Electron-electron interaction effects in Floquet topological superconducting chains: Suppression of topological edge states and crossover from weak to strong chaos

We study the effects of electron-electron interaction in a Floquet topological superconducting chain using the exact diagonalization and the dynamics simulation of quenching process. It is found that a weak interaction can lead to a multitude of tiny avoided crossings in quasienergy spectra, indicating the emergence of chaos, and that the Floquet topological superconducting state can be robust against the weak chaos. With increasing the interaction strength, there is a crossover from weak to strong chaos. On the other hand, the survival probability of Majorana edge modes vanishes and so the Floquet system is transformed into a topologically trivial phase.

Figure 1

Quasienergy spectra versus interaction strength $V$ for an open chain with even and odd fermion numbers. Here, in order to show clearly the details of the degeneracy between odd and even sectors and the avoided crossings, only a few energy levels are displayed. The spectra of even and odd sectors are almost identical to each other for $V\lesssim 1$, suggesting a topologically nontrivial phase. The other parameters are set to $L=8,\mathrm{\Delta }=1,{\mu }_0=0,\delta \mu =2.5$, and $T=1$.

Figure 2

${\tilde{N}}_{\beta }$ as a function of $V$ after slowly quenching from $V=0$. (a) For different $N_T$'s and $L=17$; (b) for different $L$'s and $N_T=2000$. Here $V_M=3.0$ and other parameters are the same as in Fig. 1.

Figure 3

Probability distribution $p_j={\sum }_a{\left|{\varphi }_j^a\right|}^2$ of the single-particle wave function of the Majorana edge state. Here, $L=17,N_T=32000,V_M=3.0$, and other parameters are the same as in Fig. 1

Figure 4

$\overline{r}$ as a function of $V$ for three configurations. The inset shows $P\left(r\right)$ for three values of $V$ for a chain of $L=15$ with OBC. Here ${\mu }_0=0.2$ and the other parameters are the same as in Fig. 1.

Figure 5

Phase diagram in the ${\mu }_0\text{−}V$ plane. The solid blue (red) line is the boundary below which the Floquet Majorana zero $\left(\pi \right)$ mode is preserved. The black dots constitute the boundary of crossover from weak to strong chaotic state. Regions I and II bounded by the blue and red lines are topologically nontrivial phases, while other regions are trivial. Region III bounded by dots is strong chaotic, while other regions are weak chaotic. The inset shows the one-particle spectrum at $V=0$. The other parameters are the same as in Fig. 1.

Figure 6

Statistics of the quasienergy gaps between even and odd sectors for $L=8,10,12,14,16$ with $V=1.0$. The other parameters are the same as in Fig. 1.

Figure 7

Majorana survival probability and ${\tilde{N}}_{\beta }$ for (a) the zero mode and (b) the $\pi$ mode, respectively. For (a), the driving potential is initially set to ${\mu }_0=0$, and slowly changed to ${\mu }_0=3.0$ in $N_T=2000$ periods. For (b), the driving potential is varied just in the opposite manner. The other parameters are taken to be $V=0,L=20,\delta \mu =2.5$, and $T=1$.