Emergence of Kugel-Khomskii physics in quarter-filled bilayer correlated systems

TL;DR

The paper addresses how Kugel–Khomskii physics can emerge in a quarter-filled bilayer with orbital-selective interlayer hybridization. It builds a low-energy effective Hamiltonian by projecting a bilayer two-orbital Hubbard model into a molecular orbital basis for the dz^2 sector via a Schrieffer–Wolff transformation, yielding an anisotropic Kugel–Khomskii model that couples spin and a layer pseudospin. The authors map the ground-state phase diagram into four phases, including a novel spin–layer entangled (SLE) phase that hosts maximal on-site spin–layer entanglement and emergent O(4) symmetry breaking to O(3) with three gapless Goldstone modes. The excitation spectrum reveals distinct spin and layer modes, a nearly flat composite mode controlling transitions, and hybridized excitations in the SLE phase, highlighting a geometrically driven route to composite entanglement with potential relevance to bilayer nickelates and engineered platforms. This framework provides a concrete theoretical avenue to explore intertwined spin, orbital, and layer dynamics in multi-component correlated materials.

Abstract

We present a theoretical study of the low-energy physics of a quarter-hole-filled two-orbital bilayer Hubbard model motivated by transition-metal bilayer systems with strong orbital-selective interlayer hybridization. By explicitly treating the strong interlayer bonding of dz2 orbitals within a molecular orbital basis and projecting out high-energy electronic states, we derive a low-energy effective Kugel-Khomskii Hamiltonian describing the interplay between electron spin and emergent layer pseudospin degrees of freedom. We map out a rich ground state phase diagram featuring diverse spin and charge ordered states. These include conventional ferromagnetic and antiferromagnetic phases with layer staggered charge densities, a layer-coherent phase characterized by spontaneous interlayer quantum coherence, and a novel maximally spin-layer-entangled phase with a hidden composite spin-layer order. We show that this exotic hidden ordered phase arises from the spontaneous breaking of an emergent O(4) symmetry down to a O(3), manifesting a unique excitation spectrum with three entangled gapless Goldstone modes. Our results uncover a geometrically driven mechanism for realizing composite entanglement in strongly correlated bilayer systems and provide a concrete theoretical framework relevant to bilayer nickelate superconductors and other multi-component correlated materials.

Figures (6)

• Figure 1: (a) Schematic illustration of the minimal bilayer two-orbital tight-binding model where $t^{zz}_{\perp}$ and $t^{xx}_{\parallel}$ denote the orbital dependent hopping parameters. (b) Sketch of the crystal splitting of $e_g$ orbitals and the formation of the bonding-antibonding molecular orbital (MO) states between Ni $z^2$ orbitals in the top and bottom layers. (c) One representative ground-state configuration where the bonding $d_{z^{2}}$ orbital is doubly occupied, while the remaining electron resides in one of the $d_{x^{2}-y^{2}}$ orbitals. The other configurations can be obtained by reversing the spin direction or the layer occupation from those presented in (c).
• Figure 2: Four degenerate ground-state configurations $|S^{z},\ \tau^{z}\rangle$ labeled with the spin and layer quantum numbers $S^{z}=\pm\frac{1}{2}$ and $\tau^{z}=\pm\frac{1}{2}$ in the $d_{x^{2}-y^{2}}$ orbital subspace.
• Figure 3: Schematic illustration of the major virtual hopping processes in the second-order perturbation expansion. For clarity, the bonding $d_{z^{2}}$ orbitals are omitted as they remain fully occupied throughout the process. (a) Type-I: Processes involving electrons in the same layer. (b) Type-II: Processes involving electrons in different layers.
• Figure 4: (a) Ground-state phase diagram of the effective Kugel-Khomskii model in Eq. \ref{['eq:H_eff']}. The axes represent $K_{s}^{zz}$ and $\Delta_{s}=K_{s}^{xx}/K_{s}^{zz}$ (assuming $K_{s}^{xx}=K_{s}^{yy}$). The other parameters are fixed to $J_{s}=0.2$, $K_{t}^{zz}=1.5$, and $K_{t}^{xx}=K_{t}^{yy}=1.2$. Four phases are stabilized, which are denoted as FM-LS (spin-ferromagnetic and layer-staggered), AFM-LS (spin-antiferromagnetic and layer-staggered), AFM-LC (spin-antiferromagnetic and interlayer coherent), and SLE (spin-layer-entangled), respectively. Here, $\mathbf{Q}_S$ and $\mathbf{Q}_{\tau^\alpha}$ denote the ordering momenta where the corresponding spin and layer structure factors exhibit Bragg peaks, respectively. (b) Configuration of the FM-LS phase, where a spin-ferromagnetic state coexists with spatially staggered layer occupation, manifesting as a checkerboard charge pattern. (c) Configuration of the AFM– LS phase, where the spin-antiferromagnetic state retains a staggered layer occupation. (d) AFM– LC phase, where the spin-antiferromagnetic state is accompanied by a layer-coherent order. Here, the pseudospins lie in the $xy$-plane, representing spontaneous quantum coherence between the top and bottom layers.
• Figure 5: (a), (b) Averaged local moments in the spin and pseudospin sectors along the dashed line in Fig. \ref{['fig4']} (a), defined as $\langle O_{s}\rangle=\frac{1}{L^{2}}\sum_{i,\alpha}\langle S_{i}^{\alpha}\rangle^{2}$ and $\langle O_{\tau}\rangle=\frac{1}{L^{2}}\sum_{i,\alpha}\langle\Gamma_{i}^{\alpha}\rangle^{2}$, respectively. (c), (d) Momentum-space distribution of spin-layer correlations $S_{\mu}(\mathbf{k})=\frac{1}{L^{2}}\sum_{i}e^{i\mathbf{k}\cdot\mathbf{r}_{i}}\sum_{\alpha,\beta}\mathcal{Q}_{i}^{\beta\alpha}$, where $\beta\in\{x,y,z\}$. (c) shows the transverse component ($\mu=\perp$) summing over $\alpha\in\{x,y\}$, and (d) shows the longitudinal component ($\mu=\parallel$) with $\alpha=z$.