Gapless Floquet topology

TL;DR

This paper extends the concept of topological edge states to gapless Floquet systems with chiral symmetry, introducing new invariants and demonstrating the existence of zero- and pi-modes without a bulk gap.

Contribution

It develops a framework for topological invariants in gapless Floquet systems that do not rely on a bulk gap, applicable to chiral symmetric chains and models.

Findings

Topological edge zero- and pi-modes exist in gapless Floquet systems.

New invariants are constructed from half-period evolution, bypassing the Floquet Hamiltonian.

Interactions cause finite lifetime of edge modes, consistent with Fermi's Golden Rule.

Abstract

Symmetry-protected topological (SPT) phases in insulators and superconductors are known for their robust edge modes, linked to bulk invariants through the bulk-boundary correspondence. While this principle traditionally applies to gapped phases, recent advances have extended it to gapless systems, where topological edge states persist even in the absence of a bulk gap. We extend this framework to periodically driven chains with chiral symmetry, revealing the existence of topological edge zero- and pi-modes despite the lack of bulk gaps in the quasienergy spectrum. By examining the half-period decomposition of chiral evolutions, we construct topological invariants that circumvent the need to define the Floquet Hamiltonian, making them more suitable for generalization to the gapless regime. We provide explicit examples, including generalizations of the Kitaev chain and related spin models, where localized pi-modes emerge even when the bulk is gapless at the same quasi-energy as the edge modes. We numerically study the effect of interactions, which give a finite lifetime to the edge modes in the thermodynamic limit with the decay rate consistent with Fermi's Golden Rule.