Irreducible Projective Representations and Their Physical Applications

TL;DR

This paper develops a method to derive irreducible projective representations of finite groups, including anti-unitary groups, and explores their applications in many-body physics, especially in topological phases and quantum simulations.

Contribution

It introduces a novel eigenfunction approach for reducing regular projective representations and extends the theory to anti-unitary groups with modified Schur's lemma, applying these to physical systems.

Findings

Irreducible projective reps classify symmetry defects in topological phases

Projective reps relate to spectrum degeneracy in quantum models

Method aids in identifying models without sign problems in quantum Monte Carlo

Abstract

An eigenfunction method is applied to reduce the regular projective representations (Reps) of finite groups to obtain their irreducible projective Reps. Anti-unitary groups are treated specially, where the decoupled factor systems and modified Schur's lemma are introduced. We discuss the applications of irreducible Reps in many-body physics. It is shown that in symmetry protected topological phases, geometric defects or symmetry defects may carry projective Rep of the symmetry group; while in symmetry enriched topological phases, intrinsic excitations (such as spinons or visons) may carry projective Rep of the symmetry group. We also discuss the applications of projective Reps in problems related to spectrum degeneracy, such as in search of models without sign problem in quantum Monte Carlo Simulations.