Rigorous Index Theory for One-Dimensional Interacting Topological Insulators

Abstract

We present a rigorous but elementary index theory for a class of one-dimensional systems of interacting fermions that includes the Su-Schrieffer-Heeger (SSH) model as a special case. We prove that the sign of the expectation value of the local twist operator gives a topological Z2 index for a unique gapped ground state on the infinite chain. This establishes that any path of interacting disordered models (in the class) that connects the two extreme cases of the SSH model must go through a phase transition. We also prove that any unique gapped ground state in the class is accompanied by a gapless edge mode when defined on a suitable half-infinite chain. (There are two lecture videos in which the main results of the paper are discussed [1,2].