Curved domain-wall fermion and its anomaly inflow

TL;DR

This paper studies how curved domain-wall fermions on lattices exhibit anomaly inflow and edge modes influenced by gauge fields, gravitational effects, and magnetic flux, revealing phenomena like anomaly cancellation and the Witten effect.

Contribution

It introduces a lattice model of curved domain-wall fermions with gauge and gravitational fields, demonstrating anomaly inflow and edge mode behavior consistent with index theorems and revealing new flux-induced phenomena.

Findings

Edge modes localized at curved domain walls are affected by gauge and gravitational fields.

Anomaly inflow is confirmed to match the Atiyah-Patodi-Singer index theorem.

Quantized flux induces additional domain walls and zero modes, explaining the Witten effect.

Abstract

We investigate the effect of $U (1)$ gauge field on lattice fermion systems with a curved domain-wall mass term. In the same way as the conventional flat domain-wall fermion, the chiral edge modes appear localized at the wall, whose Dirac operator contains the induced gravitational potential as well as the $U(1)$ vector potential. In the case of $S^1$ domain-wall fermion on a two-dimensional flat lattice, we find a competition between the Aharonov-Bohm(AB) effect and gravitational gap in the Dirac eigenvalue spectrum, which leads to anomaly of the time-reversal ($T$) symmetry. Our numerical result shows a good consistency with the Atiyah-Patodi-Singer index theorem on a disk inside the $S^1$ domain-wall, which describes the cancellation of the $T$ anomaly between the bulk and edge. When the $U(1)$ flux is squeezed inside one plaquette, and the AB phase takes a quantized value $\pi$ mod $2\pi\mathbb{Z}$, the anomaly inflow drastically changes: the strong flux creates another domain-wall around the flux to make the two zero modes coexist. This phenomenon is also observed in the $S^2$ domain-wall fermion in the presence of a magnetic monopole. We find that the domain-wall creation around the monopole microscopically explains the Witten effect.