Twisted Partition Functions as Order Parameters

TL;DR

This paper explores how twisted partition functions serve as order parameters to distinguish various quantum phases, including symmetry-breaking and topological states, and applies this to 4d Yang-Mills theory and U(1) symmetry breaking.

Contribution

It introduces the use of symmetry-twisted partition functions as diagnostic tools for quantum phases and connects them to Wilson-'t Hooft classification and anomaly considerations.

Findings

Twisted partition functions differentiate SSB, SPT, and SET states.

Application to 4d Yang-Mills links twisted partition functions with topological classifications.

Analysis of U(1) symmetry breaking reveals insights from mixed anomalies.

Abstract

For quantum field theories with global symmetry, we can study the behavior of the partition function with the background gauge field to diagnose different quantum phases. For the case of discrete symmetries, we find that the symmetry-twisted partition function works as an order parameter that discriminates spontaneous symmetry breaking (SSB), symmetry-protected topological (SPT) states, and symmetry-enriched topological (SET) states. We then consider its application to the case of 4d Yang-Mills theory with adjoint matters to understand the relation between the twisted partition function and the Wilson-'t Hooft classification. We also study its behavior for the spontaneously broken U(1) symmetry and interpret the result from the viewpoint of the mixed anomaly with the emergent solitonic symmetry.