Rigorous Index Theory for One-Dimensional Interacting Topological Insulators

We present a rigorous but elementary index theory for a class of one-dimensional systems of interacting (and possibly disordered) fermions with $\Uone\rtimes\bbZ_2$ symmetry defined on the infinite chain. The class includes the Su-Schrieffer-Heeger (SSH) model as a special case. For any locally-unique gapped (fixed-charge) ground state of a model in the class, we define a $\bbZ_2$ index in terms of the sign of the expectation value of the local twist operator. We prove that the index is topological in the sense that it is invariant under continuous modification of models in the class with a locally-unique (fixed-charge) gapped ground state. This establishes that any path of models in the class that connects the two extreme cases of the SSH model must go through a phase transition. Our rigorous $\bbZ_2$ classification is believed to be optimal for the class of models considered here. We also show an interesting duality of the index, and prove that any topologically nontrivial model in the class has a gapless edge excitation above the ground state when defined on the half-infinite chain. The results extend to other classes of models, including the extended Hubbard model. Our strategy to focus on the expectation value of local unitary operators makes the theory intuitive and conceptually simple. The paper also contains a careful discussion about the notion of unique gapped ground states of a particle system on the infinite chain.