Ferromagnetic spin-orbital liquid of dipolar fermions in zigzag lattices

Two-component dipolar fermions in zigzag optical lattices allow for the engineering of spin-orbital models. We show that dipolar lattice fermions permit the exploration of a regime typically unavailable in solid-state compounds that is characterized by a spin-liquid phase with a finite magnetization and spontaneously broken SU(2) symmetry. This peculiar spin liquid may be understood as the Luttinger liquid of composite particles consisting of bound states of spin waves and orbital domain walls moving in an unsaturated ferromagnetic background. In addition, we show that the system exhibits a boundary phase transitions involving nonlocal entanglement of edge spins.

Figure 1

(a) Spin-orbital model on a quasi-1D zigzag lattice with weak interchain hoppings $\varepsilon \ll t$. The model (4) corresponds to $\varepsilon =0$. (b) Phase diagram of the model for $\lambda =0$. (c) A sketch of the kink-magnon bound state in the (F,AF) phase, where $J$ denotes the effective spin exchange on the corresponding link. A magnon binds only to the orbital kink, but not to the antikink. Open and solid circles in (b) and (c) show the occupied orbital states ${\sigma }^z=\pm 1$.

Figure 2

Phase diagram of the 1D spin-orbital model for $\alpha =3$ and $L=96$ sites (for open boundaries). Symbols denote numerical results (solid and dotted lines are a guide to the eye), whereas dashed lines correspond to the analytical estimates ${\lambda }_F$ and ${\lambda }_c$. Curves $b$1 and $b$2 mark the boundary phase transitions that involve nonlocal entanglement between edge spins ${\mathit{\tau }}_1={\mathit{S}}_{\mathbf{1}}+{\mathit{S}}_{\mathbf{2}}+{\mathit{S}}_{\mathbf{3}}$ and ${\mathit{\tau }}_N={\mathit{S}}_{\mathbf{N}}+{\mathit{S}}_{\mathbf{N}-\mathbf{1}}+{\mathit{S}}_{\mathbf{N}-\mathbf{2}}$. The inset shows the ground-state correlation between edge spins $\langle {\mathit{\tau }}_1{\mathit{\tau }}_N\rangle$ as a function of $\Delta$ for $\lambda =4$.

Figure 3

Properties of the FSOL phase for $\alpha =3$ and $\lambda =4$: (a) Magnetization curve for different system sizes $L$, ${\sum }_i\langle S_i^z\rangle /L$, calculated in the highest weight states of ground-state SU(2) multiplets; (b) magnon density (circles) and orbital domain wall density (triangles) in the ground state at $\Delta =1.02$, $S^z=44$, $L=96$ (open boundary conditions break translation symmetry and pin kink magnons); finite-size scaling of the particle-hole (c) and magnon (d) excitation gaps at $S^z=N/3$; (e) ground-state correlators for $\Delta =0.86$, $L=48$. See text for details.

Figure 4

Excitation gap for open boundary conditions (OBCs) and periodic boundary conditions (PBCs) as a function of $\Delta$ and the system size $L$. The inset zooms in the region $0.36<\Delta <0.6$, revealing the existence of the two boundary transitions at $b1$ and $b2$.

Figure 5

Spin and orbital distribution profiles in the ground states of the $S^z=1$ and $S^z=2$ sectors, at two points of the $\alpha =3$ phase diagram, along the $\lambda =4$ line: (a) At a point between the $b1$ and $b2$ boundary transition lines, the $S^z=1$ excitation is localized at the system edges, while the $S^z=2$ excitation belongs to the bulk; (b) at another point between the $b2$ line and the boundary of the FSOL phase, both $S^z=1$ and $S^z=2$ excitations are localized at the edges.