Green Function Invariants for Floquet Topological Superconductivity Induced by Proximity Effects

TL;DR

This work develops a Green function framework to predict Floquet topological phases in driven superconductor–semiconductor hybrids by constructing a quasi-energy operator from the Hermitian part of the semiconductor self-energy and extracting level broadening from the anti-Hermitian part. The authors derive explicit self-energy forms for typical drives, introduce a rotated-frame Floquet formulation, and define robust topological invariants in Sambe space, applying them to a Rashba nanowire proximitized by a superconductor under a time-periodic Zeeman drive. They demonstrate that at low frequencies Majorana zero and Majorana $\pi$ modes can appear (potentially with multiple edge modes), while intermediate frequencies introduce significant broadening that suppresses $\pi$-modes; at high frequencies the topology remains essentially that of the undriven system with small broadening. Overall, the paper provides a transferable, quantitative method to predict Floquet topological phases and assess their observability in proximitized hybrids, highlighting the critical role of self-energy effects in driven topological superconductivity.

Abstract

We bring forward a Green function approach for the prediction of Floquet topological phases in driven superconductor-semiconductor hybrids. Although it is common to treat the superconducting component as a mere Cooper-pair reservoir, it was recently pointed out that such an approximation breaks down in the presence of driving, due to the emergence of level broadening. Here, we go beyond these recent works and prescribe how to construct the Floquet topological invariants for such driven hybrids. Specifically, we propose to first obtain the midgap quasi-energy spectra by including the hermitian part of the semiconductor's self-energy and, subsequently, read out the respective level broadenings by projecting the anti-hermitian part of the self-energy onto the quasi-energy eigenvectors. We exemplify our approach for a Rashba nanowire coupled to a superconductor and a time-dependent Zeeman field. Using our method, we obtain the Floquet band structure, the respective level broadenings, and the topological invariants. Our analysis reinforces the need to properly account for the self-energy, and corroborates that broadening effects can hinder the observation of the Floquet topological phases and especially of those harboring Majorana $π$ modes.

Figures (7)

• Figure 1: Band diagram of the superconductor-semiconductor hybrid structure after the two materials get in contact and equilibrium is established. Occupied (empty) bands are denoted with shaded (empty) boxes. The Fermi energy of the superconductor also sets the Fermi level of the entire hybrid system, since it is here considered to constitute a good metal and, therefore, it can be viewed as a particle bath. Additionally, the pro-xi-mi-ty to the superconductor induces a pairing gap in the conduction band of the semiconductor, which is of focus in this work. The behavior of the driven system depends on the hier-archy of the frequency of the periodic driving $\omega$ and the superconducting gap $\Delta\geq0$. For frequency values $\omega_1\ll\Delta$ and $\omega_3\gg\Delta$, the energy level broadening is negligible. In stark contrast, for intermediate frequencies $\omega_2\sim\Delta$, the broadening becomes substantial and predominantly hinders the emergence of Majorana $\pi$ modes. Note that the downfolding to the first Floquet zone is not considered here.
• Figure 2: Schematic illustration of the concrete driven superconductor-semiconductor hybrid investigated in this work. The semiconductor component consists of an ideal one-dimensional nanowire which is coupled to a conventional superconductor that plays the role of the bath. The nanowire is simultaneously subject to a time-periodic magnetic field $B+{\cal B}\cos(\omega t)$ that is oriented along the nanowire's axis. Majorana zero and $\pi$ edge modes emerge at the nanowire ends.
• Figure 3: Quasi-energy spectra obtained numerically for the following values of che-mi-cal potential and magnetic field, i.e., $(\mu/\Delta, B ({\rm T}))=\{(0.6,0.6),(0.3,0.5),(0.4,0.7),(0.2,0.7)\}$ and driving frequency $\omega/\Delta=0.3$. While in the absence of the drive, the superconductor-induced self-energy on the semiconductor solely renormalizes the band structure of individual Floquet modes, in the presence of the drive, the self-energy also acquires nonzero off-diagonal elements in Floquet space, thus, further contributing to the hybridization of the Floquet bands. Note that the gaps around zero and $\pi$ quasi-energies are protected by the presence of a chiral symmetry which is preserved even when the hybrid is driven.
• Figure 4: Numerically-evaluated topological phase diagrams using the self-energy formalism presented in this work. (a) and (b) depict the winding numbers correspon-ding to MZMs and MPMs, respectively, for a driving frequency $\omega/\Delta=0.3$. The numerical results show that both MZMs and MPMs exist in the low driving frequency regime. In addition, we predict phases with multiple Majorana edge modes, which are otherwise inaccessible in the undriven hybrid.
• Figure 5: (a) and (b) present our numerical calculations of the topological phase diagrams for MZMs and MPMs, as functions of the chemical potential and the Zeeman field for the intermediate driving frequency $\omega/\Delta=0.8$. Quite remar-kably, in (a) we observe the emergence of topologically non-trivial phases harboring multiple MZMs per edge already in the range of zero and generally low Zeeman field values. Such a trend was also observed for lower frequencies. See Fig. \ref{['fig:Figure3']}(a). In contrast, a threshold Zeeman field is required for MPMs in both low and intermediate values of driving frequency.