Floquet Second Order Topological Superconductor based on Unconventional Pairing

TL;DR

This paper proposes a method to generate and control second-order topological superconducting phases in 2D and 3D using Floquet engineering on unconventional pairing, revealing weak and strong phases with Majorana corner modes.

Contribution

It introduces a Floquet-based approach to realize and distinguish weak and strong second-order topological superconductors with Majorana modes, including effects of TRS breaking and disorder.

Findings

Floquet driving induces weak SOTSC with eight Majorana modes at corners.

Explicit TRS breaking transforms weak into strong SOTSC with one Majorana mode per corner.

Corner modes are robust against moderate disorder and topological invariants remain quantized.

Abstract

We theoretically investigate the Floquet generation of second-order topological superconducting (SOTSC) phase in the high-temperature platform both in two-dimension (2D) and three-dimension (3D). Starting from a $d$-wave superconducting pairing gap, we periodically kick the mass term to engineer the dynamical SOTSC phase within a specific range of the strength of the drive. Under such dynamical breaking of time-reversal symmetry (TRS), we show the emergence of the \textit{weak} SOTSC phase, harboring eight corner modes \ie two zero-energy Majorana per corner, with vanishing Floquet quadrupole moment. On the other hand, our study interestingly indicates that upon the introduction of an explicit TRS breaking Zeeman field, the \textit{weak} SOTSC phase can be transformed into \textit{strong} SOTSC phase, hosting one zero-energy Majorana mode per corner, with quantized quadrupole moment. We also compute the Floquet Wannier spectra that further establishes the \textit{weak} and \textit{strong} nature of these phases. We numerically verify our protocol computing the exact Floquet operator in open boundary condition and then analytically validate our findings with the low energy effective theory (in the high-frequency limit). The above protocol is applicable for 3D as well where we find one dimensional (1D) hinge mode in the SOTSC phase. We then show that these corner modes are robust against moderate disorder and the topological invariants continue to exhibit quantized nature until disorder becomes substantially strong. The existence of zero-energy Majorana modes in these higher-order phases is guaranteed by the anti-unitary spectral symmetry.