Topological transitions in two-dimensional Floquet superconductors

We demonstrate the occurrence of a topological phase transition induced by an effective magnetic field in a two-dimensional electron gas with spin-orbit coupling and in proximity to an $s$-wave superconductor. The effective, perpendicular magnetic field is generated by an in plane, off-resonant ac-magnetic field or by circularly polarized light. The conditions for entering the topological phase do not rely on fine parameter tuning: For fixed frequency, one requires a minimal amplitude of the effective field which can be evaluated analytically. In this phase, chiral edge states generally emerge for a system in stripe geometry unless the Rashba and Dresselhaus coupling have the same magnitude. In this special case, for magnetic field driving the edge states become Majorana flat bands, due to the presence of a chiral symmetry; the light irradiated system is a trivial superconductor.

Figure 1

A 2DEG in proximity to a conventional superconductor can become itself a (synthetic) superconductor. The combined action of spin-orbit coupling (SOC) and a time-dependent magnetic (a) or electric (b) field can drive the system to a topologically nontrivial phase.

Figure 2

Topological phase transition induced by an in plane time-dependent magnetic field. (a), (b) Numerical quasienergy spectrum of $H_F$ in stripe geometry for the two values of chemical potential and Zeeman field shown with red dots in panel (d). We considered $N_y=3600$ transverse channels and retained $N_F=3$ Floquet modes. In the topological phase, panel (b), chiral mode crossing at the $\mathrm{\Gamma }$ point emerge. (c) The two chiral edge modes decay exponentially towards the device interior and are localized at opposite edges. Here one of them is shown for $k_{x}a=0.1$. (d) Chern number $C_2$ of the bulk Floquet-BdG Hamiltonian for different values of $h_Z$ and $\mu$. It holds $C_2=1$ in the nontrivial region; its boundary, depicted by the red solid line, is given by Eq. (14).

Figure 3

Majorana flat bands for the symmetric case with SOC parameters $\alpha =-\beta$. (a), (b) Quasienergy spectrum of $H_F$ in stripe geometry with Zeeman amplitude and chemical potential corresponding to the two red dots on the left (a) and right (b) of the topological phase boundary [solid red line in panel (d)]. (c) The spatial profile of the Majorana modes is shown for $k_{x}a=0.1$. (d) The partial Berry-phase sum parity $P_{B,k_x=0}=P_{B,k_x=\pi /2}$ is shown as a function of $h_Z/\mathrm{\Delta }$ and $\mu /\mathrm{\Delta }$. The dashed vertical line separates the region with and without flat bands.

Figure 4

Stability of flat bands. Midgap states and Majorana oscillations are still seen by lowering the driving frequency from $\hslash \mathrm{\Omega }=7.5\xi$ in panel (a) to $\hslash \mathrm{\Omega }=5\xi$ in panel (b).

Figure 5

As expected from a Floquet Hamiltonian, replicas appear which are separated by the horizontal red dashed lines.

Figure 6

Chern numbers of the second (a) and third (b) band of the first Floquet Brillouin zone of $H_F$. (c) Energy gap between the second and third band of the central Floquet mode of $H_F$ in units of $\xi$. Values are calculated using the tight-binding formulation and shown for different field strengths $h_{\text{Z}}$ and chemical potential $\mu$ in units of the superconducting gap $\mathrm{\Delta }$. The red line indicates $|h_{\text{Z},c}|$, Eq. (A50). Parameters used: $N_F=5$ and the frequency is $\hslash \mathrm{\Omega }/\xi =20, q_x=q_z=0, r_z=1\xi , \alpha =0.1\xi a, \beta =0.37\xi a$.

Figure 7

Smallest energy gap in units of $\xi$. (a) Shown is the gap between the second and third band of the central Floquet mode of $H_F$. (b) Smallest energy gap between different Floquet replica, of $H_F$ for different values of field strength $h_{\text{Z}}$ and chemical potential $\mu$. Parameters used: $N_F=5$ and the frequency is $\hslash \mathrm{\Omega }/\xi =16, q_x=q_z=0, r_z=1\xi , \alpha =0.1\xi a, \beta =0.37\xi a$.

Figure 8

Central gap of the 2D Floquet spectrum in units of $\xi$ for $\alpha =\beta$. The gap closes for $h_{\text{Z}}=\sqrt{{\mathrm{\Delta }}^2+{\mu }^2}$ at $k=0$. However, there is already a gap closing at finite wave vectors for $h_{\text{Z}}\ge \mathrm{\Delta }$, see discussion in Sec. 3. The parameters are $\alpha =\beta =0.37\xi a, N_F=3, \mathrm{\Delta }=0.1\xi , r_x=1\xi , r_y=0.2\xi , r_z=q_x=q_y=0$.

Figure 9

Quasienergy spectrum of $H_F\left(k\right)$ in stripe geometry with a width of $N_y=200$ transverse channels and for SOC parameters $\alpha =\beta$. The three panels show different field strengths. The field is changed via $q_y$ (for fixed frequency $\hslash \mathrm{\Omega }=20\xi$ and $N_F=4$) for $\alpha =\beta =0.1\xi a$. $[\mathrm{\Delta }=0.1\xi ,\mu =0.1\xi ,r_x=2\xi ,r_y=0.2\xi ,q_x=0,h_{\text{Z},c}/\mathrm{\Delta }=\sqrt{2}]$: (a) $q_y=1.2\xi$, thus $h_{\text{Z}}/h_{\text{Z},c}=0.85$, (b) $q_y=1.4\xi$, thus $h_{\text{Z}}/h_{\text{Z},c}=0.99$, (c) $q_y=1.6\xi$, thus $h_{\text{Z}}/h_{\text{Z},c}=1.13$.

Figure 10

Quasienergy spectrum of $H_F\left(k\right)$ in stripe geometry with a width of $N_y=200$ transverse channels and for SOC parameters $\alpha =-\beta$. The three panels show different field strengths. The field is changed via $q_y$ (for fixed frequency $\mathrm{\Omega }=20\xi$ and $N_F=4$) for $\alpha =-\beta =0.1\xi a$. $[\mathrm{\Delta }=0.1\xi ,\mu =0.1\xi ,r_x=2\xi ,r_y=0.2\xi ,q_x=0,h_{\text{Z},c}/\mathrm{\Delta }=\sqrt{2}]$: (a) $q_y=1.2\xi$, thus $h_{\text{Z}}/h_{\text{Z},c}=0.85$, (b) $q_y=1.4\xi$, thus $h_{\text{Z}}/h_{\text{Z},c}=0.99$, (c) $q_y=1.6\xi$, thus $h_{\text{Z}}/h_{\text{Z},c}=1.13$.

Figure 11

Central gap, in units of $\hslash \mathrm{\Omega }$, plotted for $\alpha =-\beta =0.1\xi$ with (a) $\mu =0$ and (b) $\mu =0.1\xi$. In (a) the vertical line indicates the critical field $h_{\text{Z}}/\mathrm{\Delta }=1$ below which one finds a gapped system. In (b) the second vertical line indicates $h_{\text{Z},c}$. Other parameters: $N_F=2, \mathrm{\Delta }=0.1\xi , r_x=1\xi , r_y=0.2\xi , r_z=q_x=q_y=0$.