Intertwined order of generalized global symmetries

TL;DR

This paper explores the complex interplay of generalized global symmetries in a 2+1D lattice model, revealing how their coupling leads to novel phases with intertwined orders, anomalies, and boundary phenomena.

Contribution

It introduces a lattice model coupling a $ ext{Z}_N$ clock and gauge theory to study emergent symmetries and anomalies, extending to non-invertible symmetries and boundary states.

Findings

Identification of phases with mixed 't Hooft anomalies

Construction of boundary states for exotic phases

Extension to non-invertible global symmetries with domain walls

Abstract

We investigate the interplay of generalized global symmetries in 2+1 dimensions in a lattice model that couples a $\mathbb{Z}_N$ clock model to a $\mathbb{Z}_N$ gauge theory via a topological interaction. This coupling binds the charges of one symmetry to the disorder operators of the other, and when these composite objects condense, they give rise to emergent generalized symmetries with mixed 't Hooft anomalies. These anomalies result in phases with ordinary symmetry breaking, topological order, and symmetry-protected topological (SPT) order, where the different types of order are not independent but intimately related. We further explore the gapped boundary states of these exotic phases and develop theories for phase transitions between them. Additionally, we extend this lattice model to incorporate a non-invertible global symmetry, which can be spontaneously broken, leading to domain walls with non-trivial fusion rules.