arXiv subjects

A. R. Its

Publications and source records attributed to A. R. Its.

At least 19 recordsLinked to original sources

arXiv2024

On some Hamiltonian properties of the isomonodromic tau functions

We discuss some new aspects of the theory of the Jimbo-Miwa-Ueno tau function which have come to light within the recent developments in the global asymptotic analysis of the tau functions related to the Painlevé equations. Specifically, we show that up to the total differentials the logarithmic derivatives of the Painlevé tau functions coincide with the corresponding classical action differential. This fact simplifies considerably the evaluation of the constant factors in the asymptotics of tau-functions, which has been a long-standing problem of the asymptotic theory of Painlevé equations. Furthermore, we believe that this observation is yet another manifestation of L. D. Faddeev's emphasis of the key role which the Hamiltonian aspects play in the theory of integrable system. This article will appear in the WSPC memorial volume dedicated to Ludwig Faddeev.

math-ph↗

arXiv2020

Entanglement entropy of two disjoint intervals separated by one spin in a chain of free fermion

We calculate the entanglement entropy of a non-contiguous subsystem of a chain of free fermions. The starting point is a formula suggested by Jin and Korepin, \texttt{arXiv:1104.1004}, for the reduced density of states of two disjoint intervals with lattice sites $P=\{1,2,\dots,m\}\cup\{2m+1,2m+2,\dots, 3m\}$, which applies to this model. As a first step in the asymptotic analysis of this system, we consider its simplification to two disjoint intervals separated just by one site, and we rigorously calculate the mutual information between these two blocks and the rest of the chain. In order to compute the entropy we need to study the asymptotic behaviour of an inverse Toeplitz matrix with Fisher-Hartwig symbol using the the Riemann--Hilbert method.

math-ph↗

arXiv2019

Riemann-Hilbert approach to a generalised sine kernel

We derive the large distance asymptotics of the Fredholm determinant of the so-called generalised sine kernel at the critical point. This kernel corresponds to a generalisation of the pure sine kernel arising in the theory of random matrices and has potential applications to the analysis of the large-distance asymptotic behaviour of the so-called emptiness formation probability for various quantum integrable models away from their free fermion point.

math-ph↗

arXiv2014

Large-x analysis of an operator valued Riemann-Hilbert problem

The purpose of this paper is to push forward the theory of operator-valued Riemann Hilbert problems and demonstrate their effectiveness in respect to the implementation of a non-linear steepest descent method \textit{á la} Deift-Zhou. In the present paper, we demonstrate that the operator-valued Riemann--Hilbert problem arising in the characterisation of so-called $c$-shifted integrable integral operators allows one to extract the large-$x$ asymptotics of the Fredholm determinant associated with such operators.

math-ph↗

arXiv2013

On determinants of integrable operators with shifts

Integrable integral operator can be studied by means of a matrix Riemann--Hilbert problem. However, in the case of so-called integrable operators with shifts, the associated Riemann--Hilbert problem becomes operator valued and this complicates strongly the analysis. In this note, we show how to circumvent, in a very simple way, the use of such a setting while still being able to characterize the large-$x$ asymptotic behavior of the determinant associated with the operator.

math.FA↗

arXiv2010

Entanglement Spectrum for the XY Model in One Dimension

We consider the reduced density matrix of a large block of consecutive spins in the ground states of the XY spin chain on an infinite lattice. We derive the spectrum of the density matrix using the expression of the Renyi entropy in terms of modular functions. The eigenvalues λ_n form an exact geometric sequence. For example, for strong magnetic field λ_n = C \exp{(-πτ_0 n)}, here τ_0>0 and C > 0 depend on the anisotropy and the magnetic field. Different eigenvalues are degenerated differently. The largest eigenvalue is unique, but the degeneracy g_n increases sub-exponentially as eigenvalues diminish: g_n \sim \exp{(π\sqrt{n/3})}. For weak magnetic field expressions are similar.

quant-ph↗

arXiv2009

The Fisher-Hartwig Formula and Generalized Entropies in XY Spin Chain

Toeplitz matrices have applications to different problems of statistical mechanics. Recently they were used for calculation of entanglement entropy in spin chains. We use the Fisher-Hartwig formula to calculate entanglement entropy of large block of spins in the ground state of XY spin chain. We also calculate Renyi entropy and prove that the spectrum of the density matrix of a block of spins is exact geometric sequence [also different eigenvalues are degenerated differently].

math-ph↗

arXiv2008

Asymptotics for a special solution of the thirty fourth Painleve equation

In a previous paper we studied the double scaling limit of unitary random matrix ensembles of the form Z_{n,N}^{-1} |\det M|^{2α} e^{-N \Tr V(M)} dM with α> -1/2. The factor |\det M|^{2α} induces critical eigenvalue behavior near the origin. Under the assumption that the limiting mean eigenvalue density associated with V is regular, and that the origin is a right endpoint of its support, we computed the limiting eigenvalue correlation kernel in the double scaling limit as n, N \to \infty such that n^{2/3}(n/N-1) = O(1) by using the Deift-Zhou steepest descent method for the Riemann-Hilbert problem for polynomials on the line orthogonal with respect to the weight |x|^{2α} e^{-NV(x)}. Our main attention was on the construction of a local parametrix near the origin by means of the ψ-functions associated with a distinguished solution u_α of the Painleve XXXIV equation. This solution is related to a particular solution of the Painleve II equation, which however is different from the usual Hastings-McLeod solution. In this paper we compute the asymptotic behavior of u_α(s) as s \to \pm \infty. We conjecture that this asymptotics characterizes u_α and we present supporting arguments based on the asymptotic analysis of a one-parameter family of solutions of the Painleve XXXIV equation which includes u_α. We identify this family as the family of tronquee solutions of the thirty fourth Painleve equation.

math.CA↗

arXiv2008

Entanglement entropy in quantum spin chains with finite range interaction

We study the entropy of entanglement of the ground state in a wide family of one-dimensional quantum spin chains whose interaction is of finite range and translation invariant. Such systems can be thought of as generalizations of the XY model. The chain is divided in two parts: one containing the first consecutive L spins; the second the remaining ones. In this setting the entropy of entanglement is the von Neumann entropy of either part. At the core of our computation is the explicit evaluation of the leading order term as L tends to infinity of the determinant of a block-Toeplitz matrix whose symbol belongs to a general class of 2 x 2 matrix functions. The asymptotics of such determinant is computed in terms of multi-dimensional theta-functions associated to a hyperelliptic curve of genus g >= 1, which enter into the solution of a Riemann-Hilbert problem. Phase transitions for thes systems are characterized by the branch points of the hyperelliptic curve approaching the unit circle. In these circumstances the entropy diverges logarithmically. We also recover, as particular cases, the formulae for the entropy discovered by Jin and Korepin (2004) for the XX model and Its, Jin and Korepin (2005,2006) for the XY model.

math-ph↗

arXiv2007

Renyi Entropy of the XY Spin Chain

We consider the one-dimensional XY quantum spin chain in a transverse magnetic field. We are interested in the Renyi entropy of a block of L neighboring spins at zero temperature on an infinite lattice. The Renyi entropy is essentially the trace of some power $α$ of the density matrix of the block. We calculate the asymptotic for $L \to \infty$ analytically in terms of Klein's elliptic $λ$ - function. We study the limiting entropy as a function of its parameter $α$. We show that up to the trivial addition terms and multiplicative factors, and after a proper re-scaling, the Renyi entropy is an automorphic function with respect to a certain subgroup of the modular group; moreover, the subgroup depends on whether the magnetic field is above or below its critical value. Using this fact, we derive the transformation properties of the Renyi entropy under the map $α\to α^{-1}$ and show that the entropy becomes an elementary function of the magnetic field and the anisotropy when $α$ is a integer power of 2, this includes the purity $tr ρ^2$. We also analyze the behavior of the entropy as $α\to 0$ and $\infty$ and at the critical magnetic field and in the isotropic limit [XX model].

quant-ph↗

arXiv2007

Ellipses of Constant Entropy in the XY Spin Chain

Entanglement in the ground state of the XY model on the infinite chain can be measured by the von Neumann entropy of a block of neighboring spins. We study a double scaling limit: the size of the block is much larger then 1 but much smaller then the length of the whole chain. The entropy of the block has an asymptotic limit. We study this limiting entropy as a function of the anisotropy and of the magnetic field. We identify its minima at product states and its divergencies at the quantum phase transitions. We find that the curves of constant entropy are ellipses and hyperbolas and that they all meet at one point (essential critical point). Depending on the approach to the essential critical point the entropy can take any value between 0 and infinity. In the vicinity of this point small changes in the parameters cause large change of the entropy.

quant-ph↗

arXiv2007

Critical edge behavior in unitary random matrix ensembles and the thirty fourth Painleve transcendent

We describe a new universality class for unitary invariant random matrix ensembles. It arises in the double scaling limit of ensembles of random $n \times n$ Hermitian matrices $Z_{n,N}^{-1} |\det M|^{2α} e^{-N \Tr V(M)} dM$ with $α> -1/2$, where the factor $|\det M|^{2α}$ induces critical eigenvalue behavior near the origin. Under the assumption that the limiting mean eigenvalue density associated with $V$ is regular, and that the origin is a right endpoint of its support, we compute the limiting eigenvalue correlation kernel in the double scaling limit as $n, N \to \infty$ such that $n^{2/3}(n/N-1) = O(1)$. We use the Deift-Zhou steepest descent method for the Riemann-Hilbert problem for polynomials on the line orthogonal with respect to the weight $|x|^{2α} e^{-NV(x)}$. Our main attention is on the construction of a local parametrix near the origin by means of the $ψ$-functions associated with a distinguished solution of the Painleve XXXIV equation. This solution is related to a particular solution of the Painleve II equation, which however is different from the usual Hastings-McLeod solution.

math.CA↗

arXiv2006

Entropy of XY Spin Chain and Block Toeplitz Determinants

We consider entanglement in the ground state of the XY spin model on infinite chain. We use von Neumann entropy of a sub-system as a measure of entanglement. The entropy of a large block of neighboring spins approaches a constant as the size of the block increases. We prove rigorously expression for limiting entropy which was published before. We observe that the entropy reaches minimum at product states but increases boundlessly at phase transitions.

quant-ph↗

arXiv2006

Analysis of entropy of XY Spin Chain

quant-ph↗

arXiv2005

The Nonlinear Schrödinger Equation on the Interval

Let $q(x,t)$ satisfy the Dirichlet initial-boundary value problem for the nonlinear Schrödinger equation on the finite interval, $0 < x < L$, with $q_{0}(x) = q(x,0)$, $g_{0}(t) = q(0,t)$, $f_{0}(t) = q(L,t)$. Let $g_{1}(t)$ and $f_{1}(t)$ denote the {\it unknown} boundary values $q_{x}(0,t)$ and $q_{x}(L,t)$, respectively. We first show that these unknown functions can be expressed in terms of the given initial and boundary conditions through the solution of a system of nonlinear ODEs. Although the question of the global existence of solution of this system remains open, it appears that this is the first time in the literature that such a characterization is explicitely described for a nonlinear evolution PDE defined on the interval; this result is the extension of the analogous result of [4] and [6] from the half-line to the interval. We then show that $q(x,t)$ can be expressed in terms of the solution of a $2\times 2$ matrix Riemann-Hilbert problem formulated in the complex $k$ - plane. This problem has explicit $(x,t)$ dependence in the form $\exp[2ikx + 4ik^2t]$, and it has jumps across the real and imaginary axes. The relevant jump matrices are explicitely given in terms of the spectral data $\{a(k), b(k)\}$, $\{A(k), B(k)\}$, and $\{\A(k), \B(k)\}$, which in turn are defined in terms of $q_{0}(x)$, $\{g_{0}(t), g_{1}(t)\}$, and $\{f_{0}(t), f_{1}(t)\}$, espectively.

nlin.SI↗

arXiv2005

Effective integration of the Nonlinear Vector Schrödinger Equation

A comprehensive algebro-geometric integration of the two component Nonlinear Vector Schrödinger equation (Manakov system) is developed. The allied spectral variety is a trigonal Riemann surface, which is described explicitly and the solutions of the equations are given in terms of theta-functions of the surface. The final formulae are effective in that sense that all entries like transcendental constants in exponentials, winding vectors etc. are expressed in terms of prime-form of the curve and well algorithmized operations on them. That made the result available for direct calculations in applied problems implementing the Manakov system. The simplest solutions in Jacobian theta-functions are given as particular case of general formulae and discussed in details.

math-ph↗

arXiv2004

Entanglement in XY Spin Chain

We consider the ground state of the XY model on an infinite chain at zero temperature. Following Bennett, Bernstein, Popescu, and Schumacher we use entropy of a sub-system as a measure of entanglement. Vidal, Latorre, Rico and Kitaev conjectured that von Neumann entropy of a large block of neighboring spins approaches a constant as the size of the block increases. We evaluated this limiting entropy as a function of anisotropy and transverse magnetic field. We used the methods based on integrable Fredholm operators and Riemann-Hilbert problem. The entropy is singular at phase transitions.

quant-ph↗

arXiv2001

The nonlinear steepest descent approach to the asymptotics of the second Painleve transcendent in the complex domain

The asymptotics of the generic second Painleve transcendent in the complex domain is found and justified via the direct asymptotic analysis of the associated Riemann-Hilbert problem based on the Deift-Zhou nonlinear steepest descent method. The asymptotics is proved of the Boutroux type, i.e. it is expressed in terms of the elliptic functions. Explicit connection formulae between the asymptotic phases in the different sectors are obtained as well.

nlin.SI↗