Topological invariants and interacting one-dimensional fermionic systems

TL;DR

This paper investigates one-dimensional interacting fermionic systems, using topological invariants derived from Green's functions to identify topologically protected boundary states, combining analytical and numerical methods to map phase diagrams.

Contribution

It introduces a topological invariant based on Green's functions for interacting fermionic systems and links it to boundary states, extending topological classification to interacting regimes.

Findings

Topologically protected zero-energy boundary states are characterized by the invariant.

Numerical calculations map phase diagrams and identify topological phases.

Spin systems in the Mott regime exhibit topological states linked to fermionic invariants.

Abstract

We study one-dimensional, interacting, gapped fermionic systems described by variants of the Peierls-Hubbard model and characterize their phases via a topological invariant constructed out of their Green's functions. We demonstrate that the existence of topologically protected, zero-energy states at the boundaries of these systems can be tied to the values of their topological invariant, just like when working with the conventional, noninteracting topological insulators. We use a combination of analytical methods and the numerical density matrix renormalization group method to calculate the values of the topological invariant throughout the phase diagrams of these systems, thus deducing when topologically protected boundary states are present. We are also able to study topological states in spin systems because, deep in the Mott insulating regime, these fermionic systems reduce to spin chains. In this way, we associate the zero-energy states at the end of an antiferromagnetic spin-one Heisenberg chain with the topological invariant 2.