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arXiv:2601.06440

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

Guijing Duan, Yunlong Wang, Zhiguang Liao, Changle Liu, Rong Yu

Published Jan 10, 2026• 0 citations
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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.

DOI
10.48550/arxiv.2601.06440

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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.
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 d z 2 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.
Quantum materials featuring both itinerant and localized degrees of freedom exhibit numerous exotic phases and transitions that deviate from the Ginzburg-Landau paradigm. This paper uses the composite operator formalism to examine the bilayer strongly correlated Hubbard model. We observe the spontaneous breaking of layer symmetry, where the electron density in one of the layer reaches half-filling, resulting in a layer selective Mott phase (LSMP). This broken symmetry phase becomes unstable at a critical average electronic density away from half-filling. Furthermore, significant layer differentiation persists up to a moderate interlayer hopping, beyond which the system abruptly transitions to a layer uniform phase. In the LSMP phase, the electrons in the two layers are weakly hybridized, resulting in a small Fermi surface. The volume of the Fermi surface jumps at the transition from the LSMP to the uniform phase. We also discuss the physical mechanisms leading to the collapse of the LSMP phase under different perturbations.
While superconductors are conventionally established by attractive interactions, higher-temperature mechanisms for emergent electronic pairing from strong repulsive electron-electron interactions remain under considerable scrutiny. Here, we establish a strong-coupling mechanism for intertwined superconductivity and magnetic order from purely repulsive interactions in a Kondo-like bilayer system, composed of a two-dimensional Mott insulator coupled to a layer of weakly-interacting itinerant electrons. Combining large scale DMRG and Monte Carlo simulations, we find that superconductivity persists and coexists with magnetism over a wide range of interlayer couplings. We classify the resulting rich phase diagram and find 2-rung antiferromagnetic and 4-rung antiferromagnetic order in one-dimensional systems along with a phase separation regime, while finding that superconductivity coexists with either antiferromagnetic or ferromagnetic order in two dimensions. Remarkably, the model permits a rigorous strong-coupling analysis via localized spins coupled to charge-2e bosons through Kugel-Khomskii interactions, capturing the pairing mechanism in the presence of magnetism due to emergent attractive interactions. Our numerical analysis reveals that pairing remains robust well beyond the strong-coupling regime, establishing a new mechanism for superconductivity in coupled weakly- and strongly-interacting electron systems, relevant for infinite-layer nickelates and superconductivity in moire multilayer heterostructures.
By following the ideas of Emery and Noguera, a recent study revealed the dynamics of the charge sector of a one-dimensional quarter-filled electronic system with extended Hubbard interactions to be that of an effective pseudospin transverse-field Ising model (TFIM) in the strong-coupling limit. With the twin motivations of studying the co-existing charge and spin order found in strongly correlated chain systems and the effects of interchain couplings, we investigate the phase diagram of coupled effective (TFIM) systems. A bosonization and renormalization group (RG) analysis for a two-leg TFIM ladder yields a rich phase diagram showing Wigner/Peierls charge order and Néel/dimer spin order. In a broad parameter regime, the orbital antiferromagnetic phase is found to be stable. An intermediate gapless phase of finite width is found to lie in between two charge-ordered gapped phases. Kosterlitz-Thouless transitions are found to lead from the gapless phase to either of the charge-ordered phases. A detailed analysis is also carried out for the dimensional crossover physics when many such pseudospin systems are coupled to one another. Importantly, the analysis reveals the key role of critical quantum fluctuations in driving the strong dispersion in the transverse directions, as well as a T = 0 deconfinement transition. Our work is potentially relevant for a unified description of a class of strongly correlated, quarter-filled chain and ladder systems.
A recent study revealed the dynamics of the charge sector of a one-dimensional quarterfilled electronic system with extended Hubbard interactions to be that of an effective pseudospin transverse-field Ising model (TFIM) in the strong coupling limit. With the twin motivations of studying the co-existing charge and spin order found in strongly correlated chain systems and the effects of inter-chain couplings, we investigate the phase diagram of coupled effective (TFIM) systems. A bosonisation and RG analysis for a two-leg TFIM ladder yields a rich phase diagram showing Wigner/Peierls charge order and Neel/dimer spin order. In a broad parameter regime, the orbital antiferromagnetic phase is found to be stable. An intermediate gapless phase of finite width is found to lie in between two charge-ordered gapped phases. Kosterlitz-Thouless transitions are found to lead from the gapless phase to either of the charge-ordered phases. Low energy effective Hamiltonian analyses of a strongly coupled 2-chain ladder system confirm a phase diagram with in-chain CO, rung-dimer, and orbital antiferromagnetic ordered phases with varying interchain couplings as well as superconductivity upon hole-doping. Our work is potentially relevant for a unified description of a class of strongly correlated, quarter-filled chain and ladder systems.
Strongly correlated electron systems have the potential to host very exoticphases of matter. In order to have relevance to real materials, this exoticphysics often must emerge from relatively simple models. The quantumwavefunctions which describe such phases may bear little resemblance to theoriginal microscopic models. In these cases a variety of complex analytictools often must be supplemented with controlled numerical calculations tofully understand the essential behavior of these models. In this dissertation,we study such quantum phases of matter and their relationship to real materials.We focus on three main problems. First, we explore the relationship betweenstrong spin-orbit coupling and spin liquid physics by studying a very generalmodel on the triangular lattice where spin-orbit coupling leads to the presenceof highly anisotropic interactions. We use variational Monte Carlo to study bothU(1) quantum spin liquid states and ordered ones, via the Gutzwiller projectedfermion construction. We thereby obtain the ground state phase diagram in thisphase space. We furthermore consider effects beyond the Gutzwiller wavefunctionsfor the spinon Fermi surface quantum spin liquid, which are of particularimportance when spin-orbit coupling is present.Second we show that the interplay between a high density two-dimensional electrongas and localized electrons in a neighboring Mott insulator leads to kineticmagnetism unique to the Mott/band insulator interface. Our study is based upona bilayer Hubbard model at U=inftyU=\\inftyU=infty with a potential difference between thetwo layers. We combine analytic results with DMRG simulations to show thatmagnetism, and especially kinetic ferromagnetism, is greatly enhanced relative by theproximity of the two subsystems. Third we study the effect of interactions on the properties of a model 2Dtopological Kondo insulator phase. We introduce a model Hamiltonian which webelieve captures the essential physics of the different competing phases.Perhaps the most dramatic example of many-body physics in symmetry protectedtopological phases is the existence of exotic gapless edge states. We comment onthe potentially dramatic effects that interactions can have on such edge states.