Exact analytical edge states in the extended Su-Schrieffer-Heeger model

TL;DR

This paper derives exact analytical expressions for edge states in the extended SSH model, linking topological phases to boundary states and providing precise formulas for finite chains.

Contribution

It introduces exact analytical solutions for edge states in the extended SSH model, expanding understanding of topological phases beyond nearest-neighbor interactions.

Findings

Edge states decay exponentially with a specific unit-cell decay factor z.

Bulk-boundary correspondence is confirmed via winding number and |z|=1 condition.

Analytical expressions for finite chain edge states are highly accurate.

Abstract

We investigate the topology of the different phases of the extended Su-Schrieffer-Heeger (eSSH) model, which includes hopping processes between translationally inequivalent atoms beyond nearest neighbors. Exact analytical expressions for the edge states of a semi-infinite eSSH chain are derived, with wave functions that decay exponentially from the boundary with a unit-cell decay factor z. From the winding number of the bulk Hamiltonian under periodic boundary conditions, we determine the topological phase diagram and establish the bulk-boundary correspondence: changes in the winding number coincide with bulk gap closings and with the condition |z|=1 for the edge-state solutions. For finite chains, we further obtain analytical, approximate expressions for the low-energy edge states, which are shown to be highly accurate.