Real-space topological invariant for time-quasiperiodic Majoranas

TL;DR

This paper introduces a real-space topological invariant to identify time-quasiperiodic Majorana modes in superconducting systems driven by incommensurate frequencies, overcoming challenges posed by dense spectra.

Contribution

It develops a novel localizer-based topological invariant suitable for dense spectra in quasiperiodically driven systems, inspired by non-Hermitian physics.

Findings

The invariant successfully detects Majorana modes in a driven Kitaev chain.

Numerical simulations confirm the robustness of the invariant against dense spectra.

The approach extends topological characterization to non-static, quasiperiodic systems.

Abstract

When subjected to quasiperiodic driving protocols, superconducting systems have been found to harbor robust time-quasiperiodic Majorana modes, extending the concept beyond static and Floquet systems. However, the presence of incommensurate driving frequencies results in dense energy spectra, rendering conventional methods of defining topological invariants based on band structure inadequate. In this work, we introduce a real-space topological invariant capable of identifying time-quasiperiodic Majoranas by leveraging the system's spectral localizer, which integrates information from both Hamiltonian and position operators. Drawing insights from non-Hermitian physics, we establish criteria for constructing the localizer and elucidate the robustness of this invariant in the presence of dense spectra. Our numerical simulations, focusing on a Kitaev chain driven by two incommensurate frequencies, validate the efficacy of our approach.