Non-Hermitian higher-order topological superconductors in two dimensions: Statics and dynamics

Being motivated by intriguing phenomena, such as the breakdown of conventional bulk boundary correspondence and the emergence of skin modes in the context of non-Hermitian (NH) topological insulators, we here propose a NH second-order topological superconductor (SOTSC) model that hosts Majorana zero modes (MZMs). Employing the non-Bloch form of the NH Hamiltonian, we topologically characterize the above modes by biorthogonal nested polarization and resolve the apparent breakdown of the bulk boundary correspondence. Unlike the Hermitian SOTSC, we note that the MZMs inhabit only one corner out of four in the two-dimensional NH SOTSCs. Such a localization profile of MZMs is protected by mirror rotation symmetry and remains robust under on-site random disorder. We extend the static MZMs into the realm of the Floquet drive. We find the anomalous $\ensuremath{\pi}$ mode following low-frequency mass kick in addition to the regular 0 mode that is usually engineered in a high-frequency regime. We further characterize the regular 0 mode with biorthogonal Floquet nested polarization. Our proposal is not limited to the $d$-wave superconductivity only and can be realized in the experiment with strongly correlated optical lattice platforms.