Floquet engineering of long-range $p$-wave superconductivity

Floquet Majorana fermions appear as steady states at the boundary of time-periodic topological phases of matter. In this work, we theoretically study the main features of these exotic topological phases in the periodically driven one-dimensional Kitaev model. By controlling the ac fields, we can predict topological phase transitions that should give rise to signatures of Majorana states in experiments. Moreover, the knowledge of the time dependence of these Majorana states allows one to manipulate them. Our work contains a complete analysis of the monochromatic driving in different frequency regimes.

Figure 1

Regions of convergence of the Magnus expansion of ${\tilde{H}}_k^{\alpha }\left(t\right)$ for $\alpha \in \{0,1,2,3,4\}$ as a function of ${\mu }_0$ and $\omega$. To obtain a convergent Magnus expansion, the Hamiltonian ${\tilde{H}}_k^{\alpha }\left(t\right)$ has to fulfill the convergence condition Eq. (11) for all the values of $k$. At high frequency $\omega >2\tau$, the series of ${\tilde{H}}_k^0\left(t\right)$ converges. For $\omega <2\tau$, the successive ${\tilde{H}}_k^{\alpha }\left(t\right)$ have convergent series for some values of ${\mu }_0$. We consider a fixed bandwidth $\tau ={\mu }_0+w_0$.

Figure 2

Phase diagram of the Kitaev Hamiltonian of Eq. (1) with time-dependent chemical potential $\mu \left(t\right)={\mu }_0+\frac{{\mu }_1}{2}cos\omega t$. The white region is topologically trivial ($W=0$), while the other ones are nontrivial ($W\ne 0$). (a) In the frequency regime $\omega >2\tau$, ${\tilde{H}}_k^{\text{eff},0}$ is used to calculate the bulk invariant $W_0$ for all values of ${\mu }_0$, because the series of ${\tilde{H}}_k^0\left(t\right)$ converges. Transitions between the topological phases $W_0=+1$ and $-1$ occur at zeros of ${\mathcal{J}}_0\left(\frac{{\mu }_1}{\omega }\right)$. (b) For $\omega =1.5\tau$, the series of ${\tilde{H}}_k^1\left(t\right)$ converges for ${\mu }_0>0.5\tau$. The trivial phase appears for ${\mu }_0>0.875\tau$ and transitions between phases $W_1=1$ and $-1$ take place at zeros of ${\mathcal{J}}_1\left(\frac{{\mu }_1}{\omega }\right)$. (c) Depicts the extension of the phase diagram shown in (b) for $\omega =1.5\tau$ to smaller values of ${\mu }_0$. Besides the phases $W_1=0,\pm 1$, we find new topological phases that are described by two topological invariants $(W_0,W_1)$, corresponding to the effective Hamiltonians ${\tilde{H}}_k^{\text{eff},0}$ and ${\tilde{H}}_k^{\text{eff},1}$. We consider a fixed bandwidth $\tau ={\mu }_0+w_0$.

Figure 3

(a,b) Temporal stroboscopic evolution of the stationary left end state for a finite chain with $N=60$ sites. The color indicates the value of coefficients $u_i$ and $v_i$ in Eq. (24) ($l$ corresponding to the left Majorana end mode) for the first five sites. For a frequency $\omega =1.5\tau$, we have performed numerical calculations in the case of a driven chemical potential $\mu \left(t\right)={\mu }_0+\frac{{\mu }_1}{2}cos\omega t$ with ${\mu }_0=0.75\tau$. (a) Depicts the evolution for $\frac{{\mu }_1}{\omega }=1$, being ${\mathcal{J}}_{-1}\left(\frac{{\mu }_1}{\omega }\right)<0$, and (b) for $\frac{{\mu }_1}{\omega }=4.5$, when ${\mathcal{J}}_{-1}\left(\frac{{\mu }_1}{\omega }\right)>0$. The stroboscopic dynamic agrees with the predicted one. (c) Shows the evolution of coefficients ${\tilde{u}}_i$ and ${\tilde{v}}_i$ [Eq. (26)] in case ${\mu }_0=0.3\tau$ and $\frac{{\mu }_1}{\omega }=1$ imposing the initial condition $\Gamma \left(0\right)=f_1$ at $t=0$. The predicted double period electron-hole oscillations are observed. We consider a fixed bandwidth $\tau ={\mu }_0+w_0$.

Figure 4

Phase diagram for $\omega >2\tau$ in the case of $\mu \left(t\right)={\mu }_0$, (a) $\Delta \left(t\right)=w\left(t\right)=J\left(t\right)=J_0+\frac{J_1}{2}cos\omega t$, (b) $w\left(t\right)=w_0+\frac{w_1}{2}cos\omega t$, and $\Delta \left(t\right)={\Delta }_0$. The white region is topologically trivial ($W=0$), the light-orange ($W=1$) and blue ($W=-1$) regions are nontrivial phases with one end state, and the brown region ($W=2$) is a nontrivial phase with two end states.

Figure 5

BCS interaction strength $g_l$ as a function of $w_1/\omega$ for neighbors $l=1,3,5,7,9$. Notice that the sign of the interaction is ${(-1)}^{\frac{l-1}{2}}$.

Figure 6

Phase diagram for $\omega =1.5\tau$ in the case of a driven chemical potential. The dark lines show the closings of (a) the 0 gap, (b) the $\frac{\omega }{2}$ gap, and (c) both gaps. The whole topological phase diagram is given by (c), which agrees perfectly with the analytical result in Fig. 2.

Figure 7

Phase diagram for $\omega =0.9\tau$ in the case of a driven chemical potential. The dark lines show the closings of (a) the 0 gap, (b) the $\frac{\omega }{2}$ gap, and (c) both gaps. The diagram (c) in the right part is the same as the diagram for $\omega =1.5\tau$ but changing the zeros of ${\mathcal{J}}_0$ by zeros of ${\mathcal{J}}_1$ and the zeros of ${\mathcal{J}}_1$ by zeros of ${\mathcal{J}}_2$ ($\tau =1$).