Time-induced second-order topological superconductors

Higher-order topological materials with topologically protected states at the boundaries of their boundaries (hinges or corners) have attracted attention in recent years. In this paper, we utilize time-periodic driving to generate second-order topological superconductors out of systems which otherwise do not even allow second-order topological characterization. This is made possible by the design of periodic drives which inherently exhibit nontrivial winding in the time domain. Through the interplay of topology in both spatial and temporal dimensions, nonchiral Majorana modes may emerge at the systems' corners and sometimes even coexist with chiral Majorana modes. Our proposal thus presents an opportunity for Floquet engineering with minimal system complexity and its application in quantum information processing.

Figure 1

Due to $\mathcal{P}$ and $\tilde{\mathcal{P}}$ symmetries, the diagonal (antidiagonal) line in the 2D Brillouin zone can be further divided into two equivalent subregions marked by the blue and red (green and purple) colored lines. The inset illustrates two representative many-body quasienergy bands along the blue diagonal line. There, the bands' colors (green and yellow) label the two different eigenstate parities defined in Eq. (11).

Figure 2

Quasienergy spectrum of Eq. (1) under (a)–(c) OBC and (d)–(f) PBC in both directions. In panels (a) and (d), only $J_{0,x}=J_{0,y}=m\hslash \omega$ is varied, while $\mathrm{\Delta }=\frac{\hslash \omega }{2\pi }$ is fixed. In panels (b) and (e), only $\mathrm{\Delta }=\frac{m}{2}\hslash \omega$ is varied, while $J_{0,x}=J_{0,y}=\frac{\hslash \omega }{\pi }$ is fixed. In panels (c) and (f), $J_{0,x}=J_{0,y}=2\mathrm{\Delta }=m\hslash \omega$ is varied. In all panels, we set $J_{s,x}=J_{s,y}=0$ and $\mu =\frac{0.1}{2\pi }\hslash \omega$, include up to $\pm 3$ photon sectors of the Floquet Hamiltonian (i.e., $n_{\text{max}}=3$) in our numerics, and take the system size to be $15\times 15$.

Figure 3

Quasienergy spectrum of Eq. (1) under (a) OBC in the $y$ direction and PBC in the $x$ direction and (b) OBC in the $x$ direction and PBC in the $y$ direction. In both panels, 40 sites are taken in the direction where OBC are applied and up to $\pm 3$ photon sectors of the Floquet Hamiltonian are included (i.e., $n_{\text{max}}=3$). The other parameters are set as $\mu =\frac{0.1}{2\pi }\hslash \omega$, $J_{0,x}=J_{0,y}=2\mathrm{\Delta }=\frac{2}{\pi }\hslash \omega$, and $J_{s,x}=J_{s,y}=0$.

Figure 4

Support of each corner MPM on Majorana operators representing the system's $20\times 20$ lattice sites [see Eq. (27)]. Here, $i$ and $j$ represent the Majorana indices in the $x$ and $y$ directions, respectively [see Eq. (10)]. While the corner MPM solutions are obtained by numerically diagonalizing the truncated Floquet Hamiltonian containing up to $\pm 3$ photon sectors, only the zeroth photon sector contributions are shown. System parameters are chosen as $\mathrm{\Delta }=\frac{\hslash \omega }{\pi }$, $\delta =\frac{0.1}{2\pi }\hslash \omega$, $J_{0,x}=\frac{3.4}{2\pi }\hslash \omega$, $J_{0,y}=\frac{3.6}{2\pi }\hslash \omega$, and $\mu =\frac{0.1}{2\pi }\hslash \omega$.

Figure 5

(a) SIPRs of all quasienergy eigenmodes as $J_{0,x}=J_{0,y}=2\mathrm{\Delta }=m\hslash \omega$ is varied, while other system parameters are fixed at $J_{s,x}=J_{s,y}=0$, $\mu =\frac{0.1}{2\pi }\hslash \omega$ (those of MPMs are marked by green circles). (b) The associated quasienergy spectrum under the same parameter values as panel (a).

Figure 6

Quasienergy spectrum of Eq. (1) under (a) OBC and (b) PBC in both directions as $J_{s,x}=J_{s,y}=J$ is varied. In panel (a) $15\times 15$ lattice sites are taken, and the other parameter values are $J_{0,x}=J_{0,y}=\frac{\hslash \omega }{\pi }$, $\mathrm{\Delta }=\frac{1.5}{2\pi }\hslash \omega$, and $\mu =\frac{1}{4\pi }\hslash \omega$ in both panels.

Figure 7

Quasienergy spectrum of Eq. (1) under (a), (b) OBC in the $x$ direction with 55 sites and PBC in the $y$ direction, (c), (d) PBC in both directions, and (e) OBC in both directions with $15\times 15$ sites. Periodic driving parameters are set to (a), (c) $J_{0,x}=J_{0,y}=0$ (static limit) and (b), (d), (e) $J_{0,x}=J_{0,y}=\frac{\hslash \omega }{\pi }$. The other parameter values are $\mathrm{\Delta }=\frac{1.5}{2\pi }\hslash \omega$, and $\mu =J_{s,x}=J_{s,y}=\frac{1}{4\pi }\hslash \omega$ in all panels.

Figure 8

(a) Quasienergy levels of the system under OBC in both directions with $20\times 20$ sites under disorder parameters $\delta J_{0,x}=\delta J_{0,y}=\delta \mu =10\delta {\mathrm{\Delta }}_x=10\delta {\mathrm{\Delta }}_y=10\delta J_{s,x}=10\delta J_{s,y}=\frac{\hslash \omega }{20\pi }$, averaged over 50 disorder realizations. (b), (c) Support of the time evolved ${\psi }_c(t=10T)$ on Majorana operators ${\gamma }_{i,j}$ representing the system's $20\times 20$ sites [see Eq. (27)] in the (b) absence and (c) presence of temporal disorders with $2\delta J_{s,x}=2\delta J_{s,y}=\delta J_{0,x}=\delta J_{0,y}=2\delta {\mathrm{\Delta }}_x=2\delta {\mathrm{\Delta }}_y=2\delta \mu =\frac{\hslash \omega }{20\pi }$. All other system parameters are set to $J_{0,x}=J_{0,y}=\frac{\hslash \omega }{\pi }$, $\mathrm{\Delta }=\frac{1.5\hslash \omega }{2\pi }$, and $\mu =J_{s,x}=J_{s,y}=\frac{\hslash \omega }{4\pi }$.

Figure 9

Support of a corner MPM (higher values correspond to brighter colors) on Majorana operators representing the system's $20\times 20$ sites in the spirit of Eq. (27) under the presence of (a) no defect, (b) a square-shaped defect of size $3\times 3$ sites, (c) a rectangle-shaped defect of size $3\times 5$ sites, and (d) a rectangle-shaped defect of size $5\times 3$ sites. System parameters are set to $J_{0,x}=J_{0,y}=\frac{\hslash \omega }{\pi }$, $\mathrm{\Delta }=\frac{1.5\hslash \omega }{2\pi }$, and $\mu =J_{s,x}=J_{s,y}=\frac{\hslash \omega }{4\pi }$.

Figure 10

Quasienergy spectrum of Eq. (1) as a function of the relative phase $\xi$ between $J_x\left(t\right)$ and $J_y\left(t\right)$ driving under (a) OBC and (b) PBC in both directions. All system parameters are set to $J_{0,x}=J_{0,y}=\frac{\hslash \omega }{\pi }$, $\mathrm{\Delta }=\frac{1.5\hslash \omega }{2\pi }$, and $\mu =J_{s,x}=J_{s,y}=\frac{\hslash \omega }{4\pi }$.

Figure 11

The system's quasienergy spectrum under the modified time periodicity described in Appendix pp3 where (a)–(c) OBC with $20\times 20$ sites and (d)–(f) PBC are applied in both directions. In panels (a) and (d), $J_{0,x}=J_{0,y}=2m\hslash$ is varied while $\mathrm{\Delta }=\frac{\hslash \omega }{4\pi }$ is fixed. In panels (b) and (e), $\mathrm{\Delta }=\frac{m}{2\pi }\hslash \omega$ is varied while $J_{0,x}=J_{0,y}=0.4\hslash$ is fixed. In panels (c) and (f), $J_{0,x}=J_{0,y}=\frac{4\pi \mathrm{\Delta }}{\omega }=2m\hslash$ is varied. We take $\mu =\frac{0.1}{2\pi }\hslash \omega$ in all panels.