Floquet generation of Majorana end modes and topological invariants

Abstract

We show how Majorana end modes can be generated in a one-dimensional system by varying some of the parameters in the Hamiltonian periodically in time. The specific model we consider is a chain containing spinless electrons with a nearest-neighbor hopping amplitude, a $p$-wave superconducting term, and a chemical potential; this is equivalent to a spin-$\frac{1}{2}$ chain with anisotropic $XY$ couplings between nearest neighbors and a magnetic field applied in the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$ direction. We show that varying the chemical potential (or magnetic field) periodically in time can produce Majorana modes at the ends of a long chain. We discuss two kinds of periodic driving, periodic $\ensuremath{\delta}$-function kicks, and a simple harmonic variation with time. We discuss some distinctive features of the end modes such as the inverse participation ratio of their wave functions and their Floquet eigenvalues which are always equal to $\ifmmode\pm\else\textpm\fi{}1$ for time-reversal-symmetric systems. For the case of periodic $\ensuremath{\delta}$-function kicks, we use the effective Hamiltonian of a system with periodic boundary conditions to define two topological invariants. The first invariant is a well-known winding number, while the second invariant has not appeared in the literature before. The second invariant is more powerful in that it always correctly predicts the numbers of end modes with Floquet eigenvalues equal to $+1$ and $\ensuremath{-}1$, while the first invariant does not. We find that the number of end modes can become very large as the driving frequency decreases. We show that periodic $\ensuremath{\delta}$-function kicks in the hopping and superconducting terms can also produce end modes. Finally, we study the effect of electron-phonon interactions (which are relevant at finite temperatures) and a random noise in the chemical potential on the Majorana modes.