Higher dimensional abelian quantum double models have been shown to be well defined in any finite dimension and exhibit the characteristic behavior of SPT phases models. In this talk, we will introduce the formalism of these models in a pedagogical manner, focusing on the characterization of the topological ground state subspace and briefly presenting its classification scheme. We will discuss the connection of these models with pressing problems in condensed matter physics and quantum computation.

I will first show that for general 2d random ergodic one-body magnetic Schrödinger operators the bulk magnetization equals the total edge current at any temperature. Moreover, the celebrated bulk-edge correspondence between quantum transport indices will be obtained as a corollary of our result by imposing a gap condition and by taking a "zero temperature" limit. After that, I will show how to extend the equality of bulk magnetization and total edge current to lattice fermion systems with finite-range interactions satisfying local indistinguishability of the Gibbs state, a condition known to hold for sufficiently high temperatures. In the interacting framework, an important intermediate result is a new version of Bloch's theorem for two-dimensional systems, stating that persistent currents vanish in the bulk.

The talk is based on joint works with Horia Cornean, Jonas Lampart, Stefan Teufel and Tom Wessel.

For a one-dimensional random discrete Schrödinger operator, the energies at which all transfer matrices commute and have their spectrum off the unit circle are called critical hyperbolic. Disorder driven topological phase transitions in such models are characterized by a vanishing Lyapunov exponent at the critical energy. It is shown that the density of states away from a transition has pseudogap with an explicitly computable Hölder exponent, while it has a logarithmic divergence (Dyson spike) at the transition points. The proof is based on renewal theory for the Prüfer phase dynamics and the optional stopping theorem for suitably constructed comparison martingales.