Floquet generation of Majorana end modes and topological invariants

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 $\widehat{z}$ 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 $\delta$-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 $\pm 1$ for time-reversal-symmetric systems. For the case of periodic $\delta$-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 $-1$, 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 $\delta$-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.

Figure 1

Phase diagram of the model in Eq. (1) as a function of $\mu /\gamma$ and $\Delta /\gamma$. Phases I and II are topological, while III is nontopological.

Figure 2

IPRs of different eigenvectors of the Floquet operator for a 200-site system with a periodic $\delta$-function kick with $\gamma =1$, $\Delta =-1$, $c_0=2.5$, $c_1=0.2$, and $\omega =12$. The two eigenvectors with the largest IPRs both have an IPR equal to $0.142$ and Floquet eigenvalue equal to $-1$.

Figure 3

Majorana end states for a 200-site system with a periodic $\delta$-function kick with $\gamma =1$, $\Delta =-1$, $c_0=2.5$, $c_1=0.2$, and $\omega =12$. These states correspond to the two eigenvectors with the largest IPRs in Fig. 2.

Figure 4

Plot of the number of end states versus $\omega$ for a 200-site system, with Floquet eigenvalues $+1$ ($N_{+}$, blue squares) and $-1$ ($N_{-}$, red stars), for a periodic $\delta$-function kick with $\gamma =1$, $\Delta =-1$, $c_0=2.5$, and $c_1=0.2$.

Figure 5

Closed curves in the $(a_{2,k},a_{3,k})$ plane for $\omega =3,7,12$, and 17 for a 200-site system with $\gamma =1$, $\Delta =-1$, $c_0=2.5$, and a periodic $\delta$-function kick with $c_1=0.2$. The corresponding winding numbers around the origin [the point $(0,0)$ shown by a red dot] are given by 2, 2, 1, and 0, respectively.

Figure 6

Comparison of the number of Majorana modes at each end of a 200-site system (black solid, $y$ axis on left) and the winding number (magenta dashed, $y$ axis on right) as a function of $\omega$ from 1 to 18, for $\gamma =1$, $\Delta =-1$, $c_0=2.5$, and a periodic $\delta$-function kick with $c_1=0.2$. The inset shows a range of $\omega$ from $0.2$ to 1 where there is a large number of Majorana modes.

Figure 7

Plot of $b_0$ and $b_{\pi }$ as a function of $\omega$ for a system with $\gamma =1$, $\Delta =-1$, $c_0=2.5$, and a periodic $\delta$-function kick with $c_1=0.2$. For each value of $\omega$, the number of even and odd integers lying in the shaded region between $b_0$ and $b_{\pi }$ gives the number of Majorana modes at each end of a chain with Floquet eigenvalues equal to $+1$ and $-1$, respectively.

Figure 8

Floquet eigenvalues close to $\pm 1$ for a 1000-site system with $\gamma =1$, $\Delta =-1$, $c_0=2.5$, and a $\delta$-function kick with $c_1=0.2$ at $t=0$ which is repeated with a time period $T=2\pi$. For this time-reversal-symmetric case, there are four eigenvalues at exactly $+1$ and $-1$ each, separated by a gap from all other eigenvalues; these are shown more clearly in the inset.

Figure 9

Floquet eigenvalues close to $\pm 1$ for a 1000-site system with $\gamma =1$, $\Delta =-1$, $c_0=2.5$, and two $\delta$-function kicks with $c_1=0.2$ and $c_2=0.1$ at $t=0$ and $T/4$ which are repeated with a time period $T=2\pi$. For this case with no time-reversal symmetry, there are four eigenvalues close to but not exactly at $+1$ and $-1$, separated by a gap from all other eigenvalues; these are shown more clearly in the inset.

Figure 10

Plot of the probabilities of the two end states for a 100-site system with $\gamma =1$, $\Delta =-1$, $c_0=2.5$, and a simple harmonic driving with $c_1=0.3$, $\omega =14$, and $\varphi =0$.

Figure 11

Plot of the probabilities of the two end states for a 100-site system with $\gamma =1$, $\Delta =-1$, $c_0=2.5$, and a simple harmonic driving with $c_1=0.3$, $\omega =14$, and $\varphi =\pi /2$.