Entanglement asymmetry in the ordered phase of many-body systems: the Ising Field Theory

TL;DR

This paper investigates entanglement asymmetry in the ordered phase of many-body systems, extending the concept to arbitrary groups, and demonstrates the theory in the Ising field model with spontaneous symmetry breaking.

Contribution

It introduces a field theoretic framework for entanglement asymmetry applicable to arbitrary finite groups and applies it to the Ising model with spontaneous symmetry breaking.

Findings

Analytical predictions for entanglement asymmetry in large intervals

Extension of entanglement asymmetry to arbitrary finite groups

A conjecture relating asymmetry to the number of degenerate vacua

Abstract

Global symmetries of quantum many-body systems can be spontaneously broken. Whenever this mechanism happens, the ground state is degenerate and one encounters an ordered phase. In this study, our objective is to investigate this phenomenon by examining the entanglement asymmetry of a specific region. This quantity, which has recently been introduced in the context of $U(1)$ symmetry breaking, is extended to encompass arbitrary finite groups $G$. We also establish a field theoretic framework in the replica theory using twist operators. We explicitly demonstrate our construction in the ordered phase of the Ising field theory in 1+1 dimensions, where a $\mathbb{Z}_2$ symmetry is spontaneously broken, and we employ a form factor bootstrap approach to characterise a family of composite twist fields. Analytical predictions are provided for the entanglement asymmetry of an interval in the Ising model as the length of the interval becomes large. We also propose a general conjecture relating the entanglement asymmetry and the number of degenerate vacua, expected to be valid for a large class of states, and we prove it explicitly in some cases.