Parity anomaly from LSM: exact valley symmetries on the lattice

TL;DR

This paper demonstrates that the honeycomb lattice model exhibits an exact SU(2) valley symmetry and parity anomaly, with implications for understanding topological phases and anomalies in condensed matter systems.

Contribution

It reveals an exact microscopic realization of valley symmetry and parity anomaly on the lattice, connecting lattice symmetries with continuum anomalies.

Findings

SU(2) valley symmetry arises from conserved charge operators

Lattice reflection and time-reversal symmetries form an LSM anomaly

No trivial gapped ground state exists with these symmetries

Abstract

We show that the honeycomb tight-binding model hosts an exact microscopic avatar of its low-energy SU(2) valley symmetry and parity anomaly. Specifically, the SU(2) valley symmetry arises from a collection of conserved, integer quantized charge operators that obey the Onsager algebra. Along with lattice reflection and time-reversal symmetries, this Onsager symmetry has a Lieb-Schultz-Mattis (LSM) anomaly that matches the parity anomaly in the IR. Indeed, we show that any local Hamiltonian commuting with these symmetries cannot have a trivial unique gapped ground state. We study the phase diagram of the simplest symmetric model and survey various deformations, including Haldane's mass term, which preserves only the Onsager symmetry. Our results place the parity anomaly in ${2+1}$D alongside Schwinger's anomaly in ${1+1}$D and Witten's SU(2) anomaly in ${3+1}$D as 't Hooft anomalies that can arise from the Onsager symmetry on the lattice.