Entanglement Entropy in Quantum Spin Chains with Finite Range Interaction

Abstract

We study the entropy of entanglement of the ground state in a wide family of onedimensional 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 → ∞ of the determinant of a block-Toeplitz matrix with symbolwhere g(z) is the square root of a rational function and g(1/z) = g −1 (z). The asymptotics of such determinant is computed in terms of multi-dimensional thetafunctions associated to a hyperelliptic curve L of genus g ≥ 1, which enter into the solution of a Riemann-Hilbert problem. Phase transitions for these systems are characterized by the branch points of L 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 [14] for the XX model and Its, Jin and Korepin [12,13] for the XY model.

Block Toeplitz Determinants, Constrained KP and Gelfand-Dickey Hierarchies

“…For simplicity we reduce to the case n = 2; the procedure used here is equivalent to the one used by Its, Jin and Korepin in [25] and generalized by Its, Mezzadri and Mo in [26]. Suppose we want to solve the factorization problem W(t; z) := exp ξ(t, Λ) W(z) = T − (t; z)T + (t; z) for our GD symbol with W(z) = diag(w 1 (z), w 2 (z)) as in example 6.10; since it will appear many times we denote A the matrix…”

“…Asymptotics of block Toeplitz determinants and their applications to physics is a developing field of research; in recent years it has been shown how to compute some physically relevant quantities (e.g. correlation functions) studying asymptotics of some block Toeplitz determinants (see [25], [26], [27]). In particular in [25] and [26] authors, for the first time, showed effective computations for the case of block Toeplitz determinants with symbols that do not have half truncated Fourier series.…”

“…correlation functions) studying asymptotics of some block Toeplitz determinants (see [25], [26], [27]). In particular in [25] and [26] authors, for the first time, showed effective computations for the case of block Toeplitz determinants with symbols that do not have half truncated Fourier series. This is of particular interest for us as, with our approach, we will be able to do the same for certain block Toeplitz determinants associated to algebro-geometric solutions of Gelfand Dickey hierarchies.…”

“…Also we formulate an alternative Riemann-Hilbert problem equivalent to (2) in analogy with what has been done in [25] and [26]. We explain how to solve it using θ-functions.…”

“…In this way we give concrete formulas for a wide class of symbols that do not have half truncated Fourier series. We think this is quite remarkable since concrete results for non half truncated symbols were available, till now, just for the concrete cases presented in [25] and [26]. The paper is organized as follows:…”