Spontaneous edge currents for the Dirac equation in two space dimensions

TL;DR

This paper demonstrates the existence of spontaneous edge currents in a two-dimensional Dirac system without magnetic fields, driven by symmetry-breaking mass terms, and explores their topological stability and relation to chemical potential differences.

Contribution

It introduces a novel effect of edge currents in a field-free Dirac system with broken time reversal symmetry, analyzing their properties and topological invariance.

Findings

Edge currents are proportional to chemical potential differences.

Edge conductivity depends on the choice of selfadjoint extension.

Edge conductivity remains invariant under small perturbations.

Abstract

Spontaneous edge currents are known to occur in systems of two space dimensions in a strong magnetic field. The latter creates chirality and determines the direction of the currents. Here we show that an analogous effect occurs in a field-free situation when time reversal symmetry is broken by the mass term of the Dirac equation in two space dimensions. On a half plane, one sees explicitly that the strength of the edge current is proportional to the difference between the chemical potentials at the edge and in the bulk, so that the effect is analogous to the Hall effect, but with an internal potential. The edge conductivity differs from the bulk (Hall) conductivity on the whole plane. This results from the dependence of the edge conductivity on the choice of a selfadjoint extension of the Dirac Hamiltonian. The invariance of the edge conductivity with respect to small perturbations is studied in this example by topological techniques.