene, including the latter) and systems in interactive network panels that share a common interface (Figure 8). This interface provides a button to download an image in PNG format with transparency containing the network displayed, which is free for use in papers and other academic/ non-commercial uses as long as proper citation is provided (see 'Availability and requirements' section). It is also possible to remove global regulators, disconnected nodes and weakly supported interactions temporally. A dropdown list enables the user to apply a layout to the network, and a button reapplies the selected layout. A checkbox controls if layout transitions are animated or not, this is a useful control when the number of nodes could slow the web browser. The full quality checkbox controls whether full quality is used during network manipulation and animated layout transitions. The animation is off by default, and we suggest this for improved responsiveness. The configuration of all the checkboxes and selected layout is preserved across a session in the same web browser tab. When the user closes the tab, the configurations are restored to default.
We fine-tuned the parameters for rendering the network in order to optimize responsiveness and user experience. Besides, we developed an adaptive layer running on top of the algorithms rendering the network. Before a network is rendered, this adaptive layer assesses the performance of the user's computer and, if necessary, overrides user's selections for a smoother experience. If the adaptive layer found that the user's computer will exhibit degraded performance for a particular network, then layout animation is disabled and the network is always initially displayed using the grid layout despite previous user selections. The user can always reenable layout animation or change the layout upon accepting a warning regarding the risk of degraded performance.
In the upper-right corner of the network panel, a widget similar to that used by Google Maps is available. This widget enables panning and zooming the network. The user may also use the mouse wheel for zooming the network. Inside the interactive panel, the network is displayed by using a color code for nodes and interactions. We encoded interaction effects in line colors and arrowhead shapes as follows: red and T shape for repressions, green and arrow shape for activations, orange and diamond shape for dual and grey and arrow shape for unknown effect. Evidences supporting interactions, if available, are encoded as different line styles: solid for strong evidences and dashed for weak evidences. Genes belonging to the same system or class of systems-level element have the same color. In the gene details section, the central gene is highlighted in yellow. If the central gene product is a subunit shared by several complexes, all these complexes are also highlighted. We represented heteromeric regulatory complexes as pentagons, whereas genes are circular nodes.
All the nodes can be dragged around and repositioned, and clicking on an interaction switches its state between normal and highlighted. If the mouse hovers over a node or interaction, a tooltip displays providing information such as local (neighborhood dependent) out-and in-degrees, subunits if the node is a regulatory complex and links to other genes and systems. The user also may explore the RN model by graphical browsing. If the user clicks in a node, he is redirected to the details section of the corresponding gene.
Use case 1: Global regulators of C. glutamicum ATCC 13032
The NDA defines an equilibrium point (j-value) between two apparently contradictory behaviors occurring in hierarchical-modular networks: hubness and modularity. In these networks, modularity is inversely proportional to hubness (Figure 1). The j-value is defined as the connectivity value for which the variation of the clustering coefficient (modularity) equals the variation of out-connectivity (hubness) but with the opposite sign (dC/dk out ¼ À1) (9,10). This is a network-dependent parameter that allows the identification of global regulators as those nodes with connectivity greater than j. We identified 19 global regulators in the C. glutamicum RN model (j-value ¼ 11): sigA, glxR, sigH, dtxR, ramA, ramB, lexA, mcbR, cg2115 (sugR), amtR, mtrA, cgtR11 (hrrA), ripA, cgtR3 (phoR), cg0156 (cysR), sigM, cg1324 (rosR), cg0196 (pckR) and sigB. These global regulators comprise four out of the seven sigma factors encoded in the genome of C. glutamicum (31). All of these regulators have been jointly described as global regulators (32) except for mtrA, cgtR11 (hrrA), ripA, cg0156 (cysR), sigM, cg1324 (rosR), and cg0196 (pckR). Nevertheless, cgtR11 (hrrA) (33) has also been individually reported as global regulator. In our set of global regulators we only missed arnR (cg1340) that has been reported as a global regulator (32), but it is a modular gene belonging to a system annotated as 'nitrate metabolic process' according to the NDA prediction.
Use case 2: Guilt by association, the case of B. subtilis 168 hemD, yppF, yptA and yycS involved in the biosynthesis of antibiotics and protoporphyrinogen IX Genomes are functionally annotated by using homology analysis. Annotation is conducted by sequence similarity as assessed by BLAST-based tools, and it is complemented with structural features and comparative genomic context analysis. Nowadays, genome annotation has reached a mature development but yet there are no functional predictions for hundreds of genes (e.g. those annotated as hypothetical predicted/conserved protein) even in the best characterized genomes because no homologs have been characterized (34). Genomic context analysis is based on the idea of exploiting all types of functional associations between genes in the same or in different genomes that may indicate a common function justifying a verdict of guilt by association.
Abasy Atlas goes beyond genomic functional associations providing systems-level functional associations allowing identification of the function of genes whose annotation by homology analysis is not possible. In Abasy Atlas, 682 genes annotated as hypothetical or with missing annotation belong to functionally annotated systems. This provides a way to assign functional annotations to these genes using a guilt-by-association strategy: if gene G belongs to a system with function X, then G is also involved in X.
For example, hemD, yppF, yptA and yycS are genes belonging to the system 1.19 of our meta-curated model for B. subtilis 168 (Figure 8). This system is annotated with the GO terms 'Protoporphyrinogen IX biosynthetic process' (GO:0006782), 'Metal ion transport' (GO:0030001) and 'Antibiotic biosynthetic process' (GO:0017000). This provides evidence from a systems-level context to propose that these genes are also involved in these functions.
Data available for download
To facilitate the access to and analysis of the RN models and the systems predicted, we provide various datasets including the compendium of genes names, synonyms and accessions IDs as flat files for download. The datasets available are:
RN models in JSON data-interchange format, including NDA predictions and, if available, effect and evidences supporting regulatory interactions.
Gene information as follows: canonical gene name, locus tag, NCBI GeneID, UniProt ID, synonyms, product function and class of systems-level element predicted by the NDA.
Module annotation comprising module ID, gene ontology term and Q-value supporting the annotation.
Future development plans
The long-term goal of Abasy Atlas is to provide an integrated platform to perform comparative large-scale studies of bacterial regulatory systems and the functional architectures governing them from a large-scale systems biology perspective. To accomplish this goal, we envision several improvements and extensions:
Improvements to the search interface and engine to provide support for using regular expressions and searching by function.
Expansion to include the regulators and systems controlling the intermodular genes, and the regulators of the basal machinery genes.
Expanding the global properties section to include new interesting properties.
Introduction of an interactive network panel to browse a RN model as a functional architecture displaying the hierarchy governing the systems-level elements.
Providing an online tool for applying the NDA to a RN uploaded by the user to obtain its system-level elements and functionally annotated systems.
Classification of the force of evidences supporting regulatory interactions in other organisms.
Updating and expanding our RN models by metacuration of new datasets and inclusion of predicted regulatory datasets as those from RegPrecise (35), PRODORIC (36) and CoryneRegNet (4).
Enabling the user to explore gene expression in the interactive network panel at the level of gene neighborhood, system, and functional architecture by mapping gene expression high-throughput data uploaded by the user onto the RN model.
Conclusions
Abasy Atlas takes the first step towards a global understanding of the RNs organization by providing a comprehensive inventory of annotated functional systems, global network properties and systems-level elements across a large range of bacteria, including organisms of medical and biotechnological relevance. In addition, meta-curation of regulatory datasets provides the most complete and reliable set of regulatory interactions currently available, which can be projected into subsets by considering the force or weight of evidence supporting them or the systems that they belong to.
The NDA provides formal and quantitative criteria to identify the systems and systems-level elements shaping the functional architecture of RNs. The decomposition of diverse RNs into their systems and systems-level elements beyond the regulon allows unraveling the complexity of these networks and provides new insights into the organizational principles governing them. The prevalence of intermodular genes (a novel system-level element first identified by the NDA) across a large range of prokaryotes support the diamond-shaped, three-layer, hierarchy unraveled by the NDA as a universal organizational principle among bacteria.
Abasy Atlas provides a standardized framework for biological networks annotation and meta-curation. For example, the gene-name disambiguation algorithm may be applied to other projects to reduce the influence of this confounding factor. Abasy Atlas also provides data enabling future large-scale comparative systems biology studies to understand the common principles and particular lifestyle adaptations of regulatory systems across bacteria. All this brings together a large body of data that will surely inspire studies to generate hypothesis regarding the principles underpinning the evolution and organization of systems and the functional architectures controlling them.
Availability and requirements
Abasy Atlas is available for web access at http://abasy.ccg. unam.mx. If you use any material from Abasy Atlas please cite properly. Use of Abasy Atlas and each downloaded network PNG image by Ibarra-Arellano et al. is licensed under a Creative Commons Attribution 4.0 International License. Permissions beyond the scope of this license may be available at [email protected]. Disclaimer: Please note that original data contained in Abasy Atlas may be subject to rights claimed by third parties. It is the responsibility of users of Abasy Atlas to ensure that their exploitation of the data does not infringe any of the rights of such third parties.
Supplementary data
Supplementary data are available at Database Online.
Figure 1 .
1The natural decomposition approach.
Figure 3 .
3Gene names disambiguation algorithm. A bipartite graph models the equivalence relations between locus tags (upper nodes) and gene names (lower nodes). Canonical gene names and locus tags relate via vertical links, whereas diagonal links connect synonyms and locus tags.
Figure 4 .
4Meta-curation of regulatory datasets. Poor overlap between different datasets for (A) nodes and (B) interactions in B. subtilis 168, (C) nodes and (D) interactions in P. aeruginosa PAO1, and (E) nodes and (F) interactions in M. tuberculosis H37Rv.
Figure 5 .
5Pipeline for data integration.
Figure 6 .
6Biological diversity in Abasy Atlas. regulatory interactions in a given RN, we evaluated the completeness of each RN model in terms of its percentage of genomic coverage. The two most-complete (genomic coverage >75%) RN models in Abasy Atlas are our metacurations for M. tuberculosis H37Rv (77%) and B. subtilis 168 (75%). Other RN models having a high genomic coverage are E. coli K-12 including regulatory RNAs (73%), E. coli K-12 all evidences (72%), B. subtilis 168 reconstructed from Arrieta et al. (70%) and M. tuberculosis H37Rv reconstructed from Minch et al. (62%)
Figure 7 .
7Listing of the systems identified in our meta-curation of M. tuberculosis H37Rv.
Figure 8 .
8Interactive network panel for module 1.19 of our meta-curation of B. subtilis 168. For a description of the meaning of different line-styles and colors, please refer to the section 'Interactive network panel' in the main text.
Table 1 .
1Regulatory datasets used for meta-curationRegulatory dataset
Organism
Genomic coverage
Balazsi et al. (2008)
M. tuberculosis H37Rv
22.9%
Sanz et al. (2011)
M. tuberculosis H37Rv
39.7%
Rohde et al. (2012)
M. tuberculosis H37Rv
27.8%
Minch et al. (2015)
M. tuberculosis H37Rv
62.1%
Galan-Vasquez et al. (2011)
P. aeruginosa PAO1
11.6%
RegTransBase-Cipriano et al. (2013)
P. aeruginosa PAO1
11.6%
DBTBS-Sierro et al. (2008)
B. subtilis 168
38.2%
SubtiWiki-Michna et al. (2016)
B. subtilis 168
42.4%
Arrieta-Ortiz et al. (2015)
B. subtilis 168
69.8%
Database, Vol. 2016, Article ID baw089
AcknowledgementsWe thank Josue Jiménez for drafting the first version of the web interface and database, and José Enrique Le on-Burguete and Aldo Carmona for technical support and manual curation. We thank Ricardo Ram ırez-Flores and Patricia Romero-N ajera for technical support and fruitful discussions. We also thank two anonymous reviewers for helpful suggestions.Authors' informationMAI-A holds a BSc in Genomic Sciences and is research intern at University of Florida. AIC-G is senior student at the Undergraduate Program in Genomic Sciences, UNAM.LGT-Q holds a PhD in Biochemistry and a BSc in Pharmaco-Biological Chemistry and is professor of Environmental Microbiology and head of the group of Contamination and Sustainability at the Polytechnic University of the State of Morelos. AT holds a PhD in Genetics and is the cocoordinator of the German Network for Bioinformatics Infrastructure -de.NBI. JAF-G holds a PhD in Biochemistry, an MSc in Computer Science, and a BSc in Computer Systems Engineering and is associate professor of systems biology and head of the group of Regulatory Systems Biology at the Center for Genomic Sciences, UNAM.Authors' contributionsMAI-A and AIC-G developed software for the automated data extraction, parsers and converters and carried out the meta-curation of RNs. MAI-A and JAF-G developed software for NDA predictions and automated functional annotation. AIC-G developed software for the disambiguation and mapping of gene symbols into canonical gene names. AT provided C. glutamicum regulatory interaction data.LGT-Q and AT provided biological expertise. JAF-G conceived and designed Abasy Atlas, coordinated its development, developed the interactive network panel and the final version of the web interface and database and wrote the manuscript. All authors read and approved the final manuscript.
Programa de Apoyo a Proyectos de Investigaci on e Innovaci on Tecnol ogica. PAPIIT-UNAMIA200614 and IA200616 to JAF-GPrograma de Apoyo a Proyectos de Investigaci on e Innovaci on Tecnol ogica (PAPIIT-UNAM) [IA200614 and IA200616 to JAF-G].
Funding for open access charge: Universidad Nacional Aut onoma de México Conflict of interest: The authors declare that they have not competing interest. Funding for open access charge: Universidad Nacional Aut onoma de México Conflict of interest: The authors declare that they have not compet- ing interest.
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[
"MAXIMAL GREEN SEQUENCES OF SKEW-SYMMETRIZABLE 3 × 3 MATRICES",
"MAXIMAL GREEN SEQUENCES OF SKEW-SYMMETRIZABLE 3 × 3 MATRICES"
]
| [
"Ahmet I Seven "
]
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| Maximal green sequences are particular sequences of mutations of skew-symmetrizable matrices which were introduced by Keller in the context of quantum dilogarithm identities and independently by Cecotti-Córdova-Vafa in the context of supersymmetric gauge theory. In this paper we study maximal green sequences of skew-symmetrizable 3 × 3 matrices. We show that such a matrix with a mutation-cyclic diagram does not have any maximal green sequences. We also obtain some properties of maximal green sequences of skew-symmetrizable matrices with mutation-acyclic diagrams. | 10.1016/j.laa.2013.10.018 | [
"https://arxiv.org/pdf/1207.6265v1.pdf"
]
| 119,322,743 | 1207.6265 | eff435922ab61eedf8031066049eef8818129dc2 |
MAXIMAL GREEN SEQUENCES OF SKEW-SYMMETRIZABLE 3 × 3 MATRICES
26 Jul 2012
Ahmet I Seven
MAXIMAL GREEN SEQUENCES OF SKEW-SYMMETRIZABLE 3 × 3 MATRICES
26 Jul 2012
Maximal green sequences are particular sequences of mutations of skew-symmetrizable matrices which were introduced by Keller in the context of quantum dilogarithm identities and independently by Cecotti-Córdova-Vafa in the context of supersymmetric gauge theory. In this paper we study maximal green sequences of skew-symmetrizable 3 × 3 matrices. We show that such a matrix with a mutation-cyclic diagram does not have any maximal green sequences. We also obtain some properties of maximal green sequences of skew-symmetrizable matrices with mutation-acyclic diagrams.
Introduction
Maximal green sequences are particular sequences of mutations of skew-symmetrizable matrices. They were used in [5] to obtain quantum dilogarithm identities. Moreover, the same sequences appeared in theoretical physics where they yield the complete spectrum of a BPS particle, see [2,Section 4.2]. In this paper we study the maximal green sequences of skew-symmetrizable 3×3 matrices. We show that those matrices with a mutation-cyclic diagram do not have any maximal green sequences. We also obtain some properties of maximal green sequences of skew-symmetrizable matrices with mutation-acyclic diagrams.
To be more specific, we need some terminology. Let us recall that a skewsymmetrizable matrix B is an n×n integer matrix such that DB is skew-symmetric for some diagonal matrix D with positive diagonal entries. We consider pairs (c, B), where B is a skew-symmetrizable integer matrix and c = (c 1 , ..., c n ) such that each c i = (c 1 , ..., c n ) ∈ Z n is non-zero. Motivated by the structural theory of cluster algebras, we call such a pair (c, B) a Y -seed. Then, for k = 1, . . . , n, the Y -seed mutation µ k transforms (c, B) into the Y -seed µ k (c, B) = (c ′ , B ′ ) defined as follows [4,Equation (5.9)], where we use the notation [b] + = max(b, 0):
• The entries of the exchange matrix B ′ = (B ′ ij ) are given by
(1.1) B ′ ij = −B ij if i = k or j = k; B ij + [B ik ] + [B kj ] + − [−B ik ] + [−B kj ] + otherwise.
• The tuple c ′ = (c ′ 1 , . . . , c ′ n ) is given by This transformation is involutive; furthermore, B ′ is skew-symmetrizable with the same choice of D. We also use the notation B ′ = µ k (B) (in (1.1)) and call the transformation B → B ′ the matrix mutation. This operation is involutive, so it defines a mutation-equivalence relation on skew-symmetrizable matrices. We use the Y -seeds in association with the vertices of a regular tree. To be more precise, let T n be an n-regular tree whose edges are labeled by the numbers 1, . . . , n, so that the n edges emanating from each vertex receive different labels.
(1.2) c ′ i = −c i if i = k; c i + [sgn(c k )B k,i ] + c k if i = k.
We write t k −−− t ′ to indicate that vertices t, t ′ ∈ T n are joined by an edge labeled by k. Let us fix an initial seed at a vertex t 0 in T n and assign the (initial) Y -seed (c 0 , B 0 ), where c 0 is the tuple of standard basis. This defines a Y -seed pattern on T n , i.e. an assignment of a seed (c t , B t ) to every vertex t ∈ T n , such that the seeds assigned to the endpoints of any edge t k −−− t ′ are obtained from each other by the seed mutation µ k . We write:
(1.3) c t = c = (c 1 , . . . , c n ) , B t = B = (B ij ) .
We refer to B as the exchange matrix and c as the c-vector tuple of the Y -seed. It is conjectured that c-vectors have the following sign coherence property:
(1.4) each vector c j has either all entries nonnegative or all entries nonpositive.
This conjectural property (1.4) has been proved in [3] for the case of skew-symmetric exchange matrices, using quivers with potentials and their representations.
We need a bit more terminology. The diagram of a skew-symmetrizable n × n matrix B is the directed graph Γ(B) defined as follows: the vertices of Γ(B) are the indices 1, 2, ..., n such that there is a directed edge from i to j if and only if B j,i > 0, and this edge is assigned the weight |B ij B ji | . By a subdiagram of Γ(B), we always mean a diagram obtained from Γ(B) by taking an induced (full) directed subgraph on a subset of vertices and keeping all its edge weights the same as in Γ(B). By a cycle in Γ(B) we mean a subdiagram whose vertices can be labeled by elements of Z/mZ so that the edges betweeen them are precisely {i, i + 1} for i ∈ Z/mZ. Let us also note that if B is skew-symmetric then it is also represented, alternatively, by a quiver whose vertices are the indices 1, 2, ..., n and there are B j,i > 0 many arrows from i to j. This quiver uniquely determines the corresponding skew-symmetric matrix, so mutation of skew-symmetric matrices can be viewed as a "quiver mutation". We call a diagram Γ mutation-acyclic if it is mutation-equivalent to an acyclic diagram (i.e. a diagram which has no oriented cycles at all); otherwise we call it mutation-cyclic. Now we can recall the notion of a green sequence [5]:
Definition 1.1. Let B 0 be a skew-symmetrizable n × n matrix. A green sequence for B 0 is a sequence i = (i 1 , . . . , i l ) such that, for any 1 ≤ k ≤ l with (c, B) = µ i k−1 •· · ·•µ i1 (c 0 , B 0 ), we have c i k > 0 ; here if k = 1, then we take (c, B) = (c 0 , B 0 ). A green sequence i = (i 1 , . . . , i l ) is maximal if, for (c, B) = µ i l • · · · • µ i1 (c 0 , B 0 ), we have c k < 0 for all k = 1, ..., n.
In this paper, we study the maximal green sequences in the basic case of size 3 skew-symmetrizable matrices. Our first result is the following:
Theorem 1.2. Suppose that B is a skew-symmetrizable 3 × 3 matrix. Under the assumption (1.4), if Γ(B)
is mutation-cyclic, then B does not have any maximal green sequences.
For skew-symmetrizable matrices with mutation-acyclic diagrams, we have the following result:
Theorem 1.3. Suppose that B is a skew-symmetrizable 3 × 3 matrix. Suppose also that Γ(B) is mutation-acyclic and i = (i 1 , . . . , i l ) is a maximal green sequence for B. Let B = B 0 and, for j = 1, ..., l, let B j = µ ij • · · · • µ i1 (B). Then, under the assumption (1.4), the diagram Γ(B j ) is acyclic for some 0 ≤ j ≤ l .
We also have the following result, which writes the initial exchange matrix in terms of a Y -seed:
Theorem 1.4. For any skew-symmetrizable 3×3 matrix B, let u = u(B) = (a, b, c) be defined as follows: if Γ(B) is cyclic, then (a, b, c) = (d 2 |B 2,3 |, d 3 |B 3,1 |, d 1 |B 1,2 |); if Γ(B) is acyclic, then (a, b, c) = (±d 2 |B 2,3 |, ±d 3 |B 3,1 |, ±d 1 |B 1,2 |)
such that the coordinates corresponding to the source and sink have the same sign and the remaining coordinate has the opposite sign.
Suppose now that (c ′ , B ′ ) is a Y -seed with respect to the initial seed (c 0 , B) and let u ′ = u(B ′ ) = (a ′ , b ′ , c ′ ). Then, under the assumption (1.4), we have a ′ c ′ 1 +b ′ c ′ 2 + c ′ c ′ 3 = ±(a, b, c). Furthermore, if Γ(B) is mutation-cyclic then a ′ c ′ 1 + b ′ c ′ 2 + c ′ c ′ 3 = (a, b, c).
Proofs of main results
The matrices in this section are skew-symmetrizable 3 × 3 matrices. We also assume that (1.4) is satisfied. First we note the following two properties, which can be easily checked using the definitions:
Proposition 2.1. Suppose that (c, B)
is a Y -seed (with respect to an initial Yseed). Suppose also that the coordinate vector of u with respect to c is (a 1 , ..., a n ). We also need the following two lemmas to prove our results: that (c, B) is a Y -seed (with respect to an initial Y -seed) such that Γ(B) is cyclic and let u be the vector whose coordinate vector with respect to the basis c is ( Proof. Assume without loss of generality that sgn(c k ) = sgn(B k,i ). Then c ′ i = c i +|B k,i |c k , c ′ j = c j , c ′ k = −c k . Thus the k-th coordinate of u with respect to c ′ will
Lemma 2.3. Supposed 2 |B 2,3 |, d 3 |B 3,1 |, d 1 |B 1,2 |). Let (c ′ , B ′ ) = µ k (c, Bbe −d i |B i,j | + |B k,i |d k |B k,j | = −d i |B i,j | + |B i,k |d i |B k,j | = d i (−|B i,j | + |B i,k ||B k,j |) (Proposition 2.1 )
, the other coordinates are the same (note that the k-th coordinate of u with respect to c is d i |B i,j | = d j |B j,i |). Thus, to prove the statement in the first part, it is enough to show |B ′ i,j | = −|B i,j | + |B i,k ||B k,j | (Recall that the skew-symmetrizing matrix D is preserved under mutations). For convenience, we investigate in cases:
Case 1. B k,i > 0. Then B i,j > 0 and B j,k > 0 (so B i,k < 0, B j,i < 0, B k,j < 0). Then, since Γ(B ′ ) is cyclic, we have B ′ k,i < 0, B ′ i,j < 0, B ′ j,k < 0. By (1.1), we have B ′ ij = B ij + [B ik ] + [B kj ] + − [−B ik ] + [−B kj ] + = B ij − [−B ik ][−B kj ] = B ij − B ik B kj . Since B ′ i,j < 0 by assumption in this case, we have |B ′ ij | = −B ′ ij = −B ij +B ik B kj = −|B ij | + |B ik ||B kj | (note B i,j > 0, B ik B kj > 0 by assumption) as required. Case 2. B k,i < 0. Then B i,j < 0 and B j,k < 0 (so B i,k > 0, B j,i > 0, B k,j > 0). Then, since Γ(B ′ ) is cyclic, we have B ′ k,i > 0, B ′ i,j > 0, B ′ j,k > 0. By (1.1), we have B ′ ij = B ij + [B ik ] + [B kj ] + − [−B ik ] + [−B kj ] + = B ij + B ik B kj . Thus |B ′ ij | = B ′ ij = B ij + B ik B kj = −|B ij | + |B ik B kj | as required.
For the second part, suppose Γ(B ′ ) is acyclic. We assume, without loss of generality, that sgn(c k ) = sgn(B k,i ). To prove the statement (of the second part), it is enough to show that −|B ′ i,j | = −|B i,j | + |B i,k ||B k,j |. For convenience, we investigate in cases:
Case 1. B k,i > 0. Then B i,j > 0 and B j,k > 0 (so B i,k < 0, B j,i < 0, B k,j < 0). Then, since Γ(B ′ ) is acyclic, we have B ′ k,i < 0, B ′ j,k < 0 but B ′ i,j > 0. By (1.1), we have B ′ ij = B ij + [B ik ] + [B kj ] + − [−B ik ] + [−B kj ] + = B ij − [−B ik ][−B kj ] = B ij − B ik B kj . Thus, we have −|B ′ ij | = −B ′ ij = −B ij + B ik B kj = −|B ij | + |B ik ||B kj | as required. Case 2. B k,i < 0. Then B i,j < 0 and B j,k < 0 (so B i,k > 0, B j,i > 0, B k,j > 0). Then, since Γ(B ′ ) is acyclic, we have B ′ k,i > 0, B ′ j,k > 0 but B ′ i,j < 0. By (1.1), B ′ ij = B ij + [B ik ] + [B kj ] + − [−B ik ] + [−B kj ] + = B ij + B ik B kj . Thus −|B ′ ij | = B ′ ij = B ij + B ik B kj = −|B ij | + |B ik B kj | as required.
Lemma 2.4. Suppose that (c, B) is a seed such that Γ(B) is acyclic and u be the vector whose coordinate vector with respect to the basis c is (±d 2 |B 2,3 |, ±d 3 |B 3,1 |, ±d 1 |B 1,2 |) such that the coordinates corresponding to the source and sink have the same sign and the remaining coordinate has the opposite sign. Let (c ′ , B ′ ) = µ k (c, B). Then we have the following: If k is a source or sink in Γ(B) (so the diagram Γ(B ′ ) is also acyclic), then the coordinate vector of u with with respect to the basis c ′ is obtained from the one for c by multiplying the k-th coordinate by −1.
If k is neither a source nor a sink in Γ(B) (so the diagram Γ(B ′ ) is cyclic), then the coordinate vector of u with with respect to the basis
c ′ is (d 2 |B ′ 2,3 |, d 3 |B ′ 3,1 |, d 1 |B ′ 1,2 |) or its negative.
Proof. Let i, j be the remaining vertices (so {i, j, k} = {1, 2, 3}). For the first part, suppose that k is a source or sink, so sgn(B k,i ) = sgn(B k,j ). Let us denote the i-th,j-th and k-th coordinates of u by a i , a j , a k respectively, so |a i | = d k |B k,j | = d j |B j,k |, |a j | = d k |B k,i | = d i |B i,k |. Then, in particular, (*) |a i ||B k,i | = |a j ||B k,j |. Let us assume, without loss of generality, that i is neither a source nor a sink. Then, by the condition on the signs, the numbers a i and a j have opposite signs, so (*) implies that (**) a i |B k,i | = −a j |B k,j |.
Let us assume first that sgn(c k ) = sgn(B k,i ) = sgn(B k,j ). Then
c ′ i = c i + |B k,i |c k , c ′ j = c j + |B k,j |c k , c ′ k = −c k .
Then the k-th coordinate of u with respect to c ′ will be −a k and the other coordinates are the same because u = a k c k + a i c i + a j c j = −a k (−c k ) + a i (c i + |B k,i |c k ) + a j (c j + |B k,j |c k ) because a i |B k,i | = −a j |B k,j | by (**).
Let us now assume that sgn(
c k ) = −sgn(B k,i ) = −sgn(B k,j ). Then c ′ k = −c k and c ′ i = c i , c ′ j = c j .
Then the k-th coordinate of u with respect to c ′ will be −a k and the other coordinates are the same.
For the second part, suppose that k is neither a source nor a sink, so sgn(B k,i ) = −sgn(B k,j ). We may assume, without loss of generality, that sgn(c k ) = sgn(B k,i ).
Then c ′ i = c i + |B k,i |c k , c ′ j = c j , c ′ k = −c k .
Let us denote the i-th,j-th and k-th coordinates of u (with respect to c) by a i , a j , a k respectively, so
|a i | = d k |B k,j | = d j |B j,k |, |a j | = d k |B k,i | = d i |B i,k |, |a k | = d i |B i,j | = d j |B j,i | such that
sgn(a i ) = sgn(a j ) = −sgn(a k ) (***). Then the k-th coordinate of u with respect to c ′ will be a ′ k = −a k + a i |B k,i | and the other coordinates a ′ i , a ′ j are the same because u = a k c k + a i c i + a j c j = (−a k + a i |B k,i |)(−c k ) + a i (c i + |B k,i |c k ) + a j (c j ). Note that sgn(−a k + a i |B k,i |) = sgn(a i ) = sgn(a j ) by (***). Thus we may assume, without loss of generality that, sgn(a i ) = sgn(a j ) = +1 = −sgn(a k ), so
a i = d k |B k,j | = d j |B j,k |, a j = d k |B k,i | = d i |B i,k |, a k = −d i |B i,j | = −d j |B j,i | and show that a ′ k = −a k + a i |B k,i | = d i |B i,j |+d k |B k,j ||B k,i | = d i |B i,j |+d i |B k,j ||B i,k | = d i (|B i,j |+|B k,j ||B i,k |) = d i |B ′ i,j |, i.e. show that |B ′ i,j | = |B i,j | + |B k,j ||B i,k |.
This will complete the proof. For convenience, we investigate in cases. Case 1. B k,i > 0. Then B j,i > 0 and B j,k > 0 (so B i,k < 0, B i,j < 0, B k,j < 0). Note then that, since Γ(B ′ ) is cyclic, we have B ′ k,i < 0, B ′ i,j < 0, B ′ j,k < 0. By (1.1),
B ′ ij = B ij + [B ik ] + [B kj ] + − [−B ik ] + [−B kj ] + = B ij − [−B ik ][−B kj ] = B ij − B ik B kj . Thus |B ′ ij | = −B ′ ij = −B ij + B ik B kj = |B ij | + |B ik ||B kj | as required. Case 2. B k,i < 0. Then B j,i < 0 and B j,k < 0 (so B i,k > 0, B i,j > 0, B k,j > 0). Then, since Γ(B ′ ) is cyclic, we have B ′ k,i > 0, B ′ i,j > 0, B ′ j,k > 0. By (1.1), B ′ ij = B ij + [B ik ] + [B kj ] + − [−B ik ] + [−B kj ] + = B ij − [B ik ][B kj ] = B ij + B ik B kj , so |B ′ ij | = B ′ ij = B ij + B ik B kj = |B ij | + |B ik ||B kj | as required.
We can now prove our results. Proof of Theorem 1.2: Suppose that i = (i 1 , . . . , i l ) is a maximal green sequence for B. Let (c ′ , B ′ ) = µ i l • · · · • µ i1 (c, B). Then c ′ j < 0 for all j = 1, ..., n. Let u 0 be the vector (whose coordinate vector with respect to the initial basis c is) (d 2 |B 2,3 |, d 3 |B 3,1 |, d 1 |B 1,2 |). Let (a 1 , a 2 , a 3 ) be the coordinates of u 0 with respect to c ′ . By Lemma 2.3, the coordinates a 1 , a 2 , a 3 > 0. This implies that, since u 0 = a 1 c ′ 1 + a 2 c ′ 2 + a 3 c ′ 3 , the coordinates of u 0 with respect to c are non-positive; which is a contradiction.
Proof of Theorem 1.3: Suppose that Γ(B j ) is cyclic for all 0 ≤ j ≤ l. Let (c ′ , B ′ ) = µ i l • · · · • µ i1 (c, B). Then c ′ j < 0 for all j = 1, ..., n. Let u 0 be the vector (whose coordinate vector with respect to the initial basis c is) (d 2 |B 2,3 |, d 3 |B 3,1 |, d 1 |B 1,2 |). Let (a 1 , a 2 , a 3 ) be the coordinates of u 0 with respect to c ′ . By the first part of Lemma 2.3, the coordinates a 1 , a 2 , a 3 > 0. This implies that, since u 0 = a 1 c ′ 1 + a 2 c ′ 2 + a 3 c ′ 3 , the coordinates of u 0 with respect to c are non-positive; which is a contradiction. Proof of Theorem 1.4: By Lemmas 2.3 and 2.4, the coordinate vector of u with respect to c ′ is u ′ or −u ′ . Then the conclusions follow.
Date:
July 26, 2012. The author's research was supported in part by the Turkish Research Council (TUBITAK).
Let (c ′ , B ′ ) = µ k (c, B) and (a ′ 1 , ..., a ′ n ) be the coordinates of u with respect to c ′ .Then a i = a ′ i if i = k and a ′ k = −a k + a i [sgn(c k )B k,i ] + , where the sum is over all i = k. Proposition 2.2. Suppose that B is a skew-symmetrizable 3 × 3 matrix B. Let u = u(B) = (a, b, c) be defined as follows: if Γ(B) is cyclic, then (a, b, c) = (d 2 |B 2,3 |, d 3 |B 3,1 |, d 1 |B 1,2 |); if Γ(B)is acyclic, then (a, b, c) = (±d 2 |B 2,3 |, ±d 3 |B 3,1 |, ±d 1 |B 1,2 |) such that the coordinates corresponding to the source and sink have the same sign and the remaining coordinate has the opposite sign. Then the vector u is a radical vector for B, i.e. Bu = 0.
If the diagram Γ(B ′ ) is acyclic, then the coordinate vector of u with respect to the basis c ′ is obtained from (d 2 |B ′ |) by multiplying the k-th coordinate by −1. (Note that the vertex k is neither a source nor a sink in Γ(B ′ ).)). Then we have
the following:
If the diagram Γ(B ′ ) is also cyclic, then the coordinate vector of u with respect
to the basis c ′ is (d 2 |B ′
2,3 |, d 3 |B ′
3,1 |, d 1 |B ′
1,2 |).
2,3 |, d 3 |B ′
3,1 |, d 1 |B ′
1,2
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"Vladimir Andrievskii "
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| We prove an analogue of the classical Bernstein theorem concerning the rate of polynomial approximation of piecewise analytic functions on a compact subset of the real line. | 10.1016/j.jat.2018.04.003 | [
"https://arxiv.org/pdf/1712.07054v1.pdf"
]
| 44,089,010 | 1712.07054 | 848f0a4ed9fe67826c94572edc6ee3513cd3626b |
19 Dec 2017
Vladimir Andrievskii 19 Dec 2017Polynomial approximation on a compact subset of the real linePolynomial approximationGreen's functionPiecewise analytic function MSC: 30C1030E10
We prove an analogue of the classical Bernstein theorem concerning the rate of polynomial approximation of piecewise analytic functions on a compact subset of the real line.
Introduction and the main result
Let E ⊂ R be a compact subset of the real line R and let P n be the set of all (real) polynomials of degree at most n ∈ N := {1, 2, . . .}. Also, let for x 0 ∈ E and α > 0,
E n (|x − x 0 | α , E) := inf p∈Pn sup x∈E ||x − x 0 | α − p(x)|.
The starting point of our analysis is the classical Bernstein theory [6,7,8]. According to this theory, for any x 0 ∈ (−1, 1) and α > 0, where α is not an even integer, there exists a finite nonzero limit
σ α := lim n→∞ n α E n (|x − x 0 | α , [−1, 1]).
The question as to what happens to the best polynomial approximations for a general set E ⊂ R is investigated in monographs [15] and [14,Chapter 10] where the reader can also find a comprehensive survey of this subject. Now, we consider E to be a set in the complex plane C and use the notions of potential theory in the plane (see [12,13] for details). Let E be non-polar, i.e., be of positive (logarithmic) capacity cap(E) > 0 and let g C\E (z) = g C\E (z, ∞), z ∈ C \ E be the Green function of C \ E with pole at infinity, where C := C ∪ {∞} is the extended complex plane.
Our main objective is to prove the following result.
Theorem 1 Let x 0 ∈ E. If for some α > 0, lim sup n→∞ n α E n (|x − x 0 | α , E) > 0, (1.1) then sup z∈C\E g C\E (z) |z − x 0 | < ∞. (1.2)
Comparing Theorem 1 with [4, Corollary 1] we obtain the following result.
Theorem 2 Let x 0 ∈ E. Then for any α > 0, which is not even integer,
lim inf n→∞ n α E n (|x − x 0 | α , E) > 0 if and only if (1.2) holds.
For the geometry of E satisfying (1.2), we refer the reader to [9,14,10,3] and the many references therein.
Auxiliary results
In this section we assume that
E = m j=1 [a j , b j ], x 0 ∈ m j=1 (a j , b j ) =: Int(E), where −1 = a 1 < b 1 < a 2 < . . . < b m−1 < a m < b m = 1, m > 1.
It is known (for example, see [16, pp. 224-226], [5, pp. 409-412] or [2]) that there exists a conformal mapping w = F (z) = F E (z) of the upper half-plane H := {z : ℑz > 0} onto the domain
G = G E = {z : 0 < ℜz < π, ℑz > 0} \ ∪ m−1 j=1 [u j , u j + iv j ]
, where 0 =: u 0 < u 1 < u 2 < . . . < u m−1 < u m := π and v j > 0, j = 1, . . . , m − 1, which can be extended continuously to H satisfying the following boundary correspondence
F (∞) = ∞, F ((−∞, −1]) = {z : ℜz = 0, ℑz ≥ 0}, F ([1, ∞)) = {z : ℜz = π, ℑz ≥ 0}, F ([a j , b j ]) = [u j−1 , u j ], j = 1, . . . , m, F ([b j , a j+1 ]) = [u j , u j + iv j ], j = 1, . . . , m − 1. Moreover, g C\E (z) = ℑ(F (z)), z ∈ H, πµ E ([a, b]) = |F ([a, b] ∩ E)|, [a, b] ⊂ [−1, 1],
where µ E is the equilibrium measure for E and |S| means the linear Lebesgue measure (length) of S ⊂ C.
Since, by the reflection principle, F can be extended analytically in a neighborhood of x 0 , we can consider function
h(x 0 ) = h E (x 0 ) := F ′ (x 0 ) = πω E (x 0 ), where ω E (x 0 ) is the density of µ E at x 0 .
Our interest in this function lies in the fact that by the Vasiliev -Totik theorem (see [15,p. 163,Corollary 11] and [14,Theorem 10.5
]) for α > 0, lim n→∞ n α E n (|x − x 0 | α , E) = h(x 0 ) −α σ α . (2.1)
We set a sufficiently large constant c > 2 such that
h(x 0 ) ≤ c and cap(E) > 1 c . (2.2)
In what follows, we denote by c 1 , c 2 , . . . positive constants which depend only on c.
By virtue of [2, p. 39, (1.8)], for j = 1, . . . , m − 1,
v j ≤ log 1 + (1 − 2cap(E)) 1/2 1 − (1 − 2cap(E)) 1/2 ≤ log √ c + √ c − 2 √ c − √ c − 2 < log(2c).
For the fixed point z 0 := x 0 + 2ce 4π i ∈ H, w 0 := F (z 0 ), and D := {z : |z| < 1}, we have
ℑ(w 0 ) = g C\E (z 0 ) ≥ g C\D (z 0 ) = log |z 0 | ≥ log |z 0 − x 0 | = 4π + log(2c) =: c 1 > v j + 4π, j = 1, . . . , m − 1. (2.3)
Meanwhile, by virtue of [13, p. 53] and (2.2),
ℑ(w 0 ) = g C\E (z 0 ) = log |z 0 − x|dµ E (x) − log cap(E) ≤ log(|z 0 − x 0 | + 2) + log c < log(4c 2 ) + 4π < 2c 1 . (2.4)
We use the notion of the module of a family of curves. We refer to [1,Chapter 4], [11,Chapter 9] or [5, pp. 341-360] for the definition and basic properties of the module (such as conformal invariance, comparison principle, composition laws, etc.) We use this properties without further citation. The curves under consideration are crosscuts (see [11]) either of H or of G.
For z 1 , z 2 ∈ H denote by Γ(x 0 , z 1 ; z 2 , ∞; H) the family of all crosscuts of H that separate points x 0 and z 1 from
z 2 and ∞ in H. Note that if |z 1 − x 0 | < |z 2 − x 0 |, then for the module of Γ := Γ(x 0 , z 1 ; z 2 , ∞; H) we have 0 ≤ m(Γ) − 1 π log z 2 − x 0 z 1 − x 0 ≤ 2. (2.5) Indeed, let Γ 0 ⊂ Γ denote the family of all crosscuts of A := {z ∈ H : |z 1 − x 0 | < |z − x 0 | < |z 2 − x 0 |}m(Γ) ≥ m(Γ 0 ) = 1 π log z 2 − x 0 z 1 − x 0 ,
which implies the left hand side of (2.5).
Next, consider the metric
ρ(z) := (π|z − x 0 |) −1 if z ∈ H, e −π |z 1 − x 0 | ≤ |z − x 0 | ≤ e π |z 2 − x 0 |, 0 elsewhere in C.
The analysis analogous to the proof of [5, p. 349, (1.11)] yields γ ρ(z)|dz| ≥ 1, γ ∈ Γ,
i.e., ρ is admissible (in the L-definition) of m(Γ). Therefore,
m(Γ) ≤ ρ(z) 2 dm(z) = 1 π log z 2 − x 0 z 1 − x 0 + 2,
where dm means integration with respect to the two-dimensional Lebesgue measure (area), which implies the right-hand side inequality in (2.5).
In a similar way, for η 0 := F (x 0 ) and w 1 , w 2 ∈ G, we introduce the family Γ(η 0 , w 1 ; w 2 , ∞; G) of all crosscuts of G that separate points η 0 and w 1 from w 2 and ∞ in G.
For r > 0, denote by γ(r) = γ(η 0 , G, r) ⊂ {w ∈ G : |w −η 0 | = r} the crosscut of G which has nonempty intersection with the ray {w ∈ H : ℜ(w) = η 0 }. For 0 < r < R, denote by Q(r, R) = Q(η 0 , G, r, R) ⊂ G the bounded simply connected domain whose boundary consists of γ(r), γ(R), and two connected parts of ∂G. Let m(r, R) = m(η 0 , G, r, R) be the module of the family Γ(r, R) = Γ(η 0 , G, r, R) of all crosscuts of Q(r, R) which separate circular arcs γ(r) and γ(R) in Q(r, R).
Lemma 1 For 0 < r < R ≤ R 0 := |w 0 − η 0 |, m(r, R) ≤ 1 π log R r + c 2 , c 2 := 1 π log c 2 c 1 + 7. (2.6)
Proof. From the right-hand side of (2.5) we conclude that for Γ 1 := Γ(x 0 , z; z 0 , ∞; H) and z ∈ H with |z −
x 0 | < |z 0 − x 0 |, m(Γ 1 ) ≤ 1 π log z 0 − x 0 z − x 0 + 2.
While, for Γ ′ 1 := F (Γ 1 ) = Γ(η 0 , w; w 0 , ∞; G) and w := F (z), where |w − η 0 | is sufficiently small, we have
m(Γ ′ 1 ) ≥ m(|w − η 0 |, R 0 ) ≥ m(|w − η 0 |, r) + m(r, R) + m(R, R 0 ) ≥ m({{ξ ∈ H : |ξ − x 0 | = t} : |w − η 0 | < t < r}) + m(r, R) +m({{ξ ∈ H : |ξ − x 0 | = t} : R < t < R 0 }) = 1 π log r |w − η 0 | + m(r, R) + 1 π log R 0 R .
Since m(Γ 1 ) = m(Γ ′ 1 ), comparing above inequalities, we obtain
m(r, R) ≤ 1 π log z 0 − x 0 z − x 0 + 2 − 1 π log r |w − η 0 | − 1 π log R 0 R = 1 π log w − η 0 z − x 0 + 1 π log |z 0 − x 0 | R 0 + 2 + 1 π log R r .
Taking limit as z → x 0 we have
m(r, R) ≤ 1 π log |F ′ (x 0 )| + 1 π log |z 0 − x 0 | R 0 + 2 + 1 π log R r which,v j |η 0 − u j | < (η 0 − u j ) 2 + v 2 j |η 0 − u j | ≤ c 3 := 4 exp(3πc 2 ). (2.7)
Proof. Let r j := |η 0 − u j |, R j := |η 0 − u j | 2 + v 2 j . By Lemma 1
m(r j , R j ) ≤ 1 π log R j r j + c 2 . (2.8) Furthermore, we claim that m(r j , R j ) ≥ 4 3π log R j √ 2r j . (2.9)
Indeed, according to the nonnegativity of the module of a family of curves, there is no loss of generality in assuming that R j > √ 2r j . Since
|γ(r)| ≤ 3 4 πr, √ 2r j < r < R j , we have m(r j , R j ) ≥ m( √ 2r j , R j ) ≥ m({γ(r) : √ 2r j < r < R j }) ≥ R j √ 2r j dr |γ(r)| ≥ 4 3 R j √ 2r j dr πr = 4 3π log R j √ 2r j .
Making use of (2.8) and (2.9), we obtain
R j r j ≤ 4 exp(3πc 2 ),
which implies (2.7). Proof. First, we claim that for w := F (z),
|w − η 0 | ≤ |w 0 − η 0 |. (2.11)
To prove (2.11), we assume that |w − η 0 | > |w 0 − η 0 |. Then, for the module of
Γ ′ 2 := Γ(η 0 , w; w 0 , ∞; G) we have m(Γ ′ 2 ) ≤ 4. (2.12) Indeed, since for γ ∈ Γ ′ 2 , γ ∩ {ξ ∈ G : |ξ − η 0 | = |w 0 − η 0 } = ∅, the metric ρ 1 (ξ) := π −1 if 0 ≤ ℜ(ξ) ≤ π, |ℑ(ξ − w 0 )| ≤ 2π, 0 elsewhere in C is admissible for Γ ′ 2 . Therefore, m(Γ ′ 2 ) ≤ ρ 1 (ξ) 2 dm(ξ) = 4.
This proves (2.12).
Moreover, according to our assumption |z−x 0 | < e −4π |z 0 −x 0 | and the left-hand side of (2.5), for the module of
Γ 2 := F −1 (Γ ′ 2 ) = Γ(x 0 , z; z 0 , ∞; H) we obtain m(Γ 2 ) ≥ 1 π log z 0 − x 0 z − x 0 > 4,(2.13)
which contradicts (2.12). Hence, (2.11) holds.
Our next objective is to estimate from above the module of Γ ′ 2 under the assumption (2.11). Denote by r(w) the supremum of values of r > 0 such that γ(r) separates η 0 and w in G. By Lemma 2
|w − η 0 | ≤ r(w) ≤ c 3 |w − η 0 |.
(2.14)
Let ρ 2 be the extremal metric for the family Γ 3 := Γ (r(w), |η 0 − w 0 |), i.e.,
γ ρ 2 (ξ)|dξ| ≥ 1, γ ∈ Γ 3 , m (r(w), |η 0 − w 0 |) = ρ 2 (ξ) 2 dm(ξ).
Since for γ ∈ Γ ′ 2 with γ ∩ γ(r(w)) = ∅ we have |γ| ≥ r(w)/c 3 , for such γ and the metric
ρ 3 (ξ) := c 3 r(w) if ξ ∈ G, |ξ − η 0 | ≤ 2r(w), 0 elsewhere in C we obtain γ ρ 3 (ξ)|dξ| ≥ 1.
From what we already proved, we conclude that the metric
ρ(ξ) := max k=1,2,3 ρ k (ξ), ξ ∈ C,
is admissible for Γ ′ 2 and, by (2.6) and (2.14),
m(Γ ′ 2 ) ≤ 3 k=1 ρ k (ξ) 2 dm(ξ) ≤ 4 + 1 π log |w 0 − η 0 | r(w) + c 2 + 2πc 2 3 ≤ 1 π log |w 0 − η 0 | |w − η 0 | + c 5 , c 5 := c 2 + 4 + 2πc 2 3 .
Comparing the above inequality with (2.13) and using (2.4), we obtain
|w − η 0 | ≤ |w 0 − η 0 | |z 0 − x 0 | e c 5 π |z − x 0 | < c 1 + 1 c e c 5 π |z − x 0 |,
which implies (2.10).
✷
Proof of Theorem 1
Neither the hypothesis nor the conclusion is affected if we assume that E ⊂ [−1, 1], ±1 ∈ E and −1 < x 0 < 1. Next, we construct a sequence E j , j ∈ N of compact sets each of which consists of a finite number of intervals such that x 0 ∈ Int(E j ) for each j and
E j+1 ⊂ E j ⊂ [−1, 1], ∞ j=1 E j = E.g C\E j (z) = g C\E (z), z ∈ C \ E. (3.1)
Since E n (|x − x 0 | α , E) ≤ E n (|x − x 0 | α , E j ), by assumption (1.1) and the Vasiliev-Totik theorem (2.1), it follows that for some α > 0, h E j (x 0 ) −α σ α = lim n→∞ n α E n (|x − x 0 | α , E j ) ≥ lim sup n→∞ n α E n (|x − x 0 | α , E) =: β > 0.
Hence,
h E j (x 0 ) ≤ σ α β 1/α , cap(E j ) ≥ cap(E),
and (2.2) holds for each E j instead of E with a constant c > 2 independent of j. Referring to Lemma 3, we find that for z ∈ C \ E j with |z − x 0 | ≤ 2c,
g C\E j (z) ≤ c 4 |z − x 0 |,
where a constant c 4 also does not depend on j. A passage to the limit as j → ∞ and (3.1) yield g C\E (z) ≤ c 4 |z − x 0 |.
Acknowledgements
The author is grateful to M. Nesterenko for his helpful comments.
✷ Lemma 3
3For z ∈ H with |z − x 0 | < 2c, |F (z) − η 0 | ≤ c 4 |z − x 0 |, c 4
consider the case where |z − x 0 | ≥ 2c > 4. By[13, p. 53], in this caseg C\E (z) = log |z − x|dµ E (x) − log cap(E) ≤ log(2c|z − x 0 |)which, together with (3.2), implies(1.2) This concludes the proof of Theorem 1.
✷
separating the circular parts of the boundary ∂A in A.Then,
by [5, p. 347, Example 1.8],
By[12, p. 108, Theorem 4.4.6] lim
j→∞
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| []
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[
"Quantisation of the effective string with TBA",
"Quantisation of the effective string with TBA"
]
| [
"Michele Caselle \nDipartimento di Fisica\nIstituto Nazionale di Fisica Nucleare -sezione di Bologna and Dipartimento di Fisica e Astronomia\nUniversità di Torino and Istituto Nazionale di Fisica Nucleare -sezione di Torino\nVia P. Giuria 1I-10125TorinoItaly\n\nUniversità di Bologna\nVia Irnerio 46I-40127BolognaItaly\n",
"Davide Fioravanti [email protected] ",
"Ferdinando Gliozzi \nDipartimento di Fisica\nIstituto Nazionale di Fisica Nucleare -sezione di Bologna and Dipartimento di Fisica e Astronomia\nUniversità di Torino and Istituto Nazionale di Fisica Nucleare -sezione di Torino\nVia P. Giuria 1I-10125TorinoItaly\n\nUniversità di Bologna\nVia Irnerio 46I-40127BolognaItaly\n",
"Roberto Tateo [email protected] \nDipartimento di Fisica\nIstituto Nazionale di Fisica Nucleare -sezione di Bologna and Dipartimento di Fisica e Astronomia\nUniversità di Torino and Istituto Nazionale di Fisica Nucleare -sezione di Torino\nVia P. Giuria 1I-10125TorinoItaly\n\nUniversità di Bologna\nVia Irnerio 46I-40127BolognaItaly\n"
]
| [
"Dipartimento di Fisica\nIstituto Nazionale di Fisica Nucleare -sezione di Bologna and Dipartimento di Fisica e Astronomia\nUniversità di Torino and Istituto Nazionale di Fisica Nucleare -sezione di Torino\nVia P. Giuria 1I-10125TorinoItaly",
"Università di Bologna\nVia Irnerio 46I-40127BolognaItaly",
"Dipartimento di Fisica\nIstituto Nazionale di Fisica Nucleare -sezione di Bologna and Dipartimento di Fisica e Astronomia\nUniversità di Torino and Istituto Nazionale di Fisica Nucleare -sezione di Torino\nVia P. Giuria 1I-10125TorinoItaly",
"Università di Bologna\nVia Irnerio 46I-40127BolognaItaly",
"Dipartimento di Fisica\nIstituto Nazionale di Fisica Nucleare -sezione di Bologna and Dipartimento di Fisica e Astronomia\nUniversità di Torino and Istituto Nazionale di Fisica Nucleare -sezione di Torino\nVia P. Giuria 1I-10125TorinoItaly",
"Università di Bologna\nVia Irnerio 46I-40127BolognaItaly"
]
| []
| In presence of a static pair of sources, the spectrum of low-lying states of whatever confining gauge theory in D space-time dimensions is described, at large source separations, by an effective string theory. In the far infrared the latter flows, in the static gauge, to a two-dimensional massless free-field theory. It is known that the Lorentz invariance of the gauge theory fixes uniquely the first few subleading corrections of this free-field limit. We point out that the first allowed correction -a quartic polynomial in the field derivatives -is exactly the composite field TT , built with the chiral components, T andT , of the energy-momentum tensor. This irrelevant perturbation is quantum integrable and yields, through the thermodynamic Bethe Ansatz (TBA), the energy levels of the string which exactly coincide with the Nambu-Goto spectrum. We obtain this way the results recently found by Dubovsky, Flauger and Gorbenko. This procedure easily generalizes to any two-dimensional CFT. It is known that the leading deviation of the Nambu-Goto spectrum comes from the boundary terms of the string action. We solve the TBA equations on an infinite strip, identify the relevant boundary parameter and verify that it modifies the string spectrum as expected. | 10.1007/jhep07(2013)071 | [
"https://arxiv.org/pdf/1305.1278v3.pdf"
]
| 119,331,201 | 1305.1278 | a41c1c4bb5cebdfa7a2f440461c52ee146048b5b |
Quantisation of the effective string with TBA
19 Jun 2013
Michele Caselle
Dipartimento di Fisica
Istituto Nazionale di Fisica Nucleare -sezione di Bologna and Dipartimento di Fisica e Astronomia
Università di Torino and Istituto Nazionale di Fisica Nucleare -sezione di Torino
Via P. Giuria 1I-10125TorinoItaly
Università di Bologna
Via Irnerio 46I-40127BolognaItaly
Davide Fioravanti [email protected]
Ferdinando Gliozzi
Dipartimento di Fisica
Istituto Nazionale di Fisica Nucleare -sezione di Bologna and Dipartimento di Fisica e Astronomia
Università di Torino and Istituto Nazionale di Fisica Nucleare -sezione di Torino
Via P. Giuria 1I-10125TorinoItaly
Università di Bologna
Via Irnerio 46I-40127BolognaItaly
Roberto Tateo [email protected]
Dipartimento di Fisica
Istituto Nazionale di Fisica Nucleare -sezione di Bologna and Dipartimento di Fisica e Astronomia
Università di Torino and Istituto Nazionale di Fisica Nucleare -sezione di Torino
Via P. Giuria 1I-10125TorinoItaly
Università di Bologna
Via Irnerio 46I-40127BolognaItaly
Quantisation of the effective string with TBA
19 Jun 2013Preprint typeset in JHEP style. -PAPER VERSIONBosonic StringsLattice Gauge Field TheoriesThermodynamic Bethe Ansatz
In presence of a static pair of sources, the spectrum of low-lying states of whatever confining gauge theory in D space-time dimensions is described, at large source separations, by an effective string theory. In the far infrared the latter flows, in the static gauge, to a two-dimensional massless free-field theory. It is known that the Lorentz invariance of the gauge theory fixes uniquely the first few subleading corrections of this free-field limit. We point out that the first allowed correction -a quartic polynomial in the field derivatives -is exactly the composite field TT , built with the chiral components, T andT , of the energy-momentum tensor. This irrelevant perturbation is quantum integrable and yields, through the thermodynamic Bethe Ansatz (TBA), the energy levels of the string which exactly coincide with the Nambu-Goto spectrum. We obtain this way the results recently found by Dubovsky, Flauger and Gorbenko. This procedure easily generalizes to any two-dimensional CFT. It is known that the leading deviation of the Nambu-Goto spectrum comes from the boundary terms of the string action. We solve the TBA equations on an infinite strip, identify the relevant boundary parameter and verify that it modifies the string spectrum as expected.
Introduction
Even though a rigorous proof of quark confinement in Yang-Mills theories is still missing, numerical experiments and theoretical arguments leave little doubt that this phenomenon is associated to the formation of a thin string-like flux tube, the confining string, which generates, for large quark separations, a linearly rising of confining potential.
The string-like nature of the flux tube is particularly evident in the strong coupling region of lattice gauge theories, where the vacuum expectation value of large Wilson loops is given by a sum over certain lattice surfaces which can be considered as the world-sheet of the underlying confining string. When the coupling constant decreases, this two-dimensional system undergoes a roughening transition [1] where the sum of these surfaces diverges and the colour flux tube of whatever lattice gauge theory undergoes a transition towards a rough phase, which is connected to the continuum limit. It is widely believed that such a phase transition belongs to the Kosterlitz-Thouless universality class [2]. Accordingly, the renormalisation group equations imply that the effective string action S describing the dynamics of the flux tube in the whole rough phase flows at large scales towards a massless free-field theory. Thus, for large enough inter-quark separations it is not necessary to know explicitly the specific form of the effective string action S, but only its infrared limit S[X] = S cl + S 0 [X] + . . . , (1.1) where the classical action S cl describes the usual perimeter-area term, X denotes the twodimensional bosonic fields X i (ξ 1 , ξ 2 ), with i = 1, 2, . . . , D − 2, describing the transverse displacements of the string with respect the configuration of minimal energy, ξ 1 , ξ 2 are the coordinates on the world-sheet and S 0 [X] is the Gaussian action
S 0 [X] = σ 2 d 2 ξ (∂ α X · ∂ α X) . (1.2)
This action is written in the physical or static gauge, where the only degrees of freedom taken into account are the physical ones i.e. the transverse displacements X i . Even in this infrared approximation the effective string is highly predictive, indeed it predicts the leading correction to the linear quark-anti-quark potential, known as Lüscher term [3,4] V (R) = σR − π(D − 2)
24R + O(1/R 2 ) . (1.3)
Accurate numerical simulations have shown the validity of that expectation [5,6,7,8,9,10]. This infrared limit also accounts for the logarithmic broadening of the string width as a function of the inter-quark separation [11]. This phenomenon was first observed long time ago in the Z 2 3D gauge theory [12] and only recently, using the very efficient Lüscher-Weisz algorithm [9], has also been observed in a non-abelian Yang-Mills theory [13,14].
In the last few years there has been a substantial progress in lattice simulations in measuring various properties of the flux tube, in particular the interquark potential and the energy of the excited string states (see e.g. [15,16,17,18,19,20,21,22,23,24,25,26]) which is now sensible to the first few subleading corrections of the free-field infrared limit of the effective action. The latter is the sum of all the terms respecting the symmetry of the system, which in an Euclidean space is SO(2) × ISO(D − 2). The first few terms are S = S cl +S 0 [X]+σ d 2 ξ c 2 (∂ α X · ∂ α X) 2 + c 3 (∂ α X · ∂ β X)(∂ β X · ∂ α X) +S b +. . . , (1.4) where S b is the boundary action characterizing the open string. Indeed quantum field theories on space-time manifolds with boundaries require, in general, the inclusion in the action of contributions localized at the boundary. If the boundary is a Polyakov line in the ξ 0 direction, on which we assume Dirichlet boundary conditions, the first few terms are
S b = dξ 0 b 1 (∂ 1 X · ∂ 1 X) + b 2 (∂ 1 ∂ 0 X · ∂ 1 ∂ 0 X) + b 3 (∂ 1 X · ∂ 1 X) 2 + . . . . (1.5)
Of course the addition of these terms modifies the spectrum of the physical states. For instance the interquark potential (1.3) becomes, at first order in b 1 [9],
V (R) = σR − π(D − 2) 24R − b 1 π(D − 2) 6R 2 + O(1/R 3 ) . (1.6)
In 2004 Lüscher and Weisz [27] noted that the comparison of the string partition function on a cylinder (Polyakov correlator) with the sum over closed string states in a Lorentz (or rotation) invariant theory yields strong constraints (called open-closed string duality). In particular they showed in this way that b 1 = 0. This property was then further generalized in [28]. It was subsequently recognized that an essential ingredient of these constraints is the Lorentz invariance of the bulk space-time [29,30,31]. The confining string action could be regarded as the effective action obtained from the underlying Yang-Mills theory of the confining vacuum in presence of a large Wilson loop by integrating out all the massive degrees of freedom [31]. This integration does not spoil the original Poincaré invariance of the underlying gauge theory, however this symmetry is no longer manifest, being spontaneously broken. As expected, it is realized through non-linear transformations of the X i 's. The effective string action (1.4) should be invariant under the infinitesimal Lorentz transformation in the plane (α, j)
δX i = − αj δ ij ξ α − αj X j ∂ α X i . (1.7)
For instance, if we apply the transformation (1.7) to the term S b 1 proportional to b 1 in the boundary action S b = S b 1 + S b2 + . . . we get at once
δ(S b 1 ) = −b 1 1i dξ 0 ∂ 1 X i + higher order terms = 0 , (1.8)
thus such a term breaks explicitly Lorentz invariance, hence b 1 = 0. In a similar way [31] it is possible to show that b 3 = 0 . On the contrary the b 2 term is compatible with Lorentz invariance provided we add an infinite sequence of terms generated by the non-linearity of the transformation. The associated recursion relations can be easily solved and the final expression can be written in a closed form [32]
S 2 = b 2 dξ 0 ∂ 1 ∂ 0 X · ∂ 1 ∂ 0 X 1 + ∂ 1 X · ∂ 1 X + (∂ 1 ∂ 0 X · ∂ 1 X) 2 (1 + ∂ 1 X · ∂ 1 X) 2 . (1.9)
It is easy to construct in this way boundary terms of higher order [32]. Actually this procedure was first applied to the bulk action and it was shown that the requirement of Lorentz invariance of the infrared free-field limit (1.1) generates the whole Nambu-Goto (NG) action [30,31,33]. In the latter reference this method was generalized to the construction of the effective action of higher dimensional extended objects as D-branes on which other massless modes, besides the X i 's, are propagating. It can also be used to construct the allowed bulk corrections to the Nambu-Goto action [34,35,36]; further informations on the bulk corrections of the NG can be found working in a gauge where the Lorentz invariance is manifest [37,38] (see also a general discussion on this argument in [39]). A formal light-cone quantisation of the Nambu-Goto action [40] suggested a simple Ansatz for the energy spectrum of the closed string of length R
E (n,n) (R) = σ 2 R 2 + 4πσ n +n − D − 2 12 + 2π(n −n) R 2 ,(1.10)
where the integers n,n define the total energy 2πn/R (2πn/R) of the left (right) moving massless phonons. Similarly for the open string with fixed ends, where n =n, one has
E n (R) = σ 2 R 2 + 2πσ n − D − 2 24 .
(1.11)
We refer to (1.10) and (1.11) as the exact Nambu-Goto spectrum even if we know that this Ansatz is incompatible with Lorentz invariance in D < 26 [41], except maybe in D = 3 [42]. For large enough R one can expand (1.10) and (1.11) in powers of 1/σR 2 . Lattice simulations of confining gauge theories in 2+1 and 3+1 dimensions show that the ground state and the first excitations of the confining string have an energy spectrum very close to that of NG. This suggests that the effective action constructed along the lines illustrated above can be considered as small perturbations of NG action that can be evaluated using a suitable regularization and a standard expansion in perturbative diagrams [27,43]. It turns out that only first order calculations on the parameters c i and b i are practically feasible.
Recently, a new powerful non-perturbative method has been introduced in this context [38]. It is based on the study of the S matrix describing the scattering of the quanta of the string excitations, the phonons, in the word-sheet. Assuming a reasonably simple form for S it turns out that the system is quantum integrable [44], hence one can apply the method of thermodynamic Bethe Ansatz (TBA) to calculate the non-perturbative spectrum of the effective string. Taking the simplest form of the S matrix and some assumptions on the way of interacting of the phonons with different flavours (i.e. transverse indices), it turns out the spectrum coincides with the exact NG spectrum of the closed string [44]. This method was also applied to describe some apparent deviations of the spectrum of the closed string [45].
In the present paper we re-derive the NG spectrum starting from the observation that the first non-Gaussian correction of the string action (1.4), once the coefficients c i assume the values required by Lorentz invariance of the target space, namely c 2 = 1 8 and c 3 = − 1 2 , coincide with the composite field TT , where T andT are the chiral components of the energy momentum tensor T αβ . Thus the effective string action is, at this perturbative order, a two-dimensional integrable quantum field theory formed by a conformal field theory (the infrared Gaussian limit) perturbed by TT . We do not need any further assumption to derive, through the TBA, the NG spectrum.
We also show that a similar spectrum emerges from a general class of CFT's perturbed by TT . The energy levels for the identity primary field and its descendents coincides with (1.10) and (1.11) once one replaces D − 2 with the central charge c. The level degeneracy differs in an intriguing way: it is know that the degeneracy of the closed string grows exponentially for large E as ∼ exp(E/T H ), where T H = 3σ/π(D − 2) -the Hagedorn temperature-coincides with the inverse of the distance R c where the ground state of the NG spectrum develops a tachyon, i.e. where the argument of the square root in (1.10) vanishes. A similar relationship between the position of the tachyonic singularity of the ground state and the degeneracy of highly excited states holds for general CFT's. We shall check it in the critical Ising model where this degeneracy can be calculated exactly.
The interplay between quantum integrability and Lorentz invariance in the target space is an intriguing issue: the first non-Gaussian contribution of the string action is an integrable perturbation only if c 2 /c 3 = − 1 4 as required by Lorentz invariance, however this does not imply that the generated NG spectrum agrees with that of a Lorentz-invariant string theory. As already mentioned, a pure NG spectrum is compatible with Lorentz invariance in the Minkowski target space only for D = 26 (and perhaps D = 3) and perturbative calculations show that the NG spectrum deviates from that of a Lorentzinvariant string already at the order 1/R 5 in D > 3 dimensions , and starting at the order 1/R 7 non-universal terms are expected to contribute to the energy levels [43,38,39]. These contributions could be inserted in the TBA approach by assuming that at shorter distances other perturbations contribute, besides TT .
Actually the leading deviation of the NG spectrum comes from the boundary action (1.9). This is also the most significant from a phenomenological point of view, being associated with the first non-NG correction of the interquark potential. We have indeed [27,43]
V (R) = σR − π(D − 2) 24R − 1 2σR 3 π(D − 2) 24 2 − b 2 π 3 (D − 2) 60R 4 + O(1/R 5 ) ,(1.12)
where the third term comes from the c 2 and c 3 terms in (1.4) [27]. This deviation of the interquark potential has been already observed in lattice gauge theories and the b 2 parameter has been evaluated [46,32].
In this paper we introduce the TBA equations describing these boundary effects by generalizing the approach of [38] to an infinite strip with Dirichlet boundary conditions. We find that also in this case the equations are solvable explicitly and the energy spectrum is given by (1.11). It is also easy to derive the corrections of the open string NG spectrum due to the boundary constants b i and we recover in particular eq. (1.12).
Even though the non-perturbative method we used is very powerful and the calculations of the level corrections due to these coupling constants could be easily pushed to any perturbative order, we do not think that this formulation of the effective string is ultraviolet complete. The reason is that the whole spectrum of the theory includes an infinite set of negative energy levels: because of the square root in eq. (1.10), the complete energy spectrum is actually {±E (n,n) (R)}. The theory can be fermionized and one may assume that the sea of negative energy levels is completely filled, however we did not succeeded in finding the zero-point energy produced by this sea, because of the huge degeneracy of the string states. As far as this problem is not completely solved, one should regard this formulation as an effective theory.
In the next Section we provide an elementary calculation of the TT nature of the quartic term along with some quantum check. In Section 3 we describe in detail the exact S matrix for a critical RG flow of the Ising model in the limit of the massless phonons, following the method of Aliosha Zamoldchikov in the study of the flux between the tricritical Ising model and the critical Ising model and obtain the spectrum of the TT perturbed Ising model in a closed form. In Section 4 we study the degeneracy of the spectrum and compare it with the degeneracy of the string. Then in Section 5 we put the theory in an infinite strip, define a consistent reflection factor and, in Section 6, solve the boundary TBA for the open string in the case D = 3 and compare it with the perturbative calculations. Finally, in Appendix A, we show that Nambu-Goto like spectra emerge from a large class of TT perturbed CFT's and Section 7 contains our conclusions with a summary of the main results.
The composite perturbation TT and its expectation value
The energy-momentum tensor of the free-field theory (1.2) can be written as
T αβ = ∂ α X · ∂ β X − 1 2 δ αβ (∂ γ X · ∂ γ X) . (2.1)
Note that this tensor is symmetric, traceless and conserved, as it should. Once we put in eq. (1.4) the values of c 2 and c 3 prescribed by Lorentz invariance, we have, as anticipated in the Introduction,
S = S cl + S 0 [X] − σ 4 d 2 ξ T αβ T αβ + S b + . . . (2.2)
In two-dimensional CFT it is useful to introduce the chiral components T zz = 1 2 (T 11 − iT 12 ) and Tzz = 1 2 (T 11 + iT 12 ) and use the normalized quantities T = −2πσT zz ,T = −2πσTzz in such a way the operator product expansion begins with
T (z)T (w) = D − 2 2 1 (z − w) 4 + . . . (2.3)
and similarly forT . Thus at the end we have
S = S cl + S 0 [X] − 1 2π 2 σ d 2 ξ TT + S b + . . . (2.4)
There is a consistency check based on some general properties of the expectation value of the composite field TT pointed out by Sasha Zamolodchikov [47] for any two-dimensional quantum field theory. In the particular case of a CFT on a infinite strip we have
TT = T T . (2.5)
Under a conformal mapping z → w = f (z), T transforms as
T (z) = df dz 2 T (w) + c 12 {f, z} , (2.6) where {f, z} = −2 √ f d 2 dz 2 1 √
f is the Schwarzian derivative and c the central charge. Starting from the observation that T = 0 in the complex w plane and that the transformation f (z) = exp(πz/R) maps conformally the infinite strip into the upper half plane m(w) ≥ 0, we get, as it is well known,
T strip = T strip = c 12 {f, z} = − c 24 π R 2 , (2.7)
therefore in the infinite strip limit L → ∞ we should find
d 2 ξ TT = RL T 2 strip . (2.8)
On the other hand, the vacuum expectation value of the quartic term of the string action on a cylinder -i.e. the correlator of two Polyakov lines-has been calculated many years ago [48] in the ζ-function regularization and more recently [27] in the dimensional regularization. The result is
− 1 2π 2 σ d 2 ξ TT = − 1 2π 2 σ (D − 2)π 4 L 24 2 R 3 (D − 4)E 2 (τ ) 2 + 2E 4 (τ ) , τ = i L 2R , (2.9)
where L is the circumference of the cylinder. We do not need the explicit expression of the Eisenstein series E 2 and E 4 because in the infinite strip limit L → ∞ they become E 2 = E 4 = 1. In this limit we recover eq. (2.8).
We conclude this Section with the following remark. The free-field action (2.2), once perturbed with TT , has a new energy-momentum tensor which is no longer the one defined in (2.1) as it includes a quartic polynomial in the derivatives of X i . Inserting this new TT perturbation generates a new energy-momentum tensor made with a polynomial of higher degree, and so on. It would be interesting to see whether this kind of recursion generates the same sequence produced by the request of Lorentz invariance in the target space.
The exact S matrix for massless flows
More than twenty years ago, Aliosha Zamolodchikov [49] has proposed an interesting variant of the thermodynamic Bethe Ansatz [50] and the exact S matrix approach to twodimensional quantum field theory describing interpolating trajectories among pairs of nontrivial CFTs. The simplest instance discussed in [49], concerns the line of second order phase transitions connecting the tricritical Ising model (TIM) -identified with the conformal minimal model M 4,5 perturbed by φ 13 -to the Ising model (IM). The latter system corresponding to the CFT M 3,4 underlying the infrared fixed point of the RG flow.
Massless excitations confined on a infinite line or a ring naturally separate into right and left movers. In this simple example, only one species of particles is present. The rightright and left-left mover scattering is trivial, while the left-right scattering is described by the amplitude
S(p, q) = 2σ + ipq 2σ − ipq , (3.1)
where the real parameter σ sets the scale; it plays the role of string tension in the energy spectrum. In (3.1) p is the momentum of the right mover and −q the momentum of the left mover. In the limit σ → ∞, S(p, q) → 1, right and left mover excitations decouple and the scale invariance of the model is fully restored at σ = ∞. Starting from the S matrix (3.1), Zamolodchikov was then able to derive the thermodynamic Bethe Ansatz equations for the vacuum energy of the theory defined on a infinite cylinder with circumference R. The relevant equations are
(p) = Rp − ∞ 0 dq 2π φ(p, q)L(q),¯ (p) = Rp − ∞ 0 dq 2π φ(p, q) L(q) , (3.2)
where (p) and¯ (p) are the pseudoenergies for the right and the left movers, respectively,
φ(p, q) = −i∂ q ln S(p, q) ,(3.3)
and
L(p) = ln(1 + e − (p) ) ,L(p) = ln(1 + e −¯ (p) ) . (3.4)
The ground state energy is
E (TBA) (R) = − π 6R c( √ σR) = − ∞ 0 dp 2π (L(p) +L(p)) , (3.5) where c(R √ σ) is the (flowing) effective central charge with c UV = c TIM = c(0) = 7 10 , c IR = c IM = c(∞) = 1 2 . (3.6)
The energy levels obtained through the TBA method are automatically defined with respect to the vacuum energy in infinite space,
lim R→∞ E (TBA) 0 (R) = 0 . (3.7)
Within a pure two-dimensional setup, the normalization (3.7) is perfectly acceptable but it differs from the perturbative definition about the ultraviolet fixed point by a bulk vacuum contribution F 0 R which is not analytic in the perturbing parameter [50] E (TBA) 0
(R) = − πc UV 6R − F 0 R + regular terms , (R 0) . (3.8) (For the TIM → IM massless flow, F 0 = 2σ.)
The alternative normalisation for the energy levels
E n (R) = E (TBA) n (R) + F 0 R ,(3.9)
agrees with the definition coming from the short distance expansion, it seems to be a more natural choice in view of a possible embedding in an higher dimensional space and it highlights the similarity between the bulk energy, exactly computable within the TBA scheme, and the linear term in the quark-anti-quark potential (1.3), whose origin traces back to the classical contribution S cl to the effective string action (1.4)
E 0 (R) = F 0 R − πc IR 6R + . . . , ( √ σR 1) .
(3.10)
In the far infrared limit, equations (3.2) lead to an exact asymptotic expansion for E (TBA) (R) [49,51] where f (TBA) (t) is the scaling function [49] and t = π/(12σR 2 ). The first three terms in (3.11) reproduce the large σ expansion of the effective action
f (TBA) (t) = 1 2π RE (TBA) 0 (R) = − 1 24 − 1 48 t − 1 48 t 2 + −5S σ = S IM − 1 2π 2 σ d 2 ξ TT , (3.12)
where S IM is the Ising model CFT action. Correspondingly, the scaling function on the cylinder admits the perturbative expansion However, what had not been realised until the work [38] was that certain TBA equations lead to the full Nambu-Goto closed string spectrum. One of the main achievements of the present paper is to generalize the results of [38] to other important families of models by showing that the corresponding TBA spectra are a direct generalization of (1.10) and that the leading part S 1 (p, q) of the Zamolodchikov's S matrix (3.1) at large σ
f (pert) (t) = − c IR 12 + c IR 24 2 α − c IR 24 3 α 2 + O(α 3 ) = − 1 24 − 1 48 t − 1 48 t 2 + O(t 3 ) , (α = −48t) .S(p, q) = e ipq/σ−i(pq/σ) 3 /12+... = S 1 (p, q)e −i(pq/σ) 3 /12+...
(3.14)
selects precisely the zero trascendentality terms in (3.11) which form the large R expansion of the NG ground state energy. Therefore, following [38], we replace the kernel in (3.2) with φ(p, q) = −i∂ q ln S 1 (p, q) = p/σ , (3.15) the resulting TBA equations are
(p) = Rp − p σ C dq 2πL (q) = Rp + p σ C dq 2π q∂ qL (q) , (p) = Rp − p σ C dq 2π L(q) = Rp + p σ C dq 2π q∂ q L(q) ,(3.16)
with L(q) = ln C (1 + λ ± e − (q) ) ,L(q) = lnC(1 + λ ± e −¯ (q) ) .
(3.17)
In (3.17) λ + = 1 selects the descendents of the identity and energy primary fields, while λ − = −1 selects the conformal family of the spin field [52]. ln C is the continuous branch logarithm, C andC are certain integration contours running from q = 0 to q = ∞ on the real axis for the ground states in each subsector λ ± , but for excited states they circle around a finite number of poles {q i } and {q i } of ∂ q L(q) and ∂ qL (q) (see Figure 1) : we find the constraints
(q j ) = iπ(2n j + (1 − λ ± )/2) ,¯ (q j ) = −iπ(2n j + (1 − λ ± )/2) , (n j ,n j ∈ N) .κ = 1 + 4 πκσR 2 Li 2 (−λ ± ,C) ,κ = 1 + 4 πκσR 2 Li 2 (−λ ± , C) .
(3.20)
In (3.20) Li 2 (z, C) denotes the continuous branch dilogarithm (see e.g. [53]): The result (3.25) reproduces precisely the zero trascendentality coefficients appearing in (3.11) Although, as it will be discussed in greater detail in Section 4 below and similarly to the cases studied in [38], this simple model of quantum field theory does not possess a standard ultraviolet fixed point, it still seems reasonable to identify the bulk contribution with the linear term in the short distance expansion
Li 2 (z, C) = − C dq 2π ln C (1 − ze −q ) = Li 2 (z) + 4π 2 m − i2πnln(z) , (m, n ∈ N) (3.21) with Li 2 (−1) = − π 2 12 , Li 2 (1) = π 2 6 , Li 2 (−1, C) = − π 2 6 (c IR − 24h (0,(R) + σR = σR(κ +κ − 1) = σ 2 R 2 + σ π (Li 2 (−λ ± , C) + Li 2 (−λ ± ,C)) + Li 2 (−λ ± , C) − Li 2 (−λ ± ,C) 2πR 2 = σ 2 R 2 + 4πσ n +n −c IR 12 + 2π(n −n) R 2 ,(3.f (TBA) (t) = 1 2π RE (TBA) (0,0) (R) = 1 24t − 1 (24t) 2 − c IR 144t = − 1 24 − 1 48 t − 1 48 t 2 − 5 192 t 3 − 7 192 t 4 − 7 128 t 5 − 11 128 t 6 + . . .∂ R E (TBA) (0,0) (R) −F 0 + · · · = −σ + . . . , (forc IR = 0) . (3.28)
Adding F 0 R = σR, a nice match with the Nambu-Goto formula (1.10) at D − 2 =c IR is finally obtained:
E (n,n) (R) = E (TBA) (n,n) (R) + σR . (3.29)
Naively, we may be tempted to discard completely the negative energy sectorẼ (n,n) (R) = −E (n,n) (R), however, the two branches are not completely disconnected as there is a spectral singularity (exceptional point) at the tachyonic critical point
R c = πc IR 3σ ,(3.30)
in the ground state n =n = 0 and, for the other levels, singular points at complex values of R. The appearance of the negative energy sector, absent in the original work [49], is a signal of the somehow pathological nature of the -explicitly solvable-CFT perturbation considered. At the level of the TBA this fact is a direct consequence of the non-localized form of the scattering amplitude S 1 (p, q) used for the kernel (3.15). Still, even with a range of validity restricted to low energy, the appearance of the NG spectrum in the framework of two-dimensional integrable modes is a very striking result that may have a highly non trivial impact to the study of effective strings in confining gauge theory. The results obtained in this Section, although discussed from a slightly different perspective, heavily rely on [38]. Our model differs from those studied in [38] in two ways:
• In [38] Bose statistics was used for the derivation of the TBA equations. The Ising model and almost all the known integrable models obey instead an exclusion principle in the momentum space even when they are unmistakably associated to Bose type Lagrangians as, for instance, in the quantum affine Toda field models [54]. A well known exception is the theory of a single (non compactified) free Bose field which indeed corresponds to infrared limit of the the D = 3 case of [38]. However, even this example does not really represent an exception since an alternative Fermi type TBA with an additional delta-function in the kernels may fully replace the original equations. The change of variable is
B (p) → F (p) + ln(1 + e − F (p) ) , − ln(1 − e − B (p) ) → ln(1 + e − F (p) ) . (3.31)
• The general case discussed in [38], consists in D − 2 species of particles with left-right mover scattering amplitude S ij (p, q) = S 1 (p, q), (i, j = 1, 2, . . . D − 2). Thus microscopically the D − 2 species are actually indistinguishable as their mutual interaction is totally independent from their flavour i and j, a property difficult to understand on physical grounds.
In the following Section we study the degeneracy of the string levels and in Appendix A, starting from a more general family of exact scattering theories, we shall describe the generalization of these results to more complicated conformal field theories.
Spectrum degeneracy
The degeneracy of the energy levels (3.29) is given by the number of ways of writing n and n in (3.24) i.e. the number of decompositions of n andn into distinct integer summands without regard to the order. This is of course the degeneracy of a free fermionic system on a circle. The generating function is
∞ n=0 ϕ(n)q n = ∞ n=1 (1 + q n ) = 1 ∞ n=1 (1 − q 2n−1 )
.
(4.1)
The asymptotic behaviour of the level degeneracy for large n andn is known to be
ϕ(n)ϕ(n) ϕ(n) 2 = 1 16 √ 3n 3 e 2π √ n/3 . (4.2)
For large n n the energy (3.29) is E √ 8πσn, so the Ising degeneracy ρ I (n) is
ρ I (n) = 3 πT H 3E 3 e E/T H = ρ I (E) dE dn , (4.3)
where
T H = 3σ πc IR , (4.4)
is the Hagedorn's temperature T H = sup(T (E)) with T (E) = 1/∂ E lnρ I (E) and c IR = 1 2 . Comparison with the tachyonic singularity (3.30) at R c we obtain, as anticipated in the Introduction, R c T H = 1. Notice that the degeneracy of the energy levels of the closed string in D dimensions is asymptotically
ρ D (n) = 12(D − 2) D πT H 3E D+1 e E/T H , (4.5)
where T H differs from T H by the substitution c IR → D − 2. Further details on this degeneracy as well as its thermodynamic implications in lattice gauge theory at finite temperature can be found in Appendix B of [55].
The infinite strip
Consider a single quantum particle confined on a segment of length R. The standard quantization condition for the momentum p of the particle is
e i2p i R R α (p i )R β (p i ) = 1 , (5.1)
where δ α (p) = −ilnR α (p) and δ β (p) = −ilnR β (p) are the contributions to the total phase shift from the reflections on the left and right boundary, respectively. Although, for an 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 interacting integrable two-dimensional quantum field theory confined on a space segment of length R equation (5.1) becomes exact only in the asymptotically large R limit, it reveals that one of the main ingredients for the computation of the spectrum is, beside the exact bulk S matrix a consistent reflection factor R(p). For massive integrable quantum field theories, the basic axiomatic constraints linking the boundary reflection factor R(p) to the two body S matrix amplitude were discussed in [66,67,68]. The generalization of these results to a generic massless perturbed conformal field theories is a nice and partially open problem that certainly deserves further attention. However a full discussion of this topic would take us too far afield, and we shall postpone it to the future. For the time being, we restrict the discussion to the D = 3 case of [44], i.e. the theory of a single Bose field in two dimensions with left-right scattering amplitude
11S 1 (p, q) = e ipq/σ . (5.2)
Partially based on TIM → IM boundary flows discussed in [69], we identify the relevant constraints to be R(p)R * (p) = 1 ,
R(p) = R 0 (p) ∞ j=1 R (2j−1) δ j (p) , (5.6) with R (m) δ (p) = e (ip) m δ , (m odd) . (5.7)
The functions (5.7) satisfy the simpler equation
R (m) δ (p)R (m) δ (−p) = 1 . (5.8)
The signs of the real parameters δ's appearing (5.6) are not constrained by (5.3) or (5.4). However, as it will clear from the results described in Section 6.2 below, for left-right symmetrical boundary conditions, the terms with the largest power 2j − 1 ≥ 3 of p and δ j = 0 appearing in (5.6) should have a coefficient δ j > 0, to ensure a proper convergence of the TBA integrals and consequently also the validity of the boundary TBA approach itself.
In conclusion, the number of possible reflection factors is infinite. Since, even for a simple conformal field theory such as the M 3,4 model, the set of all possible boundary conditions corresponding to the superposition of Cardy's boundary (pure) states [70] is infinite dimensional, such a proliferation of free parameter reflection factors is not totally surprising.
The boundary TBA
The relevant TBA equation for our simple system, obeying Bose type statistics 3 and defined on a infinite strip of size R and boundary conditions (α, β), is [71,72] (p) = 2Rp + Λ(p)
+ p σ C dq 2π L(q) ,(6.
Basic boundary conditions
Let us first consider the basic boundary conditions
Λ(p) = ln (R 0 (−ip)/R 0 (ip)) = 0 ,(6.4)
the TBA equation reduces to
(p) = 2Rp + p σ C dq 2π L(q) . (6.5)
The same logical steps described in Section A can be repeated to find the following simple algebraic constraint
E ((p) = 2Rp + lnS 1 (p, −ip j ) + p σ ∞ 0 dq 2π L(q) ,(6.8)
where q = −ip j is the branch point of L(q) corresponding to (−ip j ) = −i2πn j , (n j = 1, 2, . . . ). At large R, setting (ip j ) = i2πn j on the LHS of (6.8) and dropping the exponentially subdominant term on the RHS
i2πn j i2Rp j + lnS 1 (ip j , −ip j ) = i2Rp j + lnS 1 (p j , p j ) , (6.9) or e i2Rp j S 1 (p j , p j ) = e i2Rp j R 0 (p j )R 0 (p j ) 1 . (6.10)
The result (6.10) nicely fits equation (5.1).
More general boundary conditions
Let us now consider the case
R(p) = R 0 (p)R(1)
δ 1 (p) = e ip 2 /2σ e iδ 1 p , (6.11) since Λ(p) = ln (R(−ip)/R(ip)) = 2pδ 1 , (6.12)
we see that these boundary conditions simply correspond to a shift R → R + δ 1 , the resulting exact spectrum is
E (TBA) n (R, δ 1 ) = E (TBA) n (R + δ 1 ) . (6.13)
Note that this case corresponds to the b 1 term in the boundary action (1.5). This correction was first calculated at first order in b 1 in [9] using the ζ−function regularization and in [27] using dimensional regularization. In the latter reference it was also noted that this boundary term corresponds to a shift in R and that this rule actually extends to the next order in b 1 . In our approach this "shift" property is valid at any order of δ 1 and the precise relation between δ 1 and b 1 is δ 1 = −4b 1 .
Further, we know that Lorentz symmetry of the target space, as pointed out in (1.8), makes this term inconsistent, yet the TBA approach is perfectly consistent. This clearly shows that the quantum integrability does not imply Lorentz invariance on the target space, as anticipated in the Introduction.
Consider now the case
R(p) = R 0 (p)R (3) δ 2 (p) = e ip 2 /(2σ) e −ip 3 δ 2 , Λ(p) = ln (R(−ip)/R(ip)) = 2p 3 δ 2 . (6.
14)
The corresponding TBA equations are
(p) = 2Rp + 2p 3 δ 2 + p σ C dq 2π L(q) . (6.15)
The solution to (6.15) is of the form
(p) = 2Rκp + 2p 3 β , (6.16) with κ + p 2 R β = 1 + p 2 R δ 2 + 1 2Rσ C dq 2π ln 1 − e −2Rκq−2q 3 β . (6.17)
Then it must be exactly β = δ 2 and the TBA yields an exact equation for the energy in terms of κ = κ(
√ σR, δ 2 /R 3 ) κ = 1 + 1 2Rσ C dq 2π ln 1 − e −2Rκq−2q 3 δ 2 ,(6.18)
which can be easily solved numerically for R > 0 and δ 2 > 0. Yet, in the literature there are interesting, large R, asymptotic corrections to the NG formula which we can match very easily. In fact, we can expand in the form ln 1 − e −2Rκp−2p 3 δ 2 = ln 1 − e −2Rκp + 2p 3 δ 2 e 2κpR − 1 + . . . (6.19) obtaining, for δ 2 /R 3 small,
κ = 1 + 1 4πσR − Li 2 (1,C) 2Rκ + 3δ 2 4R 4 Li 4 (1,C) + . . . ,(6.20)
which has solution
κ = 1 2 1 + 1 − 1 2πσR 2 Li 2 (1,C) + 1 4πσR 3δ 2 4R 4 Li 4 (1,C) + . . . . (6.21)
The continuous branch extension of Li 4 (1,C) is given by [53] Li 4 (1,C) = Li 4 (1) Table on page 9 of [43], provided we set D = 3 and identify δ 2 = −4b 2 .
+ 8 3 π 4 i (n i ) 3 = π 4 90 + 8 3 π 4 i (n i ) 3 , n i ∈ N ,(6.
Returning to the positivity issue, briefly mentioned at the end of Section 5, notice that the integral appearing in equation (6.15) is divergent for δ 2 < 0. It is therefore important to check the sign of the latter coefficient obtained from numerical simulations.
The n = 0 state of the spectrum (6.23) was compared with numerics in the case of the three dimensional SU (2) lattice gauge theory in [46]: (σ) 3/2 b 2 −0.015(6)(6). b 2 was also evaluated in [31] for a class of holographic confining gauge theories and also in this case, with Dirichlet's boundary conditions, the b 2 coefficient is negative. Thus, in both cases, the TBA equation (6.15) should provide a qualitatively good description of the deviation of the Nambu-Goto spectrum (1.11), caused by the presence of b 2 -perturbed boundaries, over a wide range of 1/σR 2 about the infrared fixed point.
Further, an high precision numerical simulation of the three dimensional Ising gauge model was reported very recently in [32]. The agreement with the theoretical prediction turned out to be very good allowing a precise estimate of the boundary parameter b 2 . The numerical outcome, (σ) 3/2 b 2 0.032 (2), leads now to a negative sign for δ 2 . We interpret this result as a clear signal of the presence of additional (infrared subleading) contributions associated to extra terms with higher powers of p in the boundary factor Λ(p) and/or in the TBA convolution kernel. We shall postpone a more complete discussion on this issue and a comparison between TBA and Monte Carlo results to the future.
Conclusions
In this paper we pointed out that the effective string theory describing the confining colour flux tube which joins a static quark-antiquark pair can be seen as a two-dimensional CFT of central charge D−2 perturbed by the composite field TT made with the energy momentum tensor T . This perturbation is quantum integrable and the spectrum can be calculated with the TBA, as first noted in [44]. We generalized this result to a large class of conformal models. In the case of periodic boundary conditions the energy levels E (n,n) (R) are labeled by two integers n andn which depend on the monodromy of the dilogarithm in the complex plane of the momentum p. In a generic ADE system these energies can be parametrised in the form E (n,n) (R) = σR + E +Ē, where the two quantities E andĒ obey the following consistency conditions
E = − π(c IR − 24n) 12(R +Ē/σ) ,Ē = − π(c IR − 24n) 12(R + E/σ) ,(7.1)
wherec IR is the effective central charge. The solution of these two algebraic equations is exactly the NG spectrum. We then discussed the degeneracy of these states which is growing exponentially for large n andn. Similar conclusions can be drawn for the open string case, where we wrote and solved the boundary TBA. We found that the reflection factor may depend on a set of arbitrary parameters which are associated to the coupling constants of the boundary string action. The deviation of the NG spectrum due to these terms can be easily calculated, at least at the first order in these coupling constants, and it turns out that the results coincide with those of the standard perturbative calculations. These results give a novel perspective on the TBA, and will hopefully lead to a new way to study the interquark potential by means of nonlinear integral equations, exact S matrices and form-factors for correlation functions. One of the most interesting open questions of the present approach is tied to the fact that the above equations for each pair n,n admit two solutions ±E (n,n) (R). Thus this theory has an infinite set of negative energy levels. Even if the theory can be fermionized and we may assume that the sea of negative energy states is completely filled we did not succeed in evaluating the zero-point energy associated to it and its possible effect on the NG spectrum. One possible way out is to assume that at a given perturbative order in the 1/σR 2 expansion of the NG spectrum some other irrelevant operator starts to contribute. This is in particular what happens in the massless flow from the tricritical Ising model to the critical one, which was the starting point of our analysis.
As a final remark we notice that TBA equations -both with and without boundarieshave emerged in the context of N = 4 super Yang-Mills for the study of the quark-antiquark potential [73,74] and gluon scattering amplitudes [75]. The latter quantities are equivalent to light-like polygonal Wilson loops and thus correspond to the area of minimal surfaces in AdS 5 in the classical string theory limit. Although there are some similarities between the current setup and those of [73,74] and [75], there are also important differences in the underlying physics and the analytic properties of the corresponding TBA equations and we are currently unable to identify a precise link between the results of [38], further developed here, and these important preceding works on AdS 5 /CF T 4 .
Acknowledgments -
A. The ADE general case
The analysis described in Section 3 can be immediately generalised to any perturbed conformal field theory with known exact S matrix description, as for example, the massive theories represented by the diagonal reflectionless ADE-related scattering models of [50,56]. The TBA equations are
i (p) = Re i (p) − N j=1 C j dq 2π φ ij (p, q) L j (q) ,¯ i (p) = Re i (p) − N j=1 C j dq 2π φ ij (p, q)L j (q) , (A.1)
where e i (p) = p 2 + m 2 i is the dispersion relation of the i-th particle and N is the rank of the corresponding ADE algebra. The kernels are
φ ij (p) = −i∂ q ln S ij (p, q) , (A.2)
where S ij (p, q) are the S matrix amplitudes of [57,54,58,59] parametrised using the momenta p and q of the two particles involved in the scattering. In the ultraviolet regime m i R → 0 the TBA equations (A.1) show the decoupling of the pseudoenergies for right movers from the left mover ones
i (p) = Rm i p − N j=1 C j dq 2π φ ij (p, q) L j (q) ,¯ i (p) = Rm i p − N j=1 C j dq 2π φ ij (p, q)L j (q) .
(A.3) In this limit, the energy can be found exactly [50]. The set of pure numbersm i ≡ m i /m 1 and i = 1, . . . , N fix a relative scale among the particle species. They cannot be arbitrary numbers, but must be proportional to the components of the Perron-Frobeniuns eigenvector of the corresponding Cartan matrix. For the vacuum states all this has been analysed in [50,56], furthermore here we wish to take into consideration excited states as well [61,52,60] by introducing complex contours C i andC i for the continuous branch dilogarithm and fugacities {λ i } inside the statistical functions
L i (p) = ln C i (1 + λ i e − i (p) ) ,L i (p) = lnC i (1 + λ i e −¯ i (p) ) . (A.4)
The fugacities are all equal to unity for sectors related to the CFT identity operator, while they may assume different values for conformal families of other primary fields [62,63,52].
In the ultraviolet limit the energy is
E (TBA) (n,n) (R) = E +Ē , E = − π 12R c(C) ,Ē = − π 12R c(C) , (A.5)
where the constants
c(C) = 12 π N i=1 Rm i C i dp 2π L i (p) , c(C) = 12 π N i=1
Rm i C i dp 2πL i (p) , (A. 6) can be written in terms of the solutions to (A.1) and computed exactly using the dilogarithm trick [50,64]. Besides, they are easily related to the conformal central charge c IR and the conformal weights (h,h) of the primary fields The latter pair of algebraic equations for E andĒ can be easily solved, giving E (n,n) (R) = E +Ē + σR = ± σ 2 R 2 + 4πσ n +n −c IR 12 + 2π(n −n) R 2 . (A.14)
In conclusion, we have shown that the spectrum of Nambu-Goto with D − 2 =c IR emerges from a wide class of TBA models. Actually, we can generalize this analysis to many other interesting models, including infinite families of perturbed CFT theories described by nondiagonal S matrices and we suspect that (A.14) can be obtained for any CFT. We end this Section with a further observation. It was noticed in [47] that for any two-dimensional quantum field theory the expectation values of the composite field TT admits an exact representation in terms of the expectation value of the energy-momentum tensor itself. Using the standard CFT convention and setting T = −2πT zz ,T = −2πTzz , Θ = 2πT zz = 2πTz z , (A.15) the result of [47], written using the double integer labeling introduced in the previous sections, on the cylinder is n,n|TT |n,n = n,n|T |n,n n,n|T |n,n − n,n|Θ|n,n 2 . (A. 16) With the help of the following relations linking the expectation values of the energy-tensor components with the energy eigenvalues E (n,n) (R) and the total momentum of the state P (n,n) (R) = 2π(n −n)/R n,n|T yy |n,n = − 1 R E (n,n) (R) , n,n|T xx |n,n = −∂ R E (n,n) (R) , n,n|T xy |n,n = − i R P (n,n) (R) , (A. 17) we have [47] ∂ R (E 2 (n,n) (R) − P 2 (n,n) (R)) = − 2R π 2 n,n|TT |n,n . (A.18)
Here, we would like to remark that inserting espression (A.14) for the energy levels in (A.18) leads to n,n|TT |n,n = −π 2 σ 2 , (A. 19) exactly and independently from the particular state (n,n) under consideration. Finally, using (A. 19) in (A.16) gives n,n|Θ|n,n = π 2 σ 2 + n,n|T |n,n n,n|T |n,n .
(A.20)
Since, Θ = 0 corresponds to a conformal invariant theory, the field Θ can be identified with the CFT perturbing operator, thus the exact result (A.20) should contain fundamental information on the further contributions needed in ( 2.4) and (3.12), or in an arbitray TT perturbed CFT, to build the full action associated to the Nambu-Goto like spectrum (A.14).
in (3.11) of coefficients with nonzero trascendentality 1 at order O(t 3 ) and greater is a clear signal of contributions from other irrelevant operators [49] 2 .
Figure 1 :
1.e. with a higher power of π.2 It is interesting to observe that this new operator contributes exactly at the same order where the Lorentz-invariant effective string theory admits new non-NG terms[43,38,39] Possible excited state integration contours for the λ + sector.(p) = Rpκ ,¯ (p) = Rpκ ,(3.19)
( 1 ,(0, 0 ) = 0, h (1, 2 ) = 1 16 , h (1, 3 ) = 1 2
102320) − 24n) , (n = 0, 2, 3, . . . ) − π 2 6 (c IR − 24h (1,3) − 24n) , (n = 0, 1, 2, . . . C) = Li 2 (1) + 4π 2 n = − π 2 6 (c IR − 24h (1,2) − 24n) , (n = 0, 1, 2, . . . are the conformal dimensions of the primary fields of the minimal model M 3,4 . Therefore
with unitary constraint for R(p), andR(p)R(−p) = S 1 (p, p) ,(5.4)corresponding to the equality between the two scattering diagrams represented inFigure 2.Given the S matrix (5.2), the minimal solution to equations (5.3) and (5.4) is R 0 (p) = S 1 (p, p) = e ip 2 /(2σ) , (5.5) while multi parameter solutions have the general form
q) = lnC 1 − e − (q) , Λ(p) = ln (R α (−ip)/R β (ip)) ,
(C) = 1 − 24n, (n ∈ N). Considering only the positive energy solutions, we have with the NG open string spectrum (1.11) at D = 3. To check the full consistency with the BA equation (5.1), let us consider the single particle excited state
22) and the resulting modification to the Nambu-Goto spectrum turns out to beE (TBA) {n i } (R, δ 2 ) = σR(2κ − 2) = E (TBA) i ) 3 + . . . (6.23)with n i = 0, 1, . . . , n = i n i . Equation (6.23) matches the results displayed in the
This project was partially supported by INFN grants IS FI11, P14, PI11, the Italian MIUR-PRIN contract 2009KHZKRX-007 "Symmetries of the Universe and of the Fundamental Interactions", the UniTo-SanPaolo research grant Nr TO-Call3-2012-0088 "Modern Applications of String Theory" (MAST), the ESF Network "Holographic methods for strongly coupled systems" (HoloGrav) (09-RNP-092 (PESC)) and MPNS-COST Action MP1210 "The String Theory Universe".
r
c(C) =c IR − 24n , c(C) =c IR − 24n , (n,n ∈ N) , (A.7) andc IR = c IR − 24(h +h). The central charge and the conformal dimensions are those for the coset models[65]Ĝ 1 ×Ĝ 1 G 2 , G ∈ A N , D N , E N . (A.8)Let us now come to the announced generalisation of the analysis described in Section 3, and introduce the following variant of massless TBA equation for ADE systemsi (p) = Rpm i − pm φ ij (p, q) L j (q) , = R +Ē/σ ,r = R + E/σ ,(A.11) allows to recast equations (A.9) and (A.10) in the form (A.i (p) , (A.12) where, importantly, the c(C) and c(C) coincide with those already introduced in (A.7) and computable via equations (A.3) and (A.6). Finally, the following self-consistent constraints must hold E = − π(c IR − 24n) 12(R +Ē/σ) ,Ē = − π(c IR − 24n) 12(R + E/σ) . (A.13)
This is a conventional choice. The equivalent Fermi statistics TBA, obtained through the change of variable (3.31), would be fine as well.
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| []
|
[
"SYMMETRIES IN DIRECTED GAUSSIAN GRAPHICAL MODELS",
"SYMMETRIES IN DIRECTED GAUSSIAN GRAPHICAL MODELS"
]
| [
"Visu Makam ",
"ANDPhilipp Reichenbach ",
"Anna Seigal "
]
| []
| []
| We define Gaussian graphical models on directed acyclic graphs with coloured vertices and edges, calling them RDAG (restricted directed acyclic graph) models. If two vertices or edges have the same colour, their parameters in the model must be the same. We present an algorithm to find the maximum likelihood estimate (MLE) in an RDAG model, and characterise when the MLE exists, via linear independence conditions. We relate properties of a graph, and its colouring, to the number of samples needed for the MLE to exist and to be unique. We also characterise when an RDAG model is equal to an associated undirected graphical model and study connections to groups and invariant theory. We provide examples and simulations to study the benefits of RDAGs over uncoloured DAGs. | null | [
"https://arxiv.org/pdf/2108.10058v2.pdf"
]
| 237,266,460 | 2108.10058 | 2511c95c828b290afb1b81752992dd76f4de0253 |
SYMMETRIES IN DIRECTED GAUSSIAN GRAPHICAL MODELS
Visu Makam
ANDPhilipp Reichenbach
Anna Seigal
SYMMETRIES IN DIRECTED GAUSSIAN GRAPHICAL MODELS
We define Gaussian graphical models on directed acyclic graphs with coloured vertices and edges, calling them RDAG (restricted directed acyclic graph) models. If two vertices or edges have the same colour, their parameters in the model must be the same. We present an algorithm to find the maximum likelihood estimate (MLE) in an RDAG model, and characterise when the MLE exists, via linear independence conditions. We relate properties of a graph, and its colouring, to the number of samples needed for the MLE to exist and to be unique. We also characterise when an RDAG model is equal to an associated undirected graphical model and study connections to groups and invariant theory. We provide examples and simulations to study the benefits of RDAGs over uncoloured DAGs.
Introduction
The concept of a graph is widely used across the sciences [Fou12]. Graphs are a framework to relate entities: the vertices are the entities of interest, and the edges encode connections between them. A graph is given a statistical meaning in the study of graphical models [Lau96,MDLW18]. Each vertex represents a random variable, and the edges between variables reflect their statistical dependence [VP90]. In this paper, we study directed Gaussian graphical models, also called Gaussian Bayesian networks, or linear structural equation models with independent errors [Sul18]. Such models have been applied to cell signalling [SPP + 05], gene interactions [FLNP00], causal inference [Pea09], and many other contexts.
We define graphical models on directed acyclic graphs (DAGs) with a colouring of their vertices and edges. Vertices or edges with the same colour must have the same parameter values. Thus, the graph colouring imposes symmetries in the model. We call such models RDAG models, where the 'R' stands for restricted, cf. [HL08].
Our first motivation for RDAG models is that vertex and edge symmetries appear in various applications, such as in the study of longitudinal data [AFS16,VAAW16], or clustered variables [GM15,HL08]. The coloured directed graph gives an intuitive pictorial description of the symmetry conditions in the model.
Our second motivation is to decrease the maximum likelihood threshold, the minimum sample size required for the maximum likelihood estimate (MLE) to exist almost surely, see [Dem72]. In applications, it is desirable for the MLE to exist when there is only a small number of samples; i.e., for the maximum likelihood threshold to be small. Innovative ideas have been used to find maximum likelihood thresholds in graphical models [Buh93,DFKP19,GS18,Uhl12] and for estimating the MLE from too few samples [FHT08, WZV + 04]. Removing edges from a graph can lower the threshold [Uhl12,Lau96], but there is a trade-off: removing edges imposes more conditional independence among the variables. This is why, instead, we aim to decrease the maximum likelihood threshold by introducing symmetries.
We will use the following as our running example throughout the paper.
Example 0.1. Consider the coloured graph 1 3 2 , with blue (circular) vertices {1, 2}, black (square) vertex 3 and two red edges. The RDAG model is y 1 = λy 3 + 1 , y 2 = λy 3 + 2 ,
y 3 = 3 ,
where 1 , 2 ∼ N (0, ω) and 3 ∼ N (0, ω ), i.e. ω is the variance of blue vertices 1 and 2 and ω is the variance of black vertex 3. The third parameter λ is the regression coefficient given by a red edge. We will see that the MLE exists uniquely (almost surely) given one sample. For comparison, if we remove the colours the resulting model needs two samples for the MLE to exist. We use this example to model the dependence of two daughters' heights on the height of their mother, and we compute the MLE given some sample data, in Section 4.3.
As far as we are aware, RDAG models have not been defined before in the literature; we comment on some related models. The assumption of equal variances from [PB14] is the special case of an RDAG model, where all vertex colours are the same. Special colourings encode exchangeability between variables, or invariance under a group of permutations. A graphical model is combined with group symmetries in the directed setting in [Mad00] and in the undirected setting in [AM98,SC12]. RDAG models also relate to the fused graphical lasso [DWW14], which penalises differences between parameters on different edges, whereas in an RDAG model the parameters on edges of the same colour must be equal.
In this paper, we give a closed-form formula for the MLE in an RDAG model, as a collection of least squares estimators, see Algorithm 1. We characterise the existence and uniqueness of the MLE via linear algebraic properties of the sample data, see Theorem 4.4. We give upper and lower bounds on the threshold number of samples required for existence and uniqueness of the MLE in Theorem 5.3. Our results show that RDAG thresholds are less or equal to the DAG thresholds, and that high symmetry decreases the thresholds. Finally, we compare RDAG MLEs to uncoloured DAG MLEs via simulations in Section 6. Our results hold with an assumption on the graph colouring, which we call compatibility (Defintion 2.5). It is an open problem to extend our results to the non-compatible setting, as well as to directed graphs with cycles. It is also an open problem to find the exact maximum likelihood thresholds, see Problem 5.4.
The undirected analogue to RDAG models are the RCON models from [HL08]. Although a motivation for the graph colouring in RCON models is to lower the maximum likelihood threshold, there are relatively few graphs for which the threshold is known: colourings of the four cycle are studied in [Uhl12,§6], [SU10,§5], while an example with five vertices is [Uhl12, Example 3.2]. In certain cases, RDAG models are equivalent to RCON models. We determine precisely the conditions under which this occurs in Theorem 3.4. As a consequence, we obtain an entire class of RCON models where conditions for MLE existence and uniqueness can be found by appealing to our results on RDAGs. This paper has two appendices, where we explain some connections to invariant theory. A Gaussian group model [AKRS21] is parametrised by a group. In [AKRS21], the authors draw a dictionary between maximum likelihood estimation and stability notions in invariant theory. This dictionary allows for the transfer of tools from the algebraic subjects of representation theory and invariant theory to statistics: maximum likelihood thresholds were computed for matrix normal models in [DM21] and for tensor normal models in [DMW20].
We extend the dictionary between maximum likelihood estimation and stability notions to RDAGs in Theorem A.2. This requires us to extend the definitions of stability beyond the setting of a group action, see Definition A.1. While not evident in our final presentation, this perspective gave us the understanding needed to obtain many of the results in this paper and we would like to stress its importance for future work. We have far more tools at our disposal when a model is backed by a group action, i.e., when it is a Gaussian group model. We identify RDAGs that are Gaussian group models in Proposition B.2 and exhibit additional tools that one can use in such cases. The two appendices offer two alternative descriptions of the set of MLEs, see Propositions A.3 and B.6.
Contents
f Ψ (y) = 1 det(2πΨ −1 ) exp − 1 2 y T Ψy , y ∈ R m .
We refer to a multivariate Gaussian model by the set of concentration matrices in the model. So, a model is a subset of PD m , the cone of m × m positive definite matrices. We study statistical models in PD m via a set of invertible matrices. We define
(1)
M A := {a T a | a ∈ A},L Y (Ψ) = n i=1 f Ψ (Y i ),
where Y i is the ith column of Y . We work with the log-likelihood function log L Y , which has the same maximisers as L Y . The log-likelihood function can be written, up to additive and multiplicative constants, as
(2) Y (Ψ) = log det(Ψ) − tr(ΨS Y ), where S Y = 1 n n i=1 Y i Y T
i is the sample covariance matrix. Four possibilities arise when maximising the log-likelihood:
(a) Y unbounded from above (b) Y bounded from above (c) the MLE exists (i.e. Y is bounded from above and attains its supremum) (d) the MLE exists and is unique. The minimal number of samples needed for the MLE to exist almost surely is the MLE existence threshold; the number of samples for the MLE to exist uniquely almost surely is the uniqueness threshold.
Example 1.1. Let M = PD m . The unique maximiser of the likelihood isΨ = S −1 Y , if S Y is invertible. If S Y is not invertible, the likelihood function is unbounded from above, see [Sul18,Proposition 5.3.7]. The existence and uniqueness thresholds are therefore both m, since with m samples the matrix S Y will almost surely be invertible.
For a model M A as in (1), we can rewrite the log-likelihood (2) at sample matrix Y ∈ R m×n as a function of a matrix a ∈ A.
(3) Y (a T a) = log det(a T a) − 1 n a · Y 2 .
1.3. Directed Gaussian graphical models (DAG models). A directed acyclic graph (DAG) is G = (I, E), where I is a set of vertices, and E a set of directed edges. We write j → i for an edge from j to i; the absence of such an edge is denoted j → i. The parents and children of i in G are, respectively, the vertex sets
pa(i) = {j ∈ I | (j → i) ∈ E} ch(i) = {k ∈ I | (i → k) ∈ E}.
We often take the vertex set I to be [m] = {1, 2, . . . , m}.
We call a directed Gaussian graphical model on G a DAG model. A DAG model is defined by the linear structural equation
(4) y = Λy + ε, i.e. y i = j∈pa(i) λ ij y j + ε i ,
where y ∈ R m , λ ij = 0 for j → i in G, and ∼ N (0, Ω) with Ω diagonal. The coefficient λ ij is a regression coefficient, the effect of parent j on child i. The model encodes conditional independence: a node is independent of its non-descendants, after conditioning on its parents [VP90].
Remark 1.2 (The matrix Λ is strictly upper triangular). Throughout the paper, we choose an ordering on the vertices of G so that Λ is upper triangular. That is, if edge j → i is in E then j > i. Such an ordering is possible because G is acyclic. Thinking of a vertex label as its age, the ordering makes parents older than children.
Solving (4) for y gives y = (id − Λ) −1 ε, where id denotes the m × m identity matrix, and the acyclicity of G ensures that (id − Λ) is invertible. Hence y is multivariate Gaussian with mean zero and concentration matrix
(5) Ψ = (id − Λ) T Ω −1 (id − Λ).
We define a set of matrices associated to the DAG G
(6) A(G) = {a ∈ GL m | a ij = 0 for i = j with j → i in G}.
Recall from (1) the notation M A(G) = {a T a : a ∈ A(G)}. The set of concentration matrices of the form (5) is equal to the set M A(G) . We prove this in Lemma 2.9.
1.4. Undirected Gaussian graphical models. Multivariate Gaussian models can also be obtained from undirected graphs. An undirected graph G = (I, E) is a set of vertices I and undirected edges E. The model is the set of distributions with mean zero and concentration matrix Ψ with Ψ ij = 0 whenever edge i j is not in E. That is, the variables at nodes i and j are independent after conditioning on all others, see [Sul18, Proposition 13.1.5].
1.5. Restricted concentration (RCON) models. In [HL08], the authors introduce restricted concentration (RCON) models, which impose symmetries on the concentration matrix Ψ according to a graph colouring. A coloured undirected graph is a tuple (G, c), where G = (I, E) is an undirected graph and the map c : I ∪ E → C assigns a colour to each vertex and to each edge. The vertex i ∈ I has colour c(i) ∈ C, and edge i j has colour c(ij) ∈ C.
Definition 1.3 (see [HL08,§3]). The RCON model on the coloured undirected graph (G, c) consists of concentration matrices with
(1) Ψ ij = 0 whenever i j is not in E (2) Ψ ii = Ψ jj whenever c(i) = c(j), (3) Ψ ij = Ψ kl whenever c(ij) = c(kl).
Introducing RDAG models
A colouring of a DAG assigns colours to the vertices and edges. A coloured DAG is a tuple (G, c), where G = (I, E) is a DAG on vertices I and edges E, and c : I ∪ E → C is a colouring of the vertices and edges. Vertex i ∈ I has colour c(i) ∈ C, and edge j → i has colour c(ij) ∈ C. We sometimes denote the vertex colour c(i) by c(ii), with no ambiguity because a DAG cannot have loops.
Definition 2.1. The restricted DAG (RDAG) model on the coloured DAG (G, c) is the set of concentration matrices Ψ = (id − Λ) T Ω −1 (id − Λ), where Λ ∈ R m×m satisfies (1) λ ij = 0 unless j → i in G
(2) λ ij = λ kl whenever edges j → i and l → k have the same colour and the diagonal matrix Ω ∈ R m×m has positive entries and satisfies
(3) ω ii = ω jj if vertices i and j have the same colour. The model is given by the linear structural equation y = Λy + ε, where ε ∼ N (0, Ω).
Example 2.2. Consider the coloured graph 1 3 2 from Example 0.1. The RDAG model is parametrised by matrices
Λ ∈ 0 0 λ 0 0 λ 0 0 0 : λ ∈ R and Ω ∈ ω 0 0 0 ω 0 0 0 ω : ω, ω > 0 .
We will parametrise the RDAG model on (G, c) via the set
(7) A(G, c) := a ∈ GL m a ij = 0 for i = j with j → i in G a ij = a kl whenever c(ij) = c(kl) .
Note that A(G, c) is contained in the set A(G) from (6): the zero patterns of A(G) and A(G, c) are the same, and A(G, c) has further equalities imposed by the colouring c. Before we characterise which RDAG models can be parametrised by A(G, c), we pause to motivate the use of this alternative parametrisation. The alternative parametrisation has useful consequences. First, it leads to a condition on the graph colouring, called compatibility, which is indispensable in our results of Sections 4 and 5. Second, it is helpful when comparing directed and undirected models in Section 3. Finally, it enables us to generalise the connections to invariant theory from [AKRS21] to the setting of RDAGs, see Appendices A and B.
Example 2.4. Returning to the example 1 3 2 , we have
A(G, c) = d 1 0 r 0 d 1 r 0 0 d 2 : d 1 , d 2 = 0, r ∈ R .
We now introduce a natural assumption on a colouring.
Definition 2.5. A colouring c of a directed graph is compatible, if:
(i) Vertex and edge colours are disjoint; and (ii) If edges j → i and l → k have the same colour, then the child vertices i and k also have the same colour, i.e. c(ij) = c(kl) =⇒ c(i) = c(k).
Note that compatibility does not impose equality of parent colours c(j) and c(l).
Remark 2.6 (Motivation for compatibility). In an RDAG model, we do not impose equalities between Ω and Λ. The entry ω ii is a variance, while λ kl is a regression coefficient, so setting them to be equal would be difficult to interpret. Hence the vertex and edge colours can always be thought of as disjoint, as in compatibility condition (i). Compatibility condition (ii) has the statistical interpretation that the same regression coefficient appearing in an expression for two variables implies that their error variances agree. This extra assumption is indispensable in many of our results and proofs. It is a directed analogue to the condition appearing in [HL08, Proposition 1].
The first use of compatibility condition (ii) is in relating an RDAG model on (G, c) to the set A(G, c). As in (1), we consider the model Before proving the proposition, we recall two matrix decompositions. The LDL decomposition writes a positive definite matrix as Ψ = LDL T , where D is diagonal with positive entries, and L is lower triangular and unipotent (i.e. has ones on the diagonal). The LDL decomposition is closely related to the factorisation Ψ = (id − Λ) T Ω −1 (id − Λ) from (5): the LDL decomposition is D = Ω −1 and L = (id − Λ) T . Hence an RDAG model imposes zeros and symmetries in the LDL decomposition.
M A(G,c) = a T a | a ∈ A(G, c) .
The second matrix decomposition is the Cholesky decomposition. It writes a postive definite matrix as the product Ψ = a T a, where a is upper triangular with positive diagonal entries. The model M A(G,c) imposes zeros and symmetries in the Cholesky decomposition, as follows. Proof. The set M A(G,c) consists of all matrices Ψ of the form a T a for some a ∈ A(G, c), see (1). The matrix a is upper triangular by the structure of G. To get the Cholesky decomposition, it remains to modify a to have positive diagonal entries. We replace a by ka, where k is the diagonal matrix with k ii = 1 if a ii > 0 and k ii = −1 if a ii < 0. Then ka flips the sign of all rows of a with negative diagonal entry, hence it has all diagonal entries strictly positive. The compatibility of the colouring ensures that a ij = a kl can only hold in A(G, c) if a ii = a kk . Hence multiplying by k doesn't break any edge compatibility conditions, and ka ∈ A(G, c).
The LDL and Cholesky decompositions are both unique, since Ψ is (strictly) positive definite. They are related by:
Cholesky from LDL: a = D 1/2 L T , LDL from Cholesky: D = diag(a 2 11 , . . . , a 2 mm ), L T = D −1/2 a. The following lemma is proved by comparing zero patterns in the two decompositions.
(8) a = D 1/2 L T = Ω −1/2 (id − Λ), i.e. a ij = ω −1/2 ii if i = j −ω −1/2 ii λ ij if i = j.
We have containment of the DAG model inside M A(G) , because if j → i in G, then λ ij = 0 and therefore a ij = 0. Conversely, given Ψ ∈ M A(G) , its Cholesky decomposition is a T a for some a ∈ A(G) by Lemma 2.8, since the colouring that assigns all vertices and edges different colours satisfies A(G) = A(G, c) and is compatible. Hence a ij = 0 for i = j unless j → i in G. We therefore have (id − Λ) ij = L ji = 1 a ii a ij = 0, where a ii = 0, and hence λ ij = 0.
We prove Proposition 2.7, by comparing symmetries in the two decompositions.
Proof of Proposition 2.7. Given Ψ in the RDAG model, we show that its Cholesky decomposition is Ψ = a T a with a ∈ A(G, c). By Lemma 2.9, a ∈ A(G) and it remains to show that the colour conditions hold, i.e. that a ij = a kl whenever c(ij) = c(kl). If c(ii) = c(kk), then ω ii = ω kk since Ω respects the colouring. This shows that a ii = a kk , using (8). If c(ij) = c(kl) for edges j → i and l → k, then λ ij = λ kl since Λ respects the colouring and, moreover, ω ii = ω kk by compatibility. This implies a ij = a kl , by (8).
Conversely, given Ψ ∈ M A(G,c) , we show that Ψ is in the RDAG model. The Cholesky decomposition is Ψ = a T a for a ∈ A(G, c), by Lemma 2.8. The entries of Ω and Λ are ω ii = a −2 ii and λ ij = −a −1 ii a ij , by (8), which satisfy the RDAG model conditions. Hence a compatible colouring implies the equivalence of the RDAG model on (G, c) and M A(G,c) .
If the colouring is not compatible, we exhibit some Ψ in the RDAG model that is not in M A(G,c) . Let Ψ = a T a be the Cholesky decomposition. If there is some a ∈ A(G, c) with Ψ = (a ) T a then, similar to the proof of Lemma 2.8, there is a diagonal matrix o with entries ±1 with oa = a. First, if Definition 2.5(i) does not hold, there is a vertex k and an edge j → i with c(k) = c(ij). The RDAG model imposes no relation between ω kk and λ ij , so let Ψ be given by some Ω and Λ with ω kk = 1 and λ ij = 0. Then o kk a kk = a kk = 1 and o ii a ij = a ij = 0, by (8). Hence, a kk = 0 = a ij and therefore Ψ / ∈ M A(G,c) . Second, if Definition 2.5(ii) does not hold, then there exist edges j → i and l → k with c(ij) = c(kl) but c(i) = c(k). We choose Ψ given by some Ω and Λ with ω ii = 1, ω kk = 1 4 and λ ij = λ kl = −1. Then o ii a ij = a ij = 1 and o kk a kl = a kl = 2, by (8). Thus, |a ij | = 1 = 2 = |a kl | and, again, we cannot have Ψ ∈ M A(G,c) .
Example 2.10. We return to the graph 1 3 2 from Examples 2.2 and 2.4. The colouring is compatible, because the set of vertex colours {blue, black} is disjoint from the edge colour set {red}, and the children of both red edges have the same colour. Hence Proposition 2.7 shows that the RDAG model is equal to
(9) M A(G,c) = d 2 1 0 rd 1 0 d 2 1 rd 1 rd 1 rd 1 2r 2 + d 2 2 d 1 , d 2 = 0 .
Remark 2.11. RDAG models can also be defined over the complex numbers. Here, the parameters Λ can be complex, and we obtain a subset of PD m by taking conjugate
transposes, Ψ = (id − Λ) † Ω −1 (id − Λ)
. For the M A characterisation, we replace a T a by a † a. Many of our results and proofs can be modified to hold in the complex setting. We return to complex RDAGs in Section B.2.
Comparison of RDAG and RCON models
Given a directed graph, we can forget the direction of each edge to give an undirected graph. We characterise when the RDAG model on the coloured directed graph is equal to the undirected model on the corresponding coloured undirected graph in Theorem 3.4. We begin by comparing RDAG and RCON models in two examples.
Example 3.1 (RDAG = RCON). We revisit our running example 1 3 2 . The corresponding RCON model has coloured undirected graph 1 3 2 , with blue (circular) vertices {1, 2}, black (square) vertex 3, and red edges. By Definition 1.3, the RCON model is the set of positive definite matrices of the form
Ψ = δ 1 0 0 δ 1 δ 2 .
Since the colouring is compatible, the RDAG model is equal to M A(G,c) from (9). Any matrix in M A(G,c) satisfies the equalities for the RCON model, so we have containment of the RDAG model in the RCON model. Conversely, given Ψ in the RCON model,
det(Ψ) = δ 2 1 (δ 2 − 2 2 δ −1 1 ) > 0 hence δ 2 − 2 2 δ −1 1 > 0, since Ψ is positive-definite.
Positive definiteness of Ψ also implies δ 1 , δ 2 > 0. We obtain real numbers d 1 := √ δ 1 , r := /d 1 and d 2 := δ 2 − 2 2 δ −1 1 , which shows that Ψ is of the form in (9).
Example 3.2 (RDAG = RCON). Consider the RDAG model on 1 2 , the graph with two blue (circular) vertices {1, 2} and a red edge. The colouring is compatible, so by Proposition 2.7 the RDAG model is
M A(G,c) , where A(G, c) = d r 0 d d = 0 .
The corresponding RCON model is on the coloured undirected graph 1 2 . The RCON model consists of all Ψ ∈ PD 2 with Ψ 11 = Ψ 22 and Ψ 12 = Ψ 21 , by Definition 1.3. Neither model is contained in the other: the RCON model contains 4 2 2 4 = 2 0 1 √ 3 2 1 0 √ 3 but the diagonal entries 2 and √ 3 in the Cholesky decomposition do not satisfy the conditions for A(G, c). Conversely, the matrix 1 0 2 1
1 2 0 1 = 1 2 2 5
is in the RDAG model, but not the RCON model, since Ψ 11 = Ψ 22 .
To characterize when an RDAG model is equal to its corresponding RCON model, we give two constructions of coloured graphs obtained from some (G, c), one that is built from a vertex and the other from an edge. As before, G = (I, E).
Fix a vertex i ∈ I. Recall that the children of i are the vertices k with (i → k) ∈ E. Consider the subgraph on vertex set {i} ∪ ch(i) with edges i → k for each k ∈ ch(i), and colours inherited from (G, c). We denote the subgraph by G i . Now fix an edge (j → i) ∈ E. Consider vertices {i} ∪ (ch(i) ∩ ch(j)) with vertex colours inherited from (G, c). For each k ∈ ch(i) ∩ ch(j), we introduce two edges i → k, one with colour c(ki) and the other with colour c(kj). We denote this graph by G (j→i) . The vertex construction at vertex 5 and edge construction at edge 5 → 4 are:
G 5 = 5 1 2 3 4 G (5→4) = 1 2 3 4
Given a DAG G, its corresponding undirected graph is denoted G u . Similarly, given a coloured DAG (G, c), its corresponding undirected coloured graph (where the colour of a directed edge becomes the colour of the undirected edge) is (G u , c). Two coloured graphs (G, c) and (G , c ) are isomorphic if the coloured graphs are the same up to relabelling vertices. We denote an isomorphism by G G when the colouring is clear. Example 3.5. Our running example 1 3 2 satisfies the conditions of Theorem 3.4: it has no unshielded colliders, the graphs G 1 and G 2 both consist of a single blue vertex, and G (3→1) and G (3→2) are both isomorphic to 1 2 . The RDAG and RCON models are therefore equivalent, as we saw in Example 3.1.
Example 3.6. The following graph also satisfies the conditions of Theorem 3.4: The purple (triangular) vertices have G i isomorphic to G 5 from Example 3.3. (c) All edges j → i have ch(j) ∩ ch(i) = ∅, except for the two brown edges. For these, G (10→8) and G (9→7) are both isomorphic to G (5→4) from Example 3.3. Hence the RDAG model on this coloured graph is equal to the RCON model on the underlying undirected graph. Note that the two connected components of (G, c) are not isomorphic. We will see why this is not required for the proof of Theorem 3.4, i.e. why we can collapse vertices i and j in the definition of G (j→i) .
One ingredient to our proof of Theorem 3.4 is the condition for a DAG model to be equal to its corresponding undirected graphical model. Proof. Since the colouring is compatible, the RDAG model is M A(G,c) , by Proposition 2.7. Let a ∈ A(G, c) be a general matrix. We think of it as having indeterminate entries, one for each vertex colour and one for each edge colour. If the RDAG model is contained in the RCON model, we must have (a T a) ij = 0 whenever a ij = a ji = 0. This holds iff there are no unshielded colliders in G, by Theorem 3.7. Moreover, certain equalities must hold on a T a. We have vertex colour condition (a T a) ii = (a T a) jj whenever c(i) = c(j) and edge colour condition (a T a) ij = (a T a) kl whenever c(ij) = c(kl). These give the polynomial identities
a 2 ii + k∈ch(i) a 2 ki = a 2 jj + l∈ch(j) a 2 lj whenever c(i) = c(j) (10) a ii a ij + m p =i,j a pi a pj = a kk a kl + m q =k,l a qk a ql whenever c(ij) = c(kl).(11)
We show that (10) is equivalent to (b) and that (11) is equivalent to (c).
Given two vertices i, j with c(i) = c(j), we have a 2 ii = a 2 jj . The sums in (10) are equal if and only if |ch(i)| = |ch(j)| and the edge colours in G i and G j agree (counted with multiplicity). By compatibility, the vertex colours also agree, hence (10) is equivalent to G i G j .
Next, we consider (11). No terms a ji a jj and a lk a ll appear, since there is no edge i → j or k → l. The compatibility of the colouring gives a ii = a kk . Hence a ii a ij = a kk a kl . A summand in (11) vanishes unless p ∈ ch(i) ∩ ch(j) or q ∈ ch(k) ∩ ch(l). The sums are equal if and only if |ch(i) ∩ ch(j)| = |ch(k) ∩ ch(l)| and the graphs G (j→i) and G (l→k) are isomorphic on their edge colours. By compatibility of the colouring, the vertex colours of the children also agree and Proof. Given a matrix Ψ in the RCON model on (G u , c), we show that it is contained in the RDAG model on (G, c) by showing that the Cholesky decomposition Ψ = a T a satisfies the conditions of the set A(G, c) in (7). The matrix a is upper triangular. Completing it one column at a time shows that its entries are
c(i) = c(k), hence (11) is equivalent to G (j→i) G (l→k) .a l,l = Ψ l,l − p∈ch(l) a 2 p,l 1/2 (12) a i,j = Ψ i,j − p∈ch(i)∩ch(j) a p,i a p,j a −1 i,i .(13)
Note that the expression under the square root in (12) is a positive real number, see [TB97,Lecture 23]. We need to show that a ij = 0 for i = j with j → i (the support conditions) and that a ij = a kl whenever c(ij) = c(kl) (the colour conditions).
First we show that a satisfies the support conditions of A(G, c). If there is no edge from j to i in G, then G u has no edge between i and j, so Ψ i,j = 0. Moreover, all products a p,i a p,j vanish, otherwise i → p ← j would be an unshielded collider. Hence
a ij = 0 if j → i in G.
Next, we show that a satisfies the colour conditions of A(G, c). We prove this inductively over the top left k × k blocks of a. If k = 1 there are no symmetries to check. We assume that the top left k ×k submatrix of a satisfies the symmetries. For the induction step, we compare a 1,k+1 , a 2,k+1 , . . . , a k+1,k+1 with each other and with a i,j , i, j ∈ [k].
If there is an edge (k + 1) → 1 with same colour as j → i for i, j ∈ [k], we need to show that a 1,k+1 = a i,j . First, a 11 = a ii by compatibility and the induction hypothesis, and Ψ i,j = Ψ 1,k+1 since Ψ is in the RCON model. Moreover, all a pq for p, q ∈ [k] respect the symmetries. Since G (j→i) G (k+1→1) , the expressions (13) for a i,j and a 1,k+1 are equal. Proceeding inductively, we show that all entries a 2,k+1 , . . . , a k,k+1 respect the symmetries of c.
Finally, if vertex k + 1 has same colour as vertex l ∈ [k], we show a k+1,k+1 = a l,l . We have G l G k+1 by assumption (b) and Ψ l,l = Ψ k+1,k+1 , since Ψ is in the RCON model. Furthermore, we have shown that all a p,q , where p ∈ [k] and q ∈ [k + 1], satisfy the colouring c. Altogether, we conclude a l,l = a k+1,k+1 using (12).
MLE: existence, uniqueness, and an algorithm
In this section we characterise the existence and uniqueness of the MLE in an RDAG model via linear dependence conditions on certain matrices. Our description specialises to give the characterisation of existence and uniqueness of the MLE in a usual DAG model in terms of linear dependence among the rows of the sample matrix.
Let α s be the number of vertices of colour s. For an edge j → i, vertex j is called the source of the edge and i is called the target. The parent relationship colours are the colours of all edges with a target of colour s:
prc(s) = {c(ij) | there exists j → i in G with c(i) = s}, β s := |prc(s)|.
A sample matrix is Y ∈ R m×n ; its m rows index the vertices in G, and its columns are the n samples. For each vertex colour in G we define an augmented sample matrix.
Definition 4.1. The augmented sample matrix of sample matrix Y and vertex colour s, denoted M Y,s , has size (β s + 1) × α s n. We construct it row by row. Each row consists of α s blocks, each a vector of length n. For notational simplicity, we assume for now that the vertices of colour s are the set {1, 2, . . . , α s } ⊂ I. Then the top row of M Y,s is
Y (1) Y (2) . . . Y (αs) ∈ R αsn , where Y (i) ∈ R n is the ith row of sample matrix Y ∈ R m×n .
The other rows of M Y,s are indexed by the parent relationship colours prc(s). The row indexed by t ∈ prc(s) is
1←j c(1j)=t Y (j) 2←j c(2j)=t Y (j) · · · αs←j c(αsj)=t Y (j) .
Note that the sum at the kth block is zero if there are no j → k of colour t. Let M (i) Y,s denote the ith row of M Y,s , where we index from 0 to β s .
Example 4.2. Our running example 1 3 2 has two augmented sample matrices, one for each vertex colour:
(14) M Y,• = Y (1) Y (2) Y (3) Y (3) • → ∈ R 2×2n and M Y, = Y (3) ∈ R 1×n .
Example 4.3. The RDAG model on the coloured DAG
1 3 4 5 6 7 2 has M Y,• = Y (1) Y (2) Y (3) 0 0 Y (3) + Y (5) + Y (6) Y (5) Y (4) Y (6) 0 Y (4) + Y (7) 0 • Theorem 4.4. Consider the RDAG model on (G, c)
where colouring c is compatible, and fix sample matrix Y ∈ R m×n . The following possibilities characterise maximum likelihood estimation given Y :
(a) Y unbounded from above ⇔ M (0) Y,s ∈ span M (i) Y,s : i ∈ [β s ] for some s ∈ c(I) (b) MLE exists ⇔ M (0) Y,s / ∈ span M (i) Y,s : i ∈ [β s ] for all s ∈ c(I) (c)
MLE exists uniquely ⇔ M Y,s has full row rank for all s ∈ c(I). 4.1. Usual DAG models. We recall the characterisation of MLE existence and uniqueness for usual Gaussian graphical models, as linear dependence conditions on the sample matrix. For a DAG G on m nodes, and n data samples, the sample matrix is Y ∈ R m×n . For a node i in G we denote by Y (i) the ith row of Y , by Y (pa(i)) the sub-matrix of Y with rows indexed by the parents of i in G, and by Y (pa(i)∪i) the sub-matrix of Y with rows indexed by node i and its parents.
Theorem 4.9. Consider the DAG model on G, with m nodes, and fix sample matrix Y ∈ R m×n . The following possibilities characterise maximum likelihood estimation given Y :
(a) Y unbounded from above ⇔ Y (i) ∈ span Y (j) : j ∈ pa(i) for some i ∈ [m] (b) MLE exists ⇔ Y (i) / ∈ span Y (j) : j ∈ pa(i) for all i ∈ [m] (c) MLE exists uniquely ⇔ Y (pa(i)∪i) has full row rank for all i ∈ [m].
This result follows from viewing maximum likelihood estimation in a DAG as a sequence of regression problems. The acyclicity ensures that the sub-problems are uncoupled. We give a proof, so as to see its generalisation in Theorem 4.4.
Proof. We denote the entries of the MLEs Λ and Ω byλ ij andω kk . The negative of the log-likelihood Y , in terms of the parameters ω ii and λ ij , is
m i=1 log ω ii + 1 nω ii Y i − j∈pa(i) λ ij Y j 2 .
We minimise the above expression. Each parameter only appears in one summand.
In the ith summand, theλ ij always exist: they are coefficients of each Y (j) in the orthogonal projection of Y (i) onto the span of {Y (j) : j ∈ pa(i)}. Theλ ij are unique if and only if Y (pa(i)) has full row rank. Let ζ i be the residual
Y (i) − j∈pa(i)λ ij Y (j) 2 . Letω ii minimise log(ω ii ) + ζ i nω ii . If ζ i = 0, equivalently if Y (i) ∈ span Y (j)
: j ∈ pa(i) , then in the limit ω ii → 0, the log-likelihood tends to infinity andω ii does not exist. Otherwise, the minimum is attained uniquely atω ii = ζ i /n. Combining these cases gives the theorem.
4.2.
Proof of Theorem 4.4. The proof of Theorem 4.4 will be similar to the proof for uncoloured models in Theorem 4.9. We start by proving the following lemma.
f γ : R >0 → R, x → α log(x) + γ x .
(i) If γ = 0, then f γ is neither bounded from below nor bounded from above.
(ii) If γ > 0, then f γ attains a global minimum at x 0 = γ α with function value f γ ( γ α ) = α(log(γ) − log(α) + 1). (iii) Given γ 1 ≥ γ 2 > 0, we have f γ 1 ( γ 1 α ) ≥ f γ 2 ( γ 2 α )
at the global minima. Proof. Part (i) follows from the properties of log. To prove part (ii), one computes
f γ (x) = α x − γ x 2 for x > 0. For x > 0 we have f γ (x) = 0 ⇔ α x = γ x 2 ⇔ αx = γ ⇔ x = γ α . Thus x 0 := γ α is the only possible local extremum of f γ . For x > 0, f γ (x) > 0 ⇔ α x > γ x 2 ⇔ αx > γ ⇔ x > γ α .
and similarly one has f γ (x) < 0 if and only if x < γ α = x 0 . Therefore, x 0 is a global minimum of f γ . One directly verifies the function value for f γ (x 0 ), and so part (iii) follows from the monotonicity of the logarithm.
An MLE is a minimiser of the above expression. Each parameter occurs in exactly one of the summands over s ∈ c(I), because the prc(s) partition the edge colours by compatibility. We therefore minimise each summand separately. By Lemma 4.10(iii) we can first determineλ s,t , t ∈ [β s ] that minimise
(16) M (0) Y,s − t∈[βs] λ s,t M (t) Y,s 2 .
Suchλ s,t always exist: they are coefficients in the orthogonal projection P Y,s of M Y,s , t ∈ [β s ] are linearly independent. Denote the minimum value of (16) by ζ s . We will apply Lemma 4.10 several times with γ s := ζ s /n.
If
M (0) Y,s ∈ span M (t)
Y,s : t ∈ [β s ] for some s, then ζ s = 0 and the summand α s log(ω ss )+ ζ s /(nω ss ) is not bounded from below for ω ss > 0, by Lemma 4.10(i). Hence Y is not bounded from above, e.g. by setting ω s ,s = 1 and λ s ,t = 0 for all s ∈ c(I) \ {s} and all t ∈ [β s ]. This proves "⇐" of (a).
If
M (0) Y,s / ∈ span M (t)
Y,s : t ∈ [β s ] , equivalently ζ s > 0, then the summand α s log(ω ss ) + ζ s /(nω ss ) has unique minimiserω ss = ζ s /(nα s ), by Lemma 4.10(ii). Hence, an MLE exists if ζ s > 0 for all s ∈ c(I), which proves "⇐" in (b). As the right-hand sides of (a) and (b) are opposites, we have proved (a) and (b).
Since Remark 4.11 (Comparison with usual DAG models). The framework of RDAG models includes usual DAG models as a special case; namely, when each colour is used only once. In this case, Theorem 4.4 specialises to Theorem 4.9. We see in the next section that imposing colours in a DAG reduces the threshold number of samples required for existence and uniqueness of the MLE.
Illustrative examples.
In this section we apply RDAG models to some small illustrative examples. We first apply our running example to model the effect of a mother's height on her two daughters' heights.
Example 4.12. The RDAG model on coloured graph 1 3 2 is given by y 1 = λy 3 + 1 , y 2 = λy 3 + 2 , y 3 = 3 , 1 , 2 ∼ N (0, ω), 3 ∼ N (0, ω ). Let variable 3 be the height (in cm) of a woman and let variables 1 and 2 be, respectively, the heights of her younger and older daughter. Vertices 1 and 2 both being blue indicates that, conditional on the mother's height, the variance of the daughter's heights is the same. The edges both being red encodes that the dependence of a daughter's height on the mother's height is the same for both daughters.
We saw in Example 4.5 that the MLE exists almost surely given one sample. We use Algorithm 1 to find the MLE, given one sample where the the younger daughter's height is 159.75cm, the older daughter's height is 161.56, the mother's height is 155.32, and the population mean height is 163.83cm. Mean-centring the data gives
Y (1) Y (2) Y (3) = −4.08 −2.27 −8.51 .
The only black vertex is 3, and it has no parents, hence ω = Y (3) 2 = 72.42. The orthogonal projection of Y (1) Y (2) onto span Y (3) Y (3) has coefficient λ = 0.37 and residual ω = (−3.175 + 4.08) 2 + (−3.175 + 2.27) 2 /2 = 0.82. As we would expect, the regression coefficient λ is positive and the variance of the daughters' heights conditional on the mother's height is lower than the variance of the mother's height.
We now consider multiple measurements taken in each generation. The vertex colours in an RDAG model could correspond to colours in an experiment, as follows. Fluorescent reporters can be used to take measurements in a cell. In [SRA + 21], the authors quantify data in a single living bacteria using fluorescent reporters in red, cyan, yellow, and green, see [SRA + 21, Figure 2a]. Given such measurements taken for a parent cell and its daughter cells, we could consider analogues of Example 4.13 in which, for example, the red fluorescence in a daughter cell only depends on the red fluorescence in the parent, or larger models in which there can also be dependence on the fluorescence of other colours.
Maximum likelihood thresholds
In the previous section we gave a characterisation of the existence and uniqueness of the MLE based on linear independence conditions. Here we turn this into results that depend only on the coloured graph, and that hold for a generic sample matrix. We give upper and lower bounds on the thresholds for almost sure existence and uniqueness of the MLE. The bounds hold whenever the sample matrix doesn't have extra linear dependencies among its rows.
For a fixed number of samples, the MLE in an RDAG model may exist but not be unique almost surely, which cannot happen in an uncoloured model. In fact, Example 5.5 gives a family of models for which the gap between the existence and uniqueness thresholds becomes arbitrarily large.
As well as assuming compatibility of the colouring, we often assume in this section that there are no edges between vertices of the same colour. Some of our bounds involve the following notion of generic rank.
Definition 5.1. Let M Y be a matrix whose entries are linear combinations of the entries of a matrix Y . The generic rank of M Y is its rank for generic Y .
For Y ∈ R m×n , we often study the generic rank of M Y by considering it as a symbolic matrix whose entries are linear forms in the mn indeterminates Y ij .
M Y,• = Y (1) Y (2) Y (3) Y (3) Y (4) Y (4) Y (5) Y (5) Y (6) Y (6) Y (7) Y (7) •
When n = 1, the matrix M Y,• has generic rank two. Removing its top row gives a 5 × 2 matrix of generic rank one.
Define the augmented sample matrix M Y,s ∈ R (βs+1)×αsn as in Definition 4.1. We let M Y,s ∈ R βs×αsn be obtained from M Y,s by removing its top row. As before, α s is the number of vertices of colour s and β s is the number of edge colours of edges to a vertex of colour s. Let r s be the generic rank of M Y,s when n = 1. Let mlt e (resp. mlt u ) denote the minimal number of samples needed for almost sure existence (resp. uniqueness) of the MLE.
r s − 1 α s − 1 + 1 ≤ mlt e ≤ max s β s α s + 1, (17) max s β s α s + 1 ≤ mlt u ≤ max s (β s + 2 − r s ) .(18)
It remains an open problem to turn the bounds in Theorem 5.3 into formulae for mlt e and mlt u .
Problem 5.4. Determine the maximum likelihood thresholds mlt e and mlt u of an RDAG model, as formulae involving properties of the graph G and its colouring c.
Note that the upper bounds for existence and uniqueness are both at most max s β s +1, which is the threshold for uniqueness in the (uncoloured) DAG case, see [DFKP19]. Hence the RDAG thresholds are always at least as small as the DAG threshold.
Example 5.5. We find the existence and uniqueness thresholds for the RDAG on the graph in Example 5.2. The black (square) vertices have no children, so the matrix M Y, is full rank as soon as n ≥ 1. The thresholds are therefore determined by the matrix M Y,• . The generic rank of M Y,• is one when n = 1, i.e. r • = 1. Theorem 5.3 and Proposition 5.10 then give bounds
r • − 1 α • − 1 + 1 = 1 ≤ mlt e and mlt u ≤ β • + 1 − r • = 5 + 1 − 1 = 5.
In fact, both bounds are attained, as follows. When n = 1, the row M This example extends to an arbitrary number of vertices, i.e. the graph with k + 2 vertices, 2 blue/circular and k black/square, and 2k edges of k colours (arranged as in the k = 5 case above). Repeating the above argument gives mlt e = 1 and mlt u = k.
Remark 5.6. Theorem 5.3 applies to all RDAG models with compatible colouring that are equal to some RCON model, because such models never have edges between vertices of the same colour, as follows. Take some vertex i of minimal index with i ← j in G having c(i) = c(j). Then no children of i have colour c(i), therefore G i = G j , a contradiction to Theorem 3.4(b).
We modify the edges from Examples 5.2 and 5.5 to see how the thresholds change. Given sample matrix Y ∈ R 7×n , we respectively obtain
M Y,• = Y (1) Y (2) Y (3) 0 Y (4) 0 Y (5) 0 Y (6) 0 Y (7) Y (4) • M Y,• = Y (1) Y (2) Y (3) 0 0 Y (3) + Y (5) + Y (6) Y (5) Y (4) Y (6) 0 Y (4) + Y (7) 0 •
In both cases we have α • = 2, β • = 5, and r • = 2. Thus, Theorem 5.3 gives the bounds
2 = r • − 1 α • − 1 + 1 ≤ mlt e ≤ β • α • + 1 = 3 3 = β • α • + 1 ≤ mlt u ≤ β • + 2 − r • = 5.
In fact, Proposition 5.10 gives an upper bound of 4 on mlt u , since r
• = β • + 1 − (β • /α • ).
First, we study the left-hand RDAG. When n = 2 the row Y (2) ∈ R 1×2 is generically not in the span of Y (4) , hence M
Y (2) Y (4) ∈ R 2×n
generically has rank two. Therefore, M Y,• has rank at most five if n = 3, but n = 4 suffices for M Y,• generically having full row rank. We conclude mlt u = 4.
Next, we study the right-hand RDAG. For n = 2, M
Y (3) Y (6) Y (4) + Y (7) , Y (2) Y (3) + Y (5) + Y (6) Y (4) ∈ R 3×3
of M Y,• generically have rank three, and the zero pattern then ensures that M Y,• has full row rank six. Combining with the lower bound 3 ≤ mlt u gives mlt u = 3.
5.1. Proof of Theorem 5.3. For fixed vertex colour s, we define mlt e (s) to be the smallest n such that the top row of M Y,s is almost surely not in the span of the other β s rows, and define mlt u (s) to be the smallest n such that the matrix M Y,s is almost surely of full row rank β s + 1. To prove Theorem 5.3, we use the following lemma. Without loss of generality, let 1, 2, . . . , α s be the vertices of colour s. The matrix M Y,s has α s many β s × n blocks. For each parent relationship colour p t , t ∈ [β s ] there is some vertex i = i(t) ∈ [α s ] such that there is an edge of colour p t pointing towards vertex i = i(t). That is, the i th block of M Y,s has non-zero entries in the t th row. Let C t be the t th column of that block, which exists as n ≥ β s . By construction, the t th entry of C t is non-zero. We show that the matrix C = C 1 C 2 . . . C βs , is invertible.
An entry of C is either a sum of variables or it is zero. By construction, column C t only contains (sums of) elements of the t th column of Y . The same variable Y j,t cannot occur in two different entries of C t , because there is at most one edge from j to vertex i(t). Altogether, the entries of C are (possible empty) sums of variables and each variable occurs in at most one entry of C. The determinant is an alternating sum over products of permutations, and it is enough to show that one product is non-zero. By construction, C 11 C 22 · · · C βsβs = 0. Thus, M Y,s has rank β s for n ≥ β s .
+ 1, while if α s ≥ 2 we have r s − 1 α s − 1 + 1 ≤ mlt e (s) ≤ β s α s + 1.
Proof. If α s = 1, the equality mlt e (s) = β s + 1 is known from the uncoloured case. It remains to consider α s ≥ 2. For the upper bound, we show that if n > βs αs , then the top row of M Y,s is generically not in the span of the other rows. Since there are no edges between two vertices of colour s, the nα s entries of the top row M (0) Y,s are all independent, from each other and from the entries of the other rows. If β s < α s n, the other β s rows do not span R nαs , so a generic choice of top row will not lie in their span.
For the lower bound, consider the β s × α s matrix M Y,s for n = 1. Its generic rank is denoted r s . We consider the 1 × n matrix blow up, where the scalar variables Y (i) are replaced by generic row vectors of length n, to give a β s × α s n matrix. We consider the rank of this matrix as n increases. The maximum rank is β s , which occurs for n ≥ β s by Lemma 5.8. Moreover, the rank is a (weakly) concave function in n [DM17, Corollary 2.8]. Since the rank is positive integer valued, it cannot be the same at two distinct n unless it is at its maximum. Hence the rank for fixed n is at least min(r s + n − 1, β s ). Therefore, the top row is in the span of the others whenever min(r s + n − 1, β s ) ≥ nα s , in particular, whenever α s n ≤ r s + n − 1 ≤ β s , i.e. for n ≤ min rs−1 αs−1 , β s + 1 − r s , so mlt e (s) ≥ min rs−1 αs−1 + 1, β s + 2 − r s . We exclude the possibility that the smaller of the two arguments in the minimum is β s + 2 − r s by appealing to Proposition 5.10. Proof. For the lower bound, we observe that if α s n ≤ β s , the β s + 1 rows of M Y,s will be linearly dependent. Hence, we need n > βs αs . For the upper bound, let M Y,s and r s be as above. Recall from the proof of Proposition 5.9 that, for n samples, rk(M Y,s ) ≥ min(r s + n − 1, β s ) generically. Thus, for n = β s + 1 − r s the matrix M Y,s generically has full row rank β s . It remains to consider the top row of M Y,s . We must have β s ≤ α s n, otherwise the β s × α s n matrix could not have full row rank. We look separately at the possible cases: β s < nα s and β s = nα s . If β s < nα s , the row vector M Proof. It suffices to give a randomised algorithm to compute mlt e (s) and mlt u (s) for fixed vertex colour s. The rank of a symbolic matrix can be computed (efficiently) by a randomised algorithm, see e.g. [Lov79,Sch80]. Hence, thinking of the entries of Y ∈ R m×n as indeterminates, we can compute for any n ≥ 1 the rank of the symbolic (β s + 1) × α s n matrix M Y,s as well as the rank of the symbolic β s × α s n matrix M Y,s . We obtain mlt e (s) as the smallest n such that rk(M Y,s ) > rk(M Y,s ) and mlt u (s) as the smallest n such that rk(M Y,s ) = β s + 1. The algorithm terminates by the upper bound of β s + 1 for both mlt e (s) and mlt u (s).
Simulations
In the previous section, we gave upper and lower bounds for the maximum likelihood thresholds for RDAG models, see Theorem 5.3. Our bounds quantify how the graph colouring serves to decrease the number of samples needed for existence and uniqueness of the MLE. In this section, we assume that the number of samples is above the maximum likelihood threshold. We explore via simulations the distance of an MLE to the true model parameters. We compare the RDAG model estimate from Algorithm 1 to the usual (uncoloured) DAG model MLE.
The details of our simulations are as follows. We used the NetworkX Python package [HSSC08] to build an RDAG model via the following steps. We first build a DAG by generating an undirected graph according to an Erdős-Rényi model that includes each edge with fixed probability, and then directing the edges so that j → i implies j > i. We assign edge colours randomly, after fixing the total number of possible edge colours. We choose the unique vertex colouring with the largest number of vertex colours that satisfies the compatibility assumption from Definition 2.5. We sample edge weights λ st from a uniform distribution on [−1, −0.25] ∪ [0.25, 1] and we sample noise variances ω ss uniformly from [0, 1]. Our code is available at https://github.com/seigal/rdag.
The RDAG MLE is generally closer to the true model parameters than the DAG MLE, see Figure 1. As we would expect, both estimates get closer to the true parameters as the number of samples from the distribution increases. At a high number of samples, the difference between the RDAG MLE and the DAG MLE is smaller than at a low number of samples. Figure 1. We generated RDAGs on 10 vertices, with each edge present with probability 0.5 and 5 edge colours. We sampled from the distribution n ∈ {5, 10, 100, 1000, 10000} times. For each n we generated 50 random graphs and computed the RDAG MLE and the DAG MLE, comparing them to the true parameter values on a log scale. The results are displayed in a violin plot, with blue for the RDAG MLE and orange for the DAG MLE.
Next we examined how the RDAG MLE was affected by the number of edge colours, see Figure 2. The RDAG MLE is closest to the true parameters when the number of edge colours is small; i.e., when there are fewer parameters to learn. As the number of edge colours increases, the difference between the RDAG MLE and the DAG MLE decreases. Note that the DAG model is the setting where each vertex and edge has its own colour.
Finally, we looked at how the RDAG MLE and DAG MLE are affected by the edge density of the graph, see Figure 3. The RDAG MLEs get closer to the true parameter values as the edge density increases: more edges have the same weight, so more samples contribute to estimating each edge weight. By comparison, the DAG MLEs get further from the true parameters as the edge density increases, because there are more parameters to learn. . We generated RDAGs on 10 vertices, each edge present with probability in {0.1, 0.3, 0.5, 0.7, 0.9} and 5 edge colours. For each edge probability we generated 50 random graphs, sampled from each one 100 times, and compared the RDAG and DAG MLEs. As above, blue is the RDAG MLE and orange is the DAG MLE.
(iii) polystable if Y = 0 and the set A · Y is (Euclidean) closed (iv) stable if Y is polystable and A Y is finite.
The above notions of stability are usually studied for A a reductive group [Dol03,MFK94]. They were studied for reductive and non-reductive groups in [AKRS21]. We are not aware of these definitions being used before without any group structure on A.
We relate maximum likelihood estimation for an RDAG model to these stability notions. For coloured graph (G, c), we recall the definition of A(G, c) from (7). Given a set A ⊆ GL m , we define A SL = {a ∈ A | det(a) = 1} and A ± SL = {a ∈ A | det(A) = ±1}. Theorem A.2. Consider an RDAG model on (G, c) with compatible colouring c and sample matrix Y ∈ R m×n . Then stability under A(G, c) SL relates to maximum likelihood estimation:
(a) Y unstable ⇔ Y unbounded from above (b) Y semistable ⇔ Y bounded from above (c) Y polystable ⇔ MLE exists (d) Y stable ⇔ MLE exists uniquely.b ∈ A Y to λ(a + b − id) T (a + b − id).
Theorem A.2 applies to any DAG model, see Remark 4.11. Therefore, Theorem A.2 generalises [AKRS21, Theorem 5.3] in multiple ways. First, it extends from transitive DAGs 3 to all DAGs. Second, it extends from uncoloured DAG models to RDAG models. Third, it adds part (d) about stable samples. Moreover, Proposition A.3 gives a bijection between the MLEs and the stabilising set.
To prove Theorem A.2 and Proposition A.3, we first we generalise [AKRS21, Proposition 3.4 and Theorem 3.6] to no longer require that A is a group. We say that a set A is closed under non-zero scalar multiples if a ∈ A implies ta ∈ A for all t ∈ R × . Proposition A.4. Let A be a set of real invertible m × m matrices, closed under non-zero scalar multiples. There is a correspondence between stability under A ± SL and maximum likelihood estimation in the model M A given sample matrix Y ∈ R m×n :
(a) Y unstable ⇔ likelihood Y unbounded from above (b) Y semistable ⇔ likelihood Y bounded from above (c) Y polystable ⇒ MLE exists.
The MLEs, if they exist, are the matrices λa T a, where a · Y > 0 is minimal in A ± SL · Y and λ ∈ R >0 is the unique global minimum of
R >0 → R, x → x n a · Y 2 − m log(x).
If A contains an orthogonal matrix of determinant −1, then A ± SL can be replaced by A SL .
Proof. This is proved using the same argument as [AKRS21, Proposition 3.4 and Theorem 3.6]. Maximising Y over M A is equivalent to minimising
f : A → R, a → 1 n a · Y 2 − log det(a T a).
We write a ∈ A as τ b, where τ ∈ R >0 and b ∈ A ± SL . Setting x := τ 2 we compute
f (a) = τ 2 n b · Y 2 − log det(τ 2 b T b) = x n b · Y 2 − m log(x).
The infimum of the function x → xC − m log(x) increases as C ≥ 0 increases, hence
(19) sup a∈A Y (a T a) = − inf a∈A f (a) = − inf x∈R >0 x n inf b∈A ± SL b · Y 2 − m log(x) .
We To prove (c) assume that Y is polystable under A ± SL . Then C : for all s ∈ c(I). Then Y is polystable under A and A · Y is Zariski closed.
= inf b∈A ± SL b · Y 2 is strictly positive, as Y is semistable. Since A ± SL · Y is closed in R m×n , we see that C is attained in the compact set (A ± SL · Y ) ∩ {Z ∈ R m×n | Z 2 ≤ C +
Proof. The hypotheses in the statement imply that the log-likelihood Y is bounded from above, by Theorem 4.4(b). Since A(G, c) is closed under non-zero scalar multiples, Y is semistable under A = A(G, c) SL by Proposition A.4(b) and Remark A.5. We now study the orbit A · Y . Let T be the set of diagonal matrices in A and U the unipotent matrices in A. Then A = T · U by compatibility and, in fact, any a ∈ A admits a unique decomposition a = tu with t ∈ T , u ∈ U . For s ∈ c(I), recall the construction of M Y,s ∈ R (βs+1)×αsn from Definition 4.1. Setting V s := R 1×αsn we can identify R m×n ∼ = s V s such that the rows of vertex colour s belong to V s . By definition of M Y,s , and since the prc(s) partition c(E), the set U · Y is H := s H s with
H s = M (0) Y,s + a s,1 M (1) Y,s + . . . + a s,βs M (βs) Y,s | a s,t ∈ R . The affine space H s is M (0) Y,s + X s , where X s := span M (1) Y,s , . . . , M (βs) Y,s . Since M (0)
Y,s / ∈ X s for all s ∈ c(I) by assumption, we have 0 / ∈ H s and hence H s has at least codimension one in V s . We define the linear subspace K s := RM (0) Y,s ⊕ X s of V s . Since T acts on each V s with the non-zero scalar of colour s, we have
A · Y = T · (U · Y ) = T · H = T · s H s ⊆ s K s ⊆ s V s .
It suffices to show that A · Y is Zariski-closed in s K s . Each H s is an affine subspace of K s with codimension one. Therefore, there exists a linear form p s ∈ K * s such that H s = V Ks (p s − 1), where V(·) denotes the vanishing locus.
We finish the proof by showing that A · Y = V s p αs s − 1 in s K s . First, given W = (W s ) s ∈ A · Y = T · H we can write W = a · Z with a ∈ T and Z = (Z s ) s ∈ H. Then Y,s = 0 has infinitely many solutions. Distinct solutions give distinct unipotent matrices a ∈ A by using a s,t as the entry for edge colour t ∈ prc(s), and setting all other off-diagonal entries of a to zero. Such a unipotent matrix a ∈ A satisfies aY = Y , since the sets prc(s) are disjoint, so the a s,t do not affect any rows of Y with a different vertex colour. In conclusion, A Y is infinite if M We now turn to Proposition A.3. As above, A := A(G, c) SL for an RDAG with compatible colouring. Denote the set of diagonal (respectively unipotent) matrices in A by T (respectively U ). By compatibility of the colouring, A = T · U and in fact any a ∈ A admits a unique factorisation a = tu with t ∈ T , u ∈ U .
(a) Y unstable ⇔ M (0) Y,s ∈ span M (i) Y,s : i ∈ [β s ] for some s ∈ c(I) (b) Y semistable ⇔ Y polystable (c) Y polystable ⇔ M (0) Y,s / ∈ span M (i) Y,s : i ∈ [β s ] for all s ∈ c(I) (d) Y stable ⇔ M Y,
Lemma A.8. Consider the RDAG model on (G, c) where colouring c is compatible. For A := A(G, c) SL write A = T · U as above. If Y ∈ R m×n is polystable under A, then the following hold:
(a) U · Y contains a unique element Y of minimal norm. (b) For t ∈ T and u ∈ U , tu · Y ≥ t · Y with equality if and only if u · Y = Y .
(c) Let a, a ∈ A be such that a · Y and a · Y are of minimal norm in A · Y . Then there is an orthogonal t ∈ T such that ta · Y = a · Y .
Proof. For part (a), we recall that the prc(s), s ∈ c(I), partition the edge colours c(E). Therefore, when minimising For (c), write a = tu with t ∈ T and u ∈ U . Since aY is of minimal norm in A · Y , we deduce uY = Y using (b). Thus, aY ∈ T · Y and similarly aY ∈ T · Y . As T · Y ⊆ A · Y the matrices aY and aY are also of minimal norm in T · Y . We note that T is a group isomorphic to the reductive group {(t s ) s∈c(I) | t s ∈ R × , s t αs s = 1}. Hence the Kempf-Ness theorem, see [AKRS21, Theorem 2.2], for the action of T implies that there is some orthogonal t ∈ T that relates the minimal norm elements aY and aY in T · Y .
We conclude this appendix with a proof of Proposition A.3.
Proof of Proposition A.3. Recall that A = A(G, c) SL = T · U , where T is the diagonal matrices in A, and U the unipotent matrices in A. If aY = Y , then for all s ∈ c(I),
M 0 Y,s = a s M (0) Y,s + t∈[βs] a s,t M (t) Y,s . We have M (0) Y,s / ∈ span M (t)
Y,s : t ∈ [β s ] , since Y is polystable. Hence a s = 1 for all s, i.e. a ∈ U and therefore A Y = U Y . We set N Y := U Y − id, which consists of strictly upper triangular matrices. It suffices to show that for fixed MLE λa T a the map
ϕ : N Y → {MLEs given Y } b → λ(a + b) T (a + b)
is well-defined and bijective. Note that bY = 0 for any b ∈ N Y . Therefore, (a+b)Y = aY is of minimal norm in A · Y and thus ϕ(b) is an MLE by Proposition A.4.
For surjectivity, let λ a T a be another MLE given Y . Then aY and aY are of minimal norm in A·Y , hence there is an orthogonal t ∈ T with aY = t aY by Lemma A.8(c). We set b := t a − a so that b · Y = 0 and (id + b)Y = Y . By compatibility of the colouring we have t a ∈ A and thus all entries of b = t a − a obey the colouring c. However, bY = 0 implies b s = 0 for all s by polystability of Y , hence b ∈ N Y . We compute ϕ(b) = λ(t a) T (t a) = λ a T a using orthogonality of t.
For injectivity, let b, b ∈ N Y be such that ϕ(b) = ϕ(b ). Let t ∈ T be defined by t s = 1 if a s > 0 and t s = −1 if a s < 0. Then t is orthogonal and thus (ta + tb) T (ta + tb) = (a + b) T (a + b).
Similarly, (ta + tb ) T (ta + tb ) = (a + b ) T (a + b ). Then ϕ(b) = ϕ(b ) implies (22) (ta + tb) T (ta + tb) = (ta + tb ) T (ta + tb ).
Moreover, tb, tb are strictly upper triangular and ta ∈ A has positive diagonal entries, by construction of t. Hence, applying uniqueness of the Cholesky decomposition to (22) gives ta + tb = ta + tb , and we deduce b = b .
Appendix B. Connections to Gaussian group models
A model is a Gaussian group model if it is equal to M A , see (1), where the set A a group. In this case, the second term in the log-likelihood (3) is the minimisation of the norm over a group orbit. This perspective was used in [AKRS21] to relate existence of the MLE to notions of stability under a group action. In this appendix, we characterise when the set of matrices A(G, c) from (7) is a group. We use Popov's criteron to study stability, and give our third and final description of the set of MLEs in an RDAG model. k ∈ I with j → k and k → i in G, i.e. G must be transitive, by contraposition. We have shown that (1), (2) and (3) Example B.3. Surprisingly, two graphs can have all the same butterfly graphs without being isomorphic. We present an example. Consider the following coloured graph with 10 black (square) vertices, and edges that are red (solid), green (squiggly), orange (dashed) or brown (dotted).
c 1 b 1 c 2 b 2 d 1 a 1 c 3 b 3 c 4 b 4
We add some further edges: four purple edges a 1 → c i , four blue edges b i → d 1 , and a yellow edge a 1 → d 1 . Now consider the graph obtained by exchanging the green (squiggly) and orange (dashed) edges. The butterfly graphs for the two graphs are the same, as follows. On the yellow edge, the butterfly graphs both have four paths consisting of a brown edge followed by a blue edge, and four that are a purple edge followed by a brown edge. Similarly, we can check the butterfly graphs at the other edge colours.
However, the two coloured graphs are not isomorphic. Indeed, the only way to get an isomorphism is to permute the b-layer and the c-layer. The red (solid) edges give the identity permutation, the orange (dashed) edges give the cycle σ = (1 4 3 2), and the green (squiggly) edges give σ 2 . Hence an isomorphism would need to consist of permutations µ 1 and µ 2 of {1, 2, 3, 4} with µ 1 idµ 2 = id, µ 1 σµ 2 = σ 2 , µ 1 σ 2 µ 2 = σ. The first condition implies µ 2 = µ −1 1 , hence σ and σ 2 need to be simultaneously conjugate to σ 2 and σ respectively. This implies (σ 2 ) 2 = σ, a contradiction because σ 4 = id.
B.2. Popov's criterion. If A(G, c) is a group, we can prove the important Lemma A.6 differently, via a criterion of Popov [Pop89, Theorem 4]. The criterion characterises when an orbit under a connected solvable group is closed, provided the underlying field is algebraically closed. Due to the latter assumption, we work in this subsection with RDAG models defined over the complex numbers, and note that many of our results and proofs carry over to the complex case, see Remark 2.11. We start by describing Popov's criterion for the group G := A(G, c) SL .
Since G ⊆ GL m (C) is a group of invertible upper triangular matrices, it is solvable. We decompose G = T · U ⊆ GL m as a semi-direct product, where T is the subgroup of diagonal matrices in G, and U is the subgroup of unipotent matrices in G. The group G acts on C m×n by left-multiplication. Let f k,l ∈ C[C m×n ], k ∈ [m], l ∈ [n], be the coordinate functions on C m×n . Let x s be the coordinate function corresponding to vertex colour s ∈ c(I), and let x s,t be the coordinate function for the edge colour t ∈ prc(s). Given a tuple of samples Y ∈ C m×n we consider the orbit map µ G·Y : G → C m×n g → g · Y or, on coordinate rings, µ * G·Y (f k,l ) = m j=1
x c(kj) Y j,l .
We define
R Y := µ * G·Y C[C m×n ] = C m j=1 Y j,l x c(kj) k ∈ [m], l ∈ [n] ⊆ C[G].
Since R Y is a C-algebra, we obtain the semigroup we have ker(M † Y,s ) ⊆ span{e 1 , . . . , e βs } ⊆ C βs+1 . Therefore, e 0 is in the orthogonal complement of ker(M † Y,s ), i.e. in the image of M Y,s , so there is some z ∈ C αsn with M Y,s z = e 0 . By construction of the matrix M Y,s , the equation
x s x s,1 x s,2 · · · x s,βs M Y,s z = x s x s,1 x s,2 · · · x s,βs e 0 = x s shows that x s is a C-linear combination of the m j=1 x c(kj) Y j,l , where k ∈ c −1 (s) and l ∈ [n]; the coefficients are given by z ∈ C αsn . In particular, x s ∈ R Y . Since the coordinate functions x s , s ∈ c(I) generate the character group X(T ) (thinking of characters as algebraic group morphisms G → C × ), we conclude X G·Y = X(T ). Hence, X G·Y is a group and G · Y is Zariski closed by Popov's criterion, Theorem B.4. B.3. Bijection between the stabiliser and the set of MLEs. So far we have given two descriptions of the set of MLEs given Y in an RDAG model. Corollary 4.8 gives a linear space of possible Λ, while Proposition A.3 gives an additive bijection between the MLEs and the stabiliser. Here we give an alternative (multiplicative) bijection between the set of MLEs and the stabiliser, when A(G, c) is a group. This is similar to [AKRS21, Proposition 3.9], which gives a multiplicative surjection between the MLEs and the stabiliser, for a Gaussian group model on a reductive group.
Proposition B.6. Consider the RDAG model on (G, c) where colouring c is compatible and A(G, c) is a group. Set A := A(G, c) SL and let Y ∈ R m×n be polystable under A. Given an MLE λa T a, where a ∈ A and λ is as in Proposition A.4, we have a bijection ϕ : A Y → {MLEs given Y } g → λg T a T ag.
Proof. For g ∈ A Y we have ag · Y = aY , which is of minimal norm in A · Y as λa T a is an MLE. Hence, ϕ(g) = λ(ag) T (ag) is another MLE given Y , by Proposition A.4, and we see that ϕ is well-defined. For surjectivity, let λ a T a be another MLE given Y . Then aY and aY are of minimal norm in A · Y , hence there is an orthogonal t ∈ T with taY = aY by Lemma A.8(c). We obtain g := a −1 t −1 a ∈ A Y , where we crucially used that A(G, c) (and hence A) is a group. The orthogonality of t gives ϕ(g) = λ a T a.
To prove injectivity, let g, g ∈ A Y be such that ϕ(g) = ϕ(g ). The latter implies g T a T ag = g T a T ag , which shows that h := ag g −1 a −1 is orthogonal. Therefore, h is diagonal, because any orthogonal upper triangular matrix is diagonal. Moreover, using g, g ∈ A Y we have haY = aY , i.e. h ∈ A aY . Note Y and aY have the same orbit (closure), where we again use that A is a group. Thus, aY is polystable as Y is polystable. Combining this with h(aY ) = aY and h diagonal implies h = id. Finally, id = h = ag g −1 a −1 shows g = g .
Remark 2. 3 (
3Motivation for the parametrisation via A(G, c)). The RDAG model on (G, c) will be the set M A as in (1), where A := A(G, c). This point of view is motivated by connections to invariant theory for transitive DAG models in [AKRS21, Section 5].
Proposition 2 . 7 .
27Fix a coloured DAG (G, c). The RDAG model on (G, c) is equal to M A(G,c) if and only if the colouring c is compatible.
Lemma 2. 8 .
8Fix a coloured DAG (G, c) with compatible colouring c. Then M A(G,c) is the set of matrices with Cholesky decomposition a T a for some a ∈ A(G, c).
Lemma 2. 9 .
9The DAG model on G is the model M A(G) . Proof. The LDL decomposition of Ψ = (id − Λ) T Ω −1 (id − Λ) is given by D = Ω −1 and L = (id − Λ) T . The Cholesky decomposition has
Theorem 3 . 4 .
34Consider the RDAG model on (G, c) where colouring c is compatible. The RDAG model on (G, c) is equal to the RCON model on (G u , c) if and only if: (a) G has no unshielded colliders; (b) G i G j for every pair of vertices i, j of the same colour; and (c) G (j→i) G (l→k) for every pair of edges j → i and l → k of the same colour.
has no unshielded colliders. (b) For the black (square) vertices, the graphs G i consist of one black vertex. For the blue (circular) vertices, the G i
Theorem 3 . 7 (
37Gaussian special case of [AMP97, Theorem 3.1], [Fry90, Theorem 5.6].). The DAG model on G is equal to the undirected Gaussian graphical model on G u if and only if G has no unshielded colliders.We now prove Theorem 3.4 via two propositions.
Proposition 3. 8 .
8Let (G, c) be a coloured DAG with compatible colouring c. The RDAG model on (G, c) is contained in the RCON model on (G u , c) if and only if (a) G has no unshielded colliders; (b) G i G j for every pair of vertices i, j of the same colour; and (c) G (j→i) G (l→k) for every pair of edges j → i and l → k of the same colour.
Proposition 3 . 9 .
39Let (G, c) be a coloured DAG with compatible colouring c such that (a) G has no unshielded colliders; (b) G i G j for every pair of vertices i, j of the same colour; and (c) G (j→i) G (l→k) for every pair of edges j → i and l → k of the same colour. Then the RCON model on (G u , c) is contained in the RDAG model on (G, c).
Proof of Theorem 3.4. If any of conditions (a),(b), and (c) do not hold, this rules out containment of the RDAG model in the RCON model, by Proposition 3.8, and hence rules out the two models being equal. If conditions (a),(b),(c) hold, we have containment of the RDAG model inside the RCON model (by Proposition 3.8) and the reverse containment (by Proposition 3.9).
Example 4. 5 .
5For our running example 1 3 2 , Theorem 4.4 says that the MLE exists uniquely provided Y (3) = 0 and Y (1) Y (2) is not parallel to Y (3) Y (3) . This holds almost surely as soon as we have one sample, as we mentioned in Example 0.1.
Example 4. 6 .
6Returning to Example 4.3, the MLE given Y exists provided M Y, = Y (3) · · · Y (7) = 0, and Y (1) Y (2) is not in the span of the other rows of M Y,• . The MLE is unique if and only if M Y,• is full row rank, since this also implies M Y, = 0.The proof of Theorem 4.4 gives Algorithm 1 for finding the MLE in an RDAG model with compatible colouring. The MLE is returned as entries of the matrices Λ and Ω. We give the MLE in a closed-form formula, as a collection of least squares estimators.
Remark 4. 7 .
7In Algorithm 1, the entries of Ω are given as {ω ss : s ∈ c(I)}. The entries of Λ are returned as {λ s,t : s ∈ c(I), t ∈ prc(s)}, which equals the set of edge colours by compatibility, since edge colour t only appears in prc(s) for one s.The proof of Theorem 4.4 directly gives a description of the set of MLEs.
Corollary 4. 8 .
8Consider the RDAG model on (G, c) where colouring c is compatible, with sample matrix Y ∈ R m×n . If (Λ, Ω) and (Λ , Ω ) are two MLEs, then Ω = Ω and t∈prc(s) (λ s,t − λ s,t )M (t) Y,s = 0, for all s ∈ c(I). Algorithm 1: Computing the MLE for an RDAG model input : A coloured DAG (G, c) and sample matrix Y output: An MLE given Y in the RDAG model on (G, c), if one exists for s ∈ c(I) do α s := |c −1 (s)|; β s := |prc(s)|; construct matrix M Y,s ∈ R (βs+1)×αsn ; P Y,s := orthogonal projection of M s : t ∈ [β s ]}; if P Y,s = M (0) Y,s then return MLE does not exist; else coefficients λ s,t are such that P Y,s = t∈prc(s) λ s,t M (t) Y,s ; ω ss := (α s n) −1 P Y,s − M return MLE for Λ and Ω
Lemma 4 . 10 .
410Fix α > 0 and, for γ ≥ 0, consider the family of functions
Proof of Theorem 4.4. Since colouring c is compatible, the RDAG model equals M A(G,c) , by Proposition 2.7. That is, for Ψ = (id − Λ) T Ω −1 (id − Λ) in the RDAG model, the matrix a = Ω −1/2 (id − Λ) is in M A(G,c) and satisfies Ψ = a T a. As usual, let α s := |c −1 (s)| and β s := |prc(s)|. The entry of Ω at vertex colour s is denoted ω ss and the edge colour entries of Λ that point towards colour s are labelled by λ s,t , t ∈ [β s ]. Using the construction of the matrices M Y,s ∈ R (βs+1)×αsn and that det(id − Λ) = 1, we compute − Y (Ψ) = − log det(Ψ) + tr(ΨS Y ) = log det(Ω
Furthermore
,λ s,t , t ∈ [β s ] are unique if and only if the vectors M (t)
theω ss are uniquely determined (if they exist), an MLE (if it exists) is unique if and only if for all s ∈ c(I) the vectors M (t) Y,s , t ∈ [β s ] are linearly independent. In combination with the condition for MLE existence from part (b) we deduce (c).
Example 4 . 13 .
413We consider measurements of the snout length and head length of dogs. These are the first two of the seven morphometric parameters in the study of clinical measurements of dog breeds in [MMB + 20]. We compare two RDAG models/square vertices 1 and 3 are the snout lengths of the two offspring. Blue/circular vertices 2 and 4 are their head lengths. The purple/triangular vertex 5 is the snout length of the parent and grey/pentagonal vertex 6 is the head length of the parent. The edges encode the dependence of the offsprings' traits on those of the parents.Maximum likelihood estimation in the left hand model is two copies of Example 4.12, one on the three odd variables, and one on the three even variables. Thus, given one sample a unique MLE exists almost surely. For the right hand model, Theorem 4.4 says that an MLE exists provided Y (5) = 0, Y (6) = 0 and neither Y (1) Y (3) nor Y (2) Y (4) are in span Y (5) Y (5) , Y (6) Y (6) . Hence an MLE exists almost surely with one sample. Moreover, the augmented sample matrices M Y,• and M Y, have full row rank almost surely provided n ≥ 2, hence the MLE exists uniquely with two samples, by Theorem 4.4.
Theorem 5. 3 .
3Consider the RDAG model on (G, c) where colouring c is compatible, and (G, c) has no edges between vertices of the same colour. We have the following bounds on the thresholds mlt e and mlt u max s
• = (Y (1) , Y (2) ) is almost surely not contained in the span of the other rows of M Y,• , hence mlt e = 1.On the other hand, we need at least n = 5 samples for generic linear independence of the rows (Y (3) , Y (3) ), . . . , (Y (7) , Y (7) ).
• = (Y (1) , Y (2) ) is not in the linear span of the other five rows of M Y,• , and we deduce mlt e = 2. For generic uniqueness of an MLE we need M Y,• ∈ R 6×2n to have full row rank six. For n ≥ 2, the submatrix
• = (Y (1) , Y (2) ) is generically contained in the linear span of the other rows of M Y,• . From the bounds we conclude that mlt e = 3. For uniqueness, when n = 3 the submatrices
Lemma 5. 8 .
8Consider the RDAG model on (G, c) where colouring c is compatible, and fix a vertex colour s. For n ≥ β s and generic Y ∈ R m×n the row vectors M We think of the mn entries of Y as indeterminates. Let M Y,s ∈ R βs×αsn be the matrix with rows M (1) Y,s , . . . , M (βs) Y,s . We construct an invertible β s ×β s submatrix of M Y,s .
Proposition 5 . 9 .
59Consider the RDAG model on (G, c) where colouring c is compatible, with no edges between any vertices of colour s. If α s = 1, then mlt e (s) = β s
Proposition 5 . 10 .
510Consider the RDAG model on (G, c) where colouring c is compatible, with no edges between any vertices of colour s. Then β s α s + 1 ≤ mlt u (s) ≤ β s + 2 − r s .Moreover, if r s = β s + 1 − (β s /α s ) then mlt u (s) ≤ β s + 1 − r s .
s ∈ R nαs is generically not in the span of the β s rows of M Y,s , because there are no edges between vertices of colour s. Thus, M Y,s generically has full row rank β s + 1, and mlt u (s) ≤ n = β s + 1 − r s . If β s = nα s , equivalently if r s = β s + 1 − (β s /α s ), an additional sample ensures rk(M Y,s ) = β s + 1 generically. Proof of Theorem 5.3. We have mlt e = max s mlt e (s) and mlt u = max s mlt u (s) by Theorem 4.4 parts (b) and (c). Taking the maximum of the lower and upper bounds in Propositions 5.9 and 5.10, over all vertex colours, gives the stated bounds. 5.2. A randomised algorithm. Proposition 5.11. For an RDAG model on (G, c), where colouring c is compatible, there is a randomised algorithm for computing the thresholds mlt e and mlt u .
Figure 2 .
2We generated RDAGs on 10 vertices, each edge present with probability 0.5 and number of edge colours in {2, 5, 10, 50, 100}. We sampled from the distribution 100 times and compared the MLE to the true parameter values on a log scale. The DAG MLE is shown in orange for comparison.
Figure 3
3Figure 3. We generated RDAGs on 10 vertices, each edge present with probability in {0.1, 0.3, 0.5, 0.7, 0.9} and 5 edge colours. For each edge probability we generated 50 random graphs, sampled from each one 100 times, and compared the RDAG and DAG MLEs. As above, blue is the RDAG MLE and orange is the DAG MLE.
Proposition A. 3 .
3Fix the RDAG model on (G, c) and set A := A(G, c) SL . If λa T a is an MLE given Y , where a ∈ A and λ > 0 is a scalar, then the set of MLEs given Y are in bijection with A Y under mapping
have inf a∈A f (a) = −∞ if and only if inf b∈A ± SL b · Y 2 = 0, i.e. if and only if Y is unstable. This shows parts (a) and (b).
1}. Thus, an MLE given Y exists. If an MLE exists, then the inner and outer infima in (19) are attained, and any MLE has the form in the statement.If A contains an orthogonal matrix of determinant −1, we can write from the outset a = τ ob with τ ∈ R >0 , b ∈ A SL and o orthogonal of determinant ±1. Setting x := τ 2 , we derive the same computation for f (a) (now with b ∈ A SL ), since o is orthogonal. The rest of the proof then works with A SL instead of A ± SL . Remark A.5. Given an RDAG model M A(G,c) with compatible colouring, we can always apply Proposition A.4 using stability under A(G, c) SL instead of the bigger set A(G, c) ± SL . Indeed, if the number of vertices of colour s, α s , is even for all s ∈ c(I), then A(G, c) only contains matrices of positive determinant, so A(G, c) ± SL = A(G, c) SL . If α s is odd for some vertex colour s, then A(G, c) contains an orthogonal matrix with determinant −1.Next, we return to the linear independence condition in Theorem 4.4(b).Lemma A.6. Consider the RDAG model on (G, c) where colouring c is compatible, and set A := A(G, c) SL . Assume for a non-zero Y ∈ R m×n that M
(W s ) αs = s a s p s (Z s ) αs = s (a s ) αs = 1 by the choice of p s ∈ K * s and since det(a) = s a αs s = 1. On the other hand, suppose W = (W s ) s ∈ V s p αs s − 1 ⊆ s K s . Set a s := p s (W s ), then we have s a αs s = 1, so the a s define some a ∈ T . Moreover, W := (a −1 s W s ) s ∈ H by definition of the a s and hence W = a · W is contained in T · H = A · Y . Proposition A.7. Consider an RDAG model on (G, c) with compatible colouring c and A := A(G, c) SL . Let Y ∈ R m×n be a sample matrix. Stability under A relates to linear independence conditions on the matrices M Y,s :
s has full row rank for all s ∈ c(I)Proof. Proposition A.4 in combination with Theorem 4.4 yields part (a) and the forwards direction of (c), while Lemma A.6 gives the backwards direction of (c). We obtain part (b) as a direct consequence of (a) and (c).For part (d), it suffices to see that a polystable Y has a trivial stabiliser A Y if and only if for all s ∈ c(I) the rows M s are linearly independent. So let Y be polystable. By construction of M Y,s , a matrix a ∈ A satisfies aY = Y if and all s ∈ c(I), where a s ∈ R × is the entry of a for vertex colour s, and a s,t ∈ R is the entry of a for the parent relationship colour encoded by (s, t), where t ∈ [β s ]. We note that (20) implies a s = 1 and t∈[βs] a s,t M(t) Y,s = 0, by polystability of Y and part (c). are linearly independent, then (20) has exactly one solution, namely a s = 1 and a s,t = 0 for all t ∈ [β s ]. Thus, if M s ∈ c(I), then A Y = {id}. On the other hand, if for some s ∈ c(I) s are linearly dependent, then t∈[βs] a s,t M (t)
s are linearly dependent for some s ∈ c(I). Proof of Theorem A.2. Combine Proposition A.7 with Theorem 4.4.
u ∈ U we can minimise each summand separately. For each s ∈ c(I), the affine space M s : t ∈ [β s ] has a unique element of minimal norm, call it M s . Hence, U · Y has a unique element of minimal norm, Y , determined by M(0) Y ,s = M s for all s ∈ c(I). (Note that there may be several u ∈ U with uY = Y .) To prove part (b), we use (the proof of) part (s ∈ c(I), hence tu · Y ≥ t Y . The latter inequality is strict if and only if there is strict inequality in (21) for at least one s. By uniqueness of Y , this is the case if and only if uY = Y .
are satisfied if and only if conditions (a) and (b) hold.
X
G·Y := (d s ) s∈c(I) ∈ X(T ) s∈c(I) x ds s ∈ R Y , where X(T ) ∼ = Z |c(I)| / Z · (α s ) s∈c(I) is the character group of T . Theorem B.4 (Popov's Criterion, [Pop89, Theorem 4]). Let G and Y be as above. The orbit G · Y is Zariski closed if and only if X G·Y is a group. Remark B.5. The group G = A(G, c) SL may not be connected as required in [Pop89, Theorem 4]. However, the orbit G · Y is Zariski-closed if and only if G • · Y is Zariskiclosed, where G • is the identity component of G. Thus, after restricting to G • = T • U we may assume that G is connected. Restricting to T • amounts to restricting to the torsion-free part of X(T ): if α is the greatest common divisor of all α s , s ∈ c(I), then T • ∼ = (g s ) s∈c(I) | s g αs/α s = 1 and X(T • ) = Z |c(I)| / Z · (α s /α) s∈c(I) . Second Proof of Lemma A.6. The matrix Y is semistable by Proposition A.4(b) and Theorem 4.4(b). Fix s ∈ c(I) and let M † Y,s be the Hermitian transpose of M Y,s . Since M
Multivariate Gaussian models. We consider m-dimensional Gaussian distributions with mean zero. Such a distribution is determined by its concentration (inverse covariance) matrix Ψ, a real m × m positive definite matrix. The density function is1. Preliminaries
3
2. Introducing RDAG models
6
3. Comparison of RDAG and RCON models
9
4. MLE: existence, uniqueness, and an algorithm
13
5. Maximum likelihood thresholds
18
6. Simulations
23
Appendix A. Connections to stability
27
Appendix B. Connections to Gaussian group models
32
1. Preliminaries
1.1.
where A is a subset of GL m , the real invertible m × m matrices. Many sets A can correspond to the same model M A . For instance, the full cone PD m is M A whenever A contains all invertible upper triangular matrices. When the set A is a group, the model M A is called a Gaussian group model[AKRS21].1.2. Maximum likelihood estimation. A maximum likelihood estimate (MLE) isa point in the model that maximizes the likelihood of observing some data. For n samples from a Gaussian model M ⊆ PD m , the data samples are the columns of a matrix Y ∈ R m×n . Assuming independent samples, the likelihood function is
with three vertex colours (blue/circular, black/square, and purple/triangular) and four edge colours (red/solid, green/squiggly, orange/dashed, and brown/dotted)
with two vertex colours (blue/circular and black/square) and five edge colours (red/solid, orange/dashed, green/squiggly, purple/zigzag, and brown/dotted)
A DAG is transitive if whenever k → j and j → i are in G then so is k → i. It is exactly the condition to make A(G) a group, see [AKRS21, Proposition 5.1].
Acknowledgements. We are grateful to Dominic Bunnett, Mathias Drton, Robin Evans, Caroline Uhler, and Piotr Zwiernik for helpful discussions. VM was supported by the University of Melbourne and NSF Grant CCF 1900460. PR was funded by the European Research Council (ERC) under the European's Horizon 2020 research and innovation programme (grant agreement no. 787840).Appendix A. Connections to stabilityWe give a characterization of maximum likelihood estimation for RDAG models, via stability under a group action. We generalise the definitions of stability to a set rather than a group. This offers an alternative to Corollary 4.8 for the set of MLEs.Fix a set of invertible m×m matrices A, with entries in R. Consider some Y ∈ R m×n . By analogy to an orbit and stabiliser under a group action, we define the orbit and stabiliser under a set A to be, respectively,This allows us to define stability notions analogous to the group situation.Definition A.1. We say that the matrix Y ∈ R m×n , under the set A, is (i) unstable if there exist a n ∈ A with a n Y → 0 as n → ∞, i. (2) (gh) ij = (gh) kl whenever j → i, l → k in G have c(ij) = c(kl); and (3) (gh) ij = 0 whenever j → i in G. For (1), observe that (gh) ii = g ii h ii . Thus, if c(i) = c(j) then (gh) ii = (gh) jj . For (2), take j → i, l → k in G with c(ij) = c(kl). Then. Conversely, assume (gh) ij = (gh) kl as a polynomial identity in the unknown entries of matrices g and h. By compatibility, c(i) = c(k), so g ii h ij = g kk h kl . Vertex and edge colours are disjoint and the sums over b(ij) and b(kl) only involve edge colours. Thus, (gh) ij = (gh) kl implies g ij h jj = g kl h ll , so h jj = h ll , and the sum over b(ij) must equal the sum over b(kl). This means c(j) = c(l), and the two collections (c(ip), c(pj)), p ∈ b(ij) and (c(kq), c(ql)), q ∈ b(kl) of ordered pairs counted with multiplicity agree. Compatibility ensures the correct colours on the vertices in b(ij) and b(kl) as well, hence G b(ij) G b(kl) .For(3), observe that if j → i in G then g ij = 0 = h ij and therefore (gh) ij = k∈b(ij) g ik h kj . The latter is zero for all g, h ∈ A(G, c) if and only if b(ij) = ∅. Thus, condition (3) is equivalent to the following: if j → i in G, then there does not exist
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Suppose (G, c) is a coloured DAG with compatible c. We give necessary and sufficient conditions for A(G, c) to be a group. B , We call a subset A ⊆ GL m linear if A = L ∩ GL m , where L is a linear subspace of m × m matrices. The butterfly criterionB.1. The butterfly criterion. Suppose (G, c) is a coloured DAG with compatible c. We give necessary and sufficient conditions for A(G, c) to be a group. We call a subset A ⊆ GL m linear if A = L ∩ GL m , where L is a linear subspace of m × m matrices.
1. Let A ⊆ GL m be a linear subspace of matrices containing the identity matrix. Then A is a group if and only if it is closed under multiplication. B Lemma, Lemma B.1. Let A ⊆ GL m be a linear subspace of matrices containing the identity matrix. Then A is a group if and only if it is closed under multiplication.
A group is closed under multiplication. Conversely, if A is closed under multiplication, to be a group it must also be closed under inverses. For a matrix a ∈ A, its = A, since id ∈ A. For a pair of vertices i, j ∈ I, define b(ij) = {k | i ← k, k ← j ∈ G}. Let G b(ij) denote the coloured subgraph on {i} ∪ {j} ∪ b(ij), with edges i ← k, k ← j for each k ∈ b(ij), and colours inherited from c. We call G b(ij) a butterfly graphProof. A group is closed under multiplication. Conversely, if A is closed under multi- plication, to be a group it must also be closed under inverses. For a matrix a ∈ A, its = A, since id ∈ A. For a pair of vertices i, j ∈ I, define b(ij) = {k | i ← k, k ← j ∈ G}. Let G b(ij) denote the coloured subgraph on {i} ∪ {j} ∪ b(ij), with edges i ← k, k ← j for each k ∈ b(ij), and colours inherited from c. We call G b(ij) a butterfly graph.
| [
"https://github.com/seigal/rdag."
]
|
[
"QCD CONDENSATES AND HADRON PARAMETERS IN NUCLEAR MATTER: SELF-CONSISTENT TREATMENT, SUM RULES AND ALL THAT",
"QCD CONDENSATES AND HADRON PARAMETERS IN NUCLEAR MATTER: SELF-CONSISTENT TREATMENT, SUM RULES AND ALL THAT"
]
| [
"E G Drukarev \nPetersburg Nuclear Physics Institute Gatchina\n188300St. PetersburgRussia\n",
"M G Ryskin \nPetersburg Nuclear Physics Institute Gatchina\n188300St. PetersburgRussia\n",
"V A Sadovnikova \nPetersburg Nuclear Physics Institute Gatchina\n188300St. PetersburgRussia\n"
]
| [
"Petersburg Nuclear Physics Institute Gatchina\n188300St. PetersburgRussia",
"Petersburg Nuclear Physics Institute Gatchina\n188300St. PetersburgRussia",
"Petersburg Nuclear Physics Institute Gatchina\n188300St. PetersburgRussia"
]
| []
| We review various approaches to the calculation of QCD condensates and of the nucleon characteristics in nuclear matter. We show the importance of their self-consistent treatment. The first steps in such treatment appeared to be very instructive. It is shown that the alleged pion condensation anyway can not take place earlier than the restoration of the chiral symmetry. We demonstrate how the finite density QCD sum rules for nucleons work and advocate their possible role in providing an additional bridge between the condensate and hadron physics. | 10.1016/s0146-6410(01)00153-3 | [
"https://arxiv.org/pdf/nucl-th/0106049v1.pdf"
]
| 17,917,279 | nucl-th/0106049 | 9d0494d53b88d4cfec559cace07efc0d1317d686 |
QCD CONDENSATES AND HADRON PARAMETERS IN NUCLEAR MATTER: SELF-CONSISTENT TREATMENT, SUM RULES AND ALL THAT
21 Jun 2001
E G Drukarev
Petersburg Nuclear Physics Institute Gatchina
188300St. PetersburgRussia
M G Ryskin
Petersburg Nuclear Physics Institute Gatchina
188300St. PetersburgRussia
V A Sadovnikova
Petersburg Nuclear Physics Institute Gatchina
188300St. PetersburgRussia
QCD CONDENSATES AND HADRON PARAMETERS IN NUCLEAR MATTER: SELF-CONSISTENT TREATMENT, SUM RULES AND ALL THAT
21 Jun 2001
We review various approaches to the calculation of QCD condensates and of the nucleon characteristics in nuclear matter. We show the importance of their self-consistent treatment. The first steps in such treatment appeared to be very instructive. It is shown that the alleged pion condensation anyway can not take place earlier than the restoration of the chiral symmetry. We demonstrate how the finite density QCD sum rules for nucleons work and advocate their possible role in providing an additional bridge between the condensate and hadron physics.
Introduction
The nuclear matter, i.e. the infinite system of interacting nucleons was introduced in order to simplify the problem of investigation of finite nuclei. By introducing the nuclear matter the problems of NN interaction in medium with non-zero baryon density and those of individual features of specific nuclei were separated. However, the problem of the nuclear matter is far from being solved. As we understand now, it cannot be solved in consistent way, being based on conception of NN interactions only. This is because the short distances, where we cannot help considering nucleons as composite particles, are very important.
There is limited data on the in-medium values of nucleon parameters. These are the quenching of the nucleon mass m and of the axial coupling constant g A at the saturation value ρ 0 with respect to their vacuum values. The very fact of existing of the saturation point ρ 0 is also the "experimental data", which is the characteristics of the matter as a whole. The nowadays models succeeded in reproducing the phenomena although the quantitative results differ very often.
On the other hand, the knowledge about the evolution of hadron parameters is important for understanding the evolution of the medium as a whole while the density ρ of distribution of the baryon charge number increases. (When ρ is small enough, it is just the density of the distribution of nucleons). There can be numerous phase transitions. At certain value of density ρ = ρ a the Fermi momenta of the nucleons will be so large, that it will be energetically favourable to increase ρ by adding heavier baryons instead of new nucleons. The nuclear or, more generally, hadronic matter may accumulate excitations with the pion quantum numbers, known as pion (or even kaon) condensations. Also the matter can transform to the mixture of hadrons and quark-gluon phase or totally to the quark-gluon plasma, converting thus to baryon matter. The last but not the least is the chiral phase transition. The chiral invariance is assumed to be one of the fundamental symmetries of the strong interactions.
The chiral invariance means that the Lagrangian as well as the characteristics of the system are not altered by the transformation ψ → ψe iαγ 5 of the fermion fields ψ. The model, suggested by Nambu and Jona-Lasinio (NJL) [1] provides a well-known example. The model describes the massless fermions with the four-particle interactions. In the simplest version of NJL model the Lagrangian is
L N JL =ψi∂ µ γ µ ψ + G 2 (ψψ) 2 + (ψiγ 5 ψ) 2 .(1)
If the coupling constant G is large enough, the mathematical (empty) vacuum is not the ground state of the system. Due to the strong four-fermion interaction in the Dirac sea the minimum of the energy of the system is reached at a nonzero value of the fermion density. This is the physical vacuum corresponding to the expectation value 0|ψψ|0 = 0.
This phenomenon is called "spontaneous chiral symmetry breaking". In the physical vacuum the fermion obtains the mass m = −2G 0|ψψ|0 (2) caused by the interaction with the condensate. On the other hand, the expectation value 0|ψψ|0 is expressed through the integral over the Dirac sea of the fermions. Of course, we have to introduce a cutoff Λ to prevent the ultraviolet divergence caused by the four-fermion interaction 0|ψψ|0 = − m π 2 Λ 0 dp p 2 (p 2 + m 2 ) 1/2 .
Thus Eqs. (2) Originally the NJL model was suggested for the description of the nucleons. Nowadays it is used for the quarks. The quark in the mathematical vacuum, having either vanishing or very small mass is called the "current" quark. The quark which obtained the mass, following Eq.(2) is called the "constituent" quark. In the nonrelativistic quark model the nucleon consists of three constituent quarks only.
Return to the nuclear matter. To understand, which of the hadron parameters are important, note that we believe nowadays most of the strong interaction phenomena at low and intermediate energies to be described by using effective low-energy pion-nucleon or pion-constituent quark Lagrangians. The πN coupling constant is:
g 2m = g A 2f π(4)
with f π being the pion decay constant. This is the well-known Goldberger-Treiman (GT) relation [2]. It means, that the neutron beta decay can be viewed as successive strong decay of neutron to π − p system and the decay of the pion. Thus, except the nuclear mass m * (ρ), the most important parameters will be the in-medium values g * A (ρ), f * π (ρ) and m * π (ρ). On the other hand, the baryonic matter as a whole is characterised by the values of the condensates, i.e. by the expectation values of quark and gluon operators. Even at ρ = 0 some of the condensates do not vanish, due to the complicated structure of QCD vacuum. The nonzero value of the scalar quark condensate 0|ψψ|0 reflects the violation of the chiral symmetry. In the exact chiral limit, when 0|ψψ|0 = 0 (and the current quark masses vanish also), the nucleon mass vanishes too. Thus, it is reasonable to think about the effective nucleon mass m * (ρ) and about the other parameters as the functions of the condensates. Of course, the values of the condensates change in medium. Also, some condensates which vanish in vacuum may have the nonzero value at finite density.
At the same time, while calculating the expectation value of the quark operatorψψ in medium, one finds that the contribution of the pion cloud depends on the in-medium values of hadron parameters. Hence, the parameters depend on condensates and vice versa. Thus we came to the idea of selfconsistent calculation of hadron parameters and of the values of condensates in medium. The idea of self-consistency is, of course, not a new one. We have seen just now, how NJL model provides an example. We shall try to apply the self-consistent approach to the analysis of more complicated systems.
The paper is organized as follows. In Sec.2 we review the present knowledge on the in-medium condensates. In Sec.3 we present the ideas and results of various approaches to calculation of the hadron parameters in medium. We review briefly the possible saturation mechanisms provided by these models. In Sec. 4 we consider the first steps to self-consistent calculation of scalar condensate and hadron parameters. The experience appeared to be very instructive. Say, the analysis led to the conclusion, that in any case the chiral phase transition takes place at the smaller values of density than the pion condensation. Hence, the Goldstone pions never condense. However, analysis of the behaviour of the solutions of the corresponding dispersion equation at larger densities appears to be useful.
Suggesting QCD sum rules at finite density as a tool for a future complete self-consistent investigation, we show first how the method works. This is done in Sec. 5. In Sec. 6 we present more detailed self-consistent scenario.
We present the results for symmetric matter, with equal densities of protons and neutrons.
Everywhere through the paper we denote quark field of the flavour i and colour a as ψ a i . We shall omit the colour indices in most of the cases, having in mind averaging over the colours for colourless objects. As usually, σ i , τ j and γ µ are spin and isospin Pauli matrices and 4 Dirac matrices correspondingly. For any four-vector A µ we denote A µ γ µ = A µ γ µ = A. The system of units with h = c = 1 is used.
Condensates in nuclear matter
Lowest order condensates in vacuum
The quark scalar operatorψψ is the only operator, containing minimal number of the field operators ψ, for which the expectation value, in vacuum has a nonzero value. One can find in the textbooks a remarkable relation, based on partial conservation of axial current (PCAC) and on the soft-pion theorems
m 2 πb f 2 π = − 1 3 0| F 5 b (0)[F 5 b (0), H(0)] |0(5)
with m πb , f π standing for the mass and the decay constant of pion, H being the density of the Hamiltonian of the system, while F 5 b are the charge operators, corresponding to the axial currents, b is the isospin index.
Presenting (effective) Hamiltonian
H = H 0 + H b(6)
with H 0 (H b ) conserving (explicitly breaking) the chiral symmetry, one finds that only H b piece contributes to Eq.(4). In pure QCD
H b = H QCD b = m uū u + m dd d ,(7)
with m u,d standing for the current quark masses. This leads to well known Gell-Mann-Oakes-Renner relation (GMOR) [3] 0|ūu +dd|0 = − 2f 2 π m 2 π m u + m d .
Of course, assuming SU(2) symmetry, which is true with the high accuracy, one finds 0|ūu|0 = 0|dd|0 . Numerical value 0|ūu|0 = (−240 MeV) 3 can be obtained from Eq. (8).
The quark masses can be obtained from the hadron spectroscopy relations and from QCD sum rules -see the review of Gasser and Leutwyler [4]. Thus the value of the quark condensate was calculated by using Eq. (8). The data on the lowest order gluon condensate (a is the colour index, α s is the QCD coupling constant) 0| α s π G a µν G µν a |0 ≈ (0.33 GeV) 4 (9) was extracted by Vainshtein et al. [5] from the analysis of leptonic decays of ρ and ϕ mesons and from QCD sum-rules analysis of charmonium spectrum [6].
Gas approximation
In this approximation the nuclear matter is treated as ideal Fermi gas of the nucleons. For the spin-dependent operators A s the expectation value in the matter M|A s |M = 0, although for the separate polarized nucleons N ↑ |A s |N ↑ may have a nonzero value. For the operators A which do not depend on spin the deviation of the expectation values M|A|M from 0|A|0 is determined by incoherent sum of the contributions of the nucleons. Thus for any SU(2) symmetric spin-independent operator A M|A|M = 0|A|0 + ρ N|A|N (10) with ρ standing for the density of nuclear matter and
N|A|N = d 3 x N|A(x)|N − 0|A(x)|0 .(11)
Since 0|A(x)|0 does not depend on x, Eq. (11) can be presented as
N|A|N = d 3 x N|A(x)|N − 0|A|0 · V N(12)
with V N being the volume of the nucleon.
The quark condensates of the same dimension d = 3 can be built by averaging of the expression ψBψ with B being an arbitrary 4 × 4 matrix over the ground state of the matter. However, any of such matrices can be presented as the linear combination of 5 basic matrices Γ A :
Γ 1 = I, Γ 2 = γ µ , Γ 3 = γ 5 , Γ 4 = γ µ γ 5 , Γ 5 = σ µν = 1 2 (γ µ γ ν − γ ν γ µ )(13)
with I being the unit matrix. One can see, that expectation valueψΓ 5 ψ vanishes in any uniform system, while those ofψΓ 3,4 ψ vanish due to conservation of parity.
The expectation value i M|ψ i γ µ ψ i |M = v µ (ρ) (14) takes the form v µ (ρ) = v(ρ)δ µ0 in the rest frame of the matter. It can be presented as v(ρ) = i n p i + n n
i 2 · ρ = i v i(15)
with n p(n) i standing for the number of the valence quarks of the flavour "i" in the proton (neutron). Due to conservation of the vector current Eq. (14) presents exact dependence of this condensate on ρ. For the same reason the linear dependence on ρ is true in more general case of the baryon matter v i (ρ) = 3 2 · ρ, v(ρ) = 3 · ρ.
As to the expectation value M|ψψ|M , it is quite obvious that Eq.(10) is true for the operator A =ψ i ψ i if the nucleon density is small enough. The same refers to the condensates of higher dimension. The question is: when will the terms nonlinear in ρ become important ?
Before discussing the problem we consider the lowest dimension condensates in the gas approximation.
Physical meaning of the scalar condensate in a hadron
It has been suggested by Weinberg [7] that the matrix element of the operatorψ i ψ i in a hadron is proportional to the total number of the quarks and antiquarks of flavour "i" in that hadron. The quantitative interpretation is, however, not straightforward. It was noticed by Donoghue and Nappi [8] that such identification cannot be exact, since the operatorψ i ψ i is not diagonal and can add quark-antiquark pair to the hadron. It was shown by Anselmino and Forte [9,10] that reasonable assumptions on the quark distribution inside the hadron eliminate the non-diagonal matrix elements. However there are still problems of interpretation of the diagonal matrix elements.
Present the quark field of any flavour
ψ(x) = s d 3 p (2π) 3 (2E) 1/2 b s (p)u s (p)e −i(px) + d + s (p)v s (p)e +i(px)(17)
with b s (p) and d + s (p) eliminating quarks and creating antiquarks with spin projection s, correspondingly. This leads to
h|ψψ|h = s d 3 p ū s (p)u s (p) 2E i (p) N + s (p) +v s (p)v s (p) 2E i (p) N − s (p) .(18)
Here N + s and N − s stand for the number of quarks and antiquarks. In the works [9,10] this formula was analysed for the nucleon in framework of quasi-free parton model for the quark dynamics. In this case the normalization conditions areū s (p)u s (p) =v s (p)v s (p) = 2m i with m i standing for the current mass. The further analysis required further assumptions.
In the nowadays picture of the nucleon its mass m is mostly composed of the masses of three valence quarks which are caused by the interactions inside the nucleon. In the orthodox nonrelativistic quark model, in which possible quark-antiquark pairs are ignored, we put E i = m i and find N| iψi ψ i |N = 3. In more realistic, relativistic models, there is also the contribution of the quark-antiquark pairs. Note also that in some approaches, say, in the bag models [11] or in the soliton model [12] the motion of the valence quarks is relativistic. This reduces their contribution to the expectation value N| iψi ψ i |N by about 30%, since m i /E i < 1.
The conventional nowadays picture of the nucleon is that it is the system of three valence quarks with the constituent masses M i ≈ m/3 and the number of quark-antiquark pairs
N| iψ i ψ i |N = 3 + s d 3 p a s (p) 2E(p) N s (p)(19)
with a s (p) =ū s (p)u s (p) =v s (p)v s (p), while N s (p) stands for the number of quark-antiquark pairs with momentum p. Thus, the right-hand side (rhs) of Eq. (19) can be treated as the total number of quarks and antiquarks only under certain assumptions about the dynamics of the constituents of qq pairs. They should remain light and their motion should be nonrelativistic, with a s ≈ 2m ≈ 2E.
In other models the deviation of the left-hand side (lhs) from the number 3 is a characteristic of the role of quark-antiquark pairs in the nucleon.
The value of N| iψi ψ i |N is related to the observables. The pion-nucleon σ-term, defined by analogy with Eq.(5) [13]
σ = 1 3 b N| F 5 b (0)[F 5 b (0), H(0)] |N(20)
provides by using Eq.(7)
N|qq|N = 2σ m u + m d(21)
withq q =ūu +dd .
On the other hand, [14,15] the σ-term is connected to the pion-nucleon elastic scattering amplitude T . Denote p, k(p ′ , k ′ ) as momenta of the nucleon and pion before (after) scattering. Introducing the Mandelstam variables s = (p + k) 2 , t = (k ′ − k) 2 we find the amplitude T (s, t, k 2 , k ′2 ) in the unphysical point to be
T (m 2 , 0, 0; 0) = − σ f 2 π .(23)
The experiments provide the data on the physical amplitude
T (m + m π ) 2 , 2m 2 π , m 2 π , m 2 π = − Σ f 2 π(24)
leading to [16,17]
Σ = (60 ± 7) MeV .(25)
The method of extrapolation of observable on-mass shell-amplitude to the unphysical point was developed by Gasser et al. [18,19]. They found
σ = (45 ± 7) MeV .(26)
Note that from the point of chiral expansion, the difference Σ − σ is of higher order, i.e. (Σ − σ)/σ ∼ m π .
The value σ = 45 MeV corresponds to N|qq|N ≈ 8. This is the strong support of the presence ofqq pairs inside the nucleon. However direct identification of the value N|qq|N with the total number of quarks and antiquarks is possible only under the assumptions, described above.
Quark scalar condensate in gas approximation
The formula for the scalar condensate in the gas approximation
M|qq|M = 0|qq|0 + 2σ m u + m d ρ(27)
or
M|qq|M = 0|qq|0 1 − σ f 2 π m 2 π ρ(28)
was obtained by Drukarev and Levin [20,21]. Of course, one can just substitute the semi-experimental value of σ, given by Eq. (26). However for the further discussion it is instructive to give a brief review of the calculations of the sigma-term.
Most of the early calculations of σ-term were carried out in the framework of NJL modelsee Eq.(1). The results were reviewed by Vogl and Weise [22]. In this approach the quarks with initially very small "current" masses m u ≈ 4 MeV, m d ≈ 7 MeV obtain relatively large "constituent" masses M i ∼ 300 − 400 MeV by four-fermion interaction, -Eq.(1). If the nucleon is treated as the weakly bound system of three constituent quarks, the σ-term can be calculated as the sum of those of three constituent quarks. The early calculations provided the value of σ ≈ 34 MeV, being somewhat smaller, than the one, determined by Eq. (26). The latter can be reproduced by assuming rather large content of strange quarks in the nucleon [8] or by inclusion of possible coupling of the quarks to diquarks [22,23].
In effective Lagrangian approach the Hamiltonian of the system is presented by Eq.(6) with H b determined by Eq. (7) while H 0 is written in terms of nucleon (or constituent quarks) and meson degrees of freedom. It was found by Gasser [24] that
σ =m dm dm (29) withm = m u + m d 2 .(30)
The derivation of Eq.(29) is based in Feynman-Hellmann theorem [25]. The nontrivial point of Eq. (29) is that the derivatives of the state vectors in the equation
m = N|H|N(31)
cancel.
Recently Becher and Leutwyler [26] reviewed investigations, based on pion-nucleon nonlinear Lagrangian. In this approach the contribution ofqq pairs is
σq q =m ∂m ∂m 2 π ∂m 2 π ∂m(32)
with the last factor in rhs
∂m 2 π ∂m = m 2 π m(33)
as follows from Eq. (8). The calculations in this approach reproduce the value σ ≈ 45 MeV.
Similar calculations [27] were carried out in framework of perturbative chiral quark model of Gutsche and Robson [28] which is based on the effective chiral Lagrangian describing quarks as relativistic fermions moving in effective self-consistent field. Theqq pairs are contained in pions. The value of the σ-term obtained in this model is also σ ≈ 45 MeV.
The Skyrme-type models provide somewhat larger values σ = 50 MeV [29] and σ = 59.6 MeV [30]. The chiral soliton model calculation gave σ = 54.3 MeV [31].
The results obtained in other approaches are more controversial. Two latest lattice QCD calculations gave σ = (18 ± 5) MeV [32] and σ = (50 ± 5) MeV [33]. The attempts to extract the value of σ-term directly from QCD sum rules underestimate it, providing σ = (25 ± 15) MeV [34] and σ = (36 ± 5) MeV [35].
We shall analyse the scalar condensate beyond the gas approximation expressed by Eq. (27), in Subsection 2.7.
Gluon condensate
Following Subsec.2.2 we write in the gas approximation
M| α s π G 2 |M = 0| α s π G 2 |0 + ρ N| α s π G 2 |N(34)
with notation G 2 = G a µν G µν a . Fortunately, the expectation value N| αs π G 2 |N can be calculated. This was done [36] by averaging of the trace of QCD energy-momentum tensor, including the anomaly, over the nucleon state. The trace is
θ µ µ = i m iψi ψ i − bα s 8π G 2(35)
with b = 11 − 2 3 n, while n stands for the total number of flavours. However, N|θ µ µ |N does not depend on n due to remarkable cancellation obtained by Shifman et al. [36]
N| h m hψh ψ h |N − 2 3 n h N| α s 8π G a µν G µν a |N = 0 .(36)
Here "h" denotes "heavy" quarks, i.e. the quarks, whose masses m h are much larger than the inverse confinement radius µ.
N|θ µ µ |N = − 9 8 N| α s π G 2 |N + Σm i N|ψ i ψ i |N(37)
with i standing for u, d and s. Since on the other hand N|θ µ µ |N = m one comes to
N| α s π G 2 |N = − 8 9 m − i m i N|ψ i ψ i |N .(38)
For the condensate
g(ρ) = M| α s π G 2 |M(39)
Drukarev and Levin [20,21] obtained in the gas approximation
g(ρ) = g(0) − 8 9 ρ m − i m i N|ψ i ψ i |N .(40)
In the chiral limit m u = m d = 0 and
g(ρ) = g(0) − 8 9 ρ m − m s N|ss|N .(41)
The expectation value N|ss|N is not known definitely. Donoghue and Nappi [8] obtained N|ss|N ≈ 1 assuming, that the hyperon mass splitting in SU(3) octet, is described by the lowest order perturbation theory in m s . Approximately the same result N|ss|N ≈ 0.8 was obtained in various versions of chiral perturbation theory with nonlinear Lagrangians [19]. The lattice calculations provide larger values, e.g. N|ss|N ≈ 1.6 [37]. On the contrary, the Skyrme model [12] and perturbative chiral quark model [27] lead to smaller values, N|ss|N ≈ 0.3. Since m ≫ m s it is reasonable to treat the second term in the brackets of rhs of Eq.(41) as a small correction. Thus we can put
g(ρ) = g(0) − 8 9 ρm ,(42)
which is exact in chiral SU(3) limit in gas approximation.
One can estimate the magnitude of nonlinear contributions to the condensate g(ρ). Averaging θ µ µ over the ground state of the matter one finds
g(ρ) = g(0) − 8 9 (m − m s N|ss|N )ρ − 8 9 ε(ρ)ρ + 8 9 m s S q (ρ)(43)
with ε(ρ) standing for the binding energy of the nucleon in medium, while S q (ρ) denotes nonlinear part of the condensate M|ss|M . One can expect the last factor to be small (otherwise we should accept that strange meson exchange plays large role in N − N interaction). Hence we can assume
g(ρ) = g(0) − 8 9 (m + ε(ρ))ρ + 8 9 m s N|ss|N ρ(44)
with nonlinear terms caused by the binding energy ε(ρ).
Thus, at least at the densities close to saturation value, corrections to the gas approximation are small. At ρ ≈ ρ 0 the value of the condensate g(ρ) differs from the vacuum value by about 6%.
Analysis of more complicated condensates
The condensates of higher dimension come from averaging of the products of larger number of operators of quark and (or) gluon fields. Such condensates appear also from the expansion of bilocal operators of lower dimension. Say, the simplest bilocal condensate C(x) = 0|ψ(0)ψ(x)|0 is gaugedependent (recall that the quarks interact with the vacuum gluon fields). To obtain the gaugeinvariant expression one can substitute
ψ(x) = ψ(0) + x µ D µ ψ(0) + 1 2 x µ x ν D µ D ν ψ(0) + · · ·(45)
with D µ being the covariant derivatives, which replaced the usual partial derivatives ∂ µ [38]. Due to the Lorentz invariance the expectation value C(x) depends on x 2 only. Hence, only the terms with even powers of x survive, providing in the chiral limit m q = 0
C(x) = C(0) + x 2 · 1 16 0|ψ α s π λ a 2 G µν a σ µν ψ|0 + . . . ,(46)
where λ a are Gell-Mann SU(3) basic matrices. The second term in right-hand side (rhs) of Eq. (46) can be obtained by noticing that 0|ψD µ D ν ψ|0 = 1 4 g µν 0|ψD 2 ψ|0 and by applying the QCD equation of motion in the form
D 2 − 1 2 α s π G µν a σ µν · λ a 2 − m 2 q ψ = 0 .(47)
The condensate 0|ψ αs π G µν a σ µν λ a 2 ψ|0 is usually presented "in units" of 0|ψψ|0 , i.e.
0|ψ α s π G µν a σ µν λ a 2 ψ|0 = m 2 0 0|ψψ|0(48)
with m 0 having the dimension of the mass. The QCD sum rules analysis of Belyaev and Ioffe [39] gives m 2 0 ≈ 0.8 GeV 2 for u and d quarks. However instanton liquid model estimation made by Shuryak [40] provides about three times larger value.
The situation with expectation values averaged over the nucleon is more complicated. There is infinite number of condensates of each dimension. This happens because the nonlocal condensates depend on two variables x 2 and (P x) with P being the four-dimensional momentum of the nucleon. Thus, even the lowest order term of expansion in powers of x 2 (x 2 = 0) contains infinite number of condensates. Say,
N|ψ(0)γ µ ψ(x)|N = P µ mφ a ((P x), x 2 ) + ix µ mφ b ((P x), x 2 )(49)
withφ(x) defined by expansion, presented by Eq. (45). The functionφ a (0, 0) is the number of the valence quarks of the fixed flavour in the nucleon. Presenting
ϕ a,b ((P x), 0) = ϕ a,b ((P x)); ϕ a,b ((P x)) = 1 0 dαe −iα(P x) φ a,b (α)(50)
we find the function φ a (α) to be the asymptotics of the nucleon structure function [41] and the expansion of ϕ a in powers of (P x) is expressed through expansion in the moments of the structure function. The next to leading order of the expansion ofφ a in powers of x 2 leads to the condensate
N|ψ(0) G µν γ ν γ 5 ψ(0)|N = 2P µ m · ξ a(51)
with G µν = 1 2 ε µναβ G a αβ · 1 2 λ a and
ξ a(b) = 1 0 dαθ a(b) (α, 0); θ a(b) (α, x 2 ) = ∂φ a(b) (α, x 2 ) ∂x 2 .(52)
The QCD sum rules analysis of Braun and Kolesnichenko [42] gave the value ξ a = −0.33 GeV 2 .
Using QCD equations of motion we obtain relations between the moments of the functions φ a and φ b . Denoting F = 1 0 dαF (α) for any function F we find, following Drukarev and Ryskin [43]
φ b = 1 4 φ a α ; φ b α = 1 5 φ a α 2 − 1 4 θ a ; θ b = 1 6 θ a α .(53)
Situation with the nonlocal scalar condensate is somewhat simpler, since all the matrix elements of the odd order derivatives are proportional to the current masses of the quarks. This can be shown by presenting D µ = 1 2 (γ µ D + Dγ µ ) followed by using the QCD equations of motion. Hence, in the chiral limit such condensates vanish for u and d quarks. The condensate containing one derivative can be expressed through the vector condensate and thus can be obtained beyond the gas approximation
M|ψ i D µ ψ i |M = m i v µ (ρ) .(54)
In the chiral limit m u = m d = 0 this condensate vanishes for u and d quarks. The even order derivatives contain the matrix elements corresponding to expansion in powers of x 2 which do not contain masses. In the lowest order there is the expectation value N|ψ αs π λ a 2 G µν a σ µν ψ|N -compare Eq. (48). It was estimated by Jin et al. [44] in framework of the bag model
N|ψ α s π λ a 2 G µν a σ µν ψ|N ≈ 0.6 GeV 2(55)
together with another condensate of the mass dimension 5
N|ψ α s π λ a 2 γ 0 G µν a σ µν ψ|N ≈ 0.66 GeV 2 .(56)
Considering the four-quark condensates, we limit ourselves to those with colourless diquarks with fixed flavours. The general formula for such expectation values is
Q AB ij = M|ψ i Γ A ψ iψj Γ B ψ j |M(57)
with A, B = 1 . . . 5, matrices Γ A,B are introduced in Eq. (13). For two lightest flavours there are thus 5 · 5 · 4 = 100 condensates. Due to SU(2) symmetry Q AB uu = Q AB dd = Q AB . Due to parity conservation only the diagonal condensates Q AA ij and also Q 12 ij = Q 21 ij and Q 34 ij = Q 43 ij have nonzero value in uniform matter. Since the matter is the eigenstate of the operatorψ i Γ 2 ψ i , we immediately find
Q 12 ij = ρ i M|ψ j ψ j |M(58)
with ρ i standing for the density of the quarks of i-th flavour. In the case, when the matter is composed of nucleons distributed with the density ρ, we put ρ i = n i ρ with n i being the number of quarks per nucleon.
For the four-quark scalar condensate Q 11 we can try the gas approximation as the first stepsee Eq.(10). Using Eq.(12) we find for each flavour
N|ψψψψ|N = d 3 x N|[ψ(x)ψ(x) − 0|ψψ|0 ] 2 |N + + 2 0|ψψ|0 N|ψψ|N + V N ( 0|ψψ|0 ) 2 − 0|ψψψψ|0 .(59)
One can immediately estimate the second term to be about −0.09 GeV 3 . This makes the problem of exact vacuum expectation value to be very important. Indeed, one of the usual assumptions is that [6] 0|ψψψψ|0 ≃ ( 0|ψψ|0 ) 2 .
This means that we assume the vacuum state |0 0| to give the leading contribution to the sum
0|ψψψψ|0 = n 0|ψψ|n n|ψψ|0(61)
over the complete set of the states |n with the quantum numbers of vacuum. Novikov et al. [45] showed, that Eq.(60) becomes exact in the limit of large number of colours N c → ∞. However, the contribution of excited states, e.g. of the σ-meson |σ σ| can increase the rhs of Eq. (59). Assuming the nucleon radius to be of the order of 1 Fm we find the second and the third terms of the rhs of Eq.(59) to be of the comparable magnitude. This becomes increasingly important in view of the only calculation of the 4-quarks condensate in the nucleon, carried out by Celenza et al. [46]. In this paper the calculations in the framework of NJL model show that about 75% of the contribution of the second term of rhs of Eq.(59) is cancelled by the other ones.
Quark scalar condensate beyond the gas approximation
Now we denote M|qq|M = κ(ρ) (62) and try to find the last term in the rhs of the equation
κ(ρ) = κ(0) + 2σ m u + m d · ρ + S(ρ) .(63)
The first attempt was made by Drukarev and Levin [20,21] in the framework of the meson-exchange model of nucleon-nucleon (NN) interactions. In the chiral limit m 2 π → 0 (neglecting also the finite size of the nucleons) one obtains the function S(ρ) as the power series in Fermi momenta p F . The lowest order term comes from Fock one-pion exchange diagram (Fig.1). The result beyond the chiral limit was presented in [43] In spite of the fact that the contribution of such mechanism to the interaction energy is a minor one, this contribution to the scalar condensate is quite important, since it is enhanced by the large factor (about 12) in the expectation value obtained by averaging the QCD Hamiltonian over the pion state. Using the lowest order πN coupling terms of the πN Lagrangian, we obtain in the chiral limit
π|qq|π = 2m 2 π m u + m d ≈ 2m π · 12(64)S(ρ) = −3.2 p F p F 0 ρ(65)
with p F being Fermi momentum of the nucleons, related to the density as
ρ = 2 3π 2 p 3 F ;(66)
p F 0 ≈ 268 MeV is Fermi momentum at saturation point. Of course, the chiral limit makes sense only for p 2 F ≫ m 2 π . This puts the lower limit for the densities, when Eq.(65) is true. The value, provided by one-pion exchange depends on the values of πN coupling g = g A /2f π and of the nucleon mass in medium. If we assume that these parameters are presented as power series in ρ (but not in p F ) at low densities, the contribution of the order ρ 5/3 comes from two-pion exchange with two nucleons in the two-baryon intermediate state - Fig.2.
In our paper [47] we found for p 2
F ≫ m 2 π S(ρ) = −3.2 p F p F 0 ρ − 3.1 p F p F 0 2 ρ + O(ρ 2 ).(67)
Although at saturation point m 2 π /p 2 F 0 ≈ 1/4, the discrepancy between the results of calculation of one-pion exchange term in the chiral limit and that with account of finite value of m 2 π is rather large [43]. However, working in the chiral limit one should use rather the value of Σ, defined by Eq. (25) for the sigma-term, since the difference between Σ and σ terms contains additional powers of m π [18]. This diminishes the difference of the two results strongly. Additional arguments in support of the use of the chiral limit at ρ close to ρ 0 were given recently by Bulgac et al. [48].
The higher order terms of the expansion, coming from the NN, N∆ and ∆∆ intermediate states, compensate the terms, presented in rhs of Eq.(67) to large extent. However these contributions are much more model-dependent. The finite size of the nucleons should be taken into account to regularize the logarithmic divergence. Some of the convergent terms are saturated by the pion momenta of the order k ∼ (m(m ∆ − m)) 1/2 ∼ 530 MeV, corresponding to the distances of the order [20], [21], the long-dashed and dashed curves present the pure [59] and modified [50] NJL model results. The dotted and dash-dotted lines present the result of calculation in hadronic model approach [47]. The dotted line corresponds to the physical value of the pion mass. The dash-dotted line shows the result in the chiral limit m 2 π =0.
0.4 Fm, where the finite size of the nucleons should be included as well. Also, the result are sensitive to the density dependence of the effective nucleon mass m * . This prompts, that a more rigorous analysis with the proper treatment of multi-nucleon configurations and of short distance correlations is needed. We shall return to the problem in Sec.4.
Anyway, the results for the calculation of the scalar condensate with the account of the pion cloud, produced by one-and two-pion exchanges looks as following. At very small values of density ρ < ∼ ρ 0 /8, e.g. p 2 F < ∼ m 2 π , only the two-pion exchanges contribute and
S(ρ) = 0.8ρ · ρ ρ 0 .(68)
Hence, S is positive for very small densities. However, for ρ > ∼ ρ 0 we found S < 0. The numerical results are presented in Fig.3. One can see the effects of interaction to slow down the tendency of restoration of the chiral symmetry, in any case requiring κ(ρ) = 0. There is also the negative [49] contribution to κ(ρ) of the vector meson field. The sign of this term can be understood in the following way. It was noticed by Cohen et al. [50] that the Gasser theorem [24] expressed by Eq.(29), can be generalized for the case of the finite densities with
S(ρ) = dε(ρ) dm ,(69)
while ε(ρ) is the binding energy. The contribution of vector mesons to rhs of Eq.
(67) is dV dm V dm V dm
with m V standing for the vector meson mass. Since the energy caused by the vector meson exchange V > 0 drops with growing m V , the contribution is negative indeed.
Another approach to calculation of the scalar condensate based on the soft pion technique was developed by Lyon group. Chanfray and Ericson [51] expressed the contribution of the pion cloud to κ(ρ) through the pion number excess in nuclei [52]. The calculation of Chanfray et al. [53] was based on the assumption, that GMOR relation holds in medium
f * 2 π m * 2 π = −m M|qq|M .(70)
This is true indeed, as long as the pion remains to be much lighter than the other bosonic states of unnatural parity. Under several assumptions on the properties of the amplitude of πN scattering in medium, the authors found κ(ρ) κ(0)
= 1 1 + ρσ/f 2 π m 2 π(71)
and κ(ρ) turns to zero at asymptotically large ρ only. This formula was obtained also by Ericson [54] by attributing the deviations from the linear law to the distortion factor, emerging because of the coherent rescattering of pions by the nucleons.
However, Birse and McGovern [55] and Birse [56] argued, that Eq.(71) is not an exact relation and results from the simplified model which accounts only for nucleon-nucleon interaction, mediated by one pion. In framework of linear sigma model, which accounts for the ππ interaction and the σ-meson exchange, the higher order terms of ρ expansion differ from those, provided by Eq. (71). The further development of calculation of the scalar condensate in the linear sigma model was made by Dmitrašinović [57].
In several works the function κ(ρ) was obtained in framework of NJL model. In the papers of Bernard et al. [58,59] the function κ(ρ) was calculated for purely quark matter. The approach was improved by Jaminon et al. [60] who combined the Dirac sea of quark-antiquark pairs with Fermi sea of nucleons. In all these papers there is a region of small values of ρ, where the interaction inside the matter is negligibly small and thus κ(ρ) changes linearly. However, the slope is smaller than the one, predicted by Eq. (27). In the modified treatment Cohen et al. [50] fixed the parameters of NJL model to reproduce the linear term. All the NJL approaches provide S > 0.
Recently Lutz et.al [61] suggested another hadronic model, based on chiral effective Lagrangian. The authors calculated the nonlinear contribution to the scalar condensate, provided by one-pion exchange. The value of S(ρ 0 ) appeared to be close to that, obtained in one-pion approximation of [43]. Hence, all the considered hadronic models provide S < 0 except for very small values of ρ.
There is a common feature of all the described results. Near the saturation point the nonlinear term S(ρ) is much smaller than the linear contribution. Thus, Eq. (27) can be used for obtaining the numerical values of κ(ρ) at ρ close to ρ 0 . Hence, the condensate |κ(ρ 0 )| drops by about 30% with respect to |κ(0)|.
Hadron parameters in nuclear matter
Nuclear many-body theory
Until mid 70-th the analysis of nuclear matter was based on nonrelativistic approach. The Schrödinger phenomenology for the nucleon in nuclear matter employed the Hamiltonian
H N R = − ∆ 2 2m * N R + U(ρ)(72)
and the problem was to find the realistic potential energy U(ρ). The deviation of nonrelativistic effective mass m * N R from the vacuum value m can be viewed as the dependence of potential energy on the value of three-dimensional momenta or "velocity dependent forces" [62]. The results of nonrelativistic approach were reviewed by Bethe [63] and by Day [64].
Since the pioneering paper of Walecka [65] the nucleon in nuclear matter is treated as a relativistic particle, moving in superposition of vector and scalar fields V µ (ρ) and Φ(ρ). In the rest frame of the matter V µ = δ µ0 V 0 and Hamiltonian of the nucleon with the three-dimensional momentump is
H = (ᾱp) + β(m + Φ(ρ)) + V 0 (ρ) · I(73)
withᾱ = 0σ σ 0 and β = 1 0 0 −1 being standard Dirac matrices. Since the scalar "σ-meson" is rather an effective way to describe the system of two correlated pions, its mass, as well as the coupling constants of interaction between these mesons and the nucleons are the free parameters. They can be adjusted to fit either nuclear data or to reproduce the data on nucleon-nucleon scattering in vacuum. The numerous references can be found, e.g. in [66]. In both cases the values of V 0 and Φ appeared to be of the order of 300-400 MeV at the density saturation point.
The large values of the fields V 0 and Φ require the relativistic kinematics to be applied for the description of the motion of the nucleons.
In the nonrelativistic limit the Hamiltonian (73) takes the form of Eq.(72) with m * N R being replaced by Dirac effective mass m * , defined as
m * = m + Φ(74)
and
U = V 0 + Φ .(75)
At the saturation point the fields V 0 and Φ compensate each other to large extent, providing U ≈ −60 MeV. This explains the relative success of Schrödinger phenomenology. However, as shown by Brockmann and Weise [67], the quantitative description of the large magnitude of spin-orbit forces in finite nuclei requires rather large values of both Φ and V 0 .
In the meson exchange picture the scalar and vector fields originate from the meson exchange between the nucleons of the matter. The model is known as quantum hadrodynamics -QHD. In the simplest version (QHD-1) only scalar σ-mesons and vector ω mesons are involved. In somewhat more complicated version, known as QHD-2 [68] some other mesons, e.g. the pions, are included. The matching of QHD-2 Lagrangian with low energy effective Lagrangian was done by Furnstahl and Serot [69].
The vector and scalar fields, generated by nucleons, depend on density in different ways. For the vector field
V (ρ) = 4 d 3 p (2π) 3 N V (p)g V θ(p F − p)(76)
with g V the coupling constant, while
N V = (ū N γ 0 u N )/2E with u N (p) standing for nucleon bispinors while ε(p, ρ) = V 0 (ρ) + p 2 + m * 2 (ρ) 1/2 .(77)
One finds immediately that N V = 1 and thus V (ρ) is exactly proportional to the density ρ. On the other hand, in the expression for the scalar field
Φ(ρ) = 4 d 3 p (2π) 3 N s (p) · g s θ(p F − p)(78)
the factor N s = m * /E. Thus, the scalar field is a complicated function of density ρ.
The saturation value of density ρ 0 can be found by minimization of the energy functional
E(ρ) = 1 ρ ρ 0 ε F (ρ)dρ(79)
with ε F (ρ) = ε(p F , ρ) being the single-particle energy at the Fermi surface. Thus, in QHD the saturation is caused by nonlinear dependence of the scalar field Φ on density.
The understanding of behaviour of axial coupling constant in nuclear matter g A (ρ) requires explicit introduction of pionic degrees of freedom. The quenching of g A at finite densities was predicted by Ericson [70] from the analysis of the dispersion relations for πN scattering. The result was confirmed by the analysis of experimental data on Gamow-Teller β-decay of a number of nuclei carried out by Wilkinson [71] and by investigation of beta decay of heavier nuclei -see, e.g., [72]
g A (0) = 1.25 ; g A (ρ 0 ) = 1.0 .(80)
The quenching of g A as the result of polarization of medium by the pions was considered by Ericson et al. [73]. The crucial role of isobar-hole excitations in this phenomena was described by Rho [74].
Turning to the characteristics of the pions, one can introduce effective pion mass m * π by considering the dispersion equation for the pion in nuclear matter (see, e.g., the book of Ericson and Weise [75]):
ω 2 − k 2 − Π p (ω, k) − m * 2 π = 0 .(81)
Here ω and k are the pion energy and three-dimensional momenta, Π p is the p-wave part of the pion polarization operator. Hence, Π p contains the factor k 2 . The pion effective mass is
m * 2 π = m 2 π + Π s (ω, k)(82)
with Π s being the s-wave part of polarization operator.
Polarization operator Π s (as well as Π p ) is influenced strongly by the nucleon interactions at the distances, which are much smaller than the average inter-nucleon distances ≈ m −1 π . Strictly speaking, here one should consider the nucleon as a composite particle. However, there is a possibility to consider such correlations in framework of hadron picture of strong interactions by using Finite Fermi System Theory (FFST), introduced by Migdal [76]. In framework of FFST the amplitudes of short-range baryon (nucleons and isobars) interactions are replaced by certain constant parameters. Hence, behaviour of m * π can be described in terms of QHD and FFST approaches. As well as any model based on conception of NN interaction, QHD faces difficulties at small distances. The weak points of the approach were reviewed by Negele [77] and by Sliv et al. [78]. Account of the composite structure of nucleon leads to the change of some qualitative results. Say, basing on the straightforward treatment of the Dirac Hamiltonian Brown et al. [79] found a significant term in the equation of state, arising from virtual NN pairs, generated by vector fields. The term would have been important for saturation. However, Jaroszewicz and Brodsky [80] and also Cohen [81] found that the composite nature of nucleon suppresses such contributions.
Anyway, to obtain the complete description, we need a complementary approach, accounting for the composite structure of hadrons. For pions it is reasonable to try NJL model.
Calculations in Nambu-Jona-Lasinio model
In NJL model the pion is the Goldstone meson, corresponding to the breaking of the chiral symmetry. The pion can be viewed as the solution of Bethe-Salpeter equation in the pseudoscalar quarkantiquark channel. The pion properties at finite density were investigated in frameworks of SU (2) and SU(3) flavour NJL model [58,59]. It was found that the pion mass m * π (ρ) is practically constant at ρ < ∼ ρ 0 , increasing rapidly at larger densities, while f * π (ρ) drops rapidly. These results were obtained rather for the quark matter. Anyway, as we mentioned in Subsec.2.7, at small ρ the condensate κ(ρ), obtained in this approach, does not satisfy the limiting law, presented by Eq. (27).
However, the qualitatively similar results were obtained in another NJL analysis, carried out by Lutz et al. [82]. The slope of the function κ(ρ) satisfied Eq. (27). The pion mass m * π (ρ) increased with ρ slowly, while f * π (ρ) dropped rapidly. The in-medium GMOR relation, expressed by Eq.(70) was satisfied as well.
Jaminon and Ripka [83] considered the modified version of NJL model, which includes the dilaton fields. This is the way to include effectively the gluon degrees of freedom. The results appeared to depend qualitatively on the way, the dilation fields are included into the Lagrangian. The pion mass can either increase or drop with growing density. Also the value of the slope of κ(ρ) differs strongly in different versions of the approach. In the version, which is consistent with Eq. (27) the behaviour of f * π (ρ) and m * π (ρ) is similar to the one, obtained in the other papers, mentioned in this subsection. Note, however, that the results which predict the fast drop of f * π (ρ) have, at best, a limited region of validity. This is because the pion charge radius r π is connected to the pion decay constant by the relation obtained by Carlitz and Creamer [84]
r 2 π 1/2 = √ 3 2πf π(83)
providing r 2 π 1/2 ≈ 0.6 Fm. Identifying the size of the pion with its charge radius, we find that at r 2 π (ρ) 1/2 becoming of the order of the confinement radius r c ∼ 1 Fm, the confinement forces should be included and straightforward using of NJL is not possible any more. Thus, NJL is definitely not true for the densities, when the ratio f * π (ρ)/f π becomes too small. Anyway, one needs
f * π (ρ) f π > ∼ 0.6 .(84)
For the results, obtained in [82] this means that they can be true for ρ ≤ 1.3ρ 0 only.
Quark-meson models
This class of models, reviewed by Thomas [11] is the result of development of MIT bag model, considering the nucleon as the system of three quarks in a potential well. One of the weak points of the bag-model approach is the absence of long-ranged forces in NN interactions. In the chiral bag model (CBM) the long-ranged tail is caused by the pions which are introduced into the model by requirement of chiral invariance. In the framework of CBM the pions are as fundamental degrees of freedom as quarks. In the cloudy bag model these pions are considered as the bound states ofqq pairs. The model succeeded in describing the static properties of free nucleons.
Another model, suggested by Guichon [85] is a more straightforward hybrid of QHD and QCD. The nucleon is considered as a three-quark system in a bag. The quarks are coupled to σ-and ωmesons directly. Although this quark-meson coupling model (QMC) was proposed by its author as "a caricature of nuclear matter", it was widely used afterwards. The parameters of σ-and ω-mesons and the bag radius, which are the free parameters of the model were adjusted to describe the saturation parameters of the matter. The fields Φ and V appear to be somewhat smaller than in QHD. Thus, the values of m * /m and g * A /g are quenched less than in QHD [86]. On the other hand, the unwanted NN pairs are suppressed. The nonlinearity of the scalar field is the source of saturation.
The common weak point of these models are well known. Say, there is no consistent procedure to describe the overlapping of the bags. It is also unclear, how to make their Lorentz transformations.
Skyrmion models
This is the class of models with much better theoretical foundation. They originate from the old model, suggested by Skyrme [87]. The model included the pions only, and the nucleon was the soliton. Later Wess and Zumino [88] added the specific term to the Lagrangian, which provided the current with the non-vanishing integral of the three-dimensional divergence. That was the way, how the baryon charge manifested itself.
Thus, in framework of the approach most of the nucleon characteristics are determined by Dirac sea of quarks and by the quark-antiquark pairs, which are coupled into the pions. The model can be viewed as the limiting case R → 0 of the chiral bag model, where the description in terms of the mesons at r > R is replaced by description in terms of the quarks at r < R [89].
In the framework of the Skyrme model Adkins et al. [90,91] calculated the static characteristics of isolated nucleons. A little later Jackson et al. [92] investigated NN interaction in this model. The model did not reproduce the attraction in NN potential. It was included into modified Skyrme Lagrangian by Rakhimov et al. [93] in order to calculate the renormalization of g A , m and f π in nuclear matter. The magnitude of renormalization appeared to be somewhat smaller than in QHD.
The approach was improved by Diakonov and Petrov -see a review paper [94] and references therein. The authors build the chiral quark-soliton model of the nucleon. It is based on quarkpion Lagrangian with the Wess-Zumino term and with spontaneous chiral symmetry breaking. The nucleon appeared to be a system of three quarks, moving in a classical self-consistent pion field. The approach succeeded in describing the static characteristics of nucleon. It provided the proper results for the parton distributions as well. However, the application of the approach to description of the values of nucleon parameters in medium is still ahead.
Brown-Rho scaling
Brown and Rho [95] assumed that all the hadron characteristics, which have the dimension of the mass change in medium in the same manner. The universal scale was assumed to be
χ(ρ) = (−κ(ρ)) 1/3 .(85)
Thus, the scaling which we refer to as BR1 is
m * (ρ) m = f * π (ρ) f π = χ(ρ) χ(0) .(86)
The pion mass was assumed to be an exception, scaling as
m * π (ρ) m π = χ(ρ) χ(0) 1/2 .(87)
Thus, BR1 is consistent with in-medium GMOR relation. Also, in contrast to NJL, the pion mass drops with density.
Another point of BR1 scaling is the behaviour
g * A (ρ) = g A (0) = const .(88)
Consistency of Eqs. (80) and (88) can be explained in such a way. Renormalization expressed by Eq. (80) is due to ∆-hole polarization of medium. It takes place at moderate distances of the order m −1 π , reflecting rather the properties of the medium, but not the intrinsic properties of the nucleon, which are discussed here.
Another version of Brown-Rho scaling [96], which we call BR2 is based on the in-medium GMOR relation, expressed by Eq. (70). It is still assumed that
m * m = f * π f π ,(89)
but the pion mass is assumed to be constant
m * π ≈ m π ,(90)
and thus f *
π f π = κ(ρ) κ(0) 1/2(91)
instead of 1/3 law in BR1 version -Eq. (86). Note, however, that assuming m * π (ρ 0 ) = 1.05m π [96] we find, using the results of subsection 2.6
f * π (ρ 0 ) f π = 0.76 .(92)
This is not far the limit determined by Eq. (84). At larger densities the size of the pion becomes of the order of the confinement radius. Here the pion does not exist as a Goldstone boson any more. In any case, some new physics should be included at larger densities. If Eq.(91) is assumed to be true, this happens at ρ ≈ 1.6ρ 0 .
QCD sum rules
In this approach we hope to establish some general relations between the in-medium values of QCD condensates and the characteristics of nucleons.
The QCD sum rules were invented by Shifman et al. [6] and applied for the description of the mesonic properties in vacuum. Later Ioffe [97] expanded the method for the description of the characteristics of nucleons in vacuum. The main idea is to build the function G(q 2 ) which describes the propagation of the system ("current") with the quantum numbers of the proton. (The usual notation is Π(q 2 ). We used another one to avoid confusion with pion polarization operator, expressed by Eq. (81)). The dispersion relation
G(q 2 ) = 1 π Im G(k 2 ) k 2 − q 2 dk 2(93)
is considered at q 2 → −∞. Imaginary part in the rhs is expressed through parameters of observable hadrons. Due to asymptotic freedom of QCD lhs of Eq.(93) can be presented as perturbative series in −q 2 with QCD vacuum condensates as coefficients of the expansion. Convergence of the series means that the condensates of lower dimension are the most important ones.
The method was used for the calculation of characteristics of the lowest lying hadron states. This is why the "pole+continuum" model was employed for the description of Im G(k 2 ) in the rhs of Eq. (93). This means that the contribution of the lowest lying hadron was treated explicitly, while all the other excitations were approximated by continuum. In order to emphasise the contribution of the pole inverse Laplace (Borel) transform was applied to both sides of Eq.(93) in the papers mentioned above. The Borel transform also removes the polynomial divergent terms.
Using QCD sum rules Ioffe [97] found that the nucleon mass vanishes if the scalar condensate turns to zero. Numerically, [39,97,98]
m = −2(2π) 2 0|qq|0 1/3 .(94)
Later the method was applied by Drukarev and Levin [20,21,99] for investigation of properties of nucleons in the nuclear matter. The idea was to express the change of nucleon characteristics through the in-medium change of the values of QCD condensates. The generalization for the case of finite densities was not straightforward. Since the Lorentz invariance is lost, the function G m (q) describing the propagation of the system in medium depends on two variables, e.g. G m = G(q 2 , q 0 ). Thus, each term of expansion of G m in powers of q −2 may contain infinite number of local condensates. In the rhs of dispersion relation it is necessary to separate the singularities, connected with the nucleon from those, connected with excitation of the matter itself.
We shall return to these points in Sec.5. Here we present the main results. The method provided the result for the shift of the position of the nucleon pole. The new value is expressed as a linear combination of several condensates with vector condensate v(ρ) and scalar condensate κ(ρ) being most important [20,21,99]
m m = m + C 1 κ(ρ) + C 2 v(ρ) .(95)
On the other hand,
m m − m = U 1 + 0 U m(96)
with U being single-particle potential energy of the nucleon. Hence, the scalar forces are to large extent determined by the σ-term.
The Dirac effective mass was found to be proportional to the scalar condensate
m * (ρ) = κ(ρ)F (ρ)(97)
with F (ρ) containing the dependence on the other condensates, e.g. on vector condensate v(ρ).
Using Eqs. (15) and (16) we see, that v(ρ) is linear in ρ. Thus, the main nonlinear contributions to the energy E(ρ) presented by Eq.(79) come from nonlinearities in the function κ(ρ). For the saturation properties of the matter the sign of the contribution S(ρ) becomes important. The nonlinearities of the condensate κ(ρ) can be responsible for the saturation if S < 0. Calculations of Drukarev and Ryskin [43] show that the saturation can be obtained at reasonable values of density with reasonable value of the binding energy. Of course, this result should not be taken too seriously, since it is very sensitive to the exact values of σ-term. It can be altered also by the account of higher order terms. (However, as noted by Birse [56], the QHD saturation picture is also very sensitive to the values of the parameters). Similar saturation mechanism was obtained recently in the approach developed by Lutz et.al [61]. Anyway, it can be a good starting point to analyse the problem.
First step to self-consistent treatment
As we have seen in Sec.2 in the gas approximation the scalar condensate κ(ρ) is expressed through the observables. However, beyond the gas approximation it depends on a set of other parameters. Here we show how such dependence manifests itself in a more rigorous treatment of the hadronic presentation of nuclear matter.
Account of multi-nucleon effects in the quark scalar condensate
Now we present the main equations, which describe the contribution of the pion cloud to the condensate κ(ρ). Recall, that the pions are expected to give the leading contribution to the nonlinear part S(ρ) due to the large expectation value π|qq|π -Eq. (64).
In order to calculate the contribution we employ the quasiparticle theory, developed by Migdal for the propagation of pions in matter [100]. Using Eq.(69), we present S(ρ) through the derivative of the nucleon self-energy with respect to m 2 π :
S = B S B ; S B = −C B Υ d 3 p (2π) 3 d 3 kdω (2π) 4 · i Γ 2 B D 2 (ω, k)g B (p − k) − Γ 02 B D 2 0 (ω, k)g 0 B (p − k) .(98)
Here B labels the excited baryon states with propagators g B and πNB vertices Γ B . The pion propagator D includes the multi-nucleon effects (D −1 is the lhs of Eq. (81)). The second term of the rhs of Eq. (98) , with the index "0" corresponding to the vacuum values, subtracts the terms, which are included into the expectation value already. The coefficient C B comes from summation over the spin and the isospin variables. Integration over nucleon momenta p is limited by the condition p ≤ p F . The factor Υ stands for the expectation value of the operatorqq in pion, i.e. Υ = π|qq|π = m 2 π /m. Of course, Eq. while A(p ′ ; ω, k) stands for the amplitude of the forward πN scattering (all the summation over the spin and isospin variables is assumed to be carried out) on the nucleon of the matter with the threedimensional momentum p ′ . Of course, the pion is not on the mass-shell.
D = D 0 + D 0 ΠD (99) with Π(ω, k) = 4π p F d 3 p ′ (2π) 3 A(p ′ ; ω, k) ,(100)
Neglecting the interactions inside the bubbles of Fig.5 (this is denoted by the upper index "(0)") we can present
A (0) = A (0) B ; Π (0) = Π (0) B A 0 B = c B Γ 2 B (k)Λ B (p ′ ; ω, k)(101)
with c B being a numerical coefficient,
Λ B (p ′ ; ω, k) = g B (ε ′ + ω,p ′ +k) + g B (ε ′ − ω,p ′ −k) .(102)
The factors Γ 2 B (k) come from the vertex functions. Considering p-wave part of polarization operator only (the s-wave part is expressed through the pion effective mass m * π -Eq.(82)), we present
Γ 2 B (k) =g 2 πN B k 2 d 2 N B (k)(103)
with d 2 N B accounting for the finite size of the baryons,g πN B is the coupling constant. Starting the analysis with the contribution of the nucleon intermediate state (B = N) to Eqs. (101) and (102), we see that in the nonrelativistic limit we can present the first term in rhs of Eq.(102) as
g N (ε ′ + ω,p ′ +k) = θ(|p ′ +k| − p F ) ω + ε p ′ − ε p ′ +k(104)
(similar presentation can be written for the second term) with ε q = q 2 /2m * + U, while U stands for the potential energy. Hence, the terms, containing U cancel and all the dependence on the properties of the matter enters through the effective mass m * . This enables to obtain the contribution to the polarization operator
Π (0) N = −4g 2 πN N k 2 d 2 N N (k) m * p F 2π 2 φ (0) N (ω, k)(105)
with explicit analytical expression for φ (0) N (ω, k) presented in [75,104], the static long-wave limit is φ Such approach does not include the particle-hole interactions in the bubble diagram of Fig.5. The short-range correlations can be described with the help of effective FFST constants, as it was mentioned above. Using the Dyson equation for the short-range amplitude of nucleon-hole scattering one finds
Π N = −4g 2 πN N k 2 d 2 N N (k) m * p F 2π 2 φ N (ω, k)(106)
with
φ N (ω, k) = φ (0) N (ω, k) 1 + g ′ N N φ (0) N (ω, k) ,(107)
if only the nucleon intermediate states are included.
The long-ranged correlations inside the bubbles were analysed by Dickhoff et al. [101]. It was shown, that exchange by the renormalized pions inside the bubbles ("bubbles in bubbles") can be accounted for by the altering of the values of FFST constants. The change in the numerical values does not appear to be large.
The usual approach includes also the ∆-isobar states in the sums in Eq. (98). Until the particlehole correlations are included, the total p-wave operator Π (0) is just the additive sum of the nucleon and isobar terms, i.e. Π (0) = Π ∆ under a reasonable assumption on the propagation of ∆-isobar in medium (see below). However, account of the short-range correlations makes the expression for the total p-wave polarization operator more complicated. We use the explicit form presented by Dickhoff et al. [102] Π
= Π N + Π ∆ with Π N = Π (0) N 1 − (γ ∆ − γ ∆∆ ) Π (0) ∆ k 2 E(108)Π ∆ = Π (0) ∆ 1 + (γ ∆ − γ N N ) Π (0) N k 2 E .(109)
Denominator E has the form
E = 1 − γ N N Π (0) N k 2 − γ ∆∆ Π (0) ∆ k 2 + γ N N γ ∆∆ − γ 2 ∆ Π (0) N Π (0) ∆ k 4 .(110)
The effective constants Γ are related to FFST parameters g ′ as follows:
γ N N = C 0 g ′ N Ñ g 2 πN N ; γ ∆ = C 0 g ′ N ∆ g πN NgπN ∆ , γ ∆∆ = C 0 g ′ ∆∆ g 2 πN ∆ ,(111)
where C 0 is the normalization factor for the effective particle-hole interaction in nuclear matter. We use C 0 = π 2 /p F m * , following [76]. (Note, that there is some discrepancy in the notations used by different authors. Our parameters γ coincide with those, used in [102]. We use the original FFST parameters g ′ of [76], which are related to the constants G ′ 0 of [102] as g ′ = G ′ /2). The short-range interactions require also renormalization of the vertices Γ 2
πN B → Γ 2 πN B x 2 πN B with x πN N = 1 + (γ ∆ − γ ∆∆ ) Π (0) ∆ k 2 E ; x πN ∆ = 1 + (γ ∆ − γ N N ) Π (0) N k 2 E .(112)
In our paper [103] we calculated the contribution S(ρ), presented by Eq.
Interpretation of the pion condensate
The pion dispersion equation [75,104] is
ω 2 = m * 2 π + k 2 1 + χ(ω, k)(113)
with the function χ introduced as Π p (ω, k) = −k 2 χ(ω, k). It is known to have three branches of solutions ω i (k) (classified by the behaviour of the functions ω i (k) at k → 0). If the function χ(ω, k) includes nucleons only as intermediate states and does not include correlations, we find k 2 χ → 0 at k → 0. This is the pion branch for which ω π (0) = m * π . If the correlations are included, the denominator in the rhs of Eq.(107) may turn to zero at k → 0, providing the sound branch with ω s (0) = 0. Inclusion of ∆-isobars causes the contribution to χ(ω, k), proportional to [m ∆ − m − ω] −1 . Thus, there is a solution with ω ∆ (0) = m ∆ − m, called the isobar branch.
The trajectories of the solutions of Eq.(81) on the physical sheet of Riemann surface were studied by Migdal [104]. Their behaviour on the unphysical sheets was investigated recently by Sadovnikova [105] and by Sadovnikova and Ryskin [106]. In these papers it was shown, that besides the branches, mentioned above, there is one more branch starting from the value ω c (0) = m * π and moving on the unphysical sheet for larger k > 0. The branch comes to the physical sheet at certain value of k if the density exceeds certain critical value ρ C . Here ω c is either zero or purely imaginary and thus ω 2 c ≤ 0. (However, this is true if the isobar width Γ ∆ = 0, for the finite values of Γ ∆ we find ω c to be complex and Reω 2 c ≤ 0.) This corresponds to the instability of the system first found by Migdal [100] and called the "pion condensation". On the physical sheet ω c (k) coincides with the solutions, obtained in [100], [104]. However, contrary to [100], [104], the ω c (k) is not the part of zero-sound branch.
To follow the solution ω c (k), let us present the function Φ
Φ (0) N (ω, k) = ϕ (0) N (ω, k) + ϕ (0) N (−ω, k)(114)
with the explicit expression for 0
< k < 2p F ϕ (0) N (ω, k) = 1 p F k −ωm * + kp F 2 + (kp F ) 2 − (ωm * − k 2 /2) 2 2k 2 × × ln ωm * − kp F − k 2 /2 ωm * − kp F + k 2 /2 − ωm * ln ωm * ωm − kp F + k 2 /2 .(115)
At k > 2p F the expression for ϕ (0) N (ω, k) takes another form (see [104]) but we shall not need it here. It was shown in [105], [106] that, if the density ρ is large enough (ρ ≥ ρ C ), there is a branch of solutions ω 2 c (k) ≤ 0, which is on the physical sheet for certain interval k 1 < k < k 2 of the values of k. At smaller values k < k 1 the branch goes to the unphysical sheet through the cut
0 ≤ ω ≤ k m * p F − k 2 ,(116)
generated by the third term in the rhs of Eq. (115). At larger values of k > k 2 the solution ω c goes away to the unphysical sheet through the same cut. The zero-sound wave goes to the unphysical sheet through another cut:
k m * p F − k 2 ≤ ω ≤ k m * p F + k 2 ,(117)
caused by the second term in the rhs of Eq. (115).
The value of the density ρ C , for which the solution ω c penetrates to the physical sheet, depends strongly on the model assumptions. Say, if the contribution of isobar intermediate states is ignored, the value of ρ C is shifted to unrealistically large values ρ C > 25ρ 0 . Inclusion of both nucleon and isobar states and employing of realistic values of FFST constants leads to ρ C ≈ 1.4ρ 0 under additional assumption m * π (ρ) = m π (0). The zero values of ω c (k) at certain nonzero values of k signals on the instability of the ground state . New components, like baryon-hole excitations with the pion quantum numbers emerge in the ground state of nuclear matter. Thus, the appearance of the singularity corresponding to ω 2 c = 0 shows, that the phase transition takes place.
Note, however, that the imaginary part of the solution ω c (k) is negative. Thus, there is no "accumulation of pions" in the symmetric nuclear matter, contrary to the naive interpretation of the pion condensation.
The situation is much more complicated in the case of asymmetric nuclear matter. In the neutron matter the instability of the system emerges at finite values of ω, because of the conversion n → p + π − [75]. This process leads to the real accumulation of pions in the ground state. In the charged matter with the non-zero value of the difference between the neutron and proton densities there is an interplay of the reactions n ⇀ ↽ p + π − and beta decays of nucleons.
Quark scalar condensate in the presence of the pion condensate
Now we turn back to the calculation of the condensate κ(ρ). Note first, that if the isobar width is neglected, we find κ(ρ) → +∞ at ρ → ρ C . The reason is trivial. When ρ → ρ C , the contribution S(ρ), described by Eq.(98) becomes
S ∼ dωd 3 k [ω 2 − ω 2 c (ρ, k)] 2 .(118)
The curve ω c (ρ c , k) turns to zero at certain k = k c , being ω c = a(k − k c ) 2 at |k − k c | ≪ k c . Thus,
S ∼ dωk 2 c d|k − k c | [ω 2 − a 2 (k − k c ) 2 ] 2 → ∞ .(119)
Hence, S(ρ C ) = +∞ and κ(ρ C ) = +∞. Once κ(0) < 0, we find that at certain ρ ch < ρ C the scalar condensate κ(ρ) turns to zero. This means that the chiral phase transition takes place before the pion condensation. (We shall not discuss more complicated models, for which the condition κ(ρ ch ) = 0 is not sufficient for the chiral symmetry restoration). At larger densities the pion does not exist any more as a collective Goldstone degree of freedom. Also the baryon mass vanishes (if very small current quark masses is neglected), and we have to stop our calculations, based on the selected set of Feynman diagrams (Fig.4) with the exact pion propagator.
The in-medium width of delta isobar Γ ∆ (the probability of the decay) depends strongly on the kinematics of the process. We can put Γ ∆ (ω, k) = 0 due to the limitation on the phase space of the possible decay process [103]. Thus, following the paper of Sadovnikova and Ryskin [106], we find that the chiral symmetry restoration takes place at the densities, which are smaller, than those, corresponding to the pion condensation:
ρ ch < ρ C .(120)
This means, that the pion condensation point cannot be reached in framework of the models, which do not describe the physics after the restoration of the chiral symmetry.
g πN N = g πN N 2m = g A 2f π ,(121)
-see Eq.(4). The πN∆ coupling constant is
g πN ∆ = c ∆gπN N(122)
with the experiments providing c ∆ ≈ 2 [75]. This is supported by the value c ∆ ≈ 1.7, calculated in the framework of Additive Quark Model (AQM).
The form factor d N B (k) which enters Eq. (103) is taken in a simple pole form [75]
d N B = 1 − m 2 π /Λ 2 B 1 + k 2 /Λ 2 B(123)
with Λ N = 0.67 GeV, Λ ∆ = 1.0 GeV.
We use mostly the values of FFSI parameters, presented in [107]: g ′ N N = 1.0, g ′ N ∆ = 0.2, g ′ ∆∆ = 0.8 -referring to these values as to set "a". We shall also check the sensitivity of the results to the variation of these parameters.
It is know from QHD approach that the nucleon effective mass may drop with density very rapidly. Thus, we must adjust our equations for description of the case, when the relativistic kinematics should be employed. We still include only the positive energy of the nucleon propagator, presented by Eq.(104). However we use the relativistic expression for
ε p − ε p+k = p 2 + m * 2 − (p + k) 2 + m * 2 .(124)
The propagator of ∆-isobar is modified in the same way. The explicit equations for the functions Φ
Fixing the dependence m * (ρ)
As we have seen above, the contribution of nucleon-hole excitations to S(ρ) depends explicitly on the nucleon effective mass m * (ρ). Here we shall try the models, used in nuclear physics, which determine the direct dependence m * (ρ). One of them is the Fermi liquid model with the effective mass described by Landau formula [108], [76], [109] m * (ρ)
m = 1 1 + 2mp F π 2 f 1 .(125)
In QHD approach the effective mass m * is the solution of the equation [68] m * = m − cm * p F (p 2 F + m * 2 ) 1/2 − m * 2 ln
p F + (p 2 F + m * 2 ) 1/2 m * ,(126)
corresponding to the behaviour m * = m(1 − f 2 ρ) (127) in the lowest order of expansion in powers of Fermi momentum p F . The coefficients f 1,2 in Eqs. (125), (127) can be determined by fixing the value m * (ρ 0 ).
Assuming, that all the other parameters are not altered in medium:
f * π = f π , m * ∆ −m * = m ∆ −m, m * π = m π , c * ∆ = c ∆ , g * A = g A (ρ 0 ) ≈ 1.
0, we find the point of chiral symmetry restoration to depend strongly on the value m * (ρ 0 ), being stable enough under the variation of FFST parameters and of the parameter c ∆ - Fig.6. Fixing m * (ρ 0 ) = 0.8m, we find ρ ch < ρ 0 , in contradiction to experimental data. Even in a simplified model with the width of ∆-isobar being accepted to coincide with its vacuum value Γ ∆ = 115 MeV, we find ρ ch ≈ 1.15ρ 0 . The value |κ(ρ 0 )| ≪ |κ(0)| looks unrealistic, since there are practically no strong unambiguous signals on partial restoration of the chiral symmetry at the saturation value of density ρ 0 [56]. Hence, here we also come to contradiction with the experimental data.
The situation is less critical for the smaller values of m * (ρ 0 )/m. Say, for Γ ∆ = 115 MeV we find ρ ch = 1.7ρ 0 , assuming m * (ρ 0 )/m = 0.7. However, under realistic assumption Γ ∆ = 0 we come to ρ ch < ρ 0 .
Note, that there is another reason for the point of the pion condensation to be unaccessible by our approach. The perturbative treatment of πN interaction becomes invalid for large pion fields. In the chiral πN Lagrangians the πN interaction is described by the terms of the type
L πN =ψU + (iγ µ ∂ µ )Uψ(128)
with U = exp i 2f π γ 5 (τ ϕ) .
The conventional version of pseudovector πNN Lagrangian employed above may be treated either as the lowest term of expansion of the matrix U in powers of the ratio ϕ/f π (identifying the pion with ϕ-field) or as the interaction with the fieldφ = f π sin((τ ϕ)/f π ). In any case, the whole approach is valid only, when the pion field is not too strong (ϕ ≤ f π orφ ≤ f π correspondingly). However, the strict quantitative criteria for the region of validity of Eq.(98) is still obscure.
The strong dependence of the results on the value of m * (ρ 0 )/m forces us to turn to self-consistent treatment of the hadron parameters and the quark condensates.
Self-consistent treatment of nucleon mass and the condensate
Now we shall carry out the calculations in framework of the model, where the nucleon parameters depend on the values of condensates. In other words, instead of the attempt to calculate the condensate κ(ρ, y i (ρ)) with y i standing for the hadron parameters (y i = m * N , m * ∆ , f * π , . . .), we shall try
κ(ρ) = K ρ, y i (κ(ρ), c j (ρ))(129)
with c j (ρ) standing for the other QCD condensates. Here K is the rhs of Eq. (62). Strictly speaking, we should try to obtain similar equations for the condensates c j (ρ).
We shall assume the physics of nuclear matter to be determined by the condensates of lowest dimension. In other words, we expect that only the condensates, containing the minimal powers of quark and gluon fields are important. The condensates of the lowest dimension are the vector and scalar condensates, determined by Eqs. (15) and (62) and also the gluon condensate -Eq.(34). As we saw in Subsect.2.7, the relative change of the gluon condensate in matter is much smaller than that of the quark scalar condensate. Thus we assume it to play a minor role. Hence, the in-medium values of κ(ρ) and v(ρ) will be most important for us, and we must solve the set of equations
κ(ρ) = K(ρ, y i ) y i = y i (v(ρ), κ(ρ)) .(130)
Fortunately, the vector condensate v(ρ) is expressed by simple formulas (15) and (16) due to the baryon current conservation.
The idea of self-consistent treatment is not a new one. Indeed, Eqs. (2) and (3) provide an example of Eq.(130) for NJL model in vacuum, with the only parameter y i = m.
As to parameters y i , which are m * , m * ∆ , f * π , etc., there are several relations which are, to large extent, model-independent. Besides the in-medium GMOR relation -Eq.(70), we can present inmedium GT relationg *
πN N = g * A 2f * π .(131)
Recalling that GT relation means, that the neutron beta decay can be viewed as the strong decay of neutron to π − p + system followed by the decay of the pion, we see that Eq. (131) is true under the same assumption as Eq. (70). Namely, the pion should be much lighter than any other state with unnatural parity and zero baryon charge. Also the expectation value of the quark scalar operator averaged over pion is
Υ * = π * |qq|π * = m * 2 π m .(132)
The other relations depend on the additional model assumptions. Starting with the ratio m * (ρ)/m we find in the straightforward generalization of NJL model
m * (ρ) m = κ(ρ) κ(0) .(133)
This relation is referred to in the paper [96] as Nambu scaling. The QCD sum rules prompt a more complicated dependence, presented by Eq.(97) with the function
F (ρ) = 1 1 + av(ρ)/ρ 0(134)
where a ≈ −0.2 [21,43,99] -see also Sec.5. Another assumption, expressed by Brown-Rho scaling equation (85) predicts a slower decrease of m * (ρ). Note, that Eq.(85) is based on existence of a single length scale, while there are two at least: p −1 F and Λ −1 QCD . The experimental situation with ∆-isobar mass in nuclear matter is not quite clear at the moment. The result on the total photon-nucleus cross section indicates that the mass m * ∆ does not decrease in the medium [110], while the nucleon mass m * (ρ) diminishes with ρ. On the other hand, the experimental data for total pion-nucleus cross sections are consistent with the mass m * ∆ decreasing in the matter [111]. As to calculations, the description within the Skyrmion model [93] predicts that m * In Fig.7 we show the results with BR1 scaling of the nucleon mass -Eq.(86) for different sets of FFST parameters. Following [107] we try the values which were obtained at saturation density ρ = ρ 0 -set"a" defined in 4.4.1. We assume that they do not change with density. We use also another set of parameters γ N = γ N ∆ = γ ∆∆ = 0.7 (set "b"), presented in [75]. The dependence on the behaviour m * ∆ (ρ) appears to be more pronounced for set "a" of the FFST parameters. The calculations were carried out under the assumption f * π = f π . Thus, the pion mass drops somewhat faster, than in BR1 scaling with decreasing f * π , in order to save in-medium GMOR relation -Eq. (70). The nucleon effective mass at saturation density appears to be quenched somewhat less, than in QHD models, being closer to the value, preferred by FFST approaches [76,107].
Assuming that the pion mass does not change in medium we find strong dependence on the values of FFST parameters. For the choice "a" the self-consistent solution disappears before the density reaches the saturation value-see dotted curve "2" in Fig. 7. We explained in our paper [103], how it happens technically.
The results, obtained with Nambu scaling for the nucleon mass being assumed -Eq. (133), are shown in Fig.8. We put m * π = m π and thus f * π ∼ |κ(ρ)| 1/2 , following GMOR. The three curves illustrate the dependence of the results on the assumption of the in-medium behaviour of the isobar mass.
One of the results of this subsection is that the improved (self-consistent) approach excludes the possibility of the pion condensation at relatively small densities. On the other hand Dickhoff et al. [101] carried out self-consistent description of the particle-hole interactions by inclusion of the induced interactions to all orders. This shifts the point of the pion condensation to the higher densities. The rigorous analysis should include both aspects of self-consistency.
Accumulation of isobars as a possible first phase transition
While the density increases, the Fermi momentum and the energy of the nucleon at the Fermi surface increase too. At some value of ρ it becomes energetically favourable to start the formation of the Fermi sea of the baryons of another sort instead of adding new nucleons. This phase transition takes
with p F a being the value of Fermi momentum, corresponding to ρ a , while m * B is the mass of the second lightest baryon at ρ = ρ a .
The vacuum values of Λ and Σ + hyperon masses are respectively 115 MeV and 43 MeV smaller than that of ∆-isobar. However, both experimental and theoretical data confirm that the hyperons interact with the scalar fields much weaker than the nucleons. Thus, at least in framework of certain assumptions on the behaviour of m * ∆ (ρ), the hierarchy of the baryon masses changes in medium. (The investigations of the problem are devoted mostly to the case of neutron or strongly asymmetric matter because of the astrophysical applications. See, however, the paper of Pandharipande [112]). The delta isobar can become the second lightest baryon state. In this case accumulation of ∆-isobars in the ground state is the first phase transition in nuclear matter. Such possibility was considered in several papers [113]- [116].
Under the assumption of BR1 scaling the accumulation of ∆-isobar takes place at ρ a ≈ 3ρ 0 , being the first phase transition. The value of ρ a is consistent with the result of Boguta [114].
QCD sum rules
QCD sum rules in vacuum
Here we review briefly the main ideas of the method. There are several detailed reviews on the subject -see, e.g., [117]. Here we focus on the points, which will be needed for the application of the approach to the case of nuclear matter.
The main idea is to establish a correspondence between descriptions of the function G, introduced in Subsec.3.6 in terms of hadronic and quark-gluon degrees of freedom. (Recall that G describes the propagation of the system with the quantum numbers of the nucleon). The method is based on the fundamental feature of QCD, known as the asymptotic freedom. This means, that at q 2 → −∞ the function G(q 2 ) can be presented as the power series of q −2 and QCD coupling α s . The coefficients of the expansion are the expectation values of local operators constructed of quark and gluon fields, which are called "condensates". Thus such presentation, known as operator product expansion (OPE) [118], provides the perturbative expansion of short-distance effects, while all the nonperturbative physics is contained in the condensates.
The correspondence between the hadron and quark-gluon descriptions is based on Eq. (93). The empirical data are used for the spectral function Im G(k 2 ) in the rhs of Eq. (93). Namely, we know, that the lowest lying state is the bound state of three quarks, which manifests itself as a pole in the (unknown) point k 2 = m 2 . Assuming, that the next singularity is the branching point k 2 = W 2 ph = (m + m π ) 2 , one can write exact presentation
Im G(k 2 ) =λ 2 δ(k 2 − m 2 ) + f (k 2 )θ(k 2 − W 2 ph )(136)
one finds certain connections between quark-gluon and hadron presentations
G OP E (q 2 ) =λ 2 m 2 − q 2 + 1 π ∞ W 2 ph f (k 2 ) k 2 − q 2 dk 2 .(138)
Of course, the detailed structure of the spectral density f (k 2 ) cannot be resolved in such approach. The further approximations can be prompted by asymptotic behaviour
f (k 2 ) = 1 2i ∆G OP E (k 2 )(139)
at k 2 ≫ |q 2 | with ∆ denoting the discontinuity. The discontinuity is caused by the logarithmic contributions of the perturbative OPE terms. The usual ansatz consist in extrapolation of Eq. (139) to the lower values of k 2 , replacing also the physical threshold W 2 ph by the unknown effective threshold
W 2 , i.e. 1 π ∞ W 2 ph f (k 2 ) k 2 − q 2 dk 2 = 1 2πi ∞ W 2 ∆G OP E (k 2 ) k 2 − q 2 dk 2(140)
and thus
G OP E (q 2 ) =λ 2 m 2 − q 2 + 1 2πi ∞ W 2 ∆G OP E (k 2 ) k 2 − q 2 dk 2 .(141)
The lhs of Eq.(141) contains QCD condensates. The rhs of Eq.(141) contains three unknown parameters: m,λ 2 and W 2 . Of course, Eq.(141) makes sense only if the first term of the rhs, treated exactly is larger than the second term, treated approximately.
The approximation G(q 2 ) ≈ G OP E (q 2 ) becomes increasingly true while the value |q 2 | increases. On the contrary, the "pole+continuum" model in the rhs of Eq.(141) becomes more accurate while |q 2 | decreases. The analytical dependence of the lhs and rhs of Eq.(141) on q 2 is quite different. The important assumption is that they are close in certain intermediate region of the values of q 2 , being close also to the true function G(q 2 ).
To improve the overlap of the QCD and phenomenological descriptions, one usually applies the Borel transform, defined as
Bf (Q 2 ) = lim Q 2 ,n→∞ (Q 2 ) n+1 n! − d dQ 2 n f (Q 2 ) ≡f (M 2 )(142)Q 2 = −q 2 ; M 2 = Q 2 /n
with M called the Borel mass. There are several useful features of the Borel transform.
1. It removes the divergent terms in the lhs of Eqs. (138) and (141) which are caused by the free quark loops. This happens, since the Borel transform eliminates all the polynomials in q 2 .
2. It emphasise the contribution of the lowest lying states in rhs of Eq.(141) due to the relation
B 1 Q 2 + m 2 = e −m 2 /M 2 .(143)
3. It improves the OPE series, since
B (Q 2 ) −n = 1 (n − 1)! (M 2 ) 1−n .(144)
Applying Borel transform to both sides of Eq.(141) one finds
G OP E (M 2 ) =λ 2 e −m 2 /M 2 + 1 2πi ∞ W 2 dk 2 e −k 2 /M 2 · ∆G OP E (k 2 ) .(145)
Such relations are known as QCD sum rules. If both rhs and lhs of Eq.(141) were calculated exactly, the relation would be independent on M 2 . However, certain approximations are made in both sides. The basic assumption is that there exists a range of M 2 for which the two sides have a good overlap, approximating also the true function G(M 2 ).
The lhs of Eq.(145) can be obtained by presenting the function G(q 2 ), which is often called "correlation function" or "correlator" as (strictly speaking, G depends on the components of vector q also through the trivial termq)
G(q 2 ) = i d 4 xe i(qx) 0|T {η(x)η(0)}|0(146)
with η being the local operator with the proton quantum numbers. It was shown in [119] that there are three independent operators η
η 1 = u T a Cγ µ u b γ 5 γ µ d c · ε abc , η 2 = u T a Cσ µν u b σ µν γ 5 d c ε abc , η 3µ = (u T a Cγ µ u b )γ 5 d c − (u T a Cγ µ d b )γ 5 u c ε abc ,(147)
where T denotes the transpose in Dirac space and C is the charge conjugation matrix. However, the operator η 2 provides strong admixture of the states with negative parity [119]. As to the operator η 3 , it provides large contribution of the states with spin 3/2 [119]. Thus, the calculations with η = η 1 are most convincing. We shall assume η = η 1 in the further analysis.
The correlation function has the form
G(q) = G q (q 2 ) ·q + G s (q 2 ) · I(148)
with I standing for the unit 4 × 4 matrix. The leading OPE contribution to G q comes from the loop with three free quarks. If the quark masses m u,d are neglected, the leading OPE term in G s comes from the exchange by the quarks between the system described by operator η and vacuum. Technically this means, that the contribution comes from the second term of the quark propagator in vacuum where only the contribution with A = 1 (see Eq. (13)) survives. This is illustrated by Fig.9. The higher order terms come from exchange by soft gluons between vacuum and free quarks carrying hard momenta. Next comes the four-quark condensate which can be viewed as the expansion of the two-quark propagator, similar to Eq. (149).
0|T q α (x)q β (0)|0 = i 2π 2x αβ x 4 − 1 4 A Γ A αβ 0|qΓ A q|0 + 0(x 2 ) ,(149)
Direct calculation provides for massless quarks [98]
G OP E q = − 1 64π 4 (q 2 ) 2 ln(−q 2 ) − 1 32π 2 ln(−q 2 )g(0) − 2 3q 2 h 0 ,(150)G OP E s = 1 8π 2 q 2 ln(−q 2 ) − 1 48q 2 g(0) κ(0)(151)
with the condensates g(0) = 0| αs π G 2 |0 , κ(0) = 0|qq|0 and h 0 = 0|ūuūu|0 . The terms, containing polynomials of q 2 are omitted, since they will be eliminated by the Borel transform. This leads to the sum rules [98]
M 6 E 2 W 2 M 2 + 1 4 bM 2 E 0 W 2 M 2 + 4 3 C 0 = λ 2 e −m 2 /M 2 ,(152)
2a
M 4 E 1 W 2 M 2 − b 24 = mλ 2 e −m 2 /M 2 (153) with traditional notations a = −2π 2 κ(0), b = (2π) 2 g(0), λ 2 = 32π 4λ2 , E 0 (x) = 1 − e −x , E 1 (x) = 1 − (1 + x)e −x , E 2 (x) = 1 − x 2 2 + x + 1 e −x .
Also C 0 = (2π) 4 h 0 . Here we omitted the anomalous dimensions, which account for the most important corrections of the order α s , enhanced by the "large logarithms". The radiative corrections were shown to provide smaller contributions,as well as the higher order power corrections [98]. The matching of the lhs and rhs of Eqs. (152),(153) was found in [98] for the domain 0.8 GeV 2 < M 2 < 1.4 GeV 2 .
As one can see from Eq.(153), the nucleon mass turns to zero if 0|qq|0 = 0. Hence, the mass is determined by the exchange by quarks between the our system and vacuum.
The method was applied successfully to calculation of the static characteristics of the nucleons reproducing the values of its mass [39,97,98] as well as of magnetic moment [98] and of the axial coupling constant [121]. The proton structure functions also were analysed in framework of the approach [122]
Proton dynamics in nuclear matter
Now we extent the sum-rule approach to the investigation of the characteristics of the proton in nuclear matter. The extension is not straightforward. This is mostly because the spectrum of correlation function in medium
G m (q) = i d 4 xe i(qx) M|T {η(x)η(0)}|M(155)
is much more complicated, than that of the vacuum correlator G(q 2 ). The singularities of the correlator can be connected with the proton placed into the matter, as well as with the matter itself.
One of the problems is to find the proper variables, which would enable us to focus on the properties of our probe proton.
Choice of the variables
Searching for the analogy in the earlier investigations one can find two different approaches. Basing on the analogy with the QCD sum rules in vacuum, one should build the dispersion relation in the variable q 2 . The physical meaning of the shift of the position of the proton pole is expressed by Eq. (96). Another analogy is the Lehmann representation [120], which is dispersion relation for the nucleon propagator in medium g N (q 0 , |q|) in the time component q 0 . Such dispersion relation would contain all possible excited states of the matter in rhs. Thus, we expect the dispersion relations in q 2 to be a more reasonable choice in our case.
It is instructive to adduce the propagation of the photon with the energy ω and three dimensional momentum k in medium. The vacuum propagator is D γ ∼ [ω 2 − k 2 ] −1 . Being considered as the function of q 2 = ω 2 − k 2 it has a pole at q 2 = 0. The propagator in medium is D m γ ∼ [ω 2 ε(ω) − k 2 ] −1 . The dielectric function ε(ω) depends on the structure of the matter, making D m γ (ω) a complicated function. However, the function D m γ (q 2 ) still has a simple pole, shifted to the value q 2 m = ω 2 (1−ε(ω)). A straightforward calculation of the new value q 2 m is a complicated problem. The same refers to the proton in-medium. The sum rules are expected to provide the value in some indirect way.
Thus, we try to build the dispersion relations in q 2 . Since the Lorentz invariance is lost, the correlator G m (q) depends on two variables. Considering the matter as the system of A nucleons with momenta p i , introduce vector
p = Σp i A ,(156)
which is thus p ≈ (m, 0) in the rest frame of the matter. The correlator can be presented as G m (q) = G m (q 2 , ϕ(p, q)) with the arbitrary choice of the function ϕ(p, q), which is kept constant in the dispersion relations. This is rather formal statement, and there should be physical reasons for the choice.
To make the proper choice of the function ϕ(p, q), let us consider the matrix element, which enters Eq.(155)
M|T {η(x)η(0)}|M = M A |η(x)|M A+1 M A+1 |η(0)|M A θ(x 0 ) − − M A |η(0)|M A−1 M A−1 |η(x)|M A θ(−x 0 )(157)
with |M A standing for the ground state of the matter, while |M A±1 are the systems with baryon numbers A ± 1. The summation over these states is implied. The matrix element M A+1 |η|M A contains the term N|η|0 which adds the nucleon to the Fermi surface of the state |M A . If the interactions between this nucleon and the other ones are neglected, it is just the pole at q 2 = m 2 . Now we include the interactions. The amplitudes of the nucleon interactions with the nucleons of the matter are known to have singularities in variables s i = (p i + q) 2 . These singularities correspond to excitation of two nucleons in the state |M A+1 . Thus, they are connected with the properties of the matter itself. To avoid these singularities we fix
ϕ(p, q) = s = max s i = 4E 2 0F(158)
with E 0F being the relativistic value of nucleon energy at the Fermi surface. Neglecting the terms of the order p 2 F /m 2 we can assume p i = (m, 0) and thus
s = 4m 2 .(159)
Our choice of the value of s corresponds to |q| = p F (in the simplified case, expressed by Eq.(159) |q| = 0). However, varying the value of s we can find the position of the nucleon poles, corresponding to other values of |q|.
Let us look at what happens to the nucleon pole q 2 = m 2 after we included the interactions with the matter. The self-energy insertions Σ modify the free nucleon propagator g 0 N to g N with
(g N ) −1 = (g 0 N ) −1 − Σ(160)
-see Fig.10.
In the mean field approximation (Fig.10a) the function Σ does not contain additional intermediate states. It does not cause additional singularities in the correlator G m (q 2 , s). The position of the pole is just shifted by the value, which does not depend on s. (Note that this does not mean that in the mean field approximation the condition s = const can be dropped. Some other contributions to the matrix element M A+1 |η|M A are singular in s. Say, there is the term B|η|0 with B standing for the system, containing the nucleon and mesons. If the mesons are absorbed by the state |M A , we come to the box diagram ( Fig.11) with the branching point, starting at s = 4m 2 ).
Leaving the framework of the mean-field approximation we find Hartree self-energy diagrams (Fig.10b) depending on s. The latter is kept constant in our approach. Hence, no additional singularities emerge in this case as well.
The situation becomes more complicated if we take into account the Fock (exchange) diagrams (Fig.10c). The self-energy insertions depend on the variable u = (p − q) 2 . The contribution of these Figure 10: Self-energy insertions to the nucleon pole contribution to the correlator. The solid lines denote the nucleons, the helix line stands for the correlator. Fig.10a corresponds to mean-field approximation. Fig.10b shows the self-energy in the direct channel. Fig.10c presents the exchange contribution. terms shifts the nucleon pole and gives birth to additional singularities corresponding to real states with baryon number equal to zero. They are the poles at the points u = m 2 x , with m x denoting the masses of the mesons (π, ω, etc.), and the cuts running to the right from the point q 2 = m 2 + 2m 2 π . The latter value is the position of the branching point corresponding to the real two-pion state in the u-channel.
Thus the single-nucleon states |B ±1 cause the pole q 2 = m 2 m , a set of poles corresponding to the states with baryon number B = 0 and a set of branching points. The lowest-lying one is q 2 = m 2 + 2m 2 π . Note that the antinucleon corresponding to q 0 = −m generates the pole q 2 = 5m 2 shifted far to the right from the lowest-lying one.
The lowest-lying branching point q 2 = m 2 +2m 2 π is separated from the position of the pole q 2 = m 2 by a much smaller distance than in the case of the vacuum (q 2 = m 2 +2mm π in the latter case). Note, however, that at the very threshold the discounting is quenched since the vertices contain moments of the intermediate pions. Thus the branching points can be considered as a separated from the pole q 2 = m 2 . Note also that for the same reason the residue at the pole q 2 = m 2 + 2m 2 π in the u-channel vanishes.
As it was shown in the work [43], all the other singularities of the correlator G m (q 2 , s) in q 2 are lying to the right from the nucleon pole until we include the three-nucleon terms. Thus, they are accounted for by the continuum and suppressed by Borel transform. To prove the dispersion relation we must be sure of the possibility of contour integration in the complex q 2 plane. This cannot be done on an axiomatic level. However a strong argument in support of the possibility is the analytical continuation from the region of large real q 2 . At these values of q 2 the asymptotic freedom of QCD enables one to find an explicit expression of the integrand. The integral over large circle gives a non-vanishing contribution. However, the latter contains only the finite polynomials in q 2 which are eliminated by the Borel transform.
Thus we expect "pole+continuum" model to be valid for the spectrum of the correlator G m (q 2 , s).
The situation becomes more complicated if we include the three-nucleon interactions [99]. The probe proton, created by the operator η can interact with n nucleons of the matter. The corresponding amplitudes depend on the variable s n = (np + q) 2 . For n ≥ 2 this causes the cuts, running to the left from the point q 2 = m 2 . This requires somewhat more complicated model of the spectrum. From the point of view of expansion in powers of ρ this means, that "pole + continuum" model is legitimate until the terms of the order ρ 2 are included.
Operator expansion
Following our general strategy, we shall try to obtain the leading terms of expansion of the correlator
G m (q) = G m q (q)q + G m p (q)p + G m s (q)I ,(161)
in powers of q −2 . Note, that the condition s = const, which we needed for separation of the singularities, connected with our probe proton, provides
(pq) q 2 → const(162)
at q 2 → −∞. This is just the condition which insures the operator expansion in deep-inelastic scattering (see, e.g., the book of Ioffe et al. [123]). It is not necessary in our case. However, the physical meaning of some of the condensates, say, that of ϕ a (α)α n -Eq.(50) becomes most transparent in this very kinematics.
The problem is more complicated than in vacuum, since each of the terms of the expansion in powers of q 2 provides, generally speaking, infinite number of the condensates. Present each of the components G m i (i = q, p, s) of the correlator G m
G m i = d 4 xe i(qx) T i (x) .(163)
The function T i (x) contains in-medium expectation values of the products of QCD operators in space-time points "0" and "x" with an operator at the point x defined by Eq. (45). Each in-medium expectation value, containing covariant derivative D µ is proportional to the vector p µ . This can be easily generalized for the case of the larger number of derivatives. Thus the correlators take the form G m i = n C n (p∇ q ) n f i (q 2 ). For the contributions f i (q 2 ) ∼ (q 2 ) −k the terms (p∇ q ) n f i (q 2 ) are of the same order. This is the "price" for the choice of kinematics s = const. Fortunately, the leading terms of the operator expansion contain the logarithmic loops and thus can be expressed through the finite number of the condensates [99].
The leading terms of the operator expansion can be obtained by replacing the free quark propagators by those in medium
M|T ψ α (x)ψ β (0)|M = i 2π 2x x 4 − A 1 4 Γ A αβ M|ψ(0)Γ A ψ(x)|M(164)
with the matrices Γ A being defined by Eq. (13). Operator ψ(x) is defined by Eq. (45). While looking for the lowest order term of the operator expansion we can put x 2 = 0 in the second term of the rhs of Eq.(164). In the sum over A the contributions with A = 3, 4 vanish due to the parity conservation by strong interactions, the one with A = 5 turns to zero in any uniform system. Thus, only the terms with A = 1, 2 survive. Looking for the lowest order density effects, we assume that propagation of one of the quarks of the correlator G m is influenced by the medium. Hence,the term with A = 1 contributes to the scalar structure G m s , while that with A = 2 -to the vector structures G m q and G m p .
G m s = 1 2π 2 q 2 ln(−q 2 ) κ(ρ)(165)G m q = − 1 64π 4 (q 2 ) 2 ln(−q 2 ) + 1 6π 2 (s − m 2 − q 2 ) ln(−q 2 )v(ρ) (166) G m p = 2 3π 2 q 2 ln(−q 2 ) v(ρ) .(167)
Thus the correlator G m s can be just obtained from the vacuum correlator G s by replacing of κ(0) by κ(ρ). The correlator G m q obtains additional contribution proportional to the vector condensate v(ρ). Also, the correlator G m p , which vanishes in vacuum is proportional to v(ρ). These terms are illustrated by Fig.12 a,b. Turn now to the next OPE terms. Start with the structure G m s . In the case of vacuum there is a contribution which behaves as ln(−q 2 ), which is proportional to the condensate 0|ψ αs π G a µν σ µν λ a 2 ψ|0 . The others notations coincide with those of Fig.9.
However, similar term comes from expansion of expectation value 0|q(0)q(x)|0 in powers of x 2 . The two terms cancel [39]. Similar cancellation takes place in medium [99]. However there is a contribution, caused by the second term of rhs of Eq.(45). It does not vanish identically, but it can be neglected due to Eq. (54). Hence, the next OPE term in rhs of Eq.(165) can be obtained by simple replacement of the condensates κ(0) and g(0) in the second term of Eq.(151) by κ(ρ) and g(ρ).
The next-to-leading order corrections to the correlators G m q,p come from expansion of the expectation value M|ψ(0)γ 0 ψ(x)|M . In the lowest order of x 2 expansion the matrix element can be presented through the moments of the deep-inelastic scattering (DIS) nucleon structure functions -Eqs. (49) and (50). Since the medium effects in DIS are known to be small we limit ourselves to the gas approximation at this point. The main contributions to q −2 expansion compose the series of the terms [(s − m 2 )/q 2 ] n α n with α n denoting n-th moment of the structure function. Being expressed in a closed form, they change Eqs. (166) and (167) to
G m q = − 1 64π 4 (q 2 ) 2 ln(−q 2 ) + 1 12π 2 (s − m 2 − q 2 ) m 1 0 dαF (α) ln(q − pα) 2 · ρ(168)+ m 3π 2 1 0 dαφ b (α) ln(q − pα) 2 · ρ G m p = 2 3π 2 q 2 1 0 dαF (α) ln(q − pα) 2 · ρ(169)
with F (α) being structure function, normalized as 1 0 dαF (α) = 3. The function F (α) can be presented also as F (α) = φ a (α) with φ a defined by Eq. (50). Another leading OPE term is caused by modification of the value of gluon condensate. This is expressed by changing of the value g(0) in the second term in rhs of Eq. (150) to g(ρ) - Fig.12b.
The higher order OPE terms lead to the contributions which decrease as q −2 . One of them is caused by the lowest order power correction to the first moment of the structure function. This term is expressed through the factor ξ which is determined by Eq.(51), being calculated in [42]. The other corrections of this order come from the four-quark condensates Q AB , defined by Eq. (57). The correlator G m s contains the condensate Q 12 . The vector part includes the condensates Q 11 and Q 22 . Another contribution of this order comes from the replacement of the condensates g(0) and κ(0) in the last term in rhs of Eq.(151) by their in-medium values - Fig.12c,d.
Building up the sum rules
To construct the rhs of the sum rules, consider the nucleon propagator g N = (H − E) −1 with E standing for the nucleon energy, while the Hamiltonian H in the mean field approximation is presented by Eq. (73). Beyond the mean field approximation the potentials V µ and Φ should be replaced by vector and scalar self-energies
V µ = Σ V µ ; Σ V µ = p µ Σ p + q µ Σ q ; Φ = Σ s .(170)
Thus, under condition s = 4m 2 -Eq.(159)
g N = Zq (1 − Σ q ) −pΣ p + m + Σ s q 2 − m 2 m(171)
with
m m = m + U ; U = m(Σ q + Σ p ) + Σ s ,(172)while Z = 1 (1 − Σ q )(1 − Σ q + Σ p ) .(173)
Of course, Σ q = 0 in mean field approximation.
The Borel-transformed sum rules for in-medium correlators in the assumed "pole+continuum" model for the spectrum are:
L m q (M 2 ) = λ 2 m e −m 2 m /M 2 (1 − Σ q ) (174) L m p (M 2 ) = −λ 2 m e −m 2 m /M 2 Σ p (175) L m s (M 2 ) = λ 2 m e −m 2 m /M 2 (m + Σ s )(176)
with λ 2 m = 32π 4λ2 m Z withλ 2 m standing for the residue in the nucleon pole (see similar definition in vacuum -Eqs. (152) and (153)). The lhs of Eqs. (174)-(176) are:
L m q (M 2 ) = M 6 E 2 W 2 m M 2 L −4/9 − 8π 2 3 (s − m 2 )M 2 E 0 W 2 m M 2 − M 4 E 1 W 2 m M 2 × F µ(α) − 2m 2 M 2 E 0 W 2 m M 2 F µ(α)α + 4m 2 M 2 E 0 W 2 m M 2 φ b µ(α) ρL −4/9 +π 2 M 2 E 0 W 2 m M 2 g(ρ) + 3 4 m 2 (s − m 2 ) θ a µ + 2m 4 θ b µ + 4 3 (2π) 4 Q 11 uu (ρ)L 4 9 (177) L m p (M 2 ) = − 8π 2 3 M 4 E 1 W 2 m M 2 F µ(α) ρL −4/9 − (2π) 4 Q 22 uu (178) L m s (M 2 ) = (2π 2 ) 2 M 4 E 1 W 2 m M 2 − (2π) 2 12 g(ρ) κ(ρ) + 8(2π 2 ) 3 3 Q 12 ud (ρ) .(179)
In Eqs. (177)-(179) ψ = 1 0 dαψ(α) for any function ψ, F is the structure function, the function Recall, that "pole+continuum" model is true until we do not touch the terms of the order ρ 2 . Thus, in the sum rules for the difference between in-medium and vacuum correlators we must limit ourselves to linear shifts of the parameters
µ(α) = exp −(s − m 2 )α + m 2 α 2 M 2 (1 + α)(180)∆L m q (M 2 ) = λ 2 e −m 2 /M 2 ∆λ 2 λ 2 − Σ q − 2m∆m M 2 − W 4 2L 4/9 exp − W 2 M 2 ∆W 2 (182) L m p (M 2 ) = −λ 2 e −m 2 /M 2 Σ p (183) ∆L m s (M 2 ) = λ 2 e −m 2 /M 2 m ∆λ 2 λ 2 + Σ s − 2m 2 ∆m M 2 − 2aW 2 exp − W 2 M 2 ∆W 2 . (184)
Here ∆ denotes the difference between in-medium and vacuum values. However the self-energy Σ q and Σ s cannot be determined separately, since only the sum Σ q + Σ s can be extracted.
Anyway
Since the value (W 2 /2L 4/9 )m − 2a is numerically small, one can write approximate sum rule, neglecting the second term in rhs of Eq.(185)
U = e m 2 /M 2 λ 2 ∆L m s − mL m p − m∆L m q ,(186)
or, assuming the sum rules in vacuum to be perfect
U = e m 2 /M 2 λ 2 (L s − mL p − mL q )(187)
with the vacuum part cancelling exactly.
The two lowest order OPE terms ( without perturbative expansion in parameter s−m 2 M 2 ) are presented by the first two terms of rhs of Eqs. (177), (179) and by the first term of rhs of Eq.(178). They are expressed through the condensates v(ρ), κ(ρ) and g(ρ) and through the moments of the nucleon functions φ a,b introduced in Subsec. 2.6. The values of the lowest moments of the structure function F (α) = φ a (α) are well known from experimental data. By using the value of ξ a and employing relations, presented by Eq.(53) one can find the lowest moments of the function φ b . Only the first moment of the function φ b and thus the first and second moments of the function φ a appeared to be numerically important. Thus, at least in the gas approximation all the contributions to the lhs of the sum rules can be either calculated in the model-independent way or related to the observables [43]. The scalar condensate is the most important parameter beyond the gas approximation [20]. The model calculations have been carried out in this case.
The next order of OPE includes explicitly the moments of the functions θ a,b defined in Subsec.2.6. It includes also the four-quark condensates Q 11 , Q 12 and Q 22 . Using Eqs. (53) one can find that only the first moment of the function θ a is numerically important while the moments of the function θ b can be neglected. The condensate Q 12 can be obtained easily by using Eq. (58). The uncertainties of the values of the other four-quark condensates Q 11 is the main obstacle for decisive quantitative predictions, based on Eqs. (182)-(187). The scalar four-quark condensate Q 11 may appear to be a challenge for the convergence of OPE due to the large value of the second term in rhs of Eq. (59). This may be a signal that large numbers are involved. Fortunately, the only calculation of Q 11 carried out in [46] demonstrated that there is a large cancellation between the model-dependent first term in rhs of Eq.(59) and the second one, which is to large extent model-independent. However, assuming the result presented in [46], we still find this contribution to be numerically important.
We can try (at least for illustrative reasons) to get rid of this term in two ways. One of them is to ignore its contribution. The reason is that it corresponds to exchange by a quark system with the quantum numbers of a scalar channel between our probe proton and the matter. On the other hand, it contributes to the vector structure of the correlator G m q , and thus to the vector structureq of the propagator of the nucleon with the momentum q. Such terms are not forbidden by any physical law. However most of QHD calculations are successful without such contributions. Thus the appearance of the terms with such structure, having a noticeable magnitude is unlikely. (Of course, this is not a physical argument, but rather an excuse for trying this version). The other possibility is to eliminate the contribution by calculation of the derivative with respect to M 2 . The two ways provide relatively close results.
The structure of the potential energy
Under the conditions, described above, we find that the rhs of Eq.(186) is a slowly varying function of M 2 in the interval, defined by Eq.(154). Among the moments of the structure function the two first ones appeared to be numerically important. Thus we find
U(ρ) = 66 v(ρ) + 70 v 2 (ρ) − 10∆g(ρ) m − 32 ∆κ(ρ) GeV −2 .(188)
Here v(ρ) = 3ρ is the vector condensate -Eq.(16), ∆κ(ρ) = κ(ρ) − κ(0) is the in-medium change of the scalar condensate -Eq. (62). The condensate v 2 (ρ), determined as
M|ψγ µ D ν ψ|M = g µν − 4p µ p ν p 2 v 2 (ρ)
is connected to the second moment of the nucleon structure function. Numerically v 2 (ρ) ≈ 0.3ρ. Finally, ∆g(ρ) is the shift of the gluon condensate, expressed by Eq. (39).
Thus the problem of presenting the nucleon potential energy through in-medium condensates is solved. At the saturation value ρ = ρ 0 we find U = −36 MeV in the gas approximation. This should be considered as a satisfactory result for such a rough model. This is increasingly true, since there is a compensation of large positive and negative values in rhs of Eq.(186).
Note that the simplest account of nonlinear terms signals on the possible saturation mechanism. Following the discussion of Subsection 2.7 and assuming the chiral limit, present ∆κ(ρ) = Σ/m − 3.2(p F /p F 0 )ρ. Thus we obtain the potential
U(ρ) = 198 − 42 Σ m ρ ρ 0 + 133 ρ ρ 0 4/3 MeV .(189)
After adding the kinetic energy it provides the minimum of the functional E(ρ) defined by Eq.(79) at Σ = 62.8 MeV, which is consistent with experimental data -Eq. (25). The binding energy is E = −9 MeV. The incompressibility coefficient K = 9ρ 0 (d 2 ε/dρ 2 ) also has a reasonable value K = 182 MeV.
Of course, the results for the saturation should not be taken too seriously. As we have seen in Sect.3, the structure of the nonlinear terms of the condensate is much more complicated. Also, the result is very sensitive to the exact value of Σ-term. Say, assuming it to be larger by the magnitude of 2 MeV, we find the Fermi momentum at the saturation point about 1/3 larger than p F 0 . Thus, the value of the saturation density becomes about 2.5 times larger than ρ 0 . Such sharp dependence is caused by the form of the nonlinear term in the potential energy equation -Eq.(189). The form of the term is due to oversimplified treatment of nonlinear effects. However the result can be the sign, that further development of the approach may appear to be fruitful.
Relation to conventional models and new points
We obtained a simple mechanism of formation of the potential energy. Recall that Ioffe analysis of QCD vacuum sum rules [97] provided the mechanism of formation of nucleon mass as due to the Figure 13: Contribution to the scalar structure of the correlator from the exchange ofqq pairs with quantum numbers of vector mesons between the correlator and the matter. The dark blob stands for the vector condensate. The light circles denote the vacuum expectation value. The generated term is unusual for QHD. exchange by quarks between the probe nucleon and the quark-antiquark pairs of vacuum. In the nuclear matter the new mass is formed by the exchange with the modified distribution of the quarkantiquark pairs and with the valence quarks. The modified distribution ofqq pairs is described by the condensate κ(ρ). At ρ close to ρ 0 the modification is mostly due to the difference of the densities ofqq pairs inside the free nucleons and in the free space. Similar exchange with the valence quarks is determined by the vector condensate v(ρ) and is described by the first term of rhs of Eq.(188). The second term describes additional interaction which takes place during such exchange. These exchanges cause the shift of the position of the pole m m − m. While the interactions of the nucleons depend on the condensates ∆κ(ρ) and v(ρ), these condensates emerge due to the presence of the nucleons. Also, the nonlinear part of ∆κ is determined by NN interactions. Thus, there is certain analogy between QCD sum rules picture and NJL mechanism.
As we have seen, the QCD sum rules can be viewed as connection between exchange by uncorrelatedqq pairs and exchange by strongly correlated pairs with the same quantum numbers (mesons). This results in connection between the Lorentz structures of correlators and in-medium nucleon propagators. In the leading terms of OPE the vector (scalar) structure is determined by vector (scalar) condensate. The large values (of about 250-300 MeV) of the first and the fourth terms in rhs of Eq.(188) provide thus the direct analogy with QHD picture.
Note, however, that the sum rules, presented by Eqs. (177)-(179) contain also the terms, which are unusual for QHD approach. Indeed, the term Q 11 in Eq.(177) enters the vector structure of the correlator (and thus, of the propagator of the nucleon) corresponding, however, to exchange by the vacuum quantum numbers with the matter. On the other hand, the last term of rhs of Eq.(179) treated in the gas approximation, corresponds to exchange by the quantum numbers of vector mesons. However, it appears in the scalar structure. This term originated from the four-quark condensate Q 12 . While the exact value of the condensate Q 11 is still obscure, the condensate Q 12 is easily calculated. This OPE term is shown in Fig. 13. It provides a noticeable contribution.
Such terms do not emerge in the mean-filed approximation of QHD. They can be originated by more complicated structure of the nucleon-meson vertices. (Note that if the nucleons interacted through the four-fermion interaction, such terms would have emerged from the exchange interaction due to Fierz transform).
Another approach, developed by Maryland group, was reviewed by Cohen et al. [124]. In most of the papers (except [125]) the Lehmann representation was a departure point. In framework of this approach the authors investigated the Lorentz structure of QCD sum rules [126]. They analysed detaily the dependence of the self-energies on the in-medium value of the scalar four-quark condensate [127]. The approach was used for investigation of hyperons in nuclear matter [128].
The approach is based on the dispersion relations in the time component q 0 at fixed threedimensional momentum |q|. It is not clear, if in this case the singularities, connected with the probe proton are separated from those of the matter itself. The fixed value of three-dimensional momenta is a proper characteristics for in-medium nucleon. That is why this was the choice of variables in Lehmann dispersion relation with the Fermi energy as a typical scale. The sum rules are dispersion relations rather for the correlation function with the possible states N, N+ pions, N * , etc. The scale of the energy is a different one and it is not clear, if this choice of variables is reasonable for QCD sum rules.
Charge -symmetry breaking phenomena
Nolen-Schiffer anomaly
The nuclei consisting of equal numbers of protons and neutrons with one more proton or neutron added are known as the mirror nuclei. If the charge symmetry (known also as isospin symmetry) of strong interactions is assumed, the binding energy difference of mirror nuclei is determined by electromagnetic interactions only, the main contribution being caused by the interaction of the odd nucleon. Nolen and Schiffer [129] found the discrepancy between the experimental data and theoretical results on the electromagnetic contribution to the energy difference. This discrepancy appeared to be a growing function of atomic number A. It reaches the value of about 0.5 MeV at A = 40. Later the effect became known as Nolen-Schiffer anomaly (NSA).
The NSA stimulated more detailed analysis of electromagnetic interactions in such systems. Auerbach et al. [130] studied the influence of Coulomb forces on core polarization. However, this did not explain the NSA. Bulgac and Shaginyan [131] attributed the whole NSA phenomena to the influence of the nuclear surface on the electromagnetic interactions. Thus they predict NSA to vanish in infinite medium.
However most of the publications on the subject contain the attempts to explain NSA by the charge symmetry breaking (CSB) by the strong interactions at the hadronic level. The CSB potentials of NN interactions were reviewed by Miller et al. [132]. Some of phenomenological potentials described the NSA, but contradicted the experimental data on CSB effects in NN scattering. The meson-exchange potentials contain CSB effects by inclusion of ρ − ω mixing. This explains the large part of NSA, but not the whole effect.
On the quark level the CSB effects in the strong interactions are due to the nonzero value of the difference of the quark masses
m d ≈ 7 MeV ; m u ≈ 4 MeV ; µ = m d − m u ≈ 3 MeV .(190)
Several quark models have been used for investigation of NSA by calculation of neutron-proton mass difference in nuclear matter (recall that in vacuum m n − m p = 1.8 MeV, while the Coulomb energy difference is -0.5 MeV. Hence, the shift caused by strong interactions is δm = 2.3 MeV). Henley and Krein [133] used the NJL model for the quarks with the finite values of the current masses. The calculated neutron-proton mass difference appeared to be strongly density dependent. The result overestimated the value of NSA. The application of the bag models were considered by Hatsuda et al. [134]. The chiral bag model provided the proper sign of the effect, but underestimated its magnitude.
QCD sum rules view
The QCD sum rules look to be a reasonable tool for the calculation of neutron-proton binding energy difference for the nucleons, placed into the isotope-symmetric nuclear matter. Denoting
δx = x n − x p(191)
for the strong interaction contribution to the neutron-proton difference of any parameter x, we present δε = δU + δT
with U(T ) -the potential (kinetic) energy of the nucleon. To the lowest order in the powers of density one finds
δT = − p 2 F m 2 δm ,(193)
while the value δU = δm m can be obtained from the sum rules.
The attempts to apply QCD sum rules for solving the NSA problem were made in several papers [134]- [137]. We shall follow the papers of Drukarev and Ryskin [137] which present the direct extension of the approach, discussed above. It is based on Eqs. (182)-(184) with the terms L m i being calculated with the account of the finite values of the current quark masses. Besides the quark mass difference the CSB effects manifest themselves through isospin breaking condensates
γ 0 = 0|dd −ūu|0 0|ūu|0 ; γ m = M|dd −ūu|M M|ūu|M .(194)
The characteristics γ 0 and γ m are not independent degrees of freedom. They turn to zero, at µ = 0, being certain (unknown) functions of µ. The dependence γ 0 (µ), γ m (µ) can be obtained in framework of the specific models. Anyway, due to the small values of m u,d we expect |γ 0 |, |γ m | ≪ 1. Following the strategy and keeping only the leading terms, which are linear in µ, we shall obtain the energy shift in the form δε = a 1 µ + a 2 γ 0 + a 3 γ m
with a i being the functions of the density ρ (the contribution δT is included into a 1 ).
To calculate the mass-dependent terms in lhs of Eqs. (182)-(184) one should include the quark masses to the in-medium quark propagator -Eq. (149). The first term in rhs of Eq. (149), which is just the free quark propagator should be modified into (ix αβ )/(2π 2 x 4 ) − m q /(2π 2 x 2 ). This provides the contribution to the scalar structure of the correlator in the lowest order of OPE. Account of the finite quark masses in the second term of Eq.(149) manifest themselves in the next to leading orders of OPE. Say, during evaluation of the second term in Subsection 5.2, we used the relations expressed by Eq. (53), which were obtained for the massless quarks. Now the first of them takes the form
φ i b = 1 4 φ i a α − m i 4m N|ψ i ψ i |N(196)
for the flavour i = u, d. This leads to the contribution to the vector structure of the correlator proportional to the scalar condensate. Also, the second moment of the scalar distribution is proportional to the vector condensate, contributing to the scalar structure of the correlator.
The leading contribution caused by the scalar structure of the correlator
(δU) 1 = 0.18 µ m v(ρ) ρ 0 GeV ,(197)
while the CSB term, originated by the vector structure is
(δU) 2 = − 0.031 µ m κ(ρ) − κ(ρ 0 ) ρ 0 GeV(198)
with ρ 0 = 0.17 Fm −3 being the value of saturation density. The term ∆L m s of Eq.(186) provides the contribution containing the CSB condensate γ m -Eq.(194)
(δU) 3 = 32 (γ m (κ(ρ) − κ(0)) + (γ m − γ 0 )κ(0)) / GeV 2 .(199)
For the complete calculation one needs the isospin breaking shifts of vacuum parameters δλ 2 and δW 2 while the empirical value of δm can be used. Thus, the analysis of CSB effects in vacuum should be carried out in framework of the method as well. This was done by Adami et al. [135]. The values of δλ 2 , δW 2 and of the vacuum isospin breaking value γ 0 were obtained through the quark mass difference and the empirical values of the shifts of the baryon masses. This prompts another form of Eq.(195)
δε = b 1 µ + b 2 γ m(200)
with b 1 = a 1 + a 2 (γ 0 /µ), b 2 = a 3 . The expressions for the contributions to δε, caused by the shifts of the vacuum values δm, δλ 2 and δW 2 are rather complicated. At ρ = ρ 0 the corresponding contribution is
(δU) 4 = −0.4 MeV .(201)
For the sum i (δU) i we find, after adding the contribution δT
b 1 (ρ 0 ) = −0.73 , b 2 (ρ 0 ) = −1.0 GeV .(202)
The numerical results can be obtained if the value of γ m is calculated. This can be done in framework of certain models. However, even now we can make some conclusions. If we expect the increasing restoration of the isospin symmetry with growing density, it is reasonable to assume that |γ m | < |γ 0 |. Also, all the model calculations provide γ 0 < 0. Thus we expect γ 0 < γ m < 0. If γ m = 0 we find δε = −2.4 MeV, eliminating the vacuum value δm = 2.3 MeV. Hence, the isospin invariance appears to be restored for both the condensates and nucleon masses.
The present analysis enables also to clarify the role of the CSB effects in the scalar channel. Indeed, neglecting these effects, i.e. putting (δU) 2 = (δU) 3 = 0 we obtain δε > 0. This contradicts both experimental values and general theoretical expectations. Thus we came to the importance of CSB effects in the scalar channel.
Adami and Brown [135] used NJL model, combined with BR1 scaling for calculation of parameter γ m . They found γ m /γ 0 = (κ(ρ)/κ(0)) 1/3 . Substituting this value into Eq.(200) we find δε = (−0.9 ± 0.6)MeV
with the errors caused mostly by uncertainties of the value of γ 0 . A more rapid decrease of the ratio γ m /γ 0 would lead to larger values |δε| with δε < 0. Putting γ m = γ 0 provides δε = −0.3 MeV.
Of course, Eq.(203) is obtained for infinite nuclear matter and it is not clear, if it can be extrapolated for the case A = 40. We can state that at least qualitative explanation of the NSA is achieved.
New knowledge
As we have stated earlier, the QCD sum rules can be viewed as a connection between exchange of uncorrelatedqq pairs between our probe nucleon and the matter and the exchange by strongly correlated pairs with the same quantum numbers (the mesons). In the conventional QHD picture this means that in the Dirac equation for the nucleon in the nuclear matter
(q −V )ψ = (m + Φ)ψ(204)
the vector interaction V corresponds to exchange by the vector mesons with the matter while the scalar interaction Φ is caused by the scalar mesons exchange. In the mean field approximation the vector interaction V is proportional to density ρ, while the scalar interaction is proportional to the "scalar density"
ρ s = d 3 p (2π) 3 m * ε(p) ,(205)
which is a more complicated function of density ρ -see Eqs. (76), (78). Thus V = V (ρ), while Φ = Φ(ρ s ). We have seen that QCD sum rules provide similar picture in the lowest orders of OPE: vector and scalar parts of the correlator G m depend on vector and scalar condensates correspondingly: G m q,p = G m q,p (v(ρ)); G m s = G m s (κ(ρ)). However as we have seen in Subsection 5.2.6, we find a somewhat more complicated dependence in the higher order OPE terms, say, G m s = G m s (κ(ρ), v(ρ)), depending on both scalar and vector condensates. This means that the corresponding scalar interaction Φ = Φ(ρ s , ρ), requiring analysis beyond the mean field approximation.
As one can see from Eqs. (197) and (198) in the case of CSB interactions such complications emerge in the sum rules approach in the leading orders of OPE.
Thus the QCD sum rules motivated CSB nuclear forces V and Φ in Eq.(204) are expected to contain dependence on both "vector" and "scalar" densities, i.e. V = V (ρ, ρ s ) and Φ = Φ(ρ, ρ s ). As we said above, such potentials can emerge due to complicated structure of nucleon-meson vertices. This can provide the guide-lines for building up the CSB nucleon-nucleon potentials.
Another new point is the importance of the CSB in the scalar channel. Neglecting the scalar channel CSB interactions we obtain the wrong sign of the effects, i.e. δε > 0. This contradicts the earlier belief that the vector channel ω − ρ mixing is the main mechanism of the effect [132]. Our result is supported by the analysis of Hatsuda et al. [139] who found that the ω − ρ mixing changes sign for the off-shell mesons. This can also help in constructing the CSB nuclear forces.
EMC effect
The experiments carried out by EMC collaboration [140] demonstrated that deep inelastic scattering function F A 2 (x B ) of nucleus with atomic number A ( x B stands for Bjorken variable) differs from the sum of those of free nucleons. Most of the data were obtained for iron (Fe). The structure function was compared to that of deuteron, which imitates the system of free nucleons. The deviation of the ratio
R A (x B ) = F A 2 (x B ) A F D 2 (x B ) 2(206)
from unity is caused by deviation of a nucleus from the system of free nucleons. The ratio R(x B ) appeared to be the function of x B indeed. Exceeding unity at x B < 0.2 it drops at larger x B reaching the minimum value R Fe (x B ) ≈ 0.85 at x B ≈ 0.7. This behaviour of the ratio was called the EMC effect.
There are several mechanisms which may cause the deviation of the ratio R(x B ) from unity. These are the contribution of quark-antiquark pairs, hidden in pions, originated by the nucleon-nucleon interactions, possible formation of multiquark clusters inside the nucleus, etc. Here we shall try to find how the difference of the quark distributions inside the in-medium and free nucleons changes the ratio R(x B ).
The QCD sum rules method was applied to investigation of the proton deep inelastic structure functions in vacuum in several papers. The second moments of the structure functions were obtained by Kolesnichenko [141] and by Belyaev and Block [142]. The structure function F 2 (x B ) at moderate values of x B was calculated by Belyaev and Ioffe [122]. Here we shall rely on the approach, developed by Braun et al. [143] which can be generalized for the case of finite densities in a natural way. On the other hand, such generalization is the extension of the approach discussed in this section.
To obtain the structure function of the proton, the authors of [143] considered the correlation function G, describing the system with the quantum numbers of proton, interacting twice with a strongly virtual hard photon G(q, k) = i 2 d 4 xd 4 ye i(qx)+i(ky) 0|T [η(x)η(0)]H(y, ∆)|0 .
Here q and q +k are the momenta carried by the correlator in initial and final states, k = k 1 −k 2 is the momentum transferred by the photon scattering. The incoming (outgoing) photon carries momentum k 1 (k 2 ), interacting with the correlator in the point y − ∆/2 (y + ∆/2). The quark-photon interaction is presented by the function H(y, ∆). In the next step the double dispersion relation in variables q 2 1 = q 2 and q 2 2 = (q + k) 2 is considered. The crucial point is the operator expansion in terms of the nonlocal operators depending on the light-like (∆ 2 = 0) vector ∆ [143]. After the Borel transform in both q 2 1 and q 2 2 is carried out and the equal Borel masses M 2 1 = M 2 2 are considered, the Fourier transform in ∆ provides the momentum distribution of the quarks. This approach was applied by Drukarev and Ryskin [144] for calculation of the quark distributions in the proton, placed into the nuclear matter. The two types of contributions to the correlator should be considered - Fig.14a,b. In the diagram of Fig.14a the photon interacts with the quark of the free loop. In the diagram of Fig.14b it interacts with the quark exchanging with the matter. The modification of the distributions of the quarks was expressed through the vector condensate, which vanishes in vacuum and through the in-medium shifts of the other condensates and of the nucleon parameters m, λ 2 and W 2 . The result appeared to be less sensitive to the value of the four-quark condensate than the characteristics of the nucleon considered in Subsection 5.2.
Note that the results are true for the moderate values of x only and cannot be extended to the region x ≪ 1. This is because the OPE diverges at small x [122].
Omitting the details of calculation, provided in [144] we present the results in Fig.15. One can see that the distributions of u and d quarks in fraction of the target momentum x are modified in a different way and there is no common scale. The fraction of the momentum carried by the u quarks x u decreases by about 4%. The ratio R, determined by Eq.(206) has a typical EMC shape.
The technique used in [144] can be expanded for the calculation of the quark distributions at 1 < x < 2. Thus the approach enables to describe the cumulative aspects of the problem as well.
The difficulties
In spite of the relative success, described above, the approach faces a number of difficulties. Some of them take place in vacuum as well. The other ones emerge in the case of the finite density.
The first problem is the convergence of OPE in the lhs of the sum rules. Fortunately, the condensates which contribute to the lowest order OPE terms can be either calculated in a modelindependent way, or expressed through the observables. This is true for both vacuum and nuclear matter -at least, for the values of density which are close to the saturation values ρ 0 . However the situation is not so simple for the higher order OPE contributions. The four-quark condensate is the well known headache of all the QCD sum rules practitioners. The problem becomes more complicated at finite density, since the conventional form of presentation of this condensate contains the strongly cancelling contributions.
In order to include the higher OPE terms one needs the additional model assumptions. The same is true for the attempts to go beyond the gas approximation at finite densities. Recently Kisslinger [145] suggested a hybrid of QCD sum rules and of the cloudy bag model. Note, however that QCD sum rules is not a universal tool and there are the cases when OPE does not converge. We mentioned earlier that this takes place for the nucleon structure functions at small x - [122]. Some time ago Eletsky and Ioffe [147] adduced the case when the short distance physics plays important role, making the OPE convergence assumption less convincing. Recently Dutt-Mazumder et al. [146] faced the situation when the ratio of two successive terms of q −2 expansion is not quenched.
There are also problems with the rhs of the sum rules. The "pole+continuum" model is a very simple ansatz, and it may appear to be oversimplified even in vacuum. The spectrum of the nucleon correlator in-medium is much more complicated than in vacuum. The problem is to separate the singularities of the correlator, connected with the nucleon from those of the medium itself. As we have seen, the "pole+continuum" model can be justified to the same extent as in vacuum until we do not include the three-nucleon interactions. We do not have a simple and convincing model of the spectrum which would include such interactions.
Anyway, the success of vacuum sum rules [117] and reasonable results for the nucleons at finite densities described in this Section prompt that the further development of the approach is worth while.
A possible scenario
The shape of the density dependence of the quark scalar condensate in the baryon matter κ(ρ) appears to be very important for hadronic physics. It is the characteristics of the matter as a whole, describing the degree of restoration of the chiral symmetry with growing density. On the other hand, the dependence κ(ρ) is believed to determine the change of the nucleon effective mass m * (ρ). The shape of the dependence m * (κ(ρ)) differs in the different models.
The lowest order density dependence term in the expansion of the function κ(ρ) is modelindependent. However for the rigorous calculation of the higher order terms one needs to know the density dependence of the hadron parameters m * (ρ), m * ∆ (ρ), f * π (ρ), etc. In Sec.4 we presented the results of the calculations of the condensate κ(ρ) under certain model assumptions. A more detailed analysis requires the investigation of the dependence of these parameters on QCD condensates.
Such dependence can be obtained by using the approach, based on the in-medium QCD sum rules. In Sec.5 we show how in-medium QCD sum rules for the nucleons work. Even in a somewhat skeptical review of Leinweber [148], where the present state of art of applications of the QCD sum rules is criticised, the method is referred to as "the best fundamentally based approach for investigations of hadrons in nuclear matter". Of course, to proceed further one must try to overcome the difficulties, discussed in Subsec.5.5.
The lowest order condensates can be either calculated, or connected directly to the observables. This is true for both vacuum and nuclear mater. However, neither in vacuum nor in medium the higher order condensates can be obtained without applications of certain models. Thus in further steps we shall need a composition of QCD sum rules with model assumptions.
We have seen that the density dependence of the delta isobar effective mass is important for the calculation of the nonlinear contribution to the scalar condensate κ(ρ). The shape of this dependence is still obscure. Thus the extension of the QCD sum rules method for the description of ∆-isobars inmedium dynamics is needed. Such work is going on -see, e.g., the paper of Johnson and Kisslinger [149].
The fundamental in-medium Goldberger-Treiman and Gell-Mann-Oakes-Renner relations are expected to be the other milestones of the approach. The agreement with the results with those of conventional nuclear physics at ρ ∼ ρ 0 would be the test of the approach.
Further development of the approach would require inclusion of the vector mesons. The vector meson physics at finite densities is widely studied nowadays. Say, various aspects of QCD sum rules application where considered in recent papers [146,150,151,152] while the earlier works are cited in reviews [124], [148].
We expect the investigation in framework of this scenario to clarify the features of baryon parameters and of the condensates in nuclear matter.
and (3) compose self-consistent set of equations which determine the values of the condensate and of fermion mass m in the physical vacuum.
Figure 1 :
1The interaction of the operatorqq (the dark blob) with the pion field emerging in the single-pion exchange. The solid lines denote the nucleons; wavy line denotes the pion.
Figure 2 :
2The interaction of the operatorqq with the pion field created by the two-pion exchange between the two nucleons, denoted by the solid lines. The solid lines in the intermediate states stand for the nucleons or for delta isobars. The other notations are the same as inFig.1.
Figure 3 :
3The behaviour of the quark scalar condensate κ(ρ)/|κ(0)| as function of the ratio ρ/ρ 0 obtained in framework of various models. The solid line shows the gas approximation
(98), illustrated by Fig.4 corresponds to the Lagrangian which includes the lowest order πN interactions only. The pion propagator in medium can be viewed as the solution of the Dyson equation [75, 76] -Fig.5:
Figure 4 :
4a)The interaction of the operatorqq (the dark blob) with the pion field. The solid line denotes the nucleon; the wavy line stands for the pion; b,c) The diagrammatic presentation of Eq.(98) with the nucleon in the intermediate state. The bold wavy line denotes the pion propagator renormalized due to baryon-hole excitations in the framework of FFST; d,e)The diagrammatic presentation of Eq.(98) with the ∆-isobar (double solid line)in the intermediate state.
Figure 5 :
5The Dyson equation(99) for the pion propagator in medium in the quasiparticle-hole formalism. Wavy line denotes the vacuum pion propagator, bold wavy line stands for the propagator in matter. The dark angle denotes the correlations.
(98), using nucleons and ∆-isobars as intermediate states. The integration over ω requires investigation of the solutions of the pion dispersion equation-Eq.(81).
N
, which enters Eq.(105) as
4. 4
4Calculation of the scalar condensate 4.4.1 Parameters of the model Now we must specify the functional dependence and the values of the parameters which are involved into the calculations. The πNN coupling constant is
∆ , accounting for the relativistic kinematics are presented in[103].
Figure 6 :
6The function κ(ρ)/|κ(0)|. The solid curve presents the result obtained with g ′ N N = 1.0, g ′ N ∆ = 0.2, g ′ ∆∆ = 0.8, c ∆ = 2.0, Γ ∆ = 0.115 GeV and nucleon effective mass given by Eq.(125) The other curves are obtained with the values of some of the parameters or the shape of the density dependence of the effective mass being modified: a) Dependence of κ(ρ)/|κ(0)| on the variation of nuclear parameters. The dashed curve corresponds to the calculation with c ∆ = 1.7, the dotted curve -to g ′ ∆∆ =1.2, dot-dashed curve -to g ′ N N = 0.7. b) Dependence of κ(ρ)/|κ(0)| on the isobar width. Dotted curve corresponds to the calculation with Γ ∆ = 0.07 GeV, dot-dashed curve -to Γ ∆ = 0.05 GeV, dashed curve -to Γ ∆ = 0.01 GeV. c) Dependence of κ(ρ)/|κ(0)| on the shape of m * (ρ). Dashed curve corresponds to Walecka formula (127) with m * (m * (ρ = ρ 0 ) = 0.8m). Dot-dashed curve is obtained in framework of Walecka model with m * (ρ = ρ 0 ) = 0.7m. to solve the equation
∆ decreases in nuclear matter and m * ∆ −m * < m ∆ −m. Assuming the Additive Quark Model prediction for the scalar field-baryon couplings g sN N = g s∆∆ we come to the equation m * ∆ − m * = m ∆ − m. The Brown-Rho scaling leads to still smaller shift m * ∆ − m * = [m * (m ∆ − m)]/m. Now we present the results of the self-consistent calculations of the condensate under various assumptions on the dependence y i (κ(ρ)) -Eq.(130).
Figure 7 :
7The quark scalar condensate κ(ρ) and nucleon effective mass m * (ρ), calculated in framework of BR1 scaling of the nucleon mass. The dash-double-dotted line shows the gas approximation. The dotted lines 1 and 2 present the results under assumption m * π (ρ) = m π for the sets "b" and "a" of FFST parameters. The other curves present the results obtained under assumption f * π (ρ) = f π and illustrate dependence on the choice of the values of FFST parameters and on the assumed behaviour m * ∆ (ρ). The solid and dashed curves are obtained for the set "a" of FFST parameters with the BR assumption m * ∆ − m ∆ = (m * − m) m ∆ m and for m * ∆ = m ∆ correspondingly. The two other curves are obtained for the set "b" under BR scaling assumption for the isobar mass (dot-dashed curve) and under assumption that the isobar mass does not change in medium (long-dashed curve).
Figure 8 :
8The dependence κ(ρ) and m * (ρ) under the assumption of Nambu scaling of the nucleon mass and m * π (ρ) = m π . The three curves illustrate the dependence on the behaviour of m * ∆ (ρ). Solid line corresponds to BR scaling m *∆ − m ∆ = (m * − m) m ∆ m , dashed curve to m * ∆ − m ∆ = m * − m while the dashed-dotted curve is obtained for m * ∆ = m ∆ .The calculations were carried out with the choice "b" of FFST parameters. Long-dashed line inFig.8ashows the gas approximation. place at the value ρ a , determined by
withλ 2 being the residue at the pole. Substituting rhs of Eq.(136) into Eq.(93) and employing q −2 power expansion in lhs, i.e. putting G(q 2 ) = G OP E (q 2 )
Figure 9 :
9The Feynman diagrams, describing the lowest order OPE contribution to the nucleon correlator in vacuum. The helix line stands for the system with the quantum numbers of proton. The solid lines denote the quarks. The light circles denote the vacuum expectation value.
Figure 11 :
11One of the contributions, providing the branching point at s = 4m 2 . The bold solid line denotes the nucleon of the matter. The dashed lines stands for the meson systems.
Figure 12 :
12The Feynman diagrams, contributing to the leading terms of OPE of the nucleon correlator in medium. The dark blob denotes in-medium expectation values. The dotted lines stand for gluons.
takes into account the terms [(s − m 2 )/M 2 ] n . The factorL = ln M 2 /Λ 2 ln ν 2 /Λ 2(181)accounts for the anomalous dimensions. Here Λ = 0.15 GeV is the QCD parameter while ν = 0.5 GeV is the normalization point of the characteristics involved.
, the shift of the position of the nucleon pole m m − m can be obtained: using Eq.(172) we find ∆L m s − mL m p − m∆L m q = ∆mλ 2 e −m 2 /M 2 + W 2 e −m 2 /M 2 W 2 2L 4/9 m − 2a ∆W 2 .
Figure 14 :
14Second order interaction of the hard photon (dashed line) with the correlator. The other notations are the same as in the previous pictures.
Figure 15 :
15The in-medium changes of the d quark distribution (dashed curve) and of the u quark distribution (dash-dotted curve) of the fraction x of the momentum of the target nucleon. The solid curve presents the function R − 1 with the ratio R, defined by Eq.(206).
The accuracy of Eq.(35) is (µ/m h ) 2 . Thus we only have to consider the light flavors u, d, s to give a reasonable approximation since m c ≈ 1.5 GeV ≈ 0.3 Fm −1 . This leads to
We thank V. Braun
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| []
|
[
"Extraction of Positional Player Data from Broadcast Soccer Videos",
"Extraction of Positional Player Data from Broadcast Soccer Videos"
]
| [
"Jonas Theiner [email protected] \nL3S Research Center\nLeibniz University Hannover\nHannoverGermany\n",
"Wolfgang Gritz \nL3S Research Center\nLeibniz University Hannover\nHannoverGermany\n",
"Eric Müller-Budack \nTIB -Leibniz Information Centre for Science and Technology\nHannoverGermany\n",
"Robert Rein [email protected] \nInstitute of Exercise and Sport Informatics\nGerman Sport University Cologne\n\n",
"Daniel Memmert [email protected] \nInstitute of Exercise and Sport Informatics\nGerman Sport University Cologne\n\n",
"Ralph Ewerth [email protected] \nL3S Research Center\nLeibniz University Hannover\nHannoverGermany\n\nTIB -Leibniz Information Centre for Science and Technology\nHannoverGermany\n"
]
| [
"L3S Research Center\nLeibniz University Hannover\nHannoverGermany",
"L3S Research Center\nLeibniz University Hannover\nHannoverGermany",
"TIB -Leibniz Information Centre for Science and Technology\nHannoverGermany",
"Institute of Exercise and Sport Informatics\nGerman Sport University Cologne\n",
"Institute of Exercise and Sport Informatics\nGerman Sport University Cologne\n",
"L3S Research Center\nLeibniz University Hannover\nHannoverGermany",
"TIB -Leibniz Information Centre for Science and Technology\nHannoverGermany"
]
| []
| Computer-aided support and analysis are becoming increasingly important in the modern world of sports. The scouting of potential prospective players, performance as well as match analysis, and the monitoring of training programs rely more and more on data-driven technologies to ensure success. Therefore, many approaches require large amounts of data, which are, however, not easy to obtain in general. In this paper, we propose a pipeline for the fully-automated extraction of positional data from broadcast video recordings of soccer matches. In contrast to previous work, the system integrates all necessary sub-tasks like sports field registration, player detection, or team assignment that are crucial for player position estimation. The quality of the modules and the entire system is interdependent. A comprehensive experimental evaluation is presented for the individual modules as well as the entire pipeline to identify the influence of errors to subsequent modules and the overall result. In this context, we propose novel evaluation metrics to compare the output with ground-truth positional data. | 10.1109/wacv51458.2022.00153 | [
"https://arxiv.org/pdf/2110.11107v1.pdf"
]
| 239,050,096 | 2110.11107 | 6f4894c09d7452cbe9a45446a9edfec63054a53f |
Extraction of Positional Player Data from Broadcast Soccer Videos
Jonas Theiner [email protected]
L3S Research Center
Leibniz University Hannover
HannoverGermany
Wolfgang Gritz
L3S Research Center
Leibniz University Hannover
HannoverGermany
Eric Müller-Budack
TIB -Leibniz Information Centre for Science and Technology
HannoverGermany
Robert Rein [email protected]
Institute of Exercise and Sport Informatics
German Sport University Cologne
Daniel Memmert [email protected]
Institute of Exercise and Sport Informatics
German Sport University Cologne
Ralph Ewerth [email protected]
L3S Research Center
Leibniz University Hannover
HannoverGermany
TIB -Leibniz Information Centre for Science and Technology
HannoverGermany
Extraction of Positional Player Data from Broadcast Soccer Videos
Computer-aided support and analysis are becoming increasingly important in the modern world of sports. The scouting of potential prospective players, performance as well as match analysis, and the monitoring of training programs rely more and more on data-driven technologies to ensure success. Therefore, many approaches require large amounts of data, which are, however, not easy to obtain in general. In this paper, we propose a pipeline for the fully-automated extraction of positional data from broadcast video recordings of soccer matches. In contrast to previous work, the system integrates all necessary sub-tasks like sports field registration, player detection, or team assignment that are crucial for player position estimation. The quality of the modules and the entire system is interdependent. A comprehensive experimental evaluation is presented for the individual modules as well as the entire pipeline to identify the influence of errors to subsequent modules and the overall result. In this context, we propose novel evaluation metrics to compare the output with ground-truth positional data.
Introduction
Match analysis in soccer is very complex and many different factors can affect the outcome of a match. The question is which so-called key performance parameters allow for the characterization of successful teams [27,34,40,44]. While team behavior can be differentiated into a hierarchical scheme consisting of individual, group, and team tactics, different metrics are necessary to capture behavior at each level [13,40]. Researchers have recognized that game plays should be segmented into different phases since tactics vary greatly [31] across them. Performance in soccer is also determined by physiological factors [10] such as running distance [8]. For this reason, it has been suggested to link such information to tactical parameters [3].
To carry out such analyses, the player positions on the field are required. Current tracking technologies allow the recording of several million data points representing player and ball positions during a match by using additional hardware, e.g., multiple static cameras or sensors on players. However, they are difficult to obtain, for instance, due to licensing, financial restrictions, or competitive concerns, i.e., a club normally does not want or disclose its own team's data. In contrast, broadcast video recordings of soccer matches can be accessed more easily. In this paper, we introduce a modular pipeline to extract the two-dimensional positions of the visible players from ordinary broadcast recordings. As illustrated in Figure 1, the system involves sports field registration, shot boundary detection, shot type classification, player detection, and team assignment.
Application Novelty: While commercial approaches like [51,52,56] primarily use multiple static cameras for position data generation from video data, the TV feed is concretely used by SkillCorner [49] and Track160 [57]. However, only the final output of such systems is accessible [49,57]. To the best of our knowledge, neither their quality, nor used architectures or even information about training data and applicability to own data is publicly reported. While individual sub-tasks were tackled in research, its combination for the joint real-world task of player position estimation has not been studied yet (also not beyond soccer). Even individual sub-modules have not been sufficiently evaluated in terms of applicability to real-world data. For the essential step of sports field registration, recent approaches [37,46] are evaluated only on a single smallscale dataset [16]. Potential for generalization were mentioned [5,37] with the use of many cost-intensive annota- Figure 1. Proposed pipeline to extract positional data with team assignment from broadcast videos: The video is pre-processed to segment the field and detect shot boundaries. The camera type is estimated to extract shots from the main camera. Subsequently, the sports field is registrated and the extracted homography matrix is used to transform the sport field and player detections in order to obtain two-dimensional coordinates for each player. Team assignment is performed by clustering the player's bounding boxes.
tions from various data sources. Furthermore, the influence of errors in individual modules and their connections has not been explored. To tackle this demanding real-world task is of interest for the computer vision community as well for sports science, and has direct applications.
Contributions: In contrast to commercial systems and related work, we provide the first transparent baseline for player position estimation with interchangeable modules, that relies on state-of-the-art techniques and freely available data, while evaluating each module. We demonstrate the generalizability on multiple datasets where the applied models were not originally trained on. The proposed pipeline is also applicable to the so-called "tactic-cam" that is located next to the main camera. It usually covers the entire soccer field (without any cuts) and is consequently of interest for video analysts. To evaluate the global task, estimated positions are compared to ground-truth positional data. This comparison is not trivial due to non-visible players in the video and the influence of errors of individual modules. Therefore, we propose novel evaluation metrics and identify the impact of errors on the final system output.
The remainder of the paper is organized as follows. Section 2 gives a brief overview of related work.The pipeline itself is introduced in Section 3. In Section 4, the different system components and the accuracy of the extracted positional data are evaluated. Finally, Section 5 discusses the results and describes possible areas of future research.
Related Work
Since the global task of player position estimation has not yet been addressed, we briefly review related work for all individual sub-tasks in this section.
Great progress has been made in recent years for sports field registration with monocular non-static cameras. Cuevas et al. [7] trained a probabilistic decision tree to classify specific line segments as an intermediate step for homography estimation and integrated a self-verification step to judge whether a predicted homography matrix is cor-rect. Homayounfar et al. [16] propose a solution that relies on field segmentation and Markov random fields. Sharma et al. [47] and Chen and Little [4] propose the nearest neighbor search from a synthetic dataset of pairs of edge images and camera images for fully-automated registration. Jiang et al. [19] present a two-step deep learning approach that initially estimates a homography and minimizes the error using another deep network instead of the Lucas-Kanade algorithm [1]. Citraro et al. [6] suggest an approach that also takes into account the position of players and is trained on a separate dataset for uncalibrated cameras. Sha et al. [46] propose an end-to-end approach for area-based field segmentation, camera pose estimation, and online homography refinement that allows end-to-end training and efficient inference. Nie et al. [37] tackle the challenge when no prior knowledge about the camera is available and propose a multi-task network to simultaneously detect a grid of key points and dense field features to estimate and refine a homography matrix end-to-end. This approach seems suitable since also temporal consistency is verified for successive frames. However, a very large number of training samples is required to achieve the desired accuracy and generalizability, but training data are not publicly available except for the WorldCup2014 dataset (WC14 [16]).
Shot boundary detection (e.g., [14,26,53,58]) and shot type classification (e.g., [45,54]) are necessary preprocessing steps for many tasks of video analysis. It enables the distinction between different camera shot types. Related work in the context of soccer distinguishes between three [48], four [35] or five [59] different camera shot types. For the extraction of positional data, the main camera (with the largest distance) offers the most useful information, because it normally covers a larger part of the field depicting several players.
There are several approaches for the detection of players in sports analysis [21,28,43,60]. Although generalpurpose approaches for object detection [25,42] are also able to detect persons, sports offer specific challenges. For example, the players are usually small, they differ in scale due to the distance from the camera, they can occlude one another, and there is blur caused by camera movement. Nevertheless, specialized approaches [21,60] compare themselves to general-purpose detectors such as the Single Shot Detector (SSD) [25] or Faster R-CNN [42]. Komorowski et al. [21] have recently introduced a computationally much more efficient method with results similar to a fine-tuned Faster R-CNN. In team sports, the jerseys of the teams are designed so that they can be easily recognized by their color. Thus, for team assignment of the (detected) players, color information can be used as a discriminant feature. Hand-crafted (color) features ( [9,30,55]) or features from convolutional neural networks (CNNs) ( [18,22,29]) are exploited and clustered by these approaches for team assignment. An approach for player detection and team discrimination [18] addresses the problem of occlusions and errors in object detection [32].
Player Position Estimation in Soccer Videos
A frequent problem in the field of automatic sports analysis is the lack of publicly available datasets. Currently, there is no public dataset that provides positional data for given broadcast soccer videos. Besides, related work solely considered sub-problems of the overall task of player position estimation. This section describes a pipeline as well as the choice and modifications of individual components that solve all required sub-tasks for player position estimation to predict the two-dimensional player positions on the field given an input (broadcast) video ( Figure 1).
After all relevant (main camera) shots are identified (Section 3.1), the step of sports field registration is essential to extract position data (Section 3.2). A homography matrix is determined and used to transform the positions of the players from the image plane into world coordinates (Section 3.3).
Shot Boundary and Shot Type Detection
We aim at estimating player positions in frames recorded by the main camera since it is most frequently used and shows the area of the game that is relevant for tactical analysis, as shown in Figure 1. We first extract shots from the television (TV) broadcast using the widely applied TransNet [26,50] for shot boundary detection. Since our objective is to gather only valuable positional data, we subsequently apply shot type classification to identify shots captured by the main camera. We exploit the homography matrices estimated by the sports field registration approach presented in Section 3.2. We found that the homography matrices do not change fundamentally in successive frames captured by the main camera. On the other hand, all other cameras that, for example, capture player close-ups or actions depict no or only small fractions of the sports field causing large errors and consequently inconsistencies in the predicted homography matrices. For this reason, we calculate the average L H of the homography changes for each shot. The homography change for two successive frames t is defined as L H (H t , H t+1 ) = H t − H t+1 2 where each entry in H is (min-max) normalized for each shot. Finally, we classify each shot as the main camera shot if the condition L H ≤ τ is fulfilled.
Sports Field Registration
The task of sports field registration aims at determining a homography matrix H for the transformation of an image from the (main) camera into two-dimensional sports field coordinates. Formally, the matrix H defines a twodimensional projective transformation and is defined by a 3 × 3 matrix with eight degrees of freedom. We use Chen and Little's approach [4] as the basis for sports field registration. The camera calibration is defined as the nearest neighbor search in a synthetic dataset of edge map camera pairs. We choose this approach for multiple reasons: (1) It obtains almost state-of-the-art performance on the only test set for soccer [16], (2) it does not rely on manual annotations to obtain training data [5,19,37], and (3) is adaptable to other environments (e.g., stadiums and camera parameters) by changing only a few hyper-parameters, as shown in our experiments (Section 4.4).
Chen and Little [4] adopt a pix2pix [17] model for field segmentation and the subsequent detection of the field markings. The edge images generated in this way are compared with a dataset of synthetic edge images for which the camera parameters are known (x, y, z position, focal length, pan, tilt). This comparison is based on a Siamese CNN [15], which takes two edge images as input. Feature vectors are used to construct the reference database. The nearest neighbor search on the feature vectors is then applied by computing the L2 distance over all pairs. The camera parameters of the nearest neighbor in the synthetic dataset are used to determine an initial homography matrix. This initial estimation is refined using the Lucas-Kanade algorithm [1].
Player Detection and Position Estimation
Sports analysis offers some specific challenges for the task of object detection and tracking, e.g., the objects (like players) are often small because they are far away from the camera. Camera motion causes blur in the players' silhouettes. But the movement with unpredictable changes of players' direction and pace poses problems also for welltested approaches. Therefore, some approaches address these problems in the architectural design [21,28]. Zhou et al. [61] solves object detection and tracking based on the object center and should therefore be less susceptible to movements of the players. In Section 4.2, a comparison of three approaches is performed.
To determine the actual position of each player on the field, we can utilize the predicted homography matrix H, which maps pixel coordinates to sports field coordinates. We define the image positionp ∈ R 2 of players as the center of the bottom of the detected bounding box, which usually corresponds to the feet of the player. The predicted positionp ∈ R 2 of the player on the field is then calculated with the inverse homography matrix and the detected image positions of the players:p = H −1p .
Self-Verification (sv): The predicted positions can be used to verify the homography matrix extracted by the sports field registration. Assuming that most player positions should be assigned to a coordinate within the sports field, the system can automatically discard individual frames where the sports field registration is obviously erroneous. If one of the projected player positions is far outside the dimensions of the field including a tolerance distance ρ in meter, then normally there is an error in the homography estimation. The smaller the value ρ is chosen, the more frames are discarded, because only smaller errors in the homography estimation are being tolerated. Intuitively, a tolerance distance between two and five meters seems reasonable which is proven experimentally (Section 4.5).
Team Assignment: Assuming that for some sports analytic tasks the position of the goalkeeper is of minor relevance (e.g., formation or movement analysis) and it is extremely rare that both goalkeepers are visible in the video at the same time, they are ignored in the team assignment step. Due to the different jersey type and color it requires context information (i.e. the location) to correctly assign the team.
Another problem is that coaches and attendants also protrude onto the sports field with their bodies due to the perspective of the camera so that the number of visible classes which appear in a frame cannot be predetermined. We present a simple approach that provides a differentiation between only two classes (team A and B) based on the object detection and assumes that the use of an unsupervised clustering method is more appropriate in this domain since it does not rely on any training data and the player detection results are already available with high quality. We apply DBScan [11] to determine two dominant clusters representing the field players of both teams. Any unassigned detection, which should include goalkeepers, referees, and other persons, is discarded. The feature vectors are formed based on the player detection results, i.e., the bounding boxes. We use the upper half of a bounding box since it usually covers the torso of a player. Each bounding box is first uniformly scaled to 20 × 20 and then the center of size 16 × 16 is cropped. This should reduce the influence of the surrounding grass in the considered area. Since the jersey colors differ greatly, it is sufficient to use the average over color channels (HSV color space). It can be assumed that field players are most frequently detected and that this is roughly balanced between both teams. Furthermore, due to the previous segmentation of the playing field, only a few detections are expected which are not field players. DB-Scan requires two parameters: , which is the maximum normalized (color) L2 distance between two detections to be assigned to the same cluster, and n cls ∈ [0, 0.5], which specifies how many of all detections must belong together to form a cluster (maximum of 0.5 due to two main clusters). Since the optimal value for will be different for each match, a grid search for randomly selected frames of each sequence of the match is performed to determine the parameter. In contrast to previous work that generally utilizes color histograms [30,55] to reduce the input feature space, we apply the average over pixels without any performance decline. The value for is selected, for which the cost function c( ) = |X (O)ther | + ||X A | − |X B || is minimal and restricted to form exactly two clusters (X A and X B ). The cost function should ensure that the clusters A and B, which represent the two teams, are about the same size and that there are as few as possible unassigned detections (X (O)ther ).
Experimental Results
All individual components are evaluated individually, while the main task of player position estimation is evaluated at the end. The main test sets that are used both to evaluate the sports field registration and player position estimation are introduced in Section 4.1. As shot boundary and shot type classification (Section 3.1) are common preprocessing steps in video data, we refer to the supplemental material A. Section 4.2 and 4.3 focus on the evaluation of player detection and team assignment, while the evaluation of sports field registration is reported in Section 4.4. Finally, the main task is evaluated by comparing the estimated positional data with the ground-truth data (Section 4.5).
Main Datasets
To evaluate the main task of player position estimation, synchronized video and positional data are needed. To indicate the generalizability, we use a total of four datasets that are primarily designed only for testing, i.e., no training nor fine-tuning of individual modules is performed on this or closely related data.
Common broadcast videos of different resolutions (SD and HD) and seasons (2012, 2014) are available as well another type of video -the tactic-cam (TC): this camera recording is without any cuts and usually covers a wider range of the pitch. Since the tactic-cam is located next to the main TV camera and usually covers the majority of players, it is usually used for video analysis. In general, each dataset contains four halves from four matches from the German Bundesliga in 25 Hz temporal resolution with synchronized positional data. Our datasets are referred to as TV12 (2012, SD resolution), TV14 (2014, HD), TC14 (HD), and TV14-S that covers the broadcast videos of the same matches as TC14. Due to temporal inconsistencies in the raw video of TV14-S to the positional data, these videos are synchronized using the visible game clock. The position data are considered as ground truth since they are generated by a calibrated (multi-)camera system that covers the entire field. However, this system can be inaccurate in some cases [39]. An error of one meter is to be assumed in the data provided to us.
The quality of the field registration is essential for the accurate prediction of the player positions, but as there is only one limited dataset for sports field registration in soccer [16], we manually estimate ground-truth homography matrices for a subset of our datasets. In particular, 25 representative and challenging images per match are chosen to cover a wide range of camera settings resulting in 100 annotated images per test set. The remaining modules, i.e., shot boundary and shot type classification, player detection, and team assignment are trained and evaluated on other publicly available datasets and introduced in their respective sections.
Evaluating Player Detection
Player detection and the usage of the homography matrix enable the extraction of two-dimensional coordinates for players. While a general object detector like Faster R-CNN [42] localizes the bounding box for each object, this information is not necessarily needed, rather the exact position is of interest. To assess the performance of Center-Track [61] on soccer data, we compare it to another specialized network [21] for this domain and to a general object detection framework that is fine-tuned [42] for the soccer domain. We note that alternative solutions such as [28] exist and a comparison is generally possible. However, it is out of scope of our paper to re-implement and test several variants especially if a satisfactory quality is achieved with the selected solution.
Datasets & Setup: Due to the lack of publicly available datasets for training and evaluation, Komorowski et al. [21] train their network on two small-scale datasets [9,28] where the training and test data is separated by frame-wise shuffling and subsequential random selection (80% training, 20% test). CenterTrack can exploit temporal information to track players. However, to the best of our knowledge, there exists only one dataset in the domain of soccer with tracking information (ISSIA-CNR [9]), but it contains a very limited number of scene perspectives from multiple static cameras and is thus inappropriate for our system. For a fair comparison with the alternative approach, we follow the train-test split of Komorowski et al. [21] where individual frames are used for training. The publicly available ISSIA-CNR [9] dataset contains annotated sequences from several matches captured by six static cameras (in 30 Hz and ISSIA-CNR [9] Soccer Player [28] Faster R-CNN [42] 87 Table 1.
Since Faster R-CNN and FootAndBall perform well on only one test set and perform significantly worse on the other, this suggests a lack of generalizability whereas CenterTrack achieves good results on both data sets. As CenterTrack benefits from training with tracking data [61], we are confident that results can further be improved, but choose this model for our pipeline as it already provides good results.
Evaluating Team Assignment
In this experiment, we evaluate the team assignment that relies on detected bounding boxes.
Dataset & Setup: In contrast to very small datasets [9,28], Yu et al. [59]'s dataset provides a good diversity regarding the environmental setting (camera movements, lighting conditions, different matches, jersey colors, etc.). Therefore, a subset from their dataset is manually annotated with respect to team assignment. To bypass errors in the player detection, bounding boxes and player assignment are manually annotated for a set of frames containing multiple shot perspectives and matches. Team assignment is annotated for three categories, team A, team B, and other including referees and goalkeepers (due to its sparsity). We randomly select one frame for a total of ten shots captured by the main camera for each match. We took 20 matches that were already used to evaluate the temporal segmentation resulting in 200 frames for evaluation. As mentioned before, the aim is to find two main clusters, and we found empirically that n cls = 0.2 provides good results for this task.
Metrics: Istasse et al. [18] proposed micro accuracy for this task, but this metric only considers labels from both teams and is insufficient in our case, since it can be misleading when the algorithm assigns uncertain associations to the class other. To prevent this, referees and goalkeepers must be excluded from the object detection or an alternative metric needs to be defined. For this reason, we additionally consider the macro accuracy for all three classes.
Results: Our simple method performs well, both in terms of macro accuracy (0.91) for the three classes and micro accuracy (0.93) for the two team classes. We found that most errors are players that are assigned to other (goalkeepers, referees). This leads to the conclusion that field players are assigned correctly with a high probability in most cases. In comparison to an end-to-end approach for team assignment of Istasse et al. [18], where the overall performance is evaluated on basketball data, a similar micro accuracy (0.91) is reported. However, the domain basketball differs much from soccer making a direct comparison difficult.
Importance of Sports Field Registration
As already introduced, many approaches rely on manually annotated ground-truth data for training. There exist only one public benchmark dataset (World-Cup2014 (WC14) [16]). While the test set follows the same data distribution as the training data, in particular, the camera hyper-parameters (location, focal-length, etc.), generalization capabilities are not investigated by existing solutions [4,19,37,46]. Primarily, to indicate the adaptability of Chen and Little [4]'s approach (Section 3.2) to different environmental settings, we explore several hyper-parameters on our target test sets (see Section 4.1). Additionally we compare them with recent work.
Metrics: Since the visible part of the pitch is of interest for application, we report the intersection over union (IoU part ) score to measure the calibration accuracy. It is computed between the two edge images using the predicted homography and the ground-truth homography on the visible part of the image.
Camera Hyper-parameters: In general, we assume that the recommended parameters [4] (derived from WC14 [16]) for generating synthetic training data fit for many soccer stadiums. However, we also evaluate slight modifications of the base camera parameters which are available in WC14: camera location distri- [4] and F for implementation details.
Results: As reported in Table 2 the reproduced results (base parameters) from Chen and Little [4] at WC14 are of similar quality compared to other methods [19,37,46]. We observe a noticeable drop in IoU part on our test sets where the camera parameters (especially the camera position (x, y, z)) are unknown. For the TV12 test set all configurations fail on challenging images. This further indicates that the original parameters are optimized for the camera dataset distribution in WC14. However, on the remaining three test sets, the approach of Chen and Little [4] is able to generalize, whereas an alternative solution [20] fails. Due to the non-availability of (private) training data a comparison with [37,46] is not fair (colored gray). Yet, these approaches seem to yield comparable results. A (student) CCBV [46] model from [5] is trained on the output of a teacher model. As it was originally trained on a large-scale and private dataset, noticeable lower transfer performance is observed on WC14. In summary, with slight changes in the hyper-parameters, the approach from Chen and Little [4] is suitable for the applicability to new data without fine-tuning by human annotations.
Player Position Estimation
This section investigates the performance for player position estimation. Besides, errors of individual modules, i.e., sports field registration, player detection, and team assignment as well as compounding errors of the system are discussed. We choose the full datasets as introduced in Section 4.1. Despite the shot boundary and shot type classification provide good results (see Appendix A), we eliminate their influence by considering manually annotated shots as the results for position estimation depend on this pre-processing step. False-negative errors lead to a lower number of relevant frames for the system's output and for evaluation, while false-positive errors (e.g., close-ups) primarily produce erroneous output for homography estimation.
Metrics: We measure the distance (in meters) between the estimated positions and the actual positions by taking the mean and median over all individual frames (d mean , d med. ) and additionally report how many frames have an error of less or equal than l ∈ {2.0, 3.0} meters (a l ). As previously mentioned, the sensor devices that capture position data (used as ground-truth also in other works [33]) can be slightly inaccurate. A domain expert confirmed, that errors in our system of less or equal than l ≤ 2 m can be considered as correct results and that errors of less than 3 m can still be meaningful for some sports analysis applications.
Matching estimated positions to ground-truth: Most of the time only a subset of players is visible in the broadcast videos and there is no information about which player is visible at a certain frame -making evaluation complex. As there is no direct mapping between predicted and groundtruth positions and the number of detections may vary, the resulting linear sum assignment problem first minimized using the Hungarian Method [23]. Its solution provides a set of distances for each field player visible in the frame t, formally D t = {d 1 , . . . , d n } where n is the number of players and d i = p−p 2 is the distance between the estimated positionp ∈ R 2 for the i-th player to its actual (ground-truth) position p. To aggregate the player distances of one frame, the use of the average distance as an error metric can be misleading as an outlier, e.g., a false-positive player detection (like a substitute player or goalkeeper) can be matched to a ground truth position with high distance (Fig. 2 D,E,H). These outliers can drastically affect the average distance and lead to wrong impressions. To efficiently reject outliers without using an error threshold as an additional system parameter, we propose to report the average distances of the best 80-percent position estimates. Detailed results for this aggregation are included in the Appendix E.
Player mismatch (pm) due to homography estimation & player detection errors: Despite the self-verification (sv) step (Section 3.3) that discards erroneous homography estimations, we cannot directly evaluate whether the remaining homography matrices are correct, since some errors are not considered (e.g., wrong focal length as in Fig. 2 D).
To analyze the impact of very inaccurate homography estimations and major errors in player detection, we utilize ground-truth data to isolate these types of failures. We reproject all ground-truth positions to the image space accord- Table 3. Results regarding mean (dmean) and median error (dmed) in meters and fraction of frames with an error of less or equal than l meters (acc l ) of the total system on several datasets. Ratio indicates how many frames are kept for evaluation after applying different criteria (system output: only with sv).
ing to the estimated homography matrix. If the number of detected players differs significantly from the actual players then the homography is probably erroneous (called player mismatch: pm). We also define a tolerance range of 5% of the image borders to include players that are at the boundary to avoid penalizing smallest discrepancies in the estimation of the homography matrix. Finally, we discard all frames for evaluation that do not satisfy the following condition:
α t := 1 − ζ < |D real t | |D gt t | < 1 + ζ (1)
α t is the indicator function whether a frame t is discarded based on the ratio of detected players |D real t | and expected players |D gt t |. For example, assuming that ten players are claimed to be visible, but only six are detected by the system, then we want to discard such discrepancies and set ζ = 0.3. Furthermore, we incorporate a constraint to measure the results after team assignment by differentiating between the teams before the linear sum assignment and report the mean distance over both teams per frame. Table 3 shows the results for each dataset while taking the best performing models from Table 2. The results are summarized for all matches by taking the mean of the per match results. The results after self-verification sv are the output of our system and set its tolerance area to ρ = 3 m. The results for other thresholds are reported in the supplemental material (??). With the pm criteria the impact of erroneous sports field registration is analyzed. As evaluated, the applied model for sports field registration provides good results, however, for a couple of frames the IoU part is below 90 %. Our sv-process is able to discard some of these frames as the error drops significantly. For the re- maining frames, the pipeline provides promising results on all datasets. The pm criterion demonstrates the high impact of the sports field registration. Even for marginal errors in the homography estimation, i.e., ca. 95% IoU, the absolute error in meter (mean) is about 1 m when backprojecting known keypoints [37]. Hence, the applied reduction of bounding boxes to one point does not substantially affect the error in meters. The qualitative examples in Figure 2 (with applied sv) primarily show the output of the pipeline and support the choice of our metrics. In Figure 2 (I, J), the output is obviously erroneous, but not discarded in the sv process demonstrating the importance of an accurate sports field registration. Furthermore, the influence of false-positive field players (A,D,E), and incorrect team identification is visible (G,H,I). Since the quantitative results are weaker with team assignment, this suggests a lack of generalizability to the test data for the player detection and team assignment module. Indeed, player tracking is not covered which would lead to more stable predictions across multiple frames, and temporal consistency of the sports field registration is not evaluated quantitatively. However, the sports field registration appears to provide stable results even without explicitly treating the temporal component but could be post-processed in an additional step [24,47].
Results:
In summary, we claim that our system outputs promising results in many cases providing a first baseline to conduct various automatic analyses, for instance, regarding forma-tion detection [2,36] or space control [12,41].
Conclusions & Future Work
In this paper, we have presented a fully-automated system for the extraction of positional data from broadcast soccer videos with interchangeable modules for shot boundary detection, shot type classification, player detection, field registration, and team assignment. All components as well as their impact on the overall performance were evaluated. We investigated which parts of the pipeline influence each other and how they could be improved, e.g., by fine-tuning a specific module with more appropriate data. A relatively small error in meters should allow sports analysts to study team behavior. Indeed, the adaptation to other sports would definitely be interesting. In the future, we also plan to integrate a tracking module. However, additional steps for player re-identification (within and across shots) are necessary to allow player-based analysis across a match.
In this experiment, we evaluate the performance of the pre-processing module on soccer data that includes (1) shot boundary detection and (2) shot type detection where its output is essential for the success of the following modules.
Dataset: For this purpose, we select a subset from SSET [59] where the shot boundaries and camera types (main camera or other) are manually annotated for each frame. Since a general method for shot boundary detection is applied where no extra training is needed, 20 matches are randomly selected that contain different challenges like varying resolutions (ten HD (1280 × 720) and ten SD (640 × 360)), light conditions (artificial vs. daylight) or camera movement.
Shot Boundary Detection: To evaluate the shot boundary detection, we report recall, precision, and the F 1 score for pre-trained 1 models of TransNet [26] and TransNetV2 [50]. However, minor temporal offsets of detected cuts (e.g., a cut was detected a few frames earlier or later than the ground truth) are treated as missing cuts. Therefore we consider three different tolerance offsets with ∆ = {0, 1, 2} frames. The results for TransNet on soccer data are presented in Table 4. With a tolerance of only two frames, the improved TransNetV2 [50] achieves useful results (F 1 = 0.89). Because this model is a general detector, fine-tuning is a way to improve the results. But this would be beyond the scope of this work and we believe that the performance is already sufficient.
Shot Type Detection: Please recall, that it is reasonable to consider only shots captured by the main camera since it shows most of the players on the field. To obtain these shots, a simple but effective classification method was introduced 1 https://github.com/soCzech/TransNetV2
B. Runtime of the Pipeline
All components are real-time capable (> 25 f ps) on common hardware (we used a 8x 2.9 GHz CPU and one Nvidia 2080Ti to run each module) except for the field registration, especially the homography refinement [1] 2 . Even an alternative approximated solution [20] is not able to improve the runtime. However, this step is fundamental for the quality of the field registration. If computation time is a requirement, the entire module could be replaced for instance by Nie et al. [37] that was recently published, but unfortunately, the authors do not provide data, nor pre-trained models.
C. Temporal Consistency
In the last two experiments (Section 4.4 and 4.5 of the paper) we evaluated only single frames and did not consider temporal consistency of successive frames. A previously evaluated approach [47] that detects individual outliers in the homography estimation and makes the estimates consistent across multiple frames (a kind of smoothing) was not implemented for two reasons. On the one hand, the runtime would have dropped to less than 1 f ps and on the other hand the authors have already shown that there are only minor improvements. Table 5. Results of the actual error made in meters (dmean) and fraction of frames smaller than l meters on TV12 while varying the self-verification parameter ρ. Ratio indicates how many frames are discarded for evaluation (∞ corresponds to no self-verification.) D. Self-Verification (sv) Parameter ρ
In Section 3.3 of the paper we have introduced the selfverification (sv) criterion that allows the system to automatically discard individual frames where the sports field registration is erroneous. Here, we vary the control-able parameter ρ, and show its influence on the TV12 dataset in Table 5 according to the final result table (Table 3). The stricter (smaller) the value is chosen, the more frames are discarded without much improvement of the results. Thus, we have used ρ = 3 m as it provides a good trade-off.
E. Per-Frame Distances Aggregation
As shortly explained in the main paper, a set of per-frame distances D t = {d 1 , . . . , d n } is aggregated to calculate a global error in meters over all frames and matches. The use of the average distance as error metric can be misleading as an outlier, i.e., a false positive player detection (e.g. substitute player) can be matched to a ground truth player position with high distance. These outliers can drastically affect the average distance and lead to wrong impressions. To efficiently reject outliers without using an error threshold as additional system parameter, we proposed to report the average distances of the best q-percent position estimates. The choice of q is intuitively chosen with 80% to allow minor false-positive player detection errors.
The influence of q to the global error in meter over all frames (d mean ) of one randomly chosen match is addressed in Figure 3 and Figure 4 after applying both filtering criteria (sv and pm) and without considering team assignment. This influence of q could be ideally explained for the match in Figure 3 where no ground-truth positions for the referees are given, but for most of time one referee is visible in the image and has a bounding box according to Center-Track [61]. This bounding will be assigned to one of the ground-truth player positions which is just an issue in evaluation metric.
If we choose a q between 0.8 and 0.9, these kinds of errors can be ignored in the evaluation because the focus is on the distances of the significant player positions. A value of q = 0.8 is also according to Figure 4 not too strict. The influence on the global error in meter using mean, median and our proposed per-frame aggregation function is shown for TV12 without team assignment in Table 6. Table 6. The influence of the per-frame aggregation function on the actual error made in meters is shown on the TV12 dataset.
F. Implementation Details
F.1. Sports Field Registration
For sports field registration and line segmentation we have modified the officially available source code 3 [4].
During training, we sample 50000 camera poses using the provided camera pose engine. Camera parameters are set as explained in our paper. Following [4] we generate edges images (1280 × 720 resolution) for sampled camera poses. Then, the edge images are resized to 320 × 180 and are used to train the two pix2pix models with underlying U-Net architecture. We refer to the implementation section of [4] for hyper-parameters as these parameters have remained unchanged. These generated edge images are the input of the siamese network (structure as in [4]) and produces a 16-dimensional deep feature for each edge images used for retrieval during inference.
F.2. Player Detection
As a pre-trained CenterTrack [61] model is fine-tuned on the described data as in the paper, we use the fine-tuning script from the official implementation 4 with default parameters with the except that the number of epochs is set to 70. In this context, we would like to point out that providing all details and hyperparameters in this document is not purposeful.
bution N (µ =[52, −45, 17] T , σ =[2,9,3] T ) in meters, i.e., the average location from all stadiums (origin is the lower left corner flag of the pitch); focal length (N (3018, 716) mm) and pan (U(−35 • , 35 • )), tilt (U(−15 • , −5 • )) ranges. We extend the pan and tilt range to (−40 • , 40 • ) and (−20 • , −5 • ), respectively, in all models. As the tactic-cam obviously covers a wider range (especially focal length as seen inFigure 2A,D,E), we also
Figure 2 .
2Qualitative results of the proposed system for the extraction of positional player data ordered from low (left) to high error (right):The top row presents the output without considering teams. The green triangles correspond to the predicted positions ( ) of players and the black points to the ground-truth positions (•); team assignments are colorized red and blue. For the input image the ground-truth positions are re-projected according to the estimated homography matrix; in the sports field some grid points are highlighted.
Figure 3 .
3Error in meter over all frames for one match by varying q when the quantile is used to aggregate per-frame distances: Ground-truth positions of the referees are not included which results in a lot of false-positive player detections.
Figure 4 .
4Error in meter over all frames for one match by varying q when the quantile is used to aggregate per-frame distances.
The results on the test set for our fine-tuned Faster R-CNN[42], Komorowski et al.[21]'s model and the fine-tuned CenterTrack[61] are reported in.4
92.8
FootAndBall [21]
92.1
88.5
CenterTrack [61]
90.1
90.2
Table 1. Performance evaluation for player detection: The average
precision in percent is measured on two subsets from the ISSIA-
CNR and Soccer Player dataset.
FHD resolution) comprising 3000 frames per camera. Soc-
cer Player [28] is a dataset created from two professional
matches where each match is recorded by three HD broad-
cast cameras with 30 Hz and bounding boxes are annotated
for approximately 2000 frames. For evaluation, we report
the average precision (AP) according to [38]. In the final
step of CenterTrack bounding boxes are estimated, which
makes AP a suitable metric to compare the performance of
object detectors, even though the size of the bounding box
is not relevant to extract positional data. We refer to the sup-
plemental material (F) for details about the training process.
Results:
Table 4. Evaluation of the shot boundary detection with ∆ frames tolerance for the two methods.in Section 3.1 of the main paper where only a pre-defined parameter τ and the model for sports field registration are involved. We have determined this value experimentally using five randomly sampled matches from SSET[59] without intersections to the test set und using the reproduced model from Chen and Little[4] (Table 2second row of the main paper). For each match, a grid search is conducted to determine a corresponding threshold and the average overall matches provide a final value of τ = 0.35. The F 1 score is optimized as it provides the harmonic mean between precision and recall. Applied on the test set, an overall F 1 score of 0.88 indicates that almost all relevant shots are correctly classified with a precision of 0.90 and recall of 0.87.TransNet [26] TransNetV2 [50]
∆
0
1
2
0
1
2
precision
0.65 0.81 0.88 0.59 0.76 0.86
recall
0.80 0.83 0.84 0.88 0.91 0.92
F1
0.72 0.82 0.86 0.71 0.83 0.89
Ratio dmean d ≤2 m d ≤3 m d ≤4 m d ≤5 mρ
1.0
0.81 2.12
0.70
0.84
0.91
0.94
2.0
0.87 2.12
0.70
0.84
0.91
0.94
3.0
0.90 2.15
0.69
0.84
0.90
0.93
4.0
0.91 2.18
0.69
0.83
0.90
0.93
5.0
0.92 2.20
0.68
0.83
0.90
0.93
∞
1.0 3.69
0.64
0.78
0.84
0.88
Aggregation sv pm Ratio dmean d ≤2 m d ≤3 mMean
1.00 8.90
0.48
0.69
0.90 2.67
0.53
0.75
0.75 2.27
0.56
0.79
Median
1.00 3.70
0.62
0.78
0.90 2.24
0.66
0.83
0.75 1.83
0.71
0.88
proposed
1.00 3.69
0.64
0.78
0.90 2.15
0.69
0.84
0.75 1.77
0.73
0.88
https://docs.opencv.org/4.5.1/dc/d6b/group__video_ _track.html#ga1aa357007eaec11e9ed03500ecbcbe47
https://github.com/lood339/SCCvSD
AcknowledgementThis project has received funding from the German Federal Ministry of Education and Research (BMBF -Bundesministerium für Bildung und Forschung) under 01IS20021B and 01IS20021A.AppendixIn the supplemental material the shot boundary detection and shot type classification are evaluated in Section A and contains additional dataset description for the evaluation of the team assignment in Section 4.3 of the main paper. The computation costs to run the entire pipeline is roughly specified in Section B whereas related limitations regarding the temporal consistency is addressed in Section C. Section D provides additional information to the self-verification filtering to discard obviously errors in the sports field registration, especially the influence of the controllable parameter ρ. The choice of the function to aggregate per-frame distances (predicted positions matched to ground-truth) is described in Section E.
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"Y.-C Li \nDepartment of Physics\nShanghai University\n200444ShanghaiPeople's Republic of China\n",
"D Martínez \nDepartment of Physical Chemistry\nUniversidad del País Vasco -Euskal Herriko Unibertsitatea\nApdo. 644BilbaoSpain\n",
"S Martínez-Garaot \nDepartment of Physical Chemistry\nUniversidad del País Vasco -Euskal Herriko Unibertsitatea\nApdo. 644BilbaoSpain\n",
"X Chen \nDepartment of Physics\nShanghai University\n200444ShanghaiPeople's Republic of China\n",
"J G Muga \nDepartment of Physical Chemistry\nUniversidad del País Vasco -Euskal Herriko Unibertsitatea\nApdo. 644BilbaoSpain\n"
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"Department of Physics\nShanghai University\n200444ShanghaiPeople's Republic of China",
"Department of Physical Chemistry\nUniversidad del País Vasco -Euskal Herriko Unibertsitatea\nApdo. 644BilbaoSpain",
"Department of Physical Chemistry\nUniversidad del País Vasco -Euskal Herriko Unibertsitatea\nApdo. 644BilbaoSpain",
"Department of Physics\nShanghai University\n200444ShanghaiPeople's Republic of China",
"Department of Physical Chemistry\nUniversidad del País Vasco -Euskal Herriko Unibertsitatea\nApdo. 644BilbaoSpain"
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| []
| We propose a method to manipulate, possibly faster than adiabatically, four-level systems with time-dependent couplings and constant energy shifts (detunings in quantum-optical realizations). We inversely engineer the Hamiltonian, in ladder, tripod, or diamond configurations, to prepare arbitrary states using the geometry of four-dimensional rotations to set the state populations, specifically we use Cayley's factorization of a general rotation into right-and left-isoclinic rotations. | 10.1103/physreva.97.013830 | [
"https://arxiv.org/pdf/1710.10207v1.pdf"
]
| 59,246,423 | 1710.10207 | 73c93b057bb3776558266a925e0717db43aa40f1 |
Hamiltonian design to prepare arbitrary states of four-level systems
27 Oct 2017
Y.-C Li
Department of Physics
Shanghai University
200444ShanghaiPeople's Republic of China
D Martínez
Department of Physical Chemistry
Universidad del País Vasco -Euskal Herriko Unibertsitatea
Apdo. 644BilbaoSpain
S Martínez-Garaot
Department of Physical Chemistry
Universidad del País Vasco -Euskal Herriko Unibertsitatea
Apdo. 644BilbaoSpain
X Chen
Department of Physics
Shanghai University
200444ShanghaiPeople's Republic of China
J G Muga
Department of Physical Chemistry
Universidad del País Vasco -Euskal Herriko Unibertsitatea
Apdo. 644BilbaoSpain
Hamiltonian design to prepare arbitrary states of four-level systems
27 Oct 2017
We propose a method to manipulate, possibly faster than adiabatically, four-level systems with time-dependent couplings and constant energy shifts (detunings in quantum-optical realizations). We inversely engineer the Hamiltonian, in ladder, tripod, or diamond configurations, to prepare arbitrary states using the geometry of four-dimensional rotations to set the state populations, specifically we use Cayley's factorization of a general rotation into right-and left-isoclinic rotations.
I. INTRODUCTION
The coherent state manipulation and control of multiple-level quantum systems plays a significant role in atomic, molecular and optical physics, with applications in existing or developing quantum technologies and quantum information processing [1]. Slow adiabatic protocols may be used but they require long times, and detrimental effects of noise and perturbations accumulate. This has motivated the development of a set of techniques denominated "shortcuts to adiabaticity" to speed up the processes, which include counter-diabatic driving [2,3], inverse engineering based on invariants [4], Lie algebraic methods [5,6], fast quasi-adiabatic approaches [7], or fast-forward approaches [8][9][10].
Some of these methods require to add terms in the Hamiltonian which are not easy or possible to implement in practice [4,[11][12][13]. This problem has been addressed in specific systems by optimizing physically available terms [13], or by unitary transformations making use of the Lie algebraic structure of the dynamics [12,[14][15][16][17]. However, generic solutions are not known and, as the system complexity and number of generators increase, the Lie algebraic methods may become numerically unstable or cumbersome to apply. These difficulties may be already noticed in three-level or four-level systems, so alternative or complementary approaches are currently being explored.
In Ref. [18] the authors proposed a scheme to control three-level system dynamics by separating the evolution into population changes, which may be parameterized using Rodrigues' rotation formula, and phase changes. This separation was used to inversely construct the Hamiltonian of the three-level system so as to drive a given transition with allowed couplings and vanishing forbidden couplings. Our goal here is to explore the extension of this concept to four-level systems. Certain couplings should not appear in the final Hamiltonian to implement specific 4-level configurations such as a "diamond", a "tripod", or a "ladder". The population dynamics is now represented by rotations in a four dimensional space, which are considerably more complex and less intuitive than in three dimensions. We have found a description of the rotation in terms of isoclinic matrices and quaternions, making use of Cayley's factorization, more convenient to perform the inversion than a generalized Rodrigues' formula, see Sec. II. In Sec. III we find the Hamiltonian for the different configurations and provide examples. The appendices address technical points: long formulae in Appendix A, a short account of quaternions for 4D rotations in Appendix B, and details of quantum optical realizations in Appendix C.
Four-level systems are widely found and used in different contexts such as atomic physics, optical lattices [19][20][21] or waveguides [22][23][24], with applications such as electromagnetically induced transparency (EIT) [19,25,26], electromagnetically induced absorption [20], or beam splitting [22,23]. Most of the results in this paper are set in an abstract way, without specifying necessarily the physical system, but the notation is chosen as in a quantum-optical realization where atomic internal levels are coupled by laser fields, consistent with Rabi frequencies or detunings as matrix elements of the Hamiltonian. An explicit connection for the diamond configuration is worked out in Appendix C.
II. 4D ROTATIONS
Consider a four-level system in the state |ψ(t) = c 1 (t)|1 + c 2 (t)e iϕ2(t) |2 + c 3 (t)e iϕ3(t) |3 + c 4 (t)e iϕ4(t) |4 , where c n (t) are real probability amplitudes of bare states |n satisfiying the normalization c 2 1 (t) + c 2 2 (t) + c 2 3 (t) + c 2 4 (t) = 1, and the ϕ n (t) are relative phases. Following [18], we separate phase and amplitude information by writing
|ψ(t) = K(t)|ψ r (t) , where K(t) = |1 1| + e iϕ2(t) |2 2| + e iϕ3(t) |3 3| + e iϕ4(t) |4 4| and |ψ r (t) = c 1 (t)|1 + c 2 (t)|2 + c 3 (t)|3 + c 4 (t)|4 . K(t)
is a unitary transformation that contains the phases and |ψ r (t) represents a 4-dimensional (4D) vector on the surface of a 4D sphere. The states |ψ(t) and |ψ r (t) evolve via time-evolution operators U (t) and U r (t) related by U r (t) = K † (t)U (t)K(0),
|ψ(t) = U (t)|ψ(0) , |ψ r (t) = U r (t)|ψ r (0) ,(1)
where we set initial time as 0. U r (t) represents a 4D rotation displacing points on the surface of the 4D sphere.
In the four-dimension real space, we define the rotation Hamiltonian as
H r (t) = i U r (t)U † r (t),(2)
such that i U r (t) = H r (t)U r (t), whereas the total Hamiltonian is
H(t) = i U (t)U † (t) = i K (t)K † (t) + K(t)H r (t)K † (t). (3) A. Rotations in E 4
In four dimensional Euclidean space E 4 , a 4D rotation with centre O can be expressed by a rotation matrix [27][28][29]
cos α − sin α 0 0 sin α cos α 0 0 0 0 cos β − sin β 0 0 sin β cos β (4)
in some appropriate orthogonal coordinateswxỹz. Instead of having an axis of rotation as in 3D, 4D rotations are defined by a pair of completely orthogonal planes of rotation (w-x andỹ-z in the example), α and β are the angles of rotation with respect to the origin of any point on thew-x andỹ-z planes, respectively. More details can be found e.g. in [27][28][29]. We may classify the rotations based on the α and β angles:
If α = β = 0, the rotation is a double rotation. There are two completely orthogonal (invariant) planes of rotation, with just the point O in common. Points in the first plane rotate through α with respect to the origin, and in the second plane rotate through β. For a general double rotation the planes of rotation and angles are unique. Points which are not in the two planes rotate with respect to the origin through an angle between α and β.
If either of α or β are zero, the rotation is a simple rotation about the rotation center O: There is a fixed plane whose points do not change, whereas half-lines from O orthogonal to this plane are displaced through the nonzero angle (α or β).
If α = ±β the rotation is isoclinic and all non-zero points are rotated through the same angle. Then there are infinitely many pairs of orthogonal planes that can be treated as planes of rotation [27]. An isoclinic rotation can be left-or right-isoclinic (depending on whether α=β or α=-β) [30]. According to Cayley's factorization [31,32], any 4D rotation matrix can be decomposed into the product of a right-and a left-isoclinic matrix. This decomposition is also conveniently expressed in terms of quaternions, as discussed in the following subsection.
B. Isoclinic rotations and quaternions
In 4D Euclidean space, an arbitrary point C can be represented as a column vector (w, x, y, z) T or as C = w + xi + yj + zk [33,34]. If |C| 2 = w 2 + x 2 + y 2 + z 2 = 1 we call it unit quaternion. A general 4D rotation takes C to C ′ , according to
C ′ = qCp,(5)
where q = q w + q x i + q y j + q z k and p = p w + p x i + p y j + p z k are two unit quaternions. See the Appendix A for a minimal introduction to quaternion algebra. In more common matrix language, the rotation reads
C ′ = M L M R C,(6) w ′ x ′ y ′ z ′ = q w −q x −q y −q z q x q w −q z q y q y q z q w −q x q z −q y q x q w p w −p x −p y −p z p x p w p z −p y p y −p z p w p x p z p y −p x p w w x y z ,(7)
a formula due to Van Elfrinkhof [30,31]. M L and M R are isoclinic matrices [33,34], so R = M L M R = M R M L is a 4D rotation matrix without loss of generality. Furthermore, R † R = RR † = I due to |q| 2 = q 2 w + q 2 x + q 2 y + q 2 z = 1 and |p| 2 = p 2 w + p 2 x + p 2 y + p 2 z = 1. A summary of further relations between quaternions and 4D-rotations, such as the relation between the isoclinic matrices and the orthogonal rotation planes and corresponding rotation angles, may be found in Appendix A.
III. HAMILTONIAN INVERSE ENGINEERING
In this section, we will make use of the rotation formula (7) to engineer the Hamiltonian and dynamics to drive a four-level system from an initial state to a final state. We substitute U r (t) = R(t) in Eq. (2), where the quaternion components are generally time dependent. The corresponding rotation Hamiltonian has the follow- ing structure
H r (t) = i U r (t)U † r (t) = i 0 Ω 12 (t) Ω 13 (t) Ω 14 (t) −Ω 12 (t) 0 Ω 23 (t) Ω 24 (t) −Ω 13 (t) −Ω 23 (t) 0 Ω 34 (t) −Ω 14 (t) −Ω 24 (t) −Ω 34 (t) 0 ,(8)
where the real elements Ω nm (t) are functions of the unit quaternion components (the explicit expression is given in Appendix B).
Taking the relative phases into account, the total Hamiltonian (3) is
H(t) = i U (t)U † (t) = i K (t)K † (t) + K(t)H r (t)K † (t) = [−φ 2 (t)|2 2| −φ 3 (t)|3 3| −φ 4 (t)|4 4| + i(e −iϕ2(t) Ω 12 (t)|1 2| + e −iϕ3(t) Ω 13 (t)|1 3| + e −iϕ4(t) Ω 14 (t)|1 4| + e i[ϕ2(t)−ϕ3(t)]) Ω 23 (t)|2 3| + e i[ϕ2(t)−ϕ4(t)] Ω 24 (t)|2 4| + e i[ϕ3(t)−ϕ4(t)] Ω 34 (t)|3 4|)] + H.c.(9)
The physical interpretation of this Hamiltonian depends on the system considered. In quantum optics this is to be interpreted as an interaction picture Hamiltonian where the diagonal terms are not energies of the bare levels, as depicted e.g. in Fig. 1, but detunings, see Appendix C. It proves useful to parameterize the quaternion components in terms of generalized spherical angles [35,36],
q w = cos γ 1 , q x = sin γ 1 cos θ 1 , q y = sin γ 1 sin θ 1 cos φ 1 , q z = sin γ 1 sin θ 1 sin φ 1 , p w = cos γ 2 , p x = sin γ 2 cos θ 2 , p y = sin γ 2 sin θ 2 cos φ 2 , p z = sin γ 2 sin θ 2 sin φ 2 ,(10)
where 0 ≤ φ 1,2 ≤ 2π, 0 ≤ θ 1,2 , γ 1,2 ≤ π, and all angles may be time dependent. The explicit expression of the Hamiltonian (8) in terms of these angles is in Appendix B. We denote the initial and final states, at time t = T , as |ψ r (0) = a 1 |1 +a 2 |2 +a 3 |3 +a 4 |4 (a 2 1 +a 2 2 +a 2 3 +a 2 4 = 1) and |ψ r (
T ) = b 1 |1 +b 2 |2 +b 3 |3 +b 4 |4 (b 2 1 +b 2 2 +b 2 3 +b 2 4 = 1), with phases ϕ k (0) = ǫ k and ϕ k (T ) = ǫ ′ k (k = 2, 3, 4). Since |ψ r (T ) = U r (T )|ψ r (0) , we have four equations b 1 b 2 b 3 b 4 = U r (T ) a 1 a 2 a 3 a 4 .(11)
If the angles at time T and the initial a j components are fixed, these equations specify the final coefficients b j . Alternatively, if both initial and final coefficients are given, we have four equations and six variables to play with. The additional freedom may be used to cancel certain terms in the Hamiltonian as demonstrated below.
A. The inverse tripod configuration As a first four-level system, we consider the "Inverse Tripod" configuration in Fig. 1. The three excited states (|2 , |3 and |4 ) are coupled to the ground state |1 by three couplings Ω 12 , Ω 13 and Ω 14 , respectively [25,37,38]. In this configuration, the transitions |2 ↔ |3 , |2 ↔ |4 and |3 ↔ |4 , are not allowed so we want to cancel these couplings in the Hamiltonian (8). One posible choice to set Ω 23 (t) = Ω 34 (t) = Ω 24 (t) = 0 is
φ 2 = φ 1 = φ, θ 2 = θ 1 = θ, γ 2 = γ 1 = γ(t),(12)
see Eq. (B1), where φ and θ are constants and γ(t) may generally depend on time. The angles are equal for both isoclinic matrices, so the evolution operator
U r (t) = cos [2γ(t)]|1 1| − 2 cos γ(t) sin γ(t) cos θ|1 2| − 2 cos γ(t) sin γ(t) sin θ cos φ|1 3| − 2 cos γ(t) sin γ(t) sin θ sin φ|1 4| + sin [2γ(t)] cos θ|2 1| + {[cos γ(t)] 2 − cos (2θ)[sin γ(t)] 2 }|2 2| − 2[sin γ(t)] 2 cos θ sin θ cos φ|2 3| − 2[sin γ(t)] 2 cos θ sin θ sin φ|2 4| + sin [2γ(t)] sin θ cos φ|3 1| − [sin γ(t)] 2 cos θ sin θ cos φ|3 2| + {[cos γ(t)] 2 + [sin γ(t)] 2 [(cos θ) 2 − cos (2φ)(sin θ) 2 ]}|3 3| − 2[sin γ(t)] 2 (sin θ) 2 cos φ sin φ|3 4| + sin [2γ(t)] sin θ sin φ|4 1| − 2[sin γ(t)] 2 cos θ sin θ sin φ|4 2| − 2[sin γ(t)] 2 (sin θ) 2 cos φ sin φ|4 3| + {[cos γ(t)] 2 + [sin γ(t)] 2 [(cos θ) 2 + cos (2φ)(sin θ) 2 ]}|4 4|(13)
is a simple rotation (see a geometrical explanation in Appendix A), and the rotation Hamiltonian reduces to
H r (t) = − 2i {[γ(t) cos θ]|1 2| + [γ(t) cos φ sin θ]|1 3| + [γ(t) sin φ sin θ]|1 4|} + H.c..(14)
For this particular case, the couplings
Ω 12 (t) =γ(t) cos θ, Ω 13 (t) =γ(t) sin θ cos φ, Ω 14 (t) =γ(t) sin θ sin φ,(15)
take the form of cartesian coordinates of a point on a sphere in terms of spherical coordinates. Starting from the ground state |1 we have freedom to achieve any final state. Setting a 1 = 1, a 2 = a 3 = a 4 = 0 and substituting Eq. (13) in Eq. (11) we get
b 1 = cos [2γ(T )], b 2 = sin [2γ(T )] cos θ, b 3 = sin [2γ(T )] sin θ cos φ, b 4 = sin [2γ(T )] sin θ sin φ,(16)
which we rewrite as
b 1 = A, b 2 = BC, b 3 = BDE, b 4 = BDF,(17)
with
A = cos [2γ(T )], B = sin [2γ(T )], C = cos θ, D = sin θ, E = cos φ, F = sin φ,(18)
obeying the conditions A 2 + B 2 = 1, C 2 + D 2 = 1 and E 2 + F 2 = 1. The system in Eq. (17) with the above conditions has solution
A = b 1 , B = b 2 2 + b 2 3 + b 2 4 , C = b2 √ b 2 2 +b 2 3 +b 2 4 , D = √ b 2 3 +b 2 4 √ b 2 2 +b 2 3 +b 2 4 , E = b3 √ b 2 3 +b 2 4 , F = b4 √ b 2 3 +b 2 4 ,(19)
where we take positive square roots, so it is possible to drive population transfers between the ground state and any final state. To exemplify the method, let us implement the transition |1 → 1
√ 3 (|2 +|3 +|4 ). Substituting b 1 = 0, b 2 = 1/ √ 3, b 3 = 1/ √ 3 and b 4 = 1/ √ 3 in Eq.(19)
and using Eq. (18) we get four equations for γ(T ), θ and φ with solutions
γ(T ) = π 4 , θ = arctan √ 2, φ = π 4 .(20)
We now use an ansatz for γ(t) consistent with γ(T ),
γ(t) = π 8 [1 − cos( πt T )
]. It will determine the timedependence of the Hamiltonian by Eq. (15). Notice that this is just a simple choice, we could use different functions, e.g. to optimize some physically relevant variable or improve robustness.
For the phases we use simple linear interpolation ansatzes,
ϕ k (t) = ǫ k + ∆ k t,(21)
where
∆ k = (ǫ ′ k − ǫ k )/T, (k = 2, 3, 4)(22)
may be interpreted as constant detunings in a quantumoptical realization, see Appendix C. Substituting them in Eq. (9), the total Hamiltonian is
H(t) = − 4 k=2 ∆ k |k k| + i[e −i(ǫ2+∆2t) Ω 12 (t)|1 2| + e −i(ǫ3+∆3t) Ω 13 (t)|1 3| + e −i(ǫ4+∆4t) Ω 14 (t)|1 4| + e i[(ǫ2−ǫ3)+(∆2−∆3)t] Ω 23 (t)|2 3| + e i[(ǫ2−ǫ4)+(∆2−∆4)t] Ω 24 (t)|2 4| + e i[(ǫ3−ǫ4)+(∆3−∆4)t] Ω 34 (t)|3 4|] + H.c..(23)
As an example, let us choose the following boundary conditions, k = 2, 3, 4, to set the phases ϕ k (t). Fig. 2 (a) shows the common smooth amplitude of the couplings, and Fig. 2 (b) demonstrates the perfect population transfer.
ǫ k = 0, ǫ ′ k = π 3 ,(24)
B. The diamond configuration
Now we will focus on the diamond configuration shown in Fig. 3. In this configuration one ground state |1 is coupled in a V -type structure to two intermediate states |2 , |3 , which are themselves coupled to a common excited state |4 in a λ-type structure (see examples in atomic systems in Refs. [26,39,40] and in optical lattices in [21]). Figure 3 shows that the transitions |1 ↔ |4 and |2 ↔ |3 are not allowed so, they must be cancelled in the Hamiltonian (8). To remove the unwanted terms we proceed similarly as in the inverse tripod, taking now
φ 1 = φ 2 = 0, θ 1 =θ 2 =φ 1 =φ 2 = 0,(25)
to achieve Ω 14 (t) = Ω 23 (t) = 0, which gives for the other couplings Ω 12 (t) = −[γ 1 (t) cos θ 1 +γ 2 (t) cos θ 2 ], Ω 13 (t) = −[γ 1 (t) sin θ 1 +γ 2 (t) sin θ 2 ], Ω 24 (t) =γ 1 (t) sin θ 1 −γ 2 (t) sin θ 2 ,
Ω 34 (t) = −[γ 1 (t) cos θ 1 −γ 2 (t) cos θ 2 ].(26)
The evolution operator becomes (27) and the rotating Hamiltonian is
Ur(t) = [cos γ1(t) cos γ2(t) − cos (θ1 − θ2) sin γ1(t) sin γ2(t)]|1 1| − [sin γ1(t) cos γ2(t) cos θ1 + cos γ1(t) sin γ2(t) cos θ2]|1 2| − [sin γ1(t) cos γ2(t) sin θ1 + cos γ1(t) sin γ2(t) sin θ2]|1 3| − [sin γ1(t) sin γ2(t) sin (θ1 − θ2)]|1 4| + [sin γ1(t) cos γ2(t) cos θ1 + cos γ1(t) sin γ2(t) cos θ2]|2 1| + [cos γ1(t) cos γ2(t) − cos (θ1 + θ2) sin γ1(t) sin γ2(t)]|2 2| − [sin γ1(t) sin γ2(t) sin (θ1 + θ2)]|2 3| + [sin γ1(t) cos γ2(t) sin θ1 − cos γ1(t) sinH r (t) = − i {[γ 1 (t) cos θ 1 +γ 2 (t) cos θ 2 ]|1 2| + [γ 1 (t) sin θ 1 +γ 2 (t) sin θ 2 ]|1 3| + [−γ 1 (t) sin θ 1 +γ 2 (t) sin θ 2 ]|2 4| + [γ 1 (t) cos θ 1 −γ 2 (t) cos θ 2 ]|3 4|} + H.c..(28)
To design the Hamiltonian for a transition from |ψ r (0) = |1 , we set a 1 = 1, a 2 = a 3 = a 4 = 0 and substitute Eq. (27) in Eq. (11),
b 1 = cos γ 1 (T )cos γ 2 (T ) − cos (θ 1 −θ 2 )sin γ 1 (T )sin γ 2 (T ), b 2 = sin γ 1 (T )cos γ 2 (T )cos θ 1 + cos γ 1 (T )sin γ 2 (T )cos θ 2 , b 3 = sin γ 1 (T )cos γ 2 (T ) sin θ 1 + cos γ 1 (T )sin γ 2 (T )sin θ 2 , b 4 = − sin γ 1 (T ) sin γ 2 (T ) sin (θ 1 − θ 2 ).(29)
Using the change of variables A = cos γ 1 (T ), B = cos γ 2 (T ), C = sin γ 1 (T ), D = sin γ 2 (T ),
E = cos θ 1 , F = cos θ 2 , G = sin θ 1 , H = sin θ 2 ,(30)
the equations in (29) become
b 1 = AB − CD(EF + GH), b 2 = CBE + ADF, b 3 = CBG + ADH, b 4 = CD(HE − GF ),(31)
where A 2 + C 2 = 1, B 2 + D 2 = 1, E 2 + G 2 = 1 and F 2 + H 2 . The solution in terms of the final state coefficients is
A = b 3 E − b 2 G b 2 4 + (b 3 E − b 2 G) 2 , B = [(b 1 b 3 + b 2 b 4 )E + (b 3 b 4 − b 1 b 2 )G] b 2 4 + (b 3 E − b 2 G) 2 (b 2 3 + b 2 4 )E 2 − 2b 2 b 3 EG + (b 2 2 + b 2 4 )G 2 , C = b 4 b 2 4 + (b 3 E − b 2 G) 2 , D = 1 − [(b 1 b 3 + b 2 b 4 )E + (b 3 b 4 − b 1 b 2 )G] 2 [b 2 4 + (b 3 E − b 2 G) 2 ] [(b 2 3 + b 2 4 )E 2 − 2b 2 b 3 EG + (b 2 2 + b 2 4 )G 2 ] 2 , F = − [(b 4 b 1 − b 2 b 3 )E + (b 2 2 + b 2 4 )G] b 2 4 + (b 3 E − b 2 G) 2 [(b 2 3 + b 2 4 )E 2 − 2b 2 b 3 EG + (b 2 2 + b 2 4 )G 2 ]D , H = [(b 2 3 + b 2 4 )E − (b 2 b 3 + b 1 b 4 )G] b 2 4 + (b 3 E − b 2 G) 2 [(b 2 3 + b 2 4 )E 2 − 2b 2 b 3 EG + (b 2 2 + b 2 4 )G 2 ]D ,(32)
where E and G must obey E 2 + G 2 = 1, so there is freedom to fix the value of the angle θ 1 , see Eq. (30). The other angles, γ 1,2 (T ) and θ 2 , are found from Eq. (30). As an example, we study the population transfer from |1 to the final state |ψ(T ) = 1 √ 2 (|2 ± i|3 ). Substituting b 1 = 0, b 2 = 1/ √ 2, b 3 = 1/ √ 2 and b 4 = 0 in Eq. (32), choosing θ 1 = π/2 and using Eq. (30) we find for the angles the values
γ 1 (T ) = π, γ 2 (T ) = π 2 , θ 2 = − 3π 4 .(33)
For γ 1 (t) and γ 2 (t) we pick out smooth functions consistent with the values at T ,
γ 1 (t) = π 2 1 − cos πt T , γ 2 (t) = π 4 1 − cos πt T .(34)
To find the full Hamiltonian we use Eq. (9) with ϕ k (t) = ǫ k + ∆ k t, k = 2, 3, 4, where the ∆ k are chosen to satisfy the boundary conditions of the example, The results are shown in Fig. (4). Figure 4 (b) shows the perfect population transfer.
ǫ k = 0, ǫ ′ 2 = 0, ǫ ′ 3 = ±π/2, ǫ ′ 4 = 0.(35)φ1 = φ2 = 0, θ1 = π 2 , θ2 = − 3π 4 , ǫ k = 0, ǫ ′ 2 = ǫ ′ 4 = 0 and ǫ ′ 3 = ±π/2.
C. The N-type configuration
The last four-level structure we study is the N-type level scheme [19], with three non-zero couplings Ω 12 , Ω 23 and Ω 34 , see Fig. 5 (A ladder configuration would be treated similarly.). This configuration is applied, for example, to realize the phenomenon of EIT and population transfers in optical lattice systems [19,20,41]. To eliminate the unwanted terms, i.e., to have Ω 13 (t) = Ω 14 (t) = Ω 24 (t) = 0 in Eq. (8) one possible solution iṡ
φ 1 =φ 2 =θ 1 =θ 2 = 0,(36)φ 1 = φ 2 = π 2 ,(37)γ 1 = − sin θ 2 sin θ 1γ 2 .(38)
The Hamiltonian H r (t) becomes
H r (t) = i[(cot θ 1 sin θ 2 − cos θ 2 )γ 2 (t)|1 2| + 2 sin θ 2γ2 (t)|2 3| + (cot θ 1 sin θ 2 + cos θ 2 )γ 2 (t)|3 4|] + H.c.(39)
and the couplings are
Ω 12 (t) =γ 2 (t)(sin θ 2 cot θ 1 − cos θ 2 )
,
Ω 23 (t) = 2γ 2 (t) sin θ 2 , Ω 34 (t) =γ 2 (t)(sin θ 2 cot θ 1 + cos θ 2 ).(40)
Unlike the previous cases, we do not find an analytical expression for the general solution of U r (T ) in Eq. (11) for the initial state |ψ r (0) = |1 . However, for a given final state the system can be solved to get the needed angles. As an example, let us engineer the interaction to go from |ψ r (0) = |1 to |ψ r (T ) = |4 . From Eq. (11) and Eq. (B2), we get four equations for γ 1 (T ), γ 2 (T ), The parameters are θ1 = π/6, θ2 = π/2, γ1(T ) = π, γ2(T ) = −π/2, ǫ2 = 0, ǫ3 = 0, ǫ4 = 0, ǫ ′ 2 = 0, ǫ ′ 3 = 0, ǫ ′ 4 = π/6.
[note that γ 1 = − sin θ2 sin θ1 γ 2 + c, see Eq. (38)], θ 1 , and θ 2 with solutions θ 1 = π/6, θ 2 = π/2, γ 2 (T ) = −π/2, γ 1 (T ) = π. We choose again γ 2 (t) = π 4 [cos( πt T ) − 1] as a smooth ansatz, so H r (t) takes the form H r (t) = i(Ω 12 (t)|1 2|+Ω 23 (t)|2 3|+Ω 34 (t)|3 4|)+H.c., (41) where
Ω 12 (t) = Ω 34 (t) = − √ 3π 2 4T sin πt T , Ω 23 (t) = − π 2 2T sin πt T .(42)
We may use the simple linear interpolation (21) for the phases. For an example with boundary conditions Fig. 6 shows the couplings (a) and population transfer (b) from state |1 to the desired state e iǫ ′ 4 |4 .
ǫ k = 0, ǫ ′ 2 = ǫ ′ 3 = 0, ǫ ′ 4 = π/6,(43)
IV. DISCUSSION
We have set a method to design four-level Hamiltonians so as to drive, in principle in an arbitrary time, specific transitions for different, preselected configurations of the couplings. For arbitrary final states, the method requires full control of the real and imaginary parts of the couplings, and of constant energy shifts. The possibility to realize this level of control will depend on the specific system and physical realization of the Hamiltonian (9). In an atomic system subjected to optical laser fields, this is an interaction picture Hamiltonian after applying the rotating wave approximation, see Appendix C, where the diagonal terms can be interpreted as detunings, and the non-diagonal terms as complex Rabi frequencies. Independent control may be required of the real and imaginary parts of the Rabi frequencies for final states with non-zero phases.
We intend to apply these results in different scenarios. For example, to manipulate the spin state in quantum dots with spin-orbit coupling and electric field control [42]. As for generalizations, the geometry of rotations in higher dimensions has been much less studied that in 3D or 4D, but there are different approaches available [43,44] that could be used to generalize the current scheme to systems with more levels.
Hr(t) = i U r (t)U † r (t) = i sin γ1 sin θ1(θ1 cos γ1 −φ1 sin γ1 sin θ1) + sin γ2 sin θ2(θ2 cos γ2 +φ2 sin γ2 sin θ2) −γ1 cos θ1 −γ2 cos θ2 |1 2| +i θ 1 sin γ1(sin γ1 sin φ1 − cos γ1 cos θ1 cos φ1) −θ2 sin γ2(sin γ2 sin φ2 + cos γ2 cos θ2 cos φ2) −γ1 sin θ1 cos φ1 −γ2 sin θ2 cos φ2 +φ1 sin γ1 sin θ1(cos γ1 sin φ1 + sin γ1 cos θ1 cos φ1) +φ2 sin γ2 sin θ2(cos γ2 sin φ2 − sin γ2 cos θ2 cos φ2) |1 3| +i −θ1 sin γ1(sin γ1 cos φ1 + cos γ1 cos θ1 sin φ1) +θ2 sin γ2(sin γ2 cos φ2 − cos γ2 cos θ2 sin φ2) −γ1 sin θ1 sin φ1 −γ2 sin θ2 sin φ2 −φ1 sin γ1 sin θ1(cos γ1 cos φ1 − sin γ1 cos θ1 sin φ1) −φ2 sin γ2 sin θ2(cos γ2 cos φ2 + sin γ2 cos θ2 sin φ2) |1 4| +i −θ1 sin γ1(sin γ1 cos φ1 + cos γ1 cos θ1 sin φ1) −θ2 sin γ2(sin γ2 cos φ2 − cos γ2 cos θ2 sin φ2) −γ1 sin θ1 sin φ1 +γ2 sin θ2 sin φ2 −φ1 sin γ1 sin θ1(cos γ1 cos φ1 − sin γ1 cos θ1 sin φ1) +φ2 sin γ2 sin θ2(cos γ2 cos φ2 + sin γ2 cos θ2 sin φ2) |2 3| +i −θ1 sin γ1(sin γ1 sin φ1 − cos γ1 cos θ1 cos φ1) −θ2 sin γ2(sin γ2 sin φ2 + cos γ2 cos θ2 cos φ2) +γ1 sin θ1 cos φ1 −γ2 sin θ2 cos φ2 −φ1 sin γ1 sin θ1(cos γ1 sin φ1 + sin γ1 cos θ1 cos φ1) +φ2 sin γ2 sin θ2(cos γ2 sin φ2 − sin γ2 cos θ2 cos φ2) |2 4| = i sin γ1 sin θ1(θ1 cos γ1 −φ1 sin γ1 sin θ1) − sin γ2 sin θ2(θ2 cos γ2 +φ2 sin γ2 sin θ2) −γ1 cos θ1 +γ2 cos θ2 |3 4| (B1)
The time-dependent evolution operator parameterized by the generalized spherical angles in Eq. (10) is
U r (t) ={cos γ 1 cos γ 2 − sin γ 1 sin γ 2 [sin θ 1 sin θ 2 cos(φ 1 − φ 2 ) + cos θ 1 cos θ 2 ]}|1 1|, {sin γ 2 [cos θ 2 cos γ 1 − sin θ 1 sin θ 2 sin γ 1 sin(φ 1 − φ 2 )] + cos θ 1 sin γ 1 cos γ 2 }|2 1|,
{sin γ 2 [sin γ 1 (sin θ 1 cos θ 2 sin φ 1 − cos θ 1 sin θ 2 sin φ 2 ) + sin θ 2 cos γ 1 cos φ 2 ] + sin θ 1 sin γ 1 cos γ 2 cos φ 1 }|3 1| {sin γ 2 [sin γ 1 (cos θ 1 sin θ 2 cos φ 2 − sin θ 1 cos θ 2 cos φ 1 ) + sin θ 2 cos γ 1 sin φ 2 ] + sin θ 1 sin γ 1 cos γ 2 sin φ 1 }|4 1| {− sin γ 2 [sin θ 1 sin θ 2 sin γ 1 sin(φ 1 − φ 2 ) + cos θ 2 cos γ 1 ] − cos θ 1 sin γ 1 cos γ 2 }|1 2| {sin γ 1 sin γ 2 [sin θ 1 sin θ 2 cos(φ 1 − φ 2 ) − cos θ 1 cos θ 2 ] + cos γ 1 cos γ 2 }|2 2| {sin θ 1 sin γ 1 cos γ 2 sin φ 1 − sin γ 2 [sin γ 1 (sin θ 1 cos θ 2 cos φ 1 + cos θ 1 sin θ 2 cos φ 2 ) + sin θ 2 cos γ 1 sin φ 2 ]}|3 2| {sin θ 2 cos γ 1 sin γ 2 cos φ 2 − sin γ 1 [sin γ 2 (sin θ 1 cos θ 2 sin φ 1 + cos θ 1 sin θ 2 sin φ 2 ) + sin θ 1 cos γ 2 cos φ 1 ]}|4 2| {sin θ 1 sin γ 1 cos γ 2 (− cos φ 1 ) − sin γ 2 [sin γ 1 (cos θ 1 sin θ 2 sin φ 2 − sin θ 1 cos θ 2 sin φ 1 ) + sin θ 2 cos γ 1 cos φ 2 ]}|1 3| {sin θ 2 cos γ 1 sin γ 2 sin φ 2 − sin γ 1 [sin γ 2 (sin θ 1 cos θ 2 cos φ 1 + cos θ 1 sin θ 2 cos φ 2 ) + sin θ 1 cos γ 2 sin φ 1 ]}|2 3| {sin γ 1 sin γ 2 [cos θ 1 cos θ 2 − sin θ 1 sin θ 2 cos(φ 1 + φ 2 )] + cos γ 1 cos γ 2 }|3 3| {cos θ 1 sin γ 1 cos γ 2 − sin γ 2 [sin θ 1 sin θ 2 sin γ 1 sin(φ 1 + φ 2 ) + cos θ 2 cos γ 1 ]}|4 3| {sin θ 1 sin γ 1 (− cos γ 2 ) sin φ 1 − sin γ 2 [sin γ 1 (sin θ 1 cos θ 2 cos φ 1 − cos θ 1 sin θ 2 cos φ 2 ) + sin θ 2 cos γ 1 sin φ 2 ]}|1 4| {sin θ 1 sin γ 1 cos γ 2 cos φ 1 − sin γ 2 [sin γ 1 (sin θ 1 cos θ 2 sin φ 1 + cos θ 1 sin θ 2 sin φ 2 ) + sin θ 2 cos γ 1 cos φ 2 ]}|2 4| {cos θ 2 cos γ 1 sin γ 2 − sin γ 1 [sin θ 1 sin θ 2 sin γ 2 sin(φ 1 + φ 2 ) + cos θ 1 cos γ 2 ]}|3 4| {sin γ 1 sin γ 2 [sin θ 1 sin θ 2 cos(φ 1 + φ 2 ) + cos θ 1 cos θ 2 ] + cos γ 1 cos γ 2 }|4 4|. (B2)
Appendix C: Connection with quantum optics (diamond configuration)
To relate the Hamiltonian of the inverse engineering approach, Eq. (9), to an interaction picture Hamiltonian for a four-level atom illuminated by laser fields, we as-sume a semiclassical description of the interaction of the atom with coupling laser fields. Neglecting atomic motion, the Hamiltonian in the Schrödinger picture for the diamond configuration and fields composed by combinations of out-of-phase quadrature components is
H(t) = Ω 12 (t) [|1 2| + |2+ 4 i=2 ω i |i i| } ,(C1)
where we use the vector basis |1 =
1 0 0 0 , |2 = 0 2 0 0 , |3 = 0 0 1 0 , |4 = 0 0 0 1 .Ω ij (t),Ω ′ ij (t) are
the atom-field coupling strengths (Rabi frequencies), assumed real for simplicity, and φ ij the phases of the coher-ent driving fields. The atomic levels |i have energies ω i and the fields have angular frequencies ω ij . We choose the energy zero to match that of level |1 (ω 1 = 0). To transform the system into a laser-adapted interaction picture (rotating frame), we define the unitary operator
U 0 (t) = 1 0 0 0 0 e i(ω12t+φ12) 0 0 0 0 e i(ω13t+φ13) 0 0 0 0 e i[(ω12+ω24)t+φ12+φ24] .(C2)
Using H I (t) = U 0 (t)H(t)U † 0 (t) + i U 0 (t)U † 0 (t),
and imposing the four-photon resonance condition [46][47][48] ω 13 + ω 34 = ω 12 + ω 24 ,
the Hamiltonian in the interacting picture is H I (t) = 2 {2(ω 2 − ω 12 )|2 2| + 2(ω 3 − ω 13 )|3 3| + 2(ω 4 − ω 12 − ω 24 )|4 4| +Ω 12 (t) (1 + e −2i(ω12t+φ12) )|1 2| + (1 + e 2i(ω12t+φ12) )|2 1| + iΩ ′ 12 (t) (1 − e −2i(ω12t+φ12) )|1 2| − (1 − e 2i(ω12t+φ12) )|2 1| +Ω 13 (t) (1 + e −2i(ω13t+φ13) )|1 3| + (1 + e 2i(ω13t+φ13) )|3 1| + iΩ ′ 13 (t) (1 − e −2i(ω13t+φ13) )|1 3| − (1 − e 2i(ω13t+φ13) )|3 1| +Ω 24 (t) (1 + e −2i(ω24t+φ24) )|2 4| + (1 + e 2i(ω24t+φ24) )|4 2| + iΩ ′ 24 (t) (1 − e −2i(ω24t+φ24) )|2 4| − (1 − e 2i(ω24t+φ24) )|4 2| +Ω 34 (t) (1 + e −2i(ω34t+φ34) )e −iΦ |3 4| + (1 + e 2i(ω34t+φ34) )e iΦ |4 3| + iΩ ′ 34 (t) (1 − e −2i(ω34t+φ34) )e −iΦ |3 4| − (1 − e 2i(ω34t+φ34) )e iΦ |4 3| (C5)
where Φ = φ 12 − φ 13 + φ 24 − φ 34 .(C6)
Applying now a rotating wave approximation (RWA) to get rid of the counter-rotating terms we end up with
HI,RW A(t) = 2 0Ω12(t) + iΩ ′ 12 (t)Ω13(t) + iΩ ′ 13 (t) 0 Ω12(t) − iΩ ′ 12 (t)∆2 0Ω24(t) + iΩ ′ 24 (t) Ω13(t) − iΩ ′ 13 (t) 0∆3 (Ω34(t) + iΩ ′ 34 (t))e −iΦ 0Ω24(t) − iΩ ′ 24 (t) (Ω34(t) − iΩ ′ 34 (t))e iΦ∆ 4 ,(C7)
where∆ i (i = 2, 3, 4) are the detunings defined as ∆ 2 = 2(ω 2 − ω 12 ), ∆ 3 = 2(ω 3 − ω 13 ),
∆ 4 = 2(ω 4 − ω 12 − ω 24 ).(C8)
Assuming that the phases of the coherent driving fields can be manipulated to satisfy
φ 12 − φ 13 + φ 24 − φ 34 = 0,(C9)
the Hamiltonian in Eq. (C7) has the structure of the one in Eq. (9). Notice that, the four-photon resonance condition (C4) is key to find a simple Hamiltonian structure in terms of the Rabi frequencies for closed-loop configurations. Equating the diagonal terms, −∆ i =∆ i /2, the laser (angular) frequencies are
ω 12 = ω 2 − ǫ ′ 2 − ǫ 2 2T , ω 13 = ω 3 − ǫ ′ 3 − ǫ 3 2T , ω 24 = ω 4 − ω 2 + ǫ ′ 2 − ǫ 2 2T − ǫ ′ 4 − ǫ 4 2T ,(C10)
and, to satisfy the four-photon resonance condition,
ω 34 = ω 4 − ω 3 − ǫ ′ 4 − ǫ 4 2T + ǫ ′ 3 − ǫ 3 2T .(C11)
Comparing the non-diagonal terms in Eqs. (C7) and (9) we find the form of the Rabi frequencies,
Ω jk = 2e i(φj−φ k )t Ω jk ,(C12)
with φ 1 = 0, φ k (k = 2, 3, 4) given by Eqs. (21,22), and Ω jk =Ω jk + iΩ ′ jk .
For other configurations that do not form a closed loop, similar steps may be followed, but the four-photon resonance condition is not imposed.
FIG. 1 :
1Energy level scheme for the inverse-tripod configuration with three non-zero couplings Ω12, Ω13 and Ω14.
FIG. 2 :
2(Color online) (a) Overlapping couplings Ω12(t) (solid black line), Ω13(t) (green dots) and Ω14(t) (red triangles). (b) Populations of |1 (solid black line), |2 (long-dashed blue line), |3 (green dots) and |4 (red triangles). Parameters: φ = π 4 , θ = arctan √ 2, ǫ k = 0 and ǫ ′ k = π/3, for k = 2, 3, 4.
FIG. 3 :
3Energy level scheme for the diamond-type configuration with four couplings Ω12, Ω13, Ω24 and Ω34.
FIG. 4 :
4(Color online) (a) Couplings Ω12(t) (solid black line), Ω13(t) (long-dashed blue line), Ω24(t) (green dots) and Ω34(t) (red triangles), Ω12(t) = Ω34(t). (b) Populations of |1 (solid black line), |2 (long-dashed blue line), |3 (green dots) and |4 (red triangles). The parameters are:
FIG. 5 :
5Energy level scheme for the four-level Nconfiguration. There are three allowed couplings, Ω12, Ω23, and Ω34.
FIG. 6 :
6(Color online) (a) Couplings Ω12(t) (solid black line), Ω23(t) (green dots), and Ω34(t) (red triangles). (b) Populations of |1 (solid black line), |2 (long-dashed blue line), |3 (green dots) and |4 (red triangles).
1|] cos (ω12 t + φ 12 ) −Ω ′ 12 (t) [|1 2| + |2 1|] sin (ω 12 t + φ 12 ) +Ω 13 (t) [|1 3| + |3 1|] cos (ω 13 t + φ 13 ) −Ω ′ 13 (t) [|1 3| + |3 1|] sin (ω 13 t + φ 13 ) +Ω 24 (t) [|2 4| + |4 2|] cos (ω 24 t + φ 24 ) −Ω ′ 24 (t) [|2 4| + |4 2|] sin (ω 24 t + φ 24 ) +Ω 34 (t) [|3 4| + |4 3|] cos (ω 34 t + φ 34 ) −Ω ′ 34 (t) [|3 4| + |4 3|] sin (ω 34 t + φ 34 )
AcknowledgmentsWe are grateful to E. Y. Sherman and D. Guéry-Odelin for useful comments on the manuscript. This work was supported by the Basque Country Government (Grant No. IT986-16), and MINECO/FEDER,UE (Grant No. FIS2015-67161-P).Appendix A: Quaternions and 4D rotationsA quaternion q can be defined as the sum of a scalar q w and a vector q, namely[45]The rule of product of two quaternions is defined byIf |q| 2 = 1, namely, q 2 w + q 2 x + q 2 y + q 2 z = 1, q is a unit quaternion and q −1 =q. If u = u and |u| 2 = 1, u is a pure unit quaternion, and every pure unit quaternion is a square root of -1. A unit quaternion can be expressed in terms of a real number γ and a pure unit quaternion u as q = e uγ = cos γ + u sin γ.(A3)Consider two arbitrary unit quaternions p and q. We may choose proper pure unit quaternions u and v with corresponding real numbers γ 1 and γ 2 , so that p = e uγ1 and q = e vγ2 . As noted in Sec. II A, an arbitrary rotation R in E 4 of a 4-vector C can be represented by the product qCp, associated with left and right isoclinic rotations with rotation angles γ 1 and γ 2 . R also corresponds to a product of rotations in two mutually orthogonal planes[30,31,33,34]. If u = ±v, R rotates the plane spanned by u + v and uv − 1 through the angle |γ 1 + γ 2 |, and the plane spanned by v − u and uv + 1 through the angle |γ 1 − γ 2 |, respectively[45]. If u = ±v, the planes are spanned by 1 and u and its orthogonal complement, and the rotation angles are as well |γ 1 + γ 2 | and |γ 1 − γ 2 |[45].Appendix B: Hamiltonian and evolutionUsing Eqs.(7,8,10), the parameterized Hamiltonian is given by
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A Sommerfeld, Partial Differential Equations in Physics. New YorkAcademic227A. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1949), p. 227.
. J G Muga, D M Wardlaw, Phys. Rev. E. 515377J. G. Muga and D. M. Wardlaw, Phys. Rev. E 51, 5377 (1995).
. R Unanyan, M Fleischhauer, B W Shore, K Bergmann, Opt. Commun. 155144R. Unanyan, M. Fleischhauer, B.W. Shore, and K. Bergmann, Opt. Commun. 155, 144 (1998).
. E Paspalakis, N J Kylstra, P L Knight, Phys. Rev. A. 6553808E. Paspalakis, N. J. Kylstra, and P. L. Knight, Phys. Rev. A 65, 053808 (2002).
. H Suchowski, Y Silberberg, D B Uskov, Phys. Rev. A. 8413414H. Suchowski, Y. Silberberg, and D. B. Uskov, Phys. Rev. A 84, 013414 (2011).
. G Morigi, S Franke-Arnold, G.-L Oppo, Phys. Rev. A. 6653409G. Morigi, S. Franke-Arnold, and G.-L. Oppo, Phys. Rev. A 66, 053409 (2002).
. D Zhang, Z Wang, S Zhu, Front. Phys. 731D.W Zhang, Z. D Wang, and S.L Zhu, Front. Phys., 7, 31 (2012).
. Y Ban, X Chen, E Y Sherman, J G Muga, Phys. Rev. Lett. 109206602Y. Ban, X. Chen, E. Y. Sherman, and J. G. Muga, Phys. Rev. Lett. 109, 206602 (2012).
. O I Zhelezov, Am. J. Comp. Appl. Math. 751O. I. Zhelezov, Am. J. Comp. Appl. Math. 7, 51 (2017).
. A Richard, L Fuchs, E Andres, G Largeteau-Skapin, A. Richard, L. Fuchs, E. Andres, and G. Largeteau- Skapin, hal.archives-ouvertes.fr/hal-00713697
. J L Weiner, G R Wilkens, Am. Math. Mon. 11269J. L. Weiner and G. R. Wilkens, Am. Math. Mon, 112, 69 (2005).
. S J Buckle, S M Barnett, P L Knight, M A Lauder, D T Pegg, Opt. Acta. 331129S. J. Buckle, S. M. Barnett, P. L. Knight, M. A. Lauder, and D. T. Pegg, Opt. Acta 33, 1129 (1986).
. G Morigi, S Franke-Arnold, G.-L Oppo, Phys. Rev. A. 6653409G. Morigi, S. Franke-Arnold, and G.-L. Oppo, Phys. Rev. A 66, 053409 (2002).
. E Arimondo, Appl. Phys. B. 122293E. Arimondo, Appl. Phys. B 122 293 (2016).
| []
|
[
"The Sooner: a Large Robotic Telescope",
"The Sooner: a Large Robotic Telescope"
]
| [
"G Chincarini \nDipartimento di Fisica G. Occhialini\nUniversità degli Studi di Milano -Bicocca\nPiazza della Scienza 320126MilanoItaly\n\nIstituto Nazionale di Astrofisica\nOsservatorio Astronomico di Brera\nvia Bianchi 4623807MerateItaly\n",
"M Zannoni \nDipartimento di Fisica G. Occhialini\nUniversità degli Studi di Milano -Bicocca\nPiazza della Scienza 320126MilanoItaly\n",
"S Covino \nIstituto Nazionale di Astrofisica\nOsservatorio Astronomico di Brera\nvia Bianchi 4623807MerateItaly\n",
"E Molinari \nIstituto Nazionale di Astrofisica\nOsservatorio Astronomico di Brera\nvia Bianchi 4623807MerateItaly\n\nIstituto Nazionale di Astrofisica\nTelescopio Nazionale Galileo\nRambla José Ana Fernández Pérez\n38712Breña BajaSpain\n",
"S Benetti \nIstituto Nazionale di Astrofisica\nOsservatorio Astronomico di Padova\nVicolo dell'Osservatorio, 535122PadovaItaly\n",
"F Vitali \nIstituto Nazionale di Astrofisica\nOsservatorio Astronomico di Roma\nVia Frascati, 3300040RomaMonte Porzio CatoneItaly\n",
"C Bonoli \nIstituto Nazionale di Astrofisica\nOsservatorio Astronomico di Padova\nVicolo dell'Osservatorio, 535122PadovaItaly\n",
"F Bortoletto \nIstituto Nazionale di Astrofisica\nOsservatorio Astronomico di Padova\nVicolo dell'Osservatorio, 535122PadovaItaly\n",
"E Cascone \nIstituto Nazionale di Astrofisica\nOsservatorio Astronomico di Capodimonte\nSalita Moiariello, 1680131NapoliItaly\n",
"R Cosentino \nIstituto Nazionale di Astrofisica\nTelescopio Nazionale Galileo\nRambla José Ana Fernández Pérez\n38712Breña BajaSpain\n",
"F D'alessio \nIstituto Nazionale di Astrofisica\nOsservatorio Astronomico di Roma\nVia Frascati, 3300040RomaMonte Porzio CatoneItaly\n",
"P D'avanzo \nDipartimento di Fisica G. Occhialini\nUniversità degli Studi di Milano -Bicocca\nPiazza della Scienza 320126MilanoItaly\n\nIstituto Nazionale di Astrofisica\nOsservatorio Astronomico di Brera\nvia Bianchi 4623807MerateItaly\n",
"V De Caprio \nIstituto Nazionale di Astrofisica\nOsservatorio Astronomico di Brera\nvia Bianchi 4623807MerateItaly\n",
"M Della Valle \nIstituto Nazionale di Astrofisica\nOsservatorio Astronomico di Capodimonte\nSalita Moiariello, 1680131NapoliItaly\n",
"A Fernandez-Soto \nInstituto de Fisica de Cantabria (CSIC-UC)\nEdificio Juan Jorda, Av. de los Castros s/n39005SantanderSpain\n",
"D Fugazza \nIstituto Nazionale di Astrofisica\nOsservatorio Astronomico di Brera\nvia Bianchi 4623807MerateItaly\n",
"E Giro \nIstituto Nazionale di Astrofisica\nOsservatorio Astronomico di Padova\nVicolo dell'Osservatorio, 535122PadovaItaly\n",
"A Gomboc \nFakulteta za matematiko in fiziko\nUniverza v Ljubljani\nJadranska 191000LjubljanaSlovenia\n",
"C Guidorzi \nDipartimento di Fisica\nUniversità di Ferrara\nvia Saragat 144122FerraraItaly\n",
"D Magrin \nIstituto Nazionale di Astrofisica\nOsservatorio Astronomico di Padova\nVicolo dell'Osservatorio, 535122PadovaItaly\n",
"G Malaspina \nIstituto Nazionale di Astrofisica\nOsservatorio Astronomico di Brera\nvia Bianchi 4623807MerateItaly\n",
"L Mankiewicz \nCenter for Theoretical Physics of Polish Academy of Science\nAl. Lotnikow 32/4602-668WarsawPoland\n",
"R Margutti \nDipartimento di Fisica G. Occhialini\nUniversità degli Studi di Milano -Bicocca\nPiazza della Scienza 320126MilanoItaly\n",
"R Mazzoleni \nIstituto Nazionale di Astrofisica\nOsservatorio Astronomico di Brera\nvia Bianchi 4623807MerateItaly\n",
"L Nicastro \nIstituto Nazionale di Astrofisica\nIASF Bologna\nvia Gobetti 10140129BolognaItaly\n",
"A Riva \nIstituto Nazionale di Astrofisica\nOsservatorio Astronomico di Torino\nVia Osservatorio, 20, 10025 Pino TorineseItaly\n",
"M Riva \nDipartimento Ingegneria Aerospaziale\nPolitecnico Milano\nVia La Masa, 3420156MilanoItaly\n",
"R Salvaterra \nDipartimento di Fisica G. Occhialini\nUniversità degli Studi di Milano -Bicocca\nPiazza della Scienza 320126MilanoItaly\n",
"P Spanò \nIstituto Nazionale di Astrofisica\nOsservatorio Astronomico di Brera\nvia Bianchi 4623807MerateItaly\n",
"M Sperandio \nIstituto Nazionale di Astrofisica\nOsservatorio Astronomico di Brera\nvia Bianchi 4623807MerateItaly\n",
"M Stefanon \nInstituto de Fisica de Cantabria (CSIC-UC)\nEdificio Juan Jorda, Av. de los Castros s/n39005SantanderSpain\n",
"G Tosti \nDipartimento di Fisica\nFacoltà di Scienze MM. FF. NN\nUniversità degli Studi di Perugia\nvia A. Pascoli06123PerugiaItaly\n",
"V Testa \nIstituto Nazionale di Astrofisica\nOsservatorio Astronomico di Roma\nVia Frascati, 3300040RomaMonte Porzio CatoneItaly\n"
]
| [
"Dipartimento di Fisica G. Occhialini\nUniversità degli Studi di Milano -Bicocca\nPiazza della Scienza 320126MilanoItaly",
"Istituto Nazionale di Astrofisica\nOsservatorio Astronomico di Brera\nvia Bianchi 4623807MerateItaly",
"Dipartimento di Fisica G. Occhialini\nUniversità degli Studi di Milano -Bicocca\nPiazza della Scienza 320126MilanoItaly",
"Istituto Nazionale di Astrofisica\nOsservatorio Astronomico di Brera\nvia Bianchi 4623807MerateItaly",
"Istituto Nazionale di Astrofisica\nOsservatorio Astronomico di Brera\nvia Bianchi 4623807MerateItaly",
"Istituto Nazionale di Astrofisica\nTelescopio Nazionale Galileo\nRambla José Ana Fernández Pérez\n38712Breña BajaSpain",
"Istituto Nazionale di Astrofisica\nOsservatorio Astronomico di Padova\nVicolo dell'Osservatorio, 535122PadovaItaly",
"Istituto Nazionale di Astrofisica\nOsservatorio Astronomico di Roma\nVia Frascati, 3300040RomaMonte Porzio CatoneItaly",
"Istituto Nazionale di Astrofisica\nOsservatorio Astronomico di Padova\nVicolo dell'Osservatorio, 535122PadovaItaly",
"Istituto Nazionale di Astrofisica\nOsservatorio Astronomico di Padova\nVicolo dell'Osservatorio, 535122PadovaItaly",
"Istituto Nazionale di Astrofisica\nOsservatorio Astronomico di Capodimonte\nSalita Moiariello, 1680131NapoliItaly",
"Istituto Nazionale di Astrofisica\nTelescopio Nazionale Galileo\nRambla José Ana Fernández Pérez\n38712Breña BajaSpain",
"Istituto Nazionale di Astrofisica\nOsservatorio Astronomico di Roma\nVia Frascati, 3300040RomaMonte Porzio CatoneItaly",
"Dipartimento di Fisica G. Occhialini\nUniversità degli Studi di Milano -Bicocca\nPiazza della Scienza 320126MilanoItaly",
"Istituto Nazionale di Astrofisica\nOsservatorio Astronomico di Brera\nvia Bianchi 4623807MerateItaly",
"Istituto Nazionale di Astrofisica\nOsservatorio Astronomico di Brera\nvia Bianchi 4623807MerateItaly",
"Istituto Nazionale di Astrofisica\nOsservatorio Astronomico di Capodimonte\nSalita Moiariello, 1680131NapoliItaly",
"Instituto de Fisica de Cantabria (CSIC-UC)\nEdificio Juan Jorda, Av. de los Castros s/n39005SantanderSpain",
"Istituto Nazionale di Astrofisica\nOsservatorio Astronomico di Brera\nvia Bianchi 4623807MerateItaly",
"Istituto Nazionale di Astrofisica\nOsservatorio Astronomico di Padova\nVicolo dell'Osservatorio, 535122PadovaItaly",
"Fakulteta za matematiko in fiziko\nUniverza v Ljubljani\nJadranska 191000LjubljanaSlovenia",
"Dipartimento di Fisica\nUniversità di Ferrara\nvia Saragat 144122FerraraItaly",
"Istituto Nazionale di Astrofisica\nOsservatorio Astronomico di Padova\nVicolo dell'Osservatorio, 535122PadovaItaly",
"Istituto Nazionale di Astrofisica\nOsservatorio Astronomico di Brera\nvia Bianchi 4623807MerateItaly",
"Center for Theoretical Physics of Polish Academy of Science\nAl. Lotnikow 32/4602-668WarsawPoland",
"Dipartimento di Fisica G. Occhialini\nUniversità degli Studi di Milano -Bicocca\nPiazza della Scienza 320126MilanoItaly",
"Istituto Nazionale di Astrofisica\nOsservatorio Astronomico di Brera\nvia Bianchi 4623807MerateItaly",
"Istituto Nazionale di Astrofisica\nIASF Bologna\nvia Gobetti 10140129BolognaItaly",
"Istituto Nazionale di Astrofisica\nOsservatorio Astronomico di Torino\nVia Osservatorio, 20, 10025 Pino TorineseItaly",
"Dipartimento Ingegneria Aerospaziale\nPolitecnico Milano\nVia La Masa, 3420156MilanoItaly",
"Dipartimento di Fisica G. Occhialini\nUniversità degli Studi di Milano -Bicocca\nPiazza della Scienza 320126MilanoItaly",
"Istituto Nazionale di Astrofisica\nOsservatorio Astronomico di Brera\nvia Bianchi 4623807MerateItaly",
"Istituto Nazionale di Astrofisica\nOsservatorio Astronomico di Brera\nvia Bianchi 4623807MerateItaly",
"Instituto de Fisica de Cantabria (CSIC-UC)\nEdificio Juan Jorda, Av. de los Castros s/n39005SantanderSpain",
"Dipartimento di Fisica\nFacoltà di Scienze MM. FF. NN\nUniversità degli Studi di Perugia\nvia A. Pascoli06123PerugiaItaly",
"Istituto Nazionale di Astrofisica\nOsservatorio Astronomico di Roma\nVia Frascati, 3300040RomaMonte Porzio CatoneItaly"
]
| []
| 1The approach of Observational Astronomy is mainly aimed at the construction of larger aperture telescopes, more sensitive detectors and broader wavelength coverage. Certainly fruitful, this approach turns out to be not completely fulfilling the needs when phenomena related to the formation of black holes (BH), neutron stars (NS) and relativistic stars in general are concerned. Indeed they manifest themselves through highly variable emission of electromagnetic energy and quite often via sudden bursts of electromagnetic energy possibly accompanied (or preceded) by the emission of gravitational waves and neutrinos. These are expected to occur in the collapse of massive stars into GRBs or SNe to produce a BH or a NS, and in the merging of such relativistic objects (NS+NS and BH+NS). Radio observations and later X ray observations showed that we are living in a violent and very dynamic highly variable Universe where the energy involved in explosive phenomena may be as large as 10 52 -10 54 erg (isotropic energy). Since then the field developed theoretically thanks to the contribution of many gifted theoreticians and observationally thanks to a huge development in the area of instrumentation and detectors. Recently, mainly through the Vela, Beppo-SAX and Swift satellites, we reached a reasonable knowledge of the most violent events in the Universe and of some of the processes we believe are leading to the formation of black holes (BH). Massive BHs are believed to exist in AGN and in the nuclear region of the galaxies in general.We plan to open a new window of opportunity to study the variegated physics of very fast astronomical transients, particularly the one related to extreme compact objects. The innovative approach is based on three cornerstones: 1) the design (the conceptual design has been already completed) of a 3m robotic telescope and related focal plane instrumentation characterized by the unique features: "No telescope points faster"; 2) simultaneous multi-wavelengths observations (photometry, spectroscopy & polarimetry); 3) high time resolution observations. The conceptual design of the telescope and related instrumentation is optimized to address the following topics: High frequency a-periodic variability, Polarization, High z GRBs, Short GRBs, GRB-Supernovae association, Multi-wavelengths simultaneous photometry and rapid low dispersion spectroscopy. This experiment will turn the "exception" (like the optical observations of GRB 080319B) to "routine". In GRB 080319B (the naked eye GRB [1]) we observed rapid variability both at high energy and in the optical. The optical light curve variability on a scale of 5 to 10 s follows the hard X ray variability with a delay of about 6 seconds. Differences between the optical and the high energy exist (here however it is essential to plan for higher time resolution at optical wavelengths) at higher frequency (variability). Invoking an inverse Compton for the high-energy emission the differences in variability and the delay call for a revised model[2]. In other words since the inverse Compton on the synchrotron photons occurs with no observable delay, the high energy signal should follow the optical prompt emission light curve in all the details and without any delay unless the sources are in different locations. The high speed and multi-wavelengths photometry at optical wavelengths during the early phase of the prompt emission will lead to a major Figure 1: Top: Focal plane instrumentation with all the instruments developed perpendicular to the optical axis of the telescope. Details on the single instruments and mounting of the optics are given in the exploded figures on the top right part of the figure.Each instrument has its own detector at the focal plane and all of them cover an optimum range of wavelengths in order to have the best coverage possible. The spectrograph in particular covers the wavelength range 370 -920 nm to identify easily and quickly the high z objects. The polarimeter will use either o double Wollaston prism or a system as the one developed for the Liverpool telescope by[3]. The telescope, bottom part of the figure, has an aperture of 300 cm and will carry all the instrumentation at the Cassegrain focus. The multi-wavelength photometry will be carried out simultaneously with four CCD working at optical wavelengths (g, r, I, z) and three in the near infrared (NIR, J, H, K). The field of view of the cameras is of 10 x 10 arcmin 2 while with the spectrograph we plan to over a field of 2 x 8 arcmin 2 . For further details see[4]. advance of our understanding of gamma-ray burst, and help answer one of the fundamental unanswered questions as to how the radiation is produced in these explosions.We need furthermore to understand about the magnetic field. Polarimetry during the prompt emission would be exceedingly important to determine the geometry and origin of magnetic fields in GRB shocks.One of the long-standing issues with our current understanding of long GRBs is that the supernovae associated with these bursts are type Ic supernovae (suggesting that long GRBs lack both hydrogen and helium atmospheres). The lack of a hydrogen atmosphere was expected. Hydrogen atmospheres are too extended for the jet to propagate through the star during the disk accretion timescale, leading to mildly relativistic jets. But it is believed that helium atmospheres are compact and most of the progenitors of long GRBs predict the outburst to be associated with both type Ib and type Ic supernovae. Either the progenitor for long GRBs requires a more exotic model than many of the current proposals, or the nature of the explosion has hidden the helium, making what would normally be a type Ib supernova appear as a type Ic supernova. One clue to this long GRB progenitor problem lies in understanding shock break out.This could have been observed in GRB 060218 and in 2008D/XRF 080109[5,6,7]. On the other hand this interpretation is being debated since a similar phenomenon could be caused by a shockwave interacting with gas shells ejected by luminous blue variable outbursts. The complexity of the problem requires full radiation-hydrodynamics calculations as those carried out by C. Fryer at Los Alamos [8]; however for these calculations it is critical to have an observational counterpart to constrain the timing of such phenomena. One of the major goals justifying the search of high z galaxies is, in addition to the understanding of the formation and evolution of Pop III stars, the understanding of the sources that reionize the Universe at that epoch. The most distant galaxy has been detected at z = 6.96 [9] while the most distant AGN has been detected at z ∼ 6.43[10]. Photometric indications (these galaxies and AGN are too faint to get a spectrum even with the very large telescopes) exist of objects with 7 < z < 10; what is really needed is the spectrum in order to have not only a certain identification but also the possibility to measure continuum and lines to estimate the population and the metal abundance. Swift detected three objects for which the optical follow up evidenced through their spectra very high z objects: GRB 050904 at z = 6.29[11], GRB 080913 at z=6.7[12]and GRB 090423 at z = 8.2[13,14]. The latter hold the record for any celestial object so far observed.The host galaxy of GRB 050904 [15] indicate a mass smaller than a few 10 9 solar masses while the metal lines [11] call for a rather low metallicity Z ∼ 0.05Z ⊙ . Unfortunately the spectrum of GRB 090423 does not show any detectable emission or absorption line due to the very small signal to noise ratio. To make any progress in this field we need to get to the target as soon as possible and obtain reasonably high S/N ratio spectroscopic observations. Finally we should clarify the morphology between short and long gamma ray bursts in connection with the physics of their formation and the identification of the progenitors. They both have similar characteristics on the decaying light curve, naturally referring to those cases in which the light curve of shorts has been observed and can clearly be distinguished as two well separated classes only in the Amati relation. Fast follow up will enable (1) collection of the first optical afterglow spectrum of a short burst, providing an in-situ probe of short burst environs and possible progenitor signatures; (2) searches for the predicted signature of the decay of radioactive sub-relativistic material at early times; and (3) collection of extended afterglow light curves to probe the beaming angle distribution of the short bursts via "jet break" analyses, a crucial input in estimating merger event rates. So far the spectra of these GRBs have been elusive due to their faintness and extremely rapid optical decay. In conclusion to make any further significant progress on the GRB physics and related modeling we need a medium size robotic telescope described above. | null | [
"https://arxiv.org/pdf/1005.1569v1.pdf"
]
| 118,185,627 | 1005.1569 | 7dbcf9875cc163db2cf8fc2665abc7dcd95a0d34 |
The Sooner: a Large Robotic Telescope
10 May 2010
G Chincarini
Dipartimento di Fisica G. Occhialini
Università degli Studi di Milano -Bicocca
Piazza della Scienza 320126MilanoItaly
Istituto Nazionale di Astrofisica
Osservatorio Astronomico di Brera
via Bianchi 4623807MerateItaly
M Zannoni
Dipartimento di Fisica G. Occhialini
Università degli Studi di Milano -Bicocca
Piazza della Scienza 320126MilanoItaly
S Covino
Istituto Nazionale di Astrofisica
Osservatorio Astronomico di Brera
via Bianchi 4623807MerateItaly
E Molinari
Istituto Nazionale di Astrofisica
Osservatorio Astronomico di Brera
via Bianchi 4623807MerateItaly
Istituto Nazionale di Astrofisica
Telescopio Nazionale Galileo
Rambla José Ana Fernández Pérez
38712Breña BajaSpain
S Benetti
Istituto Nazionale di Astrofisica
Osservatorio Astronomico di Padova
Vicolo dell'Osservatorio, 535122PadovaItaly
F Vitali
Istituto Nazionale di Astrofisica
Osservatorio Astronomico di Roma
Via Frascati, 3300040RomaMonte Porzio CatoneItaly
C Bonoli
Istituto Nazionale di Astrofisica
Osservatorio Astronomico di Padova
Vicolo dell'Osservatorio, 535122PadovaItaly
F Bortoletto
Istituto Nazionale di Astrofisica
Osservatorio Astronomico di Padova
Vicolo dell'Osservatorio, 535122PadovaItaly
E Cascone
Istituto Nazionale di Astrofisica
Osservatorio Astronomico di Capodimonte
Salita Moiariello, 1680131NapoliItaly
R Cosentino
Istituto Nazionale di Astrofisica
Telescopio Nazionale Galileo
Rambla José Ana Fernández Pérez
38712Breña BajaSpain
F D'alessio
Istituto Nazionale di Astrofisica
Osservatorio Astronomico di Roma
Via Frascati, 3300040RomaMonte Porzio CatoneItaly
P D'avanzo
Dipartimento di Fisica G. Occhialini
Università degli Studi di Milano -Bicocca
Piazza della Scienza 320126MilanoItaly
Istituto Nazionale di Astrofisica
Osservatorio Astronomico di Brera
via Bianchi 4623807MerateItaly
V De Caprio
Istituto Nazionale di Astrofisica
Osservatorio Astronomico di Brera
via Bianchi 4623807MerateItaly
M Della Valle
Istituto Nazionale di Astrofisica
Osservatorio Astronomico di Capodimonte
Salita Moiariello, 1680131NapoliItaly
A Fernandez-Soto
Instituto de Fisica de Cantabria (CSIC-UC)
Edificio Juan Jorda, Av. de los Castros s/n39005SantanderSpain
D Fugazza
Istituto Nazionale di Astrofisica
Osservatorio Astronomico di Brera
via Bianchi 4623807MerateItaly
E Giro
Istituto Nazionale di Astrofisica
Osservatorio Astronomico di Padova
Vicolo dell'Osservatorio, 535122PadovaItaly
A Gomboc
Fakulteta za matematiko in fiziko
Univerza v Ljubljani
Jadranska 191000LjubljanaSlovenia
C Guidorzi
Dipartimento di Fisica
Università di Ferrara
via Saragat 144122FerraraItaly
D Magrin
Istituto Nazionale di Astrofisica
Osservatorio Astronomico di Padova
Vicolo dell'Osservatorio, 535122PadovaItaly
G Malaspina
Istituto Nazionale di Astrofisica
Osservatorio Astronomico di Brera
via Bianchi 4623807MerateItaly
L Mankiewicz
Center for Theoretical Physics of Polish Academy of Science
Al. Lotnikow 32/4602-668WarsawPoland
R Margutti
Dipartimento di Fisica G. Occhialini
Università degli Studi di Milano -Bicocca
Piazza della Scienza 320126MilanoItaly
R Mazzoleni
Istituto Nazionale di Astrofisica
Osservatorio Astronomico di Brera
via Bianchi 4623807MerateItaly
L Nicastro
Istituto Nazionale di Astrofisica
IASF Bologna
via Gobetti 10140129BolognaItaly
A Riva
Istituto Nazionale di Astrofisica
Osservatorio Astronomico di Torino
Via Osservatorio, 20, 10025 Pino TorineseItaly
M Riva
Dipartimento Ingegneria Aerospaziale
Politecnico Milano
Via La Masa, 3420156MilanoItaly
R Salvaterra
Dipartimento di Fisica G. Occhialini
Università degli Studi di Milano -Bicocca
Piazza della Scienza 320126MilanoItaly
P Spanò
Istituto Nazionale di Astrofisica
Osservatorio Astronomico di Brera
via Bianchi 4623807MerateItaly
M Sperandio
Istituto Nazionale di Astrofisica
Osservatorio Astronomico di Brera
via Bianchi 4623807MerateItaly
M Stefanon
Instituto de Fisica de Cantabria (CSIC-UC)
Edificio Juan Jorda, Av. de los Castros s/n39005SantanderSpain
G Tosti
Dipartimento di Fisica
Facoltà di Scienze MM. FF. NN
Università degli Studi di Perugia
via A. Pascoli06123PerugiaItaly
V Testa
Istituto Nazionale di Astrofisica
Osservatorio Astronomico di Roma
Via Frascati, 3300040RomaMonte Porzio CatoneItaly
The Sooner: a Large Robotic Telescope
10 May 2010
1The approach of Observational Astronomy is mainly aimed at the construction of larger aperture telescopes, more sensitive detectors and broader wavelength coverage. Certainly fruitful, this approach turns out to be not completely fulfilling the needs when phenomena related to the formation of black holes (BH), neutron stars (NS) and relativistic stars in general are concerned. Indeed they manifest themselves through highly variable emission of electromagnetic energy and quite often via sudden bursts of electromagnetic energy possibly accompanied (or preceded) by the emission of gravitational waves and neutrinos. These are expected to occur in the collapse of massive stars into GRBs or SNe to produce a BH or a NS, and in the merging of such relativistic objects (NS+NS and BH+NS). Radio observations and later X ray observations showed that we are living in a violent and very dynamic highly variable Universe where the energy involved in explosive phenomena may be as large as 10 52 -10 54 erg (isotropic energy). Since then the field developed theoretically thanks to the contribution of many gifted theoreticians and observationally thanks to a huge development in the area of instrumentation and detectors. Recently, mainly through the Vela, Beppo-SAX and Swift satellites, we reached a reasonable knowledge of the most violent events in the Universe and of some of the processes we believe are leading to the formation of black holes (BH). Massive BHs are believed to exist in AGN and in the nuclear region of the galaxies in general.We plan to open a new window of opportunity to study the variegated physics of very fast astronomical transients, particularly the one related to extreme compact objects. The innovative approach is based on three cornerstones: 1) the design (the conceptual design has been already completed) of a 3m robotic telescope and related focal plane instrumentation characterized by the unique features: "No telescope points faster"; 2) simultaneous multi-wavelengths observations (photometry, spectroscopy & polarimetry); 3) high time resolution observations. The conceptual design of the telescope and related instrumentation is optimized to address the following topics: High frequency a-periodic variability, Polarization, High z GRBs, Short GRBs, GRB-Supernovae association, Multi-wavelengths simultaneous photometry and rapid low dispersion spectroscopy. This experiment will turn the "exception" (like the optical observations of GRB 080319B) to "routine". In GRB 080319B (the naked eye GRB [1]) we observed rapid variability both at high energy and in the optical. The optical light curve variability on a scale of 5 to 10 s follows the hard X ray variability with a delay of about 6 seconds. Differences between the optical and the high energy exist (here however it is essential to plan for higher time resolution at optical wavelengths) at higher frequency (variability). Invoking an inverse Compton for the high-energy emission the differences in variability and the delay call for a revised model[2]. In other words since the inverse Compton on the synchrotron photons occurs with no observable delay, the high energy signal should follow the optical prompt emission light curve in all the details and without any delay unless the sources are in different locations. The high speed and multi-wavelengths photometry at optical wavelengths during the early phase of the prompt emission will lead to a major Figure 1: Top: Focal plane instrumentation with all the instruments developed perpendicular to the optical axis of the telescope. Details on the single instruments and mounting of the optics are given in the exploded figures on the top right part of the figure.Each instrument has its own detector at the focal plane and all of them cover an optimum range of wavelengths in order to have the best coverage possible. The spectrograph in particular covers the wavelength range 370 -920 nm to identify easily and quickly the high z objects. The polarimeter will use either o double Wollaston prism or a system as the one developed for the Liverpool telescope by[3]. The telescope, bottom part of the figure, has an aperture of 300 cm and will carry all the instrumentation at the Cassegrain focus. The multi-wavelength photometry will be carried out simultaneously with four CCD working at optical wavelengths (g, r, I, z) and three in the near infrared (NIR, J, H, K). The field of view of the cameras is of 10 x 10 arcmin 2 while with the spectrograph we plan to over a field of 2 x 8 arcmin 2 . For further details see[4]. advance of our understanding of gamma-ray burst, and help answer one of the fundamental unanswered questions as to how the radiation is produced in these explosions.We need furthermore to understand about the magnetic field. Polarimetry during the prompt emission would be exceedingly important to determine the geometry and origin of magnetic fields in GRB shocks.One of the long-standing issues with our current understanding of long GRBs is that the supernovae associated with these bursts are type Ic supernovae (suggesting that long GRBs lack both hydrogen and helium atmospheres). The lack of a hydrogen atmosphere was expected. Hydrogen atmospheres are too extended for the jet to propagate through the star during the disk accretion timescale, leading to mildly relativistic jets. But it is believed that helium atmospheres are compact and most of the progenitors of long GRBs predict the outburst to be associated with both type Ib and type Ic supernovae. Either the progenitor for long GRBs requires a more exotic model than many of the current proposals, or the nature of the explosion has hidden the helium, making what would normally be a type Ib supernova appear as a type Ic supernova. One clue to this long GRB progenitor problem lies in understanding shock break out.This could have been observed in GRB 060218 and in 2008D/XRF 080109[5,6,7]. On the other hand this interpretation is being debated since a similar phenomenon could be caused by a shockwave interacting with gas shells ejected by luminous blue variable outbursts. The complexity of the problem requires full radiation-hydrodynamics calculations as those carried out by C. Fryer at Los Alamos [8]; however for these calculations it is critical to have an observational counterpart to constrain the timing of such phenomena. One of the major goals justifying the search of high z galaxies is, in addition to the understanding of the formation and evolution of Pop III stars, the understanding of the sources that reionize the Universe at that epoch. The most distant galaxy has been detected at z = 6.96 [9] while the most distant AGN has been detected at z ∼ 6.43[10]. Photometric indications (these galaxies and AGN are too faint to get a spectrum even with the very large telescopes) exist of objects with 7 < z < 10; what is really needed is the spectrum in order to have not only a certain identification but also the possibility to measure continuum and lines to estimate the population and the metal abundance. Swift detected three objects for which the optical follow up evidenced through their spectra very high z objects: GRB 050904 at z = 6.29[11], GRB 080913 at z=6.7[12]and GRB 090423 at z = 8.2[13,14]. The latter hold the record for any celestial object so far observed.The host galaxy of GRB 050904 [15] indicate a mass smaller than a few 10 9 solar masses while the metal lines [11] call for a rather low metallicity Z ∼ 0.05Z ⊙ . Unfortunately the spectrum of GRB 090423 does not show any detectable emission or absorption line due to the very small signal to noise ratio. To make any progress in this field we need to get to the target as soon as possible and obtain reasonably high S/N ratio spectroscopic observations. Finally we should clarify the morphology between short and long gamma ray bursts in connection with the physics of their formation and the identification of the progenitors. They both have similar characteristics on the decaying light curve, naturally referring to those cases in which the light curve of shorts has been observed and can clearly be distinguished as two well separated classes only in the Amati relation. Fast follow up will enable (1) collection of the first optical afterglow spectrum of a short burst, providing an in-situ probe of short burst environs and possible progenitor signatures; (2) searches for the predicted signature of the decay of radioactive sub-relativistic material at early times; and (3) collection of extended afterglow light curves to probe the beaming angle distribution of the short bursts via "jet break" analyses, a crucial input in estimating merger event rates. So far the spectra of these GRBs have been elusive due to their faintness and extremely rapid optical decay. In conclusion to make any further significant progress on the GRB physics and related modeling we need a medium size robotic telescope described above.
The approach of Observational Astronomy is mainly aimed at the construction of larger aperture telescopes, more sensitive detectors and broader wavelength coverage. Certainly fruitful, this approach turns out to be not completely fulfilling the needs when phenomena related to the formation of black holes (BH), neutron stars (NS) and relativistic stars in general are concerned. Indeed they manifest themselves through highly variable emission of electromagnetic energy and quite often via sudden bursts of electromagnetic energy possibly accompanied (or preceded) by the emission of gravitational waves and neutrinos. These are expected to occur in the collapse of massive stars into GRBs or SNe to produce a BH or a NS, and in the merging of such relativistic objects (NS+NS and BH+NS). Radio observations and later X ray observations showed that we are living in a violent and very dynamic highly variable Universe where the energy involved in explosive phenomena may be as large as 10 52 -10 54 erg (isotropic energy). Since then the field developed theoretically thanks to the contribution of many gifted theoreticians and observationally thanks to a huge development in the area of instrumentation and detectors. Recently, mainly through the Vela, Beppo-SAX and Swift satellites, we reached a reasonable knowledge of the most violent events in the Universe and of some of the processes we believe are leading to the formation of black holes (BH). Massive BHs are believed to exist in AGN and in the nuclear region of the galaxies in general.
We plan to open a new window of opportunity to study the variegated physics of very fast astronomical transients, particularly the one related to extreme compact objects. The innovative approach is based on three cornerstones: 1) the design (the conceptual design has been already completed) of a 3m robotic telescope and related focal plane instrumentation characterized by the unique features: "No telescope points faster"; 2) simultaneous multi-wavelengths observations (photometry, spectroscopy & polarimetry); 3) high time resolution observations. The conceptual design of the telescope and related instrumentation is optimized to address the following topics: High frequency a-periodic variability, Polarization, High z GRBs, Short GRBs, GRB-Supernovae association, Multi-wavelengths simultaneous photometry and rapid low dispersion spectroscopy. This experiment will turn the "exception" (like the optical observations of GRB 080319B) to "routine". In GRB 080319B (the naked eye GRB [1]) we observed rapid variability both at high energy and in the optical. The optical light curve variability on a scale of 5 to 10 s follows the hard X ray variability with a delay of about 6 seconds. Differences between the optical and the high energy exist (here however it is essential to plan for higher time resolution at optical wavelengths) at higher frequency (variability). Invoking an inverse Compton for the high-energy emission the differences in variability and the delay call for a revised model [2]. In other words since the inverse Compton on the synchrotron photons occurs with no observable delay, the high energy signal should follow the optical prompt emission light curve in all the details and without any delay unless the sources are in different locations. The high speed and multi-wavelengths photometry at optical wavelengths during the early phase of the prompt emission will lead to a major Each instrument has its own detector at the focal plane and all of them cover an optimum range of wavelengths in order to have the best coverage possible. The spectrograph in particular covers the wavelength range 370 -920 nm to identify easily and quickly the high z objects. The polarimeter will use either o double Wollaston prism or a system as the one developed for the Liverpool telescope by [3]. The telescope, bottom part of the figure, has an aperture of 300 cm and will carry all the instrumentation at the Cassegrain focus. The multi-wavelength photometry will be carried out simultaneously with four CCD working at optical wavelengths (g, r, I, z) and three in the near infrared (NIR, J, H, K). The field of view of the cameras is of 10 x 10 arcmin 2 while with the spectrograph we plan to over a field of 2 x 8 arcmin 2 . For further details see [4]. advance of our understanding of gamma-ray burst, and help answer one of the fundamental unanswered questions as to how the radiation is produced in these explosions.
We need furthermore to understand about the magnetic field. Polarimetry during the prompt emission would be exceedingly important to determine the geometry and origin of magnetic fields in GRB shocks.
One of the long-standing issues with our current understanding of long GRBs is that the supernovae associated with these bursts are type Ic supernovae (suggesting that long GRBs lack both hydrogen and helium atmospheres). The lack of a hydrogen atmosphere was expected. Hydrogen atmospheres are too extended for the jet to propagate through the star during the disk accretion timescale, leading to mildly relativistic jets. But it is believed that helium atmospheres are compact and most of the progenitors of long GRBs predict the outburst to be associated with both type Ib and type Ic supernovae. Either the progenitor for long GRBs requires a more exotic model than many of the current proposals, or the nature of the explosion has hidden the helium, making what would normally be a type Ib supernova appear as a type Ic supernova. One clue to this long GRB progenitor problem lies in understanding shock break out.
This could have been observed in GRB 060218 and in 2008D/XRF 080109 [5,6,7]. On the other hand this interpretation is being debated since a similar phenomenon could be caused by a shockwave interacting with gas shells ejected by luminous blue variable outbursts. The complexity of the problem requires full radiation-hydrodynamics calculations as those carried out by C. Fryer at Los Alamos [8]; however for these calculations it is critical to have an observational counterpart to constrain the timing of such phenomena. One of the major goals justifying the search of high z galaxies is, in addition to the understanding of the formation and evolution of Pop III stars, the understanding of the sources that reionize the Universe at that epoch. The most distant galaxy has been detected at z = 6.96 [9] while the most distant AGN has been detected at z ∼ 6.43 [10]. Photometric indications (these galaxies and AGN are too faint to get a spectrum even with the very large telescopes) exist of objects with 7 < z < 10; what is really needed is the spectrum in order to have not only a certain identification but also the possibility to measure continuum and lines to estimate the population and the metal abundance. Swift detected three objects for which the optical follow up evidenced through their spectra very high z objects: GRB 050904 at z = 6.29 [11], GRB 080913 at z=6.7 [12] and GRB 090423 at z = 8.2 [13,14]. The latter hold the record for any celestial object so far observed.
The host galaxy of GRB 050904 [15] indicate a mass smaller than a few 10 9 solar masses while the metal lines [11] call for a rather low metallicity Z ∼ 0.05Z ⊙ . Unfortunately the spectrum of GRB 090423 does not show any detectable emission or absorption line due to the very small signal to noise ratio. To make any progress in this field we need to get to the target as soon as possible and obtain reasonably high S/N ratio spectroscopic observations. Finally we should clarify the morphology between short and long gamma ray bursts in connection with the physics of their formation and the identification of the progenitors. They both have similar characteristics on the decaying light curve, naturally referring to those cases in which the light curve of shorts has been observed and can clearly be distinguished as two well separated classes only in the Amati relation. Fast follow up will enable (1) collection of the first optical afterglow spectrum of a short burst, providing an in-situ probe of short burst environs and possible progenitor signatures; (2) searches for the predicted signature of the decay of radioactive sub-relativistic material at early times; and (3) collection of extended afterglow light curves to probe the beaming angle distribution of the short bursts via "jet break" analyses, a crucial input in estimating merger event rates. So far the spectra of these GRBs have been elusive due to their faintness and extremely rapid optical decay. In conclusion to make any further significant progress on the GRB physics and related modeling we need a medium size robotic telescope described above.
Figure 1 :
1Top: Focal plane instrumentation with all the instruments developed perpendicular to the optical axis of the telescope. Details on the single instruments and mounting of the optics are given in the exploded figures on the top right part of the figure.
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| []
|
[
"Discriminating and Constraining the Synchrotron and Inverse Compton Radiations from Primordial Black Hole and Dark Matter at the Galactic Centre Region",
"Discriminating and Constraining the Synchrotron and Inverse Compton Radiations from Primordial Black Hole and Dark Matter at the Galactic Centre Region"
]
| [
"Upala Mukhopadhyay [email protected] \nTheory Division\nSaha Institute of Nuclear Physics\n1/AF Bidhannagar700064KolkataIndia\n\nHomi Bhabha National Institute\nTraining school complex400094AnushaktinagarMumbaiIndia\n",
"Debasish Majumdar †e-mail:[email protected] \nTheory Division\nSaha Institute of Nuclear Physics\n1/AF Bidhannagar700064KolkataIndia\n\nHomi Bhabha National Institute\nTraining school complex400094AnushaktinagarMumbaiIndia\n",
"Avik Paul \nTheory Division\nSaha Institute of Nuclear Physics\n1/AF Bidhannagar700064KolkataIndia\n\nHomi Bhabha National Institute\nTraining school complex400094AnushaktinagarMumbaiIndia\n"
]
| [
"Theory Division\nSaha Institute of Nuclear Physics\n1/AF Bidhannagar700064KolkataIndia",
"Homi Bhabha National Institute\nTraining school complex400094AnushaktinagarMumbaiIndia",
"Theory Division\nSaha Institute of Nuclear Physics\n1/AF Bidhannagar700064KolkataIndia",
"Homi Bhabha National Institute\nTraining school complex400094AnushaktinagarMumbaiIndia",
"Theory Division\nSaha Institute of Nuclear Physics\n1/AF Bidhannagar700064KolkataIndia",
"Homi Bhabha National Institute\nTraining school complex400094AnushaktinagarMumbaiIndia"
]
| []
| The evaporations of Primordial Black Holes (PBH) (via Hawking radiation) can produce electrons/positrons (e − /e + ) in the Galactic Centre (GC) region which under the influence of the magnetic field of Centre region can emit synchrotron radiation. These e − /e + can also induce Inverse Compton radiation due to the scattering with ambient photons. In this work three different PBH mass distributions namely, monochromatic, power law and lognormal distributions are considered to calculate such radiation fluxes. On the other hand, annihilation or decay of dark matter in the Galactic Centre region can also yield e − /e + as the end product which again may emit synchrotron radiation in the Galactic magnetic field and also induce Inverse Compton scattering. In this work a comparative study is made for these radiation fluxes from both PBH evaporations and from dark matter origins and their detectabilities are addressed in various ongoing and other telescopes as well as in upcoming telescopes such as SKA. Moreover, constraints on the model parameters are obtained from these experimental predictions. The variations of these radiation fluxes with the distance from the Galactic Centre are also computed and it is found that such variations could be a useful probe to determine the mass of PBH or the mass of dark matter. * | null | [
"https://arxiv.org/pdf/2109.14955v3.pdf"
]
| 238,226,699 | 2109.14955 | c81124ed1524ed881666daed35cbec4deb4bf2f8 |
Discriminating and Constraining the Synchrotron and Inverse Compton Radiations from Primordial Black Hole and Dark Matter at the Galactic Centre Region
7 Apr 2022
Upala Mukhopadhyay [email protected]
Theory Division
Saha Institute of Nuclear Physics
1/AF Bidhannagar700064KolkataIndia
Homi Bhabha National Institute
Training school complex400094AnushaktinagarMumbaiIndia
Debasish Majumdar †e-mail:[email protected]
Theory Division
Saha Institute of Nuclear Physics
1/AF Bidhannagar700064KolkataIndia
Homi Bhabha National Institute
Training school complex400094AnushaktinagarMumbaiIndia
Avik Paul
Theory Division
Saha Institute of Nuclear Physics
1/AF Bidhannagar700064KolkataIndia
Homi Bhabha National Institute
Training school complex400094AnushaktinagarMumbaiIndia
Discriminating and Constraining the Synchrotron and Inverse Compton Radiations from Primordial Black Hole and Dark Matter at the Galactic Centre Region
7 Apr 20222
The evaporations of Primordial Black Holes (PBH) (via Hawking radiation) can produce electrons/positrons (e − /e + ) in the Galactic Centre (GC) region which under the influence of the magnetic field of Centre region can emit synchrotron radiation. These e − /e + can also induce Inverse Compton radiation due to the scattering with ambient photons. In this work three different PBH mass distributions namely, monochromatic, power law and lognormal distributions are considered to calculate such radiation fluxes. On the other hand, annihilation or decay of dark matter in the Galactic Centre region can also yield e − /e + as the end product which again may emit synchrotron radiation in the Galactic magnetic field and also induce Inverse Compton scattering. In this work a comparative study is made for these radiation fluxes from both PBH evaporations and from dark matter origins and their detectabilities are addressed in various ongoing and other telescopes as well as in upcoming telescopes such as SKA. Moreover, constraints on the model parameters are obtained from these experimental predictions. The variations of these radiation fluxes with the distance from the Galactic Centre are also computed and it is found that such variations could be a useful probe to determine the mass of PBH or the mass of dark matter. *
I. INTRODUCTION
Primordial Black Holes (PBHs) [1,2] are massive compact halo objects that could have been created shortly after Big Bang. Their masses depend on the time at which they are created and therefore they are likely to have a range of masses or a mass distribution instead of a single mass for all the PBHs. The PBHs being nearly collisionless and could be stable if sufficiently massive can well qualify to be candidates for dark matter [3][4][5][6]. In fact it is suggested from observational hints [7], found from the abundance studies of PBHs, that the PBHs may constitute the dark matter content or a component of the dark matter content. In fact several observational results such as Voyager 1 data [8,9], strong lensing [10,11] measurements, background gamma ray data [12,13] etc. indicate a fractional mass ratio f PBH of the total PBHs to the total dark matter [12][13][14][15][16]. This indicates that PBHs may constitute a fraction of total dark matter content and that may vary with the PBH masses. PBHs are believed to undergo evaporation via Hawking radiation. If PBH mass is sufficiently small ( 10 14 g), they would be withered away completely via Hawking radiation. The Standard Model particles such as electrons/positrons may be emitted via Hawking radiation and these may interact with surrounding medium. These electrons/positrons could produce synchrotron radiation by the influence of the magnetic field present in the medium.
These particles may also induce Inverse Compton scattering with the ambient photons and these IC scattered photons also carry information regarding the nature and evaporation of PBHs.
On the other hand the indirect detection of dark matter [17][18][19][20][21] is related to detecting the signal from the products of the dark matter annihilation or decay. The gravity of massive astrophysical objects may trap dark matter from escaping those objects and when they are trapped in substantial magnitude inside the core of the massive objects such as Galactic Centre, Solar core etc. they can undergo the process of self annihilation to produce standard model particles such as fermions, photons etc. as the end products. The excess signal of these fermions, photons etc. could be a probable signature of dark matter indirect detections. Possible decay of dark matter can also lead to fermions as one of the final products. If electrons/positrons are the products of such dark matter annihilations or dark matter decays then these charged particles under the influence of Galactic magnetic field can also emit synchrotron radiation which may be detected by terrestrial radio telescopes. On the other hand these electrons/positrons can also induce Inverse Compton (IC) scattering with ambient photons and the IC scattered photons may again be a useful probe for these phenomena.
Thus synchrotron radiations from distant cosmos could be an effective probe to understand unknown phenomena in the Universe. The present and future radio telescopes such as Square Kilometer Array (SKA) [22][23][24][25], GMRT [26], uGMRT [26], MeerKAT [27], LOFAR [28], Jodrell Bank [29], JVLA [30], ASKAP [22] etc. provide new possibilities of probing the evidence and physics of dark matter and PBHs among other cosmological phenomena.
In the present work we address these issues of emission of synchrotron and IC radiations from the products of PBH evaporation in our Galaxy as well as the synchrotron and IC emissions resulting from the end products of possible dark matter self annihilation or dark matter decay.
The synchrotron radiation fluxes are computed for PBH decays and for both dark matter cases (annihilation and evaporation as well as dark matter decay). The IC radiation flux for these cases are also calculated. The results are then compared. Various observational results of existing radio telescopes and the sensitivity estimates given by SKA for 100 hour and 1000 hour runs are then addressed to asses the detectabilities of the computed signals. The first category includes constant PBH masses (M PBH ) and the constant masses are chosen to be the two extremum masses, namely 10 14 g and 10 17 g, of the range of PBH masses considered in this work. For the other two categories, we consider mass distributions for PBHs. In the second category, a power law mass distribution [31,32] has been adopted while we analyse with a lognormal distribution of PBH masses [31,32] in the third category. In Ref. [32] the authors have addressed the synchrotron emissions originated from the PBH evaporation at the Galactic Centre and have constrained the PBH to dark matter ratio in case of three above mentioned PBH mass distributions using the radio observational data of the inner Galactic Centre. In this work, we compute not only the synchrotron emission but also the IC radiation from evaporation of PBHs (for the three possible PBH mass distributions mentioned above) at the Galactic Centre region. We also compute the flux densities of synchrotron and IC radiations originated from the decay/annihilation of dark matter. In addition, the parameters of PBH mass distributions as well as dark matter mass and decay/annihilation rates are constrained in this work from the observational data namely SKA, GMRT, uGMRT, Jodrell Bank, MeerKat, VLA, JVLA, LOFAR, ASAP. Moreover, for all these cases, the variations of synchrotron flux (and IC radiations) with the distance from the Galactic Centre are also estimated.
The paper is organised as follows. In Sect. II, we provide detailed theoretical framework for computations of synchrotron radiations and IC radiations for the cases of dark matter and PBHs as described above. Sect. II also furnishes the analytical expressions for the estimations magnetic field in and around the Galactic Centre. Sect. II therefore consists of various subsections. In Sect. III and in the subsections within, we furnish detailed calculational procedure and present our results. Finally in Sect. IV, we conclude with a summary.
II. CALCULATIONS FOR SYNCHROTRON RADIATION AND INVERSE COMPTON
RADIATION
In this section, we estimate synchrotron flux and Inverse Compton (IC) flux originated due to possible decay of primordial black holes (PBH) or the decay/ annihilation of dark matter (DM) at the Galactic Centre. The possible decay of PBH or the possible decay/annihilation of DM in the Galactic Centre may produce electrons (e − ) and positrons (e + ) as the final products. In the presence of large magnetic field in the Galactic Centre region these e − /e + produce synchrotron radiations while the scattering of e − /e + with surrounding photons gives rise to IC radiations.
A. Diffusion Equation
In order to obtain synchrotron flux and IC flux as mentioned above, we need to solve the diffusion equation with the form given by [33],
K(E)∇ 2 dn e (E, r) dE + ∂ ∂E b(E, r) dn e (E, r) dE + Q(E, r) = 0 .(1)
Here, dne(E,r) dE is the number density of electron (positron) per unit energy interval at position r, K(E) and b(E, r) represent diffusion coefficient and energy loss coefficient respectively while Q(E, r) denotes the electron (positron) source term. We assume the system is in equilibrium and a steady state solution of the above equation is being sought. To this end we consider dne(E,r) dE is independent of time.
The produced e − and e + can lose their energy via different processes such as synchrotron, IC, bremsstrahlung and Coulomb. Energy loss depends on the magnetic field strength (synchrotron loss), CMB photon spectrum (IC loss), the electron density n e and hydrogen density n H (bremsstrahlung and Coulomb losses) as well the energies of e − and e + . Total energy loss term b(E, r) can be expressed as [34],
b(E, r) = b synchrotron (E, r) + b IC (E, r) + b bremsstrahlung (E, r) + b coulomb (E, r) ,(2)
and the function f (x) is defined as [40],
f (x) = x ∞ x dx K 5 3 (x ) ,(11)
with K 5 3 (x ) is the modified Bessel's function of order 5 3 . The synchrotron flux density can be obtained by taking the line of sight (l.o.s) integral of the synchrotron power density j sync (ν, r) and then subsequently taking integral over the solid angle (δΩ) of the observed range. The average synchrotron flux density over the solid angle (δΩ) is written as [39,42]
F sync = 1 4π dΩ l.o.s dlj sync (ν, r) .(12)
The integral over the line of sight l and the solid angle δΩ can be calculated from the relations r = (r 2 + l 2 − 2r lcosθ) 1 2 and δΩ = 2π θmax θ min dθsinθ respectively. Here, r is the distance between the Sun and the Galactic Centre while r represents the distance of the site of decay/annihilation from the Galactic Centre and θ denotes the angle between the direction of the l.o.s and the line joining the Earth and the Galactic Centre.
C. Inverse Compton Radiation
The electrons and positrons originated from the decay of PBHs or the decay/annihilation of DM can produce radiations from the process of Inverse Compton scattering with the background photons, dominantly with Cosmic Microwave Background photons with temperature 2.73 K. The low energy photons gain energy through IC scattering from the kinetic energy of e − or e + .
The IC power can be written as [39],
P IC (E, E γ ) = cE γ d n( )σ(E, E γ , ) ,(13)
where σ(E, E γ , ) and n( ) respectively denote IC scattering cross section and number density of photon. In the above equation, , E and E γ represent energy of the target photons, energy of electrons or positrons and energy of the up-scattered photons respectively.
IC scattering cross section σ(E, E γ , ) is calculated from the Klein-Nishina formula [43] σ(E,
E γ , ) = 3m 2 e c 4 σ T 4 E 2 G(q, Γ ) ,(14)
where σ T is the Thompson scattering cross section and G(q, Γ ) is defined as [44],
G(q, Γ ) = 2qlnq + (1 + 2q)(1 − q) + (2q) 2 (1 − q) 2(1 + Γ q) ,(15)
with Γ = 4 E m 2 e c 4 and q = Eγ Γ (E−Eγ ) . Now, the IC power density per unit frequency can be calculated as [42],
j IC (E γ , r) = 2 dE dn e dE P IC (E, E γ ) ,(16)
and the average IC flux density over the solid angle is derived as [42], [45,46],
F IC = 1 4π dΩ l.o.s dlj IC (E γ , r) .(17)Q decay (E, r) = Γ DM m DM ρ(r) dN e dE ,(18)Q annihilation (E, r) = σv 2m 2 DM ρ 2 (r) dN e dE .(19)
Here, Γ DM is the decay rate of DM and σv is the annihilation rate of DM while DM mass is denoted by m DM . In the above equations dNe dE represents e − /e + energy distribution originated from the decay or annihilation of each DM.
For the decay of PBH the source term Q(E, r) is written as [47], [32], [48],
Q PBH (E, r) = f PBH M PBH ρ(r) dṄ e dE(20)
where dṄe dE , the e − /e + energy distribution produced per unit time from the decay of a single PBH is given as,
dṄ e dE = 1 2π Γ PBH exp(E/k B T PBH ) + 1 .(21)
Here, Γ PBH is the absorption coefficient of electron like particles (spin 1 2 particles) and approximately expressed as [49] Γ PBH
27G 2 M 2 PBH E 2 2 c 6 3.82 × 10 −2 M PBH 10 13 g 2 E GeV 2 .
Temperature of a PBH (T PBH ) is linked to its mass (M PBH ) through the relation [50] k B T PBH 1.06 10 13 g M PBH GeV. In Eq. 20, f PBH represents the fraction of the PBH density to the total DM density ρ(r) (i.e., ρ PBH (r) = f PBH ρ(r)).
In order to compute the Q PBH (E, r) of Eq. 20 it is assumed that all the PBHs have equal mass value. But recent studies suggest that realistic production mechanisms could be due to an extended mass distribution of the PBHs. In this scenario Q PBH (E, r) can be written as [16], [31],
Q PBH (E, r) = f PBH ρ(r) ρ ∀M PBH dM PBH g(M PBH ) M PBH dṄ e dE ,(22)
with g(M PBH ) denotes mass distribution of PBHs normalised to ρ . In this work, we have considered two types of extended mass distribution of PBHs namely, power law distribution and lognormal distribution.
The power law mass distribution of PBHs with power law index p (p = 0), maximum mass limit M max and minimum mass limit M min can be written as [31,32],
g(M PBH ) = pρ M p max − M p min M p−1 PBH ,(23)
while the lognormal distribution with mean µ and standard deviation σ is defined as [31,32],
g(M PBH ) = ρ √ 2πσM PBH exp − ln 2 (M PBH /µ) 2σ 2 .(24)
The lognormal distribution of PBH mass function is first mentioned in Ref. [51] to discuss a mechanism of PBH formation for a model of baryogenesis which gives rise to large fluctuations (of baryon number) on small scales and small fluctuations on large scales. This lognormal distribution is a good approximation when it is considered that PBHs originate from a smooth symmetric peak in the inflationary power spectrum. For example in Ref. [52] it is shown that for single field double inflation theory, the associated PBH mass function is approximately lognormal. Numerical demonstration of this type of mass functions of PBH are given in Ref. [53].
On the other hand, power law mass function of PBHs results from scale invariant density fluctuations or from the collapse of cosmic strings [31]. In both the cases p = − 2ω 1+ω where ω describes the equation of state at the time of PBH formation [54]. Since PBHs form after inflation due to the inflation -generated density fluctuations and for the non inflationary Universe ω ∈ {−1/3, 1}, the range of power index p would be p ∈ {−1, 1}. Therefore, in this work two values p, namely p = −0.5 and p = 0.5 are considered.
E. Magnetic Field and Dark Matter Halo Profile
In this case, the assumption has been made that the magnetic field near the Galactic Centre can be determined using the form [32,36],
B 0 r c r 5/4 for r r c B = (25) B 0 r c r 2 for r > r c .
Here, r c is the accretion radius of the supermassive black hole (SMBH) at the central region and r c = 0.04 pc. While the strength of the magnetic field B 0 is assumed to be equal to 7.2 mG.
For DM halo profile, we have primarily adopted the Navarro -Frenk -White or NFW profile of DM halo to compute the results. The NFW profile can be written as [55],
ρ(r) = ρ s r rs 1 + r rs 2 ,(26)
where r s and ρ s are scale radius and scale density respectively. We have taken r s = 20 kpc and have chosen ρ s in such a way that it can provide the local DM density ρ = 0.4 GeV cm −3 [56,57] at a distance r = 8.5 kpc from the Galactic Centre.
III. CALCULATIONS AND RESULTS
In this section, using the formalism discussed in Sect. II, we compute the synchrotron flux and IC flux for five different cases related to PBH decay and decay/annihilation of dark matter. These five cases are i) decay of PBH with monochromatic mass distribution, ii) decay of PBH with power law mass distribution, iii) decay of PBH with lognormal mass distribution iv) decay of DM and v) annihilation of DM. We then compare the computed results for all these five cases. Further, the detectability of such fluxes in the present and upcoming radio telescopes are also probed by comparing these with and bounds on the model parameters are obtained from the given SKA sensitivity [22] and Jodrell Bank data [29]. Variations of the fluxes with the distance from the Galactic Centre are also calculated. Each of the cases is addressed for two ranges of e − /e + energy;
high enrgy (order of GeV) e − /e + and low energy (order of MeV) e − /e + . In the present calculations flux density is always higher when decaying DM mass is 50 GeV than when the same is 100 GeV. This is expected since from Eq. 18 it can be understood that smaller values of DM mass will lead to higher values of flux densities. The flux density also depends on the decay rate (larger the decay rate, larger would be the flux density). To this end, later in this work (in section III A 2) we obtain bounds on the DM decay rate and DM mass in the present analysis using the experimental results.
Similar computations are made for the synchrotron and IC fluxes with the consideration that In this high energy (∼ 0.1 GeV) e − /e + case, we have computed that the synchrotron emissions induced by the decay products (e − /e + ) of PBH or the decay/annihilation products (e − /e + ) of DM produce radio waves. Therefore in Fig. 4, the synchrotron flux densities are compared with the present and future radio telescopes' data.
A versatile next generation radio telescope to observe a large area of the sky is Square Kilometre Array (SKA) [22][23][24][25]. It operates in the frequency range of 70 MHz to 10 GHz and predicts the flux density limit even in the order of µJy. The upgraded version of GMRT, the uGMRT [26] is expected to be sensitive in the frequency ranges 130 -260 MHz, 250 -500 MHz, 550 -900 MHz, 1000 -1450 MHz and it is more sensitive in the low frequency band ranges of 250 to 1000 MHz. In addition a new radio telescope MerrKAT [27], a precursor to SKA telescope, will be operated in the frequency bands 0.7 -2.5 GHz, 0.7-10 GHz. A low frequency radio interferometer is the LOw-Frequency ARray or LOFAR [28] is designed to cover the low-frequency range from 10 -240 MHz. The Jansky Very Large Array (JVLA) [30] is another new generation radio telescope that operates between 1 -50 GHz while the Very Large Array or the VLA operates around 1.4 GHz and 5 GHz [58]. Operational frequency range of another upcoming radio telescope ASKAP is 0.7 GHz -1.8 GHz [22]. The radio telescope Jodrell Bank [29] measures radio flux within 4 arcsec at the Galactic Centre and it measurement reveals that at 408 MHz the upper bound on the radio flux is 50 mJy.
In Fig. 4, we have compared all our results for the synchrotron flux densities with the data/sensitivity of the above mentioned radio telescopes. In the left panel of the case of DM decay/annihilation products. It can be noted from the figure that DM annihilation products yield a larger flux than DM decay. A point corresponding to "upper limit" shown in Fig. 4 indicates that upper limit of the radio flux densities at the Galactic Centre at frequency 408 MHz is 50 mJy. This provides tight constraints on the parameter f PBH of PBH distributions and DM annihilation cross sections or decay rates as well as on the masses of PBH and DM. The bounds on these quantities from experimental results will be discussed later in section III A 2.
Variation of the Flux with Distance from the Galactic Centre
This can be noted from the formalism in Sect. II that the synchrotron and IC flux densities would vary with the distance r from the Galactic Centre. The variations are principally caused due to the variations of magnetic field B(r) with r and due to the variations of DM densities with r.
In this section we compute synchrotron flux densities at different distances (r) from the Galactic Centre for the cases considered earlier i.e., PBH evaporation and DM annihilation/decay. In this work we have considered a region within 2 pc (4 arcsec) from the Galactic Centre. Moreover, it is also observed from Fig. 5 that larger the PBH masses (corresponds to smaller flux densities) nearer are the distances r from the Galactic Centre at which the synchrotron radiation peaks. Therefore, the different peak positions for different PBH masses indicate that for larger masses of PBH, the flux decreases with r much faster than the case for smaller PBH masses. While for the case of heavier PBHs, significant synchrotron flux may not be obtained at larger distances from the Galactic Centre but for smaller PBH masses the chances to obtain detectable PBH induced flux at extended region are larger. It indicates that more massive the PBH is more probable they are to be present closer to the Centre of the Galaxy and therefore, the synchrotron flux caused by the evaporation of PBH in and around the Galactic Centre region is likely to attempt its peak value at a distance closer to the Centre of the Galaxy. Moreover, the shape of the curves in Fig. 5 can be understood from the form of the magnetic field B(r) given in Eq. 25, the shape of DM halo profile ρ(r) (Eq. 26) and total energy loss term b(E, r) (Eq. 2). The r dependence of synchrotron flux density can be approximately of the form of ∼ 1 B(r) ρ(r) giving rise to such bell shaped curve on integration over energy.
In Fig. 5(b), similar plots as in Fig. 5(a) For p = −0.5, maximum flux density (∼ 0.18 Jy) is obtained at r ∼ 0.12 pc while for p = 0.5, the maximum value (∼ 19 × 10 −3 Jy) is obtained at r ∼ 0.1 pc. Here also it can be observed that for p = −0.5 (which corresponds to larger flux density), the peak position of the flux density is located at a distance r from the Galactic Centre which is further from the peak value of lower synchrotron flux when p = 0.5 is adopted for power law distribution of PBH masses. It is already discussed that for p = −0.5 case, there are more contributions from smaller masses of PBH and flux densities, originated due to the evaporation of PBH with monochromatic mass distribution, with the distance r from the Galactic Centre. This is shown in Fig. 8. From Fig. 8 it can be seen that IC flux density increases with r since the IC flux density approximately varies with distance r as ∼ 1 B(r) 2 ρ(r) (note that this is different from the dependence of synchrotron flux density on r (∼ 1 B(r) ρ(r))). Hence, the variations of synchrotron flux densities and IC flux densities with r show different behaviour. Variations of IC flux densities with r for the cases of power law or lognormal mass distributions of PBH or annihilation/decay of DM are also computed and similar behaviours of flux densities (as in Fig. 8) are obtained. These are however not shown here.
The computations for Figs. 5 -8 are performed by adopting NFW profile for DM halo density ρ(r) (Eq. 26). We now perform the same calculations but with a different DM density profile namely, EINASTO profile for ρ(r). In order to demonstrate the dependence of flux density on DM halo profile, we have plotted in Fig. 9, variations of flux density with r for the case of PBH evaporation with monochromatic mass distribution by considering EINASTO profile for DM halo density. The EINASTO profile is given by [59],
ρ(r) = ρ s exp 2 α r rs α − 1 .(27)
For this calculations α = 0.17 has been adopted while other symbols have similar meaning as in Eq. 26. The evaporation of PBH with monochromatic mass distribution with two PBH masses namely, M PBH = 4 × 10 14 g and 10 17 g are considered and the results are plotted in Fig. 9. By comparing Fig. 5 (a) and Fig. 9 it can be noted that the variations of the flux densities with r are different for the two DM halo profiles (NFW and EINASTO). Hence, the peak positions and peak values of the flux densities depend on the profile of the DM halo.
Constraints on the Model Parameters from Experimental Limits for High Energy Electrons and Positrons
From the above discussions it may be evident that certain demonstrative values have been adopted for the quantities f PBH , DM decay constant Γ, DM annihilation cross section σv in order to investigate the possible nature and variations of synchrotron radiations that could have been originated due to the influence of Galactic magnetic field on e + /e − produced from the evaporation of PBH or from the decay/annihilation of DM. In this section we make an attempt to constrain these quantities or parameters by the SKA sensitivities and from the Jodrell Bank telescope's result that radio flux within 4 arcsec at the Galactic Centre should not be greater than 50 mJy at 408
MHz. We furnish our results in Fig. 10 and Fig. 11.
In Fig. 10(a) the region above the orange line labeled as "greater than SKA limit" is for the values of f PBH and M PBH for which the intensity of the synchrotron radiation signal would be in the detectable range of SKA (i.e., the synchrotron flux density is above the order of µJy). It can be observed from Fig. 10(a) that for smaller values of M PBH , the allowed range for f PBH is smaller than that for higher values of M PBH . This is expected because a PBH with smaller mass (M PBH ) evaporates at a higher rate resulting in larger e − /e + fluxes and hence detectable radio signals through synchrotron emission even for smaller PBH densities (or f PBH ) may be detected.
From the figure it can be found that for M PBH = 10 17 g, the minimum value of f PBH is 10 −9 while for M PBH = 4 × 10 14 g, it can probe the f PBH down to 10 −14 . Moreover, in Fig. 10(a) we have also
shown the upper limits of f PBH for different M PBH that satisfy the highest limit of the flux (50 mJy) at 408 MHz. This is shown in Fig. 10(a) by the dark green line in the plot labeled as "less than 50 mJy". Hence the region below the dark green line is allowed by Jodrell Bank's prediction.
In Fig. 10(b), we constrain the parameter f PBH , using SKA sensitivity, for the case of power law mass distribution of the PBH. To this end, f PBH − M max parameter space (M max is the maximum mass in the PBH distribution) is constrained for two values of power law index p namely, p = 0.5 and p = −0.5. The radio flux upper limit at 408 MHz for the power law mass distribution of PBH is also used. In Fig. 10(b), the M min (minimum value of PBH maa) is fixed at 4 × 10 14 g. It can be noted from Fig. 10(b) that even for p = 0.5 and M max = 10 17 g, f PBH does not go beyond ∼ 10 −8 in order to satisfy the experimental constraints but it can be as low as ∼ 10 −14 to produce detectable radio signals in SKA. We then constrain f PBH if the PBH masses follow a lognormal distribution.
Here the parameter space is f PBH − µ is constrained for two values of σ -the variance of the mass distribution. The results are plotted in Fig. 10(c). In Fig. 10(c) the region above the orange line labeled as "greater than SKA limit" describes the values of f PBH to obtain detectable flux density in SKA and the dark green line labeled as "less than 50 mJy" indicates the upper limit of f PBH to satisfy the highest limit of the flux (50 mJy) at 408 MHz for lognormal mass distribution of PBH.
The bounds on f PBH and the mean values of the mass distribution µ are plotted for σ = 0.1, 1.
In this case f PBH can be ∼ 10 −5 for µ = 10 17 g, σ = 0.1 and f PBH ∼ 10 −6 for µ = 10 17 g, σ = 1.
Moreover, for µ = 10 15 g, SKA can detect the radio fluxes even for f PBH as small as ∼ 10 −14 (σ = 0.1) and ∼ 10 −15 (σ = 1).
In this context it may be mentioned that the constraints on f PBH as a function of mass of the PBH for monochromatic mass distribution or for the extended mass functions of the PBH are discussed in literature [31]. Also, different other authors have used different cosmic-ray, gamma- ray or X-ray data [16] to provide the upper limit of the f PBH values. Moreover, in Ref. [14] the constraints are given for PBH mass range of 10 13 − 10 17 g with CMB anisotropy damping limit and the Galactic positron limit. In this work we have provided another tight constraints on PBH mass function parameters with the help of SKA sensitivity and Jodrell Bank's data. We now constrain the parameters, DM decay width Γ and DM annihilation cross section σv , using the SKA limits. These are shown in Fig. 11(a) and 11(b). In Fig. 11(a) the constrained parameter space Γ − m DM (m DM being the DM mass) is shown while in Fig. 11(b) the DM annihilation cross section σv is constrained for different values of m DM . In both Fig. 11(a) and Fig. 11(b) the regions above the orange line labeled as "greater than SKA limit" describe the values of the decay rate Γ and annihilation cross section σv respectively for which the flux of the synchrotron radiation signal would be in the detectable range of SKA (flux density is above the order of µJy).
It can be seen from Fig. 11(a) GeV). Similarly in Fig. 11(b), σv is smaller for smaller m DM in order to satisfy the SKA limit with the peak frequency at ∼ 10 −1 MHz while the IC radiations produce radio signals with the peak frequency at ∼ 10 5 MHz. Therefore, for low energy e − /e + the frequencies of the synchrotron signals are very low and not in the detectable range of present or upcoming radio telescopes but IC radiation appears to produce detectable radio signals. Hence, in this section, for the case of e − /e + with energy in MeV range we have compared the flux densities from IC process only with different In the left panel of Fig. 12 variations of the flux for synchrotron radiation (the purple line in the plot) and IC radiation (the green line in the plot) with frequency ν, generated due to the evaporation of PBH with mass M PBH =10 17 g are shown while in the right panel of Fig. 12 variations of the flux for synchrotron radiation and IC radiation with frequency ν, originated due to the evaporation of PBH with masses M PBH =10 17 g and 4× 10 14 g are compared. Sensitivities of SKA for 100 hrs and 1000 hrs are also indicated in the figure for comparison. As in Fig. 1, Fig. 12 also shows smaller flux densities for larger PBH mass 10 17 g than that for PBH mass of 4×10 14 g as expected. But comparing Figs. 1 and 12 it can be noted that the peak frequencies for both the synchrotron flux density and IC flux density are different for the two cases (as already discussed in the previous paragraph). It can also be noted that flux densities for Fig. 12 are somewhat smaller than the flux densities in Fig. 1.
The flux calculations (synchrotron and IC) related to the PBH evaporation for power law and lognormal distributions of PBH are shown in Fig. 13. In the left panel of Fig. 13 variations of the flux for synchrotron radiation and IC radiation with frequency ν, produced from the evaporation shown by the black and red curves respectively. One observes from this figure that larger flux densities are obtained when σ = 1 (as in the case of Fig. 2 (right panel)). Comparing Fig. 2 and Also by comparing Fig. 4 and Fig. 15 it can be stated that the flux densities for the low energy e − /e + case are smaller than the flux densities for the high energy e + /e − and thus larger number of plots lie below the upper limit provided by Jodrell Bank data (the "upper limit" point in Fig. 15).
Constraints on the Parameters from Experimental Limits for Low Energy Electrons and Positrons
In this section, we have constrained the parameters f PBH , M PBH , Γ, σv and m DM from the IC scattering of low energy e − /e + emitted from the decay of PBH or decay/annihilation of DM.
These constraints are obtained using the sensitivity of SKA and applying the condition that the experimental prediction for radio flux with 4 arcsec of the Galactic Centre can not be greater than for these PBH masses. In Fig. 16(b), we constrain the parameter space f PBH with M max with the condition described earlier, for two power law index values namely p = −0.5 and p = 0.5 while M min is kept fixed at M min = 4 × 10 14 g. In Fig. 16(c) the constraints on f PBH are plotted for lognormal mass distribution of PBH with mean value of the mass µ for σ = 0.1, 1 are shown. By comparing Fig. 10 with Fig. 16 it can be noted that the allowed range of f PBH would be modified when the emission of low energy e + , e − with energy of the order of MeV are considered (low energy case) than the case of e + , e − with high energy (order of GeV). Thus, the allowed range of PBH densities depend on the energy of produced e + , e − .
In Fig. 17(a) the constraints are shown for DM decay rate Γ and DM mass m DM for loe energy e − /e + case. It is such that SKA would be able to detect the emitted radio flux for DM mass 1
GeV and 100 GeV even when decay rate is very small say 10 −32 s −1 or 10 −30 s −1 respectively. But
IV. SUMMARY AND DISCUSSIONS
The primordial black hole or PBH that are believed to have been produced due to perturbation in very early Universe may evaporate via Hawking radiation and leave fermions, photons etc.
The charged e − /e + thus produced may emit synchrotron radiation while propagating through the Galactic magnetic field. These charge particles on the other hand may induce Inverse Compton (IC) scattering and emit radio signal. Similar situation may arise for the decay or self annihilation of DM producing e − /e + as the end products. The synchrotron radiation or IC radiation can also be induced in such scenario. In this work these emissions are addressed and their detectabilities are explored. Variations of such signals with the distance from the Galactic Centre are also observed. Additionally, attempts have been made to constrain the parameters associated with the above processes using experimental limits or available data. In this work we have considered five cases to compute such synchrotron radiations and IC radiations. These are i) decay of PBH with monochromatic mass distribution, ii) decay of PBH with power law mass distribution, iii) decay of PBH with lognormal mass distribution iv) decay of DM and v) annihilation of DM. A comparison has also been made for these five cases. Further, the detectability of such fluxes in the present and upcoming radio wave experiments are discussed and bounds on the model parameters are obtained.
Constraints on the model parameters are calculated for two energy ranges of electrons, high energy (∼ 0.1 GeV) e − /e + and low energy (∼ 2 MeV) e − /e + . Variations of the fluxes with the distance from the Galactic Centre are also produced.
It is found that larger flux densities are obtained when contributions from smaller mass PBHs become more significant. This is also the case when DM with smaller masses decay or annihilate to produce such flux densities. It is noted that for high energy e − /e + emission, synchrotron radiations produce radio signals with the peak frequency at ∼ 338 MHz while the IC radiations give rise to ultraviolet to X-ray signals. Therefore, the calculated synchrotron flux densities for different masses of PBH, mass functions of PBH and also for different masses of DM are compared with data from upcoming and present radio telescopes like GMRT, uGMRT, MeerKat, VLA, JVLA, LOFAR, ASKAP, SKA (100 hrs), SKA (1000 hrs). It can also be predicted from the comparison that parameters used to compute the synchrotron radiations from the decay of PBH and decay/annihilation of DM would be tightly constrained as the upper limit of the radio flux densities at the Galactic Centre at frequency 408 MHz is 50 mJy.
Variations of the synchrotron flux with the distance from the Galactic Centre (r) are also calculated for different PBH masses, different PBH mass distributions and different DM masses.
It is found that the synchrotron emission is highest at a particular distance and that peak distance The synchrotron radiation and IC radiation arises due to the low energy e − , e + originated from the above mentioned five scenarios are also computed in this work. In this case synchrotron emissions produce signals with the peak frequency at ∼ 10 −1 MHz while the IC radiations produce radio signals with the peak frequency at ∼ 10 5 MHz. Therefore, for low energy e − , e + , we have compared the IC radiation flux densities with radio telescopes data. It is found that for e − , e + with low energy, the flux densities are smaller than the flux densities originated from high energy e − , e + .
Therefore, the constraints on the upper limit of f PBH , Γ or σv are comparatively less severe.
For both the cases (synchrotron and IC), two possibilities are considered for electrons/positrons -high energy electrons/positrons (of the order of GeV) and low energy electrons/positrons (of the order of MeV). For the case of dark matter, two dark matter masses namely 100 GeV and 50 GeV are considered for demonstrative purpose and the dark matter here is considered to WIMPs or Weakly Interacting Massive Particles. For the PBH case, three categories of PBHs are considered.
Figure 1 .
1Left Panel: Variation of the flux with frequency ν for synchrotron radiation (the purple line in the plot) and for IC radiation (the green line in the plot), originated due to the e − /e + evaporation of PBH with mass M PBH =10 17 g. Right Panel: Variation of the flux for synchrotron radiation and for IC radiation with frequency ν, generated due to the e − /e + evaporation of PBH with masses M PBH =10 17 g and 4× 10 14 g.Sensitivity of SKA for 100 hrs and 1000 hrs are shown by the black and red curves respectively. Sensitivity of experiments in higher frequencies like Spitzer, JWST, EELT, Hubble, Chandra are also shown.A. Synchrotron and IC Radiation From the Produced High Energy Electrons/PositronsIn this scenario of high energy e − /e + , the synchrotron peak frequency and IC peak frequencyare computed for electrons/positrons with energy E ∼ 0.1 GeV. For these calculations, unless mentioned otherwise, f PBH = 10 −7 and NFW profile of DM Halo have been adopted. The results are plotted in Figs. (1 -4). It can be seen from Figs. 1-4 that in this case synchrotron emissions give rise to radio signals with the peak frequency at ∼ 338 MHz while for the IC radiations ultraviolet to X-ray signals are obtained with the peak frequency at ∼ 1.2 × 10 9 MHz. The values of the synchrotron peak frequency depend on the energy of the electrons (positrons) and on the strength of the magnetic field while the IC peak frequencies depend on the energy of electrons (positrons) and photons. The sensitivities of SKA, Spitzer, JWST, EELT, Hubble and Chandra telescopes are also plotted for comparison. From Fig. 1 one notes that the intensities of the signals (for both the cases, i.e., synchrotron and IC) depend on the mass of the PBHs and the signal strengths are higher for M PBH = 4 × 10 14 g than for M PBH = 10 17 g. This indicates that PBHs with smaller mass (or with higher temperature) evaporate in a higher rate than a PBH with larger mass. Moreover, it can be observed from Fig. 1 that for both the values of PBH mass (4 × 10 14 g and 10 17 g) the synchrotron radiations are in the detectable range of SKA data but the detectability of the IC fluxes are observed on the basis of the experiments in higher frequency like Spitzer, JWST, EELT, Hubble and Chandra. In Fig. 2, variations of the flux for synchrotron radiations and IC radiations with frequency ν, originated due to the evaporation of PBH with power law mass distribution (left panel) and lognormal mass distribution (right panel) are shown. For the power law distribution two values of p, namely p = −0.5, 0.5, are considered with PBH masses within the range M min =4×10 14 g to M max =10 17 g. From Fig. 2, it is observed that both the synchrotron and IC radiation fluxes are higher when the power index p of the PBH power law mass distribution is taken to be p = −0.5 than when p = 0.5 is considered. It can be noted from Eq. 23 that for p = −0.5 the g(M PBH ) respectively. Hence, for the case of p = −0.5 the weight factor of PBH mass function g(M PBH ) increases more rapidly with smaller values of M PBH than the case when p = 0.5 and as PBHs with smaller mass values provide larger e − /e + fluxes the flux densities with p = −0.5 are greater than the same with p = 0.5. The right panel of Fig. 2 shows similar results but for lognormal distribution of PBH masses with µ = 5 × 10 15 g and for two values of the variance, namely σ = 0.1 and 1. It can be noted from Fig. 2 that a larger variance (σ = 1) yields larger fluxes for synchrotron radiations and IC radiations. This is however expected since a larger value of variance σ includes the contribution of larger range of PBH masses to the flux densities. This may be understood that larger the mass ranges of PBHs smaller are the range of values of M PBH (with larger evaporation rate) that can contribute to the fluxes. Hence the flux densities for σ = 1 are larger than the flux densities when σ = 0.1 is adopted. Similar comments as in Fig. 1 in relation to the peak frequencies and detectability of the signals for SKA data and Spitzer, JWST, EELT, Hubble, Chandra data can also be made from Fig: 2. We now calculate the fluxes for synchrotron and IC radiations that could be originated from the possible decay products (e + , e − ) of DM. For demonstrative purpose we consider two values of masses of the decaying DM. The results are plotted in left panel of Fig. 3. In Fig. 3 (left panel) variations of the flux with frequency ν are plotted for synchrotron radiations and IC radiations originated due to the decay of DM with mass m DM = 50 GeV and 100 GeV and decay rate Γ = 10 −25 s −1 .
Figure 2 .
2Left Panel: Variation of the flux for synchrotron radiation and for IC radiation with frequency ν, originated due to the evaporation of PBH with power law distribution of PBH mass where M min =4×10 14 g, M max =10 17 g and p = −0.5, 0.5. Right Panel: Variation of the flux for synchrotron radiation and for IC radiation with frequency ν, originated due to the evaporation of PBH with lognormal distribution of PBH mass where average of the mass distribution µ = 5 × 10 15 g and variance σ = 0.1 and σ = 1. Sensitivity of SKA for 100 hrs and 1000 hrs are shown by the black and red curves respectively. Sensitivity of experiments in higher frequencies such as Spitzer, JWST, EELT, Hubble, Chandra are also shown. these fluxes are originated due to the motion of the products of the DM annihilation in Galactic magnetic field and the scattering of these products. The fluxes are obtained for two annihilating DM masses, namely 50 GeV and 100 GeV with annihilation cross section σv = 10 −26 cm 3 s −1 . The results are plotted in the right panel of Fig. 3. It can again be noted from the figure that values of flux density depend on DM mass and it is higher for smaller value of DM mass (i.e., 50 GeV). This can be explained from Eq. 18 that smaller values of DM mass would lead to higher values of the flux density. Moreover, the values of the flux density also depend on the annihilation rate of DM (larger annihilation rates provide larger flux densities). In section III A 2 we discuss the bounds on the annihilation rates and DM mass on the basis of the experimental results. For both the plots (left panel and right panel) of Fig. 3 the flux densities are compared with experimental results of SKA and Spitzer, JWST, EELT, Hubble, Chandra.
Figure 3 .
3Left Panel: Variation of the flux for synchrotron radiation and for IC radiation with frequency ν, originated due to the decay of DM with mass m DM =50 GeV, 100 GeV and decay rate Γ = 10 −25 s −1 . Right Panel: Variation of the flux for synchrotron radiation and for IC radiation with frequency ν, originated due to the annihilation of DM with mass m DM =50 GeV and 100 GeV and annihilation rate σv = 10 −26 cm 3 s −1 . Sensitivity of SKA for 100 hrs and 1000 hrs are shown by the black and red curves respectively. Sensitivity of experiments in higher frequency domains such as Spitzer, JWST, EELT, Hubble, Chandra are also shown. Another low frequency radio telescope is Giant Metrewave Radio Telescope (GMRT) [26] located in India. It operates in the frequency range of 150 -1500 MHz with five discrete bands namely 130 -170 MHz, 225 -245 MHz, 300 -360 MHz, 580 -600 MHz, 1000 -1450 MHz and the r.m.s sensitivities of these five discrete bands are 0.7, 0.25, 0.04, 0.02, 0.03 respectively in the mJy unit.
Fig. 4 flux densities due to the decay of PBHs with different masses and mass distributions are compared with the experimental data while in the right panel of Fig. 4 similar comparisons have been made for
Figure 4 .
4Left panel: Variation of synchrotron flux with frequency generated from the evaporation of PBH with different mass values and different mass distributions.Right panel: Variation of synchrotron flux with frequency generated from the decay of DM and annihilation of DM with different masses. The fluxes are compared with the sensitivities of different radio frequency experiments i.e., GMRT, uGMRT, MeerKat, VLA, JVLA, LOFAR, ASKAP, SKA (100 hrs), SKA (1000 hrs). These data are also shown in both the left and right panels.
Fig. 5
5(a) shows variations of the synchrotron flux densities with the distance from the Galactic Centre when the evaporation of PBHs with monochromatic mass distribution is considered. Variations are plotted for three PBH masses namely, 4 × 10 14 g, 10 16 g and 10 17 g. It can be noted that for each of these cases the flux density shows a bell shaped curve i.e., it increases with the distance from the Galactic Centre until it achieves the largest flux density at a fixed distance and then it starts to decrease. But the position of a peak depends on the PBH mass values. For larger values of M PBH the flux densities attain their peak value at smaller distances r. It can be seen from Fig. 5(a) that the positions of the highest flux densities for M PBH = 4 × 10 14 g, 10 16 g and 10 17 g are at ∼ 0.11, ∼ 0.09 and ∼ 0.08 respectively. Therefore, the positions of such peaks could be a possible probe to determine the value of the PBH mass. In order to compare the peak positions of the flux densities for three different values of M PBH , we have plotted the fluxes for M PBH = 4 × 10 14 g, 10 16 g and 10 17 g in the units of 10 −1 , 10 −4 and 10 −6 Jy respectively. From Fig. 5(a) one finds that the peak values of the flux density for these three cases are ∼ 9 × 10 −1 Jy, 7 × 10 −4 Jy and 4 × 10 −6Jy respectively.
are shown but for the power law mass distribution of the PBHs. Similar bell shaped curves are obtained with different peak positions depending upon the values of the power index p (here we have considered M min = 4 × 10 14 g and M max = 10 17 g).
Figure 5 .
5(a) Variations of the synchrotron flux densities with the distance from the Galactic Centre are plotted due to the evaporation of PBHs with monochromatic mass distributions. Variations are plotted for three PBH masses 4 × 10 14 g, 10 16 g and 10 17 g. (b) Variations of the synchrotron flux densities with the distance from the Galactic Centre are plotted due to the evaporation of PBH with power law mass distributions for two values of p, namely -0.5 and 0.5. (c) Variations of the synchrotron flux densities with the distance from the Galactic Centre are plotted due to the evaporation of PBH with lognormal mass distributions for two values of σ, namely 0.1 and 1. hence significant flux densities at larger distances from the Galactic Centre can be obtained. In Fig. 5(c) variations of synchrotron flux densities at different positions from the Galactic Centre for the case of lognormal mass distribution of PBHs are computed and plotted. The lognormal distribution of PBHs is considered for this computation for mean µ = 5 × 10 15 g and two values of variance σ, namely σ = 0.1 and σ = 1. It can be seen from the Fig. 2 that σ = 1 yields larger flux densities. It is noted from Fig. 5(c) that for σ = 1 the peak position of the flux density is obtained at a larger distance r than the flux density calculated with σ = 0.1. In Fig. 6 the variations of the synchrotron flux densities with distance from the Galactic Centre are plotted for the decay/annihilation of DM. The results for the decay of DM and the annihilation of DM are plotted in Fig. 6(a) and Fig. 6(b) respectively. For both the cases, flux densities are computed for two mass values (m DM ) of DM namely, 50 GeV and 100 GeV. It can be noted from
Figure 6 .
6(a) The variations of the synchrotron flux densities from the Galactic Centre are plotted for the decay of DM. (b) The variations of the flux densities from the Galactic Centre are plotted for the annihilation of DM. For both the cases flux densities are compared for two mass values of DM, namely m DM = 50 GeV, 100 GeV. the figure that the peak positions of the flux densities depend on the DM mass. The position of the peak flux densities from the Galactic Centre could be a useful probe to determine the mass of the DM at the Galactic Centre. The plots in Fig. 6 are obtained by choosing fixed values of decay rate Γ = 10 −25 s −1 and annihilation cross section σv = 10 −26 cm 3 s −1 . It can also be observed from the figure that as 100 GeV DM produces lower flux densities than 50 GeV DM (both for the decay or the annihilation), the peaks of flux densities when decay or annihilation of 100 GeV DM is considered, occur nearer to the Galactic Centre i.e., at a smaller distance r than the case when DM mass is considered to be 50 GeV. In Fig. 7 variations of the synchrotron flux with r are compared for five cases considered so far namely, decay of PBH with M PBH = 10 17 g (the dark green line in the figure), decay of PBH with power law distribution for p = −0.5 (the dark blue line in the figure), decay of PBH with lognormal mass distribution for the case σ = 0.1 (the magenta line in the figure), decay of 50 GeV DM (the brown line in the figure) and annihilation of DM (the yellow line in Fig. 7). It is to be understood that all the plots in Fig. 7 have already been showm in Fig. 5 -6 but here they are plotted together for a direct comparison. In order to compare these five cases, the flux densities are plotted in the units of 10 −6 , 10 −2 , 10 −2 , 10 −3 and 10 −2 Jy respectively. It can again be observed (from Fig. 7) that different cases show different peak values and peak positions of synchrotron flux densities. Thus such peak values and peak positions could be a possible probe to determine the mass of PBH or the index parameters of the PBH mass distributions and mass of DM as well. Now, we focus on IC flux densities for the case of PBH evaporations and decay and annihilations of DM similar to what considered for the synchrotron case. First we consider the variations of IC
Figure 7 .
7Variations of the synchrotron flux with r are compared for four cases, decay of PBH with M PBH = 10 17 g, decay of PBH with power law distribution with p = −0.5, decay of PBH with lognormal mass distribution with σ = 0.1, decay of 50 GeV DM, annihilation of 50 GeV DM.
Figure 8 .
8Variations of the IC flux densities, originated due to the evaporation of PBH with monochromatic mass distribution, with the distance from the Galactic Centre.
Figure 9 .
9Variations of the synchrotron flux densities, originated due to the evaporation of PBH with monochromatic mass distribution (M PBH = 4 × 10 14 g, 10 17 g), with the distance from the Galactic Centre with EINASTO DM halo profile.
Figure 10 .
10(a) The upper limits and lower limit of f PBH as a function of the PBH monochromatic mass. (b) The upper limits and lower limit of f PBH as a function of the maximum mass in power law mass distribution of PBH. (c) The upper limits and lower limit of f PBH as a function of the mean value of lognormal mass distribution of PBH.
that the allowed values of Γ are smaller for smaller m DM (minimum value of Γ is 10 −32 s −1 for m DM = 1 GeV while minimum value of Γ is 10 −30 s −1 for m DM = 100
Figure 11 .
11(a) The upper limit and lower limit of Γ as a function of the DM mass m DM for the decay of DM. (b) The upper limit and lower limit of σv as a function of the DM mass m DM for the annihilation of DM.(minimum value of σv is 10 −36 cm 3 s −1 for m DM = 1 GeV while minimum value of σv is 10 −32 cm 3 s −1 for m DM = 100 GeV). This is also expected fromFig. 4. As larger is the DM mass smaller is the flux densities, larger decay rates or larger annihilation cross sections are required to satisfy the detectable criteria of SKA in this situations. Tight constraints on the upper limits of Γ and σv arise from the fact that radio flux densities can not be grater than 50 mJy at 408 MHz within 2 pc of the Galactic Centre. Thus the dark green lines inFig. 11labeled "less than 50 mJy" represent the upper limits.B. Synchrotron Radiation and IC Radiation from the Produced Low EnergyElectrons/PositronsSo far, we are assumed the e − /e + produced as a result of PBH evaporation and DM decay or annihilation to be particles with energy in GeV order (high enrgy e − /e + case). In this section, we focus on the synchrotron or IC radiation from such e − /e + if their energies are in the order of MeV. To this end low energy electrons/positrons with energy E ∼ 2 MeV are considered to compute as before, the synchrotron peak frequency and IC peak frequency. For all our calculations in this section, we have considered f PBH = 10 −7 and NFW profile of DM Halo unless otherwise mentioned. In the present calculations, synchrotron emissions are found to yield possible signals
Figure 12 .
12Left Panel: Variation of the flux of synchrotron radiation (the purple line in the plot) and of IC radiation (the green line in the plot) with frequency ν, generated due to the evaporation of PBH with mass M PBH =10 17 g. Right Panel: Variation of the flux for synchrotron radiation and Inverse Compton with frequency ν, originated due to the evaporation of PBH with masses M PBH =10 17 g and 4× 10 14 g. Sensitivity of SKA for 100 hrs and 1000 hrs are shown by the black and red curves respectively. experimental data. Likewise the previous sections, here too five processes are considered, the end product e − /e + of which may induce radio flux densities after being undergone IC scattering. These five processes are i) decay of PBH with monochromatic mass distribution, ii) decay of PBH with power law distribution of mass, iii) decay of PBH with lognormal mass distribution iv) decay of DM, v) annihilation of DM. The flux are then computed for each of the five cases and the results are plotted in Figs. 12 -15.
Figure 13 .
13Left Panel: Variation of the flux for synchrotron radiation and for IC radiation with frequency ν, originated due to the evaporation of PBH with power law distribution of PBH mass where M min =4×10 14 g, M max =10 17 g and p = −0.5, 0.5. Sensitivity of SKA for 100 hrs and 1000 hrs are shown by the black and red curves respectively. Right Panel: Variation of the flux for synchrotron radiation and for IC radiation with frequency ν, originated due to the evaporation of PBH with lognormal distribution of PBH mass where average of the mass distribution µ = 5 × 10 15 gm and variance σ = 0.1 and 1. Sensitivity of SKA for 100 hrs and 1000 hrs are shown by the black and red curves respectively. of PBH with power law distribution of PBH mass where M min =4×10 14 g, M max =10 17 g and p = −0.5, 0.5 are plotted and sensitivity of SKA for 100 hrs and 1000 hrs are also shown by the black and red curves respectively. It can be compared withFig. 2 (left panel). Note that in this case (Fig. 13 (left panel)) also larger flux densities arise for the case p = −0.5 as expected. The right panel ofFig. 13shows variation of the flux for synchrotron radiation and IC radiation with frequency ν, from the evaporation of PBH with lognormal distribution of PBH mass. The average of the mass distribution µ is taken to be µ = 5 × 10 15 g and the computations are made for two values of variances namely, σ = 0.1 and 1. Sensitivity of SKA for 100 hrs and 1000 hrs are also
Fig. 12 Figure 14 .
1214it can again be noted that the peak frequencies for synchrotron and IC radiation fluxes are different for the two figures and for low energy e − /e + scenario flux densities have smaller values.The low energy e − /e + that may produce due to decay or annihilation of DM will now beconsidered. The synchrotron and IC fluxes from such e − /e + are computed and the results are plotted in Fig. 14. In the left panel of Fig. 14, variations of the flux for synchrotron radiation and IC with frequency ν, originated due to the decay of DM with mass m DM = 50 GeV, 100 GeV and decay rate Γ = 10 −25 s −1 are shown while in the right panel of Fig. 14 variations of the same with frequency ν, originated due to the annihilation of DM with mass m DM = 50 GeV and 100 GeV Left Panel: Variation of the flux for synchrotron radiation and IC radiation with frequency ν, originated due to the decay of DM with mass m DM = 50 GeV, 100 GeV and decay rate Γ = 10 −25 s −1 . Sensitivity of SKA for 100 hrs and 1000 hrs are shown by the black and red curves respectively. Right Panel: Variation of the flux for synchrotron radiation and IC scattering with frequency ν, originated due to the annihilation of DM with mass m DM = 50 GeV and 100 GeV and annihilation rate σv = 10 −26 cm 3 s −1 .Sensitivity of SKA for 100hrs and 1000hrs are shown by the black and red curves respectively.
Figure 15 .
15Left panel: Variation of IC flux with frequency generated from the evaporation of PBH with different mass values and different mass distributions. Right panel: Variation of IC flux with frequency generated from the decay of DM and annihilation of DM with mass 50 GeV and 100 GeV. The fluxes are compared with the sensitivity of different radio frequency experiments i.e., GMRT, uGMRT, MeerKat, VLA, JVLA, LOFAR, ASAP, SKA (100 hrs), SKA (1000 hrs). and annihilation rate σv = 10 −26 cm 3 s −1 are plotted. Sensitivity of SKA for 100 hrs and 1000 hrs are shown by the black and red curves respectively. For both DM decay and DM annihilation cases flux densities for m DM = 50 GeV are larger than the case when DM mass m DM = 100 GeV is considered. Here too, it can be stated by comparing Fig. 3 and Fig. 14 that the peak frequencies are different for both the sets and flux densities due to e − /e + with energy in the order of MeV are smaller than that for high energy e − /e + case (∼ GeV).As demonstrated inFig. 4(comparison of synchrotron flux densities with experimental data) in Sect. III A, inFig. 15, we have compared the IC flux densities with the results (expected results) of present and upcoming radio telescopes. It is to be mentioned again that this calculations are for low energy e + /e − which we consider to be decay/annihilation product of DM. In the left panel ofFig. 15, flux densities due to the decay of PBHs with different masses and mass distributions are compared and in the right panel of Fig. 15 flux densities produced from DM decay/annihilation for m DM = 50 GeV, 100 GeV are compared. Similar to what have been shown inFig. 4, a point labeled "upper limit" has also been shown inFig. 15to indicate the upper limit of the radio flux densities at the Galactic Centre at frequency 408 MHz(50 mJy). This can also be noted fromFig. 15that all the computed flux for DM decay/annihilation case lie below the "upper limit" (right panel ofFig. 15) while some of the computed fluxes related to PBH have not crossed the "upper limit". In contrast, in similar plots of synchrotron flux densities with high energy e + /e − inFig. 4(left and right panels), all the calculated fluxes lie above the "upper limit". This shows that the IC induced fluxes inspired by PBH evaporation and DM decay/annihilation are in general lower for low energy e + /e − .
50 mJy at 408 MHz. The allowed limits of the parameters obtained using SKA sensitivity has been shown in Figs. 16 and 17 (the region above the orange lines) whereas the later condition is shown by the dark green lines in both Figs. 16 and 17. In Fig. 16(a), (b) and (c) the constraints are shown for the case of monochromatic mass distribution of PBH, power law mass function of PBH and lognormal mass function of PBH respectively. FromFig. 16(a)it can be observed that to produce detectable flux density in SKA, f PBH can be as small as 10 −14 and 10 −9 respectively for PBH mass 4 × 10 14 g and 10 17 g but f PBH should not be greater than 10 −8 and 10 −2 respectively
Figure 16 .
16(a) The upper limit and lower limit of f PBH as a function of the PBH monochromatic mass. (b) The upper limit and lower limit of f PBH as a function of the maximum mass for power law mass distribution of PBH. (c) The upper limit and lower limit of f PBH as a function of the mean value of lognormal mass distribution of PBH.
Figure 17 .
17(a) The upper limit and lower limit of Γ as a function of the DM mass m DM for the case of DM decay. (b) The upper limit and lower limit of σv as a function of the DM mass m DM for the case of DM annihilation. the upper limits of Γ are 10 −26 s −1 and 10 −24 s −1 for mass values 1 GeV and 100 GeV. In Fig. 17(b) the constraints are shown for the parameter space of DM annihilation rate σv and DM mass m DM . It may be seen that SKA would be able to detect the emitted radio flux for DM mass 1 GeV and 100 GeV for annihilation rate 10 −35 cm 3 s −1 or 10 −30 cm 3 s −1 respectively. But the upper limits of σv are 10 −29 cm 3 s −1 and 10 −25 cm 3 s −1 for mass values 1 GeV and 100 GeV.
depends on the PBH mass or DM mass or on the type of PBH distributions. For heavier PBHs (or DMs) one would obtain peak flux densities at smaller distances (or nearer to the Galactic Centre). Similarly, for power law mass distribution or lognormal mass distribution of PBH the peak flux densities are situated nearer to the Galactic Centre when contributions of the decay of lighter PBHs are negligible. Therefore, the measurement of the peak positions of such fluxes can be useful to determine the PBH mass or DM mass at the Galactic Centre region. Moreover, variations of the IC flux densities, originated due to the evaporation of PBHs with monochromatic mass distribution, with the distance from the Galactic Centre are computed and it is found that variations of the synchrotron flux densities and IC flux densities with r (the distance from the Galactic Centre) show different behaviour and hence fluxes produced from these two processes can be distinguishable. We have also shown such variations of synchrotron flux with r for a different DM profile namely, EINASTO profile of DM and observed that the variations depend on the form of DM halo profile.Additionally, limits on the model parameters are calculated from the SKA sensitivity and from the Jodrell Bank's experimental result (radio flux within 4 arcsec at the Galactic Centre should not be greater than 50 mJy at 408 MHz). It can be observed from the present calculations that SKA would be able to detect the radio signals even if energy density of PBH Ω PBH is ∼ 10 −14 times Ω DM for PBH mass distributions. But the upper limits on the f PBH (PBH fraction) are tight, which means amount of PBHs is a small component of DM at the Centre of Galaxy. The upper limit and lower limit of decay rates (Γ) of DM as a function of the DM mass and annihilation rates of DM ( σv ) as a function of the DM mass are also obtained. It is noted that SKA would be able to detect the signals even if Γ lies between 10 −32 − 10 −30 s −1 for mass range 1 -100 GeV respectively but the upper limit of Γ should be between 10 −27 − 10 −25 s −1 for such mass range. Similarly, SKA will be able to detect the signals even if σv lies between 10 −36 − 10 −33 cm 3 s −1 for mass range 1 -100 GeV respectively but the upper limit of σv should be between 10 −29 − 10 −27 cm 3 s −1 for such mass range.
For lognormal mass distribution and mochromatic mass distribution of PBH, f PBH , fraction of PBH in the Centre of Galaxy, can be upto 10 −2 and for power law mass distribution it is upto 10 −4 . The decay rate of DM can take values between 10 −26 − 10 −24 s −1 and annihilation rate can lie between 10 −29 − 10 −25 cm 3 s −1 for DM mass of 1 -100 GeV.In summary, in this work we have observed synchrotron flux densities and IC flux densities originated from the five above mentioned process at the Galactic Centre region. Fluxes computed for each process are compared. Further, the fluxes are compared with present and upcoming radio telescopes data and bounds on the model parameters are obtained from experimental predictions.The cases are compared for the two energy ranges of produced e − /e + , high energy (order of GeV) and low energy (order of MeV). It is also found that variations of synchrotron fluxes with the distance r can be a useful probe to determine the mass of the PBH (or parameters of PBH mass functions) and mass of the DM.It can be mentioned here that observations in the Galactic Centre region in radio wavelengths are complicated due to the presence of other astrophysical sources besides PBHs and DM. One of the most significant radio sources of the GC is Sagittarius A * (Sgr A * ). The measured integrated flux densities of Sgr A * by the VLA radio telescope are 0.71 Jy to 1.53 Jy for the frequencies 1.5 GHz to 41 GHz respectively [30]. On the other hand, the GMRT has observed the central region of the Milky Way at lower frequencies namely 580 MHz, 620 MHz and 1010 MHz for the detection of emission from Sgr A * [60]. From the latter observations the flux density at 620 MHz is estimated to be 0.5±0.1 Jy while at 1010 MHz the same is estimated to be 0.6±0.12 Jy. Flux density at 580 MHz matches with that at 620 MHz within the error bar. In our work, we have calculated the synchrotron and IC flux densities for the case when e − /e + are produced due to the evaporation of PBH or as the possible end products of annihilation or decay of DM at the central region of our Galaxy. For high energy e − /e + (E ∼ 0.1 GeV), it is found in our calculation that the synchrotron emission gives radio signals with peak frequency at ∼ 338 MHz and the IC emission gives UV to X-ray emission while for low energy e − /e + (E∼ 2 MeV), the peak frequencies of synchrotron and IC emissions are ∼ 10 −1 MHz and ∼ 10 5 MHz respectively. Here, the calculated flux densities primarily depend on the PBH (or DM) mass and mass distributions. For PBH mass distributions (monochromatic, power law and log normal) the flux densities are found to vary from O(10 −5 ) Jy to O(1) Jy and for the case of DM annihilation/decay the variations for the same are obtained from O(10 −3 ) Jy to O(1) Jy. Thus, while observational results indicate particular flux densities of Sgr A * at particular frequencies, in our calculations a range of flux densities are obtained at a particular frequency depending on the mass values and mass distributions of PBH (or DM).Therefore, improvement in measuring the flux densities in the future observations could be useful to discriminate different radio sources at the GC. Moreover, in Ref.[61,62] the variabilities of the emission of Sgr A * are observed with both time and frequency and it is found that Sgr A * flux density is highly variable with time (on an hourly time scale). But the flux densities originated either from PBH evaporation or from DM annihilation/decay would remain unchanged for such small time periods and thus would show constant flux densities with time in contrast to Sgr A * .
D. Source Term of Electrons and Positrons SpectraIn order to calculate synchrotron flux density or IC flux density from Eq. 12 and Eq. 17,
one needs to obtain dne(E,r)
dE
(number density of electrons (positrons) per unit energy interval at
position r) by solving Eq. 7. This can be done by defining e − /e + source term Q(E, r) for the
decay of PBH or the decay/annihilation of DM. For the decay and annihilation of DM, Q(E, r) are
defined as
AcknowledgementsThe authors would like to thank K. K. Datta for useful initial suggestions. One of the au-
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| []
|
[
"Travelling waves and light-front approach in relativistic electrodynamics",
"Travelling waves and light-front approach in relativistic electrodynamics"
]
| [
"Gaetano Fiore \nDip. di Matematica e Applicazioni\nUniversità \"Federico II\"\nComplesso MSA\n80126NapoliV. CintiaItaly\n\nI.N.F.N\nSezione di Napoli\nComplesso MSA\n80126NapoliV. CintiaItaly\n",
"Paolo Catelan \nCentro de Energías Alternativas y Ambiente\nEscuela Superior Politécnica del Chimborazo\nRiobambaEcuador\n\nDip. di Matematica ed Informatica\nUniversitá della Calabria\nArcavacata, RendeItaly\n"
]
| [
"Dip. di Matematica e Applicazioni\nUniversità \"Federico II\"\nComplesso MSA\n80126NapoliV. CintiaItaly",
"I.N.F.N\nSezione di Napoli\nComplesso MSA\n80126NapoliV. CintiaItaly",
"Centro de Energías Alternativas y Ambiente\nEscuela Superior Politécnica del Chimborazo\nRiobambaEcuador",
"Dip. di Matematica ed Informatica\nUniversitá della Calabria\nArcavacata, RendeItaly"
]
| []
| We briefly report on a recent proposal[1]for simplifying the equations of motion of charged particles in an electromagnetic (EM) field F µν that is the sum of a plane travelling wave F µν t (ct − z) and a static part F µν s (x, y, z); it adopts the light-like coordinate ξ = ct − z instead of time t as an independent variable. We illustrate it in a few cases of extreme acceleration, first of an isolated particle, then of electrons in a plasma in plane hydrodynamic conditions: the Lorentz-Maxwell & continuity PDEs can be simplified or sometimes even completely reduced to a family of decoupled systems of ordinary ones; this occurs e.g. with the impact of the travelling wave on a vacuum-plasma interface (what may produce plasma waves or the slingshot effect). | 10.1007/s11587-018-0411-y | [
"https://arxiv.org/pdf/1807.00667v1.pdf"
]
| 119,096,929 | 1807.00667 | 3318a5525681344bc58fd89db9c27293cda53950 |
Travelling waves and light-front approach in relativistic electrodynamics
Gaetano Fiore
Dip. di Matematica e Applicazioni
Università "Federico II"
Complesso MSA
80126NapoliV. CintiaItaly
I.N.F.N
Sezione di Napoli
Complesso MSA
80126NapoliV. CintiaItaly
Paolo Catelan
Centro de Energías Alternativas y Ambiente
Escuela Superior Politécnica del Chimborazo
RiobambaEcuador
Dip. di Matematica ed Informatica
Universitá della Calabria
Arcavacata, RendeItaly
Travelling waves and light-front approach in relativistic electrodynamics
We briefly report on a recent proposal[1]for simplifying the equations of motion of charged particles in an electromagnetic (EM) field F µν that is the sum of a plane travelling wave F µν t (ct − z) and a static part F µν s (x, y, z); it adopts the light-like coordinate ξ = ct − z instead of time t as an independent variable. We illustrate it in a few cases of extreme acceleration, first of an isolated particle, then of electrons in a plasma in plane hydrodynamic conditions: the Lorentz-Maxwell & continuity PDEs can be simplified or sometimes even completely reduced to a family of decoupled systems of ordinary ones; this occurs e.g. with the impact of the travelling wave on a vacuum-plasma interface (what may produce plasma waves or the slingshot effect).
Introduction
The equation of motion of a charged particle in an external EM fielḋ
p(t) = qE[t, x(t)] + p(t) m 2 c 2 +p 2 (t) ∧ qB[t, x(t)],
x(t) = cp(t) m 2 c 2 +p 2 (t) (1) in its general form is non-autonomous and highly nonlinear in the unknowns x(t), p(t).
Here m, q, x, p are the rest mass, electric charge, position and relativistic momentum of the particle, E = −∂ t A/c − ∇A 0 and B = ∇ ∧ A are the electric and magnetic field,
(A µ ) = (A 0 , −A)
is the electromagnetic (EM) potential 4-vector (E i = F i0 , B 1 = F 32 , etc.; we use Gauss CGS units). We decompose x = xi+yj+zk = x ⊥ +zk, etc, in the cartesian coordinates of the laboratory frame, and often use the dimensionless variables β ≡ v/c =ẋ/c, γ ≡ 1/ 1−β 2 = √ 1+u 2 and the 4-velocity u = (u 0 , u) ≡ (γ, γβ), i.e. the dimensionless version of the 4-momentum p. Usually, (1) is simplified assuming:
1. E, B are constant or vary "slowly" in space/time; or 2. E, B are "small" (so that nonlinear effects in E, B are negligible); or 3. E, B are monochromatic waves, or slow modulations of; or 4. the motion of the particle keeps non-relativistic.
The on-going, astonishing developments of laser technologies today allow the construction of compact sources of extremely intense coherent EM waves, possibly concentrated in very short laser pulses. Chirped Pulse Amplification [2,3] allows the production of pulses of intensity up to 10 23 Watt per square centimeter and duration down to 10 −15 seconds. Huge investments in new technologies (thin film compression, relativistic mirror compression, etc. [4,5]) will soon allow to produce even more intense/short (or cheaper) pulses. For instance, 850 MEuro have been allocated for the Extreme Light Infrastructure (ELI) program within the European Union ESFRI roadmap, with three of the planned four sites already under construction in Czech Republic, Hungary, Romania. One major motivation is the quest for table-top particle accelerators based on Laser Wake Field Acceleration (LWFA) [6] in plasmas. Among the possible applications of such accelerators we mention:
• Medicine: inspection (PET,...), cancer therapy by accelerated particles (electrons, protons, ions) or radioisotope production,...;
• Research: particle physics, materials science, structural biology, (inertial) nuclear fusion, X-ray free electron laser,...;
• Industry: atomic scale lithography, surface treatment of materials, sterilization, energy efficient manufacturing, detection systems,...;
• Environmental remediation: flue gas cleanup, petroleum cracking, transmutation of nuclear wastes,....
These and other applications of small accelerators were discussed e.g. at the "Big Idea Summit" organized by the US Department of Energy (Washington, 2016). In Europe the large network of research centers "European Plasma Research Accelerator with eXcellence In Applications" (EUPRAXIA) has been recently created to develop the associated technologies. Extremely intense and rapidly varying electromagnetic fields are present also in several violent astrophysical processes (see e.g. [5] and references therein). In either case the effects are so fast, huge, highly nonlinear, ultra-relativistic that conditions 1-4 are not fulfilled. Alternative simplifying approaches are therefore welcome.
Here we summarize an approach [1] that systematically applies the light-front formalism [7]; it is especially fruitful if in the spacetime region of interest (where we wish to follow the particles' worldlines) E, B are the sum of static parts and plane transverse travelling waves propagating in the z direction:
E(t, x) = ⊥ (ct−z) travelling wave + E s (x) static , B(t, x) = k ∧ ⊥ (ct−z) travelling wave + B s (x) static .(2)
The starting point is: as no particle can reach the speed of light c, thenξ(t) = ct−z(t) is strictly growing and we can make the change t → ξ = ct−z of independent parameter along the worldline λ ( fig. 1) of the particle; then the term ⊥ [ct−z(t)], where the unknown z(t) is in the argument of the highly nonlinear and rapidly varying ⊥ , becomes the known forcing term ⊥ (ξ). We apply the approach first to an isolated particle (sections 2, 3), then to a cold diluted plasma initially at rest and hit by a plane EM wave (section 4). The fields (2) can be obtained from an EM potential of the same form, A µ (x) = α µ (ct− z) + A µ s (x); in the Landau gauges (∂ µ A µ = 0) A s must fufill the Coulomb gauges (∇·A s = 0), and it must be α z = α 0 , ⊥ = −α ⊥ , E s = −∇A 0 s , B s = ∇∧A s . We shall set α z = α 0 = 0, as they appear neither in the observables E, B nor in the equations of motion. Assuming only that ⊥ (ξ) is piecewise continuous and
a)
⊥ has a compact support [0, l],
or a') ⊥ ∈ L 1 (R),(3)
we can fix α ⊥ (ξ) uniquely by requiring that it vanish as ξ → −∞:
α ⊥ (ξ) ≡ − ξ −∞ dy ⊥ (y); (4) in case a) α ⊥ (ξ) = 0 if ξ ≤ 0, α ⊥ (ξ) = α ⊥ (l) if ξ ≥ l.
We can treat on the same footing all ⊥ fulfilling (3) regardless of their Fourier analysis, in particular:
1. A modulated monochromatic wave:
⊥ (ξ) = (ξ) modulation [ia 1 cos(kξ +ϕ)+ja 2 sin(kξ)] carrier wave ⊥ o (ξ)(5)
(with a 2 1 +a 2 2 = 1). Under rather general assumptions
α ⊥ (ξ) = − (ξ) k ⊥ p (ξ) + O 1 k 2 − (ξ) k ⊥ p (ξ),(6)
⊥ p (ξ) := ⊥ o (ξ+π/2k); in the appendix we recall upper bounds for the remainder O(1/k 2 ). For slow modulations (i.e. | | |k |) -like the ones characterizing most conventional applications (radio broadcasting, ordinary laser beams, etc.) -the right estimate is very good.
2.
A superposition of waves of type 1.
3. An 'impulse' (few, one, or even a fraction of oscillation) [8,4]. 2 Set-up and general results for a single particle Letx(ξ) be the position as a function of ξ; it is determined byx(ξ) = x(t). More generally, for any given function f (t, x) we denotef (ξ,x) ≡ f [(ξ +ẑ)/c,x], abbreviateḟ ≡ df /dt, f ≡ df /dξ (total derivatives). Also the change of dependent (and unknown) variable u z → s is convenient, where the s-factor [1]
s ≡ γ − u z = u − = γ(1 − β z ) = γ c dξ dt > 0(7)
is the light-like component of u (the dimensionless version of p), as well as the Doppler factor of the particle. In fact, γ, u, β are rational functions of u ⊥ , s:
γ = 1+u ⊥2 +s 2 2s , u z = 1+u ⊥2 −s 2 2s , β = u γ(8)
(these relations hold also with the carets); so, replacing d/dt → (cs/γ)d/dξ and putting carets on all variables (1) becomes rational in the unknownsû ⊥ ,ŝ:
x =û s ,û ⊥ = q mc 2ŝ γÊ+û∧B ⊥ , s = q mc 2 û ⊥ s ·Ê ⊥ −Ê z − (û ⊥ ∧B ⊥ ) ẑ s(9)
withû z ,γ expressed as in (8). These equations amount [1] to the Euler-Lagrange equations d dξ ∂L ∂x = ∂L ∂x , that are obtained applying Hamilton's principle to the action functional S(λ) with λ parametrized by ξ (instead of t), as well as to the Hamilton equationsx = ∂Ĥ ∂Π ,Π = − ∂Ĥ ∂x , where the Hamiltonian
H(x,Π;ξ) = mc 2 1+ŝ 2 +û ⊥2 2ŝ +q 0 (ξ,x), with û ⊥ =Π ⊥ −q ⊥ (ξ,x) mc 2 s = −Π z +q[ 0 − z ](ξ,x) mc 2 ,(10)
is obtained by Legendre transform from L and again is rational inΠ ≡ ∂L ∂x , or equivalently inû ⊥ ,ŝ. Along the solutionsĤ gives the particle energy as a function of ξ, and
dĤ dξ = ∂Ĥ ∂ξ .(11)
Under the EM field (2) equations (9) amount tô
x =û s ,û ⊥ = q mc 2 (1+ẑ )Ê ⊥ s +(x ∧B s ) ⊥ + ⊥ (ξ) , s = −q mc 2 Ê z s −x ⊥ ·Ê ⊥ s + (x ⊥ ∧B ⊥ s ) z ,(12)
while the energy gain (normalized to the rest energy
mc 2 ) in the interval [ξ 0 , ξ 1 ] is E :=Ĥ (ξ 1 )−Ĥ(ξ 0 ) mc 2 = ξ 1 ξ 0 dξ q ⊥ ·û ⊥ mc 2ŝ (ξ).(13)
In particular, under assumption (3a) we obtain the total energy gain choosing ξ 0 = 0, ξ 1 = l, which are the values of the lightlike coordinate at the beginning and at the end of the interaction, see fig. 1. If we used parameter t, to compute E we should first determine the time t f when the pulse-particle interaction fineshes. Once solved (12), analytically or numerically, to obtain the solution as a function of t we just need to invertt(ξ) = ξ +ẑ(ξ) and set
x(t) =x[ξ(t)].
Contrary to (12), (1) is not rational in u, and the unknown z(t) appears in the argument of the rapidly varying functions ⊥ , α ⊥ in (1a), which now reads:
mc qu (t) = E s + u∧B s √ 1+u 2 + u· ⊥ [ct−z(t)] √ 1+u 2 k+ 1− u z √ 1+u 2 ⊥ [ct−z(t)].
H(x, P, t) is not rational in P := ∂L ∂ẋ , and also determining E(t) is more complicated.
Dynamics under a A µ independent of the transverse coordinates
Eq. (9) are further simplified if A µ = A µ (t,z). This applies in particular if E s = E z s (z)k, B s = B ⊥ s (z), choosing A 0 = − z dζE z s (ζ), A ⊥ = α ⊥ −k∧ z dζB ⊥ (ζ), A z ≡ 0. As ∂Ĥ/∂x ⊥ = 0, we findΠ ⊥ = qK ⊥ = const, i.e. the known result mc 2 qû ⊥ = K ⊥ −Â ⊥ (ξ,ẑ).
Setting v :=û ⊥2 and replacing in (12) we obtain
z = 1+v 2ŝ 2 − 1 2 ,ŝ = −q mc 2 E z s (ẑ)− 1 2ŝ ∂v ∂ẑ .(14)
Once solved system (14) forẑ(ξ),ŝ(ξ), the other unknowns are obtained from
x(ξ) = x 0 + ξ ξ 0 dyû (y) s(y)
.
[the z-component of (15) amounts to (14a) with initial conditionẑ
(ξ 0 ) = z 0 ]. If in addition B s ≡ 0, then A s ≡ 0, implying thatû ⊥ (ξ) = q mc 2 [K ⊥ − α ⊥ (ξ)
] andv =û ⊥2 are already known. The system (14) to be solved simplifies tô
z = 1+v 2ŝ 2 − 1 2 ,ŝ = −q mc 2 E z s (ẑ).(16)
Remarks. Some remarkables properties of the corresponding solutions are [1]: (16) is solved by quadrature.
1. Where ⊥ (ξ) = 0 thenv(ξ) = v c =const,Ĥ is conserved,
2. In case (3a) the final transverse momentum is mcû ⊥ (l). If of (5) varies slowly and u ⊥ (0) = 0, then by (6)û ⊥ (l) 0.
3. Fast oscillations of ⊥ makeẑ(ξ) oscillate much less thanx ⊥ (ξ), andŝ(ξ) even less: asŝ > 0,v =û ⊥2 ≥ 0, integrating (16a) averages the fast oscillations of u ⊥ to yield much smaller relative oscillations ofẑ, while integrating (16b) averages the residual small oscillations of E z s [ẑ(ξ)] to yield an essentially smoothŝ(ξ). On the contrary,γ(ξ),β(ξ),û(ξ), ..., which are recovered via (8), oscillate fast, and so do also γ(t), β(t), u(t), ... See e.g. fig. 2,4,6.
4. If u ⊥ (0) = 0 and the EM wave is a slowly modulated (5)-(3a), integrating (13) by parts across [0, l] and using (6) we find E l 0 dξv(ξ)ŝ (ξ)/2ŝ 2 (ξ): the energy gain will be automatically positive (resp. negative) ifŝ(ξ) is growing (resp. decreasing) in all of [0, l]. Correspondingly, the interaction with the EM wave can be used to accelerate (resp. decelerate) the particle.
Some solutions in closed form under constant B s , E s
Assume B s , E s are constants, and let b := qB s /mc 2 , e ≡ qE s /mc 2 . Upon integration over ξ and use of (12a) equations (12b-c) yield
u x = (e x −b y )ẑ +b zŷ +w x (ξ), u y = (e y +b x )ẑ −b zx +w y (ξ), s = (e x −b y )x+(e y +b x )ŷ−w z (ξ),(17)
where w(ξ) ≡ q[K−α ⊥ (ξ)+ ξE s ]/mc 2 (w is known and dimensionless), and K is an integra-
tion constant. For any E z s , B z s , E ⊥ s , if B ⊥ s = k ∧ E ⊥ s , then e x = b y , e y = −b x , and (17c) is solved: s = −w z (ξ)
. Then we solve in closed form the rest of the system (17), (12a) first forx ⊥ (ξ), then forû ⊥ (ξ),û z (ξ),ẑ(ξ). Assuming for simplicity the initial conditions x(0) = 0 = u(0) we find
(x + iŷ)(ξ) = (1−e z ξ) ib z /e z ξ 0 dζ (w x + iw y )(ζ) (1−e z ζ) 1+ib z /e z , z(ξ) = ξ 0 dζ 2 1 (1−e z ζ) 2 +x ⊥ 2 (ζ)−1 ,ŝ(ξ) = 1−e z ξ, u ⊥ (ξ) = (1−e z ξ)x ⊥ (ξ),γ(ξ) = 1−e z ξ +û z (ξ) u z (ξ) = 1 2(1−e z ξ) + (1−e z ξ)x ⊥ 2 (ξ)−1 2(18)
if e z = 0 and
(x+iŷ)(ξ) = ξ 0 dζ e −ib(ξ−ζ) (w x +iw y )(ζ),û ⊥ =x ⊥ , u z =ẑ =û ⊥2 2 = E(ξ) =γ(ξ) − 1,ẑ(ξ) = ξ 0 dζû ⊥2 (ζ) 2 .(19)
if e z = 0. As fas as we now, such general solutions have not appeared in the literature before Ref. [1]. We next analyze a few special cases (the first two have already appeared in the literature).
3.1 Case E s = B s = 0 (zero static fields). Then (18) becomes [9,10]:
s ≡ 1,û ⊥ = −qα ⊥ mc 2 ,û z =û ⊥2 2 ,γ = 1+û ẑ z(ξ) = ξ 0 dyû ⊥2 (y) 2 ,x ⊥ (ξ) = ξ 0 dyû ⊥ (y).(20)
The solutions (20) induced by two x-polarized pulses and the corresponding electron trajectories in the zx plane are shown in fig. 2. Note that:
• The maxima of γ, α ⊥ coincide (and approximately also of (ξ), if (ξ) is slowly varying).
• Since u z ≥ 0, the z-drift is nonnegative-definite. If we rescale ⊥ → a ⊥ thenx ⊥ ,û ⊥ scale like a, whereasẑ,û z scale like a 2 ; hence the trajectory goes to a straight line in the limit a → ∞. This is due to magnetic force qβ ∧ B.
• Corollary [1] The final u and energy gain read [in case (3a) it is also u ⊥ f =û ⊥ (l)]. By (6), both are very small if the pulse modulation is slow [extremely small if ∈ S(R) or ∈ C ∞ c (R)]. This can be seen as a rigouros version of the Lawson-Woodward Theorem [11,12,13,14] (an outgrowth of the original Woodward-Lawson Theorem [15,16]): this theorem states that, in spite of large energy variations during the interaction, the final energy gain E f of a charged particle interacting with an EM field is zero if:
u ⊥ f =û ⊥ (∞), u z f = E f = 1 2 u ⊥ f 2 = γ f −1(21)
i) the interaction occurs in R 3 vacuum (no boundaries); ii) E s = B s = 0 and ⊥ is slowly modulated; iii) v z c along the whole acceleration path; iv) nonlinear (in ⊥ ) effects qβ∧B are negligible; v) the power radiated by the particle is negligible. Our Corollary, as Ref. [17], states that the same result holds if we relax iii), iv), but the EM field is a plane travelling wave. To obtain a non-zero E f one has to violate some other conditions of the theorem, as e.g. we consider in next cases.
Case
E s = 0, B s = B z s k.
Then the solution (18) becomes (see fig. 3)
(x+iŷ)(ξ) = ξ 0 dζ e ib(ζ−ξ) (w x +iw y )(ζ),û ⊥ =x ⊥ , s ≡ 1,û z =ẑ =û ⊥2 2 = E =γ −1,ẑ(ξ) = ξ 0 dζû ⊥2 (ζ) 2 .(22)
For monochromatic ⊥ it reduces to the solution of [18,19,20] and leads to cyclotron autoresonance if −b = k 1 l : assuming for simplicity circular polarization [a 1 = a 2 = 1 in (5)], by (6) it is w x (ξ)+iw y (ξ) e ikξ w(ξ), whence
(x+iŷ)(ξ) iW(ξ)e ikξ ,ẑ(ξ) ξ 0 dζ k 2 W 2 (ζ) 2 , W (ξ) := ξ 0 dζ q (ζ) kmc 2 > 0;
clearly W (ξ) grows with ξ. In particular if ⊥ (ξ) = 0 for ξ ≥ l, then for such ξ
z (ξ) k 2 2 W 2 (l) 2E f , |x ⊥ (ξ)| z (ξ) 2 kW(l) 1;
the final energy gain is noteworthy by the first formula, the final collimation is very good by the second. by Remark 2.4, if ⊥ is slowly modulated the energy gain E f is negative if e z ≡ qE z s /mc 2 > 0, is positive if e z < 0, and has a unique maximum at some point e z M < 0 if (ξ) fulfills (3a) with a unique maximum. An acceleration device based on this solution would consist of the following: at t = 0 the particle is initially at rest with z 0 0, just at the left of a metallic grating G contained in the z = 0 plane and set at zero electric potential (see fig. 4); another metallic plate P contained in a plane z = z p > 0 is set at electric potential V = V p . A short laser pulse ⊥ travelling in the positive z-direction hits and boosts the particle into the latter region (see section 3.1); choosing qV p > 0 implies e z < 0, and a backward longitudinal electric force qE z s . If qV p is large enough, then z(t) reaches a maximum smaller than z p , then is accelerated backwards and exits the grating with energy E f and negligible transverse momentum. A large E f requires extremely large |V p |, far beyond the material breakdown threshold, what prevents its realization by a static potential (sparks between G, P would arise and rapidly reduce |V p |). A way out is to make the pulse itself generate such large |E z s | within a plasma just at the right time, so as to induce the slingshot effect, as sketchily explained at the end of next section.
Plane plasma problems
Assume that the plasma is initially in hydrodynamic conditions with all initial data [Eulerian velocities v h and densities n h of the h-th fluid, EM fields of the form (2); h enumerates electrons and kinds of ions composing the plasma, q h , m h are their charge, mass] not de- pending on x ⊥ . Then also the solutions of the Lorentz-Maxwell and continuity equations for B, E, u h , n h do not depend on x ⊥ , nor the displacements ∆x h ≡ x h (t, X)−X on X ⊥ . Here x h (t,X) is the position at t of the material element of the h-th fluid with initial position X ≡ (X,Y,Z); X h (t, x) is the inverse of x h (t,X) (at fixed t); β h = v h /c, etc. More specifically, we consider ( fig. 5) a very short and intense EM plane wave (3a) hitting normally a cold plasma (or a gas that is locally ionized into a plasma by the very high electric field of the pulse itself) initially in equilibrium, possibly in a static and uniform magnetic field B s ; the initial conditions are:
n h (0, x) = 0 if z ≤ 0, u h (0, x) = 0, j 0 (0, x) = h q h n h (0, x) ≡ 0, E(0, x) = ⊥ (−z), B(0, x) = k ∧ ⊥ (−z) + B s ,(24)
whence the 4-current density j = (j 0 , j) = h q h n h (1, β h ) is zero at t = 0. Then the Maxwell (whence the average pulse intensity is 10 19 W/cm 2 ); the associated u ⊥ e is painted purple. l is the length of the z-interval where the amplitude overcomes all ionization thresholds of the atoms of the gas yielding the plasma; here we have chosen helium, whence l 27µm, and the thresholds for 1 st and 2 nd ionization are overcome almost simultaneously.
equations ∇·E = 4πj 0 , ∂ t E z /c +4πj z = (∇∧B) z = 0 imply [10] E z (t,z) = 4π h q h N h [Z h (t, z)], N h (Z) := Z 0 dζ n h (0,ζ);(25)
using (25) to express E z in terms of the (still unknown) longitudinal motion [Z h (t, ·) is the inverse of z h (t, ·)] we reduce the number of unknowns by one.
Define α ⊥ as in (4); α ⊥ (ξ) = 0 if ξ ≤ 0. In the Landau gauges (24) are compatible with the following initial conditions for the gauge potential:
A(0, x) = α ⊥ (−z)+B s ∧x/2, ∂ t A(0, x) = −c ⊥ (−z),(26)
A 0 (0,x) = ∂ t A 0 (0,x) = 0. By (24)(25)(26) and causality x h (t, X) = X, A ⊥ (t, x) ≡ B s ∧x/2 if ct ≤ z, j ≡ 0 if ct ≤ |z|. A ⊥ is coupled to the current through A ⊥ = 4πj ⊥ . Including (26) the latter amounts to the integral equation
A ⊥ −α ⊥ − 1 2 (B s ∧x) ⊥ = 2π dsdζ θ(ct−s−|z−ζ|)θ(s) j ⊥ s c , ζ ;(27)
here we have used the Green function of the d'Alembertian ∂ 2 t /c 2 −∂ 2 z in dimension 2. The right-hand side (rhs) is zero for t ≤ 0 (t = 0 is the beginning of the laser-plasma interaction). Within short time intervals [0, t ] (to be determined a posteriori) we can thus: approximate A ⊥ (t, z) α ⊥ (ct−z)+ Bs 2 ∧x ⊥ ; also neglect the motion of ions with respect to the motion of the (much lighter) electrons. Hence it is z p (t, Z) ≡ Z, and the proton density n p (due to ions of all kinds) equals the initial one and therefore the initial electron density n 0 (z) := n e (0, z), by the initial electric neutrality of the plasma. Then the equations (
∆x e (0,X) = 0,û e (0,X) = 0 ⇒ŝ e (0,X) = 1.
(28) is a family parametrized by Z of decoupled ODEs in the unknowns ∆x e ,ŝ e ,û ⊥ e , which can be solved numerically. The approximation on A ⊥ (t,z) is acceptable as long as the so determined motion makes |rhs(27)| |α ⊥ + Bs 2 ∧ x|; otherwise rhs (27) determines the first correction to A ⊥ ; and so on.
If B s = 0, again (28b) is solved byû ⊥ e (ξ) = eα ⊥ (ξ)/mc 2 , while, setting v =û ⊥2 , (28a) and the z-component of (28c) take [21,22] the form of (16),
∆ẑ e = 1+v 2ŝ 2 − 1 2 ,ŝ e = 4πe 2 mc 2 N [ẑ e ]− N (Z) .(30)
If n e (0,X) = n 0 θ(Z) (with a constant electron density n 0 ), then as long asẑ e (ξ, Z) > 0 (30), (29) reduce to the same Cauchy problem for all Z:
∆ = 1 + v 2s 2 − 1 2 , s = M ∆, M := 4πe 2 n 0 mc 2 ≡ ω 2 p c 2 ,(31)∆(0) = 0, s(0) = 1.(32)
These are the equations of motion of a relativistic harmonic oscillator with a forcing term v. In fig. 6 we depict the solution corresponding to the pulse of fig. 5-right (with l 27µm) and to n 0 = 2×10 18 cm −3 ; s(ξ) is indeed insensitive to the fast oscillations of ⊥ (see remark 2.3), ∆(ξ) grows positive for small ξ. The other unknowns are obtained through (15). After the pulse is passed the solution becomes periodic with period ξ H 49µm. These l, n 0 fulfill
2l ξ H = ct H ,(33)
where t H is the plasma period associated to n 0 (recall that t H ≥ t nr H ≡ 2π/ω p , the nonrelativistic limit of t H 1 ). For all layers of electrons with initial Z > ∆ M (∆ M is the oscillation amplitude) it isẑ e (ξ, Z) = Z +∆(ξ) for all ξ (because this keeps positive for all ξ), u(t, z) = u(ct − z), and similarly for all other Eulerian fields: a plasma wave with spacial period ξ H and phase velocity c trails the pulse [23,24]. On the other hand, if Z < ∆ M then z e (ξ, Z) = Z +∆(ξ) becomes negative at some ξ = ξ e , namely the layers of electrons with such initial Z exit the plasma bulk; in the ξ-intervals where Z+∆(ξ) < 0 the ruling equation (30b) becomesŝ e (ξ, Z) = −M Z. Condition (33) secures both that the pulse is completely inside the bulk before any electron gets out of it, and that the spacial period of the plasma wave is larger that the pulse length.
Replacing these solutions in the rhs (27) we find that A ⊥ α ⊥ is indeed verified at least for t < t c 5ξ H /c. On the other hand we find [23,24] that, while the map z e (t, ·) : Z → z is indeed one-to-one everywhere for t < t c , at later times wave-breaking [25] (due to crossing of different Z-layers) occurs near the vacuum-plasma interface Z ∼ 0. This implies that the hydrodynamic description is globally self-consistent for t < t c , whereas the use of kinetic theory (i.e of a statistical description in phase space taking collisions into account, e.g. by BGK [26] equations or effective linear inheritance relations [27]) is necessary if t > t c , starting from a region near the vacuum-plasma interface. But as its effects can propagate only with . Figure 6: Down: Solution of corresponding to the pulse of fig. 5-right, and to the initial electron density n 0 (Z) = n 0 θ(Z), with n 0 = 2×10 18 cm −3 . a velocity smaller than c, they will not affect the plasma wave trailing the pulse with phase velocity c.
The above predictions are based on idealizing the laser pulse as a plane EM wave. In a more realistic picture the laser pulse is cylindrically symmetric around the z-axis and has a finite spot radius R. Using causality and heuristic arguments we can compute [21] rough R < ∞ corrections to the above predictions: as a result, the impact of a very short and intense laser pulse on the surface of a cold low-density plasma (or gas, ionized into a plasma by the pulse itself), as considered e.g. in fig. 5-right, may induce [for carefully tuned R, n 0 (Z)], beside a plasma traveling-wave propagating behind the pulse, also the slingshot effect [21,22,28], i.e. the backward acceleration and expulsion from the plasma of some surface electrons (those with smallest Z and closest to the z-axis) with remarkable energy. For reviews see also [29,30,31].
Appendix: some useful estimates of oscillatory integrals
Given a function f ∈ S(R), integrating by parts we find for all n ∈ N
Hence we find the following upper bounds for the remainders R f n :
R f 1 (ξ) ≤ 1 |k| 2 |f (ξ)| + ξ −∞ dζ |f (ζ)| ≤ f ∞ + f 1 |k| 2 ,(37)R f n (ξ) ≤ 1 |k| n+1 f (n) (ξ)+ ξ −∞ dζ |f (n+1) (ζ)| ≤ f (n) ∞ + f (n+1) 1 |k| n+1 .(38)
It follows R f 1 = O(1/k 2 ), and more generally R f n = O(1/k n+1 ), so that (35) are asymptotic expansions in 1/k. All inequalities in (37-38) are useful: the left inequalities are more stringent, while the right ones are ξ-independent.
Equations (34), (37) and R f 1 = O(1/k 2 ) hold also if f ∈ W 2,1 (R) (a Sobolev space), in particular if f ∈ C 2 (R) and f, f , f ∈ L 1 (R), because the previous steps can be done also under such assumptions. Equations (34) will hold with a remainder R f 1 = O(1/k 2 ) also under weaker assumptions, e.g. if f is bounded and piecewise continuous and f, f , f ∈ L 1 (R), but R f 1 will be a sum of contributions like (36) for every interval in which f is continuous. Similarly, (35), (38) and/or R f n = O(1/k n+1 ) hold also under analogous weaker conditions. Letting ξ → ∞ in (34), (37) we find for the Fourier transformf (k) =
∞ −∞ dζ f (ζ)e −iky of f (ξ) |f (k)| ≤ f ∞ + f 1 |k| 2 ,(39)
hencef (k) = O(1/k 2 ) as well. Actually, for functions f ∈ S(R) the decay off (k) as |k| → ∞ is much faster, sincef ∈ S(R) as well. For instance, for the gaussian f (ξ) = exp[−ξ 2 /2σ] it isf (k) = √ πσ exp[−k 2 σ/2]. To prove approximation (6) now we just need to choose f = and note that every component of α ⊥ will be a combination of (35) and (35) k →−k .
Figure 1 :
1Every worldline λ and hyperplane ξ = ξ in Minkowski space intersect once. The wave-particle interaction occurs only along the intersection of λ with the support of the EM wave (painted pink), which assuming(3)is delimited by the ξ = 0 and ξ = l hyperplanes.
Figure 2 :
2Solutions (20) (up and center) and corresponding electron trajectories in the zx plane (down) induced by two x-polarized pulses with carrier wavelength λ = .8µm, gaussian modulation (ξ) = a exp[−ξ 2 /2σ], σ = 20µm 2 , |q|aλ/mc 2 = 4, 15 (left, right).
3. 3 Figure 3 :
33Case E s = E z s k, B s = 0.The solution (18) reduces tô s(ξ) = 1−e z ξ,(x + iŷ)(ξ) = ξ 0 dy(w x +iw y )(y) 1−e z y ,ẑ(ξ) The electron motion(22)(up) and the zx-projection of the corresponding trajectory (down) induced in a longitudinal magnetic field B z = 10 5 G by a circularly polarized modulated EM wave (5) with wavelength λ ≡ 2π/k = 1mm, b = k = 58.6cm −1 , gaussian enveloping amplitude (ξ) = a exp[−ξ 2 /2σ] with σ = 3cm 2 and ea/kmc 2 = 0.15, trivial initial conditions (x 0 = u 0 = 0), giving E f 28.5.
Figure 4 :
4Left: the motion (23) induced by a linearly polarized modulated EM wave(5) with wavelength λ = 2π/k = 0.8µm, gaussian enveloping amplitude (ξ) = a exp[−ξ 2 /2σ] with σ = 20µm 2 and |q|a √ 2/kmc 2 = 6.6, trivial initial conditions, B s = 0, E s = kE z M , where E z M q 37GeV/m (this yields the maximum energy gain E f 1.5 with such a wave). Right: the corresponding trajectory in the zx plane within an hypothetical acceleration device based on a laser pulse and metallic gratings G, P at potentials V = 0, V p , with qV p /z p 37GeV/m.
Figure 5 :
5Left: a plane EM wave of finite length approaching normally a plasma in equilibrium. Right: normalized EM wave ⊥ (blue) with carrier wavelength λ = 0.8µm, linear polarization, gaussian modulation (ξ) = a exp[−ξ 2 /2σ] with σ = 20µm 2 , eaλ/mc 2 = 15
9) & initial conditions for the electron fluid amount to mc 2ŝ e (ξ, Z) = 4πe 2 N (ẑ e )− N (Z) + e(∆x ⊥ e ∧B ⊥ s ) z , mc 2û⊥ e (ξ, Z) = eα ⊥ − e(∆x e ∧B s ) ⊥ , ∆x e = ue(ξ,Z) se(ξ,Z)
(n+1) (ζ) e ikζ .
Whenv = 0 then (31) implies ∆ = −M ∆/ŝ 3 . In the nonrelativistic regimeŝ 1,ξ(t) ct, cd/dξ d/dt, and this becomes the nonrelativistic harmonic equation∆ = −ω 2 p ∆.
Acknowledgments. The results contained in the present paper have been partially presented in the international conference "Wascom 2017". Devoted to Tommaso Ruggeri on the occasion of his 70th birthday.
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| The Data Access System (DAS) is a metadata and data management software system, providing a reusable solution for the storage of data acquired both from telescopes and auxiliary data sources during the instrument development phases and operations. It is part of the Customizable Instrument WorkStation system (CIWS-FW), a framework for the storage, processing and quick-look at the data acquired from scientific instruments. The DAS provides a data access layer mainly targeted to software applications: quick-look displays, pre-processing pipelines and scientific workflows. It is logically organized in three main components: an intuitive and compact Data Definition Language (DAS DDL) in XML format, aimed for user-defined data types; an Application Programming Interface (DAS API), automatically adding classes and methods supporting the DDL data types, and providing an object-oriented query language; a data management component, which maps the metadata of the DDL data types in a relational Data Base Management System (DBMS), and stores the data in a shared (network) file system. With the DAS DDL, developers define the data model for a particular project, specifying for each data type the metadata attributes, the data format and layout (if applicable), and named references to related or aggregated data types. Together with the DDL user-defined data types, the DAS API acts as the only interface to store, query and retrieve the metadata and data in the DAS system, providing both an abstract interface and a data model specific one in C, C++ and Python. The mapping of metadata in the back-end database is automatic and supports several relational DBMSs, including MySQL, Oracle and PostgreSQL. | null | [
"https://arxiv.org/pdf/1405.7584v1.pdf"
]
| 204,989 | 1405.7584 | 29fac46dfccfc9c65082c5082960d8e7d7ef0ab9 |
**Volume Title** ASP Conference Series, Vol. **Volume Number** **Author** c **Copyright Year** Astronomical Society of the Pacific DAS: a data management system for instrument tests and operations
29 May 2014
Marco Frailis
INAF-Osservatorio Astronomico di Trieste
Via G.B. Tiepolo 11TriesteItaly
Stefano Sartor
INAF-Osservatorio Astronomico di Trieste
Via G.B. Tiepolo 11TriesteItaly
Andrea Zacchei
INAF-Osservatorio Astronomico di Trieste
Via G.B. Tiepolo 11TriesteItaly
Marcello Lodi
FGG -INAF
Telescopio Nazionale Galileo
Rambla José Ana Férnandez Pérez 7, 38712 Breña bajaEspaña
Roberto Cirami
INAF-Osservatorio Astronomico di Trieste
Via G.B. Tiepolo 11TriesteItaly
Fabio Pasian
INAF-Osservatorio Astronomico di Trieste
Via G.B. Tiepolo 11TriesteItaly
Massimo Trifoglio
INAF-IASF Bologna
Via P. Gobetti 101I40129BolognaItaly
Andrea Bulgarelli
INAF-IASF Bologna
Via P. Gobetti 101I40129BolognaItaly
Fulvio Gianotti
INAF-IASF Bologna
Via P. Gobetti 101I40129BolognaItaly
Enrico Franceschi
INAF-IASF Bologna
Via P. Gobetti 101I40129BolognaItaly
Luciano Nicastro
INAF-IASF Bologna
Via P. Gobetti 101I40129BolognaItaly
Vito Conforti
INAF-IASF Bologna
Via P. Gobetti 101I40129BolognaItaly
Andrea Zoli
INAF-IASF Bologna
Via P. Gobetti 101I40129BolognaItaly
Ricky Smart
INAF-Osservatorio Astrofisico di Torino
Strada Osservatorio 2010025Pino TorineseItaly
Roberto Morbidelli
INAF-Osservatorio Astrofisico di Torino
Strada Osservatorio 2010025Pino TorineseItaly
Mauro Dadina
INAF-IASF Bologna
Via P. Gobetti 101I40129BolognaItaly
**Volume Title** ASP Conference Series, Vol. **Volume Number** **Author** c **Copyright Year** Astronomical Society of the Pacific DAS: a data management system for instrument tests and operations
29 May 2014
The Data Access System (DAS) is a metadata and data management software system, providing a reusable solution for the storage of data acquired both from telescopes and auxiliary data sources during the instrument development phases and operations. It is part of the Customizable Instrument WorkStation system (CIWS-FW), a framework for the storage, processing and quick-look at the data acquired from scientific instruments. The DAS provides a data access layer mainly targeted to software applications: quick-look displays, pre-processing pipelines and scientific workflows. It is logically organized in three main components: an intuitive and compact Data Definition Language (DAS DDL) in XML format, aimed for user-defined data types; an Application Programming Interface (DAS API), automatically adding classes and methods supporting the DDL data types, and providing an object-oriented query language; a data management component, which maps the metadata of the DDL data types in a relational Data Base Management System (DBMS), and stores the data in a shared (network) file system. With the DAS DDL, developers define the data model for a particular project, specifying for each data type the metadata attributes, the data format and layout (if applicable), and named references to related or aggregated data types. Together with the DDL user-defined data types, the DAS API acts as the only interface to store, query and retrieve the metadata and data in the DAS system, providing both an abstract interface and a data model specific one in C, C++ and Python. The mapping of metadata in the back-end database is automatic and supports several relational DBMSs, including MySQL, Oracle and PostgreSQL.
Introduction
The Data Access System (DAS) is a reusable software system that allows storage, retrieval and management of metadata and data acquired from telescopes and auxiliary data sources or produced by their subsequent levels of data processing. It is part of the Customizable Instrument WorkStation software project (see Conforti et al. 2013), 2 Frailis, Sartor et al.
aimed at providing a framework, named CIWS-FW, for the storage, processing and quick-look at data acquired from space-borne and ground-based telescope observatories, to support all their development phases. The DAS inherits several high-level design concepts from the Planck-LFI Data Management Component (Hazell et al. 2002), a system designed and developed by the MPA (Max Planck institute for Astrophysics) Planck Analysis Centre and successfully deployed by the Planck-LFI (Low Frequency Instrument) Data Processing Centre for the instrument tests and flight operations and used by all its processing levels (Zacchei et al. 2011). The DAS provides a data access layer mainly targeted to software applications: quick-look displays for the near real-time or off-line analysis of the instrument raw data, pre-processing and calibration pipelines, data analysis tasks or data reduction workflows. It is logically organized in three main components. A Data Definition Language (DAS DDL) in XML format, consisting of an intuitive and compact grammar that the developers use to define the data structures that will be archived and retrieved in a specific project. Each DDL type is defined by specifying its metadata attributes, data format and layout -binary table or image -and associations (references) to other related types. An Application Programming Interface (DAS API), automatically mapping each user-defined DDL type to a number of supporting classes and methods, and providing an object-oriented query language on the metadata attributes and the associations of the DDL types. A Data Management Component (DMC), which maps the metadata and the associations of the DDL data types to a relational DBMS, with the aid of an Object-to-Relational Mapping system (ORM), and stores the data in a shared (network) file system. The first DAS user task consists in the specification of the data model for a particular project, by creating a DDL file that collects all the data structure definitions. Starting from this file, the DAS building system automatically generates classes that correspond to the DDL data types and updates the DAS API library and the DBMS database schema with the new types (see fig. 1). Then, the DAS user can code the program module, using the DAS API to persist, query and access the data.
DAS Data Definition Language
The DDL provides an XML grammar, formalized in the XML Schema Definition language (XSD), to define new data types that will be stored in the DAS system. DAS data management system 3 Figure 2. DDL grammar elements A DDL data type definition includes two main components: metadata and data (see fig. 2). The metadata section defines a list of keywords describing the data. The data section can define either a binary table or an image. These two components are logically analogous to the header block and data block of the FITS standard. Since the data section is not mandatory, it is possible to define metadata-only data types. Additionally, a data type can be associated to other correlated data types. A single association with a data type can also specify: i) a multiplicity, when many instances of the same type are involved in an association; ii) the relation type (shared, exclusive, extend), that affects the mapping of the association in the database schema. In order to define new data types by extending existing ones, in a data type definition it is possible to add a single parent type (ancestor).
By default, the DAS system stores the metadata in a back-end relational DBMS; the data is transparently persisted in a binary format (analogous to the VOTable binary serialization, Ochsenbein et al. (2011)) in a shared file system. However, for small data sizes (small arrays or images), a DDL type definition can specify to store the data as BLOBs (Binary Large Objects) in the DBMS. Figure 1 shows the high-level components of the DAS system. The core of the system is developed in C++. One component is the ORM system, that allows to persist C++ objects to a relational DBMS, automatically handling the conversion between C++ and SQL types. We are using an existing open-source ORM, ODB (www.codesynthesis.com), which currently supports several DBMSs, including MySQL, Oracle, PostreSQL and SQLite. For each supported DBMS, ODB uses the native C API instead of ODBC to reduce the overhead.
DAS Architecture and API
From the DDL file provided by the user, the DAS automatically generates the C++ classes corresponding to each type and the ORM directives. The database support code is then built through the ORM system. The DMC implements the DAS API, using some concepts taken from the JPA specification (Java Persistency API), in particular the so called persistence context (Bauer & King 2006), which simplifies the automatic synchronization of an in-memory data object and all its associated instances with the persistent counterparts. The Data Storage Engine component implements the data serialization mechanism. Currently, only one binary serialization is provided; however base classes are available to support other possible types of serialization. Figure 3 shows a short example of DDL data type definition in XML and the client code in C++ using the class automatically generated for that type by the DAS system. The DAS core API, in C++, provides both a template based and a polymorphic interface (for run-time type inference). Currently, a Python binding, based on SWIG, is also available.
Conclusions
The DAS requirements and subsequent design and development is based on the experience gained by the team in monitoring, control and analysis software for space-borne and ground observatories (Planck, AGILE, TNG, GSCII, REM). It will soon reach a first stable release in the upcoming months. Additional information on the CIWS-FW and the DAS system can be found at the following links: http://ciws-fw.iasfbo.inaf.it/ciws-fw, http://redmine.iasfbo.inaf.it/projects/ciwsfw/repository/gitdas.
Figure 1 .
1DAS system overview
Figure 3 .
3DDL type definition and client code example
Java Persistence with Hibernate. C Bauer, G King, Manning Pubns CoShelter Island, NY1st edBauer, C., & King, G. 2006, Java Persistence with Hibernate (Shelter Island, NY: Manning Pubns Co), 1st ed.
V Conforti, TBD of ASP Conf. Ser., TBD. N. Manset, & P. ForshaySan FranciscoADASS XXIIIConforti, V., et al. 2013, in ADASS XXIII, edited by N. Manset, & P. Forshay (San Francisco: ASP), vol. TBD of ASP Conf. Ser., TBD
A Hazell, K Bennett, O Williams, ADASS XI. D. Bohlender, D. Durand, & T. H. HandleyVictoria, BC, Canada281125ASP)Hazell, A., Bennett, K., & Williams, O. 2002, in ADASS XI, edited by D. Bohlender, D. Du- rand, & T. H. Handley (Victoria, BC, Canada: ASP), vol. 281 of ASP Conf. Ser., 125
. F Ochsenbein, A&A. 5365Ochsenbein, F., et al. 2011, ArXiv e-prints Zacchei, A., et al. 2011, A&A, 536, A5
| []
|
[
"Advantages of using YBCO-Nanowire-YBCO heterostructures in the search for Majorana Fermions",
"Advantages of using YBCO-Nanowire-YBCO heterostructures in the search for Majorana Fermions"
]
| [
"P Lucignano \nCNR-ISC\nvia Fosso del Cavaliere 100I-00133RomaItaly\n\nDipartimento di Scienze Fisiche\nUniversità di Napoli \"Federico II\"\nMonte S.AngeloI-80126NapoliItaly\n",
"A Mezzacapo \nDepartamento de Química Física\nUniversidad del País Vasco -Euskal Herriko Unibertsitatea\nApdo. 64448080BilbaoSpain\n",
"F Tafuri \nDipartimento Ingegneria dell'Informazione\nSeconda Università di Napoli\nI-81031Aversa (CE)Italy\n\nCNR-SPIN\nMonte S.Angelo -via CinthiaI-80126NapoliItaly\n",
"A Tagliacozzo \nDipartimento di Scienze Fisiche\nUniversità di Napoli \"Federico II\"\nMonte S.AngeloI-80126NapoliItaly\n\nCNR-SPIN\nMonte S.Angelo -via CinthiaI-80126NapoliItaly\n"
]
| [
"CNR-ISC\nvia Fosso del Cavaliere 100I-00133RomaItaly",
"Dipartimento di Scienze Fisiche\nUniversità di Napoli \"Federico II\"\nMonte S.AngeloI-80126NapoliItaly",
"Departamento de Química Física\nUniversidad del País Vasco -Euskal Herriko Unibertsitatea\nApdo. 64448080BilbaoSpain",
"Dipartimento Ingegneria dell'Informazione\nSeconda Università di Napoli\nI-81031Aversa (CE)Italy",
"CNR-SPIN\nMonte S.Angelo -via CinthiaI-80126NapoliItaly",
"Dipartimento di Scienze Fisiche\nUniversità di Napoli \"Federico II\"\nMonte S.AngeloI-80126NapoliItaly",
"CNR-SPIN\nMonte S.Angelo -via CinthiaI-80126NapoliItaly"
]
| []
| We propose an alternative platform to observe Majorana bound states in solid state systems. High critical temperature cuprate superconductors can induce superconductivity, by proximity effect, in quasi one dimensional nanowires with strong spin orbit coupling. They favor a wider and more robust range of conditions to stabilize Majorana fermions due to the large gap values, and offer novel functionalities in the design of the experiments determined by different dispersion for Andreev bound states as a function of the phase difference.Recently there is an increasing interest in topological quantum computation based on Majorana Bound States (MBS's)[1,2]. Majorana Fermions have been predicted in a wide class of low-dimensional solid state devices. Many of these proposals make use of quasi one dimensional superconductors in contact with topological insulators[3]or quasi one-dimensional materials with strong spin orbit interactions[4][5][6][7]. Also helical magnets[8]and other materials [9-13] are considered. In this paper we propose a quite distinctive heterostructure to observe topologically protected MBS's in a solid state device. Our work rests on the physics of S/R/S hybrid structures in which "R" is a quasi one dimensional semiconductor nanowire (NW) with strong Rashba spin orbit coupling (e.g. InAs or InSb) electrically connected to two conventional low T c superconductor leads ( "S" )[4,5]. Superconductivity is induced in the spin-orbit coupled semiconductor by proximity effect due to the superconducting electrodes. The coexistence of superconductivity and spin-orbit coupling is a key ingredient for the existence of MBS's at the interfaces between the R region and the superconducting S regions.However, despite the considerable theoretical and experimental [14] efforts, some challenges still remain before a real device allowing isolation and manipulation of MBS's in such geometry can be realized. In particular the difficulties of tuning the chemical potential of the semiconductor region µ, controlling the disorder on the bulk gap as well as optimizing the coupling between the different materials [15-17] make the realization of such devices extremely difficult.All schemes proposed up to now to generate MBS's substantially use conventional s-wave superconductors to induce superconductivity and a gap ∆ in the R nanowire[1]. The role of superconducting pairing is to relax number conservation, thus allowing for the mixing of particle and hole degrees of freedom. Zeeman spin splitting is required to halve the number of degrees of freedom at low energies, thus generating the elusive neutral (Ma-jorana) excitation. A simple criterion to induce MBS's at S/R interfaces is given in terms of the applied magnetic field B x oriented along the wire, µ and ∆. The inequality to be satisfied can be stated as: B 2x > µ 2 + ∆ 2 [4]. Low critical magnetic fields (H c ) and low gap values characteristic of conventional low Tc superconductors substantially define the limits of the nominal range of dynamical parameters required to observe MBS's. Not only do H c and ∆ enter into the criteria to stabilize MBS's, but they also endanger the feasibility of the experiment in case high magnetic fields are required. High critical temperature superconductors (HTS) may favor a completely different approach to experiments on MBS's, since HTS plaquettes/contacts (even a few micron square) sustain superconductivity up to a few tenths of Tesla and induce robust superconductivity in a wide range of barrier materials. When conventional low-Tc superconductors (e.g. Nb) are considered, the large difference in the g-factors for Nb (g N b ∼ 1) and InAs (g InAs ∼ 35) implies that the in-plane magnetic field B ∼ 0.1T can open a sizable Zeeman gap in InAs (V x ≤ 1K) without substantially suppressing the superconductivity in Nb. However these conditions holds even more firmly in YBCO contacts because the YBCO gap is very stable w.r.t. magnetic fields, despite a doubling of the g−factor (g Y BCO ∼ 2).The induced gap in the NW can be considered of the order of the bare gap of the superconductor, projected along the wire direction, provided that the radius of the wire is negligibly small with respect to the coherence length ξ of the superconducting material and that no sizeable barriers are present at the interfaces. As recently pointed out in Ref.[15], the interface tunnelling between different materials renormalizes the induced gap to∆ i = (1 − Z)∆ i where Z ∼ (1 + πρ 0 |V hop | 2 /∆ i ) −1 is the quasiparticle weight, V hop mimicks the electron hopping between the superconductor and the NW and ρ 0 is the density of states of the superconductor at the Fermi energy. The better is the coupling with a larger V hop , the smaller becomes Z and the larger is the induced | 10.1103/physrevb.86.144513 | [
"https://arxiv.org/pdf/1209.5562v1.pdf"
]
| 118,405,228 | 1209.5562 | 7b658c90829975a27f8b1dc530f101a76e852126 |
Advantages of using YBCO-Nanowire-YBCO heterostructures in the search for Majorana Fermions
25 Sep 2012 (Dated: May 2, 2014)
P Lucignano
CNR-ISC
via Fosso del Cavaliere 100I-00133RomaItaly
Dipartimento di Scienze Fisiche
Università di Napoli "Federico II"
Monte S.AngeloI-80126NapoliItaly
A Mezzacapo
Departamento de Química Física
Universidad del País Vasco -Euskal Herriko Unibertsitatea
Apdo. 64448080BilbaoSpain
F Tafuri
Dipartimento Ingegneria dell'Informazione
Seconda Università di Napoli
I-81031Aversa (CE)Italy
CNR-SPIN
Monte S.Angelo -via CinthiaI-80126NapoliItaly
A Tagliacozzo
Dipartimento di Scienze Fisiche
Università di Napoli "Federico II"
Monte S.AngeloI-80126NapoliItaly
CNR-SPIN
Monte S.Angelo -via CinthiaI-80126NapoliItaly
Advantages of using YBCO-Nanowire-YBCO heterostructures in the search for Majorana Fermions
25 Sep 2012 (Dated: May 2, 2014)
We propose an alternative platform to observe Majorana bound states in solid state systems. High critical temperature cuprate superconductors can induce superconductivity, by proximity effect, in quasi one dimensional nanowires with strong spin orbit coupling. They favor a wider and more robust range of conditions to stabilize Majorana fermions due to the large gap values, and offer novel functionalities in the design of the experiments determined by different dispersion for Andreev bound states as a function of the phase difference.Recently there is an increasing interest in topological quantum computation based on Majorana Bound States (MBS's)[1,2]. Majorana Fermions have been predicted in a wide class of low-dimensional solid state devices. Many of these proposals make use of quasi one dimensional superconductors in contact with topological insulators[3]or quasi one-dimensional materials with strong spin orbit interactions[4][5][6][7]. Also helical magnets[8]and other materials [9-13] are considered. In this paper we propose a quite distinctive heterostructure to observe topologically protected MBS's in a solid state device. Our work rests on the physics of S/R/S hybrid structures in which "R" is a quasi one dimensional semiconductor nanowire (NW) with strong Rashba spin orbit coupling (e.g. InAs or InSb) electrically connected to two conventional low T c superconductor leads ( "S" )[4,5]. Superconductivity is induced in the spin-orbit coupled semiconductor by proximity effect due to the superconducting electrodes. The coexistence of superconductivity and spin-orbit coupling is a key ingredient for the existence of MBS's at the interfaces between the R region and the superconducting S regions.However, despite the considerable theoretical and experimental [14] efforts, some challenges still remain before a real device allowing isolation and manipulation of MBS's in such geometry can be realized. In particular the difficulties of tuning the chemical potential of the semiconductor region µ, controlling the disorder on the bulk gap as well as optimizing the coupling between the different materials [15-17] make the realization of such devices extremely difficult.All schemes proposed up to now to generate MBS's substantially use conventional s-wave superconductors to induce superconductivity and a gap ∆ in the R nanowire[1]. The role of superconducting pairing is to relax number conservation, thus allowing for the mixing of particle and hole degrees of freedom. Zeeman spin splitting is required to halve the number of degrees of freedom at low energies, thus generating the elusive neutral (Ma-jorana) excitation. A simple criterion to induce MBS's at S/R interfaces is given in terms of the applied magnetic field B x oriented along the wire, µ and ∆. The inequality to be satisfied can be stated as: B 2x > µ 2 + ∆ 2 [4]. Low critical magnetic fields (H c ) and low gap values characteristic of conventional low Tc superconductors substantially define the limits of the nominal range of dynamical parameters required to observe MBS's. Not only do H c and ∆ enter into the criteria to stabilize MBS's, but they also endanger the feasibility of the experiment in case high magnetic fields are required. High critical temperature superconductors (HTS) may favor a completely different approach to experiments on MBS's, since HTS plaquettes/contacts (even a few micron square) sustain superconductivity up to a few tenths of Tesla and induce robust superconductivity in a wide range of barrier materials. When conventional low-Tc superconductors (e.g. Nb) are considered, the large difference in the g-factors for Nb (g N b ∼ 1) and InAs (g InAs ∼ 35) implies that the in-plane magnetic field B ∼ 0.1T can open a sizable Zeeman gap in InAs (V x ≤ 1K) without substantially suppressing the superconductivity in Nb. However these conditions holds even more firmly in YBCO contacts because the YBCO gap is very stable w.r.t. magnetic fields, despite a doubling of the g−factor (g Y BCO ∼ 2).The induced gap in the NW can be considered of the order of the bare gap of the superconductor, projected along the wire direction, provided that the radius of the wire is negligibly small with respect to the coherence length ξ of the superconducting material and that no sizeable barriers are present at the interfaces. As recently pointed out in Ref.[15], the interface tunnelling between different materials renormalizes the induced gap to∆ i = (1 − Z)∆ i where Z ∼ (1 + πρ 0 |V hop | 2 /∆ i ) −1 is the quasiparticle weight, V hop mimicks the electron hopping between the superconductor and the NW and ρ 0 is the density of states of the superconductor at the Fermi energy. The better is the coupling with a larger V hop , the smaller becomes Z and the larger is the induced
We propose an alternative platform to observe Majorana bound states in solid state systems. High critical temperature cuprate superconductors can induce superconductivity, by proximity effect, in quasi one dimensional nanowires with strong spin orbit coupling. They favor a wider and more robust range of conditions to stabilize Majorana fermions due to the large gap values, and offer novel functionalities in the design of the experiments determined by different dispersion for Andreev bound states as a function of the phase difference.
Recently there is an increasing interest in topological quantum computation based on Majorana Bound States (MBS's) [1,2]. Majorana Fermions have been predicted in a wide class of low-dimensional solid state devices. Many of these proposals make use of quasi one dimensional superconductors in contact with topological insulators [3] or quasi one-dimensional materials with strong spin orbit interactions [4][5][6][7]. Also helical magnets [8] and other materials [9][10][11][12][13] are considered. In this paper we propose a quite distinctive heterostructure to observe topologically protected MBS's in a solid state device. Our work rests on the physics of S/R/S hybrid structures in which "R" is a quasi one dimensional semiconductor nanowire (NW) with strong Rashba spin orbit coupling (e.g. InAs or InSb) electrically connected to two conventional low T c superconductor leads ( "S" ) [4,5]. Superconductivity is induced in the spin-orbit coupled semiconductor by proximity effect due to the superconducting electrodes. The coexistence of superconductivity and spin-orbit coupling is a key ingredient for the existence of MBS's at the interfaces between the R region and the superconducting S regions.
However, despite the considerable theoretical and experimental [14] efforts, some challenges still remain before a real device allowing isolation and manipulation of MBS's in such geometry can be realized. In particular the difficulties of tuning the chemical potential of the semiconductor region µ, controlling the disorder on the bulk gap as well as optimizing the coupling between the different materials [15][16][17] make the realization of such devices extremely difficult.
All schemes proposed up to now to generate MBS's substantially use conventional s-wave superconductors to induce superconductivity and a gap ∆ in the R nanowire [1]. The role of superconducting pairing is to relax number conservation, thus allowing for the mixing of particle and hole degrees of freedom. Zeeman spin splitting is required to halve the number of degrees of freedom at low energies, thus generating the elusive neutral (Ma-jorana) excitation. A simple criterion to induce MBS's at S/R interfaces is given in terms of the applied magnetic field B x oriented along the wire, µ and ∆. The inequality to be satisfied can be stated as: B 2
x > µ 2 + ∆ 2 [4]. Low critical magnetic fields (H c ) and low gap values characteristic of conventional low Tc superconductors substantially define the limits of the nominal range of dynamical parameters required to observe MBS's. Not only do H c and ∆ enter into the criteria to stabilize MBS's, but they also endanger the feasibility of the experiment in case high magnetic fields are required. High critical temperature superconductors (HTS) may favor a completely different approach to experiments on MBS's, since HTS plaquettes/contacts (even a few micron square) sustain superconductivity up to a few tenths of Tesla and induce robust superconductivity in a wide range of barrier materials. When conventional low-Tc superconductors (e.g. Nb) are considered, the large difference in the g-factors for Nb (g N b ∼ 1) and InAs (g InAs ∼ 35) implies that the in-plane magnetic field B ∼ 0.1T can open a sizable Zeeman gap in InAs (V x ≤ 1K) without substantially suppressing the superconductivity in Nb. However these conditions holds even more firmly in YBCO contacts because the YBCO gap is very stable w.r.t. magnetic fields, despite a doubling of the g−factor (g Y BCO ∼ 2).
The induced gap in the NW can be considered of the order of the bare gap of the superconductor, projected along the wire direction, provided that the radius of the wire is negligibly small with respect to the coherence length ξ of the superconducting material and that no sizeable barriers are present at the interfaces. As recently pointed out in Ref. [15], the interface tunnelling between different materials renormalizes the induced gap
to∆ i = (1 − Z)∆ i where Z ∼ (1 + πρ 0 |V hop | 2 /∆ i ) −1 is the quasiparticle weight, V hop
mimicks the electron hopping between the superconductor and the NW and ρ 0 is the density of states of the superconductor at the Fermi energy. The better is the coupling with a larger V hop , the smaller becomes Z and the larger is the induced gap. Z also renormalizes the whole NW Hamiltonian H N W →H N W = ZH N W which means that, by the same token, all the NW Hamiltonian parameters are effectively reduced when V hop increases. When taking the renormalization into account in the model that we discuss below, the criterion for the appearance of the topologically non trivial phase given by Eq.5 becomes
Z 2 (B 2 − µ 2 ) > (1 − Z) 2 × max(|∆ L | 2 , |∆ R | 2 ) .(1)
This renormalization effect requires caution in the nanostrucure design and, interestingly enough, it can be fruitfully exploited in the case of HTS proximity. A convenient tradeoff can be found by accepting a rather poor intermaterial coupling V hop , due to the very large bare superconducting gap along the lobe direction, which is almost one order of magnitude larger than in conventional low Tc superconductors. The InAs nanowire mostly rules the scaling of the proximity effect [18,20] once good interface conditions are guaranteed between the HTS material and the InAs nanowire [21]. The magnetic field can be very high with negligible effects both on the superconducting properties of the HTS electrode and on the interface transparency.
Here we focus on other functionalities of HTS hybrid devices which are offered by an anisotropic d-wave order parameter symmetry [19]. In d-wave systems lobes in the excitation gap of amplitude 20 meV coexist with nodes, while in conventional s-wave superconductors the gap value is about or less than 1 meV and uniform in all directions. In HTS contacts, the crystal axes' orientations with respect to the nanowire can be chosen in order to maximize the proximity induced ∆. Different crystal orientations can be currently achieved by bicrystal or biepitaxial techniques [20].
For sake of simplicity, we model the system as an effective one-dimensional device composed by the NW of length L N and two superconducting regions (see Fig.1b), whose effective gaps differ not only in phase but also in their modulus, depending on the relative crystal orientation (see Fig.2). Effective Hamiltonian parameters, including the interface renormalization, will be disregarded, for the moment, and reintroduced in a second step.
In the superconducting regions of the nanowire, spinorbit interaction and superconductivity coexist. We assume that L N << ξ << L T OT to have penetration of superconductivity in the whole nanowire.
A Bogoliubov-De Gennes mean field Hamiltonian fully accounts for superconductivity induced in the normal material by proximity effect:
H S = (H 0 − µ N ) + ∞ −∞ dx ∆(x)ψ † ↑ (x)ψ † ↓ (x) + h.c. , H 0 − µ N = ∞ −∞ dx ψ † α (x) − ∂ 2 x 2m * − µ I 2 + iη σ y ∂ x + B x σ x αβ ψ β (x)(2)
where x is the coordinate along the wire and α, β =↑, ↓ denote the two components of the electronic fermionic fields. m * and η are the effective mass and the Rashba spin orbit coupling strength, respectively. B x = gµ B B/2 is the effective Zeeman spin splitting energy. It is assumed that the magnetic field, chosen in the direction of the wire, does not induce any undesired orbital effect . The d x 2 −y 2 superconductivity pairing is modeled as [28]:
∆(x) = ∆ L = ∆ 0 cos(2(ϑ − α L )) f or x < −L N /2 , 0 f or − L N /2 ≤ x ≤ L N /2 , ∆ R = ∆ 0 e −iφ cos(2(ϑ − α R )) f or x > L N /2 ,(3)
Angles α R,L , ϑ are defined in Fig.2. φ is an U (1) phase difference across the junction. Let us choose ϑ zero. Depending on the relative orientation of the order parameters in the L,R regions with respect to the nanowire, a wealth of possibilities occur. For an effectively onedimensional wire we can set α L = 0 with no loss of generality. By rotating α R from 0 to π/2 we can continuously explore all the configurations from lobe-lobe (+/+) to lobe-antilobe(+/-). Nodal configurations are not interesting here, as we need large superconducting gaps. As we are searching for MBS's, we will choose only a few angle configurations to demonstrate the main concepts. The Hamiltonian operator in H S can be recast in the compact form, in the basisψ(
x) = [ψ ↑ (x), ψ ↓ (x), ψ ↓ † (x), −ψ ↑ † (x)]
: for i = 0 and the I 2 identity matrix for i = 0. They refer to the Nambu and spin degrees of freedom, respectively. A new space scale x → η m * x has been introduced, as well as energy scale:
H S /η = − 1 2 ∂ 2 x − µ σ 0 + i∂ x σ y τ z +B x σ x τ 0 +∆(x)τ x .(4)µ, B x , ∆ → µ, B x , ∆ /(m * η 2 ).
The Hamiltonian of Eq.4 has a topologically non-trivial phase whose boundary states are Majorana Fermions [4,5], provided
B 2 x − µ 2 > max(|∆ L | 2 , |∆ R | 2 ) ,(5)
where the hamiltonian parameters used here, effectively include the quasiparticle renormalization weight Z. The topologically trivial phase, is adiabatically deformable to the usual Andreev physics [22]. We calculate numerically the low lying part of the energy spectrum by matching the eigenfunctions. In order to simplify calculations, we take the limit L N ∼ 0 by matching the wavefunction and its derivative at x = 0. As shown in [4], this assumption does not alter the generality of our results as interaction terms among Majorana end states are neglected in our approach. The effects of a finite size wire are shown for example in Ref. [23]. In Fig. 3 the dispersion relation of MBS's is shown as a function of the phase difference, φ, between the superconducting pads. For α R < π/4 the Andreev levels show a single crossing at φ = π. The odd number of crossings in the Andreev spectrum is the characteristic signature of the topological non trivial phase, consistent with that found with conventional s-wave superconductivity. However, the Andreev spectrum shows an unexpected behavior when α R > π/4 i.e. when the effective induced gaps have opposite signs. In this case, the crossing, which features the zero energy MBS, is still present, but located at φ = 0. This is specific of the d-wave order parameter. When α R < π/4 the gaps ∆ L and ∆ R have the same sign. Therefore, a phase difference of π between the two order parameters is required, in order to have an inversion of the sign of the gap between the two regions S 1 and S 2 . Provided that the appropriate condition for the parameters is met, the sign inversion, irrespective of the relative strength of the two gaps (and of the actual value of α R ), enforces the crossing to be localized at φ = 0, and the Majorana excitation with it. Together with this change, the shape of the dispersion relation changes by changing α R , with an increase of the current I(φ) = ∂E(φ)/∂φ up to a maximum, when the gaps reach their maximum at α R = 0 or π/2. The crossing only appears at φ = 0, π, because only at these points the Hamiltonian is real. Moreover, depending on the crystal relative arrangements we can have a different dispersion for Andreev Bound states as a function of the phase difference φ. In both cases a single crossing at zero energy appears, which reveals the presence of the MBS, at φ = 0 or π depending on the sign of the product ∆ L ∆ R .
At present, the race to detect signatures of the elusive Majorana Fermions in an S/NW/S structure is quite exciting [24][25][26][27]. A system exploiting d-wave electrodes, as the one proposed in this work, can inspire hallmark experiments in the search for Majorana excitations.
Despite the fact that the magnetic field should dominate over the superconductivity, still, a sizable superconducting gap is needed, as the smaller energy between V z and ∆ sets the minimum energy sufficient to wash out the topological protection of the Majorana excitation. In this respect, HTS's appear to offer more chances in stabilizing MBS's. Question arises whether nodal quasiparticles in the d-wave [28] topologically trivial superconductor could affect their stability. As the MBS's imply strong non local correlations, one is inclined to conclude that local nodal quasiparticle should be inefficient in producing a decay of the Majorana zero energy excitation. Besides, nodal quasiparticles are strictly at zero energy if travelling along given directions in an uniform system d-wave ordering. The presence of the Josephson barrier, inhomogeinity or confined geometry, should move those states to finite energy. In the different context of YBCO grain boundary Josephson Junctions [29,30], we have experienced a long lasting quantum coherence of antinodal quasiparticles, while an efficient relaxation mechanism could have been the production of nodal quasiparticles [30,31]. This is a conforting piece of evidence, but, of course, not a proof, though.
A d-wave induced superconductivity encompasses a wider range of opportunities to discriminate the presence of the MBS. Andreev states induced by d-wave pairing are strongly sensitive to the geometry of the device. The characteristic increase of the I c at the lower temperatures, used as a benchmark for the existence of the Andreev midgap state in HTS junctions [19], is strongly suppressed when the width of the junction is reduced toward a quasi 1D device. However, in our devices, an anomalous increase of I c at low temperatures would persist in the one-dimensional limit and would be even sharper, the lower the barrier transparency is. This would unambiguously signals feature that can be only correlated to the presence of Majorana fermions [32][33][34]. Prototype structures using HTS electrodes are being already tested [21].
Moreover, it is a distinctive property of ring structures with appropriate multi crystal arrangements to entail frustrated d-wave pairing ordering with trapped fractional fluxes in the ground state [28]. The possibility highlighted in this work, to have MBS localized at 0− and π− junctions, depending on the phase configuration, can be exploited in the design of quantum coherent, topologically protected devices, which go beyond the simple experimental confirmation of this amazing new physics, to enter the field of applications.
We envisage the possibility of engineering quasidegenerate odd fermionic parity states, using a mesoscopic, charge isolated island, formed by a d-wave tricrystal [32] topologically protected with respect to the excitations.
We acknowledge important discussions with D.
It is a tensor product of matrices τ i × σ j with {i, j} ∈ {0, 1, 2, 3}, where τ i and σ i are the usual Pauli matricesFIG. 1: a)Side view of the Superconductor-InAs nanowire-Superconductor heterostructure. b) Scheme of the structure used for the effective one-dimensional model.
FIG. 2 :
2The top view sketch for different geometries. Configurations of the order parameter are determined by a suitable orientation of the electrodes and of the nanowire.
Bercioux, P. Brouwer, M. Cuoco, P. Gentile, D. Urban, F. von Oppen. P.L. acknowledges F. Dalton for a critical proofreading of the manuscript. Financial support from FP7/2007-2013 under the grant N. 264098 -MAMA (Multifunctioned Advanced Materials and Nanoscale Phenomena), MIUR-Italy by Prin-project 2009 "Nanowire high critical temperature superconductor field-effect devices" , and European "SOLID" Project are gratefully acknowledged.
FIG. 3 :
3Energy spectrum of zero energy Majorana bound states in the case of equal (opposite) sign gaps (in top and bottom panel respectively).
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| []
|
[
"Exact Calculation of the Capacitance and the Electrostatic Potential Energy for a Nonlinear Parallel-Plate Capacitor in a Two-Parameter Modification of Born-Infeld Electrodynamics",
"Exact Calculation of the Capacitance and the Electrostatic Potential Energy for a Nonlinear Parallel-Plate Capacitor in a Two-Parameter Modification of Born-Infeld Electrodynamics"
]
| [
"S K Moayedi \nDepartment of Physics\nFaculty of Sciences\nArak University\n38156-8-8349ArakIran\n",
"F Fathi \nDepartment of Physics\nFaculty of Sciences\nArak University\n38156-8-8349ArakIran\n"
]
| [
"Department of Physics\nFaculty of Sciences\nArak University\n38156-8-8349ArakIran",
"Department of Physics\nFaculty of Sciences\nArak University\n38156-8-8349ArakIran"
]
| []
| The nonlinear capacitors are important devices in modern technologies and applied physics. The aim of this paper is to calculate exactly the capacitance and the electrostatic potential energy of a nonlinear parallelplate capacitor by using a two-parameter modification of Born-Infeld electrodynamics. Our calculations show that the capacitance and the electrostatic potential energy of a nonlinear parallel-plate capacitor in modified Born-Infeld theory have the weak field expansions Cwhere q is the amount of electric charge on each plate of the capacitor. It is demonstrated that the results of this paper are in agreement with the results of Maxwell electrodynamics for weak electric fields. Numerical evaluations show that the nonlinear electrodynamical effects in modified Born-Infeld theory are negligible in the weak field regime. | null | [
"https://arxiv.org/pdf/1704.03425v1.pdf"
]
| 119,433,094 | 1704.03425 | 41c0a00a247548efdd0e2bc7d02962eeecdd5030 |
Exact Calculation of the Capacitance and the Electrostatic Potential Energy for a Nonlinear Parallel-Plate Capacitor in a Two-Parameter Modification of Born-Infeld Electrodynamics
S K Moayedi
Department of Physics
Faculty of Sciences
Arak University
38156-8-8349ArakIran
F Fathi
Department of Physics
Faculty of Sciences
Arak University
38156-8-8349ArakIran
Exact Calculation of the Capacitance and the Electrostatic Potential Energy for a Nonlinear Parallel-Plate Capacitor in a Two-Parameter Modification of Born-Infeld Electrodynamics
arXiv:1704.03425v1 [physics.class-ph] 9 Apr 2017Classical field theoriesApplied classical electromagnetismOther special classical field theoriesNonlinear or nonlocal theories and modelsNonlinear Capacitor PACS: 0350-z, 0350De, 0350Kk, 1110Lm
The nonlinear capacitors are important devices in modern technologies and applied physics. The aim of this paper is to calculate exactly the capacitance and the electrostatic potential energy of a nonlinear parallelplate capacitor by using a two-parameter modification of Born-Infeld electrodynamics. Our calculations show that the capacitance and the electrostatic potential energy of a nonlinear parallel-plate capacitor in modified Born-Infeld theory have the weak field expansions Cwhere q is the amount of electric charge on each plate of the capacitor. It is demonstrated that the results of this paper are in agreement with the results of Maxwell electrodynamics for weak electric fields. Numerical evaluations show that the nonlinear electrodynamical effects in modified Born-Infeld theory are negligible in the weak field regime.
Introduction
The nonlinear capacitors have found a wide range of applications in circuit theory in electrical engineering and different branches of applied physics [1][2][3][4][5][6][7]. In a nonlinear capacitor, in contrast with the ordinary capacitors, the capacitance is a function of voltage, i.e., C = f (△φ) where △φ is the potential difference between the plates of the capacitor and the function f (△φ) can be determined empirically or from a fundamental electromagnetic theory like Maxwell electrodynamics or Born-Infeld theory [1,2,[8][9][10]. In Ref. [2], it has been shown that for a nonlinear capacitor which its capacitance depends linearly on the voltage, the usual relation U = 1 2 C(△φ) 2 is not satisfied. The authors of Ref. [6] have shown that the electrostatic potential energy of a nonlinear capacitor can be expanded as follows: U = αq 2 + βq 4 + γq 6 + ...,
where α, β, γ, and ... are material dependent constants. Today we know that the interaction between the charged bodies can be described by Maxwell equations classically [8]. On the other hand, Maxwell electrodynamics suffers from serious difficulties such as infinite self-energy of the point charges [8]. In 1934 Born and Infeld introduced the following Lagrangian density 1 [9]:
L BI = ǫ 0 β 2 1 − 1 + c 2 2β 2 F µν F µν − c 4 16β 4 (F µν ⋆ F µν ) 2 ,(2)
where F µν = ∂ µ A ν − ∂ ν A µ is the Faraday tensor, A µ = ( 1 c φ, A) is the gauge potential, ⋆F µν = 1 2 ǫ µναβ F αβ is the dual field tensor, and β is the maximum value of the electric field in Born-Infeld theory. In string theory the dynamics of electromagnetic fields on D-branes can be represented by a Born-Infeld type theory [10]. The solutions of Born-Infeld equations for an infinite charged line and an infinitely long cylinder have been obtained in Ref. [11]. In 1930s, Heisenberg, Euler, and Kockel studied the scattering of light by light according to Dirac's hole theory [12][13][14]. They showed that Maxwell electrodynamics should be corrected by adding nonlinear terms due to the quantum electrodynamical effects [12][13][14]. It must be emphasized that, for weak electromagnetic fields, the Lagrangian density and the energy density of nonlinear electrodynamics have the following explicit expressions:
L = ∞ i=0 ∞ j=0 c i,j F i G j ,(3)u = ∞ i=0 ∞ j=0 c i,j 2ǫ 0 iF (i−1) G j E 2 + (j − 1)F i G j ,(4)
where c i,j are field-independent parameters, F := ǫ 0 E 2 − B 2 µ0 , and G := ǫ0 µ0 (E.B) [15,16]. Note that in the weak field regime (2) is a particular example of (3). The effect of nonlinear corrections on the electric field between the plates of a parallel-plate capacitor has been studied in the framework of Heisenberg-Euler-Kockel electrostatics [17]. In a recent paper, the capacitance and the electrostatic potential energy for a parallel-plate and spherical capacitors have been computed in ordinary Born-Infeld theory [18]. Iacopini and Zavattini suggested and developed a (p, τ )-two-parameter modification of Born-Ifeld electrodynamics, in which the electrostatic self-energy of a point charge becomes a finite value for p < 1 [19]. The most important aim of this paper is to calculate the capacitance of a nonlinear parallel-plate capacitor from the viewpoint of Iacopini-Zavattini modification of Born-Infeld electrodynamics [19]. Another aim is to show that the nonlinear phenomena in electrodynamics are negligible for weak electric fields. This paper is organized as follows. In Section 2, the formulation of modified Born-Infeld electrodynamics coupled to an external current is presented. In Section 3, the symmetric energy-momentum tensor for modified Born-Infeld electrodynamics is constructed from the canonical energy-momentum tensor by using Belinfante's procedure. In Section 4, we show that the capacitance and the electrostatic potential energy for a nonlinear parallel-plate capacitor in modified Iacopini-Zavattini electrodynamics can be calculated exactly when the parameter p takes the values { 1 2 , 2 3 , 3 4 , 5 6 }. It is verified that the results of Section 4 for p = 1 2 are compatible with those obtained previously in [18]. Numerical evaluations in summary and conclusions indicate that the nonlinear corrections to the electrostatic potential energy of a parallel-plate capacitor in modified Born-Infeld electrostatics are not important in the weak field regime. 1 We use SI units in this paper. The space-time metric has the signature (+, −, −, −).
A Brief Review of Modified Born-Infeld Electrodynamics
The modified Born-Infeld electrodynamics in a (3+1)-dimensional Minkowski space-time is described by the following Lagrangian density (see Eq. (B.10) in Ref. [19]):
L p,τ = 1 2p ǫ 0 β 2 1 − 1 + c 2 2β 2 F µν F µν − τ c 4 16β 4 F µν ⋆ F µν 2 p − J µ A µ ,(5)
where p < 1 is a real dimensionless constant, τ is another dimensionless constant, and J µ = (cρ, J) is an external current for the U (1) gauge field A µ . The parameter β in Eq. (5) is called the Born-Infeld parameter and shows the upper limit of the electric field in modified Born-Infeld electrodynamics. It is necessary to note that for p = 1 2 , τ = 1 the Lagrangian density in Eq. (5) becomes the standard Born-Infeld Lagrangian density [9], while for p = 1, τ = 0 we obtain the Maxwell Lagrangian density. The equation of motion for the vector field A λ is
∂L p,τ ∂A λ − ∂ σ ∂L p,τ ∂(∂ σ A λ ) = 0.(6)
If we put Eq. (5) into Eq. (6), we will get the inhomogeneous modified Born-Infeld equations as follows:
∂ σ F σλ − τ c 2 4β 2 F µν ⋆ F µν ⋆ F σλ 1 + c 2 2β 2 F µν F µν − τ c 4 16β 4 F µν ⋆ F µν 2 1−p = µ 0 J λ .(7)
The dual field tensor ⋆F µν satisfies the following Bianchi identity:
∂ µ ⋆ F µν = 0.(8)
In (3+1)-dimensional space-time, the Faraday 2-form F and its dual ⋆F have the following expressions [20]:
F = F 0i dx 0 ∧ dx i + 1 2 F ij dx i ∧ dx j = E i dt ∧ dx i − 1 2 ǫ ijk B i dx j ∧ dx k ,(9)⋆F = −B i dt ∧ dx i − 1 2 ǫ ijk E i dx j ∧ dx k ,(10)
where i, j, k = 1, 2, 3, and
{E i } = {E x , E y , E z }, {B i } = {B x , B y , B z }.
Using Eqs. (9) and (10), Eqs. (7) and (8) take the following vector forms:
∇ · D(x, t) = ρ(x, t),(11)∇ × H(x, t) = J(x, t) + ∂D(x, t) ∂t ,(12)∇ · B(x, t) = 0,(13)∇ × E(x, t) = − ∂B(x, t) ∂t ,(14)
where D(x, t) and H(x, t) are given by
D(x, t) = ǫ 0 E(x, t) + τ c 2 β 2 E(x, t).B(x, t) B(x, t) Ω p,τ 1 p −1 ,(15)H(x, t) = 1 µ 0 B(x, t) − τ 1 β 2 E(x, t).B(x, t) E(x, t) Ω p,τ 1 p −1 ,(16)
and Ω p,τ is defined as follows:
Ω p,τ := 1 + c 2 2β 2 F µν F µν − τ c 4 16β 4 F µν ⋆ F µν 2 p .(17)
Now, let us study the electrostatic case where B = J = 0 and all other physical quantities are time independent.
In this case the modified Born-Infeld equations (11)-(14) are
∇ · E(x) [1 − E 2 (x) β 2 ] 1−p = ρ(x) ǫ 0 ,(18)∇ × E(x) = 0.(19)
The above equations are basic equations of modified Born-Infeld electrostatics [19]. By using the divergence theorem in vector calculus, we get the integral form of Eq. (18) as follows:
C 2 1 [1 − E 2 (x) β 2 ] 1−p E(x).n da = 1 ǫ 0 C 3 ρ(x)d 3 x,(20)
where C 2 is a 2-chain which is the boundary of a 3-chain C 3 , i.e., C 2 = ∂C 3 [20]. Equation (20) is Gauss's law in modified Born-Infeld electrostatics.
The Symmetric Energy-Momentum Tensor for Modified Born-Infeld Electrodynamics
In this section, we obtain the symmetric energy-momentum tensor for modified Born-Infeld electrodynamics. According to Eq. (5), the Lagrangian density for modified Born-Infeld electrodynamics in the absence of external current J µ is
L p,τ = 1 2p ǫ 0 β 2 1 − 1 + c 2 2β 2 F µν F µν − τ c 4 16β 4 F µν ⋆ F µν 2 p .(21)
From (21), we derive the following classical field equation:
∂ σ F σλ − τ c 2 4β 2 F µν ⋆ F µν ⋆ F σλ Ω p,τ 1 p −1 = 0.(22)
The canonical energy-momentum tensor for Eq. (21) is [21][22][23]
Θ σ η = ∂L p,τ ∂(∂ σ A λ ) (∂ η A λ ) − δ σ η L p,τ .(23)
If we substitute (21) into (23) and use (22), we will obtain the following expression for the canonical energymomentum tensor Θ σ η :
Θ σ η = 1 µ 0 F σλ − τ c 2 4β 2 F µν ⋆ F µν ⋆ F σλ Ω p,τ 1 p −1 F λη + 1 2p ǫ 0 β 2 (Ω p,τ − 1) δ σ η + ∂ λ R λσ η ,(24)
where
R λσ η := 1 µ 0 F λσ − τ c 2 4β 2 F µν ⋆ F µν ⋆ F λσ Ω p,τ 1 p −1 A η ,(25)R σλ η = −R λσ η .(26)
It is well known that the canonical energy-momentum tensor Θ σ η in (23) is generally not symmetric [21][22][23][24]. Belinfante showed that the canonical energy-momentum tensor Θ σ η in Eq. (23) can be written as follows [21]:
Θ σ η = T σ η + ∂ λ R λσ η ,(27)
where the second-and third-order tensors T σ η and R λσ η must satisfy the following conditions:
T ση = T ησ ,(28a)R λσ η = −R σλ η .(28b)
The second-order tensor T ση in the above equations is called the symmetric energy-momentum tensor [22]. A comparison between Eqs. (24) and (27) clearly shows that the symmetric energy-momentum tensor for modified Born-Infeld electrodynamics is
T σ η = 1 µ 0 F σλ − τ c 2 4β 2 F µν ⋆ F µν ⋆ F σλ Ω p,τ 1 p −1 F λη + 1 2p ǫ 0 β 2 (Ω p,τ − 1) δ σ η .(29)
After straightforward but tedious calculations, one finds that in the presence of an external current the symmetric energy-momentum tensor T σ η in (29) satisfies the following equation
: 2 ∂ σ T σ η = J σ F ση .(30)
Using Eqs. (9) and (10) together with Eq. (29), the energy density of modified Born-Infeld electrodynamics is given by
u(x, t) = T 0 0 (x, t) = 1 2p ǫ 0 β 2 (2p − 1) E 2 (x,t) β 2 + τ E(x,t).c B(x,t) 2 β 4 + c 2 B 2 (x,t) β 2 + 1 1 − E 2 (x,t)−c 2 B 2 (x,t) β 2 − τ E(x,t).c B(x,t) 2 β 4 1−p − 1 .(31)
According to Eq. (31), the energy density of an electrostatic field in modified Born-Infeld electrodynamics becomes
u(x) = T 0 0 (x) = 1 2p ǫ 0 β 2 (2p − 1) E 2 (x) β 2 + 1 1 − E 2 (x) β 2 1−p − 1 .(32)
For p = 1 2 , the modified electrostatic energy density in Eq. (32) becomes the energy density of an electrostatic field in Born-Infeld electrodynamics, i.e.,
u(x) = ǫ 0 β 2 1 1 − E 2 (x) β 2 − 1 .(33)
Calculation of the Capacitance and the Electrostatic Potential Energy of a Nonlinear Parallel-Plate Capacitor in Modified Born-Infeld Theory
In order to calculate the capacitance of a nonlinear parallel-plate capacitor in modified Born-Infeld electrostatics, we assume a capacitor composed of two large parallel conducting plates with area A and separation d (see Figure 1). By applying modified Gauss's law in (20) to the Gaussian surface in Figure 1, we obtain the following equation
for E z E z Γ + q β 2 Γ ǫ 0 A Γ E z 2 − q ǫ 0 A Γ = 0,(34)
where Γ := 1 1−p . Now, let us obtain the exact solutions of Eq. (34) for p ∈ { 1 2 , 2 3 , 3 4 , 5 6 }. For p = 1 2 (Γ = 2), Eq. (34) becomes the quadratic equation
1 + q βǫ 0 A 2 E z 2 − q ǫ 0 A 2 = 0.
For p = 2 3 (Γ = 3), Eq. (34) becomes the cubic equation
E z 3 + q β 2 3 ǫ 0 A 3 E z 2 − q ǫ 0 A 3 = 0.
For p = 3 4 (Γ = 4), Eq. (34) becomes the following quartic equation
E z 4 + q β 1 2 ǫ 0 A 4 E z 2 − q ǫ 0 A 4 = 0.
Finally, for p = 5 6 (Γ = 6), (34) becomes the following sextic equation
E z 6 + q β 1 3 ǫ 0 A 6 E z 2 − q ǫ 0 A 6 = 0.
Note that the above sextic equation by a suitable change of variable reduces to a cubic equation. In Galois theory, the Abel-Ruffini theorem or Abel's impossibility theorem says that: for all n ≥ 5, there is a polynomial in Q[x] of degree n that is not solvable by irreducible radicals over Q [25,26]. 3 According to Abel's impossibility theorem, equation (34) has the following exact solutions for
p ∈ { 1 2 , 2 3 , 3 4 , 5 6 } E z p= 1 2 = q ǫ 0 A Π p= 1 2 (q), (35a) E z p= 2 3 = q ǫ 0 A Π p= 2 3 (q), (35b) E z p= 3 4 = q ǫ 0 A Π p= 3 4 (q), (35c) E z p= 5 6 = q ǫ 0 A Π p= 5 6 (q), (35d) where Π p= 1 2 (q) := 1 1 + ( q βǫ0A ) 2 , (36a) Π p= 2 3 (q) := 3 1 2 − 1 27β 6 q ǫ 0 A 6 + 1 4 − 1 27β 6 q ǫ 0 A 6 + 3 1 2 − 1 27β 6 q ǫ 0 A 6 − 1 4 − 1 27β 6 q ǫ 0 A 6 − 1 3β 2 q ǫ 0 A 2 , (36b) Π p= 3 4 (q) := 1 + 1 4β 4 q ǫ 0 A 4 − 1 2β 2 q ǫ 0 A 2 ,(36c)
Π p= 5 6 (q) := 3 1 2
+ 1 4 + 1 27β 6 q ǫ 0 A 6 + 3 1 2 − 1 4 + 1 27β 6 q ǫ 0 A 6 .
(36d)
When the Born-Infeld parameter β takes the large values, the behavior of the electric fields in (35a)-(35d) are given by
E z p= 1 2 = q ǫ 0 A 1 − 1 2 β −2 q ǫ 0 A 2 + 3 8 β −4 q ǫ 0 A 4 + O β −6 , (37a) E z p= 2 3 = q ǫ 0 A 1 − 1 3 β −2 q ǫ 0 A 2 + 1 9 β −4 q ǫ 0 A 4 + O β −6 , (37b) E z p= 3 4 = q ǫ 0 A 1 − 1 4 β −2 q ǫ 0 A 2 + 1 32 β −4 q ǫ 0 A 4 + O β −6 , (37c) E z p= 5 6 = q ǫ 0 A 1 − 1 6 β −2 q ǫ 0 A 2 − 1 72 β −4 q ǫ 0 A 4 + O β −6 .(37d)
It must be emphasized that in obtaining the above results, the Mathematica software has been used [27]. All of the electric fields (37a)-(37d), have the Maxwellian limit in the weak field regime. The first term on the righthand side of (37a)-(37d) represent the electric field between the plates of a parallel-plate capacitor in Maxwell theory, while the higher-order terms represent the effect of nonlinear corrections. Using (19), it is obvious that we can write E(x) in the following way:
E(x) = −∇φ(x),(38)
where φ(x) is the electrostatic potential. From (38) we obtain the following relation:
△φ = − f i E(x).dl,(39)
where △φ = φ f − φ i is the potential difference between the initial and final points, and dl is an infinitesimal displacement vector. If we use Eqs. (35) and (39), we will get the following expressions for the potential difference between the plates of a nonlinear parallel-plate capacitor for As is well known, in electrostatics the capacitance C of a capacitor is the ratio of the amount of charge on each plate of a capacitor to the potential difference between the plates of the capacitor, i.e.,
p ∈ { 1 2 , 2 3 , 3 4 , 5 6 } △φ p= 1 2 = − + − q ǫ 0 A Π p= 1 2 (q)(−ê z ).(ê z dz) = qd ǫ 0 A Π p= 1 2 (q), (40a) △φ p= 2 3 = − + − q ǫ 0 A Π p= 2 3 (q)(−ê z ).(ê z dz) = qd ǫ 0 A Π p= 2 3 (q), (40b) △φ p= 3 4 = − + − q ǫ 0 A Π p= 3 4 (q)(−ê z ).(ê z dz) = qd ǫ 0 A Π p= 3 4 (q),(40c)C = q △φ .(41)
It is necessary to note that Eq. (41) is also applicable for determination of the capacitance of nonlinear capacitors [2,17,18]. After inserting (40) into (41), the capacitance of a nonlinear parallel-plate capacitor in modified Born-Infeld electrostatics for p ∈ { 1 2 , 2 3 , 3 4 , 5 6 } becomes
C p= 1 2 = ǫ 0 A d 1 Π p= 1 2 (q) ,(42a)C p= 2 3 = ǫ 0 A d 1 Π p= 2 3 (q) , (42b) C p= 3 4 = ǫ 0 A d 1 Π p= 3 4 (q) ,(42c)C p= 5 6 = ǫ 0 A d 1 Π p= 5 6 (q) .(42d)
The above equations have the following weak field expansions:
C p= 1 2 = ǫ 0 A d 1 + 1 2 β −2 q ǫ 0 A 2 − 1 8 β −4 q ǫ 0 A 4 + O β −6 ,(42e)C p= 2 3 = ǫ 0 A d 1 + 1 3 β −2 q ǫ 0 A 2 − 1 81 β −6 q ǫ 0 A 6 + O β −8 , (42f) C p= 3 4 = ǫ 0 A d 1 + 1 4 β −2 q ǫ 0 A 2 + 1 32 β −4 q ǫ 0 A 4 + O β −6 ,(42g)C p= 5 6 = ǫ 0 A d 1 + 1 6 β −2 q ǫ 0 A 2 + 1 24 β −4 q ǫ 0 A 4 + O β −6 .(42h)
Equations (42e)-(42h) show that the capacitance of a nonlinear parallel-plate capacitor in modified Born-Infeld electrostatics is a function of the amount of charge on each plate of the capacitor. Now, let us compute the energy density between the plates of a nonlinear parallel-plate capacitor in modified Born-Infeld electrostatics.
If we put (35) into (32), we will get the following results:
u(x) p= 1 2 = ǫ 0 β 2 1 Π p= 1 2 (q) − 1 , (43a) u(x) p= 2 3 = 3 4 ǫ 0 β 2 1 3β 2 q ǫ0A Π p= 2 3 (q) 2 + 1 Π p= 2 3 (q) − 1 , (43b) u(x) p= 3 4 = 2 3 ǫ 0 β 2 1 2β 2 q ǫ0A Π p= 3 4 (q) 2 + 1 Π p= 3 4 (q) − 1 ,(43c)
u(x) (43d)
Using Eq. (43), the electrostatic potential energy of a nonlinear parallel-plate capacitor in modified Born-Infeld electrostatics according to Figure 1 becomes
U p= 1 2 = area of a plate da d 0 dz u(x) p= 1 2 = ǫ 0 β 2 1 + ( q βǫ 0 A ) 2 − 1 Ad, (44a) U p= 2 3 = area of a plate da d 0 dz u(x) p= 2 3 = 3 4 ǫ 0 β 2 1 3β 2 q ǫ0A Π p= 2 3 (q) 2 + 1 Π p= 2 3 (q) − 1 Ad, (44b) U p= 3 4 = area of a plate da d 0 dz u(x) p= 3 4 = 2 3 ǫ 0 β 2 1 2β 2 q ǫ0A Π p= 3 4 (q) 2 + 1 Π p= 3 4 (q) − 1 Ad,(44c)
For the large values of the Born-Infeld parameter β, the behavior of the electrostatic potential energies in (44a)-(44d) are given by
U p= 1 2 | large β = U M 1 − 1 4 β −2 q ǫ 0 A 2 + 1 8 β −4 q ǫ 0 A 4 + O β −6 ,(45a)U p= 2 3 | large β = U M 1 − 1 6 β −2 q ǫ 0 A 2 + 1 27 β −4 q ǫ 0 A 4 + O β −6 ,(45b)U p= 3 4 | large β = U M 1 − 1 8 β −2 q ǫ 0 A 2 + 1 96 β −4 q ǫ 0 A 4 + O β −6 ,(45c)U p= 5 6 | large β = U M 1 − 1 12 β −2 q ǫ 0 A 2 − 1 216 β −4 q ǫ 0 A 4 + O β −6 ,(45d)
where U M = q 2 2C M and C M = ǫ0A d are the electrostatic potential energy and the capacitance of a parallel-plate capacitor in Maxwell electrostatics respectively. Equation (45) shows that the relation U M = q 2 2C M is not true for a parallel-plate capacitor in modified Born-Infeld electrostatics. In the limit of β → ∞, equations (45a)-(45d) reduce to the following equation:
Summary and Conclusions
In 1930s Born-Infeld theory was introduced in order to remove the infinite self-energy of the electron in Maxwell electrodynamics [9]. In Born-Infeld electrodynamics the absolute value of the electric field has an upper limit β, i.e., |E| ≤ β. In Born-Infeld paper the numerical value of β was [9,28]:
β Born−Inf eld = 1.2 × 10 20 V m .(47)
It must be noted that in (53d)-(53g) the minimum value of β in Eqs. (48) and (49) has been used. Equations (53a)-(53g) tell us that the nonlinear corrections to electrostatic potential energy in a parallel-plate capacitor are not important in the weak field limit. For p = 7 8 (Γ = 8), Eq. (34) becomes
E z 8 + q β 1 4 ǫ 0 A 8 E z 2 − q ǫ 0 A 8 = 0.(54)
Equation (54) is an eight-order equation which by a suitable change of variable reduces to a quartic equation.
In future studies we want to obtain the exact solutions of (54) in order to calculate the capacitance and the electrostatic potential energy of a nonlinear parallel-plate capacitor in modified Born-Infeld electrostatics for p = 7 8 . More recently, a new modification of Born-Infeld electrodynamics has been developed which includes three independent parameters [32]. We hope to study the problems discussed in our work from the viewpoint of [32] in an independent research.
Figure 1 :
1A parallel-plate capacitor. The Gaussian surface is represented by dashed lines. The symmetry of the problem implies that E(x) = Ez(−êz), whereêz is the unit vector in the z-direction.
6
| β=∞ = U M .
It is obvious that for source-free modified Born-Infeld theory the right-hand side of (30) vanishes, i.e., ∂σ T σ η = 0.
Q is the field of rational numbers (see page 116 in Ref.[26]).
Soff, Rafelski, and Greiner obtained the following lower bound on β[29]:In a paper about photon-photon scattering and photon splitting in a magnetic field in Born-Infeld theory, Davila and his coworkers obtained the following new lower bound on β[30]:In 1983, E. Iacopini and E. Zavattini introduced a (p, τ )-two-parameter modification of Born-Infeld electrodynamics, in which the self-energy of a point-like charge becomes finite for p < 1[19]. In our paper, after a brief formulation of Born-Infeld-Iacopini-Zavattini electrodynamics (modified Born-Infeld electrodynamics) in the presence of an external current, the capacitance and the electrostatic potential energy of a nonlinear parallel-plate capacitor have been calculated exactly in the framework of modified Born-Infeld electrostatics for p ∈ { 1 2 , 2 3 , 3 4 , 5 6 }. In order to have a deeper understanding of nonlinear effects in modified Born-Infeld electrostatics, let us rewrite (45a) as follows:
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"\nBOUNDS FOR SMOOTH FANO WEIGHTED COMPLETE INTERSECTIONS VICTOR PRZYJALKOWSKI AND CONSTANTIN SHRAMOV\n\n"
]
| [
"BOUNDS FOR SMOOTH FANO WEIGHTED COMPLETE INTERSECTIONS VICTOR PRZYJALKOWSKI AND CONSTANTIN SHRAMOV\n"
]
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| We prove that if a smooth variety with non-positive canonical class can be embedded into a weighted projective space of dimension n as a well formed complete intersection and it is not an intersection with a linear cone therein, then the weights of the weighted projective space do not exceed n + 1. Based on this bound we classify all smooth Fano complete intersections of dimensions 4 and 5, and compute their invariants. | 10.4310/cntp.2020.v14.n3.a3 | [
"https://arxiv.org/pdf/1611.09556v2.pdf"
]
| 119,133,280 | 1611.09556 | 2c7251b57526819720490d0f289b4a904f95bbba |
29 Nov 2016
BOUNDS FOR SMOOTH FANO WEIGHTED COMPLETE INTERSECTIONS VICTOR PRZYJALKOWSKI AND CONSTANTIN SHRAMOV
29 Nov 2016
We prove that if a smooth variety with non-positive canonical class can be embedded into a weighted projective space of dimension n as a well formed complete intersection and it is not an intersection with a linear cone therein, then the weights of the weighted projective space do not exceed n + 1. Based on this bound we classify all smooth Fano complete intersections of dimensions 4 and 5, and compute their invariants.
Introduction
Fano varieties are one of the important classes of algebraic varieties, both from birational and biregular points of view. It is known that smooth Fano varieties of a given dimension are bounded, see [KMM92,Theorem 0.2], so that one can hope for their explicit classification (actually this is known also for ε-log terminal Fano varieties, see [Bi16], but in this case any kind of explicit classification is hardly possible even in dimension 3). The only smooth Fano curve is P 1 . Smooth Fano varieties of dimension 2 are known as del Pezzo surfaces, and they were classified long ago. Smooth Fano threefolds were classified by V. Iskovskikh (see [Is77], [Is78], or [IP99,§12]), and S. Mori and S. Mukai (see, [MM82] and [Mu88]). The most important and hard part of this classification concerns Fano varieties with Picard rank 1. In dimension 3 such varieties (at least if they are general in the corresponding deformation family) appear to be either complete intersections in a weighted projective space, or zero loci of sections of homogeneous vector bundles on Grassmannians. In dimensions 4 and higher no complete classification is known, and at the moment no reasonable approach to the problem is yet in sight. Still there are partial classification results, including the list of all smooth Fano fourfolds of index at least 2 or Picard rank greater than one and Fano varieties of high coindex (see [Fu80], [Fu81], [Fu84], [Mu84], [Wi87], [Mu89], [Wi90]), smooth Fano fourfolds that are zero loci of sections of homogeneous vector bundles on Grassmannians (see [Kü95], [Kü97,§4], [Ku16]), smooth Fano fourfolds that are weighted complete intersections (see [Kü97, Proposition 2.2.1]), and some other sporadic results (see [Kü97,§3]). The purpose of this paper is to study smooth Fano weighted complete intersections, and give effective numerical bounds that allow to classify them.
To be able to classify weighted complete intersections of a given dimension satisfying some nice properties, one needs an effective bound on the corresponding discrete parameters. In [CCC11, Theorem 1.3] such bound was obtained for codimension of a quasi-smooth (see Definition 2.4 below) weighted complete intersection. In [Ch15, Theorem 1.3] the degrees were bounded in terms of canonical volume and discrepancies.
We will be interested in the case of smooth Fano varieties that can be described as complete intersections in weighted projective spaces. Also, we will deal with the case when our weighted complete intersection is Fano or Calabi-Yau. In both cases by adjunction formula it is actually enough to bound the weights of the corresponding weighted projective space.
Let P = P(a 0 , . . . , a n ) be a weighted projective space, and X ⊂ P be a weighted complete intersection of multidegree (d 1 , . . . , d c ) for some c 0. We will usually assume that X is not an intersection with a linear cone, i.e. one has d j = a i for all i and j, cf. Remark 2.8. Finally, it is convenient to assume that X is well formed, see Definition 2.3 and Theorem 2.9 below.
The main result of this paper is the following.
Theorem 1.1. Let X ⊂ P(a 0 , . . . , a n ), n 2, be a smooth well formed weighted complete intersection of multidegree (d 1 , . . . , d c ). Suppose that X is not an intersection with a linear cone. If X is Fano, then for every 0 i n one has a i n, and for every 1 j c one has d j n(n + 1). Similarly, if X is Calabi-Yau, then for every 0 i n one has a i n + 1, and for every 1 j c one has d j (n + 1) 2 .
To prove Theorem 1.1 we use the following approach. Exploiting smoothness assumption, we write down a bunch of necessary conditions on the parameters a i and d j , which appear to be inequalities (sometimes involving products of weights or degrees). On the other hand, Fano or Calabi-Yau condition implies an inequality between the sums of a i and d j . Then we treat all these inequalities as if a i and d j were arbitrary real numbers, and solve the corresponding optimization problem using the standard down-to-earth method of Lagrange multipliers.
The bounds given by Theorem 1.1 are sharp for an infinite set of dimensions, see Remark 3.2 below.
Using Theorem 1.1, we will give a classification of weighted Fano complete intersections of dimensions 4 and 5, see §5 below. Note that a classification of four-dimensional weighted Fano complete intersections was already obtained by O. Küchle in [Kü97, Proposition 2.2.1]; his method builds on a classification of weighted homogeneous polarized Calabi-Yau complete intersections, see [Og91,Main Theorem II]. In a way this is closer to the methods that were classically used in a classification of Fano threefolds, but one can hardly expect that it can be easily generalized to higher dimensions.
The plan of the paper is as follows. In §2 we recall some basic properties of weighted complete intersections. In §3 we prove Theorem 1.1. In §4 we explain the (well known) method that can be used to compute Hodge numbers of smooth weighted complete intersections. In §5 we provide a classification of smooth Fano weighted complete intersections of dimensions 4 and 5. Finally, in Appendix A we collect some (nearly elementary) auxiliary material used in §3.
Notation and conventions. All varieties are compact and are defined over the field of complex numbers C. For a bigraded ring R we denote its (p, q)-component by R (p,q) . For a weighted complete intersection X in P(a 0 , . . . , a n ) of multidegree (d 1 , . . . , d c ) the number a i − d j is denoted by I(X).
Smoothness
We recall here some basic properties of weighted complete intersections. We refer the reader to [Do82] and [IF00] for more details. Put P = P(a 0 , . . . , a n ) = ProjC[x 0 , . . . , x n ], where the weight of x i equals a i .
Definition 2.1 (see [IF00,Definition 5.11]). The weighted projective space P is said to be well formed if the greatest common divisor of any n of the weights a i is 1.
Any weighted projective space is isomorphic to a well formed one, see [Do82,1.3.1].
Lemma 2.2 (see [IF00,5.15]). The singular locus of P is a union of strata
Λ J = {(x 0 : . . . : x n ) | x j = 0 for all j / ∈ J}
for all subsets J ⊂ {0, . . . , n} such that the greatest common divisor of the weights a j for j ∈ J is greater than 1.
Definition 2.3 (see [IF00, Definition 6.9]). A subvariety X ⊂ P of codimension c is said to be well formed if P is well formed and codim X (X ∩ Sing P) 2.
The following notion is a replacement of smoothness suitable for subvarieties of weighted projective spaces.
Definition 2.4 (see [IF00, Definition 6.3]). Let p : A n+1 \ {0} → P be the natural projection. A subvariety X ⊂ P is said to be quasi-smooth if p −1 (X) is smooth.
We say that a variety X ⊂ P of codimension c is a weighted complete intersection of multidegree (d 1 , . . . , d c ) if its weighted homogeneous ideal in C[x 0 , . . . , x n ] is generated by a regular sequence of c homogeneous elements of degrees d 1 , . . . , d c . Note that in general a weighted complete intersection is not even locally a complete intersection in the usual sense.
Remark 2.5. It is possible that a weighted complete intersection of a given multidegree in P does not exist, even if c is small. For example, there is no such thing as a hypersurface of degree d < min(a 0 , . . . , a n ) in P, or a weighted complete intersection of multidegree (2, 2) in P(1, 3, 4, 5).
Singularities of quasi-smooth well formed weighted complete intersections can be easily described.
Proposition 2.6 (see [Di86,Proposition 8]). Let X ⊂ P be a quasi-smooth well formed weighted complete intersection. Then the singular locus of X is the intersection of X with the singular locus of P.
Remark 2.7. Note that the definition of "general position" in [Di86] coincides with our definition of well formedness.
Recall that the weighted complete intersection X is said to be an intersection with a linear cone if one has d j = a i for some i and j.
Remark 2.8. If this condition fails, one can exclude the i-th weighted homogeneous coordinate and think about X as a weighted complete intersection in a weighted projective space of lower dimension, provided that X is general enough, cf. Remark 5.2 below. Note however that in general this new weighted projective space may fail to be well formed, and the new weighted complete intersection may fail to be nice in other ways as well.
It appears that the assumptions that the complete intersection is well formed, quasismooth, and is not an intersection with a linear cone are not always independent. In principle it can allow us to drop some of the assumptions in the rest of the paper, but we will refrain from doing so to keep the assertions more explicit.
Theorem 2.9 (see [IF00, Theorem 6.17]). Suppose that the weighted projective space P is well formed. Then any quasi-smooth complete intersection of dimension at least 3 in P is either an intersection with a linear cone or well formed.
There is the following version of the adjunction formula that holds for quasi-smooth well formed weighted complete intersections.
Lemma 2.10 (see [Do82,Theorem 3.3.4], [IF00, 6.14]). Let X ⊂ P be a quasi-smooth well formed weighted complete intersection of multidegree (d 1 , . . . , d c ).
Then
ω X = O X d i − a j .
Along with quasi-smooth weighted complete intersections one may consider those weighted complete intersections that are smooth in the usual sense.
Proposition 2.11. Let X ⊂ P be a smooth well formed weighted complete intersection. Then X does not pass through singular points of P.
Proof. Suppose that X contains a singular point P of P. Let U ⊂ P be an affine neighborhood of P , and π :Ũ → U be its natural finite cover (see [IF00,5.3]), so thatŨ is isomorphic to an open subset of A n , and π is a quotient by a group Z/rZ for some r > 1. Put V = X ∩U. Let Σ be the singular locus of U. Since X is well formed, the intersection of Σ with V has codimension at least 2 in V . LetṼ be the preimage of V with respect to π, and let π V :Ṽ → V be the corresponding finite cover. ThenṼ is a complete intersection in U. Note also that π V isétale outside of Σ, and thusṼ is smooth in codimension 1. In particular,Ṽ is Cohen-Macaulay, see [Ei95,§18.5].
Put V o = V \ Σ, and letṼ o be the preimage of V o with respect to π V . Since V is smooth and dim V > 1, there exists a simply connected punctured neighborhood of P in V , and hence there also exists a simply connected punctured neighborhood of P in V o . Thus the morphism π V isétale in some punctured neighborhoodṼ of the point π −1 (P ) inṼ o whose image with respect to π is simply connected. The complex spaceṼ splits into a union of its connected componentsṼ 1 , . . . ,Ṽ r . Thus in a neighborhood of the point π −1 (P ) the varietyṼ splits into a union of its irreducible componentsṼ 1 , . . . ,Ṽ r . The irreducible componentṼ 1 intersects all other irreducible components (in particular) at the point π −1 (P ), so thatṼ is connected. SinceṼ is Cohen-Macaulay, by [Ei95,Theorem 18.12] there is an irreducible componentṼ k such that the intersection Z =Ṽ 1 ∩Ṽ k has codimension 1 inṼ 1 . The varietyṼ is singular along Z. Since π V isétale at a general point of Z, we conclude that V is singular at a general point of π V (Z), which is a contradiction.
Remark 2.12. A. Kuznetsov pointed out that there is an alternative proof of Proposition 2.11 that is purely algebraic and does not depend on the base field. Namely, in the notation of our proof of Proposition 2.11 the varietyṼ is normal (since it is a locally complete intersection smooth in codimension 1). Since V is smooth by assumption, the branch locus R of π V has codimension 1 in V by the purity of the branch locus, see [Au62,Theorem 1.4]. Now we can obtain a contradiction as above. Still we prefer to keep the original proof of Proposition 2.11, since we believe that it makes the geometric reason explaining why this property holds more evident, and our base field is C anyway.
Remark 2.13. The assertion of Proposition 2.11 fails without the assumption that X is well formed. Indeed, suppose that a 0 = a 1 = 1 and a 2 = . . . = a n = 2; put c = 1 and d 1 = 2. Then X is not well formed, but it is smooth since it is isomorphic to P n−1 . However, it passes through singular points of P. For n = 2 this example gives a line on a usual quadratic cone.
As an application of Proposition 2.11 one can show that being smooth is a stronger condition than being quasi-smooth, provided that we work with well formed weighted complete intersections.
Corollary 2.14. Let X ⊂ P be a smooth well formed weighted complete intersection. Then X is quasi-smooth.
Proof. The morphism p : A n+1 \ {0} → P is a locally trivial C * -bundle over the nonsingular part U of P, while X is contained in U by Proposition 2.11. Hence p is a locally trivial C * -bundle over X, and thus the preimage of X with respect to p is smooth, which means that X itself is quasi-smooth.
Another consequence of Proposition 2.11 is the following result.
Lemma 2.15 (cf. [IF00, 6.12]). Let X ⊂ P be a smooth well formed weighted complete intersection of multidegree (d 1 , . . . , d c ). Then for every k and every choice of k weights a i 1 , . . . , a i k , i 1 < . . . < i k , such that their greatest common divisor δ is greater than 1 there exist k degrees d s 1 , . . . , d s k , s 1 < . . . < s k , such that their greatest common divisor is divisible by δ.
Proof. Choose a positive integer k, and suppose that there are k weights a i 1 , . . . , a i k with i 1 < . . . < i k , such that their greatest common divisor δ is greater than 1. Let t be the number of degrees d j that are divisible by δ. Suppose that t < k. We claim that in this case X is singular. Indeed, let J be the set of indices j such that d j is divisible by δ, and let Λ be the subvariety in P given by equations x j = 0 for j ∈ J. Then for any j ′ ∈ J the polynomial F j ′ does not contain monomials that depend only on x j with j ∈ J. On the other hand, the equations F j = 0 for j ∈ J cut out a non-empty subset of Λ. Since δ > 1, the weighted projective space P is singular along Λ, see Lemma 2.2. This gives a contradiction with Proposition 2.11.
Remark 2.16. The condition provided by Lemma 2.15 is only necessary for the weighted complete intersection X to be smooth, but not sufficient. For example, assume that a 0 = . . . = a r = 1, while 2 < a r+1 . . . a n , and a j are pairwise coprime. Choose d 1 and d 2 so that d 2 is divisible by all a j , and 2 d 1 < a r+1 . Then a general weighted complete intersection X of multidegree (d 1 , d 2 ) in P is not smooth provided that r n − 2; moreover, it is reducible if r = 1, and non-reduced if r = 0. Another way how smoothness may fail is illustrated by an example of a weighted complete intersection X of multidegree (2, 30) in P when a 0 = . . . = a n−2 = 1, a n−1 = 6, and a n = 10; in this case we see that the assertion of Proposition 2.11 does not hold, so that X is singular. See also Lemma 3.1(i) below.
Weight bound
In this section we derive Theorem 1.1 from elementary results of Appendix A. The method we use here is somewhat similar to [CS13,§3].
Let X ⊂ P = P(a 0 , . . . , a n ), n 2, be a smooth well formed weighted complete intersection of multidegree (d 1 , . . . , d c ) that is not an intersection with a linear cone. We can assume that X is normalized, i.e. that inequalities a 0 . . . a n and d 1 . . . d c hold. Moreover, if c = 0, then one has X ∼ = P ∼ = P n , and there is nothing to prove; therefore, we will always assume that c 1.
We need an auxiliary result that is easy to establish and well known to experts.
Lemma 3.1. Let X ⊂ P = P(a 0 , . . . , a n ), n 2, be a smooth well formed normalized weighted complete intersection of multidegree (d 1 , . . . , d c ) that is not an intersection with a linear cone. Then the following assertions hold.
(i) One has d c−k > a n−k for all 1 k c − 1.
(ii) One has d c 2a n . (iii) The integer a 0 · . . . · a n divides the integer d 1 · . . . · d c .
Proof. Assertion (i) is given by [IF00, Lemma 18.14] and holds under a weaker assumption of quasi-smoothness. If a n = 1, then the remaining assertions of the lemma obviously hold, and thus we can assume that a n > 1. Let x 0 , . . . , x n be homogeneous coordinates on P of weights a 0 , . . . , a n , respectively, and let f 1 = . . . = f c = 0 be equations of X in P.
Suppose that d c < 2a n . Then none of f j contains a monomial x r n with non-zero coefficient if r 2. Also, since X is not an intersection with a linear cone, none of f j contains a monomial x n with non-zero coefficient either. Therefore, we see that every f j vanishes at the point P given by x 0 = . . . = x n−1 = 0, so that X passes through P . On the other hand, P is a singular point of P by Lemma 2.2 because a n > 1. Thus Proposition 2.6 implies that X is singular at P , which is a contradiction. This gives assertion (ii).
To prove assertion (iii), choose a prime number p, and denote by ν
(r)
p (a 0 , . . . , a n ) the number of the weights a i that are divisible by p r . Similarly, denote by ν
ν (r) p (a 0 , . . . , a n ) ν (r) p (d 1 , . . . , d c )
. This implies that the p-adic valuation of the integer a 0 · . . . · a n does not exceed the p-adic valuation of the integer d 1 · . . . · d c . Since this holds for an arbitrary prime p, we obtain assertion (iii). Now we are ready to prove Theorem 1.1.
Proof of Theorem 1.1. Put N = n+1. Denote A i+1 = a n−i for 0 i n, and D j = d c−j+1 for 1 j c. Then one has A 1 . . . A N and D 1 . . . D c . Moreover, by Lemma 3.1(i) one has D 2 > A 2 , . . . , D c > A c . By Lemma 3.1(ii) we also have D 1 2A 1 , and by Lemma 3.1(iii) we have
A 1 · . . . · A N D 1 · . . . · D c . Put L = a 0 + . . . + a n − d 1 − . . . − d c = A 1 + . . . + A N − D 1 − . . . − D c .
Then L > 0 provided that X is Fano, and L 0 provided that X is Calabi-Yau. This follows from Lemma 2.10.
Suppose that X is a Fano variety. Then N 2c + 1 by [CCC11, Theorem 1.3]. Therefore, Proposition A.11 implies that A 1 N − 1, which can be rewritten as a n n.
Now suppose that X is Calabi-Yau. Then a general weighted complete intersection of multidegree d 1 , . . . , d c in P(1, a 0 , . . . , a n ) is a smooth well formed Fano weighted complete intersection that is not an intersection with a linear cone. Thus one has a n n + 1.
Since X is normalized, we obtain similar inequalities for all other weights a i . Finally, the inequalities for the degrees d j follow from the fact that L is positive if X is Fano, and non-negative if X is Calabi-Yau. This completes the proof of Theorem 1.1.
Remark 3.2. The bounds for a n given by Theorem 1.1 are sharp for an infinite set of dimensions. Indeed, if n is odd, a 0 = . . . = a n−2 = 1, a n−1 = 2, and a n = n, then a general hypersurface of weighted degree 2n in P is a smooth well formed Fano weighted complete intersection. Similarly, if n is even, a 0 = . . . = a n−2 = 1, a n−1 = 2, and a n = n + 1, then a general hypersurface of weighted degree 2n + 2 in P is a smooth well formed Calabi-Yau weighted complete intersection.
Although Remark 3.2 shows that the bound for the maximal weight a n given by Theorem 1.1 is more or less sharp, there are stronger bounds for some other weights a i in certain cases.
Lemma 3.3. Let X ⊂ P = P(a 0 , . . . , a n ), n 2, be a smooth well formed normalized weighted complete intersection of multidegree (d 1 , . . . , d c ) that is not an intersection with a linear cone. Suppose that X is Fano. Then for every 1 k dim X one has
a k 2 dim X dim X−k+1 .
Moreover, one has a 0 = a 1 = 1.
Proof. By [CCC11, (2.6)] and Lemma 3.1(iii) one has
c+dim(X)+1−I(X) c c dim(X) i=0 a i c j=1 d j n k=0 a k 1.
Since X is Fano, one has I(X) > 0, so that
1 + dim(X) c c = 1 + dim(X) c c dim(X) dim(X) dim(X) i=0 a i .
Suppose that for some 1 k dim X the inequality
a k > 2 dim X dim X−k+1 holds. Then dim(X) i=0 a i a dim X−k+1 k > 2 dim(X) , and thus 1 + dim(X) c c dim(X) > 2.
The latter means that c dim(X) > 1 and gives a contradiction with [CCC11, Theorem 1.3]. Now suppose that a 1 > 1. To avoid a contradiction with [CCC11, Theorem 1.3] one must have a 0 = 1 and a 1 = . . . = a dim(X) = 2, and the inequalities in the above argument must become equalities. In particular, one has dim(X) = c. Moreover, Lemma 2.15 implies that all d j are divisible by 2, and a i for i > dim(X) are not. Now Lemma 3.
1(i),(ii) implies that for all 0 k c − 1 one has d c−k 2a n−k , so that c j=1 d j n i=dim(X)+1 2a i n i=dim(X)+1 a i + 2c + 1 = n i=0 a i .
This contradicts the assumption that X is Fano.
Hodge numbers
The idea of description of Hodge numbers for complete intersections in weighted projective spaces as dimensions of graded components of particular (bi)graded rings goes back to [Gr69], [St77], [Do82], [PSt83]; another approach, due to Hirzebruch, can be found in [SGA7, Exp. XI, Theorem 2.2]. For complete intersections in toric varieties one can look at [BC94]. The way of the computation called the Cayley trick is to relate the Hodge structure of a complete intersection to the Hodge structure of some higher-dimensional hypersurface. We describe this approach following [Ma99].
Let Y be a simplicial toric variety of dimension n. Let D 1 , . . . , D b be its prime boundary divisors. Denote the group of r-cycles on Y modulo rational equivalence by A r (Y ).
Consider an
A n−1 (Y )-graded ring R 0 = C[x 1 , . . . , x b ] with grading defined by deg A b i=1 x r i i = b i=1 r i D i .
One has Spec (R 0 ) ∼ = A b , and there is a natural correspondence between rays e i of a fan of Y and variables x i . Define a subvariety Z in Spec(R 0 ) as a union of hypersur-
faces { x i = 0 | e i / ∈ σ} over all cones σ of a fan of Y . Then Y is a geometric quotient of Spec (R 0 ) \ Z ⊂ A b by the torus D = Hom Z (A n−1 (Y ), C * ).
We call a polynomial f ∈ R 0 homogeneous if all its monomials are of degree d for some d ∈ A n−1 (Y ). For any homogeneous polynomials f 1 , . . . , f c their common zero set intersected with U is stable under the action of D so they determine a closed subset X in Y .
Consider a ring R = C[x 1 , . . . , x b , y 1 , . . . , y c ]. Choose c homogeneous polynomi-
als f 1 , . . . , f c ∈ R 0 ⊂ R with deg A (f i ) = d i ∈ A n−1 (Y )
. Define a bigrading on R with values in A n−1 (Y ) × Z by bideg(x i ) = (D i , 0) and bideg(y i ) = (−d i , 1). Consider the polynomial F = y 1 f 1 + . . . + y c f c . Obviously, one has bideg(F ) = (0, 1). Define a Jacobian ideal J = ∂F ∂x 1 , . . . , ∂F ∂x b , ∂F ∂y 1 , . . . , ∂F ∂y c and a bigraded ring R(F ) = R/J.
We will assume that the subvariety X defined by the polynomials f 1 , . . . , f c is quasismooth. Recall from [Ma99, Definition 1.1] that this means that a common zero set of f 1 , . . . , f c inside U is a smooth subvariety of codimension c, cf. Definition 2.4. In this case X has a pure Hodge structure on its cohomology, see [Ma99,§3]. In particular, one can speak about Hodge numbers h p,q (X).
Define c k as a number of cones of dimension k in the fan of Y . Put There is an analog of Lefschetz hyperplane section theorem for complete intersections in simplicial toric varieties, see e.g. [Ma99, Proposition 1.4]. In particular, the only Hodge numbers of such complete intersection X that are not inherited from the ambient toric variety are h p,q (X) with p + q = dim(X).
l k = n i=k (−1) i−k i k c n−i .h p−1,p (X) = dim R(F ) (−i(X),p) + l p−c − l p . For p = n−c 2 one has h p,p (X) = dim R(F ) (−i(X),p) + l p .
In a particular case of complete intersections in a weighted projective space the even cohomology spaces H 2k (Y, C) are one-dimensional, see [Do82, Corollary 2.3.6]. This allows to simplify Theorem 4.2 in this case. Recall that for a weighted complete intersection X in P(a 0 , . . . , a n ) of multidegree (d 1 , . . . , d c ) we denote the number a i − d j by I(X). and for p = n−c 2 one has h p,p (X) = dim R(F ) (−I(X),p) + 1.
Example 4.4. Consider the weighted projective space P = P (1, 1, 1, 1, 1, 1, 3) with weighted homogeneous coordinates x 0 , . . . , x 6 , where the weights of x 0 , . . . , x 5 equal 1, and the weight of x 6 equals 3. Let X be a (general) weighted complete intersection of hypersurfaces of degrees 2 and 6 in P given by polynomials f 1 and f 2 , respectively. Thus F = y 1 f 1 + y 2 f 2 and
J = ∂F ∂x 0 , . . . , ∂F ∂x 6 , f 1 , f 2 .
One has bideg(x 0 ) = . . . = bideg(x 5 ) = (1, 0), bideg(x 6 ) = (3, 0), bideg(y 1 ) = (−2, 1), bideg(y 2 ) = (−6, 1).
Since I(X) = 1, one gets h 1,3 (X) = dim R(F ) (−1,1) . The component R(F ) (−1,1) is generated by polynomials of type g 1 y 1 + g 5 y 2 , where g s are polynomials of degrees s in x i . There are 6 parameters for g 1 and 10 5 + 7 5 = 273 parameters for g 5 , so R (−1,1) = 279. There are no polynomials from R (−1,1) that are divisible by f 2 , and 8 5 = 56 parameters for polynomials in x 0 , . . . , x 5 , y 2 that are divisible by f 1 . Up to scaling there is a unique polynomial that is divisible by f 1 and x 6 . Moreover, one has bideg( ∂F ∂x i ) = (−1, 1) for i = 1, . . . , 6, so there are 6 parameters for polynomials from R (−1,1) that are divisible by ∂F ∂x i . Similarly, one has bideg( ∂F ∂x 6 ) = (−3, 1), so there are 7 5 = 21 parameters for polynomials from R (−1,1) that are divisible by ∂F ∂x 6 . One of them, namely ∂F ∂x 6 f 1 , is already taken into account. Thus One can obtain the following elegant formula for dim H 0 (X, O X (k)). . Let X be a quasi-smooth well formed weighted complete intersection of multidegree (d 1 , . . . , d c ) in P(a 0 , . . . , a n ). Then
∞ k=0 dim H 0 (X, O X (k)) t k = c s=0 1 − t d i n r=0 (1 − t a i )
.
For a weighted projective space P(a 0 , . . . , a n ) with weighted homogeneous coordinates x 0 , . . . , x n denote by P (r) the dimension of the vector space of (weighted) homogeneous polynomials in x 0 , . . . , x n of (weighted) degree r. Applying Theorem 4.5 together with Lemma 2.10 one gets the following. Another approach to compute Hodge numbers for complete intersections in (usual) projective spaces is due to F. Hirzebruch. Define
H(d) = (y + 1) d−1 − (z + 1) d−1 (z + 1) d y − (y + 1) d z = d − 1 + d−1 2 (y + z) + d−1 3 (y 2 + yz + z 2 ) + . . . 1 − d 2 yz − d 3 yz(y + z) + . . . and H(d 1 , . . . , d c ) = Q⊂[1,c],Q =∅ ((y + 1)(z + 1)) |Q|−1 i∈Q H(d i ),
where |Q| is a number of elements of Q. If F is a formal series in two variables y and z, we denote by F (m) the sum of monomials in F of homogeneous degree m. Remark 4.8. There is a conjectural approach to description of Hodge numbers of Fano varieties via their Landau-Ginzburg models, see [KKP14]. It was verified for del Pezzo surfaces, see [LP16]; for smooth toric varieties see [Ha16]. Its reformulation in terms of toric Landau-Ginzburg models for one of the Hodge numbers was checked for Picard rank one Fano threefolds (see [Prz13]) and partially for complete intersections in projective spaces (see [PS15]).
Dimensions 4 and 5
In this section we provide a classification of smooth well formed Fano weighted complete intersections of dimensions 4 and 5. This is done by a straightforward check using the bounds given by Theorem 1.1 and Lemma 3.3. To simplify conventions, we exclude the projective space (which is a codimension 0 smooth Fano complete intersection in itself) from our lists. Note that at this step we obtain only necessary conditions on the weights and degrees, and in each case one has to check that there actually exists a weighted complete intersection with the corresponding parameters, and that it is smooth. The Hodge numbers are computed using Corollary 4.3, and sometimes Theorem 4.7 when the latter is more convenient to apply. Since the Hodge numbers are constant in smooth families, for these computations it is enough to pick up one example in each of the families. In principle, all this can be done in an arbitrary given dimension.
Let X be a smooth well formed Fano weighted complete intersection of multidegree (d 1 , . . . , d c ) in P = P(a 0 , . . . , a n ) of dimension n − c 3. Important invariants of X are its anticanonical degree (−K X ) dim(X) , the dimension h 0 (−K X ), and the index I(X), which is defined as the maximal number i such that K X is divisible by i in Pic (X). Since dim X 3, the class of the line bundle O P (1)| X is not divisible in Pic (X), see [Ok16b, Remark 4.2]. Therefore, by Lemma 2.10 one has
I(X) = a i − d j .
For the anticanonical degree of X one has
(−K X ) dim(X) = d j a i · I(X) dim(X) .
The number h 0 (−K X ) can be computed by Corollary 4.6. We will use the abbreviation where k 0 , . . . , k m will be allowed to be any positive integers. If is some of k i is equal to 1 we drop it for simplicity. , 2 2 , 3 2 ). This corresponds to family No. 1 in Table 1.
Remark 5.2. The numerical data listed in Table 1 does not describe every variety in the corresponding deformation family. For example, a quartic in P 5 can be seen as a complete intersection of bidegree (2, 4) in P(1 6 , 2), that is an intersection with a linear cone. A nongeneral variety of the latter type can be contained in a hypersurface of weighted degree 2 whose equation does not depend on the variable of weight 2; such complete intersections cannot be embedded as quartics in P 5 .
Looking at the anticanonical degrees and dimensions of anticanonical linear systems of varieties from There are also special rational varieties in families No. 17 and No. 20, but rationality questions for general varieties from these families are far from being settled. Similarly to Remark 5.2, the numerical data listed in Table 2 does not describe every variety in the corresponding deformation family, but only a general one.
Looking at the anticanonical degrees and dimensions of anticanonical linear systems of varieties from Table 2, we see that varieties from different families are never isomorphic to each other.
Remark 5.4. By [Pu98] a general variety from family No. 5 in Table 2 It would be interesting to study birational geometry of weighted complete intersections from Tables 1 and 2 that are not covered by Remarks 5.3 and 5.4. Also, it would be interesting to study automorphism groups of Fano varieties from Tables 1 and 2, cf. [PrSh16, § A.2]. In particular, it would be interesting to find weighted Fano complete intersections acted on by relatively large automorphism groups, cf. [PS16].
Using the list of index 1 Fano fivefolds provided in Table 2 together with the same trick as in §3, one can compile a list of smooth well formed Calabi-Yau weighted complete intersections of dimension 4 that are not intersections with linear cones. Namely, if a weighted complete intersection of multidegree d 1 , . . . , d c in P(a 0 , . . . , a n ) is Calabi-Yau, then a general complete intersection of multidegree d 1 , . . . , d c in P (1, a 0 , . . . , a n ) is Fano. For partial classification results concerning Calabi-Yau threefolds see [Og91], [AESZ05], [IMOU16], [IIM16], and references therein.
Appendix A. Optimization
The purpose of this section is to prove some bounds on the values of linear functions on special subsets of R m .
Let L be a real-valued function on a set Ω. We say that L attains its maximum in Ω if L is bounded and there is a point P in Ω such that L(P ) = sup P ′ ∈Ω L(P ′ ); in this case we also say that L attains its maximum in Ω at P .
Lemma A.1. Let N and c be positive integers such that N > c, and α be a real number.
Let Ω be a subset of R N +c with coordinates A 1 , . . . , A N , D 1 , . . . , D c defined by inequalities
A 1 · . . . · A N D 1 · . . . · D c , A 1 . . . A N 1, D 1 . . . D c , D 1 2A 1 , D 2 A 2 , . . . , D c A c . Put L(A 1 , . . . , A N , D 1 , . . . , D c ) = N i=1 A i − c j=1 D j + α.
Let Ω ⊂ Ω be a non-empty closed subset. Then the function L attains its maximum in Ω.
Proof. It is enough to prove the assertion for α = 0.
Let M be a real number, and let
Ω ′ ⊂ Ω ∩ {(A 1 , . . . , D c ) | A 1 M}
be a non-empty closed subset. We claim that L attains its maximum in Ω ′ . Indeed, letP be some point of Ω, and putL = L(P ). If D 1 > NM −L, then
L(A 1 , . . . , D c ) = N i=1 A i − c j=1 D j NA 1 − D 1 < N(A 1 − M) +L L .
Thus L attains its maximum in Ω ′ if and only if it attains its maximum in the closed subset Ω ′′ = Ω ′ ∩ {(A 1 , . . . , D c ) | D 1 NM −L} containingP . It remains to notice that Ω ′′ is a compact subset of R N +c .
We conclude from the above argument that L attains its maximum in Ω if and only if it attains its maximum in the closed subset
Ω (1) = Ω ∩ {(A 1 , . . . , D c ) | A 1 −L 0 }.
Let P 0 be some point of Ω, and put L 0 = L(P 0 ).
If A c+1 < (A 1 + L 0 ) · (N − c) −1 , then L(A 1 , . . . , D c ) = N i=1 A i − c j=1 D j = (A 1 − D 1 ) + c j=2 (A j − D j ) + N i=c+1 A i −A 1 + (N − c)A c+1 < −A 1 + A 1 + L 0 = L 0 .
Thus L attains its maximum in Ω (1) if and only if it attains its maximum in the closed subset
Ω (2) = Ω (1) ∩ (A 1 , . . . , D c ) | A c+1 A 1 + L 0 N − c containing P 0 . Suppose that (A 1 , . . . , D c ) ∈ Ω (2) . Then D c 1 D 1 · . . . · D c A 1 · . . . · A N A 1 · . . . · A c+1 A 1 + L 0 N − c c+1 , so that D 1 1 N − c c+1 c · (A 1 + L 0 ) c+1 c .
One has
(A.1) L(A 1 , . . . , D c ) = N i=1 A i − c j=1 D j N · A 1 − D 1 = N(A 1 + L 0 ) − D 1 − NL 0 (A 1 + L 0 ) · N − 1 N − c c+1 c · (A 1 + L 0 ) 1 c − NL 0 . Put M = max |(N + 1)L 0 | − L 0 , (N + 1) c · (N − c) c+1 − L 0 .
If A 1 > M, then the right hand side of (A.1) is less than L 0 . Thus L attains its maximum in Ω (2) if and only if it attains its maximum in the closed subset
Ω (3) = Ω (2) ∩ {(A 1 , . . . , D c ) | A 1 M},
and we are done by the argument from the beginning of the proof.
Remark A.2. One can easily modify the assertion of Lemma A.1 so that the condition N > c will not be important.
The following theorem will be our main technical tool to find the points where certain functions attain their maximal values. It is well known as the method of Lagrange multipliers, or sometimes the Kuhn-Tusker Theorem. Let G 1 , . . . , G p be differentiable functions on R m with coordinates x 1 , . . . , x m . Let Ω ⊂ R m be a subset defined by inequalities G i 0, 1 i p. Let L be a differentiable function on R m . Suppose that L attains its maximum in Ω at a point P . Then
∂L ∂x 1 , . . . , ∂L ∂x m [P ] = p i=1 λ i ∂G i ∂x 1 , . . . , ∂G i ∂x m [P ]
for some non-negative numbers λ i . Moreover, if for some j one has λ j = 0, then G j (P ) = 0.
To proceed we need to establish some elementary inequalities that will be used in the proof of Proposition A.11. The function L attains its maximum in Ω at some point P . To see this apply Lemma A.1 with c = r + 1, an arbitrary N max{l + 1, c + 1}, and the closed subset defined by conditions
A 1 = . . . = A l+1 , A l+2 = . . . = A N = 1, D 1 = 2A 1 , D 2 = . . . = D c .
Abusing notation a little bit, we write P = (B, D) and put M = L(P ).
If B = 1, then L(B, D) = l − rD − (l + 1) = −rD − 1 < 0. Thus we will assume that B > 1.
If l r, then
M = lB − rD − (l + 1) (l − r)B − (l + 1) < 0.
Thus we will assume that l > r. Suppose that r = 0. Then B l 2. Since l > 1, this implies that B < 1 + 1 l . This gives M = lB − rD − (l + 1) < l · 1 + 1 l − (l + 1) = 0.
Thus we will assume that r > 0. Similarly, suppose that B = D. Then B l−r 2, which implies B 1 + 1 l−r . This gives M = lB − rD − (l + 1) = (l − r)B − (l + 1) (l − r) · 1 + 1 l − r − (l + 1) = −r < 0.
Thus we will assume that D > B.
By Theorem A.3 one has (A.2) (l, −r) = λ(lB l−1 , −2rD r−1 )
for some non-negative number λ. Actually, (A.2) implies that λ = 1 B l−1 , so that λ is positive and B l = 2D r by Theorem A.3. Also, since r > 0, equation (A.2) implies that
1 = 2D r−1 · λ = 2D r−1 B l−1 = B D < 1,
which is a contradiction.
Lemma A.5. Let k be a positive integer, l be a non-negative integer, and p be a nonnegative real number such that k+l+p > 2; let r be a non-negative integer. Let Ω ⊂ R 3 Proof. The function L attains its maximum in Ω at some point P . To see this apply Lemma A.1 with c = r + 1, an arbitrary N max{k + l, c + 1}, and the closed subset defined by conditions
A,B,D be defined by inequalities A k−1 B l 2D r , A B 1, A k + l + p,A 1 = . . . = A k , A k+1 = . . . = A k+l , A k+l+1 = . . . = A N = 1, D 1 = 2A 1 , D 2 = .
. . = D c , A 1 k + l + p.
Abusing notation, we write P = (A, B, D) and put M = L(P ). If k = 1, we have B l 2D r , so that
M = lB − rD + (p − A) lB − rD − (l + 1) 0
by Lemma A.4. Moreover, if M = 0, then l = 1, r = 0, and B = 2, so M = p + 2 − A and A = p + 2. In particular, condition k + l + p > 2 implies that p > 0. Thus we will assume that k 2. Note that r > 0. Indeed, otherwise one has
2 A k−1 B l A k−1 A > 2,
which is absurd. If l = 0, then A k−1 2D r , so that
M = (k − 2)A − rD + p = ((k − 1)A − rD − k) + (k + p − A) < 0
by Lemma A.4. Thus we will assume that l > 0. If B = 1 then A k−1 2D r , so that
M = (k − 1)A + l − rD − A + p = ((k − 1)A − rD − k) + (k + l + p − A) < 0
by Lemma A.4. Thus we will assume that B > 1. If A = B then A k+l−1 2D r , so that M = (k + l − 2)A − rD + p = ((k + l − 1)A − rD − (k + l)) + (k + l + p − A) < 0 by Lemma A.4. Thus we will assume that A > B. In particular, we have D > B.
Suppose that D = A. If r < k − 1, then A k−1−r B l 2, so that A 2 < k + l + p, a contradiction. Hence r k − 1, and B l 2D r−k+1 . We have M = lB + (k − r − 2)D + p = (lB − (r − k + 1)D − (l + 1)) + (l + p + 1 − D) < 0 by Lemma A.4. Thus we will assume that D > A.
By Theorem A.3 one has
(A.3) (k − 2, l, −r) = λ 1 ((k − 1)A k−2 B l , lA k−1 B l−1 , −2rD r−1 ) + λ 2 (−1, 0, 0)
for some non-negative numbers λ 1 and λ 2 . Since l > 0, equation (A.3) implies that
λ 1 = 1 A k−1 B l−1 ,
so that λ 1 is positive and A k−1 B l = 2D r by Theorem A.3. Finally, since r > 0, equation (A.3) implies that
1 = 2D r−1 · λ 1 = 2D r−1 A k−1 B l−1 = B D < 1,
which is a contradiction.
Lemma A.6. Let k and c be positive integers, and l be a non-negative real number such that k + l > 2.
Let Ω be a subset of R 2 with coordinates A, D defined by inequalities
A k D c , D 2A, and A k + l. Put L(A, D) = kA − cD + l.
Then L is negative on Ω.
Proof. The function L attains its maximum in Ω at some point P . To see this apply Lemma A.1 with an arbitrary N max{k, c + 1}, and the closed subset defined by conditions
A 1 = . . . = A k , A k+1 = . . . = A N = 1, D 1 = . . . = D c , A 1 k + l.
Abusing notation, we write P = (A, D). Note that if (k + 1)A cD, then L(A, D) = kA − cD + l l − A < 0.
In particular, this happens if k < 2c, since k and c are integers.
By Theorem A.3 one has
(k, −c) = λ 1 (kA k−1 , −cD c−1 ) + λ 2 (2, −1) + λ 3 (−1, 0)
for some non-negative numbers λ 1 , λ 2 , and λ 3 . Suppose that λ 1 > 0. By Theorem A.3 this means that A k = D c . We can assume that k 2c. Thus D c A 2c , and D A 2 .
Assume that c 2. Then k + 1 ck c(k + l) cA, so that 2c. If k = 2c, then A 2c = (2A) c , which means A = 2 < k + l, a contradiction. Thus, we have k < 2c, which implies the assertion of the lemma.
Lemma A.7. Let k be a positive integer, let l be a non-negative integer such that k+l > 2, and let c be a non-negative integer. Let Ω ⊂ R 3 A,B,D be defined by inequalities A k B l D c , D 2A, A B 1, and A k + l. Put
L(A, B, D) = kA + lB − cD.
Then L is negative on Ω.
Proof. The function L attains its maximum in Ω at some point P . To see this apply Lemma A.1 with an arbitrary N max{k + l, c + 1}, and the closed subset defined by conditions
A 1 = . . . = A k , A k+1 = . . . = A k+l , A k+l+1 = . . . = A N = 1, D 1 = . . . = D c , A 1 k + l.
Abusing notation, we write P = (A, B, D) and put M = L(P ). Note that c > 0. Indeed, otherwise one has
1 A k B l A k A > 2,
which is absurd. If l = 0, then A k D c , so that M = kA − cD < 0 by Lemma A.6. Thus we will assume that l > 0. If A = B, then A k+l D c , so that M = (k + l)A − cD < 0 by Lemma A.6. Thus we will assume that A > B. If B = 1, then A k D c , so that M = kA − cD + l < 0 by Lemma A.6. Thus we will assume that B > 1. If D = 2A, then
M = kA + lB − c · 2A = (k − 2)A + lB − (c − 1) · 2A < 0
by Lemma A.5. Thus we will assume that D > 2A. By Theorem A.3 one has
(A.5) (k, l, −c) = λ 1 (kA k−1 B l , lA k B l−1 , −cD c−1 ) + λ 2 (−1, 0, 0)
for some non-negative numbers λ 1 and λ 2 . Since l > 0, equation (A.5) implies that λ 1 = 1 A k B l−1 , so that λ 1 is positive and A k B l = D c by Theorem A.3. Finally, since c > 0, equation (A.5) implies that
1 = D c−1 · λ 1 = D c−1 A k B l−1 = B D < 1,
which is a contradiction.
Consider a real vector space R m . For a vector v ∈ R m we denote by v (i) its i-th coordinate. Denote Proof. Note that for the vector λu the same assumptions hold as for the vector u itself provided that λ 0. Thus we will replace u by λu and assume that λ = 1. In other words we have a system of equations
u (1) − λ −1 + λ 0 − λ 1 = 1, u (2) + λ 1 − λ 2 = 1, . . . u (m−1) + λ m−2 − λ m−1 = 1, u (m) + λ m−1 − λ m = 1.
Choose the indices p and q so that u (p) < u (p+1) = . . . = u (q) = 1 < u (q+1) .
In particular, we put p = q if one has u (p) < 1 < u (p+1) , we put p = q = 0 if 1 < u (1) , and we put p = q = m if u (m) < 1. For 1 < i p we have
1 = u (i) + λ i−1 − λ i < 1 + λ i−1 , so λ i−1 > 0. Moreover, if p > 0, then 1 = u (1) + λ −1 + λ 0 − λ 1 < 1 + λ 0 ,
so λ 0 > 0. In the same way for j > q we have
1 = u (j) + λ j−1 − λ j > 1 − λ j ,
so λ j > 0. This exactly gives the assertion of the lemma. Proof. Note that for the vector λu the same assumptions hold as for the vector u itself provided that λ 0. Thus we will replace u by λu and assume that λ = 1. In other words we have a system of equations
u (1) − λ 0 − λ 1 = −1, u (2) + λ 1 − λ 2 = −1, . . . u (m−1) + λ m−2 − λ m−1 = −1, u (m) + λ m−1 = −1.
Suppose that u (m) = −1. Let q be the minimal index such that u (q) = −1. Then, considering equations from the last one to the q-th one by one we have
λ m−1 = . . . = λ q−1 = 0.
Moreover, if q > 2, then
u (q−1) + λ q−2 − λ q−1 = u (q−1) + λ q−2 > −1,
which is impossible. Thus either q = 2, so u (1) > −1 and λ 0 > 0, which corresponds to case (i), or q = 1, which corresponds to case (ii). Now suppose that u (m) < −1. Choose the indices 1 p q < m such that
u (p) > u (p+1) = . . . = u (q) = −1 > u (q+1) .
Then for i > q one has
−1 = u (i) + λ i−1 − λ i < −1 + λ i−1 ,
so λ i−1 > 0. Moreover, for p j q one has
−1 = u (j) + λ j−1 − λ j = −1 + λ j−1 − λ j < −1 + λ j−1 ,
so we also have λ j−1 > 0. If one has λ i > 0 for all 1 i p − 1, then we obtain case (iii). Otherwise take the maximal number s with 1 s p − 1 such that λ s = 0. If s > 1 then
u (s) + λ s−1 − λ s > −1,
which is impossible. Thus s = 1 and λ 0 > 0, which gives us case (iv). Proof. Apply Lemma A.8 to the first N coordinates of the vector u and Lemma A.9 to its last c coordinates. Define numbers p and q following the notation of Lemma A.8. The only possibility to have 0 ∈ Λ is to have p = 0. Then {1, . . . , q − 1} ⊂ Λ ′′ and {q + 1, . . . , N} ⊂ Λ ′ . Moreover, for the last c coordinates only cases (ii) or (iii) from Lemma A.9 can occur, which gives us case (I).
So we can assume that 0 ∈ Λ ′ . From Lemma A.9 one can easily see that The function L attains its maximum in Ω at some point P ∈ Ω by Lemma A.1. Abusing notation a little bit, we write P = (A 1 , . . . , A N , D 1 , . . . , D c ). If for some 2 i c one has A i = D i , we cancel A i and D i from the inequalities defining Ω and from the definition of L and arrive to the same assertion with a smaller number of parameters. Therefore, we assume that for all 2 i c one has A i < D i (in particular, this is the case when c = 1). Note that after such cancelation the condition N 2c + 1 is preserved.
Denote Π A = A 1 · . . . · A N and Π D = D 1 · . . . · D c . Applying Theorem A.3 and keeping in mind that A i < D i for 2 i c, we obtain an equality for some k 1, l 1, and p 1. In all cases one has c 1 and N 2c + 1 3.
In case (I) we have L(P ) < 0 provided that A 1 N; to see this apply Lemma A.7. In case (II) we have L(P ) < 0 provided that A 1 N; to see this apply Lemma A.5 with r = c − 1 and p = 0. Finally, in case (III) we have L(P ) 0 provided that A 1 N; to see this apply Lemma A.5 with r = c − 1.
p
(d 1 , . . . , d c ) the number of the degrees d j that are divisible by p r . By Lemma 2.15 for every r one has
Theorem 4. 1
1(see[Da78, Theorem 10.8 and Remark 10.9]). Let Y be a simplicial toric variety of dimension n. Then dim H 2k (Y, C) = l k for all 0 k n.
Theorem 4.2 (see[Ma99, Theorem 3.6]). Let Y be a simplicial toric variety of dimension n and let D 1 , . . . , D b be its boundary divisors. Let X ⊂ Y be a quasi-smooth complete intersection of ample divisors defined by homogeneous polynomials f 1 , . . . , f c with deg A (f i ) = d i ∈ A n−1 . Suppose that dim X = n − c 3. ∈ A n−1 (X) by i(X). Then for p = n−c+1 2 and p = n−c 2 one has h n−c−p,p (X) = dim R(F ) (−i(X),p) .
Corollary 4. 3 .
3Let P = P(a 0 , . . . , a n ) be a well formed weighted projective space, and X ⊂ P be a quasi-smooth weighted complete intersection defined by homogeneous polynomials f 1 , . . . , f c with deg(f i ) = d i . Suppose that dim X = n − c 3. Then for p = n−c 2 one has h n−c−p,p (X) = dim R(F ) (−I(X),p)
h 1,3 (X) = dim R(F ) (−1,1) = dim R (−1,1) −56−1−6−21+1 = 279−56−1−6−21+1 = 196.
Corollary 4. 6 .
6Let X be a quasi-smooth well formed weighted complete intersection of multidegree (d 1 , . . . , d c ) in P(a 0 , . . . , a n ).Then dim H 0 (X, −K X ) = c s=0 (−1) s 1 k 1 <...<ks c P (I(X) − d k 1 − . . . − d ks ).
Theorem 4. 7
7(see [SGA7, Exp. XI, Théorème 2.3]). Let X be a smooth complete intersection of hypersurfaces of degrees d 1 , . . . , d c in P n . Put m = dim X = n − c. Then h p,m−p (X)y p z m−p = H(d 1 , . . . , d c δ = 1 if m is even and δ = 0 if m is odd.
Lemma A. 4 .
4Let l 1 be an integer, and r be a non-negative integer. Let Ω ⊂ R 2 B,D be defined by inequalities B l 2D r and D B 1. PutL(B, D) = lB − rD − (l + 1).Then L is non-positive on Ω. Moreover, L(B, D) is negative unless l = 1, r = 0, and B = 2.Proof. Suppose that l = 1. If r > 0, thenL(B, D) = B − rD − 2 B − D − 2 −2 < 0.If r = 0, then B 2, and L(B, D) = B − 2 0. Moreover, in this case L(B, D) = 0 if and only if B = 2. Thus we will assume that l 2.
D B, and an additional inequality D A in the case when k 2. Put L(A, B, D) = (k − 2)A + lB − rD + p. Then L is non-positive on Ω. Moreover, L(A, B, D) is negative unless A = p+2 and p > 0.
(k + 1 )
1A cA 2 cD, and we are done. Hence we have c = 1, and L(A, D) = kA − D + l kA − A 2 + l l − lA < 0. Therefore, we may suppose that λ 1 = 0. Then (A.4) (k, −c) = (2λ 2 − λ 3 , −λ 2 ), so that λ 2 = c and D = 2A by Theorem A.3. Also, we see from (A.4) that k
. . . , 0) ∈ R m , 1 i m.Lemma A.8. Let u ∈ R m be a vector such that0 u (1) . . . u (m) . Put u −1 = −e 1 , put u 0 = e 1 , put u i = −e i + ei+1 for 1 i m − 1, and put u m = −e m . Choose a subset Λ ⊂ {−1, 0, . . . , m}. Suppose that λu + i∈Λ λ i u i = (1, . . . , 1) for some non-negative number λ and some positive numbers λ i . Then there exist two indices 0 p q m such that u (p+1) = . . . = u (q) and {0, . . . , p − 1} ∪ {q + 1, . . . , m} ⊂ Λ.
Lemma A. 9 .
9Let u ∈ R m be a vector such that 0 u (1) . . . u (m) . Put u 0 = −e 1 , and put u i = −e i + e i+1 for 1 i m − 1. Choose a subset Λ ⊂ {0, . . . , m − 1}. Suppose that λu + i∈Λ λ i u i = (−1, . . . , −1) for some non-negative number λ and some positive numbers λ i . Then one of the following possibilities occurs: (i) 0 ∈ Λ and u (2) = . . . = u (m) ; (ii) u (1) = . . . = u (m) ; (iii) {1, 2, . . . , m − 1} ⊂ Λ; (iv) {0, 2, 3, . . . , m − 1} ⊂ Λ.
Lemma A. 10 .
10Choose a vector u = u (1) , . . . , u (N +c) ∈ R N +c such that 0 u (1) . . . u (N ) , 0 u (N +1) . . . u (N +c) .Put u −1 = −e 1 and u 0 = 2e 1 − e N +1 . Furthermore, put u i = −e i + e i+1 for 1 i N − 1 and for N + 1 i < N + c. Finally, put u N = −e N . Choose a subset Λ ′ ⊂ {−1, 0, . . . , N + c − 1}. non-negative number λ and some positive numbers λ i . DefineΛ ′′ = i | u (i) = u (i+1) .Then one of the following possibilities occurs:(I) there is an index 0 q N such that{1, . . . , q − 1, q + 1, . . . , N, N + 1, . . . , N + c − 1} ⊂ Λ ′ ∪ Λ ′′ . and N ∈ Λ ′ ; (II) there is an index 1 p N such that {0, . . . , p − 1, p + 1, . . . , N − 1, N + 2, . . . , N + c − 1} ⊂ Λ ′ ∪ Λ ′′ and 0 ∈ Λ ′ ; (III) there are indices 1 p q N − 1 such that {0, . . . , p − 1, p + 1, . . . , q − 1, q + 1, . . . , N, N + 2, . . . , N + c − 1} ⊂ Λ ′ ∪ Λ ′′ and {0, N} ⊂ Λ ′ .
{N + 2 ,
2. . . , N + c − 1} ⊂ Λ ′ ∪ Λ ′′ . If q = N then {0, . . . , p − 1} ⊂ Λ ′ and {p + 1, . . . , N − 1} ⊂ Λ ′′ , and we obtain case (II). If q < N then {0, . . . , p − 1, q + 1, . . . , N} ⊂ Λ ′ and {p + 1, . . . , q − 1} ⊂ Λ ′′ , and we obtain case (III). Now we are ready to prove the main result of this section. Proposition A.11. Let c and N be positive integers such that N 2c + 1. Let Ω be a subset of R N +c with coordinates A 1 , . . . , A N , D 1 , . . . , D c defined by inequalities A 1 · . . . · A N D 1 · . . . · D c , A 1 . . . A N 1, D 1 . . . D c , D 1 2A 1 , D 2 A 2 , . . . , D c A c . Put L(A 1 , . . . , A N , D 1 , . . . , D c ) = A 1 + . . . + A N − D 1 − . . . − D c .Then L is non-positive on the subset of Ω where A 1 N.Proof. Rewrite the inequalities defining Ω asA 1 · . . . · A N − D 1 · . . . · D c 0, A 2 − A 1 0, . . . , A N − A N −1 0, 1 − A N 0, D 2 − D 1 0, . . . , D c − D c−1 0, A 2 − D 2 0, . . . , A c − D c 0, 2A 1 − D 1 0.
, 0 N − 1 ,,, 0 N − 1 ,
0101−1, 1, 0, . . . , 0) + λ N (0, . . . , −1, 0, . . . , −1, 1, 0, . . . , 0) + λ 0 (2, 0, . . . −1, 0, . . . , 0) + λ −1 (−1, 0, . . . , 0) for some non-negative numbers λ and λ i , where −1 i N + c − 1. Let Λ ′ ⊂ {−1, 0, 1 . . . , N + c − 1} be the set of indices such that for any i ∈ Λ ′ one has λ i > 0, and let Λ ′′ = {i | u (i) = u (i+1) }. Keeping in mind that by Theorem A.3 for any i ∈ Λ ′ the corresponding inequality turns into equality, we see that by Lemma A.10 we have the following possibilities: (I) (A 1 , . . . , A N , D 1 , . . . , D c ) = ( 1 , . . . , A N , D 1 , . . . , D c ) = ( 1 , . . . , A N , D 1 , . . . , D c ) = ( k A, . . . , A, l B, . . . , B, p 1, . . . , 1 N , 2A, D, . . . , D c−1 )
Table 1
1contains a list of all smooth well formed Fano weighted complete intersections of dimension 4 that are not intersections with linear cones. This list was obtained in [Kü97, Proposition 2.2.1], cf. [BK16, §1.3].Remark 5.1. Note that there is a misprint in the first line of the table on [Kü97, p. 50]: the varieties described there should be understood as complete intersections of two hypersurfaces of weighted degree 6 in P(1 3No. I
P
Degrees K 4 h 0 (−K) h 1,3 h 2,2
1 1 P(1 3 , 2 2 , 3 2 )
6,6
1
3
107 503
2 1 P(1 4 , 2, 5)
10
1
4
412 1801
3 1 P(1 4 , 2 2 , 3)
4,6
2
4
121 572
4 1
P(1 5 , 4)
8
2
5
325 1452
5 1
P(1 5 , 2)
6
3
5
156 731
6 1 P(1 5 , 2 2 )
4,4
4
5
75 378
7 1
P(1 6 , 3)
2,6
4
6
196 912
8 1
P 5
5
5
6
120 581
9 1
P(1 6 , 2)
3,4
6
6
71 364
10 1
P 6
2,4
8
7
77 394
11 1
P 6
3,3
9
7
49 267
12 1
P 7
2,2,3
12
8
42 236
13 1
P 8
2,2,2,2 16
9
27 166
14 2
P(1 5 , 3)
6
32
15
70 382
15 2
P 5
4
64
21
21 142
16 2
P 6
2,3
96
27
8
70
17 2
P 7
2,2,2 128
33
3
38
18 3 P(1 4 , 2, 3)
6
81
25
24 161
19 3
P(1 5 , 2)
4
162
40
5
52
20 3
P 5
3
243
55
1
21
21 3
P 6
2,2
324
70
0
8
22 4
P 5
2
512
105
0
2
Table 1: Fourfold Fano weighted complete intersections.
Table 1 ,
1we see that varieties from different families are never isomorphic to each other. Similarly, none of them is isomorphic to any of the smooth Fano fourfolds that are zero loci of sections of homogeneous vector bundles on Grassmannians, see [Kü97, Theorem 4.2.1]. Remark 5.3. By [Pu98] a general variety from family No. 8 in Table 1 is non-rational. By [To16] a very general variety from families No. 8 and No. 15 is not stably rational. By [HPT16] this is also the case for a very general variety from family No. 19. By [Ok16a, Theorem 1.1] the same holds for very general varieties from families No. 4 and No. 14, by [Ok16a, Corollary 1.4] this holds for very general varieties from families No. 2 and No. 5, and by [Ok16a, Theorem 1.3] this holds for very general varieties from family No. 18. Every variety from families No. 21 and No. 22 is rational for obvious reasons.
Table 2
2contains a list of all smooth well formed Fano weighted complete intersections of dimension 5 that are not intersections with linear cones.No. I
P
Degrees −K 5 h 0 (−K) h 1,4
h 2,3
1 1 P(1 5 , 2, 3, 3)
6, 6
2
5
354 4594
2 1
P(1 6 , 5)
10
2
6
1996 24576
3 1
P(1 6 , 2, 3)
4, 6
4
6
359 4758
4 1
P(1 7 , 4)
2, 8
4
7
1386 15771
5 1
P 6
6
6
7
455 6055
6 1
P(1 7 , 2)
4, 4
8
7
168 2383
7 1
P(1 8 , 3)
2,2,6
8
8
568 7571
8 1
P 7
2, 5
10
8
294 4074
9 1
P 7
3, 4
12
8
147 2142
10 1
P 8
2,2,4
16
9
156 2295
11 1
P 8
2,3,3
18
9
88
1364
12 1
P 9
2,2,2,3
24
10
72
1155
13 1
P 10
2,2,2,2,2 32
11
44
759
Table 2 :
2Fivefold Fano weighted complete intersections.
is non-rational. By[To16] a very general variety from this family is not stably rational. By[Pu04] there exist non-rational varieties in family No. 9. By [Ok16a, Theorem 1.1] very general varieties from families No. 2, No. 17, and No. 27 are not stably rational. By [Ok16a, Theorem 1.3] very general varieties from families No. 15 and No. 18 are not stably rational. Every variety from families No. 34 and No. 35 is rational for obvious reasons. By [Tyu75, Corollary 5.1] every variety from family No. 30 is rational as well.
Acknowledgments. We are grateful to K. Besov, A. Corti, A. Kuznetsov, T. Okada, and Yu. Prokhorov for useful discussions, and to A. Harder who helped us to compute Hodge numbers in §5.
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HSE, 7 Vavilova str., Moscow, 117312, Russia. [email protected], victorprz@gmail. 117312Russian Federation, AG Laboratory, HSE, 7 Vavilova str; MoscowVictor Przyjalkowski National Research University Higher School of Economics, Russian Federation, AG Laboratory ; Constantin Shramov National Research University Higher School of EconomicsVictor Przyjalkowski National Research University Higher School of Economics, Russian Federation, AG Laboratory, HSE, 7 Vavilova str., Moscow, 117312, Russia. [email protected], [email protected] Constantin Shramov National Research University Higher School of Economics, Russian Federation, AG Laboratory, HSE, 7 Vavilova str., Moscow, 117312, Russia. [email protected]
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[
"Ketamine-Medetomidine General Anesthesia Occurs With Alternation of Cortical Electrophysiological Activity Among High and Low Complex States",
"Ketamine-Medetomidine General Anesthesia Occurs With Alternation of Cortical Electrophysiological Activity Among High and Low Complex States"
]
| [
"Eduardo C Padovani "
]
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| []
| Anesthetic agents are known to induce a series of alterations in cortical electrophysiological activity, such as the rise of signature patterns, changes in statistical properties, and altered dynamic behavior of neural records. Plenty of methods can be used to trace and monitor these changes, among them complexity metrics demonstrated to have the power to discriminate states involving distinct levels of awareness. There is a consensus that anesthetic drugs are capable of interfering with neural activities at different levels and time scales, being able to induce alterations both locally as well as on the spatiotemporal patterns established throughout the whole cortex. However, it is still unclear how such changes in the complexity of cortical activity are supposed to occur, and experimental evidence is still needed. To this purpose, we have analyzed an ECoG records database of a Ketamine-Medetomidine anesthetic induction experiment in a non-human primate subject. The MDR-ECoG technique provided records of cortical activity with both high temporal and spatial resolution allied with extensive coverage of the cortical surface. The Permutation Entropy and the Fractal Dimension were employed to evaluate the complexity of the neural time series. It was found that the complexity of cortical activity was relatively constant during awakened conditions. The transition to unconsciousness occurred in a relatively fast manner, it required about 30 to 40 seconds for the first remarkable changes to take place. During general anesthesia, the complexity assumed a considerable variation at local levels and fluctuated apparently without a defined period. On a global scale, the cortex dynamically alternated among high and low complex states. This study brings novel evidence of the effects of anesthetics on neural activity and cortical dynamics, complementing the actual scenario for the elucidation of anesthesia mechanisms in terms of circuits, pathways, and global brain functioning.* | null | [
"https://arxiv.org/pdf/2202.04320v1.pdf"
]
| 246,680,245 | 2202.04320 | 067aea5f052136f6cfcc67e52339bb025e0b861c |
Ketamine-Medetomidine General Anesthesia Occurs With Alternation of Cortical Electrophysiological Activity Among High and Low Complex States
Eduardo C Padovani
Ketamine-Medetomidine General Anesthesia Occurs With Alternation of Cortical Electrophysiological Activity Among High and Low Complex States
Padovani, E. C. -arXiv preprint -Neurons and Cognition • February 2022 •
Anesthetic agents are known to induce a series of alterations in cortical electrophysiological activity, such as the rise of signature patterns, changes in statistical properties, and altered dynamic behavior of neural records. Plenty of methods can be used to trace and monitor these changes, among them complexity metrics demonstrated to have the power to discriminate states involving distinct levels of awareness. There is a consensus that anesthetic drugs are capable of interfering with neural activities at different levels and time scales, being able to induce alterations both locally as well as on the spatiotemporal patterns established throughout the whole cortex. However, it is still unclear how such changes in the complexity of cortical activity are supposed to occur, and experimental evidence is still needed. To this purpose, we have analyzed an ECoG records database of a Ketamine-Medetomidine anesthetic induction experiment in a non-human primate subject. The MDR-ECoG technique provided records of cortical activity with both high temporal and spatial resolution allied with extensive coverage of the cortical surface. The Permutation Entropy and the Fractal Dimension were employed to evaluate the complexity of the neural time series. It was found that the complexity of cortical activity was relatively constant during awakened conditions. The transition to unconsciousness occurred in a relatively fast manner, it required about 30 to 40 seconds for the first remarkable changes to take place. During general anesthesia, the complexity assumed a considerable variation at local levels and fluctuated apparently without a defined period. On a global scale, the cortex dynamically alternated among high and low complex states. This study brings novel evidence of the effects of anesthetics on neural activity and cortical dynamics, complementing the actual scenario for the elucidation of anesthesia mechanisms in terms of circuits, pathways, and global brain functioning.*
I. Introduction E lectrophysiological cortical records are known to exhibit highly complex patterns, and it is a fact that changes in levels of awareness, whether caused by anesthetic agents, epileptiform activity, or sleep, reflect in signals with characteristic patterns that are distinguished from those typically observed under alert conditions. Since electrophysiological activity signals contain relevant information about the physiological state of individuals, a variety of methods can be used to evaluate changes in behavior, patterns, and statistical properties of neural time series. In particular, when we are interested in making inferences about different levels of awareness, it is possible to highlight complexity-related metrics, which have proven to have the power to discriminate physiological states involving distinct levels of consciousness. Because of this, complexity metrics are even used in clinical medicine to monitor patients during procedures involving sedation and general anesthesia. In these applications and neuroscience ex-periments, most of the time, a reduced number of electrodes are used, thus giving information about changes that occur locally at certain cortical areas, without providing detailed information about what is happening in the cortex as a whole.
Since anesthetic agents can concomitantly interfere with local molecular environments, neuronal circuits in the thalamic nuclei, circuits widespread throughout the cortex, and also with the communication established between the thalamus and the cortex, anesthetic drugs are able not only to affect the activity of certain cortical regions but in fact can interfere with the dynamics of neural activities in the brain as a whole, at a systems level. In this manner, we assert that the elucidation of the spatiotemporal organization of the complexity of neural activity over extensive cortical areas, may bring new relevant information to complement the actual scenario, and contribute with novel experimental evidence for the comprehension of the effects of general anesthesia in terms of neural circuits, pathways, and global brain functioning.
The objective of the present research was to verify and describe how statistical properties associated with the complexity of cortical electrophysiological activity change as soon as an anesthetic induction agent is administered to a subject. Particularly verifying the changes in patterns and dynamics that occur along with the unfolding of the transition to unconsciousness, as well as the spatiotemporal organization of the cortical activity's complexity during general anesthesia.
For this objective, a database respective to an experiment that involved Ketamine-Medetomidine anesthetic induction in an old-world monkey was analyzed. In this database, electrophysiological neural activity was recorded by the use of a technique that offered simultaneously extensive coverage of cortical surfaces along with high spatial and temporal resolution. Two different complexitymetrics methods, the Fractal Dimension, and the Permutation Entropy were used to infer the complexity of the neural time series.
In this research, we have characterized the complexity of the electrophysiological neural activity in the awakened resting-state, Ketamine-Medetomidine general anesthesia, and also during the transition to induced unconsiousness. We have analyzed the complexity over distinct cortical regions at a local level, and also the spatiotemporal patterns established throughout the cortex at a systems level. Under resting-state conditions, the complexity of cortical activity demonstrated to be relatively constant along time, despite some events of reduced complexity that appeared and vanished quite frequently, taking place mainly over the temporal and occipital lobes. The transition to unconsciousness demonstrated to be quite fast, requiring about 30 to 40 seconds for the first remarkable alterations to take place. First, the complexity decreased at the occipital and temporal lobes, later diminished over the parietal and frontal lobes. During the transition, the central sulcus and areas nearby continued to display high complexity. We contemplated that the administration of the anesthetics promoted a novel specific dynamics and not a single overall decrease effect on the complexity of brain activity. Locally, the complexity assumed a higher variation and randomly fluctuated without having a definite period. During general induced anesthesia the whole cortical spatiotemporal patterns dynamically alternated among high and low complex states, the states of high complexity observed had some features that resembled those respective to the awakened resting-state, nonetheless, we verified that both conditions were distinguishable at all times.
II. Methods
I. Neural Records Database
In the present study, a database of cortical electrophysiological activity was analyzed. The database came from the laboratory of adaptive intelligence at the Riken Brain Science Institute, Japan. All surgical and experimental procedures were idealized and performed by the re-searchers of the corresponding institution. Experiments were approved by the Riken's scientific ethics committee following the experimental protocols ) and the recommendations of the Weatherall report: "The use of non-human primates in research". Detailed information regarding methodology, subjects and materials is available on (Nagasaka et al., 2011) and (http://neurotycho.org).
The electrophysiological activity data analyzed is respective to an experiment that involved anesthetic induction in a non-human primate subject of the species Macaca fuscata. The data recording technique adopted was the MDR-ECoG, which is considered to be one of the most advanced technologies available. This technique provided a large coverage of the cortical surface and offered concomitant high spatial (5 mm) and high temporal resolution (1 KHz). The animal subject had an array of 128 ECoG electrodes chronically implanted at the subdural space covering the lateral cortical surface of the right brain hemisphere, some electrodes were also implanted at frontal and occipital medial walls.
In the experiment, the macaque was blindfolded and restrained in an appropriate chair. Neural activity was recorded for approximately 10 minutes under these conditions. After that, a cocktail of Ketamine and Medetomidine was administered for the induction of general anesthesia; thereafter neural activity continued to be monitored for the next 25 minutes.
II. Complexity-Metrics Methods:
II.1 Permutation Entropy
For the calculation of the Permutation Entropy, we have used the algorithm presented and described in (Bandt and Pompe, 2002). For each of the 128 time series from the ECoG electrodes array, the Permutation Entropy was calculated serially over time throughout the experiment as a sliding window. For each calculation, 2000 points of the time series were used, equivalent to 2 seconds of recording of neural activity, with the order parameter of the Permutation Entropy m = 4, and the time delay t = 15.
II.2 Fractal Dimension
For each of the 128 time series from the ECoG electrodes array, the Fractal Dimension values were calculated serially over time as a sliding window. To decrease the number of measurements, the Fractal Dimension was calculated once every 2.5 seconds respective to the recording time of the experiment. Higuchi's algorithm (Higuchi, 1988) was used to calculate the Fractal Dimension. For each calculation, 1000 points of the time series were used (equivalent to 1 second of recording the neural activities), with the time interval parameter k = 100.
III. Statistical Analysis Wilcoxon Signed-Rank Test
In aiming to verify whether the decrease observed in the complexity during general anesthesia compared to the awake resting state conditions was statistically significant, the Wilcoxon signed-rank test was applied. The resulting p-values of the statistical test are shown in Table ·
IV. t-SNE Plots
We have defined the time-resolved vector states, as vectors containing as entries the values of complexity at a given time of each one of the 128 electrodes, resulting in 128-Dimensional vectors that were estimated serially over time throughout the experiment. For the permutation entropy, the time-resolved vector states were estimated at every 2.0 seconds of the recording experiment, and for the Fractal Dimension at every 2.5 seconds. The timeresolved vector states (128-D) were used as input features into the t-SNE algorithm (Van der Maaten and Hinton, 2008) implemented in the R-CRAN package Rtsne (Krijthe et al., 2018), with the parameters: perplexity = 30, exaggeration factor = 12, and the maximum number of interactions = 500. This analysis was independently performed for the Fractal Dimension and for the Permutation Entropy.
III. Results
In this research, we were able to trace the spatiotemporal patterns of the cortical electrophysiological activity's complexity from the macaque under study. The dynamic patterns were characterized in awakened resting-state conditions and Ketamine-Medetomidine general anesthesia. A substantial difference between these two states has been verified. The unfolding dynamics of the loss of consciousness have also been observed, and we were able to follow the alterations that occurred over specific cortical areas as well as in the cortex at a global level.
Two different methodologies were employed to estimate the complexity of neural time series: the Fractal Dimension and the Permutation Entropy. Although some distinctions were observed among the results from each methodology, in a broad sense the findings obtained were fairly equivalent, and we were able to draw the same conclusions from both approaches.
I. Awakened Resting State
By analyzing the figures of the Permutation Entropy (Figure · 1) and the Fractal Dimension (Figure · 6) along time throughout the anesthetic induction, we verified that during resting state conditions, the complexity of cortical activity did not present an expressive and remarkable variation, in general being considerably constant over time (see Figure · 1 and Figure · 6, the values obtained before the vertical red line at 10.5 minutes in both figures). Regarding the magnitude of the values, during the resting state, the Permutation Entropy varied among the range of ≈ 3.1 to 3.2 (see Figure · 1), and the Fractal Dimension among the range of ≈ 1.7 to 1.8 (see Figure · 6). We also noticed some differences in the dynamic behavior according to the position of the electrodes over the cortex. Among all electrodes, those positioned over the frontal lobe (electrodes A to L) were the ones that exhibited the lowest variation, whereas electrodes located over the occipital and temporal lobes (electrodes N to V) were more prone to display some variation (see Figure · 1 and Table · 1 for the Permutation Entropy; Figure · 6 and Table · 2 for the Fractal Dimension). By analyzing the values of complexity over the coordinates of the electrodes, we verified the spatiotemporal patterns respective to the complexity of neural activity over distinct cortical regions, as well as the behavior of the cortex at a system's level during the awakened resting-state conditions (see Figure · 3, and Figure · 8). We have noticed that the complexity of the cortical activity was approximately the same over the whole cortex, no particular region that stood out for presenting complexity intrinsically significantly higher or inferior all the time was verified. We observe in Figure · 3 and also in Figure · 8, that the general aspect and most frequent patterns consisted of the majority of the electrodes displaying a predominantly red color according to the gradient chart. Localized events characterized by a reduction in the complexity of cortical activity were quite common although, these consisted of events that constantly appeared and vanished, not being present all the time. These events occurred mainly at the occipital and temporal lobes (see Figure Figures), the complexity of the electrophysiological activity remained the same without presenting noticeable changes for approximately 1.5 to 2.0 minutes. After this time interval, we were able to contemplate an abrupt and expressive transition, which, due to its rapid occurrence, seemed to be consistent with a single-step process. The previously higher and approximately constant values presented a decrease and started to exhibit a considerably larger variation (see Figure · By analyzing the values of complexity over the coordinates of the electrodes, we could contemplate how distinct specialized areas, as well as the cortex at a systems level, behaved along with the transition (see Figure Figure · 9). Considering that the time interval between each Sub-Figure and its subsequent is 2.5 seconds, we thereby estimate that the change between the patterns occurred within approximately 30 to 40 seconds (see Figure · 9). We thus arrive at the same conclusion from both figures.
The unfolding of the transition over different cortical regions can be ascertained by observing the consecutive Sub-Figures of Figure · 9 respective to the Fractal Dimension. First, a substantial decrease in the Fractal Dimension values in the occipital, temporal and parietal areas occurred without a significant reduction in the electrodes of the frontal region, (see Figure · 9, Sub- Figures 23, 24, 28, 29, 31 and 32). Later on, the complexity of the electrodes located over frontal areas also started to decline, (see Figure · 9, 40,42,43 and 48). A remarkable feature verified along the transition was that electrodes located over the central sulcus and areas nearby showed a tendency to display higher complexity when compared to the rest of the frontal and parietal lobes (see Figure · 9, Sub- Figures 37 to 48). Another recurrent pattern observed was that both the central sulcus and the occipital lobe displayed a higher complexity in comparison with other areas of the cortex during the transition, (see Figure · 9, 43,44,45 and 46). All these changes observed on the spatiotemporal patterns contrast the features found during awaken conditions. However, the recurrent spatiotemporal patterns characteristic of the general anesthesia that are represented on ) over time throughout the anesthetic induction, we have noticed that the first remarkable effects on the electrophysiological cortical activity complexity occurred within about 1.5 to 2.0 minutes after the administration of the anesthetics. After a few minutes following these first substantial changes, the dynamics of the complexity of cortical activity assumed a regime that prevailed during general anesthesia. It was observed on both Figure · 1 and Figure · 6, that the cortical activity's complexity started to assume a wider variation, and seemed to fluctuate without possessing a definite period, having the upper limits about the same as the values found during the resting state, and the magnitude of the lower values was much smaller. Specifically, the Permutation Entropy varied in the range of ≈ 2.2 to 3.2, and the Fractal Dimension in the interval of ≈ 1.3 to 1.8. By applying the Wilcoxon signed-rank test with a p-value of 5%, we confirmed that the general decrease observed on the magnitude values of complexity over all the electrodes of the ECoG matrix 1 was statistically significant on both Permutation Entropy and Fractal Dimension.
Examining the histograms (see Figure · 2 and Figure · 7) we can find a considerable distinction among the distributions of the two conditions. During general anesthesia, the distributions tended to be lower and substantially more widespread than during resting-state conditions. This divergence was markedly pronounced in the electrodes located over the frontal lobe.
When we analyze Figure · 1 and Figure · 6 that comprise the complexity of neural activity over time, we may have the impression that during general anesthesia, the complexity of each electrode fluctuated in a seemingly random manner. Although when we have plotted the complexity values over the coordinates of the electrodes by using a color gradient, a remarkable phenomenon has been observed. We have found that, during general anesthesia, the complexity of cortical activity alternates among high and low complex states. There were times when the majority of the electrodes presented reduced complexity and thus assumed a predominantly blue color. And there were periods in which most of the electrodes showed high complexity and had mainly red coloring, apparently resembling the patterns and features observed during alert resting-state conditions (see Figure · 5 and Figure · 10).
To visualize how the time-resolved complexity vector states (high dimensional 128-D, see methods section -IV) were spatially distributed, we have used the t-SNE algorithm (Van der Maaten and Hinton, 2008) to project them into a bidimensional map. We have verified that points respective to each experimental condition, awakened resting state, and general anesthesia positioned spread at distinct regions of the plane without mixing among themselves (see Figure · 11). We have verified these results in both complexity measures used, the Permutation Entropy and the Fractal Dimension. From these findings, we can conclude that re-garding the time-resolved complexity vector states, the awakened resting-state and general anesthesia compromise two distinct states, that are distinguishable from each other at all times. Although some patterns found during general anesthesia (see Figure · 5 and Figure · 6) seemed to resemble those found during alert conditions (see Figure · 3 and Figure · 8), they were in reality distinct.
IV. Conclusions
In the present research, we inferred the complexity of electrophysiological neural records of a database respective to a Ketamine-Medetomidine anesthetic induction experiment in a macaque subject chronically implanted with a dense ECoG electrodes array at the subdural space along the right brain hemisphere. We have inferred the dynamics of the complexity of the neural activity over distinct cortical areas and the spatial-temporal patterns established along the cortex, during awakened resting state and general anesthesia conditions. The unfolding of the transition to induced unconsciousness has also been followed. We have verified that during awakened resting-state conditions, the complexity is relatively uniform across the cortex, despite some events of localized reduction of complexity frequently occurring at the occipital and temporal lobes. We have found that the first remarkable changes in the complexity of neural activity occurred within about ≈ 1.5 to 2.0 minutes after the Ketamine-Medetomidine cocktail was administrated, and unrolled as a considerably rapid process that took about 30 to 40 seconds to succeed. During general anesthesia, at a local level, the complexity of neural activity assumed a remarkable higher variation having the maximum values about the same as those found during the resting state, and the lower values much smaller. The frontal region was the one that presented the most prominent alterations. Regarding the spatiotemporal patterns assumed by the cortex as a whole, we verified that during Ketamine-Medetomidine general anesthesia the cortex alternated among high and low complex states. Those high complex states displayed features that resembled the patterns found during the awakened resting-state conditions, nonetheless, they were still distinguishable at all times. In general terms, it was noted that in alert conditions, on the majority of the electrodes, the Permutation Entropy of the neural records presented relatively small variation over time. After the administration of the cocktail of anesthetics, the Permutation Entropy remained about the same for about 1.5 to 2.0 minutes, then abrupt changes were verified. It was noticed that the complexity of the neural records assumed a considerable greater variation, being the upper values about the same as those found during alert conditions, and the bottom values were fairly lower. It was also possible to observe that not all electrodes behaved the same, evidencing that the effects of the anesthetics on electrophysiological activities present some variation throughout the cortex. Figure, it is possible to visualize the characteristic pattern of the complexity in awakened resting-state conditions. It is noticeable that under these conditions the majority of the electrodes present a Permutation Entropy value of around 3.0 and over, and the whole cortex did not display an expressive variation over time, as the Sub-Figures display one main similar pattern. Nonetheless, it is possible to observe that electrodes located in the occipital regions showed some tendency to variation (see 20,21,and 30). Considering the cortex as a whole, the region that was less prone to variation was the frontal lobe. Figure, it is possible to visualize the first alterations that happened in the patterns of the awake resting-state following the administration of the anesthetics. Primarily some regions of the temporal and occipital lobes presented a reduction while frontal electrodes did not display expressive alterations (see Sub- Figures 10, 14, 15, 16, and 19). Thereafter, from Sub- Figure 26 onwards, a reduction in the electrodes of the frontal lobe was verified. A remarkable characteristic observed during these first moments of the transition was that the central sulcus and nearby areas showed a tendency to exhibit high Permutation Entropy (see Sub- Figures 15 to 36). Sub- Figures 8 and 9 had patterns characteristic of awake conditions, whereas Sub- Figures 28 and 29 already displayed substantially different features. Considering that the time interval among consecutive Sub-Figures is 2.0 seconds, we infer that the first abrupt changes on the complexity of the electrophysiological activity of the cortex as a whole took place within about 30 to 40 seconds. It is noticeable that the cortex alternates between high complexity states that resemble those observed during alert conditions (see 5,10,16,24,27 and 29), and low complexity states in which the vast majority of cortical areas display reduced Permutation Entropy values (see 6,9,11,12,14,17,20,22,23,25,26,28 and 30). It was also observed that even at periods of low complexity there was still a tendency for the temporal lobe and the central sulcus to assume slightly higher values than the rest of the cortex (see 7,12,15,18,19,and 26), being a recurrent pattern that appeared with a relative frequency over general anesthesia conditions, although not always present throughout all the time. Brain activity started to be recorded in awakening resting-state conditions. The anesthetic drugs cocktail was administrated at 10.5 minutes, this event is indicated in each Sub-Figure by a vertical red line. The Fractal Dimension was calculated at every 2.5 seconds throughout the experiment, being this the time interval among each point and its subsequent one. In this Figure, it is possible to observe the dynamics of the Fractal Dimension throughout the anesthetic induction experiment. During awakened alert conditions, there was a tendency to remain approximately constant around ≈ 1.7 to 1.8 in most electrodes. Within about 1.5 to 2.0 minutes after the anesthetic agents administration, the Fractal Dimension values presented an abrupt change, it decreased and started to assume a considerably larger variation than in alert conditions, apparently fluctuating randomly between a minimum of approximately 1.3 and a maximum of about 1.8. It was noticeable that the behavior of the Fractal Dimension was not the same over different cortical regions, evidencing that the effects of the anesthetics on the electrophysiological activity are not the same throughout the whole cortex. Figure, it is possible to visualize the characteristic patterns of the complexity in awakened resting-state conditions. It is noticeable that under these conditions the majority of the electrodes present Fractal Dimension values of around 1.6 and over. It is possible to observe that electrodes located at the occipital regions showed some tendency to display variation (see 10,11,12,19,29,and 30). Figure, it is possible to visualize the first alterations that happened in the patterns of the awake resting-state condition following the administration of the anesthetics. Primarily some regions of the temporal and occipital lobes presented a reduction while frontal electrodes did not display expressive alterations (see 30,31,and 32). Thereafter, a reduction in the electrodes of the frontal lobe was verified (see 40,42,43,46,48,50,51,and 53 It is possible to visualize the characteristic patterns as well as the dynamics of the Fractal Dimension during the state of general anesthesia, it is noticeable that the cortex alternates between high complexity states that appear similar to those observed during alert conditions (see 6,14,17,23,29,32,35,38,and 41 ) and low complexity states in which the vast majority of cortical areas display reduced Fractal Dimension values (see 4,10,13,15,18,19,21,24,25,27,28,30,31,33,34,36,37,39,40,and 42 ). It was also observed that even at periods of low complexity there was still a tendency for the occipital-frontal and central sulcus to assume slightly higher values than the rest of the cortex (see 8,9,11,12,17,19,20,22,26,and 27).
(A) Electrode A (B) Electrode B (C) Electrode C (D) Electrode D (E) Electrode E (F) Electrode F (G) Electrode G((A) Electrode A (B) Electrode B (C) Electrode C (D) Electrode D (E) Electrode E (F) Electrode F (G) Electrode G((A) Electrode A (B) Electrode B (C) Electrode C (D) Electrode D (E) Electrode E (F) Electrode F (G) Electrode G(
That was a recurrent pattern that has shown up with a relative frequency over general anesthesia conditions, although not always present throughout all the time. Figure B) onto a bi-dimensional map by the use of the t-SNE algorithm. Alert resting-state conditions are indicated in blue, and general anesthesia in red, see legends. By examining this Figure it is verified that in both methods, the Permutation Entropy and the Fractal Dimension, the projections of the alert state (indicated in blue) and general anesthesia (indicated in red) occupy distinct regions of the map. This result thus evidences that concerning the complexity of the electrophysiological cortical activity, alert resting-state and general anesthesia constitute two different states and are distinguishable at every moment.
· 3 and Figure · 8). II. Transition By analyzing the Figures of the Permutation Entropy and the Fractal Dimension over time throughout the anesthetic induction (see Figure · 1, and Figure · 6), it was verified that after the administration of the anesthetics (indicated by the vertical red line at 10.5 minutes in both
1 ,
1Sub-Figures A to V; and Figure · 6, Sub-Figures A to L).
· 4 ,
4and Figure · 9). Changes were noticeable in both measures of complexity evaluated, (see Figure · 4) for the Permutation Entropy and (see Figure · 9) for the Fractal Dimension. Out of these two pictures, the one concerning the Fractal Dimension (Figure · 9) had the changes on the spatiotemporal patterns quite more apparent. The first information we can infer from the Figures is the approximate time needed for the changes to take place. In the Permutation Entropy Figure · 4, we verify that Sub-Figures 8 and 9 presented patterns characteristic of the alert condition, while Sub-Figures 28 and 29 already displayed very distinct patterns from those observed previously (see Figure · 4). Since the time interval among consecutive Sub-Figures is 2.0 seconds, it is possible to infer that the change in the state occurred within approximately 30 to 40 seconds (see Figure · 4). Regarding the Fractal Dimension Figure · 9, we verified that Sub-Figures 26 and 27 presented patterns characteristic of the alert condition, while Sub-Figures 39 and 40 displayed considerably different from the previous ones (see
Figure · 5 and
5Figure · 10, only showed up after a few minutes following the occurrence of the first abrupt changes. III. General Anesthesia By analyzing the Figures of the Permutation Entropy (Figure · 9) and the Fractal Dimension (Figure ·
Figure 1 :Figure 1 :
11H) Electrode H (I) Electrode I (J) Electrode J (K) Electrode K (L) Electrode L (M) Electrode M (N) Electrode N (O) Electrode O Permutation Entropy of the electrophysiological time series over time along with the anesthetic induction experiment. (P) Electrode P (Q) Electrode Q (R) Electrode R (S) Electrode S (T) Electrode T (U) Electrode U (V) Electrode V Permutation Entropy of the electrophysiological time series over the time throughout the anesthetic induction experiment. Each Sub-Figure is respective to an electrode which was positioned over a specific cortical area, for the exact location of each electrode (see Supplementary Figure · 1, the corresponding letters from A to V). Brain activity started to be recorded in awakening resting-state conditions. At 10.5 minutes the anesthetic drugs were administrated, in each Sub-Figure this event is represented by a vertical red line. Permutation Entropy was calculated at every 2.0 seconds throughout the experiment, being this the time interval among each point and its subsequent one.
Figure 2 :
2H) Electrode H (I) Electrode I (J) Electrode J (K) Electrode K (L) Electrode L (M) Electrode M (N) Electrode N (O) Electrode O (P) Electrode P (Q) Electrode Q (R) Electrode R (S) Electrode S (T) Electrode T (U) Electrode U (V) Electrode V Overlapping histogram plots of the Permutation Entropy. Overlapping histogram plots of the electrophysiological time series's Permutation Entropy respective to the awakened (indicated in yellow) and general anesthesia conditions (indicated in red) see color chart. Sub-Figures correspond to electrodes positioned over a specific cortical region, for the location of each electrode (see SupplementaryFigure · 1, the corresponding letters from A to V). In general lines, the Permutation Entropy values of the macaque's cortical electrophysiological activity under general anesthesia tended to be smaller and more widespread when compared to awaken conditions, despite some variations being noticeable over distinct cortical areas. Considering the anesthetic induction, the ECoG electrodes that exhibited the most prominent changes were those positioned over the frontoparietal regions (electrodes A to L, see Sub-Figures A to L) and the electrodes G and H of the parietal lobe (seeSub-Figures G and H). These electrodes presented the most constant values and had a magnitude around ≈ 3.1 to 3.2 in awake resting-state conditions. While during general anesthesia the Permutation Entropy was widespread over the range from ≈ 2.2 to 3.2, not being noticed a prominent overlap among the distributions of both conditions.
Figure 3 :
3Permutation Entropy during awake resting-state conditions. In this Figure, it is shown the permutation Entropy values over the locations of their respective electrodes. Sub-Figures were estimated sequentially throughout time at every 2.0 seconds, being this the time interval between each Sub-Figure and its subsequent one. The magnitude of the Permutation Entropy is indicated by the color gradient. In this
Figure 4 :
4Permutation Entropy during the transition to Ketamine-Medetomidine induced unconsciousness. This Figure shows the Permutation Entropy values over the locations of their respective electrodes. Sub-Figures were estimated sequentially throughout time at every 2.0 seconds, being this the time interval between each Sub-Figure and its subsequent one. The magnitude of the Permutation Entropy is indicated by the color gradient. In this
Figure 5 :
5Permutation Entropy during general anesthesia. This Figure shows the Permutation Entropy values over the locations of their respective electrodes. Sub-Figures were estimated sequentially throughout time at every 2.0 seconds, being this the time interval between each Sub-Figure and its subsequent one. The magnitude of the Permutation Entropy is indicated by the color gradient. It is possible to visualize the characteristic patterns as well as the dynamics of Permutation Entropy during the state of general anesthesia.
( A )Figure 6 :Figure 6 :
A66Electrode A (B) Electrode B (C) Electrode C (D) Electrode D (E) Electrode E (F) Electrode F (G) Electrode G (H) Electrode H (I) Electrode I (J) Electrode J (K) Electrode K (L) Electrode L (M) Electrode M (N) Electrode N (O) Electrode O Fractal Dimension of the electrophysiological cortical activity throughout the anesthetic induction. (P) Electrode P (Q) Electrode Q (R) Electrode R (S) Electrode S (T) Electrode T (U) Electrode U (V) Electrode V Fractal Dimension of the electrophysiological cortical activity throughout the anesthetic induction. Each Sub-Figure is respective to an electrode which was positioned over a specific cortical area, for the exact location of each electrode (see Supplementary Figure · 1, the corresponding letters from A to V).
Figure 7 :
7H) Electrode H (I) Electrode I (J) Electrode J (K) Electrode K (L) Electrode L (M) Electrode M (N) Electrode N (O) Electrode O (P) Electrode P (Q) Electrode Q (R) Electrode R (S) Electrode S (T) Electrode T (U) Electrode U (V) Electrode V Overlapping histogram plots of the Fractal Dimension. Overlapping histogram plots of the electrophysiological time series's Fractal Dimension respective to the awakened (indicated in yellow) and general anesthesia conditions (indicated in red), see color chart. Each Sub-Figure corresponds to an electrode that was positioned over a specific cortical region, for the location of each electrode, (see SupplementaryFigure · 1, the corresponding letters from A to V). In general lines, the Fractal Dimension values of the macaque's cortical electrophysiological activity under general anesthesia tended to be smaller and more widespread than during awaken conditions. Some distinctions were observed according to the location of the electrodes over the cortex. The ECoG electrodes positioned over the frontoparietal regions (electrodes A to L, see Sub-Figures A to L), had considerably constant values and presented magnitude around 1.8 in awake resting-state conditions. While during general anesthesia the Fractal Dimension was most widespread over the range from 1.1 to 1.8, existing a relatively small overlap between the distributions of the two conditions. Among the electrodes located in the occipital-temporal areas (electrodes M to V, see Sub-Figures M to V) presented more variation during the awakened resting-state when compared to the frontoparietal ones.
Figure 8 :
8Fractal Dimension during awake resting-state conditions. In this Figure, it is shown the Fractal Dimension values over the locations of their respective electrodes. Sub-Figures were estimated sequentially throughout time at every 2.5 seconds, being this the time interval between each Sub-Figure and its subsequent one. The magnitude of the Fractal Dimension is indicated by the color gradient. In this
Figure 9 :
9Fractal Dimension during the transition to Ketamine-Medetomidine induced unconsciousness. This Figure shows the Fractal Dimension values over the locations of their respective electrodes. Sub-Figures were estimated sequentially throughout time at every 2.5 seconds, being this the time interval between each Sub-Figure and its subsequent one. The magnitude of the Fractal Dimension is indicated by the color gradient. In this
Figure 10 :
10Fractal Dimension during general anesthesia. This Figure shows the Fractal Dimension values over the locations of their respective electrodes. Sub-Figures were estimated sequentially throughout time at every 2.5 seconds, being this the time interval between each Sub-Figure and its subsequent one. The magnitude of the Fractal Dimension is indicated by the color gradient.
Figure 11 :
11Projections of the time-resolved vector states onto a 2-D map. Projections of the time-resolved vector states (128-D) respective to the Permutation Entropy (Sub-Figure A), and the Fractal Dimension (Sub-
1 for the Permutation Entropy and Table · 2 for the Fractal Dimension.
Table 1 :
1Mean and standard deviation of the Permutation Entropy values in alert and general anesthesia.This table presents the mean and standard deviation of the Permutation Entropy values in alert and general anesthesia conditions, from the electrodes located at distinct cortical regions. For the location of each electrode (see Supplementary Figure · 1, the corresponding letters from A to V). The p-values of the Wilcoxon signedrank test are shown in the condition that the Permutation Entropy values found during the state of anesthesia are smaller than those observed during the alert resting state. It was verified for all the electrodes that the average of the Permutation Entropy during alert conditions is higher than the average found in general anesthesia and that the standard deviation is smaller during alertness than during anesthesia. For all electrodes, the Wilcoxon test at a p-value of 5% confirmed that the Permutation Entropy of the cortical records decreased during the Ketamine-Medetomidine induced general anesthesia.Permutation Entropy
Awake
Anesthesia
Wilcoxon Test
Electrode:
Mean
SD Mean
SD P-Value [Anesthesia < Awake]
Electrode A
3.15
0.015 2.68
0.251
4.7e-130
Electrode B
3.15
0.017 2.70
0.224
7.9e-131
Electrode C
3.12
0.036 2.87
0.196
7.8e-95
Electrode D
3.15
0.012 2.93
0.149
4.1e-124
Electrode E
3.13
0.025 2.75
0.223
1.3e-127
Electrode F
3.11
0.034 2.81
0.176
2e-121
Electrode G
3.13
0.029 2.77
0.201
8.3e-128
Electrode H
3.15
0.015 2.81
0.188
4.2e-130
Electrode I
3.11
0.043 2.86
0.188
5.4e-104
Electrode J
3.12
0.029 2.97
0.121
2.3e-99
Electrode K
2.98
0.101 2.79
0.166
2.2e-58
Electrode L
3.12
0.037 2.84
0.198
1.3e-115
Electrode M
3.03
0.073 2.70
0.224
2.6e-99
Electrode N
3.07
0.050 2.91
0.134
1e-86
Electrode O
3.02
0.090 2.86
0.146
4e-64
Electrode P
3.14
0.026 2.92
0.125
5.4e-123
Electrode Q
3.00
0.075 2.68
0.199
2.9e-110
Electrode R
3.04
0.062 2.64
0.189
6.6e-128
Electrode S
3.06
0.057 2.83
0.145
1.9e-109
Electrode T
3.03
0.072 2.72
0.173
3.6e-117
Electrode U
3.05
0.062 2.77
0.189
1.4e-103
Electrode V
3.04
0.096 2.84
0.152
1.9e-71
Table 2 :
2Mean and standard deviation of the Fractal Dimension values in alert and general anesthesia. This table presents the mean and standard deviation of the Fractal Dimension values in alert and general anesthesia conditions, from the electrodes located at distinct cortical regions. For the location of each electrode (see SupplementaryFigure ·1, the corresponding letters from A to V). The p-values of the Wilcoxon signed-rank test are shown in the condition that the Fractal Dimension values found during the state of anesthesia are smaller than those observed during the alert resting state. It was verified for all the electrodes that the average of the Fractal Dimension during alert conditions is higher than the average found in general anesthesia and that the standard deviation is smaller during alertness than during anesthesia. For all electrodes, the Wilcoxon test at a p-value of 5% confirmed that the Fractal Dimension of the cortical records decreased during the Ketamine-Medetomidine-induced general anesthesia.Fractal Dimension
Awake
Anesthesia
Wilcoxon Test
Electrode:
Mean
SD Mean
SD P-Value [Anesthesia < Awake]
Electrode A
1.75
0.071 1.32
0.17
5.3e-97
Electrode B
1.80
0.047 1.32
0.156
2.8e-104
Electrode C
1.82
0.035 1.47
0.2
2.4e-76
Electrode D
1.81
0.038 1.52
0.191
1.4e-66
Electrode E
1.74
0.055 1.38
0.193
3.8e-80
Electrode F
1.68
0.085 1.41
0.148
3.5e-79
Electrode G
1.81
0.036 1.39
0.173
5.6e-96
Electrode H
1.83
0.034 1.41
0.177
8.6e-99
Electrode I
1.8
0.04 1.46
0.203
1.1e-73
Electrode J
1.77
0.04 1.50
0.156
2.4e-79
Electrode K
1.74
0.056 1.43
0.172
4.4e-82
Electrode L
1.78
0.033 1.42
0.186
2.7e-95
Electrode M
1.70
0.07 1.35
0.154
4.6e-92
Electrode N
1.64
0.096 1.51
0.161
1.5e-27
Electrode O
1.54
0.149 1.50
0.160
0.003
Electrode P
1.74
0.063 1.52
0.157
7.4e-71
Electrode Q
1.58
0.089 1.33
0.129
1.3e-81
Electrode R
1.61
0.084 1.31
0.115
8.2e-96
Electrode S
1.63
0.108 1.43
0.136
1.3e-61
Electrode T
1.61
0.094 1.36
0.126
3.4e-81
Electrode U
1.65
0.076 1.39
0.148
3.4e-79
Electrode V
1.72
0.052 1.49
0.157
8.7e-75
). A remarkable characteristic observed during these first moments of the transition was that the central sulcus and nearby areas showed a tendency to exhibit high Fractal Dimension (see Sub-Figures 33, 34, 36, 37, 38, 39, 41, 44, 46, 47, 48, 49, 50, 51, and 52). Sub-Figures 26 and 27 had patterns characteristic of awake conditions, whereas Sub-Figures 39 and 40 already displayed substantially different features. Considering that the time interval among consecutive Sub-Figures is 2.5 seconds, we infer that the first abrupt changes on the complexity of cortical activity as a whole took place within about 30 to 40 seconds.
This manuscript does not display the statistical test for all 128 electrodes of the ECoG matrix but displays the most representative ones according to their location over the cortical surface (see Supplementary Figure · 1).
Permutation entropy: a natural complexity measure for time series. Christoph Bandt, Bernd Pompe, Physical review letters. 8817174102Christoph Bandt and Bernd Pompe. Permuta- tion entropy: a natural complexity measure for time series. Physical review letters, 88(17): 174102, 2002.
Approach to an irregular time series on the basis of the fractal theory. Tomoyuki Higuchi, Physica D: Nonlinear Phenomena. 312Tomoyuki Higuchi. Approach to an irregular time series on the basis of the fractal the- ory. Physica D: Nonlinear Phenomena, 31(2): 277-283, 1988.
Laurens van der Maaten, and Maintainer Jesse Krijthe. Package 'rtsne'. Jesse Krijthe, Jesse Krijthe, Laurens van der Maaten, and Maintainer Jesse Krijthe. Package 'rtsne', 2018.
Multidimensional recording (mdr) and data sharing: an ecological open research and educational platform for neuroscience. Yasuo Nagasaka, Kentaro Shimoda, Naotaka Fujii, PloS one. 6722561Yasuo Nagasaka, Kentaro Shimoda, and Nao- taka Fujii. Multidimensional recording (mdr) and data sharing: an ecological open re- search and educational platform for neuro- science. PloS one, 6(7):e22561, 2011.
Laurens Van der Maaten and Geoffrey Hinton. Visualizing data using t-sne. Journal of machine learning research. 911Laurens Van der Maaten and Geoffrey Hin- ton. Visualizing data using t-sne. Journal of machine learning research, 9(11), 2008.
| []
|
[
"Contracting isometries of CAT(0) cube complexes and acylindrical hyperbolicity of diagram groups",
"Contracting isometries of CAT(0) cube complexes and acylindrical hyperbolicity of diagram groups"
]
| [
"Anthony Genevois "
]
| []
| []
| The main technical result of this paper is to characterize the contracting isometries of a CAT(0) cube complex without any assumption on its local finiteness. Afterwards, we introduce the combinatorial boundary of a CAT(0) cube complex, and we show that contracting isometries are strongly related to isolated points at infinity, when the complex is locally finite. This boundary turns out to appear naturally in the context of Guba and Sapir's diagram groups, and we apply our main criterion to determine precisely when an element of a diagram group induces a contracting isometry on the associated Farley cube complex. As a consequence, in some specific case, we are able to deduce a criterion to determine precisely when a diagram group is acylindrically hyperbolic. | 10.2140/agt.2020.20.49 | [
"https://arxiv.org/pdf/1610.07791v1.pdf"
]
| 119,178,956 | 1610.07791 | 811250db407c6b8a18b2e992467da5ccd620e8ac |
Contracting isometries of CAT(0) cube complexes and acylindrical hyperbolicity of diagram groups
October 26, 2016
Anthony Genevois
Contracting isometries of CAT(0) cube complexes and acylindrical hyperbolicity of diagram groups
October 26, 2016
The main technical result of this paper is to characterize the contracting isometries of a CAT(0) cube complex without any assumption on its local finiteness. Afterwards, we introduce the combinatorial boundary of a CAT(0) cube complex, and we show that contracting isometries are strongly related to isolated points at infinity, when the complex is locally finite. This boundary turns out to appear naturally in the context of Guba and Sapir's diagram groups, and we apply our main criterion to determine precisely when an element of a diagram group induces a contracting isometry on the associated Farley cube complex. As a consequence, in some specific case, we are able to deduce a criterion to determine precisely when a diagram group is acylindrically hyperbolic.
Introduction
Given a metric space X, an isometry g ∈ X is contracting if • g is loxodromic, ie., there exists x 0 ∈ X such that n → g n · x 0 defines a quasiisometric embedding Z → X;
• if C g = {g n · x 0 | n ∈ Z}, then the diameter of the nearest-point projection of any ball disjoint from C g onto C g is uniformly bounded.
For instance, any loxodromic isometry of a Gromov-hyperbolic space is contracting. In fact, if a group G acts by isometries on a metric space, the existence of a contracting isometry seems to confer to G a "hyperbolic behaviour". To make this statement more precise, one possibility is to introduce acylindrically hyperbolic groups as defined in [Osi13].
Definition 1.1. Let G be a group acting on a metric space X. The action G X is acylindrical if, for every d ≥ 0, there exist some constants R, N ≥ 0 such that, for every x, y ∈ X,
d(x, y) ≥ R ⇒ #{g ∈ G | d(x, gx), d(y, gy) ≤ d} ≤ N .
A group is acylindrically hyperbolic if it admits a non-elementary (ie., with an infinite limit set) acylindrical action on a Gromov-hyperbolic space.
Acylindrically hyperbolic groups may be thought of as a generalisation of relatively hyperbolic groups. See for example [Osi13,Appendix] and references therein for more details. The link between contracting isometries and acylindrically hyperbolic groups is made explicit by the following result, which is a consequence of [BBF14,Theorem H] and [Osi13, Theorem 1.2]: Theorem 1.2. If a group acts by isometries on a geodesic space with a WPD contracting isometry, then it is either virtually cyclic or acylindrically hyperbolic.
We do not give the precise definition of a WPD isometry. The only thing to know in our paper is that any isometry turns out to be WPD if the action of our group is properly discontinuous.
In this paper, we focus on a specific class of geodesic spaces: the CAT(0) cube complexes, ie., simply connected cellular complexes obtained by gluing cubes of different dimensions by isometries along their faces so that the link of every vertex is a simplicial flag complex. See Section 2 for more details. The first question we are interested in is: when is an isometry of a CAT(0) cube complex contracting? The answer we give to this question is the main technical result of our paper. Our criterion essentially deals with hyperplane configurations in the set H(γ) of the hyperlanes intersecting a combinatorial axis γ of a given loxodromic isometry g. We refer to Section 3 for precise definitions. (iv) g skewers a pair of well-separated hyperplanes.
Remark 1.4. The equivalence (i) ⇔ (iv) generalises a similar criterion established in [CS15,Theorem 4.2], where the cube complex is supposed uniformly locally finite.
In view of our application to diagram groups, it turned out to be natural to introduce a new boundary of a CAT(0) cube complex; in the definition below, if r is a combinatorial ray, H(r) denotes the set of the hyperplanes which intersect r. Definition 1.5. Let X be a CAT(0) cube complex. Its combinatorial boundary is the poset (∂ c X, ≺), where ∂ c X is the set of the combinatorial rays modulo the relation: r 1 ∼ r 2 if H(r 1 ) = a H(r 2 ); and where the partial order ≺ is defined by: r 1 ≺ r 2 whenever H(r 1 ) ⊂ a H(r 2 ).
Here, we used the following notation: given two sets X, Y , we say that X and Y are almost-equal, denoted by X = a Y , if the symmetric difference between X and Y is finite; we say that X is almost-included into Y , denoted by X ⊂ a Y , if X is almost-equal to a subset of Y . In fact, this boundary is not completely new, since it admits strong relations with other boundaries; see Appendix A for more details.
A point in the combinatorial boundary is isolated if it is not comparable with any other point. Now, thanks to Theorem 1.3, it is possible to read at infinity when an isometry is contracting: Theorem 1.6. Let X be a locally finite CAT(0) cube complex and g ∈ Isom(X) an isometry with a combinatorial axis γ. Then g is a contracting isometry if and only if γ(+∞) is isolated in ∂ c X.
Thus, the existence of a contracting isometry implies the existence of an isolated point in the combinatorial boundary. Of course, the converse cannot hold without an additional hypothesis on the action, but in some specific cases we are able to prove partial converses. For instance: Theorem 1.7. Let G be a group acting on a locally finite CAT(0) cube complex X with finitely many orbits of hyperplanes. Then G X contains a contracting isometry if and only if ∂ c X has an isolated vertex. Theorem 1.8. Let G be a countable group acting on a locally finite CAT(0) cube complex X. Suppose that the action G X is minimal (ie., X does not contain a proper Ginvariant combinatorially convex subcomplex) and G does not fix a point of ∂ c X. Then G contains a contracting isometry if and only if ∂ c X contains an isolated point.
As mentionned above, the main reason we introduce combinatorial boundaries of CAT(0) cube complexes is to apply our criteria to Guba and Sapir's diagram groups. Loosely speaking, diagram groups are "two-dimensional free groups": in the same way that free groups are defined by concatenating and reducing words, diagram groups are defined by concatenating and reducing some two-dimensional objets, called semigroup diagrams. See Section 5.1 for precise definitions. Although these two classes of groups turn out to be quite different, the previous analogy can be pushed further. On the one hand, free groups act on their canonical Cayley graphs, which are simplicial trees; on the other hand, diagram groups act on the natural Cayley graphs of their associated groupoids, which are CAT(0) cube complexes, called Farley complexes. Moreover, in the same way that the boundary of a free group may be thought of as a set of infinite reduced words, the combinatorial boundary of a Farley cube complex may be thought of as a set of infinite reduced diagrams. See Section 5.2 for a precise description.
If g is an element of a diagram group, which is absolutely reduced (ie., the product g n is reduced for every n ≥ 1), let g ∞ denote the infinite diagram obtained by concatenating infinitely many copies of g. Then, thanks to Theorem 1.6 and a precise description of the combinatorial boundaries of Farley cube complexes, we are able to deduce the following criterion: Theorem 1.9. Let G be a diagram group and g ∈ G\{1} be absolutely reduced. Then g is a contracting isometry of the associated Farley complex if and only if the following two conditions are satisfied:
• g ∞ does not contain any infinite proper prefix;
• for any infinite reduced diagram ∆ containing g ∞ as a prefix, all but finitely many cells of ∆ belong to g ∞ .
Of course, although it is sufficient for elements with few cells, this criterion may be difficult to apply in practice, because we cannot draw the whole g ∞ . This why we give a more "algorithmic" criterion in Section 5.3.
Thus, we are able to determine precisely when a given element of a diagram group is a contracting isometry. In general, if there exist such isometries, it is not too difficult to find one of them. Otherwise, it is possible to apply Theorem 1.7 or Theorem 1.8; we emphasize that Theorem 1.7 may be particularly useful since diagram groups often act on their cube complexes with finitely many orbits of hyperplanes. Conversely, we are able to state that no contracting isometry exists if the combinatorial boundary does not contain isolated points, see for instance Example 5.23.
Therefore, we get powerful tools to pursue the cubical study of negatively-curved properties of diagram groups we initialized in [Gen15a]. To be precise, the question we are interested in is:
Question 1.10. When is a diagram group acylindrically hyperbolic?
Thanks to Theorem 1.2, we are able to deduce that a diagram group is acylindrically hyperbolic if it contains a contracting isometry and if it is not cyclic. Section 5.4 provides families of acylindrically hyperbolic diagram groups which we think to be of interest. On the other hand, a given group may be represented as a diagram group in different ways, and an acylindrically hyperbolic group (eg., a non-abelian free group) may be represented as a diagram group so that there is no contracting isometry on the associated cube complex. We do not know if it is always possible to a find a "good" representation.
Nevertheless, a good class of diagram groups, where the problem mentionned above does not occur, corresponds to the case where the action on the associated complex is cocompact. Using the notation introduced below, they correspond exactly to the diagram groups D(P, w) where the class of w modulo the semigroup P is finite. We call them cocompact diagram groups. Focusing on this class of groups, we prove (see Theorem 5.45 for a precise statement):
Theorem 1.11. A cocompact diagram group decomposes naturally as a direct product of a finitely generated free abelian group and finitely many acylindrically hyperbolic diagram groups.
On the other hand, acylindrically hyperbolic groups cannot split as a direct product of two infinite groups (see [Osi13,Corollary 7.3]), so we deduce a complete answer of Question 1.10 in the cocompact case:
Corollary 1.12. A cocompact diagram group is acylindrically hyperbolic if and only if it is not cyclic and it does not split as a non-trivial direct product.
Notice that this statement is no longer true without the cocompact assumption. Indeed, Thompson's group F and the lamplighter group Z Z are diagram groups which are not acylindrically hyperbolic (since they do not contain a non-abelian free group) and they do not split non-trivially as a direct product. Another consequence of Theorem 1.11 is that a cocompact diagram group either is free abelian or has a quotient which is acylindrically hyperbolic. Because acylindrically hyperbolic groups are SQ-universal (ie., any countable group can be embedded into a quotient of the given group; see [Osi13,Theorem 8.1]), we get the following dichotomy:
Corollary 1.13. A cocompact diagram group is either free abelian or SQ-universal.
In our opinion, cocompact diagram groups turn out to be quite similar to (finitely generated) right-angled Artin groups. In this context, Theorem 1.11 should be compared to the similar statement, but more general, [BC12,Theorem 5.2] (it must be noticed that, although the statement is correct, the proof contains a mistake; see [MO13,Remark 6.21]). Compare also [BC12,Lemma 5.1] with Proposition 5.50. Notice that there exist right-angled Artin groups which are not (cocompact) diagram groups (see [GS99, Theorem 30]), and conversely there exist cocompact diagram groups which are not rightangled Artin groups (see Example 5.43).
The paper is organized as follows. In Section 2, we give the prerequisites on CAT(0) cube complexes needed in the rest of the paper. Section 3 is essentially dedicated to the proof of Theorem 1.3, and in Section 4, we introduce combinatorial boundaries of CAT(0) cube complexes and we prove Theorem 1.6, as well as Theorem 1.7 and Theorem 1.8. Finally, in Section 5, we introduce diagram groups and we apply the results of the previous sections to deduce the various statements mentionned above. We added an appendix at the end of this paper to compare the combinatorial boundary with other known boundaries of CAT(0) cube complexes.
Acknowledgement. I am grateful to Jean Lécureux for having suggested me a link between the combinatorial boundary and the Roller boundary, and to my advisor, Peter Haïssinsky, for all our discussions.
Preliminaries
A cube complex is a CW complex constructed by gluing together cubes of arbitrary (finite) dimension by isometries along their faces. Furthermore, it is nonpositively curved if the link of any of its vertices is a simplicial flag complex (ie., n + 1 vertices span a n-simplex if and only if they are pairwise adjacent), and CAT(0) if it is nonpositively curved and simply-connected. See [BH99, page 111] for more information.
Alternatively, CAT(0) cube complexes may be described by their 1-skeletons. Indeed, Chepoi notices in [Che00] that the class of graphs appearing as 1-skeletons of CAT(0) cube complexes coincides with the class of median graphs, which we now define.
Let Γ be a graph. If x, y, z ∈ Γ are three vertices, a vertex m is called a median point of x, y, z whenever The graph Γ is median if every triple (x, y, z) of pairwise distinct vertices admits a unique median point, denoted by m(x, y, z).
Theorem 2.1. [Che00,Theorem 6
.1] A graph is median if and only if it is the 1-skeleton of a CAT(0) cube complex.
A fundamental feature of cube complexes is the notion of hyperplane. Let X be a nonpositively curved cube complex. Formally, a hyperplane J is an equivalence class of edges, where two edges e and f are equivalent whenever there exists a sequence of edges e = e 0 , e 1 , . . . , e n−1 , e n = f where e i and e i+1 are parallel sides of some square in X. Notice that a hyperplane is uniquely determined by one of its edges, so if e ∈ J we say that J is the hyperplane dual to e. Geometrically, a hyperplane J is rather thought of as the union of the midcubes transverse to the edges belonging to J.
The neighborhood N (J) of a hyperplane J is the smallest subcomplex of X containing J, i.e., the union of the cubes intersecting J. In the following, ∂N (J) will denote the union of the cubes of X contained in N (J) but not intersecting J, and X\\J = (X\N (J)) ∪ ∂N (J). Notice that N (J) and X\\J are subcomplexes of X.
Theorem 2.2. [Sag95,Theorem 4.10] Let X be a CAT(0) cube complex and J a hyperplane. Then X\\J has exactly two connected components.
The two connected components of X\\J will be refered to as the halfspaces associated to the hyperplane J.
Distances p . There exist several natural metrics on a CAT(0) cube complex. For example, for any p ∈ (0, +∞), the p -norm defined on each cube can be extended to a distance defined on the whole complex, the p -metric. Usually, the 1 -metric is referred to as the combinatorial distance and the 2 -metric as the CAT(0) distance. Indeed, a CAT(0) cube complex endowed with its CAT(0) distance turns out to be a CAT(0) space [Lea13, Theorem C.9], and the combinatorial distance between two vertices corresponds to the graph metric associated to the 1-skeleton X (1) . In particular, combinatorial geodesics are edge-paths of minimal length, and a subcomplex is combinatorially convex if it contains any combinatorial geodesic between two of its points.
In fact, the combinatorial metric and the hyperplanes are strongly linked together: the combinatorial distance between two vertices corresponds exactly to the number of hyperplanes separating them [ Combinatorial projection. In CAT(0) spaces, and so in particular in CAT(0) cube complexes with respect to the CAT(0) distance, the existence of a well-defined projection onto a given convex subspace provides a useful tool. Similarly, with respect to the combinatorial distance, it is possible to introduce a combinatorial projection onto a combinatorially convex subcomplex, defined by the following result.
Proposition 2.5. [Gen15a, Lemma 1.2.3] Let X be a CAT(0) cube complex, C ⊂ X be a combinatorially convex subspace and x ∈ X\C be a vertex. Then there exists a unique vertex y ∈ C minimizing the distance to x. Moreover, for any vertex of C, there exists a combinatorial geodesic from it to x passing through y.
The following result makes precise how the combinatorial projection behaves with respect to the hyperplanes. Proposition 2.6. Let X be a CAT(0) cube complex, C a combinatorially convex subspace, p : X → C the combinatorial projection onto C and x, y ∈ X two vertices. The hyperplanes separating p(x) and p(y) are precisely the hyperplanes separating x and y which intersect C. In particular, d(p(x), p(y)) ≤ d(x, y).
Proof of Proposition 2.6. A hyperplane separating p(x) and p(y) separates x and y according to [Gen16, Lemma 2.10]. Conversely, let J be a hyperplane separating x and y and intersecting C. Notice that, according to Lemma 2.7 below, if J separates x and p(x), or y and p(y), necessarily J must be disjoint from C. Therefore, J has to separate p(x) and p(y).
Lemma 2.7. [Gen16, Lemma 2.8] Let X be a CAT(0) cube complex and N ⊂ X a combinatorially convex subspace. Let p : X → N denote the combinatorial projection onto N . Then every hyperplane separating x and p(x) separates x and N .
The following lemma will be particularly useful in this paper.
Lemma 2.8. Let X be a CAT(0) cube complex and C 1 ⊂ C 2 two subcomplexes with C 2 combinatorially convex. Let p 2 : X → C 2 denote the combinatorial projection onto C 2 , and p 1 the nearest-point projection onto C 1 , which associates to any vertex the set of the vertices of C 1 which minimize the distance from it. Then p 1 • p 2 = p 1 .
Proof. Let x ∈ X. If x ∈ C 2 , clearly p 1 (p 2 (x)) = p 1 (x), so we suppose that x / ∈ C 2 . If z ∈ C 1 , then according to Proposition 2.5, there exists a combinatorial geodesic between x and z passing through p 2 (x). Thus,
d(x, z) = d(x, p 2 (x)) + d(p 2 (x), z).
In particular, if y ∈ p 1 (x),
d(x, C 1 ) = d(x, y) = d(x, p 2 (x)) + d(p 2 (x), y).
The previous two equalities give:
d(p 2 (x), z) − d(p 2 (x), y) = d(x, z) − d(x, C 1 ) ≥ 0.
Therefore, d(p 2 (x), y) = d(p 2 (x), C 1 ), ie., y ∈ p 1 (p 2 (x)). We have proved p 1 (x) ⊂ p 1 (p 2 (x)).
It is worth noticing that we have also proved that d(x, C 1 ) = d(x, p 2 (x)) + d(p 2 (x), C 1 ). Thus, for every y ∈ p 1 (p 2 (x)), once again because there exists a combinatorial geodesic between x and y passing through p 2 (x) according to Proposition 2.5,
d(x, y) = d(x, p 2 (x)) + d(p 2 (x), y) = d(x, p 2 (x)) + d(p 2 (x), C 1 ) = d(x, C 1 ),
ie., y ∈ p 1 (x). We have proved that p 1 (p 2 (x)) ⊂ p 1 (x), concluding the proof.
We conclude with a purely technical lemma which will be used in Section 3.3.
d(z, N ) = ∆(z, N ) = ∆({x, y, z}, N ) + ∆({x, z}, N ∪ {y}) + ∆({y, z}, N ∪ {x}), and d(x, N ) = ∆(x, N ) = ∆(x, N ∪ {y, z}) + ∆({x, z}, N ∪ {y}) + ∆({x, y, z}, N ), and d(y, N ) = ∆(y, N ) = ∆(y, N ∪ {x, z}) + ∆({y, z}, N ∪ {x}) + ∆({x, y, z}, N ). Now, it is clear that d(z, N ) ≤ d(x, N ) + d(y, N ).
Disc diagrams.
A fundamental tool to study CAT(0) cube complexes is the theory of disc diagrams. For example, they were extensively used by Sageev in [Sag95] and by Wise in [Wis12]. The rest of this section is dedicated to basic definitions and properties of disc diagrams.
Definition 2.10. Let X be a nonpostively curved cube complex. A disc diagram is a continuous combinatorial map D → X, where D is a finite contractible square complex with a fixed topological embedding into S 2 ; notice that D may be non-degenerate, ie., homeomorphic to a disc, or may be degenerate. In particular, the complement of D in S 2 is a 2-cell, whose attaching map will be refered to as the boundary path ∂D → X of D → X; it is a combinatorial path. The area of D → X, denoted by Area(D), corresponds to the number of squares of D.
Given a combinatorial closed path P → X, we say that a disc diagram D → X is bounded by P → X if there exists an isomorphism P → ∂D such that the following diagram is commutative:
∂D / / X P O O 7 7
According to a classical argument due to van Kampen [VK33] (see also [MW02,Lemma 2.17]), there exists a disc diagram bounded by a given combinatorial closed path if and only if this path is null-homotopic. Thus, if X is a CAT(0) cube complex, then any combinatorial closed path bounds a disc diagram.
As a square complex, a disc diagram contains hyperplanes: they are called dual curves. Equivalently, they correspond to the connected components of the reciprocal images of the hyperplanes of X. Given a disc diagram D → X, a nogon is a dual curve homeomorphic to a circle; a monogon is a subpath, of a self-intersecting dual curve, homeomorphic to a circle; an oscugon is a subpath of a dual curve whose endpoints are the midpoints of two adjacent edges; a bigon is a pair of dual curves intersecting into two different squares. Theorem 2.11. [Wis12, Lemma 2.2] Let X be a nonpositively curved cube complex and D → X a disc diagram. If D contains a nogon, a monogon, a bigon or an oscugon, then there exists a new disc diagram D → X such that:
(i) D is bounded by ∂D, (ii) Area(D ) ≤ Area(D) − 2. Let X be a CAT(0) cube complex. A cycle of subcomplexes C is a sequence of subcomplexes C 1 , . . . , C r such that C 1 ∩C r = ∅ and C i ∩C i+1 = ∅ for every 1 ≤ i ≤ r −1. A disc diagram D → X is bounded by C if ∂D → X
can be written as the concatenation of r combinatorial geodesics P 1 , . . . , P r → X such that P i ⊂ C i for every 1 ≤ i ≤ r.
The complexity of such a disc diagram is defined by the couple (Area(D), length(∂D)), and a disc diagram bounded by C will be of minimal complexity if its complexity is minimal with respect to the lexicographic order among all the possible disc diagrams bounded by C (allowing modifications of the paths P i ). It is worth noticing that such a disc diagram does not exist if our subcomplexes contain no combinatorial geodesics. On the other hand, if our subcomplexes are combinatorially geodesic, then a disc diagram always exists.
Our main technical result on disc diagrams is the following, which we proved in [Gen16, Theorem 2.13].
Theorem 2.12. Let X be a CAT(0) cube complex, C = (C 1 , . . . , C r ) a cycle of subcomplexes, and D → X a disc diagram bounded by C. For convenience, write ∂D as the concatenation of r combinatorial geodesics P 1 , . . . , P r → X with P i ⊂ C i for every 1 ≤ i ≤ r. If the complexity of D → X is minimal, then: We mention two particular cases below.
(i) if C i is
Corollary 2.14. Let C = (C 1 , C 2 , C 3 , C 4 ) be a cycle of four subcomplexes. Suppose that C 2 , C 3 , C 4 are combinatorially convex subcomplexes. If D → X is a disc diagram of minimal complexity bounded by C, then D → X is an isometric embedding.
Proof. First, we write ∂D → X as the concatenation of four combinatorial geodesics P 1 , P 2 , P 3 , P 4 → X, where P i ⊂ C i for every 1 ≤ i ≤ 4. Suppose that there exist two dual curves c 1 , c 2 of D induced by the same hyperplane J. Because a combinatorial geodesic cannot intersect a hyperplane twice, the four endpoints of c 1 and c 2 belong to four different sides of ∂D. On the other hand, because C 2 , C 3 , C 4 are combinatorially convex, it follows from Theorem 2.12 that any dual curve intersecting P 3 must intersect P 1 ; say that c 1 intersects P 1 and P 3 . Therefore, c 2 must intersect P 2 and P 4 , and we deduce that c 1 and c 2 are transverse. But this implies that J self-intersects in X, which is impossible. Conversely, suppose that S is not contracting, ie., for all n ≥ 0, there exists a ball B(x n , r n ), with r n < d(x n , S), whose projection onto S has diameter at least n. Let p : X → S denote the combinatorial projection onto S. If for all y ∈ B(x n , r n ) we had d(p(y), p(x n )) < n/2, then the projection of B(x n , r n ) onto S would have diameter at most n. Therefore, there exists y n ∈ B(x n , r n ) satisfying d(p(x n ), p(y n )) ≥ n/2. In particular, the projection of {x n , y n } onto S has diameter at least n/2, with d(x n , y n ) < d(x n , S).
If d(x n , y n ) < d(x n , S) − L,
there is nothing to prove. Ohterwise, two cases may happen. If d(x n , S) ≤ L, then, because p is 1-Lipschitz according to Proposition 2.6, the projection of B(x n , r n ) onto S has diameter at most 2L. Without loss of generality, we may suppose n > 2L, so that this case cannot occur. From now on, suppose d(x n , y n ) ≥ d(x n , S) − L. Then, along a combinatorial geodesic [x n , y n ] between x n and y n , there exists a vertex z n such that d(z n , y n ) ≤ L and d(x n , z n ) ≤ d(x n , S) − L because we saw that d(x n , y n ) < d(x n , S). Now,
d(p(x n ), p(z n )) ≥ d(p(x n ), p(y n )) − d(p(y n ), p(z n )) ≥ n 2 − L,
so that the projection of {x n , z n } onto S has diameter at least n 2 − L. Because this diameter may be chosen arbitrarily large, this concludes the proof. However, the previous criterion only holds for combinatorially convex subspaces. Of course, it is always possible to look at the combinatorial convex hull of the subspace which is considered, and the following lemma essentially states when it is possible to deduce that the initial subspace is contracting.
Lemma 2.18. Let X be a CAT(0) cube complex, S ⊂ X a subspace and N ⊃ S a combinatorially convex subspace. Suppose that the Hausdorff distance d H (S, N ) between S and N is finite. Then S is contracting if and only if N is contracting as well.
Proof. We know from Lemma 2.8 that the projection onto S is the composition of the projection of X onto N with the projection of N onto S. Consequently, the Hausdorff distance between the projections onto N and onto S is at most d H (N, S). Our lemma follows immediately.
Typically, the previous lemma applies to quasiconvex combinatorial geodesics: Proof. We claim that (the 1-skeleton of) N (γ) is the union L of the combinatorial geodesics whose endpoints are on γ. First, it is clear that any such geodesic must be included into N (γ), whence L ⊂ N (γ). Conversely, given some x ∈ N (γ), we want to prove that there exists a geodesic between two vertices of γ passing through x. Fix some y ∈ γ. Notice that, because x belongs to N (γ), no hyperplane separates x from γ, so that any hyperplane separating x and y must intersect γ. As a consequence, for some n ≥ 1 sufficiently large, every hyperplane separating x and y separates γ(−n) and γ(n); furthermore, we may suppose without loss of generality that y belongs to the subsegment of γ between γ(−n) and γ(n). We claim that the concatenation of a geodesic from γ(−n) to x with a geodesic from x to γ(n) defines a geodesic between γ(−n) and γ(n), which proves that x ∈ L. Indeed, if this concatenation defines a path which is not a geodesic, then there must exist a hyperplane J intersecting it twice. Notice that J must separate x from {γ(n), γ(−n)}, so that we deduce from the convexity of half-spaces that J must separate x and y. By our choice of n, we deduce that J separates γ(−n) and γ(n). Finally, we obtain a contradiction since we already know that J separates x from γ(−n) and γ(n). Thus, we have proved that N (γ) ⊂ L.
It follows that γ is quasiconvex if and only if the Hausdorff distance between γ and N (γ) is finite.
It is worth noticing that any contracting geodesic is quasiconvex. This is a particular case of [Sul14, Lemma 3.3]; we include a proof for completeness.
Lemma 2.20. Let X be a geodesic metric space. Any contracting geodesic of X is quasiconvex.
Proof. Let ϕ be a geodesic between two points of γ. Let ϕ(t) be a point of ϕ. We want to prove that d(ϕ(t), γ) ≤ 11B.
If d(ϕ(t), γ) < 3B, there is nothing to prove. Otherwise, ϕ(t) belongs to a maximal subsegment [ϕ(r), ϕ(s)] of ϕ outside the 3B-neighborhood of γ, ie. d(ϕ(r), γ), d(ϕ(s), γ) ≤ 3B but d(ϕ(p), γ) ≥ 3B for every p ∈ [r, s]. Let
r = t 0 < t 1 < · · · < t k−1 < t k = s (1) with t i+1 − t i = 2B for 1 ≤ i ≤ k − 2 and t k − t k−1 ≤ 2B. Observe that d(ϕ(r), ϕ(s)) = s − r = k−1 i=0 (t i+1 − t i ) ≥ 2(k − 1)B,
but on the other hand, if p i belongs to the projection of ϕ(t i ) on γ,
d(ϕ(r), ϕ(s)) ≤ d(ϕ(r), p 0 ) + d(p 0 , p 1 ) + · · · + d(p k−1 , p k ) + d(p k , ϕ(s)). Noticing that d(ϕ(t i ), ϕ(t i+1 )) = t i+1 − t i ≤ 2B < 3B ≤ d(ϕ(t i ), γ), we deduce from the fact that γ is B-contracting that d(ϕ(r), ϕ(s)) ≤ 6B + kB.(2)
Finally, combining 1 and 2, we get k ≤ 8. Thus,
d(ϕ(r), ϕ(s)) = k−1 i=0 (t i+1 − t i ) ≤ 2kB ≤ 16B. Now, since d(ϕ(t), ϕ(r)) or d(ϕ(t), ϕ(s))
, say the first one, is bounded above by d(ϕ(r), ϕ(s))/2 ≤ 8B, we deduce
d(ϕ(t), γ) ≤ d(ϕ(t), ϕ(r)) + d(ϕ(r), γ) ≤ 8B + 3B = 11B.
This concludes the proof.
Contracting isometries from hyperplane configurations
Quasiconvex geodesics I
Given a subcomplex Y of some CAT(0) cube complex, we denote by H(Y ) the set of hyperplanes intersecting Y . In this section, the question we are interested in is to determine when a combinatorial geodesic γ is quasiconvex just from the set H(γ). We begin by introducing some definitions.
Definition 3.1. A facing triple in a CAT(0) cube complex is the data of three hyperplanes such that no two of them are separated by the third hyperplane.
Definition 3.2. A join of hyperplanes is the data of two families (H = (H 1 , . . . , H r ), V = (V 1 , . . . , V s )) of hyperplanes which do not contain facing triple and such that any hyperplane of H is transverse to any hyperplane of V. If H i (resp. V j ) separates H i−1 and H i+1 (resp. V j−1 and V j+1 ) for every 2 ≤ i ≤ r − 1 (resp. 2 ≤ j ≤ s − 1), we say that (H, V) is a grid of hyperplanes.
If (H, V) is a join or a grid of hyperplanes satisfying min(#H, #V) ≤ C, we say that
(H, V) is C-thin.
Our main criterion is: It will be a direct consequence of the following lemma and a criterion of hyperbolicity we proved in [Gen16]:
Lemma
Contracting convex subcomplexes I
In the previous section, we showed how to recognize the quasiconvexity of a combinatorial geodesic from the set of the hyperplanes it intersects. Therefore, according to Proposition 3.3, we reduced the problem of determining when a combinatorial geodesic is contracting to the problem of determining when a combinatorially convex subcomplex is contracting. This is the main criterion of this section:
) ≤ C 1 + C 2 , ie., (H, V) is (C 1 + C 2 )-thin.
Conversely, suppose that there exists a constant C ≥ 1 such that any join of hyperplanes (H, V) satisfying H ⊂ H(γ) is necessarily C-thin. According to Proposition 3.3, γ is quasiconvex, so that it is sufficient to prove that N (γ) is contracting to conclude that γ is contracting as well according to Lemma 2.18. Finally, it follows from Theorem 3.6 that N (γ) is contracting.
Well-separated hyperplanes
Separation properties of hyperplanes play a central role in the study of contracting isometries. Strongly separated hyperplanes were introduced in [BC12] in order to characterize the rank-one isometries of right-angled Artin groups; they were also crucial in the proof of the Rank Rigidity Theorem for CAT(0) cube complexes in [CS11]. In [CS15], Charney and Sultan used contracting isometries to distinguish quasi-isometrically some cubulable groups, and they introduced k-separated hyperplanes in order to characterize contracting isometries in uniformly locally finite CAT(0) cube complexes [CS15, Theorem 4.2]. In this section, we introduce well-separated hyperplanes in order to generalize this charaterization to CAT(0) cube complexes without any local finiteness condition.
Definition 3.8. Let J and B be two disjoint hyperplanes in some CAT(0) cube complex and L ≥ 0. We say that J and H are L-well separated if any family of hyperplanes transverse to both J and H, which does not contain any facing triple, has cardinality at most L. Two hyperplanes are well-separated if they are L-well-separated for some L ≥ 1.
• d(x i , x i+1 ) ≤ r,
• the hyperplanes H i and H i+1 are L-well separated.
The following lemma is a combinatorial analogue to [CS15,Lemma 4.3], although our proof is essentially different. This is the key technical lemma needed to prove Proposition 3.11, which is in turn the first step toward the proof of Theorem 3.9.
Lemma 3.10. Let γ be an infinite quasiconvex combinatorial geodesic. There exists a constant C depending only on γ such that, if two vertices x, y ∈ γ satisfy d(x, y) > C, then there exists a hyperplane J separating x and y such that the projection of
N (J) onto γ is included into [x, y] ⊂ γ.
Proof. Let J be a hyperplane separating x and y whose projection onto γ is not included into [x, y] ⊂ γ. Say that this projection contains a vertex at the right of y; the case where it contains a vertex at the left of x is completely symmetric. Thus, we have the following configuration:
where b / ∈ [x, y] is a projection onto γ of some vertex a ∈ N (J), z ∈ γ is a vertex adjacent to J, and m = m(a, b, z) the median point of {a, b, z}. Because m belongs to a combinatorial geodesic between z and b, and that these two points belong to γ, necessarily m ∈ N (γ). On the other hand, m belongs to a combinatorial geodesic between a and b, and by definition b is a vertex of γ minimizing the distance to a, so d(m, b) = d(m, γ); because, by the quasiconvexity of γ, the Hausdorff distance between γ and N (γ) is finite, say L, we deduce that d(m, b) = d(m, γ) ≤ L since m belongs to a combinatorial geodesic between b and z (which implies that m ∈ N (γ)). Using Lemma 2.9, we get
d(y, N (J)) ≤ d(z, N (J)) + d(b, N (J)) ≤ d(b, m) ≤ L.
Thus, J intersects the ball B(y, L).
Let H be the set of hyperplanes separating x and y, and intersecting B(y, L). Of course, because any hyperplane of H intersects γ, the collection H contains no facing triple. On the other hand, it follows from Proposition 3.3 that dim N (γ) is finite, so that if #H ≥ Ram(s) for some s ≥ dim N (γ) then the collection H must contain at least s pairwise disjoint hyperplanes. Since this collection of hyperplanes does not contain facing triples and intersects the ball B(y, L), we deduce that s ≤ 2L, hence #H ≤ Ram(max(dim N (γ), 2L)).
Finally, we have proved that there are at most 2Ram(max(dim N (γ), 2L)) hyperplanes separating x and y such that the projections of their neighborhoods onto γ are not included into [x, y] ⊂ γ. It clearly follows that d(x, y) > 2Ram(max(dim N (γ), 2L)) implies that there is a hyperplane separating x and y such that the projection of its neighborhood onto γ is included into [x, y] ⊂ γ.
Proposition 3.11. Let γ be an infinite combinatorial geodesic. If γ is quasiconvex then there exist constants r,
L ≥ 1, hyperplanes {H i , i ∈ Z}, and vertices {x i ∈ γ ∩ N (H i ), i ∈ Z} such that, for every i ∈ Z: • d(x i , x i+1 ) ≤ r,
• the hyperplanes H i and H i+1 are disjoint.
Proof. Because γ is quasiconvex, we may apply the previous lemma: let C be the constant it gives. Let . . . , y −1 , y 0 , y 1 , . . . ∈ γ be vertices along γ satisfying d(y i , y i+1 ) = C + 1 for all i ∈ Z. According to the previous lemma, for every k ∈ Z, there exists a hyperplane J k separating y 2k and y 2k+1 whose projection onto γ is included into
[y 2k , y 2k+1 ] ⊂ γ; let x k be one of the two vertices in γ ∩ N (J k ). Notice that d(x i , x i+1 ) ≤ d(y 2i , y 2i+3 ) ≤ d(y 2i , y 2i+1 ) + d(y 2i+1 , y 2i+2 ) + d(y 2i+2 , y 2i+3 ) = 3(C + 1).
Finally, it is clear that, for every k ∈ Z, the hyperplanes J k and J k+1 are disjoint: if it was not the case, there would exist a vertex of
N (J k ) ∩ N (J k+1 ) whose projection onto γ would belong to [y 2k , y 2k+1 ] ∩ [y 2k+2 , y 2k+3 ] = ∅.
Proof of Theorem 3.9. Suppose γ contracting. In particular, γ is quasiconvex, so, by applying Proposition 3.11, we find a constant L ≥ 1, a collection of pairwise disjoint hyperplanes {J i , i ∈ Z}, and a collection of vertices
{y i ∈ γ ∩ N (J i ), i ∈ Z}, such that d(y i , y i+1 ) ≤ L for all i ∈ Z.
Let C be the constant given by Corollary 3.7 and set
x i = y iC for all i ∈ Z. Notice that d(x i , x i+1 ) = d(y iC , y iC+C ) ≤ d(y iC , y iC+1 ) + · · · + d(y iC+C−1 , y iC+C ) ≤ (C + 1)L.
Now, we want to prove that the hyperplanes J nC and J (n+1)C are C-well separated for every n ∈ Z. So let H be a collection of hyperplanes transverse to both J nC and J (n+1)C , which contains no facing triple. Because every hyperplane H ∈ H is transverse to any J nC+k , for 0 ≤ k ≤ C, we obtain a join of hyperplanes (V C+1 , V #H ) satisfying V C+1 ⊂ H(γ). By definition of C, we deduce #H ≤ C.
Conversely, suppose that there exist constants , L ≥ 1, hyperplanes {J i , i ∈ Z}, and vertices {x i ∈ γ ∩ N (J i ), i ∈ Z} such that, for every i ∈ Z:
• d(x i , x i+1 ) ≤ ,
• the hyperplanes J i and J i+1 are L-well separated.
Let (V p , V q ) be a join of hyperplanes with V p = {V 1 , . . . , V p } ⊂ H(γ)
. For convenience, we may suppose without loss of generality that each V i intersects γ at y i with the property that y i separates y i−1 and y i+1 along γ for all 2 ≤ i ≤ p − 1. If p > 3 + 2L, there exist L < r < s < p−L such that y r and y s are separated by J k , J k+1 , J k+2 and J k+3 for some k ∈ Z. Because J k and J k+1 are L-well separated, the hyperplanes {V 1 , . . . , V r } cannot be transverse to both J k and J k+1 since r > L, so there exists 1 ≤ α ≤ r such that V α and J k+1 are disjoint. Similarly, the hyperplanes {V s , . . . , V p } cannot be transverse to both J k+2 and J k+3 since p − s > L, so there exists 1 ≤ ω ≤ r such that V ω and J k+2 are disjoint. Notice that the hyperplanes J k+1 and J k+2 separate V α and V ω , so that the hyperplanes of V q , which are all transverse to both V α and V ω , are transverse to both J k+1 and J k+2 . But J k+1 and J k+2 are L-well separated, hence q ≤ L.
Thus, we have proved that min(p, q) ≤ 3 + 2L. Finally, Corollary 3.7 implies that γ is contracting.
Contracting isometries I
Finally, we apply our criteria of contractivity to the axis of some isometry, providing a characterization of contracting isometries. We first need the following definition:
Definition 3.12. Let X be a CAT(0) cube complex and g ∈ Isom(X) an isometry. We say that g skewers a pair of hyperplanes (J 1 , J 2 ) if g n D 1 D 2 D 1 for some n ∈ Z\{0} and for some half-spaces D 1 , D 2 delimited by J 1 , J 2 respectively.
Our main criterion is:
Theorem 3.13. Let X be a CAT(0) cube complex and g ∈ Isom(X) an isometry with a combinatorial axis γ. The following statements are equivalent:
(i) g is a contracting isometry; (ii) there exists C ≥ 1 such that any join of hyperplanes (H, V) with H ⊂ H(γ) is C-thin; (iii) there exists C ≥ 1 such that: -H(γ) does not contain C pairwise transverse hyperplanes, -any grid of hyperplanes (H, V) with H ⊂ H(γ) is C-thin;
(iv) g skewers a pair of well-separated hyperplanes.
Proof. The equivalence (i) ⇔ (ii) is a direct consequence of Corollary 3.7. Then, because a grid of hyperplanes or a collection of pairwise transverse hyperplanes gives rise to a join of hyperplanes, (ii) clearly implies (iii). Now, we want to prove (iii) ⇒ (ii). Let C denote the constant given by (iii) and let (H, V) be a join of hyperplanes satisfying H ⊂ H(γ). If #H, #V ≥ Ram(C), then H and V each contain a subfamily of at least C pairwise disjoint hyperplanes, say H and V respectively. But now (H , V ) defines a grid of hyperplanes satisfying #H , #V ≥ C, contradicting the definition of C. Thus, the join (H, V) is necessarily Ram(C)-thin. Now, we want to prove (iv) ⇒ (i). So suppose that there exist two half-spaces D 1 , D 2 respectively delimited by two well separated hyperplanes J 1 , J 2 such that g n D 1 D 2 D 1 for some n ∈ Z. Notice that
· · · g 2n D 1 g n D 2 g n D 1 D 2 D 1 g −n D 2 g −n D 1 g −2n D 2 · · · .
We claim that J 1 intersects γ. Suppose by contradiction that it is not the case.
Then, because γ is g -invariant, g k J 1 does not intersect γ for every k ∈ Z. As a consequence, there exists some m ∈ Z such that γ ⊂ g mn D 1 \g (m+1)n D 1 . A fortiori, g m γ ⊂ g (m+1)n D 1 \g (m+2)n D 1 , and because γ is g -invariant, we deduce that γ ⊂ g mn D 1 \g (m+1)n D 1 ∩ g (m+1)n D 1 \g (m+2)n D 1 = ∅,
a contradiction. So J 1 intersects γ, and a fortiori, g kn J 1 intersects γ for every k ∈ Z.
For every k ∈ Z, the intersection between g kn N (J 1 ) and γ contains exacly two vertices; fix an orientation along γ and denote by y k the first vertex along γ which belongs to g kn N (J 1 ). Clearly, d(y k , y k+1 ) = d(g kn y 0 , g (k+1)n y 0 ) = d(y 0 , g n y 0 ) = g n since y 0 belongs to the combinatorial axis of g, where g denotes the combinatorial translation length of g (see [Hag07]). Furthermore, J 1 and g n J 1 are well separated: indeed, because J 2 separates J 1 and g n J 1 , we deduce that any collection of hyperplanes transverse to both J 1 and g n J 1 must also be transverse to both J 1 and J 2 , so that we conclude that J 1 and g n J 1 are well separated since J 1 and J 2 are themselves well separated. Therefore, {g kn J 1 | k ∈ N} defines a family of pairwise well separated hyperplanes intersecting γ at uniformly separated points. Theorem 3.9 implies that γ is contracting.
Conversely, suppose that g is contracting. According to Theorem 3.9, there exist three pairwise well separated hyperplanes J 1 , J 2 , J 3 intersecting γ. Say that they respectively delimit three half-spaces D 1 , D 2 , D 3 satisfying D 3 D 2 D 1 , where D 3 contains γ(+∞). We claim that g skewers the pair (J 1 , J 2 ). Fix two vertices x ∈ N (J 1 ) ∩ γ and y ∈ D 3 ∩ γ. Notice that, if I denotes the set of n ∈ N such that g n J 1 and J 1 are transverse and g n J 1 does not separate x and y, then {g n J 1 | n ∈ I} defines a set of hyperplanes (without facing triples, since they all intersect γ) transverse to both J 2 and J 3 . Because J 2 and J 3 are well separated, necessarily I has to be finite. A fortiori, there must exist at most #I + d(x, y) integers n ∈ N such that g n J 1 and J 1 are transverse. As a consequence, there exists an integer n ∈ N such that g n J 1 ⊂ D 2 . Finally, we have proved (i) ⇒ (iv).
Combinatorial boundary 4.1 Generalities
In this section, we will use the following notation: given two subsets of hyperplanes H 1 , H 2 , we define the almost inclusion H 1 ⊂ a H 2 if all but finitely many hyperplanes of H 1 belong to H 2 . In particular, H 1 and H 2 are commensurable provided that H 1 ⊂ a H 2 and H 2 ⊂ a H 1 . Notice that commensurability defines an equivalence relation on the set of all the collections of hyperplanes, and the almost-inclusion induces a partial order on the associated quotient set. This allows the following definition.
Definition 4.1. Let X be a CAT(0) cube complex. Its combinatorial boundary is the poset (∂ c X, ≺), where ∂ c X is the set of the combinatorial rays modulo the relation: r 1 ∼ r 2 if H(r 1 ) and H(r 2 ) are commensurable; and where the partial order ≺ is defined by: r 1 ≺ r 2 whenever H(r 1 ) ⊂ a H(r 2 ).
Notice that this construction is equivariant, ie., if a group G acts by isometries on X, then G acts on ∂ c X by ≺-isomorphisms.
The following two lemmas essentially state that we will be able to choose a "nice" combinatorial ray representing a given point in the combinatorial boundary. They will be useful in the next sections.
Lemma 4.2. Let X be a CAT(0) cube complex, x 0 ∈ X a base vertex and ξ ∈ ∂ c X.
There exists a combinatorial ray r with r(0) = x 0 and r = ξ in ∂ c X.
Proof. Let r n be a combinatorial path which is the concatenation of a combinatorial geodesic [x 0 , ξ(n)] between x 0 and ξ(n) with the subray ξ n of ξ starting at ξ(n). If r n is not geodesic, then there exists a hyperplane J intersecting both [x 0 , ξ(n)] and ξ n . Thus, J separates x 0 and ξ(n) but cannot separate ξ(0) and ξ(n) since otherwise J would intersect the ray ξ twice. We deduce that J necessarily separates x 0 and ξ(0). Therefore, if we choose n large enough so that the hyperplanes separating x 0 and ξ(0) do not intersect the subray ξ n , then r n is a combinatorial ray. By construction, we have r n (0) = x 0 , and H(r n ) and H(ξ) are clearly commensurable so r n = ξ in ∂ c X. Lemma 4.3. Let r, ρ be two combinatorial rays satisfying r ≺ ρ. There exists a combinatorial ray p equivalent to r satisfying p(0) = ρ(0) and H(p) ⊂ H(ρ).
Proof. According to the previous lemma, we may suppose that r(0) = ρ(0). By assumption, H(r) ⊂ a H(ρ). Let J be the last hyperplane of H(r)\H(ρ) intersected by r, ie., there exists some k ≥ 1 such that J is dual to the edge [r(k), r(k + 1)] and the hyperplanes intersecting the subray r k ⊂ r starting at r(k + 1) all belong to H(ρ). We claim that r k ⊂ ∂N (J). Indeed, if there exists some j ≥ k +1 such that d(r k (j), N (J)) ≥ 1, then some hyperplane H would separate r k (j) and N (J) according to Lemma 2.7; a fortiori, H would intersect r but not ρ, contradicting the choice of J. Let r denote the combinatorial ray obtained from r by replacing the subray r k−1 by the symmetric of r k with respect to J in N (J) J × [0, 1]. The set of hyperplanes intersecting r is precisely H(r)\{J}. Because |H(r )\H(ρ)| < |H(r)\H(ρ)|, iterating the process will produce after finitely many steps a combinatorial ray p satisfying p(0) = r(0) = ρ(0) and H(p) ⊂ H(ρ).
Definition 4.4. Let X be a CAT(0) cube complex and Y ⊂ X a subcomplex. We define the relative combinatorial boundary ∂ c Y of Y as the subset of ∂ c X corresponding the set of the combinatorial rays included into Y .
Lemma 4.5. Let r be a combinatorial ray. Then
r ∈ ∂ c Y if and only if H(r) ⊂ a H(Y ). Proof. If r ∈ ∂ c Y , then by definition there exists a combinatorial ray ρ ⊂ Y equivalent to r. Then H(r) ⊂ a H(ρ) ⊂ H(Y ).
Conversely, suppose that H(r) ⊂ a H(Y ). According to Lemma 4.2, we may suppose that r(0) ∈ Y . Using the same argument as in the previous proof, we find an equivalent combinatorial ray ρ with H(ρ) ⊂ H(Y ) and
ρ(0) = r(0) ∈ Y . Because Y is combinatorially convex, it follows that ρ ⊂ Y , hence r ∈ ∂ c Y .
Quasiconvex geodesics II
The goal of this section is to prove the following criterion, determining when a combinatorial axis is quasiconvex just from its endpoints in the combinatorial boundary.
Proposition 4.6. Let X be a locally finite CAT(0) cube complex and g ∈ Isom(X) an isometry with a combinatorial axis γ. A subray r ⊂ γ is quasiconvex if and only if r is minimal in ∂ c X.
Proof. Suppose that r is not quasiconvex. According to Proposition 3.3, for every n ≥ 1, there exist a join of hyperplanes (H n , V n ) with H n , V n ⊂ H(r) and #H n , #V n = n. The hyperplanes of H n ∪V n are naturally ordered depending on the order of their intersections along r. Without loss of generality, we may suppose that the hyperplane of H n ∪ V n which is closest to r(0) belongs to H n ; because g acts on H(γ) with finitely many orbits, up to translating by powers of g and taking a subsequence, we may suppose that this hyperplane J ∈ H n does not depend on n. Finally, let V n denote the hyperplane of V n which is farthest from r(0).
Thus, if J − (resp. V + n ) denotes the halfspace delimited by J (resp. V n ) which does not contain r(+∞) (resp. which contains r(+∞)), then C n = (J − , V + n , r) defines a cycle of three subcomplexes. Let D n → X be a disc diagram of minimal complexity bounded by C n . Because no hyperplane can intersect J, V n or r twice, it follows from Proposition 2.13 that D n → X is an isometric embedding, so we will identify D n with its image in X. Let us write the boundary ∂D n as a concatenation of three combinatorial geodesics
∂D n = µ n ∪ ν n ∪ ρ n ,
where µ n ⊂ J − , ν n ⊂ V + n and ρ n ⊂ r. Because X is locally finite, up to taking a subsequence, we may suppose that the sequence of combinatorial geodesics (µ n ) converges to a combinatorial ray µ ∞ ⊂ J − . Notice that, if H ∈ H(µ ∞ ), then H ∈ H(µ k ) for some k ≥ 1, it follows from Theorem 2.12 that H does not intersect ν k in D k , hence H ∈ H(ρ k ). Therefore, H(µ ∞ ) ⊂ H(r).
According to Lemma 4.8 below, there exists an infinite collection of pairwise disjoint hyperplanes J 1 , J 2 , . . . ∈ H(r). Because g acts on H(γ) with finitely many orbits, necessarily there exist some r, s ≥ 1 and some m ≥ 1 such that g m J r = J s . Thus, {J r , g m J r , g 2m J r , . . .} defines a collection of infinitely many pairwise disjoint hyperplanes in H(r) stable by the semigroup generated by g m . For convenience, let {H 0 , H 1 , . . .} denote this collection.
If J is disjoint from some H k , then H k , H k+1 , . . . / ∈ H(µ ∞ ) since µ ∞ ⊂ J − . On the other hand, H k , H k+1 , . . . ∈ H(r), so we deduce that µ ∞ ≺ r with µ ∞ = r in ∂ c X: it precisely means that r is not minimal in ∂ c X.
From now on, suppose that J is transverse to all the H k 's. As a consequence, g km J is transverse to H n for every n ≥ k. For every k ≥ 1, let H + k denote the halfspace delimited by H k which contains r(+∞) and let p k : X → H + k be the combinatorial projection onto H + k . We define a sequence of vertices (x n ) by:
x 0 = r(0) and x n+1 = p n+1 (x n ) for every n ≥ 0.
Because of Lemma 2.8, we have x n = p n (x 0 ) for every n ≥ 1. Therefore, it follows from Proposition 2.5 that, for every n ≥ 1, there exists a combinatorial geodesic between x 0 and x n passing through x k for k ≤ n. Thus, there exists a combinatorial ray ρ passing through all the x k 's.
Let H ∈ H(ρ). Then H separates x k and x k+1 for some k ≥ 0, and it follows from Lemma 2.7 that H separates x k and H k . As a first consequence, we deduce that, for every k ≥ 1, g km J cannot belong to H(ρ) since g km J is transverse to H n for every n ≥ k, hence r = ρ in ∂ c X. On the other hand, H separates x 0 = r(0) and H k , and r(n) ∈ H + k for n large enough, so H ∈ H(r). We have proved that H(ρ) ⊂ H(r), ie., ρ ≺ r. Thus, r is not minimal in ∂ c X.
Conversely, suppose that r is not minimal in ∂ c X. Thus, there exists a combinatorial ray ρ with ρ(0) = r(0) such that H(ρ) ⊂ H(r) and |H(r)\H(ρ)| = +∞. Let J 1 , J 2 , . . . ∈ H(r)\H(ρ) be an infinite collection. Without loss of generality, suppose that, for every Since N can be chosen arbitrarily large, it follows from Proposition 3.3 that r is not quasiconvex.
i > j ≥ 1, the edge J j ∩ r is closer to r(0) than the edge J i ∩ r. Let N ≥ 1, and let d(N ) denote the distance d(r(0), ω) where ω is the endpoint of the edge J N ∩r which is farthest from r(0). Choose any collection of hyperplanes V ⊂ H(ρ) with #V ≥ d(N ) + N . Then V contains a subcollection {V 1 , . . . , V N } such that the edge V i ∩ r is farther from r(0) than the edge J N ∩ r. We know that each V j separates {r(0),
Lemma 4.8. Let X be a complete CAT(0) cube complex and r a combinatorial ray. Then H(r) contains an infinite sequence of pairwise disjoint hyperplanes.
Proof. We begin our proof with a completely general argument, without assuming that X is complete. First, we decompose H(r) as the disjoint union Suppose by contradiction that J 1 and J 2 are disjoint, and, for convenience, say that J 1 separates r(0) and
H 0 H 1 · · · where we define H i = {J ∈ H(r) | d ∞ (r(0), J) = i} for all i ≥ 0. A priori,J 2 . Because d ∞ (r(0), J 1 ) = i, there exists i pairwise disjoint hyper- planes V 1 , . . . , V i separating r(0) and J 1 . Then V 1 , . . . , V i , J 1 are i+1 pairwise transverse hyperplanes separating r(0) and J 2 , hence d ∞ (r(0), J 2 ) ≥ i + 1, a contradiction. Claim 4.10. Let J ∈ H i , ie., there exists a collection of i pairwise disjoint hyperplanes V 0 , . . . , V i−1 separating r(0) and J. Then V j ∈ H j for every 0 ≤ j ≤ i − 1. First, because V 0 , . . . , V j−1 separate r(0) and V j , necessarily we have d ∞ (r(0), V j ) ≥ j. Let H 1 . . . , H k be k pairwise disjoint hyperplanes separating r(0) and V j . Then H 1 , . . . , H k , V j , . . . , V i−1 define k + i − j pairwise disjoint hyperplanes separating r(0) and J, hence k + i − j ≤ d ∞ (r(0), J) = i, ie., k ≤ j. We deduce that d ∞ (r(0), V j ) ≤ j. Finally, we conclude that d ∞ (r(0), V j ) = j, that is V j ∈ H j .
Our third and last claim states that, if the first collections H 0 , . . . , H p are finite, then there exists a sequence of cubes Q 0 , . . . , Q p such that the intersection between two successive cubes is a single vertex and the hyperplanes dual to the cube Q i are precisely the hyperplanes of H i . For this purpose, we define by induction the sequence of vertices x 0 , x 1 , . . . by:
• x 0 = r(0); • if x j is defined and H j is finite, x j+1 is the projection of x j onto C j := J∈H j J + ,
where J + denotes the halfspace delimited by J which does not contain r(0).
Notice that the intersection C j is non-empty precisely because of Claim 4.9 and because we assumed that H j is finite. We argue by induction.
First, we notice that any hyperplane J ∈ H 0 is adjacent to x 0 . Suppose by contradiction that this not the case, and let p denote the combinatorial projection of x 0 onto J + . By assumption, d(x 0 , p) ≥ 2 so there exists a hyperplane H separating x 0 and p which is different from J. According to Lemma 2.7, H separates x 0 and J. This contradicts J ∈ H 0 . Therefore, any hyperplane J ∈ H 0 is dual to an edge e(J) with x 0 as an endpoint.
If H 0 is infinite, there is nothing to prove. Otherwise, {e(J) | J ∈ H 0 } is a finite set of edges with a common endpoint and whose associated hyperplanes are pairwise transverse. Thus, because a CAT(0) cube complex does not contain inter-osculating hyperplanes, these edges span a cube Q 0 . By construction, the set of hyperplanes intersecting Q 0 is H 0 . Furthermore, because the vertex of Q 0 opposite to x 0 belongs to C 0 and is at distance #H 0 from x 0 , we deduce that it is precisely x 1 . Now suppose that x 0 , . . . , x i are well-defined that there exist some cubes Q 0 , . . . , Q i−1 satisfying our claim. We first want to prove that any hyperplane J ∈ H i is adjacent to x i . Suppose by contradiction that this is not the case, ie., there exists some J ∈ H i which is not adjacent to x i . As a consequence, there must exist a hyperplane H separating x i and J. We have proved that any hyperplane of H i is adjacent to x i . Thus, if H i is finite, we can construct our cube Q i as above. This concludes the proof of our claim.
≤ j ≤ i − 1, ie., H ∈ H j . Let p 0 (resp. p i ) denote the combinatorial projection of x 0 (resp. x i ) onto J + . Notice that, because H is disjoint from J,
From now on, we assume that X is a complete CAT(0) cube complex. First, as a consequence of the previous claim, we deduce that each H i is finite. Indeed, suppose by contradiction that some H i is infinite. Without loss of generality, we may suppose that H j is finite for every j < i, so that the points x 0 , . . . , x i and the cubes Q 0 , . . . , Q i−1 are well-defined. According to our previous claim, any hyperplane of H i is adjacent to x i : thus, the infinite set of edges adjacent to x i and dual to the hyperplanes of H i define an infinite cube, contradicting the completeness of X.
Therefore, we have defined an infinite sequence of vertices x 0 , x 1 , . . . and an infinite sequence of cubes Q 0 , Q 1 , . . .. Thanks to Claim 4.10, for every i ≥ 0, there exists a sequence of pairwise disjoint hyperplanes
V i 0 , . . . , V i i with V i j ∈ H j .
Because each H k is finite, up to taking a subsequence, we may suppose that, for every k ≥ 0, the sequence (V i k ) is eventually constant to some hyperplane V k ∈ H k . By construction, our sequence V 0 , V 1 , . . . defines a collection of pairwise disjoint hyperplanes in H(r), concluding the proof of our lemma.
Contracting convex subcomplexes II
Once again, according to Proposition 3.3, the previous section reduces the problem of determining when a combinatorial axis is contracting to the problem of determining when a (hyperbolic, cocompact) combinatorially convex subcomplex is contracting. To do so, we need the following definition:
Definition 4.12. A subset S ⊂ ∂ c X is full if any element of ∂ c X which is comparable with an element of S necessarily belongs to S.
Our main criterion is:
Theorem 4.13. Let X be a locally finite CAT(0) cube complex and Y ⊂ X a hyperbolic combinatorially convex Aut(X)-cocompact subcomplex. Then Y is contracting if and only if ∂ c Y is full in ∂ c X.
Proof. Suppose that Y is not contracting. Notice that Y is a cocompact subcomplex, so its dimension, say d, must be finite. According to Theorem 3.6, for every n ≥ d, there exists a join of hyperplanes (H n , V n ) with H n ⊂ H(Y ), V n ∩ H(Y ) = ∅ and #H n = #V n ≥ Ram(n). Next, up to taking subcollections of H n and V n , we may suppose thanks to Ramsey's theorem that H n and V n are collections of pairwise disjoint hyperplanes of cardinalities exactly n, ie., (H n , V n ) is a grid of hyperplanes. For convenience, write
H n = (H n 1 , . . . , H n n ) (resp. V n = (V n 1 , . . . , V n n ) where H n i (resp. V n i ) separates H n i−1 and H n i+1 (resp. V n i−1 and V n i+1 ) for every 2 ≤ i ≤ n − 1; we also suppose that V n 1 separates Y and V n n . Let C n be the cycle of subcomplexes (N (H n 1 ), N (V n n ), N (H n n ), Y )
. According to Corollary 2.15, a disc diagram of minimal complexity bounded by C n defines a flat rectangle D n . Say that a hyperplane intersecting the H n i 's in D n is vertical, and a hyperplane intersecting the V n i 's in D n is horizontal. Claim 4.14. If the grids of hyperplanes of Y are all C-thin and n > C, then at most
Ram(C) horizontal hyperplanes intersect Y .
Let V be a collection of horizontal hyperplanes which intersect Y ; notice that V does not contain any facing triple since there exists a combinatorial geodesic in D n (which is a combinatorial geodesic in X) intersecting all the hyperplanes of V. If #V ≥ Ram(s) for some s ≥ dim Y , then V contains a subcollection V with s pairwise disjoint hyperplanes, so that (H n , V ) defines a (n, s)-grid of hyperplanes. If n > C, this implies s ≤ C. Therefore, #V ≤ Ram(C). This proves the claim. Now, because Y is a cocompact subcomplex, up to translating the D n 's, we may suppose that a corner of each D n belongs to a fixed compact fundamental domain C; and because X is locally finite, up to taking a subsequence, we may suppose that, for every ball B centered in C, (D n ∩ B) is eventually constant, so that (D n ) converges naturally to a subcomplex D ∞ . Notice that D ∞ is isomorphic to the square complex [0, +∞)×[0, +∞) with [0, +∞) × {0} ⊂ Y ; let ρ denote this combinatorial ray. Clearly, if r ⊂ D ∞ is a diagonal combinatorial ray starting from (0, 0), then H(ρ) ⊂ H(r), ie. r and ρ are comparable in ∂ c X. Furthermore, the hyperplanes intersecting [0, +∞) × {0} ⊂ D ∞ are horizontal hyperplanes in some D n , and so infinitely many of them are disjoint from Y according to our previous claim: this implies that r / ∈ ∂ c Y . We conclude that ∂ c Y is not full in ∂ c X.
Conversely, suppose that ∂ c Y is not full in ∂ c X, ie., there exist two ≺-comparable combinatorial rays r, ρ satisfying ρ ⊂ Y and r / ∈ ∂ c Y .
Suppose that r ≺ ρ. According to Lemma 4.3, we may suppose that r(0) = ρ(0) and H(r) ⊂ H(ρ). But r(0) ∈ Y and H(r) ⊂ H(Y ) imply r ⊂ Y , contradicting the assumption r / ∈ ∂ c Y .
Suppose that ρ ≺ r. According to Lemma 4.2, we may suppose that r(0) = ρ(0). Because r / ∈ ∂ c Y , there exists an infinite collection J ⊂ H(r)\H(Y ); a fortiori, J ⊂ H(r)\H(ρ) since ρ ⊂ Y . It follows from Fact 4.7 that, for every N ≥ 1, there exists a join of hypergraphs (H, V) with H ⊂ J , V ⊂ H(ρ) and #H, #V ≥ N . Therefore, Theorem 3.6 implies that Y is not contracting.
Remark 4.15. Notice that neither the hyperbolicity of the subcomplex Y nor its cocompactness was necessary to prove the converse of the previous theorem. Thus, the relative combinatorial boundary of a combinatorially convex subcomplex is always full.
Contracting isometries II
Finally, we want to apply the results we have found in the previous sections to characterize contracting isometries from the combinatorial boundary. We begin by introducing the following definition:
Definition 4.16. A point ξ ∈ ∂ c X is isolated if it is not comparable with any other element of ∂ c X.
Our main criterion is the following:
Theorem 4.17. Let X be a locally finite CAT(0) cube complex and g ∈ Isom(X) an isometry with a combinatorial axis γ. Then g is a contracting isometry if and only if γ(+∞) is isolated in ∂ c X.
Proof. If g is a contracting isometry then a subray r ⊂ γ containing γ(+∞) is contracting. Because N (r) is a combinatorially convex subcomplex quasi-isometric to a line, it follows that ∂ c N (r) = {r}. On the other hand, since N (r) is contracting, it follows from Theorem 4.13 that ∂ c N (r) is full in ∂ c X. This precisely means that r is isolated.
Conversely, suppose that r is isolated in ∂ c X. It follows from Proposition 4.6 that r is quasiconvex. Thus, r is contracting if and only if N (r) is contracting. The quasiconvexity of r also implies ∂ c N (r) = {r}. Because r is isolated in ∂ c X, ∂ c N (r) is full in ∂ c X, and so N (r) is contracting according to Theorem 4.13.
Thus, containing an isolated point in the combinatorial boundary is a necessary condition in order to admit a contracting isometry. The converse is also true if we strengthen our assumptions on the action of our group. The first result in this direction is the following theorem, where we assume that there exist only finitely many orbits of hyperplanes:
Theorem 4.18. Let G be a group acting on a locally finite CAT(0) cube complex X with finitely many orbits of hyperplanes. Then G X contains a contracting isometry if and only if ∂ c X has an isolated vertex. This result will be essentially a direct consequence of our main technical lemma:
Suppose by contradiction that
H(V k ) ∩ H(r) is infinite for some k ≥ 0. Fix a vertex x 0 ∈ r adjacent to V k and H 1 , H 2 , . . . ∈ H(V k ) ∩ H(r). Now, set x i the combinatorial projection of x 0 onto N (H i ) for every i ≥ 1; notice that d(x 0 , x i ) −→ i→+∞ +∞ since X is
locally finite so that only finitely many hyperplanes intersect a given ball centered at x 0 . Then, once again because X is locally finite, if we choose a combinatorial geodesic [x 0 , x i ] between x 0 and x i for every i ≥ 1, up to taking a subsequence, we may suppose that the sequence ([x 0 , x i ]) converges to some combinatorial ray ρ.
Let J ∈ H(ρ). Then J separates x 0 and x i for some i ≥ 1. From Lemma 2.7, we deduce that J separates x 0 and N (H i ). On the other hand, we know that H i intersects the combinatorial ray r which starts from x 0 , so necessarily J ∈ H(r). We have proved that H(ρ) ⊂ H(r)holds.
Next, notice that x i ∈ N (V k ) for every i ≥ 1. Indeed, if p i denotes the combinatorial projection of x 0 onto N (V k ) ∩ N (H i ) and m the median vertex of {x 0 , x i , p i }, then x 0 , p i ∈ N (V k ) implies m ∈ N (V k ) and x i , p i ∈ N (H i ) implies m ∈ N (H i ), hence m ∈ N (H i ) ∩ N (V k )
. Since x i minimizes the distance to x 0 in N (H i ) and that m belongs to to a geodesic between x 0 and x i , we conclude that
x i = m ∈ N (V k ). A fortiori, ρ ⊂ N (V k ). As a consequence, V k+1 , V k+2 , . . . /
∈ H(ρ), so ρ defines a point of ∂ c X, different from r, which is ≺-comparable with r: this contradicts the fact that r is isolated in ∂ c X. Our claim is proved.
C i = J∈K i {halfspace delimited by J containing x i }.
Let C i be the cycle of subcomplexes (N (V k ), C i , N (V k+i ), r), and let D i → X be a disc diagram of minimal complexity bounded by C i . According to Corollary 2.14, D i → X is an isometric embedding so that we will identify D i with its image in X. Because H(V k ) ∩ H(r) is finite according to our previous claim, we know that, for sufficently large i ≥ 1, K i will contain a hyperplane disjoint from r, so that C i ∩ r = ∅. Up to taking a subsequence, we will suppose that this is always the case. In particular, ∂D i can be decomposed as the concatenation of four non trivial combinatorial geodesics
∂D i = α i ∪ β i ∪ γ i ∪ r i , where α i ⊂ N (V k ), β i ⊂ C i , γ i ⊂ N (V k+i ) and r i ⊂ r.
Notice that the intersection between α i and r i is a vertex adjacent to x 0 , so, if we use the local finiteness of X to define, up to taking a subsequence, a subcomplex D ∞ as the limit of (D i ), then D ∞ is naturally bounded by two combinatorial rays α ∞ and r ∞ which are the limits of (α i ) and (r i ) respectively; in particular, α ∞ ⊂ N (V k ) and r ∞ is a subray of r.
For each i ≥ 1, we will say that a hyperplane of D i intersecting β i is vertical, and a hyperplane of Notice that ρ ∞ is naturally the limit of (ρ i ) := (∂D i ∩ N (H i )). Thus, any hyperplane intersecting ρ ∞ is a vertical hyperplane in some D i . It follows from Fact 4.22 that
D i intersecting α i is horizontal; a horizontal hyperplane J ∈ H(D i ) is high if d(x 0 , J ∩ α i ) > #H(V k ) ∩ H(r).H(ρ ∞ ) ⊂ H(r). Because r is isolated in ∂ c X, we deduce that r = ρ ∞ in ∂ c X. Now, we clearly have H(ρ ∞ ) ⊂ H(µ), hence H(r) ⊂ a H(µ).
On the other hand, µ intersects infinitely many high horizontal hyperplanes, which are disjoint from r according to Fact 4.22. Therefore, µ is a point of ∂ c X which is ≺-comparable with r, but different from it, contradicting the assumption that r is isolated in ∂ c X. This concludes the proof of our claim. Now, we are able to conclude the proof of our lemma. Applying Claim 4.21 N − 1 times, we find a sequence of hyperplanes Proof of Theorem 4.18. If G contains a contracting isometry, then ∂ c X contains an isolated point according to Theorem 4.17. Conversely, suppose that ∂ c X contains an isolated point. Let N denote the number of orbits of hyperplanes for the action G X. According to Lemma 4.19, X contains a family of 3N pairwise well-separated hyperplanes which does not contain any facing triple. By our choice of N , we deduce that there exist a hyperplane J and g, h ∈ G such that J, gJ, hgJ are pairwise well-separated. Without loss of generality, suppose that hgJ separates J and gJ, and let J + be the halfspace delimited by J which contains gJ and hgJ. If gJ + ⊂ J + or hgJ + ⊂ J + , we deduce that g or hg skewers a pair of well-separated hyperplanes, and we deduce from Theorem 3.13 that G contains a contracting isometry. Otherwise, hgJ + ⊂ gJ + since hgJ separates J and gJ. Therefore, h skewers a pair of well-separated hyperplanes, and we deduce from Theorem 3.13 that G contains a contracting isometry.
V i(1) , . . . , V i(N ) such that V i(k) and V i(k+1) are L(k)-well-separated for some L(k) ≥ 1. Thus, {V i(1) , . . . , V i(N ) }
Our second result stating that the existence of an isolated point in the combinatorial boundary assures the existence of a contracting isometry is the following: Our main tool to prove Theorem 4.23 is:
Lemma 4.25. Let G be a countable group acting minimally on a CAT(0) cube complex X without fixing any point of ∂ c X. Then any hyperplane of X is G-flippable.
Proof. Suppose that there exists a hyperplane J which is not G-flippable, ie., J admits an orientation (J − , J + ) such that gJ + ∩ J + = ∅ for every g ∈ G. Thus, {gJ + | g ∈ G} defines a family of pairwise intersecting halfspaces. If the intersection g∈G gJ + is nonempty, it defines a proper G-invariant combinatorially convex subcomplex, so that the action G X is not minimal.
From now on, suppose that g∈G gJ + = ∅. Fix an enumeration G = {g 1 , g 2 , . . .} and let I n denote the non-empty intersection n i=1 g i J + . In particular, we have I 1 ⊃ I 2 ⊃ · · · and n≥1 I n = ∅. Define a sequence of vertices (x n ) by:
x 0 / ∈ I 1 and x n+1 = p n+1 (x n ) for every n ≥ 0,
where p k : X → I k denotes the combinatorial projection onto I k . According to Proposition 2.5, for every n ≥ 1, there exists a combinatorial geodesic between x 0 and x n passing through x k for any 1 ≤ k ≤ n, so there exists a combinatorial ray r passing through all the x n 's. Notice that, because we also have x n = p n (x 0 ) for every n ≥ 1 according to Lemma 2.8, H(r) is precisely the set of the hyperplanes separating x 0 from some I n . In particular, if ρ is any combinatorial ray starting from x 0 and intersecting all the I n 's, then H(r) ⊂ H(ρ).
Therefore, k≥1 ∂ c I k is a non-empty (since it contains r) subset of ∂ c X with r as a minimal element, ie., for any ρ ∈ k≥1 ∂ c I k necessarily r ≺ ρ.
Let us prove that k≥1 ∂ c I k is G-invariant. Fix some g ∈ G, and, for every k ≥ 1, let L(k, g) be a sufficiently large integer so that {gg 1 , . . . , gg k } ⊂ {g 1 , . . . , g L(k,g) }. Of course, L(k, g) ≥ k and gI k =
k i=1 gg i J + ⊃ I L(k,g) . Consequently, g k≥1 ∂ c I k = k≥1 ∂ c (gI k ) ⊃ k≥1 ∂ c I L(k,g) = k≥1 ∂ c I k , because L(k, g) −→ k→∞ +∞. Thus, we have proved that g k≥1 ∂ c I k ⊃ k≥1 ∂ c I k for every g ∈ G.
It follows that our intersection is G-invariant.
Finally, because G acts on ∂ c X by order-automorphisms, we conclude that G fixes r ∈ ∂ c X.
hgH + gH − = X\gH + X\gJ + = gJ − J + .
We deduce that hg skewers the pair (J, H).
Remark 4.27. Lemma 4.25 may be thought of as a generalisation of the Flipping Lemma proved in [CS11], where the Tits boundary is replaced with the combinatorial boundary in possibly infinite dimension. Corollary 4.26 corresponds to the Double Skewering Lemma [CS11].
Proof of Theorem 4.23. If G contains a contracting isometry, then ∂ c X contains an isolated point according to Theorem 4.17. Conversely, suppose that ∂ c X contains an isolated point. It follows from Lemma 4.19 that X contains two well-separated hyperplanes, and it suffices to invoke Corollary 4.26 to find an element g ∈ G which skewers this pair of hyperplanes: according to Theorem 3.13, this is a contracting isometry.
Acylindrical hyperbolicity of diagram groups
Diagram groups and their cube complexes
We refer to [GS97, §3 and §5] for a detailed introduction to semigroup diagrams and diagram groups.
For an alphabet Σ, let Σ + denote the free semigroup over Σ. If P = Σ | R is a semigroup presentation, where R is a set of pairs of words in Σ + , the semigroup associated to P is the one given by the factor-semigroup Σ + / ∼ where ∼ is the smallest equivalent relation on Σ + containing R. We will always assume that if u = v ∈ R then v = u / ∈ R; in particular, u = u / ∈ R. A semigroup diagram over P is the analogue for semigroups to van Kampen diagrams for group presentations. Formally, it is a finite connected planar graph ∆ whose edges are oriented and labelled by the alphabet Σ, satisfying the following properties:
• ∆ has exactly one vertex-source ι (which has no incoming edges) and exactly one vertex-sink τ (which has no outgoing edges);
• the boundary of each cell has the form pq −1 where p = q or q = p ∈ R;
• every vertex belongs to a positive path connecting ι and τ ;
• every positive path in ∆ is simple.
In particular, ∆ is bounded by two positive paths: the top path, denoted top(∆), and the bottom path, denoted bot(∆). By extension, we also define top(Γ) and bot(Γ) for every subdiagram Γ. In the following, the notations top(·) and bot(·) will refer to the paths and to their labels. Also, a (u, v)-cell (resp. a (u, v)-diagram) will refer to a cell (resp. a semigroup diagram) whose top path is labelled by u and whose bottom path is labelled by v. Two words w 1 , w 2 in Σ + are equal modulo P if their images in the semigroup associated to P coincide. In particular, there exists a derivation from w 1 to w 2 , i.e., a sequence of relations of R allowing us to transform w 1 into w 2 . To any such derivation is associated a semigroup diagram, or more precisely a (w 1 , w 2 )-diagram, whose construction is clear from the example below. As in the case for groups, the words w 1 , w 2 are equal modulo P if and only if there exists a (w 1 , w 2 )-diagram.
Example 5.1. Let P = a, b, c | ab = ba, ac = ca, bc = cb be a presentation of the free Abelian semigroup of rank three. In particular, the words a 2 bc and caba are equal modulo P, with the following possible derivation: Then, the associated (a 2 bc, caba)-diagram ∆ is:
On such a graph, the edges are supposed oriented from left to right. Here, the diagram ∆ has nine vertices, twelve edges and four cells; notice that the number of cells of a diagram corresponds to the length of the associated derivation. The paths top(∆) and bot(∆) are labelled respectively by a 2 bc and caba.
Since we are only interested in the combinatorics of semigroup diagrams, we will not distinguish isotopic diagrams. For example, the two diagrams below will be considered as equal.
If w ∈ Σ + , we define the trivial diagram (w) as the semigroup diagram without cells whose top and bottom paths, labelled by w, coincide. Any diagram without cells is trivial. A diagram with exactly one cell is atomic.
If ∆ 1 is a (w 1 , w 2 )-diagram and ∆ 2 a (w 2 , w 3 )-diagram, we define the concatenation ∆ 1 • ∆ 2 as the semigroup diagram obtained by identifying the bottom path of ∆ 1 with the top path of ∆ 2 . In particular, ∆ 1 •∆ 2 is a (w 1 , w 3 )-diagram. Thus, • defines a partial operation on the set of semigroup diagrams over P. However, restricted to the subset of (w, w)-diagrams for some w ∈ Σ + , it defines a semigroup operation; such diagrams are called spherical with base w. We also define the sum ∆ 1 + ∆ 2 of two diagrams ∆ 1 , ∆ 2 as the diagram obtained by identifying the rightmost vertex of ∆ 1 with the leftmost vertex of ∆ 2 .
Notice that any semigroup diagram can be viewed as a concatenation of atomic diagrams. In the following, if ∆ 1 , ∆ 2 are two diagrams, we will say that ∆ 1 is a prefix (resp. a suffix) of ∆ 2 if there exists a diagram ∆ 3 satisfying ∆ 2 = ∆ 1 • ∆ 3 (resp. ∆ 2 = ∆ 3 • ∆ 1 ). Throughout this paper, the fact that ∆ is a prefix of Γ will be denoted by ∆ ≤ Γ.
Suppose that a diagram ∆ contains a (u, v)-cell and a (v, u)-cell such that the top path of the first cell is the bottom path of the second cell. Then, we say that these two cells form a dipole. In this case, we can remove these two cells by first removing their common path, and then identifying the bottom path of the first cell with the top path of the second cell; thus, we reduce the dipole. A diagram is called reduced if it does not contain dipoles. By reducing dipoles, a diagram can be transformed into a reduced diagram, and a result of Kilibarda [Kil94] proves that this reduced form is unique. If ∆ 1 , ∆ 2 are two diagrams for which ∆ 1 • ∆ 2 is well defined, let us denote by ∆ 1 · ∆ 2 the reduced form of ∆ 1 • ∆ 2 .
Thus, the set of reduced semigroup diagrams, endowed with the product · we defined above, naturally defines a groupoid G(P), ie., loosely speaking, a "group" where the product is only partially defined. The neutral elements correspond to the trivial diagrams, and the inverse of a diagram is constructed in the following way: if ∆ is a (w 1 , w 2 )-diagram, its inverse ∆ −1 is the (w 2 , w 1 )-diagram obtained from ∆ by a mirror reflection with respect to top(∆). A natural way to deduce a group from this groupoid is to fix a base word w ∈ Σ + , and to define the diagram group D(P, w) as the set of reduced (w, w)-diagrams endowed with the product · we defined above.
Farley cube complexes. The groupoid G(P) has a natural generating set, namely the set of atomic diagrams, and the group D(P, w) acts by left-multiplication on the connected component of the associated Cayley graph which contains (w). Surprisingly, this Cayley graph turns out to be naturally the 1-skeletton of a CAT(0) cube complex. Below, we detail the construction of this cube complex as explained in [Far03].
A semigroup diagram is thin whenever it can be written as a sum of atomic diagrams. We define the Farley complex X(P, w) as the cube complex whose vertices are the reduced semigroup diagrams ∆ over P satisfying top(∆) = w, and whose n-cubes are spanned by the vertices {∆ · P | P ≤ Γ} for some vertex ∆ and some thin diagram Γ with n cells. In particular, two diagrams ∆ 1 and ∆ 2 are linked by an edge if and only if there exists an atomic diagram A such that ∆ 1 = ∆ 2 · A. There is a natural action of D(P, w) on X(P, w), namely (g, ∆) → g · ∆. Then
Proposition 5.3. [Far03, Theorem 3.13] The action D(P, w) X(P, w) is free. Moreover, it is cocompact if and only if the class [w] P of words equal to w modulo P is finite.
In this paper, we will always suppose that the cube complex X(P, w) is locally finite; in particular, this implies that the action D(P, w) X(P, w) is properly discontinuous. For instance, this is case if P is a finite presentation, and because we may always suppose that P is a finite presentation if our diagram group is finitely generated, this assumption is not really restrictive. However, notice that X(P, w) may be locally finite even if P is an infinite presentation. For example, the derived subgroup [F, F ] of Thompson's group F is isomorphic to the diagram group D (P, a 0
b 0 ), where P = x, a i , b i , i ≥ 0 | x = x 2 , a i = a i+1 x, b i = xb i+1 , i ≥ 0 .
See [GS99, Theorem 26] for more details. In this situation, X(P, a 0 b 0 ) is nevertheless locally finite.
Because X(P, w) is essentially a Cayley graph, the combinatorial geodesics behave as expected:
Lemma 5.4. [Gen15a, Lemma 2.3] Let ∆ be a reduced diagram. If ∆ = A 1 • · · · • A n where each A i is an atomic diagram, then the path (w), A 1 , A 1 • A 2 , . . . , A 1 • · · · • A n = ∆ defines a combinatorial geodesic from (w) to ∆. Furthermore, every combinatorial geodesic from (w) to ∆ has this form.
Corollary 5.5. [Gen15a, Corollary 2.4] Let A, B ∈ X(P, w) be two reduced diagrams. Then we have
d(A, B) = #(A −1 · B),
where #(·) denotes the number of cells of a diagram. In particular, d( (w), A) = #(A).
In [Gen15a], we describe the hyperplanes of X(P, w). We recal this description below. For any hyperplane J of X(P, w), we introduce the following notation:
• J + is the halfspace associated to J not containing (w),
• J − is the halfspace associated to J containing (w),
• ∂ ± J is the intersection ∂N (J) ∩ J ± .
Definition 5.6. A diagram ∆ is minimal if its maximal thin suffix F has exactly one cell. (The existence and uniqueness of the maximal thin suffix is given by [Far03, Lemma 2.3].)
In the following, ∆ will denote the diagram ∆ · F −1 , obtained from ∆ by removing the suffix F . The following result uses minimal diagrams to describe hyperplanes in Farley complexes.
Proposition 5.7. [Gen15a, Proposition 2.2] Let J be a hyperplane of X(P, w). Then there exists a unique minimal diagram ∆ such that
J + = {D diagram | ∆ ≤ D}.
Conversely, if ∆ is a minimal diagram and J the hyperplane dual to the edge [∆, ∆],
then J + = {D diagram | ∆ ≤ D}.
Thanks to the geometry of CAT(0) cube complexes, we will use this description of the hyperplanes of X(P, w) in order to define the supremum, when it exists, of a collection of minimal diagrams. If ∆ is a diagram, let M(∆) denote the set of its minimal prefixes. A set of minimal diagrams D is consistent if, for every D 1 , D 2 ∈ D, there exists some D 3 ∈ D satisfying D 1 , D 2 ≤ D 3 .
Proof. Let
D = {D 1 , . . . , D r }. For every 1 ≤ i ≤ r, let J i denote the hyperplane associated to D i . Because D is consistent, for every 1 ≤ i, j ≤ r the intersection J + i ∩ J + j is non-empty, hence C = r i=1 J + i = ∅.
Let ∆ denote the combinatorial projection of (w) onto C. For every 1 ≤ i ≤ r, ∆ belongs to J + i , hence D i ≤ ∆. Then, if D is another diagram admitting any element of D as a prefix, necessarily D ∈ C, and it follows from Proposition 2.5 that there exists a combinatorial geodesic between (w) and D passing through ∆, hence ∆ ≤ D. This proves the ≤-minimality and the uniqueness of ∆. Now, let D ∈ M(∆). If J denotes the hyperplane associated to D, then J separates (w) and ∆, and according to Lemma 2.7 it must separate (w) and C. Therefore, either J belongs to {J 1 , . . . , J r } or it separates (w) and some J k ; equivalently, either D belongs to {D 1 , . . . , D r } or it is a prefix of some D k . Hence D ∈ As a consequence, we deduce that a diagram is determined by the set of its minimal prefixes.
Lemma 5.9. Let ∆ 1 and ∆ 2 be two diagrams. If M(∆ 1 ) = M(∆ 2 ) then ∆ 1 = ∆ 2 .
Proof. We begin with some general results. Let ∆ be a diagram. Finally, our lemma follows easily, since ∆ 1 = sup(M(∆ 1 )) = sup(M(∆ 2 )) = ∆ 2 .
Squier cube complexes. Because the action D(P, w)
X(P, w) is free and properly discontinuous, we know that D(P, w) is isomorphic to the fundamental group of the quotient space X(P, w)/D(P, w). We give now a description of this quotient. We define the Squier complex S(P) as the cube complex whose vertices are the words in Σ + ; whose (oriented) edges can be written as (a, u → v, b), where u = v or v = u belongs to R, linking the vertices aub and avb; and whose n-cubes similarly can be written as (a 1 , u 1 → v 1 , . . . , a n , u n → v n , a n+1 ), spanned by the set of vertices {a 1 w 1 · · · a n w n a n+1 | w i = v i or u i }.
Then, there is a natural morphism from the fundamental group of S(P) based at w to the diagram group D(P, w). Indeed, a loop in S(P) based at w is just a series of relations of R applied to the word w so that the final word is again w, and such a derivation may be encoded into a semigroup diagram. The figure below shows an example, where the semigroup presentation is P = a, b, c | ab = ba, bc = cb, ac = ca :
Thus, this defines a map from the set of loops of S(P) based at w to the set of spherical semigroup diagrams. In fact, the map extends to a morphism which turns out to be an isomorphism:
Theorem 5.12. [GS97,Theorem 6.1] D(P, w) π 1 (S(P), w).
For convenience, S(P, w) will denote the connected component of S(P) containing w. Notice that two words w 1 , w 2 ∈ Σ + are equal modulo P if and only if they belong to the same connected component of S(P). Therefore, a consequence of Theorem 5.12 is:
Corollary 5.13. If w 1 , w 2 ∈ Σ + are equal modulo P, then there exists a (w 2 , w 1 )diagram Γ and the map ∆ → Γ · ∆ · Γ −1 induces an isomorphism from D(P, w 1 ) to D(P, w 2 ).
As claimed, we notice that the map ∆ → bot(∆) induces a universal covering X(P, w) → S(P, w) so that the action of π 1 (S(P, w)) on X(P, w) coincides with the natural action of D(P, w).
Lemma 5.14. [Gen15a, Lemma 1.3.5] The map ∆ → bot(∆) induces a cellular isomorphism from the quotient X(P, w)/D(P, w) to S(P, w).
Changing the base word. When we work on the Cayley graph of a group, because the action of the group is vertex-transitive, we may always suppose that a fixed base point is the identity element up to a conjugation. In our situation, where X(P, w) is the Cayley graph of a groupoid, the situation is slightly different but it is nevertheless possible to make something similar. Indeed, if ∆ ∈ X(P, w) is a base point, then conjugating by ∆ sends ∆ to a trivial diagram, but which does not necessarily belong to X(P, w): in general, it will belong to X(P, bot(∆)). In the same time, D(P, w) becomes D(P, bot(∆)), which is naturally isomorphic to D(P, w) according to Corollary 5.13. Finally, we get a commutative diagram
D(P, w) isomorphism / / D(P, bot(∆))
X(P, w)
isometry / / X(P, bot(∆)) Thus, we may always suppose that a fixed base point is a trivial diagram up to changing the base word, which does not disturb either the group or the cube complex. In particular, a contracting isometry stays a contracting isometry after the process.
Infinite diagrams.
Basically, an infinite diagram is just a diagram with infinitely many cells.
To be more precise, if finite diagrams are thought of as specific planar graphs, then an infinite diagram is the union of an increasing sequence finite diagrams with the same top path. Formally, Definition 5.15. An infinite diagram is a formal concatenation ∆ 1 •∆ 2 •· · · of infinitely many diagrams ∆ 1 , ∆ 2 , . . . up to the following identification: two formal concatenations A 1 • A 2 • · · · and B 1 • B 2 • · · · define the same infinite diagram if, for every i ≥ 1, there exists some j ≥ 1 such that A 1 • · · · • A i is a prefix of B 1 • · · · • B j , and conversely, for every i ≥ 1, there exists some j ≥ 1 such that B 1 • · · · • B i is a prefix of A 1 • · · · • A j . An infinite diagram ∆ = ∆ 1 • ∆ 2 • · · · is reduced if ∆ 1 • · · · • ∆ n is reduced for every n ≥ 1. If top(∆ 1 ) = w, we say that ∆ is an infinite w-diagram.
Notice that, according to Lemma 5.4, a combinatorial ray is naturally labelled by a reduced infinite diagram. This explains why infinite diagrams will play a central role in the description of the combinatorial boundaries of Farley cube complexes.
Definition 5.16. Let ∆ = ∆ 1 • ∆ 2 • · · · and Ξ = Ξ 1 • Ξ 2 • · · · be two infinite diagrams. We say that ∆ is a prefix of Ξ if, for every i ≥ 1, there exists some j ≥ 1 such that ∆ 1 • · · · • ∆ i is a prefix of Ξ 1 • · · · Ξ j .
Combinatorial boundaries of Farley complexes
The goal of this section is to describe the combinatorial boundaries of Farley cube complexes as a set of infinite reduced diagrams. First, we need to introduce a partial order on the set of infinite diagrams. Basically, we will say that a diagram ∆ 1 is almost a prefix of ∆ 2 if there exists a third diagram ∆ 0 which is a prefix of ∆ 1 and ∆ 2 such that all but finitely many cells of ∆ 1 belong to ∆ 0 . Notice that, if we fix a decomposition of a diagram ∆ as a concatenation of atomic diagrams, then to any cell π of ∆ corresponds naturally a minimal prefix of ∆, namely the smallest prefix containing π (which is precisely the union of all the cells of ∆ which are "above" π; more formally, using the relation introduced by Definition 5.32, this is the union of all the cells π of ∆ satisfying π ≺≺ π). Therefore, alternatively we can describe our order in terms of minimal prefixes.
Definition 5.17. Let ∆ 1 and ∆ 2 be two infinite reduced diagrams. Then,
• ∆ 1 is almost a prefix of ∆ 2 if M(∆ 1 ) ⊂ a M(∆ 2 ); • ∆ 1 and ∆ 2 are commensurable if M(∆ 1 ) = a M(∆ 2 ).
Notice that two infinite reduced diagrams are commensurable if and only if each one is almost a prefix of the other. Our model for the combinatorial boundaries of Farley cube complexes will be the following:
Definition 5.18. If P = Σ | R is semigroup presentation and w ∈ Σ + a base word, let (∂(P, w), < a ) denote the poset of the infinite reduced (w, * )-diagrams, up to commensurability, ordered by the relation < a of being almost a prefix.
Recall that to any combinatorial ray r ⊂ X(P, w) starting from (w) is naturally associated an infinite reduced (w, * )-diagram Ψ(r).
Theorem 5.19. The map Ψ induces a poset-isomorphism ∂ c X(P, w) → ∂(P, w).
The main tool to prove our theorem is provided by the following proposition, which is an infinite analogue of Proposition 5.8. Recall that a collection of (finite) diagrams D is consistent if, for every ∆ 1 , ∆ 2 ∈ D, there exists a third diagram ∆ satisfying ∆ 1 , ∆ 2 ≤ ∆.
Proof. Let D = {D 1 , D 2 , . . .}, with D i = D j if i = j.
For each i ≥ 1, let H i denote the hyperplane of X(P, w) associated to D i . Because D is consistent, the halfspaces H + i and H + j intersect for every i ≥ 1. Therefore, for every k ≥ 1, the intersection
I k = k i=1
H + i is non-empty; let ∆ k denote the combinatorial projection of (w) onto I k . As a consequence of Proposition 2.5, there exists a combinatorial ray starting from (w) and passing through each ∆ k . In particular, for every k ≥ 1, ∆ k is a prefix of ∆ k+1 , so that it makes sense to define the infinite diagram ∆ as the limit of the sequence (∆ k ). Notice that, for every k ≥ 1, ∆ k ∈ H + k so D k ≤ ∆ k and a fortiori D k ≤ ∆. Now, let Ξ be another reduced w-diagram admitting each D k as a prefix. Fixing a decomposition of Ξ as an infinite concatenation of atomic diagrams, say A 1 • A 2 • · · · , we associate to Ξ a combinatorial ray r starting from (w) defined as the path
(w), A 1 , A 1 • A 2 , A 1 • A 2 • A 3 . . .
Let n ≥ 1. Because D 1 , . . . , D n are prefixes of Ξ, there exists some k ≥ 1 such that r(k) ∈ I n . On the other hand, according to Proposition 2.5, there exists a combinatorial geodesic between (w) and r(k) passing through ∆ n , hence ∆ n ≤ r(k). Therefore, we have proved that, for every n ≥ 1, ∆ n is a prefix of Ξ: this precisely means that ∆ is a prefix of Ξ. The ≤-minimality and the uniqueness of ∆ is thus proved.
Because the D k 's are prefixes of ∆, the inclusion D∈D M(D) ⊂ M(∆) is clear. Conversely, let D ∈ M(∆); let J denote the associated hyperplane. Because D is a prefix of ∆, it is a prefix of ∆ k for some k ≥ 1; and because ∆ k is the combinatorial projection of (w) onto I k , we deduce that J separates (w) and I k . On the other hand,
I k = k i=1 H + i so there exists some 1 ≤ i ≤ k such that either J = H i or J separates (w) and H i ; equivalently, D is a prefix of D i . We have proved M(∆) ⊂ D∈D M(D).
The following lemma states how the operation sup behaves with respect to the inclusion, the almost-inclusion and the almost-equality.
Lemma 5.21. Let D 1 and D 2 be two countable consistent collections of reduced (w, * )diagrams. If D 1 = a D 2 , we deduce by applying the previous point twice that sup(D 1 ) is almost a prefix of sup(D 2 ) and vice-versa. Therefore, sup(D 1 ) and sup(D 2 ) are commensurable.
(i) If D 1 ⊂ D 2 then sup(D 1 ) ≤ sup(D 2 ); (ii) if D 1 ⊂ a D 2 then sup(D 1 ) is almost a prefix of sup(D 2 ); (iii) if D 1 = a D 2
Finally, Theorem 5.19 will be essentially a consequence of the previous lemma and the following observation. If r is a combinatorial ray in a Farley complex, let D(r) denote the set of the minimal diagrams which are associated to the hyperplanes of H(r). Then Lemma 5.22. Ψ(r) = sup(D(r)). A 1 , A 2 , A 3 . . . be the sequence of atomic diagrams such that r = ( (w), A 1 , A 1 • A 2 , A 1 • A 2 • A 3 , . . .).
Proof. Let
In particular, Ψ(r) = A 1 • A 2 • · · · and M(Ψ(r)) = n≥1 M(A 1 • · · · • A n ). On the other hand, any D ∈ D(r) must be a prefix of A 1 • · · · • A n for some n ≥ 1, and conversely, any minimal prefix of some A 1 • · · · • A n must belong to D(r). Therefore,
M(Ψ(r)) = n≥1 M(A 1 • · · · • A n ) = M(sup(D(r))).
If ∆ is a finite prefix of Ψ(r), then M(∆) ⊂ M(Ψ(r)) = M(sup(D(r))), so ∆ = sup(M(∆)) must be a prefix of sup(D(r)) as well. Therefore, Ψ(r) ≤ M(sup(D(r))). Similarly, we prove that M(sup(D(r))) ≤ Ψ(r), hence Ψ(r) = M(sup(D(r))).
Proof of Theorem 5.19. First, we have to verify that, if r 1 and r 2 are two equivalent combinatorial rays starting from (w), then Ψ(r 1 ) = a Ψ(r 2 ). Because r 1 and r 2 are equivalent, i.e., H(r 1 ) = a H(r 2 ), we know that D(r 1 ) = a D(r 2 ), which implies Ψ(r 1 ) = sup(D(r 1 )) = a sup(D(r 2 )) = Ψ(r 2 ) according to Lemma 5.21 and Lemma 5.22. Therefore, Ψ induces to a map ∂ c X(P, w) → ∂(P, w). For convenience, this map will also be denoted by Ψ.
If r 1 ≺ r 2 , then H(r 1 ) ⊂ a H(r 2 ) hence Ψ(r 1 ) = sup(D(r 1 )) < a sup(D(r 2 )) = Ψ(r 2 ). Therefore, Ψ is a poset-morphism.
Let ∆ be an infinite reduced (w, * )-diagram. Let A 1 , A 2 , . . . be a sequence of atomic diagrams such that ∆ = A 1 •A 2 •· · · . Let ρ denote the combinatorial ray ( (w),
A 1 , A 1 • A 2 , . . .). Now it is clear that Ψ(ρ) = A 1 • A 2 • · · · = ∆, so Ψ is surjective.
Let r 1 and r 2 be two combinatorial rays starting from (w) and satisfying Ψ(r 1 ) = a Ψ(r 2 ). It follows from Lemma 5.22 that sup(D(r 1 )) = Notice that D(r 2 ) is stable under taking a prefix. Indeed, if P is a prefix of some ∆ ∈ D(r 2 ) then the hyperplane J associated to P separates (w) and the hyperplane H associated to ∆; because H ∈ H(r 2 ) and r 2 (0) = (w), necessarily J ∈ H(r 2 ), hence P ∈ D(r 2 ). As a consequence, we deduce that
D(r 1 )\D(r 2 ) ⊂ D∈D(r 1 ) M(D) \ D∈D(r 2 ) M(D) .
We conclude that D(r 1 )\D(r 2 ) is finite. We prove similarly that D(r 2 )\D(r 1 ) is finite. Therefore, we have D(r 1 ) = a D(r 2 ), which implies H(r 1 ) = a H(r 2 ), so that r 1 and r 2 are equivalent. We have proved that Ψ is injective.
Example 5.23. Let P = x | x = x 2 be the semigroup presentation usually associated to Thompson's group F . An atomic diagram is said positive if the associated relation is x → x 2 , and negative if the associated relation is x 2 → x; by extension, a diagram is positive (resp. negative) if it is a concatenation of positive (resp. negative) atomic diagrams. Notice that any reduced infinite (x, * )-diagram has an infinite positive prefix. On the other hand, there exists a reduced infinite postive (x, * )-diagram containing all the possible reduced infinite positive (x, * )-diagrams as prefixes: it corresponds to the derivation obtained by replacing at each step each x by x 2 . Therefore, ∂(P, x) does not contain any isolated vertex. We deduce from Theorem 4.17 that the automorphism group Aut(X(P, x)) does not contain any contracting isometry.
Example 5.24. Let P = a, b, p 1 , p 2 , p 3 | a = ap 1 , b = p 1 b, p 1 = p 2 , p 2 = p 3 , p 3 = p 1 be the semigroup presentation usually associated to the lamplighter group Z Z. We leave as an exercice to notice that ∂(P, ab) has no isolated point. Therefore, Aut(X(P, ab)) does not contain any contracting isometry according to Theorem 4.17.
Contracting isometries
In this section, we fix a semigroup presentation P = Σ | R and base word w ∈ Σ + . Our goal is to determine precisely when a spherical diagram g ∈ D(P, w) induces a contracting isometry on the Farley complex X(P, w). For convenience, we will suppose that g is absolutely reduced, ie., for every n ≥ 1 the concatenation of n copies of g is reduced; this assumption is not really restrictive since, according to [GS97,Lemma 15.10], any spherical diagram is conjugated to an absolutely reduced spherical diagram (possibly for a different word base). In particular, we may define the infinite reduced (w, * )-diagram g ∞ as g • g • · · · . Our main criterion is: (ii) if ∆ is a reduced diagram with g ∞ as a prefix, then ∆ is commensurable to g ∞ . This theorem will be essentially a consequence of the following two lemmas. Proof. Notice that γ is a bi-infinite combinatorial path passing through g n for every n ∈ Z. Therefore, in order to deduce that γ is a geodesic, it is sufficient to show that, for every n, m ∈ Z, the length of γ between g n and g n+m is equal to d(g n , g n+m ); but this length is precisely m · length([ (w), g]) = m · d( (w), g) = m · #(g), and, on the other hand, because g is absolutely reduced, d(g n , g n+m ) = #(g −n · g n+m ) = #(g m ) = m · #(g).
We conclude that γ is a combinatorial geodesic. The fact that g acts on γ by translation is clear. Finally, we deduce that γ(+∞) = g ∞ from the fact that, for any k ≥ 1, we have γ(k · #(g)) = g k .
Definition 5.27. Let ∆ be a possibly infinite (w, * )-diagram. The support of ∆ is the set of maximal subpaths of w whose edges belong to the top path of some cell of ∆.
Lemma 5.28. Let ∆ 1 , . . . , ∆ n be copies of an absolutely reduced spherical (w, w)-diagram ∆, where n > |w|. If e ⊂ ∆ 1 • · · · • ∆ n which belongs to the bottom path of some cell of ∆ 1 , then e belongs to the top path of some cell in ∆ 1 • · · · • ∆ n .
Proof. Say that an edge of ∆ i is a cutting edge if it belongs to the bottom path of some cell of ∆ i but does not belong to the top path of some cell in ∆ 1 • · · · • ∆ n . Suppose that there exists a cutting edge e 1 in ∆ 1 .
In particular, e 1 is an edge of the top path of ∆ 2 , so e 1 does not belong to the support of ∆ 2 . We deduce that ∆ 2 decomposes as a sum Φ 2 + + Ψ 2 , where is a trivial diagram with top and bottom paths equal to e 1 . Let Φ 1 + + Ψ 1 denote the same decomposition of ∆ 1 . Because e 1 belongs to the bottom path of a cell of ∆ 1 , two cases may happen: either e 1 belongs to bot(Φ 1 ), so that top(Φ 2 ) bot(Φ 1 ), or e 1 belongs to bot(Ψ 1 ), so that top(Ψ 2 ) bot(Ψ 1 ). Say we are in the former case, the latter case being similar. Now, because ∆ 2 and ∆ 1 are two copies of the same diagram, to the edge e 1 of ∆ 1 corresponds an edge e 2 of ∆ 2 . Notice that, because e 1 belongs to bot(Φ 1 ), necessarily e 2 = e 1 . Moreover, the product ∆ −1 • ∆ 1 • · · · • ∆ n naturally reduces to a copy of ∆ 1 • · · · • ∆ n−1 , and this process sends the edge e 2 to the edge e 1 . Therefore, because e 1 is a cutting edge in ∆ 1 , we deduce that e 2 is a cutting edge in ∆ 2 .
By iterating this construction, we find a cutting edge e i in ∆ i for every 1 ≤ i ≤ n, where e i = e j provided that i = j. On the other hand, the path bot(∆ 1 • · · · • ∆ n ) necessarily contains all these edges, hence n ≤ |w|. Consequently, ∆ 1 cannot contain a cutting edge if n > |w|.
Proof of Theorem 5.25. Let γ be the combinatorial axis of g given by Lemma 5.26. According to Theorem 4.17, g is a contracting isometry if and only if γ(+∞) is isolated in ∂ c X(P, w). Thus, using the isomorphism given by Theorem 5.19, g is a contracting isometry if and only if g ∞ is isolated in ∂(P, w), ie.,
• every infinite prefix of g ∞ is commensurable to g ∞ ;
• if ∆ is a reduced diagram with g ∞ as a prefix, then ∆ is commensurable to g ∞ . Therefore, to conclude it is sufficient to prove that a proper infinite prefix ∆ of g ∞ cannot be commensurable to g ∞ . Because ∆ is a proper prefix, there exists a cell π 1 of g ∞ which does not belong to ∆. Now, by applying Lemma 5.28 to a given edge of the bottom path of π 1 , we find a cell π 2 whose top path intersects the bottom path of π 1 along at least one edge; by applying Lemma 5.28 to a given edge of the bottom path of π 2 , we find a cell π 3 whose top path intersects the bottom path of π 2 along at least one edge; and so on. Finally, we find an infinite sequence of cells π 2 , π 3 , . . . such that, for every i ≥ 2, the top path of π i has a common edge with the bottom path of π i−1 . For every i ≥ 1, let Ξ i denote the smallest (finite) prefix of g ∞ which contains the cell π i . This is a minimal diagram, and by construction Ξ 1 , Ξ 2 , . . . / ∈ M(∆), which proves that ∆ and g ∞ are not commensurable.
Example 5.29. Let P = a, b, p | a = ap, b = pb and let g ∈ D(P, ab) be the following spherical diagram:
Then g ∞ clearly contains a proper infinite prefix. Therefore, g is not a contracting isometry of X(P, ab).
Example 5.30. Let P = a, b, c | a = b, b = c, c = a and let g ∈ D(P, a 2 ) be the following spherical diagram:
Then g ∞ is a prefix of the diagram ∆ below, but g ∞ and ∆ are not commensurable. Therefore, g is not a contracting isometry of X(P, a 2 ).
Example 5.31. Let P = a, b, c, d | ab = ac, cd = bd and let g ∈ D(P, ab) be the following spherical diagram:
So the infinite diagram g ∞ looks like
We can notice that g ∞ does not contain a proper infinite prefix that any diagram containing g ∞ as a prefix is necessarily equal to g ∞ .
The criterion provided by Theorem 5.25 may be considered as unsatisfactory, because we cannot draw an infinite diagram in practice (although it seems to be sufficient for spherical diagrams with only few cells, as suggested by the above examples). We conclude this section by stating and proving equivalent conditions to the assertions (i) and (ii) of Theorem 5.25. For convenience, we will suppose our spherical diagram absolutely reduced and normal, ie., the factors of every decomposition as a sum are spherical; this assumption is not really restrictive since, according to [GS97,Lemma 15.13], every spherical diagram is conjugated to a normal absolutely reduced spherical diagram (possibly for a different word base), and this new diagram can be found "effectively" [GS97, Lemma 15.14]. We begin with the condition (i). The following definition will be needed.
Definition 5.32. Given a diagram ∆, we introduce a partial relation ≺ on the set of its cells, defined by: if π, π are two cells of ∆, then π ≺ π holds if the intersection between the bottom path of π and the top path of π contains at least one edge. Let ≺≺ denote the transitive closure of ≺. (ii) if ∆ is a reduced diagram with g ∞ as a prefix, then ∆ is commensurable to g ∞ .
Proof. First, notice that g = (x) + g 0 + (z) for some (y, y)-diagram g 0 . This assertion clearly holds for some (y, * )-diagram g 0 , but, because g is a (w, w)-diagram, the equality xyz = w = top(g) = bot(g) = x · bot(g 0 ) · z holds in Σ + . Therefore, bot(g 0 ) = y, and we conclude that g 0 is indeed a (y, y)-diagram. Now, we prove (i) ⇒ (ii). Let ∆ be a reduced diagram containing g ∞ as a prefix. Suppose that π is a cell of ∆ such that there exists a cell π 0 of g ∞ satisfying π 0 ≺≺ π. So there exists a chain of cells π 0 ≺ π 1 ≺ · · · ≺ π n = π. By definition, π 1 share an edge with π 0 . On the other hand, it follows from Lemma 5.28 that this edge belongs to the top path of a cell of g ∞ , hence π 1 ⊂ g ∞ . By iterating this argument, we conclude that π belongs to g ∞ . Therefore, if Ξ denotes the smallest prefix of ∆ containing a given cell which does not belong to g ∞ , then supp(Ξ) is included into x or z.
As a consequence, there exist a (x, * )-diagram Φ and (z, * )-diagram Ψ such that ∆ = Φ + g ∞ 0 + Ψ. On the other hand, because ∂(P, x) and ∂(P, y) are empty, necessarily #Φ, #Ψ < +∞. Therefore, |M(∆)\M(g ∞ )| = |M(Φ + (yz)) M( (xy) + Ψ)| = #Φ + #Ψ < +∞, where we used Fact 5.10. Therefore, ∆ is commensurable to g ∞ .
Conversely, we prove (ii) ⇒ (i). Let Φ be a reduced (x, * )-diagram and Ψ a reduced (z, * )-diagram. Set ∆ = Φ + g ∞ 0 + Ψ. This is a reduced diagram containing g ∞ as a prefix. As above, we have |M(∆)\M(g ∞ )| = #Φ + #Ψ.
Because ∆ must be commensurable to g ∞ , we deduce that Φ and Ψ are necessarily finite. We conclude that ∂(P, x) and ∂(P, z) are empty. Now, we focus on the condition (ii) of Theorem 5.25. We first introduce the following definition; explicit examples are given at the end of this section.
Definition 5.34. Let g ∈ D(P, w) − { (w)} be absolutely reduced. A subpath u of w is admissible if there exists a prefix ∆ ≤ g such that supp(∆) = {u} and bot(∆) ∩ bot(g) is a non-empty connected subpath of bot(g); the subpath of w corresponding to bot(∆) ∩ bot(g) is a final subpath associated to u; it is proper if this is not the whole bot(g). Given a subpath u, we define its admissible tree T g (u) as the rooted tree constructed inductively as follows:
• the root of T g (u) is labeled by u;
• if v is a non-admissible subpath of w labelling a vertex of T g (u), then v has only one descendant, labeled by the symbol ∅;
• if v is an admissible subpath of w labelling a vertex of T g (u), then the descendants of v correspond to the proper final subpaths associated to u.
If w has length n, we say that T g (u) is deep if it contains at least α(w) + 1 generations, where α(w) = n(n+1) 2 is chosen so that w contains at most α(w) subpaths.
Lemma 5.35. Let g ∈ D(P, w) − { (w)} be absolutely reduced. The following two conditions are equivalent:
(i) g ∞ contain an infinite proper prefix;
(ii) there exists a subpath u of w such that T g (u) is deep.
Proof. For convenience, let g ∞ = g 1 • g 2 • · · · where each g i is a copy of g.
We first prove (i) ⇒ (ii). Suppose that g ∞ contains an infinite propre prefix Ξ. For every i ≥ 1, Ξ induces a prefix Ξ i of g i . Up to taking an infinite prefix of Ξ, we can suppose that supp(Ξ i ) and bot(Ξ i ) ∩ bot(g i ) are connected subpaths of top(g i ) and bot(g i ) respectively. Thus, if ξ i denotes the word labeling supp(Ξ i ) for every i ≥ 1, then the sequence (ξ i ) defines an infinite descending ray in the rooted tree T g (ξ 1 ). A fortiori,
T g (ξ 1 ) is deep.
Conversely, we prove (ii) ⇒ (i). Suppose that there exists a subword u of w such that T g (u) is deep. Because w contains at most α(w) subwords, necessarily T g (u) contains twice a proper subword ξ of supp(g) in a same lineage. Let ξ = ξ 0 , . . . , ξ n = ξ be the geodesic between these two vertices in T g (u). For every 0 ≤ i ≤ n − 1, there exists a prefix Ξ i of g such that supp(Ξ i ) = {ξ i } and bot(X i ) ∩ bot(g) is a connected path labeled by ξ i+1 . In particular,
Ξ i = (x i ) + Ξ i + (y i ) for some words x i , y i ∈ Σ + and some (ξ i , p i ξ i+1 q i )-diagram Ξ i . Now, define Ξ i+1 = (x 0 p 0 · · · p i ) + Ξ i+1 + (q i · · · q 0 y 0 ) for i ≥ 0.
Notice that Ξ i+1 is a (x 0 p 0 · · · p i ξ i q i · · · q 0 y 0 , x 0 p 0 · · · p i+1 ξ i+1 q i+1 · · · q 0 y 0 )-diagram, so that the concatenation Ξ = Ξ 0 • Ξ 1 • · · · • Ξ n−1 is well-defined.
We have proved that g n contains a prefix Ξ such that supp(Ξ) = {ξ} and bot(Ξ)∩bot(g n ) is the same subpath ξ.
Say that Ξ is a (xξy, xpξqy)-diagram, for some words x, y, p, q ∈ Σ + . In particular, Ξ decomposes as a sum (x) + Ξ + (y) where Ξ is a (ξ, pξq)-diagram. For every i ≥ 0, set
∆ i = (xp i ) + Ξ + (q i y).
Finally, the concatenation ∆ = ∆ 0 • ∆ 1 • · · · defines an infinite prefix of g ∞ . Moreover,
supp(∆) = supp(Ξ) = {ξ} supp(g) ⊂ supp(g ∞ ),
so ∆ is a proper prefix of g ∞ .
By combining Lemma 5.33 and Lemma 5.35 with Theorem 5.25, we obtain the following new criterion.
Proposition 5.36. Let g ∈ D(P, w) − { (w)} be normal and absolutely reduced. Then g is a contracting isometry of X(P, w) if and only if the following two conditions are satisfied:
• the support of g is reduced to {w 1 } for some subword w 1 of w, and, if we write w = xw 1 y, then ∂(P, x) and ∂(P, y) are empty;
• for every subpath u of w, T g (u) is not deep.
Proof. According to Theorem 5.25, g is contracting if and only if (i) g ∞ does not contain any infinite proper prefix;
(ii) if ∆ is a reduced diagram with g ∞ as a prefix, then ∆ is commensurable to g ∞ . Now, Lemma 5.35 states that the condition (i) is equivalent to (iii) for every subpath u of w, T g (u) is not deep;
and Lemma 5.33 implies that the condition (ii) is a consequence of (iv) the support of g is reduced to {w 1 } for some subword w 1 of w, and, if we write w = xw 1 y, then ∂(P, x) and ∂(P, y) are empty.
To conclude the proof, it is sufficient to show that (iv) is a consequence of (i) and (ii). So suppose that (i) and (ii) hold. If supp(g) is not reduced to a single subpath, then g decomposes as a sum of non-trivial diagrams, and because g is normal, it decomposes as a sum of non-trivial spherical diagrams, say g = g 1 + g 2 . Now, since g 1 and g 2 are spherical, g ∞ = g ∞ 1 + g ∞ 2 , so g ∞ clearly contains an infinite proper prefix, contradicting (i). Therefore, supp(g) reduces to a single subpath, and Lemma 5.33 implies that (iv) holds.
Remark 5.37. The first condition of Proposition 5.36 is obviously satisfied if supp(g) = {w}. Because the case often happen, we will say that spherical diagrams satisfying this property are full.
Example 5.38. The group Z • Z = a, b, t | [a, b t n ] = 1, n ≥ 0 , introduced in [GS97,
Section 8], is isomorphic to the diagram group D(P, ab), where P = a 1 , a 2 , a 3 , p, b 1 , b 2
, b 3 a 1 = a 1 p b 1 = pb 1 , a 1 = a 2 , a 2 = a 3 , a 3 = a 1 b 1 = b 2 , b 2 = b 3 , b 3 = b 1 .
Let g ∈ D(P, ab) be the following spherical diagram group:
This is a full, normal, and absolutely reduced diagram. Moreover, a 1 b 1 is the only admissible subpath, and T g (a 1 b 1 ) is
a 1 b 1 b 1
Therefore, g is a contracting isometry of X(P, ab). In particular, Z • Z is acylindrically hyperbolic.
Example 5.39. Let P = a, b | ab = ba, ab 2 = b 2 a and let g ∈ D(P, ab 4 ) be the following spherical diagram:
This is a full, normal, and absolutely reduced diagram. Morever, ab 4 is the only admissible subpath, and T g (ab 4 ) is
ab 4 b 4 b ∅ ∅
We conclude that g is a contracting isometry of X(P, ab 4 ).
Some examples
In this section, we exhibit some interesting classes of acylindrically hyperbolic diagram groups by applying the results proved in the previous section. In Example 5.40, we consider a family of groups U (m 1 , . . . , m n ) already studied in [GS97] and prove that they are acylindrically hyperbolic. Morever, we notice that each L n = U (1, . . . , 1) turns out to be naturally a subgroup of a right-angled Coxeter group. In Example 5.41, we show that the •-product, as defined in [GS97], of two non trivial diagram groups is acylindrically hyperbolic but not relatively hyperbolic. Finally, in Example 5.43, we prove that some diagram products, as defined in [GS99], of diagram groups turn out to be acylindrically hyperbolic. As a by-product, we also exhibit a cocompact diagram group which is not isomorphic to a right-angled Artin group.
Example 5.40. Let P n = x 1 , . . . , x n | x i x j = x j x i , 1 ≤ i < j ≤ n and let U (m 1 , . . . , m n ) denote the diagram group D(P, x m 1 1 · · · x mn n ), where m i ≥ 1 for every 1 ≤ i ≤ n. Notice that, for every permutation σ ∈ S n , U (m σ(1) , . . . , m σ(n) ) is isomorphic to U (m 1 , . . . , m n ); therefore, we will always suppose that m 1 ≤ · · · ≤ m n . For example, the groups U (p, q, r) were considered in [GS97, Example 10.2] (see also [Gen15a,Example 5.3] and [Gen15b, Example 5.11]). In particular, we know that We claim that, whenever it is not cyclic, the group U (m 1 , . . . , m n ) is acylindrically hyperbolic. For instance, if g ∈ U (1, 1, 2) denotes the spherical diagram associated to the following derivation
x 1 x 2 x 3 x 3 → x 2 x 1 x 3 x 3 → x 2 x 3 x 1 x 3 → x 3 x 2 x 1 x 3 → x 3 x 1 x 2 x 3 → x 1 x 3 x 2 x 3 → x 1 x 2 x 3 x 3 ,
then g is a contracting isometry of the associated Farley cube complex. (The example generalizes easily to other cases although writing an argument in full generality is laborious.) In particular, we deduce that U (m 1 , . . . , m n ) does not split as non trivial direct product.
An interesting particular case is the group L n , corresponding to U (1, . . . , 1) with n ones. A first observation is that L n is naturally a subgroup of U (m 1 , . . . , m n ), and conversely, U (m 1 , . . . , m n ) is naturally a subgroup of L m 1 +···+mn . Secondly, L n can be described as a group of pseudo-braids; this description is made explicit by the following example of a diagram with its associated pseudo-braid:
Of course, we look at our pseudo-braids up to the following move, corresponding to the reduction of a dipole in the associated diagram:
Finally, if C n denotes the group of all pictures of pseudo-braids, endowed with the obvious concatenation, then L n corresponds to "pure subgroup" of C n . It is clear that, if σ i corresponds to the element of C n switching the i-th and (i + 1)-th braids, then a presentation of C n is σ 1 , . . . , σ n−1 | σ 2 i = 1, [σ i , σ j ] = 1 if |i − j| ≥ 2 . Alternatively, C n is isomorphic to the right-angled Coxeter group C(P n−1 ), where P n−1 is the complement of the graph P n−1 which is a segment with n − 1 vertices, i.e., P n−1 is the graph with the same vertices as P n−1 but whose edges link two vertices precisely when they are not adjacent in P n−1 . In particular, L n is naturally a finite-index subgroup of C(P n−1 ).
With this interpretation, the group U (m 1 , . . . , m n ) corresponds to the subgroup of L m , where m = m 1 + · · · + m n , defined as follows: Color the m braids with n colors, such that, for every 1 ≤ i ≤ n, there are m i braids with the color i. Now, U (m 1 , . . . , m n ) is the subgroup of the pure pseudo-braids where two braids of the same color are never switched.
Therefore, the groups U (m 1 , . . . , m n ) turn out to be natural subgroups of right-angled Coxeter groups.
Example 5.41. Let G and H be two groups. We define the product G • H by the relative presentation
G, H, t | [g, h t ] = 1, g ∈ G, h ∈ H .
As proved in [GS97,Theorem 8.6], the class of diagram groups is stable under the •-product. More precisely, let P 1 = Σ 1 | R 1 , P 2 = Σ 2 | R 2 be two semigroup presentations and w 1 , w 2 two base words. For i = 1, 2, suppose that there does not exist two words x and y such that xy is non-empty and w i = xw i y; as explained in during the proof of [GS97,Theorem 8.6], we may suppose that this condition holds up to a small modification of P i which does not modify the diagram group. Then, if
P = Σ 1 Σ 2 {p} | R 1 R 2 {w 1 = w 1 p, w 2 = pw 2 } ,
then D(P, w 1 w 2 ) is isomorphic to D(P 1 , w 1 ) • D(P 2 , w 2 ) [GS97, Lemma 8.5]. Now, we claim that, whenever D(P 1 , w 1 ) and D(P 2 , w 2 ) are non-trivial, the product D(P 1 , w 1 ) • D(P 2 , w 2 ) is acylindrically hyperbolic. Indeed, if Γ ∈ D(P 1 , w 1 ) and ∆ ∈ D(P 2 , w 2 ) are non-trivial, then the spherical diagram of D(P, w 1 w 2 ) is a contracting isometry. On the other hand, D(P 1 , w 1 ) • D(P 2 , w 2 ) contains a copy of Z • Z, so it cannot be cyclic. We conclude that D(P 1 , w 1 ) • D(P 2 , w 2 ) is indeed acylindrically hyperbolic. Indeed, let K be a malnormal subgroup of G • H. Suppose that K contains a non-trivial element of G × H ∞ , say gh. Then h ∈ K and g ∈ K since
gh = gh ∩ gh h ⊂ K ∩ K h and gh = gh ∩ gh g ⊂ K ∩ K g . If h is not trivial, then G ⊂ K since h = h ∩ h G ⊂ K ∩ K G ;
then, we deduce that H ∞ ⊂ K since
G = G ∩ G H ∞ ⊂ K ∩ K H ∞ . Therefore, G × H ∞ ⊂ K.
Similarly, if h is trivial then g is non-trivial, and we deduce that H ∞ ⊂ K from
g = g ∩ g H ∞ ⊂ K ∩ K H ∞ .
We also notice that G ⊂ K since
H ∞ = H ∞ ∩ (H ∞ ) G ⊂ K ∩ K G .
Consequently, G × H ∞ ⊂ K. Thus, we have proved that a malnormal subgroup of G • H either intersects trivially G × H ∞ or contains G × H ∞ . If K intersects trivially G × H ∞ , the action of K on the Bass-Serre tree of G • H, associated to its decomposition as an HNN extension we mentionned above, is free, so K is necessarily free. Otherwise, K contains G × H ∞ . If t denotes the stable letter of our HNN extension, then
H 2 * H 3 * · · · = H ∞ ∩ (H ∞ ) t ⊂ K ∩ K t ,
where we used the notation H ∞ = H 1 * H 2 * H 3 * · · · , such that each H i is a copy of H. Therefore, t ∈ K, and we conclude that K is not a proper subgroup. The fact is proved.
As a consequence, •-products of two non-trivial diagram groups are good examples of acylindrically hyperbolic groups in the sense that they are not relatively hyperbolic. Indeed, if a group is hyperbolic relatively to a collection of subgroups, then this collection must be malnormal (see for instance [Osi06, Theorem 1.4]), so that a relatively hyperbolic •-product of two torsion-free groups must hyperbolic relatively to a collection of free groups according to the previous fact. On the other hand, such a group must be hyperbolic (see for instance [Osi06, Corollary 2.41]), which is impossible since we know that a •-product of two non trivial torsion-free groups contains a subgroup isomorphic to Z 2 .
Example 5.43. Let P = Σ | R be a semigroup presentation and w ∈ Σ + a base word. Fixing a family of groups G Σ = {G s | s ∈ Σ}, we define the diagram product D(G Σ , P, w) as the fundamental group of the following 2-complex of groups:
• the underlying 2-complex is the 2-skeleton of the Squier complex S(P, w);
• to any vertex u = s 1 · · · s r ∈ Σ + is associated the group G u = G s 1 × · · · × G sr ;
• to any edge e = (a, u → v, b) is associated the group G e = G a × G b ;
• to any square is associated the trivial group;
• for every edge e = (a, u → v, b), the monomorphisms G e → G aub and G e → G avb are the canonical maps
G a × G b → G a × G u × G b and G a × G b → G a × G v × G b .
Guba and Sapir proved that the class of diagram groups is stable under diagram product [GS99,Theorem 4]. Explicitely, Theorem 5.44. If G s = D(P s , w s ) where P s = Σ s | R s is a semigroup presentation and w s ∈ Σ + s a base word, then the diagram product D(G Σ , P, w) is the diagram group D(P, w), whereP is the semigroup presentation Of course, any semigroup diagram with respect to P is a semigroup diagram with respect toP. Thus, if D(P, w) contains a normal, absolutely reduced, spherical diagram which does not contains a proper infinite prefix and whose support is {w}, then any diagram product over (P, w) will be acylindrically hyperbolic or cyclic. (Notice that such a diagram product contains D(P, w) as a subgroup, so it cannot be cyclic if D(P, w) is not cyclic itself.)
Σ Σ Σ 0 | R R S where Σ 0 = {a s | s ∈ Σ}
For instance, let P = x, y, z, a, b | yz = az, xa = xb, b = y . Then D(P, xyz) contains the following spherical diagram, which is a contracting isometry of X(P, xyz):
By the above observation, if G denotes the diagram product over (P, apb) with G x = G z = Z and G y = G a = G b = {1}, then G is acylindrically hyperbolic. In this case, the Squier complex S(P, xyz) is just a cycle of length three. By computing a presentation of the associated graph of groups, we find
G = x, y, t | [x, y] = [x, y t ] = 1 .
This example is interesting because this is a cocompact diagram group which is not a right-angled Artin group. Let us prove this assertion. From the presentation of G, it is clear that its abelianization is Z 3 and we deduce from [Bro82, Exercice II.5.5] that the rank of H 2 (G) is at most two (the number of relations of our presentation). On the other hand, if A(Γ) denotes the right-angled Artin group associated to a given graph Γ, then H 1 (A(Γ)) = Z #V (Γ) and H 2 (A(Γ)) = Z #E(Γ) , where V (Γ) and E(Γ) denote the set of vertices and edges of Γ respectively. Therefore, the only right-angled Artin groups which might be isomorphic to G must correspond to a graph with three vertices and at most two edges. We distinguish two cases: either this graph is not connected or it is a segment of length two. In the former case, the right-angled Artin group decomposes as a free product, and in the latter, it is isomorphic to F 2 × Z. Thus, it is sufficient to show that G is freely irreducible and is not isomorphic to F 2 × Z in order to deduce that G cannot be isomorphic to a right-angled Artin group. First, notice that, because G is acylindrically hyperbolic, its center must be finite (see [Osi13,Corollary 7.3]), so G cannot be isomorphic to F 2 × Z. Next, suppose that G decomposes as a free product. Because the centralizer of x is not virtually cyclic, we deduce that x belongs to a conjugate A of some free factor; in fact, A must contain the whole centralizer of x, hence y, y t ∈ A. As a consequence, y t ∈ A ∩ A t , so that A ∩ A t must be infinite. Since a free factor is a malnormal subgroup, we deduce that t ∈ A. Finally, x, y, t ∈ A hence A = G. Therefore, G does not contain any proper free factor, i.e., G is freely irreducible. This concludes the proof of our assertion.
To conclude, it is worth noticing that the whole diagram product may contain contracting isometries even if the underlying diagram group (i.e., the diagram group obtained from the semigroup presentation and the word base, forgetting the factor-groups) does not contain such isometries on its own Farley cube complex. For instance, if P = a, b, p | a = ap, b = pb , w = ab, G p = {1} and G a = D(P 1 , w 1 ), G b = D(P 2 , w 2 ) are non-trivial, then our diagram product is isomorphic to G a • G b (see [GS99,Example 7]), and the description of G a •G b as a diagram group is precisely the one we mentionned in the previous example, where we saw that the action on the associated Farley cube complex admits contracting isometries. On the other hand, X(P, ab) is a "diagonal half-plane", so that its combinatorial boundary does not contain any isolated point: a fortiori, X(P, ab) cannot admit a contracting isometry.
Cocompact case
In this section, we focus on cocompact diagram groups, ie., we will consider a semigroup presentation P = Σ | R and a word base w ∈ Σ + whose class modulo P is finite.
Roughly speaking, we want to prove that D(P, w) contains a contracting isometry if and only if it does not split as a direct product in a natural way, which we describe now.
If there exists a (w, u 1 v 1 · · · u n v n u n+1 )-diagram Γ, for some words u 1 , v 1 , . . . , u n , v n , u n+1 ∈ Σ + , then we have the natural map
D(P, v 1 ) × · · · × D(P, v n ) → D(P, w) (V 1 , . . . , V n ) → Γ · ( (u 1 ) + V 1 + · · · + (u n ) + V n + (u n+1 )) · Γ −1
This is a monomorphism, and we denote by Γ · D(P, v 1 ) × · · · × D(P, v n ) · Γ −1 its image. A subgroup of this form will be referred to as a canonical subgroup; if furthermore at least two factors are non trivial, then it will be a large canonical subgroup.
The main result of this section is the following decomposition theorem (cf. Theorem 1.11 in the introduction).
Theorem 5.45. Let P = Σ | R be a semigroup presentation and w ∈ Σ + a base word whose class modulo P is finite. Then there exist some words u 0 , u 1 , . . . , u m ∈ Σ + and a (w, u 0 u 1 · · · u m )-diagram Γ such that
D(P, w) = Γ · (D(P, u 0 ) × D(P, u 1 ) × · · · × D(P, u m )) · Γ −1 ,
where D(P, u i ) is either infinite cyclic or acylindrically hyperbolic for every 1 ≤ i ≤ m, and m ≤ dim alg D(P, w) ≤ dim X(P, w).
The acylindrical hyperbolicity will follow from the existence of a contracting isometry in the corresponding Farley cube complex. In order to understand what happens when such an isometry does not exist, we will use the following dichotomy, which is a slight modification, in the cocompact case, of the Rank Rigidity Theorem proved in [CS11].
Theorem 5.46. Let G be a group acting geometrically on an unbounded CAT(0) cube complex X. Either G contains a contracting isometry of X or there exists a G-invariant convex subcomplex Y ⊂ X which splits as the cartesian product of two unbounded subcomplexes.
Proof. Recall that a hyperplane of X is essential if no G-orbit lies in a neighborhood of some halfspace delimited by this hyperplane; we will denote by E(X) the set of the essential hyperplanes of X. Let Y ⊂ X denote the essential core associated to the action G X, which is defined as the restriction quotient of X with respect to E(X). See [CS11, Section 3.3] for more details. According to [CS11, Proposition 3.5], Y is a G-invariant convex subcomplex of X.
It is worth noticing that, because the action G X is cocompact, a hyperplane J of X does not belong to E(X) if and only if one of the halfspaces it delimits lies in the R(J)-neighborhood of N (J) for some R(J) ≥ 1. Set R = max J∈H(X) R(J). Because there exist only finitely many orbits of hyperplanes, R is finite.
We claim that two hyperplanes of Y are well-separated in Y if and only if they are well-separated in X.
Of course, two hyperplanes of Y which are well-separated in X are well-separated in Y . Conversely, let J 1 , J 2 ∈ H(Y ) be two hyperplanes and fix some finite collection H ⊂ H(X) of hyperplanes intersecting both J 1 , J 2 which does not contain any facing triple. We write H = {H 0 , . . . , H n } and we fix a halfspace H + i delimited by H i for every 0 ≤ i ≤ n, so that H + 0 ⊃ H + 1 ⊃ · · · ⊃ H + n . If n ≤ 2R, there is nothing to prove. Otherwise, by our choice of R, it is clear that the hyperplanes H R , . . . , H n−R belong to E(X), and a fortiori to H(Y ). Consequently, if J 1 and J 2 are L-well-separated in Y , then they are (L + 2R)-well-separated in X. Now we are ready to prove our theorem. Because the action G Y is geometric and essential (ie., every hyperplane of Y is essential), it follows from [CS11, Theorem 6.3] that either Y splits non trivially as a cartesian product of two subcomplexes or G contains a contracting isometry of Y . In the former case, notice that the factors cannot be bounded because the action G Y is essential; in the latter case, we deduce that the contracting isometry of Y induces a contracting isometry of X by combining our previous claim with Theorem 3.13. This concludes the proof.
Therefore, the problem reduces to understand the subproducts of X(P, w). This purpose is achieved by the next proposition.
Proposition 5.47. Let P = Σ | R be a semigroup presentation and w ∈ Σ + a base word. Let Y ⊂ X(P, w) be a convex subcomplex which splits as the cartesian product of two subcomplexes A, B ⊂ Y . Then there exist a word w ∈ Σ + , a (w, w )-diagram Γ and words u 1 , v 1 , . . . , u k , v k ∈ Σ + for some k ≥ 1 such that w = u 1 v 1 · · · u k v k in Σ + and A ⊂ Γ · X(P, u 1 ) × · · · × X(P, u k ) · Γ −1 and B ⊂ Γ · X(P, v 1 ) × · · · × X(P, v k ) · Γ −1 .
Furthermore,
Y ⊂ Γ · X(P, u 1 ) × X(P, v 1 ) × · · · × X(P, u k ) × X(P, v k ) · Γ −1 .
In the previous statement, Y is a Cartesian product of the subcomplexes A and B in the sense that there exists an isomorphism ϕ : A × B → Y which commutes with the inclusions A, B, Y → X. More explicitely, we have the following commutative diagram:
A M m { { o A × B ϕ / / Y / / X B 1 Q c c / > >
Proof of Proposition 5.47. Up to conjugating by a diagram Γ of A ∩ B, we may assume without loss of generality that (w) ∈ A ∩ B. First, we claim that, for every ∆ 1 ∈ A and ∆ 2 ∈ B, supp(∆ 1 ) ∩ supp(∆ 2 ) = ∅.
Suppose by contradiction that there exist some ∆ 1 ∈ A and ∆ 2 ∈ B satisfying supp(∆ 1 )∩ supp(∆ 2 ) = ∅. Without loss of generality, we may suppose that we chose a counterexample minimizing #∆ 1 + #∆ 2 . Because supp(∆ 1 ) ∩ supp(∆ 2 ) = ∅, there exist two cells π 1 ⊂ ∆ 1 , π 2 ⊂ ∆ 2 and an edge e ⊂ w, where w is thought of as a segment representing both top(∆ 1 ) and top(∆ 2 ), such that e ⊂ top(π 1 ) ∩ top(π 2 ). Notice that π i is the only cell of the maximal thin suffix of ∆ i for i = 1, 2. Indeed, if this was not the case, taking the minimal prefixes P 1 , P 2 of ∆ 1 , ∆ 2 containing π 1 , π 2 respectively, we would produce two diagrams satisfying supp(P 1 ) ∩ supp(P 2 ) = ∅ and #P 1 + #P 2 < #∆ 1 + #∆ 2 . Moreover, because Y is convex necessarily the factors A, B have to be convex as well, so that A and B, as sets of diagrams, have to be closed by taking prefixes since (w) ∈ A ∩ B and according to the description of the combinatorial geodesics given by Lemma 5.4; this implies that P 1 ∈ A and P 2 ∈ B, contradicting the choice of our counterexample ∆ 1 , ∆ 2 . For i = 1, 2, let ∆ i denote the diagram obtained from ∆ i by removing the cell π i . We claim that supp(∆ 1 ) ∩ suppp(∆ 2 ) = supp(∆ 1 ) ∩ suppp(∆ 2 ) = ∅.
Indeed, since we know that A and B are stable by taking prefixes, ∆ 1 belongs to A and ∆ 2 belongs to B. Therefore, because #∆ 1 + #∆ 2 = #∆ 1 + #∆ 2 < #∆ 1 + #∆ 2 , having supp(∆ 1 ) ∩ suppp(∆ 2 ) = ∅ or supp(∆ 1 ) ∩ suppp(∆ 2 ) = ∅ would contradict our choice of ∆ 1 and ∆ 2 .
Because supp(∆ 1 ) ∩ suppp(∆ 2 ) = ∅, it is possible to write w = a 1 b 1 · · · a p b p in Σ + so that
∆ 1 = A 1 + (b 1 ) + · · · + A p + (b p ) ∆ 2 = (a 1 ) + B 1 + · · · + (a p ) + B p for some (a i , * )-diagram A i and (b i , * )-diagram B i . Set ∆ 0 = A 1 + B 1 + · · · + A p + B p .
Notice that ∆ 1 and ∆ 2 are prefixes of ∆ 0 .
For i = 1, 2, let E i denote the atomic diagram such that the concatenation ∆ 0 • E i corresponds to gluing π i on ∆ i as a prefix of ∆ 0 . Notice that the diagrams E 1 , E 2 exist precisely because
supp(∆ 1 ) ∩ suppp(∆ 2 ) = supp(∆ 1 ) ∩ suppp(∆ 2 ) = ∅.
As top(π 1 ) ∩ top(π 2 ) = ∅, since the intersection contains e, the two edges (∆ 0 , ∆ 0 • E 1 ) and (∆ 0 , ∆ 0 • E 2 ) of X(P, w) do not span a square. Therefore, the hyperplanes J 1 , J 2 dual to these two edges respectively are tangent. For i = 1, 2, noticing that ∆ i is a prefix of ∆ 0 • E i whereas it is not a prefix of ∆ 0 , we deduce from Proposition 5.7 that the minimal diagram associated to J i is precisely ∆ i . On the other hand, the hyperplanes corresponding to ∆ 1 , ∆ 2 are clearly dual edges of A, B respectively. A fortiori, J 1 and J 2 must be transverse (in Y = A × B). Therefore, we have found an inter-osculation in the CAT(0) cube complex X(P, w), which is impossible.
Thus, we have proved that, for every ∆ 1 ∈ A and ∆ 2 ∈ B, supp(∆ 1 ) ∩ supp(∆ 2 ) = ∅.
In particular, we can write w = u 1 v 1 · · · u k v k in Σ + , for some u 1 , v 1 , . . . , u k , v k ∈ Σ + , so that supp(∆ 1 ) ⊂ u 1 ∪ · · · ∪ u k and supp(∆ 2 ) ⊂ v 1 ∪ · · · ∪ v k for every ∆ 1 ∈ A and ∆ 2 ∈ B. By construction, it is clear that A ⊂ X(P, u 1 ) × · · · × X(P, u k ) and B ⊂ X(P, v 1 ) × · · · × X(P, v k ). Now, let ∆ ∈ Y = A × B be vertex and let ∆ 1 ∈ A, ∆ 2 ∈ B denote its coordinates; they are also the projections of ∆ onto A, B respectively. By the previous remark, we can write
∆ 1 = U 1 + (v 1 ) + · · · + U k + (v k ) ∆ 2 = (u 1 ) + V 1 + · · · + (u k ) + V k for some (u i , * )-diagram U i and some (v i , * )-diagram V i . Set Ξ = U 1 + V 1 + · · · U k + V k .
Noticing that d(Ξ, A) ≥ #V 1 + · · · + #V k and d(Ξ, ∆ 1 ) = #V 1 + · · · + #V k , we deduce that ∆ 1 is the projection of Ξ onto A. Similarly, we show that ∆ 2 is the projection of Ξ onto B. Consequently,
∆ = Ξ ∈ X(P, u 1 ) × X(P, v 1 ) × · · · × X(P, u k ) × X(P, v k ),
which concludes the proof.
By combining Theorem 5.46 and Proposition 5.47, we are finally able to prove:
Corollary 5.48. Let P = Σ | R be a semigroup presentation and w ∈ Σ + a base word whose class modulo P is finite. Then either D(P, w) contains a contracting isometry of X(P, w) or there exists a (w, uv)-diagram Γ such that
D(P, w) = Γ · D(P, u) × D(P, v) · Γ −1 ,
where D(P, u) and D(P, v) are non trivial.
Proof. According to Theorem 5.46, if D(P, w) does not contain a contracting isometry of X(P, w), then there exists a convex D(P, w)-invariant subcomplex Y ⊂ X(P, w) which splits as the cartesian product of two unbounded subcomplexes A, B. Up to conjugating by the smallest diagram of Y , we may assume without loss of generality that Y contains (w). Let Γ be the (w, u 1 v 1 · · · u k v k )-diagram given by Proposition 5.47. Notice that, because (w) ∈ Y and that Y is D(P, w)-invariant, necessarily any spherical diagram belongs to Y . Thus, we deduce from Lemma 5.49 below that
D(P, w) = Γ · D(P, u 1 ) × D(P, v 1 ) × · · · × D(P, u k ) × D(P, v k ) · Γ −1 .
Because A ⊂ Γ · X(P, u 1 ) × · · · × X(P, u k ) · Γ −1 , we deduce from the unboundedness of A that X(P, u 1 ) × · · · × X(P, u k ) is infinite; since the class of w modulo P is finite, this implies that there exists some 1 ≤ i ≤ k such that D(P, u i ) is non trivial. Similarly, we show that there exists some 1 ≤ j ≤ k such that D(P, v j ) is non trivial. Now, write u 1 v 1 · · · u k v k = uv in Σ + where u, v ∈ Σ + are two subwords such that one contains u i and the other v j . Then
D(P, w) = Γ · D(P, u) × D(P, v) · Γ −1 ,
where D(P, u) and D(P, v) are non trivial.
Lemma 5.49. Let P = Σ | R be a semigroup presentation and w ∈ Σ + a base word whose class modulo P is finite. If a spherical diagram ∆ decomposes as a sum ∆ 1 + · · · + ∆ n , then each ∆ i is a spherical diagram.
Proof. Suppose by contradiction that one factor is not spherical. Let ∆ k be the leftmost factor which is not spherical, ie., ∆ is spherical, say a (x , x )-diagram, for every < k.
For ≥ k, say that ∆ is a (x , y )-diagram. In Σ + , we have the equality
x 1 · · · x k−1 x k x k+1 · · · x n = top(∆) = bot(∆) = x 1 · · · x k−1 y k y k+1 · · · y n , hence x k · · · x n = y k · · · y n . Because the equality holds in Σ + , notice that lg(x k ) = lg(y k ) implies x k = y k , which is impossible since we supposed that ∆ k is not spherical. Therefore, two cases may happen: either lg(x k ) > lg(y k ) or lg(x k ) < lg(y k ). Replacing ∆ with ∆ −1 if necessary, we may suppose without loss of generality that we are in the latter case. Thus, x k is a prefix of y k : there exists some y ∈ Σ + such that y k = x k y. On the other hand, because ∆ k is a (x k , y k )-diagram, the equality x k = y k holds modulo P. A fortiori, x k = x k y modulo P. We deduce that
x 1 · · · x k−1 x k y n x k+1 · · · x n ∈ [w] P for every n ≥ 1. This implies that [w] P is infinite, contradicting our hypothesis.
Proof of Theorem 5.45. We argue by induction on the algebraic dimension of the considered diagram group, ie., the maximal rank of a free abelian subgroup. If the algebraic dimension is zero, there is nothing to prove since the diagram group is trivial in this case. From now on, suppose that our diagram group D(P, w) has algebraic dimension at least one. By applying Corollary 5.48, we deduce that either D(P, w) contains a contracting isometry of X(P, w), and we conclude that D(P, w) is either cyclic or acylindrically hyperbolic, or there exist two words u, v ∈ Σ + and a (w, uv)diagram Γ such that D(P, w) = Γ · (D(P, u) × D(P, v)) · Γ −1 where D(P, u) and D(P, v) are non-trivial. In the first case, we are done; in the latter case, because D(P, u) and D(P, v) are non-trivial (and torsion-free, as any diagram group), their algebraic dimensions are strictly less than the one of D(P, w), so that we can apply our induction hypothesis to find a (u, u 1 · · · u r )-diagram Φ and a (v, v 1 · · · v s )-diagram Ψ such that D(P, u) = Φ · D(P, u 1 ) × · · · × D(P, u r ) · Φ −1 and similarly
D(P, v) = Ψ · D(P, v 1 ) × · · · × D(P, v s ) · Ψ −1 ,
where, for every 1 ≤ i ≤ r and 1 ≤ j ≤ s, D(P, u i ) and D(P, v j ) are either infinite cyclic or acylindrically hyperbolic. Now, if we set Ξ = Γ · (Φ + Ψ), then D(P, w) = Ξ · D(P, u 1 ) × · · · × D(P, u r ) × D(P, v 1 ) × · · · × D(P, v s ) · Ξ −1 is the decomposition we are looking for.
Finally, the inequality n ≤ dim alg D(P, w) is clear since diagram groups are torsion-free, and the inequality dim alg D(P, w) ≤ dim X(P, w) is a direct consequence of [Gen15a, Corollary 5.2].
We conclude this section by showing that, in the cocompact case, being a contracting isometry can be characterized algebraically. For convenience, we introduce the following definition.
Proposition 5.50. Let P = Σ | R be a semigroup presentation and w ∈ Σ + a baseword whose class modulo P is finite. If g ∈ D(P, w) − { (w)}, the following statements are equivalent:
(i) g is a contracting isometry of X(P, w);
(ii) the centraliser of g is infinite cyclic;
(iii) g does not belong to a large canonical subgroup.
Proof. It is well-known that the centraliser of a contracting isometry of a group acting properly is virtually cyclic. Because diagram groups are torsion-free, we deduce the implication (i) ⇒ (ii).
Suppose that there exists a (w, uv)-diagram Γ such that g ∈ Γ · D(P, u) × D(P, v) · Γ −1 , where D(P, u) and D(P, v) are non trivial. Let g 1 ∈ D(P, u) and g 2 ∈ D(P, v) be two spherical diagrams such that g = Γ · (g 1 + g 2 ) · Γ −1 . In particular, the centraliser of g contains: Γ · ( g 1 × g 2 ) · Γ −1 if g 1 and g 2 are non trivial; Γ · (D(P, u) × g 2 ) · Γ −1 if g 1 trivial and g 2 is not; Γ · ( g 1 × D(P, v)) · Γ −1 if g 2 is trivial and g 1 is not. We deduce that (ii) ⇒ (iii).
Finally, suppose that g is not a contracting isometry. Up to taking a conjugate of g, we may suppose without loss of generality that g is absolutely reduced. According to Theorem 5.25 and Lemma 5.33, three cases may happen:
(a) supp(g) has cardinality at least two;
(b) w = xyz in Σ + with supp(g) = {y} and X(P, x) or X(P, z) infinite;
(c) g ∞ contains a proper infinite prefix.
In case (a), g decomposes as a sum with at least two non-trivial factors, and we deduce from Lemma 5.49 that g belongs to a large canonical subgroup. In case (b), if X(P, x) is infinite then D(P, x) is non-trivial since [x] P is finite, and we deduce that g belongs to the large canonical subgroup D(P, x)×D(P, yz); we argue similarly if X(P, z) is infinite.
In case (c), suppose that g ∞ contains a proper infinite prefix Ξ. For convenience, write g ∞ = g 1 • g 2 • · · · , where each g i is a copy of g. Suppose by contradiction that g cannot be decomposed as a sum with at least two non-trivial factors. As a consequence, for each i ≥ 1, the subdiagram g i either is contained into Ξ or it contains a cell which does not belong to Ξ but whose top path intersects Ξ. Because Ξ is a proper prefix of g ∞ , there exists some index j ≥ 1 such that g j is not included into Ξ, and it follows from Lemma 5.28 that, for every i ≥ j, g i satisfies the same property. Let Ξ j+r denote the prefix of g 1 • · · · • g j+r induced by Ξ. We know that, for every 0 ≤ s ≤ r, the subdiagram g j+s contains a cell which does not belong to Ξ but whose top path intersects Ξ. This implies that bot(Ξ j+r ) has length at least r. On the other hand, the finiteness of [w] P implies that the cardinality of [bot(Ξ j+r )] P is bounded by a constant which does not depend on r. Thus, we get a contradiction if r is chosen sufficiently large. We have proved that g decomposes as a sum with at least two non-trivial factors. As in case (a), we deduce from Lemma 5.49 that g belongs to a large canonical subgroup.
Remark 5.51. The implication (iii) ⇒ (ii) also follows from the description of centralisers in diagram groups [GS97,Theorem 15.35], since they are canonical subgroups.
A Combinatorial boundary vs. other boundaries
In this appendix, we compare the combinatorial boundary of a CAT(0) cube complex with three other boundaries. Namely, the simplicial boundary, introduced in [Hag13]; the Roller boundary, introduced in [Rol98]; and its variation studied in [Gur08]. In what follows, we will always assume that our CAT(0) cube complexes have countably many hyperplanes. For instance, this happens when they are locally finite.
A.1 Simplicial boundary
For convenience, we will say that a CAT(0) cube complex is ω-dimensional if it does not contain an infinite collection of pairwise transverse hyperplanes.
Let X be an ω-dimensional CAT(0) cube complex. A Unidirection Boundary Set (or UBS for short) is an infinite collection of hyperplanes U not containing any facing triple which is:
• inseparable, ie., each hyperplane separating two elements of U belongs to U;
• unidirectional, ie., any hyperplane of U bounds a halfspace containing only finitely many elements of U.
According to [Hag13,Theorem 3.10], any UBS is commensurable to a disjoint union of minimal UBS, where a UBS U is minimal if any UBS V satisfying V ⊂ U must be commensurable to U; the number of these minimal UBS is the dimension of the UBS we are considering. Then, to any commensurability class of a k-dimensional UBS is associated a k-simplex at infinity whose faces correspond to its subUBS. All together, these simplices at infinity define a simplicial complex, called the simplicial boundary ∂ X of X. See [Hag13] for more details.
The typical example of a UBS is the set of the hyperplans intersecting a given combinatorial ray. A simplex arising in this way is said visible. Therefore, to any point in the combinatorial boundary naturally corresponds a simplex of the simplicial boundary.
The following proposition is clear from the definitions:
Proposition A.1. Let X be an ω-dimensional CAT(0) cube complex. Then (∂ c X, ≤) is isomorphic to the face-poset of the visible part of ∂ X.
In particular, when X is fully visible, ie., when every simplex of the simplicial boundary is visible, the two boundaries essentially define the same objet. Notice however that Farley cube complexes are not necessarily fully visible. For example, if P = a, b, p | a = ap, b = pb , then X(P, ab) is a "diagonal half-plane" and is not fully visible. Moreover, although Farley cube complexes are complete, they are not necessarily ω-dimensional, so that the simplicial boundary may not be well-defined. This is the case for the cube complex associated to Thompson's group F , namely X(P, x) where P = x | x = x 2 ; or the cube complex associated to the lamplighter group Z Z, namely X(P, ab) where P = a, b, p, q, r | a = ap, b = pb, p = q, q = r, r = p .
A.2 Roller boundary
A pocset (Σ, ≤, * ) is a partially-ordered set (Σ, ≤) with an order-reversing involution * such that a and a * are not ≤-comparable for every a ∈ Σ. A subset α ⊂ P(Σ) is an ultrafilter if:
• for every a ∈ Σ, exactly one element of {a, a * } belongs to α;
• if a, b ∈ Σ satisfy a ≤ b and a ∈ α, then b ∈ α.
Naturally, to every CAT(0) cube complex X is associated a pocset: the set of half-spaces U(X) with the inclusion and the complementary operation. In this way, every vertex can be thought of as an ultrafilter, called a principal ultrafilter; see for example [Sag14] and references therein. For convenience, let X • denote the set of ultrafilters of the pocset associated to our cube complex. Notice that X • is naturally a subset of 2 U (X) , and can be endowed with the Tykhonov topology. Now, the Roller boundary RX of X is defined as the space of non principal ultrafilters of X • endowed with the Tykhonov topology. Below, we give an alternative description of RX.
Given a CAT(0) cube complex X, fix a base vertex x ∈ X, and let S x X denote the set of combinatorial rays starting from x up to the following equivalence relation: two rays r 1 , r 2 are equivalent if H(r 1 ) = H(r 2 ). Now, if ξ 1 , ξ 2 ∈ S x X, we define the distance between ξ 1 and ξ 2 in S x X by d(ξ 1 , ξ 2 ) = 2 − min{d(x,J) | J∈H(r 1 )⊕H(r 2 )} ,
where ⊕ denotes the symmetric difference. In fact, as a consequence of the inclusion A ⊕ B ⊂ (A ⊕ B) ∪ (B ⊕ C) for every sets A, B, C, the distance d turns out to be ultrametric, i.e., d(ξ 1 , ξ 3 ) ≤ max (d(ξ 1 , ξ 2 ), d(ξ 2 , ξ 3 )) for every ξ 1 , ξ 2 , ξ 3 ∈ S x X. Also, notice that this distance is well-defined since its expression does not depend on the representatives we chose.
Given a combinatorial ray r, let α(r) denote the set of half-spaces U such that r lies eventually in U .
Proposition A.2. The map r → α(r) induces a homeomorphism S x X → RX.
Proof. If r is a combinatorial ray starting from x, it is clear that α(r) is a non-principal ultrafilter, ie., α(r) ∈ RX. Now, if for every hyperplane J we denote by J + (resp. J − ) the halfspace delimited by J which does not contain x (resp. which contains x), then we have α(r) = {J + | J ∈ H(r)} {J − | J / ∈ H(r)}.
We first deduce that α(r 1 ) = α(r 2 ) for any pair of equivalent combinatorial rays r 1 , r 2 starting from x, so that we get a well-defined induced map α : S x X → RX; secondly, the injectivity of α follows, since α(r 1 ) = α(r 2 ) clearly implies H(r 1 ) = H(r 2 ). Now, let η ∈ RX be a non-principal ultrafilter, and let H + (η) denote the set of the hyperplanes J satisfying J + ∈ η. Notice that, if H + (η) is finite, then all but finitely many halfspaces of η contain x, which implies that η is principal; therefore, H + (η) is infinite, say H + (η) = {J 1 , J 2 , . . .}. On the other hand, J + i ∩ J + j is non-empty for every i, j ≥ 1, so, for every k ≥ 1, the intersection C k = k i=1 J + i is non-empty; let x k denote the combinatorial projection of x onto C k . According to Lemma 2.8, x k+1 is the combinatorial projection of x k onto C k+1 for every k ≥ 1. Therefore, by applying Proposition 2.5, we know that there exists a combinatorial ray ρ starting from x and passing through all the x k 's. Now, if J ∈ H(ρ), there exists some k ≥ 1 such that J separates x and x k , and a fortiori x and C k according to Lemma 2.7. We deduce that either J belongs to {J 1 , . . . , J k } or J separates x and some hyperplane of {J 1 , . . . , J k }; equivalently, there exists some 1 ≤ ≤ k such that J + ⊂ J + . Because J + ∈ η, we deduce that J + ∈ η. Thus, we have proved that H(ρ) = H + (η). It follows that α(ρ) = η. The surjectivity of α is proved.
Finally, it is an easy exercice of general topology to verify that α is indeed a homeomorphism. The details are left to the reader. Therefore, the combinatorial boundary ∂ c X may be thought of as a quotient of the Roller boundary. This is the point of view explained in the next section.
It is worth noticing that this description of the Roller boundary allows us to give a precise description of the Roller boundary of Farley cube complexes: Proposition A.3. Let P = Σ | R be a semigroup presentation and w ∈ Σ + a base word. The Roller boundary of X(P, w) is homeomorphic to the space of the infinite reduced w-diagrams endowed with the metric (∆ 1 , ∆ 2 ) → 2 − min{#∆ | ∆ is not a prefix of both ∆ 1 and ∆ 2 } , where #(·) denotes the number of cells of a diagram.
A.3 Guralnik boundary
Given a pocset (Σ, ≤, * ), Guralnik defines the boundary R * Σ as the set of almost-equality classes of ultrafilters, partially ordered by the relation: Σ 1 ≤ Σ 2 if Σ 2 ∩ Σ 1 = ∅, where · denotes the closure in RX with respect to the Tykhonov topology. See [Gur08] for more details.
In particular, if X is a CAT(0) cube complex, it makes sense to define the boundary R * X as the previous boundary of the associated pocset. Notice that R * X contains a particular point, corresponding to the almost-equality class of the principal ultrafilters; let p denote this point. Then R * X\{p} is naturally the quotient of the Roller boundary RX by the almost-equality relation.
Remark A.4. Although Guralnik considers only ω-dimensional pocsets in [Gur08], the definition makes sense for every pocset, and in particular is well-defined for every CAT(0) cube complex. Notice also that the boundary R * is called the Roller boundary there, whereas the terminology now usually refers to the boundary defined in the previous section.
Proposition A.5. The posets (R * X\{p}, ≤) and (∂ c X, ≤) are isomorphic.
Proof. If r is an arbitrary combinatorial ray, choose a second ray ρ ∈ S x X equivalent to it, and set ϕ(r) ∈ R * X as the almost-equality class of α(ρ), where α : S x X → R(X) is the isomorphism given by Proposition A.2. Notice that, if ρ ∈ S x X is equivalent to r, then α(ρ) and α(ρ ) are almost-equal, so we deduce that ϕ(r) does not depend on our hence r 1 ≺ r 2 in ∂ c X. We have proved that ϕ −1 is a poset-morphism as well.
Lemma A.6. Let r ∈ S x X be a combinatorial ray. There exists r 0 ∈ S x X equivalent to r such that, for any combinatorial ray ρ ∈ S x X equivalent to r, we have H(r 0 ) ⊂ H(ρ).
Proof. Let H ⊂ H(r) be the set of the hyperplanes J ∈ H(r) such that r does not stay in a neighborhood of J; say H = {J 1 , J 2 , . . .}. For every k ≥ 1, let C k = k i=1 J + i and let x k denote the combinatorial projection of x onto C k . By combining Lemma 2.8 and Proposition 2.5, we know that there exists a combinatorial ray r 0 starting from x and passing through all the x k 's. We claim that r 0 is the ray we are looking for.
Let ρ ∈ S x X be a combinatorial ray equivalent to r, and let J ∈ H(r 0 ) be a hyperplane. By construction, J separates x and some x k ; a fortiori, it follows from Lemma 2.7 that J separates x and C k . On the other hand, given some 1 ≤ ≤ k, because r does not stay in neighborhood of J , there exist infinitely many hyperplanes of H(r) which are included into J + ; therefore, since ρ is equivalent to r, we deduce that ρ eventually lies in J + . A fortiori, ρ eventually lies in C k . Consequently, because J separates x = ρ(0) and C k , we conclude that J ∈ H(ρ).
Theorem 1. 3 .
3Let X be a CAT(0) cube complex and g ∈ Isom(X) an isometry with a combinatorial axis γ. The following statements are equivalent:(i) g is a contracting isometry; (ii) joins of hyperplanes (H, V) with H ⊂ H(γ) are uniformly thin; (iii) there exists C ≥ 1 such that: -H(γ) does not contain C pairwise transverse hyperplanes, any grid of hyperplanes (H, V) with H ⊂ H(γ) is C-thin;
d(x, y) = d(x, m) + d(m, y), d(x, z) = d(x, m) + d(m, z), d(y, z) = d(y, m) + d(m, z). Notice that, for every geodesics [x, m], [y, m] and [z, m], the concatenations [x, m]∪[m, y], [x, m] ∪ [m, z] and [y, m] ∪ [m, z] are also geodesics; furthermore, if [x, y], [y, z] and [x, z] are geodesics, then any vertex of [x, y] ∩ [y, z] ∩ [x, z] is a median point of x, y, z.
Figure 1 :
1From left to right: a nogon, a monogon, an oscugon and a bigon.
Lemma 2 . 19 .
219Let X be a CAT(0) cube complex and γ an infinite combinatorial geodesic. Let N (γ) denote the combinatorial convex hull of γ. Then γ is quasiconvex if and only if the Hausdorff distance between γ and N (γ) is finite.
Proposition 3. 3 .
3Let X be a CAT(0) cube complex and γ an infinite combinatorial geodesic. The following statements are equivalent: (i) γ is quasiconvex; (ii) there exists C ≥ 1 satisfying: -H(γ) does not contain C pairwise transverse hyperplanes, any grid of hyperplanes (H, V) with H, V ⊂ H(γ) is C-thin; (iii) there exists some constant C ≥ 1 such that any join of hyperplanes (H, V) satisfying H, V ⊂ H(γ) is C-thin.
ω} and {r(k), ρ(k)} for k large enough, and each J i separates {r(k), ω} and {r(0), ρ(k)} for k large enough (since J i and ρ are disjoint); we deduce that J i and V j are transverse for any 1 ≤ i, j ≤ N . Therefore,({J 1 , . . . , J N }, {V 1 , .. . , V N }) defines a join of hyperplanes in H(r). In fact, we have proved
Fact 4. 7 .
7Let r ≺ ρ be two combinatorial rays with the same origin. If J ⊂ H(r)\H(ρ) is an infinite collection, then for every N ≥ 1 there exists a join of hyperplanes (H, V) with H ⊂ J , V ⊂ H(ρ) and #H, #V ≥ n.
Claim
Lemma 4. 19 .
19Let X be a locally finite CAT(0) cube complex. If ∂ c X has an isolated point then, for every N ≥ 2, there exists a collection of N pairwise well-separated hyperplanes of X which does not contain any facing triple.Proof. Let r ∈ ∂ c X be an isolated point. According to Lemma 4.8, there exists an infinite collection V 0 , V 1 , . . . ∈ H(r) of pairwise disjoint hyperplanes.
Claim 4. 20 .
20For every k ≥ 0, H(V k ) ∩ H(r) is finite.
Claim 4. 21 .
21For every k ≥ 0, there exists some i ≥ 1 such that V k and V k+i are (2i)-well-separated.Suppose by contradiction that V k and V k+i are not well-separated for every i ≥ 1, so that there exists a collection of hyperplanes H i ⊂ H(V k ) ∩ H(V k+i ) which does not contain any facing triple and satisfying #H i ≥ 2i. Let x 0 ∈ r be a vertex adjacent to V k . Because H i does not contain any facing triple and that H i ⊂ H(V k ), there exists a vertex x i ∈ N (V k ) such that x 0 and x i are separated by at least i hyperplanes of H i ; let K i denote this set of hyperplanes. For every i ≥ 1, let
Fact 4. 22 .
22Let i ≥ 1. The vertical hyperplanes of D i are pairwise disjoint and intersect r i . The horizontal hyperplanes of D i are pairwise disjoint. The high horizontal hyperplanes of D i intersect γ i .It follows directly from Theorem 2.12 that the vertical hyperplanes of D i are pairwise disjoint and intersect r i ; and the horizontal hyperplanes of D i are pairwise disjoint. Suppose by contradiction that a high horizontal hyperplane H does not intersect γ i . According to Theorem 2.12, necessarily H intersects r. On the other hand, because d(x 0 , H∩α i ) > #H(V k )∩H(r), there must exist a horizontal hyperplane J intersecting α i between H and x 0 which does not intersect r i ; once again, it follows from Theorem 2.12 that J has to intersect γ i . A fortiori, J and H are necessarily transverse, contradicting the fact that two horizontal hyperplanes are disjoint. This proves the fact. Now, we want to define a particular subdiagram D i ⊂ D i . Let H i be the nearest high horizontal hyperplane of D i from α i ∩ r i . According to the previous fact, H i intersects γ i . Let D i be the subdiagram delimited by H i which contains β i . Notice that the hyperplanes intersecting D i are either vertical or high horizontal. Thus, from the decomposition of H(D i ) as horizontal and vertical hyperplanes given by Fact 4.22, we deduce that D i is isometric to some square complex [0,a i ] × [0, b i ]. Notice that b i −→ i→+∞ +∞ since V k+1 , .. . , V k+i−1 separate α i and γ i , and similarly a i −→ i→+∞ +∞ since a i = length(α i ) − |H(V k ) ∩ H(r)| and #K i ≥ i. Therefore, if D ∞ ⊂ D ∞ denotes the limit of (D i ), D ∞ is naturally isomorphic to the square complex [0, +∞) × [0, +∞). Let ρ ∞ ⊂ D ∞ be the combinatorial ray associated to [0, +∞) × {0}, and µ ⊂ D ∞ be a diagonal combinatorial ray, ie., a ray passing through the vertices {(i, i) | i ≥ 0}.
defines a collection of N pairwise L-well-separated hyperplanes, with L = max(L(1), . . . , L(N − 1)), which does not contain any facing triple.
Theorem 4. 23 .
23Let G be a countable group acting on a locally finite CAT(0) cube complex X. Suppose that the action G X is minimal (ie., X does not contain a proper G-invariant combinatorially convex subcomplex) and G does not fix a point of ∂ c X. Then G contains a contracting isometry if and only if ∂ c X contains an isolated point.Given a hyperplane J, delimiting two haflspaces J − and J + , an orientation of J is the choice of an ordered couple (J − , J + ). The following terminology was introduced in [CS11]: Definition 4.24. Let G be a group acting on a CAT(0) cube complex. A hyperplane J is G-flippable if, for every orientation (J − , J + ) of J, there exists some g ∈ G such that gJ − J + .
Corollary 4. 26 .
26Let G be a countable group acting minimally on a CAT(0) cube complex X without fixing any point of ∂ c X. Then any pair of disjoint hyperplanes is skewered by an element of G.Proof. Let (J, H) be a pair of disjoint hyperplanes. Fix two orientations J = (J − , J + ) and H = (H − , H + ) so that J + H + . Now, by applying the previous lemma twice, we find two elements g, h ∈ G such that gJ − J + and hgH + gH − . Thus,
Theorem 5.2.[Far03, Theorem 3.13] X(P, w) is a CAT(0) cube complex. Moreover it is complete, i.e., there are no infinite increasing sequences of cubes in X(P, w).
Proposition 5. 8 .
8Let D be a finite consistent collection of reduced minimal (w, * )diagrams. Then there exists a unique ≤-minimal reduced (w, * )-diagram admitting any element of D as a prefix; let sup(D) denote this diagram. Furthermore, M(sup(D)) = D∈D M(D).
Fact 5. 10 .
10#M(∆) = #∆. Indeed, there exists a bijection between M(∆) and the hyperplanes separating (w) and ∆, hence #M(∆) = d( (w), ∆) = #∆. We deduce that Fact 5.11. sup(M(∆)) = ∆. Indeed, by definition any element of M(∆) is a prefix of ∆, so that sup(M(∆)) exists (because M(∆) is consistent) and sup(M(∆)) ≤ ∆ by the ≤-minimality of sup(M(∆)). On the other hand, we deduce from the previous fact that #sup(M(∆)) = |M(sup(M(∆)))| = D∈M(∆) M(D) = #M(∆) = #∆.Therefore, we necessarily have sup(M(∆)) = ∆.
Proposition 5 . 20 .
520Let D be a countable consistent collection of finite reduced minimal (w, * )-diagrams. Then there exists a unique ≤-minimal reduced (w, * )-diagram admitting any element of D as a prefix; let sup(D) denote this diagram. Furthermore, M(sup(D)) = D∈D M(D).
then sup(D 1 ) and sup(D 2 ) are commensurable. Proof. If D 1 ⊂ D 2 then sup(D 2 ) admits any element of D 1 as a prefix, hence sup(D 1 ) ≤ sup(D 2 ) by the ≤-minimality of sup(D 1 ). other hand, D 1 \D 2 is finite, because D 1 ⊂ a D 2 , and M(D) is finite for any finite diagram D, so D∈D 1 \D 2 M(D) is finite. This proves that M(sup(D 1 )) ⊂ a M(sup(D 2 )), ie., sup(D 1 ) is almost a prefix of sup(D 2 ).
Theorem 5 . 25 .
525Let g ∈ D(P, w)−{ (w)} be absolutely reduced. Then g is a contracting isometry of X(P, w) if and only if the following two conditions are satisfied: (i) g ∞ does not contain any infinite proper prefix;
Lemma 5 . 26 .
526Let g ∈ D(P, w) − { (w)} be absolutely reduced. Choose a combinatorial geodesic [ (w), g] between (w) and g, and set γ = n∈Z g n ·[ (w), g]. Then γ is a bi-infinite combinatorial geodesic on which g acts by translation. Moreover, γ(+∞) = g ∞ .
Lemma 5 . 33 .
533Let g ∈ D(P, w) − { (w)} be absolutely reduced. Suppose that we can write w = xyz where supp(g) = {y}. The following two conditions are equivalent: (i) ∂(P, x) and ∂(P, z) are empty;
•
U (m 1 , . . . , m n ) is trivial if and only if n ≤ 2; • U (m 1 , . . . , m n ) is infinite cyclic if and only if n = 3 and m 1 = m 2 = m 3 = 1; • U (m 1 , . . . , m n ) is a non-abelian free group if and only if n = 3 and m 1 = 1, or n = 4 and m 1 = m 2 = m 3 = 1, or n = 5 and m 1 = m 2 = m 3 = m 4 = m 5 = 1.
is a copy of Σ,Σ = s∈Σ Σ s ,R = s∈Σ R s and finally S = s∈Σ {s = a s w s as}.
Hag08, Theorem 2.7], and Theorem 2.3. [Hag08, Corollary 2.16] Let X be a CAT(0) cube complex and J a hyperplane. The two components of X\\J are combinatorially convex, as well as the components of ∂N (J). This result is particularly useful when it is combined with the following well-known Helly property; see for example [Rol98, Theorem 2.2]. Theorem 2.4. If C is a finite collection of pairwise intersecting combinatorially subcomplexes, then the intersection C∈C C is non-empty.
Lemma 2.9. Let X be a CAT(0) cube complex, N ⊂ X a combinatorially convex subspace and x, y / ∈ N two vertices. Fix a combinatorial geodesic [x, y] and choose a vertex z ∈ [x, y]. Then d(z, N ) ≤ d(x, N ) + d(y, N ). Proof. For convenience, if A, B ⊂ X are two sets of vertices, let ∆(A, B) denote the number of hyperplanes separating A and B. According to Lemma 2.7, ∆({a}, N ) =d(a, N )for every vertex a ∈ X. Notice that, because z ∈ [x, y] implies that no hyperplane can separate z and {x, y} ∪ N , we have
combinatorially convex, two dual curves intersecting P i are disjoint;(ii) if C i and C i+1 are combinatorially convex, no dual curve intersects both P i and P i+1 .In general, a disc diagram D → X is not an embedding. However, the proposition
below, which we proved in [Gen16, Proposition 2.15], determines precisely when it is an
isometric embedding.
Proposition 2.13. Let X be a CAT(0) cube complex and D → X a disc diagram which
does not contain any bigon. With respect to the combinatorial metrics, ϕ : D → X is
an isometric embedding if and only if every hyperplane of X induces at most one dual
curve of D.
Contracting isometries. We gave in the introduction the definition of a contracting isometry. It turns out that an isometry is contracting if and only if its axis is contracting, in the following sense:Definition 2.16. Let (S, d) be a metric space, Y ⊂ C a subspace and B ≥ 0 a constant.We say that Y is B-contracting if, for every ball disjoint from Y , the diameter of its nearest-point projection onto Y is at most B. A subspace will be contracting if it is B-contracting for some B ≥ 0.In fact, in the context of CAT(0) cube complexes, we may slightly modify this definition in the following way:Proof. The implication is clear: if S is B-contracting, {x, y} is included into the ball B(x, d(x, S)) whose projection onto S has diameter at most B.Corollary 2.15. [Gen16, Corollary 2.17] Let X be a CAT(0) cube complex and C a
cycle of four combinatorially convex subcomplexes. If D → X is a disc diagram of
minimal complexity bounded by C, then D is combinatorially isometric to a rectangle
[0, a] × [0, b] ⊂ R 2 and D → X is an isometric embedding.
Lemma 2.17. Let X be a CAT(0) cube complex, S ⊂ X a combinatorially convex
subspace and L ≥ 0 a constant. Then S is contracting if and only if there exists C ≥ 0
such that, for all vertices x, y ∈ X satisfying d(x, y) < d(x, S) − L, the projection of
{x, y} onto S has diameter at most C.
Theorem 3.9. Let γ be an infinite combinatorial geodesic. Then γ is contracting if and only if there exist constants r, L ≥ 1, hyperplanes {H i , i ∈ Z}, and vertices {x i ∈ γ ∩ N (H i ), i ∈ Z} such that, for every i ∈ Z:
some H i might be empty or infinite; moreover, H i could be infinite and H i+1 non-empty. We want to understand the behaviour of this sequence of collections of hyperplanes. Our first two claims state that each H i is a collection of pairwise transverse hyperplanes and that H i non-empty implies H j non-empty for every j < i. Claim 4.9. For all i ≥ 0 and J 1 , J 2 ∈ H i , J 1 and J 2 are transverse.
4.11. For every i ≥ 0, if H i−1 is finite then any hyperplane of H i is adjacent to x i ; and, if H i is finite, there exists a cube Q i containing x i and x i+1 as opposite vertices such that H i is the set of hyperplanes intersecting Q i .
Case 1: H does not separate x 0 and x i . So H separates x 0 and J, and H does not belong to H k for k ≤ i − 1. Noticing that r(k) ∈ J + for sufficiently large k ≥ 0, we deduce that H necessarily intersects r, hence H ∈ H j for some j ≥ i. Therefore, there exist i pairwise disjoint hyperplanes V 1 , . . . , V i separating r(0) and H. A fortiori, V 1 , .. . , V i , H define i + 1 pairwise disjoint hyperplanes separating r(0) and J, contradicting J ∈ H i .Case 2: H separates x 0 and x i . In particular, H intersects a cube Q j for some 0
H does not separate p 0 and p i : these two vertices belong to the same half-space delimited by H, say H + . Because H separates x i and J, it separates x i and p i , so x i belongs to the second half-space delimited by H, say H − . Then, since H separates x 0 and x i by hypothesis, we deduce that x 0 belongs to H + ; in particular, H does not separate x 0 and p 0 . On the other hand, if k ≥ 0 is sufficiently large so that r(k) ∈ J + , H must separate r(0) and r(k) since H ∈ H(r). According to Proposition 2.5, there exists a combinatorial geodesic between x 0 and r(k) passing through p 0 . Because H is disjoint from J + , we conclude that H must separate x 0 and p 0 , a contradiction.
In fact, the product G • H can be described as an HNN extension: if H ∞ denote the free product of infinitely many copies of H, then G • H is isomorphic to the HNN extension of G × H ∞ over H ∞ with respect to two monomorphismsH ∞ → G × H ∞ associated to H i → H i and H i → H i+1 . Now, if tdenotes the stable letter of the HNN extension and if h ∈ H ∞ is a non-trivial element of the first copy of H in H ∞ , then it follows from Britton's lemma that H ∞ ∩ (H ∞ ) t −1 ht = {1}. Therefore, H ∞ is a weakly malnormal subgroup, and it follows from [MO13, Corollary 2.3] that, if G and H are both non-trivial, the product G • H is acylindrically hyperbolic. It is interesting that: Fact 5.42. If G and H are torsion-free, then any proper malnormal subgroup of G • H is a free group.
choice of ρ. As another consequence, ϕ must be constant on the equivalence classes of rays, so that we get a well-defined map ϕ : ∂ c X → R * X\{p}.Clearly, ϕ is surjective. Now, if r 1 , r 2 ∈ ∂ c X satisfy ϕ(r 1 ) = ϕ(r 2 ), then α(r 1 ) = a α(r 2 ); using the notation of the proof of Proposition A.2, this implies that H(r 1 ) = H + (α(r 1 )) = a H + (α(r 2 )) = H(r 2 ), hence r 1 = r 2 in ∂ c X. Thus, ϕ is injective.Let r 1 , r 2 ∈ ∂ c X be two combinatorial rays satisfying r 1 ≺ r 2 , ie., H(r 1 ) ⊂ a H(r 2 ). According to Lemma 4.2 and Lemma 4.3, we may suppose without loss of generality that r 1 (0) = r 2 (0) = x and H(r 1 ) ⊂ H(r 2 ). Let H(r 2 ) = {J 1 , J 2 , . . .}. For everyFinally, for every i ≥ 1, let y i denote the combinatorial projection of x k onto K i . By combining Lemma 2.8 and Proposition 2.5, we know that there exists a combinatorial ray starting from x k and passing through all the y i 's; let ρ k denote the concatenation of a combinatorial geodesic between x and x k with the previous ray.If ρ k is not a combinatorial ray, then there exists a hyperplane J separating x and x k , and x k and some y j . On the other hand, it follows from Lemma 2.7 that J must separate x and C k , so that J cannot separate x k and y j since x k , y j ∈ C k . Therefore, ρ k is a combinatorial ray.If H k denotes the set of the hyperplanes separating x and x k , thenIndeed, if J is a hyperplane intersecting ρ k which does not separate x and x k , then J must separate x k and some y j ; a fortiori, J must be disjoint from K j according to Lemma 2.7. Thus, if z is any vertex of r 1 ∩ K j , we know that J separates x k and y j , but cannot separate x and x k , or y j and z; therefore, necessarily J separates x and z, hence J ∈ H(r 1 ). Conversely, let J be a hyperplane of H(r 1 ). In particular, J ∈ H(r 2 ), so ρ k eventually lies in J + . Because ρ k (0) = x / ∈ J + , we conclude that J ∈ H(ρ k ).In particular, because H k is finite, we deduce that H(ρ k ) = a H(r 1 ), ie., r 1 = ρ k in ∂ c X. On the other hand, it is clear that, for any finite subset H ⊂ H(r 2 ), there exists some k ≥ 1 sufficiently large so that H ⊂ H k . Therefore, the sequence of ultrafilters (α(ρ k )) converges to α(r 2 ) in the Roller boundary.Consequently, α(r 2 ) belongs to the closure of the almost-equality class of α(r 1 ). This precisely means that ϕ(r 1 ) ≤ ϕ(r 2 ) in R * X. We have proved that ϕ is a poset-morphism. Now, let r 1 , r 2 ∈ ∂ c X such that ϕ(r 1 ) ≤ ϕ(r 2 ) in R * X. This means that there exists a combinatorial ray ρ such that α(ρ) belongs to the intersection between the almostequality class of α(r 2 ) and the closure of the almost-equality class of α(r 1 ) in the Roller boundary. So there exists a sequence of combinatorial rays (ρ k ) such that ρ k = r 1 in ∂ c X for every k ≥ 1 and α(ρ k ) → α(ρ). Let r 0 be the minimal element of the class of r 1 in S x X which is given by Lemma A.6 below. Then H(r 0 ) ⊂ H(ρ k ) for every k ≥ 1, hence H(r 0 ) ⊂ H(ρ). Therefore, H(r 1 ) ⊂ a H(r 0 ) ⊂ H(ρ) = a H(r 2 ),
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| []
|
[
"Prepared for submission to JCAP On the importance of nonlinear couplings in large-scale neutrino streams",
"Prepared for submission to JCAP On the importance of nonlinear couplings in large-scale neutrino streams"
]
| [
"Hélène Dupuy [email protected] \nInstitut d'Astrophysique de Paris\nUMR-7095\nCNRS\nUniversité Pierre et Marie Curie\n98 bis bd Arago75014ParisFrance\n\nInstitut de Physique Théorique\nCEA\nF-91191Gif-sur-YvetteIPhT\n\nCNRS\n2306, F-91191Gif-sur-YvetteURAFrance\n",
"Francis Bernardeau [email protected] \nInstitut d'Astrophysique de Paris\nUMR-7095\nCNRS\nUniversité Pierre et Marie Curie\n98 bis bd Arago75014ParisFrance\n\nInstitut de Physique Théorique\nCEA\nF-91191Gif-sur-YvetteIPhT\n\nCNRS\n2306, F-91191Gif-sur-YvetteURAFrance\n"
]
| [
"Institut d'Astrophysique de Paris\nUMR-7095\nCNRS\nUniversité Pierre et Marie Curie\n98 bis bd Arago75014ParisFrance",
"Institut de Physique Théorique\nCEA\nF-91191Gif-sur-YvetteIPhT",
"CNRS\n2306, F-91191Gif-sur-YvetteURAFrance",
"Institut d'Astrophysique de Paris\nUMR-7095\nCNRS\nUniversité Pierre et Marie Curie\n98 bis bd Arago75014ParisFrance",
"Institut de Physique Théorique\nCEA\nF-91191Gif-sur-YvetteIPhT",
"CNRS\n2306, F-91191Gif-sur-YvetteURAFrance"
]
| []
| We propose a procedure to evaluate the impact of nonlinear couplings on the evolution of massive neutrino streams in the context of large-scale structure growth. Such streams can be described by general nonlinear conservation equations, derived from a multipleflow perspective, which generalize the conservation equations of non-relativistic pressureless fluids. The relevance of the nonlinear couplings is quantified with the help of the eikonal approximation applied to the subhorizon limit of this system. It highlights the role played by the relative displacements of different cosmic streams and it specifies, for each flow, the spatial scales at which the growth of structure is affected by nonlinear couplings. We found that, at redshift zero, such couplings can be significant for wavenumbers as small as k = 0.2 h/Mpc for most of the neutrino streams. | 10.1088/1475-7516/2015/08/053 | [
"https://arxiv.org/pdf/1503.05707v2.pdf"
]
| 118,022,593 | 1503.05707 | a154b5865cba408d55521a88d11b43fe97c2a24a |
Prepared for submission to JCAP On the importance of nonlinear couplings in large-scale neutrino streams
19 Mar 2015
Hélène Dupuy [email protected]
Institut d'Astrophysique de Paris
UMR-7095
CNRS
Université Pierre et Marie Curie
98 bis bd Arago75014ParisFrance
Institut de Physique Théorique
CEA
F-91191Gif-sur-YvetteIPhT
CNRS
2306, F-91191Gif-sur-YvetteURAFrance
Francis Bernardeau [email protected]
Institut d'Astrophysique de Paris
UMR-7095
CNRS
Université Pierre et Marie Curie
98 bis bd Arago75014ParisFrance
Institut de Physique Théorique
CEA
F-91191Gif-sur-YvetteIPhT
CNRS
2306, F-91191Gif-sur-YvetteURAFrance
Prepared for submission to JCAP On the importance of nonlinear couplings in large-scale neutrino streams
19 Mar 2015
We propose a procedure to evaluate the impact of nonlinear couplings on the evolution of massive neutrino streams in the context of large-scale structure growth. Such streams can be described by general nonlinear conservation equations, derived from a multipleflow perspective, which generalize the conservation equations of non-relativistic pressureless fluids. The relevance of the nonlinear couplings is quantified with the help of the eikonal approximation applied to the subhorizon limit of this system. It highlights the role played by the relative displacements of different cosmic streams and it specifies, for each flow, the spatial scales at which the growth of structure is affected by nonlinear couplings. We found that, at redshift zero, such couplings can be significant for wavenumbers as small as k = 0.2 h/Mpc for most of the neutrino streams.
Introduction
Statistical properties of the large-scale structure of the universe have long been proposed as an efficient instrument to constrain cosmological parameters. In this context, a careful account of the role played by massive neutrinos is crucial. So far, it has often been overlooked in the nonlinear or quasilinear regime because of the technical complexity specific to the study of massive neutrinos (see recent attempts in [1][2][3]). This is all the more unfortunate that cosmological observations can fruitfully improve our knowledge of those particles. The signature of neutrino masses on cosmological observables is indeed expected to be significant enough for those masses to be constrained observationally [4][5][6][7][8][9].
In the linear regime, the effect of neutrinos is now well understood (see refs. [10][11][12]). The need for nonlinear corrections in their equations of motion has been raised because the cosmological observations that are the most sensitive to neutrinos masses, i.e. for wavenumbers in the 0.1-0.2 h/Mpc range, precisely correspond to the mildly nonlinear regime. To deal with this issue, several strategies can be adopted. Until recently, in analytical works, the neutrino fluids had always been treated in the linear regime, nonlinear couplings being introduced in the dark matter description only. This nonlinear treatment can be implemented with the help of the Renormalization Group time-flow approach [1,13]. Improvements upon such schemes have been proposed in [3], which consists in a hybrid approach that matches the full Boltzmann hierarchy to an effective two-fluid description at an intermediate redshift. [14] is, for its part, a systematic perturbative expansion of the Vlasov equation in which highorder corrections to the neutrino density contrast are computed without the explicit need to track the perturbed neutrino momentum distribution.
Ideally, however, the fully nonlinear evolution of the neutrino fluid should be depicted. A natural way to do so would be to take inspiration of the standard linear description, which relies on the Boltzmann equation, and extend the harmonic decomposition of the phase-space distribution function to the nonlinear regime. This has been done in [15] but this method turned out to be particularly difficult to handle. In [16,17], we proposed to describe massive neutrinos as a superposition of single-flow fluids, the equations of motion of each of them being written in the nonlinear regime.
In this paper, we are interested in exploiting those theoretical developments in order to identify the scales at which nonlinear couplings in the neutrino fluids are expected to play a significant role. In order to do so, we apply the eikonal approximation to the nonlinear equations of motion. Note that this approximation had already been exploited in the literature to develop a Perturbation Theory approach for cold dark matter ( [18]) and had proved to be able to capture the leading coupling effects.
The article is structured as follows. In section 2, we recall the form of the nonlinear equations of motion describing the time evolution of non-interacting fluids, relativistic or not, when using a multi-fluid approach. Section 3 explains in detail how the eikonal approximation can be implemented in those equations and emphasizes the key role of relative displacement fields. Finally, in section 4, power spectra of the relative displacements between neutrino fluids and the cold dark matter component are presented. The impact on the growth of large-scale structure is then discussed in a quantitative way.
Nonlinear equations of motion (multifluid description)
Following [16,17], it is now clear that any non-interacting relativistic fluid can be divided into several flows, each of them evolving then independently until first shell-crossings. In cosmology, this approach obviously applies to massive neutrinos since they are free-streaming. In this framework, each flow can be defined as the collection of all the particles (with a mass m) having the same comoving momentum. They are entirely characterized by two coupled fields, namely the comoving number density n c and momentum P µ . Those fields obey the following general equations,
∂ ∂η n c + ∂ ∂x i P i P 0 n c = 0, (2.1)
with P µ (η, x i ) = g µν P ν (η, x i ) and P µ P µ = −m 2 , g µν being the metric, and P ν P µ,ν = 1 2 P σ P ν g σν,µ .
(2.2)
These equations directly ensue from the matter and momentum conservation equations, applied to each flow. At this stage, no perturbative expansion of the metric is involved. The properties of the whole fluid are then inferred by examining an appropriate number of such flows, each of them being labeled by the initial value of its field P i , denoted τ i (found to be constant at zeroth order in Perturbation Theory). The initial number density of particles in each flow is constrained by the choice of initial conditions. For instance, the case of adiabatic initial conditions is described in details in [16]. We are interested here in equations i) involving linearized metric perturbations (but non-linearized fields) and ii) rid of the coupling terms that are subdominant at subhorizon scales. They can be written in terms of the number density contrast
δ τ i (η, x i ) = n c (η, x i ; τ i ) n (0) c (τ i ) − 1 (2.3)
and of the velocity divergence field (in units of −H, H being the conformal Hubble constant)
θ τ i (η, x i ) = − ∂ i P i (η, x i ; τ i ) maH (2.4)
since the field P i has found to be potential in this regime 1 . For a generic perturbed Friedmann-Lemaître metric 2 , whose time variable is the conformal time η,
ds 2 = a 2 (η) −(1 + 2A)dη 2 + 2B i dx i dη + (δ ij + h ij )dx i dx j , (2.5)
one has in Fourier space for the mode characterized by the wave vector k ( [17])
a∂ a − i µkτ Hτ 0 δ τ i (k) + ma τ 0 1 − µ 2 τ 2 τ 2 0 θ τ i (k) = − ma τ 0 d 3 k 1 d 3 k 2 α R (k 1 , k 2 ; τ i )δ τ i (k 1 )θ τ i (k 2 ), (2.6) 1 + a ∂ a H H + a∂ a − i µkτ Hτ 0 θ τ i (k) − k 2 maH 2 S τ i (k) = − ma τ 0 d 3 k 1 d 3 k 2 β R (k 1 , k 2 ; τ i )θ τ i (k 1 )θ τ i (k 2 ). (2.7)
We have introduced here
µ = k i τ i kτ , τ 2 = τ i τ i and τ 0 = − m 2 a 2 + τ 2 i . (2.8) Besides, S τ i (k) is a source term given by S τ i (k) = τ 0 A(k) + τ · B(k) − 1 2 τ i τ j τ 0 h ij (k). (2.9)
These equations contain also the generalized kernel functions, adapted to relativistic flows,
α R (k 1 , k 2 ; τ ) = δ Dirac (k − k 1 − k 2 ) (k 1 + k 2 ) k 2 2 · k 2 − τ k 2 · τ τ 2 0 , (2.10) β R (k 1 , k 2 ; τ ) = δ Dirac (k − k 1 − k 2 ) (k 1 + k 2 ) 2 2k 2 1 k 2 2 k 1 · k 2 − k 1 · τ k 2 · τ τ 2 0 . (2.11)
As mentioned in [17], they are extensions of the kernel functions found for pressureless fluids of non-relativistic species (see [19] for details in this context.). In any practical implementation, it is necessary to consider a collection of N streams. In that case, the general equation of motion can conveniently be written in terms of the time-dependent 2N -uplet,
Ψ a (k) = (δ τ 1 (k), θ τ 1 (k), . . . , δ τ N (k), θ τ N (k)) T . (2.12)
Before shell-crossing, it can incorporate all the relevant species (neutrinos, dark matter, baryons) as long as they interact only via gravitation. Then the equations (2.6) and (2.7) of all the flows can formally be recast in the form 3
∂ z Ψ a (k) + Ω b a Ψ b (k) = γ bc a (k 1 , k 2 )Ψ b (k 1 )Ψ c (k 2 ),(2.13)
where ∂/∂z ≡ a∂/∂a and the indices a and b run from 1 to 2N . The matrix elements Ω b a gather the linear couplings. They contain in particular the way in which the source terms S τ i (k) can be re-expressed as a function of the 2N -uplet elements. The left hand side of this equation is nothing but the linear field evolution. The right hand side contains the coupling terms. More precisely, the symmetrized vertex matrix γ bc a (k 1 , k 2 ) describes the nonlinear interactions between the Fourier modes. It is given by
γ 2p−1 2p 2p−1 (k 1 , k 2 ) = − ma 2(τ p ) 0 α R (k 1 , k 2 , τ p ), (2.14) γ 2p 2p 2p (k 1 , k 2 ) = − ma (τ p ) 0 β R (k 1 , k 2 , τ p ), (2.15)
with γ bc a (k 1 , k 2 ) = γ cb a (k 2 , k 1 ) and γ bc a = 0 otherwise.
The eikonal approximation
The eikonal approximation 4 , developed in [18], is based on the observation that the amplitudes of the kernel functions describing mode couplings, α R and β R , significantly depend on the ratio between the wave numbers at play. This has been observed for non-relativistic fluids and we show here that it is the case for relativistic fluids too. This property leads to the idea that the right hand side of eq. (2.13) can be splitted into two integration domains. One is called the hard domain and encompasses modes whose wavelengths are of the same order (and for which the coupling functions are always finite). The other one is referred to as the soft domain. It is made of modes of very different wavelengths for which the coupling functions, of the order of the wavelength ratio, are large. The main idea is that there are regimes in which the dominant coupling structure is in the soft domain. This is the case for instance in baryoncold dark matter fluids at high redshift and large wave numbers. This leads to a damping of the matter power spectrum due to the relative motion between cold dark matter and baryons, [21,22]. The same formalism can be used to obtain the large k behavior of the propagators in case of a single pressureless fluid, reproducing the results obtained in [23,24]. Here we propose to make the very same use of the eikonal approximation to infer the amplitude of mode coupling effects in a similar regime. For convenience, let us assume that the soft domain is obtained for k 2 k 1 in eq. (2.13). We have then k = k 1 and the contribution corresponding to the soft domain can be viewed as a mere corrective term in the linear equation describing the evolution of the mode k. In other words, eq. (2.13) can be rewritten
∂ ∂z Ψ a (z, k) + Ω b a (z, k)Ψ b (z, k) − Ξ b a (z, k)Ψ b (z, k) = γ bc a (k 1 , k 2 ) Ψ b (k 1 , z) Ψ c (k 2 , z) H , (3.1) with Ξ b a (k, z) ≡ 2 S d 3 q eik. γ bc a (k, q)Ψ c (q, z) . (3.2)
The soft momenta q (i.e. q k) at play in eq. (3.2) are integrated over so that Ξ b a (z, k) is independent on Ψ a (z, k). It is a time-dependent coefficient that depends on the large wave number k and that can be treated as a mode coupling matrix element at linear level. The fact that the integration domain is restricted to the soft wave numbers in eq. (3.2) is the key element. Conversely, in the right-hand side of eq. (3.1), the implicit convolution product excludes the soft domain (i.e. all the modes concerned have comparable wavelengths). When the contribution of the hard domain is negligible, eq. (3.1) can then be viewed as the equation of motion of the mode k evolving in a medium perturbed by large-scale modes. It therefore encodes the way in which long-wave modes alter the growth of structure. Once Ξ b a (z, k) is given, eq. (3.1) can be solved at linear level. This is precisely the eikonal approximation of the global equation of motion.
In practice, applying the eikonal approximation to Ξ b a (k, z) means that the vertex values that appear in this quantity have to be computed assuming k 2 k 1 . In this framework, one deduces from eqs. (2.10), (2.11), (2.14) and (2.15) that the eikonal limit of the vertex elements is
eik. γ bc 2p (k, k 2 ) = −δ b 2p δ c 2p ma 2k 2 2 (τ p ) 0 k · k 2 − k 2 · τ p (τ p ) 2 0 τ p , (3.3) eik. γ bc 2p−1 (k, k 2 ) = −δ b 2p−1 δ c 2p ma 2k 2 2 (τ p ) 0 k · k 2 − k 2 · τ p (τ p ) 2 0 τ p . (3.4)
This expression depends on each flow through its initial momentum τ p , which vanishes in the standard non-relativistic equations. Besides, (τ p ) 0 → −ma in the non-relativistic limit so that one recovers the expected formulae in this limit (see [21,25]). In the following, we exploit the consequences of this approximation in order to evaluate the relevance of the coupling terms. As a first step, we can notice from eqs. (3.2), (3.3) and (3.4) that the two non-zero elements coming from the flow labeled by τ p in the Ξ matrix, Ξ 2p 2p and Ξ 2p−1 2p−1 , are proportional to the velocity divergence of large-scale modes. Thus we can write
z z 0 Ξ b a (z , k)dz = i k.d p (z 0 , z) δ b a ,(3.5)
where a and b are either 2p or 2p − 1 and where d p is the total displacement field induced by the large-scale modes in the fluid labeled by τ p . It reads necessarily
d p (z, z 0 ) = i z z 0 dz d 3 q ma q 2 (τ p ) 0 q − q · τ p (τ p ) 2 0 τ p Ψ 2p (z , q). (3.6)
Note that this displacement is superimposed on the zeroth order displacement field induced by the homogeneous momentum of the flow. This background displacement field is given by (see eqs. (2.6) and (2.7))
d (0) p = − z z 0 τ p H(τ p ) 0 . (3.7)
Consequently, the impact of the eikonal correction introduced in the equation of motion depends on the way in which the various large-scale modes contribute to the displacement fields d p . For non-relativistic species in the linear regime, it is known that the most growing modes correspond to adiabatic modes, i.e. to particular modes inducing the same displacement for all species. We expect it to be the case also for neutrinos since they become nonrelativistic at late time. This property is useful because, when all the displacement fields are identical, the solution of the eikonal equation of motion is easy to find. It is related to the extended Galilean invariance, well established for non-relativistic species and uncovered for the system (2.6)-(2.7) in [17].
So let us assume as a first approximation that
d p (z, z 0 ) = d adiab (z, z 0 ). (3.8)
In that case, it is possible to explicitly solve the linear equation of motion in the eikonal limit. Indeed, when adiabaticity is satisfied, the appropriate Green functions ξ b a are obtained by introducing a mere phase shift in the Green function of the naked theory, denoted g b a (see appendix A for details):
ξ b a (z, z 0 ; k) = g b a (z, z 0 ; k) exp(ik.d adiab (z, z 0 )). (3.9)
Interestingly, equal time correlators such as spectra or poly-spectra are not sensitive to such a phase shift. This is a direct consequence of the extended Galilean invariance of the equations of motion (see [22]). Yet, the displacement fields d p are in general expected to be sourced by a superposition of modes, including some that induce large motions between species. In particular, when considering an early-time baryon-cold dark matter mixture at small scales, those relative displacements are the leading contribution to the nonlinear evolution of the power spectrum. More precisely, it has been shown in [21] that the presence of non-adiabatic large-scale modes induces a damping of the small-scale fluctuations, due to large relative displacements between those two species. To sketch the impact of massive neutrinos on structure formation, we evaluate in the next section the amplitude of the relative displacements involving them and we compute the corresponding power spectra.
Relative displacements and power spectra: quantitative results
We compute here on the one hand the power spectrum of the total displacement field of cold dark matter and, on the other hand, the power spectra of the relative displacement fields of relativistic flows (with respect to the dark component). To do so, we choose the conformal Newtonian gauge. Introducing the two potentials φ and ψ, the metric reads
ds 2 = a 2 (η) − (1 + 2ψ) dη 2 + (1 − 2φ) dx i dx j δ ij . (4.1)
This choice will allow us to take advantage of the numerical work presented in [16]. First, let us define two transfer functions D cdm and D
(α) p as z z 0 dz ma −(τ p ) 0 τ p (τ p ) 0 α θ τp (z , q) = D (α) p (z, z 0 , q)ψ init (q), (4.2) z z 0 dz θ cdm (z , q) = D cdm (z, z 0 , q)ψ init (q),(4.3)
where ψ init is the initial value of the potential ψ. The statistical properties of those quantities are entirely encoded in their initial power spectra P ψ (q), defined so that
ψ init (q)ψ init (q ) = (2π) 3 δ Dirac (q + q ) P ψ (q). (4.4)
Using the expressions of the transfer functions, the contribution of each mode to the displacement field of each relativistic flow reads
d p (z, z 0 ; q) = −i q q 2 D (0) p (z, z 0 , q) − τ p .q q 2 (τ p ) 2 τ p D (2) p (z, z 0 , q) ψ init . (4.5)
Besides, for the cold dark matter component, one simply has
d cdm (z, z 0 ; q) = −i q q 2 D cdm (z, z 0 , q)ψ init (q). (4.6)
Furthermore, one can notice that the displacement field of a relativistic flow along an arbitrary direction k depends on the angles between both q and τ and k and τ . After integration over the other angles, one finds for the variances of respectively k.d cdm and k.(d p − d cdm ),
(k.d cdm ) 2 = 4πk 2 dq P ψ (q) 1 3 |D cdm (q)| 2 (4.7)
and
(k.(d p − d cdm )) 2 = 2πk 2 × (4.8) dq P ψ (q) 1 −1 dµ 1 2 (1 − µ 2 k )(1 − µ 2 )|D (0) p (q) − D cdm (q) − D (2) p (q)| 2 + µ 2 µ 2 k |D (2) p (q)| 2 ,
where µ k is the cosine of the angle between the initial momentum of the flow τ p and k and where an integration is made over µ, cosine of the angle between τ p and q. In eq. (4.9), one can notice that the dependence of the r.m.s. with respect to µ k is such that it does not vanish neither for an initial momentum orthogonal to k (i.e. when µ k = 0) nor for an initial momentum along k (i.e. when µ k = 1). On Figure 1, we present the per mode contribution to the right hand side of eq. (4.9) in the particular case of neutrino fluids for µ k = 0 on the left panel and µ k = 1 on the right panel. The results have been computed using realistic cosmological parameters and assuming a single species of neutrinos whose mass is m ν = 0.3 eV. Besides, the values of the initial power spectra have been obtained under the assumption that, in the cosmological model we adopt, metric fluctuations are initially adiabatic and characterized by the scalar spectral index n s ≈ 0.96. We can see that the resulting neutrino power spectra are comparable in amplitude to the cold dark matter one (represented by a thick dashed line). Denoting σ d cdm and σ dτ the square roots of the right hand sides of eqs. (4.7) and (4.9) and using a model in which σ d cdm ≈ 6h −1 Mpc, we find that σ dτ is of the order of 2 to about 5h −1 Mpc. This result depends of course on the flow considered and is actually of the same order as the cold dark matter value. What does it mean? Similarly to what happens in mixtures of baryons and cold dark matter, one expects the perturbation growth to be damped for wave numbers larger than or comparable to 1/σ dτ . This damping is potentially larger than the damping due to the homogeneous displacements d (0) p of each flow. Indeed, the latter can be the most efficient at early time but, unlike what we found at higher order, it is null for flows evolving perpendicularly to k. A precise determination of the amplitude of these effects would require a full analysis of the nonlinear evolution of the system. We leave this for a future study.
Conclusion
Describing neutrinos as a collection of single-stream fluids is an efficient strategy to infer their impact on the growth of the cosmic structure. Using this approach, one indeed gets a complete set of equations of motion that incorporate all the nonlinear effects of relativistic or non-relativistic particles. In the subhorizon limit, the system takes the form eq. (2.13), which can be easily handled by a formalism originally developed to depict non-relativistic species.
In this paper, we evaluated the amplitudes of the nonlinear couplings and determined for each flow the scales at which they are expected to impact significantly on the structure growth. For that purpose, we implemented the eikonal approximation into the general equation of motion. We concluded that the impact of large-scale modes on an arbitrary mode are entirely driven by large-scale displacement fields whose expressions are given in eq. (3.6). The comparison between the displacement field associated with each flow of neutrinos and the one associated with cold dark matter makes easy the comparison between the power spectra of the relative displacements between neutrinos and cold dark matter and the power spectrum of the displacement of cold dark matter alone. We found as a preliminary result that couplings involving massive neutrinos (with a 0.3 eV mass) are expected to induce a damping of the perturbation growth in neutrino flows for wave numbers larger than (or of the order of) about 0.2 to 0.5 h/Mpc. A detailed quantitative analysis of the consequences of this phenomenon is yet to be done but those findings confirm the significance of nonlinear couplings in the dynamical evolution of neutrino fluids. This sets the stage for further numerical studies beyond the linear regime.
A Integral form of the overall equation of motion
For a finite number of flows, the overall equation of motion (2.13) can formally be written in an integral form. It requires the use of the associated Green operator, g b a (z, z 0 ; k), of the linear system. This operator satisfies Ψ a (z, k) = g b a (z, z 0 ; k)Ψ b (z 0 , k), (A.1)
with z and z 0 two arbitrary times. It is besides solution of the differential equation where the variables c b (α) (z 0 , k) are set so that (A.3) is satisfied. Studying in detail the Green operator of such a system is beyond the scope of this appendix. Suffice to note here that, unlike the case of a single pressureless flow, the Green operator generally depends on the wave mode k. This dependence is expected to gradually decay over time and to disappear at very late time, when all the flows have become nonrelativistic. At this stage, the situation is then identical to the one of a collection of cold dark matter fluids.
As for the standard system of non-relativistic particles, the knowledge of the Green operator of the equation of motion allows to write a formal solution (see [26][27][28]), which is given by Ψ a (k, z) = g b a (k, z, z 0 ) Ψ b (k, z 0 ) + + z z 0 dz g b a (k, z, z ) γ cd b (k 1 , k 2 )Ψ c (k 1 , z )Ψ d (k 2 , z ), (A.5)
with Ψ a (k, z 0 ) the initial conditions. Many of the approaches developed in order to improve upon standard Perturbation Theory rely on an accurate description of the Green functions beyond the linear regime. This is the purpose for instance of RPT and RegPT methods ( [28][29][30]).
Figure 1 .
1Power spectrum of the relative displacement as a function of the mode q for different neutrino flows. The values of τ range from 2.25 k B T 0 (bottom line) to 18 k B T 0 (top line). On the left panel µ k is set to 0, on the right panel µ k is set to 1 and the neutrino mass is set to 0.3 eV. The gray dashed line represents the power spectrum of the cold dark matter displacement.
the identity matrix. Formally, the Green function is the ensemble of all the independent linear solutions of the system 5 . Denoting u
This property, rigorously demonstrated in[17], generalizes that of non-relativistic species.2 Units are chosen so that the speed of light in vacuum is equal to unity.3 The Einstein notation for the summation over repeated indices is adopted and, in the right hand side of this equation, it is assumed that the wave modes are integrated over.
In this context, the term refers to diagram resummations performed in quantum electrodynamic field equations,[20].
A priori the number of solutions is equal to twice the number of flows considered.
Acknowledgements: This work is partially supported by the grant ANR-12-BS05-0002 of the French Agence Nationale de la Recherche.
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| []
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[
"ON GENUS-CHANGE IN ALGEBRAIC CURVES OVER NONPERFECT FIELDS",
"ON GENUS-CHANGE IN ALGEBRAIC CURVES OVER NONPERFECT FIELDS"
]
| [
"Stefan Schröer "
]
| []
| []
| I give a new proof, in scheme-theoretic language, of Tate's old result on genus-change over nonperfect fields in characteristic p > 0. Namely, for normal geometrically integral curves, the difference between arithmetic and geometric genus over the algebraic closure is divisible by (p − 1)/2. | 10.1090/s0002-9939-08-09712-8 | [
"https://arxiv.org/pdf/math/0703122v1.pdf"
]
| 11,291,895 | math/0703122 | e52075109f1f69e41bb81ed0963e76f7ffd409fd |
ON GENUS-CHANGE IN ALGEBRAIC CURVES OVER NONPERFECT FIELDS
5 Mar 2007 5 March 2007
Stefan Schröer
ON GENUS-CHANGE IN ALGEBRAIC CURVES OVER NONPERFECT FIELDS
5 Mar 2007 5 March 2007arXiv:math/0703122v1 [math.AG]
I give a new proof, in scheme-theoretic language, of Tate's old result on genus-change over nonperfect fields in characteristic p > 0. Namely, for normal geometrically integral curves, the difference between arithmetic and geometric genus over the algebraic closure is divisible by (p − 1)/2.
Introduction
A distinctive feature of geometry in characteristic p > 0 is that a regular scheme X of finite type over a nonperfect field K may cease to be regular after purely inseparable base change. This striking behavior easily appears for the generic fiber of morphisms f : S → B between smooth schemes over algebraically closed ground fields: Here K = κ(B) is the function field of B, and X = S K is the generic fiber in the sense of scheme theory. When it comes to classification of fibrations, for example in the Enriques classification of surfaces, the theory of Albanese maps, or the minimal model program, it is crucial to understand this behavior.
The simplest situation is that X is a proper normal curve over a nonperfect field K. If K ⊂ K ′ is a purely inseparable field extension, the induced curve X ′ = X ⊗ K K ′ is not necessarily normal. LetX ′ be its normalization. Then the genusg = h 1 (OX′ ) may be strictly smaller that the genus g = h 1 (O X ) of our original curve. Tate [6] proved that such genus-change is not arbitrary:
Theorem. (Tate) The difference g −g is divisible by (p − 1)/2.
In particular, this puts an upper bound on the characteristic in terms of the possible genera occurring in genus-change situations. This is a prominent manifestation of the intuitive principle that a given geometrical deviation in positive characteristics in a fixed dimension should occur only at finitely many primes. Example: Quasielliptic fibrations (the case g = 1,g = 0) are possible only at prime p = 2 and p = 3.
Back in 1952, Tate naturally stated and proved his result in the language of function fields and repartitions. In my opinion, it is desirable to have a proof in the modern language of schemes as well. In the special caseg = 0, Shepherd-Barron [5] found such a proof for the inequality g ≥ (p − 1)/2, using vector bundles on algebraic surfaces. The goal of this paper is to give an easy direct proof of Tate's result, using relative dualizing sheaves and relative Frobenius maps for curves. The result essentially takes the following form: Theorem. Let Y be the normalization of the Frobenius pullback X (p) . Then the degree of the relative dualizing sheaf ω Y /X (p) is divisible by p − 1.
Our proof hinges on a result of Kiehl and Kunz [4], which implies that a finite universal homeomorphism between regular curves admits locally p-bases. I expect that this approach should yield result in higher dimensions as well. The paper also contains some results on normalization of geometrically integral schemes after Frobenius pullbacks.
Acknowledgement. I wish to thank Igor Dolgachev for stimulating discussions.
Normalization after Frobenius pullback
Let K be a field of characteristic p > 0, and X be a normal K-scheme of finite type. Throughout, we assume that X geometrically integral, that is, the induced schemes X ′ = X ⊗ K K ′ remain integral for all base field extensions K ⊂ K ′ . The scheme X ′ , however, is not necessarily normal. In this section, we shall collect some useful facts about the normalization of X ′ . Our first observation is:
Lemma 1.1. Let K ⊂ K ′ be a base field extension. Set X ′ = X ⊗ K K ′ , and let Y ′ → X ′ be the normalization. Then the K ′ -scheme Y ′ is geometrically integral.
Proof. Geometric irreducibility and geometric reducedness easily follow from [2], Proposition 4.5.9, and Proposition 4.6.1, respectively.
Next, we consider the Frobenius pullback X (p) , which is defined by the cartesian square
X (p) − −−− → X Spec(K) − −−− → FK Spec(K).
Here F K denotes the absolute Frobenius morphism, which corresponds to the Frobenius map Fr : K → K, λ → λ p . The K-scheme X (p) is of finite type and geometrically integral, but not necessarily normal. In any case, the Frobenius pullback is closely related to the original normal scheme X via the relative Frobenius morphisms F X/K : X → X (p) . This is a finite universal homeomorphism, coming from the commutative square
X FX − −−− → X Spec(K) − −−− → FK Spec(K).
Using iterated Frobenius maps, we obtain similarly the iterated Frobenius pullback X (p n ) , together with the iterated relative Frobenius morphism F n X/K : X → X (p n ) . The following observation will be useful:
Lemma 1.2.
There is an integer n 0 ≥ 0 such that for all integers n ≥ n 0 the normalization of X (p n ) is geometrically normal.
Proof. Clearly, it suffices to find one integer n ≥ 0 so that the normalization of X (p n ) is geometrically normal. To do so, choose a perfect closure K ⊂ K p −∞ . Set Z = X ⊗ K K p −∞ , and let ν :Z → Z be the normalization. The schemeZ is geometrically normal over K p −∞ , because the latter is perfect. According to [3], Theorem 8.8.2, there is an intermediate field K ⊂ K ′ ⊂ K p −∞ that is finite over K, so that the schemeZ and the morphism ν :Z → Z over K p −∞ are induced from a schemeX ′ and a morphism ν ′ :X ′ → X ′ over K ′ . Here of course we write X ′ = X ⊗ K K ′ . By [2], Corollary 6.7.8, the K ′ -schemeX ′ is geometrically normal. Since ν ′ is birational, ν ′ must be the normalization map of X ′ , and remains so after any base field extension of K ′ .
Since the field extension K ⊂ K ′ is finite and purely inseparable, there is an integer n ≥ 0 with the property λ p n ∈ K for all λ ∈ K ′ . By the universal property of splittings fields, there exists a homomorphism i : K ′ → K so that the composite K ⊂ K ′ i ֒→ K equals the n-fold Frobenius map. Consequently,X ′ ⊗ K ′ K is the normalization of X (p n ) = X ′ ⊗ K ′ K, where the tensor products are with respect to i : K ′ → K. This concludes the proof, since we saw in the preceding paragraph thatX ′ is geometrically normal.
Genus-change for algebraic curves
Now let X be a proper normal curve over K. As in the preceding section, we assume that X is geometrically integral. The degree of an invertible sheaf L on X is defined as the integer deg(L) = χ(L) − χ(O X ). The main result of this paper relates the degrees of the dualizing sheaves on the Frobenius pullback and its normalization. I formulate it in terms of the relative dualizing sheaf: Theorem 2.1. Let ν : Y → X (p) be the normalization map. Then the degree of the relative dualizing sheaf ω Y /X (p) is divisible by p − 1.
Proof. The idea is to compute with relative dualizing sheaves on X. Since X is normal, there is a unique morphism f : X → Y with F X/K = ν • f . The various relative dualizing sheaves satisfy
(1) ω X/X (p) = ω X/Y ⊗ f * (ω Y /X (p) ).
Similarly we have ω X/K = ω X/X (p) ⊗ F * X/K (ω X (p) /K ). Together with the formula F * X/K (ω X (p) /K ) = F * X (ω X/K ) = ω ⊗p X/K , this yields
(2) ω X/X (p) = ω ⊗(1−p) X/K .
On the other hand, Kiehl and Kunz proved that the morphism f : X → Y admits locally p-bases ( [4], Korollar 2 of Satz 5). Therefore the sheaf of relative Kähler differentials Ω 1 X/Y is locally free of finite rank. It is related to the relative dualizing sheaf by
(3) ω X/Y = det(Ω 1 X/Y ) ⊗(1−p)
, according to loc. cit., Satz 9. Substituting formula (3) and (2) into (1), we infer that the degree of f * (ω Y /X (p) ) is divisible by p − 1. Finally, observe that deg(f * (ω Y /X (p) )) = deg(f ) · deg(ω Y /X (p) ), by the projection formula. Clearly, the surjection f : X → Y is purely inseparable, hence its degree is a p-power. From this we infer that the degree of ω Y /X (p) must be divisible by p − 1.
Actually, the preceding result is equivalent to the following seemingly stronger statement: Theorem 2.2. Let K ⊂ L be an arbitrary field extension, and Y be the normalization of X ⊗ K L. Then the degree of the relative dualizing sheaf ω Y /X⊗K L is divisible by p − 1.
Proof. First note the following transitivity property: Suppose K ⊂ L ′ ⊂ L is an intermediate field. Let Y ′ be the normalization of X ⊗ K L ′ . Then Y is the normalization of both X ⊗ K L and Y ′ ⊗ L ′ L, and we have
ω Y /X⊗K L = ω Y /Y ′ ⊗ L ′ L ⊗ ϕ * (ω Y ′ /X⊗K L ′ ⊗ L ′ L),
where ϕ : Y → Y ′ ⊗ L ′ L is the normalization map. Clearly, if two of the three dualizing sheaves have degree divisible by p − 1, so has the third.
Using this transitivity property, we now settle the special case that the field L in our field extension K ⊂ L is perfect. Choose an integer n ≥ 0 so that the normalization Y ′ of the Frobenius pullback X (p n ) is geometrically normal, as in Lemma 1.2. Since the field L is perfect, there exists precisely one homomorphism i : K → L so that the composition i • Fr n is our given extension K ⊂ L, according to [1], Chap. V, §5, No. 2, Proposition 3. Consider the intermediate field L ′ = i(F ). Induction on n, together with the transitivity property and Theorem 2.1, shows that the degree of ω Y ′ /X⊗K L ′ is divisible by p − 1. Since Y ′ is geometrically normal, we have Y = Y ′ ⊗ L ′ L. Another application of the transitivity property yields that the degree of ω Y /X⊗K L is divisible by p − 1.
It remains to treat the general case. Choose a perfect closure L ⊂ L p −∞ . According to the preceding paragraph, the theorem holds true for the field extensions K ⊂ L p −∞ and L ⊂ L p −∞ . By transitivity, it must hold for K ⊂ L as well.
We now may retrieve Tate's result: Proof. According to Lemma 1.1, the L-scheme Y is geometrically integral. In particular, we have K = H 0 (X, O X ) and L = H 0 (Y, O Y ). Whence
h 1 (O X ) − h 1 (O Y ) = χ(O Y ) − χ(O X ) = 1 2 (deg(ω X/K ) − deg(ω Y /K )),
the latter by Serre duality and Riemann-Roch. The term on the right is nothing but − 1 2 deg(ω Y /X⊗K L ), so the statement follows from Theorem 2.2.
2000
Mathematics Subject Classification. 14H20.
Corollary 2.3. (Tate) Let K ⊂ L be an arbitrary field extension, and Y be the normalization of X ⊗ K L. Then the difference h 1 (O X ) − h 1 (O Y ) is divisible by (p − 1)/2.
N Bourbaki, Algebra II. BerlinSpringerN. Bourbaki: Algebra II. Chapters 4-7. Springer, Berlin, 1990.
Éléments de géométrie algébrique IV:Étude locale des schémas et des morphismes de schémas. A Grothendieck, Publ. Math., Inst. HautesÉtud. Sci. 24A. Grothendieck:Éléments de géométrie algébrique IV:Étude locale des schémas et des morphismes de schémas. Publ. Math., Inst. HautesÉtud. Sci. 24 (1965).
Éléments de géométrie algébrique IV:Étude locale des schémas et des morphismes de schémas. A Grothendieck, Publ. Math., Inst. HautesÉtud. Sci. 28A. Grothendieck:Éléments de géométrie algébrique IV:Étude locale des schémas et des morphismes de schémas. Publ. Math., Inst. HautesÉtud. Sci. 28 (1966).
Vollständige Durchschnitte und p-Basen. R Kiehl, E Kunz, Arch. Math. 16R. Kiehl, E. Kunz: Vollständige Durchschnitte und p-Basen. Arch. Math. 16 (1965), 348- 362.
Geography for surfaces of general type in positive characteristic. N Shepherd-Barron, Invent. Math. 106N. Shepherd-Barron: Geography for surfaces of general type in positive characteristic. Invent. Math. 106 (1991), 263-274.
Genus change in inseparable extensions of function fields. J Tate, Proc. Am. Math. Soc. 3J. Tate: Genus change in inseparable extensions of function fields. Proc. Am. Math. Soc. 3 (1952), 400-406.
Germany E-mail address: [email protected]. Mathematisches Institut, Heinrich-Heine-Universität, de40225DüsseldorfMathematisches Institut, Heinrich-Heine-Universität, 40225 Düsseldorf, Germany E-mail address: [email protected]
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