Entanglement Entropy and Entanglement Spectrum for Two-Dimensional Classical Spin Configuration

TL;DR

This paper introduces a novel entanglement entropy and spectrum for 2D classical spin configurations, revealing their scaling properties, relation to correlation length, and insights into symmetry breaking at criticality.

Contribution

It defines new entanglement measures for classical spin snapshots and explores their properties, linking them to critical phenomena and fractal structures in 2D models.

Findings

Scaling relations at critical temperature T_c

Singular values decompose snapshots into different length scales

Multiple gaps in the entanglement spectrum indicating symmetry breaking

Abstract

In quantum spin chains at criticality, two types of scaling for the entanglement entropy exist: one comes from conformal field theory (CFT), and the other is for entanglement support of matrix product state (MPS) approximation. They indicates that the matrix dimension of the MPS represents a length scale of spin correlation. On the other hand, the quantum spin-chain models can be mapped onto two-dimensional (2D) classical ones. Motivated by the scaling and the mapping, we introduce new entanglement entropy for 2D classical spin configuration as well as entanglement spectrum, and examine their basic properties in Ising and 3-state Potts models on the square lattice. They are defined by the singular values of the reduced density matrix for a Monte Carlo snapshot. We find scaling relations concerned with length scales in the snapshot at $T_{c}$. There, the spin configuration is fractal, and various sizes of ordered clusters coexist. Then, the singular values automatically decompose the original snapshot into a set of images with different length scale. This is the origin of the scaling. In contrast to the MPS scaling, long-range spin correlation can be described by only few singular values. Furthermore, we find multiple gaps in the entanglement spectrum, and in contrast to standard topological phases, the low-lying entanglement levels below the gap represent spontaneous symmetry breaking. Based on these observations, we discuss about the amount of information contained in one snapshot in a viewpoint of the CFT scaling.