Tori, Klein Bottles, and Modulo 8 Parity/Time-reversal Anomalies of 2+1d Staggered Fermions

TL;DR

This paper investigates the symmetries and 't Hooft anomalies of 2+1d lattice staggered fermions on various topological backgrounds, establishing a correspondence with continuum theories and developing a formalism for lattice models on nontrivial spaces.

Contribution

It introduces a formalism to analyze lattice fermions on complex topologies and maps lattice symmetries to continuum anomalies, enhancing understanding of symmetry and anomaly matching.

Findings

Successfully placed lattice models on sheared tori and Klein bottles.

Matched lattice and continuum 't Hooft anomalies via a nontrivial symmetry map.

Developed a general approach for Hamiltonian models on flat, nontrivial spaces.

Abstract

We study the symmetries of lattice staggered fermions in 2+1d. Using the symmetries, we can place the system on any sheared torus or Klein bottle. These different backgrounds provide diagnostics of various 't Hooft anomalies associated with the crystalline symmetries. We then compare the lattice model to its continuum limit. The symmetries of the lattice system are mapped in a nontrivial way to the symmetries of the continuum theories. Using this map, we match the 't Hooft anomalies on the lattice and the continuum. Along the way, we develop a general formalism to study Hamiltonian lattice models on nontrivial, compact, flat spaces.