Curved domain-wall fermion and its anomaly inflow

Abstract

AbstractWe investigate the effect of a 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 anS1 domain-wall fermion on a two-dimensional flat lattice, we find a competition between the Aharonov–Bohm(AB) effect and a gravitational gap in the Dirac eigenvalue spectrum, which leads to an anomaly inthe time-reversal (T) symmetry. Our numerical result shows a good consistency with the Atiyah–Patodi–Singer index theorem on a disk inside the S1 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 π 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 S2 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.