Entanglement Entropy of Systems with Spontaneously Broken Continuous Symmetry

Abstract

We study entanglement properties of systems with spontaneously broken continuous symmetry. We find that in addition to the expected area law behavior, the entanglement entropy contains a subleading contribution which diverges logarithmically with the subsystem size in agreement with the Monte Carlo simulations of A. Kallin et. al. (Phys. Rev. B 84, 165134 (2011)). The coefficient of the logarithm is a universal number given simply by $N_G (d-1)/2$, where $N_G$ is the number of Goldstone modes and $d$ is the spatial dimension. This term is present even when the subsystem boundary is straight and contains no corners, and its origin lies in the interplay of Goldstone modes and restoration of symmetry in a finite volume. We also compute the "low-energy" part of the entanglement spectrum and show that it has the same characteristic "tower of states" form as the physical low-energy spectrum obtained when a system with spontaneously broken continuous symmetry is placed in a finite volume.

Figures (1)

• Figure 1: The universal constant contribution $\gamma_{\mathrm{ord}}$, Eq. (\ref{['DeltaS']}), to the Renyi entropy $S_2$ of the $O(3)$ non-linear $\sigma$-model in dimension $d = 2$ (describing e.g. the square lattice Heisenberg model). In the geometry studied here, the total system is a torus of size $L \times L$, while the subsystem is a cylinder of size $\ell \times L$. The details of the calculation are described in section \ref{['sec:eewf']}. The solid line is a guide to eye.