Spin-Orbit-Coupled Correlated Metal Phase in Kondo Lattices: An Implementation with Alkaline-Earth Atoms

We show that an interplay between quantum effects, strong on-site ferromagnetic exchange interaction, and antiferromagnetic correlations in Kondo lattices can give rise to an exotic spin-orbit coupled metallic state in regimes where classical treatments predict a trivial insulating behavior. This phenomenon can be simulated with ultracold alkaline-earth fermionic atoms subject to a laser-induced magnetic field by observing dynamics of spin-charge excitations in quench experiments.

Figure 1

(a) Schematic phase diagram of Eq. (1) with $V\gg J_0$ ($\mathrm{\Omega }$ is the AFM interaction strength), featuring the correlated metal state for strong coupling $\mathrm{\Omega }\sim V$. The conventional ferromagnetic (DE) metal occurs at $\mathrm{\Omega }\lesssim J_0$. Bottom row: allowed and forbidden (indicated by red crosses) hopping processes. Blue (red) color marks mobile (local) fermions. Light blue spins show the final states of itinerant fermions. Gray ellipses denote local entangled singlet-triplet (s-t) states [$|\mathrm{s}⟩$ in (d)]. (b) The electronic model of Eq. (1). Color notations are as in (a). Local spins feel a staggered mean-field $\mathrm{\Omega }\sim I_H⟨S^z⟩$ due to the AFM background. (c) Optical lattice of the AEA setup Eq. (5). Band 1 (2) is localized (itinerant). Gray arrows indicate the laser-induced staggered Zeeman field $\mathrm{\Omega }$ that splits pseudospin $|\uparrow ,\downarrow ⟩$ states. Gray ellipse is a spin-singlet state. Other notations are as in (b). (d) Energies on an isolated site with one fermion. Blue lines (red circle) indicate the s-t subspace (resonance). Inset: Hopping of s-t excitations $d_{i\alpha }$ (3). Other notations are as in (a). (e) Same as in (d), but for the AEA model (5). The s-t subspace is an excited manifold.

Figure 2

Propagation of wave packets with 5 particles. (a) and (b) show total density $⟨n_i^d(t)⟩$; (c) and (d) contain the transverse local-spin polarization $⟨T_i^x⟩$. Blue numbers in the top right corners indicate the detuning $\mathrm{\Delta }/J$. Inset: comparison between density evolution at a fixed $x_i=i_0$ [black arrow in panel (a)] in the full model (4) (blue line) and in the case of noninteracting (nint) $d$ fermions (black line). The parameters are $N=101$, $A={10}^{-3}J$, and $k_0=-20\pi /N$.

Figure 3

Phase diagram of the model (4) on a $N=40$ site chain with periodic boundary conditions. Black and blue lines mark 1st order transitions. For the correlated metal phase, the Drude weight $\mathcal{D}>0$, while in all other states $\mathcal{D}=0$. The dashed red line separates metallic (below) and band insulator (above) states in a model with noninteracting $d$ fermions. At $\mathrm{\Delta }=0$, $\mathcal{D}=(1/\pi )\sqrt{(2J{)}^2-{\mu }^2}$ for $|\mu |<2J$ and 0 otherwise. On the left: Drude weight and ground-state energy ${\mathcal{E}}_0$ for $\mathrm{\Delta }=J$ plotted along the arrow in the main panel. Notations are as in Fig. 2. Notice jumps in $\partial {\mathcal{E}}_0/\partial \mu$ at phase transitions. For $\mu \approx -0.5$ (CDW state), ${\mathcal{E}}_0$ decreases with increasing $n_t^d$. For noninteracting fermions, one has to multiply $\mathcal{D}$ and ${\mathcal{E}}_0$ (black curves) by 2 due to spin degeneracy, absent for constrained $d$ fermions.