Spontaneous Edge Currents for the Dirac Equation in Two Space Dimensions

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.

On spectral flow and Fermi arcs

“…In full generality, γ need only be measurable, see, e.g., [10] in the context of Fourier transforming half-plane Dirac Hamiltonians, and [9]. Typically, in a very symmetric (say rotationally) problem, continuity of γ might be lost at special momentum space points such as the origin, and could affect the spectral flow, as we saw from Cor.…”

“…Example 4. This is essentially the massive Dirac operator on a half-plane, also studied in [10]. Fix some a > 0.…”

Boundary states and edge currents for free fermions

“…on the open half-plane {x = (x, y) ∈ R 2 | x > 0}, as in [7]. This operator is a closable symmetric operator on the domain D(H) = C ∞ 0 (R + × R, C 2 ) of smooth functions with compact support vanishing in a neighbourhood of x = 0.…”

Spectral transition for Dirac operators with electrostatic $δ$-shell potentials supported on the straight line

“…We note that the method of direct integrals decomposing A η in its fibers would yield a similar result; cf. [24] for a related problem on Dirac operators with Robin type boundary conditions. In this note we prefer to work with the boundary triple technique, since it allows generalizations for more complicated curves Σ in a natural way, which we plan to study in the near future.…”