Transport signatures of a Floquet topological transition at the helical edge

Recently, the experimental realization of a quantum point contact (QPC) between helical edges has been accomplished. We predict that such a QPC, in the presence of a time periodic applied electric field, is characterized by a topological quantum phase transition in the Floquet spectrum. Crucially, the fact that the QPC is naturally attached to helical leads allows for the unambiguous detection of the topological Floquet quantum phase transition by means of multiterminal transport experiments. Moreover, such experiments enable the characterization of the topological bound states associated with the transition.

Figure 1

(a) Schematic of the system: QPC based on a QSHI under the influence of electromagnetic radiation and Zeeman field. (b) QPC eigenvalue spectrum including all TR invariant static perturbations. (c) QPC eigenvalue spectrum with additional Zeeman field. The coupling induced by external electromagnetic radiation with resonant frequency $\omega =2\sqrt{{\gamma }_c^2+{\gamma }_0^2}$ (${\gamma }_0={\gamma }_c$) is also shown by means of the arrow. (d)–(f) Quasienergy spectrum ${\epsilon }_r=\epsilon -\omega /2$ of the effective quasienergy operator based on the extended Floquet-Hilbert space with (d) $eA/B_z=0.5$, (e) $eA/B_z=1$, (f) $eA/B_z=2$. The topological phase is characterized by $0<eA/B_z<1$. The colors in the spectra (b)–(f) function as a guide to the eye to visualize band inversion.

Figure 2

(a) Two-terminal conductance evaluated from Eq. (11) and (b) four-terminal conductance from Eq. (10), for quasienergy $\epsilon =\sqrt{{\gamma }_0^2+{\gamma }_c^2}-\delta \omega$ and driving frequency $\omega =2\left(\sqrt{{\gamma }_0^2+{\gamma }_c^2}-\delta \omega \right)$. Additional parameters are $L=27\hslash v_F/{\gamma }_0$, $\overline{m}=10, v_F=1$, and ${\gamma }_0={\gamma }_c$. The red line corresponds to the gap closing and reopening of the effective quasienergy operator (see, inline equation in Ref. [58]), indicating the topological phase transition.

Figure 3

Schematic of a QPC with barriers.

Figure 4

(a), (b) Two-terminal and (c), (d) four-terminal conductance through a decoupled QPC under the influence of driven electromagnetic vector field $eA$. ${\epsilon }_r=\epsilon -\omega /2, \omega =2\sqrt{{\gamma }_0^2+{\gamma }_c^2}, L=120\hslash v_F/{\gamma }_0, L_{\mathrm{bar}}=1/2\hslash v_F/{\gamma }_0, v_F=1, \overline{m}=5, {\gamma }_0={\gamma }_c, {\gamma }_0=0.5, eA=0.1$. Further parameter choices are (a) $\lambda /{\gamma }_0=10$, (b) $\lambda /{\gamma }_0=6$, (c) $\lambda /{\gamma }_0=6$, and (d) $\lambda /{\gamma }_0=4$.

Figure 5

(a) Probability density, in units of ${\gamma }_0/\left(\hslash v_F\right)$ according to Eq. (13) with $\overline{m}=5, L=60\hslash v_F/{\gamma }_0, {\epsilon }_r=\epsilon -\omega /2, \omega =2\sqrt{{\gamma }_0^2+{\gamma }_c^2}$ (${\gamma }_0={\gamma }_c$), and $[eA\left(x\right)-B_z\left(x\right)]/\left[eA\left(x\right)\right]=1/2\mathrm{sgn}(L/2-x)$. (b)–(d) Expectation value of the spin density $\langle {\sigma }_i\rangle$, in units of ${\gamma }_0/\left(\hslash v_F\right)$, with $i\in \{x,y,z\}$ (blue, red, green) evaluated for the system shown in Fig. 3 at quasienergy $\epsilon =\omega /2$ and $\omega =2\sqrt{{\gamma }_0^2+{\gamma }_c^2}$. (b) Total spin density of the scattering state. (c), (d) Spin density for (c) upper edge and (d) lower edge, separately. Other parameters are chosen as in (a).