We present a rigorous but elementary index theory for a class of one-dimensional systems of interacting (and possibly disordered) fermions with symmetry defined on the infinite chain. The class includes the Su–Schrieffer–Heeger (SSH) model [Su et al., “Solitons in polyacetylene,” Phys. Rev. Lett. 42, 1698 (1979); Su et al., “Soliton excitations in polyacetylene,” Phys. Rev. B 22, 2099 (1983); and Asbóth et al., A Short Course on Topological Insulators: Band-Structure Topology and Edge States in One and Two Dimensions, Lecture Notes in Physics (Springer, 2016)] as a special case. For any locally unique gapped (fixed-charge) ground state of a model in the class, we define a 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 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. This paper also contains a careful discussion about the notion of unique gapped ground states of a particle system on the infinite chain. (There are two lecture videos in which the main results of this paper are discussed [H. Tasaki, “Rigorous index theory for one-dimensional interacting topological insulators: A brief introduction,” online lecture (21:41), November, 2021, seehttps://www.gakushuin.ac.jp/~881791/OL/#Index1DTI2021S and https://youtu.be/ypGVb3eYrpg and H. Tasaki, “Rigorous index theory for one-dimensional interacting topological insulators: With a pedagogical introduction to the topological phase transition in the SSH model,” online lecture (49:07), November, 2021, see https://www.gakushuin.ac.jp/~881791/OL/#Index1DTI2021L and https://youtu.be/yxZYOevV2Y].
To be rigorous, a state is a linear map from the C-algebra , the norm completion of , to .
We expect that a pure fixed-charge ground state ω of is a ground state of with suitable μ, except at phase transition points.
The converse is much more delicate in this case since a locally unique gapped ground state should have a gap for all possible excitations, while a locally unique gapped fixed-charge ground state is only required to have a gap for particle number preserving excitations.
This is a formal result that follows from an abstract theorem (the Banach–Alaoglu theorem) in functional analysis. See, e.g., Theorem A. 24 (p. 488) of Ref. 52 for more details.
Note that a locally unique gapped (fixed-charge) ground state of a Γphg-invariant Hamiltonian is not necessarily Γphg-invariant.
Note that this is true only for the infinite chain. In this sense, our index theory relies essentially on the fact that the chain is infinite.
If we use the duality stated in Theorem 3.4, then Ind0 = 1 implies Ind1 = −1. In this sense, the above (slightly complicated) evaluation of is not necessary.
The spin operators are defined in terms of the fermion operators as , , and .
To characterize fixed-charge ground states, we use perturbations such that . Note that we are making a slight abuse of notation here since not only the total charge but also the total is conserved.