Is the continuum SSH model topological?

TL;DR

This paper investigates whether the topological properties of the SSH model are fundamental or emergent by analyzing continuum Hamiltonians whose tight-binding limits are SSH models with different topological indices.

Contribution

It demonstrates that the topological character of the SSH model is emergent, arising from specific symmetries in the tight-binding limit, not an inherent property of the continuum Hamiltonian.

Findings

Two homotopically equivalent continuum Hamiltonians have SSH limits with different topological indices.

The topological nature of SSH is linked to emergent symmetries in the tight-binding approximation.

Extended tight-binding approximation results to continuum lattices depending on a parameter.

Abstract

The discrete Hamiltonian of Su, Schrieffer and Heeger (SSH) is a well-known one-dimensional translation-invariant model in condensed matter physics. The model consists of two atoms per unit cell and describes in-cell and out-of-cell electron-hopping between two sub-lattices. It is among the simplest models exhibiting a non-trivial topological phase; to the SSH Hamiltonian one can associate a winding number, the Zak phase, which depends on the ratio of hopping coefficients and takes on the values $0$ and $1$ labeling the two distinct phases. We display two homotopically equivalent continuum Hamiltonians whose tight binding limits are SSH models with different topological indices. The topological character of the SSH model is therefore an emergent rather than fundamental property, associated with emergent chiral or sublattice symmetry in the tight-binding limit. In order to establish that the tight-binding limit of these continuum Hamiltonians is the SSH model, we extend our recent results on the tight-binding approximation to lattices which depend on the tight-binding asymptotic parameter.