Physics (Internal)

2D TOPOLOGICAL INSULATORS: from 2D to 1D; from 1D to 2D

This event is part of the Preliminary Oral Exam.

Examining Committee: Claudio Chamon, Michael El-Batanouny, Emanuel Katz, and Anatoli Polkovnikov

Abstract: Topological insulators are materials that have a bulk band gap like normal insulators, but have symmetry protected gapless edge states. A 2D topological insulator is a quantum spin Hall insulator protected by time reversal symmetry. From the perspective of field theory, 2D topological insulators are described by double Chern-Simons theory. The 2+1D Chern-Simons theory actually reduces to 1+1D Wess-Zumino-Witten(WZW) model on the boundary which describes the edge states. However, the symmetry protection of the edge states are unclear in this language. We present a bulk-edge correspondence when certain perturbations are added into the Chern-Simons theory which may be generalized to address the symmetry protection. On the other hand, the fact that 1D fermionic systems can be bosonized and described by WZW models urges us to construct 2D topological insulators by coupling an array of 1D wires. With designed coupling, we can gap out the bulk of the array and leave desired robust edge states. We reconstruct all known short-range-entangled topological states and search for long-range-entangled states that have novel physics such as non-Abelian statistics and fractional central charge.