Real Edge Modes in a Floquet-modulated 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 𝒫𝒯 symmetry, it is possible to find a regime in which the eigenspectrum is completely real. In the case of static 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 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 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.