Higher-form entanglement asymmetry. Part I. The limits of symmetry breaking

TL;DR

This paper extends the concept of entanglement asymmetry to higher-form symmetries, establishing an entropic theorem that limits symmetry breaking in certain dimensions and quantifies Goldstone modes through entanglement measures.

Contribution

It introduces a generalized entropic Coleman-Mermin-Wagner theorem for higher-form symmetries, applicable to subregions, and provides explicit formulas for entanglement asymmetries in various theories.

Findings

Spontaneous breaking of continuous p-form symmetries is forbidden in dimensions d ≤ p+2.

Spontaneous symmetry breaking leads to a nonzero entanglement asymmetry that increases towards the IR.

Derived closed-form expressions for Rènyi asymmetries of scalar and gauge fields in specific dimensions.

Abstract

Entanglement asymmetry is a relative entropy that faithfully diagnoses symmetry breaking in quantum states, possibly within a spatial subregion. In this work, we extend such framework to higher-form symmetries and compute entanglement asymmetry in theories with spontaneously-broken continuous zero- and higher-form symmetries. One of our central results is an entropic Coleman-Mermin-Wagner theorem, for 0- and $p$-form symmetries, valid also on subregions, which forbids spontaneous breaking of continuous $p$-form symmetries in spacetime dimensions $d\leq p+2$. Our theorem not only qualifies symmetry breaking, it also quantifies it: spontaneous breaking triggers a nonvanishing entanglement asymmetry that grows monotonically towards the infrared, and counts the number of Goldstone fields. Along the way, we derive standalone results concerning the entanglement entropy and asymmetry of Goldstone bosons and gauge fields. In particular, we find a closed-form expression for the R\'enyi asymmetries of a compact scalar field on spherical subregions in three and four spacetime dimensions, and for higher-form gauge fields in higher dimensions.