QLunch: Horia Cornean

We consider 2d random ergodic magnetic Schrödinger operators on domains with and without boundary.  By extending the gauge covariant magnetic perturbation theory to infinite domains with boundary, we prove that the celebrated bulk-edge correspondence of systems with a (mobility) gap at zero temperature, i.e. the equality of the transversal bulk conductivity and the edge conductance, holds also at all positive temperatures and irrespective of a (mobility) gap in the bulk.  While the quantization of the transverse bulk conductivity and the edge conductivity at zero temperature and in the presence of a (mobility) gap is a topological feature, their equality is not, but applies much more generally.

Moreover, we obtain a formula which states that at any positive temperature, the derivative of a large class of bulk partition functions with respect to the external constant magnetic field is equal to the expectation of a corresponding edge distribution function of the velocity component which is parallel to the edge.  Physically,  our formula implies not only equality between transverse bulk conductivity and edge conductance, but also equality between bulk magnetization density and edge current at all temperatures. As a corollary of our purely analytical arguments, we find that in gapped systems the transverse bulk conductivity and the edge conductance approach their quantized (integer) values at a rate that is exponential in the inverse temperature. This is joint work with M. Moscolari and S. Teufel.