Projective Representations, Bogomolov Multiplier, and Their Applications in Physics

TL;DR

This paper reviews projective representations of finite groups, emphasizing the Bogomolov multiplier, and explores their physical implications in quantum systems, including new results on symmetry-protected phases and lattice models.

Contribution

It introduces new physical insights into the role of the Bogomolov multiplier in quantum phases and constructs explicit lattice models illustrating these phenomena.

Findings

Characterization of (1+1)D SPT phases via Bogomolov multiplier

Construction of lattice models with distinct gapped phases

Demonstration of nontrivial interface modes in symmetry-breaking phases

Abstract

We present a pedagogical review of projective representations of finite groups and their physical applications in quantum many-body systems. Some of our physical results are new. We begin with a self-contained introduction to projective representations, highlighting the role of group cohomology, representation theory, and classification of irreducible projective representations. We then focus on a special subset of cohomology classes, known as the Bogomolov multiplier, which consists of cocycles that are symmetric on commuting pairs but remain nontrivial in group cohomology. Such cocycles have important physical implications: they characterize (1+1)D SPT phases that cannot be detected by string order parameters and give rise, upon gauging, to distinct gapped phases with completely broken non-invertible $\mathrm{Rep}(G)$ symmetry. We construct explicit lattice models for these phases and demonstrate how they are distinguished by the fusion rules of local order parameters. We show that a pair of completely broken $\mathrm{Rep}(G)$ SSB phases host nontrivial interface modes at their domain walls. As an example, we construct a lattice model where the ground state degeneracy on a ring increases from 32 without interfaces to 56 with interfaces.