Sequential topology: iterative topological phase transitions in finite chiral structures

TL;DR

This paper explores the complex topological phases of finite disordered chiral Hamiltonians, introducing the concept of sequential topology and experimentally verifying how zero mode degeneracies influence localization and transport.

Contribution

It extends topological classification by analyzing phase boundaries and introduces a protocol to identify maximal subspaces with specific zero mode degeneracies in disordered systems.

Findings

Degeneracy of zero modes increases at phase boundaries.

Sequential topology affects localization and transport.

Experimental validation using coaxial cable platform.

Abstract

We present theoretical and experimental results probing the rich topological structure of arbitrarily disordered finite tight binding Hamiltonians with chiral symmetry. We extend the known classification by considering the topological properties of phase boundaries themselves. That is, can Hamiltonians that are confined to being topologically marginal, also have distinct topological phases? For chiral structures, we answer this in the affirmative, where we define topological phase boundaries as having an unavoidable increase in the degeneracy of real space zero modes. By iterating this question, and considering how to enforce a Hamiltonian to a phase boundary, we give a protocol to find the largest dimension subspace of a disordered parameter space that has a certain order degeneracy of zero energy states, which we call \textit{sequential topology}. We show such degeneracy alters localisation and transport properties of zero modes, allowing us to experimentally corroborate our theory using a state-of-the-art coaxial cable platform. Our theory applies to systems with an arbitrary underlying connectivity or disorder, and so can be calculated for any finite chiral structure. Technological and theoretical applications of our work are discussed.