R\'enyi entropy of a line in two-dimensional Ising models

TL;DR

This paper investigates the Rényi entropy of a line in the 2D Ising model on an infinite cylinder, revealing universal, step-like behavior at criticality and connections to boundary entropies, with implications for theoretical approaches.

Contribution

It provides numerical evidence of the universal, discontinuous behavior of the subleading Rényi entropy at the critical point and links it to boundary entropies, challenging existing field theory methods.

Findings

The subleading constant of Rényi entropy is universal and step-like at criticality.

Discontinuity at n=1 (Shannon point) affects replica trick calculations.

Connections established between Rényi parameters and boundary entropies.

Abstract

We consider the two-dimensional (2d) Ising model on a infinitely long cylinder and study the probabilities $p_i$ to observe a given spin configuration $i$ along a circular section of the cylinder. These probabilities also occur as eigenvalues of reduced density matrices in some Rokhsar-Kivelson wave-functions. We analyze the subleading constant to the R\'enyi entropy $R_n=1/(1-n) \ln (\sum_i p_i^n)$ and discuss its scaling properties at the critical point. Studying three different microscopic realizations, we provide numerical evidence that it is universal and behaves in a step-like fashion as a function of $n$, with a discontinuity at the Shannon point $n=1$. As a consequence, a field theoretical argument based on the replica trick would fail to give the correct value at this point. We nevertheless compute it numerically with high precision. Two other values of the R\'enyi parameter are of special interest: $n=1/2$ and $n=\infty$ are related in a simple way to the Affleck-Ludwig boundary entropies associated to free and fixed boundary conditions respectively.