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+{"text":"\\section{Introduction}\nVarious problems in science and engineering lead to\nnonlocal operators and corresponding operator equations.\nExamples arise from physical problems like field calculations\nand Riesz energy problems, from machine learning, and also \nfrom stochastic simulations and uncertainty quantification.\n\nTraditional discretizations of nonlocal operators result in densely \npopulated system matrices. This feature renders the computation very costly in \nboth respects, the computation time and computer memory requirements. \nTherefore, over recent decades, different ideas for the data sparse approximation \nof nonlocal operators have been developed. Most prominent examples of \nsuch methods are the fast multipole method \\cite{GR}, the panel clustering \n\\cite{HN}, the wavelet matrix compression \\cite{BCR,DHS1}, and the hierarchical \nmatrix format \\cite{HA2}. These techniques are able to represent nonlocal \noperators in linear or almost linear cost with respect to the \nnumber of degrees of freedom used for their discretization.\n\nThe present article relies on a compression of the system matrix by\nwavelets. Especially, the matrix representation of the nonlocal operator \nin wavelet coordinates is quasi-sparse, i.e.\\ most matrix entries are \nnegligible and can be treated as zero without compromising the overall\naccuracy. Discarding the non-relevant matrix entries is called matrix \ncompression. Roughly speaking, nonlocal operators become local operators \nin wavelet coordinates. A fully discrete version of the wavelet \nmatrix compression has been developed in \\cite{HS2}. It computes the \ncompressed operator within discretization accuracy with linear cost.\n\nBased on the sparsity pattern of the system matrix which is \nsolely determined by the order of the underlying operator, we\nemploy a fill-in reducing reordering of the matrix entries by\nmeans of nested dissection, see \\cite{Geo73,LRT79}. This \nreordering in turn allows for the rapid inversion of the system \nmatrix by the Cholesky decomposition or more generally by \nthe LU decomposition. In particular, besides the rigorously\ncontrollable error for the matrix compression in the wavelet \nformat and the roundoff errors in the computation of the matrix\nfactorization, no additional approximation errors are introduced.\nThis is a major difference to other approaches for the discretization \nand the arithmetics of nonlocal operators, e.g.\\ by means of hierarchical \nmatrices. As the hierarchical matrix format is not closed under \narithmetic operations, a recompression step after each arithmetic \n(block) operation has to be performed, which results into \naccumulating and hardly controllable consistency errors \nfor matrix factorizations, see \\cite{HA1,HA2}. \n\nIn order to demonstrate the efficiency of the suggested approach, \nwe consider applications from different fields. Namely, we consider \n(i) a boundary integral equation arising from the polarizable\ncontinuum model in quantum chemistry as a classical example \nfor a nonlocal operator equation, (ii) a parabolic problem for the \nfractional Laplacian, and finally (iii) the fast numerical simulation of \nGaussian random fields as an important example from computational \nuncertainty quantification.\n\nOne of the most widespread methods to include solvent effects in \nquantum chemistry is by making use of a continuum dielectric: the \nsolvent is represented by a continuum which surrounds the molecule.\nSolute-solvent interactions are then described through appropriate \nfunctions supported on the molecule's surface. For an overview of \ncontinuum solvation models, we refer the reader to \\cite{TMC}, and \nin particular for the polarizable continuum model to \\cite{CM,CMT,MST}.\nWavelet matrix compression for the polarizable continuum model has been\nconsidered in \\cite{WaveletPCM2,WaveletPCM1}. Especially, in \\cite{ILU}, \nthe use of an incomplete Cholesky decomposition based preconditioner has \nbeen suggested, which is however inferior to the approach presented here. \n\nThe fractional Laplacian is an operator which generalizes the notion of \nspatial derivatives to fractional orderss. It appears in image asnalysis, \nkinetic equations, phase transitions and nonlocal heat conduction, \njust to mention some applications. We refer to the review article\n\\cite{DGZ} and the references therein for further details. In particular,\nwe will focus here on the definition of the fractional Laplacian in its\nintegral form, as it can be found in \\cite{EG} and also in \\cite{DGZ}. \nTo the best of our knowledge, the numerical treatment of the parabolic \nproblem for the fractional Laplacian by means of wavelets has not \nbeen addressed in literature yet. \n\nThe rapid simulation of (Gaussian) random fields with a prescribed \ncovariance structure is of paramount importance in computational \nuncertainty quantification. The fast methods, which have been \nsuggested so far are based on the computation of matrix square \nroots employing low-rank factorizations, block-wise low-rank \nfactorizations, such as obtained by hierarchical matrices, or the \ndiscretization of the action of the matrix square root on a given \nvector by Krylov subspace methods. Other approaches compute the \nKarhunen-Lo\\`eve expansion by circulant embeddings and fast Fourier \ntechniques or employ the contour integral method. For more details on \nthese methods, we refer to \\cite{FKS18,GKNS+18,HPS15,HKS20,RH}.\nIn contrast to the previously mentioned approaches, we consider here \nthe direct simulation of the random field by the Cholesky decomposition \nof the covariance matrix, which is very sparse in wavelet coordinates.\n\nThis article is organized as follows. Wavelet bases and \ntheir properties are specified in Section~\\ref{sec:wavelets}.\nSection~\\ref{sec:WEM} briefly repeats the main features of the \nfully discrete wavelet matrix compression scheme from \\cite{HS2}. \nThen, in Section~\\ref{sect:ND}, for the sake of completeness,\nthe idea of nested dissection is briefly outlined.\nSection~\\ref{sct:applications} presents the three different \napplications considered in this article, while Section~\\ref{sec:numerix} \nis devoted to related numerical experiments. Finally, Section~\\ref{sec:conclusion} \ncontains some concluding remarks.\n\nIn the following, in order to avoid the repeated use of generic but\nunspecified constants, we write $C \\lesssim D$ to indicate that $C$ \ncan be bounded by a multiple of $D$, independently of parameters\nwhich $C$ and $D$ may depend on. Then, $C \\gtrsim D$\nis defined as $D \\lesssim C$, while we write $C \\sim D$, iff\n$C \\lesssim D$ and $C \\gtrsim D$.\n\n\\section{Wavelets and multiresolution Analysis}\n\t\\label{sec:wavelets}\nLet $D$ denote a domain in $\\mathbb{R}^n$ or \na manifold in $\\mathbb{R}^{n+1}$. A \n\\emph{multiresolution analysis} consists of a nested \nfamily of finite dimensional approximation spaces\n\\begin{equation}\t\\label{eq:hierarchy}\n\\{0\\} = V_{-1} \\subset V_0 \\subset V_1 \\subset \\cdots \n\\subset V_j\\subset \\cdots \\subset L^2(D),\n\\end{equation}\nsuch that\n\\[ \\overline{\\bigcup_{j\\ge0} V_j} = L^2(D)\\quad\\text{and}\\quad\n\\dim V_j\\sim 2^{jn}.\n\\] We will\nrefer to \\(j\\) as the \\emph{level} of \\(V_j\\) in the\nmultiresolution analysis.\nEach space $V_j$ is endowed with a \\emph{single-scale basis}\n\\[\n\\Phi_j = \\{\\varphi_{j,{\\bs k}}:{\\bs k}\\in\\Delta_j\\},\n\\]\ni.e.\\ $V_j=\\operatorname{span}\\Phi_j$, where $\\Delta_j$ denotes a suitable \nindex set with cardinality $|\\Delta_j| \\sim 2^{jn}$. For convenience, \nwe shall in the sequel write bases on the form of row vectors, such \nthat, for ${\\bs v} = [v_k]_{k\\in\\Delta_j}\\in\\ell^2(\\Delta_j)$, the \ncorresponding function can simply be written as a dot product according\nto\n\\[\nv_j = \\Phi_j{\\bs v}=\\sum_{k\\in\\Delta_j} v_k\\varphi_{j,k}. \n\\]\nIn addition, we shall assume that the single-scale \nbases $\\Phi_j$ are \\emph{uniformly stable}, this means that \n\\[\n\\|{\\bs v}\\|_{\\ell^2(\\Delta_j)}\n\\sim \\|\\Phi_j{\\bs v}\\|_{L^2(D)}\\quad\\text{for all }\n{\\bs v}\\in\\ell^2(\\Delta_j)\n\\]\nuniformly in $j$, and that they satisfy the locality condition \n\\[\n\\operatorname{diam}(\\operatorname{supp}\\varphi_{j,{\\bs k}})\\sim 2^{-j}.\n\\]\n\nAdditional properties of the spaces $V_j$ are required\nfor using them as trial spaces in a Galerkin scheme.\nThe approximation spaces shall have the \\emph{regularity} \n\\[\n\\gamma\\mathrel{\\mathrel{\\mathop:}=}\\sup\\{s\\in\\mathbb{R}: V_j\\subset H^s(D)\\}\n\\]\nand the \\emph{approximation order} $d\\in\\mathbb{N}$, \nthat is\n\\[\n       d = \\sup\\Big\\{s\\in\\mathbb{R}: \\inf_{v_j\\in V_j}\\|v-v_j\\|_{L^2(D)} \n       \t\t\\lesssim 2^{-js}\\|v\\|_{H^s(D)}\\Big\\}.\n\\]\n\nRather than using the multiresolution analysis corresponding to the \nhierarchy in \\eqref{eq:hierarchy}, the pivotal idea \nof wavelets is to keep track of the increment of information \nbetween two consecutive levels $j-1$ and $j$. Since we have\n$V_{j-1}\\subset V_j$, we may decompose \n\\[\nV_j = V_{j-1}\\oplus W_j,\\ \\text{i.e.}\\ V_{j-1}\\cup W_j = V_{j}\\ \\text{and}\\  V_{j-1}\\cap W_j = \\{0\\},\n\\]\nwith an appropriate \\emph{detail space} $W_j$. Of practical interest \nis the particular choice of the basis of the detail space  $W_j$ in $V_j$. This basis \nwill be denoted by\n\\[\n  \\Psi_j = \\{\\psi_{j,{\\bs k}}:\n  \t{\\bs k} \\in \\nabla_j\\mathrel{\\mathrel{\\mathop:}=} \\Delta_j\\setminus \\Delta_{j-1}\\}.\n\\]\nIn particular, we shall assume that the collections $\\Phi_{j-1}\\cup\\Psi_j$\nform uniformly stable bases of $V_j$, as well. If $\\Psi = \n\\bigcup_{j\\ge 0}\\Psi_j$, where $\\Psi_0\\mathrel{\\mathrel{\\mathop:}=} \\Phi_0$, is even a \nRiesz-basis of $L_2(D)$, then it is called a \\emph{wavelet \nbasis}. We require the functions $\\psi_{j,{\\bs k}}$ to be \nlocalized with respect to the corresponding level $j$, i.e.\\ \n\\[\n\\operatorname{diam}(\\operatorname{supp}\\psi_{j,{\\bs k}}) \\sim 2^{-j}, \n\\]\nand we normalize them such that \n\\[\n\\|\\psi_{j,{\\bs k}}\\|_{L_2(D)}\\sim 1.\n\\]\n\nAt first glance it would be very convenient to deal with a single \northonormal system of wavelets. However, it has been shown\nin \\cite{DHS1,DPS4,S} that orthogonal wavelets are not optimal \nfor the efficient approximation nonlocal operator equations. For this \nreason, we rather use \\emph{biorthogonal wavelet bases}.\nIn this case, we also have a dual, multiresolution\nanalysis, i.e.\\ dual single-scale bases and wavelets\n\\[\n\\widetilde{\\Phi}_j = \\{\\widetilde{\\varphi}_{j,{\\bs k}}:{\\bs k}\\in\\Delta_j\\},\\quad\n\\widetilde{\\Psi}_j = \\{\\widetilde{\\psi}_{j,{\\bs k}}:{\\bs k}\\in\\nabla_j\\},\n\\] \nwhich are coupled to the primal ones by the orthogonality \ncondition \n\\[\n(\\Phi_j,\\widetilde{\\Phi}_j)_{L^2(D)} = {\\bs I},\\quad\n(\\Psi_j,\\widetilde{\\Psi}_j)_{L^2(D)} = {\\bs I}. \n\\]\nThe corresponding spaces $\\widetilde{V}_j\\mathrel{\\mathrel{\\mathop:}=} \\operatorname{span}\\widetilde{\\Phi}_j$\nand $\\widetilde{W}_j\\mathrel{\\mathrel{\\mathop:}=}\\operatorname{span}\\widetilde{\\Psi}_j$ satisfy\n\\begin{equation}\t\\label{eq:space-coupling}\n  V_{j-1}\\perp \\widetilde{W}_j, \\quad \\widetilde{V}_{j-1}\\perp W_j.\n\\end{equation}\nMoreover, the dual spaces are supposed to exhibit some approximation order\n$\\widetilde{d}\\in\\mathbb{N}$ and regularity $\\widetilde{\\gamma}>0$.\n\nDenoting in complete analogy to the primal basis $\\widetilde{\\Psi} = \n\\bigcup_{j\\ge 0}\\widetilde{\\Psi}_j$, where $\\widetilde{\\Psi}_0 \n\\mathrel{\\mathrel{\\mathop:}=}\\widetilde{\\Phi}_0$, then every $v \\in L^2(D)$ has\nunique representations \n\\[\nv = \\widetilde{\\Psi} (v,\\Psi)_{L^2(D)} = \\Psi (v,\\widetilde{\\Psi})_{L^2(D)}\n\\]\nsuch that\n\\[\n \\|v\\|_{L^2(D)}^2 \\sim \\sum_{j\\ge 0}\\sum_{k\\in\\nabla_j}\n\t\\big\\|(v,\\widetilde\\psi_{j,{\\bs k}})_{L^2(D)}\\big\\|_{\\ell^2(\\nabla_j)}^2\n \\sim \\sum_{j\\ge 0}\\sum_{k\\in\\nabla_j}\n\t\\big\\|(v,\\psi_{j,{\\bs k}})_{L^2(D)}\\big\\|_{\\ell^2(\\nabla_j)}^2.\n\\]\nIn particular, relation \\eqref{eq:space-coupling} implies that the \nwavelets exhibit {\\em vanishing moments} of order $\\widetilde{d}$, i.e.\\\n\\begin{equation}\t\\label{eq:cancellation}\n  \\big|(v,\\psi_{j,{\\bs k}})_{L^2(D)}\\big| \\lesssim 2^{-j(1+\\widetilde{d})}\n\t|v|_{W^{\\widetilde{d},\\infty}(\\operatorname{supp}\\psi_{j,{\\bs k}})}.\n\\end{equation}\nHerein, the quantity $|v|_{W^{\\widetilde{d},\\infty}(D)}\n\\mathrel{\\mathrel{\\mathop:}=} \\sup_{|\\boldsymbol\\alpha|=\n\\widetilde{d}}\\|\\partial^{\\boldsymbol\\alpha}v\\|_{L^\\infty(D)}$\nis the semi-norm in $W^{\\widetilde{d},\\infty}(D)$.\nWe refer to \\cite{DA1} for further details.\n\nPiecewise constant and bilinear wavelets which \nprovide the above properties have been constructed in \n\\cite{HS1,HS3}. In what follows, we will refer to the\nwavelet basis of $V_J$ by $\\Psi_J = \\{\\psi_\\lambda:\n\\lambda\\in\\nabla_J\\}$, where the multi-index \n$\\lambda = (j,{\\bs k})$ incorporates the scale \n$j=|\\lambda|$ and the spatial location \n${\\bs k} = {\\bs k}(\\lambda)$.\n\n\\section{Wavelet Matrix Compression}\t\t\\label{sec:WEM}\nFor a given domain or manifold $D$ and $q\\in\\mathbb{R}$, let\n\\[\n\\mathcal{A}:H^q(D)\\to H^{-q}(D)\n\\]\ndenote a given (continuous and bijective) nonlocal operator of\norder $2q$. According to the Schwartz kernel theorem, it can \nbe represented in accordance with\n\\begin{equation}\t \\label{eq:integral equation}\n (\\mathcal{A}u)({\\bs x}) = \\int_D k({\\bs x},{\\bs y})\n \tu({\\bs y})\\d{\\bs y},\n \\quad {\\bs x}\\in D,\n\\end{equation}\nfor a suitable kernel function $k\\colon D\\times D\\to\\mathbb{R}$.\nThe kernel functions under consideration are supposed \nto be smooth as functions in the variables ${\\bs x}$ and \n${\\bs y}$, apart from the diagonal $\\{({\\bs x},{\\bs y})\\in\nD\\times D: {\\bs x}={\\bs y}\\}$ and may exhibit a singularity \non the diagonal. Such kernel functions arise, for instance, from \napplying a boundary integral formulation to a second order elliptic \nproblem \\cite{SS,ST}. Typically, they decay like a negative power \nof the distance of the arguments which depends on the order \n$2q$ of the operator. More precisely, there holds\n\\begin{equation}\t\\label{eq:decay}\n  \\big|\\partial_{\\bs x}^{\\boldsymbol\\alpha}\\partial_{\\bs y}^{\\boldsymbol\\beta} \n  \tk({\\bs x},{\\bs y})\\big| \\le c_{\\boldsymbol\\alpha,\\boldsymbol\\beta}\n\t\t\\|{\\bs x}-{\\bs y}\\|^{-n-2q-|\\boldsymbol\\alpha|-|\\boldsymbol\\beta|}.\n\\end{equation}\nWe emphasize that this estimate remains valid for the kernels of\narbitrary pseudodifferential operators, see \\cite{DHS2} for the details.\n\nCorresponding to the nonlocal operator from \\eqref{eq:integral equation},\nwe may consider the operator equation \\[\\mathcal{A}u=f\\] which gives rise\nto the Galerkin approach:\n\\begin{align*}\n&\\text{find $u_J\\in V_J$ such that}\\\\\n&\\quad (\\mathcal{A}u_J,v_J)_{L^2(D)} = (f,v_J)_{L^2(D)}\n\\quad\\text{for all $v_J \\in V_J$}.\n\\end{align*}\nTraditionally, this equation is discretized \nemploying the single-scale basis of $V_J$ which results in densely\npopulated system matrices. If $N_J\\sim 2^{Jn}$ denotes the \nnumber of basis functions in the space $V_J$, then the system \nmatrix contains $\\mathcal{O}(N_J^2)$ nonzero matrix entries. In \ncontrast, by utilizing a wavelet basis in the Galerkin discretization,\nwe end up with a matrix that is quasi-sparse, i.e.\\ it is compressible \nto $\\mathcal{O}(N_J)$ nonzero matrix entries without compromising \nthe overall accuracy. More precisely, by combining \\eqref{eq:cancellation}\nand \\eqref{eq:decay}, we arrive at the decay estimate\n\\begin{equation}\t\\label{eq:decay estimate}\n  (\\mathcal{A}\\psi_{\\lambda'},\\psi_\\lambda)_{L^2(D)}\n  \t\\lesssim \\frac{2^{-(|\\lambda|+|\\lambda'|)(\\widetilde{d}+n\/2)}}\n\t{\\operatorname{dist}(D_\\lambda,D_{\\lambda'})^{n+2q+2\\widetilde{d}}}\n\\end{equation}\nwhich is the foundation of the compression estimates in \\cite{DHS1}. \nHerein, $D_\\lambda\\mathrel{\\mathrel{\\mathop:}=}\\operatorname{supp}\\psi_\\lambda$ and \n$D_{\\lambda'}\\mathrel{\\mathrel{\\mathop:}=}\\operatorname{supp}\\psi_\\lambda$ denote the \nconvex hulls of the supports of the wavelets $\\psi_\\lambda$ and\n$\\psi_{\\lambda'}$.\n\nBased on \\eqref{eq:decay estimate}, we shall neglect\nall matrix entries for which the distance of \nthe supports between the associated ansatz and test \nwavelets is larger than a level dependent cut-off \nparameter $\\mathcal{B}_{j,j'}$. An additional compression, \nreflected by a cut-off parameter $\\mathcal{B}_{j,j'}^s$, \nis achieved by neglecting several of those matrix entries, \nfor which the corresponding trial and test functions \nhave overlapping supports. \n\nTo formulate this result, we introduce the abbreviation\n$D_\\lambda^s\\mathrel{\\mathrel{\\mathop:}=} \\operatorname{sing}\\operatorname{supp}\\psi_\\lambda$\nwhich denotes the {\\em singular support} of the wavelet \n$\\psi_\\lambda$, i.e.\\ that subset of $D$ where \nthe wavelet is non-smooth.\n\n\\begin{theorem}[A-priori compression \\cite{DHS1}]\\label{thm:a-priori}\nLet $D_\\lambda$ and $D_\\lambda^s$ be given as \nabove and define the compressed system matrix ${\\bs A}_J$, \ncorresponding to the boundary integral operator $\\mathcal{A}$, by\n\\begin{equation}\t\t\\label{eq:a-priori}\n  [{\\bs A}_J]_{\\lambda,\\lambda'}\\mathrel{\\mathrel{\\mathop:}=} \\begin{cases}\n  \\qquad \\quad 0,& \\operatorname{dist}(D_\\lambda,D_{\\lambda'}) > \\mathcal{B}_{|\\lambda|,|\\lambda'|}\\ \\text{and $|\\lambda|,|\\lambda'|> 0$}, \\\\\n  \\qquad \\quad 0,& \\operatorname{dist}(D_\\lambda,D_{\\lambda'}) \\le 2^{-\\min\\{|\\lambda|,|\\lambda'|\\}}\\ \\text{and}\\\\\n                 & \\operatorname{dist}(D_\\lambda^s,D_{\\lambda'}) >\n\t\t \\mathcal{B}_{|\\lambda|,|\\lambda'|}^s\\ \\text{if $|\\lambda'| > |\\lambda|\\geq 0$}, \\\\\n\t\t & \\operatorname{dist}(D_\\lambda,D_{\\lambda'}^s) >\n\t\t \\mathcal{B}_{|\\lambda|,|\\lambda'|}^s\\ \\text{if $|\\lambda| > |\\lambda'|\\geq 0$}, \\\\\n  (\\mathcal{A}\\psi_{\\lambda'},\\psi_\\lambda)_{L^2(D)}, &\\text{otherwise}. \\end{cases}\n\\end{equation}\nFixing\n\\begin{equation}\t\\label{eq:parameters}\n  a > 1, \\qquad d < \\delta < \\widetilde{d} + 2q,\n\\end{equation}\nthe cut-off parameters $\\mathcal{B}_{j,j'}$ and\n$\\mathcal{B}_{j,j'}^s$ are set according to\n\\begin{equation}\t\\label{eq:cut-off parameters}\n  \\begin{aligned}\n  \\mathcal{B}_{j,j'} \\mathrel{\\mathrel{\\mathop:}=} a \\phantom{'}\\max\\left\\{ 2^{-\\min\\{j,j'\\}},\n\t2^{\\frac{2J(\\delta-q)-(j+j')(\\delta+\\widetilde{d})}{2(\\widetilde{d}+q)}}\\right\\}, \\vspace*{2mm} \\\\\n  \\mathcal{B}_{j,j'}^s \\mathrel{\\mathrel{\\mathop:}=} a \\max\\left\\{ 2^{-\\max\\{j,j'\\}},\n\t2^{\\frac{2J(\\delta-q)-(j+j')\\delta-\\max\\{j,j'\\}\n\t\\widetilde{d}}{\\widetilde{d}+2q}}\\right\\}.\n  \\end{aligned}\n\\end{equation}\nThen, the system matrix ${\\bs A}_J$ only has $\\mathcal{O}(N_J)$\nnonzero entries. In addition, the error estimate\n\\begin{equation}\t\\label{eq:errest}\n    \\|u-u_J\\|_{H^{2q-d}(D)} \\lesssim 2^{-2J(d-q)}\\|u\\|_{H^d(D)}\n\\end{equation}\nholds for the solution $u_J$ of the compressed Galerkin system\nprovided that $u$ and $D$ are sufficiently regular.\n\\end{theorem}\n\nThe compressed system matrix can be assembled with linear \ncost if the exponentially convergent \n$hp$--quadrature method proposed in \\cite{HS2} is employed for\nthe computation of matrix entries. Moreover, \nfor performing faster matrix-vector multiplications, an\nadditional a-posteriori compression might be applied\nwhich reduces again the number of nonzero entries \nby a factor 2--5, see \\cite{DHS1}. The pattern of the compressed \nsystem matrix shows the typical {\\em finger structure}, see \nthe left hand side of Figure~\\ref{fig:compression}.\n\n\\begin{figure}\n\\begin{center}\n\\begin{minipage}{0.49\\textwidth}\n\\begin{center}\n\\includegraphics[scale=0.55,clip,trim=0 0 0 13,frame]{SWpattern_PCM}\n$\\nnz({\\bs V}_J) = 15\\,363\\,141$\n\\end{center}\n\\end{minipage}\n\\begin{minipage}{0.49\\textwidth}\n\\begin{center}\n\\includegraphics[scale=0.55,clip,trim=0 0 0 13,frame]{Spattern_PCM}\n$\\nnz({\\bs V}_{J,\\text{ND}}) = 15\\,363\\,141$\n\\end{center}\n\\end{minipage}\n\\caption{\\label{fig:compression}Sparsity patterns of \\({\\bs V}\\) (left) and its nested\ndissection reordering\n\\({\\bs V}_{J,\\text{ND}}\\) (right) for the single layer operator on the benzene \ngeometry and \\(N_J=93184\\). Each dot corresponds to a submatrix of size \n\\(64\\times 64\\). Lighter blocks have less entries than darker blocks.}\n\\end{center}\n\\end{figure}\n\n\\section{Nested dissection}\\label{sect:ND}\nThe representation of the system matrix corresponding to\na nonlocal operator with respect to an appropriate wavelet\nbasis leads to a quasi-sparse matrix, i.e.\\ a matrix with many\nsmall entries which can be neglected without compromising\naccuracy. Performing a thresholding procedure as discussed \nin the previous section then yields a sparse system matrix\nwhose symmetric sparsity pattern is solely determined by the \norder of the underlying operator, see the left hand side of \nFigure~\\ref{fig:compression}.\n\nThe factorization of the system matrix represented \nin the canoncial levelwise ordering leads to a massive \nfill-in. This means that a huge amount of nonzero \nentries is generated by a Cholesky decomposition or an LU decomposition, \ntypically resulting in dense matrix factors. In order to obtain \nmuch sparser factorizations, we employ a nested dissection \nordering, cf.\\ \\cite{Geo73,LRT79}, see the right hand side of \nFigure~\\ref{fig:compression}.\n\nNested dissection is a divide and conquer algorithm whose foundation\nis a graph theoretical observation. To each matrix\n\\({\\bs A}\\in\\mathbb{R}^{N\\times N}\\) with a symmetric sparsity\npattern, we may assign an undirected graph \\(G=(V,E)\\) with\nvertices \\(V=\\{1,2,\\ldots,N\\}\\) and edges\n\\(E=\\big\\{\\{i,j\\}:a_{i,j}\\neq 0\\big\\}\\). Then, a symmetric permutation\n\\({\\bs P}{\\bs A}{\\bs P}^\\intercal\\) of the rows and columns of \\({\\bs A}\\)\namounts to a permutation \\(\\pi(V)\\) of the nodes in \\(V\\).\nIn particular, we have the following important result from \\cite{RTL76},\nsee also \\cite{Par61}, which we formulate here only for the Cholesky decomposition\n\\({\\bs P}{\\bs A}{\\bs P}^\\intercal={\\bs L}{\\bs L}^\\intercal\\).\n\n\\begin{lemma}[\\cite{RTL76}]\nAssuming that no cancellation of nonzero entries\nin the Cholesky decomposition\nof \\({\\bs P}{\\bs A}{\\bs P}^\\intercal\\) takes place, then \n\\(\\ell_{i,j}\\neq 0\\) for \\(i>j\\), iff there is a path\n\\(i=v_1,v_2,\\ldots, v_{k+1}=j\\), \\(k\\geq 0\\), in \\(G\\)\nsuch that \\(\\pi(v_t)<\\min\\{\\pi(i),\\pi(j)\\}\\) for \\(2\\leq t\\leq k\\).\n\\end{lemma}\n\nThe lemma states that the Cholesky decomposition connects all\nnodes \\(i\\) and \\(j\\), resulting in a nonzero entry \\(\\ell_{i,j}\\),\nfor which there exists a path of nodes that have been eliminated\nbefore \\(i\\) and \\(j\\).\n\nFinding an optimal ordering is a hard problem in general. Therefore, we \nresort to the following strategy, which is known as nested dissection ordering:\nWe split \\(V=V_1\\cup V_2\\cup S\\)\nsuch that \\(E\\cap\\big\\{\\{v_1,v_2\\}: v_1\\in V_1,v_2\\in V_2\\big\\}=\\emptyset\\),\ni.e.\\ the removal of the vertices of the \\emph{separator} \\(S\\) and \nits adjacent edges results into two disjoint subgraphs.\nHence, employing an ordering which puts first the nodes into \\(V_1\\) and \\(V_2\\)\nand afterwards the nodes in \\(S\\), leads to a matrix structure of the form\n\\[\n{\\bs P}{\\bs A}{\\bs P}^\\intercal=\n\\begin{bmatrix}{\\bs A}_{V_1,V_1} & & {\\bs A}_{V_1,S}\\\\\n & {\\bs A}_{V_2,V_2}& {\\bs A}_{V_2,S}\\\\\n{\\bs A}_{S,V_1} & {\\bs A}_{S,V_2}& {\\bs A}_{S,S}\\\\\\end{bmatrix}.\n\\]\nRecursively applying this procudeure then yields a structure similar to\nthe one on the right hand side of Figure~\\ref{fig:compression}.\nFor obvious reasons, it is desirable to have a minimal separator\n\\(S\\), which evenly splits the \\(V\\) into two subsets, we refer to \n\\cite{LRT79} and the references therein for a comprehensive \ndiscussion of this topic. In order to obtain suitable separators \nfor the computations in this article, we will adopt the strategy \nfrom \\cite{KK98}, which performs very well in terms of reducing \nthe fill-in.\n\n\\section{Applications}\\label{sct:applications}\n\\subsection{Polarizable continuum model}\nContinuum solvation models are widely used to model\nquantum effects of molecules in liquid solutions, compare\n\\cite{TMC} for an overview. In the {\\em polarizable continuum \nmodel} (PCM), introduced in \\cite{MST}, the molecule under \nstudy (the solute) is located inside a cavity $D$, surrounded \nby a homogeneous dielectric (the solvent) with dielectric \nconstant $\\epsilon\\ge 1$. The solute-solvent interactions \nbetween the charge distributions which compose the solute \nand the dielectric are reduced to those of electrostatic origin.\n\nFor a given charge $\\rho\\in H^{-1}(D)$, located inside\nthe cavity, the solute-solvent interaction is expressed by the\n{\\em apparent surface charge} $\\sigma\\in H^{-1\/2}(\\partial D)$.\nIt is given by the integral equation\n\\begin{equation}\t\\label{eq:PCM}\n \\mathcal{V}\\sigma = \\bigg(\\frac{1+\\epsilon}{2}+(1-\\epsilon)\\mathcal{K}\\bigg)^{-1}\n\t\t\\mathcal{N}_\\rho-\\mathcal{N}_\\rho\\quad\\text{on $\\partial D$},\n\\end{equation}\nwhere $\\mathcal{V}$ is the \\emph{single layer operator}\n\\[\n  (\\mathcal{V}u)({\\bs x}) = \\int_{\\partial D}\n  \t\\frac{u({\\bs y})}{4\\pi\\|{\\bs x}-{\\bs y}\\|^3}\\d\\sigma_{\\bs y},\n\\]\n$\\mathcal{K}$ is the \\emph{double layer operator}\n\\[\n  (\\mathcal{K}u)({\\bs x}) = \\int_{\\partial D} u({\\bs y})\n  \t\\frac{\\langle{\\bs n}({\\bs y}),{\\bs x}-{\\bs y}\\rangle}{4\\pi\\|{\\bs x}-{\\bs y}\\|^3}\\d\\sigma_{\\bs y},\n\\]\nand $\\mathcal{N}_\\rho$ denotes the \\emph{Newton potential} of the \ngiven charge\n\\[\n  \\mathcal{N}_\\rho({\\bs x})\\mathrel{\\mathrel{\\mathop:}=} \\int_{\\partial D}\\frac{\\rho({\\bs y})}{4\\pi\\|{\\bs x}-{\\bs y}\\|}\n  \t\\d{\\bs y}.\n\\]\n\nThe discretization of the boundary integral equation\n\\eqref{eq:PCM} by means of a Galerkin scheme is as \nfollows, compare \\cite{HM1,HM2}: We make the ansatz \n\\[\n  \\sigma_J = \\sum_\\lambda\\sigma_\\lambda\\psi_\\lambda\n\\]\nand introduce the mass matrix\n\\[\n{\\bs G}_J = [(\\psi_{\\lambda'},\\psi_\\lambda)_{L^2(D)}]_{\\lambda,\\lambda'}\n\\]\nand the system matrices\n\\[\n {\\bs V}_J = [(\\mathcal{V}\\psi_{\\lambda'},\\psi_\\lambda)_{L^2(D)}]_{\\lambda,\\lambda'},\\quad\n {\\bs K}_J = [(\\mathcal{K}\\psi_{\\lambda'},\\psi_\\lambda)_{L^2(D)}]_{\\lambda,\\lambda'}.\n\\]\nThen, for a given data vector ${\\bs f}_J = [(\\mathcal{N}_\\rho,\n\\psi_\\lambda)_{L^2(\\partial D)}]_\\lambda$, we need to solve the \nlinear system of equations\n\\begin{equation}\\label{eq:PCM2}\n  {\\bs V}_J\\boldsymbol\\sigma_J = {\\bs G}_J\n  \t\\bigg(\\frac{1+\\epsilon}{2}{\\bs G}_J+(1-\\epsilon) {\\bs K}_J\\bigg)^{-1}\n  \t{\\bs f}_J-{\\bs f}_J\n\\end{equation}\nin order to determine the sought apparent surface charge.\n\nIn quantum chemical simulations, for example when \nsolving the Hartree-Fock equations in a self consistent \nfield approximation, one has to compute the interaction \nenergies between the different particles. This amounts to \nthe determination of different apparent surface charges. \nTherefore, the fast solution of \\eqref{eq:PCM2} for multiple \nright hand sides is indispensable for fast simulations \nin quantum chemistry.\n\n\\subsection{Parabolic diffusion problem for the fractional Laplacian}\nFor a given domain $D\\subset\\mathbb{R}^n$ and $0<s<1\/2$,\nthe fractional Laplacian $\\mathcal{L}_s\\colon H^s(D)\/\\mathbb{R}\n\\to H^{-s}(D)\/\\mathbb{R}$ is given by\n\\[\n(\\mathcal{L}_su)({\\bs x})\\mathrel{\\mathrel{\\mathop:}=} 2 \\int_{D} \\frac{u({\\bs y}) - u({\\bs x})}{\\|{\\bs x}-{\\bs y}\\|^{n+2s}}\\d{\\bs y},\n\t\\quad {\\bs x} \\in D,\n\\]\ncompare \\cite{EG,DGZ}. We intent to solve the following \nparabolic diffusion problem\n\\[\n \\partial_t u - \\mathcal{L}_s u = f\\quad\\text{in $D$}\n\\] \nfor the fractional Laplacian.\nTo this end, we employ the $\\theta$-scheme in time and a \nGalerkin discretization of the problem in space. This leads to the\nlinear system of equations\n\\begin{equation}\\label{eq:theta}\n {\\bs G}_J\\frac{{\\bs u}_J(t_{i+1})-{\\bs u}_J(t_i)}{t_{i+1}-t_i} \n - {\\bs L}_J \\big\\{(1-\\theta) {\\bs u}_J(t_i) + \\theta{\\bs u}_J(t_{i+1})\\big\\}\n \t= (1-\\theta){\\bs f}_J(t_i) + \\theta{\\bs f}_J(t_{i+1}).\n\\end{equation}\nHere, \n\\[\n {\\bs L}_J = \\big[(\\mathcal{L}_s\\psi_{\\lambda'},\n \t\\psi_\\lambda)_{L^2(D)}\\big]_{\\lambda,\\lambda'},\\quad\n {\\bs G}_J = \\big[(\\psi_{\\lambda'},\\psi_\\lambda)_{L^2(D)}\\big]_{\\lambda,\\lambda'},\n\\]\nare the system matrix of the fractional Laplacian and the\nmass matrix, respectively. In each time step, we have hence to invert the\nmatrix ${\\bs A}_J={\\bs G}_J - \\theta(t_{i+1}-t_i) {\\bs L}_J$ for\ncomputing the new solution ${\\bs u}_J(t_{i+1})$ from the\nsolution  ${\\bs u}_J(t_i)$ of the previous time step $t_i$.\nA factorization of the system matrix \\({\\bs A}_J\\)\nis favourable in this situation since \nthe system matrix does not change with time.\n\n\\subsection{Gaussian random fields}\nLet \\((\\Omega,\\mathcal{F},\\mathbb{P})\\) denote a complete\nand separable probability space. \nWe consider a Gaussian random field\n\\[\na\\colon D\\times\\Omega\\to\\mathbb{R}\n\\]\nwith \\emph{expectation} \n\\[\n\\Ebb[a]({\\bs x})\\mathrel{\\mathrel{\\mathop:}=}\\int_\\Omega a({\\bs x},\\omega)\\d\\Pbb(\\omega)\n\\] and \\emph{covariance}\n\\[\n\\operatorname{Cov}[a]({\\bs x},{\\bs x}')\\mathrel{\\mathrel{\\mathop:}=}\\int_\\Omega\n\\big(a({\\bs x},\\omega)-\\Ebb[a]({\\bs x})\\big)\n\\big(a({\\bs x}',\\omega)-\\Ebb[a]({\\bs x}')\\big)\\d\\Pbb(\\omega).\n\\]\nIf the expectation and the covariance are known, we may represent\n \\(a\\) by its \\emph{Karhunen-Lo\\`eve expansion}\n\\[\na({\\bs x},\\omega)=\\Ebb[a]({\\bs x})+\\sum_{k=1}^\\infty\\sqrt{\\mu_k} a_k({\\bs x})Y_k(\\omega)\n\\]\nHerein,  \\((\\mu_k,a_k)\\), \\(k=1,2,\\ldots\\),\ndenote the eigen pairs of the Hilbert-Schmidt operator\n\\[\n(\\mathcal{C}v)({\\bs x})\\mathrel{\\mathrel{\\mathop:}=}\\int_D\\operatorname{Cov}[a]({\\bs x},{\\bs x}')v({\\bs x}')\\d{\\bs x}',\n\\]\nwhile \\(Y_1,Y_2,\\ldots\\) are independent and standard normally distributed random variables.\n\nIn order to discretize the Karhunen-Loeve expansion, we proceed in complete\nanalogy to \\cite{HPS15} and compute the orthogonal projection of \\(\\mathcal{C}\\) onto \\(V_J\\).\nLet \\({\\bs{C}}_J\\in\\mathbb{R}^{N_J\\times N_J}\\) denote the corresponding coefficient matrix. Obviously, a suitable\nbasis for the orthogonal projection is given by the orthogonalized wavelet basis\n\\(\\Psi_J{\\bs G}^{-1\/2}_J\\), where\n\\[\n{\\bs G}_J = \\big[(\\psi_{\\lambda'},\\psi_\\lambda)_{L^2(D)}\\big]_{\\lambda,\\lambda'}\n\\]\ndenotes the mass matrix and \\({\\bs G}^{-1\/2}_J\\) is an inverse matrix root.\nThus, in accordance with \\cite{HPS15}, we arrive at the discretized random field\n\\begin{equation}\\label{eq:KLexp}\na_J({\\bs x},\\omega)=\\Psi_J({\\bs x}){\\bs G}^{-1\/2}_J\\big(\\overline{\\bs a}_J + {\\bs V}{\\bs\\Sigma}{\\bs Y}(\\omega)\\big),\n\\end{equation}\nwhere \\({\\bs Y}\\mathrel{\\mathrel{\\mathop:}=}[Y_1,Y_2,\\ldots, Y_{N_J}]^\\intercal\\) is a standard normally\ndistributed Gaussian vector, \\(\\overline{\\bs a}_J\\) is the orthogonal projection of\n\\(\\mathbb{E}[a]\\) onto \\(V_J\\) and \\({\\bs V}{\\bs\\Sigma}{\\bs\\Sigma}{\\bs V}^\\intercal={\\bs C}_J\\)\nis the spectral decomposition of \\({\\bs C}_J\\). \n\nHence, for the Galerkin projection, we find\nthe identities\n\\[\n\\begin{aligned}\n\\overline{\\bs a}^{G}_J&\\mathrel{\\mathrel{\\mathop:}=} \\big[(\\mathbb{E}[a],\\psi_{\\lambda})_{L^2(D)}\\big]_\\lambda={\\bs G}^{1\/2}_J\n\\overline{\\bs a}_J,\\\\\n{\\bs C}^G_J&\\mathrel{\\mathrel{\\mathop:}=}\\big[(\\Ccal\\psi_{\\lambda'},\\psi_{\\lambda})_{L^2(D)}\\big]_{\\lambda,\\lambda'}\n={\\bs G}^{1\/2}_J{\\bs C}_J{\\bs G}^{1\/2}_J.\n\\end{aligned}\\] \nHence, we obtain for the expectation\n\\[\n\\overline{\\bs a}_J={\\bs G}^{-1\/2}_J\\overline{\\bs a}^{G}_J,\n\\]\nwhile the covariance satisfies\n\\[\n{\\bs C}_J =({\\bs V}{\\bs\\Sigma})({\\bs V}{\\bs\\Sigma})^\\intercal= {\\bs G}^{-1\/2}_J{\\bs C}^G_J{\\bs G}_J^{-1\/2}=({\\bs G}^{-1\/2}_J{\\bs L})({\\bs G}^{-1\/2}_J{\\bs L})^\\intercal,\n\\]\nwhere \\({\\bs L}{\\bs L}^\\intercal={\\bs C}^G_J\\) is the Cholesky decomposition of \\({\\bs C}^G_J\\).\n\nIt is\nwell known that any two matrix square roots\n\\({\\bs R}{\\bs R}^\\intercal=\\widehat{\\bs R}\\widehat{\\bs R}^\\intercal={\\bs C}_J\\)\nonly differ by an isometry. Consequently,\nthere exists an orthogonal matrix \\({\\bs Q}\\)\nsuch that \\({\\bs V}{\\bs\\Sigma}={\\bs G}^{-1\/2}_J{\\bs L}{\\bs Q}\\).\nIntroducing the transformed random vector \n\\(\n{\\bs X}(\\omega)\\mathrel{\\mathrel{\\mathop:}=}{\\bs Q}{\\bs Y}(\\omega)\n\\), we infer\n\\[\n\\int_\\Omega{\\bs X}(\\omega){\\bs X}^\\intercal({\\omega})\\d\\mathbb{P}(\\omega)\n={\\bs Q}\\int_\\Omega{\\bs X}(\\omega){\\bs X}^\\intercal({\\omega})\\d\\mathbb{P}(\\omega){\\bs Q}^\\intercal={\\bs Q}{\\bs I}{\\bs Q}^\\intercal={\\bs I}.\n\\]\nTherefore, \\({\\bs X}(\\omega)\\) is a standard normally\ndistributed Gaussian vector as well and we end up with the representation\n\\begin{equation}\\label{eq:CholFastField}\na_J({\\bs x},\\omega)=\\Psi_J({\\bs x}){\\bs G}^{-1}_J\n\\big(\\overline{\\bs a}^{G}_J + {\\bs L}{\\bs X}(\\omega)\\big).\n\\end{equation}\n\nIf the singular values in the Karhunen-Loeve expansion \\eqref{eq:KLexp} \ndecay only slowly, as it is typically the case for non-smooth\ncovariance functions, the numerical solution of the associated\neigenvalue problem becomes prohibitive. In such cases, the direct \nsimulation of the random field by means of a sparse Cholesky \ndecomposition in \\eqref{eq:CholFastField} is computationally \nsuperior. Therefore, the proposed method is in particular useful for\nrough random fields, which issue from non-smooth covariance \nfunctions. \n\nWe remark that in case of a smooth covariance function, the \nmatrix \\({\\bs C}_J^G\\) becomes numerically positive semidefinite.\nIn such cases, the pivoted Cholesky decomposition, see \\cite{HPS}, \nimmediately yields a low-rank decomposition of \\({\\bs C}_J^G\\) and \nis computationally superior to other data sparse representations,\nwe refer to \\cite{HPS15} for a comprehensive discussion of this \ncomplementary approach.\n\n\\section{Numerical Results}\\label{sec:numerix}\nIn order to obtain consistent computation times,\nall computations reported in this section have been\ncarried out on a single core of a compute server with \nIntel Xeon E5-2650 v3 @2.30GHz CPUs and 512GB DDR4 \n@2133MHz main memory. The implementation of the wavelet matrix\ncompression has been performed in ANSI C, while nested dissection\norderings, Cholesky decompositions and LU decompositions\nhave been computed using \\textsc{Matlab} 2020a, compare \n\\cite{MLB20}.\n\n\\subsection{Polarizable continuum model}\nFor PCM, we consider the assembly and the factorization of the\nmatrices\n\\[\n  {\\bs V}_J\\quad\\text{and}\\quad{\\bs A}_J\\mathrel{\\mathrel{\\mathop:}=}\n  \t\\frac{1+\\epsilon}{2}{\\bs G}_J+(1-\\epsilon) {\\bs K}_J,\n\\]\nwhich are required to set up the linear system\nof equations \\eqref{eq:PCM2}. The dielectric constant is chosen as \\(\\epsilon=78.39\\), \ncorresponding to the solvent water. The molecule under study is \nbenzene, whose solvent excluding surface is depicted in Figure\n\\ref{fig:benzene}. For the wavelet matrix compression, we use \npiecewise constant wavelets with three vanishing moments, \nas developed in \\cite{HS1}. \n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[scale=0.16,clip,trim=290 30 280 110]{benzene}\n\\caption{\\label{fig:benzene}Solvent excluding surface of benzene.}\n\\end{center}\n\\end{figure}\n\n\\begin{table}[htb]\n\\begin{center}\n\\begin{tabular}{|r|r|r|r|r|r|r|}\\hline\n$N_J$ & $t_\\text{WEM}$ &  $t_\\text{ND}$ &   $t_\\text{Chol}({\\bs V}_J)$ &\n$t_\\text{LU}({\\bs A}_J)$ & $\\anz({\\bs V}_J)$  & $\\anz({\\bs L}_J)$ \\\\\\hline\n1456  &   0.96  &  0.032    &  0.19  &  0.22  & 134  &  209\\\\\n5824  &   7.10  &  0.15    &   0.19  &  1.59  & 163  &  328\\\\\n23296  &  52.32  &  0.74  &  1.05  & 14.03  & 159  &  419 \\\\\n93184  &  309.35 &   3.91   &  7.66  & 189.70  & 165  &  639\\\\\n372736  &  2265.91  &  19.38   & 58.28  & 2211.57  & 173  &  890\\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{tab:PCM}Computation times and \nnumbers of nonzero entries in case of PCM.}\n\\end{center}\n\\end{table}\n\nTable~\\ref{tab:PCM} shows the numerical results. The first \ncolumn labelled \\(N_J\\) corresponds to the number of surface\nelements. The second column labelled \\(t_\\text{WEM}\\) contains\nthe combined computation times in seconds for the assembly of \\({\\bs V}_J\\)\nand \\({\\bs A}_J\\) by the wavelet matrix compression. The third column, \nlabelled by \\(t_\\text{ND}\\), provides the computation times in \nseconds for the nested dissection ordering, as \\({\\bs V}_J\\)\nand \\({\\bs A}_J\\) have identical sparsity patterns, the same\nreordering can be applied to both of them. The fourth\ncolumn labelled by \\(t_\\text{Chol}({\\bs V}_J)\\) denotes\nthe computation times in seconds for the Cholesky factorization \nof \\({\\bs V}_J\\), while the fifth column shows the computation \ntimes in seconds of the LU decomposition of \\({\\bs A}_J\\). For \nthe purpose of measuring the fill-in relative to the matrix size, we\nintroduce for a sparse matrix \\({\\bs A}\\in\\mathbb{R}^{N\\times N}\\)\nthe \\emph{average number of nonzeros per row}\n\\[\n\\anz({\\bs A})\\mathrel{\\mathrel{\\mathop:}=} \\frac{\\nnz({\\bs A})}{N}.\n\\]\nThe values for \\(\\anz({\\bs V}_J)\\) and \\(\\anz({\\bs A}_J)\\) are \nidentical and can be found in the sixth row of Table~\\ref{tab:PCM},\nwhile the values \\(\\anz({\\bs L}_J)\\) for the Cholesky factor of \n\\(\\anz({\\bs V}_J)\\) are given in the last column of the table. The \nsparsity pattern of the\nL-factor of the LU decomposition of \\({\\bs A}_J\\) coincides with that\nof ${\\bs L}_J$, while the U-factor has somewhat less coefficients compared \nto the L-factor. Hence, the average number of nonzeros per row for \nthe LU factorization of \\({\\bs A}_J\\) matches that of \\(\\anz({\\bs L}_J)\\). \n\n\\begin{figure}[htb]\n\\begin{center}\n\\setlength{\\fboxrule}{0.2pt}\n\\begin{minipage}{0.31\\textwidth}\n\\begin{center}\n\\includegraphics[scale=0.4,clip,trim=0 0 0 13,frame]{Spattern_PCM}\n$\\nnz({\\bs V}_J) = 15\\,363\\,141$\n\\end{center}\n\\end{minipage}\n\\begin{minipage}{0.31\\textwidth}\n\\begin{center}\n\\includegraphics[scale=0.4,clip,trim=0 0 0 13,frame]{Lpattern_PCM}\n$\\nnz({\\bs L}_J) = 59\\,484\\,864$\n\\end{center}\n\\end{minipage}\n\\begin{minipage}{0.31\\textwidth}\n\\begin{center}\n\\includegraphics[scale=0.4,clip,trim=0 0 0 13,frame]{UKpattern_PCM}\n$\\nnz({\\bs U}_J) = 59\\,295\\,122$\n\\end{center}\n\\end{minipage}\n\\caption{\\label{fig:PCM}Sparsity patterns of \\({\\bs V}_J\\) as well as \\({\\bs A}_J\\) (left), \nthe Cholesky factor \\({\\bs L}_J\\) (middle), and the upper triangular matrix \\({\\bs U}_J\\)\n(right) for PCM and \\(N_J=93\\,184\\). Each dot corresponds to a submatrix of size \n\\(64\\times 64\\). Lighter blocks have less entries than darker blocks. }\n\\end{center}\n\\end{figure}\n\nThe sparsity patterns for \\({\\bs V}_J\\), \\({\\bs L}_J\\) and for the U-factor \nare shown in Figure~\\ref{fig:PCM} for \\(N_J=93\\,184\\). In order \nto obtain a neat representation in this figure, we have merged\nmatrix blocks of size \\(64\\times 64\\). Darker blocks have a higher\ndensity of entries, while lower blocks have a lower density of entries. \n\nAs can clearly be inferred from Table~\\ref{tab:PCM}, the \ntimes for the computing the nested dissection ordering and the \nsubsequent Cholesky decomposition are negligible with respect \nto the wavelet matrix compression. The average number of nonzero \nentries per row stays rather low and only increases for increasing \nnumbers of unknowns up to a factor of approximately 5 for \\(N_J=372\\,736\\).\n\nIt can be seen from the fifth column of Table~\\ref{tab:PCM} that\nthe LU decomposition is significantly slower than the Cholesky\ndecomposition. This issues from the fact that all matrices are \nstored in a sparse column major format, resulting in a large \noverhead for the access of matrix rows. We remark that, in \nprinciple, the LU decomposition could be accelerated by an \nappropriate data structure, which enables direct access \nto the rows of the matrix as well.\n \n\\subsection{Parabolic diffusion problem for the fractional Laplacian}\nWe consider the parabolic problem for the fractional Laplacian with \n\\(s=\\nicefrac{3}{8}\\). The right hand side is a Gaussian heat\nspot moving on a circular trajectory, given by\n\\[\nf({\\bs x},t)=100\\exp\\Big(-40\\big(x_1-\\cos(2\\pi t)\\big)^2-40\\big(x_2-\\sin(2\\pi t)\\big)^2\\Big),\n\\]\nwhile the initial condition is set to 0.\nFor the solution of the ordinary differential equation in time, \nwe employ the \\(\\theta\\)-scheme with \\(\\theta=\\nicefrac{1}{2}\\), which \nyields the Crank-Nicolson method \\cite{CN}. The time interval \nis given by \\([0,T]=[0,3]\\) and we discretize this time interval\nby 150 equidistant time steps. \n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[scale=0.14,clip,trim=510 160 510 160]{disk}\n\\caption{\\label{fig:disc}Computational geometry for the fractional Laplacian.}\n\\end{center}\n\\end{figure}\n\nIn the \\(\\theta\\)-scheme \\eqref{eq:theta}, we need to invert the matrix\n\\[\n{\\bs A}_J={\\bs G}_J + \\theta(t_{i+1}-t_i) {\\bs L}_J,\n\\]\nwhich is assembled with the help of wavelet matrix compression\nusing Haar wavelets. The computational geometry is the unit disc \ndepicted in Figure~\\ref{fig:disc}. The solution on the space-time \ncylinder is visualized for \\(N_J= 20\\,480\\) in Figure~\\ref{fig:solFL}.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\begin{tikzpicture}\n\\draw(0,0) node{\\includegraphics[scale=0.1,clip,trim=1050 120 100 120]{fractionalNew.png}};\n\\fill[color=white] (1.3,-2) rectangle (3,2);\n\\draw(1.45,1.2) node {$14$};\n\\draw(1.45,0.6) node {$10$};\n\\draw(1.45,0.25) node {$\\phantom{0}8$};\n\\draw(1.45,-0.1) node {$\\phantom{0}6$};\n\\draw(1.45,-0.45) node {$\\phantom{0}4$};\n\\draw(1.45,-0.8) node {$\\phantom{0}2$};\n\\draw(1.45,-1.15) node {$\\phantom{0}0$};\n\\draw(2.4,0) node[rotate=90] {Temperature};\n\n\\end{tikzpicture}\n\\caption{\\label{fig:solFL}Solution of the fractional Laplacian on the space-time\ncylinder.}\n\\end{center}\n\\end{figure}\n\n\\begin{table}[htb]\n\\begin{center}\n\\begin{tabular}{|r|r|r|r|r|r|r|}\\hline\n$N_J$ & $t_\\text{WEM}$ &  $t_\\text{ND}$ & $t_\\text{Chol}({\\bs A}_J)$ & \n\\(t_{\\theta}\\)  & $\\anz({\\bs A}_J)$  & $\\anz({\\bs L}_J)$ \\\\\\hline\n   1280 & 0.63       &  0.02   &   0.01  & 0.19     &      88   &  140\\\\\n   5120 &  3.35      &  0.11   &   0.10  & 0.96     &     106   &  222\\\\\n  20480 &  22.20     &  0.53   &   0.59  & 5.34     &     117   &  315\\\\\n  81920 &  161.83    &  2.71   &   3.89  & 27.94    &     129   &  474\\\\\n 327680 &   724.60   & 12.87   &  23.49  & 131.30   &     135   &  576\\\\\n1310720 &  3875.30   & 62.62   & 161.65  & 633.50   &     142   &  743\\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{tab:FLap}Computation times and numbers of \nnonzero entries in case of the fractional Laplacian.}\n\\end{center}\n\\end{table}\n\nWe have tabulated the similar characteristics from the previous \nexample for the matrix \\({\\bs A}_J\\) in Table~\\ref{tab:FLap}, while\nthe sparsity patterns of \\({\\bs A}_J\\) and \\({\\bs L}_J\\) for \\(N_J=81\\,920\\)\nafter reordering are shown in Figure~\\ref{fig:FractionalS}. We observe \na similar behaviour as for PCM, the computation times for nested \ndissection and the Cholesky decomposition are negligible compared \nto the wavelet matrix compression. In view of the fill-in, there is \nan increase of approximately a factor of 5 for the average number \nof nonzero entries per row for \\(N_J=1\\,310\\,720\\). The computation \ntimes for the \\(\\theta\\)-scheme in seconds for the 150 time steps are shown\nin the column labelled by \\(t_{\\theta}\\). As can be seen, we obtain a\nsolution time of roughly 11 minutes for \\(N_J=1\\,310\\,720\\) unknowns\nin the spatial discretization.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\setlength{\\fboxrule}{0.2pt}\n\\begin{minipage}{0.49\\textwidth}\n\\begin{center}\n\\includegraphics[scale=0.55,clip,trim=0 0 0 13,frame]{Spattern_Frac}\n$\\nnz({\\bs A}_J) = 10\\,518\\,832$\n\\end{center}\n\\end{minipage}\n\\begin{minipage}{0.49\\textwidth}\n\\begin{center}\n\\includegraphics[scale=0.55,clip,trim=0 0 0 13,frame]{Lpattern_Frac}\n$\\nnz({\\bs L}_J) = 38\\,819\\,059$\n\\end{center}\n\\end{minipage}\n\\caption{\\label{fig:FractionalS}Sparsity patterns of \\({\\bs A}_J\\) (left) and the Cholesky factor\n\\({\\bs L}_J\\) (right) for the fractional Laplacian and \\(N_J=81\\,920\\). Each dot corresponds to \na submatrix of size \\(64\\times 64\\). Lighter blocks have less entries than darker blocks.}\n\\end{center}\n\\end{figure}\n\n\\subsection{Gaussian random field}\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[scale=0.14,clip,trim= 360 75 360 75]{plate}\n\\caption{\\label{fig:randomFields}Computational geometry for the Gaussian random field.}\n\\end{center}\n\\end{figure}\n\nFor the simulation of a Gaussian random field, we consider an\nL-shaped domain with three holes, compare Figure~\\ref{fig:randomFields}.\nThe domain has a side length of 4, while the holes have a diameter \nof 0.8. For the wavelet matrix compression, we use piecewise bilinear \nbiorthogonal wavelets with four vanishing moments, see \\cite{HS1}.\nThe expectation is set to \\(\\mathbb{E}[a]({\\bs x})\\equiv 0\\), while the\ncovariance is given by the exponential kernel\n\\[\n\\operatorname{Cov}[a]({\\bs x},{\\bs x}')=\\exp(-\\|{\\bs x}-{\\bs x}'\\|_2).\n\\]\nFour different realizations of the corresponding field in case of \n\\(N_J=792\\,688\\) nodes are depicted in Figure~\\ref{fig:FieldReals}.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\begin{tikzpicture}\n\\draw(0,0) node {\\includegraphics[scale=0.11,clip,trim= 380 120 610 60]{field3}};\n\\draw(5,0) node {\\includegraphics[scale=0.11,clip,trim= 380 120 610 60]{field4}};\n\\draw(5,5) node {\\includegraphics[scale=0.11,clip,trim= 380 120 610 60]{field2}};\n\\draw(0,5) node {\\includegraphics[scale=0.11,clip,trim= 380 120 610 60]{field1}};\n\\draw(9.2,2.25) node {\\includegraphics[scale=0.194,clip,trim= 1600 400 50 420]{field1}};\n\\fill[color=white] (8.5,0.5) rectangle (10.8,4);\n\\draw(9,3.8) node {$\\phantom{-}3.8$};\n\\draw(9,2.98) node {$\\phantom{-}2.0$};\n\\draw(9,2.54) node {$\\phantom{-}1.0$};\n\\draw(9,2.1) node {$\\phantom{-}0\\phantom{.0}$};\n\\draw(9,1.65) node {$-1.0$};\n\\draw(9,1.22) node {$-2.0$};\n\\draw(9,0.75) node {$-3.0$};\n\\draw(10.1,2.25) node[rotate=90] {Value};\n\\end{tikzpicture}\n\\caption{\\label{fig:FieldReals}Four different realizations of the Gaussian random field.}\n\\end{center}\n\\end{figure}\n\nThe pattern of the covariance matrix \\({\\bs C}_J^G\\) is shown in \nFigure~\\ref{fig:RandomFieldC}, while the patterns of the reordered \nmatrix and its Cholesky factor are provided in Figure~\\ref{fig:CholRF}.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\setlength{\\fboxrule}{0.2pt}\n\\begin{minipage}{0.49\\textwidth}\n\\begin{center}\n\\includegraphics[scale=0.55,clip,trim=0 0 0 13,frame]{SWpattern_RF}\n$\\nnz({\\bs C}_J^G) = 122\\,684\\,812$\n\\end{center}\n\\end{minipage}\n\\caption{\\label{fig:RandomFieldC}Sparsity patterns of \\({\\bs C}_J^G\\) \nof the random field for \\(N_J=199\\,692\\). Each dot corresponds to a submatrix\nof size \\(108\\times 108\\). Lighter blocks have less entries than darker blocks.}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\begin{center}\n\\setlength{\\fboxrule}{0.2pt}\n\\begin{minipage}{0.49\\textwidth}\n\\begin{center}\n\\includegraphics[scale=0.55,clip,trim=0 0 0 13,frame]{Spattern_RF}\n$\\nnz({\\bs C}_J^G) = 122\\,684\\,812$\n\\end{center}\n\\end{minipage}\n\\begin{minipage}{0.49\\textwidth}\n\\begin{center}\n\\includegraphics[scale=0.55,clip,trim=0 0 0 13,frame]{Lpattern_RF}\n$\\nnz({\\bs L}_J) = 722\\,864\\,637$\n\\end{center}\n\\end{minipage}\n\\caption{\\label{fig:CholRF}Sparsity patterns of \\({\\bs C}_J^G\\) (left) and \nthe Cholesky factor \\({\\bs L}_J\\) (right) for the random field and \\(N_J=199\\,692\\). \nEach dot corresponds to a submatrix of size \\(108\\times 108\\). Lighter blocks \nhave less entries than darker blocks.}\n\\end{center}\n\\end{figure}\n\nTable~\\ref{tab:RF} shows the computation times and the average \nnumbers of nonzero entries as in the previous examples. In addition,\nwe have the column \\(t_{\\text{Sample}}\\), which contains the times for \ncomputing a single realization of the random field in seconds. These \ntimes have been computed by averaging the computation times over 1000 samples.\nIn order to compute the inverse of the mass matrix, we could in\nprinciple reuse the nested dissection ordering which has been computed for\nthe system matrix, as the pattern of the mass matrix is a subset of the \npattern of the system matrix. However, this will result in a fill-in similar \nto the system matrix. Hence, it is favourable to use a different ordering \nfor the mass matrix, resulting in much less fill-in.\n\n\\begin{table}[htb]\n\\begin{center}\n\\scalebox{0.93}{\n\\begin{tabular}{|r|r|r|r|r|r|r|r|r|}\\hline\n$N_J$ & $t_\\text{WEM}$ &  $t_\\text{ND}$ &   $t_\\text{Chol}({\\bs C}_J^G)$ &\n$t_\\text{Sample}$ & $\\anz({\\bs C}_J^G)$  & $\\anz({\\bs L}_J)$ &$\\anz({\\bs G}_J)$  &\n$\\anz(\\widehat{\\bs L}_J)$\\\\\\hline\n    972 &  1.50 & 0.04 &  0.23 &$6.25\\!\\cdot\\!10^{-4}$  &      249  &     251\n&  53 & 31\\\\ \n   3468 & 14.49 & 0.16 &  0.26 &$3.97\\!\\cdot\\!10^{-3}$  &      338  &     498\n&  91 & 72\\\\\n  13068 &  119.44 & 0.83 &  2.22 &$3.13\\!\\cdot\\!10^{-2}$  &      429  &    1096\n& 126 &124\\\\\n  50700 & 777.69 & 4.45 & 21.45 &$2.18\\!\\cdot\\!10^{-1} $  &     517   &   1998\n&156  &196\\\\\n 199692 &  4368.40 &24.34 & 222.72 &  $1.51\\!\\cdot\\!10^{0\\phantom{-}} $  &     615   &   3620\n&184  &283\\\\\n792588 &   29376.29 &   130.23 & 4290.29 & $1.61\\!\\cdot\\!10^{1\\phantom{-}}$&   714 &   8336 &   211 &   390\\\\\n\\hline\n\\end{tabular}}\n\\caption{\\label{tab:RF}Computation times and numbers of \nnonzero entries in case of the Gaussian random field.}\n\\end{center}\n\\end{table}\n\nAs can be seen from Table~\\ref{tab:RF}, the times for sampling \nthe random field only increase moderately. We remark that, due to the \nlarger supports of the bilinear wavelets and the higher precision of the \ndiscretization, the system matrix contains more entries per row on average.\nThis also leads to higher computation times for the matrix assembly.\nHowever, as before the increase of nonzero entries in the Cholesky factor \nremains very moderate. In addition, we have provided the average number \nof nonzeros per row for the mass matrix in the column labelled by \n$\\anz({\\bs G}_J)$. The number of nonzeros for the corresponding \nCholesky factor of the mass matrix is given in the column labelled \nby $\\anz(\\widehat{\\bs L}_J)$.\n\n\\section{Conclusion}\\label{sec:conclusion}\nIn this article, we have proposed a very efficient direct\nsolver for nonlocal operators. The pivotal idea is to\ncombine the wavelet matrix compression with the nested dissection\nordering. Thereby, the fill-in resulting from a Cholesky\ndecomposition or an LU decomposition is drastically reduced. \nThis has been numerically investigated into depth for three \nrelevant applications, namely the polarizable continuum model, \na parabolic problem for the fractional Laplacian in integral \nform, and the fast simulation of Gaussian random fields.\nIn all three cases, wavelet matrix compression yields sparse \nsystem matrices while the fill-in in the matrix factorization \nremains very low thanks to the nested dissection ordering. \nThis behaviour could be observed for more then \\(10^6\\) \nunknowns in the discretization. \n\nA formidable application of the presented approach is the\nfast simulation of rough (Gaussian) random fields. In such \ncases, the numerical solution of the eigenvalue problem for \nthe covariance is computationally prohibitive and, hence, the \ncomputation of a Karhunen-Lo\\`eve expansion is not feasible. \nIn turn, the use of a wavelet basis yields a sparse representation \nof the covariance operator and a matrix root is rapidly computable \nby employing the nested dissection ordering and the Cholesky \ndecomposition.\n\n\\subsection*{Acknowledgement}\nThis research has been funded in parts by the Swiss National Science \nFoundation (SNF) under project number $407540\\_167186$ NFP 75 Big Data.\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\label{sec_intro}\n\nDirect imaging of exoplanets is a field of research in rapid development. The past year has seen a number of interesting planet candidates imaged directly around stars. Most notable is arguably HR 8799, showing three planetary companions (see Marois et al. 2008) so far. These planets have been shown to exhibit Keplerian motion around the star, and have estimated masses in the range of 7-10 $M_{\\rm jup}$ from theoretical models (Baraffe et a. 2003). The system is known to also host a debris disk (e.g. Moor et al. 2006; Rhee et al. 2007). Along with the fact that there are three almost equal-mass companions, all orbiting the star in what appears to be a co-planar manner, this implies that the objects most likely formed in a circumstellar disk, by a process distinct from the star or brown dwarf formation process. This can be contrasted with the case of the 2M1207 system (Chauvin et al. 2005), where the low primary-to-secondary mass ratio is more reminiscent of a brown dwarf binary system than a star-planet system. Two other intriguing planet candidates in systems with debris disks were reported around the same time as HR 8799 b\/c\/d: Fomalhaut b (Kalas et al. 2008) and Beta Pic b (Lagrange et al. 2009), though additional follow-up observations would be desirable to provide more information on these systems. \n\n$\\epsilon$ Ind A is a K4V-type southern sky star located at a distance of 3.6 pc, with a very high proper motion of 4.7$''$ yr$^{-1}$ (Perryman et al. 1997), see Table \\ref{indtab0}. Comoving on the sky at the same rate, at a separation of 6.7$'$, is $\\epsilon$ Ind B, which was first detected by Scholz et al. (2003), and shortly thereafter resolved into the brown dwarf binary $\\epsilon$ Ind Ba\/Bb (McCaughrean et al. 2004) as it is known today. Being the most nearby binary brown dwarf, and with a physical separation small enough to determine its dynamical mass within a reasonable timeframe, $\\epsilon$ Ind Ba\/Bb will be a benchmark object for the physical understanding of brown dwarfs. However, in addition to the Ba\/Bb components, the $\\epsilon$ Ind system may provide an additional possibility to study an even lower-mass and cooler object. $\\epsilon$ Ind A displays a linear radial velocity trend (Endl et al. 2002; Zechmeister et al., in prep.) which, unless due to some exotic stellar remnant, is indicative of a giant planet or very low-mass brown dwarf companion. If this object could be directly imaged, it would constitute yet another important benchmark object, given its probable low mass, low temperature, and the possibility to estimate both its luminosity as well as its dynamical mass within a reasonable timeframe. It would also be the closest planet or very low-mass brown dwarf companion directly detected outside of our Solar System.\n\n\\begin{table}\n   \\centering\n\\caption[]{Summary of $\\epsilon$ Ind A properties.}\n         \\label{indtab0}\n\\begin{tabular}{ll}\n            \\noalign{\\smallskip}\n            \\hline\n            \\noalign{\\smallskip}\n                 Property & Value \\\\\n            \\noalign{\\smallskip}\n            \\hline\n            \\noalign{\\smallskip}\nRight Ascension  & 22 03 21.66 \\\\\nDeclination   & -56 47 09.5 \\\\\nSpectral Type  & K4.5V \\\\\nDistance & $3.626 \\pm 0.009$ pc \\\\\nProper Motion  & 4705 mas yr$^{-1}$  \\\\\nMass & 0.7 $M_{\\rm sun}$  \\\\\nAge & 1-5 Gyr \\\\\n            \\noalign{\\smallskip}\n            \\hline\n\\end{tabular}\n\\end{table}\n\nHere, we will present our deep imaging campaign of $\\epsilon$ Ind, using narrow-band 4$\\mu$m imaging with two different high-contrast techniques. One technique is pure active angular differential imaging (aADI) as already implemented and demonstrated as a powerful technique for detecting close companions to bright nearby stars (Janson et al. 2008). In aADI (also known as roll subtraction), images are taken at two different instrument rotator angles, and one is subtracted from the other, thereby removing the bulk of the stellar PSF including static instrumental speckles. The other technique is a combination of PSF reference subtraction and aADI (PSFR+aADI), using the LOCI (locally optimized combination of images) algorithm developed by Lafreniere et al. (2007). We also compare the performance of aADI at 4$\\mu$m with the same technique in the L'-band. The concept of using aADI with L' was proven by Kasper et al. (2007; the usefulness of L' for high-contrast imaging purposes was also independently demonstrated by Hinz et al. 2006). The concept of using NB4.05 to enhance the physical contrast was introduced in Janson et al. (2008).\n\nThe outline is as follows: In Sect. \\ref{sec_obs}, we summarize the observational parameters and ambient conditions of the observing runs. The two paths of data reduction employed are described in Sect. \\ref{sec_datared}. This is followed by a presentation of the results and the associated analysis in Sect. \\ref{sec_results}, including a comparison between filters (Sect. \\ref{sec_filters}), a comparison between techniques (Sect. \\ref{sec_methods}), a discussion of what we can learn from the dynamical input in Sect. \\ref{sec_dynamics}, and the final detection limits and their interpretation in Sect. \\ref{sec_detlimits}. Finally, we conclude in Sect. \\ref{sec_conclusions}.\n\n\\section{Observations}\n\n\\label{sec_obs}\n\nThe data presented here are based on two different sets of VLT\/NACO observations of $\\epsilon$ Ind. One set of observations consisted of deep imaging with aADI in the NB4.05 filter, executed in service mode and split into two equal observing blocks (henceforth observations A1 and A2), on 31 Oct 2008 and 2 Nov 2008. The other set of observations were taken in visitor mode on 3 Jul 2008, as part of a larger survey searching for planets around a volume-limited sample of nearby stars (Apai et al., in prep.). Those observations were less deep, but consisted of aADI imaging in both the NB4.05 and broad-band L' filters (henceforth observations B1 and B2, respectively). For observations A1 and A2, the same strategy was used as in previous observations (see Janson et al. 2008): Jittering was applied to enable a good subtraction of the thermal background, and the aADI was performed at two different instrument rotator angles, using a differential angle of 33 degrees. For B1 and B2, a four-point large-throw dithering scheme was applied for the background subtraction purposes, and the differential angle used for aADI was 20 degrees. All observations were taken with the L27 objective, providing a pixel scale of 27 mas\/pixel, and a field of view of 28$''$ by 28$\"$. The weather conditions and observational parameters of each run are listed in Table \\ref{indtab1}.\n\n\\begin{table}\n   \\centering\n\\caption[]{Observational conditions for the four runs.}\n         \\label{indtab1}\n\\begin{tabular}{lllll}\n            \\noalign{\\smallskip}\n            \\hline\n            \\noalign{\\smallskip}\n                 ~ & A1   & A2   & B1 & B2\\\\\n            \\noalign{\\smallskip}\n            \\hline\n            \\noalign{\\smallskip}\nDate (2008)         & 31 Oct & 2 Nov & 3 Jul & 3 Jul \\\\\nFilter          & NB4.05 & NB4.05 & NB4.05 & L'  \\\\\nSeeing$^a$    & 0.9$''$    & 0.7$''$    & 1.1$''$ & 1.0$''$  \\\\\nStrehl$^b$    & 83\\%      & 79\\%      & 85\\% & 83\\% \\\\\nHumidity      & 8\\%         & 13\\%        & 3\\% & 3\\%  \\\\\nCoh. time$^a$ & 2.6 ms      & 3.4 ms      & 2.3 ms & 2.4 ms \\\\\nFrames             & 19          & 19$^c$          & 30 & 33 \\\\\n(per angle)        &      ~      &     ~       &       ~  & ~  \\\\\nDIT                & 1.0 s       & 1.0 s       & 0.2 s & 0.2 s \\\\\nNDIT               & 61          & 61          & 150  & 150  \\\\\nTot. time          & 1159 s      & 1159 s$^c$      & 900 s & 990 s  \\\\\n(per angle)        &      ~      &     ~       &       ~  & ~  \\\\\n\n            \\noalign{\\smallskip}\n            \\hline\n\\end{tabular}\n\\begin{list}{}{}\n\\item[$^{\\mathrm{a}}$] Values given by the atmospheric seeing monitor at a wavelength of 500 nm.\n\\item[$^{\\mathrm{b}}$] Strehl ratio given by the AO system, rescaled to the observing wavelengths.\n\\item[$^{\\mathrm{c}}$] Two frames were de-selected for 0$^{\\rm o}$, hence the effective time is 1037 s for that case.\n\\end{list}\n\\end{table}\n\n\\section{Data reduction}\n\n\\label{sec_datared}\n\nThe data reduction for A1 and A2 was done differently with respect to the two different techniques applied, hence separate descriptions are provided below. The reduction of B1 and B2 was only done with aADI.\n\n\\subsection{Reduction for the purpose of aADI}\n\n\\label{sec_aadired}\n\nSince observations A1 and A2 were taken in the same way as the observations of Janson et al. (2008), largely the same data reduction could be applied as in the case for aADI purposes: The most basic reduction steps (e.g. flat fielding, bad pixel removal, background subtraction) were provided by the ESO automatic pipeline. The subsequent steps were performed with our dedicated IDL pipeline: The images were shifted using bilinear interpolation to a common center determined through cross-correlation, and to an absolute center using center of gravity. Low-frequency filtering was applied by subtracting a smoothed counterpart of each image produced by convolution with a Gaussian kernel with 0.5$''$ FWHM (the FWHM of the stellar PSF was about 120 mas). All the images corresponding to different rotation angles were subtracted from each other, and the results from the two nights were coadded. As an alternative analysis, all images were also de-rotated back to a common angle and coadded. This procedure yields a single co-added signature of any companion, which provides a useful alternative way to look at the data in the background-limited regime, with respect to the aADI-subtracted data which instead produces two independent signatures of half the amplitude each.\n\nFor observations B1 and B2, the data reduction was performed with the IDL routines as described above, with the exception that the centering was determined based on a sub-frame of 200x200 pixels around the star instead of the whole frame. This was done in order to avoid influence from residual features from the sky subtraction and the different dithering scheme.\n\nThe residuals as function of separation from the star were calculated from the standard deviation of all pixels in an annulus corresponding to each separation step. The physical brightness contrast was derived from the 3$\\sigma$ residuals by division of the peak value of the primary. Since the primary was saturated in the science images (the saturation radius was about 5-6 pixels), this had to be calibrated. For runs A1 and A2, this was done by introducing a neutral density filter into the optical path during acquisition, thus getting non-saturated images of the primary, allowing to determine a renormalized peak value. For run B1, the same non-saturated images as for A1 and A2 were used, which could be done since the Strehl ratio was stable and almost equal between the epochs. For B2, an image was taken of a fainter photometric standard star during the night.\n\n\\subsection{Reduction for the purpose of PSFR+aADI}\n\n\\label{sec_psfrred}\n\nIn applications involving LOCI, it is preferable to maximize the number of PSF representations of a target or reference star, hence for this case, reduction was done on individual frames, with the combination of frames only performed at the very end. The PSF star used was $\\epsilon$ Eri, which is practically ideal for the purpose given the similar spectral type, brightness, and the fact that observations exist taken under almost identical circumstances. All individual target and reference star frames were manually subjected to flat fielding, dark subtraction and bad pixel correction using calibration frames provided by ESO. Low-frequency filtering was applied as described above. A master sky frame was then produced by taking the median of the frames, where the stellar image is randomly placed in each frame, thereby removing the star altogether. The individual frames revealed ring-like structures in the background that could be reproduced in the master sky frame. By subtracting the master sky frame from each individual frame, the pattern could be removed. The pattern was found to be constant during the extent of an observation, but variable between observations (e.g., the frames corresponding to $\\epsilon$ Ind and $\\epsilon$ Eri were different from each other), and is probably related to instrumental dust emitting at 4$\\mu$m. The full background subtraction obtained in this way was found to be equally good as that delivered by the ESO 'jitter' routine. It is interesting to note, that given the fact that the pattern is constant during an observation, it should be possible to calibrate it out of a generic observation by making a master sky observation directly before or after the target observation. Hence, for any observation dedicated to the detection of point sources, it should be possible to achieve the same degree of background subtraction with and without jittering. This is an important realization with respect to high-contrast imaging at these wavelengths using techniques such as passive ADI (pADI) or coronagraphy, where it is desirable to maintain the stellar primary at a fixed position on the detector, and simultaneously achieve the best possible background subtraction. In summary, there appears to be no conflict between these two requirements, as long as the master sky calibration step is performed during observations.\n\nPSFR and aADI were performed separately, in sequence. For PSFR, every target and reference image was de-rotated such that the spider patterns were aligned. For each target frame, an optimized PSF reference frame was then produced from the full set of reference frames using the LOCI (Lafreniere et al. 2007) algorithm and subtracted from the target frame. The optimization was performed in 10 regions, five for the image range contaminated by the four spiders, covering different radial sections of the PSF, and five for the image range not contaminated by spiders, also covering different radial sections. The spider optimization areas were rectangular with a fixed width of 25 pixels, inner radii of 10, 20, 40, 70, and 120 pixels, and outer radii of 60, 70, 90, 150, and 200 pixels. The remaining areas were annuli excluding the spider regions, between inner radii of 20, 30 40, 50, and 60 pixels and outer radii of 50, 60, 70, 80, and 100 pixels. The subtractions were performed sequentially outwards with the subtraction zone defined from the inner radius of the optimization zone and outwards. Following this procedure, each of the target frames were re-rotated to their true parallactic angle. The aADI step was then performed through a second LOCI PSF construction, using all 33$^{\\rm o}$ frames as PSF library for each 0$^{\\rm o}$ frame, and vice versa. For this case, the optimization regions were simply five annuli between inner radii of 10, 20, 30, 50, and 70 pixels and outer radii of 40, 50, 60, 70 and 100 pixels. The optimization regions were chosen to provide a good balance between the two main criteria of LOCI: to maximize the efficiency of stellar PSF structure subtraction, and minimize subtraction of actual companions. The latter was tested by generating a series of runs where false companions had been introduced in the target frame -- in total 3600 companions distributed between 10 and 100 pixels separation from the center of the star, and over all azimuthal angles. The partial subtractions in each case were used to construct a radial profile of conserved companion flux fractions. As expected, a significant flux loss occurs at 10 pixels, but decreases rapidly outwards. At 100 pixels, the fraction of restored companion flux approaches unity, as indeed expected, given that the LOCI optimization is not applied beyond 100 pixels for the vast majority of the image space. \n\nFinally, all frames corresponding to each rotator angle were combined using 3$\\sigma$-clipping. The radial profile of residuals was created in the same way as for aADI, but with the additional step that it was normalized by the radial profile of conserved companion flux fraction to provide an accurate measure of the actual achieved contrast.\n\n\\section{Results and discussion}\n\n\\label{sec_results}\n\nThe output images from runs A1+A2 from each of the two reduction paths are shown in Fig. \\ref{ei_imgs_mrg}. No companion candidates were detected in the images. In the following, we discuss the implications of this result, and compare the methods used.\n\n   \\begin{figure*\n   \\centering\n  \n   \\includegraphics[width=15cm]{mj_fig1.eps}\n\\caption{Final output images for aADI (top) and PSFR+aADI (bottom) to the same flux scale, for the images A1+A2. The most obvious difference between the methods is in the treatment of the spiders.}\n\\label{ei_imgs_mrg}\n    \\end{figure*}\n\n\\subsection{Comparison between filters}\n\n\\label{sec_filters}\n\nAlthough the B1 and B2 images are less deep than A1+A2, the fact that they were obtained for the same target at about the same time, and with an almost identical observational setup, makes them ideal for comparing L' and NB4.05 imaging for planet detection purposes around bright stars. A comparison was already made in Janson et al. (2008) between L' aADI, NB4.05 aADI, and SDI+aADI (from Janson et al. 2007). While a fully relevant comparison could be made between SDI+aADI and NB4.05 aADI, where NB4.05 aADI was found to perform better under all circumstances, the comparison with L' was preliminary, since no comparable data was available. Instead, the comparison between L' and NB4.05 was based entirely on physical contrast given by the theoretical models, and the instrumental contrast was assumed to be the same. While this is relevant for a large part of the parameter space, there will in reality be differences in instrumental contrast due to differences in Strehl ratio, PSF diffraction, and thermal background between the filters. Using the B1 and B2 observations, we can now provide a comparison that takes all these issues into account.\n\nThe comparison was done by translating the brightness contrasts into mass detection limits using the spectral and photometric evolutionary models of Baraffe et al. (2003) and Burrows et al. (2003) for various ages. The method is described in detail in Janson et al. (2008). Note that the comparison is done for almost identical observing time, and with virtually no difference in overheads, i.e. the telescope time investment is also the same in both cases. As expected, the instrumental contrast is almost identical in the contrast-linited range, confirming the assumptions of the previous analysis, and thus the difference in the inner range is almost entirely set by the expected flux distribution of the companion. We show an example that demonstrates the favourable spectral range of NB4.05 in Fig. \\ref{xmpl_spec_sm}, for 10$M_{\\rm jup}$ and 15$M_{\\rm jup}$ objects, at an age of 1 Gyr. The flux density is higher in NB4.05 than in both L'-band and M-band. For cooler objects, the bulk of the flux moves redward, hence M-band becomes better in terms of flux density, but the thermal background is also much worse in M-band. The improvement of NB4.05 over L'  increases further for cooler objects.\n\n   \\begin{figure\n   \\centering\n  \n   \\includegraphics[width=8.0cm]{mj_fig2.eps}\n\\caption{Example of two model spectra from Burrows et al. (2003), and the corresponding flux densities in filters L', NB4.05, and M. Upper lines: A 15$M_{\\rm jup}$ object. Lower lines: A 10$M_{\\rm jup}$ object. The age is 1 Gyr in both cases.}\n\\label{xmpl_spec_sm}\n    \\end{figure}\n\nWe show the results of the detection limit comparison for 1 Gyr, 3 Gyr, and 5 Gyr in Fig. \\ref{brg_vs_lband}. It is seen that for all these ages, NB4.05 performs better in the contrast-limited inner part, and L' performs better in the outer background-limited part, as expected. The crossover point for $\\epsilon$ Ind A in our dataset is at about 4$''$. The position of the crossover point will vary as a function of stellar brightness and integration time. The brighter the star and the longer the integration time, the larger the parameter range where NB4.05 will be favourable,  and vice versa. We conclude that NB4.05 is likely to be an excellent choice for very deep planet search imaging close to bright stars, although it should be noted that this depends on the validity of the theoretical models. A first test of the models could be provided by the HR 8799 system.\n\n   \\begin{figure\n   \\centering\n  \n   \\includegraphics[width=8.0cm]{mj_fig3.eps}\n\\caption{Comparison between L' (dashed line) and NB4.05 (solid line) imaging for high-contrast purposes, for 1 Gyr (top panel), 3 Gyr, (middle panel), and 5 Gyr (bottom panel). As expected, NB4.05 provides a better performance than L' in the inner image range, and the opposite is true in the outer range. The comparison is based on sets B1 and B2, note that the A1+A2 detection limits are better.}\n\\label{brg_vs_lband}\n    \\end{figure}\n\n\\subsection{Comparison of aADI and PSFR+aADI}\n\n\\label{sec_methods}\n\nAs can be seen in the images (Fig. \\ref{ei_imgs_mrg}), the main difference between aADI and PSFR+aADI is that spiders are more efficiently removed in the latter case. However, the impact of this is largely cosmetic, as a comparable amount of flux is lost from the companion in the spider regions. This can be clearly seen in a comparison of the respective contrast curves for the two methods (see Fig. \\ref{comp_aadi_loci}), which have been normalized with respect to flux losses. PSFR+aADI slightly improves the performance at large separations, but provides no improvement at all for small separations. It should be noted that the results are based on a single PSF reference star (though with multiple representations) -- it would be preferable to use multiple reference stars, and doing so might substantially improve the performance. In any case, we do not reach as promising results as those achieved with PSFR using LOCI on space-based HST data (see Lafreniere et al. 2009), where a significant improvement over aADI is readily seen. As we have demonstrated, 4 $\\mu$m imaging provides a very high Strehl ratio, so if this was the limiting PSF stability factor at this level of contrast, we should have expected an improvement in the inner image range. Hence, the results imply that other PSF effects become dominant once the Strehl ratio is high enough, such as low-order aberrations arising in the telescope, and differences in PSF representation resulting from dithering. This in turn implies that a stable telescope configuration is the best way forward for improving the contrast in 4 $\\mu$m imaging even further. There is an obvious and well-tested technique for achieving this, called passive ADI, in which the pupil is stabilized during observations, while the field is allowed to rotate (see Marois et al. 2006). Indeed, the LOCI algorithm was originally designed for this purpose (Lafreniere et al. 2007). In fact, we have a passive ADI sequence at 4 $\\mu$m showing exquisite performance at small separations, but those data are taken with a different telescope and of a different target, so a rigorous comparison can not be made. The passive ADI data will be part of a separate publication.\n\n   \\begin{figure\n   \\centering\n  \n   \\includegraphics[width=8.0cm]{mj_fig4.eps}\n\\caption{Comparison of contrast for aADI (solid line) and PSRF+aADI (dashed line). The performance is generally very similar.}\n\\label{comp_aadi_loci}\n    \\end{figure}\n\n\\subsection{Input from dynamics}\n\n\\label{sec_dynamics}\n\nThere exist extensive radial velocity measurements of $\\epsilon$ Ind A, as well as some limited astrometric information, which can be used to constrain the properties of any sufficiently massive companion, as discussed in the following.\n\n\\subsubsection{The radial velocity linear trend}\n\n\\label{sec_rvtrend}\n\nThe linear radial velocity trend of $\\epsilon$ Ind A was first reported by Endl et al. (2002). The original dataset covered an observational baseline of about 5.2 years, taken with the ESO CES instrument in the period 1992-1998. Since then, HARPS data have been taken from 2003 to 2008 (Zechmeister et al., in prep.). The linear trend of the HARPS data is consistent with that of the aforementioned CES data, with a slope of 4.4 m s$^{-1}$ yr$^{-1}$. Hence, we adopt this slope over the 16 year total baseline. One contributor to the linear trend is secular acceleration. This is the apparent acceleration that an observer measures over time in projected motion (in this case along the line of site) of an object with constant velocity, due to the actual motion in 3D space. Using all the measured spatial coordinates and velocity components of $\\epsilon$ Ind, the secular acceleration can be calculated to 1.8 m s$^{-1}$ yr$^{-1}$. This is quite large, due to the fast motion of $\\epsilon$ Ind in the plane of the sky, but still leaves a 2.6 m s$^{-1}$ yr$^{-1}$ trend that must be due to actual acceleration.\n\nSince $\\epsilon$ Ind Ba\/Bb is known to be physically bound to $\\epsilon$ Ind A, it needs to be tested whether it could be responsible for the observed trend. We do this with the following order-of-magnitude estimate: The projected separation between A and Ba\/Bb is about 1500 AU, hence for masses of 0.7$M_{\\rm sun}$, 0.047$M_{\\rm sun}$, and 0.028$M_{\\rm sun}$ respectively (see McCaughrean et al. 2004), the orbital period of the A\/B system is at least 66000 years for a circular Keplerian orbit. Such an orbit would lead to a radial velocity semi-amplitude for $\\epsilon$ Ind A of 62 m s$^{-1}$, which in turn gives an average peak-to-peak acceleration of $4*10^{-3}$ m s$^{-1}$ yr$^{-1}$. Thus, the gravitational influence of $\\epsilon$ Ind Ba\/Bb is several orders of magnitude too small to make any significant contribution to the observed trend.\n\nWith $\\epsilon$ Ind Ba\/Bb out of the picture, we are left with closer, as of yet unseen companions. A previous H- and K-band imaging campaign (Gei{\\ss}ler et al. 2007) has excluded the presence of stellar and massive brown dwarf companions, down to 53 $M_{\\rm jup}$ outside of a projected separation of 1.5 AU and 21 $M_{\\rm jup}$ outside of 4.7 AU. This also excludes white dwarfs, since at ages up to several Gyrs, they are much brighter in H-and K-band than a 50$M_{\\rm jup}$ object (see e.g. Holberg \\& Bergeron 2006 and Baraffe et al. 2003). Stellar objects outside of the field of view can be excluded, as they would be detectable with wide-field or all-sky surveys such as 2MASS (Skrutskie et al. 2006). While more exotic forms of stellar remnants (e.g. neutron stars) can perhaps not be categorically excluded, for the remainder of this paper we will assume that the observed acceleration is due to a low-mass brown dwarf or giant planet. The combined constraints from the imaging and the radial velocity trend are given in Sect. \\ref{sec_detlimits}.\n\n\\subsubsection{Astrometry}\n\n\\label{sec_astrometry}\n\nAs will be seen in the following, the companion is expected to have a mass in the range of $\\sim$5-20$M_{\\rm jup}$, and an orbital semi-major axis in the range of $\\sim$10-20 AU. At the distance of the  $\\epsilon$ Ind system, this corresponds to a strong astrometric amplitude signature imposed on the primary of about 15-60 mas. However, with an orbital period of a few decades, it would not be possible to detect orbital motion with, e.g., \\textit{Hipparcos} data alone. On the other hand, one might expect a systematic difference between the proper motion as measured by \\textit{Hipparcos} versus that measured in long-term ground based monitoring, such as from the Fifth Fundamental Catalog (FK5). This type of signature is referred to as $\\Delta \\mu$ binarity, see Wielen et al. (2001). For $\\epsilon$ Ind, an approximate conversion between the FK5 and HIP systems implies that there is a difference between the \\textit{Hipparcos} and FK5 proper motions of $\\Delta \\mu _{\\alpha} = -0.23 \\pm 1.68$ mas yr$^{-1}$, and $\\Delta \\mu _{\\delta} = -2.5 \\pm 0.98$ mas yr$^{-1}$. This corresponds in total to a significance level of $F=2.54$, where the $F$ value is roughly to the same level of confidence as the equivalent $\\sigma$-number for Gaussian statistics (Wielen et al. 2001). In other words, there is an indication of a companion in the data, but not at a very high level of significance. We can make an order-of-magnitude estimation of whether these numbers are consistent with the RV companion by assuming that the orbital motion is completely averaged out in the FK5 data, that the orbit is circular, and that a sufficiently small fraction of the orbit is covered by \\textit{Hipparcos} such that local curvature in the motion during that period is negligible. The limiting cases quoted above then yield astrometric motions of $\\pi*15$ mas in 32 years and $\\pi*60$ mas in 89 years respectively, i.e. 1.5 mas yr$^{-1}$ and 2.0 mas yr$^{-1}$, both of which are consistent with the given $\\Delta \\mu$ within the errors. Hence, the astrometry is indeed consistent with the RV trend, though we reiterate that the significance is rather limited for the astrometry.\n\n\\subsection{Detection limits}\n\n\\label{sec_detlimits}\n\nSince A1+A2 are the deepest images, they provide the strongest detection limits, and therefore we concentrate on them in this section. The instrumental contrast for A1+A2 is determined at each separation as the maximum performance out of the aADI and PSFR+aADI contrasts at that separation. The corresponding mass limits for A1+A2, calculated in the same way as for B1 and B2 in section \\ref{sec_filters}, are shown in Fig. \\ref{mlim_rv} for ages of 1, 3, and 5 Gyr. Several age determinations exist pointing to an age in the range of 1 Gyr for $\\epsilon$ Ind (e.g. Lachaume et al. 1999; Barnes 2007). However, preliminary analysis of the astrometric masses of $\\epsilon$ Ind Ba\/Bb (Cardoso et al. 2008) implies that the components are probably under-luminous with respect to model predictions (Baraffe et al. 2003) at 1 Gyr, such that the $\\epsilon$ Ind system has to be significantly older, perhaps up to 5 Gyr, if the models are accurate (which may not be the case, see e.g. Dupuy et al. 2009). On the other hand, such an old age would be incompatible with the observed spectra of $\\epsilon$ Ind Ba and Bb according to the analysis of Kasper et al. (2009). It is with these uncertainties in mind that we consider the full range of 1 to 5 Gyr in our analysis.\n\n   \\begin{figure\n   \\centering\n  \n   \\includegraphics[width=8.0cm]{mj_fig5.eps}\n\\caption{Detection limits at 1, 3, and 5 Gyr for $\\epsilon$ Ind A. The solid line that increases outwards is the mass as function of semi-major axis corresponding to the 2.6 m s$^{-1}$ yr$^{-1}$ slope of the observed RV trend, and the dashed line is the reference for minimum projected separation at a typical inclination of 60$^{\\rm o}$, both under the assumption of a circular orbit. The dotted vertical line is the minimum semi-major axis from the RV baseline at $q=2$.}\n\\label{mlim_rv}\n    \\end{figure}\n\nAlso plotted is the mass as function of semi-major axis derived from the slope of the linear trend, under the assumption that the inclination is $60^{\\rm o}$ (the mean inclination of randomly oriented orbits). The minimum possible semi-major axis is set by the minimum possible period, which in turn is some multiple $q$ of the observational baseline. The exact value of $q$ depends on the amount of curvature present in the trend, the determination of which would be an over-interpretation of the data at hand. As discussed in Janson et al. (2008), $q=1$ would be the most conservative limit possible to set, but it is unrealistic, since it would require a discrete change in velocity state. Here, we set $q=2$, which is still conservative, and more realistic.\n\nThe mass limits and RV trend shown in Fig. \\ref{mlim_rv} provide a good illustration of the detectability of the dynamical companion under the assumption of a circular orbit. However, given the large eccentricity spread of the exoplanet population outside of 0.1 AU, it is necessary to perform more detailed simulations in order to constrain the possible physical and orbital parameters of the companion. The method for doing so is described in detail in Janson et al. (2008), and we follow it here for $q=2$. In brief, based on the empirical distribution of eccentricities for known exoplanets outside of 0.1 AU, we simulate all possible orbits and orbital phases and test whether they are consistent with the observed linear trend. The fraction of such orbits as function of semi-major axis is named $\\phi$. Out of these allowed orbits, we test what fraction would lead to a detectable companion. This fraction as a function of semi-major axis is termed $\\chi$. One addition has been made to this procedure with respect to what was presented in Janson et al. (2008): In the case of $\\epsilon$ Eri, the plane of the disk, the rotational plane of the star, and the orbital plane of the planet candidate $\\epsilon$ Eri b all gave a consistent orbital inclination of about 30$^{\\rm o}$, hence this number was fixed in the simulations. In the case of $\\epsilon$ Ind, we have no prior information of the inclination, hence it is treated as a free parameter in the simulations. This is done by performing the simulations over several different inclination angles and averaging the results. The input inclination angles are set to correspond to the actual probability of a given inclination occurring (i.e., accurately taking into account that the inclination is more likely to be edge-on than face-on). \n\n   \\begin{figure\n   \\centering\n  \n   \\includegraphics[width=8.0cm]{mj_fig6.eps}\n\\caption{Detection probability in our images as function of semi-major axis for 1, 3, and 5 Gyr. Also plotted is $\\phi$, the fraction of orbits at a given semi-major axis that are consistent with the linear RV trend, denoted with the subscript \"All\" to signify that it is independent of age, in contrast to $\\chi$.}\n\\label{probfig_q2}\n    \\end{figure}\n\nThe results of the simulation are shown in Fig. \\ref{probfig_q2}. It can be seen that if the age is 1 Gyr, the probability of detecting the companion is always about 90\\% or higher for any semi-major axis, hence since no companion is detected, it is quite unlikely that the system is that young. On the other hand, if the age is 3 Gyr, or even 5 Gyr as discussed above, there is still a substantial parameter range in which the companion could hide. Given these results, in approximate numbers we can constrain the planet or brown dwarf mass to about 5-20 $M_{\\rm jup}$ and its semi-major axis to about 10-20 AU. Also, the inclination must be larger than at least 20$^{\\rm o}$, otherwise the projection effects could never bring the companion close enough to the star to hide it, and the actual mass would be sufficiently larger than the projected mass to make it brighter than the background at any reasonable age.\n\n\\section{Conclusions}\n\n\\label{sec_conclusions}\n\nWe have attempted to image the indirectly discovered companion to $\\epsilon$ Ind A, using imaging in the 4$\\mu$m filter as well as the L'-band. As expected, 4$\\mu$m imaging was found to be a preferable choice over L'-band in the inner, contrast-limited regime, whereas the opposite was found to be true in the outer, background-limited range. This conclusion is based on theoretical models that ultimately need to be confirmed through observations of known planets. The overlap occurs at a radius of 4$''$ in our images, a number that will depend on target brightness and integration time. Two PSF subtraction techniques were employed: regular active ADI as used previously, and a new combination of techniques, using PSF reference subtraction and aADI with the LOCI algorithm. While PSFR+aADI performs slightly better at large separations, the techniques are virtually indistinguishable for most of the contrast-limited regime. Using more than one PSF reference star may change this picture. In addition, the method of combining 4$\\mu$m imaging and LOCI is also well suited for passive ADI, which has the potential to substantially enhance the performance even further.\n\nIn spite of the high sensitivities achieved in our images, we did not detect any potential companion candidate. Unless the known radial velocity companion to $\\epsilon$ Ind A is a neutron star or even more exotic stellar remnant, the non-detection in all images implies that the system is probably older than 1 Gyr, possibly consistent with preliminary results presented by Cardoso et al. (2008). Furthermore, we can constrain the planet or brown dwarf mass to within approximately 5-20 $M_{\\rm jup}$, the semi-major axis to $\\sim$10-20 AU, and the inclination to $>$20$^{\\rm o}$. An analysis based on astrometry from FK5 and \\textit{Hipparcos} is consistent with such a companion. Given the high significance of the RV trend, the fact that we can exclude all stellar, white dwarf and high-mass brown dwarf companions, and the fact that exotic stellar remnants are rare, it seems very plausible that $\\epsilon$ Ind A is one of the nearest stars to host a massive giant planet or very low-mass object. Furthermore, it is likely that this companion would be detectable through further imaging with either the presently available facilities, or facilities that come online in the relatively near future. Hence, $\\epsilon$ Ind is a high-profile target for the study of substellar objects, even aside from the fact that it hosts the nearest binary brown dwarf.\n\nFinally, we note that no sophisticated coronagraph adapted for observations beyond 3$\\mu$m presently exists on any of the 8m-class or larger AO-assisted telescopes (although simple coronagraphs do exist, e.g. a Lyot coronagraph for NACO). The potential coronagraphic performance is intimately connected to the adaptive optics performance, which leads to an interest in coronagraphs in the context of 'extreme AO' facilities currently in development (e.g. Petit et al. 2008). However, given the fact that a demonstrated Strehl ratio in the range of 85\\% can be reached even with NACO at 4$\\mu$m, an 'extreme AO'-type performance in this particular wavelength range is available already today. The development of a coronagraph for this wavelength range could therefore be another promising avenue to further increase the near-future capacity of detecting extrasolar planets through direct imaging.\n\n\\section*{Acknowledgments}\n\nThe authors wish to thank Marten van Kerkwijk and Yanqin Wu for useful discussion. The study made use of the CDS and SAO\/NASA ADS online services. M.J. is supported through the Reinhardt postdoctoral fellowship from the University of Toronto.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nMany phenomena may be described through networks, including investment interactions between bidders and firms in venture capital (VC) markets \\cite{liang2016} and professional relationships among firms \\cite{bonaventura2020}.\nRisk capital is an essential resource for the formation and growth of entrepreneurial venture and venture capital firms are often linked together in a network by their joint investments in portfolio companies \\cite{bygrave1988}. Through connections in such a network, they exchange resources and investment opportunities with one another. Many studies show the impact of network dynamics on investments, raising efficiency \\cite{wetzel1987informal} and providing precious information when there is a great level of information asymmetry \\cite{fiet1995reliance}. Also, differentiating connection types and avoiding tight cliques appear to help the success of an investor by providing more diverse information and reducing confirmation bias \\cite{bygrave1988}.\n\nCB Insights \\cite{cbins} provides records of all transactions in venture capital markets from 1948. Since data until 2000 are partial and discontinuous, we focus on the period 2000-2020, in order to minimize the impact of missing data on our analysis. Additionally, since different sectors may be characterized by different investment dynamics \\cite{dushnitsky2006does}, we focus on the healthcare sector, which is of great importance and has shown to be less sensitive to market oscillations \\cite{pisano2006}. This stability is also shared by returns of life science VC, where investments have a lower failure rate but are at the same time less likely to generate \"black-swan\" returns \\cite{booth2011defense}, offering more consistency but a lower likelihood of achieving billion-dollars evaluations. \n\nWhile the number of exits through an IPO or through a trade sale can be seen as a proxy for the success of an investor \\cite{hege2003determinants}, there are instead different definitions of \"success\" for startups, but a common factor seems to be the growth rate of the company \\cite{santi2017lit}. Our work aims to understand whether network features may affect \"success\" of investments in healthcare firms. In order to investigate this, we introduce progressively more nuanced definitions of \"success\", and analyze them with increasingly sophisticated statistical tools. \n\nThe paper is organized as follows. Section \\ref{sec:char} introduces and characterizes a network of investors and firms, describing its structure and salient properties, including the communities emerging from its topology. Then, Section \\ref{sec:success} focuses on the definition and analysis of \"successful\" firms. We first characterize \"success\" by looking at the funding trajectories of each firm, clustering these trajectories into two broad groups capturing a high and a low funding regime. The binary cluster membership labels provide a first, rough definition of \"success\". We run a logistic regression in order to explain \"success\" defined in this fashion with statistics computed on the network itself. We then move to more complex characterizations of \"success\": the total amount of money raised (a scalar) and the funding trajectory itself (a functional outcome). We run regressions also on these outcomes, to validate and refine our previous results. Finally, we discuss main findings and provide some concluding remarks in Section \\ref{sec:disc}.\n\n\\section{Network characterization}\n\\label{sec:char}\n\nThe 83258 agents in the healthcare sector are divided into two broad categories: 32796 bidders, or investors, and 50462 firms. Companies open investment calls in order to collect funds; investors answer such calls and finance firms. Each deal, i.e. each transaction from an investor to a company, is recorded in the CB Insights' database. This market dynamics can be described by a \\textit{bipartite network}, which indeed is built on the notion of dichotomous heterogeneity among its nodes. In our case, each node may be a firm or an investor, respectively. An undirected link exists between two nodes of different kinds when a bidder has invested into a firm. Of course, given the possibility for an investor to finance the same firm twice, the bipartite network is also a \\textit{multi-graph}. By knowing the date in which investments are made, we can produce yearly snapshots of the bipartite network. A company (investor) is included in a snapshot of a certain year only when it receives (makes) an investment that year. By projecting the bipartite network onto investors and firms, we produce the two projected graphs which are used to compute all the node statistics described in Table \\ref{tab:variables}. As the bipartite network is a multi-graph, defining projections on a subset of nodes requires an additional assumption. Specifically, we project the bipartite graph onto firms by linking them in a cumulative fashion: we iteratively add to each yearly projected snapshot a link between two companies in which a bidder has invested during that year. Concerning the projection of the bipartite network onto investors, we link two bidders whenever they invest in the same company in the same financing round. \n\n\\begin{table}[t]\n    \\centering\n    \\caption{Statistics computed on the projected graphs of investors and firms. Before running regressions in Section~\\ref{sec:success}, left-skewed variables are normalized through log-transformation.}\n    \\label{tab:variables}\n    \\begin{tabularx}{\\textwidth}{XX}\n    \\toprule\n    \\textbf{Variable} & \\textbf{Network meaning}\\\\\n    \\midrule\n    Degree centrality & Influence \\\\\n    Betweenness centrality \\cite{hannan1977population} & Role within flow of information \\\\\n    Eigenvector centrality \\cite{bonacich1987power} & Influence \\\\\n    VoteRank  \\cite{zhang2016identifying} & Best spreading ability \\\\\n    PageRank \\cite{page1999pagerank}& Influence \\\\\n    Closeness centrality \\cite{freeman1978centrality} & Spreading power (short average distance from all other nodes) \\\\\n    Subgraph centrality \\cite{estrada2005subgraph} & Participation in subgraphs across the network \\\\\n    Average neighbor degree \\cite{barrat2004architecture} & Affinity between neighbor nodes \\\\\n    Current flow betweenness centrality \\cite{newman2005measure} & Role within flow of information\\\\\n    \\bottomrule\n    \\end{tabularx}\n\\end{table}\n\nRoughly 75\\% of the companies in the network projected onto firms are North American and European (around 55\\% belong to the US market), while the remaining 25\\% is mostly composed of Asian companies. Around 60\\% of the companies operate within the sub-sectors of medical devices, medical facilities and biotechnology -- the pharmaceutical sub-sector alone accounts for 20\\% of the network. As of August, 2021, roughly of 80\\% the companies in the network are either active or acquired, with the remaining portion being inactive or having completed an IPO. We witness turnover of the active companies through the years, but this is expected: a company's status is evaluated as of 2021, and it is more likely to observe a dead company among those that received investments in 1999 than in 2018. Indeed, both death and IPO represent the final stage of the evolution of a company, so those that received funding in earlier years are more likely to have already reached their final stage. Finally, we do not observe marked changes in terms of graph sub-sectoral composition: the relative share of each sub-sector is rather stable through the years, with the exception of an increase in the shares of the internet software and mobile software sub-sectors (from 1\\% in 1999 to 8\\% in 2019 and from 0\\% in 1999 to 5\\% in 2019, respectively). \n\n\\subsection{Communities}\n\nBy employing the Louvain method \\cite{blondel2008fast}, we identify meso-scale structures for each yearly snapshot of the network projected onto firms. For each year, we rank communities by their size, from the largest to singletons. We then compare the largest communities across years, by looking at their relative sub-sectors, status and geographical composition.\n\nWhile the specific nodes in the biggest communities may vary throughout the years, we notice a relative stability in their features. The largest communities (which contain between 13\\% and 20\\% of the nodes) reflect the status composition of the general network, downplaying unsuccessful companies and giving higher relative weight to IPO ones, showing just a variation between acquired and active companies across years (i.e.~active companies are relatively over-represented in more recent largest communities than in older ones). Considering geographical information, the largest communities comprise mainly US companies, with an under-representation of other continents. This trait is quite consistent through the years, with the exception of\ntwo years (2013-2014). With respect to sub-sectors, the largest communities mainly contain medical device and biotechnology companies, and they are quite consistent through the years in terms of sub-sectoral composition. \n\nThe second largest communities (containing between 10\\% and 14\\% of nodes in the network) have a less consistent sub-sectoral composition through the years, although it is worth highlighting that they comprise companies operating within software and technology. Geographically, we are still witnessing communities of mostly US-based companies, although 5 years out of 20 show a remarkable (roughly 80\\%) presence of European companies. Finally, status composition is balanced between active and acquired until the later years, when active companies predominate within the second largest communities. IPOs are not present, while there are, in a small percentage (between 5\\% and 20\\%), dead startups.\n\nFinally, the third largest communities (containing between 7\\% and 12\\% of the nodes) present a clear change within the period considered: in the first ten years, they mostly comprise failed or acquired European companies within the fields of biotechnology and drug development, while, in the second decade, they comprise active US companies within the fields of medical devices and medical facilities. \n\n\\section{Success analysis}\n\\label{sec:success}\nGiven the bipartite network and its projections, we now turn to the analysis of success and of its main drivers. Because of the elusiveness of the definition of \"success\", we proceed in stages -- considering progressively more refined outcomes and comparing our findings. Moreover, since many of the records available in the CB Insights' data set are incomplete, and our aim is to capture the temporal dynamics leading a firm to succeed, we further restrict attention to those companies for which full information is available on birth year, healthcare market sub-sector and investment history for the first 10 years from founding. Although this filtering may introduce some biases, it still leaves us with a sizeable set of 3663 firms belonging to 22 different sub-sectors.\n\nNotably, we restrict our focus also in terms of potential predictors, due to the fact that our collection of network features exhibits strong multicollinearities. By building a feature dendrogram (Pearson correlation distance, complete linkage) and by evaluating the correlation matrix, we reduce the initial set to four representatives. In particular, we select two features related to the investors' projection (the maximum among the degree centralities of the investors in a company and the maximum among their current flow betweenness centralities, both computed in the company's birth year) and two features computed on the firms' projection (a company's eigenvector and closeness centralities, computed in the year in which the company received its first funding). \n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.8\\textwidth]{aligned_example_pharma.eps}\n    \\caption{Money raised cumulatively as a function of time, shown for 319 firms in the pharmaceuticals and drugs sub-sector. Funding trajectories are constructed over a period of $10$ years since birth, and aligned using birth years as registration landmarks.}\n    \\label{fig:aligned_pharma}\n\\end{figure}\n\nEach firm has its own funding history: after its birth, it collects funds over the years, building a \\textit{trajectory} of the amount of money it is able to attract. We treat these trajectories as a specific kind of structured data, by exploiting tools from a field of statistics called \\textit{Functional Data Analysis} (FDA) \\cite{ramsey2005functional}, which studies observations that come in the form of functions taking shape over a continuous domain. In particular, we focus on the \\textit{cumulative} function of the money raised over time by each company. As an example, Figure \\ref{fig:aligned_pharma} shows 319 such cumulative functions, for the firms belonging to the pharmaceuticals and drugs sub-sector. Trajectories are \\textit{aligned}, so that their domain (\"time\") starts at each company's birth (regardless of the calendar year it corresponds to). By construction, these functions exhibit two characterizing properties: first, they are monotonically non-decreasing; second, they are step functions, with jumps indicating investment events. \n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{clustering_pharma.eps}\n    \\caption{$k$-means clustering ($k=2$) of the funding trajectories of firms belonging to the pharmaceuticals and drugs sub-sector. The green and red dashed lines represent firms in the high (\"successful\") and low regimes, respectively. Bold curves represent cluster centroids. To aid their visualization, centroids are shown again in the right panel with individual trajectories in gray.}\n    \\label{fig:clustered_pharma}\n\\end{figure}\n\nOur first definition of success is based on separating these trajectories into two regimes characterized by high (successful) vs. low investment patterns: the first runs at high levels, indicating successful patterns, and the second at low levels. Because of heterogeneity among healthcare sub-sectors, we accomplish this by running a \\textit{functional k-means clustering} algorithm \\cite{jacques2014func, hartigan1979} with $k=2$, separately on firms belonging to each sub-sector.\nAs an example, companies belonging to the sub-sector of pharmaceuticals and drugs are clustered in Figure \\ref{fig:clustered_pharma}. Throughout all sub-sectors, the algorithm clusters $89$ firms in the high-regime group and $3574$ in the low-regime one. \n\nThis binary definition of \"success\" turns out to be rather conservative; very few firms are labeled as belonging to the high investment regime. Consider the logistic regression\n\\begin{equation}\n\\label{eq:log}\n    \\log\\left(\\frac{P(y_i=1)}{1-P(y_i=1)}\\right)=\\beta_0+\\sum_{j=1}^{p} \\beta_j x_{ij} \\quad i=1,\\dots n\n\\end{equation}\nwhere $n$ is the number of observations, $y_i$, $i=1,\\dots n$, are the binary responses indicating membership to the high ($y_i=1$) or low ($y_i=0$) regime clusters; $\\beta_0$ is an intercept and $x_{ij}$, $i=1,\\dots n$ and $j=1,\\dots,p$ ($p=4$), are the previously selected scalar covariates.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\textwidth]{significance.eps}\n    \\caption{Scatter plots of logistic regression coefficient estimates (horizontal) and significance (vertical; $-log(p$-$value)$). Each point represents one of $1000$ fits run on data balanced by subsampling the most abundant class. Orange solid line mark averages across the fits, and orange dashed lines $\\pm1$ standard deviations about them. Green solid lines mark $0$ on horizontal axes. Blue line mark significance values associated to a p-value of $0.1$.}\n    \\label{fig:est_coef}\n\\end{figure}\n\nIf we fit this regression on our unbalanced data, results are bound to be unsatisfactory and driven by the most abundant class. Running such a fit, one obtains an explained deviance of only $0.10$. To mitigate the effects of unbalanced data \\cite{haibo2009}, we randomly subsample the most abundant class (the low-regime firms) as to enforce balance between the two classes, and then run the logistic regression in Equation \\ref{eq:log}. We repeat this procedure 1000 times, recording estimated coefficients, associated p-values and explained deviances. The average of the latter across the 1000 replications is substantially higher than on the unbalanced fit, reaching $0.18$\n(some fits produce deviance explained as high as $0.45$). Moreover, we can investigate significance and stability of the coefficient estimates through their distribution across the repetitions. Figure \\ref{fig:est_coef} shows scatter-plots of these quantities, suggesting that the two variables related to the firms' centrality have a modest yet stable, positive impact on the probability of belonging to the high-regime cluster. This is not the case for the variables related to the investors' centrality.\n\nThis first evidence of a positive relationship between the success of a firm and its centrality, or importance (in a network sense) is promising. However, the binary definition of \"success\" we employed is very rough -- and the unbalance in the data forced us to run the analysis relying on reduced sample sizes ($89+89=178$ observations in each repeated run). Thus, we next consider a scalar proxy for \"success\", which may provide a different and potentially richer perspective. Specifically, we consider the cumulative end point of a firm's funding trajectory, i.e. the total value of the investment received through its temporal domain.\n\n\nFor this scalar response, we run a \\textit{best subset selection} \\cite{friedman2005elements} considering all the network features in our initial set -- not just the $4$ selected to mitigate multicollinearity prior to the logistic regression exercise. Notably, despite the substantial change in the definition of \"success\", results are in line with those from the logistic regression. Indeed, the first selected variable, when the predictor subset is forced to contain only one feature, is the eigenvector centrality of firms. When the predictor subset size is allowed to reach $4$, the features selected are the closeness and the VoteRank of the firm, and the maximum current flow betweenness centrality among its investors (computed on the firm's birth year). Thus, the only difference compared to our previous choice is the selection of the firms' VoteRank centrality instead of the maximum among the investors' degree centrality. We compare the two alternative selections of four features as predictors of the scalar \"success\" response fitting two linear models of the form:\n\\begin{equation}\n    y_i =\\beta_0+\\sum_{j=1}^{p} \\beta_j x_{ij} +\\epsilon_i \\quad i=1,\\dots n\n\\end{equation}\nwhere $n$ is the number of observations, $y_i$, $i=1,\\dots n$, are the scalar responses (aggregate amount of money raised); $\\beta_0$ is an intercept; $x_{ij}$, $i=1,\\dots n$ and $j=1,\\dots,p$ ($p=4$), are the scalar covariates belonging to one or the other subset and $\\epsilon_i$, $i=1,\\dots n$, are i.i.d. Gaussian model errors. As shown in Table \\ref{tab:linreg}, the maximum degree centrality among a firm's investors is not statistically significant.\nSurprisingly, the maximum among investors' current flow betweenness centralities is significantly negative,  but its magnitude is close to 0. In contrast, the firms' closeness and eigenvector centralities are positive, statistically significant and sizeable. This is in line with what we expected, since it is reasonable to think that knowledge may indirectly flow from other startups through common investors, increasing the expected aggregate money raised. Finally, the firms' VoteRank centrality appears to have a negative, statistically significant impact on the aggregate money raised. This should not be surprising, given that the higher the VoteRank centrality is, the less influential the node will be. The variance explained by the two models is similar and still relatively low ($R^2 \\approx 0.13)$, which may be simply due to the fact that network characteristics are only one among the many factors involved in a firm's success \\cite{dosilimmancabile1994}. Nevertheless, the results obtained here through the scalar \"success\" outcome are consistent with those obtained through the binary one and logistic regression.\n\n\\begin{table}[t]\n\\centering \n\\caption{Linear regressions of aggregate money raised on \ntwo sets of predictors. All variables are scaled and some are log-transformed (as indicated parenthetically).} \n\\label{tab:linreg} \n\\begin{tabular}{@{\\extracolsep{5pt}}lcc} \n\\toprule\n & \\multicolumn{2}{c}{\\textit{Dependent variable:}} \\\\ \n\\cline{2-3} \n\\\\[-1.8ex] & \\multicolumn{2}{c}{Aggregate money raised (log)} \\\\ \n\\\\[-1.8ex] & (1) & (2)\\\\ \n\\midrule\n newman\\_max & $-$0.065$^{**}$ & $-$0.072$^{*}$ \\\\ \n  & (0.030) & (0.041) \\\\ \n  & & \\\\ \n voterank (log) & $-$0.140$^{***}$ &  \\\\ \n  & (0.033) &  \\\\ \n  & & \\\\ \n degcen\\_max (log) &  & 0.050 \\\\ \n  &  & (0.040) \\\\ \n  & & \\\\ \n closeness & 0.126$^{***}$ & 0.130$^{***}$ \\\\ \n  & (0.037) & (0.030) \\\\ \n  & & \\\\ \n eigenvector (log) & 0.214$^{***}$ & 0.255$^{***}$ \\\\ \n  & (0.034) & (0.028) \\\\ \n  & & \\\\ \n Constant & 0.113$^{***}$ & 0.062$^{**}$ \\\\ \n  & (0.030) & (0.025) \\\\ \n  & & \\\\ \n\\midrule \\\\[-1.8ex] \nObservations & 1,118 & 1,364 \\\\ \nR$^{2}$ & 0.136 & 0.127 \\\\ \nAdjusted R$^{2}$ & 0.133 & 0.125 \\\\ \nResidual Std. Error & 0.992 (df = 1113) & 0.923 (df = 1359) \\\\ \nF Statistic & 43.951$^{***}$ (df = 4; 1113) & 49.458$^{***}$ (df = 4; 1359) \\\\ \n\\bottomrule\n\\textit{Note:}  & \\multicolumn{2}{r}{$^{*}$p$<$0.1; $^{**}$p$<$0.05; $^{***}$p$<$0.01} \\\\ \n\\end{tabular} \n\\end{table} \n\nOur scalar outcome (aggregate money raised) has its own drawbacks. In particular, it implicitly assumes that the right time to evaluate success and investigate its dependence on network features is, cumulatively, at the end of the period considered (10 years). Note that this translates into a 10-year gap between the measurement of network features and financial success. \n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\textwidth]{fos_plot.eps}\n    \\caption{Function-on-scalar regression, coefficient curve estimates. (a) intercept function (this can be interpreted as the sheer effect of time on the response); (b) maximum degree centrality among investors (company's birth year); (c) maximum across investors' current flow betweenness centrality (company's birth year); (d) company's eigenvector centrality; (e) company's closeness centrality. Dotted lines represent confidence bands. All the covariates are standardized.}\n    \\label{fig:fos_coef}\n\\end{figure}\n\nAlthough this issue could be approached relying on additional economic assumptions, we tackle it refining the target outcome and considering the full funding trajectories -- instead of just their end point. This requires the use of a more sophisticated regression framework from FDA; that is, function-on-scalar regression \\cite{kokoszka2017introduction}. In particular, we regress the funding trajectories on the same two sets of covariates considered in the scalar case above. The equation used for function-on-scalar regression is:\n\n\\begin{equation}\n    \\label{eq:fun_reg}\n        Y_i(t) = \\beta_0(t) + \\sum_{j=1}^{p} \\beta_j(t) x_{ij} + \\epsilon_i(t) \\quad i=1,\\dots n\n    \\end{equation}\nwhere $n$ is the number of observations; $Y_i(t)$, $i=1,\\dots n$, are the aligned funding trajectories; $\\beta_0(t)$ is a functional intercept; $x_{ij}$, $i=1,\\dots n$ and $j=1,\\dots,p$ ($p=4$), are the scalar covariates belonging to the one or the other set, and $\\epsilon_i(t)$, $i=1,\\dots n$, are i.i.d. Gaussian model errors.\n\nThe regression coefficient of a scalar covariate in this model, $\\beta_j(t)$, is itself a curve describing the time-varying relationship between the covariate and the functional response along its domain. Together with the functional coefficients, we also estimate their standard errors, which we use to build confidence bands around the estimated functional coefficients \\cite{refund2016}. Coefficient curve estimates for the covariate set including the maximum investors' degree centrality are shown in Figure~\\ref{fig:fos_coef} (results are very similar with the other set of covariates). The impacts of an increase in the maximum among the degree centralities and in the maximum among the current flow betweenness centralities of the investors in a firm are not statistically significant. Conversely, eigenvector and closeness centralities of firms have positive and significant impacts. The impact of the eigenvector centrality seems to be increasing during the first five years, reaching a \"plateau\" in the second half of the domain. These findings reinforce those obtained with the binary and scalar outcomes previously considered, confirming a role for firms' centrality in shaping their success.\n\n\\section{Discussion}\n\\label{sec:disc}\nThis paper exploits techniques from the fields of network and functional data analysis. We build a network of investors and firms in the healthcare sector and characterize its largest communities. Next, we progressively shape the concept of a firm's \"success\" using various definitions, and associate it to different network features. Our findings show a persistent positive relationship between the importance of a firm (measured by its centrality in the network) and various (binary, scalar and functional) definitions of \"success\". In particular, we cluster funding trajectories into a high (\"successful\") and a low regime, and find significant associations between the cluster memberships and firms' centrality measures. Then, we switch from this binary outcome to a scalar and then a functional one, which allow us to confirm and enrich the previous findings. Among centralities computed on the two network projections, our results suggest a preeminent role for those computed in the companies' projection. In particular, both a firm high closeness centrality, indicating a small shortest distances to other firms, and its eigenvector centrality, which may account for a firm's reputation, seem to be related to the propensity to concentrate capital.\n\nOur analysis can be expanded in several ways. First, we limit our study to the healthcare sector, while it may be interesting to investigate other fields, or more healthcare firms based on the availability of more complete records. It would also be interesting to account for external data (e.g. country, sub-sector, etc.) in two ways. One the one hand, these information would be useful as to compute more informative statistics on the network topology. On the other hand, they may be used in our regression, to control for these factors.\nMoreover, meso-scale communities may be analyzed in terms of their longitudinal evolution, as to characterize \"successful\" clusters of firms from a topological point of view. \n\n\\section*{Acknowledgments}\nF.C., C.E., G.F., A.M. and L.T. acknowledge support from the Sant'Anna School of Advanced Studies. \nF.C. acknowledges support from Penn State University. \nG.R. acknowledges support from the scheme \"INFRAIA-01-2018-2019: Research and Innovation action\", Grant Agreement n. 871042 \"SoBigData++: European Integrated Infrastructure for Social Mining and Big Data Analytics\".\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{}}\n\n\nThe $b\\to c$ transitions have been very important in the extraction of the $V_{cb}$ CKM matrix element. \nIn experiments aimed to determine $V_{cb}$, actually the product\n$|V_{cb} {\\cal F } (1) |$ is extracted, where  ${\\cal F } (1)$ denotes\nthe $B \\to D$ or $B \\to D^*$ hadronic form factors at zero recoil. A\nlack of precise information about the shapes of these form factors\naway from zero recoil is one of the main sources of uncertainties.\nFurthermore recent spectroscopic discoveries of low lying orbitally excited charmed meson states at the charm and B factories have prompted reevaluation of their contributions to B meson decays. \nThe leading order (LO) $SU(3)$ chiral corrections at leading and next to leading order (NLO) in $1\/m_Q$ to the $B_q\\to D_q^{(*)}$ semileptonic form factors have previously been computed~\\cite{Boyd:1995pq}. Also virtual effects of positive parity heavy meson states in these transitions have been considered before~\\cite{Falk:1993iu}.\nHowever, phenomenological discussion at the time was limited by the lack of experimental and lattice  QCD (LQCD) information available on the relevant phenomenological parameters. We are now able to provide the first more reliable estimates of these contributions, but we also extend the program to B meson transitions to positive parity charmed mesons~\\cite{EFK1}. In particular we address the issue of extrapolation of LQCD results on these quantities to the chiral regime.\nIn the $B_q\\to D_q \\tau\\nu$ decay mode due to the large tau mass one can discuss helicity suppressed contributions as such coming from the exchange of charged scalars~\\cite{Tanaka:1994ay,Kiers:1997zt,Nierste:2008qe,Kamenik:2008tj}. \nThis calls for high precision theoretical estimates for both the dominating vector\nas well as the subleading relative scalar form factor contribution.\nWe have estimated the leading chiral symmetry breaking corrections, governing the differences between the vector and scalar form factors in $B_q\\to D_q$ transitions~\\cite{EFK2}. These corrections can on the one hand be used to guide LQCD studies in their chiral extrapolations. On the other hand we use them to estimate qualitatively the relative scalar form factor values in $B_s\\to D_s$ transitions.\n\nFirst we present the most important results of our calculation~\\cite{EFK1} of\nleading chiral loop corrections to the form factors governing transitions of B mesons to positive and negative parity charmed mesons. In our heavy meson chiral perturbation theory (HM$\\chi$PT) description, heavy-light mesons appear in velocity ($v$) dependent spin-parity doublets due to heavy quark spin symmetry, while chiral $SU(N_f)$ symmetry determines their interactions with the multiplet of light pseudo-Goldstone bosons. Such a framework provides a systematic expansion terms of light and inverse heavy quark masses (for details c.f. ref.~\\cite{Fajfer:2006hi}).  \nThe relevant form factor contributions in the effective theory are obtained using operator bosonization procedure introducing the universal Isgur-Wise (IW) functions $\\xi(w)$, $\\tilde \\xi(w)$ and $\\tau_{1\/2}(w)$ (where $w=v\\cdot v'$ is the product of velocities of initial and final heavy-light states) governing transitions within the negative, positive parity doublets, and between states of opposite parity respectively. We calculate loop corrections to these effective weak vertices coming from the one loop diagram topologies shown in Fig. 1.\n\\psfrag{pi}[bl]{\\footnotesize $\\Red{ {\\pi^i(q)}}$}\n\\psfrag{Ha}[cc]{\\footnotesize $\\Red{H_a(v)}$}\n\\psfrag{Hb}[cc]{\\footnotesize $\\Red{H_b(v')}$}\n\\psfrag{Hc}[cc]{\\footnotesize $~\\Red{H_c(v)}$}\n\\psfrag{Hd}[cc]{\\footnotesize $~~\\Red{H_c(v')}$}\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=145mm]{xi_loop.eps}\n\\caption{Weak vertex correction diagrams. Crossed boxes represent effective weak vertices while filled circles represent LO strong vertices. Only diagrams of the utmost left topology contribute to the amplitude at the leading chiral log order.} \\label{JACpic2-f2}\n\\end{figure*}\nIn addition, wave function renormalization of the heavy-light fields needs to be taken into account. This has been done e.g. in ref.~\\cite{Fajfer:2006hi} and non-zero contributions come from the self energy diagrams with LO effective strong couplings in the loop.\nPresence of two opposite parity heavy-light meson multiplets in the theory introduces a new scale -- the mass gap between the ground and excited heavy meson states $\\Delta_{SH}$.\nIn order to tame the chiral behavior of the\namplitudes containing the mass gap we use the $1\/\\Delta_{SH}$\nexpansion of the chiral loop integrals~\\cite{Fajfer:2006hi}.\nWe have argued~\\cite{Becirevic:2006me}\nthat the expansion works well in a $SU(2)$ theory where kaons and etas,\nwhose masses would compete with $\\Delta_{SH}$, do not\npropagate in the loops. Therefore we can write down explicit expressions\nfor the chiral loop corrected IW functions specifically for\nthe strangeless states in the $SU(2)$ theory\\footnote{Following~\\cite{EFK1} we absorb the infinite and scale dependent pieces from one loop amplitudes into the appropriate counterterms at order $\\mathcal O (m_q)$.}:\n\\begin{eqnarray}\n[\\xi(w)]^{\\mathrm{Loop}}  &=&   \\xi(w) \\Bigg\\{ 1 + \\frac{3}{32\\pi^2 f^2} m^2_{\\pi} \\log \\frac{m^2_{\\pi}}{\\mu^2}  \\Bigg[ g^2 2 (r(w)-1)\n - h^2  \\frac{m^2_{\\pi}}{4\\Delta_{SH}^2} \\left(1-w\\frac{\\tilde \\xi(w)}{\\xi(w)}\\right) - h g \\frac{m^2_{\\pi}}{\\Delta_{SH}^2} w(w-1)\\frac{\\tau_{1\/2}(w)}{\\xi(w)}\\Bigg] \\Bigg\\},\\nonumber\\\\\n\\label{eq:5}\n\\end{eqnarray}\nwhere\n$r(x) = {\\log(x+\\sqrt{x^2-1})}\/{\\sqrt{x^2-1}}$ and all IW functions on the rhs should be considered tree-level, similarly\n\\begin{eqnarray}\n[\\tau_{1\/2 }(w)]^{\\mathrm{Loop}} &=&  \\tau_{1\/2}(w) \\Bigg\\{ 1 + \\frac{3}{32\\pi^2 f^2} m^2_{\\pi} \\log \\frac{m^2_{\\pi}}{\\mu^2}  \\Bigg[ - g\\tilde g(2r(w)-1) - \\frac{3}{2} (g^2+\\tilde g^2)\\nonumber\\\\\n&&\n\\hskip -0cm + h^2  \\frac{m^2_{\\pi}}{4\\Delta_{SH}^2} \\left(w-1\\right) - h g \\frac{m^2_{\\pi}}{2\\Delta_{SH}^2} \\frac{\\xi(w)}{\\tau_{1\/2}(w)} w(1+w) + h\\tilde g \\frac{m^2_{\\pi}}{2\\Delta_{SH}^2} \\frac{\\tilde \\xi(w)}{\\tau_{1\/2}(w)} w(1+w) \\Bigg] \\Bigg\\}.\n\\label{eq:6}\n\\end{eqnarray}\nThe first parts of Eqs.~(\\ref{eq:5}) and~(\\ref{eq:6}) contain the leading contributions while the calculated  $1\/\\Delta_{SH}$ corrections are contained in the second parts. Definitions and values for the LO strong couplings $g$, $\\tilde g$ and $h$ can be found in ref.~\\cite{EFK1}.\nWe present\nthe chiral behavior of the IW functions\nin the chiral limit below the $\\Delta_{SH}$ scale in Fig. 2.\n\\psfrag{xk1}[bc]{{$r\\sim m_{u,d}\/m_s$}}\n\\psfrag{xm1}[tc][tc][1][90]{{${\\xi'(1)}^{\\mathrm{Loop}}\/\\xi'(1)^{\\mathrm{Tree}}$}}\n\\psfrag{s1}[cl]{\\footnotesize{$(1\/2)^-$ contributions}}\n\\psfrag{s3}[cl]{\\footnotesize{$\\xi'(1)-\\tilde \\xi'(1)=1$}}\n\\psfrag{s4}[cl]{\\footnotesize{$\\xi'(1)-\\tilde \\xi'(1)=-1$}}\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=85mm]{plot1.eps}\n\\caption{Chiral extrapolation of the slope of the IW\nfunction at $w=1$ ($\\xi'(1)$). Negative parity heavy states'\ncontributions (black line) and a range of possible positive parity\nheavy states' contribution effects when the difference of slopes of\n$\\xi(1)$ and $\\tilde \\xi(1)$ is varied between $1$ (red dashed line)\nand $-1$ (blue dash-dotted line)} \\label{JACpic2-f3}\n\\end{figure*}\n\n\n\n\nNext we focus particularly on the matrix element of the vector $b\\to c$ quark current between a B and a D meson of velocity $v$ and $v'$ respectively, which can be parametrized in terms of two form factors:\n\\begin{eqnarray}\n&& \\bra{D(v')}\\bar c \\gamma_{\\mu} b \\ket{B(v)} = \\sqrt{m_B m_D} [ h_+(w) (v+v')_{\\mu} \n\n+ h_-(w) (v-v')_{\\mu} ]\\,.\\label{eq:h+expansion}\n\\end{eqnarray}\nIt is important to know that in heavy quark expansion \n$h_+(w)$ is the leading order term in $1\/m_Q$ (proportional to the IW function $\\xi(w)$) while $h_-(w) \\sim 1\/m_c - 1\/m_b$. \nAfter the bosonization of the relevant new operators which appear at $1\/m_Q$~\\cite{EFK2}, we determine leading logarithmic chiral corrections relevant for LQCD extraction of the $h_+(w)$ and $h_-(w)$ form factors in the $SU(2)$ theory:\n\\begin{eqnarray}\n\\label{eq:hpchi}\n\th_+ &=& h_+^{\\mathrm{Tree}} \\left[ 1 + 3 g^2 \\frac{r(w)-1}{(4\\pi f_{\\pi})^2}  m_{\\pi}^2\\log \\frac{m_{\\pi}^2}{\\mu^2} + m_{\\pi}^2 c_{+}(\\mu,w) \\right]\\,, \\\\\n\\label{eq:hmchi}\n\th_- &=&  h_-^{\\mathrm{Tree}} \\left\\{ 1 - 3 g^2 \\frac{2 - Y^{*}_+ [r(w)+1]}{(4\\pi f_{\\pi})^2}  m_{\\pi}^2\\log \\frac{m_{\\pi}^2}{\\mu^2} \\right\n+ m_{\\pi}^2 c_{-}(\\mu,w) \\Big\\}\\,,\n\\end{eqnarray}\nwhere $Y^{*}_+= h_-^{*\\mathrm{Tree}}\/h_-^{\\mathrm{Tree}}$ is the ratio of vector current matrix elements proportional to $v-v'$ between vector and pseudoscalar states respectively (see ref.~\\cite{EFK1} for details and other possible choices of parameterizing this quantity), and the $w$ dependence is implicit. Here and in the rest of the text $c_i(\\mu,w)$ denote the sums of local analytic counter-terms, which cancel the $\\mu$ dependence of the chiral log pieces. As stressed by the notation, they will in general have a non-trivial $w$ dependence, originating both from the finite analytic residuals of chiral loops as well as from local NLO chiral current operators, needed to cancel the UV divergences of chiral loops. \n\nThe ratio $h_-\/h_+$ is particularly important since it enters in the multiplicative contribution to the differential decay rate distribution distinguishing $B\\to D \\tau\\nu$ from $B \\to D e\\nu$ decays. The chiral corrections in this case read:\n\\begin{eqnarray}\n\t\\frac{h_-}{h_+} &=&  \\left(\\frac{h_-}{h_+}\\right)^{\\mathrm{Tree}} \\left[ 1 - 3 g^2 \\frac{r(w)+1}{(4\\pi f_{\\pi})^2}  m_{\\pi}^2\\log \\frac{m_{\\pi}^2}{\\mu^2}  \\right\n+ \\left(\\frac{h^*_-}{h_+}\\right)^{\\mathrm{Tree}} 3 g^2 \\frac{r(w)+1}{(4\\pi f_{\\pi})^2}  m_{\\pi}^2\\log \\frac{m_{\\pi}^2}{\\mu^2} + m_{\\pi}^2 c_{{\\pm}}(\\mu,w)\\,,\n\\end{eqnarray}\nas written in ~\\cite{EFK2} and where relevant quantities are defined. \n\n\nWe can summarized our results:\nwithin a HM$\\chi$PT  framework, which includes even and odd parity\nheavy meson interactions with light pseudoscalars as\npseudo-Goldstone bosons, we have calculated chiral loop corrections\nto the functions $\\xi$ and $\\tau_{1\/2}$. As in previous cases~\\cite{Fajfer:2006hi,Becirevic:2004uv}  we have  shown that\nthe leading pionic chiral logarithms are not changed by the\ninclusion of even parity heavy meson states we consider chiral\nextrapolation of IW functions. Our results are particularly\nimportant for the LQCD extraction of the form factors. The\npresent errors on the $V_{cb}$ parameter in the exclusive channels\nare of the order few percent. This calls for careful control over\ntheoretical uncertainties in its extraction. \nWe have found also~\\cite{EFK2} how calculating the ratio of current matrix elements in $B\\to D$ and $B^*\\to D^*$ transitions on the lattice can improve the extraction of the scalar form factor. \n\n\n\n\n\\begin{acknowledgments}\nThis work is supported in part by the European Commission RTN network, \nContract No. MRTN-CT-2006-035482 (FLAVIAnet).\nThe work of S.F. and J.K. is supported in part by the Slovenian\nResearch Agency. J.O.E. is supported in part by the Research Council\nof Norway. \n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:intro}Introduction and Summary}\nThermalization in unitary quantum field theories has been a topic of great significance. Using AdS\/CFT correspondence, it has also been linked to black hole formation \\cite{Bhattacharyya:2009uu, Chesler:2008hg}.\nOne of the current views of thermalization is that of the thermalization of a finite subsystem, in which the conjugate subsystem is considered the heat bath.\nIn other words, it is the thermalization observed by an observer who has access to only a subsystem of the full system. It can also be considered as if the `fine-grained' observables{\\footnote{Observables which show the non-thermal behaviour of the pure state, in contrast to `coarse-grained' observables which cannot distinguish between the pure state and the thermal ensemble.}} are spatially widely separated bilocal or higher point observables. Starting from a pure state, in the high energy (high effective temperature) limit, the final thermal entropy observed by such an observer is actually the entanglement entropy of the subsystem with its conjugate. Obviously, the pure state has to be a time-dependent state. Closely related to thermalization(equilibration in general), the study of time-dependent states after a quantum quench has also been of great interest\\cite{Polkovnikov:2010yn, gogolin2015equilibration}. Quantum quench is the process in which the parameters of the Hamiltonian of a system in a certain state are changed with time. After the quantum quench, in the long time limit, if the subsystem of our interest looks like a thermal ensemble, in the sense that the expectation values of observables in the finite subsystem have the same expectation values as in a thermal ensemble, then we say that the system has thermalized. In this paper, we will be mainly considering quantum quench as the preparation of the time-dependent states of our interest.\n\nWe will also restrict ourself to critical quantum quenches, in which the final Hamiltonian is a critical Hamiltonian, i.e., the corresponding theory is a conformal field theory (CFT). More specifically, we will be considering free fermions in which starting from a certain state in the massive theory, the mass is set to zero gradually or suddenly. In more general theories, starting from the ground state of a gapped theory, it has been proposed \\cite{Calabrese:2005in, Calabrese:2006rx} that the state obtained after the critical quench is a Calabrese-Cardy(CC) state which has the form $e^{-\\kappa_2 H}|B\\rangle$, where $\\kappa_2$ is a scale given by the initial gap and the other scales of the quench process, $H$ is the Hamiltonian of the CFT and $|B\\rangle$ is a conformally invariant boundary state. It has been shown that such a state thermalizes to a thermal ensemble with temperature $T=1\/\\beta=1\/(4\\kappa_2)$. This result has also been generalised to the case in which the final theory has other conserved charges of local currents \\cite{Mandal:2015jla}. The corresponding ansatz for the state after quench from ground state is a generalized Calabrese-Cardy(gCC) states which have the form $e^{-\\kappa_2 H-\\kappa_4 W_4-\\kappa_6 W_6-...}|D\\rangle$ where again the parameters $\\kappa_2, \\kappa_4, \\kappa_6,\\cdots$ are given by the initial gap and other scales in the quench process, e.g. $\\delta t=1\/\\rho$ the time taken to set the mass to zero, and $W_4, W_6, \\cdots$ are the conserved charges of local currents. In this case also, it has been shown that the state thermalizes into a generalized Gibb's Ensemble(GGE) with the density matrix $e^{-\\beta H -\\mu_4 W_4 - \\mu_6 W_6-...}$ where the corresponding temperature and chemical potentials are $T=1\/\\beta=1\/(4\\kappa_2)$, $\\mu_4=4\\kappa_4$, $\\mu_6=4\\kappa_6, \\cdots$. \n\nThe gCC state ansatz has been shown to be true for mass quenches in free scalar and free fermion theories in a recent paper(MPS) \\cite{Mandal:2015kxi}. Starting from the ground state of the massive theories, the quenched states obtained are of the gCC form with infinite number of charges $W_{2n}$ with $n\\in\\mathbb{N}$ ($W_2=H$). For the scalar theory, it was also found that naively taking the sudden limit when the mass profile is taken to be a step function, the final state is non-normalizable. For massless free scalar theory, $W_{2n}=\\sum |k|^{2n-1}d^\\dagger_kd_k$, where $d^\\dagger_k$ and $d_k$ are the bosonic annihilation and creation operators.\\footnote{The normalization of the charges differ from the normalization in \\cite{Bakas:1990ry, Pope:1991ig}.} It was also shown that starting from specially prepared squeezed states of the massive scalar theory, CC state and gCC state with finite number of charges can also be created. By calculating correlators, thermalization of these states were explicitly shown.\n\nIn this paper, we find similar results for the fermionic mass quench. In the sudden limit, starting from the ground state, we observe that the final state has divergent energy density, $W_4, W_6, \\cdots$. Again, as in the case of scalar fields in MPS, starting from specially prepared squeezed states using the sudden quench limit, we can prepare CC state and gCC state with a finite number of charges of our choice. For the CC state and the gCC state with finite number of charges, we calculate correlators and explicitly show thermalization to thermal ensemble and GGE respectively.\n\nAmong the other calculable quantities, entanglement entropy(EE) is the most interesting one. The EE growth has been calculated(mostly numerically) in many dynamical systems, see for e.g. \\cite{Calabrese:2005in, Calabrese:2007-local, PhysRevA.78.010306, PhysRevX.3.031015, Nezhadhaghighi:2014pwa, Nahum:2016muy, Cotler:2016acd}. It has also been extensively examined in holographic systems \\cite{AbajoArrastia:2010yt, Hartman:2013qma, Caputa:2013eka, Liu:2013qca, Kundu:2016cgh}. Recently, non-monotonic EE growth consisting of an initial dip around the quench time has also been observed in a holographic set-up in \\cite{Bai:2014tla}.\n\nSince our final theory consists of only massless Dirac fermions, so using bosonization, we could calculate EE in some of our time-dependent states. We are interested in EE of a single interval only. For CC states, we find that EE grows monotonically. The asymptotic time limit is given by the well-known expression from CFT in a thermal ensemble, $S_A=\\frac{c}{3}\\log(\\sinh(\\frac{\\pi r}{\\beta}))$, where for Dirac fermions $c=1$ and the effective temperature $1\/\\beta =1\/4\\kappa$. In case of gCC states, we are not able to calculate EE with the charges of the usual fermionic bilinear $\\mathcal{W}_{1+\\infty}$ currents. But we are able to calculate the EE with the fermionic charge corresponding to the bosonic charges $W_{2n}=\\sum |k|^{2n-1}d^\\dagger_kd_k$. These are the charges of bosonic bilinear $\\mathcal{W}_{2n}$ currents for $n>1$. For such gCC states with the $W_4$ charge, we found a dynamical phase transition in which EE grows non-monotonically when the effective chemical potential $\\mu_4$ is greater than a critical value. Below this critical value, the EE growth is strictly monotonic.\n\n\\gap1\n\\noindent In summary, the key results of the present work are:\n\\begin{enumerate}\n \\item For ground state quench, similar to the scalar quench, a naive sudden quench limit gives divergent conserved charges. Calculation of the correlators show equilibration explicitly. But the long distance and time and ultimately the stationary limit is significantly different from thermalization to a thermal ensemble. This is the same manifestation of the UV\/IR mixing found in MPS.\n \\item Starting from appropriately prepared squeezed states of the massive theory, we can prepare CC and gCC states with specific $W_{2n}$ charges using quench. Calculation of correlators in CC state and gCC states explicitly show thermalization to thermal ensemble and GGE respectively. Here again, for gCC state, the long time and long distance limit of the correlators have significant dependence on the chemical potentials. This is again another avatar of the UV\/IR mixing.\n \\item For CC state, we are able to calculate the growth of entanglement entropy of a single interval explicitly in analytic form. The EE growth is strictly monotonically increasing for CC state. The stationary limit is, as expected, the entanglement entropy of a single interval in thermal ensemble.\n \\item We also calculate the EE growth of a single interval in gCC state with $W_4$ charge of the $\\mathcal{W}_{2n}$ representation of fermion corresponding to the $\\mathcal{W}_{2n}$ bilinear bosonic representation. We find dynamical phase transition in which the EE growth is monotonically increasing upto a critical value of $\\kappa_4$. Beyond the critical value, the EE growth is non-monotonic.\n\\end{enumerate}\n\n\n\\gap1\n\\noindent The outline of the paper is as follows:\n\n\\noindent In section \\ref{theo}, we solve the Dirac equation with time-dependent mass and from explicit solutions for a specific mass profile, we calculate the Bogoliubov coefficients for the transformation between the massive and massless modes. In section \\ref{qstates}, we find the final state after the quench starting from the ground state and a few squeezed states of our interest. In sections \\ref{ed} and \\ref{crf}, we calculate energy density and some correlators in the different quenched states that we obtained. The EE growth of a single subsystem in CC state is explicitly calculated in section \\ref{secEE_CC}. In section \\ref{secEE_gCC}, we show the dynamical phase transition in the EE growth of a subsystem in a particular gCC state. Section \\ref{cond} contains some discussions. The appendix contains details that we have omitted in the main sections.\n\n\\section{\\label{theo}Free Dirac fermions with time-dependent mass}\nThe action for Dirac fermions with time-dependent mass is\n\\begin{eqnarray}\nS=-\\int dx^2\\left[i\\bar{\\Psi}\\gamma^\\mu\\partial_\\mu\\Psi-m(t)\\bar{\\Psi}\\Psi\\right]\\\n\\label{action}\n\\end{eqnarray}\nThe equation of motion (EOM) is\n\\begin{eqnarray}\n \\left[ i\\gamma^0\\partial_t-i\\gamma^1\\partial_x-m(t)\\right]\\Psi(x,t)=0\\\n \\label{eom}\n\\end{eqnarray}\nand we are interested in the solvable mass profile\\cite{Duncan:1977fc, Das:2014hqa}\n\\begin{eqnarray}\nm(t)=m[1-\\tanh(\\rho t)]\/2\\\n \\label{mass}\n\\end{eqnarray}\n$m$ is the initial mass and $\\rho$ is the only scale of the quench process. $\\rho\\to\\infty$ is the sudden limit in which the mass is set to zero suddenly - much faster than any other length scale in the theory.\nIt is easier to solve (\\ref{eom}) in the Dirac basis in which $\\gamma_0$ is diagonal. Since the system is translation invariant in the  spatial $x$-direction, the solution ansatz is\n\\begin{eqnarray}\n \\Psi(x,t)=\\left[\\gamma^0\\partial_t-\\gamma^1\\partial_x-im(t)\\right]e^{\\pm i kx}\\Phi(t)\\\n \\label{spinor_antz}\n\\end{eqnarray}\nSubstitution in the EOM gives,\n\\begin{eqnarray}\n \\left[\\partial^2_t+k^2+m(t)^2-i\\gamma^0\\dot{m}(t)\\right]e^{\\pm i kx}\\Phi(t)=0\\nonumber\\\n\\end{eqnarray}\nwhere $\\dot{m}(t)=\\partial_tm(t)$.\n\n$\\Phi(t)$ is solved in the eigenbasis of $\\gamma^0$. For the two eigenvalues of $\\gamma^0$ (1 and -1), the two solutions $\\phi_+(t)$ and $\\phi_-(t)$ are given by,\n\\begin{eqnarray}\n &&\\left[\\partial^2_t+k^2+m(t)^2-i\\dot{m}(t)\\right]\\phi_+(t)=0\\nonumber\\\\\n &&\\left[\\partial^2_t+k^2+m(t)^2+i\\dot{m}(t)\\right]\\phi_-(t)=0\\\n \\label{pppm}\n\\end{eqnarray}\nwhere $\\Phi(t)=\\begin{bmatrix} \\phi_+(t) & \\phi_-(t)  \\end{bmatrix}^\\text{T}$. The eigenstates of $\\gamma^0$ are $u_0=\\begin{bmatrix} 1 \\\\ 0  \\end{bmatrix}$ and $v_0=\\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix}$, they are the spinors in the rest frame.\n\nFor the mass profile (\\ref{mass}), there are two important bases of solutions in which we are interested in. The first one is the `in' basis in which the two independent solutions of the second order linear differential equations become different single frequency modes in the $t\\to -\\infty$ limit. In other words, one solution becomes the negative energy mode and the other solution becomes the positive energy mode. Similarly, there is also an `out' basis of solutions in which one solution becomes the negative energy mode and the other becomes the positive energy mode in the $t\\to\\infty$ limit. Accordingly, we will also have different `in' and `out' creation and annihilation operators.\nConsider the solutions of (\\ref{pppm}) in the two bases to be\n\\begin{eqnarray}\n&&\\phi_{\\pm}(t,k)=\\phi_{in,\\pm p}(t,k)+\\phi_{in,\\pm m}(t,k)\\\\\n&&\\phi_{\\pm}(t,k)=\\phi_{out,\\pm p}(t,k)+\\phi_{out,\\pm m}(t,k)\\\n\\end{eqnarray}\nwhere the limits are\n\\begin{eqnarray}\n&&\\lim_{t\\to-\\infty} \\phi_{in,\\pm p}(t,k)=e^{-i\\omega_{in} t}, \\quad \\lim_{t\\to-\\infty}\\phi_{in,\\pm m}(t,k)=e^{i\\omega_{in} t}\\nonumber\\\\\n&&\\lim_{t\\to\\infty} \\phi_{out,\\pm p}(t,k)=e^{-i\\omega_{out} t}, \\quad \\lim_{t\\to\\infty}\\phi_{out,\\pm m}(t,k)=e^{i\\omega_{out} t}\\nonumber\\\n\\end{eqnarray}\nwhere `p' means {\\it positive energy} and `m' means {\\it negative energy}. The above solutions are the same but written in two different bases for simplicity in the appropriate time limits, they are related by Bogoliubov transformations.\\\\\nBut from (\\ref{pppm}), we see that the equations of $\\phi_+$ and $\\phi_-$ are the complex conjugates of each other, so\n\\begin{eqnarray}\n &&\\phi_{in,+p}(t,k)=\\phi^*_{in,-m}(t,k), \\quad \\phi_{in,+m}(t,k)=\\phi^*_{in,-p}(t,k)\\\\\n \\label{conj1}\n &&\\phi_{out,+p}(t,k)=\\phi^*_{out,-m}(t,k), \\quad \\phi_{out,+m}(t,k)=\\phi^*_{out,-p}(t,k)\\\n \\label{conj2}\n\\end{eqnarray}\nThe Bogoliubov transformations are\n\\begin{eqnarray}\n \\phi_{in,+p}(t,k)&=&\\alpha'_+(k)\\phi_{out,+p}(t,k)+\\beta'_+(k)\\phi_{out,+m}(t,k)\\nonumber\\\\\n \\label{bop}&=&\\alpha'_+(k)\\phi_{out,+p}(t,k)+\\beta'_+(k)\\phi^*_{out,-p}(t,k)\\\\\n \\phi_{in,-p}(t,k)&=&\\alpha'_-(k)\\phi_{out,-p}(t,k)+\\beta'_-(k)\\phi_{out,-m}(t,k)\\nonumber\\\\\n &=&\\alpha'_-(k)\\phi_{out,-p}(t,k)+\\beta'_-(k)\\phi^*_{out,+p}(t,k)\\ \\label{bom}\n\\end{eqnarray}\nwhere $\\alpha'_{\\pm}(k)$ an $\\beta'_{\\pm}(k)$ are actually functions of $|k|$, since the equations of motion have only $k^2$ terms.\n\nNow, suppressing the basis labels `in' and `out' since they apply to both bases, we write the $u_0$ part of $\\Psi(x,t)$ as (upto normalization)\n\\begin{eqnarray}\n \\tilde{U}(x,t;k)=\\left[ \\gamma^0\\partial_t+\\gamma^1\\partial_x -im(t)\\right]e^{ikx}\\phi_{+p}(t)\\begin{bmatrix} 1 \\\\ 0  \\end{bmatrix}\\\n \\label{Uxt}\n\\end{eqnarray}\nAnd the $v_0$ part of $\\Psi(x,t)$ as\n\\begin{eqnarray}\n \\tilde{V}(x,t;k)&=&\\left[ \\gamma^0\\partial_t+\\gamma^1\\partial_x -im(t)\\right]e^{-ikx}\\phi_{-m}(t)\\begin{bmatrix} 0 \\\\ 1  \\end{bmatrix}\\nonumber\\\\\n &=&\\left[ \\gamma^0\\partial_t+\\gamma^1\\partial_x -im(t)\\right]e^{-ikx}\\phi^*_{+p}(t)\\begin{bmatrix} 0 \\\\ 1  \\end{bmatrix}\\\n \\label{Vxt}\n\\end{eqnarray}\nWe can define the spinors as (upto normalization)\n\\begin{eqnarray}\n \\tilde{u}(t,k)&=&\\frac{1}{e^{ikx}\\phi_{+p}(t)}\\tilde{U}(x,t;k)\\nonumber\\\\\n \\tilde{v}(t,k)&=&\\frac{1}{e^{-ikx}\\phi_{-m}(t)}\\tilde{V}(x,t;k)\\nonumber\\\n\\end{eqnarray}\nWith proper normalization, the final Dirac fermion mode expansion is\n\\begin{eqnarray}\n \\Psi(x,t)&=&\\int \\frac{dk}{2\\pi}\\frac{1}{\\sqrt{2\\omega}}\\left[a_k U(x,t;k)+b^\\dagger_k V(x,t;k)\\right]\\nonumber\\\\\n &=&\\int \\frac{dk}{2\\pi}\\frac{1}{\\sqrt{2\\omega}}\\left[a_k u(t;k)e^{ikx}\\phi_{+p}(t)+b^\\dagger_k v(t;k)e^{-ikx}\\phi_{-m}(t)\\right]\\\n \\label{modes}\n\\end{eqnarray}\n\n\\subsection{Bogoliubov transformation of oscillators}\nThe initial mass is taken to be $\\lim_{t\\to -\\infty}m(t)=m$. It is convenient to take the final mass $\\lim_{t\\to\\infty}m(t)$ to be some $m_{out}$, because of the spinor convention (in P\\&S), although we are interested in $m_{out}=0$.\n\nWith time-dependent mass, as mentioned above, the spinors are functionals of $m(t)$, but their normalizations are constants or else they will not solve the Dirac equations. So, we have to differentiate between `in' spinors and `out' spinors. Taking this into account, the mode expansion of $\\Psi(x,t)$ starting from `in' basis to `out' basis is\n\\begin{eqnarray}\n \\Psi(x,t)&=&\\int \\frac{dk}{2\\pi}\\frac{1}{\\sqrt{2\\omega_{in}}}\\left[a_{in,k} u_{in}(k,m) \\phi_{in,+p}(t,k) e^{ikx}+b^\\dagger_{in,k} v_{in}(k,m)\\phi^*_{in,+p}(t,k)e^{-ikx}\\right]\\nonumber\\\\\n &=&\\int \\frac{dk}{2\\pi}\\frac{1}{\\sqrt{2\\omega_{in}}}\\Huge{[} \\{\\alpha'_+(k)a_{in,k} u_{in}(k,m)\\phi_{out,+p}(t,k)+b^\\dagger_{in,-k} v_{in}(-k,m)\\beta'^*_+(k)\\phi_{out,-p}(t,k)\\} e^{ikx}\\nonumber\\\\\n &&\\quad+\\{\\alpha'^*_+(k)b^\\dagger_{in,k} v_{in}(k,m)\\phi^*_{out,+p}(t,k)+a_{in,-k} u_{in}(-k,m)\\beta'_+(k)\\phi^*_{out,-p}(t,k)\\}e^{-ikx}\\huge{]}\\nonumber\\\\\n\\end{eqnarray}\nwhere we have used the facts that the $k$ integral is from $-\\infty$ to $\\infty$ and $\\alpha'_{\\pm}$, $\\beta'_{\\pm}$ and $\\phi_{\\pm p}$ are functions of $|k|$. In $t\\to\\infty$ limit, $m(t)\\to m_{out}$, so,\n\\begin{eqnarray}\n \\lim_{t\\to\\infty}\\Psi(x,t)&=&\\int \\frac{dk}{2\\pi}\\frac{1}{\\sqrt{2\\omega_{out}}}\\sqrt{\\frac{\\omega_{out}}{\\omega_{in}}}\\huge{[} \\{\\alpha'_+(k)a_{in,k} u_{in}(k,m_{out})+b^\\dagger_{in,-k} v_{in}(-k,m_{out})\\beta'^*_+(k)\\} e^{-ik\\cdot x}\\nonumber\\\\\n &&\\quad+\\{\\alpha'^*_+(k)b^\\dagger_{in,k} v_{in}(k,m_{out})+a_{in,-k} u_{in}(-k,m_{out})\\beta'_+(k)\\}e^{ik\\cdot x}\\huge{]}\\nonumber\\\n\\end{eqnarray}\nComparing with the mode expansion in the `out' solution basis in the same limit $t\\to\\infty$,\n\\begin{eqnarray}\n \\lim_{t\\to\\infty}\\Psi(x,t)&=&\\int \\frac{dk}{2\\pi}\\frac{1}{\\sqrt{2\\omega_{out}}}\\left[ a_{out,k} u_{out}(k,m_{out}) \\phi_{out,+p}(t,k) e^{ikx}+b^\\dagger_{out,k} v_{out}(k,m_{out})\\phi^*_{out,+p}(t,k)e^{-ikx}\\right]\\nonumber\\\n\\end{eqnarray}\nwe get the Bogoliubov transformations of the creation and annihilation operators.\n\\begin{eqnarray}\n\\label{aout} a_{out,k}&=&\\alpha_+(k)a_{in,k}+b^\\dagger_{in,-k}\\, \\chi(k,m_{out})\\beta^*_+(k)\\\\\n b^\\dagger_{out,k}&=&\\alpha^*_+(k)b^\\dagger_{in,k} +a_{in,-k}\\, \\tilde{\\chi}(k,m_{out})\\beta_+(k)\\\n \\label{bdout}\n\\end{eqnarray}\nwhere $\\alpha_+(k)=\\sqrt{\\frac{\\omega_{out}(\\omega_{out}+m_{out})}{\\omega_{in}(\\omega_{in}+m_{in})}}\\,\\alpha'_+$ and $\\beta_+(k)=\\sqrt{\\frac{\\omega_{out}(\\omega_{out}+m_{out})}{\\omega_{in}(\\omega_{in}+m_{in})}}\\,\\beta'_+(k)$. Using (\\ref{norspinor})\n\\begin{eqnarray}\n \\chi(k,m_{out}) &=& \\frac{1}{2m_{out}}\\sqrt{\\frac{\\omega_{in}+m_{in}}{\\omega_{out}+m_{out}}}\\,\\bar{u}_{out}(k,m_{out},\\omega_{out})v_{in}(-k,m_{out},-\\omega_{out})\\nonumber\\\\\n &=&\\text{sgn}(k)\\qquad\\qquad \\text{when}\\quad m_{out}\\to0\\\n \\label{chi}\n\\end{eqnarray}\nwhere we have to be careful that $v_{in}(k,m_{out})$ is a functional of the accompanying mode, which is $\\sim e^{-i\\omega_{out}t}$ in the above case. Similarly, with $m_{out}\\to0$,\n\\begin{eqnarray}\n\\tilde{\\chi}(k)&=&-\\frac{1}{2m_{out}}\\sqrt{\\frac{\\omega_{in}+m_{in}}{\\omega_{out}+m_{out}}}\\,\\bar{v}(k,m_{out},\\omega_{out})u(-k,m_{out},\\omega_{out})=\\text{sgn}(k) \\\n\\label{chit}\n\\end{eqnarray}\ntaking into account the normalization of $\\bar{v}_{out}v_{out}=-2m_{out}$. Inverting (\\ref{aout}) and (\\ref{bdout}), we get\n\\begin{eqnarray}\n\\label{ain} a_{in,k}&=&\\alpha^*_+(k)a_{out,k}-\\text{sgn}(k)\\beta^*_+(k)b^\\dagger_{out,-k}\\\\\nb^\\dagger_{in,-k}&=&\\alpha_+(k)b^\\dagger_{out,-k}+\\text{sgn}(k)\\beta_+(k)a_{out,k}\\\n\\label{bdin}\n\\end{eqnarray}\nFrom here on, we will suppress the subscript `out' on creation and annihilation operators, so $a_{out,k}=a_k$, similarly for $b_{out,k}$ and their Hermitian conjugates. Also, since $\\chi(k)$ and $\\tilde{\\chi}(k)$ are simple sign functions, with a slight abuse of the nomenclature, we will call $\\alpha_+(k)$ and $\\beta_+(k)$ as the Bogoluibov coefficients.\nMoreover, $\\chi(k)^2$ and $\\tilde{\\chi}(k)^2$ are identically equal to 1. So, the fermionic anti-commutation relations of the `in' and `out' operators constraint the Bogoluibov coefficients as\n\\begin{eqnarray}\n\\label{ab1}|\\alpha_+(k)|^2+|\\beta_+(k)|^2=1\\ \n\\end{eqnarray}\n\n\\subsection{Explicit solutions}\nIn the `in' basis, for our choice of mass profile, the solutions are\n\\begin{eqnarray}\n\\phi_{in,+p} &=& e^{-i t \\left(\\omega+m\\right)} \\left(e^{-2 \\rho  t}+1\\right)^{-\\frac{i m}{2 \\rho }} \\ _2F_1\\left(\\frac{i\\left(|k|-m-\\omega\\right)}{2 \\rho },-\\frac{i \\left(|k|+m+\\omega\\right)}{2 \\rho };1-\\frac{i \\omega}{\\rho };-e^{2 t\\rho }\\right)\\nonumber\\\\\n\\phi_{in,-m} &=& e^{i t \\left(\\omega-m\\right)} \\left(e^{-2 \\rho  t}+1\\right)^{-\\frac{i m}{2 \\rho }}\\ _2F_1\\left(\\frac{i\\left(-|k|-m+\\omega\\right)}{2 \\rho },\\frac{i \\left(|k|-m+\\omega\\right)}{2 \\rho };\\frac{i \\omega}{\\rho }+1; -e^{2 t\\rho}\\right)\\nonumber\\\n\\end{eqnarray}\nwhere $\\omega = \\sqrt{k^2+m^2}$. While in the `out' basis, the solutions are\n\\begin{eqnarray}\n\\phi_{out,+p} &=& e^{-i |k| t} \\left(e^{-2\\rho t}+1\\right)^{-\\frac{i m}{2 \\rho }} \\ _2F_1\\left(\\frac{i |k|-i m+i \\omega}{2 \\rho },\\frac{i |k|-i m-i \\omega}{2 \\rho };1+\\frac{i |k|}{\\rho };-e^{-2\\rho t}\\right)\\nonumber\\\\\n\\phi_{out,-m} &=& e^{i |k| t} \\left(e^{-2\\rho t}+1\\right)^{-\\frac{i m}{2 \\rho }} \\ _2F_1\\left(\\frac{-i |k|-i m+i \\omega}{2 \\rho },\\frac{-i|k|-i m-i \\omega}{2 \\rho};1-\\frac{i |k|}{\\rho };-e^{-2\\rho t}\\right)\\nonumber\\\n\\end{eqnarray}\n\nUsing the properties of confluent hypergeometric functions $_2F_1$ given in \\cite{Abramowitz}, the Bogoliubov coefficients of the frequency modes as defined in (\\ref{bop}) are\n\\begin{eqnarray}\n\\label{alphap}\n\\alpha'_+ &=& \\frac{\\Gamma \\left(-\\frac{i |k|}{\\rho }\\right) \\Gamma \\left(1-\\frac{i \\omega}{\\rho }\\right)}{\\Gamma \\left(-\\frac{i\\left(|k|+m+\\omega\\right)}{2 \\rho }\\right) \\Gamma \\left(1+\\frac{-i |k|+i m -i \\omega}{2 \\rho }\\right)}\\\\\n\\label{betap}\n\\beta'_+ &=& \\frac{\\Gamma \\left(\\frac{i |k|}{\\rho }\\right) \\Gamma \\left(1-\\frac{i \\omega}{\\rho }\\right)}{\\Gamma \\left(1-\\frac{i\\left(-|k|-m+\\omega\\right)}{2 \\rho }\\right) \\Gamma \\left(-\\frac{i \\left(-|k|+m+\\omega\\right)}{2 \\rho }\\right)}\n\\end{eqnarray}\nIn the sudden limit($\\rho\\to\\infty$). the Bogoliubov coefficients of the frequency modes are\n\\begin{eqnarray}\n\\alpha'_+(k) &=& \\frac{|k|+m_{in}+\\sqrt{k^2+m^2}}{2|k|}\\\\\n\\beta'_+(k) &=& \\frac{|k|-m_{in}-\\sqrt{k^2+m^2}}{2|k|}\\\n\\end{eqnarray}\nAs mentioned above, for a quench starting from the ground state of the massive theory, the naive sudden limit gives a non-normalizable state in the massless theory \\cite{Mandal:2015kxi}. The problem arises only for a quench starting from the ground state. In case the quench is starting from squeezed states of our interest, the naive sudden limit given above works well.\nAs defined in (\\ref{aout}) and (\\ref{bdout}), the Bogoluibov coefficients of the oscillator modes differ from $\\alpha'_+(k)$ and $\\beta'_+(k)$ by an overall factor.\n\\begin{eqnarray}\n\\label{alpha}\n\\alpha_+ &=& \\sqrt{1-\\frac{m}{\\sqrt{k^2+m^2}}}\\frac{\\Gamma \\left(-\\frac{i |k|}{\\rho }\\right) \\Gamma \\left(1-\\frac{i \\omega}{\\rho }\\right)}{\\Gamma \\left(-\\frac{i\\left(|k|+m+\\omega\\right)}{2 \\rho }\\right) \\Gamma \\left(1+\\frac{-i |k|+i m -i \\omega}{2 \\rho }\\right)}\\\\\n\\label{beta}\n\\beta_+ &=& \\sqrt{1-\\frac{m}{\\sqrt{k^2+m^2}}}\\frac{\\Gamma \\left(\\frac{i |k|}{\\rho }\\right) \\Gamma \\left(1-\\frac{i \\omega}{\\rho }\\right)}{\\Gamma \\left(1-\\frac{i\\left(-|k|-m+\\omega\\right)}{2 \\rho }\\right) \\Gamma \\left(-\\frac{i \\left(-|k|+m+\\omega\\right)}{2 \\rho }\\right)}\n\\end{eqnarray}\nIn the sudden limit, they are\n\\begin{eqnarray}\n\\label{alphas} \\alpha_+(k) &=& \\sqrt{1-\\frac{m}{\\sqrt{k^2+m^2}}}\\, \\frac{|k|+m+\\sqrt{k^2+m^2}}{2|k|}\\\\\n\\label{betas} \\beta_+(k) &=& \\sqrt{1-\\frac{m}{\\sqrt{k^2+m^2}}}\\, \\frac{|k|-m-\\sqrt{k^2+m^2}}{2|k|}\\\n\\end{eqnarray}\n\nFor completeness, the expressions of $\\alpha'_-$ and $\\beta'_-$ in (\\ref{bom}) for our particular quench protocol are\n\\begin{eqnarray}\n\\alpha'_- &=& \\frac{\\Gamma \\left(-\\frac{i \\left| k\\right| }{\\rho }\\right) \\Gamma\\left(1-\\frac{i \\omega}{\\rho }\\right)}{\\Gamma \\left(-\\frac{i\\left(\\left| k\\right| -m+\\omega\\right)}{2 \\rho }\\right) \\Gamma \\left(1-\\frac{i \\left(\\left| k\\right| +m+\\omega \\right)}{2 \\rho }\\right)}\\\\\n\\beta'_-&=& \\frac{\\Gamma \\left(\\frac{i \\left| k\\right| }{\\rho }\\right) \\Gamma \\left(1-\\frac{i \\omega}{\\rho }\\right)}{\\Gamma \\left(\\frac{i\\left(\\left| k\\right| +m-\\omega\\right)}{2 \\rho }\\right) \\Gamma \\left(1-\\frac{i \\left(-\\left| k\\right|+m+\\omega\\right)}{2 \\rho }\\right)}\n\\end{eqnarray}\n\n\\section{\\label{qstates}Quenched states}\n\n\\subsection{From ground state}\nStarting from the ground state of the massive theory $|\\Psi\\rangle=|0,in\\rangle$, using Eq (\\ref{ain}), the state in terms of `out' operators is given by\n\\begin{align}\na_{in,k}|\\Psi\\rangle=0 \\quad&\\Rightarrow \\left[\\alpha^*_+(k)a_{k}-\\text{sgn}(k)\\beta^*_+(k)b^\\dagger_{-k}\\right]|\\Psi\\rangle=0\\nonumber\\\\\n\\label{psig}&\\Rightarrow |\\Psi\\rangle = e^{\\sum_k \\text{sgn(k)}\\gamma(k) a^\\dagger_{k}b^\\dagger_{-k}}|0\\rangle\\\\\n\\text{where} &\\quad\\gamma(k)=\\frac{\\alpha^*_+(k)}{\\beta^*_+(k)}\\\n\\label{gg}\n\\end{align}\nwhere we have taken $|0\\rangle$ to be the ground state of `out' oscillators. Using the Baker-Campbell-Hausdorff(BCH) formula derived in appendix (\\ref{bch}), the above state can be written in gCC form. For the particular mass profile (\\ref{mass}), $\\alpha_+(k)$ and $\\beta_+(k)$ are given in (\\ref{alpha}) and (\\ref{beta}). The gCC form which was first obtained in MPS is\n\\begin{eqnarray}\n\\label{psigg}|\\Psi\\rangle =e^{-\\kappa_2 H-\\kappa_4W_4-\\kappa_6W_6-...}|D\\rangle\\\n\\end{eqnarray}\nwhere\n\\begin{gather}\n\\kappa_2=\\frac{1}{2 m}+\\frac{\\pi ^2 m}{12 \\rho ^2}+\\frac{1}{m}\\mathcal{O}\\left(\\frac{m}{\\rho}\\right)^3, \\quad \\kappa_4=-\\frac{1}{12 m^3}+\\frac{\\pi ^2}{24 m \\rho ^2}+\\frac{1}{m^3}\\mathcal{O}\\left(\\frac{m}{\\rho}\\right)^3,\\nonumber\\\\\n\\kappa_6=\\frac{3}{80 m^5}-\\frac{\\pi ^2}{96 m^3 \\rho ^2}+\\frac{1}{m^5}\\mathcal{O}\\left(\\frac{m}{\\rho}\\right)^3,...\\\n\\label{kgs}\n\\end{gather}\nand $|D\\rangle$ is the Dirichelet state and the explicit expression is in Appendix \\ref{bstate}. It should be noted that since the mass does not go to zero at any finite time, the above state should is only valid in sufficiently long time limit and the correction due to the non-vanishing mass is $\\mathcal{O}(e^{-\\rho t})$.\n\n\\subsection{From squeezed states: CC state and gCC states}\nWe could start with specially prepared squeezed states so that after the quench, the states become CC states or gCC states. Here, we will consider only the simple case of sudden quench ($\\rho\\to\\infty$). For our aim of creating a CC state or a gCC state, finite `$\\rho$' quenches are an unnecessary complication.\\\\\n\nWe start with a squeezed state of `in' modes\n\\begin{eqnarray}\n |S\\rangle = \\exp\\left(\\sum_{k=-\\infty}^{\\infty}f(k)a^\\dagger_{in,k}b^\\dagger_{in,-k}\\right)|0,in\\rangle\\\n \\label{sqg}\n\\end{eqnarray}\nwhere unlike $\\gamma(k)$, $f(k)$ need not be an even function of $k$, but $|f(k)|^2$ is an even function of $k$.\n\nIt is easier to work with $|S\\rangle$ as an operator relation. $|S\\rangle$ can also be defined as\n\\begin{eqnarray}\n \\label{sstate} \\tilde{a}_{k}|S\\rangle=\\tilde{b}_{k}|S\\rangle=0\\\n \\text{and} && \\left\\{\\tilde{a}_k,\\tilde{a}^\\dagger_{k'}\\right\\}=\\left\\{\\tilde{b}_{-k},\\tilde{b}^\\dagger_{-k'}\\right\\}=\\delta(k-k')\\\n \\label{tacom}\n\\end{eqnarray}\nwhere the new operators in terms of the out modes using (\\ref{ain}) and (\\ref{bdin}) are\n\\begin{align}\n\\tilde{a}_{k}&=\\frac{1}{\\sqrt{(1+|f(k)|^2)}}a_{in,k}-\\frac{f(k)}{\\sqrt{(1+|f(k)|^2)}}b^\\dagger_{in,-k}\\nonumber\\\\\n&=A^*(k)a_{out,k}-\\text{sgn}(k)B^*(k)b_{out,-k}\\nonumber\\\\\n\\tilde{b}_{-k}&=\\frac{1}{\\sqrt{(1+|f(k)|^2)}}b_{in,-k}+\\frac{f(k)}{\\sqrt{(1+|f(k)|^2)}}a^\\dagger_{in,k}\\nonumber\\\\\n&=A^*(k)b_{out,-k}+\\text{sgn}(k)B^*(k)a^\\dagger_{out,k}\\\n\\label{tab}\n\\end{align}\nwhere $A(k)$ and $B(k)$ are the Bogoliubov coefficients for the transformation from `{\\it{tilde}}' operators to `out' operators and are given by\n\\begin{gather}\n\\label{AnB}A(k)=\\frac{\\alpha_+(k)-\\text{sgn}(k)\\beta_+^*(k)f^*(k)}{\\sqrt{(1+|f(k)|^2)}},\\quad B(k)=\\frac{\\beta_+(k)+\\text{sgn}(k)\\alpha_+^*(k)f^*(k)}{\\sqrt{(1+|f(k)|^2)}}\\\\\n|A(k)|^2+|B(k)|^2=1\\\n\\label{AB1}\n\\end{gather}\nNow using the BCH formula (\\ref{BCHeqn}) from appendix (\\ref{bch}),\n\\begin{gather}\n\\label{sqgo}|S\\rangle=\\exp\\left\\{-\\sum_k \\tilde{\\kappa}(k)\\left(a^\\dagger_{out,k}a_{out,k}+b^\\dagger_{out,k}b_{out,k}\\right)\\right\\}|D\\rangle\\\\\n\\text{where} \\quad \\tilde{\\gamma}(k)=\\frac{B^*(k)}{A^*(k)}, \\quad\\text{and}\\quad \\tilde{\\kappa}(k)=-\\frac{1}{2}\\log(\\tilde{\\gamma}(k))\\nonumber\\\n\\end{gather}\nFor a CC state, i.e., so that $|S\\rangle$ in eqn (\\ref{sqgo}) is $e^{-\\kappa_2 H}|D\\rangle$, $f(k)$ should be tuned as\n\\begin{eqnarray}\n f(k)=\\frac{\\left(\\sqrt{k^2+m^2}+m\\right) \\cosh (\\kappa_2 k )-k \\sinh (\\kappa_2 k)}{\\left(\\sqrt{k^2+m^2}+m\\right) \\sinh (\\kappa_2 k)+k\\cosh (\\kappa_2 k )}\\\n \\label{fcc}\n\\end{eqnarray}\nStarting with\n\\begin{eqnarray}\n f(k)=\\frac{k-k\\, e^{2 \\left| k\\right|  \\left(\\kappa_2+\\kappa_4 k^2\\right)}+\\text{sgn}(k)\\left(\\sqrt{k^2+m^2}+m\\right)\\left(e^{2 \\left| k\\right|  \\left(\\kappa_2+\\kappa_4k^2\\right)}+1\\right)}{\\left| k\\right|  \\left(e^{2 \\left| k\\right|  \\left(\\kappa_2 +\\kappa_4 k^2 \\right)}+1\\right)+\\left(\\sqrt{k^2+m^2}+m\\right) \\left(e^{2 \\left| k\\right|  \\left(\\kappa_2+\\kappa_4 k^2\\right)}-1\\right)}\\\n \\label{fgcc}\n\\end{eqnarray}\nwe get a gCC state of the form $e^{-\\kappa_2 H-\\kappa_4 W_4}|D\\rangle$, where as mentioned earlier, $W_4$ is the conserved charge of the $\\mathcal{W}_4$ current of free Dirac fermions{\\footnote{For the action (\\ref{action}), $H=\\sum_k |k|(a^\\dagger_k a_k+b^\\dagger_k b_k)$ or $H=\\int\\frac{dk}{2\\pi} |k|(a^\\dagger_k a_k+b^\\dagger_k b_k)$. $\\mathcal{W}_4$ has been normalized so that $W_4=\\sum_k |k|^3(a^\\dagger_k a_k+b^\\dagger_k b_k)$ or $W_4=\\int \\frac{dk}{(2\\pi)^3}|k|^3(a^\\dagger_k a_k+b^\\dagger_k b_k)$ in the continuum limit.}. Note that $f(k)$ are odd functions of $k$.\nFor future reference, we can invert Eq (\\ref{tab}) and we write down the `in' and `out' operators in terms of the `{\\it{tilde}}' operators.\n\\begin{gather}\n \\label{aintota}a_{in,k}=\\frac{1}{\\sqrt{(1+|f(k)|^2)}}\\tilde{a}_k+\\frac{f(k)}{\\sqrt{(1+|f(k)|^2)}}\\tilde{b}^\\dagger_{-k}\\\\\n \\label{bintotb} b^\\dagger_{in,-k}=\\frac{1}{\\sqrt{(1+|f(k)|^2)}}\\tilde{b}^\\dagger_{-k}-\\frac{f^*(k)}{\\sqrt{(1+|f(k)|^2)}}\\tilde{a}_k\\\\\n \\label{aouttota} a_{out,k}=A(k)\\tilde{a}_k+\\text{sgn}(k)B^*(k)\\tilde{b}^\\dagger_{-k}\\\\\n \\label{bdouttotbd}b^\\dagger_{out,-k}=A^*(k)\\tilde{b}^\\dagger_{-k}-\\text{sgn}(k)B(k)\\,\\tilde{a}_k\\\n\\end{gather}\n \n\\section{\\label{ed}Energy density}\nIn the post-quench theory, the occupation number is given by\n\\begin{eqnarray}\n\\hat{N}_k=a_k^\\dagger a_k +b_k^\\dagger b_k\\ \n\\end{eqnarray}\nusing the Bogoliubov transformations (\\ref{aouttota}) and (\\ref{bdouttotbd}) and definition of $|\\tilde{0}\\rangle$ in (\\ref{sstate}), the expectation value of the occupation number is given by\n\\begin{eqnarray}\n N_k&=&\\lim_{t\\to\\infty}\\langle\\tilde{0}|a_k^\\dagger a_k +b_k^\\dagger b_k|\\tilde{0}\\rangle\\nonumber\\\\\n &=&B^*(k)B(k)\\langle\\tilde{0}|\\tilde{b}_{-k}^\\dagger\\tilde{b}_{-k}|\\tilde{0}\\rangle+B^*(-k)B(-k)\\langle\\tilde{0}|\\tilde{a}_{-k}^\\dagger\\tilde{a}_{-k}|\\tilde{0}\\rangle+...\\nonumber\\\\\n &=& B^*(k)B(k)+B^*(-k)B(-k)\\\n\\end{eqnarray}\nThe expression of $B(k)$ is given in (\\ref{AnB}). For ground state, we have to use $f(k)=0$ in the expression of $B(k)$. So, energy density of the post-quench state is given by\n\\begin{eqnarray}\n E=\\int_{-\\infty}^\\infty\\frac{dk}{2\\pi}|k|\\left[B^*(k)B(k)+B^*(-k)B(-k)\\right]\\\n\\end{eqnarray}\n\n\\subsection*{Ground state quench}\nFor ground state quench, the occupation number is given by\n\\begin{eqnarray}\n N_k&=&\\lim_{t\\to\\infty}\\langle 0,in|\\hat{N}_k|0,in\\rangle=|\\beta_+(k)|^2+|\\beta_+(-k)|^2\\nonumber\\\n\\end{eqnarray}\nSince $\\alpha_+(k)$ and $\\beta_+(k)$ are even functions of $k$. Using (\\ref{ab1}), (\\ref{gg}) and (\\ref{forbch}), we have\n\\begin{eqnarray}\n|\\beta_+(k)|^2=\\frac{|\\gamma(k)|^2}{1+|\\gamma(k)|^2},\\qquad |\\gamma(k)|^2=e^{-4\\kappa(k)}\\\n\\end{eqnarray}\nHence, the occupation number in the ground state in the asymptotically long time limit is given by\n\\begin{eqnarray}\n N_k=\\frac{2}{e^{4\\kappa(k)}+1}\\\n\\end{eqnarray}\nThis is the occupation number in a GGE defined as\n\\begin{eqnarray}\n\\text{Tr}\\,e^{-\\sum_k 4\\kappa(k)\\hat{N}_k}=\\text{Tr} e^{- 4\\kappa_2 H-4\\kappa_4 W_4-\\kappa_6 W_6-\\cdots}\\\n\\end{eqnarray}\nwhere the $\\kappa$'s are given in (\\ref{kgs}).\nUsing the expressions of $\\beta_+(k)$ from (\\ref{beta}), the explicit expression of the occupation number is \n\\begin{eqnarray}\n N_k&=&\\text{csch}\\left(\\frac{\\pi  k}{\\rho }\\right) \\left(\\cosh \\left(\\frac{\\pi  m}{\\rho }\\right)-\\cosh \\left(\\frac{\\pi  \\left(k-\\sqrt{k^2+m^2}\\right)}{\\rho }\\right)\\right) \\text{csch}\\left(\\frac{\\pi  \\sqrt{k^2+m^2}}{\\rho }\\right)\\nonumber\\\\\n &\\xrightarrow{\\rho\\to\\infty}&1-\\frac{k}{\\sqrt{k^2+m^2}}\\nonumber\\\n\\end{eqnarray}\nIt is interesting that in $m\\to\\infty$ limit, $N_k\\to1$, not 2. This is because $\\lim_{\\rho\\to\\infty}|\\alpha_+(k)|^2=1\/2$ and we have the constraint $|\\alpha_+(k)|^2+|\\beta_+(k)|^2=1$.\\\\\nFor arbitrary $\\rho$, the energy density cannot be calculated in closed form. In the sudden limit $\\rho\\to\\infty$, the energy density diverges as $\\log(\\Lambda)$ where $\\Lambda$ is the UV cutoff. Hence, all other $W$ charges also diverge in the sudden limit. Hence, naively taking $\\rho\\to\\infty$ produce a non-renormalizable state. So, the sudden limit has to be taken as in MPS where $m\/\\Lambda\\to0$ while $m\/\\rho\\to\\epsilon^+$. Simply put, the quench rate parameter $\\rho$ should be much small than the UV cut-off.\n\n\\subsection*{Squeezed state quench: CC and gCC states}\nFor CC state given by (\\ref{fcc}), the expectation value of occupation number is given by\n\\begin{eqnarray}\n N_k=\\frac{2}{1+e^{4 \\kappa_2\\left|k\\right|}}\\\n\\end{eqnarray}\nThis is the occupation number of fermions in a thermal ensemble of temperature $1\/\\beta=1\/4\\kappa_2$. The enengy density is\n\\begin{eqnarray}\n E=\\int_{-\\infty}^{\\infty}\\frac{dk}{2\\pi}N_k=\\frac{\\pi }{96 \\kappa_2 ^2}\\\n\\end{eqnarray}\nSimilarly, for gCC state given by (\\ref{fgcc}), the expectation value of occupation number is given by\n\\begin{eqnarray}\n N_k=\\langle gCC|\\hat{N}_k|gCC\\rangle=\\frac{2}{1+e^{4 \\kappa_2\\left|k\\right|+4\\kappa_4\\left|k\\right|^3}}\\\n\\end{eqnarray}\nThis is same as the occupation number of fermions in a generalised Gibbs ensemble of temperature $1\/\\beta=4\\kappa_2$ and chemical potential $\\mu_4=4\\kappa_4$ of $W_4$ charge. The enengy density cannot be calculated in closed form.\n\n\\section{\\label{crf}Correlation functions}\nSince our theory is a free theory, all the observables can be explicitly calculated. In the following subsections we calculate $\\langle\\psi^\\dagger(r,t)\\psi(0,t)\\rangle$ correlation functions for the three different states obtained above. The quench process cannot differentiate between holomorphic dof(`left-movers') and anti-holomorphic dof(`right-movers'), so $\\langle\\bar{\\psi}^\\dagger(0,t)\\bar{\\psi}(r,t)\\rangle$ is equal to $\\langle\\psi^\\dagger(r,t)\\psi(0,t)\\rangle$ and they are time independent quantities.{\\footnote{A simple reason why these quantities are time independent is the fact that they are holomorphic-holomorphic and antiholomorphic-antiholomorphic quantities and they cannot `see' the presence of the boundary state $|D\\rangle$. They are already thermalized\/equilibrated.}} We also calculated $\\langle\\bar{\\psi}^\\dagger(r,t)\\psi(0,t)\\rangle$ which has non-trivial time-dependence. Also as expected, $-\\langle\\psi^\\dagger(0,t)\\bar{\\psi}(r,t)\\rangle$ is the complex conjugate of $\\langle\\bar{\\psi}^\\dagger(r,t)\\psi(0,t)\\rangle$. Since, we are calculating equal-time correllation functions, so for example for $\\langle\\psi^\\dagger(r,t)\\psi(0,t)\\rangle$, we would rather be calculating $\\frac{1}{2}\\langle\\psi^\\dagger(r,t)\\psi(0,t)-\\psi(0,t)\\psi^\\dagger(r,t)\\rangle$.\\\\\nUsing the Bogoluibov transformations (\\ref{aouttota}) and (\\ref{bdouttotbd}) in the chiral mode expansions (\\ref{crep}) and (\\ref{crepb})  we get \n\\begin{align}\n\\label{psioutt}\\psi(w)&=\\int_0^\\infty \\frac{dk}{2\\pi} \\Big[A(k) \\tilde{a}_{k}e^{-ikw}+\\text{sgn}(k)B^*(k)\\tilde{b}^\\dagger_{-k}e^{-ikw}+A^*(-k)\\tilde{b}^\\dagger_{k}e^{ikw}+\\text{sgn}(k)B(-k)\\tilde{a}_{-k}e^{ikw}\\Big]\\\\\n\\bar{\\psi}(\\bar{w})&=\\int^{\\infty}_0 \\frac{dk}{2\\pi} \\Big[A(-k) \\tilde{a}_{-k}e^{-ik\\bar{w}}-\\text{sgn}(k)B^*(-k)\\tilde{b}^\\dagger_{k}e^{-ik\\bar{w}}-A^*(k)\\tilde{b}^\\dagger_{-k}e^{ik\\bar{w}}+\\text{sgn}(k)B(k)\\tilde{a}_{k}e^{ik\\bar{w}}\\Big]\\\n\\label{bpsioutt}\n\\end{align}\nwhere $w=t-x$ and $\\bar{w}=t+x$. For the ground state quench, $f(k)=0,\\,\\tilde{a}_{k}=a_{in,k},\\,\\tilde{b}=b_{in,k}$ and $|\\tilde{0}\\rangle=|0,in\\rangle$.\\\\\nFor a general $f(k)$ corresponding to some $|\\tilde{0}\\rangle$, the correlation functions are\n \\begin{eqnarray}\n \\label{psidpsiS}\\langle\\tilde{0}|\\psi^\\dagger(0,t)\\psi(r,t)|\\tilde{0}\\rangle &=& \\frac{1}{2}\\int_{0}^{\\infty}\\frac{dk}{2\\pi}\\left[(2|B(k)|^2-1)e^{ikr}-(2|B(-k)|^2-1)e^{-ikr}\\right]\\\\\n \\label{bpsidpsiS}\\langle\\tilde{0}|\\bar{\\psi}^\\dagger(0,t)\\psi(r,t)|\\tilde{0}\\rangle&=& -\\int_0^{\\infty}\\frac{dk}{2\\pi}\\left[\\text{sgn}(k)A^*(-k)B(-k)e^{ik(2t-r)}+\\text{sgn}(k)A(k)B^*(k)e^{-ik(2t-r)}\\right]\\nonumber\\\\\n \\end{eqnarray}\nwhere we have used (\\ref{AB1}) to write $A(k)$ in terms of $B(k)$ in the first equation.\\\\\n\\subsection*{Ground state quench:}\nTaking careful limit, for ground state quench, we have\n\\begin{eqnarray}\n \\langle 0,in|\\psi^\\dagger(0,t)\\psi(r,t)|0,in\\rangle &&= -\\frac{1}{2}\\int_{-\\infty}^{\\infty}\\frac{dk}{2\\pi}\\frac{\\left| k\\right| }{\\sqrt{k^2+m^2}}\\nonumber\\\\\n &&=\\frac{1}{4} m \\left[\\pmb{L}_{-1}(m r)-I_1(m r)\\right]\\nonumber\\\\\n &&\\xrightarrow{m\\to\\infty}\\frac{1}{2 \\pi  m r^2}+\\frac{3}{2 \\pi  m^3 r^4}+O\\left(\\frac{1}{m^4}\\right)\\\\\n \\langle\\tilde{0}|\\bar{\\psi}^\\dagger(0,t)\\psi(r,t)|\\tilde{0}\\rangle&&=\\int_0^{\\infty}\\frac{dk}{2\\pi}\\,\\frac{i \\,\\text{sgn}(k) \\,m\\sin (k (2 t-r))}{\\sqrt{k^2+m^2}}\\nonumber\\\\\n &&=-\\frac{i m}{4} \\, \\left[\\text{sgn}(r-2 t) I_0(m (r-2 t))-\\pmb{L}_0(m (r-2 t))\\right]\\nonumber\\\\\n &&\\xrightarrow[t>r\/2]{m\\to\\infty}\\frac{i}{2 \\pi  (2 t-r)}+\\frac{i}{2 \\pi  m^2 (2 t-r)^3}+O\\left(\\frac{1}{m^4}\\right)\\\n\\end{eqnarray}\nwhere $I_\\nu(x)$ is Modified Bessel Function of the First Kind and $\\pmb{L}_\\nu(x)$ is Modified Struve Function.\n\\subsection*{Quenched squeezed state - CC state:}\nFor CC state, all the calculations are done in $|S_{CC}\\rangle$ defined as the state (\\ref{sqg}) with the expression of $f(k)$ given in (\\ref{fcc}).\n\\begin{eqnarray}\n \\langle CC|\\psi^\\dagger(0,t)\\psi(r,t)|CC\\rangle&=&-i\\int_{0}^{\\infty}\\frac{dk}{2\\pi}\\,\\tanh (2\\kappa_2 |k|)\\sin(kr)\\\\\n \\label{thermform}&=&-i\\int_{0}^{\\infty}\\frac{dk}{2\\pi}\\,\\sin(kr)\\left[\\frac{1}{e^{4\\kappa_2 |k|}+1}-\\frac{1}{2}\\right]\\\\\n &=& -\\frac{i\\, \\text{csch}\\left(\\frac{\\pi  r}{4 \\kappa _2}\\right)}{8 \\kappa _2}\\\\\n\\label{ccr=0} \\langle CC|\\bar{\\psi}^\\dagger(0,t)\\psi(r,t)|CC\\rangle&=&-i\\int_0^{\\infty}\\frac{dk}{2\\pi}\\,\\text{sech}(2 k \\kappa_2 ) \\cos (k(2 t-r))\\\\\n &=&-\\frac{i\\, \\text{sech}\\left(\\frac{\\pi  (2t-r)}{4 \\kappa_2 }\\right)}{8 \\kappa_2 }\\\n\\end{eqnarray}\nThese are exactly what have been calculated using BCFT techniques \\cite{Calabrese:2006quench}. It is evident from (\\ref{thermform}) that $\\psi^\\dagger\\psi$ expectation value is already the thermal expectation value at temperature $T=1\/\\beta =1\/(4\\kappa_2)$, i.e., it is already thermalized.\n\\subsection*{Quenched squeezed state - gCC state with $W_4$:}\nSimilarly, for gCC state, all the calculations are done in $|S_{fCC}\\rangle$ defined as the state (\\ref{sqg}) with the expression of $f(k)$ given in (\\ref{fgcc}).\n\\begin{eqnarray}\n \\langle\\psi^\\dagger(0,t)\\psi(r,t)\\rangle_{gCC}&=&-i\\int_{0}^{\\infty}\\frac{dk}{2\\pi}\\,\\tanh\\left(2\\kappa_2 |k| +2\\kappa_4 |k|^3\\right)\\,\\sin(kr)\\\\\n\\label{thermform2} &=&-i\\int_{0}^{\\infty}\\frac{dk}{2\\pi}\\,\\sin(kr)\\left[\\frac{1}{e^{4\\kappa_2 |k|+4\\kappa_4 |k|^3}+1}-\\frac{1}{2}\\right]\\\\\n \\langle\\bar{\\psi}^\\dagger(r,t)\\psi(0,t)\\rangle_{gCC}&=&-i\\int_0^{\\infty}\\frac{dk}{2\\pi}\\,\\text{sech}(2 \\kappa_2 k+2\\kappa_4 k^3 ) \\cos (k(2 t-r))\\\n \\end{eqnarray}\nAgain, it is evident from (\\ref{thermform2}) that $\\psi^\\dagger\\psi$ expectation value is already thermalized into the expectation value in a GGE with $T=1\/\\beta =1\/4\\kappa_2$ and $\\mu=4\\kappa_4$. A possible way of evaluating these integrals (which yield no closed form answer) is via the residue theorem. The integrands in both cases, have poles at the solutions of $2\\kappa_2 k +2\\kappa_4 k^3= \\frac{2n+1}{2} i\\pi$, where $n\\in\\mathbb{Z}$. These poles and their residues have been treated in detail in \\cite{Mandal:2015kxi}. The sum of residues is still an infinite sum which cannot be performed. In the perturbative regime ($\\kappa_4\/\\kappa_2^3<<1$), we see that our correlators match the general form presented in \\cite{Mandal:2015jla} with $h=1\/2$ as expected.\n\n\\gap3\n\nAs expected form MPS, here in the fermionic theory also we see the UV\/IR mixing. For the ground state quench, all the charges affect the long distance and long time limit of the correlators. This is explicit seen in the case of gCC state with $W_4$ charge only. The long time and large distance limit or the correlators are very much dependent upon $k_4$, although a naive Wilsonian RG argument would show that $k_4$ is an irrelevant coupling. \n\n\\section{\\label{secEE_CC}Exact Growth of Entanglement in CC state}\nWe will consider only a finite single interval or subsystem A, with its endpoints at $(w_1,\\bar{w}_1)$ and $(w_2,\\bar{w}_2)$ in light-cone coordinates, or $(0,t)$ and $(r,t)$ in space and time coordinates. Using the replica trick (\\cite{Holzhey:1994we}, \\cite{Calabrese:2004eu}), the $n\\textsuperscript{th}$ R\\'enyi entropy $S_n(A)$ of the interval is given by the logarithm of the expectation value of twist and antitwist operators inserted at the end-points.\n\\begin{eqnarray}\n\\label{sna} S_n(A)=\\frac{1}{1-n}\\log\\langle\\Psi(t)|\\mathcal{T}_{n}(w_1,\\bar{w}_1)\\tilde{\\mathcal{T}}_{n}(w_2,\\bar{w}_2)|\\Psi(t)\\rangle\\\n\\end{eqnarray}\nThe entanglement entropy(EE) $S_{A}$ is given by $\\lim_{n\\to1}S_n(A)$.\nWe can diagonalize the twist operators and write them as products of twist fields. Hence,\n\\begin{eqnarray}\n\\label{tntkn} \\mathcal{T}_{n}(w,\\bar{w})=\\prod_{k=-(n-1)\/2}^{k=(n-1)\/2}\\mathcal{T}_{k,n}(w,\\bar{w}), \\quad  \\tilde{\\mathcal{T}}_{n}(w,\\bar{w})=\\prod_{k=-(n-1)\/2}^{k=(n-1)\/2}\\tilde{\\mathcal{T}}_{k,n}(w,\\bar{w})\\\n\\end{eqnarray}\nIn CC state, in Heisenberg picture, the quantity of our interest is\n\\begin{eqnarray}\n\\label{tttf} Z_{k}=\\langle D_{f}|e^{-\\kappa_2 H_{f}}\\mathcal{T}_{k,n}(0,t)\\tilde{\\mathcal{T}}_{k,n}(r,t)e^{-\\kappa_2 H_{f}}|D_{f}\\rangle\\\n\\end{eqnarray}\nThe subscript `f' means we are working in the fermionic theory and the subscript `b' would mean we are working in the bosonic theory. To find the exact expression of the entanglement entropy of a spatial region in our free fermionic CFT, we will use the method using bosonization described in \\cite{Casini:2005rm}. Moreover, as shown in Appendix(\\ref{bbs}), Dirichlet state $|D_{f}\\rangle$ in fermionic theory corresponds to a Dirichlet state in the bosonic theory $|D_{b}\\rangle$ and $H_{f}$ corresponds to $H_{b}$. So, we get\n\\begin{eqnarray}\n\\label{tttf} Z_{k}&=&\\langle D_{b}|e^{-\\kappa_2 H_{b}}e^{i\\sqrt{4\\pi}\\frac{k}{n}\\left(\\phi(0,t)-\\phi(r,t)\\right)}e^{-\\kappa_2 H_{b}}|D_{b}\\rangle\\\n\\end{eqnarray}\nThis is a free scalar theory in a strip geometry with Dirichlet boundary conditions and operator insertions at $(0,t)$ and $(r,t)$. It can be calculated explicitly \n\\begin{eqnarray}\n \\log\\left[Z_{k}\\right]&=& -4\\pi\\,\\frac{2k^2}{n^2}\\left(\\langle\\phi(0,t)\\phi(0,t)\\rangle-\\langle\\phi(0,t)\\phi(r,t)\\rangle\\right)\\\n\\end{eqnarray}\n\\noindent The $n\\textsuperscript{th}$ R\\'enyi entropy of interval A is given by\n\\begin{eqnarray}\n  S_{n}(A)&=&-4\\pi\\frac{1}{1-n}\\sum_{k=-(n-1)\/2}^{k=(n-1)\/2}\\frac{2k^2}{n^2}\\left(\\langle\\phi(0,t)\\phi(0,t)\\rangle-\\langle\\phi(0,t)\\phi(r,t)\\rangle\\right)\\nonumber\\\\\n \\label{RE} &=& 4\\pi\\,\\frac{n+1}{6 n}\\,\\left(\\langle\\phi(0,t)\\phi(0,t)\\rangle-\\langle\\phi(0,t)\\phi(r,t)\\rangle\\right)\\\n\\end{eqnarray}\nTaking $n\\to1$ limit, we get the entanglement entropy,\n\\begin{eqnarray}\n\\label{EE} S_{A}=4\\pi\\,\\frac{1}{3}\\,\\left(\\langle\\phi(0,t)\\phi(0,t)\\rangle-\\langle\\phi(0,t)\\phi(r,t)\\rangle\\right)\\\n\\end{eqnarray}\n\n\\gap3\n\\noindent{\\small{\\bf{Remark on winding number:}} While the free boson considered in MPS \\cite{Mandal:2015kxi} is the uncompactified free boson, the boson in (\\ref{tttf}) is a compactified free boson. So, Hamiltonian of the compactifed boson has zero mode terms but the winding number is not important for our analysis.\nIn the large system size limit($L\\to\\infty$), the zero modes vanished.\nEven if we are taking the limiting case of a finite size system, the zero momentum modes do not play any role in our calculation. Using the mode expansion of the boson $\\phi(w,\\bar{w})=\\varphi(w)+\\bar{\\varphi}(\\bar{w})$ in \\cite{Senechal:1999us},\n\\begin{eqnarray}\n\\varphi(w)&=&Q+\\frac{P}{2L}\\,w +\\sum_{n>0}\\frac{1}{\\sqrt{4\\pi n}}\\left(d_n e^{-inw}+d^{\\dagger}_ne^{inw}\\right)\\\\\n\\bar{\\varphi}(\\bar{w})&=&\\bar{Q}+\\frac{\\bar{P}}{2L}\\,\\bar{w}+\\sum_{n>0}\\frac{1}{\\sqrt{4\\pi n}}\\left(d_{-n} e^{-in\\bar{w}}+d^{\\dagger}_{-n}e^{in\\bar{w}}\\right)\\\n\\end{eqnarray}\nFirst, $Q$ and $\\bar{Q}$ are cancelled identically in (\\ref{tttf}).\nMoreover, by bosonization formulae \\cite{Senechal:1999us, vonDelft:1998pk},\n\\begin{align}\n P&=\\sqrt{4\\pi}N_f & \\bar{P}&=\\sqrt{4\\pi}\\bar{N}_f\\\\\n N_f&=J_0=-\\sum_{k=0}^{\\infty}\\left[a^\\dagger_ka_k-b^\\dagger_kb_k\\right] & \\bar{N}_f&=\\bar{J}_0=-\\sum_{k=0}^{\\infty}\\left[a^\\dagger_{-k}a_{-k}-b^\\dagger_{-k}b_{-k}\\right]\\\n\\end{align}\n But for our particular CC state, from (\\ref{JbJ0}), $N_f|CC_f\\rangle = 0$ and $\\bar{N}_f|CC_f\\rangle = 0$. Now, $P$ and $\\bar{P}$ commute with all the other bosonic creation and annihilation operators of non-zero momentum, hence they don't play any role in the calculation of (\\ref{tttf}).\n If we still keep the system size finite, the winding number would be important to interpret the stationary limit as a thermal ensemble. But we must take the $L\\to\\infty$ limit, if we want to examine the stationary limit. In other words, $L$ is the largest length scale in our theory and time $t<<L$.}\n\\gap3\n\nThe bosonic propagator in CC state has been calculated in \\cite{Mandal:2015kxi}. It is given by\n\\begin{eqnarray}\n\\label{phiprop}\\langle CC|\\phi(0,t)\\phi(r,t)|CC\\rangle=-\\frac{1}{8\\pi}\\log \\left(\\frac{2\\sinh ^2\\left(\\frac{\\pi  r}{4 \\kappa_2 }\\right)}{\\cosh \\left(\\frac{\\pi  r}{2 \\kappa_2 }\\right)+\\cosh \\left(\\frac{\\pi  t}{\\kappa_2 }\\right)}\\right)\\\n\\end{eqnarray}\n$r\\to 0$ gives the UV divergence of scalar field theory in 2D spacetime. The $n\\textsuperscript{th}$ R\\'enyi entropy and entanglement entropy of interval A in CC state is given by\n\\begin{eqnarray}\n \\label{CC_RE} S_{n}(A)&=&\\frac{n+1}{12 n}\\,\\left[\\log \\left(\\frac{\\sinh ^2\\left(\\frac{\\pi  r}{4 \\kappa_2 }\\right)\\left(1+\\cosh \\left(\\frac{\\pi  t}{\\kappa_2 }\\right)\\right)}{\\cosh \\left(\\frac{\\pi  r}{2 \\kappa_2 }\\right)+\\cosh \\left(\\frac{\\pi  t}{\\kappa_2 }\\right)}\\right) -\\lim_{\\epsilon\\to 0+}2 \\log (\\epsilon )- \\log \\left(\\frac{\\pi^2}{16 \\kappa_2^2}\\right)\\right]\\nonumber\\\\\n \\\\\n\\label{CC_EE} S_{A}&=&\\frac{1}{3}\\,\\left[\\frac{1}{2}\\log \\left(\\frac{\\sinh ^2\\left(\\frac{\\pi  r}{4 \\kappa_2 }\\right)\\left(1+\\cosh \\left(\\frac{\\pi  t}{\\kappa_2 }\\right)\\right)}{\\cosh \\left(\\frac{\\pi  r}{2 \\kappa_2 }\\right)+\\cosh \\left(\\frac{\\pi  t}{\\kappa_2 }\\right)}\\right) -\\lim_{\\epsilon\\to 0+}\\log (\\epsilon )-\\frac{1}{2} \\log \\left(\\frac{\\pi^2}{16 \\kappa_2 ^2}\\right)\\right]\\\n\\end{eqnarray}\nTaking the stationary limit $t\\to\\infty$ gives the entanglement entropy of A in a thermal ensemble at temperature $T=1\/\\beta=1\/(4\\kappa_2)$.\n\\begin{eqnarray}\n S_{A}&=&\\frac{1}{3}\\,\\left[\\log \\left(\\sinh \\left(\\frac{\\pi  r}{4 \\kappa_2 }\\right)\\right)-\\lim_{\\epsilon\\to 0+}\\log (\\epsilon )-\\frac{1}{2} \\log \\left(\\frac{\\pi^2}{16 \\kappa_2^2}\\right)\\right]\\\n\\end{eqnarray}\nThis exactly matches the thermal value which has been calculated using CFT techniques \\cite{Calabrese:2004eu}. It is fixed only by the temperature, the central charge of the CFT($c=1$ for dirac fermion), and `$r$' the length of the interval.\n\\begin{figure}%\n    \\centering\n    \\subfloat[Low effective temperature, $\\kappa_2=1$]{{\\includegraphics[width=7cm]{CC_EEkap1.eps} }}%\n    \\qquad\n    \\subfloat[High effective temperature, $\\kappa_2=0.2$]{{\\includegraphics[width=7cm]{CC_EEkapdot2.eps} }}%\n    \\caption{Entanglement entropy growth of an interval(r=5) in CC state.}%\n    \\label{CC_EE}%\n\\end{figure}\nTaking the high temperature limit $\\kappa_2\\to 0$, we get the extensive thermal entropy formula $S_{therm}=\\frac{1}{3}\\frac{\\pi r}{\\beta}$.\n\nBesides the thermalization, the most interesting aspect of figure (\\ref{CC_EE}) is that the entanglement entropy grows monotonically. The first derivative of $S_A$ w.r.t. time is\n\\begin{eqnarray}\n \\left<\\frac{\\partial S_A}{\\partial t}\\right>_{CC} &=& \\frac{\\pi  \\sinh ^2\\left(\\frac{\\pi  r}{4 \\kappa_2 }\\right) \\tanh \\left(\\frac{\\pi  t}{2 \\kappa_2 }\\right)}{3 \\kappa_2 \\left[ \\cosh \\left(\\frac{\\pi  r}{2 \\kappa_2 }\\right)+  \\cosh \\left(\\frac{\\pi  t}{\\kappa_2 }\\right)\\right]}\\\\\n \\label{scctan} &=&\\frac{\\pi}{12 \\kappa_2 }  \\left[2 \\tanh \\left(\\frac{\\pi  t}{2 \\kappa_2 }\\right)-\\tanh \\left(\\frac{\\pi  (r+2 t)}{4 \\kappa_2 }\\right)+\\tanh \\left(\\frac{\\pi  (r-2 t)}{4 \\kappa_2 }\\right)\\right]\\\n\\end{eqnarray}\nFrom the first expression, as a function of time $t>0$, it is clear that there are no finite zero. Hence, the EE growth of CC state is always monotonically increasing. Also note that in the high effective temperature limit $\\kappa_2\\to 0$, the approach to thermal value is sharper. In the limiting case, from the second expression, it is clear that the thermalization time is \n\\begin{equation}\n\\label{thermtime}t=\\frac{r}{2}\\ \n\\end{equation}\nwhich has also been calculated using BCFT techniques in \\cite{Calabrese:2005in}.\n\nIt would be interesting to check the monotonicity of EE growth in gCC states. Unfortunately, even for the free fermions with explicit twist operators, the entanglement entropy in gCC state with $W_4$ charge cannot be explicitly calculated. The bilinear fermionic $\\mathcal{W}_4(w)$ current when bosonized gives $\\phi^4$ terms\\cite{Pope:1991ig}, so the bosonized theory is an intereacting theory.\n\n\\section{\\label{secEE_gCC}Non-Monotonic EE Growth and Dynamical Phase Transition}\nAlthough we could not calculate EE in gCC state with $W_4$ charge of the fermionic bilinear $\\mathcal{W}_4$ current, we can still calculate entanglement entropy explicitly with the fermionic charge corresponding to the  bosonic charge $W_4(w)=\\sum_k |k|^3 d^\\dagger_kd_k$, where $d^\\dagger_k$ and $d_k$ are the bosonic annihilation and creation operators. As mentioned above, the zero modes do not play any role. Refermionization of the bosonic bilinear $\\mathcal{W}_4$ is done in Appendix \\ref{refW4}.\\footnote{We would like to thank Justin David for informing us that this refermionization could be done in principle using U(1) currents and it has not been done anywhere.} So, the fermionic state that we are considering is\n\\begin{eqnarray}\n |\\Psi\\rangle=\\text{e}^{-\\kappa_2 H_f-\\kappa_4 \\tilde{W}_4}|D_f\\rangle\\\n\\end{eqnarray}\nwhere the expression for $\\tilde{W}_4$ is given in (\\ref{newW4ch}).\n\nAgain, the R\\'enyi and entanglement entropies are given by the expression (\\ref{RE}) and (\\ref{EE}). The scalar propagator with the bosonic $W_4$ charge has also been calculated in MPS.\n\\begin{eqnarray}\n \\langle\\phi(0,t)\\phi(r,t)\\rangle=\\int_{-\\infty}^{\\infty}\\frac{dk}{8\\pi}\\frac{e^{ikr}}{k} \\left[\\coth \\left(2 k \\left(\\kappa_2 +\\kappa_4  k^2\\right)\\right)-\\cos (2 k t) {\\rm cosech}\\left(2 k \\left(\\kappa_2 +\\kappa_4  k^2\\right)\\right)\\right]\\\n\\end{eqnarray}\nThe momentum integral cannot be done explicitly. But we still can plot the entanglement entropy numerically. Figure (\\ref{gCC_EE}) are the plots of EE growth with `small' and `large' values of $\\kappa_4$. As expected, the entanglement entropy reaches an equilibrium quickly.\n\\begin{figure}%\n    \\centering\n    \\subfloat[Monotonic behavior, $\\kappa_4=0.01$]{{\\includegraphics[width=7cm]{bp} }}%\n    \\qquad\n    \\subfloat[Non-monotonic behaviour, $\\kappa_4=0.30$]{{\\includegraphics[width=7cm]{ap} }}%\n    \\caption{Entanglement entropy growth of an interval(r=5) for different choice of $\\kappa_4$ and $\\kappa_2=1$.}%\n    \\label{gCC_EE}%\n\\end{figure}\n\nThe most interesting aspect of Figure (\\ref{gCC_EE}) is the non-monotonic growth of EE in the gCC state with `large' $\\kappa_4$. As in case of CC state, to study the monotonic or non-monotonic behaviour of $S_A$, the more appropriate quantity is not $S_A$ but rather $\\frac{\\partial S_A}{\\partial t}$, the expression also simplifies tremendously.\n\\begin{eqnarray}\n\\left<\\frac{\\partial S_A}{\\partial t}\\right>_{gCC}&=&\\frac{1}{3}\\int_{-\\infty}^{\\infty}\\,dk\\,(1-e^{ikr})\\text{cosech}(2\\kappa_2 k + 2\\kappa_4 k^3)\\sin(2kt)\\nonumber\\\\\n&=&\\frac{1}{3}\\int_{-\\infty}^{\\infty}\\,dk\\,(1-\\cos(kr))\\text{cosech}(2\\kappa_2 k + 2\\kappa_4 k^3)\\sin(2kt)\\\n\\label{dS_gCC}\n\\end{eqnarray}\nUnfortunately, the above integral still cannot be done in closed form. The objective is to find finite positive real zeroes of the above expression as a function of time $t$. But, calculating zeroes of Fourier transforms, unless it can be done in closed form, is notoriously hard, the most famous example being the Riemann hypothesis.\n\nThe most interesting question that can be asked in Figure (\\ref{gCC_EE}) is whether even a small infinitesimal $\\kappa_4$, although not visible in the numerical plot, gives rise to the non-monotonic EE growth or whether the non-monotonic behaviour starts from a sharp finite value of $\\kappa_4$. If it is the second case, then it is a dynamical phase transition. In other words, the question is whether (\\ref{dS_gCC}) has finite zeroes as a function of time even for an infinitesimal $\\kappa_4$ or do the finite zeroes appear for $\\kappa_4$ greater than a critical value.\n\n\\gap1\n\n{\\it{We found that the non-monotonic behaviour starts abruptly at a critical value of $\\kappa_4=16 \\kappa_2^3\/27 \\pi ^2$, i.e., it is a dynamical phase transition. In terms of the effective temperature and chemical potential in the stationary limit, $\\beta = 4\\kappa_2$ and $\\mu_4=4\\kappa_4$, the critical value is $\\mu_4=\\beta^3\/27 \\pi ^2$.}}\n\n\\gap1\n\nAlthought the integral (\\ref{dS_gCC}) cannot be done in closed form, we can take advantage of the fact that for our question we do not need to know the precise zeroes. Using contour integration, the integral is given by the sum of residues of the poles given by $2\\kappa_2 k+2\\kappa_4 k^3 =in\\pi$ where $n\\in\\mathbb{Z}-\\{0\\}$. $n=0$ is not a pole of (\\ref{dS_gCC}). The expressions of the poles(from MPS)\\footnote{The numerical values of the poles may get interchanged for specific values of the parameters but the result will always be the same set of roots. This arises from the particular method used for solving the cubic equation.} are\n\\begin{eqnarray}\nk_1&=&\\frac{-2\\ 6^{2\/3} \\kappa_2+\\sqrt[3]{6} \\left(\\sqrt{48 \\kappa_2^3-81 \\pi ^2 \\kappa_4 n^2}+9 i \\pi\\sqrt{\\kappa_4} n\\right)^{2\/3}}{6 \\sqrt[3]{\\sqrt{3} \\sqrt{\\kappa_4^3 \\left(16 \\kappa_2^3-27 \\pi ^2\\kappa_4 n^2\\right)}+9 i \\pi \\kappa_4^2 n}}\\\\\nk_2&=&\\frac{4 \\sqrt[3]{-6} \\kappa_2+i \\left(\\sqrt{3}+i\\right) \\left(\\sqrt{48 \\kappa_2^3-81 \\pi ^2 \\kappa_4 n^2}+9 i\\pi\\sqrt{\\kappa_4} n\\right)^{2\/3}}{2\\ 6^{2\/3} \\sqrt[3]{\\sqrt{3} \\sqrt{\\kappa_4^3 \\left(16 \\kappa_2^3-27\\pi^2 \\kappa_4 n^2\\right)}+9 i \\pi\\kappa_4^2 n}}\\\\\nk_3&=&-\\frac{\\sqrt[3]{-1} \\left(2 \\sqrt[3]{-6} \\kappa_2+\\left(\\sqrt{48 \\kappa_2^3-81 \\pi ^2 \\kappa_4 n^2}+9 i \\pi \\sqrt{\\kappa_4} n\\right)^{2\/3}\\right)}{6^{2\/3} \\sqrt{\\kappa_4} \\sqrt[3]{\\sqrt{48 \\kappa_2^3-81 \\pi ^2\\kappa_4 n^2}+9 i \\pi  \\sqrt{\\kappa_4} n}}\\\n\\label{poles}\n\\end{eqnarray}\nOut of the three poles, only one is perturbative. In $\\kappa_4\\to 0$ series expansion, the other two start with $\\mathcal{O}(\\frac{1}{\\sqrt{\\kappa_4}})$. One of the three poles is always imaginary for arbitrary $n$ and arbitrary positive $\\kappa_4$.\n\nThere are three important ingredients for the proof of the dynamical phase transition:\n\\begin{enumerate}\n\\item All three $n^\\text{th}$ poles become purely imaginary when $16 \\kappa_2^3-27 \\pi ^2\\kappa_4 n^2$ is negative, or $\\kappa_4$ is greater than $16 \\kappa_2^3\/27 \\pi ^2 n^2$, we will call this the $n^{\\text{th}}$ critical value $\\kappa_{4c,n}$,\n\\begin{align}\n\\label{kappa4cn} \\kappa_{4c,n}&= \\frac{16 \\kappa_2^3}{27 \\pi ^2 n^2}\\\n\\end{align}\nBelow this value, the residues of the $n^{\\text{th}}$ poles are exponential decaying functions of time $t$, with no oscillatory factor. Obviously, ($n=\\pm1$) critical\\footnote{We will call this value just `critical value' without the `$n^{\\text{th}}$' specification because, as shown below, this is the critical value of $\\kappa_4$ where the dynamical phase transition happens.} value $\\kappa_{4c}$ is larger than $\\kappa_{4c,n}$ for $|n|>1$. With $\\kappa$ scaled to $1$, $\\kappa_{4c}$ is $16 \\kappa_2^3\/27 \\pi ^2\\sim 0.0600422$.\n\\item With $\\kappa_4$ less than $(n=\\pm1)$ critical value, the sum of the residues of ($n=\\pm 1$) poles is larger than the sum of the residues of all the other ($|n|>1$) poles. Hence, the behaviour of the first poles of $n=\\pm1$ dictate the behaviour of the integral (\\ref{dS_gCC}) when $\\kappa_4<16 \\kappa_2^3\/27 \\pi ^2$.\n\\item Above this critical value, for each $n$, two of the poles have real parts while one of them, say $k_1$, is imaginary. The poles are\n\\begin{gather}\n\\label{poleR} k_1= -2i\\,\\text{sgn}(n)\\,b, \\qquad k_2=a+i\\,\\text{sgn}(n)\\,b, \\qquad k_3=-a+i\\,\\text{sgn}(n)\\,b\\\\\na=\\frac{B^{2\/3}-2 \\sqrt[3]{6} \\kappa _2}{2\\ 2^{2\/3} \\sqrt[6]{3} \\sqrt[3]{B} \\sqrt{\\kappa _4}}, \\qquad b= \\frac{B^{2\/3}+2 \\sqrt[3]{6} \\kappa_2}{2\\ 6^{2\/3} \\sqrt[3]{B} \\sqrt{\\kappa_4}}\\nonumber\\\\\nB=\\sqrt{81 \\pi ^2 \\kappa_4 n^2-48 \\kappa_2^3}+9 \\pi |n|\\sqrt{\\kappa_4}\\nonumber\\\n\\end{gather}\nwhere we have to take the real roots of the radicals. $k_1$'s have the largest imaginary parts and the exponential decay of their residues as a function of time are faster while the other poles $k_2$ and $k_3$ have comparatively large magnitudes and ocsillations.\\footnote{This competition between poles of each $n$ might be important, if we have turned on $W_6$ chemical potential instead of $W_4$, in which case there will be five poles, or $W_8$ in which case there will be seven poles and so on.} In the total integral, the contributions of the imaginary poles $k_1$'s cannot compete with the contributions of the oscillating poles.  Lastly, it would be a very special arrangment if all ocsillating terms conspire to give a non-oscillatory sum. Hence, the total integral is oscillatory as a function of time and the EE growth is non-monotonic.\n\\end{enumerate}\nFor future reference, we also note that the expansion of the real part `$a$' in (\\ref{poleR}) around the $n^\\text{th}$ critical value $\\kappa_{4c,n}$ is\n\\begin{multline}\n a=\\frac{\\sqrt[3]{\\pi } \\sqrt[3]{|n|} \\sqrt{\\kappa _4-\\kappa _{4 c,n}}}{2^{2\/3} \\sqrt{3} \\kappa _{4 c,n}^{5\/6}}-\\frac{35 \\left(\\sqrt[3]{\\pi } \\sqrt[3]{|n|}\\right) (\\kappa_4-\\kappa _{4 c,n})^{3\/2}}{54 \\left(2^{2\/3} \\sqrt{3} \\kappa _{4 c,n}^{11\/6}\\right)}\\\\\n +\\frac{1001 \\sqrt[3]{\\pi } \\sqrt[3]{|n|} (\\kappa_4-\\kappa _{4 c,n})^{5\/2}}{1944\\ 2^{2\/3} \\sqrt{3} \\kappa _{4 c,n}^{17\/6}}+\\mathcal{O}(\\kappa_4-\\kappa_{4c,n})^{7\/2}\n \\label{aexp}\n\\end{multline}\n\nFor all our calculations below, we have scaled $\\kappa_2$ to be 1. The first point is clear from figure (\\ref{RePol}). The real parts of ($n=\\pm1$) poles vanish at $\\kappa_4\\sim 0.060$, which is the critical value found above. The critical value of $(n=\\pm2$) poles is $\\kappa_4\\sim 0.015$.\n\n\\begin{figure}[H]\n\\centerline{\\includegraphics[scale=.5]{RePole}}\n\\caption{Real parts of poles of $n\\in\\{\\pm1, \\pm2\\}$ as a function of $\\kappa_4$ with $\\kappa_2$ scaled to $1$.}\n\\label{RePol}\n\\end{figure}\n\nBelow the critical value, we will show that the total contributions from $n=\\pm 1$ poles is larger than the sum of all residues of $|n|>1$ poles. We will concentrate on the late time period, $t>r\/2$. For $e^{i2kt}$ of $\\sin(2kt)$ factor in (\\ref{dS_gCC}), the contour is closed upward encircling the upper half plane, and for $e^{-i2kt}$, the contour is closed downward encircling the lower half plane. From the expansion of ${\\rm cosech}\\left(2 \\kappa _4 \\left(k-k_1\\right)\n\\left(k-k_2\\right) \\left(k-k_3\\right)+i \\pi n\\right)$ around $k_1$, the contribution from $k_1$ poles for arbitrary $n$ are the real parts of \n\\begin{align}\nP_n(k_1)&=2\\pi iR_1(k_1)=\\frac{(-1)^n}{6\\kappa_4(k_1-k_2)(k_1-k_3)}\\,\\left(e^{i2k_1t}-\\frac{e^{ik_1(r+2t)}+e^{ik_1(-r+2t)}}{2}\\right) \\; &\\text{if} \\;\\text{Im}[k_1]>0\\nonumber\\\\\n\\label{I1}\\\\\nQ_n(k_1)&=-2\\pi iR_2(k_1)=\\frac{(-1)^n}{6\\kappa_4(k_1-k_2)(k_1-k_3)}\\,\\left(e^{-i2k_1t}-\\frac{e^{ik_1(r-2t)}+e^{-ik_1(r+2t)}}{2}\\right)\\; &\\text{if} \\;\\text{Im}[k_1]<0\\nonumber\\\\\n\\label{I2}\\\n\\end{align}\nwhere $R_1$ and $R_2$ denote the residues. Similarly, cyclic replacements of $k_1$ with $k_2$ and $k_3$ give the contributions of $k_2$ and $k_3$ poles. For the poles in the lower half of the complex plane, since the contour is anticlockwise, $Q_n$ have an extra minus sign in the residue.\nWe will call the contributions to the integral form $n=\\pm1$ poles as $I_0(t)$ and the contributions of the $|n|>1$ poles as $I_1(t)$. The other parameters ($\\kappa_4$, $r$ and $\\kappa$ which is already scaled to 1) are suppressed.\n\nAs a first visual evidence, Figure (\\ref{ccritNnR}) is the comparison of numerical integration of (\\ref{dS_gCC}) and $I_0(t)$. It is evident that the residues of ($n=\\pm1$) poles dominate the contour integration. We have chosen $\\kappa_4=0.0600420$ which is very close to the critical value. As mentioned above, with this choice, all the poles except the $n=\\pm 1$ poles give ocsillating residues as a function of time. Although it is not very conspicuous, it is also evident from the graph that $I_1(t)$ is oscillating around $I_0(t)$, the value of the numerical integration is above the $I_0(t)$ curve in some regions and below in other regions of time $t$.\n\\begin{figure}[H]\n\\centerline{\\includegraphics[scale=.5]{NintRes}}\n\\caption{Comparison of numerical integration of $\\left<\\partial{S_A}\/\\partial t\\right>_{gCC}$ (blue curve) and $I_0(t)$ (purple curve) as a function of time $t$. The parameters are $\\kappa_4=0.0600420$, $r=5$.}\n\\label{ccritNnR}\n\\end{figure}\nThe numerical integration is unreliable in the long time limit. So, to complete our argument, we will calculate an upper bound of $I_1(t)$ and compare it with $I_0(t)$ for a specific time $t$. The choice of the parameters are\n\\begin{eqnarray}\n\\kappa=1, \\; \\kappa_4=0.0600420, \\; r=5,\\; t=4r=20,\\\n\\label{para1}\n\\end{eqnarray}\nWith these parameters, the $n=1$ and $n=-1$ poles are\n\\begin{align}\n k_1&= 2.3538234 i & k_2&=2.3585719 i& k_3&=-4.7123954 i &;\\; n&=1\\\\\n k_1&= 4.7123954 i& k_2&=-2.3538234 i& k_3&=-2.3585719 i &;\\; n&=-1\\\n\\end{align}\nand $I_0(t)$ is given by\n\\begin{eqnarray}\n I_0(t)|_{t=20}&=&P(k_1)|_{n=1}+P(k_2)|_{n=1}+Q(k_3)|_{n=1}+P(k_1)|_{n=-1}+Q(k_2)|_{n=-1}+Q(k_3)|_{n=-1}\\nonumber\\\\\n &=&6.646589\\times10^{-35}\\\n \\label{Inpm1}\n\\end{eqnarray}\nWe can show that $I_1(t)|_{t=20}$ is less than $I_0(t)|_{t=20}$. The first few poles are\n\\begin{align*}\nk_1&=5.5495551 i& k_2&=2.5383386 - 2.7747775 i & k_3&=-2.5383386 - 2.7747775 i  &;\\; n&=-3\\\\\nk_1&=5.1737935 i, & k_2&=1.8496206 - 2.5868967i & k_3&= -1.8496206 - 2.5868967 i &;\\; n&=-2\\\\\nk_1&=-5.1737935 i & k_2&= -1.8496206 + 2.5868967 i & k_3&=1.8496206 + 2.5868967 i &;\\; n&=2\\\\\nk_1&= -5.54955505 i & k_2&= -2.5383386 + 2.7747775 i & k_3&=2.5383386 +2.7747775 i &;\\; n&=3\\\n\\end{align*}\nThe residues of these ($|n|>1$) poles cannot be summed up into a closed form, as that would amount to doing the integral in closed form. We are interested in an upper bound. The residues of two of the three poles of every ($|n|>1$) have an oscillation factor. As we saw, even each residue has a separate 3-6 real oscillating terms as a function of time. So, we can represent the sum of the modulus (absolute value of the amplitude) of the oscillating terms of the three residues for each $n$, by a bigger function which has the analytic sum from $|n|>1$ to infinity. And if the sum is less $I_0(t)$, then $I_0(t)$ dominates the contribution from all the other poles.\\footnote{A simplified example of our strategy is the comparison between say $X$ and $a\\sin(x)+b\\cos(y)$ where $\\{X,a,b,x,y\\}\\in\\mathcal{R}$, while $A>|a|$ and $B>|b|$ and $\\{A,B\\}\\in\\mathcal{R}^+$, then $A+B>|a|+|b|>a\\sin(x)+b\\cos(y)$ and if $X>A+B$ then $X>a\\sin(x)+b\\cos(y)$.}\n\\begin{figure}[H]\n\\centerline{\\includegraphics[scale=.6]{upperbound}}\n\\caption{Comparison of sum of modulus of residues of ($|n|>1$) poles with the approximating function $f(n)=10^{-39}\/n^2$. The dots are the discrete $n$ values of the corresponding functions.}\n\\label{ResComp}\n\\end{figure}\nFigure (\\ref{ResComp}) are the plots of the sum of the moduli separately for the oscillating terms of the three residues as a function of $n$ and the approximating function $f(n)=10^{-39}\/n^2$. Now, we have\n\\begin{eqnarray}\n \\sum_{-\\infty}^{n=-2}\\frac{10^{-39}}{n^2}+ \\sum_{n=2}^{\\infty}\\frac{10^{-39}}{n^2} =1.289868\\times10^{-39}\\\n \\label{rescompsum}\n\\end{eqnarray}\nThis is much less than $I_0(t)|_{t=20}$ in (\\ref{Inpm1}) and is of the order of $10^{-5}$ of $I_0(t)|_{t=20}$. So, the non-oscillating $I_0(t)$ dominates $I_1(t)$, the contribution from the other poles. Hence, below $\\kappa_4=16 \\kappa_2^3\/27 \\pi ^2$, the EE growth is monotonic.\n\nVisually from figure (\\ref{ccritNnR}), $t=3.7$ is a time-slice where the difference between $I_0(t)$ and the numerical integration has a local maxima. At this time slice, repeating the above exercise, $I_0(t)|_{t=3.7}=0.109727$ and repeating the same exercise of estimating the upper bound of $I_1(t)|_{t=3.7}$ with the same parameters as (\\ref{para1}) except the change in $t$, we get a good upper bound to be $0.0064493$ which is less than $I_0(t)|_{t=3.7}$ and is of the order of $60\\%$ of $I_0(t)|_{t=3.7}$. So, the approximation of the full integral by $I_0(t)$ gets better with increasing time. In the long time limit, we can effectively take the only time-dependence to be the time-dependence of $I_0(t)$. {\\it{It is worth mentioning here that even $(n=\\pm1)$ pole calculations take into account $\\kappa_4$ non-perturbatively, since two of the poles of each $n$ are non-perturbative in $\\kappa_4$.}}\n \nAs listed above as one of the main points, above the critcal value, each $n$ has an imaginary pole but the other two poles have real parts and also have larger magnitudes so the total residue of the three poles of each $n$ is oscillatory. It would also be a very special arrangement if all the oscillatory contributions of each $n$ conspire to give a non-oscillatory $\\partial S_A\/\\partial t$. Hence, we conclude that the EE growth is non-monotonic above the critical value.\n\nNear the critical point $(\\kappa_4-\\kappa_{4c})\\to 0^+$, we can try to estimate an upper bound of the time upto which the EE growth is monotonic. The upper bound is half of the longest time period. Using the leading term in expansion of `$a$' from (\\ref{aexp}) and the expressions of the residues (\\ref{I1}) and (\\ref{I2}), the lowest frequency($|n|=1$) gives the upper bound as\n\\begin{align}\n\\frac{\\sqrt[3]{\\pi } \\sqrt{\\kappa _4-\\kappa _{4 c}}}{2^{2\/3} \\sqrt{3} \\kappa _{4 c}^{5\/6}}(2t-r)=\\pi\\quad\\Rightarrow\\quad t= \\frac{(2\\pi)^{2\/3} \\sqrt{3} \\kappa _{4 c}^{5\/6}}{2\\sqrt{\\kappa _4-\\kappa _{4 c}}}+\\frac{r}{2}\\;\\sim\\; \\frac{2.95\\,\\kappa _{4 c}^{5\/6}}{\\sqrt{\\kappa _4-\\kappa _{4 c}}}\\     \n\\end{align}\nwhere finite `$r$' can be neglected in the limit $(\\kappa_4-\\kappa_{4c})\\to 0^+$.\n\nThe critical value in terms the effective temperature $\\beta=4\\kappa_2$ and chemical potential $\\mu_4=4\\kappa^4$ in the stationary limit is\n\\begin{eqnarray}\n\\mu_4=\\frac{\\beta^3}{27 \\pi ^2}\\\n\\label{mucrit}\n\\end{eqnarray}\n\\begin{figure}[H]\n\\centerline{\\includegraphics[scale=.4]{pd_f}}\n\\caption{The critical curve $\\mu_4=\\beta^3\/27 \\pi ^2$ in terms of the effective temperature and chemical potential in the stationary limit and the phase diagram.}\n\\label{pd}\n\\end{figure}\nFor the early times $t<r\/2$, in the residue calculations (\\ref{I1}) and (\\ref{I2}), we have to replace the sign of the exponents with $r-2t$ so the magnitudes of the exponentials decreases as time increases. Upto the critical value of $\\kappa_4$, the EE growth is always monotonic for this time period.\n\n\\subsection{Turning on other charges}\nWe could also calculate the EE growth of gCC states with other charges of the fermionic theory corresponding to bosonic bilinear $W_{2n}=\\sum |k|^{2n-1}d^\\dagger_kd_k$ where $n=3,4,5,...$. Repeating the exercise of quenching tuned squeezed states of scalar field theory in MPS, the propagator with these charges are simply given by\n\\begin{multline}\n\\langle\\phi(0,t)\\phi(r,t)\\rangle=\\int\\frac{dk}{4\\pi}\\frac{e^{ikr}}{k} \\left(\\coth \\left(2 \\kappa_2 k +\\sum_{n=2}^{\\infty}\\kappa_{2n}  k^{2n-1}\\right)\\right.\\\\\n\\left.-\\cos (2 k t) {\\rm cosech}\\left(2 \\kappa_2 k +\\sum_{n=2}^{\\infty}\\kappa_{2n}  k^{2n-1}\\right)-1\\right)\n\\end{multline}\nSubstituting this propagator in the general formula (\\ref{RE}) and (\\ref{EE}) give the R\\'enyi entropy  and entanglement entropy. The first derivative of EE w.r.t. time is\n\\begin{eqnarray}\n\\left<\\frac{\\partial S_A}{\\partial t}\\right>_{gCC}=\\frac{1}{3}\\int_{-\\infty}^{\\infty}\\,dk\\,(1-\\cos(kr))\\text{cosech}\\left(2\\kappa k + \\sum_{n=2}^{\\infty}\\kappa_{2n}  k^{2n-1}\\right)\\sin(2kt)\\\n\\label{dS_gCCn} \n\\end{eqnarray}\nWe believe the dynamics will be much richer with these other charges, with much more complex phase diagrams which can be in a $n-1$ dimensional space.\nBut the general poles analysis cannot be done in these cases because the poles will be given by quintic and higher order equations.\nConsidering gCC states with $W_4$ and $W_6$ charges, the numerical plots of EE growth looks the same as (\\ref{gCC_EE}) where by trial and error method, some parameter subspace gives monotonic growth and some subspaces do not give monotonic growth. Considering $n=\\pm1$, the poles are given by $2\\kappa_2 k+2\\kappa_4 k^3 + 2\\kappa_6 k^5=i\\pi$. For $\\kappa_2=1$ and $\\kappa_4=0.06$, numerically we find two interesting parameter subspaces of $\\kappa_6$. The first one is when all the poles become imaginary when $\\kappa_4$ is decreased.\n\\begin{align}\n &k_1 =2.0887597\\,\\text{sgn}(n)\\,i, \\quad  k_2=2.9527785\\,\\text{sgn}(n)\\,i, \\quad k_3=-6.5425830 \\,\\text{sgn}(n)\\,i,\\nonumber\\\\\n &k_4= -6.6158300 \\,\\text{sgn}(n)\\,i, \\quad k_5=8.1168748 \\,\\text{sgn}(n)\\,i      \\qquad\\text{for}\\qquad\\kappa_6=0.0007249\\\\\n \\nonumber\\\\\n &k_1=-0.0076887-6.5788763 \\,\\text{sgn}(n)\\,i, \\quad k_2=2.0887456 \\,\\text{sgn}(n)\\,i, \\quad k_3=2.9528549 \\,\\text{sgn}(n)\\,i,\\nonumber\\\\\n &k_4=8.1161520 \\,\\text{sgn}(n)\\,i, \\quad k_5=0.0076887-6.5788763 \\,\\text{sgn}(n)\\,i     \\qquad\\text{for}\\qquad\\kappa_6=0.0007250\\\n\\end{align}\nThis looks like the same transition if $n=\\pm1$ dominates, but the poles with real parts have large imaginary part also, so they would be highly damped.\nThe  other case is\n\\begin{align}\n &k_1 =-0.8215058 + 1.9681831\\,\\text{sgn}(n)\\,i, \\quad  k_2=-5.2389645\\,\\text{sgn}(n)\\,i, \\quad k_3=-5.2472000 \\,\\text{sgn}(n)\\,i,\\nonumber\\\\\n &k_4= 6.5497983\\,\\text{sgn}(n)\\,i, \\quad k_5=0.8215058 + 1.9681831 \\,\\text{sgn}(n)\\,i    \\qquad\\text{for}\\qquad\\kappa_6=0.0019179\\\\\n &k_1=-0.8215060 + 1.9681836 \\,\\text{sgn}(n)\\,i, \\quad k_2=-0.0040372 - 5.2430724\\,\\text{sgn}(n)\\,i, \\nonumber\\\\\n &k_3=6.5497775 \\,\\text{sgn}(n)\\,i, \\quad k_4=0.0040372 - 5.2430724\\,\\text{sgn}(n)\\,i, \\nonumber\\\\\n &k_5=0.8215060 + 1.9681836\\,\\text{sgn}(n)\\,i    \\qquad\\text{for}\\qquad\\kappa_6=0.0019180\\\n\\end{align}\nfor the smaller $\\kappa_4$, although two of the poles have real parts, they have to compete with the three imaginary poles. So, this could also be phase transition.\n\n\\section{\\label{cond}Discussion}\nIn this paper, we have examined free fermionic mass quench. We find that the ground state quench equilibrates but not to a thermal ensemble. Starting from specially prepared squeezed states, we get CC and gCC states with fermionic bilinear $W_{2n}$ charges. Calculation of correlators in CC and gCC states explicitly shows thermalization to thermal emsemble and GGE respectively.\n\nFor CC state, we calculate EE growth exactly. The EE growth is strictly monotonically increasing. For gCC state with a particular charge, we find dynamical phase transition in which the EE growth is monotonic upto a critical value of the effective chemical potential. In the pure state, the effective chemical potential is the coupling constant of the current corresponding to the charge. Above the critical value, the EE growth is non-monotonic. It would be interesting to reproduce our result in large $c$ holographic CFTs and examine what it would mean for Black hole physics.\n\n\\section*{Acknowledgement}\nWe are especially grateful to Prof. Gautam Mandal for many discussions on this work and for proofreading the draft. This work is possible because of scholarship grants from the Government of India. This work was also partly supported by Infosys Endowment for the study of the Quantum Structure of Space Time.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}