Emergent Kitaev materials in synthetic Fermi-Hubbard bilayers

Abstract

We investigate the emergence of bond-directional spin-spin interactions in a synthetic Fermi-Hubbard bilayer that can be realized with ultracold fermions in Raman optical lattices. The model exploits synthetic dimensions to couple two honeycomb layers, each corresponding to a different hyperfine atomic state, via Raman-assisted tunneling and, moreover, via an inter-layer Hubbard repulsion due to the cold-atom scattering. In the strong-coupling regime at half filling, we derive effective spin Hamiltonians for the kinetic exchange featuring Kitaev, Heisenberg, off-diagonal exchange ($Γ$-couplings), as well as tunable Dzyaloshinskii-Moriya interactions. We identify specific configurations that generate both ferromagnetic and antiferromagnetic Kitaev couplings with various perturbations of relevance to Kitaev materials, providing a tunable platform that can explore how quantum spin liquids emerge from itinerant fermion systems. We analyze the Fermi-liquid and Mott-insulating phases, highlighting a correspondence between Dirac and Majorana quasi-particles, with possible phase transitions thereof. In an extreme anisotropic limit, we show that the model reduces to an inter-layer ribbon in a quasi-1D ladder, allowing for a numerical study of the correlated ground state using matrix product states. We find a transition from a symmetry-protected topological insulator to a Kitaev-like regime characterized by nonlocal string order. Our results establish that cold-atom quantum simulators based on Raman optical lattices can be a playground for extended Kitaev models, bridging itinerant fermionic systems and spin-liquid physics.

Figures (12)

• Figure 1: Fermi-Hubbard bilayer (FHB): Sketch of the synthetic model defined on two vertically stacked honeycomb layers, each with a two-site unit cell $A$ and $B$, and two (three) unit (bond) vectors $\boldsymbol{e}_1,\boldsymbol{e}_2$ ($\boldsymbol{u}_1,\boldsymbol{u}_2,\boldsymbol{u}_3$). In addition to fermionic tunnelings $\mathsf{T}_i$, we consider onsite interaction $U$ coupling the two layers. Each lattice site hosts two fermionic states (u, d), and the atoms can tunnel between neighboring sites within each layer via complex, bond-dependent tunnelings.
• Figure 2: intra- and inter-layer tunnelings: Illustration of the different types of tunneling processes used to construct bond-dependent exchange interactions. The tunneling matrices $\mathsf{T}_{i}$ and their time-reversed counterparts $\mathsf{T}^\dagger_{i}$ allow for encoding a bond-directional hopping, which has inter-layer $t^{\rm uu}_i, \, t^{\rm dd}_i$ and intra-layer $t^{\rm ud}_i, \, t^{\rm du}_i$ tunnelings.
• Figure 3: Strong-coupling Kitaev compass model: Illustration of the different types of bond-dependent exchange interactions. The interplay fo the intra- and inter-layer tunnelings can lead to interference pathways that allow for control of the bond-exchange anisotropy, such that spins only interact though $xx$ interactions among the $\boldsymbol{u}_1$ nearest neighbors, $yy$ interactions among the $\boldsymbol{u}_2$ nearest neighbors, and $zz$ interactions among the $\boldsymbol{u}_3$ nearest neighbors.
• Figure 4: Kitaev's material corrections: Illustration of representative kinetic exchange processes and the resulting spin-spin interactions on the honeycomb lattice. (Left panel) Standard Heisenberg-type exchange interaction arising from spin-independent tunneling, which leads to isotropic couplings that are identical on all bonds. The corresponding exchange tensor is characterized by $J_1^x = J + K_x$, $J_1^y = J$, $J_1^z = J$, with cyclic permutations for $\{J_2^\alpha, J_3^\alpha\}$. (Right panel) Symmetric off-diagonal exchange terms $\{\Gamma_i^\alpha\}$ that result from spin-dependent tunneling processes. These include both the $\Gamma$ and $\Gamma'$ interactions commonly used for extended Kitaev models in spin-orbit-coupled Mott insulators.
• Figure 5: Numerical validation of tailored kinetic exchange: We perform exact numerical simulations (circles) of a Hubbard bi-dimmer with various tunneling matrices $\mathsf{T}_1$ and strong Hubbard interactions $U=15t_1$, and compare to the effective exchange dynamics (solid lines) predicted by Eq. \ref{['eq:exchangle_J']}-\ref{['eq:exchangle_Gamma']}. a Standard Heisenberg exchange leading a flip-flop dynamics for $\ket{\uparrow_1\downarrow_2}\leftrightarrow\ket{\downarrow_1\uparrow_2}$ but no dynamics for parallel spins. (b) Kitaev $xx$ ferro coupling along $\boldsymbol{e}_1$, leading to a flip-flop of both $\ket{\uparrow_1\downarrow_2}\leftrightarrow\ket{\downarrow_1\uparrow_2}$ and $\ket{\uparrow_1\uparrow_2}\leftrightarrow\ket{\downarrow_1\downarrow_2}$. (c) Heisenberg-Kitaev coupling along $\boldsymbol{e}_1$, leading to a two-fold increase of the flip-flop dynamics of $\ket{\uparrow_1\downarrow_2}\leftrightarrow\ket{\downarrow_1\uparrow_2}$. (d) Kitaev materials coupling with competing bond-dependent and symmetric off-diagonal couplings, which leads to only partial exchange of both $\ket{\uparrow_1\downarrow_2}\leftrightarrow\ket{\downarrow_1\uparrow_2}$ and $\ket{\uparrow_1\uparrow_2}\leftrightarrow\ket{\downarrow_1\downarrow_2}$.