Real-space topological invariant for time-quasiperiodic Majoranas

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.

10 pages, 4 figures

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• Tutorial: Classifying Photonic Topology Using the Spectral Localizer and Numerical $K$-Theory
• Self-similar phase diagram of the Fibonacci-driven quantum Ising model
• Local and energy-resolved topological invariants for Floquet systems
• Gapless superconductivity and its real-space topology in quasicrystals
• Majorana-metal transition in a disordered superconductor: percolation in a landscape of topological domain walls
• Emergent topological phases and coexistence of gapless and spectral-localized Floquet quantum spin Hall states via electron-phonon interaction
• Quantitative measure of topological protection in Floquet systems through the spectral localizer
• Fourier Neural Operators for Time-Periodic Quantum Systems: Learning Floquet Hamiltonians, Observable Dynamics, and Operator Growth
• Dissipative Yao-Lee Spin-Orbital Model: Exact Solvability and $\mathcal{P}\mathcal{T}$ Symmetry Breaking
• Dimensional crossover of class D real-space topological invariants
• Topological energy pumping in a quasi-periodically driven four-level system