Exact chiral symmetry, non-Hermiticity, and the axial anomaly on the lattice: A symmetry-class criterion for disorder-induced undoubling
Abstract
A lattice Dirac operator designed to preserve exact chiral symmetry by weighting the derivative with a quenched random field---the \emph{chiral disordered fermions} (\CDF) of [L.\ Jacobs, Phys.\ Rev.\ Lett.\ \textbf{51}, 172 (1983)]---is reexamined with modern tools. We show that the disorder-averaged operator satisfies anticommDeffgfive=0 exactly, to all orders in disorder strength. Disorder lifts the doubler degeneracy, driving doublers off the real axis into a complex mass while the physical mode remains protected, but does so through non-Hermiticity rather than a Ginsparg--Wilson relation. Consequently the axial index vanishes in every formulation: the Hermitian trace Tr(gfive D), the bi-orthogonal index for non-Hermitian operators, and the algebraic Riesz-projector index, including at exceptional points. The \CDF operator therefore cannot reproduce the axial anomaly: its failure is enforced by the exact chiral symmetry it preserves. We classify neighboring constructions by the Dirac channel carrying the lattice structure. Disorder in the gμ channel preserves chirality, realizes an Altland--Zirnbauer chiral class, and cannot carry an index; disorder in the 𝟙 mass channel breaks chiral symmetry and yields Callan--Harvey/domain-wall physics. A one-parameter family interpolates between them, with the 1983 operator and Kaplan's domain wall as opposite limits, distinguished not by disorder itself but by the channel it occupies. This unifies lattice-QCD undoubling with the tenfold-way classification of disordered Dirac fermions and settles the disorder route to chiral lattice fermions.
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BibTeX
@misc{jacobs2026exact,
title = {{Exact chiral symmetry, non-Hermiticity, and the axial anomaly on the lattice: A symmetry-class criterion for disorder-induced undoubling}},
author = {Laurence A. Jacobs},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05395},
eprint = {2611.05395},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05395},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exact chiral symmetry, non-Hermiticity, and the axial anomaly on the lattice: A symmetry-class criterion for disorder-induced undoubling
AU - Laurence A. Jacobs
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.05395
UR - https://arxiv.org/abs/2611.05395
AB - A lattice Dirac operator designed to preserve exact chiral symmetry by weighting the derivative with a quenched random field---the \emph{chiral disordered fermions} (\CDF) of [L.\ Jacobs, Phys.\ Rev.\ Lett.\ \textbf{51}, 172 (1983)]---is reexamined with modern tools. We show that the disorder-averaged operator satisfies anticommDeffgfive=0 exactly, to all orders in disorder strength. Disorder lifts the doubler degeneracy, driving doublers off the real axis into a complex mass while the physical mode remains protected, but does so through non-Hermiticity rather than a Ginsparg--Wilson relation. Consequently the axial index vanishes in every formulation: the Hermitian trace Tr(gfive D), the bi-orthogonal index for non-Hermitian operators, and the algebraic Riesz-projector index, including at exceptional points. The \CDF operator therefore cannot reproduce the axial anomaly: its failure is enforced by the exact chiral symmetry it preserves. We classify neighboring constructions by the Dirac channel carrying the lattice structure. Disorder in the g_μ channel preserves chirality, realizes an Altland--Zirnbauer chiral class, and cannot carry an index; disorder in the 𝟙 mass channel breaks chiral symmetry and yields Callan--Harvey/domain-wall physics. A one-parameter family interpolates between them, with the 1983 operator and Kaplan's domain wall as opposite limits, distinguished not by disorder itself but by the channel it occupies. This unifies lattice-QCD undoubling with the tenfold-way classification of disordered Dirac fermions and settles the disorder route to chiral lattice fermions.
ER -