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 $\anticomm{\Deff}{\gfive}=0$ 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.