Emergent Kinetics and Fractionalized Charge in 1D Spin-Orbit Coupled Flatband Optical Lattices

Recent ultracold atomic gas experiments implementing synthetic spin-orbit coupling allow access to flatbands that emphasize interactions. We model spin-orbit coupled fermions in a one-dimensional flatband optical lattice. We introduce an effective Luttinger-liquid theory to show that interactions generate collective excitations with emergent kinetics and fractionalized charge, analogous to properties found in the two-dimensional fractional quantum Hall regime. Observation of these excitations would provide an important platform for exploring exotic quantum states derived solely from interactions.

Figure 1

(Main panels) Single-particle energy $\omega (k)$ of several lowest Bloch bands due to SOC for $s=4$ (left) and $s=10$ (right). The ratio of the lowest energy gap to the bandwidth is tuned to $\approx 8$ [27] in both panels by setting $k_R=k_L/2$, $\mathrm{\Omega }=0.22E_R$ for $s=4$, and $\mathrm{\Omega }=0.05E_R$ for $s=10$. (Inset) Diagonal interaction between $\chi$ particles as a function of intersite distance, $d$, for $s=4E_R$ and $s=10E_R$ yielding $V_1/U\approx 0.0529$.

Figure 2

(Left panel) The many-body energy dispersion $E(K)$ (scattered symbols) for the flatband-projected Hamiltonian $H$ on various lattice sizes, $L$. The solid curve is the fit of $E(K)$ with the effective extended Hubbard model [Eq. (4)] that highlights a dispersive collective mode. (Right panel) The energy dispersion of a similar classical model [Eq. (3)] with $V_1/U=0.0529$, showing no dispersive collective modes. The four schematics denote representative ground and excited state configurations of spinor $\chi$ particles (encircled arrows) in real space.

Figure 3

(Main panel) Emergent single-particle energy dispersion [Eq. (5)]. The dashed and dotted lines cross at Fermi points, $k_F=\pm \pi /4$. Linearization is shown as solid straight lines. Differing slopes indicate asymmetric bands, i.e., $v_{1F}\ne v_{2F}$. (Inset) Schematic of hopping terms used in Eq. (4).