Study of Curved Domain-wall Fermions on a Lattice

TL;DR

This thesis investigates fermion systems with curved domain-wall mass terms on lattices, revealing localized chiral states, effects of gravity and gauge fields, and applications to topological insulators and anomalies.

Contribution

It introduces a lattice model with curved domain-wall fermions, analyzing their properties, gauge interactions, and topological effects, connecting lattice results with continuum theories and condensed matter phenomena.

Findings

Localized massless chiral states at curved domain walls

Gauge fields induce Aharonov-Bohm effects and T-symmetry anomalies

Novel localized modes at fluxes cancel T anomalies

Abstract

In this thesis, we consider fermion systems on square lattice spaces with a curved domain-wall mass term. In a similar way to the flat case, we find massless and chiral states localized at the wall. In the case of $S^1$ and $S^2$ domain-wall embedded into a square lattice, we find that these edge states feel gravity through the induced spin connection. In the conventional continuum limit of the higher dimensional lattice, we find a good consistency with the analytic results in the continuum theory. We also confirm that the rotational symmetry is recovered automatically. We also discuss the effect of a $U(1)$ gauge connection on a two-dimensional lattice fermion with the $S^1$ domain-wall mass term. We find that the gauge field changes the eigenvalue spectrum of the boundary system by the Aharanov-Bohm effect and generates an anomaly of the time-reversal ($T$) symmetry. Our numerical evaluation is consistent with the Atiyah-Patodi-Singer index, which describes the cancellation of the $T$ anomaly by the topological term on the bulk system. When we squeeze the flux inside one plaquette while keeping the total flux unchanged, the anomaly inflow undergoes a drastic change. The intense flux gives rise to an additional domain wall around the flux. We observe a novel localized mode at the flux, canceling the $T$ anomaly on the wall instead of the topological term in the bulk. We apply the study to a problem in condensed matter physics. It is known that inside topological insulators, a vortex or monopole acquires a fractional electric charge and turns into a dyon. Describing the topological insulator as a negative mass region of a Dirac fermion, we provide a microscopic description of this phenomenon in terms of the dynamical domain-wall creation.