The Axial Charge in Hilbert Space and the Role in Chiral Gauge Theories

TL;DR

This paper develops a Hamiltonian framework for lattice fermions that preserves axial symmetry exactly, enabling the study of chiral gauge theories and symmetric mass generation mechanisms.

Contribution

It reconstructs axial and vector charge operators within Wilson fermions, enabling exact axial symmetry on the lattice and applications to SMG models.

Findings

Axial charge operators act locally with quantized eigenvalues.

Framework allows symmetry-preserving interactions with quantized chiral charges.

Potential to realize symmetric mass generation in lattice models.

Abstract

We investigate the Hamiltonian formulation of 1+1-dimensional staggered fermions and reconstruct the vector and axial charge operators, originally identified by Arkya Chatterjee et al., within the Wilson fermion formalism. These operators commute with the Hamiltonian and reduce, in the continuum limit, to the generators of the vector and axial $\mathrm{U}(1)$ symmetries. A notable feature of the axial charge operator is that it acts locally on operators and possesses quantized eigenvalues. Its eigenstates can therefore be interpreted as fermion states with well-defined integer chirality, analogous to those in the continuum theory. This structure enables the formulation of a gauge theory in which the axial $\mathrm{U}(1)_A$ symmetry is promoted to a gauge symmetry. We construct a Hamiltonian in terms of the eigenstates of the axial charge operator, thereby preserving exact axial symmetry on the lattice while recovering vector symmetry in the continuum limit. As applications, we study the implementation of the Symmetric Mass Generation (SMG) mechanism in the 3-4-5-0 models. Our framework admits symmetry-preserving interaction terms with quantized chiral charges, although further numerical investigation is required to confirm the realization of the SMG mechanism in interacting systems.