Topological characterization and stability of Floquet Majorana modes in Rashba nanowires

We theoretically investigate a practically realizable Floquet topological superconductor model based on a one-dimensional Rashba nanowire and proximity-induced $s$-wave superconductivity in the presence of a Zeeman field. The driven system hosts regular 0-Majorana end modes and anomalous $\pi$-Majorana end modes (MEMs). By tuning the chemical potential and the frequency of the drive, we illustrate the generation of multiple MEMs in our theoretical setup. We utilize the chiral symmetry operator to topologically characterize these MEMs via a dynamical winding number constructed out of the periodized evolution operator. Interestingly, the robustness of the 0- and $\pi$-MEMs is established in the presence of on-site time-independent random disorder potential. We employ the twisted boundary condition to define the dynamical topological invariant for this translational-symmetry broken system. The interplay between the Floquet driving and the weak disorder can stabilize the MEMs, giving rise to a quantized value of the dynamical winding number for a finite range of drive parameters. This observation might be experimentally helpful in scrutinizing the topological nature of the Floquet MEMs. We showcase another driving protocol, namely, a periodic kick in the chemical potential, to study the generation of Floquet MEMs in our setup. Our work paves a realistic way to engineer multiple MEMs in a driven system.

Figure 1

(a) Schematic representation of our setup is shown here. One-dimensional nanowire (NW) (blue) with strong spin-orbit coupling is placed on top of an $s$-wave superconducting slab (green). A magnetic field $B_x$ is applied along the $x$ direction and a gate voltage $V_{\mathrm{g}}$ is applied across the cross section of the NW to control the chemical potential. The Majorana end modes (MEMs) [red] appear at the two ends of the NW. (b) Schematic demonstration of our periodic three-step drive protocol [see Eq. (5)] is presented.

Figure 2

(a) Energy eigenvalue spectra for the static Hamiltonian [Eq. (1)], employing the open boundary condition, are shown as a function of the magnetic field $B_x$. The corresponding winding number $W$ is depicted as a function of $B_x$ in panel (b) for a fixed value of $c_0=1.0$. (c) We demonstrate $W$ in the $c_0-B_x$ plane to illustrate the topological phase diagram. Other model parameters are chosen as $(t,u,\mathrm{\Delta })=(1.0,0.5,1.0)$. MZMs are obtained for $|B_x^1|\le B_x\le \left|B_x^2\right|$ (see text for details).

Figure 3

Bulk quasienergy spectra for the three-step drive protocol [Eq. (5)] are shown for $\mathrm{\Omega }=1.5$ and $\mathrm{\Omega }=2.5$ in panels (a) and (b), respectively. In the insets I1 and I2 of panel (a) [inset of panel (b)], we depict the bulk gap around quasienergy $E\left(k\right)=\pi$ and $E\left(k\right)=0$, respectively [quasienergy $E\left(k\right)=\pi ]$. In panels (c) and (d), we illustrate the quasienergy spectra $E_m$ as a function of the state index $m$ for our system obeying OBC, corresponding to panels (a) and (b), respectively, while in the insets, we portray the zoomed-in quasienergies for better clarity. The appearance of 0- and $\pi$-Majorana end modes (MEMs) are evident from these figures. We show the normalized site-resolved probability (${\left|\mathrm{\Psi }\right|}^2$) corresponding to the 0- and $\pi$-MEMs in panels (e) and (f), respectively. Here, we consider $N=4000$ lattice sites to obtain sharp Majorana localization at individual ends of the NW. All the other model parameters are chosen as $(c_0,c_1,\mathrm{\Delta },B_x,t,u)=(1.0,0.2,1.0,1.0,1.0,0.5)$.

Figure 4

We demonstrate the dynamical winding numbers $W_0$ and $W_{\pi }$ (quasienergy gap corresponding to 0-energy, $G_0$ and $\pi$-energy, $G_{\pi }$) in the $c_0-\mathrm{\Omega }$ plane in panels (a) and (b) [panels (c) and (d)], respectively. Here the color bar represents the winding number (energy gap) for panels (a) and (b) [panels (c) and (d)]. All the other parameters are chosen to be the same as those mentioned in Fig. 3.

Figure 5

In panels (a) and (b), we depict the dynamical winding numbers $W_0$ and $W_{\pi }$, respectively, in the presence of small $(w=0.1)$ on-site disorder as a function of the driving frequency $\mathrm{\Omega }$. We repeat panels (a) and (b) in panels (c) and (d) and panels (e) and (f) for moderate ($w=0.5$) and strong ($w=0.8$) disorder strength, respectively. Here, green (dashed), red (solid), and blue (dotted-dashed) lines represent the winding number ($W_{\epsilon }^{\mathrm{c}}$) for the corresponding clean system, the disorder-averaged winding number ($W_{\epsilon }^{\mathrm{d}}$) for the disordered system, and the variance (${\sigma }^2$) of the winding number of the disordered system from the mean value $W_{\epsilon }^{\mathrm{d}}$, respectively, for quasienergy $\epsilon$. We choose $c_0=0.2$, while all the other model parameter values remain the same as those mentioned in Fig. 3.

Figure 6

Panels (a) and (b) represent the bulk quasienergy spectra for the periodic kick drive protocol [Eq. (17)] for $\mathrm{\Omega }=3.0$ and $\mathrm{\Omega }=5.0$, respectively. The insets I1 and I2 of panel (a) [inset of panel (b)] exhibit(s) the bulk gap around quasienergies $E\left(k\right)=\pi$ and $E\left(k\right)=0$, respectively [quasienergy $E\left(k\right)=\pi ]$. In panels (c) and (d), we illustrate the quasienergy spectra $E_m$ as a function of the state index $m$ while system obeys the OBC, corresponding to panels (a) and (b), respectively. In the insets of these panels, we depict the zoomed-in quasienergies for better clarity. Here, we consider $N=4000$ lattice sites. All the other model parameters are chosen as $(c_0,c_1,\mathrm{\Delta },B_x,t,u)=(1.0,0.2,1.0,1.0,1.0,0.5)$.

Figure 7

We demonstrate the dynamical winding numbers $W_0$ and $W_{\pi }$ in the $c_0-\mathrm{\Omega }$ plane in panels (a) and (b), respectively, for periodic-kick drive protocol. All the model parameter values remain the same as those mentioned in Fig. 6.