CM - AMO SEMINAR | When Do Crystalline Insulators Exhibit Quantized Hall Effects?

Quantum Hall effects of 2-dimensional electron gas (2DEG) under a magnetic field exemplify the first class of topological phases. Galilean invariance plays an important role in 2DEG, which determines the Hall conductivity as the ratio of electron and flux density in unit of a universal constant. On the other hand, for electrons moving under a periodic potential, in lack of Galilean invariance, can the electron and flux density determines the Hall conductivity of a crystalline insulator? We show that even without Galilean invariance, discrete lattice translations under a magnetic field gives rise to a generic relation between charge/flux density and the Hall conductance of any crystalline insulator. We discuss how this relation applies to quantum Hall effects on lattices and certain magnetic insulators.