Zero Field Hall Effect in (2+1)-dimensional QED

Abstract

In QED of two space dimensions, a quantum Hall effect occurs in the absence of any magnetic field. We give a simple and transparent explanation. In solid state physics, the Hall conductivity for non-degenerate ground state is expected to be given by an integer, the Chern number. In our field-free situation, however, the conductivity is $\pm 1/2$ in natural units. We fit this half-integral result into the topological setting and give a geometric explanation reconciling the points of view of QFT and solid state physics. For quasi-periodic boundary conditions, we calculate the finite size correction to the Hall conductivity. Applications to graphene and similar materials are discussed.