Entanglement asymmetry in the critical XXZ spin chain

TL;DR

This paper investigates how the entanglement asymmetry can quantify the explicit breaking of $SU(2)$ symmetry to $U(1)$ in the critical XXZ spin chain, combining conformal perturbation theory and numerical simulations.

Contribution

It introduces the entanglement asymmetry as a tool to measure symmetry breaking in quantum systems and provides an asymptotic expression for it in the XXZ model.

Findings

Entanglement asymmetry effectively captures $SU(2)$ symmetry breaking.

Derived an asymptotic formula for entanglement asymmetry.

Numerical simulations support the theoretical predictions.

Abstract

We study the explicit breaking of a $SU(2)$ symmetry to a $U(1)$ subgroup employing the entanglement asymmetry, a recently introduced observable that measures how much symmetries are broken in a part of extended quantum systems. We consider as specific model the critical XXZ spin chain, which breaks the $SU(2)$ symmetry of spin rotations except at the isotropic point, and is described by the massless compact boson in the continuum limit. We examine the $U(1)$ subgroup of $SU(2)$ that is broken outside the isotropic point by applying conformal perturbation theory, which we complement with numerical simulations on the lattice. We also analyse the entanglement asymmetry of the full $SU(2)$ group. By relying on very generic scaling arguments, we derive an asymptotic expression for it.