Renyi Entropy of the XY Spin Chain

TL;DR

This paper analytically computes the asymptotic Renyi entropy of a block of spins in the XY quantum chain, revealing its automorphic properties and transformation behaviors under parameter changes, especially near critical points.

Contribution

It provides an exact asymptotic expression for the Renyi entropy in the XY chain using elliptic functions and uncovers its automorphic nature related to modular groups.

Findings

Renyi entropy expressed via Klein's elliptic λ-function

Entropy exhibits automorphic properties under modular subgroup transformations

Special simplifications occur when the Renyi parameter is a power of two

Abstract

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 $\alpha$ of the density matrix of the block. We calculate the asymptotic for $L \to \infty$ analytically in terms of Klein's elliptic $\lambda$ - function. We study the limiting entropy as a function of its parameter $\alpha$. 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 $\alpha \to \alpha^{-1}$ and show that the entropy becomes an elementary function of the magnetic field and the anisotropy when $\alpha$ is a integer power of 2, this includes the purity $tr \rho^2$. We also analyze the behavior of the entropy as $\alpha \to 0$ and $\infty$ and at the critical magnetic field and in the isotropic limit [XX model].