Topological phases, edge modes, and the Hofstadter butterfly in coupled Su-Schrieffer-Heeger systems

TL;DR

This paper explores a coupled SSH ladder system revealing a complex phase diagram with topological, trivial, and weak topological regimes, and demonstrates how quasiperiodic couplings produce Hofstadter's butterfly pattern, suggesting experimental realizations.

Contribution

It introduces a novel SSH ladder model exhibiting rich topological phases and fractal Hofstadter butterfly patterns, expanding understanding of topological matter in coupled chain systems.

Findings

Identifies three distinct topological regimes in the SSH ladder.

Shows Hofstadter butterfly pattern emerges with quasiperiodic couplings.

Demonstrates potential for experimental observation of fractal topological phases.

Abstract

Motivated by recent experimental realizations of topological edge states in Su-Schrieffer-Heeger (SSH) chains, we theoretically study a ladder system whose legs are comprised of two such chains. We show that the ladder hosts a rich phase diagram and related edge mode structure dictated by choice of inter-chain and intra-chain couplings. Namely, we exhibit three distinct physical regimes: a topological hosting localized zero energy edge modes, a topologically trivial phase having no edge mode structure, and a regime reminiscent of a weak topological insulator having unprotected edge modes resembling a "twin-SSH" construction. In the topological phase, the SSH ladder system acts as an analog of the Kitaev chain, which is known to support localized Majorana fermion end modes, with the difference that bound states of the SSH ladder having the same spatial wavefunction profiles correspond to Dirac fermion modes. Further, inhomogeneity in the couplings can have a drastic effect on the topological phase diagram of the ladder system. In particular for quasiperiodic variations of the inter-chain coupling, the phase diagram reproduces Hofstadter's butterfly pattern. We thus identify the SSH ladder system as a potential candidate for experimental observation of such fractal structure.