Bilayer Two-Orbital Kanamori-Hubbard Model

• The bilayer two-orbital Kanamori-Hubbard model is a correlated-electron lattice framework that integrates bilayer geometry with two active orbital flavors and Kanamori-type local interactions.
• It unifies diverse constructions—from explicit bilayer square-lattice formulations to moiré-derived effective models—revealing rich phases including superconductivity, magnetism, and charge density waves.
• The model's adjustable parameters, such as interlayer hopping and orbital hybridization, enable systematic studies of orbital selectivity and the interplay between kinetic energy and local correlations.

The bilayer two-orbital Kanamori-Hubbard model denotes a family of correlated-electron lattice theories in which bilayer geometry and two active orbital flavors are treated on comparable footing, while local interactions are organized in the Kanamori form or in controlled truncations of it. In current usage, the term covers several distinct but related constructions: explicit bilayer square-lattice $e_g$ models with layer labels $\alpha=1,2$ and orbitals $d_{x^2-y^2}$ and $d_{z^2}$; strong-coupling descendants such as bilayer $t$-$J$ and Kugel-Khomskii models; and bilayer-derived moiré theories in which the microscopic layer degree of freedom has been integrated into effective Wannier orbitals, so that the final Hamiltonian is no longer an explicit bilayer lattice model even though its origin is bilayer physics (Duan et al., 10 Jan 2026, Mo et al., 6 Aug 2025, Kaushal et al., 2023).

1. Scope of the model class

In the explicit nickelate-oriented formulations, each in-plane lattice site carries two layers and two orbitals per layer, so the local flavor structure before spin is $(x_1,z_1,x_2,z_2)$, with $x\equiv d_{x^2-y^2}$ and $z\equiv d_{z^2}$ or $d_{3z^2-r^2}$. This is the setting adopted in bilayer square-lattice models motivated by $\alpha=1,2$0, where interlayer $\alpha=1,2$1-$\alpha=1,2$2 hopping and intralayer $\alpha=1,2$3-$\alpha=1,2$4 hybridization are central one-body ingredients (Zheng et al., 2023, Maier et al., 9 Jun 2025).

A narrower usage reserves the phrase for models with a fully rotationally invariant onsite Kanamori interaction. In that strict sense, the quarter-hole-filled bilayer $\alpha=1,2$5 model studied in a molecular-orbital basis is an explicit bilayer two-orbital Kanamori-Hubbard model: it contains two layers $\alpha=1,2$6, two orbitals $\alpha=1,2$7, crystal-field splitting, and onsite $\alpha=1,2$8, $\alpha=1,2$9, Hund exchange, spin-flip, and pair-hopping terms with $d_{x^2-y^2}$0 (Duan et al., 10 Jan 2026). The same strict classification applies to the bilayer nickelate model used to study two intertwined $d_{x^2-y^2}$1-wave superconductivities, which employs the onsite multiorbital Kanamori form with $d_{x^2-y^2}$2 and $d_{x^2-y^2}$3 scanned over $d_{x^2-y^2}$4 (Mo et al., 6 Aug 2025).

A broader usage includes density-density or longitudinal-Hund truncations. This is the case for the bilayer two-orbital Hubbard model studied by cluster DMFT for nickelate superconductivity, where the main calculations retain only $d_{x^2-y^2}$5 on the $d_{x^2-y^2}$6 orbital and later add $d_{x^2-y^2}$7, $d_{x^2-y^2}$8, and a longitudinal Hund coupling $d_{x^2-y^2}$9, but no spin-flip or pair-hopping (Zheng et al., 2023). It is also the case for realistic DCA and DMRG studies of bilayer nickelates that keep $d_{z^2}$0, $d_{z^2}$1, and density-density Hund terms but omit the full rotational structure for sign-problem or tractability reasons (Maier et al., 9 Jun 2025, Chen et al., 3 Nov 2025).

An important limiting case is the bilayer-derived moiré construction. In twisted $d_{z^2}$2-valley TMD homobilayers, Wannierization of the continuum model produces a two-orbital honeycomb theory with $d_{z^2}$3- and $d_{z^2}$4-like Wannier states on the moiré sublattices. The microscopic origin is bilayer, but the low-energy Hamiltonian is not an explicit bilayer lattice Hamiltonian; instead, it is an effective honeycomb model with two orbitals per site and a local sector that resembles Kanamori interactions while also containing substantial nonlocal Coulomb, exchange, pair-hopping, and interaction-assisted hopping terms. For that reason the authors term it a Kanamori-Moiré-Hubbard model rather than a standard bilayer Kanamori-Hubbard model (Kaushal et al., 2023).

2. Interaction structure and the meaning of “Kanamori”

A canonical two-orbital Kanamori interaction is written as

$d_{z^2}$5

often together with the rotationally invariant constraint $d_{z^2}$6. In the current literature, explicit bilayer two-orbital models span the full range from this form to drastic truncations (Kaushal et al., 2023).

Model family	Bilayer representation	Interaction sector
Explicit bilayer $d_{z^2}$7 Kanamori models	Layers retained explicitly	Full local Kanamori with spin-flip and pair-hopping
Bilayer two-orbital density-density models	Layers retained explicitly	$d_{z^2}$8, $d_{z^2}$9, longitudinal or Ising-like Hund terms
Kanamori-Moiré-Hubbard models	Bilayer encoded implicitly after Wannierization	Local Kanamori-like terms plus strong nonlocal terms

The fully rotationally invariant case is represented by the quarter-hole-filled bilayer model in which

$t$0

together with explicit spin-flip and pair-hopping terms, again with $t$1 (Duan et al., 10 Jan 2026). The Kanamori model for $t$2 used in CDMFT likewise includes intraorbital Hubbard repulsion, interorbital density-density interactions, Hund exchange, spin-flip, and pair-hopping (Mo et al., 6 Aug 2025).

By contrast, several influential nickelate studies are only Kanamori-like. One adopts

$t$3

for the main calculations, and later extends it to

$t$4

with $t$5, $t$6, and $t$7, but explicitly without spin-flip and pair-hopping (Zheng et al., 2023). A realistic DCA study of pressurized $t$8 similarly uses a density-density two-orbital Hubbard-Hund interaction with $t$9, $J$0, and $J$1, consistent with $J$2, while omitting spin-flip and pair-hopping (Maier et al., 9 Jun 2025). The zero-temperature CDMFT+NORG work on La327 goes further toward an Ising-Hund representation by separating interorbital couplings into $J$3 and $J$4 and identifying the local interorbital spin coupling $J$5 as the density-density proxy for Hund physics (Chen et al., 2024).

The moiré case is more general still. Its intra-unit-cell interaction $J$6 contains the usual intra-orbital Hubbard term $J$7, interorbital density-density terms $J$8, Hund/direct-exchange terms $J$9, and pair-hopping, but also interaction-assisted hoppings, spin-flip hopping accompanied by local spin flip, and doublon scattering processes parameterized by $(x_1,z_1,x_2,z_2)$0, $(x_1,z_1,x_2,z_2)$1, and $(x_1,z_1,x_2,z_2)$2. The authors state that these additional terms are precisely what place the model beyond the standard multi-orbital Hubbard description (Kaushal et al., 2023).

A persistent source of confusion is therefore terminological. “Bilayer two-orbital Kanamori-Hubbard model” may refer either to a fully SU(2)-invariant local interaction with explicit layer operators, or to a broader family of bilayer two-orbital Hubbard-Hund models in which only the density-density subset is kept. The literature represented here contains both usages (Duan et al., 10 Jan 2026, Zheng et al., 2023).

3. Bilayer geometry, orbital selectivity, and one-body structure

The defining one-body feature of the nickelate models is strongly orbital-selective bilayer hopping. In the minimal bilayer square-lattice formulation motivated by $(x_1,z_1,x_2,z_2)$3, the noninteracting Hamiltonian contains intralayer $(x_1,z_1,x_2,z_2)$4 and $(x_1,z_1,x_2,z_2)$5 hoppings, interlayer hopping $(x_1,z_1,x_2,z_2)$6 only between $(x_1,z_1,x_2,z_2)$7 orbitals on the same in-plane site, and intralayer interorbital hybridization $(x_1,z_1,x_2,z_2)$8. One representative parametrization fixes $(x_1,z_1,x_2,z_2)$9, $x\equiv d_{x^2-y^2}$0, $x\equiv d_{x^2-y^2}$1, $x\equiv d_{x^2-y^2}$2, with total filling $x\equiv d_{x^2-y^2}$3 and typical orbital occupancies $x\equiv d_{x^2-y^2}$4 and $x\equiv d_{x^2-y^2}$5 (Zheng et al., 2023).

The realistic DCA model for pressurized $x\equiv d_{x^2-y^2}$6 sharpens this hierarchy. Using parameters for 25 GPa, it takes $x\equiv d_{x^2-y^2}$7 eV, $x\equiv d_{x^2-y^2}$8 eV, $x\equiv d_{x^2-y^2}$9 eV, $z\equiv d_{z^2}$0 eV, and crystal-field splitting $z\equiv d_{z^2}$1 eV at filling $z\equiv d_{z^2}$2. The largest hopping is $z\equiv d_{z^2}$3, so the dominant bonding-antibonding splitting is in the $z\equiv d_{z^2}$4 sector, and the low-energy superconducting physics becomes overwhelmingly $z\equiv d_{z^2}$5-orbital dominated despite the presence of both $z\equiv d_{z^2}$6 orbitals (Maier et al., 9 Jun 2025).

Pressure-dependent DFT+Wannier downfolding reaches the same conclusion from a different direction. In the bilayer ladder model derived for $z\equiv d_{z^2}$7, $z\equiv d_{z^2}$8 increases by about $z\equiv d_{z^2}$9 and $d_{3z^2-r^2}$0 by about $d_{3z^2-r^2}$1 between 21.6 and 100 GPa, while the effective interaction-to-hopping ratio $d_{3z^2-r^2}$2 decreases from $d_{3z^2-r^2}$3 to $d_{3z^2-r^2}$4 by 80.3 GPa. This explicitly identifies pressure as simultaneously enhancing bilayer/interorbital kinetic couplings and reducing relative correlation strength (Chen et al., 3 Nov 2025).

The strong-coupling bilayer Kanamori model at quarter-hole filling isolates the same orbital selectivity in molecular-orbital language. There the relevant hierarchy is

$d_{3z^2-r^2}$5

and the $d_{3z^2-r^2}$6 sector is reorganized into bonding and antibonding molecular orbitals. The low-energy manifold on each rung consists of a doubly occupied bonding $d_{3z^2-r^2}$7 singlet plus one remaining $d_{3z^2-r^2}$8 electron, giving four states $d_{3z^2-r^2}$9 with $\alpha=1,2$00 and a layer pseudospin $\alpha=1,2$01 that tracks whether the $\alpha=1,2$02 electron sits on the top or bottom layer (Duan et al., 10 Jan 2026).

The moiré realization changes the geometry but preserves the multiorbital logic. Starting from a continuum moiré Hamiltonian

$\alpha=1,2$03

Wannierization yields four well-localized $\alpha=1,2$04- and $\alpha=1,2$05-like Wannier functions on a moiré honeycomb lattice. The resulting tight-binding description lives on a single effective honeycomb lattice with two orbitals per site and hoppings up to third nearest neighbors; the bilayer origin survives only implicitly through the continuum bands and the Wannier orbitals (Kaushal et al., 2023).

A plausible implication is that “bilayer” is not a unique microscopic category but a projection-dependent one. When interlayer dimerization is retained explicitly, the natural degrees of freedom are layers, orbitals, and sometimes bonding-antibonding combinations. When the same physics is downfolded into extended Wannier states, the low-energy theory may instead look like a two-orbital single-layer model, even though its parameters remain bilayer controlled (Duan et al., 10 Jan 2026, Kaushal et al., 2023).

4. Theoretical methods and effective reductions

The model class has been studied with a correspondingly heterogeneous methodological toolkit. Weak-to-intermediate-coupling superconductivity in explicit bilayer two-orbital lattice models has mainly been addressed by cluster DMFT and DCA. The nickelate study built around $\alpha=1,2$06, $\alpha=1,2$07, and $\alpha=1,2$08 uses single-site DMFT, four-site CDMFT, and eight-site DCA with CTQMC and HFQMC impurity solvers, plus high-temperature DQMC benchmarking on $\alpha=1,2$09 orbitals (Zheng et al., 2023). The realistic bilayer two-orbital Hubbard-Hund study of pressurized $\alpha=1,2$10 employs DCA with a continuous-time auxiliary-field QMC solver on clusters $\alpha=1,2$11, $\alpha=1,2$12, and $\alpha=1,2$13 (Maier et al., 9 Jun 2025).

Zero-temperature normal-state correlation physics has been treated by CDMFT with a natural orbitals renormalization group impurity solver. In the La327 study this choice is tailored to a two-site bilayer cluster containing the two layers $\alpha=1,2$14 and both orbitals on each site, thereby resolving interlayer bonding/antibonding structure and Hund-mediated magnetic transfer exactly at the impurity level (Chen et al., 2024). Sign-problem-free determinant QMC becomes available when the interaction sector is simplified sufficiently; this route was used both in the half-filled bilayer two-orbital Hubbard benchmark and in a related bilayer $\alpha=1,2$15-symmetric model with an Ising-like ferromagnetic interlayer coupling (Yang et al., 2024, Caplan et al., 2023).

Strong-coupling reductions have been central to the subject. The quarter-hole-filled bilayer Kanamori model is reduced by a Schrieffer-Wolff transformation to an anisotropic Kugel-Khomskii Hamiltonian,

$\alpha=1,2$16

and its ordered phases are then analyzed by Weiss mean-field theory and generalized flavor-wave theory (Duan et al., 10 Jan 2026). A related derivation from a two-orbital Hubbard model with $\alpha=1,2$17, $\alpha=1,2$18, crystal-field splitting, and Hund coupling yields a projected multicomponent $\alpha=1,2$19-$\alpha=1,2$20 theory with local spin-$\alpha=1,2$21 states, spin-1 triplets, and additional orbital-conversion and Kugel-Khomskii-type terms; this was proposed as the appropriate low-energy model for bilayer nickelates (Kaneko et al., 14 Apr 2025).

An alternative strong-coupling route starts directly from a two-orbital bilayer $\alpha=1,2$22-$\alpha=1,2$23 model with onsite ferromagnetic Hund’s-rule coupling and intersite interorbital hybridization. In that framework, $\alpha=1,2$24 behaves as a localized bilayer-rung subsystem with large $\alpha=1,2$25 and $\alpha=1,2$26, whereas $\alpha=1,2$27 is the more itinerant pairing-active band (Qu et al., 2023). The moiré counterpart combines projection-based Wannierization, unrestricted Hartree-Fock, and Lanczos exact diagonalization to expose how broad moiré Wannier functions invalidate a purely local Kanamori approximation (Kaushal et al., 2023).

Methodological pluralism is therefore not incidental. It reflects the fact that the same nominal model class supports weak-coupling Fermi-surface competition, strong-coupling spin-orbital-layer exchange, and bilayer-to-effective-single-orbital reductions, depending on filling, hopping anisotropy, and the degree of interaction truncation (Zheng et al., 2023, Duan et al., 10 Jan 2026, Kaneko et al., 14 Apr 2025).

5. Correlated phases outside superconductivity

At half filling in the bilayer-derived Kanamori-Moiré-Hubbard model, the phase diagram in dielectric constant $\alpha=1,2$28 and twist angle $\alpha=1,2$29 contains three principal phases: antiferromagnetism, an $\alpha=1,2$30 ferromagnetic insulator, and a charge density wave. For $\alpha=1,2$31 and lower screening the dominant phase is the $\alpha=1,2$32 ferromagnetic insulator, stabilized by competition between nonlocal ferromagnetic direct exchange and antiferromagnetic superexchange; the AFM state appears only for rather large dielectric constant, $\alpha=1,2$33. For $\alpha=1,2$34, lower $\alpha=1,2$35 instead favors a CDW due primarily to nonlocal density-density repulsion. A simplified interaction retaining only density-density, direct-exchange, and pair-hopping terms reproduces nearly the same three-phase structure, indicating that nonlocal density repulsion and nonlocal direct exchange are the essential half-filled ingredients (Kaushal et al., 2023).

The explicit bilayer Kanamori model in the quarter-hole-filled molecular-orbital regime yields a different but equally rich phase structure. With couplings $\alpha=1,2$36, $\alpha=1,2$37, $\alpha=1,2$38, and variable $\alpha=1,2$39, Weiss mean-field theory finds four phases: FM-LS, AFM-LS, AFM-LC, and SLE. The LS states exhibit layer-staggered pseudospin order, the AFM-LC phase has spontaneous interlayer coherence with $\alpha=1,2$40, and the SLE phase has no ordinary dipolar spin or layer order but instead a composite order parameter $\alpha=1,2$41. The SLE phase is described as arising from emergent $\alpha=1,2$42 symmetry breaking and supports three entangled gapless Goldstone modes (Duan et al., 10 Jan 2026).

Normal-state bilayer nickelate models reveal yet another sequence. In zero-temperature CDMFT+NORG, the realistic La327-motivated density-density Kanamori-Hubbard model displays

$\alpha=1,2$43

with increasing hole doping. The nominal La327 point lies at $\alpha=1,2$44, $\alpha=1,2$45, $\alpha=1,2$46, $\alpha=1,2$47, inside the NFL regime. A central result is that Hund spin correlation transmits strong interlayer AFM correlations from the $\alpha=1,2$48 orbital sector into the $\alpha=1,2$49 orbital sector; when the bonding $\alpha=1,2$50 band is no longer fully filled, these interlayer AFM correlations weaken rapidly (Chen et al., 2024).

At half filling in a sign-problem-free simplified bilayer two-orbital Hubbard model, DQMC reveals a nonmagnetic weakly insulating phase at weak $\alpha=1,2$51, followed by a transition to an antiferromagnetic Mott insulator at

$\alpha=1,2$52

The transition is consistent with the 3D $\alpha=1,2$53 Heisenberg universality class. The weakly insulating phase shows a full single-particle gap, a small but finite spin gap, no extrapolated CDW, PDW, excitonic, or bond-order parameter, and pronounced interlayer $\alpha=1,2$54 correlations. The same calculations show a downward shift and extended flatness of the $\alpha=1,2$55 band, mirroring ARPES observations (Yang et al., 2024).

These results collectively undercut a simplified expectation that bilayer two-orbital models generically interpolate only between weak itinerant antiferromagnetism and a standard Mott insulator. Nonlocal exchange can stabilize ferromagnetic or CDW phases in moiré realizations; strong interlayer dimerization can generate nonmagnetic weak insulators; and Hund-coupled bilayer geometries can produce composite spin-layer order not visible in conventional dipolar observables (Kaushal et al., 2023, Yang et al., 2024, Duan et al., 10 Jan 2026).

6. Superconductivity, pairing mechanisms, and model reductions

The superconducting literature on bilayer two-orbital Hubbard and Kanamori-Hubbard models is dominated by $\alpha=1,2$56-type interlayer pairing, but the microscopic meaning of that state varies substantially across models. In the cluster-DMFT study of the bilayer two-orbital Hubbard model with $\alpha=1,2$57, superconductivity is diagnosed from the interlayer $\alpha=1,2$58 spin-singlet susceptibility

$\alpha=1,2$59

and the resulting state is identified as $\alpha=1,2$60-wave. Its $\alpha=1,2$61 is nonmonotonic in both $\alpha=1,2$62 and $\alpha=1,2$63: a threshold $\alpha=1,2$64 is required, the optimal regime is around $\alpha=1,2$65, and for $\alpha=1,2$66, $\alpha=1,2$67, $\alpha=1,2$68, the calculated $\alpha=1,2$69 becomes almost independent of $\alpha=1,2$70 for $\alpha=1,2$71, corresponding to about $\alpha=1,2$72 K when $\alpha=1,2$73 eV is used for conversion. The physical interpretation is a two-component mechanism in which $\alpha=1,2$74 provides pairing glue through $\alpha=1,2$75, while $\alpha=1,2$76 supplies phase coherence via the itinerant $\alpha=1,2$77 sector (Zheng et al., 2023).

A more explicitly Kanamori-based CDMFT study argues that bilayer $\alpha=1,2$78 supports two intertwined $\alpha=1,2$79-wave superconductivities. SC I is strongest at $\alpha=1,2$80, with the $\alpha=1,2$81 bonding band below $\alpha=1,2$82, and is strongly enhanced by $\alpha=1,2$83 but only weakly dependent on $\alpha=1,2$84; at $\alpha=1,2$85 and $\alpha=1,2$86, it gives $\alpha=1,2$87, and it survives even at $\alpha=1,2$88. SC II becomes dominant at finite $\alpha=1,2$89 hole doping, is relatively insensitive to $\alpha=1,2$90, depends critically on hybridization with $\alpha=1,2$91, and peaks near $\alpha=1,2$92 with $\alpha=1,2$93. Because SC I weakens while SC II strengthens over comparable $\alpha=1,2$94 intervals, the combined $\alpha=1,2$95 evolves smoothly even though the underlying pairing mechanism changes (Mo et al., 6 Aug 2025).

The realistic DCA study of the two-orbital Hubbard-Hund model for pressurized $\alpha=1,2$96 finds a leading $\alpha=1,2$97 instability with form factor

$\alpha=1,2$98

so the sign change is between bonding and antibonding bilayer sectors rather than between distinct in-plane pockets. Real-space eigenvector analysis shows that the leading pair is primarily interlayer, intraorbital, local in-plane, and overwhelmingly $\alpha=1,2$99-dominated; interorbital pair correlations do not control the leading instability. The same study identifies interlayer antiferromagnetic spin fluctuations in the $d_{x^2-y^2}$00 orbital as the pairing driver and argues that a single-orbital bilayer Hubbard model for $d_{x^2-y^2}$01 is therefore an excellent effective low-energy description of the superconducting behavior (Maier et al., 9 Jun 2025).

The DMRG study of a DFT-parameterized bilayer two-orbital Hubbard ladder adds a pressure axis. Near $d_{x^2-y^2}$02, the dominant instability is an SDW with $d_{x^2-y^2}$03; at $d_{x^2-y^2}$04, superconducting correlations appear with interlayer singlet pairing in both $d_{x^2-y^2}$05 and $d_{x^2-y^2}$06, plus in-plane $d_{x^2-y^2}$07 singlet pairing. At 21.6 GPa the exponents are approximately $d_{x^2-y^2}$08 and $d_{x^2-y^2}$09, while increasing pressure weakens superconductivity and drives an SC-to-Luttinger-liquid crossover near $d_{x^2-y^2}$10 GPa. The comparative analysis in that work attributes the suppression primarily to the reduction of $d_{x^2-y^2}$11 and $d_{x^2-y^2}$12, rather than to large changes in hopping anisotropy ratios (Chen et al., 3 Nov 2025).

Auxiliary models clarify what is generic and what is not. In the single-orbital bilayer Hubbard model, DCA finds a crossover from $d_{x^2-y^2}$13 $d_{x^2-y^2}$14-wave pairing at small $d_{x^2-y^2}$15 to $d_{x^2-y^2}$16 $d_{x^2-y^2}$17-wave pairing at larger $d_{x^2-y^2}$18, with interlayer spin fluctuations favoring the latter (Maier et al., 2011). In a sign-problem-free bilayer $d_{x^2-y^2}$19-symmetric Hubbard model with an Ising-like ferromagnetic interlayer interaction, the favored superconducting state is not singlet $d_{x^2-y^2}$20 but equal-spin interlayer pairing $d_{x^2-y^2}$21, again coexisting with stripe order and forming a dome in doping (Caplan et al., 2023). These analogues suggest that bilayer geometry robustly promotes interlayer pairing channels, while the spin structure of the condensate depends sensitively on whether the interorbital coupling is antiferromagnetic, ferromagnetic, SU(2)-invariant, or Ising-like.

A recurrent conceptual outcome is model reduction. Realistic two-orbital bilayer models often flow toward smaller effective descriptions: to a $d_{x^2-y^2}$22-only bilayer Hubbard model in DCA (Maier et al., 9 Jun 2025), to a $d_{x^2-y^2}$23-based bilayer $d_{x^2-y^2}$24-$d_{x^2-y^2}$25-$d_{x^2-y^2}$26 model after Hund-mediated transfer of interlayer AF correlations from a localized $d_{x^2-y^2}$27 sector (Qu et al., 2023), or to a projected $d_{x^2-y^2}$28-$d_{x^2-y^2}$29 theory with local spin-$d_{x^2-y^2}$30 and spin-1 states once the on-site two-orbital multiplets are treated exactly in strong coupling (Kaneko et al., 14 Apr 2025). This suggests that the encyclopedic subject is best understood not as a single Hamiltonian but as a hierarchy of related models whose common core is the competition between bilayer bonding, orbital differentiation, and multiorbital local interactions.