Symmetry Resolved Measures in Quantum Field Theory: a Short Review

TL;DR

This review surveys symmetry resolved entanglement entropy (SREE) in 1+1D quantum field theories with internal symmetries, detailing definitions, construction via composite branch point twist fields, and the replica-based charged moments that isolate symmetry sectors. It emphasizes two concrete realizations, the U(1) symmetry in the sine-Gordon model and the Z2 symmetry in the Ising field theory, and explains how the form factor program extends to composite twist fields to compute SREE in massive integrable QFTs as well as excited states. Key insights include the logarithmic scaling and equipartition of SREE at criticality, constant saturation in gapped theories, and universal finite-interval corrections captured by form factors. The paper also situates SREE within broader contexts including lattice models, holography, and noninvertible symmetries, and outlines open directions such as other symmetry-resolved measures and dynamical settings.

Abstract

In this short review we present the key definitions, ideas and techniques involved in the study of symmetry resolved entanglement measures, with a focus on the symmetry resolved entanglement entropy. In order to be able to define such entanglement measures, it is essential that the theory under study possess an internal symmetry. Then, symmetry resolved entanglement measures quantify the contribution to a particular entanglement measure that can be associated to a chosen symmetry sector. Our review focuses on conformal (gapless/massless/critical) and integrable (gapped/massive) quantum field theories, where the leading computational technique employs symmetry fields known as (composite) branch point twist fields.

Figures (1)

• Figure 1: Relationship between the replica partition function and the correlation function of branch point twist fields for $n=3$ and a compact region of length $|a-b|$. $|\Psi\rangle_n$ is a pure state in the replica theory.