Unpaired Floquet Majorana fermions without magnetic fields

Quantum wires subject to the combined action of spin-orbit and Zeeman coupling in the presence of $s$-wave pairing potentials (superconducting proximity effect in semiconductors or superfluidity in cold atoms) are one of the most promising systems for the developing of topological phases hosting Majorana fermions. The breaking of time-reversal symmetry is essential for the appearance of unpaired Majorana fermions. By implementing a time-dependent spin rotation, we show that the standard magnetostatic model maps into a nonmagnetic one where the breaking of time-reversal symmetry is guaranteed by a periodical change of the spin-orbit coupling axis as a function of time. This suggests the possibility of developing the topological superconducting state of matter driven by external forces in the absence of magnetic fields and magnetic elements. From a practical viewpoint, the scheme avoids the disadvantages of conjugating magnetism and superconductivity, even though the need of a high-frequency driving of spin-orbit coupling may represent a technological challenge. We describe the basic properties of this Floquet system by showing that finite samples host unpaired Majorana fermions at their edges despite the fact that the bulk Floquet quasienergies are gapless and that the Hamiltonian at each instant of time preserves time-reversal symmetry. Remarkably, we identify the mean energy of the Floquet states as a topological indicator. We additionally show that the localized Floquet Majorana fermions are robust under local perturbations. Our results are supported by complementary numerical Floquet simulations.

Figure 1

Ideal rotating (IR) case of Eq. (14): Floquet quasienergies and mean energies. Reference energy and momentum are ${\Delta }_0$ and $k_{\Delta }=\sqrt{2m^{*}{\Delta }_0}/\hslash$, respectively. We set $\mu =2{\Delta }_0$ and thus $\hslash {\Omega }_{\mathrm{c}}=2\sqrt{5}{\Delta }_0$. Top (T), center (C), and bottom (B) panels show results for $\Omega =0.9{\Omega }_{\mathrm{c}}$, $\Omega ={\Omega }_{\mathrm{c}}$, and $\Omega =1.1{\Omega }_{\mathrm{c}}$, respectively. Dotted horizontal lines at $n\hslash \Omega$ are included in all panels. Panels (a) show the spin-degenerated dispersion relations $E_{\pm ,k\sigma }$ of Eq. (22) in the absence of the IR SOC field (solid lines). Dashed lines correspond to $E_{\pm ,k\sigma }+m\hslash \Omega$ branches which are important for the mixing due to the periodic driving; see Eq. (A4). Panels (b) and (c) present Floquet quasienergies $\varepsilon$ for finite IR SOC field of intensity ${\tilde{E}}_{\mathrm{so}}\equiv {\tilde{\alpha }}^2{m}^{*}/\left(2{\hslash }^2\right)=0.25{\Delta }_0$ and $1.75{\Delta }_0$, respectively. Numerical results show an excellent agreement with analytical calculations obtained by mapping the time-dependent problem to the static system having a constant SOC and a finite Zeeman field: The solid black lines depict the four bands of the corresponding static system (shifted in energy by $+\hslash \Omega /2$; see text). Panels (d) show the mean energies $\overline{E}$ of the Floquet states for the simulated IR SOC-field amplitudes. In (C.b) and (C.c), two branches of Floquet solutions close in a Dirac-like quasienergy cone for small $k$. The reopening of the Dirac cone for $\Omega >{\Omega }_{\mathrm{c}}$, (B.b) and (B.c) indicates a topological phase distinct from that for $\Omega <{\Omega }_{\mathrm{c}}$. The mean energy shows a distinct pattern on each phase (see text and Fig. 2).

Figure 2

(a) Quasienergies ${\varepsilon }^{\prime }=\varepsilon -{\varepsilon }_{0,n}\left(\Omega \right)$ and (g) mean energies $\overline{E}$ of Floquet states as a function on the driving frequency $\Omega$ and the linear momentum $\hslash k$. In the plot ${\Delta }_{\mu }\equiv \hslash {\Omega }_{\mathrm{c}}/2=\sqrt{{\Delta }_0^2+{\mu }^2}$. We focus on the solutions that are affected by the ideal rotating SOC driving in the small $k$ limit for $\mu \ll {\Delta }_0$, given in Eq. (42). For the sake of clarity, surface plots show the lower energy bands only [see ${\varepsilon }_3\left(k\right)$ and ${\overline{E}}_3\left(k\right)$ in the text]: upper bands follow by symmetry [${\varepsilon }_4\left(k\right)=-{\varepsilon }_3\left(k\right)$ and ${\overline{E}}_4\left(k\right)=-{\overline{E}}_3\left(k\right)$]. In panels (b)–(f), the driving frequency $\Omega$ is fixed and both bands are shown. At the critical frequency—panel (d)—the quasienergies form a Dirac-cone while the mean energies show a similar behavior except for a discontinuity at $k=0$. Just away from ${\Omega }_{\mathrm{c}}$, sub- and supercritical regimes cannot be distinguished from the quasienergy bands alone. In contrast, thanks to the discontinuity developed at the critical point, the mean-energy bands show remarkable features distinguishing one regime from another. Such a contrast is clearly seen already in the surface plots of panels (a) and (g).

Figure 3

Sketch of three different cases for the periodic SOC driving $\Lambda \left(t\right)$ of Eq. (13). (a) Ideally rotating (IR) driving ${\Lambda }_{\mathrm{IR}}\left(t\right)$ of Eq. (14): The SOC axis rotates uniformly in the $1-2$ plane with constant coupling strength. (b) Linear (L) driving ${\Lambda }_{\mathrm{L}}\left(t\right)$ of Eq. (48): The orientation of the SOC is fixed while the coupling strength oscillates harmonically. (c) Ramp (R) rotation driving ${\Lambda }_{\mathrm{R}}\left(t\right)$ of Eq. (50): The SOC axis rotates anharmonically in the $1-2$ plane with varying coupling strength driven by symmetrical triangular waves.

Figure 4

Floquet quasienergies $\varepsilon \left(k\right)$ and mean energies $\overline{E}\left(k\right)$ for the situations listed in Table with $\mu =1.35{\Delta }_0$, $\hslash {\Omega }_{\mathrm{c}}\approx 3.36{\Delta }_0$, and ${\tilde{E}}_{\mathrm{so}}={\tilde{\alpha }}^2{m}^{*}/\left(2{\hslash }^2\right)=0.75{\Delta }_0$. Whenever present (lower panels), the static SOC component is set ${\alpha }_0=\tilde{\alpha }/5$. Labels subindices “$<$”, “$\mathrm{c}$”, and “$>$” stand for $\Omega =0.9{\Omega }_{\mathrm{c}}$, ${\Omega }_{\mathrm{c}}$, and $1.1{\Omega }_{\mathrm{c}}$, respectively. Floquet quasienergies are shown from 0 to $\hslash {\Omega }_{\mathrm{c}}$. Floquet bands are symmetrical with respect to $\varepsilon =\hslash \Omega /2$ (horizontal gray line). We do not show the $-k$ quasienergies and mean energies whenever they are identical to the $+k$ ones. This is not the case for finite static SOC orthogonal to the periodic SOC driving. For vanishing static SOC (upper panels), solid black lines depict IR-quasienergy Floquet solutions, in particular, the bands associated with the topological transition [quasienergies from Eq. (29) with $n=0$ generated by the energies of the static problem $E_{+,-}\left(k\right)$ and $E_{-,+}\left(k\right)$ given in Eq. (27)]. For L driving, the solutions with ${\alpha }_0=0$ are doubly degenerated. For R driving, higher-frequency driving components generate new transitions. Floquet transitions introduce avoided crossings in the quasienergy spectrum and strong discontinuities in the mean energies (Refs. 53 and 54). See text for further discussion.

Figure 5

Floquet Majorana states in a finite sample subject to the IR excitation in the $\Omega >{\Omega }_{\mathrm{c}}$ phase with ${\tilde{E}}_{\mathrm{so}}={\tilde{\alpha }}^2{m}^{*}/\left(2{\hslash }^2\right)=0.5{\Delta }_0$. Reference quantities are ${\Delta }_0$ and the wavelength ${\lambda }_{\Delta }=2\pi /k_{\Delta }$, with $k_{\Delta }=\sqrt{2m^{*}{\Delta }_0}/\hslash$. The lattice spacing is $a_0=0.0225{\lambda }_{\Delta }$ (see text) with sample length $300a_0=6.75{\lambda }_{\Delta }$. We choose $\mu =2{\Delta }_0$ such that $\hslash {\Omega }_{\mathrm{c}}=2\sqrt{5}{\Delta }_0$, with driving frequency $\Omega =1.15{\Omega }_{\mathrm{c}}$. (a) Floquet solutions ordered by growing quasienergy (left) and mean energy (right). We plot ${\varepsilon }^{\prime }\equiv \varepsilon -\hslash \Omega /2$ for $0\le \varepsilon \le \hslash \Omega$. Two edge states, $|a\rangle$ and $|b\rangle$, are found with $\varepsilon \approx \hslash \Omega /2$ and mean energy $\overline{E}\approx 0$. Isolated FMFs at each edge, $|{\varphi }^T\rangle$, are extracted from the bonding and antibonding combinations of $|a\rangle$ and $|b\rangle$ (see text). For the two FMF states at $t=0$ we plot, as a function of the position $x$: (b) the probability density, $|{\varphi }_0^T{|}^2\equiv {\left|{\varphi }^T(x,t=0)\right|}^2$; (c) the electron spin density (ESD) along ${\sigma }_{\perp \left(0\right)}={\sigma }_2$, $S_{\perp \left(0\right)}^{{\varphi }_0^T}\equiv S_{\perp \left(t\right)}^{{\varphi }^T}(x,t=0)$; and (d) the ESD along ${\sigma }_3$, $S_3^{{\varphi }_0^T}\equiv S_3^{{\varphi }^T}(x,t=0)$. The ESD along ${\sigma }_{\parallel \left(0\right)}={\sigma }_1$ (not shown) is zero. (e) We include uncorrelated disorder $E_i\in [-0.75{\Delta }_0,0.75{\Delta }_0]$. FMFs are found with probability density similar to that of clean systems shown in (b). FMFs' densities are plotted as a function of time along one driving period in the presence of disorder: (f) $|{\varphi }^T{(x,t)|}^2$, (g) $S_{\perp \left(t\right)}^{{\varphi }^T}(x,t)$, and (h) $S_3^{{\varphi }^T}(x,t)$. The in-plane spin component rotates while the FMF state conserves its location (see text).

Figure 6

Floquet Majorana states in finite samples subject to different types of SOC drivings and/or static SOC. All samples include disorder as the one given in Fig. 5e. Unless otherwise stated the parameters $a_0$, $\mu$, ${\Omega }_{\mathrm{c}}$, $\Omega =1.15{\Omega }_{\mathrm{c}}$, and ${\tilde{E}}_{\mathrm{so}}$ are those given in Fig. 5. As in Fig. 5, decoupled solutions are obtained by combining Floquet states $|a\rangle$ and $|b\rangle$ with $\varepsilon \approx \hslash \Omega /2$ and $\overline{E}\approx 0$. In all cases we plot the probability density $|{\varphi }_0^T{|}^2\equiv {\left|{\varphi }^T(x,t=0)\right|}^2$ of the two (or more) FMF-candidate states. We rescale some curves ($\times 5$) to improve visibility. (a) Full sample with R driving. (b) Left ($l$) section of the sample is subject to IR driving, whereas the right section ($r$) is L-driven. (c),(d) IR-driven sample in the presence of a static SOC with ${\alpha }_0=0.2\tilde{\alpha }$ either parallel (c) or perpendicular (d) to the plane of the IR driving. (e) IR-driven sample with inhomogeneous chemical potential on the left and right sections, ${\mu }^l=2{\Delta }_0={\mu }^r/2$. The driving frequency is $1.15{\Omega }_{\mathrm{c}}^l$ but smaller than the ${\Omega }_{\mathrm{c}}^r$. (f) IR-driven sample on the left section in contact with a zero-SOC section on the right. (g) IR-driven sample subject to different driving frequencies on left and right sections, ${\Omega }_l=2{\Omega }_r=1.15{\Omega }_{\mathrm{c}}$: The left (right) section is in the supercritical (subcritical) regime. (h) Sample subject to counterrotating IR drivings with $\Omega =1.15{\Omega }_{\mathrm{c}}$, clockwise (counterclokwise) on the left (right) section of the sample. Four Floquet states arise in this case.