Line nodes and surface Majorana flat bands in static and kicked $p$-wave superconducting Harper model

We investigate the effect of introducing nearest-neighbor $p$-wave superconducting pairing to both the static and kicked extended Harper model with two periodic phase parameters acting as artificial dimensions to simulate three-dimensional systems. It is found that in both the static model and the kicked model, by varying the $p$-wave pairing order parameter, the system can switch between a fully gapped phase and a gapless phase with point nodes or line nodes. The topological property of both the static and kicked model is revealed by calculating corresponding topological invariants defined in the one-dimensional lattice dimension. Under open boundary conditions along the physical dimension, Majorana flat bands at energy zero (quasienergy zero and $\pi$) emerge in the static (kicked) model at the two-dimensional surface Brillouin zone. For certain values of pairing order parameter, (Floquet) Su-Schrieffer-Heeger-like edge modes appear in the form of arcs connecting different (Floquet) Majorana flat bands. Finally, we find that in the kicked model, it is possible to generate two controllable Floquet Majorana modes, one at quasienergy zero and the other at quasienergy $\pi$, at the same parameter values.

Figure 1

(a) Nodal lines as a function of ${\varphi }_y$ and ${\varphi }_z$ in the $k=\frac{\pi }{2}$ plane for $J=1,\lambda =0.5,V=2$ with different values of $\mathrm{\Delta }$. (b) The $Z_2$ index, (c) the chiral index $N_1$, and (d) $N_2$ for $J=1,\lambda =0.5,V=2$, and $\mathrm{\Delta }=0.3$.

Figure 2

For fixed $J=1,\lambda =0.5,V=2$, and $\mathrm{\Delta }=0.3$ under OBCs in the lattice dimension with 60 lattice sites, (a) the lower one of the middle two energy bands around zero energy as a function of ${\varphi }_y$ and ${\varphi }_z$, where Majorana zero-mode flat bands as well as Dirac zero-mode arcs connecting them can be seen clearly. Inset: The modular square of the wave function's particle component against the lattice sites for the state $({\varphi }_y,{\varphi }_z)=(\frac{\pi }{2},\frac{\pi }{2})$ inside the flat bands. (b) The energy spectrum as a function of ${\varphi }_z$ for fixed ${\varphi }_y=0$, where counterpropagating chiral edge states appear around ${\varphi }_z=\pm \frac{\pi }{2}$.

Figure 3

The quasienergy spectrum as a function of ${\varphi }_y$ and ${\varphi }_z$ for $J=1,\lambda =0.5,V=2,k=\frac{\pi }{2}$, (a) $\mathrm{\Delta }=0$ with nodal points at quasienergy zero and (b) $\mathrm{\Delta }=0.3$ where the nodal points evolve into nodal lines. Nodal lines with representative values of $\mathrm{\Delta }$, (c) at quasienergy zero and (d) at quasienergy $\pi$.

Figure 4

The $Z_2\times Z_2$ index, $Q_0$ (upper row) and $Q_{\pi }$ (lower row) as a function of ${\varphi }_y/\pi$ and ${\varphi }_z/\pi$, for fixed $J=1,\lambda =0.5$, with $V=2,\mathrm{\Delta }=0.3$ (first column), $V=2,\mathrm{\Delta }=1.5$ (second column), $V=2,\mathrm{\Delta }=2.5$ (third column), and $V=8,\mathrm{\Delta }=0.3$ (last column). The $Q_{0,\pi }$ index inside (outside) the blue regions equals $-1\left(1\right)$, which indicates a nontrivial (trivial) phase.

Figure 5

Quasienergy spectrum for fixed $J=1,\lambda =0.5$ and different $V,\mathrm{\Delta }$ under OBCs in the lattice dimension with 60 lattice sites. For $V=8$ and $\mathrm{\Delta }=0.3$, (a) the uppermost band around quasienergy $\pi$ as a function of ${\varphi }_y$ and ${\varphi }_z$, where Floquet Majorana $\pi$-mode flat bands as well as Floquet Dirac $\pi$-mode arcs connecting them can be seen. Inset: the modular square of the particle component of the wave function against lattice sites for $({\varphi }_y,{\varphi }_z)=\left(\frac{\pi }{2},arccos\frac{\pi }{4}\right)$ inside the $\pi$-mode flat bands. (b) The quasienergy spectrum as a function of ${\varphi }_z$ for fixed ${\varphi }_y=\frac{\pi }{2}$. (c) The quasienergy spectrum for $V=8$ and $\mathrm{\Delta }=0$ as a function of ${\varphi }_z$ with fixed ${\varphi }_y=0$, where counterpropagating chiral edge states appear in both Floquet gaps near $\epsilon =0$ and $\epsilon =\pi$, which still exist in (d) with a small $\mathrm{\Delta }=0.3$. (e) For $V=2,\mathrm{\Delta }=2.5$ and (f) $V=8,\mathrm{\Delta }=2.2$, the quasienergy spectrum as a function of ${\varphi }_z$ for fixed ${\varphi }_y=\frac{\pi }{2}$, where edge modes emerge at quasienergy zero and $\pi$ simultaneously under the same ${\varphi }_y$ and ${\varphi }_z$.