Exact Chiral Symmetries of 3+1D Hamiltonian Lattice Fermions

TL;DR

The paper constructs ultralocal 3+1D lattice Hamiltonians realizing exact chiral and SU(2) anomalies via not-on-site emanant symmetries. It presents a single Weyl fermion protected by a non-quantized, non-on-site $U(1)$ chiral symmetry and a symmetry-protected double Weyl fermion whose two generators generate the Onsager algebra, yielding an $SU(2)$ flavor anomaly that protects gaplessness even when translations are broken. A complementary no-go theorem demonstrates that quantized on-site $U(1)$ symmetries cannot realize the single-Weyl anomaly, motivating the not-on-site approach, while a 2+1D parity-anomaly example with an almost-local chiral symmetry extends the framework. The results illustrate sharp, ultralocal lattice realizations of chiral and flavor anomalies, with implications for gauging these symmetries and for exploring lattice models of chiral gauge theories.

Abstract

We construct Hamiltonian models on a 3+1d cubic lattice for a single Weyl fermion and for a single Weyl doublet protected by exact (as opposed to emergent) chiral symmetries. In the former, we find a not-on-site, non-compact chiral symmetry which can be viewed as a Hamiltonian analog of the Ginsparg-Wilson symmetry in Euclidean lattice models of Weyl fermions. In the latter, we combine an on-site $U(1)$ symmetry with a not-on-site $U(1)$ symmetry, which together generate the $SU(2)$ flavor symmetry of the doublet at low energies, while in the UV they generate an algebra known in integrability as the Onsager algebra. This latter model is in fact the celebrated magnetic Weyl semimetal which is known to have a chiral anomaly from the action of $U(1)$ and crystalline translation, that gives rise to an anomalous Hall response - however reinterpreted in our language, it has two exact $U(1)$ symmetries that gives rise to the global $SU(2)$ anomaly which protects the gaplessness even when crystalline translations are broken. We also construct an exact symmetry-protected single Dirac cone in 2+1d with the $U(1) \rtimes T$ parity anomaly. Our constructions evade both old and recently-proven no-go theorems by using not-on-siteness in a crucial way, showing our results are sharp.