Helical Floquet Channels in 1D Lattices

We show how dispersionless channels exhibiting perfect spin-momentum locking can arise in a 1D lattice model. While such spectra are forbidden by fermion doubling in static 1D systems, here we demonstrate their appearance in the stroboscopic dynamics of a periodically driven system. Remarkably, this phenomenon does not rely on any adiabatic assumptions, in contrast to the well known Thouless pump and related models of adiabatic spin pumps. The proposed setup is shown to be experimentally feasible with state-of-the-art techniques used to control ultracold alkaline earth atoms in optical lattices.

Figure 1

(a) Illustration of basic constraints by fermion doubling in 1D lattice systems: The left plot shows an ordinary metallic band which must be periodic in the first Brillouin zone, while the unidirectional channel in the right plot violates this periodicity and, hence, is forbidden by fermion doubling. (b) Schematic of the proposed driving protocol. The spin-flip hopping [see $H_1$ in Eq. (2)] acts during the first half-period $[0,T/2)$, while on-site spin flips [see $H_2$ in Eq. (2)] characterize the second half-period $[T/2,T)$. (c) Floquet band structure of the proposed lattice model [see Eqs. (2), (3)] with perfect spin-momentum locking. Parameters are $\alpha =\beta =\pi /T$. (d) Illustration of the toroidal time-momentum space ${\mathcal{T}}^2$.

Figure 2

Lower panel: Topologically nontrivial spin micromotion of the Bloch states $|u_k^{\uparrow }(t)⟩=U_k(t,0)|u_k^{\uparrow }(0)⟩$ that are eigenstates of the spin-up Floquet operator $U_k^{\uparrow }(T,0)$ at stroboscopic times $t=0(\text{mod}T)$. Upper panel: Berry curvature ${\mathcal{F}}_{k,t}^{\uparrow }=2\mathrm{Im}\{⟨{\partial }_k{u}_k^{\uparrow }(t)|{\partial }_t{u}_k^{\uparrow }(t)⟩\}$ in combined time-momentum space. Parameters are $\alpha =\beta =\pi /T=\pi$ in all plots.

Figure 3

Top: Gap around $k=0$ for $\alpha =1.1\pi /T$, $\beta =0.9\pi /T$. From left to right, the three plots show the Floquet spectrum of the system, the total $S_z$ polarization as a function of time, and the spatially resolved $S_z$ polarization as a function of time. Bottom: Gap around $k=\pi$ for $\alpha =\beta =0.92\pi /T$. The plots are analogous to those in the top panel.

Figure 4

Left: Scattering due to ${\sigma }_z$ impurity at site $x=6$. The plot shows the spatial distribution of the $S_z$ polarization as a function of time. Right: Scattering due to ${\sigma }_0$ impurity at site $x=6$. The bulk parameters are $\alpha =\beta =\pi /T$ and $V_d=1.5/T$ in both plots.