Topological phases and spontaneous symmetry breaking: the revenge of the original Su-Schrieffer-Heeger model

TL;DR

This paper investigates how spontaneous symmetry breaking affects topological phases in one-dimensional models, revealing that the original SSH model is topologically trivial when symmetry breaking is considered, unlike its mean-field approximation.

Contribution

It demonstrates that spontaneous symmetry breaking can alter the topological classification of the SSH model, and clarifies the topological nature of the spinful version with symmetry considerations.

Findings

Original SSH model is topologically trivial under symmetry breaking.

Spinful SSH model exhibits a topologically non-trivial phase.

Topological phases are protected by chiral symmetry.

Abstract

We study the interplay of spontaneous symmetry breaking and topological properties in interacting one-dimensional models. We solve these models using bozonization and identify topologically non-trivial phases by counting the additional degeneracy (affiliated with the edge modes) of a finite-size system relative to the infinite one. We find even if the mean-field solution is topological, this may not be true when it arises from spontaneous symmetry breaking, including in the Su-Schrieffer-Heeger (SSH) model. This implies that the original SSH model, as presented by Su, Schrieffer, and Heeger, is topologically trivial, as opposed to its mean-field version. A spinful version, on the other hand, does exhibit a topologically non-trivial phase. In that state, both mean-field solutions are topologically non-trivial and correspond to non-interacting SSH chains in the opposite phases with the winding number $\nu=1$. We show that this phase is protected by a chiral symmetry, similar to the non-interacting phases.