Geometric response of FQH states

Abstract

Two-dimensional interacting electron gas in strong transverse magnetic field forms a collective state -- incompressible electron liquid, known as fractional quantum Hall (FQH) state. FQH states are genuinely new states of matter with long range topological order. Their primary observable characteristics are the absence of dissipation and quantization of the transverse electro-magnetic response known Hall conductance. In addition to quantized electromagnetic response FQH states are characterized by quantized geometric responses such as Hall viscosity and thermal Hall conductance.
  I will show how to derive the effective action for various Abelian and non-Abelian FQH states on a curved space. In particular, I will derive the quantized universal responses to the changes in geometry of space. These responses are described by Chern-Simons-type terms. It will be shown that in order to obtain the responses in a self consistent way one has to take into account the framing anomaly of the quantum Chern-Simons(-Witten) theory. This peculiar phenomenon illustrates the failure of a classically topological theory to remain topological at the quantum level.
   If time permits I will comment on the coupling of non-relativistic systems to the space-time geometry. Using the appropriate geometry I will write an effective action describing the bulk energy and thermal Hall conductances. From this effective action it will be clear that these response functions are neither universal nor topologically protected.