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arXiv:2606.12510

Infinite-Order Lattice Chiral Anomalies and CPT

Elijah Lew-Smith, Salvatore D. Pace, Shu-Heng Shao

Published Jun 10, 2026• 0 citations
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Abstract

A key property of a global symmetry's anomaly is its order: the smallest integer $n$ for which the diagonal symmetry of the $n$-copy system is anomaly-free. While many familiar lattice anomalies have finite order, perturbative anomalies in the continuum$-$those captured by Feynman diagrams$-$have infinite order. In this paper, we show that the Onsager symmetry, a lattice realization of the chiral symmetry of a 1+1d massless Dirac fermion, has an order-two anomaly. However, imposing lattice CPT symmetry enhances this anomaly from order two to infinite order, yielding a lattice chiral symmetry structure that more faithfully matches the continuum chiral anomaly. We also discuss the corresponding 2+1d symmetry-protected topological phases for these infinite-order lattice anomalies.

DOI
10.48550/arxiv.2606.12510

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A key property of a global symmetry's anomaly is its order: the smallest integer nnn for which the diagonal symmetry of the nnn-copy system is anomaly-free. While many familiar lattice anomalies have finite order, perturbative anomalies in the continuum−-−those captured by Feynman diagrams−-−have infinite order. In this paper, we show that the Onsager symmetry, a lattice realization of the chiral symmetry of a 1+1d massless Dirac fermion, has an order-two anomaly. However, imposing lattice CPT symmetry enhances this anomaly from order two to infinite order, yielding a lattice chiral symmetry structure that more faithfully matches the continuum chiral anomaly. We also discuss the corresponding 2+1d symmetry-protected topological phases for these infinite-order lattice anomalies.
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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 U(1){\mathrm U}(1)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 Z2{\mathbb Z}_2Z2​ fermion parity.