Chiral and Geometric Anomalies in Finite Systems

Abstract: Dirac fermions coupled to gauge fields can exhibit the chiral anomaly even on a finite spatial lattice. A careful description of this phenomenon yields new insights into the nature of spin-charge relations and on-site symmetries (symmetries that are gauged by placing gauge fields on all links of the lattice). One notable result is that only sufficiently small symmetry groups can act on-site in a system with finitely many degrees of freedom. Symmetries that break this rule either cannot be gauged on an arbitrary lattice, or the gauging decouples the matter fields from gauge fields. These "anomalies" are not quantum in nature, and they are diagnosed geometrically, by the volume of the spatial manifold. The familiar particle number ${\mathrm U}(1)$ exhibits this kind of anomalous behavior in any finite fermion theory. The chiral anomaly in such a finite system instead manifests itself most simply after gauging the ${\mathbb Z}_2$ fermion parity.