Real Edge Modes in a Floquet-modulated $\mathcal{PT}$-symmetric SSH model

Non-Hermitian Hamiltonians provide a simple picture for analyzing systems with natural or induced gain and loss; however, in general, such Hamiltonians feature complex energies and a corresponding non-orthonormal eigenbasis. Provided that the Hamiltonian has $\mathcal{PT}$ symmetry, it is possible to find a regime in which the eigenspectrum is completely real. In the case of static $\mathcal{PT}$-symmetric extensions of the simple Su-Schrieffer-Heeger model, it has been shown that the energies associated with any edge states are guaranteed to be complex. Moving to a time-dependent system means that treatment of the Hamiltonian must be done at the effective time-scale of the modulation itself, allowing for more intricate phases to occur than in the static case. It has been demonstrated that with particular classes of periodic driving, achieving a real topological phase at high driving frequency is possible. In the present paper, we show the details of this process by using a simple two-step periodic modulation. We obtain a rigorous expression for the effective Floquet Hamiltonian and compare its symmetries to those of the original Hamiltonians which comprise the modulation steps. The $\mathcal{PT}$ phase of the effective Hamiltonian is dependent on the modulation frequency as well as the gain/loss strength. Furthermore, the topologically nontrivial regime of the $\mathcal{PT}$-unbroken phase admits highly-localized edge states with real eigenvalues in both the high frequency case and below it, albeit within a smaller extent of the parameter space.