Symmetries and Anomalies of Hamiltonian Staggered Fermions

TL;DR

The paper investigates the symmetry structure of Hamiltonian staggered fermions, showing that lattice shift symmetries realize a discrete axial-flavor subgroup while odd shifts anticommute with time reversal, hinting at mixed ’t Hooft anomalies. By constructing explicit lattice operators and decomposing complex fields into real halves, the authors connect lattice shifts to continuum axial and vector transformations and classify mass-like operators under lattice symmetries, demonstrating that the massless theory remains free of additive mass terms. They propose anomaly cancellation via symmetric mass generation with four-fermion interactions, achieving a gapped, invariant ground state consistent with a $Z_{16}$ condition in $d=3$, and they identify a rich algebra of conserved charges generated by half-shifts that signals lattice anomalies when gauged. The work thereby links Hamiltonian lattice formulations to continuum anomaly physics, providing tools to study gauging, symmetry breaking, and the continuum limit in staggered fermion systems with potential applications to lattice gauge theories and chiral constructions.

Abstract

We review the shift (translation) and time reversal symmetries of Hamiltonian staggered fermions and their connection to continuum symmetries concentrating in particular on the case of massless fermions and (3+1) dimensions. We construct operators using the staggered fields that implement these symmetries on finite lattices. We show that shifts composed of an odd multiple of the elementary shift anti-commute with time reversal and are related to continuum axial transformations. We argue that the presence of these non-trivial commutation relations implies the existence of lattice 't Hooft anomalies. From the shifts we also construct a set of conserved, quantized charges that generate continuous symmetries of the lattice theory. In general these do not commute with the vector charge signaling further 't Hooft anomalies.