Anomaly of conserved and nonconserved axial charges in Hamiltonian lattice gauge theory

TL;DR

The paper studies how the axial anomaly arises in Hamiltonian lattice gauge theory and identifies a crucial ambiguity in defining axial charges in the 1+1D U(1) setting. By comparing a nonconserved axial charge $Q_5'$ with a conserved one $Q_5$ under the Wilson-Dirac Hamiltonian, it shows that $Q_5'$ fails to reproduce the anomaly while $Q_5$ yields the correct relation $∂_μ j_5^μ = (e/π) E$ in the continuum, aided by a spectral-flow interpretation that isolates physical zero modes from doublers. Through analytic derivations and classical-background simulations, the work demonstrates that doubler artifacts can contaminate time evolution in Hamiltonian lattice theories and that using the conserved axial charge provides a robust route to the anomaly, with practical implications for quantum simulations. Overall, the study provides a concrete, operational lesson: to correctly capture the axial anomaly in real-time Hamiltonian lattices, one must employ the conserved axial charge to avoid doubler-induced artifacts and ensure the continuum limit reproduces the correct anomaly structure.

Abstract

We investigate the axial anomaly in Hamiltonian lattice gauge theory. The definition of axial charge operators is ambiguous, especially between conserved and nonconserved axial charges. While these charges appear to differ only by a higher-order term in lattice spacing, they do not coincide in the continuum limit. We demonstrate, through analytical and numerical calculations in 1+1 dimensions, that the conserved axial charge correctly reproduces the axial anomaly relation in continuous spacetime. Our finding would serve as a valuable lesson about doubler artifact in Hamiltonian time evolution of lattice gauge theory.

Figures (2)

• Figure 1: Left: anomalous production of the nonconserved axial charge $\langle Q_5'\rangle a$ and the conserved axial charge $\langle Q_5 \rangle a$ against time $t$. An electric field is switched on at $t=0$. Right: same as the left panel but the initial region is enlarged.
• Figure 2: Single-particle spectrum of the energy $\varepsilon$ vs the axial charge $q_5$. The left is the vacuum state ($t=0$) and the right is an evolved state ($t>0$).