Non-linear Optical Response of an Exactly Soluble Model of Topological Insulators

We examine an exactly soluble model of time-reversal and inversion symmetric Topological Insulators and calculate the linear optical conductivity. For frequencies smaller than the bulk gap, the response is quantized and yields information on the surface Weyl cones and their energies. The inclusion of many-body interactions changes the result and may cause an instability due to the spontaneous emission of spin-one k=0 excitations, if the Fermi-energy is in close proximity to the vertex of the Weyl cone. For topological insulators, the low-frequency conductivity is caused by metallic states and the conductivity is highly non-local. The non-linear response functions are shown to diverge at low-frequencies, due to the 1/√ω coupling that lead to the well-known infra-red divergences found in Brehmstrahlung. We show that the infra-red divergences in the response functions due to the interaction with the surface states at both surfaces of the slab, cancel analogously to Furry's Theorem. Cancellation of the divergences seems to requires that the non-local Maxwell's equations are solved and the systematic inclusion many-body interactions.