Entanglement in XY Spin Chain

TL;DR

The paper analyzes the ground-state entanglement entropy of an infinite XY spin chain by framing the problem in terms of a block Toeplitz determinant derived from Majorana correlations. It employs integrable Fredholm operators and a Riemann-Hilbert approach to compute the large-block entropy, obtaining a closed-form expression in terms of Jacobi theta-functions with elliptic parameter $\tau$ and providing rigorous error bounds. The main result shows how $S(\rho_A)$ depends on the anisotropy $\gamma$ and transverse field $h$, and it exhibits singular behavior at quantum phase transitions, consistent with conformal-field theory predictions. The Appendix connects the findings to Cardy’s results by presenting Peschel’s closed-form simplifications in special cases, expressed through complete elliptic integrals.

Abstract

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.

Figures (2)

• Figure 1: Contours $\Gamma'$ (smaller one) and $\Gamma$ (larger one). Bold lines $(-\infty, -1-\epsilon)$ and $(1+\epsilon,\infty)$ are the cuts of integrand $e(1+\epsilon,\lambda)$. Zeros of $D_{L}(\lambda)$ (Eq. \ref{['exd']}) are located on bold line $(-1, 1)$. The arrow is the direction of the route of integral we take and $\mathrm{r}$ and $\mathrm{R}$ are the radius of circles. $\P$
• Figure 2: Polygonal line $\Sigma$ (direction as labeled) separates the complex $z$ plane into the two parts: the part $\Omega_{+}$ which lies to the left of $\Sigma$, and the part $\Omega_{-}$ which lies to the right of $\Sigma$. Curve $\Xi$ is the unit circle in anti-clockwise direction. Cuts $J_1, J_2$ for functions $\phi(z),w(z)$ are labeled by bold on line $\Sigma$. Definition of the end points of the cuts $\lambda_{\ldots}$ depends on the case: Case$1$a: $\lambda_A=\lambda_1$ and $\lambda_B=\lambda_2^{-1}$, $\lambda_C= \lambda_2$ and $\lambda_D= \lambda_1^{-1}$. Case$1$b: $\lambda_A=\lambda_1$ and $\lambda_B=\lambda_2^{-1}$, $\lambda_C= \lambda_1^{-1}$ and $\lambda_D= \lambda_2$. Case $2$: $\lambda_A=\lambda_1$ and $\lambda_B=\lambda_2$, $\lambda_C= \lambda_2^{-1}$ and $\lambda_D= \lambda_1^{-1}$. $\P$