Odd-frequency superconducting pairing due to multiple Majorana edge modes in driven topological superconductors

Majorana zero modes have been shown to be the simplest quasiparticles exhibiting pure odd-frequency pairing, an effect that has so far been theoretically established in the static regime. In this paper, we investigate the formation of Majorana modes and odd-frequency pairing in $p$-wave spin-polarized superconductors under a time-dependent drive. We first show that the driven system hosts multiple Majorana modes emerging at zero and $\ensuremath{\pi}$, whose formation can be controlled by an appropriate tuning of the drive frequency and chemical potential, in agreement with previous studies. Then we explore the induced pair correlations and find that odd-frequency spin-polarized $s$-wave pairing is broadly induced, acquiring large values in the presence of Majorana modes. We discover that, while odd-frequency pairing is proportional to $\ensuremath{\sim}1/\ensuremath{\omega}$ in the presence of Majorana zero modes, it is proportional to $\ensuremath{\sim}1/(\ensuremath{\omega}\ensuremath{-}\ensuremath{\pi}\ensuremath{\hbar}/T)$ in the presence of Majorana $\ensuremath{\pi}$ modes, where $T$ is the periodicity of the drive. Furthermore, we find that the amount of odd-frequency pairing becomes larger when multiple Majorana modes appear but the overall divergent profile as a function of frequency remains. We also show that the divergent odd-frequency pairing is robust against scalar disorder. Notably, we establish a bulk-boundary correspondence between the amount of boundary odd-$\ensuremath{\omega}$ pairing and the bulk topological invariants in driven chiral systems, which we show to be protected by chiral symmetry and is thus robust against disorder. Our paper thus paves the way for understanding the emergent pair correlations in driven topological superconductors.