Universal logarithmic corrections to entanglement entropies in two dimensions with spontaneously broken continuous symmetries

TL;DR

This paper demonstrates that in two-dimensional quantum spin models with spontaneously broken continuous symmetries, the entanglement entropy exhibits a universal logarithmic correction proportional to the number of Goldstone modes, confirmed through quantum Monte Carlo and spin-wave analysis.

Contribution

The study provides the first numerical verification of the universal logarithmic correction to entanglement entropy related to Goldstone modes in 2D systems, aligning with theoretical predictions.

Findings

Universal logarithmic correction with coefficient ~1 for SU(2) symmetry breaking.

Confirmation of the relation l_q = n_G/2 between correction coefficient and Goldstone modes.

Agreement between quantum Monte Carlo results and spin-wave analysis.

Abstract

We explore the R\'enyi entanglement entropies of a one-dimensional (line) subsystem of length $L$ embedded in two-dimensional $L\times L$ square lattice for quantum spin models whose ground-state breaks a continuous symmetry in the thermodynamic limit. Using quantum Monte Carlo simulations, we first study the $J_1 - J_2$ Heisenberg model with antiferromagnetic nearest-neighbor $J_1>0$ and ferromagnetic second-neighbor couplings $J_2\le 0$. The signature of SU(2) symmetry breaking on finite size systems, ranging from $L=4$ up to $L=40$ clearly appears as a universal additive logarithmic correction to the R\'enyi entanglement entropies: $l_q \ln L$ with $l_q\simeq 1$, independent of the R\'enyi index and values of $J_2$. We confirm this result using a high precision spin-wave analysis (with restored spin rotational symmetry) on finite lattices up to $10^5\times 10^5$ sites, allowing to explore further non-universal finite size corrections and study in addition the case of U(1) symmetry breaking. Our results fully agree with the prediction $l_q=n_G/2$ where $n_G$ is the number of Goldstone modes, by Metlitski and Grover [arXiv:1112.5166].