Two-Orbital Trilayer Hubbard Model

• The two-orbital trilayer Hubbard model is defined by describing three NiO2 layers with two active orbitals per layer, distinguishing between Ni eₙ₋g and charge-transfer constructions.
• It employs DFT-derived tight-binding parameters to detail interlayer hopping, orbital hybridization, and magnetic exchange, linking pressure dependence to superconductivity.
• Magnetic correlations and superconducting phenomenology emerge from exchange-driven pairing, with cross-layer antiferromagnetic interactions dominating the superconducting channel.

Searching arXiv for papers on trilayer nickelates and two-orbital Hubbard models to ground the article in current literature. The two-orbital trilayer Hubbard model denotes a class of layer-resolved low-energy theories developed for pressurized trilayer nickelate La$_4$Ni$_3$O$_{10}$, in which three coupled NiO$_2$ layers are described by two active orbital channels per layer. In the current nickelate literature, the same label covers two distinct constructions: a Ni $e_g$ model built from $d_{x^2-y^2}$ and $d_{z^2}$ orbitals on each layer (Chen et al., 9 Aug 2025), and a charge-transfer formulation in which the active itinerant states are oxygen-derived $p_W$ and interlayer $p_z$ orbitals coupled to Hund-stabilized Ni $S=1$ moments (Oh et al., 2024). The distinction is substantive rather than terminological, because it changes the local Hilbert space, the dominant hopping channels, the form of the strong-coupling reduction, and the microscopic interpretation of pressure-tuned superconductivity.

1. Orbital content and local degrees of freedom

In the charge-transfer formulation, the starting point is $_3$0, so doped holes reside predominantly on O $_3$1 orbitals rather than on Ni $_3$2 orbitals. Each Ni is pinned near the $_3$3 configuration with a spin-one moment arising from the two $_3$4 orbitals $_3$5 and $_3$6, strongly coupled by Hund’s $_3$7. A doped oxygen hole screens the Ni $_3$8 only down to $_3$9, producing a $_{10}$0 “Zhang–Rice spin-half” rather than the Zhang–Rice singlet familiar from cuprates (Oh et al., 2024).

In the Ni $_{10}$1 trilayer Hubbard construction, the active orbitals are instead taken directly as $_{10}$2 and $_{10}$3 on each of three layers $_{10}$4, with outer layers $_{10}$5 and inner layer $_{10}$6. The filling is set to an average $_{10}$7 electrons per orbital per site, corresponding to nominal Ni $_{10}$8 in La$_{10}$9Ni$_2$0O$_2$1 under pressure (Chen et al., 9 Aug 2025).

Aspect	Charge-transfer formulation	Ni $_2$2 formulation
Active orbitals	In-plane $_2$3 and interlayer $_2$4	$_2$5 and $_2$6
Local Ni state	Ni near $_2$7, $_2$8; doped hole gives $_2$9 Zhang–Rice spin-half	Filling $e_g$0 electrons per orbital per site
Interlayer kinetics	$e_g$1 at lowest order	$e_g$2 large, $e_g$3 negligible
Pairing emphasis	Exchange-driven interlayer pairing	Cross-layer singlet pairing in outer $e_g$4 layers

This suggests that “two-orbital” in the trilayer nickelate context labels the number of retained low-energy channels, not a unique microscopic basis.

2. Layer-resolved Hamiltonian structure

For the Ni $e_g$5 description, the full trilayer Hamiltonian is written as

$e_g$6

with intralayer kinetic energy

$e_g$7

interlayer inner–outer hopping

$e_g$8

and cross-layer outer–outer hopping

$e_g$9

The model also contains nearest-neighbor intralayer interorbital hybridization $d_{x^2-y^2}$0, orbital- and layer-dependent onsite energies $d_{x^2-y^2}$1, and Kanamori interactions with intra-orbital $d_{x^2-y^2}$2, inter-orbital $d_{x^2-y^2}$3, Hund spin-exchange, and pair-hopping (Chen et al., 9 Aug 2025).

In the charge-transfer construction, the low-energy oxygen-based trilayer Hubbard model retains $d_{x^2-y^2}$4 and is written as

$d_{x^2-y^2}$5

$d_{x^2-y^2}$6

Here the in-plane Wannier orbital $d_{x^2-y^2}$7 has $d_{x^2-y^2}$8 symmetry matching $d_{x^2-y^2}$9, the interlayer $d_{z^2}$0 orbital bridges adjacent layers, and the itinerant oxygen sector couples to the local Ni background through Kondo-like exchange. Two independent $d_{z^2}$1 orbitals reside between $d_{z^2}$2 and $d_{z^2}$3, there is no direct top–bottom $d_{z^2}$4 path, and direct interlayer $d_{z^2}$5–$d_{z^2}$6 hopping is symmetry suppressed, $d_{z^2}$7 at lowest order (Oh et al., 2024).

The conceptual difference is therefore sharp. In the Ni $d_{z^2}$8 model, the two orbitals are local Ni states and the trilayer structure is encoded directly in the kinetic tensor. In the charge-transfer model, the two active orbitals are oxygen-derived, while Ni enters as a correlated spin background that imposes a constrained local manifold.

3. Parameterization, pressure dependence, and mixed dimensionality

For pressurized La$d_{z^2}$9Ni$p_W$0O$p_W$1, the Ni $p_W$2 model is parameterized from DFT-derived tight binding. In the inner layer, $p_W$3 eV, $p_W$4 eV, $p_W$5 eV, $p_W$6 eV, and $p_W$7 eV. In the outer layers, $p_W$8 eV, $p_W$9 eV, $p_z$0 eV, $p_z$1 eV, and $p_z$2 eV. The dominant interlayer amplitude is $p_z$3 eV, while $p_z$4 eV is negligible; the cross-layer terms are $p_z$5 eV and $p_z$6. The interaction scales are $p_z$7 eV and $p_z$8 eV, with $p_z$9 (Chen et al., 9 Aug 2025).

These numbers define a mixed-dimensional regime. The $S=1$0 orbital is quasi-two-dimensional, with large in-plane hopping and negligible interlayer motion, whereas $S=1$1 is markedly more three-dimensional, with strong inner–outer tunneling, smaller in-plane hopping, and finite outer–outer coupling. Pressure stabilizes this anisotropy by enhancing interlayer tunneling and layer-dependent onsite energies.

In the charge-transfer formulation, moderate pressure of roughly $S=1$2 GPa selects the in-plane oxygen Wannier orbital $S=1$3 as the lowest-energy doped-hole state, while at higher pressure the interlayer oxygen $S=1$4 becomes competitive and can overtake $S=1$5. A representative DFT/Wannier-consistent parameter set gives $S=1$6 eV, $S=1$7 eV, and $S=1$8 eV at $S=1$9 GPa, together with $_3$00 eV, $_3$01 eV, $_3$02 eV, and $_3$03 eV. These parameters yield $_3$04 eV and $_3$05 eV, so $_3$06. The effective in-plane oxygen-band hopping lies in the $_3$07–$_3$08 eV range, with typical exchange scales $_3$09 eV and $_3$10 eV (Oh et al., 2024).

A plausible implication is that pressure acts differently in the two constructions. In the Ni $_3$11 model it sharpens the quasi-2D versus 3D orbital anisotropy, whereas in the charge-transfer model it changes the identity of the active oxygen orbital sector itself.

4. Strong-coupling reduction and effective low-energy models

The charge-transfer trilayer theory is explicitly reduced by a Schrieffer–Wolff transformation to a projected Hilbert space containing three spin-one $_3$12 states and two Zhang–Rice spin-half $_3$13 states per Ni site. In this subspace, the low-energy Hamiltonian becomes a trilayer type II $_3$14-$_3$15 model,

$_3$16

with exchange terms $_3$17, $_3$18, and $_3$19 coupling the Zhang–Rice spin-half $_3$20 and local Ni spin-one $_3$21. For adjacent layers,

$_3$22

$_3$23

with $_3$24. In this formulation, interlayer couplings exist only between adjacent layers, direct $_3$25–$_3$26 hopping remains suppressed, and pairing is exchange-driven rather than hopping-driven (Oh et al., 2024).

The Ni $_3$27 trilayer model is simplified in a different way. Integrating out high-energy charge fluctuations and the $_3$28 orbital yields an effective bilayer mixed-dimensional $_3$29-$_3$30 model for the outer $_3$31 layers,

$_3$32

with no-double-occupancy constraint on $_3$33. The in-plane superexchange obeys $_3$34, while the induced cross-layer exchange satisfies schematically $_3$35, where $_3$36 is an antiferromagnetic susceptibility of the $_3$37 sector across the outer layers (Chen et al., 9 Aug 2025).

The two reductions are structurally different but converge on a common motif: the dominant superconducting channel is not set by direct interlayer charge motion in the nominally active pairing orbital, but by magnetically mediated exchange transmitted through the trilayer structure.

5. Magnetic correlations and superconducting phenomenology

Large-scale DMRG on the Ni $_3$38 trilayer model was carried out on a $_3$39 trilayer ladder of length $_3$40 sites per layer, with open ends, maximum bond dimension $_3$41 multiplets (about $_3$42 $_3$43 states), truncation errors $_3$44, and correlators extrapolated versus bond dimension $_3$45–$_3$46 to $_3$47 (Chen et al., 9 Aug 2025). The magnetic correlations are strongly orbital selective. In the $_3$48 orbital, both interlayer $_3$49 and cross-layer $_3$50 antiferromagnetic correlations are present, with cross-layer slightly stronger under realistic parameters. In the $_3$51 orbital, by contrast, the average interlayer spin correlation is tiny, about $_3$52, while the cross-layer correlation is sizable and antiferromagnetic, about $_3$53. The mechanism proposed is “kinetic AFM correlation”: even at $_3$54, hopping processes favor AFM alignment because Pauli blocking suppresses the ferromagnetic channel, and for realistic $_3$55 eV the three-site $_3$56 problem produces both interlayer and cross-layer AFM, with cross-layer AFM strengthened for $_3$57 eV.

The leading superconducting response also resides in the $_3$58 sector. The cross-layer singlet correlator $_3$59 displays quasi-long-range order with $_3$60. By contrast, $_3$61 interlayer and intralayer pairing decay exponentially, and all reported $_3$62 channels have weaker power-law decay: $_3$63 for cross-layer $_3$64, $_3$65 for interlayer $_3$66, $_3$67 for outer-layer intralayer $_3$68, and $_3$69 for inner-layer intralayer $_3$70. The $_3$71 single-particle Green’s function decays exponentially with $_3$72 in the outer layer and $_3$73 in the inner layer, and the corresponding spin correlations also decay exponentially, consistent with a spin-gapped superconducting channel. Hund’s coupling is essential: $_3$74 eV is required for quasi-long-range superconductivity, and increasing $_3$75 strengthens cross-layer AFM and lowers $_3$76. Intermediate $_3$77 eV is optimal; for $_3$78 eV, a period-3 stripe-ordered state emerges in the outer $_3$79 layer and suppresses superconductivity.

The related charge-transfer type II $_3$80-$_3$81 model supplies a complementary superconducting phenomenology. DMRG on two-leg ladders finds a pairing dome with optimal hole doping $_3$82–$_3$83, distinct from hole-doped cuprates where optimal doping occurs around $_3$84. The low-energy state is a Luther–Emery liquid with finite spin gap, power-law pair correlations, and central charge $_3$85. Pressure initially increases $_3$86 by enhancing $_3$87 and therefore $_3$88, but beyond an optimal pressure $_3$89 the hole shifts from in-plane $_3$90 to interlayer $_3$91, and the binding energy and pairing decrease (Oh et al., 2024). This provides a pressure-dome mechanism that the trilayer construction inherits as an extension.

6. Relation to alternative descriptions, experimental implications, and open problems

A central issue in the nickelate literature is whether La$_3$92Ni$_3$93O$_3$94 under pressure should be viewed primarily through a charge-transfer or a Mott–Hubbard lens. In the charge-transfer regime $_3$95, the active carriers reside on O $_3$96 orbitals, the local Ni state remains $_3$97, the doped state is $_3$98, and interlayer coupling arises through $_3$99–$_{10}$00 Kondo-mediated superexchange rather than direct $_{10}$01–$_{10}$02 hopping. In the opposite limit $_{10}$03, holes enter Ni $_{10}$04 orbitals and a $_{10}$05-only two-orbital model may become appropriate, with small but nonzero $_{10}$06 and $_{10}$07 transmitted via Hund’s coupling on $_{10}$08 (Oh et al., 2024). The disagreement is therefore not merely about parametrization; it concerns which degrees of freedom should be integrated out at all.

The Ni $_{10}$09 trilayer study gives several experimentally oriented predictions. Neutron scattering should detect enhanced outer–outer magnetic correlations and suppressed inner–outer correlations in the superconducting regime. Josephson tunneling is predicted to be stronger across outer–outer pathways than inner–outer ones. ARPES should show an orbital-selective gap opening predominantly on outer-layer $_{10}$10 bands, while $_{10}$11 bands should show weaker gap signatures. A charge modulation at wave vector $_{10}$12 is present in $_{10}$13, and suppressing this CDW is predicted to enhance superconductivity. Transport anisotropy reflecting quasi-2D $_{10}$14 carriers and more 3D-coupled $_{10}$15 carriers is also expected, as are layer-selective NMR or $_{10}$16SR signatures in relaxation rates and Knight shifts (Chen et al., 9 Aug 2025).

The present formulations also have explicit limitations. The DMRG study of the Ni $_{10}$17 model is quasi-one-dimensional, since it uses a $_{10}$18 trilayer ladder and cannot directly access true two-dimensional order or finite-temperature $_{10}$19. It includes only Ni $_{10}$20 orbitals and neglects oxygen $_{10}$21 states, longer-range Coulomb interactions, and electron–phonon couplings. Quantitative uncertainties remain in $_{10}$22, $_{10}$23, and $_{10}$24, and intertrilayer couplings beyond a single trilayer are omitted. Conversely, the charge-transfer construction emphasizes that, when oxygen-hole character is substantial, integrating out oxygen is not appropriate and the $_{10}$25 sector must remain explicit (Oh et al., 2024).

Taken together, these works define the two-orbital trilayer Hubbard model as a family of closely related but microscopically distinct descriptions for pressurized La$_{10}$26Ni$_{10}$27O$_{10}$28. In both, superconductivity is tied to interlayer or cross-layer antiferromagnetic exchange, but the route to that exchange differs: oxygen-mediated type II $_{10}$29-$_{10}$30 physics in the charge-transfer picture, and Hund-transferred cross-layer pairing in a mixed-dimensional $_{10}$31/$_{10}$32 framework in the Ni $_{10}$33 picture.