Anomaly diagnosis via symmetry restriction in two-dimensional lattice systems

TL;DR

This paper introduces a method to compute anomalies of finite symmetry groups in 2D lattice systems, revealing obstructions to trivial gapped phases and onsite symmetry realization, with a diagrammatic and higher categorical perspective.

Contribution

It develops a cohomological index for symmetry anomalies in 2D lattice systems, generalizing previous indices and providing a diagrammatic interpretation.

Findings

Nontrivial anomalies prevent trivial gapped symmetric Hamiltonians.

The method applies to complex symmetry groups like ^4.

A diagrammatic and higher categorical understanding of anomalies is proposed.

Abstract

We describe a method for computing the anomaly of any finite unitary symmetry group $G$ acting by finite-depth quantum circuits on a two-dimensional lattice system. The anomaly is characterized by an index valued in the cohomology group $H^4(G,U(1))$, which generalizes the Else-Nayak index for locality preserving symmetries of quantum spin chains. We show that a nontrivial index precludes the existence of a trivially gapped symmetric Hamiltonian; it is also an obstruction to ``onsiteability" of the symmetry action. We illustrate our method via a simple example with $G=\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2$. Finally, we provide a diagrammatic interpretation of the anomaly formula which hints at a higher categorical structure.